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<!DOCTYPE html> <html lang="en"> <head> <meta content="text/html; charset=utf-8" http-equiv="content-type"/> <title>Streaming Algorithms for Network Design</title>  <meta content="width=device-width, initial-scale=1, shrink-to-fit=no" name="viewport"/> <link href="https://cdn.jsdelivr.net/npm/bootstrap@5.3.0/dist/css/bootstrap.min.css" rel="stylesheet" type="text/css"/> <link href="/static/browse/0.3.4/css/ar5iv.0.7.9.min.css" rel="stylesheet" type="text/css"/> <link href="/static/browse/0.3.4/css/ar5iv-fonts.0.7.9.min.css" rel="stylesheet" type="text/css"/> <link href="/static/browse/0.3.4/css/latexml_styles.css" rel="stylesheet" type="text/css"/> <script src="https://cdn.jsdelivr.net/npm/bootstrap@5.3.0/dist/js/bootstrap.bundle.min.js"></script> <script src="https://cdnjs.cloudflare.com/ajax/libs/html2canvas/1.3.3/html2canvas.min.js"></script> <script src="/static/browse/0.3.4/js/addons_new.js"></script> <script src="/static/browse/0.3.4/js/feedbackOverlay.js"></script> <base href="/html/2503.00712v1/"/></head> <body> <nav class="ltx_page_navbar"> <nav class="ltx_TOC"> <ol class="ltx_toclist"> <li class="ltx_tocentry ltx_tocentry_section"> <a class="ltx_ref" href="https://arxiv.org/html/2503.00712v1#S1" title="In Streaming Algorithms for Network Design"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">1 </span>Introduction</span></a> <ol class="ltx_toclist ltx_toclist_section"> <li class="ltx_tocentry ltx_tocentry_subsection"> <a class="ltx_ref" href="https://arxiv.org/html/2503.00712v1#S1.SS1" title="In 1 Introduction ‣ Streaming Algorithms for Network Design"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">1.1 </span>Results and Techniques</span></a> <ol class="ltx_toclist ltx_toclist_subsection"> <li class="ltx_tocentry ltx_tocentry_paragraph"><a class="ltx_ref" href="https://arxiv.org/html/2503.00712v1#S1.SS1.SSS0.Px1" title="In 1.1 Results and Techniques ‣ 1 Introduction ‣ Streaming Algorithms for Network Design"><span class="ltx_text ltx_ref_title">A framework for SNDP:</span></a></li> <li class="ltx_tocentry ltx_tocentry_paragraph"><a class="ltx_ref" href="https://arxiv.org/html/2503.00712v1#S1.SS1.SSS0.Px2" title="In 1.1 Results and Techniques ‣ 1 Introduction ‣ Streaming Algorithms for Network Design"><span class="ltx_text ltx_ref_title">Algorithms for <math alttext="k" class="ltx_Math" display="inline"><semantics><mi>k</mi><annotation-xml encoding="MathML-Content"><ci>𝑘</ci></annotation-xml><annotation encoding="application/x-tex">k</annotation><annotation encoding="application/x-llamapun">italic_k</annotation></semantics></math>-VC-CAP:</span></a></li> <li class="ltx_tocentry ltx_tocentry_paragraph"><a class="ltx_ref" href="https://arxiv.org/html/2503.00712v1#S1.SS1.SSS0.Px3" title="In 1.1 Results and Techniques ‣ 1 Introduction ‣ Streaming Algorithms for Network Design"><span class="ltx_text ltx_ref_title">Techniques:</span></a></li> </ol> </li> <li class="ltx_tocentry ltx_tocentry_subsection"> <a class="ltx_ref" href="https://arxiv.org/html/2503.00712v1#S1.SS2" title="In 1 Introduction ‣ Streaming Algorithms for Network Design"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">1.2 </span>Related Work</span></a> <ol class="ltx_toclist ltx_toclist_subsection"> <li class="ltx_tocentry ltx_tocentry_paragraph"><a class="ltx_ref" href="https://arxiv.org/html/2503.00712v1#S1.SS2.SSS0.Px1" title="In 1.2 Related Work ‣ 1 Introduction ‣ Streaming Algorithms for Network Design"><span class="ltx_text ltx_ref_title">Offline Network Design:</span></a></li> <li class="ltx_tocentry ltx_tocentry_paragraph"><a class="ltx_ref" href="https://arxiv.org/html/2503.00712v1#S1.SS2.SSS0.Px2" title="In 1.2 Related Work ‣ 1 Introduction ‣ Streaming Algorithms for Network Design"><span class="ltx_text ltx_ref_title">Streaming Graph Algorithms:</span></a></li> </ol> </li> </ol> </li> <li class="ltx_tocentry ltx_tocentry_section"> <a class="ltx_ref" href="https://arxiv.org/html/2503.00712v1#S2" title="In Streaming Algorithms for Network Design"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">2 </span>Preliminaries</span></a> <ol class="ltx_toclist ltx_toclist_section"> <li class="ltx_tocentry ltx_tocentry_subsection"> <a class="ltx_ref" href="https://arxiv.org/html/2503.00712v1#S2.SS1" title="In 2 Preliminaries ‣ Streaming Algorithms for Network Design"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">2.1 </span>Fault-Tolerant Spanners in Streaming</span></a> <ol class="ltx_toclist ltx_toclist_subsection"> <li class="ltx_tocentry ltx_tocentry_paragraph"><a class="ltx_ref" href="https://arxiv.org/html/2503.00712v1#S2.SS1.SSS0.Px1" title="In 2.1 Fault-Tolerant Spanners in Streaming ‣ 2 Preliminaries ‣ Streaming Algorithms for Network Design"><span class="ltx_text ltx_ref_title">Weighted graphs.</span></a></li> </ol> </li> </ol> </li> <li class="ltx_tocentry ltx_tocentry_section"> <a class="ltx_ref" href="https://arxiv.org/html/2503.00712v1#S3" title="In Streaming Algorithms for Network Design"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">3 </span>Generic Framework for Streaming Algorithms for Network Design</span></a> <ol class="ltx_toclist ltx_toclist_section"> <li class="ltx_tocentry ltx_tocentry_subsection"> <a class="ltx_ref" href="https://arxiv.org/html/2503.00712v1#S3.SS1" title="In 3 Generic Framework for Streaming Algorithms for Network Design ‣ Streaming Algorithms for Network Design"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">3.1 </span>Vertex Connectivity Network Design</span></a> <ol class="ltx_toclist ltx_toclist_subsection"> <li class="ltx_tocentry ltx_tocentry_subsubsection"><a class="ltx_ref" href="https://arxiv.org/html/2503.00712v1#S3.SS1.SSS1" title="In 3.1 Vertex Connectivity Network Design ‣ 3 Generic Framework for Streaming Algorithms for Network Design ‣ Streaming Algorithms for Network Design"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">3.1.1 </span>A Simple Analysis Based on Integral Solutions</span></a></li> <li class="ltx_tocentry ltx_tocentry_subsubsection"> <a class="ltx_ref" href="https://arxiv.org/html/2503.00712v1#S3.SS1.SSS2" title="In 3.1 Vertex Connectivity Network Design ‣ 3 Generic Framework for Streaming Algorithms for Network Design ‣ Streaming Algorithms for Network Design"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">3.1.2 </span>An Improved Analysis via Fractional Solutions</span></a> <ol class="ltx_toclist ltx_toclist_subsubsection"> <li class="ltx_tocentry ltx_tocentry_paragraph"><a class="ltx_ref" href="https://arxiv.org/html/2503.00712v1#S3.SS1.SSS2.Px1" title="In 3.1.2 An Improved Analysis via Fractional Solutions ‣ 3.1 Vertex Connectivity Network Design ‣ 3 Generic Framework for Streaming Algorithms for Network Design ‣ Streaming Algorithms for Network Design"><span class="ltx_text ltx_ref_title">Implications for VC-SNDP and special cases:</span></a></li> </ol> </li> </ol> </li> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2503.00712v1#S3.SS2" title="In 3 Generic Framework for Streaming Algorithms for Network Design ‣ Streaming Algorithms for Network Design"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">3.2 </span>EC-SNDP and ELC-SNDP</span></a></li> </ol> </li> <li class="ltx_tocentry ltx_tocentry_section"> <a class="ltx_ref" href="https://arxiv.org/html/2503.00712v1#S4" title="In Streaming Algorithms for Network Design"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">4 </span>Vertex Connectivity Augmentation in Link-Arrival Model</span></a> <ol class="ltx_toclist ltx_toclist_section"> <li class="ltx_tocentry ltx_tocentry_subsection"> <a class="ltx_ref" href="https://arxiv.org/html/2503.00712v1#S4.SS1" title="In 4 Vertex Connectivity Augmentation in Link-Arrival Model ‣ Streaming Algorithms for Network Design"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">4.1 </span>One-to-Two Augmentation</span></a> <ol class="ltx_toclist ltx_toclist_subsection"> <li class="ltx_tocentry ltx_tocentry_subsubsection"><a class="ltx_ref" href="https://arxiv.org/html/2503.00712v1#S4.SS1.SSS1" title="In 4.1 One-to-Two Augmentation ‣ 4 Vertex Connectivity Augmentation in Link-Arrival Model ‣ Streaming Algorithms for Network Design"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">4.1.1 </span>The Streaming Algorithm</span></a></li> <li class="ltx_tocentry ltx_tocentry_subsubsection"><a class="ltx_ref" href="https://arxiv.org/html/2503.00712v1#S4.SS1.SSS2" title="In 4.1 One-to-Two Augmentation ‣ 4 Vertex Connectivity Augmentation in Link-Arrival Model ‣ Streaming Algorithms for Network Design"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">4.1.2 </span>Bounding Approximation Ratio</span></a></li> </ol> </li> <li class="ltx_tocentry ltx_tocentry_subsection"> <a class="ltx_ref" href="https://arxiv.org/html/2503.00712v1#S4.SS2" title="In 4 Vertex Connectivity Augmentation in Link-Arrival Model ‣ Streaming Algorithms for Network Design"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">4.2 </span>Two-to-Three Augmentation</span></a> <ol class="ltx_toclist ltx_toclist_subsection"> <li class="ltx_tocentry ltx_tocentry_subsubsection"><a class="ltx_ref" href="https://arxiv.org/html/2503.00712v1#S4.SS2.SSS1" title="In 4.2 Two-to-Three Augmentation ‣ 4 Vertex Connectivity Augmentation in Link-Arrival Model ‣ Streaming Algorithms for Network Design"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">4.2.1 </span>SPQR Trees</span></a></li> <li class="ltx_tocentry ltx_tocentry_subsubsection"> <a class="ltx_ref" href="https://arxiv.org/html/2503.00712v1#S4.SS2.SSS2" title="In 4.2 Two-to-Three Augmentation ‣ 4 Vertex Connectivity Augmentation in Link-Arrival Model ‣ Streaming Algorithms for Network Design"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">4.2.2 </span>The Streaming Algorithm</span></a> <ol class="ltx_toclist ltx_toclist_subsubsection"> <li class="ltx_tocentry ltx_tocentry_paragraph"><a class="ltx_ref" href="https://arxiv.org/html/2503.00712v1#S4.SS2.SSS2.Px1" title="In 4.2.2 The Streaming Algorithm ‣ 4.2 Two-to-Three Augmentation ‣ 4 Vertex Connectivity Augmentation in Link-Arrival Model ‣ Streaming Algorithms for Network Design"><span class="ltx_text ltx_ref_title">“Tree” Cuts:</span></a></li> <li class="ltx_tocentry ltx_tocentry_paragraph"><a class="ltx_ref" href="https://arxiv.org/html/2503.00712v1#S4.SS2.SSS2.Px2" title="In 4.2.2 The Streaming Algorithm ‣ 4.2 Two-to-Three Augmentation ‣ 4 Vertex Connectivity Augmentation in Link-Arrival Model ‣ Streaming Algorithms for Network Design"><span class="ltx_text ltx_ref_title">“Cycle” Cuts:</span></a></li> </ol> </li> <li class="ltx_tocentry ltx_tocentry_subsubsection"> <a class="ltx_ref" href="https://arxiv.org/html/2503.00712v1#S4.SS2.SSS3" title="In 4.2 Two-to-Three Augmentation ‣ 4 Vertex Connectivity Augmentation in Link-Arrival Model ‣ Streaming Algorithms for Network Design"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">4.2.3 </span>Bounding the Approximation Ratio</span></a> <ol class="ltx_toclist ltx_toclist_subsubsection"> <li class="ltx_tocentry ltx_tocentry_paragraph"><a class="ltx_ref" href="https://arxiv.org/html/2503.00712v1#S4.SS2.SSS3.Px1" title="In 4.2.3 Bounding the Approximation Ratio ‣ 4.2 Two-to-Three Augmentation ‣ 4 Vertex Connectivity Augmentation in Link-Arrival Model ‣ Streaming Algorithms for Network Design"><span class="ltx_text ltx_ref_title">Case 1: <math alttext="\boldsymbol{e=ab}" class="ltx_Math" display="inline"><semantics><mrow><mi>𝒆</mi><mo class="ltx_mathvariant_bold" mathvariant="bold">=</mo><mrow><mi>𝒂</mi><mo>⁢</mo><mi>𝒃</mi></mrow></mrow><annotation-xml encoding="MathML-Content"><apply><eq></eq><ci>𝒆</ci><apply><times></times><ci>𝒂</ci><ci>𝒃</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex">\boldsymbol{e=ab}</annotation><annotation encoding="application/x-llamapun">bold_italic_e bold_= bold_italic_a bold_italic_b</annotation></semantics></math> is a virtual edge of an R-node and an R or S-node:</span></a></li> <li class="ltx_tocentry ltx_tocentry_paragraph"><a class="ltx_ref" href="https://arxiv.org/html/2503.00712v1#S4.SS2.SSS3.Px2" title="In 4.2.3 Bounding the Approximation Ratio ‣ 4.2 Two-to-Three Augmentation ‣ 4 Vertex Connectivity Augmentation in Link-Arrival Model ‣ Streaming Algorithms for Network Design"><span class="ltx_text ltx_ref_title">Case 2: <math alttext="\boldsymbol{V(G_{x})=\{a,b\}}" class="ltx_Math" display="inline"><semantics><mrow><mrow><mi>𝑽</mi><mo>⁢</mo><mrow><mo class="ltx_mathvariant_bold" mathvariant="bold" stretchy="false">(</mo><msub><mi>𝑮</mi><mi>𝒙</mi></msub><mo class="ltx_mathvariant_bold" mathvariant="bold" stretchy="false">)</mo></mrow></mrow><mo class="ltx_mathvariant_bold" mathvariant="bold">=</mo><mrow><mo class="ltx_mathvariant_bold" mathvariant="bold" stretchy="false">{</mo><mi>𝒂</mi><mo class="ltx_mathvariant_bold" mathvariant="bold">,</mo><mi>𝒃</mi><mo class="ltx_mathvariant_bold" mathvariant="bold" stretchy="false">}</mo></mrow></mrow><annotation-xml encoding="MathML-Content"><apply><eq></eq><apply><times></times><ci>𝑽</ci><apply><csymbol cd="ambiguous">subscript</csymbol><ci>𝑮</ci><ci>𝒙</ci></apply></apply><set><ci>𝒂</ci><ci>𝒃</ci></set></apply></annotation-xml><annotation encoding="application/x-tex">\boldsymbol{V(G_{x})=\{a,b\}}</annotation><annotation encoding="application/x-llamapun">bold_italic_V bold_( bold_italic_G start_POSTSUBSCRIPT bold_italic_x end_POSTSUBSCRIPT bold_) bold_= bold_{ bold_italic_a bold_, bold_italic_b bold_}</annotation></semantics></math> for a P-node <math alttext="\boldsymbol{x}" class="ltx_Math" display="inline"><semantics><mi>𝒙</mi><annotation-xml encoding="MathML-Content"><ci>𝒙</ci></annotation-xml><annotation encoding="application/x-tex">\boldsymbol{x}</annotation><annotation encoding="application/x-llamapun">bold_italic_x</annotation></semantics></math>:</span></a></li> <li class="ltx_tocentry ltx_tocentry_paragraph"><a class="ltx_ref" href="https://arxiv.org/html/2503.00712v1#S4.SS2.SSS3.Px3" title="In 4.2.3 Bounding the Approximation Ratio ‣ 4.2 Two-to-Three Augmentation ‣ 4 Vertex Connectivity Augmentation in Link-Arrival Model ‣ Streaming Algorithms for Network Design"><span class="ltx_text ltx_ref_title">Case 3: <math alttext="\boldsymbol{a}" class="ltx_Math" display="inline"><semantics><mi>𝒂</mi><annotation-xml encoding="MathML-Content"><ci>𝒂</ci></annotation-xml><annotation encoding="application/x-tex">\boldsymbol{a}</annotation><annotation encoding="application/x-llamapun">bold_italic_a</annotation></semantics></math> and <math alttext="\boldsymbol{b}" class="ltx_Math" display="inline"><semantics><mi>𝒃</mi><annotation-xml encoding="MathML-Content"><ci>𝒃</ci></annotation-xml><annotation encoding="application/x-tex">\boldsymbol{b}</annotation><annotation encoding="application/x-llamapun">bold_italic_b</annotation></semantics></math> are non-adjacent nodes of <math alttext="\boldsymbol{G_{x}}" class="ltx_Math" display="inline"><semantics><msub><mi>𝑮</mi><mi>𝒙</mi></msub><annotation-xml encoding="MathML-Content"><apply><csymbol cd="ambiguous">subscript</csymbol><ci>𝑮</ci><ci>𝒙</ci></apply></annotation-xml><annotation encoding="application/x-tex">\boldsymbol{G_{x}}</annotation><annotation encoding="application/x-llamapun">bold_italic_G start_POSTSUBSCRIPT bold_italic_x end_POSTSUBSCRIPT</annotation></semantics></math> for an S-node <math alttext="\boldsymbol{x}" class="ltx_Math" display="inline"><semantics><mi>𝒙</mi><annotation-xml encoding="MathML-Content"><ci>𝒙</ci></annotation-xml><annotation encoding="application/x-tex">\boldsymbol{x}</annotation><annotation encoding="application/x-llamapun">bold_italic_x</annotation></semantics></math>:</span></a></li> </ol> </li> </ol> </li> </ol> </li> <li class="ltx_tocentry ltx_tocentry_section"><a class="ltx_ref" href="https://arxiv.org/html/2503.00712v1#S5" title="In Streaming Algorithms for Network Design"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">5 </span>Lower Bounds for Streaming Network Design</span></a></li> </ol></nav> </nav> <div class="ltx_page_main"> <div class="ltx_page_content"> <article class="ltx_document ltx_authors_1line"> <h1 class="ltx_title ltx_title_document">Streaming Algorithms for Network Design</h1> <div class="ltx_authors"> <span class="ltx_creator ltx_role_author"> <span class="ltx_personname">Chandra Chekuri </span><span class="ltx_author_notes">University of Illinois at Urbana-Champaign: <a class="ltx_ref ltx_href" href="mailto:chekuri@illinois.edu" title=""><span class="ltx_ref ltx_nolink">chekuri@illinois.edu</span></a>. Supported in part by NSF grant CCF-2402667.</span></span> <span class="ltx_author_before"> </span><span class="ltx_creator ltx_role_author"> <span class="ltx_personname">Rhea Jain </span><span class="ltx_author_notes">University of Illinois at Urbana-Champaign: <a class="ltx_ref ltx_href" href="mailto:rheaj3@illinois.edu" title=""><span class="ltx_ref ltx_nolink">rheaj3@illinois.edu</span></a>. Supported in part by NSF grant CCF-2402667.</span></span> <span class="ltx_author_before"> </span><span class="ltx_creator ltx_role_author"> <span class="ltx_personname">Sepideh Mahabadi </span><span class="ltx_author_notes">Microsoft Research: <a class="ltx_ref ltx_href" href="mailto:smahabadi@microsoft.com" title=""><span class="ltx_ref ltx_nolink">smahabadi@microsoft.com</span></a>.</span></span> <span class="ltx_author_before"> </span><span class="ltx_creator ltx_role_author"> <span class="ltx_personname">Ali Vakilian </span><span class="ltx_author_notes">Toyota Technological Institute at Chicago (TTIC): <a class="ltx_ref ltx_href" href="mailto:vakilian@ttic.edu" title=""><span class="ltx_ref ltx_nolink">vakilian@ttic.edu</span></a>.</span></span> </div> <div class="ltx_abstract"> <h6 class="ltx_title ltx_title_abstract">Abstract</h6> <p class="ltx_p" id="id8.8">We consider the Survivable Network Design problem (SNDP) in the single-pass insertion-only streaming model. The input to SNDP is an edge-weighted graph <math alttext="G=(V,E)" class="ltx_Math" display="inline" id="id1.1.m1.2"><semantics id="id1.1.m1.2a"><mrow id="id1.1.m1.2.3" xref="id1.1.m1.2.3.cmml"><mi id="id1.1.m1.2.3.2" xref="id1.1.m1.2.3.2.cmml">G</mi><mo id="id1.1.m1.2.3.1" xref="id1.1.m1.2.3.1.cmml">=</mo><mrow id="id1.1.m1.2.3.3.2" xref="id1.1.m1.2.3.3.1.cmml"><mo id="id1.1.m1.2.3.3.2.1" stretchy="false" xref="id1.1.m1.2.3.3.1.cmml">(</mo><mi id="id1.1.m1.1.1" xref="id1.1.m1.1.1.cmml">V</mi><mo id="id1.1.m1.2.3.3.2.2" xref="id1.1.m1.2.3.3.1.cmml">,</mo><mi id="id1.1.m1.2.2" xref="id1.1.m1.2.2.cmml">E</mi><mo id="id1.1.m1.2.3.3.2.3" stretchy="false" xref="id1.1.m1.2.3.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="id1.1.m1.2b"><apply id="id1.1.m1.2.3.cmml" xref="id1.1.m1.2.3"><eq id="id1.1.m1.2.3.1.cmml" xref="id1.1.m1.2.3.1"></eq><ci id="id1.1.m1.2.3.2.cmml" xref="id1.1.m1.2.3.2">𝐺</ci><interval closure="open" id="id1.1.m1.2.3.3.1.cmml" xref="id1.1.m1.2.3.3.2"><ci id="id1.1.m1.1.1.cmml" xref="id1.1.m1.1.1">𝑉</ci><ci id="id1.1.m1.2.2.cmml" xref="id1.1.m1.2.2">𝐸</ci></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="id1.1.m1.2c">G=(V,E)</annotation><annotation encoding="application/x-llamapun" id="id1.1.m1.2d">italic_G = ( italic_V , italic_E )</annotation></semantics></math> and an integer connectivity requirement <math alttext="r(uv)" class="ltx_Math" display="inline" id="id2.2.m2.1"><semantics id="id2.2.m2.1a"><mrow id="id2.2.m2.1.1" xref="id2.2.m2.1.1.cmml"><mi id="id2.2.m2.1.1.3" xref="id2.2.m2.1.1.3.cmml">r</mi><mo id="id2.2.m2.1.1.2" xref="id2.2.m2.1.1.2.cmml">⁢</mo><mrow id="id2.2.m2.1.1.1.1" xref="id2.2.m2.1.1.1.1.1.cmml"><mo id="id2.2.m2.1.1.1.1.2" stretchy="false" xref="id2.2.m2.1.1.1.1.1.cmml">(</mo><mrow id="id2.2.m2.1.1.1.1.1" xref="id2.2.m2.1.1.1.1.1.cmml"><mi id="id2.2.m2.1.1.1.1.1.2" xref="id2.2.m2.1.1.1.1.1.2.cmml">u</mi><mo id="id2.2.m2.1.1.1.1.1.1" xref="id2.2.m2.1.1.1.1.1.1.cmml">⁢</mo><mi id="id2.2.m2.1.1.1.1.1.3" xref="id2.2.m2.1.1.1.1.1.3.cmml">v</mi></mrow><mo id="id2.2.m2.1.1.1.1.3" stretchy="false" xref="id2.2.m2.1.1.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="id2.2.m2.1b"><apply id="id2.2.m2.1.1.cmml" xref="id2.2.m2.1.1"><times id="id2.2.m2.1.1.2.cmml" xref="id2.2.m2.1.1.2"></times><ci id="id2.2.m2.1.1.3.cmml" xref="id2.2.m2.1.1.3">𝑟</ci><apply id="id2.2.m2.1.1.1.1.1.cmml" xref="id2.2.m2.1.1.1.1"><times id="id2.2.m2.1.1.1.1.1.1.cmml" xref="id2.2.m2.1.1.1.1.1.1"></times><ci id="id2.2.m2.1.1.1.1.1.2.cmml" xref="id2.2.m2.1.1.1.1.1.2">𝑢</ci><ci id="id2.2.m2.1.1.1.1.1.3.cmml" xref="id2.2.m2.1.1.1.1.1.3">𝑣</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="id2.2.m2.1c">r(uv)</annotation><annotation encoding="application/x-llamapun" id="id2.2.m2.1d">italic_r ( italic_u italic_v )</annotation></semantics></math> for each <math alttext="u,v\in V" class="ltx_Math" display="inline" id="id3.3.m3.2"><semantics id="id3.3.m3.2a"><mrow id="id3.3.m3.2.3" xref="id3.3.m3.2.3.cmml"><mrow id="id3.3.m3.2.3.2.2" xref="id3.3.m3.2.3.2.1.cmml"><mi id="id3.3.m3.1.1" xref="id3.3.m3.1.1.cmml">u</mi><mo id="id3.3.m3.2.3.2.2.1" xref="id3.3.m3.2.3.2.1.cmml">,</mo><mi id="id3.3.m3.2.2" xref="id3.3.m3.2.2.cmml">v</mi></mrow><mo id="id3.3.m3.2.3.1" xref="id3.3.m3.2.3.1.cmml">∈</mo><mi id="id3.3.m3.2.3.3" xref="id3.3.m3.2.3.3.cmml">V</mi></mrow><annotation-xml encoding="MathML-Content" id="id3.3.m3.2b"><apply id="id3.3.m3.2.3.cmml" xref="id3.3.m3.2.3"><in id="id3.3.m3.2.3.1.cmml" xref="id3.3.m3.2.3.1"></in><list id="id3.3.m3.2.3.2.1.cmml" xref="id3.3.m3.2.3.2.2"><ci id="id3.3.m3.1.1.cmml" xref="id3.3.m3.1.1">𝑢</ci><ci id="id3.3.m3.2.2.cmml" xref="id3.3.m3.2.2">𝑣</ci></list><ci id="id3.3.m3.2.3.3.cmml" xref="id3.3.m3.2.3.3">𝑉</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="id3.3.m3.2c">u,v\in V</annotation><annotation encoding="application/x-llamapun" id="id3.3.m3.2d">italic_u , italic_v ∈ italic_V</annotation></semantics></math>. The objective is to find a minimum-weight subgraph <math alttext="H\subseteq G" class="ltx_Math" display="inline" id="id4.4.m4.1"><semantics id="id4.4.m4.1a"><mrow id="id4.4.m4.1.1" xref="id4.4.m4.1.1.cmml"><mi id="id4.4.m4.1.1.2" xref="id4.4.m4.1.1.2.cmml">H</mi><mo id="id4.4.m4.1.1.1" xref="id4.4.m4.1.1.1.cmml">⊆</mo><mi id="id4.4.m4.1.1.3" xref="id4.4.m4.1.1.3.cmml">G</mi></mrow><annotation-xml encoding="MathML-Content" id="id4.4.m4.1b"><apply id="id4.4.m4.1.1.cmml" xref="id4.4.m4.1.1"><subset id="id4.4.m4.1.1.1.cmml" xref="id4.4.m4.1.1.1"></subset><ci id="id4.4.m4.1.1.2.cmml" xref="id4.4.m4.1.1.2">𝐻</ci><ci id="id4.4.m4.1.1.3.cmml" xref="id4.4.m4.1.1.3">𝐺</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="id4.4.m4.1c">H\subseteq G</annotation><annotation encoding="application/x-llamapun" id="id4.4.m4.1d">italic_H ⊆ italic_G</annotation></semantics></math> such that, for every pair of vertices <math alttext="u,v\in V" class="ltx_Math" display="inline" id="id5.5.m5.2"><semantics id="id5.5.m5.2a"><mrow id="id5.5.m5.2.3" xref="id5.5.m5.2.3.cmml"><mrow id="id5.5.m5.2.3.2.2" xref="id5.5.m5.2.3.2.1.cmml"><mi id="id5.5.m5.1.1" xref="id5.5.m5.1.1.cmml">u</mi><mo id="id5.5.m5.2.3.2.2.1" xref="id5.5.m5.2.3.2.1.cmml">,</mo><mi id="id5.5.m5.2.2" xref="id5.5.m5.2.2.cmml">v</mi></mrow><mo id="id5.5.m5.2.3.1" xref="id5.5.m5.2.3.1.cmml">∈</mo><mi id="id5.5.m5.2.3.3" xref="id5.5.m5.2.3.3.cmml">V</mi></mrow><annotation-xml encoding="MathML-Content" id="id5.5.m5.2b"><apply id="id5.5.m5.2.3.cmml" xref="id5.5.m5.2.3"><in id="id5.5.m5.2.3.1.cmml" xref="id5.5.m5.2.3.1"></in><list id="id5.5.m5.2.3.2.1.cmml" xref="id5.5.m5.2.3.2.2"><ci id="id5.5.m5.1.1.cmml" xref="id5.5.m5.1.1">𝑢</ci><ci id="id5.5.m5.2.2.cmml" xref="id5.5.m5.2.2">𝑣</ci></list><ci id="id5.5.m5.2.3.3.cmml" xref="id5.5.m5.2.3.3">𝑉</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="id5.5.m5.2c">u,v\in V</annotation><annotation encoding="application/x-llamapun" id="id5.5.m5.2d">italic_u , italic_v ∈ italic_V</annotation></semantics></math>, <math alttext="u" class="ltx_Math" display="inline" id="id6.6.m6.1"><semantics id="id6.6.m6.1a"><mi id="id6.6.m6.1.1" xref="id6.6.m6.1.1.cmml">u</mi><annotation-xml encoding="MathML-Content" id="id6.6.m6.1b"><ci id="id6.6.m6.1.1.cmml" xref="id6.6.m6.1.1">𝑢</ci></annotation-xml><annotation encoding="application/x-tex" id="id6.6.m6.1c">u</annotation><annotation encoding="application/x-llamapun" id="id6.6.m6.1d">italic_u</annotation></semantics></math> and <math alttext="v" class="ltx_Math" display="inline" id="id7.7.m7.1"><semantics id="id7.7.m7.1a"><mi id="id7.7.m7.1.1" xref="id7.7.m7.1.1.cmml">v</mi><annotation-xml encoding="MathML-Content" id="id7.7.m7.1b"><ci id="id7.7.m7.1.1.cmml" xref="id7.7.m7.1.1">𝑣</ci></annotation-xml><annotation encoding="application/x-tex" id="id7.7.m7.1c">v</annotation><annotation encoding="application/x-llamapun" id="id7.7.m7.1d">italic_v</annotation></semantics></math> are <math alttext="r(uv)" class="ltx_Math" display="inline" id="id8.8.m8.1"><semantics id="id8.8.m8.1a"><mrow id="id8.8.m8.1.1" xref="id8.8.m8.1.1.cmml"><mi id="id8.8.m8.1.1.3" xref="id8.8.m8.1.1.3.cmml">r</mi><mo id="id8.8.m8.1.1.2" xref="id8.8.m8.1.1.2.cmml">⁢</mo><mrow id="id8.8.m8.1.1.1.1" xref="id8.8.m8.1.1.1.1.1.cmml"><mo id="id8.8.m8.1.1.1.1.2" stretchy="false" xref="id8.8.m8.1.1.1.1.1.cmml">(</mo><mrow id="id8.8.m8.1.1.1.1.1" xref="id8.8.m8.1.1.1.1.1.cmml"><mi id="id8.8.m8.1.1.1.1.1.2" xref="id8.8.m8.1.1.1.1.1.2.cmml">u</mi><mo id="id8.8.m8.1.1.1.1.1.1" xref="id8.8.m8.1.1.1.1.1.1.cmml">⁢</mo><mi id="id8.8.m8.1.1.1.1.1.3" xref="id8.8.m8.1.1.1.1.1.3.cmml">v</mi></mrow><mo id="id8.8.m8.1.1.1.1.3" stretchy="false" xref="id8.8.m8.1.1.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="id8.8.m8.1b"><apply id="id8.8.m8.1.1.cmml" xref="id8.8.m8.1.1"><times id="id8.8.m8.1.1.2.cmml" xref="id8.8.m8.1.1.2"></times><ci id="id8.8.m8.1.1.3.cmml" xref="id8.8.m8.1.1.3">𝑟</ci><apply id="id8.8.m8.1.1.1.1.1.cmml" xref="id8.8.m8.1.1.1.1"><times id="id8.8.m8.1.1.1.1.1.1.cmml" xref="id8.8.m8.1.1.1.1.1.1"></times><ci id="id8.8.m8.1.1.1.1.1.2.cmml" xref="id8.8.m8.1.1.1.1.1.2">𝑢</ci><ci id="id8.8.m8.1.1.1.1.1.3.cmml" xref="id8.8.m8.1.1.1.1.1.3">𝑣</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="id8.8.m8.1c">r(uv)</annotation><annotation encoding="application/x-llamapun" id="id8.8.m8.1d">italic_r ( italic_u italic_v )</annotation></semantics></math>-edge/vertex-connected. Recent work by <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.00712v1#bib.bibx54" title="">JKMV24</a>]</cite> obtained approximation algorithms for edge-connectivity augmentation, and via that, also derived algorithms for edge-connectivity SNDP (EC-SNDP). In this work we consider <em class="ltx_emph ltx_font_italic" id="id8.8.1">vertex-connectivity</em> setting (VC-SNDP) and obtain several results for it as well as improved results for EC-SNDP.</p> <ul class="ltx_itemize" id="S0.I1"> <li class="ltx_item" id="S0.I1.i1" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">•</span> <div class="ltx_para" id="S0.I1.i1.p1"> <p class="ltx_p" id="S0.I1.i1.p1.8">We provide a general framework for solving connectivity problems including SNDP and others in streaming; this is based on a connection to <em class="ltx_emph ltx_font_italic" id="S0.I1.i1.p1.8.1">fault-tolerant spanners</em>. For VC-SNDP we provide an <math alttext="O(tk)" class="ltx_Math" display="inline" id="S0.I1.i1.p1.1.m1.1"><semantics id="S0.I1.i1.p1.1.m1.1a"><mrow id="S0.I1.i1.p1.1.m1.1.1" xref="S0.I1.i1.p1.1.m1.1.1.cmml"><mi id="S0.I1.i1.p1.1.m1.1.1.3" xref="S0.I1.i1.p1.1.m1.1.1.3.cmml">O</mi><mo id="S0.I1.i1.p1.1.m1.1.1.2" xref="S0.I1.i1.p1.1.m1.1.1.2.cmml">⁢</mo><mrow id="S0.I1.i1.p1.1.m1.1.1.1.1" xref="S0.I1.i1.p1.1.m1.1.1.1.1.1.cmml"><mo id="S0.I1.i1.p1.1.m1.1.1.1.1.2" stretchy="false" xref="S0.I1.i1.p1.1.m1.1.1.1.1.1.cmml">(</mo><mrow id="S0.I1.i1.p1.1.m1.1.1.1.1.1" xref="S0.I1.i1.p1.1.m1.1.1.1.1.1.cmml"><mi id="S0.I1.i1.p1.1.m1.1.1.1.1.1.2" xref="S0.I1.i1.p1.1.m1.1.1.1.1.1.2.cmml">t</mi><mo id="S0.I1.i1.p1.1.m1.1.1.1.1.1.1" xref="S0.I1.i1.p1.1.m1.1.1.1.1.1.1.cmml">⁢</mo><mi id="S0.I1.i1.p1.1.m1.1.1.1.1.1.3" xref="S0.I1.i1.p1.1.m1.1.1.1.1.1.3.cmml">k</mi></mrow><mo id="S0.I1.i1.p1.1.m1.1.1.1.1.3" stretchy="false" xref="S0.I1.i1.p1.1.m1.1.1.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml 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xref="S0.I1.i1.p1.2.m2.1.1.1.1.1.2.3.3.1"></divide><cn id="S0.I1.i1.p1.2.m2.1.1.1.1.1.2.3.3.2.cmml" type="integer" xref="S0.I1.i1.p1.2.m2.1.1.1.1.1.2.3.3.2">1</cn><ci id="S0.I1.i1.p1.2.m2.1.1.1.1.1.2.3.3.3.cmml" xref="S0.I1.i1.p1.2.m2.1.1.1.1.1.2.3.3.3">𝑡</ci></apply></apply></apply><apply id="S0.I1.i1.p1.2.m2.1.1.1.1.1.3.cmml" xref="S0.I1.i1.p1.2.m2.1.1.1.1.1.3"><csymbol cd="ambiguous" id="S0.I1.i1.p1.2.m2.1.1.1.1.1.3.1.cmml" xref="S0.I1.i1.p1.2.m2.1.1.1.1.1.3">superscript</csymbol><ci id="S0.I1.i1.p1.2.m2.1.1.1.1.1.3.2.cmml" xref="S0.I1.i1.p1.2.m2.1.1.1.1.1.3.2">𝑛</ci><apply id="S0.I1.i1.p1.2.m2.1.1.1.1.1.3.3.cmml" xref="S0.I1.i1.p1.2.m2.1.1.1.1.1.3.3"><plus id="S0.I1.i1.p1.2.m2.1.1.1.1.1.3.3.1.cmml" xref="S0.I1.i1.p1.2.m2.1.1.1.1.1.3.3.1"></plus><cn id="S0.I1.i1.p1.2.m2.1.1.1.1.1.3.3.2.cmml" type="integer" xref="S0.I1.i1.p1.2.m2.1.1.1.1.1.3.3.2">1</cn><apply id="S0.I1.i1.p1.2.m2.1.1.1.1.1.3.3.3.cmml" xref="S0.I1.i1.p1.2.m2.1.1.1.1.1.3.3.3"><divide id="S0.I1.i1.p1.2.m2.1.1.1.1.1.3.3.3.1.cmml" xref="S0.I1.i1.p1.2.m2.1.1.1.1.1.3.3.3.1"></divide><cn id="S0.I1.i1.p1.2.m2.1.1.1.1.1.3.3.3.2.cmml" type="integer" xref="S0.I1.i1.p1.2.m2.1.1.1.1.1.3.3.3.2">1</cn><ci id="S0.I1.i1.p1.2.m2.1.1.1.1.1.3.3.3.3.cmml" xref="S0.I1.i1.p1.2.m2.1.1.1.1.1.3.3.3.3">𝑡</ci></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S0.I1.i1.p1.2.m2.1c">\tilde{O}(k^{1-1/t}n^{1+1/t})</annotation><annotation encoding="application/x-llamapun" id="S0.I1.i1.p1.2.m2.1d">over~ start_ARG italic_O end_ARG ( italic_k start_POSTSUPERSCRIPT 1 - 1 / italic_t end_POSTSUPERSCRIPT italic_n start_POSTSUPERSCRIPT 1 + 1 / italic_t end_POSTSUPERSCRIPT )</annotation></semantics></math> space, where <math alttext="k" class="ltx_Math" display="inline" id="S0.I1.i1.p1.3.m3.1"><semantics id="S0.I1.i1.p1.3.m3.1a"><mi id="S0.I1.i1.p1.3.m3.1.1" xref="S0.I1.i1.p1.3.m3.1.1.cmml">k</mi><annotation-xml encoding="MathML-Content" id="S0.I1.i1.p1.3.m3.1b"><ci id="S0.I1.i1.p1.3.m3.1.1.cmml" xref="S0.I1.i1.p1.3.m3.1.1">𝑘</ci></annotation-xml><annotation encoding="application/x-tex" id="S0.I1.i1.p1.3.m3.1c">k</annotation><annotation encoding="application/x-llamapun" id="S0.I1.i1.p1.3.m3.1d">italic_k</annotation></semantics></math> is the maximum connectivity requirement, assuming an exact algorithm at the end of the stream. Using a refined LP-based analysis, we provide an <math alttext="O(\beta t)" class="ltx_Math" display="inline" id="S0.I1.i1.p1.4.m4.1"><semantics id="S0.I1.i1.p1.4.m4.1a"><mrow id="S0.I1.i1.p1.4.m4.1.1" xref="S0.I1.i1.p1.4.m4.1.1.cmml"><mi id="S0.I1.i1.p1.4.m4.1.1.3" xref="S0.I1.i1.p1.4.m4.1.1.3.cmml">O</mi><mo id="S0.I1.i1.p1.4.m4.1.1.2" xref="S0.I1.i1.p1.4.m4.1.1.2.cmml">⁢</mo><mrow id="S0.I1.i1.p1.4.m4.1.1.1.1" xref="S0.I1.i1.p1.4.m4.1.1.1.1.1.cmml"><mo id="S0.I1.i1.p1.4.m4.1.1.1.1.2" stretchy="false" xref="S0.I1.i1.p1.4.m4.1.1.1.1.1.cmml">(</mo><mrow id="S0.I1.i1.p1.4.m4.1.1.1.1.1" xref="S0.I1.i1.p1.4.m4.1.1.1.1.1.cmml"><mi id="S0.I1.i1.p1.4.m4.1.1.1.1.1.2" xref="S0.I1.i1.p1.4.m4.1.1.1.1.1.2.cmml">β</mi><mo id="S0.I1.i1.p1.4.m4.1.1.1.1.1.1" xref="S0.I1.i1.p1.4.m4.1.1.1.1.1.1.cmml">⁢</mo><mi id="S0.I1.i1.p1.4.m4.1.1.1.1.1.3" xref="S0.I1.i1.p1.4.m4.1.1.1.1.1.3.cmml">t</mi></mrow><mo id="S0.I1.i1.p1.4.m4.1.1.1.1.3" stretchy="false" xref="S0.I1.i1.p1.4.m4.1.1.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S0.I1.i1.p1.4.m4.1b"><apply id="S0.I1.i1.p1.4.m4.1.1.cmml" xref="S0.I1.i1.p1.4.m4.1.1"><times id="S0.I1.i1.p1.4.m4.1.1.2.cmml" xref="S0.I1.i1.p1.4.m4.1.1.2"></times><ci id="S0.I1.i1.p1.4.m4.1.1.3.cmml" xref="S0.I1.i1.p1.4.m4.1.1.3">𝑂</ci><apply id="S0.I1.i1.p1.4.m4.1.1.1.1.1.cmml" xref="S0.I1.i1.p1.4.m4.1.1.1.1"><times id="S0.I1.i1.p1.4.m4.1.1.1.1.1.1.cmml" xref="S0.I1.i1.p1.4.m4.1.1.1.1.1.1"></times><ci id="S0.I1.i1.p1.4.m4.1.1.1.1.1.2.cmml" xref="S0.I1.i1.p1.4.m4.1.1.1.1.1.2">𝛽</ci><ci id="S0.I1.i1.p1.4.m4.1.1.1.1.1.3.cmml" xref="S0.I1.i1.p1.4.m4.1.1.1.1.1.3">𝑡</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S0.I1.i1.p1.4.m4.1c">O(\beta t)</annotation><annotation encoding="application/x-llamapun" id="S0.I1.i1.p1.4.m4.1d">italic_O ( italic_β italic_t )</annotation></semantics></math>-approximation in polynomial time, where <math alttext="\beta" class="ltx_Math" display="inline" id="S0.I1.i1.p1.5.m5.1"><semantics id="S0.I1.i1.p1.5.m5.1a"><mi id="S0.I1.i1.p1.5.m5.1.1" xref="S0.I1.i1.p1.5.m5.1.1.cmml">β</mi><annotation-xml encoding="MathML-Content" id="S0.I1.i1.p1.5.m5.1b"><ci id="S0.I1.i1.p1.5.m5.1.1.cmml" xref="S0.I1.i1.p1.5.m5.1.1">𝛽</ci></annotation-xml><annotation encoding="application/x-tex" id="S0.I1.i1.p1.5.m5.1c">\beta</annotation><annotation encoding="application/x-llamapun" id="S0.I1.i1.p1.5.m5.1d">italic_β</annotation></semantics></math> is the best polynomial time approximation with respect to the optimal <em class="ltx_emph ltx_font_italic" id="S0.I1.i1.p1.8.2">fractional</em> solution to a natural LP relaxation. These are the first approximation algorithms in the streaming model for VC-SNDP. 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end_ARG ( italic_k start_POSTSUPERSCRIPT 1 - 1 / italic_t end_POSTSUPERSCRIPT italic_n start_POSTSUPERSCRIPT 1 + 1 / italic_t end_POSTSUPERSCRIPT )</annotation></semantics></math> space, improving the <math alttext="O(t\log k)" class="ltx_Math" display="inline" id="S0.I1.i1.p1.8.m8.1"><semantics id="S0.I1.i1.p1.8.m8.1a"><mrow id="S0.I1.i1.p1.8.m8.1.1" xref="S0.I1.i1.p1.8.m8.1.1.cmml"><mi id="S0.I1.i1.p1.8.m8.1.1.3" xref="S0.I1.i1.p1.8.m8.1.1.3.cmml">O</mi><mo id="S0.I1.i1.p1.8.m8.1.1.2" xref="S0.I1.i1.p1.8.m8.1.1.2.cmml">⁢</mo><mrow id="S0.I1.i1.p1.8.m8.1.1.1.1" xref="S0.I1.i1.p1.8.m8.1.1.1.1.1.cmml"><mo id="S0.I1.i1.p1.8.m8.1.1.1.1.2" stretchy="false" xref="S0.I1.i1.p1.8.m8.1.1.1.1.1.cmml">(</mo><mrow id="S0.I1.i1.p1.8.m8.1.1.1.1.1" xref="S0.I1.i1.p1.8.m8.1.1.1.1.1.cmml"><mi id="S0.I1.i1.p1.8.m8.1.1.1.1.1.2" xref="S0.I1.i1.p1.8.m8.1.1.1.1.1.2.cmml">t</mi><mo id="S0.I1.i1.p1.8.m8.1.1.1.1.1.1" lspace="0.167em" xref="S0.I1.i1.p1.8.m8.1.1.1.1.1.1.cmml">⁢</mo><mrow 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xref="S0.I1.i1.p1.8.m8.1.1.1.1.1.2">𝑡</ci><apply id="S0.I1.i1.p1.8.m8.1.1.1.1.1.3.cmml" xref="S0.I1.i1.p1.8.m8.1.1.1.1.1.3"><log id="S0.I1.i1.p1.8.m8.1.1.1.1.1.3.1.cmml" xref="S0.I1.i1.p1.8.m8.1.1.1.1.1.3.1"></log><ci id="S0.I1.i1.p1.8.m8.1.1.1.1.1.3.2.cmml" xref="S0.I1.i1.p1.8.m8.1.1.1.1.1.3.2">𝑘</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S0.I1.i1.p1.8.m8.1c">O(t\log k)</annotation><annotation encoding="application/x-llamapun" id="S0.I1.i1.p1.8.m8.1d">italic_O ( italic_t roman_log italic_k )</annotation></semantics></math>-approximation of <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.00712v1#bib.bibx54" title="">JKMV24</a>]</cite>; this also extends to element-connectivity SNDP.</p> </div> </li> <li class="ltx_item" id="S0.I1.i2" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">•</span> <div class="ltx_para" id="S0.I1.i2.p1"> <p class="ltx_p" id="S0.I1.i2.p1.8">We consider vertex connectivity-augmentation in the link-arrival model. The input is a <math alttext="k" class="ltx_Math" display="inline" id="S0.I1.i2.p1.1.m1.1"><semantics id="S0.I1.i2.p1.1.m1.1a"><mi id="S0.I1.i2.p1.1.m1.1.1" xref="S0.I1.i2.p1.1.m1.1.1.cmml">k</mi><annotation-xml encoding="MathML-Content" id="S0.I1.i2.p1.1.m1.1b"><ci id="S0.I1.i2.p1.1.m1.1.1.cmml" xref="S0.I1.i2.p1.1.m1.1.1">𝑘</ci></annotation-xml><annotation encoding="application/x-tex" id="S0.I1.i2.p1.1.m1.1c">k</annotation><annotation encoding="application/x-llamapun" id="S0.I1.i2.p1.1.m1.1d">italic_k</annotation></semantics></math>-vertex-connected spanning subgraph <math alttext="G" class="ltx_Math" display="inline" id="S0.I1.i2.p1.2.m2.1"><semantics id="S0.I1.i2.p1.2.m2.1a"><mi id="S0.I1.i2.p1.2.m2.1.1" xref="S0.I1.i2.p1.2.m2.1.1.cmml">G</mi><annotation-xml encoding="MathML-Content" id="S0.I1.i2.p1.2.m2.1b"><ci id="S0.I1.i2.p1.2.m2.1.1.cmml" xref="S0.I1.i2.p1.2.m2.1.1">𝐺</ci></annotation-xml><annotation encoding="application/x-tex" id="S0.I1.i2.p1.2.m2.1c">G</annotation><annotation encoding="application/x-llamapun" id="S0.I1.i2.p1.2.m2.1d">italic_G</annotation></semantics></math>, and additional weighted links <math alttext="L" class="ltx_Math" display="inline" id="S0.I1.i2.p1.3.m3.1"><semantics id="S0.I1.i2.p1.3.m3.1a"><mi id="S0.I1.i2.p1.3.m3.1.1" xref="S0.I1.i2.p1.3.m3.1.1.cmml">L</mi><annotation-xml encoding="MathML-Content" id="S0.I1.i2.p1.3.m3.1b"><ci id="S0.I1.i2.p1.3.m3.1.1.cmml" xref="S0.I1.i2.p1.3.m3.1.1">𝐿</ci></annotation-xml><annotation encoding="application/x-tex" id="S0.I1.i2.p1.3.m3.1c">L</annotation><annotation encoding="application/x-llamapun" id="S0.I1.i2.p1.3.m3.1d">italic_L</annotation></semantics></math> arrive in the stream; the goal is to store the min-weight set of links such that <math alttext="G\cup L" class="ltx_Math" display="inline" id="S0.I1.i2.p1.4.m4.1"><semantics id="S0.I1.i2.p1.4.m4.1a"><mrow id="S0.I1.i2.p1.4.m4.1.1" xref="S0.I1.i2.p1.4.m4.1.1.cmml"><mi id="S0.I1.i2.p1.4.m4.1.1.2" xref="S0.I1.i2.p1.4.m4.1.1.2.cmml">G</mi><mo id="S0.I1.i2.p1.4.m4.1.1.1" xref="S0.I1.i2.p1.4.m4.1.1.1.cmml">∪</mo><mi id="S0.I1.i2.p1.4.m4.1.1.3" xref="S0.I1.i2.p1.4.m4.1.1.3.cmml">L</mi></mrow><annotation-xml encoding="MathML-Content" id="S0.I1.i2.p1.4.m4.1b"><apply id="S0.I1.i2.p1.4.m4.1.1.cmml" xref="S0.I1.i2.p1.4.m4.1.1"><union id="S0.I1.i2.p1.4.m4.1.1.1.cmml" xref="S0.I1.i2.p1.4.m4.1.1.1"></union><ci id="S0.I1.i2.p1.4.m4.1.1.2.cmml" xref="S0.I1.i2.p1.4.m4.1.1.2">𝐺</ci><ci id="S0.I1.i2.p1.4.m4.1.1.3.cmml" xref="S0.I1.i2.p1.4.m4.1.1.3">𝐿</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S0.I1.i2.p1.4.m4.1c">G\cup L</annotation><annotation encoding="application/x-llamapun" id="S0.I1.i2.p1.4.m4.1d">italic_G ∪ italic_L</annotation></semantics></math> is <math alttext="(k+1)" class="ltx_Math" display="inline" id="S0.I1.i2.p1.5.m5.1"><semantics id="S0.I1.i2.p1.5.m5.1a"><mrow id="S0.I1.i2.p1.5.m5.1.1.1" xref="S0.I1.i2.p1.5.m5.1.1.1.1.cmml"><mo id="S0.I1.i2.p1.5.m5.1.1.1.2" stretchy="false" xref="S0.I1.i2.p1.5.m5.1.1.1.1.cmml">(</mo><mrow id="S0.I1.i2.p1.5.m5.1.1.1.1" xref="S0.I1.i2.p1.5.m5.1.1.1.1.cmml"><mi id="S0.I1.i2.p1.5.m5.1.1.1.1.2" xref="S0.I1.i2.p1.5.m5.1.1.1.1.2.cmml">k</mi><mo id="S0.I1.i2.p1.5.m5.1.1.1.1.1" xref="S0.I1.i2.p1.5.m5.1.1.1.1.1.cmml">+</mo><mn id="S0.I1.i2.p1.5.m5.1.1.1.1.3" xref="S0.I1.i2.p1.5.m5.1.1.1.1.3.cmml">1</mn></mrow><mo id="S0.I1.i2.p1.5.m5.1.1.1.3" stretchy="false" xref="S0.I1.i2.p1.5.m5.1.1.1.1.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S0.I1.i2.p1.5.m5.1b"><apply id="S0.I1.i2.p1.5.m5.1.1.1.1.cmml" xref="S0.I1.i2.p1.5.m5.1.1.1"><plus id="S0.I1.i2.p1.5.m5.1.1.1.1.1.cmml" xref="S0.I1.i2.p1.5.m5.1.1.1.1.1"></plus><ci id="S0.I1.i2.p1.5.m5.1.1.1.1.2.cmml" xref="S0.I1.i2.p1.5.m5.1.1.1.1.2">𝑘</ci><cn id="S0.I1.i2.p1.5.m5.1.1.1.1.3.cmml" type="integer" xref="S0.I1.i2.p1.5.m5.1.1.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S0.I1.i2.p1.5.m5.1c">(k+1)</annotation><annotation encoding="application/x-llamapun" id="S0.I1.i2.p1.5.m5.1d">( italic_k + 1 )</annotation></semantics></math>-vertex-connected. We obtain constant-factor approximations in near-linear space for <math alttext="k=1,2" class="ltx_Math" display="inline" id="S0.I1.i2.p1.6.m6.2"><semantics id="S0.I1.i2.p1.6.m6.2a"><mrow id="S0.I1.i2.p1.6.m6.2.3" xref="S0.I1.i2.p1.6.m6.2.3.cmml"><mi id="S0.I1.i2.p1.6.m6.2.3.2" xref="S0.I1.i2.p1.6.m6.2.3.2.cmml">k</mi><mo id="S0.I1.i2.p1.6.m6.2.3.1" xref="S0.I1.i2.p1.6.m6.2.3.1.cmml">=</mo><mrow id="S0.I1.i2.p1.6.m6.2.3.3.2" xref="S0.I1.i2.p1.6.m6.2.3.3.1.cmml"><mn id="S0.I1.i2.p1.6.m6.1.1" xref="S0.I1.i2.p1.6.m6.1.1.cmml">1</mn><mo id="S0.I1.i2.p1.6.m6.2.3.3.2.1" xref="S0.I1.i2.p1.6.m6.2.3.3.1.cmml">,</mo><mn id="S0.I1.i2.p1.6.m6.2.2" xref="S0.I1.i2.p1.6.m6.2.2.cmml">2</mn></mrow></mrow><annotation-xml encoding="MathML-Content" id="S0.I1.i2.p1.6.m6.2b"><apply id="S0.I1.i2.p1.6.m6.2.3.cmml" xref="S0.I1.i2.p1.6.m6.2.3"><eq id="S0.I1.i2.p1.6.m6.2.3.1.cmml" xref="S0.I1.i2.p1.6.m6.2.3.1"></eq><ci id="S0.I1.i2.p1.6.m6.2.3.2.cmml" xref="S0.I1.i2.p1.6.m6.2.3.2">𝑘</ci><list id="S0.I1.i2.p1.6.m6.2.3.3.1.cmml" xref="S0.I1.i2.p1.6.m6.2.3.3.2"><cn id="S0.I1.i2.p1.6.m6.1.1.cmml" type="integer" xref="S0.I1.i2.p1.6.m6.1.1">1</cn><cn id="S0.I1.i2.p1.6.m6.2.2.cmml" type="integer" xref="S0.I1.i2.p1.6.m6.2.2">2</cn></list></apply></annotation-xml><annotation encoding="application/x-tex" id="S0.I1.i2.p1.6.m6.2c">k=1,2</annotation><annotation encoding="application/x-llamapun" id="S0.I1.i2.p1.6.m6.2d">italic_k = 1 , 2</annotation></semantics></math>. Our result for <math alttext="k=2" class="ltx_Math" display="inline" id="S0.I1.i2.p1.7.m7.1"><semantics id="S0.I1.i2.p1.7.m7.1a"><mrow id="S0.I1.i2.p1.7.m7.1.1" xref="S0.I1.i2.p1.7.m7.1.1.cmml"><mi id="S0.I1.i2.p1.7.m7.1.1.2" xref="S0.I1.i2.p1.7.m7.1.1.2.cmml">k</mi><mo id="S0.I1.i2.p1.7.m7.1.1.1" xref="S0.I1.i2.p1.7.m7.1.1.1.cmml">=</mo><mn id="S0.I1.i2.p1.7.m7.1.1.3" xref="S0.I1.i2.p1.7.m7.1.1.3.cmml">2</mn></mrow><annotation-xml encoding="MathML-Content" id="S0.I1.i2.p1.7.m7.1b"><apply id="S0.I1.i2.p1.7.m7.1.1.cmml" xref="S0.I1.i2.p1.7.m7.1.1"><eq id="S0.I1.i2.p1.7.m7.1.1.1.cmml" xref="S0.I1.i2.p1.7.m7.1.1.1"></eq><ci id="S0.I1.i2.p1.7.m7.1.1.2.cmml" xref="S0.I1.i2.p1.7.m7.1.1.2">𝑘</ci><cn id="S0.I1.i2.p1.7.m7.1.1.3.cmml" type="integer" xref="S0.I1.i2.p1.7.m7.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S0.I1.i2.p1.7.m7.1c">k=2</annotation><annotation encoding="application/x-llamapun" id="S0.I1.i2.p1.7.m7.1d">italic_k = 2</annotation></semantics></math> is based on using the SPQR tree, a novel application for this well-known representation of <math alttext="2" class="ltx_Math" display="inline" id="S0.I1.i2.p1.8.m8.1"><semantics id="S0.I1.i2.p1.8.m8.1a"><mn id="S0.I1.i2.p1.8.m8.1.1" xref="S0.I1.i2.p1.8.m8.1.1.cmml">2</mn><annotation-xml encoding="MathML-Content" id="S0.I1.i2.p1.8.m8.1b"><cn id="S0.I1.i2.p1.8.m8.1.1.cmml" type="integer" xref="S0.I1.i2.p1.8.m8.1.1">2</cn></annotation-xml><annotation encoding="application/x-tex" id="S0.I1.i2.p1.8.m8.1c">2</annotation><annotation encoding="application/x-llamapun" id="S0.I1.i2.p1.8.m8.1d">2</annotation></semantics></math>-connected graphs.</p> </div> </li> </ul> </div> <div class="ltx_pagination ltx_role_newpage"></div> <section class="ltx_section" id="S1"> <h2 class="ltx_title ltx_title_section"> <span class="ltx_tag ltx_tag_section">1 </span>Introduction</h2> <div class="ltx_para" id="S1.p1"> <p class="ltx_p" id="S1.p1.1">Network design is a classical area of research in combinatorial optimization that has been instrumental in the development and advancement of several important algorithmic techniques. Moreover, it has practical applications across a wide variety of domains. In many modern real-world settings, graphs are extremely large, making it impractical to run algorithms that require access to the entire graph at once. This motivates the study of graph algorithms in the streaming model of computation; this is a commonly used model for handling large-scale or real-time data. There has been an extensive line of work on graph algorithms in the streaming setting. In addition to its practical utility, this line of work has led to a variety of new theoretical advances that have had auxiliary benefits well-beyond what was initially anticipated. Some well-studied problems in the streaming model include matching <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.00712v1#bib.bibx66" title="">McG05</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.00712v1#bib.bibx47" title="">GKK12</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.00712v1#bib.bibx11" title="">AKLY16</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.00712v1#bib.bibx10" title="">AKL17</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.00712v1#bib.bibx56" title="">Kap21</a>]</cite>, max-cut <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.00712v1#bib.bibx57" title="">KKS14</a>]</cite>, spanners <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.00712v1#bib.bibx15" title="">Bas08</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.00712v1#bib.bibx38" title="">Elk11</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.00712v1#bib.bibx7" title="">AGM12</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.00712v1#bib.bibx63" title="">KW14</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.00712v1#bib.bibx44" title="">FVWY20</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.00712v1#bib.bibx42" title="">FKN21</a>]</cite>, sparsifiers <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.00712v1#bib.bibx7" title="">AGM12</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.00712v1#bib.bibx63" title="">KW14</a>]</cite>, shortest paths <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.00712v1#bib.bibx41" title="">FKM<sup class="ltx_sup"><span class="ltx_text ltx_font_italic">+</span></sup>08</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.00712v1#bib.bibx50" title="">GO16</a>]</cite>, and the minimum spanning tree problem <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.00712v1#bib.bibx7" title="">AGM12</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.00712v1#bib.bibx77" title="">SW15</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.00712v1#bib.bibx76" title="">NY19</a>]</cite>, among many others.</p> </div> <div class="ltx_para" id="S1.p2"> <p class="ltx_p" id="S1.p2.12">We consider the Survivable Network Design problem (SNDP). The input to this problem is an undirected graph <math alttext="G=(V,E)" class="ltx_Math" display="inline" id="S1.p2.1.m1.2"><semantics id="S1.p2.1.m1.2a"><mrow id="S1.p2.1.m1.2.3" xref="S1.p2.1.m1.2.3.cmml"><mi id="S1.p2.1.m1.2.3.2" xref="S1.p2.1.m1.2.3.2.cmml">G</mi><mo id="S1.p2.1.m1.2.3.1" xref="S1.p2.1.m1.2.3.1.cmml">=</mo><mrow id="S1.p2.1.m1.2.3.3.2" xref="S1.p2.1.m1.2.3.3.1.cmml"><mo id="S1.p2.1.m1.2.3.3.2.1" stretchy="false" xref="S1.p2.1.m1.2.3.3.1.cmml">(</mo><mi id="S1.p2.1.m1.1.1" xref="S1.p2.1.m1.1.1.cmml">V</mi><mo id="S1.p2.1.m1.2.3.3.2.2" xref="S1.p2.1.m1.2.3.3.1.cmml">,</mo><mi id="S1.p2.1.m1.2.2" xref="S1.p2.1.m1.2.2.cmml">E</mi><mo id="S1.p2.1.m1.2.3.3.2.3" stretchy="false" xref="S1.p2.1.m1.2.3.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.p2.1.m1.2b"><apply id="S1.p2.1.m1.2.3.cmml" xref="S1.p2.1.m1.2.3"><eq id="S1.p2.1.m1.2.3.1.cmml" xref="S1.p2.1.m1.2.3.1"></eq><ci id="S1.p2.1.m1.2.3.2.cmml" xref="S1.p2.1.m1.2.3.2">𝐺</ci><interval closure="open" id="S1.p2.1.m1.2.3.3.1.cmml" xref="S1.p2.1.m1.2.3.3.2"><ci id="S1.p2.1.m1.1.1.cmml" xref="S1.p2.1.m1.1.1">𝑉</ci><ci id="S1.p2.1.m1.2.2.cmml" xref="S1.p2.1.m1.2.2">𝐸</ci></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p2.1.m1.2c">G=(V,E)</annotation><annotation encoding="application/x-llamapun" id="S1.p2.1.m1.2d">italic_G = ( italic_V , italic_E )</annotation></semantics></math> with non-negative edge-weights <math alttext="w:E\to\mathbb{R}_{\geq 0}" class="ltx_Math" display="inline" id="S1.p2.2.m2.1"><semantics id="S1.p2.2.m2.1a"><mrow id="S1.p2.2.m2.1.1" xref="S1.p2.2.m2.1.1.cmml"><mi id="S1.p2.2.m2.1.1.2" xref="S1.p2.2.m2.1.1.2.cmml">w</mi><mo id="S1.p2.2.m2.1.1.1" lspace="0.278em" rspace="0.278em" xref="S1.p2.2.m2.1.1.1.cmml">:</mo><mrow id="S1.p2.2.m2.1.1.3" xref="S1.p2.2.m2.1.1.3.cmml"><mi id="S1.p2.2.m2.1.1.3.2" xref="S1.p2.2.m2.1.1.3.2.cmml">E</mi><mo id="S1.p2.2.m2.1.1.3.1" stretchy="false" xref="S1.p2.2.m2.1.1.3.1.cmml">→</mo><msub id="S1.p2.2.m2.1.1.3.3" xref="S1.p2.2.m2.1.1.3.3.cmml"><mi id="S1.p2.2.m2.1.1.3.3.2" xref="S1.p2.2.m2.1.1.3.3.2.cmml">ℝ</mi><mrow id="S1.p2.2.m2.1.1.3.3.3" xref="S1.p2.2.m2.1.1.3.3.3.cmml"><mi id="S1.p2.2.m2.1.1.3.3.3.2" xref="S1.p2.2.m2.1.1.3.3.3.2.cmml"></mi><mo id="S1.p2.2.m2.1.1.3.3.3.1" xref="S1.p2.2.m2.1.1.3.3.3.1.cmml">≥</mo><mn id="S1.p2.2.m2.1.1.3.3.3.3" xref="S1.p2.2.m2.1.1.3.3.3.3.cmml">0</mn></mrow></msub></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.p2.2.m2.1b"><apply id="S1.p2.2.m2.1.1.cmml" xref="S1.p2.2.m2.1.1"><ci id="S1.p2.2.m2.1.1.1.cmml" xref="S1.p2.2.m2.1.1.1">:</ci><ci id="S1.p2.2.m2.1.1.2.cmml" xref="S1.p2.2.m2.1.1.2">𝑤</ci><apply id="S1.p2.2.m2.1.1.3.cmml" xref="S1.p2.2.m2.1.1.3"><ci id="S1.p2.2.m2.1.1.3.1.cmml" xref="S1.p2.2.m2.1.1.3.1">→</ci><ci id="S1.p2.2.m2.1.1.3.2.cmml" xref="S1.p2.2.m2.1.1.3.2">𝐸</ci><apply id="S1.p2.2.m2.1.1.3.3.cmml" xref="S1.p2.2.m2.1.1.3.3"><csymbol cd="ambiguous" id="S1.p2.2.m2.1.1.3.3.1.cmml" xref="S1.p2.2.m2.1.1.3.3">subscript</csymbol><ci id="S1.p2.2.m2.1.1.3.3.2.cmml" xref="S1.p2.2.m2.1.1.3.3.2">ℝ</ci><apply id="S1.p2.2.m2.1.1.3.3.3.cmml" xref="S1.p2.2.m2.1.1.3.3.3"><geq id="S1.p2.2.m2.1.1.3.3.3.1.cmml" xref="S1.p2.2.m2.1.1.3.3.3.1"></geq><csymbol cd="latexml" id="S1.p2.2.m2.1.1.3.3.3.2.cmml" xref="S1.p2.2.m2.1.1.3.3.3.2">absent</csymbol><cn id="S1.p2.2.m2.1.1.3.3.3.3.cmml" type="integer" xref="S1.p2.2.m2.1.1.3.3.3.3">0</cn></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p2.2.m2.1c">w:E\to\mathbb{R}_{\geq 0}</annotation><annotation encoding="application/x-llamapun" id="S1.p2.2.m2.1d">italic_w : italic_E → blackboard_R start_POSTSUBSCRIPT ≥ 0 end_POSTSUBSCRIPT</annotation></semantics></math> and an integer connectivity requirement <math alttext="r(uv)" class="ltx_Math" display="inline" id="S1.p2.3.m3.1"><semantics id="S1.p2.3.m3.1a"><mrow id="S1.p2.3.m3.1.1" xref="S1.p2.3.m3.1.1.cmml"><mi id="S1.p2.3.m3.1.1.3" xref="S1.p2.3.m3.1.1.3.cmml">r</mi><mo id="S1.p2.3.m3.1.1.2" xref="S1.p2.3.m3.1.1.2.cmml">⁢</mo><mrow id="S1.p2.3.m3.1.1.1.1" xref="S1.p2.3.m3.1.1.1.1.1.cmml"><mo id="S1.p2.3.m3.1.1.1.1.2" stretchy="false" xref="S1.p2.3.m3.1.1.1.1.1.cmml">(</mo><mrow id="S1.p2.3.m3.1.1.1.1.1" xref="S1.p2.3.m3.1.1.1.1.1.cmml"><mi id="S1.p2.3.m3.1.1.1.1.1.2" xref="S1.p2.3.m3.1.1.1.1.1.2.cmml">u</mi><mo id="S1.p2.3.m3.1.1.1.1.1.1" xref="S1.p2.3.m3.1.1.1.1.1.1.cmml">⁢</mo><mi id="S1.p2.3.m3.1.1.1.1.1.3" xref="S1.p2.3.m3.1.1.1.1.1.3.cmml">v</mi></mrow><mo id="S1.p2.3.m3.1.1.1.1.3" stretchy="false" xref="S1.p2.3.m3.1.1.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.p2.3.m3.1b"><apply id="S1.p2.3.m3.1.1.cmml" xref="S1.p2.3.m3.1.1"><times id="S1.p2.3.m3.1.1.2.cmml" xref="S1.p2.3.m3.1.1.2"></times><ci id="S1.p2.3.m3.1.1.3.cmml" xref="S1.p2.3.m3.1.1.3">𝑟</ci><apply id="S1.p2.3.m3.1.1.1.1.1.cmml" xref="S1.p2.3.m3.1.1.1.1"><times id="S1.p2.3.m3.1.1.1.1.1.1.cmml" xref="S1.p2.3.m3.1.1.1.1.1.1"></times><ci id="S1.p2.3.m3.1.1.1.1.1.2.cmml" xref="S1.p2.3.m3.1.1.1.1.1.2">𝑢</ci><ci id="S1.p2.3.m3.1.1.1.1.1.3.cmml" xref="S1.p2.3.m3.1.1.1.1.1.3">𝑣</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p2.3.m3.1c">r(uv)</annotation><annotation encoding="application/x-llamapun" id="S1.p2.3.m3.1d">italic_r ( italic_u italic_v )</annotation></semantics></math> for each unordered pair of vertices <math alttext="u,v\in V" class="ltx_Math" display="inline" id="S1.p2.4.m4.2"><semantics id="S1.p2.4.m4.2a"><mrow id="S1.p2.4.m4.2.3" xref="S1.p2.4.m4.2.3.cmml"><mrow id="S1.p2.4.m4.2.3.2.2" xref="S1.p2.4.m4.2.3.2.1.cmml"><mi id="S1.p2.4.m4.1.1" xref="S1.p2.4.m4.1.1.cmml">u</mi><mo id="S1.p2.4.m4.2.3.2.2.1" xref="S1.p2.4.m4.2.3.2.1.cmml">,</mo><mi id="S1.p2.4.m4.2.2" xref="S1.p2.4.m4.2.2.cmml">v</mi></mrow><mo id="S1.p2.4.m4.2.3.1" xref="S1.p2.4.m4.2.3.1.cmml">∈</mo><mi id="S1.p2.4.m4.2.3.3" xref="S1.p2.4.m4.2.3.3.cmml">V</mi></mrow><annotation-xml encoding="MathML-Content" id="S1.p2.4.m4.2b"><apply id="S1.p2.4.m4.2.3.cmml" xref="S1.p2.4.m4.2.3"><in id="S1.p2.4.m4.2.3.1.cmml" xref="S1.p2.4.m4.2.3.1"></in><list id="S1.p2.4.m4.2.3.2.1.cmml" xref="S1.p2.4.m4.2.3.2.2"><ci id="S1.p2.4.m4.1.1.cmml" xref="S1.p2.4.m4.1.1">𝑢</ci><ci id="S1.p2.4.m4.2.2.cmml" xref="S1.p2.4.m4.2.2">𝑣</ci></list><ci id="S1.p2.4.m4.2.3.3.cmml" xref="S1.p2.4.m4.2.3.3">𝑉</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p2.4.m4.2c">u,v\in V</annotation><annotation encoding="application/x-llamapun" id="S1.p2.4.m4.2d">italic_u , italic_v ∈ italic_V</annotation></semantics></math>. The objective is to find a minimum-weight subgraph <math alttext="H\subseteq G" class="ltx_Math" display="inline" id="S1.p2.5.m5.1"><semantics id="S1.p2.5.m5.1a"><mrow id="S1.p2.5.m5.1.1" xref="S1.p2.5.m5.1.1.cmml"><mi id="S1.p2.5.m5.1.1.2" xref="S1.p2.5.m5.1.1.2.cmml">H</mi><mo id="S1.p2.5.m5.1.1.1" xref="S1.p2.5.m5.1.1.1.cmml">⊆</mo><mi id="S1.p2.5.m5.1.1.3" xref="S1.p2.5.m5.1.1.3.cmml">G</mi></mrow><annotation-xml encoding="MathML-Content" id="S1.p2.5.m5.1b"><apply id="S1.p2.5.m5.1.1.cmml" xref="S1.p2.5.m5.1.1"><subset id="S1.p2.5.m5.1.1.1.cmml" xref="S1.p2.5.m5.1.1.1"></subset><ci id="S1.p2.5.m5.1.1.2.cmml" xref="S1.p2.5.m5.1.1.2">𝐻</ci><ci id="S1.p2.5.m5.1.1.3.cmml" xref="S1.p2.5.m5.1.1.3">𝐺</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p2.5.m5.1c">H\subseteq G</annotation><annotation encoding="application/x-llamapun" id="S1.p2.5.m5.1d">italic_H ⊆ italic_G</annotation></semantics></math> such that, for every pair of vertices <math alttext="u,v\in V" class="ltx_Math" display="inline" id="S1.p2.6.m6.2"><semantics id="S1.p2.6.m6.2a"><mrow id="S1.p2.6.m6.2.3" xref="S1.p2.6.m6.2.3.cmml"><mrow id="S1.p2.6.m6.2.3.2.2" xref="S1.p2.6.m6.2.3.2.1.cmml"><mi id="S1.p2.6.m6.1.1" xref="S1.p2.6.m6.1.1.cmml">u</mi><mo id="S1.p2.6.m6.2.3.2.2.1" xref="S1.p2.6.m6.2.3.2.1.cmml">,</mo><mi id="S1.p2.6.m6.2.2" xref="S1.p2.6.m6.2.2.cmml">v</mi></mrow><mo id="S1.p2.6.m6.2.3.1" xref="S1.p2.6.m6.2.3.1.cmml">∈</mo><mi id="S1.p2.6.m6.2.3.3" xref="S1.p2.6.m6.2.3.3.cmml">V</mi></mrow><annotation-xml encoding="MathML-Content" id="S1.p2.6.m6.2b"><apply id="S1.p2.6.m6.2.3.cmml" xref="S1.p2.6.m6.2.3"><in id="S1.p2.6.m6.2.3.1.cmml" xref="S1.p2.6.m6.2.3.1"></in><list id="S1.p2.6.m6.2.3.2.1.cmml" xref="S1.p2.6.m6.2.3.2.2"><ci id="S1.p2.6.m6.1.1.cmml" xref="S1.p2.6.m6.1.1">𝑢</ci><ci id="S1.p2.6.m6.2.2.cmml" xref="S1.p2.6.m6.2.2">𝑣</ci></list><ci id="S1.p2.6.m6.2.3.3.cmml" xref="S1.p2.6.m6.2.3.3">𝑉</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p2.6.m6.2c">u,v\in V</annotation><annotation encoding="application/x-llamapun" id="S1.p2.6.m6.2d">italic_u , italic_v ∈ italic_V</annotation></semantics></math>, there exist <math alttext="r(uv)" class="ltx_Math" display="inline" id="S1.p2.7.m7.1"><semantics id="S1.p2.7.m7.1a"><mrow id="S1.p2.7.m7.1.1" xref="S1.p2.7.m7.1.1.cmml"><mi id="S1.p2.7.m7.1.1.3" xref="S1.p2.7.m7.1.1.3.cmml">r</mi><mo id="S1.p2.7.m7.1.1.2" xref="S1.p2.7.m7.1.1.2.cmml">⁢</mo><mrow id="S1.p2.7.m7.1.1.1.1" xref="S1.p2.7.m7.1.1.1.1.1.cmml"><mo id="S1.p2.7.m7.1.1.1.1.2" stretchy="false" xref="S1.p2.7.m7.1.1.1.1.1.cmml">(</mo><mrow id="S1.p2.7.m7.1.1.1.1.1" xref="S1.p2.7.m7.1.1.1.1.1.cmml"><mi id="S1.p2.7.m7.1.1.1.1.1.2" xref="S1.p2.7.m7.1.1.1.1.1.2.cmml">u</mi><mo id="S1.p2.7.m7.1.1.1.1.1.1" xref="S1.p2.7.m7.1.1.1.1.1.1.cmml">⁢</mo><mi id="S1.p2.7.m7.1.1.1.1.1.3" xref="S1.p2.7.m7.1.1.1.1.1.3.cmml">v</mi></mrow><mo id="S1.p2.7.m7.1.1.1.1.3" stretchy="false" xref="S1.p2.7.m7.1.1.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.p2.7.m7.1b"><apply id="S1.p2.7.m7.1.1.cmml" xref="S1.p2.7.m7.1.1"><times id="S1.p2.7.m7.1.1.2.cmml" xref="S1.p2.7.m7.1.1.2"></times><ci id="S1.p2.7.m7.1.1.3.cmml" xref="S1.p2.7.m7.1.1.3">𝑟</ci><apply id="S1.p2.7.m7.1.1.1.1.1.cmml" xref="S1.p2.7.m7.1.1.1.1"><times id="S1.p2.7.m7.1.1.1.1.1.1.cmml" xref="S1.p2.7.m7.1.1.1.1.1.1"></times><ci id="S1.p2.7.m7.1.1.1.1.1.2.cmml" xref="S1.p2.7.m7.1.1.1.1.1.2">𝑢</ci><ci id="S1.p2.7.m7.1.1.1.1.1.3.cmml" xref="S1.p2.7.m7.1.1.1.1.1.3">𝑣</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p2.7.m7.1c">r(uv)</annotation><annotation encoding="application/x-llamapun" id="S1.p2.7.m7.1d">italic_r ( italic_u italic_v )</annotation></semantics></math> disjoint <math alttext="uv" class="ltx_Math" display="inline" id="S1.p2.8.m8.1"><semantics id="S1.p2.8.m8.1a"><mrow id="S1.p2.8.m8.1.1" xref="S1.p2.8.m8.1.1.cmml"><mi id="S1.p2.8.m8.1.1.2" xref="S1.p2.8.m8.1.1.2.cmml">u</mi><mo id="S1.p2.8.m8.1.1.1" xref="S1.p2.8.m8.1.1.1.cmml">⁢</mo><mi id="S1.p2.8.m8.1.1.3" xref="S1.p2.8.m8.1.1.3.cmml">v</mi></mrow><annotation-xml encoding="MathML-Content" id="S1.p2.8.m8.1b"><apply id="S1.p2.8.m8.1.1.cmml" xref="S1.p2.8.m8.1.1"><times id="S1.p2.8.m8.1.1.1.cmml" xref="S1.p2.8.m8.1.1.1"></times><ci id="S1.p2.8.m8.1.1.2.cmml" xref="S1.p2.8.m8.1.1.2">𝑢</ci><ci id="S1.p2.8.m8.1.1.3.cmml" xref="S1.p2.8.m8.1.1.3">𝑣</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p2.8.m8.1c">uv</annotation><annotation encoding="application/x-llamapun" id="S1.p2.8.m8.1d">italic_u italic_v</annotation></semantics></math>-paths in <math alttext="H" class="ltx_Math" display="inline" id="S1.p2.9.m9.1"><semantics id="S1.p2.9.m9.1a"><mi id="S1.p2.9.m9.1.1" xref="S1.p2.9.m9.1.1.cmml">H</mi><annotation-xml encoding="MathML-Content" id="S1.p2.9.m9.1b"><ci id="S1.p2.9.m9.1.1.cmml" xref="S1.p2.9.m9.1.1">𝐻</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.p2.9.m9.1c">H</annotation><annotation encoding="application/x-llamapun" id="S1.p2.9.m9.1d">italic_H</annotation></semantics></math>. If the paths for the pairs are required to be <span class="ltx_text ltx_font_italic" id="S1.p2.12.1">edge-disjoint</span>, the problem is referred to as <span class="ltx_text ltx_font_italic" id="S1.p2.12.2">edge-connectivity</span> SNDP (EC-SNDP), and if they are required to be <span class="ltx_text ltx_font_italic" id="S1.p2.12.3">vertex-disjoint</span>, it is known as the <span class="ltx_text ltx_font_italic" id="S1.p2.12.4">vertex-connectivity</span> SNDP (VC-SNDP). We refer to the maximum connectivity requirement as <math alttext="k:=\max_{uv}r(uv)" class="ltx_Math" display="inline" id="S1.p2.10.m10.1"><semantics id="S1.p2.10.m10.1a"><mrow id="S1.p2.10.m10.1.1" xref="S1.p2.10.m10.1.1.cmml"><mi id="S1.p2.10.m10.1.1.3" xref="S1.p2.10.m10.1.1.3.cmml">k</mi><mo id="S1.p2.10.m10.1.1.2" lspace="0.278em" rspace="0.278em" xref="S1.p2.10.m10.1.1.2.cmml">:=</mo><mrow id="S1.p2.10.m10.1.1.1" xref="S1.p2.10.m10.1.1.1.cmml"><mrow id="S1.p2.10.m10.1.1.1.3" xref="S1.p2.10.m10.1.1.1.3.cmml"><msub id="S1.p2.10.m10.1.1.1.3.1" xref="S1.p2.10.m10.1.1.1.3.1.cmml"><mi id="S1.p2.10.m10.1.1.1.3.1.2" xref="S1.p2.10.m10.1.1.1.3.1.2.cmml">max</mi><mrow id="S1.p2.10.m10.1.1.1.3.1.3" xref="S1.p2.10.m10.1.1.1.3.1.3.cmml"><mi id="S1.p2.10.m10.1.1.1.3.1.3.2" xref="S1.p2.10.m10.1.1.1.3.1.3.2.cmml">u</mi><mo id="S1.p2.10.m10.1.1.1.3.1.3.1" xref="S1.p2.10.m10.1.1.1.3.1.3.1.cmml">⁢</mo><mi id="S1.p2.10.m10.1.1.1.3.1.3.3" xref="S1.p2.10.m10.1.1.1.3.1.3.3.cmml">v</mi></mrow></msub><mo id="S1.p2.10.m10.1.1.1.3a" lspace="0.167em" xref="S1.p2.10.m10.1.1.1.3.cmml">⁡</mo><mi id="S1.p2.10.m10.1.1.1.3.2" xref="S1.p2.10.m10.1.1.1.3.2.cmml">r</mi></mrow><mo id="S1.p2.10.m10.1.1.1.2" xref="S1.p2.10.m10.1.1.1.2.cmml">⁢</mo><mrow id="S1.p2.10.m10.1.1.1.1.1" xref="S1.p2.10.m10.1.1.1.1.1.1.cmml"><mo id="S1.p2.10.m10.1.1.1.1.1.2" stretchy="false" xref="S1.p2.10.m10.1.1.1.1.1.1.cmml">(</mo><mrow id="S1.p2.10.m10.1.1.1.1.1.1" xref="S1.p2.10.m10.1.1.1.1.1.1.cmml"><mi id="S1.p2.10.m10.1.1.1.1.1.1.2" xref="S1.p2.10.m10.1.1.1.1.1.1.2.cmml">u</mi><mo id="S1.p2.10.m10.1.1.1.1.1.1.1" xref="S1.p2.10.m10.1.1.1.1.1.1.1.cmml">⁢</mo><mi id="S1.p2.10.m10.1.1.1.1.1.1.3" xref="S1.p2.10.m10.1.1.1.1.1.1.3.cmml">v</mi></mrow><mo id="S1.p2.10.m10.1.1.1.1.1.3" stretchy="false" xref="S1.p2.10.m10.1.1.1.1.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.p2.10.m10.1b"><apply id="S1.p2.10.m10.1.1.cmml" xref="S1.p2.10.m10.1.1"><csymbol cd="latexml" id="S1.p2.10.m10.1.1.2.cmml" xref="S1.p2.10.m10.1.1.2">assign</csymbol><ci id="S1.p2.10.m10.1.1.3.cmml" xref="S1.p2.10.m10.1.1.3">𝑘</ci><apply id="S1.p2.10.m10.1.1.1.cmml" xref="S1.p2.10.m10.1.1.1"><times id="S1.p2.10.m10.1.1.1.2.cmml" xref="S1.p2.10.m10.1.1.1.2"></times><apply id="S1.p2.10.m10.1.1.1.3.cmml" xref="S1.p2.10.m10.1.1.1.3"><apply id="S1.p2.10.m10.1.1.1.3.1.cmml" xref="S1.p2.10.m10.1.1.1.3.1"><csymbol cd="ambiguous" id="S1.p2.10.m10.1.1.1.3.1.1.cmml" xref="S1.p2.10.m10.1.1.1.3.1">subscript</csymbol><max id="S1.p2.10.m10.1.1.1.3.1.2.cmml" xref="S1.p2.10.m10.1.1.1.3.1.2"></max><apply id="S1.p2.10.m10.1.1.1.3.1.3.cmml" xref="S1.p2.10.m10.1.1.1.3.1.3"><times id="S1.p2.10.m10.1.1.1.3.1.3.1.cmml" xref="S1.p2.10.m10.1.1.1.3.1.3.1"></times><ci id="S1.p2.10.m10.1.1.1.3.1.3.2.cmml" xref="S1.p2.10.m10.1.1.1.3.1.3.2">𝑢</ci><ci id="S1.p2.10.m10.1.1.1.3.1.3.3.cmml" xref="S1.p2.10.m10.1.1.1.3.1.3.3">𝑣</ci></apply></apply><ci id="S1.p2.10.m10.1.1.1.3.2.cmml" xref="S1.p2.10.m10.1.1.1.3.2">𝑟</ci></apply><apply id="S1.p2.10.m10.1.1.1.1.1.1.cmml" xref="S1.p2.10.m10.1.1.1.1.1"><times id="S1.p2.10.m10.1.1.1.1.1.1.1.cmml" xref="S1.p2.10.m10.1.1.1.1.1.1.1"></times><ci id="S1.p2.10.m10.1.1.1.1.1.1.2.cmml" xref="S1.p2.10.m10.1.1.1.1.1.1.2">𝑢</ci><ci id="S1.p2.10.m10.1.1.1.1.1.1.3.cmml" xref="S1.p2.10.m10.1.1.1.1.1.1.3">𝑣</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p2.10.m10.1c">k:=\max_{uv}r(uv)</annotation><annotation encoding="application/x-llamapun" id="S1.p2.10.m10.1d">italic_k := roman_max start_POSTSUBSCRIPT italic_u italic_v end_POSTSUBSCRIPT italic_r ( italic_u italic_v )</annotation></semantics></math>. SNDP is a fundamental problem that generalizes many well-known polynomial-time solvable problems, including minimum spanning tree (MST) and <math alttext="s" class="ltx_Math" display="inline" id="S1.p2.11.m11.1"><semantics id="S1.p2.11.m11.1a"><mi id="S1.p2.11.m11.1.1" xref="S1.p2.11.m11.1.1.cmml">s</mi><annotation-xml encoding="MathML-Content" id="S1.p2.11.m11.1b"><ci id="S1.p2.11.m11.1.1.cmml" xref="S1.p2.11.m11.1.1">𝑠</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.p2.11.m11.1c">s</annotation><annotation encoding="application/x-llamapun" id="S1.p2.11.m11.1d">italic_s</annotation></semantics></math>-<math alttext="t" class="ltx_Math" display="inline" id="S1.p2.12.m12.1"><semantics id="S1.p2.12.m12.1a"><mi id="S1.p2.12.m12.1.1" xref="S1.p2.12.m12.1.1.cmml">t</mi><annotation-xml encoding="MathML-Content" id="S1.p2.12.m12.1b"><ci id="S1.p2.12.m12.1.1.cmml" xref="S1.p2.12.m12.1.1">𝑡</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.p2.12.m12.1c">t</annotation><annotation encoding="application/x-llamapun" id="S1.p2.12.m12.1d">italic_t</annotation></semantics></math> shortest path, as well as several NP-hard problems, including Steiner Tree and Steiner Forest. We define some special cases of interest:</p> <ul class="ltx_itemize" id="S1.I1"> <li class="ltx_item" id="S1.I1.i1" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">•</span> <div class="ltx_para" id="S1.I1.i1.p1"> <p class="ltx_p" id="S1.I1.i1.p1.7"><em class="ltx_emph ltx_font_italic" id="S1.I1.i1.p1.1.1"><math alttext="k" class="ltx_Math" display="inline" id="S1.I1.i1.p1.1.1.m1.1"><semantics id="S1.I1.i1.p1.1.1.m1.1a"><mi id="S1.I1.i1.p1.1.1.m1.1.1" xref="S1.I1.i1.p1.1.1.m1.1.1.cmml">k</mi><annotation-xml encoding="MathML-Content" id="S1.I1.i1.p1.1.1.m1.1b"><ci id="S1.I1.i1.p1.1.1.m1.1.1.cmml" xref="S1.I1.i1.p1.1.1.m1.1.1">𝑘</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.I1.i1.p1.1.1.m1.1c">k</annotation><annotation encoding="application/x-llamapun" id="S1.I1.i1.p1.1.1.m1.1d">italic_k</annotation></semantics></math>-Connected Subgraph:</em> This is the special case in which <math alttext="r(uv)=k" class="ltx_Math" display="inline" id="S1.I1.i1.p1.2.m1.1"><semantics id="S1.I1.i1.p1.2.m1.1a"><mrow id="S1.I1.i1.p1.2.m1.1.1" xref="S1.I1.i1.p1.2.m1.1.1.cmml"><mrow id="S1.I1.i1.p1.2.m1.1.1.1" xref="S1.I1.i1.p1.2.m1.1.1.1.cmml"><mi id="S1.I1.i1.p1.2.m1.1.1.1.3" xref="S1.I1.i1.p1.2.m1.1.1.1.3.cmml">r</mi><mo id="S1.I1.i1.p1.2.m1.1.1.1.2" xref="S1.I1.i1.p1.2.m1.1.1.1.2.cmml">⁢</mo><mrow id="S1.I1.i1.p1.2.m1.1.1.1.1.1" xref="S1.I1.i1.p1.2.m1.1.1.1.1.1.1.cmml"><mo id="S1.I1.i1.p1.2.m1.1.1.1.1.1.2" stretchy="false" xref="S1.I1.i1.p1.2.m1.1.1.1.1.1.1.cmml">(</mo><mrow id="S1.I1.i1.p1.2.m1.1.1.1.1.1.1" xref="S1.I1.i1.p1.2.m1.1.1.1.1.1.1.cmml"><mi id="S1.I1.i1.p1.2.m1.1.1.1.1.1.1.2" xref="S1.I1.i1.p1.2.m1.1.1.1.1.1.1.2.cmml">u</mi><mo id="S1.I1.i1.p1.2.m1.1.1.1.1.1.1.1" xref="S1.I1.i1.p1.2.m1.1.1.1.1.1.1.1.cmml">⁢</mo><mi id="S1.I1.i1.p1.2.m1.1.1.1.1.1.1.3" xref="S1.I1.i1.p1.2.m1.1.1.1.1.1.1.3.cmml">v</mi></mrow><mo id="S1.I1.i1.p1.2.m1.1.1.1.1.1.3" stretchy="false" xref="S1.I1.i1.p1.2.m1.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S1.I1.i1.p1.2.m1.1.1.2" xref="S1.I1.i1.p1.2.m1.1.1.2.cmml">=</mo><mi id="S1.I1.i1.p1.2.m1.1.1.3" xref="S1.I1.i1.p1.2.m1.1.1.3.cmml">k</mi></mrow><annotation-xml encoding="MathML-Content" id="S1.I1.i1.p1.2.m1.1b"><apply id="S1.I1.i1.p1.2.m1.1.1.cmml" xref="S1.I1.i1.p1.2.m1.1.1"><eq id="S1.I1.i1.p1.2.m1.1.1.2.cmml" xref="S1.I1.i1.p1.2.m1.1.1.2"></eq><apply id="S1.I1.i1.p1.2.m1.1.1.1.cmml" xref="S1.I1.i1.p1.2.m1.1.1.1"><times id="S1.I1.i1.p1.2.m1.1.1.1.2.cmml" xref="S1.I1.i1.p1.2.m1.1.1.1.2"></times><ci id="S1.I1.i1.p1.2.m1.1.1.1.3.cmml" xref="S1.I1.i1.p1.2.m1.1.1.1.3">𝑟</ci><apply id="S1.I1.i1.p1.2.m1.1.1.1.1.1.1.cmml" xref="S1.I1.i1.p1.2.m1.1.1.1.1.1"><times id="S1.I1.i1.p1.2.m1.1.1.1.1.1.1.1.cmml" xref="S1.I1.i1.p1.2.m1.1.1.1.1.1.1.1"></times><ci id="S1.I1.i1.p1.2.m1.1.1.1.1.1.1.2.cmml" xref="S1.I1.i1.p1.2.m1.1.1.1.1.1.1.2">𝑢</ci><ci id="S1.I1.i1.p1.2.m1.1.1.1.1.1.1.3.cmml" xref="S1.I1.i1.p1.2.m1.1.1.1.1.1.1.3">𝑣</ci></apply></apply><ci id="S1.I1.i1.p1.2.m1.1.1.3.cmml" xref="S1.I1.i1.p1.2.m1.1.1.3">𝑘</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.I1.i1.p1.2.m1.1c">r(uv)=k</annotation><annotation encoding="application/x-llamapun" id="S1.I1.i1.p1.2.m1.1d">italic_r ( italic_u italic_v ) = italic_k</annotation></semantics></math> for all <math alttext="u,v\in V" class="ltx_Math" display="inline" id="S1.I1.i1.p1.3.m2.2"><semantics id="S1.I1.i1.p1.3.m2.2a"><mrow id="S1.I1.i1.p1.3.m2.2.3" xref="S1.I1.i1.p1.3.m2.2.3.cmml"><mrow id="S1.I1.i1.p1.3.m2.2.3.2.2" xref="S1.I1.i1.p1.3.m2.2.3.2.1.cmml"><mi id="S1.I1.i1.p1.3.m2.1.1" xref="S1.I1.i1.p1.3.m2.1.1.cmml">u</mi><mo id="S1.I1.i1.p1.3.m2.2.3.2.2.1" xref="S1.I1.i1.p1.3.m2.2.3.2.1.cmml">,</mo><mi id="S1.I1.i1.p1.3.m2.2.2" xref="S1.I1.i1.p1.3.m2.2.2.cmml">v</mi></mrow><mo id="S1.I1.i1.p1.3.m2.2.3.1" xref="S1.I1.i1.p1.3.m2.2.3.1.cmml">∈</mo><mi id="S1.I1.i1.p1.3.m2.2.3.3" xref="S1.I1.i1.p1.3.m2.2.3.3.cmml">V</mi></mrow><annotation-xml encoding="MathML-Content" id="S1.I1.i1.p1.3.m2.2b"><apply id="S1.I1.i1.p1.3.m2.2.3.cmml" xref="S1.I1.i1.p1.3.m2.2.3"><in id="S1.I1.i1.p1.3.m2.2.3.1.cmml" xref="S1.I1.i1.p1.3.m2.2.3.1"></in><list id="S1.I1.i1.p1.3.m2.2.3.2.1.cmml" xref="S1.I1.i1.p1.3.m2.2.3.2.2"><ci id="S1.I1.i1.p1.3.m2.1.1.cmml" xref="S1.I1.i1.p1.3.m2.1.1">𝑢</ci><ci id="S1.I1.i1.p1.3.m2.2.2.cmml" xref="S1.I1.i1.p1.3.m2.2.2">𝑣</ci></list><ci id="S1.I1.i1.p1.3.m2.2.3.3.cmml" xref="S1.I1.i1.p1.3.m2.2.3.3">𝑉</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.I1.i1.p1.3.m2.2c">u,v\in V</annotation><annotation encoding="application/x-llamapun" id="S1.I1.i1.p1.3.m2.2d">italic_u , italic_v ∈ italic_V</annotation></semantics></math>. We denote the edge version as <math alttext="k" class="ltx_Math" display="inline" id="S1.I1.i1.p1.4.m3.1"><semantics id="S1.I1.i1.p1.4.m3.1a"><mi id="S1.I1.i1.p1.4.m3.1.1" xref="S1.I1.i1.p1.4.m3.1.1.cmml">k</mi><annotation-xml encoding="MathML-Content" id="S1.I1.i1.p1.4.m3.1b"><ci id="S1.I1.i1.p1.4.m3.1.1.cmml" xref="S1.I1.i1.p1.4.m3.1.1">𝑘</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.I1.i1.p1.4.m3.1c">k</annotation><annotation encoding="application/x-llamapun" id="S1.I1.i1.p1.4.m3.1d">italic_k</annotation></semantics></math>-ECSS and the vertex version as <math alttext="k" class="ltx_Math" display="inline" id="S1.I1.i1.p1.5.m4.1"><semantics id="S1.I1.i1.p1.5.m4.1a"><mi id="S1.I1.i1.p1.5.m4.1.1" xref="S1.I1.i1.p1.5.m4.1.1.cmml">k</mi><annotation-xml encoding="MathML-Content" id="S1.I1.i1.p1.5.m4.1b"><ci id="S1.I1.i1.p1.5.m4.1.1.cmml" xref="S1.I1.i1.p1.5.m4.1.1">𝑘</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.I1.i1.p1.5.m4.1c">k</annotation><annotation encoding="application/x-llamapun" id="S1.I1.i1.p1.5.m4.1d">italic_k</annotation></semantics></math>-VCSS; the goal is to find a min-weight <math alttext="k" class="ltx_Math" display="inline" id="S1.I1.i1.p1.6.m5.1"><semantics id="S1.I1.i1.p1.6.m5.1a"><mi id="S1.I1.i1.p1.6.m5.1.1" xref="S1.I1.i1.p1.6.m5.1.1.cmml">k</mi><annotation-xml encoding="MathML-Content" id="S1.I1.i1.p1.6.m5.1b"><ci id="S1.I1.i1.p1.6.m5.1.1.cmml" xref="S1.I1.i1.p1.6.m5.1.1">𝑘</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.I1.i1.p1.6.m5.1c">k</annotation><annotation encoding="application/x-llamapun" id="S1.I1.i1.p1.6.m5.1d">italic_k</annotation></semantics></math>-edge-connected and <math alttext="k" class="ltx_Math" display="inline" id="S1.I1.i1.p1.7.m6.1"><semantics id="S1.I1.i1.p1.7.m6.1a"><mi id="S1.I1.i1.p1.7.m6.1.1" xref="S1.I1.i1.p1.7.m6.1.1.cmml">k</mi><annotation-xml encoding="MathML-Content" id="S1.I1.i1.p1.7.m6.1b"><ci id="S1.I1.i1.p1.7.m6.1.1.cmml" xref="S1.I1.i1.p1.7.m6.1.1">𝑘</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.I1.i1.p1.7.m6.1c">k</annotation><annotation encoding="application/x-llamapun" id="S1.I1.i1.p1.7.m6.1d">italic_k</annotation></semantics></math>-vertex-connected spanning subgraph respectively.</p> </div> </li> <li class="ltx_item" id="S1.I1.i2" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">•</span> <div class="ltx_para" id="S1.I1.i2.p1"> <p class="ltx_p" id="S1.I1.i2.p1.12"><em class="ltx_emph ltx_font_italic" id="S1.I1.i2.p1.12.1">Connectivity Augmentation:</em> In this problem, we are given a partial solution <math alttext="G^{\prime}=(V,E^{\prime})\subseteq G" class="ltx_Math" display="inline" id="S1.I1.i2.p1.1.m1.2"><semantics id="S1.I1.i2.p1.1.m1.2a"><mrow id="S1.I1.i2.p1.1.m1.2.2" xref="S1.I1.i2.p1.1.m1.2.2.cmml"><msup id="S1.I1.i2.p1.1.m1.2.2.3" xref="S1.I1.i2.p1.1.m1.2.2.3.cmml"><mi id="S1.I1.i2.p1.1.m1.2.2.3.2" xref="S1.I1.i2.p1.1.m1.2.2.3.2.cmml">G</mi><mo id="S1.I1.i2.p1.1.m1.2.2.3.3" xref="S1.I1.i2.p1.1.m1.2.2.3.3.cmml">′</mo></msup><mo id="S1.I1.i2.p1.1.m1.2.2.4" xref="S1.I1.i2.p1.1.m1.2.2.4.cmml">=</mo><mrow id="S1.I1.i2.p1.1.m1.2.2.1.1" xref="S1.I1.i2.p1.1.m1.2.2.1.2.cmml"><mo id="S1.I1.i2.p1.1.m1.2.2.1.1.2" stretchy="false" xref="S1.I1.i2.p1.1.m1.2.2.1.2.cmml">(</mo><mi id="S1.I1.i2.p1.1.m1.1.1" xref="S1.I1.i2.p1.1.m1.1.1.cmml">V</mi><mo id="S1.I1.i2.p1.1.m1.2.2.1.1.3" xref="S1.I1.i2.p1.1.m1.2.2.1.2.cmml">,</mo><msup id="S1.I1.i2.p1.1.m1.2.2.1.1.1" xref="S1.I1.i2.p1.1.m1.2.2.1.1.1.cmml"><mi id="S1.I1.i2.p1.1.m1.2.2.1.1.1.2" xref="S1.I1.i2.p1.1.m1.2.2.1.1.1.2.cmml">E</mi><mo id="S1.I1.i2.p1.1.m1.2.2.1.1.1.3" xref="S1.I1.i2.p1.1.m1.2.2.1.1.1.3.cmml">′</mo></msup><mo id="S1.I1.i2.p1.1.m1.2.2.1.1.4" stretchy="false" xref="S1.I1.i2.p1.1.m1.2.2.1.2.cmml">)</mo></mrow><mo id="S1.I1.i2.p1.1.m1.2.2.5" xref="S1.I1.i2.p1.1.m1.2.2.5.cmml">⊆</mo><mi id="S1.I1.i2.p1.1.m1.2.2.6" xref="S1.I1.i2.p1.1.m1.2.2.6.cmml">G</mi></mrow><annotation-xml encoding="MathML-Content" id="S1.I1.i2.p1.1.m1.2b"><apply id="S1.I1.i2.p1.1.m1.2.2.cmml" xref="S1.I1.i2.p1.1.m1.2.2"><and id="S1.I1.i2.p1.1.m1.2.2a.cmml" xref="S1.I1.i2.p1.1.m1.2.2"></and><apply id="S1.I1.i2.p1.1.m1.2.2b.cmml" xref="S1.I1.i2.p1.1.m1.2.2"><eq id="S1.I1.i2.p1.1.m1.2.2.4.cmml" xref="S1.I1.i2.p1.1.m1.2.2.4"></eq><apply id="S1.I1.i2.p1.1.m1.2.2.3.cmml" xref="S1.I1.i2.p1.1.m1.2.2.3"><csymbol cd="ambiguous" id="S1.I1.i2.p1.1.m1.2.2.3.1.cmml" xref="S1.I1.i2.p1.1.m1.2.2.3">superscript</csymbol><ci id="S1.I1.i2.p1.1.m1.2.2.3.2.cmml" xref="S1.I1.i2.p1.1.m1.2.2.3.2">𝐺</ci><ci id="S1.I1.i2.p1.1.m1.2.2.3.3.cmml" xref="S1.I1.i2.p1.1.m1.2.2.3.3">′</ci></apply><interval closure="open" id="S1.I1.i2.p1.1.m1.2.2.1.2.cmml" xref="S1.I1.i2.p1.1.m1.2.2.1.1"><ci id="S1.I1.i2.p1.1.m1.1.1.cmml" xref="S1.I1.i2.p1.1.m1.1.1">𝑉</ci><apply id="S1.I1.i2.p1.1.m1.2.2.1.1.1.cmml" xref="S1.I1.i2.p1.1.m1.2.2.1.1.1"><csymbol cd="ambiguous" id="S1.I1.i2.p1.1.m1.2.2.1.1.1.1.cmml" xref="S1.I1.i2.p1.1.m1.2.2.1.1.1">superscript</csymbol><ci id="S1.I1.i2.p1.1.m1.2.2.1.1.1.2.cmml" xref="S1.I1.i2.p1.1.m1.2.2.1.1.1.2">𝐸</ci><ci id="S1.I1.i2.p1.1.m1.2.2.1.1.1.3.cmml" xref="S1.I1.i2.p1.1.m1.2.2.1.1.1.3">′</ci></apply></interval></apply><apply id="S1.I1.i2.p1.1.m1.2.2c.cmml" xref="S1.I1.i2.p1.1.m1.2.2"><subset id="S1.I1.i2.p1.1.m1.2.2.5.cmml" xref="S1.I1.i2.p1.1.m1.2.2.5"></subset><share href="https://arxiv.org/html/2503.00712v1#S1.I1.i2.p1.1.m1.2.2.1.cmml" id="S1.I1.i2.p1.1.m1.2.2d.cmml" xref="S1.I1.i2.p1.1.m1.2.2"></share><ci id="S1.I1.i2.p1.1.m1.2.2.6.cmml" xref="S1.I1.i2.p1.1.m1.2.2.6">𝐺</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.I1.i2.p1.1.m1.2c">G^{\prime}=(V,E^{\prime})\subseteq G</annotation><annotation encoding="application/x-llamapun" id="S1.I1.i2.p1.1.m1.2d">italic_G start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT = ( italic_V , italic_E start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) ⊆ italic_G</annotation></semantics></math> for “free”, with the guarantee that each <math alttext="u,v\in V" class="ltx_Math" display="inline" id="S1.I1.i2.p1.2.m2.2"><semantics id="S1.I1.i2.p1.2.m2.2a"><mrow id="S1.I1.i2.p1.2.m2.2.3" xref="S1.I1.i2.p1.2.m2.2.3.cmml"><mrow id="S1.I1.i2.p1.2.m2.2.3.2.2" xref="S1.I1.i2.p1.2.m2.2.3.2.1.cmml"><mi id="S1.I1.i2.p1.2.m2.1.1" xref="S1.I1.i2.p1.2.m2.1.1.cmml">u</mi><mo id="S1.I1.i2.p1.2.m2.2.3.2.2.1" xref="S1.I1.i2.p1.2.m2.2.3.2.1.cmml">,</mo><mi id="S1.I1.i2.p1.2.m2.2.2" xref="S1.I1.i2.p1.2.m2.2.2.cmml">v</mi></mrow><mo id="S1.I1.i2.p1.2.m2.2.3.1" xref="S1.I1.i2.p1.2.m2.2.3.1.cmml">∈</mo><mi id="S1.I1.i2.p1.2.m2.2.3.3" xref="S1.I1.i2.p1.2.m2.2.3.3.cmml">V</mi></mrow><annotation-xml encoding="MathML-Content" id="S1.I1.i2.p1.2.m2.2b"><apply id="S1.I1.i2.p1.2.m2.2.3.cmml" xref="S1.I1.i2.p1.2.m2.2.3"><in id="S1.I1.i2.p1.2.m2.2.3.1.cmml" xref="S1.I1.i2.p1.2.m2.2.3.1"></in><list id="S1.I1.i2.p1.2.m2.2.3.2.1.cmml" xref="S1.I1.i2.p1.2.m2.2.3.2.2"><ci id="S1.I1.i2.p1.2.m2.1.1.cmml" xref="S1.I1.i2.p1.2.m2.1.1">𝑢</ci><ci id="S1.I1.i2.p1.2.m2.2.2.cmml" xref="S1.I1.i2.p1.2.m2.2.2">𝑣</ci></list><ci id="S1.I1.i2.p1.2.m2.2.3.3.cmml" xref="S1.I1.i2.p1.2.m2.2.3.3">𝑉</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.I1.i2.p1.2.m2.2c">u,v\in V</annotation><annotation encoding="application/x-llamapun" id="S1.I1.i2.p1.2.m2.2d">italic_u , italic_v ∈ italic_V</annotation></semantics></math> is at least <math alttext="r(uv)-1" class="ltx_Math" display="inline" id="S1.I1.i2.p1.3.m3.1"><semantics id="S1.I1.i2.p1.3.m3.1a"><mrow id="S1.I1.i2.p1.3.m3.1.1" xref="S1.I1.i2.p1.3.m3.1.1.cmml"><mrow id="S1.I1.i2.p1.3.m3.1.1.1" xref="S1.I1.i2.p1.3.m3.1.1.1.cmml"><mi id="S1.I1.i2.p1.3.m3.1.1.1.3" xref="S1.I1.i2.p1.3.m3.1.1.1.3.cmml">r</mi><mo id="S1.I1.i2.p1.3.m3.1.1.1.2" xref="S1.I1.i2.p1.3.m3.1.1.1.2.cmml">⁢</mo><mrow id="S1.I1.i2.p1.3.m3.1.1.1.1.1" xref="S1.I1.i2.p1.3.m3.1.1.1.1.1.1.cmml"><mo id="S1.I1.i2.p1.3.m3.1.1.1.1.1.2" stretchy="false" xref="S1.I1.i2.p1.3.m3.1.1.1.1.1.1.cmml">(</mo><mrow id="S1.I1.i2.p1.3.m3.1.1.1.1.1.1" xref="S1.I1.i2.p1.3.m3.1.1.1.1.1.1.cmml"><mi id="S1.I1.i2.p1.3.m3.1.1.1.1.1.1.2" xref="S1.I1.i2.p1.3.m3.1.1.1.1.1.1.2.cmml">u</mi><mo id="S1.I1.i2.p1.3.m3.1.1.1.1.1.1.1" xref="S1.I1.i2.p1.3.m3.1.1.1.1.1.1.1.cmml">⁢</mo><mi id="S1.I1.i2.p1.3.m3.1.1.1.1.1.1.3" xref="S1.I1.i2.p1.3.m3.1.1.1.1.1.1.3.cmml">v</mi></mrow><mo id="S1.I1.i2.p1.3.m3.1.1.1.1.1.3" stretchy="false" xref="S1.I1.i2.p1.3.m3.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S1.I1.i2.p1.3.m3.1.1.2" xref="S1.I1.i2.p1.3.m3.1.1.2.cmml">−</mo><mn id="S1.I1.i2.p1.3.m3.1.1.3" xref="S1.I1.i2.p1.3.m3.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S1.I1.i2.p1.3.m3.1b"><apply id="S1.I1.i2.p1.3.m3.1.1.cmml" xref="S1.I1.i2.p1.3.m3.1.1"><minus id="S1.I1.i2.p1.3.m3.1.1.2.cmml" xref="S1.I1.i2.p1.3.m3.1.1.2"></minus><apply id="S1.I1.i2.p1.3.m3.1.1.1.cmml" xref="S1.I1.i2.p1.3.m3.1.1.1"><times id="S1.I1.i2.p1.3.m3.1.1.1.2.cmml" xref="S1.I1.i2.p1.3.m3.1.1.1.2"></times><ci id="S1.I1.i2.p1.3.m3.1.1.1.3.cmml" xref="S1.I1.i2.p1.3.m3.1.1.1.3">𝑟</ci><apply id="S1.I1.i2.p1.3.m3.1.1.1.1.1.1.cmml" xref="S1.I1.i2.p1.3.m3.1.1.1.1.1"><times id="S1.I1.i2.p1.3.m3.1.1.1.1.1.1.1.cmml" xref="S1.I1.i2.p1.3.m3.1.1.1.1.1.1.1"></times><ci id="S1.I1.i2.p1.3.m3.1.1.1.1.1.1.2.cmml" xref="S1.I1.i2.p1.3.m3.1.1.1.1.1.1.2">𝑢</ci><ci id="S1.I1.i2.p1.3.m3.1.1.1.1.1.1.3.cmml" xref="S1.I1.i2.p1.3.m3.1.1.1.1.1.1.3">𝑣</ci></apply></apply><cn id="S1.I1.i2.p1.3.m3.1.1.3.cmml" type="integer" xref="S1.I1.i2.p1.3.m3.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.I1.i2.p1.3.m3.1c">r(uv)-1</annotation><annotation encoding="application/x-llamapun" id="S1.I1.i2.p1.3.m3.1d">italic_r ( italic_u italic_v ) - 1</annotation></semantics></math> connected in <math alttext="G^{\prime}" class="ltx_Math" display="inline" id="S1.I1.i2.p1.4.m4.1"><semantics id="S1.I1.i2.p1.4.m4.1a"><msup id="S1.I1.i2.p1.4.m4.1.1" xref="S1.I1.i2.p1.4.m4.1.1.cmml"><mi id="S1.I1.i2.p1.4.m4.1.1.2" xref="S1.I1.i2.p1.4.m4.1.1.2.cmml">G</mi><mo id="S1.I1.i2.p1.4.m4.1.1.3" xref="S1.I1.i2.p1.4.m4.1.1.3.cmml">′</mo></msup><annotation-xml encoding="MathML-Content" id="S1.I1.i2.p1.4.m4.1b"><apply id="S1.I1.i2.p1.4.m4.1.1.cmml" xref="S1.I1.i2.p1.4.m4.1.1"><csymbol cd="ambiguous" id="S1.I1.i2.p1.4.m4.1.1.1.cmml" xref="S1.I1.i2.p1.4.m4.1.1">superscript</csymbol><ci id="S1.I1.i2.p1.4.m4.1.1.2.cmml" xref="S1.I1.i2.p1.4.m4.1.1.2">𝐺</ci><ci id="S1.I1.i2.p1.4.m4.1.1.3.cmml" xref="S1.I1.i2.p1.4.m4.1.1.3">′</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.I1.i2.p1.4.m4.1c">G^{\prime}</annotation><annotation encoding="application/x-llamapun" id="S1.I1.i2.p1.4.m4.1d">italic_G start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math>. The goal is to find a min-weight set of edges <math alttext="F\subseteq E\setminus E^{\prime}" class="ltx_Math" display="inline" id="S1.I1.i2.p1.5.m5.1"><semantics id="S1.I1.i2.p1.5.m5.1a"><mrow id="S1.I1.i2.p1.5.m5.1.1" xref="S1.I1.i2.p1.5.m5.1.1.cmml"><mi id="S1.I1.i2.p1.5.m5.1.1.2" xref="S1.I1.i2.p1.5.m5.1.1.2.cmml">F</mi><mo id="S1.I1.i2.p1.5.m5.1.1.1" xref="S1.I1.i2.p1.5.m5.1.1.1.cmml">⊆</mo><mrow id="S1.I1.i2.p1.5.m5.1.1.3" xref="S1.I1.i2.p1.5.m5.1.1.3.cmml"><mi id="S1.I1.i2.p1.5.m5.1.1.3.2" xref="S1.I1.i2.p1.5.m5.1.1.3.2.cmml">E</mi><mo id="S1.I1.i2.p1.5.m5.1.1.3.1" xref="S1.I1.i2.p1.5.m5.1.1.3.1.cmml">∖</mo><msup id="S1.I1.i2.p1.5.m5.1.1.3.3" xref="S1.I1.i2.p1.5.m5.1.1.3.3.cmml"><mi id="S1.I1.i2.p1.5.m5.1.1.3.3.2" xref="S1.I1.i2.p1.5.m5.1.1.3.3.2.cmml">E</mi><mo id="S1.I1.i2.p1.5.m5.1.1.3.3.3" xref="S1.I1.i2.p1.5.m5.1.1.3.3.3.cmml">′</mo></msup></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.I1.i2.p1.5.m5.1b"><apply id="S1.I1.i2.p1.5.m5.1.1.cmml" xref="S1.I1.i2.p1.5.m5.1.1"><subset id="S1.I1.i2.p1.5.m5.1.1.1.cmml" xref="S1.I1.i2.p1.5.m5.1.1.1"></subset><ci id="S1.I1.i2.p1.5.m5.1.1.2.cmml" xref="S1.I1.i2.p1.5.m5.1.1.2">𝐹</ci><apply id="S1.I1.i2.p1.5.m5.1.1.3.cmml" xref="S1.I1.i2.p1.5.m5.1.1.3"><setdiff id="S1.I1.i2.p1.5.m5.1.1.3.1.cmml" xref="S1.I1.i2.p1.5.m5.1.1.3.1"></setdiff><ci id="S1.I1.i2.p1.5.m5.1.1.3.2.cmml" xref="S1.I1.i2.p1.5.m5.1.1.3.2">𝐸</ci><apply id="S1.I1.i2.p1.5.m5.1.1.3.3.cmml" xref="S1.I1.i2.p1.5.m5.1.1.3.3"><csymbol cd="ambiguous" id="S1.I1.i2.p1.5.m5.1.1.3.3.1.cmml" xref="S1.I1.i2.p1.5.m5.1.1.3.3">superscript</csymbol><ci id="S1.I1.i2.p1.5.m5.1.1.3.3.2.cmml" xref="S1.I1.i2.p1.5.m5.1.1.3.3.2">𝐸</ci><ci id="S1.I1.i2.p1.5.m5.1.1.3.3.3.cmml" xref="S1.I1.i2.p1.5.m5.1.1.3.3.3">′</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.I1.i2.p1.5.m5.1c">F\subseteq E\setminus E^{\prime}</annotation><annotation encoding="application/x-llamapun" id="S1.I1.i2.p1.5.m5.1d">italic_F ⊆ italic_E ∖ italic_E start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math> that <em class="ltx_emph ltx_font_italic" id="S1.I1.i2.p1.12.2">augments</em> the connectivity of each <math alttext="u,v" class="ltx_Math" display="inline" id="S1.I1.i2.p1.6.m6.2"><semantics id="S1.I1.i2.p1.6.m6.2a"><mrow id="S1.I1.i2.p1.6.m6.2.3.2" xref="S1.I1.i2.p1.6.m6.2.3.1.cmml"><mi id="S1.I1.i2.p1.6.m6.1.1" xref="S1.I1.i2.p1.6.m6.1.1.cmml">u</mi><mo id="S1.I1.i2.p1.6.m6.2.3.2.1" xref="S1.I1.i2.p1.6.m6.2.3.1.cmml">,</mo><mi id="S1.I1.i2.p1.6.m6.2.2" xref="S1.I1.i2.p1.6.m6.2.2.cmml">v</mi></mrow><annotation-xml encoding="MathML-Content" id="S1.I1.i2.p1.6.m6.2b"><list id="S1.I1.i2.p1.6.m6.2.3.1.cmml" xref="S1.I1.i2.p1.6.m6.2.3.2"><ci id="S1.I1.i2.p1.6.m6.1.1.cmml" xref="S1.I1.i2.p1.6.m6.1.1">𝑢</ci><ci id="S1.I1.i2.p1.6.m6.2.2.cmml" xref="S1.I1.i2.p1.6.m6.2.2">𝑣</ci></list></annotation-xml><annotation encoding="application/x-tex" id="S1.I1.i2.p1.6.m6.2c">u,v</annotation><annotation encoding="application/x-llamapun" id="S1.I1.i2.p1.6.m6.2d">italic_u , italic_v</annotation></semantics></math> pair to <math alttext="r(uv)" class="ltx_Math" display="inline" id="S1.I1.i2.p1.7.m7.1"><semantics id="S1.I1.i2.p1.7.m7.1a"><mrow id="S1.I1.i2.p1.7.m7.1.1" xref="S1.I1.i2.p1.7.m7.1.1.cmml"><mi id="S1.I1.i2.p1.7.m7.1.1.3" xref="S1.I1.i2.p1.7.m7.1.1.3.cmml">r</mi><mo id="S1.I1.i2.p1.7.m7.1.1.2" xref="S1.I1.i2.p1.7.m7.1.1.2.cmml">⁢</mo><mrow id="S1.I1.i2.p1.7.m7.1.1.1.1" xref="S1.I1.i2.p1.7.m7.1.1.1.1.1.cmml"><mo id="S1.I1.i2.p1.7.m7.1.1.1.1.2" stretchy="false" xref="S1.I1.i2.p1.7.m7.1.1.1.1.1.cmml">(</mo><mrow id="S1.I1.i2.p1.7.m7.1.1.1.1.1" xref="S1.I1.i2.p1.7.m7.1.1.1.1.1.cmml"><mi id="S1.I1.i2.p1.7.m7.1.1.1.1.1.2" xref="S1.I1.i2.p1.7.m7.1.1.1.1.1.2.cmml">u</mi><mo id="S1.I1.i2.p1.7.m7.1.1.1.1.1.1" xref="S1.I1.i2.p1.7.m7.1.1.1.1.1.1.cmml">⁢</mo><mi id="S1.I1.i2.p1.7.m7.1.1.1.1.1.3" xref="S1.I1.i2.p1.7.m7.1.1.1.1.1.3.cmml">v</mi></mrow><mo id="S1.I1.i2.p1.7.m7.1.1.1.1.3" stretchy="false" xref="S1.I1.i2.p1.7.m7.1.1.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.I1.i2.p1.7.m7.1b"><apply id="S1.I1.i2.p1.7.m7.1.1.cmml" xref="S1.I1.i2.p1.7.m7.1.1"><times id="S1.I1.i2.p1.7.m7.1.1.2.cmml" xref="S1.I1.i2.p1.7.m7.1.1.2"></times><ci id="S1.I1.i2.p1.7.m7.1.1.3.cmml" xref="S1.I1.i2.p1.7.m7.1.1.3">𝑟</ci><apply id="S1.I1.i2.p1.7.m7.1.1.1.1.1.cmml" xref="S1.I1.i2.p1.7.m7.1.1.1.1"><times id="S1.I1.i2.p1.7.m7.1.1.1.1.1.1.cmml" xref="S1.I1.i2.p1.7.m7.1.1.1.1.1.1"></times><ci id="S1.I1.i2.p1.7.m7.1.1.1.1.1.2.cmml" xref="S1.I1.i2.p1.7.m7.1.1.1.1.1.2">𝑢</ci><ci id="S1.I1.i2.p1.7.m7.1.1.1.1.1.3.cmml" xref="S1.I1.i2.p1.7.m7.1.1.1.1.1.3">𝑣</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.I1.i2.p1.7.m7.1c">r(uv)</annotation><annotation encoding="application/x-llamapun" id="S1.I1.i2.p1.7.m7.1d">italic_r ( italic_u italic_v )</annotation></semantics></math>. The edges <math alttext="E\setminus E^{\prime}" class="ltx_Math" display="inline" id="S1.I1.i2.p1.8.m8.1"><semantics id="S1.I1.i2.p1.8.m8.1a"><mrow id="S1.I1.i2.p1.8.m8.1.1" xref="S1.I1.i2.p1.8.m8.1.1.cmml"><mi id="S1.I1.i2.p1.8.m8.1.1.2" xref="S1.I1.i2.p1.8.m8.1.1.2.cmml">E</mi><mo id="S1.I1.i2.p1.8.m8.1.1.1" xref="S1.I1.i2.p1.8.m8.1.1.1.cmml">∖</mo><msup id="S1.I1.i2.p1.8.m8.1.1.3" xref="S1.I1.i2.p1.8.m8.1.1.3.cmml"><mi id="S1.I1.i2.p1.8.m8.1.1.3.2" xref="S1.I1.i2.p1.8.m8.1.1.3.2.cmml">E</mi><mo id="S1.I1.i2.p1.8.m8.1.1.3.3" xref="S1.I1.i2.p1.8.m8.1.1.3.3.cmml">′</mo></msup></mrow><annotation-xml encoding="MathML-Content" id="S1.I1.i2.p1.8.m8.1b"><apply id="S1.I1.i2.p1.8.m8.1.1.cmml" xref="S1.I1.i2.p1.8.m8.1.1"><setdiff id="S1.I1.i2.p1.8.m8.1.1.1.cmml" xref="S1.I1.i2.p1.8.m8.1.1.1"></setdiff><ci id="S1.I1.i2.p1.8.m8.1.1.2.cmml" xref="S1.I1.i2.p1.8.m8.1.1.2">𝐸</ci><apply id="S1.I1.i2.p1.8.m8.1.1.3.cmml" xref="S1.I1.i2.p1.8.m8.1.1.3"><csymbol cd="ambiguous" id="S1.I1.i2.p1.8.m8.1.1.3.1.cmml" xref="S1.I1.i2.p1.8.m8.1.1.3">superscript</csymbol><ci id="S1.I1.i2.p1.8.m8.1.1.3.2.cmml" xref="S1.I1.i2.p1.8.m8.1.1.3.2">𝐸</ci><ci id="S1.I1.i2.p1.8.m8.1.1.3.3.cmml" xref="S1.I1.i2.p1.8.m8.1.1.3.3">′</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.I1.i2.p1.8.m8.1c">E\setminus E^{\prime}</annotation><annotation encoding="application/x-llamapun" id="S1.I1.i2.p1.8.m8.1d">italic_E ∖ italic_E start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math> are often referred to as <em class="ltx_emph ltx_font_italic" id="S1.I1.i2.p1.12.3">links</em>. We refer to the augmentation version of the edge/vertex spanning problems as <math alttext="k" class="ltx_Math" display="inline" id="S1.I1.i2.p1.9.m9.1"><semantics id="S1.I1.i2.p1.9.m9.1a"><mi id="S1.I1.i2.p1.9.m9.1.1" xref="S1.I1.i2.p1.9.m9.1.1.cmml">k</mi><annotation-xml encoding="MathML-Content" id="S1.I1.i2.p1.9.m9.1b"><ci id="S1.I1.i2.p1.9.m9.1.1.cmml" xref="S1.I1.i2.p1.9.m9.1.1">𝑘</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.I1.i2.p1.9.m9.1c">k</annotation><annotation encoding="application/x-llamapun" id="S1.I1.i2.p1.9.m9.1d">italic_k</annotation></semantics></math>-EC-CAP and <math alttext="k" class="ltx_Math" display="inline" id="S1.I1.i2.p1.10.m10.1"><semantics id="S1.I1.i2.p1.10.m10.1a"><mi id="S1.I1.i2.p1.10.m10.1.1" xref="S1.I1.i2.p1.10.m10.1.1.cmml">k</mi><annotation-xml encoding="MathML-Content" id="S1.I1.i2.p1.10.m10.1b"><ci id="S1.I1.i2.p1.10.m10.1.1.cmml" xref="S1.I1.i2.p1.10.m10.1.1">𝑘</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.I1.i2.p1.10.m10.1c">k</annotation><annotation encoding="application/x-llamapun" id="S1.I1.i2.p1.10.m10.1d">italic_k</annotation></semantics></math>-VC-CAP respectively: here we are given a <math alttext="k" class="ltx_Math" display="inline" id="S1.I1.i2.p1.11.m11.1"><semantics id="S1.I1.i2.p1.11.m11.1a"><mi id="S1.I1.i2.p1.11.m11.1.1" xref="S1.I1.i2.p1.11.m11.1.1.cmml">k</mi><annotation-xml encoding="MathML-Content" id="S1.I1.i2.p1.11.m11.1b"><ci id="S1.I1.i2.p1.11.m11.1.1.cmml" xref="S1.I1.i2.p1.11.m11.1.1">𝑘</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.I1.i2.p1.11.m11.1c">k</annotation><annotation encoding="application/x-llamapun" id="S1.I1.i2.p1.11.m11.1d">italic_k</annotation></semantics></math>-connected graph and the goal is to increase its connectivity to <math alttext="k+1" class="ltx_Math" display="inline" id="S1.I1.i2.p1.12.m12.1"><semantics id="S1.I1.i2.p1.12.m12.1a"><mrow id="S1.I1.i2.p1.12.m12.1.1" xref="S1.I1.i2.p1.12.m12.1.1.cmml"><mi id="S1.I1.i2.p1.12.m12.1.1.2" xref="S1.I1.i2.p1.12.m12.1.1.2.cmml">k</mi><mo id="S1.I1.i2.p1.12.m12.1.1.1" xref="S1.I1.i2.p1.12.m12.1.1.1.cmml">+</mo><mn id="S1.I1.i2.p1.12.m12.1.1.3" xref="S1.I1.i2.p1.12.m12.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S1.I1.i2.p1.12.m12.1b"><apply id="S1.I1.i2.p1.12.m12.1.1.cmml" xref="S1.I1.i2.p1.12.m12.1.1"><plus id="S1.I1.i2.p1.12.m12.1.1.1.cmml" xref="S1.I1.i2.p1.12.m12.1.1.1"></plus><ci id="S1.I1.i2.p1.12.m12.1.1.2.cmml" xref="S1.I1.i2.p1.12.m12.1.1.2">𝑘</ci><cn id="S1.I1.i2.p1.12.m12.1.1.3.cmml" type="integer" xref="S1.I1.i2.p1.12.m12.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.I1.i2.p1.12.m12.1c">k+1</annotation><annotation encoding="application/x-llamapun" id="S1.I1.i2.p1.12.m12.1d">italic_k + 1</annotation></semantics></math>.</p> </div> </li> </ul> </div> <div class="ltx_para" id="S1.p3"> <p class="ltx_p" id="S1.p3.8">In this work, we study SNDP in the <span class="ltx_text ltx_font_italic" id="S1.p3.8.1">insertion-only</span> streaming model. Formally, the algorithm reads the edges of a graph sequentially in an arbitrary order, processing each edge as it arrives. The goal is to solve graph problems over the streamed edges in a single pass, using a memory significantly smaller than storing the entire set of edges. While there has been significant recent progress for EC-SNDP in the streaming model, VC-SNDP remains essentially unexplored. One reason for this is that vertex-connectivity network design problems often do not share the same structural properties as their edge-connectivity counterparts. For instance, in the offline setting, EC-SNDP admits a <math alttext="2" class="ltx_Math" display="inline" id="S1.p3.1.m1.1"><semantics id="S1.p3.1.m1.1a"><mn id="S1.p3.1.m1.1.1" xref="S1.p3.1.m1.1.1.cmml">2</mn><annotation-xml encoding="MathML-Content" id="S1.p3.1.m1.1b"><cn id="S1.p3.1.m1.1.1.cmml" type="integer" xref="S1.p3.1.m1.1.1">2</cn></annotation-xml><annotation encoding="application/x-tex" id="S1.p3.1.m1.1c">2</annotation><annotation encoding="application/x-llamapun" id="S1.p3.1.m1.1d">2</annotation></semantics></math>-approximation via a seminal iterated rounding algorithm of Jain <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.00712v1#bib.bibx53" title="">Jai01</a>]</cite>. However, these techniques fail to extend to VC-SNDP. The best known approximation for VC-SNDP is <math alttext="O(k^{3}\log n)" class="ltx_Math" display="inline" id="S1.p3.2.m2.1"><semantics id="S1.p3.2.m2.1a"><mrow id="S1.p3.2.m2.1.1" xref="S1.p3.2.m2.1.1.cmml"><mi id="S1.p3.2.m2.1.1.3" xref="S1.p3.2.m2.1.1.3.cmml">O</mi><mo id="S1.p3.2.m2.1.1.2" xref="S1.p3.2.m2.1.1.2.cmml">⁢</mo><mrow id="S1.p3.2.m2.1.1.1.1" xref="S1.p3.2.m2.1.1.1.1.1.cmml"><mo id="S1.p3.2.m2.1.1.1.1.2" stretchy="false" xref="S1.p3.2.m2.1.1.1.1.1.cmml">(</mo><mrow id="S1.p3.2.m2.1.1.1.1.1" xref="S1.p3.2.m2.1.1.1.1.1.cmml"><msup id="S1.p3.2.m2.1.1.1.1.1.2" xref="S1.p3.2.m2.1.1.1.1.1.2.cmml"><mi id="S1.p3.2.m2.1.1.1.1.1.2.2" xref="S1.p3.2.m2.1.1.1.1.1.2.2.cmml">k</mi><mn id="S1.p3.2.m2.1.1.1.1.1.2.3" xref="S1.p3.2.m2.1.1.1.1.1.2.3.cmml">3</mn></msup><mo id="S1.p3.2.m2.1.1.1.1.1.1" lspace="0.167em" xref="S1.p3.2.m2.1.1.1.1.1.1.cmml">⁢</mo><mrow id="S1.p3.2.m2.1.1.1.1.1.3" xref="S1.p3.2.m2.1.1.1.1.1.3.cmml"><mi id="S1.p3.2.m2.1.1.1.1.1.3.1" xref="S1.p3.2.m2.1.1.1.1.1.3.1.cmml">log</mi><mo id="S1.p3.2.m2.1.1.1.1.1.3a" lspace="0.167em" xref="S1.p3.2.m2.1.1.1.1.1.3.cmml">⁡</mo><mi id="S1.p3.2.m2.1.1.1.1.1.3.2" xref="S1.p3.2.m2.1.1.1.1.1.3.2.cmml">n</mi></mrow></mrow><mo id="S1.p3.2.m2.1.1.1.1.3" stretchy="false" xref="S1.p3.2.m2.1.1.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.p3.2.m2.1b"><apply id="S1.p3.2.m2.1.1.cmml" xref="S1.p3.2.m2.1.1"><times id="S1.p3.2.m2.1.1.2.cmml" xref="S1.p3.2.m2.1.1.2"></times><ci id="S1.p3.2.m2.1.1.3.cmml" xref="S1.p3.2.m2.1.1.3">𝑂</ci><apply id="S1.p3.2.m2.1.1.1.1.1.cmml" xref="S1.p3.2.m2.1.1.1.1"><times id="S1.p3.2.m2.1.1.1.1.1.1.cmml" xref="S1.p3.2.m2.1.1.1.1.1.1"></times><apply id="S1.p3.2.m2.1.1.1.1.1.2.cmml" xref="S1.p3.2.m2.1.1.1.1.1.2"><csymbol cd="ambiguous" id="S1.p3.2.m2.1.1.1.1.1.2.1.cmml" xref="S1.p3.2.m2.1.1.1.1.1.2">superscript</csymbol><ci id="S1.p3.2.m2.1.1.1.1.1.2.2.cmml" xref="S1.p3.2.m2.1.1.1.1.1.2.2">𝑘</ci><cn id="S1.p3.2.m2.1.1.1.1.1.2.3.cmml" type="integer" xref="S1.p3.2.m2.1.1.1.1.1.2.3">3</cn></apply><apply id="S1.p3.2.m2.1.1.1.1.1.3.cmml" xref="S1.p3.2.m2.1.1.1.1.1.3"><log id="S1.p3.2.m2.1.1.1.1.1.3.1.cmml" xref="S1.p3.2.m2.1.1.1.1.1.3.1"></log><ci id="S1.p3.2.m2.1.1.1.1.1.3.2.cmml" xref="S1.p3.2.m2.1.1.1.1.1.3.2">𝑛</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p3.2.m2.1c">O(k^{3}\log n)</annotation><annotation encoding="application/x-llamapun" id="S1.p3.2.m2.1d">italic_O ( italic_k start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT roman_log italic_n )</annotation></semantics></math> <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.00712v1#bib.bibx26" title="">CK09</a>]</cite>; moreover, the dependence on <math alttext="k" class="ltx_Math" display="inline" id="S1.p3.3.m3.1"><semantics id="S1.p3.3.m3.1a"><mi id="S1.p3.3.m3.1.1" xref="S1.p3.3.m3.1.1.cmml">k</mi><annotation-xml encoding="MathML-Content" id="S1.p3.3.m3.1b"><ci id="S1.p3.3.m3.1.1.cmml" xref="S1.p3.3.m3.1.1">𝑘</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.p3.3.m3.1c">k</annotation><annotation encoding="application/x-llamapun" id="S1.p3.3.m3.1d">italic_k</annotation></semantics></math> is known to be necessary <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.00712v1#bib.bibx22" title="">CCK08</a>]</cite>. Another example highlighting the difficulty of vertex-connectivity problems is <math alttext="k" class="ltx_Math" display="inline" id="S1.p3.4.m4.1"><semantics id="S1.p3.4.m4.1a"><mi id="S1.p3.4.m4.1.1" xref="S1.p3.4.m4.1.1.cmml">k</mi><annotation-xml encoding="MathML-Content" id="S1.p3.4.m4.1b"><ci id="S1.p3.4.m4.1.1.cmml" xref="S1.p3.4.m4.1.1">𝑘</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.p3.4.m4.1c">k</annotation><annotation encoding="application/x-llamapun" id="S1.p3.4.m4.1d">italic_k</annotation></semantics></math>-CAP: <math alttext="k" class="ltx_Math" display="inline" id="S1.p3.5.m5.1"><semantics id="S1.p3.5.m5.1a"><mi id="S1.p3.5.m5.1.1" xref="S1.p3.5.m5.1.1.cmml">k</mi><annotation-xml encoding="MathML-Content" id="S1.p3.5.m5.1b"><ci id="S1.p3.5.m5.1.1.cmml" xref="S1.p3.5.m5.1.1">𝑘</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.p3.5.m5.1c">k</annotation><annotation encoding="application/x-llamapun" id="S1.p3.5.m5.1d">italic_k</annotation></semantics></math>-EC-CAP is known to reduce to <math alttext="1" class="ltx_Math" display="inline" id="S1.p3.6.m6.1"><semantics id="S1.p3.6.m6.1a"><mn id="S1.p3.6.m6.1.1" xref="S1.p3.6.m6.1.1.cmml">1</mn><annotation-xml encoding="MathML-Content" id="S1.p3.6.m6.1b"><cn id="S1.p3.6.m6.1.1.cmml" type="integer" xref="S1.p3.6.m6.1.1">1</cn></annotation-xml><annotation encoding="application/x-tex" id="S1.p3.6.m6.1c">1</annotation><annotation encoding="application/x-llamapun" id="S1.p3.6.m6.1d">1</annotation></semantics></math> or <math alttext="2" class="ltx_Math" display="inline" id="S1.p3.7.m7.1"><semantics id="S1.p3.7.m7.1a"><mn id="S1.p3.7.m7.1.1" xref="S1.p3.7.m7.1.1.cmml">2</mn><annotation-xml encoding="MathML-Content" id="S1.p3.7.m7.1b"><cn id="S1.p3.7.m7.1.1.cmml" type="integer" xref="S1.p3.7.m7.1.1">2</cn></annotation-xml><annotation encoding="application/x-tex" id="S1.p3.7.m7.1c">2</annotation><annotation encoding="application/x-llamapun" id="S1.p3.7.m7.1d">2</annotation></semantics></math>-EC-CAP for all <math alttext="k" class="ltx_Math" display="inline" id="S1.p3.8.m8.1"><semantics id="S1.p3.8.m8.1a"><mi id="S1.p3.8.m8.1.1" xref="S1.p3.8.m8.1.1.cmml">k</mi><annotation-xml encoding="MathML-Content" id="S1.p3.8.m8.1b"><ci id="S1.p3.8.m8.1.1.cmml" xref="S1.p3.8.m8.1.1">𝑘</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.p3.8.m8.1c">k</annotation><annotation encoding="application/x-llamapun" id="S1.p3.8.m8.1d">italic_k</annotation></semantics></math> <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.00712v1#bib.bibx37" title="">DKL73</a>]</cite> (see also <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.00712v1#bib.bibx61" title="">KT93</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.00712v1#bib.bibx25" title="">CJR99</a>]</cite>), but no such structure exists for the vertex-connectivity setting. Thus the main motivating question for this paper is the following:</p> </div> <div class="ltx_theorem ltx_theorem_question" id="Thmquestion1"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="Thmquestion1.1.1.1">Question 1</span></span><span class="ltx_text ltx_font_bold" id="Thmquestion1.2.2">.</span> </h6> <div class="ltx_para" id="Thmquestion1.p1"> <p class="ltx_p" id="Thmquestion1.p1.1">What is the approximability of vertex-connectivity network design problems in the streaming setting?</p> </div> </div> <div class="ltx_para" id="S1.p4"> <p class="ltx_p" id="S1.p4.9">We briefly discuss prior work on streaming algorithms for EC-SNDP and several of its special cases. As mentioned earlier, streaming algorithms for shortest path and MST are both well-studied. We note an important distinction between these two problems: while MST admits an exact algorithm in <math alttext="O(n)" class="ltx_Math" display="inline" id="S1.p4.1.m1.1"><semantics id="S1.p4.1.m1.1a"><mrow id="S1.p4.1.m1.1.2" xref="S1.p4.1.m1.1.2.cmml"><mi id="S1.p4.1.m1.1.2.2" xref="S1.p4.1.m1.1.2.2.cmml">O</mi><mo id="S1.p4.1.m1.1.2.1" xref="S1.p4.1.m1.1.2.1.cmml">⁢</mo><mrow id="S1.p4.1.m1.1.2.3.2" xref="S1.p4.1.m1.1.2.cmml"><mo id="S1.p4.1.m1.1.2.3.2.1" stretchy="false" xref="S1.p4.1.m1.1.2.cmml">(</mo><mi id="S1.p4.1.m1.1.1" xref="S1.p4.1.m1.1.1.cmml">n</mi><mo id="S1.p4.1.m1.1.2.3.2.2" stretchy="false" xref="S1.p4.1.m1.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.p4.1.m1.1b"><apply id="S1.p4.1.m1.1.2.cmml" xref="S1.p4.1.m1.1.2"><times id="S1.p4.1.m1.1.2.1.cmml" xref="S1.p4.1.m1.1.2.1"></times><ci id="S1.p4.1.m1.1.2.2.cmml" xref="S1.p4.1.m1.1.2.2">𝑂</ci><ci id="S1.p4.1.m1.1.1.cmml" xref="S1.p4.1.m1.1.1">𝑛</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p4.1.m1.1c">O(n)</annotation><annotation encoding="application/x-llamapun" id="S1.p4.1.m1.1d">italic_O ( italic_n )</annotation></semantics></math> words of space, the current best known streaming algorithm for <math alttext="s" class="ltx_Math" display="inline" id="S1.p4.2.m2.1"><semantics id="S1.p4.2.m2.1a"><mi id="S1.p4.2.m2.1.1" xref="S1.p4.2.m2.1.1.cmml">s</mi><annotation-xml encoding="MathML-Content" id="S1.p4.2.m2.1b"><ci id="S1.p4.2.m2.1.1.cmml" xref="S1.p4.2.m2.1.1">𝑠</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.p4.2.m2.1c">s</annotation><annotation encoding="application/x-llamapun" id="S1.p4.2.m2.1d">italic_s</annotation></semantics></math>-<math alttext="t" class="ltx_Math" display="inline" id="S1.p4.3.m3.1"><semantics id="S1.p4.3.m3.1a"><mi id="S1.p4.3.m3.1.1" xref="S1.p4.3.m3.1.1.cmml">t</mi><annotation-xml encoding="MathML-Content" 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id="S1.p4.5.m5.1.1.1.1.1.3.3.3.cmml" xref="S1.p4.5.m5.1.1.1.1.1.3.3.3">𝑡</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p4.5.m5.1c">\tilde{O}(n^{1+1/t})</annotation><annotation encoding="application/x-llamapun" id="S1.p4.5.m5.1d">over~ start_ARG italic_O end_ARG ( italic_n start_POSTSUPERSCRIPT 1 + 1 / italic_t end_POSTSUPERSCRIPT )</annotation></semantics></math>-space. Thus, any problem that contains <math alttext="s" class="ltx_Math" display="inline" id="S1.p4.6.m6.1"><semantics id="S1.p4.6.m6.1a"><mi id="S1.p4.6.m6.1.1" xref="S1.p4.6.m6.1.1.cmml">s</mi><annotation-xml encoding="MathML-Content" id="S1.p4.6.m6.1b"><ci id="S1.p4.6.m6.1.1.cmml" xref="S1.p4.6.m6.1.1">𝑠</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.p4.6.m6.1c">s</annotation><annotation encoding="application/x-llamapun" id="S1.p4.6.m6.1d">italic_s</annotation></semantics></math>-<math alttext="t" class="ltx_Math" display="inline" id="S1.p4.7.m7.1"><semantics id="S1.p4.7.m7.1a"><mi id="S1.p4.7.m7.1.1" xref="S1.p4.7.m7.1.1.cmml">t</mi><annotation-xml encoding="MathML-Content" id="S1.p4.7.m7.1b"><ci id="S1.p4.7.m7.1.1.cmml" xref="S1.p4.7.m7.1.1">𝑡</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.p4.7.m7.1c">t</annotation><annotation encoding="application/x-llamapun" id="S1.p4.7.m7.1d">italic_t</annotation></semantics></math> shortest path as a special case incurs this limitation, while global connectivity problems such as MST are likely to have better trade-offs. Some other special cases of SNDP that have been studied in the streaming model include the Steiner forest problem in geometric setting <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.00712v1#bib.bibx24" title="">CJKV22</a>]</cite>, and testing <math alttext="k" class="ltx_Math" display="inline" id="S1.p4.8.m8.1"><semantics id="S1.p4.8.m8.1a"><mi id="S1.p4.8.m8.1.1" xref="S1.p4.8.m8.1.1.cmml">k</mi><annotation-xml encoding="MathML-Content" id="S1.p4.8.m8.1b"><ci id="S1.p4.8.m8.1.1.cmml" xref="S1.p4.8.m8.1.1">𝑘</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.p4.8.m8.1c">k</annotation><annotation encoding="application/x-llamapun" id="S1.p4.8.m8.1d">italic_k</annotation></semantics></math>-connectivity <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.00712v1#bib.bibx81" title="">Zel06</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.00712v1#bib.bibx82" title="">Zel11</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.00712v1#bib.bibx77" title="">SW15</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.00712v1#bib.bibx1" title="">AD21</a>]</cite>. EC-SNDP in the streaming model was first studied in generality very recently in the work of <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.00712v1#bib.bibx54" title="">JKMV24</a>]</cite>. Their work primarily focused on <math alttext="k" class="ltx_Math" display="inline" id="S1.p4.9.m9.1"><semantics id="S1.p4.9.m9.1a"><mi id="S1.p4.9.m9.1.1" xref="S1.p4.9.m9.1.1.cmml">k</mi><annotation-xml encoding="MathML-Content" id="S1.p4.9.m9.1b"><ci id="S1.p4.9.m9.1.1.cmml" xref="S1.p4.9.m9.1.1">𝑘</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.p4.9.m9.1c">k</annotation><annotation encoding="application/x-llamapun" id="S1.p4.9.m9.1d">italic_k</annotation></semantics></math>-EC-CAP in two models:</p> <ul class="ltx_itemize" id="S1.I2"> <li class="ltx_item" id="S1.I2.i1" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">•</span> <div class="ltx_para" id="S1.I2.i1.p1"> <p class="ltx_p" id="S1.I2.i1.p1.2"><em class="ltx_emph ltx_font_italic" id="S1.I2.i1.p1.2.1">Link arrival:</em> In this model, the partial solution <math alttext="G^{\prime}" class="ltx_Math" display="inline" id="S1.I2.i1.p1.1.m1.1"><semantics id="S1.I2.i1.p1.1.m1.1a"><msup id="S1.I2.i1.p1.1.m1.1.1" xref="S1.I2.i1.p1.1.m1.1.1.cmml"><mi id="S1.I2.i1.p1.1.m1.1.1.2" xref="S1.I2.i1.p1.1.m1.1.1.2.cmml">G</mi><mo id="S1.I2.i1.p1.1.m1.1.1.3" xref="S1.I2.i1.p1.1.m1.1.1.3.cmml">′</mo></msup><annotation-xml encoding="MathML-Content" id="S1.I2.i1.p1.1.m1.1b"><apply id="S1.I2.i1.p1.1.m1.1.1.cmml" xref="S1.I2.i1.p1.1.m1.1.1"><csymbol cd="ambiguous" id="S1.I2.i1.p1.1.m1.1.1.1.cmml" xref="S1.I2.i1.p1.1.m1.1.1">superscript</csymbol><ci id="S1.I2.i1.p1.1.m1.1.1.2.cmml" xref="S1.I2.i1.p1.1.m1.1.1.2">𝐺</ci><ci id="S1.I2.i1.p1.1.m1.1.1.3.cmml" xref="S1.I2.i1.p1.1.m1.1.1.3">′</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.I2.i1.p1.1.m1.1c">G^{\prime}</annotation><annotation encoding="application/x-llamapun" id="S1.I2.i1.p1.1.m1.1d">italic_G start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math> is given up-front and does not count towards the space complexity of the algorithm. The weighted links <math alttext="E\setminus E(G^{\prime})" class="ltx_Math" display="inline" id="S1.I2.i1.p1.2.m2.1"><semantics id="S1.I2.i1.p1.2.m2.1a"><mrow id="S1.I2.i1.p1.2.m2.1.1" xref="S1.I2.i1.p1.2.m2.1.1.cmml"><mi id="S1.I2.i1.p1.2.m2.1.1.3" xref="S1.I2.i1.p1.2.m2.1.1.3.cmml">E</mi><mo id="S1.I2.i1.p1.2.m2.1.1.2" xref="S1.I2.i1.p1.2.m2.1.1.2.cmml">∖</mo><mrow id="S1.I2.i1.p1.2.m2.1.1.1" xref="S1.I2.i1.p1.2.m2.1.1.1.cmml"><mi id="S1.I2.i1.p1.2.m2.1.1.1.3" xref="S1.I2.i1.p1.2.m2.1.1.1.3.cmml">E</mi><mo id="S1.I2.i1.p1.2.m2.1.1.1.2" xref="S1.I2.i1.p1.2.m2.1.1.1.2.cmml">⁢</mo><mrow id="S1.I2.i1.p1.2.m2.1.1.1.1.1" xref="S1.I2.i1.p1.2.m2.1.1.1.1.1.1.cmml"><mo id="S1.I2.i1.p1.2.m2.1.1.1.1.1.2" stretchy="false" xref="S1.I2.i1.p1.2.m2.1.1.1.1.1.1.cmml">(</mo><msup id="S1.I2.i1.p1.2.m2.1.1.1.1.1.1" xref="S1.I2.i1.p1.2.m2.1.1.1.1.1.1.cmml"><mi id="S1.I2.i1.p1.2.m2.1.1.1.1.1.1.2" xref="S1.I2.i1.p1.2.m2.1.1.1.1.1.1.2.cmml">G</mi><mo id="S1.I2.i1.p1.2.m2.1.1.1.1.1.1.3" xref="S1.I2.i1.p1.2.m2.1.1.1.1.1.1.3.cmml">′</mo></msup><mo id="S1.I2.i1.p1.2.m2.1.1.1.1.1.3" stretchy="false" xref="S1.I2.i1.p1.2.m2.1.1.1.1.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.I2.i1.p1.2.m2.1b"><apply id="S1.I2.i1.p1.2.m2.1.1.cmml" xref="S1.I2.i1.p1.2.m2.1.1"><setdiff id="S1.I2.i1.p1.2.m2.1.1.2.cmml" xref="S1.I2.i1.p1.2.m2.1.1.2"></setdiff><ci id="S1.I2.i1.p1.2.m2.1.1.3.cmml" xref="S1.I2.i1.p1.2.m2.1.1.3">𝐸</ci><apply id="S1.I2.i1.p1.2.m2.1.1.1.cmml" xref="S1.I2.i1.p1.2.m2.1.1.1"><times id="S1.I2.i1.p1.2.m2.1.1.1.2.cmml" xref="S1.I2.i1.p1.2.m2.1.1.1.2"></times><ci id="S1.I2.i1.p1.2.m2.1.1.1.3.cmml" xref="S1.I2.i1.p1.2.m2.1.1.1.3">𝐸</ci><apply id="S1.I2.i1.p1.2.m2.1.1.1.1.1.1.cmml" xref="S1.I2.i1.p1.2.m2.1.1.1.1.1"><csymbol cd="ambiguous" id="S1.I2.i1.p1.2.m2.1.1.1.1.1.1.1.cmml" xref="S1.I2.i1.p1.2.m2.1.1.1.1.1">superscript</csymbol><ci id="S1.I2.i1.p1.2.m2.1.1.1.1.1.1.2.cmml" xref="S1.I2.i1.p1.2.m2.1.1.1.1.1.1.2">𝐺</ci><ci id="S1.I2.i1.p1.2.m2.1.1.1.1.1.1.3.cmml" xref="S1.I2.i1.p1.2.m2.1.1.1.1.1.1.3">′</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.I2.i1.p1.2.m2.1c">E\setminus E(G^{\prime})</annotation><annotation encoding="application/x-llamapun" id="S1.I2.i1.p1.2.m2.1d">italic_E ∖ italic_E ( italic_G start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT )</annotation></semantics></math> arrive in the stream.</p> </div> </li> <li class="ltx_item" id="S1.I2.i2" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">•</span> <div class="ltx_para" id="S1.I2.i2.p1"> <p class="ltx_p" id="S1.I2.i2.p1.1"><em class="ltx_emph ltx_font_italic" id="S1.I2.i2.p1.1.1">Fully streaming:</em> In this model, the edges of the partial solution, along with the additional weighted links used in augmentation, both arrive in the stream. They may arrive in an interleaved fashion; that is, the algorithm may not know the full partial solution when processing some links in <math alttext="E\setminus E(G^{\prime})" class="ltx_Math" display="inline" id="S1.I2.i2.p1.1.m1.1"><semantics id="S1.I2.i2.p1.1.m1.1a"><mrow id="S1.I2.i2.p1.1.m1.1.1" xref="S1.I2.i2.p1.1.m1.1.1.cmml"><mi id="S1.I2.i2.p1.1.m1.1.1.3" xref="S1.I2.i2.p1.1.m1.1.1.3.cmml">E</mi><mo id="S1.I2.i2.p1.1.m1.1.1.2" xref="S1.I2.i2.p1.1.m1.1.1.2.cmml">∖</mo><mrow id="S1.I2.i2.p1.1.m1.1.1.1" xref="S1.I2.i2.p1.1.m1.1.1.1.cmml"><mi id="S1.I2.i2.p1.1.m1.1.1.1.3" xref="S1.I2.i2.p1.1.m1.1.1.1.3.cmml">E</mi><mo id="S1.I2.i2.p1.1.m1.1.1.1.2" xref="S1.I2.i2.p1.1.m1.1.1.1.2.cmml">⁢</mo><mrow id="S1.I2.i2.p1.1.m1.1.1.1.1.1" xref="S1.I2.i2.p1.1.m1.1.1.1.1.1.1.cmml"><mo id="S1.I2.i2.p1.1.m1.1.1.1.1.1.2" stretchy="false" xref="S1.I2.i2.p1.1.m1.1.1.1.1.1.1.cmml">(</mo><msup id="S1.I2.i2.p1.1.m1.1.1.1.1.1.1" xref="S1.I2.i2.p1.1.m1.1.1.1.1.1.1.cmml"><mi id="S1.I2.i2.p1.1.m1.1.1.1.1.1.1.2" xref="S1.I2.i2.p1.1.m1.1.1.1.1.1.1.2.cmml">G</mi><mo id="S1.I2.i2.p1.1.m1.1.1.1.1.1.1.3" xref="S1.I2.i2.p1.1.m1.1.1.1.1.1.1.3.cmml">′</mo></msup><mo id="S1.I2.i2.p1.1.m1.1.1.1.1.1.3" stretchy="false" xref="S1.I2.i2.p1.1.m1.1.1.1.1.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.I2.i2.p1.1.m1.1b"><apply id="S1.I2.i2.p1.1.m1.1.1.cmml" xref="S1.I2.i2.p1.1.m1.1.1"><setdiff id="S1.I2.i2.p1.1.m1.1.1.2.cmml" xref="S1.I2.i2.p1.1.m1.1.1.2"></setdiff><ci id="S1.I2.i2.p1.1.m1.1.1.3.cmml" xref="S1.I2.i2.p1.1.m1.1.1.3">𝐸</ci><apply id="S1.I2.i2.p1.1.m1.1.1.1.cmml" xref="S1.I2.i2.p1.1.m1.1.1.1"><times id="S1.I2.i2.p1.1.m1.1.1.1.2.cmml" xref="S1.I2.i2.p1.1.m1.1.1.1.2"></times><ci id="S1.I2.i2.p1.1.m1.1.1.1.3.cmml" xref="S1.I2.i2.p1.1.m1.1.1.1.3">𝐸</ci><apply id="S1.I2.i2.p1.1.m1.1.1.1.1.1.1.cmml" xref="S1.I2.i2.p1.1.m1.1.1.1.1.1"><csymbol cd="ambiguous" id="S1.I2.i2.p1.1.m1.1.1.1.1.1.1.1.cmml" xref="S1.I2.i2.p1.1.m1.1.1.1.1.1">superscript</csymbol><ci id="S1.I2.i2.p1.1.m1.1.1.1.1.1.1.2.cmml" xref="S1.I2.i2.p1.1.m1.1.1.1.1.1.1.2">𝐺</ci><ci id="S1.I2.i2.p1.1.m1.1.1.1.1.1.1.3.cmml" xref="S1.I2.i2.p1.1.m1.1.1.1.1.1.1.3">′</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.I2.i2.p1.1.m1.1c">E\setminus E(G^{\prime})</annotation><annotation encoding="application/x-llamapun" id="S1.I2.i2.p1.1.m1.1d">italic_E ∖ italic_E ( italic_G start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT )</annotation></semantics></math>.</p> </div> </li> </ul> <p class="ltx_p" id="S1.p4.15">Both models are practically useful in their own right. Note that an <math alttext="\alpha" class="ltx_Math" display="inline" id="S1.p4.10.m1.1"><semantics id="S1.p4.10.m1.1a"><mi id="S1.p4.10.m1.1.1" xref="S1.p4.10.m1.1.1.cmml">α</mi><annotation-xml encoding="MathML-Content" id="S1.p4.10.m1.1b"><ci id="S1.p4.10.m1.1.1.cmml" xref="S1.p4.10.m1.1.1">𝛼</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.p4.10.m1.1c">\alpha</annotation><annotation encoding="application/x-llamapun" id="S1.p4.10.m1.1d">italic_α</annotation></semantics></math>-approximation for <math alttext="k" class="ltx_Math" display="inline" id="S1.p4.11.m2.1"><semantics id="S1.p4.11.m2.1a"><mi id="S1.p4.11.m2.1.1" xref="S1.p4.11.m2.1.1.cmml">k</mi><annotation-xml encoding="MathML-Content" id="S1.p4.11.m2.1b"><ci id="S1.p4.11.m2.1.1.cmml" xref="S1.p4.11.m2.1.1">𝑘</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.p4.11.m2.1c">k</annotation><annotation encoding="application/x-llamapun" id="S1.p4.11.m2.1d">italic_k</annotation></semantics></math>-CAP in the link-arrival model implies an <math alttext="(\alpha k)" class="ltx_Math" display="inline" id="S1.p4.12.m3.1"><semantics id="S1.p4.12.m3.1a"><mrow id="S1.p4.12.m3.1.1.1" xref="S1.p4.12.m3.1.1.1.1.cmml"><mo id="S1.p4.12.m3.1.1.1.2" stretchy="false" xref="S1.p4.12.m3.1.1.1.1.cmml">(</mo><mrow id="S1.p4.12.m3.1.1.1.1" xref="S1.p4.12.m3.1.1.1.1.cmml"><mi id="S1.p4.12.m3.1.1.1.1.2" xref="S1.p4.12.m3.1.1.1.1.2.cmml">α</mi><mo id="S1.p4.12.m3.1.1.1.1.1" xref="S1.p4.12.m3.1.1.1.1.1.cmml">⁢</mo><mi id="S1.p4.12.m3.1.1.1.1.3" xref="S1.p4.12.m3.1.1.1.1.3.cmml">k</mi></mrow><mo id="S1.p4.12.m3.1.1.1.3" stretchy="false" xref="S1.p4.12.m3.1.1.1.1.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S1.p4.12.m3.1b"><apply id="S1.p4.12.m3.1.1.1.1.cmml" xref="S1.p4.12.m3.1.1.1"><times id="S1.p4.12.m3.1.1.1.1.1.cmml" xref="S1.p4.12.m3.1.1.1.1.1"></times><ci id="S1.p4.12.m3.1.1.1.1.2.cmml" xref="S1.p4.12.m3.1.1.1.1.2">𝛼</ci><ci id="S1.p4.12.m3.1.1.1.1.3.cmml" xref="S1.p4.12.m3.1.1.1.1.3">𝑘</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p4.12.m3.1c">(\alpha k)</annotation><annotation encoding="application/x-llamapun" id="S1.p4.12.m3.1d">( italic_α italic_k )</annotation></semantics></math>-approximation for <math alttext="k" class="ltx_Math" display="inline" id="S1.p4.13.m4.1"><semantics id="S1.p4.13.m4.1a"><mi id="S1.p4.13.m4.1.1" xref="S1.p4.13.m4.1.1.cmml">k</mi><annotation-xml encoding="MathML-Content" id="S1.p4.13.m4.1b"><ci id="S1.p4.13.m4.1.1.cmml" xref="S1.p4.13.m4.1.1">𝑘</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.p4.13.m4.1c">k</annotation><annotation encoding="application/x-llamapun" id="S1.p4.13.m4.1d">italic_k</annotation></semantics></math>-ECSS if we make <math alttext="k" class="ltx_Math" display="inline" id="S1.p4.14.m5.1"><semantics id="S1.p4.14.m5.1a"><mi id="S1.p4.14.m5.1.1" xref="S1.p4.14.m5.1.1.cmml">k</mi><annotation-xml encoding="MathML-Content" id="S1.p4.14.m5.1b"><ci id="S1.p4.14.m5.1.1.cmml" xref="S1.p4.14.m5.1.1">𝑘</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.p4.14.m5.1c">k</annotation><annotation encoding="application/x-llamapun" id="S1.p4.14.m5.1d">italic_k</annotation></semantics></math> passes over the stream, as we can augment the connectivity of the graph by one in each pass. This is particularly useful in situations where <math alttext="k" class="ltx_Math" display="inline" id="S1.p4.15.m6.1"><semantics id="S1.p4.15.m6.1a"><mi id="S1.p4.15.m6.1.1" xref="S1.p4.15.m6.1.1.cmml">k</mi><annotation-xml encoding="MathML-Content" id="S1.p4.15.m6.1b"><ci id="S1.p4.15.m6.1.1.cmml" xref="S1.p4.15.m6.1.1">𝑘</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.p4.15.m6.1c">k</annotation><annotation encoding="application/x-llamapun" id="S1.p4.15.m6.1d">italic_k</annotation></semantics></math> is a small constant and the best approximation ratio for link-arrival is significantly better than that of fully-streaming.</p> </div> <div class="ltx_para" id="S1.p5"> <p class="ltx_p" id="S1.p5.15">In the link-arrival model, <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.00712v1#bib.bibx54" title="">JKMV24</a>]</cite> obtained a tight <math alttext="(2+\epsilon)" class="ltx_Math" display="inline" id="S1.p5.1.m1.1"><semantics id="S1.p5.1.m1.1a"><mrow id="S1.p5.1.m1.1.1.1" xref="S1.p5.1.m1.1.1.1.1.cmml"><mo id="S1.p5.1.m1.1.1.1.2" stretchy="false" xref="S1.p5.1.m1.1.1.1.1.cmml">(</mo><mrow id="S1.p5.1.m1.1.1.1.1" xref="S1.p5.1.m1.1.1.1.1.cmml"><mn id="S1.p5.1.m1.1.1.1.1.2" xref="S1.p5.1.m1.1.1.1.1.2.cmml">2</mn><mo id="S1.p5.1.m1.1.1.1.1.1" xref="S1.p5.1.m1.1.1.1.1.1.cmml">+</mo><mi id="S1.p5.1.m1.1.1.1.1.3" xref="S1.p5.1.m1.1.1.1.1.3.cmml">ϵ</mi></mrow><mo id="S1.p5.1.m1.1.1.1.3" stretchy="false" xref="S1.p5.1.m1.1.1.1.1.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S1.p5.1.m1.1b"><apply id="S1.p5.1.m1.1.1.1.1.cmml" xref="S1.p5.1.m1.1.1.1"><plus id="S1.p5.1.m1.1.1.1.1.1.cmml" xref="S1.p5.1.m1.1.1.1.1.1"></plus><cn id="S1.p5.1.m1.1.1.1.1.2.cmml" type="integer" xref="S1.p5.1.m1.1.1.1.1.2">2</cn><ci id="S1.p5.1.m1.1.1.1.1.3.cmml" xref="S1.p5.1.m1.1.1.1.1.3">italic-ϵ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p5.1.m1.1c">(2+\epsilon)</annotation><annotation encoding="application/x-llamapun" id="S1.p5.1.m1.1d">( 2 + italic_ϵ )</annotation></semantics></math>-approximation for <math alttext="k" class="ltx_Math" display="inline" id="S1.p5.2.m2.1"><semantics id="S1.p5.2.m2.1a"><mi id="S1.p5.2.m2.1.1" xref="S1.p5.2.m2.1.1.cmml">k</mi><annotation-xml encoding="MathML-Content" id="S1.p5.2.m2.1b"><ci id="S1.p5.2.m2.1.1.cmml" xref="S1.p5.2.m2.1.1">𝑘</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.p5.2.m2.1c">k</annotation><annotation encoding="application/x-llamapun" id="S1.p5.2.m2.1d">italic_k</annotation></semantics></math>-EC-CAP in <math alttext="O(\frac{n\log n}{\epsilon})" class="ltx_Math" display="inline" id="S1.p5.3.m3.1"><semantics id="S1.p5.3.m3.1a"><mrow id="S1.p5.3.m3.1.2" xref="S1.p5.3.m3.1.2.cmml"><mi id="S1.p5.3.m3.1.2.2" xref="S1.p5.3.m3.1.2.2.cmml">O</mi><mo id="S1.p5.3.m3.1.2.1" xref="S1.p5.3.m3.1.2.1.cmml">⁢</mo><mrow id="S1.p5.3.m3.1.2.3.2" xref="S1.p5.3.m3.1.1.cmml"><mo id="S1.p5.3.m3.1.2.3.2.1" stretchy="false" xref="S1.p5.3.m3.1.1.cmml">(</mo><mfrac id="S1.p5.3.m3.1.1" xref="S1.p5.3.m3.1.1.cmml"><mrow id="S1.p5.3.m3.1.1.2" xref="S1.p5.3.m3.1.1.2.cmml"><mi id="S1.p5.3.m3.1.1.2.2" xref="S1.p5.3.m3.1.1.2.2.cmml">n</mi><mo id="S1.p5.3.m3.1.1.2.1" lspace="0.167em" xref="S1.p5.3.m3.1.1.2.1.cmml">⁢</mo><mrow id="S1.p5.3.m3.1.1.2.3" xref="S1.p5.3.m3.1.1.2.3.cmml"><mi id="S1.p5.3.m3.1.1.2.3.1" xref="S1.p5.3.m3.1.1.2.3.1.cmml">log</mi><mo id="S1.p5.3.m3.1.1.2.3a" lspace="0.167em" xref="S1.p5.3.m3.1.1.2.3.cmml">⁡</mo><mi id="S1.p5.3.m3.1.1.2.3.2" xref="S1.p5.3.m3.1.1.2.3.2.cmml">n</mi></mrow></mrow><mi id="S1.p5.3.m3.1.1.3" xref="S1.p5.3.m3.1.1.3.cmml">ϵ</mi></mfrac><mo id="S1.p5.3.m3.1.2.3.2.2" stretchy="false" xref="S1.p5.3.m3.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.p5.3.m3.1b"><apply id="S1.p5.3.m3.1.2.cmml" xref="S1.p5.3.m3.1.2"><times id="S1.p5.3.m3.1.2.1.cmml" xref="S1.p5.3.m3.1.2.1"></times><ci id="S1.p5.3.m3.1.2.2.cmml" xref="S1.p5.3.m3.1.2.2">𝑂</ci><apply id="S1.p5.3.m3.1.1.cmml" xref="S1.p5.3.m3.1.2.3.2"><divide id="S1.p5.3.m3.1.1.1.cmml" xref="S1.p5.3.m3.1.2.3.2"></divide><apply id="S1.p5.3.m3.1.1.2.cmml" xref="S1.p5.3.m3.1.1.2"><times id="S1.p5.3.m3.1.1.2.1.cmml" xref="S1.p5.3.m3.1.1.2.1"></times><ci id="S1.p5.3.m3.1.1.2.2.cmml" xref="S1.p5.3.m3.1.1.2.2">𝑛</ci><apply id="S1.p5.3.m3.1.1.2.3.cmml" xref="S1.p5.3.m3.1.1.2.3"><log id="S1.p5.3.m3.1.1.2.3.1.cmml" xref="S1.p5.3.m3.1.1.2.3.1"></log><ci id="S1.p5.3.m3.1.1.2.3.2.cmml" xref="S1.p5.3.m3.1.1.2.3.2">𝑛</ci></apply></apply><ci id="S1.p5.3.m3.1.1.3.cmml" xref="S1.p5.3.m3.1.1.3">italic-ϵ</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p5.3.m3.1c">O(\frac{n\log n}{\epsilon})</annotation><annotation encoding="application/x-llamapun" id="S1.p5.3.m3.1d">italic_O ( divide start_ARG italic_n roman_log italic_n end_ARG start_ARG italic_ϵ end_ARG )</annotation></semantics></math> space; note that while one can obtain a better than <math alttext="2" class="ltx_Math" display="inline" id="S1.p5.4.m4.1"><semantics id="S1.p5.4.m4.1a"><mn id="S1.p5.4.m4.1.1" xref="S1.p5.4.m4.1.1.cmml">2</mn><annotation-xml encoding="MathML-Content" id="S1.p5.4.m4.1b"><cn id="S1.p5.4.m4.1.1.cmml" type="integer" xref="S1.p5.4.m4.1.1">2</cn></annotation-xml><annotation encoding="application/x-tex" id="S1.p5.4.m4.1c">2</annotation><annotation encoding="application/x-llamapun" id="S1.p5.4.m4.1d">2</annotation></semantics></math> in the offline setting, there is a lower bound of <math alttext="2" class="ltx_Math" display="inline" id="S1.p5.5.m5.1"><semantics id="S1.p5.5.m5.1a"><mn id="S1.p5.5.m5.1.1" xref="S1.p5.5.m5.1.1.cmml">2</mn><annotation-xml encoding="MathML-Content" id="S1.p5.5.m5.1b"><cn id="S1.p5.5.m5.1.1.cmml" type="integer" xref="S1.p5.5.m5.1.1">2</cn></annotation-xml><annotation encoding="application/x-tex" id="S1.p5.5.m5.1c">2</annotation><annotation encoding="application/x-llamapun" id="S1.p5.5.m5.1d">2</annotation></semantics></math> in the semi-streaming setting. In the fully streaming model, they obtained an <math alttext="O(t)" class="ltx_Math" display="inline" id="S1.p5.6.m6.1"><semantics id="S1.p5.6.m6.1a"><mrow id="S1.p5.6.m6.1.2" xref="S1.p5.6.m6.1.2.cmml"><mi id="S1.p5.6.m6.1.2.2" xref="S1.p5.6.m6.1.2.2.cmml">O</mi><mo id="S1.p5.6.m6.1.2.1" xref="S1.p5.6.m6.1.2.1.cmml">⁢</mo><mrow id="S1.p5.6.m6.1.2.3.2" xref="S1.p5.6.m6.1.2.cmml"><mo id="S1.p5.6.m6.1.2.3.2.1" stretchy="false" xref="S1.p5.6.m6.1.2.cmml">(</mo><mi id="S1.p5.6.m6.1.1" xref="S1.p5.6.m6.1.1.cmml">t</mi><mo id="S1.p5.6.m6.1.2.3.2.2" stretchy="false" xref="S1.p5.6.m6.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.p5.6.m6.1b"><apply id="S1.p5.6.m6.1.2.cmml" xref="S1.p5.6.m6.1.2"><times id="S1.p5.6.m6.1.2.1.cmml" xref="S1.p5.6.m6.1.2.1"></times><ci id="S1.p5.6.m6.1.2.2.cmml" xref="S1.p5.6.m6.1.2.2">𝑂</ci><ci id="S1.p5.6.m6.1.1.cmml" xref="S1.p5.6.m6.1.1">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p5.6.m6.1c">O(t)</annotation><annotation encoding="application/x-llamapun" id="S1.p5.6.m6.1d">italic_O ( italic_t )</annotation></semantics></math>-approximation in <math alttext="\tilde{O}(kn+n^{1+1/t})" class="ltx_Math" display="inline" id="S1.p5.7.m7.1"><semantics id="S1.p5.7.m7.1a"><mrow id="S1.p5.7.m7.1.1" xref="S1.p5.7.m7.1.1.cmml"><mover accent="true" id="S1.p5.7.m7.1.1.3" xref="S1.p5.7.m7.1.1.3.cmml"><mi id="S1.p5.7.m7.1.1.3.2" xref="S1.p5.7.m7.1.1.3.2.cmml">O</mi><mo id="S1.p5.7.m7.1.1.3.1" xref="S1.p5.7.m7.1.1.3.1.cmml">~</mo></mover><mo id="S1.p5.7.m7.1.1.2" xref="S1.p5.7.m7.1.1.2.cmml">⁢</mo><mrow id="S1.p5.7.m7.1.1.1.1" xref="S1.p5.7.m7.1.1.1.1.1.cmml"><mo id="S1.p5.7.m7.1.1.1.1.2" stretchy="false" xref="S1.p5.7.m7.1.1.1.1.1.cmml">(</mo><mrow id="S1.p5.7.m7.1.1.1.1.1" xref="S1.p5.7.m7.1.1.1.1.1.cmml"><mrow id="S1.p5.7.m7.1.1.1.1.1.2" xref="S1.p5.7.m7.1.1.1.1.1.2.cmml"><mi id="S1.p5.7.m7.1.1.1.1.1.2.2" xref="S1.p5.7.m7.1.1.1.1.1.2.2.cmml">k</mi><mo 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stretchy="false" xref="S1.p5.7.m7.1.1.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.p5.7.m7.1b"><apply id="S1.p5.7.m7.1.1.cmml" xref="S1.p5.7.m7.1.1"><times id="S1.p5.7.m7.1.1.2.cmml" xref="S1.p5.7.m7.1.1.2"></times><apply id="S1.p5.7.m7.1.1.3.cmml" xref="S1.p5.7.m7.1.1.3"><ci id="S1.p5.7.m7.1.1.3.1.cmml" xref="S1.p5.7.m7.1.1.3.1">~</ci><ci id="S1.p5.7.m7.1.1.3.2.cmml" xref="S1.p5.7.m7.1.1.3.2">𝑂</ci></apply><apply id="S1.p5.7.m7.1.1.1.1.1.cmml" xref="S1.p5.7.m7.1.1.1.1"><plus id="S1.p5.7.m7.1.1.1.1.1.1.cmml" xref="S1.p5.7.m7.1.1.1.1.1.1"></plus><apply id="S1.p5.7.m7.1.1.1.1.1.2.cmml" xref="S1.p5.7.m7.1.1.1.1.1.2"><times id="S1.p5.7.m7.1.1.1.1.1.2.1.cmml" xref="S1.p5.7.m7.1.1.1.1.1.2.1"></times><ci id="S1.p5.7.m7.1.1.1.1.1.2.2.cmml" xref="S1.p5.7.m7.1.1.1.1.1.2.2">𝑘</ci><ci id="S1.p5.7.m7.1.1.1.1.1.2.3.cmml" xref="S1.p5.7.m7.1.1.1.1.1.2.3">𝑛</ci></apply><apply id="S1.p5.7.m7.1.1.1.1.1.3.cmml" xref="S1.p5.7.m7.1.1.1.1.1.3"><csymbol cd="ambiguous" id="S1.p5.7.m7.1.1.1.1.1.3.1.cmml" xref="S1.p5.7.m7.1.1.1.1.1.3">superscript</csymbol><ci id="S1.p5.7.m7.1.1.1.1.1.3.2.cmml" xref="S1.p5.7.m7.1.1.1.1.1.3.2">𝑛</ci><apply id="S1.p5.7.m7.1.1.1.1.1.3.3.cmml" xref="S1.p5.7.m7.1.1.1.1.1.3.3"><plus id="S1.p5.7.m7.1.1.1.1.1.3.3.1.cmml" xref="S1.p5.7.m7.1.1.1.1.1.3.3.1"></plus><cn id="S1.p5.7.m7.1.1.1.1.1.3.3.2.cmml" type="integer" xref="S1.p5.7.m7.1.1.1.1.1.3.3.2">1</cn><apply id="S1.p5.7.m7.1.1.1.1.1.3.3.3.cmml" xref="S1.p5.7.m7.1.1.1.1.1.3.3.3"><divide id="S1.p5.7.m7.1.1.1.1.1.3.3.3.1.cmml" xref="S1.p5.7.m7.1.1.1.1.1.3.3.3.1"></divide><cn id="S1.p5.7.m7.1.1.1.1.1.3.3.3.2.cmml" type="integer" xref="S1.p5.7.m7.1.1.1.1.1.3.3.3.2">1</cn><ci id="S1.p5.7.m7.1.1.1.1.1.3.3.3.3.cmml" xref="S1.p5.7.m7.1.1.1.1.1.3.3.3.3">𝑡</ci></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p5.7.m7.1c">\tilde{O}(kn+n^{1+1/t})</annotation><annotation encoding="application/x-llamapun" id="S1.p5.7.m7.1d">over~ start_ARG italic_O end_ARG ( italic_k italic_n + italic_n start_POSTSUPERSCRIPT 1 + 1 / italic_t end_POSTSUPERSCRIPT )</annotation></semantics></math> space for the connectivity augmentation problem using a <em class="ltx_emph ltx_font_italic" id="S1.p5.15.1">spanner</em> approach (a <math alttext="t" class="ltx_Math" display="inline" id="S1.p5.8.m8.1"><semantics id="S1.p5.8.m8.1a"><mi id="S1.p5.8.m8.1.1" xref="S1.p5.8.m8.1.1.cmml">t</mi><annotation-xml encoding="MathML-Content" id="S1.p5.8.m8.1b"><ci id="S1.p5.8.m8.1.1.cmml" xref="S1.p5.8.m8.1.1">𝑡</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.p5.8.m8.1c">t</annotation><annotation encoding="application/x-llamapun" id="S1.p5.8.m8.1d">italic_t</annotation></semantics></math>-spanner is a sparse subgraph that preserves all distances to within a factor of <math alttext="t" class="ltx_Math" display="inline" id="S1.p5.9.m9.1"><semantics id="S1.p5.9.m9.1a"><mi id="S1.p5.9.m9.1.1" xref="S1.p5.9.m9.1.1.cmml">t</mi><annotation-xml encoding="MathML-Content" id="S1.p5.9.m9.1b"><ci id="S1.p5.9.m9.1.1.cmml" xref="S1.p5.9.m9.1.1">𝑡</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.p5.9.m9.1c">t</annotation><annotation encoding="application/x-llamapun" id="S1.p5.9.m9.1d">italic_t</annotation></semantics></math>). By building on this, and using the reverse augmentation framework of Goemans <span class="ltx_text ltx_font_italic" id="S1.p5.15.2">et al.</span> <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.00712v1#bib.bibx46" title="">GGP<sup class="ltx_sup"><span class="ltx_text ltx_font_italic">+</span></sup>94</a>]</cite>, they achieved an <math alttext="O(t\log k)" class="ltx_Math" display="inline" id="S1.p5.10.m10.1"><semantics id="S1.p5.10.m10.1a"><mrow id="S1.p5.10.m10.1.1" xref="S1.p5.10.m10.1.1.cmml"><mi id="S1.p5.10.m10.1.1.3" xref="S1.p5.10.m10.1.1.3.cmml">O</mi><mo id="S1.p5.10.m10.1.1.2" xref="S1.p5.10.m10.1.1.2.cmml">⁢</mo><mrow id="S1.p5.10.m10.1.1.1.1" xref="S1.p5.10.m10.1.1.1.1.1.cmml"><mo id="S1.p5.10.m10.1.1.1.1.2" stretchy="false" xref="S1.p5.10.m10.1.1.1.1.1.cmml">(</mo><mrow id="S1.p5.10.m10.1.1.1.1.1" xref="S1.p5.10.m10.1.1.1.1.1.cmml"><mi id="S1.p5.10.m10.1.1.1.1.1.2" xref="S1.p5.10.m10.1.1.1.1.1.2.cmml">t</mi><mo id="S1.p5.10.m10.1.1.1.1.1.1" lspace="0.167em" xref="S1.p5.10.m10.1.1.1.1.1.1.cmml">⁢</mo><mrow id="S1.p5.10.m10.1.1.1.1.1.3" xref="S1.p5.10.m10.1.1.1.1.1.3.cmml"><mi id="S1.p5.10.m10.1.1.1.1.1.3.1" xref="S1.p5.10.m10.1.1.1.1.1.3.1.cmml">log</mi><mo id="S1.p5.10.m10.1.1.1.1.1.3a" lspace="0.167em" xref="S1.p5.10.m10.1.1.1.1.1.3.cmml">⁡</mo><mi id="S1.p5.10.m10.1.1.1.1.1.3.2" xref="S1.p5.10.m10.1.1.1.1.1.3.2.cmml">k</mi></mrow></mrow><mo id="S1.p5.10.m10.1.1.1.1.3" stretchy="false" xref="S1.p5.10.m10.1.1.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.p5.10.m10.1b"><apply id="S1.p5.10.m10.1.1.cmml" xref="S1.p5.10.m10.1.1"><times id="S1.p5.10.m10.1.1.2.cmml" xref="S1.p5.10.m10.1.1.2"></times><ci id="S1.p5.10.m10.1.1.3.cmml" xref="S1.p5.10.m10.1.1.3">𝑂</ci><apply id="S1.p5.10.m10.1.1.1.1.1.cmml" xref="S1.p5.10.m10.1.1.1.1"><times id="S1.p5.10.m10.1.1.1.1.1.1.cmml" xref="S1.p5.10.m10.1.1.1.1.1.1"></times><ci id="S1.p5.10.m10.1.1.1.1.1.2.cmml" xref="S1.p5.10.m10.1.1.1.1.1.2">𝑡</ci><apply id="S1.p5.10.m10.1.1.1.1.1.3.cmml" xref="S1.p5.10.m10.1.1.1.1.1.3"><log id="S1.p5.10.m10.1.1.1.1.1.3.1.cmml" xref="S1.p5.10.m10.1.1.1.1.1.3.1"></log><ci id="S1.p5.10.m10.1.1.1.1.1.3.2.cmml" xref="S1.p5.10.m10.1.1.1.1.1.3.2">𝑘</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p5.10.m10.1c">O(t\log k)</annotation><annotation encoding="application/x-llamapun" id="S1.p5.10.m10.1d">italic_O ( italic_t roman_log italic_k )</annotation></semantics></math>-approximation for EC-SNDP with maximum connectivity requirement <math alttext="k" class="ltx_Math" display="inline" id="S1.p5.11.m11.1"><semantics id="S1.p5.11.m11.1a"><mi id="S1.p5.11.m11.1.1" xref="S1.p5.11.m11.1.1.cmml">k</mi><annotation-xml encoding="MathML-Content" id="S1.p5.11.m11.1b"><ci id="S1.p5.11.m11.1.1.cmml" xref="S1.p5.11.m11.1.1">𝑘</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.p5.11.m11.1c">k</annotation><annotation encoding="application/x-llamapun" id="S1.p5.11.m11.1d">italic_k</annotation></semantics></math>; the space usage is <math alttext="\tilde{O}(kn^{1+1/t})" class="ltx_Math" display="inline" id="S1.p5.12.m12.1"><semantics id="S1.p5.12.m12.1a"><mrow id="S1.p5.12.m12.1.1" xref="S1.p5.12.m12.1.1.cmml"><mover accent="true" id="S1.p5.12.m12.1.1.3" xref="S1.p5.12.m12.1.1.3.cmml"><mi id="S1.p5.12.m12.1.1.3.2" xref="S1.p5.12.m12.1.1.3.2.cmml">O</mi><mo id="S1.p5.12.m12.1.1.3.1" xref="S1.p5.12.m12.1.1.3.1.cmml">~</mo></mover><mo id="S1.p5.12.m12.1.1.2" xref="S1.p5.12.m12.1.1.2.cmml">⁢</mo><mrow id="S1.p5.12.m12.1.1.1.1" xref="S1.p5.12.m12.1.1.1.1.1.cmml"><mo id="S1.p5.12.m12.1.1.1.1.2" stretchy="false" xref="S1.p5.12.m12.1.1.1.1.1.cmml">(</mo><mrow id="S1.p5.12.m12.1.1.1.1.1" xref="S1.p5.12.m12.1.1.1.1.1.cmml"><mi id="S1.p5.12.m12.1.1.1.1.1.2" xref="S1.p5.12.m12.1.1.1.1.1.2.cmml">k</mi><mo id="S1.p5.12.m12.1.1.1.1.1.1" xref="S1.p5.12.m12.1.1.1.1.1.1.cmml">⁢</mo><msup id="S1.p5.12.m12.1.1.1.1.1.3" xref="S1.p5.12.m12.1.1.1.1.1.3.cmml"><mi id="S1.p5.12.m12.1.1.1.1.1.3.2" xref="S1.p5.12.m12.1.1.1.1.1.3.2.cmml">n</mi><mrow id="S1.p5.12.m12.1.1.1.1.1.3.3" xref="S1.p5.12.m12.1.1.1.1.1.3.3.cmml"><mn id="S1.p5.12.m12.1.1.1.1.1.3.3.2" xref="S1.p5.12.m12.1.1.1.1.1.3.3.2.cmml">1</mn><mo id="S1.p5.12.m12.1.1.1.1.1.3.3.1" xref="S1.p5.12.m12.1.1.1.1.1.3.3.1.cmml">+</mo><mrow id="S1.p5.12.m12.1.1.1.1.1.3.3.3" xref="S1.p5.12.m12.1.1.1.1.1.3.3.3.cmml"><mn id="S1.p5.12.m12.1.1.1.1.1.3.3.3.2" xref="S1.p5.12.m12.1.1.1.1.1.3.3.3.2.cmml">1</mn><mo id="S1.p5.12.m12.1.1.1.1.1.3.3.3.1" xref="S1.p5.12.m12.1.1.1.1.1.3.3.3.1.cmml">/</mo><mi id="S1.p5.12.m12.1.1.1.1.1.3.3.3.3" xref="S1.p5.12.m12.1.1.1.1.1.3.3.3.3.cmml">t</mi></mrow></mrow></msup></mrow><mo id="S1.p5.12.m12.1.1.1.1.3" stretchy="false" xref="S1.p5.12.m12.1.1.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.p5.12.m12.1b"><apply id="S1.p5.12.m12.1.1.cmml" xref="S1.p5.12.m12.1.1"><times id="S1.p5.12.m12.1.1.2.cmml" xref="S1.p5.12.m12.1.1.2"></times><apply id="S1.p5.12.m12.1.1.3.cmml" xref="S1.p5.12.m12.1.1.3"><ci id="S1.p5.12.m12.1.1.3.1.cmml" xref="S1.p5.12.m12.1.1.3.1">~</ci><ci id="S1.p5.12.m12.1.1.3.2.cmml" xref="S1.p5.12.m12.1.1.3.2">𝑂</ci></apply><apply id="S1.p5.12.m12.1.1.1.1.1.cmml" xref="S1.p5.12.m12.1.1.1.1"><times id="S1.p5.12.m12.1.1.1.1.1.1.cmml" xref="S1.p5.12.m12.1.1.1.1.1.1"></times><ci id="S1.p5.12.m12.1.1.1.1.1.2.cmml" xref="S1.p5.12.m12.1.1.1.1.1.2">𝑘</ci><apply id="S1.p5.12.m12.1.1.1.1.1.3.cmml" xref="S1.p5.12.m12.1.1.1.1.1.3"><csymbol cd="ambiguous" id="S1.p5.12.m12.1.1.1.1.1.3.1.cmml" xref="S1.p5.12.m12.1.1.1.1.1.3">superscript</csymbol><ci id="S1.p5.12.m12.1.1.1.1.1.3.2.cmml" xref="S1.p5.12.m12.1.1.1.1.1.3.2">𝑛</ci><apply id="S1.p5.12.m12.1.1.1.1.1.3.3.cmml" xref="S1.p5.12.m12.1.1.1.1.1.3.3"><plus id="S1.p5.12.m12.1.1.1.1.1.3.3.1.cmml" xref="S1.p5.12.m12.1.1.1.1.1.3.3.1"></plus><cn id="S1.p5.12.m12.1.1.1.1.1.3.3.2.cmml" type="integer" xref="S1.p5.12.m12.1.1.1.1.1.3.3.2">1</cn><apply id="S1.p5.12.m12.1.1.1.1.1.3.3.3.cmml" xref="S1.p5.12.m12.1.1.1.1.1.3.3.3"><divide id="S1.p5.12.m12.1.1.1.1.1.3.3.3.1.cmml" xref="S1.p5.12.m12.1.1.1.1.1.3.3.3.1"></divide><cn id="S1.p5.12.m12.1.1.1.1.1.3.3.3.2.cmml" type="integer" xref="S1.p5.12.m12.1.1.1.1.1.3.3.3.2">1</cn><ci id="S1.p5.12.m12.1.1.1.1.1.3.3.3.3.cmml" xref="S1.p5.12.m12.1.1.1.1.1.3.3.3.3">𝑡</ci></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p5.12.m12.1c">\tilde{O}(kn^{1+1/t})</annotation><annotation encoding="application/x-llamapun" id="S1.p5.12.m12.1d">over~ start_ARG italic_O end_ARG ( italic_k italic_n start_POSTSUPERSCRIPT 1 + 1 / italic_t end_POSTSUPERSCRIPT )</annotation></semantics></math>. They further showed that any <math alttext="O(t)" class="ltx_Math" display="inline" id="S1.p5.13.m13.1"><semantics id="S1.p5.13.m13.1a"><mrow id="S1.p5.13.m13.1.2" xref="S1.p5.13.m13.1.2.cmml"><mi id="S1.p5.13.m13.1.2.2" xref="S1.p5.13.m13.1.2.2.cmml">O</mi><mo id="S1.p5.13.m13.1.2.1" xref="S1.p5.13.m13.1.2.1.cmml">⁢</mo><mrow id="S1.p5.13.m13.1.2.3.2" xref="S1.p5.13.m13.1.2.cmml"><mo id="S1.p5.13.m13.1.2.3.2.1" stretchy="false" xref="S1.p5.13.m13.1.2.cmml">(</mo><mi id="S1.p5.13.m13.1.1" xref="S1.p5.13.m13.1.1.cmml">t</mi><mo id="S1.p5.13.m13.1.2.3.2.2" stretchy="false" xref="S1.p5.13.m13.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.p5.13.m13.1b"><apply id="S1.p5.13.m13.1.2.cmml" xref="S1.p5.13.m13.1.2"><times id="S1.p5.13.m13.1.2.1.cmml" xref="S1.p5.13.m13.1.2.1"></times><ci id="S1.p5.13.m13.1.2.2.cmml" xref="S1.p5.13.m13.1.2.2">𝑂</ci><ci id="S1.p5.13.m13.1.1.cmml" xref="S1.p5.13.m13.1.1">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p5.13.m13.1c">O(t)</annotation><annotation encoding="application/x-llamapun" id="S1.p5.13.m13.1d">italic_O ( italic_t )</annotation></semantics></math>-approximation for this problem requires at least <math alttext="\tilde{\Omega}(kn+n^{1+1/t})" class="ltx_Math" display="inline" id="S1.p5.14.m14.1"><semantics id="S1.p5.14.m14.1a"><mrow id="S1.p5.14.m14.1.1" xref="S1.p5.14.m14.1.1.cmml"><mover accent="true" id="S1.p5.14.m14.1.1.3" xref="S1.p5.14.m14.1.1.3.cmml"><mi id="S1.p5.14.m14.1.1.3.2" mathvariant="normal" xref="S1.p5.14.m14.1.1.3.2.cmml">Ω</mi><mo id="S1.p5.14.m14.1.1.3.1" xref="S1.p5.14.m14.1.1.3.1.cmml">~</mo></mover><mo id="S1.p5.14.m14.1.1.2" xref="S1.p5.14.m14.1.1.2.cmml">⁢</mo><mrow id="S1.p5.14.m14.1.1.1.1" xref="S1.p5.14.m14.1.1.1.1.1.cmml"><mo id="S1.p5.14.m14.1.1.1.1.2" stretchy="false" xref="S1.p5.14.m14.1.1.1.1.1.cmml">(</mo><mrow id="S1.p5.14.m14.1.1.1.1.1" xref="S1.p5.14.m14.1.1.1.1.1.cmml"><mrow id="S1.p5.14.m14.1.1.1.1.1.2" xref="S1.p5.14.m14.1.1.1.1.1.2.cmml"><mi id="S1.p5.14.m14.1.1.1.1.1.2.2" xref="S1.p5.14.m14.1.1.1.1.1.2.2.cmml">k</mi><mo id="S1.p5.14.m14.1.1.1.1.1.2.1" xref="S1.p5.14.m14.1.1.1.1.1.2.1.cmml">⁢</mo><mi id="S1.p5.14.m14.1.1.1.1.1.2.3" xref="S1.p5.14.m14.1.1.1.1.1.2.3.cmml">n</mi></mrow><mo id="S1.p5.14.m14.1.1.1.1.1.1" xref="S1.p5.14.m14.1.1.1.1.1.1.cmml">+</mo><msup id="S1.p5.14.m14.1.1.1.1.1.3" xref="S1.p5.14.m14.1.1.1.1.1.3.cmml"><mi id="S1.p5.14.m14.1.1.1.1.1.3.2" xref="S1.p5.14.m14.1.1.1.1.1.3.2.cmml">n</mi><mrow id="S1.p5.14.m14.1.1.1.1.1.3.3" xref="S1.p5.14.m14.1.1.1.1.1.3.3.cmml"><mn id="S1.p5.14.m14.1.1.1.1.1.3.3.2" xref="S1.p5.14.m14.1.1.1.1.1.3.3.2.cmml">1</mn><mo id="S1.p5.14.m14.1.1.1.1.1.3.3.1" xref="S1.p5.14.m14.1.1.1.1.1.3.3.1.cmml">+</mo><mrow id="S1.p5.14.m14.1.1.1.1.1.3.3.3" xref="S1.p5.14.m14.1.1.1.1.1.3.3.3.cmml"><mn id="S1.p5.14.m14.1.1.1.1.1.3.3.3.2" xref="S1.p5.14.m14.1.1.1.1.1.3.3.3.2.cmml">1</mn><mo id="S1.p5.14.m14.1.1.1.1.1.3.3.3.1" xref="S1.p5.14.m14.1.1.1.1.1.3.3.3.1.cmml">/</mo><mi id="S1.p5.14.m14.1.1.1.1.1.3.3.3.3" xref="S1.p5.14.m14.1.1.1.1.1.3.3.3.3.cmml">t</mi></mrow></mrow></msup></mrow><mo id="S1.p5.14.m14.1.1.1.1.3" stretchy="false" xref="S1.p5.14.m14.1.1.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.p5.14.m14.1b"><apply id="S1.p5.14.m14.1.1.cmml" xref="S1.p5.14.m14.1.1"><times id="S1.p5.14.m14.1.1.2.cmml" xref="S1.p5.14.m14.1.1.2"></times><apply id="S1.p5.14.m14.1.1.3.cmml" xref="S1.p5.14.m14.1.1.3"><ci id="S1.p5.14.m14.1.1.3.1.cmml" xref="S1.p5.14.m14.1.1.3.1">~</ci><ci id="S1.p5.14.m14.1.1.3.2.cmml" xref="S1.p5.14.m14.1.1.3.2">Ω</ci></apply><apply id="S1.p5.14.m14.1.1.1.1.1.cmml" xref="S1.p5.14.m14.1.1.1.1"><plus id="S1.p5.14.m14.1.1.1.1.1.1.cmml" xref="S1.p5.14.m14.1.1.1.1.1.1"></plus><apply id="S1.p5.14.m14.1.1.1.1.1.2.cmml" xref="S1.p5.14.m14.1.1.1.1.1.2"><times id="S1.p5.14.m14.1.1.1.1.1.2.1.cmml" xref="S1.p5.14.m14.1.1.1.1.1.2.1"></times><ci id="S1.p5.14.m14.1.1.1.1.1.2.2.cmml" xref="S1.p5.14.m14.1.1.1.1.1.2.2">𝑘</ci><ci id="S1.p5.14.m14.1.1.1.1.1.2.3.cmml" xref="S1.p5.14.m14.1.1.1.1.1.2.3">𝑛</ci></apply><apply id="S1.p5.14.m14.1.1.1.1.1.3.cmml" xref="S1.p5.14.m14.1.1.1.1.1.3"><csymbol cd="ambiguous" id="S1.p5.14.m14.1.1.1.1.1.3.1.cmml" xref="S1.p5.14.m14.1.1.1.1.1.3">superscript</csymbol><ci id="S1.p5.14.m14.1.1.1.1.1.3.2.cmml" xref="S1.p5.14.m14.1.1.1.1.1.3.2">𝑛</ci><apply id="S1.p5.14.m14.1.1.1.1.1.3.3.cmml" xref="S1.p5.14.m14.1.1.1.1.1.3.3"><plus id="S1.p5.14.m14.1.1.1.1.1.3.3.1.cmml" xref="S1.p5.14.m14.1.1.1.1.1.3.3.1"></plus><cn id="S1.p5.14.m14.1.1.1.1.1.3.3.2.cmml" type="integer" xref="S1.p5.14.m14.1.1.1.1.1.3.3.2">1</cn><apply id="S1.p5.14.m14.1.1.1.1.1.3.3.3.cmml" xref="S1.p5.14.m14.1.1.1.1.1.3.3.3"><divide id="S1.p5.14.m14.1.1.1.1.1.3.3.3.1.cmml" xref="S1.p5.14.m14.1.1.1.1.1.3.3.3.1"></divide><cn id="S1.p5.14.m14.1.1.1.1.1.3.3.3.2.cmml" type="integer" xref="S1.p5.14.m14.1.1.1.1.1.3.3.3.2">1</cn><ci id="S1.p5.14.m14.1.1.1.1.1.3.3.3.3.cmml" xref="S1.p5.14.m14.1.1.1.1.1.3.3.3.3">𝑡</ci></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p5.14.m14.1c">\tilde{\Omega}(kn+n^{1+1/t})</annotation><annotation encoding="application/x-llamapun" id="S1.p5.14.m14.1d">over~ start_ARG roman_Ω end_ARG ( italic_k italic_n + italic_n start_POSTSUPERSCRIPT 1 + 1 / italic_t end_POSTSUPERSCRIPT )</annotation></semantics></math> space.<span class="ltx_note ltx_role_footnote" id="footnote1"><sup class="ltx_note_mark">1</sup><span class="ltx_note_outer"><span class="ltx_note_content"><sup class="ltx_note_mark">1</sup><span class="ltx_tag ltx_tag_note">1</span>While they only mentioned <math alttext="\tilde{\Omega}(n^{1+1/t})" class="ltx_Math" display="inline" id="footnote1.m1.1"><semantics id="footnote1.m1.1b"><mrow id="footnote1.m1.1.1" xref="footnote1.m1.1.1.cmml"><mover accent="true" id="footnote1.m1.1.1.3" xref="footnote1.m1.1.1.3.cmml"><mi id="footnote1.m1.1.1.3.2" mathvariant="normal" xref="footnote1.m1.1.1.3.2.cmml">Ω</mi><mo id="footnote1.m1.1.1.3.1" xref="footnote1.m1.1.1.3.1.cmml">~</mo></mover><mo id="footnote1.m1.1.1.2" xref="footnote1.m1.1.1.2.cmml">⁢</mo><mrow id="footnote1.m1.1.1.1.1" xref="footnote1.m1.1.1.1.1.1.cmml"><mo id="footnote1.m1.1.1.1.1.2" stretchy="false" xref="footnote1.m1.1.1.1.1.1.cmml">(</mo><msup id="footnote1.m1.1.1.1.1.1" xref="footnote1.m1.1.1.1.1.1.cmml"><mi id="footnote1.m1.1.1.1.1.1.2" xref="footnote1.m1.1.1.1.1.1.2.cmml">n</mi><mrow id="footnote1.m1.1.1.1.1.1.3" xref="footnote1.m1.1.1.1.1.1.3.cmml"><mn id="footnote1.m1.1.1.1.1.1.3.2" xref="footnote1.m1.1.1.1.1.1.3.2.cmml">1</mn><mo id="footnote1.m1.1.1.1.1.1.3.1" xref="footnote1.m1.1.1.1.1.1.3.1.cmml">+</mo><mrow id="footnote1.m1.1.1.1.1.1.3.3" xref="footnote1.m1.1.1.1.1.1.3.3.cmml"><mn id="footnote1.m1.1.1.1.1.1.3.3.2" xref="footnote1.m1.1.1.1.1.1.3.3.2.cmml">1</mn><mo id="footnote1.m1.1.1.1.1.1.3.3.1" xref="footnote1.m1.1.1.1.1.1.3.3.1.cmml">/</mo><mi id="footnote1.m1.1.1.1.1.1.3.3.3" xref="footnote1.m1.1.1.1.1.1.3.3.3.cmml">t</mi></mrow></mrow></msup><mo id="footnote1.m1.1.1.1.1.3" stretchy="false" xref="footnote1.m1.1.1.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="footnote1.m1.1c"><apply id="footnote1.m1.1.1.cmml" xref="footnote1.m1.1.1"><times id="footnote1.m1.1.1.2.cmml" xref="footnote1.m1.1.1.2"></times><apply id="footnote1.m1.1.1.3.cmml" xref="footnote1.m1.1.1.3"><ci id="footnote1.m1.1.1.3.1.cmml" xref="footnote1.m1.1.1.3.1">~</ci><ci id="footnote1.m1.1.1.3.2.cmml" xref="footnote1.m1.1.1.3.2">Ω</ci></apply><apply id="footnote1.m1.1.1.1.1.1.cmml" xref="footnote1.m1.1.1.1.1"><csymbol cd="ambiguous" id="footnote1.m1.1.1.1.1.1.1.cmml" xref="footnote1.m1.1.1.1.1">superscript</csymbol><ci id="footnote1.m1.1.1.1.1.1.2.cmml" xref="footnote1.m1.1.1.1.1.1.2">𝑛</ci><apply id="footnote1.m1.1.1.1.1.1.3.cmml" xref="footnote1.m1.1.1.1.1.1.3"><plus id="footnote1.m1.1.1.1.1.1.3.1.cmml" xref="footnote1.m1.1.1.1.1.1.3.1"></plus><cn id="footnote1.m1.1.1.1.1.1.3.2.cmml" type="integer" xref="footnote1.m1.1.1.1.1.1.3.2">1</cn><apply id="footnote1.m1.1.1.1.1.1.3.3.cmml" xref="footnote1.m1.1.1.1.1.1.3.3"><divide id="footnote1.m1.1.1.1.1.1.3.3.1.cmml" xref="footnote1.m1.1.1.1.1.1.3.3.1"></divide><cn id="footnote1.m1.1.1.1.1.1.3.3.2.cmml" type="integer" xref="footnote1.m1.1.1.1.1.1.3.3.2">1</cn><ci id="footnote1.m1.1.1.1.1.1.3.3.3.cmml" xref="footnote1.m1.1.1.1.1.1.3.3.3">𝑡</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="footnote1.m1.1d">\tilde{\Omega}(n^{1+1/t})</annotation><annotation encoding="application/x-llamapun" id="footnote1.m1.1e">over~ start_ARG roman_Ω end_ARG ( italic_n start_POSTSUPERSCRIPT 1 + 1 / italic_t end_POSTSUPERSCRIPT )</annotation></semantics></math>, it is straightforward to show that in general, the number of edges in an feasible solution of an SNDP instance with maximum connectivity requirement <math alttext="k" class="ltx_Math" display="inline" id="footnote1.m2.1"><semantics id="footnote1.m2.1b"><mi id="footnote1.m2.1.1" xref="footnote1.m2.1.1.cmml">k</mi><annotation-xml encoding="MathML-Content" id="footnote1.m2.1c"><ci id="footnote1.m2.1.1.cmml" xref="footnote1.m2.1.1">𝑘</ci></annotation-xml><annotation encoding="application/x-tex" id="footnote1.m2.1d">k</annotation><annotation encoding="application/x-llamapun" id="footnote1.m2.1e">italic_k</annotation></semantics></math> can be as large as <math alttext="nk" class="ltx_Math" display="inline" id="footnote1.m3.1"><semantics id="footnote1.m3.1b"><mrow id="footnote1.m3.1.1" xref="footnote1.m3.1.1.cmml"><mi id="footnote1.m3.1.1.2" xref="footnote1.m3.1.1.2.cmml">n</mi><mo id="footnote1.m3.1.1.1" xref="footnote1.m3.1.1.1.cmml">⁢</mo><mi id="footnote1.m3.1.1.3" xref="footnote1.m3.1.1.3.cmml">k</mi></mrow><annotation-xml encoding="MathML-Content" id="footnote1.m3.1c"><apply id="footnote1.m3.1.1.cmml" xref="footnote1.m3.1.1"><times id="footnote1.m3.1.1.1.cmml" xref="footnote1.m3.1.1.1"></times><ci id="footnote1.m3.1.1.2.cmml" xref="footnote1.m3.1.1.2">𝑛</ci><ci id="footnote1.m3.1.1.3.cmml" xref="footnote1.m3.1.1.3">𝑘</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="footnote1.m3.1d">nk</annotation><annotation encoding="application/x-llamapun" id="footnote1.m3.1e">italic_n italic_k</annotation></semantics></math>. Hence, <math alttext="nk" class="ltx_Math" display="inline" id="footnote1.m4.1"><semantics id="footnote1.m4.1b"><mrow id="footnote1.m4.1.1" xref="footnote1.m4.1.1.cmml"><mi id="footnote1.m4.1.1.2" xref="footnote1.m4.1.1.2.cmml">n</mi><mo id="footnote1.m4.1.1.1" xref="footnote1.m4.1.1.1.cmml">⁢</mo><mi id="footnote1.m4.1.1.3" xref="footnote1.m4.1.1.3.cmml">k</mi></mrow><annotation-xml encoding="MathML-Content" id="footnote1.m4.1c"><apply id="footnote1.m4.1.1.cmml" xref="footnote1.m4.1.1"><times id="footnote1.m4.1.1.1.cmml" xref="footnote1.m4.1.1.1"></times><ci id="footnote1.m4.1.1.2.cmml" xref="footnote1.m4.1.1.2">𝑛</ci><ci id="footnote1.m4.1.1.3.cmml" xref="footnote1.m4.1.1.3">𝑘</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="footnote1.m4.1d">nk</annotation><annotation encoding="application/x-llamapun" id="footnote1.m4.1e">italic_n italic_k</annotation></semantics></math> is a trivial lower bound too.</span></span></span> Although the space complexity of their algorithm nearly matches the lower bound, their approximation for EC-SNDP is worse by a <math alttext="\log k" class="ltx_Math" display="inline" id="S1.p5.15.m15.1"><semantics id="S1.p5.15.m15.1a"><mrow id="S1.p5.15.m15.1.1" xref="S1.p5.15.m15.1.1.cmml"><mi id="S1.p5.15.m15.1.1.1" xref="S1.p5.15.m15.1.1.1.cmml">log</mi><mo id="S1.p5.15.m15.1.1a" lspace="0.167em" xref="S1.p5.15.m15.1.1.cmml">⁡</mo><mi id="S1.p5.15.m15.1.1.2" xref="S1.p5.15.m15.1.1.2.cmml">k</mi></mrow><annotation-xml encoding="MathML-Content" id="S1.p5.15.m15.1b"><apply id="S1.p5.15.m15.1.1.cmml" xref="S1.p5.15.m15.1.1"><log id="S1.p5.15.m15.1.1.1.cmml" xref="S1.p5.15.m15.1.1.1"></log><ci id="S1.p5.15.m15.1.1.2.cmml" xref="S1.p5.15.m15.1.1.2">𝑘</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p5.15.m15.1c">\log k</annotation><annotation encoding="application/x-llamapun" id="S1.p5.15.m15.1d">roman_log italic_k</annotation></semantics></math> factor. A natural question is the following.</p> </div> <div class="ltx_theorem ltx_theorem_question" id="Thmquestion2"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="Thmquestion2.1.1.1">Question 2</span></span><span class="ltx_text ltx_font_bold" id="Thmquestion2.2.2">.</span> </h6> <div class="ltx_para" id="Thmquestion2.p1"> <p class="ltx_p" id="Thmquestion2.p1.2">Is there an <math alttext="O(t)" class="ltx_Math" display="inline" id="Thmquestion2.p1.1.m1.1"><semantics id="Thmquestion2.p1.1.m1.1a"><mrow id="Thmquestion2.p1.1.m1.1.2" xref="Thmquestion2.p1.1.m1.1.2.cmml"><mi id="Thmquestion2.p1.1.m1.1.2.2" xref="Thmquestion2.p1.1.m1.1.2.2.cmml">O</mi><mo id="Thmquestion2.p1.1.m1.1.2.1" xref="Thmquestion2.p1.1.m1.1.2.1.cmml">⁢</mo><mrow id="Thmquestion2.p1.1.m1.1.2.3.2" xref="Thmquestion2.p1.1.m1.1.2.cmml"><mo id="Thmquestion2.p1.1.m1.1.2.3.2.1" stretchy="false" xref="Thmquestion2.p1.1.m1.1.2.cmml">(</mo><mi id="Thmquestion2.p1.1.m1.1.1" xref="Thmquestion2.p1.1.m1.1.1.cmml">t</mi><mo id="Thmquestion2.p1.1.m1.1.2.3.2.2" stretchy="false" xref="Thmquestion2.p1.1.m1.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="Thmquestion2.p1.1.m1.1b"><apply id="Thmquestion2.p1.1.m1.1.2.cmml" xref="Thmquestion2.p1.1.m1.1.2"><times id="Thmquestion2.p1.1.m1.1.2.1.cmml" xref="Thmquestion2.p1.1.m1.1.2.1"></times><ci id="Thmquestion2.p1.1.m1.1.2.2.cmml" xref="Thmquestion2.p1.1.m1.1.2.2">𝑂</ci><ci id="Thmquestion2.p1.1.m1.1.1.cmml" xref="Thmquestion2.p1.1.m1.1.1">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmquestion2.p1.1.m1.1c">O(t)</annotation><annotation encoding="application/x-llamapun" id="Thmquestion2.p1.1.m1.1d">italic_O ( italic_t )</annotation></semantics></math>-approximation for EC-SNDP using <math alttext="\tilde{O}(kn+n^{1+1/t})" class="ltx_Math" display="inline" id="Thmquestion2.p1.2.m2.1"><semantics id="Thmquestion2.p1.2.m2.1a"><mrow 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xref="Thmquestion2.p1.2.m2.1.1.1.1.1.2"><times id="Thmquestion2.p1.2.m2.1.1.1.1.1.2.1.cmml" xref="Thmquestion2.p1.2.m2.1.1.1.1.1.2.1"></times><ci id="Thmquestion2.p1.2.m2.1.1.1.1.1.2.2.cmml" xref="Thmquestion2.p1.2.m2.1.1.1.1.1.2.2">𝑘</ci><ci id="Thmquestion2.p1.2.m2.1.1.1.1.1.2.3.cmml" xref="Thmquestion2.p1.2.m2.1.1.1.1.1.2.3">𝑛</ci></apply><apply id="Thmquestion2.p1.2.m2.1.1.1.1.1.3.cmml" xref="Thmquestion2.p1.2.m2.1.1.1.1.1.3"><csymbol cd="ambiguous" id="Thmquestion2.p1.2.m2.1.1.1.1.1.3.1.cmml" xref="Thmquestion2.p1.2.m2.1.1.1.1.1.3">superscript</csymbol><ci id="Thmquestion2.p1.2.m2.1.1.1.1.1.3.2.cmml" xref="Thmquestion2.p1.2.m2.1.1.1.1.1.3.2">𝑛</ci><apply id="Thmquestion2.p1.2.m2.1.1.1.1.1.3.3.cmml" xref="Thmquestion2.p1.2.m2.1.1.1.1.1.3.3"><plus id="Thmquestion2.p1.2.m2.1.1.1.1.1.3.3.1.cmml" xref="Thmquestion2.p1.2.m2.1.1.1.1.1.3.3.1"></plus><cn id="Thmquestion2.p1.2.m2.1.1.1.1.1.3.3.2.cmml" type="integer" xref="Thmquestion2.p1.2.m2.1.1.1.1.1.3.3.2">1</cn><apply id="Thmquestion2.p1.2.m2.1.1.1.1.1.3.3.3.cmml" xref="Thmquestion2.p1.2.m2.1.1.1.1.1.3.3.3"><divide id="Thmquestion2.p1.2.m2.1.1.1.1.1.3.3.3.1.cmml" xref="Thmquestion2.p1.2.m2.1.1.1.1.1.3.3.3.1"></divide><cn id="Thmquestion2.p1.2.m2.1.1.1.1.1.3.3.3.2.cmml" type="integer" xref="Thmquestion2.p1.2.m2.1.1.1.1.1.3.3.3.2">1</cn><ci id="Thmquestion2.p1.2.m2.1.1.1.1.1.3.3.3.3.cmml" xref="Thmquestion2.p1.2.m2.1.1.1.1.1.3.3.3.3">𝑡</ci></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmquestion2.p1.2.m2.1c">\tilde{O}(kn+n^{1+1/t})</annotation><annotation encoding="application/x-llamapun" id="Thmquestion2.p1.2.m2.1d">over~ start_ARG italic_O end_ARG ( italic_k italic_n + italic_n start_POSTSUPERSCRIPT 1 + 1 / italic_t end_POSTSUPERSCRIPT )</annotation></semantics></math>-space?</p> </div> </div> <section class="ltx_subsection" id="S1.SS1"> <h3 class="ltx_title ltx_title_subsection"> <span class="ltx_tag ltx_tag_subsection">1.1 </span>Results and Techniques</h3> <div class="ltx_para" id="S1.SS1.p1"> <p class="ltx_p" id="S1.SS1.p1.7">We make significant progress towards our motivating questions and obtain a number of results. We refer the reader to a summary of our results and prior work on streaming network design problems, provided in Tables <a class="ltx_ref" href="https://arxiv.org/html/2503.00712v1#S1.T1" title="Table 1 ‣ Algorithms for 𝑘-VC-CAP: ‣ 1.1 Results and Techniques ‣ 1 Introduction ‣ Streaming Algorithms for Network Design"><span class="ltx_text ltx_ref_tag">1</span></a> and <a class="ltx_ref" href="https://arxiv.org/html/2503.00712v1#S1.T2" title="Table 2 ‣ Algorithms for 𝑘-VC-CAP: ‣ 1.1 Results and Techniques ‣ 1 Introduction ‣ Streaming Algorithms for Network Design"><span class="ltx_text ltx_ref_tag">2</span></a>. Note that all results in the table assume only one pass over the stream. In all weighted graphs <math alttext="G=(V,E)" class="ltx_Math" display="inline" id="S1.SS1.p1.1.m1.2"><semantics id="S1.SS1.p1.1.m1.2a"><mrow id="S1.SS1.p1.1.m1.2.3" xref="S1.SS1.p1.1.m1.2.3.cmml"><mi id="S1.SS1.p1.1.m1.2.3.2" xref="S1.SS1.p1.1.m1.2.3.2.cmml">G</mi><mo id="S1.SS1.p1.1.m1.2.3.1" xref="S1.SS1.p1.1.m1.2.3.1.cmml">=</mo><mrow id="S1.SS1.p1.1.m1.2.3.3.2" xref="S1.SS1.p1.1.m1.2.3.3.1.cmml"><mo id="S1.SS1.p1.1.m1.2.3.3.2.1" stretchy="false" xref="S1.SS1.p1.1.m1.2.3.3.1.cmml">(</mo><mi id="S1.SS1.p1.1.m1.1.1" xref="S1.SS1.p1.1.m1.1.1.cmml">V</mi><mo id="S1.SS1.p1.1.m1.2.3.3.2.2" xref="S1.SS1.p1.1.m1.2.3.3.1.cmml">,</mo><mi id="S1.SS1.p1.1.m1.2.2" xref="S1.SS1.p1.1.m1.2.2.cmml">E</mi><mo id="S1.SS1.p1.1.m1.2.3.3.2.3" stretchy="false" xref="S1.SS1.p1.1.m1.2.3.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.SS1.p1.1.m1.2b"><apply id="S1.SS1.p1.1.m1.2.3.cmml" xref="S1.SS1.p1.1.m1.2.3"><eq id="S1.SS1.p1.1.m1.2.3.1.cmml" xref="S1.SS1.p1.1.m1.2.3.1"></eq><ci id="S1.SS1.p1.1.m1.2.3.2.cmml" xref="S1.SS1.p1.1.m1.2.3.2">𝐺</ci><interval closure="open" id="S1.SS1.p1.1.m1.2.3.3.1.cmml" xref="S1.SS1.p1.1.m1.2.3.3.2"><ci id="S1.SS1.p1.1.m1.1.1.cmml" xref="S1.SS1.p1.1.m1.1.1">𝑉</ci><ci id="S1.SS1.p1.1.m1.2.2.cmml" xref="S1.SS1.p1.1.m1.2.2">𝐸</ci></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS1.p1.1.m1.2c">G=(V,E)</annotation><annotation encoding="application/x-llamapun" id="S1.SS1.p1.1.m1.2d">italic_G = ( italic_V , italic_E )</annotation></semantics></math>, we assume that the weight function <math alttext="w:E\rightarrow\{0,1,\dots,W\}" class="ltx_Math" display="inline" id="S1.SS1.p1.2.m2.4"><semantics id="S1.SS1.p1.2.m2.4a"><mrow id="S1.SS1.p1.2.m2.4.5" xref="S1.SS1.p1.2.m2.4.5.cmml"><mi id="S1.SS1.p1.2.m2.4.5.2" xref="S1.SS1.p1.2.m2.4.5.2.cmml">w</mi><mo id="S1.SS1.p1.2.m2.4.5.1" lspace="0.278em" rspace="0.278em" xref="S1.SS1.p1.2.m2.4.5.1.cmml">:</mo><mrow id="S1.SS1.p1.2.m2.4.5.3" xref="S1.SS1.p1.2.m2.4.5.3.cmml"><mi id="S1.SS1.p1.2.m2.4.5.3.2" xref="S1.SS1.p1.2.m2.4.5.3.2.cmml">E</mi><mo id="S1.SS1.p1.2.m2.4.5.3.1" stretchy="false" xref="S1.SS1.p1.2.m2.4.5.3.1.cmml">→</mo><mrow id="S1.SS1.p1.2.m2.4.5.3.3.2" xref="S1.SS1.p1.2.m2.4.5.3.3.1.cmml"><mo id="S1.SS1.p1.2.m2.4.5.3.3.2.1" stretchy="false" xref="S1.SS1.p1.2.m2.4.5.3.3.1.cmml">{</mo><mn id="S1.SS1.p1.2.m2.1.1" xref="S1.SS1.p1.2.m2.1.1.cmml">0</mn><mo id="S1.SS1.p1.2.m2.4.5.3.3.2.2" xref="S1.SS1.p1.2.m2.4.5.3.3.1.cmml">,</mo><mn id="S1.SS1.p1.2.m2.2.2" xref="S1.SS1.p1.2.m2.2.2.cmml">1</mn><mo id="S1.SS1.p1.2.m2.4.5.3.3.2.3" xref="S1.SS1.p1.2.m2.4.5.3.3.1.cmml">,</mo><mi id="S1.SS1.p1.2.m2.3.3" mathvariant="normal" xref="S1.SS1.p1.2.m2.3.3.cmml">…</mi><mo id="S1.SS1.p1.2.m2.4.5.3.3.2.4" xref="S1.SS1.p1.2.m2.4.5.3.3.1.cmml">,</mo><mi id="S1.SS1.p1.2.m2.4.4" xref="S1.SS1.p1.2.m2.4.4.cmml">W</mi><mo id="S1.SS1.p1.2.m2.4.5.3.3.2.5" stretchy="false" xref="S1.SS1.p1.2.m2.4.5.3.3.1.cmml">}</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.SS1.p1.2.m2.4b"><apply id="S1.SS1.p1.2.m2.4.5.cmml" xref="S1.SS1.p1.2.m2.4.5"><ci id="S1.SS1.p1.2.m2.4.5.1.cmml" xref="S1.SS1.p1.2.m2.4.5.1">:</ci><ci id="S1.SS1.p1.2.m2.4.5.2.cmml" xref="S1.SS1.p1.2.m2.4.5.2">𝑤</ci><apply id="S1.SS1.p1.2.m2.4.5.3.cmml" xref="S1.SS1.p1.2.m2.4.5.3"><ci id="S1.SS1.p1.2.m2.4.5.3.1.cmml" xref="S1.SS1.p1.2.m2.4.5.3.1">→</ci><ci id="S1.SS1.p1.2.m2.4.5.3.2.cmml" xref="S1.SS1.p1.2.m2.4.5.3.2">𝐸</ci><set id="S1.SS1.p1.2.m2.4.5.3.3.1.cmml" xref="S1.SS1.p1.2.m2.4.5.3.3.2"><cn id="S1.SS1.p1.2.m2.1.1.cmml" type="integer" xref="S1.SS1.p1.2.m2.1.1">0</cn><cn id="S1.SS1.p1.2.m2.2.2.cmml" type="integer" xref="S1.SS1.p1.2.m2.2.2">1</cn><ci id="S1.SS1.p1.2.m2.3.3.cmml" xref="S1.SS1.p1.2.m2.3.3">…</ci><ci id="S1.SS1.p1.2.m2.4.4.cmml" xref="S1.SS1.p1.2.m2.4.4">𝑊</ci></set></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS1.p1.2.m2.4c">w:E\rightarrow\{0,1,\dots,W\}</annotation><annotation encoding="application/x-llamapun" id="S1.SS1.p1.2.m2.4d">italic_w : italic_E → { 0 , 1 , … , italic_W }</annotation></semantics></math> where <math alttext="W=n^{\operatorname{polylog}(n)}" class="ltx_Math" display="inline" id="S1.SS1.p1.3.m3.2"><semantics id="S1.SS1.p1.3.m3.2a"><mrow id="S1.SS1.p1.3.m3.2.3" xref="S1.SS1.p1.3.m3.2.3.cmml"><mi id="S1.SS1.p1.3.m3.2.3.2" xref="S1.SS1.p1.3.m3.2.3.2.cmml">W</mi><mo id="S1.SS1.p1.3.m3.2.3.1" xref="S1.SS1.p1.3.m3.2.3.1.cmml">=</mo><msup id="S1.SS1.p1.3.m3.2.3.3" xref="S1.SS1.p1.3.m3.2.3.3.cmml"><mi id="S1.SS1.p1.3.m3.2.3.3.2" xref="S1.SS1.p1.3.m3.2.3.3.2.cmml">n</mi><mrow id="S1.SS1.p1.3.m3.2.2.2.4" xref="S1.SS1.p1.3.m3.2.2.2.3.cmml"><mi id="S1.SS1.p1.3.m3.1.1.1.1" xref="S1.SS1.p1.3.m3.1.1.1.1.cmml">polylog</mi><mo id="S1.SS1.p1.3.m3.2.2.2.4a" xref="S1.SS1.p1.3.m3.2.2.2.3.cmml">⁡</mo><mrow id="S1.SS1.p1.3.m3.2.2.2.4.1" xref="S1.SS1.p1.3.m3.2.2.2.3.cmml"><mo id="S1.SS1.p1.3.m3.2.2.2.4.1.1" stretchy="false" xref="S1.SS1.p1.3.m3.2.2.2.3.cmml">(</mo><mi id="S1.SS1.p1.3.m3.2.2.2.2" xref="S1.SS1.p1.3.m3.2.2.2.2.cmml">n</mi><mo id="S1.SS1.p1.3.m3.2.2.2.4.1.2" stretchy="false" xref="S1.SS1.p1.3.m3.2.2.2.3.cmml">)</mo></mrow></mrow></msup></mrow><annotation-xml encoding="MathML-Content" id="S1.SS1.p1.3.m3.2b"><apply id="S1.SS1.p1.3.m3.2.3.cmml" xref="S1.SS1.p1.3.m3.2.3"><eq id="S1.SS1.p1.3.m3.2.3.1.cmml" xref="S1.SS1.p1.3.m3.2.3.1"></eq><ci id="S1.SS1.p1.3.m3.2.3.2.cmml" xref="S1.SS1.p1.3.m3.2.3.2">𝑊</ci><apply id="S1.SS1.p1.3.m3.2.3.3.cmml" xref="S1.SS1.p1.3.m3.2.3.3"><csymbol cd="ambiguous" id="S1.SS1.p1.3.m3.2.3.3.1.cmml" xref="S1.SS1.p1.3.m3.2.3.3">superscript</csymbol><ci id="S1.SS1.p1.3.m3.2.3.3.2.cmml" xref="S1.SS1.p1.3.m3.2.3.3.2">𝑛</ci><apply id="S1.SS1.p1.3.m3.2.2.2.3.cmml" xref="S1.SS1.p1.3.m3.2.2.2.4"><ci id="S1.SS1.p1.3.m3.1.1.1.1.cmml" xref="S1.SS1.p1.3.m3.1.1.1.1">polylog</ci><ci id="S1.SS1.p1.3.m3.2.2.2.2.cmml" xref="S1.SS1.p1.3.m3.2.2.2.2">𝑛</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS1.p1.3.m3.2c">W=n^{\operatorname{polylog}(n)}</annotation><annotation encoding="application/x-llamapun" id="S1.SS1.p1.3.m3.2d">italic_W = italic_n start_POSTSUPERSCRIPT roman_polylog ( italic_n ) end_POSTSUPERSCRIPT</annotation></semantics></math>. This, in particular, will imply that <math alttext="\log W=\mathrm{poly}(\log n)" class="ltx_Math" display="inline" id="S1.SS1.p1.4.m4.1"><semantics id="S1.SS1.p1.4.m4.1a"><mrow id="S1.SS1.p1.4.m4.1.1" xref="S1.SS1.p1.4.m4.1.1.cmml"><mrow id="S1.SS1.p1.4.m4.1.1.3" xref="S1.SS1.p1.4.m4.1.1.3.cmml"><mi id="S1.SS1.p1.4.m4.1.1.3.1" xref="S1.SS1.p1.4.m4.1.1.3.1.cmml">log</mi><mo id="S1.SS1.p1.4.m4.1.1.3a" lspace="0.167em" xref="S1.SS1.p1.4.m4.1.1.3.cmml">⁡</mo><mi id="S1.SS1.p1.4.m4.1.1.3.2" xref="S1.SS1.p1.4.m4.1.1.3.2.cmml">W</mi></mrow><mo id="S1.SS1.p1.4.m4.1.1.2" xref="S1.SS1.p1.4.m4.1.1.2.cmml">=</mo><mrow id="S1.SS1.p1.4.m4.1.1.1" xref="S1.SS1.p1.4.m4.1.1.1.cmml"><mi id="S1.SS1.p1.4.m4.1.1.1.3" xref="S1.SS1.p1.4.m4.1.1.1.3.cmml">poly</mi><mo id="S1.SS1.p1.4.m4.1.1.1.2" xref="S1.SS1.p1.4.m4.1.1.1.2.cmml">⁢</mo><mrow id="S1.SS1.p1.4.m4.1.1.1.1.1" xref="S1.SS1.p1.4.m4.1.1.1.1.1.1.cmml"><mo id="S1.SS1.p1.4.m4.1.1.1.1.1.2" stretchy="false" xref="S1.SS1.p1.4.m4.1.1.1.1.1.1.cmml">(</mo><mrow id="S1.SS1.p1.4.m4.1.1.1.1.1.1" xref="S1.SS1.p1.4.m4.1.1.1.1.1.1.cmml"><mi id="S1.SS1.p1.4.m4.1.1.1.1.1.1.1" xref="S1.SS1.p1.4.m4.1.1.1.1.1.1.1.cmml">log</mi><mo id="S1.SS1.p1.4.m4.1.1.1.1.1.1a" lspace="0.167em" xref="S1.SS1.p1.4.m4.1.1.1.1.1.1.cmml">⁡</mo><mi id="S1.SS1.p1.4.m4.1.1.1.1.1.1.2" xref="S1.SS1.p1.4.m4.1.1.1.1.1.1.2.cmml">n</mi></mrow><mo id="S1.SS1.p1.4.m4.1.1.1.1.1.3" stretchy="false" xref="S1.SS1.p1.4.m4.1.1.1.1.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.SS1.p1.4.m4.1b"><apply id="S1.SS1.p1.4.m4.1.1.cmml" xref="S1.SS1.p1.4.m4.1.1"><eq id="S1.SS1.p1.4.m4.1.1.2.cmml" xref="S1.SS1.p1.4.m4.1.1.2"></eq><apply id="S1.SS1.p1.4.m4.1.1.3.cmml" xref="S1.SS1.p1.4.m4.1.1.3"><log id="S1.SS1.p1.4.m4.1.1.3.1.cmml" xref="S1.SS1.p1.4.m4.1.1.3.1"></log><ci id="S1.SS1.p1.4.m4.1.1.3.2.cmml" xref="S1.SS1.p1.4.m4.1.1.3.2">𝑊</ci></apply><apply id="S1.SS1.p1.4.m4.1.1.1.cmml" xref="S1.SS1.p1.4.m4.1.1.1"><times id="S1.SS1.p1.4.m4.1.1.1.2.cmml" xref="S1.SS1.p1.4.m4.1.1.1.2"></times><ci id="S1.SS1.p1.4.m4.1.1.1.3.cmml" xref="S1.SS1.p1.4.m4.1.1.1.3">poly</ci><apply id="S1.SS1.p1.4.m4.1.1.1.1.1.1.cmml" xref="S1.SS1.p1.4.m4.1.1.1.1.1"><log id="S1.SS1.p1.4.m4.1.1.1.1.1.1.1.cmml" xref="S1.SS1.p1.4.m4.1.1.1.1.1.1.1"></log><ci id="S1.SS1.p1.4.m4.1.1.1.1.1.1.2.cmml" xref="S1.SS1.p1.4.m4.1.1.1.1.1.1.2">𝑛</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS1.p1.4.m4.1c">\log W=\mathrm{poly}(\log n)</annotation><annotation encoding="application/x-llamapun" id="S1.SS1.p1.4.m4.1d">roman_log italic_W = roman_poly ( roman_log italic_n )</annotation></semantics></math> and we will not mention <math alttext="\log W" class="ltx_Math" display="inline" id="S1.SS1.p1.5.m5.1"><semantics id="S1.SS1.p1.5.m5.1a"><mrow id="S1.SS1.p1.5.m5.1.1" xref="S1.SS1.p1.5.m5.1.1.cmml"><mi id="S1.SS1.p1.5.m5.1.1.1" xref="S1.SS1.p1.5.m5.1.1.1.cmml">log</mi><mo id="S1.SS1.p1.5.m5.1.1a" lspace="0.167em" xref="S1.SS1.p1.5.m5.1.1.cmml">⁡</mo><mi id="S1.SS1.p1.5.m5.1.1.2" xref="S1.SS1.p1.5.m5.1.1.2.cmml">W</mi></mrow><annotation-xml encoding="MathML-Content" id="S1.SS1.p1.5.m5.1b"><apply id="S1.SS1.p1.5.m5.1.1.cmml" xref="S1.SS1.p1.5.m5.1.1"><log id="S1.SS1.p1.5.m5.1.1.1.cmml" xref="S1.SS1.p1.5.m5.1.1.1"></log><ci id="S1.SS1.p1.5.m5.1.1.2.cmml" xref="S1.SS1.p1.5.m5.1.1.2">𝑊</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS1.p1.5.m5.1c">\log W</annotation><annotation encoding="application/x-llamapun" id="S1.SS1.p1.5.m5.1d">roman_log italic_W</annotation></semantics></math> explicitly in the space complexity. While all of our algorithms have a <math alttext="\log W" class="ltx_Math" display="inline" id="S1.SS1.p1.6.m6.1"><semantics id="S1.SS1.p1.6.m6.1a"><mrow id="S1.SS1.p1.6.m6.1.1" xref="S1.SS1.p1.6.m6.1.1.cmml"><mi id="S1.SS1.p1.6.m6.1.1.1" xref="S1.SS1.p1.6.m6.1.1.1.cmml">log</mi><mo id="S1.SS1.p1.6.m6.1.1a" lspace="0.167em" xref="S1.SS1.p1.6.m6.1.1.cmml">⁡</mo><mi id="S1.SS1.p1.6.m6.1.1.2" xref="S1.SS1.p1.6.m6.1.1.2.cmml">W</mi></mrow><annotation-xml encoding="MathML-Content" id="S1.SS1.p1.6.m6.1b"><apply id="S1.SS1.p1.6.m6.1.1.cmml" xref="S1.SS1.p1.6.m6.1.1"><log id="S1.SS1.p1.6.m6.1.1.1.cmml" xref="S1.SS1.p1.6.m6.1.1.1"></log><ci id="S1.SS1.p1.6.m6.1.1.2.cmml" xref="S1.SS1.p1.6.m6.1.1.2">𝑊</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS1.p1.6.m6.1c">\log W</annotation><annotation encoding="application/x-llamapun" id="S1.SS1.p1.6.m6.1d">roman_log italic_W</annotation></semantics></math> dependence in their space complexity (in terms of words of space), designing streaming algorithms where the total number of stored edges is independent of <math alttext="W" class="ltx_Math" display="inline" id="S1.SS1.p1.7.m7.1"><semantics id="S1.SS1.p1.7.m7.1a"><mi id="S1.SS1.p1.7.m7.1.1" xref="S1.SS1.p1.7.m7.1.1.cmml">W</mi><annotation-xml encoding="MathML-Content" id="S1.SS1.p1.7.m7.1b"><ci id="S1.SS1.p1.7.m7.1.1.cmml" xref="S1.SS1.p1.7.m7.1.1">𝑊</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.SS1.p1.7.m7.1c">W</annotation><annotation encoding="application/x-llamapun" id="S1.SS1.p1.7.m7.1d">italic_W</annotation></semantics></math>, as explored in <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.00712v1#bib.bibx54" title="">JKMV24</a>]</cite>, is an interesting technical question in its own right.</p> </div> <section class="ltx_paragraph" id="S1.SS1.SSS0.Px1"> <h5 class="ltx_title ltx_title_paragraph">A framework for SNDP:</h5> <div class="ltx_para" id="S1.SS1.SSS0.Px1.p1"> <p class="ltx_p" id="S1.SS1.SSS0.Px1.p1.3">Our first contribution is a general and broadly applicable framework to obtain streaming algorithms with low space and good approximation ratios for SNDP; this is provided in Section <a class="ltx_ref" href="https://arxiv.org/html/2503.00712v1#S3" title="3 Generic Framework for Streaming Algorithms for Network Design ‣ Streaming Algorithms for Network Design"><span class="ltx_text ltx_ref_tag">3</span></a>. This framework applies to both edge and vertex connectivity, as well as an intermediate setting known as element-connectivity (formally defined in Section <a class="ltx_ref" href="https://arxiv.org/html/2503.00712v1#S2" title="2 Preliminaries ‣ Streaming Algorithms for Network Design"><span class="ltx_text ltx_ref_tag">2</span></a>). This framework addresses Question <a class="ltx_ref" href="https://arxiv.org/html/2503.00712v1#Thmquestion1" title="Question 1. ‣ 1 Introduction ‣ Streaming Algorithms for Network Design"><span class="ltx_text ltx_ref_tag">1</span></a> and partially resolves Question <a class="ltx_ref" href="https://arxiv.org/html/2503.00712v1#Thmquestion2" title="Question 2. ‣ 1 Introduction ‣ Streaming Algorithms for Network Design"><span class="ltx_text ltx_ref_tag">2</span></a> affirmatively.<span class="ltx_note ltx_role_footnote" id="footnote2"><sup class="ltx_note_mark">2</sup><span class="ltx_note_outer"><span class="ltx_note_content"><sup class="ltx_note_mark">2</sup><span class="ltx_tag ltx_tag_note">2</span>The upper and lower bounds match for several interesting/practical values of <math alttext="k" class="ltx_Math" display="inline" id="footnote2.m1.1"><semantics id="footnote2.m1.1b"><mi id="footnote2.m1.1.1" xref="footnote2.m1.1.1.cmml">k</mi><annotation-xml encoding="MathML-Content" id="footnote2.m1.1c"><ci id="footnote2.m1.1.1.cmml" xref="footnote2.m1.1.1">𝑘</ci></annotation-xml><annotation encoding="application/x-tex" id="footnote2.m1.1d">k</annotation><annotation encoding="application/x-llamapun" id="footnote2.m1.1e">italic_k</annotation></semantics></math>, e.g. <math alttext="k=O(\operatorname{polylog}n)" class="ltx_Math" display="inline" id="footnote2.m2.1"><semantics id="footnote2.m2.1b"><mrow id="footnote2.m2.1.1" xref="footnote2.m2.1.1.cmml"><mi id="footnote2.m2.1.1.3" xref="footnote2.m2.1.1.3.cmml">k</mi><mo id="footnote2.m2.1.1.2" xref="footnote2.m2.1.1.2.cmml">=</mo><mrow id="footnote2.m2.1.1.1" xref="footnote2.m2.1.1.1.cmml"><mi id="footnote2.m2.1.1.1.3" xref="footnote2.m2.1.1.1.3.cmml">O</mi><mo id="footnote2.m2.1.1.1.2" xref="footnote2.m2.1.1.1.2.cmml">⁢</mo><mrow id="footnote2.m2.1.1.1.1.1" xref="footnote2.m2.1.1.1.1.1.1.cmml"><mo id="footnote2.m2.1.1.1.1.1.2" stretchy="false" xref="footnote2.m2.1.1.1.1.1.1.cmml">(</mo><mrow id="footnote2.m2.1.1.1.1.1.1" xref="footnote2.m2.1.1.1.1.1.1.cmml"><mi id="footnote2.m2.1.1.1.1.1.1.1" xref="footnote2.m2.1.1.1.1.1.1.1.cmml">polylog</mi><mo id="footnote2.m2.1.1.1.1.1.1b" lspace="0.167em" xref="footnote2.m2.1.1.1.1.1.1.cmml">⁡</mo><mi id="footnote2.m2.1.1.1.1.1.1.2" xref="footnote2.m2.1.1.1.1.1.1.2.cmml">n</mi></mrow><mo id="footnote2.m2.1.1.1.1.1.3" stretchy="false" xref="footnote2.m2.1.1.1.1.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="footnote2.m2.1c"><apply id="footnote2.m2.1.1.cmml" xref="footnote2.m2.1.1"><eq id="footnote2.m2.1.1.2.cmml" xref="footnote2.m2.1.1.2"></eq><ci id="footnote2.m2.1.1.3.cmml" xref="footnote2.m2.1.1.3">𝑘</ci><apply id="footnote2.m2.1.1.1.cmml" xref="footnote2.m2.1.1.1"><times id="footnote2.m2.1.1.1.2.cmml" xref="footnote2.m2.1.1.1.2"></times><ci id="footnote2.m2.1.1.1.3.cmml" xref="footnote2.m2.1.1.1.3">𝑂</ci><apply id="footnote2.m2.1.1.1.1.1.1.cmml" xref="footnote2.m2.1.1.1.1.1"><ci id="footnote2.m2.1.1.1.1.1.1.1.cmml" xref="footnote2.m2.1.1.1.1.1.1.1">polylog</ci><ci id="footnote2.m2.1.1.1.1.1.1.2.cmml" xref="footnote2.m2.1.1.1.1.1.1.2">𝑛</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="footnote2.m2.1d">k=O(\operatorname{polylog}n)</annotation><annotation encoding="application/x-llamapun" id="footnote2.m2.1e">italic_k = italic_O ( roman_polylog italic_n )</annotation></semantics></math> or <math alttext="k=\Omega(n)" class="ltx_Math" display="inline" id="footnote2.m3.1"><semantics id="footnote2.m3.1b"><mrow id="footnote2.m3.1.2" xref="footnote2.m3.1.2.cmml"><mi id="footnote2.m3.1.2.2" xref="footnote2.m3.1.2.2.cmml">k</mi><mo id="footnote2.m3.1.2.1" xref="footnote2.m3.1.2.1.cmml">=</mo><mrow id="footnote2.m3.1.2.3" xref="footnote2.m3.1.2.3.cmml"><mi id="footnote2.m3.1.2.3.2" mathvariant="normal" xref="footnote2.m3.1.2.3.2.cmml">Ω</mi><mo id="footnote2.m3.1.2.3.1" xref="footnote2.m3.1.2.3.1.cmml">⁢</mo><mrow id="footnote2.m3.1.2.3.3.2" xref="footnote2.m3.1.2.3.cmml"><mo id="footnote2.m3.1.2.3.3.2.1" stretchy="false" xref="footnote2.m3.1.2.3.cmml">(</mo><mi id="footnote2.m3.1.1" xref="footnote2.m3.1.1.cmml">n</mi><mo id="footnote2.m3.1.2.3.3.2.2" stretchy="false" xref="footnote2.m3.1.2.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="footnote2.m3.1c"><apply id="footnote2.m3.1.2.cmml" xref="footnote2.m3.1.2"><eq id="footnote2.m3.1.2.1.cmml" xref="footnote2.m3.1.2.1"></eq><ci id="footnote2.m3.1.2.2.cmml" xref="footnote2.m3.1.2.2">𝑘</ci><apply id="footnote2.m3.1.2.3.cmml" xref="footnote2.m3.1.2.3"><times id="footnote2.m3.1.2.3.1.cmml" xref="footnote2.m3.1.2.3.1"></times><ci id="footnote2.m3.1.2.3.2.cmml" xref="footnote2.m3.1.2.3.2">Ω</ci><ci id="footnote2.m3.1.1.cmml" xref="footnote2.m3.1.1">𝑛</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="footnote2.m3.1d">k=\Omega(n)</annotation><annotation encoding="application/x-llamapun" id="footnote2.m3.1e">italic_k = roman_Ω ( italic_n )</annotation></semantics></math>. However, there is still a small gap between the bounds in general.</span></span></span> All the results below use <math alttext="\tilde{O}(k^{1-1/t}n^{1+1/t})" class="ltx_Math" display="inline" id="S1.SS1.SSS0.Px1.p1.1.m1.1"><semantics id="S1.SS1.SSS0.Px1.p1.1.m1.1a"><mrow id="S1.SS1.SSS0.Px1.p1.1.m1.1.1" xref="S1.SS1.SSS0.Px1.p1.1.m1.1.1.cmml"><mover accent="true" id="S1.SS1.SSS0.Px1.p1.1.m1.1.1.3" xref="S1.SS1.SSS0.Px1.p1.1.m1.1.1.3.cmml"><mi id="S1.SS1.SSS0.Px1.p1.1.m1.1.1.3.2" xref="S1.SS1.SSS0.Px1.p1.1.m1.1.1.3.2.cmml">O</mi><mo id="S1.SS1.SSS0.Px1.p1.1.m1.1.1.3.1" xref="S1.SS1.SSS0.Px1.p1.1.m1.1.1.3.1.cmml">~</mo></mover><mo id="S1.SS1.SSS0.Px1.p1.1.m1.1.1.2" xref="S1.SS1.SSS0.Px1.p1.1.m1.1.1.2.cmml">⁢</mo><mrow id="S1.SS1.SSS0.Px1.p1.1.m1.1.1.1.1" xref="S1.SS1.SSS0.Px1.p1.1.m1.1.1.1.1.1.cmml"><mo id="S1.SS1.SSS0.Px1.p1.1.m1.1.1.1.1.2" stretchy="false" xref="S1.SS1.SSS0.Px1.p1.1.m1.1.1.1.1.1.cmml">(</mo><mrow id="S1.SS1.SSS0.Px1.p1.1.m1.1.1.1.1.1" xref="S1.SS1.SSS0.Px1.p1.1.m1.1.1.1.1.1.cmml"><msup id="S1.SS1.SSS0.Px1.p1.1.m1.1.1.1.1.1.2" xref="S1.SS1.SSS0.Px1.p1.1.m1.1.1.1.1.1.2.cmml"><mi id="S1.SS1.SSS0.Px1.p1.1.m1.1.1.1.1.1.2.2" xref="S1.SS1.SSS0.Px1.p1.1.m1.1.1.1.1.1.2.2.cmml">k</mi><mrow id="S1.SS1.SSS0.Px1.p1.1.m1.1.1.1.1.1.2.3" xref="S1.SS1.SSS0.Px1.p1.1.m1.1.1.1.1.1.2.3.cmml"><mn id="S1.SS1.SSS0.Px1.p1.1.m1.1.1.1.1.1.2.3.2" xref="S1.SS1.SSS0.Px1.p1.1.m1.1.1.1.1.1.2.3.2.cmml">1</mn><mo id="S1.SS1.SSS0.Px1.p1.1.m1.1.1.1.1.1.2.3.1" xref="S1.SS1.SSS0.Px1.p1.1.m1.1.1.1.1.1.2.3.1.cmml">−</mo><mrow id="S1.SS1.SSS0.Px1.p1.1.m1.1.1.1.1.1.2.3.3" xref="S1.SS1.SSS0.Px1.p1.1.m1.1.1.1.1.1.2.3.3.cmml"><mn id="S1.SS1.SSS0.Px1.p1.1.m1.1.1.1.1.1.2.3.3.2" xref="S1.SS1.SSS0.Px1.p1.1.m1.1.1.1.1.1.2.3.3.2.cmml">1</mn><mo id="S1.SS1.SSS0.Px1.p1.1.m1.1.1.1.1.1.2.3.3.1" xref="S1.SS1.SSS0.Px1.p1.1.m1.1.1.1.1.1.2.3.3.1.cmml">/</mo><mi id="S1.SS1.SSS0.Px1.p1.1.m1.1.1.1.1.1.2.3.3.3" xref="S1.SS1.SSS0.Px1.p1.1.m1.1.1.1.1.1.2.3.3.3.cmml">t</mi></mrow></mrow></msup><mo id="S1.SS1.SSS0.Px1.p1.1.m1.1.1.1.1.1.1" xref="S1.SS1.SSS0.Px1.p1.1.m1.1.1.1.1.1.1.cmml">⁢</mo><msup id="S1.SS1.SSS0.Px1.p1.1.m1.1.1.1.1.1.3" xref="S1.SS1.SSS0.Px1.p1.1.m1.1.1.1.1.1.3.cmml"><mi id="S1.SS1.SSS0.Px1.p1.1.m1.1.1.1.1.1.3.2" xref="S1.SS1.SSS0.Px1.p1.1.m1.1.1.1.1.1.3.2.cmml">n</mi><mrow id="S1.SS1.SSS0.Px1.p1.1.m1.1.1.1.1.1.3.3" xref="S1.SS1.SSS0.Px1.p1.1.m1.1.1.1.1.1.3.3.cmml"><mn id="S1.SS1.SSS0.Px1.p1.1.m1.1.1.1.1.1.3.3.2" xref="S1.SS1.SSS0.Px1.p1.1.m1.1.1.1.1.1.3.3.2.cmml">1</mn><mo id="S1.SS1.SSS0.Px1.p1.1.m1.1.1.1.1.1.3.3.1" xref="S1.SS1.SSS0.Px1.p1.1.m1.1.1.1.1.1.3.3.1.cmml">+</mo><mrow id="S1.SS1.SSS0.Px1.p1.1.m1.1.1.1.1.1.3.3.3" xref="S1.SS1.SSS0.Px1.p1.1.m1.1.1.1.1.1.3.3.3.cmml"><mn id="S1.SS1.SSS0.Px1.p1.1.m1.1.1.1.1.1.3.3.3.2" xref="S1.SS1.SSS0.Px1.p1.1.m1.1.1.1.1.1.3.3.3.2.cmml">1</mn><mo id="S1.SS1.SSS0.Px1.p1.1.m1.1.1.1.1.1.3.3.3.1" xref="S1.SS1.SSS0.Px1.p1.1.m1.1.1.1.1.1.3.3.3.1.cmml">/</mo><mi id="S1.SS1.SSS0.Px1.p1.1.m1.1.1.1.1.1.3.3.3.3" xref="S1.SS1.SSS0.Px1.p1.1.m1.1.1.1.1.1.3.3.3.3.cmml">t</mi></mrow></mrow></msup></mrow><mo id="S1.SS1.SSS0.Px1.p1.1.m1.1.1.1.1.3" stretchy="false" xref="S1.SS1.SSS0.Px1.p1.1.m1.1.1.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.SS1.SSS0.Px1.p1.1.m1.1b"><apply id="S1.SS1.SSS0.Px1.p1.1.m1.1.1.cmml" xref="S1.SS1.SSS0.Px1.p1.1.m1.1.1"><times id="S1.SS1.SSS0.Px1.p1.1.m1.1.1.2.cmml" xref="S1.SS1.SSS0.Px1.p1.1.m1.1.1.2"></times><apply id="S1.SS1.SSS0.Px1.p1.1.m1.1.1.3.cmml" xref="S1.SS1.SSS0.Px1.p1.1.m1.1.1.3"><ci id="S1.SS1.SSS0.Px1.p1.1.m1.1.1.3.1.cmml" xref="S1.SS1.SSS0.Px1.p1.1.m1.1.1.3.1">~</ci><ci id="S1.SS1.SSS0.Px1.p1.1.m1.1.1.3.2.cmml" xref="S1.SS1.SSS0.Px1.p1.1.m1.1.1.3.2">𝑂</ci></apply><apply id="S1.SS1.SSS0.Px1.p1.1.m1.1.1.1.1.1.cmml" xref="S1.SS1.SSS0.Px1.p1.1.m1.1.1.1.1"><times id="S1.SS1.SSS0.Px1.p1.1.m1.1.1.1.1.1.1.cmml" xref="S1.SS1.SSS0.Px1.p1.1.m1.1.1.1.1.1.1"></times><apply id="S1.SS1.SSS0.Px1.p1.1.m1.1.1.1.1.1.2.cmml" xref="S1.SS1.SSS0.Px1.p1.1.m1.1.1.1.1.1.2"><csymbol cd="ambiguous" id="S1.SS1.SSS0.Px1.p1.1.m1.1.1.1.1.1.2.1.cmml" xref="S1.SS1.SSS0.Px1.p1.1.m1.1.1.1.1.1.2">superscript</csymbol><ci id="S1.SS1.SSS0.Px1.p1.1.m1.1.1.1.1.1.2.2.cmml" xref="S1.SS1.SSS0.Px1.p1.1.m1.1.1.1.1.1.2.2">𝑘</ci><apply id="S1.SS1.SSS0.Px1.p1.1.m1.1.1.1.1.1.2.3.cmml" xref="S1.SS1.SSS0.Px1.p1.1.m1.1.1.1.1.1.2.3"><minus id="S1.SS1.SSS0.Px1.p1.1.m1.1.1.1.1.1.2.3.1.cmml" xref="S1.SS1.SSS0.Px1.p1.1.m1.1.1.1.1.1.2.3.1"></minus><cn id="S1.SS1.SSS0.Px1.p1.1.m1.1.1.1.1.1.2.3.2.cmml" type="integer" xref="S1.SS1.SSS0.Px1.p1.1.m1.1.1.1.1.1.2.3.2">1</cn><apply id="S1.SS1.SSS0.Px1.p1.1.m1.1.1.1.1.1.2.3.3.cmml" xref="S1.SS1.SSS0.Px1.p1.1.m1.1.1.1.1.1.2.3.3"><divide id="S1.SS1.SSS0.Px1.p1.1.m1.1.1.1.1.1.2.3.3.1.cmml" xref="S1.SS1.SSS0.Px1.p1.1.m1.1.1.1.1.1.2.3.3.1"></divide><cn id="S1.SS1.SSS0.Px1.p1.1.m1.1.1.1.1.1.2.3.3.2.cmml" type="integer" xref="S1.SS1.SSS0.Px1.p1.1.m1.1.1.1.1.1.2.3.3.2">1</cn><ci id="S1.SS1.SSS0.Px1.p1.1.m1.1.1.1.1.1.2.3.3.3.cmml" xref="S1.SS1.SSS0.Px1.p1.1.m1.1.1.1.1.1.2.3.3.3">𝑡</ci></apply></apply></apply><apply id="S1.SS1.SSS0.Px1.p1.1.m1.1.1.1.1.1.3.cmml" xref="S1.SS1.SSS0.Px1.p1.1.m1.1.1.1.1.1.3"><csymbol cd="ambiguous" id="S1.SS1.SSS0.Px1.p1.1.m1.1.1.1.1.1.3.1.cmml" xref="S1.SS1.SSS0.Px1.p1.1.m1.1.1.1.1.1.3">superscript</csymbol><ci id="S1.SS1.SSS0.Px1.p1.1.m1.1.1.1.1.1.3.2.cmml" xref="S1.SS1.SSS0.Px1.p1.1.m1.1.1.1.1.1.3.2">𝑛</ci><apply id="S1.SS1.SSS0.Px1.p1.1.m1.1.1.1.1.1.3.3.cmml" xref="S1.SS1.SSS0.Px1.p1.1.m1.1.1.1.1.1.3.3"><plus id="S1.SS1.SSS0.Px1.p1.1.m1.1.1.1.1.1.3.3.1.cmml" xref="S1.SS1.SSS0.Px1.p1.1.m1.1.1.1.1.1.3.3.1"></plus><cn id="S1.SS1.SSS0.Px1.p1.1.m1.1.1.1.1.1.3.3.2.cmml" type="integer" xref="S1.SS1.SSS0.Px1.p1.1.m1.1.1.1.1.1.3.3.2">1</cn><apply id="S1.SS1.SSS0.Px1.p1.1.m1.1.1.1.1.1.3.3.3.cmml" xref="S1.SS1.SSS0.Px1.p1.1.m1.1.1.1.1.1.3.3.3"><divide id="S1.SS1.SSS0.Px1.p1.1.m1.1.1.1.1.1.3.3.3.1.cmml" xref="S1.SS1.SSS0.Px1.p1.1.m1.1.1.1.1.1.3.3.3.1"></divide><cn id="S1.SS1.SSS0.Px1.p1.1.m1.1.1.1.1.1.3.3.3.2.cmml" type="integer" xref="S1.SS1.SSS0.Px1.p1.1.m1.1.1.1.1.1.3.3.3.2">1</cn><ci id="S1.SS1.SSS0.Px1.p1.1.m1.1.1.1.1.1.3.3.3.3.cmml" xref="S1.SS1.SSS0.Px1.p1.1.m1.1.1.1.1.1.3.3.3.3">𝑡</ci></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS1.SSS0.Px1.p1.1.m1.1c">\tilde{O}(k^{1-1/t}n^{1+1/t})</annotation><annotation encoding="application/x-llamapun" id="S1.SS1.SSS0.Px1.p1.1.m1.1d">over~ start_ARG italic_O end_ARG ( italic_k start_POSTSUPERSCRIPT 1 - 1 / italic_t end_POSTSUPERSCRIPT italic_n start_POSTSUPERSCRIPT 1 + 1 / italic_t end_POSTSUPERSCRIPT )</annotation></semantics></math>-space where <math alttext="k" class="ltx_Math" display="inline" id="S1.SS1.SSS0.Px1.p1.2.m2.1"><semantics id="S1.SS1.SSS0.Px1.p1.2.m2.1a"><mi id="S1.SS1.SSS0.Px1.p1.2.m2.1.1" xref="S1.SS1.SSS0.Px1.p1.2.m2.1.1.cmml">k</mi><annotation-xml encoding="MathML-Content" id="S1.SS1.SSS0.Px1.p1.2.m2.1b"><ci id="S1.SS1.SSS0.Px1.p1.2.m2.1.1.cmml" xref="S1.SS1.SSS0.Px1.p1.2.m2.1.1">𝑘</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.SS1.SSS0.Px1.p1.2.m2.1c">k</annotation><annotation encoding="application/x-llamapun" id="S1.SS1.SSS0.Px1.p1.2.m2.1d">italic_k</annotation></semantics></math> is the maximum connectivity requirement and <math alttext="t" class="ltx_Math" display="inline" id="S1.SS1.SSS0.Px1.p1.3.m3.1"><semantics id="S1.SS1.SSS0.Px1.p1.3.m3.1a"><mi id="S1.SS1.SSS0.Px1.p1.3.m3.1.1" xref="S1.SS1.SSS0.Px1.p1.3.m3.1.1.cmml">t</mi><annotation-xml encoding="MathML-Content" id="S1.SS1.SSS0.Px1.p1.3.m3.1b"><ci id="S1.SS1.SSS0.Px1.p1.3.m3.1.1.cmml" xref="S1.SS1.SSS0.Px1.p1.3.m3.1.1">𝑡</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.SS1.SSS0.Px1.p1.3.m3.1c">t</annotation><annotation encoding="application/x-llamapun" id="S1.SS1.SSS0.Px1.p1.3.m3.1d">italic_t</annotation></semantics></math> is a parameter that allows for an approximation vs. space tradeoff.</p> </div> <div class="ltx_para" id="S1.SS1.SSS0.Px1.p2"> <ul class="ltx_itemize" id="S1.I3"> <li class="ltx_item" id="S1.I3.i1" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">•</span> <div class="ltx_para" id="S1.I3.i1.p1"> <p class="ltx_p" id="S1.I3.i1.p1.2">For EC-SNDP, the framework yields <math alttext="8t" class="ltx_Math" display="inline" id="S1.I3.i1.p1.1.m1.1"><semantics id="S1.I3.i1.p1.1.m1.1a"><mrow id="S1.I3.i1.p1.1.m1.1.1" xref="S1.I3.i1.p1.1.m1.1.1.cmml"><mn id="S1.I3.i1.p1.1.m1.1.1.2" xref="S1.I3.i1.p1.1.m1.1.1.2.cmml">8</mn><mo id="S1.I3.i1.p1.1.m1.1.1.1" xref="S1.I3.i1.p1.1.m1.1.1.1.cmml">⁢</mo><mi id="S1.I3.i1.p1.1.m1.1.1.3" xref="S1.I3.i1.p1.1.m1.1.1.3.cmml">t</mi></mrow><annotation-xml encoding="MathML-Content" id="S1.I3.i1.p1.1.m1.1b"><apply id="S1.I3.i1.p1.1.m1.1.1.cmml" xref="S1.I3.i1.p1.1.m1.1.1"><times id="S1.I3.i1.p1.1.m1.1.1.1.cmml" xref="S1.I3.i1.p1.1.m1.1.1.1"></times><cn id="S1.I3.i1.p1.1.m1.1.1.2.cmml" type="integer" xref="S1.I3.i1.p1.1.m1.1.1.2">8</cn><ci id="S1.I3.i1.p1.1.m1.1.1.3.cmml" xref="S1.I3.i1.p1.1.m1.1.1.3">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.I3.i1.p1.1.m1.1c">8t</annotation><annotation encoding="application/x-llamapun" id="S1.I3.i1.p1.1.m1.1d">8 italic_t</annotation></semantics></math>-approximation, improving upon the <math alttext="O(t\log k)" class="ltx_Math" display="inline" id="S1.I3.i1.p1.2.m2.1"><semantics id="S1.I3.i1.p1.2.m2.1a"><mrow id="S1.I3.i1.p1.2.m2.1.1" xref="S1.I3.i1.p1.2.m2.1.1.cmml"><mi id="S1.I3.i1.p1.2.m2.1.1.3" xref="S1.I3.i1.p1.2.m2.1.1.3.cmml">O</mi><mo id="S1.I3.i1.p1.2.m2.1.1.2" xref="S1.I3.i1.p1.2.m2.1.1.2.cmml">⁢</mo><mrow id="S1.I3.i1.p1.2.m2.1.1.1.1" xref="S1.I3.i1.p1.2.m2.1.1.1.1.1.cmml"><mo id="S1.I3.i1.p1.2.m2.1.1.1.1.2" stretchy="false" xref="S1.I3.i1.p1.2.m2.1.1.1.1.1.cmml">(</mo><mrow 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xref="S1.I3.i1.p1.2.m2.1.1.3">𝑂</ci><apply id="S1.I3.i1.p1.2.m2.1.1.1.1.1.cmml" xref="S1.I3.i1.p1.2.m2.1.1.1.1"><times id="S1.I3.i1.p1.2.m2.1.1.1.1.1.1.cmml" xref="S1.I3.i1.p1.2.m2.1.1.1.1.1.1"></times><ci id="S1.I3.i1.p1.2.m2.1.1.1.1.1.2.cmml" xref="S1.I3.i1.p1.2.m2.1.1.1.1.1.2">𝑡</ci><apply id="S1.I3.i1.p1.2.m2.1.1.1.1.1.3.cmml" xref="S1.I3.i1.p1.2.m2.1.1.1.1.1.3"><log id="S1.I3.i1.p1.2.m2.1.1.1.1.1.3.1.cmml" xref="S1.I3.i1.p1.2.m2.1.1.1.1.1.3.1"></log><ci id="S1.I3.i1.p1.2.m2.1.1.1.1.1.3.2.cmml" xref="S1.I3.i1.p1.2.m2.1.1.1.1.1.3.2">𝑘</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.I3.i1.p1.2.m2.1c">O(t\log k)</annotation><annotation encoding="application/x-llamapun" id="S1.I3.i1.p1.2.m2.1d">italic_O ( italic_t roman_log italic_k )</annotation></semantics></math>-approximation from <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.00712v1#bib.bibx54" title="">JKMV24</a>]</cite>. The trade-off we obtain is tight within small constant factors. These bounds also hold for element-connectivity, providing the first such results for this variant.</p> </div> </li> <li class="ltx_item" id="S1.I3.i2" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">•</span> <div class="ltx_para" id="S1.I3.i2.p1"> <p class="ltx_p" id="S1.I3.i2.p1.3">For VC-SNDP, the framework yields an <math alttext="O(tk)" class="ltx_Math" display="inline" id="S1.I3.i2.p1.1.m1.1"><semantics id="S1.I3.i2.p1.1.m1.1a"><mrow id="S1.I3.i2.p1.1.m1.1.1" xref="S1.I3.i2.p1.1.m1.1.1.cmml"><mi id="S1.I3.i2.p1.1.m1.1.1.3" xref="S1.I3.i2.p1.1.m1.1.1.3.cmml">O</mi><mo id="S1.I3.i2.p1.1.m1.1.1.2" xref="S1.I3.i2.p1.1.m1.1.1.2.cmml">⁢</mo><mrow id="S1.I3.i2.p1.1.m1.1.1.1.1" xref="S1.I3.i2.p1.1.m1.1.1.1.1.1.cmml"><mo id="S1.I3.i2.p1.1.m1.1.1.1.1.2" stretchy="false" xref="S1.I3.i2.p1.1.m1.1.1.1.1.1.cmml">(</mo><mrow id="S1.I3.i2.p1.1.m1.1.1.1.1.1" xref="S1.I3.i2.p1.1.m1.1.1.1.1.1.cmml"><mi id="S1.I3.i2.p1.1.m1.1.1.1.1.1.2" xref="S1.I3.i2.p1.1.m1.1.1.1.1.1.2.cmml">t</mi><mo id="S1.I3.i2.p1.1.m1.1.1.1.1.1.1" xref="S1.I3.i2.p1.1.m1.1.1.1.1.1.1.cmml">⁢</mo><mi id="S1.I3.i2.p1.1.m1.1.1.1.1.1.3" xref="S1.I3.i2.p1.1.m1.1.1.1.1.1.3.cmml">k</mi></mrow><mo id="S1.I3.i2.p1.1.m1.1.1.1.1.3" stretchy="false" xref="S1.I3.i2.p1.1.m1.1.1.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.I3.i2.p1.1.m1.1b"><apply id="S1.I3.i2.p1.1.m1.1.1.cmml" xref="S1.I3.i2.p1.1.m1.1.1"><times id="S1.I3.i2.p1.1.m1.1.1.2.cmml" xref="S1.I3.i2.p1.1.m1.1.1.2"></times><ci id="S1.I3.i2.p1.1.m1.1.1.3.cmml" xref="S1.I3.i2.p1.1.m1.1.1.3">𝑂</ci><apply id="S1.I3.i2.p1.1.m1.1.1.1.1.1.cmml" xref="S1.I3.i2.p1.1.m1.1.1.1.1"><times id="S1.I3.i2.p1.1.m1.1.1.1.1.1.1.cmml" xref="S1.I3.i2.p1.1.m1.1.1.1.1.1.1"></times><ci id="S1.I3.i2.p1.1.m1.1.1.1.1.1.2.cmml" xref="S1.I3.i2.p1.1.m1.1.1.1.1.1.2">𝑡</ci><ci id="S1.I3.i2.p1.1.m1.1.1.1.1.1.3.cmml" xref="S1.I3.i2.p1.1.m1.1.1.1.1.1.3">𝑘</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.I3.i2.p1.1.m1.1c">O(tk)</annotation><annotation encoding="application/x-llamapun" id="S1.I3.i2.p1.1.m1.1d">italic_O ( italic_t italic_k )</annotation></semantics></math>-approximation assuming exact algorithm at the end of the stream or an <math alttext="O(\alpha tk)" class="ltx_Math" display="inline" id="S1.I3.i2.p1.2.m2.1"><semantics id="S1.I3.i2.p1.2.m2.1a"><mrow id="S1.I3.i2.p1.2.m2.1.1" xref="S1.I3.i2.p1.2.m2.1.1.cmml"><mi id="S1.I3.i2.p1.2.m2.1.1.3" xref="S1.I3.i2.p1.2.m2.1.1.3.cmml">O</mi><mo id="S1.I3.i2.p1.2.m2.1.1.2" xref="S1.I3.i2.p1.2.m2.1.1.2.cmml">⁢</mo><mrow id="S1.I3.i2.p1.2.m2.1.1.1.1" xref="S1.I3.i2.p1.2.m2.1.1.1.1.1.cmml"><mo id="S1.I3.i2.p1.2.m2.1.1.1.1.2" stretchy="false" xref="S1.I3.i2.p1.2.m2.1.1.1.1.1.cmml">(</mo><mrow id="S1.I3.i2.p1.2.m2.1.1.1.1.1" xref="S1.I3.i2.p1.2.m2.1.1.1.1.1.cmml"><mi id="S1.I3.i2.p1.2.m2.1.1.1.1.1.2" xref="S1.I3.i2.p1.2.m2.1.1.1.1.1.2.cmml">α</mi><mo id="S1.I3.i2.p1.2.m2.1.1.1.1.1.1" xref="S1.I3.i2.p1.2.m2.1.1.1.1.1.1.cmml">⁢</mo><mi id="S1.I3.i2.p1.2.m2.1.1.1.1.1.3" xref="S1.I3.i2.p1.2.m2.1.1.1.1.1.3.cmml">t</mi><mo id="S1.I3.i2.p1.2.m2.1.1.1.1.1.1a" xref="S1.I3.i2.p1.2.m2.1.1.1.1.1.1.cmml">⁢</mo><mi id="S1.I3.i2.p1.2.m2.1.1.1.1.1.4" xref="S1.I3.i2.p1.2.m2.1.1.1.1.1.4.cmml">k</mi></mrow><mo id="S1.I3.i2.p1.2.m2.1.1.1.1.3" stretchy="false" xref="S1.I3.i2.p1.2.m2.1.1.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.I3.i2.p1.2.m2.1b"><apply id="S1.I3.i2.p1.2.m2.1.1.cmml" xref="S1.I3.i2.p1.2.m2.1.1"><times id="S1.I3.i2.p1.2.m2.1.1.2.cmml" xref="S1.I3.i2.p1.2.m2.1.1.2"></times><ci id="S1.I3.i2.p1.2.m2.1.1.3.cmml" xref="S1.I3.i2.p1.2.m2.1.1.3">𝑂</ci><apply id="S1.I3.i2.p1.2.m2.1.1.1.1.1.cmml" xref="S1.I3.i2.p1.2.m2.1.1.1.1"><times id="S1.I3.i2.p1.2.m2.1.1.1.1.1.1.cmml" xref="S1.I3.i2.p1.2.m2.1.1.1.1.1.1"></times><ci id="S1.I3.i2.p1.2.m2.1.1.1.1.1.2.cmml" xref="S1.I3.i2.p1.2.m2.1.1.1.1.1.2">𝛼</ci><ci id="S1.I3.i2.p1.2.m2.1.1.1.1.1.3.cmml" xref="S1.I3.i2.p1.2.m2.1.1.1.1.1.3">𝑡</ci><ci id="S1.I3.i2.p1.2.m2.1.1.1.1.1.4.cmml" xref="S1.I3.i2.p1.2.m2.1.1.1.1.1.4">𝑘</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.I3.i2.p1.2.m2.1c">O(\alpha tk)</annotation><annotation encoding="application/x-llamapun" id="S1.I3.i2.p1.2.m2.1d">italic_O ( italic_α italic_t italic_k )</annotation></semantics></math>-approximation in polynomial time, where <math alttext="\alpha" class="ltx_Math" display="inline" id="S1.I3.i2.p1.3.m3.1"><semantics id="S1.I3.i2.p1.3.m3.1a"><mi id="S1.I3.i2.p1.3.m3.1.1" xref="S1.I3.i2.p1.3.m3.1.1.cmml">α</mi><annotation-xml encoding="MathML-Content" id="S1.I3.i2.p1.3.m3.1b"><ci id="S1.I3.i2.p1.3.m3.1.1.cmml" xref="S1.I3.i2.p1.3.m3.1.1">𝛼</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.I3.i2.p1.3.m3.1c">\alpha</annotation><annotation encoding="application/x-llamapun" id="S1.I3.i2.p1.3.m3.1d">italic_α</annotation></semantics></math> is the best poly-time approximation.</p> </div> </li> <li class="ltx_item" id="S1.I3.i3" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">•</span> <div class="ltx_para" id="S1.I3.i3.p1"> <p class="ltx_p" id="S1.I3.i3.p1.5">For VC-SNDP, the framework yields an <math alttext="O(\beta t)" class="ltx_Math" display="inline" id="S1.I3.i3.p1.1.m1.1"><semantics id="S1.I3.i3.p1.1.m1.1a"><mrow id="S1.I3.i3.p1.1.m1.1.1" xref="S1.I3.i3.p1.1.m1.1.1.cmml"><mi id="S1.I3.i3.p1.1.m1.1.1.3" xref="S1.I3.i3.p1.1.m1.1.1.3.cmml">O</mi><mo id="S1.I3.i3.p1.1.m1.1.1.2" xref="S1.I3.i3.p1.1.m1.1.1.2.cmml">⁢</mo><mrow id="S1.I3.i3.p1.1.m1.1.1.1.1" xref="S1.I3.i3.p1.1.m1.1.1.1.1.1.cmml"><mo id="S1.I3.i3.p1.1.m1.1.1.1.1.2" stretchy="false" xref="S1.I3.i3.p1.1.m1.1.1.1.1.1.cmml">(</mo><mrow id="S1.I3.i3.p1.1.m1.1.1.1.1.1" xref="S1.I3.i3.p1.1.m1.1.1.1.1.1.cmml"><mi id="S1.I3.i3.p1.1.m1.1.1.1.1.1.2" xref="S1.I3.i3.p1.1.m1.1.1.1.1.1.2.cmml">β</mi><mo id="S1.I3.i3.p1.1.m1.1.1.1.1.1.1" xref="S1.I3.i3.p1.1.m1.1.1.1.1.1.1.cmml">⁢</mo><mi id="S1.I3.i3.p1.1.m1.1.1.1.1.1.3" xref="S1.I3.i3.p1.1.m1.1.1.1.1.1.3.cmml">t</mi></mrow><mo id="S1.I3.i3.p1.1.m1.1.1.1.1.3" stretchy="false" xref="S1.I3.i3.p1.1.m1.1.1.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.I3.i3.p1.1.m1.1b"><apply id="S1.I3.i3.p1.1.m1.1.1.cmml" xref="S1.I3.i3.p1.1.m1.1.1"><times id="S1.I3.i3.p1.1.m1.1.1.2.cmml" xref="S1.I3.i3.p1.1.m1.1.1.2"></times><ci id="S1.I3.i3.p1.1.m1.1.1.3.cmml" xref="S1.I3.i3.p1.1.m1.1.1.3">𝑂</ci><apply id="S1.I3.i3.p1.1.m1.1.1.1.1.1.cmml" xref="S1.I3.i3.p1.1.m1.1.1.1.1"><times id="S1.I3.i3.p1.1.m1.1.1.1.1.1.1.cmml" xref="S1.I3.i3.p1.1.m1.1.1.1.1.1.1"></times><ci id="S1.I3.i3.p1.1.m1.1.1.1.1.1.2.cmml" xref="S1.I3.i3.p1.1.m1.1.1.1.1.1.2">𝛽</ci><ci id="S1.I3.i3.p1.1.m1.1.1.1.1.1.3.cmml" xref="S1.I3.i3.p1.1.m1.1.1.1.1.1.3">𝑡</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.I3.i3.p1.1.m1.1c">O(\beta t)</annotation><annotation encoding="application/x-llamapun" id="S1.I3.i3.p1.1.m1.1d">italic_O ( italic_β italic_t )</annotation></semantics></math>-approximation in polynomial time, where <math alttext="\beta" class="ltx_Math" display="inline" id="S1.I3.i3.p1.2.m2.1"><semantics id="S1.I3.i3.p1.2.m2.1a"><mi id="S1.I3.i3.p1.2.m2.1.1" xref="S1.I3.i3.p1.2.m2.1.1.cmml">β</mi><annotation-xml encoding="MathML-Content" id="S1.I3.i3.p1.2.m2.1b"><ci id="S1.I3.i3.p1.2.m2.1.1.cmml" xref="S1.I3.i3.p1.2.m2.1.1">𝛽</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.I3.i3.p1.2.m2.1c">\beta</annotation><annotation encoding="application/x-llamapun" id="S1.I3.i3.p1.2.m2.1d">italic_β</annotation></semantics></math> is the best polynomial time approximation with respect to the optimal <em class="ltx_emph ltx_font_italic" id="S1.I3.i3.p1.5.1">fractional</em> solution to a natural LP relaxation. Using this, we obtain improved algorithms for several important special cases including VC-SNDP when <math alttext="k\leq 2" class="ltx_Math" display="inline" id="S1.I3.i3.p1.3.m3.1"><semantics id="S1.I3.i3.p1.3.m3.1a"><mrow id="S1.I3.i3.p1.3.m3.1.1" xref="S1.I3.i3.p1.3.m3.1.1.cmml"><mi id="S1.I3.i3.p1.3.m3.1.1.2" xref="S1.I3.i3.p1.3.m3.1.1.2.cmml">k</mi><mo id="S1.I3.i3.p1.3.m3.1.1.1" xref="S1.I3.i3.p1.3.m3.1.1.1.cmml">≤</mo><mn id="S1.I3.i3.p1.3.m3.1.1.3" xref="S1.I3.i3.p1.3.m3.1.1.3.cmml">2</mn></mrow><annotation-xml encoding="MathML-Content" id="S1.I3.i3.p1.3.m3.1b"><apply id="S1.I3.i3.p1.3.m3.1.1.cmml" xref="S1.I3.i3.p1.3.m3.1.1"><leq id="S1.I3.i3.p1.3.m3.1.1.1.cmml" xref="S1.I3.i3.p1.3.m3.1.1.1"></leq><ci id="S1.I3.i3.p1.3.m3.1.1.2.cmml" xref="S1.I3.i3.p1.3.m3.1.1.2">𝑘</ci><cn id="S1.I3.i3.p1.3.m3.1.1.3.cmml" type="integer" xref="S1.I3.i3.p1.3.m3.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.I3.i3.p1.3.m3.1c">k\leq 2</annotation><annotation encoding="application/x-llamapun" id="S1.I3.i3.p1.3.m3.1d">italic_k ≤ 2</annotation></semantics></math>, <math alttext="k" class="ltx_Math" display="inline" id="S1.I3.i3.p1.4.m4.1"><semantics id="S1.I3.i3.p1.4.m4.1a"><mi id="S1.I3.i3.p1.4.m4.1.1" xref="S1.I3.i3.p1.4.m4.1.1.cmml">k</mi><annotation-xml encoding="MathML-Content" id="S1.I3.i3.p1.4.m4.1b"><ci id="S1.I3.i3.p1.4.m4.1.1.cmml" xref="S1.I3.i3.p1.4.m4.1.1">𝑘</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.I3.i3.p1.4.m4.1c">k</annotation><annotation encoding="application/x-llamapun" id="S1.I3.i3.p1.4.m4.1d">italic_k</annotation></semantics></math>-VCSS, and <math alttext="k" class="ltx_Math" display="inline" id="S1.I3.i3.p1.5.m5.1"><semantics id="S1.I3.i3.p1.5.m5.1a"><mi id="S1.I3.i3.p1.5.m5.1.1" xref="S1.I3.i3.p1.5.m5.1.1.cmml">k</mi><annotation-xml encoding="MathML-Content" id="S1.I3.i3.p1.5.m5.1b"><ci id="S1.I3.i3.p1.5.m5.1.1.cmml" xref="S1.I3.i3.p1.5.m5.1.1">𝑘</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.I3.i3.p1.5.m5.1c">k</annotation><annotation encoding="application/x-llamapun" id="S1.I3.i3.p1.5.m5.1d">italic_k</annotation></semantics></math>-VC-CAP.</p> </div> </li> </ul> </div> <div class="ltx_theorem ltx_theorem_remark" id="S1.Thmtheorem1"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="S1.Thmtheorem1.1.1.1">Remark 1.1</span></span><span class="ltx_text ltx_font_bold" id="S1.Thmtheorem1.2.2">.</span> </h6> <div class="ltx_para" id="S1.Thmtheorem1.p1"> <p class="ltx_p" id="S1.Thmtheorem1.p1.2">Extending the lower bound construction from <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.00712v1#bib.bibx54" title="">JKMV24</a>]</cite> to the vertex connectivity setting, we show that approximating VC-SNDP within a factor better than <math alttext="2t+1" class="ltx_Math" display="inline" id="S1.Thmtheorem1.p1.1.m1.1"><semantics id="S1.Thmtheorem1.p1.1.m1.1a"><mrow id="S1.Thmtheorem1.p1.1.m1.1.1" xref="S1.Thmtheorem1.p1.1.m1.1.1.cmml"><mrow id="S1.Thmtheorem1.p1.1.m1.1.1.2" xref="S1.Thmtheorem1.p1.1.m1.1.1.2.cmml"><mn id="S1.Thmtheorem1.p1.1.m1.1.1.2.2" xref="S1.Thmtheorem1.p1.1.m1.1.1.2.2.cmml">2</mn><mo id="S1.Thmtheorem1.p1.1.m1.1.1.2.1" xref="S1.Thmtheorem1.p1.1.m1.1.1.2.1.cmml">⁢</mo><mi id="S1.Thmtheorem1.p1.1.m1.1.1.2.3" xref="S1.Thmtheorem1.p1.1.m1.1.1.2.3.cmml">t</mi></mrow><mo id="S1.Thmtheorem1.p1.1.m1.1.1.1" xref="S1.Thmtheorem1.p1.1.m1.1.1.1.cmml">+</mo><mn id="S1.Thmtheorem1.p1.1.m1.1.1.3" xref="S1.Thmtheorem1.p1.1.m1.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S1.Thmtheorem1.p1.1.m1.1b"><apply id="S1.Thmtheorem1.p1.1.m1.1.1.cmml" xref="S1.Thmtheorem1.p1.1.m1.1.1"><plus id="S1.Thmtheorem1.p1.1.m1.1.1.1.cmml" xref="S1.Thmtheorem1.p1.1.m1.1.1.1"></plus><apply id="S1.Thmtheorem1.p1.1.m1.1.1.2.cmml" xref="S1.Thmtheorem1.p1.1.m1.1.1.2"><times id="S1.Thmtheorem1.p1.1.m1.1.1.2.1.cmml" xref="S1.Thmtheorem1.p1.1.m1.1.1.2.1"></times><cn id="S1.Thmtheorem1.p1.1.m1.1.1.2.2.cmml" type="integer" xref="S1.Thmtheorem1.p1.1.m1.1.1.2.2">2</cn><ci id="S1.Thmtheorem1.p1.1.m1.1.1.2.3.cmml" xref="S1.Thmtheorem1.p1.1.m1.1.1.2.3">𝑡</ci></apply><cn id="S1.Thmtheorem1.p1.1.m1.1.1.3.cmml" type="integer" xref="S1.Thmtheorem1.p1.1.m1.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.Thmtheorem1.p1.1.m1.1c">2t+1</annotation><annotation encoding="application/x-llamapun" id="S1.Thmtheorem1.p1.1.m1.1d">2 italic_t + 1</annotation></semantics></math> requires <math alttext="\Omega(nk+n^{1+1/t})" class="ltx_Math" display="inline" id="S1.Thmtheorem1.p1.2.m2.1"><semantics id="S1.Thmtheorem1.p1.2.m2.1a"><mrow id="S1.Thmtheorem1.p1.2.m2.1.1" xref="S1.Thmtheorem1.p1.2.m2.1.1.cmml"><mi id="S1.Thmtheorem1.p1.2.m2.1.1.3" mathvariant="normal" 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xref="S1.Thmtheorem1.p1.2.m2.1.1.1.1.1.3.cmml"><mi id="S1.Thmtheorem1.p1.2.m2.1.1.1.1.1.3.2" xref="S1.Thmtheorem1.p1.2.m2.1.1.1.1.1.3.2.cmml">n</mi><mrow id="S1.Thmtheorem1.p1.2.m2.1.1.1.1.1.3.3" xref="S1.Thmtheorem1.p1.2.m2.1.1.1.1.1.3.3.cmml"><mn id="S1.Thmtheorem1.p1.2.m2.1.1.1.1.1.3.3.2" xref="S1.Thmtheorem1.p1.2.m2.1.1.1.1.1.3.3.2.cmml">1</mn><mo id="S1.Thmtheorem1.p1.2.m2.1.1.1.1.1.3.3.1" xref="S1.Thmtheorem1.p1.2.m2.1.1.1.1.1.3.3.1.cmml">+</mo><mrow id="S1.Thmtheorem1.p1.2.m2.1.1.1.1.1.3.3.3" xref="S1.Thmtheorem1.p1.2.m2.1.1.1.1.1.3.3.3.cmml"><mn id="S1.Thmtheorem1.p1.2.m2.1.1.1.1.1.3.3.3.2" xref="S1.Thmtheorem1.p1.2.m2.1.1.1.1.1.3.3.3.2.cmml">1</mn><mo id="S1.Thmtheorem1.p1.2.m2.1.1.1.1.1.3.3.3.1" xref="S1.Thmtheorem1.p1.2.m2.1.1.1.1.1.3.3.3.1.cmml">/</mo><mi id="S1.Thmtheorem1.p1.2.m2.1.1.1.1.1.3.3.3.3" xref="S1.Thmtheorem1.p1.2.m2.1.1.1.1.1.3.3.3.3.cmml">t</mi></mrow></mrow></msup></mrow><mo id="S1.Thmtheorem1.p1.2.m2.1.1.1.1.3" stretchy="false" xref="S1.Thmtheorem1.p1.2.m2.1.1.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.Thmtheorem1.p1.2.m2.1b"><apply id="S1.Thmtheorem1.p1.2.m2.1.1.cmml" xref="S1.Thmtheorem1.p1.2.m2.1.1"><times id="S1.Thmtheorem1.p1.2.m2.1.1.2.cmml" xref="S1.Thmtheorem1.p1.2.m2.1.1.2"></times><ci id="S1.Thmtheorem1.p1.2.m2.1.1.3.cmml" xref="S1.Thmtheorem1.p1.2.m2.1.1.3">Ω</ci><apply id="S1.Thmtheorem1.p1.2.m2.1.1.1.1.1.cmml" xref="S1.Thmtheorem1.p1.2.m2.1.1.1.1"><plus id="S1.Thmtheorem1.p1.2.m2.1.1.1.1.1.1.cmml" xref="S1.Thmtheorem1.p1.2.m2.1.1.1.1.1.1"></plus><apply id="S1.Thmtheorem1.p1.2.m2.1.1.1.1.1.2.cmml" xref="S1.Thmtheorem1.p1.2.m2.1.1.1.1.1.2"><times id="S1.Thmtheorem1.p1.2.m2.1.1.1.1.1.2.1.cmml" xref="S1.Thmtheorem1.p1.2.m2.1.1.1.1.1.2.1"></times><ci id="S1.Thmtheorem1.p1.2.m2.1.1.1.1.1.2.2.cmml" xref="S1.Thmtheorem1.p1.2.m2.1.1.1.1.1.2.2">𝑛</ci><ci id="S1.Thmtheorem1.p1.2.m2.1.1.1.1.1.2.3.cmml" xref="S1.Thmtheorem1.p1.2.m2.1.1.1.1.1.2.3">𝑘</ci></apply><apply id="S1.Thmtheorem1.p1.2.m2.1.1.1.1.1.3.cmml" xref="S1.Thmtheorem1.p1.2.m2.1.1.1.1.1.3"><csymbol cd="ambiguous" id="S1.Thmtheorem1.p1.2.m2.1.1.1.1.1.3.1.cmml" xref="S1.Thmtheorem1.p1.2.m2.1.1.1.1.1.3">superscript</csymbol><ci id="S1.Thmtheorem1.p1.2.m2.1.1.1.1.1.3.2.cmml" xref="S1.Thmtheorem1.p1.2.m2.1.1.1.1.1.3.2">𝑛</ci><apply id="S1.Thmtheorem1.p1.2.m2.1.1.1.1.1.3.3.cmml" xref="S1.Thmtheorem1.p1.2.m2.1.1.1.1.1.3.3"><plus id="S1.Thmtheorem1.p1.2.m2.1.1.1.1.1.3.3.1.cmml" xref="S1.Thmtheorem1.p1.2.m2.1.1.1.1.1.3.3.1"></plus><cn id="S1.Thmtheorem1.p1.2.m2.1.1.1.1.1.3.3.2.cmml" type="integer" xref="S1.Thmtheorem1.p1.2.m2.1.1.1.1.1.3.3.2">1</cn><apply id="S1.Thmtheorem1.p1.2.m2.1.1.1.1.1.3.3.3.cmml" xref="S1.Thmtheorem1.p1.2.m2.1.1.1.1.1.3.3.3"><divide id="S1.Thmtheorem1.p1.2.m2.1.1.1.1.1.3.3.3.1.cmml" xref="S1.Thmtheorem1.p1.2.m2.1.1.1.1.1.3.3.3.1"></divide><cn id="S1.Thmtheorem1.p1.2.m2.1.1.1.1.1.3.3.3.2.cmml" type="integer" xref="S1.Thmtheorem1.p1.2.m2.1.1.1.1.1.3.3.3.2">1</cn><ci id="S1.Thmtheorem1.p1.2.m2.1.1.1.1.1.3.3.3.3.cmml" xref="S1.Thmtheorem1.p1.2.m2.1.1.1.1.1.3.3.3.3">𝑡</ci></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.Thmtheorem1.p1.2.m2.1c">\Omega(nk+n^{1+1/t})</annotation><annotation encoding="application/x-llamapun" id="S1.Thmtheorem1.p1.2.m2.1d">roman_Ω ( italic_n italic_k + italic_n start_POSTSUPERSCRIPT 1 + 1 / italic_t end_POSTSUPERSCRIPT )</annotation></semantics></math> space, which point out that our upper bounds are nearly tight. The formal statement of the lower bound is in Section <a class="ltx_ref" href="https://arxiv.org/html/2503.00712v1#S5" title="5 Lower Bounds for Streaming Network Design ‣ Streaming Algorithms for Network Design"><span class="ltx_text ltx_ref_tag">5</span></a>.</p> </div> </div> <div class="ltx_theorem ltx_theorem_remark" id="S1.Thmtheorem2"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="S1.Thmtheorem2.1.1.1">Remark 1.2</span></span><span class="ltx_text ltx_font_bold" id="S1.Thmtheorem2.2.2">.</span> </h6> <div class="ltx_para" id="S1.Thmtheorem2.p1"> <p class="ltx_p" id="S1.Thmtheorem2.p1.1">This framework also extends to <em class="ltx_emph ltx_font_italic" id="S1.Thmtheorem2.p1.1.1">non-uniform</em> network design models. By Menger’s theorem (see Section <a class="ltx_ref" href="https://arxiv.org/html/2503.00712v1#S2" title="2 Preliminaries ‣ Streaming Algorithms for Network Design"><span class="ltx_text ltx_ref_tag">2</span></a>), the goal in SNDP is to construct graphs that maintain connectivity between terminal pairs despite the failure of <em class="ltx_emph ltx_font_italic" id="S1.Thmtheorem2.p1.1.2">any</em> <math alttext="k-1" class="ltx_Math" display="inline" id="S1.Thmtheorem2.p1.1.m1.1"><semantics id="S1.Thmtheorem2.p1.1.m1.1a"><mrow id="S1.Thmtheorem2.p1.1.m1.1.1" xref="S1.Thmtheorem2.p1.1.m1.1.1.cmml"><mi id="S1.Thmtheorem2.p1.1.m1.1.1.2" xref="S1.Thmtheorem2.p1.1.m1.1.1.2.cmml">k</mi><mo id="S1.Thmtheorem2.p1.1.m1.1.1.1" xref="S1.Thmtheorem2.p1.1.m1.1.1.1.cmml">−</mo><mn id="S1.Thmtheorem2.p1.1.m1.1.1.3" xref="S1.Thmtheorem2.p1.1.m1.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S1.Thmtheorem2.p1.1.m1.1b"><apply id="S1.Thmtheorem2.p1.1.m1.1.1.cmml" xref="S1.Thmtheorem2.p1.1.m1.1.1"><minus id="S1.Thmtheorem2.p1.1.m1.1.1.1.cmml" xref="S1.Thmtheorem2.p1.1.m1.1.1.1"></minus><ci id="S1.Thmtheorem2.p1.1.m1.1.1.2.cmml" xref="S1.Thmtheorem2.p1.1.m1.1.1.2">𝑘</ci><cn id="S1.Thmtheorem2.p1.1.m1.1.1.3.cmml" type="integer" xref="S1.Thmtheorem2.p1.1.m1.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.Thmtheorem2.p1.1.m1.1c">k-1</annotation><annotation encoding="application/x-llamapun" id="S1.Thmtheorem2.p1.1.m1.1d">italic_k - 1</annotation></semantics></math> edges/vertices. Non-uniform network design models scenarios in which only certain specified subsets of edges can fail. This model is relevant in settings where edges failures are correlated in some way. For brevity we omit a detailed discussion here, and instead refer the reader to the following works on some problems non-uniform network design for which our framework holds: Bulk-Robust Network Design <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.00712v1#bib.bibx13" title="">ASZ15</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.00712v1#bib.bibx4" title="">Adj15</a>]</cite>, Flexible Graph Connectivity <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.00712v1#bib.bibx3" title="">Adj13</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.00712v1#bib.bibx8" title="">AHM20</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.00712v1#bib.bibx9" title="">AHMS22</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.00712v1#bib.bibx17" title="">BCHI24</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.00712v1#bib.bibx16" title="">BCGI23</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.00712v1#bib.bibx14" title="">Ban23</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.00712v1#bib.bibx23" title="">CJ23</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.00712v1#bib.bibx75" title="">Nut24</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.00712v1#bib.bibx51" title="">HDJAS24</a>]</cite>, and Relative Survivable Network Design <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.00712v1#bib.bibx35" title="">DKK22</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.00712v1#bib.bibx36" title="">DKKN23</a>]</cite>. We note that while these problems have mostly been studied in edge-connectivity settings, our streaming framework can also handle vertex-connectivity versions assuming exact algorithms at the end of the stream.</p> </div> </div> </section> <section class="ltx_paragraph" id="S1.SS1.SSS0.Px2"> <h5 class="ltx_title ltx_title_paragraph">Algorithms for <math alttext="k" class="ltx_Math" display="inline" id="S1.SS1.SSS0.Px2.1.m1.1"><semantics id="S1.SS1.SSS0.Px2.1.m1.1b"><mi id="S1.SS1.SSS0.Px2.1.m1.1.1" xref="S1.SS1.SSS0.Px2.1.m1.1.1.cmml">k</mi><annotation-xml encoding="MathML-Content" id="S1.SS1.SSS0.Px2.1.m1.1c"><ci id="S1.SS1.SSS0.Px2.1.m1.1.1.cmml" xref="S1.SS1.SSS0.Px2.1.m1.1.1">𝑘</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.SS1.SSS0.Px2.1.m1.1d">k</annotation><annotation encoding="application/x-llamapun" id="S1.SS1.SSS0.Px2.1.m1.1e">italic_k</annotation></semantics></math>-VC-CAP:</h5> <div class="ltx_para" id="S1.SS1.SSS0.Px2.p1"> <p class="ltx_p" id="S1.SS1.SSS0.Px2.p1.5">Our second contribution is for <math alttext="k" class="ltx_Math" display="inline" id="S1.SS1.SSS0.Px2.p1.1.m1.1"><semantics id="S1.SS1.SSS0.Px2.p1.1.m1.1a"><mi id="S1.SS1.SSS0.Px2.p1.1.m1.1.1" xref="S1.SS1.SSS0.Px2.p1.1.m1.1.1.cmml">k</mi><annotation-xml encoding="MathML-Content" id="S1.SS1.SSS0.Px2.p1.1.m1.1b"><ci id="S1.SS1.SSS0.Px2.p1.1.m1.1.1.cmml" xref="S1.SS1.SSS0.Px2.p1.1.m1.1.1">𝑘</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.SS1.SSS0.Px2.p1.1.m1.1c">k</annotation><annotation encoding="application/x-llamapun" id="S1.SS1.SSS0.Px2.p1.1.m1.1d">italic_k</annotation></semantics></math>-VC-CAP in the link-arrival model; this is provided in Section <a class="ltx_ref" href="https://arxiv.org/html/2503.00712v1#S4" title="4 Vertex Connectivity Augmentation in Link-Arrival Model ‣ Streaming Algorithms for Network Design"><span class="ltx_text ltx_ref_tag">4</span></a>. Note that this is a global connectivity problem and does not include the <math alttext="s" class="ltx_Math" display="inline" id="S1.SS1.SSS0.Px2.p1.2.m2.1"><semantics id="S1.SS1.SSS0.Px2.p1.2.m2.1a"><mi id="S1.SS1.SSS0.Px2.p1.2.m2.1.1" xref="S1.SS1.SSS0.Px2.p1.2.m2.1.1.cmml">s</mi><annotation-xml encoding="MathML-Content" id="S1.SS1.SSS0.Px2.p1.2.m2.1b"><ci id="S1.SS1.SSS0.Px2.p1.2.m2.1.1.cmml" xref="S1.SS1.SSS0.Px2.p1.2.m2.1.1">𝑠</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.SS1.SSS0.Px2.p1.2.m2.1c">s</annotation><annotation encoding="application/x-llamapun" id="S1.SS1.SSS0.Px2.p1.2.m2.1d">italic_s</annotation></semantics></math>-<math alttext="t" class="ltx_Math" display="inline" id="S1.SS1.SSS0.Px2.p1.3.m3.1"><semantics id="S1.SS1.SSS0.Px2.p1.3.m3.1a"><mi id="S1.SS1.SSS0.Px2.p1.3.m3.1.1" xref="S1.SS1.SSS0.Px2.p1.3.m3.1.1.cmml">t</mi><annotation-xml encoding="MathML-Content" id="S1.SS1.SSS0.Px2.p1.3.m3.1b"><ci id="S1.SS1.SSS0.Px2.p1.3.m3.1.1.cmml" xref="S1.SS1.SSS0.Px2.p1.3.m3.1.1">𝑡</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.SS1.SSS0.Px2.p1.3.m3.1c">t</annotation><annotation encoding="application/x-llamapun" id="S1.SS1.SSS0.Px2.p1.3.m3.1d">italic_t</annotation></semantics></math>-shortest path problem as a special case; hence we hope to obtain better bounds that avoid the overhead of using spanners. We obtain the following approximation ratios for <math alttext="k" class="ltx_Math" display="inline" id="S1.SS1.SSS0.Px2.p1.4.m4.1"><semantics id="S1.SS1.SSS0.Px2.p1.4.m4.1a"><mi id="S1.SS1.SSS0.Px2.p1.4.m4.1.1" xref="S1.SS1.SSS0.Px2.p1.4.m4.1.1.cmml">k</mi><annotation-xml encoding="MathML-Content" id="S1.SS1.SSS0.Px2.p1.4.m4.1b"><ci id="S1.SS1.SSS0.Px2.p1.4.m4.1.1.cmml" xref="S1.SS1.SSS0.Px2.p1.4.m4.1.1">𝑘</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.SS1.SSS0.Px2.p1.4.m4.1c">k</annotation><annotation encoding="application/x-llamapun" id="S1.SS1.SSS0.Px2.p1.4.m4.1d">italic_k</annotation></semantics></math>-VC-CAP; both algorithms use <math alttext="\tilde{O}(n/\epsilon)" class="ltx_Math" display="inline" id="S1.SS1.SSS0.Px2.p1.5.m5.1"><semantics id="S1.SS1.SSS0.Px2.p1.5.m5.1a"><mrow id="S1.SS1.SSS0.Px2.p1.5.m5.1.1" xref="S1.SS1.SSS0.Px2.p1.5.m5.1.1.cmml"><mover accent="true" id="S1.SS1.SSS0.Px2.p1.5.m5.1.1.3" xref="S1.SS1.SSS0.Px2.p1.5.m5.1.1.3.cmml"><mi id="S1.SS1.SSS0.Px2.p1.5.m5.1.1.3.2" xref="S1.SS1.SSS0.Px2.p1.5.m5.1.1.3.2.cmml">O</mi><mo id="S1.SS1.SSS0.Px2.p1.5.m5.1.1.3.1" xref="S1.SS1.SSS0.Px2.p1.5.m5.1.1.3.1.cmml">~</mo></mover><mo id="S1.SS1.SSS0.Px2.p1.5.m5.1.1.2" xref="S1.SS1.SSS0.Px2.p1.5.m5.1.1.2.cmml">⁢</mo><mrow id="S1.SS1.SSS0.Px2.p1.5.m5.1.1.1.1" xref="S1.SS1.SSS0.Px2.p1.5.m5.1.1.1.1.1.cmml"><mo id="S1.SS1.SSS0.Px2.p1.5.m5.1.1.1.1.2" stretchy="false" xref="S1.SS1.SSS0.Px2.p1.5.m5.1.1.1.1.1.cmml">(</mo><mrow id="S1.SS1.SSS0.Px2.p1.5.m5.1.1.1.1.1" xref="S1.SS1.SSS0.Px2.p1.5.m5.1.1.1.1.1.cmml"><mi id="S1.SS1.SSS0.Px2.p1.5.m5.1.1.1.1.1.2" xref="S1.SS1.SSS0.Px2.p1.5.m5.1.1.1.1.1.2.cmml">n</mi><mo id="S1.SS1.SSS0.Px2.p1.5.m5.1.1.1.1.1.1" xref="S1.SS1.SSS0.Px2.p1.5.m5.1.1.1.1.1.1.cmml">/</mo><mi id="S1.SS1.SSS0.Px2.p1.5.m5.1.1.1.1.1.3" xref="S1.SS1.SSS0.Px2.p1.5.m5.1.1.1.1.1.3.cmml">ϵ</mi></mrow><mo id="S1.SS1.SSS0.Px2.p1.5.m5.1.1.1.1.3" stretchy="false" xref="S1.SS1.SSS0.Px2.p1.5.m5.1.1.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.SS1.SSS0.Px2.p1.5.m5.1b"><apply id="S1.SS1.SSS0.Px2.p1.5.m5.1.1.cmml" xref="S1.SS1.SSS0.Px2.p1.5.m5.1.1"><times id="S1.SS1.SSS0.Px2.p1.5.m5.1.1.2.cmml" xref="S1.SS1.SSS0.Px2.p1.5.m5.1.1.2"></times><apply id="S1.SS1.SSS0.Px2.p1.5.m5.1.1.3.cmml" xref="S1.SS1.SSS0.Px2.p1.5.m5.1.1.3"><ci id="S1.SS1.SSS0.Px2.p1.5.m5.1.1.3.1.cmml" xref="S1.SS1.SSS0.Px2.p1.5.m5.1.1.3.1">~</ci><ci id="S1.SS1.SSS0.Px2.p1.5.m5.1.1.3.2.cmml" xref="S1.SS1.SSS0.Px2.p1.5.m5.1.1.3.2">𝑂</ci></apply><apply id="S1.SS1.SSS0.Px2.p1.5.m5.1.1.1.1.1.cmml" xref="S1.SS1.SSS0.Px2.p1.5.m5.1.1.1.1"><divide id="S1.SS1.SSS0.Px2.p1.5.m5.1.1.1.1.1.1.cmml" xref="S1.SS1.SSS0.Px2.p1.5.m5.1.1.1.1.1.1"></divide><ci id="S1.SS1.SSS0.Px2.p1.5.m5.1.1.1.1.1.2.cmml" xref="S1.SS1.SSS0.Px2.p1.5.m5.1.1.1.1.1.2">𝑛</ci><ci id="S1.SS1.SSS0.Px2.p1.5.m5.1.1.1.1.1.3.cmml" xref="S1.SS1.SSS0.Px2.p1.5.m5.1.1.1.1.1.3">italic-ϵ</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS1.SSS0.Px2.p1.5.m5.1c">\tilde{O}(n/\epsilon)</annotation><annotation encoding="application/x-llamapun" id="S1.SS1.SSS0.Px2.p1.5.m5.1d">over~ start_ARG italic_O end_ARG ( italic_n / italic_ϵ )</annotation></semantics></math> space.</p> <ul class="ltx_itemize" id="S1.I4"> <li class="ltx_item" id="S1.I4.i1" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">•</span> <div class="ltx_para" id="S1.I4.i1.p1"> <p class="ltx_p" id="S1.I4.i1.p1.5">For <math alttext="k=1" class="ltx_Math" display="inline" id="S1.I4.i1.p1.1.m1.1"><semantics id="S1.I4.i1.p1.1.m1.1a"><mrow id="S1.I4.i1.p1.1.m1.1.1" xref="S1.I4.i1.p1.1.m1.1.1.cmml"><mi id="S1.I4.i1.p1.1.m1.1.1.2" xref="S1.I4.i1.p1.1.m1.1.1.2.cmml">k</mi><mo id="S1.I4.i1.p1.1.m1.1.1.1" xref="S1.I4.i1.p1.1.m1.1.1.1.cmml">=</mo><mn id="S1.I4.i1.p1.1.m1.1.1.3" xref="S1.I4.i1.p1.1.m1.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S1.I4.i1.p1.1.m1.1b"><apply id="S1.I4.i1.p1.1.m1.1.1.cmml" xref="S1.I4.i1.p1.1.m1.1.1"><eq id="S1.I4.i1.p1.1.m1.1.1.1.cmml" xref="S1.I4.i1.p1.1.m1.1.1.1"></eq><ci id="S1.I4.i1.p1.1.m1.1.1.2.cmml" xref="S1.I4.i1.p1.1.m1.1.1.2">𝑘</ci><cn id="S1.I4.i1.p1.1.m1.1.1.3.cmml" type="integer" xref="S1.I4.i1.p1.1.m1.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.I4.i1.p1.1.m1.1c">k=1</annotation><annotation encoding="application/x-llamapun" id="S1.I4.i1.p1.1.m1.1d">italic_k = 1</annotation></semantics></math>, we obtain a <math alttext="(3+\epsilon)" class="ltx_Math" display="inline" id="S1.I4.i1.p1.2.m2.1"><semantics id="S1.I4.i1.p1.2.m2.1a"><mrow id="S1.I4.i1.p1.2.m2.1.1.1" xref="S1.I4.i1.p1.2.m2.1.1.1.1.cmml"><mo id="S1.I4.i1.p1.2.m2.1.1.1.2" stretchy="false" xref="S1.I4.i1.p1.2.m2.1.1.1.1.cmml">(</mo><mrow id="S1.I4.i1.p1.2.m2.1.1.1.1" xref="S1.I4.i1.p1.2.m2.1.1.1.1.cmml"><mn id="S1.I4.i1.p1.2.m2.1.1.1.1.2" xref="S1.I4.i1.p1.2.m2.1.1.1.1.2.cmml">3</mn><mo id="S1.I4.i1.p1.2.m2.1.1.1.1.1" xref="S1.I4.i1.p1.2.m2.1.1.1.1.1.cmml">+</mo><mi id="S1.I4.i1.p1.2.m2.1.1.1.1.3" xref="S1.I4.i1.p1.2.m2.1.1.1.1.3.cmml">ϵ</mi></mrow><mo id="S1.I4.i1.p1.2.m2.1.1.1.3" stretchy="false" xref="S1.I4.i1.p1.2.m2.1.1.1.1.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S1.I4.i1.p1.2.m2.1b"><apply id="S1.I4.i1.p1.2.m2.1.1.1.1.cmml" xref="S1.I4.i1.p1.2.m2.1.1.1"><plus id="S1.I4.i1.p1.2.m2.1.1.1.1.1.cmml" xref="S1.I4.i1.p1.2.m2.1.1.1.1.1"></plus><cn id="S1.I4.i1.p1.2.m2.1.1.1.1.2.cmml" type="integer" xref="S1.I4.i1.p1.2.m2.1.1.1.1.2">3</cn><ci id="S1.I4.i1.p1.2.m2.1.1.1.1.3.cmml" xref="S1.I4.i1.p1.2.m2.1.1.1.1.3">italic-ϵ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.I4.i1.p1.2.m2.1c">(3+\epsilon)</annotation><annotation encoding="application/x-llamapun" id="S1.I4.i1.p1.2.m2.1d">( 3 + italic_ϵ )</annotation></semantics></math>-approximation algorithm assuming an exact algorithm at the end of the stream. This yields a <math alttext="(6+\epsilon)" class="ltx_Math" display="inline" id="S1.I4.i1.p1.3.m3.1"><semantics id="S1.I4.i1.p1.3.m3.1a"><mrow id="S1.I4.i1.p1.3.m3.1.1.1" xref="S1.I4.i1.p1.3.m3.1.1.1.1.cmml"><mo id="S1.I4.i1.p1.3.m3.1.1.1.2" stretchy="false" xref="S1.I4.i1.p1.3.m3.1.1.1.1.cmml">(</mo><mrow id="S1.I4.i1.p1.3.m3.1.1.1.1" xref="S1.I4.i1.p1.3.m3.1.1.1.1.cmml"><mn id="S1.I4.i1.p1.3.m3.1.1.1.1.2" xref="S1.I4.i1.p1.3.m3.1.1.1.1.2.cmml">6</mn><mo id="S1.I4.i1.p1.3.m3.1.1.1.1.1" xref="S1.I4.i1.p1.3.m3.1.1.1.1.1.cmml">+</mo><mi id="S1.I4.i1.p1.3.m3.1.1.1.1.3" xref="S1.I4.i1.p1.3.m3.1.1.1.1.3.cmml">ϵ</mi></mrow><mo id="S1.I4.i1.p1.3.m3.1.1.1.3" stretchy="false" xref="S1.I4.i1.p1.3.m3.1.1.1.1.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S1.I4.i1.p1.3.m3.1b"><apply id="S1.I4.i1.p1.3.m3.1.1.1.1.cmml" xref="S1.I4.i1.p1.3.m3.1.1.1"><plus id="S1.I4.i1.p1.3.m3.1.1.1.1.1.cmml" xref="S1.I4.i1.p1.3.m3.1.1.1.1.1"></plus><cn id="S1.I4.i1.p1.3.m3.1.1.1.1.2.cmml" type="integer" xref="S1.I4.i1.p1.3.m3.1.1.1.1.2">6</cn><ci id="S1.I4.i1.p1.3.m3.1.1.1.1.3.cmml" xref="S1.I4.i1.p1.3.m3.1.1.1.1.3">italic-ϵ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.I4.i1.p1.3.m3.1c">(6+\epsilon)</annotation><annotation encoding="application/x-llamapun" id="S1.I4.i1.p1.3.m3.1d">( 6 + italic_ϵ )</annotation></semantics></math>-approximation in polynomial time using the offline <math alttext="2" class="ltx_Math" display="inline" id="S1.I4.i1.p1.4.m4.1"><semantics id="S1.I4.i1.p1.4.m4.1a"><mn id="S1.I4.i1.p1.4.m4.1.1" xref="S1.I4.i1.p1.4.m4.1.1.cmml">2</mn><annotation-xml encoding="MathML-Content" id="S1.I4.i1.p1.4.m4.1b"><cn id="S1.I4.i1.p1.4.m4.1.1.cmml" type="integer" xref="S1.I4.i1.p1.4.m4.1.1">2</cn></annotation-xml><annotation encoding="application/x-tex" id="S1.I4.i1.p1.4.m4.1c">2</annotation><annotation encoding="application/x-llamapun" id="S1.I4.i1.p1.4.m4.1d">2</annotation></semantics></math>-approximation for <math alttext="1" class="ltx_Math" display="inline" id="S1.I4.i1.p1.5.m5.1"><semantics id="S1.I4.i1.p1.5.m5.1a"><mn id="S1.I4.i1.p1.5.m5.1.1" xref="S1.I4.i1.p1.5.m5.1.1.cmml">1</mn><annotation-xml encoding="MathML-Content" id="S1.I4.i1.p1.5.m5.1b"><cn id="S1.I4.i1.p1.5.m5.1.1.cmml" type="integer" xref="S1.I4.i1.p1.5.m5.1.1">1</cn></annotation-xml><annotation encoding="application/x-tex" id="S1.I4.i1.p1.5.m5.1c">1</annotation><annotation encoding="application/x-llamapun" id="S1.I4.i1.p1.5.m5.1d">1</annotation></semantics></math>-VC-CAP <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.00712v1#bib.bibx39" title="">FJW06</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.00712v1#bib.bibx31" title="">CVV06</a>]</cite>.</p> </div> </li> <li class="ltx_item" id="S1.I4.i2" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">•</span> <div class="ltx_para" id="S1.I4.i2.p1"> <p class="ltx_p" id="S1.I4.i2.p1.5">For <math alttext="k=2" class="ltx_Math" display="inline" id="S1.I4.i2.p1.1.m1.1"><semantics id="S1.I4.i2.p1.1.m1.1a"><mrow id="S1.I4.i2.p1.1.m1.1.1" xref="S1.I4.i2.p1.1.m1.1.1.cmml"><mi id="S1.I4.i2.p1.1.m1.1.1.2" xref="S1.I4.i2.p1.1.m1.1.1.2.cmml">k</mi><mo id="S1.I4.i2.p1.1.m1.1.1.1" xref="S1.I4.i2.p1.1.m1.1.1.1.cmml">=</mo><mn id="S1.I4.i2.p1.1.m1.1.1.3" xref="S1.I4.i2.p1.1.m1.1.1.3.cmml">2</mn></mrow><annotation-xml encoding="MathML-Content" id="S1.I4.i2.p1.1.m1.1b"><apply id="S1.I4.i2.p1.1.m1.1.1.cmml" xref="S1.I4.i2.p1.1.m1.1.1"><eq id="S1.I4.i2.p1.1.m1.1.1.1.cmml" xref="S1.I4.i2.p1.1.m1.1.1.1"></eq><ci id="S1.I4.i2.p1.1.m1.1.1.2.cmml" xref="S1.I4.i2.p1.1.m1.1.1.2">𝑘</ci><cn id="S1.I4.i2.p1.1.m1.1.1.3.cmml" type="integer" xref="S1.I4.i2.p1.1.m1.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.I4.i2.p1.1.m1.1c">k=2</annotation><annotation encoding="application/x-llamapun" id="S1.I4.i2.p1.1.m1.1d">italic_k = 2</annotation></semantics></math>, we obtain a <math alttext="(7+\epsilon)" class="ltx_Math" display="inline" id="S1.I4.i2.p1.2.m2.1"><semantics id="S1.I4.i2.p1.2.m2.1a"><mrow id="S1.I4.i2.p1.2.m2.1.1.1" xref="S1.I4.i2.p1.2.m2.1.1.1.1.cmml"><mo id="S1.I4.i2.p1.2.m2.1.1.1.2" stretchy="false" xref="S1.I4.i2.p1.2.m2.1.1.1.1.cmml">(</mo><mrow id="S1.I4.i2.p1.2.m2.1.1.1.1" xref="S1.I4.i2.p1.2.m2.1.1.1.1.cmml"><mn id="S1.I4.i2.p1.2.m2.1.1.1.1.2" xref="S1.I4.i2.p1.2.m2.1.1.1.1.2.cmml">7</mn><mo id="S1.I4.i2.p1.2.m2.1.1.1.1.1" xref="S1.I4.i2.p1.2.m2.1.1.1.1.1.cmml">+</mo><mi id="S1.I4.i2.p1.2.m2.1.1.1.1.3" xref="S1.I4.i2.p1.2.m2.1.1.1.1.3.cmml">ϵ</mi></mrow><mo id="S1.I4.i2.p1.2.m2.1.1.1.3" stretchy="false" xref="S1.I4.i2.p1.2.m2.1.1.1.1.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S1.I4.i2.p1.2.m2.1b"><apply id="S1.I4.i2.p1.2.m2.1.1.1.1.cmml" xref="S1.I4.i2.p1.2.m2.1.1.1"><plus id="S1.I4.i2.p1.2.m2.1.1.1.1.1.cmml" xref="S1.I4.i2.p1.2.m2.1.1.1.1.1"></plus><cn id="S1.I4.i2.p1.2.m2.1.1.1.1.2.cmml" type="integer" xref="S1.I4.i2.p1.2.m2.1.1.1.1.2">7</cn><ci id="S1.I4.i2.p1.2.m2.1.1.1.1.3.cmml" xref="S1.I4.i2.p1.2.m2.1.1.1.1.3">italic-ϵ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.I4.i2.p1.2.m2.1c">(7+\epsilon)</annotation><annotation encoding="application/x-llamapun" id="S1.I4.i2.p1.2.m2.1d">( 7 + italic_ϵ )</annotation></semantics></math>-approximation algorithm assuming an exact algorithm at the end of the stream. This yields a <math alttext="(14+\epsilon)" class="ltx_Math" display="inline" id="S1.I4.i2.p1.3.m3.1"><semantics id="S1.I4.i2.p1.3.m3.1a"><mrow id="S1.I4.i2.p1.3.m3.1.1.1" xref="S1.I4.i2.p1.3.m3.1.1.1.1.cmml"><mo id="S1.I4.i2.p1.3.m3.1.1.1.2" stretchy="false" xref="S1.I4.i2.p1.3.m3.1.1.1.1.cmml">(</mo><mrow id="S1.I4.i2.p1.3.m3.1.1.1.1" xref="S1.I4.i2.p1.3.m3.1.1.1.1.cmml"><mn id="S1.I4.i2.p1.3.m3.1.1.1.1.2" xref="S1.I4.i2.p1.3.m3.1.1.1.1.2.cmml">14</mn><mo id="S1.I4.i2.p1.3.m3.1.1.1.1.1" xref="S1.I4.i2.p1.3.m3.1.1.1.1.1.cmml">+</mo><mi id="S1.I4.i2.p1.3.m3.1.1.1.1.3" xref="S1.I4.i2.p1.3.m3.1.1.1.1.3.cmml">ϵ</mi></mrow><mo id="S1.I4.i2.p1.3.m3.1.1.1.3" stretchy="false" xref="S1.I4.i2.p1.3.m3.1.1.1.1.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S1.I4.i2.p1.3.m3.1b"><apply id="S1.I4.i2.p1.3.m3.1.1.1.1.cmml" xref="S1.I4.i2.p1.3.m3.1.1.1"><plus id="S1.I4.i2.p1.3.m3.1.1.1.1.1.cmml" xref="S1.I4.i2.p1.3.m3.1.1.1.1.1"></plus><cn id="S1.I4.i2.p1.3.m3.1.1.1.1.2.cmml" type="integer" xref="S1.I4.i2.p1.3.m3.1.1.1.1.2">14</cn><ci id="S1.I4.i2.p1.3.m3.1.1.1.1.3.cmml" xref="S1.I4.i2.p1.3.m3.1.1.1.1.3">italic-ϵ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.I4.i2.p1.3.m3.1c">(14+\epsilon)</annotation><annotation encoding="application/x-llamapun" id="S1.I4.i2.p1.3.m3.1d">( 14 + italic_ϵ )</annotation></semantics></math>-approximation in polynomial time using the offline <math alttext="2" class="ltx_Math" display="inline" id="S1.I4.i2.p1.4.m4.1"><semantics id="S1.I4.i2.p1.4.m4.1a"><mn id="S1.I4.i2.p1.4.m4.1.1" xref="S1.I4.i2.p1.4.m4.1.1.cmml">2</mn><annotation-xml encoding="MathML-Content" id="S1.I4.i2.p1.4.m4.1b"><cn id="S1.I4.i2.p1.4.m4.1.1.cmml" type="integer" xref="S1.I4.i2.p1.4.m4.1.1">2</cn></annotation-xml><annotation encoding="application/x-tex" id="S1.I4.i2.p1.4.m4.1c">2</annotation><annotation encoding="application/x-llamapun" id="S1.I4.i2.p1.4.m4.1d">2</annotation></semantics></math>-approximation for <math alttext="2" class="ltx_Math" display="inline" id="S1.I4.i2.p1.5.m5.1"><semantics id="S1.I4.i2.p1.5.m5.1a"><mn id="S1.I4.i2.p1.5.m5.1.1" xref="S1.I4.i2.p1.5.m5.1.1.cmml">2</mn><annotation-xml encoding="MathML-Content" id="S1.I4.i2.p1.5.m5.1b"><cn id="S1.I4.i2.p1.5.m5.1.1.cmml" type="integer" xref="S1.I4.i2.p1.5.m5.1.1">2</cn></annotation-xml><annotation encoding="application/x-tex" id="S1.I4.i2.p1.5.m5.1c">2</annotation><annotation encoding="application/x-llamapun" id="S1.I4.i2.p1.5.m5.1d">2</annotation></semantics></math>-VC-CAP <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.00712v1#bib.bibx5" title="">ADNP99</a>]</cite>.</p> </div> </li> </ul> <p class="ltx_p" id="S1.SS1.SSS0.Px2.p1.8">Following the earlier discussion, this implies that with <math alttext="k" class="ltx_Math" display="inline" id="S1.SS1.SSS0.Px2.p1.6.m1.1"><semantics id="S1.SS1.SSS0.Px2.p1.6.m1.1a"><mi id="S1.SS1.SSS0.Px2.p1.6.m1.1.1" xref="S1.SS1.SSS0.Px2.p1.6.m1.1.1.cmml">k</mi><annotation-xml encoding="MathML-Content" id="S1.SS1.SSS0.Px2.p1.6.m1.1b"><ci id="S1.SS1.SSS0.Px2.p1.6.m1.1.1.cmml" xref="S1.SS1.SSS0.Px2.p1.6.m1.1.1">𝑘</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.SS1.SSS0.Px2.p1.6.m1.1c">k</annotation><annotation encoding="application/x-llamapun" id="S1.SS1.SSS0.Px2.p1.6.m1.1d">italic_k</annotation></semantics></math> passes of the stream, we obtain a constant-factor approximation in near-linear space for <math alttext="k" class="ltx_Math" display="inline" id="S1.SS1.SSS0.Px2.p1.7.m2.1"><semantics id="S1.SS1.SSS0.Px2.p1.7.m2.1a"><mi id="S1.SS1.SSS0.Px2.p1.7.m2.1.1" xref="S1.SS1.SSS0.Px2.p1.7.m2.1.1.cmml">k</mi><annotation-xml encoding="MathML-Content" id="S1.SS1.SSS0.Px2.p1.7.m2.1b"><ci id="S1.SS1.SSS0.Px2.p1.7.m2.1.1.cmml" xref="S1.SS1.SSS0.Px2.p1.7.m2.1.1">𝑘</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.SS1.SSS0.Px2.p1.7.m2.1c">k</annotation><annotation encoding="application/x-llamapun" id="S1.SS1.SSS0.Px2.p1.7.m2.1d">italic_k</annotation></semantics></math>-VCSS for <math alttext="k\leq 3" class="ltx_Math" display="inline" id="S1.SS1.SSS0.Px2.p1.8.m3.1"><semantics id="S1.SS1.SSS0.Px2.p1.8.m3.1a"><mrow id="S1.SS1.SSS0.Px2.p1.8.m3.1.1" xref="S1.SS1.SSS0.Px2.p1.8.m3.1.1.cmml"><mi id="S1.SS1.SSS0.Px2.p1.8.m3.1.1.2" xref="S1.SS1.SSS0.Px2.p1.8.m3.1.1.2.cmml">k</mi><mo id="S1.SS1.SSS0.Px2.p1.8.m3.1.1.1" xref="S1.SS1.SSS0.Px2.p1.8.m3.1.1.1.cmml">≤</mo><mn id="S1.SS1.SSS0.Px2.p1.8.m3.1.1.3" xref="S1.SS1.SSS0.Px2.p1.8.m3.1.1.3.cmml">3</mn></mrow><annotation-xml encoding="MathML-Content" id="S1.SS1.SSS0.Px2.p1.8.m3.1b"><apply id="S1.SS1.SSS0.Px2.p1.8.m3.1.1.cmml" xref="S1.SS1.SSS0.Px2.p1.8.m3.1.1"><leq id="S1.SS1.SSS0.Px2.p1.8.m3.1.1.1.cmml" xref="S1.SS1.SSS0.Px2.p1.8.m3.1.1.1"></leq><ci id="S1.SS1.SSS0.Px2.p1.8.m3.1.1.2.cmml" xref="S1.SS1.SSS0.Px2.p1.8.m3.1.1.2">𝑘</ci><cn id="S1.SS1.SSS0.Px2.p1.8.m3.1.1.3.cmml" type="integer" xref="S1.SS1.SSS0.Px2.p1.8.m3.1.1.3">3</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS1.SSS0.Px2.p1.8.m3.1c">k\leq 3</annotation><annotation encoding="application/x-llamapun" id="S1.SS1.SSS0.Px2.p1.8.m3.1d">italic_k ≤ 3</annotation></semantics></math>.</p> </div> <figure class="ltx_table" id="S1.T1"> <table class="ltx_tabular ltx_centering ltx_align_middle" id="S1.T1.21"> <tbody class="ltx_tbody"> <tr class="ltx_tr" id="S1.T1.21.22.1"> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_tt" id="S1.T1.21.22.1.1" style="padding-top:2.5pt;padding-bottom:2.5pt;"><span class="ltx_text ltx_font_bold" id="S1.T1.21.22.1.1.1">Problem</span></td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_tt" id="S1.T1.21.22.1.2" style="padding-top:2.5pt;padding-bottom:2.5pt;"><span class="ltx_text ltx_font_bold" id="S1.T1.21.22.1.2.1">Approx.</span></td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_tt" id="S1.T1.21.22.1.3" style="padding-top:2.5pt;padding-bottom:2.5pt;"><span class="ltx_text ltx_font_bold" id="S1.T1.21.22.1.3.1">Space</span></td> <td class="ltx_td ltx_align_center ltx_border_tt" id="S1.T1.21.22.1.4" style="padding-top:2.5pt;padding-bottom:2.5pt;"><span class="ltx_text ltx_font_bold" id="S1.T1.21.22.1.4.1">Note</span></td> </tr> <tr class="ltx_tr" id="S1.T1.3.3"> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_tt" id="S1.T1.1.1.1" rowspan="4" style="padding-top:2.5pt;padding-bottom:2.5pt;"><span class="ltx_text" id="S1.T1.1.1.1.1"><math alttext="k" class="ltx_Math" display="inline" id="S1.T1.1.1.1.1.m1.1"><semantics id="S1.T1.1.1.1.1.m1.1a"><mi id="S1.T1.1.1.1.1.m1.1.1" xref="S1.T1.1.1.1.1.m1.1.1.cmml">k</mi><annotation-xml encoding="MathML-Content" id="S1.T1.1.1.1.1.m1.1b"><ci id="S1.T1.1.1.1.1.m1.1.1.cmml" xref="S1.T1.1.1.1.1.m1.1.1">𝑘</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.T1.1.1.1.1.m1.1c">k</annotation><annotation encoding="application/x-llamapun" id="S1.T1.1.1.1.1.m1.1d">italic_k</annotation></semantics></math>-EC-CAP</span></td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_tt" id="S1.T1.2.2.2" style="padding-top:2.5pt;padding-bottom:2.5pt;"><math alttext="2+\epsilon" class="ltx_Math" display="inline" id="S1.T1.2.2.2.m1.1"><semantics id="S1.T1.2.2.2.m1.1a"><mrow id="S1.T1.2.2.2.m1.1.1" xref="S1.T1.2.2.2.m1.1.1.cmml"><mn id="S1.T1.2.2.2.m1.1.1.2" xref="S1.T1.2.2.2.m1.1.1.2.cmml">2</mn><mo id="S1.T1.2.2.2.m1.1.1.1" xref="S1.T1.2.2.2.m1.1.1.1.cmml">+</mo><mi id="S1.T1.2.2.2.m1.1.1.3" xref="S1.T1.2.2.2.m1.1.1.3.cmml">ϵ</mi></mrow><annotation-xml encoding="MathML-Content" id="S1.T1.2.2.2.m1.1b"><apply id="S1.T1.2.2.2.m1.1.1.cmml" xref="S1.T1.2.2.2.m1.1.1"><plus id="S1.T1.2.2.2.m1.1.1.1.cmml" xref="S1.T1.2.2.2.m1.1.1.1"></plus><cn id="S1.T1.2.2.2.m1.1.1.2.cmml" type="integer" xref="S1.T1.2.2.2.m1.1.1.2">2</cn><ci id="S1.T1.2.2.2.m1.1.1.3.cmml" xref="S1.T1.2.2.2.m1.1.1.3">italic-ϵ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.T1.2.2.2.m1.1c">2+\epsilon</annotation><annotation encoding="application/x-llamapun" id="S1.T1.2.2.2.m1.1d">2 + italic_ϵ</annotation></semantics></math></td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_tt" id="S1.T1.3.3.3" style="padding-top:2.5pt;padding-bottom:2.5pt;"> <math alttext="O(\frac{n}{\epsilon}\log n)" class="ltx_Math" display="inline" id="S1.T1.3.3.3.m1.1"><semantics id="S1.T1.3.3.3.m1.1a"><mrow id="S1.T1.3.3.3.m1.1.1" xref="S1.T1.3.3.3.m1.1.1.cmml"><mi id="S1.T1.3.3.3.m1.1.1.3" xref="S1.T1.3.3.3.m1.1.1.3.cmml">O</mi><mo id="S1.T1.3.3.3.m1.1.1.2" xref="S1.T1.3.3.3.m1.1.1.2.cmml">⁢</mo><mrow id="S1.T1.3.3.3.m1.1.1.1.1" xref="S1.T1.3.3.3.m1.1.1.1.1.1.cmml"><mo id="S1.T1.3.3.3.m1.1.1.1.1.2" stretchy="false" xref="S1.T1.3.3.3.m1.1.1.1.1.1.cmml">(</mo><mrow id="S1.T1.3.3.3.m1.1.1.1.1.1" xref="S1.T1.3.3.3.m1.1.1.1.1.1.cmml"><mfrac 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xref="S1.T1.3.3.3.m1.1.1.2"></times><ci id="S1.T1.3.3.3.m1.1.1.3.cmml" xref="S1.T1.3.3.3.m1.1.1.3">𝑂</ci><apply id="S1.T1.3.3.3.m1.1.1.1.1.1.cmml" xref="S1.T1.3.3.3.m1.1.1.1.1"><times id="S1.T1.3.3.3.m1.1.1.1.1.1.1.cmml" xref="S1.T1.3.3.3.m1.1.1.1.1.1.1"></times><apply id="S1.T1.3.3.3.m1.1.1.1.1.1.2.cmml" xref="S1.T1.3.3.3.m1.1.1.1.1.1.2"><divide id="S1.T1.3.3.3.m1.1.1.1.1.1.2.1.cmml" xref="S1.T1.3.3.3.m1.1.1.1.1.1.2"></divide><ci id="S1.T1.3.3.3.m1.1.1.1.1.1.2.2.cmml" xref="S1.T1.3.3.3.m1.1.1.1.1.1.2.2">𝑛</ci><ci id="S1.T1.3.3.3.m1.1.1.1.1.1.2.3.cmml" xref="S1.T1.3.3.3.m1.1.1.1.1.1.2.3">italic-ϵ</ci></apply><apply id="S1.T1.3.3.3.m1.1.1.1.1.1.3.cmml" xref="S1.T1.3.3.3.m1.1.1.1.1.1.3"><log id="S1.T1.3.3.3.m1.1.1.1.1.1.3.1.cmml" xref="S1.T1.3.3.3.m1.1.1.1.1.1.3.1"></log><ci id="S1.T1.3.3.3.m1.1.1.1.1.1.3.2.cmml" xref="S1.T1.3.3.3.m1.1.1.1.1.1.3.2">𝑛</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.T1.3.3.3.m1.1c">O(\frac{n}{\epsilon}\log n)</annotation><annotation encoding="application/x-llamapun" id="S1.T1.3.3.3.m1.1d">italic_O ( divide start_ARG italic_n end_ARG start_ARG italic_ϵ end_ARG roman_log italic_n )</annotation></semantics></math> <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.00712v1#bib.bibx54" title="">JKMV24</a>]</cite> </td> <td class="ltx_td ltx_align_center ltx_border_tt" id="S1.T1.3.3.4" style="padding-top:2.5pt;padding-bottom:2.5pt;">link arrival</td> </tr> <tr class="ltx_tr" id="S1.T1.5.5"> <td class="ltx_td ltx_align_center ltx_border_r" id="S1.T1.4.4.1" style="padding-top:2.5pt;padding-bottom:2.5pt;"><math alttext="2-\epsilon" class="ltx_Math" display="inline" id="S1.T1.4.4.1.m1.1"><semantics id="S1.T1.4.4.1.m1.1a"><mrow id="S1.T1.4.4.1.m1.1.1" xref="S1.T1.4.4.1.m1.1.1.cmml"><mn id="S1.T1.4.4.1.m1.1.1.2" xref="S1.T1.4.4.1.m1.1.1.2.cmml">2</mn><mo id="S1.T1.4.4.1.m1.1.1.1" xref="S1.T1.4.4.1.m1.1.1.1.cmml">−</mo><mi id="S1.T1.4.4.1.m1.1.1.3" xref="S1.T1.4.4.1.m1.1.1.3.cmml">ϵ</mi></mrow><annotation-xml encoding="MathML-Content" id="S1.T1.4.4.1.m1.1b"><apply id="S1.T1.4.4.1.m1.1.1.cmml" xref="S1.T1.4.4.1.m1.1.1"><minus id="S1.T1.4.4.1.m1.1.1.1.cmml" xref="S1.T1.4.4.1.m1.1.1.1"></minus><cn id="S1.T1.4.4.1.m1.1.1.2.cmml" type="integer" xref="S1.T1.4.4.1.m1.1.1.2">2</cn><ci id="S1.T1.4.4.1.m1.1.1.3.cmml" xref="S1.T1.4.4.1.m1.1.1.3">italic-ϵ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.T1.4.4.1.m1.1c">2-\epsilon</annotation><annotation encoding="application/x-llamapun" id="S1.T1.4.4.1.m1.1d">2 - italic_ϵ</annotation></semantics></math></td> <td class="ltx_td ltx_align_center ltx_border_r" id="S1.T1.5.5.2" style="padding-top:2.5pt;padding-bottom:2.5pt;"> <math alttext="\Omega(n^{2})" class="ltx_Math" display="inline" id="S1.T1.5.5.2.m1.1"><semantics id="S1.T1.5.5.2.m1.1a"><mrow id="S1.T1.5.5.2.m1.1.1" xref="S1.T1.5.5.2.m1.1.1.cmml"><mi id="S1.T1.5.5.2.m1.1.1.3" mathvariant="normal" 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id="S1.T1.5.5.2.m1.1.1.1.1.1.1.cmml" xref="S1.T1.5.5.2.m1.1.1.1.1">superscript</csymbol><ci id="S1.T1.5.5.2.m1.1.1.1.1.1.2.cmml" xref="S1.T1.5.5.2.m1.1.1.1.1.1.2">𝑛</ci><cn id="S1.T1.5.5.2.m1.1.1.1.1.1.3.cmml" type="integer" xref="S1.T1.5.5.2.m1.1.1.1.1.1.3">2</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.T1.5.5.2.m1.1c">\Omega(n^{2})</annotation><annotation encoding="application/x-llamapun" id="S1.T1.5.5.2.m1.1d">roman_Ω ( italic_n start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT )</annotation></semantics></math> bits <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.00712v1#bib.bibx54" title="">JKMV24</a>]</cite> </td> <td class="ltx_td ltx_align_center" id="S1.T1.5.5.3" style="padding-top:2.5pt;padding-bottom:2.5pt;">link arrival lower bound</td> </tr> <tr class="ltx_tr" id="S1.T1.7.7"> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S1.T1.6.6.1" rowspan="2" 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ltx_border_t" id="S1.T1.7.7.3" style="padding-top:2.5pt;padding-bottom:2.5pt;">fully streaming</td> </tr> <tr class="ltx_tr" id="S1.T1.8.8"> <td class="ltx_td ltx_align_center ltx_border_r" id="S1.T1.8.8.1" style="padding-top:2.5pt;padding-bottom:2.5pt;"> <math alttext="\Omega(kn+n^{1+\frac{1}{t}})" class="ltx_Math" display="inline" id="S1.T1.8.8.1.m1.1"><semantics id="S1.T1.8.8.1.m1.1a"><mrow id="S1.T1.8.8.1.m1.1.1" xref="S1.T1.8.8.1.m1.1.1.cmml"><mi id="S1.T1.8.8.1.m1.1.1.3" mathvariant="normal" xref="S1.T1.8.8.1.m1.1.1.3.cmml">Ω</mi><mo id="S1.T1.8.8.1.m1.1.1.2" xref="S1.T1.8.8.1.m1.1.1.2.cmml">⁢</mo><mrow id="S1.T1.8.8.1.m1.1.1.1.1" xref="S1.T1.8.8.1.m1.1.1.1.1.1.cmml"><mo id="S1.T1.8.8.1.m1.1.1.1.1.2" stretchy="false" xref="S1.T1.8.8.1.m1.1.1.1.1.1.cmml">(</mo><mrow id="S1.T1.8.8.1.m1.1.1.1.1.1" xref="S1.T1.8.8.1.m1.1.1.1.1.1.cmml"><mrow id="S1.T1.8.8.1.m1.1.1.1.1.1.2" xref="S1.T1.8.8.1.m1.1.1.1.1.1.2.cmml"><mi id="S1.T1.8.8.1.m1.1.1.1.1.1.2.2" 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xref="S1.T1.8.8.1.m1.1.1.1.1.1.3.3.3.3">𝑡</ci></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.T1.8.8.1.m1.1c">\Omega(kn+n^{1+\frac{1}{t}})</annotation><annotation encoding="application/x-llamapun" id="S1.T1.8.8.1.m1.1d">roman_Ω ( italic_k italic_n + italic_n start_POSTSUPERSCRIPT 1 + divide start_ARG 1 end_ARG start_ARG italic_t end_ARG end_POSTSUPERSCRIPT )</annotation></semantics></math> bits <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.00712v1#bib.bibx54" title="">JKMV24</a>]</cite> </td> <td class="ltx_td ltx_align_center" id="S1.T1.8.8.2" style="padding-top:2.5pt;padding-bottom:2.5pt;">fully streaming lower bound</td> </tr> <tr class="ltx_tr" id="S1.T1.10.10"> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_tt" id="S1.T1.10.10.3" rowspan="3" style="padding-top:2.5pt;padding-bottom:2.5pt;"><span class="ltx_text" id="S1.T1.10.10.3.1">EC-SNDP</span></td> <td 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xref="S1.T1.9.9.1.m1.1.1.1.1.1.3.1"></log><ci id="S1.T1.9.9.1.m1.1.1.1.1.1.3.2.cmml" xref="S1.T1.9.9.1.m1.1.1.1.1.1.3.2">𝑘</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.T1.9.9.1.m1.1c">O(t\log k)</annotation><annotation encoding="application/x-llamapun" id="S1.T1.9.9.1.m1.1d">italic_O ( italic_t roman_log italic_k )</annotation></semantics></math></td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_tt" id="S1.T1.10.10.2" style="padding-top:2.5pt;padding-bottom:2.5pt;"> <math alttext="\tilde{O}(kn^{1+\frac{1}{t}})" class="ltx_Math" display="inline" id="S1.T1.10.10.2.m1.1"><semantics id="S1.T1.10.10.2.m1.1a"><mrow id="S1.T1.10.10.2.m1.1.1" xref="S1.T1.10.10.2.m1.1.1.cmml"><mover accent="true" id="S1.T1.10.10.2.m1.1.1.3" xref="S1.T1.10.10.2.m1.1.1.3.cmml"><mi id="S1.T1.10.10.2.m1.1.1.3.2" xref="S1.T1.10.10.2.m1.1.1.3.2.cmml">O</mi><mo id="S1.T1.10.10.2.m1.1.1.3.1" xref="S1.T1.10.10.2.m1.1.1.3.1.cmml">~</mo></mover><mo id="S1.T1.10.10.2.m1.1.1.2" xref="S1.T1.10.10.2.m1.1.1.2.cmml">⁢</mo><mrow id="S1.T1.10.10.2.m1.1.1.1.1" xref="S1.T1.10.10.2.m1.1.1.1.1.1.cmml"><mo id="S1.T1.10.10.2.m1.1.1.1.1.2" stretchy="false" xref="S1.T1.10.10.2.m1.1.1.1.1.1.cmml">(</mo><mrow id="S1.T1.10.10.2.m1.1.1.1.1.1" xref="S1.T1.10.10.2.m1.1.1.1.1.1.cmml"><mi id="S1.T1.10.10.2.m1.1.1.1.1.1.2" xref="S1.T1.10.10.2.m1.1.1.1.1.1.2.cmml">k</mi><mo id="S1.T1.10.10.2.m1.1.1.1.1.1.1" xref="S1.T1.10.10.2.m1.1.1.1.1.1.1.cmml">⁢</mo><msup id="S1.T1.10.10.2.m1.1.1.1.1.1.3" xref="S1.T1.10.10.2.m1.1.1.1.1.1.3.cmml"><mi id="S1.T1.10.10.2.m1.1.1.1.1.1.3.2" xref="S1.T1.10.10.2.m1.1.1.1.1.1.3.2.cmml">n</mi><mrow id="S1.T1.10.10.2.m1.1.1.1.1.1.3.3" xref="S1.T1.10.10.2.m1.1.1.1.1.1.3.3.cmml"><mn id="S1.T1.10.10.2.m1.1.1.1.1.1.3.3.2" xref="S1.T1.10.10.2.m1.1.1.1.1.1.3.3.2.cmml">1</mn><mo id="S1.T1.10.10.2.m1.1.1.1.1.1.3.3.1" xref="S1.T1.10.10.2.m1.1.1.1.1.1.3.3.1.cmml">+</mo><mfrac id="S1.T1.10.10.2.m1.1.1.1.1.1.3.3.3" 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id="S1.T1.10.10.2.m1.1.1.1.1.1.3.3.3.3.cmml" xref="S1.T1.10.10.2.m1.1.1.1.1.1.3.3.3.3">𝑡</ci></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.T1.10.10.2.m1.1c">\tilde{O}(kn^{1+\frac{1}{t}})</annotation><annotation encoding="application/x-llamapun" id="S1.T1.10.10.2.m1.1d">over~ start_ARG italic_O end_ARG ( italic_k italic_n start_POSTSUPERSCRIPT 1 + divide start_ARG 1 end_ARG start_ARG italic_t end_ARG end_POSTSUPERSCRIPT )</annotation></semantics></math> <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.00712v1#bib.bibx54" title="">JKMV24</a>]</cite> </td> <td class="ltx_td ltx_border_tt" id="S1.T1.10.10.4" style="padding-top:2.5pt;padding-bottom:2.5pt;"></td> </tr> <tr class="ltx_tr" id="S1.T1.12.12"> <td class="ltx_td ltx_align_center ltx_border_r" id="S1.T1.11.11.1" style="padding-top:2.5pt;padding-bottom:2.5pt;"><math alttext="8t" class="ltx_Math" display="inline" 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id="S1.T1.12.12.2" style="padding-top:2.5pt;padding-bottom:2.5pt;"> <math alttext="\tilde{O}(k^{1-\frac{1}{t}}n^{1+\frac{1}{t}})" class="ltx_Math" display="inline" id="S1.T1.12.12.2.m1.1"><semantics id="S1.T1.12.12.2.m1.1a"><mrow id="S1.T1.12.12.2.m1.1.1" xref="S1.T1.12.12.2.m1.1.1.cmml"><mover accent="true" id="S1.T1.12.12.2.m1.1.1.3" xref="S1.T1.12.12.2.m1.1.1.3.cmml"><mi id="S1.T1.12.12.2.m1.1.1.3.2" xref="S1.T1.12.12.2.m1.1.1.3.2.cmml">O</mi><mo id="S1.T1.12.12.2.m1.1.1.3.1" xref="S1.T1.12.12.2.m1.1.1.3.1.cmml">~</mo></mover><mo id="S1.T1.12.12.2.m1.1.1.2" xref="S1.T1.12.12.2.m1.1.1.2.cmml">⁢</mo><mrow id="S1.T1.12.12.2.m1.1.1.1.1" xref="S1.T1.12.12.2.m1.1.1.1.1.1.cmml"><mo id="S1.T1.12.12.2.m1.1.1.1.1.2" stretchy="false" xref="S1.T1.12.12.2.m1.1.1.1.1.1.cmml">(</mo><mrow id="S1.T1.12.12.2.m1.1.1.1.1.1" xref="S1.T1.12.12.2.m1.1.1.1.1.1.cmml"><msup id="S1.T1.12.12.2.m1.1.1.1.1.1.2" xref="S1.T1.12.12.2.m1.1.1.1.1.1.2.cmml"><mi id="S1.T1.12.12.2.m1.1.1.1.1.1.2.2" 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id="S1.T1.12.12.2.m1.1.1.1.1.1.3.3.3.1.cmml" xref="S1.T1.12.12.2.m1.1.1.1.1.1.3.3.3"></divide><cn id="S1.T1.12.12.2.m1.1.1.1.1.1.3.3.3.2.cmml" type="integer" xref="S1.T1.12.12.2.m1.1.1.1.1.1.3.3.3.2">1</cn><ci id="S1.T1.12.12.2.m1.1.1.1.1.1.3.3.3.3.cmml" xref="S1.T1.12.12.2.m1.1.1.1.1.1.3.3.3.3">𝑡</ci></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.T1.12.12.2.m1.1c">\tilde{O}(k^{1-\frac{1}{t}}n^{1+\frac{1}{t}})</annotation><annotation encoding="application/x-llamapun" id="S1.T1.12.12.2.m1.1d">over~ start_ARG italic_O end_ARG ( italic_k start_POSTSUPERSCRIPT 1 - divide start_ARG 1 end_ARG start_ARG italic_t end_ARG end_POSTSUPERSCRIPT italic_n start_POSTSUPERSCRIPT 1 + divide start_ARG 1 end_ARG start_ARG italic_t end_ARG end_POSTSUPERSCRIPT )</annotation></semantics></math> <span class="ltx_text" id="S1.T1.12.12.2.1" style="color:#0000FF;">[Here]</span> </td> <td class="ltx_td" id="S1.T1.12.12.3" style="padding-top:2.5pt;padding-bottom:2.5pt;"></td> </tr> <tr class="ltx_tr" id="S1.T1.14.14"> <td class="ltx_td ltx_align_center ltx_border_r" id="S1.T1.13.13.1" style="padding-top:2.5pt;padding-bottom:2.5pt;"><math alttext="O(t)" class="ltx_Math" display="inline" id="S1.T1.13.13.1.m1.1"><semantics id="S1.T1.13.13.1.m1.1a"><mrow id="S1.T1.13.13.1.m1.1.2" xref="S1.T1.13.13.1.m1.1.2.cmml"><mi id="S1.T1.13.13.1.m1.1.2.2" xref="S1.T1.13.13.1.m1.1.2.2.cmml">O</mi><mo id="S1.T1.13.13.1.m1.1.2.1" xref="S1.T1.13.13.1.m1.1.2.1.cmml">⁢</mo><mrow id="S1.T1.13.13.1.m1.1.2.3.2" xref="S1.T1.13.13.1.m1.1.2.cmml"><mo id="S1.T1.13.13.1.m1.1.2.3.2.1" stretchy="false" xref="S1.T1.13.13.1.m1.1.2.cmml">(</mo><mi id="S1.T1.13.13.1.m1.1.1" xref="S1.T1.13.13.1.m1.1.1.cmml">t</mi><mo id="S1.T1.13.13.1.m1.1.2.3.2.2" stretchy="false" xref="S1.T1.13.13.1.m1.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.T1.13.13.1.m1.1b"><apply id="S1.T1.13.13.1.m1.1.2.cmml" xref="S1.T1.13.13.1.m1.1.2"><times id="S1.T1.13.13.1.m1.1.2.1.cmml" xref="S1.T1.13.13.1.m1.1.2.1"></times><ci id="S1.T1.13.13.1.m1.1.2.2.cmml" xref="S1.T1.13.13.1.m1.1.2.2">𝑂</ci><ci id="S1.T1.13.13.1.m1.1.1.cmml" xref="S1.T1.13.13.1.m1.1.1">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.T1.13.13.1.m1.1c">O(t)</annotation><annotation encoding="application/x-llamapun" id="S1.T1.13.13.1.m1.1d">italic_O ( italic_t )</annotation></semantics></math></td> <td class="ltx_td ltx_align_center ltx_border_r" id="S1.T1.14.14.2" style="padding-top:2.5pt;padding-bottom:2.5pt;"> <math alttext="\Omega(kn+n^{1+\frac{1}{t}})" class="ltx_Math" display="inline" id="S1.T1.14.14.2.m1.1"><semantics id="S1.T1.14.14.2.m1.1a"><mrow id="S1.T1.14.14.2.m1.1.1" xref="S1.T1.14.14.2.m1.1.1.cmml"><mi id="S1.T1.14.14.2.m1.1.1.3" mathvariant="normal" xref="S1.T1.14.14.2.m1.1.1.3.cmml">Ω</mi><mo id="S1.T1.14.14.2.m1.1.1.2" xref="S1.T1.14.14.2.m1.1.1.2.cmml">⁢</mo><mrow id="S1.T1.14.14.2.m1.1.1.1.1" xref="S1.T1.14.14.2.m1.1.1.1.1.1.cmml"><mo id="S1.T1.14.14.2.m1.1.1.1.1.2" stretchy="false" xref="S1.T1.14.14.2.m1.1.1.1.1.1.cmml">(</mo><mrow id="S1.T1.14.14.2.m1.1.1.1.1.1" xref="S1.T1.14.14.2.m1.1.1.1.1.1.cmml"><mrow id="S1.T1.14.14.2.m1.1.1.1.1.1.2" xref="S1.T1.14.14.2.m1.1.1.1.1.1.2.cmml"><mi id="S1.T1.14.14.2.m1.1.1.1.1.1.2.2" xref="S1.T1.14.14.2.m1.1.1.1.1.1.2.2.cmml">k</mi><mo id="S1.T1.14.14.2.m1.1.1.1.1.1.2.1" xref="S1.T1.14.14.2.m1.1.1.1.1.1.2.1.cmml">⁢</mo><mi id="S1.T1.14.14.2.m1.1.1.1.1.1.2.3" xref="S1.T1.14.14.2.m1.1.1.1.1.1.2.3.cmml">n</mi></mrow><mo id="S1.T1.14.14.2.m1.1.1.1.1.1.1" xref="S1.T1.14.14.2.m1.1.1.1.1.1.1.cmml">+</mo><msup id="S1.T1.14.14.2.m1.1.1.1.1.1.3" xref="S1.T1.14.14.2.m1.1.1.1.1.1.3.cmml"><mi id="S1.T1.14.14.2.m1.1.1.1.1.1.3.2" xref="S1.T1.14.14.2.m1.1.1.1.1.1.3.2.cmml">n</mi><mrow id="S1.T1.14.14.2.m1.1.1.1.1.1.3.3" xref="S1.T1.14.14.2.m1.1.1.1.1.1.3.3.cmml"><mn id="S1.T1.14.14.2.m1.1.1.1.1.1.3.3.2" 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id="S1.T1.14.14.2.m1.1.1.1.1.1.3.3.3.cmml" xref="S1.T1.14.14.2.m1.1.1.1.1.1.3.3.3"><divide id="S1.T1.14.14.2.m1.1.1.1.1.1.3.3.3.1.cmml" xref="S1.T1.14.14.2.m1.1.1.1.1.1.3.3.3"></divide><cn id="S1.T1.14.14.2.m1.1.1.1.1.1.3.3.3.2.cmml" type="integer" xref="S1.T1.14.14.2.m1.1.1.1.1.1.3.3.3.2">1</cn><ci id="S1.T1.14.14.2.m1.1.1.1.1.1.3.3.3.3.cmml" xref="S1.T1.14.14.2.m1.1.1.1.1.1.3.3.3.3">𝑡</ci></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.T1.14.14.2.m1.1c">\Omega(kn+n^{1+\frac{1}{t}})</annotation><annotation encoding="application/x-llamapun" id="S1.T1.14.14.2.m1.1d">roman_Ω ( italic_k italic_n + italic_n start_POSTSUPERSCRIPT 1 + divide start_ARG 1 end_ARG start_ARG italic_t end_ARG end_POSTSUPERSCRIPT )</annotation></semantics></math> bits <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.00712v1#bib.bibx54" title="">JKMV24</a>]</cite> </td> <td class="ltx_td ltx_align_center" id="S1.T1.14.14.3" style="padding-top:2.5pt;padding-bottom:2.5pt;">lower bound</td> </tr> <tr class="ltx_tr" id="S1.T1.17.17"> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S1.T1.15.15.1" rowspan="2" style="padding-top:2.5pt;padding-bottom:2.5pt;"><span class="ltx_text" id="S1.T1.15.15.1.1"><math alttext="k" class="ltx_Math" display="inline" id="S1.T1.15.15.1.1.m1.1"><semantics id="S1.T1.15.15.1.1.m1.1a"><mi id="S1.T1.15.15.1.1.m1.1.1" xref="S1.T1.15.15.1.1.m1.1.1.cmml">k</mi><annotation-xml encoding="MathML-Content" id="S1.T1.15.15.1.1.m1.1b"><ci id="S1.T1.15.15.1.1.m1.1.1.cmml" xref="S1.T1.15.15.1.1.m1.1.1">𝑘</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.T1.15.15.1.1.m1.1c">k</annotation><annotation encoding="application/x-llamapun" id="S1.T1.15.15.1.1.m1.1d">italic_k</annotation></semantics></math>-ECSS</span></td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S1.T1.16.16.2" style="padding-top:2.5pt;padding-bottom:2.5pt;"><math alttext="2" class="ltx_Math" display="inline" id="S1.T1.16.16.2.m1.1"><semantics id="S1.T1.16.16.2.m1.1a"><mn id="S1.T1.16.16.2.m1.1.1" xref="S1.T1.16.16.2.m1.1.1.cmml">2</mn><annotation-xml encoding="MathML-Content" id="S1.T1.16.16.2.m1.1b"><cn id="S1.T1.16.16.2.m1.1.1.cmml" type="integer" xref="S1.T1.16.16.2.m1.1.1">2</cn></annotation-xml><annotation encoding="application/x-tex" id="S1.T1.16.16.2.m1.1c">2</annotation><annotation encoding="application/x-llamapun" id="S1.T1.16.16.2.m1.1d">2</annotation></semantics></math></td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S1.T1.17.17.3" style="padding-top:2.5pt;padding-bottom:2.5pt;"> <math alttext="O(kn)" class="ltx_Math" display="inline" id="S1.T1.17.17.3.m1.1"><semantics id="S1.T1.17.17.3.m1.1a"><mrow id="S1.T1.17.17.3.m1.1.1" xref="S1.T1.17.17.3.m1.1.1.cmml"><mi id="S1.T1.17.17.3.m1.1.1.3" xref="S1.T1.17.17.3.m1.1.1.3.cmml">O</mi><mo id="S1.T1.17.17.3.m1.1.1.2" xref="S1.T1.17.17.3.m1.1.1.2.cmml">⁢</mo><mrow id="S1.T1.17.17.3.m1.1.1.1.1" xref="S1.T1.17.17.3.m1.1.1.1.1.1.cmml"><mo id="S1.T1.17.17.3.m1.1.1.1.1.2" stretchy="false" xref="S1.T1.17.17.3.m1.1.1.1.1.1.cmml">(</mo><mrow id="S1.T1.17.17.3.m1.1.1.1.1.1" xref="S1.T1.17.17.3.m1.1.1.1.1.1.cmml"><mi id="S1.T1.17.17.3.m1.1.1.1.1.1.2" xref="S1.T1.17.17.3.m1.1.1.1.1.1.2.cmml">k</mi><mo id="S1.T1.17.17.3.m1.1.1.1.1.1.1" xref="S1.T1.17.17.3.m1.1.1.1.1.1.1.cmml">⁢</mo><mi id="S1.T1.17.17.3.m1.1.1.1.1.1.3" xref="S1.T1.17.17.3.m1.1.1.1.1.1.3.cmml">n</mi></mrow><mo id="S1.T1.17.17.3.m1.1.1.1.1.3" stretchy="false" xref="S1.T1.17.17.3.m1.1.1.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.T1.17.17.3.m1.1b"><apply id="S1.T1.17.17.3.m1.1.1.cmml" xref="S1.T1.17.17.3.m1.1.1"><times id="S1.T1.17.17.3.m1.1.1.2.cmml" xref="S1.T1.17.17.3.m1.1.1.2"></times><ci id="S1.T1.17.17.3.m1.1.1.3.cmml" xref="S1.T1.17.17.3.m1.1.1.3">𝑂</ci><apply id="S1.T1.17.17.3.m1.1.1.1.1.1.cmml" xref="S1.T1.17.17.3.m1.1.1.1.1"><times id="S1.T1.17.17.3.m1.1.1.1.1.1.1.cmml" xref="S1.T1.17.17.3.m1.1.1.1.1.1.1"></times><ci id="S1.T1.17.17.3.m1.1.1.1.1.1.2.cmml" xref="S1.T1.17.17.3.m1.1.1.1.1.1.2">𝑘</ci><ci id="S1.T1.17.17.3.m1.1.1.1.1.1.3.cmml" xref="S1.T1.17.17.3.m1.1.1.1.1.1.3">𝑛</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.T1.17.17.3.m1.1c">O(kn)</annotation><annotation encoding="application/x-llamapun" id="S1.T1.17.17.3.m1.1d">italic_O ( italic_k italic_n )</annotation></semantics></math> <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.00712v1#bib.bibx81" title="">Zel06</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.00712v1#bib.bibx27" title="">CKT93</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.00712v1#bib.bibx69" title="">NI92</a>]</cite> </td> <td class="ltx_td ltx_align_center ltx_border_t" id="S1.T1.17.17.4" style="padding-top:2.5pt;padding-bottom:2.5pt;">for unweighted graphs</td> </tr> <tr class="ltx_tr" id="S1.T1.19.19"> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S1.T1.18.18.1" style="padding-top:2.5pt;padding-bottom:2.5pt;"><math alttext="8t" class="ltx_Math" display="inline" id="S1.T1.18.18.1.m1.1"><semantics id="S1.T1.18.18.1.m1.1a"><mrow id="S1.T1.18.18.1.m1.1.1" xref="S1.T1.18.18.1.m1.1.1.cmml"><mn id="S1.T1.18.18.1.m1.1.1.2" xref="S1.T1.18.18.1.m1.1.1.2.cmml">8</mn><mo id="S1.T1.18.18.1.m1.1.1.1" xref="S1.T1.18.18.1.m1.1.1.1.cmml">⁢</mo><mi id="S1.T1.18.18.1.m1.1.1.3" xref="S1.T1.18.18.1.m1.1.1.3.cmml">t</mi></mrow><annotation-xml encoding="MathML-Content" id="S1.T1.18.18.1.m1.1b"><apply id="S1.T1.18.18.1.m1.1.1.cmml" xref="S1.T1.18.18.1.m1.1.1"><times id="S1.T1.18.18.1.m1.1.1.1.cmml" xref="S1.T1.18.18.1.m1.1.1.1"></times><cn id="S1.T1.18.18.1.m1.1.1.2.cmml" type="integer" xref="S1.T1.18.18.1.m1.1.1.2">8</cn><ci id="S1.T1.18.18.1.m1.1.1.3.cmml" xref="S1.T1.18.18.1.m1.1.1.3">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.T1.18.18.1.m1.1c">8t</annotation><annotation encoding="application/x-llamapun" id="S1.T1.18.18.1.m1.1d">8 italic_t</annotation></semantics></math></td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S1.T1.19.19.2" style="padding-top:2.5pt;padding-bottom:2.5pt;"> <math alttext="\tilde{O}(k^{1-\frac{1}{t}}n^{1+\frac{1}{t}})" class="ltx_Math" display="inline" id="S1.T1.19.19.2.m1.1"><semantics id="S1.T1.19.19.2.m1.1a"><mrow id="S1.T1.19.19.2.m1.1.1" xref="S1.T1.19.19.2.m1.1.1.cmml"><mover accent="true" id="S1.T1.19.19.2.m1.1.1.3" xref="S1.T1.19.19.2.m1.1.1.3.cmml"><mi id="S1.T1.19.19.2.m1.1.1.3.2" xref="S1.T1.19.19.2.m1.1.1.3.2.cmml">O</mi><mo id="S1.T1.19.19.2.m1.1.1.3.1" xref="S1.T1.19.19.2.m1.1.1.3.1.cmml">~</mo></mover><mo id="S1.T1.19.19.2.m1.1.1.2" xref="S1.T1.19.19.2.m1.1.1.2.cmml">⁢</mo><mrow id="S1.T1.19.19.2.m1.1.1.1.1" xref="S1.T1.19.19.2.m1.1.1.1.1.1.cmml"><mo id="S1.T1.19.19.2.m1.1.1.1.1.2" stretchy="false" xref="S1.T1.19.19.2.m1.1.1.1.1.1.cmml">(</mo><mrow id="S1.T1.19.19.2.m1.1.1.1.1.1" xref="S1.T1.19.19.2.m1.1.1.1.1.1.cmml"><msup id="S1.T1.19.19.2.m1.1.1.1.1.1.2" xref="S1.T1.19.19.2.m1.1.1.1.1.1.2.cmml"><mi id="S1.T1.19.19.2.m1.1.1.1.1.1.2.2" xref="S1.T1.19.19.2.m1.1.1.1.1.1.2.2.cmml">k</mi><mrow id="S1.T1.19.19.2.m1.1.1.1.1.1.2.3" xref="S1.T1.19.19.2.m1.1.1.1.1.1.2.3.cmml"><mn id="S1.T1.19.19.2.m1.1.1.1.1.1.2.3.2" xref="S1.T1.19.19.2.m1.1.1.1.1.1.2.3.2.cmml">1</mn><mo id="S1.T1.19.19.2.m1.1.1.1.1.1.2.3.1" xref="S1.T1.19.19.2.m1.1.1.1.1.1.2.3.1.cmml">−</mo><mfrac id="S1.T1.19.19.2.m1.1.1.1.1.1.2.3.3" xref="S1.T1.19.19.2.m1.1.1.1.1.1.2.3.3.cmml"><mn id="S1.T1.19.19.2.m1.1.1.1.1.1.2.3.3.2" xref="S1.T1.19.19.2.m1.1.1.1.1.1.2.3.3.2.cmml">1</mn><mi id="S1.T1.19.19.2.m1.1.1.1.1.1.2.3.3.3" xref="S1.T1.19.19.2.m1.1.1.1.1.1.2.3.3.3.cmml">t</mi></mfrac></mrow></msup><mo id="S1.T1.19.19.2.m1.1.1.1.1.1.1" xref="S1.T1.19.19.2.m1.1.1.1.1.1.1.cmml">⁢</mo><msup id="S1.T1.19.19.2.m1.1.1.1.1.1.3" xref="S1.T1.19.19.2.m1.1.1.1.1.1.3.cmml"><mi id="S1.T1.19.19.2.m1.1.1.1.1.1.3.2" xref="S1.T1.19.19.2.m1.1.1.1.1.1.3.2.cmml">n</mi><mrow id="S1.T1.19.19.2.m1.1.1.1.1.1.3.3" xref="S1.T1.19.19.2.m1.1.1.1.1.1.3.3.cmml"><mn id="S1.T1.19.19.2.m1.1.1.1.1.1.3.3.2" xref="S1.T1.19.19.2.m1.1.1.1.1.1.3.3.2.cmml">1</mn><mo id="S1.T1.19.19.2.m1.1.1.1.1.1.3.3.1" xref="S1.T1.19.19.2.m1.1.1.1.1.1.3.3.1.cmml">+</mo><mfrac id="S1.T1.19.19.2.m1.1.1.1.1.1.3.3.3" xref="S1.T1.19.19.2.m1.1.1.1.1.1.3.3.3.cmml"><mn id="S1.T1.19.19.2.m1.1.1.1.1.1.3.3.3.2" xref="S1.T1.19.19.2.m1.1.1.1.1.1.3.3.3.2.cmml">1</mn><mi id="S1.T1.19.19.2.m1.1.1.1.1.1.3.3.3.3" xref="S1.T1.19.19.2.m1.1.1.1.1.1.3.3.3.3.cmml">t</mi></mfrac></mrow></msup></mrow><mo id="S1.T1.19.19.2.m1.1.1.1.1.3" stretchy="false" xref="S1.T1.19.19.2.m1.1.1.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.T1.19.19.2.m1.1b"><apply id="S1.T1.19.19.2.m1.1.1.cmml" xref="S1.T1.19.19.2.m1.1.1"><times id="S1.T1.19.19.2.m1.1.1.2.cmml" xref="S1.T1.19.19.2.m1.1.1.2"></times><apply id="S1.T1.19.19.2.m1.1.1.3.cmml" xref="S1.T1.19.19.2.m1.1.1.3"><ci id="S1.T1.19.19.2.m1.1.1.3.1.cmml" xref="S1.T1.19.19.2.m1.1.1.3.1">~</ci><ci id="S1.T1.19.19.2.m1.1.1.3.2.cmml" xref="S1.T1.19.19.2.m1.1.1.3.2">𝑂</ci></apply><apply id="S1.T1.19.19.2.m1.1.1.1.1.1.cmml" xref="S1.T1.19.19.2.m1.1.1.1.1"><times id="S1.T1.19.19.2.m1.1.1.1.1.1.1.cmml" xref="S1.T1.19.19.2.m1.1.1.1.1.1.1"></times><apply id="S1.T1.19.19.2.m1.1.1.1.1.1.2.cmml" xref="S1.T1.19.19.2.m1.1.1.1.1.1.2"><csymbol cd="ambiguous" id="S1.T1.19.19.2.m1.1.1.1.1.1.2.1.cmml" xref="S1.T1.19.19.2.m1.1.1.1.1.1.2">superscript</csymbol><ci id="S1.T1.19.19.2.m1.1.1.1.1.1.2.2.cmml" xref="S1.T1.19.19.2.m1.1.1.1.1.1.2.2">𝑘</ci><apply id="S1.T1.19.19.2.m1.1.1.1.1.1.2.3.cmml" xref="S1.T1.19.19.2.m1.1.1.1.1.1.2.3"><minus id="S1.T1.19.19.2.m1.1.1.1.1.1.2.3.1.cmml" xref="S1.T1.19.19.2.m1.1.1.1.1.1.2.3.1"></minus><cn id="S1.T1.19.19.2.m1.1.1.1.1.1.2.3.2.cmml" type="integer" xref="S1.T1.19.19.2.m1.1.1.1.1.1.2.3.2">1</cn><apply id="S1.T1.19.19.2.m1.1.1.1.1.1.2.3.3.cmml" xref="S1.T1.19.19.2.m1.1.1.1.1.1.2.3.3"><divide id="S1.T1.19.19.2.m1.1.1.1.1.1.2.3.3.1.cmml" xref="S1.T1.19.19.2.m1.1.1.1.1.1.2.3.3"></divide><cn id="S1.T1.19.19.2.m1.1.1.1.1.1.2.3.3.2.cmml" type="integer" xref="S1.T1.19.19.2.m1.1.1.1.1.1.2.3.3.2">1</cn><ci id="S1.T1.19.19.2.m1.1.1.1.1.1.2.3.3.3.cmml" xref="S1.T1.19.19.2.m1.1.1.1.1.1.2.3.3.3">𝑡</ci></apply></apply></apply><apply id="S1.T1.19.19.2.m1.1.1.1.1.1.3.cmml" xref="S1.T1.19.19.2.m1.1.1.1.1.1.3"><csymbol cd="ambiguous" id="S1.T1.19.19.2.m1.1.1.1.1.1.3.1.cmml" xref="S1.T1.19.19.2.m1.1.1.1.1.1.3">superscript</csymbol><ci id="S1.T1.19.19.2.m1.1.1.1.1.1.3.2.cmml" xref="S1.T1.19.19.2.m1.1.1.1.1.1.3.2">𝑛</ci><apply id="S1.T1.19.19.2.m1.1.1.1.1.1.3.3.cmml" xref="S1.T1.19.19.2.m1.1.1.1.1.1.3.3"><plus id="S1.T1.19.19.2.m1.1.1.1.1.1.3.3.1.cmml" xref="S1.T1.19.19.2.m1.1.1.1.1.1.3.3.1"></plus><cn id="S1.T1.19.19.2.m1.1.1.1.1.1.3.3.2.cmml" type="integer" xref="S1.T1.19.19.2.m1.1.1.1.1.1.3.3.2">1</cn><apply id="S1.T1.19.19.2.m1.1.1.1.1.1.3.3.3.cmml" xref="S1.T1.19.19.2.m1.1.1.1.1.1.3.3.3"><divide id="S1.T1.19.19.2.m1.1.1.1.1.1.3.3.3.1.cmml" xref="S1.T1.19.19.2.m1.1.1.1.1.1.3.3.3"></divide><cn id="S1.T1.19.19.2.m1.1.1.1.1.1.3.3.3.2.cmml" type="integer" xref="S1.T1.19.19.2.m1.1.1.1.1.1.3.3.3.2">1</cn><ci id="S1.T1.19.19.2.m1.1.1.1.1.1.3.3.3.3.cmml" xref="S1.T1.19.19.2.m1.1.1.1.1.1.3.3.3.3">𝑡</ci></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.T1.19.19.2.m1.1c">\tilde{O}(k^{1-\frac{1}{t}}n^{1+\frac{1}{t}})</annotation><annotation encoding="application/x-llamapun" id="S1.T1.19.19.2.m1.1d">over~ start_ARG italic_O end_ARG ( italic_k start_POSTSUPERSCRIPT 1 - divide start_ARG 1 end_ARG start_ARG italic_t end_ARG end_POSTSUPERSCRIPT italic_n start_POSTSUPERSCRIPT 1 + divide start_ARG 1 end_ARG start_ARG italic_t end_ARG end_POSTSUPERSCRIPT )</annotation></semantics></math> <span class="ltx_text" id="S1.T1.19.19.2.1" style="color:#0000FF;">[Here]</span> </td> <td class="ltx_td ltx_border_t" id="S1.T1.19.19.3" style="padding-top:2.5pt;padding-bottom:2.5pt;"></td> </tr> <tr class="ltx_tr" id="S1.T1.21.21"> <td class="ltx_td ltx_border_bb ltx_border_r" id="S1.T1.21.21.3" style="padding-top:2.5pt;padding-bottom:2.5pt;"></td> <td class="ltx_td ltx_align_center ltx_border_bb ltx_border_r" id="S1.T1.20.20.1" style="padding-top:2.5pt;padding-bottom:2.5pt;"><math alttext="O(t)" class="ltx_Math" display="inline" id="S1.T1.20.20.1.m1.1"><semantics id="S1.T1.20.20.1.m1.1a"><mrow id="S1.T1.20.20.1.m1.1.2" xref="S1.T1.20.20.1.m1.1.2.cmml"><mi id="S1.T1.20.20.1.m1.1.2.2" xref="S1.T1.20.20.1.m1.1.2.2.cmml">O</mi><mo id="S1.T1.20.20.1.m1.1.2.1" xref="S1.T1.20.20.1.m1.1.2.1.cmml">⁢</mo><mrow id="S1.T1.20.20.1.m1.1.2.3.2" xref="S1.T1.20.20.1.m1.1.2.cmml"><mo id="S1.T1.20.20.1.m1.1.2.3.2.1" stretchy="false" xref="S1.T1.20.20.1.m1.1.2.cmml">(</mo><mi id="S1.T1.20.20.1.m1.1.1" xref="S1.T1.20.20.1.m1.1.1.cmml">t</mi><mo id="S1.T1.20.20.1.m1.1.2.3.2.2" stretchy="false" xref="S1.T1.20.20.1.m1.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.T1.20.20.1.m1.1b"><apply id="S1.T1.20.20.1.m1.1.2.cmml" xref="S1.T1.20.20.1.m1.1.2"><times id="S1.T1.20.20.1.m1.1.2.1.cmml" xref="S1.T1.20.20.1.m1.1.2.1"></times><ci id="S1.T1.20.20.1.m1.1.2.2.cmml" xref="S1.T1.20.20.1.m1.1.2.2">𝑂</ci><ci id="S1.T1.20.20.1.m1.1.1.cmml" xref="S1.T1.20.20.1.m1.1.1">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.T1.20.20.1.m1.1c">O(t)</annotation><annotation encoding="application/x-llamapun" id="S1.T1.20.20.1.m1.1d">italic_O ( italic_t )</annotation></semantics></math></td> <td class="ltx_td ltx_align_center ltx_border_bb ltx_border_r" id="S1.T1.21.21.2" style="padding-top:2.5pt;padding-bottom:2.5pt;"> <math alttext="\Omega(kn+n^{1+\frac{1}{t}})" class="ltx_Math" display="inline" id="S1.T1.21.21.2.m1.1"><semantics id="S1.T1.21.21.2.m1.1a"><mrow id="S1.T1.21.21.2.m1.1.1" xref="S1.T1.21.21.2.m1.1.1.cmml"><mi id="S1.T1.21.21.2.m1.1.1.3" mathvariant="normal" xref="S1.T1.21.21.2.m1.1.1.3.cmml">Ω</mi><mo id="S1.T1.21.21.2.m1.1.1.2" xref="S1.T1.21.21.2.m1.1.1.2.cmml">⁢</mo><mrow id="S1.T1.21.21.2.m1.1.1.1.1" xref="S1.T1.21.21.2.m1.1.1.1.1.1.cmml"><mo id="S1.T1.21.21.2.m1.1.1.1.1.2" stretchy="false" xref="S1.T1.21.21.2.m1.1.1.1.1.1.cmml">(</mo><mrow id="S1.T1.21.21.2.m1.1.1.1.1.1" xref="S1.T1.21.21.2.m1.1.1.1.1.1.cmml"><mrow id="S1.T1.21.21.2.m1.1.1.1.1.1.2" xref="S1.T1.21.21.2.m1.1.1.1.1.1.2.cmml"><mi id="S1.T1.21.21.2.m1.1.1.1.1.1.2.2" xref="S1.T1.21.21.2.m1.1.1.1.1.1.2.2.cmml">k</mi><mo id="S1.T1.21.21.2.m1.1.1.1.1.1.2.1" xref="S1.T1.21.21.2.m1.1.1.1.1.1.2.1.cmml">⁢</mo><mi id="S1.T1.21.21.2.m1.1.1.1.1.1.2.3" xref="S1.T1.21.21.2.m1.1.1.1.1.1.2.3.cmml">n</mi></mrow><mo id="S1.T1.21.21.2.m1.1.1.1.1.1.1" xref="S1.T1.21.21.2.m1.1.1.1.1.1.1.cmml">+</mo><msup id="S1.T1.21.21.2.m1.1.1.1.1.1.3" xref="S1.T1.21.21.2.m1.1.1.1.1.1.3.cmml"><mi id="S1.T1.21.21.2.m1.1.1.1.1.1.3.2" xref="S1.T1.21.21.2.m1.1.1.1.1.1.3.2.cmml">n</mi><mrow id="S1.T1.21.21.2.m1.1.1.1.1.1.3.3" xref="S1.T1.21.21.2.m1.1.1.1.1.1.3.3.cmml"><mn id="S1.T1.21.21.2.m1.1.1.1.1.1.3.3.2" xref="S1.T1.21.21.2.m1.1.1.1.1.1.3.3.2.cmml">1</mn><mo id="S1.T1.21.21.2.m1.1.1.1.1.1.3.3.1" xref="S1.T1.21.21.2.m1.1.1.1.1.1.3.3.1.cmml">+</mo><mfrac id="S1.T1.21.21.2.m1.1.1.1.1.1.3.3.3" xref="S1.T1.21.21.2.m1.1.1.1.1.1.3.3.3.cmml"><mn id="S1.T1.21.21.2.m1.1.1.1.1.1.3.3.3.2" xref="S1.T1.21.21.2.m1.1.1.1.1.1.3.3.3.2.cmml">1</mn><mi id="S1.T1.21.21.2.m1.1.1.1.1.1.3.3.3.3" xref="S1.T1.21.21.2.m1.1.1.1.1.1.3.3.3.3.cmml">t</mi></mfrac></mrow></msup></mrow><mo id="S1.T1.21.21.2.m1.1.1.1.1.3" stretchy="false" xref="S1.T1.21.21.2.m1.1.1.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.T1.21.21.2.m1.1b"><apply id="S1.T1.21.21.2.m1.1.1.cmml" xref="S1.T1.21.21.2.m1.1.1"><times id="S1.T1.21.21.2.m1.1.1.2.cmml" xref="S1.T1.21.21.2.m1.1.1.2"></times><ci id="S1.T1.21.21.2.m1.1.1.3.cmml" xref="S1.T1.21.21.2.m1.1.1.3">Ω</ci><apply id="S1.T1.21.21.2.m1.1.1.1.1.1.cmml" xref="S1.T1.21.21.2.m1.1.1.1.1"><plus id="S1.T1.21.21.2.m1.1.1.1.1.1.1.cmml" xref="S1.T1.21.21.2.m1.1.1.1.1.1.1"></plus><apply id="S1.T1.21.21.2.m1.1.1.1.1.1.2.cmml" xref="S1.T1.21.21.2.m1.1.1.1.1.1.2"><times id="S1.T1.21.21.2.m1.1.1.1.1.1.2.1.cmml" xref="S1.T1.21.21.2.m1.1.1.1.1.1.2.1"></times><ci id="S1.T1.21.21.2.m1.1.1.1.1.1.2.2.cmml" xref="S1.T1.21.21.2.m1.1.1.1.1.1.2.2">𝑘</ci><ci id="S1.T1.21.21.2.m1.1.1.1.1.1.2.3.cmml" xref="S1.T1.21.21.2.m1.1.1.1.1.1.2.3">𝑛</ci></apply><apply id="S1.T1.21.21.2.m1.1.1.1.1.1.3.cmml" xref="S1.T1.21.21.2.m1.1.1.1.1.1.3"><csymbol cd="ambiguous" id="S1.T1.21.21.2.m1.1.1.1.1.1.3.1.cmml" xref="S1.T1.21.21.2.m1.1.1.1.1.1.3">superscript</csymbol><ci id="S1.T1.21.21.2.m1.1.1.1.1.1.3.2.cmml" xref="S1.T1.21.21.2.m1.1.1.1.1.1.3.2">𝑛</ci><apply id="S1.T1.21.21.2.m1.1.1.1.1.1.3.3.cmml" xref="S1.T1.21.21.2.m1.1.1.1.1.1.3.3"><plus id="S1.T1.21.21.2.m1.1.1.1.1.1.3.3.1.cmml" xref="S1.T1.21.21.2.m1.1.1.1.1.1.3.3.1"></plus><cn id="S1.T1.21.21.2.m1.1.1.1.1.1.3.3.2.cmml" type="integer" xref="S1.T1.21.21.2.m1.1.1.1.1.1.3.3.2">1</cn><apply id="S1.T1.21.21.2.m1.1.1.1.1.1.3.3.3.cmml" xref="S1.T1.21.21.2.m1.1.1.1.1.1.3.3.3"><divide id="S1.T1.21.21.2.m1.1.1.1.1.1.3.3.3.1.cmml" xref="S1.T1.21.21.2.m1.1.1.1.1.1.3.3.3"></divide><cn id="S1.T1.21.21.2.m1.1.1.1.1.1.3.3.3.2.cmml" type="integer" xref="S1.T1.21.21.2.m1.1.1.1.1.1.3.3.3.2">1</cn><ci id="S1.T1.21.21.2.m1.1.1.1.1.1.3.3.3.3.cmml" xref="S1.T1.21.21.2.m1.1.1.1.1.1.3.3.3.3">𝑡</ci></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.T1.21.21.2.m1.1c">\Omega(kn+n^{1+\frac{1}{t}})</annotation><annotation encoding="application/x-llamapun" id="S1.T1.21.21.2.m1.1d">roman_Ω ( italic_k italic_n + italic_n start_POSTSUPERSCRIPT 1 + divide start_ARG 1 end_ARG start_ARG italic_t end_ARG end_POSTSUPERSCRIPT )</annotation></semantics></math> bits <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.00712v1#bib.bibx54" title="">JKMV24</a>]</cite> </td> <td class="ltx_td ltx_align_center ltx_border_bb" id="S1.T1.21.21.4" style="padding-top:2.5pt;padding-bottom:2.5pt;">lower bound</td> </tr> </tbody> </table> <figcaption class="ltx_caption ltx_centering"><span class="ltx_tag ltx_tag_table"><span class="ltx_text" id="S1.T1.23.1.1" style="font-size:90%;">Table 1</span>: </span><span class="ltx_text" id="S1.T1.24.2" style="font-size:90%;">Summary of results for edge-connectivity network design. </span></figcaption> </figure> <figure class="ltx_table" id="S1.T2"> <table class="ltx_tabular ltx_centering ltx_guessed_headers ltx_align_middle" id="S1.T2.27"> <thead class="ltx_thead"> <tr class="ltx_tr" id="S1.T2.27.28.1"> <th class="ltx_td ltx_align_center ltx_th ltx_th_column ltx_th_row ltx_border_r ltx_border_tt" id="S1.T2.27.28.1.1" style="padding-top:2.5pt;padding-bottom:2.5pt;"><span class="ltx_text ltx_font_bold" id="S1.T2.27.28.1.1.1">Problem</span></th> <th class="ltx_td ltx_align_center ltx_th ltx_th_column ltx_border_r ltx_border_tt" id="S1.T2.27.28.1.2" style="padding-top:2.5pt;padding-bottom:2.5pt;"><span class="ltx_text ltx_font_bold" id="S1.T2.27.28.1.2.1">Approx.</span></th> <th class="ltx_td ltx_align_center ltx_th ltx_th_column ltx_border_r ltx_border_tt" id="S1.T2.27.28.1.3" style="padding-top:2.5pt;padding-bottom:2.5pt;"><span class="ltx_text ltx_font_bold" id="S1.T2.27.28.1.3.1">Space</span></th> <th class="ltx_td ltx_align_center ltx_th ltx_th_column ltx_border_tt" id="S1.T2.27.28.1.4" style="padding-top:2.5pt;padding-bottom:2.5pt;"><span class="ltx_text ltx_font_bold" id="S1.T2.27.28.1.4.1">Note</span></th> </tr> </thead> <tbody class="ltx_tbody"> <tr class="ltx_tr" id="S1.T2.4.4"> <th class="ltx_td ltx_align_center ltx_th ltx_th_row ltx_border_r ltx_border_tt" id="S1.T2.1.1.1" rowspan="4" style="padding-top:2.5pt;padding-bottom:2.5pt;"><span class="ltx_text" id="S1.T2.1.1.1.1"><math alttext="k" class="ltx_Math" display="inline" id="S1.T2.1.1.1.1.m1.1"><semantics id="S1.T2.1.1.1.1.m1.1a"><mi id="S1.T2.1.1.1.1.m1.1.1" xref="S1.T2.1.1.1.1.m1.1.1.cmml">k</mi><annotation-xml encoding="MathML-Content" id="S1.T2.1.1.1.1.m1.1b"><ci id="S1.T2.1.1.1.1.m1.1.1.cmml" xref="S1.T2.1.1.1.1.m1.1.1">𝑘</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.T2.1.1.1.1.m1.1c">k</annotation><annotation encoding="application/x-llamapun" id="S1.T2.1.1.1.1.m1.1d">italic_k</annotation></semantics></math>-VC-CAP</span></th> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_tt" id="S1.T2.2.2.2" style="padding-top:2.5pt;padding-bottom:2.5pt;"><math alttext="O(t)" class="ltx_Math" display="inline" id="S1.T2.2.2.2.m1.1"><semantics id="S1.T2.2.2.2.m1.1a"><mrow id="S1.T2.2.2.2.m1.1.2" xref="S1.T2.2.2.2.m1.1.2.cmml"><mi id="S1.T2.2.2.2.m1.1.2.2" xref="S1.T2.2.2.2.m1.1.2.2.cmml">O</mi><mo id="S1.T2.2.2.2.m1.1.2.1" xref="S1.T2.2.2.2.m1.1.2.1.cmml">⁢</mo><mrow id="S1.T2.2.2.2.m1.1.2.3.2" xref="S1.T2.2.2.2.m1.1.2.cmml"><mo id="S1.T2.2.2.2.m1.1.2.3.2.1" stretchy="false" xref="S1.T2.2.2.2.m1.1.2.cmml">(</mo><mi id="S1.T2.2.2.2.m1.1.1" xref="S1.T2.2.2.2.m1.1.1.cmml">t</mi><mo id="S1.T2.2.2.2.m1.1.2.3.2.2" stretchy="false" xref="S1.T2.2.2.2.m1.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.T2.2.2.2.m1.1b"><apply id="S1.T2.2.2.2.m1.1.2.cmml" xref="S1.T2.2.2.2.m1.1.2"><times id="S1.T2.2.2.2.m1.1.2.1.cmml" xref="S1.T2.2.2.2.m1.1.2.1"></times><ci id="S1.T2.2.2.2.m1.1.2.2.cmml" xref="S1.T2.2.2.2.m1.1.2.2">𝑂</ci><ci id="S1.T2.2.2.2.m1.1.1.cmml" xref="S1.T2.2.2.2.m1.1.1">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.T2.2.2.2.m1.1c">O(t)</annotation><annotation encoding="application/x-llamapun" id="S1.T2.2.2.2.m1.1d">italic_O ( italic_t )</annotation></semantics></math></td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_tt" id="S1.T2.3.3.3" style="padding-top:2.5pt;padding-bottom:2.5pt;"> <math alttext="\tilde{O}(k^{1-\frac{1}{t}}n^{1+\frac{1}{t}})" class="ltx_Math" display="inline" id="S1.T2.3.3.3.m1.1"><semantics id="S1.T2.3.3.3.m1.1a"><mrow id="S1.T2.3.3.3.m1.1.1" xref="S1.T2.3.3.3.m1.1.1.cmml"><mover accent="true" id="S1.T2.3.3.3.m1.1.1.3" xref="S1.T2.3.3.3.m1.1.1.3.cmml"><mi id="S1.T2.3.3.3.m1.1.1.3.2" xref="S1.T2.3.3.3.m1.1.1.3.2.cmml">O</mi><mo id="S1.T2.3.3.3.m1.1.1.3.1" xref="S1.T2.3.3.3.m1.1.1.3.1.cmml">~</mo></mover><mo id="S1.T2.3.3.3.m1.1.1.2" xref="S1.T2.3.3.3.m1.1.1.2.cmml">⁢</mo><mrow id="S1.T2.3.3.3.m1.1.1.1.1" 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xref="S1.T2.3.3.3.m1.1.1.1.1.1.1.cmml">⁢</mo><msup id="S1.T2.3.3.3.m1.1.1.1.1.1.3" xref="S1.T2.3.3.3.m1.1.1.1.1.1.3.cmml"><mi id="S1.T2.3.3.3.m1.1.1.1.1.1.3.2" xref="S1.T2.3.3.3.m1.1.1.1.1.1.3.2.cmml">n</mi><mrow id="S1.T2.3.3.3.m1.1.1.1.1.1.3.3" xref="S1.T2.3.3.3.m1.1.1.1.1.1.3.3.cmml"><mn id="S1.T2.3.3.3.m1.1.1.1.1.1.3.3.2" xref="S1.T2.3.3.3.m1.1.1.1.1.1.3.3.2.cmml">1</mn><mo id="S1.T2.3.3.3.m1.1.1.1.1.1.3.3.1" xref="S1.T2.3.3.3.m1.1.1.1.1.1.3.3.1.cmml">+</mo><mfrac id="S1.T2.3.3.3.m1.1.1.1.1.1.3.3.3" xref="S1.T2.3.3.3.m1.1.1.1.1.1.3.3.3.cmml"><mn id="S1.T2.3.3.3.m1.1.1.1.1.1.3.3.3.2" xref="S1.T2.3.3.3.m1.1.1.1.1.1.3.3.3.2.cmml">1</mn><mi id="S1.T2.3.3.3.m1.1.1.1.1.1.3.3.3.3" xref="S1.T2.3.3.3.m1.1.1.1.1.1.3.3.3.3.cmml">t</mi></mfrac></mrow></msup></mrow><mo id="S1.T2.3.3.3.m1.1.1.1.1.3" stretchy="false" xref="S1.T2.3.3.3.m1.1.1.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.T2.3.3.3.m1.1b"><apply id="S1.T2.3.3.3.m1.1.1.cmml" 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xref="S1.T2.3.3.3.m1.1.1.1.1.1.3.3.2">1</cn><apply id="S1.T2.3.3.3.m1.1.1.1.1.1.3.3.3.cmml" xref="S1.T2.3.3.3.m1.1.1.1.1.1.3.3.3"><divide id="S1.T2.3.3.3.m1.1.1.1.1.1.3.3.3.1.cmml" xref="S1.T2.3.3.3.m1.1.1.1.1.1.3.3.3"></divide><cn id="S1.T2.3.3.3.m1.1.1.1.1.1.3.3.3.2.cmml" type="integer" xref="S1.T2.3.3.3.m1.1.1.1.1.1.3.3.3.2">1</cn><ci id="S1.T2.3.3.3.m1.1.1.1.1.1.3.3.3.3.cmml" xref="S1.T2.3.3.3.m1.1.1.1.1.1.3.3.3.3">𝑡</ci></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.T2.3.3.3.m1.1c">\tilde{O}(k^{1-\frac{1}{t}}n^{1+\frac{1}{t}})</annotation><annotation encoding="application/x-llamapun" id="S1.T2.3.3.3.m1.1d">over~ start_ARG italic_O end_ARG ( italic_k start_POSTSUPERSCRIPT 1 - divide start_ARG 1 end_ARG start_ARG italic_t end_ARG end_POSTSUPERSCRIPT italic_n start_POSTSUPERSCRIPT 1 + divide start_ARG 1 end_ARG start_ARG italic_t end_ARG end_POSTSUPERSCRIPT )</annotation></semantics></math> <span class="ltx_text" id="S1.T2.3.3.3.1" style="color:#0000FF;">[Here]</span> </td> <td class="ltx_td ltx_align_center ltx_border_tt" id="S1.T2.4.4.4" style="padding-top:2.5pt;padding-bottom:2.5pt;">fully streaming (<math alttext="n=\Omega(k^{3})" class="ltx_Math" display="inline" id="S1.T2.4.4.4.m1.1"><semantics id="S1.T2.4.4.4.m1.1a"><mrow id="S1.T2.4.4.4.m1.1.1" xref="S1.T2.4.4.4.m1.1.1.cmml"><mi id="S1.T2.4.4.4.m1.1.1.3" xref="S1.T2.4.4.4.m1.1.1.3.cmml">n</mi><mo id="S1.T2.4.4.4.m1.1.1.2" xref="S1.T2.4.4.4.m1.1.1.2.cmml">=</mo><mrow id="S1.T2.4.4.4.m1.1.1.1" xref="S1.T2.4.4.4.m1.1.1.1.cmml"><mi id="S1.T2.4.4.4.m1.1.1.1.3" mathvariant="normal" xref="S1.T2.4.4.4.m1.1.1.1.3.cmml">Ω</mi><mo id="S1.T2.4.4.4.m1.1.1.1.2" xref="S1.T2.4.4.4.m1.1.1.1.2.cmml">⁢</mo><mrow id="S1.T2.4.4.4.m1.1.1.1.1.1" xref="S1.T2.4.4.4.m1.1.1.1.1.1.1.cmml"><mo id="S1.T2.4.4.4.m1.1.1.1.1.1.2" stretchy="false" xref="S1.T2.4.4.4.m1.1.1.1.1.1.1.cmml">(</mo><msup id="S1.T2.4.4.4.m1.1.1.1.1.1.1" xref="S1.T2.4.4.4.m1.1.1.1.1.1.1.cmml"><mi 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id="S1.T2.4.4.4.m1.1.1.1.1.1.1.2.cmml" xref="S1.T2.4.4.4.m1.1.1.1.1.1.1.2">𝑘</ci><cn id="S1.T2.4.4.4.m1.1.1.1.1.1.1.3.cmml" type="integer" xref="S1.T2.4.4.4.m1.1.1.1.1.1.1.3">3</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.T2.4.4.4.m1.1c">n=\Omega(k^{3})</annotation><annotation encoding="application/x-llamapun" id="S1.T2.4.4.4.m1.1d">italic_n = roman_Ω ( italic_k start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT )</annotation></semantics></math>)</td> </tr> <tr class="ltx_tr" id="S1.T2.6.6"> <td class="ltx_td ltx_align_center ltx_border_r" id="S1.T2.5.5.1" style="padding-top:2.5pt;padding-bottom:2.5pt;"><math alttext="O(t)" class="ltx_Math" display="inline" id="S1.T2.5.5.1.m1.1"><semantics id="S1.T2.5.5.1.m1.1a"><mrow id="S1.T2.5.5.1.m1.1.2" xref="S1.T2.5.5.1.m1.1.2.cmml"><mi id="S1.T2.5.5.1.m1.1.2.2" xref="S1.T2.5.5.1.m1.1.2.2.cmml">O</mi><mo id="S1.T2.5.5.1.m1.1.2.1" xref="S1.T2.5.5.1.m1.1.2.1.cmml">⁢</mo><mrow id="S1.T2.5.5.1.m1.1.2.3.2" xref="S1.T2.5.5.1.m1.1.2.cmml"><mo id="S1.T2.5.5.1.m1.1.2.3.2.1" stretchy="false" xref="S1.T2.5.5.1.m1.1.2.cmml">(</mo><mi id="S1.T2.5.5.1.m1.1.1" xref="S1.T2.5.5.1.m1.1.1.cmml">t</mi><mo id="S1.T2.5.5.1.m1.1.2.3.2.2" stretchy="false" xref="S1.T2.5.5.1.m1.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.T2.5.5.1.m1.1b"><apply id="S1.T2.5.5.1.m1.1.2.cmml" xref="S1.T2.5.5.1.m1.1.2"><times id="S1.T2.5.5.1.m1.1.2.1.cmml" xref="S1.T2.5.5.1.m1.1.2.1"></times><ci id="S1.T2.5.5.1.m1.1.2.2.cmml" xref="S1.T2.5.5.1.m1.1.2.2">𝑂</ci><ci id="S1.T2.5.5.1.m1.1.1.cmml" xref="S1.T2.5.5.1.m1.1.1">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.T2.5.5.1.m1.1c">O(t)</annotation><annotation encoding="application/x-llamapun" id="S1.T2.5.5.1.m1.1d">italic_O ( italic_t )</annotation></semantics></math></td> <td class="ltx_td ltx_align_center ltx_border_r" id="S1.T2.6.6.2" style="padding-top:2.5pt;padding-bottom:2.5pt;"> <math alttext="\Omega(n^{1+\frac{1}{t}})" class="ltx_Math" display="inline" id="S1.T2.6.6.2.m1.1"><semantics id="S1.T2.6.6.2.m1.1a"><mrow id="S1.T2.6.6.2.m1.1.1" xref="S1.T2.6.6.2.m1.1.1.cmml"><mi id="S1.T2.6.6.2.m1.1.1.3" mathvariant="normal" xref="S1.T2.6.6.2.m1.1.1.3.cmml">Ω</mi><mo id="S1.T2.6.6.2.m1.1.1.2" xref="S1.T2.6.6.2.m1.1.1.2.cmml">⁢</mo><mrow id="S1.T2.6.6.2.m1.1.1.1.1" xref="S1.T2.6.6.2.m1.1.1.1.1.1.cmml"><mo id="S1.T2.6.6.2.m1.1.1.1.1.2" stretchy="false" xref="S1.T2.6.6.2.m1.1.1.1.1.1.cmml">(</mo><msup id="S1.T2.6.6.2.m1.1.1.1.1.1" xref="S1.T2.6.6.2.m1.1.1.1.1.1.cmml"><mi id="S1.T2.6.6.2.m1.1.1.1.1.1.2" xref="S1.T2.6.6.2.m1.1.1.1.1.1.2.cmml">n</mi><mrow id="S1.T2.6.6.2.m1.1.1.1.1.1.3" xref="S1.T2.6.6.2.m1.1.1.1.1.1.3.cmml"><mn id="S1.T2.6.6.2.m1.1.1.1.1.1.3.2" xref="S1.T2.6.6.2.m1.1.1.1.1.1.3.2.cmml">1</mn><mo id="S1.T2.6.6.2.m1.1.1.1.1.1.3.1" xref="S1.T2.6.6.2.m1.1.1.1.1.1.3.1.cmml">+</mo><mfrac id="S1.T2.6.6.2.m1.1.1.1.1.1.3.3" xref="S1.T2.6.6.2.m1.1.1.1.1.1.3.3.cmml"><mn id="S1.T2.6.6.2.m1.1.1.1.1.1.3.3.2" xref="S1.T2.6.6.2.m1.1.1.1.1.1.3.3.2.cmml">1</mn><mi id="S1.T2.6.6.2.m1.1.1.1.1.1.3.3.3" xref="S1.T2.6.6.2.m1.1.1.1.1.1.3.3.3.cmml">t</mi></mfrac></mrow></msup><mo id="S1.T2.6.6.2.m1.1.1.1.1.3" stretchy="false" xref="S1.T2.6.6.2.m1.1.1.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.T2.6.6.2.m1.1b"><apply id="S1.T2.6.6.2.m1.1.1.cmml" xref="S1.T2.6.6.2.m1.1.1"><times id="S1.T2.6.6.2.m1.1.1.2.cmml" xref="S1.T2.6.6.2.m1.1.1.2"></times><ci id="S1.T2.6.6.2.m1.1.1.3.cmml" xref="S1.T2.6.6.2.m1.1.1.3">Ω</ci><apply id="S1.T2.6.6.2.m1.1.1.1.1.1.cmml" xref="S1.T2.6.6.2.m1.1.1.1.1"><csymbol cd="ambiguous" id="S1.T2.6.6.2.m1.1.1.1.1.1.1.cmml" xref="S1.T2.6.6.2.m1.1.1.1.1">superscript</csymbol><ci id="S1.T2.6.6.2.m1.1.1.1.1.1.2.cmml" xref="S1.T2.6.6.2.m1.1.1.1.1.1.2">𝑛</ci><apply id="S1.T2.6.6.2.m1.1.1.1.1.1.3.cmml" xref="S1.T2.6.6.2.m1.1.1.1.1.1.3"><plus id="S1.T2.6.6.2.m1.1.1.1.1.1.3.1.cmml" xref="S1.T2.6.6.2.m1.1.1.1.1.1.3.1"></plus><cn id="S1.T2.6.6.2.m1.1.1.1.1.1.3.2.cmml" type="integer" xref="S1.T2.6.6.2.m1.1.1.1.1.1.3.2">1</cn><apply id="S1.T2.6.6.2.m1.1.1.1.1.1.3.3.cmml" xref="S1.T2.6.6.2.m1.1.1.1.1.1.3.3"><divide id="S1.T2.6.6.2.m1.1.1.1.1.1.3.3.1.cmml" xref="S1.T2.6.6.2.m1.1.1.1.1.1.3.3"></divide><cn id="S1.T2.6.6.2.m1.1.1.1.1.1.3.3.2.cmml" type="integer" xref="S1.T2.6.6.2.m1.1.1.1.1.1.3.3.2">1</cn><ci id="S1.T2.6.6.2.m1.1.1.1.1.1.3.3.3.cmml" xref="S1.T2.6.6.2.m1.1.1.1.1.1.3.3.3">𝑡</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.T2.6.6.2.m1.1c">\Omega(n^{1+\frac{1}{t}})</annotation><annotation encoding="application/x-llamapun" id="S1.T2.6.6.2.m1.1d">roman_Ω ( italic_n start_POSTSUPERSCRIPT 1 + divide start_ARG 1 end_ARG start_ARG italic_t end_ARG end_POSTSUPERSCRIPT )</annotation></semantics></math> bits <span class="ltx_text" id="S1.T2.6.6.2.1" style="color:#0000FF;">[Here]</span> </td> <td class="ltx_td ltx_align_center" id="S1.T2.6.6.3" style="padding-top:2.5pt;padding-bottom:2.5pt;">fully streaming lower bound</td> </tr> <tr class="ltx_tr" id="S1.T2.9.9"> <td class="ltx_td ltx_align_center ltx_border_r" id="S1.T2.7.7.1" style="padding-top:2.5pt;padding-bottom:2.5pt;"><math alttext="3+\epsilon" class="ltx_Math" display="inline" id="S1.T2.7.7.1.m1.1"><semantics id="S1.T2.7.7.1.m1.1a"><mrow id="S1.T2.7.7.1.m1.1.1" xref="S1.T2.7.7.1.m1.1.1.cmml"><mn id="S1.T2.7.7.1.m1.1.1.2" xref="S1.T2.7.7.1.m1.1.1.2.cmml">3</mn><mo id="S1.T2.7.7.1.m1.1.1.1" xref="S1.T2.7.7.1.m1.1.1.1.cmml">+</mo><mi id="S1.T2.7.7.1.m1.1.1.3" xref="S1.T2.7.7.1.m1.1.1.3.cmml">ϵ</mi></mrow><annotation-xml encoding="MathML-Content" id="S1.T2.7.7.1.m1.1b"><apply id="S1.T2.7.7.1.m1.1.1.cmml" xref="S1.T2.7.7.1.m1.1.1"><plus id="S1.T2.7.7.1.m1.1.1.1.cmml" xref="S1.T2.7.7.1.m1.1.1.1"></plus><cn id="S1.T2.7.7.1.m1.1.1.2.cmml" type="integer" xref="S1.T2.7.7.1.m1.1.1.2">3</cn><ci id="S1.T2.7.7.1.m1.1.1.3.cmml" xref="S1.T2.7.7.1.m1.1.1.3">italic-ϵ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.T2.7.7.1.m1.1c">3+\epsilon</annotation><annotation encoding="application/x-llamapun" id="S1.T2.7.7.1.m1.1d">3 + italic_ϵ</annotation></semantics></math></td> <td class="ltx_td ltx_align_center ltx_border_r" id="S1.T2.8.8.2" style="padding-top:2.5pt;padding-bottom:2.5pt;"> <math alttext="\tilde{O}(n/\epsilon)" class="ltx_Math" display="inline" id="S1.T2.8.8.2.m1.1"><semantics id="S1.T2.8.8.2.m1.1a"><mrow id="S1.T2.8.8.2.m1.1.1" xref="S1.T2.8.8.2.m1.1.1.cmml"><mover accent="true" id="S1.T2.8.8.2.m1.1.1.3" xref="S1.T2.8.8.2.m1.1.1.3.cmml"><mi id="S1.T2.8.8.2.m1.1.1.3.2" xref="S1.T2.8.8.2.m1.1.1.3.2.cmml">O</mi><mo id="S1.T2.8.8.2.m1.1.1.3.1" xref="S1.T2.8.8.2.m1.1.1.3.1.cmml">~</mo></mover><mo id="S1.T2.8.8.2.m1.1.1.2" xref="S1.T2.8.8.2.m1.1.1.2.cmml">⁢</mo><mrow id="S1.T2.8.8.2.m1.1.1.1.1" xref="S1.T2.8.8.2.m1.1.1.1.1.1.cmml"><mo id="S1.T2.8.8.2.m1.1.1.1.1.2" stretchy="false" xref="S1.T2.8.8.2.m1.1.1.1.1.1.cmml">(</mo><mrow id="S1.T2.8.8.2.m1.1.1.1.1.1" xref="S1.T2.8.8.2.m1.1.1.1.1.1.cmml"><mi id="S1.T2.8.8.2.m1.1.1.1.1.1.2" xref="S1.T2.8.8.2.m1.1.1.1.1.1.2.cmml">n</mi><mo id="S1.T2.8.8.2.m1.1.1.1.1.1.1" xref="S1.T2.8.8.2.m1.1.1.1.1.1.1.cmml">/</mo><mi id="S1.T2.8.8.2.m1.1.1.1.1.1.3" xref="S1.T2.8.8.2.m1.1.1.1.1.1.3.cmml">ϵ</mi></mrow><mo id="S1.T2.8.8.2.m1.1.1.1.1.3" stretchy="false" xref="S1.T2.8.8.2.m1.1.1.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.T2.8.8.2.m1.1b"><apply id="S1.T2.8.8.2.m1.1.1.cmml" xref="S1.T2.8.8.2.m1.1.1"><times id="S1.T2.8.8.2.m1.1.1.2.cmml" xref="S1.T2.8.8.2.m1.1.1.2"></times><apply id="S1.T2.8.8.2.m1.1.1.3.cmml" xref="S1.T2.8.8.2.m1.1.1.3"><ci id="S1.T2.8.8.2.m1.1.1.3.1.cmml" xref="S1.T2.8.8.2.m1.1.1.3.1">~</ci><ci id="S1.T2.8.8.2.m1.1.1.3.2.cmml" xref="S1.T2.8.8.2.m1.1.1.3.2">𝑂</ci></apply><apply id="S1.T2.8.8.2.m1.1.1.1.1.1.cmml" xref="S1.T2.8.8.2.m1.1.1.1.1"><divide id="S1.T2.8.8.2.m1.1.1.1.1.1.1.cmml" xref="S1.T2.8.8.2.m1.1.1.1.1.1.1"></divide><ci id="S1.T2.8.8.2.m1.1.1.1.1.1.2.cmml" xref="S1.T2.8.8.2.m1.1.1.1.1.1.2">𝑛</ci><ci id="S1.T2.8.8.2.m1.1.1.1.1.1.3.cmml" xref="S1.T2.8.8.2.m1.1.1.1.1.1.3">italic-ϵ</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.T2.8.8.2.m1.1c">\tilde{O}(n/\epsilon)</annotation><annotation encoding="application/x-llamapun" id="S1.T2.8.8.2.m1.1d">over~ start_ARG italic_O end_ARG ( italic_n / italic_ϵ )</annotation></semantics></math> <span class="ltx_text" id="S1.T2.8.8.2.1" style="color:#0000FF;">[Here]</span> </td> <td class="ltx_td ltx_align_center" id="S1.T2.9.9.3" style="padding-top:2.5pt;padding-bottom:2.5pt;"> <math alttext="k=1" class="ltx_Math" display="inline" id="S1.T2.9.9.3.m1.1"><semantics id="S1.T2.9.9.3.m1.1a"><mrow id="S1.T2.9.9.3.m1.1.1" xref="S1.T2.9.9.3.m1.1.1.cmml"><mi id="S1.T2.9.9.3.m1.1.1.2" xref="S1.T2.9.9.3.m1.1.1.2.cmml">k</mi><mo id="S1.T2.9.9.3.m1.1.1.1" xref="S1.T2.9.9.3.m1.1.1.1.cmml">=</mo><mn id="S1.T2.9.9.3.m1.1.1.3" xref="S1.T2.9.9.3.m1.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S1.T2.9.9.3.m1.1b"><apply id="S1.T2.9.9.3.m1.1.1.cmml" xref="S1.T2.9.9.3.m1.1.1"><eq id="S1.T2.9.9.3.m1.1.1.1.cmml" xref="S1.T2.9.9.3.m1.1.1.1"></eq><ci id="S1.T2.9.9.3.m1.1.1.2.cmml" xref="S1.T2.9.9.3.m1.1.1.2">𝑘</ci><cn id="S1.T2.9.9.3.m1.1.1.3.cmml" type="integer" xref="S1.T2.9.9.3.m1.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.T2.9.9.3.m1.1c">k=1</annotation><annotation encoding="application/x-llamapun" id="S1.T2.9.9.3.m1.1d">italic_k = 1</annotation></semantics></math> (link arrival)</td> </tr> <tr class="ltx_tr" id="S1.T2.12.12"> <td class="ltx_td ltx_align_center ltx_border_r" id="S1.T2.10.10.1" style="padding-top:2.5pt;padding-bottom:2.5pt;"><math alttext="7+\epsilon" class="ltx_Math" display="inline" id="S1.T2.10.10.1.m1.1"><semantics id="S1.T2.10.10.1.m1.1a"><mrow id="S1.T2.10.10.1.m1.1.1" xref="S1.T2.10.10.1.m1.1.1.cmml"><mn id="S1.T2.10.10.1.m1.1.1.2" xref="S1.T2.10.10.1.m1.1.1.2.cmml">7</mn><mo id="S1.T2.10.10.1.m1.1.1.1" xref="S1.T2.10.10.1.m1.1.1.1.cmml">+</mo><mi id="S1.T2.10.10.1.m1.1.1.3" xref="S1.T2.10.10.1.m1.1.1.3.cmml">ϵ</mi></mrow><annotation-xml encoding="MathML-Content" id="S1.T2.10.10.1.m1.1b"><apply id="S1.T2.10.10.1.m1.1.1.cmml" xref="S1.T2.10.10.1.m1.1.1"><plus id="S1.T2.10.10.1.m1.1.1.1.cmml" xref="S1.T2.10.10.1.m1.1.1.1"></plus><cn id="S1.T2.10.10.1.m1.1.1.2.cmml" type="integer" xref="S1.T2.10.10.1.m1.1.1.2">7</cn><ci id="S1.T2.10.10.1.m1.1.1.3.cmml" xref="S1.T2.10.10.1.m1.1.1.3">italic-ϵ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.T2.10.10.1.m1.1c">7+\epsilon</annotation><annotation encoding="application/x-llamapun" id="S1.T2.10.10.1.m1.1d">7 + italic_ϵ</annotation></semantics></math></td> <td class="ltx_td ltx_align_center ltx_border_r" id="S1.T2.11.11.2" style="padding-top:2.5pt;padding-bottom:2.5pt;"> <math alttext="\tilde{O}(n/\epsilon)" class="ltx_Math" display="inline" id="S1.T2.11.11.2.m1.1"><semantics id="S1.T2.11.11.2.m1.1a"><mrow id="S1.T2.11.11.2.m1.1.1" xref="S1.T2.11.11.2.m1.1.1.cmml"><mover accent="true" id="S1.T2.11.11.2.m1.1.1.3" xref="S1.T2.11.11.2.m1.1.1.3.cmml"><mi id="S1.T2.11.11.2.m1.1.1.3.2" xref="S1.T2.11.11.2.m1.1.1.3.2.cmml">O</mi><mo id="S1.T2.11.11.2.m1.1.1.3.1" xref="S1.T2.11.11.2.m1.1.1.3.1.cmml">~</mo></mover><mo id="S1.T2.11.11.2.m1.1.1.2" xref="S1.T2.11.11.2.m1.1.1.2.cmml">⁢</mo><mrow id="S1.T2.11.11.2.m1.1.1.1.1" xref="S1.T2.11.11.2.m1.1.1.1.1.1.cmml"><mo id="S1.T2.11.11.2.m1.1.1.1.1.2" stretchy="false" xref="S1.T2.11.11.2.m1.1.1.1.1.1.cmml">(</mo><mrow id="S1.T2.11.11.2.m1.1.1.1.1.1" xref="S1.T2.11.11.2.m1.1.1.1.1.1.cmml"><mi id="S1.T2.11.11.2.m1.1.1.1.1.1.2" xref="S1.T2.11.11.2.m1.1.1.1.1.1.2.cmml">n</mi><mo id="S1.T2.11.11.2.m1.1.1.1.1.1.1" xref="S1.T2.11.11.2.m1.1.1.1.1.1.1.cmml">/</mo><mi id="S1.T2.11.11.2.m1.1.1.1.1.1.3" xref="S1.T2.11.11.2.m1.1.1.1.1.1.3.cmml">ϵ</mi></mrow><mo id="S1.T2.11.11.2.m1.1.1.1.1.3" stretchy="false" xref="S1.T2.11.11.2.m1.1.1.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.T2.11.11.2.m1.1b"><apply id="S1.T2.11.11.2.m1.1.1.cmml" xref="S1.T2.11.11.2.m1.1.1"><times id="S1.T2.11.11.2.m1.1.1.2.cmml" xref="S1.T2.11.11.2.m1.1.1.2"></times><apply id="S1.T2.11.11.2.m1.1.1.3.cmml" xref="S1.T2.11.11.2.m1.1.1.3"><ci id="S1.T2.11.11.2.m1.1.1.3.1.cmml" xref="S1.T2.11.11.2.m1.1.1.3.1">~</ci><ci id="S1.T2.11.11.2.m1.1.1.3.2.cmml" xref="S1.T2.11.11.2.m1.1.1.3.2">𝑂</ci></apply><apply id="S1.T2.11.11.2.m1.1.1.1.1.1.cmml" xref="S1.T2.11.11.2.m1.1.1.1.1"><divide id="S1.T2.11.11.2.m1.1.1.1.1.1.1.cmml" xref="S1.T2.11.11.2.m1.1.1.1.1.1.1"></divide><ci id="S1.T2.11.11.2.m1.1.1.1.1.1.2.cmml" xref="S1.T2.11.11.2.m1.1.1.1.1.1.2">𝑛</ci><ci id="S1.T2.11.11.2.m1.1.1.1.1.1.3.cmml" xref="S1.T2.11.11.2.m1.1.1.1.1.1.3">italic-ϵ</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.T2.11.11.2.m1.1c">\tilde{O}(n/\epsilon)</annotation><annotation encoding="application/x-llamapun" id="S1.T2.11.11.2.m1.1d">over~ start_ARG italic_O end_ARG ( italic_n / italic_ϵ )</annotation></semantics></math> <span class="ltx_text" id="S1.T2.11.11.2.1" style="color:#0000FF;">[Here]</span> </td> <td class="ltx_td ltx_align_center" id="S1.T2.12.12.3" style="padding-top:2.5pt;padding-bottom:2.5pt;"> <math alttext="k=2" class="ltx_Math" display="inline" id="S1.T2.12.12.3.m1.1"><semantics id="S1.T2.12.12.3.m1.1a"><mrow id="S1.T2.12.12.3.m1.1.1" xref="S1.T2.12.12.3.m1.1.1.cmml"><mi id="S1.T2.12.12.3.m1.1.1.2" xref="S1.T2.12.12.3.m1.1.1.2.cmml">k</mi><mo id="S1.T2.12.12.3.m1.1.1.1" xref="S1.T2.12.12.3.m1.1.1.1.cmml">=</mo><mn id="S1.T2.12.12.3.m1.1.1.3" xref="S1.T2.12.12.3.m1.1.1.3.cmml">2</mn></mrow><annotation-xml encoding="MathML-Content" id="S1.T2.12.12.3.m1.1b"><apply id="S1.T2.12.12.3.m1.1.1.cmml" xref="S1.T2.12.12.3.m1.1.1"><eq id="S1.T2.12.12.3.m1.1.1.1.cmml" xref="S1.T2.12.12.3.m1.1.1.1"></eq><ci id="S1.T2.12.12.3.m1.1.1.2.cmml" xref="S1.T2.12.12.3.m1.1.1.2">𝑘</ci><cn id="S1.T2.12.12.3.m1.1.1.3.cmml" type="integer" xref="S1.T2.12.12.3.m1.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.T2.12.12.3.m1.1c">k=2</annotation><annotation encoding="application/x-llamapun" id="S1.T2.12.12.3.m1.1d">italic_k = 2</annotation></semantics></math> (link arrival)</td> </tr> <tr class="ltx_tr" id="S1.T2.14.14"> <th class="ltx_td ltx_align_center ltx_th ltx_th_row ltx_border_r ltx_border_tt" id="S1.T2.14.14.3" rowspan="3" style="padding-top:2.5pt;padding-bottom:2.5pt;"><span class="ltx_text" id="S1.T2.14.14.3.1">VC-SNDP</span></th> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_tt" id="S1.T2.13.13.1" style="padding-top:2.5pt;padding-bottom:2.5pt;"><math alttext="2tk" class="ltx_Math" display="inline" id="S1.T2.13.13.1.m1.1"><semantics id="S1.T2.13.13.1.m1.1a"><mrow id="S1.T2.13.13.1.m1.1.1" xref="S1.T2.13.13.1.m1.1.1.cmml"><mn id="S1.T2.13.13.1.m1.1.1.2" xref="S1.T2.13.13.1.m1.1.1.2.cmml">2</mn><mo id="S1.T2.13.13.1.m1.1.1.1" xref="S1.T2.13.13.1.m1.1.1.1.cmml">⁢</mo><mi id="S1.T2.13.13.1.m1.1.1.3" xref="S1.T2.13.13.1.m1.1.1.3.cmml">t</mi><mo id="S1.T2.13.13.1.m1.1.1.1a" xref="S1.T2.13.13.1.m1.1.1.1.cmml">⁢</mo><mi id="S1.T2.13.13.1.m1.1.1.4" xref="S1.T2.13.13.1.m1.1.1.4.cmml">k</mi></mrow><annotation-xml encoding="MathML-Content" id="S1.T2.13.13.1.m1.1b"><apply id="S1.T2.13.13.1.m1.1.1.cmml" xref="S1.T2.13.13.1.m1.1.1"><times id="S1.T2.13.13.1.m1.1.1.1.cmml" xref="S1.T2.13.13.1.m1.1.1.1"></times><cn id="S1.T2.13.13.1.m1.1.1.2.cmml" type="integer" xref="S1.T2.13.13.1.m1.1.1.2">2</cn><ci id="S1.T2.13.13.1.m1.1.1.3.cmml" xref="S1.T2.13.13.1.m1.1.1.3">𝑡</ci><ci id="S1.T2.13.13.1.m1.1.1.4.cmml" xref="S1.T2.13.13.1.m1.1.1.4">𝑘</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.T2.13.13.1.m1.1c">2tk</annotation><annotation encoding="application/x-llamapun" id="S1.T2.13.13.1.m1.1d">2 italic_t italic_k</annotation></semantics></math></td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_tt" id="S1.T2.14.14.2" style="padding-top:2.5pt;padding-bottom:2.5pt;"> <math alttext="\tilde{O}(k^{1-\frac{1}{t}}n^{1+\frac{1}{t}})" class="ltx_Math" display="inline" id="S1.T2.14.14.2.m1.1"><semantics id="S1.T2.14.14.2.m1.1a"><mrow id="S1.T2.14.14.2.m1.1.1" xref="S1.T2.14.14.2.m1.1.1.cmml"><mover accent="true" id="S1.T2.14.14.2.m1.1.1.3" xref="S1.T2.14.14.2.m1.1.1.3.cmml"><mi id="S1.T2.14.14.2.m1.1.1.3.2" xref="S1.T2.14.14.2.m1.1.1.3.2.cmml">O</mi><mo id="S1.T2.14.14.2.m1.1.1.3.1" xref="S1.T2.14.14.2.m1.1.1.3.1.cmml">~</mo></mover><mo id="S1.T2.14.14.2.m1.1.1.2" xref="S1.T2.14.14.2.m1.1.1.2.cmml">⁢</mo><mrow id="S1.T2.14.14.2.m1.1.1.1.1" xref="S1.T2.14.14.2.m1.1.1.1.1.1.cmml"><mo id="S1.T2.14.14.2.m1.1.1.1.1.2" stretchy="false" xref="S1.T2.14.14.2.m1.1.1.1.1.1.cmml">(</mo><mrow id="S1.T2.14.14.2.m1.1.1.1.1.1" xref="S1.T2.14.14.2.m1.1.1.1.1.1.cmml"><msup id="S1.T2.14.14.2.m1.1.1.1.1.1.2" xref="S1.T2.14.14.2.m1.1.1.1.1.1.2.cmml"><mi id="S1.T2.14.14.2.m1.1.1.1.1.1.2.2" xref="S1.T2.14.14.2.m1.1.1.1.1.1.2.2.cmml">k</mi><mrow id="S1.T2.14.14.2.m1.1.1.1.1.1.2.3" xref="S1.T2.14.14.2.m1.1.1.1.1.1.2.3.cmml"><mn id="S1.T2.14.14.2.m1.1.1.1.1.1.2.3.2" xref="S1.T2.14.14.2.m1.1.1.1.1.1.2.3.2.cmml">1</mn><mo id="S1.T2.14.14.2.m1.1.1.1.1.1.2.3.1" xref="S1.T2.14.14.2.m1.1.1.1.1.1.2.3.1.cmml">−</mo><mfrac id="S1.T2.14.14.2.m1.1.1.1.1.1.2.3.3" xref="S1.T2.14.14.2.m1.1.1.1.1.1.2.3.3.cmml"><mn id="S1.T2.14.14.2.m1.1.1.1.1.1.2.3.3.2" xref="S1.T2.14.14.2.m1.1.1.1.1.1.2.3.3.2.cmml">1</mn><mi id="S1.T2.14.14.2.m1.1.1.1.1.1.2.3.3.3" xref="S1.T2.14.14.2.m1.1.1.1.1.1.2.3.3.3.cmml">t</mi></mfrac></mrow></msup><mo id="S1.T2.14.14.2.m1.1.1.1.1.1.1" xref="S1.T2.14.14.2.m1.1.1.1.1.1.1.cmml">⁢</mo><msup id="S1.T2.14.14.2.m1.1.1.1.1.1.3" xref="S1.T2.14.14.2.m1.1.1.1.1.1.3.cmml"><mi id="S1.T2.14.14.2.m1.1.1.1.1.1.3.2" xref="S1.T2.14.14.2.m1.1.1.1.1.1.3.2.cmml">n</mi><mrow id="S1.T2.14.14.2.m1.1.1.1.1.1.3.3" xref="S1.T2.14.14.2.m1.1.1.1.1.1.3.3.cmml"><mn id="S1.T2.14.14.2.m1.1.1.1.1.1.3.3.2" xref="S1.T2.14.14.2.m1.1.1.1.1.1.3.3.2.cmml">1</mn><mo id="S1.T2.14.14.2.m1.1.1.1.1.1.3.3.1" xref="S1.T2.14.14.2.m1.1.1.1.1.1.3.3.1.cmml">+</mo><mfrac id="S1.T2.14.14.2.m1.1.1.1.1.1.3.3.3" xref="S1.T2.14.14.2.m1.1.1.1.1.1.3.3.3.cmml"><mn id="S1.T2.14.14.2.m1.1.1.1.1.1.3.3.3.2" xref="S1.T2.14.14.2.m1.1.1.1.1.1.3.3.3.2.cmml">1</mn><mi id="S1.T2.14.14.2.m1.1.1.1.1.1.3.3.3.3" xref="S1.T2.14.14.2.m1.1.1.1.1.1.3.3.3.3.cmml">t</mi></mfrac></mrow></msup></mrow><mo id="S1.T2.14.14.2.m1.1.1.1.1.3" stretchy="false" xref="S1.T2.14.14.2.m1.1.1.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.T2.14.14.2.m1.1b"><apply id="S1.T2.14.14.2.m1.1.1.cmml" xref="S1.T2.14.14.2.m1.1.1"><times id="S1.T2.14.14.2.m1.1.1.2.cmml" xref="S1.T2.14.14.2.m1.1.1.2"></times><apply id="S1.T2.14.14.2.m1.1.1.3.cmml" xref="S1.T2.14.14.2.m1.1.1.3"><ci id="S1.T2.14.14.2.m1.1.1.3.1.cmml" xref="S1.T2.14.14.2.m1.1.1.3.1">~</ci><ci id="S1.T2.14.14.2.m1.1.1.3.2.cmml" xref="S1.T2.14.14.2.m1.1.1.3.2">𝑂</ci></apply><apply id="S1.T2.14.14.2.m1.1.1.1.1.1.cmml" xref="S1.T2.14.14.2.m1.1.1.1.1"><times id="S1.T2.14.14.2.m1.1.1.1.1.1.1.cmml" xref="S1.T2.14.14.2.m1.1.1.1.1.1.1"></times><apply id="S1.T2.14.14.2.m1.1.1.1.1.1.2.cmml" xref="S1.T2.14.14.2.m1.1.1.1.1.1.2"><csymbol cd="ambiguous" id="S1.T2.14.14.2.m1.1.1.1.1.1.2.1.cmml" xref="S1.T2.14.14.2.m1.1.1.1.1.1.2">superscript</csymbol><ci id="S1.T2.14.14.2.m1.1.1.1.1.1.2.2.cmml" xref="S1.T2.14.14.2.m1.1.1.1.1.1.2.2">𝑘</ci><apply id="S1.T2.14.14.2.m1.1.1.1.1.1.2.3.cmml" xref="S1.T2.14.14.2.m1.1.1.1.1.1.2.3"><minus id="S1.T2.14.14.2.m1.1.1.1.1.1.2.3.1.cmml" xref="S1.T2.14.14.2.m1.1.1.1.1.1.2.3.1"></minus><cn id="S1.T2.14.14.2.m1.1.1.1.1.1.2.3.2.cmml" type="integer" xref="S1.T2.14.14.2.m1.1.1.1.1.1.2.3.2">1</cn><apply id="S1.T2.14.14.2.m1.1.1.1.1.1.2.3.3.cmml" xref="S1.T2.14.14.2.m1.1.1.1.1.1.2.3.3"><divide id="S1.T2.14.14.2.m1.1.1.1.1.1.2.3.3.1.cmml" xref="S1.T2.14.14.2.m1.1.1.1.1.1.2.3.3"></divide><cn id="S1.T2.14.14.2.m1.1.1.1.1.1.2.3.3.2.cmml" type="integer" xref="S1.T2.14.14.2.m1.1.1.1.1.1.2.3.3.2">1</cn><ci id="S1.T2.14.14.2.m1.1.1.1.1.1.2.3.3.3.cmml" xref="S1.T2.14.14.2.m1.1.1.1.1.1.2.3.3.3">𝑡</ci></apply></apply></apply><apply id="S1.T2.14.14.2.m1.1.1.1.1.1.3.cmml" xref="S1.T2.14.14.2.m1.1.1.1.1.1.3"><csymbol cd="ambiguous" id="S1.T2.14.14.2.m1.1.1.1.1.1.3.1.cmml" xref="S1.T2.14.14.2.m1.1.1.1.1.1.3">superscript</csymbol><ci id="S1.T2.14.14.2.m1.1.1.1.1.1.3.2.cmml" xref="S1.T2.14.14.2.m1.1.1.1.1.1.3.2">𝑛</ci><apply id="S1.T2.14.14.2.m1.1.1.1.1.1.3.3.cmml" xref="S1.T2.14.14.2.m1.1.1.1.1.1.3.3"><plus id="S1.T2.14.14.2.m1.1.1.1.1.1.3.3.1.cmml" xref="S1.T2.14.14.2.m1.1.1.1.1.1.3.3.1"></plus><cn id="S1.T2.14.14.2.m1.1.1.1.1.1.3.3.2.cmml" type="integer" xref="S1.T2.14.14.2.m1.1.1.1.1.1.3.3.2">1</cn><apply id="S1.T2.14.14.2.m1.1.1.1.1.1.3.3.3.cmml" xref="S1.T2.14.14.2.m1.1.1.1.1.1.3.3.3"><divide id="S1.T2.14.14.2.m1.1.1.1.1.1.3.3.3.1.cmml" xref="S1.T2.14.14.2.m1.1.1.1.1.1.3.3.3"></divide><cn id="S1.T2.14.14.2.m1.1.1.1.1.1.3.3.3.2.cmml" type="integer" xref="S1.T2.14.14.2.m1.1.1.1.1.1.3.3.3.2">1</cn><ci id="S1.T2.14.14.2.m1.1.1.1.1.1.3.3.3.3.cmml" xref="S1.T2.14.14.2.m1.1.1.1.1.1.3.3.3.3">𝑡</ci></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.T2.14.14.2.m1.1c">\tilde{O}(k^{1-\frac{1}{t}}n^{1+\frac{1}{t}})</annotation><annotation encoding="application/x-llamapun" id="S1.T2.14.14.2.m1.1d">over~ start_ARG italic_O end_ARG ( italic_k start_POSTSUPERSCRIPT 1 - divide start_ARG 1 end_ARG start_ARG italic_t end_ARG end_POSTSUPERSCRIPT italic_n start_POSTSUPERSCRIPT 1 + divide start_ARG 1 end_ARG start_ARG italic_t end_ARG end_POSTSUPERSCRIPT )</annotation></semantics></math> <span class="ltx_text" id="S1.T2.14.14.2.1" style="color:#0000FF;">[Here]</span> </td> <td class="ltx_td ltx_border_tt" id="S1.T2.14.14.4" style="padding-top:2.5pt;padding-bottom:2.5pt;"></td> </tr> <tr class="ltx_tr" id="S1.T2.17.17"> <td class="ltx_td ltx_align_center ltx_border_r" id="S1.T2.15.15.1" style="padding-top:2.5pt;padding-bottom:2.5pt;"><math alttext="8t" class="ltx_Math" display="inline" id="S1.T2.15.15.1.m1.1"><semantics id="S1.T2.15.15.1.m1.1a"><mrow id="S1.T2.15.15.1.m1.1.1" xref="S1.T2.15.15.1.m1.1.1.cmml"><mn id="S1.T2.15.15.1.m1.1.1.2" xref="S1.T2.15.15.1.m1.1.1.2.cmml">8</mn><mo id="S1.T2.15.15.1.m1.1.1.1" xref="S1.T2.15.15.1.m1.1.1.1.cmml">⁢</mo><mi id="S1.T2.15.15.1.m1.1.1.3" xref="S1.T2.15.15.1.m1.1.1.3.cmml">t</mi></mrow><annotation-xml encoding="MathML-Content" id="S1.T2.15.15.1.m1.1b"><apply id="S1.T2.15.15.1.m1.1.1.cmml" xref="S1.T2.15.15.1.m1.1.1"><times id="S1.T2.15.15.1.m1.1.1.1.cmml" xref="S1.T2.15.15.1.m1.1.1.1"></times><cn id="S1.T2.15.15.1.m1.1.1.2.cmml" type="integer" xref="S1.T2.15.15.1.m1.1.1.2">8</cn><ci id="S1.T2.15.15.1.m1.1.1.3.cmml" xref="S1.T2.15.15.1.m1.1.1.3">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.T2.15.15.1.m1.1c">8t</annotation><annotation encoding="application/x-llamapun" id="S1.T2.15.15.1.m1.1d">8 italic_t</annotation></semantics></math></td> <td class="ltx_td ltx_align_center ltx_border_r" id="S1.T2.16.16.2" style="padding-top:2.5pt;padding-bottom:2.5pt;"> <math alttext="\tilde{O}(n^{1+\frac{1}{t}})" class="ltx_Math" display="inline" id="S1.T2.16.16.2.m1.1"><semantics id="S1.T2.16.16.2.m1.1a"><mrow id="S1.T2.16.16.2.m1.1.1" xref="S1.T2.16.16.2.m1.1.1.cmml"><mover accent="true" id="S1.T2.16.16.2.m1.1.1.3" xref="S1.T2.16.16.2.m1.1.1.3.cmml"><mi id="S1.T2.16.16.2.m1.1.1.3.2" xref="S1.T2.16.16.2.m1.1.1.3.2.cmml">O</mi><mo id="S1.T2.16.16.2.m1.1.1.3.1" xref="S1.T2.16.16.2.m1.1.1.3.1.cmml">~</mo></mover><mo id="S1.T2.16.16.2.m1.1.1.2" xref="S1.T2.16.16.2.m1.1.1.2.cmml">⁢</mo><mrow id="S1.T2.16.16.2.m1.1.1.1.1" xref="S1.T2.16.16.2.m1.1.1.1.1.1.cmml"><mo id="S1.T2.16.16.2.m1.1.1.1.1.2" stretchy="false" xref="S1.T2.16.16.2.m1.1.1.1.1.1.cmml">(</mo><msup id="S1.T2.16.16.2.m1.1.1.1.1.1" xref="S1.T2.16.16.2.m1.1.1.1.1.1.cmml"><mi id="S1.T2.16.16.2.m1.1.1.1.1.1.2" xref="S1.T2.16.16.2.m1.1.1.1.1.1.2.cmml">n</mi><mrow id="S1.T2.16.16.2.m1.1.1.1.1.1.3" xref="S1.T2.16.16.2.m1.1.1.1.1.1.3.cmml"><mn id="S1.T2.16.16.2.m1.1.1.1.1.1.3.2" xref="S1.T2.16.16.2.m1.1.1.1.1.1.3.2.cmml">1</mn><mo id="S1.T2.16.16.2.m1.1.1.1.1.1.3.1" xref="S1.T2.16.16.2.m1.1.1.1.1.1.3.1.cmml">+</mo><mfrac id="S1.T2.16.16.2.m1.1.1.1.1.1.3.3" xref="S1.T2.16.16.2.m1.1.1.1.1.1.3.3.cmml"><mn id="S1.T2.16.16.2.m1.1.1.1.1.1.3.3.2" xref="S1.T2.16.16.2.m1.1.1.1.1.1.3.3.2.cmml">1</mn><mi id="S1.T2.16.16.2.m1.1.1.1.1.1.3.3.3" xref="S1.T2.16.16.2.m1.1.1.1.1.1.3.3.3.cmml">t</mi></mfrac></mrow></msup><mo id="S1.T2.16.16.2.m1.1.1.1.1.3" stretchy="false" xref="S1.T2.16.16.2.m1.1.1.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.T2.16.16.2.m1.1b"><apply id="S1.T2.16.16.2.m1.1.1.cmml" xref="S1.T2.16.16.2.m1.1.1"><times id="S1.T2.16.16.2.m1.1.1.2.cmml" xref="S1.T2.16.16.2.m1.1.1.2"></times><apply id="S1.T2.16.16.2.m1.1.1.3.cmml" xref="S1.T2.16.16.2.m1.1.1.3"><ci id="S1.T2.16.16.2.m1.1.1.3.1.cmml" xref="S1.T2.16.16.2.m1.1.1.3.1">~</ci><ci id="S1.T2.16.16.2.m1.1.1.3.2.cmml" xref="S1.T2.16.16.2.m1.1.1.3.2">𝑂</ci></apply><apply id="S1.T2.16.16.2.m1.1.1.1.1.1.cmml" xref="S1.T2.16.16.2.m1.1.1.1.1"><csymbol cd="ambiguous" id="S1.T2.16.16.2.m1.1.1.1.1.1.1.cmml" xref="S1.T2.16.16.2.m1.1.1.1.1">superscript</csymbol><ci id="S1.T2.16.16.2.m1.1.1.1.1.1.2.cmml" xref="S1.T2.16.16.2.m1.1.1.1.1.1.2">𝑛</ci><apply id="S1.T2.16.16.2.m1.1.1.1.1.1.3.cmml" xref="S1.T2.16.16.2.m1.1.1.1.1.1.3"><plus id="S1.T2.16.16.2.m1.1.1.1.1.1.3.1.cmml" xref="S1.T2.16.16.2.m1.1.1.1.1.1.3.1"></plus><cn id="S1.T2.16.16.2.m1.1.1.1.1.1.3.2.cmml" type="integer" xref="S1.T2.16.16.2.m1.1.1.1.1.1.3.2">1</cn><apply id="S1.T2.16.16.2.m1.1.1.1.1.1.3.3.cmml" xref="S1.T2.16.16.2.m1.1.1.1.1.1.3.3"><divide id="S1.T2.16.16.2.m1.1.1.1.1.1.3.3.1.cmml" xref="S1.T2.16.16.2.m1.1.1.1.1.1.3.3"></divide><cn id="S1.T2.16.16.2.m1.1.1.1.1.1.3.3.2.cmml" type="integer" xref="S1.T2.16.16.2.m1.1.1.1.1.1.3.3.2">1</cn><ci id="S1.T2.16.16.2.m1.1.1.1.1.1.3.3.3.cmml" xref="S1.T2.16.16.2.m1.1.1.1.1.1.3.3.3">𝑡</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.T2.16.16.2.m1.1c">\tilde{O}(n^{1+\frac{1}{t}})</annotation><annotation encoding="application/x-llamapun" id="S1.T2.16.16.2.m1.1d">over~ start_ARG italic_O end_ARG ( italic_n start_POSTSUPERSCRIPT 1 + divide start_ARG 1 end_ARG start_ARG italic_t end_ARG end_POSTSUPERSCRIPT )</annotation></semantics></math> <span class="ltx_text" id="S1.T2.16.16.2.1" style="color:#0000FF;">[Here]</span> </td> <td class="ltx_td ltx_align_center" id="S1.T2.17.17.3" style="padding-top:2.5pt;padding-bottom:2.5pt;"> <math alttext="\{0,1,2\}" class="ltx_Math" display="inline" id="S1.T2.17.17.3.m1.3"><semantics id="S1.T2.17.17.3.m1.3a"><mrow id="S1.T2.17.17.3.m1.3.4.2" xref="S1.T2.17.17.3.m1.3.4.1.cmml"><mo id="S1.T2.17.17.3.m1.3.4.2.1" stretchy="false" xref="S1.T2.17.17.3.m1.3.4.1.cmml">{</mo><mn id="S1.T2.17.17.3.m1.1.1" xref="S1.T2.17.17.3.m1.1.1.cmml">0</mn><mo id="S1.T2.17.17.3.m1.3.4.2.2" xref="S1.T2.17.17.3.m1.3.4.1.cmml">,</mo><mn id="S1.T2.17.17.3.m1.2.2" xref="S1.T2.17.17.3.m1.2.2.cmml">1</mn><mo id="S1.T2.17.17.3.m1.3.4.2.3" xref="S1.T2.17.17.3.m1.3.4.1.cmml">,</mo><mn id="S1.T2.17.17.3.m1.3.3" xref="S1.T2.17.17.3.m1.3.3.cmml">2</mn><mo id="S1.T2.17.17.3.m1.3.4.2.4" stretchy="false" xref="S1.T2.17.17.3.m1.3.4.1.cmml">}</mo></mrow><annotation-xml encoding="MathML-Content" id="S1.T2.17.17.3.m1.3b"><set id="S1.T2.17.17.3.m1.3.4.1.cmml" xref="S1.T2.17.17.3.m1.3.4.2"><cn id="S1.T2.17.17.3.m1.1.1.cmml" type="integer" xref="S1.T2.17.17.3.m1.1.1">0</cn><cn id="S1.T2.17.17.3.m1.2.2.cmml" type="integer" xref="S1.T2.17.17.3.m1.2.2">1</cn><cn id="S1.T2.17.17.3.m1.3.3.cmml" type="integer" xref="S1.T2.17.17.3.m1.3.3">2</cn></set></annotation-xml><annotation encoding="application/x-tex" id="S1.T2.17.17.3.m1.3c">\{0,1,2\}</annotation><annotation encoding="application/x-llamapun" id="S1.T2.17.17.3.m1.3d">{ 0 , 1 , 2 }</annotation></semantics></math>-VC-SNDP</td> </tr> <tr class="ltx_tr" id="S1.T2.19.19"> <td class="ltx_td ltx_align_center ltx_border_r" id="S1.T2.18.18.1" style="padding-top:2.5pt;padding-bottom:2.5pt;"><math alttext="O(t)" class="ltx_Math" display="inline" id="S1.T2.18.18.1.m1.1"><semantics id="S1.T2.18.18.1.m1.1a"><mrow id="S1.T2.18.18.1.m1.1.2" xref="S1.T2.18.18.1.m1.1.2.cmml"><mi id="S1.T2.18.18.1.m1.1.2.2" xref="S1.T2.18.18.1.m1.1.2.2.cmml">O</mi><mo id="S1.T2.18.18.1.m1.1.2.1" xref="S1.T2.18.18.1.m1.1.2.1.cmml">⁢</mo><mrow id="S1.T2.18.18.1.m1.1.2.3.2" xref="S1.T2.18.18.1.m1.1.2.cmml"><mo id="S1.T2.18.18.1.m1.1.2.3.2.1" stretchy="false" xref="S1.T2.18.18.1.m1.1.2.cmml">(</mo><mi id="S1.T2.18.18.1.m1.1.1" xref="S1.T2.18.18.1.m1.1.1.cmml">t</mi><mo id="S1.T2.18.18.1.m1.1.2.3.2.2" stretchy="false" xref="S1.T2.18.18.1.m1.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.T2.18.18.1.m1.1b"><apply id="S1.T2.18.18.1.m1.1.2.cmml" xref="S1.T2.18.18.1.m1.1.2"><times id="S1.T2.18.18.1.m1.1.2.1.cmml" xref="S1.T2.18.18.1.m1.1.2.1"></times><ci id="S1.T2.18.18.1.m1.1.2.2.cmml" xref="S1.T2.18.18.1.m1.1.2.2">𝑂</ci><ci id="S1.T2.18.18.1.m1.1.1.cmml" xref="S1.T2.18.18.1.m1.1.1">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.T2.18.18.1.m1.1c">O(t)</annotation><annotation encoding="application/x-llamapun" id="S1.T2.18.18.1.m1.1d">italic_O ( italic_t )</annotation></semantics></math></td> <td class="ltx_td ltx_align_center ltx_border_r" id="S1.T2.19.19.2" style="padding-top:2.5pt;padding-bottom:2.5pt;"> <math 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id="S1.T2.19.19.2.m1.1b"><apply id="S1.T2.19.19.2.m1.1.1.cmml" xref="S1.T2.19.19.2.m1.1.1"><times id="S1.T2.19.19.2.m1.1.1.2.cmml" xref="S1.T2.19.19.2.m1.1.1.2"></times><ci id="S1.T2.19.19.2.m1.1.1.3.cmml" xref="S1.T2.19.19.2.m1.1.1.3">Ω</ci><apply id="S1.T2.19.19.2.m1.1.1.1.1.1.cmml" xref="S1.T2.19.19.2.m1.1.1.1.1"><plus id="S1.T2.19.19.2.m1.1.1.1.1.1.1.cmml" xref="S1.T2.19.19.2.m1.1.1.1.1.1.1"></plus><apply id="S1.T2.19.19.2.m1.1.1.1.1.1.2.cmml" xref="S1.T2.19.19.2.m1.1.1.1.1.1.2"><times id="S1.T2.19.19.2.m1.1.1.1.1.1.2.1.cmml" xref="S1.T2.19.19.2.m1.1.1.1.1.1.2.1"></times><ci id="S1.T2.19.19.2.m1.1.1.1.1.1.2.2.cmml" xref="S1.T2.19.19.2.m1.1.1.1.1.1.2.2">𝑛</ci><ci id="S1.T2.19.19.2.m1.1.1.1.1.1.2.3.cmml" xref="S1.T2.19.19.2.m1.1.1.1.1.1.2.3">𝑘</ci></apply><apply id="S1.T2.19.19.2.m1.1.1.1.1.1.3.cmml" xref="S1.T2.19.19.2.m1.1.1.1.1.1.3"><csymbol cd="ambiguous" id="S1.T2.19.19.2.m1.1.1.1.1.1.3.1.cmml" xref="S1.T2.19.19.2.m1.1.1.1.1.1.3">superscript</csymbol><ci id="S1.T2.19.19.2.m1.1.1.1.1.1.3.2.cmml" xref="S1.T2.19.19.2.m1.1.1.1.1.1.3.2">𝑛</ci><apply id="S1.T2.19.19.2.m1.1.1.1.1.1.3.3.cmml" xref="S1.T2.19.19.2.m1.1.1.1.1.1.3.3"><plus id="S1.T2.19.19.2.m1.1.1.1.1.1.3.3.1.cmml" xref="S1.T2.19.19.2.m1.1.1.1.1.1.3.3.1"></plus><cn id="S1.T2.19.19.2.m1.1.1.1.1.1.3.3.2.cmml" type="integer" xref="S1.T2.19.19.2.m1.1.1.1.1.1.3.3.2">1</cn><apply id="S1.T2.19.19.2.m1.1.1.1.1.1.3.3.3.cmml" xref="S1.T2.19.19.2.m1.1.1.1.1.1.3.3.3"><divide id="S1.T2.19.19.2.m1.1.1.1.1.1.3.3.3.1.cmml" xref="S1.T2.19.19.2.m1.1.1.1.1.1.3.3.3"></divide><cn id="S1.T2.19.19.2.m1.1.1.1.1.1.3.3.3.2.cmml" type="integer" xref="S1.T2.19.19.2.m1.1.1.1.1.1.3.3.3.2">1</cn><ci id="S1.T2.19.19.2.m1.1.1.1.1.1.3.3.3.3.cmml" xref="S1.T2.19.19.2.m1.1.1.1.1.1.3.3.3.3">𝑡</ci></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.T2.19.19.2.m1.1c">\Omega(nk+n^{1+\frac{1}{t}})</annotation><annotation encoding="application/x-llamapun" id="S1.T2.19.19.2.m1.1d">roman_Ω ( italic_n italic_k + italic_n start_POSTSUPERSCRIPT 1 + divide start_ARG 1 end_ARG start_ARG italic_t end_ARG end_POSTSUPERSCRIPT )</annotation></semantics></math> bits <span class="ltx_text" id="S1.T2.19.19.2.1" style="color:#0000FF;">[Here]</span> </td> <td class="ltx_td ltx_align_center" id="S1.T2.19.19.3" style="padding-top:2.5pt;padding-bottom:2.5pt;">lower bound</td> </tr> <tr class="ltx_tr" id="S1.T2.22.22"> <th class="ltx_td ltx_align_center ltx_th ltx_th_row ltx_border_bb ltx_border_r ltx_border_t" id="S1.T2.20.20.1" rowspan="4" style="padding-top:2.5pt;padding-bottom:2.5pt;"><span class="ltx_text" id="S1.T2.20.20.1.1"><math alttext="k" class="ltx_Math" display="inline" id="S1.T2.20.20.1.1.m1.1"><semantics id="S1.T2.20.20.1.1.m1.1a"><mi id="S1.T2.20.20.1.1.m1.1.1" xref="S1.T2.20.20.1.1.m1.1.1.cmml">k</mi><annotation-xml encoding="MathML-Content" id="S1.T2.20.20.1.1.m1.1b"><ci id="S1.T2.20.20.1.1.m1.1.1.cmml" xref="S1.T2.20.20.1.1.m1.1.1">𝑘</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.T2.20.20.1.1.m1.1c">k</annotation><annotation encoding="application/x-llamapun" id="S1.T2.20.20.1.1.m1.1d">italic_k</annotation></semantics></math>-VCSS</span></th> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S1.T2.21.21.2" style="padding-top:2.5pt;padding-bottom:2.5pt;"><math alttext="2" class="ltx_Math" display="inline" id="S1.T2.21.21.2.m1.1"><semantics id="S1.T2.21.21.2.m1.1a"><mn id="S1.T2.21.21.2.m1.1.1" xref="S1.T2.21.21.2.m1.1.1.cmml">2</mn><annotation-xml encoding="MathML-Content" id="S1.T2.21.21.2.m1.1b"><cn id="S1.T2.21.21.2.m1.1.1.cmml" type="integer" xref="S1.T2.21.21.2.m1.1.1">2</cn></annotation-xml><annotation encoding="application/x-tex" id="S1.T2.21.21.2.m1.1c">2</annotation><annotation encoding="application/x-llamapun" id="S1.T2.21.21.2.m1.1d">2</annotation></semantics></math></td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S1.T2.22.22.3" style="padding-top:2.5pt;padding-bottom:2.5pt;"> <math alttext="O(nk)" class="ltx_Math" display="inline" id="S1.T2.22.22.3.m1.1"><semantics id="S1.T2.22.22.3.m1.1a"><mrow id="S1.T2.22.22.3.m1.1.1" xref="S1.T2.22.22.3.m1.1.1.cmml"><mi id="S1.T2.22.22.3.m1.1.1.3" xref="S1.T2.22.22.3.m1.1.1.3.cmml">O</mi><mo id="S1.T2.22.22.3.m1.1.1.2" xref="S1.T2.22.22.3.m1.1.1.2.cmml">⁢</mo><mrow id="S1.T2.22.22.3.m1.1.1.1.1" xref="S1.T2.22.22.3.m1.1.1.1.1.1.cmml"><mo id="S1.T2.22.22.3.m1.1.1.1.1.2" stretchy="false" xref="S1.T2.22.22.3.m1.1.1.1.1.1.cmml">(</mo><mrow id="S1.T2.22.22.3.m1.1.1.1.1.1" xref="S1.T2.22.22.3.m1.1.1.1.1.1.cmml"><mi id="S1.T2.22.22.3.m1.1.1.1.1.1.2" xref="S1.T2.22.22.3.m1.1.1.1.1.1.2.cmml">n</mi><mo id="S1.T2.22.22.3.m1.1.1.1.1.1.1" xref="S1.T2.22.22.3.m1.1.1.1.1.1.1.cmml">⁢</mo><mi id="S1.T2.22.22.3.m1.1.1.1.1.1.3" xref="S1.T2.22.22.3.m1.1.1.1.1.1.3.cmml">k</mi></mrow><mo id="S1.T2.22.22.3.m1.1.1.1.1.3" stretchy="false" xref="S1.T2.22.22.3.m1.1.1.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.T2.22.22.3.m1.1b"><apply id="S1.T2.22.22.3.m1.1.1.cmml" xref="S1.T2.22.22.3.m1.1.1"><times id="S1.T2.22.22.3.m1.1.1.2.cmml" xref="S1.T2.22.22.3.m1.1.1.2"></times><ci id="S1.T2.22.22.3.m1.1.1.3.cmml" xref="S1.T2.22.22.3.m1.1.1.3">𝑂</ci><apply id="S1.T2.22.22.3.m1.1.1.1.1.1.cmml" xref="S1.T2.22.22.3.m1.1.1.1.1"><times id="S1.T2.22.22.3.m1.1.1.1.1.1.1.cmml" xref="S1.T2.22.22.3.m1.1.1.1.1.1.1"></times><ci id="S1.T2.22.22.3.m1.1.1.1.1.1.2.cmml" xref="S1.T2.22.22.3.m1.1.1.1.1.1.2">𝑛</ci><ci id="S1.T2.22.22.3.m1.1.1.1.1.1.3.cmml" xref="S1.T2.22.22.3.m1.1.1.1.1.1.3">𝑘</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.T2.22.22.3.m1.1c">O(nk)</annotation><annotation encoding="application/x-llamapun" id="S1.T2.22.22.3.m1.1d">italic_O ( italic_n italic_k )</annotation></semantics></math> <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.00712v1#bib.bibx81" title="">Zel06</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.00712v1#bib.bibx27" title="">CKT93</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.00712v1#bib.bibx69" title="">NI92</a>]</cite> </td> <td class="ltx_td ltx_align_center ltx_border_t" id="S1.T2.22.22.4" style="padding-top:2.5pt;padding-bottom:2.5pt;">for unweighted graphs</td> </tr> <tr class="ltx_tr" id="S1.T2.25.25"> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S1.T2.23.23.1" style="padding-top:2.5pt;padding-bottom:2.5pt;"><math alttext="O(t)" class="ltx_Math" display="inline" id="S1.T2.23.23.1.m1.1"><semantics id="S1.T2.23.23.1.m1.1a"><mrow id="S1.T2.23.23.1.m1.1.2" xref="S1.T2.23.23.1.m1.1.2.cmml"><mi id="S1.T2.23.23.1.m1.1.2.2" xref="S1.T2.23.23.1.m1.1.2.2.cmml">O</mi><mo id="S1.T2.23.23.1.m1.1.2.1" xref="S1.T2.23.23.1.m1.1.2.1.cmml">⁢</mo><mrow id="S1.T2.23.23.1.m1.1.2.3.2" xref="S1.T2.23.23.1.m1.1.2.cmml"><mo id="S1.T2.23.23.1.m1.1.2.3.2.1" stretchy="false" xref="S1.T2.23.23.1.m1.1.2.cmml">(</mo><mi id="S1.T2.23.23.1.m1.1.1" xref="S1.T2.23.23.1.m1.1.1.cmml">t</mi><mo id="S1.T2.23.23.1.m1.1.2.3.2.2" stretchy="false" xref="S1.T2.23.23.1.m1.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.T2.23.23.1.m1.1b"><apply id="S1.T2.23.23.1.m1.1.2.cmml" xref="S1.T2.23.23.1.m1.1.2"><times id="S1.T2.23.23.1.m1.1.2.1.cmml" xref="S1.T2.23.23.1.m1.1.2.1"></times><ci id="S1.T2.23.23.1.m1.1.2.2.cmml" xref="S1.T2.23.23.1.m1.1.2.2">𝑂</ci><ci id="S1.T2.23.23.1.m1.1.1.cmml" xref="S1.T2.23.23.1.m1.1.1">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.T2.23.23.1.m1.1c">O(t)</annotation><annotation encoding="application/x-llamapun" id="S1.T2.23.23.1.m1.1d">italic_O ( italic_t )</annotation></semantics></math></td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S1.T2.24.24.2" style="padding-top:2.5pt;padding-bottom:2.5pt;"> <math alttext="\tilde{O}(k^{1-\frac{1}{t}}n^{1+\frac{1}{t}})" class="ltx_Math" display="inline" id="S1.T2.24.24.2.m1.1"><semantics id="S1.T2.24.24.2.m1.1a"><mrow id="S1.T2.24.24.2.m1.1.1" xref="S1.T2.24.24.2.m1.1.1.cmml"><mover accent="true" id="S1.T2.24.24.2.m1.1.1.3" xref="S1.T2.24.24.2.m1.1.1.3.cmml"><mi id="S1.T2.24.24.2.m1.1.1.3.2" xref="S1.T2.24.24.2.m1.1.1.3.2.cmml">O</mi><mo id="S1.T2.24.24.2.m1.1.1.3.1" xref="S1.T2.24.24.2.m1.1.1.3.1.cmml">~</mo></mover><mo id="S1.T2.24.24.2.m1.1.1.2" xref="S1.T2.24.24.2.m1.1.1.2.cmml">⁢</mo><mrow id="S1.T2.24.24.2.m1.1.1.1.1" xref="S1.T2.24.24.2.m1.1.1.1.1.1.cmml"><mo id="S1.T2.24.24.2.m1.1.1.1.1.2" stretchy="false" xref="S1.T2.24.24.2.m1.1.1.1.1.1.cmml">(</mo><mrow id="S1.T2.24.24.2.m1.1.1.1.1.1" xref="S1.T2.24.24.2.m1.1.1.1.1.1.cmml"><msup id="S1.T2.24.24.2.m1.1.1.1.1.1.2" xref="S1.T2.24.24.2.m1.1.1.1.1.1.2.cmml"><mi id="S1.T2.24.24.2.m1.1.1.1.1.1.2.2" xref="S1.T2.24.24.2.m1.1.1.1.1.1.2.2.cmml">k</mi><mrow id="S1.T2.24.24.2.m1.1.1.1.1.1.2.3" 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xref="S1.T2.24.24.2.m1.1.1.1.1.1.2.3.3.2">1</cn><ci id="S1.T2.24.24.2.m1.1.1.1.1.1.2.3.3.3.cmml" xref="S1.T2.24.24.2.m1.1.1.1.1.1.2.3.3.3">𝑡</ci></apply></apply></apply><apply id="S1.T2.24.24.2.m1.1.1.1.1.1.3.cmml" xref="S1.T2.24.24.2.m1.1.1.1.1.1.3"><csymbol cd="ambiguous" id="S1.T2.24.24.2.m1.1.1.1.1.1.3.1.cmml" xref="S1.T2.24.24.2.m1.1.1.1.1.1.3">superscript</csymbol><ci id="S1.T2.24.24.2.m1.1.1.1.1.1.3.2.cmml" xref="S1.T2.24.24.2.m1.1.1.1.1.1.3.2">𝑛</ci><apply id="S1.T2.24.24.2.m1.1.1.1.1.1.3.3.cmml" xref="S1.T2.24.24.2.m1.1.1.1.1.1.3.3"><plus id="S1.T2.24.24.2.m1.1.1.1.1.1.3.3.1.cmml" xref="S1.T2.24.24.2.m1.1.1.1.1.1.3.3.1"></plus><cn id="S1.T2.24.24.2.m1.1.1.1.1.1.3.3.2.cmml" type="integer" xref="S1.T2.24.24.2.m1.1.1.1.1.1.3.3.2">1</cn><apply id="S1.T2.24.24.2.m1.1.1.1.1.1.3.3.3.cmml" xref="S1.T2.24.24.2.m1.1.1.1.1.1.3.3.3"><divide id="S1.T2.24.24.2.m1.1.1.1.1.1.3.3.3.1.cmml" xref="S1.T2.24.24.2.m1.1.1.1.1.1.3.3.3"></divide><cn id="S1.T2.24.24.2.m1.1.1.1.1.1.3.3.3.2.cmml" type="integer" xref="S1.T2.24.24.2.m1.1.1.1.1.1.3.3.3.2">1</cn><ci id="S1.T2.24.24.2.m1.1.1.1.1.1.3.3.3.3.cmml" xref="S1.T2.24.24.2.m1.1.1.1.1.1.3.3.3.3">𝑡</ci></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.T2.24.24.2.m1.1c">\tilde{O}(k^{1-\frac{1}{t}}n^{1+\frac{1}{t}})</annotation><annotation encoding="application/x-llamapun" id="S1.T2.24.24.2.m1.1d">over~ start_ARG italic_O end_ARG ( italic_k start_POSTSUPERSCRIPT 1 - divide start_ARG 1 end_ARG start_ARG italic_t end_ARG end_POSTSUPERSCRIPT italic_n start_POSTSUPERSCRIPT 1 + divide start_ARG 1 end_ARG start_ARG italic_t end_ARG end_POSTSUPERSCRIPT )</annotation></semantics></math> <span class="ltx_text" id="S1.T2.24.24.2.1" style="color:#0000FF;">[Here]</span> </td> <td class="ltx_td ltx_align_center ltx_border_t" id="S1.T2.25.25.3" style="padding-top:2.5pt;padding-bottom:2.5pt;"><math alttext="n=\Omega(k^{3})" class="ltx_Math" display="inline" id="S1.T2.25.25.3.m1.1"><semantics id="S1.T2.25.25.3.m1.1a"><mrow id="S1.T2.25.25.3.m1.1.1" xref="S1.T2.25.25.3.m1.1.1.cmml"><mi id="S1.T2.25.25.3.m1.1.1.3" xref="S1.T2.25.25.3.m1.1.1.3.cmml">n</mi><mo id="S1.T2.25.25.3.m1.1.1.2" xref="S1.T2.25.25.3.m1.1.1.2.cmml">=</mo><mrow id="S1.T2.25.25.3.m1.1.1.1" xref="S1.T2.25.25.3.m1.1.1.1.cmml"><mi id="S1.T2.25.25.3.m1.1.1.1.3" mathvariant="normal" xref="S1.T2.25.25.3.m1.1.1.1.3.cmml">Ω</mi><mo id="S1.T2.25.25.3.m1.1.1.1.2" xref="S1.T2.25.25.3.m1.1.1.1.2.cmml">⁢</mo><mrow id="S1.T2.25.25.3.m1.1.1.1.1.1" xref="S1.T2.25.25.3.m1.1.1.1.1.1.1.cmml"><mo id="S1.T2.25.25.3.m1.1.1.1.1.1.2" stretchy="false" xref="S1.T2.25.25.3.m1.1.1.1.1.1.1.cmml">(</mo><msup id="S1.T2.25.25.3.m1.1.1.1.1.1.1" xref="S1.T2.25.25.3.m1.1.1.1.1.1.1.cmml"><mi id="S1.T2.25.25.3.m1.1.1.1.1.1.1.2" xref="S1.T2.25.25.3.m1.1.1.1.1.1.1.2.cmml">k</mi><mn id="S1.T2.25.25.3.m1.1.1.1.1.1.1.3" xref="S1.T2.25.25.3.m1.1.1.1.1.1.1.3.cmml">3</mn></msup><mo id="S1.T2.25.25.3.m1.1.1.1.1.1.3" stretchy="false" xref="S1.T2.25.25.3.m1.1.1.1.1.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.T2.25.25.3.m1.1b"><apply id="S1.T2.25.25.3.m1.1.1.cmml" xref="S1.T2.25.25.3.m1.1.1"><eq id="S1.T2.25.25.3.m1.1.1.2.cmml" xref="S1.T2.25.25.3.m1.1.1.2"></eq><ci id="S1.T2.25.25.3.m1.1.1.3.cmml" xref="S1.T2.25.25.3.m1.1.1.3">𝑛</ci><apply id="S1.T2.25.25.3.m1.1.1.1.cmml" xref="S1.T2.25.25.3.m1.1.1.1"><times id="S1.T2.25.25.3.m1.1.1.1.2.cmml" xref="S1.T2.25.25.3.m1.1.1.1.2"></times><ci id="S1.T2.25.25.3.m1.1.1.1.3.cmml" xref="S1.T2.25.25.3.m1.1.1.1.3">Ω</ci><apply id="S1.T2.25.25.3.m1.1.1.1.1.1.1.cmml" xref="S1.T2.25.25.3.m1.1.1.1.1.1"><csymbol cd="ambiguous" id="S1.T2.25.25.3.m1.1.1.1.1.1.1.1.cmml" xref="S1.T2.25.25.3.m1.1.1.1.1.1">superscript</csymbol><ci id="S1.T2.25.25.3.m1.1.1.1.1.1.1.2.cmml" xref="S1.T2.25.25.3.m1.1.1.1.1.1.1.2">𝑘</ci><cn id="S1.T2.25.25.3.m1.1.1.1.1.1.1.3.cmml" type="integer" xref="S1.T2.25.25.3.m1.1.1.1.1.1.1.3">3</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.T2.25.25.3.m1.1c">n=\Omega(k^{3})</annotation><annotation encoding="application/x-llamapun" id="S1.T2.25.25.3.m1.1d">italic_n = roman_Ω ( italic_k start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT )</annotation></semantics></math></td> </tr> <tr class="ltx_tr" id="S1.T2.27.27"> <td class="ltx_td ltx_align_center ltx_border_bb ltx_border_r" id="S1.T2.26.26.1" style="padding-top:2.5pt;padding-bottom:2.5pt;"><math alttext="O(t)" class="ltx_Math" display="inline" id="S1.T2.26.26.1.m1.1"><semantics id="S1.T2.26.26.1.m1.1a"><mrow id="S1.T2.26.26.1.m1.1.2" xref="S1.T2.26.26.1.m1.1.2.cmml"><mi id="S1.T2.26.26.1.m1.1.2.2" xref="S1.T2.26.26.1.m1.1.2.2.cmml">O</mi><mo id="S1.T2.26.26.1.m1.1.2.1" xref="S1.T2.26.26.1.m1.1.2.1.cmml">⁢</mo><mrow id="S1.T2.26.26.1.m1.1.2.3.2" xref="S1.T2.26.26.1.m1.1.2.cmml"><mo id="S1.T2.26.26.1.m1.1.2.3.2.1" stretchy="false" xref="S1.T2.26.26.1.m1.1.2.cmml">(</mo><mi id="S1.T2.26.26.1.m1.1.1" xref="S1.T2.26.26.1.m1.1.1.cmml">t</mi><mo id="S1.T2.26.26.1.m1.1.2.3.2.2" stretchy="false" xref="S1.T2.26.26.1.m1.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.T2.26.26.1.m1.1b"><apply id="S1.T2.26.26.1.m1.1.2.cmml" xref="S1.T2.26.26.1.m1.1.2"><times id="S1.T2.26.26.1.m1.1.2.1.cmml" xref="S1.T2.26.26.1.m1.1.2.1"></times><ci id="S1.T2.26.26.1.m1.1.2.2.cmml" xref="S1.T2.26.26.1.m1.1.2.2">𝑂</ci><ci id="S1.T2.26.26.1.m1.1.1.cmml" xref="S1.T2.26.26.1.m1.1.1">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.T2.26.26.1.m1.1c">O(t)</annotation><annotation encoding="application/x-llamapun" id="S1.T2.26.26.1.m1.1d">italic_O ( italic_t )</annotation></semantics></math></td> <td class="ltx_td ltx_align_center ltx_border_bb ltx_border_r" id="S1.T2.27.27.2" style="padding-top:2.5pt;padding-bottom:2.5pt;"> <math alttext="\Omega(nk+n^{1+\frac{1}{t}})" class="ltx_Math" 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italic_n start_POSTSUPERSCRIPT 1 + divide start_ARG 1 end_ARG start_ARG italic_t end_ARG end_POSTSUPERSCRIPT )</annotation></semantics></math> bits <span class="ltx_text" id="S1.T2.27.27.2.1" style="color:#0000FF;">[Here]</span> </td> <td class="ltx_td ltx_align_center ltx_border_bb" id="S1.T2.27.27.3" style="padding-top:2.5pt;padding-bottom:2.5pt;">lower bound</td> </tr> </tbody> </table> <figcaption class="ltx_caption ltx_centering"><span class="ltx_tag ltx_tag_table"><span class="ltx_text" id="S1.T2.29.1.1" style="font-size:90%;">Table 2</span>: </span><span class="ltx_text" id="S1.T2.30.2" style="font-size:90%;">Summary of our results for vertex-connectivity network design.</span></figcaption> </figure> </section> <section class="ltx_paragraph" id="S1.SS1.SSS0.Px3"> <h5 class="ltx_title ltx_title_paragraph">Techniques:</h5> <div class="ltx_para" id="S1.SS1.SSS0.Px3.p1"> <p class="ltx_p" id="S1.SS1.SSS0.Px3.p1.4">Our general framework for SNDP and related problems is based on a connection to <em class="ltx_emph ltx_font_italic" id="S1.SS1.SSS0.Px3.p1.4.1">fault-tolerant spanners</em> which were first introduced in <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.00712v1#bib.bibx65" title="">LNS98</a>]</cite> in geometric settings and then extensively studied in the graph setting. These objects generalize the notion of spanners to allow faults and are surprisingly powerful. Recent work has shown that, like ordinary spanners, optimal fault-tolerant spanners can be constructed via a simple greedy algorithm which enables their use in streaming setting <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.00712v1#bib.bibx21" title="">BP19</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.00712v1#bib.bibx18" title="">BDPW18</a>]</cite>. Using these spanners alone suffices to obtain an <math alttext="O(tk)" class="ltx_Math" display="inline" id="S1.SS1.SSS0.Px3.p1.1.m1.1"><semantics id="S1.SS1.SSS0.Px3.p1.1.m1.1a"><mrow id="S1.SS1.SSS0.Px3.p1.1.m1.1.1" xref="S1.SS1.SSS0.Px3.p1.1.m1.1.1.cmml"><mi id="S1.SS1.SSS0.Px3.p1.1.m1.1.1.3" xref="S1.SS1.SSS0.Px3.p1.1.m1.1.1.3.cmml">O</mi><mo id="S1.SS1.SSS0.Px3.p1.1.m1.1.1.2" xref="S1.SS1.SSS0.Px3.p1.1.m1.1.1.2.cmml">⁢</mo><mrow id="S1.SS1.SSS0.Px3.p1.1.m1.1.1.1.1" xref="S1.SS1.SSS0.Px3.p1.1.m1.1.1.1.1.1.cmml"><mo id="S1.SS1.SSS0.Px3.p1.1.m1.1.1.1.1.2" stretchy="false" xref="S1.SS1.SSS0.Px3.p1.1.m1.1.1.1.1.1.cmml">(</mo><mrow id="S1.SS1.SSS0.Px3.p1.1.m1.1.1.1.1.1" xref="S1.SS1.SSS0.Px3.p1.1.m1.1.1.1.1.1.cmml"><mi id="S1.SS1.SSS0.Px3.p1.1.m1.1.1.1.1.1.2" xref="S1.SS1.SSS0.Px3.p1.1.m1.1.1.1.1.1.2.cmml">t</mi><mo id="S1.SS1.SSS0.Px3.p1.1.m1.1.1.1.1.1.1" xref="S1.SS1.SSS0.Px3.p1.1.m1.1.1.1.1.1.1.cmml">⁢</mo><mi id="S1.SS1.SSS0.Px3.p1.1.m1.1.1.1.1.1.3" xref="S1.SS1.SSS0.Px3.p1.1.m1.1.1.1.1.1.3.cmml">k</mi></mrow><mo id="S1.SS1.SSS0.Px3.p1.1.m1.1.1.1.1.3" stretchy="false" xref="S1.SS1.SSS0.Px3.p1.1.m1.1.1.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.SS1.SSS0.Px3.p1.1.m1.1b"><apply id="S1.SS1.SSS0.Px3.p1.1.m1.1.1.cmml" xref="S1.SS1.SSS0.Px3.p1.1.m1.1.1"><times id="S1.SS1.SSS0.Px3.p1.1.m1.1.1.2.cmml" xref="S1.SS1.SSS0.Px3.p1.1.m1.1.1.2"></times><ci id="S1.SS1.SSS0.Px3.p1.1.m1.1.1.3.cmml" xref="S1.SS1.SSS0.Px3.p1.1.m1.1.1.3">𝑂</ci><apply id="S1.SS1.SSS0.Px3.p1.1.m1.1.1.1.1.1.cmml" xref="S1.SS1.SSS0.Px3.p1.1.m1.1.1.1.1"><times id="S1.SS1.SSS0.Px3.p1.1.m1.1.1.1.1.1.1.cmml" xref="S1.SS1.SSS0.Px3.p1.1.m1.1.1.1.1.1.1"></times><ci id="S1.SS1.SSS0.Px3.p1.1.m1.1.1.1.1.1.2.cmml" xref="S1.SS1.SSS0.Px3.p1.1.m1.1.1.1.1.1.2">𝑡</ci><ci id="S1.SS1.SSS0.Px3.p1.1.m1.1.1.1.1.1.3.cmml" xref="S1.SS1.SSS0.Px3.p1.1.m1.1.1.1.1.1.3">𝑘</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS1.SSS0.Px3.p1.1.m1.1c">O(tk)</annotation><annotation encoding="application/x-llamapun" id="S1.SS1.SSS0.Px3.p1.1.m1.1d">italic_O ( italic_t italic_k )</annotation></semantics></math>-approximation assuming an exact algorithm at the end of the stream. Obtaining our refined approximation bounds requires additional insight: we combine the use of fault-tolerant spanners with an analysis via natural LP relaxations for SNDP. This improved analysis has two key benefits. First, it improves the approximation ratio by a factor of <math alttext="k" class="ltx_Math" display="inline" id="S1.SS1.SSS0.Px3.p1.2.m2.1"><semantics id="S1.SS1.SSS0.Px3.p1.2.m2.1a"><mi id="S1.SS1.SSS0.Px3.p1.2.m2.1.1" xref="S1.SS1.SSS0.Px3.p1.2.m2.1.1.cmml">k</mi><annotation-xml encoding="MathML-Content" id="S1.SS1.SSS0.Px3.p1.2.m2.1b"><ci id="S1.SS1.SSS0.Px3.p1.2.m2.1.1.cmml" xref="S1.SS1.SSS0.Px3.p1.2.m2.1.1">𝑘</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.SS1.SSS0.Px3.p1.2.m2.1c">k</annotation><annotation encoding="application/x-llamapun" id="S1.SS1.SSS0.Px3.p1.2.m2.1d">italic_k</annotation></semantics></math> in regimes requiring polynomial-time algorithms, provided that the corresponding network design problem has a good approximation algorithm with respect to the LP. For many of the problems considered in this paper, the best known approximation algorithms are in fact LP-based (see Section <a class="ltx_ref" href="https://arxiv.org/html/2503.00712v1#S1.SS2" title="1.2 Related Work ‣ 1 Introduction ‣ Streaming Algorithms for Network Design"><span class="ltx_text ltx_ref_tag">1.2</span></a> for details), highlighting the utility of this analysis. Second, it allows us to directly improve the factor <math alttext="O(tk)" class="ltx_Math" display="inline" id="S1.SS1.SSS0.Px3.p1.3.m3.1"><semantics id="S1.SS1.SSS0.Px3.p1.3.m3.1a"><mrow id="S1.SS1.SSS0.Px3.p1.3.m3.1.1" xref="S1.SS1.SSS0.Px3.p1.3.m3.1.1.cmml"><mi id="S1.SS1.SSS0.Px3.p1.3.m3.1.1.3" xref="S1.SS1.SSS0.Px3.p1.3.m3.1.1.3.cmml">O</mi><mo id="S1.SS1.SSS0.Px3.p1.3.m3.1.1.2" xref="S1.SS1.SSS0.Px3.p1.3.m3.1.1.2.cmml">⁢</mo><mrow id="S1.SS1.SSS0.Px3.p1.3.m3.1.1.1.1" xref="S1.SS1.SSS0.Px3.p1.3.m3.1.1.1.1.1.cmml"><mo id="S1.SS1.SSS0.Px3.p1.3.m3.1.1.1.1.2" stretchy="false" xref="S1.SS1.SSS0.Px3.p1.3.m3.1.1.1.1.1.cmml">(</mo><mrow id="S1.SS1.SSS0.Px3.p1.3.m3.1.1.1.1.1" xref="S1.SS1.SSS0.Px3.p1.3.m3.1.1.1.1.1.cmml"><mi id="S1.SS1.SSS0.Px3.p1.3.m3.1.1.1.1.1.2" xref="S1.SS1.SSS0.Px3.p1.3.m3.1.1.1.1.1.2.cmml">t</mi><mo id="S1.SS1.SSS0.Px3.p1.3.m3.1.1.1.1.1.1" xref="S1.SS1.SSS0.Px3.p1.3.m3.1.1.1.1.1.1.cmml">⁢</mo><mi id="S1.SS1.SSS0.Px3.p1.3.m3.1.1.1.1.1.3" xref="S1.SS1.SSS0.Px3.p1.3.m3.1.1.1.1.1.3.cmml">k</mi></mrow><mo id="S1.SS1.SSS0.Px3.p1.3.m3.1.1.1.1.3" stretchy="false" xref="S1.SS1.SSS0.Px3.p1.3.m3.1.1.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.SS1.SSS0.Px3.p1.3.m3.1b"><apply id="S1.SS1.SSS0.Px3.p1.3.m3.1.1.cmml" xref="S1.SS1.SSS0.Px3.p1.3.m3.1.1"><times id="S1.SS1.SSS0.Px3.p1.3.m3.1.1.2.cmml" xref="S1.SS1.SSS0.Px3.p1.3.m3.1.1.2"></times><ci id="S1.SS1.SSS0.Px3.p1.3.m3.1.1.3.cmml" xref="S1.SS1.SSS0.Px3.p1.3.m3.1.1.3">𝑂</ci><apply id="S1.SS1.SSS0.Px3.p1.3.m3.1.1.1.1.1.cmml" xref="S1.SS1.SSS0.Px3.p1.3.m3.1.1.1.1"><times id="S1.SS1.SSS0.Px3.p1.3.m3.1.1.1.1.1.1.cmml" xref="S1.SS1.SSS0.Px3.p1.3.m3.1.1.1.1.1.1"></times><ci id="S1.SS1.SSS0.Px3.p1.3.m3.1.1.1.1.1.2.cmml" xref="S1.SS1.SSS0.Px3.p1.3.m3.1.1.1.1.1.2">𝑡</ci><ci id="S1.SS1.SSS0.Px3.p1.3.m3.1.1.1.1.1.3.cmml" xref="S1.SS1.SSS0.Px3.p1.3.m3.1.1.1.1.1.3">𝑘</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS1.SSS0.Px3.p1.3.m3.1c">O(tk)</annotation><annotation encoding="application/x-llamapun" id="S1.SS1.SSS0.Px3.p1.3.m3.1d">italic_O ( italic_t italic_k )</annotation></semantics></math> for problems with corresponding integrality gap better than <math alttext="k" class="ltx_Math" display="inline" id="S1.SS1.SSS0.Px3.p1.4.m4.1"><semantics id="S1.SS1.SSS0.Px3.p1.4.m4.1a"><mi id="S1.SS1.SSS0.Px3.p1.4.m4.1.1" xref="S1.SS1.SSS0.Px3.p1.4.m4.1.1.cmml">k</mi><annotation-xml encoding="MathML-Content" id="S1.SS1.SSS0.Px3.p1.4.m4.1b"><ci id="S1.SS1.SSS0.Px3.p1.4.m4.1.1.cmml" xref="S1.SS1.SSS0.Px3.p1.4.m4.1.1">𝑘</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.SS1.SSS0.Px3.p1.4.m4.1c">k</annotation><annotation encoding="application/x-llamapun" id="S1.SS1.SSS0.Px3.p1.4.m4.1d">italic_k</annotation></semantics></math>—this applies to several of the problems we consider.</p> </div> <div class="ltx_para" id="S1.SS1.SSS0.Px3.p2"> <p class="ltx_p" id="S1.SS1.SSS0.Px3.p2.11">For <math alttext="1" class="ltx_Math" display="inline" id="S1.SS1.SSS0.Px3.p2.1.m1.1"><semantics id="S1.SS1.SSS0.Px3.p2.1.m1.1a"><mn id="S1.SS1.SSS0.Px3.p2.1.m1.1.1" xref="S1.SS1.SSS0.Px3.p2.1.m1.1.1.cmml">1</mn><annotation-xml encoding="MathML-Content" id="S1.SS1.SSS0.Px3.p2.1.m1.1b"><cn id="S1.SS1.SSS0.Px3.p2.1.m1.1.1.cmml" type="integer" xref="S1.SS1.SSS0.Px3.p2.1.m1.1.1">1</cn></annotation-xml><annotation encoding="application/x-tex" id="S1.SS1.SSS0.Px3.p2.1.m1.1c">1</annotation><annotation encoding="application/x-llamapun" id="S1.SS1.SSS0.Px3.p2.1.m1.1d">1</annotation></semantics></math>-VC-CAP, we aim to augment a spanning tree to a <math alttext="2" class="ltx_Math" display="inline" id="S1.SS1.SSS0.Px3.p2.2.m2.1"><semantics id="S1.SS1.SSS0.Px3.p2.2.m2.1a"><mn id="S1.SS1.SSS0.Px3.p2.2.m2.1.1" xref="S1.SS1.SSS0.Px3.p2.2.m2.1.1.cmml">2</mn><annotation-xml encoding="MathML-Content" id="S1.SS1.SSS0.Px3.p2.2.m2.1b"><cn id="S1.SS1.SSS0.Px3.p2.2.m2.1.1.cmml" type="integer" xref="S1.SS1.SSS0.Px3.p2.2.m2.1.1">2</cn></annotation-xml><annotation encoding="application/x-tex" id="S1.SS1.SSS0.Px3.p2.2.m2.1c">2</annotation><annotation encoding="application/x-llamapun" id="S1.SS1.SSS0.Px3.p2.2.m2.1d">2</annotation></semantics></math>-vertex-connected graph. To do so, we borrow ideas from the edge-connectivity setting: we root the tree arbitrarily, and for each node <math alttext="u\in V" class="ltx_Math" display="inline" id="S1.SS1.SSS0.Px3.p2.3.m3.1"><semantics id="S1.SS1.SSS0.Px3.p2.3.m3.1a"><mrow id="S1.SS1.SSS0.Px3.p2.3.m3.1.1" xref="S1.SS1.SSS0.Px3.p2.3.m3.1.1.cmml"><mi id="S1.SS1.SSS0.Px3.p2.3.m3.1.1.2" xref="S1.SS1.SSS0.Px3.p2.3.m3.1.1.2.cmml">u</mi><mo id="S1.SS1.SSS0.Px3.p2.3.m3.1.1.1" xref="S1.SS1.SSS0.Px3.p2.3.m3.1.1.1.cmml">∈</mo><mi id="S1.SS1.SSS0.Px3.p2.3.m3.1.1.3" xref="S1.SS1.SSS0.Px3.p2.3.m3.1.1.3.cmml">V</mi></mrow><annotation-xml encoding="MathML-Content" id="S1.SS1.SSS0.Px3.p2.3.m3.1b"><apply id="S1.SS1.SSS0.Px3.p2.3.m3.1.1.cmml" xref="S1.SS1.SSS0.Px3.p2.3.m3.1.1"><in id="S1.SS1.SSS0.Px3.p2.3.m3.1.1.1.cmml" xref="S1.SS1.SSS0.Px3.p2.3.m3.1.1.1"></in><ci id="S1.SS1.SSS0.Px3.p2.3.m3.1.1.2.cmml" xref="S1.SS1.SSS0.Px3.p2.3.m3.1.1.2">𝑢</ci><ci id="S1.SS1.SSS0.Px3.p2.3.m3.1.1.3.cmml" xref="S1.SS1.SSS0.Px3.p2.3.m3.1.1.3">𝑉</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS1.SSS0.Px3.p2.3.m3.1c">u\in V</annotation><annotation encoding="application/x-llamapun" id="S1.SS1.SSS0.Px3.p2.3.m3.1d">italic_u ∈ italic_V</annotation></semantics></math>, we store the link that “covers” the most edges in the path from <math alttext="u" class="ltx_Math" display="inline" id="S1.SS1.SSS0.Px3.p2.4.m4.1"><semantics id="S1.SS1.SSS0.Px3.p2.4.m4.1a"><mi id="S1.SS1.SSS0.Px3.p2.4.m4.1.1" xref="S1.SS1.SSS0.Px3.p2.4.m4.1.1.cmml">u</mi><annotation-xml encoding="MathML-Content" id="S1.SS1.SSS0.Px3.p2.4.m4.1b"><ci id="S1.SS1.SSS0.Px3.p2.4.m4.1.1.cmml" xref="S1.SS1.SSS0.Px3.p2.4.m4.1.1">𝑢</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.SS1.SSS0.Px3.p2.4.m4.1c">u</annotation><annotation encoding="application/x-llamapun" id="S1.SS1.SSS0.Px3.p2.4.m4.1d">italic_u</annotation></semantics></math> to the root. This does not quite suffice in the vertex-connectivity setting, thus some additional care is required. Our main contribution is an algorithm for <math alttext="2" class="ltx_Math" display="inline" id="S1.SS1.SSS0.Px3.p2.5.m5.1"><semantics id="S1.SS1.SSS0.Px3.p2.5.m5.1a"><mn id="S1.SS1.SSS0.Px3.p2.5.m5.1.1" xref="S1.SS1.SSS0.Px3.p2.5.m5.1.1.cmml">2</mn><annotation-xml encoding="MathML-Content" id="S1.SS1.SSS0.Px3.p2.5.m5.1b"><cn id="S1.SS1.SSS0.Px3.p2.5.m5.1.1.cmml" type="integer" xref="S1.SS1.SSS0.Px3.p2.5.m5.1.1">2</cn></annotation-xml><annotation encoding="application/x-tex" id="S1.SS1.SSS0.Px3.p2.5.m5.1c">2</annotation><annotation encoding="application/x-llamapun" id="S1.SS1.SSS0.Px3.p2.5.m5.1d">2</annotation></semantics></math>-VC-CAP. Here we need to augment a <math alttext="2" class="ltx_Math" display="inline" id="S1.SS1.SSS0.Px3.p2.6.m6.1"><semantics id="S1.SS1.SSS0.Px3.p2.6.m6.1a"><mn id="S1.SS1.SSS0.Px3.p2.6.m6.1.1" xref="S1.SS1.SSS0.Px3.p2.6.m6.1.1.cmml">2</mn><annotation-xml encoding="MathML-Content" id="S1.SS1.SSS0.Px3.p2.6.m6.1b"><cn id="S1.SS1.SSS0.Px3.p2.6.m6.1.1.cmml" type="integer" xref="S1.SS1.SSS0.Px3.p2.6.m6.1.1">2</cn></annotation-xml><annotation encoding="application/x-tex" id="S1.SS1.SSS0.Px3.p2.6.m6.1c">2</annotation><annotation encoding="application/x-llamapun" id="S1.SS1.SSS0.Px3.p2.6.m6.1d">2</annotation></semantics></math>-vertex-connected graph to a <math alttext="3" class="ltx_Math" display="inline" id="S1.SS1.SSS0.Px3.p2.7.m7.1"><semantics id="S1.SS1.SSS0.Px3.p2.7.m7.1a"><mn id="S1.SS1.SSS0.Px3.p2.7.m7.1.1" xref="S1.SS1.SSS0.Px3.p2.7.m7.1.1.cmml">3</mn><annotation-xml encoding="MathML-Content" id="S1.SS1.SSS0.Px3.p2.7.m7.1b"><cn id="S1.SS1.SSS0.Px3.p2.7.m7.1.1.cmml" type="integer" xref="S1.SS1.SSS0.Px3.p2.7.m7.1.1">3</cn></annotation-xml><annotation encoding="application/x-tex" id="S1.SS1.SSS0.Px3.p2.7.m7.1c">3</annotation><annotation encoding="application/x-llamapun" id="S1.SS1.SSS0.Px3.p2.7.m7.1d">3</annotation></semantics></math>-vertex-connected graph. In edge-connectivity, this reduces to cactus-augmentation (which can essentially be reduced to augmenting a cycle). However this does not hold for the vertex-connectivity setting. Instead, we use the SPQR-representation of a <math alttext="2" class="ltx_Math" display="inline" id="S1.SS1.SSS0.Px3.p2.8.m8.1"><semantics id="S1.SS1.SSS0.Px3.p2.8.m8.1a"><mn id="S1.SS1.SSS0.Px3.p2.8.m8.1.1" xref="S1.SS1.SSS0.Px3.p2.8.m8.1.1.cmml">2</mn><annotation-xml encoding="MathML-Content" id="S1.SS1.SSS0.Px3.p2.8.m8.1b"><cn id="S1.SS1.SSS0.Px3.p2.8.m8.1.1.cmml" type="integer" xref="S1.SS1.SSS0.Px3.p2.8.m8.1.1">2</cn></annotation-xml><annotation encoding="application/x-tex" id="S1.SS1.SSS0.Px3.p2.8.m8.1c">2</annotation><annotation encoding="application/x-llamapun" id="S1.SS1.SSS0.Px3.p2.8.m8.1d">2</annotation></semantics></math>-vertex-connected graph: this is a compact tree-like data structure that captures all the 2-node cuts of a graph and was introduced by Di Battista and Tamassia <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.00712v1#bib.bibx33" title="">DBT96a</a>]</cite> that dynamically maintains the triconnected components (and thus all <math alttext="2" class="ltx_Math" display="inline" id="S1.SS1.SSS0.Px3.p2.9.m9.1"><semantics id="S1.SS1.SSS0.Px3.p2.9.m9.1a"><mn id="S1.SS1.SSS0.Px3.p2.9.m9.1.1" xref="S1.SS1.SSS0.Px3.p2.9.m9.1.1.cmml">2</mn><annotation-xml encoding="MathML-Content" id="S1.SS1.SSS0.Px3.p2.9.m9.1b"><cn id="S1.SS1.SSS0.Px3.p2.9.m9.1.1.cmml" type="integer" xref="S1.SS1.SSS0.Px3.p2.9.m9.1.1">2</cn></annotation-xml><annotation encoding="application/x-tex" id="S1.SS1.SSS0.Px3.p2.9.m9.1c">2</annotation><annotation encoding="application/x-llamapun" id="S1.SS1.SSS0.Px3.p2.9.m9.1d">2</annotation></semantics></math>-node cuts) of a graph. It was initially developed in the context of planar graphs <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.00712v1#bib.bibx32" title="">DBT89</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.00712v1#bib.bibx34" title="">DBT96b</a>]</cite>. We use this tree-like structure to combine ideas from <math alttext="1" class="ltx_Math" display="inline" id="S1.SS1.SSS0.Px3.p2.10.m10.1"><semantics id="S1.SS1.SSS0.Px3.p2.10.m10.1a"><mn id="S1.SS1.SSS0.Px3.p2.10.m10.1.1" xref="S1.SS1.SSS0.Px3.p2.10.m10.1.1.cmml">1</mn><annotation-xml encoding="MathML-Content" id="S1.SS1.SSS0.Px3.p2.10.m10.1b"><cn id="S1.SS1.SSS0.Px3.p2.10.m10.1.1.cmml" type="integer" xref="S1.SS1.SSS0.Px3.p2.10.m10.1.1">1</cn></annotation-xml><annotation encoding="application/x-tex" id="S1.SS1.SSS0.Px3.p2.10.m10.1c">1</annotation><annotation encoding="application/x-llamapun" id="S1.SS1.SSS0.Px3.p2.10.m10.1d">1</annotation></semantics></math>-VC-CAP with ideas from cactus augmentation in the edge-connectivity setting to obtain our result. The algorithm is tailored to the particulars of the SPQR representation and is technically quite involved. We refer the reader to Figure <a class="ltx_ref" href="https://arxiv.org/html/2503.00712v1#S4.F5" title="Figure 5 ‣ 4.2.1 SPQR Trees ‣ 4.2 Two-to-Three Augmentation ‣ 4 Vertex Connectivity Augmentation in Link-Arrival Model ‣ Streaming Algorithms for Network Design"><span class="ltx_text ltx_ref_tag">5</span></a> for an example of a <math alttext="2" class="ltx_Math" display="inline" id="S1.SS1.SSS0.Px3.p2.11.m11.1"><semantics id="S1.SS1.SSS0.Px3.p2.11.m11.1a"><mn id="S1.SS1.SSS0.Px3.p2.11.m11.1.1" xref="S1.SS1.SSS0.Px3.p2.11.m11.1.1.cmml">2</mn><annotation-xml encoding="MathML-Content" id="S1.SS1.SSS0.Px3.p2.11.m11.1b"><cn id="S1.SS1.SSS0.Px3.p2.11.m11.1.1.cmml" type="integer" xref="S1.SS1.SSS0.Px3.p2.11.m11.1.1">2</cn></annotation-xml><annotation encoding="application/x-tex" id="S1.SS1.SSS0.Px3.p2.11.m11.1c">2</annotation><annotation encoding="application/x-llamapun" id="S1.SS1.SSS0.Px3.p2.11.m11.1d">2</annotation></semantics></math>-connected graph and its SPQR tree.</p> </div> </section> </section> <section class="ltx_subsection" id="S1.SS2"> <h3 class="ltx_title ltx_title_subsection"> <span class="ltx_tag ltx_tag_subsection">1.2 </span>Related Work</h3> <section class="ltx_paragraph" id="S1.SS2.SSS0.Px1"> <h5 class="ltx_title ltx_title_paragraph">Offline Network Design:</h5> <div class="ltx_para" id="S1.SS2.SSS0.Px1.p1"> <p class="ltx_p" id="S1.SS2.SSS0.Px1.p1.37">There is substantial literature on network design; here we briefly discuss some relevant literature. EC-SNDP admits a <math alttext="2" class="ltx_Math" display="inline" id="S1.SS2.SSS0.Px1.p1.1.m1.1"><semantics id="S1.SS2.SSS0.Px1.p1.1.m1.1a"><mn id="S1.SS2.SSS0.Px1.p1.1.m1.1.1" xref="S1.SS2.SSS0.Px1.p1.1.m1.1.1.cmml">2</mn><annotation-xml encoding="MathML-Content" id="S1.SS2.SSS0.Px1.p1.1.m1.1b"><cn id="S1.SS2.SSS0.Px1.p1.1.m1.1.1.cmml" type="integer" xref="S1.SS2.SSS0.Px1.p1.1.m1.1.1">2</cn></annotation-xml><annotation encoding="application/x-tex" id="S1.SS2.SSS0.Px1.p1.1.m1.1c">2</annotation><annotation encoding="application/x-llamapun" id="S1.SS2.SSS0.Px1.p1.1.m1.1d">2</annotation></semantics></math>-approximation via the seminal work on iterated rounding by Jain <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.00712v1#bib.bibx53" title="">Jai01</a>]</cite> and this has been extended to ELC-SNDP <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.00712v1#bib.bibx39" title="">FJW06</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.00712v1#bib.bibx31" title="">CVV06</a>]</cite>. We note that the best known approximation for Steiner Forest, the special case with connectivity requirements in <math alttext="\{0,1\}" class="ltx_Math" display="inline" id="S1.SS2.SSS0.Px1.p1.2.m2.2"><semantics id="S1.SS2.SSS0.Px1.p1.2.m2.2a"><mrow id="S1.SS2.SSS0.Px1.p1.2.m2.2.3.2" xref="S1.SS2.SSS0.Px1.p1.2.m2.2.3.1.cmml"><mo id="S1.SS2.SSS0.Px1.p1.2.m2.2.3.2.1" stretchy="false" xref="S1.SS2.SSS0.Px1.p1.2.m2.2.3.1.cmml">{</mo><mn id="S1.SS2.SSS0.Px1.p1.2.m2.1.1" xref="S1.SS2.SSS0.Px1.p1.2.m2.1.1.cmml">0</mn><mo id="S1.SS2.SSS0.Px1.p1.2.m2.2.3.2.2" xref="S1.SS2.SSS0.Px1.p1.2.m2.2.3.1.cmml">,</mo><mn id="S1.SS2.SSS0.Px1.p1.2.m2.2.2" xref="S1.SS2.SSS0.Px1.p1.2.m2.2.2.cmml">1</mn><mo id="S1.SS2.SSS0.Px1.p1.2.m2.2.3.2.3" stretchy="false" xref="S1.SS2.SSS0.Px1.p1.2.m2.2.3.1.cmml">}</mo></mrow><annotation-xml encoding="MathML-Content" id="S1.SS2.SSS0.Px1.p1.2.m2.2b"><set id="S1.SS2.SSS0.Px1.p1.2.m2.2.3.1.cmml" xref="S1.SS2.SSS0.Px1.p1.2.m2.2.3.2"><cn id="S1.SS2.SSS0.Px1.p1.2.m2.1.1.cmml" type="integer" xref="S1.SS2.SSS0.Px1.p1.2.m2.1.1">0</cn><cn id="S1.SS2.SSS0.Px1.p1.2.m2.2.2.cmml" type="integer" xref="S1.SS2.SSS0.Px1.p1.2.m2.2.2">1</cn></set></annotation-xml><annotation encoding="application/x-tex" id="S1.SS2.SSS0.Px1.p1.2.m2.2c">\{0,1\}</annotation><annotation encoding="application/x-llamapun" id="S1.SS2.SSS0.Px1.p1.2.m2.2d">{ 0 , 1 }</annotation></semantics></math>, is <math alttext="2" class="ltx_Math" display="inline" id="S1.SS2.SSS0.Px1.p1.3.m3.1"><semantics id="S1.SS2.SSS0.Px1.p1.3.m3.1a"><mn id="S1.SS2.SSS0.Px1.p1.3.m3.1.1" xref="S1.SS2.SSS0.Px1.p1.3.m3.1.1.cmml">2</mn><annotation-xml encoding="MathML-Content" id="S1.SS2.SSS0.Px1.p1.3.m3.1b"><cn id="S1.SS2.SSS0.Px1.p1.3.m3.1.1.cmml" type="integer" xref="S1.SS2.SSS0.Px1.p1.3.m3.1.1">2</cn></annotation-xml><annotation encoding="application/x-tex" id="S1.SS2.SSS0.Px1.p1.3.m3.1c">2</annotation><annotation encoding="application/x-llamapun" id="S1.SS2.SSS0.Px1.p1.3.m3.1d">2</annotation></semantics></math>. Steiner tree is another important special case, and for this there is a <math alttext="(\ln 4+\epsilon)" class="ltx_Math" display="inline" id="S1.SS2.SSS0.Px1.p1.4.m4.1"><semantics id="S1.SS2.SSS0.Px1.p1.4.m4.1a"><mrow id="S1.SS2.SSS0.Px1.p1.4.m4.1.1.1" xref="S1.SS2.SSS0.Px1.p1.4.m4.1.1.1.1.cmml"><mo id="S1.SS2.SSS0.Px1.p1.4.m4.1.1.1.2" stretchy="false" xref="S1.SS2.SSS0.Px1.p1.4.m4.1.1.1.1.cmml">(</mo><mrow id="S1.SS2.SSS0.Px1.p1.4.m4.1.1.1.1" xref="S1.SS2.SSS0.Px1.p1.4.m4.1.1.1.1.cmml"><mrow id="S1.SS2.SSS0.Px1.p1.4.m4.1.1.1.1.2" xref="S1.SS2.SSS0.Px1.p1.4.m4.1.1.1.1.2.cmml"><mi id="S1.SS2.SSS0.Px1.p1.4.m4.1.1.1.1.2.1" xref="S1.SS2.SSS0.Px1.p1.4.m4.1.1.1.1.2.1.cmml">ln</mi><mo id="S1.SS2.SSS0.Px1.p1.4.m4.1.1.1.1.2a" lspace="0.167em" xref="S1.SS2.SSS0.Px1.p1.4.m4.1.1.1.1.2.cmml">⁡</mo><mn id="S1.SS2.SSS0.Px1.p1.4.m4.1.1.1.1.2.2" xref="S1.SS2.SSS0.Px1.p1.4.m4.1.1.1.1.2.2.cmml">4</mn></mrow><mo id="S1.SS2.SSS0.Px1.p1.4.m4.1.1.1.1.1" xref="S1.SS2.SSS0.Px1.p1.4.m4.1.1.1.1.1.cmml">+</mo><mi id="S1.SS2.SSS0.Px1.p1.4.m4.1.1.1.1.3" xref="S1.SS2.SSS0.Px1.p1.4.m4.1.1.1.1.3.cmml">ϵ</mi></mrow><mo id="S1.SS2.SSS0.Px1.p1.4.m4.1.1.1.3" stretchy="false" xref="S1.SS2.SSS0.Px1.p1.4.m4.1.1.1.1.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S1.SS2.SSS0.Px1.p1.4.m4.1b"><apply id="S1.SS2.SSS0.Px1.p1.4.m4.1.1.1.1.cmml" xref="S1.SS2.SSS0.Px1.p1.4.m4.1.1.1"><plus id="S1.SS2.SSS0.Px1.p1.4.m4.1.1.1.1.1.cmml" xref="S1.SS2.SSS0.Px1.p1.4.m4.1.1.1.1.1"></plus><apply id="S1.SS2.SSS0.Px1.p1.4.m4.1.1.1.1.2.cmml" xref="S1.SS2.SSS0.Px1.p1.4.m4.1.1.1.1.2"><ln id="S1.SS2.SSS0.Px1.p1.4.m4.1.1.1.1.2.1.cmml" xref="S1.SS2.SSS0.Px1.p1.4.m4.1.1.1.1.2.1"></ln><cn id="S1.SS2.SSS0.Px1.p1.4.m4.1.1.1.1.2.2.cmml" type="integer" xref="S1.SS2.SSS0.Px1.p1.4.m4.1.1.1.1.2.2">4</cn></apply><ci id="S1.SS2.SSS0.Px1.p1.4.m4.1.1.1.1.3.cmml" xref="S1.SS2.SSS0.Px1.p1.4.m4.1.1.1.1.3">italic-ϵ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS2.SSS0.Px1.p1.4.m4.1c">(\ln 4+\epsilon)</annotation><annotation encoding="application/x-llamapun" id="S1.SS2.SSS0.Px1.p1.4.m4.1d">( roman_ln 4 + italic_ϵ )</annotation></semantics></math>-approximation <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.00712v1#bib.bibx20" title="">BGRS13</a>]</cite>. Even for <math alttext="2" class="ltx_Math" display="inline" id="S1.SS2.SSS0.Px1.p1.5.m5.1"><semantics id="S1.SS2.SSS0.Px1.p1.5.m5.1a"><mn id="S1.SS2.SSS0.Px1.p1.5.m5.1.1" xref="S1.SS2.SSS0.Px1.p1.5.m5.1.1.cmml">2</mn><annotation-xml encoding="MathML-Content" id="S1.SS2.SSS0.Px1.p1.5.m5.1b"><cn id="S1.SS2.SSS0.Px1.p1.5.m5.1.1.cmml" type="integer" xref="S1.SS2.SSS0.Px1.p1.5.m5.1.1">2</cn></annotation-xml><annotation encoding="application/x-tex" id="S1.SS2.SSS0.Px1.p1.5.m5.1c">2</annotation><annotation encoding="application/x-llamapun" id="S1.SS2.SSS0.Px1.p1.5.m5.1d">2</annotation></semantics></math>-ECSS the best known approximation is <math alttext="2" class="ltx_Math" display="inline" id="S1.SS2.SSS0.Px1.p1.6.m6.1"><semantics id="S1.SS2.SSS0.Px1.p1.6.m6.1a"><mn id="S1.SS2.SSS0.Px1.p1.6.m6.1.1" xref="S1.SS2.SSS0.Px1.p1.6.m6.1.1.cmml">2</mn><annotation-xml encoding="MathML-Content" id="S1.SS2.SSS0.Px1.p1.6.m6.1b"><cn id="S1.SS2.SSS0.Px1.p1.6.m6.1.1.cmml" type="integer" xref="S1.SS2.SSS0.Px1.p1.6.m6.1.1">2</cn></annotation-xml><annotation encoding="application/x-tex" id="S1.SS2.SSS0.Px1.p1.6.m6.1c">2</annotation><annotation encoding="application/x-llamapun" id="S1.SS2.SSS0.Px1.p1.6.m6.1d">2</annotation></semantics></math>, although better bounds are known for the unweighted case. Recently there has been exciting progress on <math alttext="k" class="ltx_Math" display="inline" id="S1.SS2.SSS0.Px1.p1.7.m7.1"><semantics id="S1.SS2.SSS0.Px1.p1.7.m7.1a"><mi id="S1.SS2.SSS0.Px1.p1.7.m7.1.1" xref="S1.SS2.SSS0.Px1.p1.7.m7.1.1.cmml">k</mi><annotation-xml encoding="MathML-Content" id="S1.SS2.SSS0.Px1.p1.7.m7.1b"><ci id="S1.SS2.SSS0.Px1.p1.7.m7.1.1.cmml" xref="S1.SS2.SSS0.Px1.p1.7.m7.1.1">𝑘</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.SS2.SSS0.Px1.p1.7.m7.1c">k</annotation><annotation encoding="application/x-llamapun" id="S1.SS2.SSS0.Px1.p1.7.m7.1d">italic_k</annotation></semantics></math>-EC-CAP, starting with progress on the special case of weighted TAP (tree augmentation problem which corresponds to <math alttext="k=1" class="ltx_Math" display="inline" id="S1.SS2.SSS0.Px1.p1.8.m8.1"><semantics id="S1.SS2.SSS0.Px1.p1.8.m8.1a"><mrow id="S1.SS2.SSS0.Px1.p1.8.m8.1.1" xref="S1.SS2.SSS0.Px1.p1.8.m8.1.1.cmml"><mi id="S1.SS2.SSS0.Px1.p1.8.m8.1.1.2" xref="S1.SS2.SSS0.Px1.p1.8.m8.1.1.2.cmml">k</mi><mo id="S1.SS2.SSS0.Px1.p1.8.m8.1.1.1" xref="S1.SS2.SSS0.Px1.p1.8.m8.1.1.1.cmml">=</mo><mn id="S1.SS2.SSS0.Px1.p1.8.m8.1.1.3" xref="S1.SS2.SSS0.Px1.p1.8.m8.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S1.SS2.SSS0.Px1.p1.8.m8.1b"><apply id="S1.SS2.SSS0.Px1.p1.8.m8.1.1.cmml" xref="S1.SS2.SSS0.Px1.p1.8.m8.1.1"><eq id="S1.SS2.SSS0.Px1.p1.8.m8.1.1.1.cmml" xref="S1.SS2.SSS0.Px1.p1.8.m8.1.1.1"></eq><ci id="S1.SS2.SSS0.Px1.p1.8.m8.1.1.2.cmml" xref="S1.SS2.SSS0.Px1.p1.8.m8.1.1.2">𝑘</ci><cn id="S1.SS2.SSS0.Px1.p1.8.m8.1.1.3.cmml" type="integer" xref="S1.SS2.SSS0.Px1.p1.8.m8.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS2.SSS0.Px1.p1.8.m8.1c">k=1</annotation><annotation encoding="application/x-llamapun" id="S1.SS2.SSS0.Px1.p1.8.m8.1d">italic_k = 1</annotation></semantics></math>). For all <math alttext="k" class="ltx_Math" display="inline" id="S1.SS2.SSS0.Px1.p1.9.m9.1"><semantics id="S1.SS2.SSS0.Px1.p1.9.m9.1a"><mi id="S1.SS2.SSS0.Px1.p1.9.m9.1.1" xref="S1.SS2.SSS0.Px1.p1.9.m9.1.1.cmml">k</mi><annotation-xml encoding="MathML-Content" id="S1.SS2.SSS0.Px1.p1.9.m9.1b"><ci id="S1.SS2.SSS0.Px1.p1.9.m9.1.1.cmml" xref="S1.SS2.SSS0.Px1.p1.9.m9.1.1">𝑘</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.SS2.SSS0.Px1.p1.9.m9.1c">k</annotation><annotation encoding="application/x-llamapun" id="S1.SS2.SSS0.Px1.p1.9.m9.1d">italic_k</annotation></semantics></math>, <math alttext="k" class="ltx_Math" display="inline" id="S1.SS2.SSS0.Px1.p1.10.m10.1"><semantics id="S1.SS2.SSS0.Px1.p1.10.m10.1a"><mi id="S1.SS2.SSS0.Px1.p1.10.m10.1.1" xref="S1.SS2.SSS0.Px1.p1.10.m10.1.1.cmml">k</mi><annotation-xml encoding="MathML-Content" id="S1.SS2.SSS0.Px1.p1.10.m10.1b"><ci id="S1.SS2.SSS0.Px1.p1.10.m10.1.1.cmml" xref="S1.SS2.SSS0.Px1.p1.10.m10.1.1">𝑘</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.SS2.SSS0.Px1.p1.10.m10.1c">k</annotation><annotation encoding="application/x-llamapun" id="S1.SS2.SSS0.Px1.p1.10.m10.1d">italic_k</annotation></semantics></math>-EC-CAP admits a (<math alttext="1.5+\epsilon)" class="ltx_math_unparsed" display="inline" id="S1.SS2.SSS0.Px1.p1.11.m11.1"><semantics id="S1.SS2.SSS0.Px1.p1.11.m11.1a"><mrow id="S1.SS2.SSS0.Px1.p1.11.m11.1b"><mn id="S1.SS2.SSS0.Px1.p1.11.m11.1.1">1.5</mn><mo id="S1.SS2.SSS0.Px1.p1.11.m11.1.2">+</mo><mi id="S1.SS2.SSS0.Px1.p1.11.m11.1.3">ϵ</mi><mo id="S1.SS2.SSS0.Px1.p1.11.m11.1.4" stretchy="false">)</mo></mrow><annotation encoding="application/x-tex" id="S1.SS2.SSS0.Px1.p1.11.m11.1c">1.5+\epsilon)</annotation><annotation encoding="application/x-llamapun" id="S1.SS2.SSS0.Px1.p1.11.m11.1d">1.5 + italic_ϵ )</annotation></semantics></math>-approximation <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.00712v1#bib.bibx80" title="">TZ23</a>]</cite>—see also <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.00712v1#bib.bibx78" title="">TZ22a</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.00712v1#bib.bibx79" title="">TZ22b</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.00712v1#bib.bibx19" title="">BGA20</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.00712v1#bib.bibx29" title="">CTZ21</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.00712v1#bib.bibx45" title="">GGJA23</a>]</cite>. In contrast to the constant factor approximation results for edge-connectivity, the best known approximation for VC-SNDP is <math alttext="O(k^{3}\log n)" class="ltx_Math" display="inline" id="S1.SS2.SSS0.Px1.p1.12.m12.1"><semantics id="S1.SS2.SSS0.Px1.p1.12.m12.1a"><mrow id="S1.SS2.SSS0.Px1.p1.12.m12.1.1" xref="S1.SS2.SSS0.Px1.p1.12.m12.1.1.cmml"><mi id="S1.SS2.SSS0.Px1.p1.12.m12.1.1.3" xref="S1.SS2.SSS0.Px1.p1.12.m12.1.1.3.cmml">O</mi><mo id="S1.SS2.SSS0.Px1.p1.12.m12.1.1.2" xref="S1.SS2.SSS0.Px1.p1.12.m12.1.1.2.cmml">⁢</mo><mrow id="S1.SS2.SSS0.Px1.p1.12.m12.1.1.1.1" xref="S1.SS2.SSS0.Px1.p1.12.m12.1.1.1.1.1.cmml"><mo id="S1.SS2.SSS0.Px1.p1.12.m12.1.1.1.1.2" stretchy="false" xref="S1.SS2.SSS0.Px1.p1.12.m12.1.1.1.1.1.cmml">(</mo><mrow id="S1.SS2.SSS0.Px1.p1.12.m12.1.1.1.1.1" xref="S1.SS2.SSS0.Px1.p1.12.m12.1.1.1.1.1.cmml"><msup id="S1.SS2.SSS0.Px1.p1.12.m12.1.1.1.1.1.2" xref="S1.SS2.SSS0.Px1.p1.12.m12.1.1.1.1.1.2.cmml"><mi id="S1.SS2.SSS0.Px1.p1.12.m12.1.1.1.1.1.2.2" xref="S1.SS2.SSS0.Px1.p1.12.m12.1.1.1.1.1.2.2.cmml">k</mi><mn id="S1.SS2.SSS0.Px1.p1.12.m12.1.1.1.1.1.2.3" xref="S1.SS2.SSS0.Px1.p1.12.m12.1.1.1.1.1.2.3.cmml">3</mn></msup><mo id="S1.SS2.SSS0.Px1.p1.12.m12.1.1.1.1.1.1" lspace="0.167em" xref="S1.SS2.SSS0.Px1.p1.12.m12.1.1.1.1.1.1.cmml">⁢</mo><mrow id="S1.SS2.SSS0.Px1.p1.12.m12.1.1.1.1.1.3" xref="S1.SS2.SSS0.Px1.p1.12.m12.1.1.1.1.1.3.cmml"><mi id="S1.SS2.SSS0.Px1.p1.12.m12.1.1.1.1.1.3.1" xref="S1.SS2.SSS0.Px1.p1.12.m12.1.1.1.1.1.3.1.cmml">log</mi><mo id="S1.SS2.SSS0.Px1.p1.12.m12.1.1.1.1.1.3a" lspace="0.167em" xref="S1.SS2.SSS0.Px1.p1.12.m12.1.1.1.1.1.3.cmml">⁡</mo><mi id="S1.SS2.SSS0.Px1.p1.12.m12.1.1.1.1.1.3.2" xref="S1.SS2.SSS0.Px1.p1.12.m12.1.1.1.1.1.3.2.cmml">n</mi></mrow></mrow><mo id="S1.SS2.SSS0.Px1.p1.12.m12.1.1.1.1.3" stretchy="false" xref="S1.SS2.SSS0.Px1.p1.12.m12.1.1.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.SS2.SSS0.Px1.p1.12.m12.1b"><apply id="S1.SS2.SSS0.Px1.p1.12.m12.1.1.cmml" xref="S1.SS2.SSS0.Px1.p1.12.m12.1.1"><times id="S1.SS2.SSS0.Px1.p1.12.m12.1.1.2.cmml" xref="S1.SS2.SSS0.Px1.p1.12.m12.1.1.2"></times><ci id="S1.SS2.SSS0.Px1.p1.12.m12.1.1.3.cmml" xref="S1.SS2.SSS0.Px1.p1.12.m12.1.1.3">𝑂</ci><apply id="S1.SS2.SSS0.Px1.p1.12.m12.1.1.1.1.1.cmml" xref="S1.SS2.SSS0.Px1.p1.12.m12.1.1.1.1"><times id="S1.SS2.SSS0.Px1.p1.12.m12.1.1.1.1.1.1.cmml" xref="S1.SS2.SSS0.Px1.p1.12.m12.1.1.1.1.1.1"></times><apply id="S1.SS2.SSS0.Px1.p1.12.m12.1.1.1.1.1.2.cmml" xref="S1.SS2.SSS0.Px1.p1.12.m12.1.1.1.1.1.2"><csymbol cd="ambiguous" id="S1.SS2.SSS0.Px1.p1.12.m12.1.1.1.1.1.2.1.cmml" xref="S1.SS2.SSS0.Px1.p1.12.m12.1.1.1.1.1.2">superscript</csymbol><ci id="S1.SS2.SSS0.Px1.p1.12.m12.1.1.1.1.1.2.2.cmml" xref="S1.SS2.SSS0.Px1.p1.12.m12.1.1.1.1.1.2.2">𝑘</ci><cn id="S1.SS2.SSS0.Px1.p1.12.m12.1.1.1.1.1.2.3.cmml" type="integer" xref="S1.SS2.SSS0.Px1.p1.12.m12.1.1.1.1.1.2.3">3</cn></apply><apply id="S1.SS2.SSS0.Px1.p1.12.m12.1.1.1.1.1.3.cmml" xref="S1.SS2.SSS0.Px1.p1.12.m12.1.1.1.1.1.3"><log id="S1.SS2.SSS0.Px1.p1.12.m12.1.1.1.1.1.3.1.cmml" xref="S1.SS2.SSS0.Px1.p1.12.m12.1.1.1.1.1.3.1"></log><ci id="S1.SS2.SSS0.Px1.p1.12.m12.1.1.1.1.1.3.2.cmml" xref="S1.SS2.SSS0.Px1.p1.12.m12.1.1.1.1.1.3.2">𝑛</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS2.SSS0.Px1.p1.12.m12.1c">O(k^{3}\log n)</annotation><annotation encoding="application/x-llamapun" id="S1.SS2.SSS0.Px1.p1.12.m12.1d">italic_O ( italic_k start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT roman_log italic_n )</annotation></semantics></math> due to Chuzhoy and Khanna <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.00712v1#bib.bibx26" title="">CK09</a>]</cite>. Moreover, even for the single-source setting it is known that the approximation ratio needs to depend on <math alttext="k" class="ltx_Math" display="inline" id="S1.SS2.SSS0.Px1.p1.13.m13.1"><semantics id="S1.SS2.SSS0.Px1.p1.13.m13.1a"><mi id="S1.SS2.SSS0.Px1.p1.13.m13.1.1" xref="S1.SS2.SSS0.Px1.p1.13.m13.1.1.cmml">k</mi><annotation-xml encoding="MathML-Content" id="S1.SS2.SSS0.Px1.p1.13.m13.1b"><ci id="S1.SS2.SSS0.Px1.p1.13.m13.1.1.cmml" xref="S1.SS2.SSS0.Px1.p1.13.m13.1.1">𝑘</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.SS2.SSS0.Px1.p1.13.m13.1c">k</annotation><annotation encoding="application/x-llamapun" id="S1.SS2.SSS0.Px1.p1.13.m13.1d">italic_k</annotation></semantics></math> for sufficiently large <math alttext="k" class="ltx_Math" display="inline" id="S1.SS2.SSS0.Px1.p1.14.m14.1"><semantics id="S1.SS2.SSS0.Px1.p1.14.m14.1a"><mi id="S1.SS2.SSS0.Px1.p1.14.m14.1.1" xref="S1.SS2.SSS0.Px1.p1.14.m14.1.1.cmml">k</mi><annotation-xml encoding="MathML-Content" id="S1.SS2.SSS0.Px1.p1.14.m14.1b"><ci id="S1.SS2.SSS0.Px1.p1.14.m14.1.1.cmml" xref="S1.SS2.SSS0.Px1.p1.14.m14.1.1">𝑘</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.SS2.SSS0.Px1.p1.14.m14.1c">k</annotation><annotation encoding="application/x-llamapun" id="S1.SS2.SSS0.Px1.p1.14.m14.1d">italic_k</annotation></semantics></math> <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.00712v1#bib.bibx22" title="">CCK08</a>]</cite>. The single-source problem admits an <math alttext="O(k^{2})" class="ltx_Math" display="inline" id="S1.SS2.SSS0.Px1.p1.15.m15.1"><semantics id="S1.SS2.SSS0.Px1.p1.15.m15.1a"><mrow id="S1.SS2.SSS0.Px1.p1.15.m15.1.1" xref="S1.SS2.SSS0.Px1.p1.15.m15.1.1.cmml"><mi id="S1.SS2.SSS0.Px1.p1.15.m15.1.1.3" xref="S1.SS2.SSS0.Px1.p1.15.m15.1.1.3.cmml">O</mi><mo id="S1.SS2.SSS0.Px1.p1.15.m15.1.1.2" xref="S1.SS2.SSS0.Px1.p1.15.m15.1.1.2.cmml">⁢</mo><mrow id="S1.SS2.SSS0.Px1.p1.15.m15.1.1.1.1" xref="S1.SS2.SSS0.Px1.p1.15.m15.1.1.1.1.1.cmml"><mo id="S1.SS2.SSS0.Px1.p1.15.m15.1.1.1.1.2" stretchy="false" xref="S1.SS2.SSS0.Px1.p1.15.m15.1.1.1.1.1.cmml">(</mo><msup id="S1.SS2.SSS0.Px1.p1.15.m15.1.1.1.1.1" xref="S1.SS2.SSS0.Px1.p1.15.m15.1.1.1.1.1.cmml"><mi id="S1.SS2.SSS0.Px1.p1.15.m15.1.1.1.1.1.2" xref="S1.SS2.SSS0.Px1.p1.15.m15.1.1.1.1.1.2.cmml">k</mi><mn id="S1.SS2.SSS0.Px1.p1.15.m15.1.1.1.1.1.3" xref="S1.SS2.SSS0.Px1.p1.15.m15.1.1.1.1.1.3.cmml">2</mn></msup><mo id="S1.SS2.SSS0.Px1.p1.15.m15.1.1.1.1.3" stretchy="false" xref="S1.SS2.SSS0.Px1.p1.15.m15.1.1.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.SS2.SSS0.Px1.p1.15.m15.1b"><apply id="S1.SS2.SSS0.Px1.p1.15.m15.1.1.cmml" xref="S1.SS2.SSS0.Px1.p1.15.m15.1.1"><times id="S1.SS2.SSS0.Px1.p1.15.m15.1.1.2.cmml" xref="S1.SS2.SSS0.Px1.p1.15.m15.1.1.2"></times><ci id="S1.SS2.SSS0.Px1.p1.15.m15.1.1.3.cmml" xref="S1.SS2.SSS0.Px1.p1.15.m15.1.1.3">𝑂</ci><apply id="S1.SS2.SSS0.Px1.p1.15.m15.1.1.1.1.1.cmml" xref="S1.SS2.SSS0.Px1.p1.15.m15.1.1.1.1"><csymbol cd="ambiguous" id="S1.SS2.SSS0.Px1.p1.15.m15.1.1.1.1.1.1.cmml" xref="S1.SS2.SSS0.Px1.p1.15.m15.1.1.1.1">superscript</csymbol><ci id="S1.SS2.SSS0.Px1.p1.15.m15.1.1.1.1.1.2.cmml" xref="S1.SS2.SSS0.Px1.p1.15.m15.1.1.1.1.1.2">𝑘</ci><cn id="S1.SS2.SSS0.Px1.p1.15.m15.1.1.1.1.1.3.cmml" type="integer" xref="S1.SS2.SSS0.Px1.p1.15.m15.1.1.1.1.1.3">2</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS2.SSS0.Px1.p1.15.m15.1c">O(k^{2})</annotation><annotation encoding="application/x-llamapun" id="S1.SS2.SSS0.Px1.p1.15.m15.1d">italic_O ( italic_k start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT )</annotation></semantics></math>-approximation <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.00712v1#bib.bibx70" title="">Nut12</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.00712v1#bib.bibx72" title="">Nut18a</a>]</cite> and subset <math alttext="k" class="ltx_Math" display="inline" id="S1.SS2.SSS0.Px1.p1.16.m16.1"><semantics id="S1.SS2.SSS0.Px1.p1.16.m16.1a"><mi id="S1.SS2.SSS0.Px1.p1.16.m16.1.1" xref="S1.SS2.SSS0.Px1.p1.16.m16.1.1.cmml">k</mi><annotation-xml encoding="MathML-Content" id="S1.SS2.SSS0.Px1.p1.16.m16.1b"><ci id="S1.SS2.SSS0.Px1.p1.16.m16.1.1.cmml" xref="S1.SS2.SSS0.Px1.p1.16.m16.1.1">𝑘</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.SS2.SSS0.Px1.p1.16.m16.1c">k</annotation><annotation encoding="application/x-llamapun" id="S1.SS2.SSS0.Px1.p1.16.m16.1d">italic_k</annotation></semantics></math>-connectivity admits an <math alttext="O(k\log^{2}k)" class="ltx_Math" display="inline" id="S1.SS2.SSS0.Px1.p1.17.m17.1"><semantics id="S1.SS2.SSS0.Px1.p1.17.m17.1a"><mrow id="S1.SS2.SSS0.Px1.p1.17.m17.1.1" xref="S1.SS2.SSS0.Px1.p1.17.m17.1.1.cmml"><mi id="S1.SS2.SSS0.Px1.p1.17.m17.1.1.3" xref="S1.SS2.SSS0.Px1.p1.17.m17.1.1.3.cmml">O</mi><mo id="S1.SS2.SSS0.Px1.p1.17.m17.1.1.2" xref="S1.SS2.SSS0.Px1.p1.17.m17.1.1.2.cmml">⁢</mo><mrow id="S1.SS2.SSS0.Px1.p1.17.m17.1.1.1.1" xref="S1.SS2.SSS0.Px1.p1.17.m17.1.1.1.1.1.cmml"><mo id="S1.SS2.SSS0.Px1.p1.17.m17.1.1.1.1.2" stretchy="false" xref="S1.SS2.SSS0.Px1.p1.17.m17.1.1.1.1.1.cmml">(</mo><mrow id="S1.SS2.SSS0.Px1.p1.17.m17.1.1.1.1.1" xref="S1.SS2.SSS0.Px1.p1.17.m17.1.1.1.1.1.cmml"><mi id="S1.SS2.SSS0.Px1.p1.17.m17.1.1.1.1.1.2" xref="S1.SS2.SSS0.Px1.p1.17.m17.1.1.1.1.1.2.cmml">k</mi><mo id="S1.SS2.SSS0.Px1.p1.17.m17.1.1.1.1.1.1" lspace="0.167em" xref="S1.SS2.SSS0.Px1.p1.17.m17.1.1.1.1.1.1.cmml">⁢</mo><mrow id="S1.SS2.SSS0.Px1.p1.17.m17.1.1.1.1.1.3" xref="S1.SS2.SSS0.Px1.p1.17.m17.1.1.1.1.1.3.cmml"><msup id="S1.SS2.SSS0.Px1.p1.17.m17.1.1.1.1.1.3.1" xref="S1.SS2.SSS0.Px1.p1.17.m17.1.1.1.1.1.3.1.cmml"><mi id="S1.SS2.SSS0.Px1.p1.17.m17.1.1.1.1.1.3.1.2" xref="S1.SS2.SSS0.Px1.p1.17.m17.1.1.1.1.1.3.1.2.cmml">log</mi><mn id="S1.SS2.SSS0.Px1.p1.17.m17.1.1.1.1.1.3.1.3" xref="S1.SS2.SSS0.Px1.p1.17.m17.1.1.1.1.1.3.1.3.cmml">2</mn></msup><mo id="S1.SS2.SSS0.Px1.p1.17.m17.1.1.1.1.1.3a" lspace="0.167em" xref="S1.SS2.SSS0.Px1.p1.17.m17.1.1.1.1.1.3.cmml">⁡</mo><mi id="S1.SS2.SSS0.Px1.p1.17.m17.1.1.1.1.1.3.2" xref="S1.SS2.SSS0.Px1.p1.17.m17.1.1.1.1.1.3.2.cmml">k</mi></mrow></mrow><mo id="S1.SS2.SSS0.Px1.p1.17.m17.1.1.1.1.3" stretchy="false" xref="S1.SS2.SSS0.Px1.p1.17.m17.1.1.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.SS2.SSS0.Px1.p1.17.m17.1b"><apply id="S1.SS2.SSS0.Px1.p1.17.m17.1.1.cmml" xref="S1.SS2.SSS0.Px1.p1.17.m17.1.1"><times id="S1.SS2.SSS0.Px1.p1.17.m17.1.1.2.cmml" xref="S1.SS2.SSS0.Px1.p1.17.m17.1.1.2"></times><ci id="S1.SS2.SSS0.Px1.p1.17.m17.1.1.3.cmml" xref="S1.SS2.SSS0.Px1.p1.17.m17.1.1.3">𝑂</ci><apply id="S1.SS2.SSS0.Px1.p1.17.m17.1.1.1.1.1.cmml" xref="S1.SS2.SSS0.Px1.p1.17.m17.1.1.1.1"><times id="S1.SS2.SSS0.Px1.p1.17.m17.1.1.1.1.1.1.cmml" xref="S1.SS2.SSS0.Px1.p1.17.m17.1.1.1.1.1.1"></times><ci id="S1.SS2.SSS0.Px1.p1.17.m17.1.1.1.1.1.2.cmml" xref="S1.SS2.SSS0.Px1.p1.17.m17.1.1.1.1.1.2">𝑘</ci><apply id="S1.SS2.SSS0.Px1.p1.17.m17.1.1.1.1.1.3.cmml" xref="S1.SS2.SSS0.Px1.p1.17.m17.1.1.1.1.1.3"><apply id="S1.SS2.SSS0.Px1.p1.17.m17.1.1.1.1.1.3.1.cmml" xref="S1.SS2.SSS0.Px1.p1.17.m17.1.1.1.1.1.3.1"><csymbol cd="ambiguous" id="S1.SS2.SSS0.Px1.p1.17.m17.1.1.1.1.1.3.1.1.cmml" xref="S1.SS2.SSS0.Px1.p1.17.m17.1.1.1.1.1.3.1">superscript</csymbol><log id="S1.SS2.SSS0.Px1.p1.17.m17.1.1.1.1.1.3.1.2.cmml" xref="S1.SS2.SSS0.Px1.p1.17.m17.1.1.1.1.1.3.1.2"></log><cn id="S1.SS2.SSS0.Px1.p1.17.m17.1.1.1.1.1.3.1.3.cmml" type="integer" xref="S1.SS2.SSS0.Px1.p1.17.m17.1.1.1.1.1.3.1.3">2</cn></apply><ci id="S1.SS2.SSS0.Px1.p1.17.m17.1.1.1.1.1.3.2.cmml" xref="S1.SS2.SSS0.Px1.p1.17.m17.1.1.1.1.1.3.2">𝑘</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS2.SSS0.Px1.p1.17.m17.1c">O(k\log^{2}k)</annotation><annotation encoding="application/x-llamapun" id="S1.SS2.SSS0.Px1.p1.17.m17.1d">italic_O ( italic_k roman_log start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_k )</annotation></semantics></math>-approximation <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.00712v1#bib.bibx64" title="">Lae15</a>]</cite>. Several special cases have better approximation bounds. When <math alttext="k\leq 2" class="ltx_Math" display="inline" id="S1.SS2.SSS0.Px1.p1.18.m18.1"><semantics id="S1.SS2.SSS0.Px1.p1.18.m18.1a"><mrow id="S1.SS2.SSS0.Px1.p1.18.m18.1.1" xref="S1.SS2.SSS0.Px1.p1.18.m18.1.1.cmml"><mi id="S1.SS2.SSS0.Px1.p1.18.m18.1.1.2" xref="S1.SS2.SSS0.Px1.p1.18.m18.1.1.2.cmml">k</mi><mo id="S1.SS2.SSS0.Px1.p1.18.m18.1.1.1" xref="S1.SS2.SSS0.Px1.p1.18.m18.1.1.1.cmml">≤</mo><mn id="S1.SS2.SSS0.Px1.p1.18.m18.1.1.3" xref="S1.SS2.SSS0.Px1.p1.18.m18.1.1.3.cmml">2</mn></mrow><annotation-xml encoding="MathML-Content" id="S1.SS2.SSS0.Px1.p1.18.m18.1b"><apply id="S1.SS2.SSS0.Px1.p1.18.m18.1.1.cmml" xref="S1.SS2.SSS0.Px1.p1.18.m18.1.1"><leq id="S1.SS2.SSS0.Px1.p1.18.m18.1.1.1.cmml" xref="S1.SS2.SSS0.Px1.p1.18.m18.1.1.1"></leq><ci id="S1.SS2.SSS0.Px1.p1.18.m18.1.1.2.cmml" xref="S1.SS2.SSS0.Px1.p1.18.m18.1.1.2">𝑘</ci><cn id="S1.SS2.SSS0.Px1.p1.18.m18.1.1.3.cmml" type="integer" xref="S1.SS2.SSS0.Px1.p1.18.m18.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS2.SSS0.Px1.p1.18.m18.1c">k\leq 2</annotation><annotation encoding="application/x-llamapun" id="S1.SS2.SSS0.Px1.p1.18.m18.1d">italic_k ≤ 2</annotation></semantics></math>, VC-SNDP admits a <math alttext="2" class="ltx_Math" display="inline" id="S1.SS2.SSS0.Px1.p1.19.m19.1"><semantics id="S1.SS2.SSS0.Px1.p1.19.m19.1a"><mn id="S1.SS2.SSS0.Px1.p1.19.m19.1.1" xref="S1.SS2.SSS0.Px1.p1.19.m19.1.1.cmml">2</mn><annotation-xml encoding="MathML-Content" id="S1.SS2.SSS0.Px1.p1.19.m19.1b"><cn id="S1.SS2.SSS0.Px1.p1.19.m19.1.1.cmml" type="integer" xref="S1.SS2.SSS0.Px1.p1.19.m19.1.1">2</cn></annotation-xml><annotation encoding="application/x-tex" id="S1.SS2.SSS0.Px1.p1.19.m19.1c">2</annotation><annotation encoding="application/x-llamapun" id="S1.SS2.SSS0.Px1.p1.19.m19.1d">2</annotation></semantics></math>-approximation <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.00712v1#bib.bibx39" title="">FJW06</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.00712v1#bib.bibx31" title="">CVV06</a>]</cite> which also implies that <math alttext="k" class="ltx_Math" display="inline" id="S1.SS2.SSS0.Px1.p1.20.m20.1"><semantics id="S1.SS2.SSS0.Px1.p1.20.m20.1a"><mi id="S1.SS2.SSS0.Px1.p1.20.m20.1.1" xref="S1.SS2.SSS0.Px1.p1.20.m20.1.1.cmml">k</mi><annotation-xml encoding="MathML-Content" id="S1.SS2.SSS0.Px1.p1.20.m20.1b"><ci id="S1.SS2.SSS0.Px1.p1.20.m20.1.1.cmml" xref="S1.SS2.SSS0.Px1.p1.20.m20.1.1">𝑘</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.SS2.SSS0.Px1.p1.20.m20.1c">k</annotation><annotation encoding="application/x-llamapun" id="S1.SS2.SSS0.Px1.p1.20.m20.1d">italic_k</annotation></semantics></math>-VCSS for <math alttext="k=2" class="ltx_Math" display="inline" id="S1.SS2.SSS0.Px1.p1.21.m21.1"><semantics id="S1.SS2.SSS0.Px1.p1.21.m21.1a"><mrow id="S1.SS2.SSS0.Px1.p1.21.m21.1.1" xref="S1.SS2.SSS0.Px1.p1.21.m21.1.1.cmml"><mi id="S1.SS2.SSS0.Px1.p1.21.m21.1.1.2" xref="S1.SS2.SSS0.Px1.p1.21.m21.1.1.2.cmml">k</mi><mo id="S1.SS2.SSS0.Px1.p1.21.m21.1.1.1" xref="S1.SS2.SSS0.Px1.p1.21.m21.1.1.1.cmml">=</mo><mn id="S1.SS2.SSS0.Px1.p1.21.m21.1.1.3" xref="S1.SS2.SSS0.Px1.p1.21.m21.1.1.3.cmml">2</mn></mrow><annotation-xml encoding="MathML-Content" id="S1.SS2.SSS0.Px1.p1.21.m21.1b"><apply id="S1.SS2.SSS0.Px1.p1.21.m21.1.1.cmml" xref="S1.SS2.SSS0.Px1.p1.21.m21.1.1"><eq id="S1.SS2.SSS0.Px1.p1.21.m21.1.1.1.cmml" xref="S1.SS2.SSS0.Px1.p1.21.m21.1.1.1"></eq><ci id="S1.SS2.SSS0.Px1.p1.21.m21.1.1.2.cmml" xref="S1.SS2.SSS0.Px1.p1.21.m21.1.1.2">𝑘</ci><cn id="S1.SS2.SSS0.Px1.p1.21.m21.1.1.3.cmml" type="integer" xref="S1.SS2.SSS0.Px1.p1.21.m21.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS2.SSS0.Px1.p1.21.m21.1c">k=2</annotation><annotation encoding="application/x-llamapun" id="S1.SS2.SSS0.Px1.p1.21.m21.1d">italic_k = 2</annotation></semantics></math> and <math alttext="k" class="ltx_Math" display="inline" id="S1.SS2.SSS0.Px1.p1.22.m22.1"><semantics id="S1.SS2.SSS0.Px1.p1.22.m22.1a"><mi id="S1.SS2.SSS0.Px1.p1.22.m22.1.1" xref="S1.SS2.SSS0.Px1.p1.22.m22.1.1.cmml">k</mi><annotation-xml encoding="MathML-Content" id="S1.SS2.SSS0.Px1.p1.22.m22.1b"><ci id="S1.SS2.SSS0.Px1.p1.22.m22.1.1.cmml" xref="S1.SS2.SSS0.Px1.p1.22.m22.1.1">𝑘</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.SS2.SSS0.Px1.p1.22.m22.1c">k</annotation><annotation encoding="application/x-llamapun" id="S1.SS2.SSS0.Px1.p1.22.m22.1d">italic_k</annotation></semantics></math>-VC-CAP for <math alttext="k=1" class="ltx_Math" display="inline" id="S1.SS2.SSS0.Px1.p1.23.m23.1"><semantics id="S1.SS2.SSS0.Px1.p1.23.m23.1a"><mrow id="S1.SS2.SSS0.Px1.p1.23.m23.1.1" xref="S1.SS2.SSS0.Px1.p1.23.m23.1.1.cmml"><mi id="S1.SS2.SSS0.Px1.p1.23.m23.1.1.2" xref="S1.SS2.SSS0.Px1.p1.23.m23.1.1.2.cmml">k</mi><mo id="S1.SS2.SSS0.Px1.p1.23.m23.1.1.1" xref="S1.SS2.SSS0.Px1.p1.23.m23.1.1.1.cmml">=</mo><mn id="S1.SS2.SSS0.Px1.p1.23.m23.1.1.3" xref="S1.SS2.SSS0.Px1.p1.23.m23.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S1.SS2.SSS0.Px1.p1.23.m23.1b"><apply id="S1.SS2.SSS0.Px1.p1.23.m23.1.1.cmml" xref="S1.SS2.SSS0.Px1.p1.23.m23.1.1"><eq id="S1.SS2.SSS0.Px1.p1.23.m23.1.1.1.cmml" xref="S1.SS2.SSS0.Px1.p1.23.m23.1.1.1"></eq><ci id="S1.SS2.SSS0.Px1.p1.23.m23.1.1.2.cmml" xref="S1.SS2.SSS0.Px1.p1.23.m23.1.1.2">𝑘</ci><cn id="S1.SS2.SSS0.Px1.p1.23.m23.1.1.3.cmml" type="integer" xref="S1.SS2.SSS0.Px1.p1.23.m23.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS2.SSS0.Px1.p1.23.m23.1c">k=1</annotation><annotation encoding="application/x-llamapun" id="S1.SS2.SSS0.Px1.p1.23.m23.1d">italic_k = 1</annotation></semantics></math> admit a <math alttext="2" class="ltx_Math" display="inline" id="S1.SS2.SSS0.Px1.p1.24.m24.1"><semantics id="S1.SS2.SSS0.Px1.p1.24.m24.1a"><mn id="S1.SS2.SSS0.Px1.p1.24.m24.1.1" xref="S1.SS2.SSS0.Px1.p1.24.m24.1.1.cmml">2</mn><annotation-xml encoding="MathML-Content" id="S1.SS2.SSS0.Px1.p1.24.m24.1b"><cn id="S1.SS2.SSS0.Px1.p1.24.m24.1.1.cmml" type="integer" xref="S1.SS2.SSS0.Px1.p1.24.m24.1.1">2</cn></annotation-xml><annotation encoding="application/x-tex" id="S1.SS2.SSS0.Px1.p1.24.m24.1c">2</annotation><annotation encoding="application/x-llamapun" id="S1.SS2.SSS0.Px1.p1.24.m24.1d">2</annotation></semantics></math>-approximation. For <math alttext="k" class="ltx_Math" display="inline" id="S1.SS2.SSS0.Px1.p1.25.m25.1"><semantics id="S1.SS2.SSS0.Px1.p1.25.m25.1a"><mi id="S1.SS2.SSS0.Px1.p1.25.m25.1.1" xref="S1.SS2.SSS0.Px1.p1.25.m25.1.1.cmml">k</mi><annotation-xml encoding="MathML-Content" id="S1.SS2.SSS0.Px1.p1.25.m25.1b"><ci id="S1.SS2.SSS0.Px1.p1.25.m25.1.1.cmml" xref="S1.SS2.SSS0.Px1.p1.25.m25.1.1">𝑘</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.SS2.SSS0.Px1.p1.25.m25.1c">k</annotation><annotation encoding="application/x-llamapun" id="S1.SS2.SSS0.Px1.p1.25.m25.1d">italic_k</annotation></semantics></math>-VCSS a <math alttext="(4+\epsilon)" class="ltx_Math" display="inline" id="S1.SS2.SSS0.Px1.p1.26.m26.1"><semantics id="S1.SS2.SSS0.Px1.p1.26.m26.1a"><mrow id="S1.SS2.SSS0.Px1.p1.26.m26.1.1.1" xref="S1.SS2.SSS0.Px1.p1.26.m26.1.1.1.1.cmml"><mo id="S1.SS2.SSS0.Px1.p1.26.m26.1.1.1.2" stretchy="false" xref="S1.SS2.SSS0.Px1.p1.26.m26.1.1.1.1.cmml">(</mo><mrow id="S1.SS2.SSS0.Px1.p1.26.m26.1.1.1.1" xref="S1.SS2.SSS0.Px1.p1.26.m26.1.1.1.1.cmml"><mn id="S1.SS2.SSS0.Px1.p1.26.m26.1.1.1.1.2" xref="S1.SS2.SSS0.Px1.p1.26.m26.1.1.1.1.2.cmml">4</mn><mo id="S1.SS2.SSS0.Px1.p1.26.m26.1.1.1.1.1" xref="S1.SS2.SSS0.Px1.p1.26.m26.1.1.1.1.1.cmml">+</mo><mi id="S1.SS2.SSS0.Px1.p1.26.m26.1.1.1.1.3" xref="S1.SS2.SSS0.Px1.p1.26.m26.1.1.1.1.3.cmml">ϵ</mi></mrow><mo id="S1.SS2.SSS0.Px1.p1.26.m26.1.1.1.3" stretchy="false" xref="S1.SS2.SSS0.Px1.p1.26.m26.1.1.1.1.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S1.SS2.SSS0.Px1.p1.26.m26.1b"><apply id="S1.SS2.SSS0.Px1.p1.26.m26.1.1.1.1.cmml" xref="S1.SS2.SSS0.Px1.p1.26.m26.1.1.1"><plus id="S1.SS2.SSS0.Px1.p1.26.m26.1.1.1.1.1.cmml" xref="S1.SS2.SSS0.Px1.p1.26.m26.1.1.1.1.1"></plus><cn id="S1.SS2.SSS0.Px1.p1.26.m26.1.1.1.1.2.cmml" type="integer" xref="S1.SS2.SSS0.Px1.p1.26.m26.1.1.1.1.2">4</cn><ci id="S1.SS2.SSS0.Px1.p1.26.m26.1.1.1.1.3.cmml" xref="S1.SS2.SSS0.Px1.p1.26.m26.1.1.1.1.3">italic-ϵ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS2.SSS0.Px1.p1.26.m26.1c">(4+\epsilon)</annotation><annotation encoding="application/x-llamapun" id="S1.SS2.SSS0.Px1.p1.26.m26.1d">( 4 + italic_ϵ )</annotation></semantics></math>-approximation is known when <math alttext="n" class="ltx_Math" display="inline" id="S1.SS2.SSS0.Px1.p1.27.m27.1"><semantics id="S1.SS2.SSS0.Px1.p1.27.m27.1a"><mi id="S1.SS2.SSS0.Px1.p1.27.m27.1.1" xref="S1.SS2.SSS0.Px1.p1.27.m27.1.1.cmml">n</mi><annotation-xml encoding="MathML-Content" id="S1.SS2.SSS0.Px1.p1.27.m27.1b"><ci id="S1.SS2.SSS0.Px1.p1.27.m27.1.1.cmml" xref="S1.SS2.SSS0.Px1.p1.27.m27.1.1">𝑛</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.SS2.SSS0.Px1.p1.27.m27.1c">n</annotation><annotation encoding="application/x-llamapun" id="S1.SS2.SSS0.Px1.p1.27.m27.1d">italic_n</annotation></semantics></math> is large compared to <math alttext="k" class="ltx_Math" display="inline" id="S1.SS2.SSS0.Px1.p1.28.m28.1"><semantics id="S1.SS2.SSS0.Px1.p1.28.m28.1a"><mi id="S1.SS2.SSS0.Px1.p1.28.m28.1.1" xref="S1.SS2.SSS0.Px1.p1.28.m28.1.1.cmml">k</mi><annotation-xml encoding="MathML-Content" id="S1.SS2.SSS0.Px1.p1.28.m28.1b"><ci id="S1.SS2.SSS0.Px1.p1.28.m28.1.1.cmml" xref="S1.SS2.SSS0.Px1.p1.28.m28.1.1">𝑘</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.SS2.SSS0.Px1.p1.28.m28.1c">k</annotation><annotation encoding="application/x-llamapun" id="S1.SS2.SSS0.Px1.p1.28.m28.1d">italic_k</annotation></semantics></math> <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.00712v1#bib.bibx74" title="">Nut22</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.00712v1#bib.bibx30" title="">CV14</a>]</cite>. For smaller value of <math alttext="k\leq 6" class="ltx_Math" display="inline" id="S1.SS2.SSS0.Px1.p1.29.m29.1"><semantics id="S1.SS2.SSS0.Px1.p1.29.m29.1a"><mrow id="S1.SS2.SSS0.Px1.p1.29.m29.1.1" xref="S1.SS2.SSS0.Px1.p1.29.m29.1.1.cmml"><mi id="S1.SS2.SSS0.Px1.p1.29.m29.1.1.2" xref="S1.SS2.SSS0.Px1.p1.29.m29.1.1.2.cmml">k</mi><mo id="S1.SS2.SSS0.Px1.p1.29.m29.1.1.1" xref="S1.SS2.SSS0.Px1.p1.29.m29.1.1.1.cmml">≤</mo><mn id="S1.SS2.SSS0.Px1.p1.29.m29.1.1.3" xref="S1.SS2.SSS0.Px1.p1.29.m29.1.1.3.cmml">6</mn></mrow><annotation-xml encoding="MathML-Content" id="S1.SS2.SSS0.Px1.p1.29.m29.1b"><apply id="S1.SS2.SSS0.Px1.p1.29.m29.1.1.cmml" xref="S1.SS2.SSS0.Px1.p1.29.m29.1.1"><leq id="S1.SS2.SSS0.Px1.p1.29.m29.1.1.1.cmml" xref="S1.SS2.SSS0.Px1.p1.29.m29.1.1.1"></leq><ci id="S1.SS2.SSS0.Px1.p1.29.m29.1.1.2.cmml" xref="S1.SS2.SSS0.Px1.p1.29.m29.1.1.2">𝑘</ci><cn id="S1.SS2.SSS0.Px1.p1.29.m29.1.1.3.cmml" type="integer" xref="S1.SS2.SSS0.Px1.p1.29.m29.1.1.3">6</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS2.SSS0.Px1.p1.29.m29.1c">k\leq 6</annotation><annotation encoding="application/x-llamapun" id="S1.SS2.SSS0.Px1.p1.29.m29.1d">italic_k ≤ 6</annotation></semantics></math> improved bounds are known—see <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.00712v1#bib.bibx73" title="">Nut18b</a>]</cite>. These are also the best known bounds for <math alttext="k" class="ltx_Math" display="inline" id="S1.SS2.SSS0.Px1.p1.30.m30.1"><semantics id="S1.SS2.SSS0.Px1.p1.30.m30.1a"><mi id="S1.SS2.SSS0.Px1.p1.30.m30.1.1" xref="S1.SS2.SSS0.Px1.p1.30.m30.1.1.cmml">k</mi><annotation-xml encoding="MathML-Content" id="S1.SS2.SSS0.Px1.p1.30.m30.1b"><ci id="S1.SS2.SSS0.Px1.p1.30.m30.1.1.cmml" xref="S1.SS2.SSS0.Px1.p1.30.m30.1.1">𝑘</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.SS2.SSS0.Px1.p1.30.m30.1c">k</annotation><annotation encoding="application/x-llamapun" id="S1.SS2.SSS0.Px1.p1.30.m30.1d">italic_k</annotation></semantics></math>-VC-CAP when <math alttext="n" class="ltx_Math" display="inline" id="S1.SS2.SSS0.Px1.p1.31.m31.1"><semantics id="S1.SS2.SSS0.Px1.p1.31.m31.1a"><mi id="S1.SS2.SSS0.Px1.p1.31.m31.1.1" xref="S1.SS2.SSS0.Px1.p1.31.m31.1.1.cmml">n</mi><annotation-xml encoding="MathML-Content" id="S1.SS2.SSS0.Px1.p1.31.m31.1b"><ci id="S1.SS2.SSS0.Px1.p1.31.m31.1.1.cmml" xref="S1.SS2.SSS0.Px1.p1.31.m31.1.1">𝑛</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.SS2.SSS0.Px1.p1.31.m31.1c">n</annotation><annotation encoding="application/x-llamapun" id="S1.SS2.SSS0.Px1.p1.31.m31.1d">italic_n</annotation></semantics></math> is large compared to <math alttext="k" class="ltx_Math" display="inline" id="S1.SS2.SSS0.Px1.p1.32.m32.1"><semantics id="S1.SS2.SSS0.Px1.p1.32.m32.1a"><mi id="S1.SS2.SSS0.Px1.p1.32.m32.1.1" xref="S1.SS2.SSS0.Px1.p1.32.m32.1.1.cmml">k</mi><annotation-xml encoding="MathML-Content" id="S1.SS2.SSS0.Px1.p1.32.m32.1b"><ci id="S1.SS2.SSS0.Px1.p1.32.m32.1.1.cmml" xref="S1.SS2.SSS0.Px1.p1.32.m32.1.1">𝑘</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.SS2.SSS0.Px1.p1.32.m32.1c">k</annotation><annotation encoding="application/x-llamapun" id="S1.SS2.SSS0.Px1.p1.32.m32.1d">italic_k</annotation></semantics></math>. For large <math alttext="k" class="ltx_Math" display="inline" id="S1.SS2.SSS0.Px1.p1.33.m33.1"><semantics id="S1.SS2.SSS0.Px1.p1.33.m33.1a"><mi id="S1.SS2.SSS0.Px1.p1.33.m33.1.1" xref="S1.SS2.SSS0.Px1.p1.33.m33.1.1.cmml">k</mi><annotation-xml encoding="MathML-Content" id="S1.SS2.SSS0.Px1.p1.33.m33.1b"><ci id="S1.SS2.SSS0.Px1.p1.33.m33.1.1.cmml" xref="S1.SS2.SSS0.Px1.p1.33.m33.1.1">𝑘</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.SS2.SSS0.Px1.p1.33.m33.1c">k</annotation><annotation encoding="application/x-llamapun" id="S1.SS2.SSS0.Px1.p1.33.m33.1d">italic_k</annotation></semantics></math>, <math alttext="k" class="ltx_Math" display="inline" id="S1.SS2.SSS0.Px1.p1.34.m34.1"><semantics id="S1.SS2.SSS0.Px1.p1.34.m34.1a"><mi id="S1.SS2.SSS0.Px1.p1.34.m34.1.1" xref="S1.SS2.SSS0.Px1.p1.34.m34.1.1.cmml">k</mi><annotation-xml encoding="MathML-Content" id="S1.SS2.SSS0.Px1.p1.34.m34.1b"><ci id="S1.SS2.SSS0.Px1.p1.34.m34.1.1.cmml" xref="S1.SS2.SSS0.Px1.p1.34.m34.1.1">𝑘</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.SS2.SSS0.Px1.p1.34.m34.1c">k</annotation><annotation encoding="application/x-llamapun" id="S1.SS2.SSS0.Px1.p1.34.m34.1d">italic_k</annotation></semantics></math>-VC-CAP and <math alttext="k" class="ltx_Math" display="inline" id="S1.SS2.SSS0.Px1.p1.35.m35.1"><semantics id="S1.SS2.SSS0.Px1.p1.35.m35.1a"><mi id="S1.SS2.SSS0.Px1.p1.35.m35.1.1" xref="S1.SS2.SSS0.Px1.p1.35.m35.1.1.cmml">k</mi><annotation-xml encoding="MathML-Content" id="S1.SS2.SSS0.Px1.p1.35.m35.1b"><ci id="S1.SS2.SSS0.Px1.p1.35.m35.1.1.cmml" xref="S1.SS2.SSS0.Px1.p1.35.m35.1.1">𝑘</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.SS2.SSS0.Px1.p1.35.m35.1c">k</annotation><annotation encoding="application/x-llamapun" id="S1.SS2.SSS0.Px1.p1.35.m35.1d">italic_k</annotation></semantics></math>-VCSS admit an <math alttext="O(\log\frac{n}{n-k})" class="ltx_Math" display="inline" id="S1.SS2.SSS0.Px1.p1.36.m36.1"><semantics id="S1.SS2.SSS0.Px1.p1.36.m36.1a"><mrow id="S1.SS2.SSS0.Px1.p1.36.m36.1.1" xref="S1.SS2.SSS0.Px1.p1.36.m36.1.1.cmml"><mi id="S1.SS2.SSS0.Px1.p1.36.m36.1.1.3" xref="S1.SS2.SSS0.Px1.p1.36.m36.1.1.3.cmml">O</mi><mo id="S1.SS2.SSS0.Px1.p1.36.m36.1.1.2" xref="S1.SS2.SSS0.Px1.p1.36.m36.1.1.2.cmml">⁢</mo><mrow id="S1.SS2.SSS0.Px1.p1.36.m36.1.1.1.1" xref="S1.SS2.SSS0.Px1.p1.36.m36.1.1.1.1.1.cmml"><mo id="S1.SS2.SSS0.Px1.p1.36.m36.1.1.1.1.2" stretchy="false" xref="S1.SS2.SSS0.Px1.p1.36.m36.1.1.1.1.1.cmml">(</mo><mrow id="S1.SS2.SSS0.Px1.p1.36.m36.1.1.1.1.1" xref="S1.SS2.SSS0.Px1.p1.36.m36.1.1.1.1.1.cmml"><mi id="S1.SS2.SSS0.Px1.p1.36.m36.1.1.1.1.1.1" xref="S1.SS2.SSS0.Px1.p1.36.m36.1.1.1.1.1.1.cmml">log</mi><mo id="S1.SS2.SSS0.Px1.p1.36.m36.1.1.1.1.1a" lspace="0.167em" xref="S1.SS2.SSS0.Px1.p1.36.m36.1.1.1.1.1.cmml">⁡</mo><mfrac id="S1.SS2.SSS0.Px1.p1.36.m36.1.1.1.1.1.2" xref="S1.SS2.SSS0.Px1.p1.36.m36.1.1.1.1.1.2.cmml"><mi id="S1.SS2.SSS0.Px1.p1.36.m36.1.1.1.1.1.2.2" xref="S1.SS2.SSS0.Px1.p1.36.m36.1.1.1.1.1.2.2.cmml">n</mi><mrow id="S1.SS2.SSS0.Px1.p1.36.m36.1.1.1.1.1.2.3" xref="S1.SS2.SSS0.Px1.p1.36.m36.1.1.1.1.1.2.3.cmml"><mi id="S1.SS2.SSS0.Px1.p1.36.m36.1.1.1.1.1.2.3.2" xref="S1.SS2.SSS0.Px1.p1.36.m36.1.1.1.1.1.2.3.2.cmml">n</mi><mo id="S1.SS2.SSS0.Px1.p1.36.m36.1.1.1.1.1.2.3.1" xref="S1.SS2.SSS0.Px1.p1.36.m36.1.1.1.1.1.2.3.1.cmml">−</mo><mi id="S1.SS2.SSS0.Px1.p1.36.m36.1.1.1.1.1.2.3.3" xref="S1.SS2.SSS0.Px1.p1.36.m36.1.1.1.1.1.2.3.3.cmml">k</mi></mrow></mfrac></mrow><mo id="S1.SS2.SSS0.Px1.p1.36.m36.1.1.1.1.3" stretchy="false" xref="S1.SS2.SSS0.Px1.p1.36.m36.1.1.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.SS2.SSS0.Px1.p1.36.m36.1b"><apply id="S1.SS2.SSS0.Px1.p1.36.m36.1.1.cmml" xref="S1.SS2.SSS0.Px1.p1.36.m36.1.1"><times id="S1.SS2.SSS0.Px1.p1.36.m36.1.1.2.cmml" xref="S1.SS2.SSS0.Px1.p1.36.m36.1.1.2"></times><ci id="S1.SS2.SSS0.Px1.p1.36.m36.1.1.3.cmml" xref="S1.SS2.SSS0.Px1.p1.36.m36.1.1.3">𝑂</ci><apply id="S1.SS2.SSS0.Px1.p1.36.m36.1.1.1.1.1.cmml" xref="S1.SS2.SSS0.Px1.p1.36.m36.1.1.1.1"><log id="S1.SS2.SSS0.Px1.p1.36.m36.1.1.1.1.1.1.cmml" xref="S1.SS2.SSS0.Px1.p1.36.m36.1.1.1.1.1.1"></log><apply id="S1.SS2.SSS0.Px1.p1.36.m36.1.1.1.1.1.2.cmml" xref="S1.SS2.SSS0.Px1.p1.36.m36.1.1.1.1.1.2"><divide id="S1.SS2.SSS0.Px1.p1.36.m36.1.1.1.1.1.2.1.cmml" xref="S1.SS2.SSS0.Px1.p1.36.m36.1.1.1.1.1.2"></divide><ci id="S1.SS2.SSS0.Px1.p1.36.m36.1.1.1.1.1.2.2.cmml" xref="S1.SS2.SSS0.Px1.p1.36.m36.1.1.1.1.1.2.2">𝑛</ci><apply id="S1.SS2.SSS0.Px1.p1.36.m36.1.1.1.1.1.2.3.cmml" xref="S1.SS2.SSS0.Px1.p1.36.m36.1.1.1.1.1.2.3"><minus id="S1.SS2.SSS0.Px1.p1.36.m36.1.1.1.1.1.2.3.1.cmml" xref="S1.SS2.SSS0.Px1.p1.36.m36.1.1.1.1.1.2.3.1"></minus><ci id="S1.SS2.SSS0.Px1.p1.36.m36.1.1.1.1.1.2.3.2.cmml" xref="S1.SS2.SSS0.Px1.p1.36.m36.1.1.1.1.1.2.3.2">𝑛</ci><ci id="S1.SS2.SSS0.Px1.p1.36.m36.1.1.1.1.1.2.3.3.cmml" xref="S1.SS2.SSS0.Px1.p1.36.m36.1.1.1.1.1.2.3.3">𝑘</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS2.SSS0.Px1.p1.36.m36.1c">O(\log\frac{n}{n-k})</annotation><annotation encoding="application/x-llamapun" id="S1.SS2.SSS0.Px1.p1.36.m36.1d">italic_O ( roman_log divide start_ARG italic_n end_ARG start_ARG italic_n - italic_k end_ARG )</annotation></semantics></math> and <math alttext="O(\log k\log\frac{n}{n-k})" class="ltx_Math" display="inline" id="S1.SS2.SSS0.Px1.p1.37.m37.1"><semantics id="S1.SS2.SSS0.Px1.p1.37.m37.1a"><mrow id="S1.SS2.SSS0.Px1.p1.37.m37.1.1" xref="S1.SS2.SSS0.Px1.p1.37.m37.1.1.cmml"><mi id="S1.SS2.SSS0.Px1.p1.37.m37.1.1.3" xref="S1.SS2.SSS0.Px1.p1.37.m37.1.1.3.cmml">O</mi><mo id="S1.SS2.SSS0.Px1.p1.37.m37.1.1.2" xref="S1.SS2.SSS0.Px1.p1.37.m37.1.1.2.cmml">⁢</mo><mrow id="S1.SS2.SSS0.Px1.p1.37.m37.1.1.1.1" xref="S1.SS2.SSS0.Px1.p1.37.m37.1.1.1.1.1.cmml"><mo id="S1.SS2.SSS0.Px1.p1.37.m37.1.1.1.1.2" stretchy="false" 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id="S1.SS2.SSS0.Px1.p1.37.m37.1c">O(\log k\log\frac{n}{n-k})</annotation><annotation encoding="application/x-llamapun" id="S1.SS2.SSS0.Px1.p1.37.m37.1d">italic_O ( roman_log italic_k roman_log divide start_ARG italic_n end_ARG start_ARG italic_n - italic_k end_ARG )</annotation></semantics></math>-approximation respectively with more precise bounds known in various regimes <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.00712v1#bib.bibx73" title="">Nut18b</a>]</cite>. We note that many of the approximation results (but not all) are with respect to a natural cut-cover LP relaxation. This is important to our analysis since some of our results are based in exploiting the integrality gap of this LP relaxation. Many improved results are known for special cases of graphs including unweighted graphs, planar and graphs from proper minor-closed families, and graphs arising from geometric instances. We focus on general graphs in this work.</p> </div> </section> <section class="ltx_paragraph" id="S1.SS2.SSS0.Px2"> <h5 class="ltx_title ltx_title_paragraph">Streaming Graph Algorithms:</h5> <div class="ltx_para" id="S1.SS2.SSS0.Px2.p1"> <p class="ltx_p" id="S1.SS2.SSS0.Px2.p1.4">Graph problems in the streaming model have been studied extensively, particularly in the context of compression methods that reduce the graph size and preserve connectivity within a factor of <math alttext="t" class="ltx_Math" display="inline" id="S1.SS2.SSS0.Px2.p1.1.m1.1"><semantics id="S1.SS2.SSS0.Px2.p1.1.m1.1a"><mi id="S1.SS2.SSS0.Px2.p1.1.m1.1.1" xref="S1.SS2.SSS0.Px2.p1.1.m1.1.1.cmml">t</mi><annotation-xml encoding="MathML-Content" id="S1.SS2.SSS0.Px2.p1.1.m1.1b"><ci id="S1.SS2.SSS0.Px2.p1.1.m1.1.1.cmml" xref="S1.SS2.SSS0.Px2.p1.1.m1.1.1">𝑡</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.SS2.SSS0.Px2.p1.1.m1.1c">t</annotation><annotation encoding="application/x-llamapun" id="S1.SS2.SSS0.Px2.p1.1.m1.1d">italic_t</annotation></semantics></math> (i.e., <span class="ltx_text ltx_font_italic" id="S1.SS2.SSS0.Px2.p1.4.1">spanners</span>) <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.00712v1#bib.bibx40" title="">FKM<sup class="ltx_sup"><span class="ltx_text ltx_font_italic">+</span></sup>05</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.00712v1#bib.bibx15" title="">Bas08</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.00712v1#bib.bibx38" title="">Elk11</a>]</cite> or approximate cuts within a <math alttext="(1+\epsilon)" class="ltx_Math" display="inline" id="S1.SS2.SSS0.Px2.p1.2.m2.1"><semantics id="S1.SS2.SSS0.Px2.p1.2.m2.1a"><mrow id="S1.SS2.SSS0.Px2.p1.2.m2.1.1.1" xref="S1.SS2.SSS0.Px2.p1.2.m2.1.1.1.1.cmml"><mo id="S1.SS2.SSS0.Px2.p1.2.m2.1.1.1.2" stretchy="false" xref="S1.SS2.SSS0.Px2.p1.2.m2.1.1.1.1.cmml">(</mo><mrow id="S1.SS2.SSS0.Px2.p1.2.m2.1.1.1.1" xref="S1.SS2.SSS0.Px2.p1.2.m2.1.1.1.1.cmml"><mn id="S1.SS2.SSS0.Px2.p1.2.m2.1.1.1.1.2" xref="S1.SS2.SSS0.Px2.p1.2.m2.1.1.1.1.2.cmml">1</mn><mo id="S1.SS2.SSS0.Px2.p1.2.m2.1.1.1.1.1" xref="S1.SS2.SSS0.Px2.p1.2.m2.1.1.1.1.1.cmml">+</mo><mi id="S1.SS2.SSS0.Px2.p1.2.m2.1.1.1.1.3" xref="S1.SS2.SSS0.Px2.p1.2.m2.1.1.1.1.3.cmml">ϵ</mi></mrow><mo id="S1.SS2.SSS0.Px2.p1.2.m2.1.1.1.3" stretchy="false" xref="S1.SS2.SSS0.Px2.p1.2.m2.1.1.1.1.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S1.SS2.SSS0.Px2.p1.2.m2.1b"><apply id="S1.SS2.SSS0.Px2.p1.2.m2.1.1.1.1.cmml" xref="S1.SS2.SSS0.Px2.p1.2.m2.1.1.1"><plus id="S1.SS2.SSS0.Px2.p1.2.m2.1.1.1.1.1.cmml" xref="S1.SS2.SSS0.Px2.p1.2.m2.1.1.1.1.1"></plus><cn id="S1.SS2.SSS0.Px2.p1.2.m2.1.1.1.1.2.cmml" type="integer" xref="S1.SS2.SSS0.Px2.p1.2.m2.1.1.1.1.2">1</cn><ci id="S1.SS2.SSS0.Px2.p1.2.m2.1.1.1.1.3.cmml" xref="S1.SS2.SSS0.Px2.p1.2.m2.1.1.1.1.3">italic-ϵ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS2.SSS0.Px2.p1.2.m2.1c">(1+\epsilon)</annotation><annotation encoding="application/x-llamapun" id="S1.SS2.SSS0.Px2.p1.2.m2.1d">( 1 + italic_ϵ )</annotation></semantics></math> factor (i.e., <span class="ltx_text ltx_font_italic" id="S1.SS2.SSS0.Px2.p1.4.2">cut sparsifiers</span> and related problems) <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.00712v1#bib.bibx6" title="">AG09</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.00712v1#bib.bibx58" title="">KL13</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.00712v1#bib.bibx59" title="">KLM<sup class="ltx_sup"><span class="ltx_text ltx_font_italic">+</span></sup>17</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.00712v1#bib.bibx60" title="">KMM<sup class="ltx_sup"><span class="ltx_text ltx_font_italic">+</span></sup>20</a>]</cite>. These problems have also been widely explored in the <span class="ltx_text ltx_font_italic" id="S1.SS2.SSS0.Px2.p1.4.3">dynamic</span> streaming model, where edges are both inserted and deleted. While graph sketching approaches have sufficed to yield near-optimal algorithms for sparsifiers <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.00712v1#bib.bibx7" title="">AGM12</a>]</cite>, the state-of-the-art for spanners in dynamic streams remained multi-pass algorithms until recently <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.00712v1#bib.bibx7" title="">AGM12</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.00712v1#bib.bibx57" title="">KKS14</a>]</cite>. Filtser <span class="ltx_text ltx_font_italic" id="S1.SS2.SSS0.Px2.p1.4.4">et al.</span> <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.00712v1#bib.bibx42" title="">FKN21</a>]</cite> developed the first single-pass algorithm for <math alttext="\tilde{O}(n^{2/3})" class="ltx_Math" display="inline" id="S1.SS2.SSS0.Px2.p1.3.m3.1"><semantics id="S1.SS2.SSS0.Px2.p1.3.m3.1a"><mrow id="S1.SS2.SSS0.Px2.p1.3.m3.1.1" xref="S1.SS2.SSS0.Px2.p1.3.m3.1.1.cmml"><mover accent="true" id="S1.SS2.SSS0.Px2.p1.3.m3.1.1.3" xref="S1.SS2.SSS0.Px2.p1.3.m3.1.1.3.cmml"><mi id="S1.SS2.SSS0.Px2.p1.3.m3.1.1.3.2" xref="S1.SS2.SSS0.Px2.p1.3.m3.1.1.3.2.cmml">O</mi><mo id="S1.SS2.SSS0.Px2.p1.3.m3.1.1.3.1" xref="S1.SS2.SSS0.Px2.p1.3.m3.1.1.3.1.cmml">~</mo></mover><mo id="S1.SS2.SSS0.Px2.p1.3.m3.1.1.2" xref="S1.SS2.SSS0.Px2.p1.3.m3.1.1.2.cmml">⁢</mo><mrow id="S1.SS2.SSS0.Px2.p1.3.m3.1.1.1.1" xref="S1.SS2.SSS0.Px2.p1.3.m3.1.1.1.1.1.cmml"><mo 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Though this result has a large approximation factor, Filtser <span class="ltx_text ltx_font_italic" id="S1.SS2.SSS0.Px2.p1.4.5">et al.</span> conjectured that it may represent an optimal trade-off.</p> </div> <div class="ltx_para" id="S1.SS2.SSS0.Px2.p2"> <p class="ltx_p" id="S1.SS2.SSS0.Px2.p2.2">Another line of research related to our work is testing connectivity in graph algorithms. This problem has been studied for both edge-connectivity and vertex-connectivity <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.00712v1#bib.bibx40" title="">FKM<sup class="ltx_sup"><span class="ltx_text ltx_font_italic">+</span></sup>05</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.00712v1#bib.bibx81" title="">Zel06</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.00712v1#bib.bibx77" title="">SW15</a>]</cite>, as well as in dynamic settings <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.00712v1#bib.bibx7" title="">AGM12</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.00712v1#bib.bibx28" title="">CMS13</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.00712v1#bib.bibx49" title="">GMT15</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.00712v1#bib.bibx12" title="">AS23</a>]</cite>. Specifically, <math alttext="k" class="ltx_Math" display="inline" id="S1.SS2.SSS0.Px2.p2.1.m1.1"><semantics id="S1.SS2.SSS0.Px2.p2.1.m1.1a"><mi id="S1.SS2.SSS0.Px2.p2.1.m1.1.1" xref="S1.SS2.SSS0.Px2.p2.1.m1.1.1.cmml">k</mi><annotation-xml encoding="MathML-Content" id="S1.SS2.SSS0.Px2.p2.1.m1.1b"><ci id="S1.SS2.SSS0.Px2.p2.1.m1.1.1.cmml" xref="S1.SS2.SSS0.Px2.p2.1.m1.1.1">𝑘</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.SS2.SSS0.Px2.p2.1.m1.1c">k</annotation><annotation encoding="application/x-llamapun" id="S1.SS2.SSS0.Px2.p2.1.m1.1d">italic_k</annotation></semantics></math>-connectivity for both edge- and vertex-connectivity can be tested using <math alttext="\tilde{O}(nk)" class="ltx_Math" display="inline" id="S1.SS2.SSS0.Px2.p2.2.m2.1"><semantics id="S1.SS2.SSS0.Px2.p2.2.m2.1a"><mrow id="S1.SS2.SSS0.Px2.p2.2.m2.1.1" xref="S1.SS2.SSS0.Px2.p2.2.m2.1.1.cmml"><mover accent="true" id="S1.SS2.SSS0.Px2.p2.2.m2.1.1.3" xref="S1.SS2.SSS0.Px2.p2.2.m2.1.1.3.cmml"><mi 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xref="S1.SS2.SSS0.Px2.p2.2.m2.1.1.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.SS2.SSS0.Px2.p2.2.m2.1b"><apply id="S1.SS2.SSS0.Px2.p2.2.m2.1.1.cmml" xref="S1.SS2.SSS0.Px2.p2.2.m2.1.1"><times id="S1.SS2.SSS0.Px2.p2.2.m2.1.1.2.cmml" xref="S1.SS2.SSS0.Px2.p2.2.m2.1.1.2"></times><apply id="S1.SS2.SSS0.Px2.p2.2.m2.1.1.3.cmml" xref="S1.SS2.SSS0.Px2.p2.2.m2.1.1.3"><ci id="S1.SS2.SSS0.Px2.p2.2.m2.1.1.3.1.cmml" xref="S1.SS2.SSS0.Px2.p2.2.m2.1.1.3.1">~</ci><ci id="S1.SS2.SSS0.Px2.p2.2.m2.1.1.3.2.cmml" xref="S1.SS2.SSS0.Px2.p2.2.m2.1.1.3.2">𝑂</ci></apply><apply id="S1.SS2.SSS0.Px2.p2.2.m2.1.1.1.1.1.cmml" xref="S1.SS2.SSS0.Px2.p2.2.m2.1.1.1.1"><times id="S1.SS2.SSS0.Px2.p2.2.m2.1.1.1.1.1.1.cmml" xref="S1.SS2.SSS0.Px2.p2.2.m2.1.1.1.1.1.1"></times><ci id="S1.SS2.SSS0.Px2.p2.2.m2.1.1.1.1.1.2.cmml" xref="S1.SS2.SSS0.Px2.p2.2.m2.1.1.1.1.1.2">𝑛</ci><ci id="S1.SS2.SSS0.Px2.p2.2.m2.1.1.1.1.1.3.cmml" xref="S1.SS2.SSS0.Px2.p2.2.m2.1.1.1.1.1.3">𝑘</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS2.SSS0.Px2.p2.2.m2.1c">\tilde{O}(nk)</annotation><annotation encoding="application/x-llamapun" id="S1.SS2.SSS0.Px2.p2.2.m2.1d">over~ start_ARG italic_O end_ARG ( italic_n italic_k )</annotation></semantics></math> space even in dynamic streams <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.00712v1#bib.bibx7" title="">AGM12</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.00712v1#bib.bibx12" title="">AS23</a>]</cite>.</p> </div> <div class="ltx_para" id="S1.SS2.SSS0.Px2.p3"> <p class="ltx_p" id="S1.SS2.SSS0.Px2.p3.4">Finally, for the problem of <math alttext="k" class="ltx_Math" display="inline" id="S1.SS2.SSS0.Px2.p3.1.m1.1"><semantics id="S1.SS2.SSS0.Px2.p3.1.m1.1a"><mi id="S1.SS2.SSS0.Px2.p3.1.m1.1.1" xref="S1.SS2.SSS0.Px2.p3.1.m1.1.1.cmml">k</mi><annotation-xml encoding="MathML-Content" id="S1.SS2.SSS0.Px2.p3.1.m1.1b"><ci id="S1.SS2.SSS0.Px2.p3.1.m1.1.1.cmml" xref="S1.SS2.SSS0.Px2.p3.1.m1.1.1">𝑘</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.SS2.SSS0.Px2.p3.1.m1.1c">k</annotation><annotation encoding="application/x-llamapun" id="S1.SS2.SSS0.Px2.p3.1.m1.1d">italic_k</annotation></semantics></math>-ECSS, <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.00712v1#bib.bibx54" title="">JKMV24</a>]</cite> designed a <math alttext="k" class="ltx_Math" display="inline" id="S1.SS2.SSS0.Px2.p3.2.m2.1"><semantics id="S1.SS2.SSS0.Px2.p3.2.m2.1a"><mi id="S1.SS2.SSS0.Px2.p3.2.m2.1.1" xref="S1.SS2.SSS0.Px2.p3.2.m2.1.1.cmml">k</mi><annotation-xml encoding="MathML-Content" id="S1.SS2.SSS0.Px2.p3.2.m2.1b"><ci id="S1.SS2.SSS0.Px2.p3.2.m2.1.1.cmml" xref="S1.SS2.SSS0.Px2.p3.2.m2.1.1">𝑘</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.SS2.SSS0.Px2.p3.2.m2.1c">k</annotation><annotation encoding="application/x-llamapun" id="S1.SS2.SSS0.Px2.p3.2.m2.1d">italic_k</annotation></semantics></math>-pass algorithm within the augmentation framework <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.00712v1#bib.bibx46" title="">GGP<sup class="ltx_sup"><span class="ltx_text ltx_font_italic">+</span></sup>94</a>]</cite> that finds an <math alttext="O(\log k)" class="ltx_Math" display="inline" id="S1.SS2.SSS0.Px2.p3.3.m3.1"><semantics id="S1.SS2.SSS0.Px2.p3.3.m3.1a"><mrow id="S1.SS2.SSS0.Px2.p3.3.m3.1.1" xref="S1.SS2.SSS0.Px2.p3.3.m3.1.1.cmml"><mi id="S1.SS2.SSS0.Px2.p3.3.m3.1.1.3" xref="S1.SS2.SSS0.Px2.p3.3.m3.1.1.3.cmml">O</mi><mo id="S1.SS2.SSS0.Px2.p3.3.m3.1.1.2" xref="S1.SS2.SSS0.Px2.p3.3.m3.1.1.2.cmml">⁢</mo><mrow id="S1.SS2.SSS0.Px2.p3.3.m3.1.1.1.1" xref="S1.SS2.SSS0.Px2.p3.3.m3.1.1.1.1.1.cmml"><mo id="S1.SS2.SSS0.Px2.p3.3.m3.1.1.1.1.2" stretchy="false" xref="S1.SS2.SSS0.Px2.p3.3.m3.1.1.1.1.1.cmml">(</mo><mrow id="S1.SS2.SSS0.Px2.p3.3.m3.1.1.1.1.1" xref="S1.SS2.SSS0.Px2.p3.3.m3.1.1.1.1.1.cmml"><mi id="S1.SS2.SSS0.Px2.p3.3.m3.1.1.1.1.1.1" xref="S1.SS2.SSS0.Px2.p3.3.m3.1.1.1.1.1.1.cmml">log</mi><mo id="S1.SS2.SSS0.Px2.p3.3.m3.1.1.1.1.1a" lspace="0.167em" xref="S1.SS2.SSS0.Px2.p3.3.m3.1.1.1.1.1.cmml">⁡</mo><mi id="S1.SS2.SSS0.Px2.p3.3.m3.1.1.1.1.1.2" xref="S1.SS2.SSS0.Px2.p3.3.m3.1.1.1.1.1.2.cmml">k</mi></mrow><mo id="S1.SS2.SSS0.Px2.p3.3.m3.1.1.1.1.3" stretchy="false" xref="S1.SS2.SSS0.Px2.p3.3.m3.1.1.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.SS2.SSS0.Px2.p3.3.m3.1b"><apply id="S1.SS2.SSS0.Px2.p3.3.m3.1.1.cmml" xref="S1.SS2.SSS0.Px2.p3.3.m3.1.1"><times id="S1.SS2.SSS0.Px2.p3.3.m3.1.1.2.cmml" xref="S1.SS2.SSS0.Px2.p3.3.m3.1.1.2"></times><ci id="S1.SS2.SSS0.Px2.p3.3.m3.1.1.3.cmml" xref="S1.SS2.SSS0.Px2.p3.3.m3.1.1.3">𝑂</ci><apply id="S1.SS2.SSS0.Px2.p3.3.m3.1.1.1.1.1.cmml" xref="S1.SS2.SSS0.Px2.p3.3.m3.1.1.1.1"><log id="S1.SS2.SSS0.Px2.p3.3.m3.1.1.1.1.1.1.cmml" xref="S1.SS2.SSS0.Px2.p3.3.m3.1.1.1.1.1.1"></log><ci id="S1.SS2.SSS0.Px2.p3.3.m3.1.1.1.1.1.2.cmml" xref="S1.SS2.SSS0.Px2.p3.3.m3.1.1.1.1.1.2">𝑘</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS2.SSS0.Px2.p3.3.m3.1c">O(\log k)</annotation><annotation encoding="application/x-llamapun" id="S1.SS2.SSS0.Px2.p3.3.m3.1d">italic_O ( roman_log italic_k )</annotation></semantics></math>-approximate solution using <math alttext="\tilde{O}(nk)" class="ltx_Math" display="inline" id="S1.SS2.SSS0.Px2.p3.4.m4.1"><semantics id="S1.SS2.SSS0.Px2.p3.4.m4.1a"><mrow id="S1.SS2.SSS0.Px2.p3.4.m4.1.1" xref="S1.SS2.SSS0.Px2.p3.4.m4.1.1.cmml"><mover accent="true" id="S1.SS2.SSS0.Px2.p3.4.m4.1.1.3" xref="S1.SS2.SSS0.Px2.p3.4.m4.1.1.3.cmml"><mi id="S1.SS2.SSS0.Px2.p3.4.m4.1.1.3.2" xref="S1.SS2.SSS0.Px2.p3.4.m4.1.1.3.2.cmml">O</mi><mo id="S1.SS2.SSS0.Px2.p3.4.m4.1.1.3.1" xref="S1.SS2.SSS0.Px2.p3.4.m4.1.1.3.1.cmml">~</mo></mover><mo id="S1.SS2.SSS0.Px2.p3.4.m4.1.1.2" xref="S1.SS2.SSS0.Px2.p3.4.m4.1.1.2.cmml">⁢</mo><mrow id="S1.SS2.SSS0.Px2.p3.4.m4.1.1.1.1" xref="S1.SS2.SSS0.Px2.p3.4.m4.1.1.1.1.1.cmml"><mo id="S1.SS2.SSS0.Px2.p3.4.m4.1.1.1.1.2" stretchy="false" xref="S1.SS2.SSS0.Px2.p3.4.m4.1.1.1.1.1.cmml">(</mo><mrow id="S1.SS2.SSS0.Px2.p3.4.m4.1.1.1.1.1" xref="S1.SS2.SSS0.Px2.p3.4.m4.1.1.1.1.1.cmml"><mi id="S1.SS2.SSS0.Px2.p3.4.m4.1.1.1.1.1.2" xref="S1.SS2.SSS0.Px2.p3.4.m4.1.1.1.1.1.2.cmml">n</mi><mo id="S1.SS2.SSS0.Px2.p3.4.m4.1.1.1.1.1.1" xref="S1.SS2.SSS0.Px2.p3.4.m4.1.1.1.1.1.1.cmml">⁢</mo><mi id="S1.SS2.SSS0.Px2.p3.4.m4.1.1.1.1.1.3" xref="S1.SS2.SSS0.Px2.p3.4.m4.1.1.1.1.1.3.cmml">k</mi></mrow><mo id="S1.SS2.SSS0.Px2.p3.4.m4.1.1.1.1.3" stretchy="false" xref="S1.SS2.SSS0.Px2.p3.4.m4.1.1.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.SS2.SSS0.Px2.p3.4.m4.1b"><apply id="S1.SS2.SSS0.Px2.p3.4.m4.1.1.cmml" xref="S1.SS2.SSS0.Px2.p3.4.m4.1.1"><times id="S1.SS2.SSS0.Px2.p3.4.m4.1.1.2.cmml" xref="S1.SS2.SSS0.Px2.p3.4.m4.1.1.2"></times><apply id="S1.SS2.SSS0.Px2.p3.4.m4.1.1.3.cmml" xref="S1.SS2.SSS0.Px2.p3.4.m4.1.1.3"><ci id="S1.SS2.SSS0.Px2.p3.4.m4.1.1.3.1.cmml" xref="S1.SS2.SSS0.Px2.p3.4.m4.1.1.3.1">~</ci><ci id="S1.SS2.SSS0.Px2.p3.4.m4.1.1.3.2.cmml" xref="S1.SS2.SSS0.Px2.p3.4.m4.1.1.3.2">𝑂</ci></apply><apply id="S1.SS2.SSS0.Px2.p3.4.m4.1.1.1.1.1.cmml" xref="S1.SS2.SSS0.Px2.p3.4.m4.1.1.1.1"><times id="S1.SS2.SSS0.Px2.p3.4.m4.1.1.1.1.1.1.cmml" xref="S1.SS2.SSS0.Px2.p3.4.m4.1.1.1.1.1.1"></times><ci id="S1.SS2.SSS0.Px2.p3.4.m4.1.1.1.1.1.2.cmml" xref="S1.SS2.SSS0.Px2.p3.4.m4.1.1.1.1.1.2">𝑛</ci><ci id="S1.SS2.SSS0.Px2.p3.4.m4.1.1.1.1.1.3.cmml" xref="S1.SS2.SSS0.Px2.p3.4.m4.1.1.1.1.1.3">𝑘</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS2.SSS0.Px2.p3.4.m4.1c">\tilde{O}(nk)</annotation><annotation encoding="application/x-llamapun" id="S1.SS2.SSS0.Px2.p3.4.m4.1d">over~ start_ARG italic_O end_ARG ( italic_n italic_k )</annotation></semantics></math> space.</p> </div> </section> </section> </section> <section class="ltx_section" id="S2"> <h2 class="ltx_title ltx_title_section"> <span class="ltx_tag ltx_tag_section">2 </span>Preliminaries</h2> <div class="ltx_para" id="S2.p1"> <p class="ltx_p" id="S2.p1.17">In this section, we provide preliminary background on connectivity. In a graph <math alttext="G" class="ltx_Math" display="inline" id="S2.p1.1.m1.1"><semantics id="S2.p1.1.m1.1a"><mi id="S2.p1.1.m1.1.1" xref="S2.p1.1.m1.1.1.cmml">G</mi><annotation-xml encoding="MathML-Content" id="S2.p1.1.m1.1b"><ci id="S2.p1.1.m1.1.1.cmml" xref="S2.p1.1.m1.1.1">𝐺</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.p1.1.m1.1c">G</annotation><annotation encoding="application/x-llamapun" id="S2.p1.1.m1.1d">italic_G</annotation></semantics></math>, two vertices <math alttext="s" class="ltx_Math" display="inline" id="S2.p1.2.m2.1"><semantics id="S2.p1.2.m2.1a"><mi id="S2.p1.2.m2.1.1" xref="S2.p1.2.m2.1.1.cmml">s</mi><annotation-xml encoding="MathML-Content" id="S2.p1.2.m2.1b"><ci id="S2.p1.2.m2.1.1.cmml" xref="S2.p1.2.m2.1.1">𝑠</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.p1.2.m2.1c">s</annotation><annotation encoding="application/x-llamapun" id="S2.p1.2.m2.1d">italic_s</annotation></semantics></math> and <math alttext="t" class="ltx_Math" display="inline" id="S2.p1.3.m3.1"><semantics id="S2.p1.3.m3.1a"><mi id="S2.p1.3.m3.1.1" xref="S2.p1.3.m3.1.1.cmml">t</mi><annotation-xml encoding="MathML-Content" id="S2.p1.3.m3.1b"><ci id="S2.p1.3.m3.1.1.cmml" xref="S2.p1.3.m3.1.1">𝑡</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.p1.3.m3.1c">t</annotation><annotation encoding="application/x-llamapun" id="S2.p1.3.m3.1d">italic_t</annotation></semantics></math> are <math alttext="k" class="ltx_Math" display="inline" id="S2.p1.4.m4.1"><semantics id="S2.p1.4.m4.1a"><mi id="S2.p1.4.m4.1.1" xref="S2.p1.4.m4.1.1.cmml">k</mi><annotation-xml encoding="MathML-Content" id="S2.p1.4.m4.1b"><ci id="S2.p1.4.m4.1.1.cmml" xref="S2.p1.4.m4.1.1">𝑘</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.p1.4.m4.1c">k</annotation><annotation encoding="application/x-llamapun" id="S2.p1.4.m4.1d">italic_k</annotation></semantics></math>-edge (<math alttext="k" class="ltx_Math" display="inline" id="S2.p1.5.m5.1"><semantics id="S2.p1.5.m5.1a"><mi id="S2.p1.5.m5.1.1" xref="S2.p1.5.m5.1.1.cmml">k</mi><annotation-xml encoding="MathML-Content" id="S2.p1.5.m5.1b"><ci id="S2.p1.5.m5.1.1.cmml" xref="S2.p1.5.m5.1.1">𝑘</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.p1.5.m5.1c">k</annotation><annotation encoding="application/x-llamapun" id="S2.p1.5.m5.1d">italic_k</annotation></semantics></math>-vertex) connected if <math alttext="G" class="ltx_Math" display="inline" id="S2.p1.6.m6.1"><semantics id="S2.p1.6.m6.1a"><mi id="S2.p1.6.m6.1.1" xref="S2.p1.6.m6.1.1.cmml">G</mi><annotation-xml encoding="MathML-Content" id="S2.p1.6.m6.1b"><ci id="S2.p1.6.m6.1.1.cmml" xref="S2.p1.6.m6.1.1">𝐺</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.p1.6.m6.1c">G</annotation><annotation encoding="application/x-llamapun" id="S2.p1.6.m6.1d">italic_G</annotation></semantics></math> contains <math alttext="k" class="ltx_Math" display="inline" id="S2.p1.7.m7.1"><semantics id="S2.p1.7.m7.1a"><mi id="S2.p1.7.m7.1.1" xref="S2.p1.7.m7.1.1.cmml">k</mi><annotation-xml encoding="MathML-Content" id="S2.p1.7.m7.1b"><ci id="S2.p1.7.m7.1.1.cmml" xref="S2.p1.7.m7.1.1">𝑘</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.p1.7.m7.1c">k</annotation><annotation encoding="application/x-llamapun" id="S2.p1.7.m7.1d">italic_k</annotation></semantics></math> edge-disjoint (vertex-disjoint) <math alttext="st" class="ltx_Math" display="inline" id="S2.p1.8.m8.1"><semantics id="S2.p1.8.m8.1a"><mrow id="S2.p1.8.m8.1.1" xref="S2.p1.8.m8.1.1.cmml"><mi id="S2.p1.8.m8.1.1.2" xref="S2.p1.8.m8.1.1.2.cmml">s</mi><mo id="S2.p1.8.m8.1.1.1" xref="S2.p1.8.m8.1.1.1.cmml">⁢</mo><mi id="S2.p1.8.m8.1.1.3" xref="S2.p1.8.m8.1.1.3.cmml">t</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.p1.8.m8.1b"><apply id="S2.p1.8.m8.1.1.cmml" xref="S2.p1.8.m8.1.1"><times id="S2.p1.8.m8.1.1.1.cmml" xref="S2.p1.8.m8.1.1.1"></times><ci id="S2.p1.8.m8.1.1.2.cmml" xref="S2.p1.8.m8.1.1.2">𝑠</ci><ci id="S2.p1.8.m8.1.1.3.cmml" xref="S2.p1.8.m8.1.1.3">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p1.8.m8.1c">st</annotation><annotation encoding="application/x-llamapun" id="S2.p1.8.m8.1d">italic_s italic_t</annotation></semantics></math>-paths. There is a close relation between the maximum connectivity of a pair of vertices <math alttext="s,t" class="ltx_Math" display="inline" id="S2.p1.9.m9.2"><semantics id="S2.p1.9.m9.2a"><mrow id="S2.p1.9.m9.2.3.2" xref="S2.p1.9.m9.2.3.1.cmml"><mi id="S2.p1.9.m9.1.1" xref="S2.p1.9.m9.1.1.cmml">s</mi><mo id="S2.p1.9.m9.2.3.2.1" xref="S2.p1.9.m9.2.3.1.cmml">,</mo><mi id="S2.p1.9.m9.2.2" xref="S2.p1.9.m9.2.2.cmml">t</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.p1.9.m9.2b"><list id="S2.p1.9.m9.2.3.1.cmml" xref="S2.p1.9.m9.2.3.2"><ci id="S2.p1.9.m9.1.1.cmml" xref="S2.p1.9.m9.1.1">𝑠</ci><ci id="S2.p1.9.m9.2.2.cmml" xref="S2.p1.9.m9.2.2">𝑡</ci></list></annotation-xml><annotation encoding="application/x-tex" id="S2.p1.9.m9.2c">s,t</annotation><annotation encoding="application/x-llamapun" id="S2.p1.9.m9.2d">italic_s , italic_t</annotation></semantics></math>, and the minimum cut separating <math alttext="s" class="ltx_Math" display="inline" id="S2.p1.10.m10.1"><semantics id="S2.p1.10.m10.1a"><mi id="S2.p1.10.m10.1.1" xref="S2.p1.10.m10.1.1.cmml">s</mi><annotation-xml encoding="MathML-Content" id="S2.p1.10.m10.1b"><ci id="S2.p1.10.m10.1.1.cmml" xref="S2.p1.10.m10.1.1">𝑠</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.p1.10.m10.1c">s</annotation><annotation encoding="application/x-llamapun" id="S2.p1.10.m10.1d">italic_s</annotation></semantics></math> and <math alttext="t" class="ltx_Math" display="inline" id="S2.p1.11.m11.1"><semantics id="S2.p1.11.m11.1a"><mi id="S2.p1.11.m11.1.1" xref="S2.p1.11.m11.1.1.cmml">t</mi><annotation-xml encoding="MathML-Content" id="S2.p1.11.m11.1b"><ci id="S2.p1.11.m11.1.1.cmml" xref="S2.p1.11.m11.1.1">𝑡</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.p1.11.m11.1c">t</annotation><annotation encoding="application/x-llamapun" id="S2.p1.11.m11.1d">italic_t</annotation></semantics></math>, as characterized by Menger’s theorem. Let <math alttext="S\subset V(G)" class="ltx_Math" display="inline" id="S2.p1.12.m12.1"><semantics id="S2.p1.12.m12.1a"><mrow id="S2.p1.12.m12.1.2" xref="S2.p1.12.m12.1.2.cmml"><mi id="S2.p1.12.m12.1.2.2" xref="S2.p1.12.m12.1.2.2.cmml">S</mi><mo id="S2.p1.12.m12.1.2.1" xref="S2.p1.12.m12.1.2.1.cmml">⊂</mo><mrow id="S2.p1.12.m12.1.2.3" xref="S2.p1.12.m12.1.2.3.cmml"><mi id="S2.p1.12.m12.1.2.3.2" xref="S2.p1.12.m12.1.2.3.2.cmml">V</mi><mo id="S2.p1.12.m12.1.2.3.1" xref="S2.p1.12.m12.1.2.3.1.cmml">⁢</mo><mrow id="S2.p1.12.m12.1.2.3.3.2" xref="S2.p1.12.m12.1.2.3.cmml"><mo id="S2.p1.12.m12.1.2.3.3.2.1" stretchy="false" xref="S2.p1.12.m12.1.2.3.cmml">(</mo><mi id="S2.p1.12.m12.1.1" xref="S2.p1.12.m12.1.1.cmml">G</mi><mo id="S2.p1.12.m12.1.2.3.3.2.2" stretchy="false" xref="S2.p1.12.m12.1.2.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.p1.12.m12.1b"><apply id="S2.p1.12.m12.1.2.cmml" xref="S2.p1.12.m12.1.2"><subset id="S2.p1.12.m12.1.2.1.cmml" xref="S2.p1.12.m12.1.2.1"></subset><ci id="S2.p1.12.m12.1.2.2.cmml" xref="S2.p1.12.m12.1.2.2">𝑆</ci><apply id="S2.p1.12.m12.1.2.3.cmml" xref="S2.p1.12.m12.1.2.3"><times id="S2.p1.12.m12.1.2.3.1.cmml" xref="S2.p1.12.m12.1.2.3.1"></times><ci id="S2.p1.12.m12.1.2.3.2.cmml" xref="S2.p1.12.m12.1.2.3.2">𝑉</ci><ci id="S2.p1.12.m12.1.1.cmml" xref="S2.p1.12.m12.1.1">𝐺</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p1.12.m12.1c">S\subset V(G)</annotation><annotation encoding="application/x-llamapun" id="S2.p1.12.m12.1d">italic_S ⊂ italic_V ( italic_G )</annotation></semantics></math> be a subset of vertices in <math alttext="G" class="ltx_Math" display="inline" id="S2.p1.13.m13.1"><semantics id="S2.p1.13.m13.1a"><mi id="S2.p1.13.m13.1.1" xref="S2.p1.13.m13.1.1.cmml">G</mi><annotation-xml encoding="MathML-Content" id="S2.p1.13.m13.1b"><ci id="S2.p1.13.m13.1.1.cmml" xref="S2.p1.13.m13.1.1">𝐺</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.p1.13.m13.1c">G</annotation><annotation encoding="application/x-llamapun" id="S2.p1.13.m13.1d">italic_G</annotation></semantics></math>. We denote the set of edges crossing <math alttext="S" class="ltx_Math" display="inline" id="S2.p1.14.m14.1"><semantics id="S2.p1.14.m14.1a"><mi id="S2.p1.14.m14.1.1" xref="S2.p1.14.m14.1.1.cmml">S</mi><annotation-xml encoding="MathML-Content" id="S2.p1.14.m14.1b"><ci id="S2.p1.14.m14.1.1.cmml" xref="S2.p1.14.m14.1.1">𝑆</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.p1.14.m14.1c">S</annotation><annotation encoding="application/x-llamapun" id="S2.p1.14.m14.1d">italic_S</annotation></semantics></math> by <math alttext="\delta_{G}(S)" class="ltx_Math" display="inline" id="S2.p1.15.m15.1"><semantics id="S2.p1.15.m15.1a"><mrow id="S2.p1.15.m15.1.2" xref="S2.p1.15.m15.1.2.cmml"><msub id="S2.p1.15.m15.1.2.2" xref="S2.p1.15.m15.1.2.2.cmml"><mi id="S2.p1.15.m15.1.2.2.2" xref="S2.p1.15.m15.1.2.2.2.cmml">δ</mi><mi id="S2.p1.15.m15.1.2.2.3" xref="S2.p1.15.m15.1.2.2.3.cmml">G</mi></msub><mo id="S2.p1.15.m15.1.2.1" xref="S2.p1.15.m15.1.2.1.cmml">⁢</mo><mrow id="S2.p1.15.m15.1.2.3.2" xref="S2.p1.15.m15.1.2.cmml"><mo id="S2.p1.15.m15.1.2.3.2.1" stretchy="false" xref="S2.p1.15.m15.1.2.cmml">(</mo><mi id="S2.p1.15.m15.1.1" xref="S2.p1.15.m15.1.1.cmml">S</mi><mo id="S2.p1.15.m15.1.2.3.2.2" stretchy="false" xref="S2.p1.15.m15.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.p1.15.m15.1b"><apply id="S2.p1.15.m15.1.2.cmml" xref="S2.p1.15.m15.1.2"><times id="S2.p1.15.m15.1.2.1.cmml" xref="S2.p1.15.m15.1.2.1"></times><apply id="S2.p1.15.m15.1.2.2.cmml" xref="S2.p1.15.m15.1.2.2"><csymbol cd="ambiguous" id="S2.p1.15.m15.1.2.2.1.cmml" xref="S2.p1.15.m15.1.2.2">subscript</csymbol><ci id="S2.p1.15.m15.1.2.2.2.cmml" xref="S2.p1.15.m15.1.2.2.2">𝛿</ci><ci id="S2.p1.15.m15.1.2.2.3.cmml" xref="S2.p1.15.m15.1.2.2.3">𝐺</ci></apply><ci id="S2.p1.15.m15.1.1.cmml" xref="S2.p1.15.m15.1.1">𝑆</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p1.15.m15.1c">\delta_{G}(S)</annotation><annotation encoding="application/x-llamapun" id="S2.p1.15.m15.1d">italic_δ start_POSTSUBSCRIPT italic_G end_POSTSUBSCRIPT ( italic_S )</annotation></semantics></math>; <math alttext="\delta_{G}(S)=\{uv\in E(G)\;|\;u\in S,v\in V\setminus S\}" class="ltx_Math" display="inline" id="S2.p1.16.m16.4"><semantics id="S2.p1.16.m16.4a"><mrow id="S2.p1.16.m16.4.4" xref="S2.p1.16.m16.4.4.cmml"><mrow id="S2.p1.16.m16.4.4.4" xref="S2.p1.16.m16.4.4.4.cmml"><msub id="S2.p1.16.m16.4.4.4.2" xref="S2.p1.16.m16.4.4.4.2.cmml"><mi id="S2.p1.16.m16.4.4.4.2.2" xref="S2.p1.16.m16.4.4.4.2.2.cmml">δ</mi><mi id="S2.p1.16.m16.4.4.4.2.3" xref="S2.p1.16.m16.4.4.4.2.3.cmml">G</mi></msub><mo id="S2.p1.16.m16.4.4.4.1" xref="S2.p1.16.m16.4.4.4.1.cmml">⁢</mo><mrow id="S2.p1.16.m16.4.4.4.3.2" xref="S2.p1.16.m16.4.4.4.cmml"><mo id="S2.p1.16.m16.4.4.4.3.2.1" stretchy="false" xref="S2.p1.16.m16.4.4.4.cmml">(</mo><mi id="S2.p1.16.m16.1.1" xref="S2.p1.16.m16.1.1.cmml">S</mi><mo id="S2.p1.16.m16.4.4.4.3.2.2" stretchy="false" xref="S2.p1.16.m16.4.4.4.cmml">)</mo></mrow></mrow><mo id="S2.p1.16.m16.4.4.3" xref="S2.p1.16.m16.4.4.3.cmml">=</mo><mrow id="S2.p1.16.m16.4.4.2.2" xref="S2.p1.16.m16.4.4.2.3.cmml"><mo id="S2.p1.16.m16.4.4.2.2.3" stretchy="false" xref="S2.p1.16.m16.4.4.2.3.1.cmml">{</mo><mrow id="S2.p1.16.m16.3.3.1.1.1" xref="S2.p1.16.m16.3.3.1.1.1.cmml"><mrow id="S2.p1.16.m16.3.3.1.1.1.2" xref="S2.p1.16.m16.3.3.1.1.1.2.cmml"><mi id="S2.p1.16.m16.3.3.1.1.1.2.2" xref="S2.p1.16.m16.3.3.1.1.1.2.2.cmml">u</mi><mo id="S2.p1.16.m16.3.3.1.1.1.2.1" xref="S2.p1.16.m16.3.3.1.1.1.2.1.cmml">⁢</mo><mi id="S2.p1.16.m16.3.3.1.1.1.2.3" xref="S2.p1.16.m16.3.3.1.1.1.2.3.cmml">v</mi></mrow><mo id="S2.p1.16.m16.3.3.1.1.1.1" xref="S2.p1.16.m16.3.3.1.1.1.1.cmml">∈</mo><mrow id="S2.p1.16.m16.3.3.1.1.1.3" xref="S2.p1.16.m16.3.3.1.1.1.3.cmml"><mi id="S2.p1.16.m16.3.3.1.1.1.3.2" xref="S2.p1.16.m16.3.3.1.1.1.3.2.cmml">E</mi><mo id="S2.p1.16.m16.3.3.1.1.1.3.1" xref="S2.p1.16.m16.3.3.1.1.1.3.1.cmml">⁢</mo><mrow 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id="S2.p1.16.m16.4.4.2.2.2.2.2.1.cmml" xref="S2.p1.16.m16.4.4.2.2.2.2.2.1"></in><ci id="S2.p1.16.m16.4.4.2.2.2.2.2.2.cmml" xref="S2.p1.16.m16.4.4.2.2.2.2.2.2">𝑣</ci><apply id="S2.p1.16.m16.4.4.2.2.2.2.2.3.cmml" xref="S2.p1.16.m16.4.4.2.2.2.2.2.3"><setdiff id="S2.p1.16.m16.4.4.2.2.2.2.2.3.1.cmml" xref="S2.p1.16.m16.4.4.2.2.2.2.2.3.1"></setdiff><ci id="S2.p1.16.m16.4.4.2.2.2.2.2.3.2.cmml" xref="S2.p1.16.m16.4.4.2.2.2.2.2.3.2">𝑉</ci><ci id="S2.p1.16.m16.4.4.2.2.2.2.2.3.3.cmml" xref="S2.p1.16.m16.4.4.2.2.2.2.2.3.3">𝑆</ci></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p1.16.m16.4c">\delta_{G}(S)=\{uv\in E(G)\;|\;u\in S,v\in V\setminus S\}</annotation><annotation encoding="application/x-llamapun" id="S2.p1.16.m16.4d">italic_δ start_POSTSUBSCRIPT italic_G end_POSTSUBSCRIPT ( italic_S ) = { italic_u italic_v ∈ italic_E ( italic_G ) | italic_u ∈ italic_S , italic_v ∈ italic_V ∖ italic_S }</annotation></semantics></math>. We drop <math alttext="G" class="ltx_Math" display="inline" id="S2.p1.17.m17.1"><semantics id="S2.p1.17.m17.1a"><mi id="S2.p1.17.m17.1.1" xref="S2.p1.17.m17.1.1.cmml">G</mi><annotation-xml encoding="MathML-Content" id="S2.p1.17.m17.1b"><ci id="S2.p1.17.m17.1.1.cmml" xref="S2.p1.17.m17.1.1">𝐺</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.p1.17.m17.1c">G</annotation><annotation encoding="application/x-llamapun" id="S2.p1.17.m17.1d">italic_G</annotation></semantics></math> when it is implicit from the context. Menger’s theorem is a key result in problems involving connectivity requirements. Its formulation in terms of edge connectivity is as follows:</p> </div> <div class="ltx_theorem ltx_theorem_theorem" id="S2.Thmtheorem1"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="S2.Thmtheorem1.1.1.1">Theorem 2.1</span></span><span class="ltx_text ltx_font_bold" id="S2.Thmtheorem1.2.2"> </span>(Edge-connectivity Menger’s theorem)<span class="ltx_text ltx_font_bold" id="S2.Thmtheorem1.3.3">.</span> </h6> <div class="ltx_para" id="S2.Thmtheorem1.p1"> <p class="ltx_p" id="S2.Thmtheorem1.p1.7">Let <math alttext="G=(V,E)" class="ltx_Math" display="inline" id="S2.Thmtheorem1.p1.1.m1.2"><semantics id="S2.Thmtheorem1.p1.1.m1.2a"><mrow id="S2.Thmtheorem1.p1.1.m1.2.3" xref="S2.Thmtheorem1.p1.1.m1.2.3.cmml"><mi id="S2.Thmtheorem1.p1.1.m1.2.3.2" xref="S2.Thmtheorem1.p1.1.m1.2.3.2.cmml">G</mi><mo id="S2.Thmtheorem1.p1.1.m1.2.3.1" xref="S2.Thmtheorem1.p1.1.m1.2.3.1.cmml">=</mo><mrow id="S2.Thmtheorem1.p1.1.m1.2.3.3.2" xref="S2.Thmtheorem1.p1.1.m1.2.3.3.1.cmml"><mo id="S2.Thmtheorem1.p1.1.m1.2.3.3.2.1" stretchy="false" xref="S2.Thmtheorem1.p1.1.m1.2.3.3.1.cmml">(</mo><mi id="S2.Thmtheorem1.p1.1.m1.1.1" xref="S2.Thmtheorem1.p1.1.m1.1.1.cmml">V</mi><mo id="S2.Thmtheorem1.p1.1.m1.2.3.3.2.2" xref="S2.Thmtheorem1.p1.1.m1.2.3.3.1.cmml">,</mo><mi id="S2.Thmtheorem1.p1.1.m1.2.2" xref="S2.Thmtheorem1.p1.1.m1.2.2.cmml">E</mi><mo id="S2.Thmtheorem1.p1.1.m1.2.3.3.2.3" stretchy="false" xref="S2.Thmtheorem1.p1.1.m1.2.3.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem1.p1.1.m1.2b"><apply id="S2.Thmtheorem1.p1.1.m1.2.3.cmml" xref="S2.Thmtheorem1.p1.1.m1.2.3"><eq id="S2.Thmtheorem1.p1.1.m1.2.3.1.cmml" xref="S2.Thmtheorem1.p1.1.m1.2.3.1"></eq><ci id="S2.Thmtheorem1.p1.1.m1.2.3.2.cmml" xref="S2.Thmtheorem1.p1.1.m1.2.3.2">𝐺</ci><interval closure="open" id="S2.Thmtheorem1.p1.1.m1.2.3.3.1.cmml" xref="S2.Thmtheorem1.p1.1.m1.2.3.3.2"><ci id="S2.Thmtheorem1.p1.1.m1.1.1.cmml" xref="S2.Thmtheorem1.p1.1.m1.1.1">𝑉</ci><ci id="S2.Thmtheorem1.p1.1.m1.2.2.cmml" xref="S2.Thmtheorem1.p1.1.m1.2.2">𝐸</ci></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem1.p1.1.m1.2c">G=(V,E)</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem1.p1.1.m1.2d">italic_G = ( italic_V , italic_E )</annotation></semantics></math> be an undirected graph. Two vertices <math alttext="s,t\in V" class="ltx_Math" display="inline" id="S2.Thmtheorem1.p1.2.m2.2"><semantics id="S2.Thmtheorem1.p1.2.m2.2a"><mrow id="S2.Thmtheorem1.p1.2.m2.2.3" xref="S2.Thmtheorem1.p1.2.m2.2.3.cmml"><mrow id="S2.Thmtheorem1.p1.2.m2.2.3.2.2" xref="S2.Thmtheorem1.p1.2.m2.2.3.2.1.cmml"><mi id="S2.Thmtheorem1.p1.2.m2.1.1" xref="S2.Thmtheorem1.p1.2.m2.1.1.cmml">s</mi><mo id="S2.Thmtheorem1.p1.2.m2.2.3.2.2.1" xref="S2.Thmtheorem1.p1.2.m2.2.3.2.1.cmml">,</mo><mi id="S2.Thmtheorem1.p1.2.m2.2.2" xref="S2.Thmtheorem1.p1.2.m2.2.2.cmml">t</mi></mrow><mo id="S2.Thmtheorem1.p1.2.m2.2.3.1" xref="S2.Thmtheorem1.p1.2.m2.2.3.1.cmml">∈</mo><mi id="S2.Thmtheorem1.p1.2.m2.2.3.3" xref="S2.Thmtheorem1.p1.2.m2.2.3.3.cmml">V</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem1.p1.2.m2.2b"><apply id="S2.Thmtheorem1.p1.2.m2.2.3.cmml" xref="S2.Thmtheorem1.p1.2.m2.2.3"><in id="S2.Thmtheorem1.p1.2.m2.2.3.1.cmml" xref="S2.Thmtheorem1.p1.2.m2.2.3.1"></in><list id="S2.Thmtheorem1.p1.2.m2.2.3.2.1.cmml" xref="S2.Thmtheorem1.p1.2.m2.2.3.2.2"><ci id="S2.Thmtheorem1.p1.2.m2.1.1.cmml" xref="S2.Thmtheorem1.p1.2.m2.1.1">𝑠</ci><ci id="S2.Thmtheorem1.p1.2.m2.2.2.cmml" xref="S2.Thmtheorem1.p1.2.m2.2.2">𝑡</ci></list><ci id="S2.Thmtheorem1.p1.2.m2.2.3.3.cmml" xref="S2.Thmtheorem1.p1.2.m2.2.3.3">𝑉</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem1.p1.2.m2.2c">s,t\in V</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem1.p1.2.m2.2d">italic_s , italic_t ∈ italic_V</annotation></semantics></math> are <math alttext="k" class="ltx_Math" display="inline" id="S2.Thmtheorem1.p1.3.m3.1"><semantics id="S2.Thmtheorem1.p1.3.m3.1a"><mi id="S2.Thmtheorem1.p1.3.m3.1.1" xref="S2.Thmtheorem1.p1.3.m3.1.1.cmml">k</mi><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem1.p1.3.m3.1b"><ci id="S2.Thmtheorem1.p1.3.m3.1.1.cmml" xref="S2.Thmtheorem1.p1.3.m3.1.1">𝑘</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem1.p1.3.m3.1c">k</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem1.p1.3.m3.1d">italic_k</annotation></semantics></math>-edge connected iff for each set <math alttext="S\subset V" class="ltx_Math" display="inline" id="S2.Thmtheorem1.p1.4.m4.1"><semantics id="S2.Thmtheorem1.p1.4.m4.1a"><mrow id="S2.Thmtheorem1.p1.4.m4.1.1" xref="S2.Thmtheorem1.p1.4.m4.1.1.cmml"><mi id="S2.Thmtheorem1.p1.4.m4.1.1.2" xref="S2.Thmtheorem1.p1.4.m4.1.1.2.cmml">S</mi><mo id="S2.Thmtheorem1.p1.4.m4.1.1.1" xref="S2.Thmtheorem1.p1.4.m4.1.1.1.cmml">⊂</mo><mi id="S2.Thmtheorem1.p1.4.m4.1.1.3" xref="S2.Thmtheorem1.p1.4.m4.1.1.3.cmml">V</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem1.p1.4.m4.1b"><apply id="S2.Thmtheorem1.p1.4.m4.1.1.cmml" xref="S2.Thmtheorem1.p1.4.m4.1.1"><subset id="S2.Thmtheorem1.p1.4.m4.1.1.1.cmml" xref="S2.Thmtheorem1.p1.4.m4.1.1.1"></subset><ci id="S2.Thmtheorem1.p1.4.m4.1.1.2.cmml" xref="S2.Thmtheorem1.p1.4.m4.1.1.2">𝑆</ci><ci id="S2.Thmtheorem1.p1.4.m4.1.1.3.cmml" xref="S2.Thmtheorem1.p1.4.m4.1.1.3">𝑉</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem1.p1.4.m4.1c">S\subset V</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem1.p1.4.m4.1d">italic_S ⊂ italic_V</annotation></semantics></math> such that <math alttext="s\in S" class="ltx_Math" display="inline" id="S2.Thmtheorem1.p1.5.m5.1"><semantics id="S2.Thmtheorem1.p1.5.m5.1a"><mrow id="S2.Thmtheorem1.p1.5.m5.1.1" xref="S2.Thmtheorem1.p1.5.m5.1.1.cmml"><mi id="S2.Thmtheorem1.p1.5.m5.1.1.2" xref="S2.Thmtheorem1.p1.5.m5.1.1.2.cmml">s</mi><mo id="S2.Thmtheorem1.p1.5.m5.1.1.1" xref="S2.Thmtheorem1.p1.5.m5.1.1.1.cmml">∈</mo><mi id="S2.Thmtheorem1.p1.5.m5.1.1.3" xref="S2.Thmtheorem1.p1.5.m5.1.1.3.cmml">S</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem1.p1.5.m5.1b"><apply id="S2.Thmtheorem1.p1.5.m5.1.1.cmml" xref="S2.Thmtheorem1.p1.5.m5.1.1"><in id="S2.Thmtheorem1.p1.5.m5.1.1.1.cmml" xref="S2.Thmtheorem1.p1.5.m5.1.1.1"></in><ci id="S2.Thmtheorem1.p1.5.m5.1.1.2.cmml" xref="S2.Thmtheorem1.p1.5.m5.1.1.2">𝑠</ci><ci id="S2.Thmtheorem1.p1.5.m5.1.1.3.cmml" xref="S2.Thmtheorem1.p1.5.m5.1.1.3">𝑆</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem1.p1.5.m5.1c">s\in S</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem1.p1.5.m5.1d">italic_s ∈ italic_S</annotation></semantics></math> and <math alttext="t\in V\setminus S" class="ltx_Math" display="inline" id="S2.Thmtheorem1.p1.6.m6.1"><semantics id="S2.Thmtheorem1.p1.6.m6.1a"><mrow id="S2.Thmtheorem1.p1.6.m6.1.1" xref="S2.Thmtheorem1.p1.6.m6.1.1.cmml"><mi id="S2.Thmtheorem1.p1.6.m6.1.1.2" xref="S2.Thmtheorem1.p1.6.m6.1.1.2.cmml">t</mi><mo id="S2.Thmtheorem1.p1.6.m6.1.1.1" xref="S2.Thmtheorem1.p1.6.m6.1.1.1.cmml">∈</mo><mrow id="S2.Thmtheorem1.p1.6.m6.1.1.3" xref="S2.Thmtheorem1.p1.6.m6.1.1.3.cmml"><mi id="S2.Thmtheorem1.p1.6.m6.1.1.3.2" xref="S2.Thmtheorem1.p1.6.m6.1.1.3.2.cmml">V</mi><mo id="S2.Thmtheorem1.p1.6.m6.1.1.3.1" xref="S2.Thmtheorem1.p1.6.m6.1.1.3.1.cmml">∖</mo><mi id="S2.Thmtheorem1.p1.6.m6.1.1.3.3" xref="S2.Thmtheorem1.p1.6.m6.1.1.3.3.cmml">S</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem1.p1.6.m6.1b"><apply id="S2.Thmtheorem1.p1.6.m6.1.1.cmml" xref="S2.Thmtheorem1.p1.6.m6.1.1"><in id="S2.Thmtheorem1.p1.6.m6.1.1.1.cmml" xref="S2.Thmtheorem1.p1.6.m6.1.1.1"></in><ci id="S2.Thmtheorem1.p1.6.m6.1.1.2.cmml" xref="S2.Thmtheorem1.p1.6.m6.1.1.2">𝑡</ci><apply id="S2.Thmtheorem1.p1.6.m6.1.1.3.cmml" xref="S2.Thmtheorem1.p1.6.m6.1.1.3"><setdiff id="S2.Thmtheorem1.p1.6.m6.1.1.3.1.cmml" xref="S2.Thmtheorem1.p1.6.m6.1.1.3.1"></setdiff><ci id="S2.Thmtheorem1.p1.6.m6.1.1.3.2.cmml" xref="S2.Thmtheorem1.p1.6.m6.1.1.3.2">𝑉</ci><ci id="S2.Thmtheorem1.p1.6.m6.1.1.3.3.cmml" xref="S2.Thmtheorem1.p1.6.m6.1.1.3.3">𝑆</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem1.p1.6.m6.1c">t\in V\setminus S</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem1.p1.6.m6.1d">italic_t ∈ italic_V ∖ italic_S</annotation></semantics></math>, <math alttext="|{\delta(S)}|\geq k" class="ltx_Math" display="inline" id="S2.Thmtheorem1.p1.7.m7.2"><semantics id="S2.Thmtheorem1.p1.7.m7.2a"><mrow id="S2.Thmtheorem1.p1.7.m7.2.2" xref="S2.Thmtheorem1.p1.7.m7.2.2.cmml"><mrow id="S2.Thmtheorem1.p1.7.m7.2.2.1.1" xref="S2.Thmtheorem1.p1.7.m7.2.2.1.2.cmml"><mo id="S2.Thmtheorem1.p1.7.m7.2.2.1.1.2" stretchy="false" xref="S2.Thmtheorem1.p1.7.m7.2.2.1.2.1.cmml">|</mo><mrow id="S2.Thmtheorem1.p1.7.m7.2.2.1.1.1" xref="S2.Thmtheorem1.p1.7.m7.2.2.1.1.1.cmml"><mi id="S2.Thmtheorem1.p1.7.m7.2.2.1.1.1.2" xref="S2.Thmtheorem1.p1.7.m7.2.2.1.1.1.2.cmml">δ</mi><mo id="S2.Thmtheorem1.p1.7.m7.2.2.1.1.1.1" xref="S2.Thmtheorem1.p1.7.m7.2.2.1.1.1.1.cmml">⁢</mo><mrow id="S2.Thmtheorem1.p1.7.m7.2.2.1.1.1.3.2" xref="S2.Thmtheorem1.p1.7.m7.2.2.1.1.1.cmml"><mo id="S2.Thmtheorem1.p1.7.m7.2.2.1.1.1.3.2.1" stretchy="false" xref="S2.Thmtheorem1.p1.7.m7.2.2.1.1.1.cmml">(</mo><mi id="S2.Thmtheorem1.p1.7.m7.1.1" xref="S2.Thmtheorem1.p1.7.m7.1.1.cmml">S</mi><mo id="S2.Thmtheorem1.p1.7.m7.2.2.1.1.1.3.2.2" stretchy="false" xref="S2.Thmtheorem1.p1.7.m7.2.2.1.1.1.cmml">)</mo></mrow></mrow><mo id="S2.Thmtheorem1.p1.7.m7.2.2.1.1.3" stretchy="false" xref="S2.Thmtheorem1.p1.7.m7.2.2.1.2.1.cmml">|</mo></mrow><mo id="S2.Thmtheorem1.p1.7.m7.2.2.2" xref="S2.Thmtheorem1.p1.7.m7.2.2.2.cmml">≥</mo><mi id="S2.Thmtheorem1.p1.7.m7.2.2.3" xref="S2.Thmtheorem1.p1.7.m7.2.2.3.cmml">k</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem1.p1.7.m7.2b"><apply id="S2.Thmtheorem1.p1.7.m7.2.2.cmml" xref="S2.Thmtheorem1.p1.7.m7.2.2"><geq id="S2.Thmtheorem1.p1.7.m7.2.2.2.cmml" xref="S2.Thmtheorem1.p1.7.m7.2.2.2"></geq><apply id="S2.Thmtheorem1.p1.7.m7.2.2.1.2.cmml" xref="S2.Thmtheorem1.p1.7.m7.2.2.1.1"><abs id="S2.Thmtheorem1.p1.7.m7.2.2.1.2.1.cmml" xref="S2.Thmtheorem1.p1.7.m7.2.2.1.1.2"></abs><apply id="S2.Thmtheorem1.p1.7.m7.2.2.1.1.1.cmml" xref="S2.Thmtheorem1.p1.7.m7.2.2.1.1.1"><times id="S2.Thmtheorem1.p1.7.m7.2.2.1.1.1.1.cmml" xref="S2.Thmtheorem1.p1.7.m7.2.2.1.1.1.1"></times><ci id="S2.Thmtheorem1.p1.7.m7.2.2.1.1.1.2.cmml" xref="S2.Thmtheorem1.p1.7.m7.2.2.1.1.1.2">𝛿</ci><ci id="S2.Thmtheorem1.p1.7.m7.1.1.cmml" xref="S2.Thmtheorem1.p1.7.m7.1.1">𝑆</ci></apply></apply><ci id="S2.Thmtheorem1.p1.7.m7.2.2.3.cmml" xref="S2.Thmtheorem1.p1.7.m7.2.2.3">𝑘</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem1.p1.7.m7.2c">|{\delta(S)}|\geq k</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem1.p1.7.m7.2d">| italic_δ ( italic_S ) | ≥ italic_k</annotation></semantics></math>.</p> </div> </div> <div class="ltx_para" id="S2.p2"> <p class="ltx_p" id="S2.p2.6">To state Menger’s theorem for vertex-connectivity requirements, we first introduce some notation known as a <span class="ltx_text ltx_font_italic" id="S2.p2.6.1">biset</span>, following prior work on vertex connectivity <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.00712v1#bib.bibx70" title="">Nut12</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.00712v1#bib.bibx31" title="">CVV06</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.00712v1#bib.bibx39" title="">FJW06</a>]</cite>. A biset <math alttext="\hat{X}=(X,X^{+})" class="ltx_Math" display="inline" id="S2.p2.1.m1.2"><semantics id="S2.p2.1.m1.2a"><mrow id="S2.p2.1.m1.2.2" xref="S2.p2.1.m1.2.2.cmml"><mover accent="true" id="S2.p2.1.m1.2.2.3" xref="S2.p2.1.m1.2.2.3.cmml"><mi id="S2.p2.1.m1.2.2.3.2" xref="S2.p2.1.m1.2.2.3.2.cmml">X</mi><mo id="S2.p2.1.m1.2.2.3.1" xref="S2.p2.1.m1.2.2.3.1.cmml">^</mo></mover><mo id="S2.p2.1.m1.2.2.2" xref="S2.p2.1.m1.2.2.2.cmml">=</mo><mrow id="S2.p2.1.m1.2.2.1.1" xref="S2.p2.1.m1.2.2.1.2.cmml"><mo id="S2.p2.1.m1.2.2.1.1.2" stretchy="false" xref="S2.p2.1.m1.2.2.1.2.cmml">(</mo><mi id="S2.p2.1.m1.1.1" xref="S2.p2.1.m1.1.1.cmml">X</mi><mo id="S2.p2.1.m1.2.2.1.1.3" xref="S2.p2.1.m1.2.2.1.2.cmml">,</mo><msup id="S2.p2.1.m1.2.2.1.1.1" xref="S2.p2.1.m1.2.2.1.1.1.cmml"><mi id="S2.p2.1.m1.2.2.1.1.1.2" xref="S2.p2.1.m1.2.2.1.1.1.2.cmml">X</mi><mo id="S2.p2.1.m1.2.2.1.1.1.3" xref="S2.p2.1.m1.2.2.1.1.1.3.cmml">+</mo></msup><mo id="S2.p2.1.m1.2.2.1.1.4" stretchy="false" xref="S2.p2.1.m1.2.2.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.p2.1.m1.2b"><apply id="S2.p2.1.m1.2.2.cmml" xref="S2.p2.1.m1.2.2"><eq id="S2.p2.1.m1.2.2.2.cmml" xref="S2.p2.1.m1.2.2.2"></eq><apply id="S2.p2.1.m1.2.2.3.cmml" xref="S2.p2.1.m1.2.2.3"><ci id="S2.p2.1.m1.2.2.3.1.cmml" xref="S2.p2.1.m1.2.2.3.1">^</ci><ci id="S2.p2.1.m1.2.2.3.2.cmml" xref="S2.p2.1.m1.2.2.3.2">𝑋</ci></apply><interval closure="open" id="S2.p2.1.m1.2.2.1.2.cmml" xref="S2.p2.1.m1.2.2.1.1"><ci id="S2.p2.1.m1.1.1.cmml" xref="S2.p2.1.m1.1.1">𝑋</ci><apply id="S2.p2.1.m1.2.2.1.1.1.cmml" xref="S2.p2.1.m1.2.2.1.1.1"><csymbol cd="ambiguous" id="S2.p2.1.m1.2.2.1.1.1.1.cmml" xref="S2.p2.1.m1.2.2.1.1.1">superscript</csymbol><ci id="S2.p2.1.m1.2.2.1.1.1.2.cmml" xref="S2.p2.1.m1.2.2.1.1.1.2">𝑋</ci><plus id="S2.p2.1.m1.2.2.1.1.1.3.cmml" xref="S2.p2.1.m1.2.2.1.1.1.3"></plus></apply></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p2.1.m1.2c">\hat{X}=(X,X^{+})</annotation><annotation encoding="application/x-llamapun" id="S2.p2.1.m1.2d">over^ start_ARG italic_X end_ARG = ( italic_X , italic_X start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT )</annotation></semantics></math> is a pair of sets where <math alttext="X\subseteq X^{+}\subseteq V" class="ltx_Math" display="inline" id="S2.p2.2.m2.1"><semantics id="S2.p2.2.m2.1a"><mrow id="S2.p2.2.m2.1.1" xref="S2.p2.2.m2.1.1.cmml"><mi id="S2.p2.2.m2.1.1.2" xref="S2.p2.2.m2.1.1.2.cmml">X</mi><mo id="S2.p2.2.m2.1.1.3" xref="S2.p2.2.m2.1.1.3.cmml">⊆</mo><msup id="S2.p2.2.m2.1.1.4" xref="S2.p2.2.m2.1.1.4.cmml"><mi id="S2.p2.2.m2.1.1.4.2" xref="S2.p2.2.m2.1.1.4.2.cmml">X</mi><mo id="S2.p2.2.m2.1.1.4.3" xref="S2.p2.2.m2.1.1.4.3.cmml">+</mo></msup><mo id="S2.p2.2.m2.1.1.5" xref="S2.p2.2.m2.1.1.5.cmml">⊆</mo><mi id="S2.p2.2.m2.1.1.6" xref="S2.p2.2.m2.1.1.6.cmml">V</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.p2.2.m2.1b"><apply id="S2.p2.2.m2.1.1.cmml" xref="S2.p2.2.m2.1.1"><and id="S2.p2.2.m2.1.1a.cmml" xref="S2.p2.2.m2.1.1"></and><apply id="S2.p2.2.m2.1.1b.cmml" xref="S2.p2.2.m2.1.1"><subset id="S2.p2.2.m2.1.1.3.cmml" xref="S2.p2.2.m2.1.1.3"></subset><ci id="S2.p2.2.m2.1.1.2.cmml" xref="S2.p2.2.m2.1.1.2">𝑋</ci><apply id="S2.p2.2.m2.1.1.4.cmml" xref="S2.p2.2.m2.1.1.4"><csymbol cd="ambiguous" id="S2.p2.2.m2.1.1.4.1.cmml" xref="S2.p2.2.m2.1.1.4">superscript</csymbol><ci id="S2.p2.2.m2.1.1.4.2.cmml" xref="S2.p2.2.m2.1.1.4.2">𝑋</ci><plus id="S2.p2.2.m2.1.1.4.3.cmml" xref="S2.p2.2.m2.1.1.4.3"></plus></apply></apply><apply id="S2.p2.2.m2.1.1c.cmml" xref="S2.p2.2.m2.1.1"><subset id="S2.p2.2.m2.1.1.5.cmml" xref="S2.p2.2.m2.1.1.5"></subset><share href="https://arxiv.org/html/2503.00712v1#S2.p2.2.m2.1.1.4.cmml" id="S2.p2.2.m2.1.1d.cmml" xref="S2.p2.2.m2.1.1"></share><ci id="S2.p2.2.m2.1.1.6.cmml" xref="S2.p2.2.m2.1.1.6">𝑉</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p2.2.m2.1c">X\subseteq X^{+}\subseteq V</annotation><annotation encoding="application/x-llamapun" id="S2.p2.2.m2.1d">italic_X ⊆ italic_X start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT ⊆ italic_V</annotation></semantics></math>. We say that an edge crosses a biset <math alttext="\hat{S}" class="ltx_Math" display="inline" id="S2.p2.3.m3.1"><semantics id="S2.p2.3.m3.1a"><mover accent="true" id="S2.p2.3.m3.1.1" xref="S2.p2.3.m3.1.1.cmml"><mi id="S2.p2.3.m3.1.1.2" xref="S2.p2.3.m3.1.1.2.cmml">S</mi><mo id="S2.p2.3.m3.1.1.1" xref="S2.p2.3.m3.1.1.1.cmml">^</mo></mover><annotation-xml encoding="MathML-Content" id="S2.p2.3.m3.1b"><apply id="S2.p2.3.m3.1.1.cmml" xref="S2.p2.3.m3.1.1"><ci id="S2.p2.3.m3.1.1.1.cmml" xref="S2.p2.3.m3.1.1.1">^</ci><ci id="S2.p2.3.m3.1.1.2.cmml" xref="S2.p2.3.m3.1.1.2">𝑆</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p2.3.m3.1c">\hat{S}</annotation><annotation encoding="application/x-llamapun" id="S2.p2.3.m3.1d">over^ start_ARG italic_S end_ARG</annotation></semantics></math> if one of its endpoints is in <math alttext="S" class="ltx_Math" display="inline" id="S2.p2.4.m4.1"><semantics id="S2.p2.4.m4.1a"><mi id="S2.p2.4.m4.1.1" xref="S2.p2.4.m4.1.1.cmml">S</mi><annotation-xml encoding="MathML-Content" id="S2.p2.4.m4.1b"><ci id="S2.p2.4.m4.1.1.cmml" xref="S2.p2.4.m4.1.1">𝑆</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.p2.4.m4.1c">S</annotation><annotation encoding="application/x-llamapun" id="S2.p2.4.m4.1d">italic_S</annotation></semantics></math> and the other is in <math alttext="V\setminus S^{+}" class="ltx_Math" display="inline" id="S2.p2.5.m5.1"><semantics id="S2.p2.5.m5.1a"><mrow id="S2.p2.5.m5.1.1" xref="S2.p2.5.m5.1.1.cmml"><mi id="S2.p2.5.m5.1.1.2" xref="S2.p2.5.m5.1.1.2.cmml">V</mi><mo id="S2.p2.5.m5.1.1.1" xref="S2.p2.5.m5.1.1.1.cmml">∖</mo><msup id="S2.p2.5.m5.1.1.3" xref="S2.p2.5.m5.1.1.3.cmml"><mi id="S2.p2.5.m5.1.1.3.2" xref="S2.p2.5.m5.1.1.3.2.cmml">S</mi><mo id="S2.p2.5.m5.1.1.3.3" xref="S2.p2.5.m5.1.1.3.3.cmml">+</mo></msup></mrow><annotation-xml encoding="MathML-Content" id="S2.p2.5.m5.1b"><apply id="S2.p2.5.m5.1.1.cmml" xref="S2.p2.5.m5.1.1"><setdiff id="S2.p2.5.m5.1.1.1.cmml" xref="S2.p2.5.m5.1.1.1"></setdiff><ci id="S2.p2.5.m5.1.1.2.cmml" xref="S2.p2.5.m5.1.1.2">𝑉</ci><apply id="S2.p2.5.m5.1.1.3.cmml" xref="S2.p2.5.m5.1.1.3"><csymbol cd="ambiguous" id="S2.p2.5.m5.1.1.3.1.cmml" xref="S2.p2.5.m5.1.1.3">superscript</csymbol><ci id="S2.p2.5.m5.1.1.3.2.cmml" xref="S2.p2.5.m5.1.1.3.2">𝑆</ci><plus id="S2.p2.5.m5.1.1.3.3.cmml" xref="S2.p2.5.m5.1.1.3.3"></plus></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p2.5.m5.1c">V\setminus S^{+}</annotation><annotation encoding="application/x-llamapun" id="S2.p2.5.m5.1d">italic_V ∖ italic_S start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT</annotation></semantics></math>. We define <math alttext="\delta(\hat{S})=\{uv\in E(G)\mid u\in S,v\in V\setminus S^{+}\}" class="ltx_Math" display="inline" id="S2.p2.6.m6.4"><semantics id="S2.p2.6.m6.4a"><mrow id="S2.p2.6.m6.4.4" xref="S2.p2.6.m6.4.4.cmml"><mrow id="S2.p2.6.m6.4.4.4" xref="S2.p2.6.m6.4.4.4.cmml"><mi id="S2.p2.6.m6.4.4.4.2" xref="S2.p2.6.m6.4.4.4.2.cmml">δ</mi><mo id="S2.p2.6.m6.4.4.4.1" xref="S2.p2.6.m6.4.4.4.1.cmml">⁢</mo><mrow id="S2.p2.6.m6.4.4.4.3.2" xref="S2.p2.6.m6.1.1.cmml"><mo id="S2.p2.6.m6.4.4.4.3.2.1" stretchy="false" xref="S2.p2.6.m6.1.1.cmml">(</mo><mover accent="true" id="S2.p2.6.m6.1.1" xref="S2.p2.6.m6.1.1.cmml"><mi id="S2.p2.6.m6.1.1.2" xref="S2.p2.6.m6.1.1.2.cmml">S</mi><mo id="S2.p2.6.m6.1.1.1" xref="S2.p2.6.m6.1.1.1.cmml">^</mo></mover><mo id="S2.p2.6.m6.4.4.4.3.2.2" stretchy="false" xref="S2.p2.6.m6.1.1.cmml">)</mo></mrow></mrow><mo id="S2.p2.6.m6.4.4.3" xref="S2.p2.6.m6.4.4.3.cmml">=</mo><mrow id="S2.p2.6.m6.4.4.2.2" xref="S2.p2.6.m6.4.4.2.3.cmml"><mo id="S2.p2.6.m6.4.4.2.2.3" stretchy="false" xref="S2.p2.6.m6.4.4.2.3.1.cmml">{</mo><mrow id="S2.p2.6.m6.3.3.1.1.1" xref="S2.p2.6.m6.3.3.1.1.1.cmml"><mrow id="S2.p2.6.m6.3.3.1.1.1.2" xref="S2.p2.6.m6.3.3.1.1.1.2.cmml"><mi id="S2.p2.6.m6.3.3.1.1.1.2.2" xref="S2.p2.6.m6.3.3.1.1.1.2.2.cmml">u</mi><mo id="S2.p2.6.m6.3.3.1.1.1.2.1" xref="S2.p2.6.m6.3.3.1.1.1.2.1.cmml">⁢</mo><mi id="S2.p2.6.m6.3.3.1.1.1.2.3" xref="S2.p2.6.m6.3.3.1.1.1.2.3.cmml">v</mi></mrow><mo id="S2.p2.6.m6.3.3.1.1.1.1" xref="S2.p2.6.m6.3.3.1.1.1.1.cmml">∈</mo><mrow id="S2.p2.6.m6.3.3.1.1.1.3" xref="S2.p2.6.m6.3.3.1.1.1.3.cmml"><mi id="S2.p2.6.m6.3.3.1.1.1.3.2" xref="S2.p2.6.m6.3.3.1.1.1.3.2.cmml">E</mi><mo id="S2.p2.6.m6.3.3.1.1.1.3.1" xref="S2.p2.6.m6.3.3.1.1.1.3.1.cmml">⁢</mo><mrow id="S2.p2.6.m6.3.3.1.1.1.3.3.2" xref="S2.p2.6.m6.3.3.1.1.1.3.cmml"><mo id="S2.p2.6.m6.3.3.1.1.1.3.3.2.1" stretchy="false" xref="S2.p2.6.m6.3.3.1.1.1.3.cmml">(</mo><mi id="S2.p2.6.m6.2.2" xref="S2.p2.6.m6.2.2.cmml">G</mi><mo id="S2.p2.6.m6.3.3.1.1.1.3.3.2.2" stretchy="false" xref="S2.p2.6.m6.3.3.1.1.1.3.cmml">)</mo></mrow></mrow></mrow><mo fence="true" id="S2.p2.6.m6.4.4.2.2.4" lspace="0em" rspace="0em" xref="S2.p2.6.m6.4.4.2.3.1.cmml">∣</mo><mrow id="S2.p2.6.m6.4.4.2.2.2.2" xref="S2.p2.6.m6.4.4.2.2.2.3.cmml"><mrow id="S2.p2.6.m6.4.4.2.2.2.1.1" xref="S2.p2.6.m6.4.4.2.2.2.1.1.cmml"><mi id="S2.p2.6.m6.4.4.2.2.2.1.1.2" xref="S2.p2.6.m6.4.4.2.2.2.1.1.2.cmml">u</mi><mo id="S2.p2.6.m6.4.4.2.2.2.1.1.1" xref="S2.p2.6.m6.4.4.2.2.2.1.1.1.cmml">∈</mo><mi id="S2.p2.6.m6.4.4.2.2.2.1.1.3" xref="S2.p2.6.m6.4.4.2.2.2.1.1.3.cmml">S</mi></mrow><mo id="S2.p2.6.m6.4.4.2.2.2.2.3" xref="S2.p2.6.m6.4.4.2.2.2.3a.cmml">,</mo><mrow id="S2.p2.6.m6.4.4.2.2.2.2.2" xref="S2.p2.6.m6.4.4.2.2.2.2.2.cmml"><mi id="S2.p2.6.m6.4.4.2.2.2.2.2.2" xref="S2.p2.6.m6.4.4.2.2.2.2.2.2.cmml">v</mi><mo id="S2.p2.6.m6.4.4.2.2.2.2.2.1" xref="S2.p2.6.m6.4.4.2.2.2.2.2.1.cmml">∈</mo><mrow id="S2.p2.6.m6.4.4.2.2.2.2.2.3" xref="S2.p2.6.m6.4.4.2.2.2.2.2.3.cmml"><mi 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id="S2.p2.6.m6.1.1.1.cmml" xref="S2.p2.6.m6.1.1.1">^</ci><ci id="S2.p2.6.m6.1.1.2.cmml" xref="S2.p2.6.m6.1.1.2">𝑆</ci></apply></apply><apply id="S2.p2.6.m6.4.4.2.3.cmml" xref="S2.p2.6.m6.4.4.2.2"><csymbol cd="latexml" id="S2.p2.6.m6.4.4.2.3.1.cmml" xref="S2.p2.6.m6.4.4.2.2.3">conditional-set</csymbol><apply id="S2.p2.6.m6.3.3.1.1.1.cmml" xref="S2.p2.6.m6.3.3.1.1.1"><in id="S2.p2.6.m6.3.3.1.1.1.1.cmml" xref="S2.p2.6.m6.3.3.1.1.1.1"></in><apply id="S2.p2.6.m6.3.3.1.1.1.2.cmml" xref="S2.p2.6.m6.3.3.1.1.1.2"><times id="S2.p2.6.m6.3.3.1.1.1.2.1.cmml" xref="S2.p2.6.m6.3.3.1.1.1.2.1"></times><ci id="S2.p2.6.m6.3.3.1.1.1.2.2.cmml" xref="S2.p2.6.m6.3.3.1.1.1.2.2">𝑢</ci><ci id="S2.p2.6.m6.3.3.1.1.1.2.3.cmml" xref="S2.p2.6.m6.3.3.1.1.1.2.3">𝑣</ci></apply><apply id="S2.p2.6.m6.3.3.1.1.1.3.cmml" xref="S2.p2.6.m6.3.3.1.1.1.3"><times id="S2.p2.6.m6.3.3.1.1.1.3.1.cmml" xref="S2.p2.6.m6.3.3.1.1.1.3.1"></times><ci id="S2.p2.6.m6.3.3.1.1.1.3.2.cmml" xref="S2.p2.6.m6.3.3.1.1.1.3.2">𝐸</ci><ci id="S2.p2.6.m6.2.2.cmml" xref="S2.p2.6.m6.2.2">𝐺</ci></apply></apply><apply id="S2.p2.6.m6.4.4.2.2.2.3.cmml" xref="S2.p2.6.m6.4.4.2.2.2.2"><csymbol cd="ambiguous" id="S2.p2.6.m6.4.4.2.2.2.3a.cmml" xref="S2.p2.6.m6.4.4.2.2.2.2.3">formulae-sequence</csymbol><apply id="S2.p2.6.m6.4.4.2.2.2.1.1.cmml" xref="S2.p2.6.m6.4.4.2.2.2.1.1"><in id="S2.p2.6.m6.4.4.2.2.2.1.1.1.cmml" xref="S2.p2.6.m6.4.4.2.2.2.1.1.1"></in><ci id="S2.p2.6.m6.4.4.2.2.2.1.1.2.cmml" xref="S2.p2.6.m6.4.4.2.2.2.1.1.2">𝑢</ci><ci id="S2.p2.6.m6.4.4.2.2.2.1.1.3.cmml" xref="S2.p2.6.m6.4.4.2.2.2.1.1.3">𝑆</ci></apply><apply id="S2.p2.6.m6.4.4.2.2.2.2.2.cmml" xref="S2.p2.6.m6.4.4.2.2.2.2.2"><in id="S2.p2.6.m6.4.4.2.2.2.2.2.1.cmml" xref="S2.p2.6.m6.4.4.2.2.2.2.2.1"></in><ci id="S2.p2.6.m6.4.4.2.2.2.2.2.2.cmml" xref="S2.p2.6.m6.4.4.2.2.2.2.2.2">𝑣</ci><apply id="S2.p2.6.m6.4.4.2.2.2.2.2.3.cmml" xref="S2.p2.6.m6.4.4.2.2.2.2.2.3"><setdiff id="S2.p2.6.m6.4.4.2.2.2.2.2.3.1.cmml" xref="S2.p2.6.m6.4.4.2.2.2.2.2.3.1"></setdiff><ci id="S2.p2.6.m6.4.4.2.2.2.2.2.3.2.cmml" xref="S2.p2.6.m6.4.4.2.2.2.2.2.3.2">𝑉</ci><apply id="S2.p2.6.m6.4.4.2.2.2.2.2.3.3.cmml" xref="S2.p2.6.m6.4.4.2.2.2.2.2.3.3"><csymbol cd="ambiguous" id="S2.p2.6.m6.4.4.2.2.2.2.2.3.3.1.cmml" xref="S2.p2.6.m6.4.4.2.2.2.2.2.3.3">superscript</csymbol><ci id="S2.p2.6.m6.4.4.2.2.2.2.2.3.3.2.cmml" xref="S2.p2.6.m6.4.4.2.2.2.2.2.3.3.2">𝑆</ci><plus id="S2.p2.6.m6.4.4.2.2.2.2.2.3.3.3.cmml" xref="S2.p2.6.m6.4.4.2.2.2.2.2.3.3.3"></plus></apply></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p2.6.m6.4c">\delta(\hat{S})=\{uv\in E(G)\mid u\in S,v\in V\setminus S^{+}\}</annotation><annotation encoding="application/x-llamapun" id="S2.p2.6.m6.4d">italic_δ ( over^ start_ARG italic_S end_ARG ) = { italic_u italic_v ∈ italic_E ( italic_G ) ∣ italic_u ∈ italic_S , italic_v ∈ italic_V ∖ italic_S start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT }</annotation></semantics></math>.</p> </div> <div class="ltx_theorem ltx_theorem_theorem" id="S2.Thmtheorem2"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="S2.Thmtheorem2.1.1.1">Theorem 2.2</span></span><span class="ltx_text ltx_font_bold" id="S2.Thmtheorem2.2.2"> </span>(Vertex-connectivity Menger’s theorem)<span class="ltx_text ltx_font_bold" id="S2.Thmtheorem2.3.3">.</span> </h6> <div class="ltx_para" id="S2.Thmtheorem2.p1"> <p class="ltx_p" id="S2.Thmtheorem2.p1.7">Let <math alttext="G=(V,E)" class="ltx_Math" display="inline" id="S2.Thmtheorem2.p1.1.m1.2"><semantics id="S2.Thmtheorem2.p1.1.m1.2a"><mrow id="S2.Thmtheorem2.p1.1.m1.2.3" xref="S2.Thmtheorem2.p1.1.m1.2.3.cmml"><mi id="S2.Thmtheorem2.p1.1.m1.2.3.2" xref="S2.Thmtheorem2.p1.1.m1.2.3.2.cmml">G</mi><mo id="S2.Thmtheorem2.p1.1.m1.2.3.1" xref="S2.Thmtheorem2.p1.1.m1.2.3.1.cmml">=</mo><mrow id="S2.Thmtheorem2.p1.1.m1.2.3.3.2" xref="S2.Thmtheorem2.p1.1.m1.2.3.3.1.cmml"><mo id="S2.Thmtheorem2.p1.1.m1.2.3.3.2.1" stretchy="false" xref="S2.Thmtheorem2.p1.1.m1.2.3.3.1.cmml">(</mo><mi id="S2.Thmtheorem2.p1.1.m1.1.1" xref="S2.Thmtheorem2.p1.1.m1.1.1.cmml">V</mi><mo id="S2.Thmtheorem2.p1.1.m1.2.3.3.2.2" xref="S2.Thmtheorem2.p1.1.m1.2.3.3.1.cmml">,</mo><mi id="S2.Thmtheorem2.p1.1.m1.2.2" xref="S2.Thmtheorem2.p1.1.m1.2.2.cmml">E</mi><mo id="S2.Thmtheorem2.p1.1.m1.2.3.3.2.3" stretchy="false" xref="S2.Thmtheorem2.p1.1.m1.2.3.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem2.p1.1.m1.2b"><apply id="S2.Thmtheorem2.p1.1.m1.2.3.cmml" xref="S2.Thmtheorem2.p1.1.m1.2.3"><eq id="S2.Thmtheorem2.p1.1.m1.2.3.1.cmml" xref="S2.Thmtheorem2.p1.1.m1.2.3.1"></eq><ci id="S2.Thmtheorem2.p1.1.m1.2.3.2.cmml" xref="S2.Thmtheorem2.p1.1.m1.2.3.2">𝐺</ci><interval closure="open" id="S2.Thmtheorem2.p1.1.m1.2.3.3.1.cmml" xref="S2.Thmtheorem2.p1.1.m1.2.3.3.2"><ci id="S2.Thmtheorem2.p1.1.m1.1.1.cmml" xref="S2.Thmtheorem2.p1.1.m1.1.1">𝑉</ci><ci id="S2.Thmtheorem2.p1.1.m1.2.2.cmml" xref="S2.Thmtheorem2.p1.1.m1.2.2">𝐸</ci></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem2.p1.1.m1.2c">G=(V,E)</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem2.p1.1.m1.2d">italic_G = ( italic_V , italic_E )</annotation></semantics></math> be an undirected graph. Two vertices <math alttext="s,t\in V" class="ltx_Math" display="inline" id="S2.Thmtheorem2.p1.2.m2.2"><semantics id="S2.Thmtheorem2.p1.2.m2.2a"><mrow id="S2.Thmtheorem2.p1.2.m2.2.3" xref="S2.Thmtheorem2.p1.2.m2.2.3.cmml"><mrow id="S2.Thmtheorem2.p1.2.m2.2.3.2.2" xref="S2.Thmtheorem2.p1.2.m2.2.3.2.1.cmml"><mi id="S2.Thmtheorem2.p1.2.m2.1.1" xref="S2.Thmtheorem2.p1.2.m2.1.1.cmml">s</mi><mo id="S2.Thmtheorem2.p1.2.m2.2.3.2.2.1" xref="S2.Thmtheorem2.p1.2.m2.2.3.2.1.cmml">,</mo><mi id="S2.Thmtheorem2.p1.2.m2.2.2" xref="S2.Thmtheorem2.p1.2.m2.2.2.cmml">t</mi></mrow><mo id="S2.Thmtheorem2.p1.2.m2.2.3.1" xref="S2.Thmtheorem2.p1.2.m2.2.3.1.cmml">∈</mo><mi id="S2.Thmtheorem2.p1.2.m2.2.3.3" xref="S2.Thmtheorem2.p1.2.m2.2.3.3.cmml">V</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem2.p1.2.m2.2b"><apply id="S2.Thmtheorem2.p1.2.m2.2.3.cmml" xref="S2.Thmtheorem2.p1.2.m2.2.3"><in id="S2.Thmtheorem2.p1.2.m2.2.3.1.cmml" xref="S2.Thmtheorem2.p1.2.m2.2.3.1"></in><list id="S2.Thmtheorem2.p1.2.m2.2.3.2.1.cmml" xref="S2.Thmtheorem2.p1.2.m2.2.3.2.2"><ci id="S2.Thmtheorem2.p1.2.m2.1.1.cmml" xref="S2.Thmtheorem2.p1.2.m2.1.1">𝑠</ci><ci id="S2.Thmtheorem2.p1.2.m2.2.2.cmml" xref="S2.Thmtheorem2.p1.2.m2.2.2">𝑡</ci></list><ci id="S2.Thmtheorem2.p1.2.m2.2.3.3.cmml" xref="S2.Thmtheorem2.p1.2.m2.2.3.3">𝑉</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem2.p1.2.m2.2c">s,t\in V</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem2.p1.2.m2.2d">italic_s , italic_t ∈ italic_V</annotation></semantics></math> are <math alttext="k" class="ltx_Math" display="inline" id="S2.Thmtheorem2.p1.3.m3.1"><semantics id="S2.Thmtheorem2.p1.3.m3.1a"><mi id="S2.Thmtheorem2.p1.3.m3.1.1" xref="S2.Thmtheorem2.p1.3.m3.1.1.cmml">k</mi><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem2.p1.3.m3.1b"><ci id="S2.Thmtheorem2.p1.3.m3.1.1.cmml" xref="S2.Thmtheorem2.p1.3.m3.1.1">𝑘</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem2.p1.3.m3.1c">k</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem2.p1.3.m3.1d">italic_k</annotation></semantics></math>-vertex connected iff for each biset <math alttext="\hat{S}\subset V\times V" class="ltx_Math" display="inline" id="S2.Thmtheorem2.p1.4.m4.1"><semantics id="S2.Thmtheorem2.p1.4.m4.1a"><mrow id="S2.Thmtheorem2.p1.4.m4.1.1" xref="S2.Thmtheorem2.p1.4.m4.1.1.cmml"><mover accent="true" id="S2.Thmtheorem2.p1.4.m4.1.1.2" xref="S2.Thmtheorem2.p1.4.m4.1.1.2.cmml"><mi id="S2.Thmtheorem2.p1.4.m4.1.1.2.2" xref="S2.Thmtheorem2.p1.4.m4.1.1.2.2.cmml">S</mi><mo id="S2.Thmtheorem2.p1.4.m4.1.1.2.1" xref="S2.Thmtheorem2.p1.4.m4.1.1.2.1.cmml">^</mo></mover><mo id="S2.Thmtheorem2.p1.4.m4.1.1.1" xref="S2.Thmtheorem2.p1.4.m4.1.1.1.cmml">⊂</mo><mrow id="S2.Thmtheorem2.p1.4.m4.1.1.3" xref="S2.Thmtheorem2.p1.4.m4.1.1.3.cmml"><mi id="S2.Thmtheorem2.p1.4.m4.1.1.3.2" xref="S2.Thmtheorem2.p1.4.m4.1.1.3.2.cmml">V</mi><mo id="S2.Thmtheorem2.p1.4.m4.1.1.3.1" lspace="0.222em" rspace="0.222em" xref="S2.Thmtheorem2.p1.4.m4.1.1.3.1.cmml">×</mo><mi id="S2.Thmtheorem2.p1.4.m4.1.1.3.3" xref="S2.Thmtheorem2.p1.4.m4.1.1.3.3.cmml">V</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem2.p1.4.m4.1b"><apply id="S2.Thmtheorem2.p1.4.m4.1.1.cmml" xref="S2.Thmtheorem2.p1.4.m4.1.1"><subset id="S2.Thmtheorem2.p1.4.m4.1.1.1.cmml" xref="S2.Thmtheorem2.p1.4.m4.1.1.1"></subset><apply id="S2.Thmtheorem2.p1.4.m4.1.1.2.cmml" xref="S2.Thmtheorem2.p1.4.m4.1.1.2"><ci id="S2.Thmtheorem2.p1.4.m4.1.1.2.1.cmml" xref="S2.Thmtheorem2.p1.4.m4.1.1.2.1">^</ci><ci id="S2.Thmtheorem2.p1.4.m4.1.1.2.2.cmml" xref="S2.Thmtheorem2.p1.4.m4.1.1.2.2">𝑆</ci></apply><apply id="S2.Thmtheorem2.p1.4.m4.1.1.3.cmml" xref="S2.Thmtheorem2.p1.4.m4.1.1.3"><times id="S2.Thmtheorem2.p1.4.m4.1.1.3.1.cmml" xref="S2.Thmtheorem2.p1.4.m4.1.1.3.1"></times><ci id="S2.Thmtheorem2.p1.4.m4.1.1.3.2.cmml" xref="S2.Thmtheorem2.p1.4.m4.1.1.3.2">𝑉</ci><ci id="S2.Thmtheorem2.p1.4.m4.1.1.3.3.cmml" xref="S2.Thmtheorem2.p1.4.m4.1.1.3.3">𝑉</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem2.p1.4.m4.1c">\hat{S}\subset V\times V</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem2.p1.4.m4.1d">over^ start_ARG italic_S end_ARG ⊂ italic_V × italic_V</annotation></semantics></math> such that <math alttext="s\in S" class="ltx_Math" display="inline" id="S2.Thmtheorem2.p1.5.m5.1"><semantics id="S2.Thmtheorem2.p1.5.m5.1a"><mrow id="S2.Thmtheorem2.p1.5.m5.1.1" xref="S2.Thmtheorem2.p1.5.m5.1.1.cmml"><mi id="S2.Thmtheorem2.p1.5.m5.1.1.2" xref="S2.Thmtheorem2.p1.5.m5.1.1.2.cmml">s</mi><mo id="S2.Thmtheorem2.p1.5.m5.1.1.1" xref="S2.Thmtheorem2.p1.5.m5.1.1.1.cmml">∈</mo><mi id="S2.Thmtheorem2.p1.5.m5.1.1.3" xref="S2.Thmtheorem2.p1.5.m5.1.1.3.cmml">S</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem2.p1.5.m5.1b"><apply id="S2.Thmtheorem2.p1.5.m5.1.1.cmml" xref="S2.Thmtheorem2.p1.5.m5.1.1"><in id="S2.Thmtheorem2.p1.5.m5.1.1.1.cmml" xref="S2.Thmtheorem2.p1.5.m5.1.1.1"></in><ci id="S2.Thmtheorem2.p1.5.m5.1.1.2.cmml" xref="S2.Thmtheorem2.p1.5.m5.1.1.2">𝑠</ci><ci id="S2.Thmtheorem2.p1.5.m5.1.1.3.cmml" xref="S2.Thmtheorem2.p1.5.m5.1.1.3">𝑆</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem2.p1.5.m5.1c">s\in S</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem2.p1.5.m5.1d">italic_s ∈ italic_S</annotation></semantics></math> and <math alttext="t\in V\setminus S^{+}" class="ltx_Math" display="inline" id="S2.Thmtheorem2.p1.6.m6.1"><semantics id="S2.Thmtheorem2.p1.6.m6.1a"><mrow id="S2.Thmtheorem2.p1.6.m6.1.1" xref="S2.Thmtheorem2.p1.6.m6.1.1.cmml"><mi id="S2.Thmtheorem2.p1.6.m6.1.1.2" xref="S2.Thmtheorem2.p1.6.m6.1.1.2.cmml">t</mi><mo id="S2.Thmtheorem2.p1.6.m6.1.1.1" xref="S2.Thmtheorem2.p1.6.m6.1.1.1.cmml">∈</mo><mrow 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xref="S2.Thmtheorem2.p1.6.m6.1.1.3.1"></setdiff><ci id="S2.Thmtheorem2.p1.6.m6.1.1.3.2.cmml" xref="S2.Thmtheorem2.p1.6.m6.1.1.3.2">𝑉</ci><apply id="S2.Thmtheorem2.p1.6.m6.1.1.3.3.cmml" xref="S2.Thmtheorem2.p1.6.m6.1.1.3.3"><csymbol cd="ambiguous" id="S2.Thmtheorem2.p1.6.m6.1.1.3.3.1.cmml" xref="S2.Thmtheorem2.p1.6.m6.1.1.3.3">superscript</csymbol><ci id="S2.Thmtheorem2.p1.6.m6.1.1.3.3.2.cmml" xref="S2.Thmtheorem2.p1.6.m6.1.1.3.3.2">𝑆</ci><plus id="S2.Thmtheorem2.p1.6.m6.1.1.3.3.3.cmml" xref="S2.Thmtheorem2.p1.6.m6.1.1.3.3.3"></plus></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem2.p1.6.m6.1c">t\in V\setminus S^{+}</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem2.p1.6.m6.1d">italic_t ∈ italic_V ∖ italic_S start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT</annotation></semantics></math>, <math alttext="|\delta(\hat{S})|+|S^{+}\setminus S|\geq k" class="ltx_Math" display="inline" id="S2.Thmtheorem2.p1.7.m7.3"><semantics id="S2.Thmtheorem2.p1.7.m7.3a"><mrow id="S2.Thmtheorem2.p1.7.m7.3.3" xref="S2.Thmtheorem2.p1.7.m7.3.3.cmml"><mrow id="S2.Thmtheorem2.p1.7.m7.3.3.2" xref="S2.Thmtheorem2.p1.7.m7.3.3.2.cmml"><mrow id="S2.Thmtheorem2.p1.7.m7.2.2.1.1.1" xref="S2.Thmtheorem2.p1.7.m7.2.2.1.1.2.cmml"><mo id="S2.Thmtheorem2.p1.7.m7.2.2.1.1.1.2" stretchy="false" xref="S2.Thmtheorem2.p1.7.m7.2.2.1.1.2.1.cmml">|</mo><mrow id="S2.Thmtheorem2.p1.7.m7.2.2.1.1.1.1" xref="S2.Thmtheorem2.p1.7.m7.2.2.1.1.1.1.cmml"><mi id="S2.Thmtheorem2.p1.7.m7.2.2.1.1.1.1.2" xref="S2.Thmtheorem2.p1.7.m7.2.2.1.1.1.1.2.cmml">δ</mi><mo id="S2.Thmtheorem2.p1.7.m7.2.2.1.1.1.1.1" xref="S2.Thmtheorem2.p1.7.m7.2.2.1.1.1.1.1.cmml">⁢</mo><mrow id="S2.Thmtheorem2.p1.7.m7.2.2.1.1.1.1.3.2" xref="S2.Thmtheorem2.p1.7.m7.1.1.cmml"><mo id="S2.Thmtheorem2.p1.7.m7.2.2.1.1.1.1.3.2.1" stretchy="false" xref="S2.Thmtheorem2.p1.7.m7.1.1.cmml">(</mo><mover accent="true" id="S2.Thmtheorem2.p1.7.m7.1.1" xref="S2.Thmtheorem2.p1.7.m7.1.1.cmml"><mi id="S2.Thmtheorem2.p1.7.m7.1.1.2" xref="S2.Thmtheorem2.p1.7.m7.1.1.2.cmml">S</mi><mo id="S2.Thmtheorem2.p1.7.m7.1.1.1" xref="S2.Thmtheorem2.p1.7.m7.1.1.1.cmml">^</mo></mover><mo id="S2.Thmtheorem2.p1.7.m7.2.2.1.1.1.1.3.2.2" stretchy="false" xref="S2.Thmtheorem2.p1.7.m7.1.1.cmml">)</mo></mrow></mrow><mo id="S2.Thmtheorem2.p1.7.m7.2.2.1.1.1.3" stretchy="false" xref="S2.Thmtheorem2.p1.7.m7.2.2.1.1.2.1.cmml">|</mo></mrow><mo id="S2.Thmtheorem2.p1.7.m7.3.3.2.3" xref="S2.Thmtheorem2.p1.7.m7.3.3.2.3.cmml">+</mo><mrow id="S2.Thmtheorem2.p1.7.m7.3.3.2.2.1" xref="S2.Thmtheorem2.p1.7.m7.3.3.2.2.2.cmml"><mo id="S2.Thmtheorem2.p1.7.m7.3.3.2.2.1.2" stretchy="false" xref="S2.Thmtheorem2.p1.7.m7.3.3.2.2.2.1.cmml">|</mo><mrow id="S2.Thmtheorem2.p1.7.m7.3.3.2.2.1.1" xref="S2.Thmtheorem2.p1.7.m7.3.3.2.2.1.1.cmml"><msup id="S2.Thmtheorem2.p1.7.m7.3.3.2.2.1.1.2" xref="S2.Thmtheorem2.p1.7.m7.3.3.2.2.1.1.2.cmml"><mi id="S2.Thmtheorem2.p1.7.m7.3.3.2.2.1.1.2.2" xref="S2.Thmtheorem2.p1.7.m7.3.3.2.2.1.1.2.2.cmml">S</mi><mo id="S2.Thmtheorem2.p1.7.m7.3.3.2.2.1.1.2.3" xref="S2.Thmtheorem2.p1.7.m7.3.3.2.2.1.1.2.3.cmml">+</mo></msup><mo id="S2.Thmtheorem2.p1.7.m7.3.3.2.2.1.1.1" xref="S2.Thmtheorem2.p1.7.m7.3.3.2.2.1.1.1.cmml">∖</mo><mi id="S2.Thmtheorem2.p1.7.m7.3.3.2.2.1.1.3" xref="S2.Thmtheorem2.p1.7.m7.3.3.2.2.1.1.3.cmml">S</mi></mrow><mo id="S2.Thmtheorem2.p1.7.m7.3.3.2.2.1.3" stretchy="false" xref="S2.Thmtheorem2.p1.7.m7.3.3.2.2.2.1.cmml">|</mo></mrow></mrow><mo id="S2.Thmtheorem2.p1.7.m7.3.3.3" xref="S2.Thmtheorem2.p1.7.m7.3.3.3.cmml">≥</mo><mi id="S2.Thmtheorem2.p1.7.m7.3.3.4" xref="S2.Thmtheorem2.p1.7.m7.3.3.4.cmml">k</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem2.p1.7.m7.3b"><apply id="S2.Thmtheorem2.p1.7.m7.3.3.cmml" xref="S2.Thmtheorem2.p1.7.m7.3.3"><geq id="S2.Thmtheorem2.p1.7.m7.3.3.3.cmml" xref="S2.Thmtheorem2.p1.7.m7.3.3.3"></geq><apply id="S2.Thmtheorem2.p1.7.m7.3.3.2.cmml" xref="S2.Thmtheorem2.p1.7.m7.3.3.2"><plus id="S2.Thmtheorem2.p1.7.m7.3.3.2.3.cmml" xref="S2.Thmtheorem2.p1.7.m7.3.3.2.3"></plus><apply id="S2.Thmtheorem2.p1.7.m7.2.2.1.1.2.cmml" xref="S2.Thmtheorem2.p1.7.m7.2.2.1.1.1"><abs id="S2.Thmtheorem2.p1.7.m7.2.2.1.1.2.1.cmml" xref="S2.Thmtheorem2.p1.7.m7.2.2.1.1.1.2"></abs><apply id="S2.Thmtheorem2.p1.7.m7.2.2.1.1.1.1.cmml" xref="S2.Thmtheorem2.p1.7.m7.2.2.1.1.1.1"><times id="S2.Thmtheorem2.p1.7.m7.2.2.1.1.1.1.1.cmml" xref="S2.Thmtheorem2.p1.7.m7.2.2.1.1.1.1.1"></times><ci id="S2.Thmtheorem2.p1.7.m7.2.2.1.1.1.1.2.cmml" xref="S2.Thmtheorem2.p1.7.m7.2.2.1.1.1.1.2">𝛿</ci><apply id="S2.Thmtheorem2.p1.7.m7.1.1.cmml" xref="S2.Thmtheorem2.p1.7.m7.2.2.1.1.1.1.3.2"><ci id="S2.Thmtheorem2.p1.7.m7.1.1.1.cmml" xref="S2.Thmtheorem2.p1.7.m7.1.1.1">^</ci><ci id="S2.Thmtheorem2.p1.7.m7.1.1.2.cmml" xref="S2.Thmtheorem2.p1.7.m7.1.1.2">𝑆</ci></apply></apply></apply><apply id="S2.Thmtheorem2.p1.7.m7.3.3.2.2.2.cmml" xref="S2.Thmtheorem2.p1.7.m7.3.3.2.2.1"><abs id="S2.Thmtheorem2.p1.7.m7.3.3.2.2.2.1.cmml" xref="S2.Thmtheorem2.p1.7.m7.3.3.2.2.1.2"></abs><apply id="S2.Thmtheorem2.p1.7.m7.3.3.2.2.1.1.cmml" xref="S2.Thmtheorem2.p1.7.m7.3.3.2.2.1.1"><setdiff id="S2.Thmtheorem2.p1.7.m7.3.3.2.2.1.1.1.cmml" xref="S2.Thmtheorem2.p1.7.m7.3.3.2.2.1.1.1"></setdiff><apply id="S2.Thmtheorem2.p1.7.m7.3.3.2.2.1.1.2.cmml" xref="S2.Thmtheorem2.p1.7.m7.3.3.2.2.1.1.2"><csymbol cd="ambiguous" id="S2.Thmtheorem2.p1.7.m7.3.3.2.2.1.1.2.1.cmml" xref="S2.Thmtheorem2.p1.7.m7.3.3.2.2.1.1.2">superscript</csymbol><ci id="S2.Thmtheorem2.p1.7.m7.3.3.2.2.1.1.2.2.cmml" xref="S2.Thmtheorem2.p1.7.m7.3.3.2.2.1.1.2.2">𝑆</ci><plus id="S2.Thmtheorem2.p1.7.m7.3.3.2.2.1.1.2.3.cmml" xref="S2.Thmtheorem2.p1.7.m7.3.3.2.2.1.1.2.3"></plus></apply><ci id="S2.Thmtheorem2.p1.7.m7.3.3.2.2.1.1.3.cmml" xref="S2.Thmtheorem2.p1.7.m7.3.3.2.2.1.1.3">𝑆</ci></apply></apply></apply><ci id="S2.Thmtheorem2.p1.7.m7.3.3.4.cmml" xref="S2.Thmtheorem2.p1.7.m7.3.3.4">𝑘</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem2.p1.7.m7.3c">|\delta(\hat{S})|+|S^{+}\setminus S|\geq k</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem2.p1.7.m7.3d">| italic_δ ( over^ start_ARG italic_S end_ARG ) | + | italic_S start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT ∖ italic_S | ≥ italic_k</annotation></semantics></math>.</p> </div> </div> <div class="ltx_para" id="S2.p3"> <p class="ltx_p" id="S2.p3.7">An intermediate connectivity notion between edge-connectivity and vertex-connectivity, proposed by Jain <span class="ltx_text ltx_font_italic" id="S2.p3.7.1">et al.</span> <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.00712v1#bib.bibx55" title="">JMVW02</a>]</cite>, is <span class="ltx_text ltx_font_italic" id="S2.p3.7.2">element-connectivity</span>, in which the input set of vertices <math alttext="V" class="ltx_Math" display="inline" id="S2.p3.1.m1.1"><semantics id="S2.p3.1.m1.1a"><mi id="S2.p3.1.m1.1.1" xref="S2.p3.1.m1.1.1.cmml">V</mi><annotation-xml encoding="MathML-Content" id="S2.p3.1.m1.1b"><ci id="S2.p3.1.m1.1.1.cmml" xref="S2.p3.1.m1.1.1">𝑉</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.p3.1.m1.1c">V</annotation><annotation encoding="application/x-llamapun" id="S2.p3.1.m1.1d">italic_V</annotation></semantics></math> is divided into two types: <span class="ltx_text ltx_font_italic" id="S2.p3.7.3">reliable</span> (<math alttext="R" class="ltx_Math" display="inline" id="S2.p3.2.m2.1"><semantics id="S2.p3.2.m2.1a"><mi id="S2.p3.2.m2.1.1" xref="S2.p3.2.m2.1.1.cmml">R</mi><annotation-xml encoding="MathML-Content" id="S2.p3.2.m2.1b"><ci id="S2.p3.2.m2.1.1.cmml" xref="S2.p3.2.m2.1.1">𝑅</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.p3.2.m2.1c">R</annotation><annotation encoding="application/x-llamapun" id="S2.p3.2.m2.1d">italic_R</annotation></semantics></math>) and <span class="ltx_text ltx_font_italic" id="S2.p3.7.4">non-reliable</span> (<math alttext="V\setminus R" class="ltx_Math" display="inline" id="S2.p3.3.m3.1"><semantics id="S2.p3.3.m3.1a"><mrow id="S2.p3.3.m3.1.1" xref="S2.p3.3.m3.1.1.cmml"><mi id="S2.p3.3.m3.1.1.2" xref="S2.p3.3.m3.1.1.2.cmml">V</mi><mo id="S2.p3.3.m3.1.1.1" xref="S2.p3.3.m3.1.1.1.cmml">∖</mo><mi id="S2.p3.3.m3.1.1.3" xref="S2.p3.3.m3.1.1.3.cmml">R</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.p3.3.m3.1b"><apply id="S2.p3.3.m3.1.1.cmml" xref="S2.p3.3.m3.1.1"><setdiff id="S2.p3.3.m3.1.1.1.cmml" xref="S2.p3.3.m3.1.1.1"></setdiff><ci id="S2.p3.3.m3.1.1.2.cmml" xref="S2.p3.3.m3.1.1.2">𝑉</ci><ci id="S2.p3.3.m3.1.1.3.cmml" xref="S2.p3.3.m3.1.1.3">𝑅</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p3.3.m3.1c">V\setminus R</annotation><annotation encoding="application/x-llamapun" id="S2.p3.3.m3.1d">italic_V ∖ italic_R</annotation></semantics></math>). Elements denote the set of edges <math alttext="E" class="ltx_Math" display="inline" id="S2.p3.4.m4.1"><semantics id="S2.p3.4.m4.1a"><mi id="S2.p3.4.m4.1.1" xref="S2.p3.4.m4.1.1.cmml">E</mi><annotation-xml encoding="MathML-Content" id="S2.p3.4.m4.1b"><ci id="S2.p3.4.m4.1.1.cmml" xref="S2.p3.4.m4.1.1">𝐸</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.p3.4.m4.1c">E</annotation><annotation encoding="application/x-llamapun" id="S2.p3.4.m4.1d">italic_E</annotation></semantics></math> and the set of non-reliable vertices <math alttext="V\setminus R" class="ltx_Math" display="inline" id="S2.p3.5.m5.1"><semantics id="S2.p3.5.m5.1a"><mrow id="S2.p3.5.m5.1.1" xref="S2.p3.5.m5.1.1.cmml"><mi id="S2.p3.5.m5.1.1.2" xref="S2.p3.5.m5.1.1.2.cmml">V</mi><mo id="S2.p3.5.m5.1.1.1" xref="S2.p3.5.m5.1.1.1.cmml">∖</mo><mi id="S2.p3.5.m5.1.1.3" xref="S2.p3.5.m5.1.1.3.cmml">R</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.p3.5.m5.1b"><apply id="S2.p3.5.m5.1.1.cmml" xref="S2.p3.5.m5.1.1"><setdiff id="S2.p3.5.m5.1.1.1.cmml" xref="S2.p3.5.m5.1.1.1"></setdiff><ci id="S2.p3.5.m5.1.1.2.cmml" xref="S2.p3.5.m5.1.1.2">𝑉</ci><ci id="S2.p3.5.m5.1.1.3.cmml" xref="S2.p3.5.m5.1.1.3">𝑅</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p3.5.m5.1c">V\setminus R</annotation><annotation encoding="application/x-llamapun" id="S2.p3.5.m5.1d">italic_V ∖ italic_R</annotation></semantics></math>. For a pair of vertices in <math alttext="u,v\in R" class="ltx_Math" display="inline" id="S2.p3.6.m6.2"><semantics id="S2.p3.6.m6.2a"><mrow id="S2.p3.6.m6.2.3" xref="S2.p3.6.m6.2.3.cmml"><mrow id="S2.p3.6.m6.2.3.2.2" xref="S2.p3.6.m6.2.3.2.1.cmml"><mi id="S2.p3.6.m6.1.1" xref="S2.p3.6.m6.1.1.cmml">u</mi><mo id="S2.p3.6.m6.2.3.2.2.1" xref="S2.p3.6.m6.2.3.2.1.cmml">,</mo><mi id="S2.p3.6.m6.2.2" xref="S2.p3.6.m6.2.2.cmml">v</mi></mrow><mo id="S2.p3.6.m6.2.3.1" xref="S2.p3.6.m6.2.3.1.cmml">∈</mo><mi id="S2.p3.6.m6.2.3.3" xref="S2.p3.6.m6.2.3.3.cmml">R</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.p3.6.m6.2b"><apply id="S2.p3.6.m6.2.3.cmml" xref="S2.p3.6.m6.2.3"><in id="S2.p3.6.m6.2.3.1.cmml" xref="S2.p3.6.m6.2.3.1"></in><list id="S2.p3.6.m6.2.3.2.1.cmml" xref="S2.p3.6.m6.2.3.2.2"><ci id="S2.p3.6.m6.1.1.cmml" xref="S2.p3.6.m6.1.1">𝑢</ci><ci id="S2.p3.6.m6.2.2.cmml" xref="S2.p3.6.m6.2.2">𝑣</ci></list><ci id="S2.p3.6.m6.2.3.3.cmml" xref="S2.p3.6.m6.2.3.3">𝑅</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p3.6.m6.2c">u,v\in R</annotation><annotation encoding="application/x-llamapun" id="S2.p3.6.m6.2d">italic_u , italic_v ∈ italic_R</annotation></semantics></math>, a set of <math alttext="uv" class="ltx_Math" display="inline" id="S2.p3.7.m7.1"><semantics id="S2.p3.7.m7.1a"><mrow id="S2.p3.7.m7.1.1" xref="S2.p3.7.m7.1.1.cmml"><mi id="S2.p3.7.m7.1.1.2" xref="S2.p3.7.m7.1.1.2.cmml">u</mi><mo id="S2.p3.7.m7.1.1.1" xref="S2.p3.7.m7.1.1.1.cmml">⁢</mo><mi id="S2.p3.7.m7.1.1.3" xref="S2.p3.7.m7.1.1.3.cmml">v</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.p3.7.m7.1b"><apply id="S2.p3.7.m7.1.1.cmml" xref="S2.p3.7.m7.1.1"><times id="S2.p3.7.m7.1.1.1.cmml" xref="S2.p3.7.m7.1.1.1"></times><ci id="S2.p3.7.m7.1.1.2.cmml" xref="S2.p3.7.m7.1.1.2">𝑢</ci><ci id="S2.p3.7.m7.1.1.3.cmml" xref="S2.p3.7.m7.1.1.3">𝑣</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p3.7.m7.1c">uv</annotation><annotation encoding="application/x-llamapun" id="S2.p3.7.m7.1d">italic_u italic_v</annotation></semantics></math>-paths are <em class="ltx_emph ltx_font_italic" id="S2.p3.7.5">element-disjoint</em> if they are disjoint on elements (i.e., edges and non-reliable vertices). Note that the main distinction between element and vertex connectivity is that element-disjoint paths are not necessarily disjoint in the reliable vertices, and the requirements are only on reliable vertices.</p> </div> <div class="ltx_para" id="S2.p4"> <p class="ltx_p" id="S2.p4.1">Menger’s theorem for element-connectivity can be stated as follows:</p> </div> <div class="ltx_theorem ltx_theorem_theorem" id="S2.Thmtheorem3"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="S2.Thmtheorem3.1.1.1">Theorem 2.3</span></span><span class="ltx_text ltx_font_bold" id="S2.Thmtheorem3.2.2"> </span>(Element-connectivity Menger’s theorem)<span class="ltx_text ltx_font_bold" id="S2.Thmtheorem3.3.3">.</span> </h6> <div class="ltx_para" id="S2.Thmtheorem3.p1"> <p class="ltx_p" id="S2.Thmtheorem3.p1.11">Let <math alttext="G=(V,E)" class="ltx_Math" display="inline" id="S2.Thmtheorem3.p1.1.m1.2"><semantics id="S2.Thmtheorem3.p1.1.m1.2a"><mrow id="S2.Thmtheorem3.p1.1.m1.2.3" xref="S2.Thmtheorem3.p1.1.m1.2.3.cmml"><mi id="S2.Thmtheorem3.p1.1.m1.2.3.2" xref="S2.Thmtheorem3.p1.1.m1.2.3.2.cmml">G</mi><mo id="S2.Thmtheorem3.p1.1.m1.2.3.1" xref="S2.Thmtheorem3.p1.1.m1.2.3.1.cmml">=</mo><mrow id="S2.Thmtheorem3.p1.1.m1.2.3.3.2" xref="S2.Thmtheorem3.p1.1.m1.2.3.3.1.cmml"><mo id="S2.Thmtheorem3.p1.1.m1.2.3.3.2.1" stretchy="false" xref="S2.Thmtheorem3.p1.1.m1.2.3.3.1.cmml">(</mo><mi id="S2.Thmtheorem3.p1.1.m1.1.1" xref="S2.Thmtheorem3.p1.1.m1.1.1.cmml">V</mi><mo id="S2.Thmtheorem3.p1.1.m1.2.3.3.2.2" xref="S2.Thmtheorem3.p1.1.m1.2.3.3.1.cmml">,</mo><mi id="S2.Thmtheorem3.p1.1.m1.2.2" xref="S2.Thmtheorem3.p1.1.m1.2.2.cmml">E</mi><mo id="S2.Thmtheorem3.p1.1.m1.2.3.3.2.3" stretchy="false" xref="S2.Thmtheorem3.p1.1.m1.2.3.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem3.p1.1.m1.2b"><apply id="S2.Thmtheorem3.p1.1.m1.2.3.cmml" xref="S2.Thmtheorem3.p1.1.m1.2.3"><eq id="S2.Thmtheorem3.p1.1.m1.2.3.1.cmml" xref="S2.Thmtheorem3.p1.1.m1.2.3.1"></eq><ci id="S2.Thmtheorem3.p1.1.m1.2.3.2.cmml" xref="S2.Thmtheorem3.p1.1.m1.2.3.2">𝐺</ci><interval closure="open" id="S2.Thmtheorem3.p1.1.m1.2.3.3.1.cmml" xref="S2.Thmtheorem3.p1.1.m1.2.3.3.2"><ci id="S2.Thmtheorem3.p1.1.m1.1.1.cmml" xref="S2.Thmtheorem3.p1.1.m1.1.1">𝑉</ci><ci id="S2.Thmtheorem3.p1.1.m1.2.2.cmml" xref="S2.Thmtheorem3.p1.1.m1.2.2">𝐸</ci></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem3.p1.1.m1.2c">G=(V,E)</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem3.p1.1.m1.2d">italic_G = ( italic_V , italic_E )</annotation></semantics></math> be an undirected graph, whose vertices are partitioned into reliable (<math alttext="R" class="ltx_Math" display="inline" id="S2.Thmtheorem3.p1.2.m2.1"><semantics id="S2.Thmtheorem3.p1.2.m2.1a"><mi id="S2.Thmtheorem3.p1.2.m2.1.1" xref="S2.Thmtheorem3.p1.2.m2.1.1.cmml">R</mi><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem3.p1.2.m2.1b"><ci id="S2.Thmtheorem3.p1.2.m2.1.1.cmml" xref="S2.Thmtheorem3.p1.2.m2.1.1">𝑅</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem3.p1.2.m2.1c">R</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem3.p1.2.m2.1d">italic_R</annotation></semantics></math>) and non-reliable (<math alttext="V\setminus R" class="ltx_Math" display="inline" id="S2.Thmtheorem3.p1.3.m3.1"><semantics id="S2.Thmtheorem3.p1.3.m3.1a"><mrow id="S2.Thmtheorem3.p1.3.m3.1.1" xref="S2.Thmtheorem3.p1.3.m3.1.1.cmml"><mi id="S2.Thmtheorem3.p1.3.m3.1.1.2" xref="S2.Thmtheorem3.p1.3.m3.1.1.2.cmml">V</mi><mo id="S2.Thmtheorem3.p1.3.m3.1.1.1" xref="S2.Thmtheorem3.p1.3.m3.1.1.1.cmml">∖</mo><mi id="S2.Thmtheorem3.p1.3.m3.1.1.3" xref="S2.Thmtheorem3.p1.3.m3.1.1.3.cmml">R</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem3.p1.3.m3.1b"><apply id="S2.Thmtheorem3.p1.3.m3.1.1.cmml" xref="S2.Thmtheorem3.p1.3.m3.1.1"><setdiff id="S2.Thmtheorem3.p1.3.m3.1.1.1.cmml" xref="S2.Thmtheorem3.p1.3.m3.1.1.1"></setdiff><ci id="S2.Thmtheorem3.p1.3.m3.1.1.2.cmml" xref="S2.Thmtheorem3.p1.3.m3.1.1.2">𝑉</ci><ci id="S2.Thmtheorem3.p1.3.m3.1.1.3.cmml" xref="S2.Thmtheorem3.p1.3.m3.1.1.3">𝑅</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem3.p1.3.m3.1c">V\setminus R</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem3.p1.3.m3.1d">italic_V ∖ italic_R</annotation></semantics></math>). Two vertices <math alttext="s,t\in R" class="ltx_Math" display="inline" id="S2.Thmtheorem3.p1.4.m4.2"><semantics id="S2.Thmtheorem3.p1.4.m4.2a"><mrow id="S2.Thmtheorem3.p1.4.m4.2.3" xref="S2.Thmtheorem3.p1.4.m4.2.3.cmml"><mrow id="S2.Thmtheorem3.p1.4.m4.2.3.2.2" xref="S2.Thmtheorem3.p1.4.m4.2.3.2.1.cmml"><mi id="S2.Thmtheorem3.p1.4.m4.1.1" xref="S2.Thmtheorem3.p1.4.m4.1.1.cmml">s</mi><mo id="S2.Thmtheorem3.p1.4.m4.2.3.2.2.1" xref="S2.Thmtheorem3.p1.4.m4.2.3.2.1.cmml">,</mo><mi id="S2.Thmtheorem3.p1.4.m4.2.2" xref="S2.Thmtheorem3.p1.4.m4.2.2.cmml">t</mi></mrow><mo id="S2.Thmtheorem3.p1.4.m4.2.3.1" xref="S2.Thmtheorem3.p1.4.m4.2.3.1.cmml">∈</mo><mi id="S2.Thmtheorem3.p1.4.m4.2.3.3" xref="S2.Thmtheorem3.p1.4.m4.2.3.3.cmml">R</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem3.p1.4.m4.2b"><apply id="S2.Thmtheorem3.p1.4.m4.2.3.cmml" xref="S2.Thmtheorem3.p1.4.m4.2.3"><in id="S2.Thmtheorem3.p1.4.m4.2.3.1.cmml" xref="S2.Thmtheorem3.p1.4.m4.2.3.1"></in><list id="S2.Thmtheorem3.p1.4.m4.2.3.2.1.cmml" xref="S2.Thmtheorem3.p1.4.m4.2.3.2.2"><ci id="S2.Thmtheorem3.p1.4.m4.1.1.cmml" xref="S2.Thmtheorem3.p1.4.m4.1.1">𝑠</ci><ci id="S2.Thmtheorem3.p1.4.m4.2.2.cmml" xref="S2.Thmtheorem3.p1.4.m4.2.2">𝑡</ci></list><ci id="S2.Thmtheorem3.p1.4.m4.2.3.3.cmml" xref="S2.Thmtheorem3.p1.4.m4.2.3.3">𝑅</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem3.p1.4.m4.2c">s,t\in R</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem3.p1.4.m4.2d">italic_s , italic_t ∈ italic_R</annotation></semantics></math> are <math alttext="k" class="ltx_Math" display="inline" id="S2.Thmtheorem3.p1.5.m5.1"><semantics id="S2.Thmtheorem3.p1.5.m5.1a"><mi id="S2.Thmtheorem3.p1.5.m5.1.1" xref="S2.Thmtheorem3.p1.5.m5.1.1.cmml">k</mi><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem3.p1.5.m5.1b"><ci id="S2.Thmtheorem3.p1.5.m5.1.1.cmml" xref="S2.Thmtheorem3.p1.5.m5.1.1">𝑘</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem3.p1.5.m5.1c">k</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem3.p1.5.m5.1d">italic_k</annotation></semantics></math>-element connected iff for each biset <math alttext="\hat{S}\subset V\times V" class="ltx_Math" display="inline" id="S2.Thmtheorem3.p1.6.m6.1"><semantics id="S2.Thmtheorem3.p1.6.m6.1a"><mrow id="S2.Thmtheorem3.p1.6.m6.1.1" xref="S2.Thmtheorem3.p1.6.m6.1.1.cmml"><mover accent="true" id="S2.Thmtheorem3.p1.6.m6.1.1.2" xref="S2.Thmtheorem3.p1.6.m6.1.1.2.cmml"><mi id="S2.Thmtheorem3.p1.6.m6.1.1.2.2" xref="S2.Thmtheorem3.p1.6.m6.1.1.2.2.cmml">S</mi><mo id="S2.Thmtheorem3.p1.6.m6.1.1.2.1" xref="S2.Thmtheorem3.p1.6.m6.1.1.2.1.cmml">^</mo></mover><mo id="S2.Thmtheorem3.p1.6.m6.1.1.1" xref="S2.Thmtheorem3.p1.6.m6.1.1.1.cmml">⊂</mo><mrow id="S2.Thmtheorem3.p1.6.m6.1.1.3" xref="S2.Thmtheorem3.p1.6.m6.1.1.3.cmml"><mi id="S2.Thmtheorem3.p1.6.m6.1.1.3.2" xref="S2.Thmtheorem3.p1.6.m6.1.1.3.2.cmml">V</mi><mo id="S2.Thmtheorem3.p1.6.m6.1.1.3.1" lspace="0.222em" rspace="0.222em" xref="S2.Thmtheorem3.p1.6.m6.1.1.3.1.cmml">×</mo><mi id="S2.Thmtheorem3.p1.6.m6.1.1.3.3" xref="S2.Thmtheorem3.p1.6.m6.1.1.3.3.cmml">V</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem3.p1.6.m6.1b"><apply id="S2.Thmtheorem3.p1.6.m6.1.1.cmml" xref="S2.Thmtheorem3.p1.6.m6.1.1"><subset id="S2.Thmtheorem3.p1.6.m6.1.1.1.cmml" xref="S2.Thmtheorem3.p1.6.m6.1.1.1"></subset><apply id="S2.Thmtheorem3.p1.6.m6.1.1.2.cmml" xref="S2.Thmtheorem3.p1.6.m6.1.1.2"><ci id="S2.Thmtheorem3.p1.6.m6.1.1.2.1.cmml" xref="S2.Thmtheorem3.p1.6.m6.1.1.2.1">^</ci><ci id="S2.Thmtheorem3.p1.6.m6.1.1.2.2.cmml" xref="S2.Thmtheorem3.p1.6.m6.1.1.2.2">𝑆</ci></apply><apply id="S2.Thmtheorem3.p1.6.m6.1.1.3.cmml" xref="S2.Thmtheorem3.p1.6.m6.1.1.3"><times id="S2.Thmtheorem3.p1.6.m6.1.1.3.1.cmml" xref="S2.Thmtheorem3.p1.6.m6.1.1.3.1"></times><ci id="S2.Thmtheorem3.p1.6.m6.1.1.3.2.cmml" xref="S2.Thmtheorem3.p1.6.m6.1.1.3.2">𝑉</ci><ci id="S2.Thmtheorem3.p1.6.m6.1.1.3.3.cmml" xref="S2.Thmtheorem3.p1.6.m6.1.1.3.3">𝑉</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem3.p1.6.m6.1c">\hat{S}\subset V\times V</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem3.p1.6.m6.1d">over^ start_ARG italic_S end_ARG ⊂ italic_V × italic_V</annotation></semantics></math> such that <math alttext="s\in S" class="ltx_Math" display="inline" id="S2.Thmtheorem3.p1.7.m7.1"><semantics id="S2.Thmtheorem3.p1.7.m7.1a"><mrow id="S2.Thmtheorem3.p1.7.m7.1.1" xref="S2.Thmtheorem3.p1.7.m7.1.1.cmml"><mi id="S2.Thmtheorem3.p1.7.m7.1.1.2" xref="S2.Thmtheorem3.p1.7.m7.1.1.2.cmml">s</mi><mo id="S2.Thmtheorem3.p1.7.m7.1.1.1" xref="S2.Thmtheorem3.p1.7.m7.1.1.1.cmml">∈</mo><mi id="S2.Thmtheorem3.p1.7.m7.1.1.3" xref="S2.Thmtheorem3.p1.7.m7.1.1.3.cmml">S</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem3.p1.7.m7.1b"><apply id="S2.Thmtheorem3.p1.7.m7.1.1.cmml" xref="S2.Thmtheorem3.p1.7.m7.1.1"><in id="S2.Thmtheorem3.p1.7.m7.1.1.1.cmml" xref="S2.Thmtheorem3.p1.7.m7.1.1.1"></in><ci id="S2.Thmtheorem3.p1.7.m7.1.1.2.cmml" xref="S2.Thmtheorem3.p1.7.m7.1.1.2">𝑠</ci><ci id="S2.Thmtheorem3.p1.7.m7.1.1.3.cmml" xref="S2.Thmtheorem3.p1.7.m7.1.1.3">𝑆</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem3.p1.7.m7.1c">s\in S</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem3.p1.7.m7.1d">italic_s ∈ italic_S</annotation></semantics></math>, <math alttext="t\in V\setminus S^{+}" class="ltx_Math" display="inline" id="S2.Thmtheorem3.p1.8.m8.1"><semantics id="S2.Thmtheorem3.p1.8.m8.1a"><mrow id="S2.Thmtheorem3.p1.8.m8.1.1" xref="S2.Thmtheorem3.p1.8.m8.1.1.cmml"><mi id="S2.Thmtheorem3.p1.8.m8.1.1.2" xref="S2.Thmtheorem3.p1.8.m8.1.1.2.cmml">t</mi><mo id="S2.Thmtheorem3.p1.8.m8.1.1.1" xref="S2.Thmtheorem3.p1.8.m8.1.1.1.cmml">∈</mo><mrow id="S2.Thmtheorem3.p1.8.m8.1.1.3" xref="S2.Thmtheorem3.p1.8.m8.1.1.3.cmml"><mi id="S2.Thmtheorem3.p1.8.m8.1.1.3.2" xref="S2.Thmtheorem3.p1.8.m8.1.1.3.2.cmml">V</mi><mo id="S2.Thmtheorem3.p1.8.m8.1.1.3.1" xref="S2.Thmtheorem3.p1.8.m8.1.1.3.1.cmml">∖</mo><msup id="S2.Thmtheorem3.p1.8.m8.1.1.3.3" xref="S2.Thmtheorem3.p1.8.m8.1.1.3.3.cmml"><mi id="S2.Thmtheorem3.p1.8.m8.1.1.3.3.2" xref="S2.Thmtheorem3.p1.8.m8.1.1.3.3.2.cmml">S</mi><mo id="S2.Thmtheorem3.p1.8.m8.1.1.3.3.3" xref="S2.Thmtheorem3.p1.8.m8.1.1.3.3.3.cmml">+</mo></msup></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem3.p1.8.m8.1b"><apply id="S2.Thmtheorem3.p1.8.m8.1.1.cmml" xref="S2.Thmtheorem3.p1.8.m8.1.1"><in id="S2.Thmtheorem3.p1.8.m8.1.1.1.cmml" xref="S2.Thmtheorem3.p1.8.m8.1.1.1"></in><ci id="S2.Thmtheorem3.p1.8.m8.1.1.2.cmml" xref="S2.Thmtheorem3.p1.8.m8.1.1.2">𝑡</ci><apply id="S2.Thmtheorem3.p1.8.m8.1.1.3.cmml" xref="S2.Thmtheorem3.p1.8.m8.1.1.3"><setdiff id="S2.Thmtheorem3.p1.8.m8.1.1.3.1.cmml" xref="S2.Thmtheorem3.p1.8.m8.1.1.3.1"></setdiff><ci id="S2.Thmtheorem3.p1.8.m8.1.1.3.2.cmml" xref="S2.Thmtheorem3.p1.8.m8.1.1.3.2">𝑉</ci><apply id="S2.Thmtheorem3.p1.8.m8.1.1.3.3.cmml" xref="S2.Thmtheorem3.p1.8.m8.1.1.3.3"><csymbol cd="ambiguous" id="S2.Thmtheorem3.p1.8.m8.1.1.3.3.1.cmml" xref="S2.Thmtheorem3.p1.8.m8.1.1.3.3">superscript</csymbol><ci id="S2.Thmtheorem3.p1.8.m8.1.1.3.3.2.cmml" xref="S2.Thmtheorem3.p1.8.m8.1.1.3.3.2">𝑆</ci><plus id="S2.Thmtheorem3.p1.8.m8.1.1.3.3.3.cmml" xref="S2.Thmtheorem3.p1.8.m8.1.1.3.3.3"></plus></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem3.p1.8.m8.1c">t\in V\setminus S^{+}</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem3.p1.8.m8.1d">italic_t ∈ italic_V ∖ italic_S start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT</annotation></semantics></math>, and <math alttext="S^{+}\setminus S" class="ltx_Math" display="inline" id="S2.Thmtheorem3.p1.9.m9.1"><semantics id="S2.Thmtheorem3.p1.9.m9.1a"><mrow id="S2.Thmtheorem3.p1.9.m9.1.1" xref="S2.Thmtheorem3.p1.9.m9.1.1.cmml"><msup id="S2.Thmtheorem3.p1.9.m9.1.1.2" xref="S2.Thmtheorem3.p1.9.m9.1.1.2.cmml"><mi id="S2.Thmtheorem3.p1.9.m9.1.1.2.2" xref="S2.Thmtheorem3.p1.9.m9.1.1.2.2.cmml">S</mi><mo id="S2.Thmtheorem3.p1.9.m9.1.1.2.3" xref="S2.Thmtheorem3.p1.9.m9.1.1.2.3.cmml">+</mo></msup><mo id="S2.Thmtheorem3.p1.9.m9.1.1.1" xref="S2.Thmtheorem3.p1.9.m9.1.1.1.cmml">∖</mo><mi id="S2.Thmtheorem3.p1.9.m9.1.1.3" xref="S2.Thmtheorem3.p1.9.m9.1.1.3.cmml">S</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem3.p1.9.m9.1b"><apply id="S2.Thmtheorem3.p1.9.m9.1.1.cmml" xref="S2.Thmtheorem3.p1.9.m9.1.1"><setdiff id="S2.Thmtheorem3.p1.9.m9.1.1.1.cmml" xref="S2.Thmtheorem3.p1.9.m9.1.1.1"></setdiff><apply id="S2.Thmtheorem3.p1.9.m9.1.1.2.cmml" xref="S2.Thmtheorem3.p1.9.m9.1.1.2"><csymbol cd="ambiguous" id="S2.Thmtheorem3.p1.9.m9.1.1.2.1.cmml" xref="S2.Thmtheorem3.p1.9.m9.1.1.2">superscript</csymbol><ci id="S2.Thmtheorem3.p1.9.m9.1.1.2.2.cmml" xref="S2.Thmtheorem3.p1.9.m9.1.1.2.2">𝑆</ci><plus id="S2.Thmtheorem3.p1.9.m9.1.1.2.3.cmml" xref="S2.Thmtheorem3.p1.9.m9.1.1.2.3"></plus></apply><ci id="S2.Thmtheorem3.p1.9.m9.1.1.3.cmml" xref="S2.Thmtheorem3.p1.9.m9.1.1.3">𝑆</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem3.p1.9.m9.1c">S^{+}\setminus S</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem3.p1.9.m9.1d">italic_S start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT ∖ italic_S</annotation></semantics></math> containing only reliable vertices (i.e., <math alttext="S^{+}\setminus S\subseteq R" class="ltx_Math" display="inline" id="S2.Thmtheorem3.p1.10.m10.1"><semantics id="S2.Thmtheorem3.p1.10.m10.1a"><mrow id="S2.Thmtheorem3.p1.10.m10.1.1" xref="S2.Thmtheorem3.p1.10.m10.1.1.cmml"><mrow id="S2.Thmtheorem3.p1.10.m10.1.1.2" xref="S2.Thmtheorem3.p1.10.m10.1.1.2.cmml"><msup id="S2.Thmtheorem3.p1.10.m10.1.1.2.2" xref="S2.Thmtheorem3.p1.10.m10.1.1.2.2.cmml"><mi id="S2.Thmtheorem3.p1.10.m10.1.1.2.2.2" xref="S2.Thmtheorem3.p1.10.m10.1.1.2.2.2.cmml">S</mi><mo id="S2.Thmtheorem3.p1.10.m10.1.1.2.2.3" xref="S2.Thmtheorem3.p1.10.m10.1.1.2.2.3.cmml">+</mo></msup><mo id="S2.Thmtheorem3.p1.10.m10.1.1.2.1" xref="S2.Thmtheorem3.p1.10.m10.1.1.2.1.cmml">∖</mo><mi id="S2.Thmtheorem3.p1.10.m10.1.1.2.3" xref="S2.Thmtheorem3.p1.10.m10.1.1.2.3.cmml">S</mi></mrow><mo id="S2.Thmtheorem3.p1.10.m10.1.1.1" xref="S2.Thmtheorem3.p1.10.m10.1.1.1.cmml">⊆</mo><mi id="S2.Thmtheorem3.p1.10.m10.1.1.3" xref="S2.Thmtheorem3.p1.10.m10.1.1.3.cmml">R</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem3.p1.10.m10.1b"><apply id="S2.Thmtheorem3.p1.10.m10.1.1.cmml" xref="S2.Thmtheorem3.p1.10.m10.1.1"><subset id="S2.Thmtheorem3.p1.10.m10.1.1.1.cmml" xref="S2.Thmtheorem3.p1.10.m10.1.1.1"></subset><apply id="S2.Thmtheorem3.p1.10.m10.1.1.2.cmml" xref="S2.Thmtheorem3.p1.10.m10.1.1.2"><setdiff id="S2.Thmtheorem3.p1.10.m10.1.1.2.1.cmml" xref="S2.Thmtheorem3.p1.10.m10.1.1.2.1"></setdiff><apply id="S2.Thmtheorem3.p1.10.m10.1.1.2.2.cmml" xref="S2.Thmtheorem3.p1.10.m10.1.1.2.2"><csymbol cd="ambiguous" id="S2.Thmtheorem3.p1.10.m10.1.1.2.2.1.cmml" xref="S2.Thmtheorem3.p1.10.m10.1.1.2.2">superscript</csymbol><ci id="S2.Thmtheorem3.p1.10.m10.1.1.2.2.2.cmml" xref="S2.Thmtheorem3.p1.10.m10.1.1.2.2.2">𝑆</ci><plus id="S2.Thmtheorem3.p1.10.m10.1.1.2.2.3.cmml" xref="S2.Thmtheorem3.p1.10.m10.1.1.2.2.3"></plus></apply><ci id="S2.Thmtheorem3.p1.10.m10.1.1.2.3.cmml" xref="S2.Thmtheorem3.p1.10.m10.1.1.2.3">𝑆</ci></apply><ci id="S2.Thmtheorem3.p1.10.m10.1.1.3.cmml" xref="S2.Thmtheorem3.p1.10.m10.1.1.3">𝑅</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem3.p1.10.m10.1c">S^{+}\setminus S\subseteq R</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem3.p1.10.m10.1d">italic_S start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT ∖ italic_S ⊆ italic_R</annotation></semantics></math>), we have <math alttext="|\delta(\hat{S})|+|S^{+}\setminus S|\geq k" class="ltx_Math" display="inline" id="S2.Thmtheorem3.p1.11.m11.3"><semantics id="S2.Thmtheorem3.p1.11.m11.3a"><mrow id="S2.Thmtheorem3.p1.11.m11.3.3" xref="S2.Thmtheorem3.p1.11.m11.3.3.cmml"><mrow id="S2.Thmtheorem3.p1.11.m11.3.3.2" xref="S2.Thmtheorem3.p1.11.m11.3.3.2.cmml"><mrow id="S2.Thmtheorem3.p1.11.m11.2.2.1.1.1" xref="S2.Thmtheorem3.p1.11.m11.2.2.1.1.2.cmml"><mo id="S2.Thmtheorem3.p1.11.m11.2.2.1.1.1.2" stretchy="false" xref="S2.Thmtheorem3.p1.11.m11.2.2.1.1.2.1.cmml">|</mo><mrow id="S2.Thmtheorem3.p1.11.m11.2.2.1.1.1.1" xref="S2.Thmtheorem3.p1.11.m11.2.2.1.1.1.1.cmml"><mi id="S2.Thmtheorem3.p1.11.m11.2.2.1.1.1.1.2" xref="S2.Thmtheorem3.p1.11.m11.2.2.1.1.1.1.2.cmml">δ</mi><mo id="S2.Thmtheorem3.p1.11.m11.2.2.1.1.1.1.1" xref="S2.Thmtheorem3.p1.11.m11.2.2.1.1.1.1.1.cmml">⁢</mo><mrow id="S2.Thmtheorem3.p1.11.m11.2.2.1.1.1.1.3.2" xref="S2.Thmtheorem3.p1.11.m11.1.1.cmml"><mo id="S2.Thmtheorem3.p1.11.m11.2.2.1.1.1.1.3.2.1" stretchy="false" xref="S2.Thmtheorem3.p1.11.m11.1.1.cmml">(</mo><mover accent="true" id="S2.Thmtheorem3.p1.11.m11.1.1" xref="S2.Thmtheorem3.p1.11.m11.1.1.cmml"><mi id="S2.Thmtheorem3.p1.11.m11.1.1.2" xref="S2.Thmtheorem3.p1.11.m11.1.1.2.cmml">S</mi><mo id="S2.Thmtheorem3.p1.11.m11.1.1.1" xref="S2.Thmtheorem3.p1.11.m11.1.1.1.cmml">^</mo></mover><mo id="S2.Thmtheorem3.p1.11.m11.2.2.1.1.1.1.3.2.2" stretchy="false" xref="S2.Thmtheorem3.p1.11.m11.1.1.cmml">)</mo></mrow></mrow><mo id="S2.Thmtheorem3.p1.11.m11.2.2.1.1.1.3" stretchy="false" xref="S2.Thmtheorem3.p1.11.m11.2.2.1.1.2.1.cmml">|</mo></mrow><mo id="S2.Thmtheorem3.p1.11.m11.3.3.2.3" xref="S2.Thmtheorem3.p1.11.m11.3.3.2.3.cmml">+</mo><mrow id="S2.Thmtheorem3.p1.11.m11.3.3.2.2.1" xref="S2.Thmtheorem3.p1.11.m11.3.3.2.2.2.cmml"><mo id="S2.Thmtheorem3.p1.11.m11.3.3.2.2.1.2" stretchy="false" xref="S2.Thmtheorem3.p1.11.m11.3.3.2.2.2.1.cmml">|</mo><mrow id="S2.Thmtheorem3.p1.11.m11.3.3.2.2.1.1" xref="S2.Thmtheorem3.p1.11.m11.3.3.2.2.1.1.cmml"><msup id="S2.Thmtheorem3.p1.11.m11.3.3.2.2.1.1.2" xref="S2.Thmtheorem3.p1.11.m11.3.3.2.2.1.1.2.cmml"><mi id="S2.Thmtheorem3.p1.11.m11.3.3.2.2.1.1.2.2" xref="S2.Thmtheorem3.p1.11.m11.3.3.2.2.1.1.2.2.cmml">S</mi><mo id="S2.Thmtheorem3.p1.11.m11.3.3.2.2.1.1.2.3" xref="S2.Thmtheorem3.p1.11.m11.3.3.2.2.1.1.2.3.cmml">+</mo></msup><mo id="S2.Thmtheorem3.p1.11.m11.3.3.2.2.1.1.1" xref="S2.Thmtheorem3.p1.11.m11.3.3.2.2.1.1.1.cmml">∖</mo><mi id="S2.Thmtheorem3.p1.11.m11.3.3.2.2.1.1.3" xref="S2.Thmtheorem3.p1.11.m11.3.3.2.2.1.1.3.cmml">S</mi></mrow><mo id="S2.Thmtheorem3.p1.11.m11.3.3.2.2.1.3" stretchy="false" xref="S2.Thmtheorem3.p1.11.m11.3.3.2.2.2.1.cmml">|</mo></mrow></mrow><mo id="S2.Thmtheorem3.p1.11.m11.3.3.3" xref="S2.Thmtheorem3.p1.11.m11.3.3.3.cmml">≥</mo><mi id="S2.Thmtheorem3.p1.11.m11.3.3.4" xref="S2.Thmtheorem3.p1.11.m11.3.3.4.cmml">k</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem3.p1.11.m11.3b"><apply id="S2.Thmtheorem3.p1.11.m11.3.3.cmml" xref="S2.Thmtheorem3.p1.11.m11.3.3"><geq id="S2.Thmtheorem3.p1.11.m11.3.3.3.cmml" xref="S2.Thmtheorem3.p1.11.m11.3.3.3"></geq><apply id="S2.Thmtheorem3.p1.11.m11.3.3.2.cmml" xref="S2.Thmtheorem3.p1.11.m11.3.3.2"><plus id="S2.Thmtheorem3.p1.11.m11.3.3.2.3.cmml" xref="S2.Thmtheorem3.p1.11.m11.3.3.2.3"></plus><apply id="S2.Thmtheorem3.p1.11.m11.2.2.1.1.2.cmml" xref="S2.Thmtheorem3.p1.11.m11.2.2.1.1.1"><abs id="S2.Thmtheorem3.p1.11.m11.2.2.1.1.2.1.cmml" xref="S2.Thmtheorem3.p1.11.m11.2.2.1.1.1.2"></abs><apply id="S2.Thmtheorem3.p1.11.m11.2.2.1.1.1.1.cmml" xref="S2.Thmtheorem3.p1.11.m11.2.2.1.1.1.1"><times id="S2.Thmtheorem3.p1.11.m11.2.2.1.1.1.1.1.cmml" xref="S2.Thmtheorem3.p1.11.m11.2.2.1.1.1.1.1"></times><ci id="S2.Thmtheorem3.p1.11.m11.2.2.1.1.1.1.2.cmml" xref="S2.Thmtheorem3.p1.11.m11.2.2.1.1.1.1.2">𝛿</ci><apply id="S2.Thmtheorem3.p1.11.m11.1.1.cmml" xref="S2.Thmtheorem3.p1.11.m11.2.2.1.1.1.1.3.2"><ci id="S2.Thmtheorem3.p1.11.m11.1.1.1.cmml" xref="S2.Thmtheorem3.p1.11.m11.1.1.1">^</ci><ci id="S2.Thmtheorem3.p1.11.m11.1.1.2.cmml" xref="S2.Thmtheorem3.p1.11.m11.1.1.2">𝑆</ci></apply></apply></apply><apply id="S2.Thmtheorem3.p1.11.m11.3.3.2.2.2.cmml" xref="S2.Thmtheorem3.p1.11.m11.3.3.2.2.1"><abs id="S2.Thmtheorem3.p1.11.m11.3.3.2.2.2.1.cmml" xref="S2.Thmtheorem3.p1.11.m11.3.3.2.2.1.2"></abs><apply id="S2.Thmtheorem3.p1.11.m11.3.3.2.2.1.1.cmml" xref="S2.Thmtheorem3.p1.11.m11.3.3.2.2.1.1"><setdiff id="S2.Thmtheorem3.p1.11.m11.3.3.2.2.1.1.1.cmml" xref="S2.Thmtheorem3.p1.11.m11.3.3.2.2.1.1.1"></setdiff><apply id="S2.Thmtheorem3.p1.11.m11.3.3.2.2.1.1.2.cmml" xref="S2.Thmtheorem3.p1.11.m11.3.3.2.2.1.1.2"><csymbol cd="ambiguous" id="S2.Thmtheorem3.p1.11.m11.3.3.2.2.1.1.2.1.cmml" xref="S2.Thmtheorem3.p1.11.m11.3.3.2.2.1.1.2">superscript</csymbol><ci id="S2.Thmtheorem3.p1.11.m11.3.3.2.2.1.1.2.2.cmml" xref="S2.Thmtheorem3.p1.11.m11.3.3.2.2.1.1.2.2">𝑆</ci><plus id="S2.Thmtheorem3.p1.11.m11.3.3.2.2.1.1.2.3.cmml" xref="S2.Thmtheorem3.p1.11.m11.3.3.2.2.1.1.2.3"></plus></apply><ci id="S2.Thmtheorem3.p1.11.m11.3.3.2.2.1.1.3.cmml" xref="S2.Thmtheorem3.p1.11.m11.3.3.2.2.1.1.3">𝑆</ci></apply></apply></apply><ci id="S2.Thmtheorem3.p1.11.m11.3.3.4.cmml" xref="S2.Thmtheorem3.p1.11.m11.3.3.4">𝑘</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem3.p1.11.m11.3c">|\delta(\hat{S})|+|S^{+}\setminus S|\geq k</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem3.p1.11.m11.3d">| italic_δ ( over^ start_ARG italic_S end_ARG ) | + | italic_S start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT ∖ italic_S | ≥ italic_k</annotation></semantics></math>.</p> </div> </div> <section class="ltx_subsection" id="S2.SS1"> <h3 class="ltx_title ltx_title_subsection"> <span class="ltx_tag ltx_tag_subsection">2.1 </span>Fault-Tolerant Spanners in Streaming</h3> <div class="ltx_theorem ltx_theorem_definition" id="S2.Thmtheorem4"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="S2.Thmtheorem4.1.1.1">Definition 2.4</span></span><span class="ltx_text ltx_font_bold" id="S2.Thmtheorem4.2.2"> </span>(Fault-Tolerant Spanners)<span class="ltx_text ltx_font_bold" id="S2.Thmtheorem4.3.3">.</span> </h6> <div class="ltx_para" id="S2.Thmtheorem4.p1"> <p class="ltx_p" id="S2.Thmtheorem4.p1.10">A subgraph <math alttext="H\subseteq G" class="ltx_Math" display="inline" id="S2.Thmtheorem4.p1.1.m1.1"><semantics id="S2.Thmtheorem4.p1.1.m1.1a"><mrow id="S2.Thmtheorem4.p1.1.m1.1.1" xref="S2.Thmtheorem4.p1.1.m1.1.1.cmml"><mi id="S2.Thmtheorem4.p1.1.m1.1.1.2" xref="S2.Thmtheorem4.p1.1.m1.1.1.2.cmml">H</mi><mo id="S2.Thmtheorem4.p1.1.m1.1.1.1" xref="S2.Thmtheorem4.p1.1.m1.1.1.1.cmml">⊆</mo><mi id="S2.Thmtheorem4.p1.1.m1.1.1.3" xref="S2.Thmtheorem4.p1.1.m1.1.1.3.cmml">G</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem4.p1.1.m1.1b"><apply id="S2.Thmtheorem4.p1.1.m1.1.1.cmml" xref="S2.Thmtheorem4.p1.1.m1.1.1"><subset id="S2.Thmtheorem4.p1.1.m1.1.1.1.cmml" xref="S2.Thmtheorem4.p1.1.m1.1.1.1"></subset><ci id="S2.Thmtheorem4.p1.1.m1.1.1.2.cmml" xref="S2.Thmtheorem4.p1.1.m1.1.1.2">𝐻</ci><ci id="S2.Thmtheorem4.p1.1.m1.1.1.3.cmml" xref="S2.Thmtheorem4.p1.1.m1.1.1.3">𝐺</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem4.p1.1.m1.1c">H\subseteq G</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem4.p1.1.m1.1d">italic_H ⊆ italic_G</annotation></semantics></math> is an <math alttext="f" class="ltx_Math" display="inline" id="S2.Thmtheorem4.p1.2.m2.1"><semantics id="S2.Thmtheorem4.p1.2.m2.1a"><mi id="S2.Thmtheorem4.p1.2.m2.1.1" xref="S2.Thmtheorem4.p1.2.m2.1.1.cmml">f</mi><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem4.p1.2.m2.1b"><ci id="S2.Thmtheorem4.p1.2.m2.1.1.cmml" xref="S2.Thmtheorem4.p1.2.m2.1.1">𝑓</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem4.p1.2.m2.1c">f</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem4.p1.2.m2.1d">italic_f</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S2.Thmtheorem4.p1.4.2">-vertex-fault-tolerant (<math alttext="f" class="ltx_Math" display="inline" id="S2.Thmtheorem4.p1.3.1.m1.1"><semantics id="S2.Thmtheorem4.p1.3.1.m1.1a"><mi id="S2.Thmtheorem4.p1.3.1.m1.1.1" xref="S2.Thmtheorem4.p1.3.1.m1.1.1.cmml">f</mi><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem4.p1.3.1.m1.1b"><ci id="S2.Thmtheorem4.p1.3.1.m1.1.1.cmml" xref="S2.Thmtheorem4.p1.3.1.m1.1.1">𝑓</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem4.p1.3.1.m1.1c">f</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem4.p1.3.1.m1.1d">italic_f</annotation></semantics></math>-VFT) <math alttext="t" class="ltx_Math" display="inline" id="S2.Thmtheorem4.p1.4.2.m2.1"><semantics id="S2.Thmtheorem4.p1.4.2.m2.1a"><mi id="S2.Thmtheorem4.p1.4.2.m2.1.1" xref="S2.Thmtheorem4.p1.4.2.m2.1.1.cmml">t</mi><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem4.p1.4.2.m2.1b"><ci id="S2.Thmtheorem4.p1.4.2.m2.1.1.cmml" xref="S2.Thmtheorem4.p1.4.2.m2.1.1">𝑡</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem4.p1.4.2.m2.1c">t</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem4.p1.4.2.m2.1d">italic_t</annotation></semantics></math>-spanner</span> of <math alttext="G" class="ltx_Math" display="inline" id="S2.Thmtheorem4.p1.5.m3.1"><semantics id="S2.Thmtheorem4.p1.5.m3.1a"><mi id="S2.Thmtheorem4.p1.5.m3.1.1" xref="S2.Thmtheorem4.p1.5.m3.1.1.cmml">G</mi><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem4.p1.5.m3.1b"><ci id="S2.Thmtheorem4.p1.5.m3.1.1.cmml" xref="S2.Thmtheorem4.p1.5.m3.1.1">𝐺</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem4.p1.5.m3.1c">G</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem4.p1.5.m3.1d">italic_G</annotation></semantics></math> if, for every subset of vertices <math alttext="F_{V}\subseteq V" class="ltx_Math" display="inline" id="S2.Thmtheorem4.p1.6.m4.1"><semantics id="S2.Thmtheorem4.p1.6.m4.1a"><mrow id="S2.Thmtheorem4.p1.6.m4.1.1" xref="S2.Thmtheorem4.p1.6.m4.1.1.cmml"><msub id="S2.Thmtheorem4.p1.6.m4.1.1.2" xref="S2.Thmtheorem4.p1.6.m4.1.1.2.cmml"><mi id="S2.Thmtheorem4.p1.6.m4.1.1.2.2" xref="S2.Thmtheorem4.p1.6.m4.1.1.2.2.cmml">F</mi><mi id="S2.Thmtheorem4.p1.6.m4.1.1.2.3" xref="S2.Thmtheorem4.p1.6.m4.1.1.2.3.cmml">V</mi></msub><mo id="S2.Thmtheorem4.p1.6.m4.1.1.1" xref="S2.Thmtheorem4.p1.6.m4.1.1.1.cmml">⊆</mo><mi id="S2.Thmtheorem4.p1.6.m4.1.1.3" xref="S2.Thmtheorem4.p1.6.m4.1.1.3.cmml">V</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem4.p1.6.m4.1b"><apply id="S2.Thmtheorem4.p1.6.m4.1.1.cmml" xref="S2.Thmtheorem4.p1.6.m4.1.1"><subset id="S2.Thmtheorem4.p1.6.m4.1.1.1.cmml" xref="S2.Thmtheorem4.p1.6.m4.1.1.1"></subset><apply id="S2.Thmtheorem4.p1.6.m4.1.1.2.cmml" xref="S2.Thmtheorem4.p1.6.m4.1.1.2"><csymbol cd="ambiguous" id="S2.Thmtheorem4.p1.6.m4.1.1.2.1.cmml" xref="S2.Thmtheorem4.p1.6.m4.1.1.2">subscript</csymbol><ci id="S2.Thmtheorem4.p1.6.m4.1.1.2.2.cmml" xref="S2.Thmtheorem4.p1.6.m4.1.1.2.2">𝐹</ci><ci id="S2.Thmtheorem4.p1.6.m4.1.1.2.3.cmml" xref="S2.Thmtheorem4.p1.6.m4.1.1.2.3">𝑉</ci></apply><ci id="S2.Thmtheorem4.p1.6.m4.1.1.3.cmml" xref="S2.Thmtheorem4.p1.6.m4.1.1.3">𝑉</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem4.p1.6.m4.1c">F_{V}\subseteq V</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem4.p1.6.m4.1d">italic_F start_POSTSUBSCRIPT italic_V end_POSTSUBSCRIPT ⊆ italic_V</annotation></semantics></math> of size at most <math alttext="f" class="ltx_Math" display="inline" id="S2.Thmtheorem4.p1.7.m5.1"><semantics id="S2.Thmtheorem4.p1.7.m5.1a"><mi id="S2.Thmtheorem4.p1.7.m5.1.1" xref="S2.Thmtheorem4.p1.7.m5.1.1.cmml">f</mi><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem4.p1.7.m5.1b"><ci id="S2.Thmtheorem4.p1.7.m5.1.1.cmml" xref="S2.Thmtheorem4.p1.7.m5.1.1">𝑓</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem4.p1.7.m5.1c">f</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem4.p1.7.m5.1d">italic_f</annotation></semantics></math>, all pairwise distances in <math alttext="V\setminus F_{V}" class="ltx_Math" display="inline" id="S2.Thmtheorem4.p1.8.m6.1"><semantics id="S2.Thmtheorem4.p1.8.m6.1a"><mrow id="S2.Thmtheorem4.p1.8.m6.1.1" xref="S2.Thmtheorem4.p1.8.m6.1.1.cmml"><mi id="S2.Thmtheorem4.p1.8.m6.1.1.2" xref="S2.Thmtheorem4.p1.8.m6.1.1.2.cmml">V</mi><mo id="S2.Thmtheorem4.p1.8.m6.1.1.1" xref="S2.Thmtheorem4.p1.8.m6.1.1.1.cmml">∖</mo><msub id="S2.Thmtheorem4.p1.8.m6.1.1.3" xref="S2.Thmtheorem4.p1.8.m6.1.1.3.cmml"><mi id="S2.Thmtheorem4.p1.8.m6.1.1.3.2" xref="S2.Thmtheorem4.p1.8.m6.1.1.3.2.cmml">F</mi><mi id="S2.Thmtheorem4.p1.8.m6.1.1.3.3" xref="S2.Thmtheorem4.p1.8.m6.1.1.3.3.cmml">V</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem4.p1.8.m6.1b"><apply id="S2.Thmtheorem4.p1.8.m6.1.1.cmml" xref="S2.Thmtheorem4.p1.8.m6.1.1"><setdiff id="S2.Thmtheorem4.p1.8.m6.1.1.1.cmml" xref="S2.Thmtheorem4.p1.8.m6.1.1.1"></setdiff><ci id="S2.Thmtheorem4.p1.8.m6.1.1.2.cmml" xref="S2.Thmtheorem4.p1.8.m6.1.1.2">𝑉</ci><apply id="S2.Thmtheorem4.p1.8.m6.1.1.3.cmml" xref="S2.Thmtheorem4.p1.8.m6.1.1.3"><csymbol cd="ambiguous" id="S2.Thmtheorem4.p1.8.m6.1.1.3.1.cmml" xref="S2.Thmtheorem4.p1.8.m6.1.1.3">subscript</csymbol><ci id="S2.Thmtheorem4.p1.8.m6.1.1.3.2.cmml" xref="S2.Thmtheorem4.p1.8.m6.1.1.3.2">𝐹</ci><ci id="S2.Thmtheorem4.p1.8.m6.1.1.3.3.cmml" xref="S2.Thmtheorem4.p1.8.m6.1.1.3.3">𝑉</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem4.p1.8.m6.1c">V\setminus F_{V}</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem4.p1.8.m6.1d">italic_V ∖ italic_F start_POSTSUBSCRIPT italic_V end_POSTSUBSCRIPT</annotation></semantics></math> are preserved within a factor of <math alttext="t" class="ltx_Math" display="inline" id="S2.Thmtheorem4.p1.9.m7.1"><semantics id="S2.Thmtheorem4.p1.9.m7.1a"><mi id="S2.Thmtheorem4.p1.9.m7.1.1" xref="S2.Thmtheorem4.p1.9.m7.1.1.cmml">t</mi><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem4.p1.9.m7.1b"><ci id="S2.Thmtheorem4.p1.9.m7.1.1.cmml" xref="S2.Thmtheorem4.p1.9.m7.1.1">𝑡</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem4.p1.9.m7.1c">t</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem4.p1.9.m7.1d">italic_t</annotation></semantics></math>. That is, for all <math alttext="u,v\in V\setminus F_{V}" class="ltx_Math" display="inline" id="S2.Thmtheorem4.p1.10.m8.2"><semantics id="S2.Thmtheorem4.p1.10.m8.2a"><mrow id="S2.Thmtheorem4.p1.10.m8.2.3" xref="S2.Thmtheorem4.p1.10.m8.2.3.cmml"><mrow id="S2.Thmtheorem4.p1.10.m8.2.3.2.2" xref="S2.Thmtheorem4.p1.10.m8.2.3.2.1.cmml"><mi id="S2.Thmtheorem4.p1.10.m8.1.1" xref="S2.Thmtheorem4.p1.10.m8.1.1.cmml">u</mi><mo id="S2.Thmtheorem4.p1.10.m8.2.3.2.2.1" xref="S2.Thmtheorem4.p1.10.m8.2.3.2.1.cmml">,</mo><mi id="S2.Thmtheorem4.p1.10.m8.2.2" xref="S2.Thmtheorem4.p1.10.m8.2.2.cmml">v</mi></mrow><mo id="S2.Thmtheorem4.p1.10.m8.2.3.1" xref="S2.Thmtheorem4.p1.10.m8.2.3.1.cmml">∈</mo><mrow id="S2.Thmtheorem4.p1.10.m8.2.3.3" xref="S2.Thmtheorem4.p1.10.m8.2.3.3.cmml"><mi id="S2.Thmtheorem4.p1.10.m8.2.3.3.2" xref="S2.Thmtheorem4.p1.10.m8.2.3.3.2.cmml">V</mi><mo id="S2.Thmtheorem4.p1.10.m8.2.3.3.1" xref="S2.Thmtheorem4.p1.10.m8.2.3.3.1.cmml">∖</mo><msub id="S2.Thmtheorem4.p1.10.m8.2.3.3.3" xref="S2.Thmtheorem4.p1.10.m8.2.3.3.3.cmml"><mi id="S2.Thmtheorem4.p1.10.m8.2.3.3.3.2" xref="S2.Thmtheorem4.p1.10.m8.2.3.3.3.2.cmml">F</mi><mi id="S2.Thmtheorem4.p1.10.m8.2.3.3.3.3" xref="S2.Thmtheorem4.p1.10.m8.2.3.3.3.3.cmml">V</mi></msub></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem4.p1.10.m8.2b"><apply id="S2.Thmtheorem4.p1.10.m8.2.3.cmml" xref="S2.Thmtheorem4.p1.10.m8.2.3"><in id="S2.Thmtheorem4.p1.10.m8.2.3.1.cmml" xref="S2.Thmtheorem4.p1.10.m8.2.3.1"></in><list id="S2.Thmtheorem4.p1.10.m8.2.3.2.1.cmml" xref="S2.Thmtheorem4.p1.10.m8.2.3.2.2"><ci id="S2.Thmtheorem4.p1.10.m8.1.1.cmml" xref="S2.Thmtheorem4.p1.10.m8.1.1">𝑢</ci><ci id="S2.Thmtheorem4.p1.10.m8.2.2.cmml" xref="S2.Thmtheorem4.p1.10.m8.2.2">𝑣</ci></list><apply id="S2.Thmtheorem4.p1.10.m8.2.3.3.cmml" xref="S2.Thmtheorem4.p1.10.m8.2.3.3"><setdiff id="S2.Thmtheorem4.p1.10.m8.2.3.3.1.cmml" xref="S2.Thmtheorem4.p1.10.m8.2.3.3.1"></setdiff><ci id="S2.Thmtheorem4.p1.10.m8.2.3.3.2.cmml" xref="S2.Thmtheorem4.p1.10.m8.2.3.3.2">𝑉</ci><apply id="S2.Thmtheorem4.p1.10.m8.2.3.3.3.cmml" xref="S2.Thmtheorem4.p1.10.m8.2.3.3.3"><csymbol cd="ambiguous" id="S2.Thmtheorem4.p1.10.m8.2.3.3.3.1.cmml" xref="S2.Thmtheorem4.p1.10.m8.2.3.3.3">subscript</csymbol><ci id="S2.Thmtheorem4.p1.10.m8.2.3.3.3.2.cmml" xref="S2.Thmtheorem4.p1.10.m8.2.3.3.3.2">𝐹</ci><ci id="S2.Thmtheorem4.p1.10.m8.2.3.3.3.3.cmml" xref="S2.Thmtheorem4.p1.10.m8.2.3.3.3.3">𝑉</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem4.p1.10.m8.2c">u,v\in V\setminus F_{V}</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem4.p1.10.m8.2d">italic_u , italic_v ∈ italic_V ∖ italic_F start_POSTSUBSCRIPT italic_V end_POSTSUBSCRIPT</annotation></semantics></math>,</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="Sx1.EGx1"> <tbody id="S2.Ex1"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle d_{H\setminus F_{V}}(v,u)\leq t\cdot d_{G\setminus F_{V}}(v,u)," class="ltx_Math" display="inline" id="S2.Ex1.m1.5"><semantics id="S2.Ex1.m1.5a"><mrow id="S2.Ex1.m1.5.5.1" xref="S2.Ex1.m1.5.5.1.1.cmml"><mrow id="S2.Ex1.m1.5.5.1.1" xref="S2.Ex1.m1.5.5.1.1.cmml"><mrow id="S2.Ex1.m1.5.5.1.1.2" xref="S2.Ex1.m1.5.5.1.1.2.cmml"><msub id="S2.Ex1.m1.5.5.1.1.2.2" xref="S2.Ex1.m1.5.5.1.1.2.2.cmml"><mi id="S2.Ex1.m1.5.5.1.1.2.2.2" xref="S2.Ex1.m1.5.5.1.1.2.2.2.cmml">d</mi><mrow id="S2.Ex1.m1.5.5.1.1.2.2.3" xref="S2.Ex1.m1.5.5.1.1.2.2.3.cmml"><mi id="S2.Ex1.m1.5.5.1.1.2.2.3.2" xref="S2.Ex1.m1.5.5.1.1.2.2.3.2.cmml">H</mi><mo id="S2.Ex1.m1.5.5.1.1.2.2.3.1" xref="S2.Ex1.m1.5.5.1.1.2.2.3.1.cmml">∖</mo><msub id="S2.Ex1.m1.5.5.1.1.2.2.3.3" xref="S2.Ex1.m1.5.5.1.1.2.2.3.3.cmml"><mi id="S2.Ex1.m1.5.5.1.1.2.2.3.3.2" xref="S2.Ex1.m1.5.5.1.1.2.2.3.3.2.cmml">F</mi><mi id="S2.Ex1.m1.5.5.1.1.2.2.3.3.3" xref="S2.Ex1.m1.5.5.1.1.2.2.3.3.3.cmml">V</mi></msub></mrow></msub><mo id="S2.Ex1.m1.5.5.1.1.2.1" xref="S2.Ex1.m1.5.5.1.1.2.1.cmml">⁢</mo><mrow id="S2.Ex1.m1.5.5.1.1.2.3.2" xref="S2.Ex1.m1.5.5.1.1.2.3.1.cmml"><mo id="S2.Ex1.m1.5.5.1.1.2.3.2.1" stretchy="false" xref="S2.Ex1.m1.5.5.1.1.2.3.1.cmml">(</mo><mi id="S2.Ex1.m1.1.1" xref="S2.Ex1.m1.1.1.cmml">v</mi><mo id="S2.Ex1.m1.5.5.1.1.2.3.2.2" xref="S2.Ex1.m1.5.5.1.1.2.3.1.cmml">,</mo><mi id="S2.Ex1.m1.2.2" xref="S2.Ex1.m1.2.2.cmml">u</mi><mo id="S2.Ex1.m1.5.5.1.1.2.3.2.3" stretchy="false" xref="S2.Ex1.m1.5.5.1.1.2.3.1.cmml">)</mo></mrow></mrow><mo id="S2.Ex1.m1.5.5.1.1.1" xref="S2.Ex1.m1.5.5.1.1.1.cmml">≤</mo><mrow id="S2.Ex1.m1.5.5.1.1.3" xref="S2.Ex1.m1.5.5.1.1.3.cmml"><mrow id="S2.Ex1.m1.5.5.1.1.3.2" xref="S2.Ex1.m1.5.5.1.1.3.2.cmml"><mi id="S2.Ex1.m1.5.5.1.1.3.2.2" xref="S2.Ex1.m1.5.5.1.1.3.2.2.cmml">t</mi><mo id="S2.Ex1.m1.5.5.1.1.3.2.1" lspace="0.222em" rspace="0.222em" xref="S2.Ex1.m1.5.5.1.1.3.2.1.cmml">⋅</mo><msub id="S2.Ex1.m1.5.5.1.1.3.2.3" xref="S2.Ex1.m1.5.5.1.1.3.2.3.cmml"><mi id="S2.Ex1.m1.5.5.1.1.3.2.3.2" xref="S2.Ex1.m1.5.5.1.1.3.2.3.2.cmml">d</mi><mrow id="S2.Ex1.m1.5.5.1.1.3.2.3.3" xref="S2.Ex1.m1.5.5.1.1.3.2.3.3.cmml"><mi id="S2.Ex1.m1.5.5.1.1.3.2.3.3.2" xref="S2.Ex1.m1.5.5.1.1.3.2.3.3.2.cmml">G</mi><mo id="S2.Ex1.m1.5.5.1.1.3.2.3.3.1" xref="S2.Ex1.m1.5.5.1.1.3.2.3.3.1.cmml">∖</mo><msub id="S2.Ex1.m1.5.5.1.1.3.2.3.3.3" xref="S2.Ex1.m1.5.5.1.1.3.2.3.3.3.cmml"><mi id="S2.Ex1.m1.5.5.1.1.3.2.3.3.3.2" xref="S2.Ex1.m1.5.5.1.1.3.2.3.3.3.2.cmml">F</mi><mi id="S2.Ex1.m1.5.5.1.1.3.2.3.3.3.3" xref="S2.Ex1.m1.5.5.1.1.3.2.3.3.3.3.cmml">V</mi></msub></mrow></msub></mrow><mo id="S2.Ex1.m1.5.5.1.1.3.1" xref="S2.Ex1.m1.5.5.1.1.3.1.cmml">⁢</mo><mrow id="S2.Ex1.m1.5.5.1.1.3.3.2" xref="S2.Ex1.m1.5.5.1.1.3.3.1.cmml"><mo id="S2.Ex1.m1.5.5.1.1.3.3.2.1" stretchy="false" xref="S2.Ex1.m1.5.5.1.1.3.3.1.cmml">(</mo><mi id="S2.Ex1.m1.3.3" xref="S2.Ex1.m1.3.3.cmml">v</mi><mo id="S2.Ex1.m1.5.5.1.1.3.3.2.2" xref="S2.Ex1.m1.5.5.1.1.3.3.1.cmml">,</mo><mi id="S2.Ex1.m1.4.4" xref="S2.Ex1.m1.4.4.cmml">u</mi><mo id="S2.Ex1.m1.5.5.1.1.3.3.2.3" stretchy="false" xref="S2.Ex1.m1.5.5.1.1.3.3.1.cmml">)</mo></mrow></mrow></mrow><mo id="S2.Ex1.m1.5.5.1.2" xref="S2.Ex1.m1.5.5.1.1.cmml">,</mo></mrow><annotation-xml encoding="MathML-Content" id="S2.Ex1.m1.5b"><apply id="S2.Ex1.m1.5.5.1.1.cmml" xref="S2.Ex1.m1.5.5.1"><leq id="S2.Ex1.m1.5.5.1.1.1.cmml" xref="S2.Ex1.m1.5.5.1.1.1"></leq><apply id="S2.Ex1.m1.5.5.1.1.2.cmml" xref="S2.Ex1.m1.5.5.1.1.2"><times id="S2.Ex1.m1.5.5.1.1.2.1.cmml" xref="S2.Ex1.m1.5.5.1.1.2.1"></times><apply id="S2.Ex1.m1.5.5.1.1.2.2.cmml" xref="S2.Ex1.m1.5.5.1.1.2.2"><csymbol cd="ambiguous" id="S2.Ex1.m1.5.5.1.1.2.2.1.cmml" xref="S2.Ex1.m1.5.5.1.1.2.2">subscript</csymbol><ci id="S2.Ex1.m1.5.5.1.1.2.2.2.cmml" xref="S2.Ex1.m1.5.5.1.1.2.2.2">𝑑</ci><apply id="S2.Ex1.m1.5.5.1.1.2.2.3.cmml" xref="S2.Ex1.m1.5.5.1.1.2.2.3"><setdiff id="S2.Ex1.m1.5.5.1.1.2.2.3.1.cmml" xref="S2.Ex1.m1.5.5.1.1.2.2.3.1"></setdiff><ci id="S2.Ex1.m1.5.5.1.1.2.2.3.2.cmml" xref="S2.Ex1.m1.5.5.1.1.2.2.3.2">𝐻</ci><apply id="S2.Ex1.m1.5.5.1.1.2.2.3.3.cmml" xref="S2.Ex1.m1.5.5.1.1.2.2.3.3"><csymbol cd="ambiguous" id="S2.Ex1.m1.5.5.1.1.2.2.3.3.1.cmml" xref="S2.Ex1.m1.5.5.1.1.2.2.3.3">subscript</csymbol><ci id="S2.Ex1.m1.5.5.1.1.2.2.3.3.2.cmml" xref="S2.Ex1.m1.5.5.1.1.2.2.3.3.2">𝐹</ci><ci id="S2.Ex1.m1.5.5.1.1.2.2.3.3.3.cmml" xref="S2.Ex1.m1.5.5.1.1.2.2.3.3.3">𝑉</ci></apply></apply></apply><interval closure="open" id="S2.Ex1.m1.5.5.1.1.2.3.1.cmml" xref="S2.Ex1.m1.5.5.1.1.2.3.2"><ci id="S2.Ex1.m1.1.1.cmml" xref="S2.Ex1.m1.1.1">𝑣</ci><ci id="S2.Ex1.m1.2.2.cmml" xref="S2.Ex1.m1.2.2">𝑢</ci></interval></apply><apply id="S2.Ex1.m1.5.5.1.1.3.cmml" xref="S2.Ex1.m1.5.5.1.1.3"><times id="S2.Ex1.m1.5.5.1.1.3.1.cmml" xref="S2.Ex1.m1.5.5.1.1.3.1"></times><apply id="S2.Ex1.m1.5.5.1.1.3.2.cmml" xref="S2.Ex1.m1.5.5.1.1.3.2"><ci id="S2.Ex1.m1.5.5.1.1.3.2.1.cmml" xref="S2.Ex1.m1.5.5.1.1.3.2.1">⋅</ci><ci id="S2.Ex1.m1.5.5.1.1.3.2.2.cmml" xref="S2.Ex1.m1.5.5.1.1.3.2.2">𝑡</ci><apply id="S2.Ex1.m1.5.5.1.1.3.2.3.cmml" xref="S2.Ex1.m1.5.5.1.1.3.2.3"><csymbol cd="ambiguous" id="S2.Ex1.m1.5.5.1.1.3.2.3.1.cmml" xref="S2.Ex1.m1.5.5.1.1.3.2.3">subscript</csymbol><ci id="S2.Ex1.m1.5.5.1.1.3.2.3.2.cmml" xref="S2.Ex1.m1.5.5.1.1.3.2.3.2">𝑑</ci><apply id="S2.Ex1.m1.5.5.1.1.3.2.3.3.cmml" xref="S2.Ex1.m1.5.5.1.1.3.2.3.3"><setdiff id="S2.Ex1.m1.5.5.1.1.3.2.3.3.1.cmml" xref="S2.Ex1.m1.5.5.1.1.3.2.3.3.1"></setdiff><ci id="S2.Ex1.m1.5.5.1.1.3.2.3.3.2.cmml" xref="S2.Ex1.m1.5.5.1.1.3.2.3.3.2">𝐺</ci><apply id="S2.Ex1.m1.5.5.1.1.3.2.3.3.3.cmml" xref="S2.Ex1.m1.5.5.1.1.3.2.3.3.3"><csymbol cd="ambiguous" id="S2.Ex1.m1.5.5.1.1.3.2.3.3.3.1.cmml" xref="S2.Ex1.m1.5.5.1.1.3.2.3.3.3">subscript</csymbol><ci id="S2.Ex1.m1.5.5.1.1.3.2.3.3.3.2.cmml" xref="S2.Ex1.m1.5.5.1.1.3.2.3.3.3.2">𝐹</ci><ci id="S2.Ex1.m1.5.5.1.1.3.2.3.3.3.3.cmml" xref="S2.Ex1.m1.5.5.1.1.3.2.3.3.3.3">𝑉</ci></apply></apply></apply></apply><interval closure="open" id="S2.Ex1.m1.5.5.1.1.3.3.1.cmml" xref="S2.Ex1.m1.5.5.1.1.3.3.2"><ci id="S2.Ex1.m1.3.3.cmml" xref="S2.Ex1.m1.3.3">𝑣</ci><ci id="S2.Ex1.m1.4.4.cmml" xref="S2.Ex1.m1.4.4">𝑢</ci></interval></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Ex1.m1.5c">\displaystyle d_{H\setminus F_{V}}(v,u)\leq t\cdot d_{G\setminus F_{V}}(v,u),</annotation><annotation encoding="application/x-llamapun" id="S2.Ex1.m1.5d">italic_d start_POSTSUBSCRIPT italic_H ∖ italic_F start_POSTSUBSCRIPT italic_V end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_v , italic_u ) ≤ italic_t ⋅ italic_d start_POSTSUBSCRIPT italic_G ∖ italic_F start_POSTSUBSCRIPT italic_V end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_v , italic_u ) ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S2.Thmtheorem4.p1.14">where <math alttext="G\setminus F_{V}" class="ltx_Math" display="inline" id="S2.Thmtheorem4.p1.11.m1.1"><semantics id="S2.Thmtheorem4.p1.11.m1.1a"><mrow id="S2.Thmtheorem4.p1.11.m1.1.1" xref="S2.Thmtheorem4.p1.11.m1.1.1.cmml"><mi id="S2.Thmtheorem4.p1.11.m1.1.1.2" xref="S2.Thmtheorem4.p1.11.m1.1.1.2.cmml">G</mi><mo id="S2.Thmtheorem4.p1.11.m1.1.1.1" xref="S2.Thmtheorem4.p1.11.m1.1.1.1.cmml">∖</mo><msub id="S2.Thmtheorem4.p1.11.m1.1.1.3" xref="S2.Thmtheorem4.p1.11.m1.1.1.3.cmml"><mi id="S2.Thmtheorem4.p1.11.m1.1.1.3.2" xref="S2.Thmtheorem4.p1.11.m1.1.1.3.2.cmml">F</mi><mi id="S2.Thmtheorem4.p1.11.m1.1.1.3.3" xref="S2.Thmtheorem4.p1.11.m1.1.1.3.3.cmml">V</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem4.p1.11.m1.1b"><apply id="S2.Thmtheorem4.p1.11.m1.1.1.cmml" xref="S2.Thmtheorem4.p1.11.m1.1.1"><setdiff id="S2.Thmtheorem4.p1.11.m1.1.1.1.cmml" xref="S2.Thmtheorem4.p1.11.m1.1.1.1"></setdiff><ci id="S2.Thmtheorem4.p1.11.m1.1.1.2.cmml" xref="S2.Thmtheorem4.p1.11.m1.1.1.2">𝐺</ci><apply id="S2.Thmtheorem4.p1.11.m1.1.1.3.cmml" xref="S2.Thmtheorem4.p1.11.m1.1.1.3"><csymbol cd="ambiguous" id="S2.Thmtheorem4.p1.11.m1.1.1.3.1.cmml" xref="S2.Thmtheorem4.p1.11.m1.1.1.3">subscript</csymbol><ci id="S2.Thmtheorem4.p1.11.m1.1.1.3.2.cmml" xref="S2.Thmtheorem4.p1.11.m1.1.1.3.2">𝐹</ci><ci id="S2.Thmtheorem4.p1.11.m1.1.1.3.3.cmml" xref="S2.Thmtheorem4.p1.11.m1.1.1.3.3">𝑉</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem4.p1.11.m1.1c">G\setminus F_{V}</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem4.p1.11.m1.1d">italic_G ∖ italic_F start_POSTSUBSCRIPT italic_V end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="H\setminus F_{V}" class="ltx_Math" display="inline" id="S2.Thmtheorem4.p1.12.m2.1"><semantics id="S2.Thmtheorem4.p1.12.m2.1a"><mrow id="S2.Thmtheorem4.p1.12.m2.1.1" xref="S2.Thmtheorem4.p1.12.m2.1.1.cmml"><mi id="S2.Thmtheorem4.p1.12.m2.1.1.2" xref="S2.Thmtheorem4.p1.12.m2.1.1.2.cmml">H</mi><mo id="S2.Thmtheorem4.p1.12.m2.1.1.1" xref="S2.Thmtheorem4.p1.12.m2.1.1.1.cmml">∖</mo><msub id="S2.Thmtheorem4.p1.12.m2.1.1.3" xref="S2.Thmtheorem4.p1.12.m2.1.1.3.cmml"><mi id="S2.Thmtheorem4.p1.12.m2.1.1.3.2" xref="S2.Thmtheorem4.p1.12.m2.1.1.3.2.cmml">F</mi><mi id="S2.Thmtheorem4.p1.12.m2.1.1.3.3" xref="S2.Thmtheorem4.p1.12.m2.1.1.3.3.cmml">V</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem4.p1.12.m2.1b"><apply id="S2.Thmtheorem4.p1.12.m2.1.1.cmml" xref="S2.Thmtheorem4.p1.12.m2.1.1"><setdiff id="S2.Thmtheorem4.p1.12.m2.1.1.1.cmml" xref="S2.Thmtheorem4.p1.12.m2.1.1.1"></setdiff><ci id="S2.Thmtheorem4.p1.12.m2.1.1.2.cmml" xref="S2.Thmtheorem4.p1.12.m2.1.1.2">𝐻</ci><apply id="S2.Thmtheorem4.p1.12.m2.1.1.3.cmml" xref="S2.Thmtheorem4.p1.12.m2.1.1.3"><csymbol cd="ambiguous" id="S2.Thmtheorem4.p1.12.m2.1.1.3.1.cmml" xref="S2.Thmtheorem4.p1.12.m2.1.1.3">subscript</csymbol><ci id="S2.Thmtheorem4.p1.12.m2.1.1.3.2.cmml" xref="S2.Thmtheorem4.p1.12.m2.1.1.3.2">𝐹</ci><ci id="S2.Thmtheorem4.p1.12.m2.1.1.3.3.cmml" xref="S2.Thmtheorem4.p1.12.m2.1.1.3.3">𝑉</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem4.p1.12.m2.1c">H\setminus F_{V}</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem4.p1.12.m2.1d">italic_H ∖ italic_F start_POSTSUBSCRIPT italic_V end_POSTSUBSCRIPT</annotation></semantics></math> denote the induced subgraphs <math alttext="G[V\setminus F_{V}]" class="ltx_Math" display="inline" id="S2.Thmtheorem4.p1.13.m3.1"><semantics id="S2.Thmtheorem4.p1.13.m3.1a"><mrow id="S2.Thmtheorem4.p1.13.m3.1.1" xref="S2.Thmtheorem4.p1.13.m3.1.1.cmml"><mi id="S2.Thmtheorem4.p1.13.m3.1.1.3" xref="S2.Thmtheorem4.p1.13.m3.1.1.3.cmml">G</mi><mo id="S2.Thmtheorem4.p1.13.m3.1.1.2" xref="S2.Thmtheorem4.p1.13.m3.1.1.2.cmml">⁢</mo><mrow id="S2.Thmtheorem4.p1.13.m3.1.1.1.1" xref="S2.Thmtheorem4.p1.13.m3.1.1.1.2.cmml"><mo id="S2.Thmtheorem4.p1.13.m3.1.1.1.1.2" stretchy="false" xref="S2.Thmtheorem4.p1.13.m3.1.1.1.2.1.cmml">[</mo><mrow id="S2.Thmtheorem4.p1.13.m3.1.1.1.1.1" xref="S2.Thmtheorem4.p1.13.m3.1.1.1.1.1.cmml"><mi id="S2.Thmtheorem4.p1.13.m3.1.1.1.1.1.2" xref="S2.Thmtheorem4.p1.13.m3.1.1.1.1.1.2.cmml">V</mi><mo id="S2.Thmtheorem4.p1.13.m3.1.1.1.1.1.1" xref="S2.Thmtheorem4.p1.13.m3.1.1.1.1.1.1.cmml">∖</mo><msub id="S2.Thmtheorem4.p1.13.m3.1.1.1.1.1.3" xref="S2.Thmtheorem4.p1.13.m3.1.1.1.1.1.3.cmml"><mi id="S2.Thmtheorem4.p1.13.m3.1.1.1.1.1.3.2" xref="S2.Thmtheorem4.p1.13.m3.1.1.1.1.1.3.2.cmml">F</mi><mi id="S2.Thmtheorem4.p1.13.m3.1.1.1.1.1.3.3" xref="S2.Thmtheorem4.p1.13.m3.1.1.1.1.1.3.3.cmml">V</mi></msub></mrow><mo id="S2.Thmtheorem4.p1.13.m3.1.1.1.1.3" stretchy="false" xref="S2.Thmtheorem4.p1.13.m3.1.1.1.2.1.cmml">]</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem4.p1.13.m3.1b"><apply id="S2.Thmtheorem4.p1.13.m3.1.1.cmml" xref="S2.Thmtheorem4.p1.13.m3.1.1"><times id="S2.Thmtheorem4.p1.13.m3.1.1.2.cmml" xref="S2.Thmtheorem4.p1.13.m3.1.1.2"></times><ci id="S2.Thmtheorem4.p1.13.m3.1.1.3.cmml" xref="S2.Thmtheorem4.p1.13.m3.1.1.3">𝐺</ci><apply id="S2.Thmtheorem4.p1.13.m3.1.1.1.2.cmml" xref="S2.Thmtheorem4.p1.13.m3.1.1.1.1"><csymbol cd="latexml" id="S2.Thmtheorem4.p1.13.m3.1.1.1.2.1.cmml" xref="S2.Thmtheorem4.p1.13.m3.1.1.1.1.2">delimited-[]</csymbol><apply id="S2.Thmtheorem4.p1.13.m3.1.1.1.1.1.cmml" xref="S2.Thmtheorem4.p1.13.m3.1.1.1.1.1"><setdiff id="S2.Thmtheorem4.p1.13.m3.1.1.1.1.1.1.cmml" xref="S2.Thmtheorem4.p1.13.m3.1.1.1.1.1.1"></setdiff><ci id="S2.Thmtheorem4.p1.13.m3.1.1.1.1.1.2.cmml" xref="S2.Thmtheorem4.p1.13.m3.1.1.1.1.1.2">𝑉</ci><apply id="S2.Thmtheorem4.p1.13.m3.1.1.1.1.1.3.cmml" xref="S2.Thmtheorem4.p1.13.m3.1.1.1.1.1.3"><csymbol cd="ambiguous" id="S2.Thmtheorem4.p1.13.m3.1.1.1.1.1.3.1.cmml" xref="S2.Thmtheorem4.p1.13.m3.1.1.1.1.1.3">subscript</csymbol><ci id="S2.Thmtheorem4.p1.13.m3.1.1.1.1.1.3.2.cmml" xref="S2.Thmtheorem4.p1.13.m3.1.1.1.1.1.3.2">𝐹</ci><ci id="S2.Thmtheorem4.p1.13.m3.1.1.1.1.1.3.3.cmml" xref="S2.Thmtheorem4.p1.13.m3.1.1.1.1.1.3.3">𝑉</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem4.p1.13.m3.1c">G[V\setminus F_{V}]</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem4.p1.13.m3.1d">italic_G [ italic_V ∖ italic_F start_POSTSUBSCRIPT italic_V end_POSTSUBSCRIPT ]</annotation></semantics></math> and <math alttext="H[V\setminus F_{V}]" class="ltx_Math" display="inline" id="S2.Thmtheorem4.p1.14.m4.1"><semantics id="S2.Thmtheorem4.p1.14.m4.1a"><mrow id="S2.Thmtheorem4.p1.14.m4.1.1" xref="S2.Thmtheorem4.p1.14.m4.1.1.cmml"><mi id="S2.Thmtheorem4.p1.14.m4.1.1.3" xref="S2.Thmtheorem4.p1.14.m4.1.1.3.cmml">H</mi><mo id="S2.Thmtheorem4.p1.14.m4.1.1.2" xref="S2.Thmtheorem4.p1.14.m4.1.1.2.cmml">⁢</mo><mrow id="S2.Thmtheorem4.p1.14.m4.1.1.1.1" xref="S2.Thmtheorem4.p1.14.m4.1.1.1.2.cmml"><mo id="S2.Thmtheorem4.p1.14.m4.1.1.1.1.2" stretchy="false" xref="S2.Thmtheorem4.p1.14.m4.1.1.1.2.1.cmml">[</mo><mrow id="S2.Thmtheorem4.p1.14.m4.1.1.1.1.1" xref="S2.Thmtheorem4.p1.14.m4.1.1.1.1.1.cmml"><mi id="S2.Thmtheorem4.p1.14.m4.1.1.1.1.1.2" xref="S2.Thmtheorem4.p1.14.m4.1.1.1.1.1.2.cmml">V</mi><mo id="S2.Thmtheorem4.p1.14.m4.1.1.1.1.1.1" xref="S2.Thmtheorem4.p1.14.m4.1.1.1.1.1.1.cmml">∖</mo><msub id="S2.Thmtheorem4.p1.14.m4.1.1.1.1.1.3" xref="S2.Thmtheorem4.p1.14.m4.1.1.1.1.1.3.cmml"><mi id="S2.Thmtheorem4.p1.14.m4.1.1.1.1.1.3.2" xref="S2.Thmtheorem4.p1.14.m4.1.1.1.1.1.3.2.cmml">F</mi><mi id="S2.Thmtheorem4.p1.14.m4.1.1.1.1.1.3.3" xref="S2.Thmtheorem4.p1.14.m4.1.1.1.1.1.3.3.cmml">V</mi></msub></mrow><mo id="S2.Thmtheorem4.p1.14.m4.1.1.1.1.3" stretchy="false" xref="S2.Thmtheorem4.p1.14.m4.1.1.1.2.1.cmml">]</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem4.p1.14.m4.1b"><apply id="S2.Thmtheorem4.p1.14.m4.1.1.cmml" xref="S2.Thmtheorem4.p1.14.m4.1.1"><times id="S2.Thmtheorem4.p1.14.m4.1.1.2.cmml" xref="S2.Thmtheorem4.p1.14.m4.1.1.2"></times><ci id="S2.Thmtheorem4.p1.14.m4.1.1.3.cmml" xref="S2.Thmtheorem4.p1.14.m4.1.1.3">𝐻</ci><apply id="S2.Thmtheorem4.p1.14.m4.1.1.1.2.cmml" xref="S2.Thmtheorem4.p1.14.m4.1.1.1.1"><csymbol cd="latexml" id="S2.Thmtheorem4.p1.14.m4.1.1.1.2.1.cmml" xref="S2.Thmtheorem4.p1.14.m4.1.1.1.1.2">delimited-[]</csymbol><apply id="S2.Thmtheorem4.p1.14.m4.1.1.1.1.1.cmml" xref="S2.Thmtheorem4.p1.14.m4.1.1.1.1.1"><setdiff id="S2.Thmtheorem4.p1.14.m4.1.1.1.1.1.1.cmml" xref="S2.Thmtheorem4.p1.14.m4.1.1.1.1.1.1"></setdiff><ci id="S2.Thmtheorem4.p1.14.m4.1.1.1.1.1.2.cmml" xref="S2.Thmtheorem4.p1.14.m4.1.1.1.1.1.2">𝑉</ci><apply id="S2.Thmtheorem4.p1.14.m4.1.1.1.1.1.3.cmml" xref="S2.Thmtheorem4.p1.14.m4.1.1.1.1.1.3"><csymbol cd="ambiguous" id="S2.Thmtheorem4.p1.14.m4.1.1.1.1.1.3.1.cmml" xref="S2.Thmtheorem4.p1.14.m4.1.1.1.1.1.3">subscript</csymbol><ci id="S2.Thmtheorem4.p1.14.m4.1.1.1.1.1.3.2.cmml" xref="S2.Thmtheorem4.p1.14.m4.1.1.1.1.1.3.2">𝐹</ci><ci id="S2.Thmtheorem4.p1.14.m4.1.1.1.1.1.3.3.cmml" xref="S2.Thmtheorem4.p1.14.m4.1.1.1.1.1.3.3">𝑉</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem4.p1.14.m4.1c">H[V\setminus F_{V}]</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem4.p1.14.m4.1d">italic_H [ italic_V ∖ italic_F start_POSTSUBSCRIPT italic_V end_POSTSUBSCRIPT ]</annotation></semantics></math>, respectively.</p> </div> <div class="ltx_para" id="S2.Thmtheorem4.p2"> <p class="ltx_p" id="S2.Thmtheorem4.p2.10">Similarly, a subgraph <math alttext="H\subseteq G" class="ltx_Math" display="inline" id="S2.Thmtheorem4.p2.1.m1.1"><semantics id="S2.Thmtheorem4.p2.1.m1.1a"><mrow id="S2.Thmtheorem4.p2.1.m1.1.1" xref="S2.Thmtheorem4.p2.1.m1.1.1.cmml"><mi id="S2.Thmtheorem4.p2.1.m1.1.1.2" xref="S2.Thmtheorem4.p2.1.m1.1.1.2.cmml">H</mi><mo id="S2.Thmtheorem4.p2.1.m1.1.1.1" xref="S2.Thmtheorem4.p2.1.m1.1.1.1.cmml">⊆</mo><mi id="S2.Thmtheorem4.p2.1.m1.1.1.3" xref="S2.Thmtheorem4.p2.1.m1.1.1.3.cmml">G</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem4.p2.1.m1.1b"><apply id="S2.Thmtheorem4.p2.1.m1.1.1.cmml" xref="S2.Thmtheorem4.p2.1.m1.1.1"><subset id="S2.Thmtheorem4.p2.1.m1.1.1.1.cmml" xref="S2.Thmtheorem4.p2.1.m1.1.1.1"></subset><ci id="S2.Thmtheorem4.p2.1.m1.1.1.2.cmml" xref="S2.Thmtheorem4.p2.1.m1.1.1.2">𝐻</ci><ci id="S2.Thmtheorem4.p2.1.m1.1.1.3.cmml" xref="S2.Thmtheorem4.p2.1.m1.1.1.3">𝐺</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem4.p2.1.m1.1c">H\subseteq G</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem4.p2.1.m1.1d">italic_H ⊆ italic_G</annotation></semantics></math> is an <math alttext="f" class="ltx_Math" display="inline" id="S2.Thmtheorem4.p2.2.m2.1"><semantics id="S2.Thmtheorem4.p2.2.m2.1a"><mi id="S2.Thmtheorem4.p2.2.m2.1.1" xref="S2.Thmtheorem4.p2.2.m2.1.1.cmml">f</mi><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem4.p2.2.m2.1b"><ci id="S2.Thmtheorem4.p2.2.m2.1.1.cmml" xref="S2.Thmtheorem4.p2.2.m2.1.1">𝑓</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem4.p2.2.m2.1c">f</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem4.p2.2.m2.1d">italic_f</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S2.Thmtheorem4.p2.4.2">-vertex-fault-tolerant (<math alttext="f" class="ltx_Math" display="inline" id="S2.Thmtheorem4.p2.3.1.m1.1"><semantics id="S2.Thmtheorem4.p2.3.1.m1.1a"><mi id="S2.Thmtheorem4.p2.3.1.m1.1.1" xref="S2.Thmtheorem4.p2.3.1.m1.1.1.cmml">f</mi><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem4.p2.3.1.m1.1b"><ci id="S2.Thmtheorem4.p2.3.1.m1.1.1.cmml" xref="S2.Thmtheorem4.p2.3.1.m1.1.1">𝑓</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem4.p2.3.1.m1.1c">f</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem4.p2.3.1.m1.1d">italic_f</annotation></semantics></math>-VFT) <math alttext="t" class="ltx_Math" display="inline" id="S2.Thmtheorem4.p2.4.2.m2.1"><semantics id="S2.Thmtheorem4.p2.4.2.m2.1a"><mi id="S2.Thmtheorem4.p2.4.2.m2.1.1" xref="S2.Thmtheorem4.p2.4.2.m2.1.1.cmml">t</mi><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem4.p2.4.2.m2.1b"><ci id="S2.Thmtheorem4.p2.4.2.m2.1.1.cmml" xref="S2.Thmtheorem4.p2.4.2.m2.1.1">𝑡</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem4.p2.4.2.m2.1c">t</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem4.p2.4.2.m2.1d">italic_t</annotation></semantics></math>-spanner</span> of <math alttext="G" class="ltx_Math" display="inline" id="S2.Thmtheorem4.p2.5.m3.1"><semantics id="S2.Thmtheorem4.p2.5.m3.1a"><mi id="S2.Thmtheorem4.p2.5.m3.1.1" xref="S2.Thmtheorem4.p2.5.m3.1.1.cmml">G</mi><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem4.p2.5.m3.1b"><ci id="S2.Thmtheorem4.p2.5.m3.1.1.cmml" xref="S2.Thmtheorem4.p2.5.m3.1.1">𝐺</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem4.p2.5.m3.1c">G</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem4.p2.5.m3.1d">italic_G</annotation></semantics></math> if, for every subset of vertices <math alttext="F_{V}\subseteq V" class="ltx_Math" display="inline" id="S2.Thmtheorem4.p2.6.m4.1"><semantics id="S2.Thmtheorem4.p2.6.m4.1a"><mrow id="S2.Thmtheorem4.p2.6.m4.1.1" xref="S2.Thmtheorem4.p2.6.m4.1.1.cmml"><msub id="S2.Thmtheorem4.p2.6.m4.1.1.2" xref="S2.Thmtheorem4.p2.6.m4.1.1.2.cmml"><mi id="S2.Thmtheorem4.p2.6.m4.1.1.2.2" xref="S2.Thmtheorem4.p2.6.m4.1.1.2.2.cmml">F</mi><mi id="S2.Thmtheorem4.p2.6.m4.1.1.2.3" xref="S2.Thmtheorem4.p2.6.m4.1.1.2.3.cmml">V</mi></msub><mo id="S2.Thmtheorem4.p2.6.m4.1.1.1" xref="S2.Thmtheorem4.p2.6.m4.1.1.1.cmml">⊆</mo><mi id="S2.Thmtheorem4.p2.6.m4.1.1.3" xref="S2.Thmtheorem4.p2.6.m4.1.1.3.cmml">V</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem4.p2.6.m4.1b"><apply id="S2.Thmtheorem4.p2.6.m4.1.1.cmml" xref="S2.Thmtheorem4.p2.6.m4.1.1"><subset id="S2.Thmtheorem4.p2.6.m4.1.1.1.cmml" xref="S2.Thmtheorem4.p2.6.m4.1.1.1"></subset><apply id="S2.Thmtheorem4.p2.6.m4.1.1.2.cmml" xref="S2.Thmtheorem4.p2.6.m4.1.1.2"><csymbol cd="ambiguous" id="S2.Thmtheorem4.p2.6.m4.1.1.2.1.cmml" xref="S2.Thmtheorem4.p2.6.m4.1.1.2">subscript</csymbol><ci id="S2.Thmtheorem4.p2.6.m4.1.1.2.2.cmml" xref="S2.Thmtheorem4.p2.6.m4.1.1.2.2">𝐹</ci><ci id="S2.Thmtheorem4.p2.6.m4.1.1.2.3.cmml" xref="S2.Thmtheorem4.p2.6.m4.1.1.2.3">𝑉</ci></apply><ci id="S2.Thmtheorem4.p2.6.m4.1.1.3.cmml" xref="S2.Thmtheorem4.p2.6.m4.1.1.3">𝑉</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem4.p2.6.m4.1c">F_{V}\subseteq V</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem4.p2.6.m4.1d">italic_F start_POSTSUBSCRIPT italic_V end_POSTSUBSCRIPT ⊆ italic_V</annotation></semantics></math> of size at most <math alttext="f" class="ltx_Math" display="inline" id="S2.Thmtheorem4.p2.7.m5.1"><semantics id="S2.Thmtheorem4.p2.7.m5.1a"><mi id="S2.Thmtheorem4.p2.7.m5.1.1" xref="S2.Thmtheorem4.p2.7.m5.1.1.cmml">f</mi><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem4.p2.7.m5.1b"><ci id="S2.Thmtheorem4.p2.7.m5.1.1.cmml" xref="S2.Thmtheorem4.p2.7.m5.1.1">𝑓</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem4.p2.7.m5.1c">f</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem4.p2.7.m5.1d">italic_f</annotation></semantics></math>, all pairwise distances in <math alttext="V\setminus F_{V}" class="ltx_Math" display="inline" id="S2.Thmtheorem4.p2.8.m6.1"><semantics id="S2.Thmtheorem4.p2.8.m6.1a"><mrow id="S2.Thmtheorem4.p2.8.m6.1.1" xref="S2.Thmtheorem4.p2.8.m6.1.1.cmml"><mi id="S2.Thmtheorem4.p2.8.m6.1.1.2" xref="S2.Thmtheorem4.p2.8.m6.1.1.2.cmml">V</mi><mo id="S2.Thmtheorem4.p2.8.m6.1.1.1" xref="S2.Thmtheorem4.p2.8.m6.1.1.1.cmml">∖</mo><msub id="S2.Thmtheorem4.p2.8.m6.1.1.3" xref="S2.Thmtheorem4.p2.8.m6.1.1.3.cmml"><mi id="S2.Thmtheorem4.p2.8.m6.1.1.3.2" xref="S2.Thmtheorem4.p2.8.m6.1.1.3.2.cmml">F</mi><mi id="S2.Thmtheorem4.p2.8.m6.1.1.3.3" xref="S2.Thmtheorem4.p2.8.m6.1.1.3.3.cmml">V</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem4.p2.8.m6.1b"><apply id="S2.Thmtheorem4.p2.8.m6.1.1.cmml" xref="S2.Thmtheorem4.p2.8.m6.1.1"><setdiff id="S2.Thmtheorem4.p2.8.m6.1.1.1.cmml" xref="S2.Thmtheorem4.p2.8.m6.1.1.1"></setdiff><ci id="S2.Thmtheorem4.p2.8.m6.1.1.2.cmml" xref="S2.Thmtheorem4.p2.8.m6.1.1.2">𝑉</ci><apply id="S2.Thmtheorem4.p2.8.m6.1.1.3.cmml" xref="S2.Thmtheorem4.p2.8.m6.1.1.3"><csymbol cd="ambiguous" id="S2.Thmtheorem4.p2.8.m6.1.1.3.1.cmml" xref="S2.Thmtheorem4.p2.8.m6.1.1.3">subscript</csymbol><ci id="S2.Thmtheorem4.p2.8.m6.1.1.3.2.cmml" xref="S2.Thmtheorem4.p2.8.m6.1.1.3.2">𝐹</ci><ci id="S2.Thmtheorem4.p2.8.m6.1.1.3.3.cmml" xref="S2.Thmtheorem4.p2.8.m6.1.1.3.3">𝑉</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem4.p2.8.m6.1c">V\setminus F_{V}</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem4.p2.8.m6.1d">italic_V ∖ italic_F start_POSTSUBSCRIPT italic_V end_POSTSUBSCRIPT</annotation></semantics></math> are preserved within a factor of <math alttext="t" class="ltx_Math" display="inline" id="S2.Thmtheorem4.p2.9.m7.1"><semantics id="S2.Thmtheorem4.p2.9.m7.1a"><mi id="S2.Thmtheorem4.p2.9.m7.1.1" xref="S2.Thmtheorem4.p2.9.m7.1.1.cmml">t</mi><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem4.p2.9.m7.1b"><ci id="S2.Thmtheorem4.p2.9.m7.1.1.cmml" xref="S2.Thmtheorem4.p2.9.m7.1.1">𝑡</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem4.p2.9.m7.1c">t</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem4.p2.9.m7.1d">italic_t</annotation></semantics></math>. That is, for all <math alttext="u,v\in V\setminus F_{V}" class="ltx_Math" display="inline" id="S2.Thmtheorem4.p2.10.m8.2"><semantics id="S2.Thmtheorem4.p2.10.m8.2a"><mrow id="S2.Thmtheorem4.p2.10.m8.2.3" xref="S2.Thmtheorem4.p2.10.m8.2.3.cmml"><mrow id="S2.Thmtheorem4.p2.10.m8.2.3.2.2" xref="S2.Thmtheorem4.p2.10.m8.2.3.2.1.cmml"><mi id="S2.Thmtheorem4.p2.10.m8.1.1" xref="S2.Thmtheorem4.p2.10.m8.1.1.cmml">u</mi><mo id="S2.Thmtheorem4.p2.10.m8.2.3.2.2.1" xref="S2.Thmtheorem4.p2.10.m8.2.3.2.1.cmml">,</mo><mi id="S2.Thmtheorem4.p2.10.m8.2.2" xref="S2.Thmtheorem4.p2.10.m8.2.2.cmml">v</mi></mrow><mo id="S2.Thmtheorem4.p2.10.m8.2.3.1" xref="S2.Thmtheorem4.p2.10.m8.2.3.1.cmml">∈</mo><mrow id="S2.Thmtheorem4.p2.10.m8.2.3.3" xref="S2.Thmtheorem4.p2.10.m8.2.3.3.cmml"><mi id="S2.Thmtheorem4.p2.10.m8.2.3.3.2" xref="S2.Thmtheorem4.p2.10.m8.2.3.3.2.cmml">V</mi><mo id="S2.Thmtheorem4.p2.10.m8.2.3.3.1" xref="S2.Thmtheorem4.p2.10.m8.2.3.3.1.cmml">∖</mo><msub id="S2.Thmtheorem4.p2.10.m8.2.3.3.3" xref="S2.Thmtheorem4.p2.10.m8.2.3.3.3.cmml"><mi id="S2.Thmtheorem4.p2.10.m8.2.3.3.3.2" xref="S2.Thmtheorem4.p2.10.m8.2.3.3.3.2.cmml">F</mi><mi id="S2.Thmtheorem4.p2.10.m8.2.3.3.3.3" xref="S2.Thmtheorem4.p2.10.m8.2.3.3.3.3.cmml">V</mi></msub></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem4.p2.10.m8.2b"><apply id="S2.Thmtheorem4.p2.10.m8.2.3.cmml" xref="S2.Thmtheorem4.p2.10.m8.2.3"><in id="S2.Thmtheorem4.p2.10.m8.2.3.1.cmml" xref="S2.Thmtheorem4.p2.10.m8.2.3.1"></in><list id="S2.Thmtheorem4.p2.10.m8.2.3.2.1.cmml" xref="S2.Thmtheorem4.p2.10.m8.2.3.2.2"><ci id="S2.Thmtheorem4.p2.10.m8.1.1.cmml" xref="S2.Thmtheorem4.p2.10.m8.1.1">𝑢</ci><ci id="S2.Thmtheorem4.p2.10.m8.2.2.cmml" xref="S2.Thmtheorem4.p2.10.m8.2.2">𝑣</ci></list><apply id="S2.Thmtheorem4.p2.10.m8.2.3.3.cmml" xref="S2.Thmtheorem4.p2.10.m8.2.3.3"><setdiff id="S2.Thmtheorem4.p2.10.m8.2.3.3.1.cmml" xref="S2.Thmtheorem4.p2.10.m8.2.3.3.1"></setdiff><ci id="S2.Thmtheorem4.p2.10.m8.2.3.3.2.cmml" xref="S2.Thmtheorem4.p2.10.m8.2.3.3.2">𝑉</ci><apply id="S2.Thmtheorem4.p2.10.m8.2.3.3.3.cmml" xref="S2.Thmtheorem4.p2.10.m8.2.3.3.3"><csymbol cd="ambiguous" id="S2.Thmtheorem4.p2.10.m8.2.3.3.3.1.cmml" xref="S2.Thmtheorem4.p2.10.m8.2.3.3.3">subscript</csymbol><ci id="S2.Thmtheorem4.p2.10.m8.2.3.3.3.2.cmml" xref="S2.Thmtheorem4.p2.10.m8.2.3.3.3.2">𝐹</ci><ci id="S2.Thmtheorem4.p2.10.m8.2.3.3.3.3.cmml" xref="S2.Thmtheorem4.p2.10.m8.2.3.3.3.3">𝑉</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem4.p2.10.m8.2c">u,v\in V\setminus F_{V}</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem4.p2.10.m8.2d">italic_u , italic_v ∈ italic_V ∖ italic_F start_POSTSUBSCRIPT italic_V end_POSTSUBSCRIPT</annotation></semantics></math>,</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="Sx1.EGx2"> <tbody id="S2.Ex2"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle d_{H\setminus F_{V}}(v,u)\leq t\cdot d_{G\setminus F_{V}}(v,u)," class="ltx_Math" display="inline" id="S2.Ex2.m1.5"><semantics id="S2.Ex2.m1.5a"><mrow id="S2.Ex2.m1.5.5.1" xref="S2.Ex2.m1.5.5.1.1.cmml"><mrow id="S2.Ex2.m1.5.5.1.1" xref="S2.Ex2.m1.5.5.1.1.cmml"><mrow id="S2.Ex2.m1.5.5.1.1.2" xref="S2.Ex2.m1.5.5.1.1.2.cmml"><msub id="S2.Ex2.m1.5.5.1.1.2.2" xref="S2.Ex2.m1.5.5.1.1.2.2.cmml"><mi id="S2.Ex2.m1.5.5.1.1.2.2.2" xref="S2.Ex2.m1.5.5.1.1.2.2.2.cmml">d</mi><mrow id="S2.Ex2.m1.5.5.1.1.2.2.3" xref="S2.Ex2.m1.5.5.1.1.2.2.3.cmml"><mi id="S2.Ex2.m1.5.5.1.1.2.2.3.2" xref="S2.Ex2.m1.5.5.1.1.2.2.3.2.cmml">H</mi><mo id="S2.Ex2.m1.5.5.1.1.2.2.3.1" xref="S2.Ex2.m1.5.5.1.1.2.2.3.1.cmml">∖</mo><msub id="S2.Ex2.m1.5.5.1.1.2.2.3.3" xref="S2.Ex2.m1.5.5.1.1.2.2.3.3.cmml"><mi id="S2.Ex2.m1.5.5.1.1.2.2.3.3.2" xref="S2.Ex2.m1.5.5.1.1.2.2.3.3.2.cmml">F</mi><mi id="S2.Ex2.m1.5.5.1.1.2.2.3.3.3" xref="S2.Ex2.m1.5.5.1.1.2.2.3.3.3.cmml">V</mi></msub></mrow></msub><mo id="S2.Ex2.m1.5.5.1.1.2.1" xref="S2.Ex2.m1.5.5.1.1.2.1.cmml">⁢</mo><mrow id="S2.Ex2.m1.5.5.1.1.2.3.2" xref="S2.Ex2.m1.5.5.1.1.2.3.1.cmml"><mo id="S2.Ex2.m1.5.5.1.1.2.3.2.1" stretchy="false" xref="S2.Ex2.m1.5.5.1.1.2.3.1.cmml">(</mo><mi id="S2.Ex2.m1.1.1" xref="S2.Ex2.m1.1.1.cmml">v</mi><mo id="S2.Ex2.m1.5.5.1.1.2.3.2.2" xref="S2.Ex2.m1.5.5.1.1.2.3.1.cmml">,</mo><mi id="S2.Ex2.m1.2.2" xref="S2.Ex2.m1.2.2.cmml">u</mi><mo id="S2.Ex2.m1.5.5.1.1.2.3.2.3" stretchy="false" xref="S2.Ex2.m1.5.5.1.1.2.3.1.cmml">)</mo></mrow></mrow><mo id="S2.Ex2.m1.5.5.1.1.1" xref="S2.Ex2.m1.5.5.1.1.1.cmml">≤</mo><mrow id="S2.Ex2.m1.5.5.1.1.3" xref="S2.Ex2.m1.5.5.1.1.3.cmml"><mrow id="S2.Ex2.m1.5.5.1.1.3.2" xref="S2.Ex2.m1.5.5.1.1.3.2.cmml"><mi id="S2.Ex2.m1.5.5.1.1.3.2.2" xref="S2.Ex2.m1.5.5.1.1.3.2.2.cmml">t</mi><mo id="S2.Ex2.m1.5.5.1.1.3.2.1" lspace="0.222em" rspace="0.222em" xref="S2.Ex2.m1.5.5.1.1.3.2.1.cmml">⋅</mo><msub id="S2.Ex2.m1.5.5.1.1.3.2.3" xref="S2.Ex2.m1.5.5.1.1.3.2.3.cmml"><mi id="S2.Ex2.m1.5.5.1.1.3.2.3.2" xref="S2.Ex2.m1.5.5.1.1.3.2.3.2.cmml">d</mi><mrow id="S2.Ex2.m1.5.5.1.1.3.2.3.3" xref="S2.Ex2.m1.5.5.1.1.3.2.3.3.cmml"><mi id="S2.Ex2.m1.5.5.1.1.3.2.3.3.2" xref="S2.Ex2.m1.5.5.1.1.3.2.3.3.2.cmml">G</mi><mo id="S2.Ex2.m1.5.5.1.1.3.2.3.3.1" xref="S2.Ex2.m1.5.5.1.1.3.2.3.3.1.cmml">∖</mo><msub id="S2.Ex2.m1.5.5.1.1.3.2.3.3.3" xref="S2.Ex2.m1.5.5.1.1.3.2.3.3.3.cmml"><mi id="S2.Ex2.m1.5.5.1.1.3.2.3.3.3.2" xref="S2.Ex2.m1.5.5.1.1.3.2.3.3.3.2.cmml">F</mi><mi id="S2.Ex2.m1.5.5.1.1.3.2.3.3.3.3" xref="S2.Ex2.m1.5.5.1.1.3.2.3.3.3.3.cmml">V</mi></msub></mrow></msub></mrow><mo id="S2.Ex2.m1.5.5.1.1.3.1" xref="S2.Ex2.m1.5.5.1.1.3.1.cmml">⁢</mo><mrow id="S2.Ex2.m1.5.5.1.1.3.3.2" xref="S2.Ex2.m1.5.5.1.1.3.3.1.cmml"><mo id="S2.Ex2.m1.5.5.1.1.3.3.2.1" stretchy="false" xref="S2.Ex2.m1.5.5.1.1.3.3.1.cmml">(</mo><mi id="S2.Ex2.m1.3.3" xref="S2.Ex2.m1.3.3.cmml">v</mi><mo id="S2.Ex2.m1.5.5.1.1.3.3.2.2" xref="S2.Ex2.m1.5.5.1.1.3.3.1.cmml">,</mo><mi id="S2.Ex2.m1.4.4" xref="S2.Ex2.m1.4.4.cmml">u</mi><mo id="S2.Ex2.m1.5.5.1.1.3.3.2.3" stretchy="false" xref="S2.Ex2.m1.5.5.1.1.3.3.1.cmml">)</mo></mrow></mrow></mrow><mo id="S2.Ex2.m1.5.5.1.2" xref="S2.Ex2.m1.5.5.1.1.cmml">,</mo></mrow><annotation-xml encoding="MathML-Content" id="S2.Ex2.m1.5b"><apply id="S2.Ex2.m1.5.5.1.1.cmml" xref="S2.Ex2.m1.5.5.1"><leq id="S2.Ex2.m1.5.5.1.1.1.cmml" xref="S2.Ex2.m1.5.5.1.1.1"></leq><apply id="S2.Ex2.m1.5.5.1.1.2.cmml" xref="S2.Ex2.m1.5.5.1.1.2"><times id="S2.Ex2.m1.5.5.1.1.2.1.cmml" xref="S2.Ex2.m1.5.5.1.1.2.1"></times><apply id="S2.Ex2.m1.5.5.1.1.2.2.cmml" xref="S2.Ex2.m1.5.5.1.1.2.2"><csymbol cd="ambiguous" id="S2.Ex2.m1.5.5.1.1.2.2.1.cmml" xref="S2.Ex2.m1.5.5.1.1.2.2">subscript</csymbol><ci id="S2.Ex2.m1.5.5.1.1.2.2.2.cmml" xref="S2.Ex2.m1.5.5.1.1.2.2.2">𝑑</ci><apply id="S2.Ex2.m1.5.5.1.1.2.2.3.cmml" xref="S2.Ex2.m1.5.5.1.1.2.2.3"><setdiff id="S2.Ex2.m1.5.5.1.1.2.2.3.1.cmml" xref="S2.Ex2.m1.5.5.1.1.2.2.3.1"></setdiff><ci id="S2.Ex2.m1.5.5.1.1.2.2.3.2.cmml" xref="S2.Ex2.m1.5.5.1.1.2.2.3.2">𝐻</ci><apply id="S2.Ex2.m1.5.5.1.1.2.2.3.3.cmml" xref="S2.Ex2.m1.5.5.1.1.2.2.3.3"><csymbol cd="ambiguous" id="S2.Ex2.m1.5.5.1.1.2.2.3.3.1.cmml" xref="S2.Ex2.m1.5.5.1.1.2.2.3.3">subscript</csymbol><ci id="S2.Ex2.m1.5.5.1.1.2.2.3.3.2.cmml" xref="S2.Ex2.m1.5.5.1.1.2.2.3.3.2">𝐹</ci><ci id="S2.Ex2.m1.5.5.1.1.2.2.3.3.3.cmml" xref="S2.Ex2.m1.5.5.1.1.2.2.3.3.3">𝑉</ci></apply></apply></apply><interval closure="open" id="S2.Ex2.m1.5.5.1.1.2.3.1.cmml" xref="S2.Ex2.m1.5.5.1.1.2.3.2"><ci id="S2.Ex2.m1.1.1.cmml" xref="S2.Ex2.m1.1.1">𝑣</ci><ci id="S2.Ex2.m1.2.2.cmml" xref="S2.Ex2.m1.2.2">𝑢</ci></interval></apply><apply id="S2.Ex2.m1.5.5.1.1.3.cmml" xref="S2.Ex2.m1.5.5.1.1.3"><times id="S2.Ex2.m1.5.5.1.1.3.1.cmml" xref="S2.Ex2.m1.5.5.1.1.3.1"></times><apply id="S2.Ex2.m1.5.5.1.1.3.2.cmml" xref="S2.Ex2.m1.5.5.1.1.3.2"><ci id="S2.Ex2.m1.5.5.1.1.3.2.1.cmml" xref="S2.Ex2.m1.5.5.1.1.3.2.1">⋅</ci><ci id="S2.Ex2.m1.5.5.1.1.3.2.2.cmml" xref="S2.Ex2.m1.5.5.1.1.3.2.2">𝑡</ci><apply id="S2.Ex2.m1.5.5.1.1.3.2.3.cmml" xref="S2.Ex2.m1.5.5.1.1.3.2.3"><csymbol cd="ambiguous" id="S2.Ex2.m1.5.5.1.1.3.2.3.1.cmml" xref="S2.Ex2.m1.5.5.1.1.3.2.3">subscript</csymbol><ci id="S2.Ex2.m1.5.5.1.1.3.2.3.2.cmml" xref="S2.Ex2.m1.5.5.1.1.3.2.3.2">𝑑</ci><apply id="S2.Ex2.m1.5.5.1.1.3.2.3.3.cmml" xref="S2.Ex2.m1.5.5.1.1.3.2.3.3"><setdiff id="S2.Ex2.m1.5.5.1.1.3.2.3.3.1.cmml" xref="S2.Ex2.m1.5.5.1.1.3.2.3.3.1"></setdiff><ci id="S2.Ex2.m1.5.5.1.1.3.2.3.3.2.cmml" xref="S2.Ex2.m1.5.5.1.1.3.2.3.3.2">𝐺</ci><apply id="S2.Ex2.m1.5.5.1.1.3.2.3.3.3.cmml" xref="S2.Ex2.m1.5.5.1.1.3.2.3.3.3"><csymbol cd="ambiguous" id="S2.Ex2.m1.5.5.1.1.3.2.3.3.3.1.cmml" xref="S2.Ex2.m1.5.5.1.1.3.2.3.3.3">subscript</csymbol><ci id="S2.Ex2.m1.5.5.1.1.3.2.3.3.3.2.cmml" xref="S2.Ex2.m1.5.5.1.1.3.2.3.3.3.2">𝐹</ci><ci id="S2.Ex2.m1.5.5.1.1.3.2.3.3.3.3.cmml" xref="S2.Ex2.m1.5.5.1.1.3.2.3.3.3.3">𝑉</ci></apply></apply></apply></apply><interval closure="open" id="S2.Ex2.m1.5.5.1.1.3.3.1.cmml" xref="S2.Ex2.m1.5.5.1.1.3.3.2"><ci id="S2.Ex2.m1.3.3.cmml" xref="S2.Ex2.m1.3.3">𝑣</ci><ci id="S2.Ex2.m1.4.4.cmml" xref="S2.Ex2.m1.4.4">𝑢</ci></interval></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Ex2.m1.5c">\displaystyle d_{H\setminus F_{V}}(v,u)\leq t\cdot d_{G\setminus F_{V}}(v,u),</annotation><annotation encoding="application/x-llamapun" id="S2.Ex2.m1.5d">italic_d start_POSTSUBSCRIPT italic_H ∖ italic_F start_POSTSUBSCRIPT italic_V end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_v , italic_u ) ≤ italic_t ⋅ italic_d start_POSTSUBSCRIPT italic_G ∖ italic_F start_POSTSUBSCRIPT italic_V end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_v , italic_u ) ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S2.Thmtheorem4.p2.14">where <math alttext="G\setminus F_{V}" class="ltx_Math" display="inline" id="S2.Thmtheorem4.p2.11.m1.1"><semantics id="S2.Thmtheorem4.p2.11.m1.1a"><mrow id="S2.Thmtheorem4.p2.11.m1.1.1" xref="S2.Thmtheorem4.p2.11.m1.1.1.cmml"><mi id="S2.Thmtheorem4.p2.11.m1.1.1.2" xref="S2.Thmtheorem4.p2.11.m1.1.1.2.cmml">G</mi><mo id="S2.Thmtheorem4.p2.11.m1.1.1.1" xref="S2.Thmtheorem4.p2.11.m1.1.1.1.cmml">∖</mo><msub id="S2.Thmtheorem4.p2.11.m1.1.1.3" xref="S2.Thmtheorem4.p2.11.m1.1.1.3.cmml"><mi id="S2.Thmtheorem4.p2.11.m1.1.1.3.2" xref="S2.Thmtheorem4.p2.11.m1.1.1.3.2.cmml">F</mi><mi id="S2.Thmtheorem4.p2.11.m1.1.1.3.3" xref="S2.Thmtheorem4.p2.11.m1.1.1.3.3.cmml">V</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem4.p2.11.m1.1b"><apply id="S2.Thmtheorem4.p2.11.m1.1.1.cmml" xref="S2.Thmtheorem4.p2.11.m1.1.1"><setdiff id="S2.Thmtheorem4.p2.11.m1.1.1.1.cmml" xref="S2.Thmtheorem4.p2.11.m1.1.1.1"></setdiff><ci id="S2.Thmtheorem4.p2.11.m1.1.1.2.cmml" xref="S2.Thmtheorem4.p2.11.m1.1.1.2">𝐺</ci><apply id="S2.Thmtheorem4.p2.11.m1.1.1.3.cmml" xref="S2.Thmtheorem4.p2.11.m1.1.1.3"><csymbol cd="ambiguous" id="S2.Thmtheorem4.p2.11.m1.1.1.3.1.cmml" xref="S2.Thmtheorem4.p2.11.m1.1.1.3">subscript</csymbol><ci id="S2.Thmtheorem4.p2.11.m1.1.1.3.2.cmml" xref="S2.Thmtheorem4.p2.11.m1.1.1.3.2">𝐹</ci><ci id="S2.Thmtheorem4.p2.11.m1.1.1.3.3.cmml" xref="S2.Thmtheorem4.p2.11.m1.1.1.3.3">𝑉</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem4.p2.11.m1.1c">G\setminus F_{V}</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem4.p2.11.m1.1d">italic_G ∖ italic_F start_POSTSUBSCRIPT italic_V end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="H\setminus F_{V}" class="ltx_Math" display="inline" id="S2.Thmtheorem4.p2.12.m2.1"><semantics id="S2.Thmtheorem4.p2.12.m2.1a"><mrow id="S2.Thmtheorem4.p2.12.m2.1.1" xref="S2.Thmtheorem4.p2.12.m2.1.1.cmml"><mi id="S2.Thmtheorem4.p2.12.m2.1.1.2" xref="S2.Thmtheorem4.p2.12.m2.1.1.2.cmml">H</mi><mo id="S2.Thmtheorem4.p2.12.m2.1.1.1" xref="S2.Thmtheorem4.p2.12.m2.1.1.1.cmml">∖</mo><msub id="S2.Thmtheorem4.p2.12.m2.1.1.3" xref="S2.Thmtheorem4.p2.12.m2.1.1.3.cmml"><mi id="S2.Thmtheorem4.p2.12.m2.1.1.3.2" xref="S2.Thmtheorem4.p2.12.m2.1.1.3.2.cmml">F</mi><mi id="S2.Thmtheorem4.p2.12.m2.1.1.3.3" xref="S2.Thmtheorem4.p2.12.m2.1.1.3.3.cmml">V</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem4.p2.12.m2.1b"><apply id="S2.Thmtheorem4.p2.12.m2.1.1.cmml" xref="S2.Thmtheorem4.p2.12.m2.1.1"><setdiff id="S2.Thmtheorem4.p2.12.m2.1.1.1.cmml" xref="S2.Thmtheorem4.p2.12.m2.1.1.1"></setdiff><ci id="S2.Thmtheorem4.p2.12.m2.1.1.2.cmml" xref="S2.Thmtheorem4.p2.12.m2.1.1.2">𝐻</ci><apply id="S2.Thmtheorem4.p2.12.m2.1.1.3.cmml" xref="S2.Thmtheorem4.p2.12.m2.1.1.3"><csymbol cd="ambiguous" id="S2.Thmtheorem4.p2.12.m2.1.1.3.1.cmml" xref="S2.Thmtheorem4.p2.12.m2.1.1.3">subscript</csymbol><ci id="S2.Thmtheorem4.p2.12.m2.1.1.3.2.cmml" xref="S2.Thmtheorem4.p2.12.m2.1.1.3.2">𝐹</ci><ci id="S2.Thmtheorem4.p2.12.m2.1.1.3.3.cmml" xref="S2.Thmtheorem4.p2.12.m2.1.1.3.3">𝑉</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem4.p2.12.m2.1c">H\setminus F_{V}</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem4.p2.12.m2.1d">italic_H ∖ italic_F start_POSTSUBSCRIPT italic_V end_POSTSUBSCRIPT</annotation></semantics></math> denote the induced subgraphs <math alttext="G[V\setminus F_{V}]" class="ltx_Math" display="inline" id="S2.Thmtheorem4.p2.13.m3.1"><semantics id="S2.Thmtheorem4.p2.13.m3.1a"><mrow id="S2.Thmtheorem4.p2.13.m3.1.1" xref="S2.Thmtheorem4.p2.13.m3.1.1.cmml"><mi id="S2.Thmtheorem4.p2.13.m3.1.1.3" xref="S2.Thmtheorem4.p2.13.m3.1.1.3.cmml">G</mi><mo id="S2.Thmtheorem4.p2.13.m3.1.1.2" xref="S2.Thmtheorem4.p2.13.m3.1.1.2.cmml">⁢</mo><mrow id="S2.Thmtheorem4.p2.13.m3.1.1.1.1" xref="S2.Thmtheorem4.p2.13.m3.1.1.1.2.cmml"><mo id="S2.Thmtheorem4.p2.13.m3.1.1.1.1.2" stretchy="false" xref="S2.Thmtheorem4.p2.13.m3.1.1.1.2.1.cmml">[</mo><mrow id="S2.Thmtheorem4.p2.13.m3.1.1.1.1.1" xref="S2.Thmtheorem4.p2.13.m3.1.1.1.1.1.cmml"><mi id="S2.Thmtheorem4.p2.13.m3.1.1.1.1.1.2" xref="S2.Thmtheorem4.p2.13.m3.1.1.1.1.1.2.cmml">V</mi><mo id="S2.Thmtheorem4.p2.13.m3.1.1.1.1.1.1" xref="S2.Thmtheorem4.p2.13.m3.1.1.1.1.1.1.cmml">∖</mo><msub id="S2.Thmtheorem4.p2.13.m3.1.1.1.1.1.3" xref="S2.Thmtheorem4.p2.13.m3.1.1.1.1.1.3.cmml"><mi id="S2.Thmtheorem4.p2.13.m3.1.1.1.1.1.3.2" xref="S2.Thmtheorem4.p2.13.m3.1.1.1.1.1.3.2.cmml">F</mi><mi id="S2.Thmtheorem4.p2.13.m3.1.1.1.1.1.3.3" xref="S2.Thmtheorem4.p2.13.m3.1.1.1.1.1.3.3.cmml">V</mi></msub></mrow><mo id="S2.Thmtheorem4.p2.13.m3.1.1.1.1.3" stretchy="false" xref="S2.Thmtheorem4.p2.13.m3.1.1.1.2.1.cmml">]</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem4.p2.13.m3.1b"><apply id="S2.Thmtheorem4.p2.13.m3.1.1.cmml" xref="S2.Thmtheorem4.p2.13.m3.1.1"><times id="S2.Thmtheorem4.p2.13.m3.1.1.2.cmml" xref="S2.Thmtheorem4.p2.13.m3.1.1.2"></times><ci id="S2.Thmtheorem4.p2.13.m3.1.1.3.cmml" xref="S2.Thmtheorem4.p2.13.m3.1.1.3">𝐺</ci><apply id="S2.Thmtheorem4.p2.13.m3.1.1.1.2.cmml" xref="S2.Thmtheorem4.p2.13.m3.1.1.1.1"><csymbol cd="latexml" id="S2.Thmtheorem4.p2.13.m3.1.1.1.2.1.cmml" xref="S2.Thmtheorem4.p2.13.m3.1.1.1.1.2">delimited-[]</csymbol><apply id="S2.Thmtheorem4.p2.13.m3.1.1.1.1.1.cmml" xref="S2.Thmtheorem4.p2.13.m3.1.1.1.1.1"><setdiff id="S2.Thmtheorem4.p2.13.m3.1.1.1.1.1.1.cmml" xref="S2.Thmtheorem4.p2.13.m3.1.1.1.1.1.1"></setdiff><ci id="S2.Thmtheorem4.p2.13.m3.1.1.1.1.1.2.cmml" xref="S2.Thmtheorem4.p2.13.m3.1.1.1.1.1.2">𝑉</ci><apply id="S2.Thmtheorem4.p2.13.m3.1.1.1.1.1.3.cmml" xref="S2.Thmtheorem4.p2.13.m3.1.1.1.1.1.3"><csymbol cd="ambiguous" id="S2.Thmtheorem4.p2.13.m3.1.1.1.1.1.3.1.cmml" xref="S2.Thmtheorem4.p2.13.m3.1.1.1.1.1.3">subscript</csymbol><ci id="S2.Thmtheorem4.p2.13.m3.1.1.1.1.1.3.2.cmml" xref="S2.Thmtheorem4.p2.13.m3.1.1.1.1.1.3.2">𝐹</ci><ci id="S2.Thmtheorem4.p2.13.m3.1.1.1.1.1.3.3.cmml" xref="S2.Thmtheorem4.p2.13.m3.1.1.1.1.1.3.3">𝑉</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem4.p2.13.m3.1c">G[V\setminus F_{V}]</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem4.p2.13.m3.1d">italic_G [ italic_V ∖ italic_F start_POSTSUBSCRIPT italic_V end_POSTSUBSCRIPT ]</annotation></semantics></math> and <math alttext="H[V\setminus F_{V}]" class="ltx_Math" display="inline" id="S2.Thmtheorem4.p2.14.m4.1"><semantics id="S2.Thmtheorem4.p2.14.m4.1a"><mrow id="S2.Thmtheorem4.p2.14.m4.1.1" xref="S2.Thmtheorem4.p2.14.m4.1.1.cmml"><mi id="S2.Thmtheorem4.p2.14.m4.1.1.3" xref="S2.Thmtheorem4.p2.14.m4.1.1.3.cmml">H</mi><mo id="S2.Thmtheorem4.p2.14.m4.1.1.2" xref="S2.Thmtheorem4.p2.14.m4.1.1.2.cmml">⁢</mo><mrow id="S2.Thmtheorem4.p2.14.m4.1.1.1.1" xref="S2.Thmtheorem4.p2.14.m4.1.1.1.2.cmml"><mo id="S2.Thmtheorem4.p2.14.m4.1.1.1.1.2" stretchy="false" xref="S2.Thmtheorem4.p2.14.m4.1.1.1.2.1.cmml">[</mo><mrow id="S2.Thmtheorem4.p2.14.m4.1.1.1.1.1" xref="S2.Thmtheorem4.p2.14.m4.1.1.1.1.1.cmml"><mi id="S2.Thmtheorem4.p2.14.m4.1.1.1.1.1.2" xref="S2.Thmtheorem4.p2.14.m4.1.1.1.1.1.2.cmml">V</mi><mo id="S2.Thmtheorem4.p2.14.m4.1.1.1.1.1.1" xref="S2.Thmtheorem4.p2.14.m4.1.1.1.1.1.1.cmml">∖</mo><msub id="S2.Thmtheorem4.p2.14.m4.1.1.1.1.1.3" xref="S2.Thmtheorem4.p2.14.m4.1.1.1.1.1.3.cmml"><mi id="S2.Thmtheorem4.p2.14.m4.1.1.1.1.1.3.2" xref="S2.Thmtheorem4.p2.14.m4.1.1.1.1.1.3.2.cmml">F</mi><mi id="S2.Thmtheorem4.p2.14.m4.1.1.1.1.1.3.3" xref="S2.Thmtheorem4.p2.14.m4.1.1.1.1.1.3.3.cmml">V</mi></msub></mrow><mo id="S2.Thmtheorem4.p2.14.m4.1.1.1.1.3" stretchy="false" xref="S2.Thmtheorem4.p2.14.m4.1.1.1.2.1.cmml">]</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem4.p2.14.m4.1b"><apply id="S2.Thmtheorem4.p2.14.m4.1.1.cmml" xref="S2.Thmtheorem4.p2.14.m4.1.1"><times id="S2.Thmtheorem4.p2.14.m4.1.1.2.cmml" xref="S2.Thmtheorem4.p2.14.m4.1.1.2"></times><ci id="S2.Thmtheorem4.p2.14.m4.1.1.3.cmml" xref="S2.Thmtheorem4.p2.14.m4.1.1.3">𝐻</ci><apply id="S2.Thmtheorem4.p2.14.m4.1.1.1.2.cmml" xref="S2.Thmtheorem4.p2.14.m4.1.1.1.1"><csymbol cd="latexml" id="S2.Thmtheorem4.p2.14.m4.1.1.1.2.1.cmml" xref="S2.Thmtheorem4.p2.14.m4.1.1.1.1.2">delimited-[]</csymbol><apply id="S2.Thmtheorem4.p2.14.m4.1.1.1.1.1.cmml" xref="S2.Thmtheorem4.p2.14.m4.1.1.1.1.1"><setdiff id="S2.Thmtheorem4.p2.14.m4.1.1.1.1.1.1.cmml" xref="S2.Thmtheorem4.p2.14.m4.1.1.1.1.1.1"></setdiff><ci id="S2.Thmtheorem4.p2.14.m4.1.1.1.1.1.2.cmml" xref="S2.Thmtheorem4.p2.14.m4.1.1.1.1.1.2">𝑉</ci><apply id="S2.Thmtheorem4.p2.14.m4.1.1.1.1.1.3.cmml" xref="S2.Thmtheorem4.p2.14.m4.1.1.1.1.1.3"><csymbol cd="ambiguous" id="S2.Thmtheorem4.p2.14.m4.1.1.1.1.1.3.1.cmml" xref="S2.Thmtheorem4.p2.14.m4.1.1.1.1.1.3">subscript</csymbol><ci id="S2.Thmtheorem4.p2.14.m4.1.1.1.1.1.3.2.cmml" xref="S2.Thmtheorem4.p2.14.m4.1.1.1.1.1.3.2">𝐹</ci><ci id="S2.Thmtheorem4.p2.14.m4.1.1.1.1.1.3.3.cmml" xref="S2.Thmtheorem4.p2.14.m4.1.1.1.1.1.3.3">𝑉</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem4.p2.14.m4.1c">H[V\setminus F_{V}]</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem4.p2.14.m4.1d">italic_H [ italic_V ∖ italic_F start_POSTSUBSCRIPT italic_V end_POSTSUBSCRIPT ]</annotation></semantics></math>, respectively.</p> </div> </div> <div class="ltx_para" id="S2.SS1.p1"> <p class="ltx_p" id="S2.SS1.p1.4">A natural algorithm for constructing fault-tolerant spanners, which is a straightforward adaptation of the standard greedy algorithm of <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.00712v1#bib.bibx2" title="">ADD<sup class="ltx_sup"><span class="ltx_text ltx_font_italic">+</span></sup>93</a>]</cite> for spanners, is to process the edges in an arbitrary order and add an edge <math alttext="(u,v)" class="ltx_Math" display="inline" id="S2.SS1.p1.1.m1.2"><semantics id="S2.SS1.p1.1.m1.2a"><mrow id="S2.SS1.p1.1.m1.2.3.2" xref="S2.SS1.p1.1.m1.2.3.1.cmml"><mo id="S2.SS1.p1.1.m1.2.3.2.1" stretchy="false" xref="S2.SS1.p1.1.m1.2.3.1.cmml">(</mo><mi id="S2.SS1.p1.1.m1.1.1" xref="S2.SS1.p1.1.m1.1.1.cmml">u</mi><mo id="S2.SS1.p1.1.m1.2.3.2.2" xref="S2.SS1.p1.1.m1.2.3.1.cmml">,</mo><mi id="S2.SS1.p1.1.m1.2.2" xref="S2.SS1.p1.1.m1.2.2.cmml">v</mi><mo id="S2.SS1.p1.1.m1.2.3.2.3" stretchy="false" xref="S2.SS1.p1.1.m1.2.3.1.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.p1.1.m1.2b"><interval closure="open" id="S2.SS1.p1.1.m1.2.3.1.cmml" xref="S2.SS1.p1.1.m1.2.3.2"><ci id="S2.SS1.p1.1.m1.1.1.cmml" xref="S2.SS1.p1.1.m1.1.1">𝑢</ci><ci id="S2.SS1.p1.1.m1.2.2.cmml" xref="S2.SS1.p1.1.m1.2.2">𝑣</ci></interval></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p1.1.m1.2c">(u,v)</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p1.1.m1.2d">( italic_u , italic_v )</annotation></semantics></math> in the spanner if there exists a set of vertices of size at most <math alttext="f" class="ltx_Math" display="inline" id="S2.SS1.p1.2.m2.1"><semantics id="S2.SS1.p1.2.m2.1a"><mi id="S2.SS1.p1.2.m2.1.1" xref="S2.SS1.p1.2.m2.1.1.cmml">f</mi><annotation-xml encoding="MathML-Content" id="S2.SS1.p1.2.m2.1b"><ci id="S2.SS1.p1.2.m2.1.1.cmml" xref="S2.SS1.p1.2.m2.1.1">𝑓</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p1.2.m2.1c">f</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p1.2.m2.1d">italic_f</annotation></semantics></math> such that their removal increases the distance of <math alttext="u,v" class="ltx_Math" display="inline" id="S2.SS1.p1.3.m3.2"><semantics id="S2.SS1.p1.3.m3.2a"><mrow id="S2.SS1.p1.3.m3.2.3.2" xref="S2.SS1.p1.3.m3.2.3.1.cmml"><mi id="S2.SS1.p1.3.m3.1.1" xref="S2.SS1.p1.3.m3.1.1.cmml">u</mi><mo id="S2.SS1.p1.3.m3.2.3.2.1" xref="S2.SS1.p1.3.m3.2.3.1.cmml">,</mo><mi id="S2.SS1.p1.3.m3.2.2" xref="S2.SS1.p1.3.m3.2.2.cmml">v</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.p1.3.m3.2b"><list id="S2.SS1.p1.3.m3.2.3.1.cmml" xref="S2.SS1.p1.3.m3.2.3.2"><ci id="S2.SS1.p1.3.m3.1.1.cmml" xref="S2.SS1.p1.3.m3.1.1">𝑢</ci><ci id="S2.SS1.p1.3.m3.2.2.cmml" xref="S2.SS1.p1.3.m3.2.2">𝑣</ci></list></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p1.3.m3.2c">u,v</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p1.3.m3.2d">italic_u , italic_v</annotation></semantics></math> in the so-far-constructed spanner to at least <math alttext="t+1" class="ltx_Math" display="inline" id="S2.SS1.p1.4.m4.1"><semantics id="S2.SS1.p1.4.m4.1a"><mrow id="S2.SS1.p1.4.m4.1.1" xref="S2.SS1.p1.4.m4.1.1.cmml"><mi id="S2.SS1.p1.4.m4.1.1.2" xref="S2.SS1.p1.4.m4.1.1.2.cmml">t</mi><mo id="S2.SS1.p1.4.m4.1.1.1" xref="S2.SS1.p1.4.m4.1.1.1.cmml">+</mo><mn id="S2.SS1.p1.4.m4.1.1.3" xref="S2.SS1.p1.4.m4.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.p1.4.m4.1b"><apply id="S2.SS1.p1.4.m4.1.1.cmml" xref="S2.SS1.p1.4.m4.1.1"><plus id="S2.SS1.p1.4.m4.1.1.1.cmml" xref="S2.SS1.p1.4.m4.1.1.1"></plus><ci id="S2.SS1.p1.4.m4.1.1.2.cmml" xref="S2.SS1.p1.4.m4.1.1.2">𝑡</ci><cn id="S2.SS1.p1.4.m4.1.1.3.cmml" type="integer" xref="S2.SS1.p1.4.m4.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p1.4.m4.1c">t+1</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p1.4.m4.1d">italic_t + 1</annotation></semantics></math> (refer to Algorithm <a class="ltx_ref" href="https://arxiv.org/html/2503.00712v1#algorithm1" title="In 2.1 Fault-Tolerant Spanners in Streaming ‣ 2 Preliminaries ‣ Streaming Algorithms for Network Design"><span class="ltx_text ltx_ref_tag">1</span></a>).</p> </div> <figure class="ltx_float ltx_algorithm" id="algorithm1"> <div class="ltx_listing ltx_lst_numbers_left ltx_listing" id="algorithm1.11"> <div class="ltx_listingline" id="algorithm1.3.3"> <span class="ltx_text" id="algorithm1.3.3.1"><span class="ltx_text ltx_font_bold" id="algorithm1.3.3.1.1">Input:</span> </span>Unweighted graph <math alttext="G=(V,E)" class="ltx_Math" display="inline" id="algorithm1.1.1.m1.2"><semantics id="algorithm1.1.1.m1.2a"><mrow id="algorithm1.1.1.m1.2.3" xref="algorithm1.1.1.m1.2.3.cmml"><mi id="algorithm1.1.1.m1.2.3.2" xref="algorithm1.1.1.m1.2.3.2.cmml">G</mi><mo id="algorithm1.1.1.m1.2.3.1" xref="algorithm1.1.1.m1.2.3.1.cmml">=</mo><mrow id="algorithm1.1.1.m1.2.3.3.2" xref="algorithm1.1.1.m1.2.3.3.1.cmml"><mo id="algorithm1.1.1.m1.2.3.3.2.1" stretchy="false" xref="algorithm1.1.1.m1.2.3.3.1.cmml">(</mo><mi id="algorithm1.1.1.m1.1.1" xref="algorithm1.1.1.m1.1.1.cmml">V</mi><mo id="algorithm1.1.1.m1.2.3.3.2.2" xref="algorithm1.1.1.m1.2.3.3.1.cmml">,</mo><mi id="algorithm1.1.1.m1.2.2" xref="algorithm1.1.1.m1.2.2.cmml">E</mi><mo id="algorithm1.1.1.m1.2.3.3.2.3" stretchy="false" xref="algorithm1.1.1.m1.2.3.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="algorithm1.1.1.m1.2b"><apply id="algorithm1.1.1.m1.2.3.cmml" xref="algorithm1.1.1.m1.2.3"><eq id="algorithm1.1.1.m1.2.3.1.cmml" xref="algorithm1.1.1.m1.2.3.1"></eq><ci id="algorithm1.1.1.m1.2.3.2.cmml" xref="algorithm1.1.1.m1.2.3.2">𝐺</ci><interval closure="open" id="algorithm1.1.1.m1.2.3.3.1.cmml" xref="algorithm1.1.1.m1.2.3.3.2"><ci id="algorithm1.1.1.m1.1.1.cmml" xref="algorithm1.1.1.m1.1.1">𝑉</ci><ci id="algorithm1.1.1.m1.2.2.cmml" xref="algorithm1.1.1.m1.2.2">𝐸</ci></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="algorithm1.1.1.m1.2c">G=(V,E)</annotation><annotation encoding="application/x-llamapun" id="algorithm1.1.1.m1.2d">italic_G = ( italic_V , italic_E )</annotation></semantics></math>, a fault-tolerance parameter <math alttext="f" class="ltx_Math" display="inline" id="algorithm1.2.2.m2.1"><semantics id="algorithm1.2.2.m2.1a"><mi id="algorithm1.2.2.m2.1.1" xref="algorithm1.2.2.m2.1.1.cmml">f</mi><annotation-xml encoding="MathML-Content" id="algorithm1.2.2.m2.1b"><ci id="algorithm1.2.2.m2.1.1.cmml" xref="algorithm1.2.2.m2.1.1">𝑓</ci></annotation-xml><annotation encoding="application/x-tex" id="algorithm1.2.2.m2.1c">f</annotation><annotation encoding="application/x-llamapun" id="algorithm1.2.2.m2.1d">italic_f</annotation></semantics></math>, and a stretch parameter <math alttext="t" class="ltx_Math" display="inline" id="algorithm1.3.3.m3.1"><semantics id="algorithm1.3.3.m3.1a"><mi id="algorithm1.3.3.m3.1.1" xref="algorithm1.3.3.m3.1.1.cmml">t</mi><annotation-xml encoding="MathML-Content" id="algorithm1.3.3.m3.1b"><ci id="algorithm1.3.3.m3.1.1.cmml" xref="algorithm1.3.3.m3.1.1">𝑡</ci></annotation-xml><annotation encoding="application/x-tex" id="algorithm1.3.3.m3.1c">t</annotation><annotation encoding="application/x-llamapun" id="algorithm1.3.3.m3.1d">italic_t</annotation></semantics></math>. </div> <div class="ltx_listingline" id="algorithm1.4.4"> <span class="ltx_text ltx_font_bold" id="algorithm1.4.4.1">initialize</span> <math alttext="H\leftarrow(V,\emptyset)" class="ltx_Math" display="inline" id="algorithm1.4.4.m1.2"><semantics id="algorithm1.4.4.m1.2a"><mrow id="algorithm1.4.4.m1.2.3" xref="algorithm1.4.4.m1.2.3.cmml"><mi id="algorithm1.4.4.m1.2.3.2" xref="algorithm1.4.4.m1.2.3.2.cmml">H</mi><mo id="algorithm1.4.4.m1.2.3.1" stretchy="false" xref="algorithm1.4.4.m1.2.3.1.cmml">←</mo><mrow id="algorithm1.4.4.m1.2.3.3.2" xref="algorithm1.4.4.m1.2.3.3.1.cmml"><mo id="algorithm1.4.4.m1.2.3.3.2.1" stretchy="false" xref="algorithm1.4.4.m1.2.3.3.1.cmml">(</mo><mi id="algorithm1.4.4.m1.1.1" xref="algorithm1.4.4.m1.1.1.cmml">V</mi><mo id="algorithm1.4.4.m1.2.3.3.2.2" xref="algorithm1.4.4.m1.2.3.3.1.cmml">,</mo><mi id="algorithm1.4.4.m1.2.2" mathvariant="normal" xref="algorithm1.4.4.m1.2.2.cmml">∅</mi><mo id="algorithm1.4.4.m1.2.3.3.2.3" stretchy="false" xref="algorithm1.4.4.m1.2.3.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="algorithm1.4.4.m1.2b"><apply id="algorithm1.4.4.m1.2.3.cmml" xref="algorithm1.4.4.m1.2.3"><ci id="algorithm1.4.4.m1.2.3.1.cmml" xref="algorithm1.4.4.m1.2.3.1">←</ci><ci id="algorithm1.4.4.m1.2.3.2.cmml" xref="algorithm1.4.4.m1.2.3.2">𝐻</ci><interval closure="open" id="algorithm1.4.4.m1.2.3.3.1.cmml" xref="algorithm1.4.4.m1.2.3.3.2"><ci id="algorithm1.4.4.m1.1.1.cmml" xref="algorithm1.4.4.m1.1.1">𝑉</ci><emptyset id="algorithm1.4.4.m1.2.2.cmml" xref="algorithm1.4.4.m1.2.2"></emptyset></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="algorithm1.4.4.m1.2c">H\leftarrow(V,\emptyset)</annotation><annotation encoding="application/x-llamapun" id="algorithm1.4.4.m1.2d">italic_H ← ( italic_V , ∅ )</annotation></semantics></math> </div> <div class="ltx_listingline" id="algorithm1.5.5"> <span class="ltx_text ltx_font_bold" id="algorithm1.5.5.2">for</span> <em class="ltx_emph ltx_font_italic" id="algorithm1.5.5.1"><math alttext="(u,v)\in E" class="ltx_Math" display="inline" id="algorithm1.5.5.1.m1.2"><semantics id="algorithm1.5.5.1.m1.2a"><mrow id="algorithm1.5.5.1.m1.2.3" xref="algorithm1.5.5.1.m1.2.3.cmml"><mrow id="algorithm1.5.5.1.m1.2.3.2.2" xref="algorithm1.5.5.1.m1.2.3.2.1.cmml"><mo id="algorithm1.5.5.1.m1.2.3.2.2.1" stretchy="false" xref="algorithm1.5.5.1.m1.2.3.2.1.cmml">(</mo><mi id="algorithm1.5.5.1.m1.1.1" xref="algorithm1.5.5.1.m1.1.1.cmml">u</mi><mo id="algorithm1.5.5.1.m1.2.3.2.2.2" xref="algorithm1.5.5.1.m1.2.3.2.1.cmml">,</mo><mi id="algorithm1.5.5.1.m1.2.2" xref="algorithm1.5.5.1.m1.2.2.cmml">v</mi><mo id="algorithm1.5.5.1.m1.2.3.2.2.3" stretchy="false" xref="algorithm1.5.5.1.m1.2.3.2.1.cmml">)</mo></mrow><mo id="algorithm1.5.5.1.m1.2.3.1" xref="algorithm1.5.5.1.m1.2.3.1.cmml">∈</mo><mi id="algorithm1.5.5.1.m1.2.3.3" xref="algorithm1.5.5.1.m1.2.3.3.cmml">E</mi></mrow><annotation-xml encoding="MathML-Content" id="algorithm1.5.5.1.m1.2b"><apply id="algorithm1.5.5.1.m1.2.3.cmml" xref="algorithm1.5.5.1.m1.2.3"><in id="algorithm1.5.5.1.m1.2.3.1.cmml" xref="algorithm1.5.5.1.m1.2.3.1"></in><interval closure="open" id="algorithm1.5.5.1.m1.2.3.2.1.cmml" xref="algorithm1.5.5.1.m1.2.3.2.2"><ci id="algorithm1.5.5.1.m1.1.1.cmml" xref="algorithm1.5.5.1.m1.1.1">𝑢</ci><ci id="algorithm1.5.5.1.m1.2.2.cmml" xref="algorithm1.5.5.1.m1.2.2">𝑣</ci></interval><ci id="algorithm1.5.5.1.m1.2.3.3.cmml" xref="algorithm1.5.5.1.m1.2.3.3">𝐸</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="algorithm1.5.5.1.m1.2c">(u,v)\in E</annotation><annotation encoding="application/x-llamapun" id="algorithm1.5.5.1.m1.2d">( italic_u , italic_v ) ∈ italic_E</annotation></semantics></math> in an arbitrary order</em> <span class="ltx_text ltx_font_bold" id="algorithm1.5.5.3">do</span> </div> <div class="ltx_listingline" id="algorithm1.8.8"> <span class="ltx_rule" style="width:1px;height:100%;background:black;display:inline-block;"> </span> <span class="ltx_text ltx_font_bold" id="algorithm1.8.8.4">if</span> <em class="ltx_emph ltx_font_italic" id="algorithm1.8.8.3">there exists a set of vertices (or edges) <math alttext="F" class="ltx_Math" display="inline" id="algorithm1.6.6.1.m1.1"><semantics id="algorithm1.6.6.1.m1.1a"><mi id="algorithm1.6.6.1.m1.1.1" xref="algorithm1.6.6.1.m1.1.1.cmml">F</mi><annotation-xml encoding="MathML-Content" id="algorithm1.6.6.1.m1.1b"><ci id="algorithm1.6.6.1.m1.1.1.cmml" xref="algorithm1.6.6.1.m1.1.1">𝐹</ci></annotation-xml><annotation encoding="application/x-tex" id="algorithm1.6.6.1.m1.1c">F</annotation><annotation encoding="application/x-llamapun" id="algorithm1.6.6.1.m1.1d">italic_F</annotation></semantics></math> of size <math alttext="f" class="ltx_Math" display="inline" id="algorithm1.7.7.2.m2.1"><semantics id="algorithm1.7.7.2.m2.1a"><mi id="algorithm1.7.7.2.m2.1.1" xref="algorithm1.7.7.2.m2.1.1.cmml">f</mi><annotation-xml encoding="MathML-Content" id="algorithm1.7.7.2.m2.1b"><ci id="algorithm1.7.7.2.m2.1.1.cmml" xref="algorithm1.7.7.2.m2.1.1">𝑓</ci></annotation-xml><annotation encoding="application/x-tex" id="algorithm1.7.7.2.m2.1c">f</annotation><annotation encoding="application/x-llamapun" id="algorithm1.7.7.2.m2.1d">italic_f</annotation></semantics></math> such that <math alttext="d_{H\setminus F}(u,v)>t" class="ltx_Math" display="inline" id="algorithm1.8.8.3.m3.2"><semantics id="algorithm1.8.8.3.m3.2a"><mrow id="algorithm1.8.8.3.m3.2.3" xref="algorithm1.8.8.3.m3.2.3.cmml"><mrow id="algorithm1.8.8.3.m3.2.3.2" xref="algorithm1.8.8.3.m3.2.3.2.cmml"><msub id="algorithm1.8.8.3.m3.2.3.2.2" xref="algorithm1.8.8.3.m3.2.3.2.2.cmml"><mi id="algorithm1.8.8.3.m3.2.3.2.2.2" xref="algorithm1.8.8.3.m3.2.3.2.2.2.cmml">d</mi><mrow id="algorithm1.8.8.3.m3.2.3.2.2.3" xref="algorithm1.8.8.3.m3.2.3.2.2.3.cmml"><mi id="algorithm1.8.8.3.m3.2.3.2.2.3.2" xref="algorithm1.8.8.3.m3.2.3.2.2.3.2.cmml">H</mi><mo id="algorithm1.8.8.3.m3.2.3.2.2.3.1" xref="algorithm1.8.8.3.m3.2.3.2.2.3.1.cmml">∖</mo><mi id="algorithm1.8.8.3.m3.2.3.2.2.3.3" xref="algorithm1.8.8.3.m3.2.3.2.2.3.3.cmml">F</mi></mrow></msub><mo id="algorithm1.8.8.3.m3.2.3.2.1" xref="algorithm1.8.8.3.m3.2.3.2.1.cmml">⁢</mo><mrow id="algorithm1.8.8.3.m3.2.3.2.3.2" xref="algorithm1.8.8.3.m3.2.3.2.3.1.cmml"><mo id="algorithm1.8.8.3.m3.2.3.2.3.2.1" stretchy="false" xref="algorithm1.8.8.3.m3.2.3.2.3.1.cmml">(</mo><mi id="algorithm1.8.8.3.m3.1.1" xref="algorithm1.8.8.3.m3.1.1.cmml">u</mi><mo id="algorithm1.8.8.3.m3.2.3.2.3.2.2" xref="algorithm1.8.8.3.m3.2.3.2.3.1.cmml">,</mo><mi id="algorithm1.8.8.3.m3.2.2" xref="algorithm1.8.8.3.m3.2.2.cmml">v</mi><mo id="algorithm1.8.8.3.m3.2.3.2.3.2.3" stretchy="false" xref="algorithm1.8.8.3.m3.2.3.2.3.1.cmml">)</mo></mrow></mrow><mo id="algorithm1.8.8.3.m3.2.3.1" xref="algorithm1.8.8.3.m3.2.3.1.cmml">></mo><mi id="algorithm1.8.8.3.m3.2.3.3" xref="algorithm1.8.8.3.m3.2.3.3.cmml">t</mi></mrow><annotation-xml encoding="MathML-Content" id="algorithm1.8.8.3.m3.2b"><apply id="algorithm1.8.8.3.m3.2.3.cmml" xref="algorithm1.8.8.3.m3.2.3"><gt id="algorithm1.8.8.3.m3.2.3.1.cmml" xref="algorithm1.8.8.3.m3.2.3.1"></gt><apply id="algorithm1.8.8.3.m3.2.3.2.cmml" xref="algorithm1.8.8.3.m3.2.3.2"><times id="algorithm1.8.8.3.m3.2.3.2.1.cmml" xref="algorithm1.8.8.3.m3.2.3.2.1"></times><apply id="algorithm1.8.8.3.m3.2.3.2.2.cmml" xref="algorithm1.8.8.3.m3.2.3.2.2"><csymbol cd="ambiguous" id="algorithm1.8.8.3.m3.2.3.2.2.1.cmml" xref="algorithm1.8.8.3.m3.2.3.2.2">subscript</csymbol><ci id="algorithm1.8.8.3.m3.2.3.2.2.2.cmml" xref="algorithm1.8.8.3.m3.2.3.2.2.2">𝑑</ci><apply id="algorithm1.8.8.3.m3.2.3.2.2.3.cmml" xref="algorithm1.8.8.3.m3.2.3.2.2.3"><setdiff id="algorithm1.8.8.3.m3.2.3.2.2.3.1.cmml" xref="algorithm1.8.8.3.m3.2.3.2.2.3.1"></setdiff><ci id="algorithm1.8.8.3.m3.2.3.2.2.3.2.cmml" xref="algorithm1.8.8.3.m3.2.3.2.2.3.2">𝐻</ci><ci id="algorithm1.8.8.3.m3.2.3.2.2.3.3.cmml" xref="algorithm1.8.8.3.m3.2.3.2.2.3.3">𝐹</ci></apply></apply><interval closure="open" id="algorithm1.8.8.3.m3.2.3.2.3.1.cmml" xref="algorithm1.8.8.3.m3.2.3.2.3.2"><ci id="algorithm1.8.8.3.m3.1.1.cmml" xref="algorithm1.8.8.3.m3.1.1">𝑢</ci><ci id="algorithm1.8.8.3.m3.2.2.cmml" xref="algorithm1.8.8.3.m3.2.2">𝑣</ci></interval></apply><ci id="algorithm1.8.8.3.m3.2.3.3.cmml" xref="algorithm1.8.8.3.m3.2.3.3">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="algorithm1.8.8.3.m3.2c">d_{H\setminus F}(u,v)>t</annotation><annotation encoding="application/x-llamapun" id="algorithm1.8.8.3.m3.2d">italic_d start_POSTSUBSCRIPT italic_H ∖ italic_F end_POSTSUBSCRIPT ( italic_u , italic_v ) > italic_t</annotation></semantics></math></em> <span class="ltx_text ltx_font_bold" id="algorithm1.8.8.5">then</span> </div> <div class="ltx_listingline" id="algorithm1.10.10"> <span class="ltx_rule" style="width:1px;height:100%;background:black;display:inline-block;"> </span> <span class="ltx_rule" style="width:1px;height:100%;background:black;display:inline-block;"> </span> <span class="ltx_text ltx_font_bold" id="algorithm1.10.10.1">add</span> <math alttext="(u,v)" class="ltx_Math" display="inline" id="algorithm1.9.9.m1.2"><semantics id="algorithm1.9.9.m1.2a"><mrow id="algorithm1.9.9.m1.2.3.2" xref="algorithm1.9.9.m1.2.3.1.cmml"><mo id="algorithm1.9.9.m1.2.3.2.1" stretchy="false" xref="algorithm1.9.9.m1.2.3.1.cmml">(</mo><mi id="algorithm1.9.9.m1.1.1" xref="algorithm1.9.9.m1.1.1.cmml">u</mi><mo id="algorithm1.9.9.m1.2.3.2.2" xref="algorithm1.9.9.m1.2.3.1.cmml">,</mo><mi id="algorithm1.9.9.m1.2.2" xref="algorithm1.9.9.m1.2.2.cmml">v</mi><mo id="algorithm1.9.9.m1.2.3.2.3" stretchy="false" xref="algorithm1.9.9.m1.2.3.1.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="algorithm1.9.9.m1.2b"><interval closure="open" id="algorithm1.9.9.m1.2.3.1.cmml" xref="algorithm1.9.9.m1.2.3.2"><ci id="algorithm1.9.9.m1.1.1.cmml" xref="algorithm1.9.9.m1.1.1">𝑢</ci><ci id="algorithm1.9.9.m1.2.2.cmml" xref="algorithm1.9.9.m1.2.2">𝑣</ci></interval></annotation-xml><annotation encoding="application/x-tex" id="algorithm1.9.9.m1.2c">(u,v)</annotation><annotation encoding="application/x-llamapun" id="algorithm1.9.9.m1.2d">( italic_u , italic_v )</annotation></semantics></math> to <math alttext="H" class="ltx_Math" display="inline" id="algorithm1.10.10.m2.1"><semantics id="algorithm1.10.10.m2.1a"><mi id="algorithm1.10.10.m2.1.1" xref="algorithm1.10.10.m2.1.1.cmml">H</mi><annotation-xml encoding="MathML-Content" id="algorithm1.10.10.m2.1b"><ci id="algorithm1.10.10.m2.1.1.cmml" xref="algorithm1.10.10.m2.1.1">𝐻</ci></annotation-xml><annotation encoding="application/x-tex" id="algorithm1.10.10.m2.1c">H</annotation><annotation encoding="application/x-llamapun" id="algorithm1.10.10.m2.1d">italic_H</annotation></semantics></math> </div> <div class="ltx_listingline" id="algorithm1.11.11"> <span class="ltx_text ltx_font_bold" id="algorithm1.11.11.1">return</span> <math alttext="H" class="ltx_Math" display="inline" id="algorithm1.11.11.m1.1"><semantics id="algorithm1.11.11.m1.1a"><mi id="algorithm1.11.11.m1.1.1" xref="algorithm1.11.11.m1.1.1.cmml">H</mi><annotation-xml encoding="MathML-Content" id="algorithm1.11.11.m1.1b"><ci id="algorithm1.11.11.m1.1.1.cmml" xref="algorithm1.11.11.m1.1.1">𝐻</ci></annotation-xml><annotation encoding="application/x-tex" id="algorithm1.11.11.m1.1c">H</annotation><annotation encoding="application/x-llamapun" id="algorithm1.11.11.m1.1d">italic_H</annotation></semantics></math> </div> </div> <figcaption class="ltx_caption"><span class="ltx_tag ltx_tag_float"><span class="ltx_text ltx_font_bold" id="algorithm1.13.1.1">Algorithm 1</span> </span>The greedy algorithm for unweighted VFT (EFT) spanners.</figcaption> </figure> <div class="ltx_theorem ltx_theorem_theorem" id="S2.Thmtheorem5"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="S2.Thmtheorem5.1.1.1">Theorem 2.5</span></span><span class="ltx_text ltx_font_bold" id="S2.Thmtheorem5.2.2"> </span>(<cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.00712v1#bib.bibx21" title="">BP19</a>]</cite>)<span class="ltx_text ltx_font_bold" id="S2.Thmtheorem5.3.3">.</span> </h6> <div class="ltx_para" id="S2.Thmtheorem5.p1"> <p class="ltx_p" id="S2.Thmtheorem5.p1.7">For any <math alttext="n" class="ltx_Math" display="inline" id="S2.Thmtheorem5.p1.1.m1.1"><semantics id="S2.Thmtheorem5.p1.1.m1.1a"><mi id="S2.Thmtheorem5.p1.1.m1.1.1" xref="S2.Thmtheorem5.p1.1.m1.1.1.cmml">n</mi><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem5.p1.1.m1.1b"><ci id="S2.Thmtheorem5.p1.1.m1.1.1.cmml" xref="S2.Thmtheorem5.p1.1.m1.1.1">𝑛</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem5.p1.1.m1.1c">n</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem5.p1.1.m1.1d">italic_n</annotation></semantics></math>-vertex graph <math alttext="G" class="ltx_Math" display="inline" id="S2.Thmtheorem5.p1.2.m2.1"><semantics id="S2.Thmtheorem5.p1.2.m2.1a"><mi id="S2.Thmtheorem5.p1.2.m2.1.1" xref="S2.Thmtheorem5.p1.2.m2.1.1.cmml">G</mi><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem5.p1.2.m2.1b"><ci id="S2.Thmtheorem5.p1.2.m2.1.1.cmml" xref="S2.Thmtheorem5.p1.2.m2.1.1">𝐺</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem5.p1.2.m2.1c">G</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem5.p1.2.m2.1d">italic_G</annotation></semantics></math>, the greedy algorithm for the <math alttext="f" class="ltx_Math" display="inline" id="S2.Thmtheorem5.p1.3.m3.1"><semantics id="S2.Thmtheorem5.p1.3.m3.1a"><mi id="S2.Thmtheorem5.p1.3.m3.1.1" xref="S2.Thmtheorem5.p1.3.m3.1.1.cmml">f</mi><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem5.p1.3.m3.1b"><ci id="S2.Thmtheorem5.p1.3.m3.1.1.cmml" xref="S2.Thmtheorem5.p1.3.m3.1.1">𝑓</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem5.p1.3.m3.1c">f</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem5.p1.3.m3.1d">italic_f</annotation></semantics></math>-VFT (or <math alttext="f" class="ltx_Math" display="inline" id="S2.Thmtheorem5.p1.4.m4.1"><semantics id="S2.Thmtheorem5.p1.4.m4.1a"><mi id="S2.Thmtheorem5.p1.4.m4.1.1" xref="S2.Thmtheorem5.p1.4.m4.1.1.cmml">f</mi><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem5.p1.4.m4.1b"><ci id="S2.Thmtheorem5.p1.4.m4.1.1.cmml" xref="S2.Thmtheorem5.p1.4.m4.1.1">𝑓</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem5.p1.4.m4.1c">f</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem5.p1.4.m4.1d">italic_f</annotation></semantics></math>-EFT) <math alttext="(2t-1)" class="ltx_Math" display="inline" id="S2.Thmtheorem5.p1.5.m5.1"><semantics id="S2.Thmtheorem5.p1.5.m5.1a"><mrow id="S2.Thmtheorem5.p1.5.m5.1.1.1" xref="S2.Thmtheorem5.p1.5.m5.1.1.1.1.cmml"><mo id="S2.Thmtheorem5.p1.5.m5.1.1.1.2" stretchy="false" xref="S2.Thmtheorem5.p1.5.m5.1.1.1.1.cmml">(</mo><mrow id="S2.Thmtheorem5.p1.5.m5.1.1.1.1" xref="S2.Thmtheorem5.p1.5.m5.1.1.1.1.cmml"><mrow id="S2.Thmtheorem5.p1.5.m5.1.1.1.1.2" xref="S2.Thmtheorem5.p1.5.m5.1.1.1.1.2.cmml"><mn id="S2.Thmtheorem5.p1.5.m5.1.1.1.1.2.2" xref="S2.Thmtheorem5.p1.5.m5.1.1.1.1.2.2.cmml">2</mn><mo id="S2.Thmtheorem5.p1.5.m5.1.1.1.1.2.1" xref="S2.Thmtheorem5.p1.5.m5.1.1.1.1.2.1.cmml">⁢</mo><mi id="S2.Thmtheorem5.p1.5.m5.1.1.1.1.2.3" xref="S2.Thmtheorem5.p1.5.m5.1.1.1.1.2.3.cmml">t</mi></mrow><mo id="S2.Thmtheorem5.p1.5.m5.1.1.1.1.1" xref="S2.Thmtheorem5.p1.5.m5.1.1.1.1.1.cmml">−</mo><mn id="S2.Thmtheorem5.p1.5.m5.1.1.1.1.3" xref="S2.Thmtheorem5.p1.5.m5.1.1.1.1.3.cmml">1</mn></mrow><mo id="S2.Thmtheorem5.p1.5.m5.1.1.1.3" stretchy="false" xref="S2.Thmtheorem5.p1.5.m5.1.1.1.1.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem5.p1.5.m5.1b"><apply id="S2.Thmtheorem5.p1.5.m5.1.1.1.1.cmml" xref="S2.Thmtheorem5.p1.5.m5.1.1.1"><minus id="S2.Thmtheorem5.p1.5.m5.1.1.1.1.1.cmml" xref="S2.Thmtheorem5.p1.5.m5.1.1.1.1.1"></minus><apply id="S2.Thmtheorem5.p1.5.m5.1.1.1.1.2.cmml" xref="S2.Thmtheorem5.p1.5.m5.1.1.1.1.2"><times id="S2.Thmtheorem5.p1.5.m5.1.1.1.1.2.1.cmml" xref="S2.Thmtheorem5.p1.5.m5.1.1.1.1.2.1"></times><cn id="S2.Thmtheorem5.p1.5.m5.1.1.1.1.2.2.cmml" type="integer" xref="S2.Thmtheorem5.p1.5.m5.1.1.1.1.2.2">2</cn><ci id="S2.Thmtheorem5.p1.5.m5.1.1.1.1.2.3.cmml" xref="S2.Thmtheorem5.p1.5.m5.1.1.1.1.2.3">𝑡</ci></apply><cn id="S2.Thmtheorem5.p1.5.m5.1.1.1.1.3.cmml" type="integer" xref="S2.Thmtheorem5.p1.5.m5.1.1.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem5.p1.5.m5.1c">(2t-1)</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem5.p1.5.m5.1d">( 2 italic_t - 1 )</annotation></semantics></math>-spanner (i.e., Algorithm <a class="ltx_ref" href="https://arxiv.org/html/2503.00712v1#algorithm1" title="In 2.1 Fault-Tolerant Spanners in Streaming ‣ 2 Preliminaries ‣ Streaming Algorithms for Network Design"><span class="ltx_text ltx_ref_tag">1</span></a>) returns a feasible subgraph <math alttext="H" class="ltx_Math" display="inline" id="S2.Thmtheorem5.p1.6.m6.1"><semantics id="S2.Thmtheorem5.p1.6.m6.1a"><mi id="S2.Thmtheorem5.p1.6.m6.1.1" xref="S2.Thmtheorem5.p1.6.m6.1.1.cmml">H</mi><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem5.p1.6.m6.1b"><ci id="S2.Thmtheorem5.p1.6.m6.1.1.cmml" xref="S2.Thmtheorem5.p1.6.m6.1.1">𝐻</ci></annotation-xml><annotation encoding="application/x-tex" 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xref="S2.Thmtheorem5.p1.7.m7.3.3.2.1.1.1.3">superscript</csymbol><ci id="S2.Thmtheorem5.p1.7.m7.3.3.2.1.1.1.3.2.cmml" xref="S2.Thmtheorem5.p1.7.m7.3.3.2.1.1.1.3.2">𝑛</ci><apply id="S2.Thmtheorem5.p1.7.m7.3.3.2.1.1.1.3.3.cmml" xref="S2.Thmtheorem5.p1.7.m7.3.3.2.1.1.1.3.3"><plus id="S2.Thmtheorem5.p1.7.m7.3.3.2.1.1.1.3.3.1.cmml" xref="S2.Thmtheorem5.p1.7.m7.3.3.2.1.1.1.3.3.1"></plus><cn id="S2.Thmtheorem5.p1.7.m7.3.3.2.1.1.1.3.3.2.cmml" type="integer" xref="S2.Thmtheorem5.p1.7.m7.3.3.2.1.1.1.3.3.2">1</cn><apply id="S2.Thmtheorem5.p1.7.m7.3.3.2.1.1.1.3.3.3.cmml" xref="S2.Thmtheorem5.p1.7.m7.3.3.2.1.1.1.3.3.3"><divide id="S2.Thmtheorem5.p1.7.m7.3.3.2.1.1.1.3.3.3.1.cmml" xref="S2.Thmtheorem5.p1.7.m7.3.3.2.1.1.1.3.3.3.1"></divide><cn id="S2.Thmtheorem5.p1.7.m7.3.3.2.1.1.1.3.3.3.2.cmml" type="integer" xref="S2.Thmtheorem5.p1.7.m7.3.3.2.1.1.1.3.3.3.2">1</cn><ci id="S2.Thmtheorem5.p1.7.m7.3.3.2.1.1.1.3.3.3.3.cmml" xref="S2.Thmtheorem5.p1.7.m7.3.3.2.1.1.1.3.3.3.3">𝑡</ci></apply></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem5.p1.7.m7.3c">|E(H)|=O(f^{1-1/t}\cdot n^{1+1/t})</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem5.p1.7.m7.3d">| italic_E ( italic_H ) | = italic_O ( italic_f start_POSTSUPERSCRIPT 1 - 1 / italic_t end_POSTSUPERSCRIPT ⋅ italic_n start_POSTSUPERSCRIPT 1 + 1 / italic_t end_POSTSUPERSCRIPT )</annotation></semantics></math>.</p> </div> </div> <div class="ltx_para" id="S2.SS1.p2"> <p class="ltx_p" id="S2.SS1.p2.1">It is straightforward to show that for unweighted graphs, the greedy algorithm for spanners can be implemented in insertion-only streams with a space complexity equal to the size of <math alttext="H" class="ltx_Math" display="inline" id="S2.SS1.p2.1.m1.1"><semantics id="S2.SS1.p2.1.m1.1a"><mi id="S2.SS1.p2.1.m1.1.1" xref="S2.SS1.p2.1.m1.1.1.cmml">H</mi><annotation-xml 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A standard technique to address this issue is <span class="ltx_text ltx_font_italic" id="S2.SS1.SSS0.Px1.p1.9.1">bucketing</span>. Given a weighted graph <math alttext="G=(V,E)" class="ltx_Math" display="inline" id="S2.SS1.SSS0.Px1.p1.1.m1.2"><semantics id="S2.SS1.SSS0.Px1.p1.1.m1.2a"><mrow id="S2.SS1.SSS0.Px1.p1.1.m1.2.3" xref="S2.SS1.SSS0.Px1.p1.1.m1.2.3.cmml"><mi id="S2.SS1.SSS0.Px1.p1.1.m1.2.3.2" xref="S2.SS1.SSS0.Px1.p1.1.m1.2.3.2.cmml">G</mi><mo id="S2.SS1.SSS0.Px1.p1.1.m1.2.3.1" xref="S2.SS1.SSS0.Px1.p1.1.m1.2.3.1.cmml">=</mo><mrow id="S2.SS1.SSS0.Px1.p1.1.m1.2.3.3.2" xref="S2.SS1.SSS0.Px1.p1.1.m1.2.3.3.1.cmml"><mo id="S2.SS1.SSS0.Px1.p1.1.m1.2.3.3.2.1" stretchy="false" xref="S2.SS1.SSS0.Px1.p1.1.m1.2.3.3.1.cmml">(</mo><mi id="S2.SS1.SSS0.Px1.p1.1.m1.1.1" xref="S2.SS1.SSS0.Px1.p1.1.m1.1.1.cmml">V</mi><mo id="S2.SS1.SSS0.Px1.p1.1.m1.2.3.3.2.2" xref="S2.SS1.SSS0.Px1.p1.1.m1.2.3.3.1.cmml">,</mo><mi id="S2.SS1.SSS0.Px1.p1.1.m1.2.2" xref="S2.SS1.SSS0.Px1.p1.1.m1.2.2.cmml">E</mi><mo id="S2.SS1.SSS0.Px1.p1.1.m1.2.3.3.2.3" stretchy="false" xref="S2.SS1.SSS0.Px1.p1.1.m1.2.3.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.SSS0.Px1.p1.1.m1.2b"><apply id="S2.SS1.SSS0.Px1.p1.1.m1.2.3.cmml" xref="S2.SS1.SSS0.Px1.p1.1.m1.2.3"><eq id="S2.SS1.SSS0.Px1.p1.1.m1.2.3.1.cmml" xref="S2.SS1.SSS0.Px1.p1.1.m1.2.3.1"></eq><ci id="S2.SS1.SSS0.Px1.p1.1.m1.2.3.2.cmml" xref="S2.SS1.SSS0.Px1.p1.1.m1.2.3.2">𝐺</ci><interval closure="open" id="S2.SS1.SSS0.Px1.p1.1.m1.2.3.3.1.cmml" xref="S2.SS1.SSS0.Px1.p1.1.m1.2.3.3.2"><ci id="S2.SS1.SSS0.Px1.p1.1.m1.1.1.cmml" xref="S2.SS1.SSS0.Px1.p1.1.m1.1.1">𝑉</ci><ci id="S2.SS1.SSS0.Px1.p1.1.m1.2.2.cmml" xref="S2.SS1.SSS0.Px1.p1.1.m1.2.2">𝐸</ci></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.SSS0.Px1.p1.1.m1.2c">G=(V,E)</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.SSS0.Px1.p1.1.m1.2d">italic_G = ( italic_V , italic_E )</annotation></semantics></math> with weight function <math alttext="w:E\rightarrow\{0,1,\cdots,W\}" class="ltx_Math" display="inline" id="S2.SS1.SSS0.Px1.p1.2.m2.4"><semantics id="S2.SS1.SSS0.Px1.p1.2.m2.4a"><mrow id="S2.SS1.SSS0.Px1.p1.2.m2.4.5" xref="S2.SS1.SSS0.Px1.p1.2.m2.4.5.cmml"><mi id="S2.SS1.SSS0.Px1.p1.2.m2.4.5.2" xref="S2.SS1.SSS0.Px1.p1.2.m2.4.5.2.cmml">w</mi><mo id="S2.SS1.SSS0.Px1.p1.2.m2.4.5.1" lspace="0.278em" rspace="0.278em" xref="S2.SS1.SSS0.Px1.p1.2.m2.4.5.1.cmml">:</mo><mrow id="S2.SS1.SSS0.Px1.p1.2.m2.4.5.3" xref="S2.SS1.SSS0.Px1.p1.2.m2.4.5.3.cmml"><mi id="S2.SS1.SSS0.Px1.p1.2.m2.4.5.3.2" xref="S2.SS1.SSS0.Px1.p1.2.m2.4.5.3.2.cmml">E</mi><mo id="S2.SS1.SSS0.Px1.p1.2.m2.4.5.3.1" stretchy="false" xref="S2.SS1.SSS0.Px1.p1.2.m2.4.5.3.1.cmml">→</mo><mrow id="S2.SS1.SSS0.Px1.p1.2.m2.4.5.3.3.2" xref="S2.SS1.SSS0.Px1.p1.2.m2.4.5.3.3.1.cmml"><mo id="S2.SS1.SSS0.Px1.p1.2.m2.4.5.3.3.2.1" stretchy="false" xref="S2.SS1.SSS0.Px1.p1.2.m2.4.5.3.3.1.cmml">{</mo><mn id="S2.SS1.SSS0.Px1.p1.2.m2.1.1" xref="S2.SS1.SSS0.Px1.p1.2.m2.1.1.cmml">0</mn><mo id="S2.SS1.SSS0.Px1.p1.2.m2.4.5.3.3.2.2" xref="S2.SS1.SSS0.Px1.p1.2.m2.4.5.3.3.1.cmml">,</mo><mn id="S2.SS1.SSS0.Px1.p1.2.m2.2.2" xref="S2.SS1.SSS0.Px1.p1.2.m2.2.2.cmml">1</mn><mo id="S2.SS1.SSS0.Px1.p1.2.m2.4.5.3.3.2.3" xref="S2.SS1.SSS0.Px1.p1.2.m2.4.5.3.3.1.cmml">,</mo><mi id="S2.SS1.SSS0.Px1.p1.2.m2.3.3" mathvariant="normal" xref="S2.SS1.SSS0.Px1.p1.2.m2.3.3.cmml">⋯</mi><mo id="S2.SS1.SSS0.Px1.p1.2.m2.4.5.3.3.2.4" xref="S2.SS1.SSS0.Px1.p1.2.m2.4.5.3.3.1.cmml">,</mo><mi id="S2.SS1.SSS0.Px1.p1.2.m2.4.4" xref="S2.SS1.SSS0.Px1.p1.2.m2.4.4.cmml">W</mi><mo id="S2.SS1.SSS0.Px1.p1.2.m2.4.5.3.3.2.5" stretchy="false" xref="S2.SS1.SSS0.Px1.p1.2.m2.4.5.3.3.1.cmml">}</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.SSS0.Px1.p1.2.m2.4b"><apply id="S2.SS1.SSS0.Px1.p1.2.m2.4.5.cmml" xref="S2.SS1.SSS0.Px1.p1.2.m2.4.5"><ci id="S2.SS1.SSS0.Px1.p1.2.m2.4.5.1.cmml" xref="S2.SS1.SSS0.Px1.p1.2.m2.4.5.1">:</ci><ci id="S2.SS1.SSS0.Px1.p1.2.m2.4.5.2.cmml" xref="S2.SS1.SSS0.Px1.p1.2.m2.4.5.2">𝑤</ci><apply id="S2.SS1.SSS0.Px1.p1.2.m2.4.5.3.cmml" xref="S2.SS1.SSS0.Px1.p1.2.m2.4.5.3"><ci id="S2.SS1.SSS0.Px1.p1.2.m2.4.5.3.1.cmml" xref="S2.SS1.SSS0.Px1.p1.2.m2.4.5.3.1">→</ci><ci id="S2.SS1.SSS0.Px1.p1.2.m2.4.5.3.2.cmml" xref="S2.SS1.SSS0.Px1.p1.2.m2.4.5.3.2">𝐸</ci><set id="S2.SS1.SSS0.Px1.p1.2.m2.4.5.3.3.1.cmml" xref="S2.SS1.SSS0.Px1.p1.2.m2.4.5.3.3.2"><cn id="S2.SS1.SSS0.Px1.p1.2.m2.1.1.cmml" type="integer" xref="S2.SS1.SSS0.Px1.p1.2.m2.1.1">0</cn><cn id="S2.SS1.SSS0.Px1.p1.2.m2.2.2.cmml" type="integer" xref="S2.SS1.SSS0.Px1.p1.2.m2.2.2">1</cn><ci id="S2.SS1.SSS0.Px1.p1.2.m2.3.3.cmml" xref="S2.SS1.SSS0.Px1.p1.2.m2.3.3">⋯</ci><ci id="S2.SS1.SSS0.Px1.p1.2.m2.4.4.cmml" xref="S2.SS1.SSS0.Px1.p1.2.m2.4.4">𝑊</ci></set></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.SSS0.Px1.p1.2.m2.4c">w:E\rightarrow\{0,1,\cdots,W\}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.SSS0.Px1.p1.2.m2.4d">italic_w : italic_E → { 0 , 1 , ⋯ , italic_W }</annotation></semantics></math>, we partition the edges of <math alttext="G" class="ltx_Math" display="inline" id="S2.SS1.SSS0.Px1.p1.3.m3.1"><semantics id="S2.SS1.SSS0.Px1.p1.3.m3.1a"><mi id="S2.SS1.SSS0.Px1.p1.3.m3.1.1" xref="S2.SS1.SSS0.Px1.p1.3.m3.1.1.cmml">G</mi><annotation-xml encoding="MathML-Content" id="S2.SS1.SSS0.Px1.p1.3.m3.1b"><ci id="S2.SS1.SSS0.Px1.p1.3.m3.1.1.cmml" xref="S2.SS1.SSS0.Px1.p1.3.m3.1.1">𝐺</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.SSS0.Px1.p1.3.m3.1c">G</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.SSS0.Px1.p1.3.m3.1d">italic_G</annotation></semantics></math> into <math alttext="O(\epsilon^{-1}\log W)" class="ltx_Math" display="inline" id="S2.SS1.SSS0.Px1.p1.4.m4.1"><semantics id="S2.SS1.SSS0.Px1.p1.4.m4.1a"><mrow id="S2.SS1.SSS0.Px1.p1.4.m4.1.1" xref="S2.SS1.SSS0.Px1.p1.4.m4.1.1.cmml"><mi id="S2.SS1.SSS0.Px1.p1.4.m4.1.1.3" xref="S2.SS1.SSS0.Px1.p1.4.m4.1.1.3.cmml">O</mi><mo id="S2.SS1.SSS0.Px1.p1.4.m4.1.1.2" xref="S2.SS1.SSS0.Px1.p1.4.m4.1.1.2.cmml">⁢</mo><mrow id="S2.SS1.SSS0.Px1.p1.4.m4.1.1.1.1" xref="S2.SS1.SSS0.Px1.p1.4.m4.1.1.1.1.1.cmml"><mo id="S2.SS1.SSS0.Px1.p1.4.m4.1.1.1.1.2" stretchy="false" xref="S2.SS1.SSS0.Px1.p1.4.m4.1.1.1.1.1.cmml">(</mo><mrow id="S2.SS1.SSS0.Px1.p1.4.m4.1.1.1.1.1" xref="S2.SS1.SSS0.Px1.p1.4.m4.1.1.1.1.1.cmml"><msup id="S2.SS1.SSS0.Px1.p1.4.m4.1.1.1.1.1.2" xref="S2.SS1.SSS0.Px1.p1.4.m4.1.1.1.1.1.2.cmml"><mi id="S2.SS1.SSS0.Px1.p1.4.m4.1.1.1.1.1.2.2" xref="S2.SS1.SSS0.Px1.p1.4.m4.1.1.1.1.1.2.2.cmml">ϵ</mi><mrow id="S2.SS1.SSS0.Px1.p1.4.m4.1.1.1.1.1.2.3" xref="S2.SS1.SSS0.Px1.p1.4.m4.1.1.1.1.1.2.3.cmml"><mo id="S2.SS1.SSS0.Px1.p1.4.m4.1.1.1.1.1.2.3a" xref="S2.SS1.SSS0.Px1.p1.4.m4.1.1.1.1.1.2.3.cmml">−</mo><mn id="S2.SS1.SSS0.Px1.p1.4.m4.1.1.1.1.1.2.3.2" xref="S2.SS1.SSS0.Px1.p1.4.m4.1.1.1.1.1.2.3.2.cmml">1</mn></mrow></msup><mo id="S2.SS1.SSS0.Px1.p1.4.m4.1.1.1.1.1.1" lspace="0.167em" xref="S2.SS1.SSS0.Px1.p1.4.m4.1.1.1.1.1.1.cmml">⁢</mo><mrow id="S2.SS1.SSS0.Px1.p1.4.m4.1.1.1.1.1.3" xref="S2.SS1.SSS0.Px1.p1.4.m4.1.1.1.1.1.3.cmml"><mi id="S2.SS1.SSS0.Px1.p1.4.m4.1.1.1.1.1.3.1" xref="S2.SS1.SSS0.Px1.p1.4.m4.1.1.1.1.1.3.1.cmml">log</mi><mo id="S2.SS1.SSS0.Px1.p1.4.m4.1.1.1.1.1.3a" lspace="0.167em" xref="S2.SS1.SSS0.Px1.p1.4.m4.1.1.1.1.1.3.cmml">⁡</mo><mi id="S2.SS1.SSS0.Px1.p1.4.m4.1.1.1.1.1.3.2" xref="S2.SS1.SSS0.Px1.p1.4.m4.1.1.1.1.1.3.2.cmml">W</mi></mrow></mrow><mo id="S2.SS1.SSS0.Px1.p1.4.m4.1.1.1.1.3" stretchy="false" xref="S2.SS1.SSS0.Px1.p1.4.m4.1.1.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.SSS0.Px1.p1.4.m4.1b"><apply id="S2.SS1.SSS0.Px1.p1.4.m4.1.1.cmml" xref="S2.SS1.SSS0.Px1.p1.4.m4.1.1"><times id="S2.SS1.SSS0.Px1.p1.4.m4.1.1.2.cmml" xref="S2.SS1.SSS0.Px1.p1.4.m4.1.1.2"></times><ci id="S2.SS1.SSS0.Px1.p1.4.m4.1.1.3.cmml" xref="S2.SS1.SSS0.Px1.p1.4.m4.1.1.3">𝑂</ci><apply id="S2.SS1.SSS0.Px1.p1.4.m4.1.1.1.1.1.cmml" xref="S2.SS1.SSS0.Px1.p1.4.m4.1.1.1.1"><times id="S2.SS1.SSS0.Px1.p1.4.m4.1.1.1.1.1.1.cmml" xref="S2.SS1.SSS0.Px1.p1.4.m4.1.1.1.1.1.1"></times><apply id="S2.SS1.SSS0.Px1.p1.4.m4.1.1.1.1.1.2.cmml" xref="S2.SS1.SSS0.Px1.p1.4.m4.1.1.1.1.1.2"><csymbol cd="ambiguous" id="S2.SS1.SSS0.Px1.p1.4.m4.1.1.1.1.1.2.1.cmml" xref="S2.SS1.SSS0.Px1.p1.4.m4.1.1.1.1.1.2">superscript</csymbol><ci id="S2.SS1.SSS0.Px1.p1.4.m4.1.1.1.1.1.2.2.cmml" xref="S2.SS1.SSS0.Px1.p1.4.m4.1.1.1.1.1.2.2">italic-ϵ</ci><apply id="S2.SS1.SSS0.Px1.p1.4.m4.1.1.1.1.1.2.3.cmml" xref="S2.SS1.SSS0.Px1.p1.4.m4.1.1.1.1.1.2.3"><minus id="S2.SS1.SSS0.Px1.p1.4.m4.1.1.1.1.1.2.3.1.cmml" xref="S2.SS1.SSS0.Px1.p1.4.m4.1.1.1.1.1.2.3"></minus><cn id="S2.SS1.SSS0.Px1.p1.4.m4.1.1.1.1.1.2.3.2.cmml" type="integer" xref="S2.SS1.SSS0.Px1.p1.4.m4.1.1.1.1.1.2.3.2">1</cn></apply></apply><apply id="S2.SS1.SSS0.Px1.p1.4.m4.1.1.1.1.1.3.cmml" xref="S2.SS1.SSS0.Px1.p1.4.m4.1.1.1.1.1.3"><log id="S2.SS1.SSS0.Px1.p1.4.m4.1.1.1.1.1.3.1.cmml" xref="S2.SS1.SSS0.Px1.p1.4.m4.1.1.1.1.1.3.1"></log><ci id="S2.SS1.SSS0.Px1.p1.4.m4.1.1.1.1.1.3.2.cmml" xref="S2.SS1.SSS0.Px1.p1.4.m4.1.1.1.1.1.3.2">𝑊</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.SSS0.Px1.p1.4.m4.1c">O(\epsilon^{-1}\log W)</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.SSS0.Px1.p1.4.m4.1d">italic_O ( italic_ϵ start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT roman_log italic_W )</annotation></semantics></math> buckets, where the <math alttext="i" class="ltx_Math" display="inline" id="S2.SS1.SSS0.Px1.p1.5.m5.1"><semantics id="S2.SS1.SSS0.Px1.p1.5.m5.1a"><mi id="S2.SS1.SSS0.Px1.p1.5.m5.1.1" xref="S2.SS1.SSS0.Px1.p1.5.m5.1.1.cmml">i</mi><annotation-xml encoding="MathML-Content" id="S2.SS1.SSS0.Px1.p1.5.m5.1b"><ci id="S2.SS1.SSS0.Px1.p1.5.m5.1.1.cmml" xref="S2.SS1.SSS0.Px1.p1.5.m5.1.1">𝑖</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.SSS0.Px1.p1.5.m5.1c">i</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.SSS0.Px1.p1.5.m5.1d">italic_i</annotation></semantics></math>th bucket contains edges with weights in the range <math alttext="B_{i}\coloneqq\left[(1+\epsilon)^{i-1},(1+\epsilon)^{i}\right)" class="ltx_Math" display="inline" id="S2.SS1.SSS0.Px1.p1.6.m6.2"><semantics id="S2.SS1.SSS0.Px1.p1.6.m6.2a"><mrow id="S2.SS1.SSS0.Px1.p1.6.m6.2.2" xref="S2.SS1.SSS0.Px1.p1.6.m6.2.2.cmml"><msub id="S2.SS1.SSS0.Px1.p1.6.m6.2.2.4" xref="S2.SS1.SSS0.Px1.p1.6.m6.2.2.4.cmml"><mi id="S2.SS1.SSS0.Px1.p1.6.m6.2.2.4.2" xref="S2.SS1.SSS0.Px1.p1.6.m6.2.2.4.2.cmml">B</mi><mi id="S2.SS1.SSS0.Px1.p1.6.m6.2.2.4.3" xref="S2.SS1.SSS0.Px1.p1.6.m6.2.2.4.3.cmml">i</mi></msub><mo id="S2.SS1.SSS0.Px1.p1.6.m6.2.2.3" xref="S2.SS1.SSS0.Px1.p1.6.m6.2.2.3.cmml">≔</mo><mrow id="S2.SS1.SSS0.Px1.p1.6.m6.2.2.2.2" 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xref="S2.SS1.SSS0.Px1.p1.6.m6.2.2.4"><csymbol cd="ambiguous" id="S2.SS1.SSS0.Px1.p1.6.m6.2.2.4.1.cmml" xref="S2.SS1.SSS0.Px1.p1.6.m6.2.2.4">subscript</csymbol><ci id="S2.SS1.SSS0.Px1.p1.6.m6.2.2.4.2.cmml" xref="S2.SS1.SSS0.Px1.p1.6.m6.2.2.4.2">𝐵</ci><ci id="S2.SS1.SSS0.Px1.p1.6.m6.2.2.4.3.cmml" xref="S2.SS1.SSS0.Px1.p1.6.m6.2.2.4.3">𝑖</ci></apply><interval closure="closed-open" id="S2.SS1.SSS0.Px1.p1.6.m6.2.2.2.3.cmml" xref="S2.SS1.SSS0.Px1.p1.6.m6.2.2.2.2"><apply id="S2.SS1.SSS0.Px1.p1.6.m6.1.1.1.1.1.cmml" xref="S2.SS1.SSS0.Px1.p1.6.m6.1.1.1.1.1"><csymbol cd="ambiguous" id="S2.SS1.SSS0.Px1.p1.6.m6.1.1.1.1.1.2.cmml" xref="S2.SS1.SSS0.Px1.p1.6.m6.1.1.1.1.1">superscript</csymbol><apply id="S2.SS1.SSS0.Px1.p1.6.m6.1.1.1.1.1.1.1.1.cmml" xref="S2.SS1.SSS0.Px1.p1.6.m6.1.1.1.1.1.1.1"><plus id="S2.SS1.SSS0.Px1.p1.6.m6.1.1.1.1.1.1.1.1.1.cmml" xref="S2.SS1.SSS0.Px1.p1.6.m6.1.1.1.1.1.1.1.1.1"></plus><cn id="S2.SS1.SSS0.Px1.p1.6.m6.1.1.1.1.1.1.1.1.2.cmml" type="integer" xref="S2.SS1.SSS0.Px1.p1.6.m6.1.1.1.1.1.1.1.1.2">1</cn><ci id="S2.SS1.SSS0.Px1.p1.6.m6.1.1.1.1.1.1.1.1.3.cmml" xref="S2.SS1.SSS0.Px1.p1.6.m6.1.1.1.1.1.1.1.1.3">italic-ϵ</ci></apply><apply id="S2.SS1.SSS0.Px1.p1.6.m6.1.1.1.1.1.3.cmml" xref="S2.SS1.SSS0.Px1.p1.6.m6.1.1.1.1.1.3"><minus id="S2.SS1.SSS0.Px1.p1.6.m6.1.1.1.1.1.3.1.cmml" xref="S2.SS1.SSS0.Px1.p1.6.m6.1.1.1.1.1.3.1"></minus><ci id="S2.SS1.SSS0.Px1.p1.6.m6.1.1.1.1.1.3.2.cmml" xref="S2.SS1.SSS0.Px1.p1.6.m6.1.1.1.1.1.3.2">𝑖</ci><cn id="S2.SS1.SSS0.Px1.p1.6.m6.1.1.1.1.1.3.3.cmml" type="integer" xref="S2.SS1.SSS0.Px1.p1.6.m6.1.1.1.1.1.3.3">1</cn></apply></apply><apply id="S2.SS1.SSS0.Px1.p1.6.m6.2.2.2.2.2.cmml" xref="S2.SS1.SSS0.Px1.p1.6.m6.2.2.2.2.2"><csymbol cd="ambiguous" id="S2.SS1.SSS0.Px1.p1.6.m6.2.2.2.2.2.2.cmml" xref="S2.SS1.SSS0.Px1.p1.6.m6.2.2.2.2.2">superscript</csymbol><apply id="S2.SS1.SSS0.Px1.p1.6.m6.2.2.2.2.2.1.1.1.cmml" xref="S2.SS1.SSS0.Px1.p1.6.m6.2.2.2.2.2.1.1"><plus id="S2.SS1.SSS0.Px1.p1.6.m6.2.2.2.2.2.1.1.1.1.cmml" xref="S2.SS1.SSS0.Px1.p1.6.m6.2.2.2.2.2.1.1.1.1"></plus><cn id="S2.SS1.SSS0.Px1.p1.6.m6.2.2.2.2.2.1.1.1.2.cmml" type="integer" xref="S2.SS1.SSS0.Px1.p1.6.m6.2.2.2.2.2.1.1.1.2">1</cn><ci id="S2.SS1.SSS0.Px1.p1.6.m6.2.2.2.2.2.1.1.1.3.cmml" xref="S2.SS1.SSS0.Px1.p1.6.m6.2.2.2.2.2.1.1.1.3">italic-ϵ</ci></apply><ci id="S2.SS1.SSS0.Px1.p1.6.m6.2.2.2.2.2.3.cmml" xref="S2.SS1.SSS0.Px1.p1.6.m6.2.2.2.2.2.3">𝑖</ci></apply></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.SSS0.Px1.p1.6.m6.2c">B_{i}\coloneqq\left[(1+\epsilon)^{i-1},(1+\epsilon)^{i}\right)</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.SSS0.Px1.p1.6.m6.2d">italic_B start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ≔ [ ( 1 + italic_ϵ ) start_POSTSUPERSCRIPT italic_i - 1 end_POSTSUPERSCRIPT , ( 1 + italic_ϵ ) start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT )</annotation></semantics></math>, for <math alttext="i\geq 1" class="ltx_Math" display="inline" id="S2.SS1.SSS0.Px1.p1.7.m7.1"><semantics id="S2.SS1.SSS0.Px1.p1.7.m7.1a"><mrow id="S2.SS1.SSS0.Px1.p1.7.m7.1.1" xref="S2.SS1.SSS0.Px1.p1.7.m7.1.1.cmml"><mi id="S2.SS1.SSS0.Px1.p1.7.m7.1.1.2" xref="S2.SS1.SSS0.Px1.p1.7.m7.1.1.2.cmml">i</mi><mo id="S2.SS1.SSS0.Px1.p1.7.m7.1.1.1" xref="S2.SS1.SSS0.Px1.p1.7.m7.1.1.1.cmml">≥</mo><mn id="S2.SS1.SSS0.Px1.p1.7.m7.1.1.3" xref="S2.SS1.SSS0.Px1.p1.7.m7.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.SSS0.Px1.p1.7.m7.1b"><apply id="S2.SS1.SSS0.Px1.p1.7.m7.1.1.cmml" xref="S2.SS1.SSS0.Px1.p1.7.m7.1.1"><geq id="S2.SS1.SSS0.Px1.p1.7.m7.1.1.1.cmml" xref="S2.SS1.SSS0.Px1.p1.7.m7.1.1.1"></geq><ci id="S2.SS1.SSS0.Px1.p1.7.m7.1.1.2.cmml" xref="S2.SS1.SSS0.Px1.p1.7.m7.1.1.2">𝑖</ci><cn id="S2.SS1.SSS0.Px1.p1.7.m7.1.1.3.cmml" type="integer" xref="S2.SS1.SSS0.Px1.p1.7.m7.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.SSS0.Px1.p1.7.m7.1c">i\geq 1</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.SSS0.Px1.p1.7.m7.1d">italic_i ≥ 1</annotation></semantics></math>. Furthermore, we use bucket <math alttext="B_{0}" class="ltx_Math" display="inline" id="S2.SS1.SSS0.Px1.p1.8.m8.1"><semantics id="S2.SS1.SSS0.Px1.p1.8.m8.1a"><msub id="S2.SS1.SSS0.Px1.p1.8.m8.1.1" xref="S2.SS1.SSS0.Px1.p1.8.m8.1.1.cmml"><mi id="S2.SS1.SSS0.Px1.p1.8.m8.1.1.2" xref="S2.SS1.SSS0.Px1.p1.8.m8.1.1.2.cmml">B</mi><mn id="S2.SS1.SSS0.Px1.p1.8.m8.1.1.3" xref="S2.SS1.SSS0.Px1.p1.8.m8.1.1.3.cmml">0</mn></msub><annotation-xml encoding="MathML-Content" id="S2.SS1.SSS0.Px1.p1.8.m8.1b"><apply id="S2.SS1.SSS0.Px1.p1.8.m8.1.1.cmml" xref="S2.SS1.SSS0.Px1.p1.8.m8.1.1"><csymbol cd="ambiguous" id="S2.SS1.SSS0.Px1.p1.8.m8.1.1.1.cmml" xref="S2.SS1.SSS0.Px1.p1.8.m8.1.1">subscript</csymbol><ci id="S2.SS1.SSS0.Px1.p1.8.m8.1.1.2.cmml" xref="S2.SS1.SSS0.Px1.p1.8.m8.1.1.2">𝐵</ci><cn id="S2.SS1.SSS0.Px1.p1.8.m8.1.1.3.cmml" type="integer" xref="S2.SS1.SSS0.Px1.p1.8.m8.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.SSS0.Px1.p1.8.m8.1c">B_{0}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.SSS0.Px1.p1.8.m8.1d">italic_B start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT</annotation></semantics></math> for zero-weight edges. Then, we construct a fault-tolerant spanner in each bucket (treating it as an unweighted graph) using the greedy algorithm for unweighted <math alttext="f" class="ltx_Math" display="inline" id="S2.SS1.SSS0.Px1.p1.9.m9.1"><semantics id="S2.SS1.SSS0.Px1.p1.9.m9.1a"><mi id="S2.SS1.SSS0.Px1.p1.9.m9.1.1" xref="S2.SS1.SSS0.Px1.p1.9.m9.1.1.cmml">f</mi><annotation-xml encoding="MathML-Content" id="S2.SS1.SSS0.Px1.p1.9.m9.1b"><ci id="S2.SS1.SSS0.Px1.p1.9.m9.1.1.cmml" xref="S2.SS1.SSS0.Px1.p1.9.m9.1.1">𝑓</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.SSS0.Px1.p1.9.m9.1c">f</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.SSS0.Px1.p1.9.m9.1d">italic_f</annotation></semantics></math>-FT spanners.</p> </div> <figure class="ltx_float ltx_algorithm" id="algorithm2"> <div class="ltx_listing ltx_lst_numbers_left ltx_listing" id="algorithm2.16"> <div class="ltx_listingline" id="algorithm2.3.3"> <span class="ltx_text" id="algorithm2.3.3.1"><span class="ltx_text ltx_font_bold" id="algorithm2.3.3.1.1">Input:</span> </span>Weighted graph <math alttext="(G=(V,E),w)" class="ltx_Math" display="inline" id="algorithm2.1.1.m1.4"><semantics id="algorithm2.1.1.m1.4a"><mrow id="algorithm2.1.1.m1.4.4.1" xref="algorithm2.1.1.m1.4.4.1.1.cmml"><mo id="algorithm2.1.1.m1.4.4.1.2" stretchy="false" xref="algorithm2.1.1.m1.4.4.1.1.cmml">(</mo><mrow id="algorithm2.1.1.m1.4.4.1.1" xref="algorithm2.1.1.m1.4.4.1.1.cmml"><mi id="algorithm2.1.1.m1.4.4.1.1.3" xref="algorithm2.1.1.m1.4.4.1.1.3.cmml">G</mi><mo id="algorithm2.1.1.m1.4.4.1.1.2" xref="algorithm2.1.1.m1.4.4.1.1.2.cmml">=</mo><mrow id="algorithm2.1.1.m1.4.4.1.1.1.1" xref="algorithm2.1.1.m1.4.4.1.1.1.2.cmml"><mrow id="algorithm2.1.1.m1.4.4.1.1.1.1.1.2" xref="algorithm2.1.1.m1.4.4.1.1.1.1.1.1.cmml"><mo id="algorithm2.1.1.m1.4.4.1.1.1.1.1.2.1" stretchy="false" xref="algorithm2.1.1.m1.4.4.1.1.1.1.1.1.cmml">(</mo><mi id="algorithm2.1.1.m1.1.1" xref="algorithm2.1.1.m1.1.1.cmml">V</mi><mo id="algorithm2.1.1.m1.4.4.1.1.1.1.1.2.2" xref="algorithm2.1.1.m1.4.4.1.1.1.1.1.1.cmml">,</mo><mi id="algorithm2.1.1.m1.2.2" xref="algorithm2.1.1.m1.2.2.cmml">E</mi><mo id="algorithm2.1.1.m1.4.4.1.1.1.1.1.2.3" stretchy="false" xref="algorithm2.1.1.m1.4.4.1.1.1.1.1.1.cmml">)</mo></mrow><mo id="algorithm2.1.1.m1.4.4.1.1.1.1.2" xref="algorithm2.1.1.m1.4.4.1.1.1.2.cmml">,</mo><mi id="algorithm2.1.1.m1.3.3" xref="algorithm2.1.1.m1.3.3.cmml">w</mi></mrow></mrow><mo id="algorithm2.1.1.m1.4.4.1.3" stretchy="false" xref="algorithm2.1.1.m1.4.4.1.1.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="algorithm2.1.1.m1.4b"><apply id="algorithm2.1.1.m1.4.4.1.1.cmml" xref="algorithm2.1.1.m1.4.4.1"><eq id="algorithm2.1.1.m1.4.4.1.1.2.cmml" xref="algorithm2.1.1.m1.4.4.1.1.2"></eq><ci id="algorithm2.1.1.m1.4.4.1.1.3.cmml" xref="algorithm2.1.1.m1.4.4.1.1.3">𝐺</ci><list id="algorithm2.1.1.m1.4.4.1.1.1.2.cmml" xref="algorithm2.1.1.m1.4.4.1.1.1.1"><interval closure="open" id="algorithm2.1.1.m1.4.4.1.1.1.1.1.1.cmml" xref="algorithm2.1.1.m1.4.4.1.1.1.1.1.2"><ci id="algorithm2.1.1.m1.1.1.cmml" xref="algorithm2.1.1.m1.1.1">𝑉</ci><ci id="algorithm2.1.1.m1.2.2.cmml" xref="algorithm2.1.1.m1.2.2">𝐸</ci></interval><ci id="algorithm2.1.1.m1.3.3.cmml" xref="algorithm2.1.1.m1.3.3">𝑤</ci></list></apply></annotation-xml><annotation encoding="application/x-tex" id="algorithm2.1.1.m1.4c">(G=(V,E),w)</annotation><annotation encoding="application/x-llamapun" id="algorithm2.1.1.m1.4d">( italic_G = ( italic_V , italic_E ) , italic_w )</annotation></semantics></math>, a stretch parameter <math alttext="t" class="ltx_Math" display="inline" id="algorithm2.2.2.m2.1"><semantics id="algorithm2.2.2.m2.1a"><mi id="algorithm2.2.2.m2.1.1" xref="algorithm2.2.2.m2.1.1.cmml">t</mi><annotation-xml encoding="MathML-Content" id="algorithm2.2.2.m2.1b"><ci id="algorithm2.2.2.m2.1.1.cmml" xref="algorithm2.2.2.m2.1.1">𝑡</ci></annotation-xml><annotation encoding="application/x-tex" id="algorithm2.2.2.m2.1c">t</annotation><annotation encoding="application/x-llamapun" id="algorithm2.2.2.m2.1d">italic_t</annotation></semantics></math>, and a fault tolerance parameter <math alttext="f" class="ltx_Math" display="inline" id="algorithm2.3.3.m3.1"><semantics id="algorithm2.3.3.m3.1a"><mi id="algorithm2.3.3.m3.1.1" xref="algorithm2.3.3.m3.1.1.cmml">f</mi><annotation-xml encoding="MathML-Content" id="algorithm2.3.3.m3.1b"><ci id="algorithm2.3.3.m3.1.1.cmml" xref="algorithm2.3.3.m3.1.1">𝑓</ci></annotation-xml><annotation encoding="application/x-tex" id="algorithm2.3.3.m3.1c">f</annotation><annotation encoding="application/x-llamapun" id="algorithm2.3.3.m3.1d">italic_f</annotation></semantics></math>. </div> <div class="ltx_listingline" id="algorithm2.5.5"> <span class="ltx_text ltx_font_bold" id="algorithm2.5.5.3">for</span> <em class="ltx_emph ltx_font_italic" id="algorithm2.5.5.2"><math alttext="i=1" class="ltx_Math" display="inline" id="algorithm2.4.4.1.m1.1"><semantics id="algorithm2.4.4.1.m1.1a"><mrow id="algorithm2.4.4.1.m1.1.1" xref="algorithm2.4.4.1.m1.1.1.cmml"><mi id="algorithm2.4.4.1.m1.1.1.2" xref="algorithm2.4.4.1.m1.1.1.2.cmml">i</mi><mo id="algorithm2.4.4.1.m1.1.1.1" xref="algorithm2.4.4.1.m1.1.1.1.cmml">=</mo><mn id="algorithm2.4.4.1.m1.1.1.3" xref="algorithm2.4.4.1.m1.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="algorithm2.4.4.1.m1.1b"><apply id="algorithm2.4.4.1.m1.1.1.cmml" xref="algorithm2.4.4.1.m1.1.1"><eq id="algorithm2.4.4.1.m1.1.1.1.cmml" xref="algorithm2.4.4.1.m1.1.1.1"></eq><ci id="algorithm2.4.4.1.m1.1.1.2.cmml" xref="algorithm2.4.4.1.m1.1.1.2">𝑖</ci><cn id="algorithm2.4.4.1.m1.1.1.3.cmml" type="integer" xref="algorithm2.4.4.1.m1.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="algorithm2.4.4.1.m1.1c">i=1</annotation><annotation encoding="application/x-llamapun" id="algorithm2.4.4.1.m1.1d">italic_i = 1</annotation></semantics></math> to <math alttext="T=O(\epsilon^{-1}\log W)" class="ltx_Math" display="inline" id="algorithm2.5.5.2.m2.1"><semantics id="algorithm2.5.5.2.m2.1a"><mrow id="algorithm2.5.5.2.m2.1.1" xref="algorithm2.5.5.2.m2.1.1.cmml"><mi id="algorithm2.5.5.2.m2.1.1.3" xref="algorithm2.5.5.2.m2.1.1.3.cmml">T</mi><mo id="algorithm2.5.5.2.m2.1.1.2" xref="algorithm2.5.5.2.m2.1.1.2.cmml">=</mo><mrow id="algorithm2.5.5.2.m2.1.1.1" xref="algorithm2.5.5.2.m2.1.1.1.cmml"><mi id="algorithm2.5.5.2.m2.1.1.1.3" xref="algorithm2.5.5.2.m2.1.1.1.3.cmml">O</mi><mo id="algorithm2.5.5.2.m2.1.1.1.2" xref="algorithm2.5.5.2.m2.1.1.1.2.cmml">⁢</mo><mrow id="algorithm2.5.5.2.m2.1.1.1.1.1" xref="algorithm2.5.5.2.m2.1.1.1.1.1.1.cmml"><mo id="algorithm2.5.5.2.m2.1.1.1.1.1.2" stretchy="false" xref="algorithm2.5.5.2.m2.1.1.1.1.1.1.cmml">(</mo><mrow id="algorithm2.5.5.2.m2.1.1.1.1.1.1" xref="algorithm2.5.5.2.m2.1.1.1.1.1.1.cmml"><msup id="algorithm2.5.5.2.m2.1.1.1.1.1.1.2" xref="algorithm2.5.5.2.m2.1.1.1.1.1.1.2.cmml"><mi id="algorithm2.5.5.2.m2.1.1.1.1.1.1.2.2" xref="algorithm2.5.5.2.m2.1.1.1.1.1.1.2.2.cmml">ϵ</mi><mrow id="algorithm2.5.5.2.m2.1.1.1.1.1.1.2.3" xref="algorithm2.5.5.2.m2.1.1.1.1.1.1.2.3.cmml"><mo id="algorithm2.5.5.2.m2.1.1.1.1.1.1.2.3a" xref="algorithm2.5.5.2.m2.1.1.1.1.1.1.2.3.cmml">−</mo><mn id="algorithm2.5.5.2.m2.1.1.1.1.1.1.2.3.2" xref="algorithm2.5.5.2.m2.1.1.1.1.1.1.2.3.2.cmml">1</mn></mrow></msup><mo id="algorithm2.5.5.2.m2.1.1.1.1.1.1.1" lspace="0.167em" xref="algorithm2.5.5.2.m2.1.1.1.1.1.1.1.cmml">⁢</mo><mrow id="algorithm2.5.5.2.m2.1.1.1.1.1.1.3" xref="algorithm2.5.5.2.m2.1.1.1.1.1.1.3.cmml"><mi id="algorithm2.5.5.2.m2.1.1.1.1.1.1.3.1" xref="algorithm2.5.5.2.m2.1.1.1.1.1.1.3.1.cmml">log</mi><mo id="algorithm2.5.5.2.m2.1.1.1.1.1.1.3a" lspace="0.167em" xref="algorithm2.5.5.2.m2.1.1.1.1.1.1.3.cmml">⁡</mo><mi id="algorithm2.5.5.2.m2.1.1.1.1.1.1.3.2" xref="algorithm2.5.5.2.m2.1.1.1.1.1.1.3.2.cmml">W</mi></mrow></mrow><mo id="algorithm2.5.5.2.m2.1.1.1.1.1.3" stretchy="false" xref="algorithm2.5.5.2.m2.1.1.1.1.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="algorithm2.5.5.2.m2.1b"><apply id="algorithm2.5.5.2.m2.1.1.cmml" xref="algorithm2.5.5.2.m2.1.1"><eq id="algorithm2.5.5.2.m2.1.1.2.cmml" xref="algorithm2.5.5.2.m2.1.1.2"></eq><ci id="algorithm2.5.5.2.m2.1.1.3.cmml" xref="algorithm2.5.5.2.m2.1.1.3">𝑇</ci><apply id="algorithm2.5.5.2.m2.1.1.1.cmml" xref="algorithm2.5.5.2.m2.1.1.1"><times id="algorithm2.5.5.2.m2.1.1.1.2.cmml" xref="algorithm2.5.5.2.m2.1.1.1.2"></times><ci id="algorithm2.5.5.2.m2.1.1.1.3.cmml" xref="algorithm2.5.5.2.m2.1.1.1.3">𝑂</ci><apply id="algorithm2.5.5.2.m2.1.1.1.1.1.1.cmml" xref="algorithm2.5.5.2.m2.1.1.1.1.1"><times id="algorithm2.5.5.2.m2.1.1.1.1.1.1.1.cmml" xref="algorithm2.5.5.2.m2.1.1.1.1.1.1.1"></times><apply id="algorithm2.5.5.2.m2.1.1.1.1.1.1.2.cmml" xref="algorithm2.5.5.2.m2.1.1.1.1.1.1.2"><csymbol cd="ambiguous" id="algorithm2.5.5.2.m2.1.1.1.1.1.1.2.1.cmml" xref="algorithm2.5.5.2.m2.1.1.1.1.1.1.2">superscript</csymbol><ci id="algorithm2.5.5.2.m2.1.1.1.1.1.1.2.2.cmml" xref="algorithm2.5.5.2.m2.1.1.1.1.1.1.2.2">italic-ϵ</ci><apply id="algorithm2.5.5.2.m2.1.1.1.1.1.1.2.3.cmml" xref="algorithm2.5.5.2.m2.1.1.1.1.1.1.2.3"><minus id="algorithm2.5.5.2.m2.1.1.1.1.1.1.2.3.1.cmml" xref="algorithm2.5.5.2.m2.1.1.1.1.1.1.2.3"></minus><cn id="algorithm2.5.5.2.m2.1.1.1.1.1.1.2.3.2.cmml" type="integer" xref="algorithm2.5.5.2.m2.1.1.1.1.1.1.2.3.2">1</cn></apply></apply><apply id="algorithm2.5.5.2.m2.1.1.1.1.1.1.3.cmml" xref="algorithm2.5.5.2.m2.1.1.1.1.1.1.3"><log id="algorithm2.5.5.2.m2.1.1.1.1.1.1.3.1.cmml" xref="algorithm2.5.5.2.m2.1.1.1.1.1.1.3.1"></log><ci id="algorithm2.5.5.2.m2.1.1.1.1.1.1.3.2.cmml" xref="algorithm2.5.5.2.m2.1.1.1.1.1.1.3.2">𝑊</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="algorithm2.5.5.2.m2.1c">T=O(\epsilon^{-1}\log W)</annotation><annotation encoding="application/x-llamapun" id="algorithm2.5.5.2.m2.1d">italic_T = italic_O ( italic_ϵ start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT roman_log italic_W )</annotation></semantics></math></em> <span class="ltx_text ltx_font_bold" id="algorithm2.5.5.4">do</span> </div> <div class="ltx_listingline" id="algorithm2.6.6"> <span class="ltx_rule" style="width:1px;height:100%;background:black;display:inline-block;"> </span> <span class="ltx_text ltx_font_bold" id="algorithm2.6.6.1">initialize</span> <math alttext="H_{i}\leftarrow(V,\emptyset)" class="ltx_Math" display="inline" id="algorithm2.6.6.m1.2"><semantics id="algorithm2.6.6.m1.2a"><mrow id="algorithm2.6.6.m1.2.3" xref="algorithm2.6.6.m1.2.3.cmml"><msub id="algorithm2.6.6.m1.2.3.2" xref="algorithm2.6.6.m1.2.3.2.cmml"><mi id="algorithm2.6.6.m1.2.3.2.2" xref="algorithm2.6.6.m1.2.3.2.2.cmml">H</mi><mi id="algorithm2.6.6.m1.2.3.2.3" xref="algorithm2.6.6.m1.2.3.2.3.cmml">i</mi></msub><mo id="algorithm2.6.6.m1.2.3.1" stretchy="false" xref="algorithm2.6.6.m1.2.3.1.cmml">←</mo><mrow id="algorithm2.6.6.m1.2.3.3.2" xref="algorithm2.6.6.m1.2.3.3.1.cmml"><mo id="algorithm2.6.6.m1.2.3.3.2.1" stretchy="false" xref="algorithm2.6.6.m1.2.3.3.1.cmml">(</mo><mi id="algorithm2.6.6.m1.1.1" xref="algorithm2.6.6.m1.1.1.cmml">V</mi><mo id="algorithm2.6.6.m1.2.3.3.2.2" xref="algorithm2.6.6.m1.2.3.3.1.cmml">,</mo><mi id="algorithm2.6.6.m1.2.2" mathvariant="normal" xref="algorithm2.6.6.m1.2.2.cmml">∅</mi><mo id="algorithm2.6.6.m1.2.3.3.2.3" stretchy="false" xref="algorithm2.6.6.m1.2.3.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="algorithm2.6.6.m1.2b"><apply id="algorithm2.6.6.m1.2.3.cmml" xref="algorithm2.6.6.m1.2.3"><ci id="algorithm2.6.6.m1.2.3.1.cmml" xref="algorithm2.6.6.m1.2.3.1">←</ci><apply id="algorithm2.6.6.m1.2.3.2.cmml" xref="algorithm2.6.6.m1.2.3.2"><csymbol cd="ambiguous" id="algorithm2.6.6.m1.2.3.2.1.cmml" xref="algorithm2.6.6.m1.2.3.2">subscript</csymbol><ci id="algorithm2.6.6.m1.2.3.2.2.cmml" xref="algorithm2.6.6.m1.2.3.2.2">𝐻</ci><ci id="algorithm2.6.6.m1.2.3.2.3.cmml" xref="algorithm2.6.6.m1.2.3.2.3">𝑖</ci></apply><interval closure="open" id="algorithm2.6.6.m1.2.3.3.1.cmml" xref="algorithm2.6.6.m1.2.3.3.2"><ci id="algorithm2.6.6.m1.1.1.cmml" xref="algorithm2.6.6.m1.1.1">𝑉</ci><emptyset id="algorithm2.6.6.m1.2.2.cmml" xref="algorithm2.6.6.m1.2.2"></emptyset></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="algorithm2.6.6.m1.2c">H_{i}\leftarrow(V,\emptyset)</annotation><annotation encoding="application/x-llamapun" id="algorithm2.6.6.m1.2d">italic_H start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ← ( italic_V , ∅ )</annotation></semantics></math> </div> <div class="ltx_listingline" id="algorithm2.7.7"> <span class="ltx_text ltx_font_bold" id="algorithm2.7.7.2">for</span> <em class="ltx_emph ltx_font_italic" id="algorithm2.7.7.1"><math alttext="(u,v)\in E" class="ltx_Math" display="inline" id="algorithm2.7.7.1.m1.2"><semantics id="algorithm2.7.7.1.m1.2a"><mrow id="algorithm2.7.7.1.m1.2.3" xref="algorithm2.7.7.1.m1.2.3.cmml"><mrow id="algorithm2.7.7.1.m1.2.3.2.2" xref="algorithm2.7.7.1.m1.2.3.2.1.cmml"><mo id="algorithm2.7.7.1.m1.2.3.2.2.1" stretchy="false" xref="algorithm2.7.7.1.m1.2.3.2.1.cmml">(</mo><mi id="algorithm2.7.7.1.m1.1.1" xref="algorithm2.7.7.1.m1.1.1.cmml">u</mi><mo id="algorithm2.7.7.1.m1.2.3.2.2.2" xref="algorithm2.7.7.1.m1.2.3.2.1.cmml">,</mo><mi id="algorithm2.7.7.1.m1.2.2" xref="algorithm2.7.7.1.m1.2.2.cmml">v</mi><mo id="algorithm2.7.7.1.m1.2.3.2.2.3" stretchy="false" xref="algorithm2.7.7.1.m1.2.3.2.1.cmml">)</mo></mrow><mo id="algorithm2.7.7.1.m1.2.3.1" xref="algorithm2.7.7.1.m1.2.3.1.cmml">∈</mo><mi id="algorithm2.7.7.1.m1.2.3.3" xref="algorithm2.7.7.1.m1.2.3.3.cmml">E</mi></mrow><annotation-xml encoding="MathML-Content" id="algorithm2.7.7.1.m1.2b"><apply id="algorithm2.7.7.1.m1.2.3.cmml" xref="algorithm2.7.7.1.m1.2.3"><in id="algorithm2.7.7.1.m1.2.3.1.cmml" xref="algorithm2.7.7.1.m1.2.3.1"></in><interval closure="open" id="algorithm2.7.7.1.m1.2.3.2.1.cmml" xref="algorithm2.7.7.1.m1.2.3.2.2"><ci id="algorithm2.7.7.1.m1.1.1.cmml" xref="algorithm2.7.7.1.m1.1.1">𝑢</ci><ci id="algorithm2.7.7.1.m1.2.2.cmml" xref="algorithm2.7.7.1.m1.2.2">𝑣</ci></interval><ci id="algorithm2.7.7.1.m1.2.3.3.cmml" xref="algorithm2.7.7.1.m1.2.3.3">𝐸</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="algorithm2.7.7.1.m1.2c">(u,v)\in E</annotation><annotation encoding="application/x-llamapun" id="algorithm2.7.7.1.m1.2d">( italic_u , italic_v ) ∈ italic_E</annotation></semantics></math> in the stream</em> <span class="ltx_text ltx_font_bold" id="algorithm2.7.7.3">do</span> </div> <div class="ltx_listingline" id="algorithm2.10.10"> <span class="ltx_rule" style="width:1px;height:100%;background:black;display:inline-block;"> </span> <span class="ltx_text" id="algorithm2.10.10.4" style="color:#0000FF;">/* </span><span class="ltx_text" id="algorithm2.10.10.3" style="color:#0000FF;">let <math alttext="H_{j}" class="ltx_Math" display="inline" id="algorithm2.8.8.1.m1.1"><semantics id="algorithm2.8.8.1.m1.1a"><msub id="algorithm2.8.8.1.m1.1.1" xref="algorithm2.8.8.1.m1.1.1.cmml"><mi id="algorithm2.8.8.1.m1.1.1.2" mathcolor="#0000FF" xref="algorithm2.8.8.1.m1.1.1.2.cmml">H</mi><mi id="algorithm2.8.8.1.m1.1.1.3" mathcolor="#0000FF" xref="algorithm2.8.8.1.m1.1.1.3.cmml">j</mi></msub><annotation-xml encoding="MathML-Content" id="algorithm2.8.8.1.m1.1b"><apply id="algorithm2.8.8.1.m1.1.1.cmml" xref="algorithm2.8.8.1.m1.1.1"><csymbol cd="ambiguous" id="algorithm2.8.8.1.m1.1.1.1.cmml" xref="algorithm2.8.8.1.m1.1.1">subscript</csymbol><ci id="algorithm2.8.8.1.m1.1.1.2.cmml" xref="algorithm2.8.8.1.m1.1.1.2">𝐻</ci><ci id="algorithm2.8.8.1.m1.1.1.3.cmml" xref="algorithm2.8.8.1.m1.1.1.3">𝑗</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="algorithm2.8.8.1.m1.1c">H_{j}</annotation><annotation encoding="application/x-llamapun" id="algorithm2.8.8.1.m1.1d">italic_H start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT</annotation></semantics></math> be the FT spanner corresponding to the weight class of <math alttext="(u,v)" class="ltx_Math" display="inline" id="algorithm2.9.9.2.m2.2"><semantics id="algorithm2.9.9.2.m2.2a"><mrow id="algorithm2.9.9.2.m2.2.3.2" xref="algorithm2.9.9.2.m2.2.3.1.cmml"><mo id="algorithm2.9.9.2.m2.2.3.2.1" mathcolor="#0000FF" stretchy="false" xref="algorithm2.9.9.2.m2.2.3.1.cmml">(</mo><mi id="algorithm2.9.9.2.m2.1.1" mathcolor="#0000FF" xref="algorithm2.9.9.2.m2.1.1.cmml">u</mi><mo id="algorithm2.9.9.2.m2.2.3.2.2" mathcolor="#0000FF" xref="algorithm2.9.9.2.m2.2.3.1.cmml">,</mo><mi id="algorithm2.9.9.2.m2.2.2" mathcolor="#0000FF" xref="algorithm2.9.9.2.m2.2.2.cmml">v</mi><mo id="algorithm2.9.9.2.m2.2.3.2.3" mathcolor="#0000FF" stretchy="false" xref="algorithm2.9.9.2.m2.2.3.1.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="algorithm2.9.9.2.m2.2b"><interval closure="open" id="algorithm2.9.9.2.m2.2.3.1.cmml" xref="algorithm2.9.9.2.m2.2.3.2"><ci id="algorithm2.9.9.2.m2.1.1.cmml" xref="algorithm2.9.9.2.m2.1.1">𝑢</ci><ci id="algorithm2.9.9.2.m2.2.2.cmml" xref="algorithm2.9.9.2.m2.2.2">𝑣</ci></interval></annotation-xml><annotation encoding="application/x-tex" id="algorithm2.9.9.2.m2.2c">(u,v)</annotation><annotation encoding="application/x-llamapun" id="algorithm2.9.9.2.m2.2d">( italic_u , italic_v )</annotation></semantics></math>; i.e. <math alttext="w(u,v)\in B_{j}" class="ltx_Math" display="inline" id="algorithm2.10.10.3.m3.2"><semantics id="algorithm2.10.10.3.m3.2a"><mrow id="algorithm2.10.10.3.m3.2.3" xref="algorithm2.10.10.3.m3.2.3.cmml"><mrow id="algorithm2.10.10.3.m3.2.3.2" xref="algorithm2.10.10.3.m3.2.3.2.cmml"><mi id="algorithm2.10.10.3.m3.2.3.2.2" mathcolor="#0000FF" xref="algorithm2.10.10.3.m3.2.3.2.2.cmml">w</mi><mo id="algorithm2.10.10.3.m3.2.3.2.1" xref="algorithm2.10.10.3.m3.2.3.2.1.cmml">⁢</mo><mrow id="algorithm2.10.10.3.m3.2.3.2.3.2" xref="algorithm2.10.10.3.m3.2.3.2.3.1.cmml"><mo id="algorithm2.10.10.3.m3.2.3.2.3.2.1" mathcolor="#0000FF" stretchy="false" xref="algorithm2.10.10.3.m3.2.3.2.3.1.cmml">(</mo><mi id="algorithm2.10.10.3.m3.1.1" mathcolor="#0000FF" xref="algorithm2.10.10.3.m3.1.1.cmml">u</mi><mo id="algorithm2.10.10.3.m3.2.3.2.3.2.2" mathcolor="#0000FF" xref="algorithm2.10.10.3.m3.2.3.2.3.1.cmml">,</mo><mi id="algorithm2.10.10.3.m3.2.2" mathcolor="#0000FF" xref="algorithm2.10.10.3.m3.2.2.cmml">v</mi><mo id="algorithm2.10.10.3.m3.2.3.2.3.2.3" mathcolor="#0000FF" stretchy="false" xref="algorithm2.10.10.3.m3.2.3.2.3.1.cmml">)</mo></mrow></mrow><mo id="algorithm2.10.10.3.m3.2.3.1" mathcolor="#0000FF" xref="algorithm2.10.10.3.m3.2.3.1.cmml">∈</mo><msub id="algorithm2.10.10.3.m3.2.3.3" xref="algorithm2.10.10.3.m3.2.3.3.cmml"><mi id="algorithm2.10.10.3.m3.2.3.3.2" mathcolor="#0000FF" xref="algorithm2.10.10.3.m3.2.3.3.2.cmml">B</mi><mi id="algorithm2.10.10.3.m3.2.3.3.3" mathcolor="#0000FF" xref="algorithm2.10.10.3.m3.2.3.3.3.cmml">j</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="algorithm2.10.10.3.m3.2b"><apply id="algorithm2.10.10.3.m3.2.3.cmml" xref="algorithm2.10.10.3.m3.2.3"><in id="algorithm2.10.10.3.m3.2.3.1.cmml" xref="algorithm2.10.10.3.m3.2.3.1"></in><apply id="algorithm2.10.10.3.m3.2.3.2.cmml" xref="algorithm2.10.10.3.m3.2.3.2"><times id="algorithm2.10.10.3.m3.2.3.2.1.cmml" xref="algorithm2.10.10.3.m3.2.3.2.1"></times><ci id="algorithm2.10.10.3.m3.2.3.2.2.cmml" xref="algorithm2.10.10.3.m3.2.3.2.2">𝑤</ci><interval closure="open" id="algorithm2.10.10.3.m3.2.3.2.3.1.cmml" xref="algorithm2.10.10.3.m3.2.3.2.3.2"><ci id="algorithm2.10.10.3.m3.1.1.cmml" xref="algorithm2.10.10.3.m3.1.1">𝑢</ci><ci id="algorithm2.10.10.3.m3.2.2.cmml" xref="algorithm2.10.10.3.m3.2.2">𝑣</ci></interval></apply><apply id="algorithm2.10.10.3.m3.2.3.3.cmml" xref="algorithm2.10.10.3.m3.2.3.3"><csymbol cd="ambiguous" id="algorithm2.10.10.3.m3.2.3.3.1.cmml" xref="algorithm2.10.10.3.m3.2.3.3">subscript</csymbol><ci id="algorithm2.10.10.3.m3.2.3.3.2.cmml" xref="algorithm2.10.10.3.m3.2.3.3.2">𝐵</ci><ci id="algorithm2.10.10.3.m3.2.3.3.3.cmml" xref="algorithm2.10.10.3.m3.2.3.3.3">𝑗</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="algorithm2.10.10.3.m3.2c">w(u,v)\in B_{j}</annotation><annotation encoding="application/x-llamapun" id="algorithm2.10.10.3.m3.2d">italic_w ( italic_u , italic_v ) ∈ italic_B start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT</annotation></semantics></math> */</span> </div> <div class="ltx_listingline" id="algorithm2.16.17"> <span class="ltx_rule" style="width:1px;height:100%;background:black;display:inline-block;"> </span> </div> <div class="ltx_listingline" id="algorithm2.13.13"> <span class="ltx_rule" style="width:1px;height:100%;background:black;display:inline-block;"> </span> <span class="ltx_text ltx_font_bold" id="algorithm2.13.13.4">if</span> <em class="ltx_emph ltx_font_italic" id="algorithm2.13.13.3">there exists a set of vertices (or edges) <math alttext="F" class="ltx_Math" display="inline" id="algorithm2.11.11.1.m1.1"><semantics id="algorithm2.11.11.1.m1.1a"><mi id="algorithm2.11.11.1.m1.1.1" xref="algorithm2.11.11.1.m1.1.1.cmml">F</mi><annotation-xml encoding="MathML-Content" id="algorithm2.11.11.1.m1.1b"><ci id="algorithm2.11.11.1.m1.1.1.cmml" xref="algorithm2.11.11.1.m1.1.1">𝐹</ci></annotation-xml><annotation encoding="application/x-tex" id="algorithm2.11.11.1.m1.1c">F</annotation><annotation encoding="application/x-llamapun" id="algorithm2.11.11.1.m1.1d">italic_F</annotation></semantics></math> of size <math alttext="f" class="ltx_Math" display="inline" id="algorithm2.12.12.2.m2.1"><semantics id="algorithm2.12.12.2.m2.1a"><mi id="algorithm2.12.12.2.m2.1.1" xref="algorithm2.12.12.2.m2.1.1.cmml">f</mi><annotation-xml encoding="MathML-Content" id="algorithm2.12.12.2.m2.1b"><ci id="algorithm2.12.12.2.m2.1.1.cmml" xref="algorithm2.12.12.2.m2.1.1">𝑓</ci></annotation-xml><annotation encoding="application/x-tex" id="algorithm2.12.12.2.m2.1c">f</annotation><annotation encoding="application/x-llamapun" id="algorithm2.12.12.2.m2.1d">italic_f</annotation></semantics></math> such that <math alttext="d_{H_{j}\setminus F}(u,v)>2t-1" class="ltx_Math" display="inline" id="algorithm2.13.13.3.m3.2"><semantics id="algorithm2.13.13.3.m3.2a"><mrow id="algorithm2.13.13.3.m3.2.3" xref="algorithm2.13.13.3.m3.2.3.cmml"><mrow id="algorithm2.13.13.3.m3.2.3.2" xref="algorithm2.13.13.3.m3.2.3.2.cmml"><msub id="algorithm2.13.13.3.m3.2.3.2.2" xref="algorithm2.13.13.3.m3.2.3.2.2.cmml"><mi id="algorithm2.13.13.3.m3.2.3.2.2.2" xref="algorithm2.13.13.3.m3.2.3.2.2.2.cmml">d</mi><mrow id="algorithm2.13.13.3.m3.2.3.2.2.3" xref="algorithm2.13.13.3.m3.2.3.2.2.3.cmml"><msub id="algorithm2.13.13.3.m3.2.3.2.2.3.2" xref="algorithm2.13.13.3.m3.2.3.2.2.3.2.cmml"><mi id="algorithm2.13.13.3.m3.2.3.2.2.3.2.2" xref="algorithm2.13.13.3.m3.2.3.2.2.3.2.2.cmml">H</mi><mi id="algorithm2.13.13.3.m3.2.3.2.2.3.2.3" xref="algorithm2.13.13.3.m3.2.3.2.2.3.2.3.cmml">j</mi></msub><mo id="algorithm2.13.13.3.m3.2.3.2.2.3.1" xref="algorithm2.13.13.3.m3.2.3.2.2.3.1.cmml">∖</mo><mi id="algorithm2.13.13.3.m3.2.3.2.2.3.3" xref="algorithm2.13.13.3.m3.2.3.2.2.3.3.cmml">F</mi></mrow></msub><mo id="algorithm2.13.13.3.m3.2.3.2.1" xref="algorithm2.13.13.3.m3.2.3.2.1.cmml">⁢</mo><mrow id="algorithm2.13.13.3.m3.2.3.2.3.2" xref="algorithm2.13.13.3.m3.2.3.2.3.1.cmml"><mo id="algorithm2.13.13.3.m3.2.3.2.3.2.1" stretchy="false" xref="algorithm2.13.13.3.m3.2.3.2.3.1.cmml">(</mo><mi id="algorithm2.13.13.3.m3.1.1" xref="algorithm2.13.13.3.m3.1.1.cmml">u</mi><mo id="algorithm2.13.13.3.m3.2.3.2.3.2.2" xref="algorithm2.13.13.3.m3.2.3.2.3.1.cmml">,</mo><mi id="algorithm2.13.13.3.m3.2.2" xref="algorithm2.13.13.3.m3.2.2.cmml">v</mi><mo id="algorithm2.13.13.3.m3.2.3.2.3.2.3" stretchy="false" xref="algorithm2.13.13.3.m3.2.3.2.3.1.cmml">)</mo></mrow></mrow><mo id="algorithm2.13.13.3.m3.2.3.1" xref="algorithm2.13.13.3.m3.2.3.1.cmml">></mo><mrow id="algorithm2.13.13.3.m3.2.3.3" xref="algorithm2.13.13.3.m3.2.3.3.cmml"><mrow id="algorithm2.13.13.3.m3.2.3.3.2" xref="algorithm2.13.13.3.m3.2.3.3.2.cmml"><mn id="algorithm2.13.13.3.m3.2.3.3.2.2" xref="algorithm2.13.13.3.m3.2.3.3.2.2.cmml">2</mn><mo id="algorithm2.13.13.3.m3.2.3.3.2.1" xref="algorithm2.13.13.3.m3.2.3.3.2.1.cmml">⁢</mo><mi id="algorithm2.13.13.3.m3.2.3.3.2.3" xref="algorithm2.13.13.3.m3.2.3.3.2.3.cmml">t</mi></mrow><mo id="algorithm2.13.13.3.m3.2.3.3.1" xref="algorithm2.13.13.3.m3.2.3.3.1.cmml">−</mo><mn id="algorithm2.13.13.3.m3.2.3.3.3" xref="algorithm2.13.13.3.m3.2.3.3.3.cmml">1</mn></mrow></mrow><annotation-xml encoding="MathML-Content" id="algorithm2.13.13.3.m3.2b"><apply id="algorithm2.13.13.3.m3.2.3.cmml" xref="algorithm2.13.13.3.m3.2.3"><gt id="algorithm2.13.13.3.m3.2.3.1.cmml" xref="algorithm2.13.13.3.m3.2.3.1"></gt><apply id="algorithm2.13.13.3.m3.2.3.2.cmml" xref="algorithm2.13.13.3.m3.2.3.2"><times id="algorithm2.13.13.3.m3.2.3.2.1.cmml" xref="algorithm2.13.13.3.m3.2.3.2.1"></times><apply id="algorithm2.13.13.3.m3.2.3.2.2.cmml" xref="algorithm2.13.13.3.m3.2.3.2.2"><csymbol cd="ambiguous" id="algorithm2.13.13.3.m3.2.3.2.2.1.cmml" xref="algorithm2.13.13.3.m3.2.3.2.2">subscript</csymbol><ci id="algorithm2.13.13.3.m3.2.3.2.2.2.cmml" xref="algorithm2.13.13.3.m3.2.3.2.2.2">𝑑</ci><apply id="algorithm2.13.13.3.m3.2.3.2.2.3.cmml" xref="algorithm2.13.13.3.m3.2.3.2.2.3"><setdiff id="algorithm2.13.13.3.m3.2.3.2.2.3.1.cmml" xref="algorithm2.13.13.3.m3.2.3.2.2.3.1"></setdiff><apply id="algorithm2.13.13.3.m3.2.3.2.2.3.2.cmml" xref="algorithm2.13.13.3.m3.2.3.2.2.3.2"><csymbol cd="ambiguous" id="algorithm2.13.13.3.m3.2.3.2.2.3.2.1.cmml" xref="algorithm2.13.13.3.m3.2.3.2.2.3.2">subscript</csymbol><ci id="algorithm2.13.13.3.m3.2.3.2.2.3.2.2.cmml" xref="algorithm2.13.13.3.m3.2.3.2.2.3.2.2">𝐻</ci><ci id="algorithm2.13.13.3.m3.2.3.2.2.3.2.3.cmml" xref="algorithm2.13.13.3.m3.2.3.2.2.3.2.3">𝑗</ci></apply><ci id="algorithm2.13.13.3.m3.2.3.2.2.3.3.cmml" xref="algorithm2.13.13.3.m3.2.3.2.2.3.3">𝐹</ci></apply></apply><interval closure="open" id="algorithm2.13.13.3.m3.2.3.2.3.1.cmml" xref="algorithm2.13.13.3.m3.2.3.2.3.2"><ci id="algorithm2.13.13.3.m3.1.1.cmml" xref="algorithm2.13.13.3.m3.1.1">𝑢</ci><ci id="algorithm2.13.13.3.m3.2.2.cmml" xref="algorithm2.13.13.3.m3.2.2">𝑣</ci></interval></apply><apply id="algorithm2.13.13.3.m3.2.3.3.cmml" xref="algorithm2.13.13.3.m3.2.3.3"><minus id="algorithm2.13.13.3.m3.2.3.3.1.cmml" xref="algorithm2.13.13.3.m3.2.3.3.1"></minus><apply id="algorithm2.13.13.3.m3.2.3.3.2.cmml" xref="algorithm2.13.13.3.m3.2.3.3.2"><times id="algorithm2.13.13.3.m3.2.3.3.2.1.cmml" xref="algorithm2.13.13.3.m3.2.3.3.2.1"></times><cn id="algorithm2.13.13.3.m3.2.3.3.2.2.cmml" type="integer" xref="algorithm2.13.13.3.m3.2.3.3.2.2">2</cn><ci id="algorithm2.13.13.3.m3.2.3.3.2.3.cmml" xref="algorithm2.13.13.3.m3.2.3.3.2.3">𝑡</ci></apply><cn id="algorithm2.13.13.3.m3.2.3.3.3.cmml" type="integer" xref="algorithm2.13.13.3.m3.2.3.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="algorithm2.13.13.3.m3.2c">d_{H_{j}\setminus F}(u,v)>2t-1</annotation><annotation encoding="application/x-llamapun" id="algorithm2.13.13.3.m3.2d">italic_d start_POSTSUBSCRIPT italic_H start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ∖ italic_F end_POSTSUBSCRIPT ( italic_u , italic_v ) > 2 italic_t - 1</annotation></semantics></math></em> <span class="ltx_text ltx_font_bold" id="algorithm2.13.13.5">then</span> </div> <div class="ltx_listingline" id="algorithm2.15.15"> <span class="ltx_rule" style="width:1px;height:100%;background:black;display:inline-block;"> </span> <span class="ltx_rule" style="width:1px;height:100%;background:black;display:inline-block;"> </span> <span class="ltx_text ltx_font_bold" id="algorithm2.15.15.1">add</span> <math alttext="(u,v)" class="ltx_Math" display="inline" id="algorithm2.14.14.m1.2"><semantics id="algorithm2.14.14.m1.2a"><mrow id="algorithm2.14.14.m1.2.3.2" xref="algorithm2.14.14.m1.2.3.1.cmml"><mo id="algorithm2.14.14.m1.2.3.2.1" stretchy="false" xref="algorithm2.14.14.m1.2.3.1.cmml">(</mo><mi id="algorithm2.14.14.m1.1.1" xref="algorithm2.14.14.m1.1.1.cmml">u</mi><mo id="algorithm2.14.14.m1.2.3.2.2" xref="algorithm2.14.14.m1.2.3.1.cmml">,</mo><mi id="algorithm2.14.14.m1.2.2" xref="algorithm2.14.14.m1.2.2.cmml">v</mi><mo id="algorithm2.14.14.m1.2.3.2.3" stretchy="false" xref="algorithm2.14.14.m1.2.3.1.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="algorithm2.14.14.m1.2b"><interval closure="open" id="algorithm2.14.14.m1.2.3.1.cmml" xref="algorithm2.14.14.m1.2.3.2"><ci id="algorithm2.14.14.m1.1.1.cmml" xref="algorithm2.14.14.m1.1.1">𝑢</ci><ci id="algorithm2.14.14.m1.2.2.cmml" xref="algorithm2.14.14.m1.2.2">𝑣</ci></interval></annotation-xml><annotation encoding="application/x-tex" id="algorithm2.14.14.m1.2c">(u,v)</annotation><annotation encoding="application/x-llamapun" id="algorithm2.14.14.m1.2d">( italic_u , italic_v )</annotation></semantics></math> to <math alttext="H_{j}" class="ltx_Math" display="inline" id="algorithm2.15.15.m2.1"><semantics id="algorithm2.15.15.m2.1a"><msub id="algorithm2.15.15.m2.1.1" xref="algorithm2.15.15.m2.1.1.cmml"><mi id="algorithm2.15.15.m2.1.1.2" xref="algorithm2.15.15.m2.1.1.2.cmml">H</mi><mi id="algorithm2.15.15.m2.1.1.3" xref="algorithm2.15.15.m2.1.1.3.cmml">j</mi></msub><annotation-xml encoding="MathML-Content" id="algorithm2.15.15.m2.1b"><apply id="algorithm2.15.15.m2.1.1.cmml" xref="algorithm2.15.15.m2.1.1"><csymbol cd="ambiguous" id="algorithm2.15.15.m2.1.1.1.cmml" xref="algorithm2.15.15.m2.1.1">subscript</csymbol><ci id="algorithm2.15.15.m2.1.1.2.cmml" xref="algorithm2.15.15.m2.1.1.2">𝐻</ci><ci id="algorithm2.15.15.m2.1.1.3.cmml" xref="algorithm2.15.15.m2.1.1.3">𝑗</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="algorithm2.15.15.m2.1c">H_{j}</annotation><annotation encoding="application/x-llamapun" id="algorithm2.15.15.m2.1d">italic_H start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT</annotation></semantics></math> </div> <div class="ltx_listingline" id="algorithm2.16.16"> <span class="ltx_text ltx_font_bold" id="algorithm2.16.16.1">return</span> <math alttext="H\coloneqq H_{1}\cup\cdots\cup H_{T}" class="ltx_Math" display="inline" id="algorithm2.16.16.m1.1"><semantics id="algorithm2.16.16.m1.1a"><mrow id="algorithm2.16.16.m1.1.1" xref="algorithm2.16.16.m1.1.1.cmml"><mi id="algorithm2.16.16.m1.1.1.2" xref="algorithm2.16.16.m1.1.1.2.cmml">H</mi><mo id="algorithm2.16.16.m1.1.1.1" xref="algorithm2.16.16.m1.1.1.1.cmml">≔</mo><mrow id="algorithm2.16.16.m1.1.1.3" xref="algorithm2.16.16.m1.1.1.3.cmml"><msub id="algorithm2.16.16.m1.1.1.3.2" xref="algorithm2.16.16.m1.1.1.3.2.cmml"><mi id="algorithm2.16.16.m1.1.1.3.2.2" xref="algorithm2.16.16.m1.1.1.3.2.2.cmml">H</mi><mn id="algorithm2.16.16.m1.1.1.3.2.3" xref="algorithm2.16.16.m1.1.1.3.2.3.cmml">1</mn></msub><mo id="algorithm2.16.16.m1.1.1.3.1" xref="algorithm2.16.16.m1.1.1.3.1.cmml">∪</mo><mi id="algorithm2.16.16.m1.1.1.3.3" mathvariant="normal" xref="algorithm2.16.16.m1.1.1.3.3.cmml">⋯</mi><mo id="algorithm2.16.16.m1.1.1.3.1a" xref="algorithm2.16.16.m1.1.1.3.1.cmml">∪</mo><msub id="algorithm2.16.16.m1.1.1.3.4" xref="algorithm2.16.16.m1.1.1.3.4.cmml"><mi id="algorithm2.16.16.m1.1.1.3.4.2" xref="algorithm2.16.16.m1.1.1.3.4.2.cmml">H</mi><mi id="algorithm2.16.16.m1.1.1.3.4.3" xref="algorithm2.16.16.m1.1.1.3.4.3.cmml">T</mi></msub></mrow></mrow><annotation-xml encoding="MathML-Content" id="algorithm2.16.16.m1.1b"><apply id="algorithm2.16.16.m1.1.1.cmml" xref="algorithm2.16.16.m1.1.1"><ci id="algorithm2.16.16.m1.1.1.1.cmml" xref="algorithm2.16.16.m1.1.1.1">≔</ci><ci id="algorithm2.16.16.m1.1.1.2.cmml" xref="algorithm2.16.16.m1.1.1.2">𝐻</ci><apply id="algorithm2.16.16.m1.1.1.3.cmml" xref="algorithm2.16.16.m1.1.1.3"><union id="algorithm2.16.16.m1.1.1.3.1.cmml" xref="algorithm2.16.16.m1.1.1.3.1"></union><apply id="algorithm2.16.16.m1.1.1.3.2.cmml" xref="algorithm2.16.16.m1.1.1.3.2"><csymbol cd="ambiguous" id="algorithm2.16.16.m1.1.1.3.2.1.cmml" xref="algorithm2.16.16.m1.1.1.3.2">subscript</csymbol><ci id="algorithm2.16.16.m1.1.1.3.2.2.cmml" xref="algorithm2.16.16.m1.1.1.3.2.2">𝐻</ci><cn id="algorithm2.16.16.m1.1.1.3.2.3.cmml" type="integer" xref="algorithm2.16.16.m1.1.1.3.2.3">1</cn></apply><ci id="algorithm2.16.16.m1.1.1.3.3.cmml" xref="algorithm2.16.16.m1.1.1.3.3">⋯</ci><apply id="algorithm2.16.16.m1.1.1.3.4.cmml" xref="algorithm2.16.16.m1.1.1.3.4"><csymbol cd="ambiguous" id="algorithm2.16.16.m1.1.1.3.4.1.cmml" xref="algorithm2.16.16.m1.1.1.3.4">subscript</csymbol><ci id="algorithm2.16.16.m1.1.1.3.4.2.cmml" xref="algorithm2.16.16.m1.1.1.3.4.2">𝐻</ci><ci id="algorithm2.16.16.m1.1.1.3.4.3.cmml" xref="algorithm2.16.16.m1.1.1.3.4.3">𝑇</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="algorithm2.16.16.m1.1c">H\coloneqq H_{1}\cup\cdots\cup H_{T}</annotation><annotation encoding="application/x-llamapun" id="algorithm2.16.16.m1.1d">italic_H ≔ italic_H start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ∪ ⋯ ∪ italic_H start_POSTSUBSCRIPT italic_T end_POSTSUBSCRIPT</annotation></semantics></math> </div> </div> <figcaption class="ltx_caption"><span class="ltx_tag ltx_tag_float"><span class="ltx_text ltx_font_bold" id="algorithm2.18.1.1">Algorithm 2</span> </span>The streaming greedy algorithm for VFT (or EFT) spanners on weighted graphs.</figcaption> </figure> <div class="ltx_theorem ltx_theorem_theorem" id="S2.Thmtheorem6"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="S2.Thmtheorem6.1.1.1">Theorem 2.6</span></span><span class="ltx_text ltx_font_bold" id="S2.Thmtheorem6.2.2">.</span> </h6> <div class="ltx_para" id="S2.Thmtheorem6.p1"> <p class="ltx_p" id="S2.Thmtheorem6.p1.8">There exists a single-pass streaming algorithm (i.e., Algorithm <a class="ltx_ref" href="https://arxiv.org/html/2503.00712v1#algorithm2" title="In Weighted graphs. ‣ 2.1 Fault-Tolerant Spanners in Streaming ‣ 2 Preliminaries ‣ Streaming Algorithms for Network Design"><span class="ltx_text ltx_ref_tag">2</span></a>) that uses <math alttext="O(f^{1-1/t}\cdot n^{1+1/t}\cdot\epsilon^{-1}\cdot\log W)" class="ltx_Math" display="inline" id="S2.Thmtheorem6.p1.1.m1.1"><semantics id="S2.Thmtheorem6.p1.1.m1.1a"><mrow id="S2.Thmtheorem6.p1.1.m1.1.1" xref="S2.Thmtheorem6.p1.1.m1.1.1.cmml"><mi id="S2.Thmtheorem6.p1.1.m1.1.1.3" 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xref="S2.Thmtheorem6.p1.1.m1.1.1"><times id="S2.Thmtheorem6.p1.1.m1.1.1.2.cmml" xref="S2.Thmtheorem6.p1.1.m1.1.1.2"></times><ci id="S2.Thmtheorem6.p1.1.m1.1.1.3.cmml" xref="S2.Thmtheorem6.p1.1.m1.1.1.3">𝑂</ci><apply id="S2.Thmtheorem6.p1.1.m1.1.1.1.1.1.cmml" xref="S2.Thmtheorem6.p1.1.m1.1.1.1.1"><ci id="S2.Thmtheorem6.p1.1.m1.1.1.1.1.1.1.cmml" xref="S2.Thmtheorem6.p1.1.m1.1.1.1.1.1.1">⋅</ci><apply id="S2.Thmtheorem6.p1.1.m1.1.1.1.1.1.2.cmml" xref="S2.Thmtheorem6.p1.1.m1.1.1.1.1.1.2"><csymbol cd="ambiguous" id="S2.Thmtheorem6.p1.1.m1.1.1.1.1.1.2.1.cmml" xref="S2.Thmtheorem6.p1.1.m1.1.1.1.1.1.2">superscript</csymbol><ci id="S2.Thmtheorem6.p1.1.m1.1.1.1.1.1.2.2.cmml" xref="S2.Thmtheorem6.p1.1.m1.1.1.1.1.1.2.2">𝑓</ci><apply id="S2.Thmtheorem6.p1.1.m1.1.1.1.1.1.2.3.cmml" xref="S2.Thmtheorem6.p1.1.m1.1.1.1.1.1.2.3"><minus id="S2.Thmtheorem6.p1.1.m1.1.1.1.1.1.2.3.1.cmml" xref="S2.Thmtheorem6.p1.1.m1.1.1.1.1.1.2.3.1"></minus><cn id="S2.Thmtheorem6.p1.1.m1.1.1.1.1.1.2.3.2.cmml" type="integer" xref="S2.Thmtheorem6.p1.1.m1.1.1.1.1.1.2.3.2">1</cn><apply id="S2.Thmtheorem6.p1.1.m1.1.1.1.1.1.2.3.3.cmml" xref="S2.Thmtheorem6.p1.1.m1.1.1.1.1.1.2.3.3"><divide id="S2.Thmtheorem6.p1.1.m1.1.1.1.1.1.2.3.3.1.cmml" xref="S2.Thmtheorem6.p1.1.m1.1.1.1.1.1.2.3.3.1"></divide><cn id="S2.Thmtheorem6.p1.1.m1.1.1.1.1.1.2.3.3.2.cmml" type="integer" xref="S2.Thmtheorem6.p1.1.m1.1.1.1.1.1.2.3.3.2">1</cn><ci id="S2.Thmtheorem6.p1.1.m1.1.1.1.1.1.2.3.3.3.cmml" xref="S2.Thmtheorem6.p1.1.m1.1.1.1.1.1.2.3.3.3">𝑡</ci></apply></apply></apply><apply id="S2.Thmtheorem6.p1.1.m1.1.1.1.1.1.3.cmml" xref="S2.Thmtheorem6.p1.1.m1.1.1.1.1.1.3"><csymbol cd="ambiguous" id="S2.Thmtheorem6.p1.1.m1.1.1.1.1.1.3.1.cmml" xref="S2.Thmtheorem6.p1.1.m1.1.1.1.1.1.3">superscript</csymbol><ci id="S2.Thmtheorem6.p1.1.m1.1.1.1.1.1.3.2.cmml" xref="S2.Thmtheorem6.p1.1.m1.1.1.1.1.1.3.2">𝑛</ci><apply id="S2.Thmtheorem6.p1.1.m1.1.1.1.1.1.3.3.cmml" xref="S2.Thmtheorem6.p1.1.m1.1.1.1.1.1.3.3"><plus id="S2.Thmtheorem6.p1.1.m1.1.1.1.1.1.3.3.1.cmml" xref="S2.Thmtheorem6.p1.1.m1.1.1.1.1.1.3.3.1"></plus><cn id="S2.Thmtheorem6.p1.1.m1.1.1.1.1.1.3.3.2.cmml" type="integer" xref="S2.Thmtheorem6.p1.1.m1.1.1.1.1.1.3.3.2">1</cn><apply id="S2.Thmtheorem6.p1.1.m1.1.1.1.1.1.3.3.3.cmml" xref="S2.Thmtheorem6.p1.1.m1.1.1.1.1.1.3.3.3"><divide id="S2.Thmtheorem6.p1.1.m1.1.1.1.1.1.3.3.3.1.cmml" xref="S2.Thmtheorem6.p1.1.m1.1.1.1.1.1.3.3.3.1"></divide><cn id="S2.Thmtheorem6.p1.1.m1.1.1.1.1.1.3.3.3.2.cmml" type="integer" xref="S2.Thmtheorem6.p1.1.m1.1.1.1.1.1.3.3.3.2">1</cn><ci id="S2.Thmtheorem6.p1.1.m1.1.1.1.1.1.3.3.3.3.cmml" xref="S2.Thmtheorem6.p1.1.m1.1.1.1.1.1.3.3.3.3">𝑡</ci></apply></apply></apply><apply id="S2.Thmtheorem6.p1.1.m1.1.1.1.1.1.4.cmml" xref="S2.Thmtheorem6.p1.1.m1.1.1.1.1.1.4"><csymbol cd="ambiguous" id="S2.Thmtheorem6.p1.1.m1.1.1.1.1.1.4.1.cmml" xref="S2.Thmtheorem6.p1.1.m1.1.1.1.1.1.4">superscript</csymbol><ci id="S2.Thmtheorem6.p1.1.m1.1.1.1.1.1.4.2.cmml" xref="S2.Thmtheorem6.p1.1.m1.1.1.1.1.1.4.2">italic-ϵ</ci><apply id="S2.Thmtheorem6.p1.1.m1.1.1.1.1.1.4.3.cmml" xref="S2.Thmtheorem6.p1.1.m1.1.1.1.1.1.4.3"><minus id="S2.Thmtheorem6.p1.1.m1.1.1.1.1.1.4.3.1.cmml" xref="S2.Thmtheorem6.p1.1.m1.1.1.1.1.1.4.3"></minus><cn id="S2.Thmtheorem6.p1.1.m1.1.1.1.1.1.4.3.2.cmml" type="integer" xref="S2.Thmtheorem6.p1.1.m1.1.1.1.1.1.4.3.2">1</cn></apply></apply><apply id="S2.Thmtheorem6.p1.1.m1.1.1.1.1.1.5.cmml" xref="S2.Thmtheorem6.p1.1.m1.1.1.1.1.1.5"><log id="S2.Thmtheorem6.p1.1.m1.1.1.1.1.1.5.1.cmml" xref="S2.Thmtheorem6.p1.1.m1.1.1.1.1.1.5.1"></log><ci id="S2.Thmtheorem6.p1.1.m1.1.1.1.1.1.5.2.cmml" xref="S2.Thmtheorem6.p1.1.m1.1.1.1.1.1.5.2">𝑊</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem6.p1.1.m1.1c">O(f^{1-1/t}\cdot n^{1+1/t}\cdot\epsilon^{-1}\cdot\log W)</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem6.p1.1.m1.1d">italic_O ( italic_f start_POSTSUPERSCRIPT 1 - 1 / italic_t end_POSTSUPERSCRIPT ⋅ italic_n start_POSTSUPERSCRIPT 1 + 1 / italic_t end_POSTSUPERSCRIPT ⋅ italic_ϵ start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT ⋅ roman_log italic_W )</annotation></semantics></math> words of space and returns an <math alttext="f" class="ltx_Math" display="inline" id="S2.Thmtheorem6.p1.2.m2.1"><semantics id="S2.Thmtheorem6.p1.2.m2.1a"><mi id="S2.Thmtheorem6.p1.2.m2.1.1" xref="S2.Thmtheorem6.p1.2.m2.1.1.cmml">f</mi><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem6.p1.2.m2.1b"><ci id="S2.Thmtheorem6.p1.2.m2.1.1.cmml" xref="S2.Thmtheorem6.p1.2.m2.1.1">𝑓</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem6.p1.2.m2.1c">f</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem6.p1.2.m2.1d">italic_f</annotation></semantics></math>-VFT (or <math alttext="f" class="ltx_Math" display="inline" id="S2.Thmtheorem6.p1.3.m3.1"><semantics id="S2.Thmtheorem6.p1.3.m3.1a"><mi id="S2.Thmtheorem6.p1.3.m3.1.1" xref="S2.Thmtheorem6.p1.3.m3.1.1.cmml">f</mi><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem6.p1.3.m3.1b"><ci id="S2.Thmtheorem6.p1.3.m3.1.1.cmml" xref="S2.Thmtheorem6.p1.3.m3.1.1">𝑓</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem6.p1.3.m3.1c">f</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem6.p1.3.m3.1d">italic_f</annotation></semantics></math>-EFT) <math alttext="\big{(}(1+\epsilon)(2t-1)\big{)}" class="ltx_Math" display="inline" id="S2.Thmtheorem6.p1.4.m4.1"><semantics id="S2.Thmtheorem6.p1.4.m4.1a"><mrow id="S2.Thmtheorem6.p1.4.m4.1.1.1" xref="S2.Thmtheorem6.p1.4.m4.1.1.1.1.cmml"><mo id="S2.Thmtheorem6.p1.4.m4.1.1.1.2" maxsize="120%" minsize="120%" xref="S2.Thmtheorem6.p1.4.m4.1.1.1.1.cmml">(</mo><mrow id="S2.Thmtheorem6.p1.4.m4.1.1.1.1" xref="S2.Thmtheorem6.p1.4.m4.1.1.1.1.cmml"><mrow id="S2.Thmtheorem6.p1.4.m4.1.1.1.1.1.1" xref="S2.Thmtheorem6.p1.4.m4.1.1.1.1.1.1.1.cmml"><mo id="S2.Thmtheorem6.p1.4.m4.1.1.1.1.1.1.2" stretchy="false" xref="S2.Thmtheorem6.p1.4.m4.1.1.1.1.1.1.1.cmml">(</mo><mrow id="S2.Thmtheorem6.p1.4.m4.1.1.1.1.1.1.1" xref="S2.Thmtheorem6.p1.4.m4.1.1.1.1.1.1.1.cmml"><mn id="S2.Thmtheorem6.p1.4.m4.1.1.1.1.1.1.1.2" xref="S2.Thmtheorem6.p1.4.m4.1.1.1.1.1.1.1.2.cmml">1</mn><mo id="S2.Thmtheorem6.p1.4.m4.1.1.1.1.1.1.1.1" xref="S2.Thmtheorem6.p1.4.m4.1.1.1.1.1.1.1.1.cmml">+</mo><mi id="S2.Thmtheorem6.p1.4.m4.1.1.1.1.1.1.1.3" xref="S2.Thmtheorem6.p1.4.m4.1.1.1.1.1.1.1.3.cmml">ϵ</mi></mrow><mo id="S2.Thmtheorem6.p1.4.m4.1.1.1.1.1.1.3" stretchy="false" xref="S2.Thmtheorem6.p1.4.m4.1.1.1.1.1.1.1.cmml">)</mo></mrow><mo id="S2.Thmtheorem6.p1.4.m4.1.1.1.1.3" xref="S2.Thmtheorem6.p1.4.m4.1.1.1.1.3.cmml">⁢</mo><mrow id="S2.Thmtheorem6.p1.4.m4.1.1.1.1.2.1" xref="S2.Thmtheorem6.p1.4.m4.1.1.1.1.2.1.1.cmml"><mo id="S2.Thmtheorem6.p1.4.m4.1.1.1.1.2.1.2" stretchy="false" xref="S2.Thmtheorem6.p1.4.m4.1.1.1.1.2.1.1.cmml">(</mo><mrow id="S2.Thmtheorem6.p1.4.m4.1.1.1.1.2.1.1" xref="S2.Thmtheorem6.p1.4.m4.1.1.1.1.2.1.1.cmml"><mrow id="S2.Thmtheorem6.p1.4.m4.1.1.1.1.2.1.1.2" xref="S2.Thmtheorem6.p1.4.m4.1.1.1.1.2.1.1.2.cmml"><mn id="S2.Thmtheorem6.p1.4.m4.1.1.1.1.2.1.1.2.2" xref="S2.Thmtheorem6.p1.4.m4.1.1.1.1.2.1.1.2.2.cmml">2</mn><mo id="S2.Thmtheorem6.p1.4.m4.1.1.1.1.2.1.1.2.1" xref="S2.Thmtheorem6.p1.4.m4.1.1.1.1.2.1.1.2.1.cmml">⁢</mo><mi id="S2.Thmtheorem6.p1.4.m4.1.1.1.1.2.1.1.2.3" xref="S2.Thmtheorem6.p1.4.m4.1.1.1.1.2.1.1.2.3.cmml">t</mi></mrow><mo id="S2.Thmtheorem6.p1.4.m4.1.1.1.1.2.1.1.1" xref="S2.Thmtheorem6.p1.4.m4.1.1.1.1.2.1.1.1.cmml">−</mo><mn id="S2.Thmtheorem6.p1.4.m4.1.1.1.1.2.1.1.3" xref="S2.Thmtheorem6.p1.4.m4.1.1.1.1.2.1.1.3.cmml">1</mn></mrow><mo id="S2.Thmtheorem6.p1.4.m4.1.1.1.1.2.1.3" stretchy="false" xref="S2.Thmtheorem6.p1.4.m4.1.1.1.1.2.1.1.cmml">)</mo></mrow></mrow><mo id="S2.Thmtheorem6.p1.4.m4.1.1.1.3" maxsize="120%" minsize="120%" xref="S2.Thmtheorem6.p1.4.m4.1.1.1.1.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem6.p1.4.m4.1b"><apply id="S2.Thmtheorem6.p1.4.m4.1.1.1.1.cmml" xref="S2.Thmtheorem6.p1.4.m4.1.1.1"><times id="S2.Thmtheorem6.p1.4.m4.1.1.1.1.3.cmml" xref="S2.Thmtheorem6.p1.4.m4.1.1.1.1.3"></times><apply id="S2.Thmtheorem6.p1.4.m4.1.1.1.1.1.1.1.cmml" xref="S2.Thmtheorem6.p1.4.m4.1.1.1.1.1.1"><plus id="S2.Thmtheorem6.p1.4.m4.1.1.1.1.1.1.1.1.cmml" xref="S2.Thmtheorem6.p1.4.m4.1.1.1.1.1.1.1.1"></plus><cn id="S2.Thmtheorem6.p1.4.m4.1.1.1.1.1.1.1.2.cmml" type="integer" xref="S2.Thmtheorem6.p1.4.m4.1.1.1.1.1.1.1.2">1</cn><ci id="S2.Thmtheorem6.p1.4.m4.1.1.1.1.1.1.1.3.cmml" xref="S2.Thmtheorem6.p1.4.m4.1.1.1.1.1.1.1.3">italic-ϵ</ci></apply><apply id="S2.Thmtheorem6.p1.4.m4.1.1.1.1.2.1.1.cmml" xref="S2.Thmtheorem6.p1.4.m4.1.1.1.1.2.1"><minus id="S2.Thmtheorem6.p1.4.m4.1.1.1.1.2.1.1.1.cmml" xref="S2.Thmtheorem6.p1.4.m4.1.1.1.1.2.1.1.1"></minus><apply id="S2.Thmtheorem6.p1.4.m4.1.1.1.1.2.1.1.2.cmml" xref="S2.Thmtheorem6.p1.4.m4.1.1.1.1.2.1.1.2"><times id="S2.Thmtheorem6.p1.4.m4.1.1.1.1.2.1.1.2.1.cmml" xref="S2.Thmtheorem6.p1.4.m4.1.1.1.1.2.1.1.2.1"></times><cn id="S2.Thmtheorem6.p1.4.m4.1.1.1.1.2.1.1.2.2.cmml" type="integer" xref="S2.Thmtheorem6.p1.4.m4.1.1.1.1.2.1.1.2.2">2</cn><ci id="S2.Thmtheorem6.p1.4.m4.1.1.1.1.2.1.1.2.3.cmml" xref="S2.Thmtheorem6.p1.4.m4.1.1.1.1.2.1.1.2.3">𝑡</ci></apply><cn id="S2.Thmtheorem6.p1.4.m4.1.1.1.1.2.1.1.3.cmml" type="integer" xref="S2.Thmtheorem6.p1.4.m4.1.1.1.1.2.1.1.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem6.p1.4.m4.1c">\big{(}(1+\epsilon)(2t-1)\big{)}</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem6.p1.4.m4.1d">( ( 1 + italic_ϵ ) ( 2 italic_t - 1 ) )</annotation></semantics></math>-spanner of an <math alttext="n" class="ltx_Math" display="inline" id="S2.Thmtheorem6.p1.5.m5.1"><semantics id="S2.Thmtheorem6.p1.5.m5.1a"><mi id="S2.Thmtheorem6.p1.5.m5.1.1" xref="S2.Thmtheorem6.p1.5.m5.1.1.cmml">n</mi><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem6.p1.5.m5.1b"><ci id="S2.Thmtheorem6.p1.5.m5.1.1.cmml" xref="S2.Thmtheorem6.p1.5.m5.1.1">𝑛</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem6.p1.5.m5.1c">n</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem6.p1.5.m5.1d">italic_n</annotation></semantics></math>-vertex weighted graph <math alttext="G=(V,E)" class="ltx_Math" display="inline" id="S2.Thmtheorem6.p1.6.m6.2"><semantics id="S2.Thmtheorem6.p1.6.m6.2a"><mrow id="S2.Thmtheorem6.p1.6.m6.2.3" xref="S2.Thmtheorem6.p1.6.m6.2.3.cmml"><mi id="S2.Thmtheorem6.p1.6.m6.2.3.2" xref="S2.Thmtheorem6.p1.6.m6.2.3.2.cmml">G</mi><mo id="S2.Thmtheorem6.p1.6.m6.2.3.1" xref="S2.Thmtheorem6.p1.6.m6.2.3.1.cmml">=</mo><mrow id="S2.Thmtheorem6.p1.6.m6.2.3.3.2" xref="S2.Thmtheorem6.p1.6.m6.2.3.3.1.cmml"><mo id="S2.Thmtheorem6.p1.6.m6.2.3.3.2.1" stretchy="false" xref="S2.Thmtheorem6.p1.6.m6.2.3.3.1.cmml">(</mo><mi id="S2.Thmtheorem6.p1.6.m6.1.1" xref="S2.Thmtheorem6.p1.6.m6.1.1.cmml">V</mi><mo id="S2.Thmtheorem6.p1.6.m6.2.3.3.2.2" xref="S2.Thmtheorem6.p1.6.m6.2.3.3.1.cmml">,</mo><mi id="S2.Thmtheorem6.p1.6.m6.2.2" xref="S2.Thmtheorem6.p1.6.m6.2.2.cmml">E</mi><mo id="S2.Thmtheorem6.p1.6.m6.2.3.3.2.3" stretchy="false" xref="S2.Thmtheorem6.p1.6.m6.2.3.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem6.p1.6.m6.2b"><apply id="S2.Thmtheorem6.p1.6.m6.2.3.cmml" xref="S2.Thmtheorem6.p1.6.m6.2.3"><eq id="S2.Thmtheorem6.p1.6.m6.2.3.1.cmml" xref="S2.Thmtheorem6.p1.6.m6.2.3.1"></eq><ci id="S2.Thmtheorem6.p1.6.m6.2.3.2.cmml" xref="S2.Thmtheorem6.p1.6.m6.2.3.2">𝐺</ci><interval closure="open" id="S2.Thmtheorem6.p1.6.m6.2.3.3.1.cmml" xref="S2.Thmtheorem6.p1.6.m6.2.3.3.2"><ci id="S2.Thmtheorem6.p1.6.m6.1.1.cmml" xref="S2.Thmtheorem6.p1.6.m6.1.1">𝑉</ci><ci id="S2.Thmtheorem6.p1.6.m6.2.2.cmml" xref="S2.Thmtheorem6.p1.6.m6.2.2">𝐸</ci></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem6.p1.6.m6.2c">G=(V,E)</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem6.p1.6.m6.2d">italic_G = ( italic_V , italic_E )</annotation></semantics></math> with the weight function <math alttext="w:E\rightarrow\{0,1,\cdots,W\}" class="ltx_Math" display="inline" id="S2.Thmtheorem6.p1.7.m7.4"><semantics id="S2.Thmtheorem6.p1.7.m7.4a"><mrow id="S2.Thmtheorem6.p1.7.m7.4.5" xref="S2.Thmtheorem6.p1.7.m7.4.5.cmml"><mi id="S2.Thmtheorem6.p1.7.m7.4.5.2" xref="S2.Thmtheorem6.p1.7.m7.4.5.2.cmml">w</mi><mo id="S2.Thmtheorem6.p1.7.m7.4.5.1" lspace="0.278em" rspace="0.278em" xref="S2.Thmtheorem6.p1.7.m7.4.5.1.cmml">:</mo><mrow id="S2.Thmtheorem6.p1.7.m7.4.5.3" xref="S2.Thmtheorem6.p1.7.m7.4.5.3.cmml"><mi id="S2.Thmtheorem6.p1.7.m7.4.5.3.2" xref="S2.Thmtheorem6.p1.7.m7.4.5.3.2.cmml">E</mi><mo id="S2.Thmtheorem6.p1.7.m7.4.5.3.1" stretchy="false" xref="S2.Thmtheorem6.p1.7.m7.4.5.3.1.cmml">→</mo><mrow id="S2.Thmtheorem6.p1.7.m7.4.5.3.3.2" xref="S2.Thmtheorem6.p1.7.m7.4.5.3.3.1.cmml"><mo id="S2.Thmtheorem6.p1.7.m7.4.5.3.3.2.1" stretchy="false" xref="S2.Thmtheorem6.p1.7.m7.4.5.3.3.1.cmml">{</mo><mn id="S2.Thmtheorem6.p1.7.m7.1.1" xref="S2.Thmtheorem6.p1.7.m7.1.1.cmml">0</mn><mo id="S2.Thmtheorem6.p1.7.m7.4.5.3.3.2.2" xref="S2.Thmtheorem6.p1.7.m7.4.5.3.3.1.cmml">,</mo><mn id="S2.Thmtheorem6.p1.7.m7.2.2" xref="S2.Thmtheorem6.p1.7.m7.2.2.cmml">1</mn><mo id="S2.Thmtheorem6.p1.7.m7.4.5.3.3.2.3" xref="S2.Thmtheorem6.p1.7.m7.4.5.3.3.1.cmml">,</mo><mi id="S2.Thmtheorem6.p1.7.m7.3.3" mathvariant="normal" xref="S2.Thmtheorem6.p1.7.m7.3.3.cmml">⋯</mi><mo id="S2.Thmtheorem6.p1.7.m7.4.5.3.3.2.4" xref="S2.Thmtheorem6.p1.7.m7.4.5.3.3.1.cmml">,</mo><mi id="S2.Thmtheorem6.p1.7.m7.4.4" xref="S2.Thmtheorem6.p1.7.m7.4.4.cmml">W</mi><mo id="S2.Thmtheorem6.p1.7.m7.4.5.3.3.2.5" stretchy="false" xref="S2.Thmtheorem6.p1.7.m7.4.5.3.3.1.cmml">}</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem6.p1.7.m7.4b"><apply id="S2.Thmtheorem6.p1.7.m7.4.5.cmml" xref="S2.Thmtheorem6.p1.7.m7.4.5"><ci id="S2.Thmtheorem6.p1.7.m7.4.5.1.cmml" xref="S2.Thmtheorem6.p1.7.m7.4.5.1">:</ci><ci id="S2.Thmtheorem6.p1.7.m7.4.5.2.cmml" xref="S2.Thmtheorem6.p1.7.m7.4.5.2">𝑤</ci><apply id="S2.Thmtheorem6.p1.7.m7.4.5.3.cmml" xref="S2.Thmtheorem6.p1.7.m7.4.5.3"><ci id="S2.Thmtheorem6.p1.7.m7.4.5.3.1.cmml" xref="S2.Thmtheorem6.p1.7.m7.4.5.3.1">→</ci><ci id="S2.Thmtheorem6.p1.7.m7.4.5.3.2.cmml" xref="S2.Thmtheorem6.p1.7.m7.4.5.3.2">𝐸</ci><set id="S2.Thmtheorem6.p1.7.m7.4.5.3.3.1.cmml" xref="S2.Thmtheorem6.p1.7.m7.4.5.3.3.2"><cn id="S2.Thmtheorem6.p1.7.m7.1.1.cmml" type="integer" xref="S2.Thmtheorem6.p1.7.m7.1.1">0</cn><cn id="S2.Thmtheorem6.p1.7.m7.2.2.cmml" type="integer" xref="S2.Thmtheorem6.p1.7.m7.2.2">1</cn><ci id="S2.Thmtheorem6.p1.7.m7.3.3.cmml" xref="S2.Thmtheorem6.p1.7.m7.3.3">⋯</ci><ci id="S2.Thmtheorem6.p1.7.m7.4.4.cmml" xref="S2.Thmtheorem6.p1.7.m7.4.4">𝑊</ci></set></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem6.p1.7.m7.4c">w:E\rightarrow\{0,1,\cdots,W\}</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem6.p1.7.m7.4d">italic_w : italic_E → { 0 , 1 , ⋯ , italic_W }</annotation></semantics></math>, of size <math alttext="O(f^{1-1/t}\cdot n^{1+1/t}\cdot\epsilon^{-1}\cdot\log W)" class="ltx_Math" display="inline" id="S2.Thmtheorem6.p1.8.m8.1"><semantics id="S2.Thmtheorem6.p1.8.m8.1a"><mrow id="S2.Thmtheorem6.p1.8.m8.1.1" xref="S2.Thmtheorem6.p1.8.m8.1.1.cmml"><mi id="S2.Thmtheorem6.p1.8.m8.1.1.3" xref="S2.Thmtheorem6.p1.8.m8.1.1.3.cmml">O</mi><mo id="S2.Thmtheorem6.p1.8.m8.1.1.2" xref="S2.Thmtheorem6.p1.8.m8.1.1.2.cmml">⁢</mo><mrow id="S2.Thmtheorem6.p1.8.m8.1.1.1.1" xref="S2.Thmtheorem6.p1.8.m8.1.1.1.1.1.cmml"><mo id="S2.Thmtheorem6.p1.8.m8.1.1.1.1.2" 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id="S2.Thmtheorem6.p1.8.m8.1.1.1.1.1.2.3.3.3" xref="S2.Thmtheorem6.p1.8.m8.1.1.1.1.1.2.3.3.3.cmml">t</mi></mrow></mrow></msup><mo id="S2.Thmtheorem6.p1.8.m8.1.1.1.1.1.1" lspace="0.222em" rspace="0.222em" xref="S2.Thmtheorem6.p1.8.m8.1.1.1.1.1.1.cmml">⋅</mo><msup id="S2.Thmtheorem6.p1.8.m8.1.1.1.1.1.3" xref="S2.Thmtheorem6.p1.8.m8.1.1.1.1.1.3.cmml"><mi id="S2.Thmtheorem6.p1.8.m8.1.1.1.1.1.3.2" xref="S2.Thmtheorem6.p1.8.m8.1.1.1.1.1.3.2.cmml">n</mi><mrow id="S2.Thmtheorem6.p1.8.m8.1.1.1.1.1.3.3" xref="S2.Thmtheorem6.p1.8.m8.1.1.1.1.1.3.3.cmml"><mn id="S2.Thmtheorem6.p1.8.m8.1.1.1.1.1.3.3.2" xref="S2.Thmtheorem6.p1.8.m8.1.1.1.1.1.3.3.2.cmml">1</mn><mo id="S2.Thmtheorem6.p1.8.m8.1.1.1.1.1.3.3.1" xref="S2.Thmtheorem6.p1.8.m8.1.1.1.1.1.3.3.1.cmml">+</mo><mrow id="S2.Thmtheorem6.p1.8.m8.1.1.1.1.1.3.3.3" xref="S2.Thmtheorem6.p1.8.m8.1.1.1.1.1.3.3.3.cmml"><mn id="S2.Thmtheorem6.p1.8.m8.1.1.1.1.1.3.3.3.2" xref="S2.Thmtheorem6.p1.8.m8.1.1.1.1.1.3.3.3.2.cmml">1</mn><mo 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xref="S2.Thmtheorem6.p1.8.m8.1.1.1.1.1.2.3.3.1"></divide><cn id="S2.Thmtheorem6.p1.8.m8.1.1.1.1.1.2.3.3.2.cmml" type="integer" xref="S2.Thmtheorem6.p1.8.m8.1.1.1.1.1.2.3.3.2">1</cn><ci id="S2.Thmtheorem6.p1.8.m8.1.1.1.1.1.2.3.3.3.cmml" xref="S2.Thmtheorem6.p1.8.m8.1.1.1.1.1.2.3.3.3">𝑡</ci></apply></apply></apply><apply id="S2.Thmtheorem6.p1.8.m8.1.1.1.1.1.3.cmml" xref="S2.Thmtheorem6.p1.8.m8.1.1.1.1.1.3"><csymbol cd="ambiguous" id="S2.Thmtheorem6.p1.8.m8.1.1.1.1.1.3.1.cmml" xref="S2.Thmtheorem6.p1.8.m8.1.1.1.1.1.3">superscript</csymbol><ci id="S2.Thmtheorem6.p1.8.m8.1.1.1.1.1.3.2.cmml" xref="S2.Thmtheorem6.p1.8.m8.1.1.1.1.1.3.2">𝑛</ci><apply id="S2.Thmtheorem6.p1.8.m8.1.1.1.1.1.3.3.cmml" xref="S2.Thmtheorem6.p1.8.m8.1.1.1.1.1.3.3"><plus id="S2.Thmtheorem6.p1.8.m8.1.1.1.1.1.3.3.1.cmml" xref="S2.Thmtheorem6.p1.8.m8.1.1.1.1.1.3.3.1"></plus><cn id="S2.Thmtheorem6.p1.8.m8.1.1.1.1.1.3.3.2.cmml" type="integer" xref="S2.Thmtheorem6.p1.8.m8.1.1.1.1.1.3.3.2">1</cn><apply id="S2.Thmtheorem6.p1.8.m8.1.1.1.1.1.3.3.3.cmml" xref="S2.Thmtheorem6.p1.8.m8.1.1.1.1.1.3.3.3"><divide id="S2.Thmtheorem6.p1.8.m8.1.1.1.1.1.3.3.3.1.cmml" xref="S2.Thmtheorem6.p1.8.m8.1.1.1.1.1.3.3.3.1"></divide><cn id="S2.Thmtheorem6.p1.8.m8.1.1.1.1.1.3.3.3.2.cmml" type="integer" xref="S2.Thmtheorem6.p1.8.m8.1.1.1.1.1.3.3.3.2">1</cn><ci id="S2.Thmtheorem6.p1.8.m8.1.1.1.1.1.3.3.3.3.cmml" xref="S2.Thmtheorem6.p1.8.m8.1.1.1.1.1.3.3.3.3">𝑡</ci></apply></apply></apply><apply id="S2.Thmtheorem6.p1.8.m8.1.1.1.1.1.4.cmml" xref="S2.Thmtheorem6.p1.8.m8.1.1.1.1.1.4"><csymbol cd="ambiguous" id="S2.Thmtheorem6.p1.8.m8.1.1.1.1.1.4.1.cmml" xref="S2.Thmtheorem6.p1.8.m8.1.1.1.1.1.4">superscript</csymbol><ci id="S2.Thmtheorem6.p1.8.m8.1.1.1.1.1.4.2.cmml" xref="S2.Thmtheorem6.p1.8.m8.1.1.1.1.1.4.2">italic-ϵ</ci><apply id="S2.Thmtheorem6.p1.8.m8.1.1.1.1.1.4.3.cmml" xref="S2.Thmtheorem6.p1.8.m8.1.1.1.1.1.4.3"><minus id="S2.Thmtheorem6.p1.8.m8.1.1.1.1.1.4.3.1.cmml" xref="S2.Thmtheorem6.p1.8.m8.1.1.1.1.1.4.3"></minus><cn id="S2.Thmtheorem6.p1.8.m8.1.1.1.1.1.4.3.2.cmml" type="integer" xref="S2.Thmtheorem6.p1.8.m8.1.1.1.1.1.4.3.2">1</cn></apply></apply><apply id="S2.Thmtheorem6.p1.8.m8.1.1.1.1.1.5.cmml" xref="S2.Thmtheorem6.p1.8.m8.1.1.1.1.1.5"><log id="S2.Thmtheorem6.p1.8.m8.1.1.1.1.1.5.1.cmml" xref="S2.Thmtheorem6.p1.8.m8.1.1.1.1.1.5.1"></log><ci id="S2.Thmtheorem6.p1.8.m8.1.1.1.1.1.5.2.cmml" xref="S2.Thmtheorem6.p1.8.m8.1.1.1.1.1.5.2">𝑊</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem6.p1.8.m8.1c">O(f^{1-1/t}\cdot n^{1+1/t}\cdot\epsilon^{-1}\cdot\log W)</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem6.p1.8.m8.1d">italic_O ( italic_f start_POSTSUPERSCRIPT 1 - 1 / italic_t end_POSTSUPERSCRIPT ⋅ italic_n start_POSTSUPERSCRIPT 1 + 1 / italic_t end_POSTSUPERSCRIPT ⋅ italic_ϵ start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT ⋅ roman_log italic_W )</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="S2.Thmtheorem6.p2"> <p class="ltx_p" id="S2.Thmtheorem6.p2.7">In particular, if <math alttext="W=n^{O(\log n)}" class="ltx_Math" display="inline" id="S2.Thmtheorem6.p2.1.m1.1"><semantics id="S2.Thmtheorem6.p2.1.m1.1a"><mrow id="S2.Thmtheorem6.p2.1.m1.1.2" xref="S2.Thmtheorem6.p2.1.m1.1.2.cmml"><mi id="S2.Thmtheorem6.p2.1.m1.1.2.2" xref="S2.Thmtheorem6.p2.1.m1.1.2.2.cmml">W</mi><mo id="S2.Thmtheorem6.p2.1.m1.1.2.1" xref="S2.Thmtheorem6.p2.1.m1.1.2.1.cmml">=</mo><msup id="S2.Thmtheorem6.p2.1.m1.1.2.3" xref="S2.Thmtheorem6.p2.1.m1.1.2.3.cmml"><mi id="S2.Thmtheorem6.p2.1.m1.1.2.3.2" xref="S2.Thmtheorem6.p2.1.m1.1.2.3.2.cmml">n</mi><mrow id="S2.Thmtheorem6.p2.1.m1.1.1.1" xref="S2.Thmtheorem6.p2.1.m1.1.1.1.cmml"><mi id="S2.Thmtheorem6.p2.1.m1.1.1.1.3" xref="S2.Thmtheorem6.p2.1.m1.1.1.1.3.cmml">O</mi><mo id="S2.Thmtheorem6.p2.1.m1.1.1.1.2" xref="S2.Thmtheorem6.p2.1.m1.1.1.1.2.cmml">⁢</mo><mrow id="S2.Thmtheorem6.p2.1.m1.1.1.1.1.1" xref="S2.Thmtheorem6.p2.1.m1.1.1.1.1.1.1.cmml"><mo id="S2.Thmtheorem6.p2.1.m1.1.1.1.1.1.2" stretchy="false" xref="S2.Thmtheorem6.p2.1.m1.1.1.1.1.1.1.cmml">(</mo><mrow id="S2.Thmtheorem6.p2.1.m1.1.1.1.1.1.1" xref="S2.Thmtheorem6.p2.1.m1.1.1.1.1.1.1.cmml"><mi id="S2.Thmtheorem6.p2.1.m1.1.1.1.1.1.1.1" xref="S2.Thmtheorem6.p2.1.m1.1.1.1.1.1.1.1.cmml">log</mi><mo id="S2.Thmtheorem6.p2.1.m1.1.1.1.1.1.1a" lspace="0.167em" xref="S2.Thmtheorem6.p2.1.m1.1.1.1.1.1.1.cmml">⁡</mo><mi id="S2.Thmtheorem6.p2.1.m1.1.1.1.1.1.1.2" xref="S2.Thmtheorem6.p2.1.m1.1.1.1.1.1.1.2.cmml">n</mi></mrow><mo id="S2.Thmtheorem6.p2.1.m1.1.1.1.1.1.3" stretchy="false" xref="S2.Thmtheorem6.p2.1.m1.1.1.1.1.1.1.cmml">)</mo></mrow></mrow></msup></mrow><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem6.p2.1.m1.1b"><apply id="S2.Thmtheorem6.p2.1.m1.1.2.cmml" xref="S2.Thmtheorem6.p2.1.m1.1.2"><eq id="S2.Thmtheorem6.p2.1.m1.1.2.1.cmml" xref="S2.Thmtheorem6.p2.1.m1.1.2.1"></eq><ci id="S2.Thmtheorem6.p2.1.m1.1.2.2.cmml" xref="S2.Thmtheorem6.p2.1.m1.1.2.2">𝑊</ci><apply id="S2.Thmtheorem6.p2.1.m1.1.2.3.cmml" xref="S2.Thmtheorem6.p2.1.m1.1.2.3"><csymbol cd="ambiguous" id="S2.Thmtheorem6.p2.1.m1.1.2.3.1.cmml" xref="S2.Thmtheorem6.p2.1.m1.1.2.3">superscript</csymbol><ci id="S2.Thmtheorem6.p2.1.m1.1.2.3.2.cmml" xref="S2.Thmtheorem6.p2.1.m1.1.2.3.2">𝑛</ci><apply id="S2.Thmtheorem6.p2.1.m1.1.1.1.cmml" xref="S2.Thmtheorem6.p2.1.m1.1.1.1"><times id="S2.Thmtheorem6.p2.1.m1.1.1.1.2.cmml" xref="S2.Thmtheorem6.p2.1.m1.1.1.1.2"></times><ci id="S2.Thmtheorem6.p2.1.m1.1.1.1.3.cmml" xref="S2.Thmtheorem6.p2.1.m1.1.1.1.3">𝑂</ci><apply id="S2.Thmtheorem6.p2.1.m1.1.1.1.1.1.1.cmml" xref="S2.Thmtheorem6.p2.1.m1.1.1.1.1.1"><log id="S2.Thmtheorem6.p2.1.m1.1.1.1.1.1.1.1.cmml" xref="S2.Thmtheorem6.p2.1.m1.1.1.1.1.1.1.1"></log><ci id="S2.Thmtheorem6.p2.1.m1.1.1.1.1.1.1.2.cmml" xref="S2.Thmtheorem6.p2.1.m1.1.1.1.1.1.1.2">𝑛</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem6.p2.1.m1.1c">W=n^{O(\log n)}</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem6.p2.1.m1.1d">italic_W = italic_n start_POSTSUPERSCRIPT italic_O ( roman_log italic_n ) end_POSTSUPERSCRIPT</annotation></semantics></math> by setting <math alttext="\epsilon=1/(2t-1)" class="ltx_Math" display="inline" id="S2.Thmtheorem6.p2.2.m2.1"><semantics id="S2.Thmtheorem6.p2.2.m2.1a"><mrow id="S2.Thmtheorem6.p2.2.m2.1.1" xref="S2.Thmtheorem6.p2.2.m2.1.1.cmml"><mi id="S2.Thmtheorem6.p2.2.m2.1.1.3" xref="S2.Thmtheorem6.p2.2.m2.1.1.3.cmml">ϵ</mi><mo id="S2.Thmtheorem6.p2.2.m2.1.1.2" xref="S2.Thmtheorem6.p2.2.m2.1.1.2.cmml">=</mo><mrow id="S2.Thmtheorem6.p2.2.m2.1.1.1" xref="S2.Thmtheorem6.p2.2.m2.1.1.1.cmml"><mn id="S2.Thmtheorem6.p2.2.m2.1.1.1.3" xref="S2.Thmtheorem6.p2.2.m2.1.1.1.3.cmml">1</mn><mo id="S2.Thmtheorem6.p2.2.m2.1.1.1.2" xref="S2.Thmtheorem6.p2.2.m2.1.1.1.2.cmml">/</mo><mrow id="S2.Thmtheorem6.p2.2.m2.1.1.1.1.1" xref="S2.Thmtheorem6.p2.2.m2.1.1.1.1.1.1.cmml"><mo id="S2.Thmtheorem6.p2.2.m2.1.1.1.1.1.2" stretchy="false" xref="S2.Thmtheorem6.p2.2.m2.1.1.1.1.1.1.cmml">(</mo><mrow id="S2.Thmtheorem6.p2.2.m2.1.1.1.1.1.1" xref="S2.Thmtheorem6.p2.2.m2.1.1.1.1.1.1.cmml"><mrow id="S2.Thmtheorem6.p2.2.m2.1.1.1.1.1.1.2" xref="S2.Thmtheorem6.p2.2.m2.1.1.1.1.1.1.2.cmml"><mn id="S2.Thmtheorem6.p2.2.m2.1.1.1.1.1.1.2.2" xref="S2.Thmtheorem6.p2.2.m2.1.1.1.1.1.1.2.2.cmml">2</mn><mo id="S2.Thmtheorem6.p2.2.m2.1.1.1.1.1.1.2.1" xref="S2.Thmtheorem6.p2.2.m2.1.1.1.1.1.1.2.1.cmml">⁢</mo><mi id="S2.Thmtheorem6.p2.2.m2.1.1.1.1.1.1.2.3" xref="S2.Thmtheorem6.p2.2.m2.1.1.1.1.1.1.2.3.cmml">t</mi></mrow><mo id="S2.Thmtheorem6.p2.2.m2.1.1.1.1.1.1.1" xref="S2.Thmtheorem6.p2.2.m2.1.1.1.1.1.1.1.cmml">−</mo><mn id="S2.Thmtheorem6.p2.2.m2.1.1.1.1.1.1.3" xref="S2.Thmtheorem6.p2.2.m2.1.1.1.1.1.1.3.cmml">1</mn></mrow><mo id="S2.Thmtheorem6.p2.2.m2.1.1.1.1.1.3" stretchy="false" xref="S2.Thmtheorem6.p2.2.m2.1.1.1.1.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem6.p2.2.m2.1b"><apply id="S2.Thmtheorem6.p2.2.m2.1.1.cmml" xref="S2.Thmtheorem6.p2.2.m2.1.1"><eq id="S2.Thmtheorem6.p2.2.m2.1.1.2.cmml" xref="S2.Thmtheorem6.p2.2.m2.1.1.2"></eq><ci id="S2.Thmtheorem6.p2.2.m2.1.1.3.cmml" xref="S2.Thmtheorem6.p2.2.m2.1.1.3">italic-ϵ</ci><apply id="S2.Thmtheorem6.p2.2.m2.1.1.1.cmml" xref="S2.Thmtheorem6.p2.2.m2.1.1.1"><divide id="S2.Thmtheorem6.p2.2.m2.1.1.1.2.cmml" xref="S2.Thmtheorem6.p2.2.m2.1.1.1.2"></divide><cn id="S2.Thmtheorem6.p2.2.m2.1.1.1.3.cmml" type="integer" xref="S2.Thmtheorem6.p2.2.m2.1.1.1.3">1</cn><apply id="S2.Thmtheorem6.p2.2.m2.1.1.1.1.1.1.cmml" xref="S2.Thmtheorem6.p2.2.m2.1.1.1.1.1"><minus id="S2.Thmtheorem6.p2.2.m2.1.1.1.1.1.1.1.cmml" xref="S2.Thmtheorem6.p2.2.m2.1.1.1.1.1.1.1"></minus><apply id="S2.Thmtheorem6.p2.2.m2.1.1.1.1.1.1.2.cmml" xref="S2.Thmtheorem6.p2.2.m2.1.1.1.1.1.1.2"><times id="S2.Thmtheorem6.p2.2.m2.1.1.1.1.1.1.2.1.cmml" xref="S2.Thmtheorem6.p2.2.m2.1.1.1.1.1.1.2.1"></times><cn id="S2.Thmtheorem6.p2.2.m2.1.1.1.1.1.1.2.2.cmml" type="integer" xref="S2.Thmtheorem6.p2.2.m2.1.1.1.1.1.1.2.2">2</cn><ci id="S2.Thmtheorem6.p2.2.m2.1.1.1.1.1.1.2.3.cmml" xref="S2.Thmtheorem6.p2.2.m2.1.1.1.1.1.1.2.3">𝑡</ci></apply><cn id="S2.Thmtheorem6.p2.2.m2.1.1.1.1.1.1.3.cmml" type="integer" xref="S2.Thmtheorem6.p2.2.m2.1.1.1.1.1.1.3">1</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem6.p2.2.m2.1c">\epsilon=1/(2t-1)</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem6.p2.2.m2.1d">italic_ϵ = 1 / ( 2 italic_t - 1 )</annotation></semantics></math>, the algorithm uses <math alttext="\tilde{O}(f^{1-1/t}\cdot n^{1+1/t})" class="ltx_Math" display="inline" id="S2.Thmtheorem6.p2.3.m3.1"><semantics id="S2.Thmtheorem6.p2.3.m3.1a"><mrow id="S2.Thmtheorem6.p2.3.m3.1.1" xref="S2.Thmtheorem6.p2.3.m3.1.1.cmml"><mover accent="true" id="S2.Thmtheorem6.p2.3.m3.1.1.3" xref="S2.Thmtheorem6.p2.3.m3.1.1.3.cmml"><mi id="S2.Thmtheorem6.p2.3.m3.1.1.3.2" xref="S2.Thmtheorem6.p2.3.m3.1.1.3.2.cmml">O</mi><mo id="S2.Thmtheorem6.p2.3.m3.1.1.3.1" xref="S2.Thmtheorem6.p2.3.m3.1.1.3.1.cmml">~</mo></mover><mo id="S2.Thmtheorem6.p2.3.m3.1.1.2" xref="S2.Thmtheorem6.p2.3.m3.1.1.2.cmml">⁢</mo><mrow id="S2.Thmtheorem6.p2.3.m3.1.1.1.1" xref="S2.Thmtheorem6.p2.3.m3.1.1.1.1.1.cmml"><mo id="S2.Thmtheorem6.p2.3.m3.1.1.1.1.2" stretchy="false" xref="S2.Thmtheorem6.p2.3.m3.1.1.1.1.1.cmml">(</mo><mrow id="S2.Thmtheorem6.p2.3.m3.1.1.1.1.1" xref="S2.Thmtheorem6.p2.3.m3.1.1.1.1.1.cmml"><msup id="S2.Thmtheorem6.p2.3.m3.1.1.1.1.1.2" xref="S2.Thmtheorem6.p2.3.m3.1.1.1.1.1.2.cmml"><mi id="S2.Thmtheorem6.p2.3.m3.1.1.1.1.1.2.2" xref="S2.Thmtheorem6.p2.3.m3.1.1.1.1.1.2.2.cmml">f</mi><mrow id="S2.Thmtheorem6.p2.3.m3.1.1.1.1.1.2.3" xref="S2.Thmtheorem6.p2.3.m3.1.1.1.1.1.2.3.cmml"><mn id="S2.Thmtheorem6.p2.3.m3.1.1.1.1.1.2.3.2" xref="S2.Thmtheorem6.p2.3.m3.1.1.1.1.1.2.3.2.cmml">1</mn><mo id="S2.Thmtheorem6.p2.3.m3.1.1.1.1.1.2.3.1" xref="S2.Thmtheorem6.p2.3.m3.1.1.1.1.1.2.3.1.cmml">−</mo><mrow id="S2.Thmtheorem6.p2.3.m3.1.1.1.1.1.2.3.3" xref="S2.Thmtheorem6.p2.3.m3.1.1.1.1.1.2.3.3.cmml"><mn id="S2.Thmtheorem6.p2.3.m3.1.1.1.1.1.2.3.3.2" xref="S2.Thmtheorem6.p2.3.m3.1.1.1.1.1.2.3.3.2.cmml">1</mn><mo id="S2.Thmtheorem6.p2.3.m3.1.1.1.1.1.2.3.3.1" xref="S2.Thmtheorem6.p2.3.m3.1.1.1.1.1.2.3.3.1.cmml">/</mo><mi id="S2.Thmtheorem6.p2.3.m3.1.1.1.1.1.2.3.3.3" xref="S2.Thmtheorem6.p2.3.m3.1.1.1.1.1.2.3.3.3.cmml">t</mi></mrow></mrow></msup><mo id="S2.Thmtheorem6.p2.3.m3.1.1.1.1.1.1" lspace="0.222em" rspace="0.222em" xref="S2.Thmtheorem6.p2.3.m3.1.1.1.1.1.1.cmml">⋅</mo><msup id="S2.Thmtheorem6.p2.3.m3.1.1.1.1.1.3" xref="S2.Thmtheorem6.p2.3.m3.1.1.1.1.1.3.cmml"><mi id="S2.Thmtheorem6.p2.3.m3.1.1.1.1.1.3.2" xref="S2.Thmtheorem6.p2.3.m3.1.1.1.1.1.3.2.cmml">n</mi><mrow id="S2.Thmtheorem6.p2.3.m3.1.1.1.1.1.3.3" 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xref="S2.Thmtheorem6.p2.3.m3.1.1.1.1.1.3.2">𝑛</ci><apply id="S2.Thmtheorem6.p2.3.m3.1.1.1.1.1.3.3.cmml" xref="S2.Thmtheorem6.p2.3.m3.1.1.1.1.1.3.3"><plus id="S2.Thmtheorem6.p2.3.m3.1.1.1.1.1.3.3.1.cmml" xref="S2.Thmtheorem6.p2.3.m3.1.1.1.1.1.3.3.1"></plus><cn id="S2.Thmtheorem6.p2.3.m3.1.1.1.1.1.3.3.2.cmml" type="integer" xref="S2.Thmtheorem6.p2.3.m3.1.1.1.1.1.3.3.2">1</cn><apply id="S2.Thmtheorem6.p2.3.m3.1.1.1.1.1.3.3.3.cmml" xref="S2.Thmtheorem6.p2.3.m3.1.1.1.1.1.3.3.3"><divide id="S2.Thmtheorem6.p2.3.m3.1.1.1.1.1.3.3.3.1.cmml" xref="S2.Thmtheorem6.p2.3.m3.1.1.1.1.1.3.3.3.1"></divide><cn id="S2.Thmtheorem6.p2.3.m3.1.1.1.1.1.3.3.3.2.cmml" type="integer" xref="S2.Thmtheorem6.p2.3.m3.1.1.1.1.1.3.3.3.2">1</cn><ci id="S2.Thmtheorem6.p2.3.m3.1.1.1.1.1.3.3.3.3.cmml" xref="S2.Thmtheorem6.p2.3.m3.1.1.1.1.1.3.3.3.3">𝑡</ci></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem6.p2.3.m3.1c">\tilde{O}(f^{1-1/t}\cdot n^{1+1/t})</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem6.p2.3.m3.1d">over~ start_ARG italic_O end_ARG ( italic_f start_POSTSUPERSCRIPT 1 - 1 / italic_t end_POSTSUPERSCRIPT ⋅ italic_n start_POSTSUPERSCRIPT 1 + 1 / italic_t end_POSTSUPERSCRIPT )</annotation></semantics></math> words of space and returns an <math alttext="f" class="ltx_Math" display="inline" id="S2.Thmtheorem6.p2.4.m4.1"><semantics id="S2.Thmtheorem6.p2.4.m4.1a"><mi id="S2.Thmtheorem6.p2.4.m4.1.1" xref="S2.Thmtheorem6.p2.4.m4.1.1.cmml">f</mi><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem6.p2.4.m4.1b"><ci id="S2.Thmtheorem6.p2.4.m4.1.1.cmml" xref="S2.Thmtheorem6.p2.4.m4.1.1">𝑓</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem6.p2.4.m4.1c">f</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem6.p2.4.m4.1d">italic_f</annotation></semantics></math>-VFT (or <math alttext="f" class="ltx_Math" display="inline" 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id="S2.Thmtheorem6.p2.6.m6.1d">( 2 italic_t )</annotation></semantics></math>-spanner of size <math alttext="\tilde{O}(f^{1-1/t}\cdot n^{1+1/t})" class="ltx_Math" display="inline" id="S2.Thmtheorem6.p2.7.m7.1"><semantics id="S2.Thmtheorem6.p2.7.m7.1a"><mrow id="S2.Thmtheorem6.p2.7.m7.1.1" xref="S2.Thmtheorem6.p2.7.m7.1.1.cmml"><mover accent="true" id="S2.Thmtheorem6.p2.7.m7.1.1.3" xref="S2.Thmtheorem6.p2.7.m7.1.1.3.cmml"><mi id="S2.Thmtheorem6.p2.7.m7.1.1.3.2" xref="S2.Thmtheorem6.p2.7.m7.1.1.3.2.cmml">O</mi><mo id="S2.Thmtheorem6.p2.7.m7.1.1.3.1" xref="S2.Thmtheorem6.p2.7.m7.1.1.3.1.cmml">~</mo></mover><mo id="S2.Thmtheorem6.p2.7.m7.1.1.2" xref="S2.Thmtheorem6.p2.7.m7.1.1.2.cmml">⁢</mo><mrow id="S2.Thmtheorem6.p2.7.m7.1.1.1.1" xref="S2.Thmtheorem6.p2.7.m7.1.1.1.1.1.cmml"><mo id="S2.Thmtheorem6.p2.7.m7.1.1.1.1.2" stretchy="false" xref="S2.Thmtheorem6.p2.7.m7.1.1.1.1.1.cmml">(</mo><mrow id="S2.Thmtheorem6.p2.7.m7.1.1.1.1.1" xref="S2.Thmtheorem6.p2.7.m7.1.1.1.1.1.cmml"><msup 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id="S2.Thmtheorem6.p2.7.m7.1.1.1.1.1.2.3.3.3.cmml" xref="S2.Thmtheorem6.p2.7.m7.1.1.1.1.1.2.3.3.3">𝑡</ci></apply></apply></apply><apply id="S2.Thmtheorem6.p2.7.m7.1.1.1.1.1.3.cmml" xref="S2.Thmtheorem6.p2.7.m7.1.1.1.1.1.3"><csymbol cd="ambiguous" id="S2.Thmtheorem6.p2.7.m7.1.1.1.1.1.3.1.cmml" xref="S2.Thmtheorem6.p2.7.m7.1.1.1.1.1.3">superscript</csymbol><ci id="S2.Thmtheorem6.p2.7.m7.1.1.1.1.1.3.2.cmml" xref="S2.Thmtheorem6.p2.7.m7.1.1.1.1.1.3.2">𝑛</ci><apply id="S2.Thmtheorem6.p2.7.m7.1.1.1.1.1.3.3.cmml" xref="S2.Thmtheorem6.p2.7.m7.1.1.1.1.1.3.3"><plus id="S2.Thmtheorem6.p2.7.m7.1.1.1.1.1.3.3.1.cmml" xref="S2.Thmtheorem6.p2.7.m7.1.1.1.1.1.3.3.1"></plus><cn id="S2.Thmtheorem6.p2.7.m7.1.1.1.1.1.3.3.2.cmml" type="integer" xref="S2.Thmtheorem6.p2.7.m7.1.1.1.1.1.3.3.2">1</cn><apply id="S2.Thmtheorem6.p2.7.m7.1.1.1.1.1.3.3.3.cmml" xref="S2.Thmtheorem6.p2.7.m7.1.1.1.1.1.3.3.3"><divide id="S2.Thmtheorem6.p2.7.m7.1.1.1.1.1.3.3.3.1.cmml" xref="S2.Thmtheorem6.p2.7.m7.1.1.1.1.1.3.3.3.1"></divide><cn id="S2.Thmtheorem6.p2.7.m7.1.1.1.1.1.3.3.3.2.cmml" type="integer" xref="S2.Thmtheorem6.p2.7.m7.1.1.1.1.1.3.3.3.2">1</cn><ci id="S2.Thmtheorem6.p2.7.m7.1.1.1.1.1.3.3.3.3.cmml" xref="S2.Thmtheorem6.p2.7.m7.1.1.1.1.1.3.3.3.3">𝑡</ci></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem6.p2.7.m7.1c">\tilde{O}(f^{1-1/t}\cdot n^{1+1/t})</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem6.p2.7.m7.1d">over~ start_ARG italic_O end_ARG ( italic_f start_POSTSUPERSCRIPT 1 - 1 / italic_t end_POSTSUPERSCRIPT ⋅ italic_n start_POSTSUPERSCRIPT 1 + 1 / italic_t end_POSTSUPERSCRIPT )</annotation></semantics></math>.</p> </div> </div> </section> </section> </section> <section class="ltx_section" id="S3"> <h2 class="ltx_title ltx_title_section"> <span class="ltx_tag ltx_tag_section">3 </span>Generic Framework for Streaming Algorithms for Network Design</h2> <div class="ltx_para" id="S3.p1"> <p class="ltx_p" id="S3.p1.1">In this section, we describe our generic framework for streaming algorithms for network design problems. The algorithm, described in Algorithm <a class="ltx_ref" href="https://arxiv.org/html/2503.00712v1#algorithm3" title="In 3 Generic Framework for Streaming Algorithms for Network Design ‣ Streaming Algorithms for Network Design"><span class="ltx_text ltx_ref_tag">3</span></a>, is both simple and practical.</p> </div> <figure class="ltx_float ltx_algorithm" id="algorithm3"> <div class="ltx_listing ltx_lst_numbers_left ltx_listing" id="algorithm3.14"> <div class="ltx_listingline" id="algorithm3.6.6"> <span class="ltx_text" id="algorithm3.6.6.1"><span class="ltx_text ltx_font_bold" id="algorithm3.6.6.1.1">Input:</span> </span>Stream of weighted edges <math alttext="\big{(}e,w(e)\big{)}" class="ltx_Math" display="inline" id="algorithm3.1.1.m1.3"><semantics id="algorithm3.1.1.m1.3a"><mrow id="algorithm3.1.1.m1.3.3.1" xref="algorithm3.1.1.m1.3.3.2.cmml"><mo id="algorithm3.1.1.m1.3.3.1.2" maxsize="120%" minsize="120%" xref="algorithm3.1.1.m1.3.3.2.cmml">(</mo><mi id="algorithm3.1.1.m1.2.2" xref="algorithm3.1.1.m1.2.2.cmml">e</mi><mo id="algorithm3.1.1.m1.3.3.1.3" xref="algorithm3.1.1.m1.3.3.2.cmml">,</mo><mrow id="algorithm3.1.1.m1.3.3.1.1" xref="algorithm3.1.1.m1.3.3.1.1.cmml"><mi id="algorithm3.1.1.m1.3.3.1.1.2" xref="algorithm3.1.1.m1.3.3.1.1.2.cmml">w</mi><mo id="algorithm3.1.1.m1.3.3.1.1.1" xref="algorithm3.1.1.m1.3.3.1.1.1.cmml">⁢</mo><mrow id="algorithm3.1.1.m1.3.3.1.1.3.2" xref="algorithm3.1.1.m1.3.3.1.1.cmml"><mo id="algorithm3.1.1.m1.3.3.1.1.3.2.1" stretchy="false" xref="algorithm3.1.1.m1.3.3.1.1.cmml">(</mo><mi id="algorithm3.1.1.m1.1.1" xref="algorithm3.1.1.m1.1.1.cmml">e</mi><mo id="algorithm3.1.1.m1.3.3.1.1.3.2.2" stretchy="false" xref="algorithm3.1.1.m1.3.3.1.1.cmml">)</mo></mrow></mrow><mo id="algorithm3.1.1.m1.3.3.1.4" maxsize="120%" minsize="120%" xref="algorithm3.1.1.m1.3.3.2.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="algorithm3.1.1.m1.3b"><interval closure="open" id="algorithm3.1.1.m1.3.3.2.cmml" xref="algorithm3.1.1.m1.3.3.1"><ci id="algorithm3.1.1.m1.2.2.cmml" xref="algorithm3.1.1.m1.2.2">𝑒</ci><apply id="algorithm3.1.1.m1.3.3.1.1.cmml" xref="algorithm3.1.1.m1.3.3.1.1"><times id="algorithm3.1.1.m1.3.3.1.1.1.cmml" xref="algorithm3.1.1.m1.3.3.1.1.1"></times><ci id="algorithm3.1.1.m1.3.3.1.1.2.cmml" xref="algorithm3.1.1.m1.3.3.1.1.2">𝑤</ci><ci id="algorithm3.1.1.m1.1.1.cmml" xref="algorithm3.1.1.m1.1.1">𝑒</ci></apply></interval></annotation-xml><annotation encoding="application/x-tex" id="algorithm3.1.1.m1.3c">\big{(}e,w(e)\big{)}</annotation><annotation encoding="application/x-llamapun" id="algorithm3.1.1.m1.3d">( italic_e , italic_w ( italic_e ) )</annotation></semantics></math>, network design problem <math alttext="\mathcal{M}" class="ltx_Math" display="inline" id="algorithm3.2.2.m2.1"><semantics id="algorithm3.2.2.m2.1a"><mi class="ltx_font_mathcaligraphic" id="algorithm3.2.2.m2.1.1" xref="algorithm3.2.2.m2.1.1.cmml">ℳ</mi><annotation-xml encoding="MathML-Content" id="algorithm3.2.2.m2.1b"><ci id="algorithm3.2.2.m2.1.1.cmml" xref="algorithm3.2.2.m2.1.1">ℳ</ci></annotation-xml><annotation encoding="application/x-tex" id="algorithm3.2.2.m2.1c">\mathcal{M}</annotation><annotation encoding="application/x-llamapun" id="algorithm3.2.2.m2.1d">caligraphic_M</annotation></semantics></math> with maximum connectivity requirement <math alttext="k" class="ltx_Math" display="inline" id="algorithm3.3.3.m3.1"><semantics id="algorithm3.3.3.m3.1a"><mi id="algorithm3.3.3.m3.1.1" xref="algorithm3.3.3.m3.1.1.cmml">k</mi><annotation-xml encoding="MathML-Content" id="algorithm3.3.3.m3.1b"><ci id="algorithm3.3.3.m3.1.1.cmml" xref="algorithm3.3.3.m3.1.1">𝑘</ci></annotation-xml><annotation encoding="application/x-tex" id="algorithm3.3.3.m3.1c">k</annotation><annotation encoding="application/x-llamapun" id="algorithm3.3.3.m3.1d">italic_k</annotation></semantics></math>, approximation parameter <math alttext="t" class="ltx_Math" display="inline" id="algorithm3.4.4.m4.1"><semantics id="algorithm3.4.4.m4.1a"><mi id="algorithm3.4.4.m4.1.1" xref="algorithm3.4.4.m4.1.1.cmml">t</mi><annotation-xml encoding="MathML-Content" id="algorithm3.4.4.m4.1b"><ci id="algorithm3.4.4.m4.1.1.cmml" xref="algorithm3.4.4.m4.1.1">𝑡</ci></annotation-xml><annotation encoding="application/x-tex" id="algorithm3.4.4.m4.1c">t</annotation><annotation encoding="application/x-llamapun" id="algorithm3.4.4.m4.1d">italic_t</annotation></semantics></math>, and “offline” algorithm <math alttext="\mathcal{A}" class="ltx_Math" display="inline" id="algorithm3.5.5.m5.1"><semantics id="algorithm3.5.5.m5.1a"><mi class="ltx_font_mathcaligraphic" id="algorithm3.5.5.m5.1.1" xref="algorithm3.5.5.m5.1.1.cmml">𝒜</mi><annotation-xml encoding="MathML-Content" id="algorithm3.5.5.m5.1b"><ci id="algorithm3.5.5.m5.1.1.cmml" xref="algorithm3.5.5.m5.1.1">𝒜</ci></annotation-xml><annotation encoding="application/x-tex" id="algorithm3.5.5.m5.1c">\mathcal{A}</annotation><annotation encoding="application/x-llamapun" id="algorithm3.5.5.m5.1d">caligraphic_A</annotation></semantics></math> for <math alttext="\mathcal{M}" class="ltx_Math" display="inline" id="algorithm3.6.6.m6.1"><semantics id="algorithm3.6.6.m6.1a"><mi class="ltx_font_mathcaligraphic" id="algorithm3.6.6.m6.1.1" xref="algorithm3.6.6.m6.1.1.cmml">ℳ</mi><annotation-xml encoding="MathML-Content" id="algorithm3.6.6.m6.1b"><ci id="algorithm3.6.6.m6.1.1.cmml" xref="algorithm3.6.6.m6.1.1">ℳ</ci></annotation-xml><annotation encoding="application/x-tex" id="algorithm3.6.6.m6.1c">\mathcal{M}</annotation><annotation encoding="application/x-llamapun" id="algorithm3.6.6.m6.1d">caligraphic_M</annotation></semantics></math>. </div> <div class="ltx_listingline" id="algorithm3.14.15"> </div> <div class="ltx_listingline" id="algorithm3.14.16"> <span class="ltx_text" id="algorithm3.14.16.1" style="font-size:120%;color:#0000FF;">/* </span><span class="ltx_text ltx_font_smallcaps" id="algorithm3.14.16.2" style="font-size:120%;color:#0000FF;">In Stream: */</span> </div> <div class="ltx_listingline" id="algorithm3.10.10"> <span class="ltx_text" id="algorithm3.10.10.4"> <span class="ltx_text ltx_font_bold" id="algorithm3.10.10.4.1">construct</span> <math alttext="H\leftarrow" class="ltx_Math" display="inline" id="algorithm3.7.7.1.m1.1"><semantics id="algorithm3.7.7.1.m1.1a"><mrow id="algorithm3.7.7.1.m1.1.1" xref="algorithm3.7.7.1.m1.1.1.cmml"><mi id="algorithm3.7.7.1.m1.1.1.2" xref="algorithm3.7.7.1.m1.1.1.2.cmml">H</mi><mo id="algorithm3.7.7.1.m1.1.1.1" stretchy="false" xref="algorithm3.7.7.1.m1.1.1.1.cmml">←</mo><mi id="algorithm3.7.7.1.m1.1.1.3" xref="algorithm3.7.7.1.m1.1.1.3.cmml"></mi></mrow><annotation-xml encoding="MathML-Content" id="algorithm3.7.7.1.m1.1b"><apply id="algorithm3.7.7.1.m1.1.1.cmml" xref="algorithm3.7.7.1.m1.1.1"><ci id="algorithm3.7.7.1.m1.1.1.1.cmml" xref="algorithm3.7.7.1.m1.1.1.1">←</ci><ci id="algorithm3.7.7.1.m1.1.1.2.cmml" xref="algorithm3.7.7.1.m1.1.1.2">𝐻</ci><csymbol cd="latexml" id="algorithm3.7.7.1.m1.1.1.3.cmml" xref="algorithm3.7.7.1.m1.1.1.3">absent</csymbol></apply></annotation-xml><annotation encoding="application/x-tex" id="algorithm3.7.7.1.m1.1c">H\leftarrow</annotation><annotation encoding="application/x-llamapun" id="algorithm3.7.7.1.m1.1d">italic_H ←</annotation></semantics></math> a <math alttext="O(kt)" class="ltx_Math" display="inline" id="algorithm3.8.8.2.m2.1"><semantics id="algorithm3.8.8.2.m2.1a"><mrow id="algorithm3.8.8.2.m2.1.1" xref="algorithm3.8.8.2.m2.1.1.cmml"><mi id="algorithm3.8.8.2.m2.1.1.3" xref="algorithm3.8.8.2.m2.1.1.3.cmml">O</mi><mo id="algorithm3.8.8.2.m2.1.1.2" xref="algorithm3.8.8.2.m2.1.1.2.cmml">⁢</mo><mrow id="algorithm3.8.8.2.m2.1.1.1.1" xref="algorithm3.8.8.2.m2.1.1.1.1.1.cmml"><mo id="algorithm3.8.8.2.m2.1.1.1.1.2" stretchy="false" xref="algorithm3.8.8.2.m2.1.1.1.1.1.cmml">(</mo><mrow id="algorithm3.8.8.2.m2.1.1.1.1.1" xref="algorithm3.8.8.2.m2.1.1.1.1.1.cmml"><mi id="algorithm3.8.8.2.m2.1.1.1.1.1.2" xref="algorithm3.8.8.2.m2.1.1.1.1.1.2.cmml">k</mi><mo id="algorithm3.8.8.2.m2.1.1.1.1.1.1" xref="algorithm3.8.8.2.m2.1.1.1.1.1.1.cmml">⁢</mo><mi id="algorithm3.8.8.2.m2.1.1.1.1.1.3" xref="algorithm3.8.8.2.m2.1.1.1.1.1.3.cmml">t</mi></mrow><mo id="algorithm3.8.8.2.m2.1.1.1.1.3" stretchy="false" xref="algorithm3.8.8.2.m2.1.1.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="algorithm3.8.8.2.m2.1b"><apply id="algorithm3.8.8.2.m2.1.1.cmml" xref="algorithm3.8.8.2.m2.1.1"><times id="algorithm3.8.8.2.m2.1.1.2.cmml" xref="algorithm3.8.8.2.m2.1.1.2"></times><ci id="algorithm3.8.8.2.m2.1.1.3.cmml" xref="algorithm3.8.8.2.m2.1.1.3">𝑂</ci><apply id="algorithm3.8.8.2.m2.1.1.1.1.1.cmml" xref="algorithm3.8.8.2.m2.1.1.1.1"><times id="algorithm3.8.8.2.m2.1.1.1.1.1.1.cmml" xref="algorithm3.8.8.2.m2.1.1.1.1.1.1"></times><ci id="algorithm3.8.8.2.m2.1.1.1.1.1.2.cmml" xref="algorithm3.8.8.2.m2.1.1.1.1.1.2">𝑘</ci><ci id="algorithm3.8.8.2.m2.1.1.1.1.1.3.cmml" xref="algorithm3.8.8.2.m2.1.1.1.1.1.3">𝑡</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="algorithm3.8.8.2.m2.1c">O(kt)</annotation><annotation encoding="application/x-llamapun" id="algorithm3.8.8.2.m2.1d">italic_O ( italic_k italic_t )</annotation></semantics></math>-FT <math alttext="(2t-1)" class="ltx_Math" display="inline" id="algorithm3.9.9.3.m3.1"><semantics id="algorithm3.9.9.3.m3.1a"><mrow id="algorithm3.9.9.3.m3.1.1.1" xref="algorithm3.9.9.3.m3.1.1.1.1.cmml"><mo id="algorithm3.9.9.3.m3.1.1.1.2" stretchy="false" xref="algorithm3.9.9.3.m3.1.1.1.1.cmml">(</mo><mrow id="algorithm3.9.9.3.m3.1.1.1.1" xref="algorithm3.9.9.3.m3.1.1.1.1.cmml"><mrow id="algorithm3.9.9.3.m3.1.1.1.1.2" xref="algorithm3.9.9.3.m3.1.1.1.1.2.cmml"><mn id="algorithm3.9.9.3.m3.1.1.1.1.2.2" xref="algorithm3.9.9.3.m3.1.1.1.1.2.2.cmml">2</mn><mo id="algorithm3.9.9.3.m3.1.1.1.1.2.1" xref="algorithm3.9.9.3.m3.1.1.1.1.2.1.cmml">⁢</mo><mi id="algorithm3.9.9.3.m3.1.1.1.1.2.3" xref="algorithm3.9.9.3.m3.1.1.1.1.2.3.cmml">t</mi></mrow><mo id="algorithm3.9.9.3.m3.1.1.1.1.1" xref="algorithm3.9.9.3.m3.1.1.1.1.1.cmml">−</mo><mn id="algorithm3.9.9.3.m3.1.1.1.1.3" xref="algorithm3.9.9.3.m3.1.1.1.1.3.cmml">1</mn></mrow><mo id="algorithm3.9.9.3.m3.1.1.1.3" stretchy="false" xref="algorithm3.9.9.3.m3.1.1.1.1.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="algorithm3.9.9.3.m3.1b"><apply id="algorithm3.9.9.3.m3.1.1.1.1.cmml" xref="algorithm3.9.9.3.m3.1.1.1"><minus id="algorithm3.9.9.3.m3.1.1.1.1.1.cmml" xref="algorithm3.9.9.3.m3.1.1.1.1.1"></minus><apply id="algorithm3.9.9.3.m3.1.1.1.1.2.cmml" xref="algorithm3.9.9.3.m3.1.1.1.1.2"><times id="algorithm3.9.9.3.m3.1.1.1.1.2.1.cmml" xref="algorithm3.9.9.3.m3.1.1.1.1.2.1"></times><cn id="algorithm3.9.9.3.m3.1.1.1.1.2.2.cmml" type="integer" xref="algorithm3.9.9.3.m3.1.1.1.1.2.2">2</cn><ci id="algorithm3.9.9.3.m3.1.1.1.1.2.3.cmml" xref="algorithm3.9.9.3.m3.1.1.1.1.2.3">𝑡</ci></apply><cn id="algorithm3.9.9.3.m3.1.1.1.1.3.cmml" type="integer" xref="algorithm3.9.9.3.m3.1.1.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="algorithm3.9.9.3.m3.1c">(2t-1)</annotation><annotation encoding="application/x-llamapun" id="algorithm3.9.9.3.m3.1d">( 2 italic_t - 1 )</annotation></semantics></math>-spanner of the edges in the stream using Algorithm <a class="ltx_ref" href="https://arxiv.org/html/2503.00712v1#algorithm2" title="In Weighted graphs. ‣ 2.1 Fault-Tolerant Spanners in Streaming ‣ 2 Preliminaries ‣ Streaming Algorithms for Network Design"><span class="ltx_text ltx_ref_tag">2</span></a> with <math alttext="\epsilon=\frac{1}{2t-1}" class="ltx_Math" display="inline" id="algorithm3.10.10.4.m4.1"><semantics id="algorithm3.10.10.4.m4.1a"><mrow id="algorithm3.10.10.4.m4.1.1" xref="algorithm3.10.10.4.m4.1.1.cmml"><mi id="algorithm3.10.10.4.m4.1.1.2" xref="algorithm3.10.10.4.m4.1.1.2.cmml">ϵ</mi><mo id="algorithm3.10.10.4.m4.1.1.1" xref="algorithm3.10.10.4.m4.1.1.1.cmml">=</mo><mfrac id="algorithm3.10.10.4.m4.1.1.3" xref="algorithm3.10.10.4.m4.1.1.3.cmml"><mn id="algorithm3.10.10.4.m4.1.1.3.2" xref="algorithm3.10.10.4.m4.1.1.3.2.cmml">1</mn><mrow id="algorithm3.10.10.4.m4.1.1.3.3" xref="algorithm3.10.10.4.m4.1.1.3.3.cmml"><mrow id="algorithm3.10.10.4.m4.1.1.3.3.2" xref="algorithm3.10.10.4.m4.1.1.3.3.2.cmml"><mn id="algorithm3.10.10.4.m4.1.1.3.3.2.2" xref="algorithm3.10.10.4.m4.1.1.3.3.2.2.cmml">2</mn><mo id="algorithm3.10.10.4.m4.1.1.3.3.2.1" xref="algorithm3.10.10.4.m4.1.1.3.3.2.1.cmml">⁢</mo><mi id="algorithm3.10.10.4.m4.1.1.3.3.2.3" xref="algorithm3.10.10.4.m4.1.1.3.3.2.3.cmml">t</mi></mrow><mo id="algorithm3.10.10.4.m4.1.1.3.3.1" xref="algorithm3.10.10.4.m4.1.1.3.3.1.cmml">−</mo><mn id="algorithm3.10.10.4.m4.1.1.3.3.3" xref="algorithm3.10.10.4.m4.1.1.3.3.3.cmml">1</mn></mrow></mfrac></mrow><annotation-xml encoding="MathML-Content" id="algorithm3.10.10.4.m4.1b"><apply id="algorithm3.10.10.4.m4.1.1.cmml" xref="algorithm3.10.10.4.m4.1.1"><eq id="algorithm3.10.10.4.m4.1.1.1.cmml" xref="algorithm3.10.10.4.m4.1.1.1"></eq><ci id="algorithm3.10.10.4.m4.1.1.2.cmml" xref="algorithm3.10.10.4.m4.1.1.2">italic-ϵ</ci><apply id="algorithm3.10.10.4.m4.1.1.3.cmml" xref="algorithm3.10.10.4.m4.1.1.3"><divide id="algorithm3.10.10.4.m4.1.1.3.1.cmml" xref="algorithm3.10.10.4.m4.1.1.3"></divide><cn id="algorithm3.10.10.4.m4.1.1.3.2.cmml" type="integer" xref="algorithm3.10.10.4.m4.1.1.3.2">1</cn><apply id="algorithm3.10.10.4.m4.1.1.3.3.cmml" xref="algorithm3.10.10.4.m4.1.1.3.3"><minus id="algorithm3.10.10.4.m4.1.1.3.3.1.cmml" xref="algorithm3.10.10.4.m4.1.1.3.3.1"></minus><apply id="algorithm3.10.10.4.m4.1.1.3.3.2.cmml" xref="algorithm3.10.10.4.m4.1.1.3.3.2"><times id="algorithm3.10.10.4.m4.1.1.3.3.2.1.cmml" xref="algorithm3.10.10.4.m4.1.1.3.3.2.1"></times><cn id="algorithm3.10.10.4.m4.1.1.3.3.2.2.cmml" type="integer" xref="algorithm3.10.10.4.m4.1.1.3.3.2.2">2</cn><ci id="algorithm3.10.10.4.m4.1.1.3.3.2.3.cmml" xref="algorithm3.10.10.4.m4.1.1.3.3.2.3">𝑡</ci></apply><cn id="algorithm3.10.10.4.m4.1.1.3.3.3.cmml" type="integer" xref="algorithm3.10.10.4.m4.1.1.3.3.3">1</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="algorithm3.10.10.4.m4.1c">\epsilon=\frac{1}{2t-1}</annotation><annotation encoding="application/x-llamapun" id="algorithm3.10.10.4.m4.1d">italic_ϵ = divide start_ARG 1 end_ARG start_ARG 2 italic_t - 1 end_ARG</annotation></semantics></math>. </span> </div> <div class="ltx_listingline" id="algorithm3.14.17"> <span class="ltx_text" id="algorithm3.14.17.1" style="font-size:120%;color:#0000FF;">/* </span><span class="ltx_text ltx_font_smallcaps" id="algorithm3.14.17.2" style="font-size:120%;color:#0000FF;">Postprocessing<span class="ltx_text ltx_font_upright" id="algorithm3.14.17.2.1"> <span class="ltx_text" id="algorithm3.14.17.2.1.1" style="font-size:83%;">(<em class="ltx_emph ltx_font_italic" id="algorithm3.14.17.2.1.1.1">after stream terminates</em>)</span>: */</span></span> </div> <div class="ltx_listingline" id="algorithm3.14.14"> <span class="ltx_text" id="algorithm3.14.14.4"> <span class="ltx_text ltx_font_bold" id="algorithm3.14.14.4.1">return</span> the solution <span class="ltx_text ltx_markedasmath" id="algorithm3.14.14.4.2">SOL</span> of <math alttext="\mathcal{M}" class="ltx_Math" display="inline" id="algorithm3.12.12.2.m2.1"><semantics id="algorithm3.12.12.2.m2.1a"><mi class="ltx_font_mathcaligraphic" id="algorithm3.12.12.2.m2.1.1" xref="algorithm3.12.12.2.m2.1.1.cmml">ℳ</mi><annotation-xml encoding="MathML-Content" id="algorithm3.12.12.2.m2.1b"><ci id="algorithm3.12.12.2.m2.1.1.cmml" xref="algorithm3.12.12.2.m2.1.1">ℳ</ci></annotation-xml><annotation encoding="application/x-tex" id="algorithm3.12.12.2.m2.1c">\mathcal{M}</annotation><annotation encoding="application/x-llamapun" id="algorithm3.12.12.2.m2.1d">caligraphic_M</annotation></semantics></math> on <math alttext="H" class="ltx_Math" display="inline" id="algorithm3.13.13.3.m3.1"><semantics id="algorithm3.13.13.3.m3.1a"><mi id="algorithm3.13.13.3.m3.1.1" xref="algorithm3.13.13.3.m3.1.1.cmml">H</mi><annotation-xml encoding="MathML-Content" id="algorithm3.13.13.3.m3.1b"><ci id="algorithm3.13.13.3.m3.1.1.cmml" xref="algorithm3.13.13.3.m3.1.1">𝐻</ci></annotation-xml><annotation encoding="application/x-tex" id="algorithm3.13.13.3.m3.1c">H</annotation><annotation encoding="application/x-llamapun" id="algorithm3.13.13.3.m3.1d">italic_H</annotation></semantics></math> output by the algorithm <math alttext="\mathcal{A}" class="ltx_Math" display="inline" id="algorithm3.14.14.4.m4.1"><semantics id="algorithm3.14.14.4.m4.1a"><mi class="ltx_font_mathcaligraphic" id="algorithm3.14.14.4.m4.1.1" xref="algorithm3.14.14.4.m4.1.1.cmml">𝒜</mi><annotation-xml encoding="MathML-Content" id="algorithm3.14.14.4.m4.1b"><ci id="algorithm3.14.14.4.m4.1.1.cmml" xref="algorithm3.14.14.4.m4.1.1">𝒜</ci></annotation-xml><annotation encoding="application/x-tex" id="algorithm3.14.14.4.m4.1c">\mathcal{A}</annotation><annotation encoding="application/x-llamapun" id="algorithm3.14.14.4.m4.1d">caligraphic_A</annotation></semantics></math> </span> </div> </div> <figcaption class="ltx_caption"><span class="ltx_tag ltx_tag_float"><span class="ltx_text ltx_font_bold" id="algorithm3.16.1.1">Algorithm 3</span> </span>A Generic framework for solving network design problems in streaming setting.</figcaption> </figure> <div class="ltx_para" id="S3.p2"> <p class="ltx_p" id="S3.p2.5">As the algorithm only constructs and stores a fault-tolerant spanner with the given parameters <math alttext="f=O(tk)" class="ltx_Math" display="inline" id="S3.p2.1.m1.1"><semantics id="S3.p2.1.m1.1a"><mrow id="S3.p2.1.m1.1.1" xref="S3.p2.1.m1.1.1.cmml"><mi id="S3.p2.1.m1.1.1.3" xref="S3.p2.1.m1.1.1.3.cmml">f</mi><mo id="S3.p2.1.m1.1.1.2" xref="S3.p2.1.m1.1.1.2.cmml">=</mo><mrow id="S3.p2.1.m1.1.1.1" xref="S3.p2.1.m1.1.1.1.cmml"><mi id="S3.p2.1.m1.1.1.1.3" xref="S3.p2.1.m1.1.1.1.3.cmml">O</mi><mo id="S3.p2.1.m1.1.1.1.2" xref="S3.p2.1.m1.1.1.1.2.cmml">⁢</mo><mrow id="S3.p2.1.m1.1.1.1.1.1" xref="S3.p2.1.m1.1.1.1.1.1.1.cmml"><mo id="S3.p2.1.m1.1.1.1.1.1.2" stretchy="false" xref="S3.p2.1.m1.1.1.1.1.1.1.cmml">(</mo><mrow id="S3.p2.1.m1.1.1.1.1.1.1" xref="S3.p2.1.m1.1.1.1.1.1.1.cmml"><mi id="S3.p2.1.m1.1.1.1.1.1.1.2" xref="S3.p2.1.m1.1.1.1.1.1.1.2.cmml">t</mi><mo id="S3.p2.1.m1.1.1.1.1.1.1.1" xref="S3.p2.1.m1.1.1.1.1.1.1.1.cmml">⁢</mo><mi id="S3.p2.1.m1.1.1.1.1.1.1.3" xref="S3.p2.1.m1.1.1.1.1.1.1.3.cmml">k</mi></mrow><mo id="S3.p2.1.m1.1.1.1.1.1.3" stretchy="false" xref="S3.p2.1.m1.1.1.1.1.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.p2.1.m1.1b"><apply id="S3.p2.1.m1.1.1.cmml" xref="S3.p2.1.m1.1.1"><eq id="S3.p2.1.m1.1.1.2.cmml" xref="S3.p2.1.m1.1.1.2"></eq><ci id="S3.p2.1.m1.1.1.3.cmml" xref="S3.p2.1.m1.1.1.3">𝑓</ci><apply id="S3.p2.1.m1.1.1.1.cmml" xref="S3.p2.1.m1.1.1.1"><times id="S3.p2.1.m1.1.1.1.2.cmml" xref="S3.p2.1.m1.1.1.1.2"></times><ci id="S3.p2.1.m1.1.1.1.3.cmml" xref="S3.p2.1.m1.1.1.1.3">𝑂</ci><apply id="S3.p2.1.m1.1.1.1.1.1.1.cmml" xref="S3.p2.1.m1.1.1.1.1.1"><times id="S3.p2.1.m1.1.1.1.1.1.1.1.cmml" xref="S3.p2.1.m1.1.1.1.1.1.1.1"></times><ci id="S3.p2.1.m1.1.1.1.1.1.1.2.cmml" xref="S3.p2.1.m1.1.1.1.1.1.1.2">𝑡</ci><ci id="S3.p2.1.m1.1.1.1.1.1.1.3.cmml" xref="S3.p2.1.m1.1.1.1.1.1.1.3">𝑘</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p2.1.m1.1c">f=O(tk)</annotation><annotation encoding="application/x-llamapun" id="S3.p2.1.m1.1d">italic_f = italic_O ( italic_t italic_k )</annotation></semantics></math> and <math alttext="t" class="ltx_Math" display="inline" id="S3.p2.2.m2.1"><semantics id="S3.p2.2.m2.1a"><mi id="S3.p2.2.m2.1.1" xref="S3.p2.2.m2.1.1.cmml">t</mi><annotation-xml encoding="MathML-Content" id="S3.p2.2.m2.1b"><ci id="S3.p2.2.m2.1.1.cmml" xref="S3.p2.2.m2.1.1">𝑡</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.p2.2.m2.1c">t</annotation><annotation encoding="application/x-llamapun" id="S3.p2.2.m2.1d">italic_t</annotation></semantics></math> in the stream, it follows directly from Theorem <a class="ltx_ref" href="https://arxiv.org/html/2503.00712v1#S2.Thmtheorem6" title="Theorem 2.6. ‣ Weighted graphs. ‣ 2.1 Fault-Tolerant Spanners in Streaming ‣ 2 Preliminaries ‣ Streaming Algorithms for Network Design"><span class="ltx_text ltx_ref_tag">2.6</span></a> that its space complexity is <math alttext="\tilde{O}\big{(}f^{1-1/t}\cdot n^{1+1/t}\big{)}=\tilde{O}\big{(}k^{1-1/t}\cdot n% ^{1+1/t}\big{)}" class="ltx_Math" display="inline" id="S3.p2.3.m3.2"><semantics id="S3.p2.3.m3.2a"><mrow id="S3.p2.3.m3.2.2" xref="S3.p2.3.m3.2.2.cmml"><mrow id="S3.p2.3.m3.1.1.1" xref="S3.p2.3.m3.1.1.1.cmml"><mover accent="true" id="S3.p2.3.m3.1.1.1.3" xref="S3.p2.3.m3.1.1.1.3.cmml"><mi id="S3.p2.3.m3.1.1.1.3.2" xref="S3.p2.3.m3.1.1.1.3.2.cmml">O</mi><mo id="S3.p2.3.m3.1.1.1.3.1" xref="S3.p2.3.m3.1.1.1.3.1.cmml">~</mo></mover><mo id="S3.p2.3.m3.1.1.1.2" xref="S3.p2.3.m3.1.1.1.2.cmml">⁢</mo><mrow id="S3.p2.3.m3.1.1.1.1.1" xref="S3.p2.3.m3.1.1.1.1.1.1.cmml"><mo id="S3.p2.3.m3.1.1.1.1.1.2" maxsize="120%" minsize="120%" xref="S3.p2.3.m3.1.1.1.1.1.1.cmml">(</mo><mrow id="S3.p2.3.m3.1.1.1.1.1.1" xref="S3.p2.3.m3.1.1.1.1.1.1.cmml"><msup id="S3.p2.3.m3.1.1.1.1.1.1.2" xref="S3.p2.3.m3.1.1.1.1.1.1.2.cmml"><mi id="S3.p2.3.m3.1.1.1.1.1.1.2.2" xref="S3.p2.3.m3.1.1.1.1.1.1.2.2.cmml">f</mi><mrow id="S3.p2.3.m3.1.1.1.1.1.1.2.3" xref="S3.p2.3.m3.1.1.1.1.1.1.2.3.cmml"><mn id="S3.p2.3.m3.1.1.1.1.1.1.2.3.2" xref="S3.p2.3.m3.1.1.1.1.1.1.2.3.2.cmml">1</mn><mo id="S3.p2.3.m3.1.1.1.1.1.1.2.3.1" xref="S3.p2.3.m3.1.1.1.1.1.1.2.3.1.cmml">−</mo><mrow id="S3.p2.3.m3.1.1.1.1.1.1.2.3.3" xref="S3.p2.3.m3.1.1.1.1.1.1.2.3.3.cmml"><mn 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xref="S3.p2.3.m3.2.2.2.1.1.1.2">superscript</csymbol><ci id="S3.p2.3.m3.2.2.2.1.1.1.2.2.cmml" xref="S3.p2.3.m3.2.2.2.1.1.1.2.2">𝑘</ci><apply id="S3.p2.3.m3.2.2.2.1.1.1.2.3.cmml" xref="S3.p2.3.m3.2.2.2.1.1.1.2.3"><minus id="S3.p2.3.m3.2.2.2.1.1.1.2.3.1.cmml" xref="S3.p2.3.m3.2.2.2.1.1.1.2.3.1"></minus><cn id="S3.p2.3.m3.2.2.2.1.1.1.2.3.2.cmml" type="integer" xref="S3.p2.3.m3.2.2.2.1.1.1.2.3.2">1</cn><apply id="S3.p2.3.m3.2.2.2.1.1.1.2.3.3.cmml" xref="S3.p2.3.m3.2.2.2.1.1.1.2.3.3"><divide id="S3.p2.3.m3.2.2.2.1.1.1.2.3.3.1.cmml" xref="S3.p2.3.m3.2.2.2.1.1.1.2.3.3.1"></divide><cn id="S3.p2.3.m3.2.2.2.1.1.1.2.3.3.2.cmml" type="integer" xref="S3.p2.3.m3.2.2.2.1.1.1.2.3.3.2">1</cn><ci id="S3.p2.3.m3.2.2.2.1.1.1.2.3.3.3.cmml" xref="S3.p2.3.m3.2.2.2.1.1.1.2.3.3.3">𝑡</ci></apply></apply></apply><apply id="S3.p2.3.m3.2.2.2.1.1.1.3.cmml" xref="S3.p2.3.m3.2.2.2.1.1.1.3"><csymbol cd="ambiguous" id="S3.p2.3.m3.2.2.2.1.1.1.3.1.cmml" xref="S3.p2.3.m3.2.2.2.1.1.1.3">superscript</csymbol><ci id="S3.p2.3.m3.2.2.2.1.1.1.3.2.cmml" xref="S3.p2.3.m3.2.2.2.1.1.1.3.2">𝑛</ci><apply id="S3.p2.3.m3.2.2.2.1.1.1.3.3.cmml" xref="S3.p2.3.m3.2.2.2.1.1.1.3.3"><plus id="S3.p2.3.m3.2.2.2.1.1.1.3.3.1.cmml" xref="S3.p2.3.m3.2.2.2.1.1.1.3.3.1"></plus><cn id="S3.p2.3.m3.2.2.2.1.1.1.3.3.2.cmml" type="integer" xref="S3.p2.3.m3.2.2.2.1.1.1.3.3.2">1</cn><apply id="S3.p2.3.m3.2.2.2.1.1.1.3.3.3.cmml" xref="S3.p2.3.m3.2.2.2.1.1.1.3.3.3"><divide id="S3.p2.3.m3.2.2.2.1.1.1.3.3.3.1.cmml" xref="S3.p2.3.m3.2.2.2.1.1.1.3.3.3.1"></divide><cn id="S3.p2.3.m3.2.2.2.1.1.1.3.3.3.2.cmml" type="integer" xref="S3.p2.3.m3.2.2.2.1.1.1.3.3.3.2">1</cn><ci id="S3.p2.3.m3.2.2.2.1.1.1.3.3.3.3.cmml" xref="S3.p2.3.m3.2.2.2.1.1.1.3.3.3.3">𝑡</ci></apply></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p2.3.m3.2c">\tilde{O}\big{(}f^{1-1/t}\cdot n^{1+1/t}\big{)}=\tilde{O}\big{(}k^{1-1/t}\cdot n% ^{1+1/t}\big{)}</annotation><annotation encoding="application/x-llamapun" id="S3.p2.3.m3.2d">over~ start_ARG italic_O end_ARG ( italic_f start_POSTSUPERSCRIPT 1 - 1 / italic_t end_POSTSUPERSCRIPT ⋅ italic_n start_POSTSUPERSCRIPT 1 + 1 / italic_t end_POSTSUPERSCRIPT ) = over~ start_ARG italic_O end_ARG ( italic_k start_POSTSUPERSCRIPT 1 - 1 / italic_t end_POSTSUPERSCRIPT ⋅ italic_n start_POSTSUPERSCRIPT 1 + 1 / italic_t end_POSTSUPERSCRIPT )</annotation></semantics></math>, using the fact that <math alttext="t" class="ltx_Math" display="inline" id="S3.p2.4.m4.1"><semantics id="S3.p2.4.m4.1a"><mi id="S3.p2.4.m4.1.1" xref="S3.p2.4.m4.1.1.cmml">t</mi><annotation-xml encoding="MathML-Content" id="S3.p2.4.m4.1b"><ci id="S3.p2.4.m4.1.1.cmml" xref="S3.p2.4.m4.1.1">𝑡</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.p2.4.m4.1c">t</annotation><annotation encoding="application/x-llamapun" id="S3.p2.4.m4.1d">italic_t</annotation></semantics></math> is always <math alttext="O(\log n)" class="ltx_Math" display="inline" id="S3.p2.5.m5.1"><semantics id="S3.p2.5.m5.1a"><mrow id="S3.p2.5.m5.1.1" xref="S3.p2.5.m5.1.1.cmml"><mi id="S3.p2.5.m5.1.1.3" xref="S3.p2.5.m5.1.1.3.cmml">O</mi><mo id="S3.p2.5.m5.1.1.2" xref="S3.p2.5.m5.1.1.2.cmml">⁢</mo><mrow id="S3.p2.5.m5.1.1.1.1" xref="S3.p2.5.m5.1.1.1.1.1.cmml"><mo id="S3.p2.5.m5.1.1.1.1.2" stretchy="false" xref="S3.p2.5.m5.1.1.1.1.1.cmml">(</mo><mrow id="S3.p2.5.m5.1.1.1.1.1" xref="S3.p2.5.m5.1.1.1.1.1.cmml"><mi id="S3.p2.5.m5.1.1.1.1.1.1" xref="S3.p2.5.m5.1.1.1.1.1.1.cmml">log</mi><mo id="S3.p2.5.m5.1.1.1.1.1a" lspace="0.167em" xref="S3.p2.5.m5.1.1.1.1.1.cmml">⁡</mo><mi id="S3.p2.5.m5.1.1.1.1.1.2" xref="S3.p2.5.m5.1.1.1.1.1.2.cmml">n</mi></mrow><mo id="S3.p2.5.m5.1.1.1.1.3" stretchy="false" xref="S3.p2.5.m5.1.1.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.p2.5.m5.1b"><apply id="S3.p2.5.m5.1.1.cmml" xref="S3.p2.5.m5.1.1"><times id="S3.p2.5.m5.1.1.2.cmml" xref="S3.p2.5.m5.1.1.2"></times><ci id="S3.p2.5.m5.1.1.3.cmml" xref="S3.p2.5.m5.1.1.3">𝑂</ci><apply id="S3.p2.5.m5.1.1.1.1.1.cmml" xref="S3.p2.5.m5.1.1.1.1"><log id="S3.p2.5.m5.1.1.1.1.1.1.cmml" xref="S3.p2.5.m5.1.1.1.1.1.1"></log><ci id="S3.p2.5.m5.1.1.1.1.1.2.cmml" xref="S3.p2.5.m5.1.1.1.1.1.2">𝑛</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p2.5.m5.1c">O(\log n)</annotation><annotation encoding="application/x-llamapun" id="S3.p2.5.m5.1d">italic_O ( roman_log italic_n )</annotation></semantics></math> as setting it larger does not yield saving in space complexity. However, the approximation <em class="ltx_emph ltx_font_italic" id="S3.p2.5.1">analysis</em> of the algorithms is technical.</p> </div> <div class="ltx_para" id="S3.p3"> <p class="ltx_p" id="S3.p3.1">The rest of this section is dedicated to analyzing the approximation performance of Algorithm <a class="ltx_ref" href="https://arxiv.org/html/2503.00712v1#algorithm3" title="In 3 Generic Framework for Streaming Algorithms for Network Design ‣ Streaming Algorithms for Network Design"><span class="ltx_text ltx_ref_tag">3</span></a>, and we will mainly focus on vertex-connectivity requirements. Before analyzing our framework, we present an observation on the structure of VFT spanners (or EFT spanners) that is used in our analysis for all variants.</p> </div> <div class="ltx_theorem ltx_theorem_lemma" id="S3.Thmtheorem1"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="S3.Thmtheorem1.1.1.1">Lemma 3.1</span></span><span class="ltx_text ltx_font_bold" id="S3.Thmtheorem1.2.2">.</span> </h6> <div class="ltx_para" id="S3.Thmtheorem1.p1"> <p class="ltx_p" id="S3.Thmtheorem1.p1.13">Given a weighted graph <math alttext="G=(V,E)" class="ltx_Math" display="inline" id="S3.Thmtheorem1.p1.1.m1.2"><semantics id="S3.Thmtheorem1.p1.1.m1.2a"><mrow id="S3.Thmtheorem1.p1.1.m1.2.3" xref="S3.Thmtheorem1.p1.1.m1.2.3.cmml"><mi id="S3.Thmtheorem1.p1.1.m1.2.3.2" xref="S3.Thmtheorem1.p1.1.m1.2.3.2.cmml">G</mi><mo id="S3.Thmtheorem1.p1.1.m1.2.3.1" xref="S3.Thmtheorem1.p1.1.m1.2.3.1.cmml">=</mo><mrow id="S3.Thmtheorem1.p1.1.m1.2.3.3.2" xref="S3.Thmtheorem1.p1.1.m1.2.3.3.1.cmml"><mo id="S3.Thmtheorem1.p1.1.m1.2.3.3.2.1" stretchy="false" xref="S3.Thmtheorem1.p1.1.m1.2.3.3.1.cmml">(</mo><mi id="S3.Thmtheorem1.p1.1.m1.1.1" xref="S3.Thmtheorem1.p1.1.m1.1.1.cmml">V</mi><mo id="S3.Thmtheorem1.p1.1.m1.2.3.3.2.2" xref="S3.Thmtheorem1.p1.1.m1.2.3.3.1.cmml">,</mo><mi id="S3.Thmtheorem1.p1.1.m1.2.2" xref="S3.Thmtheorem1.p1.1.m1.2.2.cmml">E</mi><mo id="S3.Thmtheorem1.p1.1.m1.2.3.3.2.3" stretchy="false" xref="S3.Thmtheorem1.p1.1.m1.2.3.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem1.p1.1.m1.2b"><apply id="S3.Thmtheorem1.p1.1.m1.2.3.cmml" xref="S3.Thmtheorem1.p1.1.m1.2.3"><eq id="S3.Thmtheorem1.p1.1.m1.2.3.1.cmml" xref="S3.Thmtheorem1.p1.1.m1.2.3.1"></eq><ci id="S3.Thmtheorem1.p1.1.m1.2.3.2.cmml" xref="S3.Thmtheorem1.p1.1.m1.2.3.2">𝐺</ci><interval closure="open" id="S3.Thmtheorem1.p1.1.m1.2.3.3.1.cmml" xref="S3.Thmtheorem1.p1.1.m1.2.3.3.2"><ci id="S3.Thmtheorem1.p1.1.m1.1.1.cmml" xref="S3.Thmtheorem1.p1.1.m1.1.1">𝑉</ci><ci id="S3.Thmtheorem1.p1.1.m1.2.2.cmml" xref="S3.Thmtheorem1.p1.1.m1.2.2">𝐸</ci></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem1.p1.1.m1.2c">G=(V,E)</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem1.p1.1.m1.2d">italic_G = ( italic_V , italic_E )</annotation></semantics></math>, let <math alttext="H" class="ltx_Math" display="inline" id="S3.Thmtheorem1.p1.2.m2.1"><semantics id="S3.Thmtheorem1.p1.2.m2.1a"><mi id="S3.Thmtheorem1.p1.2.m2.1.1" xref="S3.Thmtheorem1.p1.2.m2.1.1.cmml">H</mi><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem1.p1.2.m2.1b"><ci id="S3.Thmtheorem1.p1.2.m2.1.1.cmml" xref="S3.Thmtheorem1.p1.2.m2.1.1">𝐻</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem1.p1.2.m2.1c">H</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem1.p1.2.m2.1d">italic_H</annotation></semantics></math> denote the VFT spanner of <math alttext="G" class="ltx_Math" display="inline" id="S3.Thmtheorem1.p1.3.m3.1"><semantics id="S3.Thmtheorem1.p1.3.m3.1a"><mi id="S3.Thmtheorem1.p1.3.m3.1.1" xref="S3.Thmtheorem1.p1.3.m3.1.1.cmml">G</mi><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem1.p1.3.m3.1b"><ci id="S3.Thmtheorem1.p1.3.m3.1.1.cmml" xref="S3.Thmtheorem1.p1.3.m3.1.1">𝐺</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem1.p1.3.m3.1c">G</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem1.p1.3.m3.1d">italic_G</annotation></semantics></math> constructed by Algorithm <a class="ltx_ref" href="https://arxiv.org/html/2503.00712v1#algorithm2" title="In Weighted graphs. ‣ 2.1 Fault-Tolerant Spanners in Streaming ‣ 2 Preliminaries ‣ Streaming Algorithms for Network Design"><span class="ltx_text ltx_ref_tag">2</span></a> with parameters <math alttext="(t,f,\epsilon)" class="ltx_Math" display="inline" id="S3.Thmtheorem1.p1.4.m4.3"><semantics id="S3.Thmtheorem1.p1.4.m4.3a"><mrow id="S3.Thmtheorem1.p1.4.m4.3.4.2" xref="S3.Thmtheorem1.p1.4.m4.3.4.1.cmml"><mo id="S3.Thmtheorem1.p1.4.m4.3.4.2.1" stretchy="false" xref="S3.Thmtheorem1.p1.4.m4.3.4.1.cmml">(</mo><mi id="S3.Thmtheorem1.p1.4.m4.1.1" xref="S3.Thmtheorem1.p1.4.m4.1.1.cmml">t</mi><mo id="S3.Thmtheorem1.p1.4.m4.3.4.2.2" xref="S3.Thmtheorem1.p1.4.m4.3.4.1.cmml">,</mo><mi id="S3.Thmtheorem1.p1.4.m4.2.2" xref="S3.Thmtheorem1.p1.4.m4.2.2.cmml">f</mi><mo id="S3.Thmtheorem1.p1.4.m4.3.4.2.3" xref="S3.Thmtheorem1.p1.4.m4.3.4.1.cmml">,</mo><mi id="S3.Thmtheorem1.p1.4.m4.3.3" xref="S3.Thmtheorem1.p1.4.m4.3.3.cmml">ϵ</mi><mo id="S3.Thmtheorem1.p1.4.m4.3.4.2.4" stretchy="false" xref="S3.Thmtheorem1.p1.4.m4.3.4.1.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem1.p1.4.m4.3b"><vector id="S3.Thmtheorem1.p1.4.m4.3.4.1.cmml" xref="S3.Thmtheorem1.p1.4.m4.3.4.2"><ci id="S3.Thmtheorem1.p1.4.m4.1.1.cmml" xref="S3.Thmtheorem1.p1.4.m4.1.1">𝑡</ci><ci id="S3.Thmtheorem1.p1.4.m4.2.2.cmml" xref="S3.Thmtheorem1.p1.4.m4.2.2">𝑓</ci><ci id="S3.Thmtheorem1.p1.4.m4.3.3.cmml" xref="S3.Thmtheorem1.p1.4.m4.3.3">italic-ϵ</ci></vector></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem1.p1.4.m4.3c">(t,f,\epsilon)</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem1.p1.4.m4.3d">( italic_t , italic_f , italic_ϵ )</annotation></semantics></math>. If <math alttext="e\in E" class="ltx_Math" display="inline" id="S3.Thmtheorem1.p1.5.m5.1"><semantics id="S3.Thmtheorem1.p1.5.m5.1a"><mrow id="S3.Thmtheorem1.p1.5.m5.1.1" xref="S3.Thmtheorem1.p1.5.m5.1.1.cmml"><mi id="S3.Thmtheorem1.p1.5.m5.1.1.2" xref="S3.Thmtheorem1.p1.5.m5.1.1.2.cmml">e</mi><mo id="S3.Thmtheorem1.p1.5.m5.1.1.1" xref="S3.Thmtheorem1.p1.5.m5.1.1.1.cmml">∈</mo><mi id="S3.Thmtheorem1.p1.5.m5.1.1.3" xref="S3.Thmtheorem1.p1.5.m5.1.1.3.cmml">E</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem1.p1.5.m5.1b"><apply id="S3.Thmtheorem1.p1.5.m5.1.1.cmml" xref="S3.Thmtheorem1.p1.5.m5.1.1"><in id="S3.Thmtheorem1.p1.5.m5.1.1.1.cmml" xref="S3.Thmtheorem1.p1.5.m5.1.1.1"></in><ci id="S3.Thmtheorem1.p1.5.m5.1.1.2.cmml" xref="S3.Thmtheorem1.p1.5.m5.1.1.2">𝑒</ci><ci id="S3.Thmtheorem1.p1.5.m5.1.1.3.cmml" xref="S3.Thmtheorem1.p1.5.m5.1.1.3">𝐸</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem1.p1.5.m5.1c">e\in E</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem1.p1.5.m5.1d">italic_e ∈ italic_E</annotation></semantics></math> with <math alttext="w(e)\in B_{j}" class="ltx_Math" display="inline" id="S3.Thmtheorem1.p1.6.m6.1"><semantics id="S3.Thmtheorem1.p1.6.m6.1a"><mrow id="S3.Thmtheorem1.p1.6.m6.1.2" xref="S3.Thmtheorem1.p1.6.m6.1.2.cmml"><mrow id="S3.Thmtheorem1.p1.6.m6.1.2.2" xref="S3.Thmtheorem1.p1.6.m6.1.2.2.cmml"><mi id="S3.Thmtheorem1.p1.6.m6.1.2.2.2" xref="S3.Thmtheorem1.p1.6.m6.1.2.2.2.cmml">w</mi><mo id="S3.Thmtheorem1.p1.6.m6.1.2.2.1" xref="S3.Thmtheorem1.p1.6.m6.1.2.2.1.cmml">⁢</mo><mrow id="S3.Thmtheorem1.p1.6.m6.1.2.2.3.2" xref="S3.Thmtheorem1.p1.6.m6.1.2.2.cmml"><mo id="S3.Thmtheorem1.p1.6.m6.1.2.2.3.2.1" stretchy="false" xref="S3.Thmtheorem1.p1.6.m6.1.2.2.cmml">(</mo><mi id="S3.Thmtheorem1.p1.6.m6.1.1" xref="S3.Thmtheorem1.p1.6.m6.1.1.cmml">e</mi><mo id="S3.Thmtheorem1.p1.6.m6.1.2.2.3.2.2" stretchy="false" xref="S3.Thmtheorem1.p1.6.m6.1.2.2.cmml">)</mo></mrow></mrow><mo id="S3.Thmtheorem1.p1.6.m6.1.2.1" xref="S3.Thmtheorem1.p1.6.m6.1.2.1.cmml">∈</mo><msub id="S3.Thmtheorem1.p1.6.m6.1.2.3" xref="S3.Thmtheorem1.p1.6.m6.1.2.3.cmml"><mi id="S3.Thmtheorem1.p1.6.m6.1.2.3.2" xref="S3.Thmtheorem1.p1.6.m6.1.2.3.2.cmml">B</mi><mi id="S3.Thmtheorem1.p1.6.m6.1.2.3.3" xref="S3.Thmtheorem1.p1.6.m6.1.2.3.3.cmml">j</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem1.p1.6.m6.1b"><apply id="S3.Thmtheorem1.p1.6.m6.1.2.cmml" xref="S3.Thmtheorem1.p1.6.m6.1.2"><in id="S3.Thmtheorem1.p1.6.m6.1.2.1.cmml" xref="S3.Thmtheorem1.p1.6.m6.1.2.1"></in><apply id="S3.Thmtheorem1.p1.6.m6.1.2.2.cmml" xref="S3.Thmtheorem1.p1.6.m6.1.2.2"><times id="S3.Thmtheorem1.p1.6.m6.1.2.2.1.cmml" xref="S3.Thmtheorem1.p1.6.m6.1.2.2.1"></times><ci id="S3.Thmtheorem1.p1.6.m6.1.2.2.2.cmml" xref="S3.Thmtheorem1.p1.6.m6.1.2.2.2">𝑤</ci><ci id="S3.Thmtheorem1.p1.6.m6.1.1.cmml" xref="S3.Thmtheorem1.p1.6.m6.1.1">𝑒</ci></apply><apply id="S3.Thmtheorem1.p1.6.m6.1.2.3.cmml" xref="S3.Thmtheorem1.p1.6.m6.1.2.3"><csymbol cd="ambiguous" id="S3.Thmtheorem1.p1.6.m6.1.2.3.1.cmml" xref="S3.Thmtheorem1.p1.6.m6.1.2.3">subscript</csymbol><ci id="S3.Thmtheorem1.p1.6.m6.1.2.3.2.cmml" xref="S3.Thmtheorem1.p1.6.m6.1.2.3.2">𝐵</ci><ci id="S3.Thmtheorem1.p1.6.m6.1.2.3.3.cmml" xref="S3.Thmtheorem1.p1.6.m6.1.2.3.3">𝑗</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem1.p1.6.m6.1c">w(e)\in B_{j}</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem1.p1.6.m6.1d">italic_w ( italic_e ) ∈ italic_B start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT</annotation></semantics></math> does not belong to <math alttext="H_{j}" class="ltx_Math" display="inline" id="S3.Thmtheorem1.p1.7.m7.1"><semantics id="S3.Thmtheorem1.p1.7.m7.1a"><msub id="S3.Thmtheorem1.p1.7.m7.1.1" xref="S3.Thmtheorem1.p1.7.m7.1.1.cmml"><mi id="S3.Thmtheorem1.p1.7.m7.1.1.2" xref="S3.Thmtheorem1.p1.7.m7.1.1.2.cmml">H</mi><mi id="S3.Thmtheorem1.p1.7.m7.1.1.3" xref="S3.Thmtheorem1.p1.7.m7.1.1.3.cmml">j</mi></msub><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem1.p1.7.m7.1b"><apply id="S3.Thmtheorem1.p1.7.m7.1.1.cmml" xref="S3.Thmtheorem1.p1.7.m7.1.1"><csymbol cd="ambiguous" id="S3.Thmtheorem1.p1.7.m7.1.1.1.cmml" xref="S3.Thmtheorem1.p1.7.m7.1.1">subscript</csymbol><ci id="S3.Thmtheorem1.p1.7.m7.1.1.2.cmml" xref="S3.Thmtheorem1.p1.7.m7.1.1.2">𝐻</ci><ci id="S3.Thmtheorem1.p1.7.m7.1.1.3.cmml" xref="S3.Thmtheorem1.p1.7.m7.1.1.3">𝑗</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem1.p1.7.m7.1c">H_{j}</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem1.p1.7.m7.1d">italic_H start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT</annotation></semantics></math>, i.e., <math alttext="e\notin H_{j}" class="ltx_Math" display="inline" id="S3.Thmtheorem1.p1.8.m8.1"><semantics id="S3.Thmtheorem1.p1.8.m8.1a"><mrow id="S3.Thmtheorem1.p1.8.m8.1.1" xref="S3.Thmtheorem1.p1.8.m8.1.1.cmml"><mi id="S3.Thmtheorem1.p1.8.m8.1.1.2" xref="S3.Thmtheorem1.p1.8.m8.1.1.2.cmml">e</mi><mo id="S3.Thmtheorem1.p1.8.m8.1.1.1" xref="S3.Thmtheorem1.p1.8.m8.1.1.1.cmml">∉</mo><msub id="S3.Thmtheorem1.p1.8.m8.1.1.3" xref="S3.Thmtheorem1.p1.8.m8.1.1.3.cmml"><mi id="S3.Thmtheorem1.p1.8.m8.1.1.3.2" xref="S3.Thmtheorem1.p1.8.m8.1.1.3.2.cmml">H</mi><mi id="S3.Thmtheorem1.p1.8.m8.1.1.3.3" xref="S3.Thmtheorem1.p1.8.m8.1.1.3.3.cmml">j</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem1.p1.8.m8.1b"><apply id="S3.Thmtheorem1.p1.8.m8.1.1.cmml" xref="S3.Thmtheorem1.p1.8.m8.1.1"><notin id="S3.Thmtheorem1.p1.8.m8.1.1.1.cmml" xref="S3.Thmtheorem1.p1.8.m8.1.1.1"></notin><ci id="S3.Thmtheorem1.p1.8.m8.1.1.2.cmml" xref="S3.Thmtheorem1.p1.8.m8.1.1.2">𝑒</ci><apply id="S3.Thmtheorem1.p1.8.m8.1.1.3.cmml" xref="S3.Thmtheorem1.p1.8.m8.1.1.3"><csymbol cd="ambiguous" id="S3.Thmtheorem1.p1.8.m8.1.1.3.1.cmml" xref="S3.Thmtheorem1.p1.8.m8.1.1.3">subscript</csymbol><ci id="S3.Thmtheorem1.p1.8.m8.1.1.3.2.cmml" xref="S3.Thmtheorem1.p1.8.m8.1.1.3.2">𝐻</ci><ci id="S3.Thmtheorem1.p1.8.m8.1.1.3.3.cmml" xref="S3.Thmtheorem1.p1.8.m8.1.1.3.3">𝑗</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem1.p1.8.m8.1c">e\notin H_{j}</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem1.p1.8.m8.1d">italic_e ∉ italic_H start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT</annotation></semantics></math>, then <math alttext="H_{j}" class="ltx_Math" display="inline" id="S3.Thmtheorem1.p1.9.m9.1"><semantics id="S3.Thmtheorem1.p1.9.m9.1a"><msub id="S3.Thmtheorem1.p1.9.m9.1.1" xref="S3.Thmtheorem1.p1.9.m9.1.1.cmml"><mi id="S3.Thmtheorem1.p1.9.m9.1.1.2" xref="S3.Thmtheorem1.p1.9.m9.1.1.2.cmml">H</mi><mi id="S3.Thmtheorem1.p1.9.m9.1.1.3" xref="S3.Thmtheorem1.p1.9.m9.1.1.3.cmml">j</mi></msub><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem1.p1.9.m9.1b"><apply id="S3.Thmtheorem1.p1.9.m9.1.1.cmml" xref="S3.Thmtheorem1.p1.9.m9.1.1"><csymbol cd="ambiguous" id="S3.Thmtheorem1.p1.9.m9.1.1.1.cmml" xref="S3.Thmtheorem1.p1.9.m9.1.1">subscript</csymbol><ci id="S3.Thmtheorem1.p1.9.m9.1.1.2.cmml" xref="S3.Thmtheorem1.p1.9.m9.1.1.2">𝐻</ci><ci id="S3.Thmtheorem1.p1.9.m9.1.1.3.cmml" xref="S3.Thmtheorem1.p1.9.m9.1.1.3">𝑗</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem1.p1.9.m9.1c">H_{j}</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem1.p1.9.m9.1d">italic_H start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT</annotation></semantics></math> contains at least <math alttext="L=\lfloor f/(t-1)\rfloor+1" class="ltx_Math" display="inline" id="S3.Thmtheorem1.p1.10.m10.1"><semantics id="S3.Thmtheorem1.p1.10.m10.1a"><mrow id="S3.Thmtheorem1.p1.10.m10.1.1" xref="S3.Thmtheorem1.p1.10.m10.1.1.cmml"><mi id="S3.Thmtheorem1.p1.10.m10.1.1.3" xref="S3.Thmtheorem1.p1.10.m10.1.1.3.cmml">L</mi><mo id="S3.Thmtheorem1.p1.10.m10.1.1.2" xref="S3.Thmtheorem1.p1.10.m10.1.1.2.cmml">=</mo><mrow id="S3.Thmtheorem1.p1.10.m10.1.1.1" xref="S3.Thmtheorem1.p1.10.m10.1.1.1.cmml"><mrow id="S3.Thmtheorem1.p1.10.m10.1.1.1.1.1" xref="S3.Thmtheorem1.p1.10.m10.1.1.1.1.2.cmml"><mo id="S3.Thmtheorem1.p1.10.m10.1.1.1.1.1.2" stretchy="false" xref="S3.Thmtheorem1.p1.10.m10.1.1.1.1.2.1.cmml">⌊</mo><mrow id="S3.Thmtheorem1.p1.10.m10.1.1.1.1.1.1" xref="S3.Thmtheorem1.p1.10.m10.1.1.1.1.1.1.cmml"><mi id="S3.Thmtheorem1.p1.10.m10.1.1.1.1.1.1.3" xref="S3.Thmtheorem1.p1.10.m10.1.1.1.1.1.1.3.cmml">f</mi><mo id="S3.Thmtheorem1.p1.10.m10.1.1.1.1.1.1.2" xref="S3.Thmtheorem1.p1.10.m10.1.1.1.1.1.1.2.cmml">/</mo><mrow id="S3.Thmtheorem1.p1.10.m10.1.1.1.1.1.1.1.1" xref="S3.Thmtheorem1.p1.10.m10.1.1.1.1.1.1.1.1.1.cmml"><mo id="S3.Thmtheorem1.p1.10.m10.1.1.1.1.1.1.1.1.2" stretchy="false" xref="S3.Thmtheorem1.p1.10.m10.1.1.1.1.1.1.1.1.1.cmml">(</mo><mrow id="S3.Thmtheorem1.p1.10.m10.1.1.1.1.1.1.1.1.1" xref="S3.Thmtheorem1.p1.10.m10.1.1.1.1.1.1.1.1.1.cmml"><mi id="S3.Thmtheorem1.p1.10.m10.1.1.1.1.1.1.1.1.1.2" xref="S3.Thmtheorem1.p1.10.m10.1.1.1.1.1.1.1.1.1.2.cmml">t</mi><mo id="S3.Thmtheorem1.p1.10.m10.1.1.1.1.1.1.1.1.1.1" xref="S3.Thmtheorem1.p1.10.m10.1.1.1.1.1.1.1.1.1.1.cmml">−</mo><mn id="S3.Thmtheorem1.p1.10.m10.1.1.1.1.1.1.1.1.1.3" xref="S3.Thmtheorem1.p1.10.m10.1.1.1.1.1.1.1.1.1.3.cmml">1</mn></mrow><mo id="S3.Thmtheorem1.p1.10.m10.1.1.1.1.1.1.1.1.3" stretchy="false" xref="S3.Thmtheorem1.p1.10.m10.1.1.1.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S3.Thmtheorem1.p1.10.m10.1.1.1.1.1.3" stretchy="false" xref="S3.Thmtheorem1.p1.10.m10.1.1.1.1.2.1.cmml">⌋</mo></mrow><mo id="S3.Thmtheorem1.p1.10.m10.1.1.1.2" xref="S3.Thmtheorem1.p1.10.m10.1.1.1.2.cmml">+</mo><mn id="S3.Thmtheorem1.p1.10.m10.1.1.1.3" xref="S3.Thmtheorem1.p1.10.m10.1.1.1.3.cmml">1</mn></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem1.p1.10.m10.1b"><apply id="S3.Thmtheorem1.p1.10.m10.1.1.cmml" xref="S3.Thmtheorem1.p1.10.m10.1.1"><eq id="S3.Thmtheorem1.p1.10.m10.1.1.2.cmml" xref="S3.Thmtheorem1.p1.10.m10.1.1.2"></eq><ci id="S3.Thmtheorem1.p1.10.m10.1.1.3.cmml" xref="S3.Thmtheorem1.p1.10.m10.1.1.3">𝐿</ci><apply id="S3.Thmtheorem1.p1.10.m10.1.1.1.cmml" xref="S3.Thmtheorem1.p1.10.m10.1.1.1"><plus id="S3.Thmtheorem1.p1.10.m10.1.1.1.2.cmml" xref="S3.Thmtheorem1.p1.10.m10.1.1.1.2"></plus><apply id="S3.Thmtheorem1.p1.10.m10.1.1.1.1.2.cmml" xref="S3.Thmtheorem1.p1.10.m10.1.1.1.1.1"><floor id="S3.Thmtheorem1.p1.10.m10.1.1.1.1.2.1.cmml" xref="S3.Thmtheorem1.p1.10.m10.1.1.1.1.1.2"></floor><apply id="S3.Thmtheorem1.p1.10.m10.1.1.1.1.1.1.cmml" xref="S3.Thmtheorem1.p1.10.m10.1.1.1.1.1.1"><divide id="S3.Thmtheorem1.p1.10.m10.1.1.1.1.1.1.2.cmml" xref="S3.Thmtheorem1.p1.10.m10.1.1.1.1.1.1.2"></divide><ci id="S3.Thmtheorem1.p1.10.m10.1.1.1.1.1.1.3.cmml" xref="S3.Thmtheorem1.p1.10.m10.1.1.1.1.1.1.3">𝑓</ci><apply id="S3.Thmtheorem1.p1.10.m10.1.1.1.1.1.1.1.1.1.cmml" xref="S3.Thmtheorem1.p1.10.m10.1.1.1.1.1.1.1.1"><minus id="S3.Thmtheorem1.p1.10.m10.1.1.1.1.1.1.1.1.1.1.cmml" xref="S3.Thmtheorem1.p1.10.m10.1.1.1.1.1.1.1.1.1.1"></minus><ci id="S3.Thmtheorem1.p1.10.m10.1.1.1.1.1.1.1.1.1.2.cmml" xref="S3.Thmtheorem1.p1.10.m10.1.1.1.1.1.1.1.1.1.2">𝑡</ci><cn id="S3.Thmtheorem1.p1.10.m10.1.1.1.1.1.1.1.1.1.3.cmml" type="integer" xref="S3.Thmtheorem1.p1.10.m10.1.1.1.1.1.1.1.1.1.3">1</cn></apply></apply></apply><cn id="S3.Thmtheorem1.p1.10.m10.1.1.1.3.cmml" type="integer" xref="S3.Thmtheorem1.p1.10.m10.1.1.1.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem1.p1.10.m10.1c">L=\lfloor f/(t-1)\rfloor+1</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem1.p1.10.m10.1d">italic_L = ⌊ italic_f / ( italic_t - 1 ) ⌋ + 1</annotation></semantics></math> vertex-disjoint <math alttext="uv" class="ltx_Math" display="inline" id="S3.Thmtheorem1.p1.11.m11.1"><semantics id="S3.Thmtheorem1.p1.11.m11.1a"><mrow id="S3.Thmtheorem1.p1.11.m11.1.1" xref="S3.Thmtheorem1.p1.11.m11.1.1.cmml"><mi id="S3.Thmtheorem1.p1.11.m11.1.1.2" xref="S3.Thmtheorem1.p1.11.m11.1.1.2.cmml">u</mi><mo id="S3.Thmtheorem1.p1.11.m11.1.1.1" xref="S3.Thmtheorem1.p1.11.m11.1.1.1.cmml">⁢</mo><mi id="S3.Thmtheorem1.p1.11.m11.1.1.3" xref="S3.Thmtheorem1.p1.11.m11.1.1.3.cmml">v</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem1.p1.11.m11.1b"><apply id="S3.Thmtheorem1.p1.11.m11.1.1.cmml" xref="S3.Thmtheorem1.p1.11.m11.1.1"><times id="S3.Thmtheorem1.p1.11.m11.1.1.1.cmml" xref="S3.Thmtheorem1.p1.11.m11.1.1.1"></times><ci id="S3.Thmtheorem1.p1.11.m11.1.1.2.cmml" xref="S3.Thmtheorem1.p1.11.m11.1.1.2">𝑢</ci><ci id="S3.Thmtheorem1.p1.11.m11.1.1.3.cmml" xref="S3.Thmtheorem1.p1.11.m11.1.1.3">𝑣</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem1.p1.11.m11.1c">uv</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem1.p1.11.m11.1d">italic_u italic_v</annotation></semantics></math>-paths, each containing at most <math alttext="t" class="ltx_Math" display="inline" id="S3.Thmtheorem1.p1.12.m12.1"><semantics id="S3.Thmtheorem1.p1.12.m12.1a"><mi id="S3.Thmtheorem1.p1.12.m12.1.1" xref="S3.Thmtheorem1.p1.12.m12.1.1.cmml">t</mi><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem1.p1.12.m12.1b"><ci id="S3.Thmtheorem1.p1.12.m12.1.1.cmml" xref="S3.Thmtheorem1.p1.12.m12.1.1">𝑡</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem1.p1.12.m12.1c">t</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem1.p1.12.m12.1d">italic_t</annotation></semantics></math> edges that all have weights in <math alttext="B_{j}" class="ltx_Math" display="inline" id="S3.Thmtheorem1.p1.13.m13.1"><semantics id="S3.Thmtheorem1.p1.13.m13.1a"><msub id="S3.Thmtheorem1.p1.13.m13.1.1" xref="S3.Thmtheorem1.p1.13.m13.1.1.cmml"><mi id="S3.Thmtheorem1.p1.13.m13.1.1.2" xref="S3.Thmtheorem1.p1.13.m13.1.1.2.cmml">B</mi><mi id="S3.Thmtheorem1.p1.13.m13.1.1.3" xref="S3.Thmtheorem1.p1.13.m13.1.1.3.cmml">j</mi></msub><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem1.p1.13.m13.1b"><apply id="S3.Thmtheorem1.p1.13.m13.1.1.cmml" xref="S3.Thmtheorem1.p1.13.m13.1.1"><csymbol cd="ambiguous" id="S3.Thmtheorem1.p1.13.m13.1.1.1.cmml" xref="S3.Thmtheorem1.p1.13.m13.1.1">subscript</csymbol><ci id="S3.Thmtheorem1.p1.13.m13.1.1.2.cmml" xref="S3.Thmtheorem1.p1.13.m13.1.1.2">𝐵</ci><ci id="S3.Thmtheorem1.p1.13.m13.1.1.3.cmml" xref="S3.Thmtheorem1.p1.13.m13.1.1.3">𝑗</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem1.p1.13.m13.1c">B_{j}</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem1.p1.13.m13.1d">italic_B start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT</annotation></semantics></math>.</p> </div> </div> <div class="ltx_proof" id="S3.1"> <h6 class="ltx_title ltx_runin ltx_font_italic ltx_title_proof">Proof.</h6> <div class="ltx_para" id="S3.1.p1"> <p class="ltx_p" id="S3.1.p1.23">To find the vertex-disjoint paths <math alttext="P_{1},\dots,P_{L}" class="ltx_Math" display="inline" id="S3.1.p1.1.m1.3"><semantics id="S3.1.p1.1.m1.3a"><mrow id="S3.1.p1.1.m1.3.3.2" xref="S3.1.p1.1.m1.3.3.3.cmml"><msub id="S3.1.p1.1.m1.2.2.1.1" xref="S3.1.p1.1.m1.2.2.1.1.cmml"><mi id="S3.1.p1.1.m1.2.2.1.1.2" xref="S3.1.p1.1.m1.2.2.1.1.2.cmml">P</mi><mn id="S3.1.p1.1.m1.2.2.1.1.3" xref="S3.1.p1.1.m1.2.2.1.1.3.cmml">1</mn></msub><mo id="S3.1.p1.1.m1.3.3.2.3" xref="S3.1.p1.1.m1.3.3.3.cmml">,</mo><mi id="S3.1.p1.1.m1.1.1" mathvariant="normal" xref="S3.1.p1.1.m1.1.1.cmml">…</mi><mo id="S3.1.p1.1.m1.3.3.2.4" xref="S3.1.p1.1.m1.3.3.3.cmml">,</mo><msub id="S3.1.p1.1.m1.3.3.2.2" xref="S3.1.p1.1.m1.3.3.2.2.cmml"><mi id="S3.1.p1.1.m1.3.3.2.2.2" xref="S3.1.p1.1.m1.3.3.2.2.2.cmml">P</mi><mi id="S3.1.p1.1.m1.3.3.2.2.3" xref="S3.1.p1.1.m1.3.3.2.2.3.cmml">L</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S3.1.p1.1.m1.3b"><list id="S3.1.p1.1.m1.3.3.3.cmml" xref="S3.1.p1.1.m1.3.3.2"><apply id="S3.1.p1.1.m1.2.2.1.1.cmml" xref="S3.1.p1.1.m1.2.2.1.1"><csymbol cd="ambiguous" id="S3.1.p1.1.m1.2.2.1.1.1.cmml" xref="S3.1.p1.1.m1.2.2.1.1">subscript</csymbol><ci id="S3.1.p1.1.m1.2.2.1.1.2.cmml" xref="S3.1.p1.1.m1.2.2.1.1.2">𝑃</ci><cn id="S3.1.p1.1.m1.2.2.1.1.3.cmml" type="integer" xref="S3.1.p1.1.m1.2.2.1.1.3">1</cn></apply><ci id="S3.1.p1.1.m1.1.1.cmml" xref="S3.1.p1.1.m1.1.1">…</ci><apply id="S3.1.p1.1.m1.3.3.2.2.cmml" xref="S3.1.p1.1.m1.3.3.2.2"><csymbol cd="ambiguous" id="S3.1.p1.1.m1.3.3.2.2.1.cmml" xref="S3.1.p1.1.m1.3.3.2.2">subscript</csymbol><ci id="S3.1.p1.1.m1.3.3.2.2.2.cmml" xref="S3.1.p1.1.m1.3.3.2.2.2">𝑃</ci><ci id="S3.1.p1.1.m1.3.3.2.2.3.cmml" xref="S3.1.p1.1.m1.3.3.2.2.3">𝐿</ci></apply></list></annotation-xml><annotation encoding="application/x-tex" id="S3.1.p1.1.m1.3c">P_{1},\dots,P_{L}</annotation><annotation encoding="application/x-llamapun" id="S3.1.p1.1.m1.3d">italic_P start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , … , italic_P start_POSTSUBSCRIPT italic_L end_POSTSUBSCRIPT</annotation></semantics></math>, we work with the fault-tolerant spanner <math alttext="H_{j}" class="ltx_Math" display="inline" id="S3.1.p1.2.m2.1"><semantics id="S3.1.p1.2.m2.1a"><msub id="S3.1.p1.2.m2.1.1" xref="S3.1.p1.2.m2.1.1.cmml"><mi id="S3.1.p1.2.m2.1.1.2" xref="S3.1.p1.2.m2.1.1.2.cmml">H</mi><mi id="S3.1.p1.2.m2.1.1.3" xref="S3.1.p1.2.m2.1.1.3.cmml">j</mi></msub><annotation-xml encoding="MathML-Content" id="S3.1.p1.2.m2.1b"><apply id="S3.1.p1.2.m2.1.1.cmml" xref="S3.1.p1.2.m2.1.1"><csymbol cd="ambiguous" id="S3.1.p1.2.m2.1.1.1.cmml" xref="S3.1.p1.2.m2.1.1">subscript</csymbol><ci id="S3.1.p1.2.m2.1.1.2.cmml" xref="S3.1.p1.2.m2.1.1.2">𝐻</ci><ci id="S3.1.p1.2.m2.1.1.3.cmml" xref="S3.1.p1.2.m2.1.1.3">𝑗</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.1.p1.2.m2.1c">H_{j}</annotation><annotation encoding="application/x-llamapun" id="S3.1.p1.2.m2.1d">italic_H start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT</annotation></semantics></math> corresponding to <math alttext="B_{j}" class="ltx_Math" display="inline" id="S3.1.p1.3.m3.1"><semantics id="S3.1.p1.3.m3.1a"><msub id="S3.1.p1.3.m3.1.1" xref="S3.1.p1.3.m3.1.1.cmml"><mi id="S3.1.p1.3.m3.1.1.2" xref="S3.1.p1.3.m3.1.1.2.cmml">B</mi><mi id="S3.1.p1.3.m3.1.1.3" xref="S3.1.p1.3.m3.1.1.3.cmml">j</mi></msub><annotation-xml encoding="MathML-Content" id="S3.1.p1.3.m3.1b"><apply id="S3.1.p1.3.m3.1.1.cmml" xref="S3.1.p1.3.m3.1.1"><csymbol cd="ambiguous" id="S3.1.p1.3.m3.1.1.1.cmml" xref="S3.1.p1.3.m3.1.1">subscript</csymbol><ci id="S3.1.p1.3.m3.1.1.2.cmml" xref="S3.1.p1.3.m3.1.1.2">𝐵</ci><ci id="S3.1.p1.3.m3.1.1.3.cmml" xref="S3.1.p1.3.m3.1.1.3">𝑗</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.1.p1.3.m3.1c">B_{j}</annotation><annotation encoding="application/x-llamapun" id="S3.1.p1.3.m3.1d">italic_B start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT</annotation></semantics></math>. We perform <math alttext="L" class="ltx_Math" display="inline" id="S3.1.p1.4.m4.1"><semantics id="S3.1.p1.4.m4.1a"><mi id="S3.1.p1.4.m4.1.1" xref="S3.1.p1.4.m4.1.1.cmml">L</mi><annotation-xml encoding="MathML-Content" id="S3.1.p1.4.m4.1b"><ci id="S3.1.p1.4.m4.1.1.cmml" xref="S3.1.p1.4.m4.1.1">𝐿</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.1.p1.4.m4.1c">L</annotation><annotation encoding="application/x-llamapun" id="S3.1.p1.4.m4.1d">italic_L</annotation></semantics></math> iterations, and in each iteration <math alttext="i" class="ltx_Math" display="inline" id="S3.1.p1.5.m5.1"><semantics id="S3.1.p1.5.m5.1a"><mi id="S3.1.p1.5.m5.1.1" xref="S3.1.p1.5.m5.1.1.cmml">i</mi><annotation-xml encoding="MathML-Content" id="S3.1.p1.5.m5.1b"><ci id="S3.1.p1.5.m5.1.1.cmml" xref="S3.1.p1.5.m5.1.1">𝑖</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.1.p1.5.m5.1c">i</annotation><annotation encoding="application/x-llamapun" id="S3.1.p1.5.m5.1d">italic_i</annotation></semantics></math>, we find a <math alttext="uv" class="ltx_Math" display="inline" id="S3.1.p1.6.m6.1"><semantics id="S3.1.p1.6.m6.1a"><mrow id="S3.1.p1.6.m6.1.1" xref="S3.1.p1.6.m6.1.1.cmml"><mi id="S3.1.p1.6.m6.1.1.2" xref="S3.1.p1.6.m6.1.1.2.cmml">u</mi><mo id="S3.1.p1.6.m6.1.1.1" xref="S3.1.p1.6.m6.1.1.1.cmml">⁢</mo><mi id="S3.1.p1.6.m6.1.1.3" xref="S3.1.p1.6.m6.1.1.3.cmml">v</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.1.p1.6.m6.1b"><apply id="S3.1.p1.6.m6.1.1.cmml" xref="S3.1.p1.6.m6.1.1"><times id="S3.1.p1.6.m6.1.1.1.cmml" xref="S3.1.p1.6.m6.1.1.1"></times><ci id="S3.1.p1.6.m6.1.1.2.cmml" xref="S3.1.p1.6.m6.1.1.2">𝑢</ci><ci id="S3.1.p1.6.m6.1.1.3.cmml" xref="S3.1.p1.6.m6.1.1.3">𝑣</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.1.p1.6.m6.1c">uv</annotation><annotation encoding="application/x-llamapun" id="S3.1.p1.6.m6.1d">italic_u italic_v</annotation></semantics></math>-path <math alttext="P_{i}" class="ltx_Math" display="inline" id="S3.1.p1.7.m7.1"><semantics id="S3.1.p1.7.m7.1a"><msub id="S3.1.p1.7.m7.1.1" xref="S3.1.p1.7.m7.1.1.cmml"><mi id="S3.1.p1.7.m7.1.1.2" xref="S3.1.p1.7.m7.1.1.2.cmml">P</mi><mi id="S3.1.p1.7.m7.1.1.3" xref="S3.1.p1.7.m7.1.1.3.cmml">i</mi></msub><annotation-xml encoding="MathML-Content" id="S3.1.p1.7.m7.1b"><apply id="S3.1.p1.7.m7.1.1.cmml" xref="S3.1.p1.7.m7.1.1"><csymbol cd="ambiguous" id="S3.1.p1.7.m7.1.1.1.cmml" xref="S3.1.p1.7.m7.1.1">subscript</csymbol><ci id="S3.1.p1.7.m7.1.1.2.cmml" xref="S3.1.p1.7.m7.1.1.2">𝑃</ci><ci id="S3.1.p1.7.m7.1.1.3.cmml" xref="S3.1.p1.7.m7.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.1.p1.7.m7.1c">P_{i}</annotation><annotation encoding="application/x-llamapun" id="S3.1.p1.7.m7.1d">italic_P start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math> of length at most <math alttext="t" class="ltx_Math" display="inline" id="S3.1.p1.8.m8.1"><semantics id="S3.1.p1.8.m8.1a"><mi id="S3.1.p1.8.m8.1.1" xref="S3.1.p1.8.m8.1.1.cmml">t</mi><annotation-xml encoding="MathML-Content" id="S3.1.p1.8.m8.1b"><ci id="S3.1.p1.8.m8.1.1.cmml" xref="S3.1.p1.8.m8.1.1">𝑡</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.1.p1.8.m8.1c">t</annotation><annotation encoding="application/x-llamapun" id="S3.1.p1.8.m8.1d">italic_t</annotation></semantics></math> that is vertex-disjoint from the previously constructed <math alttext="uv" class="ltx_Math" display="inline" id="S3.1.p1.9.m9.1"><semantics id="S3.1.p1.9.m9.1a"><mrow id="S3.1.p1.9.m9.1.1" xref="S3.1.p1.9.m9.1.1.cmml"><mi id="S3.1.p1.9.m9.1.1.2" xref="S3.1.p1.9.m9.1.1.2.cmml">u</mi><mo id="S3.1.p1.9.m9.1.1.1" xref="S3.1.p1.9.m9.1.1.1.cmml">⁢</mo><mi id="S3.1.p1.9.m9.1.1.3" xref="S3.1.p1.9.m9.1.1.3.cmml">v</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.1.p1.9.m9.1b"><apply id="S3.1.p1.9.m9.1.1.cmml" xref="S3.1.p1.9.m9.1.1"><times id="S3.1.p1.9.m9.1.1.1.cmml" xref="S3.1.p1.9.m9.1.1.1"></times><ci id="S3.1.p1.9.m9.1.1.2.cmml" xref="S3.1.p1.9.m9.1.1.2">𝑢</ci><ci id="S3.1.p1.9.m9.1.1.3.cmml" xref="S3.1.p1.9.m9.1.1.3">𝑣</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.1.p1.9.m9.1c">uv</annotation><annotation encoding="application/x-llamapun" id="S3.1.p1.9.m9.1d">italic_u italic_v</annotation></semantics></math>-paths <math alttext="P_{1},\dots,P_{i-1}" class="ltx_Math" display="inline" id="S3.1.p1.10.m10.3"><semantics id="S3.1.p1.10.m10.3a"><mrow id="S3.1.p1.10.m10.3.3.2" xref="S3.1.p1.10.m10.3.3.3.cmml"><msub id="S3.1.p1.10.m10.2.2.1.1" xref="S3.1.p1.10.m10.2.2.1.1.cmml"><mi id="S3.1.p1.10.m10.2.2.1.1.2" xref="S3.1.p1.10.m10.2.2.1.1.2.cmml">P</mi><mn id="S3.1.p1.10.m10.2.2.1.1.3" xref="S3.1.p1.10.m10.2.2.1.1.3.cmml">1</mn></msub><mo id="S3.1.p1.10.m10.3.3.2.3" xref="S3.1.p1.10.m10.3.3.3.cmml">,</mo><mi id="S3.1.p1.10.m10.1.1" mathvariant="normal" xref="S3.1.p1.10.m10.1.1.cmml">…</mi><mo id="S3.1.p1.10.m10.3.3.2.4" xref="S3.1.p1.10.m10.3.3.3.cmml">,</mo><msub id="S3.1.p1.10.m10.3.3.2.2" xref="S3.1.p1.10.m10.3.3.2.2.cmml"><mi id="S3.1.p1.10.m10.3.3.2.2.2" xref="S3.1.p1.10.m10.3.3.2.2.2.cmml">P</mi><mrow id="S3.1.p1.10.m10.3.3.2.2.3" xref="S3.1.p1.10.m10.3.3.2.2.3.cmml"><mi id="S3.1.p1.10.m10.3.3.2.2.3.2" xref="S3.1.p1.10.m10.3.3.2.2.3.2.cmml">i</mi><mo id="S3.1.p1.10.m10.3.3.2.2.3.1" xref="S3.1.p1.10.m10.3.3.2.2.3.1.cmml">−</mo><mn id="S3.1.p1.10.m10.3.3.2.2.3.3" xref="S3.1.p1.10.m10.3.3.2.2.3.3.cmml">1</mn></mrow></msub></mrow><annotation-xml encoding="MathML-Content" id="S3.1.p1.10.m10.3b"><list id="S3.1.p1.10.m10.3.3.3.cmml" xref="S3.1.p1.10.m10.3.3.2"><apply id="S3.1.p1.10.m10.2.2.1.1.cmml" xref="S3.1.p1.10.m10.2.2.1.1"><csymbol cd="ambiguous" id="S3.1.p1.10.m10.2.2.1.1.1.cmml" xref="S3.1.p1.10.m10.2.2.1.1">subscript</csymbol><ci id="S3.1.p1.10.m10.2.2.1.1.2.cmml" xref="S3.1.p1.10.m10.2.2.1.1.2">𝑃</ci><cn id="S3.1.p1.10.m10.2.2.1.1.3.cmml" type="integer" xref="S3.1.p1.10.m10.2.2.1.1.3">1</cn></apply><ci id="S3.1.p1.10.m10.1.1.cmml" xref="S3.1.p1.10.m10.1.1">…</ci><apply id="S3.1.p1.10.m10.3.3.2.2.cmml" xref="S3.1.p1.10.m10.3.3.2.2"><csymbol cd="ambiguous" id="S3.1.p1.10.m10.3.3.2.2.1.cmml" xref="S3.1.p1.10.m10.3.3.2.2">subscript</csymbol><ci id="S3.1.p1.10.m10.3.3.2.2.2.cmml" xref="S3.1.p1.10.m10.3.3.2.2.2">𝑃</ci><apply id="S3.1.p1.10.m10.3.3.2.2.3.cmml" xref="S3.1.p1.10.m10.3.3.2.2.3"><minus id="S3.1.p1.10.m10.3.3.2.2.3.1.cmml" xref="S3.1.p1.10.m10.3.3.2.2.3.1"></minus><ci id="S3.1.p1.10.m10.3.3.2.2.3.2.cmml" xref="S3.1.p1.10.m10.3.3.2.2.3.2">𝑖</ci><cn id="S3.1.p1.10.m10.3.3.2.2.3.3.cmml" type="integer" xref="S3.1.p1.10.m10.3.3.2.2.3.3">1</cn></apply></apply></list></annotation-xml><annotation encoding="application/x-tex" id="S3.1.p1.10.m10.3c">P_{1},\dots,P_{i-1}</annotation><annotation encoding="application/x-llamapun" id="S3.1.p1.10.m10.3d">italic_P start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , … , italic_P start_POSTSUBSCRIPT italic_i - 1 end_POSTSUBSCRIPT</annotation></semantics></math>. To do this, we define the set of vertices <math alttext="F_{i}=\big{(}P_{1}\cup\dots\cup P_{i-1}\big{)}\setminus\{u,v\}" class="ltx_Math" display="inline" id="S3.1.p1.11.m11.3"><semantics id="S3.1.p1.11.m11.3a"><mrow id="S3.1.p1.11.m11.3.3" xref="S3.1.p1.11.m11.3.3.cmml"><msub id="S3.1.p1.11.m11.3.3.3" xref="S3.1.p1.11.m11.3.3.3.cmml"><mi id="S3.1.p1.11.m11.3.3.3.2" xref="S3.1.p1.11.m11.3.3.3.2.cmml">F</mi><mi id="S3.1.p1.11.m11.3.3.3.3" xref="S3.1.p1.11.m11.3.3.3.3.cmml">i</mi></msub><mo id="S3.1.p1.11.m11.3.3.2" xref="S3.1.p1.11.m11.3.3.2.cmml">=</mo><mrow id="S3.1.p1.11.m11.3.3.1" xref="S3.1.p1.11.m11.3.3.1.cmml"><mrow id="S3.1.p1.11.m11.3.3.1.1.1" xref="S3.1.p1.11.m11.3.3.1.1.1.1.cmml"><mo id="S3.1.p1.11.m11.3.3.1.1.1.2" maxsize="120%" minsize="120%" xref="S3.1.p1.11.m11.3.3.1.1.1.1.cmml">(</mo><mrow id="S3.1.p1.11.m11.3.3.1.1.1.1" xref="S3.1.p1.11.m11.3.3.1.1.1.1.cmml"><msub id="S3.1.p1.11.m11.3.3.1.1.1.1.2" xref="S3.1.p1.11.m11.3.3.1.1.1.1.2.cmml"><mi id="S3.1.p1.11.m11.3.3.1.1.1.1.2.2" xref="S3.1.p1.11.m11.3.3.1.1.1.1.2.2.cmml">P</mi><mn id="S3.1.p1.11.m11.3.3.1.1.1.1.2.3" xref="S3.1.p1.11.m11.3.3.1.1.1.1.2.3.cmml">1</mn></msub><mo id="S3.1.p1.11.m11.3.3.1.1.1.1.1" xref="S3.1.p1.11.m11.3.3.1.1.1.1.1.cmml">∪</mo><mi id="S3.1.p1.11.m11.3.3.1.1.1.1.3" mathvariant="normal" xref="S3.1.p1.11.m11.3.3.1.1.1.1.3.cmml">⋯</mi><mo id="S3.1.p1.11.m11.3.3.1.1.1.1.1a" xref="S3.1.p1.11.m11.3.3.1.1.1.1.1.cmml">∪</mo><msub id="S3.1.p1.11.m11.3.3.1.1.1.1.4" xref="S3.1.p1.11.m11.3.3.1.1.1.1.4.cmml"><mi id="S3.1.p1.11.m11.3.3.1.1.1.1.4.2" xref="S3.1.p1.11.m11.3.3.1.1.1.1.4.2.cmml">P</mi><mrow id="S3.1.p1.11.m11.3.3.1.1.1.1.4.3" xref="S3.1.p1.11.m11.3.3.1.1.1.1.4.3.cmml"><mi id="S3.1.p1.11.m11.3.3.1.1.1.1.4.3.2" xref="S3.1.p1.11.m11.3.3.1.1.1.1.4.3.2.cmml">i</mi><mo id="S3.1.p1.11.m11.3.3.1.1.1.1.4.3.1" xref="S3.1.p1.11.m11.3.3.1.1.1.1.4.3.1.cmml">−</mo><mn id="S3.1.p1.11.m11.3.3.1.1.1.1.4.3.3" xref="S3.1.p1.11.m11.3.3.1.1.1.1.4.3.3.cmml">1</mn></mrow></msub></mrow><mo id="S3.1.p1.11.m11.3.3.1.1.1.3" maxsize="120%" minsize="120%" xref="S3.1.p1.11.m11.3.3.1.1.1.1.cmml">)</mo></mrow><mo id="S3.1.p1.11.m11.3.3.1.2" xref="S3.1.p1.11.m11.3.3.1.2.cmml">∖</mo><mrow id="S3.1.p1.11.m11.3.3.1.3.2" xref="S3.1.p1.11.m11.3.3.1.3.1.cmml"><mo id="S3.1.p1.11.m11.3.3.1.3.2.1" stretchy="false" xref="S3.1.p1.11.m11.3.3.1.3.1.cmml">{</mo><mi id="S3.1.p1.11.m11.1.1" xref="S3.1.p1.11.m11.1.1.cmml">u</mi><mo id="S3.1.p1.11.m11.3.3.1.3.2.2" xref="S3.1.p1.11.m11.3.3.1.3.1.cmml">,</mo><mi id="S3.1.p1.11.m11.2.2" xref="S3.1.p1.11.m11.2.2.cmml">v</mi><mo id="S3.1.p1.11.m11.3.3.1.3.2.3" stretchy="false" xref="S3.1.p1.11.m11.3.3.1.3.1.cmml">}</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.1.p1.11.m11.3b"><apply id="S3.1.p1.11.m11.3.3.cmml" xref="S3.1.p1.11.m11.3.3"><eq id="S3.1.p1.11.m11.3.3.2.cmml" xref="S3.1.p1.11.m11.3.3.2"></eq><apply id="S3.1.p1.11.m11.3.3.3.cmml" xref="S3.1.p1.11.m11.3.3.3"><csymbol cd="ambiguous" id="S3.1.p1.11.m11.3.3.3.1.cmml" xref="S3.1.p1.11.m11.3.3.3">subscript</csymbol><ci id="S3.1.p1.11.m11.3.3.3.2.cmml" xref="S3.1.p1.11.m11.3.3.3.2">𝐹</ci><ci id="S3.1.p1.11.m11.3.3.3.3.cmml" xref="S3.1.p1.11.m11.3.3.3.3">𝑖</ci></apply><apply id="S3.1.p1.11.m11.3.3.1.cmml" xref="S3.1.p1.11.m11.3.3.1"><setdiff id="S3.1.p1.11.m11.3.3.1.2.cmml" xref="S3.1.p1.11.m11.3.3.1.2"></setdiff><apply id="S3.1.p1.11.m11.3.3.1.1.1.1.cmml" xref="S3.1.p1.11.m11.3.3.1.1.1"><union id="S3.1.p1.11.m11.3.3.1.1.1.1.1.cmml" xref="S3.1.p1.11.m11.3.3.1.1.1.1.1"></union><apply id="S3.1.p1.11.m11.3.3.1.1.1.1.2.cmml" xref="S3.1.p1.11.m11.3.3.1.1.1.1.2"><csymbol cd="ambiguous" id="S3.1.p1.11.m11.3.3.1.1.1.1.2.1.cmml" xref="S3.1.p1.11.m11.3.3.1.1.1.1.2">subscript</csymbol><ci id="S3.1.p1.11.m11.3.3.1.1.1.1.2.2.cmml" xref="S3.1.p1.11.m11.3.3.1.1.1.1.2.2">𝑃</ci><cn id="S3.1.p1.11.m11.3.3.1.1.1.1.2.3.cmml" type="integer" xref="S3.1.p1.11.m11.3.3.1.1.1.1.2.3">1</cn></apply><ci id="S3.1.p1.11.m11.3.3.1.1.1.1.3.cmml" xref="S3.1.p1.11.m11.3.3.1.1.1.1.3">⋯</ci><apply id="S3.1.p1.11.m11.3.3.1.1.1.1.4.cmml" xref="S3.1.p1.11.m11.3.3.1.1.1.1.4"><csymbol cd="ambiguous" id="S3.1.p1.11.m11.3.3.1.1.1.1.4.1.cmml" xref="S3.1.p1.11.m11.3.3.1.1.1.1.4">subscript</csymbol><ci id="S3.1.p1.11.m11.3.3.1.1.1.1.4.2.cmml" xref="S3.1.p1.11.m11.3.3.1.1.1.1.4.2">𝑃</ci><apply id="S3.1.p1.11.m11.3.3.1.1.1.1.4.3.cmml" xref="S3.1.p1.11.m11.3.3.1.1.1.1.4.3"><minus id="S3.1.p1.11.m11.3.3.1.1.1.1.4.3.1.cmml" xref="S3.1.p1.11.m11.3.3.1.1.1.1.4.3.1"></minus><ci id="S3.1.p1.11.m11.3.3.1.1.1.1.4.3.2.cmml" xref="S3.1.p1.11.m11.3.3.1.1.1.1.4.3.2">𝑖</ci><cn id="S3.1.p1.11.m11.3.3.1.1.1.1.4.3.3.cmml" type="integer" xref="S3.1.p1.11.m11.3.3.1.1.1.1.4.3.3">1</cn></apply></apply></apply><set id="S3.1.p1.11.m11.3.3.1.3.1.cmml" xref="S3.1.p1.11.m11.3.3.1.3.2"><ci id="S3.1.p1.11.m11.1.1.cmml" xref="S3.1.p1.11.m11.1.1">𝑢</ci><ci id="S3.1.p1.11.m11.2.2.cmml" xref="S3.1.p1.11.m11.2.2">𝑣</ci></set></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.1.p1.11.m11.3c">F_{i}=\big{(}P_{1}\cup\dots\cup P_{i-1}\big{)}\setminus\{u,v\}</annotation><annotation encoding="application/x-llamapun" id="S3.1.p1.11.m11.3d">italic_F start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT = ( italic_P start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ∪ ⋯ ∪ italic_P start_POSTSUBSCRIPT italic_i - 1 end_POSTSUBSCRIPT ) ∖ { italic_u , italic_v }</annotation></semantics></math> and find a <math alttext="uv" class="ltx_Math" display="inline" id="S3.1.p1.12.m12.1"><semantics id="S3.1.p1.12.m12.1a"><mrow id="S3.1.p1.12.m12.1.1" xref="S3.1.p1.12.m12.1.1.cmml"><mi id="S3.1.p1.12.m12.1.1.2" xref="S3.1.p1.12.m12.1.1.2.cmml">u</mi><mo id="S3.1.p1.12.m12.1.1.1" xref="S3.1.p1.12.m12.1.1.1.cmml">⁢</mo><mi id="S3.1.p1.12.m12.1.1.3" xref="S3.1.p1.12.m12.1.1.3.cmml">v</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.1.p1.12.m12.1b"><apply id="S3.1.p1.12.m12.1.1.cmml" xref="S3.1.p1.12.m12.1.1"><times id="S3.1.p1.12.m12.1.1.1.cmml" xref="S3.1.p1.12.m12.1.1.1"></times><ci id="S3.1.p1.12.m12.1.1.2.cmml" xref="S3.1.p1.12.m12.1.1.2">𝑢</ci><ci id="S3.1.p1.12.m12.1.1.3.cmml" xref="S3.1.p1.12.m12.1.1.3">𝑣</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.1.p1.12.m12.1c">uv</annotation><annotation encoding="application/x-llamapun" id="S3.1.p1.12.m12.1d">italic_u italic_v</annotation></semantics></math>-path in <math alttext="H_{j}\setminus F_{i}" class="ltx_Math" display="inline" id="S3.1.p1.13.m13.1"><semantics id="S3.1.p1.13.m13.1a"><mrow id="S3.1.p1.13.m13.1.1" xref="S3.1.p1.13.m13.1.1.cmml"><msub id="S3.1.p1.13.m13.1.1.2" xref="S3.1.p1.13.m13.1.1.2.cmml"><mi id="S3.1.p1.13.m13.1.1.2.2" xref="S3.1.p1.13.m13.1.1.2.2.cmml">H</mi><mi id="S3.1.p1.13.m13.1.1.2.3" xref="S3.1.p1.13.m13.1.1.2.3.cmml">j</mi></msub><mo id="S3.1.p1.13.m13.1.1.1" xref="S3.1.p1.13.m13.1.1.1.cmml">∖</mo><msub id="S3.1.p1.13.m13.1.1.3" xref="S3.1.p1.13.m13.1.1.3.cmml"><mi id="S3.1.p1.13.m13.1.1.3.2" xref="S3.1.p1.13.m13.1.1.3.2.cmml">F</mi><mi id="S3.1.p1.13.m13.1.1.3.3" xref="S3.1.p1.13.m13.1.1.3.3.cmml">i</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S3.1.p1.13.m13.1b"><apply id="S3.1.p1.13.m13.1.1.cmml" xref="S3.1.p1.13.m13.1.1"><setdiff id="S3.1.p1.13.m13.1.1.1.cmml" xref="S3.1.p1.13.m13.1.1.1"></setdiff><apply id="S3.1.p1.13.m13.1.1.2.cmml" xref="S3.1.p1.13.m13.1.1.2"><csymbol cd="ambiguous" id="S3.1.p1.13.m13.1.1.2.1.cmml" xref="S3.1.p1.13.m13.1.1.2">subscript</csymbol><ci id="S3.1.p1.13.m13.1.1.2.2.cmml" xref="S3.1.p1.13.m13.1.1.2.2">𝐻</ci><ci id="S3.1.p1.13.m13.1.1.2.3.cmml" xref="S3.1.p1.13.m13.1.1.2.3">𝑗</ci></apply><apply id="S3.1.p1.13.m13.1.1.3.cmml" xref="S3.1.p1.13.m13.1.1.3"><csymbol cd="ambiguous" id="S3.1.p1.13.m13.1.1.3.1.cmml" xref="S3.1.p1.13.m13.1.1.3">subscript</csymbol><ci id="S3.1.p1.13.m13.1.1.3.2.cmml" xref="S3.1.p1.13.m13.1.1.3.2">𝐹</ci><ci id="S3.1.p1.13.m13.1.1.3.3.cmml" xref="S3.1.p1.13.m13.1.1.3.3">𝑖</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.1.p1.13.m13.1c">H_{j}\setminus F_{i}</annotation><annotation encoding="application/x-llamapun" id="S3.1.p1.13.m13.1d">italic_H start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ∖ italic_F start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math>. Initially, set <math alttext="F_{1}=\emptyset" class="ltx_Math" display="inline" id="S3.1.p1.14.m14.1"><semantics id="S3.1.p1.14.m14.1a"><mrow id="S3.1.p1.14.m14.1.1" xref="S3.1.p1.14.m14.1.1.cmml"><msub id="S3.1.p1.14.m14.1.1.2" xref="S3.1.p1.14.m14.1.1.2.cmml"><mi id="S3.1.p1.14.m14.1.1.2.2" xref="S3.1.p1.14.m14.1.1.2.2.cmml">F</mi><mn id="S3.1.p1.14.m14.1.1.2.3" xref="S3.1.p1.14.m14.1.1.2.3.cmml">1</mn></msub><mo id="S3.1.p1.14.m14.1.1.1" xref="S3.1.p1.14.m14.1.1.1.cmml">=</mo><mi id="S3.1.p1.14.m14.1.1.3" mathvariant="normal" xref="S3.1.p1.14.m14.1.1.3.cmml">∅</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.1.p1.14.m14.1b"><apply id="S3.1.p1.14.m14.1.1.cmml" xref="S3.1.p1.14.m14.1.1"><eq id="S3.1.p1.14.m14.1.1.1.cmml" xref="S3.1.p1.14.m14.1.1.1"></eq><apply id="S3.1.p1.14.m14.1.1.2.cmml" xref="S3.1.p1.14.m14.1.1.2"><csymbol cd="ambiguous" id="S3.1.p1.14.m14.1.1.2.1.cmml" xref="S3.1.p1.14.m14.1.1.2">subscript</csymbol><ci id="S3.1.p1.14.m14.1.1.2.2.cmml" xref="S3.1.p1.14.m14.1.1.2.2">𝐹</ci><cn id="S3.1.p1.14.m14.1.1.2.3.cmml" type="integer" xref="S3.1.p1.14.m14.1.1.2.3">1</cn></apply><emptyset id="S3.1.p1.14.m14.1.1.3.cmml" xref="S3.1.p1.14.m14.1.1.3"></emptyset></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.1.p1.14.m14.1c">F_{1}=\emptyset</annotation><annotation encoding="application/x-llamapun" id="S3.1.p1.14.m14.1d">italic_F start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT = ∅</annotation></semantics></math>. Since <math alttext="|F_{i}|\leq(t-1)(i-1)\leq(t-1)(L-1)\leq f" class="ltx_Math" display="inline" id="S3.1.p1.15.m15.5"><semantics id="S3.1.p1.15.m15.5a"><mrow id="S3.1.p1.15.m15.5.5" xref="S3.1.p1.15.m15.5.5.cmml"><mrow id="S3.1.p1.15.m15.1.1.1.1" xref="S3.1.p1.15.m15.1.1.1.2.cmml"><mo id="S3.1.p1.15.m15.1.1.1.1.2" stretchy="false" xref="S3.1.p1.15.m15.1.1.1.2.1.cmml">|</mo><msub id="S3.1.p1.15.m15.1.1.1.1.1" xref="S3.1.p1.15.m15.1.1.1.1.1.cmml"><mi id="S3.1.p1.15.m15.1.1.1.1.1.2" xref="S3.1.p1.15.m15.1.1.1.1.1.2.cmml">F</mi><mi id="S3.1.p1.15.m15.1.1.1.1.1.3" xref="S3.1.p1.15.m15.1.1.1.1.1.3.cmml">i</mi></msub><mo id="S3.1.p1.15.m15.1.1.1.1.3" stretchy="false" xref="S3.1.p1.15.m15.1.1.1.2.1.cmml">|</mo></mrow><mo id="S3.1.p1.15.m15.5.5.7" xref="S3.1.p1.15.m15.5.5.7.cmml">≤</mo><mrow id="S3.1.p1.15.m15.3.3.3" xref="S3.1.p1.15.m15.3.3.3.cmml"><mrow id="S3.1.p1.15.m15.2.2.2.1.1" xref="S3.1.p1.15.m15.2.2.2.1.1.1.cmml"><mo id="S3.1.p1.15.m15.2.2.2.1.1.2" stretchy="false" xref="S3.1.p1.15.m15.2.2.2.1.1.1.cmml">(</mo><mrow id="S3.1.p1.15.m15.2.2.2.1.1.1" xref="S3.1.p1.15.m15.2.2.2.1.1.1.cmml"><mi id="S3.1.p1.15.m15.2.2.2.1.1.1.2" xref="S3.1.p1.15.m15.2.2.2.1.1.1.2.cmml">t</mi><mo id="S3.1.p1.15.m15.2.2.2.1.1.1.1" xref="S3.1.p1.15.m15.2.2.2.1.1.1.1.cmml">−</mo><mn id="S3.1.p1.15.m15.2.2.2.1.1.1.3" xref="S3.1.p1.15.m15.2.2.2.1.1.1.3.cmml">1</mn></mrow><mo id="S3.1.p1.15.m15.2.2.2.1.1.3" stretchy="false" xref="S3.1.p1.15.m15.2.2.2.1.1.1.cmml">)</mo></mrow><mo id="S3.1.p1.15.m15.3.3.3.3" xref="S3.1.p1.15.m15.3.3.3.3.cmml">⁢</mo><mrow id="S3.1.p1.15.m15.3.3.3.2.1" xref="S3.1.p1.15.m15.3.3.3.2.1.1.cmml"><mo id="S3.1.p1.15.m15.3.3.3.2.1.2" stretchy="false" xref="S3.1.p1.15.m15.3.3.3.2.1.1.cmml">(</mo><mrow id="S3.1.p1.15.m15.3.3.3.2.1.1" xref="S3.1.p1.15.m15.3.3.3.2.1.1.cmml"><mi id="S3.1.p1.15.m15.3.3.3.2.1.1.2" xref="S3.1.p1.15.m15.3.3.3.2.1.1.2.cmml">i</mi><mo id="S3.1.p1.15.m15.3.3.3.2.1.1.1" xref="S3.1.p1.15.m15.3.3.3.2.1.1.1.cmml">−</mo><mn id="S3.1.p1.15.m15.3.3.3.2.1.1.3" xref="S3.1.p1.15.m15.3.3.3.2.1.1.3.cmml">1</mn></mrow><mo id="S3.1.p1.15.m15.3.3.3.2.1.3" stretchy="false" xref="S3.1.p1.15.m15.3.3.3.2.1.1.cmml">)</mo></mrow></mrow><mo id="S3.1.p1.15.m15.5.5.8" xref="S3.1.p1.15.m15.5.5.8.cmml">≤</mo><mrow id="S3.1.p1.15.m15.5.5.5" xref="S3.1.p1.15.m15.5.5.5.cmml"><mrow id="S3.1.p1.15.m15.4.4.4.1.1" xref="S3.1.p1.15.m15.4.4.4.1.1.1.cmml"><mo id="S3.1.p1.15.m15.4.4.4.1.1.2" stretchy="false" xref="S3.1.p1.15.m15.4.4.4.1.1.1.cmml">(</mo><mrow id="S3.1.p1.15.m15.4.4.4.1.1.1" xref="S3.1.p1.15.m15.4.4.4.1.1.1.cmml"><mi id="S3.1.p1.15.m15.4.4.4.1.1.1.2" xref="S3.1.p1.15.m15.4.4.4.1.1.1.2.cmml">t</mi><mo id="S3.1.p1.15.m15.4.4.4.1.1.1.1" xref="S3.1.p1.15.m15.4.4.4.1.1.1.1.cmml">−</mo><mn id="S3.1.p1.15.m15.4.4.4.1.1.1.3" xref="S3.1.p1.15.m15.4.4.4.1.1.1.3.cmml">1</mn></mrow><mo id="S3.1.p1.15.m15.4.4.4.1.1.3" stretchy="false" xref="S3.1.p1.15.m15.4.4.4.1.1.1.cmml">)</mo></mrow><mo id="S3.1.p1.15.m15.5.5.5.3" xref="S3.1.p1.15.m15.5.5.5.3.cmml">⁢</mo><mrow id="S3.1.p1.15.m15.5.5.5.2.1" xref="S3.1.p1.15.m15.5.5.5.2.1.1.cmml"><mo id="S3.1.p1.15.m15.5.5.5.2.1.2" stretchy="false" xref="S3.1.p1.15.m15.5.5.5.2.1.1.cmml">(</mo><mrow id="S3.1.p1.15.m15.5.5.5.2.1.1" xref="S3.1.p1.15.m15.5.5.5.2.1.1.cmml"><mi id="S3.1.p1.15.m15.5.5.5.2.1.1.2" xref="S3.1.p1.15.m15.5.5.5.2.1.1.2.cmml">L</mi><mo id="S3.1.p1.15.m15.5.5.5.2.1.1.1" xref="S3.1.p1.15.m15.5.5.5.2.1.1.1.cmml">−</mo><mn id="S3.1.p1.15.m15.5.5.5.2.1.1.3" xref="S3.1.p1.15.m15.5.5.5.2.1.1.3.cmml">1</mn></mrow><mo id="S3.1.p1.15.m15.5.5.5.2.1.3" stretchy="false" xref="S3.1.p1.15.m15.5.5.5.2.1.1.cmml">)</mo></mrow></mrow><mo id="S3.1.p1.15.m15.5.5.9" xref="S3.1.p1.15.m15.5.5.9.cmml">≤</mo><mi id="S3.1.p1.15.m15.5.5.10" xref="S3.1.p1.15.m15.5.5.10.cmml">f</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.1.p1.15.m15.5b"><apply id="S3.1.p1.15.m15.5.5.cmml" xref="S3.1.p1.15.m15.5.5"><and id="S3.1.p1.15.m15.5.5a.cmml" xref="S3.1.p1.15.m15.5.5"></and><apply id="S3.1.p1.15.m15.5.5b.cmml" xref="S3.1.p1.15.m15.5.5"><leq id="S3.1.p1.15.m15.5.5.7.cmml" xref="S3.1.p1.15.m15.5.5.7"></leq><apply id="S3.1.p1.15.m15.1.1.1.2.cmml" xref="S3.1.p1.15.m15.1.1.1.1"><abs id="S3.1.p1.15.m15.1.1.1.2.1.cmml" xref="S3.1.p1.15.m15.1.1.1.1.2"></abs><apply id="S3.1.p1.15.m15.1.1.1.1.1.cmml" xref="S3.1.p1.15.m15.1.1.1.1.1"><csymbol cd="ambiguous" id="S3.1.p1.15.m15.1.1.1.1.1.1.cmml" xref="S3.1.p1.15.m15.1.1.1.1.1">subscript</csymbol><ci id="S3.1.p1.15.m15.1.1.1.1.1.2.cmml" xref="S3.1.p1.15.m15.1.1.1.1.1.2">𝐹</ci><ci id="S3.1.p1.15.m15.1.1.1.1.1.3.cmml" xref="S3.1.p1.15.m15.1.1.1.1.1.3">𝑖</ci></apply></apply><apply id="S3.1.p1.15.m15.3.3.3.cmml" xref="S3.1.p1.15.m15.3.3.3"><times id="S3.1.p1.15.m15.3.3.3.3.cmml" xref="S3.1.p1.15.m15.3.3.3.3"></times><apply id="S3.1.p1.15.m15.2.2.2.1.1.1.cmml" xref="S3.1.p1.15.m15.2.2.2.1.1"><minus id="S3.1.p1.15.m15.2.2.2.1.1.1.1.cmml" xref="S3.1.p1.15.m15.2.2.2.1.1.1.1"></minus><ci id="S3.1.p1.15.m15.2.2.2.1.1.1.2.cmml" xref="S3.1.p1.15.m15.2.2.2.1.1.1.2">𝑡</ci><cn id="S3.1.p1.15.m15.2.2.2.1.1.1.3.cmml" type="integer" xref="S3.1.p1.15.m15.2.2.2.1.1.1.3">1</cn></apply><apply id="S3.1.p1.15.m15.3.3.3.2.1.1.cmml" xref="S3.1.p1.15.m15.3.3.3.2.1"><minus id="S3.1.p1.15.m15.3.3.3.2.1.1.1.cmml" xref="S3.1.p1.15.m15.3.3.3.2.1.1.1"></minus><ci id="S3.1.p1.15.m15.3.3.3.2.1.1.2.cmml" xref="S3.1.p1.15.m15.3.3.3.2.1.1.2">𝑖</ci><cn id="S3.1.p1.15.m15.3.3.3.2.1.1.3.cmml" type="integer" xref="S3.1.p1.15.m15.3.3.3.2.1.1.3">1</cn></apply></apply></apply><apply id="S3.1.p1.15.m15.5.5c.cmml" xref="S3.1.p1.15.m15.5.5"><leq id="S3.1.p1.15.m15.5.5.8.cmml" xref="S3.1.p1.15.m15.5.5.8"></leq><share href="https://arxiv.org/html/2503.00712v1#S3.1.p1.15.m15.3.3.3.cmml" id="S3.1.p1.15.m15.5.5d.cmml" xref="S3.1.p1.15.m15.5.5"></share><apply id="S3.1.p1.15.m15.5.5.5.cmml" xref="S3.1.p1.15.m15.5.5.5"><times id="S3.1.p1.15.m15.5.5.5.3.cmml" xref="S3.1.p1.15.m15.5.5.5.3"></times><apply id="S3.1.p1.15.m15.4.4.4.1.1.1.cmml" xref="S3.1.p1.15.m15.4.4.4.1.1"><minus id="S3.1.p1.15.m15.4.4.4.1.1.1.1.cmml" xref="S3.1.p1.15.m15.4.4.4.1.1.1.1"></minus><ci id="S3.1.p1.15.m15.4.4.4.1.1.1.2.cmml" xref="S3.1.p1.15.m15.4.4.4.1.1.1.2">𝑡</ci><cn id="S3.1.p1.15.m15.4.4.4.1.1.1.3.cmml" type="integer" xref="S3.1.p1.15.m15.4.4.4.1.1.1.3">1</cn></apply><apply id="S3.1.p1.15.m15.5.5.5.2.1.1.cmml" xref="S3.1.p1.15.m15.5.5.5.2.1"><minus id="S3.1.p1.15.m15.5.5.5.2.1.1.1.cmml" xref="S3.1.p1.15.m15.5.5.5.2.1.1.1"></minus><ci id="S3.1.p1.15.m15.5.5.5.2.1.1.2.cmml" xref="S3.1.p1.15.m15.5.5.5.2.1.1.2">𝐿</ci><cn id="S3.1.p1.15.m15.5.5.5.2.1.1.3.cmml" type="integer" xref="S3.1.p1.15.m15.5.5.5.2.1.1.3">1</cn></apply></apply></apply><apply id="S3.1.p1.15.m15.5.5e.cmml" xref="S3.1.p1.15.m15.5.5"><leq id="S3.1.p1.15.m15.5.5.9.cmml" xref="S3.1.p1.15.m15.5.5.9"></leq><share href="https://arxiv.org/html/2503.00712v1#S3.1.p1.15.m15.5.5.5.cmml" id="S3.1.p1.15.m15.5.5f.cmml" xref="S3.1.p1.15.m15.5.5"></share><ci id="S3.1.p1.15.m15.5.5.10.cmml" xref="S3.1.p1.15.m15.5.5.10">𝑓</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.1.p1.15.m15.5c">|F_{i}|\leq(t-1)(i-1)\leq(t-1)(L-1)\leq f</annotation><annotation encoding="application/x-llamapun" id="S3.1.p1.15.m15.5d">| italic_F start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT | ≤ ( italic_t - 1 ) ( italic_i - 1 ) ≤ ( italic_t - 1 ) ( italic_L - 1 ) ≤ italic_f</annotation></semantics></math>, by the properties of the <math alttext="f" class="ltx_Math" display="inline" id="S3.1.p1.16.m16.1"><semantics id="S3.1.p1.16.m16.1a"><mi id="S3.1.p1.16.m16.1.1" xref="S3.1.p1.16.m16.1.1.cmml">f</mi><annotation-xml encoding="MathML-Content" id="S3.1.p1.16.m16.1b"><ci id="S3.1.p1.16.m16.1.1.cmml" xref="S3.1.p1.16.m16.1.1">𝑓</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.1.p1.16.m16.1c">f</annotation><annotation encoding="application/x-llamapun" id="S3.1.p1.16.m16.1d">italic_f</annotation></semantics></math>-VFT <math alttext="t" class="ltx_Math" display="inline" id="S3.1.p1.17.m17.1"><semantics id="S3.1.p1.17.m17.1a"><mi id="S3.1.p1.17.m17.1.1" xref="S3.1.p1.17.m17.1.1.cmml">t</mi><annotation-xml encoding="MathML-Content" id="S3.1.p1.17.m17.1b"><ci id="S3.1.p1.17.m17.1.1.cmml" xref="S3.1.p1.17.m17.1.1">𝑡</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.1.p1.17.m17.1c">t</annotation><annotation encoding="application/x-llamapun" id="S3.1.p1.17.m17.1d">italic_t</annotation></semantics></math>-spanner <math alttext="H_{j}" class="ltx_Math" display="inline" id="S3.1.p1.18.m18.1"><semantics id="S3.1.p1.18.m18.1a"><msub id="S3.1.p1.18.m18.1.1" xref="S3.1.p1.18.m18.1.1.cmml"><mi id="S3.1.p1.18.m18.1.1.2" xref="S3.1.p1.18.m18.1.1.2.cmml">H</mi><mi id="S3.1.p1.18.m18.1.1.3" xref="S3.1.p1.18.m18.1.1.3.cmml">j</mi></msub><annotation-xml encoding="MathML-Content" id="S3.1.p1.18.m18.1b"><apply id="S3.1.p1.18.m18.1.1.cmml" xref="S3.1.p1.18.m18.1.1"><csymbol cd="ambiguous" id="S3.1.p1.18.m18.1.1.1.cmml" xref="S3.1.p1.18.m18.1.1">subscript</csymbol><ci id="S3.1.p1.18.m18.1.1.2.cmml" xref="S3.1.p1.18.m18.1.1.2">𝐻</ci><ci id="S3.1.p1.18.m18.1.1.3.cmml" xref="S3.1.p1.18.m18.1.1.3">𝑗</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.1.p1.18.m18.1c">H_{j}</annotation><annotation encoding="application/x-llamapun" id="S3.1.p1.18.m18.1d">italic_H start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT</annotation></semantics></math>, there exists a <math alttext="uv" class="ltx_Math" display="inline" id="S3.1.p1.19.m19.1"><semantics id="S3.1.p1.19.m19.1a"><mrow id="S3.1.p1.19.m19.1.1" xref="S3.1.p1.19.m19.1.1.cmml"><mi id="S3.1.p1.19.m19.1.1.2" xref="S3.1.p1.19.m19.1.1.2.cmml">u</mi><mo id="S3.1.p1.19.m19.1.1.1" xref="S3.1.p1.19.m19.1.1.1.cmml">⁢</mo><mi id="S3.1.p1.19.m19.1.1.3" xref="S3.1.p1.19.m19.1.1.3.cmml">v</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.1.p1.19.m19.1b"><apply id="S3.1.p1.19.m19.1.1.cmml" xref="S3.1.p1.19.m19.1.1"><times id="S3.1.p1.19.m19.1.1.1.cmml" xref="S3.1.p1.19.m19.1.1.1"></times><ci id="S3.1.p1.19.m19.1.1.2.cmml" xref="S3.1.p1.19.m19.1.1.2">𝑢</ci><ci id="S3.1.p1.19.m19.1.1.3.cmml" xref="S3.1.p1.19.m19.1.1.3">𝑣</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.1.p1.19.m19.1c">uv</annotation><annotation encoding="application/x-llamapun" id="S3.1.p1.19.m19.1d">italic_u italic_v</annotation></semantics></math>-path <math alttext="P_{i}\subseteq H_{j}\setminus F_{i}" class="ltx_Math" display="inline" id="S3.1.p1.20.m20.1"><semantics id="S3.1.p1.20.m20.1a"><mrow id="S3.1.p1.20.m20.1.1" xref="S3.1.p1.20.m20.1.1.cmml"><msub id="S3.1.p1.20.m20.1.1.2" xref="S3.1.p1.20.m20.1.1.2.cmml"><mi id="S3.1.p1.20.m20.1.1.2.2" xref="S3.1.p1.20.m20.1.1.2.2.cmml">P</mi><mi id="S3.1.p1.20.m20.1.1.2.3" xref="S3.1.p1.20.m20.1.1.2.3.cmml">i</mi></msub><mo id="S3.1.p1.20.m20.1.1.1" xref="S3.1.p1.20.m20.1.1.1.cmml">⊆</mo><mrow id="S3.1.p1.20.m20.1.1.3" xref="S3.1.p1.20.m20.1.1.3.cmml"><msub id="S3.1.p1.20.m20.1.1.3.2" xref="S3.1.p1.20.m20.1.1.3.2.cmml"><mi id="S3.1.p1.20.m20.1.1.3.2.2" xref="S3.1.p1.20.m20.1.1.3.2.2.cmml">H</mi><mi id="S3.1.p1.20.m20.1.1.3.2.3" xref="S3.1.p1.20.m20.1.1.3.2.3.cmml">j</mi></msub><mo id="S3.1.p1.20.m20.1.1.3.1" xref="S3.1.p1.20.m20.1.1.3.1.cmml">∖</mo><msub id="S3.1.p1.20.m20.1.1.3.3" xref="S3.1.p1.20.m20.1.1.3.3.cmml"><mi id="S3.1.p1.20.m20.1.1.3.3.2" xref="S3.1.p1.20.m20.1.1.3.3.2.cmml">F</mi><mi id="S3.1.p1.20.m20.1.1.3.3.3" xref="S3.1.p1.20.m20.1.1.3.3.3.cmml">i</mi></msub></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.1.p1.20.m20.1b"><apply id="S3.1.p1.20.m20.1.1.cmml" xref="S3.1.p1.20.m20.1.1"><subset id="S3.1.p1.20.m20.1.1.1.cmml" xref="S3.1.p1.20.m20.1.1.1"></subset><apply id="S3.1.p1.20.m20.1.1.2.cmml" xref="S3.1.p1.20.m20.1.1.2"><csymbol cd="ambiguous" id="S3.1.p1.20.m20.1.1.2.1.cmml" xref="S3.1.p1.20.m20.1.1.2">subscript</csymbol><ci id="S3.1.p1.20.m20.1.1.2.2.cmml" xref="S3.1.p1.20.m20.1.1.2.2">𝑃</ci><ci id="S3.1.p1.20.m20.1.1.2.3.cmml" xref="S3.1.p1.20.m20.1.1.2.3">𝑖</ci></apply><apply id="S3.1.p1.20.m20.1.1.3.cmml" xref="S3.1.p1.20.m20.1.1.3"><setdiff id="S3.1.p1.20.m20.1.1.3.1.cmml" xref="S3.1.p1.20.m20.1.1.3.1"></setdiff><apply id="S3.1.p1.20.m20.1.1.3.2.cmml" xref="S3.1.p1.20.m20.1.1.3.2"><csymbol cd="ambiguous" id="S3.1.p1.20.m20.1.1.3.2.1.cmml" xref="S3.1.p1.20.m20.1.1.3.2">subscript</csymbol><ci id="S3.1.p1.20.m20.1.1.3.2.2.cmml" xref="S3.1.p1.20.m20.1.1.3.2.2">𝐻</ci><ci id="S3.1.p1.20.m20.1.1.3.2.3.cmml" xref="S3.1.p1.20.m20.1.1.3.2.3">𝑗</ci></apply><apply id="S3.1.p1.20.m20.1.1.3.3.cmml" xref="S3.1.p1.20.m20.1.1.3.3"><csymbol cd="ambiguous" id="S3.1.p1.20.m20.1.1.3.3.1.cmml" xref="S3.1.p1.20.m20.1.1.3.3">subscript</csymbol><ci id="S3.1.p1.20.m20.1.1.3.3.2.cmml" xref="S3.1.p1.20.m20.1.1.3.3.2">𝐹</ci><ci id="S3.1.p1.20.m20.1.1.3.3.3.cmml" xref="S3.1.p1.20.m20.1.1.3.3.3">𝑖</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.1.p1.20.m20.1c">P_{i}\subseteq H_{j}\setminus F_{i}</annotation><annotation encoding="application/x-llamapun" id="S3.1.p1.20.m20.1d">italic_P start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ⊆ italic_H start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ∖ italic_F start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math>, which is a path containing at most <math alttext="t" class="ltx_Math" display="inline" id="S3.1.p1.21.m21.1"><semantics id="S3.1.p1.21.m21.1a"><mi id="S3.1.p1.21.m21.1.1" xref="S3.1.p1.21.m21.1.1.cmml">t</mi><annotation-xml encoding="MathML-Content" id="S3.1.p1.21.m21.1b"><ci id="S3.1.p1.21.m21.1.1.cmml" xref="S3.1.p1.21.m21.1.1">𝑡</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.1.p1.21.m21.1c">t</annotation><annotation encoding="application/x-llamapun" id="S3.1.p1.21.m21.1d">italic_t</annotation></semantics></math> edges, each with weight belonging to <math alttext="B_{j}" class="ltx_Math" display="inline" id="S3.1.p1.22.m22.1"><semantics id="S3.1.p1.22.m22.1a"><msub id="S3.1.p1.22.m22.1.1" xref="S3.1.p1.22.m22.1.1.cmml"><mi id="S3.1.p1.22.m22.1.1.2" xref="S3.1.p1.22.m22.1.1.2.cmml">B</mi><mi id="S3.1.p1.22.m22.1.1.3" xref="S3.1.p1.22.m22.1.1.3.cmml">j</mi></msub><annotation-xml encoding="MathML-Content" id="S3.1.p1.22.m22.1b"><apply id="S3.1.p1.22.m22.1.1.cmml" xref="S3.1.p1.22.m22.1.1"><csymbol cd="ambiguous" id="S3.1.p1.22.m22.1.1.1.cmml" xref="S3.1.p1.22.m22.1.1">subscript</csymbol><ci id="S3.1.p1.22.m22.1.1.2.cmml" xref="S3.1.p1.22.m22.1.1.2">𝐵</ci><ci id="S3.1.p1.22.m22.1.1.3.cmml" xref="S3.1.p1.22.m22.1.1.3">𝑗</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.1.p1.22.m22.1c">B_{j}</annotation><annotation encoding="application/x-llamapun" id="S3.1.p1.22.m22.1d">italic_B start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT</annotation></semantics></math>, and is vertex-disjoint from <math alttext="P_{1},\dots,P_{i-1}" class="ltx_Math" display="inline" id="S3.1.p1.23.m23.3"><semantics id="S3.1.p1.23.m23.3a"><mrow id="S3.1.p1.23.m23.3.3.2" xref="S3.1.p1.23.m23.3.3.3.cmml"><msub id="S3.1.p1.23.m23.2.2.1.1" xref="S3.1.p1.23.m23.2.2.1.1.cmml"><mi id="S3.1.p1.23.m23.2.2.1.1.2" xref="S3.1.p1.23.m23.2.2.1.1.2.cmml">P</mi><mn id="S3.1.p1.23.m23.2.2.1.1.3" xref="S3.1.p1.23.m23.2.2.1.1.3.cmml">1</mn></msub><mo id="S3.1.p1.23.m23.3.3.2.3" xref="S3.1.p1.23.m23.3.3.3.cmml">,</mo><mi id="S3.1.p1.23.m23.1.1" mathvariant="normal" xref="S3.1.p1.23.m23.1.1.cmml">…</mi><mo id="S3.1.p1.23.m23.3.3.2.4" xref="S3.1.p1.23.m23.3.3.3.cmml">,</mo><msub id="S3.1.p1.23.m23.3.3.2.2" xref="S3.1.p1.23.m23.3.3.2.2.cmml"><mi id="S3.1.p1.23.m23.3.3.2.2.2" xref="S3.1.p1.23.m23.3.3.2.2.2.cmml">P</mi><mrow id="S3.1.p1.23.m23.3.3.2.2.3" xref="S3.1.p1.23.m23.3.3.2.2.3.cmml"><mi id="S3.1.p1.23.m23.3.3.2.2.3.2" xref="S3.1.p1.23.m23.3.3.2.2.3.2.cmml">i</mi><mo id="S3.1.p1.23.m23.3.3.2.2.3.1" xref="S3.1.p1.23.m23.3.3.2.2.3.1.cmml">−</mo><mn id="S3.1.p1.23.m23.3.3.2.2.3.3" xref="S3.1.p1.23.m23.3.3.2.2.3.3.cmml">1</mn></mrow></msub></mrow><annotation-xml encoding="MathML-Content" id="S3.1.p1.23.m23.3b"><list id="S3.1.p1.23.m23.3.3.3.cmml" xref="S3.1.p1.23.m23.3.3.2"><apply id="S3.1.p1.23.m23.2.2.1.1.cmml" xref="S3.1.p1.23.m23.2.2.1.1"><csymbol cd="ambiguous" id="S3.1.p1.23.m23.2.2.1.1.1.cmml" xref="S3.1.p1.23.m23.2.2.1.1">subscript</csymbol><ci id="S3.1.p1.23.m23.2.2.1.1.2.cmml" xref="S3.1.p1.23.m23.2.2.1.1.2">𝑃</ci><cn id="S3.1.p1.23.m23.2.2.1.1.3.cmml" type="integer" xref="S3.1.p1.23.m23.2.2.1.1.3">1</cn></apply><ci id="S3.1.p1.23.m23.1.1.cmml" xref="S3.1.p1.23.m23.1.1">…</ci><apply id="S3.1.p1.23.m23.3.3.2.2.cmml" xref="S3.1.p1.23.m23.3.3.2.2"><csymbol cd="ambiguous" id="S3.1.p1.23.m23.3.3.2.2.1.cmml" xref="S3.1.p1.23.m23.3.3.2.2">subscript</csymbol><ci id="S3.1.p1.23.m23.3.3.2.2.2.cmml" xref="S3.1.p1.23.m23.3.3.2.2.2">𝑃</ci><apply id="S3.1.p1.23.m23.3.3.2.2.3.cmml" xref="S3.1.p1.23.m23.3.3.2.2.3"><minus id="S3.1.p1.23.m23.3.3.2.2.3.1.cmml" xref="S3.1.p1.23.m23.3.3.2.2.3.1"></minus><ci id="S3.1.p1.23.m23.3.3.2.2.3.2.cmml" xref="S3.1.p1.23.m23.3.3.2.2.3.2">𝑖</ci><cn id="S3.1.p1.23.m23.3.3.2.2.3.3.cmml" type="integer" xref="S3.1.p1.23.m23.3.3.2.2.3.3">1</cn></apply></apply></list></annotation-xml><annotation encoding="application/x-tex" id="S3.1.p1.23.m23.3c">P_{1},\dots,P_{i-1}</annotation><annotation encoding="application/x-llamapun" id="S3.1.p1.23.m23.3d">italic_P start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , … , italic_P start_POSTSUBSCRIPT italic_i - 1 end_POSTSUBSCRIPT</annotation></semantics></math>. ∎</p> </div> </div> <section class="ltx_subsection" id="S3.SS1"> <h3 class="ltx_title ltx_title_subsection"> <span class="ltx_tag ltx_tag_subsection">3.1 </span>Vertex Connectivity Network Design</h3> <div class="ltx_para" id="S3.SS1.p1"> <p class="ltx_p" id="S3.SS1.p1.5">In this section, we consider the VC-SNDP in insertion-only streams. First, in Section <a class="ltx_ref" href="https://arxiv.org/html/2503.00712v1#S3.SS1.SSS1" title="3.1.1 A Simple Analysis Based on Integral Solutions ‣ 3.1 Vertex Connectivity Network Design ‣ 3 Generic Framework for Streaming Algorithms for Network Design ‣ Streaming Algorithms for Network Design"><span class="ltx_text ltx_ref_tag">3.1.1</span></a>, we present a simple analysis of our generic FT spanner based algorithm, Algorithm <a class="ltx_ref" href="https://arxiv.org/html/2503.00712v1#algorithm3" title="In 3 Generic Framework for Streaming Algorithms for Network Design ‣ Streaming Algorithms for Network Design"><span class="ltx_text ltx_ref_tag">3</span></a>, which yields a <math alttext="kt" class="ltx_Math" display="inline" id="S3.SS1.p1.1.m1.1"><semantics id="S3.SS1.p1.1.m1.1a"><mrow id="S3.SS1.p1.1.m1.1.1" xref="S3.SS1.p1.1.m1.1.1.cmml"><mi id="S3.SS1.p1.1.m1.1.1.2" xref="S3.SS1.p1.1.m1.1.1.2.cmml">k</mi><mo id="S3.SS1.p1.1.m1.1.1.1" xref="S3.SS1.p1.1.m1.1.1.1.cmml">⁢</mo><mi id="S3.SS1.p1.1.m1.1.1.3" xref="S3.SS1.p1.1.m1.1.1.3.cmml">t</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.p1.1.m1.1b"><apply id="S3.SS1.p1.1.m1.1.1.cmml" xref="S3.SS1.p1.1.m1.1.1"><times id="S3.SS1.p1.1.m1.1.1.1.cmml" xref="S3.SS1.p1.1.m1.1.1.1"></times><ci id="S3.SS1.p1.1.m1.1.1.2.cmml" xref="S3.SS1.p1.1.m1.1.1.2">𝑘</ci><ci id="S3.SS1.p1.1.m1.1.1.3.cmml" xref="S3.SS1.p1.1.m1.1.1.3">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p1.1.m1.1c">kt</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p1.1.m1.1d">italic_k italic_t</annotation></semantics></math>-approximation, where <math alttext="k" class="ltx_Math" display="inline" id="S3.SS1.p1.2.m2.1"><semantics id="S3.SS1.p1.2.m2.1a"><mi id="S3.SS1.p1.2.m2.1.1" xref="S3.SS1.p1.2.m2.1.1.cmml">k</mi><annotation-xml encoding="MathML-Content" id="S3.SS1.p1.2.m2.1b"><ci id="S3.SS1.p1.2.m2.1.1.cmml" xref="S3.SS1.p1.2.m2.1.1">𝑘</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p1.2.m2.1c">k</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p1.2.m2.1d">italic_k</annotation></semantics></math> is the maximum connectivity requirement in the SNDP instance and <math alttext="t" class="ltx_Math" display="inline" id="S3.SS1.p1.3.m3.1"><semantics id="S3.SS1.p1.3.m3.1a"><mi id="S3.SS1.p1.3.m3.1.1" xref="S3.SS1.p1.3.m3.1.1.cmml">t</mi><annotation-xml encoding="MathML-Content" id="S3.SS1.p1.3.m3.1b"><ci id="S3.SS1.p1.3.m3.1.1.cmml" xref="S3.SS1.p1.3.m3.1.1">𝑡</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p1.3.m3.1c">t</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p1.3.m3.1d">italic_t</annotation></semantics></math> is the stretch parameter in the VFT-spanner. This is a very general result, however, the approximation assumes that one solves the offline problem exactly after the stream ends. For polynomial-time algorithms, the approximation that can be achieved becomes <math alttext="\alpha kt" class="ltx_Math" display="inline" id="S3.SS1.p1.4.m4.1"><semantics id="S3.SS1.p1.4.m4.1a"><mrow id="S3.SS1.p1.4.m4.1.1" xref="S3.SS1.p1.4.m4.1.1.cmml"><mi id="S3.SS1.p1.4.m4.1.1.2" xref="S3.SS1.p1.4.m4.1.1.2.cmml">α</mi><mo id="S3.SS1.p1.4.m4.1.1.1" xref="S3.SS1.p1.4.m4.1.1.1.cmml">⁢</mo><mi id="S3.SS1.p1.4.m4.1.1.3" xref="S3.SS1.p1.4.m4.1.1.3.cmml">k</mi><mo id="S3.SS1.p1.4.m4.1.1.1a" xref="S3.SS1.p1.4.m4.1.1.1.cmml">⁢</mo><mi id="S3.SS1.p1.4.m4.1.1.4" xref="S3.SS1.p1.4.m4.1.1.4.cmml">t</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.p1.4.m4.1b"><apply id="S3.SS1.p1.4.m4.1.1.cmml" xref="S3.SS1.p1.4.m4.1.1"><times id="S3.SS1.p1.4.m4.1.1.1.cmml" xref="S3.SS1.p1.4.m4.1.1.1"></times><ci id="S3.SS1.p1.4.m4.1.1.2.cmml" xref="S3.SS1.p1.4.m4.1.1.2">𝛼</ci><ci id="S3.SS1.p1.4.m4.1.1.3.cmml" xref="S3.SS1.p1.4.m4.1.1.3">𝑘</ci><ci id="S3.SS1.p1.4.m4.1.1.4.cmml" xref="S3.SS1.p1.4.m4.1.1.4">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p1.4.m4.1c">\alpha kt</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p1.4.m4.1d">italic_α italic_k italic_t</annotation></semantics></math> where <math alttext="\alpha" class="ltx_Math" display="inline" id="S3.SS1.p1.5.m5.1"><semantics id="S3.SS1.p1.5.m5.1a"><mi id="S3.SS1.p1.5.m5.1.1" xref="S3.SS1.p1.5.m5.1.1.cmml">α</mi><annotation-xml encoding="MathML-Content" id="S3.SS1.p1.5.m5.1b"><ci id="S3.SS1.p1.5.m5.1.1.cmml" xref="S3.SS1.p1.5.m5.1.1">𝛼</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p1.5.m5.1c">\alpha</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p1.5.m5.1d">italic_α</annotation></semantics></math> is the best known approximation ratio for the offline problem. Then, in Section <a class="ltx_ref" href="https://arxiv.org/html/2503.00712v1#S3.SS1.SSS2" title="3.1.2 An Improved Analysis via Fractional Solutions ‣ 3.1 Vertex Connectivity Network Design ‣ 3 Generic Framework for Streaming Algorithms for Network Design ‣ Streaming Algorithms for Network Design"><span class="ltx_text ltx_ref_tag">3.1.2</span></a>, we provide a more involved analysis that achieves an approximation based on the integrality gap of the well-studied cut-based LP relaxation for VC-SNDP. This yields improved polynomial-time approximation algorithm for VC-SNDP, as well as improved results in several special cases even when compared to what is achievable via an exact algorithm. We also show that this approach yields near-tight bounds for EC-SNDP and ELC-SNDP due to the small integrality gaps these probelms have.</p> </div> <section class="ltx_subsubsection" id="S3.SS1.SSS1"> <h4 class="ltx_title ltx_title_subsubsection"> <span class="ltx_tag ltx_tag_subsubsection">3.1.1 </span>A Simple Analysis Based on Integral Solutions</h4> <div class="ltx_theorem ltx_theorem_theorem" id="S3.Thmtheorem2"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="S3.Thmtheorem2.1.1.1">Theorem 3.2</span></span><span class="ltx_text ltx_font_bold" id="S3.Thmtheorem2.2.2">.</span> </h6> <div class="ltx_para" id="S3.Thmtheorem2.p1"> <p class="ltx_p" id="S3.Thmtheorem2.p1.6">Let <math alttext="H" class="ltx_Math" display="inline" id="S3.Thmtheorem2.p1.1.m1.1"><semantics id="S3.Thmtheorem2.p1.1.m1.1a"><mi id="S3.Thmtheorem2.p1.1.m1.1.1" xref="S3.Thmtheorem2.p1.1.m1.1.1.cmml">H</mi><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem2.p1.1.m1.1b"><ci id="S3.Thmtheorem2.p1.1.m1.1.1.cmml" xref="S3.Thmtheorem2.p1.1.m1.1.1">𝐻</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem2.p1.1.m1.1c">H</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem2.p1.1.m1.1d">italic_H</annotation></semantics></math> be the VFT spanner of a weighted graph <math alttext="G" class="ltx_Math" display="inline" id="S3.Thmtheorem2.p1.2.m2.1"><semantics id="S3.Thmtheorem2.p1.2.m2.1a"><mi id="S3.Thmtheorem2.p1.2.m2.1.1" xref="S3.Thmtheorem2.p1.2.m2.1.1.cmml">G</mi><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem2.p1.2.m2.1b"><ci id="S3.Thmtheorem2.p1.2.m2.1.1.cmml" xref="S3.Thmtheorem2.p1.2.m2.1.1">𝐺</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem2.p1.2.m2.1c">G</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem2.p1.2.m2.1d">italic_G</annotation></semantics></math> as constructed in 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xref="S3.Thmtheorem2.p1.3.m3.3.3.1.1.1.1.1.1.1.1.2"><times id="S3.Thmtheorem2.p1.3.m3.3.3.1.1.1.1.1.1.1.1.2.1.cmml" xref="S3.Thmtheorem2.p1.3.m3.3.3.1.1.1.1.1.1.1.1.2.1"></times><cn id="S3.Thmtheorem2.p1.3.m3.3.3.1.1.1.1.1.1.1.1.2.2.cmml" type="integer" xref="S3.Thmtheorem2.p1.3.m3.3.3.1.1.1.1.1.1.1.1.2.2">2</cn><ci id="S3.Thmtheorem2.p1.3.m3.3.3.1.1.1.1.1.1.1.1.2.3.cmml" xref="S3.Thmtheorem2.p1.3.m3.3.3.1.1.1.1.1.1.1.1.2.3">𝑡</ci></apply><cn id="S3.Thmtheorem2.p1.3.m3.3.3.1.1.1.1.1.1.1.1.3.cmml" type="integer" xref="S3.Thmtheorem2.p1.3.m3.3.3.1.1.1.1.1.1.1.1.3">2</cn></apply><apply id="S3.Thmtheorem2.p1.3.m3.3.3.1.1.1.1.2.2.1.1.cmml" xref="S3.Thmtheorem2.p1.3.m3.3.3.1.1.1.1.2.2.1"><minus id="S3.Thmtheorem2.p1.3.m3.3.3.1.1.1.1.2.2.1.1.1.cmml" xref="S3.Thmtheorem2.p1.3.m3.3.3.1.1.1.1.2.2.1.1.1"></minus><ci id="S3.Thmtheorem2.p1.3.m3.3.3.1.1.1.1.2.2.1.1.2.cmml" xref="S3.Thmtheorem2.p1.3.m3.3.3.1.1.1.1.2.2.1.1.2">𝑘</ci><cn id="S3.Thmtheorem2.p1.3.m3.3.3.1.1.1.1.2.2.1.1.3.cmml" type="integer" xref="S3.Thmtheorem2.p1.3.m3.3.3.1.1.1.1.2.2.1.1.3">1</cn></apply></apply></apply><apply id="S3.Thmtheorem2.p1.3.m3.3.3.1.1.2.2.cmml" xref="S3.Thmtheorem2.p1.3.m3.3.3.1.1.2.2"><eq id="S3.Thmtheorem2.p1.3.m3.3.3.1.1.2.2.2.cmml" xref="S3.Thmtheorem2.p1.3.m3.3.3.1.1.2.2.2"></eq><ci id="S3.Thmtheorem2.p1.3.m3.3.3.1.1.2.2.3.cmml" xref="S3.Thmtheorem2.p1.3.m3.3.3.1.1.2.2.3">italic-ϵ</ci><apply id="S3.Thmtheorem2.p1.3.m3.3.3.1.1.2.2.1.cmml" xref="S3.Thmtheorem2.p1.3.m3.3.3.1.1.2.2.1"><divide id="S3.Thmtheorem2.p1.3.m3.3.3.1.1.2.2.1.2.cmml" xref="S3.Thmtheorem2.p1.3.m3.3.3.1.1.2.2.1.2"></divide><cn id="S3.Thmtheorem2.p1.3.m3.3.3.1.1.2.2.1.3.cmml" type="integer" xref="S3.Thmtheorem2.p1.3.m3.3.3.1.1.2.2.1.3">1</cn><apply id="S3.Thmtheorem2.p1.3.m3.3.3.1.1.2.2.1.1.1.1.cmml" xref="S3.Thmtheorem2.p1.3.m3.3.3.1.1.2.2.1.1.1"><minus id="S3.Thmtheorem2.p1.3.m3.3.3.1.1.2.2.1.1.1.1.1.cmml" xref="S3.Thmtheorem2.p1.3.m3.3.3.1.1.2.2.1.1.1.1.1"></minus><apply id="S3.Thmtheorem2.p1.3.m3.3.3.1.1.2.2.1.1.1.1.2.cmml" xref="S3.Thmtheorem2.p1.3.m3.3.3.1.1.2.2.1.1.1.1.2"><times id="S3.Thmtheorem2.p1.3.m3.3.3.1.1.2.2.1.1.1.1.2.1.cmml" xref="S3.Thmtheorem2.p1.3.m3.3.3.1.1.2.2.1.1.1.1.2.1"></times><cn id="S3.Thmtheorem2.p1.3.m3.3.3.1.1.2.2.1.1.1.1.2.2.cmml" type="integer" xref="S3.Thmtheorem2.p1.3.m3.3.3.1.1.2.2.1.1.1.1.2.2">2</cn><ci id="S3.Thmtheorem2.p1.3.m3.3.3.1.1.2.2.1.1.1.1.2.3.cmml" xref="S3.Thmtheorem2.p1.3.m3.3.3.1.1.2.2.1.1.1.1.2.3">𝑡</ci></apply><cn id="S3.Thmtheorem2.p1.3.m3.3.3.1.1.2.2.1.1.1.1.3.cmml" type="integer" xref="S3.Thmtheorem2.p1.3.m3.3.3.1.1.2.2.1.1.1.1.3">1</cn></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem2.p1.3.m3.3c">(t,f=(2t-2)(k-1),\epsilon=1/(2t-1))</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem2.p1.3.m3.3d">( italic_t , italic_f = ( 2 italic_t - 2 ) ( italic_k - 1 ) , italic_ϵ = 1 / ( 2 italic_t - 1 ) )</annotation></semantics></math>. Then an optimal solution of VC-SNDP on (<math alttext="H,r" class="ltx_Math" display="inline" id="S3.Thmtheorem2.p1.4.m4.2"><semantics id="S3.Thmtheorem2.p1.4.m4.2a"><mrow id="S3.Thmtheorem2.p1.4.m4.2.3.2" xref="S3.Thmtheorem2.p1.4.m4.2.3.1.cmml"><mi id="S3.Thmtheorem2.p1.4.m4.1.1" xref="S3.Thmtheorem2.p1.4.m4.1.1.cmml">H</mi><mo id="S3.Thmtheorem2.p1.4.m4.2.3.2.1" xref="S3.Thmtheorem2.p1.4.m4.2.3.1.cmml">,</mo><mi id="S3.Thmtheorem2.p1.4.m4.2.2" xref="S3.Thmtheorem2.p1.4.m4.2.2.cmml">r</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem2.p1.4.m4.2b"><list id="S3.Thmtheorem2.p1.4.m4.2.3.1.cmml" xref="S3.Thmtheorem2.p1.4.m4.2.3.2"><ci id="S3.Thmtheorem2.p1.4.m4.1.1.cmml" xref="S3.Thmtheorem2.p1.4.m4.1.1">𝐻</ci><ci id="S3.Thmtheorem2.p1.4.m4.2.2.cmml" xref="S3.Thmtheorem2.p1.4.m4.2.2">𝑟</ci></list></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem2.p1.4.m4.2c">H,r</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem2.p1.4.m4.2d">italic_H , italic_r</annotation></semantics></math>) is within a <math alttext="\big{(}2tk\big{)}" class="ltx_Math" display="inline" id="S3.Thmtheorem2.p1.5.m5.1"><semantics id="S3.Thmtheorem2.p1.5.m5.1a"><mrow id="S3.Thmtheorem2.p1.5.m5.1.1.1" xref="S3.Thmtheorem2.p1.5.m5.1.1.1.1.cmml"><mo id="S3.Thmtheorem2.p1.5.m5.1.1.1.2" maxsize="120%" minsize="120%" xref="S3.Thmtheorem2.p1.5.m5.1.1.1.1.cmml">(</mo><mrow id="S3.Thmtheorem2.p1.5.m5.1.1.1.1" xref="S3.Thmtheorem2.p1.5.m5.1.1.1.1.cmml"><mn id="S3.Thmtheorem2.p1.5.m5.1.1.1.1.2" xref="S3.Thmtheorem2.p1.5.m5.1.1.1.1.2.cmml">2</mn><mo id="S3.Thmtheorem2.p1.5.m5.1.1.1.1.1" xref="S3.Thmtheorem2.p1.5.m5.1.1.1.1.1.cmml">⁢</mo><mi id="S3.Thmtheorem2.p1.5.m5.1.1.1.1.3" xref="S3.Thmtheorem2.p1.5.m5.1.1.1.1.3.cmml">t</mi><mo id="S3.Thmtheorem2.p1.5.m5.1.1.1.1.1a" xref="S3.Thmtheorem2.p1.5.m5.1.1.1.1.1.cmml">⁢</mo><mi id="S3.Thmtheorem2.p1.5.m5.1.1.1.1.4" xref="S3.Thmtheorem2.p1.5.m5.1.1.1.1.4.cmml">k</mi></mrow><mo id="S3.Thmtheorem2.p1.5.m5.1.1.1.3" maxsize="120%" minsize="120%" xref="S3.Thmtheorem2.p1.5.m5.1.1.1.1.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem2.p1.5.m5.1b"><apply id="S3.Thmtheorem2.p1.5.m5.1.1.1.1.cmml" xref="S3.Thmtheorem2.p1.5.m5.1.1.1"><times id="S3.Thmtheorem2.p1.5.m5.1.1.1.1.1.cmml" xref="S3.Thmtheorem2.p1.5.m5.1.1.1.1.1"></times><cn id="S3.Thmtheorem2.p1.5.m5.1.1.1.1.2.cmml" type="integer" xref="S3.Thmtheorem2.p1.5.m5.1.1.1.1.2">2</cn><ci id="S3.Thmtheorem2.p1.5.m5.1.1.1.1.3.cmml" xref="S3.Thmtheorem2.p1.5.m5.1.1.1.1.3">𝑡</ci><ci id="S3.Thmtheorem2.p1.5.m5.1.1.1.1.4.cmml" xref="S3.Thmtheorem2.p1.5.m5.1.1.1.1.4">𝑘</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem2.p1.5.m5.1c">\big{(}2tk\big{)}</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem2.p1.5.m5.1d">( 2 italic_t italic_k )</annotation></semantics></math>-factor of an optimal solution of VC-SNDP on (<math alttext="G,r" class="ltx_Math" display="inline" id="S3.Thmtheorem2.p1.6.m6.2"><semantics id="S3.Thmtheorem2.p1.6.m6.2a"><mrow id="S3.Thmtheorem2.p1.6.m6.2.3.2" xref="S3.Thmtheorem2.p1.6.m6.2.3.1.cmml"><mi id="S3.Thmtheorem2.p1.6.m6.1.1" xref="S3.Thmtheorem2.p1.6.m6.1.1.cmml">G</mi><mo id="S3.Thmtheorem2.p1.6.m6.2.3.2.1" xref="S3.Thmtheorem2.p1.6.m6.2.3.1.cmml">,</mo><mi id="S3.Thmtheorem2.p1.6.m6.2.2" xref="S3.Thmtheorem2.p1.6.m6.2.2.cmml">r</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem2.p1.6.m6.2b"><list id="S3.Thmtheorem2.p1.6.m6.2.3.1.cmml" xref="S3.Thmtheorem2.p1.6.m6.2.3.2"><ci id="S3.Thmtheorem2.p1.6.m6.1.1.cmml" xref="S3.Thmtheorem2.p1.6.m6.1.1">𝐺</ci><ci id="S3.Thmtheorem2.p1.6.m6.2.2.cmml" xref="S3.Thmtheorem2.p1.6.m6.2.2">𝑟</ci></list></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem2.p1.6.m6.2c">G,r</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem2.p1.6.m6.2d">italic_G , italic_r</annotation></semantics></math>).</p> </div> </div> <div class="ltx_proof" id="S3.SS1.SSS1.3"> <h6 class="ltx_title ltx_runin ltx_font_italic ltx_title_proof">Proof.</h6> <div class="ltx_para" id="S3.SS1.SSS1.1.p1"> <p class="ltx_p" id="S3.SS1.SSS1.1.p1.20">Let <math alttext="H=H_{1}\uplus\cdots\uplus H_{T}" class="ltx_Math" display="inline" id="S3.SS1.SSS1.1.p1.1.m1.1"><semantics id="S3.SS1.SSS1.1.p1.1.m1.1a"><mrow id="S3.SS1.SSS1.1.p1.1.m1.1.1" xref="S3.SS1.SSS1.1.p1.1.m1.1.1.cmml"><mi id="S3.SS1.SSS1.1.p1.1.m1.1.1.2" xref="S3.SS1.SSS1.1.p1.1.m1.1.1.2.cmml">H</mi><mo id="S3.SS1.SSS1.1.p1.1.m1.1.1.1" xref="S3.SS1.SSS1.1.p1.1.m1.1.1.1.cmml">=</mo><mrow id="S3.SS1.SSS1.1.p1.1.m1.1.1.3" xref="S3.SS1.SSS1.1.p1.1.m1.1.1.3.cmml"><msub id="S3.SS1.SSS1.1.p1.1.m1.1.1.3.2" xref="S3.SS1.SSS1.1.p1.1.m1.1.1.3.2.cmml"><mi id="S3.SS1.SSS1.1.p1.1.m1.1.1.3.2.2" xref="S3.SS1.SSS1.1.p1.1.m1.1.1.3.2.2.cmml">H</mi><mn id="S3.SS1.SSS1.1.p1.1.m1.1.1.3.2.3" xref="S3.SS1.SSS1.1.p1.1.m1.1.1.3.2.3.cmml">1</mn></msub><mo id="S3.SS1.SSS1.1.p1.1.m1.1.1.3.1" xref="S3.SS1.SSS1.1.p1.1.m1.1.1.3.1.cmml">⊎</mo><mi id="S3.SS1.SSS1.1.p1.1.m1.1.1.3.3" mathvariant="normal" xref="S3.SS1.SSS1.1.p1.1.m1.1.1.3.3.cmml">⋯</mi><mo id="S3.SS1.SSS1.1.p1.1.m1.1.1.3.1a" xref="S3.SS1.SSS1.1.p1.1.m1.1.1.3.1.cmml">⊎</mo><msub id="S3.SS1.SSS1.1.p1.1.m1.1.1.3.4" xref="S3.SS1.SSS1.1.p1.1.m1.1.1.3.4.cmml"><mi id="S3.SS1.SSS1.1.p1.1.m1.1.1.3.4.2" xref="S3.SS1.SSS1.1.p1.1.m1.1.1.3.4.2.cmml">H</mi><mi id="S3.SS1.SSS1.1.p1.1.m1.1.1.3.4.3" xref="S3.SS1.SSS1.1.p1.1.m1.1.1.3.4.3.cmml">T</mi></msub></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS1.1.p1.1.m1.1b"><apply id="S3.SS1.SSS1.1.p1.1.m1.1.1.cmml" xref="S3.SS1.SSS1.1.p1.1.m1.1.1"><eq id="S3.SS1.SSS1.1.p1.1.m1.1.1.1.cmml" xref="S3.SS1.SSS1.1.p1.1.m1.1.1.1"></eq><ci id="S3.SS1.SSS1.1.p1.1.m1.1.1.2.cmml" xref="S3.SS1.SSS1.1.p1.1.m1.1.1.2">𝐻</ci><apply id="S3.SS1.SSS1.1.p1.1.m1.1.1.3.cmml" xref="S3.SS1.SSS1.1.p1.1.m1.1.1.3"><ci id="S3.SS1.SSS1.1.p1.1.m1.1.1.3.1.cmml" xref="S3.SS1.SSS1.1.p1.1.m1.1.1.3.1">⊎</ci><apply id="S3.SS1.SSS1.1.p1.1.m1.1.1.3.2.cmml" xref="S3.SS1.SSS1.1.p1.1.m1.1.1.3.2"><csymbol cd="ambiguous" id="S3.SS1.SSS1.1.p1.1.m1.1.1.3.2.1.cmml" xref="S3.SS1.SSS1.1.p1.1.m1.1.1.3.2">subscript</csymbol><ci id="S3.SS1.SSS1.1.p1.1.m1.1.1.3.2.2.cmml" xref="S3.SS1.SSS1.1.p1.1.m1.1.1.3.2.2">𝐻</ci><cn id="S3.SS1.SSS1.1.p1.1.m1.1.1.3.2.3.cmml" type="integer" xref="S3.SS1.SSS1.1.p1.1.m1.1.1.3.2.3">1</cn></apply><ci id="S3.SS1.SSS1.1.p1.1.m1.1.1.3.3.cmml" xref="S3.SS1.SSS1.1.p1.1.m1.1.1.3.3">⋯</ci><apply id="S3.SS1.SSS1.1.p1.1.m1.1.1.3.4.cmml" xref="S3.SS1.SSS1.1.p1.1.m1.1.1.3.4"><csymbol cd="ambiguous" id="S3.SS1.SSS1.1.p1.1.m1.1.1.3.4.1.cmml" xref="S3.SS1.SSS1.1.p1.1.m1.1.1.3.4">subscript</csymbol><ci id="S3.SS1.SSS1.1.p1.1.m1.1.1.3.4.2.cmml" xref="S3.SS1.SSS1.1.p1.1.m1.1.1.3.4.2">𝐻</ci><ci id="S3.SS1.SSS1.1.p1.1.m1.1.1.3.4.3.cmml" xref="S3.SS1.SSS1.1.p1.1.m1.1.1.3.4.3">𝑇</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS1.1.p1.1.m1.1c">H=H_{1}\uplus\cdots\uplus H_{T}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS1.1.p1.1.m1.1d">italic_H = italic_H start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ⊎ ⋯ ⊎ italic_H start_POSTSUBSCRIPT italic_T end_POSTSUBSCRIPT</annotation></semantics></math>, where each <math alttext="H_{j}" class="ltx_Math" display="inline" id="S3.SS1.SSS1.1.p1.2.m2.1"><semantics id="S3.SS1.SSS1.1.p1.2.m2.1a"><msub id="S3.SS1.SSS1.1.p1.2.m2.1.1" xref="S3.SS1.SSS1.1.p1.2.m2.1.1.cmml"><mi id="S3.SS1.SSS1.1.p1.2.m2.1.1.2" xref="S3.SS1.SSS1.1.p1.2.m2.1.1.2.cmml">H</mi><mi id="S3.SS1.SSS1.1.p1.2.m2.1.1.3" xref="S3.SS1.SSS1.1.p1.2.m2.1.1.3.cmml">j</mi></msub><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS1.1.p1.2.m2.1b"><apply id="S3.SS1.SSS1.1.p1.2.m2.1.1.cmml" xref="S3.SS1.SSS1.1.p1.2.m2.1.1"><csymbol cd="ambiguous" id="S3.SS1.SSS1.1.p1.2.m2.1.1.1.cmml" xref="S3.SS1.SSS1.1.p1.2.m2.1.1">subscript</csymbol><ci id="S3.SS1.SSS1.1.p1.2.m2.1.1.2.cmml" xref="S3.SS1.SSS1.1.p1.2.m2.1.1.2">𝐻</ci><ci id="S3.SS1.SSS1.1.p1.2.m2.1.1.3.cmml" xref="S3.SS1.SSS1.1.p1.2.m2.1.1.3">𝑗</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS1.1.p1.2.m2.1c">H_{j}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS1.1.p1.2.m2.1d">italic_H start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT</annotation></semantics></math> is a <math alttext="\big{(}(2t-2)(k-1)\big{)}" class="ltx_Math" display="inline" id="S3.SS1.SSS1.1.p1.3.m3.1"><semantics id="S3.SS1.SSS1.1.p1.3.m3.1a"><mrow id="S3.SS1.SSS1.1.p1.3.m3.1.1.1" xref="S3.SS1.SSS1.1.p1.3.m3.1.1.1.1.cmml"><mo id="S3.SS1.SSS1.1.p1.3.m3.1.1.1.2" maxsize="120%" minsize="120%" xref="S3.SS1.SSS1.1.p1.3.m3.1.1.1.1.cmml">(</mo><mrow id="S3.SS1.SSS1.1.p1.3.m3.1.1.1.1" xref="S3.SS1.SSS1.1.p1.3.m3.1.1.1.1.cmml"><mrow id="S3.SS1.SSS1.1.p1.3.m3.1.1.1.1.1.1" xref="S3.SS1.SSS1.1.p1.3.m3.1.1.1.1.1.1.1.cmml"><mo id="S3.SS1.SSS1.1.p1.3.m3.1.1.1.1.1.1.2" stretchy="false" xref="S3.SS1.SSS1.1.p1.3.m3.1.1.1.1.1.1.1.cmml">(</mo><mrow id="S3.SS1.SSS1.1.p1.3.m3.1.1.1.1.1.1.1" xref="S3.SS1.SSS1.1.p1.3.m3.1.1.1.1.1.1.1.cmml"><mrow id="S3.SS1.SSS1.1.p1.3.m3.1.1.1.1.1.1.1.2" xref="S3.SS1.SSS1.1.p1.3.m3.1.1.1.1.1.1.1.2.cmml"><mn id="S3.SS1.SSS1.1.p1.3.m3.1.1.1.1.1.1.1.2.2" xref="S3.SS1.SSS1.1.p1.3.m3.1.1.1.1.1.1.1.2.2.cmml">2</mn><mo id="S3.SS1.SSS1.1.p1.3.m3.1.1.1.1.1.1.1.2.1" xref="S3.SS1.SSS1.1.p1.3.m3.1.1.1.1.1.1.1.2.1.cmml">⁢</mo><mi id="S3.SS1.SSS1.1.p1.3.m3.1.1.1.1.1.1.1.2.3" xref="S3.SS1.SSS1.1.p1.3.m3.1.1.1.1.1.1.1.2.3.cmml">t</mi></mrow><mo id="S3.SS1.SSS1.1.p1.3.m3.1.1.1.1.1.1.1.1" xref="S3.SS1.SSS1.1.p1.3.m3.1.1.1.1.1.1.1.1.cmml">−</mo><mn id="S3.SS1.SSS1.1.p1.3.m3.1.1.1.1.1.1.1.3" xref="S3.SS1.SSS1.1.p1.3.m3.1.1.1.1.1.1.1.3.cmml">2</mn></mrow><mo id="S3.SS1.SSS1.1.p1.3.m3.1.1.1.1.1.1.3" stretchy="false" xref="S3.SS1.SSS1.1.p1.3.m3.1.1.1.1.1.1.1.cmml">)</mo></mrow><mo id="S3.SS1.SSS1.1.p1.3.m3.1.1.1.1.3" xref="S3.SS1.SSS1.1.p1.3.m3.1.1.1.1.3.cmml">⁢</mo><mrow id="S3.SS1.SSS1.1.p1.3.m3.1.1.1.1.2.1" xref="S3.SS1.SSS1.1.p1.3.m3.1.1.1.1.2.1.1.cmml"><mo id="S3.SS1.SSS1.1.p1.3.m3.1.1.1.1.2.1.2" stretchy="false" xref="S3.SS1.SSS1.1.p1.3.m3.1.1.1.1.2.1.1.cmml">(</mo><mrow id="S3.SS1.SSS1.1.p1.3.m3.1.1.1.1.2.1.1" xref="S3.SS1.SSS1.1.p1.3.m3.1.1.1.1.2.1.1.cmml"><mi id="S3.SS1.SSS1.1.p1.3.m3.1.1.1.1.2.1.1.2" xref="S3.SS1.SSS1.1.p1.3.m3.1.1.1.1.2.1.1.2.cmml">k</mi><mo id="S3.SS1.SSS1.1.p1.3.m3.1.1.1.1.2.1.1.1" xref="S3.SS1.SSS1.1.p1.3.m3.1.1.1.1.2.1.1.1.cmml">−</mo><mn id="S3.SS1.SSS1.1.p1.3.m3.1.1.1.1.2.1.1.3" xref="S3.SS1.SSS1.1.p1.3.m3.1.1.1.1.2.1.1.3.cmml">1</mn></mrow><mo id="S3.SS1.SSS1.1.p1.3.m3.1.1.1.1.2.1.3" stretchy="false" xref="S3.SS1.SSS1.1.p1.3.m3.1.1.1.1.2.1.1.cmml">)</mo></mrow></mrow><mo id="S3.SS1.SSS1.1.p1.3.m3.1.1.1.3" maxsize="120%" minsize="120%" xref="S3.SS1.SSS1.1.p1.3.m3.1.1.1.1.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS1.1.p1.3.m3.1b"><apply id="S3.SS1.SSS1.1.p1.3.m3.1.1.1.1.cmml" xref="S3.SS1.SSS1.1.p1.3.m3.1.1.1"><times id="S3.SS1.SSS1.1.p1.3.m3.1.1.1.1.3.cmml" xref="S3.SS1.SSS1.1.p1.3.m3.1.1.1.1.3"></times><apply id="S3.SS1.SSS1.1.p1.3.m3.1.1.1.1.1.1.1.cmml" xref="S3.SS1.SSS1.1.p1.3.m3.1.1.1.1.1.1"><minus id="S3.SS1.SSS1.1.p1.3.m3.1.1.1.1.1.1.1.1.cmml" xref="S3.SS1.SSS1.1.p1.3.m3.1.1.1.1.1.1.1.1"></minus><apply id="S3.SS1.SSS1.1.p1.3.m3.1.1.1.1.1.1.1.2.cmml" xref="S3.SS1.SSS1.1.p1.3.m3.1.1.1.1.1.1.1.2"><times id="S3.SS1.SSS1.1.p1.3.m3.1.1.1.1.1.1.1.2.1.cmml" xref="S3.SS1.SSS1.1.p1.3.m3.1.1.1.1.1.1.1.2.1"></times><cn id="S3.SS1.SSS1.1.p1.3.m3.1.1.1.1.1.1.1.2.2.cmml" type="integer" xref="S3.SS1.SSS1.1.p1.3.m3.1.1.1.1.1.1.1.2.2">2</cn><ci id="S3.SS1.SSS1.1.p1.3.m3.1.1.1.1.1.1.1.2.3.cmml" xref="S3.SS1.SSS1.1.p1.3.m3.1.1.1.1.1.1.1.2.3">𝑡</ci></apply><cn id="S3.SS1.SSS1.1.p1.3.m3.1.1.1.1.1.1.1.3.cmml" type="integer" xref="S3.SS1.SSS1.1.p1.3.m3.1.1.1.1.1.1.1.3">2</cn></apply><apply id="S3.SS1.SSS1.1.p1.3.m3.1.1.1.1.2.1.1.cmml" xref="S3.SS1.SSS1.1.p1.3.m3.1.1.1.1.2.1"><minus id="S3.SS1.SSS1.1.p1.3.m3.1.1.1.1.2.1.1.1.cmml" xref="S3.SS1.SSS1.1.p1.3.m3.1.1.1.1.2.1.1.1"></minus><ci id="S3.SS1.SSS1.1.p1.3.m3.1.1.1.1.2.1.1.2.cmml" xref="S3.SS1.SSS1.1.p1.3.m3.1.1.1.1.2.1.1.2">𝑘</ci><cn id="S3.SS1.SSS1.1.p1.3.m3.1.1.1.1.2.1.1.3.cmml" type="integer" xref="S3.SS1.SSS1.1.p1.3.m3.1.1.1.1.2.1.1.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS1.1.p1.3.m3.1c">\big{(}(2t-2)(k-1)\big{)}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS1.1.p1.3.m3.1d">( ( 2 italic_t - 2 ) ( italic_k - 1 ) )</annotation></semantics></math>-VFT <math alttext="(2t-1)" class="ltx_Math" display="inline" id="S3.SS1.SSS1.1.p1.4.m4.1"><semantics id="S3.SS1.SSS1.1.p1.4.m4.1a"><mrow id="S3.SS1.SSS1.1.p1.4.m4.1.1.1" xref="S3.SS1.SSS1.1.p1.4.m4.1.1.1.1.cmml"><mo id="S3.SS1.SSS1.1.p1.4.m4.1.1.1.2" stretchy="false" xref="S3.SS1.SSS1.1.p1.4.m4.1.1.1.1.cmml">(</mo><mrow id="S3.SS1.SSS1.1.p1.4.m4.1.1.1.1" xref="S3.SS1.SSS1.1.p1.4.m4.1.1.1.1.cmml"><mrow id="S3.SS1.SSS1.1.p1.4.m4.1.1.1.1.2" xref="S3.SS1.SSS1.1.p1.4.m4.1.1.1.1.2.cmml"><mn id="S3.SS1.SSS1.1.p1.4.m4.1.1.1.1.2.2" xref="S3.SS1.SSS1.1.p1.4.m4.1.1.1.1.2.2.cmml">2</mn><mo id="S3.SS1.SSS1.1.p1.4.m4.1.1.1.1.2.1" xref="S3.SS1.SSS1.1.p1.4.m4.1.1.1.1.2.1.cmml">⁢</mo><mi id="S3.SS1.SSS1.1.p1.4.m4.1.1.1.1.2.3" xref="S3.SS1.SSS1.1.p1.4.m4.1.1.1.1.2.3.cmml">t</mi></mrow><mo id="S3.SS1.SSS1.1.p1.4.m4.1.1.1.1.1" xref="S3.SS1.SSS1.1.p1.4.m4.1.1.1.1.1.cmml">−</mo><mn id="S3.SS1.SSS1.1.p1.4.m4.1.1.1.1.3" xref="S3.SS1.SSS1.1.p1.4.m4.1.1.1.1.3.cmml">1</mn></mrow><mo id="S3.SS1.SSS1.1.p1.4.m4.1.1.1.3" stretchy="false" xref="S3.SS1.SSS1.1.p1.4.m4.1.1.1.1.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS1.1.p1.4.m4.1b"><apply id="S3.SS1.SSS1.1.p1.4.m4.1.1.1.1.cmml" xref="S3.SS1.SSS1.1.p1.4.m4.1.1.1"><minus id="S3.SS1.SSS1.1.p1.4.m4.1.1.1.1.1.cmml" xref="S3.SS1.SSS1.1.p1.4.m4.1.1.1.1.1"></minus><apply id="S3.SS1.SSS1.1.p1.4.m4.1.1.1.1.2.cmml" xref="S3.SS1.SSS1.1.p1.4.m4.1.1.1.1.2"><times id="S3.SS1.SSS1.1.p1.4.m4.1.1.1.1.2.1.cmml" xref="S3.SS1.SSS1.1.p1.4.m4.1.1.1.1.2.1"></times><cn id="S3.SS1.SSS1.1.p1.4.m4.1.1.1.1.2.2.cmml" type="integer" xref="S3.SS1.SSS1.1.p1.4.m4.1.1.1.1.2.2">2</cn><ci id="S3.SS1.SSS1.1.p1.4.m4.1.1.1.1.2.3.cmml" xref="S3.SS1.SSS1.1.p1.4.m4.1.1.1.1.2.3">𝑡</ci></apply><cn id="S3.SS1.SSS1.1.p1.4.m4.1.1.1.1.3.cmml" type="integer" xref="S3.SS1.SSS1.1.p1.4.m4.1.1.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS1.1.p1.4.m4.1c">(2t-1)</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS1.1.p1.4.m4.1d">( 2 italic_t - 1 )</annotation></semantics></math>-spanner for the edges in <math alttext="G" class="ltx_Math" display="inline" id="S3.SS1.SSS1.1.p1.5.m5.1"><semantics id="S3.SS1.SSS1.1.p1.5.m5.1a"><mi id="S3.SS1.SSS1.1.p1.5.m5.1.1" xref="S3.SS1.SSS1.1.p1.5.m5.1.1.cmml">G</mi><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS1.1.p1.5.m5.1b"><ci id="S3.SS1.SSS1.1.p1.5.m5.1.1.cmml" xref="S3.SS1.SSS1.1.p1.5.m5.1.1">𝐺</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS1.1.p1.5.m5.1c">G</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS1.1.p1.5.m5.1d">italic_G</annotation></semantics></math> with weights in the range <math alttext="B_{j}\coloneqq((1+\epsilon)^{j-1},(1+\epsilon)^{j}]" class="ltx_Math" display="inline" id="S3.SS1.SSS1.1.p1.6.m6.2"><semantics id="S3.SS1.SSS1.1.p1.6.m6.2a"><mrow id="S3.SS1.SSS1.1.p1.6.m6.2.2" xref="S3.SS1.SSS1.1.p1.6.m6.2.2.cmml"><msub id="S3.SS1.SSS1.1.p1.6.m6.2.2.4" xref="S3.SS1.SSS1.1.p1.6.m6.2.2.4.cmml"><mi id="S3.SS1.SSS1.1.p1.6.m6.2.2.4.2" xref="S3.SS1.SSS1.1.p1.6.m6.2.2.4.2.cmml">B</mi><mi id="S3.SS1.SSS1.1.p1.6.m6.2.2.4.3" xref="S3.SS1.SSS1.1.p1.6.m6.2.2.4.3.cmml">j</mi></msub><mo id="S3.SS1.SSS1.1.p1.6.m6.2.2.3" xref="S3.SS1.SSS1.1.p1.6.m6.2.2.3.cmml">≔</mo><mrow id="S3.SS1.SSS1.1.p1.6.m6.2.2.2.2" xref="S3.SS1.SSS1.1.p1.6.m6.2.2.2.3.cmml"><mo id="S3.SS1.SSS1.1.p1.6.m6.2.2.2.2.3" stretchy="false" xref="S3.SS1.SSS1.1.p1.6.m6.2.2.2.3.cmml">(</mo><msup id="S3.SS1.SSS1.1.p1.6.m6.1.1.1.1.1" xref="S3.SS1.SSS1.1.p1.6.m6.1.1.1.1.1.cmml"><mrow id="S3.SS1.SSS1.1.p1.6.m6.1.1.1.1.1.1.1" xref="S3.SS1.SSS1.1.p1.6.m6.1.1.1.1.1.1.1.1.cmml"><mo id="S3.SS1.SSS1.1.p1.6.m6.1.1.1.1.1.1.1.2" stretchy="false" xref="S3.SS1.SSS1.1.p1.6.m6.1.1.1.1.1.1.1.1.cmml">(</mo><mrow id="S3.SS1.SSS1.1.p1.6.m6.1.1.1.1.1.1.1.1" xref="S3.SS1.SSS1.1.p1.6.m6.1.1.1.1.1.1.1.1.cmml"><mn id="S3.SS1.SSS1.1.p1.6.m6.1.1.1.1.1.1.1.1.2" xref="S3.SS1.SSS1.1.p1.6.m6.1.1.1.1.1.1.1.1.2.cmml">1</mn><mo id="S3.SS1.SSS1.1.p1.6.m6.1.1.1.1.1.1.1.1.1" xref="S3.SS1.SSS1.1.p1.6.m6.1.1.1.1.1.1.1.1.1.cmml">+</mo><mi id="S3.SS1.SSS1.1.p1.6.m6.1.1.1.1.1.1.1.1.3" xref="S3.SS1.SSS1.1.p1.6.m6.1.1.1.1.1.1.1.1.3.cmml">ϵ</mi></mrow><mo id="S3.SS1.SSS1.1.p1.6.m6.1.1.1.1.1.1.1.3" stretchy="false" xref="S3.SS1.SSS1.1.p1.6.m6.1.1.1.1.1.1.1.1.cmml">)</mo></mrow><mrow id="S3.SS1.SSS1.1.p1.6.m6.1.1.1.1.1.3" xref="S3.SS1.SSS1.1.p1.6.m6.1.1.1.1.1.3.cmml"><mi id="S3.SS1.SSS1.1.p1.6.m6.1.1.1.1.1.3.2" xref="S3.SS1.SSS1.1.p1.6.m6.1.1.1.1.1.3.2.cmml">j</mi><mo id="S3.SS1.SSS1.1.p1.6.m6.1.1.1.1.1.3.1" xref="S3.SS1.SSS1.1.p1.6.m6.1.1.1.1.1.3.1.cmml">−</mo><mn id="S3.SS1.SSS1.1.p1.6.m6.1.1.1.1.1.3.3" xref="S3.SS1.SSS1.1.p1.6.m6.1.1.1.1.1.3.3.cmml">1</mn></mrow></msup><mo id="S3.SS1.SSS1.1.p1.6.m6.2.2.2.2.4" xref="S3.SS1.SSS1.1.p1.6.m6.2.2.2.3.cmml">,</mo><msup id="S3.SS1.SSS1.1.p1.6.m6.2.2.2.2.2" xref="S3.SS1.SSS1.1.p1.6.m6.2.2.2.2.2.cmml"><mrow id="S3.SS1.SSS1.1.p1.6.m6.2.2.2.2.2.1.1" xref="S3.SS1.SSS1.1.p1.6.m6.2.2.2.2.2.1.1.1.cmml"><mo id="S3.SS1.SSS1.1.p1.6.m6.2.2.2.2.2.1.1.2" stretchy="false" 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id="S3.SS1.SSS1.1.p1.6.m6.2.2.2.2.2.1.1.1.1.cmml" xref="S3.SS1.SSS1.1.p1.6.m6.2.2.2.2.2.1.1.1.1"></plus><cn id="S3.SS1.SSS1.1.p1.6.m6.2.2.2.2.2.1.1.1.2.cmml" type="integer" xref="S3.SS1.SSS1.1.p1.6.m6.2.2.2.2.2.1.1.1.2">1</cn><ci id="S3.SS1.SSS1.1.p1.6.m6.2.2.2.2.2.1.1.1.3.cmml" xref="S3.SS1.SSS1.1.p1.6.m6.2.2.2.2.2.1.1.1.3">italic-ϵ</ci></apply><ci id="S3.SS1.SSS1.1.p1.6.m6.2.2.2.2.2.3.cmml" xref="S3.SS1.SSS1.1.p1.6.m6.2.2.2.2.2.3">𝑗</ci></apply></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS1.1.p1.6.m6.2c">B_{j}\coloneqq((1+\epsilon)^{j-1},(1+\epsilon)^{j}]</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS1.1.p1.6.m6.2d">italic_B start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ≔ ( ( 1 + italic_ϵ ) start_POSTSUPERSCRIPT italic_j - 1 end_POSTSUPERSCRIPT , ( 1 + italic_ϵ ) start_POSTSUPERSCRIPT italic_j end_POSTSUPERSCRIPT ]</annotation></semantics></math>, and <math alttext="T=\epsilon^{-1}\log W=\mathrm{poly}(\log n)" 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id="S3.SS1.SSS1.1.p1.7.m7.1.1.5.cmml" xref="S3.SS1.SSS1.1.p1.7.m7.1.1.5"><times id="S3.SS1.SSS1.1.p1.7.m7.1.1.5.1.cmml" xref="S3.SS1.SSS1.1.p1.7.m7.1.1.5.1"></times><apply id="S3.SS1.SSS1.1.p1.7.m7.1.1.5.2.cmml" xref="S3.SS1.SSS1.1.p1.7.m7.1.1.5.2"><csymbol cd="ambiguous" id="S3.SS1.SSS1.1.p1.7.m7.1.1.5.2.1.cmml" xref="S3.SS1.SSS1.1.p1.7.m7.1.1.5.2">superscript</csymbol><ci id="S3.SS1.SSS1.1.p1.7.m7.1.1.5.2.2.cmml" xref="S3.SS1.SSS1.1.p1.7.m7.1.1.5.2.2">italic-ϵ</ci><apply id="S3.SS1.SSS1.1.p1.7.m7.1.1.5.2.3.cmml" xref="S3.SS1.SSS1.1.p1.7.m7.1.1.5.2.3"><minus id="S3.SS1.SSS1.1.p1.7.m7.1.1.5.2.3.1.cmml" xref="S3.SS1.SSS1.1.p1.7.m7.1.1.5.2.3"></minus><cn id="S3.SS1.SSS1.1.p1.7.m7.1.1.5.2.3.2.cmml" type="integer" xref="S3.SS1.SSS1.1.p1.7.m7.1.1.5.2.3.2">1</cn></apply></apply><apply id="S3.SS1.SSS1.1.p1.7.m7.1.1.5.3.cmml" xref="S3.SS1.SSS1.1.p1.7.m7.1.1.5.3"><log id="S3.SS1.SSS1.1.p1.7.m7.1.1.5.3.1.cmml" xref="S3.SS1.SSS1.1.p1.7.m7.1.1.5.3.1"></log><ci id="S3.SS1.SSS1.1.p1.7.m7.1.1.5.3.2.cmml" xref="S3.SS1.SSS1.1.p1.7.m7.1.1.5.3.2">𝑊</ci></apply></apply></apply><apply id="S3.SS1.SSS1.1.p1.7.m7.1.1c.cmml" xref="S3.SS1.SSS1.1.p1.7.m7.1.1"><eq id="S3.SS1.SSS1.1.p1.7.m7.1.1.6.cmml" xref="S3.SS1.SSS1.1.p1.7.m7.1.1.6"></eq><share href="https://arxiv.org/html/2503.00712v1#S3.SS1.SSS1.1.p1.7.m7.1.1.5.cmml" id="S3.SS1.SSS1.1.p1.7.m7.1.1d.cmml" xref="S3.SS1.SSS1.1.p1.7.m7.1.1"></share><apply id="S3.SS1.SSS1.1.p1.7.m7.1.1.1.cmml" xref="S3.SS1.SSS1.1.p1.7.m7.1.1.1"><times id="S3.SS1.SSS1.1.p1.7.m7.1.1.1.2.cmml" xref="S3.SS1.SSS1.1.p1.7.m7.1.1.1.2"></times><ci id="S3.SS1.SSS1.1.p1.7.m7.1.1.1.3.cmml" xref="S3.SS1.SSS1.1.p1.7.m7.1.1.1.3">poly</ci><apply id="S3.SS1.SSS1.1.p1.7.m7.1.1.1.1.1.1.cmml" xref="S3.SS1.SSS1.1.p1.7.m7.1.1.1.1.1"><log id="S3.SS1.SSS1.1.p1.7.m7.1.1.1.1.1.1.1.cmml" xref="S3.SS1.SSS1.1.p1.7.m7.1.1.1.1.1.1.1"></log><ci id="S3.SS1.SSS1.1.p1.7.m7.1.1.1.1.1.1.2.cmml" xref="S3.SS1.SSS1.1.p1.7.m7.1.1.1.1.1.1.2">𝑛</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS1.1.p1.7.m7.1c">T=\epsilon^{-1}\log W=\mathrm{poly}(\log n)</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS1.1.p1.7.m7.1d">italic_T = italic_ϵ start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT roman_log italic_W = roman_poly ( roman_log italic_n )</annotation></semantics></math>. Let <math alttext="\textnormal{OPT}\subseteq E" class="ltx_Math" display="inline" id="S3.SS1.SSS1.1.p1.8.m8.1"><semantics id="S3.SS1.SSS1.1.p1.8.m8.1a"><mrow id="S3.SS1.SSS1.1.p1.8.m8.1.1" xref="S3.SS1.SSS1.1.p1.8.m8.1.1.cmml"><mtext id="S3.SS1.SSS1.1.p1.8.m8.1.1.2" xref="S3.SS1.SSS1.1.p1.8.m8.1.1.2a.cmml">OPT</mtext><mo id="S3.SS1.SSS1.1.p1.8.m8.1.1.1" xref="S3.SS1.SSS1.1.p1.8.m8.1.1.1.cmml">⊆</mo><mi id="S3.SS1.SSS1.1.p1.8.m8.1.1.3" xref="S3.SS1.SSS1.1.p1.8.m8.1.1.3.cmml">E</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS1.1.p1.8.m8.1b"><apply id="S3.SS1.SSS1.1.p1.8.m8.1.1.cmml" xref="S3.SS1.SSS1.1.p1.8.m8.1.1"><subset id="S3.SS1.SSS1.1.p1.8.m8.1.1.1.cmml" xref="S3.SS1.SSS1.1.p1.8.m8.1.1.1"></subset><ci id="S3.SS1.SSS1.1.p1.8.m8.1.1.2a.cmml" xref="S3.SS1.SSS1.1.p1.8.m8.1.1.2"><mtext id="S3.SS1.SSS1.1.p1.8.m8.1.1.2.cmml" xref="S3.SS1.SSS1.1.p1.8.m8.1.1.2">OPT</mtext></ci><ci id="S3.SS1.SSS1.1.p1.8.m8.1.1.3.cmml" xref="S3.SS1.SSS1.1.p1.8.m8.1.1.3">𝐸</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS1.1.p1.8.m8.1c">\textnormal{OPT}\subseteq E</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS1.1.p1.8.m8.1d">OPT ⊆ italic_E</annotation></semantics></math> be an optimal solution for the VC-SNDP instance on <math alttext="G" class="ltx_Math" display="inline" id="S3.SS1.SSS1.1.p1.9.m9.1"><semantics id="S3.SS1.SSS1.1.p1.9.m9.1a"><mi id="S3.SS1.SSS1.1.p1.9.m9.1.1" xref="S3.SS1.SSS1.1.p1.9.m9.1.1.cmml">G</mi><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS1.1.p1.9.m9.1b"><ci id="S3.SS1.SSS1.1.p1.9.m9.1.1.cmml" xref="S3.SS1.SSS1.1.p1.9.m9.1.1">𝐺</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS1.1.p1.9.m9.1c">G</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS1.1.p1.9.m9.1d">italic_G</annotation></semantics></math>. We construct a feasible solution <math alttext="\textnormal{SOL}\subseteq H" class="ltx_Math" display="inline" id="S3.SS1.SSS1.1.p1.10.m10.1"><semantics id="S3.SS1.SSS1.1.p1.10.m10.1a"><mrow id="S3.SS1.SSS1.1.p1.10.m10.1.1" xref="S3.SS1.SSS1.1.p1.10.m10.1.1.cmml"><mtext id="S3.SS1.SSS1.1.p1.10.m10.1.1.2" xref="S3.SS1.SSS1.1.p1.10.m10.1.1.2a.cmml">SOL</mtext><mo id="S3.SS1.SSS1.1.p1.10.m10.1.1.1" xref="S3.SS1.SSS1.1.p1.10.m10.1.1.1.cmml">⊆</mo><mi id="S3.SS1.SSS1.1.p1.10.m10.1.1.3" xref="S3.SS1.SSS1.1.p1.10.m10.1.1.3.cmml">H</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS1.1.p1.10.m10.1b"><apply id="S3.SS1.SSS1.1.p1.10.m10.1.1.cmml" xref="S3.SS1.SSS1.1.p1.10.m10.1.1"><subset id="S3.SS1.SSS1.1.p1.10.m10.1.1.1.cmml" xref="S3.SS1.SSS1.1.p1.10.m10.1.1.1"></subset><ci id="S3.SS1.SSS1.1.p1.10.m10.1.1.2a.cmml" xref="S3.SS1.SSS1.1.p1.10.m10.1.1.2"><mtext id="S3.SS1.SSS1.1.p1.10.m10.1.1.2.cmml" xref="S3.SS1.SSS1.1.p1.10.m10.1.1.2">SOL</mtext></ci><ci id="S3.SS1.SSS1.1.p1.10.m10.1.1.3.cmml" xref="S3.SS1.SSS1.1.p1.10.m10.1.1.3">𝐻</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS1.1.p1.10.m10.1c">\textnormal{SOL}\subseteq H</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS1.1.p1.10.m10.1d">SOL ⊆ italic_H</annotation></semantics></math> based on <span class="ltx_text ltx_markedasmath" id="S3.SS1.SSS1.1.p1.20.1">OPT</span> as follows. For every <math alttext="e=(u,v)\in\textnormal{OPT}" class="ltx_Math" display="inline" id="S3.SS1.SSS1.1.p1.12.m12.2"><semantics id="S3.SS1.SSS1.1.p1.12.m12.2a"><mrow id="S3.SS1.SSS1.1.p1.12.m12.2.3" xref="S3.SS1.SSS1.1.p1.12.m12.2.3.cmml"><mi id="S3.SS1.SSS1.1.p1.12.m12.2.3.2" xref="S3.SS1.SSS1.1.p1.12.m12.2.3.2.cmml">e</mi><mo id="S3.SS1.SSS1.1.p1.12.m12.2.3.3" xref="S3.SS1.SSS1.1.p1.12.m12.2.3.3.cmml">=</mo><mrow id="S3.SS1.SSS1.1.p1.12.m12.2.3.4.2" xref="S3.SS1.SSS1.1.p1.12.m12.2.3.4.1.cmml"><mo id="S3.SS1.SSS1.1.p1.12.m12.2.3.4.2.1" stretchy="false" xref="S3.SS1.SSS1.1.p1.12.m12.2.3.4.1.cmml">(</mo><mi id="S3.SS1.SSS1.1.p1.12.m12.1.1" xref="S3.SS1.SSS1.1.p1.12.m12.1.1.cmml">u</mi><mo id="S3.SS1.SSS1.1.p1.12.m12.2.3.4.2.2" xref="S3.SS1.SSS1.1.p1.12.m12.2.3.4.1.cmml">,</mo><mi id="S3.SS1.SSS1.1.p1.12.m12.2.2" xref="S3.SS1.SSS1.1.p1.12.m12.2.2.cmml">v</mi><mo id="S3.SS1.SSS1.1.p1.12.m12.2.3.4.2.3" stretchy="false" xref="S3.SS1.SSS1.1.p1.12.m12.2.3.4.1.cmml">)</mo></mrow><mo id="S3.SS1.SSS1.1.p1.12.m12.2.3.5" xref="S3.SS1.SSS1.1.p1.12.m12.2.3.5.cmml">∈</mo><mtext id="S3.SS1.SSS1.1.p1.12.m12.2.3.6" xref="S3.SS1.SSS1.1.p1.12.m12.2.3.6a.cmml">OPT</mtext></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS1.1.p1.12.m12.2b"><apply id="S3.SS1.SSS1.1.p1.12.m12.2.3.cmml" xref="S3.SS1.SSS1.1.p1.12.m12.2.3"><and id="S3.SS1.SSS1.1.p1.12.m12.2.3a.cmml" xref="S3.SS1.SSS1.1.p1.12.m12.2.3"></and><apply id="S3.SS1.SSS1.1.p1.12.m12.2.3b.cmml" xref="S3.SS1.SSS1.1.p1.12.m12.2.3"><eq id="S3.SS1.SSS1.1.p1.12.m12.2.3.3.cmml" xref="S3.SS1.SSS1.1.p1.12.m12.2.3.3"></eq><ci id="S3.SS1.SSS1.1.p1.12.m12.2.3.2.cmml" xref="S3.SS1.SSS1.1.p1.12.m12.2.3.2">𝑒</ci><interval closure="open" id="S3.SS1.SSS1.1.p1.12.m12.2.3.4.1.cmml" xref="S3.SS1.SSS1.1.p1.12.m12.2.3.4.2"><ci id="S3.SS1.SSS1.1.p1.12.m12.1.1.cmml" xref="S3.SS1.SSS1.1.p1.12.m12.1.1">𝑢</ci><ci id="S3.SS1.SSS1.1.p1.12.m12.2.2.cmml" xref="S3.SS1.SSS1.1.p1.12.m12.2.2">𝑣</ci></interval></apply><apply id="S3.SS1.SSS1.1.p1.12.m12.2.3c.cmml" xref="S3.SS1.SSS1.1.p1.12.m12.2.3"><in id="S3.SS1.SSS1.1.p1.12.m12.2.3.5.cmml" xref="S3.SS1.SSS1.1.p1.12.m12.2.3.5"></in><share href="https://arxiv.org/html/2503.00712v1#S3.SS1.SSS1.1.p1.12.m12.2.3.4.cmml" id="S3.SS1.SSS1.1.p1.12.m12.2.3d.cmml" xref="S3.SS1.SSS1.1.p1.12.m12.2.3"></share><ci id="S3.SS1.SSS1.1.p1.12.m12.2.3.6a.cmml" xref="S3.SS1.SSS1.1.p1.12.m12.2.3.6"><mtext id="S3.SS1.SSS1.1.p1.12.m12.2.3.6.cmml" xref="S3.SS1.SSS1.1.p1.12.m12.2.3.6">OPT</mtext></ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS1.1.p1.12.m12.2c">e=(u,v)\in\textnormal{OPT}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS1.1.p1.12.m12.2d">italic_e = ( italic_u , italic_v ) ∈ OPT</annotation></semantics></math>, if <math alttext="e\in H_{j}" class="ltx_Math" display="inline" id="S3.SS1.SSS1.1.p1.13.m13.1"><semantics id="S3.SS1.SSS1.1.p1.13.m13.1a"><mrow id="S3.SS1.SSS1.1.p1.13.m13.1.1" xref="S3.SS1.SSS1.1.p1.13.m13.1.1.cmml"><mi id="S3.SS1.SSS1.1.p1.13.m13.1.1.2" xref="S3.SS1.SSS1.1.p1.13.m13.1.1.2.cmml">e</mi><mo id="S3.SS1.SSS1.1.p1.13.m13.1.1.1" xref="S3.SS1.SSS1.1.p1.13.m13.1.1.1.cmml">∈</mo><msub id="S3.SS1.SSS1.1.p1.13.m13.1.1.3" xref="S3.SS1.SSS1.1.p1.13.m13.1.1.3.cmml"><mi id="S3.SS1.SSS1.1.p1.13.m13.1.1.3.2" xref="S3.SS1.SSS1.1.p1.13.m13.1.1.3.2.cmml">H</mi><mi id="S3.SS1.SSS1.1.p1.13.m13.1.1.3.3" xref="S3.SS1.SSS1.1.p1.13.m13.1.1.3.3.cmml">j</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS1.1.p1.13.m13.1b"><apply id="S3.SS1.SSS1.1.p1.13.m13.1.1.cmml" xref="S3.SS1.SSS1.1.p1.13.m13.1.1"><in id="S3.SS1.SSS1.1.p1.13.m13.1.1.1.cmml" xref="S3.SS1.SSS1.1.p1.13.m13.1.1.1"></in><ci id="S3.SS1.SSS1.1.p1.13.m13.1.1.2.cmml" xref="S3.SS1.SSS1.1.p1.13.m13.1.1.2">𝑒</ci><apply id="S3.SS1.SSS1.1.p1.13.m13.1.1.3.cmml" xref="S3.SS1.SSS1.1.p1.13.m13.1.1.3"><csymbol cd="ambiguous" id="S3.SS1.SSS1.1.p1.13.m13.1.1.3.1.cmml" xref="S3.SS1.SSS1.1.p1.13.m13.1.1.3">subscript</csymbol><ci id="S3.SS1.SSS1.1.p1.13.m13.1.1.3.2.cmml" xref="S3.SS1.SSS1.1.p1.13.m13.1.1.3.2">𝐻</ci><ci id="S3.SS1.SSS1.1.p1.13.m13.1.1.3.3.cmml" xref="S3.SS1.SSS1.1.p1.13.m13.1.1.3.3">𝑗</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS1.1.p1.13.m13.1c">e\in H_{j}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS1.1.p1.13.m13.1d">italic_e ∈ italic_H start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT</annotation></semantics></math>, add <math alttext="e" class="ltx_Math" display="inline" id="S3.SS1.SSS1.1.p1.14.m14.1"><semantics id="S3.SS1.SSS1.1.p1.14.m14.1a"><mi id="S3.SS1.SSS1.1.p1.14.m14.1.1" xref="S3.SS1.SSS1.1.p1.14.m14.1.1.cmml">e</mi><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS1.1.p1.14.m14.1b"><ci id="S3.SS1.SSS1.1.p1.14.m14.1.1.cmml" xref="S3.SS1.SSS1.1.p1.14.m14.1.1">𝑒</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS1.1.p1.14.m14.1c">e</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS1.1.p1.14.m14.1d">italic_e</annotation></semantics></math> to <span class="ltx_text ltx_markedasmath" id="S3.SS1.SSS1.1.p1.20.2">SOL</span>. Otherwise, add <math alttext="k" class="ltx_Math" display="inline" id="S3.SS1.SSS1.1.p1.16.m16.1"><semantics id="S3.SS1.SSS1.1.p1.16.m16.1a"><mi id="S3.SS1.SSS1.1.p1.16.m16.1.1" xref="S3.SS1.SSS1.1.p1.16.m16.1.1.cmml">k</mi><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS1.1.p1.16.m16.1b"><ci id="S3.SS1.SSS1.1.p1.16.m16.1.1.cmml" xref="S3.SS1.SSS1.1.p1.16.m16.1.1">𝑘</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS1.1.p1.16.m16.1c">k</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS1.1.p1.16.m16.1d">italic_k</annotation></semantics></math> vertex-disjoint <math alttext="uv" class="ltx_Math" display="inline" id="S3.SS1.SSS1.1.p1.17.m17.1"><semantics id="S3.SS1.SSS1.1.p1.17.m17.1a"><mrow id="S3.SS1.SSS1.1.p1.17.m17.1.1" xref="S3.SS1.SSS1.1.p1.17.m17.1.1.cmml"><mi id="S3.SS1.SSS1.1.p1.17.m17.1.1.2" xref="S3.SS1.SSS1.1.p1.17.m17.1.1.2.cmml">u</mi><mo id="S3.SS1.SSS1.1.p1.17.m17.1.1.1" xref="S3.SS1.SSS1.1.p1.17.m17.1.1.1.cmml">⁢</mo><mi id="S3.SS1.SSS1.1.p1.17.m17.1.1.3" xref="S3.SS1.SSS1.1.p1.17.m17.1.1.3.cmml">v</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS1.1.p1.17.m17.1b"><apply id="S3.SS1.SSS1.1.p1.17.m17.1.1.cmml" xref="S3.SS1.SSS1.1.p1.17.m17.1.1"><times id="S3.SS1.SSS1.1.p1.17.m17.1.1.1.cmml" xref="S3.SS1.SSS1.1.p1.17.m17.1.1.1"></times><ci id="S3.SS1.SSS1.1.p1.17.m17.1.1.2.cmml" xref="S3.SS1.SSS1.1.p1.17.m17.1.1.2">𝑢</ci><ci id="S3.SS1.SSS1.1.p1.17.m17.1.1.3.cmml" xref="S3.SS1.SSS1.1.p1.17.m17.1.1.3">𝑣</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS1.1.p1.17.m17.1c">uv</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS1.1.p1.17.m17.1d">italic_u italic_v</annotation></semantics></math>-paths in <math alttext="H_{j}" class="ltx_Math" display="inline" id="S3.SS1.SSS1.1.p1.18.m18.1"><semantics id="S3.SS1.SSS1.1.p1.18.m18.1a"><msub id="S3.SS1.SSS1.1.p1.18.m18.1.1" xref="S3.SS1.SSS1.1.p1.18.m18.1.1.cmml"><mi id="S3.SS1.SSS1.1.p1.18.m18.1.1.2" xref="S3.SS1.SSS1.1.p1.18.m18.1.1.2.cmml">H</mi><mi id="S3.SS1.SSS1.1.p1.18.m18.1.1.3" xref="S3.SS1.SSS1.1.p1.18.m18.1.1.3.cmml">j</mi></msub><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS1.1.p1.18.m18.1b"><apply id="S3.SS1.SSS1.1.p1.18.m18.1.1.cmml" xref="S3.SS1.SSS1.1.p1.18.m18.1.1"><csymbol cd="ambiguous" id="S3.SS1.SSS1.1.p1.18.m18.1.1.1.cmml" xref="S3.SS1.SSS1.1.p1.18.m18.1.1">subscript</csymbol><ci id="S3.SS1.SSS1.1.p1.18.m18.1.1.2.cmml" xref="S3.SS1.SSS1.1.p1.18.m18.1.1.2">𝐻</ci><ci id="S3.SS1.SSS1.1.p1.18.m18.1.1.3.cmml" xref="S3.SS1.SSS1.1.p1.18.m18.1.1.3">𝑗</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS1.1.p1.18.m18.1c">H_{j}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS1.1.p1.18.m18.1d">italic_H start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT</annotation></semantics></math> (as shown in Observation <a class="ltx_ref" href="https://arxiv.org/html/2503.00712v1#S3.Thmtheorem1" title="Lemma 3.1. ‣ 3 Generic Framework for Streaming Algorithms for Network Design ‣ Streaming Algorithms for Network Design"><span class="ltx_text ltx_ref_tag">3.1</span></a>) to <span class="ltx_text ltx_markedasmath" id="S3.SS1.SSS1.1.p1.20.3">SOL</span>. Note that our algorithm does not need to explicitly construct <span class="ltx_text ltx_markedasmath" id="S3.SS1.SSS1.1.p1.20.4">SOL</span>; this is only for analysis purposes.</p> </div> <div class="ltx_para" id="S3.SS1.SSS1.2.p2"> <p class="ltx_p" id="S3.SS1.SSS1.2.p2.20">We prove the feasibility of <span class="ltx_text ltx_markedasmath" id="S3.SS1.SSS1.2.p2.20.1">SOL</span> for the VC-SNDP instance via Menger’s theorem (Theorem <a class="ltx_ref" href="https://arxiv.org/html/2503.00712v1#S2.Thmtheorem2" title="Theorem 2.2 (Vertex-connectivity Menger’s theorem). ‣ 2 Preliminaries ‣ Streaming Algorithms for Network Design"><span class="ltx_text ltx_ref_tag">2.2</span></a>). Consider a biset <math alttext="\hat{S}=(S,S^{+})" class="ltx_Math" display="inline" id="S3.SS1.SSS1.2.p2.2.m2.2"><semantics id="S3.SS1.SSS1.2.p2.2.m2.2a"><mrow id="S3.SS1.SSS1.2.p2.2.m2.2.2" xref="S3.SS1.SSS1.2.p2.2.m2.2.2.cmml"><mover accent="true" id="S3.SS1.SSS1.2.p2.2.m2.2.2.3" xref="S3.SS1.SSS1.2.p2.2.m2.2.2.3.cmml"><mi id="S3.SS1.SSS1.2.p2.2.m2.2.2.3.2" xref="S3.SS1.SSS1.2.p2.2.m2.2.2.3.2.cmml">S</mi><mo id="S3.SS1.SSS1.2.p2.2.m2.2.2.3.1" xref="S3.SS1.SSS1.2.p2.2.m2.2.2.3.1.cmml">^</mo></mover><mo id="S3.SS1.SSS1.2.p2.2.m2.2.2.2" xref="S3.SS1.SSS1.2.p2.2.m2.2.2.2.cmml">=</mo><mrow id="S3.SS1.SSS1.2.p2.2.m2.2.2.1.1" xref="S3.SS1.SSS1.2.p2.2.m2.2.2.1.2.cmml"><mo id="S3.SS1.SSS1.2.p2.2.m2.2.2.1.1.2" stretchy="false" xref="S3.SS1.SSS1.2.p2.2.m2.2.2.1.2.cmml">(</mo><mi id="S3.SS1.SSS1.2.p2.2.m2.1.1" xref="S3.SS1.SSS1.2.p2.2.m2.1.1.cmml">S</mi><mo id="S3.SS1.SSS1.2.p2.2.m2.2.2.1.1.3" xref="S3.SS1.SSS1.2.p2.2.m2.2.2.1.2.cmml">,</mo><msup id="S3.SS1.SSS1.2.p2.2.m2.2.2.1.1.1" xref="S3.SS1.SSS1.2.p2.2.m2.2.2.1.1.1.cmml"><mi id="S3.SS1.SSS1.2.p2.2.m2.2.2.1.1.1.2" xref="S3.SS1.SSS1.2.p2.2.m2.2.2.1.1.1.2.cmml">S</mi><mo id="S3.SS1.SSS1.2.p2.2.m2.2.2.1.1.1.3" xref="S3.SS1.SSS1.2.p2.2.m2.2.2.1.1.1.3.cmml">+</mo></msup><mo id="S3.SS1.SSS1.2.p2.2.m2.2.2.1.1.4" stretchy="false" xref="S3.SS1.SSS1.2.p2.2.m2.2.2.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS1.2.p2.2.m2.2b"><apply id="S3.SS1.SSS1.2.p2.2.m2.2.2.cmml" xref="S3.SS1.SSS1.2.p2.2.m2.2.2"><eq id="S3.SS1.SSS1.2.p2.2.m2.2.2.2.cmml" xref="S3.SS1.SSS1.2.p2.2.m2.2.2.2"></eq><apply id="S3.SS1.SSS1.2.p2.2.m2.2.2.3.cmml" xref="S3.SS1.SSS1.2.p2.2.m2.2.2.3"><ci id="S3.SS1.SSS1.2.p2.2.m2.2.2.3.1.cmml" xref="S3.SS1.SSS1.2.p2.2.m2.2.2.3.1">^</ci><ci id="S3.SS1.SSS1.2.p2.2.m2.2.2.3.2.cmml" xref="S3.SS1.SSS1.2.p2.2.m2.2.2.3.2">𝑆</ci></apply><interval closure="open" id="S3.SS1.SSS1.2.p2.2.m2.2.2.1.2.cmml" xref="S3.SS1.SSS1.2.p2.2.m2.2.2.1.1"><ci id="S3.SS1.SSS1.2.p2.2.m2.1.1.cmml" xref="S3.SS1.SSS1.2.p2.2.m2.1.1">𝑆</ci><apply id="S3.SS1.SSS1.2.p2.2.m2.2.2.1.1.1.cmml" xref="S3.SS1.SSS1.2.p2.2.m2.2.2.1.1.1"><csymbol cd="ambiguous" id="S3.SS1.SSS1.2.p2.2.m2.2.2.1.1.1.1.cmml" xref="S3.SS1.SSS1.2.p2.2.m2.2.2.1.1.1">superscript</csymbol><ci id="S3.SS1.SSS1.2.p2.2.m2.2.2.1.1.1.2.cmml" xref="S3.SS1.SSS1.2.p2.2.m2.2.2.1.1.1.2">𝑆</ci><plus id="S3.SS1.SSS1.2.p2.2.m2.2.2.1.1.1.3.cmml" xref="S3.SS1.SSS1.2.p2.2.m2.2.2.1.1.1.3"></plus></apply></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS1.2.p2.2.m2.2c">\hat{S}=(S,S^{+})</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS1.2.p2.2.m2.2d">over^ start_ARG italic_S end_ARG = ( italic_S , italic_S start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT )</annotation></semantics></math> with connectivity requirement <math alttext="k^{\prime}=\max_{v\in S,u\in V\setminus S^{+}}r(u,v)\leq k" class="ltx_Math" display="inline" id="S3.SS1.SSS1.2.p2.3.m3.4"><semantics 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href="https://arxiv.org/html/2503.00712v1#S3.SS1.SSS1.2.p2.3.m3.4.5.4.cmml" id="S3.SS1.SSS1.2.p2.3.m3.4.5d.cmml" xref="S3.SS1.SSS1.2.p2.3.m3.4.5"></share><ci id="S3.SS1.SSS1.2.p2.3.m3.4.5.6.cmml" xref="S3.SS1.SSS1.2.p2.3.m3.4.5.6">𝑘</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS1.2.p2.3.m3.4c">k^{\prime}=\max_{v\in S,u\in V\setminus S^{+}}r(u,v)\leq k</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS1.2.p2.3.m3.4d">italic_k start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT = roman_max start_POSTSUBSCRIPT italic_v ∈ italic_S , italic_u ∈ italic_V ∖ italic_S start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT end_POSTSUBSCRIPT italic_r ( italic_u , italic_v ) ≤ italic_k</annotation></semantics></math>, where <math alttext="|S^{+}\setminus S|=\gamma<k^{\prime}" class="ltx_Math" display="inline" id="S3.SS1.SSS1.2.p2.4.m4.1"><semantics id="S3.SS1.SSS1.2.p2.4.m4.1a"><mrow id="S3.SS1.SSS1.2.p2.4.m4.1.1" xref="S3.SS1.SSS1.2.p2.4.m4.1.1.cmml"><mrow id="S3.SS1.SSS1.2.p2.4.m4.1.1.1.1" xref="S3.SS1.SSS1.2.p2.4.m4.1.1.1.2.cmml"><mo id="S3.SS1.SSS1.2.p2.4.m4.1.1.1.1.2" stretchy="false" xref="S3.SS1.SSS1.2.p2.4.m4.1.1.1.2.1.cmml">|</mo><mrow id="S3.SS1.SSS1.2.p2.4.m4.1.1.1.1.1" xref="S3.SS1.SSS1.2.p2.4.m4.1.1.1.1.1.cmml"><msup id="S3.SS1.SSS1.2.p2.4.m4.1.1.1.1.1.2" xref="S3.SS1.SSS1.2.p2.4.m4.1.1.1.1.1.2.cmml"><mi id="S3.SS1.SSS1.2.p2.4.m4.1.1.1.1.1.2.2" xref="S3.SS1.SSS1.2.p2.4.m4.1.1.1.1.1.2.2.cmml">S</mi><mo id="S3.SS1.SSS1.2.p2.4.m4.1.1.1.1.1.2.3" xref="S3.SS1.SSS1.2.p2.4.m4.1.1.1.1.1.2.3.cmml">+</mo></msup><mo id="S3.SS1.SSS1.2.p2.4.m4.1.1.1.1.1.1" xref="S3.SS1.SSS1.2.p2.4.m4.1.1.1.1.1.1.cmml">∖</mo><mi id="S3.SS1.SSS1.2.p2.4.m4.1.1.1.1.1.3" xref="S3.SS1.SSS1.2.p2.4.m4.1.1.1.1.1.3.cmml">S</mi></mrow><mo id="S3.SS1.SSS1.2.p2.4.m4.1.1.1.1.3" stretchy="false" xref="S3.SS1.SSS1.2.p2.4.m4.1.1.1.2.1.cmml">|</mo></mrow><mo id="S3.SS1.SSS1.2.p2.4.m4.1.1.3" xref="S3.SS1.SSS1.2.p2.4.m4.1.1.3.cmml">=</mo><mi id="S3.SS1.SSS1.2.p2.4.m4.1.1.4" xref="S3.SS1.SSS1.2.p2.4.m4.1.1.4.cmml">γ</mi><mo id="S3.SS1.SSS1.2.p2.4.m4.1.1.5" xref="S3.SS1.SSS1.2.p2.4.m4.1.1.5.cmml"><</mo><msup id="S3.SS1.SSS1.2.p2.4.m4.1.1.6" xref="S3.SS1.SSS1.2.p2.4.m4.1.1.6.cmml"><mi id="S3.SS1.SSS1.2.p2.4.m4.1.1.6.2" xref="S3.SS1.SSS1.2.p2.4.m4.1.1.6.2.cmml">k</mi><mo id="S3.SS1.SSS1.2.p2.4.m4.1.1.6.3" xref="S3.SS1.SSS1.2.p2.4.m4.1.1.6.3.cmml">′</mo></msup></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS1.2.p2.4.m4.1b"><apply id="S3.SS1.SSS1.2.p2.4.m4.1.1.cmml" xref="S3.SS1.SSS1.2.p2.4.m4.1.1"><and id="S3.SS1.SSS1.2.p2.4.m4.1.1a.cmml" xref="S3.SS1.SSS1.2.p2.4.m4.1.1"></and><apply id="S3.SS1.SSS1.2.p2.4.m4.1.1b.cmml" xref="S3.SS1.SSS1.2.p2.4.m4.1.1"><eq id="S3.SS1.SSS1.2.p2.4.m4.1.1.3.cmml" xref="S3.SS1.SSS1.2.p2.4.m4.1.1.3"></eq><apply id="S3.SS1.SSS1.2.p2.4.m4.1.1.1.2.cmml" xref="S3.SS1.SSS1.2.p2.4.m4.1.1.1.1"><abs id="S3.SS1.SSS1.2.p2.4.m4.1.1.1.2.1.cmml" xref="S3.SS1.SSS1.2.p2.4.m4.1.1.1.1.2"></abs><apply id="S3.SS1.SSS1.2.p2.4.m4.1.1.1.1.1.cmml" xref="S3.SS1.SSS1.2.p2.4.m4.1.1.1.1.1"><setdiff id="S3.SS1.SSS1.2.p2.4.m4.1.1.1.1.1.1.cmml" xref="S3.SS1.SSS1.2.p2.4.m4.1.1.1.1.1.1"></setdiff><apply id="S3.SS1.SSS1.2.p2.4.m4.1.1.1.1.1.2.cmml" xref="S3.SS1.SSS1.2.p2.4.m4.1.1.1.1.1.2"><csymbol cd="ambiguous" id="S3.SS1.SSS1.2.p2.4.m4.1.1.1.1.1.2.1.cmml" xref="S3.SS1.SSS1.2.p2.4.m4.1.1.1.1.1.2">superscript</csymbol><ci id="S3.SS1.SSS1.2.p2.4.m4.1.1.1.1.1.2.2.cmml" xref="S3.SS1.SSS1.2.p2.4.m4.1.1.1.1.1.2.2">𝑆</ci><plus id="S3.SS1.SSS1.2.p2.4.m4.1.1.1.1.1.2.3.cmml" xref="S3.SS1.SSS1.2.p2.4.m4.1.1.1.1.1.2.3"></plus></apply><ci id="S3.SS1.SSS1.2.p2.4.m4.1.1.1.1.1.3.cmml" xref="S3.SS1.SSS1.2.p2.4.m4.1.1.1.1.1.3">𝑆</ci></apply></apply><ci id="S3.SS1.SSS1.2.p2.4.m4.1.1.4.cmml" xref="S3.SS1.SSS1.2.p2.4.m4.1.1.4">𝛾</ci></apply><apply id="S3.SS1.SSS1.2.p2.4.m4.1.1c.cmml" xref="S3.SS1.SSS1.2.p2.4.m4.1.1"><lt id="S3.SS1.SSS1.2.p2.4.m4.1.1.5.cmml" xref="S3.SS1.SSS1.2.p2.4.m4.1.1.5"></lt><share href="https://arxiv.org/html/2503.00712v1#S3.SS1.SSS1.2.p2.4.m4.1.1.4.cmml" id="S3.SS1.SSS1.2.p2.4.m4.1.1d.cmml" xref="S3.SS1.SSS1.2.p2.4.m4.1.1"></share><apply id="S3.SS1.SSS1.2.p2.4.m4.1.1.6.cmml" xref="S3.SS1.SSS1.2.p2.4.m4.1.1.6"><csymbol cd="ambiguous" id="S3.SS1.SSS1.2.p2.4.m4.1.1.6.1.cmml" xref="S3.SS1.SSS1.2.p2.4.m4.1.1.6">superscript</csymbol><ci id="S3.SS1.SSS1.2.p2.4.m4.1.1.6.2.cmml" xref="S3.SS1.SSS1.2.p2.4.m4.1.1.6.2">𝑘</ci><ci id="S3.SS1.SSS1.2.p2.4.m4.1.1.6.3.cmml" xref="S3.SS1.SSS1.2.p2.4.m4.1.1.6.3">′</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS1.2.p2.4.m4.1c">|S^{+}\setminus S|=\gamma<k^{\prime}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS1.2.p2.4.m4.1d">| italic_S start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT ∖ italic_S | = italic_γ < italic_k start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math>. The optimal solution, <span class="ltx_text ltx_markedasmath" id="S3.SS1.SSS1.2.p2.20.2">OPT</span>, will have at least <math alttext="k^{\prime}-\gamma" class="ltx_Math" display="inline" id="S3.SS1.SSS1.2.p2.6.m6.1"><semantics id="S3.SS1.SSS1.2.p2.6.m6.1a"><mrow id="S3.SS1.SSS1.2.p2.6.m6.1.1" xref="S3.SS1.SSS1.2.p2.6.m6.1.1.cmml"><msup id="S3.SS1.SSS1.2.p2.6.m6.1.1.2" xref="S3.SS1.SSS1.2.p2.6.m6.1.1.2.cmml"><mi id="S3.SS1.SSS1.2.p2.6.m6.1.1.2.2" xref="S3.SS1.SSS1.2.p2.6.m6.1.1.2.2.cmml">k</mi><mo id="S3.SS1.SSS1.2.p2.6.m6.1.1.2.3" xref="S3.SS1.SSS1.2.p2.6.m6.1.1.2.3.cmml">′</mo></msup><mo id="S3.SS1.SSS1.2.p2.6.m6.1.1.1" xref="S3.SS1.SSS1.2.p2.6.m6.1.1.1.cmml">−</mo><mi id="S3.SS1.SSS1.2.p2.6.m6.1.1.3" xref="S3.SS1.SSS1.2.p2.6.m6.1.1.3.cmml">γ</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS1.2.p2.6.m6.1b"><apply id="S3.SS1.SSS1.2.p2.6.m6.1.1.cmml" xref="S3.SS1.SSS1.2.p2.6.m6.1.1"><minus id="S3.SS1.SSS1.2.p2.6.m6.1.1.1.cmml" xref="S3.SS1.SSS1.2.p2.6.m6.1.1.1"></minus><apply id="S3.SS1.SSS1.2.p2.6.m6.1.1.2.cmml" xref="S3.SS1.SSS1.2.p2.6.m6.1.1.2"><csymbol cd="ambiguous" id="S3.SS1.SSS1.2.p2.6.m6.1.1.2.1.cmml" xref="S3.SS1.SSS1.2.p2.6.m6.1.1.2">superscript</csymbol><ci id="S3.SS1.SSS1.2.p2.6.m6.1.1.2.2.cmml" xref="S3.SS1.SSS1.2.p2.6.m6.1.1.2.2">𝑘</ci><ci id="S3.SS1.SSS1.2.p2.6.m6.1.1.2.3.cmml" xref="S3.SS1.SSS1.2.p2.6.m6.1.1.2.3">′</ci></apply><ci id="S3.SS1.SSS1.2.p2.6.m6.1.1.3.cmml" xref="S3.SS1.SSS1.2.p2.6.m6.1.1.3">𝛾</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS1.2.p2.6.m6.1c">k^{\prime}-\gamma</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS1.2.p2.6.m6.1d">italic_k start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT - italic_γ</annotation></semantics></math> edges crossing this biset, i.e., <math alttext="k^{\prime}-\gamma" class="ltx_Math" display="inline" id="S3.SS1.SSS1.2.p2.7.m7.1"><semantics id="S3.SS1.SSS1.2.p2.7.m7.1a"><mrow id="S3.SS1.SSS1.2.p2.7.m7.1.1" xref="S3.SS1.SSS1.2.p2.7.m7.1.1.cmml"><msup id="S3.SS1.SSS1.2.p2.7.m7.1.1.2" xref="S3.SS1.SSS1.2.p2.7.m7.1.1.2.cmml"><mi id="S3.SS1.SSS1.2.p2.7.m7.1.1.2.2" xref="S3.SS1.SSS1.2.p2.7.m7.1.1.2.2.cmml">k</mi><mo id="S3.SS1.SSS1.2.p2.7.m7.1.1.2.3" xref="S3.SS1.SSS1.2.p2.7.m7.1.1.2.3.cmml">′</mo></msup><mo id="S3.SS1.SSS1.2.p2.7.m7.1.1.1" xref="S3.SS1.SSS1.2.p2.7.m7.1.1.1.cmml">−</mo><mi id="S3.SS1.SSS1.2.p2.7.m7.1.1.3" xref="S3.SS1.SSS1.2.p2.7.m7.1.1.3.cmml">γ</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS1.2.p2.7.m7.1b"><apply id="S3.SS1.SSS1.2.p2.7.m7.1.1.cmml" xref="S3.SS1.SSS1.2.p2.7.m7.1.1"><minus id="S3.SS1.SSS1.2.p2.7.m7.1.1.1.cmml" xref="S3.SS1.SSS1.2.p2.7.m7.1.1.1"></minus><apply id="S3.SS1.SSS1.2.p2.7.m7.1.1.2.cmml" xref="S3.SS1.SSS1.2.p2.7.m7.1.1.2"><csymbol cd="ambiguous" id="S3.SS1.SSS1.2.p2.7.m7.1.1.2.1.cmml" xref="S3.SS1.SSS1.2.p2.7.m7.1.1.2">superscript</csymbol><ci id="S3.SS1.SSS1.2.p2.7.m7.1.1.2.2.cmml" xref="S3.SS1.SSS1.2.p2.7.m7.1.1.2.2">𝑘</ci><ci id="S3.SS1.SSS1.2.p2.7.m7.1.1.2.3.cmml" xref="S3.SS1.SSS1.2.p2.7.m7.1.1.2.3">′</ci></apply><ci id="S3.SS1.SSS1.2.p2.7.m7.1.1.3.cmml" xref="S3.SS1.SSS1.2.p2.7.m7.1.1.3">𝛾</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS1.2.p2.7.m7.1c">k^{\prime}-\gamma</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS1.2.p2.7.m7.1d">italic_k start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT - italic_γ</annotation></semantics></math> edges with one endpoint in <math alttext="S" class="ltx_Math" display="inline" id="S3.SS1.SSS1.2.p2.8.m8.1"><semantics id="S3.SS1.SSS1.2.p2.8.m8.1a"><mi id="S3.SS1.SSS1.2.p2.8.m8.1.1" xref="S3.SS1.SSS1.2.p2.8.m8.1.1.cmml">S</mi><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS1.2.p2.8.m8.1b"><ci id="S3.SS1.SSS1.2.p2.8.m8.1.1.cmml" xref="S3.SS1.SSS1.2.p2.8.m8.1.1">𝑆</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS1.2.p2.8.m8.1c">S</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS1.2.p2.8.m8.1d">italic_S</annotation></semantics></math> and another one in <math alttext="V\setminus S^{+}" class="ltx_Math" display="inline" id="S3.SS1.SSS1.2.p2.9.m9.1"><semantics id="S3.SS1.SSS1.2.p2.9.m9.1a"><mrow id="S3.SS1.SSS1.2.p2.9.m9.1.1" xref="S3.SS1.SSS1.2.p2.9.m9.1.1.cmml"><mi id="S3.SS1.SSS1.2.p2.9.m9.1.1.2" xref="S3.SS1.SSS1.2.p2.9.m9.1.1.2.cmml">V</mi><mo id="S3.SS1.SSS1.2.p2.9.m9.1.1.1" xref="S3.SS1.SSS1.2.p2.9.m9.1.1.1.cmml">∖</mo><msup id="S3.SS1.SSS1.2.p2.9.m9.1.1.3" xref="S3.SS1.SSS1.2.p2.9.m9.1.1.3.cmml"><mi id="S3.SS1.SSS1.2.p2.9.m9.1.1.3.2" xref="S3.SS1.SSS1.2.p2.9.m9.1.1.3.2.cmml">S</mi><mo id="S3.SS1.SSS1.2.p2.9.m9.1.1.3.3" xref="S3.SS1.SSS1.2.p2.9.m9.1.1.3.3.cmml">+</mo></msup></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS1.2.p2.9.m9.1b"><apply id="S3.SS1.SSS1.2.p2.9.m9.1.1.cmml" xref="S3.SS1.SSS1.2.p2.9.m9.1.1"><setdiff id="S3.SS1.SSS1.2.p2.9.m9.1.1.1.cmml" xref="S3.SS1.SSS1.2.p2.9.m9.1.1.1"></setdiff><ci id="S3.SS1.SSS1.2.p2.9.m9.1.1.2.cmml" xref="S3.SS1.SSS1.2.p2.9.m9.1.1.2">𝑉</ci><apply id="S3.SS1.SSS1.2.p2.9.m9.1.1.3.cmml" xref="S3.SS1.SSS1.2.p2.9.m9.1.1.3"><csymbol cd="ambiguous" id="S3.SS1.SSS1.2.p2.9.m9.1.1.3.1.cmml" xref="S3.SS1.SSS1.2.p2.9.m9.1.1.3">superscript</csymbol><ci id="S3.SS1.SSS1.2.p2.9.m9.1.1.3.2.cmml" xref="S3.SS1.SSS1.2.p2.9.m9.1.1.3.2">𝑆</ci><plus id="S3.SS1.SSS1.2.p2.9.m9.1.1.3.3.cmml" xref="S3.SS1.SSS1.2.p2.9.m9.1.1.3.3"></plus></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS1.2.p2.9.m9.1c">V\setminus S^{+}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS1.2.p2.9.m9.1d">italic_V ∖ italic_S start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT</annotation></semantics></math>. Let these edges be denoted as <math alttext="e_{1},\dots,e_{k^{\prime}-\gamma}" class="ltx_Math" display="inline" id="S3.SS1.SSS1.2.p2.10.m10.3"><semantics id="S3.SS1.SSS1.2.p2.10.m10.3a"><mrow id="S3.SS1.SSS1.2.p2.10.m10.3.3.2" xref="S3.SS1.SSS1.2.p2.10.m10.3.3.3.cmml"><msub id="S3.SS1.SSS1.2.p2.10.m10.2.2.1.1" xref="S3.SS1.SSS1.2.p2.10.m10.2.2.1.1.cmml"><mi id="S3.SS1.SSS1.2.p2.10.m10.2.2.1.1.2" xref="S3.SS1.SSS1.2.p2.10.m10.2.2.1.1.2.cmml">e</mi><mn id="S3.SS1.SSS1.2.p2.10.m10.2.2.1.1.3" xref="S3.SS1.SSS1.2.p2.10.m10.2.2.1.1.3.cmml">1</mn></msub><mo id="S3.SS1.SSS1.2.p2.10.m10.3.3.2.3" xref="S3.SS1.SSS1.2.p2.10.m10.3.3.3.cmml">,</mo><mi id="S3.SS1.SSS1.2.p2.10.m10.1.1" mathvariant="normal" xref="S3.SS1.SSS1.2.p2.10.m10.1.1.cmml">…</mi><mo id="S3.SS1.SSS1.2.p2.10.m10.3.3.2.4" xref="S3.SS1.SSS1.2.p2.10.m10.3.3.3.cmml">,</mo><msub id="S3.SS1.SSS1.2.p2.10.m10.3.3.2.2" xref="S3.SS1.SSS1.2.p2.10.m10.3.3.2.2.cmml"><mi id="S3.SS1.SSS1.2.p2.10.m10.3.3.2.2.2" xref="S3.SS1.SSS1.2.p2.10.m10.3.3.2.2.2.cmml">e</mi><mrow id="S3.SS1.SSS1.2.p2.10.m10.3.3.2.2.3" xref="S3.SS1.SSS1.2.p2.10.m10.3.3.2.2.3.cmml"><msup id="S3.SS1.SSS1.2.p2.10.m10.3.3.2.2.3.2" xref="S3.SS1.SSS1.2.p2.10.m10.3.3.2.2.3.2.cmml"><mi id="S3.SS1.SSS1.2.p2.10.m10.3.3.2.2.3.2.2" xref="S3.SS1.SSS1.2.p2.10.m10.3.3.2.2.3.2.2.cmml">k</mi><mo id="S3.SS1.SSS1.2.p2.10.m10.3.3.2.2.3.2.3" xref="S3.SS1.SSS1.2.p2.10.m10.3.3.2.2.3.2.3.cmml">′</mo></msup><mo id="S3.SS1.SSS1.2.p2.10.m10.3.3.2.2.3.1" xref="S3.SS1.SSS1.2.p2.10.m10.3.3.2.2.3.1.cmml">−</mo><mi id="S3.SS1.SSS1.2.p2.10.m10.3.3.2.2.3.3" xref="S3.SS1.SSS1.2.p2.10.m10.3.3.2.2.3.3.cmml">γ</mi></mrow></msub></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS1.2.p2.10.m10.3b"><list id="S3.SS1.SSS1.2.p2.10.m10.3.3.3.cmml" xref="S3.SS1.SSS1.2.p2.10.m10.3.3.2"><apply id="S3.SS1.SSS1.2.p2.10.m10.2.2.1.1.cmml" xref="S3.SS1.SSS1.2.p2.10.m10.2.2.1.1"><csymbol cd="ambiguous" id="S3.SS1.SSS1.2.p2.10.m10.2.2.1.1.1.cmml" xref="S3.SS1.SSS1.2.p2.10.m10.2.2.1.1">subscript</csymbol><ci id="S3.SS1.SSS1.2.p2.10.m10.2.2.1.1.2.cmml" xref="S3.SS1.SSS1.2.p2.10.m10.2.2.1.1.2">𝑒</ci><cn id="S3.SS1.SSS1.2.p2.10.m10.2.2.1.1.3.cmml" type="integer" xref="S3.SS1.SSS1.2.p2.10.m10.2.2.1.1.3">1</cn></apply><ci id="S3.SS1.SSS1.2.p2.10.m10.1.1.cmml" xref="S3.SS1.SSS1.2.p2.10.m10.1.1">…</ci><apply id="S3.SS1.SSS1.2.p2.10.m10.3.3.2.2.cmml" xref="S3.SS1.SSS1.2.p2.10.m10.3.3.2.2"><csymbol cd="ambiguous" id="S3.SS1.SSS1.2.p2.10.m10.3.3.2.2.1.cmml" xref="S3.SS1.SSS1.2.p2.10.m10.3.3.2.2">subscript</csymbol><ci id="S3.SS1.SSS1.2.p2.10.m10.3.3.2.2.2.cmml" xref="S3.SS1.SSS1.2.p2.10.m10.3.3.2.2.2">𝑒</ci><apply id="S3.SS1.SSS1.2.p2.10.m10.3.3.2.2.3.cmml" xref="S3.SS1.SSS1.2.p2.10.m10.3.3.2.2.3"><minus id="S3.SS1.SSS1.2.p2.10.m10.3.3.2.2.3.1.cmml" xref="S3.SS1.SSS1.2.p2.10.m10.3.3.2.2.3.1"></minus><apply id="S3.SS1.SSS1.2.p2.10.m10.3.3.2.2.3.2.cmml" xref="S3.SS1.SSS1.2.p2.10.m10.3.3.2.2.3.2"><csymbol cd="ambiguous" id="S3.SS1.SSS1.2.p2.10.m10.3.3.2.2.3.2.1.cmml" xref="S3.SS1.SSS1.2.p2.10.m10.3.3.2.2.3.2">superscript</csymbol><ci id="S3.SS1.SSS1.2.p2.10.m10.3.3.2.2.3.2.2.cmml" xref="S3.SS1.SSS1.2.p2.10.m10.3.3.2.2.3.2.2">𝑘</ci><ci id="S3.SS1.SSS1.2.p2.10.m10.3.3.2.2.3.2.3.cmml" xref="S3.SS1.SSS1.2.p2.10.m10.3.3.2.2.3.2.3">′</ci></apply><ci id="S3.SS1.SSS1.2.p2.10.m10.3.3.2.2.3.3.cmml" xref="S3.SS1.SSS1.2.p2.10.m10.3.3.2.2.3.3">𝛾</ci></apply></apply></list></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS1.2.p2.10.m10.3c">e_{1},\dots,e_{k^{\prime}-\gamma}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS1.2.p2.10.m10.3d">italic_e start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , … , italic_e start_POSTSUBSCRIPT italic_k start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT - italic_γ end_POSTSUBSCRIPT</annotation></semantics></math>. If all these edges exist in <math alttext="H" class="ltx_Math" display="inline" id="S3.SS1.SSS1.2.p2.11.m11.1"><semantics id="S3.SS1.SSS1.2.p2.11.m11.1a"><mi id="S3.SS1.SSS1.2.p2.11.m11.1.1" xref="S3.SS1.SSS1.2.p2.11.m11.1.1.cmml">H</mi><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS1.2.p2.11.m11.1b"><ci id="S3.SS1.SSS1.2.p2.11.m11.1.1.cmml" xref="S3.SS1.SSS1.2.p2.11.m11.1.1">𝐻</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS1.2.p2.11.m11.1c">H</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS1.2.p2.11.m11.1d">italic_H</annotation></semantics></math>, then since <span class="ltx_text ltx_markedasmath" id="S3.SS1.SSS1.2.p2.20.3">SOL</span> also includes them, it trivially satisfies the connectivity requirement of <math alttext="\hat{S}" class="ltx_Math" display="inline" id="S3.SS1.SSS1.2.p2.13.m13.1"><semantics id="S3.SS1.SSS1.2.p2.13.m13.1a"><mover accent="true" id="S3.SS1.SSS1.2.p2.13.m13.1.1" xref="S3.SS1.SSS1.2.p2.13.m13.1.1.cmml"><mi id="S3.SS1.SSS1.2.p2.13.m13.1.1.2" xref="S3.SS1.SSS1.2.p2.13.m13.1.1.2.cmml">S</mi><mo id="S3.SS1.SSS1.2.p2.13.m13.1.1.1" xref="S3.SS1.SSS1.2.p2.13.m13.1.1.1.cmml">^</mo></mover><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS1.2.p2.13.m13.1b"><apply id="S3.SS1.SSS1.2.p2.13.m13.1.1.cmml" xref="S3.SS1.SSS1.2.p2.13.m13.1.1"><ci id="S3.SS1.SSS1.2.p2.13.m13.1.1.1.cmml" xref="S3.SS1.SSS1.2.p2.13.m13.1.1.1">^</ci><ci id="S3.SS1.SSS1.2.p2.13.m13.1.1.2.cmml" xref="S3.SS1.SSS1.2.p2.13.m13.1.1.2">𝑆</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS1.2.p2.13.m13.1c">\hat{S}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS1.2.p2.13.m13.1d">over^ start_ARG italic_S end_ARG</annotation></semantics></math> as well. Suppose, without loss of generality, that <math alttext="e_{1}=(u,v)\notin H" class="ltx_Math" display="inline" id="S3.SS1.SSS1.2.p2.14.m14.2"><semantics id="S3.SS1.SSS1.2.p2.14.m14.2a"><mrow id="S3.SS1.SSS1.2.p2.14.m14.2.3" xref="S3.SS1.SSS1.2.p2.14.m14.2.3.cmml"><msub id="S3.SS1.SSS1.2.p2.14.m14.2.3.2" xref="S3.SS1.SSS1.2.p2.14.m14.2.3.2.cmml"><mi id="S3.SS1.SSS1.2.p2.14.m14.2.3.2.2" xref="S3.SS1.SSS1.2.p2.14.m14.2.3.2.2.cmml">e</mi><mn id="S3.SS1.SSS1.2.p2.14.m14.2.3.2.3" xref="S3.SS1.SSS1.2.p2.14.m14.2.3.2.3.cmml">1</mn></msub><mo id="S3.SS1.SSS1.2.p2.14.m14.2.3.3" xref="S3.SS1.SSS1.2.p2.14.m14.2.3.3.cmml">=</mo><mrow id="S3.SS1.SSS1.2.p2.14.m14.2.3.4.2" xref="S3.SS1.SSS1.2.p2.14.m14.2.3.4.1.cmml"><mo id="S3.SS1.SSS1.2.p2.14.m14.2.3.4.2.1" stretchy="false" xref="S3.SS1.SSS1.2.p2.14.m14.2.3.4.1.cmml">(</mo><mi id="S3.SS1.SSS1.2.p2.14.m14.1.1" xref="S3.SS1.SSS1.2.p2.14.m14.1.1.cmml">u</mi><mo id="S3.SS1.SSS1.2.p2.14.m14.2.3.4.2.2" xref="S3.SS1.SSS1.2.p2.14.m14.2.3.4.1.cmml">,</mo><mi id="S3.SS1.SSS1.2.p2.14.m14.2.2" xref="S3.SS1.SSS1.2.p2.14.m14.2.2.cmml">v</mi><mo id="S3.SS1.SSS1.2.p2.14.m14.2.3.4.2.3" stretchy="false" xref="S3.SS1.SSS1.2.p2.14.m14.2.3.4.1.cmml">)</mo></mrow><mo id="S3.SS1.SSS1.2.p2.14.m14.2.3.5" xref="S3.SS1.SSS1.2.p2.14.m14.2.3.5.cmml">∉</mo><mi id="S3.SS1.SSS1.2.p2.14.m14.2.3.6" xref="S3.SS1.SSS1.2.p2.14.m14.2.3.6.cmml">H</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS1.2.p2.14.m14.2b"><apply id="S3.SS1.SSS1.2.p2.14.m14.2.3.cmml" xref="S3.SS1.SSS1.2.p2.14.m14.2.3"><and id="S3.SS1.SSS1.2.p2.14.m14.2.3a.cmml" xref="S3.SS1.SSS1.2.p2.14.m14.2.3"></and><apply id="S3.SS1.SSS1.2.p2.14.m14.2.3b.cmml" xref="S3.SS1.SSS1.2.p2.14.m14.2.3"><eq id="S3.SS1.SSS1.2.p2.14.m14.2.3.3.cmml" xref="S3.SS1.SSS1.2.p2.14.m14.2.3.3"></eq><apply id="S3.SS1.SSS1.2.p2.14.m14.2.3.2.cmml" xref="S3.SS1.SSS1.2.p2.14.m14.2.3.2"><csymbol cd="ambiguous" id="S3.SS1.SSS1.2.p2.14.m14.2.3.2.1.cmml" xref="S3.SS1.SSS1.2.p2.14.m14.2.3.2">subscript</csymbol><ci id="S3.SS1.SSS1.2.p2.14.m14.2.3.2.2.cmml" xref="S3.SS1.SSS1.2.p2.14.m14.2.3.2.2">𝑒</ci><cn id="S3.SS1.SSS1.2.p2.14.m14.2.3.2.3.cmml" type="integer" xref="S3.SS1.SSS1.2.p2.14.m14.2.3.2.3">1</cn></apply><interval closure="open" id="S3.SS1.SSS1.2.p2.14.m14.2.3.4.1.cmml" xref="S3.SS1.SSS1.2.p2.14.m14.2.3.4.2"><ci id="S3.SS1.SSS1.2.p2.14.m14.1.1.cmml" xref="S3.SS1.SSS1.2.p2.14.m14.1.1">𝑢</ci><ci id="S3.SS1.SSS1.2.p2.14.m14.2.2.cmml" xref="S3.SS1.SSS1.2.p2.14.m14.2.2">𝑣</ci></interval></apply><apply id="S3.SS1.SSS1.2.p2.14.m14.2.3c.cmml" xref="S3.SS1.SSS1.2.p2.14.m14.2.3"><notin id="S3.SS1.SSS1.2.p2.14.m14.2.3.5.cmml" xref="S3.SS1.SSS1.2.p2.14.m14.2.3.5"></notin><share href="https://arxiv.org/html/2503.00712v1#S3.SS1.SSS1.2.p2.14.m14.2.3.4.cmml" id="S3.SS1.SSS1.2.p2.14.m14.2.3d.cmml" xref="S3.SS1.SSS1.2.p2.14.m14.2.3"></share><ci id="S3.SS1.SSS1.2.p2.14.m14.2.3.6.cmml" xref="S3.SS1.SSS1.2.p2.14.m14.2.3.6">𝐻</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS1.2.p2.14.m14.2c">e_{1}=(u,v)\notin H</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS1.2.p2.14.m14.2d">italic_e start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT = ( italic_u , italic_v ) ∉ italic_H</annotation></semantics></math>. Then, by Theorem <a class="ltx_ref" href="https://arxiv.org/html/2503.00712v1#S3.Thmtheorem1" title="Lemma 3.1. ‣ 3 Generic Framework for Streaming Algorithms for Network Design ‣ Streaming Algorithms for Network Design"><span class="ltx_text ltx_ref_tag">3.1</span></a>, <span class="ltx_text ltx_markedasmath" id="S3.SS1.SSS1.2.p2.20.4">SOL</span> contains <math alttext="k" class="ltx_Math" display="inline" id="S3.SS1.SSS1.2.p2.16.m16.1"><semantics id="S3.SS1.SSS1.2.p2.16.m16.1a"><mi id="S3.SS1.SSS1.2.p2.16.m16.1.1" xref="S3.SS1.SSS1.2.p2.16.m16.1.1.cmml">k</mi><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS1.2.p2.16.m16.1b"><ci id="S3.SS1.SSS1.2.p2.16.m16.1.1.cmml" xref="S3.SS1.SSS1.2.p2.16.m16.1.1">𝑘</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS1.2.p2.16.m16.1c">k</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS1.2.p2.16.m16.1d">italic_k</annotation></semantics></math> vertex-disjoint <math alttext="uv" class="ltx_Math" display="inline" id="S3.SS1.SSS1.2.p2.17.m17.1"><semantics id="S3.SS1.SSS1.2.p2.17.m17.1a"><mrow id="S3.SS1.SSS1.2.p2.17.m17.1.1" xref="S3.SS1.SSS1.2.p2.17.m17.1.1.cmml"><mi id="S3.SS1.SSS1.2.p2.17.m17.1.1.2" xref="S3.SS1.SSS1.2.p2.17.m17.1.1.2.cmml">u</mi><mo id="S3.SS1.SSS1.2.p2.17.m17.1.1.1" xref="S3.SS1.SSS1.2.p2.17.m17.1.1.1.cmml">⁢</mo><mi id="S3.SS1.SSS1.2.p2.17.m17.1.1.3" xref="S3.SS1.SSS1.2.p2.17.m17.1.1.3.cmml">v</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS1.2.p2.17.m17.1b"><apply id="S3.SS1.SSS1.2.p2.17.m17.1.1.cmml" xref="S3.SS1.SSS1.2.p2.17.m17.1.1"><times id="S3.SS1.SSS1.2.p2.17.m17.1.1.1.cmml" xref="S3.SS1.SSS1.2.p2.17.m17.1.1.1"></times><ci id="S3.SS1.SSS1.2.p2.17.m17.1.1.2.cmml" xref="S3.SS1.SSS1.2.p2.17.m17.1.1.2">𝑢</ci><ci id="S3.SS1.SSS1.2.p2.17.m17.1.1.3.cmml" xref="S3.SS1.SSS1.2.p2.17.m17.1.1.3">𝑣</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS1.2.p2.17.m17.1c">uv</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS1.2.p2.17.m17.1d">italic_u italic_v</annotation></semantics></math>-paths from the same weight class as <math alttext="e" class="ltx_Math" display="inline" id="S3.SS1.SSS1.2.p2.18.m18.1"><semantics id="S3.SS1.SSS1.2.p2.18.m18.1a"><mi id="S3.SS1.SSS1.2.p2.18.m18.1.1" xref="S3.SS1.SSS1.2.p2.18.m18.1.1.cmml">e</mi><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS1.2.p2.18.m18.1b"><ci id="S3.SS1.SSS1.2.p2.18.m18.1.1.cmml" xref="S3.SS1.SSS1.2.p2.18.m18.1.1">𝑒</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS1.2.p2.18.m18.1c">e</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS1.2.p2.18.m18.1d">italic_e</annotation></semantics></math>; thus at least <math alttext="k-\gamma\geq k^{\prime}-\gamma" class="ltx_Math" display="inline" id="S3.SS1.SSS1.2.p2.19.m19.1"><semantics id="S3.SS1.SSS1.2.p2.19.m19.1a"><mrow id="S3.SS1.SSS1.2.p2.19.m19.1.1" xref="S3.SS1.SSS1.2.p2.19.m19.1.1.cmml"><mrow id="S3.SS1.SSS1.2.p2.19.m19.1.1.2" xref="S3.SS1.SSS1.2.p2.19.m19.1.1.2.cmml"><mi id="S3.SS1.SSS1.2.p2.19.m19.1.1.2.2" xref="S3.SS1.SSS1.2.p2.19.m19.1.1.2.2.cmml">k</mi><mo id="S3.SS1.SSS1.2.p2.19.m19.1.1.2.1" xref="S3.SS1.SSS1.2.p2.19.m19.1.1.2.1.cmml">−</mo><mi id="S3.SS1.SSS1.2.p2.19.m19.1.1.2.3" xref="S3.SS1.SSS1.2.p2.19.m19.1.1.2.3.cmml">γ</mi></mrow><mo id="S3.SS1.SSS1.2.p2.19.m19.1.1.1" xref="S3.SS1.SSS1.2.p2.19.m19.1.1.1.cmml">≥</mo><mrow id="S3.SS1.SSS1.2.p2.19.m19.1.1.3" xref="S3.SS1.SSS1.2.p2.19.m19.1.1.3.cmml"><msup id="S3.SS1.SSS1.2.p2.19.m19.1.1.3.2" xref="S3.SS1.SSS1.2.p2.19.m19.1.1.3.2.cmml"><mi id="S3.SS1.SSS1.2.p2.19.m19.1.1.3.2.2" xref="S3.SS1.SSS1.2.p2.19.m19.1.1.3.2.2.cmml">k</mi><mo id="S3.SS1.SSS1.2.p2.19.m19.1.1.3.2.3" xref="S3.SS1.SSS1.2.p2.19.m19.1.1.3.2.3.cmml">′</mo></msup><mo id="S3.SS1.SSS1.2.p2.19.m19.1.1.3.1" xref="S3.SS1.SSS1.2.p2.19.m19.1.1.3.1.cmml">−</mo><mi id="S3.SS1.SSS1.2.p2.19.m19.1.1.3.3" xref="S3.SS1.SSS1.2.p2.19.m19.1.1.3.3.cmml">γ</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS1.2.p2.19.m19.1b"><apply id="S3.SS1.SSS1.2.p2.19.m19.1.1.cmml" xref="S3.SS1.SSS1.2.p2.19.m19.1.1"><geq id="S3.SS1.SSS1.2.p2.19.m19.1.1.1.cmml" xref="S3.SS1.SSS1.2.p2.19.m19.1.1.1"></geq><apply id="S3.SS1.SSS1.2.p2.19.m19.1.1.2.cmml" xref="S3.SS1.SSS1.2.p2.19.m19.1.1.2"><minus id="S3.SS1.SSS1.2.p2.19.m19.1.1.2.1.cmml" xref="S3.SS1.SSS1.2.p2.19.m19.1.1.2.1"></minus><ci id="S3.SS1.SSS1.2.p2.19.m19.1.1.2.2.cmml" xref="S3.SS1.SSS1.2.p2.19.m19.1.1.2.2">𝑘</ci><ci id="S3.SS1.SSS1.2.p2.19.m19.1.1.2.3.cmml" xref="S3.SS1.SSS1.2.p2.19.m19.1.1.2.3">𝛾</ci></apply><apply id="S3.SS1.SSS1.2.p2.19.m19.1.1.3.cmml" xref="S3.SS1.SSS1.2.p2.19.m19.1.1.3"><minus id="S3.SS1.SSS1.2.p2.19.m19.1.1.3.1.cmml" xref="S3.SS1.SSS1.2.p2.19.m19.1.1.3.1"></minus><apply id="S3.SS1.SSS1.2.p2.19.m19.1.1.3.2.cmml" xref="S3.SS1.SSS1.2.p2.19.m19.1.1.3.2"><csymbol cd="ambiguous" id="S3.SS1.SSS1.2.p2.19.m19.1.1.3.2.1.cmml" xref="S3.SS1.SSS1.2.p2.19.m19.1.1.3.2">superscript</csymbol><ci id="S3.SS1.SSS1.2.p2.19.m19.1.1.3.2.2.cmml" xref="S3.SS1.SSS1.2.p2.19.m19.1.1.3.2.2">𝑘</ci><ci id="S3.SS1.SSS1.2.p2.19.m19.1.1.3.2.3.cmml" xref="S3.SS1.SSS1.2.p2.19.m19.1.1.3.2.3">′</ci></apply><ci id="S3.SS1.SSS1.2.p2.19.m19.1.1.3.3.cmml" xref="S3.SS1.SSS1.2.p2.19.m19.1.1.3.3">𝛾</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS1.2.p2.19.m19.1c">k-\gamma\geq k^{\prime}-\gamma</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS1.2.p2.19.m19.1d">italic_k - italic_γ ≥ italic_k start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT - italic_γ</annotation></semantics></math> of them cross <math alttext="\hat{S}" class="ltx_Math" display="inline" id="S3.SS1.SSS1.2.p2.20.m20.1"><semantics id="S3.SS1.SSS1.2.p2.20.m20.1a"><mover accent="true" id="S3.SS1.SSS1.2.p2.20.m20.1.1" xref="S3.SS1.SSS1.2.p2.20.m20.1.1.cmml"><mi id="S3.SS1.SSS1.2.p2.20.m20.1.1.2" xref="S3.SS1.SSS1.2.p2.20.m20.1.1.2.cmml">S</mi><mo id="S3.SS1.SSS1.2.p2.20.m20.1.1.1" xref="S3.SS1.SSS1.2.p2.20.m20.1.1.1.cmml">^</mo></mover><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS1.2.p2.20.m20.1b"><apply id="S3.SS1.SSS1.2.p2.20.m20.1.1.cmml" xref="S3.SS1.SSS1.2.p2.20.m20.1.1"><ci id="S3.SS1.SSS1.2.p2.20.m20.1.1.1.cmml" xref="S3.SS1.SSS1.2.p2.20.m20.1.1.1">^</ci><ci id="S3.SS1.SSS1.2.p2.20.m20.1.1.2.cmml" xref="S3.SS1.SSS1.2.p2.20.m20.1.1.2">𝑆</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS1.2.p2.20.m20.1c">\hat{S}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS1.2.p2.20.m20.1d">over^ start_ARG italic_S end_ARG</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="S3.SS1.SSS1.3.p3"> <p class="ltx_p" id="S3.SS1.SSS1.3.p3.10">For the cost analysis, note that for every edge <math alttext="e=(u,v)\in\textnormal{OPT}" class="ltx_Math" display="inline" id="S3.SS1.SSS1.3.p3.1.m1.2"><semantics id="S3.SS1.SSS1.3.p3.1.m1.2a"><mrow id="S3.SS1.SSS1.3.p3.1.m1.2.3" xref="S3.SS1.SSS1.3.p3.1.m1.2.3.cmml"><mi id="S3.SS1.SSS1.3.p3.1.m1.2.3.2" xref="S3.SS1.SSS1.3.p3.1.m1.2.3.2.cmml">e</mi><mo id="S3.SS1.SSS1.3.p3.1.m1.2.3.3" xref="S3.SS1.SSS1.3.p3.1.m1.2.3.3.cmml">=</mo><mrow id="S3.SS1.SSS1.3.p3.1.m1.2.3.4.2" xref="S3.SS1.SSS1.3.p3.1.m1.2.3.4.1.cmml"><mo id="S3.SS1.SSS1.3.p3.1.m1.2.3.4.2.1" stretchy="false" xref="S3.SS1.SSS1.3.p3.1.m1.2.3.4.1.cmml">(</mo><mi id="S3.SS1.SSS1.3.p3.1.m1.1.1" xref="S3.SS1.SSS1.3.p3.1.m1.1.1.cmml">u</mi><mo id="S3.SS1.SSS1.3.p3.1.m1.2.3.4.2.2" xref="S3.SS1.SSS1.3.p3.1.m1.2.3.4.1.cmml">,</mo><mi id="S3.SS1.SSS1.3.p3.1.m1.2.2" xref="S3.SS1.SSS1.3.p3.1.m1.2.2.cmml">v</mi><mo id="S3.SS1.SSS1.3.p3.1.m1.2.3.4.2.3" stretchy="false" xref="S3.SS1.SSS1.3.p3.1.m1.2.3.4.1.cmml">)</mo></mrow><mo id="S3.SS1.SSS1.3.p3.1.m1.2.3.5" xref="S3.SS1.SSS1.3.p3.1.m1.2.3.5.cmml">∈</mo><mtext id="S3.SS1.SSS1.3.p3.1.m1.2.3.6" xref="S3.SS1.SSS1.3.p3.1.m1.2.3.6a.cmml">OPT</mtext></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS1.3.p3.1.m1.2b"><apply id="S3.SS1.SSS1.3.p3.1.m1.2.3.cmml" xref="S3.SS1.SSS1.3.p3.1.m1.2.3"><and id="S3.SS1.SSS1.3.p3.1.m1.2.3a.cmml" xref="S3.SS1.SSS1.3.p3.1.m1.2.3"></and><apply id="S3.SS1.SSS1.3.p3.1.m1.2.3b.cmml" xref="S3.SS1.SSS1.3.p3.1.m1.2.3"><eq id="S3.SS1.SSS1.3.p3.1.m1.2.3.3.cmml" xref="S3.SS1.SSS1.3.p3.1.m1.2.3.3"></eq><ci id="S3.SS1.SSS1.3.p3.1.m1.2.3.2.cmml" xref="S3.SS1.SSS1.3.p3.1.m1.2.3.2">𝑒</ci><interval closure="open" id="S3.SS1.SSS1.3.p3.1.m1.2.3.4.1.cmml" xref="S3.SS1.SSS1.3.p3.1.m1.2.3.4.2"><ci id="S3.SS1.SSS1.3.p3.1.m1.1.1.cmml" xref="S3.SS1.SSS1.3.p3.1.m1.1.1">𝑢</ci><ci id="S3.SS1.SSS1.3.p3.1.m1.2.2.cmml" xref="S3.SS1.SSS1.3.p3.1.m1.2.2">𝑣</ci></interval></apply><apply id="S3.SS1.SSS1.3.p3.1.m1.2.3c.cmml" xref="S3.SS1.SSS1.3.p3.1.m1.2.3"><in id="S3.SS1.SSS1.3.p3.1.m1.2.3.5.cmml" xref="S3.SS1.SSS1.3.p3.1.m1.2.3.5"></in><share href="https://arxiv.org/html/2503.00712v1#S3.SS1.SSS1.3.p3.1.m1.2.3.4.cmml" id="S3.SS1.SSS1.3.p3.1.m1.2.3d.cmml" xref="S3.SS1.SSS1.3.p3.1.m1.2.3"></share><ci id="S3.SS1.SSS1.3.p3.1.m1.2.3.6a.cmml" xref="S3.SS1.SSS1.3.p3.1.m1.2.3.6"><mtext id="S3.SS1.SSS1.3.p3.1.m1.2.3.6.cmml" xref="S3.SS1.SSS1.3.p3.1.m1.2.3.6">OPT</mtext></ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS1.3.p3.1.m1.2c">e=(u,v)\in\textnormal{OPT}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS1.3.p3.1.m1.2d">italic_e = ( italic_u , italic_v ) ∈ OPT</annotation></semantics></math> in weight class <math alttext="B_{j}" class="ltx_Math" display="inline" id="S3.SS1.SSS1.3.p3.2.m2.1"><semantics id="S3.SS1.SSS1.3.p3.2.m2.1a"><msub id="S3.SS1.SSS1.3.p3.2.m2.1.1" xref="S3.SS1.SSS1.3.p3.2.m2.1.1.cmml"><mi id="S3.SS1.SSS1.3.p3.2.m2.1.1.2" xref="S3.SS1.SSS1.3.p3.2.m2.1.1.2.cmml">B</mi><mi id="S3.SS1.SSS1.3.p3.2.m2.1.1.3" xref="S3.SS1.SSS1.3.p3.2.m2.1.1.3.cmml">j</mi></msub><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS1.3.p3.2.m2.1b"><apply id="S3.SS1.SSS1.3.p3.2.m2.1.1.cmml" xref="S3.SS1.SSS1.3.p3.2.m2.1.1"><csymbol cd="ambiguous" id="S3.SS1.SSS1.3.p3.2.m2.1.1.1.cmml" xref="S3.SS1.SSS1.3.p3.2.m2.1.1">subscript</csymbol><ci id="S3.SS1.SSS1.3.p3.2.m2.1.1.2.cmml" xref="S3.SS1.SSS1.3.p3.2.m2.1.1.2">𝐵</ci><ci id="S3.SS1.SSS1.3.p3.2.m2.1.1.3.cmml" xref="S3.SS1.SSS1.3.p3.2.m2.1.1.3">𝑗</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS1.3.p3.2.m2.1c">B_{j}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS1.3.p3.2.m2.1d">italic_B start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT</annotation></semantics></math>, either <math alttext="e\in\textnormal{SOL}" class="ltx_Math" display="inline" id="S3.SS1.SSS1.3.p3.3.m3.1"><semantics id="S3.SS1.SSS1.3.p3.3.m3.1a"><mrow id="S3.SS1.SSS1.3.p3.3.m3.1.1" xref="S3.SS1.SSS1.3.p3.3.m3.1.1.cmml"><mi id="S3.SS1.SSS1.3.p3.3.m3.1.1.2" xref="S3.SS1.SSS1.3.p3.3.m3.1.1.2.cmml">e</mi><mo id="S3.SS1.SSS1.3.p3.3.m3.1.1.1" xref="S3.SS1.SSS1.3.p3.3.m3.1.1.1.cmml">∈</mo><mtext id="S3.SS1.SSS1.3.p3.3.m3.1.1.3" xref="S3.SS1.SSS1.3.p3.3.m3.1.1.3a.cmml">SOL</mtext></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS1.3.p3.3.m3.1b"><apply id="S3.SS1.SSS1.3.p3.3.m3.1.1.cmml" xref="S3.SS1.SSS1.3.p3.3.m3.1.1"><in id="S3.SS1.SSS1.3.p3.3.m3.1.1.1.cmml" xref="S3.SS1.SSS1.3.p3.3.m3.1.1.1"></in><ci id="S3.SS1.SSS1.3.p3.3.m3.1.1.2.cmml" xref="S3.SS1.SSS1.3.p3.3.m3.1.1.2">𝑒</ci><ci id="S3.SS1.SSS1.3.p3.3.m3.1.1.3a.cmml" xref="S3.SS1.SSS1.3.p3.3.m3.1.1.3"><mtext id="S3.SS1.SSS1.3.p3.3.m3.1.1.3.cmml" xref="S3.SS1.SSS1.3.p3.3.m3.1.1.3">SOL</mtext></ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS1.3.p3.3.m3.1c">e\in\textnormal{SOL}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS1.3.p3.3.m3.1d">italic_e ∈ SOL</annotation></semantics></math>, or <span class="ltx_text ltx_markedasmath" id="S3.SS1.SSS1.3.p3.10.1">SOL</span> adds at most <math alttext="k" class="ltx_Math" display="inline" id="S3.SS1.SSS1.3.p3.5.m5.1"><semantics id="S3.SS1.SSS1.3.p3.5.m5.1a"><mi id="S3.SS1.SSS1.3.p3.5.m5.1.1" xref="S3.SS1.SSS1.3.p3.5.m5.1.1.cmml">k</mi><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS1.3.p3.5.m5.1b"><ci id="S3.SS1.SSS1.3.p3.5.m5.1.1.cmml" xref="S3.SS1.SSS1.3.p3.5.m5.1.1">𝑘</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS1.3.p3.5.m5.1c">k</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS1.3.p3.5.m5.1d">italic_k</annotation></semantics></math> edge-disjoint <math alttext="uv" class="ltx_Math" display="inline" id="S3.SS1.SSS1.3.p3.6.m6.1"><semantics id="S3.SS1.SSS1.3.p3.6.m6.1a"><mrow id="S3.SS1.SSS1.3.p3.6.m6.1.1" xref="S3.SS1.SSS1.3.p3.6.m6.1.1.cmml"><mi id="S3.SS1.SSS1.3.p3.6.m6.1.1.2" xref="S3.SS1.SSS1.3.p3.6.m6.1.1.2.cmml">u</mi><mo id="S3.SS1.SSS1.3.p3.6.m6.1.1.1" xref="S3.SS1.SSS1.3.p3.6.m6.1.1.1.cmml">⁢</mo><mi id="S3.SS1.SSS1.3.p3.6.m6.1.1.3" xref="S3.SS1.SSS1.3.p3.6.m6.1.1.3.cmml">v</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS1.3.p3.6.m6.1b"><apply id="S3.SS1.SSS1.3.p3.6.m6.1.1.cmml" xref="S3.SS1.SSS1.3.p3.6.m6.1.1"><times id="S3.SS1.SSS1.3.p3.6.m6.1.1.1.cmml" xref="S3.SS1.SSS1.3.p3.6.m6.1.1.1"></times><ci id="S3.SS1.SSS1.3.p3.6.m6.1.1.2.cmml" xref="S3.SS1.SSS1.3.p3.6.m6.1.1.2">𝑢</ci><ci id="S3.SS1.SSS1.3.p3.6.m6.1.1.3.cmml" xref="S3.SS1.SSS1.3.p3.6.m6.1.1.3">𝑣</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS1.3.p3.6.m6.1c">uv</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS1.3.p3.6.m6.1d">italic_u italic_v</annotation></semantics></math>-paths, <math alttext="P_{1},\dots,P_{k}" class="ltx_Math" display="inline" id="S3.SS1.SSS1.3.p3.7.m7.3"><semantics id="S3.SS1.SSS1.3.p3.7.m7.3a"><mrow id="S3.SS1.SSS1.3.p3.7.m7.3.3.2" xref="S3.SS1.SSS1.3.p3.7.m7.3.3.3.cmml"><msub id="S3.SS1.SSS1.3.p3.7.m7.2.2.1.1" xref="S3.SS1.SSS1.3.p3.7.m7.2.2.1.1.cmml"><mi id="S3.SS1.SSS1.3.p3.7.m7.2.2.1.1.2" xref="S3.SS1.SSS1.3.p3.7.m7.2.2.1.1.2.cmml">P</mi><mn id="S3.SS1.SSS1.3.p3.7.m7.2.2.1.1.3" xref="S3.SS1.SSS1.3.p3.7.m7.2.2.1.1.3.cmml">1</mn></msub><mo id="S3.SS1.SSS1.3.p3.7.m7.3.3.2.3" xref="S3.SS1.SSS1.3.p3.7.m7.3.3.3.cmml">,</mo><mi id="S3.SS1.SSS1.3.p3.7.m7.1.1" mathvariant="normal" xref="S3.SS1.SSS1.3.p3.7.m7.1.1.cmml">…</mi><mo id="S3.SS1.SSS1.3.p3.7.m7.3.3.2.4" xref="S3.SS1.SSS1.3.p3.7.m7.3.3.3.cmml">,</mo><msub id="S3.SS1.SSS1.3.p3.7.m7.3.3.2.2" xref="S3.SS1.SSS1.3.p3.7.m7.3.3.2.2.cmml"><mi id="S3.SS1.SSS1.3.p3.7.m7.3.3.2.2.2" xref="S3.SS1.SSS1.3.p3.7.m7.3.3.2.2.2.cmml">P</mi><mi id="S3.SS1.SSS1.3.p3.7.m7.3.3.2.2.3" xref="S3.SS1.SSS1.3.p3.7.m7.3.3.2.2.3.cmml">k</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS1.3.p3.7.m7.3b"><list id="S3.SS1.SSS1.3.p3.7.m7.3.3.3.cmml" xref="S3.SS1.SSS1.3.p3.7.m7.3.3.2"><apply id="S3.SS1.SSS1.3.p3.7.m7.2.2.1.1.cmml" xref="S3.SS1.SSS1.3.p3.7.m7.2.2.1.1"><csymbol cd="ambiguous" id="S3.SS1.SSS1.3.p3.7.m7.2.2.1.1.1.cmml" xref="S3.SS1.SSS1.3.p3.7.m7.2.2.1.1">subscript</csymbol><ci id="S3.SS1.SSS1.3.p3.7.m7.2.2.1.1.2.cmml" xref="S3.SS1.SSS1.3.p3.7.m7.2.2.1.1.2">𝑃</ci><cn id="S3.SS1.SSS1.3.p3.7.m7.2.2.1.1.3.cmml" type="integer" xref="S3.SS1.SSS1.3.p3.7.m7.2.2.1.1.3">1</cn></apply><ci id="S3.SS1.SSS1.3.p3.7.m7.1.1.cmml" xref="S3.SS1.SSS1.3.p3.7.m7.1.1">…</ci><apply id="S3.SS1.SSS1.3.p3.7.m7.3.3.2.2.cmml" xref="S3.SS1.SSS1.3.p3.7.m7.3.3.2.2"><csymbol cd="ambiguous" id="S3.SS1.SSS1.3.p3.7.m7.3.3.2.2.1.cmml" xref="S3.SS1.SSS1.3.p3.7.m7.3.3.2.2">subscript</csymbol><ci id="S3.SS1.SSS1.3.p3.7.m7.3.3.2.2.2.cmml" xref="S3.SS1.SSS1.3.p3.7.m7.3.3.2.2.2">𝑃</ci><ci id="S3.SS1.SSS1.3.p3.7.m7.3.3.2.2.3.cmml" xref="S3.SS1.SSS1.3.p3.7.m7.3.3.2.2.3">𝑘</ci></apply></list></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS1.3.p3.7.m7.3c">P_{1},\dots,P_{k}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS1.3.p3.7.m7.3d">italic_P start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , … , italic_P start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT</annotation></semantics></math>, each of length at most <math alttext="2t-1" class="ltx_Math" display="inline" id="S3.SS1.SSS1.3.p3.8.m8.1"><semantics id="S3.SS1.SSS1.3.p3.8.m8.1a"><mrow id="S3.SS1.SSS1.3.p3.8.m8.1.1" xref="S3.SS1.SSS1.3.p3.8.m8.1.1.cmml"><mrow id="S3.SS1.SSS1.3.p3.8.m8.1.1.2" xref="S3.SS1.SSS1.3.p3.8.m8.1.1.2.cmml"><mn id="S3.SS1.SSS1.3.p3.8.m8.1.1.2.2" xref="S3.SS1.SSS1.3.p3.8.m8.1.1.2.2.cmml">2</mn><mo id="S3.SS1.SSS1.3.p3.8.m8.1.1.2.1" xref="S3.SS1.SSS1.3.p3.8.m8.1.1.2.1.cmml">⁢</mo><mi id="S3.SS1.SSS1.3.p3.8.m8.1.1.2.3" xref="S3.SS1.SSS1.3.p3.8.m8.1.1.2.3.cmml">t</mi></mrow><mo id="S3.SS1.SSS1.3.p3.8.m8.1.1.1" xref="S3.SS1.SSS1.3.p3.8.m8.1.1.1.cmml">−</mo><mn id="S3.SS1.SSS1.3.p3.8.m8.1.1.3" xref="S3.SS1.SSS1.3.p3.8.m8.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS1.3.p3.8.m8.1b"><apply id="S3.SS1.SSS1.3.p3.8.m8.1.1.cmml" xref="S3.SS1.SSS1.3.p3.8.m8.1.1"><minus id="S3.SS1.SSS1.3.p3.8.m8.1.1.1.cmml" xref="S3.SS1.SSS1.3.p3.8.m8.1.1.1"></minus><apply id="S3.SS1.SSS1.3.p3.8.m8.1.1.2.cmml" xref="S3.SS1.SSS1.3.p3.8.m8.1.1.2"><times id="S3.SS1.SSS1.3.p3.8.m8.1.1.2.1.cmml" xref="S3.SS1.SSS1.3.p3.8.m8.1.1.2.1"></times><cn id="S3.SS1.SSS1.3.p3.8.m8.1.1.2.2.cmml" type="integer" xref="S3.SS1.SSS1.3.p3.8.m8.1.1.2.2">2</cn><ci id="S3.SS1.SSS1.3.p3.8.m8.1.1.2.3.cmml" xref="S3.SS1.SSS1.3.p3.8.m8.1.1.2.3">𝑡</ci></apply><cn id="S3.SS1.SSS1.3.p3.8.m8.1.1.3.cmml" type="integer" xref="S3.SS1.SSS1.3.p3.8.m8.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS1.3.p3.8.m8.1c">2t-1</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS1.3.p3.8.m8.1d">2 italic_t - 1</annotation></semantics></math>, with weights belonging to <math alttext="B_{j}" class="ltx_Math" display="inline" id="S3.SS1.SSS1.3.p3.9.m9.1"><semantics id="S3.SS1.SSS1.3.p3.9.m9.1a"><msub id="S3.SS1.SSS1.3.p3.9.m9.1.1" xref="S3.SS1.SSS1.3.p3.9.m9.1.1.cmml"><mi id="S3.SS1.SSS1.3.p3.9.m9.1.1.2" xref="S3.SS1.SSS1.3.p3.9.m9.1.1.2.cmml">B</mi><mi id="S3.SS1.SSS1.3.p3.9.m9.1.1.3" xref="S3.SS1.SSS1.3.p3.9.m9.1.1.3.cmml">j</mi></msub><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS1.3.p3.9.m9.1b"><apply id="S3.SS1.SSS1.3.p3.9.m9.1.1.cmml" xref="S3.SS1.SSS1.3.p3.9.m9.1.1"><csymbol cd="ambiguous" id="S3.SS1.SSS1.3.p3.9.m9.1.1.1.cmml" xref="S3.SS1.SSS1.3.p3.9.m9.1.1">subscript</csymbol><ci id="S3.SS1.SSS1.3.p3.9.m9.1.1.2.cmml" xref="S3.SS1.SSS1.3.p3.9.m9.1.1.2">𝐵</ci><ci id="S3.SS1.SSS1.3.p3.9.m9.1.1.3.cmml" xref="S3.SS1.SSS1.3.p3.9.m9.1.1.3">𝑗</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS1.3.p3.9.m9.1c">B_{j}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS1.3.p3.9.m9.1d">italic_B start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT</annotation></semantics></math>. Therefore, <math alttext="w(\textnormal{SOL})\leq k\cdot(2t-1)\cdot(1+\frac{1}{2t-1})\cdot w(\textnormal% {OPT})=2tk\cdot w(\textnormal{OPT})" class="ltx_Math" display="inline" id="S3.SS1.SSS1.3.p3.10.m10.5"><semantics id="S3.SS1.SSS1.3.p3.10.m10.5a"><mrow id="S3.SS1.SSS1.3.p3.10.m10.5.5" xref="S3.SS1.SSS1.3.p3.10.m10.5.5.cmml"><mrow id="S3.SS1.SSS1.3.p3.10.m10.5.5.4" xref="S3.SS1.SSS1.3.p3.10.m10.5.5.4.cmml"><mi id="S3.SS1.SSS1.3.p3.10.m10.5.5.4.2" xref="S3.SS1.SSS1.3.p3.10.m10.5.5.4.2.cmml">w</mi><mo id="S3.SS1.SSS1.3.p3.10.m10.5.5.4.1" xref="S3.SS1.SSS1.3.p3.10.m10.5.5.4.1.cmml">⁢</mo><mrow id="S3.SS1.SSS1.3.p3.10.m10.5.5.4.3.2" xref="S3.SS1.SSS1.3.p3.10.m10.1.1a.cmml"><mo id="S3.SS1.SSS1.3.p3.10.m10.5.5.4.3.2.1" stretchy="false" xref="S3.SS1.SSS1.3.p3.10.m10.1.1a.cmml">(</mo><mtext id="S3.SS1.SSS1.3.p3.10.m10.1.1" xref="S3.SS1.SSS1.3.p3.10.m10.1.1.cmml">SOL</mtext><mo id="S3.SS1.SSS1.3.p3.10.m10.5.5.4.3.2.2" stretchy="false" xref="S3.SS1.SSS1.3.p3.10.m10.1.1a.cmml">)</mo></mrow></mrow><mo id="S3.SS1.SSS1.3.p3.10.m10.5.5.5" xref="S3.SS1.SSS1.3.p3.10.m10.5.5.5.cmml">≤</mo><mrow id="S3.SS1.SSS1.3.p3.10.m10.5.5.2" xref="S3.SS1.SSS1.3.p3.10.m10.5.5.2.cmml"><mrow id="S3.SS1.SSS1.3.p3.10.m10.5.5.2.2" xref="S3.SS1.SSS1.3.p3.10.m10.5.5.2.2.cmml"><mi id="S3.SS1.SSS1.3.p3.10.m10.5.5.2.2.4" xref="S3.SS1.SSS1.3.p3.10.m10.5.5.2.2.4.cmml">k</mi><mo id="S3.SS1.SSS1.3.p3.10.m10.5.5.2.2.3" lspace="0.222em" rspace="0.222em" xref="S3.SS1.SSS1.3.p3.10.m10.5.5.2.2.3.cmml">⋅</mo><mrow id="S3.SS1.SSS1.3.p3.10.m10.4.4.1.1.1.1" xref="S3.SS1.SSS1.3.p3.10.m10.4.4.1.1.1.1.1.cmml"><mo id="S3.SS1.SSS1.3.p3.10.m10.4.4.1.1.1.1.2" stretchy="false" xref="S3.SS1.SSS1.3.p3.10.m10.4.4.1.1.1.1.1.cmml">(</mo><mrow id="S3.SS1.SSS1.3.p3.10.m10.4.4.1.1.1.1.1" xref="S3.SS1.SSS1.3.p3.10.m10.4.4.1.1.1.1.1.cmml"><mrow id="S3.SS1.SSS1.3.p3.10.m10.4.4.1.1.1.1.1.2" xref="S3.SS1.SSS1.3.p3.10.m10.4.4.1.1.1.1.1.2.cmml"><mn id="S3.SS1.SSS1.3.p3.10.m10.4.4.1.1.1.1.1.2.2" xref="S3.SS1.SSS1.3.p3.10.m10.4.4.1.1.1.1.1.2.2.cmml">2</mn><mo id="S3.SS1.SSS1.3.p3.10.m10.4.4.1.1.1.1.1.2.1" xref="S3.SS1.SSS1.3.p3.10.m10.4.4.1.1.1.1.1.2.1.cmml">⁢</mo><mi id="S3.SS1.SSS1.3.p3.10.m10.4.4.1.1.1.1.1.2.3" xref="S3.SS1.SSS1.3.p3.10.m10.4.4.1.1.1.1.1.2.3.cmml">t</mi></mrow><mo id="S3.SS1.SSS1.3.p3.10.m10.4.4.1.1.1.1.1.1" xref="S3.SS1.SSS1.3.p3.10.m10.4.4.1.1.1.1.1.1.cmml">−</mo><mn id="S3.SS1.SSS1.3.p3.10.m10.4.4.1.1.1.1.1.3" xref="S3.SS1.SSS1.3.p3.10.m10.4.4.1.1.1.1.1.3.cmml">1</mn></mrow><mo id="S3.SS1.SSS1.3.p3.10.m10.4.4.1.1.1.1.3" rspace="0.055em" stretchy="false" xref="S3.SS1.SSS1.3.p3.10.m10.4.4.1.1.1.1.1.cmml">)</mo></mrow><mo id="S3.SS1.SSS1.3.p3.10.m10.5.5.2.2.3a" rspace="0.222em" xref="S3.SS1.SSS1.3.p3.10.m10.5.5.2.2.3.cmml">⋅</mo><mrow id="S3.SS1.SSS1.3.p3.10.m10.5.5.2.2.2.1" xref="S3.SS1.SSS1.3.p3.10.m10.5.5.2.2.2.1.1.cmml"><mo id="S3.SS1.SSS1.3.p3.10.m10.5.5.2.2.2.1.2" stretchy="false" xref="S3.SS1.SSS1.3.p3.10.m10.5.5.2.2.2.1.1.cmml">(</mo><mrow id="S3.SS1.SSS1.3.p3.10.m10.5.5.2.2.2.1.1" xref="S3.SS1.SSS1.3.p3.10.m10.5.5.2.2.2.1.1.cmml"><mn id="S3.SS1.SSS1.3.p3.10.m10.5.5.2.2.2.1.1.2" xref="S3.SS1.SSS1.3.p3.10.m10.5.5.2.2.2.1.1.2.cmml">1</mn><mo id="S3.SS1.SSS1.3.p3.10.m10.5.5.2.2.2.1.1.1" xref="S3.SS1.SSS1.3.p3.10.m10.5.5.2.2.2.1.1.1.cmml">+</mo><mfrac id="S3.SS1.SSS1.3.p3.10.m10.5.5.2.2.2.1.1.3" xref="S3.SS1.SSS1.3.p3.10.m10.5.5.2.2.2.1.1.3.cmml"><mn id="S3.SS1.SSS1.3.p3.10.m10.5.5.2.2.2.1.1.3.2" xref="S3.SS1.SSS1.3.p3.10.m10.5.5.2.2.2.1.1.3.2.cmml">1</mn><mrow id="S3.SS1.SSS1.3.p3.10.m10.5.5.2.2.2.1.1.3.3" xref="S3.SS1.SSS1.3.p3.10.m10.5.5.2.2.2.1.1.3.3.cmml"><mrow id="S3.SS1.SSS1.3.p3.10.m10.5.5.2.2.2.1.1.3.3.2" xref="S3.SS1.SSS1.3.p3.10.m10.5.5.2.2.2.1.1.3.3.2.cmml"><mn id="S3.SS1.SSS1.3.p3.10.m10.5.5.2.2.2.1.1.3.3.2.2" xref="S3.SS1.SSS1.3.p3.10.m10.5.5.2.2.2.1.1.3.3.2.2.cmml">2</mn><mo id="S3.SS1.SSS1.3.p3.10.m10.5.5.2.2.2.1.1.3.3.2.1" xref="S3.SS1.SSS1.3.p3.10.m10.5.5.2.2.2.1.1.3.3.2.1.cmml">⁢</mo><mi id="S3.SS1.SSS1.3.p3.10.m10.5.5.2.2.2.1.1.3.3.2.3" xref="S3.SS1.SSS1.3.p3.10.m10.5.5.2.2.2.1.1.3.3.2.3.cmml">t</mi></mrow><mo id="S3.SS1.SSS1.3.p3.10.m10.5.5.2.2.2.1.1.3.3.1" xref="S3.SS1.SSS1.3.p3.10.m10.5.5.2.2.2.1.1.3.3.1.cmml">−</mo><mn id="S3.SS1.SSS1.3.p3.10.m10.5.5.2.2.2.1.1.3.3.3" xref="S3.SS1.SSS1.3.p3.10.m10.5.5.2.2.2.1.1.3.3.3.cmml">1</mn></mrow></mfrac></mrow><mo id="S3.SS1.SSS1.3.p3.10.m10.5.5.2.2.2.1.3" rspace="0.055em" stretchy="false" xref="S3.SS1.SSS1.3.p3.10.m10.5.5.2.2.2.1.1.cmml">)</mo></mrow><mo id="S3.SS1.SSS1.3.p3.10.m10.5.5.2.2.3b" rspace="0.222em" xref="S3.SS1.SSS1.3.p3.10.m10.5.5.2.2.3.cmml">⋅</mo><mi id="S3.SS1.SSS1.3.p3.10.m10.5.5.2.2.5" xref="S3.SS1.SSS1.3.p3.10.m10.5.5.2.2.5.cmml">w</mi></mrow><mo id="S3.SS1.SSS1.3.p3.10.m10.5.5.2.3" xref="S3.SS1.SSS1.3.p3.10.m10.5.5.2.3.cmml">⁢</mo><mrow id="S3.SS1.SSS1.3.p3.10.m10.5.5.2.4.2" xref="S3.SS1.SSS1.3.p3.10.m10.2.2a.cmml"><mo 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xref="S3.SS1.SSS1.3.p3.10.m10.2.2">OPT</mtext></ci></apply></apply><apply id="S3.SS1.SSS1.3.p3.10.m10.5.5c.cmml" xref="S3.SS1.SSS1.3.p3.10.m10.5.5"><eq id="S3.SS1.SSS1.3.p3.10.m10.5.5.6.cmml" xref="S3.SS1.SSS1.3.p3.10.m10.5.5.6"></eq><share href="https://arxiv.org/html/2503.00712v1#S3.SS1.SSS1.3.p3.10.m10.5.5.2.cmml" id="S3.SS1.SSS1.3.p3.10.m10.5.5d.cmml" xref="S3.SS1.SSS1.3.p3.10.m10.5.5"></share><apply id="S3.SS1.SSS1.3.p3.10.m10.5.5.7.cmml" xref="S3.SS1.SSS1.3.p3.10.m10.5.5.7"><times id="S3.SS1.SSS1.3.p3.10.m10.5.5.7.1.cmml" xref="S3.SS1.SSS1.3.p3.10.m10.5.5.7.1"></times><apply id="S3.SS1.SSS1.3.p3.10.m10.5.5.7.2.cmml" xref="S3.SS1.SSS1.3.p3.10.m10.5.5.7.2"><ci id="S3.SS1.SSS1.3.p3.10.m10.5.5.7.2.1.cmml" xref="S3.SS1.SSS1.3.p3.10.m10.5.5.7.2.1">⋅</ci><apply id="S3.SS1.SSS1.3.p3.10.m10.5.5.7.2.2.cmml" xref="S3.SS1.SSS1.3.p3.10.m10.5.5.7.2.2"><times id="S3.SS1.SSS1.3.p3.10.m10.5.5.7.2.2.1.cmml" xref="S3.SS1.SSS1.3.p3.10.m10.5.5.7.2.2.1"></times><cn id="S3.SS1.SSS1.3.p3.10.m10.5.5.7.2.2.2.cmml" type="integer" xref="S3.SS1.SSS1.3.p3.10.m10.5.5.7.2.2.2">2</cn><ci id="S3.SS1.SSS1.3.p3.10.m10.5.5.7.2.2.3.cmml" xref="S3.SS1.SSS1.3.p3.10.m10.5.5.7.2.2.3">𝑡</ci><ci id="S3.SS1.SSS1.3.p3.10.m10.5.5.7.2.2.4.cmml" xref="S3.SS1.SSS1.3.p3.10.m10.5.5.7.2.2.4">𝑘</ci></apply><ci id="S3.SS1.SSS1.3.p3.10.m10.5.5.7.2.3.cmml" xref="S3.SS1.SSS1.3.p3.10.m10.5.5.7.2.3">𝑤</ci></apply><ci id="S3.SS1.SSS1.3.p3.10.m10.3.3a.cmml" xref="S3.SS1.SSS1.3.p3.10.m10.5.5.7.3.2"><mtext id="S3.SS1.SSS1.3.p3.10.m10.3.3.cmml" xref="S3.SS1.SSS1.3.p3.10.m10.3.3">OPT</mtext></ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS1.3.p3.10.m10.5c">w(\textnormal{SOL})\leq k\cdot(2t-1)\cdot(1+\frac{1}{2t-1})\cdot w(\textnormal% {OPT})=2tk\cdot w(\textnormal{OPT})</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS1.3.p3.10.m10.5d">italic_w ( SOL ) ≤ italic_k ⋅ ( 2 italic_t - 1 ) ⋅ ( 1 + divide start_ARG 1 end_ARG start_ARG 2 italic_t - 1 end_ARG ) ⋅ italic_w ( OPT ) = 2 italic_t italic_k ⋅ italic_w ( OPT )</annotation></semantics></math>. ∎</p> </div> </div> <div class="ltx_theorem ltx_theorem_corollary" id="S3.Thmtheorem3"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="S3.Thmtheorem3.1.1.1">Corollary 3.3</span></span><span class="ltx_text ltx_font_bold" id="S3.Thmtheorem3.2.2">.</span> </h6> <div class="ltx_para" id="S3.Thmtheorem3.p1"> <p class="ltx_p" id="S3.Thmtheorem3.p1.4">There exists an algorithm for VC-SNDP with edge weights <math alttext="w:E\rightarrow\{0,1,\dots,W\}" class="ltx_Math" display="inline" id="S3.Thmtheorem3.p1.1.m1.4"><semantics id="S3.Thmtheorem3.p1.1.m1.4a"><mrow id="S3.Thmtheorem3.p1.1.m1.4.5" xref="S3.Thmtheorem3.p1.1.m1.4.5.cmml"><mi id="S3.Thmtheorem3.p1.1.m1.4.5.2" xref="S3.Thmtheorem3.p1.1.m1.4.5.2.cmml">w</mi><mo id="S3.Thmtheorem3.p1.1.m1.4.5.1" lspace="0.278em" rspace="0.278em" xref="S3.Thmtheorem3.p1.1.m1.4.5.1.cmml">:</mo><mrow id="S3.Thmtheorem3.p1.1.m1.4.5.3" xref="S3.Thmtheorem3.p1.1.m1.4.5.3.cmml"><mi id="S3.Thmtheorem3.p1.1.m1.4.5.3.2" xref="S3.Thmtheorem3.p1.1.m1.4.5.3.2.cmml">E</mi><mo id="S3.Thmtheorem3.p1.1.m1.4.5.3.1" stretchy="false" xref="S3.Thmtheorem3.p1.1.m1.4.5.3.1.cmml">→</mo><mrow id="S3.Thmtheorem3.p1.1.m1.4.5.3.3.2" xref="S3.Thmtheorem3.p1.1.m1.4.5.3.3.1.cmml"><mo id="S3.Thmtheorem3.p1.1.m1.4.5.3.3.2.1" stretchy="false" xref="S3.Thmtheorem3.p1.1.m1.4.5.3.3.1.cmml">{</mo><mn id="S3.Thmtheorem3.p1.1.m1.1.1" xref="S3.Thmtheorem3.p1.1.m1.1.1.cmml">0</mn><mo id="S3.Thmtheorem3.p1.1.m1.4.5.3.3.2.2" xref="S3.Thmtheorem3.p1.1.m1.4.5.3.3.1.cmml">,</mo><mn id="S3.Thmtheorem3.p1.1.m1.2.2" xref="S3.Thmtheorem3.p1.1.m1.2.2.cmml">1</mn><mo id="S3.Thmtheorem3.p1.1.m1.4.5.3.3.2.3" xref="S3.Thmtheorem3.p1.1.m1.4.5.3.3.1.cmml">,</mo><mi id="S3.Thmtheorem3.p1.1.m1.3.3" mathvariant="normal" xref="S3.Thmtheorem3.p1.1.m1.3.3.cmml">…</mi><mo id="S3.Thmtheorem3.p1.1.m1.4.5.3.3.2.4" xref="S3.Thmtheorem3.p1.1.m1.4.5.3.3.1.cmml">,</mo><mi id="S3.Thmtheorem3.p1.1.m1.4.4" xref="S3.Thmtheorem3.p1.1.m1.4.4.cmml">W</mi><mo id="S3.Thmtheorem3.p1.1.m1.4.5.3.3.2.5" stretchy="false" xref="S3.Thmtheorem3.p1.1.m1.4.5.3.3.1.cmml">}</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem3.p1.1.m1.4b"><apply id="S3.Thmtheorem3.p1.1.m1.4.5.cmml" xref="S3.Thmtheorem3.p1.1.m1.4.5"><ci id="S3.Thmtheorem3.p1.1.m1.4.5.1.cmml" xref="S3.Thmtheorem3.p1.1.m1.4.5.1">:</ci><ci id="S3.Thmtheorem3.p1.1.m1.4.5.2.cmml" xref="S3.Thmtheorem3.p1.1.m1.4.5.2">𝑤</ci><apply id="S3.Thmtheorem3.p1.1.m1.4.5.3.cmml" xref="S3.Thmtheorem3.p1.1.m1.4.5.3"><ci id="S3.Thmtheorem3.p1.1.m1.4.5.3.1.cmml" xref="S3.Thmtheorem3.p1.1.m1.4.5.3.1">→</ci><ci id="S3.Thmtheorem3.p1.1.m1.4.5.3.2.cmml" xref="S3.Thmtheorem3.p1.1.m1.4.5.3.2">𝐸</ci><set id="S3.Thmtheorem3.p1.1.m1.4.5.3.3.1.cmml" xref="S3.Thmtheorem3.p1.1.m1.4.5.3.3.2"><cn id="S3.Thmtheorem3.p1.1.m1.1.1.cmml" type="integer" xref="S3.Thmtheorem3.p1.1.m1.1.1">0</cn><cn id="S3.Thmtheorem3.p1.1.m1.2.2.cmml" type="integer" xref="S3.Thmtheorem3.p1.1.m1.2.2">1</cn><ci id="S3.Thmtheorem3.p1.1.m1.3.3.cmml" xref="S3.Thmtheorem3.p1.1.m1.3.3">…</ci><ci id="S3.Thmtheorem3.p1.1.m1.4.4.cmml" xref="S3.Thmtheorem3.p1.1.m1.4.4">𝑊</ci></set></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem3.p1.1.m1.4c">w:E\rightarrow\{0,1,\dots,W\}</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem3.p1.1.m1.4d">italic_w : italic_E → { 0 , 1 , … , italic_W }</annotation></semantics></math> and a maximum connectivity requirement <math alttext="k" class="ltx_Math" display="inline" id="S3.Thmtheorem3.p1.2.m2.1"><semantics id="S3.Thmtheorem3.p1.2.m2.1a"><mi id="S3.Thmtheorem3.p1.2.m2.1.1" xref="S3.Thmtheorem3.p1.2.m2.1.1.cmml">k</mi><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem3.p1.2.m2.1b"><ci id="S3.Thmtheorem3.p1.2.m2.1.1.cmml" xref="S3.Thmtheorem3.p1.2.m2.1.1">𝑘</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem3.p1.2.m2.1c">k</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem3.p1.2.m2.1d">italic_k</annotation></semantics></math>, in insertion-only streams, that uses <math alttext="\tilde{O}(k^{1-1/t}\cdot n^{1+1/t})" class="ltx_Math" display="inline" id="S3.Thmtheorem3.p1.3.m3.1"><semantics id="S3.Thmtheorem3.p1.3.m3.1a"><mrow id="S3.Thmtheorem3.p1.3.m3.1.1" xref="S3.Thmtheorem3.p1.3.m3.1.1.cmml"><mover accent="true" id="S3.Thmtheorem3.p1.3.m3.1.1.3" xref="S3.Thmtheorem3.p1.3.m3.1.1.3.cmml"><mi id="S3.Thmtheorem3.p1.3.m3.1.1.3.2" xref="S3.Thmtheorem3.p1.3.m3.1.1.3.2.cmml">O</mi><mo id="S3.Thmtheorem3.p1.3.m3.1.1.3.1" xref="S3.Thmtheorem3.p1.3.m3.1.1.3.1.cmml">~</mo></mover><mo id="S3.Thmtheorem3.p1.3.m3.1.1.2" xref="S3.Thmtheorem3.p1.3.m3.1.1.2.cmml">⁢</mo><mrow id="S3.Thmtheorem3.p1.3.m3.1.1.1.1" xref="S3.Thmtheorem3.p1.3.m3.1.1.1.1.1.cmml"><mo id="S3.Thmtheorem3.p1.3.m3.1.1.1.1.2" stretchy="false" xref="S3.Thmtheorem3.p1.3.m3.1.1.1.1.1.cmml">(</mo><mrow id="S3.Thmtheorem3.p1.3.m3.1.1.1.1.1" xref="S3.Thmtheorem3.p1.3.m3.1.1.1.1.1.cmml"><msup id="S3.Thmtheorem3.p1.3.m3.1.1.1.1.1.2" xref="S3.Thmtheorem3.p1.3.m3.1.1.1.1.1.2.cmml"><mi id="S3.Thmtheorem3.p1.3.m3.1.1.1.1.1.2.2" xref="S3.Thmtheorem3.p1.3.m3.1.1.1.1.1.2.2.cmml">k</mi><mrow id="S3.Thmtheorem3.p1.3.m3.1.1.1.1.1.2.3" xref="S3.Thmtheorem3.p1.3.m3.1.1.1.1.1.2.3.cmml"><mn id="S3.Thmtheorem3.p1.3.m3.1.1.1.1.1.2.3.2" xref="S3.Thmtheorem3.p1.3.m3.1.1.1.1.1.2.3.2.cmml">1</mn><mo id="S3.Thmtheorem3.p1.3.m3.1.1.1.1.1.2.3.1" xref="S3.Thmtheorem3.p1.3.m3.1.1.1.1.1.2.3.1.cmml">−</mo><mrow id="S3.Thmtheorem3.p1.3.m3.1.1.1.1.1.2.3.3" xref="S3.Thmtheorem3.p1.3.m3.1.1.1.1.1.2.3.3.cmml"><mn id="S3.Thmtheorem3.p1.3.m3.1.1.1.1.1.2.3.3.2" xref="S3.Thmtheorem3.p1.3.m3.1.1.1.1.1.2.3.3.2.cmml">1</mn><mo id="S3.Thmtheorem3.p1.3.m3.1.1.1.1.1.2.3.3.1" xref="S3.Thmtheorem3.p1.3.m3.1.1.1.1.1.2.3.3.1.cmml">/</mo><mi id="S3.Thmtheorem3.p1.3.m3.1.1.1.1.1.2.3.3.3" xref="S3.Thmtheorem3.p1.3.m3.1.1.1.1.1.2.3.3.3.cmml">t</mi></mrow></mrow></msup><mo id="S3.Thmtheorem3.p1.3.m3.1.1.1.1.1.1" lspace="0.222em" rspace="0.222em" xref="S3.Thmtheorem3.p1.3.m3.1.1.1.1.1.1.cmml">⋅</mo><msup id="S3.Thmtheorem3.p1.3.m3.1.1.1.1.1.3" xref="S3.Thmtheorem3.p1.3.m3.1.1.1.1.1.3.cmml"><mi id="S3.Thmtheorem3.p1.3.m3.1.1.1.1.1.3.2" xref="S3.Thmtheorem3.p1.3.m3.1.1.1.1.1.3.2.cmml">n</mi><mrow id="S3.Thmtheorem3.p1.3.m3.1.1.1.1.1.3.3" xref="S3.Thmtheorem3.p1.3.m3.1.1.1.1.1.3.3.cmml"><mn id="S3.Thmtheorem3.p1.3.m3.1.1.1.1.1.3.3.2" xref="S3.Thmtheorem3.p1.3.m3.1.1.1.1.1.3.3.2.cmml">1</mn><mo id="S3.Thmtheorem3.p1.3.m3.1.1.1.1.1.3.3.1" xref="S3.Thmtheorem3.p1.3.m3.1.1.1.1.1.3.3.1.cmml">+</mo><mrow id="S3.Thmtheorem3.p1.3.m3.1.1.1.1.1.3.3.3" xref="S3.Thmtheorem3.p1.3.m3.1.1.1.1.1.3.3.3.cmml"><mn id="S3.Thmtheorem3.p1.3.m3.1.1.1.1.1.3.3.3.2" xref="S3.Thmtheorem3.p1.3.m3.1.1.1.1.1.3.3.3.2.cmml">1</mn><mo id="S3.Thmtheorem3.p1.3.m3.1.1.1.1.1.3.3.3.1" xref="S3.Thmtheorem3.p1.3.m3.1.1.1.1.1.3.3.3.1.cmml">/</mo><mi id="S3.Thmtheorem3.p1.3.m3.1.1.1.1.1.3.3.3.3" xref="S3.Thmtheorem3.p1.3.m3.1.1.1.1.1.3.3.3.3.cmml">t</mi></mrow></mrow></msup></mrow><mo id="S3.Thmtheorem3.p1.3.m3.1.1.1.1.3" stretchy="false" xref="S3.Thmtheorem3.p1.3.m3.1.1.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem3.p1.3.m3.1b"><apply id="S3.Thmtheorem3.p1.3.m3.1.1.cmml" xref="S3.Thmtheorem3.p1.3.m3.1.1"><times id="S3.Thmtheorem3.p1.3.m3.1.1.2.cmml" xref="S3.Thmtheorem3.p1.3.m3.1.1.2"></times><apply id="S3.Thmtheorem3.p1.3.m3.1.1.3.cmml" xref="S3.Thmtheorem3.p1.3.m3.1.1.3"><ci id="S3.Thmtheorem3.p1.3.m3.1.1.3.1.cmml" xref="S3.Thmtheorem3.p1.3.m3.1.1.3.1">~</ci><ci id="S3.Thmtheorem3.p1.3.m3.1.1.3.2.cmml" xref="S3.Thmtheorem3.p1.3.m3.1.1.3.2">𝑂</ci></apply><apply id="S3.Thmtheorem3.p1.3.m3.1.1.1.1.1.cmml" xref="S3.Thmtheorem3.p1.3.m3.1.1.1.1"><ci id="S3.Thmtheorem3.p1.3.m3.1.1.1.1.1.1.cmml" xref="S3.Thmtheorem3.p1.3.m3.1.1.1.1.1.1">⋅</ci><apply id="S3.Thmtheorem3.p1.3.m3.1.1.1.1.1.2.cmml" xref="S3.Thmtheorem3.p1.3.m3.1.1.1.1.1.2"><csymbol cd="ambiguous" id="S3.Thmtheorem3.p1.3.m3.1.1.1.1.1.2.1.cmml" xref="S3.Thmtheorem3.p1.3.m3.1.1.1.1.1.2">superscript</csymbol><ci id="S3.Thmtheorem3.p1.3.m3.1.1.1.1.1.2.2.cmml" xref="S3.Thmtheorem3.p1.3.m3.1.1.1.1.1.2.2">𝑘</ci><apply id="S3.Thmtheorem3.p1.3.m3.1.1.1.1.1.2.3.cmml" xref="S3.Thmtheorem3.p1.3.m3.1.1.1.1.1.2.3"><minus id="S3.Thmtheorem3.p1.3.m3.1.1.1.1.1.2.3.1.cmml" xref="S3.Thmtheorem3.p1.3.m3.1.1.1.1.1.2.3.1"></minus><cn id="S3.Thmtheorem3.p1.3.m3.1.1.1.1.1.2.3.2.cmml" type="integer" xref="S3.Thmtheorem3.p1.3.m3.1.1.1.1.1.2.3.2">1</cn><apply id="S3.Thmtheorem3.p1.3.m3.1.1.1.1.1.2.3.3.cmml" xref="S3.Thmtheorem3.p1.3.m3.1.1.1.1.1.2.3.3"><divide id="S3.Thmtheorem3.p1.3.m3.1.1.1.1.1.2.3.3.1.cmml" xref="S3.Thmtheorem3.p1.3.m3.1.1.1.1.1.2.3.3.1"></divide><cn id="S3.Thmtheorem3.p1.3.m3.1.1.1.1.1.2.3.3.2.cmml" type="integer" xref="S3.Thmtheorem3.p1.3.m3.1.1.1.1.1.2.3.3.2">1</cn><ci id="S3.Thmtheorem3.p1.3.m3.1.1.1.1.1.2.3.3.3.cmml" xref="S3.Thmtheorem3.p1.3.m3.1.1.1.1.1.2.3.3.3">𝑡</ci></apply></apply></apply><apply id="S3.Thmtheorem3.p1.3.m3.1.1.1.1.1.3.cmml" xref="S3.Thmtheorem3.p1.3.m3.1.1.1.1.1.3"><csymbol cd="ambiguous" id="S3.Thmtheorem3.p1.3.m3.1.1.1.1.1.3.1.cmml" xref="S3.Thmtheorem3.p1.3.m3.1.1.1.1.1.3">superscript</csymbol><ci id="S3.Thmtheorem3.p1.3.m3.1.1.1.1.1.3.2.cmml" xref="S3.Thmtheorem3.p1.3.m3.1.1.1.1.1.3.2">𝑛</ci><apply id="S3.Thmtheorem3.p1.3.m3.1.1.1.1.1.3.3.cmml" xref="S3.Thmtheorem3.p1.3.m3.1.1.1.1.1.3.3"><plus id="S3.Thmtheorem3.p1.3.m3.1.1.1.1.1.3.3.1.cmml" xref="S3.Thmtheorem3.p1.3.m3.1.1.1.1.1.3.3.1"></plus><cn id="S3.Thmtheorem3.p1.3.m3.1.1.1.1.1.3.3.2.cmml" type="integer" xref="S3.Thmtheorem3.p1.3.m3.1.1.1.1.1.3.3.2">1</cn><apply id="S3.Thmtheorem3.p1.3.m3.1.1.1.1.1.3.3.3.cmml" xref="S3.Thmtheorem3.p1.3.m3.1.1.1.1.1.3.3.3"><divide id="S3.Thmtheorem3.p1.3.m3.1.1.1.1.1.3.3.3.1.cmml" xref="S3.Thmtheorem3.p1.3.m3.1.1.1.1.1.3.3.3.1"></divide><cn id="S3.Thmtheorem3.p1.3.m3.1.1.1.1.1.3.3.3.2.cmml" type="integer" xref="S3.Thmtheorem3.p1.3.m3.1.1.1.1.1.3.3.3.2">1</cn><ci id="S3.Thmtheorem3.p1.3.m3.1.1.1.1.1.3.3.3.3.cmml" xref="S3.Thmtheorem3.p1.3.m3.1.1.1.1.1.3.3.3.3">𝑡</ci></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem3.p1.3.m3.1c">\tilde{O}(k^{1-1/t}\cdot n^{1+1/t})</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem3.p1.3.m3.1d">over~ start_ARG italic_O end_ARG ( italic_k start_POSTSUPERSCRIPT 1 - 1 / italic_t end_POSTSUPERSCRIPT ⋅ italic_n start_POSTSUPERSCRIPT 1 + 1 / italic_t end_POSTSUPERSCRIPT )</annotation></semantics></math> space and outputs a <math alttext="(2tk)" class="ltx_Math" display="inline" id="S3.Thmtheorem3.p1.4.m4.1"><semantics id="S3.Thmtheorem3.p1.4.m4.1a"><mrow id="S3.Thmtheorem3.p1.4.m4.1.1.1" xref="S3.Thmtheorem3.p1.4.m4.1.1.1.1.cmml"><mo id="S3.Thmtheorem3.p1.4.m4.1.1.1.2" stretchy="false" xref="S3.Thmtheorem3.p1.4.m4.1.1.1.1.cmml">(</mo><mrow id="S3.Thmtheorem3.p1.4.m4.1.1.1.1" xref="S3.Thmtheorem3.p1.4.m4.1.1.1.1.cmml"><mn id="S3.Thmtheorem3.p1.4.m4.1.1.1.1.2" xref="S3.Thmtheorem3.p1.4.m4.1.1.1.1.2.cmml">2</mn><mo id="S3.Thmtheorem3.p1.4.m4.1.1.1.1.1" xref="S3.Thmtheorem3.p1.4.m4.1.1.1.1.1.cmml">⁢</mo><mi id="S3.Thmtheorem3.p1.4.m4.1.1.1.1.3" xref="S3.Thmtheorem3.p1.4.m4.1.1.1.1.3.cmml">t</mi><mo id="S3.Thmtheorem3.p1.4.m4.1.1.1.1.1a" xref="S3.Thmtheorem3.p1.4.m4.1.1.1.1.1.cmml">⁢</mo><mi id="S3.Thmtheorem3.p1.4.m4.1.1.1.1.4" xref="S3.Thmtheorem3.p1.4.m4.1.1.1.1.4.cmml">k</mi></mrow><mo id="S3.Thmtheorem3.p1.4.m4.1.1.1.3" stretchy="false" xref="S3.Thmtheorem3.p1.4.m4.1.1.1.1.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem3.p1.4.m4.1b"><apply id="S3.Thmtheorem3.p1.4.m4.1.1.1.1.cmml" xref="S3.Thmtheorem3.p1.4.m4.1.1.1"><times id="S3.Thmtheorem3.p1.4.m4.1.1.1.1.1.cmml" xref="S3.Thmtheorem3.p1.4.m4.1.1.1.1.1"></times><cn id="S3.Thmtheorem3.p1.4.m4.1.1.1.1.2.cmml" type="integer" xref="S3.Thmtheorem3.p1.4.m4.1.1.1.1.2">2</cn><ci id="S3.Thmtheorem3.p1.4.m4.1.1.1.1.3.cmml" xref="S3.Thmtheorem3.p1.4.m4.1.1.1.1.3">𝑡</ci><ci id="S3.Thmtheorem3.p1.4.m4.1.1.1.1.4.cmml" xref="S3.Thmtheorem3.p1.4.m4.1.1.1.1.4">𝑘</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem3.p1.4.m4.1c">(2tk)</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem3.p1.4.m4.1d">( 2 italic_t italic_k )</annotation></semantics></math>-approximate solution.</p> </div> </div> <div class="ltx_proof" id="S3.SS1.SSS1.4"> <h6 class="ltx_title ltx_runin ltx_font_italic ltx_title_proof">Proof.</h6> <div class="ltx_para" id="S3.SS1.SSS1.4.p1"> <p class="ltx_p" id="S3.SS1.SSS1.4.p1.10">Theorem <a class="ltx_ref" href="https://arxiv.org/html/2503.00712v1#S3.Thmtheorem2" title="Theorem 3.2. ‣ 3.1.1 A Simple Analysis Based on Integral Solutions ‣ 3.1 Vertex Connectivity Network Design ‣ 3 Generic Framework for Streaming Algorithms for Network Design ‣ Streaming Algorithms for Network Design"><span class="ltx_text ltx_ref_tag">3.2</span></a> shows that Algorithm <a class="ltx_ref" href="https://arxiv.org/html/2503.00712v1#algorithm3" title="In 3 Generic Framework for Streaming Algorithms for Network Design ‣ Streaming Algorithms for Network Design"><span class="ltx_text ltx_ref_tag">3</span></a> with <math alttext="(t,f=(2t-2)(k-1),\epsilon=1/(2t-1))" class="ltx_Math" display="inline" id="S3.SS1.SSS1.4.p1.1.m1.3"><semantics id="S3.SS1.SSS1.4.p1.1.m1.3a"><mrow id="S3.SS1.SSS1.4.p1.1.m1.3.3.1"><mo id="S3.SS1.SSS1.4.p1.1.m1.3.3.1.2" stretchy="false">(</mo><mrow id="S3.SS1.SSS1.4.p1.1.m1.3.3.1.1.2" xref="S3.SS1.SSS1.4.p1.1.m1.3.3.1.1.3.cmml"><mrow 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id="S3.SS1.SSS1.4.p1.3.m3.1.1.1.1.1.2.3.2.cmml" type="integer" xref="S3.SS1.SSS1.4.p1.3.m3.1.1.1.1.1.2.3.2">1</cn><apply id="S3.SS1.SSS1.4.p1.3.m3.1.1.1.1.1.2.3.3.cmml" xref="S3.SS1.SSS1.4.p1.3.m3.1.1.1.1.1.2.3.3"><divide id="S3.SS1.SSS1.4.p1.3.m3.1.1.1.1.1.2.3.3.1.cmml" xref="S3.SS1.SSS1.4.p1.3.m3.1.1.1.1.1.2.3.3.1"></divide><cn id="S3.SS1.SSS1.4.p1.3.m3.1.1.1.1.1.2.3.3.2.cmml" type="integer" xref="S3.SS1.SSS1.4.p1.3.m3.1.1.1.1.1.2.3.3.2">1</cn><ci id="S3.SS1.SSS1.4.p1.3.m3.1.1.1.1.1.2.3.3.3.cmml" xref="S3.SS1.SSS1.4.p1.3.m3.1.1.1.1.1.2.3.3.3">𝑡</ci></apply></apply></apply><apply id="S3.SS1.SSS1.4.p1.3.m3.1.1.1.1.1.3.cmml" xref="S3.SS1.SSS1.4.p1.3.m3.1.1.1.1.1.3"><csymbol cd="ambiguous" id="S3.SS1.SSS1.4.p1.3.m3.1.1.1.1.1.3.1.cmml" xref="S3.SS1.SSS1.4.p1.3.m3.1.1.1.1.1.3">superscript</csymbol><ci id="S3.SS1.SSS1.4.p1.3.m3.1.1.1.1.1.3.2.cmml" xref="S3.SS1.SSS1.4.p1.3.m3.1.1.1.1.1.3.2">𝑛</ci><apply id="S3.SS1.SSS1.4.p1.3.m3.1.1.1.1.1.3.3.cmml" xref="S3.SS1.SSS1.4.p1.3.m3.1.1.1.1.1.3.3"><plus id="S3.SS1.SSS1.4.p1.3.m3.1.1.1.1.1.3.3.1.cmml" xref="S3.SS1.SSS1.4.p1.3.m3.1.1.1.1.1.3.3.1"></plus><cn id="S3.SS1.SSS1.4.p1.3.m3.1.1.1.1.1.3.3.2.cmml" type="integer" xref="S3.SS1.SSS1.4.p1.3.m3.1.1.1.1.1.3.3.2">1</cn><apply id="S3.SS1.SSS1.4.p1.3.m3.1.1.1.1.1.3.3.3.cmml" xref="S3.SS1.SSS1.4.p1.3.m3.1.1.1.1.1.3.3.3"><divide id="S3.SS1.SSS1.4.p1.3.m3.1.1.1.1.1.3.3.3.1.cmml" xref="S3.SS1.SSS1.4.p1.3.m3.1.1.1.1.1.3.3.3.1"></divide><cn id="S3.SS1.SSS1.4.p1.3.m3.1.1.1.1.1.3.3.3.2.cmml" type="integer" xref="S3.SS1.SSS1.4.p1.3.m3.1.1.1.1.1.3.3.3.2">1</cn><ci id="S3.SS1.SSS1.4.p1.3.m3.1.1.1.1.1.3.3.3.3.cmml" xref="S3.SS1.SSS1.4.p1.3.m3.1.1.1.1.1.3.3.3.3">𝑡</ci></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS1.4.p1.3.m3.1c">\tilde{O}(k^{1-1/t}\cdot n^{1+1/t})</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS1.4.p1.3.m3.1d">over~ start_ARG italic_O end_ARG ( italic_k start_POSTSUPERSCRIPT 1 - 1 / italic_t end_POSTSUPERSCRIPT ⋅ italic_n start_POSTSUPERSCRIPT 1 + 1 / italic_t end_POSTSUPERSCRIPT )</annotation></semantics></math> that contains a <math alttext="(2tk)" class="ltx_Math" display="inline" id="S3.SS1.SSS1.4.p1.4.m4.1"><semantics id="S3.SS1.SSS1.4.p1.4.m4.1a"><mrow id="S3.SS1.SSS1.4.p1.4.m4.1.1.1" xref="S3.SS1.SSS1.4.p1.4.m4.1.1.1.1.cmml"><mo id="S3.SS1.SSS1.4.p1.4.m4.1.1.1.2" stretchy="false" xref="S3.SS1.SSS1.4.p1.4.m4.1.1.1.1.cmml">(</mo><mrow id="S3.SS1.SSS1.4.p1.4.m4.1.1.1.1" xref="S3.SS1.SSS1.4.p1.4.m4.1.1.1.1.cmml"><mn id="S3.SS1.SSS1.4.p1.4.m4.1.1.1.1.2" xref="S3.SS1.SSS1.4.p1.4.m4.1.1.1.1.2.cmml">2</mn><mo id="S3.SS1.SSS1.4.p1.4.m4.1.1.1.1.1" xref="S3.SS1.SSS1.4.p1.4.m4.1.1.1.1.1.cmml">⁢</mo><mi id="S3.SS1.SSS1.4.p1.4.m4.1.1.1.1.3" xref="S3.SS1.SSS1.4.p1.4.m4.1.1.1.1.3.cmml">t</mi><mo id="S3.SS1.SSS1.4.p1.4.m4.1.1.1.1.1a" xref="S3.SS1.SSS1.4.p1.4.m4.1.1.1.1.1.cmml">⁢</mo><mi id="S3.SS1.SSS1.4.p1.4.m4.1.1.1.1.4" xref="S3.SS1.SSS1.4.p1.4.m4.1.1.1.1.4.cmml">k</mi></mrow><mo id="S3.SS1.SSS1.4.p1.4.m4.1.1.1.3" stretchy="false" xref="S3.SS1.SSS1.4.p1.4.m4.1.1.1.1.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS1.4.p1.4.m4.1b"><apply id="S3.SS1.SSS1.4.p1.4.m4.1.1.1.1.cmml" xref="S3.SS1.SSS1.4.p1.4.m4.1.1.1"><times id="S3.SS1.SSS1.4.p1.4.m4.1.1.1.1.1.cmml" xref="S3.SS1.SSS1.4.p1.4.m4.1.1.1.1.1"></times><cn id="S3.SS1.SSS1.4.p1.4.m4.1.1.1.1.2.cmml" type="integer" xref="S3.SS1.SSS1.4.p1.4.m4.1.1.1.1.2">2</cn><ci id="S3.SS1.SSS1.4.p1.4.m4.1.1.1.1.3.cmml" xref="S3.SS1.SSS1.4.p1.4.m4.1.1.1.1.3">𝑡</ci><ci id="S3.SS1.SSS1.4.p1.4.m4.1.1.1.1.4.cmml" xref="S3.SS1.SSS1.4.p1.4.m4.1.1.1.1.4">𝑘</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS1.4.p1.4.m4.1c">(2tk)</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS1.4.p1.4.m4.1d">( 2 italic_t italic_k )</annotation></semantics></math>-approximate solution for VC-SNDP on <math alttext="G" class="ltx_Math" display="inline" id="S3.SS1.SSS1.4.p1.5.m5.1"><semantics id="S3.SS1.SSS1.4.p1.5.m5.1a"><mi id="S3.SS1.SSS1.4.p1.5.m5.1.1" xref="S3.SS1.SSS1.4.p1.5.m5.1.1.cmml">G</mi><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS1.4.p1.5.m5.1b"><ci id="S3.SS1.SSS1.4.p1.5.m5.1.1.cmml" xref="S3.SS1.SSS1.4.p1.5.m5.1.1">𝐺</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS1.4.p1.5.m5.1c">G</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS1.4.p1.5.m5.1d">italic_G</annotation></semantics></math>. Then, once the stream terminates, we perform an exhaustive search, i.e., enumerating all possible solutions on <math alttext="H" class="ltx_Math" display="inline" id="S3.SS1.SSS1.4.p1.6.m6.1"><semantics id="S3.SS1.SSS1.4.p1.6.m6.1a"><mi id="S3.SS1.SSS1.4.p1.6.m6.1.1" xref="S3.SS1.SSS1.4.p1.6.m6.1.1.cmml">H</mi><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS1.4.p1.6.m6.1b"><ci id="S3.SS1.SSS1.4.p1.6.m6.1.1.cmml" xref="S3.SS1.SSS1.4.p1.6.m6.1.1">𝐻</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS1.4.p1.6.m6.1c">H</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS1.4.p1.6.m6.1d">italic_H</annotation></semantics></math> and output a <math alttext="(2tk)" class="ltx_Math" display="inline" id="S3.SS1.SSS1.4.p1.7.m7.1"><semantics id="S3.SS1.SSS1.4.p1.7.m7.1a"><mrow id="S3.SS1.SSS1.4.p1.7.m7.1.1.1" xref="S3.SS1.SSS1.4.p1.7.m7.1.1.1.1.cmml"><mo id="S3.SS1.SSS1.4.p1.7.m7.1.1.1.2" stretchy="false" xref="S3.SS1.SSS1.4.p1.7.m7.1.1.1.1.cmml">(</mo><mrow id="S3.SS1.SSS1.4.p1.7.m7.1.1.1.1" xref="S3.SS1.SSS1.4.p1.7.m7.1.1.1.1.cmml"><mn id="S3.SS1.SSS1.4.p1.7.m7.1.1.1.1.2" xref="S3.SS1.SSS1.4.p1.7.m7.1.1.1.1.2.cmml">2</mn><mo id="S3.SS1.SSS1.4.p1.7.m7.1.1.1.1.1" xref="S3.SS1.SSS1.4.p1.7.m7.1.1.1.1.1.cmml">⁢</mo><mi id="S3.SS1.SSS1.4.p1.7.m7.1.1.1.1.3" xref="S3.SS1.SSS1.4.p1.7.m7.1.1.1.1.3.cmml">t</mi><mo id="S3.SS1.SSS1.4.p1.7.m7.1.1.1.1.1a" xref="S3.SS1.SSS1.4.p1.7.m7.1.1.1.1.1.cmml">⁢</mo><mi id="S3.SS1.SSS1.4.p1.7.m7.1.1.1.1.4" xref="S3.SS1.SSS1.4.p1.7.m7.1.1.1.1.4.cmml">k</mi></mrow><mo id="S3.SS1.SSS1.4.p1.7.m7.1.1.1.3" stretchy="false" xref="S3.SS1.SSS1.4.p1.7.m7.1.1.1.1.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS1.4.p1.7.m7.1b"><apply id="S3.SS1.SSS1.4.p1.7.m7.1.1.1.1.cmml" xref="S3.SS1.SSS1.4.p1.7.m7.1.1.1"><times id="S3.SS1.SSS1.4.p1.7.m7.1.1.1.1.1.cmml" xref="S3.SS1.SSS1.4.p1.7.m7.1.1.1.1.1"></times><cn id="S3.SS1.SSS1.4.p1.7.m7.1.1.1.1.2.cmml" type="integer" xref="S3.SS1.SSS1.4.p1.7.m7.1.1.1.1.2">2</cn><ci id="S3.SS1.SSS1.4.p1.7.m7.1.1.1.1.3.cmml" xref="S3.SS1.SSS1.4.p1.7.m7.1.1.1.1.3">𝑡</ci><ci id="S3.SS1.SSS1.4.p1.7.m7.1.1.1.1.4.cmml" xref="S3.SS1.SSS1.4.p1.7.m7.1.1.1.1.4">𝑘</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS1.4.p1.7.m7.1c">(2tk)</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS1.4.p1.7.m7.1d">( 2 italic_t italic_k )</annotation></semantics></math>-approximate solution of <math alttext="G" class="ltx_Math" display="inline" id="S3.SS1.SSS1.4.p1.8.m8.1"><semantics id="S3.SS1.SSS1.4.p1.8.m8.1a"><mi id="S3.SS1.SSS1.4.p1.8.m8.1.1" xref="S3.SS1.SSS1.4.p1.8.m8.1.1.cmml">G</mi><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS1.4.p1.8.m8.1b"><ci id="S3.SS1.SSS1.4.p1.8.m8.1.1.cmml" xref="S3.SS1.SSS1.4.p1.8.m8.1.1">𝐺</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS1.4.p1.8.m8.1c">G</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS1.4.p1.8.m8.1d">italic_G</annotation></semantics></math>, supported on <math alttext="H" class="ltx_Math" display="inline" id="S3.SS1.SSS1.4.p1.9.m9.1"><semantics id="S3.SS1.SSS1.4.p1.9.m9.1a"><mi id="S3.SS1.SSS1.4.p1.9.m9.1.1" xref="S3.SS1.SSS1.4.p1.9.m9.1.1.cmml">H</mi><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS1.4.p1.9.m9.1b"><ci id="S3.SS1.SSS1.4.p1.9.m9.1.1.cmml" xref="S3.SS1.SSS1.4.p1.9.m9.1.1">𝐻</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS1.4.p1.9.m9.1c">H</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS1.4.p1.9.m9.1d">italic_H</annotation></semantics></math> only. Since the algorithm’s only space consumption is for storing the constructed VFT spanner, it requires <math alttext="\tilde{O}\big{(}k^{1-1/t}\cdot n^{1+1/t}\big{)}" class="ltx_Math" display="inline" id="S3.SS1.SSS1.4.p1.10.m10.1"><semantics id="S3.SS1.SSS1.4.p1.10.m10.1a"><mrow id="S3.SS1.SSS1.4.p1.10.m10.1.1" xref="S3.SS1.SSS1.4.p1.10.m10.1.1.cmml"><mover accent="true" id="S3.SS1.SSS1.4.p1.10.m10.1.1.3" xref="S3.SS1.SSS1.4.p1.10.m10.1.1.3.cmml"><mi id="S3.SS1.SSS1.4.p1.10.m10.1.1.3.2" xref="S3.SS1.SSS1.4.p1.10.m10.1.1.3.2.cmml">O</mi><mo id="S3.SS1.SSS1.4.p1.10.m10.1.1.3.1" xref="S3.SS1.SSS1.4.p1.10.m10.1.1.3.1.cmml">~</mo></mover><mo id="S3.SS1.SSS1.4.p1.10.m10.1.1.2" xref="S3.SS1.SSS1.4.p1.10.m10.1.1.2.cmml">⁢</mo><mrow id="S3.SS1.SSS1.4.p1.10.m10.1.1.1.1" xref="S3.SS1.SSS1.4.p1.10.m10.1.1.1.1.1.cmml"><mo id="S3.SS1.SSS1.4.p1.10.m10.1.1.1.1.2" maxsize="120%" minsize="120%" xref="S3.SS1.SSS1.4.p1.10.m10.1.1.1.1.1.cmml">(</mo><mrow id="S3.SS1.SSS1.4.p1.10.m10.1.1.1.1.1" xref="S3.SS1.SSS1.4.p1.10.m10.1.1.1.1.1.cmml"><msup id="S3.SS1.SSS1.4.p1.10.m10.1.1.1.1.1.2" xref="S3.SS1.SSS1.4.p1.10.m10.1.1.1.1.1.2.cmml"><mi id="S3.SS1.SSS1.4.p1.10.m10.1.1.1.1.1.2.2" xref="S3.SS1.SSS1.4.p1.10.m10.1.1.1.1.1.2.2.cmml">k</mi><mrow id="S3.SS1.SSS1.4.p1.10.m10.1.1.1.1.1.2.3" xref="S3.SS1.SSS1.4.p1.10.m10.1.1.1.1.1.2.3.cmml"><mn id="S3.SS1.SSS1.4.p1.10.m10.1.1.1.1.1.2.3.2" xref="S3.SS1.SSS1.4.p1.10.m10.1.1.1.1.1.2.3.2.cmml">1</mn><mo id="S3.SS1.SSS1.4.p1.10.m10.1.1.1.1.1.2.3.1" xref="S3.SS1.SSS1.4.p1.10.m10.1.1.1.1.1.2.3.1.cmml">−</mo><mrow id="S3.SS1.SSS1.4.p1.10.m10.1.1.1.1.1.2.3.3" xref="S3.SS1.SSS1.4.p1.10.m10.1.1.1.1.1.2.3.3.cmml"><mn id="S3.SS1.SSS1.4.p1.10.m10.1.1.1.1.1.2.3.3.2" xref="S3.SS1.SSS1.4.p1.10.m10.1.1.1.1.1.2.3.3.2.cmml">1</mn><mo id="S3.SS1.SSS1.4.p1.10.m10.1.1.1.1.1.2.3.3.1" xref="S3.SS1.SSS1.4.p1.10.m10.1.1.1.1.1.2.3.3.1.cmml">/</mo><mi id="S3.SS1.SSS1.4.p1.10.m10.1.1.1.1.1.2.3.3.3" xref="S3.SS1.SSS1.4.p1.10.m10.1.1.1.1.1.2.3.3.3.cmml">t</mi></mrow></mrow></msup><mo id="S3.SS1.SSS1.4.p1.10.m10.1.1.1.1.1.1" lspace="0.222em" rspace="0.222em" xref="S3.SS1.SSS1.4.p1.10.m10.1.1.1.1.1.1.cmml">⋅</mo><msup id="S3.SS1.SSS1.4.p1.10.m10.1.1.1.1.1.3" xref="S3.SS1.SSS1.4.p1.10.m10.1.1.1.1.1.3.cmml"><mi id="S3.SS1.SSS1.4.p1.10.m10.1.1.1.1.1.3.2" xref="S3.SS1.SSS1.4.p1.10.m10.1.1.1.1.1.3.2.cmml">n</mi><mrow id="S3.SS1.SSS1.4.p1.10.m10.1.1.1.1.1.3.3" xref="S3.SS1.SSS1.4.p1.10.m10.1.1.1.1.1.3.3.cmml"><mn id="S3.SS1.SSS1.4.p1.10.m10.1.1.1.1.1.3.3.2" xref="S3.SS1.SSS1.4.p1.10.m10.1.1.1.1.1.3.3.2.cmml">1</mn><mo id="S3.SS1.SSS1.4.p1.10.m10.1.1.1.1.1.3.3.1" xref="S3.SS1.SSS1.4.p1.10.m10.1.1.1.1.1.3.3.1.cmml">+</mo><mrow id="S3.SS1.SSS1.4.p1.10.m10.1.1.1.1.1.3.3.3" xref="S3.SS1.SSS1.4.p1.10.m10.1.1.1.1.1.3.3.3.cmml"><mn id="S3.SS1.SSS1.4.p1.10.m10.1.1.1.1.1.3.3.3.2" xref="S3.SS1.SSS1.4.p1.10.m10.1.1.1.1.1.3.3.3.2.cmml">1</mn><mo id="S3.SS1.SSS1.4.p1.10.m10.1.1.1.1.1.3.3.3.1" xref="S3.SS1.SSS1.4.p1.10.m10.1.1.1.1.1.3.3.3.1.cmml">/</mo><mi id="S3.SS1.SSS1.4.p1.10.m10.1.1.1.1.1.3.3.3.3" xref="S3.SS1.SSS1.4.p1.10.m10.1.1.1.1.1.3.3.3.3.cmml">t</mi></mrow></mrow></msup></mrow><mo id="S3.SS1.SSS1.4.p1.10.m10.1.1.1.1.3" maxsize="120%" minsize="120%" xref="S3.SS1.SSS1.4.p1.10.m10.1.1.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS1.4.p1.10.m10.1b"><apply id="S3.SS1.SSS1.4.p1.10.m10.1.1.cmml" xref="S3.SS1.SSS1.4.p1.10.m10.1.1"><times id="S3.SS1.SSS1.4.p1.10.m10.1.1.2.cmml" xref="S3.SS1.SSS1.4.p1.10.m10.1.1.2"></times><apply id="S3.SS1.SSS1.4.p1.10.m10.1.1.3.cmml" xref="S3.SS1.SSS1.4.p1.10.m10.1.1.3"><ci id="S3.SS1.SSS1.4.p1.10.m10.1.1.3.1.cmml" xref="S3.SS1.SSS1.4.p1.10.m10.1.1.3.1">~</ci><ci id="S3.SS1.SSS1.4.p1.10.m10.1.1.3.2.cmml" xref="S3.SS1.SSS1.4.p1.10.m10.1.1.3.2">𝑂</ci></apply><apply id="S3.SS1.SSS1.4.p1.10.m10.1.1.1.1.1.cmml" xref="S3.SS1.SSS1.4.p1.10.m10.1.1.1.1"><ci id="S3.SS1.SSS1.4.p1.10.m10.1.1.1.1.1.1.cmml" xref="S3.SS1.SSS1.4.p1.10.m10.1.1.1.1.1.1">⋅</ci><apply id="S3.SS1.SSS1.4.p1.10.m10.1.1.1.1.1.2.cmml" xref="S3.SS1.SSS1.4.p1.10.m10.1.1.1.1.1.2"><csymbol cd="ambiguous" id="S3.SS1.SSS1.4.p1.10.m10.1.1.1.1.1.2.1.cmml" xref="S3.SS1.SSS1.4.p1.10.m10.1.1.1.1.1.2">superscript</csymbol><ci id="S3.SS1.SSS1.4.p1.10.m10.1.1.1.1.1.2.2.cmml" xref="S3.SS1.SSS1.4.p1.10.m10.1.1.1.1.1.2.2">𝑘</ci><apply id="S3.SS1.SSS1.4.p1.10.m10.1.1.1.1.1.2.3.cmml" xref="S3.SS1.SSS1.4.p1.10.m10.1.1.1.1.1.2.3"><minus id="S3.SS1.SSS1.4.p1.10.m10.1.1.1.1.1.2.3.1.cmml" xref="S3.SS1.SSS1.4.p1.10.m10.1.1.1.1.1.2.3.1"></minus><cn id="S3.SS1.SSS1.4.p1.10.m10.1.1.1.1.1.2.3.2.cmml" type="integer" xref="S3.SS1.SSS1.4.p1.10.m10.1.1.1.1.1.2.3.2">1</cn><apply id="S3.SS1.SSS1.4.p1.10.m10.1.1.1.1.1.2.3.3.cmml" xref="S3.SS1.SSS1.4.p1.10.m10.1.1.1.1.1.2.3.3"><divide id="S3.SS1.SSS1.4.p1.10.m10.1.1.1.1.1.2.3.3.1.cmml" xref="S3.SS1.SSS1.4.p1.10.m10.1.1.1.1.1.2.3.3.1"></divide><cn id="S3.SS1.SSS1.4.p1.10.m10.1.1.1.1.1.2.3.3.2.cmml" type="integer" xref="S3.SS1.SSS1.4.p1.10.m10.1.1.1.1.1.2.3.3.2">1</cn><ci id="S3.SS1.SSS1.4.p1.10.m10.1.1.1.1.1.2.3.3.3.cmml" xref="S3.SS1.SSS1.4.p1.10.m10.1.1.1.1.1.2.3.3.3">𝑡</ci></apply></apply></apply><apply id="S3.SS1.SSS1.4.p1.10.m10.1.1.1.1.1.3.cmml" xref="S3.SS1.SSS1.4.p1.10.m10.1.1.1.1.1.3"><csymbol cd="ambiguous" id="S3.SS1.SSS1.4.p1.10.m10.1.1.1.1.1.3.1.cmml" xref="S3.SS1.SSS1.4.p1.10.m10.1.1.1.1.1.3">superscript</csymbol><ci id="S3.SS1.SSS1.4.p1.10.m10.1.1.1.1.1.3.2.cmml" xref="S3.SS1.SSS1.4.p1.10.m10.1.1.1.1.1.3.2">𝑛</ci><apply id="S3.SS1.SSS1.4.p1.10.m10.1.1.1.1.1.3.3.cmml" xref="S3.SS1.SSS1.4.p1.10.m10.1.1.1.1.1.3.3"><plus id="S3.SS1.SSS1.4.p1.10.m10.1.1.1.1.1.3.3.1.cmml" xref="S3.SS1.SSS1.4.p1.10.m10.1.1.1.1.1.3.3.1"></plus><cn id="S3.SS1.SSS1.4.p1.10.m10.1.1.1.1.1.3.3.2.cmml" type="integer" xref="S3.SS1.SSS1.4.p1.10.m10.1.1.1.1.1.3.3.2">1</cn><apply id="S3.SS1.SSS1.4.p1.10.m10.1.1.1.1.1.3.3.3.cmml" xref="S3.SS1.SSS1.4.p1.10.m10.1.1.1.1.1.3.3.3"><divide id="S3.SS1.SSS1.4.p1.10.m10.1.1.1.1.1.3.3.3.1.cmml" xref="S3.SS1.SSS1.4.p1.10.m10.1.1.1.1.1.3.3.3.1"></divide><cn id="S3.SS1.SSS1.4.p1.10.m10.1.1.1.1.1.3.3.3.2.cmml" type="integer" xref="S3.SS1.SSS1.4.p1.10.m10.1.1.1.1.1.3.3.3.2">1</cn><ci id="S3.SS1.SSS1.4.p1.10.m10.1.1.1.1.1.3.3.3.3.cmml" xref="S3.SS1.SSS1.4.p1.10.m10.1.1.1.1.1.3.3.3.3">𝑡</ci></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS1.4.p1.10.m10.1c">\tilde{O}\big{(}k^{1-1/t}\cdot n^{1+1/t}\big{)}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS1.4.p1.10.m10.1d">over~ start_ARG italic_O end_ARG ( italic_k start_POSTSUPERSCRIPT 1 - 1 / italic_t end_POSTSUPERSCRIPT ⋅ italic_n start_POSTSUPERSCRIPT 1 + 1 / italic_t end_POSTSUPERSCRIPT )</annotation></semantics></math> space. ∎</p> </div> </div> <div class="ltx_para" id="S3.SS1.SSS1.p1"> <p class="ltx_p" id="S3.SS1.SSS1.p1.2">Note that if we restrict the streaming algorithm to be computationally efficient, the best achievable approximation factor using the framework of Algorithm <a class="ltx_ref" href="https://arxiv.org/html/2503.00712v1#algorithm3" title="In 3 Generic Framework for Streaming Algorithms for Network Design ‣ Streaming Algorithms for Network Design"><span class="ltx_text ltx_ref_tag">3</span></a>, is <math alttext="O(t\cdot k^{4}\log n)" class="ltx_Math" display="inline" id="S3.SS1.SSS1.p1.1.m1.1"><semantics id="S3.SS1.SSS1.p1.1.m1.1a"><mrow id="S3.SS1.SSS1.p1.1.m1.1.1" xref="S3.SS1.SSS1.p1.1.m1.1.1.cmml"><mi id="S3.SS1.SSS1.p1.1.m1.1.1.3" xref="S3.SS1.SSS1.p1.1.m1.1.1.3.cmml">O</mi><mo id="S3.SS1.SSS1.p1.1.m1.1.1.2" xref="S3.SS1.SSS1.p1.1.m1.1.1.2.cmml">⁢</mo><mrow id="S3.SS1.SSS1.p1.1.m1.1.1.1.1" xref="S3.SS1.SSS1.p1.1.m1.1.1.1.1.1.cmml"><mo id="S3.SS1.SSS1.p1.1.m1.1.1.1.1.2" stretchy="false" xref="S3.SS1.SSS1.p1.1.m1.1.1.1.1.1.cmml">(</mo><mrow id="S3.SS1.SSS1.p1.1.m1.1.1.1.1.1" xref="S3.SS1.SSS1.p1.1.m1.1.1.1.1.1.cmml"><mrow 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xref="S3.SS1.SSS1.p1.1.m1.1.1.1.1.1.3.cmml">⁡</mo><mi id="S3.SS1.SSS1.p1.1.m1.1.1.1.1.1.3.2" xref="S3.SS1.SSS1.p1.1.m1.1.1.1.1.1.3.2.cmml">n</mi></mrow></mrow><mo id="S3.SS1.SSS1.p1.1.m1.1.1.1.1.3" stretchy="false" xref="S3.SS1.SSS1.p1.1.m1.1.1.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS1.p1.1.m1.1b"><apply id="S3.SS1.SSS1.p1.1.m1.1.1.cmml" xref="S3.SS1.SSS1.p1.1.m1.1.1"><times id="S3.SS1.SSS1.p1.1.m1.1.1.2.cmml" xref="S3.SS1.SSS1.p1.1.m1.1.1.2"></times><ci id="S3.SS1.SSS1.p1.1.m1.1.1.3.cmml" xref="S3.SS1.SSS1.p1.1.m1.1.1.3">𝑂</ci><apply id="S3.SS1.SSS1.p1.1.m1.1.1.1.1.1.cmml" xref="S3.SS1.SSS1.p1.1.m1.1.1.1.1"><times id="S3.SS1.SSS1.p1.1.m1.1.1.1.1.1.1.cmml" xref="S3.SS1.SSS1.p1.1.m1.1.1.1.1.1.1"></times><apply id="S3.SS1.SSS1.p1.1.m1.1.1.1.1.1.2.cmml" xref="S3.SS1.SSS1.p1.1.m1.1.1.1.1.1.2"><ci id="S3.SS1.SSS1.p1.1.m1.1.1.1.1.1.2.1.cmml" xref="S3.SS1.SSS1.p1.1.m1.1.1.1.1.1.2.1">⋅</ci><ci id="S3.SS1.SSS1.p1.1.m1.1.1.1.1.1.2.2.cmml" xref="S3.SS1.SSS1.p1.1.m1.1.1.1.1.1.2.2">𝑡</ci><apply id="S3.SS1.SSS1.p1.1.m1.1.1.1.1.1.2.3.cmml" xref="S3.SS1.SSS1.p1.1.m1.1.1.1.1.1.2.3"><csymbol cd="ambiguous" id="S3.SS1.SSS1.p1.1.m1.1.1.1.1.1.2.3.1.cmml" xref="S3.SS1.SSS1.p1.1.m1.1.1.1.1.1.2.3">superscript</csymbol><ci id="S3.SS1.SSS1.p1.1.m1.1.1.1.1.1.2.3.2.cmml" xref="S3.SS1.SSS1.p1.1.m1.1.1.1.1.1.2.3.2">𝑘</ci><cn id="S3.SS1.SSS1.p1.1.m1.1.1.1.1.1.2.3.3.cmml" type="integer" xref="S3.SS1.SSS1.p1.1.m1.1.1.1.1.1.2.3.3">4</cn></apply></apply><apply id="S3.SS1.SSS1.p1.1.m1.1.1.1.1.1.3.cmml" xref="S3.SS1.SSS1.p1.1.m1.1.1.1.1.1.3"><log id="S3.SS1.SSS1.p1.1.m1.1.1.1.1.1.3.1.cmml" xref="S3.SS1.SSS1.p1.1.m1.1.1.1.1.1.3.1"></log><ci id="S3.SS1.SSS1.p1.1.m1.1.1.1.1.1.3.2.cmml" xref="S3.SS1.SSS1.p1.1.m1.1.1.1.1.1.3.2">𝑛</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS1.p1.1.m1.1c">O(t\cdot k^{4}\log n)</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS1.p1.1.m1.1d">italic_O ( italic_t ⋅ italic_k start_POSTSUPERSCRIPT 4 end_POSTSUPERSCRIPT roman_log italic_n )</annotation></semantics></math>, based on best-known <math alttext="O(k^{3}\log n)" class="ltx_Math" display="inline" id="S3.SS1.SSS1.p1.2.m2.1"><semantics id="S3.SS1.SSS1.p1.2.m2.1a"><mrow id="S3.SS1.SSS1.p1.2.m2.1.1" xref="S3.SS1.SSS1.p1.2.m2.1.1.cmml"><mi id="S3.SS1.SSS1.p1.2.m2.1.1.3" xref="S3.SS1.SSS1.p1.2.m2.1.1.3.cmml">O</mi><mo id="S3.SS1.SSS1.p1.2.m2.1.1.2" xref="S3.SS1.SSS1.p1.2.m2.1.1.2.cmml">⁢</mo><mrow id="S3.SS1.SSS1.p1.2.m2.1.1.1.1" xref="S3.SS1.SSS1.p1.2.m2.1.1.1.1.1.cmml"><mo id="S3.SS1.SSS1.p1.2.m2.1.1.1.1.2" stretchy="false" xref="S3.SS1.SSS1.p1.2.m2.1.1.1.1.1.cmml">(</mo><mrow id="S3.SS1.SSS1.p1.2.m2.1.1.1.1.1" xref="S3.SS1.SSS1.p1.2.m2.1.1.1.1.1.cmml"><msup id="S3.SS1.SSS1.p1.2.m2.1.1.1.1.1.2" xref="S3.SS1.SSS1.p1.2.m2.1.1.1.1.1.2.cmml"><mi id="S3.SS1.SSS1.p1.2.m2.1.1.1.1.1.2.2" xref="S3.SS1.SSS1.p1.2.m2.1.1.1.1.1.2.2.cmml">k</mi><mn 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xref="S3.SS1.SSS1.p1.2.m2.1.1.3">𝑂</ci><apply id="S3.SS1.SSS1.p1.2.m2.1.1.1.1.1.cmml" xref="S3.SS1.SSS1.p1.2.m2.1.1.1.1"><times id="S3.SS1.SSS1.p1.2.m2.1.1.1.1.1.1.cmml" xref="S3.SS1.SSS1.p1.2.m2.1.1.1.1.1.1"></times><apply id="S3.SS1.SSS1.p1.2.m2.1.1.1.1.1.2.cmml" xref="S3.SS1.SSS1.p1.2.m2.1.1.1.1.1.2"><csymbol cd="ambiguous" id="S3.SS1.SSS1.p1.2.m2.1.1.1.1.1.2.1.cmml" xref="S3.SS1.SSS1.p1.2.m2.1.1.1.1.1.2">superscript</csymbol><ci id="S3.SS1.SSS1.p1.2.m2.1.1.1.1.1.2.2.cmml" xref="S3.SS1.SSS1.p1.2.m2.1.1.1.1.1.2.2">𝑘</ci><cn id="S3.SS1.SSS1.p1.2.m2.1.1.1.1.1.2.3.cmml" type="integer" xref="S3.SS1.SSS1.p1.2.m2.1.1.1.1.1.2.3">3</cn></apply><apply id="S3.SS1.SSS1.p1.2.m2.1.1.1.1.1.3.cmml" xref="S3.SS1.SSS1.p1.2.m2.1.1.1.1.1.3"><log id="S3.SS1.SSS1.p1.2.m2.1.1.1.1.1.3.1.cmml" xref="S3.SS1.SSS1.p1.2.m2.1.1.1.1.1.3.1"></log><ci id="S3.SS1.SSS1.p1.2.m2.1.1.1.1.1.3.2.cmml" xref="S3.SS1.SSS1.p1.2.m2.1.1.1.1.1.3.2">𝑛</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS1.p1.2.m2.1c">O(k^{3}\log n)</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS1.p1.2.m2.1d">italic_O ( italic_k start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT roman_log italic_n )</annotation></semantics></math>-approximation for VC-SNDP from <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.00712v1#bib.bibx26" title="">CK09</a>]</cite>.</p> </div> </section> <section class="ltx_subsubsection" id="S3.SS1.SSS2"> <h4 class="ltx_title ltx_title_subsubsection"> <span class="ltx_tag ltx_tag_subsubsection">3.1.2 </span>An Improved Analysis via Fractional Solutions</h4> <div class="ltx_para" id="S3.SS1.SSS2.p1"> <p class="ltx_p" id="S3.SS1.SSS2.p1.4">In this section we show a refined analysis of the framework which shows that an <math alttext="O(tk)" class="ltx_Math" display="inline" id="S3.SS1.SSS2.p1.1.m1.1"><semantics id="S3.SS1.SSS2.p1.1.m1.1a"><mrow id="S3.SS1.SSS2.p1.1.m1.1.1" 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xref="S3.SS1.SSS2.p1.1.m1.1.1"><times id="S3.SS1.SSS2.p1.1.m1.1.1.2.cmml" xref="S3.SS1.SSS2.p1.1.m1.1.1.2"></times><ci id="S3.SS1.SSS2.p1.1.m1.1.1.3.cmml" xref="S3.SS1.SSS2.p1.1.m1.1.1.3">𝑂</ci><apply id="S3.SS1.SSS2.p1.1.m1.1.1.1.1.1.cmml" xref="S3.SS1.SSS2.p1.1.m1.1.1.1.1"><times id="S3.SS1.SSS2.p1.1.m1.1.1.1.1.1.1.cmml" xref="S3.SS1.SSS2.p1.1.m1.1.1.1.1.1.1"></times><ci id="S3.SS1.SSS2.p1.1.m1.1.1.1.1.1.2.cmml" xref="S3.SS1.SSS2.p1.1.m1.1.1.1.1.1.2">𝑡</ci><ci id="S3.SS1.SSS2.p1.1.m1.1.1.1.1.1.3.cmml" xref="S3.SS1.SSS2.p1.1.m1.1.1.1.1.1.3">𝑘</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS2.p1.1.m1.1c">O(tk)</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS2.p1.1.m1.1d">italic_O ( italic_t italic_k )</annotation></semantics></math>-VFT <math alttext="O(t)" class="ltx_Math" display="inline" id="S3.SS1.SSS2.p1.2.m2.1"><semantics id="S3.SS1.SSS2.p1.2.m2.1a"><mrow id="S3.SS1.SSS2.p1.2.m2.1.2" xref="S3.SS1.SSS2.p1.2.m2.1.2.cmml"><mi id="S3.SS1.SSS2.p1.2.m2.1.2.2" xref="S3.SS1.SSS2.p1.2.m2.1.2.2.cmml">O</mi><mo id="S3.SS1.SSS2.p1.2.m2.1.2.1" xref="S3.SS1.SSS2.p1.2.m2.1.2.1.cmml">⁢</mo><mrow id="S3.SS1.SSS2.p1.2.m2.1.2.3.2" xref="S3.SS1.SSS2.p1.2.m2.1.2.cmml"><mo id="S3.SS1.SSS2.p1.2.m2.1.2.3.2.1" stretchy="false" xref="S3.SS1.SSS2.p1.2.m2.1.2.cmml">(</mo><mi id="S3.SS1.SSS2.p1.2.m2.1.1" xref="S3.SS1.SSS2.p1.2.m2.1.1.cmml">t</mi><mo id="S3.SS1.SSS2.p1.2.m2.1.2.3.2.2" stretchy="false" xref="S3.SS1.SSS2.p1.2.m2.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS2.p1.2.m2.1b"><apply id="S3.SS1.SSS2.p1.2.m2.1.2.cmml" xref="S3.SS1.SSS2.p1.2.m2.1.2"><times id="S3.SS1.SSS2.p1.2.m2.1.2.1.cmml" xref="S3.SS1.SSS2.p1.2.m2.1.2.1"></times><ci id="S3.SS1.SSS2.p1.2.m2.1.2.2.cmml" xref="S3.SS1.SSS2.p1.2.m2.1.2.2">𝑂</ci><ci id="S3.SS1.SSS2.p1.2.m2.1.1.cmml" xref="S3.SS1.SSS2.p1.2.m2.1.1">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS2.p1.2.m2.1c">O(t)</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS2.p1.2.m2.1d">italic_O ( italic_t )</annotation></semantics></math>-spanner contains a <math alttext="O(t)" class="ltx_Math" display="inline" id="S3.SS1.SSS2.p1.3.m3.1"><semantics id="S3.SS1.SSS2.p1.3.m3.1a"><mrow id="S3.SS1.SSS2.p1.3.m3.1.2" xref="S3.SS1.SSS2.p1.3.m3.1.2.cmml"><mi id="S3.SS1.SSS2.p1.3.m3.1.2.2" xref="S3.SS1.SSS2.p1.3.m3.1.2.2.cmml">O</mi><mo id="S3.SS1.SSS2.p1.3.m3.1.2.1" xref="S3.SS1.SSS2.p1.3.m3.1.2.1.cmml">⁢</mo><mrow id="S3.SS1.SSS2.p1.3.m3.1.2.3.2" xref="S3.SS1.SSS2.p1.3.m3.1.2.cmml"><mo id="S3.SS1.SSS2.p1.3.m3.1.2.3.2.1" stretchy="false" xref="S3.SS1.SSS2.p1.3.m3.1.2.cmml">(</mo><mi id="S3.SS1.SSS2.p1.3.m3.1.1" xref="S3.SS1.SSS2.p1.3.m3.1.1.cmml">t</mi><mo id="S3.SS1.SSS2.p1.3.m3.1.2.3.2.2" stretchy="false" xref="S3.SS1.SSS2.p1.3.m3.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS2.p1.3.m3.1b"><apply id="S3.SS1.SSS2.p1.3.m3.1.2.cmml" xref="S3.SS1.SSS2.p1.3.m3.1.2"><times id="S3.SS1.SSS2.p1.3.m3.1.2.1.cmml" xref="S3.SS1.SSS2.p1.3.m3.1.2.1"></times><ci id="S3.SS1.SSS2.p1.3.m3.1.2.2.cmml" xref="S3.SS1.SSS2.p1.3.m3.1.2.2">𝑂</ci><ci id="S3.SS1.SSS2.p1.3.m3.1.1.cmml" xref="S3.SS1.SSS2.p1.3.m3.1.1">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS2.p1.3.m3.1c">O(t)</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS2.p1.3.m3.1d">italic_O ( italic_t )</annotation></semantics></math>-approximate <em class="ltx_emph ltx_font_italic" id="S3.SS1.SSS2.p1.4.1">fractional</em> solution for the given VC-SNDP instance. This is particularly interesting as it demonstrates that a fault-tolerant spanner preserves a near-optimal fractional solution for VC-SNDP on the given graph <math alttext="G" class="ltx_Math" display="inline" id="S3.SS1.SSS2.p1.4.m4.1"><semantics id="S3.SS1.SSS2.p1.4.m4.1a"><mi id="S3.SS1.SSS2.p1.4.m4.1.1" xref="S3.SS1.SSS2.p1.4.m4.1.1.cmml">G</mi><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS2.p1.4.m4.1b"><ci id="S3.SS1.SSS2.p1.4.m4.1.1.cmml" xref="S3.SS1.SSS2.p1.4.m4.1.1">𝐺</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS2.p1.4.m4.1c">G</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS2.p1.4.m4.1d">italic_G</annotation></semantics></math>. In other words, fault-tolerant spanners serve as <span class="ltx_text ltx_font_italic" id="S3.SS1.SSS2.p1.4.2">coresets</span> for network design problems.</p> </div> <div class="ltx_theorem ltx_theorem_theorem" id="S3.Thmtheorem4"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="S3.Thmtheorem4.1.1.1">Theorem 3.4</span></span><span class="ltx_text ltx_font_bold" id="S3.Thmtheorem4.2.2">.</span> </h6> <div class="ltx_para" id="S3.Thmtheorem4.p1"> <p class="ltx_p" id="S3.Thmtheorem4.p1.6">Let <math alttext="H" class="ltx_Math" display="inline" id="S3.Thmtheorem4.p1.1.m1.1"><semantics id="S3.Thmtheorem4.p1.1.m1.1a"><mi id="S3.Thmtheorem4.p1.1.m1.1.1" xref="S3.Thmtheorem4.p1.1.m1.1.1.cmml">H</mi><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem4.p1.1.m1.1b"><ci id="S3.Thmtheorem4.p1.1.m1.1.1.cmml" xref="S3.Thmtheorem4.p1.1.m1.1.1">𝐻</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem4.p1.1.m1.1c">H</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem4.p1.1.m1.1d">italic_H</annotation></semantics></math> be the VFT spanner of a weighted graph <math alttext="G" class="ltx_Math" display="inline" id="S3.Thmtheorem4.p1.2.m2.1"><semantics id="S3.Thmtheorem4.p1.2.m2.1a"><mi id="S3.Thmtheorem4.p1.2.m2.1.1" xref="S3.Thmtheorem4.p1.2.m2.1.1.cmml">G</mi><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem4.p1.2.m2.1b"><ci id="S3.Thmtheorem4.p1.2.m2.1.1.cmml" xref="S3.Thmtheorem4.p1.2.m2.1.1">𝐺</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem4.p1.2.m2.1c">G</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem4.p1.2.m2.1d">italic_G</annotation></semantics></math> as constructed in Algorithm <a class="ltx_ref" href="https://arxiv.org/html/2503.00712v1#algorithm2" title="In Weighted graphs. ‣ 2.1 Fault-Tolerant Spanners in Streaming ‣ 2 Preliminaries ‣ Streaming Algorithms for Network Design"><span class="ltx_text ltx_ref_tag">2</span></a> with parameters <math alttext="(t,f=(2t-2)(2k-1),\epsilon=1/(2t-1))" class="ltx_Math" display="inline" id="S3.Thmtheorem4.p1.3.m3.3"><semantics id="S3.Thmtheorem4.p1.3.m3.3a"><mrow id="S3.Thmtheorem4.p1.3.m3.3.3.1"><mo id="S3.Thmtheorem4.p1.3.m3.3.3.1.2" stretchy="false">(</mo><mrow id="S3.Thmtheorem4.p1.3.m3.3.3.1.1.2" xref="S3.Thmtheorem4.p1.3.m3.3.3.1.1.3.cmml"><mrow id="S3.Thmtheorem4.p1.3.m3.3.3.1.1.1.1" xref="S3.Thmtheorem4.p1.3.m3.3.3.1.1.1.1.cmml"><mrow id="S3.Thmtheorem4.p1.3.m3.3.3.1.1.1.1.4.2" xref="S3.Thmtheorem4.p1.3.m3.3.3.1.1.1.1.4.1.cmml"><mi id="S3.Thmtheorem4.p1.3.m3.1.1" xref="S3.Thmtheorem4.p1.3.m3.1.1.cmml">t</mi><mo id="S3.Thmtheorem4.p1.3.m3.3.3.1.1.1.1.4.2.1" xref="S3.Thmtheorem4.p1.3.m3.3.3.1.1.1.1.4.1.cmml">,</mo><mi id="S3.Thmtheorem4.p1.3.m3.2.2" xref="S3.Thmtheorem4.p1.3.m3.2.2.cmml">f</mi></mrow><mo id="S3.Thmtheorem4.p1.3.m3.3.3.1.1.1.1.3" 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xref="S3.Thmtheorem4.p1.3.m3.3.3.1.1.2.3">formulae-sequence</csymbol><apply id="S3.Thmtheorem4.p1.3.m3.3.3.1.1.1.1.cmml" xref="S3.Thmtheorem4.p1.3.m3.3.3.1.1.1.1"><eq id="S3.Thmtheorem4.p1.3.m3.3.3.1.1.1.1.3.cmml" xref="S3.Thmtheorem4.p1.3.m3.3.3.1.1.1.1.3"></eq><list id="S3.Thmtheorem4.p1.3.m3.3.3.1.1.1.1.4.1.cmml" xref="S3.Thmtheorem4.p1.3.m3.3.3.1.1.1.1.4.2"><ci id="S3.Thmtheorem4.p1.3.m3.1.1.cmml" xref="S3.Thmtheorem4.p1.3.m3.1.1">𝑡</ci><ci id="S3.Thmtheorem4.p1.3.m3.2.2.cmml" xref="S3.Thmtheorem4.p1.3.m3.2.2">𝑓</ci></list><apply id="S3.Thmtheorem4.p1.3.m3.3.3.1.1.1.1.2.cmml" xref="S3.Thmtheorem4.p1.3.m3.3.3.1.1.1.1.2"><times id="S3.Thmtheorem4.p1.3.m3.3.3.1.1.1.1.2.3.cmml" xref="S3.Thmtheorem4.p1.3.m3.3.3.1.1.1.1.2.3"></times><apply id="S3.Thmtheorem4.p1.3.m3.3.3.1.1.1.1.1.1.1.1.cmml" xref="S3.Thmtheorem4.p1.3.m3.3.3.1.1.1.1.1.1.1"><minus id="S3.Thmtheorem4.p1.3.m3.3.3.1.1.1.1.1.1.1.1.1.cmml" xref="S3.Thmtheorem4.p1.3.m3.3.3.1.1.1.1.1.1.1.1.1"></minus><apply id="S3.Thmtheorem4.p1.3.m3.3.3.1.1.1.1.1.1.1.1.2.cmml" xref="S3.Thmtheorem4.p1.3.m3.3.3.1.1.1.1.1.1.1.1.2"><times id="S3.Thmtheorem4.p1.3.m3.3.3.1.1.1.1.1.1.1.1.2.1.cmml" xref="S3.Thmtheorem4.p1.3.m3.3.3.1.1.1.1.1.1.1.1.2.1"></times><cn id="S3.Thmtheorem4.p1.3.m3.3.3.1.1.1.1.1.1.1.1.2.2.cmml" type="integer" xref="S3.Thmtheorem4.p1.3.m3.3.3.1.1.1.1.1.1.1.1.2.2">2</cn><ci id="S3.Thmtheorem4.p1.3.m3.3.3.1.1.1.1.1.1.1.1.2.3.cmml" xref="S3.Thmtheorem4.p1.3.m3.3.3.1.1.1.1.1.1.1.1.2.3">𝑡</ci></apply><cn id="S3.Thmtheorem4.p1.3.m3.3.3.1.1.1.1.1.1.1.1.3.cmml" type="integer" xref="S3.Thmtheorem4.p1.3.m3.3.3.1.1.1.1.1.1.1.1.3">2</cn></apply><apply id="S3.Thmtheorem4.p1.3.m3.3.3.1.1.1.1.2.2.1.1.cmml" xref="S3.Thmtheorem4.p1.3.m3.3.3.1.1.1.1.2.2.1"><minus id="S3.Thmtheorem4.p1.3.m3.3.3.1.1.1.1.2.2.1.1.1.cmml" xref="S3.Thmtheorem4.p1.3.m3.3.3.1.1.1.1.2.2.1.1.1"></minus><apply id="S3.Thmtheorem4.p1.3.m3.3.3.1.1.1.1.2.2.1.1.2.cmml" xref="S3.Thmtheorem4.p1.3.m3.3.3.1.1.1.1.2.2.1.1.2"><times id="S3.Thmtheorem4.p1.3.m3.3.3.1.1.1.1.2.2.1.1.2.1.cmml" xref="S3.Thmtheorem4.p1.3.m3.3.3.1.1.1.1.2.2.1.1.2.1"></times><cn id="S3.Thmtheorem4.p1.3.m3.3.3.1.1.1.1.2.2.1.1.2.2.cmml" type="integer" xref="S3.Thmtheorem4.p1.3.m3.3.3.1.1.1.1.2.2.1.1.2.2">2</cn><ci id="S3.Thmtheorem4.p1.3.m3.3.3.1.1.1.1.2.2.1.1.2.3.cmml" xref="S3.Thmtheorem4.p1.3.m3.3.3.1.1.1.1.2.2.1.1.2.3">𝑘</ci></apply><cn id="S3.Thmtheorem4.p1.3.m3.3.3.1.1.1.1.2.2.1.1.3.cmml" type="integer" xref="S3.Thmtheorem4.p1.3.m3.3.3.1.1.1.1.2.2.1.1.3">1</cn></apply></apply></apply><apply id="S3.Thmtheorem4.p1.3.m3.3.3.1.1.2.2.cmml" xref="S3.Thmtheorem4.p1.3.m3.3.3.1.1.2.2"><eq id="S3.Thmtheorem4.p1.3.m3.3.3.1.1.2.2.2.cmml" xref="S3.Thmtheorem4.p1.3.m3.3.3.1.1.2.2.2"></eq><ci id="S3.Thmtheorem4.p1.3.m3.3.3.1.1.2.2.3.cmml" xref="S3.Thmtheorem4.p1.3.m3.3.3.1.1.2.2.3">italic-ϵ</ci><apply id="S3.Thmtheorem4.p1.3.m3.3.3.1.1.2.2.1.cmml" xref="S3.Thmtheorem4.p1.3.m3.3.3.1.1.2.2.1"><divide id="S3.Thmtheorem4.p1.3.m3.3.3.1.1.2.2.1.2.cmml" xref="S3.Thmtheorem4.p1.3.m3.3.3.1.1.2.2.1.2"></divide><cn id="S3.Thmtheorem4.p1.3.m3.3.3.1.1.2.2.1.3.cmml" type="integer" xref="S3.Thmtheorem4.p1.3.m3.3.3.1.1.2.2.1.3">1</cn><apply id="S3.Thmtheorem4.p1.3.m3.3.3.1.1.2.2.1.1.1.1.cmml" xref="S3.Thmtheorem4.p1.3.m3.3.3.1.1.2.2.1.1.1"><minus id="S3.Thmtheorem4.p1.3.m3.3.3.1.1.2.2.1.1.1.1.1.cmml" xref="S3.Thmtheorem4.p1.3.m3.3.3.1.1.2.2.1.1.1.1.1"></minus><apply id="S3.Thmtheorem4.p1.3.m3.3.3.1.1.2.2.1.1.1.1.2.cmml" xref="S3.Thmtheorem4.p1.3.m3.3.3.1.1.2.2.1.1.1.1.2"><times id="S3.Thmtheorem4.p1.3.m3.3.3.1.1.2.2.1.1.1.1.2.1.cmml" xref="S3.Thmtheorem4.p1.3.m3.3.3.1.1.2.2.1.1.1.1.2.1"></times><cn id="S3.Thmtheorem4.p1.3.m3.3.3.1.1.2.2.1.1.1.1.2.2.cmml" type="integer" xref="S3.Thmtheorem4.p1.3.m3.3.3.1.1.2.2.1.1.1.1.2.2">2</cn><ci id="S3.Thmtheorem4.p1.3.m3.3.3.1.1.2.2.1.1.1.1.2.3.cmml" xref="S3.Thmtheorem4.p1.3.m3.3.3.1.1.2.2.1.1.1.1.2.3">𝑡</ci></apply><cn id="S3.Thmtheorem4.p1.3.m3.3.3.1.1.2.2.1.1.1.1.3.cmml" type="integer" xref="S3.Thmtheorem4.p1.3.m3.3.3.1.1.2.2.1.1.1.1.3">1</cn></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem4.p1.3.m3.3c">(t,f=(2t-2)(2k-1),\epsilon=1/(2t-1))</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem4.p1.3.m3.3d">( italic_t , italic_f = ( 2 italic_t - 2 ) ( 2 italic_k - 1 ) , italic_ϵ = 1 / ( 2 italic_t - 1 ) )</annotation></semantics></math>. Then the weight of an optimal <em class="ltx_emph ltx_font_italic" id="S3.Thmtheorem4.p1.6.1">fractional</em> solution of VC-SNDP on (<math alttext="H,r" class="ltx_Math" display="inline" id="S3.Thmtheorem4.p1.4.m4.2"><semantics id="S3.Thmtheorem4.p1.4.m4.2a"><mrow id="S3.Thmtheorem4.p1.4.m4.2.3.2" xref="S3.Thmtheorem4.p1.4.m4.2.3.1.cmml"><mi id="S3.Thmtheorem4.p1.4.m4.1.1" xref="S3.Thmtheorem4.p1.4.m4.1.1.cmml">H</mi><mo id="S3.Thmtheorem4.p1.4.m4.2.3.2.1" xref="S3.Thmtheorem4.p1.4.m4.2.3.1.cmml">,</mo><mi id="S3.Thmtheorem4.p1.4.m4.2.2" xref="S3.Thmtheorem4.p1.4.m4.2.2.cmml">r</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem4.p1.4.m4.2b"><list id="S3.Thmtheorem4.p1.4.m4.2.3.1.cmml" xref="S3.Thmtheorem4.p1.4.m4.2.3.2"><ci id="S3.Thmtheorem4.p1.4.m4.1.1.cmml" xref="S3.Thmtheorem4.p1.4.m4.1.1">𝐻</ci><ci id="S3.Thmtheorem4.p1.4.m4.2.2.cmml" xref="S3.Thmtheorem4.p1.4.m4.2.2">𝑟</ci></list></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem4.p1.4.m4.2c">H,r</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem4.p1.4.m4.2d">italic_H , italic_r</annotation></semantics></math>) is within a <math alttext="4t" class="ltx_Math" display="inline" id="S3.Thmtheorem4.p1.5.m5.1"><semantics id="S3.Thmtheorem4.p1.5.m5.1a"><mrow id="S3.Thmtheorem4.p1.5.m5.1.1" xref="S3.Thmtheorem4.p1.5.m5.1.1.cmml"><mn id="S3.Thmtheorem4.p1.5.m5.1.1.2" xref="S3.Thmtheorem4.p1.5.m5.1.1.2.cmml">4</mn><mo id="S3.Thmtheorem4.p1.5.m5.1.1.1" xref="S3.Thmtheorem4.p1.5.m5.1.1.1.cmml">⁢</mo><mi id="S3.Thmtheorem4.p1.5.m5.1.1.3" xref="S3.Thmtheorem4.p1.5.m5.1.1.3.cmml">t</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem4.p1.5.m5.1b"><apply id="S3.Thmtheorem4.p1.5.m5.1.1.cmml" xref="S3.Thmtheorem4.p1.5.m5.1.1"><times id="S3.Thmtheorem4.p1.5.m5.1.1.1.cmml" xref="S3.Thmtheorem4.p1.5.m5.1.1.1"></times><cn id="S3.Thmtheorem4.p1.5.m5.1.1.2.cmml" type="integer" xref="S3.Thmtheorem4.p1.5.m5.1.1.2">4</cn><ci id="S3.Thmtheorem4.p1.5.m5.1.1.3.cmml" xref="S3.Thmtheorem4.p1.5.m5.1.1.3">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem4.p1.5.m5.1c">4t</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem4.p1.5.m5.1d">4 italic_t</annotation></semantics></math>-factor of the optimal solution of VC-SNDP on (<math alttext="G,r" class="ltx_Math" display="inline" id="S3.Thmtheorem4.p1.6.m6.2"><semantics id="S3.Thmtheorem4.p1.6.m6.2a"><mrow id="S3.Thmtheorem4.p1.6.m6.2.3.2" xref="S3.Thmtheorem4.p1.6.m6.2.3.1.cmml"><mi id="S3.Thmtheorem4.p1.6.m6.1.1" xref="S3.Thmtheorem4.p1.6.m6.1.1.cmml">G</mi><mo id="S3.Thmtheorem4.p1.6.m6.2.3.2.1" xref="S3.Thmtheorem4.p1.6.m6.2.3.1.cmml">,</mo><mi id="S3.Thmtheorem4.p1.6.m6.2.2" xref="S3.Thmtheorem4.p1.6.m6.2.2.cmml">r</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem4.p1.6.m6.2b"><list id="S3.Thmtheorem4.p1.6.m6.2.3.1.cmml" xref="S3.Thmtheorem4.p1.6.m6.2.3.2"><ci id="S3.Thmtheorem4.p1.6.m6.1.1.cmml" xref="S3.Thmtheorem4.p1.6.m6.1.1">𝐺</ci><ci id="S3.Thmtheorem4.p1.6.m6.2.2.cmml" xref="S3.Thmtheorem4.p1.6.m6.2.2">𝑟</ci></list></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem4.p1.6.m6.2c">G,r</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem4.p1.6.m6.2d">italic_G , italic_r</annotation></semantics></math>).</p> </div> </div> <div class="ltx_para" id="S3.SS1.SSS2.p2"> <p class="ltx_p" id="S3.SS1.SSS2.p2.2">We note a small but important difference between the algorithm implied by theorem above, and the one earlier. We increase the <math alttext="f" class="ltx_Math" display="inline" id="S3.SS1.SSS2.p2.1.m1.1"><semantics id="S3.SS1.SSS2.p2.1.m1.1a"><mi id="S3.SS1.SSS2.p2.1.m1.1.1" xref="S3.SS1.SSS2.p2.1.m1.1.1.cmml">f</mi><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS2.p2.1.m1.1b"><ci id="S3.SS1.SSS2.p2.1.m1.1.1.cmml" xref="S3.SS1.SSS2.p2.1.m1.1.1">𝑓</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS2.p2.1.m1.1c">f</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS2.p2.1.m1.1d">italic_f</annotation></semantics></math> by a factor of <math alttext="2" class="ltx_Math" display="inline" id="S3.SS1.SSS2.p2.2.m2.1"><semantics id="S3.SS1.SSS2.p2.2.m2.1a"><mn id="S3.SS1.SSS2.p2.2.m2.1.1" xref="S3.SS1.SSS2.p2.2.m2.1.1.cmml">2</mn><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS2.p2.2.m2.1b"><cn id="S3.SS1.SSS2.p2.2.m2.1.1.cmml" type="integer" xref="S3.SS1.SSS2.p2.2.m2.1.1">2</cn></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS2.p2.2.m2.1c">2</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS2.p2.2.m2.1d">2</annotation></semantics></math> which is needed for the proof below.</p> </div> <figure class="ltx_figure" id="S3.F1"><svg class="ltx_picture ltx_centering" height="96.01" id="S3.F1.pic1" overflow="visible" version="1.1" width="420"><g fill="#000000" stroke="#000000" stroke-width="0.4pt" transform="translate(0,96.01) matrix(1 0 0 -1 0 0)"><g fill="#404040" fill-opacity="1.0"><path d="M 0 5.91 L 0 90.1 C 0 93.36 2.64 96.01 5.91 96.01 L 414.09 96.01 C 417.35 96.01 420 93.36 420 90.1 L 420 5.91 C 420 2.64 417.35 0 414.09 0 L 5.91 0 C 2.64 0 0 2.64 0 5.91 Z" style="stroke:none"></path></g><g fill="#F2F2F2" fill-opacity="1.0"><path d="M 1.97 5.91 L 1.97 90.1 C 1.97 92.28 3.73 94.04 5.91 94.04 L 414.09 94.04 C 416.27 94.04 418.03 92.28 418.03 90.1 L 418.03 5.91 C 418.03 3.73 416.27 1.97 414.09 1.97 L 5.91 1.97 C 3.73 1.97 1.97 3.73 1.97 5.91 Z" style="stroke:none"></path></g><g fill-opacity="1.0" transform="matrix(1.0 0.0 0.0 1.0 21.65 13.78)"><foreignobject color="#000000" height="68.45" overflow="visible" transform="matrix(1 0 0 -1 0 16.6)" width="376.69"> <span class="ltx_inline-block ltx_minipage ltx_align_bottom" id="S3.F1.pic1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1" style="width:272.2pt;"> <span class="ltx_p" id="S3.F1.pic1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1"><span class="ltx_text ltx_ulem_uline" id="S3.F1.pic1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1"><span class="ltx_text ltx_font_sansserif ltx_font_bold" id="S3.F1.pic1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1" style="font-size:120%;">VC-SNDP-LP</span> <math alttext="\langle\mkern-4.0mu\langle\textsf{\it Input: }(G=(V,E),w,h)\rangle\mkern-4.0mu\rangle" class="ltx_Math" display="inline" id="S3.F1.pic1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.5"><semantics id="S3.F1.pic1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.5a"><mrow id="S3.F1.pic1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.5.5.1" xref="S3.F1.pic1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.5.5.2.cmml"><mpadded width="0.169em"><mo id="S3.F1.pic1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.5.5.1.2" stretchy="false" xref="S3.F1.pic1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.5.5.2.1.cmml">⟨</mo></mpadded><mrow id="S3.F1.pic1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.5.5.1.1.1" xref="S3.F1.pic1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.5.5.1.1.2.cmml"><mo id="S3.F1.pic1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.5.5.1.1.1.2" stretchy="false" xref="S3.F1.pic1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.5.5.1.1.2.1.cmml">⟨</mo><mrow id="S3.F1.pic1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.5.5.1.1.1.1" xref="S3.F1.pic1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.5.5.1.1.1.1.cmml"><mtext class="ltx_mathvariant_italic" id="S3.F1.pic1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.5.5.1.1.1.1.3" xref="S3.F1.pic1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.5.5.1.1.1.1.3a.cmml">Input: 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id="S3.F1.pic1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.2.2" xref="S3.F1.pic1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.2.2.cmml">E</mi><mo id="S3.F1.pic1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.5.5.1.1.1.1.1.1.1.1.1.1.2.3" stretchy="false" xref="S3.F1.pic1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.5.5.1.1.1.1.1.1.1.1.1.1.1.cmml">)</mo></mrow><mo id="S3.F1.pic1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.5.5.1.1.1.1.1.1.1.1.1.2" xref="S3.F1.pic1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.5.5.1.1.1.1.1.1.1.1.2.cmml">,</mo><mi id="S3.F1.pic1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.3.3" xref="S3.F1.pic1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.3.3.cmml">w</mi><mo id="S3.F1.pic1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.5.5.1.1.1.1.1.1.1.1.1.3" xref="S3.F1.pic1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.5.5.1.1.1.1.1.1.1.1.2.cmml">,</mo><mi id="S3.F1.pic1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.4.4" xref="S3.F1.pic1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.4.4.cmml">h</mi></mrow></mrow><mo 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id="S3.F1.pic1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.5.5.1.1.1.1.3.cmml" xref="S3.F1.pic1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.5.5.1.1.1.1.3">Input: </mtext></ci><apply id="S3.F1.pic1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.5.5.1.1.1.1.1.1.1.cmml" xref="S3.F1.pic1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.5.5.1.1.1.1.1.1"><eq id="S3.F1.pic1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.5.5.1.1.1.1.1.1.1.2.cmml" xref="S3.F1.pic1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.5.5.1.1.1.1.1.1.1.2"></eq><ci id="S3.F1.pic1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.5.5.1.1.1.1.1.1.1.3.cmml" xref="S3.F1.pic1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.5.5.1.1.1.1.1.1.1.3">𝐺</ci><list id="S3.F1.pic1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.5.5.1.1.1.1.1.1.1.1.2.cmml" xref="S3.F1.pic1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.5.5.1.1.1.1.1.1.1.1.1"><interval closure="open" id="S3.F1.pic1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.5.5.1.1.1.1.1.1.1.1.1.1.1.cmml" xref="S3.F1.pic1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.5.5.1.1.1.1.1.1.1.1.1.1.2"><ci id="S3.F1.pic1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1.1.cmml" xref="S3.F1.pic1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1.1">𝑉</ci><ci id="S3.F1.pic1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.2.2.cmml" xref="S3.F1.pic1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.2.2">𝐸</ci></interval><ci id="S3.F1.pic1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.3.3.cmml" xref="S3.F1.pic1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.3.3">𝑤</ci><ci id="S3.F1.pic1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.4.4.cmml" xref="S3.F1.pic1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.4.4">ℎ</ci></list></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.F1.pic1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.5c">\langle\mkern-4.0mu\langle\textsf{\it Input: }(G=(V,E),w,h)\rangle\mkern-4.0mu\rangle</annotation><annotation encoding="application/x-llamapun" id="S3.F1.pic1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.5d">⟨ ⟨ Input: ( italic_G = ( italic_V , italic_E ) , italic_w , italic_h ) ⟩ ⟩</annotation></semantics></math></span></span> <span class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="Sx1.EGx3"> <span id="S3.Ex3"><span class="ltx_equation ltx_eqn_row ltx_align_baseline"> <span class="ltx_eqn_cell ltx_eqn_center_padleft"></span> <span class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle\min" class="ltx_Math" display="inline" id="S3.Ex3.m1.1"><semantics id="S3.Ex3.m1.1a"><mi id="S3.Ex3.m1.1.1" xref="S3.Ex3.m1.1.1.cmml">min</mi><annotation-xml encoding="MathML-Content" id="S3.Ex3.m1.1b"><min id="S3.Ex3.m1.1.1.cmml" xref="S3.Ex3.m1.1.1"></min></annotation-xml><annotation encoding="application/x-tex" id="S3.Ex3.m1.1c">\displaystyle\min</annotation><annotation encoding="application/x-llamapun" id="S3.Ex3.m1.1d">roman_min</annotation></semantics></math></span> <span class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle\sum_{e\in E}\boldsymbol{x}(e)w(e)" class="ltx_Math" display="inline" id="S3.Ex3.m2.2"><semantics id="S3.Ex3.m2.2a"><mrow id="S3.Ex3.m2.2.3" xref="S3.Ex3.m2.2.3.cmml"><mstyle displaystyle="true" id="S3.Ex3.m2.2.3.1" xref="S3.Ex3.m2.2.3.1.cmml"><munder id="S3.Ex3.m2.2.3.1a" xref="S3.Ex3.m2.2.3.1.cmml"><mo id="S3.Ex3.m2.2.3.1.2" movablelimits="false" xref="S3.Ex3.m2.2.3.1.2.cmml">∑</mo><mrow id="S3.Ex3.m2.2.3.1.3" xref="S3.Ex3.m2.2.3.1.3.cmml"><mi id="S3.Ex3.m2.2.3.1.3.2" xref="S3.Ex3.m2.2.3.1.3.2.cmml">e</mi><mo id="S3.Ex3.m2.2.3.1.3.1" xref="S3.Ex3.m2.2.3.1.3.1.cmml">∈</mo><mi id="S3.Ex3.m2.2.3.1.3.3" xref="S3.Ex3.m2.2.3.1.3.3.cmml">E</mi></mrow></munder></mstyle><mrow id="S3.Ex3.m2.2.3.2" xref="S3.Ex3.m2.2.3.2.cmml"><mi id="S3.Ex3.m2.2.3.2.2" xref="S3.Ex3.m2.2.3.2.2.cmml">𝒙</mi><mo id="S3.Ex3.m2.2.3.2.1" xref="S3.Ex3.m2.2.3.2.1.cmml">⁢</mo><mrow id="S3.Ex3.m2.2.3.2.3.2" xref="S3.Ex3.m2.2.3.2.cmml"><mo id="S3.Ex3.m2.2.3.2.3.2.1" stretchy="false" xref="S3.Ex3.m2.2.3.2.cmml">(</mo><mi id="S3.Ex3.m2.1.1" xref="S3.Ex3.m2.1.1.cmml">e</mi><mo id="S3.Ex3.m2.2.3.2.3.2.2" stretchy="false" xref="S3.Ex3.m2.2.3.2.cmml">)</mo></mrow><mo id="S3.Ex3.m2.2.3.2.1a" xref="S3.Ex3.m2.2.3.2.1.cmml">⁢</mo><mi id="S3.Ex3.m2.2.3.2.4" xref="S3.Ex3.m2.2.3.2.4.cmml">w</mi><mo id="S3.Ex3.m2.2.3.2.1b" xref="S3.Ex3.m2.2.3.2.1.cmml">⁢</mo><mrow id="S3.Ex3.m2.2.3.2.5.2" xref="S3.Ex3.m2.2.3.2.cmml"><mo id="S3.Ex3.m2.2.3.2.5.2.1" stretchy="false" xref="S3.Ex3.m2.2.3.2.cmml">(</mo><mi id="S3.Ex3.m2.2.2" xref="S3.Ex3.m2.2.2.cmml">e</mi><mo id="S3.Ex3.m2.2.3.2.5.2.2" stretchy="false" xref="S3.Ex3.m2.2.3.2.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.Ex3.m2.2b"><apply id="S3.Ex3.m2.2.3.cmml" xref="S3.Ex3.m2.2.3"><apply id="S3.Ex3.m2.2.3.1.cmml" xref="S3.Ex3.m2.2.3.1"><csymbol cd="ambiguous" id="S3.Ex3.m2.2.3.1.1.cmml" xref="S3.Ex3.m2.2.3.1">subscript</csymbol><sum id="S3.Ex3.m2.2.3.1.2.cmml" xref="S3.Ex3.m2.2.3.1.2"></sum><apply id="S3.Ex3.m2.2.3.1.3.cmml" xref="S3.Ex3.m2.2.3.1.3"><in id="S3.Ex3.m2.2.3.1.3.1.cmml" xref="S3.Ex3.m2.2.3.1.3.1"></in><ci id="S3.Ex3.m2.2.3.1.3.2.cmml" xref="S3.Ex3.m2.2.3.1.3.2">𝑒</ci><ci id="S3.Ex3.m2.2.3.1.3.3.cmml" xref="S3.Ex3.m2.2.3.1.3.3">𝐸</ci></apply></apply><apply id="S3.Ex3.m2.2.3.2.cmml" xref="S3.Ex3.m2.2.3.2"><times id="S3.Ex3.m2.2.3.2.1.cmml" xref="S3.Ex3.m2.2.3.2.1"></times><ci id="S3.Ex3.m2.2.3.2.2.cmml" xref="S3.Ex3.m2.2.3.2.2">𝒙</ci><ci id="S3.Ex3.m2.1.1.cmml" xref="S3.Ex3.m2.1.1">𝑒</ci><ci id="S3.Ex3.m2.2.3.2.4.cmml" xref="S3.Ex3.m2.2.3.2.4">𝑤</ci><ci id="S3.Ex3.m2.2.2.cmml" xref="S3.Ex3.m2.2.2">𝑒</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Ex3.m2.2c">\displaystyle\sum_{e\in E}\boldsymbol{x}(e)w(e)</annotation><annotation encoding="application/x-llamapun" id="S3.Ex3.m2.2d">∑ start_POSTSUBSCRIPT italic_e ∈ italic_E end_POSTSUBSCRIPT bold_italic_x ( italic_e ) italic_w ( italic_e )</annotation></semantics></math></span> <span class="ltx_eqn_cell ltx_eqn_center_padright" colspan="2"></span></span></span> <span id="S3.Ex4"><span class="ltx_equation ltx_eqn_row ltx_align_baseline"> <span class="ltx_eqn_cell ltx_eqn_center_padleft"></span> <span class="ltx_td ltx_align_right ltx_eqn_cell"><span class="ltx_text ltx_markedasmath" id="S3.Ex4.3.1.1.1">s.t.</span></span> <span class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle\sum_{uv\in E\;:\;u\in S,v\in V\setminus S^{+}}\boldsymbol{x}(e)% \geq h(\hat{S})" class="ltx_Math" display="inline" id="S3.Ex4.m2.4"><semantics id="S3.Ex4.m2.4a"><mrow id="S3.Ex4.m2.4.5" xref="S3.Ex4.m2.4.5.cmml"><mrow id="S3.Ex4.m2.4.5.2" xref="S3.Ex4.m2.4.5.2.cmml"><mstyle displaystyle="true" id="S3.Ex4.m2.4.5.2.1" xref="S3.Ex4.m2.4.5.2.1.cmml"><munder id="S3.Ex4.m2.4.5.2.1a" xref="S3.Ex4.m2.4.5.2.1.cmml"><mo id="S3.Ex4.m2.4.5.2.1.2" movablelimits="false" xref="S3.Ex4.m2.4.5.2.1.2.cmml">∑</mo><mrow id="S3.Ex4.m2.2.2.2" 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xref="S3.Ex4.m3.2.2.2.2.2.2a.cmml">s.t. </mtext><mo id="S3.Ex4.m3.2.2.2.2.2.1" xref="S3.Ex4.m3.2.2.2.2.2.1.cmml">⁢</mo><mi id="S3.Ex4.m3.2.2.2.2.2.3" xref="S3.Ex4.m3.2.2.2.2.2.3.cmml">S</mi></mrow><mo id="S3.Ex4.m3.2.2.2.2.1" xref="S3.Ex4.m3.2.2.2.2.1.cmml">⊆</mo><msup id="S3.Ex4.m3.2.2.2.2.3" xref="S3.Ex4.m3.2.2.2.2.3.cmml"><mi id="S3.Ex4.m3.2.2.2.2.3.2" xref="S3.Ex4.m3.2.2.2.2.3.2.cmml">S</mi><mo id="S3.Ex4.m3.2.2.2.2.3.3" xref="S3.Ex4.m3.2.2.2.2.3.3.cmml">+</mo></msup></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.Ex4.m3.2b"><apply id="S3.Ex4.m3.2.2.3.cmml" xref="S3.Ex4.m3.2.2.2"><csymbol cd="ambiguous" id="S3.Ex4.m3.2.2.3a.cmml" xref="S3.Ex4.m3.2.2.2.3">formulae-sequence</csymbol><apply id="S3.Ex4.m3.1.1.1.1.cmml" xref="S3.Ex4.m3.1.1.1.1"><subset id="S3.Ex4.m3.1.1.1.1.3.cmml" xref="S3.Ex4.m3.1.1.1.1.3"></subset><list id="S3.Ex4.m3.1.1.1.1.2.3.cmml" xref="S3.Ex4.m3.1.1.1.1.2.2"><apply id="S3.Ex4.m3.1.1.1.1.1.1.1.cmml" xref="S3.Ex4.m3.1.1.1.1.1.1.1"><csymbol 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xref="S3.Ex4.m3.2.2.2.2.2.2">s.t. </mtext></ci><ci id="S3.Ex4.m3.2.2.2.2.2.3.cmml" xref="S3.Ex4.m3.2.2.2.2.2.3">𝑆</ci></apply><apply id="S3.Ex4.m3.2.2.2.2.3.cmml" xref="S3.Ex4.m3.2.2.2.2.3"><csymbol cd="ambiguous" id="S3.Ex4.m3.2.2.2.2.3.1.cmml" xref="S3.Ex4.m3.2.2.2.2.3">superscript</csymbol><ci id="S3.Ex4.m3.2.2.2.2.3.2.cmml" xref="S3.Ex4.m3.2.2.2.2.3.2">𝑆</ci><plus id="S3.Ex4.m3.2.2.2.2.3.3.cmml" xref="S3.Ex4.m3.2.2.2.2.3.3"></plus></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Ex4.m3.2c">\displaystyle\forall S,S^{+}\subseteq V,\text{s.t. }S\subseteq S^{+}</annotation><annotation encoding="application/x-llamapun" id="S3.Ex4.m3.2d">∀ italic_S , italic_S start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT ⊆ italic_V , s.t. italic_S ⊆ italic_S start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT</annotation></semantics></math></span> <span class="ltx_eqn_cell ltx_eqn_center_padright"></span></span></span> <span id="S3.Ex5"><span class="ltx_equation ltx_eqn_row ltx_align_baseline"> <span class="ltx_eqn_cell ltx_eqn_center_padleft"></span> <span class="ltx_td ltx_eqn_cell"></span> <span class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle\boldsymbol{x}(e)\geq 0" class="ltx_Math" display="inline" id="S3.Ex5.m1.1"><semantics id="S3.Ex5.m1.1a"><mrow id="S3.Ex5.m1.1.2" xref="S3.Ex5.m1.1.2.cmml"><mrow id="S3.Ex5.m1.1.2.2" xref="S3.Ex5.m1.1.2.2.cmml"><mi id="S3.Ex5.m1.1.2.2.2" xref="S3.Ex5.m1.1.2.2.2.cmml">𝒙</mi><mo id="S3.Ex5.m1.1.2.2.1" xref="S3.Ex5.m1.1.2.2.1.cmml">⁢</mo><mrow id="S3.Ex5.m1.1.2.2.3.2" xref="S3.Ex5.m1.1.2.2.cmml"><mo id="S3.Ex5.m1.1.2.2.3.2.1" stretchy="false" xref="S3.Ex5.m1.1.2.2.cmml">(</mo><mi id="S3.Ex5.m1.1.1" xref="S3.Ex5.m1.1.1.cmml">e</mi><mo id="S3.Ex5.m1.1.2.2.3.2.2" stretchy="false" xref="S3.Ex5.m1.1.2.2.cmml">)</mo></mrow></mrow><mo id="S3.Ex5.m1.1.2.1" xref="S3.Ex5.m1.1.2.1.cmml">≥</mo><mn id="S3.Ex5.m1.1.2.3" xref="S3.Ex5.m1.1.2.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.Ex5.m1.1b"><apply id="S3.Ex5.m1.1.2.cmml" xref="S3.Ex5.m1.1.2"><geq id="S3.Ex5.m1.1.2.1.cmml" xref="S3.Ex5.m1.1.2.1"></geq><apply id="S3.Ex5.m1.1.2.2.cmml" xref="S3.Ex5.m1.1.2.2"><times id="S3.Ex5.m1.1.2.2.1.cmml" xref="S3.Ex5.m1.1.2.2.1"></times><ci id="S3.Ex5.m1.1.2.2.2.cmml" xref="S3.Ex5.m1.1.2.2.2">𝒙</ci><ci id="S3.Ex5.m1.1.1.cmml" xref="S3.Ex5.m1.1.1">𝑒</ci></apply><cn id="S3.Ex5.m1.1.2.3.cmml" type="integer" xref="S3.Ex5.m1.1.2.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Ex5.m1.1c">\displaystyle\boldsymbol{x}(e)\geq 0</annotation><annotation encoding="application/x-llamapun" id="S3.Ex5.m1.1d">bold_italic_x ( italic_e ) ≥ 0</annotation></semantics></math></span> <span class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle\forall e\in E" class="ltx_Math" display="inline" id="S3.Ex5.m2.1"><semantics id="S3.Ex5.m2.1a"><mrow id="S3.Ex5.m2.1.1" xref="S3.Ex5.m2.1.1.cmml"><mrow id="S3.Ex5.m2.1.1.2" xref="S3.Ex5.m2.1.1.2.cmml"><mo id="S3.Ex5.m2.1.1.2.1" rspace="0.167em" xref="S3.Ex5.m2.1.1.2.1.cmml">∀</mo><mi id="S3.Ex5.m2.1.1.2.2" xref="S3.Ex5.m2.1.1.2.2.cmml">e</mi></mrow><mo id="S3.Ex5.m2.1.1.1" xref="S3.Ex5.m2.1.1.1.cmml">∈</mo><mi id="S3.Ex5.m2.1.1.3" xref="S3.Ex5.m2.1.1.3.cmml">E</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.Ex5.m2.1b"><apply id="S3.Ex5.m2.1.1.cmml" xref="S3.Ex5.m2.1.1"><in id="S3.Ex5.m2.1.1.1.cmml" xref="S3.Ex5.m2.1.1.1"></in><apply id="S3.Ex5.m2.1.1.2.cmml" xref="S3.Ex5.m2.1.1.2"><csymbol cd="latexml" id="S3.Ex5.m2.1.1.2.1.cmml" xref="S3.Ex5.m2.1.1.2.1">for-all</csymbol><ci id="S3.Ex5.m2.1.1.2.2.cmml" xref="S3.Ex5.m2.1.1.2.2">𝑒</ci></apply><ci id="S3.Ex5.m2.1.1.3.cmml" xref="S3.Ex5.m2.1.1.3">𝐸</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Ex5.m2.1c">\displaystyle\forall e\in E</annotation><annotation encoding="application/x-llamapun" id="S3.Ex5.m2.1d">∀ italic_e ∈ italic_E</annotation></semantics></math></span> <span class="ltx_eqn_cell ltx_eqn_center_padright"></span></span></span> </span> </span></foreignobject></g></g></svg> <figcaption class="ltx_caption ltx_centering"><span class="ltx_tag ltx_tag_figure"><span class="ltx_text" id="S3.F1.2.1.1" style="font-size:90%;">Figure 1</span>: </span><span class="ltx_text" id="S3.F1.3.2" style="font-size:90%;">Biset-based LP relaxation of VC-SNDP</span></figcaption> </figure> <div class="ltx_proof" id="S3.SS1.SSS2.8"> <h6 class="ltx_title ltx_runin ltx_font_italic ltx_title_proof">Proof.</h6> <div class="ltx_para" id="S3.SS1.SSS2.1.p1"> <p class="ltx_p" id="S3.SS1.SSS2.1.p1.9">Let <math alttext="H=H_{1}\uplus\cdots\uplus H_{T}" class="ltx_Math" display="inline" id="S3.SS1.SSS2.1.p1.1.m1.1"><semantics id="S3.SS1.SSS2.1.p1.1.m1.1a"><mrow id="S3.SS1.SSS2.1.p1.1.m1.1.1" xref="S3.SS1.SSS2.1.p1.1.m1.1.1.cmml"><mi id="S3.SS1.SSS2.1.p1.1.m1.1.1.2" xref="S3.SS1.SSS2.1.p1.1.m1.1.1.2.cmml">H</mi><mo id="S3.SS1.SSS2.1.p1.1.m1.1.1.1" xref="S3.SS1.SSS2.1.p1.1.m1.1.1.1.cmml">=</mo><mrow id="S3.SS1.SSS2.1.p1.1.m1.1.1.3" xref="S3.SS1.SSS2.1.p1.1.m1.1.1.3.cmml"><msub id="S3.SS1.SSS2.1.p1.1.m1.1.1.3.2" xref="S3.SS1.SSS2.1.p1.1.m1.1.1.3.2.cmml"><mi id="S3.SS1.SSS2.1.p1.1.m1.1.1.3.2.2" xref="S3.SS1.SSS2.1.p1.1.m1.1.1.3.2.2.cmml">H</mi><mn id="S3.SS1.SSS2.1.p1.1.m1.1.1.3.2.3" xref="S3.SS1.SSS2.1.p1.1.m1.1.1.3.2.3.cmml">1</mn></msub><mo id="S3.SS1.SSS2.1.p1.1.m1.1.1.3.1" xref="S3.SS1.SSS2.1.p1.1.m1.1.1.3.1.cmml">⊎</mo><mi id="S3.SS1.SSS2.1.p1.1.m1.1.1.3.3" mathvariant="normal" xref="S3.SS1.SSS2.1.p1.1.m1.1.1.3.3.cmml">⋯</mi><mo id="S3.SS1.SSS2.1.p1.1.m1.1.1.3.1a" xref="S3.SS1.SSS2.1.p1.1.m1.1.1.3.1.cmml">⊎</mo><msub id="S3.SS1.SSS2.1.p1.1.m1.1.1.3.4" xref="S3.SS1.SSS2.1.p1.1.m1.1.1.3.4.cmml"><mi id="S3.SS1.SSS2.1.p1.1.m1.1.1.3.4.2" xref="S3.SS1.SSS2.1.p1.1.m1.1.1.3.4.2.cmml">H</mi><mi id="S3.SS1.SSS2.1.p1.1.m1.1.1.3.4.3" xref="S3.SS1.SSS2.1.p1.1.m1.1.1.3.4.3.cmml">T</mi></msub></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS2.1.p1.1.m1.1b"><apply id="S3.SS1.SSS2.1.p1.1.m1.1.1.cmml" xref="S3.SS1.SSS2.1.p1.1.m1.1.1"><eq id="S3.SS1.SSS2.1.p1.1.m1.1.1.1.cmml" xref="S3.SS1.SSS2.1.p1.1.m1.1.1.1"></eq><ci id="S3.SS1.SSS2.1.p1.1.m1.1.1.2.cmml" xref="S3.SS1.SSS2.1.p1.1.m1.1.1.2">𝐻</ci><apply id="S3.SS1.SSS2.1.p1.1.m1.1.1.3.cmml" xref="S3.SS1.SSS2.1.p1.1.m1.1.1.3"><ci id="S3.SS1.SSS2.1.p1.1.m1.1.1.3.1.cmml" xref="S3.SS1.SSS2.1.p1.1.m1.1.1.3.1">⊎</ci><apply id="S3.SS1.SSS2.1.p1.1.m1.1.1.3.2.cmml" xref="S3.SS1.SSS2.1.p1.1.m1.1.1.3.2"><csymbol cd="ambiguous" id="S3.SS1.SSS2.1.p1.1.m1.1.1.3.2.1.cmml" xref="S3.SS1.SSS2.1.p1.1.m1.1.1.3.2">subscript</csymbol><ci id="S3.SS1.SSS2.1.p1.1.m1.1.1.3.2.2.cmml" xref="S3.SS1.SSS2.1.p1.1.m1.1.1.3.2.2">𝐻</ci><cn id="S3.SS1.SSS2.1.p1.1.m1.1.1.3.2.3.cmml" type="integer" xref="S3.SS1.SSS2.1.p1.1.m1.1.1.3.2.3">1</cn></apply><ci id="S3.SS1.SSS2.1.p1.1.m1.1.1.3.3.cmml" xref="S3.SS1.SSS2.1.p1.1.m1.1.1.3.3">⋯</ci><apply id="S3.SS1.SSS2.1.p1.1.m1.1.1.3.4.cmml" xref="S3.SS1.SSS2.1.p1.1.m1.1.1.3.4"><csymbol cd="ambiguous" id="S3.SS1.SSS2.1.p1.1.m1.1.1.3.4.1.cmml" xref="S3.SS1.SSS2.1.p1.1.m1.1.1.3.4">subscript</csymbol><ci id="S3.SS1.SSS2.1.p1.1.m1.1.1.3.4.2.cmml" xref="S3.SS1.SSS2.1.p1.1.m1.1.1.3.4.2">𝐻</ci><ci id="S3.SS1.SSS2.1.p1.1.m1.1.1.3.4.3.cmml" xref="S3.SS1.SSS2.1.p1.1.m1.1.1.3.4.3">𝑇</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS2.1.p1.1.m1.1c">H=H_{1}\uplus\cdots\uplus H_{T}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS2.1.p1.1.m1.1d">italic_H = italic_H start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ⊎ ⋯ ⊎ italic_H start_POSTSUBSCRIPT italic_T end_POSTSUBSCRIPT</annotation></semantics></math>, where <math alttext="T=\epsilon^{-1}\log W" class="ltx_Math" display="inline" id="S3.SS1.SSS2.1.p1.2.m2.1"><semantics id="S3.SS1.SSS2.1.p1.2.m2.1a"><mrow id="S3.SS1.SSS2.1.p1.2.m2.1.1" xref="S3.SS1.SSS2.1.p1.2.m2.1.1.cmml"><mi id="S3.SS1.SSS2.1.p1.2.m2.1.1.2" xref="S3.SS1.SSS2.1.p1.2.m2.1.1.2.cmml">T</mi><mo id="S3.SS1.SSS2.1.p1.2.m2.1.1.1" xref="S3.SS1.SSS2.1.p1.2.m2.1.1.1.cmml">=</mo><mrow id="S3.SS1.SSS2.1.p1.2.m2.1.1.3" xref="S3.SS1.SSS2.1.p1.2.m2.1.1.3.cmml"><msup id="S3.SS1.SSS2.1.p1.2.m2.1.1.3.2" xref="S3.SS1.SSS2.1.p1.2.m2.1.1.3.2.cmml"><mi id="S3.SS1.SSS2.1.p1.2.m2.1.1.3.2.2" xref="S3.SS1.SSS2.1.p1.2.m2.1.1.3.2.2.cmml">ϵ</mi><mrow id="S3.SS1.SSS2.1.p1.2.m2.1.1.3.2.3" xref="S3.SS1.SSS2.1.p1.2.m2.1.1.3.2.3.cmml"><mo id="S3.SS1.SSS2.1.p1.2.m2.1.1.3.2.3a" xref="S3.SS1.SSS2.1.p1.2.m2.1.1.3.2.3.cmml">−</mo><mn id="S3.SS1.SSS2.1.p1.2.m2.1.1.3.2.3.2" xref="S3.SS1.SSS2.1.p1.2.m2.1.1.3.2.3.2.cmml">1</mn></mrow></msup><mo id="S3.SS1.SSS2.1.p1.2.m2.1.1.3.1" lspace="0.167em" xref="S3.SS1.SSS2.1.p1.2.m2.1.1.3.1.cmml">⁢</mo><mrow id="S3.SS1.SSS2.1.p1.2.m2.1.1.3.3" xref="S3.SS1.SSS2.1.p1.2.m2.1.1.3.3.cmml"><mi id="S3.SS1.SSS2.1.p1.2.m2.1.1.3.3.1" xref="S3.SS1.SSS2.1.p1.2.m2.1.1.3.3.1.cmml">log</mi><mo id="S3.SS1.SSS2.1.p1.2.m2.1.1.3.3a" lspace="0.167em" xref="S3.SS1.SSS2.1.p1.2.m2.1.1.3.3.cmml">⁡</mo><mi id="S3.SS1.SSS2.1.p1.2.m2.1.1.3.3.2" xref="S3.SS1.SSS2.1.p1.2.m2.1.1.3.3.2.cmml">W</mi></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS2.1.p1.2.m2.1b"><apply id="S3.SS1.SSS2.1.p1.2.m2.1.1.cmml" xref="S3.SS1.SSS2.1.p1.2.m2.1.1"><eq id="S3.SS1.SSS2.1.p1.2.m2.1.1.1.cmml" xref="S3.SS1.SSS2.1.p1.2.m2.1.1.1"></eq><ci id="S3.SS1.SSS2.1.p1.2.m2.1.1.2.cmml" xref="S3.SS1.SSS2.1.p1.2.m2.1.1.2">𝑇</ci><apply id="S3.SS1.SSS2.1.p1.2.m2.1.1.3.cmml" xref="S3.SS1.SSS2.1.p1.2.m2.1.1.3"><times id="S3.SS1.SSS2.1.p1.2.m2.1.1.3.1.cmml" xref="S3.SS1.SSS2.1.p1.2.m2.1.1.3.1"></times><apply id="S3.SS1.SSS2.1.p1.2.m2.1.1.3.2.cmml" xref="S3.SS1.SSS2.1.p1.2.m2.1.1.3.2"><csymbol cd="ambiguous" id="S3.SS1.SSS2.1.p1.2.m2.1.1.3.2.1.cmml" xref="S3.SS1.SSS2.1.p1.2.m2.1.1.3.2">superscript</csymbol><ci id="S3.SS1.SSS2.1.p1.2.m2.1.1.3.2.2.cmml" xref="S3.SS1.SSS2.1.p1.2.m2.1.1.3.2.2">italic-ϵ</ci><apply id="S3.SS1.SSS2.1.p1.2.m2.1.1.3.2.3.cmml" xref="S3.SS1.SSS2.1.p1.2.m2.1.1.3.2.3"><minus id="S3.SS1.SSS2.1.p1.2.m2.1.1.3.2.3.1.cmml" xref="S3.SS1.SSS2.1.p1.2.m2.1.1.3.2.3"></minus><cn id="S3.SS1.SSS2.1.p1.2.m2.1.1.3.2.3.2.cmml" type="integer" xref="S3.SS1.SSS2.1.p1.2.m2.1.1.3.2.3.2">1</cn></apply></apply><apply id="S3.SS1.SSS2.1.p1.2.m2.1.1.3.3.cmml" xref="S3.SS1.SSS2.1.p1.2.m2.1.1.3.3"><log id="S3.SS1.SSS2.1.p1.2.m2.1.1.3.3.1.cmml" xref="S3.SS1.SSS2.1.p1.2.m2.1.1.3.3.1"></log><ci id="S3.SS1.SSS2.1.p1.2.m2.1.1.3.3.2.cmml" xref="S3.SS1.SSS2.1.p1.2.m2.1.1.3.3.2">𝑊</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS2.1.p1.2.m2.1c">T=\epsilon^{-1}\log W</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS2.1.p1.2.m2.1d">italic_T = italic_ϵ start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT roman_log italic_W</annotation></semantics></math>, be the output of Algorithm <a class="ltx_ref" href="https://arxiv.org/html/2503.00712v1#algorithm2" title="In Weighted graphs. ‣ 2.1 Fault-Tolerant Spanners in Streaming ‣ 2 Preliminaries ‣ Streaming Algorithms for Network Design"><span class="ltx_text ltx_ref_tag">2</span></a> with <math alttext="f=(2t-2)(2k-1)" class="ltx_Math" display="inline" id="S3.SS1.SSS2.1.p1.3.m3.2"><semantics id="S3.SS1.SSS2.1.p1.3.m3.2a"><mrow id="S3.SS1.SSS2.1.p1.3.m3.2.2" xref="S3.SS1.SSS2.1.p1.3.m3.2.2.cmml"><mi id="S3.SS1.SSS2.1.p1.3.m3.2.2.4" xref="S3.SS1.SSS2.1.p1.3.m3.2.2.4.cmml">f</mi><mo id="S3.SS1.SSS2.1.p1.3.m3.2.2.3" xref="S3.SS1.SSS2.1.p1.3.m3.2.2.3.cmml">=</mo><mrow id="S3.SS1.SSS2.1.p1.3.m3.2.2.2" xref="S3.SS1.SSS2.1.p1.3.m3.2.2.2.cmml"><mrow id="S3.SS1.SSS2.1.p1.3.m3.1.1.1.1.1" xref="S3.SS1.SSS2.1.p1.3.m3.1.1.1.1.1.1.cmml"><mo id="S3.SS1.SSS2.1.p1.3.m3.1.1.1.1.1.2" stretchy="false" xref="S3.SS1.SSS2.1.p1.3.m3.1.1.1.1.1.1.cmml">(</mo><mrow id="S3.SS1.SSS2.1.p1.3.m3.1.1.1.1.1.1" xref="S3.SS1.SSS2.1.p1.3.m3.1.1.1.1.1.1.cmml"><mrow id="S3.SS1.SSS2.1.p1.3.m3.1.1.1.1.1.1.2" xref="S3.SS1.SSS2.1.p1.3.m3.1.1.1.1.1.1.2.cmml"><mn id="S3.SS1.SSS2.1.p1.3.m3.1.1.1.1.1.1.2.2" xref="S3.SS1.SSS2.1.p1.3.m3.1.1.1.1.1.1.2.2.cmml">2</mn><mo id="S3.SS1.SSS2.1.p1.3.m3.1.1.1.1.1.1.2.1" xref="S3.SS1.SSS2.1.p1.3.m3.1.1.1.1.1.1.2.1.cmml">⁢</mo><mi id="S3.SS1.SSS2.1.p1.3.m3.1.1.1.1.1.1.2.3" xref="S3.SS1.SSS2.1.p1.3.m3.1.1.1.1.1.1.2.3.cmml">t</mi></mrow><mo id="S3.SS1.SSS2.1.p1.3.m3.1.1.1.1.1.1.1" xref="S3.SS1.SSS2.1.p1.3.m3.1.1.1.1.1.1.1.cmml">−</mo><mn id="S3.SS1.SSS2.1.p1.3.m3.1.1.1.1.1.1.3" xref="S3.SS1.SSS2.1.p1.3.m3.1.1.1.1.1.1.3.cmml">2</mn></mrow><mo id="S3.SS1.SSS2.1.p1.3.m3.1.1.1.1.1.3" stretchy="false" xref="S3.SS1.SSS2.1.p1.3.m3.1.1.1.1.1.1.cmml">)</mo></mrow><mo id="S3.SS1.SSS2.1.p1.3.m3.2.2.2.3" xref="S3.SS1.SSS2.1.p1.3.m3.2.2.2.3.cmml">⁢</mo><mrow id="S3.SS1.SSS2.1.p1.3.m3.2.2.2.2.1" xref="S3.SS1.SSS2.1.p1.3.m3.2.2.2.2.1.1.cmml"><mo id="S3.SS1.SSS2.1.p1.3.m3.2.2.2.2.1.2" stretchy="false" xref="S3.SS1.SSS2.1.p1.3.m3.2.2.2.2.1.1.cmml">(</mo><mrow id="S3.SS1.SSS2.1.p1.3.m3.2.2.2.2.1.1" xref="S3.SS1.SSS2.1.p1.3.m3.2.2.2.2.1.1.cmml"><mrow id="S3.SS1.SSS2.1.p1.3.m3.2.2.2.2.1.1.2" xref="S3.SS1.SSS2.1.p1.3.m3.2.2.2.2.1.1.2.cmml"><mn id="S3.SS1.SSS2.1.p1.3.m3.2.2.2.2.1.1.2.2" xref="S3.SS1.SSS2.1.p1.3.m3.2.2.2.2.1.1.2.2.cmml">2</mn><mo id="S3.SS1.SSS2.1.p1.3.m3.2.2.2.2.1.1.2.1" xref="S3.SS1.SSS2.1.p1.3.m3.2.2.2.2.1.1.2.1.cmml">⁢</mo><mi id="S3.SS1.SSS2.1.p1.3.m3.2.2.2.2.1.1.2.3" xref="S3.SS1.SSS2.1.p1.3.m3.2.2.2.2.1.1.2.3.cmml">k</mi></mrow><mo id="S3.SS1.SSS2.1.p1.3.m3.2.2.2.2.1.1.1" xref="S3.SS1.SSS2.1.p1.3.m3.2.2.2.2.1.1.1.cmml">−</mo><mn id="S3.SS1.SSS2.1.p1.3.m3.2.2.2.2.1.1.3" xref="S3.SS1.SSS2.1.p1.3.m3.2.2.2.2.1.1.3.cmml">1</mn></mrow><mo id="S3.SS1.SSS2.1.p1.3.m3.2.2.2.2.1.3" stretchy="false" xref="S3.SS1.SSS2.1.p1.3.m3.2.2.2.2.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS2.1.p1.3.m3.2b"><apply id="S3.SS1.SSS2.1.p1.3.m3.2.2.cmml" xref="S3.SS1.SSS2.1.p1.3.m3.2.2"><eq id="S3.SS1.SSS2.1.p1.3.m3.2.2.3.cmml" xref="S3.SS1.SSS2.1.p1.3.m3.2.2.3"></eq><ci id="S3.SS1.SSS2.1.p1.3.m3.2.2.4.cmml" xref="S3.SS1.SSS2.1.p1.3.m3.2.2.4">𝑓</ci><apply id="S3.SS1.SSS2.1.p1.3.m3.2.2.2.cmml" xref="S3.SS1.SSS2.1.p1.3.m3.2.2.2"><times id="S3.SS1.SSS2.1.p1.3.m3.2.2.2.3.cmml" xref="S3.SS1.SSS2.1.p1.3.m3.2.2.2.3"></times><apply id="S3.SS1.SSS2.1.p1.3.m3.1.1.1.1.1.1.cmml" xref="S3.SS1.SSS2.1.p1.3.m3.1.1.1.1.1"><minus id="S3.SS1.SSS2.1.p1.3.m3.1.1.1.1.1.1.1.cmml" xref="S3.SS1.SSS2.1.p1.3.m3.1.1.1.1.1.1.1"></minus><apply id="S3.SS1.SSS2.1.p1.3.m3.1.1.1.1.1.1.2.cmml" xref="S3.SS1.SSS2.1.p1.3.m3.1.1.1.1.1.1.2"><times id="S3.SS1.SSS2.1.p1.3.m3.1.1.1.1.1.1.2.1.cmml" xref="S3.SS1.SSS2.1.p1.3.m3.1.1.1.1.1.1.2.1"></times><cn id="S3.SS1.SSS2.1.p1.3.m3.1.1.1.1.1.1.2.2.cmml" type="integer" xref="S3.SS1.SSS2.1.p1.3.m3.1.1.1.1.1.1.2.2">2</cn><ci id="S3.SS1.SSS2.1.p1.3.m3.1.1.1.1.1.1.2.3.cmml" xref="S3.SS1.SSS2.1.p1.3.m3.1.1.1.1.1.1.2.3">𝑡</ci></apply><cn id="S3.SS1.SSS2.1.p1.3.m3.1.1.1.1.1.1.3.cmml" type="integer" xref="S3.SS1.SSS2.1.p1.3.m3.1.1.1.1.1.1.3">2</cn></apply><apply id="S3.SS1.SSS2.1.p1.3.m3.2.2.2.2.1.1.cmml" xref="S3.SS1.SSS2.1.p1.3.m3.2.2.2.2.1"><minus id="S3.SS1.SSS2.1.p1.3.m3.2.2.2.2.1.1.1.cmml" xref="S3.SS1.SSS2.1.p1.3.m3.2.2.2.2.1.1.1"></minus><apply id="S3.SS1.SSS2.1.p1.3.m3.2.2.2.2.1.1.2.cmml" xref="S3.SS1.SSS2.1.p1.3.m3.2.2.2.2.1.1.2"><times id="S3.SS1.SSS2.1.p1.3.m3.2.2.2.2.1.1.2.1.cmml" xref="S3.SS1.SSS2.1.p1.3.m3.2.2.2.2.1.1.2.1"></times><cn id="S3.SS1.SSS2.1.p1.3.m3.2.2.2.2.1.1.2.2.cmml" type="integer" xref="S3.SS1.SSS2.1.p1.3.m3.2.2.2.2.1.1.2.2">2</cn><ci id="S3.SS1.SSS2.1.p1.3.m3.2.2.2.2.1.1.2.3.cmml" xref="S3.SS1.SSS2.1.p1.3.m3.2.2.2.2.1.1.2.3">𝑘</ci></apply><cn id="S3.SS1.SSS2.1.p1.3.m3.2.2.2.2.1.1.3.cmml" type="integer" xref="S3.SS1.SSS2.1.p1.3.m3.2.2.2.2.1.1.3">1</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS2.1.p1.3.m3.2c">f=(2t-2)(2k-1)</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS2.1.p1.3.m3.2d">italic_f = ( 2 italic_t - 2 ) ( 2 italic_k - 1 )</annotation></semantics></math>. Recall that for every <math alttext="1\leq j\leq T" class="ltx_Math" display="inline" id="S3.SS1.SSS2.1.p1.4.m4.1"><semantics id="S3.SS1.SSS2.1.p1.4.m4.1a"><mrow id="S3.SS1.SSS2.1.p1.4.m4.1.1" xref="S3.SS1.SSS2.1.p1.4.m4.1.1.cmml"><mn id="S3.SS1.SSS2.1.p1.4.m4.1.1.2" xref="S3.SS1.SSS2.1.p1.4.m4.1.1.2.cmml">1</mn><mo id="S3.SS1.SSS2.1.p1.4.m4.1.1.3" xref="S3.SS1.SSS2.1.p1.4.m4.1.1.3.cmml">≤</mo><mi id="S3.SS1.SSS2.1.p1.4.m4.1.1.4" xref="S3.SS1.SSS2.1.p1.4.m4.1.1.4.cmml">j</mi><mo id="S3.SS1.SSS2.1.p1.4.m4.1.1.5" xref="S3.SS1.SSS2.1.p1.4.m4.1.1.5.cmml">≤</mo><mi id="S3.SS1.SSS2.1.p1.4.m4.1.1.6" xref="S3.SS1.SSS2.1.p1.4.m4.1.1.6.cmml">T</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS2.1.p1.4.m4.1b"><apply id="S3.SS1.SSS2.1.p1.4.m4.1.1.cmml" xref="S3.SS1.SSS2.1.p1.4.m4.1.1"><and id="S3.SS1.SSS2.1.p1.4.m4.1.1a.cmml" xref="S3.SS1.SSS2.1.p1.4.m4.1.1"></and><apply id="S3.SS1.SSS2.1.p1.4.m4.1.1b.cmml" xref="S3.SS1.SSS2.1.p1.4.m4.1.1"><leq id="S3.SS1.SSS2.1.p1.4.m4.1.1.3.cmml" xref="S3.SS1.SSS2.1.p1.4.m4.1.1.3"></leq><cn id="S3.SS1.SSS2.1.p1.4.m4.1.1.2.cmml" type="integer" xref="S3.SS1.SSS2.1.p1.4.m4.1.1.2">1</cn><ci id="S3.SS1.SSS2.1.p1.4.m4.1.1.4.cmml" xref="S3.SS1.SSS2.1.p1.4.m4.1.1.4">𝑗</ci></apply><apply id="S3.SS1.SSS2.1.p1.4.m4.1.1c.cmml" xref="S3.SS1.SSS2.1.p1.4.m4.1.1"><leq id="S3.SS1.SSS2.1.p1.4.m4.1.1.5.cmml" xref="S3.SS1.SSS2.1.p1.4.m4.1.1.5"></leq><share href="https://arxiv.org/html/2503.00712v1#S3.SS1.SSS2.1.p1.4.m4.1.1.4.cmml" id="S3.SS1.SSS2.1.p1.4.m4.1.1d.cmml" xref="S3.SS1.SSS2.1.p1.4.m4.1.1"></share><ci id="S3.SS1.SSS2.1.p1.4.m4.1.1.6.cmml" xref="S3.SS1.SSS2.1.p1.4.m4.1.1.6">𝑇</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS2.1.p1.4.m4.1c">1\leq j\leq T</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS2.1.p1.4.m4.1d">1 ≤ italic_j ≤ italic_T</annotation></semantics></math>, <math alttext="H_{j}" class="ltx_Math" display="inline" id="S3.SS1.SSS2.1.p1.5.m5.1"><semantics id="S3.SS1.SSS2.1.p1.5.m5.1a"><msub id="S3.SS1.SSS2.1.p1.5.m5.1.1" xref="S3.SS1.SSS2.1.p1.5.m5.1.1.cmml"><mi id="S3.SS1.SSS2.1.p1.5.m5.1.1.2" xref="S3.SS1.SSS2.1.p1.5.m5.1.1.2.cmml">H</mi><mi id="S3.SS1.SSS2.1.p1.5.m5.1.1.3" xref="S3.SS1.SSS2.1.p1.5.m5.1.1.3.cmml">j</mi></msub><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS2.1.p1.5.m5.1b"><apply id="S3.SS1.SSS2.1.p1.5.m5.1.1.cmml" xref="S3.SS1.SSS2.1.p1.5.m5.1.1"><csymbol cd="ambiguous" id="S3.SS1.SSS2.1.p1.5.m5.1.1.1.cmml" xref="S3.SS1.SSS2.1.p1.5.m5.1.1">subscript</csymbol><ci id="S3.SS1.SSS2.1.p1.5.m5.1.1.2.cmml" xref="S3.SS1.SSS2.1.p1.5.m5.1.1.2">𝐻</ci><ci id="S3.SS1.SSS2.1.p1.5.m5.1.1.3.cmml" xref="S3.SS1.SSS2.1.p1.5.m5.1.1.3">𝑗</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS2.1.p1.5.m5.1c">H_{j}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS2.1.p1.5.m5.1d">italic_H start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT</annotation></semantics></math> is a <math alttext="\big{(}(2t-2)(2k-1)\big{)}" class="ltx_Math" display="inline" id="S3.SS1.SSS2.1.p1.6.m6.1"><semantics id="S3.SS1.SSS2.1.p1.6.m6.1a"><mrow id="S3.SS1.SSS2.1.p1.6.m6.1.1.1" xref="S3.SS1.SSS2.1.p1.6.m6.1.1.1.1.cmml"><mo id="S3.SS1.SSS2.1.p1.6.m6.1.1.1.2" maxsize="120%" minsize="120%" xref="S3.SS1.SSS2.1.p1.6.m6.1.1.1.1.cmml">(</mo><mrow id="S3.SS1.SSS2.1.p1.6.m6.1.1.1.1" xref="S3.SS1.SSS2.1.p1.6.m6.1.1.1.1.cmml"><mrow id="S3.SS1.SSS2.1.p1.6.m6.1.1.1.1.1.1" xref="S3.SS1.SSS2.1.p1.6.m6.1.1.1.1.1.1.1.cmml"><mo id="S3.SS1.SSS2.1.p1.6.m6.1.1.1.1.1.1.2" stretchy="false" xref="S3.SS1.SSS2.1.p1.6.m6.1.1.1.1.1.1.1.cmml">(</mo><mrow id="S3.SS1.SSS2.1.p1.6.m6.1.1.1.1.1.1.1" xref="S3.SS1.SSS2.1.p1.6.m6.1.1.1.1.1.1.1.cmml"><mrow id="S3.SS1.SSS2.1.p1.6.m6.1.1.1.1.1.1.1.2" xref="S3.SS1.SSS2.1.p1.6.m6.1.1.1.1.1.1.1.2.cmml"><mn id="S3.SS1.SSS2.1.p1.6.m6.1.1.1.1.1.1.1.2.2" xref="S3.SS1.SSS2.1.p1.6.m6.1.1.1.1.1.1.1.2.2.cmml">2</mn><mo id="S3.SS1.SSS2.1.p1.6.m6.1.1.1.1.1.1.1.2.1" xref="S3.SS1.SSS2.1.p1.6.m6.1.1.1.1.1.1.1.2.1.cmml">⁢</mo><mi id="S3.SS1.SSS2.1.p1.6.m6.1.1.1.1.1.1.1.2.3" xref="S3.SS1.SSS2.1.p1.6.m6.1.1.1.1.1.1.1.2.3.cmml">t</mi></mrow><mo id="S3.SS1.SSS2.1.p1.6.m6.1.1.1.1.1.1.1.1" xref="S3.SS1.SSS2.1.p1.6.m6.1.1.1.1.1.1.1.1.cmml">−</mo><mn id="S3.SS1.SSS2.1.p1.6.m6.1.1.1.1.1.1.1.3" xref="S3.SS1.SSS2.1.p1.6.m6.1.1.1.1.1.1.1.3.cmml">2</mn></mrow><mo id="S3.SS1.SSS2.1.p1.6.m6.1.1.1.1.1.1.3" stretchy="false" xref="S3.SS1.SSS2.1.p1.6.m6.1.1.1.1.1.1.1.cmml">)</mo></mrow><mo id="S3.SS1.SSS2.1.p1.6.m6.1.1.1.1.3" xref="S3.SS1.SSS2.1.p1.6.m6.1.1.1.1.3.cmml">⁢</mo><mrow id="S3.SS1.SSS2.1.p1.6.m6.1.1.1.1.2.1" xref="S3.SS1.SSS2.1.p1.6.m6.1.1.1.1.2.1.1.cmml"><mo id="S3.SS1.SSS2.1.p1.6.m6.1.1.1.1.2.1.2" stretchy="false" xref="S3.SS1.SSS2.1.p1.6.m6.1.1.1.1.2.1.1.cmml">(</mo><mrow id="S3.SS1.SSS2.1.p1.6.m6.1.1.1.1.2.1.1" xref="S3.SS1.SSS2.1.p1.6.m6.1.1.1.1.2.1.1.cmml"><mrow id="S3.SS1.SSS2.1.p1.6.m6.1.1.1.1.2.1.1.2" xref="S3.SS1.SSS2.1.p1.6.m6.1.1.1.1.2.1.1.2.cmml"><mn id="S3.SS1.SSS2.1.p1.6.m6.1.1.1.1.2.1.1.2.2" xref="S3.SS1.SSS2.1.p1.6.m6.1.1.1.1.2.1.1.2.2.cmml">2</mn><mo id="S3.SS1.SSS2.1.p1.6.m6.1.1.1.1.2.1.1.2.1" xref="S3.SS1.SSS2.1.p1.6.m6.1.1.1.1.2.1.1.2.1.cmml">⁢</mo><mi id="S3.SS1.SSS2.1.p1.6.m6.1.1.1.1.2.1.1.2.3" xref="S3.SS1.SSS2.1.p1.6.m6.1.1.1.1.2.1.1.2.3.cmml">k</mi></mrow><mo id="S3.SS1.SSS2.1.p1.6.m6.1.1.1.1.2.1.1.1" xref="S3.SS1.SSS2.1.p1.6.m6.1.1.1.1.2.1.1.1.cmml">−</mo><mn id="S3.SS1.SSS2.1.p1.6.m6.1.1.1.1.2.1.1.3" xref="S3.SS1.SSS2.1.p1.6.m6.1.1.1.1.2.1.1.3.cmml">1</mn></mrow><mo id="S3.SS1.SSS2.1.p1.6.m6.1.1.1.1.2.1.3" stretchy="false" xref="S3.SS1.SSS2.1.p1.6.m6.1.1.1.1.2.1.1.cmml">)</mo></mrow></mrow><mo id="S3.SS1.SSS2.1.p1.6.m6.1.1.1.3" maxsize="120%" minsize="120%" xref="S3.SS1.SSS2.1.p1.6.m6.1.1.1.1.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS2.1.p1.6.m6.1b"><apply id="S3.SS1.SSS2.1.p1.6.m6.1.1.1.1.cmml" xref="S3.SS1.SSS2.1.p1.6.m6.1.1.1"><times id="S3.SS1.SSS2.1.p1.6.m6.1.1.1.1.3.cmml" xref="S3.SS1.SSS2.1.p1.6.m6.1.1.1.1.3"></times><apply id="S3.SS1.SSS2.1.p1.6.m6.1.1.1.1.1.1.1.cmml" xref="S3.SS1.SSS2.1.p1.6.m6.1.1.1.1.1.1"><minus id="S3.SS1.SSS2.1.p1.6.m6.1.1.1.1.1.1.1.1.cmml" xref="S3.SS1.SSS2.1.p1.6.m6.1.1.1.1.1.1.1.1"></minus><apply id="S3.SS1.SSS2.1.p1.6.m6.1.1.1.1.1.1.1.2.cmml" xref="S3.SS1.SSS2.1.p1.6.m6.1.1.1.1.1.1.1.2"><times id="S3.SS1.SSS2.1.p1.6.m6.1.1.1.1.1.1.1.2.1.cmml" xref="S3.SS1.SSS2.1.p1.6.m6.1.1.1.1.1.1.1.2.1"></times><cn id="S3.SS1.SSS2.1.p1.6.m6.1.1.1.1.1.1.1.2.2.cmml" type="integer" xref="S3.SS1.SSS2.1.p1.6.m6.1.1.1.1.1.1.1.2.2">2</cn><ci id="S3.SS1.SSS2.1.p1.6.m6.1.1.1.1.1.1.1.2.3.cmml" xref="S3.SS1.SSS2.1.p1.6.m6.1.1.1.1.1.1.1.2.3">𝑡</ci></apply><cn id="S3.SS1.SSS2.1.p1.6.m6.1.1.1.1.1.1.1.3.cmml" type="integer" xref="S3.SS1.SSS2.1.p1.6.m6.1.1.1.1.1.1.1.3">2</cn></apply><apply id="S3.SS1.SSS2.1.p1.6.m6.1.1.1.1.2.1.1.cmml" xref="S3.SS1.SSS2.1.p1.6.m6.1.1.1.1.2.1"><minus id="S3.SS1.SSS2.1.p1.6.m6.1.1.1.1.2.1.1.1.cmml" xref="S3.SS1.SSS2.1.p1.6.m6.1.1.1.1.2.1.1.1"></minus><apply id="S3.SS1.SSS2.1.p1.6.m6.1.1.1.1.2.1.1.2.cmml" xref="S3.SS1.SSS2.1.p1.6.m6.1.1.1.1.2.1.1.2"><times id="S3.SS1.SSS2.1.p1.6.m6.1.1.1.1.2.1.1.2.1.cmml" xref="S3.SS1.SSS2.1.p1.6.m6.1.1.1.1.2.1.1.2.1"></times><cn id="S3.SS1.SSS2.1.p1.6.m6.1.1.1.1.2.1.1.2.2.cmml" type="integer" xref="S3.SS1.SSS2.1.p1.6.m6.1.1.1.1.2.1.1.2.2">2</cn><ci id="S3.SS1.SSS2.1.p1.6.m6.1.1.1.1.2.1.1.2.3.cmml" xref="S3.SS1.SSS2.1.p1.6.m6.1.1.1.1.2.1.1.2.3">𝑘</ci></apply><cn id="S3.SS1.SSS2.1.p1.6.m6.1.1.1.1.2.1.1.3.cmml" type="integer" xref="S3.SS1.SSS2.1.p1.6.m6.1.1.1.1.2.1.1.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS2.1.p1.6.m6.1c">\big{(}(2t-2)(2k-1)\big{)}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS2.1.p1.6.m6.1d">( ( 2 italic_t - 2 ) ( 2 italic_k - 1 ) )</annotation></semantics></math>-VFT <math alttext="(2t-1)" class="ltx_Math" display="inline" id="S3.SS1.SSS2.1.p1.7.m7.1"><semantics id="S3.SS1.SSS2.1.p1.7.m7.1a"><mrow id="S3.SS1.SSS2.1.p1.7.m7.1.1.1" xref="S3.SS1.SSS2.1.p1.7.m7.1.1.1.1.cmml"><mo id="S3.SS1.SSS2.1.p1.7.m7.1.1.1.2" stretchy="false" xref="S3.SS1.SSS2.1.p1.7.m7.1.1.1.1.cmml">(</mo><mrow id="S3.SS1.SSS2.1.p1.7.m7.1.1.1.1" xref="S3.SS1.SSS2.1.p1.7.m7.1.1.1.1.cmml"><mrow id="S3.SS1.SSS2.1.p1.7.m7.1.1.1.1.2" xref="S3.SS1.SSS2.1.p1.7.m7.1.1.1.1.2.cmml"><mn id="S3.SS1.SSS2.1.p1.7.m7.1.1.1.1.2.2" xref="S3.SS1.SSS2.1.p1.7.m7.1.1.1.1.2.2.cmml">2</mn><mo id="S3.SS1.SSS2.1.p1.7.m7.1.1.1.1.2.1" xref="S3.SS1.SSS2.1.p1.7.m7.1.1.1.1.2.1.cmml">⁢</mo><mi id="S3.SS1.SSS2.1.p1.7.m7.1.1.1.1.2.3" xref="S3.SS1.SSS2.1.p1.7.m7.1.1.1.1.2.3.cmml">t</mi></mrow><mo id="S3.SS1.SSS2.1.p1.7.m7.1.1.1.1.1" xref="S3.SS1.SSS2.1.p1.7.m7.1.1.1.1.1.cmml">−</mo><mn id="S3.SS1.SSS2.1.p1.7.m7.1.1.1.1.3" xref="S3.SS1.SSS2.1.p1.7.m7.1.1.1.1.3.cmml">1</mn></mrow><mo id="S3.SS1.SSS2.1.p1.7.m7.1.1.1.3" stretchy="false" xref="S3.SS1.SSS2.1.p1.7.m7.1.1.1.1.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS2.1.p1.7.m7.1b"><apply id="S3.SS1.SSS2.1.p1.7.m7.1.1.1.1.cmml" xref="S3.SS1.SSS2.1.p1.7.m7.1.1.1"><minus id="S3.SS1.SSS2.1.p1.7.m7.1.1.1.1.1.cmml" xref="S3.SS1.SSS2.1.p1.7.m7.1.1.1.1.1"></minus><apply id="S3.SS1.SSS2.1.p1.7.m7.1.1.1.1.2.cmml" xref="S3.SS1.SSS2.1.p1.7.m7.1.1.1.1.2"><times id="S3.SS1.SSS2.1.p1.7.m7.1.1.1.1.2.1.cmml" xref="S3.SS1.SSS2.1.p1.7.m7.1.1.1.1.2.1"></times><cn id="S3.SS1.SSS2.1.p1.7.m7.1.1.1.1.2.2.cmml" type="integer" xref="S3.SS1.SSS2.1.p1.7.m7.1.1.1.1.2.2">2</cn><ci id="S3.SS1.SSS2.1.p1.7.m7.1.1.1.1.2.3.cmml" xref="S3.SS1.SSS2.1.p1.7.m7.1.1.1.1.2.3">𝑡</ci></apply><cn id="S3.SS1.SSS2.1.p1.7.m7.1.1.1.1.3.cmml" type="integer" xref="S3.SS1.SSS2.1.p1.7.m7.1.1.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS2.1.p1.7.m7.1c">(2t-1)</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS2.1.p1.7.m7.1d">( 2 italic_t - 1 )</annotation></semantics></math>-spanner for the edges in <math alttext="G" class="ltx_Math" display="inline" id="S3.SS1.SSS2.1.p1.8.m8.1"><semantics id="S3.SS1.SSS2.1.p1.8.m8.1a"><mi id="S3.SS1.SSS2.1.p1.8.m8.1.1" xref="S3.SS1.SSS2.1.p1.8.m8.1.1.cmml">G</mi><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS2.1.p1.8.m8.1b"><ci id="S3.SS1.SSS2.1.p1.8.m8.1.1.cmml" xref="S3.SS1.SSS2.1.p1.8.m8.1.1">𝐺</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS2.1.p1.8.m8.1c">G</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS2.1.p1.8.m8.1d">italic_G</annotation></semantics></math> with weights in <math alttext="B_{j}\coloneqq((1+\epsilon)^{j-1},(1+\epsilon)^{j}]" class="ltx_Math" display="inline" id="S3.SS1.SSS2.1.p1.9.m9.2"><semantics id="S3.SS1.SSS2.1.p1.9.m9.2a"><mrow id="S3.SS1.SSS2.1.p1.9.m9.2.2" xref="S3.SS1.SSS2.1.p1.9.m9.2.2.cmml"><msub id="S3.SS1.SSS2.1.p1.9.m9.2.2.4" xref="S3.SS1.SSS2.1.p1.9.m9.2.2.4.cmml"><mi id="S3.SS1.SSS2.1.p1.9.m9.2.2.4.2" xref="S3.SS1.SSS2.1.p1.9.m9.2.2.4.2.cmml">B</mi><mi id="S3.SS1.SSS2.1.p1.9.m9.2.2.4.3" xref="S3.SS1.SSS2.1.p1.9.m9.2.2.4.3.cmml">j</mi></msub><mo id="S3.SS1.SSS2.1.p1.9.m9.2.2.3" xref="S3.SS1.SSS2.1.p1.9.m9.2.2.3.cmml">≔</mo><mrow id="S3.SS1.SSS2.1.p1.9.m9.2.2.2.2" xref="S3.SS1.SSS2.1.p1.9.m9.2.2.2.3.cmml"><mo id="S3.SS1.SSS2.1.p1.9.m9.2.2.2.2.3" stretchy="false" xref="S3.SS1.SSS2.1.p1.9.m9.2.2.2.3.cmml">(</mo><msup id="S3.SS1.SSS2.1.p1.9.m9.1.1.1.1.1" xref="S3.SS1.SSS2.1.p1.9.m9.1.1.1.1.1.cmml"><mrow id="S3.SS1.SSS2.1.p1.9.m9.1.1.1.1.1.1.1" xref="S3.SS1.SSS2.1.p1.9.m9.1.1.1.1.1.1.1.1.cmml"><mo id="S3.SS1.SSS2.1.p1.9.m9.1.1.1.1.1.1.1.2" stretchy="false" xref="S3.SS1.SSS2.1.p1.9.m9.1.1.1.1.1.1.1.1.cmml">(</mo><mrow id="S3.SS1.SSS2.1.p1.9.m9.1.1.1.1.1.1.1.1" xref="S3.SS1.SSS2.1.p1.9.m9.1.1.1.1.1.1.1.1.cmml"><mn id="S3.SS1.SSS2.1.p1.9.m9.1.1.1.1.1.1.1.1.2" xref="S3.SS1.SSS2.1.p1.9.m9.1.1.1.1.1.1.1.1.2.cmml">1</mn><mo id="S3.SS1.SSS2.1.p1.9.m9.1.1.1.1.1.1.1.1.1" xref="S3.SS1.SSS2.1.p1.9.m9.1.1.1.1.1.1.1.1.1.cmml">+</mo><mi id="S3.SS1.SSS2.1.p1.9.m9.1.1.1.1.1.1.1.1.3" xref="S3.SS1.SSS2.1.p1.9.m9.1.1.1.1.1.1.1.1.3.cmml">ϵ</mi></mrow><mo id="S3.SS1.SSS2.1.p1.9.m9.1.1.1.1.1.1.1.3" stretchy="false" xref="S3.SS1.SSS2.1.p1.9.m9.1.1.1.1.1.1.1.1.cmml">)</mo></mrow><mrow id="S3.SS1.SSS2.1.p1.9.m9.1.1.1.1.1.3" xref="S3.SS1.SSS2.1.p1.9.m9.1.1.1.1.1.3.cmml"><mi id="S3.SS1.SSS2.1.p1.9.m9.1.1.1.1.1.3.2" xref="S3.SS1.SSS2.1.p1.9.m9.1.1.1.1.1.3.2.cmml">j</mi><mo id="S3.SS1.SSS2.1.p1.9.m9.1.1.1.1.1.3.1" xref="S3.SS1.SSS2.1.p1.9.m9.1.1.1.1.1.3.1.cmml">−</mo><mn id="S3.SS1.SSS2.1.p1.9.m9.1.1.1.1.1.3.3" xref="S3.SS1.SSS2.1.p1.9.m9.1.1.1.1.1.3.3.cmml">1</mn></mrow></msup><mo id="S3.SS1.SSS2.1.p1.9.m9.2.2.2.2.4" xref="S3.SS1.SSS2.1.p1.9.m9.2.2.2.3.cmml">,</mo><msup id="S3.SS1.SSS2.1.p1.9.m9.2.2.2.2.2" xref="S3.SS1.SSS2.1.p1.9.m9.2.2.2.2.2.cmml"><mrow id="S3.SS1.SSS2.1.p1.9.m9.2.2.2.2.2.1.1" xref="S3.SS1.SSS2.1.p1.9.m9.2.2.2.2.2.1.1.1.cmml"><mo id="S3.SS1.SSS2.1.p1.9.m9.2.2.2.2.2.1.1.2" stretchy="false" xref="S3.SS1.SSS2.1.p1.9.m9.2.2.2.2.2.1.1.1.cmml">(</mo><mrow id="S3.SS1.SSS2.1.p1.9.m9.2.2.2.2.2.1.1.1" xref="S3.SS1.SSS2.1.p1.9.m9.2.2.2.2.2.1.1.1.cmml"><mn id="S3.SS1.SSS2.1.p1.9.m9.2.2.2.2.2.1.1.1.2" xref="S3.SS1.SSS2.1.p1.9.m9.2.2.2.2.2.1.1.1.2.cmml">1</mn><mo id="S3.SS1.SSS2.1.p1.9.m9.2.2.2.2.2.1.1.1.1" xref="S3.SS1.SSS2.1.p1.9.m9.2.2.2.2.2.1.1.1.1.cmml">+</mo><mi id="S3.SS1.SSS2.1.p1.9.m9.2.2.2.2.2.1.1.1.3" xref="S3.SS1.SSS2.1.p1.9.m9.2.2.2.2.2.1.1.1.3.cmml">ϵ</mi></mrow><mo id="S3.SS1.SSS2.1.p1.9.m9.2.2.2.2.2.1.1.3" stretchy="false" xref="S3.SS1.SSS2.1.p1.9.m9.2.2.2.2.2.1.1.1.cmml">)</mo></mrow><mi id="S3.SS1.SSS2.1.p1.9.m9.2.2.2.2.2.3" xref="S3.SS1.SSS2.1.p1.9.m9.2.2.2.2.2.3.cmml">j</mi></msup><mo id="S3.SS1.SSS2.1.p1.9.m9.2.2.2.2.5" stretchy="false" xref="S3.SS1.SSS2.1.p1.9.m9.2.2.2.3.cmml">]</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS2.1.p1.9.m9.2b"><apply id="S3.SS1.SSS2.1.p1.9.m9.2.2.cmml" xref="S3.SS1.SSS2.1.p1.9.m9.2.2"><ci id="S3.SS1.SSS2.1.p1.9.m9.2.2.3.cmml" xref="S3.SS1.SSS2.1.p1.9.m9.2.2.3">≔</ci><apply id="S3.SS1.SSS2.1.p1.9.m9.2.2.4.cmml" xref="S3.SS1.SSS2.1.p1.9.m9.2.2.4"><csymbol cd="ambiguous" id="S3.SS1.SSS2.1.p1.9.m9.2.2.4.1.cmml" xref="S3.SS1.SSS2.1.p1.9.m9.2.2.4">subscript</csymbol><ci id="S3.SS1.SSS2.1.p1.9.m9.2.2.4.2.cmml" xref="S3.SS1.SSS2.1.p1.9.m9.2.2.4.2">𝐵</ci><ci id="S3.SS1.SSS2.1.p1.9.m9.2.2.4.3.cmml" xref="S3.SS1.SSS2.1.p1.9.m9.2.2.4.3">𝑗</ci></apply><interval closure="open-closed" id="S3.SS1.SSS2.1.p1.9.m9.2.2.2.3.cmml" xref="S3.SS1.SSS2.1.p1.9.m9.2.2.2.2"><apply id="S3.SS1.SSS2.1.p1.9.m9.1.1.1.1.1.cmml" xref="S3.SS1.SSS2.1.p1.9.m9.1.1.1.1.1"><csymbol cd="ambiguous" id="S3.SS1.SSS2.1.p1.9.m9.1.1.1.1.1.2.cmml" xref="S3.SS1.SSS2.1.p1.9.m9.1.1.1.1.1">superscript</csymbol><apply id="S3.SS1.SSS2.1.p1.9.m9.1.1.1.1.1.1.1.1.cmml" xref="S3.SS1.SSS2.1.p1.9.m9.1.1.1.1.1.1.1"><plus id="S3.SS1.SSS2.1.p1.9.m9.1.1.1.1.1.1.1.1.1.cmml" xref="S3.SS1.SSS2.1.p1.9.m9.1.1.1.1.1.1.1.1.1"></plus><cn id="S3.SS1.SSS2.1.p1.9.m9.1.1.1.1.1.1.1.1.2.cmml" type="integer" xref="S3.SS1.SSS2.1.p1.9.m9.1.1.1.1.1.1.1.1.2">1</cn><ci id="S3.SS1.SSS2.1.p1.9.m9.1.1.1.1.1.1.1.1.3.cmml" xref="S3.SS1.SSS2.1.p1.9.m9.1.1.1.1.1.1.1.1.3">italic-ϵ</ci></apply><apply id="S3.SS1.SSS2.1.p1.9.m9.1.1.1.1.1.3.cmml" xref="S3.SS1.SSS2.1.p1.9.m9.1.1.1.1.1.3"><minus id="S3.SS1.SSS2.1.p1.9.m9.1.1.1.1.1.3.1.cmml" xref="S3.SS1.SSS2.1.p1.9.m9.1.1.1.1.1.3.1"></minus><ci id="S3.SS1.SSS2.1.p1.9.m9.1.1.1.1.1.3.2.cmml" xref="S3.SS1.SSS2.1.p1.9.m9.1.1.1.1.1.3.2">𝑗</ci><cn id="S3.SS1.SSS2.1.p1.9.m9.1.1.1.1.1.3.3.cmml" type="integer" xref="S3.SS1.SSS2.1.p1.9.m9.1.1.1.1.1.3.3">1</cn></apply></apply><apply id="S3.SS1.SSS2.1.p1.9.m9.2.2.2.2.2.cmml" xref="S3.SS1.SSS2.1.p1.9.m9.2.2.2.2.2"><csymbol cd="ambiguous" id="S3.SS1.SSS2.1.p1.9.m9.2.2.2.2.2.2.cmml" xref="S3.SS1.SSS2.1.p1.9.m9.2.2.2.2.2">superscript</csymbol><apply id="S3.SS1.SSS2.1.p1.9.m9.2.2.2.2.2.1.1.1.cmml" xref="S3.SS1.SSS2.1.p1.9.m9.2.2.2.2.2.1.1"><plus id="S3.SS1.SSS2.1.p1.9.m9.2.2.2.2.2.1.1.1.1.cmml" xref="S3.SS1.SSS2.1.p1.9.m9.2.2.2.2.2.1.1.1.1"></plus><cn id="S3.SS1.SSS2.1.p1.9.m9.2.2.2.2.2.1.1.1.2.cmml" type="integer" xref="S3.SS1.SSS2.1.p1.9.m9.2.2.2.2.2.1.1.1.2">1</cn><ci id="S3.SS1.SSS2.1.p1.9.m9.2.2.2.2.2.1.1.1.3.cmml" xref="S3.SS1.SSS2.1.p1.9.m9.2.2.2.2.2.1.1.1.3">italic-ϵ</ci></apply><ci id="S3.SS1.SSS2.1.p1.9.m9.2.2.2.2.2.3.cmml" xref="S3.SS1.SSS2.1.p1.9.m9.2.2.2.2.2.3">𝑗</ci></apply></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS2.1.p1.9.m9.2c">B_{j}\coloneqq((1+\epsilon)^{j-1},(1+\epsilon)^{j}]</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS2.1.p1.9.m9.2d">italic_B start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ≔ ( ( 1 + italic_ϵ ) start_POSTSUPERSCRIPT italic_j - 1 end_POSTSUPERSCRIPT , ( 1 + italic_ϵ ) start_POSTSUPERSCRIPT italic_j end_POSTSUPERSCRIPT ]</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="S3.SS1.SSS2.2.p2"> <p class="ltx_p" id="S3.SS1.SSS2.2.p2.6">Let <math alttext="\textnormal{OPT}\subseteq E" class="ltx_Math" display="inline" id="S3.SS1.SSS2.2.p2.1.m1.1"><semantics id="S3.SS1.SSS2.2.p2.1.m1.1a"><mrow id="S3.SS1.SSS2.2.p2.1.m1.1.1" xref="S3.SS1.SSS2.2.p2.1.m1.1.1.cmml"><mtext id="S3.SS1.SSS2.2.p2.1.m1.1.1.2" xref="S3.SS1.SSS2.2.p2.1.m1.1.1.2a.cmml">OPT</mtext><mo id="S3.SS1.SSS2.2.p2.1.m1.1.1.1" xref="S3.SS1.SSS2.2.p2.1.m1.1.1.1.cmml">⊆</mo><mi id="S3.SS1.SSS2.2.p2.1.m1.1.1.3" xref="S3.SS1.SSS2.2.p2.1.m1.1.1.3.cmml">E</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS2.2.p2.1.m1.1b"><apply id="S3.SS1.SSS2.2.p2.1.m1.1.1.cmml" xref="S3.SS1.SSS2.2.p2.1.m1.1.1"><subset id="S3.SS1.SSS2.2.p2.1.m1.1.1.1.cmml" xref="S3.SS1.SSS2.2.p2.1.m1.1.1.1"></subset><ci id="S3.SS1.SSS2.2.p2.1.m1.1.1.2a.cmml" xref="S3.SS1.SSS2.2.p2.1.m1.1.1.2"><mtext id="S3.SS1.SSS2.2.p2.1.m1.1.1.2.cmml" xref="S3.SS1.SSS2.2.p2.1.m1.1.1.2">OPT</mtext></ci><ci id="S3.SS1.SSS2.2.p2.1.m1.1.1.3.cmml" xref="S3.SS1.SSS2.2.p2.1.m1.1.1.3">𝐸</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS2.2.p2.1.m1.1c">\textnormal{OPT}\subseteq E</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS2.2.p2.1.m1.1d">OPT ⊆ italic_E</annotation></semantics></math> be an optimal solution for the VC-SNDP instance on <math alttext="G" class="ltx_Math" display="inline" id="S3.SS1.SSS2.2.p2.2.m2.1"><semantics id="S3.SS1.SSS2.2.p2.2.m2.1a"><mi id="S3.SS1.SSS2.2.p2.2.m2.1.1" xref="S3.SS1.SSS2.2.p2.2.m2.1.1.cmml">G</mi><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS2.2.p2.2.m2.1b"><ci id="S3.SS1.SSS2.2.p2.2.m2.1.1.cmml" xref="S3.SS1.SSS2.2.p2.2.m2.1.1">𝐺</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS2.2.p2.2.m2.1c">G</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS2.2.p2.2.m2.1d">italic_G</annotation></semantics></math>. Starting from <span class="ltx_text ltx_markedasmath" id="S3.SS1.SSS2.2.p2.6.1">OPT</span>, we construct a feasible fractional solution <math alttext="\boldsymbol{x}" class="ltx_Math" display="inline" id="S3.SS1.SSS2.2.p2.4.m4.1"><semantics id="S3.SS1.SSS2.2.p2.4.m4.1a"><mi id="S3.SS1.SSS2.2.p2.4.m4.1.1" xref="S3.SS1.SSS2.2.p2.4.m4.1.1.cmml">𝒙</mi><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS2.2.p2.4.m4.1b"><ci id="S3.SS1.SSS2.2.p2.4.m4.1.1.cmml" xref="S3.SS1.SSS2.2.p2.4.m4.1.1">𝒙</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS2.2.p2.4.m4.1c">\boldsymbol{x}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS2.2.p2.4.m4.1d">bold_italic_x</annotation></semantics></math> supported only on <math alttext="H" class="ltx_Math" display="inline" id="S3.SS1.SSS2.2.p2.5.m5.1"><semantics id="S3.SS1.SSS2.2.p2.5.m5.1a"><mi id="S3.SS1.SSS2.2.p2.5.m5.1.1" xref="S3.SS1.SSS2.2.p2.5.m5.1.1.cmml">H</mi><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS2.2.p2.5.m5.1b"><ci id="S3.SS1.SSS2.2.p2.5.m5.1.1.cmml" xref="S3.SS1.SSS2.2.p2.5.m5.1.1">𝐻</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS2.2.p2.5.m5.1c">H</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS2.2.p2.5.m5.1d">italic_H</annotation></semantics></math>, for the standard (bi)cut-based relaxation (i.e., <a class="ltx_ref" href="https://arxiv.org/html/2503.00712v1#S3.F1" title="Figure 1 ‣ 3.1.2 An Improved Analysis via Fractional Solutions ‣ 3.1 Vertex Connectivity Network Design ‣ 3 Generic Framework for Streaming Algorithms for Network Design ‣ Streaming Algorithms for Network Design">VC-SNDP-LP</a>) of the VC-SNDP instance, with cost at most <math alttext="O(t)\cdot w(\textnormal{OPT})" class="ltx_Math" display="inline" id="S3.SS1.SSS2.2.p2.6.m6.2"><semantics id="S3.SS1.SSS2.2.p2.6.m6.2a"><mrow id="S3.SS1.SSS2.2.p2.6.m6.2.3" xref="S3.SS1.SSS2.2.p2.6.m6.2.3.cmml"><mrow id="S3.SS1.SSS2.2.p2.6.m6.2.3.2" xref="S3.SS1.SSS2.2.p2.6.m6.2.3.2.cmml"><mrow id="S3.SS1.SSS2.2.p2.6.m6.2.3.2.2" xref="S3.SS1.SSS2.2.p2.6.m6.2.3.2.2.cmml"><mi id="S3.SS1.SSS2.2.p2.6.m6.2.3.2.2.2" xref="S3.SS1.SSS2.2.p2.6.m6.2.3.2.2.2.cmml">O</mi><mo id="S3.SS1.SSS2.2.p2.6.m6.2.3.2.2.1" xref="S3.SS1.SSS2.2.p2.6.m6.2.3.2.2.1.cmml">⁢</mo><mrow id="S3.SS1.SSS2.2.p2.6.m6.2.3.2.2.3.2" xref="S3.SS1.SSS2.2.p2.6.m6.2.3.2.2.cmml"><mo id="S3.SS1.SSS2.2.p2.6.m6.2.3.2.2.3.2.1" stretchy="false" xref="S3.SS1.SSS2.2.p2.6.m6.2.3.2.2.cmml">(</mo><mi id="S3.SS1.SSS2.2.p2.6.m6.1.1" xref="S3.SS1.SSS2.2.p2.6.m6.1.1.cmml">t</mi><mo id="S3.SS1.SSS2.2.p2.6.m6.2.3.2.2.3.2.2" rspace="0.055em" stretchy="false" xref="S3.SS1.SSS2.2.p2.6.m6.2.3.2.2.cmml">)</mo></mrow></mrow><mo id="S3.SS1.SSS2.2.p2.6.m6.2.3.2.1" rspace="0.222em" xref="S3.SS1.SSS2.2.p2.6.m6.2.3.2.1.cmml">⋅</mo><mi id="S3.SS1.SSS2.2.p2.6.m6.2.3.2.3" xref="S3.SS1.SSS2.2.p2.6.m6.2.3.2.3.cmml">w</mi></mrow><mo id="S3.SS1.SSS2.2.p2.6.m6.2.3.1" xref="S3.SS1.SSS2.2.p2.6.m6.2.3.1.cmml">⁢</mo><mrow id="S3.SS1.SSS2.2.p2.6.m6.2.3.3.2" xref="S3.SS1.SSS2.2.p2.6.m6.2.2a.cmml"><mo id="S3.SS1.SSS2.2.p2.6.m6.2.3.3.2.1" stretchy="false" xref="S3.SS1.SSS2.2.p2.6.m6.2.2a.cmml">(</mo><mtext id="S3.SS1.SSS2.2.p2.6.m6.2.2" xref="S3.SS1.SSS2.2.p2.6.m6.2.2.cmml">OPT</mtext><mo id="S3.SS1.SSS2.2.p2.6.m6.2.3.3.2.2" stretchy="false" xref="S3.SS1.SSS2.2.p2.6.m6.2.2a.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS2.2.p2.6.m6.2b"><apply id="S3.SS1.SSS2.2.p2.6.m6.2.3.cmml" xref="S3.SS1.SSS2.2.p2.6.m6.2.3"><times id="S3.SS1.SSS2.2.p2.6.m6.2.3.1.cmml" xref="S3.SS1.SSS2.2.p2.6.m6.2.3.1"></times><apply id="S3.SS1.SSS2.2.p2.6.m6.2.3.2.cmml" xref="S3.SS1.SSS2.2.p2.6.m6.2.3.2"><ci id="S3.SS1.SSS2.2.p2.6.m6.2.3.2.1.cmml" xref="S3.SS1.SSS2.2.p2.6.m6.2.3.2.1">⋅</ci><apply id="S3.SS1.SSS2.2.p2.6.m6.2.3.2.2.cmml" xref="S3.SS1.SSS2.2.p2.6.m6.2.3.2.2"><times id="S3.SS1.SSS2.2.p2.6.m6.2.3.2.2.1.cmml" xref="S3.SS1.SSS2.2.p2.6.m6.2.3.2.2.1"></times><ci id="S3.SS1.SSS2.2.p2.6.m6.2.3.2.2.2.cmml" xref="S3.SS1.SSS2.2.p2.6.m6.2.3.2.2.2">𝑂</ci><ci id="S3.SS1.SSS2.2.p2.6.m6.1.1.cmml" xref="S3.SS1.SSS2.2.p2.6.m6.1.1">𝑡</ci></apply><ci id="S3.SS1.SSS2.2.p2.6.m6.2.3.2.3.cmml" xref="S3.SS1.SSS2.2.p2.6.m6.2.3.2.3">𝑤</ci></apply><ci id="S3.SS1.SSS2.2.p2.6.m6.2.2a.cmml" xref="S3.SS1.SSS2.2.p2.6.m6.2.3.3.2"><mtext id="S3.SS1.SSS2.2.p2.6.m6.2.2.cmml" xref="S3.SS1.SSS2.2.p2.6.m6.2.2">OPT</mtext></ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS2.2.p2.6.m6.2c">O(t)\cdot w(\textnormal{OPT})</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS2.2.p2.6.m6.2d">italic_O ( italic_t ) ⋅ italic_w ( OPT )</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="S3.SS1.SSS2.3.p3"> <p class="ltx_p" id="S3.SS1.SSS2.3.p3.7">In <a class="ltx_ref" href="https://arxiv.org/html/2503.00712v1#S3.F1" title="Figure 1 ‣ 3.1.2 An Improved Analysis via Fractional Solutions ‣ 3.1 Vertex Connectivity Network Design ‣ 3 Generic Framework for Streaming Algorithms for Network Design ‣ Streaming Algorithms for Network Design">VC-SNDP-LP</a>, <math alttext="h:2^{V}\times 2^{V}\rightarrow\{0,1,\cdots,k\}" class="ltx_Math" display="inline" id="S3.SS1.SSS2.3.p3.1.m1.4"><semantics id="S3.SS1.SSS2.3.p3.1.m1.4a"><mrow id="S3.SS1.SSS2.3.p3.1.m1.4.5" xref="S3.SS1.SSS2.3.p3.1.m1.4.5.cmml"><mi id="S3.SS1.SSS2.3.p3.1.m1.4.5.2" xref="S3.SS1.SSS2.3.p3.1.m1.4.5.2.cmml">h</mi><mo id="S3.SS1.SSS2.3.p3.1.m1.4.5.1" lspace="0.278em" rspace="0.278em" xref="S3.SS1.SSS2.3.p3.1.m1.4.5.1.cmml">:</mo><mrow id="S3.SS1.SSS2.3.p3.1.m1.4.5.3" xref="S3.SS1.SSS2.3.p3.1.m1.4.5.3.cmml"><mrow id="S3.SS1.SSS2.3.p3.1.m1.4.5.3.2" xref="S3.SS1.SSS2.3.p3.1.m1.4.5.3.2.cmml"><msup id="S3.SS1.SSS2.3.p3.1.m1.4.5.3.2.2" xref="S3.SS1.SSS2.3.p3.1.m1.4.5.3.2.2.cmml"><mn id="S3.SS1.SSS2.3.p3.1.m1.4.5.3.2.2.2" xref="S3.SS1.SSS2.3.p3.1.m1.4.5.3.2.2.2.cmml">2</mn><mi id="S3.SS1.SSS2.3.p3.1.m1.4.5.3.2.2.3" xref="S3.SS1.SSS2.3.p3.1.m1.4.5.3.2.2.3.cmml">V</mi></msup><mo id="S3.SS1.SSS2.3.p3.1.m1.4.5.3.2.1" lspace="0.222em" rspace="0.222em" xref="S3.SS1.SSS2.3.p3.1.m1.4.5.3.2.1.cmml">×</mo><msup id="S3.SS1.SSS2.3.p3.1.m1.4.5.3.2.3" xref="S3.SS1.SSS2.3.p3.1.m1.4.5.3.2.3.cmml"><mn id="S3.SS1.SSS2.3.p3.1.m1.4.5.3.2.3.2" xref="S3.SS1.SSS2.3.p3.1.m1.4.5.3.2.3.2.cmml">2</mn><mi id="S3.SS1.SSS2.3.p3.1.m1.4.5.3.2.3.3" xref="S3.SS1.SSS2.3.p3.1.m1.4.5.3.2.3.3.cmml">V</mi></msup></mrow><mo id="S3.SS1.SSS2.3.p3.1.m1.4.5.3.1" stretchy="false" xref="S3.SS1.SSS2.3.p3.1.m1.4.5.3.1.cmml">→</mo><mrow id="S3.SS1.SSS2.3.p3.1.m1.4.5.3.3.2" xref="S3.SS1.SSS2.3.p3.1.m1.4.5.3.3.1.cmml"><mo id="S3.SS1.SSS2.3.p3.1.m1.4.5.3.3.2.1" stretchy="false" xref="S3.SS1.SSS2.3.p3.1.m1.4.5.3.3.1.cmml">{</mo><mn id="S3.SS1.SSS2.3.p3.1.m1.1.1" xref="S3.SS1.SSS2.3.p3.1.m1.1.1.cmml">0</mn><mo id="S3.SS1.SSS2.3.p3.1.m1.4.5.3.3.2.2" xref="S3.SS1.SSS2.3.p3.1.m1.4.5.3.3.1.cmml">,</mo><mn 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xref="S3.SS1.SSS2.3.p3.1.m1.4.5.3.2.3.2">2</cn><ci id="S3.SS1.SSS2.3.p3.1.m1.4.5.3.2.3.3.cmml" xref="S3.SS1.SSS2.3.p3.1.m1.4.5.3.2.3.3">𝑉</ci></apply></apply><set id="S3.SS1.SSS2.3.p3.1.m1.4.5.3.3.1.cmml" xref="S3.SS1.SSS2.3.p3.1.m1.4.5.3.3.2"><cn id="S3.SS1.SSS2.3.p3.1.m1.1.1.cmml" type="integer" xref="S3.SS1.SSS2.3.p3.1.m1.1.1">0</cn><cn id="S3.SS1.SSS2.3.p3.1.m1.2.2.cmml" type="integer" xref="S3.SS1.SSS2.3.p3.1.m1.2.2">1</cn><ci id="S3.SS1.SSS2.3.p3.1.m1.3.3.cmml" xref="S3.SS1.SSS2.3.p3.1.m1.3.3">⋯</ci><ci id="S3.SS1.SSS2.3.p3.1.m1.4.4.cmml" xref="S3.SS1.SSS2.3.p3.1.m1.4.4">𝑘</ci></set></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS2.3.p3.1.m1.4c">h:2^{V}\times 2^{V}\rightarrow\{0,1,\cdots,k\}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS2.3.p3.1.m1.4d">italic_h : 2 start_POSTSUPERSCRIPT italic_V end_POSTSUPERSCRIPT × 2 start_POSTSUPERSCRIPT italic_V end_POSTSUPERSCRIPT → { 0 , 1 , ⋯ , italic_k }</annotation></semantics></math> is defined as <math alttext="h(\hat{S})\coloneqq\max(0,\max_{v\in S,u\in V\setminus S^{+}}r(u,v)-|S^{+}% \setminus S|)" class="ltx_Math" display="inline" id="S3.SS1.SSS2.3.p3.2.m2.8"><semantics id="S3.SS1.SSS2.3.p3.2.m2.8a"><mrow id="S3.SS1.SSS2.3.p3.2.m2.8.8" xref="S3.SS1.SSS2.3.p3.2.m2.8.8.cmml"><mrow id="S3.SS1.SSS2.3.p3.2.m2.8.8.3" xref="S3.SS1.SSS2.3.p3.2.m2.8.8.3.cmml"><mi id="S3.SS1.SSS2.3.p3.2.m2.8.8.3.2" xref="S3.SS1.SSS2.3.p3.2.m2.8.8.3.2.cmml">h</mi><mo id="S3.SS1.SSS2.3.p3.2.m2.8.8.3.1" xref="S3.SS1.SSS2.3.p3.2.m2.8.8.3.1.cmml">⁢</mo><mrow id="S3.SS1.SSS2.3.p3.2.m2.8.8.3.3.2" xref="S3.SS1.SSS2.3.p3.2.m2.3.3.cmml"><mo id="S3.SS1.SSS2.3.p3.2.m2.8.8.3.3.2.1" stretchy="false" xref="S3.SS1.SSS2.3.p3.2.m2.3.3.cmml">(</mo><mover accent="true" id="S3.SS1.SSS2.3.p3.2.m2.3.3" xref="S3.SS1.SSS2.3.p3.2.m2.3.3.cmml"><mi id="S3.SS1.SSS2.3.p3.2.m2.3.3.2" xref="S3.SS1.SSS2.3.p3.2.m2.3.3.2.cmml">S</mi><mo id="S3.SS1.SSS2.3.p3.2.m2.3.3.1" 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id="S3.SS1.SSS2.3.p3.2.m2.4.4.cmml" xref="S3.SS1.SSS2.3.p3.2.m2.4.4">𝑢</ci><ci id="S3.SS1.SSS2.3.p3.2.m2.5.5.cmml" xref="S3.SS1.SSS2.3.p3.2.m2.5.5">𝑣</ci></interval></apply><apply id="S3.SS1.SSS2.3.p3.2.m2.8.8.1.1.1.1.1.2.cmml" xref="S3.SS1.SSS2.3.p3.2.m2.8.8.1.1.1.1.1.1"><abs id="S3.SS1.SSS2.3.p3.2.m2.8.8.1.1.1.1.1.2.1.cmml" xref="S3.SS1.SSS2.3.p3.2.m2.8.8.1.1.1.1.1.1.2"></abs><apply id="S3.SS1.SSS2.3.p3.2.m2.8.8.1.1.1.1.1.1.1.cmml" xref="S3.SS1.SSS2.3.p3.2.m2.8.8.1.1.1.1.1.1.1"><setdiff id="S3.SS1.SSS2.3.p3.2.m2.8.8.1.1.1.1.1.1.1.1.cmml" xref="S3.SS1.SSS2.3.p3.2.m2.8.8.1.1.1.1.1.1.1.1"></setdiff><apply id="S3.SS1.SSS2.3.p3.2.m2.8.8.1.1.1.1.1.1.1.2.cmml" xref="S3.SS1.SSS2.3.p3.2.m2.8.8.1.1.1.1.1.1.1.2"><csymbol cd="ambiguous" id="S3.SS1.SSS2.3.p3.2.m2.8.8.1.1.1.1.1.1.1.2.1.cmml" xref="S3.SS1.SSS2.3.p3.2.m2.8.8.1.1.1.1.1.1.1.2">superscript</csymbol><ci id="S3.SS1.SSS2.3.p3.2.m2.8.8.1.1.1.1.1.1.1.2.2.cmml" xref="S3.SS1.SSS2.3.p3.2.m2.8.8.1.1.1.1.1.1.1.2.2">𝑆</ci><plus id="S3.SS1.SSS2.3.p3.2.m2.8.8.1.1.1.1.1.1.1.2.3.cmml" xref="S3.SS1.SSS2.3.p3.2.m2.8.8.1.1.1.1.1.1.1.2.3"></plus></apply><ci id="S3.SS1.SSS2.3.p3.2.m2.8.8.1.1.1.1.1.1.1.3.cmml" xref="S3.SS1.SSS2.3.p3.2.m2.8.8.1.1.1.1.1.1.1.3">𝑆</ci></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS2.3.p3.2.m2.8c">h(\hat{S})\coloneqq\max(0,\max_{v\in S,u\in V\setminus S^{+}}r(u,v)-|S^{+}% \setminus S|)</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS2.3.p3.2.m2.8d">italic_h ( over^ start_ARG italic_S end_ARG ) ≔ roman_max ( 0 , roman_max start_POSTSUBSCRIPT italic_v ∈ italic_S , italic_u ∈ italic_V ∖ italic_S start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT end_POSTSUBSCRIPT italic_r ( italic_u , italic_v ) - | italic_S start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT ∖ italic_S | )</annotation></semantics></math>, when <math alttext="S\subseteq S^{+}" class="ltx_Math" display="inline" id="S3.SS1.SSS2.3.p3.3.m3.1"><semantics id="S3.SS1.SSS2.3.p3.3.m3.1a"><mrow id="S3.SS1.SSS2.3.p3.3.m3.1.1" xref="S3.SS1.SSS2.3.p3.3.m3.1.1.cmml"><mi id="S3.SS1.SSS2.3.p3.3.m3.1.1.2" xref="S3.SS1.SSS2.3.p3.3.m3.1.1.2.cmml">S</mi><mo id="S3.SS1.SSS2.3.p3.3.m3.1.1.1" xref="S3.SS1.SSS2.3.p3.3.m3.1.1.1.cmml">⊆</mo><msup id="S3.SS1.SSS2.3.p3.3.m3.1.1.3" xref="S3.SS1.SSS2.3.p3.3.m3.1.1.3.cmml"><mi id="S3.SS1.SSS2.3.p3.3.m3.1.1.3.2" xref="S3.SS1.SSS2.3.p3.3.m3.1.1.3.2.cmml">S</mi><mo id="S3.SS1.SSS2.3.p3.3.m3.1.1.3.3" xref="S3.SS1.SSS2.3.p3.3.m3.1.1.3.3.cmml">+</mo></msup></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS2.3.p3.3.m3.1b"><apply id="S3.SS1.SSS2.3.p3.3.m3.1.1.cmml" xref="S3.SS1.SSS2.3.p3.3.m3.1.1"><subset id="S3.SS1.SSS2.3.p3.3.m3.1.1.1.cmml" xref="S3.SS1.SSS2.3.p3.3.m3.1.1.1"></subset><ci id="S3.SS1.SSS2.3.p3.3.m3.1.1.2.cmml" xref="S3.SS1.SSS2.3.p3.3.m3.1.1.2">𝑆</ci><apply id="S3.SS1.SSS2.3.p3.3.m3.1.1.3.cmml" xref="S3.SS1.SSS2.3.p3.3.m3.1.1.3"><csymbol cd="ambiguous" id="S3.SS1.SSS2.3.p3.3.m3.1.1.3.1.cmml" xref="S3.SS1.SSS2.3.p3.3.m3.1.1.3">superscript</csymbol><ci id="S3.SS1.SSS2.3.p3.3.m3.1.1.3.2.cmml" xref="S3.SS1.SSS2.3.p3.3.m3.1.1.3.2">𝑆</ci><plus id="S3.SS1.SSS2.3.p3.3.m3.1.1.3.3.cmml" xref="S3.SS1.SSS2.3.p3.3.m3.1.1.3.3"></plus></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS2.3.p3.3.m3.1c">S\subseteq S^{+}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS2.3.p3.3.m3.1d">italic_S ⊆ italic_S start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT</annotation></semantics></math>. Otherwise, <math alttext="h(\hat{S})=0" class="ltx_Math" display="inline" id="S3.SS1.SSS2.3.p3.4.m4.1"><semantics id="S3.SS1.SSS2.3.p3.4.m4.1a"><mrow id="S3.SS1.SSS2.3.p3.4.m4.1.2" xref="S3.SS1.SSS2.3.p3.4.m4.1.2.cmml"><mrow id="S3.SS1.SSS2.3.p3.4.m4.1.2.2" xref="S3.SS1.SSS2.3.p3.4.m4.1.2.2.cmml"><mi id="S3.SS1.SSS2.3.p3.4.m4.1.2.2.2" xref="S3.SS1.SSS2.3.p3.4.m4.1.2.2.2.cmml">h</mi><mo id="S3.SS1.SSS2.3.p3.4.m4.1.2.2.1" xref="S3.SS1.SSS2.3.p3.4.m4.1.2.2.1.cmml">⁢</mo><mrow id="S3.SS1.SSS2.3.p3.4.m4.1.2.2.3.2" xref="S3.SS1.SSS2.3.p3.4.m4.1.1.cmml"><mo id="S3.SS1.SSS2.3.p3.4.m4.1.2.2.3.2.1" stretchy="false" xref="S3.SS1.SSS2.3.p3.4.m4.1.1.cmml">(</mo><mover accent="true" id="S3.SS1.SSS2.3.p3.4.m4.1.1" xref="S3.SS1.SSS2.3.p3.4.m4.1.1.cmml"><mi id="S3.SS1.SSS2.3.p3.4.m4.1.1.2" xref="S3.SS1.SSS2.3.p3.4.m4.1.1.2.cmml">S</mi><mo id="S3.SS1.SSS2.3.p3.4.m4.1.1.1" xref="S3.SS1.SSS2.3.p3.4.m4.1.1.1.cmml">^</mo></mover><mo id="S3.SS1.SSS2.3.p3.4.m4.1.2.2.3.2.2" stretchy="false" xref="S3.SS1.SSS2.3.p3.4.m4.1.1.cmml">)</mo></mrow></mrow><mo id="S3.SS1.SSS2.3.p3.4.m4.1.2.1" xref="S3.SS1.SSS2.3.p3.4.m4.1.2.1.cmml">=</mo><mn id="S3.SS1.SSS2.3.p3.4.m4.1.2.3" xref="S3.SS1.SSS2.3.p3.4.m4.1.2.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS2.3.p3.4.m4.1b"><apply id="S3.SS1.SSS2.3.p3.4.m4.1.2.cmml" xref="S3.SS1.SSS2.3.p3.4.m4.1.2"><eq id="S3.SS1.SSS2.3.p3.4.m4.1.2.1.cmml" xref="S3.SS1.SSS2.3.p3.4.m4.1.2.1"></eq><apply id="S3.SS1.SSS2.3.p3.4.m4.1.2.2.cmml" xref="S3.SS1.SSS2.3.p3.4.m4.1.2.2"><times id="S3.SS1.SSS2.3.p3.4.m4.1.2.2.1.cmml" xref="S3.SS1.SSS2.3.p3.4.m4.1.2.2.1"></times><ci id="S3.SS1.SSS2.3.p3.4.m4.1.2.2.2.cmml" xref="S3.SS1.SSS2.3.p3.4.m4.1.2.2.2">ℎ</ci><apply id="S3.SS1.SSS2.3.p3.4.m4.1.1.cmml" xref="S3.SS1.SSS2.3.p3.4.m4.1.2.2.3.2"><ci id="S3.SS1.SSS2.3.p3.4.m4.1.1.1.cmml" xref="S3.SS1.SSS2.3.p3.4.m4.1.1.1">^</ci><ci id="S3.SS1.SSS2.3.p3.4.m4.1.1.2.cmml" xref="S3.SS1.SSS2.3.p3.4.m4.1.1.2">𝑆</ci></apply></apply><cn id="S3.SS1.SSS2.3.p3.4.m4.1.2.3.cmml" type="integer" xref="S3.SS1.SSS2.3.p3.4.m4.1.2.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS2.3.p3.4.m4.1c">h(\hat{S})=0</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS2.3.p3.4.m4.1d">italic_h ( over^ start_ARG italic_S end_ARG ) = 0</annotation></semantics></math>. In particular, observe that for every biset <math alttext="\hat{S}" class="ltx_Math" display="inline" id="S3.SS1.SSS2.3.p3.5.m5.1"><semantics id="S3.SS1.SSS2.3.p3.5.m5.1a"><mover accent="true" id="S3.SS1.SSS2.3.p3.5.m5.1.1" xref="S3.SS1.SSS2.3.p3.5.m5.1.1.cmml"><mi id="S3.SS1.SSS2.3.p3.5.m5.1.1.2" xref="S3.SS1.SSS2.3.p3.5.m5.1.1.2.cmml">S</mi><mo id="S3.SS1.SSS2.3.p3.5.m5.1.1.1" xref="S3.SS1.SSS2.3.p3.5.m5.1.1.1.cmml">^</mo></mover><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS2.3.p3.5.m5.1b"><apply id="S3.SS1.SSS2.3.p3.5.m5.1.1.cmml" xref="S3.SS1.SSS2.3.p3.5.m5.1.1"><ci id="S3.SS1.SSS2.3.p3.5.m5.1.1.1.cmml" xref="S3.SS1.SSS2.3.p3.5.m5.1.1.1">^</ci><ci id="S3.SS1.SSS2.3.p3.5.m5.1.1.2.cmml" xref="S3.SS1.SSS2.3.p3.5.m5.1.1.2">𝑆</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS2.3.p3.5.m5.1c">\hat{S}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS2.3.p3.5.m5.1d">over^ start_ARG italic_S end_ARG</annotation></semantics></math> with <math alttext="h(\hat{S})>0" class="ltx_Math" display="inline" id="S3.SS1.SSS2.3.p3.6.m6.1"><semantics id="S3.SS1.SSS2.3.p3.6.m6.1a"><mrow id="S3.SS1.SSS2.3.p3.6.m6.1.2" xref="S3.SS1.SSS2.3.p3.6.m6.1.2.cmml"><mrow id="S3.SS1.SSS2.3.p3.6.m6.1.2.2" xref="S3.SS1.SSS2.3.p3.6.m6.1.2.2.cmml"><mi id="S3.SS1.SSS2.3.p3.6.m6.1.2.2.2" xref="S3.SS1.SSS2.3.p3.6.m6.1.2.2.2.cmml">h</mi><mo id="S3.SS1.SSS2.3.p3.6.m6.1.2.2.1" xref="S3.SS1.SSS2.3.p3.6.m6.1.2.2.1.cmml">⁢</mo><mrow id="S3.SS1.SSS2.3.p3.6.m6.1.2.2.3.2" xref="S3.SS1.SSS2.3.p3.6.m6.1.1.cmml"><mo id="S3.SS1.SSS2.3.p3.6.m6.1.2.2.3.2.1" stretchy="false" xref="S3.SS1.SSS2.3.p3.6.m6.1.1.cmml">(</mo><mover accent="true" id="S3.SS1.SSS2.3.p3.6.m6.1.1" xref="S3.SS1.SSS2.3.p3.6.m6.1.1.cmml"><mi id="S3.SS1.SSS2.3.p3.6.m6.1.1.2" xref="S3.SS1.SSS2.3.p3.6.m6.1.1.2.cmml">S</mi><mo id="S3.SS1.SSS2.3.p3.6.m6.1.1.1" xref="S3.SS1.SSS2.3.p3.6.m6.1.1.1.cmml">^</mo></mover><mo id="S3.SS1.SSS2.3.p3.6.m6.1.2.2.3.2.2" stretchy="false" xref="S3.SS1.SSS2.3.p3.6.m6.1.1.cmml">)</mo></mrow></mrow><mo id="S3.SS1.SSS2.3.p3.6.m6.1.2.1" xref="S3.SS1.SSS2.3.p3.6.m6.1.2.1.cmml">></mo><mn id="S3.SS1.SSS2.3.p3.6.m6.1.2.3" xref="S3.SS1.SSS2.3.p3.6.m6.1.2.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS2.3.p3.6.m6.1b"><apply id="S3.SS1.SSS2.3.p3.6.m6.1.2.cmml" xref="S3.SS1.SSS2.3.p3.6.m6.1.2"><gt id="S3.SS1.SSS2.3.p3.6.m6.1.2.1.cmml" xref="S3.SS1.SSS2.3.p3.6.m6.1.2.1"></gt><apply id="S3.SS1.SSS2.3.p3.6.m6.1.2.2.cmml" xref="S3.SS1.SSS2.3.p3.6.m6.1.2.2"><times id="S3.SS1.SSS2.3.p3.6.m6.1.2.2.1.cmml" xref="S3.SS1.SSS2.3.p3.6.m6.1.2.2.1"></times><ci id="S3.SS1.SSS2.3.p3.6.m6.1.2.2.2.cmml" xref="S3.SS1.SSS2.3.p3.6.m6.1.2.2.2">ℎ</ci><apply id="S3.SS1.SSS2.3.p3.6.m6.1.1.cmml" xref="S3.SS1.SSS2.3.p3.6.m6.1.2.2.3.2"><ci id="S3.SS1.SSS2.3.p3.6.m6.1.1.1.cmml" xref="S3.SS1.SSS2.3.p3.6.m6.1.1.1">^</ci><ci id="S3.SS1.SSS2.3.p3.6.m6.1.1.2.cmml" xref="S3.SS1.SSS2.3.p3.6.m6.1.1.2">𝑆</ci></apply></apply><cn id="S3.SS1.SSS2.3.p3.6.m6.1.2.3.cmml" type="integer" xref="S3.SS1.SSS2.3.p3.6.m6.1.2.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS2.3.p3.6.m6.1c">h(\hat{S})>0</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS2.3.p3.6.m6.1d">italic_h ( over^ start_ARG italic_S end_ARG ) > 0</annotation></semantics></math>, <math alttext="h(\hat{S})+|S^{+}\setminus S|\leq k" class="ltx_Math" display="inline" id="S3.SS1.SSS2.3.p3.7.m7.2"><semantics id="S3.SS1.SSS2.3.p3.7.m7.2a"><mrow id="S3.SS1.SSS2.3.p3.7.m7.2.2" xref="S3.SS1.SSS2.3.p3.7.m7.2.2.cmml"><mrow id="S3.SS1.SSS2.3.p3.7.m7.2.2.1" xref="S3.SS1.SSS2.3.p3.7.m7.2.2.1.cmml"><mrow id="S3.SS1.SSS2.3.p3.7.m7.2.2.1.3" xref="S3.SS1.SSS2.3.p3.7.m7.2.2.1.3.cmml"><mi id="S3.SS1.SSS2.3.p3.7.m7.2.2.1.3.2" xref="S3.SS1.SSS2.3.p3.7.m7.2.2.1.3.2.cmml">h</mi><mo id="S3.SS1.SSS2.3.p3.7.m7.2.2.1.3.1" xref="S3.SS1.SSS2.3.p3.7.m7.2.2.1.3.1.cmml">⁢</mo><mrow id="S3.SS1.SSS2.3.p3.7.m7.2.2.1.3.3.2" xref="S3.SS1.SSS2.3.p3.7.m7.1.1.cmml"><mo id="S3.SS1.SSS2.3.p3.7.m7.2.2.1.3.3.2.1" stretchy="false" xref="S3.SS1.SSS2.3.p3.7.m7.1.1.cmml">(</mo><mover accent="true" id="S3.SS1.SSS2.3.p3.7.m7.1.1" xref="S3.SS1.SSS2.3.p3.7.m7.1.1.cmml"><mi id="S3.SS1.SSS2.3.p3.7.m7.1.1.2" xref="S3.SS1.SSS2.3.p3.7.m7.1.1.2.cmml">S</mi><mo id="S3.SS1.SSS2.3.p3.7.m7.1.1.1" xref="S3.SS1.SSS2.3.p3.7.m7.1.1.1.cmml">^</mo></mover><mo id="S3.SS1.SSS2.3.p3.7.m7.2.2.1.3.3.2.2" stretchy="false" xref="S3.SS1.SSS2.3.p3.7.m7.1.1.cmml">)</mo></mrow></mrow><mo id="S3.SS1.SSS2.3.p3.7.m7.2.2.1.2" xref="S3.SS1.SSS2.3.p3.7.m7.2.2.1.2.cmml">+</mo><mrow id="S3.SS1.SSS2.3.p3.7.m7.2.2.1.1.1" xref="S3.SS1.SSS2.3.p3.7.m7.2.2.1.1.2.cmml"><mo id="S3.SS1.SSS2.3.p3.7.m7.2.2.1.1.1.2" stretchy="false" xref="S3.SS1.SSS2.3.p3.7.m7.2.2.1.1.2.1.cmml">|</mo><mrow id="S3.SS1.SSS2.3.p3.7.m7.2.2.1.1.1.1" xref="S3.SS1.SSS2.3.p3.7.m7.2.2.1.1.1.1.cmml"><msup id="S3.SS1.SSS2.3.p3.7.m7.2.2.1.1.1.1.2" xref="S3.SS1.SSS2.3.p3.7.m7.2.2.1.1.1.1.2.cmml"><mi id="S3.SS1.SSS2.3.p3.7.m7.2.2.1.1.1.1.2.2" xref="S3.SS1.SSS2.3.p3.7.m7.2.2.1.1.1.1.2.2.cmml">S</mi><mo id="S3.SS1.SSS2.3.p3.7.m7.2.2.1.1.1.1.2.3" xref="S3.SS1.SSS2.3.p3.7.m7.2.2.1.1.1.1.2.3.cmml">+</mo></msup><mo id="S3.SS1.SSS2.3.p3.7.m7.2.2.1.1.1.1.1" xref="S3.SS1.SSS2.3.p3.7.m7.2.2.1.1.1.1.1.cmml">∖</mo><mi id="S3.SS1.SSS2.3.p3.7.m7.2.2.1.1.1.1.3" xref="S3.SS1.SSS2.3.p3.7.m7.2.2.1.1.1.1.3.cmml">S</mi></mrow><mo id="S3.SS1.SSS2.3.p3.7.m7.2.2.1.1.1.3" stretchy="false" xref="S3.SS1.SSS2.3.p3.7.m7.2.2.1.1.2.1.cmml">|</mo></mrow></mrow><mo id="S3.SS1.SSS2.3.p3.7.m7.2.2.2" xref="S3.SS1.SSS2.3.p3.7.m7.2.2.2.cmml">≤</mo><mi id="S3.SS1.SSS2.3.p3.7.m7.2.2.3" xref="S3.SS1.SSS2.3.p3.7.m7.2.2.3.cmml">k</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS2.3.p3.7.m7.2b"><apply id="S3.SS1.SSS2.3.p3.7.m7.2.2.cmml" xref="S3.SS1.SSS2.3.p3.7.m7.2.2"><leq id="S3.SS1.SSS2.3.p3.7.m7.2.2.2.cmml" xref="S3.SS1.SSS2.3.p3.7.m7.2.2.2"></leq><apply id="S3.SS1.SSS2.3.p3.7.m7.2.2.1.cmml" xref="S3.SS1.SSS2.3.p3.7.m7.2.2.1"><plus id="S3.SS1.SSS2.3.p3.7.m7.2.2.1.2.cmml" xref="S3.SS1.SSS2.3.p3.7.m7.2.2.1.2"></plus><apply id="S3.SS1.SSS2.3.p3.7.m7.2.2.1.3.cmml" xref="S3.SS1.SSS2.3.p3.7.m7.2.2.1.3"><times id="S3.SS1.SSS2.3.p3.7.m7.2.2.1.3.1.cmml" xref="S3.SS1.SSS2.3.p3.7.m7.2.2.1.3.1"></times><ci id="S3.SS1.SSS2.3.p3.7.m7.2.2.1.3.2.cmml" xref="S3.SS1.SSS2.3.p3.7.m7.2.2.1.3.2">ℎ</ci><apply id="S3.SS1.SSS2.3.p3.7.m7.1.1.cmml" xref="S3.SS1.SSS2.3.p3.7.m7.2.2.1.3.3.2"><ci id="S3.SS1.SSS2.3.p3.7.m7.1.1.1.cmml" xref="S3.SS1.SSS2.3.p3.7.m7.1.1.1">^</ci><ci id="S3.SS1.SSS2.3.p3.7.m7.1.1.2.cmml" xref="S3.SS1.SSS2.3.p3.7.m7.1.1.2">𝑆</ci></apply></apply><apply id="S3.SS1.SSS2.3.p3.7.m7.2.2.1.1.2.cmml" xref="S3.SS1.SSS2.3.p3.7.m7.2.2.1.1.1"><abs id="S3.SS1.SSS2.3.p3.7.m7.2.2.1.1.2.1.cmml" xref="S3.SS1.SSS2.3.p3.7.m7.2.2.1.1.1.2"></abs><apply id="S3.SS1.SSS2.3.p3.7.m7.2.2.1.1.1.1.cmml" xref="S3.SS1.SSS2.3.p3.7.m7.2.2.1.1.1.1"><setdiff id="S3.SS1.SSS2.3.p3.7.m7.2.2.1.1.1.1.1.cmml" xref="S3.SS1.SSS2.3.p3.7.m7.2.2.1.1.1.1.1"></setdiff><apply id="S3.SS1.SSS2.3.p3.7.m7.2.2.1.1.1.1.2.cmml" xref="S3.SS1.SSS2.3.p3.7.m7.2.2.1.1.1.1.2"><csymbol cd="ambiguous" id="S3.SS1.SSS2.3.p3.7.m7.2.2.1.1.1.1.2.1.cmml" xref="S3.SS1.SSS2.3.p3.7.m7.2.2.1.1.1.1.2">superscript</csymbol><ci id="S3.SS1.SSS2.3.p3.7.m7.2.2.1.1.1.1.2.2.cmml" xref="S3.SS1.SSS2.3.p3.7.m7.2.2.1.1.1.1.2.2">𝑆</ci><plus id="S3.SS1.SSS2.3.p3.7.m7.2.2.1.1.1.1.2.3.cmml" xref="S3.SS1.SSS2.3.p3.7.m7.2.2.1.1.1.1.2.3"></plus></apply><ci id="S3.SS1.SSS2.3.p3.7.m7.2.2.1.1.1.1.3.cmml" xref="S3.SS1.SSS2.3.p3.7.m7.2.2.1.1.1.1.3">𝑆</ci></apply></apply></apply><ci id="S3.SS1.SSS2.3.p3.7.m7.2.2.3.cmml" xref="S3.SS1.SSS2.3.p3.7.m7.2.2.3">𝑘</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS2.3.p3.7.m7.2c">h(\hat{S})+|S^{+}\setminus S|\leq k</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS2.3.p3.7.m7.2d">italic_h ( over^ start_ARG italic_S end_ARG ) + | italic_S start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT ∖ italic_S | ≤ italic_k</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="S3.SS1.SSS2.4.p4"> <p class="ltx_p" id="S3.SS1.SSS2.4.p4.2">Then, the fractional solution <math alttext="\boldsymbol{x}" class="ltx_Math" display="inline" id="S3.SS1.SSS2.4.p4.1.m1.1"><semantics id="S3.SS1.SSS2.4.p4.1.m1.1a"><mi id="S3.SS1.SSS2.4.p4.1.m1.1.1" xref="S3.SS1.SSS2.4.p4.1.m1.1.1.cmml">𝒙</mi><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS2.4.p4.1.m1.1b"><ci id="S3.SS1.SSS2.4.p4.1.m1.1.1.cmml" xref="S3.SS1.SSS2.4.p4.1.m1.1.1">𝒙</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS2.4.p4.1.m1.1c">\boldsymbol{x}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS2.4.p4.1.m1.1d">bold_italic_x</annotation></semantics></math> is constructed from <span class="ltx_text ltx_markedasmath" id="S3.SS1.SSS2.4.p4.2.1">OPT</span> as follows (in Figure <a class="ltx_ref" href="https://arxiv.org/html/2503.00712v1#S3.F2" title="Figure 2 ‣ Proof. ‣ 3.1.2 An Improved Analysis via Fractional Solutions ‣ 3.1 Vertex Connectivity Network Design ‣ 3 Generic Framework for Streaming Algorithms for Network Design ‣ Streaming Algorithms for Network Design"><span class="ltx_text ltx_ref_tag">2</span></a>).</p> </div> <figure class="ltx_figure" id="S3.F2"><svg class="ltx_picture ltx_centering" height="49.47" id="S3.F2.pic1" overflow="visible" version="1.1" width="510"><g fill="#000000" stroke="#000000" stroke-width="0.4pt" transform="translate(0,49.47) matrix(1 0 0 -1 0 0)"><g fill="#404040" fill-opacity="1.0"><path d="M 0 5.91 L 0 43.56 C 0 46.82 2.64 49.47 5.91 49.47 L 504.1 49.47 C 507.36 49.47 510 46.82 510 43.56 L 510 5.91 C 510 2.64 507.36 0 504.1 0 L 5.91 0 C 2.64 0 0 2.64 0 5.91 Z" style="stroke:none"></path></g><g fill="#FFFFFF" fill-opacity="1.0"><path d="M 1.97 5.91 L 1.97 43.56 C 1.97 45.74 3.73 47.5 5.91 47.5 L 504.1 47.5 C 506.27 47.5 508.04 45.74 508.04 43.56 L 508.04 5.91 C 508.04 3.73 506.27 1.97 504.1 1.97 L 5.91 1.97 C 3.73 1.97 1.97 3.73 1.97 5.91 Z" style="stroke:none"></path></g><g fill-opacity="1.0" transform="matrix(1.0 0.0 0.0 1.0 21.65 13.78)"><foreignobject color="#000000" height="21.91" overflow="visible" transform="matrix(1 0 0 -1 0 16.6)" width="466.7"> <span class="ltx_inline-block ltx_minipage ltx_align_bottom" id="S3.F2.pic1.2.2.2.2.2.2.2.2.2.2.2.2.2.2.2.2.2" style="width:337.3pt;"> <span class="ltx_p" id="S3.F2.pic1.2.2.2.2.2.2.2.2.2.2.2.2.2.2.2.2.2.2"><span class="ltx_text ltx_ulem_uline ltx_font_sansserif ltx_font_bold" id="S3.F2.pic1.2.2.2.2.2.2.2.2.2.2.2.2.2.2.2.2.2.2.2" style="font-size:120%;">Construction of <math alttext="\boldsymbol{x}" class="ltx_Math" display="inline" id="S3.F2.pic1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1"><semantics id="S3.F2.pic1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1a"><mi id="S3.F2.pic1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1.1" xref="S3.F2.pic1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1.1.cmml">x</mi><annotation-xml encoding="MathML-Content" id="S3.F2.pic1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1b"><ci id="S3.F2.pic1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1.1.cmml" xref="S3.F2.pic1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1.1">𝑥</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.F2.pic1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1c">\boldsymbol{x}</annotation><annotation encoding="application/x-llamapun" id="S3.F2.pic1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1d">bold_italic_x</annotation></semantics></math> from <span class="ltx_text ltx_markedasmath ltx_font_serif ltx_font_medium" id="S3.F2.pic1.2.2.2.2.2.2.2.2.2.2.2.2.2.2.2.2.2.2.2.3">OPT</span></span></span> <span class="ltx_itemize" id="S3.I1"> <span class="ltx_item" id="S3.I1.ix1" style="list-style-type:none;"><span class="ltx_tag ltx_tag_item"></span> <span class="ltx_para" id="S3.I1.ix1.p1"> <span class="ltx_p" id="S3.I1.ix1.p1.3"><span class="ltx_text ltx_font_bold" id="S3.I1.ix1.p1.3.1">Initialize</span> <math alttext="\boldsymbol{x}" class="ltx_Math" display="inline" id="S3.I1.ix1.p1.1.m1.1"><semantics id="S3.I1.ix1.p1.1.m1.1a"><mi id="S3.I1.ix1.p1.1.m1.1.1" xref="S3.I1.ix1.p1.1.m1.1.1.cmml">𝒙</mi><annotation-xml encoding="MathML-Content" id="S3.I1.ix1.p1.1.m1.1b"><ci id="S3.I1.ix1.p1.1.m1.1.1.cmml" xref="S3.I1.ix1.p1.1.m1.1.1">𝒙</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.I1.ix1.p1.1.m1.1c">\boldsymbol{x}</annotation><annotation encoding="application/x-llamapun" id="S3.I1.ix1.p1.1.m1.1d">bold_italic_x</annotation></semantics></math> as an all-zero vector, i.e., <math alttext="\boldsymbol{x}(e)=0" class="ltx_Math" display="inline" id="S3.I1.ix1.p1.2.m2.1"><semantics id="S3.I1.ix1.p1.2.m2.1a"><mrow id="S3.I1.ix1.p1.2.m2.1.2" xref="S3.I1.ix1.p1.2.m2.1.2.cmml"><mrow id="S3.I1.ix1.p1.2.m2.1.2.2" xref="S3.I1.ix1.p1.2.m2.1.2.2.cmml"><mi id="S3.I1.ix1.p1.2.m2.1.2.2.2" xref="S3.I1.ix1.p1.2.m2.1.2.2.2.cmml">𝒙</mi><mo id="S3.I1.ix1.p1.2.m2.1.2.2.1" xref="S3.I1.ix1.p1.2.m2.1.2.2.1.cmml">⁢</mo><mrow id="S3.I1.ix1.p1.2.m2.1.2.2.3.2" xref="S3.I1.ix1.p1.2.m2.1.2.2.cmml"><mo id="S3.I1.ix1.p1.2.m2.1.2.2.3.2.1" stretchy="false" xref="S3.I1.ix1.p1.2.m2.1.2.2.cmml">(</mo><mi id="S3.I1.ix1.p1.2.m2.1.1" xref="S3.I1.ix1.p1.2.m2.1.1.cmml">e</mi><mo id="S3.I1.ix1.p1.2.m2.1.2.2.3.2.2" stretchy="false" xref="S3.I1.ix1.p1.2.m2.1.2.2.cmml">)</mo></mrow></mrow><mo id="S3.I1.ix1.p1.2.m2.1.2.1" xref="S3.I1.ix1.p1.2.m2.1.2.1.cmml">=</mo><mn id="S3.I1.ix1.p1.2.m2.1.2.3" xref="S3.I1.ix1.p1.2.m2.1.2.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.I1.ix1.p1.2.m2.1b"><apply id="S3.I1.ix1.p1.2.m2.1.2.cmml" xref="S3.I1.ix1.p1.2.m2.1.2"><eq id="S3.I1.ix1.p1.2.m2.1.2.1.cmml" xref="S3.I1.ix1.p1.2.m2.1.2.1"></eq><apply id="S3.I1.ix1.p1.2.m2.1.2.2.cmml" xref="S3.I1.ix1.p1.2.m2.1.2.2"><times id="S3.I1.ix1.p1.2.m2.1.2.2.1.cmml" xref="S3.I1.ix1.p1.2.m2.1.2.2.1"></times><ci id="S3.I1.ix1.p1.2.m2.1.2.2.2.cmml" xref="S3.I1.ix1.p1.2.m2.1.2.2.2">𝒙</ci><ci id="S3.I1.ix1.p1.2.m2.1.1.cmml" xref="S3.I1.ix1.p1.2.m2.1.1">𝑒</ci></apply><cn id="S3.I1.ix1.p1.2.m2.1.2.3.cmml" type="integer" xref="S3.I1.ix1.p1.2.m2.1.2.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.I1.ix1.p1.2.m2.1c">\boldsymbol{x}(e)=0</annotation><annotation encoding="application/x-llamapun" id="S3.I1.ix1.p1.2.m2.1d">bold_italic_x ( italic_e ) = 0</annotation></semantics></math> for all <math alttext="e\in E" class="ltx_Math" display="inline" id="S3.I1.ix1.p1.3.m3.1"><semantics id="S3.I1.ix1.p1.3.m3.1a"><mrow id="S3.I1.ix1.p1.3.m3.1.1" xref="S3.I1.ix1.p1.3.m3.1.1.cmml"><mi id="S3.I1.ix1.p1.3.m3.1.1.2" xref="S3.I1.ix1.p1.3.m3.1.1.2.cmml">e</mi><mo id="S3.I1.ix1.p1.3.m3.1.1.1" xref="S3.I1.ix1.p1.3.m3.1.1.1.cmml">∈</mo><mi id="S3.I1.ix1.p1.3.m3.1.1.3" xref="S3.I1.ix1.p1.3.m3.1.1.3.cmml">E</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.I1.ix1.p1.3.m3.1b"><apply id="S3.I1.ix1.p1.3.m3.1.1.cmml" xref="S3.I1.ix1.p1.3.m3.1.1"><in id="S3.I1.ix1.p1.3.m3.1.1.1.cmml" xref="S3.I1.ix1.p1.3.m3.1.1.1"></in><ci id="S3.I1.ix1.p1.3.m3.1.1.2.cmml" xref="S3.I1.ix1.p1.3.m3.1.1.2">𝑒</ci><ci id="S3.I1.ix1.p1.3.m3.1.1.3.cmml" xref="S3.I1.ix1.p1.3.m3.1.1.3">𝐸</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.I1.ix1.p1.3.m3.1c">e\in E</annotation><annotation encoding="application/x-llamapun" id="S3.I1.ix1.p1.3.m3.1d">italic_e ∈ italic_E</annotation></semantics></math>.</span> </span></span> <span class="ltx_item" id="S3.I1.ix2" style="list-style-type:none;"><span class="ltx_tag ltx_tag_item"></span> <span class="ltx_para" id="S3.I1.ix2.p1"> <span class="ltx_p" id="S3.I1.ix2.p1.1"><span class="ltx_text ltx_font_bold" id="S3.I1.ix2.p1.1.1">for each</span> <math alttext="e=(u,v)\in\textnormal{OPT}" class="ltx_Math" display="inline" id="S3.I1.ix2.p1.1.m1.2"><semantics id="S3.I1.ix2.p1.1.m1.2a"><mrow id="S3.I1.ix2.p1.1.m1.2.3" xref="S3.I1.ix2.p1.1.m1.2.3.cmml"><mi id="S3.I1.ix2.p1.1.m1.2.3.2" xref="S3.I1.ix2.p1.1.m1.2.3.2.cmml">e</mi><mo id="S3.I1.ix2.p1.1.m1.2.3.3" xref="S3.I1.ix2.p1.1.m1.2.3.3.cmml">=</mo><mrow id="S3.I1.ix2.p1.1.m1.2.3.4.2" xref="S3.I1.ix2.p1.1.m1.2.3.4.1.cmml"><mo id="S3.I1.ix2.p1.1.m1.2.3.4.2.1" stretchy="false" xref="S3.I1.ix2.p1.1.m1.2.3.4.1.cmml">(</mo><mi id="S3.I1.ix2.p1.1.m1.1.1" xref="S3.I1.ix2.p1.1.m1.1.1.cmml">u</mi><mo id="S3.I1.ix2.p1.1.m1.2.3.4.2.2" xref="S3.I1.ix2.p1.1.m1.2.3.4.1.cmml">,</mo><mi id="S3.I1.ix2.p1.1.m1.2.2" xref="S3.I1.ix2.p1.1.m1.2.2.cmml">v</mi><mo id="S3.I1.ix2.p1.1.m1.2.3.4.2.3" stretchy="false" xref="S3.I1.ix2.p1.1.m1.2.3.4.1.cmml">)</mo></mrow><mo id="S3.I1.ix2.p1.1.m1.2.3.5" xref="S3.I1.ix2.p1.1.m1.2.3.5.cmml">∈</mo><mtext id="S3.I1.ix2.p1.1.m1.2.3.6" xref="S3.I1.ix2.p1.1.m1.2.3.6a.cmml">OPT</mtext></mrow><annotation-xml encoding="MathML-Content" id="S3.I1.ix2.p1.1.m1.2b"><apply id="S3.I1.ix2.p1.1.m1.2.3.cmml" xref="S3.I1.ix2.p1.1.m1.2.3"><and id="S3.I1.ix2.p1.1.m1.2.3a.cmml" xref="S3.I1.ix2.p1.1.m1.2.3"></and><apply id="S3.I1.ix2.p1.1.m1.2.3b.cmml" xref="S3.I1.ix2.p1.1.m1.2.3"><eq id="S3.I1.ix2.p1.1.m1.2.3.3.cmml" xref="S3.I1.ix2.p1.1.m1.2.3.3"></eq><ci id="S3.I1.ix2.p1.1.m1.2.3.2.cmml" xref="S3.I1.ix2.p1.1.m1.2.3.2">𝑒</ci><interval closure="open" id="S3.I1.ix2.p1.1.m1.2.3.4.1.cmml" xref="S3.I1.ix2.p1.1.m1.2.3.4.2"><ci id="S3.I1.ix2.p1.1.m1.1.1.cmml" xref="S3.I1.ix2.p1.1.m1.1.1">𝑢</ci><ci id="S3.I1.ix2.p1.1.m1.2.2.cmml" xref="S3.I1.ix2.p1.1.m1.2.2">𝑣</ci></interval></apply><apply id="S3.I1.ix2.p1.1.m1.2.3c.cmml" xref="S3.I1.ix2.p1.1.m1.2.3"><in id="S3.I1.ix2.p1.1.m1.2.3.5.cmml" xref="S3.I1.ix2.p1.1.m1.2.3.5"></in><share href="https://arxiv.org/html/2503.00712v1#S3.I1.ix2.p1.1.m1.2.3.4.cmml" id="S3.I1.ix2.p1.1.m1.2.3d.cmml" xref="S3.I1.ix2.p1.1.m1.2.3"></share><ci id="S3.I1.ix2.p1.1.m1.2.3.6a.cmml" xref="S3.I1.ix2.p1.1.m1.2.3.6"><mtext id="S3.I1.ix2.p1.1.m1.2.3.6.cmml" xref="S3.I1.ix2.p1.1.m1.2.3.6">OPT</mtext></ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.I1.ix2.p1.1.m1.2c">e=(u,v)\in\textnormal{OPT}</annotation><annotation encoding="application/x-llamapun" id="S3.I1.ix2.p1.1.m1.2d">italic_e = ( italic_u , italic_v ) ∈ OPT</annotation></semantics></math></span> <span class="ltx_itemize" id="S3.I1.ix2.I1"> <span class="ltx_item" id="S3.I1.ix2.I1.ix1" style="list-style-type:none;"><span class="ltx_tag ltx_tag_item"></span> <span class="ltx_para" id="S3.I1.ix2.I1.ix1.p1"> <span class="ltx_p" id="S3.I1.ix2.I1.ix1.p1.2"><span class="ltx_text ltx_font_bold" id="S3.I1.ix2.I1.ix1.p1.2.1">let</span> <math alttext="B_{j}" class="ltx_Math" display="inline" id="S3.I1.ix2.I1.ix1.p1.1.m1.1"><semantics id="S3.I1.ix2.I1.ix1.p1.1.m1.1a"><msub id="S3.I1.ix2.I1.ix1.p1.1.m1.1.1" xref="S3.I1.ix2.I1.ix1.p1.1.m1.1.1.cmml"><mi id="S3.I1.ix2.I1.ix1.p1.1.m1.1.1.2" xref="S3.I1.ix2.I1.ix1.p1.1.m1.1.1.2.cmml">B</mi><mi id="S3.I1.ix2.I1.ix1.p1.1.m1.1.1.3" xref="S3.I1.ix2.I1.ix1.p1.1.m1.1.1.3.cmml">j</mi></msub><annotation-xml encoding="MathML-Content" id="S3.I1.ix2.I1.ix1.p1.1.m1.1b"><apply id="S3.I1.ix2.I1.ix1.p1.1.m1.1.1.cmml" xref="S3.I1.ix2.I1.ix1.p1.1.m1.1.1"><csymbol cd="ambiguous" id="S3.I1.ix2.I1.ix1.p1.1.m1.1.1.1.cmml" xref="S3.I1.ix2.I1.ix1.p1.1.m1.1.1">subscript</csymbol><ci id="S3.I1.ix2.I1.ix1.p1.1.m1.1.1.2.cmml" xref="S3.I1.ix2.I1.ix1.p1.1.m1.1.1.2">𝐵</ci><ci id="S3.I1.ix2.I1.ix1.p1.1.m1.1.1.3.cmml" xref="S3.I1.ix2.I1.ix1.p1.1.m1.1.1.3">𝑗</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.I1.ix2.I1.ix1.p1.1.m1.1c">B_{j}</annotation><annotation encoding="application/x-llamapun" id="S3.I1.ix2.I1.ix1.p1.1.m1.1d">italic_B start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT</annotation></semantics></math> denote the weight class to which the weight of <math alttext="e" class="ltx_Math" display="inline" id="S3.I1.ix2.I1.ix1.p1.2.m2.1"><semantics id="S3.I1.ix2.I1.ix1.p1.2.m2.1a"><mi id="S3.I1.ix2.I1.ix1.p1.2.m2.1.1" xref="S3.I1.ix2.I1.ix1.p1.2.m2.1.1.cmml">e</mi><annotation-xml encoding="MathML-Content" id="S3.I1.ix2.I1.ix1.p1.2.m2.1b"><ci id="S3.I1.ix2.I1.ix1.p1.2.m2.1.1.cmml" xref="S3.I1.ix2.I1.ix1.p1.2.m2.1.1">𝑒</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.I1.ix2.I1.ix1.p1.2.m2.1c">e</annotation><annotation encoding="application/x-llamapun" id="S3.I1.ix2.I1.ix1.p1.2.m2.1d">italic_e</annotation></semantics></math> belongs.</span> </span></span> <span class="ltx_item" id="S3.I1.ix2.I1.ix2" style="list-style-type:none;"><span class="ltx_tag ltx_tag_item"></span> <span class="ltx_para" id="S3.I1.ix2.I1.ix2.p1"> <span class="ltx_p" id="S3.I1.ix2.I1.ix2.p1.2"><span class="ltx_text ltx_font_bold" id="S3.I1.ix2.I1.ix2.p1.2.1">if</span> <math alttext="e\in H_{j}" class="ltx_Math" display="inline" id="S3.I1.ix2.I1.ix2.p1.1.m1.1"><semantics id="S3.I1.ix2.I1.ix2.p1.1.m1.1a"><mrow id="S3.I1.ix2.I1.ix2.p1.1.m1.1.1" xref="S3.I1.ix2.I1.ix2.p1.1.m1.1.1.cmml"><mi id="S3.I1.ix2.I1.ix2.p1.1.m1.1.1.2" xref="S3.I1.ix2.I1.ix2.p1.1.m1.1.1.2.cmml">e</mi><mo id="S3.I1.ix2.I1.ix2.p1.1.m1.1.1.1" xref="S3.I1.ix2.I1.ix2.p1.1.m1.1.1.1.cmml">∈</mo><msub id="S3.I1.ix2.I1.ix2.p1.1.m1.1.1.3" xref="S3.I1.ix2.I1.ix2.p1.1.m1.1.1.3.cmml"><mi id="S3.I1.ix2.I1.ix2.p1.1.m1.1.1.3.2" xref="S3.I1.ix2.I1.ix2.p1.1.m1.1.1.3.2.cmml">H</mi><mi id="S3.I1.ix2.I1.ix2.p1.1.m1.1.1.3.3" xref="S3.I1.ix2.I1.ix2.p1.1.m1.1.1.3.3.cmml">j</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S3.I1.ix2.I1.ix2.p1.1.m1.1b"><apply id="S3.I1.ix2.I1.ix2.p1.1.m1.1.1.cmml" xref="S3.I1.ix2.I1.ix2.p1.1.m1.1.1"><in id="S3.I1.ix2.I1.ix2.p1.1.m1.1.1.1.cmml" xref="S3.I1.ix2.I1.ix2.p1.1.m1.1.1.1"></in><ci id="S3.I1.ix2.I1.ix2.p1.1.m1.1.1.2.cmml" xref="S3.I1.ix2.I1.ix2.p1.1.m1.1.1.2">𝑒</ci><apply id="S3.I1.ix2.I1.ix2.p1.1.m1.1.1.3.cmml" xref="S3.I1.ix2.I1.ix2.p1.1.m1.1.1.3"><csymbol cd="ambiguous" id="S3.I1.ix2.I1.ix2.p1.1.m1.1.1.3.1.cmml" xref="S3.I1.ix2.I1.ix2.p1.1.m1.1.1.3">subscript</csymbol><ci id="S3.I1.ix2.I1.ix2.p1.1.m1.1.1.3.2.cmml" xref="S3.I1.ix2.I1.ix2.p1.1.m1.1.1.3.2">𝐻</ci><ci id="S3.I1.ix2.I1.ix2.p1.1.m1.1.1.3.3.cmml" xref="S3.I1.ix2.I1.ix2.p1.1.m1.1.1.3.3">𝑗</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.I1.ix2.I1.ix2.p1.1.m1.1c">e\in H_{j}</annotation><annotation encoding="application/x-llamapun" id="S3.I1.ix2.I1.ix2.p1.1.m1.1d">italic_e ∈ italic_H start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT</annotation></semantics></math> <span class="ltx_text ltx_font_bold" id="S3.I1.ix2.I1.ix2.p1.2.2">then</span> <math alttext="\boldsymbol{x}(e)\leftarrow 1" class="ltx_Math" display="inline" id="S3.I1.ix2.I1.ix2.p1.2.m2.1"><semantics id="S3.I1.ix2.I1.ix2.p1.2.m2.1a"><mrow id="S3.I1.ix2.I1.ix2.p1.2.m2.1.2" xref="S3.I1.ix2.I1.ix2.p1.2.m2.1.2.cmml"><mrow id="S3.I1.ix2.I1.ix2.p1.2.m2.1.2.2" xref="S3.I1.ix2.I1.ix2.p1.2.m2.1.2.2.cmml"><mi id="S3.I1.ix2.I1.ix2.p1.2.m2.1.2.2.2" xref="S3.I1.ix2.I1.ix2.p1.2.m2.1.2.2.2.cmml">𝒙</mi><mo id="S3.I1.ix2.I1.ix2.p1.2.m2.1.2.2.1" xref="S3.I1.ix2.I1.ix2.p1.2.m2.1.2.2.1.cmml">⁢</mo><mrow id="S3.I1.ix2.I1.ix2.p1.2.m2.1.2.2.3.2" xref="S3.I1.ix2.I1.ix2.p1.2.m2.1.2.2.cmml"><mo id="S3.I1.ix2.I1.ix2.p1.2.m2.1.2.2.3.2.1" stretchy="false" xref="S3.I1.ix2.I1.ix2.p1.2.m2.1.2.2.cmml">(</mo><mi id="S3.I1.ix2.I1.ix2.p1.2.m2.1.1" xref="S3.I1.ix2.I1.ix2.p1.2.m2.1.1.cmml">e</mi><mo id="S3.I1.ix2.I1.ix2.p1.2.m2.1.2.2.3.2.2" stretchy="false" xref="S3.I1.ix2.I1.ix2.p1.2.m2.1.2.2.cmml">)</mo></mrow></mrow><mo id="S3.I1.ix2.I1.ix2.p1.2.m2.1.2.1" stretchy="false" xref="S3.I1.ix2.I1.ix2.p1.2.m2.1.2.1.cmml">←</mo><mn id="S3.I1.ix2.I1.ix2.p1.2.m2.1.2.3" xref="S3.I1.ix2.I1.ix2.p1.2.m2.1.2.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.I1.ix2.I1.ix2.p1.2.m2.1b"><apply id="S3.I1.ix2.I1.ix2.p1.2.m2.1.2.cmml" xref="S3.I1.ix2.I1.ix2.p1.2.m2.1.2"><ci id="S3.I1.ix2.I1.ix2.p1.2.m2.1.2.1.cmml" xref="S3.I1.ix2.I1.ix2.p1.2.m2.1.2.1">←</ci><apply id="S3.I1.ix2.I1.ix2.p1.2.m2.1.2.2.cmml" xref="S3.I1.ix2.I1.ix2.p1.2.m2.1.2.2"><times id="S3.I1.ix2.I1.ix2.p1.2.m2.1.2.2.1.cmml" xref="S3.I1.ix2.I1.ix2.p1.2.m2.1.2.2.1"></times><ci id="S3.I1.ix2.I1.ix2.p1.2.m2.1.2.2.2.cmml" xref="S3.I1.ix2.I1.ix2.p1.2.m2.1.2.2.2">𝒙</ci><ci id="S3.I1.ix2.I1.ix2.p1.2.m2.1.1.cmml" xref="S3.I1.ix2.I1.ix2.p1.2.m2.1.1">𝑒</ci></apply><cn id="S3.I1.ix2.I1.ix2.p1.2.m2.1.2.3.cmml" type="integer" xref="S3.I1.ix2.I1.ix2.p1.2.m2.1.2.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.I1.ix2.I1.ix2.p1.2.m2.1c">\boldsymbol{x}(e)\leftarrow 1</annotation><annotation encoding="application/x-llamapun" id="S3.I1.ix2.I1.ix2.p1.2.m2.1d">bold_italic_x ( italic_e ) ← 1</annotation></semantics></math></span> </span></span> <span class="ltx_item" id="S3.I1.ix2.I1.ix3" style="list-style-type:none;"><span class="ltx_tag ltx_tag_item"></span> <span class="ltx_para" id="S3.I1.ix2.I1.ix3.p1"> <span class="ltx_p" id="S3.I1.ix2.I1.ix3.p1.1"><span class="ltx_text ltx_font_bold" id="S3.I1.ix2.I1.ix3.p1.1.1">else</span></span> <span class="ltx_itemize" id="S3.I1.ix2.I1.ix3.I1"> <span class="ltx_item" id="S3.I1.ix2.I1.ix3.I1.ix1" style="list-style-type:none;"><span class="ltx_tag ltx_tag_item"></span> <span class="ltx_para" id="S3.I1.ix2.I1.ix3.I1.ix1.p1"> <span class="ltx_p" id="S3.I1.ix2.I1.ix3.I1.ix1.p1.4"><span class="ltx_text ltx_font_bold" id="S3.I1.ix2.I1.ix3.I1.ix1.p1.4.1">let</span> <math alttext="P_{1},\cdots,P_{2k}" class="ltx_Math" display="inline" id="S3.I1.ix2.I1.ix3.I1.ix1.p1.1.m1.3"><semantics id="S3.I1.ix2.I1.ix3.I1.ix1.p1.1.m1.3a"><mrow id="S3.I1.ix2.I1.ix3.I1.ix1.p1.1.m1.3.3.2" xref="S3.I1.ix2.I1.ix3.I1.ix1.p1.1.m1.3.3.3.cmml"><msub id="S3.I1.ix2.I1.ix3.I1.ix1.p1.1.m1.2.2.1.1" xref="S3.I1.ix2.I1.ix3.I1.ix1.p1.1.m1.2.2.1.1.cmml"><mi id="S3.I1.ix2.I1.ix3.I1.ix1.p1.1.m1.2.2.1.1.2" xref="S3.I1.ix2.I1.ix3.I1.ix1.p1.1.m1.2.2.1.1.2.cmml">P</mi><mn id="S3.I1.ix2.I1.ix3.I1.ix1.p1.1.m1.2.2.1.1.3" xref="S3.I1.ix2.I1.ix3.I1.ix1.p1.1.m1.2.2.1.1.3.cmml">1</mn></msub><mo id="S3.I1.ix2.I1.ix3.I1.ix1.p1.1.m1.3.3.2.3" xref="S3.I1.ix2.I1.ix3.I1.ix1.p1.1.m1.3.3.3.cmml">,</mo><mi id="S3.I1.ix2.I1.ix3.I1.ix1.p1.1.m1.1.1" mathvariant="normal" xref="S3.I1.ix2.I1.ix3.I1.ix1.p1.1.m1.1.1.cmml">⋯</mi><mo id="S3.I1.ix2.I1.ix3.I1.ix1.p1.1.m1.3.3.2.4" xref="S3.I1.ix2.I1.ix3.I1.ix1.p1.1.m1.3.3.3.cmml">,</mo><msub id="S3.I1.ix2.I1.ix3.I1.ix1.p1.1.m1.3.3.2.2" xref="S3.I1.ix2.I1.ix3.I1.ix1.p1.1.m1.3.3.2.2.cmml"><mi id="S3.I1.ix2.I1.ix3.I1.ix1.p1.1.m1.3.3.2.2.2" xref="S3.I1.ix2.I1.ix3.I1.ix1.p1.1.m1.3.3.2.2.2.cmml">P</mi><mrow id="S3.I1.ix2.I1.ix3.I1.ix1.p1.1.m1.3.3.2.2.3" xref="S3.I1.ix2.I1.ix3.I1.ix1.p1.1.m1.3.3.2.2.3.cmml"><mn id="S3.I1.ix2.I1.ix3.I1.ix1.p1.1.m1.3.3.2.2.3.2" xref="S3.I1.ix2.I1.ix3.I1.ix1.p1.1.m1.3.3.2.2.3.2.cmml">2</mn><mo id="S3.I1.ix2.I1.ix3.I1.ix1.p1.1.m1.3.3.2.2.3.1" xref="S3.I1.ix2.I1.ix3.I1.ix1.p1.1.m1.3.3.2.2.3.1.cmml">⁢</mo><mi id="S3.I1.ix2.I1.ix3.I1.ix1.p1.1.m1.3.3.2.2.3.3" xref="S3.I1.ix2.I1.ix3.I1.ix1.p1.1.m1.3.3.2.2.3.3.cmml">k</mi></mrow></msub></mrow><annotation-xml encoding="MathML-Content" id="S3.I1.ix2.I1.ix3.I1.ix1.p1.1.m1.3b"><list id="S3.I1.ix2.I1.ix3.I1.ix1.p1.1.m1.3.3.3.cmml" xref="S3.I1.ix2.I1.ix3.I1.ix1.p1.1.m1.3.3.2"><apply id="S3.I1.ix2.I1.ix3.I1.ix1.p1.1.m1.2.2.1.1.cmml" xref="S3.I1.ix2.I1.ix3.I1.ix1.p1.1.m1.2.2.1.1"><csymbol cd="ambiguous" id="S3.I1.ix2.I1.ix3.I1.ix1.p1.1.m1.2.2.1.1.1.cmml" xref="S3.I1.ix2.I1.ix3.I1.ix1.p1.1.m1.2.2.1.1">subscript</csymbol><ci id="S3.I1.ix2.I1.ix3.I1.ix1.p1.1.m1.2.2.1.1.2.cmml" xref="S3.I1.ix2.I1.ix3.I1.ix1.p1.1.m1.2.2.1.1.2">𝑃</ci><cn id="S3.I1.ix2.I1.ix3.I1.ix1.p1.1.m1.2.2.1.1.3.cmml" type="integer" xref="S3.I1.ix2.I1.ix3.I1.ix1.p1.1.m1.2.2.1.1.3">1</cn></apply><ci id="S3.I1.ix2.I1.ix3.I1.ix1.p1.1.m1.1.1.cmml" xref="S3.I1.ix2.I1.ix3.I1.ix1.p1.1.m1.1.1">⋯</ci><apply id="S3.I1.ix2.I1.ix3.I1.ix1.p1.1.m1.3.3.2.2.cmml" xref="S3.I1.ix2.I1.ix3.I1.ix1.p1.1.m1.3.3.2.2"><csymbol cd="ambiguous" id="S3.I1.ix2.I1.ix3.I1.ix1.p1.1.m1.3.3.2.2.1.cmml" xref="S3.I1.ix2.I1.ix3.I1.ix1.p1.1.m1.3.3.2.2">subscript</csymbol><ci id="S3.I1.ix2.I1.ix3.I1.ix1.p1.1.m1.3.3.2.2.2.cmml" xref="S3.I1.ix2.I1.ix3.I1.ix1.p1.1.m1.3.3.2.2.2">𝑃</ci><apply id="S3.I1.ix2.I1.ix3.I1.ix1.p1.1.m1.3.3.2.2.3.cmml" xref="S3.I1.ix2.I1.ix3.I1.ix1.p1.1.m1.3.3.2.2.3"><times id="S3.I1.ix2.I1.ix3.I1.ix1.p1.1.m1.3.3.2.2.3.1.cmml" xref="S3.I1.ix2.I1.ix3.I1.ix1.p1.1.m1.3.3.2.2.3.1"></times><cn id="S3.I1.ix2.I1.ix3.I1.ix1.p1.1.m1.3.3.2.2.3.2.cmml" type="integer" xref="S3.I1.ix2.I1.ix3.I1.ix1.p1.1.m1.3.3.2.2.3.2">2</cn><ci id="S3.I1.ix2.I1.ix3.I1.ix1.p1.1.m1.3.3.2.2.3.3.cmml" xref="S3.I1.ix2.I1.ix3.I1.ix1.p1.1.m1.3.3.2.2.3.3">𝑘</ci></apply></apply></list></annotation-xml><annotation encoding="application/x-tex" id="S3.I1.ix2.I1.ix3.I1.ix1.p1.1.m1.3c">P_{1},\cdots,P_{2k}</annotation><annotation encoding="application/x-llamapun" id="S3.I1.ix2.I1.ix3.I1.ix1.p1.1.m1.3d">italic_P start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , ⋯ , italic_P start_POSTSUBSCRIPT 2 italic_k end_POSTSUBSCRIPT</annotation></semantics></math> be vertex-disjoint <math alttext="uv" class="ltx_Math" display="inline" id="S3.I1.ix2.I1.ix3.I1.ix1.p1.2.m2.1"><semantics id="S3.I1.ix2.I1.ix3.I1.ix1.p1.2.m2.1a"><mrow id="S3.I1.ix2.I1.ix3.I1.ix1.p1.2.m2.1.1" xref="S3.I1.ix2.I1.ix3.I1.ix1.p1.2.m2.1.1.cmml"><mi id="S3.I1.ix2.I1.ix3.I1.ix1.p1.2.m2.1.1.2" xref="S3.I1.ix2.I1.ix3.I1.ix1.p1.2.m2.1.1.2.cmml">u</mi><mo id="S3.I1.ix2.I1.ix3.I1.ix1.p1.2.m2.1.1.1" xref="S3.I1.ix2.I1.ix3.I1.ix1.p1.2.m2.1.1.1.cmml">⁢</mo><mi id="S3.I1.ix2.I1.ix3.I1.ix1.p1.2.m2.1.1.3" xref="S3.I1.ix2.I1.ix3.I1.ix1.p1.2.m2.1.1.3.cmml">v</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.I1.ix2.I1.ix3.I1.ix1.p1.2.m2.1b"><apply id="S3.I1.ix2.I1.ix3.I1.ix1.p1.2.m2.1.1.cmml" xref="S3.I1.ix2.I1.ix3.I1.ix1.p1.2.m2.1.1"><times id="S3.I1.ix2.I1.ix3.I1.ix1.p1.2.m2.1.1.1.cmml" xref="S3.I1.ix2.I1.ix3.I1.ix1.p1.2.m2.1.1.1"></times><ci id="S3.I1.ix2.I1.ix3.I1.ix1.p1.2.m2.1.1.2.cmml" xref="S3.I1.ix2.I1.ix3.I1.ix1.p1.2.m2.1.1.2">𝑢</ci><ci id="S3.I1.ix2.I1.ix3.I1.ix1.p1.2.m2.1.1.3.cmml" xref="S3.I1.ix2.I1.ix3.I1.ix1.p1.2.m2.1.1.3">𝑣</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.I1.ix2.I1.ix3.I1.ix1.p1.2.m2.1c">uv</annotation><annotation encoding="application/x-llamapun" id="S3.I1.ix2.I1.ix3.I1.ix1.p1.2.m2.1d">italic_u italic_v</annotation></semantics></math>-paths in <math alttext="H_{j}" class="ltx_Math" display="inline" id="S3.I1.ix2.I1.ix3.I1.ix1.p1.3.m3.1"><semantics id="S3.I1.ix2.I1.ix3.I1.ix1.p1.3.m3.1a"><msub id="S3.I1.ix2.I1.ix3.I1.ix1.p1.3.m3.1.1" xref="S3.I1.ix2.I1.ix3.I1.ix1.p1.3.m3.1.1.cmml"><mi id="S3.I1.ix2.I1.ix3.I1.ix1.p1.3.m3.1.1.2" xref="S3.I1.ix2.I1.ix3.I1.ix1.p1.3.m3.1.1.2.cmml">H</mi><mi id="S3.I1.ix2.I1.ix3.I1.ix1.p1.3.m3.1.1.3" xref="S3.I1.ix2.I1.ix3.I1.ix1.p1.3.m3.1.1.3.cmml">j</mi></msub><annotation-xml encoding="MathML-Content" id="S3.I1.ix2.I1.ix3.I1.ix1.p1.3.m3.1b"><apply id="S3.I1.ix2.I1.ix3.I1.ix1.p1.3.m3.1.1.cmml" xref="S3.I1.ix2.I1.ix3.I1.ix1.p1.3.m3.1.1"><csymbol cd="ambiguous" id="S3.I1.ix2.I1.ix3.I1.ix1.p1.3.m3.1.1.1.cmml" xref="S3.I1.ix2.I1.ix3.I1.ix1.p1.3.m3.1.1">subscript</csymbol><ci id="S3.I1.ix2.I1.ix3.I1.ix1.p1.3.m3.1.1.2.cmml" xref="S3.I1.ix2.I1.ix3.I1.ix1.p1.3.m3.1.1.2">𝐻</ci><ci id="S3.I1.ix2.I1.ix3.I1.ix1.p1.3.m3.1.1.3.cmml" xref="S3.I1.ix2.I1.ix3.I1.ix1.p1.3.m3.1.1.3">𝑗</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.I1.ix2.I1.ix3.I1.ix1.p1.3.m3.1c">H_{j}</annotation><annotation encoding="application/x-llamapun" id="S3.I1.ix2.I1.ix3.I1.ix1.p1.3.m3.1d">italic_H start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT</annotation></semantics></math>, each of length at most <math alttext="t" class="ltx_Math" display="inline" id="S3.I1.ix2.I1.ix3.I1.ix1.p1.4.m4.1"><semantics id="S3.I1.ix2.I1.ix3.I1.ix1.p1.4.m4.1a"><mi id="S3.I1.ix2.I1.ix3.I1.ix1.p1.4.m4.1.1" xref="S3.I1.ix2.I1.ix3.I1.ix1.p1.4.m4.1.1.cmml">t</mi><annotation-xml encoding="MathML-Content" id="S3.I1.ix2.I1.ix3.I1.ix1.p1.4.m4.1b"><ci id="S3.I1.ix2.I1.ix3.I1.ix1.p1.4.m4.1.1.cmml" xref="S3.I1.ix2.I1.ix3.I1.ix1.p1.4.m4.1.1">𝑡</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.I1.ix2.I1.ix3.I1.ix1.p1.4.m4.1c">t</annotation><annotation encoding="application/x-llamapun" id="S3.I1.ix2.I1.ix3.I1.ix1.p1.4.m4.1d">italic_t</annotation></semantics></math></span> </span></span> <span class="ltx_item" id="S3.I1.ix2.I1.ix3.I1.ix2" style="list-style-type:none;"><span class="ltx_tag ltx_tag_item"></span> <span class="ltx_para" id="S3.I1.ix2.I1.ix3.I1.ix2.p1"> <span class="ltx_p" id="S3.I1.ix2.I1.ix3.I1.ix2.p1.2"><span class="ltx_text ltx_font_bold" id="S3.I1.ix2.I1.ix3.I1.ix2.p1.2.2">for each <math alttext="e^{\prime}\in P_{1}\cup\cdots\cup P_{2k}" class="ltx_Math" display="inline" id="S3.I1.ix2.I1.ix3.I1.ix2.p1.1.1.m1.1"><semantics id="S3.I1.ix2.I1.ix3.I1.ix2.p1.1.1.m1.1a"><mrow id="S3.I1.ix2.I1.ix3.I1.ix2.p1.1.1.m1.1.1" xref="S3.I1.ix2.I1.ix3.I1.ix2.p1.1.1.m1.1.1.cmml"><msup id="S3.I1.ix2.I1.ix3.I1.ix2.p1.1.1.m1.1.1.2" xref="S3.I1.ix2.I1.ix3.I1.ix2.p1.1.1.m1.1.1.2.cmml"><mi id="S3.I1.ix2.I1.ix3.I1.ix2.p1.1.1.m1.1.1.2.2" xref="S3.I1.ix2.I1.ix3.I1.ix2.p1.1.1.m1.1.1.2.2.cmml">e</mi><mo id="S3.I1.ix2.I1.ix3.I1.ix2.p1.1.1.m1.1.1.2.3" xref="S3.I1.ix2.I1.ix3.I1.ix2.p1.1.1.m1.1.1.2.3.cmml">′</mo></msup><mo id="S3.I1.ix2.I1.ix3.I1.ix2.p1.1.1.m1.1.1.1" xref="S3.I1.ix2.I1.ix3.I1.ix2.p1.1.1.m1.1.1.1.cmml">∈</mo><mrow id="S3.I1.ix2.I1.ix3.I1.ix2.p1.1.1.m1.1.1.3" xref="S3.I1.ix2.I1.ix3.I1.ix2.p1.1.1.m1.1.1.3.cmml"><msub id="S3.I1.ix2.I1.ix3.I1.ix2.p1.1.1.m1.1.1.3.2" xref="S3.I1.ix2.I1.ix3.I1.ix2.p1.1.1.m1.1.1.3.2.cmml"><mi id="S3.I1.ix2.I1.ix3.I1.ix2.p1.1.1.m1.1.1.3.2.2" xref="S3.I1.ix2.I1.ix3.I1.ix2.p1.1.1.m1.1.1.3.2.2.cmml">P</mi><mn id="S3.I1.ix2.I1.ix3.I1.ix2.p1.1.1.m1.1.1.3.2.3" xref="S3.I1.ix2.I1.ix3.I1.ix2.p1.1.1.m1.1.1.3.2.3.cmml">1</mn></msub><mo id="S3.I1.ix2.I1.ix3.I1.ix2.p1.1.1.m1.1.1.3.1" xref="S3.I1.ix2.I1.ix3.I1.ix2.p1.1.1.m1.1.1.3.1.cmml">∪</mo><mi id="S3.I1.ix2.I1.ix3.I1.ix2.p1.1.1.m1.1.1.3.3" mathvariant="normal" xref="S3.I1.ix2.I1.ix3.I1.ix2.p1.1.1.m1.1.1.3.3.cmml">⋯</mi><mo id="S3.I1.ix2.I1.ix3.I1.ix2.p1.1.1.m1.1.1.3.1a" xref="S3.I1.ix2.I1.ix3.I1.ix2.p1.1.1.m1.1.1.3.1.cmml">∪</mo><msub id="S3.I1.ix2.I1.ix3.I1.ix2.p1.1.1.m1.1.1.3.4" xref="S3.I1.ix2.I1.ix3.I1.ix2.p1.1.1.m1.1.1.3.4.cmml"><mi id="S3.I1.ix2.I1.ix3.I1.ix2.p1.1.1.m1.1.1.3.4.2" xref="S3.I1.ix2.I1.ix3.I1.ix2.p1.1.1.m1.1.1.3.4.2.cmml">P</mi><mrow id="S3.I1.ix2.I1.ix3.I1.ix2.p1.1.1.m1.1.1.3.4.3" xref="S3.I1.ix2.I1.ix3.I1.ix2.p1.1.1.m1.1.1.3.4.3.cmml"><mn id="S3.I1.ix2.I1.ix3.I1.ix2.p1.1.1.m1.1.1.3.4.3.2" xref="S3.I1.ix2.I1.ix3.I1.ix2.p1.1.1.m1.1.1.3.4.3.2.cmml">2</mn><mo id="S3.I1.ix2.I1.ix3.I1.ix2.p1.1.1.m1.1.1.3.4.3.1" xref="S3.I1.ix2.I1.ix3.I1.ix2.p1.1.1.m1.1.1.3.4.3.1.cmml">⁢</mo><mi id="S3.I1.ix2.I1.ix3.I1.ix2.p1.1.1.m1.1.1.3.4.3.3" xref="S3.I1.ix2.I1.ix3.I1.ix2.p1.1.1.m1.1.1.3.4.3.3.cmml">k</mi></mrow></msub></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.I1.ix2.I1.ix3.I1.ix2.p1.1.1.m1.1b"><apply id="S3.I1.ix2.I1.ix3.I1.ix2.p1.1.1.m1.1.1.cmml" xref="S3.I1.ix2.I1.ix3.I1.ix2.p1.1.1.m1.1.1"><in id="S3.I1.ix2.I1.ix3.I1.ix2.p1.1.1.m1.1.1.1.cmml" xref="S3.I1.ix2.I1.ix3.I1.ix2.p1.1.1.m1.1.1.1"></in><apply id="S3.I1.ix2.I1.ix3.I1.ix2.p1.1.1.m1.1.1.2.cmml" xref="S3.I1.ix2.I1.ix3.I1.ix2.p1.1.1.m1.1.1.2"><csymbol cd="ambiguous" 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</span></foreignobject></g></g></svg> <figcaption class="ltx_caption ltx_centering"><span class="ltx_tag ltx_tag_figure"><span class="ltx_text" id="S3.F2.7.3.1" style="font-size:90%;">Figure 2</span>: </span><span class="ltx_text" id="S3.F2.4.2" style="font-size:90%;">Construction of the fractional solution <math alttext="\boldsymbol{x}" class="ltx_Math" display="inline" id="S3.F2.3.1.m1.1"><semantics id="S3.F2.3.1.m1.1b"><mi id="S3.F2.3.1.m1.1.1" xref="S3.F2.3.1.m1.1.1.cmml">𝒙</mi><annotation-xml encoding="MathML-Content" id="S3.F2.3.1.m1.1c"><ci id="S3.F2.3.1.m1.1.1.cmml" xref="S3.F2.3.1.m1.1.1">𝒙</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.F2.3.1.m1.1d">\boldsymbol{x}</annotation><annotation encoding="application/x-llamapun" id="S3.F2.3.1.m1.1e">bold_italic_x</annotation></semantics></math> of <a class="ltx_ref" href="https://arxiv.org/html/2503.00712v1#S3.F1" title="Figure 1 ‣ 3.1.2 An Improved Analysis via Fractional Solutions ‣ 3.1 Vertex Connectivity Network Design ‣ 3 Generic Framework for Streaming Algorithms for Network Design ‣ Streaming Algorithms for Network Design">VC-SNDP-LP</a> from <span class="ltx_text ltx_markedasmath" id="S3.F2.4.2.1">OPT</span></span></figcaption> </figure> <div class="ltx_para" id="S3.SS1.SSS2.5.p5"> <p class="ltx_p" id="S3.SS1.SSS2.5.p5.1">Note that our algorithm does not need to explicitly construct <math alttext="\boldsymbol{x}" class="ltx_Math" display="inline" id="S3.SS1.SSS2.5.p5.1.m1.1"><semantics id="S3.SS1.SSS2.5.p5.1.m1.1a"><mi id="S3.SS1.SSS2.5.p5.1.m1.1.1" xref="S3.SS1.SSS2.5.p5.1.m1.1.1.cmml">𝒙</mi><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS2.5.p5.1.m1.1b"><ci id="S3.SS1.SSS2.5.p5.1.m1.1.1.cmml" xref="S3.SS1.SSS2.5.p5.1.m1.1.1">𝒙</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS2.5.p5.1.m1.1c">\boldsymbol{x}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS2.5.p5.1.m1.1d">bold_italic_x</annotation></semantics></math>; this is only for analysis purposes.</p> </div> <div class="ltx_para" id="S3.SS1.SSS2.6.p6"> <p class="ltx_p" id="S3.SS1.SSS2.6.p6.19">First, we prove the feasibility of <math alttext="\boldsymbol{x}" class="ltx_Math" display="inline" id="S3.SS1.SSS2.6.p6.1.m1.1"><semantics id="S3.SS1.SSS2.6.p6.1.m1.1a"><mi id="S3.SS1.SSS2.6.p6.1.m1.1.1" xref="S3.SS1.SSS2.6.p6.1.m1.1.1.cmml">𝒙</mi><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS2.6.p6.1.m1.1b"><ci id="S3.SS1.SSS2.6.p6.1.m1.1.1.cmml" xref="S3.SS1.SSS2.6.p6.1.m1.1.1">𝒙</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS2.6.p6.1.m1.1c">\boldsymbol{x}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS2.6.p6.1.m1.1d">bold_italic_x</annotation></semantics></math> for <a class="ltx_ref" href="https://arxiv.org/html/2503.00712v1#S3.F1" title="Figure 1 ‣ 3.1.2 An Improved Analysis via Fractional Solutions ‣ 3.1 Vertex Connectivity Network Design ‣ 3 Generic Framework for Streaming Algorithms for Network Design ‣ Streaming Algorithms for Network Design">VC-SNDP-LP</a>(<math alttext="G,w,h" class="ltx_Math" display="inline" id="S3.SS1.SSS2.6.p6.2.m2.3"><semantics id="S3.SS1.SSS2.6.p6.2.m2.3a"><mrow id="S3.SS1.SSS2.6.p6.2.m2.3.4.2" xref="S3.SS1.SSS2.6.p6.2.m2.3.4.1.cmml"><mi id="S3.SS1.SSS2.6.p6.2.m2.1.1" xref="S3.SS1.SSS2.6.p6.2.m2.1.1.cmml">G</mi><mo id="S3.SS1.SSS2.6.p6.2.m2.3.4.2.1" xref="S3.SS1.SSS2.6.p6.2.m2.3.4.1.cmml">,</mo><mi id="S3.SS1.SSS2.6.p6.2.m2.2.2" xref="S3.SS1.SSS2.6.p6.2.m2.2.2.cmml">w</mi><mo id="S3.SS1.SSS2.6.p6.2.m2.3.4.2.2" xref="S3.SS1.SSS2.6.p6.2.m2.3.4.1.cmml">,</mo><mi id="S3.SS1.SSS2.6.p6.2.m2.3.3" xref="S3.SS1.SSS2.6.p6.2.m2.3.3.cmml">h</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS2.6.p6.2.m2.3b"><list id="S3.SS1.SSS2.6.p6.2.m2.3.4.1.cmml" xref="S3.SS1.SSS2.6.p6.2.m2.3.4.2"><ci id="S3.SS1.SSS2.6.p6.2.m2.1.1.cmml" xref="S3.SS1.SSS2.6.p6.2.m2.1.1">𝐺</ci><ci id="S3.SS1.SSS2.6.p6.2.m2.2.2.cmml" xref="S3.SS1.SSS2.6.p6.2.m2.2.2">𝑤</ci><ci id="S3.SS1.SSS2.6.p6.2.m2.3.3.cmml" xref="S3.SS1.SSS2.6.p6.2.m2.3.3">ℎ</ci></list></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS2.6.p6.2.m2.3c">G,w,h</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS2.6.p6.2.m2.3d">italic_G , italic_w , italic_h</annotation></semantics></math>). Consider a biset <math alttext="\hat{S}" class="ltx_Math" display="inline" id="S3.SS1.SSS2.6.p6.3.m3.1"><semantics id="S3.SS1.SSS2.6.p6.3.m3.1a"><mover accent="true" id="S3.SS1.SSS2.6.p6.3.m3.1.1" xref="S3.SS1.SSS2.6.p6.3.m3.1.1.cmml"><mi id="S3.SS1.SSS2.6.p6.3.m3.1.1.2" xref="S3.SS1.SSS2.6.p6.3.m3.1.1.2.cmml">S</mi><mo id="S3.SS1.SSS2.6.p6.3.m3.1.1.1" xref="S3.SS1.SSS2.6.p6.3.m3.1.1.1.cmml">^</mo></mover><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS2.6.p6.3.m3.1b"><apply id="S3.SS1.SSS2.6.p6.3.m3.1.1.cmml" xref="S3.SS1.SSS2.6.p6.3.m3.1.1"><ci id="S3.SS1.SSS2.6.p6.3.m3.1.1.1.cmml" xref="S3.SS1.SSS2.6.p6.3.m3.1.1.1">^</ci><ci id="S3.SS1.SSS2.6.p6.3.m3.1.1.2.cmml" xref="S3.SS1.SSS2.6.p6.3.m3.1.1.2">𝑆</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS2.6.p6.3.m3.1c">\hat{S}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS2.6.p6.3.m3.1d">over^ start_ARG italic_S end_ARG</annotation></semantics></math> with <math alttext="h(\hat{S})>0" class="ltx_Math" display="inline" id="S3.SS1.SSS2.6.p6.4.m4.1"><semantics id="S3.SS1.SSS2.6.p6.4.m4.1a"><mrow id="S3.SS1.SSS2.6.p6.4.m4.1.2" xref="S3.SS1.SSS2.6.p6.4.m4.1.2.cmml"><mrow id="S3.SS1.SSS2.6.p6.4.m4.1.2.2" xref="S3.SS1.SSS2.6.p6.4.m4.1.2.2.cmml"><mi id="S3.SS1.SSS2.6.p6.4.m4.1.2.2.2" xref="S3.SS1.SSS2.6.p6.4.m4.1.2.2.2.cmml">h</mi><mo id="S3.SS1.SSS2.6.p6.4.m4.1.2.2.1" xref="S3.SS1.SSS2.6.p6.4.m4.1.2.2.1.cmml">⁢</mo><mrow id="S3.SS1.SSS2.6.p6.4.m4.1.2.2.3.2" xref="S3.SS1.SSS2.6.p6.4.m4.1.1.cmml"><mo id="S3.SS1.SSS2.6.p6.4.m4.1.2.2.3.2.1" stretchy="false" xref="S3.SS1.SSS2.6.p6.4.m4.1.1.cmml">(</mo><mover accent="true" id="S3.SS1.SSS2.6.p6.4.m4.1.1" xref="S3.SS1.SSS2.6.p6.4.m4.1.1.cmml"><mi id="S3.SS1.SSS2.6.p6.4.m4.1.1.2" xref="S3.SS1.SSS2.6.p6.4.m4.1.1.2.cmml">S</mi><mo id="S3.SS1.SSS2.6.p6.4.m4.1.1.1" xref="S3.SS1.SSS2.6.p6.4.m4.1.1.1.cmml">^</mo></mover><mo id="S3.SS1.SSS2.6.p6.4.m4.1.2.2.3.2.2" stretchy="false" xref="S3.SS1.SSS2.6.p6.4.m4.1.1.cmml">)</mo></mrow></mrow><mo id="S3.SS1.SSS2.6.p6.4.m4.1.2.1" xref="S3.SS1.SSS2.6.p6.4.m4.1.2.1.cmml">></mo><mn id="S3.SS1.SSS2.6.p6.4.m4.1.2.3" xref="S3.SS1.SSS2.6.p6.4.m4.1.2.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS2.6.p6.4.m4.1b"><apply id="S3.SS1.SSS2.6.p6.4.m4.1.2.cmml" xref="S3.SS1.SSS2.6.p6.4.m4.1.2"><gt id="S3.SS1.SSS2.6.p6.4.m4.1.2.1.cmml" xref="S3.SS1.SSS2.6.p6.4.m4.1.2.1"></gt><apply id="S3.SS1.SSS2.6.p6.4.m4.1.2.2.cmml" xref="S3.SS1.SSS2.6.p6.4.m4.1.2.2"><times id="S3.SS1.SSS2.6.p6.4.m4.1.2.2.1.cmml" xref="S3.SS1.SSS2.6.p6.4.m4.1.2.2.1"></times><ci id="S3.SS1.SSS2.6.p6.4.m4.1.2.2.2.cmml" xref="S3.SS1.SSS2.6.p6.4.m4.1.2.2.2">ℎ</ci><apply id="S3.SS1.SSS2.6.p6.4.m4.1.1.cmml" xref="S3.SS1.SSS2.6.p6.4.m4.1.2.2.3.2"><ci id="S3.SS1.SSS2.6.p6.4.m4.1.1.1.cmml" xref="S3.SS1.SSS2.6.p6.4.m4.1.1.1">^</ci><ci id="S3.SS1.SSS2.6.p6.4.m4.1.1.2.cmml" xref="S3.SS1.SSS2.6.p6.4.m4.1.1.2">𝑆</ci></apply></apply><cn id="S3.SS1.SSS2.6.p6.4.m4.1.2.3.cmml" type="integer" xref="S3.SS1.SSS2.6.p6.4.m4.1.2.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS2.6.p6.4.m4.1c">h(\hat{S})>0</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS2.6.p6.4.m4.1d">italic_h ( over^ start_ARG italic_S end_ARG ) > 0</annotation></semantics></math>. Note that for this to hold, it requires that <math alttext="|S^{+}\setminus S|<r(\hat{S})" class="ltx_Math" display="inline" id="S3.SS1.SSS2.6.p6.5.m5.2"><semantics id="S3.SS1.SSS2.6.p6.5.m5.2a"><mrow id="S3.SS1.SSS2.6.p6.5.m5.2.2" xref="S3.SS1.SSS2.6.p6.5.m5.2.2.cmml"><mrow id="S3.SS1.SSS2.6.p6.5.m5.2.2.1.1" xref="S3.SS1.SSS2.6.p6.5.m5.2.2.1.2.cmml"><mo id="S3.SS1.SSS2.6.p6.5.m5.2.2.1.1.2" stretchy="false" xref="S3.SS1.SSS2.6.p6.5.m5.2.2.1.2.1.cmml">|</mo><mrow id="S3.SS1.SSS2.6.p6.5.m5.2.2.1.1.1" xref="S3.SS1.SSS2.6.p6.5.m5.2.2.1.1.1.cmml"><msup id="S3.SS1.SSS2.6.p6.5.m5.2.2.1.1.1.2" xref="S3.SS1.SSS2.6.p6.5.m5.2.2.1.1.1.2.cmml"><mi id="S3.SS1.SSS2.6.p6.5.m5.2.2.1.1.1.2.2" xref="S3.SS1.SSS2.6.p6.5.m5.2.2.1.1.1.2.2.cmml">S</mi><mo id="S3.SS1.SSS2.6.p6.5.m5.2.2.1.1.1.2.3" xref="S3.SS1.SSS2.6.p6.5.m5.2.2.1.1.1.2.3.cmml">+</mo></msup><mo id="S3.SS1.SSS2.6.p6.5.m5.2.2.1.1.1.1" xref="S3.SS1.SSS2.6.p6.5.m5.2.2.1.1.1.1.cmml">∖</mo><mi id="S3.SS1.SSS2.6.p6.5.m5.2.2.1.1.1.3" xref="S3.SS1.SSS2.6.p6.5.m5.2.2.1.1.1.3.cmml">S</mi></mrow><mo id="S3.SS1.SSS2.6.p6.5.m5.2.2.1.1.3" stretchy="false" xref="S3.SS1.SSS2.6.p6.5.m5.2.2.1.2.1.cmml">|</mo></mrow><mo id="S3.SS1.SSS2.6.p6.5.m5.2.2.2" xref="S3.SS1.SSS2.6.p6.5.m5.2.2.2.cmml"><</mo><mrow id="S3.SS1.SSS2.6.p6.5.m5.2.2.3" xref="S3.SS1.SSS2.6.p6.5.m5.2.2.3.cmml"><mi id="S3.SS1.SSS2.6.p6.5.m5.2.2.3.2" xref="S3.SS1.SSS2.6.p6.5.m5.2.2.3.2.cmml">r</mi><mo id="S3.SS1.SSS2.6.p6.5.m5.2.2.3.1" xref="S3.SS1.SSS2.6.p6.5.m5.2.2.3.1.cmml">⁢</mo><mrow id="S3.SS1.SSS2.6.p6.5.m5.2.2.3.3.2" xref="S3.SS1.SSS2.6.p6.5.m5.1.1.cmml"><mo id="S3.SS1.SSS2.6.p6.5.m5.2.2.3.3.2.1" stretchy="false" xref="S3.SS1.SSS2.6.p6.5.m5.1.1.cmml">(</mo><mover accent="true" id="S3.SS1.SSS2.6.p6.5.m5.1.1" xref="S3.SS1.SSS2.6.p6.5.m5.1.1.cmml"><mi id="S3.SS1.SSS2.6.p6.5.m5.1.1.2" xref="S3.SS1.SSS2.6.p6.5.m5.1.1.2.cmml">S</mi><mo id="S3.SS1.SSS2.6.p6.5.m5.1.1.1" xref="S3.SS1.SSS2.6.p6.5.m5.1.1.1.cmml">^</mo></mover><mo id="S3.SS1.SSS2.6.p6.5.m5.2.2.3.3.2.2" stretchy="false" xref="S3.SS1.SSS2.6.p6.5.m5.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS2.6.p6.5.m5.2b"><apply id="S3.SS1.SSS2.6.p6.5.m5.2.2.cmml" xref="S3.SS1.SSS2.6.p6.5.m5.2.2"><lt id="S3.SS1.SSS2.6.p6.5.m5.2.2.2.cmml" xref="S3.SS1.SSS2.6.p6.5.m5.2.2.2"></lt><apply id="S3.SS1.SSS2.6.p6.5.m5.2.2.1.2.cmml" xref="S3.SS1.SSS2.6.p6.5.m5.2.2.1.1"><abs id="S3.SS1.SSS2.6.p6.5.m5.2.2.1.2.1.cmml" xref="S3.SS1.SSS2.6.p6.5.m5.2.2.1.1.2"></abs><apply id="S3.SS1.SSS2.6.p6.5.m5.2.2.1.1.1.cmml" xref="S3.SS1.SSS2.6.p6.5.m5.2.2.1.1.1"><setdiff id="S3.SS1.SSS2.6.p6.5.m5.2.2.1.1.1.1.cmml" xref="S3.SS1.SSS2.6.p6.5.m5.2.2.1.1.1.1"></setdiff><apply id="S3.SS1.SSS2.6.p6.5.m5.2.2.1.1.1.2.cmml" xref="S3.SS1.SSS2.6.p6.5.m5.2.2.1.1.1.2"><csymbol cd="ambiguous" id="S3.SS1.SSS2.6.p6.5.m5.2.2.1.1.1.2.1.cmml" xref="S3.SS1.SSS2.6.p6.5.m5.2.2.1.1.1.2">superscript</csymbol><ci id="S3.SS1.SSS2.6.p6.5.m5.2.2.1.1.1.2.2.cmml" xref="S3.SS1.SSS2.6.p6.5.m5.2.2.1.1.1.2.2">𝑆</ci><plus id="S3.SS1.SSS2.6.p6.5.m5.2.2.1.1.1.2.3.cmml" xref="S3.SS1.SSS2.6.p6.5.m5.2.2.1.1.1.2.3"></plus></apply><ci id="S3.SS1.SSS2.6.p6.5.m5.2.2.1.1.1.3.cmml" xref="S3.SS1.SSS2.6.p6.5.m5.2.2.1.1.1.3">𝑆</ci></apply></apply><apply id="S3.SS1.SSS2.6.p6.5.m5.2.2.3.cmml" xref="S3.SS1.SSS2.6.p6.5.m5.2.2.3"><times id="S3.SS1.SSS2.6.p6.5.m5.2.2.3.1.cmml" xref="S3.SS1.SSS2.6.p6.5.m5.2.2.3.1"></times><ci id="S3.SS1.SSS2.6.p6.5.m5.2.2.3.2.cmml" xref="S3.SS1.SSS2.6.p6.5.m5.2.2.3.2">𝑟</ci><apply id="S3.SS1.SSS2.6.p6.5.m5.1.1.cmml" xref="S3.SS1.SSS2.6.p6.5.m5.2.2.3.3.2"><ci id="S3.SS1.SSS2.6.p6.5.m5.1.1.1.cmml" xref="S3.SS1.SSS2.6.p6.5.m5.1.1.1">^</ci><ci id="S3.SS1.SSS2.6.p6.5.m5.1.1.2.cmml" xref="S3.SS1.SSS2.6.p6.5.m5.1.1.2">𝑆</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS2.6.p6.5.m5.2c">|S^{+}\setminus S|<r(\hat{S})</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS2.6.p6.5.m5.2d">| italic_S start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT ∖ italic_S | < italic_r ( over^ start_ARG italic_S end_ARG )</annotation></semantics></math>. In the optimal solution <span class="ltx_text ltx_markedasmath" id="S3.SS1.SSS2.6.p6.19.1">OPT</span>, there are at least <math alttext="h(\hat{S})" class="ltx_Math" display="inline" id="S3.SS1.SSS2.6.p6.7.m7.1"><semantics id="S3.SS1.SSS2.6.p6.7.m7.1a"><mrow id="S3.SS1.SSS2.6.p6.7.m7.1.2" xref="S3.SS1.SSS2.6.p6.7.m7.1.2.cmml"><mi id="S3.SS1.SSS2.6.p6.7.m7.1.2.2" xref="S3.SS1.SSS2.6.p6.7.m7.1.2.2.cmml">h</mi><mo id="S3.SS1.SSS2.6.p6.7.m7.1.2.1" xref="S3.SS1.SSS2.6.p6.7.m7.1.2.1.cmml">⁢</mo><mrow id="S3.SS1.SSS2.6.p6.7.m7.1.2.3.2" xref="S3.SS1.SSS2.6.p6.7.m7.1.1.cmml"><mo id="S3.SS1.SSS2.6.p6.7.m7.1.2.3.2.1" stretchy="false" xref="S3.SS1.SSS2.6.p6.7.m7.1.1.cmml">(</mo><mover accent="true" id="S3.SS1.SSS2.6.p6.7.m7.1.1" xref="S3.SS1.SSS2.6.p6.7.m7.1.1.cmml"><mi id="S3.SS1.SSS2.6.p6.7.m7.1.1.2" xref="S3.SS1.SSS2.6.p6.7.m7.1.1.2.cmml">S</mi><mo id="S3.SS1.SSS2.6.p6.7.m7.1.1.1" xref="S3.SS1.SSS2.6.p6.7.m7.1.1.1.cmml">^</mo></mover><mo id="S3.SS1.SSS2.6.p6.7.m7.1.2.3.2.2" stretchy="false" xref="S3.SS1.SSS2.6.p6.7.m7.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS2.6.p6.7.m7.1b"><apply id="S3.SS1.SSS2.6.p6.7.m7.1.2.cmml" xref="S3.SS1.SSS2.6.p6.7.m7.1.2"><times id="S3.SS1.SSS2.6.p6.7.m7.1.2.1.cmml" xref="S3.SS1.SSS2.6.p6.7.m7.1.2.1"></times><ci id="S3.SS1.SSS2.6.p6.7.m7.1.2.2.cmml" xref="S3.SS1.SSS2.6.p6.7.m7.1.2.2">ℎ</ci><apply id="S3.SS1.SSS2.6.p6.7.m7.1.1.cmml" xref="S3.SS1.SSS2.6.p6.7.m7.1.2.3.2"><ci id="S3.SS1.SSS2.6.p6.7.m7.1.1.1.cmml" xref="S3.SS1.SSS2.6.p6.7.m7.1.1.1">^</ci><ci id="S3.SS1.SSS2.6.p6.7.m7.1.1.2.cmml" xref="S3.SS1.SSS2.6.p6.7.m7.1.1.2">𝑆</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS2.6.p6.7.m7.1c">h(\hat{S})</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS2.6.p6.7.m7.1d">italic_h ( over^ start_ARG italic_S end_ARG )</annotation></semantics></math> edges in <math alttext="\delta_{G}(\hat{S})" class="ltx_Math" display="inline" id="S3.SS1.SSS2.6.p6.8.m8.1"><semantics id="S3.SS1.SSS2.6.p6.8.m8.1a"><mrow id="S3.SS1.SSS2.6.p6.8.m8.1.2" xref="S3.SS1.SSS2.6.p6.8.m8.1.2.cmml"><msub id="S3.SS1.SSS2.6.p6.8.m8.1.2.2" xref="S3.SS1.SSS2.6.p6.8.m8.1.2.2.cmml"><mi id="S3.SS1.SSS2.6.p6.8.m8.1.2.2.2" xref="S3.SS1.SSS2.6.p6.8.m8.1.2.2.2.cmml">δ</mi><mi id="S3.SS1.SSS2.6.p6.8.m8.1.2.2.3" xref="S3.SS1.SSS2.6.p6.8.m8.1.2.2.3.cmml">G</mi></msub><mo id="S3.SS1.SSS2.6.p6.8.m8.1.2.1" xref="S3.SS1.SSS2.6.p6.8.m8.1.2.1.cmml">⁢</mo><mrow id="S3.SS1.SSS2.6.p6.8.m8.1.2.3.2" xref="S3.SS1.SSS2.6.p6.8.m8.1.1.cmml"><mo id="S3.SS1.SSS2.6.p6.8.m8.1.2.3.2.1" stretchy="false" xref="S3.SS1.SSS2.6.p6.8.m8.1.1.cmml">(</mo><mover accent="true" id="S3.SS1.SSS2.6.p6.8.m8.1.1" xref="S3.SS1.SSS2.6.p6.8.m8.1.1.cmml"><mi id="S3.SS1.SSS2.6.p6.8.m8.1.1.2" xref="S3.SS1.SSS2.6.p6.8.m8.1.1.2.cmml">S</mi><mo id="S3.SS1.SSS2.6.p6.8.m8.1.1.1" xref="S3.SS1.SSS2.6.p6.8.m8.1.1.1.cmml">^</mo></mover><mo id="S3.SS1.SSS2.6.p6.8.m8.1.2.3.2.2" stretchy="false" xref="S3.SS1.SSS2.6.p6.8.m8.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS2.6.p6.8.m8.1b"><apply id="S3.SS1.SSS2.6.p6.8.m8.1.2.cmml" xref="S3.SS1.SSS2.6.p6.8.m8.1.2"><times id="S3.SS1.SSS2.6.p6.8.m8.1.2.1.cmml" xref="S3.SS1.SSS2.6.p6.8.m8.1.2.1"></times><apply id="S3.SS1.SSS2.6.p6.8.m8.1.2.2.cmml" xref="S3.SS1.SSS2.6.p6.8.m8.1.2.2"><csymbol cd="ambiguous" id="S3.SS1.SSS2.6.p6.8.m8.1.2.2.1.cmml" xref="S3.SS1.SSS2.6.p6.8.m8.1.2.2">subscript</csymbol><ci id="S3.SS1.SSS2.6.p6.8.m8.1.2.2.2.cmml" xref="S3.SS1.SSS2.6.p6.8.m8.1.2.2.2">𝛿</ci><ci id="S3.SS1.SSS2.6.p6.8.m8.1.2.2.3.cmml" xref="S3.SS1.SSS2.6.p6.8.m8.1.2.2.3">𝐺</ci></apply><apply id="S3.SS1.SSS2.6.p6.8.m8.1.1.cmml" xref="S3.SS1.SSS2.6.p6.8.m8.1.2.3.2"><ci id="S3.SS1.SSS2.6.p6.8.m8.1.1.1.cmml" xref="S3.SS1.SSS2.6.p6.8.m8.1.1.1">^</ci><ci id="S3.SS1.SSS2.6.p6.8.m8.1.1.2.cmml" xref="S3.SS1.SSS2.6.p6.8.m8.1.1.2">𝑆</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS2.6.p6.8.m8.1c">\delta_{G}(\hat{S})</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS2.6.p6.8.m8.1d">italic_δ start_POSTSUBSCRIPT italic_G end_POSTSUBSCRIPT ( over^ start_ARG italic_S end_ARG )</annotation></semantics></math>. Let <math alttext="L_{1}" class="ltx_Math" display="inline" id="S3.SS1.SSS2.6.p6.9.m9.1"><semantics id="S3.SS1.SSS2.6.p6.9.m9.1a"><msub id="S3.SS1.SSS2.6.p6.9.m9.1.1" xref="S3.SS1.SSS2.6.p6.9.m9.1.1.cmml"><mi id="S3.SS1.SSS2.6.p6.9.m9.1.1.2" xref="S3.SS1.SSS2.6.p6.9.m9.1.1.2.cmml">L</mi><mn id="S3.SS1.SSS2.6.p6.9.m9.1.1.3" xref="S3.SS1.SSS2.6.p6.9.m9.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS2.6.p6.9.m9.1b"><apply id="S3.SS1.SSS2.6.p6.9.m9.1.1.cmml" xref="S3.SS1.SSS2.6.p6.9.m9.1.1"><csymbol cd="ambiguous" id="S3.SS1.SSS2.6.p6.9.m9.1.1.1.cmml" xref="S3.SS1.SSS2.6.p6.9.m9.1.1">subscript</csymbol><ci id="S3.SS1.SSS2.6.p6.9.m9.1.1.2.cmml" xref="S3.SS1.SSS2.6.p6.9.m9.1.1.2">𝐿</ci><cn id="S3.SS1.SSS2.6.p6.9.m9.1.1.3.cmml" type="integer" xref="S3.SS1.SSS2.6.p6.9.m9.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS2.6.p6.9.m9.1c">L_{1}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS2.6.p6.9.m9.1d">italic_L start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="L_{0}" class="ltx_Math" display="inline" id="S3.SS1.SSS2.6.p6.10.m10.1"><semantics id="S3.SS1.SSS2.6.p6.10.m10.1a"><msub id="S3.SS1.SSS2.6.p6.10.m10.1.1" xref="S3.SS1.SSS2.6.p6.10.m10.1.1.cmml"><mi id="S3.SS1.SSS2.6.p6.10.m10.1.1.2" xref="S3.SS1.SSS2.6.p6.10.m10.1.1.2.cmml">L</mi><mn id="S3.SS1.SSS2.6.p6.10.m10.1.1.3" xref="S3.SS1.SSS2.6.p6.10.m10.1.1.3.cmml">0</mn></msub><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS2.6.p6.10.m10.1b"><apply id="S3.SS1.SSS2.6.p6.10.m10.1.1.cmml" xref="S3.SS1.SSS2.6.p6.10.m10.1.1"><csymbol cd="ambiguous" id="S3.SS1.SSS2.6.p6.10.m10.1.1.1.cmml" xref="S3.SS1.SSS2.6.p6.10.m10.1.1">subscript</csymbol><ci id="S3.SS1.SSS2.6.p6.10.m10.1.1.2.cmml" xref="S3.SS1.SSS2.6.p6.10.m10.1.1.2">𝐿</ci><cn id="S3.SS1.SSS2.6.p6.10.m10.1.1.3.cmml" type="integer" xref="S3.SS1.SSS2.6.p6.10.m10.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS2.6.p6.10.m10.1c">L_{0}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS2.6.p6.10.m10.1d">italic_L start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT</annotation></semantics></math> respectively denote the number of edges in <math alttext="\delta_{\textnormal{OPT}}(\hat{S})" class="ltx_Math" display="inline" id="S3.SS1.SSS2.6.p6.11.m11.1"><semantics id="S3.SS1.SSS2.6.p6.11.m11.1a"><mrow id="S3.SS1.SSS2.6.p6.11.m11.1.2" xref="S3.SS1.SSS2.6.p6.11.m11.1.2.cmml"><msub id="S3.SS1.SSS2.6.p6.11.m11.1.2.2" xref="S3.SS1.SSS2.6.p6.11.m11.1.2.2.cmml"><mi id="S3.SS1.SSS2.6.p6.11.m11.1.2.2.2" xref="S3.SS1.SSS2.6.p6.11.m11.1.2.2.2.cmml">δ</mi><mtext id="S3.SS1.SSS2.6.p6.11.m11.1.2.2.3" xref="S3.SS1.SSS2.6.p6.11.m11.1.2.2.3a.cmml">OPT</mtext></msub><mo id="S3.SS1.SSS2.6.p6.11.m11.1.2.1" xref="S3.SS1.SSS2.6.p6.11.m11.1.2.1.cmml">⁢</mo><mrow id="S3.SS1.SSS2.6.p6.11.m11.1.2.3.2" xref="S3.SS1.SSS2.6.p6.11.m11.1.1.cmml"><mo id="S3.SS1.SSS2.6.p6.11.m11.1.2.3.2.1" stretchy="false" xref="S3.SS1.SSS2.6.p6.11.m11.1.1.cmml">(</mo><mover accent="true" id="S3.SS1.SSS2.6.p6.11.m11.1.1" xref="S3.SS1.SSS2.6.p6.11.m11.1.1.cmml"><mi id="S3.SS1.SSS2.6.p6.11.m11.1.1.2" xref="S3.SS1.SSS2.6.p6.11.m11.1.1.2.cmml">S</mi><mo id="S3.SS1.SSS2.6.p6.11.m11.1.1.1" xref="S3.SS1.SSS2.6.p6.11.m11.1.1.1.cmml">^</mo></mover><mo id="S3.SS1.SSS2.6.p6.11.m11.1.2.3.2.2" stretchy="false" xref="S3.SS1.SSS2.6.p6.11.m11.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS2.6.p6.11.m11.1b"><apply id="S3.SS1.SSS2.6.p6.11.m11.1.2.cmml" xref="S3.SS1.SSS2.6.p6.11.m11.1.2"><times id="S3.SS1.SSS2.6.p6.11.m11.1.2.1.cmml" xref="S3.SS1.SSS2.6.p6.11.m11.1.2.1"></times><apply id="S3.SS1.SSS2.6.p6.11.m11.1.2.2.cmml" xref="S3.SS1.SSS2.6.p6.11.m11.1.2.2"><csymbol cd="ambiguous" id="S3.SS1.SSS2.6.p6.11.m11.1.2.2.1.cmml" xref="S3.SS1.SSS2.6.p6.11.m11.1.2.2">subscript</csymbol><ci id="S3.SS1.SSS2.6.p6.11.m11.1.2.2.2.cmml" xref="S3.SS1.SSS2.6.p6.11.m11.1.2.2.2">𝛿</ci><ci id="S3.SS1.SSS2.6.p6.11.m11.1.2.2.3a.cmml" xref="S3.SS1.SSS2.6.p6.11.m11.1.2.2.3"><mtext id="S3.SS1.SSS2.6.p6.11.m11.1.2.2.3.cmml" mathsize="70%" xref="S3.SS1.SSS2.6.p6.11.m11.1.2.2.3">OPT</mtext></ci></apply><apply id="S3.SS1.SSS2.6.p6.11.m11.1.1.cmml" xref="S3.SS1.SSS2.6.p6.11.m11.1.2.3.2"><ci id="S3.SS1.SSS2.6.p6.11.m11.1.1.1.cmml" xref="S3.SS1.SSS2.6.p6.11.m11.1.1.1">^</ci><ci id="S3.SS1.SSS2.6.p6.11.m11.1.1.2.cmml" xref="S3.SS1.SSS2.6.p6.11.m11.1.1.2">𝑆</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS2.6.p6.11.m11.1c">\delta_{\textnormal{OPT}}(\hat{S})</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS2.6.p6.11.m11.1d">italic_δ start_POSTSUBSCRIPT OPT end_POSTSUBSCRIPT ( over^ start_ARG italic_S end_ARG )</annotation></semantics></math> that belong to <math alttext="H" class="ltx_Math" display="inline" id="S3.SS1.SSS2.6.p6.12.m12.1"><semantics id="S3.SS1.SSS2.6.p6.12.m12.1a"><mi id="S3.SS1.SSS2.6.p6.12.m12.1.1" xref="S3.SS1.SSS2.6.p6.12.m12.1.1.cmml">H</mi><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS2.6.p6.12.m12.1b"><ci id="S3.SS1.SSS2.6.p6.12.m12.1.1.cmml" xref="S3.SS1.SSS2.6.p6.12.m12.1.1">𝐻</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS2.6.p6.12.m12.1c">H</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS2.6.p6.12.m12.1d">italic_H</annotation></semantics></math> and those that do not; i.e., <math alttext="L_{1}=|\delta_{\textnormal{OPT}}(\hat{S})\cap H|" class="ltx_Math" display="inline" id="S3.SS1.SSS2.6.p6.13.m13.2"><semantics id="S3.SS1.SSS2.6.p6.13.m13.2a"><mrow id="S3.SS1.SSS2.6.p6.13.m13.2.2" xref="S3.SS1.SSS2.6.p6.13.m13.2.2.cmml"><msub id="S3.SS1.SSS2.6.p6.13.m13.2.2.3" xref="S3.SS1.SSS2.6.p6.13.m13.2.2.3.cmml"><mi id="S3.SS1.SSS2.6.p6.13.m13.2.2.3.2" xref="S3.SS1.SSS2.6.p6.13.m13.2.2.3.2.cmml">L</mi><mn id="S3.SS1.SSS2.6.p6.13.m13.2.2.3.3" xref="S3.SS1.SSS2.6.p6.13.m13.2.2.3.3.cmml">1</mn></msub><mo id="S3.SS1.SSS2.6.p6.13.m13.2.2.2" xref="S3.SS1.SSS2.6.p6.13.m13.2.2.2.cmml">=</mo><mrow id="S3.SS1.SSS2.6.p6.13.m13.2.2.1.1" xref="S3.SS1.SSS2.6.p6.13.m13.2.2.1.2.cmml"><mo id="S3.SS1.SSS2.6.p6.13.m13.2.2.1.1.2" stretchy="false" xref="S3.SS1.SSS2.6.p6.13.m13.2.2.1.2.1.cmml">|</mo><mrow id="S3.SS1.SSS2.6.p6.13.m13.2.2.1.1.1" xref="S3.SS1.SSS2.6.p6.13.m13.2.2.1.1.1.cmml"><mrow id="S3.SS1.SSS2.6.p6.13.m13.2.2.1.1.1.2" xref="S3.SS1.SSS2.6.p6.13.m13.2.2.1.1.1.2.cmml"><msub id="S3.SS1.SSS2.6.p6.13.m13.2.2.1.1.1.2.2" xref="S3.SS1.SSS2.6.p6.13.m13.2.2.1.1.1.2.2.cmml"><mi id="S3.SS1.SSS2.6.p6.13.m13.2.2.1.1.1.2.2.2" xref="S3.SS1.SSS2.6.p6.13.m13.2.2.1.1.1.2.2.2.cmml">δ</mi><mtext id="S3.SS1.SSS2.6.p6.13.m13.2.2.1.1.1.2.2.3" xref="S3.SS1.SSS2.6.p6.13.m13.2.2.1.1.1.2.2.3a.cmml">OPT</mtext></msub><mo id="S3.SS1.SSS2.6.p6.13.m13.2.2.1.1.1.2.1" xref="S3.SS1.SSS2.6.p6.13.m13.2.2.1.1.1.2.1.cmml">⁢</mo><mrow id="S3.SS1.SSS2.6.p6.13.m13.2.2.1.1.1.2.3.2" xref="S3.SS1.SSS2.6.p6.13.m13.1.1.cmml"><mo id="S3.SS1.SSS2.6.p6.13.m13.2.2.1.1.1.2.3.2.1" stretchy="false" xref="S3.SS1.SSS2.6.p6.13.m13.1.1.cmml">(</mo><mover accent="true" id="S3.SS1.SSS2.6.p6.13.m13.1.1" xref="S3.SS1.SSS2.6.p6.13.m13.1.1.cmml"><mi id="S3.SS1.SSS2.6.p6.13.m13.1.1.2" xref="S3.SS1.SSS2.6.p6.13.m13.1.1.2.cmml">S</mi><mo id="S3.SS1.SSS2.6.p6.13.m13.1.1.1" xref="S3.SS1.SSS2.6.p6.13.m13.1.1.1.cmml">^</mo></mover><mo id="S3.SS1.SSS2.6.p6.13.m13.2.2.1.1.1.2.3.2.2" stretchy="false" xref="S3.SS1.SSS2.6.p6.13.m13.1.1.cmml">)</mo></mrow></mrow><mo id="S3.SS1.SSS2.6.p6.13.m13.2.2.1.1.1.1" xref="S3.SS1.SSS2.6.p6.13.m13.2.2.1.1.1.1.cmml">∩</mo><mi id="S3.SS1.SSS2.6.p6.13.m13.2.2.1.1.1.3" xref="S3.SS1.SSS2.6.p6.13.m13.2.2.1.1.1.3.cmml">H</mi></mrow><mo id="S3.SS1.SSS2.6.p6.13.m13.2.2.1.1.3" stretchy="false" xref="S3.SS1.SSS2.6.p6.13.m13.2.2.1.2.1.cmml">|</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS2.6.p6.13.m13.2b"><apply id="S3.SS1.SSS2.6.p6.13.m13.2.2.cmml" xref="S3.SS1.SSS2.6.p6.13.m13.2.2"><eq id="S3.SS1.SSS2.6.p6.13.m13.2.2.2.cmml" xref="S3.SS1.SSS2.6.p6.13.m13.2.2.2"></eq><apply id="S3.SS1.SSS2.6.p6.13.m13.2.2.3.cmml" xref="S3.SS1.SSS2.6.p6.13.m13.2.2.3"><csymbol cd="ambiguous" id="S3.SS1.SSS2.6.p6.13.m13.2.2.3.1.cmml" xref="S3.SS1.SSS2.6.p6.13.m13.2.2.3">subscript</csymbol><ci id="S3.SS1.SSS2.6.p6.13.m13.2.2.3.2.cmml" xref="S3.SS1.SSS2.6.p6.13.m13.2.2.3.2">𝐿</ci><cn id="S3.SS1.SSS2.6.p6.13.m13.2.2.3.3.cmml" type="integer" xref="S3.SS1.SSS2.6.p6.13.m13.2.2.3.3">1</cn></apply><apply id="S3.SS1.SSS2.6.p6.13.m13.2.2.1.2.cmml" xref="S3.SS1.SSS2.6.p6.13.m13.2.2.1.1"><abs id="S3.SS1.SSS2.6.p6.13.m13.2.2.1.2.1.cmml" xref="S3.SS1.SSS2.6.p6.13.m13.2.2.1.1.2"></abs><apply id="S3.SS1.SSS2.6.p6.13.m13.2.2.1.1.1.cmml" 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id="S3.SS1.SSS2.6.p6.14.m14.2.2.1.1.1.2.2.3" xref="S3.SS1.SSS2.6.p6.14.m14.2.2.1.1.1.2.2.3a.cmml">OPT</mtext></msub><mo id="S3.SS1.SSS2.6.p6.14.m14.2.2.1.1.1.2.1" xref="S3.SS1.SSS2.6.p6.14.m14.2.2.1.1.1.2.1.cmml">⁢</mo><mrow id="S3.SS1.SSS2.6.p6.14.m14.2.2.1.1.1.2.3.2" xref="S3.SS1.SSS2.6.p6.14.m14.1.1.cmml"><mo id="S3.SS1.SSS2.6.p6.14.m14.2.2.1.1.1.2.3.2.1" stretchy="false" xref="S3.SS1.SSS2.6.p6.14.m14.1.1.cmml">(</mo><mover accent="true" id="S3.SS1.SSS2.6.p6.14.m14.1.1" xref="S3.SS1.SSS2.6.p6.14.m14.1.1.cmml"><mi id="S3.SS1.SSS2.6.p6.14.m14.1.1.2" xref="S3.SS1.SSS2.6.p6.14.m14.1.1.2.cmml">S</mi><mo id="S3.SS1.SSS2.6.p6.14.m14.1.1.1" xref="S3.SS1.SSS2.6.p6.14.m14.1.1.1.cmml">^</mo></mover><mo id="S3.SS1.SSS2.6.p6.14.m14.2.2.1.1.1.2.3.2.2" stretchy="false" xref="S3.SS1.SSS2.6.p6.14.m14.1.1.cmml">)</mo></mrow></mrow><mo id="S3.SS1.SSS2.6.p6.14.m14.2.2.1.1.1.1" xref="S3.SS1.SSS2.6.p6.14.m14.2.2.1.1.1.1.cmml">∖</mo><mi id="S3.SS1.SSS2.6.p6.14.m14.2.2.1.1.1.3" xref="S3.SS1.SSS2.6.p6.14.m14.2.2.1.1.1.3.cmml">H</mi></mrow><mo id="S3.SS1.SSS2.6.p6.14.m14.2.2.1.1.3" stretchy="false" xref="S3.SS1.SSS2.6.p6.14.m14.2.2.1.2.1.cmml">|</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS2.6.p6.14.m14.2b"><apply id="S3.SS1.SSS2.6.p6.14.m14.2.2.cmml" xref="S3.SS1.SSS2.6.p6.14.m14.2.2"><eq id="S3.SS1.SSS2.6.p6.14.m14.2.2.2.cmml" xref="S3.SS1.SSS2.6.p6.14.m14.2.2.2"></eq><apply id="S3.SS1.SSS2.6.p6.14.m14.2.2.3.cmml" xref="S3.SS1.SSS2.6.p6.14.m14.2.2.3"><csymbol cd="ambiguous" id="S3.SS1.SSS2.6.p6.14.m14.2.2.3.1.cmml" xref="S3.SS1.SSS2.6.p6.14.m14.2.2.3">subscript</csymbol><ci id="S3.SS1.SSS2.6.p6.14.m14.2.2.3.2.cmml" xref="S3.SS1.SSS2.6.p6.14.m14.2.2.3.2">𝐿</ci><cn id="S3.SS1.SSS2.6.p6.14.m14.2.2.3.3.cmml" type="integer" xref="S3.SS1.SSS2.6.p6.14.m14.2.2.3.3">0</cn></apply><apply id="S3.SS1.SSS2.6.p6.14.m14.2.2.1.2.cmml" xref="S3.SS1.SSS2.6.p6.14.m14.2.2.1.1"><abs id="S3.SS1.SSS2.6.p6.14.m14.2.2.1.2.1.cmml" xref="S3.SS1.SSS2.6.p6.14.m14.2.2.1.1.2"></abs><apply id="S3.SS1.SSS2.6.p6.14.m14.2.2.1.1.1.cmml" xref="S3.SS1.SSS2.6.p6.14.m14.2.2.1.1.1"><setdiff id="S3.SS1.SSS2.6.p6.14.m14.2.2.1.1.1.1.cmml" xref="S3.SS1.SSS2.6.p6.14.m14.2.2.1.1.1.1"></setdiff><apply id="S3.SS1.SSS2.6.p6.14.m14.2.2.1.1.1.2.cmml" xref="S3.SS1.SSS2.6.p6.14.m14.2.2.1.1.1.2"><times id="S3.SS1.SSS2.6.p6.14.m14.2.2.1.1.1.2.1.cmml" xref="S3.SS1.SSS2.6.p6.14.m14.2.2.1.1.1.2.1"></times><apply id="S3.SS1.SSS2.6.p6.14.m14.2.2.1.1.1.2.2.cmml" xref="S3.SS1.SSS2.6.p6.14.m14.2.2.1.1.1.2.2"><csymbol cd="ambiguous" id="S3.SS1.SSS2.6.p6.14.m14.2.2.1.1.1.2.2.1.cmml" xref="S3.SS1.SSS2.6.p6.14.m14.2.2.1.1.1.2.2">subscript</csymbol><ci id="S3.SS1.SSS2.6.p6.14.m14.2.2.1.1.1.2.2.2.cmml" xref="S3.SS1.SSS2.6.p6.14.m14.2.2.1.1.1.2.2.2">𝛿</ci><ci id="S3.SS1.SSS2.6.p6.14.m14.2.2.1.1.1.2.2.3a.cmml" xref="S3.SS1.SSS2.6.p6.14.m14.2.2.1.1.1.2.2.3"><mtext id="S3.SS1.SSS2.6.p6.14.m14.2.2.1.1.1.2.2.3.cmml" mathsize="70%" xref="S3.SS1.SSS2.6.p6.14.m14.2.2.1.1.1.2.2.3">OPT</mtext></ci></apply><apply id="S3.SS1.SSS2.6.p6.14.m14.1.1.cmml" xref="S3.SS1.SSS2.6.p6.14.m14.2.2.1.1.1.2.3.2"><ci id="S3.SS1.SSS2.6.p6.14.m14.1.1.1.cmml" xref="S3.SS1.SSS2.6.p6.14.m14.1.1.1">^</ci><ci id="S3.SS1.SSS2.6.p6.14.m14.1.1.2.cmml" xref="S3.SS1.SSS2.6.p6.14.m14.1.1.2">𝑆</ci></apply></apply><ci id="S3.SS1.SSS2.6.p6.14.m14.2.2.1.1.1.3.cmml" xref="S3.SS1.SSS2.6.p6.14.m14.2.2.1.1.1.3">𝐻</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS2.6.p6.14.m14.2c">L_{0}=|\delta_{\textnormal{OPT}}(\hat{S})\setminus H|</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS2.6.p6.14.m14.2d">italic_L start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT = | italic_δ start_POSTSUBSCRIPT OPT end_POSTSUBSCRIPT ( over^ start_ARG italic_S end_ARG ) ∖ italic_H |</annotation></semantics></math>. Note that since <math alttext="\boldsymbol{x}(e)=1" class="ltx_Math" display="inline" id="S3.SS1.SSS2.6.p6.15.m15.1"><semantics id="S3.SS1.SSS2.6.p6.15.m15.1a"><mrow id="S3.SS1.SSS2.6.p6.15.m15.1.2" xref="S3.SS1.SSS2.6.p6.15.m15.1.2.cmml"><mrow id="S3.SS1.SSS2.6.p6.15.m15.1.2.2" xref="S3.SS1.SSS2.6.p6.15.m15.1.2.2.cmml"><mi id="S3.SS1.SSS2.6.p6.15.m15.1.2.2.2" xref="S3.SS1.SSS2.6.p6.15.m15.1.2.2.2.cmml">𝒙</mi><mo id="S3.SS1.SSS2.6.p6.15.m15.1.2.2.1" xref="S3.SS1.SSS2.6.p6.15.m15.1.2.2.1.cmml">⁢</mo><mrow id="S3.SS1.SSS2.6.p6.15.m15.1.2.2.3.2" xref="S3.SS1.SSS2.6.p6.15.m15.1.2.2.cmml"><mo id="S3.SS1.SSS2.6.p6.15.m15.1.2.2.3.2.1" stretchy="false" xref="S3.SS1.SSS2.6.p6.15.m15.1.2.2.cmml">(</mo><mi id="S3.SS1.SSS2.6.p6.15.m15.1.1" xref="S3.SS1.SSS2.6.p6.15.m15.1.1.cmml">e</mi><mo id="S3.SS1.SSS2.6.p6.15.m15.1.2.2.3.2.2" stretchy="false" xref="S3.SS1.SSS2.6.p6.15.m15.1.2.2.cmml">)</mo></mrow></mrow><mo id="S3.SS1.SSS2.6.p6.15.m15.1.2.1" xref="S3.SS1.SSS2.6.p6.15.m15.1.2.1.cmml">=</mo><mn id="S3.SS1.SSS2.6.p6.15.m15.1.2.3" xref="S3.SS1.SSS2.6.p6.15.m15.1.2.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS2.6.p6.15.m15.1b"><apply id="S3.SS1.SSS2.6.p6.15.m15.1.2.cmml" xref="S3.SS1.SSS2.6.p6.15.m15.1.2"><eq id="S3.SS1.SSS2.6.p6.15.m15.1.2.1.cmml" xref="S3.SS1.SSS2.6.p6.15.m15.1.2.1"></eq><apply id="S3.SS1.SSS2.6.p6.15.m15.1.2.2.cmml" xref="S3.SS1.SSS2.6.p6.15.m15.1.2.2"><times id="S3.SS1.SSS2.6.p6.15.m15.1.2.2.1.cmml" xref="S3.SS1.SSS2.6.p6.15.m15.1.2.2.1"></times><ci id="S3.SS1.SSS2.6.p6.15.m15.1.2.2.2.cmml" xref="S3.SS1.SSS2.6.p6.15.m15.1.2.2.2">𝒙</ci><ci id="S3.SS1.SSS2.6.p6.15.m15.1.1.cmml" xref="S3.SS1.SSS2.6.p6.15.m15.1.1">𝑒</ci></apply><cn id="S3.SS1.SSS2.6.p6.15.m15.1.2.3.cmml" type="integer" xref="S3.SS1.SSS2.6.p6.15.m15.1.2.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS2.6.p6.15.m15.1c">\boldsymbol{x}(e)=1</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS2.6.p6.15.m15.1d">bold_italic_x ( italic_e ) = 1</annotation></semantics></math> for every <math alttext="e\in H" class="ltx_Math" display="inline" id="S3.SS1.SSS2.6.p6.16.m16.1"><semantics id="S3.SS1.SSS2.6.p6.16.m16.1a"><mrow id="S3.SS1.SSS2.6.p6.16.m16.1.1" xref="S3.SS1.SSS2.6.p6.16.m16.1.1.cmml"><mi id="S3.SS1.SSS2.6.p6.16.m16.1.1.2" xref="S3.SS1.SSS2.6.p6.16.m16.1.1.2.cmml">e</mi><mo id="S3.SS1.SSS2.6.p6.16.m16.1.1.1" xref="S3.SS1.SSS2.6.p6.16.m16.1.1.1.cmml">∈</mo><mi id="S3.SS1.SSS2.6.p6.16.m16.1.1.3" xref="S3.SS1.SSS2.6.p6.16.m16.1.1.3.cmml">H</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS2.6.p6.16.m16.1b"><apply id="S3.SS1.SSS2.6.p6.16.m16.1.1.cmml" xref="S3.SS1.SSS2.6.p6.16.m16.1.1"><in id="S3.SS1.SSS2.6.p6.16.m16.1.1.1.cmml" xref="S3.SS1.SSS2.6.p6.16.m16.1.1.1"></in><ci id="S3.SS1.SSS2.6.p6.16.m16.1.1.2.cmml" xref="S3.SS1.SSS2.6.p6.16.m16.1.1.2">𝑒</ci><ci id="S3.SS1.SSS2.6.p6.16.m16.1.1.3.cmml" xref="S3.SS1.SSS2.6.p6.16.m16.1.1.3">𝐻</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS2.6.p6.16.m16.1c">e\in H</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS2.6.p6.16.m16.1d">italic_e ∈ italic_H</annotation></semantics></math>, if <math alttext="L_{1}\geq h(\hat{S})" class="ltx_Math" display="inline" id="S3.SS1.SSS2.6.p6.17.m17.1"><semantics id="S3.SS1.SSS2.6.p6.17.m17.1a"><mrow id="S3.SS1.SSS2.6.p6.17.m17.1.2" xref="S3.SS1.SSS2.6.p6.17.m17.1.2.cmml"><msub id="S3.SS1.SSS2.6.p6.17.m17.1.2.2" xref="S3.SS1.SSS2.6.p6.17.m17.1.2.2.cmml"><mi id="S3.SS1.SSS2.6.p6.17.m17.1.2.2.2" xref="S3.SS1.SSS2.6.p6.17.m17.1.2.2.2.cmml">L</mi><mn id="S3.SS1.SSS2.6.p6.17.m17.1.2.2.3" xref="S3.SS1.SSS2.6.p6.17.m17.1.2.2.3.cmml">1</mn></msub><mo id="S3.SS1.SSS2.6.p6.17.m17.1.2.1" xref="S3.SS1.SSS2.6.p6.17.m17.1.2.1.cmml">≥</mo><mrow id="S3.SS1.SSS2.6.p6.17.m17.1.2.3" xref="S3.SS1.SSS2.6.p6.17.m17.1.2.3.cmml"><mi id="S3.SS1.SSS2.6.p6.17.m17.1.2.3.2" xref="S3.SS1.SSS2.6.p6.17.m17.1.2.3.2.cmml">h</mi><mo id="S3.SS1.SSS2.6.p6.17.m17.1.2.3.1" xref="S3.SS1.SSS2.6.p6.17.m17.1.2.3.1.cmml">⁢</mo><mrow id="S3.SS1.SSS2.6.p6.17.m17.1.2.3.3.2" xref="S3.SS1.SSS2.6.p6.17.m17.1.1.cmml"><mo id="S3.SS1.SSS2.6.p6.17.m17.1.2.3.3.2.1" stretchy="false" xref="S3.SS1.SSS2.6.p6.17.m17.1.1.cmml">(</mo><mover accent="true" id="S3.SS1.SSS2.6.p6.17.m17.1.1" xref="S3.SS1.SSS2.6.p6.17.m17.1.1.cmml"><mi id="S3.SS1.SSS2.6.p6.17.m17.1.1.2" xref="S3.SS1.SSS2.6.p6.17.m17.1.1.2.cmml">S</mi><mo id="S3.SS1.SSS2.6.p6.17.m17.1.1.1" xref="S3.SS1.SSS2.6.p6.17.m17.1.1.1.cmml">^</mo></mover><mo id="S3.SS1.SSS2.6.p6.17.m17.1.2.3.3.2.2" stretchy="false" xref="S3.SS1.SSS2.6.p6.17.m17.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS2.6.p6.17.m17.1b"><apply id="S3.SS1.SSS2.6.p6.17.m17.1.2.cmml" xref="S3.SS1.SSS2.6.p6.17.m17.1.2"><geq id="S3.SS1.SSS2.6.p6.17.m17.1.2.1.cmml" xref="S3.SS1.SSS2.6.p6.17.m17.1.2.1"></geq><apply id="S3.SS1.SSS2.6.p6.17.m17.1.2.2.cmml" xref="S3.SS1.SSS2.6.p6.17.m17.1.2.2"><csymbol cd="ambiguous" id="S3.SS1.SSS2.6.p6.17.m17.1.2.2.1.cmml" xref="S3.SS1.SSS2.6.p6.17.m17.1.2.2">subscript</csymbol><ci id="S3.SS1.SSS2.6.p6.17.m17.1.2.2.2.cmml" xref="S3.SS1.SSS2.6.p6.17.m17.1.2.2.2">𝐿</ci><cn id="S3.SS1.SSS2.6.p6.17.m17.1.2.2.3.cmml" type="integer" xref="S3.SS1.SSS2.6.p6.17.m17.1.2.2.3">1</cn></apply><apply id="S3.SS1.SSS2.6.p6.17.m17.1.2.3.cmml" xref="S3.SS1.SSS2.6.p6.17.m17.1.2.3"><times id="S3.SS1.SSS2.6.p6.17.m17.1.2.3.1.cmml" xref="S3.SS1.SSS2.6.p6.17.m17.1.2.3.1"></times><ci id="S3.SS1.SSS2.6.p6.17.m17.1.2.3.2.cmml" xref="S3.SS1.SSS2.6.p6.17.m17.1.2.3.2">ℎ</ci><apply id="S3.SS1.SSS2.6.p6.17.m17.1.1.cmml" xref="S3.SS1.SSS2.6.p6.17.m17.1.2.3.3.2"><ci id="S3.SS1.SSS2.6.p6.17.m17.1.1.1.cmml" xref="S3.SS1.SSS2.6.p6.17.m17.1.1.1">^</ci><ci id="S3.SS1.SSS2.6.p6.17.m17.1.1.2.cmml" xref="S3.SS1.SSS2.6.p6.17.m17.1.1.2">𝑆</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS2.6.p6.17.m17.1c">L_{1}\geq h(\hat{S})</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS2.6.p6.17.m17.1d">italic_L start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ≥ italic_h ( over^ start_ARG italic_S end_ARG )</annotation></semantics></math> then the fractional solution <math alttext="\boldsymbol{x}" class="ltx_Math" display="inline" id="S3.SS1.SSS2.6.p6.18.m18.1"><semantics id="S3.SS1.SSS2.6.p6.18.m18.1a"><mi id="S3.SS1.SSS2.6.p6.18.m18.1.1" xref="S3.SS1.SSS2.6.p6.18.m18.1.1.cmml">𝒙</mi><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS2.6.p6.18.m18.1b"><ci id="S3.SS1.SSS2.6.p6.18.m18.1.1.cmml" xref="S3.SS1.SSS2.6.p6.18.m18.1.1">𝒙</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS2.6.p6.18.m18.1c">\boldsymbol{x}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS2.6.p6.18.m18.1d">bold_italic_x</annotation></semantics></math> satisfies the connectivity requirement of <math alttext="\hat{S}" class="ltx_Math" display="inline" id="S3.SS1.SSS2.6.p6.19.m19.1"><semantics id="S3.SS1.SSS2.6.p6.19.m19.1a"><mover accent="true" id="S3.SS1.SSS2.6.p6.19.m19.1.1" xref="S3.SS1.SSS2.6.p6.19.m19.1.1.cmml"><mi id="S3.SS1.SSS2.6.p6.19.m19.1.1.2" xref="S3.SS1.SSS2.6.p6.19.m19.1.1.2.cmml">S</mi><mo id="S3.SS1.SSS2.6.p6.19.m19.1.1.1" xref="S3.SS1.SSS2.6.p6.19.m19.1.1.1.cmml">^</mo></mover><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS2.6.p6.19.m19.1b"><apply id="S3.SS1.SSS2.6.p6.19.m19.1.1.cmml" xref="S3.SS1.SSS2.6.p6.19.m19.1.1"><ci id="S3.SS1.SSS2.6.p6.19.m19.1.1.1.cmml" xref="S3.SS1.SSS2.6.p6.19.m19.1.1.1">^</ci><ci id="S3.SS1.SSS2.6.p6.19.m19.1.1.2.cmml" xref="S3.SS1.SSS2.6.p6.19.m19.1.1.2">𝑆</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS2.6.p6.19.m19.1c">\hat{S}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS2.6.p6.19.m19.1d">over^ start_ARG italic_S end_ARG</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="S3.SS1.SSS2.7.p7"> <p class="ltx_p" id="S3.SS1.SSS2.7.p7.13">Next, consider the case in which <math alttext="L_{1}<h(\hat{S})=k^{\prime}" class="ltx_Math" display="inline" id="S3.SS1.SSS2.7.p7.1.m1.1"><semantics id="S3.SS1.SSS2.7.p7.1.m1.1a"><mrow id="S3.SS1.SSS2.7.p7.1.m1.1.2" xref="S3.SS1.SSS2.7.p7.1.m1.1.2.cmml"><msub id="S3.SS1.SSS2.7.p7.1.m1.1.2.2" xref="S3.SS1.SSS2.7.p7.1.m1.1.2.2.cmml"><mi id="S3.SS1.SSS2.7.p7.1.m1.1.2.2.2" xref="S3.SS1.SSS2.7.p7.1.m1.1.2.2.2.cmml">L</mi><mn id="S3.SS1.SSS2.7.p7.1.m1.1.2.2.3" xref="S3.SS1.SSS2.7.p7.1.m1.1.2.2.3.cmml">1</mn></msub><mo id="S3.SS1.SSS2.7.p7.1.m1.1.2.3" xref="S3.SS1.SSS2.7.p7.1.m1.1.2.3.cmml"><</mo><mrow id="S3.SS1.SSS2.7.p7.1.m1.1.2.4" xref="S3.SS1.SSS2.7.p7.1.m1.1.2.4.cmml"><mi id="S3.SS1.SSS2.7.p7.1.m1.1.2.4.2" xref="S3.SS1.SSS2.7.p7.1.m1.1.2.4.2.cmml">h</mi><mo id="S3.SS1.SSS2.7.p7.1.m1.1.2.4.1" xref="S3.SS1.SSS2.7.p7.1.m1.1.2.4.1.cmml">⁢</mo><mrow id="S3.SS1.SSS2.7.p7.1.m1.1.2.4.3.2" xref="S3.SS1.SSS2.7.p7.1.m1.1.1.cmml"><mo id="S3.SS1.SSS2.7.p7.1.m1.1.2.4.3.2.1" stretchy="false" xref="S3.SS1.SSS2.7.p7.1.m1.1.1.cmml">(</mo><mover accent="true" id="S3.SS1.SSS2.7.p7.1.m1.1.1" xref="S3.SS1.SSS2.7.p7.1.m1.1.1.cmml"><mi id="S3.SS1.SSS2.7.p7.1.m1.1.1.2" xref="S3.SS1.SSS2.7.p7.1.m1.1.1.2.cmml">S</mi><mo id="S3.SS1.SSS2.7.p7.1.m1.1.1.1" xref="S3.SS1.SSS2.7.p7.1.m1.1.1.1.cmml">^</mo></mover><mo id="S3.SS1.SSS2.7.p7.1.m1.1.2.4.3.2.2" stretchy="false" xref="S3.SS1.SSS2.7.p7.1.m1.1.1.cmml">)</mo></mrow></mrow><mo id="S3.SS1.SSS2.7.p7.1.m1.1.2.5" xref="S3.SS1.SSS2.7.p7.1.m1.1.2.5.cmml">=</mo><msup id="S3.SS1.SSS2.7.p7.1.m1.1.2.6" xref="S3.SS1.SSS2.7.p7.1.m1.1.2.6.cmml"><mi id="S3.SS1.SSS2.7.p7.1.m1.1.2.6.2" xref="S3.SS1.SSS2.7.p7.1.m1.1.2.6.2.cmml">k</mi><mo id="S3.SS1.SSS2.7.p7.1.m1.1.2.6.3" xref="S3.SS1.SSS2.7.p7.1.m1.1.2.6.3.cmml">′</mo></msup></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS2.7.p7.1.m1.1b"><apply id="S3.SS1.SSS2.7.p7.1.m1.1.2.cmml" xref="S3.SS1.SSS2.7.p7.1.m1.1.2"><and id="S3.SS1.SSS2.7.p7.1.m1.1.2a.cmml" xref="S3.SS1.SSS2.7.p7.1.m1.1.2"></and><apply id="S3.SS1.SSS2.7.p7.1.m1.1.2b.cmml" xref="S3.SS1.SSS2.7.p7.1.m1.1.2"><lt id="S3.SS1.SSS2.7.p7.1.m1.1.2.3.cmml" xref="S3.SS1.SSS2.7.p7.1.m1.1.2.3"></lt><apply id="S3.SS1.SSS2.7.p7.1.m1.1.2.2.cmml" xref="S3.SS1.SSS2.7.p7.1.m1.1.2.2"><csymbol cd="ambiguous" id="S3.SS1.SSS2.7.p7.1.m1.1.2.2.1.cmml" xref="S3.SS1.SSS2.7.p7.1.m1.1.2.2">subscript</csymbol><ci id="S3.SS1.SSS2.7.p7.1.m1.1.2.2.2.cmml" xref="S3.SS1.SSS2.7.p7.1.m1.1.2.2.2">𝐿</ci><cn id="S3.SS1.SSS2.7.p7.1.m1.1.2.2.3.cmml" type="integer" xref="S3.SS1.SSS2.7.p7.1.m1.1.2.2.3">1</cn></apply><apply id="S3.SS1.SSS2.7.p7.1.m1.1.2.4.cmml" xref="S3.SS1.SSS2.7.p7.1.m1.1.2.4"><times id="S3.SS1.SSS2.7.p7.1.m1.1.2.4.1.cmml" xref="S3.SS1.SSS2.7.p7.1.m1.1.2.4.1"></times><ci id="S3.SS1.SSS2.7.p7.1.m1.1.2.4.2.cmml" xref="S3.SS1.SSS2.7.p7.1.m1.1.2.4.2">ℎ</ci><apply id="S3.SS1.SSS2.7.p7.1.m1.1.1.cmml" xref="S3.SS1.SSS2.7.p7.1.m1.1.2.4.3.2"><ci id="S3.SS1.SSS2.7.p7.1.m1.1.1.1.cmml" xref="S3.SS1.SSS2.7.p7.1.m1.1.1.1">^</ci><ci id="S3.SS1.SSS2.7.p7.1.m1.1.1.2.cmml" xref="S3.SS1.SSS2.7.p7.1.m1.1.1.2">𝑆</ci></apply></apply></apply><apply id="S3.SS1.SSS2.7.p7.1.m1.1.2c.cmml" xref="S3.SS1.SSS2.7.p7.1.m1.1.2"><eq id="S3.SS1.SSS2.7.p7.1.m1.1.2.5.cmml" xref="S3.SS1.SSS2.7.p7.1.m1.1.2.5"></eq><share href="https://arxiv.org/html/2503.00712v1#S3.SS1.SSS2.7.p7.1.m1.1.2.4.cmml" id="S3.SS1.SSS2.7.p7.1.m1.1.2d.cmml" xref="S3.SS1.SSS2.7.p7.1.m1.1.2"></share><apply id="S3.SS1.SSS2.7.p7.1.m1.1.2.6.cmml" xref="S3.SS1.SSS2.7.p7.1.m1.1.2.6"><csymbol cd="ambiguous" id="S3.SS1.SSS2.7.p7.1.m1.1.2.6.1.cmml" xref="S3.SS1.SSS2.7.p7.1.m1.1.2.6">superscript</csymbol><ci id="S3.SS1.SSS2.7.p7.1.m1.1.2.6.2.cmml" xref="S3.SS1.SSS2.7.p7.1.m1.1.2.6.2">𝑘</ci><ci id="S3.SS1.SSS2.7.p7.1.m1.1.2.6.3.cmml" xref="S3.SS1.SSS2.7.p7.1.m1.1.2.6.3">′</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS2.7.p7.1.m1.1c">L_{1}<h(\hat{S})=k^{\prime}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS2.7.p7.1.m1.1d">italic_L start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT < italic_h ( over^ start_ARG italic_S end_ARG ) = italic_k start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math>. We select <math alttext="k^{\prime}-L_{1}" class="ltx_Math" display="inline" id="S3.SS1.SSS2.7.p7.2.m2.1"><semantics id="S3.SS1.SSS2.7.p7.2.m2.1a"><mrow id="S3.SS1.SSS2.7.p7.2.m2.1.1" xref="S3.SS1.SSS2.7.p7.2.m2.1.1.cmml"><msup id="S3.SS1.SSS2.7.p7.2.m2.1.1.2" xref="S3.SS1.SSS2.7.p7.2.m2.1.1.2.cmml"><mi id="S3.SS1.SSS2.7.p7.2.m2.1.1.2.2" xref="S3.SS1.SSS2.7.p7.2.m2.1.1.2.2.cmml">k</mi><mo id="S3.SS1.SSS2.7.p7.2.m2.1.1.2.3" xref="S3.SS1.SSS2.7.p7.2.m2.1.1.2.3.cmml">′</mo></msup><mo id="S3.SS1.SSS2.7.p7.2.m2.1.1.1" xref="S3.SS1.SSS2.7.p7.2.m2.1.1.1.cmml">−</mo><msub id="S3.SS1.SSS2.7.p7.2.m2.1.1.3" xref="S3.SS1.SSS2.7.p7.2.m2.1.1.3.cmml"><mi id="S3.SS1.SSS2.7.p7.2.m2.1.1.3.2" xref="S3.SS1.SSS2.7.p7.2.m2.1.1.3.2.cmml">L</mi><mn id="S3.SS1.SSS2.7.p7.2.m2.1.1.3.3" xref="S3.SS1.SSS2.7.p7.2.m2.1.1.3.3.cmml">1</mn></msub></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS2.7.p7.2.m2.1b"><apply id="S3.SS1.SSS2.7.p7.2.m2.1.1.cmml" xref="S3.SS1.SSS2.7.p7.2.m2.1.1"><minus id="S3.SS1.SSS2.7.p7.2.m2.1.1.1.cmml" xref="S3.SS1.SSS2.7.p7.2.m2.1.1.1"></minus><apply id="S3.SS1.SSS2.7.p7.2.m2.1.1.2.cmml" xref="S3.SS1.SSS2.7.p7.2.m2.1.1.2"><csymbol cd="ambiguous" id="S3.SS1.SSS2.7.p7.2.m2.1.1.2.1.cmml" xref="S3.SS1.SSS2.7.p7.2.m2.1.1.2">superscript</csymbol><ci id="S3.SS1.SSS2.7.p7.2.m2.1.1.2.2.cmml" xref="S3.SS1.SSS2.7.p7.2.m2.1.1.2.2">𝑘</ci><ci id="S3.SS1.SSS2.7.p7.2.m2.1.1.2.3.cmml" xref="S3.SS1.SSS2.7.p7.2.m2.1.1.2.3">′</ci></apply><apply id="S3.SS1.SSS2.7.p7.2.m2.1.1.3.cmml" xref="S3.SS1.SSS2.7.p7.2.m2.1.1.3"><csymbol cd="ambiguous" id="S3.SS1.SSS2.7.p7.2.m2.1.1.3.1.cmml" xref="S3.SS1.SSS2.7.p7.2.m2.1.1.3">subscript</csymbol><ci id="S3.SS1.SSS2.7.p7.2.m2.1.1.3.2.cmml" xref="S3.SS1.SSS2.7.p7.2.m2.1.1.3.2">𝐿</ci><cn id="S3.SS1.SSS2.7.p7.2.m2.1.1.3.3.cmml" type="integer" xref="S3.SS1.SSS2.7.p7.2.m2.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS2.7.p7.2.m2.1c">k^{\prime}-L_{1}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS2.7.p7.2.m2.1d">italic_k start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT - italic_L start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math> edges from the edge set <math alttext="\delta_{\textnormal{OPT}}(\hat{S})\setminus H" class="ltx_Math" display="inline" id="S3.SS1.SSS2.7.p7.3.m3.1"><semantics id="S3.SS1.SSS2.7.p7.3.m3.1a"><mrow id="S3.SS1.SSS2.7.p7.3.m3.1.2" xref="S3.SS1.SSS2.7.p7.3.m3.1.2.cmml"><mrow id="S3.SS1.SSS2.7.p7.3.m3.1.2.2" xref="S3.SS1.SSS2.7.p7.3.m3.1.2.2.cmml"><msub id="S3.SS1.SSS2.7.p7.3.m3.1.2.2.2" xref="S3.SS1.SSS2.7.p7.3.m3.1.2.2.2.cmml"><mi id="S3.SS1.SSS2.7.p7.3.m3.1.2.2.2.2" xref="S3.SS1.SSS2.7.p7.3.m3.1.2.2.2.2.cmml">δ</mi><mtext id="S3.SS1.SSS2.7.p7.3.m3.1.2.2.2.3" xref="S3.SS1.SSS2.7.p7.3.m3.1.2.2.2.3a.cmml">OPT</mtext></msub><mo id="S3.SS1.SSS2.7.p7.3.m3.1.2.2.1" xref="S3.SS1.SSS2.7.p7.3.m3.1.2.2.1.cmml">⁢</mo><mrow id="S3.SS1.SSS2.7.p7.3.m3.1.2.2.3.2" xref="S3.SS1.SSS2.7.p7.3.m3.1.1.cmml"><mo id="S3.SS1.SSS2.7.p7.3.m3.1.2.2.3.2.1" stretchy="false" xref="S3.SS1.SSS2.7.p7.3.m3.1.1.cmml">(</mo><mover accent="true" id="S3.SS1.SSS2.7.p7.3.m3.1.1" xref="S3.SS1.SSS2.7.p7.3.m3.1.1.cmml"><mi id="S3.SS1.SSS2.7.p7.3.m3.1.1.2" xref="S3.SS1.SSS2.7.p7.3.m3.1.1.2.cmml">S</mi><mo id="S3.SS1.SSS2.7.p7.3.m3.1.1.1" xref="S3.SS1.SSS2.7.p7.3.m3.1.1.1.cmml">^</mo></mover><mo id="S3.SS1.SSS2.7.p7.3.m3.1.2.2.3.2.2" stretchy="false" xref="S3.SS1.SSS2.7.p7.3.m3.1.1.cmml">)</mo></mrow></mrow><mo id="S3.SS1.SSS2.7.p7.3.m3.1.2.1" xref="S3.SS1.SSS2.7.p7.3.m3.1.2.1.cmml">∖</mo><mi id="S3.SS1.SSS2.7.p7.3.m3.1.2.3" xref="S3.SS1.SSS2.7.p7.3.m3.1.2.3.cmml">H</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS2.7.p7.3.m3.1b"><apply id="S3.SS1.SSS2.7.p7.3.m3.1.2.cmml" xref="S3.SS1.SSS2.7.p7.3.m3.1.2"><setdiff id="S3.SS1.SSS2.7.p7.3.m3.1.2.1.cmml" xref="S3.SS1.SSS2.7.p7.3.m3.1.2.1"></setdiff><apply id="S3.SS1.SSS2.7.p7.3.m3.1.2.2.cmml" xref="S3.SS1.SSS2.7.p7.3.m3.1.2.2"><times id="S3.SS1.SSS2.7.p7.3.m3.1.2.2.1.cmml" xref="S3.SS1.SSS2.7.p7.3.m3.1.2.2.1"></times><apply id="S3.SS1.SSS2.7.p7.3.m3.1.2.2.2.cmml" xref="S3.SS1.SSS2.7.p7.3.m3.1.2.2.2"><csymbol cd="ambiguous" id="S3.SS1.SSS2.7.p7.3.m3.1.2.2.2.1.cmml" xref="S3.SS1.SSS2.7.p7.3.m3.1.2.2.2">subscript</csymbol><ci id="S3.SS1.SSS2.7.p7.3.m3.1.2.2.2.2.cmml" xref="S3.SS1.SSS2.7.p7.3.m3.1.2.2.2.2">𝛿</ci><ci id="S3.SS1.SSS2.7.p7.3.m3.1.2.2.2.3a.cmml" xref="S3.SS1.SSS2.7.p7.3.m3.1.2.2.2.3"><mtext id="S3.SS1.SSS2.7.p7.3.m3.1.2.2.2.3.cmml" mathsize="70%" xref="S3.SS1.SSS2.7.p7.3.m3.1.2.2.2.3">OPT</mtext></ci></apply><apply id="S3.SS1.SSS2.7.p7.3.m3.1.1.cmml" xref="S3.SS1.SSS2.7.p7.3.m3.1.2.2.3.2"><ci id="S3.SS1.SSS2.7.p7.3.m3.1.1.1.cmml" xref="S3.SS1.SSS2.7.p7.3.m3.1.1.1">^</ci><ci id="S3.SS1.SSS2.7.p7.3.m3.1.1.2.cmml" xref="S3.SS1.SSS2.7.p7.3.m3.1.1.2">𝑆</ci></apply></apply><ci id="S3.SS1.SSS2.7.p7.3.m3.1.2.3.cmml" xref="S3.SS1.SSS2.7.p7.3.m3.1.2.3">𝐻</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS2.7.p7.3.m3.1c">\delta_{\textnormal{OPT}}(\hat{S})\setminus H</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS2.7.p7.3.m3.1d">italic_δ start_POSTSUBSCRIPT OPT end_POSTSUBSCRIPT ( over^ start_ARG italic_S end_ARG ) ∖ italic_H</annotation></semantics></math> and denote them as <math alttext="e_{1}\coloneqq(u_{1},v_{1}),\dots,e_{k^{\prime}-L_{1}}\coloneqq(u_{k^{\prime}-% L_{1}},v_{k^{\prime}-L_{1}})" class="ltx_Math" display="inline" id="S3.SS1.SSS2.7.p7.4.m4.3"><semantics id="S3.SS1.SSS2.7.p7.4.m4.3a"><mrow id="S3.SS1.SSS2.7.p7.4.m4.3.3.2" xref="S3.SS1.SSS2.7.p7.4.m4.3.3.3.cmml"><mrow id="S3.SS1.SSS2.7.p7.4.m4.2.2.1.1" xref="S3.SS1.SSS2.7.p7.4.m4.2.2.1.1.cmml"><msub id="S3.SS1.SSS2.7.p7.4.m4.2.2.1.1.3" xref="S3.SS1.SSS2.7.p7.4.m4.2.2.1.1.3.cmml"><mi id="S3.SS1.SSS2.7.p7.4.m4.2.2.1.1.3.2" xref="S3.SS1.SSS2.7.p7.4.m4.2.2.1.1.3.2.cmml">e</mi><mn id="S3.SS1.SSS2.7.p7.4.m4.2.2.1.1.3.3" 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cd="ambiguous" id="S3.SS1.SSS2.7.p7.4.m4.3.3.2.2.4.1.cmml" xref="S3.SS1.SSS2.7.p7.4.m4.3.3.2.2.4">subscript</csymbol><ci id="S3.SS1.SSS2.7.p7.4.m4.3.3.2.2.4.2.cmml" xref="S3.SS1.SSS2.7.p7.4.m4.3.3.2.2.4.2">𝑒</ci><apply id="S3.SS1.SSS2.7.p7.4.m4.3.3.2.2.4.3.cmml" xref="S3.SS1.SSS2.7.p7.4.m4.3.3.2.2.4.3"><minus id="S3.SS1.SSS2.7.p7.4.m4.3.3.2.2.4.3.1.cmml" xref="S3.SS1.SSS2.7.p7.4.m4.3.3.2.2.4.3.1"></minus><apply id="S3.SS1.SSS2.7.p7.4.m4.3.3.2.2.4.3.2.cmml" xref="S3.SS1.SSS2.7.p7.4.m4.3.3.2.2.4.3.2"><csymbol cd="ambiguous" id="S3.SS1.SSS2.7.p7.4.m4.3.3.2.2.4.3.2.1.cmml" xref="S3.SS1.SSS2.7.p7.4.m4.3.3.2.2.4.3.2">superscript</csymbol><ci id="S3.SS1.SSS2.7.p7.4.m4.3.3.2.2.4.3.2.2.cmml" xref="S3.SS1.SSS2.7.p7.4.m4.3.3.2.2.4.3.2.2">𝑘</ci><ci id="S3.SS1.SSS2.7.p7.4.m4.3.3.2.2.4.3.2.3.cmml" xref="S3.SS1.SSS2.7.p7.4.m4.3.3.2.2.4.3.2.3">′</ci></apply><apply id="S3.SS1.SSS2.7.p7.4.m4.3.3.2.2.4.3.3.cmml" xref="S3.SS1.SSS2.7.p7.4.m4.3.3.2.2.4.3.3"><csymbol cd="ambiguous" id="S3.SS1.SSS2.7.p7.4.m4.3.3.2.2.4.3.3.1.cmml" xref="S3.SS1.SSS2.7.p7.4.m4.3.3.2.2.4.3.3">subscript</csymbol><ci id="S3.SS1.SSS2.7.p7.4.m4.3.3.2.2.4.3.3.2.cmml" xref="S3.SS1.SSS2.7.p7.4.m4.3.3.2.2.4.3.3.2">𝐿</ci><cn id="S3.SS1.SSS2.7.p7.4.m4.3.3.2.2.4.3.3.3.cmml" type="integer" xref="S3.SS1.SSS2.7.p7.4.m4.3.3.2.2.4.3.3.3">1</cn></apply></apply></apply><interval closure="open" id="S3.SS1.SSS2.7.p7.4.m4.3.3.2.2.2.3.cmml" xref="S3.SS1.SSS2.7.p7.4.m4.3.3.2.2.2.2"><apply id="S3.SS1.SSS2.7.p7.4.m4.3.3.2.2.1.1.1.cmml" xref="S3.SS1.SSS2.7.p7.4.m4.3.3.2.2.1.1.1"><csymbol cd="ambiguous" id="S3.SS1.SSS2.7.p7.4.m4.3.3.2.2.1.1.1.1.cmml" xref="S3.SS1.SSS2.7.p7.4.m4.3.3.2.2.1.1.1">subscript</csymbol><ci id="S3.SS1.SSS2.7.p7.4.m4.3.3.2.2.1.1.1.2.cmml" xref="S3.SS1.SSS2.7.p7.4.m4.3.3.2.2.1.1.1.2">𝑢</ci><apply id="S3.SS1.SSS2.7.p7.4.m4.3.3.2.2.1.1.1.3.cmml" xref="S3.SS1.SSS2.7.p7.4.m4.3.3.2.2.1.1.1.3"><minus id="S3.SS1.SSS2.7.p7.4.m4.3.3.2.2.1.1.1.3.1.cmml" xref="S3.SS1.SSS2.7.p7.4.m4.3.3.2.2.1.1.1.3.1"></minus><apply id="S3.SS1.SSS2.7.p7.4.m4.3.3.2.2.1.1.1.3.2.cmml" xref="S3.SS1.SSS2.7.p7.4.m4.3.3.2.2.1.1.1.3.2"><csymbol cd="ambiguous" id="S3.SS1.SSS2.7.p7.4.m4.3.3.2.2.1.1.1.3.2.1.cmml" xref="S3.SS1.SSS2.7.p7.4.m4.3.3.2.2.1.1.1.3.2">superscript</csymbol><ci id="S3.SS1.SSS2.7.p7.4.m4.3.3.2.2.1.1.1.3.2.2.cmml" xref="S3.SS1.SSS2.7.p7.4.m4.3.3.2.2.1.1.1.3.2.2">𝑘</ci><ci id="S3.SS1.SSS2.7.p7.4.m4.3.3.2.2.1.1.1.3.2.3.cmml" xref="S3.SS1.SSS2.7.p7.4.m4.3.3.2.2.1.1.1.3.2.3">′</ci></apply><apply id="S3.SS1.SSS2.7.p7.4.m4.3.3.2.2.1.1.1.3.3.cmml" xref="S3.SS1.SSS2.7.p7.4.m4.3.3.2.2.1.1.1.3.3"><csymbol cd="ambiguous" id="S3.SS1.SSS2.7.p7.4.m4.3.3.2.2.1.1.1.3.3.1.cmml" xref="S3.SS1.SSS2.7.p7.4.m4.3.3.2.2.1.1.1.3.3">subscript</csymbol><ci id="S3.SS1.SSS2.7.p7.4.m4.3.3.2.2.1.1.1.3.3.2.cmml" xref="S3.SS1.SSS2.7.p7.4.m4.3.3.2.2.1.1.1.3.3.2">𝐿</ci><cn id="S3.SS1.SSS2.7.p7.4.m4.3.3.2.2.1.1.1.3.3.3.cmml" type="integer" xref="S3.SS1.SSS2.7.p7.4.m4.3.3.2.2.1.1.1.3.3.3">1</cn></apply></apply></apply><apply id="S3.SS1.SSS2.7.p7.4.m4.3.3.2.2.2.2.2.cmml" xref="S3.SS1.SSS2.7.p7.4.m4.3.3.2.2.2.2.2"><csymbol cd="ambiguous" id="S3.SS1.SSS2.7.p7.4.m4.3.3.2.2.2.2.2.1.cmml" xref="S3.SS1.SSS2.7.p7.4.m4.3.3.2.2.2.2.2">subscript</csymbol><ci id="S3.SS1.SSS2.7.p7.4.m4.3.3.2.2.2.2.2.2.cmml" xref="S3.SS1.SSS2.7.p7.4.m4.3.3.2.2.2.2.2.2">𝑣</ci><apply id="S3.SS1.SSS2.7.p7.4.m4.3.3.2.2.2.2.2.3.cmml" xref="S3.SS1.SSS2.7.p7.4.m4.3.3.2.2.2.2.2.3"><minus id="S3.SS1.SSS2.7.p7.4.m4.3.3.2.2.2.2.2.3.1.cmml" xref="S3.SS1.SSS2.7.p7.4.m4.3.3.2.2.2.2.2.3.1"></minus><apply id="S3.SS1.SSS2.7.p7.4.m4.3.3.2.2.2.2.2.3.2.cmml" xref="S3.SS1.SSS2.7.p7.4.m4.3.3.2.2.2.2.2.3.2"><csymbol cd="ambiguous" id="S3.SS1.SSS2.7.p7.4.m4.3.3.2.2.2.2.2.3.2.1.cmml" xref="S3.SS1.SSS2.7.p7.4.m4.3.3.2.2.2.2.2.3.2">superscript</csymbol><ci id="S3.SS1.SSS2.7.p7.4.m4.3.3.2.2.2.2.2.3.2.2.cmml" xref="S3.SS1.SSS2.7.p7.4.m4.3.3.2.2.2.2.2.3.2.2">𝑘</ci><ci id="S3.SS1.SSS2.7.p7.4.m4.3.3.2.2.2.2.2.3.2.3.cmml" xref="S3.SS1.SSS2.7.p7.4.m4.3.3.2.2.2.2.2.3.2.3">′</ci></apply><apply id="S3.SS1.SSS2.7.p7.4.m4.3.3.2.2.2.2.2.3.3.cmml" xref="S3.SS1.SSS2.7.p7.4.m4.3.3.2.2.2.2.2.3.3"><csymbol cd="ambiguous" id="S3.SS1.SSS2.7.p7.4.m4.3.3.2.2.2.2.2.3.3.1.cmml" xref="S3.SS1.SSS2.7.p7.4.m4.3.3.2.2.2.2.2.3.3">subscript</csymbol><ci id="S3.SS1.SSS2.7.p7.4.m4.3.3.2.2.2.2.2.3.3.2.cmml" xref="S3.SS1.SSS2.7.p7.4.m4.3.3.2.2.2.2.2.3.3.2">𝐿</ci><cn id="S3.SS1.SSS2.7.p7.4.m4.3.3.2.2.2.2.2.3.3.3.cmml" type="integer" xref="S3.SS1.SSS2.7.p7.4.m4.3.3.2.2.2.2.2.3.3.3">1</cn></apply></apply></apply></interval></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS2.7.p7.4.m4.3c">e_{1}\coloneqq(u_{1},v_{1}),\dots,e_{k^{\prime}-L_{1}}\coloneqq(u_{k^{\prime}-% L_{1}},v_{k^{\prime}-L_{1}})</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS2.7.p7.4.m4.3d">italic_e start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ≔ ( italic_u start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_v start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) , … , italic_e start_POSTSUBSCRIPT italic_k start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT - italic_L start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ≔ ( italic_u start_POSTSUBSCRIPT italic_k start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT - italic_L start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT , italic_v start_POSTSUBSCRIPT italic_k start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT - italic_L start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT )</annotation></semantics></math>. Then, for each <math alttext="i\leq k^{\prime}-L_{1}" class="ltx_Math" display="inline" id="S3.SS1.SSS2.7.p7.5.m5.1"><semantics id="S3.SS1.SSS2.7.p7.5.m5.1a"><mrow id="S3.SS1.SSS2.7.p7.5.m5.1.1" xref="S3.SS1.SSS2.7.p7.5.m5.1.1.cmml"><mi id="S3.SS1.SSS2.7.p7.5.m5.1.1.2" xref="S3.SS1.SSS2.7.p7.5.m5.1.1.2.cmml">i</mi><mo id="S3.SS1.SSS2.7.p7.5.m5.1.1.1" xref="S3.SS1.SSS2.7.p7.5.m5.1.1.1.cmml">≤</mo><mrow id="S3.SS1.SSS2.7.p7.5.m5.1.1.3" xref="S3.SS1.SSS2.7.p7.5.m5.1.1.3.cmml"><msup id="S3.SS1.SSS2.7.p7.5.m5.1.1.3.2" xref="S3.SS1.SSS2.7.p7.5.m5.1.1.3.2.cmml"><mi id="S3.SS1.SSS2.7.p7.5.m5.1.1.3.2.2" xref="S3.SS1.SSS2.7.p7.5.m5.1.1.3.2.2.cmml">k</mi><mo id="S3.SS1.SSS2.7.p7.5.m5.1.1.3.2.3" xref="S3.SS1.SSS2.7.p7.5.m5.1.1.3.2.3.cmml">′</mo></msup><mo id="S3.SS1.SSS2.7.p7.5.m5.1.1.3.1" xref="S3.SS1.SSS2.7.p7.5.m5.1.1.3.1.cmml">−</mo><msub id="S3.SS1.SSS2.7.p7.5.m5.1.1.3.3" xref="S3.SS1.SSS2.7.p7.5.m5.1.1.3.3.cmml"><mi id="S3.SS1.SSS2.7.p7.5.m5.1.1.3.3.2" xref="S3.SS1.SSS2.7.p7.5.m5.1.1.3.3.2.cmml">L</mi><mn id="S3.SS1.SSS2.7.p7.5.m5.1.1.3.3.3" xref="S3.SS1.SSS2.7.p7.5.m5.1.1.3.3.3.cmml">1</mn></msub></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS2.7.p7.5.m5.1b"><apply id="S3.SS1.SSS2.7.p7.5.m5.1.1.cmml" xref="S3.SS1.SSS2.7.p7.5.m5.1.1"><leq id="S3.SS1.SSS2.7.p7.5.m5.1.1.1.cmml" xref="S3.SS1.SSS2.7.p7.5.m5.1.1.1"></leq><ci id="S3.SS1.SSS2.7.p7.5.m5.1.1.2.cmml" xref="S3.SS1.SSS2.7.p7.5.m5.1.1.2">𝑖</ci><apply id="S3.SS1.SSS2.7.p7.5.m5.1.1.3.cmml" xref="S3.SS1.SSS2.7.p7.5.m5.1.1.3"><minus id="S3.SS1.SSS2.7.p7.5.m5.1.1.3.1.cmml" xref="S3.SS1.SSS2.7.p7.5.m5.1.1.3.1"></minus><apply id="S3.SS1.SSS2.7.p7.5.m5.1.1.3.2.cmml" xref="S3.SS1.SSS2.7.p7.5.m5.1.1.3.2"><csymbol cd="ambiguous" id="S3.SS1.SSS2.7.p7.5.m5.1.1.3.2.1.cmml" xref="S3.SS1.SSS2.7.p7.5.m5.1.1.3.2">superscript</csymbol><ci id="S3.SS1.SSS2.7.p7.5.m5.1.1.3.2.2.cmml" xref="S3.SS1.SSS2.7.p7.5.m5.1.1.3.2.2">𝑘</ci><ci id="S3.SS1.SSS2.7.p7.5.m5.1.1.3.2.3.cmml" xref="S3.SS1.SSS2.7.p7.5.m5.1.1.3.2.3">′</ci></apply><apply id="S3.SS1.SSS2.7.p7.5.m5.1.1.3.3.cmml" xref="S3.SS1.SSS2.7.p7.5.m5.1.1.3.3"><csymbol cd="ambiguous" id="S3.SS1.SSS2.7.p7.5.m5.1.1.3.3.1.cmml" xref="S3.SS1.SSS2.7.p7.5.m5.1.1.3.3">subscript</csymbol><ci id="S3.SS1.SSS2.7.p7.5.m5.1.1.3.3.2.cmml" xref="S3.SS1.SSS2.7.p7.5.m5.1.1.3.3.2">𝐿</ci><cn id="S3.SS1.SSS2.7.p7.5.m5.1.1.3.3.3.cmml" type="integer" xref="S3.SS1.SSS2.7.p7.5.m5.1.1.3.3.3">1</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS2.7.p7.5.m5.1c">i\leq k^{\prime}-L_{1}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS2.7.p7.5.m5.1d">italic_i ≤ italic_k start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT - italic_L start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math>, consider the <math alttext="2k" class="ltx_Math" display="inline" id="S3.SS1.SSS2.7.p7.6.m6.1"><semantics id="S3.SS1.SSS2.7.p7.6.m6.1a"><mrow id="S3.SS1.SSS2.7.p7.6.m6.1.1" xref="S3.SS1.SSS2.7.p7.6.m6.1.1.cmml"><mn id="S3.SS1.SSS2.7.p7.6.m6.1.1.2" xref="S3.SS1.SSS2.7.p7.6.m6.1.1.2.cmml">2</mn><mo id="S3.SS1.SSS2.7.p7.6.m6.1.1.1" xref="S3.SS1.SSS2.7.p7.6.m6.1.1.1.cmml">⁢</mo><mi id="S3.SS1.SSS2.7.p7.6.m6.1.1.3" xref="S3.SS1.SSS2.7.p7.6.m6.1.1.3.cmml">k</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS2.7.p7.6.m6.1b"><apply id="S3.SS1.SSS2.7.p7.6.m6.1.1.cmml" xref="S3.SS1.SSS2.7.p7.6.m6.1.1"><times id="S3.SS1.SSS2.7.p7.6.m6.1.1.1.cmml" xref="S3.SS1.SSS2.7.p7.6.m6.1.1.1"></times><cn id="S3.SS1.SSS2.7.p7.6.m6.1.1.2.cmml" type="integer" xref="S3.SS1.SSS2.7.p7.6.m6.1.1.2">2</cn><ci id="S3.SS1.SSS2.7.p7.6.m6.1.1.3.cmml" xref="S3.SS1.SSS2.7.p7.6.m6.1.1.3">𝑘</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS2.7.p7.6.m6.1c">2k</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS2.7.p7.6.m6.1d">2 italic_k</annotation></semantics></math> vertex-disjoint <math alttext="(u_{i},v_{i})" class="ltx_Math" display="inline" id="S3.SS1.SSS2.7.p7.7.m7.2"><semantics id="S3.SS1.SSS2.7.p7.7.m7.2a"><mrow id="S3.SS1.SSS2.7.p7.7.m7.2.2.2" xref="S3.SS1.SSS2.7.p7.7.m7.2.2.3.cmml"><mo id="S3.SS1.SSS2.7.p7.7.m7.2.2.2.3" stretchy="false" xref="S3.SS1.SSS2.7.p7.7.m7.2.2.3.cmml">(</mo><msub id="S3.SS1.SSS2.7.p7.7.m7.1.1.1.1" xref="S3.SS1.SSS2.7.p7.7.m7.1.1.1.1.cmml"><mi id="S3.SS1.SSS2.7.p7.7.m7.1.1.1.1.2" xref="S3.SS1.SSS2.7.p7.7.m7.1.1.1.1.2.cmml">u</mi><mi id="S3.SS1.SSS2.7.p7.7.m7.1.1.1.1.3" xref="S3.SS1.SSS2.7.p7.7.m7.1.1.1.1.3.cmml">i</mi></msub><mo id="S3.SS1.SSS2.7.p7.7.m7.2.2.2.4" xref="S3.SS1.SSS2.7.p7.7.m7.2.2.3.cmml">,</mo><msub id="S3.SS1.SSS2.7.p7.7.m7.2.2.2.2" xref="S3.SS1.SSS2.7.p7.7.m7.2.2.2.2.cmml"><mi id="S3.SS1.SSS2.7.p7.7.m7.2.2.2.2.2" xref="S3.SS1.SSS2.7.p7.7.m7.2.2.2.2.2.cmml">v</mi><mi id="S3.SS1.SSS2.7.p7.7.m7.2.2.2.2.3" xref="S3.SS1.SSS2.7.p7.7.m7.2.2.2.2.3.cmml">i</mi></msub><mo id="S3.SS1.SSS2.7.p7.7.m7.2.2.2.5" stretchy="false" xref="S3.SS1.SSS2.7.p7.7.m7.2.2.3.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS2.7.p7.7.m7.2b"><interval closure="open" id="S3.SS1.SSS2.7.p7.7.m7.2.2.3.cmml" xref="S3.SS1.SSS2.7.p7.7.m7.2.2.2"><apply id="S3.SS1.SSS2.7.p7.7.m7.1.1.1.1.cmml" xref="S3.SS1.SSS2.7.p7.7.m7.1.1.1.1"><csymbol cd="ambiguous" id="S3.SS1.SSS2.7.p7.7.m7.1.1.1.1.1.cmml" xref="S3.SS1.SSS2.7.p7.7.m7.1.1.1.1">subscript</csymbol><ci id="S3.SS1.SSS2.7.p7.7.m7.1.1.1.1.2.cmml" xref="S3.SS1.SSS2.7.p7.7.m7.1.1.1.1.2">𝑢</ci><ci id="S3.SS1.SSS2.7.p7.7.m7.1.1.1.1.3.cmml" xref="S3.SS1.SSS2.7.p7.7.m7.1.1.1.1.3">𝑖</ci></apply><apply id="S3.SS1.SSS2.7.p7.7.m7.2.2.2.2.cmml" xref="S3.SS1.SSS2.7.p7.7.m7.2.2.2.2"><csymbol cd="ambiguous" id="S3.SS1.SSS2.7.p7.7.m7.2.2.2.2.1.cmml" xref="S3.SS1.SSS2.7.p7.7.m7.2.2.2.2">subscript</csymbol><ci id="S3.SS1.SSS2.7.p7.7.m7.2.2.2.2.2.cmml" xref="S3.SS1.SSS2.7.p7.7.m7.2.2.2.2.2">𝑣</ci><ci id="S3.SS1.SSS2.7.p7.7.m7.2.2.2.2.3.cmml" xref="S3.SS1.SSS2.7.p7.7.m7.2.2.2.2.3">𝑖</ci></apply></interval></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS2.7.p7.7.m7.2c">(u_{i},v_{i})</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS2.7.p7.7.m7.2d">( italic_u start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT , italic_v start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT )</annotation></semantics></math>-paths <math alttext="P_{1},\dots,P_{2k}" class="ltx_Math" display="inline" id="S3.SS1.SSS2.7.p7.8.m8.3"><semantics id="S3.SS1.SSS2.7.p7.8.m8.3a"><mrow id="S3.SS1.SSS2.7.p7.8.m8.3.3.2" xref="S3.SS1.SSS2.7.p7.8.m8.3.3.3.cmml"><msub id="S3.SS1.SSS2.7.p7.8.m8.2.2.1.1" xref="S3.SS1.SSS2.7.p7.8.m8.2.2.1.1.cmml"><mi id="S3.SS1.SSS2.7.p7.8.m8.2.2.1.1.2" xref="S3.SS1.SSS2.7.p7.8.m8.2.2.1.1.2.cmml">P</mi><mn id="S3.SS1.SSS2.7.p7.8.m8.2.2.1.1.3" xref="S3.SS1.SSS2.7.p7.8.m8.2.2.1.1.3.cmml">1</mn></msub><mo id="S3.SS1.SSS2.7.p7.8.m8.3.3.2.3" xref="S3.SS1.SSS2.7.p7.8.m8.3.3.3.cmml">,</mo><mi id="S3.SS1.SSS2.7.p7.8.m8.1.1" mathvariant="normal" xref="S3.SS1.SSS2.7.p7.8.m8.1.1.cmml">…</mi><mo id="S3.SS1.SSS2.7.p7.8.m8.3.3.2.4" xref="S3.SS1.SSS2.7.p7.8.m8.3.3.3.cmml">,</mo><msub id="S3.SS1.SSS2.7.p7.8.m8.3.3.2.2" xref="S3.SS1.SSS2.7.p7.8.m8.3.3.2.2.cmml"><mi id="S3.SS1.SSS2.7.p7.8.m8.3.3.2.2.2" xref="S3.SS1.SSS2.7.p7.8.m8.3.3.2.2.2.cmml">P</mi><mrow id="S3.SS1.SSS2.7.p7.8.m8.3.3.2.2.3" xref="S3.SS1.SSS2.7.p7.8.m8.3.3.2.2.3.cmml"><mn id="S3.SS1.SSS2.7.p7.8.m8.3.3.2.2.3.2" xref="S3.SS1.SSS2.7.p7.8.m8.3.3.2.2.3.2.cmml">2</mn><mo id="S3.SS1.SSS2.7.p7.8.m8.3.3.2.2.3.1" xref="S3.SS1.SSS2.7.p7.8.m8.3.3.2.2.3.1.cmml">⁢</mo><mi id="S3.SS1.SSS2.7.p7.8.m8.3.3.2.2.3.3" xref="S3.SS1.SSS2.7.p7.8.m8.3.3.2.2.3.3.cmml">k</mi></mrow></msub></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS2.7.p7.8.m8.3b"><list id="S3.SS1.SSS2.7.p7.8.m8.3.3.3.cmml" xref="S3.SS1.SSS2.7.p7.8.m8.3.3.2"><apply id="S3.SS1.SSS2.7.p7.8.m8.2.2.1.1.cmml" xref="S3.SS1.SSS2.7.p7.8.m8.2.2.1.1"><csymbol cd="ambiguous" id="S3.SS1.SSS2.7.p7.8.m8.2.2.1.1.1.cmml" xref="S3.SS1.SSS2.7.p7.8.m8.2.2.1.1">subscript</csymbol><ci id="S3.SS1.SSS2.7.p7.8.m8.2.2.1.1.2.cmml" xref="S3.SS1.SSS2.7.p7.8.m8.2.2.1.1.2">𝑃</ci><cn id="S3.SS1.SSS2.7.p7.8.m8.2.2.1.1.3.cmml" type="integer" xref="S3.SS1.SSS2.7.p7.8.m8.2.2.1.1.3">1</cn></apply><ci id="S3.SS1.SSS2.7.p7.8.m8.1.1.cmml" xref="S3.SS1.SSS2.7.p7.8.m8.1.1">…</ci><apply id="S3.SS1.SSS2.7.p7.8.m8.3.3.2.2.cmml" xref="S3.SS1.SSS2.7.p7.8.m8.3.3.2.2"><csymbol cd="ambiguous" id="S3.SS1.SSS2.7.p7.8.m8.3.3.2.2.1.cmml" xref="S3.SS1.SSS2.7.p7.8.m8.3.3.2.2">subscript</csymbol><ci id="S3.SS1.SSS2.7.p7.8.m8.3.3.2.2.2.cmml" xref="S3.SS1.SSS2.7.p7.8.m8.3.3.2.2.2">𝑃</ci><apply id="S3.SS1.SSS2.7.p7.8.m8.3.3.2.2.3.cmml" xref="S3.SS1.SSS2.7.p7.8.m8.3.3.2.2.3"><times id="S3.SS1.SSS2.7.p7.8.m8.3.3.2.2.3.1.cmml" xref="S3.SS1.SSS2.7.p7.8.m8.3.3.2.2.3.1"></times><cn id="S3.SS1.SSS2.7.p7.8.m8.3.3.2.2.3.2.cmml" type="integer" xref="S3.SS1.SSS2.7.p7.8.m8.3.3.2.2.3.2">2</cn><ci id="S3.SS1.SSS2.7.p7.8.m8.3.3.2.2.3.3.cmml" xref="S3.SS1.SSS2.7.p7.8.m8.3.3.2.2.3.3">𝑘</ci></apply></apply></list></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS2.7.p7.8.m8.3c">P_{1},\dots,P_{2k}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS2.7.p7.8.m8.3d">italic_P start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , … , italic_P start_POSTSUBSCRIPT 2 italic_k end_POSTSUBSCRIPT</annotation></semantics></math> in <math alttext="H_{j}" class="ltx_Math" display="inline" id="S3.SS1.SSS2.7.p7.9.m9.1"><semantics id="S3.SS1.SSS2.7.p7.9.m9.1a"><msub id="S3.SS1.SSS2.7.p7.9.m9.1.1" xref="S3.SS1.SSS2.7.p7.9.m9.1.1.cmml"><mi id="S3.SS1.SSS2.7.p7.9.m9.1.1.2" xref="S3.SS1.SSS2.7.p7.9.m9.1.1.2.cmml">H</mi><mi id="S3.SS1.SSS2.7.p7.9.m9.1.1.3" xref="S3.SS1.SSS2.7.p7.9.m9.1.1.3.cmml">j</mi></msub><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS2.7.p7.9.m9.1b"><apply id="S3.SS1.SSS2.7.p7.9.m9.1.1.cmml" xref="S3.SS1.SSS2.7.p7.9.m9.1.1"><csymbol cd="ambiguous" id="S3.SS1.SSS2.7.p7.9.m9.1.1.1.cmml" xref="S3.SS1.SSS2.7.p7.9.m9.1.1">subscript</csymbol><ci id="S3.SS1.SSS2.7.p7.9.m9.1.1.2.cmml" xref="S3.SS1.SSS2.7.p7.9.m9.1.1.2">𝐻</ci><ci id="S3.SS1.SSS2.7.p7.9.m9.1.1.3.cmml" xref="S3.SS1.SSS2.7.p7.9.m9.1.1.3">𝑗</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS2.7.p7.9.m9.1c">H_{j}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS2.7.p7.9.m9.1d">italic_H start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT</annotation></semantics></math>, as shown in Observation <a class="ltx_ref" href="https://arxiv.org/html/2503.00712v1#S3.Thmtheorem1" title="Lemma 3.1. ‣ 3 Generic Framework for Streaming Algorithms for Network Design ‣ Streaming Algorithms for Network Design"><span class="ltx_text ltx_ref_tag">3.1</span></a>, and used in the construction of <math alttext="\boldsymbol{x}" class="ltx_Math" display="inline" id="S3.SS1.SSS2.7.p7.10.m10.1"><semantics id="S3.SS1.SSS2.7.p7.10.m10.1a"><mi id="S3.SS1.SSS2.7.p7.10.m10.1.1" xref="S3.SS1.SSS2.7.p7.10.m10.1.1.cmml">𝒙</mi><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS2.7.p7.10.m10.1b"><ci id="S3.SS1.SSS2.7.p7.10.m10.1.1.cmml" xref="S3.SS1.SSS2.7.p7.10.m10.1.1">𝒙</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS2.7.p7.10.m10.1c">\boldsymbol{x}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS2.7.p7.10.m10.1d">bold_italic_x</annotation></semantics></math> (see Figure <a class="ltx_ref" href="https://arxiv.org/html/2503.00712v1#S3.F2" title="Figure 2 ‣ Proof. ‣ 3.1.2 An Improved Analysis via Fractional Solutions ‣ 3.1 Vertex Connectivity Network Design ‣ 3 Generic Framework for Streaming Algorithms for Network Design ‣ Streaming Algorithms for Network Design"><span class="ltx_text ltx_ref_tag">2</span></a>). Since <math alttext="P_{1},\cdots,P_{2k}" class="ltx_Math" display="inline" id="S3.SS1.SSS2.7.p7.11.m11.3"><semantics id="S3.SS1.SSS2.7.p7.11.m11.3a"><mrow id="S3.SS1.SSS2.7.p7.11.m11.3.3.2" xref="S3.SS1.SSS2.7.p7.11.m11.3.3.3.cmml"><msub id="S3.SS1.SSS2.7.p7.11.m11.2.2.1.1" xref="S3.SS1.SSS2.7.p7.11.m11.2.2.1.1.cmml"><mi id="S3.SS1.SSS2.7.p7.11.m11.2.2.1.1.2" xref="S3.SS1.SSS2.7.p7.11.m11.2.2.1.1.2.cmml">P</mi><mn id="S3.SS1.SSS2.7.p7.11.m11.2.2.1.1.3" xref="S3.SS1.SSS2.7.p7.11.m11.2.2.1.1.3.cmml">1</mn></msub><mo id="S3.SS1.SSS2.7.p7.11.m11.3.3.2.3" xref="S3.SS1.SSS2.7.p7.11.m11.3.3.3.cmml">,</mo><mi id="S3.SS1.SSS2.7.p7.11.m11.1.1" mathvariant="normal" xref="S3.SS1.SSS2.7.p7.11.m11.1.1.cmml">⋯</mi><mo id="S3.SS1.SSS2.7.p7.11.m11.3.3.2.4" xref="S3.SS1.SSS2.7.p7.11.m11.3.3.3.cmml">,</mo><msub id="S3.SS1.SSS2.7.p7.11.m11.3.3.2.2" xref="S3.SS1.SSS2.7.p7.11.m11.3.3.2.2.cmml"><mi id="S3.SS1.SSS2.7.p7.11.m11.3.3.2.2.2" xref="S3.SS1.SSS2.7.p7.11.m11.3.3.2.2.2.cmml">P</mi><mrow id="S3.SS1.SSS2.7.p7.11.m11.3.3.2.2.3" xref="S3.SS1.SSS2.7.p7.11.m11.3.3.2.2.3.cmml"><mn id="S3.SS1.SSS2.7.p7.11.m11.3.3.2.2.3.2" xref="S3.SS1.SSS2.7.p7.11.m11.3.3.2.2.3.2.cmml">2</mn><mo id="S3.SS1.SSS2.7.p7.11.m11.3.3.2.2.3.1" xref="S3.SS1.SSS2.7.p7.11.m11.3.3.2.2.3.1.cmml">⁢</mo><mi id="S3.SS1.SSS2.7.p7.11.m11.3.3.2.2.3.3" xref="S3.SS1.SSS2.7.p7.11.m11.3.3.2.2.3.3.cmml">k</mi></mrow></msub></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS2.7.p7.11.m11.3b"><list id="S3.SS1.SSS2.7.p7.11.m11.3.3.3.cmml" xref="S3.SS1.SSS2.7.p7.11.m11.3.3.2"><apply id="S3.SS1.SSS2.7.p7.11.m11.2.2.1.1.cmml" xref="S3.SS1.SSS2.7.p7.11.m11.2.2.1.1"><csymbol cd="ambiguous" id="S3.SS1.SSS2.7.p7.11.m11.2.2.1.1.1.cmml" xref="S3.SS1.SSS2.7.p7.11.m11.2.2.1.1">subscript</csymbol><ci id="S3.SS1.SSS2.7.p7.11.m11.2.2.1.1.2.cmml" xref="S3.SS1.SSS2.7.p7.11.m11.2.2.1.1.2">𝑃</ci><cn id="S3.SS1.SSS2.7.p7.11.m11.2.2.1.1.3.cmml" type="integer" xref="S3.SS1.SSS2.7.p7.11.m11.2.2.1.1.3">1</cn></apply><ci id="S3.SS1.SSS2.7.p7.11.m11.1.1.cmml" xref="S3.SS1.SSS2.7.p7.11.m11.1.1">⋯</ci><apply id="S3.SS1.SSS2.7.p7.11.m11.3.3.2.2.cmml" xref="S3.SS1.SSS2.7.p7.11.m11.3.3.2.2"><csymbol cd="ambiguous" id="S3.SS1.SSS2.7.p7.11.m11.3.3.2.2.1.cmml" xref="S3.SS1.SSS2.7.p7.11.m11.3.3.2.2">subscript</csymbol><ci id="S3.SS1.SSS2.7.p7.11.m11.3.3.2.2.2.cmml" xref="S3.SS1.SSS2.7.p7.11.m11.3.3.2.2.2">𝑃</ci><apply id="S3.SS1.SSS2.7.p7.11.m11.3.3.2.2.3.cmml" xref="S3.SS1.SSS2.7.p7.11.m11.3.3.2.2.3"><times id="S3.SS1.SSS2.7.p7.11.m11.3.3.2.2.3.1.cmml" xref="S3.SS1.SSS2.7.p7.11.m11.3.3.2.2.3.1"></times><cn id="S3.SS1.SSS2.7.p7.11.m11.3.3.2.2.3.2.cmml" type="integer" xref="S3.SS1.SSS2.7.p7.11.m11.3.3.2.2.3.2">2</cn><ci id="S3.SS1.SSS2.7.p7.11.m11.3.3.2.2.3.3.cmml" xref="S3.SS1.SSS2.7.p7.11.m11.3.3.2.2.3.3">𝑘</ci></apply></apply></list></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS2.7.p7.11.m11.3c">P_{1},\cdots,P_{2k}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS2.7.p7.11.m11.3d">italic_P start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , ⋯ , italic_P start_POSTSUBSCRIPT 2 italic_k end_POSTSUBSCRIPT</annotation></semantics></math> are vertex-disjoint and <math alttext="u_{i}v_{i}" class="ltx_Math" display="inline" id="S3.SS1.SSS2.7.p7.12.m12.1"><semantics id="S3.SS1.SSS2.7.p7.12.m12.1a"><mrow id="S3.SS1.SSS2.7.p7.12.m12.1.1" xref="S3.SS1.SSS2.7.p7.12.m12.1.1.cmml"><msub id="S3.SS1.SSS2.7.p7.12.m12.1.1.2" xref="S3.SS1.SSS2.7.p7.12.m12.1.1.2.cmml"><mi id="S3.SS1.SSS2.7.p7.12.m12.1.1.2.2" xref="S3.SS1.SSS2.7.p7.12.m12.1.1.2.2.cmml">u</mi><mi id="S3.SS1.SSS2.7.p7.12.m12.1.1.2.3" xref="S3.SS1.SSS2.7.p7.12.m12.1.1.2.3.cmml">i</mi></msub><mo id="S3.SS1.SSS2.7.p7.12.m12.1.1.1" xref="S3.SS1.SSS2.7.p7.12.m12.1.1.1.cmml">⁢</mo><msub id="S3.SS1.SSS2.7.p7.12.m12.1.1.3" xref="S3.SS1.SSS2.7.p7.12.m12.1.1.3.cmml"><mi id="S3.SS1.SSS2.7.p7.12.m12.1.1.3.2" xref="S3.SS1.SSS2.7.p7.12.m12.1.1.3.2.cmml">v</mi><mi id="S3.SS1.SSS2.7.p7.12.m12.1.1.3.3" xref="S3.SS1.SSS2.7.p7.12.m12.1.1.3.3.cmml">i</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS2.7.p7.12.m12.1b"><apply id="S3.SS1.SSS2.7.p7.12.m12.1.1.cmml" xref="S3.SS1.SSS2.7.p7.12.m12.1.1"><times id="S3.SS1.SSS2.7.p7.12.m12.1.1.1.cmml" xref="S3.SS1.SSS2.7.p7.12.m12.1.1.1"></times><apply id="S3.SS1.SSS2.7.p7.12.m12.1.1.2.cmml" xref="S3.SS1.SSS2.7.p7.12.m12.1.1.2"><csymbol cd="ambiguous" id="S3.SS1.SSS2.7.p7.12.m12.1.1.2.1.cmml" xref="S3.SS1.SSS2.7.p7.12.m12.1.1.2">subscript</csymbol><ci id="S3.SS1.SSS2.7.p7.12.m12.1.1.2.2.cmml" xref="S3.SS1.SSS2.7.p7.12.m12.1.1.2.2">𝑢</ci><ci id="S3.SS1.SSS2.7.p7.12.m12.1.1.2.3.cmml" xref="S3.SS1.SSS2.7.p7.12.m12.1.1.2.3">𝑖</ci></apply><apply id="S3.SS1.SSS2.7.p7.12.m12.1.1.3.cmml" xref="S3.SS1.SSS2.7.p7.12.m12.1.1.3"><csymbol cd="ambiguous" id="S3.SS1.SSS2.7.p7.12.m12.1.1.3.1.cmml" xref="S3.SS1.SSS2.7.p7.12.m12.1.1.3">subscript</csymbol><ci id="S3.SS1.SSS2.7.p7.12.m12.1.1.3.2.cmml" xref="S3.SS1.SSS2.7.p7.12.m12.1.1.3.2">𝑣</ci><ci id="S3.SS1.SSS2.7.p7.12.m12.1.1.3.3.cmml" xref="S3.SS1.SSS2.7.p7.12.m12.1.1.3.3">𝑖</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS2.7.p7.12.m12.1c">u_{i}v_{i}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS2.7.p7.12.m12.1d">italic_u start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT italic_v start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math> is crossing the biset <math alttext="\hat{S}" class="ltx_Math" display="inline" id="S3.SS1.SSS2.7.p7.13.m13.1"><semantics id="S3.SS1.SSS2.7.p7.13.m13.1a"><mover accent="true" id="S3.SS1.SSS2.7.p7.13.m13.1.1" xref="S3.SS1.SSS2.7.p7.13.m13.1.1.cmml"><mi id="S3.SS1.SSS2.7.p7.13.m13.1.1.2" xref="S3.SS1.SSS2.7.p7.13.m13.1.1.2.cmml">S</mi><mo id="S3.SS1.SSS2.7.p7.13.m13.1.1.1" xref="S3.SS1.SSS2.7.p7.13.m13.1.1.1.cmml">^</mo></mover><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS2.7.p7.13.m13.1b"><apply id="S3.SS1.SSS2.7.p7.13.m13.1.1.cmml" xref="S3.SS1.SSS2.7.p7.13.m13.1.1"><ci id="S3.SS1.SSS2.7.p7.13.m13.1.1.1.cmml" xref="S3.SS1.SSS2.7.p7.13.m13.1.1.1">^</ci><ci id="S3.SS1.SSS2.7.p7.13.m13.1.1.2.cmml" xref="S3.SS1.SSS2.7.p7.13.m13.1.1.2">𝑆</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS2.7.p7.13.m13.1c">\hat{S}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS2.7.p7.13.m13.1d">over^ start_ARG italic_S end_ARG</annotation></semantics></math>, at least</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="Sx1.EGx4"> <tbody id="S3.Ex6"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle 2k-L_{1}-|S^{+}\setminus S|>2k-L_{1}-|S^{+}\setminus S|-(k^{% \prime}-L_{1})" class="ltx_Math" display="inline" id="S3.Ex6.m1.3"><semantics id="S3.Ex6.m1.3a"><mrow id="S3.Ex6.m1.3.3" xref="S3.Ex6.m1.3.3.cmml"><mrow id="S3.Ex6.m1.1.1.1" xref="S3.Ex6.m1.1.1.1.cmml"><mrow id="S3.Ex6.m1.1.1.1.3" xref="S3.Ex6.m1.1.1.1.3.cmml"><mn id="S3.Ex6.m1.1.1.1.3.2" xref="S3.Ex6.m1.1.1.1.3.2.cmml">2</mn><mo id="S3.Ex6.m1.1.1.1.3.1" xref="S3.Ex6.m1.1.1.1.3.1.cmml">⁢</mo><mi id="S3.Ex6.m1.1.1.1.3.3" xref="S3.Ex6.m1.1.1.1.3.3.cmml">k</mi></mrow><mo id="S3.Ex6.m1.1.1.1.2" xref="S3.Ex6.m1.1.1.1.2.cmml">−</mo><msub id="S3.Ex6.m1.1.1.1.4" xref="S3.Ex6.m1.1.1.1.4.cmml"><mi id="S3.Ex6.m1.1.1.1.4.2" xref="S3.Ex6.m1.1.1.1.4.2.cmml">L</mi><mn id="S3.Ex6.m1.1.1.1.4.3" xref="S3.Ex6.m1.1.1.1.4.3.cmml">1</mn></msub><mo id="S3.Ex6.m1.1.1.1.2a" xref="S3.Ex6.m1.1.1.1.2.cmml">−</mo><mrow id="S3.Ex6.m1.1.1.1.1.1" xref="S3.Ex6.m1.1.1.1.1.2.cmml"><mo id="S3.Ex6.m1.1.1.1.1.1.2" stretchy="false" xref="S3.Ex6.m1.1.1.1.1.2.1.cmml">|</mo><mrow id="S3.Ex6.m1.1.1.1.1.1.1" xref="S3.Ex6.m1.1.1.1.1.1.1.cmml"><msup id="S3.Ex6.m1.1.1.1.1.1.1.2" xref="S3.Ex6.m1.1.1.1.1.1.1.2.cmml"><mi id="S3.Ex6.m1.1.1.1.1.1.1.2.2" xref="S3.Ex6.m1.1.1.1.1.1.1.2.2.cmml">S</mi><mo id="S3.Ex6.m1.1.1.1.1.1.1.2.3" xref="S3.Ex6.m1.1.1.1.1.1.1.2.3.cmml">+</mo></msup><mo id="S3.Ex6.m1.1.1.1.1.1.1.1" xref="S3.Ex6.m1.1.1.1.1.1.1.1.cmml">∖</mo><mi id="S3.Ex6.m1.1.1.1.1.1.1.3" xref="S3.Ex6.m1.1.1.1.1.1.1.3.cmml">S</mi></mrow><mo id="S3.Ex6.m1.1.1.1.1.1.3" stretchy="false" xref="S3.Ex6.m1.1.1.1.1.2.1.cmml">|</mo></mrow></mrow><mo id="S3.Ex6.m1.3.3.4" xref="S3.Ex6.m1.3.3.4.cmml">></mo><mrow id="S3.Ex6.m1.3.3.3" xref="S3.Ex6.m1.3.3.3.cmml"><mrow id="S3.Ex6.m1.3.3.3.4" xref="S3.Ex6.m1.3.3.3.4.cmml"><mn id="S3.Ex6.m1.3.3.3.4.2" xref="S3.Ex6.m1.3.3.3.4.2.cmml">2</mn><mo id="S3.Ex6.m1.3.3.3.4.1" xref="S3.Ex6.m1.3.3.3.4.1.cmml">⁢</mo><mi id="S3.Ex6.m1.3.3.3.4.3" xref="S3.Ex6.m1.3.3.3.4.3.cmml">k</mi></mrow><mo id="S3.Ex6.m1.3.3.3.3" xref="S3.Ex6.m1.3.3.3.3.cmml">−</mo><msub id="S3.Ex6.m1.3.3.3.5" xref="S3.Ex6.m1.3.3.3.5.cmml"><mi id="S3.Ex6.m1.3.3.3.5.2" xref="S3.Ex6.m1.3.3.3.5.2.cmml">L</mi><mn id="S3.Ex6.m1.3.3.3.5.3" xref="S3.Ex6.m1.3.3.3.5.3.cmml">1</mn></msub><mo id="S3.Ex6.m1.3.3.3.3a" xref="S3.Ex6.m1.3.3.3.3.cmml">−</mo><mrow id="S3.Ex6.m1.2.2.2.1.1" xref="S3.Ex6.m1.2.2.2.1.2.cmml"><mo id="S3.Ex6.m1.2.2.2.1.1.2" stretchy="false" xref="S3.Ex6.m1.2.2.2.1.2.1.cmml">|</mo><mrow id="S3.Ex6.m1.2.2.2.1.1.1" xref="S3.Ex6.m1.2.2.2.1.1.1.cmml"><msup id="S3.Ex6.m1.2.2.2.1.1.1.2" xref="S3.Ex6.m1.2.2.2.1.1.1.2.cmml"><mi id="S3.Ex6.m1.2.2.2.1.1.1.2.2" xref="S3.Ex6.m1.2.2.2.1.1.1.2.2.cmml">S</mi><mo id="S3.Ex6.m1.2.2.2.1.1.1.2.3" xref="S3.Ex6.m1.2.2.2.1.1.1.2.3.cmml">+</mo></msup><mo id="S3.Ex6.m1.2.2.2.1.1.1.1" xref="S3.Ex6.m1.2.2.2.1.1.1.1.cmml">∖</mo><mi id="S3.Ex6.m1.2.2.2.1.1.1.3" xref="S3.Ex6.m1.2.2.2.1.1.1.3.cmml">S</mi></mrow><mo id="S3.Ex6.m1.2.2.2.1.1.3" stretchy="false" xref="S3.Ex6.m1.2.2.2.1.2.1.cmml">|</mo></mrow><mo id="S3.Ex6.m1.3.3.3.3b" xref="S3.Ex6.m1.3.3.3.3.cmml">−</mo><mrow id="S3.Ex6.m1.3.3.3.2.1" xref="S3.Ex6.m1.3.3.3.2.1.1.cmml"><mo id="S3.Ex6.m1.3.3.3.2.1.2" stretchy="false" 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xref="S3.Ex6.m1.3.3.3.2.1"><minus id="S3.Ex6.m1.3.3.3.2.1.1.1.cmml" xref="S3.Ex6.m1.3.3.3.2.1.1.1"></minus><apply id="S3.Ex6.m1.3.3.3.2.1.1.2.cmml" xref="S3.Ex6.m1.3.3.3.2.1.1.2"><csymbol cd="ambiguous" id="S3.Ex6.m1.3.3.3.2.1.1.2.1.cmml" xref="S3.Ex6.m1.3.3.3.2.1.1.2">superscript</csymbol><ci id="S3.Ex6.m1.3.3.3.2.1.1.2.2.cmml" xref="S3.Ex6.m1.3.3.3.2.1.1.2.2">𝑘</ci><ci id="S3.Ex6.m1.3.3.3.2.1.1.2.3.cmml" xref="S3.Ex6.m1.3.3.3.2.1.1.2.3">′</ci></apply><apply id="S3.Ex6.m1.3.3.3.2.1.1.3.cmml" xref="S3.Ex6.m1.3.3.3.2.1.1.3"><csymbol cd="ambiguous" id="S3.Ex6.m1.3.3.3.2.1.1.3.1.cmml" xref="S3.Ex6.m1.3.3.3.2.1.1.3">subscript</csymbol><ci id="S3.Ex6.m1.3.3.3.2.1.1.3.2.cmml" xref="S3.Ex6.m1.3.3.3.2.1.1.3.2">𝐿</ci><cn id="S3.Ex6.m1.3.3.3.2.1.1.3.3.cmml" type="integer" xref="S3.Ex6.m1.3.3.3.2.1.1.3.3">1</cn></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Ex6.m1.3c">\displaystyle 2k-L_{1}-|S^{+}\setminus S|>2k-L_{1}-|S^{+}\setminus S|-(k^{% \prime}-L_{1})</annotation><annotation encoding="application/x-llamapun" id="S3.Ex6.m1.3d">2 italic_k - italic_L start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT - | italic_S start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT ∖ italic_S | > 2 italic_k - italic_L start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT - | italic_S start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT ∖ italic_S | - ( italic_k start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT - italic_L start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT )</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle=2k-(k^{\prime}+|S^{+}\setminus S|)" class="ltx_Math" display="inline" id="S3.Ex6.m2.1"><semantics id="S3.Ex6.m2.1a"><mrow id="S3.Ex6.m2.1.1" xref="S3.Ex6.m2.1.1.cmml"><mi id="S3.Ex6.m2.1.1.3" xref="S3.Ex6.m2.1.1.3.cmml"></mi><mo id="S3.Ex6.m2.1.1.2" xref="S3.Ex6.m2.1.1.2.cmml">=</mo><mrow id="S3.Ex6.m2.1.1.1" xref="S3.Ex6.m2.1.1.1.cmml"><mrow id="S3.Ex6.m2.1.1.1.3" xref="S3.Ex6.m2.1.1.1.3.cmml"><mn id="S3.Ex6.m2.1.1.1.3.2" xref="S3.Ex6.m2.1.1.1.3.2.cmml">2</mn><mo id="S3.Ex6.m2.1.1.1.3.1" xref="S3.Ex6.m2.1.1.1.3.1.cmml">⁢</mo><mi id="S3.Ex6.m2.1.1.1.3.3" xref="S3.Ex6.m2.1.1.1.3.3.cmml">k</mi></mrow><mo id="S3.Ex6.m2.1.1.1.2" xref="S3.Ex6.m2.1.1.1.2.cmml">−</mo><mrow id="S3.Ex6.m2.1.1.1.1.1" xref="S3.Ex6.m2.1.1.1.1.1.1.cmml"><mo id="S3.Ex6.m2.1.1.1.1.1.2" stretchy="false" xref="S3.Ex6.m2.1.1.1.1.1.1.cmml">(</mo><mrow id="S3.Ex6.m2.1.1.1.1.1.1" xref="S3.Ex6.m2.1.1.1.1.1.1.cmml"><msup id="S3.Ex6.m2.1.1.1.1.1.1.3" xref="S3.Ex6.m2.1.1.1.1.1.1.3.cmml"><mi id="S3.Ex6.m2.1.1.1.1.1.1.3.2" xref="S3.Ex6.m2.1.1.1.1.1.1.3.2.cmml">k</mi><mo id="S3.Ex6.m2.1.1.1.1.1.1.3.3" xref="S3.Ex6.m2.1.1.1.1.1.1.3.3.cmml">′</mo></msup><mo id="S3.Ex6.m2.1.1.1.1.1.1.2" xref="S3.Ex6.m2.1.1.1.1.1.1.2.cmml">+</mo><mrow id="S3.Ex6.m2.1.1.1.1.1.1.1.1" xref="S3.Ex6.m2.1.1.1.1.1.1.1.2.cmml"><mo id="S3.Ex6.m2.1.1.1.1.1.1.1.1.2" stretchy="false" xref="S3.Ex6.m2.1.1.1.1.1.1.1.2.1.cmml">|</mo><mrow 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encoding="application/x-llamapun" id="S3.Ex6.m2.1d">= 2 italic_k - ( italic_k start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT + | italic_S start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT ∖ italic_S | )</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> <tbody id="S3.Ex7"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_eqn_cell"></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle=2k-(h(\hat{S})+|S^{+}\setminus S|)\geq k" class="ltx_Math" display="inline" id="S3.Ex7.m1.2"><semantics id="S3.Ex7.m1.2a"><mrow id="S3.Ex7.m1.2.2" xref="S3.Ex7.m1.2.2.cmml"><mi id="S3.Ex7.m1.2.2.3" xref="S3.Ex7.m1.2.2.3.cmml"></mi><mo id="S3.Ex7.m1.2.2.4" xref="S3.Ex7.m1.2.2.4.cmml">=</mo><mrow id="S3.Ex7.m1.2.2.1" xref="S3.Ex7.m1.2.2.1.cmml"><mrow id="S3.Ex7.m1.2.2.1.3" xref="S3.Ex7.m1.2.2.1.3.cmml"><mn id="S3.Ex7.m1.2.2.1.3.2" 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encoding="application/x-llamapun" id="S3.Ex7.m1.2d">= 2 italic_k - ( italic_h ( over^ start_ARG italic_S end_ARG ) + | italic_S start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT ∖ italic_S | ) ≥ italic_k</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S3.SS1.SSS2.7.p7.21">of the paths must have an edge crossing <math alttext="\hat{S}" class="ltx_Math" display="inline" id="S3.SS1.SSS2.7.p7.14.m1.1"><semantics id="S3.SS1.SSS2.7.p7.14.m1.1a"><mover accent="true" id="S3.SS1.SSS2.7.p7.14.m1.1.1" xref="S3.SS1.SSS2.7.p7.14.m1.1.1.cmml"><mi id="S3.SS1.SSS2.7.p7.14.m1.1.1.2" xref="S3.SS1.SSS2.7.p7.14.m1.1.1.2.cmml">S</mi><mo id="S3.SS1.SSS2.7.p7.14.m1.1.1.1" xref="S3.SS1.SSS2.7.p7.14.m1.1.1.1.cmml">^</mo></mover><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS2.7.p7.14.m1.1b"><apply id="S3.SS1.SSS2.7.p7.14.m1.1.1.cmml" xref="S3.SS1.SSS2.7.p7.14.m1.1.1"><ci id="S3.SS1.SSS2.7.p7.14.m1.1.1.1.cmml" 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xref="S3.SS1.SSS2.7.p7.15.m2.1.1.1">^</ci><ci id="S3.SS1.SSS2.7.p7.15.m2.1.1.2.cmml" xref="S3.SS1.SSS2.7.p7.15.m2.1.1.2">𝑆</ci></apply></apply><ci id="S3.SS1.SSS2.7.p7.15.m2.1.2.3.cmml" xref="S3.SS1.SSS2.7.p7.15.m2.1.2.3">𝐻</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS2.7.p7.15.m2.1c">\delta_{\textnormal{OPT}}(\hat{S})\cap H</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS2.7.p7.15.m2.1d">italic_δ start_POSTSUBSCRIPT OPT end_POSTSUBSCRIPT ( over^ start_ARG italic_S end_ARG ) ∩ italic_H</annotation></semantics></math>, where the last inequality holds because for every biset <math alttext="\hat{S}" class="ltx_Math" display="inline" id="S3.SS1.SSS2.7.p7.16.m3.1"><semantics id="S3.SS1.SSS2.7.p7.16.m3.1a"><mover accent="true" id="S3.SS1.SSS2.7.p7.16.m3.1.1" xref="S3.SS1.SSS2.7.p7.16.m3.1.1.cmml"><mi id="S3.SS1.SSS2.7.p7.16.m3.1.1.2" xref="S3.SS1.SSS2.7.p7.16.m3.1.1.2.cmml">S</mi><mo id="S3.SS1.SSS2.7.p7.16.m3.1.1.1" xref="S3.SS1.SSS2.7.p7.16.m3.1.1.1.cmml">^</mo></mover><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS2.7.p7.16.m3.1b"><apply id="S3.SS1.SSS2.7.p7.16.m3.1.1.cmml" xref="S3.SS1.SSS2.7.p7.16.m3.1.1"><ci id="S3.SS1.SSS2.7.p7.16.m3.1.1.1.cmml" xref="S3.SS1.SSS2.7.p7.16.m3.1.1.1">^</ci><ci id="S3.SS1.SSS2.7.p7.16.m3.1.1.2.cmml" xref="S3.SS1.SSS2.7.p7.16.m3.1.1.2">𝑆</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS2.7.p7.16.m3.1c">\hat{S}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS2.7.p7.16.m3.1d">over^ start_ARG italic_S end_ARG</annotation></semantics></math> with <math alttext="h(\hat{S})>0" class="ltx_Math" display="inline" id="S3.SS1.SSS2.7.p7.17.m4.1"><semantics id="S3.SS1.SSS2.7.p7.17.m4.1a"><mrow id="S3.SS1.SSS2.7.p7.17.m4.1.2" xref="S3.SS1.SSS2.7.p7.17.m4.1.2.cmml"><mrow id="S3.SS1.SSS2.7.p7.17.m4.1.2.2" xref="S3.SS1.SSS2.7.p7.17.m4.1.2.2.cmml"><mi id="S3.SS1.SSS2.7.p7.17.m4.1.2.2.2" xref="S3.SS1.SSS2.7.p7.17.m4.1.2.2.2.cmml">h</mi><mo id="S3.SS1.SSS2.7.p7.17.m4.1.2.2.1" xref="S3.SS1.SSS2.7.p7.17.m4.1.2.2.1.cmml">⁢</mo><mrow id="S3.SS1.SSS2.7.p7.17.m4.1.2.2.3.2" xref="S3.SS1.SSS2.7.p7.17.m4.1.1.cmml"><mo id="S3.SS1.SSS2.7.p7.17.m4.1.2.2.3.2.1" stretchy="false" xref="S3.SS1.SSS2.7.p7.17.m4.1.1.cmml">(</mo><mover accent="true" id="S3.SS1.SSS2.7.p7.17.m4.1.1" xref="S3.SS1.SSS2.7.p7.17.m4.1.1.cmml"><mi id="S3.SS1.SSS2.7.p7.17.m4.1.1.2" xref="S3.SS1.SSS2.7.p7.17.m4.1.1.2.cmml">S</mi><mo id="S3.SS1.SSS2.7.p7.17.m4.1.1.1" xref="S3.SS1.SSS2.7.p7.17.m4.1.1.1.cmml">^</mo></mover><mo id="S3.SS1.SSS2.7.p7.17.m4.1.2.2.3.2.2" stretchy="false" xref="S3.SS1.SSS2.7.p7.17.m4.1.1.cmml">)</mo></mrow></mrow><mo id="S3.SS1.SSS2.7.p7.17.m4.1.2.1" xref="S3.SS1.SSS2.7.p7.17.m4.1.2.1.cmml">></mo><mn id="S3.SS1.SSS2.7.p7.17.m4.1.2.3" xref="S3.SS1.SSS2.7.p7.17.m4.1.2.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS2.7.p7.17.m4.1b"><apply id="S3.SS1.SSS2.7.p7.17.m4.1.2.cmml" xref="S3.SS1.SSS2.7.p7.17.m4.1.2"><gt id="S3.SS1.SSS2.7.p7.17.m4.1.2.1.cmml" xref="S3.SS1.SSS2.7.p7.17.m4.1.2.1"></gt><apply id="S3.SS1.SSS2.7.p7.17.m4.1.2.2.cmml" xref="S3.SS1.SSS2.7.p7.17.m4.1.2.2"><times id="S3.SS1.SSS2.7.p7.17.m4.1.2.2.1.cmml" xref="S3.SS1.SSS2.7.p7.17.m4.1.2.2.1"></times><ci id="S3.SS1.SSS2.7.p7.17.m4.1.2.2.2.cmml" xref="S3.SS1.SSS2.7.p7.17.m4.1.2.2.2">ℎ</ci><apply id="S3.SS1.SSS2.7.p7.17.m4.1.1.cmml" xref="S3.SS1.SSS2.7.p7.17.m4.1.2.2.3.2"><ci id="S3.SS1.SSS2.7.p7.17.m4.1.1.1.cmml" xref="S3.SS1.SSS2.7.p7.17.m4.1.1.1">^</ci><ci id="S3.SS1.SSS2.7.p7.17.m4.1.1.2.cmml" xref="S3.SS1.SSS2.7.p7.17.m4.1.1.2">𝑆</ci></apply></apply><cn id="S3.SS1.SSS2.7.p7.17.m4.1.2.3.cmml" type="integer" xref="S3.SS1.SSS2.7.p7.17.m4.1.2.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS2.7.p7.17.m4.1c">h(\hat{S})>0</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS2.7.p7.17.m4.1d">italic_h ( over^ start_ARG italic_S end_ARG ) > 0</annotation></semantics></math>, <math alttext="h(\hat{S})+|S^{+}\setminus S|\leq k" class="ltx_Math" display="inline" id="S3.SS1.SSS2.7.p7.18.m5.2"><semantics id="S3.SS1.SSS2.7.p7.18.m5.2a"><mrow id="S3.SS1.SSS2.7.p7.18.m5.2.2" xref="S3.SS1.SSS2.7.p7.18.m5.2.2.cmml"><mrow id="S3.SS1.SSS2.7.p7.18.m5.2.2.1" xref="S3.SS1.SSS2.7.p7.18.m5.2.2.1.cmml"><mrow id="S3.SS1.SSS2.7.p7.18.m5.2.2.1.3" xref="S3.SS1.SSS2.7.p7.18.m5.2.2.1.3.cmml"><mi id="S3.SS1.SSS2.7.p7.18.m5.2.2.1.3.2" xref="S3.SS1.SSS2.7.p7.18.m5.2.2.1.3.2.cmml">h</mi><mo id="S3.SS1.SSS2.7.p7.18.m5.2.2.1.3.1" xref="S3.SS1.SSS2.7.p7.18.m5.2.2.1.3.1.cmml">⁢</mo><mrow id="S3.SS1.SSS2.7.p7.18.m5.2.2.1.3.3.2" xref="S3.SS1.SSS2.7.p7.18.m5.1.1.cmml"><mo id="S3.SS1.SSS2.7.p7.18.m5.2.2.1.3.3.2.1" stretchy="false" xref="S3.SS1.SSS2.7.p7.18.m5.1.1.cmml">(</mo><mover accent="true" id="S3.SS1.SSS2.7.p7.18.m5.1.1" xref="S3.SS1.SSS2.7.p7.18.m5.1.1.cmml"><mi id="S3.SS1.SSS2.7.p7.18.m5.1.1.2" 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id="S3.SS1.SSS2.7.p7.18.m5.2.2.1.1.1.1.1" xref="S3.SS1.SSS2.7.p7.18.m5.2.2.1.1.1.1.1.cmml">∖</mo><mi id="S3.SS1.SSS2.7.p7.18.m5.2.2.1.1.1.1.3" xref="S3.SS1.SSS2.7.p7.18.m5.2.2.1.1.1.1.3.cmml">S</mi></mrow><mo id="S3.SS1.SSS2.7.p7.18.m5.2.2.1.1.1.3" stretchy="false" xref="S3.SS1.SSS2.7.p7.18.m5.2.2.1.1.2.1.cmml">|</mo></mrow></mrow><mo id="S3.SS1.SSS2.7.p7.18.m5.2.2.2" xref="S3.SS1.SSS2.7.p7.18.m5.2.2.2.cmml">≤</mo><mi id="S3.SS1.SSS2.7.p7.18.m5.2.2.3" xref="S3.SS1.SSS2.7.p7.18.m5.2.2.3.cmml">k</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS2.7.p7.18.m5.2b"><apply id="S3.SS1.SSS2.7.p7.18.m5.2.2.cmml" xref="S3.SS1.SSS2.7.p7.18.m5.2.2"><leq id="S3.SS1.SSS2.7.p7.18.m5.2.2.2.cmml" xref="S3.SS1.SSS2.7.p7.18.m5.2.2.2"></leq><apply id="S3.SS1.SSS2.7.p7.18.m5.2.2.1.cmml" xref="S3.SS1.SSS2.7.p7.18.m5.2.2.1"><plus id="S3.SS1.SSS2.7.p7.18.m5.2.2.1.2.cmml" xref="S3.SS1.SSS2.7.p7.18.m5.2.2.1.2"></plus><apply id="S3.SS1.SSS2.7.p7.18.m5.2.2.1.3.cmml" xref="S3.SS1.SSS2.7.p7.18.m5.2.2.1.3"><times id="S3.SS1.SSS2.7.p7.18.m5.2.2.1.3.1.cmml" xref="S3.SS1.SSS2.7.p7.18.m5.2.2.1.3.1"></times><ci id="S3.SS1.SSS2.7.p7.18.m5.2.2.1.3.2.cmml" xref="S3.SS1.SSS2.7.p7.18.m5.2.2.1.3.2">ℎ</ci><apply id="S3.SS1.SSS2.7.p7.18.m5.1.1.cmml" xref="S3.SS1.SSS2.7.p7.18.m5.2.2.1.3.3.2"><ci id="S3.SS1.SSS2.7.p7.18.m5.1.1.1.cmml" xref="S3.SS1.SSS2.7.p7.18.m5.1.1.1">^</ci><ci id="S3.SS1.SSS2.7.p7.18.m5.1.1.2.cmml" xref="S3.SS1.SSS2.7.p7.18.m5.1.1.2">𝑆</ci></apply></apply><apply id="S3.SS1.SSS2.7.p7.18.m5.2.2.1.1.2.cmml" xref="S3.SS1.SSS2.7.p7.18.m5.2.2.1.1.1"><abs id="S3.SS1.SSS2.7.p7.18.m5.2.2.1.1.2.1.cmml" xref="S3.SS1.SSS2.7.p7.18.m5.2.2.1.1.1.2"></abs><apply id="S3.SS1.SSS2.7.p7.18.m5.2.2.1.1.1.1.cmml" xref="S3.SS1.SSS2.7.p7.18.m5.2.2.1.1.1.1"><setdiff id="S3.SS1.SSS2.7.p7.18.m5.2.2.1.1.1.1.1.cmml" xref="S3.SS1.SSS2.7.p7.18.m5.2.2.1.1.1.1.1"></setdiff><apply id="S3.SS1.SSS2.7.p7.18.m5.2.2.1.1.1.1.2.cmml" xref="S3.SS1.SSS2.7.p7.18.m5.2.2.1.1.1.1.2"><csymbol cd="ambiguous" id="S3.SS1.SSS2.7.p7.18.m5.2.2.1.1.1.1.2.1.cmml" xref="S3.SS1.SSS2.7.p7.18.m5.2.2.1.1.1.1.2">superscript</csymbol><ci id="S3.SS1.SSS2.7.p7.18.m5.2.2.1.1.1.1.2.2.cmml" xref="S3.SS1.SSS2.7.p7.18.m5.2.2.1.1.1.1.2.2">𝑆</ci><plus id="S3.SS1.SSS2.7.p7.18.m5.2.2.1.1.1.1.2.3.cmml" xref="S3.SS1.SSS2.7.p7.18.m5.2.2.1.1.1.1.2.3"></plus></apply><ci id="S3.SS1.SSS2.7.p7.18.m5.2.2.1.1.1.1.3.cmml" xref="S3.SS1.SSS2.7.p7.18.m5.2.2.1.1.1.1.3">𝑆</ci></apply></apply></apply><ci id="S3.SS1.SSS2.7.p7.18.m5.2.2.3.cmml" xref="S3.SS1.SSS2.7.p7.18.m5.2.2.3">𝑘</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS2.7.p7.18.m5.2c">h(\hat{S})+|S^{+}\setminus S|\leq k</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS2.7.p7.18.m5.2d">italic_h ( over^ start_ARG italic_S end_ARG ) + | italic_S start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT ∖ italic_S | ≤ italic_k</annotation></semantics></math>. Without loss of generality, let us denote <math alttext="k" class="ltx_Math" display="inline" id="S3.SS1.SSS2.7.p7.19.m6.1"><semantics id="S3.SS1.SSS2.7.p7.19.m6.1a"><mi id="S3.SS1.SSS2.7.p7.19.m6.1.1" xref="S3.SS1.SSS2.7.p7.19.m6.1.1.cmml">k</mi><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS2.7.p7.19.m6.1b"><ci id="S3.SS1.SSS2.7.p7.19.m6.1.1.cmml" xref="S3.SS1.SSS2.7.p7.19.m6.1.1">𝑘</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS2.7.p7.19.m6.1c">k</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS2.7.p7.19.m6.1d">italic_k</annotation></semantics></math> of these distinct edges by <math alttext="e^{i}_{1},\dots,e^{i}_{k}" class="ltx_Math" display="inline" id="S3.SS1.SSS2.7.p7.20.m7.3"><semantics id="S3.SS1.SSS2.7.p7.20.m7.3a"><mrow id="S3.SS1.SSS2.7.p7.20.m7.3.3.2" xref="S3.SS1.SSS2.7.p7.20.m7.3.3.3.cmml"><msubsup id="S3.SS1.SSS2.7.p7.20.m7.2.2.1.1" xref="S3.SS1.SSS2.7.p7.20.m7.2.2.1.1.cmml"><mi id="S3.SS1.SSS2.7.p7.20.m7.2.2.1.1.2.2" xref="S3.SS1.SSS2.7.p7.20.m7.2.2.1.1.2.2.cmml">e</mi><mn id="S3.SS1.SSS2.7.p7.20.m7.2.2.1.1.3" xref="S3.SS1.SSS2.7.p7.20.m7.2.2.1.1.3.cmml">1</mn><mi id="S3.SS1.SSS2.7.p7.20.m7.2.2.1.1.2.3" xref="S3.SS1.SSS2.7.p7.20.m7.2.2.1.1.2.3.cmml">i</mi></msubsup><mo id="S3.SS1.SSS2.7.p7.20.m7.3.3.2.3" xref="S3.SS1.SSS2.7.p7.20.m7.3.3.3.cmml">,</mo><mi id="S3.SS1.SSS2.7.p7.20.m7.1.1" mathvariant="normal" xref="S3.SS1.SSS2.7.p7.20.m7.1.1.cmml">…</mi><mo id="S3.SS1.SSS2.7.p7.20.m7.3.3.2.4" xref="S3.SS1.SSS2.7.p7.20.m7.3.3.3.cmml">,</mo><msubsup id="S3.SS1.SSS2.7.p7.20.m7.3.3.2.2" xref="S3.SS1.SSS2.7.p7.20.m7.3.3.2.2.cmml"><mi id="S3.SS1.SSS2.7.p7.20.m7.3.3.2.2.2.2" xref="S3.SS1.SSS2.7.p7.20.m7.3.3.2.2.2.2.cmml">e</mi><mi id="S3.SS1.SSS2.7.p7.20.m7.3.3.2.2.3" xref="S3.SS1.SSS2.7.p7.20.m7.3.3.2.2.3.cmml">k</mi><mi id="S3.SS1.SSS2.7.p7.20.m7.3.3.2.2.2.3" xref="S3.SS1.SSS2.7.p7.20.m7.3.3.2.2.2.3.cmml">i</mi></msubsup></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS2.7.p7.20.m7.3b"><list id="S3.SS1.SSS2.7.p7.20.m7.3.3.3.cmml" xref="S3.SS1.SSS2.7.p7.20.m7.3.3.2"><apply id="S3.SS1.SSS2.7.p7.20.m7.2.2.1.1.cmml" xref="S3.SS1.SSS2.7.p7.20.m7.2.2.1.1"><csymbol cd="ambiguous" id="S3.SS1.SSS2.7.p7.20.m7.2.2.1.1.1.cmml" xref="S3.SS1.SSS2.7.p7.20.m7.2.2.1.1">subscript</csymbol><apply id="S3.SS1.SSS2.7.p7.20.m7.2.2.1.1.2.cmml" xref="S3.SS1.SSS2.7.p7.20.m7.2.2.1.1"><csymbol cd="ambiguous" id="S3.SS1.SSS2.7.p7.20.m7.2.2.1.1.2.1.cmml" xref="S3.SS1.SSS2.7.p7.20.m7.2.2.1.1">superscript</csymbol><ci id="S3.SS1.SSS2.7.p7.20.m7.2.2.1.1.2.2.cmml" xref="S3.SS1.SSS2.7.p7.20.m7.2.2.1.1.2.2">𝑒</ci><ci id="S3.SS1.SSS2.7.p7.20.m7.2.2.1.1.2.3.cmml" xref="S3.SS1.SSS2.7.p7.20.m7.2.2.1.1.2.3">𝑖</ci></apply><cn id="S3.SS1.SSS2.7.p7.20.m7.2.2.1.1.3.cmml" type="integer" xref="S3.SS1.SSS2.7.p7.20.m7.2.2.1.1.3">1</cn></apply><ci id="S3.SS1.SSS2.7.p7.20.m7.1.1.cmml" xref="S3.SS1.SSS2.7.p7.20.m7.1.1">…</ci><apply id="S3.SS1.SSS2.7.p7.20.m7.3.3.2.2.cmml" xref="S3.SS1.SSS2.7.p7.20.m7.3.3.2.2"><csymbol cd="ambiguous" id="S3.SS1.SSS2.7.p7.20.m7.3.3.2.2.1.cmml" xref="S3.SS1.SSS2.7.p7.20.m7.3.3.2.2">subscript</csymbol><apply id="S3.SS1.SSS2.7.p7.20.m7.3.3.2.2.2.cmml" xref="S3.SS1.SSS2.7.p7.20.m7.3.3.2.2"><csymbol cd="ambiguous" id="S3.SS1.SSS2.7.p7.20.m7.3.3.2.2.2.1.cmml" xref="S3.SS1.SSS2.7.p7.20.m7.3.3.2.2">superscript</csymbol><ci id="S3.SS1.SSS2.7.p7.20.m7.3.3.2.2.2.2.cmml" xref="S3.SS1.SSS2.7.p7.20.m7.3.3.2.2.2.2">𝑒</ci><ci id="S3.SS1.SSS2.7.p7.20.m7.3.3.2.2.2.3.cmml" xref="S3.SS1.SSS2.7.p7.20.m7.3.3.2.2.2.3">𝑖</ci></apply><ci id="S3.SS1.SSS2.7.p7.20.m7.3.3.2.2.3.cmml" xref="S3.SS1.SSS2.7.p7.20.m7.3.3.2.2.3">𝑘</ci></apply></list></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS2.7.p7.20.m7.3c">e^{i}_{1},\dots,e^{i}_{k}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS2.7.p7.20.m7.3d">italic_e start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , … , italic_e start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT</annotation></semantics></math>. Then, by the construction of <math alttext="\boldsymbol{x}" class="ltx_Math" display="inline" id="S3.SS1.SSS2.7.p7.21.m8.1"><semantics id="S3.SS1.SSS2.7.p7.21.m8.1a"><mi id="S3.SS1.SSS2.7.p7.21.m8.1.1" xref="S3.SS1.SSS2.7.p7.21.m8.1.1.cmml">𝒙</mi><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS2.7.p7.21.m8.1b"><ci id="S3.SS1.SSS2.7.p7.21.m8.1.1.cmml" xref="S3.SS1.SSS2.7.p7.21.m8.1.1">𝒙</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS2.7.p7.21.m8.1c">\boldsymbol{x}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS2.7.p7.21.m8.1d">bold_italic_x</annotation></semantics></math>,</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="Sx1.EGx5"> <tbody id="S3.Ex8"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle\sum_{e\in\delta_{H}(\hat{S})}\boldsymbol{x}(e)\geq\sum_{e\in% \delta_{\textnormal{OPT}}(\hat{S})\cap H}\boldsymbol{x}(e)+\sum_{e\in\{e^{i}_{% j}\}_{i\in[k^{\prime}-L_{1}],j\in[k]}}\boldsymbol{x}(e)\geq L_{1}+(k^{\prime}-% L_{1})\cdot k\cdot\frac{1}{k}=k^{\prime}=h(\hat{S})." class="ltx_Math" display="inline" id="S3.Ex8.m1.11"><semantics id="S3.Ex8.m1.11a"><mrow id="S3.Ex8.m1.11.11.1" xref="S3.Ex8.m1.11.11.1.1.cmml"><mrow id="S3.Ex8.m1.11.11.1.1" xref="S3.Ex8.m1.11.11.1.1.cmml"><mrow id="S3.Ex8.m1.11.11.1.1.3" xref="S3.Ex8.m1.11.11.1.1.3.cmml"><mstyle displaystyle="true" id="S3.Ex8.m1.11.11.1.1.3.1" xref="S3.Ex8.m1.11.11.1.1.3.1.cmml"><munder id="S3.Ex8.m1.11.11.1.1.3.1a" xref="S3.Ex8.m1.11.11.1.1.3.1.cmml"><mo id="S3.Ex8.m1.11.11.1.1.3.1.2" movablelimits="false" xref="S3.Ex8.m1.11.11.1.1.3.1.2.cmml">∑</mo><mrow id="S3.Ex8.m1.1.1.1" xref="S3.Ex8.m1.1.1.1.cmml"><mi id="S3.Ex8.m1.1.1.1.3" xref="S3.Ex8.m1.1.1.1.3.cmml">e</mi><mo id="S3.Ex8.m1.1.1.1.2" xref="S3.Ex8.m1.1.1.1.2.cmml">∈</mo><mrow id="S3.Ex8.m1.1.1.1.4" xref="S3.Ex8.m1.1.1.1.4.cmml"><msub id="S3.Ex8.m1.1.1.1.4.2" xref="S3.Ex8.m1.1.1.1.4.2.cmml"><mi id="S3.Ex8.m1.1.1.1.4.2.2" xref="S3.Ex8.m1.1.1.1.4.2.2.cmml">δ</mi><mi id="S3.Ex8.m1.1.1.1.4.2.3" xref="S3.Ex8.m1.1.1.1.4.2.3.cmml">H</mi></msub><mo id="S3.Ex8.m1.1.1.1.4.1" xref="S3.Ex8.m1.1.1.1.4.1.cmml">⁢</mo><mrow id="S3.Ex8.m1.1.1.1.4.3.2" xref="S3.Ex8.m1.1.1.1.1.cmml"><mo id="S3.Ex8.m1.1.1.1.4.3.2.1" stretchy="false" xref="S3.Ex8.m1.1.1.1.1.cmml">(</mo><mover accent="true" id="S3.Ex8.m1.1.1.1.1" xref="S3.Ex8.m1.1.1.1.1.cmml"><mi id="S3.Ex8.m1.1.1.1.1.2" xref="S3.Ex8.m1.1.1.1.1.2.cmml">S</mi><mo id="S3.Ex8.m1.1.1.1.1.1" xref="S3.Ex8.m1.1.1.1.1.1.cmml">^</mo></mover><mo id="S3.Ex8.m1.1.1.1.4.3.2.2" stretchy="false" xref="S3.Ex8.m1.1.1.1.1.cmml">)</mo></mrow></mrow></mrow></munder></mstyle><mrow id="S3.Ex8.m1.11.11.1.1.3.2" xref="S3.Ex8.m1.11.11.1.1.3.2.cmml"><mi id="S3.Ex8.m1.11.11.1.1.3.2.2" xref="S3.Ex8.m1.11.11.1.1.3.2.2.cmml">𝒙</mi><mo id="S3.Ex8.m1.11.11.1.1.3.2.1" xref="S3.Ex8.m1.11.11.1.1.3.2.1.cmml">⁢</mo><mrow id="S3.Ex8.m1.11.11.1.1.3.2.3.2" xref="S3.Ex8.m1.11.11.1.1.3.2.cmml"><mo id="S3.Ex8.m1.11.11.1.1.3.2.3.2.1" stretchy="false" xref="S3.Ex8.m1.11.11.1.1.3.2.cmml">(</mo><mi id="S3.Ex8.m1.7.7" xref="S3.Ex8.m1.7.7.cmml">e</mi><mo id="S3.Ex8.m1.11.11.1.1.3.2.3.2.2" stretchy="false" xref="S3.Ex8.m1.11.11.1.1.3.2.cmml">)</mo></mrow></mrow></mrow><mo id="S3.Ex8.m1.11.11.1.1.4" xref="S3.Ex8.m1.11.11.1.1.4.cmml">≥</mo><mrow id="S3.Ex8.m1.11.11.1.1.5" xref="S3.Ex8.m1.11.11.1.1.5.cmml"><mrow id="S3.Ex8.m1.11.11.1.1.5.2" xref="S3.Ex8.m1.11.11.1.1.5.2.cmml"><mstyle displaystyle="true" id="S3.Ex8.m1.11.11.1.1.5.2.1" xref="S3.Ex8.m1.11.11.1.1.5.2.1.cmml"><munder id="S3.Ex8.m1.11.11.1.1.5.2.1a" xref="S3.Ex8.m1.11.11.1.1.5.2.1.cmml"><mo id="S3.Ex8.m1.11.11.1.1.5.2.1.2" movablelimits="false" xref="S3.Ex8.m1.11.11.1.1.5.2.1.2.cmml">∑</mo><mrow id="S3.Ex8.m1.2.2.1" xref="S3.Ex8.m1.2.2.1.cmml"><mi id="S3.Ex8.m1.2.2.1.3" xref="S3.Ex8.m1.2.2.1.3.cmml">e</mi><mo id="S3.Ex8.m1.2.2.1.2" xref="S3.Ex8.m1.2.2.1.2.cmml">∈</mo><mrow id="S3.Ex8.m1.2.2.1.4" xref="S3.Ex8.m1.2.2.1.4.cmml"><mrow id="S3.Ex8.m1.2.2.1.4.2" xref="S3.Ex8.m1.2.2.1.4.2.cmml"><msub id="S3.Ex8.m1.2.2.1.4.2.2" xref="S3.Ex8.m1.2.2.1.4.2.2.cmml"><mi id="S3.Ex8.m1.2.2.1.4.2.2.2" xref="S3.Ex8.m1.2.2.1.4.2.2.2.cmml">δ</mi><mtext id="S3.Ex8.m1.2.2.1.4.2.2.3" xref="S3.Ex8.m1.2.2.1.4.2.2.3a.cmml">OPT</mtext></msub><mo id="S3.Ex8.m1.2.2.1.4.2.1" xref="S3.Ex8.m1.2.2.1.4.2.1.cmml">⁢</mo><mrow id="S3.Ex8.m1.2.2.1.4.2.3.2" xref="S3.Ex8.m1.2.2.1.1.cmml"><mo id="S3.Ex8.m1.2.2.1.4.2.3.2.1" stretchy="false" xref="S3.Ex8.m1.2.2.1.1.cmml">(</mo><mover accent="true" id="S3.Ex8.m1.2.2.1.1" xref="S3.Ex8.m1.2.2.1.1.cmml"><mi id="S3.Ex8.m1.2.2.1.1.2" xref="S3.Ex8.m1.2.2.1.1.2.cmml">S</mi><mo id="S3.Ex8.m1.2.2.1.1.1" xref="S3.Ex8.m1.2.2.1.1.1.cmml">^</mo></mover><mo id="S3.Ex8.m1.2.2.1.4.2.3.2.2" stretchy="false" xref="S3.Ex8.m1.2.2.1.1.cmml">)</mo></mrow></mrow><mo id="S3.Ex8.m1.2.2.1.4.1" xref="S3.Ex8.m1.2.2.1.4.1.cmml">∩</mo><mi id="S3.Ex8.m1.2.2.1.4.3" xref="S3.Ex8.m1.2.2.1.4.3.cmml">H</mi></mrow></mrow></munder></mstyle><mrow id="S3.Ex8.m1.11.11.1.1.5.2.2" xref="S3.Ex8.m1.11.11.1.1.5.2.2.cmml"><mi id="S3.Ex8.m1.11.11.1.1.5.2.2.2" xref="S3.Ex8.m1.11.11.1.1.5.2.2.2.cmml">𝒙</mi><mo id="S3.Ex8.m1.11.11.1.1.5.2.2.1" xref="S3.Ex8.m1.11.11.1.1.5.2.2.1.cmml">⁢</mo><mrow id="S3.Ex8.m1.11.11.1.1.5.2.2.3.2" xref="S3.Ex8.m1.11.11.1.1.5.2.2.cmml"><mo id="S3.Ex8.m1.11.11.1.1.5.2.2.3.2.1" stretchy="false" xref="S3.Ex8.m1.11.11.1.1.5.2.2.cmml">(</mo><mi id="S3.Ex8.m1.8.8" xref="S3.Ex8.m1.8.8.cmml">e</mi><mo id="S3.Ex8.m1.11.11.1.1.5.2.2.3.2.2" stretchy="false" xref="S3.Ex8.m1.11.11.1.1.5.2.2.cmml">)</mo></mrow></mrow></mrow><mo id="S3.Ex8.m1.11.11.1.1.5.1" xref="S3.Ex8.m1.11.11.1.1.5.1.cmml">+</mo><mrow id="S3.Ex8.m1.11.11.1.1.5.3" xref="S3.Ex8.m1.11.11.1.1.5.3.cmml"><mstyle displaystyle="true" id="S3.Ex8.m1.11.11.1.1.5.3.1" xref="S3.Ex8.m1.11.11.1.1.5.3.1.cmml"><munder 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encoding="application/x-tex" id="S3.Ex8.m1.11c">\displaystyle\sum_{e\in\delta_{H}(\hat{S})}\boldsymbol{x}(e)\geq\sum_{e\in% \delta_{\textnormal{OPT}}(\hat{S})\cap H}\boldsymbol{x}(e)+\sum_{e\in\{e^{i}_{% j}\}_{i\in[k^{\prime}-L_{1}],j\in[k]}}\boldsymbol{x}(e)\geq L_{1}+(k^{\prime}-% L_{1})\cdot k\cdot\frac{1}{k}=k^{\prime}=h(\hat{S}).</annotation><annotation encoding="application/x-llamapun" id="S3.Ex8.m1.11d">∑ start_POSTSUBSCRIPT italic_e ∈ italic_δ start_POSTSUBSCRIPT italic_H end_POSTSUBSCRIPT ( over^ start_ARG italic_S end_ARG ) end_POSTSUBSCRIPT bold_italic_x ( italic_e ) ≥ ∑ start_POSTSUBSCRIPT italic_e ∈ italic_δ start_POSTSUBSCRIPT OPT end_POSTSUBSCRIPT ( over^ start_ARG italic_S end_ARG ) ∩ italic_H end_POSTSUBSCRIPT bold_italic_x ( italic_e ) + ∑ start_POSTSUBSCRIPT italic_e ∈ { italic_e start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT } start_POSTSUBSCRIPT italic_i ∈ [ italic_k start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT - italic_L start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ] , italic_j ∈ [ italic_k ] end_POSTSUBSCRIPT end_POSTSUBSCRIPT bold_italic_x ( italic_e ) ≥ italic_L start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT + ( italic_k start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT - italic_L start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) ⋅ italic_k ⋅ divide start_ARG 1 end_ARG start_ARG italic_k end_ARG = italic_k start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT = italic_h ( over^ start_ARG italic_S end_ARG ) .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S3.SS1.SSS2.7.p7.34">Note that the second inequality holds because <math alttext="\delta_{\textnormal{OPT}}(\hat{S})\cap H" class="ltx_Math" display="inline" id="S3.SS1.SSS2.7.p7.22.m1.1"><semantics id="S3.SS1.SSS2.7.p7.22.m1.1a"><mrow id="S3.SS1.SSS2.7.p7.22.m1.1.2" xref="S3.SS1.SSS2.7.p7.22.m1.1.2.cmml"><mrow id="S3.SS1.SSS2.7.p7.22.m1.1.2.2" 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xref="S3.SS1.SSS2.7.p7.23.m2.1.1.1.1">𝑘</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS2.7.p7.23.m2.4c">\{e^{i}_{j}\}_{i\in[k^{\prime}-L_{1}],j\in[k]}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS2.7.p7.23.m2.4d">{ italic_e start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT } start_POSTSUBSCRIPT italic_i ∈ [ italic_k start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT - italic_L start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ] , italic_j ∈ [ italic_k ] end_POSTSUBSCRIPT</annotation></semantics></math> are disjoint and no edge appears more than <math alttext="k^{\prime}-L_{1}" class="ltx_Math" display="inline" id="S3.SS1.SSS2.7.p7.24.m3.1"><semantics id="S3.SS1.SSS2.7.p7.24.m3.1a"><mrow id="S3.SS1.SSS2.7.p7.24.m3.1.1" xref="S3.SS1.SSS2.7.p7.24.m3.1.1.cmml"><msup id="S3.SS1.SSS2.7.p7.24.m3.1.1.2" xref="S3.SS1.SSS2.7.p7.24.m3.1.1.2.cmml"><mi id="S3.SS1.SSS2.7.p7.24.m3.1.1.2.2" xref="S3.SS1.SSS2.7.p7.24.m3.1.1.2.2.cmml">k</mi><mo id="S3.SS1.SSS2.7.p7.24.m3.1.1.2.3" xref="S3.SS1.SSS2.7.p7.24.m3.1.1.2.3.cmml">′</mo></msup><mo id="S3.SS1.SSS2.7.p7.24.m3.1.1.1" xref="S3.SS1.SSS2.7.p7.24.m3.1.1.1.cmml">−</mo><msub id="S3.SS1.SSS2.7.p7.24.m3.1.1.3" xref="S3.SS1.SSS2.7.p7.24.m3.1.1.3.cmml"><mi id="S3.SS1.SSS2.7.p7.24.m3.1.1.3.2" xref="S3.SS1.SSS2.7.p7.24.m3.1.1.3.2.cmml">L</mi><mn id="S3.SS1.SSS2.7.p7.24.m3.1.1.3.3" xref="S3.SS1.SSS2.7.p7.24.m3.1.1.3.3.cmml">1</mn></msub></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS2.7.p7.24.m3.1b"><apply id="S3.SS1.SSS2.7.p7.24.m3.1.1.cmml" xref="S3.SS1.SSS2.7.p7.24.m3.1.1"><minus id="S3.SS1.SSS2.7.p7.24.m3.1.1.1.cmml" xref="S3.SS1.SSS2.7.p7.24.m3.1.1.1"></minus><apply id="S3.SS1.SSS2.7.p7.24.m3.1.1.2.cmml" xref="S3.SS1.SSS2.7.p7.24.m3.1.1.2"><csymbol cd="ambiguous" id="S3.SS1.SSS2.7.p7.24.m3.1.1.2.1.cmml" xref="S3.SS1.SSS2.7.p7.24.m3.1.1.2">superscript</csymbol><ci id="S3.SS1.SSS2.7.p7.24.m3.1.1.2.2.cmml" xref="S3.SS1.SSS2.7.p7.24.m3.1.1.2.2">𝑘</ci><ci id="S3.SS1.SSS2.7.p7.24.m3.1.1.2.3.cmml" xref="S3.SS1.SSS2.7.p7.24.m3.1.1.2.3">′</ci></apply><apply id="S3.SS1.SSS2.7.p7.24.m3.1.1.3.cmml" xref="S3.SS1.SSS2.7.p7.24.m3.1.1.3"><csymbol cd="ambiguous" id="S3.SS1.SSS2.7.p7.24.m3.1.1.3.1.cmml" xref="S3.SS1.SSS2.7.p7.24.m3.1.1.3">subscript</csymbol><ci id="S3.SS1.SSS2.7.p7.24.m3.1.1.3.2.cmml" xref="S3.SS1.SSS2.7.p7.24.m3.1.1.3.2">𝐿</ci><cn id="S3.SS1.SSS2.7.p7.24.m3.1.1.3.3.cmml" type="integer" xref="S3.SS1.SSS2.7.p7.24.m3.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS2.7.p7.24.m3.1c">k^{\prime}-L_{1}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS2.7.p7.24.m3.1d">italic_k start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT - italic_L start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math> times in the second summation (i.e., <math 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id="S3.SS1.SSS2.7.p7.26.m5.5.5.1.1.1.1.2.2" xref="S3.SS1.SSS2.7.p7.26.m5.5.5.1.1.1.1.2.2.cmml">e</mi><mi id="S3.SS1.SSS2.7.p7.26.m5.5.5.1.1.1.1.3" xref="S3.SS1.SSS2.7.p7.26.m5.5.5.1.1.1.1.3.cmml">j</mi><mi id="S3.SS1.SSS2.7.p7.26.m5.5.5.1.1.1.1.2.3" xref="S3.SS1.SSS2.7.p7.26.m5.5.5.1.1.1.1.2.3.cmml">i</mi></msubsup><mo id="S3.SS1.SSS2.7.p7.26.m5.5.5.1.1.1.3" stretchy="false" xref="S3.SS1.SSS2.7.p7.26.m5.5.5.1.1.2.cmml">}</mo></mrow><mrow id="S3.SS1.SSS2.7.p7.26.m5.4.4.4.4" xref="S3.SS1.SSS2.7.p7.26.m5.4.4.4.5.cmml"><mrow id="S3.SS1.SSS2.7.p7.26.m5.3.3.3.3.1" xref="S3.SS1.SSS2.7.p7.26.m5.3.3.3.3.1.cmml"><mi id="S3.SS1.SSS2.7.p7.26.m5.3.3.3.3.1.2" xref="S3.SS1.SSS2.7.p7.26.m5.3.3.3.3.1.2.cmml">i</mi><mo id="S3.SS1.SSS2.7.p7.26.m5.3.3.3.3.1.1" xref="S3.SS1.SSS2.7.p7.26.m5.3.3.3.3.1.1.cmml">∈</mo><mrow id="S3.SS1.SSS2.7.p7.26.m5.3.3.3.3.1.3.2" xref="S3.SS1.SSS2.7.p7.26.m5.3.3.3.3.1.3.1.cmml"><mo id="S3.SS1.SSS2.7.p7.26.m5.3.3.3.3.1.3.2.1" stretchy="false" xref="S3.SS1.SSS2.7.p7.26.m5.3.3.3.3.1.3.1.1.cmml">[</mo><mi id="S3.SS1.SSS2.7.p7.26.m5.1.1.1.1" xref="S3.SS1.SSS2.7.p7.26.m5.1.1.1.1.cmml">L</mi><mo id="S3.SS1.SSS2.7.p7.26.m5.3.3.3.3.1.3.2.2" stretchy="false" xref="S3.SS1.SSS2.7.p7.26.m5.3.3.3.3.1.3.1.1.cmml">]</mo></mrow></mrow><mo id="S3.SS1.SSS2.7.p7.26.m5.4.4.4.4.3" xref="S3.SS1.SSS2.7.p7.26.m5.4.4.4.5a.cmml">,</mo><mrow id="S3.SS1.SSS2.7.p7.26.m5.4.4.4.4.2" xref="S3.SS1.SSS2.7.p7.26.m5.4.4.4.4.2.cmml"><mi id="S3.SS1.SSS2.7.p7.26.m5.4.4.4.4.2.2" xref="S3.SS1.SSS2.7.p7.26.m5.4.4.4.4.2.2.cmml">j</mi><mo id="S3.SS1.SSS2.7.p7.26.m5.4.4.4.4.2.1" xref="S3.SS1.SSS2.7.p7.26.m5.4.4.4.4.2.1.cmml">∈</mo><mrow id="S3.SS1.SSS2.7.p7.26.m5.4.4.4.4.2.3.2" xref="S3.SS1.SSS2.7.p7.26.m5.4.4.4.4.2.3.1.cmml"><mo id="S3.SS1.SSS2.7.p7.26.m5.4.4.4.4.2.3.2.1" stretchy="false" xref="S3.SS1.SSS2.7.p7.26.m5.4.4.4.4.2.3.1.1.cmml">[</mo><mi id="S3.SS1.SSS2.7.p7.26.m5.2.2.2.2" xref="S3.SS1.SSS2.7.p7.26.m5.2.2.2.2.cmml">k</mi><mo id="S3.SS1.SSS2.7.p7.26.m5.4.4.4.4.2.3.2.2" stretchy="false" xref="S3.SS1.SSS2.7.p7.26.m5.4.4.4.4.2.3.1.1.cmml">]</mo></mrow></mrow></mrow></msub></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS2.7.p7.26.m5.5b"><apply id="S3.SS1.SSS2.7.p7.26.m5.5.5.cmml" xref="S3.SS1.SSS2.7.p7.26.m5.5.5"><in id="S3.SS1.SSS2.7.p7.26.m5.5.5.2.cmml" xref="S3.SS1.SSS2.7.p7.26.m5.5.5.2"></in><apply id="S3.SS1.SSS2.7.p7.26.m5.5.5.3.cmml" xref="S3.SS1.SSS2.7.p7.26.m5.5.5.3"><csymbol cd="ambiguous" id="S3.SS1.SSS2.7.p7.26.m5.5.5.3.1.cmml" xref="S3.SS1.SSS2.7.p7.26.m5.5.5.3">superscript</csymbol><ci id="S3.SS1.SSS2.7.p7.26.m5.5.5.3.2.cmml" xref="S3.SS1.SSS2.7.p7.26.m5.5.5.3.2">𝑒</ci><ci id="S3.SS1.SSS2.7.p7.26.m5.5.5.3.3.cmml" xref="S3.SS1.SSS2.7.p7.26.m5.5.5.3.3">′</ci></apply><apply id="S3.SS1.SSS2.7.p7.26.m5.5.5.1.cmml" xref="S3.SS1.SSS2.7.p7.26.m5.5.5.1"><csymbol cd="ambiguous" id="S3.SS1.SSS2.7.p7.26.m5.5.5.1.2.cmml" xref="S3.SS1.SSS2.7.p7.26.m5.5.5.1">subscript</csymbol><set id="S3.SS1.SSS2.7.p7.26.m5.5.5.1.1.2.cmml" xref="S3.SS1.SSS2.7.p7.26.m5.5.5.1.1.1"><apply id="S3.SS1.SSS2.7.p7.26.m5.5.5.1.1.1.1.cmml" xref="S3.SS1.SSS2.7.p7.26.m5.5.5.1.1.1.1"><csymbol cd="ambiguous" id="S3.SS1.SSS2.7.p7.26.m5.5.5.1.1.1.1.1.cmml" xref="S3.SS1.SSS2.7.p7.26.m5.5.5.1.1.1.1">subscript</csymbol><apply id="S3.SS1.SSS2.7.p7.26.m5.5.5.1.1.1.1.2.cmml" xref="S3.SS1.SSS2.7.p7.26.m5.5.5.1.1.1.1"><csymbol cd="ambiguous" id="S3.SS1.SSS2.7.p7.26.m5.5.5.1.1.1.1.2.1.cmml" xref="S3.SS1.SSS2.7.p7.26.m5.5.5.1.1.1.1">superscript</csymbol><ci id="S3.SS1.SSS2.7.p7.26.m5.5.5.1.1.1.1.2.2.cmml" xref="S3.SS1.SSS2.7.p7.26.m5.5.5.1.1.1.1.2.2">𝑒</ci><ci id="S3.SS1.SSS2.7.p7.26.m5.5.5.1.1.1.1.2.3.cmml" xref="S3.SS1.SSS2.7.p7.26.m5.5.5.1.1.1.1.2.3">𝑖</ci></apply><ci id="S3.SS1.SSS2.7.p7.26.m5.5.5.1.1.1.1.3.cmml" xref="S3.SS1.SSS2.7.p7.26.m5.5.5.1.1.1.1.3">𝑗</ci></apply></set><apply id="S3.SS1.SSS2.7.p7.26.m5.4.4.4.5.cmml" xref="S3.SS1.SSS2.7.p7.26.m5.4.4.4.4"><csymbol cd="ambiguous" id="S3.SS1.SSS2.7.p7.26.m5.4.4.4.5a.cmml" xref="S3.SS1.SSS2.7.p7.26.m5.4.4.4.4.3">formulae-sequence</csymbol><apply id="S3.SS1.SSS2.7.p7.26.m5.3.3.3.3.1.cmml" xref="S3.SS1.SSS2.7.p7.26.m5.3.3.3.3.1"><in id="S3.SS1.SSS2.7.p7.26.m5.3.3.3.3.1.1.cmml" xref="S3.SS1.SSS2.7.p7.26.m5.3.3.3.3.1.1"></in><ci id="S3.SS1.SSS2.7.p7.26.m5.3.3.3.3.1.2.cmml" xref="S3.SS1.SSS2.7.p7.26.m5.3.3.3.3.1.2">𝑖</ci><apply id="S3.SS1.SSS2.7.p7.26.m5.3.3.3.3.1.3.1.cmml" xref="S3.SS1.SSS2.7.p7.26.m5.3.3.3.3.1.3.2"><csymbol cd="latexml" id="S3.SS1.SSS2.7.p7.26.m5.3.3.3.3.1.3.1.1.cmml" xref="S3.SS1.SSS2.7.p7.26.m5.3.3.3.3.1.3.2.1">delimited-[]</csymbol><ci id="S3.SS1.SSS2.7.p7.26.m5.1.1.1.1.cmml" xref="S3.SS1.SSS2.7.p7.26.m5.1.1.1.1">𝐿</ci></apply></apply><apply id="S3.SS1.SSS2.7.p7.26.m5.4.4.4.4.2.cmml" xref="S3.SS1.SSS2.7.p7.26.m5.4.4.4.4.2"><in id="S3.SS1.SSS2.7.p7.26.m5.4.4.4.4.2.1.cmml" xref="S3.SS1.SSS2.7.p7.26.m5.4.4.4.4.2.1"></in><ci id="S3.SS1.SSS2.7.p7.26.m5.4.4.4.4.2.2.cmml" xref="S3.SS1.SSS2.7.p7.26.m5.4.4.4.4.2.2">𝑗</ci><apply id="S3.SS1.SSS2.7.p7.26.m5.4.4.4.4.2.3.1.cmml" xref="S3.SS1.SSS2.7.p7.26.m5.4.4.4.4.2.3.2"><csymbol cd="latexml" id="S3.SS1.SSS2.7.p7.26.m5.4.4.4.4.2.3.1.1.cmml" xref="S3.SS1.SSS2.7.p7.26.m5.4.4.4.4.2.3.2.1">delimited-[]</csymbol><ci id="S3.SS1.SSS2.7.p7.26.m5.2.2.2.2.cmml" xref="S3.SS1.SSS2.7.p7.26.m5.2.2.2.2">𝑘</ci></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS2.7.p7.26.m5.5c">e^{\prime}\in\{e^{i}_{j}\}_{i\in[L],j\in[k]}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS2.7.p7.26.m5.5d">italic_e start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ∈ { italic_e start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT } start_POSTSUBSCRIPT italic_i ∈ [ italic_L ] , italic_j ∈ [ italic_k ] end_POSTSUBSCRIPT</annotation></semantics></math>, its <math alttext="\boldsymbol{x}" class="ltx_Math" display="inline" id="S3.SS1.SSS2.7.p7.27.m6.1"><semantics id="S3.SS1.SSS2.7.p7.27.m6.1a"><mi id="S3.SS1.SSS2.7.p7.27.m6.1.1" xref="S3.SS1.SSS2.7.p7.27.m6.1.1.cmml">𝒙</mi><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS2.7.p7.27.m6.1b"><ci id="S3.SS1.SSS2.7.p7.27.m6.1.1.cmml" xref="S3.SS1.SSS2.7.p7.27.m6.1.1">𝒙</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS2.7.p7.27.m6.1c">\boldsymbol{x}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS2.7.p7.27.m6.1d">bold_italic_x</annotation></semantics></math> value is at most <math alttext="\min(1,c(e^{\prime})/k)=c(e^{\prime})/k" class="ltx_Math" display="inline" id="S3.SS1.SSS2.7.p7.28.m7.4"><semantics id="S3.SS1.SSS2.7.p7.28.m7.4a"><mrow id="S3.SS1.SSS2.7.p7.28.m7.4.4" xref="S3.SS1.SSS2.7.p7.28.m7.4.4.cmml"><mrow id="S3.SS1.SSS2.7.p7.28.m7.3.3.1.1" xref="S3.SS1.SSS2.7.p7.28.m7.3.3.1.2.cmml"><mi id="S3.SS1.SSS2.7.p7.28.m7.1.1" xref="S3.SS1.SSS2.7.p7.28.m7.1.1.cmml">min</mi><mo id="S3.SS1.SSS2.7.p7.28.m7.3.3.1.1a" xref="S3.SS1.SSS2.7.p7.28.m7.3.3.1.2.cmml">⁡</mo><mrow id="S3.SS1.SSS2.7.p7.28.m7.3.3.1.1.1" xref="S3.SS1.SSS2.7.p7.28.m7.3.3.1.2.cmml"><mo id="S3.SS1.SSS2.7.p7.28.m7.3.3.1.1.1.2" stretchy="false" xref="S3.SS1.SSS2.7.p7.28.m7.3.3.1.2.cmml">(</mo><mn id="S3.SS1.SSS2.7.p7.28.m7.2.2" xref="S3.SS1.SSS2.7.p7.28.m7.2.2.cmml">1</mn><mo id="S3.SS1.SSS2.7.p7.28.m7.3.3.1.1.1.3" xref="S3.SS1.SSS2.7.p7.28.m7.3.3.1.2.cmml">,</mo><mrow id="S3.SS1.SSS2.7.p7.28.m7.3.3.1.1.1.1" xref="S3.SS1.SSS2.7.p7.28.m7.3.3.1.1.1.1.cmml"><mrow id="S3.SS1.SSS2.7.p7.28.m7.3.3.1.1.1.1.1" xref="S3.SS1.SSS2.7.p7.28.m7.3.3.1.1.1.1.1.cmml"><mi id="S3.SS1.SSS2.7.p7.28.m7.3.3.1.1.1.1.1.3" xref="S3.SS1.SSS2.7.p7.28.m7.3.3.1.1.1.1.1.3.cmml">c</mi><mo id="S3.SS1.SSS2.7.p7.28.m7.3.3.1.1.1.1.1.2" xref="S3.SS1.SSS2.7.p7.28.m7.3.3.1.1.1.1.1.2.cmml">⁢</mo><mrow id="S3.SS1.SSS2.7.p7.28.m7.3.3.1.1.1.1.1.1.1" xref="S3.SS1.SSS2.7.p7.28.m7.3.3.1.1.1.1.1.1.1.1.cmml"><mo 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xref="S3.SS1.SSS2.7.p7.28.m7.4.4.2.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S3.SS1.SSS2.7.p7.28.m7.4.4.2.2" xref="S3.SS1.SSS2.7.p7.28.m7.4.4.2.2.cmml">/</mo><mi id="S3.SS1.SSS2.7.p7.28.m7.4.4.2.3" xref="S3.SS1.SSS2.7.p7.28.m7.4.4.2.3.cmml">k</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS2.7.p7.28.m7.4b"><apply id="S3.SS1.SSS2.7.p7.28.m7.4.4.cmml" xref="S3.SS1.SSS2.7.p7.28.m7.4.4"><eq id="S3.SS1.SSS2.7.p7.28.m7.4.4.3.cmml" xref="S3.SS1.SSS2.7.p7.28.m7.4.4.3"></eq><apply id="S3.SS1.SSS2.7.p7.28.m7.3.3.1.2.cmml" xref="S3.SS1.SSS2.7.p7.28.m7.3.3.1.1"><min id="S3.SS1.SSS2.7.p7.28.m7.1.1.cmml" xref="S3.SS1.SSS2.7.p7.28.m7.1.1"></min><cn id="S3.SS1.SSS2.7.p7.28.m7.2.2.cmml" type="integer" xref="S3.SS1.SSS2.7.p7.28.m7.2.2">1</cn><apply id="S3.SS1.SSS2.7.p7.28.m7.3.3.1.1.1.1.cmml" xref="S3.SS1.SSS2.7.p7.28.m7.3.3.1.1.1.1"><divide id="S3.SS1.SSS2.7.p7.28.m7.3.3.1.1.1.1.2.cmml" xref="S3.SS1.SSS2.7.p7.28.m7.3.3.1.1.1.1.2"></divide><apply id="S3.SS1.SSS2.7.p7.28.m7.3.3.1.1.1.1.1.cmml" xref="S3.SS1.SSS2.7.p7.28.m7.3.3.1.1.1.1.1"><times id="S3.SS1.SSS2.7.p7.28.m7.3.3.1.1.1.1.1.2.cmml" xref="S3.SS1.SSS2.7.p7.28.m7.3.3.1.1.1.1.1.2"></times><ci id="S3.SS1.SSS2.7.p7.28.m7.3.3.1.1.1.1.1.3.cmml" xref="S3.SS1.SSS2.7.p7.28.m7.3.3.1.1.1.1.1.3">𝑐</ci><apply id="S3.SS1.SSS2.7.p7.28.m7.3.3.1.1.1.1.1.1.1.1.cmml" xref="S3.SS1.SSS2.7.p7.28.m7.3.3.1.1.1.1.1.1.1"><csymbol cd="ambiguous" id="S3.SS1.SSS2.7.p7.28.m7.3.3.1.1.1.1.1.1.1.1.1.cmml" xref="S3.SS1.SSS2.7.p7.28.m7.3.3.1.1.1.1.1.1.1">superscript</csymbol><ci id="S3.SS1.SSS2.7.p7.28.m7.3.3.1.1.1.1.1.1.1.1.2.cmml" xref="S3.SS1.SSS2.7.p7.28.m7.3.3.1.1.1.1.1.1.1.1.2">𝑒</ci><ci id="S3.SS1.SSS2.7.p7.28.m7.3.3.1.1.1.1.1.1.1.1.3.cmml" xref="S3.SS1.SSS2.7.p7.28.m7.3.3.1.1.1.1.1.1.1.1.3">′</ci></apply></apply><ci id="S3.SS1.SSS2.7.p7.28.m7.3.3.1.1.1.1.3.cmml" xref="S3.SS1.SSS2.7.p7.28.m7.3.3.1.1.1.1.3">𝑘</ci></apply></apply><apply id="S3.SS1.SSS2.7.p7.28.m7.4.4.2.cmml" xref="S3.SS1.SSS2.7.p7.28.m7.4.4.2"><divide id="S3.SS1.SSS2.7.p7.28.m7.4.4.2.2.cmml" xref="S3.SS1.SSS2.7.p7.28.m7.4.4.2.2"></divide><apply id="S3.SS1.SSS2.7.p7.28.m7.4.4.2.1.cmml" xref="S3.SS1.SSS2.7.p7.28.m7.4.4.2.1"><times id="S3.SS1.SSS2.7.p7.28.m7.4.4.2.1.2.cmml" xref="S3.SS1.SSS2.7.p7.28.m7.4.4.2.1.2"></times><ci id="S3.SS1.SSS2.7.p7.28.m7.4.4.2.1.3.cmml" xref="S3.SS1.SSS2.7.p7.28.m7.4.4.2.1.3">𝑐</ci><apply id="S3.SS1.SSS2.7.p7.28.m7.4.4.2.1.1.1.1.cmml" xref="S3.SS1.SSS2.7.p7.28.m7.4.4.2.1.1.1"><csymbol cd="ambiguous" id="S3.SS1.SSS2.7.p7.28.m7.4.4.2.1.1.1.1.1.cmml" xref="S3.SS1.SSS2.7.p7.28.m7.4.4.2.1.1.1">superscript</csymbol><ci id="S3.SS1.SSS2.7.p7.28.m7.4.4.2.1.1.1.1.2.cmml" xref="S3.SS1.SSS2.7.p7.28.m7.4.4.2.1.1.1.1.2">𝑒</ci><ci id="S3.SS1.SSS2.7.p7.28.m7.4.4.2.1.1.1.1.3.cmml" xref="S3.SS1.SSS2.7.p7.28.m7.4.4.2.1.1.1.1.3">′</ci></apply></apply><ci id="S3.SS1.SSS2.7.p7.28.m7.4.4.2.3.cmml" xref="S3.SS1.SSS2.7.p7.28.m7.4.4.2.3">𝑘</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS2.7.p7.28.m7.4c">\min(1,c(e^{\prime})/k)=c(e^{\prime})/k</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS2.7.p7.28.m7.4d">roman_min ( 1 , italic_c ( italic_e start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) / italic_k ) = italic_c ( italic_e start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) / italic_k</annotation></semantics></math>, because <math alttext="c(e^{\prime})" class="ltx_Math" display="inline" id="S3.SS1.SSS2.7.p7.29.m8.1"><semantics id="S3.SS1.SSS2.7.p7.29.m8.1a"><mrow id="S3.SS1.SSS2.7.p7.29.m8.1.1" xref="S3.SS1.SSS2.7.p7.29.m8.1.1.cmml"><mi id="S3.SS1.SSS2.7.p7.29.m8.1.1.3" xref="S3.SS1.SSS2.7.p7.29.m8.1.1.3.cmml">c</mi><mo id="S3.SS1.SSS2.7.p7.29.m8.1.1.2" xref="S3.SS1.SSS2.7.p7.29.m8.1.1.2.cmml">⁢</mo><mrow id="S3.SS1.SSS2.7.p7.29.m8.1.1.1.1" xref="S3.SS1.SSS2.7.p7.29.m8.1.1.1.1.1.cmml"><mo id="S3.SS1.SSS2.7.p7.29.m8.1.1.1.1.2" stretchy="false" xref="S3.SS1.SSS2.7.p7.29.m8.1.1.1.1.1.cmml">(</mo><msup id="S3.SS1.SSS2.7.p7.29.m8.1.1.1.1.1" xref="S3.SS1.SSS2.7.p7.29.m8.1.1.1.1.1.cmml"><mi id="S3.SS1.SSS2.7.p7.29.m8.1.1.1.1.1.2" xref="S3.SS1.SSS2.7.p7.29.m8.1.1.1.1.1.2.cmml">e</mi><mo id="S3.SS1.SSS2.7.p7.29.m8.1.1.1.1.1.3" xref="S3.SS1.SSS2.7.p7.29.m8.1.1.1.1.1.3.cmml">′</mo></msup><mo id="S3.SS1.SSS2.7.p7.29.m8.1.1.1.1.3" stretchy="false" xref="S3.SS1.SSS2.7.p7.29.m8.1.1.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS2.7.p7.29.m8.1b"><apply id="S3.SS1.SSS2.7.p7.29.m8.1.1.cmml" xref="S3.SS1.SSS2.7.p7.29.m8.1.1"><times id="S3.SS1.SSS2.7.p7.29.m8.1.1.2.cmml" xref="S3.SS1.SSS2.7.p7.29.m8.1.1.2"></times><ci id="S3.SS1.SSS2.7.p7.29.m8.1.1.3.cmml" xref="S3.SS1.SSS2.7.p7.29.m8.1.1.3">𝑐</ci><apply id="S3.SS1.SSS2.7.p7.29.m8.1.1.1.1.1.cmml" xref="S3.SS1.SSS2.7.p7.29.m8.1.1.1.1"><csymbol cd="ambiguous" id="S3.SS1.SSS2.7.p7.29.m8.1.1.1.1.1.1.cmml" xref="S3.SS1.SSS2.7.p7.29.m8.1.1.1.1">superscript</csymbol><ci id="S3.SS1.SSS2.7.p7.29.m8.1.1.1.1.1.2.cmml" xref="S3.SS1.SSS2.7.p7.29.m8.1.1.1.1.1.2">𝑒</ci><ci id="S3.SS1.SSS2.7.p7.29.m8.1.1.1.1.1.3.cmml" xref="S3.SS1.SSS2.7.p7.29.m8.1.1.1.1.1.3">′</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS2.7.p7.29.m8.1c">c(e^{\prime})</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS2.7.p7.29.m8.1d">italic_c ( italic_e start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT )</annotation></semantics></math> which is defined as the number of times <math alttext="e^{\prime}" class="ltx_Math" display="inline" id="S3.SS1.SSS2.7.p7.30.m9.1"><semantics id="S3.SS1.SSS2.7.p7.30.m9.1a"><msup id="S3.SS1.SSS2.7.p7.30.m9.1.1" xref="S3.SS1.SSS2.7.p7.30.m9.1.1.cmml"><mi id="S3.SS1.SSS2.7.p7.30.m9.1.1.2" xref="S3.SS1.SSS2.7.p7.30.m9.1.1.2.cmml">e</mi><mo id="S3.SS1.SSS2.7.p7.30.m9.1.1.3" xref="S3.SS1.SSS2.7.p7.30.m9.1.1.3.cmml">′</mo></msup><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS2.7.p7.30.m9.1b"><apply id="S3.SS1.SSS2.7.p7.30.m9.1.1.cmml" xref="S3.SS1.SSS2.7.p7.30.m9.1.1"><csymbol cd="ambiguous" id="S3.SS1.SSS2.7.p7.30.m9.1.1.1.cmml" xref="S3.SS1.SSS2.7.p7.30.m9.1.1">superscript</csymbol><ci id="S3.SS1.SSS2.7.p7.30.m9.1.1.2.cmml" xref="S3.SS1.SSS2.7.p7.30.m9.1.1.2">𝑒</ci><ci id="S3.SS1.SSS2.7.p7.30.m9.1.1.3.cmml" xref="S3.SS1.SSS2.7.p7.30.m9.1.1.3">′</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS2.7.p7.30.m9.1c">e^{\prime}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS2.7.p7.30.m9.1d">italic_e start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math> appears in the collection <math alttext="\{e^{i}_{j}\big{\}}_{i\in[k^{\prime}-L_{1}],j\in[k]}" class="ltx_Math" display="inline" id="S3.SS1.SSS2.7.p7.31.m10.4"><semantics id="S3.SS1.SSS2.7.p7.31.m10.4a"><msub id="S3.SS1.SSS2.7.p7.31.m10.4.4" xref="S3.SS1.SSS2.7.p7.31.m10.4.4.cmml"><mrow id="S3.SS1.SSS2.7.p7.31.m10.4.4.1.1" xref="S3.SS1.SSS2.7.p7.31.m10.4.4.1.2.cmml"><mo 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xref="S3.SS1.SSS2.7.p7.32.m11.1.1.4">superscript</csymbol><ci id="S3.SS1.SSS2.7.p7.32.m11.1.1.4.2.cmml" xref="S3.SS1.SSS2.7.p7.32.m11.1.1.4.2">𝑘</ci><ci id="S3.SS1.SSS2.7.p7.32.m11.1.1.4.3.cmml" xref="S3.SS1.SSS2.7.p7.32.m11.1.1.4.3">′</ci></apply></apply><apply id="S3.SS1.SSS2.7.p7.32.m11.1.1c.cmml" xref="S3.SS1.SSS2.7.p7.32.m11.1.1"><leq id="S3.SS1.SSS2.7.p7.32.m11.1.1.5.cmml" xref="S3.SS1.SSS2.7.p7.32.m11.1.1.5"></leq><share href="https://arxiv.org/html/2503.00712v1#S3.SS1.SSS2.7.p7.32.m11.1.1.4.cmml" id="S3.SS1.SSS2.7.p7.32.m11.1.1d.cmml" xref="S3.SS1.SSS2.7.p7.32.m11.1.1"></share><ci id="S3.SS1.SSS2.7.p7.32.m11.1.1.6.cmml" xref="S3.SS1.SSS2.7.p7.32.m11.1.1.6">𝑘</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS2.7.p7.32.m11.1c">k^{\prime}-L_{1}\leq k^{\prime}\leq k</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS2.7.p7.32.m11.1d">italic_k start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT - italic_L start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ≤ italic_k start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ≤ italic_k</annotation></semantics></math>. So, <math alttext="\boldsymbol{x}" class="ltx_Math" display="inline" id="S3.SS1.SSS2.7.p7.33.m12.1"><semantics id="S3.SS1.SSS2.7.p7.33.m12.1a"><mi id="S3.SS1.SSS2.7.p7.33.m12.1.1" xref="S3.SS1.SSS2.7.p7.33.m12.1.1.cmml">𝒙</mi><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS2.7.p7.33.m12.1b"><ci id="S3.SS1.SSS2.7.p7.33.m12.1.1.cmml" xref="S3.SS1.SSS2.7.p7.33.m12.1.1">𝒙</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS2.7.p7.33.m12.1c">\boldsymbol{x}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS2.7.p7.33.m12.1d">bold_italic_x</annotation></semantics></math> is a feasible solution for <a class="ltx_ref" href="https://arxiv.org/html/2503.00712v1#S3.F1" title="Figure 1 ‣ 3.1.2 An Improved Analysis via Fractional Solutions ‣ 3.1 Vertex Connectivity Network Design ‣ 3 Generic Framework for Streaming Algorithms for Network Design ‣ Streaming Algorithms for Network Design">VC-SNDP-LP</a>(<math alttext="G,w,h" class="ltx_Math" display="inline" id="S3.SS1.SSS2.7.p7.34.m13.3"><semantics id="S3.SS1.SSS2.7.p7.34.m13.3a"><mrow id="S3.SS1.SSS2.7.p7.34.m13.3.4.2" xref="S3.SS1.SSS2.7.p7.34.m13.3.4.1.cmml"><mi id="S3.SS1.SSS2.7.p7.34.m13.1.1" xref="S3.SS1.SSS2.7.p7.34.m13.1.1.cmml">G</mi><mo id="S3.SS1.SSS2.7.p7.34.m13.3.4.2.1" xref="S3.SS1.SSS2.7.p7.34.m13.3.4.1.cmml">,</mo><mi id="S3.SS1.SSS2.7.p7.34.m13.2.2" xref="S3.SS1.SSS2.7.p7.34.m13.2.2.cmml">w</mi><mo id="S3.SS1.SSS2.7.p7.34.m13.3.4.2.2" xref="S3.SS1.SSS2.7.p7.34.m13.3.4.1.cmml">,</mo><mi id="S3.SS1.SSS2.7.p7.34.m13.3.3" xref="S3.SS1.SSS2.7.p7.34.m13.3.3.cmml">h</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS2.7.p7.34.m13.3b"><list id="S3.SS1.SSS2.7.p7.34.m13.3.4.1.cmml" xref="S3.SS1.SSS2.7.p7.34.m13.3.4.2"><ci id="S3.SS1.SSS2.7.p7.34.m13.1.1.cmml" xref="S3.SS1.SSS2.7.p7.34.m13.1.1">𝐺</ci><ci id="S3.SS1.SSS2.7.p7.34.m13.2.2.cmml" xref="S3.SS1.SSS2.7.p7.34.m13.2.2">𝑤</ci><ci id="S3.SS1.SSS2.7.p7.34.m13.3.3.cmml" xref="S3.SS1.SSS2.7.p7.34.m13.3.3">ℎ</ci></list></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS2.7.p7.34.m13.3c">G,w,h</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS2.7.p7.34.m13.3d">italic_G , italic_w , italic_h</annotation></semantics></math>).</p> </div> <div class="ltx_para" id="S3.SS1.SSS2.8.p8"> <p class="ltx_p" id="S3.SS1.SSS2.8.p8.7">For the cost analysis, note that for every edge <math alttext="e=(u,v)\in\textnormal{OPT}" class="ltx_Math" display="inline" id="S3.SS1.SSS2.8.p8.1.m1.2"><semantics id="S3.SS1.SSS2.8.p8.1.m1.2a"><mrow id="S3.SS1.SSS2.8.p8.1.m1.2.3" xref="S3.SS1.SSS2.8.p8.1.m1.2.3.cmml"><mi id="S3.SS1.SSS2.8.p8.1.m1.2.3.2" xref="S3.SS1.SSS2.8.p8.1.m1.2.3.2.cmml">e</mi><mo id="S3.SS1.SSS2.8.p8.1.m1.2.3.3" xref="S3.SS1.SSS2.8.p8.1.m1.2.3.3.cmml">=</mo><mrow id="S3.SS1.SSS2.8.p8.1.m1.2.3.4.2" xref="S3.SS1.SSS2.8.p8.1.m1.2.3.4.1.cmml"><mo id="S3.SS1.SSS2.8.p8.1.m1.2.3.4.2.1" stretchy="false" xref="S3.SS1.SSS2.8.p8.1.m1.2.3.4.1.cmml">(</mo><mi id="S3.SS1.SSS2.8.p8.1.m1.1.1" xref="S3.SS1.SSS2.8.p8.1.m1.1.1.cmml">u</mi><mo id="S3.SS1.SSS2.8.p8.1.m1.2.3.4.2.2" xref="S3.SS1.SSS2.8.p8.1.m1.2.3.4.1.cmml">,</mo><mi id="S3.SS1.SSS2.8.p8.1.m1.2.2" xref="S3.SS1.SSS2.8.p8.1.m1.2.2.cmml">v</mi><mo id="S3.SS1.SSS2.8.p8.1.m1.2.3.4.2.3" stretchy="false" xref="S3.SS1.SSS2.8.p8.1.m1.2.3.4.1.cmml">)</mo></mrow><mo id="S3.SS1.SSS2.8.p8.1.m1.2.3.5" xref="S3.SS1.SSS2.8.p8.1.m1.2.3.5.cmml">∈</mo><mtext id="S3.SS1.SSS2.8.p8.1.m1.2.3.6" xref="S3.SS1.SSS2.8.p8.1.m1.2.3.6a.cmml">OPT</mtext></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS2.8.p8.1.m1.2b"><apply id="S3.SS1.SSS2.8.p8.1.m1.2.3.cmml" xref="S3.SS1.SSS2.8.p8.1.m1.2.3"><and id="S3.SS1.SSS2.8.p8.1.m1.2.3a.cmml" xref="S3.SS1.SSS2.8.p8.1.m1.2.3"></and><apply id="S3.SS1.SSS2.8.p8.1.m1.2.3b.cmml" xref="S3.SS1.SSS2.8.p8.1.m1.2.3"><eq id="S3.SS1.SSS2.8.p8.1.m1.2.3.3.cmml" xref="S3.SS1.SSS2.8.p8.1.m1.2.3.3"></eq><ci id="S3.SS1.SSS2.8.p8.1.m1.2.3.2.cmml" xref="S3.SS1.SSS2.8.p8.1.m1.2.3.2">𝑒</ci><interval closure="open" id="S3.SS1.SSS2.8.p8.1.m1.2.3.4.1.cmml" xref="S3.SS1.SSS2.8.p8.1.m1.2.3.4.2"><ci id="S3.SS1.SSS2.8.p8.1.m1.1.1.cmml" xref="S3.SS1.SSS2.8.p8.1.m1.1.1">𝑢</ci><ci id="S3.SS1.SSS2.8.p8.1.m1.2.2.cmml" xref="S3.SS1.SSS2.8.p8.1.m1.2.2">𝑣</ci></interval></apply><apply id="S3.SS1.SSS2.8.p8.1.m1.2.3c.cmml" xref="S3.SS1.SSS2.8.p8.1.m1.2.3"><in id="S3.SS1.SSS2.8.p8.1.m1.2.3.5.cmml" xref="S3.SS1.SSS2.8.p8.1.m1.2.3.5"></in><share href="https://arxiv.org/html/2503.00712v1#S3.SS1.SSS2.8.p8.1.m1.2.3.4.cmml" id="S3.SS1.SSS2.8.p8.1.m1.2.3d.cmml" xref="S3.SS1.SSS2.8.p8.1.m1.2.3"></share><ci id="S3.SS1.SSS2.8.p8.1.m1.2.3.6a.cmml" xref="S3.SS1.SSS2.8.p8.1.m1.2.3.6"><mtext id="S3.SS1.SSS2.8.p8.1.m1.2.3.6.cmml" xref="S3.SS1.SSS2.8.p8.1.m1.2.3.6">OPT</mtext></ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS2.8.p8.1.m1.2c">e=(u,v)\in\textnormal{OPT}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS2.8.p8.1.m1.2d">italic_e = ( italic_u , italic_v ) ∈ OPT</annotation></semantics></math> in weight class <math alttext="B_{j}" class="ltx_Math" display="inline" id="S3.SS1.SSS2.8.p8.2.m2.1"><semantics id="S3.SS1.SSS2.8.p8.2.m2.1a"><msub id="S3.SS1.SSS2.8.p8.2.m2.1.1" xref="S3.SS1.SSS2.8.p8.2.m2.1.1.cmml"><mi id="S3.SS1.SSS2.8.p8.2.m2.1.1.2" xref="S3.SS1.SSS2.8.p8.2.m2.1.1.2.cmml">B</mi><mi id="S3.SS1.SSS2.8.p8.2.m2.1.1.3" xref="S3.SS1.SSS2.8.p8.2.m2.1.1.3.cmml">j</mi></msub><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS2.8.p8.2.m2.1b"><apply id="S3.SS1.SSS2.8.p8.2.m2.1.1.cmml" xref="S3.SS1.SSS2.8.p8.2.m2.1.1"><csymbol cd="ambiguous" id="S3.SS1.SSS2.8.p8.2.m2.1.1.1.cmml" xref="S3.SS1.SSS2.8.p8.2.m2.1.1">subscript</csymbol><ci id="S3.SS1.SSS2.8.p8.2.m2.1.1.2.cmml" xref="S3.SS1.SSS2.8.p8.2.m2.1.1.2">𝐵</ci><ci id="S3.SS1.SSS2.8.p8.2.m2.1.1.3.cmml" xref="S3.SS1.SSS2.8.p8.2.m2.1.1.3">𝑗</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS2.8.p8.2.m2.1c">B_{j}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS2.8.p8.2.m2.1d">italic_B start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT</annotation></semantics></math>, either <math alttext="\boldsymbol{x}(e)=1" class="ltx_Math" display="inline" id="S3.SS1.SSS2.8.p8.3.m3.1"><semantics id="S3.SS1.SSS2.8.p8.3.m3.1a"><mrow id="S3.SS1.SSS2.8.p8.3.m3.1.2" xref="S3.SS1.SSS2.8.p8.3.m3.1.2.cmml"><mrow id="S3.SS1.SSS2.8.p8.3.m3.1.2.2" xref="S3.SS1.SSS2.8.p8.3.m3.1.2.2.cmml"><mi id="S3.SS1.SSS2.8.p8.3.m3.1.2.2.2" xref="S3.SS1.SSS2.8.p8.3.m3.1.2.2.2.cmml">𝒙</mi><mo id="S3.SS1.SSS2.8.p8.3.m3.1.2.2.1" xref="S3.SS1.SSS2.8.p8.3.m3.1.2.2.1.cmml">⁢</mo><mrow id="S3.SS1.SSS2.8.p8.3.m3.1.2.2.3.2" xref="S3.SS1.SSS2.8.p8.3.m3.1.2.2.cmml"><mo id="S3.SS1.SSS2.8.p8.3.m3.1.2.2.3.2.1" stretchy="false" xref="S3.SS1.SSS2.8.p8.3.m3.1.2.2.cmml">(</mo><mi id="S3.SS1.SSS2.8.p8.3.m3.1.1" xref="S3.SS1.SSS2.8.p8.3.m3.1.1.cmml">e</mi><mo id="S3.SS1.SSS2.8.p8.3.m3.1.2.2.3.2.2" stretchy="false" xref="S3.SS1.SSS2.8.p8.3.m3.1.2.2.cmml">)</mo></mrow></mrow><mo id="S3.SS1.SSS2.8.p8.3.m3.1.2.1" xref="S3.SS1.SSS2.8.p8.3.m3.1.2.1.cmml">=</mo><mn id="S3.SS1.SSS2.8.p8.3.m3.1.2.3" xref="S3.SS1.SSS2.8.p8.3.m3.1.2.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS2.8.p8.3.m3.1b"><apply id="S3.SS1.SSS2.8.p8.3.m3.1.2.cmml" xref="S3.SS1.SSS2.8.p8.3.m3.1.2"><eq id="S3.SS1.SSS2.8.p8.3.m3.1.2.1.cmml" xref="S3.SS1.SSS2.8.p8.3.m3.1.2.1"></eq><apply id="S3.SS1.SSS2.8.p8.3.m3.1.2.2.cmml" xref="S3.SS1.SSS2.8.p8.3.m3.1.2.2"><times id="S3.SS1.SSS2.8.p8.3.m3.1.2.2.1.cmml" xref="S3.SS1.SSS2.8.p8.3.m3.1.2.2.1"></times><ci id="S3.SS1.SSS2.8.p8.3.m3.1.2.2.2.cmml" xref="S3.SS1.SSS2.8.p8.3.m3.1.2.2.2">𝒙</ci><ci id="S3.SS1.SSS2.8.p8.3.m3.1.1.cmml" xref="S3.SS1.SSS2.8.p8.3.m3.1.1">𝑒</ci></apply><cn id="S3.SS1.SSS2.8.p8.3.m3.1.2.3.cmml" type="integer" xref="S3.SS1.SSS2.8.p8.3.m3.1.2.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS2.8.p8.3.m3.1c">\boldsymbol{x}(e)=1</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS2.8.p8.3.m3.1d">bold_italic_x ( italic_e ) = 1</annotation></semantics></math>, or it contributes in increasing the <math alttext="\boldsymbol{x}" class="ltx_Math" display="inline" id="S3.SS1.SSS2.8.p8.4.m4.1"><semantics id="S3.SS1.SSS2.8.p8.4.m4.1a"><mi id="S3.SS1.SSS2.8.p8.4.m4.1.1" xref="S3.SS1.SSS2.8.p8.4.m4.1.1.cmml">𝒙</mi><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS2.8.p8.4.m4.1b"><ci id="S3.SS1.SSS2.8.p8.4.m4.1.1.cmml" xref="S3.SS1.SSS2.8.p8.4.m4.1.1">𝒙</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS2.8.p8.4.m4.1c">\boldsymbol{x}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS2.8.p8.4.m4.1d">bold_italic_x</annotation></semantics></math> values at most <math alttext="2k\cdot(2t-1)\cdot\frac{1}{k}=2\cdot(2t-1)" class="ltx_Math" display="inline" id="S3.SS1.SSS2.8.p8.5.m5.2"><semantics id="S3.SS1.SSS2.8.p8.5.m5.2a"><mrow id="S3.SS1.SSS2.8.p8.5.m5.2.2" xref="S3.SS1.SSS2.8.p8.5.m5.2.2.cmml"><mrow id="S3.SS1.SSS2.8.p8.5.m5.1.1.1" xref="S3.SS1.SSS2.8.p8.5.m5.1.1.1.cmml"><mrow id="S3.SS1.SSS2.8.p8.5.m5.1.1.1.3" xref="S3.SS1.SSS2.8.p8.5.m5.1.1.1.3.cmml"><mn id="S3.SS1.SSS2.8.p8.5.m5.1.1.1.3.2" xref="S3.SS1.SSS2.8.p8.5.m5.1.1.1.3.2.cmml">2</mn><mo id="S3.SS1.SSS2.8.p8.5.m5.1.1.1.3.1" xref="S3.SS1.SSS2.8.p8.5.m5.1.1.1.3.1.cmml">⁢</mo><mi id="S3.SS1.SSS2.8.p8.5.m5.1.1.1.3.3" xref="S3.SS1.SSS2.8.p8.5.m5.1.1.1.3.3.cmml">k</mi></mrow><mo id="S3.SS1.SSS2.8.p8.5.m5.1.1.1.2" lspace="0.222em" rspace="0.222em" xref="S3.SS1.SSS2.8.p8.5.m5.1.1.1.2.cmml">⋅</mo><mrow id="S3.SS1.SSS2.8.p8.5.m5.1.1.1.1.1" xref="S3.SS1.SSS2.8.p8.5.m5.1.1.1.1.1.1.cmml"><mo id="S3.SS1.SSS2.8.p8.5.m5.1.1.1.1.1.2" stretchy="false" xref="S3.SS1.SSS2.8.p8.5.m5.1.1.1.1.1.1.cmml">(</mo><mrow id="S3.SS1.SSS2.8.p8.5.m5.1.1.1.1.1.1" xref="S3.SS1.SSS2.8.p8.5.m5.1.1.1.1.1.1.cmml"><mrow id="S3.SS1.SSS2.8.p8.5.m5.1.1.1.1.1.1.2" xref="S3.SS1.SSS2.8.p8.5.m5.1.1.1.1.1.1.2.cmml"><mn id="S3.SS1.SSS2.8.p8.5.m5.1.1.1.1.1.1.2.2" xref="S3.SS1.SSS2.8.p8.5.m5.1.1.1.1.1.1.2.2.cmml">2</mn><mo id="S3.SS1.SSS2.8.p8.5.m5.1.1.1.1.1.1.2.1" xref="S3.SS1.SSS2.8.p8.5.m5.1.1.1.1.1.1.2.1.cmml">⁢</mo><mi id="S3.SS1.SSS2.8.p8.5.m5.1.1.1.1.1.1.2.3" xref="S3.SS1.SSS2.8.p8.5.m5.1.1.1.1.1.1.2.3.cmml">t</mi></mrow><mo id="S3.SS1.SSS2.8.p8.5.m5.1.1.1.1.1.1.1" xref="S3.SS1.SSS2.8.p8.5.m5.1.1.1.1.1.1.1.cmml">−</mo><mn id="S3.SS1.SSS2.8.p8.5.m5.1.1.1.1.1.1.3" xref="S3.SS1.SSS2.8.p8.5.m5.1.1.1.1.1.1.3.cmml">1</mn></mrow><mo id="S3.SS1.SSS2.8.p8.5.m5.1.1.1.1.1.3" rspace="0.055em" stretchy="false" xref="S3.SS1.SSS2.8.p8.5.m5.1.1.1.1.1.1.cmml">)</mo></mrow><mo id="S3.SS1.SSS2.8.p8.5.m5.1.1.1.2a" rspace="0.222em" xref="S3.SS1.SSS2.8.p8.5.m5.1.1.1.2.cmml">⋅</mo><mfrac id="S3.SS1.SSS2.8.p8.5.m5.1.1.1.4" xref="S3.SS1.SSS2.8.p8.5.m5.1.1.1.4.cmml"><mn id="S3.SS1.SSS2.8.p8.5.m5.1.1.1.4.2" xref="S3.SS1.SSS2.8.p8.5.m5.1.1.1.4.2.cmml">1</mn><mi id="S3.SS1.SSS2.8.p8.5.m5.1.1.1.4.3" xref="S3.SS1.SSS2.8.p8.5.m5.1.1.1.4.3.cmml">k</mi></mfrac></mrow><mo id="S3.SS1.SSS2.8.p8.5.m5.2.2.3" xref="S3.SS1.SSS2.8.p8.5.m5.2.2.3.cmml">=</mo><mrow id="S3.SS1.SSS2.8.p8.5.m5.2.2.2" xref="S3.SS1.SSS2.8.p8.5.m5.2.2.2.cmml"><mn id="S3.SS1.SSS2.8.p8.5.m5.2.2.2.3" xref="S3.SS1.SSS2.8.p8.5.m5.2.2.2.3.cmml">2</mn><mo id="S3.SS1.SSS2.8.p8.5.m5.2.2.2.2" lspace="0.222em" rspace="0.222em" xref="S3.SS1.SSS2.8.p8.5.m5.2.2.2.2.cmml">⋅</mo><mrow id="S3.SS1.SSS2.8.p8.5.m5.2.2.2.1.1" xref="S3.SS1.SSS2.8.p8.5.m5.2.2.2.1.1.1.cmml"><mo id="S3.SS1.SSS2.8.p8.5.m5.2.2.2.1.1.2" stretchy="false" xref="S3.SS1.SSS2.8.p8.5.m5.2.2.2.1.1.1.cmml">(</mo><mrow id="S3.SS1.SSS2.8.p8.5.m5.2.2.2.1.1.1" xref="S3.SS1.SSS2.8.p8.5.m5.2.2.2.1.1.1.cmml"><mrow id="S3.SS1.SSS2.8.p8.5.m5.2.2.2.1.1.1.2" xref="S3.SS1.SSS2.8.p8.5.m5.2.2.2.1.1.1.2.cmml"><mn id="S3.SS1.SSS2.8.p8.5.m5.2.2.2.1.1.1.2.2" xref="S3.SS1.SSS2.8.p8.5.m5.2.2.2.1.1.1.2.2.cmml">2</mn><mo id="S3.SS1.SSS2.8.p8.5.m5.2.2.2.1.1.1.2.1" xref="S3.SS1.SSS2.8.p8.5.m5.2.2.2.1.1.1.2.1.cmml">⁢</mo><mi id="S3.SS1.SSS2.8.p8.5.m5.2.2.2.1.1.1.2.3" xref="S3.SS1.SSS2.8.p8.5.m5.2.2.2.1.1.1.2.3.cmml">t</mi></mrow><mo id="S3.SS1.SSS2.8.p8.5.m5.2.2.2.1.1.1.1" xref="S3.SS1.SSS2.8.p8.5.m5.2.2.2.1.1.1.1.cmml">−</mo><mn id="S3.SS1.SSS2.8.p8.5.m5.2.2.2.1.1.1.3" xref="S3.SS1.SSS2.8.p8.5.m5.2.2.2.1.1.1.3.cmml">1</mn></mrow><mo id="S3.SS1.SSS2.8.p8.5.m5.2.2.2.1.1.3" stretchy="false" xref="S3.SS1.SSS2.8.p8.5.m5.2.2.2.1.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS2.8.p8.5.m5.2b"><apply id="S3.SS1.SSS2.8.p8.5.m5.2.2.cmml" xref="S3.SS1.SSS2.8.p8.5.m5.2.2"><eq id="S3.SS1.SSS2.8.p8.5.m5.2.2.3.cmml" xref="S3.SS1.SSS2.8.p8.5.m5.2.2.3"></eq><apply id="S3.SS1.SSS2.8.p8.5.m5.1.1.1.cmml" xref="S3.SS1.SSS2.8.p8.5.m5.1.1.1"><ci id="S3.SS1.SSS2.8.p8.5.m5.1.1.1.2.cmml" xref="S3.SS1.SSS2.8.p8.5.m5.1.1.1.2">⋅</ci><apply id="S3.SS1.SSS2.8.p8.5.m5.1.1.1.3.cmml" xref="S3.SS1.SSS2.8.p8.5.m5.1.1.1.3"><times id="S3.SS1.SSS2.8.p8.5.m5.1.1.1.3.1.cmml" xref="S3.SS1.SSS2.8.p8.5.m5.1.1.1.3.1"></times><cn id="S3.SS1.SSS2.8.p8.5.m5.1.1.1.3.2.cmml" type="integer" xref="S3.SS1.SSS2.8.p8.5.m5.1.1.1.3.2">2</cn><ci id="S3.SS1.SSS2.8.p8.5.m5.1.1.1.3.3.cmml" xref="S3.SS1.SSS2.8.p8.5.m5.1.1.1.3.3">𝑘</ci></apply><apply id="S3.SS1.SSS2.8.p8.5.m5.1.1.1.1.1.1.cmml" xref="S3.SS1.SSS2.8.p8.5.m5.1.1.1.1.1"><minus id="S3.SS1.SSS2.8.p8.5.m5.1.1.1.1.1.1.1.cmml" xref="S3.SS1.SSS2.8.p8.5.m5.1.1.1.1.1.1.1"></minus><apply id="S3.SS1.SSS2.8.p8.5.m5.1.1.1.1.1.1.2.cmml" xref="S3.SS1.SSS2.8.p8.5.m5.1.1.1.1.1.1.2"><times id="S3.SS1.SSS2.8.p8.5.m5.1.1.1.1.1.1.2.1.cmml" xref="S3.SS1.SSS2.8.p8.5.m5.1.1.1.1.1.1.2.1"></times><cn id="S3.SS1.SSS2.8.p8.5.m5.1.1.1.1.1.1.2.2.cmml" type="integer" xref="S3.SS1.SSS2.8.p8.5.m5.1.1.1.1.1.1.2.2">2</cn><ci id="S3.SS1.SSS2.8.p8.5.m5.1.1.1.1.1.1.2.3.cmml" xref="S3.SS1.SSS2.8.p8.5.m5.1.1.1.1.1.1.2.3">𝑡</ci></apply><cn id="S3.SS1.SSS2.8.p8.5.m5.1.1.1.1.1.1.3.cmml" type="integer" xref="S3.SS1.SSS2.8.p8.5.m5.1.1.1.1.1.1.3">1</cn></apply><apply id="S3.SS1.SSS2.8.p8.5.m5.1.1.1.4.cmml" xref="S3.SS1.SSS2.8.p8.5.m5.1.1.1.4"><divide id="S3.SS1.SSS2.8.p8.5.m5.1.1.1.4.1.cmml" xref="S3.SS1.SSS2.8.p8.5.m5.1.1.1.4"></divide><cn id="S3.SS1.SSS2.8.p8.5.m5.1.1.1.4.2.cmml" type="integer" xref="S3.SS1.SSS2.8.p8.5.m5.1.1.1.4.2">1</cn><ci id="S3.SS1.SSS2.8.p8.5.m5.1.1.1.4.3.cmml" xref="S3.SS1.SSS2.8.p8.5.m5.1.1.1.4.3">𝑘</ci></apply></apply><apply id="S3.SS1.SSS2.8.p8.5.m5.2.2.2.cmml" xref="S3.SS1.SSS2.8.p8.5.m5.2.2.2"><ci id="S3.SS1.SSS2.8.p8.5.m5.2.2.2.2.cmml" xref="S3.SS1.SSS2.8.p8.5.m5.2.2.2.2">⋅</ci><cn id="S3.SS1.SSS2.8.p8.5.m5.2.2.2.3.cmml" type="integer" xref="S3.SS1.SSS2.8.p8.5.m5.2.2.2.3">2</cn><apply id="S3.SS1.SSS2.8.p8.5.m5.2.2.2.1.1.1.cmml" xref="S3.SS1.SSS2.8.p8.5.m5.2.2.2.1.1"><minus id="S3.SS1.SSS2.8.p8.5.m5.2.2.2.1.1.1.1.cmml" xref="S3.SS1.SSS2.8.p8.5.m5.2.2.2.1.1.1.1"></minus><apply id="S3.SS1.SSS2.8.p8.5.m5.2.2.2.1.1.1.2.cmml" xref="S3.SS1.SSS2.8.p8.5.m5.2.2.2.1.1.1.2"><times id="S3.SS1.SSS2.8.p8.5.m5.2.2.2.1.1.1.2.1.cmml" xref="S3.SS1.SSS2.8.p8.5.m5.2.2.2.1.1.1.2.1"></times><cn id="S3.SS1.SSS2.8.p8.5.m5.2.2.2.1.1.1.2.2.cmml" type="integer" xref="S3.SS1.SSS2.8.p8.5.m5.2.2.2.1.1.1.2.2">2</cn><ci id="S3.SS1.SSS2.8.p8.5.m5.2.2.2.1.1.1.2.3.cmml" xref="S3.SS1.SSS2.8.p8.5.m5.2.2.2.1.1.1.2.3">𝑡</ci></apply><cn id="S3.SS1.SSS2.8.p8.5.m5.2.2.2.1.1.1.3.cmml" type="integer" xref="S3.SS1.SSS2.8.p8.5.m5.2.2.2.1.1.1.3">1</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS2.8.p8.5.m5.2c">2k\cdot(2t-1)\cdot\frac{1}{k}=2\cdot(2t-1)</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS2.8.p8.5.m5.2d">2 italic_k ⋅ ( 2 italic_t - 1 ) ⋅ divide start_ARG 1 end_ARG start_ARG italic_k end_ARG = 2 ⋅ ( 2 italic_t - 1 )</annotation></semantics></math> units on edges whose weight belong to <math alttext="B_{j}" class="ltx_Math" display="inline" id="S3.SS1.SSS2.8.p8.6.m6.1"><semantics id="S3.SS1.SSS2.8.p8.6.m6.1a"><msub id="S3.SS1.SSS2.8.p8.6.m6.1.1" xref="S3.SS1.SSS2.8.p8.6.m6.1.1.cmml"><mi id="S3.SS1.SSS2.8.p8.6.m6.1.1.2" xref="S3.SS1.SSS2.8.p8.6.m6.1.1.2.cmml">B</mi><mi id="S3.SS1.SSS2.8.p8.6.m6.1.1.3" xref="S3.SS1.SSS2.8.p8.6.m6.1.1.3.cmml">j</mi></msub><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS2.8.p8.6.m6.1b"><apply id="S3.SS1.SSS2.8.p8.6.m6.1.1.cmml" xref="S3.SS1.SSS2.8.p8.6.m6.1.1"><csymbol cd="ambiguous" id="S3.SS1.SSS2.8.p8.6.m6.1.1.1.cmml" xref="S3.SS1.SSS2.8.p8.6.m6.1.1">subscript</csymbol><ci id="S3.SS1.SSS2.8.p8.6.m6.1.1.2.cmml" xref="S3.SS1.SSS2.8.p8.6.m6.1.1.2">𝐵</ci><ci id="S3.SS1.SSS2.8.p8.6.m6.1.1.3.cmml" xref="S3.SS1.SSS2.8.p8.6.m6.1.1.3">𝑗</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS2.8.p8.6.m6.1c">B_{j}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS2.8.p8.6.m6.1d">italic_B start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT</annotation></semantics></math>. Therefore, <math alttext="w(\boldsymbol{x})\leq 2\cdot(2t-1)\cdot(1+\epsilon)\cdot w(\textnormal{OPT})=4% t\cdot w(\textnormal{OPT})" class="ltx_Math" display="inline" id="S3.SS1.SSS2.8.p8.7.m7.5"><semantics id="S3.SS1.SSS2.8.p8.7.m7.5a"><mrow id="S3.SS1.SSS2.8.p8.7.m7.5.5" xref="S3.SS1.SSS2.8.p8.7.m7.5.5.cmml"><mrow id="S3.SS1.SSS2.8.p8.7.m7.5.5.4" xref="S3.SS1.SSS2.8.p8.7.m7.5.5.4.cmml"><mi id="S3.SS1.SSS2.8.p8.7.m7.5.5.4.2" xref="S3.SS1.SSS2.8.p8.7.m7.5.5.4.2.cmml">w</mi><mo id="S3.SS1.SSS2.8.p8.7.m7.5.5.4.1" xref="S3.SS1.SSS2.8.p8.7.m7.5.5.4.1.cmml">⁢</mo><mrow id="S3.SS1.SSS2.8.p8.7.m7.5.5.4.3.2" xref="S3.SS1.SSS2.8.p8.7.m7.5.5.4.cmml"><mo id="S3.SS1.SSS2.8.p8.7.m7.5.5.4.3.2.1" stretchy="false" xref="S3.SS1.SSS2.8.p8.7.m7.5.5.4.cmml">(</mo><mi id="S3.SS1.SSS2.8.p8.7.m7.1.1" xref="S3.SS1.SSS2.8.p8.7.m7.1.1.cmml">𝒙</mi><mo id="S3.SS1.SSS2.8.p8.7.m7.5.5.4.3.2.2" stretchy="false" xref="S3.SS1.SSS2.8.p8.7.m7.5.5.4.cmml">)</mo></mrow></mrow><mo id="S3.SS1.SSS2.8.p8.7.m7.5.5.5" 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xref="S3.SS1.SSS2.8.p8.7.m7.4.4.1.1.1.1.1.2.1.cmml">⁢</mo><mi id="S3.SS1.SSS2.8.p8.7.m7.4.4.1.1.1.1.1.2.3" xref="S3.SS1.SSS2.8.p8.7.m7.4.4.1.1.1.1.1.2.3.cmml">t</mi></mrow><mo id="S3.SS1.SSS2.8.p8.7.m7.4.4.1.1.1.1.1.1" xref="S3.SS1.SSS2.8.p8.7.m7.4.4.1.1.1.1.1.1.cmml">−</mo><mn id="S3.SS1.SSS2.8.p8.7.m7.4.4.1.1.1.1.1.3" xref="S3.SS1.SSS2.8.p8.7.m7.4.4.1.1.1.1.1.3.cmml">1</mn></mrow><mo id="S3.SS1.SSS2.8.p8.7.m7.4.4.1.1.1.1.3" rspace="0.055em" stretchy="false" xref="S3.SS1.SSS2.8.p8.7.m7.4.4.1.1.1.1.1.cmml">)</mo></mrow><mo id="S3.SS1.SSS2.8.p8.7.m7.5.5.2.2.3a" rspace="0.222em" xref="S3.SS1.SSS2.8.p8.7.m7.5.5.2.2.3.cmml">⋅</mo><mrow id="S3.SS1.SSS2.8.p8.7.m7.5.5.2.2.2.1" xref="S3.SS1.SSS2.8.p8.7.m7.5.5.2.2.2.1.1.cmml"><mo id="S3.SS1.SSS2.8.p8.7.m7.5.5.2.2.2.1.2" stretchy="false" xref="S3.SS1.SSS2.8.p8.7.m7.5.5.2.2.2.1.1.cmml">(</mo><mrow id="S3.SS1.SSS2.8.p8.7.m7.5.5.2.2.2.1.1" xref="S3.SS1.SSS2.8.p8.7.m7.5.5.2.2.2.1.1.cmml"><mn id="S3.SS1.SSS2.8.p8.7.m7.5.5.2.2.2.1.1.2" xref="S3.SS1.SSS2.8.p8.7.m7.5.5.2.2.2.1.1.2.cmml">1</mn><mo id="S3.SS1.SSS2.8.p8.7.m7.5.5.2.2.2.1.1.1" xref="S3.SS1.SSS2.8.p8.7.m7.5.5.2.2.2.1.1.1.cmml">+</mo><mi id="S3.SS1.SSS2.8.p8.7.m7.5.5.2.2.2.1.1.3" xref="S3.SS1.SSS2.8.p8.7.m7.5.5.2.2.2.1.1.3.cmml">ϵ</mi></mrow><mo id="S3.SS1.SSS2.8.p8.7.m7.5.5.2.2.2.1.3" rspace="0.055em" stretchy="false" xref="S3.SS1.SSS2.8.p8.7.m7.5.5.2.2.2.1.1.cmml">)</mo></mrow><mo id="S3.SS1.SSS2.8.p8.7.m7.5.5.2.2.3b" rspace="0.222em" xref="S3.SS1.SSS2.8.p8.7.m7.5.5.2.2.3.cmml">⋅</mo><mi id="S3.SS1.SSS2.8.p8.7.m7.5.5.2.2.5" xref="S3.SS1.SSS2.8.p8.7.m7.5.5.2.2.5.cmml">w</mi></mrow><mo id="S3.SS1.SSS2.8.p8.7.m7.5.5.2.3" xref="S3.SS1.SSS2.8.p8.7.m7.5.5.2.3.cmml">⁢</mo><mrow id="S3.SS1.SSS2.8.p8.7.m7.5.5.2.4.2" xref="S3.SS1.SSS2.8.p8.7.m7.2.2a.cmml"><mo id="S3.SS1.SSS2.8.p8.7.m7.5.5.2.4.2.1" stretchy="false" xref="S3.SS1.SSS2.8.p8.7.m7.2.2a.cmml">(</mo><mtext id="S3.SS1.SSS2.8.p8.7.m7.2.2" xref="S3.SS1.SSS2.8.p8.7.m7.2.2.cmml">OPT</mtext><mo 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xref="S3.SS1.SSS2.8.p8.7.m7.5.5.7.1.cmml">⁢</mo><mrow id="S3.SS1.SSS2.8.p8.7.m7.5.5.7.3.2" xref="S3.SS1.SSS2.8.p8.7.m7.3.3a.cmml"><mo id="S3.SS1.SSS2.8.p8.7.m7.5.5.7.3.2.1" stretchy="false" xref="S3.SS1.SSS2.8.p8.7.m7.3.3a.cmml">(</mo><mtext id="S3.SS1.SSS2.8.p8.7.m7.3.3" xref="S3.SS1.SSS2.8.p8.7.m7.3.3.cmml">OPT</mtext><mo id="S3.SS1.SSS2.8.p8.7.m7.5.5.7.3.2.2" stretchy="false" xref="S3.SS1.SSS2.8.p8.7.m7.3.3a.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS2.8.p8.7.m7.5b"><apply id="S3.SS1.SSS2.8.p8.7.m7.5.5.cmml" xref="S3.SS1.SSS2.8.p8.7.m7.5.5"><and id="S3.SS1.SSS2.8.p8.7.m7.5.5a.cmml" xref="S3.SS1.SSS2.8.p8.7.m7.5.5"></and><apply id="S3.SS1.SSS2.8.p8.7.m7.5.5b.cmml" xref="S3.SS1.SSS2.8.p8.7.m7.5.5"><leq id="S3.SS1.SSS2.8.p8.7.m7.5.5.5.cmml" xref="S3.SS1.SSS2.8.p8.7.m7.5.5.5"></leq><apply id="S3.SS1.SSS2.8.p8.7.m7.5.5.4.cmml" xref="S3.SS1.SSS2.8.p8.7.m7.5.5.4"><times id="S3.SS1.SSS2.8.p8.7.m7.5.5.4.1.cmml" xref="S3.SS1.SSS2.8.p8.7.m7.5.5.4.1"></times><ci id="S3.SS1.SSS2.8.p8.7.m7.5.5.4.2.cmml" xref="S3.SS1.SSS2.8.p8.7.m7.5.5.4.2">𝑤</ci><ci id="S3.SS1.SSS2.8.p8.7.m7.1.1.cmml" xref="S3.SS1.SSS2.8.p8.7.m7.1.1">𝒙</ci></apply><apply id="S3.SS1.SSS2.8.p8.7.m7.5.5.2.cmml" xref="S3.SS1.SSS2.8.p8.7.m7.5.5.2"><times id="S3.SS1.SSS2.8.p8.7.m7.5.5.2.3.cmml" xref="S3.SS1.SSS2.8.p8.7.m7.5.5.2.3"></times><apply id="S3.SS1.SSS2.8.p8.7.m7.5.5.2.2.cmml" xref="S3.SS1.SSS2.8.p8.7.m7.5.5.2.2"><ci id="S3.SS1.SSS2.8.p8.7.m7.5.5.2.2.3.cmml" xref="S3.SS1.SSS2.8.p8.7.m7.5.5.2.2.3">⋅</ci><cn id="S3.SS1.SSS2.8.p8.7.m7.5.5.2.2.4.cmml" type="integer" xref="S3.SS1.SSS2.8.p8.7.m7.5.5.2.2.4">2</cn><apply id="S3.SS1.SSS2.8.p8.7.m7.4.4.1.1.1.1.1.cmml" xref="S3.SS1.SSS2.8.p8.7.m7.4.4.1.1.1.1"><minus id="S3.SS1.SSS2.8.p8.7.m7.4.4.1.1.1.1.1.1.cmml" xref="S3.SS1.SSS2.8.p8.7.m7.4.4.1.1.1.1.1.1"></minus><apply id="S3.SS1.SSS2.8.p8.7.m7.4.4.1.1.1.1.1.2.cmml" xref="S3.SS1.SSS2.8.p8.7.m7.4.4.1.1.1.1.1.2"><times id="S3.SS1.SSS2.8.p8.7.m7.4.4.1.1.1.1.1.2.1.cmml" xref="S3.SS1.SSS2.8.p8.7.m7.4.4.1.1.1.1.1.2.1"></times><cn id="S3.SS1.SSS2.8.p8.7.m7.4.4.1.1.1.1.1.2.2.cmml" type="integer" xref="S3.SS1.SSS2.8.p8.7.m7.4.4.1.1.1.1.1.2.2">2</cn><ci id="S3.SS1.SSS2.8.p8.7.m7.4.4.1.1.1.1.1.2.3.cmml" xref="S3.SS1.SSS2.8.p8.7.m7.4.4.1.1.1.1.1.2.3">𝑡</ci></apply><cn id="S3.SS1.SSS2.8.p8.7.m7.4.4.1.1.1.1.1.3.cmml" type="integer" xref="S3.SS1.SSS2.8.p8.7.m7.4.4.1.1.1.1.1.3">1</cn></apply><apply id="S3.SS1.SSS2.8.p8.7.m7.5.5.2.2.2.1.1.cmml" xref="S3.SS1.SSS2.8.p8.7.m7.5.5.2.2.2.1"><plus id="S3.SS1.SSS2.8.p8.7.m7.5.5.2.2.2.1.1.1.cmml" xref="S3.SS1.SSS2.8.p8.7.m7.5.5.2.2.2.1.1.1"></plus><cn id="S3.SS1.SSS2.8.p8.7.m7.5.5.2.2.2.1.1.2.cmml" type="integer" xref="S3.SS1.SSS2.8.p8.7.m7.5.5.2.2.2.1.1.2">1</cn><ci id="S3.SS1.SSS2.8.p8.7.m7.5.5.2.2.2.1.1.3.cmml" xref="S3.SS1.SSS2.8.p8.7.m7.5.5.2.2.2.1.1.3">italic-ϵ</ci></apply><ci id="S3.SS1.SSS2.8.p8.7.m7.5.5.2.2.5.cmml" xref="S3.SS1.SSS2.8.p8.7.m7.5.5.2.2.5">𝑤</ci></apply><ci id="S3.SS1.SSS2.8.p8.7.m7.2.2a.cmml" xref="S3.SS1.SSS2.8.p8.7.m7.5.5.2.4.2"><mtext id="S3.SS1.SSS2.8.p8.7.m7.2.2.cmml" xref="S3.SS1.SSS2.8.p8.7.m7.2.2">OPT</mtext></ci></apply></apply><apply id="S3.SS1.SSS2.8.p8.7.m7.5.5c.cmml" xref="S3.SS1.SSS2.8.p8.7.m7.5.5"><eq id="S3.SS1.SSS2.8.p8.7.m7.5.5.6.cmml" xref="S3.SS1.SSS2.8.p8.7.m7.5.5.6"></eq><share href="https://arxiv.org/html/2503.00712v1#S3.SS1.SSS2.8.p8.7.m7.5.5.2.cmml" id="S3.SS1.SSS2.8.p8.7.m7.5.5d.cmml" xref="S3.SS1.SSS2.8.p8.7.m7.5.5"></share><apply id="S3.SS1.SSS2.8.p8.7.m7.5.5.7.cmml" xref="S3.SS1.SSS2.8.p8.7.m7.5.5.7"><times id="S3.SS1.SSS2.8.p8.7.m7.5.5.7.1.cmml" xref="S3.SS1.SSS2.8.p8.7.m7.5.5.7.1"></times><apply id="S3.SS1.SSS2.8.p8.7.m7.5.5.7.2.cmml" xref="S3.SS1.SSS2.8.p8.7.m7.5.5.7.2"><ci id="S3.SS1.SSS2.8.p8.7.m7.5.5.7.2.1.cmml" xref="S3.SS1.SSS2.8.p8.7.m7.5.5.7.2.1">⋅</ci><apply id="S3.SS1.SSS2.8.p8.7.m7.5.5.7.2.2.cmml" xref="S3.SS1.SSS2.8.p8.7.m7.5.5.7.2.2"><times id="S3.SS1.SSS2.8.p8.7.m7.5.5.7.2.2.1.cmml" xref="S3.SS1.SSS2.8.p8.7.m7.5.5.7.2.2.1"></times><cn id="S3.SS1.SSS2.8.p8.7.m7.5.5.7.2.2.2.cmml" type="integer" xref="S3.SS1.SSS2.8.p8.7.m7.5.5.7.2.2.2">4</cn><ci id="S3.SS1.SSS2.8.p8.7.m7.5.5.7.2.2.3.cmml" xref="S3.SS1.SSS2.8.p8.7.m7.5.5.7.2.2.3">𝑡</ci></apply><ci id="S3.SS1.SSS2.8.p8.7.m7.5.5.7.2.3.cmml" xref="S3.SS1.SSS2.8.p8.7.m7.5.5.7.2.3">𝑤</ci></apply><ci id="S3.SS1.SSS2.8.p8.7.m7.3.3a.cmml" xref="S3.SS1.SSS2.8.p8.7.m7.5.5.7.3.2"><mtext id="S3.SS1.SSS2.8.p8.7.m7.3.3.cmml" xref="S3.SS1.SSS2.8.p8.7.m7.3.3">OPT</mtext></ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS2.8.p8.7.m7.5c">w(\boldsymbol{x})\leq 2\cdot(2t-1)\cdot(1+\epsilon)\cdot w(\textnormal{OPT})=4% t\cdot w(\textnormal{OPT})</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS2.8.p8.7.m7.5d">italic_w ( bold_italic_x ) ≤ 2 ⋅ ( 2 italic_t - 1 ) ⋅ ( 1 + italic_ϵ ) ⋅ italic_w ( OPT ) = 4 italic_t ⋅ italic_w ( OPT )</annotation></semantics></math>. ∎</p> </div> </div> <div class="ltx_theorem ltx_theorem_corollary" id="S3.Thmtheorem5"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="S3.Thmtheorem5.1.1.1">Corollary 3.5</span></span><span class="ltx_text ltx_font_bold" id="S3.Thmtheorem5.2.2">.</span> </h6> <div class="ltx_para" id="S3.Thmtheorem5.p1"> <p class="ltx_p" id="S3.Thmtheorem5.p1.5">There exists an algorithm for VC-SNDP with edge weights <math alttext="w:E\rightarrow\{0,1,\dots,W\}" class="ltx_Math" display="inline" id="S3.Thmtheorem5.p1.1.m1.4"><semantics id="S3.Thmtheorem5.p1.1.m1.4a"><mrow id="S3.Thmtheorem5.p1.1.m1.4.5" xref="S3.Thmtheorem5.p1.1.m1.4.5.cmml"><mi id="S3.Thmtheorem5.p1.1.m1.4.5.2" xref="S3.Thmtheorem5.p1.1.m1.4.5.2.cmml">w</mi><mo id="S3.Thmtheorem5.p1.1.m1.4.5.1" lspace="0.278em" rspace="0.278em" xref="S3.Thmtheorem5.p1.1.m1.4.5.1.cmml">:</mo><mrow id="S3.Thmtheorem5.p1.1.m1.4.5.3" xref="S3.Thmtheorem5.p1.1.m1.4.5.3.cmml"><mi id="S3.Thmtheorem5.p1.1.m1.4.5.3.2" xref="S3.Thmtheorem5.p1.1.m1.4.5.3.2.cmml">E</mi><mo id="S3.Thmtheorem5.p1.1.m1.4.5.3.1" stretchy="false" xref="S3.Thmtheorem5.p1.1.m1.4.5.3.1.cmml">→</mo><mrow id="S3.Thmtheorem5.p1.1.m1.4.5.3.3.2" xref="S3.Thmtheorem5.p1.1.m1.4.5.3.3.1.cmml"><mo id="S3.Thmtheorem5.p1.1.m1.4.5.3.3.2.1" stretchy="false" xref="S3.Thmtheorem5.p1.1.m1.4.5.3.3.1.cmml">{</mo><mn id="S3.Thmtheorem5.p1.1.m1.1.1" xref="S3.Thmtheorem5.p1.1.m1.1.1.cmml">0</mn><mo id="S3.Thmtheorem5.p1.1.m1.4.5.3.3.2.2" xref="S3.Thmtheorem5.p1.1.m1.4.5.3.3.1.cmml">,</mo><mn id="S3.Thmtheorem5.p1.1.m1.2.2" xref="S3.Thmtheorem5.p1.1.m1.2.2.cmml">1</mn><mo id="S3.Thmtheorem5.p1.1.m1.4.5.3.3.2.3" xref="S3.Thmtheorem5.p1.1.m1.4.5.3.3.1.cmml">,</mo><mi id="S3.Thmtheorem5.p1.1.m1.3.3" mathvariant="normal" xref="S3.Thmtheorem5.p1.1.m1.3.3.cmml">…</mi><mo id="S3.Thmtheorem5.p1.1.m1.4.5.3.3.2.4" xref="S3.Thmtheorem5.p1.1.m1.4.5.3.3.1.cmml">,</mo><mi id="S3.Thmtheorem5.p1.1.m1.4.4" xref="S3.Thmtheorem5.p1.1.m1.4.4.cmml">W</mi><mo id="S3.Thmtheorem5.p1.1.m1.4.5.3.3.2.5" stretchy="false" xref="S3.Thmtheorem5.p1.1.m1.4.5.3.3.1.cmml">}</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem5.p1.1.m1.4b"><apply id="S3.Thmtheorem5.p1.1.m1.4.5.cmml" xref="S3.Thmtheorem5.p1.1.m1.4.5"><ci id="S3.Thmtheorem5.p1.1.m1.4.5.1.cmml" xref="S3.Thmtheorem5.p1.1.m1.4.5.1">:</ci><ci id="S3.Thmtheorem5.p1.1.m1.4.5.2.cmml" xref="S3.Thmtheorem5.p1.1.m1.4.5.2">𝑤</ci><apply id="S3.Thmtheorem5.p1.1.m1.4.5.3.cmml" xref="S3.Thmtheorem5.p1.1.m1.4.5.3"><ci id="S3.Thmtheorem5.p1.1.m1.4.5.3.1.cmml" xref="S3.Thmtheorem5.p1.1.m1.4.5.3.1">→</ci><ci id="S3.Thmtheorem5.p1.1.m1.4.5.3.2.cmml" xref="S3.Thmtheorem5.p1.1.m1.4.5.3.2">𝐸</ci><set id="S3.Thmtheorem5.p1.1.m1.4.5.3.3.1.cmml" xref="S3.Thmtheorem5.p1.1.m1.4.5.3.3.2"><cn id="S3.Thmtheorem5.p1.1.m1.1.1.cmml" type="integer" xref="S3.Thmtheorem5.p1.1.m1.1.1">0</cn><cn id="S3.Thmtheorem5.p1.1.m1.2.2.cmml" type="integer" xref="S3.Thmtheorem5.p1.1.m1.2.2">1</cn><ci id="S3.Thmtheorem5.p1.1.m1.3.3.cmml" xref="S3.Thmtheorem5.p1.1.m1.3.3">…</ci><ci id="S3.Thmtheorem5.p1.1.m1.4.4.cmml" xref="S3.Thmtheorem5.p1.1.m1.4.4">𝑊</ci></set></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem5.p1.1.m1.4c">w:E\rightarrow\{0,1,\dots,W\}</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem5.p1.1.m1.4d">italic_w : italic_E → { 0 , 1 , … , italic_W }</annotation></semantics></math> and a maximum connectivity requirement <math alttext="k" class="ltx_Math" display="inline" id="S3.Thmtheorem5.p1.2.m2.1"><semantics id="S3.Thmtheorem5.p1.2.m2.1a"><mi id="S3.Thmtheorem5.p1.2.m2.1.1" xref="S3.Thmtheorem5.p1.2.m2.1.1.cmml">k</mi><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem5.p1.2.m2.1b"><ci id="S3.Thmtheorem5.p1.2.m2.1.1.cmml" xref="S3.Thmtheorem5.p1.2.m2.1.1">𝑘</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem5.p1.2.m2.1c">k</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem5.p1.2.m2.1d">italic_k</annotation></semantics></math>, in insertion-only streams, that uses <math alttext="\tilde{O}(k^{1-1/t}\cdot n^{1+1/t})" class="ltx_Math" display="inline" id="S3.Thmtheorem5.p1.3.m3.1"><semantics id="S3.Thmtheorem5.p1.3.m3.1a"><mrow id="S3.Thmtheorem5.p1.3.m3.1.1" xref="S3.Thmtheorem5.p1.3.m3.1.1.cmml"><mover accent="true" id="S3.Thmtheorem5.p1.3.m3.1.1.3" xref="S3.Thmtheorem5.p1.3.m3.1.1.3.cmml"><mi id="S3.Thmtheorem5.p1.3.m3.1.1.3.2" xref="S3.Thmtheorem5.p1.3.m3.1.1.3.2.cmml">O</mi><mo id="S3.Thmtheorem5.p1.3.m3.1.1.3.1" xref="S3.Thmtheorem5.p1.3.m3.1.1.3.1.cmml">~</mo></mover><mo id="S3.Thmtheorem5.p1.3.m3.1.1.2" xref="S3.Thmtheorem5.p1.3.m3.1.1.2.cmml">⁢</mo><mrow id="S3.Thmtheorem5.p1.3.m3.1.1.1.1" xref="S3.Thmtheorem5.p1.3.m3.1.1.1.1.1.cmml"><mo id="S3.Thmtheorem5.p1.3.m3.1.1.1.1.2" stretchy="false" xref="S3.Thmtheorem5.p1.3.m3.1.1.1.1.1.cmml">(</mo><mrow id="S3.Thmtheorem5.p1.3.m3.1.1.1.1.1" xref="S3.Thmtheorem5.p1.3.m3.1.1.1.1.1.cmml"><msup id="S3.Thmtheorem5.p1.3.m3.1.1.1.1.1.2" xref="S3.Thmtheorem5.p1.3.m3.1.1.1.1.1.2.cmml"><mi id="S3.Thmtheorem5.p1.3.m3.1.1.1.1.1.2.2" xref="S3.Thmtheorem5.p1.3.m3.1.1.1.1.1.2.2.cmml">k</mi><mrow id="S3.Thmtheorem5.p1.3.m3.1.1.1.1.1.2.3" xref="S3.Thmtheorem5.p1.3.m3.1.1.1.1.1.2.3.cmml"><mn id="S3.Thmtheorem5.p1.3.m3.1.1.1.1.1.2.3.2" xref="S3.Thmtheorem5.p1.3.m3.1.1.1.1.1.2.3.2.cmml">1</mn><mo id="S3.Thmtheorem5.p1.3.m3.1.1.1.1.1.2.3.1" xref="S3.Thmtheorem5.p1.3.m3.1.1.1.1.1.2.3.1.cmml">−</mo><mrow id="S3.Thmtheorem5.p1.3.m3.1.1.1.1.1.2.3.3" xref="S3.Thmtheorem5.p1.3.m3.1.1.1.1.1.2.3.3.cmml"><mn id="S3.Thmtheorem5.p1.3.m3.1.1.1.1.1.2.3.3.2" xref="S3.Thmtheorem5.p1.3.m3.1.1.1.1.1.2.3.3.2.cmml">1</mn><mo 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id="S3.Thmtheorem5.p1.3.m3.1.1.1.1.1.3.3.3.2" xref="S3.Thmtheorem5.p1.3.m3.1.1.1.1.1.3.3.3.2.cmml">1</mn><mo id="S3.Thmtheorem5.p1.3.m3.1.1.1.1.1.3.3.3.1" xref="S3.Thmtheorem5.p1.3.m3.1.1.1.1.1.3.3.3.1.cmml">/</mo><mi id="S3.Thmtheorem5.p1.3.m3.1.1.1.1.1.3.3.3.3" xref="S3.Thmtheorem5.p1.3.m3.1.1.1.1.1.3.3.3.3.cmml">t</mi></mrow></mrow></msup></mrow><mo id="S3.Thmtheorem5.p1.3.m3.1.1.1.1.3" stretchy="false" xref="S3.Thmtheorem5.p1.3.m3.1.1.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem5.p1.3.m3.1b"><apply id="S3.Thmtheorem5.p1.3.m3.1.1.cmml" xref="S3.Thmtheorem5.p1.3.m3.1.1"><times id="S3.Thmtheorem5.p1.3.m3.1.1.2.cmml" xref="S3.Thmtheorem5.p1.3.m3.1.1.2"></times><apply id="S3.Thmtheorem5.p1.3.m3.1.1.3.cmml" xref="S3.Thmtheorem5.p1.3.m3.1.1.3"><ci id="S3.Thmtheorem5.p1.3.m3.1.1.3.1.cmml" xref="S3.Thmtheorem5.p1.3.m3.1.1.3.1">~</ci><ci id="S3.Thmtheorem5.p1.3.m3.1.1.3.2.cmml" xref="S3.Thmtheorem5.p1.3.m3.1.1.3.2">𝑂</ci></apply><apply 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xref="S3.Thmtheorem5.p1.3.m3.1.1.1.1.1.2.3.3.1"></divide><cn id="S3.Thmtheorem5.p1.3.m3.1.1.1.1.1.2.3.3.2.cmml" type="integer" xref="S3.Thmtheorem5.p1.3.m3.1.1.1.1.1.2.3.3.2">1</cn><ci id="S3.Thmtheorem5.p1.3.m3.1.1.1.1.1.2.3.3.3.cmml" xref="S3.Thmtheorem5.p1.3.m3.1.1.1.1.1.2.3.3.3">𝑡</ci></apply></apply></apply><apply id="S3.Thmtheorem5.p1.3.m3.1.1.1.1.1.3.cmml" xref="S3.Thmtheorem5.p1.3.m3.1.1.1.1.1.3"><csymbol cd="ambiguous" id="S3.Thmtheorem5.p1.3.m3.1.1.1.1.1.3.1.cmml" xref="S3.Thmtheorem5.p1.3.m3.1.1.1.1.1.3">superscript</csymbol><ci id="S3.Thmtheorem5.p1.3.m3.1.1.1.1.1.3.2.cmml" xref="S3.Thmtheorem5.p1.3.m3.1.1.1.1.1.3.2">𝑛</ci><apply id="S3.Thmtheorem5.p1.3.m3.1.1.1.1.1.3.3.cmml" xref="S3.Thmtheorem5.p1.3.m3.1.1.1.1.1.3.3"><plus id="S3.Thmtheorem5.p1.3.m3.1.1.1.1.1.3.3.1.cmml" xref="S3.Thmtheorem5.p1.3.m3.1.1.1.1.1.3.3.1"></plus><cn id="S3.Thmtheorem5.p1.3.m3.1.1.1.1.1.3.3.2.cmml" type="integer" xref="S3.Thmtheorem5.p1.3.m3.1.1.1.1.1.3.3.2">1</cn><apply id="S3.Thmtheorem5.p1.3.m3.1.1.1.1.1.3.3.3.cmml" xref="S3.Thmtheorem5.p1.3.m3.1.1.1.1.1.3.3.3"><divide id="S3.Thmtheorem5.p1.3.m3.1.1.1.1.1.3.3.3.1.cmml" xref="S3.Thmtheorem5.p1.3.m3.1.1.1.1.1.3.3.3.1"></divide><cn id="S3.Thmtheorem5.p1.3.m3.1.1.1.1.1.3.3.3.2.cmml" type="integer" xref="S3.Thmtheorem5.p1.3.m3.1.1.1.1.1.3.3.3.2">1</cn><ci id="S3.Thmtheorem5.p1.3.m3.1.1.1.1.1.3.3.3.3.cmml" xref="S3.Thmtheorem5.p1.3.m3.1.1.1.1.1.3.3.3.3">𝑡</ci></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem5.p1.3.m3.1c">\tilde{O}(k^{1-1/t}\cdot n^{1+1/t})</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem5.p1.3.m3.1d">over~ start_ARG italic_O end_ARG ( italic_k start_POSTSUPERSCRIPT 1 - 1 / italic_t end_POSTSUPERSCRIPT ⋅ italic_n start_POSTSUPERSCRIPT 1 + 1 / italic_t end_POSTSUPERSCRIPT )</annotation></semantics></math> space and outputs a <math alttext="(4t\beta)" class="ltx_Math" display="inline" id="S3.Thmtheorem5.p1.4.m4.1"><semantics id="S3.Thmtheorem5.p1.4.m4.1a"><mrow id="S3.Thmtheorem5.p1.4.m4.1.1.1" xref="S3.Thmtheorem5.p1.4.m4.1.1.1.1.cmml"><mo id="S3.Thmtheorem5.p1.4.m4.1.1.1.2" stretchy="false" xref="S3.Thmtheorem5.p1.4.m4.1.1.1.1.cmml">(</mo><mrow id="S3.Thmtheorem5.p1.4.m4.1.1.1.1" xref="S3.Thmtheorem5.p1.4.m4.1.1.1.1.cmml"><mn id="S3.Thmtheorem5.p1.4.m4.1.1.1.1.2" xref="S3.Thmtheorem5.p1.4.m4.1.1.1.1.2.cmml">4</mn><mo id="S3.Thmtheorem5.p1.4.m4.1.1.1.1.1" xref="S3.Thmtheorem5.p1.4.m4.1.1.1.1.1.cmml">⁢</mo><mi id="S3.Thmtheorem5.p1.4.m4.1.1.1.1.3" xref="S3.Thmtheorem5.p1.4.m4.1.1.1.1.3.cmml">t</mi><mo id="S3.Thmtheorem5.p1.4.m4.1.1.1.1.1a" xref="S3.Thmtheorem5.p1.4.m4.1.1.1.1.1.cmml">⁢</mo><mi id="S3.Thmtheorem5.p1.4.m4.1.1.1.1.4" xref="S3.Thmtheorem5.p1.4.m4.1.1.1.1.4.cmml">β</mi></mrow><mo id="S3.Thmtheorem5.p1.4.m4.1.1.1.3" stretchy="false" xref="S3.Thmtheorem5.p1.4.m4.1.1.1.1.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem5.p1.4.m4.1b"><apply id="S3.Thmtheorem5.p1.4.m4.1.1.1.1.cmml" xref="S3.Thmtheorem5.p1.4.m4.1.1.1"><times id="S3.Thmtheorem5.p1.4.m4.1.1.1.1.1.cmml" xref="S3.Thmtheorem5.p1.4.m4.1.1.1.1.1"></times><cn id="S3.Thmtheorem5.p1.4.m4.1.1.1.1.2.cmml" type="integer" xref="S3.Thmtheorem5.p1.4.m4.1.1.1.1.2">4</cn><ci id="S3.Thmtheorem5.p1.4.m4.1.1.1.1.3.cmml" xref="S3.Thmtheorem5.p1.4.m4.1.1.1.1.3">𝑡</ci><ci id="S3.Thmtheorem5.p1.4.m4.1.1.1.1.4.cmml" xref="S3.Thmtheorem5.p1.4.m4.1.1.1.1.4">𝛽</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem5.p1.4.m4.1c">(4t\beta)</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem5.p1.4.m4.1d">( 4 italic_t italic_β )</annotation></semantics></math>-approximation where <math alttext="\beta" class="ltx_Math" display="inline" id="S3.Thmtheorem5.p1.5.m5.1"><semantics id="S3.Thmtheorem5.p1.5.m5.1a"><mi id="S3.Thmtheorem5.p1.5.m5.1.1" xref="S3.Thmtheorem5.p1.5.m5.1.1.cmml">β</mi><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem5.p1.5.m5.1b"><ci id="S3.Thmtheorem5.p1.5.m5.1.1.cmml" xref="S3.Thmtheorem5.p1.5.m5.1.1">𝛽</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem5.p1.5.m5.1c">\beta</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem5.p1.5.m5.1d">italic_β</annotation></semantics></math> is the integrality gap of the cut-based LP relaxation. The running-time of the post-processing algorithm is dominated by the algorithm to round a fractional solution.</p> </div> </div> <section class="ltx_paragraph" id="S3.SS1.SSS2.Px1"> <h5 class="ltx_title ltx_title_paragraph">Implications for VC-SNDP and special cases:</h5> <div class="ltx_para" id="S3.SS1.SSS2.Px1.p1"> <p class="ltx_p" id="S3.SS1.SSS2.Px1.p1.3">The previous approach (via integral solutions) yields an <math alttext="O(tk^{4}\log n)" class="ltx_Math" display="inline" id="S3.SS1.SSS2.Px1.p1.1.m1.1"><semantics id="S3.SS1.SSS2.Px1.p1.1.m1.1a"><mrow id="S3.SS1.SSS2.Px1.p1.1.m1.1.1" xref="S3.SS1.SSS2.Px1.p1.1.m1.1.1.cmml"><mi id="S3.SS1.SSS2.Px1.p1.1.m1.1.1.3" xref="S3.SS1.SSS2.Px1.p1.1.m1.1.1.3.cmml">O</mi><mo id="S3.SS1.SSS2.Px1.p1.1.m1.1.1.2" xref="S3.SS1.SSS2.Px1.p1.1.m1.1.1.2.cmml">⁢</mo><mrow id="S3.SS1.SSS2.Px1.p1.1.m1.1.1.1.1" xref="S3.SS1.SSS2.Px1.p1.1.m1.1.1.1.1.1.cmml"><mo id="S3.SS1.SSS2.Px1.p1.1.m1.1.1.1.1.2" stretchy="false" xref="S3.SS1.SSS2.Px1.p1.1.m1.1.1.1.1.1.cmml">(</mo><mrow id="S3.SS1.SSS2.Px1.p1.1.m1.1.1.1.1.1" xref="S3.SS1.SSS2.Px1.p1.1.m1.1.1.1.1.1.cmml"><mi id="S3.SS1.SSS2.Px1.p1.1.m1.1.1.1.1.1.2" xref="S3.SS1.SSS2.Px1.p1.1.m1.1.1.1.1.1.2.cmml">t</mi><mo id="S3.SS1.SSS2.Px1.p1.1.m1.1.1.1.1.1.1" xref="S3.SS1.SSS2.Px1.p1.1.m1.1.1.1.1.1.1.cmml">⁢</mo><msup id="S3.SS1.SSS2.Px1.p1.1.m1.1.1.1.1.1.3" xref="S3.SS1.SSS2.Px1.p1.1.m1.1.1.1.1.1.3.cmml"><mi id="S3.SS1.SSS2.Px1.p1.1.m1.1.1.1.1.1.3.2" xref="S3.SS1.SSS2.Px1.p1.1.m1.1.1.1.1.1.3.2.cmml">k</mi><mn id="S3.SS1.SSS2.Px1.p1.1.m1.1.1.1.1.1.3.3" xref="S3.SS1.SSS2.Px1.p1.1.m1.1.1.1.1.1.3.3.cmml">4</mn></msup><mo id="S3.SS1.SSS2.Px1.p1.1.m1.1.1.1.1.1.1a" lspace="0.167em" xref="S3.SS1.SSS2.Px1.p1.1.m1.1.1.1.1.1.1.cmml">⁢</mo><mrow id="S3.SS1.SSS2.Px1.p1.1.m1.1.1.1.1.1.4" xref="S3.SS1.SSS2.Px1.p1.1.m1.1.1.1.1.1.4.cmml"><mi id="S3.SS1.SSS2.Px1.p1.1.m1.1.1.1.1.1.4.1" xref="S3.SS1.SSS2.Px1.p1.1.m1.1.1.1.1.1.4.1.cmml">log</mi><mo 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id="S3.SS1.SSS2.Px1.p1.1.m1.1.1.1.1.1.3.cmml" xref="S3.SS1.SSS2.Px1.p1.1.m1.1.1.1.1.1.3"><csymbol cd="ambiguous" id="S3.SS1.SSS2.Px1.p1.1.m1.1.1.1.1.1.3.1.cmml" xref="S3.SS1.SSS2.Px1.p1.1.m1.1.1.1.1.1.3">superscript</csymbol><ci id="S3.SS1.SSS2.Px1.p1.1.m1.1.1.1.1.1.3.2.cmml" xref="S3.SS1.SSS2.Px1.p1.1.m1.1.1.1.1.1.3.2">𝑘</ci><cn id="S3.SS1.SSS2.Px1.p1.1.m1.1.1.1.1.1.3.3.cmml" type="integer" xref="S3.SS1.SSS2.Px1.p1.1.m1.1.1.1.1.1.3.3">4</cn></apply><apply id="S3.SS1.SSS2.Px1.p1.1.m1.1.1.1.1.1.4.cmml" xref="S3.SS1.SSS2.Px1.p1.1.m1.1.1.1.1.1.4"><log id="S3.SS1.SSS2.Px1.p1.1.m1.1.1.1.1.1.4.1.cmml" xref="S3.SS1.SSS2.Px1.p1.1.m1.1.1.1.1.1.4.1"></log><ci id="S3.SS1.SSS2.Px1.p1.1.m1.1.1.1.1.1.4.2.cmml" xref="S3.SS1.SSS2.Px1.p1.1.m1.1.1.1.1.1.4.2">𝑛</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS2.Px1.p1.1.m1.1c">O(tk^{4}\log n)</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS2.Px1.p1.1.m1.1d">italic_O ( italic_t italic_k start_POSTSUPERSCRIPT 4 end_POSTSUPERSCRIPT roman_log italic_n )</annotation></semantics></math>-approximation in polynomial-time while by Corollary <a class="ltx_ref" href="https://arxiv.org/html/2503.00712v1#S3.Thmtheorem5" title="Corollary 3.5. ‣ 3.1.2 An Improved Analysis via Fractional Solutions ‣ 3.1 Vertex Connectivity Network Design ‣ 3 Generic Framework for Streaming Algorithms for Network Design ‣ Streaming Algorithms for Network Design"><span class="ltx_text ltx_ref_tag">3.5</span></a>, the approach via fractional solution yields an <math alttext="O(tk^{3}\log n)" class="ltx_Math" display="inline" id="S3.SS1.SSS2.Px1.p1.2.m2.1"><semantics id="S3.SS1.SSS2.Px1.p1.2.m2.1a"><mrow id="S3.SS1.SSS2.Px1.p1.2.m2.1.1" xref="S3.SS1.SSS2.Px1.p1.2.m2.1.1.cmml"><mi id="S3.SS1.SSS2.Px1.p1.2.m2.1.1.3" xref="S3.SS1.SSS2.Px1.p1.2.m2.1.1.3.cmml">O</mi><mo id="S3.SS1.SSS2.Px1.p1.2.m2.1.1.2" xref="S3.SS1.SSS2.Px1.p1.2.m2.1.1.2.cmml">⁢</mo><mrow id="S3.SS1.SSS2.Px1.p1.2.m2.1.1.1.1" 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n)</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS2.Px1.p1.2.m2.1d">italic_O ( italic_t italic_k start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT roman_log italic_n )</annotation></semantics></math>-approximation. Note that this is an improvement by a factor of <math alttext="k" class="ltx_Math" display="inline" id="S3.SS1.SSS2.Px1.p1.3.m3.1"><semantics id="S3.SS1.SSS2.Px1.p1.3.m3.1a"><mi id="S3.SS1.SSS2.Px1.p1.3.m3.1.1" xref="S3.SS1.SSS2.Px1.p1.3.m3.1.1.cmml">k</mi><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS2.Px1.p1.3.m3.1b"><ci id="S3.SS1.SSS2.Px1.p1.3.m3.1.1.cmml" xref="S3.SS1.SSS2.Px1.p1.3.m3.1.1">𝑘</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS2.Px1.p1.3.m3.1c">k</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS2.Px1.p1.3.m3.1d">italic_k</annotation></semantics></math>. The main advantage of the fractional analysis is for those cases where the integrality gap of the LP is small. We point out some of those cases and note that this is particularly useful for EC-SNDP and ELC-SNDP, which we discuss later.</p> </div> <div class="ltx_para" id="S3.SS1.SSS2.Px1.p2"> <ul class="ltx_itemize" id="S3.I2"> <li class="ltx_item" id="S3.I2.i1" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">•</span> <div class="ltx_para" id="S3.I2.i1.p1"> <p class="ltx_p" id="S3.I2.i1.p1.4">For <math alttext="\{0,1,2\}" class="ltx_Math" display="inline" id="S3.I2.i1.p1.1.m1.3"><semantics id="S3.I2.i1.p1.1.m1.3a"><mrow id="S3.I2.i1.p1.1.m1.3.4.2" xref="S3.I2.i1.p1.1.m1.3.4.1.cmml"><mo id="S3.I2.i1.p1.1.m1.3.4.2.1" stretchy="false" xref="S3.I2.i1.p1.1.m1.3.4.1.cmml">{</mo><mn id="S3.I2.i1.p1.1.m1.1.1" xref="S3.I2.i1.p1.1.m1.1.1.cmml">0</mn><mo id="S3.I2.i1.p1.1.m1.3.4.2.2" xref="S3.I2.i1.p1.1.m1.3.4.1.cmml">,</mo><mn id="S3.I2.i1.p1.1.m1.2.2" xref="S3.I2.i1.p1.1.m1.2.2.cmml">1</mn><mo id="S3.I2.i1.p1.1.m1.3.4.2.3" xref="S3.I2.i1.p1.1.m1.3.4.1.cmml">,</mo><mn id="S3.I2.i1.p1.1.m1.3.3" xref="S3.I2.i1.p1.1.m1.3.3.cmml">2</mn><mo id="S3.I2.i1.p1.1.m1.3.4.2.4" stretchy="false" xref="S3.I2.i1.p1.1.m1.3.4.1.cmml">}</mo></mrow><annotation-xml encoding="MathML-Content" id="S3.I2.i1.p1.1.m1.3b"><set id="S3.I2.i1.p1.1.m1.3.4.1.cmml" xref="S3.I2.i1.p1.1.m1.3.4.2"><cn id="S3.I2.i1.p1.1.m1.1.1.cmml" type="integer" xref="S3.I2.i1.p1.1.m1.1.1">0</cn><cn id="S3.I2.i1.p1.1.m1.2.2.cmml" type="integer" xref="S3.I2.i1.p1.1.m1.2.2">1</cn><cn id="S3.I2.i1.p1.1.m1.3.3.cmml" type="integer" xref="S3.I2.i1.p1.1.m1.3.3">2</cn></set></annotation-xml><annotation encoding="application/x-tex" id="S3.I2.i1.p1.1.m1.3c">\{0,1,2\}</annotation><annotation encoding="application/x-llamapun" id="S3.I2.i1.p1.1.m1.3d">{ 0 , 1 , 2 }</annotation></semantics></math>-VC-SNDP there is an algorithm that uses <math alttext="\tilde{O}(n^{1+1/t})" class="ltx_Math" display="inline" id="S3.I2.i1.p1.2.m2.1"><semantics id="S3.I2.i1.p1.2.m2.1a"><mrow id="S3.I2.i1.p1.2.m2.1.1" xref="S3.I2.i1.p1.2.m2.1.1.cmml"><mover accent="true" id="S3.I2.i1.p1.2.m2.1.1.3" xref="S3.I2.i1.p1.2.m2.1.1.3.cmml"><mi id="S3.I2.i1.p1.2.m2.1.1.3.2" xref="S3.I2.i1.p1.2.m2.1.1.3.2.cmml">O</mi><mo id="S3.I2.i1.p1.2.m2.1.1.3.1" xref="S3.I2.i1.p1.2.m2.1.1.3.1.cmml">~</mo></mover><mo id="S3.I2.i1.p1.2.m2.1.1.2" xref="S3.I2.i1.p1.2.m2.1.1.2.cmml">⁢</mo><mrow id="S3.I2.i1.p1.2.m2.1.1.1.1" xref="S3.I2.i1.p1.2.m2.1.1.1.1.1.cmml"><mo id="S3.I2.i1.p1.2.m2.1.1.1.1.2" stretchy="false" xref="S3.I2.i1.p1.2.m2.1.1.1.1.1.cmml">(</mo><msup id="S3.I2.i1.p1.2.m2.1.1.1.1.1" xref="S3.I2.i1.p1.2.m2.1.1.1.1.1.cmml"><mi id="S3.I2.i1.p1.2.m2.1.1.1.1.1.2" xref="S3.I2.i1.p1.2.m2.1.1.1.1.1.2.cmml">n</mi><mrow id="S3.I2.i1.p1.2.m2.1.1.1.1.1.3" xref="S3.I2.i1.p1.2.m2.1.1.1.1.1.3.cmml"><mn id="S3.I2.i1.p1.2.m2.1.1.1.1.1.3.2" xref="S3.I2.i1.p1.2.m2.1.1.1.1.1.3.2.cmml">1</mn><mo id="S3.I2.i1.p1.2.m2.1.1.1.1.1.3.1" xref="S3.I2.i1.p1.2.m2.1.1.1.1.1.3.1.cmml">+</mo><mrow id="S3.I2.i1.p1.2.m2.1.1.1.1.1.3.3" xref="S3.I2.i1.p1.2.m2.1.1.1.1.1.3.3.cmml"><mn id="S3.I2.i1.p1.2.m2.1.1.1.1.1.3.3.2" xref="S3.I2.i1.p1.2.m2.1.1.1.1.1.3.3.2.cmml">1</mn><mo id="S3.I2.i1.p1.2.m2.1.1.1.1.1.3.3.1" xref="S3.I2.i1.p1.2.m2.1.1.1.1.1.3.3.1.cmml">/</mo><mi id="S3.I2.i1.p1.2.m2.1.1.1.1.1.3.3.3" xref="S3.I2.i1.p1.2.m2.1.1.1.1.1.3.3.3.cmml">t</mi></mrow></mrow></msup><mo id="S3.I2.i1.p1.2.m2.1.1.1.1.3" stretchy="false" xref="S3.I2.i1.p1.2.m2.1.1.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.I2.i1.p1.2.m2.1b"><apply id="S3.I2.i1.p1.2.m2.1.1.cmml" xref="S3.I2.i1.p1.2.m2.1.1"><times id="S3.I2.i1.p1.2.m2.1.1.2.cmml" xref="S3.I2.i1.p1.2.m2.1.1.2"></times><apply id="S3.I2.i1.p1.2.m2.1.1.3.cmml" xref="S3.I2.i1.p1.2.m2.1.1.3"><ci id="S3.I2.i1.p1.2.m2.1.1.3.1.cmml" xref="S3.I2.i1.p1.2.m2.1.1.3.1">~</ci><ci id="S3.I2.i1.p1.2.m2.1.1.3.2.cmml" xref="S3.I2.i1.p1.2.m2.1.1.3.2">𝑂</ci></apply><apply id="S3.I2.i1.p1.2.m2.1.1.1.1.1.cmml" xref="S3.I2.i1.p1.2.m2.1.1.1.1"><csymbol cd="ambiguous" id="S3.I2.i1.p1.2.m2.1.1.1.1.1.1.cmml" xref="S3.I2.i1.p1.2.m2.1.1.1.1">superscript</csymbol><ci id="S3.I2.i1.p1.2.m2.1.1.1.1.1.2.cmml" xref="S3.I2.i1.p1.2.m2.1.1.1.1.1.2">𝑛</ci><apply id="S3.I2.i1.p1.2.m2.1.1.1.1.1.3.cmml" xref="S3.I2.i1.p1.2.m2.1.1.1.1.1.3"><plus id="S3.I2.i1.p1.2.m2.1.1.1.1.1.3.1.cmml" xref="S3.I2.i1.p1.2.m2.1.1.1.1.1.3.1"></plus><cn id="S3.I2.i1.p1.2.m2.1.1.1.1.1.3.2.cmml" type="integer" xref="S3.I2.i1.p1.2.m2.1.1.1.1.1.3.2">1</cn><apply id="S3.I2.i1.p1.2.m2.1.1.1.1.1.3.3.cmml" xref="S3.I2.i1.p1.2.m2.1.1.1.1.1.3.3"><divide id="S3.I2.i1.p1.2.m2.1.1.1.1.1.3.3.1.cmml" xref="S3.I2.i1.p1.2.m2.1.1.1.1.1.3.3.1"></divide><cn id="S3.I2.i1.p1.2.m2.1.1.1.1.1.3.3.2.cmml" type="integer" xref="S3.I2.i1.p1.2.m2.1.1.1.1.1.3.3.2">1</cn><ci id="S3.I2.i1.p1.2.m2.1.1.1.1.1.3.3.3.cmml" xref="S3.I2.i1.p1.2.m2.1.1.1.1.1.3.3.3">𝑡</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.I2.i1.p1.2.m2.1c">\tilde{O}(n^{1+1/t})</annotation><annotation encoding="application/x-llamapun" id="S3.I2.i1.p1.2.m2.1d">over~ start_ARG italic_O end_ARG ( italic_n start_POSTSUPERSCRIPT 1 + 1 / italic_t end_POSTSUPERSCRIPT )</annotation></semantics></math> space and outputs a <math alttext="8t" class="ltx_Math" display="inline" id="S3.I2.i1.p1.3.m3.1"><semantics id="S3.I2.i1.p1.3.m3.1a"><mrow id="S3.I2.i1.p1.3.m3.1.1" xref="S3.I2.i1.p1.3.m3.1.1.cmml"><mn id="S3.I2.i1.p1.3.m3.1.1.2" xref="S3.I2.i1.p1.3.m3.1.1.2.cmml">8</mn><mo id="S3.I2.i1.p1.3.m3.1.1.1" xref="S3.I2.i1.p1.3.m3.1.1.1.cmml">⁢</mo><mi id="S3.I2.i1.p1.3.m3.1.1.3" xref="S3.I2.i1.p1.3.m3.1.1.3.cmml">t</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.I2.i1.p1.3.m3.1b"><apply id="S3.I2.i1.p1.3.m3.1.1.cmml" xref="S3.I2.i1.p1.3.m3.1.1"><times id="S3.I2.i1.p1.3.m3.1.1.1.cmml" xref="S3.I2.i1.p1.3.m3.1.1.1"></times><cn id="S3.I2.i1.p1.3.m3.1.1.2.cmml" type="integer" xref="S3.I2.i1.p1.3.m3.1.1.2">8</cn><ci id="S3.I2.i1.p1.3.m3.1.1.3.cmml" xref="S3.I2.i1.p1.3.m3.1.1.3">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.I2.i1.p1.3.m3.1c">8t</annotation><annotation encoding="application/x-llamapun" id="S3.I2.i1.p1.3.m3.1d">8 italic_t</annotation></semantics></math>-approximate solution. This follows via the known integrality gap of <math alttext="2" class="ltx_Math" display="inline" id="S3.I2.i1.p1.4.m4.1"><semantics id="S3.I2.i1.p1.4.m4.1a"><mn id="S3.I2.i1.p1.4.m4.1.1" xref="S3.I2.i1.p1.4.m4.1.1.cmml">2</mn><annotation-xml encoding="MathML-Content" id="S3.I2.i1.p1.4.m4.1b"><cn id="S3.I2.i1.p1.4.m4.1.1.cmml" type="integer" xref="S3.I2.i1.p1.4.m4.1.1">2</cn></annotation-xml><annotation encoding="application/x-tex" id="S3.I2.i1.p1.4.m4.1c">2</annotation><annotation encoding="application/x-llamapun" id="S3.I2.i1.p1.4.m4.1d">2</annotation></semantics></math> for these instances <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.00712v1#bib.bibx39" title="">FJW06</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.00712v1#bib.bibx31" title="">CVV06</a>]</cite>.</p> </div> </li> <li class="ltx_item" id="S3.I2.i2" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">•</span> <div class="ltx_para" id="S3.I2.i2.p1"> <p class="ltx_p" id="S3.I2.i2.p1.7">For <math alttext="k" class="ltx_Math" display="inline" id="S3.I2.i2.p1.1.m1.1"><semantics id="S3.I2.i2.p1.1.m1.1a"><mi id="S3.I2.i2.p1.1.m1.1.1" xref="S3.I2.i2.p1.1.m1.1.1.cmml">k</mi><annotation-xml encoding="MathML-Content" id="S3.I2.i2.p1.1.m1.1b"><ci id="S3.I2.i2.p1.1.m1.1.1.cmml" xref="S3.I2.i2.p1.1.m1.1.1">𝑘</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.I2.i2.p1.1.m1.1c">k</annotation><annotation encoding="application/x-llamapun" id="S3.I2.i2.p1.1.m1.1d">italic_k</annotation></semantics></math>-VCSS there is an algorithm that uses <math alttext="\tilde{O}(k^{1-1/t}\cdot n^{1+1/t})" class="ltx_Math" display="inline" id="S3.I2.i2.p1.2.m2.1"><semantics id="S3.I2.i2.p1.2.m2.1a"><mrow id="S3.I2.i2.p1.2.m2.1.1" xref="S3.I2.i2.p1.2.m2.1.1.cmml"><mover accent="true" id="S3.I2.i2.p1.2.m2.1.1.3" xref="S3.I2.i2.p1.2.m2.1.1.3.cmml"><mi id="S3.I2.i2.p1.2.m2.1.1.3.2" xref="S3.I2.i2.p1.2.m2.1.1.3.2.cmml">O</mi><mo id="S3.I2.i2.p1.2.m2.1.1.3.1" xref="S3.I2.i2.p1.2.m2.1.1.3.1.cmml">~</mo></mover><mo id="S3.I2.i2.p1.2.m2.1.1.2" xref="S3.I2.i2.p1.2.m2.1.1.2.cmml">⁢</mo><mrow id="S3.I2.i2.p1.2.m2.1.1.1.1" xref="S3.I2.i2.p1.2.m2.1.1.1.1.1.cmml"><mo id="S3.I2.i2.p1.2.m2.1.1.1.1.2" stretchy="false" xref="S3.I2.i2.p1.2.m2.1.1.1.1.1.cmml">(</mo><mrow id="S3.I2.i2.p1.2.m2.1.1.1.1.1" xref="S3.I2.i2.p1.2.m2.1.1.1.1.1.cmml"><msup id="S3.I2.i2.p1.2.m2.1.1.1.1.1.2" xref="S3.I2.i2.p1.2.m2.1.1.1.1.1.2.cmml"><mi id="S3.I2.i2.p1.2.m2.1.1.1.1.1.2.2" xref="S3.I2.i2.p1.2.m2.1.1.1.1.1.2.2.cmml">k</mi><mrow id="S3.I2.i2.p1.2.m2.1.1.1.1.1.2.3" xref="S3.I2.i2.p1.2.m2.1.1.1.1.1.2.3.cmml"><mn id="S3.I2.i2.p1.2.m2.1.1.1.1.1.2.3.2" xref="S3.I2.i2.p1.2.m2.1.1.1.1.1.2.3.2.cmml">1</mn><mo id="S3.I2.i2.p1.2.m2.1.1.1.1.1.2.3.1" xref="S3.I2.i2.p1.2.m2.1.1.1.1.1.2.3.1.cmml">−</mo><mrow id="S3.I2.i2.p1.2.m2.1.1.1.1.1.2.3.3" xref="S3.I2.i2.p1.2.m2.1.1.1.1.1.2.3.3.cmml"><mn id="S3.I2.i2.p1.2.m2.1.1.1.1.1.2.3.3.2" 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xref="S3.I2.i2.p1.2.m2.1.1.1.1.1.3.3.3.2.cmml">1</mn><mo id="S3.I2.i2.p1.2.m2.1.1.1.1.1.3.3.3.1" xref="S3.I2.i2.p1.2.m2.1.1.1.1.1.3.3.3.1.cmml">/</mo><mi id="S3.I2.i2.p1.2.m2.1.1.1.1.1.3.3.3.3" xref="S3.I2.i2.p1.2.m2.1.1.1.1.1.3.3.3.3.cmml">t</mi></mrow></mrow></msup></mrow><mo id="S3.I2.i2.p1.2.m2.1.1.1.1.3" stretchy="false" xref="S3.I2.i2.p1.2.m2.1.1.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.I2.i2.p1.2.m2.1b"><apply id="S3.I2.i2.p1.2.m2.1.1.cmml" xref="S3.I2.i2.p1.2.m2.1.1"><times id="S3.I2.i2.p1.2.m2.1.1.2.cmml" xref="S3.I2.i2.p1.2.m2.1.1.2"></times><apply id="S3.I2.i2.p1.2.m2.1.1.3.cmml" xref="S3.I2.i2.p1.2.m2.1.1.3"><ci id="S3.I2.i2.p1.2.m2.1.1.3.1.cmml" xref="S3.I2.i2.p1.2.m2.1.1.3.1">~</ci><ci id="S3.I2.i2.p1.2.m2.1.1.3.2.cmml" xref="S3.I2.i2.p1.2.m2.1.1.3.2">𝑂</ci></apply><apply id="S3.I2.i2.p1.2.m2.1.1.1.1.1.cmml" xref="S3.I2.i2.p1.2.m2.1.1.1.1"><ci id="S3.I2.i2.p1.2.m2.1.1.1.1.1.1.cmml" xref="S3.I2.i2.p1.2.m2.1.1.1.1.1.1">⋅</ci><apply id="S3.I2.i2.p1.2.m2.1.1.1.1.1.2.cmml" xref="S3.I2.i2.p1.2.m2.1.1.1.1.1.2"><csymbol cd="ambiguous" id="S3.I2.i2.p1.2.m2.1.1.1.1.1.2.1.cmml" xref="S3.I2.i2.p1.2.m2.1.1.1.1.1.2">superscript</csymbol><ci id="S3.I2.i2.p1.2.m2.1.1.1.1.1.2.2.cmml" xref="S3.I2.i2.p1.2.m2.1.1.1.1.1.2.2">𝑘</ci><apply id="S3.I2.i2.p1.2.m2.1.1.1.1.1.2.3.cmml" xref="S3.I2.i2.p1.2.m2.1.1.1.1.1.2.3"><minus id="S3.I2.i2.p1.2.m2.1.1.1.1.1.2.3.1.cmml" xref="S3.I2.i2.p1.2.m2.1.1.1.1.1.2.3.1"></minus><cn id="S3.I2.i2.p1.2.m2.1.1.1.1.1.2.3.2.cmml" type="integer" xref="S3.I2.i2.p1.2.m2.1.1.1.1.1.2.3.2">1</cn><apply id="S3.I2.i2.p1.2.m2.1.1.1.1.1.2.3.3.cmml" xref="S3.I2.i2.p1.2.m2.1.1.1.1.1.2.3.3"><divide id="S3.I2.i2.p1.2.m2.1.1.1.1.1.2.3.3.1.cmml" xref="S3.I2.i2.p1.2.m2.1.1.1.1.1.2.3.3.1"></divide><cn id="S3.I2.i2.p1.2.m2.1.1.1.1.1.2.3.3.2.cmml" type="integer" xref="S3.I2.i2.p1.2.m2.1.1.1.1.1.2.3.3.2">1</cn><ci id="S3.I2.i2.p1.2.m2.1.1.1.1.1.2.3.3.3.cmml" xref="S3.I2.i2.p1.2.m2.1.1.1.1.1.2.3.3.3">𝑡</ci></apply></apply></apply><apply id="S3.I2.i2.p1.2.m2.1.1.1.1.1.3.cmml" xref="S3.I2.i2.p1.2.m2.1.1.1.1.1.3"><csymbol cd="ambiguous" id="S3.I2.i2.p1.2.m2.1.1.1.1.1.3.1.cmml" xref="S3.I2.i2.p1.2.m2.1.1.1.1.1.3">superscript</csymbol><ci id="S3.I2.i2.p1.2.m2.1.1.1.1.1.3.2.cmml" xref="S3.I2.i2.p1.2.m2.1.1.1.1.1.3.2">𝑛</ci><apply id="S3.I2.i2.p1.2.m2.1.1.1.1.1.3.3.cmml" xref="S3.I2.i2.p1.2.m2.1.1.1.1.1.3.3"><plus id="S3.I2.i2.p1.2.m2.1.1.1.1.1.3.3.1.cmml" xref="S3.I2.i2.p1.2.m2.1.1.1.1.1.3.3.1"></plus><cn id="S3.I2.i2.p1.2.m2.1.1.1.1.1.3.3.2.cmml" type="integer" xref="S3.I2.i2.p1.2.m2.1.1.1.1.1.3.3.2">1</cn><apply id="S3.I2.i2.p1.2.m2.1.1.1.1.1.3.3.3.cmml" xref="S3.I2.i2.p1.2.m2.1.1.1.1.1.3.3.3"><divide id="S3.I2.i2.p1.2.m2.1.1.1.1.1.3.3.3.1.cmml" xref="S3.I2.i2.p1.2.m2.1.1.1.1.1.3.3.3.1"></divide><cn id="S3.I2.i2.p1.2.m2.1.1.1.1.1.3.3.3.2.cmml" type="integer" xref="S3.I2.i2.p1.2.m2.1.1.1.1.1.3.3.3.2">1</cn><ci id="S3.I2.i2.p1.2.m2.1.1.1.1.1.3.3.3.3.cmml" xref="S3.I2.i2.p1.2.m2.1.1.1.1.1.3.3.3.3">𝑡</ci></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.I2.i2.p1.2.m2.1c">\tilde{O}(k^{1-1/t}\cdot n^{1+1/t})</annotation><annotation encoding="application/x-llamapun" id="S3.I2.i2.p1.2.m2.1d">over~ start_ARG italic_O end_ARG ( italic_k start_POSTSUPERSCRIPT 1 - 1 / italic_t end_POSTSUPERSCRIPT ⋅ italic_n start_POSTSUPERSCRIPT 1 + 1 / italic_t end_POSTSUPERSCRIPT )</annotation></semantics></math> space and outputs a <math alttext="(16+\epsilon)t" class="ltx_Math" display="inline" id="S3.I2.i2.p1.3.m3.1"><semantics id="S3.I2.i2.p1.3.m3.1a"><mrow id="S3.I2.i2.p1.3.m3.1.1" xref="S3.I2.i2.p1.3.m3.1.1.cmml"><mrow id="S3.I2.i2.p1.3.m3.1.1.1.1" xref="S3.I2.i2.p1.3.m3.1.1.1.1.1.cmml"><mo id="S3.I2.i2.p1.3.m3.1.1.1.1.2" stretchy="false" xref="S3.I2.i2.p1.3.m3.1.1.1.1.1.cmml">(</mo><mrow id="S3.I2.i2.p1.3.m3.1.1.1.1.1" xref="S3.I2.i2.p1.3.m3.1.1.1.1.1.cmml"><mn id="S3.I2.i2.p1.3.m3.1.1.1.1.1.2" xref="S3.I2.i2.p1.3.m3.1.1.1.1.1.2.cmml">16</mn><mo id="S3.I2.i2.p1.3.m3.1.1.1.1.1.1" xref="S3.I2.i2.p1.3.m3.1.1.1.1.1.1.cmml">+</mo><mi id="S3.I2.i2.p1.3.m3.1.1.1.1.1.3" xref="S3.I2.i2.p1.3.m3.1.1.1.1.1.3.cmml">ϵ</mi></mrow><mo id="S3.I2.i2.p1.3.m3.1.1.1.1.3" stretchy="false" xref="S3.I2.i2.p1.3.m3.1.1.1.1.1.cmml">)</mo></mrow><mo id="S3.I2.i2.p1.3.m3.1.1.2" xref="S3.I2.i2.p1.3.m3.1.1.2.cmml">⁢</mo><mi id="S3.I2.i2.p1.3.m3.1.1.3" xref="S3.I2.i2.p1.3.m3.1.1.3.cmml">t</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.I2.i2.p1.3.m3.1b"><apply id="S3.I2.i2.p1.3.m3.1.1.cmml" xref="S3.I2.i2.p1.3.m3.1.1"><times id="S3.I2.i2.p1.3.m3.1.1.2.cmml" xref="S3.I2.i2.p1.3.m3.1.1.2"></times><apply id="S3.I2.i2.p1.3.m3.1.1.1.1.1.cmml" xref="S3.I2.i2.p1.3.m3.1.1.1.1"><plus id="S3.I2.i2.p1.3.m3.1.1.1.1.1.1.cmml" xref="S3.I2.i2.p1.3.m3.1.1.1.1.1.1"></plus><cn id="S3.I2.i2.p1.3.m3.1.1.1.1.1.2.cmml" type="integer" xref="S3.I2.i2.p1.3.m3.1.1.1.1.1.2">16</cn><ci id="S3.I2.i2.p1.3.m3.1.1.1.1.1.3.cmml" xref="S3.I2.i2.p1.3.m3.1.1.1.1.1.3">italic-ϵ</ci></apply><ci id="S3.I2.i2.p1.3.m3.1.1.3.cmml" xref="S3.I2.i2.p1.3.m3.1.1.3">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.I2.i2.p1.3.m3.1c">(16+\epsilon)t</annotation><annotation encoding="application/x-llamapun" id="S3.I2.i2.p1.3.m3.1d">( 16 + italic_ϵ ) italic_t</annotation></semantics></math>-approximate solution when <math alttext="n" class="ltx_Math" display="inline" id="S3.I2.i2.p1.4.m4.1"><semantics id="S3.I2.i2.p1.4.m4.1a"><mi id="S3.I2.i2.p1.4.m4.1.1" xref="S3.I2.i2.p1.4.m4.1.1.cmml">n</mi><annotation-xml encoding="MathML-Content" id="S3.I2.i2.p1.4.m4.1b"><ci id="S3.I2.i2.p1.4.m4.1.1.cmml" xref="S3.I2.i2.p1.4.m4.1.1">𝑛</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.I2.i2.p1.4.m4.1c">n</annotation><annotation encoding="application/x-llamapun" id="S3.I2.i2.p1.4.m4.1d">italic_n</annotation></semantics></math> is sufficiently large compared to <math alttext="k" class="ltx_Math" display="inline" id="S3.I2.i2.p1.5.m5.1"><semantics id="S3.I2.i2.p1.5.m5.1a"><mi id="S3.I2.i2.p1.5.m5.1.1" xref="S3.I2.i2.p1.5.m5.1.1.cmml">k</mi><annotation-xml encoding="MathML-Content" id="S3.I2.i2.p1.5.m5.1b"><ci id="S3.I2.i2.p1.5.m5.1.1.cmml" xref="S3.I2.i2.p1.5.m5.1.1">𝑘</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.I2.i2.p1.5.m5.1c">k</annotation><annotation encoding="application/x-llamapun" id="S3.I2.i2.p1.5.m5.1d">italic_k</annotation></semantics></math>. This follows from the known the integrality gap results for <math alttext="k" class="ltx_Math" display="inline" id="S3.I2.i2.p1.6.m6.1"><semantics id="S3.I2.i2.p1.6.m6.1a"><mi id="S3.I2.i2.p1.6.m6.1.1" xref="S3.I2.i2.p1.6.m6.1.1.cmml">k</mi><annotation-xml encoding="MathML-Content" id="S3.I2.i2.p1.6.m6.1b"><ci id="S3.I2.i2.p1.6.m6.1.1.cmml" xref="S3.I2.i2.p1.6.m6.1.1">𝑘</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.I2.i2.p1.6.m6.1c">k</annotation><annotation encoding="application/x-llamapun" id="S3.I2.i2.p1.6.m6.1d">italic_k</annotation></semantics></math>-VCSS from <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.00712v1#bib.bibx74" title="">Nut22</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.00712v1#bib.bibx30" title="">CV14</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.00712v1#bib.bibx43" title="">FNR15</a>]</cite>. This also implies a similar result for the connectivity augmentation problem <math alttext="k" class="ltx_Math" display="inline" id="S3.I2.i2.p1.7.m7.1"><semantics id="S3.I2.i2.p1.7.m7.1a"><mi id="S3.I2.i2.p1.7.m7.1.1" xref="S3.I2.i2.p1.7.m7.1.1.cmml">k</mi><annotation-xml encoding="MathML-Content" id="S3.I2.i2.p1.7.m7.1b"><ci id="S3.I2.i2.p1.7.m7.1.1.cmml" xref="S3.I2.i2.p1.7.m7.1.1">𝑘</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.I2.i2.p1.7.m7.1c">k</annotation><annotation encoding="application/x-llamapun" id="S3.I2.i2.p1.7.m7.1d">italic_k</annotation></semantics></math>-VC-CAP.</p> </div> </li> <li class="ltx_item" id="S3.I2.i3" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">•</span> <div class="ltx_para" id="S3.I2.i3.p1"> <p class="ltx_p" id="S3.I2.i3.p1.7">For finding the cheapest <math alttext="k" class="ltx_Math" display="inline" id="S3.I2.i3.p1.1.m1.1"><semantics id="S3.I2.i3.p1.1.m1.1a"><mi id="S3.I2.i3.p1.1.m1.1.1" xref="S3.I2.i3.p1.1.m1.1.1.cmml">k</mi><annotation-xml encoding="MathML-Content" id="S3.I2.i3.p1.1.m1.1b"><ci id="S3.I2.i3.p1.1.m1.1.1.cmml" xref="S3.I2.i3.p1.1.m1.1.1">𝑘</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.I2.i3.p1.1.m1.1c">k</annotation><annotation encoding="application/x-llamapun" id="S3.I2.i3.p1.1.m1.1d">italic_k</annotation></semantics></math> vertex-disjoint <math alttext="s" class="ltx_Math" display="inline" id="S3.I2.i3.p1.2.m2.1"><semantics id="S3.I2.i3.p1.2.m2.1a"><mi id="S3.I2.i3.p1.2.m2.1.1" xref="S3.I2.i3.p1.2.m2.1.1.cmml">s</mi><annotation-xml encoding="MathML-Content" id="S3.I2.i3.p1.2.m2.1b"><ci id="S3.I2.i3.p1.2.m2.1.1.cmml" xref="S3.I2.i3.p1.2.m2.1.1">𝑠</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.I2.i3.p1.2.m2.1c">s</annotation><annotation encoding="application/x-llamapun" id="S3.I2.i3.p1.2.m2.1d">italic_s</annotation></semantics></math>-<math alttext="t" class="ltx_Math" display="inline" id="S3.I2.i3.p1.3.m3.1"><semantics id="S3.I2.i3.p1.3.m3.1a"><mi id="S3.I2.i3.p1.3.m3.1.1" xref="S3.I2.i3.p1.3.m3.1.1.cmml">t</mi><annotation-xml encoding="MathML-Content" id="S3.I2.i3.p1.3.m3.1b"><ci id="S3.I2.i3.p1.3.m3.1.1.cmml" xref="S3.I2.i3.p1.3.m3.1.1">𝑡</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.I2.i3.p1.3.m3.1c">t</annotation><annotation encoding="application/x-llamapun" id="S3.I2.i3.p1.3.m3.1d">italic_t</annotation></semantics></math> paths, there is an algorithm that uses <math alttext="\tilde{O}(k^{1-1/t}\cdot n^{1+1/t})" class="ltx_Math" display="inline" id="S3.I2.i3.p1.4.m4.1"><semantics id="S3.I2.i3.p1.4.m4.1a"><mrow id="S3.I2.i3.p1.4.m4.1.1" xref="S3.I2.i3.p1.4.m4.1.1.cmml"><mover accent="true" id="S3.I2.i3.p1.4.m4.1.1.3" xref="S3.I2.i3.p1.4.m4.1.1.3.cmml"><mi id="S3.I2.i3.p1.4.m4.1.1.3.2" xref="S3.I2.i3.p1.4.m4.1.1.3.2.cmml">O</mi><mo id="S3.I2.i3.p1.4.m4.1.1.3.1" xref="S3.I2.i3.p1.4.m4.1.1.3.1.cmml">~</mo></mover><mo 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xref="S3.I2.i3.p1.4.m4.1.1.1.1.1.3.3.3.1.cmml">/</mo><mi id="S3.I2.i3.p1.4.m4.1.1.1.1.1.3.3.3.3" xref="S3.I2.i3.p1.4.m4.1.1.1.1.1.3.3.3.3.cmml">t</mi></mrow></mrow></msup></mrow><mo id="S3.I2.i3.p1.4.m4.1.1.1.1.3" stretchy="false" xref="S3.I2.i3.p1.4.m4.1.1.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.I2.i3.p1.4.m4.1b"><apply id="S3.I2.i3.p1.4.m4.1.1.cmml" xref="S3.I2.i3.p1.4.m4.1.1"><times id="S3.I2.i3.p1.4.m4.1.1.2.cmml" xref="S3.I2.i3.p1.4.m4.1.1.2"></times><apply id="S3.I2.i3.p1.4.m4.1.1.3.cmml" xref="S3.I2.i3.p1.4.m4.1.1.3"><ci id="S3.I2.i3.p1.4.m4.1.1.3.1.cmml" xref="S3.I2.i3.p1.4.m4.1.1.3.1">~</ci><ci id="S3.I2.i3.p1.4.m4.1.1.3.2.cmml" xref="S3.I2.i3.p1.4.m4.1.1.3.2">𝑂</ci></apply><apply id="S3.I2.i3.p1.4.m4.1.1.1.1.1.cmml" xref="S3.I2.i3.p1.4.m4.1.1.1.1"><ci id="S3.I2.i3.p1.4.m4.1.1.1.1.1.1.cmml" xref="S3.I2.i3.p1.4.m4.1.1.1.1.1.1">⋅</ci><apply id="S3.I2.i3.p1.4.m4.1.1.1.1.1.2.cmml" xref="S3.I2.i3.p1.4.m4.1.1.1.1.1.2"><csymbol cd="ambiguous" 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id="S3.I2.i3.p1.4.m4.1c">\tilde{O}(k^{1-1/t}\cdot n^{1+1/t})</annotation><annotation encoding="application/x-llamapun" id="S3.I2.i3.p1.4.m4.1d">over~ start_ARG italic_O end_ARG ( italic_k start_POSTSUPERSCRIPT 1 - 1 / italic_t end_POSTSUPERSCRIPT ⋅ italic_n start_POSTSUPERSCRIPT 1 + 1 / italic_t end_POSTSUPERSCRIPT )</annotation></semantics></math> space and outputs a <math alttext="4t" class="ltx_Math" display="inline" id="S3.I2.i3.p1.5.m5.1"><semantics id="S3.I2.i3.p1.5.m5.1a"><mrow id="S3.I2.i3.p1.5.m5.1.1" xref="S3.I2.i3.p1.5.m5.1.1.cmml"><mn id="S3.I2.i3.p1.5.m5.1.1.2" xref="S3.I2.i3.p1.5.m5.1.1.2.cmml">4</mn><mo id="S3.I2.i3.p1.5.m5.1.1.1" xref="S3.I2.i3.p1.5.m5.1.1.1.cmml">⁢</mo><mi id="S3.I2.i3.p1.5.m5.1.1.3" xref="S3.I2.i3.p1.5.m5.1.1.3.cmml">t</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.I2.i3.p1.5.m5.1b"><apply id="S3.I2.i3.p1.5.m5.1.1.cmml" xref="S3.I2.i3.p1.5.m5.1.1"><times id="S3.I2.i3.p1.5.m5.1.1.1.cmml" xref="S3.I2.i3.p1.5.m5.1.1.1"></times><cn id="S3.I2.i3.p1.5.m5.1.1.2.cmml" type="integer" xref="S3.I2.i3.p1.5.m5.1.1.2">4</cn><ci id="S3.I2.i3.p1.5.m5.1.1.3.cmml" xref="S3.I2.i3.p1.5.m5.1.1.3">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.I2.i3.p1.5.m5.1c">4t</annotation><annotation encoding="application/x-llamapun" id="S3.I2.i3.p1.5.m5.1d">4 italic_t</annotation></semantics></math>-approximate solution. This follows from the fact that the flow-LP is optimal for <math alttext="s" class="ltx_Math" display="inline" id="S3.I2.i3.p1.6.m6.1"><semantics id="S3.I2.i3.p1.6.m6.1a"><mi id="S3.I2.i3.p1.6.m6.1.1" xref="S3.I2.i3.p1.6.m6.1.1.cmml">s</mi><annotation-xml encoding="MathML-Content" id="S3.I2.i3.p1.6.m6.1b"><ci id="S3.I2.i3.p1.6.m6.1.1.cmml" xref="S3.I2.i3.p1.6.m6.1.1">𝑠</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.I2.i3.p1.6.m6.1c">s</annotation><annotation encoding="application/x-llamapun" id="S3.I2.i3.p1.6.m6.1d">italic_s</annotation></semantics></math>-<math alttext="t" class="ltx_Math" display="inline" id="S3.I2.i3.p1.7.m7.1"><semantics id="S3.I2.i3.p1.7.m7.1a"><mi id="S3.I2.i3.p1.7.m7.1.1" xref="S3.I2.i3.p1.7.m7.1.1.cmml">t</mi><annotation-xml encoding="MathML-Content" id="S3.I2.i3.p1.7.m7.1b"><ci id="S3.I2.i3.p1.7.m7.1.1.cmml" xref="S3.I2.i3.p1.7.m7.1.1">𝑡</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.I2.i3.p1.7.m7.1c">t</annotation><annotation encoding="application/x-llamapun" id="S3.I2.i3.p1.7.m7.1d">italic_t</annotation></semantics></math> disjoint paths.</p> </div> </li> </ul> <p class="ltx_p" id="S3.SS1.SSS2.Px1.p2.7">We believe that the regime of <math alttext="k" class="ltx_Math" display="inline" id="S3.SS1.SSS2.Px1.p2.1.m1.1"><semantics id="S3.SS1.SSS2.Px1.p2.1.m1.1a"><mi id="S3.SS1.SSS2.Px1.p2.1.m1.1.1" xref="S3.SS1.SSS2.Px1.p2.1.m1.1.1.cmml">k</mi><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS2.Px1.p2.1.m1.1b"><ci id="S3.SS1.SSS2.Px1.p2.1.m1.1.1.cmml" xref="S3.SS1.SSS2.Px1.p2.1.m1.1.1">𝑘</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS2.Px1.p2.1.m1.1c">k</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS2.Px1.p2.1.m1.1d">italic_k</annotation></semantics></math> being small compared to <math alttext="n" class="ltx_Math" display="inline" id="S3.SS1.SSS2.Px1.p2.2.m2.1"><semantics id="S3.SS1.SSS2.Px1.p2.2.m2.1a"><mi id="S3.SS1.SSS2.Px1.p2.2.m2.1.1" xref="S3.SS1.SSS2.Px1.p2.2.m2.1.1.cmml">n</mi><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS2.Px1.p2.2.m2.1b"><ci id="S3.SS1.SSS2.Px1.p2.2.m2.1.1.cmml" xref="S3.SS1.SSS2.Px1.p2.2.m2.1.1">𝑛</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS2.Px1.p2.2.m2.1c">n</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS2.Px1.p2.2.m2.1d">italic_n</annotation></semantics></math> is the main interest in the streaming setting. For large values of <math alttext="k" class="ltx_Math" display="inline" id="S3.SS1.SSS2.Px1.p2.3.m3.1"><semantics id="S3.SS1.SSS2.Px1.p2.3.m3.1a"><mi id="S3.SS1.SSS2.Px1.p2.3.m3.1.1" xref="S3.SS1.SSS2.Px1.p2.3.m3.1.1.cmml">k</mi><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS2.Px1.p2.3.m3.1b"><ci id="S3.SS1.SSS2.Px1.p2.3.m3.1.1.cmml" xref="S3.SS1.SSS2.Px1.p2.3.m3.1.1">𝑘</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS2.Px1.p2.3.m3.1c">k</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS2.Px1.p2.3.m3.1d">italic_k</annotation></semantics></math>, <math alttext="O(\log(\frac{n}{n-k}))" class="ltx_Math" display="inline" id="S3.SS1.SSS2.Px1.p2.4.m4.3"><semantics id="S3.SS1.SSS2.Px1.p2.4.m4.3a"><mrow id="S3.SS1.SSS2.Px1.p2.4.m4.3.3" xref="S3.SS1.SSS2.Px1.p2.4.m4.3.3.cmml"><mi id="S3.SS1.SSS2.Px1.p2.4.m4.3.3.3" xref="S3.SS1.SSS2.Px1.p2.4.m4.3.3.3.cmml">O</mi><mo id="S3.SS1.SSS2.Px1.p2.4.m4.3.3.2" xref="S3.SS1.SSS2.Px1.p2.4.m4.3.3.2.cmml">⁢</mo><mrow id="S3.SS1.SSS2.Px1.p2.4.m4.3.3.1.1" xref="S3.SS1.SSS2.Px1.p2.4.m4.3.3.cmml"><mo id="S3.SS1.SSS2.Px1.p2.4.m4.3.3.1.1.2" stretchy="false" xref="S3.SS1.SSS2.Px1.p2.4.m4.3.3.cmml">(</mo><mrow id="S3.SS1.SSS2.Px1.p2.4.m4.3.3.1.1.1.2" xref="S3.SS1.SSS2.Px1.p2.4.m4.3.3.1.1.1.1.cmml"><mi id="S3.SS1.SSS2.Px1.p2.4.m4.1.1" xref="S3.SS1.SSS2.Px1.p2.4.m4.1.1.cmml">log</mi><mo id="S3.SS1.SSS2.Px1.p2.4.m4.3.3.1.1.1.2a" xref="S3.SS1.SSS2.Px1.p2.4.m4.3.3.1.1.1.1.cmml">⁡</mo><mrow id="S3.SS1.SSS2.Px1.p2.4.m4.3.3.1.1.1.2.1" xref="S3.SS1.SSS2.Px1.p2.4.m4.3.3.1.1.1.1.cmml"><mo id="S3.SS1.SSS2.Px1.p2.4.m4.3.3.1.1.1.2.1.1" stretchy="false" xref="S3.SS1.SSS2.Px1.p2.4.m4.3.3.1.1.1.1.cmml">(</mo><mfrac id="S3.SS1.SSS2.Px1.p2.4.m4.2.2" xref="S3.SS1.SSS2.Px1.p2.4.m4.2.2.cmml"><mi id="S3.SS1.SSS2.Px1.p2.4.m4.2.2.2" xref="S3.SS1.SSS2.Px1.p2.4.m4.2.2.2.cmml">n</mi><mrow id="S3.SS1.SSS2.Px1.p2.4.m4.2.2.3" xref="S3.SS1.SSS2.Px1.p2.4.m4.2.2.3.cmml"><mi id="S3.SS1.SSS2.Px1.p2.4.m4.2.2.3.2" xref="S3.SS1.SSS2.Px1.p2.4.m4.2.2.3.2.cmml">n</mi><mo id="S3.SS1.SSS2.Px1.p2.4.m4.2.2.3.1" xref="S3.SS1.SSS2.Px1.p2.4.m4.2.2.3.1.cmml">−</mo><mi id="S3.SS1.SSS2.Px1.p2.4.m4.2.2.3.3" xref="S3.SS1.SSS2.Px1.p2.4.m4.2.2.3.3.cmml">k</mi></mrow></mfrac><mo id="S3.SS1.SSS2.Px1.p2.4.m4.3.3.1.1.1.2.1.2" stretchy="false" xref="S3.SS1.SSS2.Px1.p2.4.m4.3.3.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S3.SS1.SSS2.Px1.p2.4.m4.3.3.1.1.3" stretchy="false" xref="S3.SS1.SSS2.Px1.p2.4.m4.3.3.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS2.Px1.p2.4.m4.3b"><apply id="S3.SS1.SSS2.Px1.p2.4.m4.3.3.cmml" xref="S3.SS1.SSS2.Px1.p2.4.m4.3.3"><times id="S3.SS1.SSS2.Px1.p2.4.m4.3.3.2.cmml" xref="S3.SS1.SSS2.Px1.p2.4.m4.3.3.2"></times><ci id="S3.SS1.SSS2.Px1.p2.4.m4.3.3.3.cmml" xref="S3.SS1.SSS2.Px1.p2.4.m4.3.3.3">𝑂</ci><apply id="S3.SS1.SSS2.Px1.p2.4.m4.3.3.1.1.1.1.cmml" xref="S3.SS1.SSS2.Px1.p2.4.m4.3.3.1.1.1.2"><log id="S3.SS1.SSS2.Px1.p2.4.m4.1.1.cmml" xref="S3.SS1.SSS2.Px1.p2.4.m4.1.1"></log><apply id="S3.SS1.SSS2.Px1.p2.4.m4.2.2.cmml" xref="S3.SS1.SSS2.Px1.p2.4.m4.2.2"><divide id="S3.SS1.SSS2.Px1.p2.4.m4.2.2.1.cmml" xref="S3.SS1.SSS2.Px1.p2.4.m4.2.2"></divide><ci id="S3.SS1.SSS2.Px1.p2.4.m4.2.2.2.cmml" xref="S3.SS1.SSS2.Px1.p2.4.m4.2.2.2">𝑛</ci><apply id="S3.SS1.SSS2.Px1.p2.4.m4.2.2.3.cmml" xref="S3.SS1.SSS2.Px1.p2.4.m4.2.2.3"><minus id="S3.SS1.SSS2.Px1.p2.4.m4.2.2.3.1.cmml" xref="S3.SS1.SSS2.Px1.p2.4.m4.2.2.3.1"></minus><ci id="S3.SS1.SSS2.Px1.p2.4.m4.2.2.3.2.cmml" xref="S3.SS1.SSS2.Px1.p2.4.m4.2.2.3.2">𝑛</ci><ci id="S3.SS1.SSS2.Px1.p2.4.m4.2.2.3.3.cmml" xref="S3.SS1.SSS2.Px1.p2.4.m4.2.2.3.3">𝑘</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS2.Px1.p2.4.m4.3c">O(\log(\frac{n}{n-k}))</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS2.Px1.p2.4.m4.3d">italic_O ( roman_log ( divide start_ARG italic_n end_ARG start_ARG italic_n - italic_k end_ARG ) )</annotation></semantics></math> and <math alttext="O(\log k\cdot\log(\frac{n}{n-k}))" class="ltx_Math" display="inline" id="S3.SS1.SSS2.Px1.p2.5.m5.3"><semantics id="S3.SS1.SSS2.Px1.p2.5.m5.3a"><mrow id="S3.SS1.SSS2.Px1.p2.5.m5.3.3" xref="S3.SS1.SSS2.Px1.p2.5.m5.3.3.cmml"><mi id="S3.SS1.SSS2.Px1.p2.5.m5.3.3.3" xref="S3.SS1.SSS2.Px1.p2.5.m5.3.3.3.cmml">O</mi><mo id="S3.SS1.SSS2.Px1.p2.5.m5.3.3.2" xref="S3.SS1.SSS2.Px1.p2.5.m5.3.3.2.cmml">⁢</mo><mrow id="S3.SS1.SSS2.Px1.p2.5.m5.3.3.1.1" xref="S3.SS1.SSS2.Px1.p2.5.m5.3.3.1.1.1.cmml"><mo id="S3.SS1.SSS2.Px1.p2.5.m5.3.3.1.1.2" stretchy="false" xref="S3.SS1.SSS2.Px1.p2.5.m5.3.3.1.1.1.cmml">(</mo><mrow id="S3.SS1.SSS2.Px1.p2.5.m5.3.3.1.1.1" xref="S3.SS1.SSS2.Px1.p2.5.m5.3.3.1.1.1.cmml"><mi id="S3.SS1.SSS2.Px1.p2.5.m5.3.3.1.1.1.1" xref="S3.SS1.SSS2.Px1.p2.5.m5.3.3.1.1.1.1.cmml">log</mi><mo id="S3.SS1.SSS2.Px1.p2.5.m5.3.3.1.1.1a" lspace="0.167em" xref="S3.SS1.SSS2.Px1.p2.5.m5.3.3.1.1.1.cmml">⁡</mo><mrow id="S3.SS1.SSS2.Px1.p2.5.m5.3.3.1.1.1.2" xref="S3.SS1.SSS2.Px1.p2.5.m5.3.3.1.1.1.2.cmml"><mi id="S3.SS1.SSS2.Px1.p2.5.m5.3.3.1.1.1.2.2" xref="S3.SS1.SSS2.Px1.p2.5.m5.3.3.1.1.1.2.2.cmml">k</mi><mo id="S3.SS1.SSS2.Px1.p2.5.m5.3.3.1.1.1.2.1" lspace="0.222em" rspace="0.222em" xref="S3.SS1.SSS2.Px1.p2.5.m5.3.3.1.1.1.2.1.cmml">⋅</mo><mrow id="S3.SS1.SSS2.Px1.p2.5.m5.3.3.1.1.1.2.3.2" xref="S3.SS1.SSS2.Px1.p2.5.m5.3.3.1.1.1.2.3.1.cmml"><mi id="S3.SS1.SSS2.Px1.p2.5.m5.1.1" xref="S3.SS1.SSS2.Px1.p2.5.m5.1.1.cmml">log</mi><mo id="S3.SS1.SSS2.Px1.p2.5.m5.3.3.1.1.1.2.3.2a" xref="S3.SS1.SSS2.Px1.p2.5.m5.3.3.1.1.1.2.3.1.cmml">⁡</mo><mrow id="S3.SS1.SSS2.Px1.p2.5.m5.3.3.1.1.1.2.3.2.1" xref="S3.SS1.SSS2.Px1.p2.5.m5.3.3.1.1.1.2.3.1.cmml"><mo id="S3.SS1.SSS2.Px1.p2.5.m5.3.3.1.1.1.2.3.2.1.1" stretchy="false" xref="S3.SS1.SSS2.Px1.p2.5.m5.3.3.1.1.1.2.3.1.cmml">(</mo><mfrac id="S3.SS1.SSS2.Px1.p2.5.m5.2.2" xref="S3.SS1.SSS2.Px1.p2.5.m5.2.2.cmml"><mi id="S3.SS1.SSS2.Px1.p2.5.m5.2.2.2" xref="S3.SS1.SSS2.Px1.p2.5.m5.2.2.2.cmml">n</mi><mrow id="S3.SS1.SSS2.Px1.p2.5.m5.2.2.3" xref="S3.SS1.SSS2.Px1.p2.5.m5.2.2.3.cmml"><mi id="S3.SS1.SSS2.Px1.p2.5.m5.2.2.3.2" xref="S3.SS1.SSS2.Px1.p2.5.m5.2.2.3.2.cmml">n</mi><mo id="S3.SS1.SSS2.Px1.p2.5.m5.2.2.3.1" xref="S3.SS1.SSS2.Px1.p2.5.m5.2.2.3.1.cmml">−</mo><mi id="S3.SS1.SSS2.Px1.p2.5.m5.2.2.3.3" xref="S3.SS1.SSS2.Px1.p2.5.m5.2.2.3.3.cmml">k</mi></mrow></mfrac><mo id="S3.SS1.SSS2.Px1.p2.5.m5.3.3.1.1.1.2.3.2.1.2" stretchy="false" xref="S3.SS1.SSS2.Px1.p2.5.m5.3.3.1.1.1.2.3.1.cmml">)</mo></mrow></mrow></mrow></mrow><mo id="S3.SS1.SSS2.Px1.p2.5.m5.3.3.1.1.3" stretchy="false" xref="S3.SS1.SSS2.Px1.p2.5.m5.3.3.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS2.Px1.p2.5.m5.3b"><apply id="S3.SS1.SSS2.Px1.p2.5.m5.3.3.cmml" xref="S3.SS1.SSS2.Px1.p2.5.m5.3.3"><times id="S3.SS1.SSS2.Px1.p2.5.m5.3.3.2.cmml" xref="S3.SS1.SSS2.Px1.p2.5.m5.3.3.2"></times><ci id="S3.SS1.SSS2.Px1.p2.5.m5.3.3.3.cmml" xref="S3.SS1.SSS2.Px1.p2.5.m5.3.3.3">𝑂</ci><apply id="S3.SS1.SSS2.Px1.p2.5.m5.3.3.1.1.1.cmml" xref="S3.SS1.SSS2.Px1.p2.5.m5.3.3.1.1"><log id="S3.SS1.SSS2.Px1.p2.5.m5.3.3.1.1.1.1.cmml" xref="S3.SS1.SSS2.Px1.p2.5.m5.3.3.1.1.1.1"></log><apply id="S3.SS1.SSS2.Px1.p2.5.m5.3.3.1.1.1.2.cmml" xref="S3.SS1.SSS2.Px1.p2.5.m5.3.3.1.1.1.2"><ci id="S3.SS1.SSS2.Px1.p2.5.m5.3.3.1.1.1.2.1.cmml" xref="S3.SS1.SSS2.Px1.p2.5.m5.3.3.1.1.1.2.1">⋅</ci><ci id="S3.SS1.SSS2.Px1.p2.5.m5.3.3.1.1.1.2.2.cmml" xref="S3.SS1.SSS2.Px1.p2.5.m5.3.3.1.1.1.2.2">𝑘</ci><apply id="S3.SS1.SSS2.Px1.p2.5.m5.3.3.1.1.1.2.3.1.cmml" xref="S3.SS1.SSS2.Px1.p2.5.m5.3.3.1.1.1.2.3.2"><log id="S3.SS1.SSS2.Px1.p2.5.m5.1.1.cmml" xref="S3.SS1.SSS2.Px1.p2.5.m5.1.1"></log><apply id="S3.SS1.SSS2.Px1.p2.5.m5.2.2.cmml" xref="S3.SS1.SSS2.Px1.p2.5.m5.2.2"><divide id="S3.SS1.SSS2.Px1.p2.5.m5.2.2.1.cmml" xref="S3.SS1.SSS2.Px1.p2.5.m5.2.2"></divide><ci id="S3.SS1.SSS2.Px1.p2.5.m5.2.2.2.cmml" xref="S3.SS1.SSS2.Px1.p2.5.m5.2.2.2">𝑛</ci><apply id="S3.SS1.SSS2.Px1.p2.5.m5.2.2.3.cmml" xref="S3.SS1.SSS2.Px1.p2.5.m5.2.2.3"><minus id="S3.SS1.SSS2.Px1.p2.5.m5.2.2.3.1.cmml" xref="S3.SS1.SSS2.Px1.p2.5.m5.2.2.3.1"></minus><ci id="S3.SS1.SSS2.Px1.p2.5.m5.2.2.3.2.cmml" xref="S3.SS1.SSS2.Px1.p2.5.m5.2.2.3.2">𝑛</ci><ci id="S3.SS1.SSS2.Px1.p2.5.m5.2.2.3.3.cmml" xref="S3.SS1.SSS2.Px1.p2.5.m5.2.2.3.3">𝑘</ci></apply></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS2.Px1.p2.5.m5.3c">O(\log k\cdot\log(\frac{n}{n-k}))</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS2.Px1.p2.5.m5.3d">italic_O ( roman_log italic_k ⋅ roman_log ( divide start_ARG italic_n end_ARG start_ARG italic_n - italic_k end_ARG ) )</annotation></semantics></math> approximation bounds can be derived for <math alttext="k" class="ltx_Math" display="inline" id="S3.SS1.SSS2.Px1.p2.6.m6.1"><semantics id="S3.SS1.SSS2.Px1.p2.6.m6.1a"><mi id="S3.SS1.SSS2.Px1.p2.6.m6.1.1" xref="S3.SS1.SSS2.Px1.p2.6.m6.1.1.cmml">k</mi><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS2.Px1.p2.6.m6.1b"><ci id="S3.SS1.SSS2.Px1.p2.6.m6.1.1.cmml" xref="S3.SS1.SSS2.Px1.p2.6.m6.1.1">𝑘</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS2.Px1.p2.6.m6.1c">k</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS2.Px1.p2.6.m6.1d">italic_k</annotation></semantics></math>-VC-CAP and <math alttext="k" class="ltx_Math" display="inline" id="S3.SS1.SSS2.Px1.p2.7.m7.1"><semantics id="S3.SS1.SSS2.Px1.p2.7.m7.1a"><mi id="S3.SS1.SSS2.Px1.p2.7.m7.1.1" xref="S3.SS1.SSS2.Px1.p2.7.m7.1.1.cmml">k</mi><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS2.Px1.p2.7.m7.1b"><ci id="S3.SS1.SSS2.Px1.p2.7.m7.1.1.cmml" xref="S3.SS1.SSS2.Px1.p2.7.m7.1.1">𝑘</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS2.Px1.p2.7.m7.1c">k</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS2.Px1.p2.7.m7.1d">italic_k</annotation></semantics></math>-VCSS, respectively, via known integrality gaps (see <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.00712v1#bib.bibx73" title="">Nut18b</a>]</cite> and <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.00712v1#bib.bibx71" title="">Nut14</a>]</cite>). We omit formal statements in this version.</p> </div> <div class="ltx_para" id="S3.SS1.SSS2.Px1.p3"> <p class="ltx_p" id="S3.SS1.SSS2.Px1.p3.2">Other important cases of interest are single-source VC-SNDP and subset-connectivity VC-SNDP. Known integrality gap results <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.00712v1#bib.bibx70" title="">Nut12</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.00712v1#bib.bibx64" title="">Lae15</a>]</cite> translate to efficient streaming algorithms with <math alttext="\beta=O(k^{2})" class="ltx_Math" display="inline" id="S3.SS1.SSS2.Px1.p3.1.m1.1"><semantics id="S3.SS1.SSS2.Px1.p3.1.m1.1a"><mrow id="S3.SS1.SSS2.Px1.p3.1.m1.1.1" xref="S3.SS1.SSS2.Px1.p3.1.m1.1.1.cmml"><mi id="S3.SS1.SSS2.Px1.p3.1.m1.1.1.3" xref="S3.SS1.SSS2.Px1.p3.1.m1.1.1.3.cmml">β</mi><mo id="S3.SS1.SSS2.Px1.p3.1.m1.1.1.2" xref="S3.SS1.SSS2.Px1.p3.1.m1.1.1.2.cmml">=</mo><mrow id="S3.SS1.SSS2.Px1.p3.1.m1.1.1.1" xref="S3.SS1.SSS2.Px1.p3.1.m1.1.1.1.cmml"><mi id="S3.SS1.SSS2.Px1.p3.1.m1.1.1.1.3" xref="S3.SS1.SSS2.Px1.p3.1.m1.1.1.1.3.cmml">O</mi><mo id="S3.SS1.SSS2.Px1.p3.1.m1.1.1.1.2" xref="S3.SS1.SSS2.Px1.p3.1.m1.1.1.1.2.cmml">⁢</mo><mrow 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id="S3.SS1.SSS2.Px1.p3.2.m2.1.1.1.1.1.1.3.1.2.cmml" xref="S3.SS1.SSS2.Px1.p3.2.m2.1.1.1.1.1.1.3.1.2"></log><cn id="S3.SS1.SSS2.Px1.p3.2.m2.1.1.1.1.1.1.3.1.3.cmml" type="integer" xref="S3.SS1.SSS2.Px1.p3.2.m2.1.1.1.1.1.1.3.1.3">2</cn></apply><ci id="S3.SS1.SSS2.Px1.p3.2.m2.1.1.1.1.1.1.3.2.cmml" xref="S3.SS1.SSS2.Px1.p3.2.m2.1.1.1.1.1.1.3.2">𝑘</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS2.Px1.p3.2.m2.1c">\beta=O(k^{2}\log^{2}k)</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS2.Px1.p3.2.m2.1d">italic_β = italic_O ( italic_k start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT roman_log start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_k )</annotation></semantics></math> respectively.</p> </div> </section> </section> </section> <section class="ltx_subsection" id="S3.SS2"> <h3 class="ltx_title ltx_title_subsection"> <span class="ltx_tag ltx_tag_subsection">3.2 </span>EC-SNDP and ELC-SNDP</h3> <div class="ltx_para" id="S3.SS2.p1"> <p class="ltx_p" id="S3.SS2.p1.1">The proof technique that we outlined for VC-SNDP applies very broadly and also hold for EC-SNDP and ELC-SNDP. We state below the theorem for EC-SNDP that results from the analysis with respect to the fractional solution.</p> </div> <div class="ltx_theorem ltx_theorem_theorem" id="S3.Thmtheorem6"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="S3.Thmtheorem6.1.1.1">Theorem 3.6</span></span><span class="ltx_text ltx_font_bold" id="S3.Thmtheorem6.2.2">.</span> </h6> <div class="ltx_para" id="S3.Thmtheorem6.p1"> <p class="ltx_p" id="S3.Thmtheorem6.p1.6">Let <math alttext="H" class="ltx_Math" display="inline" id="S3.Thmtheorem6.p1.1.m1.1"><semantics id="S3.Thmtheorem6.p1.1.m1.1a"><mi id="S3.Thmtheorem6.p1.1.m1.1.1" xref="S3.Thmtheorem6.p1.1.m1.1.1.cmml">H</mi><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem6.p1.1.m1.1b"><ci id="S3.Thmtheorem6.p1.1.m1.1.1.cmml" xref="S3.Thmtheorem6.p1.1.m1.1.1">𝐻</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem6.p1.1.m1.1c">H</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem6.p1.1.m1.1d">italic_H</annotation></semantics></math> be the EFT spanner of a weighted graph <math alttext="G" class="ltx_Math" display="inline" id="S3.Thmtheorem6.p1.2.m2.1"><semantics id="S3.Thmtheorem6.p1.2.m2.1a"><mi id="S3.Thmtheorem6.p1.2.m2.1.1" xref="S3.Thmtheorem6.p1.2.m2.1.1.cmml">G</mi><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem6.p1.2.m2.1b"><ci id="S3.Thmtheorem6.p1.2.m2.1.1.cmml" xref="S3.Thmtheorem6.p1.2.m2.1.1">𝐺</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem6.p1.2.m2.1c">G</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem6.p1.2.m2.1d">italic_G</annotation></semantics></math> as constructed in Algorithm <a class="ltx_ref" href="https://arxiv.org/html/2503.00712v1#algorithm2" title="In Weighted graphs. ‣ 2.1 Fault-Tolerant Spanners in Streaming ‣ 2 Preliminaries ‣ Streaming Algorithms for Network Design"><span class="ltx_text ltx_ref_tag">2</span></a> with parameters <math alttext="(t,f=(2t-1)(2k-1),\epsilon=1/(2t-1))" class="ltx_Math" display="inline" id="S3.Thmtheorem6.p1.3.m3.3"><semantics id="S3.Thmtheorem6.p1.3.m3.3a"><mrow id="S3.Thmtheorem6.p1.3.m3.3.3.1"><mo id="S3.Thmtheorem6.p1.3.m3.3.3.1.2" stretchy="false">(</mo><mrow id="S3.Thmtheorem6.p1.3.m3.3.3.1.1.2" xref="S3.Thmtheorem6.p1.3.m3.3.3.1.1.3.cmml"><mrow id="S3.Thmtheorem6.p1.3.m3.3.3.1.1.1.1" xref="S3.Thmtheorem6.p1.3.m3.3.3.1.1.1.1.cmml"><mrow id="S3.Thmtheorem6.p1.3.m3.3.3.1.1.1.1.4.2" xref="S3.Thmtheorem6.p1.3.m3.3.3.1.1.1.1.4.1.cmml"><mi id="S3.Thmtheorem6.p1.3.m3.1.1" xref="S3.Thmtheorem6.p1.3.m3.1.1.cmml">t</mi><mo id="S3.Thmtheorem6.p1.3.m3.3.3.1.1.1.1.4.2.1" xref="S3.Thmtheorem6.p1.3.m3.3.3.1.1.1.1.4.1.cmml">,</mo><mi id="S3.Thmtheorem6.p1.3.m3.2.2" xref="S3.Thmtheorem6.p1.3.m3.2.2.cmml">f</mi></mrow><mo id="S3.Thmtheorem6.p1.3.m3.3.3.1.1.1.1.3" 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id="S3.Thmtheorem6.p1.3.m3.3.3.1.1.2.2.1.1.1.1.2.1" xref="S3.Thmtheorem6.p1.3.m3.3.3.1.1.2.2.1.1.1.1.2.1.cmml">⁢</mo><mi id="S3.Thmtheorem6.p1.3.m3.3.3.1.1.2.2.1.1.1.1.2.3" xref="S3.Thmtheorem6.p1.3.m3.3.3.1.1.2.2.1.1.1.1.2.3.cmml">t</mi></mrow><mo id="S3.Thmtheorem6.p1.3.m3.3.3.1.1.2.2.1.1.1.1.1" xref="S3.Thmtheorem6.p1.3.m3.3.3.1.1.2.2.1.1.1.1.1.cmml">−</mo><mn id="S3.Thmtheorem6.p1.3.m3.3.3.1.1.2.2.1.1.1.1.3" xref="S3.Thmtheorem6.p1.3.m3.3.3.1.1.2.2.1.1.1.1.3.cmml">1</mn></mrow><mo id="S3.Thmtheorem6.p1.3.m3.3.3.1.1.2.2.1.1.1.3" stretchy="false" xref="S3.Thmtheorem6.p1.3.m3.3.3.1.1.2.2.1.1.1.1.cmml">)</mo></mrow></mrow></mrow></mrow><mo id="S3.Thmtheorem6.p1.3.m3.3.3.1.3" stretchy="false">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem6.p1.3.m3.3b"><apply id="S3.Thmtheorem6.p1.3.m3.3.3.1.1.3.cmml" xref="S3.Thmtheorem6.p1.3.m3.3.3.1.1.2"><csymbol cd="ambiguous" id="S3.Thmtheorem6.p1.3.m3.3.3.1.1.3a.cmml" xref="S3.Thmtheorem6.p1.3.m3.3.3.1.1.2.3">formulae-sequence</csymbol><apply id="S3.Thmtheorem6.p1.3.m3.3.3.1.1.1.1.cmml" xref="S3.Thmtheorem6.p1.3.m3.3.3.1.1.1.1"><eq id="S3.Thmtheorem6.p1.3.m3.3.3.1.1.1.1.3.cmml" xref="S3.Thmtheorem6.p1.3.m3.3.3.1.1.1.1.3"></eq><list id="S3.Thmtheorem6.p1.3.m3.3.3.1.1.1.1.4.1.cmml" xref="S3.Thmtheorem6.p1.3.m3.3.3.1.1.1.1.4.2"><ci id="S3.Thmtheorem6.p1.3.m3.1.1.cmml" xref="S3.Thmtheorem6.p1.3.m3.1.1">𝑡</ci><ci id="S3.Thmtheorem6.p1.3.m3.2.2.cmml" xref="S3.Thmtheorem6.p1.3.m3.2.2">𝑓</ci></list><apply id="S3.Thmtheorem6.p1.3.m3.3.3.1.1.1.1.2.cmml" xref="S3.Thmtheorem6.p1.3.m3.3.3.1.1.1.1.2"><times id="S3.Thmtheorem6.p1.3.m3.3.3.1.1.1.1.2.3.cmml" xref="S3.Thmtheorem6.p1.3.m3.3.3.1.1.1.1.2.3"></times><apply id="S3.Thmtheorem6.p1.3.m3.3.3.1.1.1.1.1.1.1.1.cmml" xref="S3.Thmtheorem6.p1.3.m3.3.3.1.1.1.1.1.1.1"><minus id="S3.Thmtheorem6.p1.3.m3.3.3.1.1.1.1.1.1.1.1.1.cmml" xref="S3.Thmtheorem6.p1.3.m3.3.3.1.1.1.1.1.1.1.1.1"></minus><apply id="S3.Thmtheorem6.p1.3.m3.3.3.1.1.1.1.1.1.1.1.2.cmml" xref="S3.Thmtheorem6.p1.3.m3.3.3.1.1.1.1.1.1.1.1.2"><times id="S3.Thmtheorem6.p1.3.m3.3.3.1.1.1.1.1.1.1.1.2.1.cmml" xref="S3.Thmtheorem6.p1.3.m3.3.3.1.1.1.1.1.1.1.1.2.1"></times><cn id="S3.Thmtheorem6.p1.3.m3.3.3.1.1.1.1.1.1.1.1.2.2.cmml" type="integer" xref="S3.Thmtheorem6.p1.3.m3.3.3.1.1.1.1.1.1.1.1.2.2">2</cn><ci id="S3.Thmtheorem6.p1.3.m3.3.3.1.1.1.1.1.1.1.1.2.3.cmml" xref="S3.Thmtheorem6.p1.3.m3.3.3.1.1.1.1.1.1.1.1.2.3">𝑡</ci></apply><cn id="S3.Thmtheorem6.p1.3.m3.3.3.1.1.1.1.1.1.1.1.3.cmml" type="integer" xref="S3.Thmtheorem6.p1.3.m3.3.3.1.1.1.1.1.1.1.1.3">1</cn></apply><apply id="S3.Thmtheorem6.p1.3.m3.3.3.1.1.1.1.2.2.1.1.cmml" xref="S3.Thmtheorem6.p1.3.m3.3.3.1.1.1.1.2.2.1"><minus id="S3.Thmtheorem6.p1.3.m3.3.3.1.1.1.1.2.2.1.1.1.cmml" xref="S3.Thmtheorem6.p1.3.m3.3.3.1.1.1.1.2.2.1.1.1"></minus><apply id="S3.Thmtheorem6.p1.3.m3.3.3.1.1.1.1.2.2.1.1.2.cmml" xref="S3.Thmtheorem6.p1.3.m3.3.3.1.1.1.1.2.2.1.1.2"><times id="S3.Thmtheorem6.p1.3.m3.3.3.1.1.1.1.2.2.1.1.2.1.cmml" xref="S3.Thmtheorem6.p1.3.m3.3.3.1.1.1.1.2.2.1.1.2.1"></times><cn id="S3.Thmtheorem6.p1.3.m3.3.3.1.1.1.1.2.2.1.1.2.2.cmml" type="integer" xref="S3.Thmtheorem6.p1.3.m3.3.3.1.1.1.1.2.2.1.1.2.2">2</cn><ci id="S3.Thmtheorem6.p1.3.m3.3.3.1.1.1.1.2.2.1.1.2.3.cmml" xref="S3.Thmtheorem6.p1.3.m3.3.3.1.1.1.1.2.2.1.1.2.3">𝑘</ci></apply><cn id="S3.Thmtheorem6.p1.3.m3.3.3.1.1.1.1.2.2.1.1.3.cmml" type="integer" xref="S3.Thmtheorem6.p1.3.m3.3.3.1.1.1.1.2.2.1.1.3">1</cn></apply></apply></apply><apply id="S3.Thmtheorem6.p1.3.m3.3.3.1.1.2.2.cmml" xref="S3.Thmtheorem6.p1.3.m3.3.3.1.1.2.2"><eq id="S3.Thmtheorem6.p1.3.m3.3.3.1.1.2.2.2.cmml" xref="S3.Thmtheorem6.p1.3.m3.3.3.1.1.2.2.2"></eq><ci id="S3.Thmtheorem6.p1.3.m3.3.3.1.1.2.2.3.cmml" xref="S3.Thmtheorem6.p1.3.m3.3.3.1.1.2.2.3">italic-ϵ</ci><apply id="S3.Thmtheorem6.p1.3.m3.3.3.1.1.2.2.1.cmml" xref="S3.Thmtheorem6.p1.3.m3.3.3.1.1.2.2.1"><divide id="S3.Thmtheorem6.p1.3.m3.3.3.1.1.2.2.1.2.cmml" xref="S3.Thmtheorem6.p1.3.m3.3.3.1.1.2.2.1.2"></divide><cn id="S3.Thmtheorem6.p1.3.m3.3.3.1.1.2.2.1.3.cmml" type="integer" xref="S3.Thmtheorem6.p1.3.m3.3.3.1.1.2.2.1.3">1</cn><apply id="S3.Thmtheorem6.p1.3.m3.3.3.1.1.2.2.1.1.1.1.cmml" xref="S3.Thmtheorem6.p1.3.m3.3.3.1.1.2.2.1.1.1"><minus id="S3.Thmtheorem6.p1.3.m3.3.3.1.1.2.2.1.1.1.1.1.cmml" xref="S3.Thmtheorem6.p1.3.m3.3.3.1.1.2.2.1.1.1.1.1"></minus><apply id="S3.Thmtheorem6.p1.3.m3.3.3.1.1.2.2.1.1.1.1.2.cmml" xref="S3.Thmtheorem6.p1.3.m3.3.3.1.1.2.2.1.1.1.1.2"><times id="S3.Thmtheorem6.p1.3.m3.3.3.1.1.2.2.1.1.1.1.2.1.cmml" xref="S3.Thmtheorem6.p1.3.m3.3.3.1.1.2.2.1.1.1.1.2.1"></times><cn id="S3.Thmtheorem6.p1.3.m3.3.3.1.1.2.2.1.1.1.1.2.2.cmml" type="integer" xref="S3.Thmtheorem6.p1.3.m3.3.3.1.1.2.2.1.1.1.1.2.2">2</cn><ci id="S3.Thmtheorem6.p1.3.m3.3.3.1.1.2.2.1.1.1.1.2.3.cmml" xref="S3.Thmtheorem6.p1.3.m3.3.3.1.1.2.2.1.1.1.1.2.3">𝑡</ci></apply><cn id="S3.Thmtheorem6.p1.3.m3.3.3.1.1.2.2.1.1.1.1.3.cmml" type="integer" xref="S3.Thmtheorem6.p1.3.m3.3.3.1.1.2.2.1.1.1.1.3">1</cn></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem6.p1.3.m3.3c">(t,f=(2t-1)(2k-1),\epsilon=1/(2t-1))</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem6.p1.3.m3.3d">( italic_t , italic_f = ( 2 italic_t - 1 ) ( 2 italic_k - 1 ) , italic_ϵ = 1 / ( 2 italic_t - 1 ) )</annotation></semantics></math>. Then, the weight of an optimal fractional solution of EC-SNDP on (<math alttext="H,r" class="ltx_Math" display="inline" id="S3.Thmtheorem6.p1.4.m4.2"><semantics id="S3.Thmtheorem6.p1.4.m4.2a"><mrow id="S3.Thmtheorem6.p1.4.m4.2.3.2" xref="S3.Thmtheorem6.p1.4.m4.2.3.1.cmml"><mi id="S3.Thmtheorem6.p1.4.m4.1.1" xref="S3.Thmtheorem6.p1.4.m4.1.1.cmml">H</mi><mo id="S3.Thmtheorem6.p1.4.m4.2.3.2.1" xref="S3.Thmtheorem6.p1.4.m4.2.3.1.cmml">,</mo><mi id="S3.Thmtheorem6.p1.4.m4.2.2" xref="S3.Thmtheorem6.p1.4.m4.2.2.cmml">r</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem6.p1.4.m4.2b"><list id="S3.Thmtheorem6.p1.4.m4.2.3.1.cmml" xref="S3.Thmtheorem6.p1.4.m4.2.3.2"><ci id="S3.Thmtheorem6.p1.4.m4.1.1.cmml" xref="S3.Thmtheorem6.p1.4.m4.1.1">𝐻</ci><ci id="S3.Thmtheorem6.p1.4.m4.2.2.cmml" xref="S3.Thmtheorem6.p1.4.m4.2.2">𝑟</ci></list></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem6.p1.4.m4.2c">H,r</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem6.p1.4.m4.2d">italic_H , italic_r</annotation></semantics></math>) is within a <math alttext="(4t)" class="ltx_Math" display="inline" id="S3.Thmtheorem6.p1.5.m5.1"><semantics id="S3.Thmtheorem6.p1.5.m5.1a"><mrow id="S3.Thmtheorem6.p1.5.m5.1.1.1" xref="S3.Thmtheorem6.p1.5.m5.1.1.1.1.cmml"><mo id="S3.Thmtheorem6.p1.5.m5.1.1.1.2" stretchy="false" xref="S3.Thmtheorem6.p1.5.m5.1.1.1.1.cmml">(</mo><mrow id="S3.Thmtheorem6.p1.5.m5.1.1.1.1" xref="S3.Thmtheorem6.p1.5.m5.1.1.1.1.cmml"><mn id="S3.Thmtheorem6.p1.5.m5.1.1.1.1.2" xref="S3.Thmtheorem6.p1.5.m5.1.1.1.1.2.cmml">4</mn><mo id="S3.Thmtheorem6.p1.5.m5.1.1.1.1.1" xref="S3.Thmtheorem6.p1.5.m5.1.1.1.1.1.cmml">⁢</mo><mi id="S3.Thmtheorem6.p1.5.m5.1.1.1.1.3" xref="S3.Thmtheorem6.p1.5.m5.1.1.1.1.3.cmml">t</mi></mrow><mo id="S3.Thmtheorem6.p1.5.m5.1.1.1.3" stretchy="false" xref="S3.Thmtheorem6.p1.5.m5.1.1.1.1.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem6.p1.5.m5.1b"><apply id="S3.Thmtheorem6.p1.5.m5.1.1.1.1.cmml" xref="S3.Thmtheorem6.p1.5.m5.1.1.1"><times id="S3.Thmtheorem6.p1.5.m5.1.1.1.1.1.cmml" xref="S3.Thmtheorem6.p1.5.m5.1.1.1.1.1"></times><cn id="S3.Thmtheorem6.p1.5.m5.1.1.1.1.2.cmml" type="integer" xref="S3.Thmtheorem6.p1.5.m5.1.1.1.1.2">4</cn><ci id="S3.Thmtheorem6.p1.5.m5.1.1.1.1.3.cmml" xref="S3.Thmtheorem6.p1.5.m5.1.1.1.1.3">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem6.p1.5.m5.1c">(4t)</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem6.p1.5.m5.1d">( 4 italic_t )</annotation></semantics></math>-factor of the weight of an optimal solution of EC-SNDP on (<math alttext="G,r" class="ltx_Math" display="inline" id="S3.Thmtheorem6.p1.6.m6.2"><semantics id="S3.Thmtheorem6.p1.6.m6.2a"><mrow id="S3.Thmtheorem6.p1.6.m6.2.3.2" xref="S3.Thmtheorem6.p1.6.m6.2.3.1.cmml"><mi id="S3.Thmtheorem6.p1.6.m6.1.1" xref="S3.Thmtheorem6.p1.6.m6.1.1.cmml">G</mi><mo id="S3.Thmtheorem6.p1.6.m6.2.3.2.1" xref="S3.Thmtheorem6.p1.6.m6.2.3.1.cmml">,</mo><mi id="S3.Thmtheorem6.p1.6.m6.2.2" xref="S3.Thmtheorem6.p1.6.m6.2.2.cmml">r</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem6.p1.6.m6.2b"><list id="S3.Thmtheorem6.p1.6.m6.2.3.1.cmml" xref="S3.Thmtheorem6.p1.6.m6.2.3.2"><ci id="S3.Thmtheorem6.p1.6.m6.1.1.cmml" xref="S3.Thmtheorem6.p1.6.m6.1.1">𝐺</ci><ci id="S3.Thmtheorem6.p1.6.m6.2.2.cmml" xref="S3.Thmtheorem6.p1.6.m6.2.2">𝑟</ci></list></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem6.p1.6.m6.2c">G,r</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem6.p1.6.m6.2d">italic_G , italic_r</annotation></semantics></math>).</p> </div> </div> <div class="ltx_para" id="S3.SS2.p2"> <p class="ltx_p" id="S3.SS2.p2.1">We omit the proof since it follows the same outline as that for VC-SNDP.</p> </div> <div class="ltx_theorem ltx_theorem_corollary" id="S3.Thmtheorem7"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="S3.Thmtheorem7.1.1.1">Corollary 3.7</span></span><span class="ltx_text ltx_font_bold" id="S3.Thmtheorem7.2.2">.</span> </h6> <div class="ltx_para" id="S3.Thmtheorem7.p1"> <p class="ltx_p" id="S3.Thmtheorem7.p1.4">There exists a streaming algorithm for EC-SNDP with edge weights <math alttext="w:E\rightarrow\{0,1,\dots,W\}" class="ltx_Math" display="inline" id="S3.Thmtheorem7.p1.1.m1.4"><semantics id="S3.Thmtheorem7.p1.1.m1.4a"><mrow id="S3.Thmtheorem7.p1.1.m1.4.5" xref="S3.Thmtheorem7.p1.1.m1.4.5.cmml"><mi id="S3.Thmtheorem7.p1.1.m1.4.5.2" xref="S3.Thmtheorem7.p1.1.m1.4.5.2.cmml">w</mi><mo id="S3.Thmtheorem7.p1.1.m1.4.5.1" lspace="0.278em" rspace="0.278em" xref="S3.Thmtheorem7.p1.1.m1.4.5.1.cmml">:</mo><mrow id="S3.Thmtheorem7.p1.1.m1.4.5.3" xref="S3.Thmtheorem7.p1.1.m1.4.5.3.cmml"><mi id="S3.Thmtheorem7.p1.1.m1.4.5.3.2" xref="S3.Thmtheorem7.p1.1.m1.4.5.3.2.cmml">E</mi><mo id="S3.Thmtheorem7.p1.1.m1.4.5.3.1" stretchy="false" xref="S3.Thmtheorem7.p1.1.m1.4.5.3.1.cmml">→</mo><mrow id="S3.Thmtheorem7.p1.1.m1.4.5.3.3.2" xref="S3.Thmtheorem7.p1.1.m1.4.5.3.3.1.cmml"><mo id="S3.Thmtheorem7.p1.1.m1.4.5.3.3.2.1" stretchy="false" xref="S3.Thmtheorem7.p1.1.m1.4.5.3.3.1.cmml">{</mo><mn id="S3.Thmtheorem7.p1.1.m1.1.1" xref="S3.Thmtheorem7.p1.1.m1.1.1.cmml">0</mn><mo id="S3.Thmtheorem7.p1.1.m1.4.5.3.3.2.2" xref="S3.Thmtheorem7.p1.1.m1.4.5.3.3.1.cmml">,</mo><mn id="S3.Thmtheorem7.p1.1.m1.2.2" xref="S3.Thmtheorem7.p1.1.m1.2.2.cmml">1</mn><mo id="S3.Thmtheorem7.p1.1.m1.4.5.3.3.2.3" xref="S3.Thmtheorem7.p1.1.m1.4.5.3.3.1.cmml">,</mo><mi id="S3.Thmtheorem7.p1.1.m1.3.3" mathvariant="normal" xref="S3.Thmtheorem7.p1.1.m1.3.3.cmml">…</mi><mo id="S3.Thmtheorem7.p1.1.m1.4.5.3.3.2.4" xref="S3.Thmtheorem7.p1.1.m1.4.5.3.3.1.cmml">,</mo><mi id="S3.Thmtheorem7.p1.1.m1.4.4" xref="S3.Thmtheorem7.p1.1.m1.4.4.cmml">W</mi><mo id="S3.Thmtheorem7.p1.1.m1.4.5.3.3.2.5" stretchy="false" xref="S3.Thmtheorem7.p1.1.m1.4.5.3.3.1.cmml">}</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem7.p1.1.m1.4b"><apply id="S3.Thmtheorem7.p1.1.m1.4.5.cmml" xref="S3.Thmtheorem7.p1.1.m1.4.5"><ci id="S3.Thmtheorem7.p1.1.m1.4.5.1.cmml" xref="S3.Thmtheorem7.p1.1.m1.4.5.1">:</ci><ci id="S3.Thmtheorem7.p1.1.m1.4.5.2.cmml" xref="S3.Thmtheorem7.p1.1.m1.4.5.2">𝑤</ci><apply id="S3.Thmtheorem7.p1.1.m1.4.5.3.cmml" xref="S3.Thmtheorem7.p1.1.m1.4.5.3"><ci id="S3.Thmtheorem7.p1.1.m1.4.5.3.1.cmml" xref="S3.Thmtheorem7.p1.1.m1.4.5.3.1">→</ci><ci id="S3.Thmtheorem7.p1.1.m1.4.5.3.2.cmml" xref="S3.Thmtheorem7.p1.1.m1.4.5.3.2">𝐸</ci><set id="S3.Thmtheorem7.p1.1.m1.4.5.3.3.1.cmml" xref="S3.Thmtheorem7.p1.1.m1.4.5.3.3.2"><cn id="S3.Thmtheorem7.p1.1.m1.1.1.cmml" type="integer" xref="S3.Thmtheorem7.p1.1.m1.1.1">0</cn><cn id="S3.Thmtheorem7.p1.1.m1.2.2.cmml" type="integer" xref="S3.Thmtheorem7.p1.1.m1.2.2">1</cn><ci id="S3.Thmtheorem7.p1.1.m1.3.3.cmml" xref="S3.Thmtheorem7.p1.1.m1.3.3">…</ci><ci id="S3.Thmtheorem7.p1.1.m1.4.4.cmml" xref="S3.Thmtheorem7.p1.1.m1.4.4">𝑊</ci></set></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem7.p1.1.m1.4c">w:E\rightarrow\{0,1,\dots,W\}</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem7.p1.1.m1.4d">italic_w : italic_E → { 0 , 1 , … , italic_W }</annotation></semantics></math> and a maximum connectivity requirement <math alttext="k" class="ltx_Math" display="inline" id="S3.Thmtheorem7.p1.2.m2.1"><semantics id="S3.Thmtheorem7.p1.2.m2.1a"><mi id="S3.Thmtheorem7.p1.2.m2.1.1" xref="S3.Thmtheorem7.p1.2.m2.1.1.cmml">k</mi><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem7.p1.2.m2.1b"><ci id="S3.Thmtheorem7.p1.2.m2.1.1.cmml" xref="S3.Thmtheorem7.p1.2.m2.1.1">𝑘</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem7.p1.2.m2.1c">k</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem7.p1.2.m2.1d">italic_k</annotation></semantics></math>, in insertion-only streams, that uses <math alttext="\tilde{O}\big{(}k^{1-1/t}\cdot n^{1+1/t}\big{)}" class="ltx_Math" display="inline" id="S3.Thmtheorem7.p1.3.m3.1"><semantics id="S3.Thmtheorem7.p1.3.m3.1a"><mrow id="S3.Thmtheorem7.p1.3.m3.1.1" xref="S3.Thmtheorem7.p1.3.m3.1.1.cmml"><mover accent="true" id="S3.Thmtheorem7.p1.3.m3.1.1.3" xref="S3.Thmtheorem7.p1.3.m3.1.1.3.cmml"><mi id="S3.Thmtheorem7.p1.3.m3.1.1.3.2" xref="S3.Thmtheorem7.p1.3.m3.1.1.3.2.cmml">O</mi><mo id="S3.Thmtheorem7.p1.3.m3.1.1.3.1" xref="S3.Thmtheorem7.p1.3.m3.1.1.3.1.cmml">~</mo></mover><mo id="S3.Thmtheorem7.p1.3.m3.1.1.2" xref="S3.Thmtheorem7.p1.3.m3.1.1.2.cmml">⁢</mo><mrow id="S3.Thmtheorem7.p1.3.m3.1.1.1.1" xref="S3.Thmtheorem7.p1.3.m3.1.1.1.1.1.cmml"><mo id="S3.Thmtheorem7.p1.3.m3.1.1.1.1.2" maxsize="120%" minsize="120%" xref="S3.Thmtheorem7.p1.3.m3.1.1.1.1.1.cmml">(</mo><mrow 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id="S3.Thmtheorem7.p1.3.m3.1.1.1.1.1.3.3.3.1.cmml" xref="S3.Thmtheorem7.p1.3.m3.1.1.1.1.1.3.3.3.1"></divide><cn id="S3.Thmtheorem7.p1.3.m3.1.1.1.1.1.3.3.3.2.cmml" type="integer" xref="S3.Thmtheorem7.p1.3.m3.1.1.1.1.1.3.3.3.2">1</cn><ci id="S3.Thmtheorem7.p1.3.m3.1.1.1.1.1.3.3.3.3.cmml" xref="S3.Thmtheorem7.p1.3.m3.1.1.1.1.1.3.3.3.3">𝑡</ci></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem7.p1.3.m3.1c">\tilde{O}\big{(}k^{1-1/t}\cdot n^{1+1/t}\big{)}</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem7.p1.3.m3.1d">over~ start_ARG italic_O end_ARG ( italic_k start_POSTSUPERSCRIPT 1 - 1 / italic_t end_POSTSUPERSCRIPT ⋅ italic_n start_POSTSUPERSCRIPT 1 + 1 / italic_t end_POSTSUPERSCRIPT )</annotation></semantics></math> space and outputs a <math alttext="(8t)" class="ltx_Math" display="inline" id="S3.Thmtheorem7.p1.4.m4.1"><semantics id="S3.Thmtheorem7.p1.4.m4.1a"><mrow id="S3.Thmtheorem7.p1.4.m4.1.1.1" xref="S3.Thmtheorem7.p1.4.m4.1.1.1.1.cmml"><mo id="S3.Thmtheorem7.p1.4.m4.1.1.1.2" stretchy="false" xref="S3.Thmtheorem7.p1.4.m4.1.1.1.1.cmml">(</mo><mrow id="S3.Thmtheorem7.p1.4.m4.1.1.1.1" xref="S3.Thmtheorem7.p1.4.m4.1.1.1.1.cmml"><mn id="S3.Thmtheorem7.p1.4.m4.1.1.1.1.2" xref="S3.Thmtheorem7.p1.4.m4.1.1.1.1.2.cmml">8</mn><mo id="S3.Thmtheorem7.p1.4.m4.1.1.1.1.1" xref="S3.Thmtheorem7.p1.4.m4.1.1.1.1.1.cmml">⁢</mo><mi id="S3.Thmtheorem7.p1.4.m4.1.1.1.1.3" xref="S3.Thmtheorem7.p1.4.m4.1.1.1.1.3.cmml">t</mi></mrow><mo id="S3.Thmtheorem7.p1.4.m4.1.1.1.3" stretchy="false" xref="S3.Thmtheorem7.p1.4.m4.1.1.1.1.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem7.p1.4.m4.1b"><apply id="S3.Thmtheorem7.p1.4.m4.1.1.1.1.cmml" xref="S3.Thmtheorem7.p1.4.m4.1.1.1"><times id="S3.Thmtheorem7.p1.4.m4.1.1.1.1.1.cmml" xref="S3.Thmtheorem7.p1.4.m4.1.1.1.1.1"></times><cn id="S3.Thmtheorem7.p1.4.m4.1.1.1.1.2.cmml" type="integer" xref="S3.Thmtheorem7.p1.4.m4.1.1.1.1.2">8</cn><ci id="S3.Thmtheorem7.p1.4.m4.1.1.1.1.3.cmml" xref="S3.Thmtheorem7.p1.4.m4.1.1.1.1.3">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem7.p1.4.m4.1c">(8t)</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem7.p1.4.m4.1d">( 8 italic_t )</annotation></semantics></math>-approximate solution.</p> </div> </div> <div class="ltx_proof" id="S3.SS2.2"> <h6 class="ltx_title ltx_runin ltx_font_italic ltx_title_proof">Proof.</h6> <div class="ltx_para" id="S3.SS2.1.p1"> <p class="ltx_p" id="S3.SS2.1.p1.6">The proof follows from Theorem <a class="ltx_ref" href="https://arxiv.org/html/2503.00712v1#S3.Thmtheorem6" title="Theorem 3.6. ‣ 3.2 EC-SNDP and ELC-SNDP ‣ 3 Generic Framework for Streaming Algorithms for Network Design ‣ Streaming Algorithms for Network Design"><span class="ltx_text ltx_ref_tag">3.6</span></a> which shows that Algorithm <a class="ltx_ref" href="https://arxiv.org/html/2503.00712v1#algorithm3" title="In 3 Generic Framework for Streaming Algorithms for Network Design ‣ Streaming Algorithms for Network Design"><span class="ltx_text ltx_ref_tag">3</span></a> with <math alttext="(t,f=(2t-1)(2k-1),\epsilon=1/(2t-1))" class="ltx_Math" display="inline" id="S3.SS2.1.p1.1.m1.3"><semantics id="S3.SS2.1.p1.1.m1.3a"><mrow id="S3.SS2.1.p1.1.m1.3.3.1"><mo id="S3.SS2.1.p1.1.m1.3.3.1.2" stretchy="false">(</mo><mrow id="S3.SS2.1.p1.1.m1.3.3.1.1.2" xref="S3.SS2.1.p1.1.m1.3.3.1.1.3.cmml"><mrow id="S3.SS2.1.p1.1.m1.3.3.1.1.1.1" xref="S3.SS2.1.p1.1.m1.3.3.1.1.1.1.cmml"><mrow id="S3.SS2.1.p1.1.m1.3.3.1.1.1.1.4.2" xref="S3.SS2.1.p1.1.m1.3.3.1.1.1.1.4.1.cmml"><mi id="S3.SS2.1.p1.1.m1.1.1" xref="S3.SS2.1.p1.1.m1.1.1.cmml">t</mi><mo id="S3.SS2.1.p1.1.m1.3.3.1.1.1.1.4.2.1" xref="S3.SS2.1.p1.1.m1.3.3.1.1.1.1.4.1.cmml">,</mo><mi id="S3.SS2.1.p1.1.m1.2.2" xref="S3.SS2.1.p1.1.m1.2.2.cmml">f</mi></mrow><mo id="S3.SS2.1.p1.1.m1.3.3.1.1.1.1.3" xref="S3.SS2.1.p1.1.m1.3.3.1.1.1.1.3.cmml">=</mo><mrow id="S3.SS2.1.p1.1.m1.3.3.1.1.1.1.2" 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xref="S3.SS2.1.p1.1.m1.3.3.1.1.1.1.2.2.1"><minus id="S3.SS2.1.p1.1.m1.3.3.1.1.1.1.2.2.1.1.1.cmml" xref="S3.SS2.1.p1.1.m1.3.3.1.1.1.1.2.2.1.1.1"></minus><apply id="S3.SS2.1.p1.1.m1.3.3.1.1.1.1.2.2.1.1.2.cmml" xref="S3.SS2.1.p1.1.m1.3.3.1.1.1.1.2.2.1.1.2"><times id="S3.SS2.1.p1.1.m1.3.3.1.1.1.1.2.2.1.1.2.1.cmml" xref="S3.SS2.1.p1.1.m1.3.3.1.1.1.1.2.2.1.1.2.1"></times><cn id="S3.SS2.1.p1.1.m1.3.3.1.1.1.1.2.2.1.1.2.2.cmml" type="integer" xref="S3.SS2.1.p1.1.m1.3.3.1.1.1.1.2.2.1.1.2.2">2</cn><ci id="S3.SS2.1.p1.1.m1.3.3.1.1.1.1.2.2.1.1.2.3.cmml" xref="S3.SS2.1.p1.1.m1.3.3.1.1.1.1.2.2.1.1.2.3">𝑘</ci></apply><cn id="S3.SS2.1.p1.1.m1.3.3.1.1.1.1.2.2.1.1.3.cmml" type="integer" xref="S3.SS2.1.p1.1.m1.3.3.1.1.1.1.2.2.1.1.3">1</cn></apply></apply></apply><apply id="S3.SS2.1.p1.1.m1.3.3.1.1.2.2.cmml" xref="S3.SS2.1.p1.1.m1.3.3.1.1.2.2"><eq id="S3.SS2.1.p1.1.m1.3.3.1.1.2.2.2.cmml" xref="S3.SS2.1.p1.1.m1.3.3.1.1.2.2.2"></eq><ci id="S3.SS2.1.p1.1.m1.3.3.1.1.2.2.3.cmml" xref="S3.SS2.1.p1.1.m1.3.3.1.1.2.2.3">italic-ϵ</ci><apply id="S3.SS2.1.p1.1.m1.3.3.1.1.2.2.1.cmml" xref="S3.SS2.1.p1.1.m1.3.3.1.1.2.2.1"><divide id="S3.SS2.1.p1.1.m1.3.3.1.1.2.2.1.2.cmml" xref="S3.SS2.1.p1.1.m1.3.3.1.1.2.2.1.2"></divide><cn id="S3.SS2.1.p1.1.m1.3.3.1.1.2.2.1.3.cmml" type="integer" xref="S3.SS2.1.p1.1.m1.3.3.1.1.2.2.1.3">1</cn><apply id="S3.SS2.1.p1.1.m1.3.3.1.1.2.2.1.1.1.1.cmml" xref="S3.SS2.1.p1.1.m1.3.3.1.1.2.2.1.1.1"><minus id="S3.SS2.1.p1.1.m1.3.3.1.1.2.2.1.1.1.1.1.cmml" xref="S3.SS2.1.p1.1.m1.3.3.1.1.2.2.1.1.1.1.1"></minus><apply id="S3.SS2.1.p1.1.m1.3.3.1.1.2.2.1.1.1.1.2.cmml" xref="S3.SS2.1.p1.1.m1.3.3.1.1.2.2.1.1.1.1.2"><times id="S3.SS2.1.p1.1.m1.3.3.1.1.2.2.1.1.1.1.2.1.cmml" xref="S3.SS2.1.p1.1.m1.3.3.1.1.2.2.1.1.1.1.2.1"></times><cn id="S3.SS2.1.p1.1.m1.3.3.1.1.2.2.1.1.1.1.2.2.cmml" type="integer" xref="S3.SS2.1.p1.1.m1.3.3.1.1.2.2.1.1.1.1.2.2">2</cn><ci id="S3.SS2.1.p1.1.m1.3.3.1.1.2.2.1.1.1.1.2.3.cmml" xref="S3.SS2.1.p1.1.m1.3.3.1.1.2.2.1.1.1.1.2.3">𝑡</ci></apply><cn id="S3.SS2.1.p1.1.m1.3.3.1.1.2.2.1.1.1.1.3.cmml" type="integer" xref="S3.SS2.1.p1.1.m1.3.3.1.1.2.2.1.1.1.1.3">1</cn></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.1.p1.1.m1.3c">(t,f=(2t-1)(2k-1),\epsilon=1/(2t-1))</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.1.p1.1.m1.3d">( italic_t , italic_f = ( 2 italic_t - 1 ) ( 2 italic_k - 1 ) , italic_ϵ = 1 / ( 2 italic_t - 1 ) )</annotation></semantics></math>, returns an EFT <math alttext="2t" class="ltx_Math" display="inline" id="S3.SS2.1.p1.2.m2.1"><semantics id="S3.SS2.1.p1.2.m2.1a"><mrow id="S3.SS2.1.p1.2.m2.1.1" xref="S3.SS2.1.p1.2.m2.1.1.cmml"><mn id="S3.SS2.1.p1.2.m2.1.1.2" xref="S3.SS2.1.p1.2.m2.1.1.2.cmml">2</mn><mo id="S3.SS2.1.p1.2.m2.1.1.1" xref="S3.SS2.1.p1.2.m2.1.1.1.cmml">⁢</mo><mi id="S3.SS2.1.p1.2.m2.1.1.3" xref="S3.SS2.1.p1.2.m2.1.1.3.cmml">t</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.SS2.1.p1.2.m2.1b"><apply id="S3.SS2.1.p1.2.m2.1.1.cmml" xref="S3.SS2.1.p1.2.m2.1.1"><times id="S3.SS2.1.p1.2.m2.1.1.1.cmml" xref="S3.SS2.1.p1.2.m2.1.1.1"></times><cn id="S3.SS2.1.p1.2.m2.1.1.2.cmml" type="integer" xref="S3.SS2.1.p1.2.m2.1.1.2">2</cn><ci id="S3.SS2.1.p1.2.m2.1.1.3.cmml" xref="S3.SS2.1.p1.2.m2.1.1.3">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.1.p1.2.m2.1c">2t</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.1.p1.2.m2.1d">2 italic_t</annotation></semantics></math>-spanner <math alttext="H" class="ltx_Math" display="inline" id="S3.SS2.1.p1.3.m3.1"><semantics id="S3.SS2.1.p1.3.m3.1a"><mi id="S3.SS2.1.p1.3.m3.1.1" xref="S3.SS2.1.p1.3.m3.1.1.cmml">H</mi><annotation-xml encoding="MathML-Content" id="S3.SS2.1.p1.3.m3.1b"><ci id="S3.SS2.1.p1.3.m3.1.1.cmml" xref="S3.SS2.1.p1.3.m3.1.1">𝐻</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.1.p1.3.m3.1c">H</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.1.p1.3.m3.1d">italic_H</annotation></semantics></math> of size <math alttext="\tilde{O}(k^{1-1/t}\cdot n^{1+1/t})" class="ltx_Math" display="inline" id="S3.SS2.1.p1.4.m4.1"><semantics id="S3.SS2.1.p1.4.m4.1a"><mrow id="S3.SS2.1.p1.4.m4.1.1" xref="S3.SS2.1.p1.4.m4.1.1.cmml"><mover accent="true" id="S3.SS2.1.p1.4.m4.1.1.3" xref="S3.SS2.1.p1.4.m4.1.1.3.cmml"><mi id="S3.SS2.1.p1.4.m4.1.1.3.2" xref="S3.SS2.1.p1.4.m4.1.1.3.2.cmml">O</mi><mo id="S3.SS2.1.p1.4.m4.1.1.3.1" xref="S3.SS2.1.p1.4.m4.1.1.3.1.cmml">~</mo></mover><mo id="S3.SS2.1.p1.4.m4.1.1.2" xref="S3.SS2.1.p1.4.m4.1.1.2.cmml">⁢</mo><mrow id="S3.SS2.1.p1.4.m4.1.1.1.1" xref="S3.SS2.1.p1.4.m4.1.1.1.1.1.cmml"><mo id="S3.SS2.1.p1.4.m4.1.1.1.1.2" stretchy="false" xref="S3.SS2.1.p1.4.m4.1.1.1.1.1.cmml">(</mo><mrow id="S3.SS2.1.p1.4.m4.1.1.1.1.1" xref="S3.SS2.1.p1.4.m4.1.1.1.1.1.cmml"><msup id="S3.SS2.1.p1.4.m4.1.1.1.1.1.2" 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xref="S3.SS2.1.p1.4.m4.1.1.1.1.1.2.3.1"></minus><cn id="S3.SS2.1.p1.4.m4.1.1.1.1.1.2.3.2.cmml" type="integer" xref="S3.SS2.1.p1.4.m4.1.1.1.1.1.2.3.2">1</cn><apply id="S3.SS2.1.p1.4.m4.1.1.1.1.1.2.3.3.cmml" xref="S3.SS2.1.p1.4.m4.1.1.1.1.1.2.3.3"><divide id="S3.SS2.1.p1.4.m4.1.1.1.1.1.2.3.3.1.cmml" xref="S3.SS2.1.p1.4.m4.1.1.1.1.1.2.3.3.1"></divide><cn id="S3.SS2.1.p1.4.m4.1.1.1.1.1.2.3.3.2.cmml" type="integer" xref="S3.SS2.1.p1.4.m4.1.1.1.1.1.2.3.3.2">1</cn><ci id="S3.SS2.1.p1.4.m4.1.1.1.1.1.2.3.3.3.cmml" xref="S3.SS2.1.p1.4.m4.1.1.1.1.1.2.3.3.3">𝑡</ci></apply></apply></apply><apply id="S3.SS2.1.p1.4.m4.1.1.1.1.1.3.cmml" xref="S3.SS2.1.p1.4.m4.1.1.1.1.1.3"><csymbol cd="ambiguous" id="S3.SS2.1.p1.4.m4.1.1.1.1.1.3.1.cmml" xref="S3.SS2.1.p1.4.m4.1.1.1.1.1.3">superscript</csymbol><ci id="S3.SS2.1.p1.4.m4.1.1.1.1.1.3.2.cmml" xref="S3.SS2.1.p1.4.m4.1.1.1.1.1.3.2">𝑛</ci><apply id="S3.SS2.1.p1.4.m4.1.1.1.1.1.3.3.cmml" xref="S3.SS2.1.p1.4.m4.1.1.1.1.1.3.3"><plus 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end_POSTSUPERSCRIPT )</annotation></semantics></math> that contains a <math alttext="(4t)" class="ltx_Math" display="inline" id="S3.SS2.1.p1.5.m5.1"><semantics id="S3.SS2.1.p1.5.m5.1a"><mrow id="S3.SS2.1.p1.5.m5.1.1.1" xref="S3.SS2.1.p1.5.m5.1.1.1.1.cmml"><mo id="S3.SS2.1.p1.5.m5.1.1.1.2" stretchy="false" xref="S3.SS2.1.p1.5.m5.1.1.1.1.cmml">(</mo><mrow id="S3.SS2.1.p1.5.m5.1.1.1.1" xref="S3.SS2.1.p1.5.m5.1.1.1.1.cmml"><mn id="S3.SS2.1.p1.5.m5.1.1.1.1.2" xref="S3.SS2.1.p1.5.m5.1.1.1.1.2.cmml">4</mn><mo id="S3.SS2.1.p1.5.m5.1.1.1.1.1" xref="S3.SS2.1.p1.5.m5.1.1.1.1.1.cmml">⁢</mo><mi id="S3.SS2.1.p1.5.m5.1.1.1.1.3" xref="S3.SS2.1.p1.5.m5.1.1.1.1.3.cmml">t</mi></mrow><mo id="S3.SS2.1.p1.5.m5.1.1.1.3" stretchy="false" xref="S3.SS2.1.p1.5.m5.1.1.1.1.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S3.SS2.1.p1.5.m5.1b"><apply id="S3.SS2.1.p1.5.m5.1.1.1.1.cmml" xref="S3.SS2.1.p1.5.m5.1.1.1"><times id="S3.SS2.1.p1.5.m5.1.1.1.1.1.cmml" xref="S3.SS2.1.p1.5.m5.1.1.1.1.1"></times><cn id="S3.SS2.1.p1.5.m5.1.1.1.1.2.cmml" type="integer" xref="S3.SS2.1.p1.5.m5.1.1.1.1.2">4</cn><ci id="S3.SS2.1.p1.5.m5.1.1.1.1.3.cmml" xref="S3.SS2.1.p1.5.m5.1.1.1.1.3">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.1.p1.5.m5.1c">(4t)</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.1.p1.5.m5.1d">( 4 italic_t )</annotation></semantics></math>-approximate fractional solution for EC-SNDP on <math alttext="G" class="ltx_Math" display="inline" id="S3.SS2.1.p1.6.m6.1"><semantics id="S3.SS2.1.p1.6.m6.1a"><mi id="S3.SS2.1.p1.6.m6.1.1" xref="S3.SS2.1.p1.6.m6.1.1.cmml">G</mi><annotation-xml encoding="MathML-Content" id="S3.SS2.1.p1.6.m6.1b"><ci id="S3.SS2.1.p1.6.m6.1.1.cmml" xref="S3.SS2.1.p1.6.m6.1.1">𝐺</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.1.p1.6.m6.1c">G</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.1.p1.6.m6.1d">italic_G</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="S3.SS2.2.p2"> <p class="ltx_p" id="S3.SS2.2.p2.4">The integrality gap of EC-SNDP-LP is <math alttext="2" class="ltx_Math" display="inline" id="S3.SS2.2.p2.1.m1.1"><semantics id="S3.SS2.2.p2.1.m1.1a"><mn id="S3.SS2.2.p2.1.m1.1.1" xref="S3.SS2.2.p2.1.m1.1.1.cmml">2</mn><annotation-xml encoding="MathML-Content" id="S3.SS2.2.p2.1.m1.1b"><cn id="S3.SS2.2.p2.1.m1.1.1.cmml" type="integer" xref="S3.SS2.2.p2.1.m1.1.1">2</cn></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.2.p2.1.m1.1c">2</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.2.p2.1.m1.1d">2</annotation></semantics></math> (cf. <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.00712v1#bib.bibx53" title="">Jai01</a>]</cite>). Once the stream terminates, we apply the LP-based algorithm of <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.00712v1#bib.bibx53" title="">Jai01</a>]</cite>, as <math alttext="\mathcal{A}" class="ltx_Math" display="inline" id="S3.SS2.2.p2.2.m2.1"><semantics id="S3.SS2.2.p2.2.m2.1a"><mi class="ltx_font_mathcaligraphic" id="S3.SS2.2.p2.2.m2.1.1" xref="S3.SS2.2.p2.2.m2.1.1.cmml">𝒜</mi><annotation-xml encoding="MathML-Content" id="S3.SS2.2.p2.2.m2.1b"><ci id="S3.SS2.2.p2.2.m2.1.1.cmml" xref="S3.SS2.2.p2.2.m2.1.1">𝒜</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.2.p2.2.m2.1c">\mathcal{A}</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.2.p2.2.m2.1d">caligraphic_A</annotation></semantics></math> in Algorithm <a class="ltx_ref" href="https://arxiv.org/html/2503.00712v1#algorithm3" title="In 3 Generic Framework for Streaming Algorithms for Network Design ‣ Streaming Algorithms for Network Design"><span class="ltx_text ltx_ref_tag">3</span></a> to find a <math alttext="(8t)" class="ltx_Math" display="inline" id="S3.SS2.2.p2.3.m3.1"><semantics id="S3.SS2.2.p2.3.m3.1a"><mrow id="S3.SS2.2.p2.3.m3.1.1.1" xref="S3.SS2.2.p2.3.m3.1.1.1.1.cmml"><mo id="S3.SS2.2.p2.3.m3.1.1.1.2" stretchy="false" xref="S3.SS2.2.p2.3.m3.1.1.1.1.cmml">(</mo><mrow id="S3.SS2.2.p2.3.m3.1.1.1.1" xref="S3.SS2.2.p2.3.m3.1.1.1.1.cmml"><mn id="S3.SS2.2.p2.3.m3.1.1.1.1.2" xref="S3.SS2.2.p2.3.m3.1.1.1.1.2.cmml">8</mn><mo id="S3.SS2.2.p2.3.m3.1.1.1.1.1" xref="S3.SS2.2.p2.3.m3.1.1.1.1.1.cmml">⁢</mo><mi id="S3.SS2.2.p2.3.m3.1.1.1.1.3" xref="S3.SS2.2.p2.3.m3.1.1.1.1.3.cmml">t</mi></mrow><mo id="S3.SS2.2.p2.3.m3.1.1.1.3" stretchy="false" xref="S3.SS2.2.p2.3.m3.1.1.1.1.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S3.SS2.2.p2.3.m3.1b"><apply id="S3.SS2.2.p2.3.m3.1.1.1.1.cmml" xref="S3.SS2.2.p2.3.m3.1.1.1"><times id="S3.SS2.2.p2.3.m3.1.1.1.1.1.cmml" xref="S3.SS2.2.p2.3.m3.1.1.1.1.1"></times><cn id="S3.SS2.2.p2.3.m3.1.1.1.1.2.cmml" type="integer" xref="S3.SS2.2.p2.3.m3.1.1.1.1.2">8</cn><ci id="S3.SS2.2.p2.3.m3.1.1.1.1.3.cmml" xref="S3.SS2.2.p2.3.m3.1.1.1.1.3">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.2.p2.3.m3.1c">(8t)</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.2.p2.3.m3.1d">( 8 italic_t )</annotation></semantics></math>-approximate solution. 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type="integer" xref="S3.SS2.2.p2.4.m4.1.1.1.1.1.2.3.2">1</cn><apply id="S3.SS2.2.p2.4.m4.1.1.1.1.1.2.3.3.cmml" xref="S3.SS2.2.p2.4.m4.1.1.1.1.1.2.3.3"><divide id="S3.SS2.2.p2.4.m4.1.1.1.1.1.2.3.3.1.cmml" xref="S3.SS2.2.p2.4.m4.1.1.1.1.1.2.3.3.1"></divide><cn id="S3.SS2.2.p2.4.m4.1.1.1.1.1.2.3.3.2.cmml" type="integer" xref="S3.SS2.2.p2.4.m4.1.1.1.1.1.2.3.3.2">1</cn><ci id="S3.SS2.2.p2.4.m4.1.1.1.1.1.2.3.3.3.cmml" xref="S3.SS2.2.p2.4.m4.1.1.1.1.1.2.3.3.3">𝑡</ci></apply></apply></apply><apply id="S3.SS2.2.p2.4.m4.1.1.1.1.1.3.cmml" xref="S3.SS2.2.p2.4.m4.1.1.1.1.1.3"><csymbol cd="ambiguous" id="S3.SS2.2.p2.4.m4.1.1.1.1.1.3.1.cmml" xref="S3.SS2.2.p2.4.m4.1.1.1.1.1.3">superscript</csymbol><ci id="S3.SS2.2.p2.4.m4.1.1.1.1.1.3.2.cmml" xref="S3.SS2.2.p2.4.m4.1.1.1.1.1.3.2">𝑛</ci><apply id="S3.SS2.2.p2.4.m4.1.1.1.1.1.3.3.cmml" xref="S3.SS2.2.p2.4.m4.1.1.1.1.1.3.3"><plus id="S3.SS2.2.p2.4.m4.1.1.1.1.1.3.3.1.cmml" xref="S3.SS2.2.p2.4.m4.1.1.1.1.1.3.3.1"></plus><cn id="S3.SS2.2.p2.4.m4.1.1.1.1.1.3.3.2.cmml" type="integer" xref="S3.SS2.2.p2.4.m4.1.1.1.1.1.3.3.2">1</cn><apply id="S3.SS2.2.p2.4.m4.1.1.1.1.1.3.3.3.cmml" xref="S3.SS2.2.p2.4.m4.1.1.1.1.1.3.3.3"><divide id="S3.SS2.2.p2.4.m4.1.1.1.1.1.3.3.3.1.cmml" xref="S3.SS2.2.p2.4.m4.1.1.1.1.1.3.3.3.1"></divide><cn id="S3.SS2.2.p2.4.m4.1.1.1.1.1.3.3.3.2.cmml" type="integer" xref="S3.SS2.2.p2.4.m4.1.1.1.1.1.3.3.3.2">1</cn><ci id="S3.SS2.2.p2.4.m4.1.1.1.1.1.3.3.3.3.cmml" xref="S3.SS2.2.p2.4.m4.1.1.1.1.1.3.3.3.3">𝑡</ci></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.2.p2.4.m4.1c">\tilde{O}\big{(}k^{1-1/t}\cdot n^{1+1/t}\big{)}</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.2.p2.4.m4.1d">over~ start_ARG italic_O end_ARG ( italic_k start_POSTSUPERSCRIPT 1 - 1 / italic_t end_POSTSUPERSCRIPT ⋅ italic_n start_POSTSUPERSCRIPT 1 + 1 / italic_t end_POSTSUPERSCRIPT )</annotation></semantics></math> space. ∎</p> </div> </div> <div class="ltx_para" id="S3.SS2.p3"> <p class="ltx_p" id="S3.SS2.p3.3">Notably, in the setting where <math alttext="W=n^{O(\log n)}" class="ltx_Math" display="inline" id="S3.SS2.p3.1.m1.1"><semantics id="S3.SS2.p3.1.m1.1a"><mrow id="S3.SS2.p3.1.m1.1.2" xref="S3.SS2.p3.1.m1.1.2.cmml"><mi id="S3.SS2.p3.1.m1.1.2.2" xref="S3.SS2.p3.1.m1.1.2.2.cmml">W</mi><mo id="S3.SS2.p3.1.m1.1.2.1" xref="S3.SS2.p3.1.m1.1.2.1.cmml">=</mo><msup id="S3.SS2.p3.1.m1.1.2.3" xref="S3.SS2.p3.1.m1.1.2.3.cmml"><mi id="S3.SS2.p3.1.m1.1.2.3.2" xref="S3.SS2.p3.1.m1.1.2.3.2.cmml">n</mi><mrow id="S3.SS2.p3.1.m1.1.1.1" xref="S3.SS2.p3.1.m1.1.1.1.cmml"><mi id="S3.SS2.p3.1.m1.1.1.1.3" xref="S3.SS2.p3.1.m1.1.1.1.3.cmml">O</mi><mo id="S3.SS2.p3.1.m1.1.1.1.2" xref="S3.SS2.p3.1.m1.1.1.1.2.cmml">⁢</mo><mrow id="S3.SS2.p3.1.m1.1.1.1.1.1" xref="S3.SS2.p3.1.m1.1.1.1.1.1.1.cmml"><mo id="S3.SS2.p3.1.m1.1.1.1.1.1.2" stretchy="false" xref="S3.SS2.p3.1.m1.1.1.1.1.1.1.cmml">(</mo><mrow id="S3.SS2.p3.1.m1.1.1.1.1.1.1" xref="S3.SS2.p3.1.m1.1.1.1.1.1.1.cmml"><mi id="S3.SS2.p3.1.m1.1.1.1.1.1.1.1" xref="S3.SS2.p3.1.m1.1.1.1.1.1.1.1.cmml">log</mi><mo id="S3.SS2.p3.1.m1.1.1.1.1.1.1a" lspace="0.167em" xref="S3.SS2.p3.1.m1.1.1.1.1.1.1.cmml">⁡</mo><mi id="S3.SS2.p3.1.m1.1.1.1.1.1.1.2" xref="S3.SS2.p3.1.m1.1.1.1.1.1.1.2.cmml">n</mi></mrow><mo id="S3.SS2.p3.1.m1.1.1.1.1.1.3" stretchy="false" xref="S3.SS2.p3.1.m1.1.1.1.1.1.1.cmml">)</mo></mrow></mrow></msup></mrow><annotation-xml encoding="MathML-Content" id="S3.SS2.p3.1.m1.1b"><apply id="S3.SS2.p3.1.m1.1.2.cmml" xref="S3.SS2.p3.1.m1.1.2"><eq id="S3.SS2.p3.1.m1.1.2.1.cmml" xref="S3.SS2.p3.1.m1.1.2.1"></eq><ci id="S3.SS2.p3.1.m1.1.2.2.cmml" xref="S3.SS2.p3.1.m1.1.2.2">𝑊</ci><apply id="S3.SS2.p3.1.m1.1.2.3.cmml" xref="S3.SS2.p3.1.m1.1.2.3"><csymbol cd="ambiguous" id="S3.SS2.p3.1.m1.1.2.3.1.cmml" xref="S3.SS2.p3.1.m1.1.2.3">superscript</csymbol><ci id="S3.SS2.p3.1.m1.1.2.3.2.cmml" xref="S3.SS2.p3.1.m1.1.2.3.2">𝑛</ci><apply id="S3.SS2.p3.1.m1.1.1.1.cmml" xref="S3.SS2.p3.1.m1.1.1.1"><times id="S3.SS2.p3.1.m1.1.1.1.2.cmml" xref="S3.SS2.p3.1.m1.1.1.1.2"></times><ci id="S3.SS2.p3.1.m1.1.1.1.3.cmml" xref="S3.SS2.p3.1.m1.1.1.1.3">𝑂</ci><apply id="S3.SS2.p3.1.m1.1.1.1.1.1.1.cmml" xref="S3.SS2.p3.1.m1.1.1.1.1.1"><log id="S3.SS2.p3.1.m1.1.1.1.1.1.1.1.cmml" xref="S3.SS2.p3.1.m1.1.1.1.1.1.1.1"></log><ci id="S3.SS2.p3.1.m1.1.1.1.1.1.1.2.cmml" xref="S3.SS2.p3.1.m1.1.1.1.1.1.1.2">𝑛</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p3.1.m1.1c">W=n^{O(\log n)}</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p3.1.m1.1d">italic_W = italic_n start_POSTSUPERSCRIPT italic_O ( roman_log italic_n ) end_POSTSUPERSCRIPT</annotation></semantics></math>, Corollary <a class="ltx_ref" href="https://arxiv.org/html/2503.00712v1#S3.Thmtheorem7" title="Corollary 3.7. ‣ 3.2 EC-SNDP and ELC-SNDP ‣ 3 Generic Framework for Streaming Algorithms for Network Design ‣ Streaming Algorithms for Network Design"><span class="ltx_text ltx_ref_tag">3.7</span></a> improves upon the algorithm of <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.00712v1#bib.bibx54" title="">JKMV24</a>]</cite>, which outputs an <math alttext="O(t\cdot\log k)" class="ltx_Math" display="inline" id="S3.SS2.p3.2.m2.1"><semantics id="S3.SS2.p3.2.m2.1a"><mrow id="S3.SS2.p3.2.m2.1.1" xref="S3.SS2.p3.2.m2.1.1.cmml"><mi id="S3.SS2.p3.2.m2.1.1.3" xref="S3.SS2.p3.2.m2.1.1.3.cmml">O</mi><mo id="S3.SS2.p3.2.m2.1.1.2" xref="S3.SS2.p3.2.m2.1.1.2.cmml">⁢</mo><mrow id="S3.SS2.p3.2.m2.1.1.1.1" xref="S3.SS2.p3.2.m2.1.1.1.1.1.cmml"><mo id="S3.SS2.p3.2.m2.1.1.1.1.2" stretchy="false" xref="S3.SS2.p3.2.m2.1.1.1.1.1.cmml">(</mo><mrow id="S3.SS2.p3.2.m2.1.1.1.1.1" xref="S3.SS2.p3.2.m2.1.1.1.1.1.cmml"><mi id="S3.SS2.p3.2.m2.1.1.1.1.1.2" xref="S3.SS2.p3.2.m2.1.1.1.1.1.2.cmml">t</mi><mo id="S3.SS2.p3.2.m2.1.1.1.1.1.1" lspace="0.222em" rspace="0.222em" xref="S3.SS2.p3.2.m2.1.1.1.1.1.1.cmml">⋅</mo><mrow id="S3.SS2.p3.2.m2.1.1.1.1.1.3" xref="S3.SS2.p3.2.m2.1.1.1.1.1.3.cmml"><mi id="S3.SS2.p3.2.m2.1.1.1.1.1.3.1" xref="S3.SS2.p3.2.m2.1.1.1.1.1.3.1.cmml">log</mi><mo id="S3.SS2.p3.2.m2.1.1.1.1.1.3a" lspace="0.167em" xref="S3.SS2.p3.2.m2.1.1.1.1.1.3.cmml">⁡</mo><mi id="S3.SS2.p3.2.m2.1.1.1.1.1.3.2" xref="S3.SS2.p3.2.m2.1.1.1.1.1.3.2.cmml">k</mi></mrow></mrow><mo id="S3.SS2.p3.2.m2.1.1.1.1.3" stretchy="false" xref="S3.SS2.p3.2.m2.1.1.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS2.p3.2.m2.1b"><apply id="S3.SS2.p3.2.m2.1.1.cmml" xref="S3.SS2.p3.2.m2.1.1"><times id="S3.SS2.p3.2.m2.1.1.2.cmml" xref="S3.SS2.p3.2.m2.1.1.2"></times><ci id="S3.SS2.p3.2.m2.1.1.3.cmml" xref="S3.SS2.p3.2.m2.1.1.3">𝑂</ci><apply id="S3.SS2.p3.2.m2.1.1.1.1.1.cmml" xref="S3.SS2.p3.2.m2.1.1.1.1"><ci id="S3.SS2.p3.2.m2.1.1.1.1.1.1.cmml" xref="S3.SS2.p3.2.m2.1.1.1.1.1.1">⋅</ci><ci id="S3.SS2.p3.2.m2.1.1.1.1.1.2.cmml" xref="S3.SS2.p3.2.m2.1.1.1.1.1.2">𝑡</ci><apply id="S3.SS2.p3.2.m2.1.1.1.1.1.3.cmml" xref="S3.SS2.p3.2.m2.1.1.1.1.1.3"><log id="S3.SS2.p3.2.m2.1.1.1.1.1.3.1.cmml" xref="S3.SS2.p3.2.m2.1.1.1.1.1.3.1"></log><ci id="S3.SS2.p3.2.m2.1.1.1.1.1.3.2.cmml" xref="S3.SS2.p3.2.m2.1.1.1.1.1.3.2">𝑘</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p3.2.m2.1c">O(t\cdot\log k)</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p3.2.m2.1d">italic_O ( italic_t ⋅ roman_log italic_k )</annotation></semantics></math>-approximation using <math alttext="\tilde{O}(k\cdot n^{1+1/t})" class="ltx_Math" display="inline" id="S3.SS2.p3.3.m3.1"><semantics id="S3.SS2.p3.3.m3.1a"><mrow id="S3.SS2.p3.3.m3.1.1" xref="S3.SS2.p3.3.m3.1.1.cmml"><mover accent="true" id="S3.SS2.p3.3.m3.1.1.3" xref="S3.SS2.p3.3.m3.1.1.3.cmml"><mi id="S3.SS2.p3.3.m3.1.1.3.2" xref="S3.SS2.p3.3.m3.1.1.3.2.cmml">O</mi><mo id="S3.SS2.p3.3.m3.1.1.3.1" xref="S3.SS2.p3.3.m3.1.1.3.1.cmml">~</mo></mover><mo id="S3.SS2.p3.3.m3.1.1.2" xref="S3.SS2.p3.3.m3.1.1.2.cmml">⁢</mo><mrow id="S3.SS2.p3.3.m3.1.1.1.1" xref="S3.SS2.p3.3.m3.1.1.1.1.1.cmml"><mo id="S3.SS2.p3.3.m3.1.1.1.1.2" stretchy="false" xref="S3.SS2.p3.3.m3.1.1.1.1.1.cmml">(</mo><mrow id="S3.SS2.p3.3.m3.1.1.1.1.1" xref="S3.SS2.p3.3.m3.1.1.1.1.1.cmml"><mi id="S3.SS2.p3.3.m3.1.1.1.1.1.2" xref="S3.SS2.p3.3.m3.1.1.1.1.1.2.cmml">k</mi><mo id="S3.SS2.p3.3.m3.1.1.1.1.1.1" lspace="0.222em" rspace="0.222em" xref="S3.SS2.p3.3.m3.1.1.1.1.1.1.cmml">⋅</mo><msup id="S3.SS2.p3.3.m3.1.1.1.1.1.3" xref="S3.SS2.p3.3.m3.1.1.1.1.1.3.cmml"><mi id="S3.SS2.p3.3.m3.1.1.1.1.1.3.2" xref="S3.SS2.p3.3.m3.1.1.1.1.1.3.2.cmml">n</mi><mrow id="S3.SS2.p3.3.m3.1.1.1.1.1.3.3" xref="S3.SS2.p3.3.m3.1.1.1.1.1.3.3.cmml"><mn id="S3.SS2.p3.3.m3.1.1.1.1.1.3.3.2" xref="S3.SS2.p3.3.m3.1.1.1.1.1.3.3.2.cmml">1</mn><mo id="S3.SS2.p3.3.m3.1.1.1.1.1.3.3.1" xref="S3.SS2.p3.3.m3.1.1.1.1.1.3.3.1.cmml">+</mo><mrow 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id="S3.SS2.p3.3.m3.1.1.1.1.1.1.cmml" xref="S3.SS2.p3.3.m3.1.1.1.1.1.1">⋅</ci><ci id="S3.SS2.p3.3.m3.1.1.1.1.1.2.cmml" xref="S3.SS2.p3.3.m3.1.1.1.1.1.2">𝑘</ci><apply id="S3.SS2.p3.3.m3.1.1.1.1.1.3.cmml" xref="S3.SS2.p3.3.m3.1.1.1.1.1.3"><csymbol cd="ambiguous" id="S3.SS2.p3.3.m3.1.1.1.1.1.3.1.cmml" xref="S3.SS2.p3.3.m3.1.1.1.1.1.3">superscript</csymbol><ci id="S3.SS2.p3.3.m3.1.1.1.1.1.3.2.cmml" xref="S3.SS2.p3.3.m3.1.1.1.1.1.3.2">𝑛</ci><apply id="S3.SS2.p3.3.m3.1.1.1.1.1.3.3.cmml" xref="S3.SS2.p3.3.m3.1.1.1.1.1.3.3"><plus id="S3.SS2.p3.3.m3.1.1.1.1.1.3.3.1.cmml" xref="S3.SS2.p3.3.m3.1.1.1.1.1.3.3.1"></plus><cn id="S3.SS2.p3.3.m3.1.1.1.1.1.3.3.2.cmml" type="integer" xref="S3.SS2.p3.3.m3.1.1.1.1.1.3.3.2">1</cn><apply id="S3.SS2.p3.3.m3.1.1.1.1.1.3.3.3.cmml" xref="S3.SS2.p3.3.m3.1.1.1.1.1.3.3.3"><divide id="S3.SS2.p3.3.m3.1.1.1.1.1.3.3.3.1.cmml" xref="S3.SS2.p3.3.m3.1.1.1.1.1.3.3.3.1"></divide><cn id="S3.SS2.p3.3.m3.1.1.1.1.1.3.3.3.2.cmml" type="integer" xref="S3.SS2.p3.3.m3.1.1.1.1.1.3.3.3.2">1</cn><ci id="S3.SS2.p3.3.m3.1.1.1.1.1.3.3.3.3.cmml" xref="S3.SS2.p3.3.m3.1.1.1.1.1.3.3.3.3">𝑡</ci></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p3.3.m3.1c">\tilde{O}(k\cdot n^{1+1/t})</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p3.3.m3.1d">over~ start_ARG italic_O end_ARG ( italic_k ⋅ italic_n start_POSTSUPERSCRIPT 1 + 1 / italic_t end_POSTSUPERSCRIPT )</annotation></semantics></math> space. More importantly, Algorithm <a class="ltx_ref" href="https://arxiv.org/html/2503.00712v1#algorithm3" title="In 3 Generic Framework for Streaming Algorithms for Network Design ‣ Streaming Algorithms for Network Design"><span class="ltx_text ltx_ref_tag">3</span></a> is a more natural and generic approach for EC-SNDP compared to the analysis in <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.00712v1#bib.bibx54" title="">JKMV24</a>]</cite>.</p> </div> <div class="ltx_para" id="S3.SS2.p4"> <p class="ltx_p" id="S3.SS2.p4.1">One can derive the following theorem for ELC-SNDP in a similar fashion. For this we rely on the fact that the integrality gap of the biset based LP for ELC-SNDP is <math alttext="2" class="ltx_Math" display="inline" id="S3.SS2.p4.1.m1.1"><semantics id="S3.SS2.p4.1.m1.1a"><mn id="S3.SS2.p4.1.m1.1.1" xref="S3.SS2.p4.1.m1.1.1.cmml">2</mn><annotation-xml encoding="MathML-Content" id="S3.SS2.p4.1.m1.1b"><cn id="S3.SS2.p4.1.m1.1.1.cmml" type="integer" xref="S3.SS2.p4.1.m1.1.1">2</cn></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p4.1.m1.1c">2</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p4.1.m1.1d">2</annotation></semantics></math> <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.00712v1#bib.bibx39" title="">FJW06</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.00712v1#bib.bibx31" title="">CVV06</a>]</cite>. We omit the formal proof since it is very similar to the ones above for VC-SNDP and EC-SNDP.</p> </div> <div class="ltx_theorem ltx_theorem_theorem" id="S3.Thmtheorem8"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="S3.Thmtheorem8.1.1.1">Theorem 3.8</span></span><span class="ltx_text ltx_font_bold" id="S3.Thmtheorem8.2.2">.</span> </h6> <div class="ltx_para" id="S3.Thmtheorem8.p1"> <p class="ltx_p" id="S3.Thmtheorem8.p1.4">There exists a streaming algorithm for ELC-SNDP with edge weights <math alttext="w:E\rightarrow\{0,1,\dots,W\}" class="ltx_Math" display="inline" id="S3.Thmtheorem8.p1.1.m1.4"><semantics id="S3.Thmtheorem8.p1.1.m1.4a"><mrow id="S3.Thmtheorem8.p1.1.m1.4.5" xref="S3.Thmtheorem8.p1.1.m1.4.5.cmml"><mi id="S3.Thmtheorem8.p1.1.m1.4.5.2" xref="S3.Thmtheorem8.p1.1.m1.4.5.2.cmml">w</mi><mo id="S3.Thmtheorem8.p1.1.m1.4.5.1" lspace="0.278em" rspace="0.278em" 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id="S3.Thmtheorem8.p1.1.m1.4.5.3.3.2.4" xref="S3.Thmtheorem8.p1.1.m1.4.5.3.3.1.cmml">,</mo><mi id="S3.Thmtheorem8.p1.1.m1.4.4" xref="S3.Thmtheorem8.p1.1.m1.4.4.cmml">W</mi><mo id="S3.Thmtheorem8.p1.1.m1.4.5.3.3.2.5" stretchy="false" xref="S3.Thmtheorem8.p1.1.m1.4.5.3.3.1.cmml">}</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem8.p1.1.m1.4b"><apply id="S3.Thmtheorem8.p1.1.m1.4.5.cmml" xref="S3.Thmtheorem8.p1.1.m1.4.5"><ci id="S3.Thmtheorem8.p1.1.m1.4.5.1.cmml" xref="S3.Thmtheorem8.p1.1.m1.4.5.1">:</ci><ci id="S3.Thmtheorem8.p1.1.m1.4.5.2.cmml" xref="S3.Thmtheorem8.p1.1.m1.4.5.2">𝑤</ci><apply id="S3.Thmtheorem8.p1.1.m1.4.5.3.cmml" xref="S3.Thmtheorem8.p1.1.m1.4.5.3"><ci id="S3.Thmtheorem8.p1.1.m1.4.5.3.1.cmml" xref="S3.Thmtheorem8.p1.1.m1.4.5.3.1">→</ci><ci id="S3.Thmtheorem8.p1.1.m1.4.5.3.2.cmml" xref="S3.Thmtheorem8.p1.1.m1.4.5.3.2">𝐸</ci><set id="S3.Thmtheorem8.p1.1.m1.4.5.3.3.1.cmml" xref="S3.Thmtheorem8.p1.1.m1.4.5.3.3.2"><cn id="S3.Thmtheorem8.p1.1.m1.1.1.cmml" type="integer" xref="S3.Thmtheorem8.p1.1.m1.1.1">0</cn><cn id="S3.Thmtheorem8.p1.1.m1.2.2.cmml" type="integer" xref="S3.Thmtheorem8.p1.1.m1.2.2">1</cn><ci id="S3.Thmtheorem8.p1.1.m1.3.3.cmml" xref="S3.Thmtheorem8.p1.1.m1.3.3">…</ci><ci id="S3.Thmtheorem8.p1.1.m1.4.4.cmml" xref="S3.Thmtheorem8.p1.1.m1.4.4">𝑊</ci></set></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem8.p1.1.m1.4c">w:E\rightarrow\{0,1,\dots,W\}</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem8.p1.1.m1.4d">italic_w : italic_E → { 0 , 1 , … , italic_W }</annotation></semantics></math> and a maximum connectivity requirement <math alttext="k" class="ltx_Math" display="inline" id="S3.Thmtheorem8.p1.2.m2.1"><semantics id="S3.Thmtheorem8.p1.2.m2.1a"><mi id="S3.Thmtheorem8.p1.2.m2.1.1" xref="S3.Thmtheorem8.p1.2.m2.1.1.cmml">k</mi><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem8.p1.2.m2.1b"><ci id="S3.Thmtheorem8.p1.2.m2.1.1.cmml" xref="S3.Thmtheorem8.p1.2.m2.1.1">𝑘</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem8.p1.2.m2.1c">k</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem8.p1.2.m2.1d">italic_k</annotation></semantics></math>, in insertion-only streams, that uses <math alttext="\tilde{O}\big{(}k^{1-1/t}\cdot n^{1+1/t}\big{)}" class="ltx_Math" display="inline" id="S3.Thmtheorem8.p1.3.m3.1"><semantics id="S3.Thmtheorem8.p1.3.m3.1a"><mrow id="S3.Thmtheorem8.p1.3.m3.1.1" xref="S3.Thmtheorem8.p1.3.m3.1.1.cmml"><mover accent="true" id="S3.Thmtheorem8.p1.3.m3.1.1.3" xref="S3.Thmtheorem8.p1.3.m3.1.1.3.cmml"><mi id="S3.Thmtheorem8.p1.3.m3.1.1.3.2" xref="S3.Thmtheorem8.p1.3.m3.1.1.3.2.cmml">O</mi><mo id="S3.Thmtheorem8.p1.3.m3.1.1.3.1" xref="S3.Thmtheorem8.p1.3.m3.1.1.3.1.cmml">~</mo></mover><mo id="S3.Thmtheorem8.p1.3.m3.1.1.2" xref="S3.Thmtheorem8.p1.3.m3.1.1.2.cmml">⁢</mo><mrow id="S3.Thmtheorem8.p1.3.m3.1.1.1.1" xref="S3.Thmtheorem8.p1.3.m3.1.1.1.1.1.cmml"><mo id="S3.Thmtheorem8.p1.3.m3.1.1.1.1.2" maxsize="120%" minsize="120%" xref="S3.Thmtheorem8.p1.3.m3.1.1.1.1.1.cmml">(</mo><mrow id="S3.Thmtheorem8.p1.3.m3.1.1.1.1.1" xref="S3.Thmtheorem8.p1.3.m3.1.1.1.1.1.cmml"><msup id="S3.Thmtheorem8.p1.3.m3.1.1.1.1.1.2" xref="S3.Thmtheorem8.p1.3.m3.1.1.1.1.1.2.cmml"><mi id="S3.Thmtheorem8.p1.3.m3.1.1.1.1.1.2.2" xref="S3.Thmtheorem8.p1.3.m3.1.1.1.1.1.2.2.cmml">k</mi><mrow id="S3.Thmtheorem8.p1.3.m3.1.1.1.1.1.2.3" xref="S3.Thmtheorem8.p1.3.m3.1.1.1.1.1.2.3.cmml"><mn id="S3.Thmtheorem8.p1.3.m3.1.1.1.1.1.2.3.2" xref="S3.Thmtheorem8.p1.3.m3.1.1.1.1.1.2.3.2.cmml">1</mn><mo id="S3.Thmtheorem8.p1.3.m3.1.1.1.1.1.2.3.1" xref="S3.Thmtheorem8.p1.3.m3.1.1.1.1.1.2.3.1.cmml">−</mo><mrow id="S3.Thmtheorem8.p1.3.m3.1.1.1.1.1.2.3.3" xref="S3.Thmtheorem8.p1.3.m3.1.1.1.1.1.2.3.3.cmml"><mn id="S3.Thmtheorem8.p1.3.m3.1.1.1.1.1.2.3.3.2" xref="S3.Thmtheorem8.p1.3.m3.1.1.1.1.1.2.3.3.2.cmml">1</mn><mo id="S3.Thmtheorem8.p1.3.m3.1.1.1.1.1.2.3.3.1" xref="S3.Thmtheorem8.p1.3.m3.1.1.1.1.1.2.3.3.1.cmml">/</mo><mi id="S3.Thmtheorem8.p1.3.m3.1.1.1.1.1.2.3.3.3" xref="S3.Thmtheorem8.p1.3.m3.1.1.1.1.1.2.3.3.3.cmml">t</mi></mrow></mrow></msup><mo id="S3.Thmtheorem8.p1.3.m3.1.1.1.1.1.1" lspace="0.222em" rspace="0.222em" xref="S3.Thmtheorem8.p1.3.m3.1.1.1.1.1.1.cmml">⋅</mo><msup id="S3.Thmtheorem8.p1.3.m3.1.1.1.1.1.3" xref="S3.Thmtheorem8.p1.3.m3.1.1.1.1.1.3.cmml"><mi id="S3.Thmtheorem8.p1.3.m3.1.1.1.1.1.3.2" xref="S3.Thmtheorem8.p1.3.m3.1.1.1.1.1.3.2.cmml">n</mi><mrow id="S3.Thmtheorem8.p1.3.m3.1.1.1.1.1.3.3" xref="S3.Thmtheorem8.p1.3.m3.1.1.1.1.1.3.3.cmml"><mn id="S3.Thmtheorem8.p1.3.m3.1.1.1.1.1.3.3.2" xref="S3.Thmtheorem8.p1.3.m3.1.1.1.1.1.3.3.2.cmml">1</mn><mo id="S3.Thmtheorem8.p1.3.m3.1.1.1.1.1.3.3.1" xref="S3.Thmtheorem8.p1.3.m3.1.1.1.1.1.3.3.1.cmml">+</mo><mrow id="S3.Thmtheorem8.p1.3.m3.1.1.1.1.1.3.3.3" xref="S3.Thmtheorem8.p1.3.m3.1.1.1.1.1.3.3.3.cmml"><mn id="S3.Thmtheorem8.p1.3.m3.1.1.1.1.1.3.3.3.2" xref="S3.Thmtheorem8.p1.3.m3.1.1.1.1.1.3.3.3.2.cmml">1</mn><mo id="S3.Thmtheorem8.p1.3.m3.1.1.1.1.1.3.3.3.1" xref="S3.Thmtheorem8.p1.3.m3.1.1.1.1.1.3.3.3.1.cmml">/</mo><mi id="S3.Thmtheorem8.p1.3.m3.1.1.1.1.1.3.3.3.3" xref="S3.Thmtheorem8.p1.3.m3.1.1.1.1.1.3.3.3.3.cmml">t</mi></mrow></mrow></msup></mrow><mo id="S3.Thmtheorem8.p1.3.m3.1.1.1.1.3" maxsize="120%" minsize="120%" xref="S3.Thmtheorem8.p1.3.m3.1.1.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem8.p1.3.m3.1b"><apply id="S3.Thmtheorem8.p1.3.m3.1.1.cmml" xref="S3.Thmtheorem8.p1.3.m3.1.1"><times id="S3.Thmtheorem8.p1.3.m3.1.1.2.cmml" xref="S3.Thmtheorem8.p1.3.m3.1.1.2"></times><apply id="S3.Thmtheorem8.p1.3.m3.1.1.3.cmml" xref="S3.Thmtheorem8.p1.3.m3.1.1.3"><ci id="S3.Thmtheorem8.p1.3.m3.1.1.3.1.cmml" xref="S3.Thmtheorem8.p1.3.m3.1.1.3.1">~</ci><ci id="S3.Thmtheorem8.p1.3.m3.1.1.3.2.cmml" xref="S3.Thmtheorem8.p1.3.m3.1.1.3.2">𝑂</ci></apply><apply id="S3.Thmtheorem8.p1.3.m3.1.1.1.1.1.cmml" xref="S3.Thmtheorem8.p1.3.m3.1.1.1.1"><ci id="S3.Thmtheorem8.p1.3.m3.1.1.1.1.1.1.cmml" xref="S3.Thmtheorem8.p1.3.m3.1.1.1.1.1.1">⋅</ci><apply id="S3.Thmtheorem8.p1.3.m3.1.1.1.1.1.2.cmml" xref="S3.Thmtheorem8.p1.3.m3.1.1.1.1.1.2"><csymbol cd="ambiguous" id="S3.Thmtheorem8.p1.3.m3.1.1.1.1.1.2.1.cmml" xref="S3.Thmtheorem8.p1.3.m3.1.1.1.1.1.2">superscript</csymbol><ci id="S3.Thmtheorem8.p1.3.m3.1.1.1.1.1.2.2.cmml" xref="S3.Thmtheorem8.p1.3.m3.1.1.1.1.1.2.2">𝑘</ci><apply id="S3.Thmtheorem8.p1.3.m3.1.1.1.1.1.2.3.cmml" xref="S3.Thmtheorem8.p1.3.m3.1.1.1.1.1.2.3"><minus id="S3.Thmtheorem8.p1.3.m3.1.1.1.1.1.2.3.1.cmml" xref="S3.Thmtheorem8.p1.3.m3.1.1.1.1.1.2.3.1"></minus><cn id="S3.Thmtheorem8.p1.3.m3.1.1.1.1.1.2.3.2.cmml" type="integer" xref="S3.Thmtheorem8.p1.3.m3.1.1.1.1.1.2.3.2">1</cn><apply id="S3.Thmtheorem8.p1.3.m3.1.1.1.1.1.2.3.3.cmml" xref="S3.Thmtheorem8.p1.3.m3.1.1.1.1.1.2.3.3"><divide id="S3.Thmtheorem8.p1.3.m3.1.1.1.1.1.2.3.3.1.cmml" xref="S3.Thmtheorem8.p1.3.m3.1.1.1.1.1.2.3.3.1"></divide><cn id="S3.Thmtheorem8.p1.3.m3.1.1.1.1.1.2.3.3.2.cmml" type="integer" xref="S3.Thmtheorem8.p1.3.m3.1.1.1.1.1.2.3.3.2">1</cn><ci id="S3.Thmtheorem8.p1.3.m3.1.1.1.1.1.2.3.3.3.cmml" xref="S3.Thmtheorem8.p1.3.m3.1.1.1.1.1.2.3.3.3">𝑡</ci></apply></apply></apply><apply id="S3.Thmtheorem8.p1.3.m3.1.1.1.1.1.3.cmml" xref="S3.Thmtheorem8.p1.3.m3.1.1.1.1.1.3"><csymbol cd="ambiguous" id="S3.Thmtheorem8.p1.3.m3.1.1.1.1.1.3.1.cmml" xref="S3.Thmtheorem8.p1.3.m3.1.1.1.1.1.3">superscript</csymbol><ci id="S3.Thmtheorem8.p1.3.m3.1.1.1.1.1.3.2.cmml" xref="S3.Thmtheorem8.p1.3.m3.1.1.1.1.1.3.2">𝑛</ci><apply id="S3.Thmtheorem8.p1.3.m3.1.1.1.1.1.3.3.cmml" xref="S3.Thmtheorem8.p1.3.m3.1.1.1.1.1.3.3"><plus id="S3.Thmtheorem8.p1.3.m3.1.1.1.1.1.3.3.1.cmml" xref="S3.Thmtheorem8.p1.3.m3.1.1.1.1.1.3.3.1"></plus><cn id="S3.Thmtheorem8.p1.3.m3.1.1.1.1.1.3.3.2.cmml" type="integer" xref="S3.Thmtheorem8.p1.3.m3.1.1.1.1.1.3.3.2">1</cn><apply id="S3.Thmtheorem8.p1.3.m3.1.1.1.1.1.3.3.3.cmml" xref="S3.Thmtheorem8.p1.3.m3.1.1.1.1.1.3.3.3"><divide id="S3.Thmtheorem8.p1.3.m3.1.1.1.1.1.3.3.3.1.cmml" xref="S3.Thmtheorem8.p1.3.m3.1.1.1.1.1.3.3.3.1"></divide><cn id="S3.Thmtheorem8.p1.3.m3.1.1.1.1.1.3.3.3.2.cmml" type="integer" xref="S3.Thmtheorem8.p1.3.m3.1.1.1.1.1.3.3.3.2">1</cn><ci id="S3.Thmtheorem8.p1.3.m3.1.1.1.1.1.3.3.3.3.cmml" xref="S3.Thmtheorem8.p1.3.m3.1.1.1.1.1.3.3.3.3">𝑡</ci></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem8.p1.3.m3.1c">\tilde{O}\big{(}k^{1-1/t}\cdot n^{1+1/t}\big{)}</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem8.p1.3.m3.1d">over~ start_ARG italic_O end_ARG ( italic_k start_POSTSUPERSCRIPT 1 - 1 / italic_t end_POSTSUPERSCRIPT ⋅ italic_n start_POSTSUPERSCRIPT 1 + 1 / italic_t end_POSTSUPERSCRIPT )</annotation></semantics></math> space and outputs a <math alttext="(8t)" class="ltx_Math" display="inline" id="S3.Thmtheorem8.p1.4.m4.1"><semantics id="S3.Thmtheorem8.p1.4.m4.1a"><mrow id="S3.Thmtheorem8.p1.4.m4.1.1.1" xref="S3.Thmtheorem8.p1.4.m4.1.1.1.1.cmml"><mo id="S3.Thmtheorem8.p1.4.m4.1.1.1.2" stretchy="false" xref="S3.Thmtheorem8.p1.4.m4.1.1.1.1.cmml">(</mo><mrow id="S3.Thmtheorem8.p1.4.m4.1.1.1.1" xref="S3.Thmtheorem8.p1.4.m4.1.1.1.1.cmml"><mn id="S3.Thmtheorem8.p1.4.m4.1.1.1.1.2" xref="S3.Thmtheorem8.p1.4.m4.1.1.1.1.2.cmml">8</mn><mo id="S3.Thmtheorem8.p1.4.m4.1.1.1.1.1" xref="S3.Thmtheorem8.p1.4.m4.1.1.1.1.1.cmml">⁢</mo><mi id="S3.Thmtheorem8.p1.4.m4.1.1.1.1.3" xref="S3.Thmtheorem8.p1.4.m4.1.1.1.1.3.cmml">t</mi></mrow><mo id="S3.Thmtheorem8.p1.4.m4.1.1.1.3" stretchy="false" xref="S3.Thmtheorem8.p1.4.m4.1.1.1.1.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem8.p1.4.m4.1b"><apply id="S3.Thmtheorem8.p1.4.m4.1.1.1.1.cmml" xref="S3.Thmtheorem8.p1.4.m4.1.1.1"><times id="S3.Thmtheorem8.p1.4.m4.1.1.1.1.1.cmml" xref="S3.Thmtheorem8.p1.4.m4.1.1.1.1.1"></times><cn id="S3.Thmtheorem8.p1.4.m4.1.1.1.1.2.cmml" type="integer" xref="S3.Thmtheorem8.p1.4.m4.1.1.1.1.2">8</cn><ci id="S3.Thmtheorem8.p1.4.m4.1.1.1.1.3.cmml" xref="S3.Thmtheorem8.p1.4.m4.1.1.1.1.3">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem8.p1.4.m4.1c">(8t)</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem8.p1.4.m4.1d">( 8 italic_t )</annotation></semantics></math>-approximate solution.</p> </div> </div> </section> </section> <section class="ltx_section" id="S4"> <h2 class="ltx_title ltx_title_section"> <span class="ltx_tag ltx_tag_section">4 </span>Vertex Connectivity Augmentation in Link-Arrival Model</h2> <div class="ltx_para" id="S4.p1"> <p class="ltx_p" id="S4.p1.11">In this section we consider <math alttext="k" class="ltx_Math" display="inline" id="S4.p1.1.m1.1"><semantics id="S4.p1.1.m1.1a"><mi id="S4.p1.1.m1.1.1" xref="S4.p1.1.m1.1.1.cmml">k</mi><annotation-xml encoding="MathML-Content" id="S4.p1.1.m1.1b"><ci id="S4.p1.1.m1.1.1.cmml" xref="S4.p1.1.m1.1.1">𝑘</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.p1.1.m1.1c">k</annotation><annotation encoding="application/x-llamapun" id="S4.p1.1.m1.1d">italic_k</annotation></semantics></math>-VC-CAP in the link arrival setting. Recall that in this problem, we are initially given an underlying <math alttext="k" class="ltx_Math" display="inline" id="S4.p1.2.m2.1"><semantics id="S4.p1.2.m2.1a"><mi id="S4.p1.2.m2.1.1" xref="S4.p1.2.m2.1.1.cmml">k</mi><annotation-xml encoding="MathML-Content" id="S4.p1.2.m2.1b"><ci id="S4.p1.2.m2.1.1.cmml" xref="S4.p1.2.m2.1.1">𝑘</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.p1.2.m2.1c">k</annotation><annotation encoding="application/x-llamapun" id="S4.p1.2.m2.1d">italic_k</annotation></semantics></math>-vertex-connected graph <math alttext="G=(V,E)" class="ltx_Math" display="inline" id="S4.p1.3.m3.2"><semantics id="S4.p1.3.m3.2a"><mrow id="S4.p1.3.m3.2.3" xref="S4.p1.3.m3.2.3.cmml"><mi id="S4.p1.3.m3.2.3.2" xref="S4.p1.3.m3.2.3.2.cmml">G</mi><mo id="S4.p1.3.m3.2.3.1" xref="S4.p1.3.m3.2.3.1.cmml">=</mo><mrow id="S4.p1.3.m3.2.3.3.2" xref="S4.p1.3.m3.2.3.3.1.cmml"><mo id="S4.p1.3.m3.2.3.3.2.1" stretchy="false" xref="S4.p1.3.m3.2.3.3.1.cmml">(</mo><mi id="S4.p1.3.m3.1.1" xref="S4.p1.3.m3.1.1.cmml">V</mi><mo id="S4.p1.3.m3.2.3.3.2.2" xref="S4.p1.3.m3.2.3.3.1.cmml">,</mo><mi id="S4.p1.3.m3.2.2" xref="S4.p1.3.m3.2.2.cmml">E</mi><mo id="S4.p1.3.m3.2.3.3.2.3" stretchy="false" xref="S4.p1.3.m3.2.3.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.p1.3.m3.2b"><apply id="S4.p1.3.m3.2.3.cmml" xref="S4.p1.3.m3.2.3"><eq id="S4.p1.3.m3.2.3.1.cmml" xref="S4.p1.3.m3.2.3.1"></eq><ci id="S4.p1.3.m3.2.3.2.cmml" xref="S4.p1.3.m3.2.3.2">𝐺</ci><interval closure="open" id="S4.p1.3.m3.2.3.3.1.cmml" xref="S4.p1.3.m3.2.3.3.2"><ci id="S4.p1.3.m3.1.1.cmml" xref="S4.p1.3.m3.1.1">𝑉</ci><ci id="S4.p1.3.m3.2.2.cmml" xref="S4.p1.3.m3.2.2">𝐸</ci></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.p1.3.m3.2c">G=(V,E)</annotation><annotation encoding="application/x-llamapun" id="S4.p1.3.m3.2d">italic_G = ( italic_V , italic_E )</annotation></semantics></math>, while additional links <math alttext="L\subseteq V\times V" class="ltx_Math" display="inline" id="S4.p1.4.m4.1"><semantics id="S4.p1.4.m4.1a"><mrow id="S4.p1.4.m4.1.1" xref="S4.p1.4.m4.1.1.cmml"><mi id="S4.p1.4.m4.1.1.2" xref="S4.p1.4.m4.1.1.2.cmml">L</mi><mo id="S4.p1.4.m4.1.1.1" xref="S4.p1.4.m4.1.1.1.cmml">⊆</mo><mrow id="S4.p1.4.m4.1.1.3" xref="S4.p1.4.m4.1.1.3.cmml"><mi id="S4.p1.4.m4.1.1.3.2" xref="S4.p1.4.m4.1.1.3.2.cmml">V</mi><mo id="S4.p1.4.m4.1.1.3.1" lspace="0.222em" rspace="0.222em" xref="S4.p1.4.m4.1.1.3.1.cmml">×</mo><mi id="S4.p1.4.m4.1.1.3.3" xref="S4.p1.4.m4.1.1.3.3.cmml">V</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.p1.4.m4.1b"><apply id="S4.p1.4.m4.1.1.cmml" xref="S4.p1.4.m4.1.1"><subset id="S4.p1.4.m4.1.1.1.cmml" xref="S4.p1.4.m4.1.1.1"></subset><ci id="S4.p1.4.m4.1.1.2.cmml" xref="S4.p1.4.m4.1.1.2">𝐿</ci><apply id="S4.p1.4.m4.1.1.3.cmml" xref="S4.p1.4.m4.1.1.3"><times id="S4.p1.4.m4.1.1.3.1.cmml" xref="S4.p1.4.m4.1.1.3.1"></times><ci id="S4.p1.4.m4.1.1.3.2.cmml" xref="S4.p1.4.m4.1.1.3.2">𝑉</ci><ci id="S4.p1.4.m4.1.1.3.3.cmml" xref="S4.p1.4.m4.1.1.3.3">𝑉</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.p1.4.m4.1c">L\subseteq V\times V</annotation><annotation encoding="application/x-llamapun" id="S4.p1.4.m4.1d">italic_L ⊆ italic_V × italic_V</annotation></semantics></math> with edge weights <math alttext="w:L\to\mathbb{R}_{\geq 0}" class="ltx_Math" display="inline" id="S4.p1.5.m5.1"><semantics id="S4.p1.5.m5.1a"><mrow id="S4.p1.5.m5.1.1" xref="S4.p1.5.m5.1.1.cmml"><mi id="S4.p1.5.m5.1.1.2" xref="S4.p1.5.m5.1.1.2.cmml">w</mi><mo id="S4.p1.5.m5.1.1.1" lspace="0.278em" rspace="0.278em" xref="S4.p1.5.m5.1.1.1.cmml">:</mo><mrow id="S4.p1.5.m5.1.1.3" xref="S4.p1.5.m5.1.1.3.cmml"><mi id="S4.p1.5.m5.1.1.3.2" xref="S4.p1.5.m5.1.1.3.2.cmml">L</mi><mo id="S4.p1.5.m5.1.1.3.1" stretchy="false" xref="S4.p1.5.m5.1.1.3.1.cmml">→</mo><msub id="S4.p1.5.m5.1.1.3.3" xref="S4.p1.5.m5.1.1.3.3.cmml"><mi id="S4.p1.5.m5.1.1.3.3.2" xref="S4.p1.5.m5.1.1.3.3.2.cmml">ℝ</mi><mrow id="S4.p1.5.m5.1.1.3.3.3" xref="S4.p1.5.m5.1.1.3.3.3.cmml"><mi id="S4.p1.5.m5.1.1.3.3.3.2" xref="S4.p1.5.m5.1.1.3.3.3.2.cmml"></mi><mo id="S4.p1.5.m5.1.1.3.3.3.1" xref="S4.p1.5.m5.1.1.3.3.3.1.cmml">≥</mo><mn id="S4.p1.5.m5.1.1.3.3.3.3" xref="S4.p1.5.m5.1.1.3.3.3.3.cmml">0</mn></mrow></msub></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.p1.5.m5.1b"><apply id="S4.p1.5.m5.1.1.cmml" xref="S4.p1.5.m5.1.1"><ci id="S4.p1.5.m5.1.1.1.cmml" xref="S4.p1.5.m5.1.1.1">:</ci><ci id="S4.p1.5.m5.1.1.2.cmml" xref="S4.p1.5.m5.1.1.2">𝑤</ci><apply id="S4.p1.5.m5.1.1.3.cmml" xref="S4.p1.5.m5.1.1.3"><ci id="S4.p1.5.m5.1.1.3.1.cmml" xref="S4.p1.5.m5.1.1.3.1">→</ci><ci id="S4.p1.5.m5.1.1.3.2.cmml" xref="S4.p1.5.m5.1.1.3.2">𝐿</ci><apply id="S4.p1.5.m5.1.1.3.3.cmml" xref="S4.p1.5.m5.1.1.3.3"><csymbol cd="ambiguous" id="S4.p1.5.m5.1.1.3.3.1.cmml" xref="S4.p1.5.m5.1.1.3.3">subscript</csymbol><ci id="S4.p1.5.m5.1.1.3.3.2.cmml" xref="S4.p1.5.m5.1.1.3.3.2">ℝ</ci><apply id="S4.p1.5.m5.1.1.3.3.3.cmml" xref="S4.p1.5.m5.1.1.3.3.3"><geq id="S4.p1.5.m5.1.1.3.3.3.1.cmml" xref="S4.p1.5.m5.1.1.3.3.3.1"></geq><csymbol cd="latexml" id="S4.p1.5.m5.1.1.3.3.3.2.cmml" xref="S4.p1.5.m5.1.1.3.3.3.2">absent</csymbol><cn id="S4.p1.5.m5.1.1.3.3.3.3.cmml" type="integer" xref="S4.p1.5.m5.1.1.3.3.3.3">0</cn></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.p1.5.m5.1c">w:L\to\mathbb{R}_{\geq 0}</annotation><annotation encoding="application/x-llamapun" id="S4.p1.5.m5.1d">italic_w : italic_L → blackboard_R start_POSTSUBSCRIPT ≥ 0 end_POSTSUBSCRIPT</annotation></semantics></math> arrive as a stream. For readability, we refer to edges of the given graph <math alttext="G" class="ltx_Math" display="inline" id="S4.p1.6.m6.1"><semantics id="S4.p1.6.m6.1a"><mi id="S4.p1.6.m6.1.1" xref="S4.p1.6.m6.1.1.cmml">G</mi><annotation-xml encoding="MathML-Content" id="S4.p1.6.m6.1b"><ci id="S4.p1.6.m6.1.1.cmml" xref="S4.p1.6.m6.1.1">𝐺</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.p1.6.m6.1c">G</annotation><annotation encoding="application/x-llamapun" id="S4.p1.6.m6.1d">italic_G</annotation></semantics></math> as “edges” and the incoming streaming edges <math alttext="L" class="ltx_Math" display="inline" id="S4.p1.7.m7.1"><semantics id="S4.p1.7.m7.1a"><mi id="S4.p1.7.m7.1.1" xref="S4.p1.7.m7.1.1.cmml">L</mi><annotation-xml encoding="MathML-Content" id="S4.p1.7.m7.1b"><ci id="S4.p1.7.m7.1.1.cmml" xref="S4.p1.7.m7.1.1">𝐿</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.p1.7.m7.1c">L</annotation><annotation encoding="application/x-llamapun" id="S4.p1.7.m7.1d">italic_L</annotation></semantics></math> as “links”. Our goal is to find the min-weight subset <math alttext="L^{\prime}\subseteq L" class="ltx_Math" display="inline" id="S4.p1.8.m8.1"><semantics id="S4.p1.8.m8.1a"><mrow id="S4.p1.8.m8.1.1" xref="S4.p1.8.m8.1.1.cmml"><msup id="S4.p1.8.m8.1.1.2" xref="S4.p1.8.m8.1.1.2.cmml"><mi id="S4.p1.8.m8.1.1.2.2" xref="S4.p1.8.m8.1.1.2.2.cmml">L</mi><mo id="S4.p1.8.m8.1.1.2.3" xref="S4.p1.8.m8.1.1.2.3.cmml">′</mo></msup><mo id="S4.p1.8.m8.1.1.1" xref="S4.p1.8.m8.1.1.1.cmml">⊆</mo><mi id="S4.p1.8.m8.1.1.3" xref="S4.p1.8.m8.1.1.3.cmml">L</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.p1.8.m8.1b"><apply id="S4.p1.8.m8.1.1.cmml" xref="S4.p1.8.m8.1.1"><subset id="S4.p1.8.m8.1.1.1.cmml" xref="S4.p1.8.m8.1.1.1"></subset><apply id="S4.p1.8.m8.1.1.2.cmml" xref="S4.p1.8.m8.1.1.2"><csymbol cd="ambiguous" id="S4.p1.8.m8.1.1.2.1.cmml" xref="S4.p1.8.m8.1.1.2">superscript</csymbol><ci id="S4.p1.8.m8.1.1.2.2.cmml" xref="S4.p1.8.m8.1.1.2.2">𝐿</ci><ci id="S4.p1.8.m8.1.1.2.3.cmml" xref="S4.p1.8.m8.1.1.2.3">′</ci></apply><ci id="S4.p1.8.m8.1.1.3.cmml" xref="S4.p1.8.m8.1.1.3">𝐿</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.p1.8.m8.1c">L^{\prime}\subseteq L</annotation><annotation encoding="application/x-llamapun" id="S4.p1.8.m8.1d">italic_L start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ⊆ italic_L</annotation></semantics></math> such that <math alttext="(V,E\cup L^{\prime})" class="ltx_Math" display="inline" id="S4.p1.9.m9.2"><semantics id="S4.p1.9.m9.2a"><mrow id="S4.p1.9.m9.2.2.1" xref="S4.p1.9.m9.2.2.2.cmml"><mo id="S4.p1.9.m9.2.2.1.2" stretchy="false" xref="S4.p1.9.m9.2.2.2.cmml">(</mo><mi id="S4.p1.9.m9.1.1" xref="S4.p1.9.m9.1.1.cmml">V</mi><mo id="S4.p1.9.m9.2.2.1.3" xref="S4.p1.9.m9.2.2.2.cmml">,</mo><mrow id="S4.p1.9.m9.2.2.1.1" xref="S4.p1.9.m9.2.2.1.1.cmml"><mi id="S4.p1.9.m9.2.2.1.1.2" xref="S4.p1.9.m9.2.2.1.1.2.cmml">E</mi><mo id="S4.p1.9.m9.2.2.1.1.1" xref="S4.p1.9.m9.2.2.1.1.1.cmml">∪</mo><msup id="S4.p1.9.m9.2.2.1.1.3" xref="S4.p1.9.m9.2.2.1.1.3.cmml"><mi id="S4.p1.9.m9.2.2.1.1.3.2" xref="S4.p1.9.m9.2.2.1.1.3.2.cmml">L</mi><mo id="S4.p1.9.m9.2.2.1.1.3.3" xref="S4.p1.9.m9.2.2.1.1.3.3.cmml">′</mo></msup></mrow><mo id="S4.p1.9.m9.2.2.1.4" stretchy="false" xref="S4.p1.9.m9.2.2.2.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S4.p1.9.m9.2b"><interval closure="open" id="S4.p1.9.m9.2.2.2.cmml" xref="S4.p1.9.m9.2.2.1"><ci id="S4.p1.9.m9.1.1.cmml" xref="S4.p1.9.m9.1.1">𝑉</ci><apply id="S4.p1.9.m9.2.2.1.1.cmml" xref="S4.p1.9.m9.2.2.1.1"><union id="S4.p1.9.m9.2.2.1.1.1.cmml" xref="S4.p1.9.m9.2.2.1.1.1"></union><ci id="S4.p1.9.m9.2.2.1.1.2.cmml" xref="S4.p1.9.m9.2.2.1.1.2">𝐸</ci><apply id="S4.p1.9.m9.2.2.1.1.3.cmml" xref="S4.p1.9.m9.2.2.1.1.3"><csymbol cd="ambiguous" id="S4.p1.9.m9.2.2.1.1.3.1.cmml" xref="S4.p1.9.m9.2.2.1.1.3">superscript</csymbol><ci id="S4.p1.9.m9.2.2.1.1.3.2.cmml" xref="S4.p1.9.m9.2.2.1.1.3.2">𝐿</ci><ci id="S4.p1.9.m9.2.2.1.1.3.3.cmml" xref="S4.p1.9.m9.2.2.1.1.3.3">′</ci></apply></apply></interval></annotation-xml><annotation encoding="application/x-tex" id="S4.p1.9.m9.2c">(V,E\cup L^{\prime})</annotation><annotation encoding="application/x-llamapun" id="S4.p1.9.m9.2d">( italic_V , italic_E ∪ italic_L start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT )</annotation></semantics></math> is <math alttext="(k+1)" class="ltx_Math" display="inline" id="S4.p1.10.m10.1"><semantics id="S4.p1.10.m10.1a"><mrow id="S4.p1.10.m10.1.1.1" xref="S4.p1.10.m10.1.1.1.1.cmml"><mo id="S4.p1.10.m10.1.1.1.2" stretchy="false" xref="S4.p1.10.m10.1.1.1.1.cmml">(</mo><mrow id="S4.p1.10.m10.1.1.1.1" xref="S4.p1.10.m10.1.1.1.1.cmml"><mi id="S4.p1.10.m10.1.1.1.1.2" xref="S4.p1.10.m10.1.1.1.1.2.cmml">k</mi><mo id="S4.p1.10.m10.1.1.1.1.1" xref="S4.p1.10.m10.1.1.1.1.1.cmml">+</mo><mn id="S4.p1.10.m10.1.1.1.1.3" xref="S4.p1.10.m10.1.1.1.1.3.cmml">1</mn></mrow><mo id="S4.p1.10.m10.1.1.1.3" stretchy="false" xref="S4.p1.10.m10.1.1.1.1.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S4.p1.10.m10.1b"><apply id="S4.p1.10.m10.1.1.1.1.cmml" xref="S4.p1.10.m10.1.1.1"><plus id="S4.p1.10.m10.1.1.1.1.1.cmml" xref="S4.p1.10.m10.1.1.1.1.1"></plus><ci id="S4.p1.10.m10.1.1.1.1.2.cmml" xref="S4.p1.10.m10.1.1.1.1.2">𝑘</ci><cn id="S4.p1.10.m10.1.1.1.1.3.cmml" type="integer" xref="S4.p1.10.m10.1.1.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.p1.10.m10.1c">(k+1)</annotation><annotation encoding="application/x-llamapun" id="S4.p1.10.m10.1d">( italic_k + 1 )</annotation></semantics></math>-vertex-connected. Note that this model is easier than the fully streaming setting, so results in Section <a class="ltx_ref" href="https://arxiv.org/html/2503.00712v1#S3" title="3 Generic Framework for Streaming Algorithms for Network Design ‣ Streaming Algorithms for Network Design"><span class="ltx_text ltx_ref_tag">3</span></a> immediately apply here as well. We show that for <math alttext="k=1,2" class="ltx_Math" display="inline" id="S4.p1.11.m11.2"><semantics id="S4.p1.11.m11.2a"><mrow id="S4.p1.11.m11.2.3" xref="S4.p1.11.m11.2.3.cmml"><mi id="S4.p1.11.m11.2.3.2" xref="S4.p1.11.m11.2.3.2.cmml">k</mi><mo id="S4.p1.11.m11.2.3.1" xref="S4.p1.11.m11.2.3.1.cmml">=</mo><mrow id="S4.p1.11.m11.2.3.3.2" xref="S4.p1.11.m11.2.3.3.1.cmml"><mn id="S4.p1.11.m11.1.1" xref="S4.p1.11.m11.1.1.cmml">1</mn><mo id="S4.p1.11.m11.2.3.3.2.1" xref="S4.p1.11.m11.2.3.3.1.cmml">,</mo><mn id="S4.p1.11.m11.2.2" xref="S4.p1.11.m11.2.2.cmml">2</mn></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.p1.11.m11.2b"><apply id="S4.p1.11.m11.2.3.cmml" xref="S4.p1.11.m11.2.3"><eq id="S4.p1.11.m11.2.3.1.cmml" xref="S4.p1.11.m11.2.3.1"></eq><ci id="S4.p1.11.m11.2.3.2.cmml" xref="S4.p1.11.m11.2.3.2">𝑘</ci><list id="S4.p1.11.m11.2.3.3.1.cmml" xref="S4.p1.11.m11.2.3.3.2"><cn id="S4.p1.11.m11.1.1.cmml" type="integer" xref="S4.p1.11.m11.1.1">1</cn><cn id="S4.p1.11.m11.2.2.cmml" type="integer" xref="S4.p1.11.m11.2.2">2</cn></list></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.p1.11.m11.2c">k=1,2</annotation><annotation encoding="application/x-llamapun" id="S4.p1.11.m11.2d">italic_k = 1 , 2</annotation></semantics></math>, we can obtain constant-factor approximations in near-linear space.</p> </div> <div class="ltx_para" id="S4.p2"> <p class="ltx_p" id="S4.p2.3">For ease of notation, we write <math alttext="k" class="ltx_Math" display="inline" id="S4.p2.1.m1.1"><semantics id="S4.p2.1.m1.1a"><mi id="S4.p2.1.m1.1.1" xref="S4.p2.1.m1.1.1.cmml">k</mi><annotation-xml encoding="MathML-Content" id="S4.p2.1.m1.1b"><ci id="S4.p2.1.m1.1.1.cmml" xref="S4.p2.1.m1.1.1">𝑘</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.p2.1.m1.1c">k</annotation><annotation encoding="application/x-llamapun" id="S4.p2.1.m1.1d">italic_k</annotation></semantics></math>-connected to mean <math alttext="k" class="ltx_Math" display="inline" id="S4.p2.2.m2.1"><semantics id="S4.p2.2.m2.1a"><mi id="S4.p2.2.m2.1.1" xref="S4.p2.2.m2.1.1.cmml">k</mi><annotation-xml encoding="MathML-Content" id="S4.p2.2.m2.1b"><ci id="S4.p2.2.m2.1.1.cmml" xref="S4.p2.2.m2.1.1">𝑘</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.p2.2.m2.1c">k</annotation><annotation encoding="application/x-llamapun" id="S4.p2.2.m2.1d">italic_k</annotation></semantics></math>-vertex-connected. In Section <a class="ltx_ref" href="https://arxiv.org/html/2503.00712v1#S4.SS1" title="4.1 One-to-Two Augmentation ‣ 4 Vertex Connectivity Augmentation in Link-Arrival Model ‣ Streaming Algorithms for Network Design"><span class="ltx_text ltx_ref_tag">4.1</span></a> we describe a simple algorithm for augmenting from 1 to 2 connectivity. This algorithm relies on the fact that the underlying 1-connected graph is a tree. Unfortunately, <math alttext="2" class="ltx_Math" display="inline" id="S4.p2.3.m3.1"><semantics id="S4.p2.3.m3.1a"><mn id="S4.p2.3.m3.1.1" xref="S4.p2.3.m3.1.1.cmml">2</mn><annotation-xml encoding="MathML-Content" id="S4.p2.3.m3.1b"><cn id="S4.p2.3.m3.1.1.cmml" type="integer" xref="S4.p2.3.m3.1.1">2</cn></annotation-xml><annotation encoding="application/x-tex" id="S4.p2.3.m3.1c">2</annotation><annotation encoding="application/x-llamapun" id="S4.p2.3.m3.1d">2</annotation></semantics></math>-VC-CAP is significantly less straightforward, since 2-connected graphs do not share the same nice structural properties as trees. To circumvent this issue, we use a data structure called an SPQR tree, which is a tree-like decomposition of a 2-connected graph into its 3-connected components. We describe SPQR trees and their key properties, along with the augmentation algorithm from 2 to 3 connectivity, in Section <a class="ltx_ref" href="https://arxiv.org/html/2503.00712v1#S4.SS2" title="4.2 Two-to-Three Augmentation ‣ 4 Vertex Connectivity Augmentation in Link-Arrival Model ‣ Streaming Algorithms for Network Design"><span class="ltx_text ltx_ref_tag">4.2</span></a>.</p> </div> <section class="ltx_subsection" id="S4.SS1"> <h3 class="ltx_title ltx_title_subsection"> <span class="ltx_tag ltx_tag_subsection">4.1 </span>One-to-Two Augmentation</h3> <div class="ltx_para" id="S4.SS1.p1"> <p class="ltx_p" id="S4.SS1.p1.1">In this section, we prove the following theorem:</p> </div> <div class="ltx_theorem ltx_theorem_theorem" id="S4.Thmtheorem1"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="S4.Thmtheorem1.1.1.1">Theorem 4.1</span></span><span class="ltx_text ltx_font_bold" id="S4.Thmtheorem1.2.2">.</span> </h6> <div class="ltx_para" id="S4.Thmtheorem1.p1"> <p class="ltx_p" id="S4.Thmtheorem1.p1.4">There exists a streaming algorithm for <math alttext="1" class="ltx_Math" display="inline" id="S4.Thmtheorem1.p1.1.m1.1"><semantics id="S4.Thmtheorem1.p1.1.m1.1a"><mn id="S4.Thmtheorem1.p1.1.m1.1.1" xref="S4.Thmtheorem1.p1.1.m1.1.1.cmml">1</mn><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem1.p1.1.m1.1b"><cn id="S4.Thmtheorem1.p1.1.m1.1.1.cmml" type="integer" xref="S4.Thmtheorem1.p1.1.m1.1.1">1</cn></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem1.p1.1.m1.1c">1</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem1.p1.1.m1.1d">1</annotation></semantics></math>-VC-CAP with edge weights <math alttext="w:E\to[1,W]" class="ltx_Math" display="inline" id="S4.Thmtheorem1.p1.2.m2.2"><semantics id="S4.Thmtheorem1.p1.2.m2.2a"><mrow id="S4.Thmtheorem1.p1.2.m2.2.3" xref="S4.Thmtheorem1.p1.2.m2.2.3.cmml"><mi id="S4.Thmtheorem1.p1.2.m2.2.3.2" xref="S4.Thmtheorem1.p1.2.m2.2.3.2.cmml">w</mi><mo id="S4.Thmtheorem1.p1.2.m2.2.3.1" lspace="0.278em" rspace="0.278em" xref="S4.Thmtheorem1.p1.2.m2.2.3.1.cmml">:</mo><mrow id="S4.Thmtheorem1.p1.2.m2.2.3.3" xref="S4.Thmtheorem1.p1.2.m2.2.3.3.cmml"><mi id="S4.Thmtheorem1.p1.2.m2.2.3.3.2" xref="S4.Thmtheorem1.p1.2.m2.2.3.3.2.cmml">E</mi><mo id="S4.Thmtheorem1.p1.2.m2.2.3.3.1" stretchy="false" xref="S4.Thmtheorem1.p1.2.m2.2.3.3.1.cmml">→</mo><mrow id="S4.Thmtheorem1.p1.2.m2.2.3.3.3.2" xref="S4.Thmtheorem1.p1.2.m2.2.3.3.3.1.cmml"><mo id="S4.Thmtheorem1.p1.2.m2.2.3.3.3.2.1" stretchy="false" xref="S4.Thmtheorem1.p1.2.m2.2.3.3.3.1.cmml">[</mo><mn id="S4.Thmtheorem1.p1.2.m2.1.1" xref="S4.Thmtheorem1.p1.2.m2.1.1.cmml">1</mn><mo id="S4.Thmtheorem1.p1.2.m2.2.3.3.3.2.2" xref="S4.Thmtheorem1.p1.2.m2.2.3.3.3.1.cmml">,</mo><mi id="S4.Thmtheorem1.p1.2.m2.2.2" xref="S4.Thmtheorem1.p1.2.m2.2.2.cmml">W</mi><mo id="S4.Thmtheorem1.p1.2.m2.2.3.3.3.2.3" stretchy="false" xref="S4.Thmtheorem1.p1.2.m2.2.3.3.3.1.cmml">]</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem1.p1.2.m2.2b"><apply id="S4.Thmtheorem1.p1.2.m2.2.3.cmml" xref="S4.Thmtheorem1.p1.2.m2.2.3"><ci id="S4.Thmtheorem1.p1.2.m2.2.3.1.cmml" xref="S4.Thmtheorem1.p1.2.m2.2.3.1">:</ci><ci id="S4.Thmtheorem1.p1.2.m2.2.3.2.cmml" xref="S4.Thmtheorem1.p1.2.m2.2.3.2">𝑤</ci><apply id="S4.Thmtheorem1.p1.2.m2.2.3.3.cmml" xref="S4.Thmtheorem1.p1.2.m2.2.3.3"><ci id="S4.Thmtheorem1.p1.2.m2.2.3.3.1.cmml" xref="S4.Thmtheorem1.p1.2.m2.2.3.3.1">→</ci><ci id="S4.Thmtheorem1.p1.2.m2.2.3.3.2.cmml" xref="S4.Thmtheorem1.p1.2.m2.2.3.3.2">𝐸</ci><interval closure="closed" id="S4.Thmtheorem1.p1.2.m2.2.3.3.3.1.cmml" xref="S4.Thmtheorem1.p1.2.m2.2.3.3.3.2"><cn id="S4.Thmtheorem1.p1.2.m2.1.1.cmml" type="integer" xref="S4.Thmtheorem1.p1.2.m2.1.1">1</cn><ci id="S4.Thmtheorem1.p1.2.m2.2.2.cmml" xref="S4.Thmtheorem1.p1.2.m2.2.2">𝑊</ci></interval></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem1.p1.2.m2.2c">w:E\to[1,W]</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem1.p1.2.m2.2d">italic_w : italic_E → [ 1 , italic_W ]</annotation></semantics></math>, in an insertion-only stream, that uses <math alttext="O(n\epsilon^{-1}\log W)" class="ltx_Math" display="inline" id="S4.Thmtheorem1.p1.3.m3.1"><semantics id="S4.Thmtheorem1.p1.3.m3.1a"><mrow id="S4.Thmtheorem1.p1.3.m3.1.1" xref="S4.Thmtheorem1.p1.3.m3.1.1.cmml"><mi id="S4.Thmtheorem1.p1.3.m3.1.1.3" xref="S4.Thmtheorem1.p1.3.m3.1.1.3.cmml">O</mi><mo id="S4.Thmtheorem1.p1.3.m3.1.1.2" xref="S4.Thmtheorem1.p1.3.m3.1.1.2.cmml">⁢</mo><mrow id="S4.Thmtheorem1.p1.3.m3.1.1.1.1" xref="S4.Thmtheorem1.p1.3.m3.1.1.1.1.1.cmml"><mo id="S4.Thmtheorem1.p1.3.m3.1.1.1.1.2" stretchy="false" xref="S4.Thmtheorem1.p1.3.m3.1.1.1.1.1.cmml">(</mo><mrow id="S4.Thmtheorem1.p1.3.m3.1.1.1.1.1" xref="S4.Thmtheorem1.p1.3.m3.1.1.1.1.1.cmml"><mi id="S4.Thmtheorem1.p1.3.m3.1.1.1.1.1.2" xref="S4.Thmtheorem1.p1.3.m3.1.1.1.1.1.2.cmml">n</mi><mo id="S4.Thmtheorem1.p1.3.m3.1.1.1.1.1.1" xref="S4.Thmtheorem1.p1.3.m3.1.1.1.1.1.1.cmml">⁢</mo><msup id="S4.Thmtheorem1.p1.3.m3.1.1.1.1.1.3" xref="S4.Thmtheorem1.p1.3.m3.1.1.1.1.1.3.cmml"><mi id="S4.Thmtheorem1.p1.3.m3.1.1.1.1.1.3.2" xref="S4.Thmtheorem1.p1.3.m3.1.1.1.1.1.3.2.cmml">ϵ</mi><mrow id="S4.Thmtheorem1.p1.3.m3.1.1.1.1.1.3.3" xref="S4.Thmtheorem1.p1.3.m3.1.1.1.1.1.3.3.cmml"><mo id="S4.Thmtheorem1.p1.3.m3.1.1.1.1.1.3.3a" xref="S4.Thmtheorem1.p1.3.m3.1.1.1.1.1.3.3.cmml">−</mo><mn id="S4.Thmtheorem1.p1.3.m3.1.1.1.1.1.3.3.2" xref="S4.Thmtheorem1.p1.3.m3.1.1.1.1.1.3.3.2.cmml">1</mn></mrow></msup><mo id="S4.Thmtheorem1.p1.3.m3.1.1.1.1.1.1a" lspace="0.167em" xref="S4.Thmtheorem1.p1.3.m3.1.1.1.1.1.1.cmml">⁢</mo><mrow id="S4.Thmtheorem1.p1.3.m3.1.1.1.1.1.4" xref="S4.Thmtheorem1.p1.3.m3.1.1.1.1.1.4.cmml"><mi id="S4.Thmtheorem1.p1.3.m3.1.1.1.1.1.4.1" xref="S4.Thmtheorem1.p1.3.m3.1.1.1.1.1.4.1.cmml">log</mi><mo id="S4.Thmtheorem1.p1.3.m3.1.1.1.1.1.4a" lspace="0.167em" xref="S4.Thmtheorem1.p1.3.m3.1.1.1.1.1.4.cmml">⁡</mo><mi id="S4.Thmtheorem1.p1.3.m3.1.1.1.1.1.4.2" xref="S4.Thmtheorem1.p1.3.m3.1.1.1.1.1.4.2.cmml">W</mi></mrow></mrow><mo id="S4.Thmtheorem1.p1.3.m3.1.1.1.1.3" stretchy="false" xref="S4.Thmtheorem1.p1.3.m3.1.1.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem1.p1.3.m3.1b"><apply id="S4.Thmtheorem1.p1.3.m3.1.1.cmml" xref="S4.Thmtheorem1.p1.3.m3.1.1"><times id="S4.Thmtheorem1.p1.3.m3.1.1.2.cmml" xref="S4.Thmtheorem1.p1.3.m3.1.1.2"></times><ci id="S4.Thmtheorem1.p1.3.m3.1.1.3.cmml" xref="S4.Thmtheorem1.p1.3.m3.1.1.3">𝑂</ci><apply id="S4.Thmtheorem1.p1.3.m3.1.1.1.1.1.cmml" xref="S4.Thmtheorem1.p1.3.m3.1.1.1.1"><times id="S4.Thmtheorem1.p1.3.m3.1.1.1.1.1.1.cmml" xref="S4.Thmtheorem1.p1.3.m3.1.1.1.1.1.1"></times><ci id="S4.Thmtheorem1.p1.3.m3.1.1.1.1.1.2.cmml" xref="S4.Thmtheorem1.p1.3.m3.1.1.1.1.1.2">𝑛</ci><apply id="S4.Thmtheorem1.p1.3.m3.1.1.1.1.1.3.cmml" xref="S4.Thmtheorem1.p1.3.m3.1.1.1.1.1.3"><csymbol cd="ambiguous" id="S4.Thmtheorem1.p1.3.m3.1.1.1.1.1.3.1.cmml" xref="S4.Thmtheorem1.p1.3.m3.1.1.1.1.1.3">superscript</csymbol><ci id="S4.Thmtheorem1.p1.3.m3.1.1.1.1.1.3.2.cmml" xref="S4.Thmtheorem1.p1.3.m3.1.1.1.1.1.3.2">italic-ϵ</ci><apply id="S4.Thmtheorem1.p1.3.m3.1.1.1.1.1.3.3.cmml" xref="S4.Thmtheorem1.p1.3.m3.1.1.1.1.1.3.3"><minus id="S4.Thmtheorem1.p1.3.m3.1.1.1.1.1.3.3.1.cmml" xref="S4.Thmtheorem1.p1.3.m3.1.1.1.1.1.3.3"></minus><cn id="S4.Thmtheorem1.p1.3.m3.1.1.1.1.1.3.3.2.cmml" type="integer" xref="S4.Thmtheorem1.p1.3.m3.1.1.1.1.1.3.3.2">1</cn></apply></apply><apply id="S4.Thmtheorem1.p1.3.m3.1.1.1.1.1.4.cmml" xref="S4.Thmtheorem1.p1.3.m3.1.1.1.1.1.4"><log id="S4.Thmtheorem1.p1.3.m3.1.1.1.1.1.4.1.cmml" xref="S4.Thmtheorem1.p1.3.m3.1.1.1.1.1.4.1"></log><ci id="S4.Thmtheorem1.p1.3.m3.1.1.1.1.1.4.2.cmml" xref="S4.Thmtheorem1.p1.3.m3.1.1.1.1.1.4.2">𝑊</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem1.p1.3.m3.1c">O(n\epsilon^{-1}\log W)</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem1.p1.3.m3.1d">italic_O ( italic_n italic_ϵ start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT roman_log italic_W )</annotation></semantics></math> space and outputs a <math alttext="(3+\epsilon)" class="ltx_Math" display="inline" id="S4.Thmtheorem1.p1.4.m4.1"><semantics id="S4.Thmtheorem1.p1.4.m4.1a"><mrow id="S4.Thmtheorem1.p1.4.m4.1.1.1" xref="S4.Thmtheorem1.p1.4.m4.1.1.1.1.cmml"><mo id="S4.Thmtheorem1.p1.4.m4.1.1.1.2" stretchy="false" xref="S4.Thmtheorem1.p1.4.m4.1.1.1.1.cmml">(</mo><mrow id="S4.Thmtheorem1.p1.4.m4.1.1.1.1" xref="S4.Thmtheorem1.p1.4.m4.1.1.1.1.cmml"><mn id="S4.Thmtheorem1.p1.4.m4.1.1.1.1.2" xref="S4.Thmtheorem1.p1.4.m4.1.1.1.1.2.cmml">3</mn><mo id="S4.Thmtheorem1.p1.4.m4.1.1.1.1.1" xref="S4.Thmtheorem1.p1.4.m4.1.1.1.1.1.cmml">+</mo><mi id="S4.Thmtheorem1.p1.4.m4.1.1.1.1.3" xref="S4.Thmtheorem1.p1.4.m4.1.1.1.1.3.cmml">ϵ</mi></mrow><mo id="S4.Thmtheorem1.p1.4.m4.1.1.1.3" stretchy="false" xref="S4.Thmtheorem1.p1.4.m4.1.1.1.1.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem1.p1.4.m4.1b"><apply id="S4.Thmtheorem1.p1.4.m4.1.1.1.1.cmml" xref="S4.Thmtheorem1.p1.4.m4.1.1.1"><plus id="S4.Thmtheorem1.p1.4.m4.1.1.1.1.1.cmml" xref="S4.Thmtheorem1.p1.4.m4.1.1.1.1.1"></plus><cn id="S4.Thmtheorem1.p1.4.m4.1.1.1.1.2.cmml" type="integer" xref="S4.Thmtheorem1.p1.4.m4.1.1.1.1.2">3</cn><ci id="S4.Thmtheorem1.p1.4.m4.1.1.1.1.3.cmml" xref="S4.Thmtheorem1.p1.4.m4.1.1.1.1.3">italic-ϵ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem1.p1.4.m4.1c">(3+\epsilon)</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem1.p1.4.m4.1d">( 3 + italic_ϵ )</annotation></semantics></math>-approximate solution.</p> </div> </div> <div class="ltx_para" id="S4.SS1.p2"> <p class="ltx_p" id="S4.SS1.p2.30">We fix a 1-connected graph <math alttext="G=(V,E)" class="ltx_Math" display="inline" id="S4.SS1.p2.1.m1.2"><semantics id="S4.SS1.p2.1.m1.2a"><mrow id="S4.SS1.p2.1.m1.2.3" xref="S4.SS1.p2.1.m1.2.3.cmml"><mi id="S4.SS1.p2.1.m1.2.3.2" xref="S4.SS1.p2.1.m1.2.3.2.cmml">G</mi><mo id="S4.SS1.p2.1.m1.2.3.1" xref="S4.SS1.p2.1.m1.2.3.1.cmml">=</mo><mrow id="S4.SS1.p2.1.m1.2.3.3.2" xref="S4.SS1.p2.1.m1.2.3.3.1.cmml"><mo id="S4.SS1.p2.1.m1.2.3.3.2.1" stretchy="false" xref="S4.SS1.p2.1.m1.2.3.3.1.cmml">(</mo><mi id="S4.SS1.p2.1.m1.1.1" xref="S4.SS1.p2.1.m1.1.1.cmml">V</mi><mo id="S4.SS1.p2.1.m1.2.3.3.2.2" xref="S4.SS1.p2.1.m1.2.3.3.1.cmml">,</mo><mi id="S4.SS1.p2.1.m1.2.2" xref="S4.SS1.p2.1.m1.2.2.cmml">E</mi><mo id="S4.SS1.p2.1.m1.2.3.3.2.3" stretchy="false" xref="S4.SS1.p2.1.m1.2.3.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.p2.1.m1.2b"><apply id="S4.SS1.p2.1.m1.2.3.cmml" xref="S4.SS1.p2.1.m1.2.3"><eq id="S4.SS1.p2.1.m1.2.3.1.cmml" xref="S4.SS1.p2.1.m1.2.3.1"></eq><ci id="S4.SS1.p2.1.m1.2.3.2.cmml" xref="S4.SS1.p2.1.m1.2.3.2">𝐺</ci><interval closure="open" id="S4.SS1.p2.1.m1.2.3.3.1.cmml" xref="S4.SS1.p2.1.m1.2.3.3.2"><ci id="S4.SS1.p2.1.m1.1.1.cmml" xref="S4.SS1.p2.1.m1.1.1">𝑉</ci><ci id="S4.SS1.p2.1.m1.2.2.cmml" xref="S4.SS1.p2.1.m1.2.2">𝐸</ci></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p2.1.m1.2c">G=(V,E)</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p2.1.m1.2d">italic_G = ( italic_V , italic_E )</annotation></semantics></math>. We can assume without loss of generality that <math alttext="G" class="ltx_Math" display="inline" id="S4.SS1.p2.2.m2.1"><semantics id="S4.SS1.p2.2.m2.1a"><mi id="S4.SS1.p2.2.m2.1.1" xref="S4.SS1.p2.2.m2.1.1.cmml">G</mi><annotation-xml encoding="MathML-Content" id="S4.SS1.p2.2.m2.1b"><ci id="S4.SS1.p2.2.m2.1.1.cmml" xref="S4.SS1.p2.2.m2.1.1">𝐺</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p2.2.m2.1c">G</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p2.2.m2.1d">italic_G</annotation></semantics></math> is a tree; if not, we can fix a spanning tree of <math alttext="G" class="ltx_Math" display="inline" id="S4.SS1.p2.3.m3.1"><semantics id="S4.SS1.p2.3.m3.1a"><mi id="S4.SS1.p2.3.m3.1.1" xref="S4.SS1.p2.3.m3.1.1.cmml">G</mi><annotation-xml encoding="MathML-Content" id="S4.SS1.p2.3.m3.1b"><ci id="S4.SS1.p2.3.m3.1.1.cmml" xref="S4.SS1.p2.3.m3.1.1">𝐺</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p2.3.m3.1c">G</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p2.3.m3.1d">italic_G</annotation></semantics></math> as the underlying graph, and consider all remaining edges as <math alttext="0" class="ltx_Math" display="inline" id="S4.SS1.p2.4.m4.1"><semantics id="S4.SS1.p2.4.m4.1a"><mn id="S4.SS1.p2.4.m4.1.1" xref="S4.SS1.p2.4.m4.1.1.cmml">0</mn><annotation-xml encoding="MathML-Content" id="S4.SS1.p2.4.m4.1b"><cn id="S4.SS1.p2.4.m4.1.1.cmml" type="integer" xref="S4.SS1.p2.4.m4.1.1">0</cn></annotation-xml></semantics></math>-weight links in the stream. It is easy to verify that this does not change the problem. We fix an arbitrary root <math alttext="r" class="ltx_Math" display="inline" id="S4.SS1.p2.5.m5.1"><semantics id="S4.SS1.p2.5.m5.1a"><mi id="S4.SS1.p2.5.m5.1.1" xref="S4.SS1.p2.5.m5.1.1.cmml">r</mi><annotation-xml encoding="MathML-Content" id="S4.SS1.p2.5.m5.1b"><ci id="S4.SS1.p2.5.m5.1.1.cmml" xref="S4.SS1.p2.5.m5.1.1">𝑟</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p2.5.m5.1c">r</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p2.5.m5.1d">italic_r</annotation></semantics></math> of the tree <math alttext="G" class="ltx_Math" display="inline" id="S4.SS1.p2.6.m6.1"><semantics id="S4.SS1.p2.6.m6.1a"><mi id="S4.SS1.p2.6.m6.1.1" xref="S4.SS1.p2.6.m6.1.1.cmml">G</mi><annotation-xml encoding="MathML-Content" id="S4.SS1.p2.6.m6.1b"><ci id="S4.SS1.p2.6.m6.1.1.cmml" xref="S4.SS1.p2.6.m6.1.1">𝐺</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p2.6.m6.1c">G</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p2.6.m6.1d">italic_G</annotation></semantics></math>. For each <math alttext="u\in V" class="ltx_Math" display="inline" id="S4.SS1.p2.7.m7.1"><semantics id="S4.SS1.p2.7.m7.1a"><mrow id="S4.SS1.p2.7.m7.1.1" xref="S4.SS1.p2.7.m7.1.1.cmml"><mi id="S4.SS1.p2.7.m7.1.1.2" xref="S4.SS1.p2.7.m7.1.1.2.cmml">u</mi><mo id="S4.SS1.p2.7.m7.1.1.1" xref="S4.SS1.p2.7.m7.1.1.1.cmml">∈</mo><mi id="S4.SS1.p2.7.m7.1.1.3" xref="S4.SS1.p2.7.m7.1.1.3.cmml">V</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.p2.7.m7.1b"><apply id="S4.SS1.p2.7.m7.1.1.cmml" xref="S4.SS1.p2.7.m7.1.1"><in id="S4.SS1.p2.7.m7.1.1.1.cmml" xref="S4.SS1.p2.7.m7.1.1.1"></in><ci id="S4.SS1.p2.7.m7.1.1.2.cmml" xref="S4.SS1.p2.7.m7.1.1.2">𝑢</ci><ci id="S4.SS1.p2.7.m7.1.1.3.cmml" xref="S4.SS1.p2.7.m7.1.1.3">𝑉</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p2.7.m7.1c">u\in V</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p2.7.m7.1d">italic_u ∈ italic_V</annotation></semantics></math>, we let <math alttext="G_{u}" class="ltx_Math" display="inline" id="S4.SS1.p2.8.m8.1"><semantics id="S4.SS1.p2.8.m8.1a"><msub id="S4.SS1.p2.8.m8.1.1" xref="S4.SS1.p2.8.m8.1.1.cmml"><mi id="S4.SS1.p2.8.m8.1.1.2" xref="S4.SS1.p2.8.m8.1.1.2.cmml">G</mi><mi id="S4.SS1.p2.8.m8.1.1.3" xref="S4.SS1.p2.8.m8.1.1.3.cmml">u</mi></msub><annotation-xml encoding="MathML-Content" id="S4.SS1.p2.8.m8.1b"><apply id="S4.SS1.p2.8.m8.1.1.cmml" xref="S4.SS1.p2.8.m8.1.1"><csymbol cd="ambiguous" id="S4.SS1.p2.8.m8.1.1.1.cmml" xref="S4.SS1.p2.8.m8.1.1">subscript</csymbol><ci id="S4.SS1.p2.8.m8.1.1.2.cmml" xref="S4.SS1.p2.8.m8.1.1.2">𝐺</ci><ci id="S4.SS1.p2.8.m8.1.1.3.cmml" xref="S4.SS1.p2.8.m8.1.1.3">𝑢</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p2.8.m8.1c">G_{u}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p2.8.m8.1d">italic_G start_POSTSUBSCRIPT italic_u end_POSTSUBSCRIPT</annotation></semantics></math> denote the subtree of <math alttext="G" class="ltx_Math" display="inline" id="S4.SS1.p2.9.m9.1"><semantics id="S4.SS1.p2.9.m9.1a"><mi id="S4.SS1.p2.9.m9.1.1" xref="S4.SS1.p2.9.m9.1.1.cmml">G</mi><annotation-xml encoding="MathML-Content" id="S4.SS1.p2.9.m9.1b"><ci id="S4.SS1.p2.9.m9.1.1.cmml" xref="S4.SS1.p2.9.m9.1.1">𝐺</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p2.9.m9.1c">G</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p2.9.m9.1d">italic_G</annotation></semantics></math> rooted at <math alttext="u" class="ltx_Math" display="inline" id="S4.SS1.p2.10.m10.1"><semantics id="S4.SS1.p2.10.m10.1a"><mi id="S4.SS1.p2.10.m10.1.1" xref="S4.SS1.p2.10.m10.1.1.cmml">u</mi><annotation-xml encoding="MathML-Content" id="S4.SS1.p2.10.m10.1b"><ci id="S4.SS1.p2.10.m10.1.1.cmml" xref="S4.SS1.p2.10.m10.1.1">𝑢</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p2.10.m10.1c">u</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p2.10.m10.1d">italic_u</annotation></semantics></math>, and we let <math alttext="C(u)" class="ltx_Math" display="inline" id="S4.SS1.p2.11.m11.1"><semantics id="S4.SS1.p2.11.m11.1a"><mrow id="S4.SS1.p2.11.m11.1.2" xref="S4.SS1.p2.11.m11.1.2.cmml"><mi id="S4.SS1.p2.11.m11.1.2.2" xref="S4.SS1.p2.11.m11.1.2.2.cmml">C</mi><mo id="S4.SS1.p2.11.m11.1.2.1" xref="S4.SS1.p2.11.m11.1.2.1.cmml">⁢</mo><mrow id="S4.SS1.p2.11.m11.1.2.3.2" xref="S4.SS1.p2.11.m11.1.2.cmml"><mo id="S4.SS1.p2.11.m11.1.2.3.2.1" stretchy="false" xref="S4.SS1.p2.11.m11.1.2.cmml">(</mo><mi id="S4.SS1.p2.11.m11.1.1" xref="S4.SS1.p2.11.m11.1.1.cmml">u</mi><mo id="S4.SS1.p2.11.m11.1.2.3.2.2" stretchy="false" xref="S4.SS1.p2.11.m11.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.p2.11.m11.1b"><apply id="S4.SS1.p2.11.m11.1.2.cmml" xref="S4.SS1.p2.11.m11.1.2"><times id="S4.SS1.p2.11.m11.1.2.1.cmml" xref="S4.SS1.p2.11.m11.1.2.1"></times><ci id="S4.SS1.p2.11.m11.1.2.2.cmml" xref="S4.SS1.p2.11.m11.1.2.2">𝐶</ci><ci id="S4.SS1.p2.11.m11.1.1.cmml" xref="S4.SS1.p2.11.m11.1.1">𝑢</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p2.11.m11.1c">C(u)</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p2.11.m11.1d">italic_C ( italic_u )</annotation></semantics></math> denote the set of children of <math alttext="u" class="ltx_Math" display="inline" id="S4.SS1.p2.12.m12.1"><semantics id="S4.SS1.p2.12.m12.1a"><mi id="S4.SS1.p2.12.m12.1.1" xref="S4.SS1.p2.12.m12.1.1.cmml">u</mi><annotation-xml encoding="MathML-Content" id="S4.SS1.p2.12.m12.1b"><ci id="S4.SS1.p2.12.m12.1.1.cmml" xref="S4.SS1.p2.12.m12.1.1">𝑢</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p2.12.m12.1c">u</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p2.12.m12.1d">italic_u</annotation></semantics></math>. For <math alttext="u,v\in V" class="ltx_Math" display="inline" id="S4.SS1.p2.13.m13.2"><semantics id="S4.SS1.p2.13.m13.2a"><mrow id="S4.SS1.p2.13.m13.2.3" xref="S4.SS1.p2.13.m13.2.3.cmml"><mrow id="S4.SS1.p2.13.m13.2.3.2.2" xref="S4.SS1.p2.13.m13.2.3.2.1.cmml"><mi id="S4.SS1.p2.13.m13.1.1" xref="S4.SS1.p2.13.m13.1.1.cmml">u</mi><mo id="S4.SS1.p2.13.m13.2.3.2.2.1" xref="S4.SS1.p2.13.m13.2.3.2.1.cmml">,</mo><mi id="S4.SS1.p2.13.m13.2.2" xref="S4.SS1.p2.13.m13.2.2.cmml">v</mi></mrow><mo id="S4.SS1.p2.13.m13.2.3.1" xref="S4.SS1.p2.13.m13.2.3.1.cmml">∈</mo><mi id="S4.SS1.p2.13.m13.2.3.3" xref="S4.SS1.p2.13.m13.2.3.3.cmml">V</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.p2.13.m13.2b"><apply id="S4.SS1.p2.13.m13.2.3.cmml" xref="S4.SS1.p2.13.m13.2.3"><in id="S4.SS1.p2.13.m13.2.3.1.cmml" xref="S4.SS1.p2.13.m13.2.3.1"></in><list id="S4.SS1.p2.13.m13.2.3.2.1.cmml" xref="S4.SS1.p2.13.m13.2.3.2.2"><ci id="S4.SS1.p2.13.m13.1.1.cmml" xref="S4.SS1.p2.13.m13.1.1">𝑢</ci><ci id="S4.SS1.p2.13.m13.2.2.cmml" xref="S4.SS1.p2.13.m13.2.2">𝑣</ci></list><ci id="S4.SS1.p2.13.m13.2.3.3.cmml" xref="S4.SS1.p2.13.m13.2.3.3">𝑉</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p2.13.m13.2c">u,v\in V</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p2.13.m13.2d">italic_u , italic_v ∈ italic_V</annotation></semantics></math>, we let <math alttext="\text{LCA}(u,v)" class="ltx_Math" display="inline" id="S4.SS1.p2.14.m14.2"><semantics id="S4.SS1.p2.14.m14.2a"><mrow id="S4.SS1.p2.14.m14.2.3" xref="S4.SS1.p2.14.m14.2.3.cmml"><mtext id="S4.SS1.p2.14.m14.2.3.2" xref="S4.SS1.p2.14.m14.2.3.2a.cmml">LCA</mtext><mo id="S4.SS1.p2.14.m14.2.3.1" xref="S4.SS1.p2.14.m14.2.3.1.cmml">⁢</mo><mrow id="S4.SS1.p2.14.m14.2.3.3.2" xref="S4.SS1.p2.14.m14.2.3.3.1.cmml"><mo id="S4.SS1.p2.14.m14.2.3.3.2.1" stretchy="false" xref="S4.SS1.p2.14.m14.2.3.3.1.cmml">(</mo><mi id="S4.SS1.p2.14.m14.1.1" xref="S4.SS1.p2.14.m14.1.1.cmml">u</mi><mo id="S4.SS1.p2.14.m14.2.3.3.2.2" xref="S4.SS1.p2.14.m14.2.3.3.1.cmml">,</mo><mi id="S4.SS1.p2.14.m14.2.2" xref="S4.SS1.p2.14.m14.2.2.cmml">v</mi><mo id="S4.SS1.p2.14.m14.2.3.3.2.3" stretchy="false" xref="S4.SS1.p2.14.m14.2.3.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.p2.14.m14.2b"><apply id="S4.SS1.p2.14.m14.2.3.cmml" xref="S4.SS1.p2.14.m14.2.3"><times id="S4.SS1.p2.14.m14.2.3.1.cmml" xref="S4.SS1.p2.14.m14.2.3.1"></times><ci id="S4.SS1.p2.14.m14.2.3.2a.cmml" xref="S4.SS1.p2.14.m14.2.3.2"><mtext id="S4.SS1.p2.14.m14.2.3.2.cmml" xref="S4.SS1.p2.14.m14.2.3.2">LCA</mtext></ci><interval closure="open" id="S4.SS1.p2.14.m14.2.3.3.1.cmml" xref="S4.SS1.p2.14.m14.2.3.3.2"><ci id="S4.SS1.p2.14.m14.1.1.cmml" xref="S4.SS1.p2.14.m14.1.1">𝑢</ci><ci id="S4.SS1.p2.14.m14.2.2.cmml" xref="S4.SS1.p2.14.m14.2.2">𝑣</ci></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p2.14.m14.2c">\text{LCA}(u,v)</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p2.14.m14.2d">LCA ( italic_u , italic_v )</annotation></semantics></math> denote the lowest common ancestor of <math alttext="u" class="ltx_Math" display="inline" id="S4.SS1.p2.15.m15.1"><semantics id="S4.SS1.p2.15.m15.1a"><mi id="S4.SS1.p2.15.m15.1.1" xref="S4.SS1.p2.15.m15.1.1.cmml">u</mi><annotation-xml encoding="MathML-Content" id="S4.SS1.p2.15.m15.1b"><ci id="S4.SS1.p2.15.m15.1.1.cmml" xref="S4.SS1.p2.15.m15.1.1">𝑢</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p2.15.m15.1c">u</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p2.15.m15.1d">italic_u</annotation></semantics></math> and <math alttext="v" class="ltx_Math" display="inline" id="S4.SS1.p2.16.m16.1"><semantics id="S4.SS1.p2.16.m16.1a"><mi id="S4.SS1.p2.16.m16.1.1" xref="S4.SS1.p2.16.m16.1.1.cmml">v</mi><annotation-xml encoding="MathML-Content" id="S4.SS1.p2.16.m16.1b"><ci id="S4.SS1.p2.16.m16.1.1.cmml" xref="S4.SS1.p2.16.m16.1.1">𝑣</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p2.16.m16.1c">v</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p2.16.m16.1d">italic_v</annotation></semantics></math> in the tree <math alttext="G" class="ltx_Math" display="inline" id="S4.SS1.p2.17.m17.1"><semantics id="S4.SS1.p2.17.m17.1a"><mi id="S4.SS1.p2.17.m17.1.1" xref="S4.SS1.p2.17.m17.1.1.cmml">G</mi><annotation-xml encoding="MathML-Content" id="S4.SS1.p2.17.m17.1b"><ci id="S4.SS1.p2.17.m17.1.1.cmml" xref="S4.SS1.p2.17.m17.1.1">𝐺</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p2.17.m17.1c">G</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p2.17.m17.1d">italic_G</annotation></semantics></math>; this is the vertex <math alttext="w" class="ltx_Math" display="inline" id="S4.SS1.p2.18.m18.1"><semantics id="S4.SS1.p2.18.m18.1a"><mi id="S4.SS1.p2.18.m18.1.1" xref="S4.SS1.p2.18.m18.1.1.cmml">w</mi><annotation-xml encoding="MathML-Content" id="S4.SS1.p2.18.m18.1b"><ci id="S4.SS1.p2.18.m18.1.1.cmml" xref="S4.SS1.p2.18.m18.1.1">𝑤</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p2.18.m18.1c">w</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p2.18.m18.1d">italic_w</annotation></semantics></math> furthest from the root such that <math alttext="u,v\in G_{w}" class="ltx_Math" display="inline" id="S4.SS1.p2.19.m19.2"><semantics id="S4.SS1.p2.19.m19.2a"><mrow id="S4.SS1.p2.19.m19.2.3" xref="S4.SS1.p2.19.m19.2.3.cmml"><mrow id="S4.SS1.p2.19.m19.2.3.2.2" xref="S4.SS1.p2.19.m19.2.3.2.1.cmml"><mi id="S4.SS1.p2.19.m19.1.1" xref="S4.SS1.p2.19.m19.1.1.cmml">u</mi><mo id="S4.SS1.p2.19.m19.2.3.2.2.1" xref="S4.SS1.p2.19.m19.2.3.2.1.cmml">,</mo><mi id="S4.SS1.p2.19.m19.2.2" xref="S4.SS1.p2.19.m19.2.2.cmml">v</mi></mrow><mo id="S4.SS1.p2.19.m19.2.3.1" xref="S4.SS1.p2.19.m19.2.3.1.cmml">∈</mo><msub id="S4.SS1.p2.19.m19.2.3.3" xref="S4.SS1.p2.19.m19.2.3.3.cmml"><mi id="S4.SS1.p2.19.m19.2.3.3.2" xref="S4.SS1.p2.19.m19.2.3.3.2.cmml">G</mi><mi id="S4.SS1.p2.19.m19.2.3.3.3" xref="S4.SS1.p2.19.m19.2.3.3.3.cmml">w</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.p2.19.m19.2b"><apply id="S4.SS1.p2.19.m19.2.3.cmml" xref="S4.SS1.p2.19.m19.2.3"><in id="S4.SS1.p2.19.m19.2.3.1.cmml" xref="S4.SS1.p2.19.m19.2.3.1"></in><list id="S4.SS1.p2.19.m19.2.3.2.1.cmml" xref="S4.SS1.p2.19.m19.2.3.2.2"><ci id="S4.SS1.p2.19.m19.1.1.cmml" xref="S4.SS1.p2.19.m19.1.1">𝑢</ci><ci id="S4.SS1.p2.19.m19.2.2.cmml" xref="S4.SS1.p2.19.m19.2.2">𝑣</ci></list><apply id="S4.SS1.p2.19.m19.2.3.3.cmml" xref="S4.SS1.p2.19.m19.2.3.3"><csymbol cd="ambiguous" id="S4.SS1.p2.19.m19.2.3.3.1.cmml" xref="S4.SS1.p2.19.m19.2.3.3">subscript</csymbol><ci id="S4.SS1.p2.19.m19.2.3.3.2.cmml" xref="S4.SS1.p2.19.m19.2.3.3.2">𝐺</ci><ci id="S4.SS1.p2.19.m19.2.3.3.3.cmml" xref="S4.SS1.p2.19.m19.2.3.3.3">𝑤</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p2.19.m19.2c">u,v\in G_{w}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p2.19.m19.2d">italic_u , italic_v ∈ italic_G start_POSTSUBSCRIPT italic_w end_POSTSUBSCRIPT</annotation></semantics></math>. We overload notation and write <math alttext="\text{LCA}(e)" class="ltx_Math" display="inline" id="S4.SS1.p2.20.m20.1"><semantics id="S4.SS1.p2.20.m20.1a"><mrow id="S4.SS1.p2.20.m20.1.2" xref="S4.SS1.p2.20.m20.1.2.cmml"><mtext id="S4.SS1.p2.20.m20.1.2.2" xref="S4.SS1.p2.20.m20.1.2.2a.cmml">LCA</mtext><mo id="S4.SS1.p2.20.m20.1.2.1" xref="S4.SS1.p2.20.m20.1.2.1.cmml">⁢</mo><mrow id="S4.SS1.p2.20.m20.1.2.3.2" xref="S4.SS1.p2.20.m20.1.2.cmml"><mo id="S4.SS1.p2.20.m20.1.2.3.2.1" stretchy="false" xref="S4.SS1.p2.20.m20.1.2.cmml">(</mo><mi id="S4.SS1.p2.20.m20.1.1" xref="S4.SS1.p2.20.m20.1.1.cmml">e</mi><mo id="S4.SS1.p2.20.m20.1.2.3.2.2" stretchy="false" xref="S4.SS1.p2.20.m20.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.p2.20.m20.1b"><apply id="S4.SS1.p2.20.m20.1.2.cmml" xref="S4.SS1.p2.20.m20.1.2"><times id="S4.SS1.p2.20.m20.1.2.1.cmml" xref="S4.SS1.p2.20.m20.1.2.1"></times><ci id="S4.SS1.p2.20.m20.1.2.2a.cmml" xref="S4.SS1.p2.20.m20.1.2.2"><mtext id="S4.SS1.p2.20.m20.1.2.2.cmml" xref="S4.SS1.p2.20.m20.1.2.2">LCA</mtext></ci><ci id="S4.SS1.p2.20.m20.1.1.cmml" xref="S4.SS1.p2.20.m20.1.1">𝑒</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p2.20.m20.1c">\text{LCA}(e)</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p2.20.m20.1d">LCA ( italic_e )</annotation></semantics></math> for an edge <math alttext="e" class="ltx_Math" display="inline" id="S4.SS1.p2.21.m21.1"><semantics id="S4.SS1.p2.21.m21.1a"><mi id="S4.SS1.p2.21.m21.1.1" xref="S4.SS1.p2.21.m21.1.1.cmml">e</mi><annotation-xml encoding="MathML-Content" id="S4.SS1.p2.21.m21.1b"><ci id="S4.SS1.p2.21.m21.1.1.cmml" xref="S4.SS1.p2.21.m21.1.1">𝑒</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p2.21.m21.1c">e</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p2.21.m21.1d">italic_e</annotation></semantics></math> to be the LCA of its endpoints. We let <math alttext="d_{G}(u,v)" class="ltx_Math" display="inline" id="S4.SS1.p2.22.m22.2"><semantics id="S4.SS1.p2.22.m22.2a"><mrow id="S4.SS1.p2.22.m22.2.3" xref="S4.SS1.p2.22.m22.2.3.cmml"><msub id="S4.SS1.p2.22.m22.2.3.2" xref="S4.SS1.p2.22.m22.2.3.2.cmml"><mi id="S4.SS1.p2.22.m22.2.3.2.2" xref="S4.SS1.p2.22.m22.2.3.2.2.cmml">d</mi><mi id="S4.SS1.p2.22.m22.2.3.2.3" xref="S4.SS1.p2.22.m22.2.3.2.3.cmml">G</mi></msub><mo id="S4.SS1.p2.22.m22.2.3.1" xref="S4.SS1.p2.22.m22.2.3.1.cmml">⁢</mo><mrow id="S4.SS1.p2.22.m22.2.3.3.2" xref="S4.SS1.p2.22.m22.2.3.3.1.cmml"><mo id="S4.SS1.p2.22.m22.2.3.3.2.1" stretchy="false" xref="S4.SS1.p2.22.m22.2.3.3.1.cmml">(</mo><mi id="S4.SS1.p2.22.m22.1.1" xref="S4.SS1.p2.22.m22.1.1.cmml">u</mi><mo id="S4.SS1.p2.22.m22.2.3.3.2.2" xref="S4.SS1.p2.22.m22.2.3.3.1.cmml">,</mo><mi id="S4.SS1.p2.22.m22.2.2" xref="S4.SS1.p2.22.m22.2.2.cmml">v</mi><mo id="S4.SS1.p2.22.m22.2.3.3.2.3" stretchy="false" xref="S4.SS1.p2.22.m22.2.3.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.p2.22.m22.2b"><apply id="S4.SS1.p2.22.m22.2.3.cmml" xref="S4.SS1.p2.22.m22.2.3"><times id="S4.SS1.p2.22.m22.2.3.1.cmml" xref="S4.SS1.p2.22.m22.2.3.1"></times><apply id="S4.SS1.p2.22.m22.2.3.2.cmml" xref="S4.SS1.p2.22.m22.2.3.2"><csymbol cd="ambiguous" id="S4.SS1.p2.22.m22.2.3.2.1.cmml" xref="S4.SS1.p2.22.m22.2.3.2">subscript</csymbol><ci id="S4.SS1.p2.22.m22.2.3.2.2.cmml" xref="S4.SS1.p2.22.m22.2.3.2.2">𝑑</ci><ci id="S4.SS1.p2.22.m22.2.3.2.3.cmml" xref="S4.SS1.p2.22.m22.2.3.2.3">𝐺</ci></apply><interval closure="open" id="S4.SS1.p2.22.m22.2.3.3.1.cmml" xref="S4.SS1.p2.22.m22.2.3.3.2"><ci id="S4.SS1.p2.22.m22.1.1.cmml" xref="S4.SS1.p2.22.m22.1.1">𝑢</ci><ci id="S4.SS1.p2.22.m22.2.2.cmml" xref="S4.SS1.p2.22.m22.2.2">𝑣</ci></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p2.22.m22.2c">d_{G}(u,v)</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p2.22.m22.2d">italic_d start_POSTSUBSCRIPT italic_G end_POSTSUBSCRIPT ( italic_u , italic_v )</annotation></semantics></math> denote the number of edges in the unique tree path between <math alttext="u" class="ltx_Math" display="inline" id="S4.SS1.p2.23.m23.1"><semantics id="S4.SS1.p2.23.m23.1a"><mi id="S4.SS1.p2.23.m23.1.1" xref="S4.SS1.p2.23.m23.1.1.cmml">u</mi><annotation-xml encoding="MathML-Content" id="S4.SS1.p2.23.m23.1b"><ci id="S4.SS1.p2.23.m23.1.1.cmml" xref="S4.SS1.p2.23.m23.1.1">𝑢</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p2.23.m23.1c">u</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p2.23.m23.1d">italic_u</annotation></semantics></math> and <math alttext="v" class="ltx_Math" display="inline" id="S4.SS1.p2.24.m24.1"><semantics id="S4.SS1.p2.24.m24.1a"><mi id="S4.SS1.p2.24.m24.1.1" xref="S4.SS1.p2.24.m24.1.1.cmml">v</mi><annotation-xml encoding="MathML-Content" id="S4.SS1.p2.24.m24.1b"><ci id="S4.SS1.p2.24.m24.1.1.cmml" xref="S4.SS1.p2.24.m24.1.1">𝑣</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p2.24.m24.1c">v</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p2.24.m24.1d">italic_v</annotation></semantics></math>. For any edge set <math alttext="E^{\prime}\subseteq E\cup L" class="ltx_Math" display="inline" id="S4.SS1.p2.25.m25.1"><semantics id="S4.SS1.p2.25.m25.1a"><mrow id="S4.SS1.p2.25.m25.1.1" xref="S4.SS1.p2.25.m25.1.1.cmml"><msup id="S4.SS1.p2.25.m25.1.1.2" xref="S4.SS1.p2.25.m25.1.1.2.cmml"><mi id="S4.SS1.p2.25.m25.1.1.2.2" xref="S4.SS1.p2.25.m25.1.1.2.2.cmml">E</mi><mo id="S4.SS1.p2.25.m25.1.1.2.3" xref="S4.SS1.p2.25.m25.1.1.2.3.cmml">′</mo></msup><mo id="S4.SS1.p2.25.m25.1.1.1" xref="S4.SS1.p2.25.m25.1.1.1.cmml">⊆</mo><mrow id="S4.SS1.p2.25.m25.1.1.3" xref="S4.SS1.p2.25.m25.1.1.3.cmml"><mi id="S4.SS1.p2.25.m25.1.1.3.2" xref="S4.SS1.p2.25.m25.1.1.3.2.cmml">E</mi><mo id="S4.SS1.p2.25.m25.1.1.3.1" xref="S4.SS1.p2.25.m25.1.1.3.1.cmml">∪</mo><mi id="S4.SS1.p2.25.m25.1.1.3.3" xref="S4.SS1.p2.25.m25.1.1.3.3.cmml">L</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.p2.25.m25.1b"><apply id="S4.SS1.p2.25.m25.1.1.cmml" xref="S4.SS1.p2.25.m25.1.1"><subset id="S4.SS1.p2.25.m25.1.1.1.cmml" xref="S4.SS1.p2.25.m25.1.1.1"></subset><apply id="S4.SS1.p2.25.m25.1.1.2.cmml" xref="S4.SS1.p2.25.m25.1.1.2"><csymbol cd="ambiguous" id="S4.SS1.p2.25.m25.1.1.2.1.cmml" xref="S4.SS1.p2.25.m25.1.1.2">superscript</csymbol><ci id="S4.SS1.p2.25.m25.1.1.2.2.cmml" xref="S4.SS1.p2.25.m25.1.1.2.2">𝐸</ci><ci id="S4.SS1.p2.25.m25.1.1.2.3.cmml" xref="S4.SS1.p2.25.m25.1.1.2.3">′</ci></apply><apply id="S4.SS1.p2.25.m25.1.1.3.cmml" xref="S4.SS1.p2.25.m25.1.1.3"><union id="S4.SS1.p2.25.m25.1.1.3.1.cmml" xref="S4.SS1.p2.25.m25.1.1.3.1"></union><ci id="S4.SS1.p2.25.m25.1.1.3.2.cmml" xref="S4.SS1.p2.25.m25.1.1.3.2">𝐸</ci><ci id="S4.SS1.p2.25.m25.1.1.3.3.cmml" xref="S4.SS1.p2.25.m25.1.1.3.3">𝐿</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p2.25.m25.1c">E^{\prime}\subseteq E\cup L</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p2.25.m25.1d">italic_E start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ⊆ italic_E ∪ italic_L</annotation></semantics></math> and any vertex sets <math alttext="A,B\subseteq V" class="ltx_Math" display="inline" id="S4.SS1.p2.26.m26.2"><semantics id="S4.SS1.p2.26.m26.2a"><mrow id="S4.SS1.p2.26.m26.2.3" xref="S4.SS1.p2.26.m26.2.3.cmml"><mrow id="S4.SS1.p2.26.m26.2.3.2.2" xref="S4.SS1.p2.26.m26.2.3.2.1.cmml"><mi id="S4.SS1.p2.26.m26.1.1" xref="S4.SS1.p2.26.m26.1.1.cmml">A</mi><mo id="S4.SS1.p2.26.m26.2.3.2.2.1" xref="S4.SS1.p2.26.m26.2.3.2.1.cmml">,</mo><mi id="S4.SS1.p2.26.m26.2.2" xref="S4.SS1.p2.26.m26.2.2.cmml">B</mi></mrow><mo id="S4.SS1.p2.26.m26.2.3.1" xref="S4.SS1.p2.26.m26.2.3.1.cmml">⊆</mo><mi id="S4.SS1.p2.26.m26.2.3.3" xref="S4.SS1.p2.26.m26.2.3.3.cmml">V</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.p2.26.m26.2b"><apply id="S4.SS1.p2.26.m26.2.3.cmml" xref="S4.SS1.p2.26.m26.2.3"><subset id="S4.SS1.p2.26.m26.2.3.1.cmml" xref="S4.SS1.p2.26.m26.2.3.1"></subset><list id="S4.SS1.p2.26.m26.2.3.2.1.cmml" xref="S4.SS1.p2.26.m26.2.3.2.2"><ci id="S4.SS1.p2.26.m26.1.1.cmml" xref="S4.SS1.p2.26.m26.1.1">𝐴</ci><ci id="S4.SS1.p2.26.m26.2.2.cmml" xref="S4.SS1.p2.26.m26.2.2">𝐵</ci></list><ci id="S4.SS1.p2.26.m26.2.3.3.cmml" xref="S4.SS1.p2.26.m26.2.3.3">𝑉</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p2.26.m26.2c">A,B\subseteq V</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p2.26.m26.2d">italic_A , italic_B ⊆ italic_V</annotation></semantics></math>, we let <math alttext="E^{\prime}[A,B]" class="ltx_Math" display="inline" id="S4.SS1.p2.27.m27.2"><semantics id="S4.SS1.p2.27.m27.2a"><mrow id="S4.SS1.p2.27.m27.2.3" xref="S4.SS1.p2.27.m27.2.3.cmml"><msup id="S4.SS1.p2.27.m27.2.3.2" xref="S4.SS1.p2.27.m27.2.3.2.cmml"><mi id="S4.SS1.p2.27.m27.2.3.2.2" xref="S4.SS1.p2.27.m27.2.3.2.2.cmml">E</mi><mo id="S4.SS1.p2.27.m27.2.3.2.3" xref="S4.SS1.p2.27.m27.2.3.2.3.cmml">′</mo></msup><mo id="S4.SS1.p2.27.m27.2.3.1" xref="S4.SS1.p2.27.m27.2.3.1.cmml">⁢</mo><mrow id="S4.SS1.p2.27.m27.2.3.3.2" xref="S4.SS1.p2.27.m27.2.3.3.1.cmml"><mo id="S4.SS1.p2.27.m27.2.3.3.2.1" stretchy="false" xref="S4.SS1.p2.27.m27.2.3.3.1.cmml">[</mo><mi id="S4.SS1.p2.27.m27.1.1" xref="S4.SS1.p2.27.m27.1.1.cmml">A</mi><mo id="S4.SS1.p2.27.m27.2.3.3.2.2" xref="S4.SS1.p2.27.m27.2.3.3.1.cmml">,</mo><mi id="S4.SS1.p2.27.m27.2.2" xref="S4.SS1.p2.27.m27.2.2.cmml">B</mi><mo id="S4.SS1.p2.27.m27.2.3.3.2.3" stretchy="false" xref="S4.SS1.p2.27.m27.2.3.3.1.cmml">]</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.p2.27.m27.2b"><apply id="S4.SS1.p2.27.m27.2.3.cmml" xref="S4.SS1.p2.27.m27.2.3"><times id="S4.SS1.p2.27.m27.2.3.1.cmml" xref="S4.SS1.p2.27.m27.2.3.1"></times><apply id="S4.SS1.p2.27.m27.2.3.2.cmml" xref="S4.SS1.p2.27.m27.2.3.2"><csymbol cd="ambiguous" id="S4.SS1.p2.27.m27.2.3.2.1.cmml" xref="S4.SS1.p2.27.m27.2.3.2">superscript</csymbol><ci id="S4.SS1.p2.27.m27.2.3.2.2.cmml" xref="S4.SS1.p2.27.m27.2.3.2.2">𝐸</ci><ci id="S4.SS1.p2.27.m27.2.3.2.3.cmml" xref="S4.SS1.p2.27.m27.2.3.2.3">′</ci></apply><interval closure="closed" id="S4.SS1.p2.27.m27.2.3.3.1.cmml" xref="S4.SS1.p2.27.m27.2.3.3.2"><ci id="S4.SS1.p2.27.m27.1.1.cmml" xref="S4.SS1.p2.27.m27.1.1">𝐴</ci><ci id="S4.SS1.p2.27.m27.2.2.cmml" xref="S4.SS1.p2.27.m27.2.2">𝐵</ci></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p2.27.m27.2c">E^{\prime}[A,B]</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p2.27.m27.2d">italic_E start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT [ italic_A , italic_B ]</annotation></semantics></math> denote the set of edges in <math alttext="E^{\prime}" class="ltx_Math" display="inline" id="S4.SS1.p2.28.m28.1"><semantics id="S4.SS1.p2.28.m28.1a"><msup id="S4.SS1.p2.28.m28.1.1" xref="S4.SS1.p2.28.m28.1.1.cmml"><mi id="S4.SS1.p2.28.m28.1.1.2" xref="S4.SS1.p2.28.m28.1.1.2.cmml">E</mi><mo id="S4.SS1.p2.28.m28.1.1.3" xref="S4.SS1.p2.28.m28.1.1.3.cmml">′</mo></msup><annotation-xml encoding="MathML-Content" id="S4.SS1.p2.28.m28.1b"><apply id="S4.SS1.p2.28.m28.1.1.cmml" xref="S4.SS1.p2.28.m28.1.1"><csymbol cd="ambiguous" id="S4.SS1.p2.28.m28.1.1.1.cmml" xref="S4.SS1.p2.28.m28.1.1">superscript</csymbol><ci id="S4.SS1.p2.28.m28.1.1.2.cmml" xref="S4.SS1.p2.28.m28.1.1.2">𝐸</ci><ci id="S4.SS1.p2.28.m28.1.1.3.cmml" xref="S4.SS1.p2.28.m28.1.1.3">′</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p2.28.m28.1c">E^{\prime}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p2.28.m28.1d">italic_E start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math> with one endpoint in <math alttext="A" class="ltx_Math" display="inline" id="S4.SS1.p2.29.m29.1"><semantics id="S4.SS1.p2.29.m29.1a"><mi id="S4.SS1.p2.29.m29.1.1" xref="S4.SS1.p2.29.m29.1.1.cmml">A</mi><annotation-xml encoding="MathML-Content" id="S4.SS1.p2.29.m29.1b"><ci id="S4.SS1.p2.29.m29.1.1.cmml" xref="S4.SS1.p2.29.m29.1.1">𝐴</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p2.29.m29.1c">A</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p2.29.m29.1d">italic_A</annotation></semantics></math> and the other in <math alttext="B" class="ltx_Math" display="inline" id="S4.SS1.p2.30.m30.1"><semantics id="S4.SS1.p2.30.m30.1a"><mi id="S4.SS1.p2.30.m30.1.1" xref="S4.SS1.p2.30.m30.1.1.cmml">B</mi><annotation-xml encoding="MathML-Content" id="S4.SS1.p2.30.m30.1b"><ci id="S4.SS1.p2.30.m30.1.1.cmml" xref="S4.SS1.p2.30.m30.1.1">𝐵</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p2.30.m30.1c">B</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p2.30.m30.1d">italic_B</annotation></semantics></math>. We use the following lemma as a subroutine:</p> </div> <div class="ltx_theorem ltx_theorem_lemma" id="S4.Thmtheorem2"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="S4.Thmtheorem2.1.1.1">Lemma 4.2</span></span><span class="ltx_text ltx_font_bold" id="S4.Thmtheorem2.2.2">.</span> </h6> <div class="ltx_para" id="S4.Thmtheorem2.p1"> <p class="ltx_p" id="S4.Thmtheorem2.p1.4"><cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.00712v1#bib.bibx67" title="">McG14</a>]</cite> Given any set of nodes <math alttext="V^{\prime}" class="ltx_Math" display="inline" id="S4.Thmtheorem2.p1.1.m1.1"><semantics id="S4.Thmtheorem2.p1.1.m1.1a"><msup id="S4.Thmtheorem2.p1.1.m1.1.1" xref="S4.Thmtheorem2.p1.1.m1.1.1.cmml"><mi id="S4.Thmtheorem2.p1.1.m1.1.1.2" xref="S4.Thmtheorem2.p1.1.m1.1.1.2.cmml">V</mi><mo id="S4.Thmtheorem2.p1.1.m1.1.1.3" xref="S4.Thmtheorem2.p1.1.m1.1.1.3.cmml">′</mo></msup><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem2.p1.1.m1.1b"><apply id="S4.Thmtheorem2.p1.1.m1.1.1.cmml" xref="S4.Thmtheorem2.p1.1.m1.1.1"><csymbol cd="ambiguous" id="S4.Thmtheorem2.p1.1.m1.1.1.1.cmml" xref="S4.Thmtheorem2.p1.1.m1.1.1">superscript</csymbol><ci id="S4.Thmtheorem2.p1.1.m1.1.1.2.cmml" xref="S4.Thmtheorem2.p1.1.m1.1.1.2">𝑉</ci><ci id="S4.Thmtheorem2.p1.1.m1.1.1.3.cmml" xref="S4.Thmtheorem2.p1.1.m1.1.1.3">′</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem2.p1.1.m1.1c">V^{\prime}</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem2.p1.1.m1.1d">italic_V start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math> with links <math alttext="L\subseteq V^{\prime}\times V^{\prime}" class="ltx_Math" display="inline" id="S4.Thmtheorem2.p1.2.m2.1"><semantics id="S4.Thmtheorem2.p1.2.m2.1a"><mrow id="S4.Thmtheorem2.p1.2.m2.1.1" xref="S4.Thmtheorem2.p1.2.m2.1.1.cmml"><mi id="S4.Thmtheorem2.p1.2.m2.1.1.2" xref="S4.Thmtheorem2.p1.2.m2.1.1.2.cmml">L</mi><mo id="S4.Thmtheorem2.p1.2.m2.1.1.1" xref="S4.Thmtheorem2.p1.2.m2.1.1.1.cmml">⊆</mo><mrow id="S4.Thmtheorem2.p1.2.m2.1.1.3" xref="S4.Thmtheorem2.p1.2.m2.1.1.3.cmml"><msup id="S4.Thmtheorem2.p1.2.m2.1.1.3.2" xref="S4.Thmtheorem2.p1.2.m2.1.1.3.2.cmml"><mi id="S4.Thmtheorem2.p1.2.m2.1.1.3.2.2" xref="S4.Thmtheorem2.p1.2.m2.1.1.3.2.2.cmml">V</mi><mo id="S4.Thmtheorem2.p1.2.m2.1.1.3.2.3" xref="S4.Thmtheorem2.p1.2.m2.1.1.3.2.3.cmml">′</mo></msup><mo id="S4.Thmtheorem2.p1.2.m2.1.1.3.1" lspace="0.222em" rspace="0.222em" xref="S4.Thmtheorem2.p1.2.m2.1.1.3.1.cmml">×</mo><msup id="S4.Thmtheorem2.p1.2.m2.1.1.3.3" xref="S4.Thmtheorem2.p1.2.m2.1.1.3.3.cmml"><mi id="S4.Thmtheorem2.p1.2.m2.1.1.3.3.2" xref="S4.Thmtheorem2.p1.2.m2.1.1.3.3.2.cmml">V</mi><mo id="S4.Thmtheorem2.p1.2.m2.1.1.3.3.3" xref="S4.Thmtheorem2.p1.2.m2.1.1.3.3.3.cmml">′</mo></msup></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem2.p1.2.m2.1b"><apply id="S4.Thmtheorem2.p1.2.m2.1.1.cmml" xref="S4.Thmtheorem2.p1.2.m2.1.1"><subset id="S4.Thmtheorem2.p1.2.m2.1.1.1.cmml" xref="S4.Thmtheorem2.p1.2.m2.1.1.1"></subset><ci id="S4.Thmtheorem2.p1.2.m2.1.1.2.cmml" xref="S4.Thmtheorem2.p1.2.m2.1.1.2">𝐿</ci><apply id="S4.Thmtheorem2.p1.2.m2.1.1.3.cmml" xref="S4.Thmtheorem2.p1.2.m2.1.1.3"><times id="S4.Thmtheorem2.p1.2.m2.1.1.3.1.cmml" xref="S4.Thmtheorem2.p1.2.m2.1.1.3.1"></times><apply id="S4.Thmtheorem2.p1.2.m2.1.1.3.2.cmml" xref="S4.Thmtheorem2.p1.2.m2.1.1.3.2"><csymbol cd="ambiguous" id="S4.Thmtheorem2.p1.2.m2.1.1.3.2.1.cmml" xref="S4.Thmtheorem2.p1.2.m2.1.1.3.2">superscript</csymbol><ci id="S4.Thmtheorem2.p1.2.m2.1.1.3.2.2.cmml" xref="S4.Thmtheorem2.p1.2.m2.1.1.3.2.2">𝑉</ci><ci id="S4.Thmtheorem2.p1.2.m2.1.1.3.2.3.cmml" xref="S4.Thmtheorem2.p1.2.m2.1.1.3.2.3">′</ci></apply><apply id="S4.Thmtheorem2.p1.2.m2.1.1.3.3.cmml" xref="S4.Thmtheorem2.p1.2.m2.1.1.3.3"><csymbol cd="ambiguous" id="S4.Thmtheorem2.p1.2.m2.1.1.3.3.1.cmml" xref="S4.Thmtheorem2.p1.2.m2.1.1.3.3">superscript</csymbol><ci id="S4.Thmtheorem2.p1.2.m2.1.1.3.3.2.cmml" xref="S4.Thmtheorem2.p1.2.m2.1.1.3.3.2">𝑉</ci><ci id="S4.Thmtheorem2.p1.2.m2.1.1.3.3.3.cmml" xref="S4.Thmtheorem2.p1.2.m2.1.1.3.3.3">′</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem2.p1.2.m2.1c">L\subseteq V^{\prime}\times V^{\prime}</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem2.p1.2.m2.1d">italic_L ⊆ italic_V start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT × italic_V start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math> appearing in an insertion-only stream, one can store a minimum spanning tree on <math alttext="V^{\prime}" class="ltx_Math" display="inline" id="S4.Thmtheorem2.p1.3.m3.1"><semantics id="S4.Thmtheorem2.p1.3.m3.1a"><msup id="S4.Thmtheorem2.p1.3.m3.1.1" xref="S4.Thmtheorem2.p1.3.m3.1.1.cmml"><mi id="S4.Thmtheorem2.p1.3.m3.1.1.2" xref="S4.Thmtheorem2.p1.3.m3.1.1.2.cmml">V</mi><mo id="S4.Thmtheorem2.p1.3.m3.1.1.3" xref="S4.Thmtheorem2.p1.3.m3.1.1.3.cmml">′</mo></msup><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem2.p1.3.m3.1b"><apply id="S4.Thmtheorem2.p1.3.m3.1.1.cmml" xref="S4.Thmtheorem2.p1.3.m3.1.1"><csymbol cd="ambiguous" id="S4.Thmtheorem2.p1.3.m3.1.1.1.cmml" xref="S4.Thmtheorem2.p1.3.m3.1.1">superscript</csymbol><ci id="S4.Thmtheorem2.p1.3.m3.1.1.2.cmml" xref="S4.Thmtheorem2.p1.3.m3.1.1.2">𝑉</ci><ci id="S4.Thmtheorem2.p1.3.m3.1.1.3.cmml" xref="S4.Thmtheorem2.p1.3.m3.1.1.3">′</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem2.p1.3.m3.1c">V^{\prime}</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem2.p1.3.m3.1d">italic_V start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math> using <math alttext="O(|V^{\prime}|)" class="ltx_Math" display="inline" id="S4.Thmtheorem2.p1.4.m4.1"><semantics id="S4.Thmtheorem2.p1.4.m4.1a"><mrow id="S4.Thmtheorem2.p1.4.m4.1.1" xref="S4.Thmtheorem2.p1.4.m4.1.1.cmml"><mi id="S4.Thmtheorem2.p1.4.m4.1.1.3" xref="S4.Thmtheorem2.p1.4.m4.1.1.3.cmml">O</mi><mo id="S4.Thmtheorem2.p1.4.m4.1.1.2" xref="S4.Thmtheorem2.p1.4.m4.1.1.2.cmml">⁢</mo><mrow id="S4.Thmtheorem2.p1.4.m4.1.1.1.1" xref="S4.Thmtheorem2.p1.4.m4.1.1.cmml"><mo id="S4.Thmtheorem2.p1.4.m4.1.1.1.1.2" stretchy="false" xref="S4.Thmtheorem2.p1.4.m4.1.1.cmml">(</mo><mrow id="S4.Thmtheorem2.p1.4.m4.1.1.1.1.1.1" xref="S4.Thmtheorem2.p1.4.m4.1.1.1.1.1.2.cmml"><mo id="S4.Thmtheorem2.p1.4.m4.1.1.1.1.1.1.2" stretchy="false" xref="S4.Thmtheorem2.p1.4.m4.1.1.1.1.1.2.1.cmml">|</mo><msup id="S4.Thmtheorem2.p1.4.m4.1.1.1.1.1.1.1" xref="S4.Thmtheorem2.p1.4.m4.1.1.1.1.1.1.1.cmml"><mi id="S4.Thmtheorem2.p1.4.m4.1.1.1.1.1.1.1.2" xref="S4.Thmtheorem2.p1.4.m4.1.1.1.1.1.1.1.2.cmml">V</mi><mo id="S4.Thmtheorem2.p1.4.m4.1.1.1.1.1.1.1.3" xref="S4.Thmtheorem2.p1.4.m4.1.1.1.1.1.1.1.3.cmml">′</mo></msup><mo id="S4.Thmtheorem2.p1.4.m4.1.1.1.1.1.1.3" stretchy="false" xref="S4.Thmtheorem2.p1.4.m4.1.1.1.1.1.2.1.cmml">|</mo></mrow><mo id="S4.Thmtheorem2.p1.4.m4.1.1.1.1.3" stretchy="false" xref="S4.Thmtheorem2.p1.4.m4.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem2.p1.4.m4.1b"><apply id="S4.Thmtheorem2.p1.4.m4.1.1.cmml" xref="S4.Thmtheorem2.p1.4.m4.1.1"><times id="S4.Thmtheorem2.p1.4.m4.1.1.2.cmml" xref="S4.Thmtheorem2.p1.4.m4.1.1.2"></times><ci id="S4.Thmtheorem2.p1.4.m4.1.1.3.cmml" xref="S4.Thmtheorem2.p1.4.m4.1.1.3">𝑂</ci><apply id="S4.Thmtheorem2.p1.4.m4.1.1.1.1.1.2.cmml" xref="S4.Thmtheorem2.p1.4.m4.1.1.1.1.1.1"><abs id="S4.Thmtheorem2.p1.4.m4.1.1.1.1.1.2.1.cmml" xref="S4.Thmtheorem2.p1.4.m4.1.1.1.1.1.1.2"></abs><apply id="S4.Thmtheorem2.p1.4.m4.1.1.1.1.1.1.1.cmml" xref="S4.Thmtheorem2.p1.4.m4.1.1.1.1.1.1.1"><csymbol cd="ambiguous" id="S4.Thmtheorem2.p1.4.m4.1.1.1.1.1.1.1.1.cmml" xref="S4.Thmtheorem2.p1.4.m4.1.1.1.1.1.1.1">superscript</csymbol><ci id="S4.Thmtheorem2.p1.4.m4.1.1.1.1.1.1.1.2.cmml" xref="S4.Thmtheorem2.p1.4.m4.1.1.1.1.1.1.1.2">𝑉</ci><ci id="S4.Thmtheorem2.p1.4.m4.1.1.1.1.1.1.1.3.cmml" xref="S4.Thmtheorem2.p1.4.m4.1.1.1.1.1.1.1.3">′</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem2.p1.4.m4.1c">O(|V^{\prime}|)</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem2.p1.4.m4.1d">italic_O ( | italic_V start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT | )</annotation></semantics></math> memory space.</p> </div> </div> <section class="ltx_subsubsection" id="S4.SS1.SSS1"> <h4 class="ltx_title ltx_title_subsubsection"> <span class="ltx_tag ltx_tag_subsubsection">4.1.1 </span>The Streaming Algorithm</h4> <div class="ltx_para" id="S4.SS1.SSS1.p1"> <p class="ltx_p" id="S4.SS1.SSS1.p1.9">For each vertex <math alttext="u\in V" class="ltx_Math" display="inline" id="S4.SS1.SSS1.p1.1.m1.1"><semantics id="S4.SS1.SSS1.p1.1.m1.1a"><mrow id="S4.SS1.SSS1.p1.1.m1.1.1" xref="S4.SS1.SSS1.p1.1.m1.1.1.cmml"><mi id="S4.SS1.SSS1.p1.1.m1.1.1.2" xref="S4.SS1.SSS1.p1.1.m1.1.1.2.cmml">u</mi><mo id="S4.SS1.SSS1.p1.1.m1.1.1.1" xref="S4.SS1.SSS1.p1.1.m1.1.1.1.cmml">∈</mo><mi id="S4.SS1.SSS1.p1.1.m1.1.1.3" xref="S4.SS1.SSS1.p1.1.m1.1.1.3.cmml">V</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.SSS1.p1.1.m1.1b"><apply id="S4.SS1.SSS1.p1.1.m1.1.1.cmml" xref="S4.SS1.SSS1.p1.1.m1.1.1"><in id="S4.SS1.SSS1.p1.1.m1.1.1.1.cmml" xref="S4.SS1.SSS1.p1.1.m1.1.1.1"></in><ci id="S4.SS1.SSS1.p1.1.m1.1.1.2.cmml" xref="S4.SS1.SSS1.p1.1.m1.1.1.2">𝑢</ci><ci id="S4.SS1.SSS1.p1.1.m1.1.1.3.cmml" xref="S4.SS1.SSS1.p1.1.m1.1.1.3">𝑉</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.SSS1.p1.1.m1.1c">u\in V</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.SSS1.p1.1.m1.1d">italic_u ∈ italic_V</annotation></semantics></math> and each weight bucket <math alttext="[(1+\epsilon)^{j},(1+\epsilon)^{j+1})" class="ltx_Math" display="inline" id="S4.SS1.SSS1.p1.2.m2.2"><semantics id="S4.SS1.SSS1.p1.2.m2.2a"><mrow id="S4.SS1.SSS1.p1.2.m2.2.2.2" xref="S4.SS1.SSS1.p1.2.m2.2.2.3.cmml"><mo id="S4.SS1.SSS1.p1.2.m2.2.2.2.3" stretchy="false" xref="S4.SS1.SSS1.p1.2.m2.2.2.3.cmml">[</mo><msup id="S4.SS1.SSS1.p1.2.m2.1.1.1.1" xref="S4.SS1.SSS1.p1.2.m2.1.1.1.1.cmml"><mrow id="S4.SS1.SSS1.p1.2.m2.1.1.1.1.1.1" xref="S4.SS1.SSS1.p1.2.m2.1.1.1.1.1.1.1.cmml"><mo id="S4.SS1.SSS1.p1.2.m2.1.1.1.1.1.1.2" stretchy="false" xref="S4.SS1.SSS1.p1.2.m2.1.1.1.1.1.1.1.cmml">(</mo><mrow id="S4.SS1.SSS1.p1.2.m2.1.1.1.1.1.1.1" xref="S4.SS1.SSS1.p1.2.m2.1.1.1.1.1.1.1.cmml"><mn id="S4.SS1.SSS1.p1.2.m2.1.1.1.1.1.1.1.2" xref="S4.SS1.SSS1.p1.2.m2.1.1.1.1.1.1.1.2.cmml">1</mn><mo id="S4.SS1.SSS1.p1.2.m2.1.1.1.1.1.1.1.1" xref="S4.SS1.SSS1.p1.2.m2.1.1.1.1.1.1.1.1.cmml">+</mo><mi id="S4.SS1.SSS1.p1.2.m2.1.1.1.1.1.1.1.3" xref="S4.SS1.SSS1.p1.2.m2.1.1.1.1.1.1.1.3.cmml">ϵ</mi></mrow><mo id="S4.SS1.SSS1.p1.2.m2.1.1.1.1.1.1.3" stretchy="false" xref="S4.SS1.SSS1.p1.2.m2.1.1.1.1.1.1.1.cmml">)</mo></mrow><mi id="S4.SS1.SSS1.p1.2.m2.1.1.1.1.3" xref="S4.SS1.SSS1.p1.2.m2.1.1.1.1.3.cmml">j</mi></msup><mo id="S4.SS1.SSS1.p1.2.m2.2.2.2.4" xref="S4.SS1.SSS1.p1.2.m2.2.2.3.cmml">,</mo><msup id="S4.SS1.SSS1.p1.2.m2.2.2.2.2" xref="S4.SS1.SSS1.p1.2.m2.2.2.2.2.cmml"><mrow id="S4.SS1.SSS1.p1.2.m2.2.2.2.2.1.1" xref="S4.SS1.SSS1.p1.2.m2.2.2.2.2.1.1.1.cmml"><mo id="S4.SS1.SSS1.p1.2.m2.2.2.2.2.1.1.2" stretchy="false" xref="S4.SS1.SSS1.p1.2.m2.2.2.2.2.1.1.1.cmml">(</mo><mrow id="S4.SS1.SSS1.p1.2.m2.2.2.2.2.1.1.1" xref="S4.SS1.SSS1.p1.2.m2.2.2.2.2.1.1.1.cmml"><mn id="S4.SS1.SSS1.p1.2.m2.2.2.2.2.1.1.1.2" xref="S4.SS1.SSS1.p1.2.m2.2.2.2.2.1.1.1.2.cmml">1</mn><mo id="S4.SS1.SSS1.p1.2.m2.2.2.2.2.1.1.1.1" xref="S4.SS1.SSS1.p1.2.m2.2.2.2.2.1.1.1.1.cmml">+</mo><mi id="S4.SS1.SSS1.p1.2.m2.2.2.2.2.1.1.1.3" xref="S4.SS1.SSS1.p1.2.m2.2.2.2.2.1.1.1.3.cmml">ϵ</mi></mrow><mo id="S4.SS1.SSS1.p1.2.m2.2.2.2.2.1.1.3" stretchy="false" xref="S4.SS1.SSS1.p1.2.m2.2.2.2.2.1.1.1.cmml">)</mo></mrow><mrow id="S4.SS1.SSS1.p1.2.m2.2.2.2.2.3" xref="S4.SS1.SSS1.p1.2.m2.2.2.2.2.3.cmml"><mi id="S4.SS1.SSS1.p1.2.m2.2.2.2.2.3.2" xref="S4.SS1.SSS1.p1.2.m2.2.2.2.2.3.2.cmml">j</mi><mo id="S4.SS1.SSS1.p1.2.m2.2.2.2.2.3.1" xref="S4.SS1.SSS1.p1.2.m2.2.2.2.2.3.1.cmml">+</mo><mn id="S4.SS1.SSS1.p1.2.m2.2.2.2.2.3.3" xref="S4.SS1.SSS1.p1.2.m2.2.2.2.2.3.3.cmml">1</mn></mrow></msup><mo id="S4.SS1.SSS1.p1.2.m2.2.2.2.5" stretchy="false" xref="S4.SS1.SSS1.p1.2.m2.2.2.3.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.SSS1.p1.2.m2.2b"><interval closure="closed-open" id="S4.SS1.SSS1.p1.2.m2.2.2.3.cmml" xref="S4.SS1.SSS1.p1.2.m2.2.2.2"><apply id="S4.SS1.SSS1.p1.2.m2.1.1.1.1.cmml" xref="S4.SS1.SSS1.p1.2.m2.1.1.1.1"><csymbol cd="ambiguous" id="S4.SS1.SSS1.p1.2.m2.1.1.1.1.2.cmml" xref="S4.SS1.SSS1.p1.2.m2.1.1.1.1">superscript</csymbol><apply id="S4.SS1.SSS1.p1.2.m2.1.1.1.1.1.1.1.cmml" xref="S4.SS1.SSS1.p1.2.m2.1.1.1.1.1.1"><plus id="S4.SS1.SSS1.p1.2.m2.1.1.1.1.1.1.1.1.cmml" xref="S4.SS1.SSS1.p1.2.m2.1.1.1.1.1.1.1.1"></plus><cn id="S4.SS1.SSS1.p1.2.m2.1.1.1.1.1.1.1.2.cmml" type="integer" xref="S4.SS1.SSS1.p1.2.m2.1.1.1.1.1.1.1.2">1</cn><ci id="S4.SS1.SSS1.p1.2.m2.1.1.1.1.1.1.1.3.cmml" xref="S4.SS1.SSS1.p1.2.m2.1.1.1.1.1.1.1.3">italic-ϵ</ci></apply><ci id="S4.SS1.SSS1.p1.2.m2.1.1.1.1.3.cmml" xref="S4.SS1.SSS1.p1.2.m2.1.1.1.1.3">𝑗</ci></apply><apply id="S4.SS1.SSS1.p1.2.m2.2.2.2.2.cmml" xref="S4.SS1.SSS1.p1.2.m2.2.2.2.2"><csymbol cd="ambiguous" id="S4.SS1.SSS1.p1.2.m2.2.2.2.2.2.cmml" xref="S4.SS1.SSS1.p1.2.m2.2.2.2.2">superscript</csymbol><apply id="S4.SS1.SSS1.p1.2.m2.2.2.2.2.1.1.1.cmml" xref="S4.SS1.SSS1.p1.2.m2.2.2.2.2.1.1"><plus id="S4.SS1.SSS1.p1.2.m2.2.2.2.2.1.1.1.1.cmml" xref="S4.SS1.SSS1.p1.2.m2.2.2.2.2.1.1.1.1"></plus><cn id="S4.SS1.SSS1.p1.2.m2.2.2.2.2.1.1.1.2.cmml" type="integer" xref="S4.SS1.SSS1.p1.2.m2.2.2.2.2.1.1.1.2">1</cn><ci id="S4.SS1.SSS1.p1.2.m2.2.2.2.2.1.1.1.3.cmml" xref="S4.SS1.SSS1.p1.2.m2.2.2.2.2.1.1.1.3">italic-ϵ</ci></apply><apply id="S4.SS1.SSS1.p1.2.m2.2.2.2.2.3.cmml" xref="S4.SS1.SSS1.p1.2.m2.2.2.2.2.3"><plus id="S4.SS1.SSS1.p1.2.m2.2.2.2.2.3.1.cmml" xref="S4.SS1.SSS1.p1.2.m2.2.2.2.2.3.1"></plus><ci id="S4.SS1.SSS1.p1.2.m2.2.2.2.2.3.2.cmml" xref="S4.SS1.SSS1.p1.2.m2.2.2.2.2.3.2">𝑗</ci><cn id="S4.SS1.SSS1.p1.2.m2.2.2.2.2.3.3.cmml" type="integer" xref="S4.SS1.SSS1.p1.2.m2.2.2.2.2.3.3">1</cn></apply></apply></interval></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.SSS1.p1.2.m2.2c">[(1+\epsilon)^{j},(1+\epsilon)^{j+1})</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.SSS1.p1.2.m2.2d">[ ( 1 + italic_ϵ ) start_POSTSUPERSCRIPT italic_j end_POSTSUPERSCRIPT , ( 1 + italic_ϵ ) start_POSTSUPERSCRIPT italic_j + 1 end_POSTSUPERSCRIPT )</annotation></semantics></math>, we store the link <math alttext="uv\in L" class="ltx_Math" display="inline" id="S4.SS1.SSS1.p1.3.m3.1"><semantics id="S4.SS1.SSS1.p1.3.m3.1a"><mrow id="S4.SS1.SSS1.p1.3.m3.1.1" xref="S4.SS1.SSS1.p1.3.m3.1.1.cmml"><mrow id="S4.SS1.SSS1.p1.3.m3.1.1.2" xref="S4.SS1.SSS1.p1.3.m3.1.1.2.cmml"><mi id="S4.SS1.SSS1.p1.3.m3.1.1.2.2" xref="S4.SS1.SSS1.p1.3.m3.1.1.2.2.cmml">u</mi><mo id="S4.SS1.SSS1.p1.3.m3.1.1.2.1" xref="S4.SS1.SSS1.p1.3.m3.1.1.2.1.cmml">⁢</mo><mi id="S4.SS1.SSS1.p1.3.m3.1.1.2.3" xref="S4.SS1.SSS1.p1.3.m3.1.1.2.3.cmml">v</mi></mrow><mo id="S4.SS1.SSS1.p1.3.m3.1.1.1" xref="S4.SS1.SSS1.p1.3.m3.1.1.1.cmml">∈</mo><mi id="S4.SS1.SSS1.p1.3.m3.1.1.3" xref="S4.SS1.SSS1.p1.3.m3.1.1.3.cmml">L</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.SSS1.p1.3.m3.1b"><apply id="S4.SS1.SSS1.p1.3.m3.1.1.cmml" xref="S4.SS1.SSS1.p1.3.m3.1.1"><in id="S4.SS1.SSS1.p1.3.m3.1.1.1.cmml" xref="S4.SS1.SSS1.p1.3.m3.1.1.1"></in><apply id="S4.SS1.SSS1.p1.3.m3.1.1.2.cmml" xref="S4.SS1.SSS1.p1.3.m3.1.1.2"><times id="S4.SS1.SSS1.p1.3.m3.1.1.2.1.cmml" xref="S4.SS1.SSS1.p1.3.m3.1.1.2.1"></times><ci id="S4.SS1.SSS1.p1.3.m3.1.1.2.2.cmml" xref="S4.SS1.SSS1.p1.3.m3.1.1.2.2">𝑢</ci><ci id="S4.SS1.SSS1.p1.3.m3.1.1.2.3.cmml" xref="S4.SS1.SSS1.p1.3.m3.1.1.2.3">𝑣</ci></apply><ci id="S4.SS1.SSS1.p1.3.m3.1.1.3.cmml" xref="S4.SS1.SSS1.p1.3.m3.1.1.3">𝐿</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.SSS1.p1.3.m3.1c">uv\in L</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.SSS1.p1.3.m3.1d">italic_u italic_v ∈ italic_L</annotation></semantics></math> in this weight bucket with <math alttext="\text{LCA}(u,v)" class="ltx_Math" display="inline" id="S4.SS1.SSS1.p1.4.m4.2"><semantics id="S4.SS1.SSS1.p1.4.m4.2a"><mrow id="S4.SS1.SSS1.p1.4.m4.2.3" xref="S4.SS1.SSS1.p1.4.m4.2.3.cmml"><mtext id="S4.SS1.SSS1.p1.4.m4.2.3.2" xref="S4.SS1.SSS1.p1.4.m4.2.3.2a.cmml">LCA</mtext><mo id="S4.SS1.SSS1.p1.4.m4.2.3.1" xref="S4.SS1.SSS1.p1.4.m4.2.3.1.cmml">⁢</mo><mrow id="S4.SS1.SSS1.p1.4.m4.2.3.3.2" xref="S4.SS1.SSS1.p1.4.m4.2.3.3.1.cmml"><mo id="S4.SS1.SSS1.p1.4.m4.2.3.3.2.1" stretchy="false" xref="S4.SS1.SSS1.p1.4.m4.2.3.3.1.cmml">(</mo><mi id="S4.SS1.SSS1.p1.4.m4.1.1" xref="S4.SS1.SSS1.p1.4.m4.1.1.cmml">u</mi><mo id="S4.SS1.SSS1.p1.4.m4.2.3.3.2.2" xref="S4.SS1.SSS1.p1.4.m4.2.3.3.1.cmml">,</mo><mi id="S4.SS1.SSS1.p1.4.m4.2.2" xref="S4.SS1.SSS1.p1.4.m4.2.2.cmml">v</mi><mo id="S4.SS1.SSS1.p1.4.m4.2.3.3.2.3" stretchy="false" xref="S4.SS1.SSS1.p1.4.m4.2.3.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.SSS1.p1.4.m4.2b"><apply id="S4.SS1.SSS1.p1.4.m4.2.3.cmml" xref="S4.SS1.SSS1.p1.4.m4.2.3"><times id="S4.SS1.SSS1.p1.4.m4.2.3.1.cmml" xref="S4.SS1.SSS1.p1.4.m4.2.3.1"></times><ci id="S4.SS1.SSS1.p1.4.m4.2.3.2a.cmml" xref="S4.SS1.SSS1.p1.4.m4.2.3.2"><mtext id="S4.SS1.SSS1.p1.4.m4.2.3.2.cmml" xref="S4.SS1.SSS1.p1.4.m4.2.3.2">LCA</mtext></ci><interval closure="open" id="S4.SS1.SSS1.p1.4.m4.2.3.3.1.cmml" xref="S4.SS1.SSS1.p1.4.m4.2.3.3.2"><ci id="S4.SS1.SSS1.p1.4.m4.1.1.cmml" xref="S4.SS1.SSS1.p1.4.m4.1.1">𝑢</ci><ci id="S4.SS1.SSS1.p1.4.m4.2.2.cmml" xref="S4.SS1.SSS1.p1.4.m4.2.2">𝑣</ci></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.SSS1.p1.4.m4.2c">\text{LCA}(u,v)</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.SSS1.p1.4.m4.2d">LCA ( italic_u , italic_v )</annotation></semantics></math> closest to the root; these will be stored in the dictionaries <math alttext="L_{u}" class="ltx_Math" display="inline" id="S4.SS1.SSS1.p1.5.m5.1"><semantics id="S4.SS1.SSS1.p1.5.m5.1a"><msub id="S4.SS1.SSS1.p1.5.m5.1.1" xref="S4.SS1.SSS1.p1.5.m5.1.1.cmml"><mi id="S4.SS1.SSS1.p1.5.m5.1.1.2" xref="S4.SS1.SSS1.p1.5.m5.1.1.2.cmml">L</mi><mi id="S4.SS1.SSS1.p1.5.m5.1.1.3" xref="S4.SS1.SSS1.p1.5.m5.1.1.3.cmml">u</mi></msub><annotation-xml encoding="MathML-Content" id="S4.SS1.SSS1.p1.5.m5.1b"><apply id="S4.SS1.SSS1.p1.5.m5.1.1.cmml" xref="S4.SS1.SSS1.p1.5.m5.1.1"><csymbol cd="ambiguous" id="S4.SS1.SSS1.p1.5.m5.1.1.1.cmml" xref="S4.SS1.SSS1.p1.5.m5.1.1">subscript</csymbol><ci id="S4.SS1.SSS1.p1.5.m5.1.1.2.cmml" xref="S4.SS1.SSS1.p1.5.m5.1.1.2">𝐿</ci><ci id="S4.SS1.SSS1.p1.5.m5.1.1.3.cmml" xref="S4.SS1.SSS1.p1.5.m5.1.1.3">𝑢</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.SSS1.p1.5.m5.1c">L_{u}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.SSS1.p1.5.m5.1d">italic_L start_POSTSUBSCRIPT italic_u end_POSTSUBSCRIPT</annotation></semantics></math> defined in Algorithm <a class="ltx_ref" href="https://arxiv.org/html/2503.00712v1#algorithm4" title="In 4.1.1 The Streaming Algorithm ‣ 4.1 One-to-Two Augmentation ‣ 4 Vertex Connectivity Augmentation in Link-Arrival Model ‣ Streaming Algorithms for Network Design"><span class="ltx_text ltx_ref_tag">4</span></a>. Furthermore, for each <math alttext="u\in G" class="ltx_Math" display="inline" id="S4.SS1.SSS1.p1.6.m6.1"><semantics id="S4.SS1.SSS1.p1.6.m6.1a"><mrow id="S4.SS1.SSS1.p1.6.m6.1.1" xref="S4.SS1.SSS1.p1.6.m6.1.1.cmml"><mi id="S4.SS1.SSS1.p1.6.m6.1.1.2" xref="S4.SS1.SSS1.p1.6.m6.1.1.2.cmml">u</mi><mo id="S4.SS1.SSS1.p1.6.m6.1.1.1" xref="S4.SS1.SSS1.p1.6.m6.1.1.1.cmml">∈</mo><mi id="S4.SS1.SSS1.p1.6.m6.1.1.3" xref="S4.SS1.SSS1.p1.6.m6.1.1.3.cmml">G</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.SSS1.p1.6.m6.1b"><apply id="S4.SS1.SSS1.p1.6.m6.1.1.cmml" xref="S4.SS1.SSS1.p1.6.m6.1.1"><in id="S4.SS1.SSS1.p1.6.m6.1.1.1.cmml" xref="S4.SS1.SSS1.p1.6.m6.1.1.1"></in><ci id="S4.SS1.SSS1.p1.6.m6.1.1.2.cmml" xref="S4.SS1.SSS1.p1.6.m6.1.1.2">𝑢</ci><ci id="S4.SS1.SSS1.p1.6.m6.1.1.3.cmml" xref="S4.SS1.SSS1.p1.6.m6.1.1.3">𝐺</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.SSS1.p1.6.m6.1c">u\in G</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.SSS1.p1.6.m6.1d">italic_u ∈ italic_G</annotation></semantics></math>, we consider a contracted graph with <math alttext="C(u)" class="ltx_Math" display="inline" id="S4.SS1.SSS1.p1.7.m7.1"><semantics id="S4.SS1.SSS1.p1.7.m7.1a"><mrow id="S4.SS1.SSS1.p1.7.m7.1.2" xref="S4.SS1.SSS1.p1.7.m7.1.2.cmml"><mi id="S4.SS1.SSS1.p1.7.m7.1.2.2" xref="S4.SS1.SSS1.p1.7.m7.1.2.2.cmml">C</mi><mo id="S4.SS1.SSS1.p1.7.m7.1.2.1" xref="S4.SS1.SSS1.p1.7.m7.1.2.1.cmml">⁢</mo><mrow id="S4.SS1.SSS1.p1.7.m7.1.2.3.2" xref="S4.SS1.SSS1.p1.7.m7.1.2.cmml"><mo id="S4.SS1.SSS1.p1.7.m7.1.2.3.2.1" stretchy="false" xref="S4.SS1.SSS1.p1.7.m7.1.2.cmml">(</mo><mi id="S4.SS1.SSS1.p1.7.m7.1.1" xref="S4.SS1.SSS1.p1.7.m7.1.1.cmml">u</mi><mo id="S4.SS1.SSS1.p1.7.m7.1.2.3.2.2" stretchy="false" xref="S4.SS1.SSS1.p1.7.m7.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.SSS1.p1.7.m7.1b"><apply id="S4.SS1.SSS1.p1.7.m7.1.2.cmml" xref="S4.SS1.SSS1.p1.7.m7.1.2"><times id="S4.SS1.SSS1.p1.7.m7.1.2.1.cmml" xref="S4.SS1.SSS1.p1.7.m7.1.2.1"></times><ci id="S4.SS1.SSS1.p1.7.m7.1.2.2.cmml" xref="S4.SS1.SSS1.p1.7.m7.1.2.2">𝐶</ci><ci id="S4.SS1.SSS1.p1.7.m7.1.1.cmml" xref="S4.SS1.SSS1.p1.7.m7.1.1">𝑢</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.SSS1.p1.7.m7.1c">C(u)</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.SSS1.p1.7.m7.1d">italic_C ( italic_u )</annotation></semantics></math> nodes, where each <math alttext="G_{v}" class="ltx_Math" display="inline" id="S4.SS1.SSS1.p1.8.m8.1"><semantics id="S4.SS1.SSS1.p1.8.m8.1a"><msub id="S4.SS1.SSS1.p1.8.m8.1.1" xref="S4.SS1.SSS1.p1.8.m8.1.1.cmml"><mi id="S4.SS1.SSS1.p1.8.m8.1.1.2" xref="S4.SS1.SSS1.p1.8.m8.1.1.2.cmml">G</mi><mi id="S4.SS1.SSS1.p1.8.m8.1.1.3" xref="S4.SS1.SSS1.p1.8.m8.1.1.3.cmml">v</mi></msub><annotation-xml encoding="MathML-Content" id="S4.SS1.SSS1.p1.8.m8.1b"><apply id="S4.SS1.SSS1.p1.8.m8.1.1.cmml" xref="S4.SS1.SSS1.p1.8.m8.1.1"><csymbol cd="ambiguous" id="S4.SS1.SSS1.p1.8.m8.1.1.1.cmml" xref="S4.SS1.SSS1.p1.8.m8.1.1">subscript</csymbol><ci id="S4.SS1.SSS1.p1.8.m8.1.1.2.cmml" xref="S4.SS1.SSS1.p1.8.m8.1.1.2">𝐺</ci><ci id="S4.SS1.SSS1.p1.8.m8.1.1.3.cmml" xref="S4.SS1.SSS1.p1.8.m8.1.1.3">𝑣</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.SSS1.p1.8.m8.1c">G_{v}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.SSS1.p1.8.m8.1d">italic_G start_POSTSUBSCRIPT italic_v end_POSTSUBSCRIPT</annotation></semantics></math> for <math alttext="v\in C(u)" class="ltx_Math" display="inline" id="S4.SS1.SSS1.p1.9.m9.1"><semantics id="S4.SS1.SSS1.p1.9.m9.1a"><mrow id="S4.SS1.SSS1.p1.9.m9.1.2" xref="S4.SS1.SSS1.p1.9.m9.1.2.cmml"><mi id="S4.SS1.SSS1.p1.9.m9.1.2.2" xref="S4.SS1.SSS1.p1.9.m9.1.2.2.cmml">v</mi><mo id="S4.SS1.SSS1.p1.9.m9.1.2.1" xref="S4.SS1.SSS1.p1.9.m9.1.2.1.cmml">∈</mo><mrow id="S4.SS1.SSS1.p1.9.m9.1.2.3" xref="S4.SS1.SSS1.p1.9.m9.1.2.3.cmml"><mi id="S4.SS1.SSS1.p1.9.m9.1.2.3.2" xref="S4.SS1.SSS1.p1.9.m9.1.2.3.2.cmml">C</mi><mo id="S4.SS1.SSS1.p1.9.m9.1.2.3.1" xref="S4.SS1.SSS1.p1.9.m9.1.2.3.1.cmml">⁢</mo><mrow id="S4.SS1.SSS1.p1.9.m9.1.2.3.3.2" xref="S4.SS1.SSS1.p1.9.m9.1.2.3.cmml"><mo id="S4.SS1.SSS1.p1.9.m9.1.2.3.3.2.1" stretchy="false" xref="S4.SS1.SSS1.p1.9.m9.1.2.3.cmml">(</mo><mi id="S4.SS1.SSS1.p1.9.m9.1.1" xref="S4.SS1.SSS1.p1.9.m9.1.1.cmml">u</mi><mo id="S4.SS1.SSS1.p1.9.m9.1.2.3.3.2.2" stretchy="false" xref="S4.SS1.SSS1.p1.9.m9.1.2.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.SSS1.p1.9.m9.1b"><apply id="S4.SS1.SSS1.p1.9.m9.1.2.cmml" xref="S4.SS1.SSS1.p1.9.m9.1.2"><in id="S4.SS1.SSS1.p1.9.m9.1.2.1.cmml" xref="S4.SS1.SSS1.p1.9.m9.1.2.1"></in><ci id="S4.SS1.SSS1.p1.9.m9.1.2.2.cmml" xref="S4.SS1.SSS1.p1.9.m9.1.2.2">𝑣</ci><apply id="S4.SS1.SSS1.p1.9.m9.1.2.3.cmml" xref="S4.SS1.SSS1.p1.9.m9.1.2.3"><times id="S4.SS1.SSS1.p1.9.m9.1.2.3.1.cmml" xref="S4.SS1.SSS1.p1.9.m9.1.2.3.1"></times><ci id="S4.SS1.SSS1.p1.9.m9.1.2.3.2.cmml" xref="S4.SS1.SSS1.p1.9.m9.1.2.3.2">𝐶</ci><ci id="S4.SS1.SSS1.p1.9.m9.1.1.cmml" xref="S4.SS1.SSS1.p1.9.m9.1.1">𝑢</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.SSS1.p1.9.m9.1c">v\in C(u)</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.SSS1.p1.9.m9.1d">italic_v ∈ italic_C ( italic_u )</annotation></semantics></math> is contracted into a node. We maintain an MST on this contracted graph. The details are formalized in Algorithm <a class="ltx_ref" href="https://arxiv.org/html/2503.00712v1#algorithm4" title="In 4.1.1 The Streaming Algorithm ‣ 4.1 One-to-Two Augmentation ‣ 4 Vertex Connectivity Augmentation in Link-Arrival Model ‣ Streaming Algorithms for Network Design"><span class="ltx_text ltx_ref_tag">4</span></a> below.</p> </div> <figure class="ltx_float ltx_algorithm" id="algorithm4"> <div class="ltx_listing ltx_lst_numbers_left ltx_listing" id="algorithm4.28"> <div class="ltx_listingline" id="algorithm4.3.3"> <span class="ltx_text" id="algorithm4.3.3.1"><span class="ltx_text ltx_font_bold" id="algorithm4.3.3.1.1">Input:</span> </span>Weighted tree <math alttext="G=(V,E)" class="ltx_Math" display="inline" id="algorithm4.1.1.m1.2"><semantics id="algorithm4.1.1.m1.2a"><mrow id="algorithm4.1.1.m1.2.3" xref="algorithm4.1.1.m1.2.3.cmml"><mi id="algorithm4.1.1.m1.2.3.2" xref="algorithm4.1.1.m1.2.3.2.cmml">G</mi><mo id="algorithm4.1.1.m1.2.3.1" xref="algorithm4.1.1.m1.2.3.1.cmml">=</mo><mrow id="algorithm4.1.1.m1.2.3.3.2" xref="algorithm4.1.1.m1.2.3.3.1.cmml"><mo id="algorithm4.1.1.m1.2.3.3.2.1" stretchy="false" xref="algorithm4.1.1.m1.2.3.3.1.cmml">(</mo><mi id="algorithm4.1.1.m1.1.1" xref="algorithm4.1.1.m1.1.1.cmml">V</mi><mo id="algorithm4.1.1.m1.2.3.3.2.2" xref="algorithm4.1.1.m1.2.3.3.1.cmml">,</mo><mi id="algorithm4.1.1.m1.2.2" xref="algorithm4.1.1.m1.2.2.cmml">E</mi><mo id="algorithm4.1.1.m1.2.3.3.2.3" stretchy="false" xref="algorithm4.1.1.m1.2.3.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="algorithm4.1.1.m1.2b"><apply id="algorithm4.1.1.m1.2.3.cmml" xref="algorithm4.1.1.m1.2.3"><eq id="algorithm4.1.1.m1.2.3.1.cmml" xref="algorithm4.1.1.m1.2.3.1"></eq><ci id="algorithm4.1.1.m1.2.3.2.cmml" xref="algorithm4.1.1.m1.2.3.2">𝐺</ci><interval closure="open" id="algorithm4.1.1.m1.2.3.3.1.cmml" xref="algorithm4.1.1.m1.2.3.3.2"><ci id="algorithm4.1.1.m1.1.1.cmml" xref="algorithm4.1.1.m1.1.1">𝑉</ci><ci id="algorithm4.1.1.m1.2.2.cmml" xref="algorithm4.1.1.m1.2.2">𝐸</ci></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="algorithm4.1.1.m1.2c">G=(V,E)</annotation><annotation encoding="application/x-llamapun" id="algorithm4.1.1.m1.2d">italic_G = ( italic_V , italic_E )</annotation></semantics></math> with root <math alttext="r\in V" class="ltx_Math" display="inline" id="algorithm4.2.2.m2.1"><semantics id="algorithm4.2.2.m2.1a"><mrow id="algorithm4.2.2.m2.1.1" xref="algorithm4.2.2.m2.1.1.cmml"><mi id="algorithm4.2.2.m2.1.1.2" xref="algorithm4.2.2.m2.1.1.2.cmml">r</mi><mo id="algorithm4.2.2.m2.1.1.1" xref="algorithm4.2.2.m2.1.1.1.cmml">∈</mo><mi id="algorithm4.2.2.m2.1.1.3" xref="algorithm4.2.2.m2.1.1.3.cmml">V</mi></mrow><annotation-xml encoding="MathML-Content" id="algorithm4.2.2.m2.1b"><apply id="algorithm4.2.2.m2.1.1.cmml" xref="algorithm4.2.2.m2.1.1"><in id="algorithm4.2.2.m2.1.1.1.cmml" xref="algorithm4.2.2.m2.1.1.1"></in><ci id="algorithm4.2.2.m2.1.1.2.cmml" xref="algorithm4.2.2.m2.1.1.2">𝑟</ci><ci id="algorithm4.2.2.m2.1.1.3.cmml" xref="algorithm4.2.2.m2.1.1.3">𝑉</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="algorithm4.2.2.m2.1c">r\in V</annotation><annotation encoding="application/x-llamapun" id="algorithm4.2.2.m2.1d">italic_r ∈ italic_V</annotation></semantics></math> and edge weights <math alttext="w:E\rightarrow[1,W]" class="ltx_Math" display="inline" id="algorithm4.3.3.m3.2"><semantics id="algorithm4.3.3.m3.2a"><mrow id="algorithm4.3.3.m3.2.3" xref="algorithm4.3.3.m3.2.3.cmml"><mi id="algorithm4.3.3.m3.2.3.2" xref="algorithm4.3.3.m3.2.3.2.cmml">w</mi><mo id="algorithm4.3.3.m3.2.3.1" lspace="0.278em" rspace="0.278em" xref="algorithm4.3.3.m3.2.3.1.cmml">:</mo><mrow id="algorithm4.3.3.m3.2.3.3" xref="algorithm4.3.3.m3.2.3.3.cmml"><mi id="algorithm4.3.3.m3.2.3.3.2" xref="algorithm4.3.3.m3.2.3.3.2.cmml">E</mi><mo id="algorithm4.3.3.m3.2.3.3.1" stretchy="false" xref="algorithm4.3.3.m3.2.3.3.1.cmml">→</mo><mrow id="algorithm4.3.3.m3.2.3.3.3.2" xref="algorithm4.3.3.m3.2.3.3.3.1.cmml"><mo id="algorithm4.3.3.m3.2.3.3.3.2.1" stretchy="false" xref="algorithm4.3.3.m3.2.3.3.3.1.cmml">[</mo><mn id="algorithm4.3.3.m3.1.1" xref="algorithm4.3.3.m3.1.1.cmml">1</mn><mo id="algorithm4.3.3.m3.2.3.3.3.2.2" xref="algorithm4.3.3.m3.2.3.3.3.1.cmml">,</mo><mi id="algorithm4.3.3.m3.2.2" xref="algorithm4.3.3.m3.2.2.cmml">W</mi><mo id="algorithm4.3.3.m3.2.3.3.3.2.3" stretchy="false" xref="algorithm4.3.3.m3.2.3.3.3.1.cmml">]</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="algorithm4.3.3.m3.2b"><apply id="algorithm4.3.3.m3.2.3.cmml" xref="algorithm4.3.3.m3.2.3"><ci id="algorithm4.3.3.m3.2.3.1.cmml" xref="algorithm4.3.3.m3.2.3.1">:</ci><ci id="algorithm4.3.3.m3.2.3.2.cmml" xref="algorithm4.3.3.m3.2.3.2">𝑤</ci><apply id="algorithm4.3.3.m3.2.3.3.cmml" xref="algorithm4.3.3.m3.2.3.3"><ci id="algorithm4.3.3.m3.2.3.3.1.cmml" xref="algorithm4.3.3.m3.2.3.3.1">→</ci><ci id="algorithm4.3.3.m3.2.3.3.2.cmml" xref="algorithm4.3.3.m3.2.3.3.2">𝐸</ci><interval closure="closed" id="algorithm4.3.3.m3.2.3.3.3.1.cmml" xref="algorithm4.3.3.m3.2.3.3.3.2"><cn id="algorithm4.3.3.m3.1.1.cmml" type="integer" xref="algorithm4.3.3.m3.1.1">1</cn><ci id="algorithm4.3.3.m3.2.2.cmml" xref="algorithm4.3.3.m3.2.2">𝑊</ci></interval></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="algorithm4.3.3.m3.2c">w:E\rightarrow[1,W]</annotation><annotation encoding="application/x-llamapun" id="algorithm4.3.3.m3.2d">italic_w : italic_E → [ 1 , italic_W ]</annotation></semantics></math>. </div> <div class="ltx_listingline" id="algorithm4.28.29"> <span class="ltx_text" id="algorithm4.28.29.1" style="color:#0000FF;">/* </span><span class="ltx_text ltx_font_smallcaps" id="algorithm4.28.29.2" style="color:#0000FF;">Preprocessing: */</span> </div> <div class="ltx_listingline" id="algorithm4.4.4"> <span class="ltx_text ltx_font_bold" id="algorithm4.4.4.2">for</span> <em class="ltx_emph ltx_font_italic" id="algorithm4.4.4.1"><math alttext="u\in V" class="ltx_Math" display="inline" id="algorithm4.4.4.1.m1.1"><semantics id="algorithm4.4.4.1.m1.1a"><mrow id="algorithm4.4.4.1.m1.1.1" xref="algorithm4.4.4.1.m1.1.1.cmml"><mi id="algorithm4.4.4.1.m1.1.1.2" xref="algorithm4.4.4.1.m1.1.1.2.cmml">u</mi><mo id="algorithm4.4.4.1.m1.1.1.1" xref="algorithm4.4.4.1.m1.1.1.1.cmml">∈</mo><mi id="algorithm4.4.4.1.m1.1.1.3" xref="algorithm4.4.4.1.m1.1.1.3.cmml">V</mi></mrow><annotation-xml encoding="MathML-Content" id="algorithm4.4.4.1.m1.1b"><apply id="algorithm4.4.4.1.m1.1.1.cmml" xref="algorithm4.4.4.1.m1.1.1"><in id="algorithm4.4.4.1.m1.1.1.1.cmml" xref="algorithm4.4.4.1.m1.1.1.1"></in><ci id="algorithm4.4.4.1.m1.1.1.2.cmml" xref="algorithm4.4.4.1.m1.1.1.2">𝑢</ci><ci id="algorithm4.4.4.1.m1.1.1.3.cmml" xref="algorithm4.4.4.1.m1.1.1.3">𝑉</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="algorithm4.4.4.1.m1.1c">u\in V</annotation><annotation encoding="application/x-llamapun" id="algorithm4.4.4.1.m1.1d">italic_u ∈ italic_V</annotation></semantics></math></em> <span class="ltx_text ltx_font_bold" id="algorithm4.4.4.3">do</span> </div> <div class="ltx_listingline" id="algorithm4.5.5"> <span class="ltx_rule" style="width:1px;height:100%;background:black;display:inline-block;"> </span> <span class="ltx_text ltx_font_bold" id="algorithm4.5.5.1">initialize</span> an empty dictionary <math alttext="L_{u}" class="ltx_Math" display="inline" id="algorithm4.5.5.m1.1"><semantics id="algorithm4.5.5.m1.1a"><msub id="algorithm4.5.5.m1.1.1" xref="algorithm4.5.5.m1.1.1.cmml"><mi id="algorithm4.5.5.m1.1.1.2" xref="algorithm4.5.5.m1.1.1.2.cmml">L</mi><mi id="algorithm4.5.5.m1.1.1.3" xref="algorithm4.5.5.m1.1.1.3.cmml">u</mi></msub><annotation-xml encoding="MathML-Content" id="algorithm4.5.5.m1.1b"><apply id="algorithm4.5.5.m1.1.1.cmml" xref="algorithm4.5.5.m1.1.1"><csymbol cd="ambiguous" id="algorithm4.5.5.m1.1.1.1.cmml" xref="algorithm4.5.5.m1.1.1">subscript</csymbol><ci id="algorithm4.5.5.m1.1.1.2.cmml" xref="algorithm4.5.5.m1.1.1.2">𝐿</ci><ci id="algorithm4.5.5.m1.1.1.3.cmml" xref="algorithm4.5.5.m1.1.1.3">𝑢</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="algorithm4.5.5.m1.1c">L_{u}</annotation><annotation encoding="application/x-llamapun" id="algorithm4.5.5.m1.1d">italic_L start_POSTSUBSCRIPT italic_u end_POSTSUBSCRIPT</annotation></semantics></math> </div> <div class="ltx_listingline" id="algorithm4.9.9"> <span class="ltx_rule" style="width:1px;height:100%;background:black;display:inline-block;"> </span> <span class="ltx_text ltx_font_bold" id="algorithm4.9.9.1">construct</span> <math alttext="C^{\prime}(u)" class="ltx_Math" display="inline" id="algorithm4.6.6.m1.1"><semantics id="algorithm4.6.6.m1.1a"><mrow id="algorithm4.6.6.m1.1.2" xref="algorithm4.6.6.m1.1.2.cmml"><msup id="algorithm4.6.6.m1.1.2.2" xref="algorithm4.6.6.m1.1.2.2.cmml"><mi id="algorithm4.6.6.m1.1.2.2.2" xref="algorithm4.6.6.m1.1.2.2.2.cmml">C</mi><mo id="algorithm4.6.6.m1.1.2.2.3" xref="algorithm4.6.6.m1.1.2.2.3.cmml">′</mo></msup><mo id="algorithm4.6.6.m1.1.2.1" xref="algorithm4.6.6.m1.1.2.1.cmml">⁢</mo><mrow id="algorithm4.6.6.m1.1.2.3.2" xref="algorithm4.6.6.m1.1.2.cmml"><mo id="algorithm4.6.6.m1.1.2.3.2.1" stretchy="false" xref="algorithm4.6.6.m1.1.2.cmml">(</mo><mi id="algorithm4.6.6.m1.1.1" xref="algorithm4.6.6.m1.1.1.cmml">u</mi><mo id="algorithm4.6.6.m1.1.2.3.2.2" stretchy="false" xref="algorithm4.6.6.m1.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="algorithm4.6.6.m1.1b"><apply id="algorithm4.6.6.m1.1.2.cmml" xref="algorithm4.6.6.m1.1.2"><times id="algorithm4.6.6.m1.1.2.1.cmml" xref="algorithm4.6.6.m1.1.2.1"></times><apply id="algorithm4.6.6.m1.1.2.2.cmml" xref="algorithm4.6.6.m1.1.2.2"><csymbol cd="ambiguous" id="algorithm4.6.6.m1.1.2.2.1.cmml" xref="algorithm4.6.6.m1.1.2.2">superscript</csymbol><ci id="algorithm4.6.6.m1.1.2.2.2.cmml" xref="algorithm4.6.6.m1.1.2.2.2">𝐶</ci><ci id="algorithm4.6.6.m1.1.2.2.3.cmml" xref="algorithm4.6.6.m1.1.2.2.3">′</ci></apply><ci id="algorithm4.6.6.m1.1.1.cmml" xref="algorithm4.6.6.m1.1.1">𝑢</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="algorithm4.6.6.m1.1c">C^{\prime}(u)</annotation><annotation encoding="application/x-llamapun" id="algorithm4.6.6.m1.1d">italic_C start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ( italic_u )</annotation></semantics></math>: for each <math alttext="v\in C(u)" class="ltx_Math" display="inline" id="algorithm4.7.7.m2.1"><semantics id="algorithm4.7.7.m2.1a"><mrow id="algorithm4.7.7.m2.1.2" xref="algorithm4.7.7.m2.1.2.cmml"><mi id="algorithm4.7.7.m2.1.2.2" xref="algorithm4.7.7.m2.1.2.2.cmml">v</mi><mo id="algorithm4.7.7.m2.1.2.1" xref="algorithm4.7.7.m2.1.2.1.cmml">∈</mo><mrow id="algorithm4.7.7.m2.1.2.3" xref="algorithm4.7.7.m2.1.2.3.cmml"><mi id="algorithm4.7.7.m2.1.2.3.2" xref="algorithm4.7.7.m2.1.2.3.2.cmml">C</mi><mo id="algorithm4.7.7.m2.1.2.3.1" xref="algorithm4.7.7.m2.1.2.3.1.cmml">⁢</mo><mrow id="algorithm4.7.7.m2.1.2.3.3.2" xref="algorithm4.7.7.m2.1.2.3.cmml"><mo id="algorithm4.7.7.m2.1.2.3.3.2.1" stretchy="false" xref="algorithm4.7.7.m2.1.2.3.cmml">(</mo><mi id="algorithm4.7.7.m2.1.1" xref="algorithm4.7.7.m2.1.1.cmml">u</mi><mo id="algorithm4.7.7.m2.1.2.3.3.2.2" stretchy="false" xref="algorithm4.7.7.m2.1.2.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="algorithm4.7.7.m2.1b"><apply id="algorithm4.7.7.m2.1.2.cmml" xref="algorithm4.7.7.m2.1.2"><in id="algorithm4.7.7.m2.1.2.1.cmml" xref="algorithm4.7.7.m2.1.2.1"></in><ci id="algorithm4.7.7.m2.1.2.2.cmml" xref="algorithm4.7.7.m2.1.2.2">𝑣</ci><apply id="algorithm4.7.7.m2.1.2.3.cmml" xref="algorithm4.7.7.m2.1.2.3"><times id="algorithm4.7.7.m2.1.2.3.1.cmml" xref="algorithm4.7.7.m2.1.2.3.1"></times><ci id="algorithm4.7.7.m2.1.2.3.2.cmml" xref="algorithm4.7.7.m2.1.2.3.2">𝐶</ci><ci id="algorithm4.7.7.m2.1.1.cmml" xref="algorithm4.7.7.m2.1.1">𝑢</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="algorithm4.7.7.m2.1c">v\in C(u)</annotation><annotation encoding="application/x-llamapun" id="algorithm4.7.7.m2.1d">italic_v ∈ italic_C ( italic_u )</annotation></semantics></math>, contract <math alttext="G_{u}" class="ltx_Math" display="inline" id="algorithm4.8.8.m3.1"><semantics id="algorithm4.8.8.m3.1a"><msub id="algorithm4.8.8.m3.1.1" xref="algorithm4.8.8.m3.1.1.cmml"><mi id="algorithm4.8.8.m3.1.1.2" xref="algorithm4.8.8.m3.1.1.2.cmml">G</mi><mi id="algorithm4.8.8.m3.1.1.3" xref="algorithm4.8.8.m3.1.1.3.cmml">u</mi></msub><annotation-xml encoding="MathML-Content" id="algorithm4.8.8.m3.1b"><apply id="algorithm4.8.8.m3.1.1.cmml" xref="algorithm4.8.8.m3.1.1"><csymbol cd="ambiguous" id="algorithm4.8.8.m3.1.1.1.cmml" xref="algorithm4.8.8.m3.1.1">subscript</csymbol><ci id="algorithm4.8.8.m3.1.1.2.cmml" xref="algorithm4.8.8.m3.1.1.2">𝐺</ci><ci id="algorithm4.8.8.m3.1.1.3.cmml" xref="algorithm4.8.8.m3.1.1.3">𝑢</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="algorithm4.8.8.m3.1c">G_{u}</annotation><annotation encoding="application/x-llamapun" id="algorithm4.8.8.m3.1d">italic_G start_POSTSUBSCRIPT italic_u end_POSTSUBSCRIPT</annotation></semantics></math> into one “supernode” and add it to <math alttext="C^{\prime}(u)" class="ltx_Math" display="inline" id="algorithm4.9.9.m4.1"><semantics id="algorithm4.9.9.m4.1a"><mrow id="algorithm4.9.9.m4.1.2" xref="algorithm4.9.9.m4.1.2.cmml"><msup id="algorithm4.9.9.m4.1.2.2" xref="algorithm4.9.9.m4.1.2.2.cmml"><mi id="algorithm4.9.9.m4.1.2.2.2" xref="algorithm4.9.9.m4.1.2.2.2.cmml">C</mi><mo id="algorithm4.9.9.m4.1.2.2.3" xref="algorithm4.9.9.m4.1.2.2.3.cmml">′</mo></msup><mo id="algorithm4.9.9.m4.1.2.1" xref="algorithm4.9.9.m4.1.2.1.cmml">⁢</mo><mrow id="algorithm4.9.9.m4.1.2.3.2" xref="algorithm4.9.9.m4.1.2.cmml"><mo id="algorithm4.9.9.m4.1.2.3.2.1" stretchy="false" xref="algorithm4.9.9.m4.1.2.cmml">(</mo><mi id="algorithm4.9.9.m4.1.1" xref="algorithm4.9.9.m4.1.1.cmml">u</mi><mo id="algorithm4.9.9.m4.1.2.3.2.2" stretchy="false" xref="algorithm4.9.9.m4.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="algorithm4.9.9.m4.1b"><apply id="algorithm4.9.9.m4.1.2.cmml" xref="algorithm4.9.9.m4.1.2"><times id="algorithm4.9.9.m4.1.2.1.cmml" xref="algorithm4.9.9.m4.1.2.1"></times><apply id="algorithm4.9.9.m4.1.2.2.cmml" xref="algorithm4.9.9.m4.1.2.2"><csymbol cd="ambiguous" id="algorithm4.9.9.m4.1.2.2.1.cmml" xref="algorithm4.9.9.m4.1.2.2">superscript</csymbol><ci id="algorithm4.9.9.m4.1.2.2.2.cmml" xref="algorithm4.9.9.m4.1.2.2.2">𝐶</ci><ci id="algorithm4.9.9.m4.1.2.2.3.cmml" xref="algorithm4.9.9.m4.1.2.2.3">′</ci></apply><ci id="algorithm4.9.9.m4.1.1.cmml" xref="algorithm4.9.9.m4.1.1">𝑢</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="algorithm4.9.9.m4.1c">C^{\prime}(u)</annotation><annotation encoding="application/x-llamapun" id="algorithm4.9.9.m4.1d">italic_C start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ( italic_u )</annotation></semantics></math> </div> <div class="ltx_listingline" id="algorithm4.12.12"> <span class="ltx_rule" style="width:1px;height:100%;background:black;display:inline-block;"> </span> <span class="ltx_text ltx_font_bold" id="algorithm4.12.12.1">define</span> the graph <math alttext="T^{\prime}_{u}" class="ltx_Math" display="inline" id="algorithm4.10.10.m1.1"><semantics id="algorithm4.10.10.m1.1a"><msubsup id="algorithm4.10.10.m1.1.1" xref="algorithm4.10.10.m1.1.1.cmml"><mi id="algorithm4.10.10.m1.1.1.2.2" xref="algorithm4.10.10.m1.1.1.2.2.cmml">T</mi><mi id="algorithm4.10.10.m1.1.1.3" xref="algorithm4.10.10.m1.1.1.3.cmml">u</mi><mo id="algorithm4.10.10.m1.1.1.2.3" xref="algorithm4.10.10.m1.1.1.2.3.cmml">′</mo></msubsup><annotation-xml encoding="MathML-Content" id="algorithm4.10.10.m1.1b"><apply id="algorithm4.10.10.m1.1.1.cmml" xref="algorithm4.10.10.m1.1.1"><csymbol cd="ambiguous" id="algorithm4.10.10.m1.1.1.1.cmml" xref="algorithm4.10.10.m1.1.1">subscript</csymbol><apply id="algorithm4.10.10.m1.1.1.2.cmml" xref="algorithm4.10.10.m1.1.1"><csymbol cd="ambiguous" id="algorithm4.10.10.m1.1.1.2.1.cmml" xref="algorithm4.10.10.m1.1.1">superscript</csymbol><ci id="algorithm4.10.10.m1.1.1.2.2.cmml" xref="algorithm4.10.10.m1.1.1.2.2">𝑇</ci><ci id="algorithm4.10.10.m1.1.1.2.3.cmml" xref="algorithm4.10.10.m1.1.1.2.3">′</ci></apply><ci id="algorithm4.10.10.m1.1.1.3.cmml" xref="algorithm4.10.10.m1.1.1.3">𝑢</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="algorithm4.10.10.m1.1c">T^{\prime}_{u}</annotation><annotation encoding="application/x-llamapun" id="algorithm4.10.10.m1.1d">italic_T start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_u end_POSTSUBSCRIPT</annotation></semantics></math> with vertex set <math alttext="C^{\prime}(u)" class="ltx_Math" display="inline" id="algorithm4.11.11.m2.1"><semantics id="algorithm4.11.11.m2.1a"><mrow id="algorithm4.11.11.m2.1.2" xref="algorithm4.11.11.m2.1.2.cmml"><msup id="algorithm4.11.11.m2.1.2.2" xref="algorithm4.11.11.m2.1.2.2.cmml"><mi id="algorithm4.11.11.m2.1.2.2.2" xref="algorithm4.11.11.m2.1.2.2.2.cmml">C</mi><mo id="algorithm4.11.11.m2.1.2.2.3" xref="algorithm4.11.11.m2.1.2.2.3.cmml">′</mo></msup><mo id="algorithm4.11.11.m2.1.2.1" xref="algorithm4.11.11.m2.1.2.1.cmml">⁢</mo><mrow id="algorithm4.11.11.m2.1.2.3.2" xref="algorithm4.11.11.m2.1.2.cmml"><mo id="algorithm4.11.11.m2.1.2.3.2.1" stretchy="false" xref="algorithm4.11.11.m2.1.2.cmml">(</mo><mi id="algorithm4.11.11.m2.1.1" xref="algorithm4.11.11.m2.1.1.cmml">u</mi><mo id="algorithm4.11.11.m2.1.2.3.2.2" stretchy="false" xref="algorithm4.11.11.m2.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="algorithm4.11.11.m2.1b"><apply id="algorithm4.11.11.m2.1.2.cmml" xref="algorithm4.11.11.m2.1.2"><times id="algorithm4.11.11.m2.1.2.1.cmml" xref="algorithm4.11.11.m2.1.2.1"></times><apply id="algorithm4.11.11.m2.1.2.2.cmml" xref="algorithm4.11.11.m2.1.2.2"><csymbol cd="ambiguous" id="algorithm4.11.11.m2.1.2.2.1.cmml" xref="algorithm4.11.11.m2.1.2.2">superscript</csymbol><ci id="algorithm4.11.11.m2.1.2.2.2.cmml" xref="algorithm4.11.11.m2.1.2.2.2">𝐶</ci><ci id="algorithm4.11.11.m2.1.2.2.3.cmml" xref="algorithm4.11.11.m2.1.2.2.3">′</ci></apply><ci id="algorithm4.11.11.m2.1.1.cmml" xref="algorithm4.11.11.m2.1.1">𝑢</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="algorithm4.11.11.m2.1c">C^{\prime}(u)</annotation><annotation encoding="application/x-llamapun" id="algorithm4.11.11.m2.1d">italic_C start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ( italic_u )</annotation></semantics></math> and edge set <math alttext="\emptyset" class="ltx_Math" display="inline" id="algorithm4.12.12.m3.1"><semantics id="algorithm4.12.12.m3.1a"><mi id="algorithm4.12.12.m3.1.1" mathvariant="normal" xref="algorithm4.12.12.m3.1.1.cmml">∅</mi><annotation-xml encoding="MathML-Content" id="algorithm4.12.12.m3.1b"><emptyset id="algorithm4.12.12.m3.1.1.cmml" xref="algorithm4.12.12.m3.1.1"></emptyset></annotation-xml><annotation encoding="application/x-tex" id="algorithm4.12.12.m3.1c">\emptyset</annotation><annotation encoding="application/x-llamapun" id="algorithm4.12.12.m3.1d">∅</annotation></semantics></math> </div> <div class="ltx_listingline" id="algorithm4.28.30"> <span class="ltx_text" id="algorithm4.28.30.1" style="color:#0000FF;">/* </span><span class="ltx_text ltx_font_smallcaps" id="algorithm4.28.30.2" style="color:#0000FF;">In Stream: */</span> </div> <div class="ltx_listingline" id="algorithm4.13.13"> <span class="ltx_text ltx_font_bold" id="algorithm4.13.13.2">for</span> <em class="ltx_emph ltx_font_italic" id="algorithm4.13.13.1"><math alttext="e=uv\in L" class="ltx_Math" display="inline" id="algorithm4.13.13.1.m1.1"><semantics id="algorithm4.13.13.1.m1.1a"><mrow id="algorithm4.13.13.1.m1.1.1" xref="algorithm4.13.13.1.m1.1.1.cmml"><mi id="algorithm4.13.13.1.m1.1.1.2" xref="algorithm4.13.13.1.m1.1.1.2.cmml">e</mi><mo id="algorithm4.13.13.1.m1.1.1.3" xref="algorithm4.13.13.1.m1.1.1.3.cmml">=</mo><mrow id="algorithm4.13.13.1.m1.1.1.4" xref="algorithm4.13.13.1.m1.1.1.4.cmml"><mi id="algorithm4.13.13.1.m1.1.1.4.2" xref="algorithm4.13.13.1.m1.1.1.4.2.cmml">u</mi><mo id="algorithm4.13.13.1.m1.1.1.4.1" xref="algorithm4.13.13.1.m1.1.1.4.1.cmml">⁢</mo><mi id="algorithm4.13.13.1.m1.1.1.4.3" xref="algorithm4.13.13.1.m1.1.1.4.3.cmml">v</mi></mrow><mo id="algorithm4.13.13.1.m1.1.1.5" xref="algorithm4.13.13.1.m1.1.1.5.cmml">∈</mo><mi id="algorithm4.13.13.1.m1.1.1.6" xref="algorithm4.13.13.1.m1.1.1.6.cmml">L</mi></mrow><annotation-xml encoding="MathML-Content" id="algorithm4.13.13.1.m1.1b"><apply id="algorithm4.13.13.1.m1.1.1.cmml" xref="algorithm4.13.13.1.m1.1.1"><and id="algorithm4.13.13.1.m1.1.1a.cmml" xref="algorithm4.13.13.1.m1.1.1"></and><apply id="algorithm4.13.13.1.m1.1.1b.cmml" xref="algorithm4.13.13.1.m1.1.1"><eq id="algorithm4.13.13.1.m1.1.1.3.cmml" xref="algorithm4.13.13.1.m1.1.1.3"></eq><ci id="algorithm4.13.13.1.m1.1.1.2.cmml" xref="algorithm4.13.13.1.m1.1.1.2">𝑒</ci><apply id="algorithm4.13.13.1.m1.1.1.4.cmml" xref="algorithm4.13.13.1.m1.1.1.4"><times id="algorithm4.13.13.1.m1.1.1.4.1.cmml" xref="algorithm4.13.13.1.m1.1.1.4.1"></times><ci id="algorithm4.13.13.1.m1.1.1.4.2.cmml" xref="algorithm4.13.13.1.m1.1.1.4.2">𝑢</ci><ci id="algorithm4.13.13.1.m1.1.1.4.3.cmml" xref="algorithm4.13.13.1.m1.1.1.4.3">𝑣</ci></apply></apply><apply id="algorithm4.13.13.1.m1.1.1c.cmml" xref="algorithm4.13.13.1.m1.1.1"><in id="algorithm4.13.13.1.m1.1.1.5.cmml" xref="algorithm4.13.13.1.m1.1.1.5"></in><share href="https://arxiv.org/html/2503.00712v1#algorithm4.13.13.1.m1.1.1.4.cmml" id="algorithm4.13.13.1.m1.1.1d.cmml" xref="algorithm4.13.13.1.m1.1.1"></share><ci id="algorithm4.13.13.1.m1.1.1.6.cmml" xref="algorithm4.13.13.1.m1.1.1.6">𝐿</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="algorithm4.13.13.1.m1.1c">e=uv\in L</annotation><annotation encoding="application/x-llamapun" id="algorithm4.13.13.1.m1.1d">italic_e = italic_u italic_v ∈ italic_L</annotation></semantics></math> in the stream</em> <span class="ltx_text ltx_font_bold" id="algorithm4.13.13.3">do</span> </div> <div class="ltx_listingline" id="algorithm4.15.15"> <span class="ltx_rule" style="width:1px;height:100%;background:black;display:inline-block;"> </span> Let <math alttext="j" class="ltx_Math" display="inline" id="algorithm4.14.14.m1.1"><semantics id="algorithm4.14.14.m1.1a"><mi id="algorithm4.14.14.m1.1.1" xref="algorithm4.14.14.m1.1.1.cmml">j</mi><annotation-xml encoding="MathML-Content" id="algorithm4.14.14.m1.1b"><ci id="algorithm4.14.14.m1.1.1.cmml" xref="algorithm4.14.14.m1.1.1">𝑗</ci></annotation-xml><annotation encoding="application/x-tex" id="algorithm4.14.14.m1.1c">j</annotation><annotation encoding="application/x-llamapun" id="algorithm4.14.14.m1.1d">italic_j</annotation></semantics></math> be weight class such that <math alttext="w(e)\in[(1+\epsilon)^{j},(1+\epsilon)^{j+1})" class="ltx_Math" display="inline" id="algorithm4.15.15.m2.3"><semantics id="algorithm4.15.15.m2.3a"><mrow id="algorithm4.15.15.m2.3.3" xref="algorithm4.15.15.m2.3.3.cmml"><mrow id="algorithm4.15.15.m2.3.3.4" xref="algorithm4.15.15.m2.3.3.4.cmml"><mi id="algorithm4.15.15.m2.3.3.4.2" xref="algorithm4.15.15.m2.3.3.4.2.cmml">w</mi><mo id="algorithm4.15.15.m2.3.3.4.1" xref="algorithm4.15.15.m2.3.3.4.1.cmml">⁢</mo><mrow id="algorithm4.15.15.m2.3.3.4.3.2" xref="algorithm4.15.15.m2.3.3.4.cmml"><mo id="algorithm4.15.15.m2.3.3.4.3.2.1" stretchy="false" xref="algorithm4.15.15.m2.3.3.4.cmml">(</mo><mi id="algorithm4.15.15.m2.1.1" xref="algorithm4.15.15.m2.1.1.cmml">e</mi><mo id="algorithm4.15.15.m2.3.3.4.3.2.2" stretchy="false" xref="algorithm4.15.15.m2.3.3.4.cmml">)</mo></mrow></mrow><mo id="algorithm4.15.15.m2.3.3.3" xref="algorithm4.15.15.m2.3.3.3.cmml">∈</mo><mrow id="algorithm4.15.15.m2.3.3.2.2" xref="algorithm4.15.15.m2.3.3.2.3.cmml"><mo id="algorithm4.15.15.m2.3.3.2.2.3" stretchy="false" xref="algorithm4.15.15.m2.3.3.2.3.cmml">[</mo><msup id="algorithm4.15.15.m2.2.2.1.1.1" xref="algorithm4.15.15.m2.2.2.1.1.1.cmml"><mrow id="algorithm4.15.15.m2.2.2.1.1.1.1.1" xref="algorithm4.15.15.m2.2.2.1.1.1.1.1.1.cmml"><mo id="algorithm4.15.15.m2.2.2.1.1.1.1.1.2" stretchy="false" xref="algorithm4.15.15.m2.2.2.1.1.1.1.1.1.cmml">(</mo><mrow id="algorithm4.15.15.m2.2.2.1.1.1.1.1.1" xref="algorithm4.15.15.m2.2.2.1.1.1.1.1.1.cmml"><mn id="algorithm4.15.15.m2.2.2.1.1.1.1.1.1.2" xref="algorithm4.15.15.m2.2.2.1.1.1.1.1.1.2.cmml">1</mn><mo id="algorithm4.15.15.m2.2.2.1.1.1.1.1.1.1" xref="algorithm4.15.15.m2.2.2.1.1.1.1.1.1.1.cmml">+</mo><mi id="algorithm4.15.15.m2.2.2.1.1.1.1.1.1.3" xref="algorithm4.15.15.m2.2.2.1.1.1.1.1.1.3.cmml">ϵ</mi></mrow><mo id="algorithm4.15.15.m2.2.2.1.1.1.1.1.3" stretchy="false" xref="algorithm4.15.15.m2.2.2.1.1.1.1.1.1.cmml">)</mo></mrow><mi id="algorithm4.15.15.m2.2.2.1.1.1.3" xref="algorithm4.15.15.m2.2.2.1.1.1.3.cmml">j</mi></msup><mo id="algorithm4.15.15.m2.3.3.2.2.4" xref="algorithm4.15.15.m2.3.3.2.3.cmml">,</mo><msup id="algorithm4.15.15.m2.3.3.2.2.2" xref="algorithm4.15.15.m2.3.3.2.2.2.cmml"><mrow id="algorithm4.15.15.m2.3.3.2.2.2.1.1" xref="algorithm4.15.15.m2.3.3.2.2.2.1.1.1.cmml"><mo id="algorithm4.15.15.m2.3.3.2.2.2.1.1.2" stretchy="false" xref="algorithm4.15.15.m2.3.3.2.2.2.1.1.1.cmml">(</mo><mrow id="algorithm4.15.15.m2.3.3.2.2.2.1.1.1" xref="algorithm4.15.15.m2.3.3.2.2.2.1.1.1.cmml"><mn id="algorithm4.15.15.m2.3.3.2.2.2.1.1.1.2" xref="algorithm4.15.15.m2.3.3.2.2.2.1.1.1.2.cmml">1</mn><mo id="algorithm4.15.15.m2.3.3.2.2.2.1.1.1.1" xref="algorithm4.15.15.m2.3.3.2.2.2.1.1.1.1.cmml">+</mo><mi id="algorithm4.15.15.m2.3.3.2.2.2.1.1.1.3" xref="algorithm4.15.15.m2.3.3.2.2.2.1.1.1.3.cmml">ϵ</mi></mrow><mo id="algorithm4.15.15.m2.3.3.2.2.2.1.1.3" stretchy="false" xref="algorithm4.15.15.m2.3.3.2.2.2.1.1.1.cmml">)</mo></mrow><mrow id="algorithm4.15.15.m2.3.3.2.2.2.3" xref="algorithm4.15.15.m2.3.3.2.2.2.3.cmml"><mi id="algorithm4.15.15.m2.3.3.2.2.2.3.2" xref="algorithm4.15.15.m2.3.3.2.2.2.3.2.cmml">j</mi><mo id="algorithm4.15.15.m2.3.3.2.2.2.3.1" xref="algorithm4.15.15.m2.3.3.2.2.2.3.1.cmml">+</mo><mn id="algorithm4.15.15.m2.3.3.2.2.2.3.3" xref="algorithm4.15.15.m2.3.3.2.2.2.3.3.cmml">1</mn></mrow></msup><mo id="algorithm4.15.15.m2.3.3.2.2.5" stretchy="false" xref="algorithm4.15.15.m2.3.3.2.3.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="algorithm4.15.15.m2.3b"><apply id="algorithm4.15.15.m2.3.3.cmml" xref="algorithm4.15.15.m2.3.3"><in id="algorithm4.15.15.m2.3.3.3.cmml" xref="algorithm4.15.15.m2.3.3.3"></in><apply id="algorithm4.15.15.m2.3.3.4.cmml" xref="algorithm4.15.15.m2.3.3.4"><times id="algorithm4.15.15.m2.3.3.4.1.cmml" xref="algorithm4.15.15.m2.3.3.4.1"></times><ci id="algorithm4.15.15.m2.3.3.4.2.cmml" xref="algorithm4.15.15.m2.3.3.4.2">𝑤</ci><ci id="algorithm4.15.15.m2.1.1.cmml" xref="algorithm4.15.15.m2.1.1">𝑒</ci></apply><interval closure="closed-open" id="algorithm4.15.15.m2.3.3.2.3.cmml" xref="algorithm4.15.15.m2.3.3.2.2"><apply id="algorithm4.15.15.m2.2.2.1.1.1.cmml" xref="algorithm4.15.15.m2.2.2.1.1.1"><csymbol cd="ambiguous" id="algorithm4.15.15.m2.2.2.1.1.1.2.cmml" xref="algorithm4.15.15.m2.2.2.1.1.1">superscript</csymbol><apply id="algorithm4.15.15.m2.2.2.1.1.1.1.1.1.cmml" xref="algorithm4.15.15.m2.2.2.1.1.1.1.1"><plus id="algorithm4.15.15.m2.2.2.1.1.1.1.1.1.1.cmml" xref="algorithm4.15.15.m2.2.2.1.1.1.1.1.1.1"></plus><cn id="algorithm4.15.15.m2.2.2.1.1.1.1.1.1.2.cmml" type="integer" xref="algorithm4.15.15.m2.2.2.1.1.1.1.1.1.2">1</cn><ci id="algorithm4.15.15.m2.2.2.1.1.1.1.1.1.3.cmml" xref="algorithm4.15.15.m2.2.2.1.1.1.1.1.1.3">italic-ϵ</ci></apply><ci id="algorithm4.15.15.m2.2.2.1.1.1.3.cmml" xref="algorithm4.15.15.m2.2.2.1.1.1.3">𝑗</ci></apply><apply id="algorithm4.15.15.m2.3.3.2.2.2.cmml" xref="algorithm4.15.15.m2.3.3.2.2.2"><csymbol cd="ambiguous" id="algorithm4.15.15.m2.3.3.2.2.2.2.cmml" xref="algorithm4.15.15.m2.3.3.2.2.2">superscript</csymbol><apply id="algorithm4.15.15.m2.3.3.2.2.2.1.1.1.cmml" xref="algorithm4.15.15.m2.3.3.2.2.2.1.1"><plus id="algorithm4.15.15.m2.3.3.2.2.2.1.1.1.1.cmml" xref="algorithm4.15.15.m2.3.3.2.2.2.1.1.1.1"></plus><cn id="algorithm4.15.15.m2.3.3.2.2.2.1.1.1.2.cmml" type="integer" xref="algorithm4.15.15.m2.3.3.2.2.2.1.1.1.2">1</cn><ci id="algorithm4.15.15.m2.3.3.2.2.2.1.1.1.3.cmml" xref="algorithm4.15.15.m2.3.3.2.2.2.1.1.1.3">italic-ϵ</ci></apply><apply id="algorithm4.15.15.m2.3.3.2.2.2.3.cmml" xref="algorithm4.15.15.m2.3.3.2.2.2.3"><plus id="algorithm4.15.15.m2.3.3.2.2.2.3.1.cmml" xref="algorithm4.15.15.m2.3.3.2.2.2.3.1"></plus><ci id="algorithm4.15.15.m2.3.3.2.2.2.3.2.cmml" xref="algorithm4.15.15.m2.3.3.2.2.2.3.2">𝑗</ci><cn id="algorithm4.15.15.m2.3.3.2.2.2.3.3.cmml" type="integer" xref="algorithm4.15.15.m2.3.3.2.2.2.3.3">1</cn></apply></apply></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="algorithm4.15.15.m2.3c">w(e)\in[(1+\epsilon)^{j},(1+\epsilon)^{j+1})</annotation><annotation encoding="application/x-llamapun" id="algorithm4.15.15.m2.3d">italic_w ( italic_e ) ∈ [ ( 1 + italic_ϵ ) start_POSTSUPERSCRIPT italic_j end_POSTSUPERSCRIPT , ( 1 + italic_ϵ ) start_POSTSUPERSCRIPT italic_j + 1 end_POSTSUPERSCRIPT )</annotation></semantics></math> </div> <div class="ltx_listingline" id="algorithm4.18.18"> <span class="ltx_rule" style="width:1px;height:100%;background:black;display:inline-block;"> </span> <span class="ltx_text ltx_font_bold" id="algorithm4.18.18.4">if</span> <em class="ltx_emph ltx_font_italic" id="algorithm4.18.18.3"><math alttext="L_{u}(j)" class="ltx_Math" display="inline" id="algorithm4.16.16.1.m1.1"><semantics id="algorithm4.16.16.1.m1.1a"><mrow id="algorithm4.16.16.1.m1.1.2" xref="algorithm4.16.16.1.m1.1.2.cmml"><msub id="algorithm4.16.16.1.m1.1.2.2" xref="algorithm4.16.16.1.m1.1.2.2.cmml"><mi id="algorithm4.16.16.1.m1.1.2.2.2" xref="algorithm4.16.16.1.m1.1.2.2.2.cmml">L</mi><mi id="algorithm4.16.16.1.m1.1.2.2.3" xref="algorithm4.16.16.1.m1.1.2.2.3.cmml">u</mi></msub><mo id="algorithm4.16.16.1.m1.1.2.1" xref="algorithm4.16.16.1.m1.1.2.1.cmml">⁢</mo><mrow id="algorithm4.16.16.1.m1.1.2.3.2" xref="algorithm4.16.16.1.m1.1.2.cmml"><mo id="algorithm4.16.16.1.m1.1.2.3.2.1" stretchy="false" xref="algorithm4.16.16.1.m1.1.2.cmml">(</mo><mi id="algorithm4.16.16.1.m1.1.1" xref="algorithm4.16.16.1.m1.1.1.cmml">j</mi><mo id="algorithm4.16.16.1.m1.1.2.3.2.2" stretchy="false" xref="algorithm4.16.16.1.m1.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="algorithm4.16.16.1.m1.1b"><apply id="algorithm4.16.16.1.m1.1.2.cmml" xref="algorithm4.16.16.1.m1.1.2"><times id="algorithm4.16.16.1.m1.1.2.1.cmml" xref="algorithm4.16.16.1.m1.1.2.1"></times><apply id="algorithm4.16.16.1.m1.1.2.2.cmml" xref="algorithm4.16.16.1.m1.1.2.2"><csymbol cd="ambiguous" id="algorithm4.16.16.1.m1.1.2.2.1.cmml" xref="algorithm4.16.16.1.m1.1.2.2">subscript</csymbol><ci id="algorithm4.16.16.1.m1.1.2.2.2.cmml" xref="algorithm4.16.16.1.m1.1.2.2.2">𝐿</ci><ci id="algorithm4.16.16.1.m1.1.2.2.3.cmml" xref="algorithm4.16.16.1.m1.1.2.2.3">𝑢</ci></apply><ci id="algorithm4.16.16.1.m1.1.1.cmml" xref="algorithm4.16.16.1.m1.1.1">𝑗</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="algorithm4.16.16.1.m1.1c">L_{u}(j)</annotation><annotation encoding="application/x-llamapun" id="algorithm4.16.16.1.m1.1d">italic_L start_POSTSUBSCRIPT italic_u end_POSTSUBSCRIPT ( italic_j )</annotation></semantics></math> is undefined <span class="ltx_text ltx_font_bold" id="algorithm4.18.18.3.1">or</span> <math alttext="L_{u}(j)=uv^{\prime}" class="ltx_Math" display="inline" id="algorithm4.17.17.2.m2.1"><semantics id="algorithm4.17.17.2.m2.1a"><mrow id="algorithm4.17.17.2.m2.1.2" xref="algorithm4.17.17.2.m2.1.2.cmml"><mrow id="algorithm4.17.17.2.m2.1.2.2" xref="algorithm4.17.17.2.m2.1.2.2.cmml"><msub id="algorithm4.17.17.2.m2.1.2.2.2" xref="algorithm4.17.17.2.m2.1.2.2.2.cmml"><mi id="algorithm4.17.17.2.m2.1.2.2.2.2" xref="algorithm4.17.17.2.m2.1.2.2.2.2.cmml">L</mi><mi id="algorithm4.17.17.2.m2.1.2.2.2.3" xref="algorithm4.17.17.2.m2.1.2.2.2.3.cmml">u</mi></msub><mo id="algorithm4.17.17.2.m2.1.2.2.1" xref="algorithm4.17.17.2.m2.1.2.2.1.cmml">⁢</mo><mrow id="algorithm4.17.17.2.m2.1.2.2.3.2" xref="algorithm4.17.17.2.m2.1.2.2.cmml"><mo id="algorithm4.17.17.2.m2.1.2.2.3.2.1" stretchy="false" xref="algorithm4.17.17.2.m2.1.2.2.cmml">(</mo><mi id="algorithm4.17.17.2.m2.1.1" xref="algorithm4.17.17.2.m2.1.1.cmml">j</mi><mo id="algorithm4.17.17.2.m2.1.2.2.3.2.2" stretchy="false" xref="algorithm4.17.17.2.m2.1.2.2.cmml">)</mo></mrow></mrow><mo id="algorithm4.17.17.2.m2.1.2.1" xref="algorithm4.17.17.2.m2.1.2.1.cmml">=</mo><mrow id="algorithm4.17.17.2.m2.1.2.3" xref="algorithm4.17.17.2.m2.1.2.3.cmml"><mi id="algorithm4.17.17.2.m2.1.2.3.2" xref="algorithm4.17.17.2.m2.1.2.3.2.cmml">u</mi><mo id="algorithm4.17.17.2.m2.1.2.3.1" xref="algorithm4.17.17.2.m2.1.2.3.1.cmml">⁢</mo><msup id="algorithm4.17.17.2.m2.1.2.3.3" xref="algorithm4.17.17.2.m2.1.2.3.3.cmml"><mi id="algorithm4.17.17.2.m2.1.2.3.3.2" xref="algorithm4.17.17.2.m2.1.2.3.3.2.cmml">v</mi><mo id="algorithm4.17.17.2.m2.1.2.3.3.3" xref="algorithm4.17.17.2.m2.1.2.3.3.3.cmml">′</mo></msup></mrow></mrow><annotation-xml encoding="MathML-Content" id="algorithm4.17.17.2.m2.1b"><apply id="algorithm4.17.17.2.m2.1.2.cmml" xref="algorithm4.17.17.2.m2.1.2"><eq id="algorithm4.17.17.2.m2.1.2.1.cmml" xref="algorithm4.17.17.2.m2.1.2.1"></eq><apply id="algorithm4.17.17.2.m2.1.2.2.cmml" xref="algorithm4.17.17.2.m2.1.2.2"><times id="algorithm4.17.17.2.m2.1.2.2.1.cmml" xref="algorithm4.17.17.2.m2.1.2.2.1"></times><apply id="algorithm4.17.17.2.m2.1.2.2.2.cmml" xref="algorithm4.17.17.2.m2.1.2.2.2"><csymbol cd="ambiguous" id="algorithm4.17.17.2.m2.1.2.2.2.1.cmml" xref="algorithm4.17.17.2.m2.1.2.2.2">subscript</csymbol><ci id="algorithm4.17.17.2.m2.1.2.2.2.2.cmml" xref="algorithm4.17.17.2.m2.1.2.2.2.2">𝐿</ci><ci id="algorithm4.17.17.2.m2.1.2.2.2.3.cmml" xref="algorithm4.17.17.2.m2.1.2.2.2.3">𝑢</ci></apply><ci id="algorithm4.17.17.2.m2.1.1.cmml" xref="algorithm4.17.17.2.m2.1.1">𝑗</ci></apply><apply id="algorithm4.17.17.2.m2.1.2.3.cmml" xref="algorithm4.17.17.2.m2.1.2.3"><times id="algorithm4.17.17.2.m2.1.2.3.1.cmml" xref="algorithm4.17.17.2.m2.1.2.3.1"></times><ci id="algorithm4.17.17.2.m2.1.2.3.2.cmml" xref="algorithm4.17.17.2.m2.1.2.3.2">𝑢</ci><apply id="algorithm4.17.17.2.m2.1.2.3.3.cmml" xref="algorithm4.17.17.2.m2.1.2.3.3"><csymbol cd="ambiguous" id="algorithm4.17.17.2.m2.1.2.3.3.1.cmml" xref="algorithm4.17.17.2.m2.1.2.3.3">superscript</csymbol><ci id="algorithm4.17.17.2.m2.1.2.3.3.2.cmml" xref="algorithm4.17.17.2.m2.1.2.3.3.2">𝑣</ci><ci id="algorithm4.17.17.2.m2.1.2.3.3.3.cmml" xref="algorithm4.17.17.2.m2.1.2.3.3.3">′</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="algorithm4.17.17.2.m2.1c">L_{u}(j)=uv^{\prime}</annotation><annotation encoding="application/x-llamapun" id="algorithm4.17.17.2.m2.1d">italic_L start_POSTSUBSCRIPT italic_u end_POSTSUBSCRIPT ( italic_j ) = italic_u italic_v start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math> such that <math alttext="d_{G}(r,\text{LCA}(u,v))<d_{G}(r,\text{LCA}(u,v^{\prime}))" class="ltx_Math" display="inline" id="algorithm4.18.18.3.m3.7"><semantics id="algorithm4.18.18.3.m3.7a"><mrow id="algorithm4.18.18.3.m3.7.7" xref="algorithm4.18.18.3.m3.7.7.cmml"><mrow id="algorithm4.18.18.3.m3.6.6.1" xref="algorithm4.18.18.3.m3.6.6.1.cmml"><msub id="algorithm4.18.18.3.m3.6.6.1.3" xref="algorithm4.18.18.3.m3.6.6.1.3.cmml"><mi id="algorithm4.18.18.3.m3.6.6.1.3.2" xref="algorithm4.18.18.3.m3.6.6.1.3.2.cmml">d</mi><mi id="algorithm4.18.18.3.m3.6.6.1.3.3" xref="algorithm4.18.18.3.m3.6.6.1.3.3.cmml">G</mi></msub><mo id="algorithm4.18.18.3.m3.6.6.1.2" xref="algorithm4.18.18.3.m3.6.6.1.2.cmml">⁢</mo><mrow id="algorithm4.18.18.3.m3.6.6.1.1.1" xref="algorithm4.18.18.3.m3.6.6.1.1.2.cmml"><mo id="algorithm4.18.18.3.m3.6.6.1.1.1.2" stretchy="false" xref="algorithm4.18.18.3.m3.6.6.1.1.2.cmml">(</mo><mi id="algorithm4.18.18.3.m3.3.3" xref="algorithm4.18.18.3.m3.3.3.cmml">r</mi><mo id="algorithm4.18.18.3.m3.6.6.1.1.1.3" xref="algorithm4.18.18.3.m3.6.6.1.1.2.cmml">,</mo><mrow id="algorithm4.18.18.3.m3.6.6.1.1.1.1" xref="algorithm4.18.18.3.m3.6.6.1.1.1.1.cmml"><mtext class="ltx_mathvariant_italic" id="algorithm4.18.18.3.m3.6.6.1.1.1.1.2" xref="algorithm4.18.18.3.m3.6.6.1.1.1.1.2a.cmml">LCA</mtext><mo id="algorithm4.18.18.3.m3.6.6.1.1.1.1.1" xref="algorithm4.18.18.3.m3.6.6.1.1.1.1.1.cmml">⁢</mo><mrow id="algorithm4.18.18.3.m3.6.6.1.1.1.1.3.2" xref="algorithm4.18.18.3.m3.6.6.1.1.1.1.3.1.cmml"><mo id="algorithm4.18.18.3.m3.6.6.1.1.1.1.3.2.1" stretchy="false" xref="algorithm4.18.18.3.m3.6.6.1.1.1.1.3.1.cmml">(</mo><mi id="algorithm4.18.18.3.m3.1.1" xref="algorithm4.18.18.3.m3.1.1.cmml">u</mi><mo id="algorithm4.18.18.3.m3.6.6.1.1.1.1.3.2.2" xref="algorithm4.18.18.3.m3.6.6.1.1.1.1.3.1.cmml">,</mo><mi id="algorithm4.18.18.3.m3.2.2" xref="algorithm4.18.18.3.m3.2.2.cmml">v</mi><mo id="algorithm4.18.18.3.m3.6.6.1.1.1.1.3.2.3" stretchy="false" xref="algorithm4.18.18.3.m3.6.6.1.1.1.1.3.1.cmml">)</mo></mrow></mrow><mo id="algorithm4.18.18.3.m3.6.6.1.1.1.4" stretchy="false" xref="algorithm4.18.18.3.m3.6.6.1.1.2.cmml">)</mo></mrow></mrow><mo id="algorithm4.18.18.3.m3.7.7.3" xref="algorithm4.18.18.3.m3.7.7.3.cmml"><</mo><mrow id="algorithm4.18.18.3.m3.7.7.2" xref="algorithm4.18.18.3.m3.7.7.2.cmml"><msub id="algorithm4.18.18.3.m3.7.7.2.3" xref="algorithm4.18.18.3.m3.7.7.2.3.cmml"><mi id="algorithm4.18.18.3.m3.7.7.2.3.2" xref="algorithm4.18.18.3.m3.7.7.2.3.2.cmml">d</mi><mi id="algorithm4.18.18.3.m3.7.7.2.3.3" xref="algorithm4.18.18.3.m3.7.7.2.3.3.cmml">G</mi></msub><mo id="algorithm4.18.18.3.m3.7.7.2.2" xref="algorithm4.18.18.3.m3.7.7.2.2.cmml">⁢</mo><mrow id="algorithm4.18.18.3.m3.7.7.2.1.1" xref="algorithm4.18.18.3.m3.7.7.2.1.2.cmml"><mo id="algorithm4.18.18.3.m3.7.7.2.1.1.2" stretchy="false" xref="algorithm4.18.18.3.m3.7.7.2.1.2.cmml">(</mo><mi id="algorithm4.18.18.3.m3.5.5" xref="algorithm4.18.18.3.m3.5.5.cmml">r</mi><mo id="algorithm4.18.18.3.m3.7.7.2.1.1.3" xref="algorithm4.18.18.3.m3.7.7.2.1.2.cmml">,</mo><mrow id="algorithm4.18.18.3.m3.7.7.2.1.1.1" xref="algorithm4.18.18.3.m3.7.7.2.1.1.1.cmml"><mtext class="ltx_mathvariant_italic" id="algorithm4.18.18.3.m3.7.7.2.1.1.1.3" xref="algorithm4.18.18.3.m3.7.7.2.1.1.1.3a.cmml">LCA</mtext><mo id="algorithm4.18.18.3.m3.7.7.2.1.1.1.2" xref="algorithm4.18.18.3.m3.7.7.2.1.1.1.2.cmml">⁢</mo><mrow id="algorithm4.18.18.3.m3.7.7.2.1.1.1.1.1" xref="algorithm4.18.18.3.m3.7.7.2.1.1.1.1.2.cmml"><mo id="algorithm4.18.18.3.m3.7.7.2.1.1.1.1.1.2" stretchy="false" xref="algorithm4.18.18.3.m3.7.7.2.1.1.1.1.2.cmml">(</mo><mi id="algorithm4.18.18.3.m3.4.4" xref="algorithm4.18.18.3.m3.4.4.cmml">u</mi><mo id="algorithm4.18.18.3.m3.7.7.2.1.1.1.1.1.3" xref="algorithm4.18.18.3.m3.7.7.2.1.1.1.1.2.cmml">,</mo><msup id="algorithm4.18.18.3.m3.7.7.2.1.1.1.1.1.1" xref="algorithm4.18.18.3.m3.7.7.2.1.1.1.1.1.1.cmml"><mi id="algorithm4.18.18.3.m3.7.7.2.1.1.1.1.1.1.2" xref="algorithm4.18.18.3.m3.7.7.2.1.1.1.1.1.1.2.cmml">v</mi><mo id="algorithm4.18.18.3.m3.7.7.2.1.1.1.1.1.1.3" xref="algorithm4.18.18.3.m3.7.7.2.1.1.1.1.1.1.3.cmml">′</mo></msup><mo id="algorithm4.18.18.3.m3.7.7.2.1.1.1.1.1.4" stretchy="false" xref="algorithm4.18.18.3.m3.7.7.2.1.1.1.1.2.cmml">)</mo></mrow></mrow><mo id="algorithm4.18.18.3.m3.7.7.2.1.1.4" stretchy="false" xref="algorithm4.18.18.3.m3.7.7.2.1.2.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="algorithm4.18.18.3.m3.7b"><apply id="algorithm4.18.18.3.m3.7.7.cmml" xref="algorithm4.18.18.3.m3.7.7"><lt id="algorithm4.18.18.3.m3.7.7.3.cmml" xref="algorithm4.18.18.3.m3.7.7.3"></lt><apply id="algorithm4.18.18.3.m3.6.6.1.cmml" xref="algorithm4.18.18.3.m3.6.6.1"><times id="algorithm4.18.18.3.m3.6.6.1.2.cmml" xref="algorithm4.18.18.3.m3.6.6.1.2"></times><apply id="algorithm4.18.18.3.m3.6.6.1.3.cmml" xref="algorithm4.18.18.3.m3.6.6.1.3"><csymbol cd="ambiguous" id="algorithm4.18.18.3.m3.6.6.1.3.1.cmml" xref="algorithm4.18.18.3.m3.6.6.1.3">subscript</csymbol><ci id="algorithm4.18.18.3.m3.6.6.1.3.2.cmml" xref="algorithm4.18.18.3.m3.6.6.1.3.2">𝑑</ci><ci id="algorithm4.18.18.3.m3.6.6.1.3.3.cmml" xref="algorithm4.18.18.3.m3.6.6.1.3.3">𝐺</ci></apply><interval closure="open" id="algorithm4.18.18.3.m3.6.6.1.1.2.cmml" xref="algorithm4.18.18.3.m3.6.6.1.1.1"><ci id="algorithm4.18.18.3.m3.3.3.cmml" xref="algorithm4.18.18.3.m3.3.3">𝑟</ci><apply id="algorithm4.18.18.3.m3.6.6.1.1.1.1.cmml" xref="algorithm4.18.18.3.m3.6.6.1.1.1.1"><times id="algorithm4.18.18.3.m3.6.6.1.1.1.1.1.cmml" xref="algorithm4.18.18.3.m3.6.6.1.1.1.1.1"></times><ci id="algorithm4.18.18.3.m3.6.6.1.1.1.1.2a.cmml" xref="algorithm4.18.18.3.m3.6.6.1.1.1.1.2"><mtext class="ltx_mathvariant_italic" id="algorithm4.18.18.3.m3.6.6.1.1.1.1.2.cmml" xref="algorithm4.18.18.3.m3.6.6.1.1.1.1.2">LCA</mtext></ci><interval closure="open" id="algorithm4.18.18.3.m3.6.6.1.1.1.1.3.1.cmml" xref="algorithm4.18.18.3.m3.6.6.1.1.1.1.3.2"><ci id="algorithm4.18.18.3.m3.1.1.cmml" xref="algorithm4.18.18.3.m3.1.1">𝑢</ci><ci id="algorithm4.18.18.3.m3.2.2.cmml" xref="algorithm4.18.18.3.m3.2.2">𝑣</ci></interval></apply></interval></apply><apply id="algorithm4.18.18.3.m3.7.7.2.cmml" xref="algorithm4.18.18.3.m3.7.7.2"><times id="algorithm4.18.18.3.m3.7.7.2.2.cmml" xref="algorithm4.18.18.3.m3.7.7.2.2"></times><apply id="algorithm4.18.18.3.m3.7.7.2.3.cmml" xref="algorithm4.18.18.3.m3.7.7.2.3"><csymbol cd="ambiguous" id="algorithm4.18.18.3.m3.7.7.2.3.1.cmml" xref="algorithm4.18.18.3.m3.7.7.2.3">subscript</csymbol><ci id="algorithm4.18.18.3.m3.7.7.2.3.2.cmml" xref="algorithm4.18.18.3.m3.7.7.2.3.2">𝑑</ci><ci id="algorithm4.18.18.3.m3.7.7.2.3.3.cmml" xref="algorithm4.18.18.3.m3.7.7.2.3.3">𝐺</ci></apply><interval closure="open" id="algorithm4.18.18.3.m3.7.7.2.1.2.cmml" xref="algorithm4.18.18.3.m3.7.7.2.1.1"><ci id="algorithm4.18.18.3.m3.5.5.cmml" xref="algorithm4.18.18.3.m3.5.5">𝑟</ci><apply id="algorithm4.18.18.3.m3.7.7.2.1.1.1.cmml" xref="algorithm4.18.18.3.m3.7.7.2.1.1.1"><times id="algorithm4.18.18.3.m3.7.7.2.1.1.1.2.cmml" xref="algorithm4.18.18.3.m3.7.7.2.1.1.1.2"></times><ci id="algorithm4.18.18.3.m3.7.7.2.1.1.1.3a.cmml" xref="algorithm4.18.18.3.m3.7.7.2.1.1.1.3"><mtext class="ltx_mathvariant_italic" id="algorithm4.18.18.3.m3.7.7.2.1.1.1.3.cmml" xref="algorithm4.18.18.3.m3.7.7.2.1.1.1.3">LCA</mtext></ci><interval closure="open" id="algorithm4.18.18.3.m3.7.7.2.1.1.1.1.2.cmml" xref="algorithm4.18.18.3.m3.7.7.2.1.1.1.1.1"><ci id="algorithm4.18.18.3.m3.4.4.cmml" xref="algorithm4.18.18.3.m3.4.4">𝑢</ci><apply id="algorithm4.18.18.3.m3.7.7.2.1.1.1.1.1.1.cmml" xref="algorithm4.18.18.3.m3.7.7.2.1.1.1.1.1.1"><csymbol cd="ambiguous" id="algorithm4.18.18.3.m3.7.7.2.1.1.1.1.1.1.1.cmml" xref="algorithm4.18.18.3.m3.7.7.2.1.1.1.1.1.1">superscript</csymbol><ci id="algorithm4.18.18.3.m3.7.7.2.1.1.1.1.1.1.2.cmml" xref="algorithm4.18.18.3.m3.7.7.2.1.1.1.1.1.1.2">𝑣</ci><ci id="algorithm4.18.18.3.m3.7.7.2.1.1.1.1.1.1.3.cmml" xref="algorithm4.18.18.3.m3.7.7.2.1.1.1.1.1.1.3">′</ci></apply></interval></apply></interval></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="algorithm4.18.18.3.m3.7c">d_{G}(r,\text{LCA}(u,v))<d_{G}(r,\text{LCA}(u,v^{\prime}))</annotation><annotation encoding="application/x-llamapun" id="algorithm4.18.18.3.m3.7d">italic_d start_POSTSUBSCRIPT italic_G end_POSTSUBSCRIPT ( italic_r , LCA ( italic_u , italic_v ) ) < italic_d start_POSTSUBSCRIPT italic_G end_POSTSUBSCRIPT ( italic_r , LCA ( italic_u , italic_v start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) )</annotation></semantics></math></em> <span class="ltx_text ltx_font_bold" id="algorithm4.18.18.5">then</span> </div> <div class="ltx_listingline" id="algorithm4.19.19"> <span class="ltx_rule" style="width:1px;height:100%;background:black;display:inline-block;"> </span> <span class="ltx_rule" style="width:1px;height:100%;background:black;display:inline-block;"> </span> <math alttext="L_{u}(j)\leftarrow e" class="ltx_Math" display="inline" id="algorithm4.19.19.m1.1"><semantics id="algorithm4.19.19.m1.1a"><mrow id="algorithm4.19.19.m1.1.2" xref="algorithm4.19.19.m1.1.2.cmml"><mrow id="algorithm4.19.19.m1.1.2.2" xref="algorithm4.19.19.m1.1.2.2.cmml"><msub id="algorithm4.19.19.m1.1.2.2.2" xref="algorithm4.19.19.m1.1.2.2.2.cmml"><mi id="algorithm4.19.19.m1.1.2.2.2.2" xref="algorithm4.19.19.m1.1.2.2.2.2.cmml">L</mi><mi id="algorithm4.19.19.m1.1.2.2.2.3" xref="algorithm4.19.19.m1.1.2.2.2.3.cmml">u</mi></msub><mo id="algorithm4.19.19.m1.1.2.2.1" xref="algorithm4.19.19.m1.1.2.2.1.cmml">⁢</mo><mrow id="algorithm4.19.19.m1.1.2.2.3.2" xref="algorithm4.19.19.m1.1.2.2.cmml"><mo id="algorithm4.19.19.m1.1.2.2.3.2.1" stretchy="false" xref="algorithm4.19.19.m1.1.2.2.cmml">(</mo><mi id="algorithm4.19.19.m1.1.1" xref="algorithm4.19.19.m1.1.1.cmml">j</mi><mo id="algorithm4.19.19.m1.1.2.2.3.2.2" stretchy="false" xref="algorithm4.19.19.m1.1.2.2.cmml">)</mo></mrow></mrow><mo id="algorithm4.19.19.m1.1.2.1" stretchy="false" xref="algorithm4.19.19.m1.1.2.1.cmml">←</mo><mi id="algorithm4.19.19.m1.1.2.3" xref="algorithm4.19.19.m1.1.2.3.cmml">e</mi></mrow><annotation-xml encoding="MathML-Content" id="algorithm4.19.19.m1.1b"><apply id="algorithm4.19.19.m1.1.2.cmml" xref="algorithm4.19.19.m1.1.2"><ci id="algorithm4.19.19.m1.1.2.1.cmml" xref="algorithm4.19.19.m1.1.2.1">←</ci><apply id="algorithm4.19.19.m1.1.2.2.cmml" xref="algorithm4.19.19.m1.1.2.2"><times id="algorithm4.19.19.m1.1.2.2.1.cmml" xref="algorithm4.19.19.m1.1.2.2.1"></times><apply id="algorithm4.19.19.m1.1.2.2.2.cmml" xref="algorithm4.19.19.m1.1.2.2.2"><csymbol cd="ambiguous" id="algorithm4.19.19.m1.1.2.2.2.1.cmml" xref="algorithm4.19.19.m1.1.2.2.2">subscript</csymbol><ci id="algorithm4.19.19.m1.1.2.2.2.2.cmml" xref="algorithm4.19.19.m1.1.2.2.2.2">𝐿</ci><ci id="algorithm4.19.19.m1.1.2.2.2.3.cmml" xref="algorithm4.19.19.m1.1.2.2.2.3">𝑢</ci></apply><ci id="algorithm4.19.19.m1.1.1.cmml" xref="algorithm4.19.19.m1.1.1">𝑗</ci></apply><ci id="algorithm4.19.19.m1.1.2.3.cmml" xref="algorithm4.19.19.m1.1.2.3">𝑒</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="algorithm4.19.19.m1.1c">L_{u}(j)\leftarrow e</annotation><annotation encoding="application/x-llamapun" id="algorithm4.19.19.m1.1d">italic_L start_POSTSUBSCRIPT italic_u end_POSTSUBSCRIPT ( italic_j ) ← italic_e</annotation></semantics></math> </div> <div class="ltx_listingline" id="algorithm4.22.22"> <span class="ltx_rule" style="width:1px;height:100%;background:black;display:inline-block;"> </span> <span class="ltx_text ltx_font_bold" id="algorithm4.22.22.4">if</span> <em class="ltx_emph ltx_font_italic" id="algorithm4.22.22.3"><math alttext="L_{v}(j)" class="ltx_Math" display="inline" id="algorithm4.20.20.1.m1.1"><semantics id="algorithm4.20.20.1.m1.1a"><mrow id="algorithm4.20.20.1.m1.1.2" xref="algorithm4.20.20.1.m1.1.2.cmml"><msub id="algorithm4.20.20.1.m1.1.2.2" xref="algorithm4.20.20.1.m1.1.2.2.cmml"><mi id="algorithm4.20.20.1.m1.1.2.2.2" xref="algorithm4.20.20.1.m1.1.2.2.2.cmml">L</mi><mi id="algorithm4.20.20.1.m1.1.2.2.3" xref="algorithm4.20.20.1.m1.1.2.2.3.cmml">v</mi></msub><mo id="algorithm4.20.20.1.m1.1.2.1" xref="algorithm4.20.20.1.m1.1.2.1.cmml">⁢</mo><mrow id="algorithm4.20.20.1.m1.1.2.3.2" xref="algorithm4.20.20.1.m1.1.2.cmml"><mo id="algorithm4.20.20.1.m1.1.2.3.2.1" stretchy="false" xref="algorithm4.20.20.1.m1.1.2.cmml">(</mo><mi id="algorithm4.20.20.1.m1.1.1" xref="algorithm4.20.20.1.m1.1.1.cmml">j</mi><mo id="algorithm4.20.20.1.m1.1.2.3.2.2" stretchy="false" xref="algorithm4.20.20.1.m1.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="algorithm4.20.20.1.m1.1b"><apply id="algorithm4.20.20.1.m1.1.2.cmml" xref="algorithm4.20.20.1.m1.1.2"><times id="algorithm4.20.20.1.m1.1.2.1.cmml" xref="algorithm4.20.20.1.m1.1.2.1"></times><apply id="algorithm4.20.20.1.m1.1.2.2.cmml" xref="algorithm4.20.20.1.m1.1.2.2"><csymbol cd="ambiguous" id="algorithm4.20.20.1.m1.1.2.2.1.cmml" xref="algorithm4.20.20.1.m1.1.2.2">subscript</csymbol><ci id="algorithm4.20.20.1.m1.1.2.2.2.cmml" xref="algorithm4.20.20.1.m1.1.2.2.2">𝐿</ci><ci id="algorithm4.20.20.1.m1.1.2.2.3.cmml" xref="algorithm4.20.20.1.m1.1.2.2.3">𝑣</ci></apply><ci id="algorithm4.20.20.1.m1.1.1.cmml" xref="algorithm4.20.20.1.m1.1.1">𝑗</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="algorithm4.20.20.1.m1.1c">L_{v}(j)</annotation><annotation encoding="application/x-llamapun" id="algorithm4.20.20.1.m1.1d">italic_L start_POSTSUBSCRIPT italic_v end_POSTSUBSCRIPT ( italic_j )</annotation></semantics></math> is undefined <span class="ltx_text ltx_font_bold" id="algorithm4.22.22.3.1">or</span> <math alttext="L_{v}(j)=u^{\prime}v" class="ltx_Math" display="inline" id="algorithm4.21.21.2.m2.1"><semantics id="algorithm4.21.21.2.m2.1a"><mrow id="algorithm4.21.21.2.m2.1.2" xref="algorithm4.21.21.2.m2.1.2.cmml"><mrow id="algorithm4.21.21.2.m2.1.2.2" xref="algorithm4.21.21.2.m2.1.2.2.cmml"><msub id="algorithm4.21.21.2.m2.1.2.2.2" xref="algorithm4.21.21.2.m2.1.2.2.2.cmml"><mi id="algorithm4.21.21.2.m2.1.2.2.2.2" xref="algorithm4.21.21.2.m2.1.2.2.2.2.cmml">L</mi><mi id="algorithm4.21.21.2.m2.1.2.2.2.3" xref="algorithm4.21.21.2.m2.1.2.2.2.3.cmml">v</mi></msub><mo id="algorithm4.21.21.2.m2.1.2.2.1" xref="algorithm4.21.21.2.m2.1.2.2.1.cmml">⁢</mo><mrow id="algorithm4.21.21.2.m2.1.2.2.3.2" xref="algorithm4.21.21.2.m2.1.2.2.cmml"><mo id="algorithm4.21.21.2.m2.1.2.2.3.2.1" stretchy="false" xref="algorithm4.21.21.2.m2.1.2.2.cmml">(</mo><mi id="algorithm4.21.21.2.m2.1.1" xref="algorithm4.21.21.2.m2.1.1.cmml">j</mi><mo id="algorithm4.21.21.2.m2.1.2.2.3.2.2" stretchy="false" xref="algorithm4.21.21.2.m2.1.2.2.cmml">)</mo></mrow></mrow><mo id="algorithm4.21.21.2.m2.1.2.1" xref="algorithm4.21.21.2.m2.1.2.1.cmml">=</mo><mrow id="algorithm4.21.21.2.m2.1.2.3" xref="algorithm4.21.21.2.m2.1.2.3.cmml"><msup id="algorithm4.21.21.2.m2.1.2.3.2" xref="algorithm4.21.21.2.m2.1.2.3.2.cmml"><mi id="algorithm4.21.21.2.m2.1.2.3.2.2" xref="algorithm4.21.21.2.m2.1.2.3.2.2.cmml">u</mi><mo id="algorithm4.21.21.2.m2.1.2.3.2.3" xref="algorithm4.21.21.2.m2.1.2.3.2.3.cmml">′</mo></msup><mo id="algorithm4.21.21.2.m2.1.2.3.1" xref="algorithm4.21.21.2.m2.1.2.3.1.cmml">⁢</mo><mi id="algorithm4.21.21.2.m2.1.2.3.3" xref="algorithm4.21.21.2.m2.1.2.3.3.cmml">v</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="algorithm4.21.21.2.m2.1b"><apply id="algorithm4.21.21.2.m2.1.2.cmml" xref="algorithm4.21.21.2.m2.1.2"><eq id="algorithm4.21.21.2.m2.1.2.1.cmml" xref="algorithm4.21.21.2.m2.1.2.1"></eq><apply id="algorithm4.21.21.2.m2.1.2.2.cmml" xref="algorithm4.21.21.2.m2.1.2.2"><times id="algorithm4.21.21.2.m2.1.2.2.1.cmml" xref="algorithm4.21.21.2.m2.1.2.2.1"></times><apply id="algorithm4.21.21.2.m2.1.2.2.2.cmml" xref="algorithm4.21.21.2.m2.1.2.2.2"><csymbol cd="ambiguous" id="algorithm4.21.21.2.m2.1.2.2.2.1.cmml" xref="algorithm4.21.21.2.m2.1.2.2.2">subscript</csymbol><ci id="algorithm4.21.21.2.m2.1.2.2.2.2.cmml" xref="algorithm4.21.21.2.m2.1.2.2.2.2">𝐿</ci><ci id="algorithm4.21.21.2.m2.1.2.2.2.3.cmml" xref="algorithm4.21.21.2.m2.1.2.2.2.3">𝑣</ci></apply><ci id="algorithm4.21.21.2.m2.1.1.cmml" xref="algorithm4.21.21.2.m2.1.1">𝑗</ci></apply><apply id="algorithm4.21.21.2.m2.1.2.3.cmml" xref="algorithm4.21.21.2.m2.1.2.3"><times id="algorithm4.21.21.2.m2.1.2.3.1.cmml" xref="algorithm4.21.21.2.m2.1.2.3.1"></times><apply id="algorithm4.21.21.2.m2.1.2.3.2.cmml" xref="algorithm4.21.21.2.m2.1.2.3.2"><csymbol cd="ambiguous" id="algorithm4.21.21.2.m2.1.2.3.2.1.cmml" xref="algorithm4.21.21.2.m2.1.2.3.2">superscript</csymbol><ci id="algorithm4.21.21.2.m2.1.2.3.2.2.cmml" xref="algorithm4.21.21.2.m2.1.2.3.2.2">𝑢</ci><ci id="algorithm4.21.21.2.m2.1.2.3.2.3.cmml" xref="algorithm4.21.21.2.m2.1.2.3.2.3">′</ci></apply><ci id="algorithm4.21.21.2.m2.1.2.3.3.cmml" xref="algorithm4.21.21.2.m2.1.2.3.3">𝑣</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="algorithm4.21.21.2.m2.1c">L_{v}(j)=u^{\prime}v</annotation><annotation encoding="application/x-llamapun" id="algorithm4.21.21.2.m2.1d">italic_L start_POSTSUBSCRIPT italic_v end_POSTSUBSCRIPT ( italic_j ) = italic_u start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT italic_v</annotation></semantics></math> such that <math alttext="d_{G}(r,\text{LCA}(u,v))<d_{G}(r,\text{LCA}(u^{\prime},v))" class="ltx_Math" display="inline" id="algorithm4.22.22.3.m3.7"><semantics id="algorithm4.22.22.3.m3.7a"><mrow id="algorithm4.22.22.3.m3.7.7" xref="algorithm4.22.22.3.m3.7.7.cmml"><mrow id="algorithm4.22.22.3.m3.6.6.1" xref="algorithm4.22.22.3.m3.6.6.1.cmml"><msub id="algorithm4.22.22.3.m3.6.6.1.3" xref="algorithm4.22.22.3.m3.6.6.1.3.cmml"><mi id="algorithm4.22.22.3.m3.6.6.1.3.2" xref="algorithm4.22.22.3.m3.6.6.1.3.2.cmml">d</mi><mi id="algorithm4.22.22.3.m3.6.6.1.3.3" xref="algorithm4.22.22.3.m3.6.6.1.3.3.cmml">G</mi></msub><mo id="algorithm4.22.22.3.m3.6.6.1.2" xref="algorithm4.22.22.3.m3.6.6.1.2.cmml">⁢</mo><mrow id="algorithm4.22.22.3.m3.6.6.1.1.1" xref="algorithm4.22.22.3.m3.6.6.1.1.2.cmml"><mo id="algorithm4.22.22.3.m3.6.6.1.1.1.2" stretchy="false" xref="algorithm4.22.22.3.m3.6.6.1.1.2.cmml">(</mo><mi id="algorithm4.22.22.3.m3.3.3" xref="algorithm4.22.22.3.m3.3.3.cmml">r</mi><mo id="algorithm4.22.22.3.m3.6.6.1.1.1.3" xref="algorithm4.22.22.3.m3.6.6.1.1.2.cmml">,</mo><mrow id="algorithm4.22.22.3.m3.6.6.1.1.1.1" xref="algorithm4.22.22.3.m3.6.6.1.1.1.1.cmml"><mtext class="ltx_mathvariant_italic" id="algorithm4.22.22.3.m3.6.6.1.1.1.1.2" xref="algorithm4.22.22.3.m3.6.6.1.1.1.1.2a.cmml">LCA</mtext><mo id="algorithm4.22.22.3.m3.6.6.1.1.1.1.1" xref="algorithm4.22.22.3.m3.6.6.1.1.1.1.1.cmml">⁢</mo><mrow id="algorithm4.22.22.3.m3.6.6.1.1.1.1.3.2" xref="algorithm4.22.22.3.m3.6.6.1.1.1.1.3.1.cmml"><mo id="algorithm4.22.22.3.m3.6.6.1.1.1.1.3.2.1" stretchy="false" xref="algorithm4.22.22.3.m3.6.6.1.1.1.1.3.1.cmml">(</mo><mi id="algorithm4.22.22.3.m3.1.1" xref="algorithm4.22.22.3.m3.1.1.cmml">u</mi><mo id="algorithm4.22.22.3.m3.6.6.1.1.1.1.3.2.2" xref="algorithm4.22.22.3.m3.6.6.1.1.1.1.3.1.cmml">,</mo><mi id="algorithm4.22.22.3.m3.2.2" xref="algorithm4.22.22.3.m3.2.2.cmml">v</mi><mo id="algorithm4.22.22.3.m3.6.6.1.1.1.1.3.2.3" stretchy="false" xref="algorithm4.22.22.3.m3.6.6.1.1.1.1.3.1.cmml">)</mo></mrow></mrow><mo id="algorithm4.22.22.3.m3.6.6.1.1.1.4" stretchy="false" xref="algorithm4.22.22.3.m3.6.6.1.1.2.cmml">)</mo></mrow></mrow><mo id="algorithm4.22.22.3.m3.7.7.3" xref="algorithm4.22.22.3.m3.7.7.3.cmml"><</mo><mrow id="algorithm4.22.22.3.m3.7.7.2" xref="algorithm4.22.22.3.m3.7.7.2.cmml"><msub id="algorithm4.22.22.3.m3.7.7.2.3" xref="algorithm4.22.22.3.m3.7.7.2.3.cmml"><mi id="algorithm4.22.22.3.m3.7.7.2.3.2" xref="algorithm4.22.22.3.m3.7.7.2.3.2.cmml">d</mi><mi id="algorithm4.22.22.3.m3.7.7.2.3.3" xref="algorithm4.22.22.3.m3.7.7.2.3.3.cmml">G</mi></msub><mo id="algorithm4.22.22.3.m3.7.7.2.2" xref="algorithm4.22.22.3.m3.7.7.2.2.cmml">⁢</mo><mrow id="algorithm4.22.22.3.m3.7.7.2.1.1" xref="algorithm4.22.22.3.m3.7.7.2.1.2.cmml"><mo id="algorithm4.22.22.3.m3.7.7.2.1.1.2" stretchy="false" xref="algorithm4.22.22.3.m3.7.7.2.1.2.cmml">(</mo><mi id="algorithm4.22.22.3.m3.5.5" xref="algorithm4.22.22.3.m3.5.5.cmml">r</mi><mo id="algorithm4.22.22.3.m3.7.7.2.1.1.3" xref="algorithm4.22.22.3.m3.7.7.2.1.2.cmml">,</mo><mrow id="algorithm4.22.22.3.m3.7.7.2.1.1.1" xref="algorithm4.22.22.3.m3.7.7.2.1.1.1.cmml"><mtext class="ltx_mathvariant_italic" id="algorithm4.22.22.3.m3.7.7.2.1.1.1.3" xref="algorithm4.22.22.3.m3.7.7.2.1.1.1.3a.cmml">LCA</mtext><mo id="algorithm4.22.22.3.m3.7.7.2.1.1.1.2" xref="algorithm4.22.22.3.m3.7.7.2.1.1.1.2.cmml">⁢</mo><mrow id="algorithm4.22.22.3.m3.7.7.2.1.1.1.1.1" xref="algorithm4.22.22.3.m3.7.7.2.1.1.1.1.2.cmml"><mo id="algorithm4.22.22.3.m3.7.7.2.1.1.1.1.1.2" stretchy="false" xref="algorithm4.22.22.3.m3.7.7.2.1.1.1.1.2.cmml">(</mo><msup id="algorithm4.22.22.3.m3.7.7.2.1.1.1.1.1.1" xref="algorithm4.22.22.3.m3.7.7.2.1.1.1.1.1.1.cmml"><mi id="algorithm4.22.22.3.m3.7.7.2.1.1.1.1.1.1.2" xref="algorithm4.22.22.3.m3.7.7.2.1.1.1.1.1.1.2.cmml">u</mi><mo id="algorithm4.22.22.3.m3.7.7.2.1.1.1.1.1.1.3" xref="algorithm4.22.22.3.m3.7.7.2.1.1.1.1.1.1.3.cmml">′</mo></msup><mo id="algorithm4.22.22.3.m3.7.7.2.1.1.1.1.1.3" xref="algorithm4.22.22.3.m3.7.7.2.1.1.1.1.2.cmml">,</mo><mi id="algorithm4.22.22.3.m3.4.4" xref="algorithm4.22.22.3.m3.4.4.cmml">v</mi><mo id="algorithm4.22.22.3.m3.7.7.2.1.1.1.1.1.4" stretchy="false" xref="algorithm4.22.22.3.m3.7.7.2.1.1.1.1.2.cmml">)</mo></mrow></mrow><mo id="algorithm4.22.22.3.m3.7.7.2.1.1.4" stretchy="false" xref="algorithm4.22.22.3.m3.7.7.2.1.2.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="algorithm4.22.22.3.m3.7b"><apply id="algorithm4.22.22.3.m3.7.7.cmml" xref="algorithm4.22.22.3.m3.7.7"><lt id="algorithm4.22.22.3.m3.7.7.3.cmml" xref="algorithm4.22.22.3.m3.7.7.3"></lt><apply id="algorithm4.22.22.3.m3.6.6.1.cmml" xref="algorithm4.22.22.3.m3.6.6.1"><times id="algorithm4.22.22.3.m3.6.6.1.2.cmml" xref="algorithm4.22.22.3.m3.6.6.1.2"></times><apply id="algorithm4.22.22.3.m3.6.6.1.3.cmml" xref="algorithm4.22.22.3.m3.6.6.1.3"><csymbol cd="ambiguous" id="algorithm4.22.22.3.m3.6.6.1.3.1.cmml" xref="algorithm4.22.22.3.m3.6.6.1.3">subscript</csymbol><ci id="algorithm4.22.22.3.m3.6.6.1.3.2.cmml" xref="algorithm4.22.22.3.m3.6.6.1.3.2">𝑑</ci><ci id="algorithm4.22.22.3.m3.6.6.1.3.3.cmml" xref="algorithm4.22.22.3.m3.6.6.1.3.3">𝐺</ci></apply><interval closure="open" id="algorithm4.22.22.3.m3.6.6.1.1.2.cmml" xref="algorithm4.22.22.3.m3.6.6.1.1.1"><ci id="algorithm4.22.22.3.m3.3.3.cmml" xref="algorithm4.22.22.3.m3.3.3">𝑟</ci><apply id="algorithm4.22.22.3.m3.6.6.1.1.1.1.cmml" xref="algorithm4.22.22.3.m3.6.6.1.1.1.1"><times id="algorithm4.22.22.3.m3.6.6.1.1.1.1.1.cmml" xref="algorithm4.22.22.3.m3.6.6.1.1.1.1.1"></times><ci id="algorithm4.22.22.3.m3.6.6.1.1.1.1.2a.cmml" xref="algorithm4.22.22.3.m3.6.6.1.1.1.1.2"><mtext class="ltx_mathvariant_italic" id="algorithm4.22.22.3.m3.6.6.1.1.1.1.2.cmml" xref="algorithm4.22.22.3.m3.6.6.1.1.1.1.2">LCA</mtext></ci><interval closure="open" id="algorithm4.22.22.3.m3.6.6.1.1.1.1.3.1.cmml" xref="algorithm4.22.22.3.m3.6.6.1.1.1.1.3.2"><ci id="algorithm4.22.22.3.m3.1.1.cmml" xref="algorithm4.22.22.3.m3.1.1">𝑢</ci><ci id="algorithm4.22.22.3.m3.2.2.cmml" xref="algorithm4.22.22.3.m3.2.2">𝑣</ci></interval></apply></interval></apply><apply id="algorithm4.22.22.3.m3.7.7.2.cmml" xref="algorithm4.22.22.3.m3.7.7.2"><times id="algorithm4.22.22.3.m3.7.7.2.2.cmml" xref="algorithm4.22.22.3.m3.7.7.2.2"></times><apply id="algorithm4.22.22.3.m3.7.7.2.3.cmml" xref="algorithm4.22.22.3.m3.7.7.2.3"><csymbol cd="ambiguous" id="algorithm4.22.22.3.m3.7.7.2.3.1.cmml" xref="algorithm4.22.22.3.m3.7.7.2.3">subscript</csymbol><ci id="algorithm4.22.22.3.m3.7.7.2.3.2.cmml" xref="algorithm4.22.22.3.m3.7.7.2.3.2">𝑑</ci><ci id="algorithm4.22.22.3.m3.7.7.2.3.3.cmml" xref="algorithm4.22.22.3.m3.7.7.2.3.3">𝐺</ci></apply><interval closure="open" id="algorithm4.22.22.3.m3.7.7.2.1.2.cmml" xref="algorithm4.22.22.3.m3.7.7.2.1.1"><ci id="algorithm4.22.22.3.m3.5.5.cmml" xref="algorithm4.22.22.3.m3.5.5">𝑟</ci><apply id="algorithm4.22.22.3.m3.7.7.2.1.1.1.cmml" xref="algorithm4.22.22.3.m3.7.7.2.1.1.1"><times id="algorithm4.22.22.3.m3.7.7.2.1.1.1.2.cmml" xref="algorithm4.22.22.3.m3.7.7.2.1.1.1.2"></times><ci id="algorithm4.22.22.3.m3.7.7.2.1.1.1.3a.cmml" xref="algorithm4.22.22.3.m3.7.7.2.1.1.1.3"><mtext class="ltx_mathvariant_italic" id="algorithm4.22.22.3.m3.7.7.2.1.1.1.3.cmml" xref="algorithm4.22.22.3.m3.7.7.2.1.1.1.3">LCA</mtext></ci><interval closure="open" id="algorithm4.22.22.3.m3.7.7.2.1.1.1.1.2.cmml" xref="algorithm4.22.22.3.m3.7.7.2.1.1.1.1.1"><apply id="algorithm4.22.22.3.m3.7.7.2.1.1.1.1.1.1.cmml" xref="algorithm4.22.22.3.m3.7.7.2.1.1.1.1.1.1"><csymbol cd="ambiguous" id="algorithm4.22.22.3.m3.7.7.2.1.1.1.1.1.1.1.cmml" xref="algorithm4.22.22.3.m3.7.7.2.1.1.1.1.1.1">superscript</csymbol><ci id="algorithm4.22.22.3.m3.7.7.2.1.1.1.1.1.1.2.cmml" xref="algorithm4.22.22.3.m3.7.7.2.1.1.1.1.1.1.2">𝑢</ci><ci id="algorithm4.22.22.3.m3.7.7.2.1.1.1.1.1.1.3.cmml" xref="algorithm4.22.22.3.m3.7.7.2.1.1.1.1.1.1.3">′</ci></apply><ci id="algorithm4.22.22.3.m3.4.4.cmml" xref="algorithm4.22.22.3.m3.4.4">𝑣</ci></interval></apply></interval></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="algorithm4.22.22.3.m3.7c">d_{G}(r,\text{LCA}(u,v))<d_{G}(r,\text{LCA}(u^{\prime},v))</annotation><annotation encoding="application/x-llamapun" id="algorithm4.22.22.3.m3.7d">italic_d start_POSTSUBSCRIPT italic_G end_POSTSUBSCRIPT ( italic_r , LCA ( italic_u , italic_v ) ) < italic_d start_POSTSUBSCRIPT italic_G end_POSTSUBSCRIPT ( italic_r , LCA ( italic_u start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT , italic_v ) )</annotation></semantics></math></em> <span class="ltx_text ltx_font_bold" id="algorithm4.22.22.5">then</span> </div> <div class="ltx_listingline" id="algorithm4.23.23"> <span class="ltx_rule" style="width:1px;height:100%;background:black;display:inline-block;"> </span> <span class="ltx_rule" style="width:1px;height:100%;background:black;display:inline-block;"> </span> <math alttext="L_{v}(j)\leftarrow e" class="ltx_Math" display="inline" id="algorithm4.23.23.m1.1"><semantics id="algorithm4.23.23.m1.1a"><mrow id="algorithm4.23.23.m1.1.2" xref="algorithm4.23.23.m1.1.2.cmml"><mrow id="algorithm4.23.23.m1.1.2.2" xref="algorithm4.23.23.m1.1.2.2.cmml"><msub id="algorithm4.23.23.m1.1.2.2.2" xref="algorithm4.23.23.m1.1.2.2.2.cmml"><mi id="algorithm4.23.23.m1.1.2.2.2.2" xref="algorithm4.23.23.m1.1.2.2.2.2.cmml">L</mi><mi id="algorithm4.23.23.m1.1.2.2.2.3" xref="algorithm4.23.23.m1.1.2.2.2.3.cmml">v</mi></msub><mo id="algorithm4.23.23.m1.1.2.2.1" xref="algorithm4.23.23.m1.1.2.2.1.cmml">⁢</mo><mrow id="algorithm4.23.23.m1.1.2.2.3.2" xref="algorithm4.23.23.m1.1.2.2.cmml"><mo id="algorithm4.23.23.m1.1.2.2.3.2.1" stretchy="false" xref="algorithm4.23.23.m1.1.2.2.cmml">(</mo><mi id="algorithm4.23.23.m1.1.1" xref="algorithm4.23.23.m1.1.1.cmml">j</mi><mo id="algorithm4.23.23.m1.1.2.2.3.2.2" stretchy="false" xref="algorithm4.23.23.m1.1.2.2.cmml">)</mo></mrow></mrow><mo id="algorithm4.23.23.m1.1.2.1" stretchy="false" xref="algorithm4.23.23.m1.1.2.1.cmml">←</mo><mi id="algorithm4.23.23.m1.1.2.3" xref="algorithm4.23.23.m1.1.2.3.cmml">e</mi></mrow><annotation-xml encoding="MathML-Content" id="algorithm4.23.23.m1.1b"><apply id="algorithm4.23.23.m1.1.2.cmml" xref="algorithm4.23.23.m1.1.2"><ci id="algorithm4.23.23.m1.1.2.1.cmml" xref="algorithm4.23.23.m1.1.2.1">←</ci><apply id="algorithm4.23.23.m1.1.2.2.cmml" xref="algorithm4.23.23.m1.1.2.2"><times id="algorithm4.23.23.m1.1.2.2.1.cmml" xref="algorithm4.23.23.m1.1.2.2.1"></times><apply id="algorithm4.23.23.m1.1.2.2.2.cmml" xref="algorithm4.23.23.m1.1.2.2.2"><csymbol cd="ambiguous" id="algorithm4.23.23.m1.1.2.2.2.1.cmml" xref="algorithm4.23.23.m1.1.2.2.2">subscript</csymbol><ci id="algorithm4.23.23.m1.1.2.2.2.2.cmml" xref="algorithm4.23.23.m1.1.2.2.2.2">𝐿</ci><ci id="algorithm4.23.23.m1.1.2.2.2.3.cmml" xref="algorithm4.23.23.m1.1.2.2.2.3">𝑣</ci></apply><ci id="algorithm4.23.23.m1.1.1.cmml" xref="algorithm4.23.23.m1.1.1">𝑗</ci></apply><ci id="algorithm4.23.23.m1.1.2.3.cmml" xref="algorithm4.23.23.m1.1.2.3">𝑒</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="algorithm4.23.23.m1.1c">L_{v}(j)\leftarrow e</annotation><annotation encoding="application/x-llamapun" id="algorithm4.23.23.m1.1d">italic_L start_POSTSUBSCRIPT italic_v end_POSTSUBSCRIPT ( italic_j ) ← italic_e</annotation></semantics></math> </div> <div class="ltx_listingline" id="algorithm4.26.26"> <span class="ltx_rule" style="width:1px;height:100%;background:black;display:inline-block;"> </span> <span class="ltx_text ltx_font_bold" id="algorithm4.26.26.1">update</span> MST <math alttext="T_{x}^{\prime}" class="ltx_Math" display="inline" id="algorithm4.24.24.m1.1"><semantics id="algorithm4.24.24.m1.1a"><msubsup id="algorithm4.24.24.m1.1.1" xref="algorithm4.24.24.m1.1.1.cmml"><mi id="algorithm4.24.24.m1.1.1.2.2" xref="algorithm4.24.24.m1.1.1.2.2.cmml">T</mi><mi id="algorithm4.24.24.m1.1.1.2.3" xref="algorithm4.24.24.m1.1.1.2.3.cmml">x</mi><mo id="algorithm4.24.24.m1.1.1.3" xref="algorithm4.24.24.m1.1.1.3.cmml">′</mo></msubsup><annotation-xml encoding="MathML-Content" id="algorithm4.24.24.m1.1b"><apply id="algorithm4.24.24.m1.1.1.cmml" xref="algorithm4.24.24.m1.1.1"><csymbol cd="ambiguous" id="algorithm4.24.24.m1.1.1.1.cmml" xref="algorithm4.24.24.m1.1.1">superscript</csymbol><apply id="algorithm4.24.24.m1.1.1.2.cmml" xref="algorithm4.24.24.m1.1.1"><csymbol cd="ambiguous" id="algorithm4.24.24.m1.1.1.2.1.cmml" xref="algorithm4.24.24.m1.1.1">subscript</csymbol><ci id="algorithm4.24.24.m1.1.1.2.2.cmml" xref="algorithm4.24.24.m1.1.1.2.2">𝑇</ci><ci id="algorithm4.24.24.m1.1.1.2.3.cmml" xref="algorithm4.24.24.m1.1.1.2.3">𝑥</ci></apply><ci id="algorithm4.24.24.m1.1.1.3.cmml" xref="algorithm4.24.24.m1.1.1.3">′</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="algorithm4.24.24.m1.1c">T_{x}^{\prime}</annotation><annotation encoding="application/x-llamapun" id="algorithm4.24.24.m1.1d">italic_T start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math> for <math alttext="x=\text{LCA}(u,v)" class="ltx_Math" display="inline" id="algorithm4.25.25.m2.2"><semantics id="algorithm4.25.25.m2.2a"><mrow id="algorithm4.25.25.m2.2.3" xref="algorithm4.25.25.m2.2.3.cmml"><mi id="algorithm4.25.25.m2.2.3.2" xref="algorithm4.25.25.m2.2.3.2.cmml">x</mi><mo id="algorithm4.25.25.m2.2.3.1" xref="algorithm4.25.25.m2.2.3.1.cmml">=</mo><mrow id="algorithm4.25.25.m2.2.3.3" xref="algorithm4.25.25.m2.2.3.3.cmml"><mtext id="algorithm4.25.25.m2.2.3.3.2" xref="algorithm4.25.25.m2.2.3.3.2a.cmml">LCA</mtext><mo id="algorithm4.25.25.m2.2.3.3.1" xref="algorithm4.25.25.m2.2.3.3.1.cmml">⁢</mo><mrow id="algorithm4.25.25.m2.2.3.3.3.2" xref="algorithm4.25.25.m2.2.3.3.3.1.cmml"><mo id="algorithm4.25.25.m2.2.3.3.3.2.1" stretchy="false" xref="algorithm4.25.25.m2.2.3.3.3.1.cmml">(</mo><mi id="algorithm4.25.25.m2.1.1" xref="algorithm4.25.25.m2.1.1.cmml">u</mi><mo id="algorithm4.25.25.m2.2.3.3.3.2.2" xref="algorithm4.25.25.m2.2.3.3.3.1.cmml">,</mo><mi id="algorithm4.25.25.m2.2.2" xref="algorithm4.25.25.m2.2.2.cmml">v</mi><mo id="algorithm4.25.25.m2.2.3.3.3.2.3" stretchy="false" xref="algorithm4.25.25.m2.2.3.3.3.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="algorithm4.25.25.m2.2b"><apply id="algorithm4.25.25.m2.2.3.cmml" xref="algorithm4.25.25.m2.2.3"><eq id="algorithm4.25.25.m2.2.3.1.cmml" xref="algorithm4.25.25.m2.2.3.1"></eq><ci id="algorithm4.25.25.m2.2.3.2.cmml" xref="algorithm4.25.25.m2.2.3.2">𝑥</ci><apply id="algorithm4.25.25.m2.2.3.3.cmml" xref="algorithm4.25.25.m2.2.3.3"><times id="algorithm4.25.25.m2.2.3.3.1.cmml" xref="algorithm4.25.25.m2.2.3.3.1"></times><ci id="algorithm4.25.25.m2.2.3.3.2a.cmml" xref="algorithm4.25.25.m2.2.3.3.2"><mtext id="algorithm4.25.25.m2.2.3.3.2.cmml" xref="algorithm4.25.25.m2.2.3.3.2">LCA</mtext></ci><interval closure="open" id="algorithm4.25.25.m2.2.3.3.3.1.cmml" xref="algorithm4.25.25.m2.2.3.3.3.2"><ci id="algorithm4.25.25.m2.1.1.cmml" xref="algorithm4.25.25.m2.1.1">𝑢</ci><ci id="algorithm4.25.25.m2.2.2.cmml" xref="algorithm4.25.25.m2.2.2">𝑣</ci></interval></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="algorithm4.25.25.m2.2c">x=\text{LCA}(u,v)</annotation><annotation encoding="application/x-llamapun" id="algorithm4.25.25.m2.2d">italic_x = LCA ( italic_u , italic_v )</annotation></semantics></math> with link <math alttext="uv" class="ltx_Math" display="inline" id="algorithm4.26.26.m3.1"><semantics id="algorithm4.26.26.m3.1a"><mrow id="algorithm4.26.26.m3.1.1" xref="algorithm4.26.26.m3.1.1.cmml"><mi id="algorithm4.26.26.m3.1.1.2" xref="algorithm4.26.26.m3.1.1.2.cmml">u</mi><mo id="algorithm4.26.26.m3.1.1.1" xref="algorithm4.26.26.m3.1.1.1.cmml">⁢</mo><mi id="algorithm4.26.26.m3.1.1.3" xref="algorithm4.26.26.m3.1.1.3.cmml">v</mi></mrow><annotation-xml encoding="MathML-Content" id="algorithm4.26.26.m3.1b"><apply id="algorithm4.26.26.m3.1.1.cmml" xref="algorithm4.26.26.m3.1.1"><times id="algorithm4.26.26.m3.1.1.1.cmml" xref="algorithm4.26.26.m3.1.1.1"></times><ci id="algorithm4.26.26.m3.1.1.2.cmml" xref="algorithm4.26.26.m3.1.1.2">𝑢</ci><ci id="algorithm4.26.26.m3.1.1.3.cmml" xref="algorithm4.26.26.m3.1.1.3">𝑣</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="algorithm4.26.26.m3.1c">uv</annotation><annotation encoding="application/x-llamapun" id="algorithm4.26.26.m3.1d">italic_u italic_v</annotation></semantics></math> using Lemma <a class="ltx_ref" href="https://arxiv.org/html/2503.00712v1#S4.Thmtheorem2" title="Lemma 4.2. ‣ 4.1 One-to-Two Augmentation ‣ 4 Vertex Connectivity Augmentation in Link-Arrival Model ‣ Streaming Algorithms for Network Design"><span class="ltx_text ltx_ref_tag">4.2</span></a> </div> <div class="ltx_listingline" id="algorithm4.28.31"> <span class="ltx_text" id="algorithm4.28.31.1" style="color:#0000FF;">/* </span><span class="ltx_text ltx_font_smallcaps" id="algorithm4.28.31.2" style="color:#0000FF;">Postprocessing: */</span> </div> <div class="ltx_listingline" id="algorithm4.27.27"> <math alttext="F=\cup_{u\in V}(E(T_{u}^{\prime})\cup L_{u})" class="ltx_Math" display="inline" id="algorithm4.27.27.m1.1"><semantics id="algorithm4.27.27.m1.1a"><mrow id="algorithm4.27.27.m1.1.1" xref="algorithm4.27.27.m1.1.1.cmml"><mi id="algorithm4.27.27.m1.1.1.3" xref="algorithm4.27.27.m1.1.1.3.cmml">F</mi><mo id="algorithm4.27.27.m1.1.1.2" rspace="0em" xref="algorithm4.27.27.m1.1.1.2.cmml">=</mo><mrow id="algorithm4.27.27.m1.1.1.1" xref="algorithm4.27.27.m1.1.1.1.cmml"><msub id="algorithm4.27.27.m1.1.1.1.2" xref="algorithm4.27.27.m1.1.1.1.2.cmml"><mo id="algorithm4.27.27.m1.1.1.1.2.2" lspace="0em" xref="algorithm4.27.27.m1.1.1.1.2.2.cmml">∪</mo><mrow id="algorithm4.27.27.m1.1.1.1.2.3" xref="algorithm4.27.27.m1.1.1.1.2.3.cmml"><mi id="algorithm4.27.27.m1.1.1.1.2.3.2" xref="algorithm4.27.27.m1.1.1.1.2.3.2.cmml">u</mi><mo id="algorithm4.27.27.m1.1.1.1.2.3.1" xref="algorithm4.27.27.m1.1.1.1.2.3.1.cmml">∈</mo><mi id="algorithm4.27.27.m1.1.1.1.2.3.3" xref="algorithm4.27.27.m1.1.1.1.2.3.3.cmml">V</mi></mrow></msub><mrow id="algorithm4.27.27.m1.1.1.1.1.1" xref="algorithm4.27.27.m1.1.1.1.1.1.1.cmml"><mo id="algorithm4.27.27.m1.1.1.1.1.1.2" stretchy="false" xref="algorithm4.27.27.m1.1.1.1.1.1.1.cmml">(</mo><mrow id="algorithm4.27.27.m1.1.1.1.1.1.1" xref="algorithm4.27.27.m1.1.1.1.1.1.1.cmml"><mrow id="algorithm4.27.27.m1.1.1.1.1.1.1.1" xref="algorithm4.27.27.m1.1.1.1.1.1.1.1.cmml"><mi id="algorithm4.27.27.m1.1.1.1.1.1.1.1.3" xref="algorithm4.27.27.m1.1.1.1.1.1.1.1.3.cmml">E</mi><mo id="algorithm4.27.27.m1.1.1.1.1.1.1.1.2" xref="algorithm4.27.27.m1.1.1.1.1.1.1.1.2.cmml">⁢</mo><mrow id="algorithm4.27.27.m1.1.1.1.1.1.1.1.1.1" xref="algorithm4.27.27.m1.1.1.1.1.1.1.1.1.1.1.cmml"><mo id="algorithm4.27.27.m1.1.1.1.1.1.1.1.1.1.2" stretchy="false" xref="algorithm4.27.27.m1.1.1.1.1.1.1.1.1.1.1.cmml">(</mo><msubsup id="algorithm4.27.27.m1.1.1.1.1.1.1.1.1.1.1" xref="algorithm4.27.27.m1.1.1.1.1.1.1.1.1.1.1.cmml"><mi id="algorithm4.27.27.m1.1.1.1.1.1.1.1.1.1.1.2.2" xref="algorithm4.27.27.m1.1.1.1.1.1.1.1.1.1.1.2.2.cmml">T</mi><mi id="algorithm4.27.27.m1.1.1.1.1.1.1.1.1.1.1.2.3" xref="algorithm4.27.27.m1.1.1.1.1.1.1.1.1.1.1.2.3.cmml">u</mi><mo id="algorithm4.27.27.m1.1.1.1.1.1.1.1.1.1.1.3" xref="algorithm4.27.27.m1.1.1.1.1.1.1.1.1.1.1.3.cmml">′</mo></msubsup><mo id="algorithm4.27.27.m1.1.1.1.1.1.1.1.1.1.3" stretchy="false" xref="algorithm4.27.27.m1.1.1.1.1.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="algorithm4.27.27.m1.1.1.1.1.1.1.2" xref="algorithm4.27.27.m1.1.1.1.1.1.1.2.cmml">∪</mo><msub id="algorithm4.27.27.m1.1.1.1.1.1.1.3" xref="algorithm4.27.27.m1.1.1.1.1.1.1.3.cmml"><mi id="algorithm4.27.27.m1.1.1.1.1.1.1.3.2" xref="algorithm4.27.27.m1.1.1.1.1.1.1.3.2.cmml">L</mi><mi id="algorithm4.27.27.m1.1.1.1.1.1.1.3.3" xref="algorithm4.27.27.m1.1.1.1.1.1.1.3.3.cmml">u</mi></msub></mrow><mo id="algorithm4.27.27.m1.1.1.1.1.1.3" stretchy="false" xref="algorithm4.27.27.m1.1.1.1.1.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="algorithm4.27.27.m1.1b"><apply id="algorithm4.27.27.m1.1.1.cmml" xref="algorithm4.27.27.m1.1.1"><eq id="algorithm4.27.27.m1.1.1.2.cmml" xref="algorithm4.27.27.m1.1.1.2"></eq><ci id="algorithm4.27.27.m1.1.1.3.cmml" xref="algorithm4.27.27.m1.1.1.3">𝐹</ci><apply id="algorithm4.27.27.m1.1.1.1.cmml" xref="algorithm4.27.27.m1.1.1.1"><apply id="algorithm4.27.27.m1.1.1.1.2.cmml" xref="algorithm4.27.27.m1.1.1.1.2"><csymbol cd="ambiguous" id="algorithm4.27.27.m1.1.1.1.2.1.cmml" xref="algorithm4.27.27.m1.1.1.1.2">subscript</csymbol><union id="algorithm4.27.27.m1.1.1.1.2.2.cmml" xref="algorithm4.27.27.m1.1.1.1.2.2"></union><apply id="algorithm4.27.27.m1.1.1.1.2.3.cmml" xref="algorithm4.27.27.m1.1.1.1.2.3"><in id="algorithm4.27.27.m1.1.1.1.2.3.1.cmml" xref="algorithm4.27.27.m1.1.1.1.2.3.1"></in><ci id="algorithm4.27.27.m1.1.1.1.2.3.2.cmml" xref="algorithm4.27.27.m1.1.1.1.2.3.2">𝑢</ci><ci id="algorithm4.27.27.m1.1.1.1.2.3.3.cmml" xref="algorithm4.27.27.m1.1.1.1.2.3.3">𝑉</ci></apply></apply><apply id="algorithm4.27.27.m1.1.1.1.1.1.1.cmml" xref="algorithm4.27.27.m1.1.1.1.1.1"><union id="algorithm4.27.27.m1.1.1.1.1.1.1.2.cmml" xref="algorithm4.27.27.m1.1.1.1.1.1.1.2"></union><apply id="algorithm4.27.27.m1.1.1.1.1.1.1.1.cmml" xref="algorithm4.27.27.m1.1.1.1.1.1.1.1"><times id="algorithm4.27.27.m1.1.1.1.1.1.1.1.2.cmml" xref="algorithm4.27.27.m1.1.1.1.1.1.1.1.2"></times><ci id="algorithm4.27.27.m1.1.1.1.1.1.1.1.3.cmml" xref="algorithm4.27.27.m1.1.1.1.1.1.1.1.3">𝐸</ci><apply id="algorithm4.27.27.m1.1.1.1.1.1.1.1.1.1.1.cmml" xref="algorithm4.27.27.m1.1.1.1.1.1.1.1.1.1"><csymbol cd="ambiguous" id="algorithm4.27.27.m1.1.1.1.1.1.1.1.1.1.1.1.cmml" xref="algorithm4.27.27.m1.1.1.1.1.1.1.1.1.1">superscript</csymbol><apply id="algorithm4.27.27.m1.1.1.1.1.1.1.1.1.1.1.2.cmml" xref="algorithm4.27.27.m1.1.1.1.1.1.1.1.1.1"><csymbol cd="ambiguous" id="algorithm4.27.27.m1.1.1.1.1.1.1.1.1.1.1.2.1.cmml" xref="algorithm4.27.27.m1.1.1.1.1.1.1.1.1.1">subscript</csymbol><ci id="algorithm4.27.27.m1.1.1.1.1.1.1.1.1.1.1.2.2.cmml" xref="algorithm4.27.27.m1.1.1.1.1.1.1.1.1.1.1.2.2">𝑇</ci><ci id="algorithm4.27.27.m1.1.1.1.1.1.1.1.1.1.1.2.3.cmml" xref="algorithm4.27.27.m1.1.1.1.1.1.1.1.1.1.1.2.3">𝑢</ci></apply><ci id="algorithm4.27.27.m1.1.1.1.1.1.1.1.1.1.1.3.cmml" xref="algorithm4.27.27.m1.1.1.1.1.1.1.1.1.1.1.3">′</ci></apply></apply><apply id="algorithm4.27.27.m1.1.1.1.1.1.1.3.cmml" xref="algorithm4.27.27.m1.1.1.1.1.1.1.3"><csymbol cd="ambiguous" id="algorithm4.27.27.m1.1.1.1.1.1.1.3.1.cmml" xref="algorithm4.27.27.m1.1.1.1.1.1.1.3">subscript</csymbol><ci id="algorithm4.27.27.m1.1.1.1.1.1.1.3.2.cmml" xref="algorithm4.27.27.m1.1.1.1.1.1.1.3.2">𝐿</ci><ci id="algorithm4.27.27.m1.1.1.1.1.1.1.3.3.cmml" xref="algorithm4.27.27.m1.1.1.1.1.1.1.3.3">𝑢</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="algorithm4.27.27.m1.1c">F=\cup_{u\in V}(E(T_{u}^{\prime})\cup L_{u})</annotation><annotation encoding="application/x-llamapun" id="algorithm4.27.27.m1.1d">italic_F = ∪ start_POSTSUBSCRIPT italic_u ∈ italic_V end_POSTSUBSCRIPT ( italic_E ( italic_T start_POSTSUBSCRIPT italic_u end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) ∪ italic_L start_POSTSUBSCRIPT italic_u end_POSTSUBSCRIPT )</annotation></semantics></math> </div> <div class="ltx_listingline" id="algorithm4.28.28"> <span class="ltx_text ltx_font_bold" id="algorithm4.28.28.1">return</span> An optimal solution on link set <math alttext="F" class="ltx_Math" display="inline" id="algorithm4.28.28.m1.1"><semantics id="algorithm4.28.28.m1.1a"><mi id="algorithm4.28.28.m1.1.1" xref="algorithm4.28.28.m1.1.1.cmml">F</mi><annotation-xml encoding="MathML-Content" id="algorithm4.28.28.m1.1b"><ci id="algorithm4.28.28.m1.1.1.cmml" xref="algorithm4.28.28.m1.1.1">𝐹</ci></annotation-xml><annotation encoding="application/x-tex" id="algorithm4.28.28.m1.1c">F</annotation><annotation encoding="application/x-llamapun" id="algorithm4.28.28.m1.1d">italic_F</annotation></semantics></math> </div> </div> <figcaption class="ltx_caption"><span class="ltx_tag ltx_tag_float"><span class="ltx_text ltx_font_bold" id="algorithm4.30.1.1">Algorithm 4</span> </span>The streaming algorithm for 1-to-2 VCSS augmentation</figcaption> </figure> <div class="ltx_theorem ltx_theorem_lemma" id="S4.Thmtheorem3"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="S4.Thmtheorem3.1.1.1">Lemma 4.3</span></span><span class="ltx_text ltx_font_bold" id="S4.Thmtheorem3.2.2">.</span> </h6> <div class="ltx_para" id="S4.Thmtheorem3.p1"> <p class="ltx_p" id="S4.Thmtheorem3.p1.1">The number of links stored in Algorithm <a class="ltx_ref" href="https://arxiv.org/html/2503.00712v1#algorithm4" title="In 4.1.1 The Streaming Algorithm ‣ 4.1 One-to-Two Augmentation ‣ 4 Vertex Connectivity Augmentation in Link-Arrival Model ‣ Streaming Algorithms for Network Design"><span class="ltx_text ltx_ref_tag">4</span></a> is <math alttext="O(n\epsilon^{-1}\log W)" class="ltx_Math" display="inline" id="S4.Thmtheorem3.p1.1.m1.1"><semantics id="S4.Thmtheorem3.p1.1.m1.1a"><mrow id="S4.Thmtheorem3.p1.1.m1.1.1" xref="S4.Thmtheorem3.p1.1.m1.1.1.cmml"><mi id="S4.Thmtheorem3.p1.1.m1.1.1.3" xref="S4.Thmtheorem3.p1.1.m1.1.1.3.cmml">O</mi><mo id="S4.Thmtheorem3.p1.1.m1.1.1.2" xref="S4.Thmtheorem3.p1.1.m1.1.1.2.cmml">⁢</mo><mrow id="S4.Thmtheorem3.p1.1.m1.1.1.1.1" xref="S4.Thmtheorem3.p1.1.m1.1.1.1.1.1.cmml"><mo id="S4.Thmtheorem3.p1.1.m1.1.1.1.1.2" stretchy="false" xref="S4.Thmtheorem3.p1.1.m1.1.1.1.1.1.cmml">(</mo><mrow id="S4.Thmtheorem3.p1.1.m1.1.1.1.1.1" xref="S4.Thmtheorem3.p1.1.m1.1.1.1.1.1.cmml"><mi id="S4.Thmtheorem3.p1.1.m1.1.1.1.1.1.2" xref="S4.Thmtheorem3.p1.1.m1.1.1.1.1.1.2.cmml">n</mi><mo id="S4.Thmtheorem3.p1.1.m1.1.1.1.1.1.1" xref="S4.Thmtheorem3.p1.1.m1.1.1.1.1.1.1.cmml">⁢</mo><msup id="S4.Thmtheorem3.p1.1.m1.1.1.1.1.1.3" xref="S4.Thmtheorem3.p1.1.m1.1.1.1.1.1.3.cmml"><mi id="S4.Thmtheorem3.p1.1.m1.1.1.1.1.1.3.2" xref="S4.Thmtheorem3.p1.1.m1.1.1.1.1.1.3.2.cmml">ϵ</mi><mrow id="S4.Thmtheorem3.p1.1.m1.1.1.1.1.1.3.3" xref="S4.Thmtheorem3.p1.1.m1.1.1.1.1.1.3.3.cmml"><mo id="S4.Thmtheorem3.p1.1.m1.1.1.1.1.1.3.3a" xref="S4.Thmtheorem3.p1.1.m1.1.1.1.1.1.3.3.cmml">−</mo><mn id="S4.Thmtheorem3.p1.1.m1.1.1.1.1.1.3.3.2" xref="S4.Thmtheorem3.p1.1.m1.1.1.1.1.1.3.3.2.cmml">1</mn></mrow></msup><mo id="S4.Thmtheorem3.p1.1.m1.1.1.1.1.1.1a" lspace="0.167em" xref="S4.Thmtheorem3.p1.1.m1.1.1.1.1.1.1.cmml">⁢</mo><mrow id="S4.Thmtheorem3.p1.1.m1.1.1.1.1.1.4" xref="S4.Thmtheorem3.p1.1.m1.1.1.1.1.1.4.cmml"><mi id="S4.Thmtheorem3.p1.1.m1.1.1.1.1.1.4.1" xref="S4.Thmtheorem3.p1.1.m1.1.1.1.1.1.4.1.cmml">log</mi><mo id="S4.Thmtheorem3.p1.1.m1.1.1.1.1.1.4a" lspace="0.167em" xref="S4.Thmtheorem3.p1.1.m1.1.1.1.1.1.4.cmml">⁡</mo><mi id="S4.Thmtheorem3.p1.1.m1.1.1.1.1.1.4.2" xref="S4.Thmtheorem3.p1.1.m1.1.1.1.1.1.4.2.cmml">W</mi></mrow></mrow><mo id="S4.Thmtheorem3.p1.1.m1.1.1.1.1.3" stretchy="false" xref="S4.Thmtheorem3.p1.1.m1.1.1.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem3.p1.1.m1.1b"><apply id="S4.Thmtheorem3.p1.1.m1.1.1.cmml" xref="S4.Thmtheorem3.p1.1.m1.1.1"><times id="S4.Thmtheorem3.p1.1.m1.1.1.2.cmml" xref="S4.Thmtheorem3.p1.1.m1.1.1.2"></times><ci id="S4.Thmtheorem3.p1.1.m1.1.1.3.cmml" xref="S4.Thmtheorem3.p1.1.m1.1.1.3">𝑂</ci><apply id="S4.Thmtheorem3.p1.1.m1.1.1.1.1.1.cmml" xref="S4.Thmtheorem3.p1.1.m1.1.1.1.1"><times id="S4.Thmtheorem3.p1.1.m1.1.1.1.1.1.1.cmml" xref="S4.Thmtheorem3.p1.1.m1.1.1.1.1.1.1"></times><ci id="S4.Thmtheorem3.p1.1.m1.1.1.1.1.1.2.cmml" xref="S4.Thmtheorem3.p1.1.m1.1.1.1.1.1.2">𝑛</ci><apply id="S4.Thmtheorem3.p1.1.m1.1.1.1.1.1.3.cmml" xref="S4.Thmtheorem3.p1.1.m1.1.1.1.1.1.3"><csymbol cd="ambiguous" id="S4.Thmtheorem3.p1.1.m1.1.1.1.1.1.3.1.cmml" xref="S4.Thmtheorem3.p1.1.m1.1.1.1.1.1.3">superscript</csymbol><ci id="S4.Thmtheorem3.p1.1.m1.1.1.1.1.1.3.2.cmml" xref="S4.Thmtheorem3.p1.1.m1.1.1.1.1.1.3.2">italic-ϵ</ci><apply id="S4.Thmtheorem3.p1.1.m1.1.1.1.1.1.3.3.cmml" xref="S4.Thmtheorem3.p1.1.m1.1.1.1.1.1.3.3"><minus id="S4.Thmtheorem3.p1.1.m1.1.1.1.1.1.3.3.1.cmml" xref="S4.Thmtheorem3.p1.1.m1.1.1.1.1.1.3.3"></minus><cn id="S4.Thmtheorem3.p1.1.m1.1.1.1.1.1.3.3.2.cmml" type="integer" xref="S4.Thmtheorem3.p1.1.m1.1.1.1.1.1.3.3.2">1</cn></apply></apply><apply id="S4.Thmtheorem3.p1.1.m1.1.1.1.1.1.4.cmml" xref="S4.Thmtheorem3.p1.1.m1.1.1.1.1.1.4"><log id="S4.Thmtheorem3.p1.1.m1.1.1.1.1.1.4.1.cmml" xref="S4.Thmtheorem3.p1.1.m1.1.1.1.1.1.4.1"></log><ci id="S4.Thmtheorem3.p1.1.m1.1.1.1.1.1.4.2.cmml" xref="S4.Thmtheorem3.p1.1.m1.1.1.1.1.1.4.2">𝑊</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem3.p1.1.m1.1c">O(n\epsilon^{-1}\log W)</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem3.p1.1.m1.1d">italic_O ( italic_n italic_ϵ start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT roman_log italic_W )</annotation></semantics></math>.</p> </div> </div> <div class="ltx_proof" id="S4.SS1.SSS1.1"> <h6 class="ltx_title ltx_runin ltx_font_italic ltx_title_proof">Proof.</h6> <div class="ltx_para" id="S4.SS1.SSS1.1.p1"> <p class="ltx_p" id="S4.SS1.SSS1.1.p1.8">The set of links stored in Algorithm <a class="ltx_ref" href="https://arxiv.org/html/2503.00712v1#algorithm4" title="In 4.1.1 The Streaming Algorithm ‣ 4.1 One-to-Two Augmentation ‣ 4 Vertex Connectivity Augmentation in Link-Arrival Model ‣ Streaming Algorithms for Network Design"><span class="ltx_text ltx_ref_tag">4</span></a> is exactly <math alttext="F" class="ltx_Math" display="inline" id="S4.SS1.SSS1.1.p1.1.m1.1"><semantics id="S4.SS1.SSS1.1.p1.1.m1.1a"><mi id="S4.SS1.SSS1.1.p1.1.m1.1.1" xref="S4.SS1.SSS1.1.p1.1.m1.1.1.cmml">F</mi><annotation-xml encoding="MathML-Content" id="S4.SS1.SSS1.1.p1.1.m1.1b"><ci id="S4.SS1.SSS1.1.p1.1.m1.1.1.cmml" xref="S4.SS1.SSS1.1.p1.1.m1.1.1">𝐹</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.SSS1.1.p1.1.m1.1c">F</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.SSS1.1.p1.1.m1.1d">italic_F</annotation></semantics></math>. For each vertex <math alttext="u\in V" class="ltx_Math" display="inline" id="S4.SS1.SSS1.1.p1.2.m2.1"><semantics id="S4.SS1.SSS1.1.p1.2.m2.1a"><mrow id="S4.SS1.SSS1.1.p1.2.m2.1.1" xref="S4.SS1.SSS1.1.p1.2.m2.1.1.cmml"><mi id="S4.SS1.SSS1.1.p1.2.m2.1.1.2" xref="S4.SS1.SSS1.1.p1.2.m2.1.1.2.cmml">u</mi><mo id="S4.SS1.SSS1.1.p1.2.m2.1.1.1" xref="S4.SS1.SSS1.1.p1.2.m2.1.1.1.cmml">∈</mo><mi id="S4.SS1.SSS1.1.p1.2.m2.1.1.3" xref="S4.SS1.SSS1.1.p1.2.m2.1.1.3.cmml">V</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.SSS1.1.p1.2.m2.1b"><apply id="S4.SS1.SSS1.1.p1.2.m2.1.1.cmml" xref="S4.SS1.SSS1.1.p1.2.m2.1.1"><in id="S4.SS1.SSS1.1.p1.2.m2.1.1.1.cmml" xref="S4.SS1.SSS1.1.p1.2.m2.1.1.1"></in><ci id="S4.SS1.SSS1.1.p1.2.m2.1.1.2.cmml" xref="S4.SS1.SSS1.1.p1.2.m2.1.1.2">𝑢</ci><ci id="S4.SS1.SSS1.1.p1.2.m2.1.1.3.cmml" xref="S4.SS1.SSS1.1.p1.2.m2.1.1.3">𝑉</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.SSS1.1.p1.2.m2.1c">u\in V</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.SSS1.1.p1.2.m2.1d">italic_u ∈ italic_V</annotation></semantics></math>, we store one link in <math alttext="L_{u}" class="ltx_Math" display="inline" id="S4.SS1.SSS1.1.p1.3.m3.1"><semantics id="S4.SS1.SSS1.1.p1.3.m3.1a"><msub id="S4.SS1.SSS1.1.p1.3.m3.1.1" xref="S4.SS1.SSS1.1.p1.3.m3.1.1.cmml"><mi id="S4.SS1.SSS1.1.p1.3.m3.1.1.2" xref="S4.SS1.SSS1.1.p1.3.m3.1.1.2.cmml">L</mi><mi id="S4.SS1.SSS1.1.p1.3.m3.1.1.3" xref="S4.SS1.SSS1.1.p1.3.m3.1.1.3.cmml">u</mi></msub><annotation-xml encoding="MathML-Content" id="S4.SS1.SSS1.1.p1.3.m3.1b"><apply id="S4.SS1.SSS1.1.p1.3.m3.1.1.cmml" xref="S4.SS1.SSS1.1.p1.3.m3.1.1"><csymbol cd="ambiguous" id="S4.SS1.SSS1.1.p1.3.m3.1.1.1.cmml" xref="S4.SS1.SSS1.1.p1.3.m3.1.1">subscript</csymbol><ci id="S4.SS1.SSS1.1.p1.3.m3.1.1.2.cmml" xref="S4.SS1.SSS1.1.p1.3.m3.1.1.2">𝐿</ci><ci id="S4.SS1.SSS1.1.p1.3.m3.1.1.3.cmml" xref="S4.SS1.SSS1.1.p1.3.m3.1.1.3">𝑢</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.SSS1.1.p1.3.m3.1c">L_{u}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.SSS1.1.p1.3.m3.1d">italic_L start_POSTSUBSCRIPT italic_u end_POSTSUBSCRIPT</annotation></semantics></math> per weight class. 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xref="S4.SS1.SSS1.1.p1.4.m4.3.3.3.2"></times><ci id="S4.SS1.SSS1.1.p1.4.m4.3.3.3.3.cmml" xref="S4.SS1.SSS1.1.p1.4.m4.3.3.3.3">𝑂</ci><apply id="S4.SS1.SSS1.1.p1.4.m4.3.3.3.1.1.1.cmml" xref="S4.SS1.SSS1.1.p1.4.m4.3.3.3.1.1"><times id="S4.SS1.SSS1.1.p1.4.m4.3.3.3.1.1.1.1.cmml" xref="S4.SS1.SSS1.1.p1.4.m4.3.3.3.1.1.1.1"></times><ci id="S4.SS1.SSS1.1.p1.4.m4.3.3.3.1.1.1.2.cmml" xref="S4.SS1.SSS1.1.p1.4.m4.3.3.3.1.1.1.2">𝑛</ci><apply id="S4.SS1.SSS1.1.p1.4.m4.3.3.3.1.1.1.3.cmml" xref="S4.SS1.SSS1.1.p1.4.m4.3.3.3.1.1.1.3"><csymbol cd="ambiguous" id="S4.SS1.SSS1.1.p1.4.m4.3.3.3.1.1.1.3.1.cmml" xref="S4.SS1.SSS1.1.p1.4.m4.3.3.3.1.1.1.3">superscript</csymbol><ci id="S4.SS1.SSS1.1.p1.4.m4.3.3.3.1.1.1.3.2.cmml" xref="S4.SS1.SSS1.1.p1.4.m4.3.3.3.1.1.1.3.2">italic-ϵ</ci><apply id="S4.SS1.SSS1.1.p1.4.m4.3.3.3.1.1.1.3.3.cmml" xref="S4.SS1.SSS1.1.p1.4.m4.3.3.3.1.1.1.3.3"><minus id="S4.SS1.SSS1.1.p1.4.m4.3.3.3.1.1.1.3.3.1.cmml" xref="S4.SS1.SSS1.1.p1.4.m4.3.3.3.1.1.1.3.3"></minus><cn id="S4.SS1.SSS1.1.p1.4.m4.3.3.3.1.1.1.3.3.2.cmml" type="integer" xref="S4.SS1.SSS1.1.p1.4.m4.3.3.3.1.1.1.3.3.2">1</cn></apply></apply><apply id="S4.SS1.SSS1.1.p1.4.m4.3.3.3.1.1.1.4.cmml" xref="S4.SS1.SSS1.1.p1.4.m4.3.3.3.1.1.1.4"><log id="S4.SS1.SSS1.1.p1.4.m4.3.3.3.1.1.1.4.1.cmml" xref="S4.SS1.SSS1.1.p1.4.m4.3.3.3.1.1.1.4.1"></log><ci id="S4.SS1.SSS1.1.p1.4.m4.3.3.3.1.1.1.4.2.cmml" xref="S4.SS1.SSS1.1.p1.4.m4.3.3.3.1.1.1.4.2">𝑊</ci></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.SSS1.1.p1.4.m4.3c">|\cup_{u\in V}L_{u}|=\sum_{u\in V}O(\epsilon^{-1}\log W)=O(n\epsilon^{-1}\log W)</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.SSS1.1.p1.4.m4.3d">| ∪ start_POSTSUBSCRIPT italic_u ∈ italic_V end_POSTSUBSCRIPT italic_L start_POSTSUBSCRIPT italic_u end_POSTSUBSCRIPT | = ∑ start_POSTSUBSCRIPT italic_u ∈ italic_V end_POSTSUBSCRIPT italic_O ( italic_ϵ start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT roman_log italic_W ) = italic_O ( italic_n italic_ϵ start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT roman_log italic_W )</annotation></semantics></math>. We bound the number of links in the spanning trees by <math alttext="|\cup_{u\in V}E(T_{v}^{\prime})|\leq\sum_{u\in V}|C(u)-1|\leq\sum_{u\in V}\deg% (u)=2|E|" class="ltx_Math" display="inline" id="S4.SS1.SSS1.1.p1.5.m5.6"><semantics id="S4.SS1.SSS1.1.p1.5.m5.6a"><mrow id="S4.SS1.SSS1.1.p1.5.m5.6.6" xref="S4.SS1.SSS1.1.p1.5.m5.6.6.cmml"><mrow id="S4.SS1.SSS1.1.p1.5.m5.5.5.1.1" xref="S4.SS1.SSS1.1.p1.5.m5.5.5.1.2.cmml"><mo id="S4.SS1.SSS1.1.p1.5.m5.5.5.1.1.2" stretchy="false" xref="S4.SS1.SSS1.1.p1.5.m5.5.5.1.2.1.cmml">|</mo><mrow id="S4.SS1.SSS1.1.p1.5.m5.5.5.1.1.1" xref="S4.SS1.SSS1.1.p1.5.m5.5.5.1.1.1.cmml"><msub id="S4.SS1.SSS1.1.p1.5.m5.5.5.1.1.1.2" xref="S4.SS1.SSS1.1.p1.5.m5.5.5.1.1.1.2.cmml"><mo id="S4.SS1.SSS1.1.p1.5.m5.5.5.1.1.1.2.2" lspace="0em" xref="S4.SS1.SSS1.1.p1.5.m5.5.5.1.1.1.2.2.cmml">∪</mo><mrow id="S4.SS1.SSS1.1.p1.5.m5.5.5.1.1.1.2.3" xref="S4.SS1.SSS1.1.p1.5.m5.5.5.1.1.1.2.3.cmml"><mi id="S4.SS1.SSS1.1.p1.5.m5.5.5.1.1.1.2.3.2" 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xref="S4.SS1.SSS1.1.p1.5.m5.6.6.2.1.1.1.2.2">𝐶</ci><ci id="S4.SS1.SSS1.1.p1.5.m5.1.1.cmml" xref="S4.SS1.SSS1.1.p1.5.m5.1.1">𝑢</ci></apply><cn id="S4.SS1.SSS1.1.p1.5.m5.6.6.2.1.1.1.3.cmml" type="integer" xref="S4.SS1.SSS1.1.p1.5.m5.6.6.2.1.1.1.3">1</cn></apply></apply></apply></apply><apply id="S4.SS1.SSS1.1.p1.5.m5.6.6c.cmml" xref="S4.SS1.SSS1.1.p1.5.m5.6.6"><leq id="S4.SS1.SSS1.1.p1.5.m5.6.6.5.cmml" xref="S4.SS1.SSS1.1.p1.5.m5.6.6.5"></leq><share href="https://arxiv.org/html/2503.00712v1#S4.SS1.SSS1.1.p1.5.m5.6.6.2.cmml" id="S4.SS1.SSS1.1.p1.5.m5.6.6d.cmml" xref="S4.SS1.SSS1.1.p1.5.m5.6.6"></share><apply id="S4.SS1.SSS1.1.p1.5.m5.6.6.6.cmml" xref="S4.SS1.SSS1.1.p1.5.m5.6.6.6"><apply id="S4.SS1.SSS1.1.p1.5.m5.6.6.6.1.cmml" xref="S4.SS1.SSS1.1.p1.5.m5.6.6.6.1"><csymbol cd="ambiguous" id="S4.SS1.SSS1.1.p1.5.m5.6.6.6.1.1.cmml" xref="S4.SS1.SSS1.1.p1.5.m5.6.6.6.1">subscript</csymbol><sum id="S4.SS1.SSS1.1.p1.5.m5.6.6.6.1.2.cmml" xref="S4.SS1.SSS1.1.p1.5.m5.6.6.6.1.2"></sum><apply 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xref="S4.SS1.SSS1.1.p1.5.m5.6.6"></share><apply id="S4.SS1.SSS1.1.p1.5.m5.6.6.8.cmml" xref="S4.SS1.SSS1.1.p1.5.m5.6.6.8"><times id="S4.SS1.SSS1.1.p1.5.m5.6.6.8.1.cmml" xref="S4.SS1.SSS1.1.p1.5.m5.6.6.8.1"></times><cn id="S4.SS1.SSS1.1.p1.5.m5.6.6.8.2.cmml" type="integer" xref="S4.SS1.SSS1.1.p1.5.m5.6.6.8.2">2</cn><apply id="S4.SS1.SSS1.1.p1.5.m5.6.6.8.3.1.cmml" xref="S4.SS1.SSS1.1.p1.5.m5.6.6.8.3.2"><abs id="S4.SS1.SSS1.1.p1.5.m5.6.6.8.3.1.1.cmml" xref="S4.SS1.SSS1.1.p1.5.m5.6.6.8.3.2.1"></abs><ci id="S4.SS1.SSS1.1.p1.5.m5.4.4.cmml" xref="S4.SS1.SSS1.1.p1.5.m5.4.4">𝐸</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.SSS1.1.p1.5.m5.6c">|\cup_{u\in V}E(T_{v}^{\prime})|\leq\sum_{u\in V}|C(u)-1|\leq\sum_{u\in V}\deg% (u)=2|E|</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.SSS1.1.p1.5.m5.6d">| ∪ start_POSTSUBSCRIPT italic_u ∈ italic_V end_POSTSUBSCRIPT italic_E ( italic_T start_POSTSUBSCRIPT italic_v end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) | ≤ ∑ start_POSTSUBSCRIPT italic_u ∈ italic_V end_POSTSUBSCRIPT | italic_C ( italic_u ) - 1 | ≤ ∑ start_POSTSUBSCRIPT italic_u ∈ italic_V end_POSTSUBSCRIPT roman_deg ( italic_u ) = 2 | italic_E |</annotation></semantics></math>. Since we assume <math alttext="G" class="ltx_Math" display="inline" id="S4.SS1.SSS1.1.p1.6.m6.1"><semantics id="S4.SS1.SSS1.1.p1.6.m6.1a"><mi id="S4.SS1.SSS1.1.p1.6.m6.1.1" xref="S4.SS1.SSS1.1.p1.6.m6.1.1.cmml">G</mi><annotation-xml encoding="MathML-Content" id="S4.SS1.SSS1.1.p1.6.m6.1b"><ci id="S4.SS1.SSS1.1.p1.6.m6.1.1.cmml" xref="S4.SS1.SSS1.1.p1.6.m6.1.1">𝐺</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.SSS1.1.p1.6.m6.1c">G</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.SSS1.1.p1.6.m6.1d">italic_G</annotation></semantics></math> is a tree, <math alttext="|E|=n-1" class="ltx_Math" display="inline" id="S4.SS1.SSS1.1.p1.7.m7.1"><semantics id="S4.SS1.SSS1.1.p1.7.m7.1a"><mrow id="S4.SS1.SSS1.1.p1.7.m7.1.2" xref="S4.SS1.SSS1.1.p1.7.m7.1.2.cmml"><mrow id="S4.SS1.SSS1.1.p1.7.m7.1.2.2.2" xref="S4.SS1.SSS1.1.p1.7.m7.1.2.2.1.cmml"><mo id="S4.SS1.SSS1.1.p1.7.m7.1.2.2.2.1" stretchy="false" xref="S4.SS1.SSS1.1.p1.7.m7.1.2.2.1.1.cmml">|</mo><mi id="S4.SS1.SSS1.1.p1.7.m7.1.1" xref="S4.SS1.SSS1.1.p1.7.m7.1.1.cmml">E</mi><mo id="S4.SS1.SSS1.1.p1.7.m7.1.2.2.2.2" stretchy="false" xref="S4.SS1.SSS1.1.p1.7.m7.1.2.2.1.1.cmml">|</mo></mrow><mo id="S4.SS1.SSS1.1.p1.7.m7.1.2.1" xref="S4.SS1.SSS1.1.p1.7.m7.1.2.1.cmml">=</mo><mrow id="S4.SS1.SSS1.1.p1.7.m7.1.2.3" xref="S4.SS1.SSS1.1.p1.7.m7.1.2.3.cmml"><mi id="S4.SS1.SSS1.1.p1.7.m7.1.2.3.2" xref="S4.SS1.SSS1.1.p1.7.m7.1.2.3.2.cmml">n</mi><mo id="S4.SS1.SSS1.1.p1.7.m7.1.2.3.1" xref="S4.SS1.SSS1.1.p1.7.m7.1.2.3.1.cmml">−</mo><mn id="S4.SS1.SSS1.1.p1.7.m7.1.2.3.3" xref="S4.SS1.SSS1.1.p1.7.m7.1.2.3.3.cmml">1</mn></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.SSS1.1.p1.7.m7.1b"><apply id="S4.SS1.SSS1.1.p1.7.m7.1.2.cmml" xref="S4.SS1.SSS1.1.p1.7.m7.1.2"><eq id="S4.SS1.SSS1.1.p1.7.m7.1.2.1.cmml" xref="S4.SS1.SSS1.1.p1.7.m7.1.2.1"></eq><apply id="S4.SS1.SSS1.1.p1.7.m7.1.2.2.1.cmml" xref="S4.SS1.SSS1.1.p1.7.m7.1.2.2.2"><abs id="S4.SS1.SSS1.1.p1.7.m7.1.2.2.1.1.cmml" xref="S4.SS1.SSS1.1.p1.7.m7.1.2.2.2.1"></abs><ci id="S4.SS1.SSS1.1.p1.7.m7.1.1.cmml" xref="S4.SS1.SSS1.1.p1.7.m7.1.1">𝐸</ci></apply><apply id="S4.SS1.SSS1.1.p1.7.m7.1.2.3.cmml" xref="S4.SS1.SSS1.1.p1.7.m7.1.2.3"><minus id="S4.SS1.SSS1.1.p1.7.m7.1.2.3.1.cmml" xref="S4.SS1.SSS1.1.p1.7.m7.1.2.3.1"></minus><ci id="S4.SS1.SSS1.1.p1.7.m7.1.2.3.2.cmml" xref="S4.SS1.SSS1.1.p1.7.m7.1.2.3.2">𝑛</ci><cn id="S4.SS1.SSS1.1.p1.7.m7.1.2.3.3.cmml" type="integer" xref="S4.SS1.SSS1.1.p1.7.m7.1.2.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.SSS1.1.p1.7.m7.1c">|E|=n-1</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.SSS1.1.p1.7.m7.1d">| italic_E | = italic_n - 1</annotation></semantics></math>, so <math alttext="|\cup_{u\in V}E(T_{v}^{\prime})|=O(n)" class="ltx_Math" display="inline" id="S4.SS1.SSS1.1.p1.8.m8.2"><semantics id="S4.SS1.SSS1.1.p1.8.m8.2a"><mrow id="S4.SS1.SSS1.1.p1.8.m8.2.2" xref="S4.SS1.SSS1.1.p1.8.m8.2.2.cmml"><mrow 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xref="S4.SS1.SSS1.1.p1.8.m8.2.2.1.1.1.1.1.1">subscript</csymbol><ci id="S4.SS1.SSS1.1.p1.8.m8.2.2.1.1.1.1.1.1.1.2.2.cmml" xref="S4.SS1.SSS1.1.p1.8.m8.2.2.1.1.1.1.1.1.1.2.2">𝑇</ci><ci id="S4.SS1.SSS1.1.p1.8.m8.2.2.1.1.1.1.1.1.1.2.3.cmml" xref="S4.SS1.SSS1.1.p1.8.m8.2.2.1.1.1.1.1.1.1.2.3">𝑣</ci></apply><ci id="S4.SS1.SSS1.1.p1.8.m8.2.2.1.1.1.1.1.1.1.3.cmml" xref="S4.SS1.SSS1.1.p1.8.m8.2.2.1.1.1.1.1.1.1.3">′</ci></apply></apply></apply></apply><apply id="S4.SS1.SSS1.1.p1.8.m8.2.2.3.cmml" xref="S4.SS1.SSS1.1.p1.8.m8.2.2.3"><times id="S4.SS1.SSS1.1.p1.8.m8.2.2.3.1.cmml" xref="S4.SS1.SSS1.1.p1.8.m8.2.2.3.1"></times><ci id="S4.SS1.SSS1.1.p1.8.m8.2.2.3.2.cmml" xref="S4.SS1.SSS1.1.p1.8.m8.2.2.3.2">𝑂</ci><ci id="S4.SS1.SSS1.1.p1.8.m8.1.1.cmml" xref="S4.SS1.SSS1.1.p1.8.m8.1.1">𝑛</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.SSS1.1.p1.8.m8.2c">|\cup_{u\in V}E(T_{v}^{\prime})|=O(n)</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.SSS1.1.p1.8.m8.2d">| ∪ start_POSTSUBSCRIPT italic_u ∈ italic_V end_POSTSUBSCRIPT italic_E ( italic_T start_POSTSUBSCRIPT italic_v end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) | = italic_O ( italic_n )</annotation></semantics></math>. ∎</p> </div> </div> </section> <section class="ltx_subsubsection" id="S4.SS1.SSS2"> <h4 class="ltx_title ltx_title_subsubsection"> <span class="ltx_tag ltx_tag_subsubsection">4.1.2 </span>Bounding Approximation Ratio</h4> <div class="ltx_para" id="S4.SS1.SSS2.p1"> <p class="ltx_p" id="S4.SS1.SSS2.p1.1">For the rest of this section, we bound the approximation ratio by proving the following lemma.</p> </div> <div class="ltx_theorem ltx_theorem_lemma" id="S4.Thmtheorem4"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="S4.Thmtheorem4.1.1.1">Lemma 4.4</span></span><span class="ltx_text ltx_font_bold" id="S4.Thmtheorem4.2.2">.</span> </h6> <div class="ltx_para" id="S4.Thmtheorem4.p1"> <p class="ltx_p" id="S4.Thmtheorem4.p1.6">Let <math alttext="F" class="ltx_Math" display="inline" id="S4.Thmtheorem4.p1.1.m1.1"><semantics id="S4.Thmtheorem4.p1.1.m1.1a"><mi id="S4.Thmtheorem4.p1.1.m1.1.1" xref="S4.Thmtheorem4.p1.1.m1.1.1.cmml">F</mi><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem4.p1.1.m1.1b"><ci id="S4.Thmtheorem4.p1.1.m1.1.1.cmml" xref="S4.Thmtheorem4.p1.1.m1.1.1">𝐹</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem4.p1.1.m1.1c">F</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem4.p1.1.m1.1d">italic_F</annotation></semantics></math> be the set of links stored in Algorithm <a class="ltx_ref" href="https://arxiv.org/html/2503.00712v1#algorithm4" title="In 4.1.1 The Streaming Algorithm ‣ 4.1 One-to-Two Augmentation ‣ 4 Vertex Connectivity Augmentation in Link-Arrival Model ‣ Streaming Algorithms for Network Design"><span class="ltx_text ltx_ref_tag">4</span></a>. The weight of an optimal solution to <math alttext="1" class="ltx_Math" display="inline" id="S4.Thmtheorem4.p1.2.m2.1"><semantics id="S4.Thmtheorem4.p1.2.m2.1a"><mn id="S4.Thmtheorem4.p1.2.m2.1.1" xref="S4.Thmtheorem4.p1.2.m2.1.1.cmml">1</mn><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem4.p1.2.m2.1b"><cn id="S4.Thmtheorem4.p1.2.m2.1.1.cmml" type="integer" xref="S4.Thmtheorem4.p1.2.m2.1.1">1</cn></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem4.p1.2.m2.1c">1</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem4.p1.2.m2.1d">1</annotation></semantics></math>-VC-CAP on <math alttext="(V,F)" class="ltx_Math" display="inline" id="S4.Thmtheorem4.p1.3.m3.2"><semantics id="S4.Thmtheorem4.p1.3.m3.2a"><mrow id="S4.Thmtheorem4.p1.3.m3.2.3.2" xref="S4.Thmtheorem4.p1.3.m3.2.3.1.cmml"><mo id="S4.Thmtheorem4.p1.3.m3.2.3.2.1" stretchy="false" xref="S4.Thmtheorem4.p1.3.m3.2.3.1.cmml">(</mo><mi id="S4.Thmtheorem4.p1.3.m3.1.1" xref="S4.Thmtheorem4.p1.3.m3.1.1.cmml">V</mi><mo id="S4.Thmtheorem4.p1.3.m3.2.3.2.2" xref="S4.Thmtheorem4.p1.3.m3.2.3.1.cmml">,</mo><mi id="S4.Thmtheorem4.p1.3.m3.2.2" xref="S4.Thmtheorem4.p1.3.m3.2.2.cmml">F</mi><mo id="S4.Thmtheorem4.p1.3.m3.2.3.2.3" stretchy="false" xref="S4.Thmtheorem4.p1.3.m3.2.3.1.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem4.p1.3.m3.2b"><interval closure="open" id="S4.Thmtheorem4.p1.3.m3.2.3.1.cmml" xref="S4.Thmtheorem4.p1.3.m3.2.3.2"><ci id="S4.Thmtheorem4.p1.3.m3.1.1.cmml" xref="S4.Thmtheorem4.p1.3.m3.1.1">𝑉</ci><ci id="S4.Thmtheorem4.p1.3.m3.2.2.cmml" xref="S4.Thmtheorem4.p1.3.m3.2.2">𝐹</ci></interval></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem4.p1.3.m3.2c">(V,F)</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem4.p1.3.m3.2d">( italic_V , italic_F )</annotation></semantics></math> is at most <math alttext="(3+\epsilon)" class="ltx_Math" display="inline" id="S4.Thmtheorem4.p1.4.m4.1"><semantics id="S4.Thmtheorem4.p1.4.m4.1a"><mrow id="S4.Thmtheorem4.p1.4.m4.1.1.1" xref="S4.Thmtheorem4.p1.4.m4.1.1.1.1.cmml"><mo id="S4.Thmtheorem4.p1.4.m4.1.1.1.2" stretchy="false" xref="S4.Thmtheorem4.p1.4.m4.1.1.1.1.cmml">(</mo><mrow id="S4.Thmtheorem4.p1.4.m4.1.1.1.1" xref="S4.Thmtheorem4.p1.4.m4.1.1.1.1.cmml"><mn id="S4.Thmtheorem4.p1.4.m4.1.1.1.1.2" xref="S4.Thmtheorem4.p1.4.m4.1.1.1.1.2.cmml">3</mn><mo id="S4.Thmtheorem4.p1.4.m4.1.1.1.1.1" xref="S4.Thmtheorem4.p1.4.m4.1.1.1.1.1.cmml">+</mo><mi id="S4.Thmtheorem4.p1.4.m4.1.1.1.1.3" xref="S4.Thmtheorem4.p1.4.m4.1.1.1.1.3.cmml">ϵ</mi></mrow><mo id="S4.Thmtheorem4.p1.4.m4.1.1.1.3" stretchy="false" xref="S4.Thmtheorem4.p1.4.m4.1.1.1.1.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem4.p1.4.m4.1b"><apply id="S4.Thmtheorem4.p1.4.m4.1.1.1.1.cmml" xref="S4.Thmtheorem4.p1.4.m4.1.1.1"><plus id="S4.Thmtheorem4.p1.4.m4.1.1.1.1.1.cmml" xref="S4.Thmtheorem4.p1.4.m4.1.1.1.1.1"></plus><cn id="S4.Thmtheorem4.p1.4.m4.1.1.1.1.2.cmml" type="integer" xref="S4.Thmtheorem4.p1.4.m4.1.1.1.1.2">3</cn><ci id="S4.Thmtheorem4.p1.4.m4.1.1.1.1.3.cmml" xref="S4.Thmtheorem4.p1.4.m4.1.1.1.1.3">italic-ϵ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem4.p1.4.m4.1c">(3+\epsilon)</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem4.p1.4.m4.1d">( 3 + italic_ϵ )</annotation></semantics></math> times the optimal solution to <math alttext="1" class="ltx_Math" display="inline" id="S4.Thmtheorem4.p1.5.m5.1"><semantics id="S4.Thmtheorem4.p1.5.m5.1a"><mn id="S4.Thmtheorem4.p1.5.m5.1.1" xref="S4.Thmtheorem4.p1.5.m5.1.1.cmml">1</mn><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem4.p1.5.m5.1b"><cn id="S4.Thmtheorem4.p1.5.m5.1.1.cmml" type="integer" xref="S4.Thmtheorem4.p1.5.m5.1.1">1</cn></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem4.p1.5.m5.1c">1</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem4.p1.5.m5.1d">1</annotation></semantics></math>-VC-CAP <math alttext="G=(V,E)" class="ltx_Math" display="inline" id="S4.Thmtheorem4.p1.6.m6.2"><semantics id="S4.Thmtheorem4.p1.6.m6.2a"><mrow id="S4.Thmtheorem4.p1.6.m6.2.3" xref="S4.Thmtheorem4.p1.6.m6.2.3.cmml"><mi id="S4.Thmtheorem4.p1.6.m6.2.3.2" xref="S4.Thmtheorem4.p1.6.m6.2.3.2.cmml">G</mi><mo id="S4.Thmtheorem4.p1.6.m6.2.3.1" xref="S4.Thmtheorem4.p1.6.m6.2.3.1.cmml">=</mo><mrow id="S4.Thmtheorem4.p1.6.m6.2.3.3.2" xref="S4.Thmtheorem4.p1.6.m6.2.3.3.1.cmml"><mo id="S4.Thmtheorem4.p1.6.m6.2.3.3.2.1" stretchy="false" xref="S4.Thmtheorem4.p1.6.m6.2.3.3.1.cmml">(</mo><mi id="S4.Thmtheorem4.p1.6.m6.1.1" xref="S4.Thmtheorem4.p1.6.m6.1.1.cmml">V</mi><mo id="S4.Thmtheorem4.p1.6.m6.2.3.3.2.2" xref="S4.Thmtheorem4.p1.6.m6.2.3.3.1.cmml">,</mo><mi id="S4.Thmtheorem4.p1.6.m6.2.2" xref="S4.Thmtheorem4.p1.6.m6.2.2.cmml">E</mi><mo id="S4.Thmtheorem4.p1.6.m6.2.3.3.2.3" stretchy="false" xref="S4.Thmtheorem4.p1.6.m6.2.3.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem4.p1.6.m6.2b"><apply id="S4.Thmtheorem4.p1.6.m6.2.3.cmml" xref="S4.Thmtheorem4.p1.6.m6.2.3"><eq id="S4.Thmtheorem4.p1.6.m6.2.3.1.cmml" xref="S4.Thmtheorem4.p1.6.m6.2.3.1"></eq><ci id="S4.Thmtheorem4.p1.6.m6.2.3.2.cmml" xref="S4.Thmtheorem4.p1.6.m6.2.3.2">𝐺</ci><interval closure="open" id="S4.Thmtheorem4.p1.6.m6.2.3.3.1.cmml" xref="S4.Thmtheorem4.p1.6.m6.2.3.3.2"><ci id="S4.Thmtheorem4.p1.6.m6.1.1.cmml" xref="S4.Thmtheorem4.p1.6.m6.1.1">𝑉</ci><ci id="S4.Thmtheorem4.p1.6.m6.2.2.cmml" xref="S4.Thmtheorem4.p1.6.m6.2.2">𝐸</ci></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem4.p1.6.m6.2c">G=(V,E)</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem4.p1.6.m6.2d">italic_G = ( italic_V , italic_E )</annotation></semantics></math>.</p> </div> </div> <div class="ltx_para" id="S4.SS1.SSS2.p2"> <p class="ltx_p" id="S4.SS1.SSS2.p2.6">Fix an optimal solution <span class="ltx_text ltx_markedasmath" id="S4.SS1.SSS2.p2.6.1">OPT</span> on <math alttext="G=(V,E)" class="ltx_Math" display="inline" id="S4.SS1.SSS2.p2.2.m2.2"><semantics id="S4.SS1.SSS2.p2.2.m2.2a"><mrow id="S4.SS1.SSS2.p2.2.m2.2.3" xref="S4.SS1.SSS2.p2.2.m2.2.3.cmml"><mi id="S4.SS1.SSS2.p2.2.m2.2.3.2" xref="S4.SS1.SSS2.p2.2.m2.2.3.2.cmml">G</mi><mo id="S4.SS1.SSS2.p2.2.m2.2.3.1" xref="S4.SS1.SSS2.p2.2.m2.2.3.1.cmml">=</mo><mrow id="S4.SS1.SSS2.p2.2.m2.2.3.3.2" xref="S4.SS1.SSS2.p2.2.m2.2.3.3.1.cmml"><mo id="S4.SS1.SSS2.p2.2.m2.2.3.3.2.1" stretchy="false" xref="S4.SS1.SSS2.p2.2.m2.2.3.3.1.cmml">(</mo><mi id="S4.SS1.SSS2.p2.2.m2.1.1" xref="S4.SS1.SSS2.p2.2.m2.1.1.cmml">V</mi><mo id="S4.SS1.SSS2.p2.2.m2.2.3.3.2.2" xref="S4.SS1.SSS2.p2.2.m2.2.3.3.1.cmml">,</mo><mi id="S4.SS1.SSS2.p2.2.m2.2.2" xref="S4.SS1.SSS2.p2.2.m2.2.2.cmml">E</mi><mo id="S4.SS1.SSS2.p2.2.m2.2.3.3.2.3" stretchy="false" xref="S4.SS1.SSS2.p2.2.m2.2.3.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.SSS2.p2.2.m2.2b"><apply id="S4.SS1.SSS2.p2.2.m2.2.3.cmml" xref="S4.SS1.SSS2.p2.2.m2.2.3"><eq id="S4.SS1.SSS2.p2.2.m2.2.3.1.cmml" xref="S4.SS1.SSS2.p2.2.m2.2.3.1"></eq><ci id="S4.SS1.SSS2.p2.2.m2.2.3.2.cmml" xref="S4.SS1.SSS2.p2.2.m2.2.3.2">𝐺</ci><interval closure="open" id="S4.SS1.SSS2.p2.2.m2.2.3.3.1.cmml" xref="S4.SS1.SSS2.p2.2.m2.2.3.3.2"><ci id="S4.SS1.SSS2.p2.2.m2.1.1.cmml" xref="S4.SS1.SSS2.p2.2.m2.1.1">𝑉</ci><ci id="S4.SS1.SSS2.p2.2.m2.2.2.cmml" xref="S4.SS1.SSS2.p2.2.m2.2.2">𝐸</ci></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.SSS2.p2.2.m2.2c">G=(V,E)</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.SSS2.p2.2.m2.2d">italic_G = ( italic_V , italic_E )</annotation></semantics></math>. To prove Lemma <a class="ltx_ref" href="https://arxiv.org/html/2503.00712v1#S4.Thmtheorem4" title="Lemma 4.4. ‣ 4.1.2 Bounding Approximation Ratio ‣ 4.1 One-to-Two Augmentation ‣ 4 Vertex Connectivity Augmentation in Link-Arrival Model ‣ Streaming Algorithms for Network Design"><span class="ltx_text ltx_ref_tag">4.4</span></a>, we provide an algorithm to construct a feasible solution <math alttext="\textnormal{SOL}\subseteq F" class="ltx_Math" display="inline" id="S4.SS1.SSS2.p2.3.m3.1"><semantics id="S4.SS1.SSS2.p2.3.m3.1a"><mrow id="S4.SS1.SSS2.p2.3.m3.1.1" xref="S4.SS1.SSS2.p2.3.m3.1.1.cmml"><mtext id="S4.SS1.SSS2.p2.3.m3.1.1.2" xref="S4.SS1.SSS2.p2.3.m3.1.1.2a.cmml">SOL</mtext><mo id="S4.SS1.SSS2.p2.3.m3.1.1.1" xref="S4.SS1.SSS2.p2.3.m3.1.1.1.cmml">⊆</mo><mi id="S4.SS1.SSS2.p2.3.m3.1.1.3" xref="S4.SS1.SSS2.p2.3.m3.1.1.3.cmml">F</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.SSS2.p2.3.m3.1b"><apply id="S4.SS1.SSS2.p2.3.m3.1.1.cmml" xref="S4.SS1.SSS2.p2.3.m3.1.1"><subset id="S4.SS1.SSS2.p2.3.m3.1.1.1.cmml" xref="S4.SS1.SSS2.p2.3.m3.1.1.1"></subset><ci id="S4.SS1.SSS2.p2.3.m3.1.1.2a.cmml" xref="S4.SS1.SSS2.p2.3.m3.1.1.2"><mtext id="S4.SS1.SSS2.p2.3.m3.1.1.2.cmml" xref="S4.SS1.SSS2.p2.3.m3.1.1.2">SOL</mtext></ci><ci id="S4.SS1.SSS2.p2.3.m3.1.1.3.cmml" xref="S4.SS1.SSS2.p2.3.m3.1.1.3">𝐹</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.SSS2.p2.3.m3.1c">\textnormal{SOL}\subseteq F</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.SSS2.p2.3.m3.1d">SOL ⊆ italic_F</annotation></semantics></math> such that <math alttext="w(\textnormal{SOL})\leq(3+\epsilon)w(\textnormal{OPT})" class="ltx_Math" display="inline" id="S4.SS1.SSS2.p2.4.m4.3"><semantics id="S4.SS1.SSS2.p2.4.m4.3a"><mrow id="S4.SS1.SSS2.p2.4.m4.3.3" xref="S4.SS1.SSS2.p2.4.m4.3.3.cmml"><mrow id="S4.SS1.SSS2.p2.4.m4.3.3.3" xref="S4.SS1.SSS2.p2.4.m4.3.3.3.cmml"><mi id="S4.SS1.SSS2.p2.4.m4.3.3.3.2" xref="S4.SS1.SSS2.p2.4.m4.3.3.3.2.cmml">w</mi><mo id="S4.SS1.SSS2.p2.4.m4.3.3.3.1" xref="S4.SS1.SSS2.p2.4.m4.3.3.3.1.cmml">⁢</mo><mrow id="S4.SS1.SSS2.p2.4.m4.3.3.3.3.2" xref="S4.SS1.SSS2.p2.4.m4.1.1a.cmml"><mo id="S4.SS1.SSS2.p2.4.m4.3.3.3.3.2.1" stretchy="false" xref="S4.SS1.SSS2.p2.4.m4.1.1a.cmml">(</mo><mtext id="S4.SS1.SSS2.p2.4.m4.1.1" xref="S4.SS1.SSS2.p2.4.m4.1.1.cmml">SOL</mtext><mo id="S4.SS1.SSS2.p2.4.m4.3.3.3.3.2.2" stretchy="false" xref="S4.SS1.SSS2.p2.4.m4.1.1a.cmml">)</mo></mrow></mrow><mo id="S4.SS1.SSS2.p2.4.m4.3.3.2" xref="S4.SS1.SSS2.p2.4.m4.3.3.2.cmml">≤</mo><mrow id="S4.SS1.SSS2.p2.4.m4.3.3.1" xref="S4.SS1.SSS2.p2.4.m4.3.3.1.cmml"><mrow id="S4.SS1.SSS2.p2.4.m4.3.3.1.1.1" xref="S4.SS1.SSS2.p2.4.m4.3.3.1.1.1.1.cmml"><mo id="S4.SS1.SSS2.p2.4.m4.3.3.1.1.1.2" stretchy="false" xref="S4.SS1.SSS2.p2.4.m4.3.3.1.1.1.1.cmml">(</mo><mrow id="S4.SS1.SSS2.p2.4.m4.3.3.1.1.1.1" xref="S4.SS1.SSS2.p2.4.m4.3.3.1.1.1.1.cmml"><mn id="S4.SS1.SSS2.p2.4.m4.3.3.1.1.1.1.2" xref="S4.SS1.SSS2.p2.4.m4.3.3.1.1.1.1.2.cmml">3</mn><mo id="S4.SS1.SSS2.p2.4.m4.3.3.1.1.1.1.1" xref="S4.SS1.SSS2.p2.4.m4.3.3.1.1.1.1.1.cmml">+</mo><mi id="S4.SS1.SSS2.p2.4.m4.3.3.1.1.1.1.3" xref="S4.SS1.SSS2.p2.4.m4.3.3.1.1.1.1.3.cmml">ϵ</mi></mrow><mo id="S4.SS1.SSS2.p2.4.m4.3.3.1.1.1.3" stretchy="false" xref="S4.SS1.SSS2.p2.4.m4.3.3.1.1.1.1.cmml">)</mo></mrow><mo id="S4.SS1.SSS2.p2.4.m4.3.3.1.2" xref="S4.SS1.SSS2.p2.4.m4.3.3.1.2.cmml">⁢</mo><mi id="S4.SS1.SSS2.p2.4.m4.3.3.1.3" xref="S4.SS1.SSS2.p2.4.m4.3.3.1.3.cmml">w</mi><mo id="S4.SS1.SSS2.p2.4.m4.3.3.1.2a" xref="S4.SS1.SSS2.p2.4.m4.3.3.1.2.cmml">⁢</mo><mrow id="S4.SS1.SSS2.p2.4.m4.3.3.1.4.2" xref="S4.SS1.SSS2.p2.4.m4.2.2a.cmml"><mo id="S4.SS1.SSS2.p2.4.m4.3.3.1.4.2.1" stretchy="false" xref="S4.SS1.SSS2.p2.4.m4.2.2a.cmml">(</mo><mtext id="S4.SS1.SSS2.p2.4.m4.2.2" xref="S4.SS1.SSS2.p2.4.m4.2.2.cmml">OPT</mtext><mo id="S4.SS1.SSS2.p2.4.m4.3.3.1.4.2.2" stretchy="false" xref="S4.SS1.SSS2.p2.4.m4.2.2a.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.SSS2.p2.4.m4.3b"><apply id="S4.SS1.SSS2.p2.4.m4.3.3.cmml" xref="S4.SS1.SSS2.p2.4.m4.3.3"><leq id="S4.SS1.SSS2.p2.4.m4.3.3.2.cmml" xref="S4.SS1.SSS2.p2.4.m4.3.3.2"></leq><apply id="S4.SS1.SSS2.p2.4.m4.3.3.3.cmml" xref="S4.SS1.SSS2.p2.4.m4.3.3.3"><times id="S4.SS1.SSS2.p2.4.m4.3.3.3.1.cmml" xref="S4.SS1.SSS2.p2.4.m4.3.3.3.1"></times><ci id="S4.SS1.SSS2.p2.4.m4.3.3.3.2.cmml" xref="S4.SS1.SSS2.p2.4.m4.3.3.3.2">𝑤</ci><ci id="S4.SS1.SSS2.p2.4.m4.1.1a.cmml" xref="S4.SS1.SSS2.p2.4.m4.3.3.3.3.2"><mtext id="S4.SS1.SSS2.p2.4.m4.1.1.cmml" xref="S4.SS1.SSS2.p2.4.m4.1.1">SOL</mtext></ci></apply><apply id="S4.SS1.SSS2.p2.4.m4.3.3.1.cmml" xref="S4.SS1.SSS2.p2.4.m4.3.3.1"><times id="S4.SS1.SSS2.p2.4.m4.3.3.1.2.cmml" xref="S4.SS1.SSS2.p2.4.m4.3.3.1.2"></times><apply id="S4.SS1.SSS2.p2.4.m4.3.3.1.1.1.1.cmml" xref="S4.SS1.SSS2.p2.4.m4.3.3.1.1.1"><plus id="S4.SS1.SSS2.p2.4.m4.3.3.1.1.1.1.1.cmml" xref="S4.SS1.SSS2.p2.4.m4.3.3.1.1.1.1.1"></plus><cn id="S4.SS1.SSS2.p2.4.m4.3.3.1.1.1.1.2.cmml" type="integer" xref="S4.SS1.SSS2.p2.4.m4.3.3.1.1.1.1.2">3</cn><ci id="S4.SS1.SSS2.p2.4.m4.3.3.1.1.1.1.3.cmml" xref="S4.SS1.SSS2.p2.4.m4.3.3.1.1.1.1.3">italic-ϵ</ci></apply><ci id="S4.SS1.SSS2.p2.4.m4.3.3.1.3.cmml" xref="S4.SS1.SSS2.p2.4.m4.3.3.1.3">𝑤</ci><ci id="S4.SS1.SSS2.p2.4.m4.2.2a.cmml" xref="S4.SS1.SSS2.p2.4.m4.3.3.1.4.2"><mtext id="S4.SS1.SSS2.p2.4.m4.2.2.cmml" xref="S4.SS1.SSS2.p2.4.m4.2.2">OPT</mtext></ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.SSS2.p2.4.m4.3c">w(\textnormal{SOL})\leq(3+\epsilon)w(\textnormal{OPT})</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.SSS2.p2.4.m4.3d">italic_w ( SOL ) ≤ ( 3 + italic_ϵ ) italic_w ( OPT )</annotation></semantics></math>. Note that this algorithm is for analysis purposes only, as it requires knowledge of an optimal solution <span class="ltx_text ltx_markedasmath" id="S4.SS1.SSS2.p2.6.2">OPT</span> to the instance on <math alttext="G=(V,E)" class="ltx_Math" display="inline" id="S4.SS1.SSS2.p2.6.m6.2"><semantics id="S4.SS1.SSS2.p2.6.m6.2a"><mrow id="S4.SS1.SSS2.p2.6.m6.2.3" xref="S4.SS1.SSS2.p2.6.m6.2.3.cmml"><mi id="S4.SS1.SSS2.p2.6.m6.2.3.2" xref="S4.SS1.SSS2.p2.6.m6.2.3.2.cmml">G</mi><mo id="S4.SS1.SSS2.p2.6.m6.2.3.1" xref="S4.SS1.SSS2.p2.6.m6.2.3.1.cmml">=</mo><mrow id="S4.SS1.SSS2.p2.6.m6.2.3.3.2" xref="S4.SS1.SSS2.p2.6.m6.2.3.3.1.cmml"><mo id="S4.SS1.SSS2.p2.6.m6.2.3.3.2.1" stretchy="false" xref="S4.SS1.SSS2.p2.6.m6.2.3.3.1.cmml">(</mo><mi id="S4.SS1.SSS2.p2.6.m6.1.1" xref="S4.SS1.SSS2.p2.6.m6.1.1.cmml">V</mi><mo id="S4.SS1.SSS2.p2.6.m6.2.3.3.2.2" xref="S4.SS1.SSS2.p2.6.m6.2.3.3.1.cmml">,</mo><mi id="S4.SS1.SSS2.p2.6.m6.2.2" xref="S4.SS1.SSS2.p2.6.m6.2.2.cmml">E</mi><mo id="S4.SS1.SSS2.p2.6.m6.2.3.3.2.3" stretchy="false" xref="S4.SS1.SSS2.p2.6.m6.2.3.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.SSS2.p2.6.m6.2b"><apply id="S4.SS1.SSS2.p2.6.m6.2.3.cmml" xref="S4.SS1.SSS2.p2.6.m6.2.3"><eq id="S4.SS1.SSS2.p2.6.m6.2.3.1.cmml" xref="S4.SS1.SSS2.p2.6.m6.2.3.1"></eq><ci id="S4.SS1.SSS2.p2.6.m6.2.3.2.cmml" xref="S4.SS1.SSS2.p2.6.m6.2.3.2">𝐺</ci><interval closure="open" id="S4.SS1.SSS2.p2.6.m6.2.3.3.1.cmml" xref="S4.SS1.SSS2.p2.6.m6.2.3.3.2"><ci id="S4.SS1.SSS2.p2.6.m6.1.1.cmml" xref="S4.SS1.SSS2.p2.6.m6.1.1">𝑉</ci><ci id="S4.SS1.SSS2.p2.6.m6.2.2.cmml" xref="S4.SS1.SSS2.p2.6.m6.2.2">𝐸</ci></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.SSS2.p2.6.m6.2c">G=(V,E)</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.SSS2.p2.6.m6.2d">italic_G = ( italic_V , italic_E )</annotation></semantics></math>.</p> </div> <figure class="ltx_float ltx_algorithm" id="algorithm5"> <div class="ltx_listing ltx_lst_numbers_left ltx_listing" id="algorithm5.20"> <div class="ltx_listingline" id="algorithm5.5.1"> <math alttext="\textnormal{SOL}\leftarrow\emptyset" class="ltx_Math" display="inline" id="algorithm5.5.1.m1.1"><semantics id="algorithm5.5.1.m1.1a"><mrow id="algorithm5.5.1.m1.1.1" xref="algorithm5.5.1.m1.1.1.cmml"><mtext id="algorithm5.5.1.m1.1.1.2" xref="algorithm5.5.1.m1.1.1.2a.cmml">SOL</mtext><mo id="algorithm5.5.1.m1.1.1.1" stretchy="false" xref="algorithm5.5.1.m1.1.1.1.cmml">←</mo><mi id="algorithm5.5.1.m1.1.1.3" mathvariant="normal" xref="algorithm5.5.1.m1.1.1.3.cmml">∅</mi></mrow><annotation-xml encoding="MathML-Content" id="algorithm5.5.1.m1.1b"><apply id="algorithm5.5.1.m1.1.1.cmml" xref="algorithm5.5.1.m1.1.1"><ci id="algorithm5.5.1.m1.1.1.1.cmml" xref="algorithm5.5.1.m1.1.1.1">←</ci><ci id="algorithm5.5.1.m1.1.1.2a.cmml" xref="algorithm5.5.1.m1.1.1.2"><mtext id="algorithm5.5.1.m1.1.1.2.cmml" xref="algorithm5.5.1.m1.1.1.2">SOL</mtext></ci><emptyset id="algorithm5.5.1.m1.1.1.3.cmml" xref="algorithm5.5.1.m1.1.1.3"></emptyset></apply></annotation-xml><annotation encoding="application/x-tex" id="algorithm5.5.1.m1.1c">\textnormal{SOL}\leftarrow\emptyset</annotation><annotation encoding="application/x-llamapun" id="algorithm5.5.1.m1.1d">SOL ← ∅</annotation></semantics></math> </div> <div class="ltx_listingline" id="algorithm5.6.2"> <span class="ltx_text ltx_font_bold" id="algorithm5.6.2.2">for</span> <em class="ltx_emph ltx_font_italic" id="algorithm5.6.2.1"><math alttext="e=uv\in\textnormal{OPT}" class="ltx_Math" display="inline" id="algorithm5.6.2.1.m1.1"><semantics id="algorithm5.6.2.1.m1.1a"><mrow id="algorithm5.6.2.1.m1.1.1" xref="algorithm5.6.2.1.m1.1.1.cmml"><mi id="algorithm5.6.2.1.m1.1.1.2" xref="algorithm5.6.2.1.m1.1.1.2.cmml">e</mi><mo id="algorithm5.6.2.1.m1.1.1.3" xref="algorithm5.6.2.1.m1.1.1.3.cmml">=</mo><mrow id="algorithm5.6.2.1.m1.1.1.4" xref="algorithm5.6.2.1.m1.1.1.4.cmml"><mi id="algorithm5.6.2.1.m1.1.1.4.2" xref="algorithm5.6.2.1.m1.1.1.4.2.cmml">u</mi><mo id="algorithm5.6.2.1.m1.1.1.4.1" xref="algorithm5.6.2.1.m1.1.1.4.1.cmml">⁢</mo><mi id="algorithm5.6.2.1.m1.1.1.4.3" xref="algorithm5.6.2.1.m1.1.1.4.3.cmml">v</mi></mrow><mo id="algorithm5.6.2.1.m1.1.1.5" xref="algorithm5.6.2.1.m1.1.1.5.cmml">∈</mo><mtext id="algorithm5.6.2.1.m1.1.1.6" xref="algorithm5.6.2.1.m1.1.1.6b.cmml"><em class="ltx_emph ltx_font_upright" id="algorithm5.6.2.1.m1.1.1.6.1nest">OPT</em></mtext></mrow><annotation-xml encoding="MathML-Content" id="algorithm5.6.2.1.m1.1b"><apply id="algorithm5.6.2.1.m1.1.1.cmml" xref="algorithm5.6.2.1.m1.1.1"><and id="algorithm5.6.2.1.m1.1.1a.cmml" xref="algorithm5.6.2.1.m1.1.1"></and><apply id="algorithm5.6.2.1.m1.1.1b.cmml" xref="algorithm5.6.2.1.m1.1.1"><eq id="algorithm5.6.2.1.m1.1.1.3.cmml" xref="algorithm5.6.2.1.m1.1.1.3"></eq><ci id="algorithm5.6.2.1.m1.1.1.2.cmml" xref="algorithm5.6.2.1.m1.1.1.2">𝑒</ci><apply id="algorithm5.6.2.1.m1.1.1.4.cmml" xref="algorithm5.6.2.1.m1.1.1.4"><times id="algorithm5.6.2.1.m1.1.1.4.1.cmml" xref="algorithm5.6.2.1.m1.1.1.4.1"></times><ci id="algorithm5.6.2.1.m1.1.1.4.2.cmml" xref="algorithm5.6.2.1.m1.1.1.4.2">𝑢</ci><ci id="algorithm5.6.2.1.m1.1.1.4.3.cmml" xref="algorithm5.6.2.1.m1.1.1.4.3">𝑣</ci></apply></apply><apply id="algorithm5.6.2.1.m1.1.1c.cmml" xref="algorithm5.6.2.1.m1.1.1"><in id="algorithm5.6.2.1.m1.1.1.5.cmml" xref="algorithm5.6.2.1.m1.1.1.5"></in><share href="https://arxiv.org/html/2503.00712v1#algorithm5.6.2.1.m1.1.1.4.cmml" id="algorithm5.6.2.1.m1.1.1d.cmml" xref="algorithm5.6.2.1.m1.1.1"></share><ci id="algorithm5.6.2.1.m1.1.1.6b.cmml" xref="algorithm5.6.2.1.m1.1.1.6"><mtext id="algorithm5.6.2.1.m1.1.1.6.cmml" xref="algorithm5.6.2.1.m1.1.1.6"><em class="ltx_emph ltx_font_upright" id="algorithm5.6.2.1.m1.1.1.6.1anest">OPT</em></mtext></ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="algorithm5.6.2.1.m1.1c">e=uv\in\textnormal{OPT}</annotation><annotation encoding="application/x-llamapun" id="algorithm5.6.2.1.m1.1d">italic_e = italic_u italic_v ∈ OPT</annotation></semantics></math></em> <span class="ltx_text ltx_font_bold" id="algorithm5.6.2.3">do</span> </div> <div class="ltx_listingline" id="algorithm5.8.4"> <span class="ltx_rule" style="width:1px;height:100%;background:black;display:inline-block;"> </span> Let <math alttext="j" class="ltx_Math" display="inline" id="algorithm5.7.3.m1.1"><semantics id="algorithm5.7.3.m1.1a"><mi id="algorithm5.7.3.m1.1.1" xref="algorithm5.7.3.m1.1.1.cmml">j</mi><annotation-xml encoding="MathML-Content" id="algorithm5.7.3.m1.1b"><ci id="algorithm5.7.3.m1.1.1.cmml" xref="algorithm5.7.3.m1.1.1">𝑗</ci></annotation-xml><annotation encoding="application/x-tex" id="algorithm5.7.3.m1.1c">j</annotation><annotation encoding="application/x-llamapun" id="algorithm5.7.3.m1.1d">italic_j</annotation></semantics></math> be weight class such that <math alttext="w(e)\in[(1+\epsilon)^{j},(1+\epsilon)^{j+1})" class="ltx_Math" display="inline" id="algorithm5.8.4.m2.3"><semantics id="algorithm5.8.4.m2.3a"><mrow id="algorithm5.8.4.m2.3.3" xref="algorithm5.8.4.m2.3.3.cmml"><mrow id="algorithm5.8.4.m2.3.3.4" xref="algorithm5.8.4.m2.3.3.4.cmml"><mi id="algorithm5.8.4.m2.3.3.4.2" xref="algorithm5.8.4.m2.3.3.4.2.cmml">w</mi><mo id="algorithm5.8.4.m2.3.3.4.1" xref="algorithm5.8.4.m2.3.3.4.1.cmml">⁢</mo><mrow id="algorithm5.8.4.m2.3.3.4.3.2" xref="algorithm5.8.4.m2.3.3.4.cmml"><mo id="algorithm5.8.4.m2.3.3.4.3.2.1" stretchy="false" xref="algorithm5.8.4.m2.3.3.4.cmml">(</mo><mi id="algorithm5.8.4.m2.1.1" xref="algorithm5.8.4.m2.1.1.cmml">e</mi><mo id="algorithm5.8.4.m2.3.3.4.3.2.2" stretchy="false" xref="algorithm5.8.4.m2.3.3.4.cmml">)</mo></mrow></mrow><mo id="algorithm5.8.4.m2.3.3.3" xref="algorithm5.8.4.m2.3.3.3.cmml">∈</mo><mrow id="algorithm5.8.4.m2.3.3.2.2" xref="algorithm5.8.4.m2.3.3.2.3.cmml"><mo id="algorithm5.8.4.m2.3.3.2.2.3" stretchy="false" xref="algorithm5.8.4.m2.3.3.2.3.cmml">[</mo><msup id="algorithm5.8.4.m2.2.2.1.1.1" xref="algorithm5.8.4.m2.2.2.1.1.1.cmml"><mrow id="algorithm5.8.4.m2.2.2.1.1.1.1.1" xref="algorithm5.8.4.m2.2.2.1.1.1.1.1.1.cmml"><mo id="algorithm5.8.4.m2.2.2.1.1.1.1.1.2" stretchy="false" xref="algorithm5.8.4.m2.2.2.1.1.1.1.1.1.cmml">(</mo><mrow id="algorithm5.8.4.m2.2.2.1.1.1.1.1.1" xref="algorithm5.8.4.m2.2.2.1.1.1.1.1.1.cmml"><mn id="algorithm5.8.4.m2.2.2.1.1.1.1.1.1.2" xref="algorithm5.8.4.m2.2.2.1.1.1.1.1.1.2.cmml">1</mn><mo id="algorithm5.8.4.m2.2.2.1.1.1.1.1.1.1" xref="algorithm5.8.4.m2.2.2.1.1.1.1.1.1.1.cmml">+</mo><mi id="algorithm5.8.4.m2.2.2.1.1.1.1.1.1.3" xref="algorithm5.8.4.m2.2.2.1.1.1.1.1.1.3.cmml">ϵ</mi></mrow><mo id="algorithm5.8.4.m2.2.2.1.1.1.1.1.3" stretchy="false" xref="algorithm5.8.4.m2.2.2.1.1.1.1.1.1.cmml">)</mo></mrow><mi id="algorithm5.8.4.m2.2.2.1.1.1.3" xref="algorithm5.8.4.m2.2.2.1.1.1.3.cmml">j</mi></msup><mo id="algorithm5.8.4.m2.3.3.2.2.4" xref="algorithm5.8.4.m2.3.3.2.3.cmml">,</mo><msup id="algorithm5.8.4.m2.3.3.2.2.2" xref="algorithm5.8.4.m2.3.3.2.2.2.cmml"><mrow id="algorithm5.8.4.m2.3.3.2.2.2.1.1" xref="algorithm5.8.4.m2.3.3.2.2.2.1.1.1.cmml"><mo id="algorithm5.8.4.m2.3.3.2.2.2.1.1.2" stretchy="false" xref="algorithm5.8.4.m2.3.3.2.2.2.1.1.1.cmml">(</mo><mrow id="algorithm5.8.4.m2.3.3.2.2.2.1.1.1" xref="algorithm5.8.4.m2.3.3.2.2.2.1.1.1.cmml"><mn id="algorithm5.8.4.m2.3.3.2.2.2.1.1.1.2" xref="algorithm5.8.4.m2.3.3.2.2.2.1.1.1.2.cmml">1</mn><mo id="algorithm5.8.4.m2.3.3.2.2.2.1.1.1.1" xref="algorithm5.8.4.m2.3.3.2.2.2.1.1.1.1.cmml">+</mo><mi id="algorithm5.8.4.m2.3.3.2.2.2.1.1.1.3" xref="algorithm5.8.4.m2.3.3.2.2.2.1.1.1.3.cmml">ϵ</mi></mrow><mo id="algorithm5.8.4.m2.3.3.2.2.2.1.1.3" stretchy="false" xref="algorithm5.8.4.m2.3.3.2.2.2.1.1.1.cmml">)</mo></mrow><mrow id="algorithm5.8.4.m2.3.3.2.2.2.3" xref="algorithm5.8.4.m2.3.3.2.2.2.3.cmml"><mi id="algorithm5.8.4.m2.3.3.2.2.2.3.2" xref="algorithm5.8.4.m2.3.3.2.2.2.3.2.cmml">j</mi><mo id="algorithm5.8.4.m2.3.3.2.2.2.3.1" xref="algorithm5.8.4.m2.3.3.2.2.2.3.1.cmml">+</mo><mn id="algorithm5.8.4.m2.3.3.2.2.2.3.3" xref="algorithm5.8.4.m2.3.3.2.2.2.3.3.cmml">1</mn></mrow></msup><mo id="algorithm5.8.4.m2.3.3.2.2.5" stretchy="false" xref="algorithm5.8.4.m2.3.3.2.3.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="algorithm5.8.4.m2.3b"><apply id="algorithm5.8.4.m2.3.3.cmml" xref="algorithm5.8.4.m2.3.3"><in id="algorithm5.8.4.m2.3.3.3.cmml" xref="algorithm5.8.4.m2.3.3.3"></in><apply id="algorithm5.8.4.m2.3.3.4.cmml" xref="algorithm5.8.4.m2.3.3.4"><times id="algorithm5.8.4.m2.3.3.4.1.cmml" xref="algorithm5.8.4.m2.3.3.4.1"></times><ci id="algorithm5.8.4.m2.3.3.4.2.cmml" xref="algorithm5.8.4.m2.3.3.4.2">𝑤</ci><ci id="algorithm5.8.4.m2.1.1.cmml" xref="algorithm5.8.4.m2.1.1">𝑒</ci></apply><interval closure="closed-open" id="algorithm5.8.4.m2.3.3.2.3.cmml" xref="algorithm5.8.4.m2.3.3.2.2"><apply id="algorithm5.8.4.m2.2.2.1.1.1.cmml" xref="algorithm5.8.4.m2.2.2.1.1.1"><csymbol cd="ambiguous" id="algorithm5.8.4.m2.2.2.1.1.1.2.cmml" xref="algorithm5.8.4.m2.2.2.1.1.1">superscript</csymbol><apply id="algorithm5.8.4.m2.2.2.1.1.1.1.1.1.cmml" xref="algorithm5.8.4.m2.2.2.1.1.1.1.1"><plus id="algorithm5.8.4.m2.2.2.1.1.1.1.1.1.1.cmml" xref="algorithm5.8.4.m2.2.2.1.1.1.1.1.1.1"></plus><cn id="algorithm5.8.4.m2.2.2.1.1.1.1.1.1.2.cmml" type="integer" xref="algorithm5.8.4.m2.2.2.1.1.1.1.1.1.2">1</cn><ci id="algorithm5.8.4.m2.2.2.1.1.1.1.1.1.3.cmml" xref="algorithm5.8.4.m2.2.2.1.1.1.1.1.1.3">italic-ϵ</ci></apply><ci id="algorithm5.8.4.m2.2.2.1.1.1.3.cmml" xref="algorithm5.8.4.m2.2.2.1.1.1.3">𝑗</ci></apply><apply id="algorithm5.8.4.m2.3.3.2.2.2.cmml" xref="algorithm5.8.4.m2.3.3.2.2.2"><csymbol cd="ambiguous" id="algorithm5.8.4.m2.3.3.2.2.2.2.cmml" xref="algorithm5.8.4.m2.3.3.2.2.2">superscript</csymbol><apply id="algorithm5.8.4.m2.3.3.2.2.2.1.1.1.cmml" xref="algorithm5.8.4.m2.3.3.2.2.2.1.1"><plus id="algorithm5.8.4.m2.3.3.2.2.2.1.1.1.1.cmml" xref="algorithm5.8.4.m2.3.3.2.2.2.1.1.1.1"></plus><cn id="algorithm5.8.4.m2.3.3.2.2.2.1.1.1.2.cmml" type="integer" xref="algorithm5.8.4.m2.3.3.2.2.2.1.1.1.2">1</cn><ci id="algorithm5.8.4.m2.3.3.2.2.2.1.1.1.3.cmml" xref="algorithm5.8.4.m2.3.3.2.2.2.1.1.1.3">italic-ϵ</ci></apply><apply id="algorithm5.8.4.m2.3.3.2.2.2.3.cmml" xref="algorithm5.8.4.m2.3.3.2.2.2.3"><plus id="algorithm5.8.4.m2.3.3.2.2.2.3.1.cmml" xref="algorithm5.8.4.m2.3.3.2.2.2.3.1"></plus><ci id="algorithm5.8.4.m2.3.3.2.2.2.3.2.cmml" xref="algorithm5.8.4.m2.3.3.2.2.2.3.2">𝑗</ci><cn id="algorithm5.8.4.m2.3.3.2.2.2.3.3.cmml" type="integer" xref="algorithm5.8.4.m2.3.3.2.2.2.3.3">1</cn></apply></apply></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="algorithm5.8.4.m2.3c">w(e)\in[(1+\epsilon)^{j},(1+\epsilon)^{j+1})</annotation><annotation encoding="application/x-llamapun" id="algorithm5.8.4.m2.3d">italic_w ( italic_e ) ∈ [ ( 1 + italic_ϵ ) start_POSTSUPERSCRIPT italic_j end_POSTSUPERSCRIPT , ( 1 + italic_ϵ ) start_POSTSUPERSCRIPT italic_j + 1 end_POSTSUPERSCRIPT )</annotation></semantics></math> </div> <div class="ltx_listingline" id="algorithm5.9.5"> <span class="ltx_rule" style="width:1px;height:100%;background:black;display:inline-block;"> </span> <math alttext="\textnormal{SOL}\leftarrow\textnormal{SOL}\cup\{L_{u}(j),L_{v}(j)\}" class="ltx_Math" display="inline" id="algorithm5.9.5.m1.4"><semantics id="algorithm5.9.5.m1.4a"><mrow id="algorithm5.9.5.m1.4.4" xref="algorithm5.9.5.m1.4.4.cmml"><mtext id="algorithm5.9.5.m1.4.4.4" xref="algorithm5.9.5.m1.4.4.4a.cmml">SOL</mtext><mo id="algorithm5.9.5.m1.4.4.3" stretchy="false" xref="algorithm5.9.5.m1.4.4.3.cmml">←</mo><mrow id="algorithm5.9.5.m1.4.4.2" xref="algorithm5.9.5.m1.4.4.2.cmml"><mtext id="algorithm5.9.5.m1.4.4.2.4" xref="algorithm5.9.5.m1.4.4.2.4a.cmml">SOL</mtext><mo id="algorithm5.9.5.m1.4.4.2.3" xref="algorithm5.9.5.m1.4.4.2.3.cmml">∪</mo><mrow id="algorithm5.9.5.m1.4.4.2.2.2" xref="algorithm5.9.5.m1.4.4.2.2.3.cmml"><mo id="algorithm5.9.5.m1.4.4.2.2.2.3" stretchy="false" xref="algorithm5.9.5.m1.4.4.2.2.3.cmml">{</mo><mrow id="algorithm5.9.5.m1.3.3.1.1.1.1" xref="algorithm5.9.5.m1.3.3.1.1.1.1.cmml"><msub id="algorithm5.9.5.m1.3.3.1.1.1.1.2" xref="algorithm5.9.5.m1.3.3.1.1.1.1.2.cmml"><mi id="algorithm5.9.5.m1.3.3.1.1.1.1.2.2" xref="algorithm5.9.5.m1.3.3.1.1.1.1.2.2.cmml">L</mi><mi id="algorithm5.9.5.m1.3.3.1.1.1.1.2.3" xref="algorithm5.9.5.m1.3.3.1.1.1.1.2.3.cmml">u</mi></msub><mo id="algorithm5.9.5.m1.3.3.1.1.1.1.1" xref="algorithm5.9.5.m1.3.3.1.1.1.1.1.cmml">⁢</mo><mrow id="algorithm5.9.5.m1.3.3.1.1.1.1.3.2" xref="algorithm5.9.5.m1.3.3.1.1.1.1.cmml"><mo id="algorithm5.9.5.m1.3.3.1.1.1.1.3.2.1" stretchy="false" xref="algorithm5.9.5.m1.3.3.1.1.1.1.cmml">(</mo><mi id="algorithm5.9.5.m1.1.1" xref="algorithm5.9.5.m1.1.1.cmml">j</mi><mo id="algorithm5.9.5.m1.3.3.1.1.1.1.3.2.2" stretchy="false" xref="algorithm5.9.5.m1.3.3.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="algorithm5.9.5.m1.4.4.2.2.2.4" xref="algorithm5.9.5.m1.4.4.2.2.3.cmml">,</mo><mrow id="algorithm5.9.5.m1.4.4.2.2.2.2" xref="algorithm5.9.5.m1.4.4.2.2.2.2.cmml"><msub id="algorithm5.9.5.m1.4.4.2.2.2.2.2" xref="algorithm5.9.5.m1.4.4.2.2.2.2.2.cmml"><mi id="algorithm5.9.5.m1.4.4.2.2.2.2.2.2" xref="algorithm5.9.5.m1.4.4.2.2.2.2.2.2.cmml">L</mi><mi id="algorithm5.9.5.m1.4.4.2.2.2.2.2.3" xref="algorithm5.9.5.m1.4.4.2.2.2.2.2.3.cmml">v</mi></msub><mo id="algorithm5.9.5.m1.4.4.2.2.2.2.1" xref="algorithm5.9.5.m1.4.4.2.2.2.2.1.cmml">⁢</mo><mrow id="algorithm5.9.5.m1.4.4.2.2.2.2.3.2" xref="algorithm5.9.5.m1.4.4.2.2.2.2.cmml"><mo id="algorithm5.9.5.m1.4.4.2.2.2.2.3.2.1" stretchy="false" xref="algorithm5.9.5.m1.4.4.2.2.2.2.cmml">(</mo><mi id="algorithm5.9.5.m1.2.2" xref="algorithm5.9.5.m1.2.2.cmml">j</mi><mo id="algorithm5.9.5.m1.4.4.2.2.2.2.3.2.2" stretchy="false" xref="algorithm5.9.5.m1.4.4.2.2.2.2.cmml">)</mo></mrow></mrow><mo id="algorithm5.9.5.m1.4.4.2.2.2.5" stretchy="false" xref="algorithm5.9.5.m1.4.4.2.2.3.cmml">}</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="algorithm5.9.5.m1.4b"><apply id="algorithm5.9.5.m1.4.4.cmml" xref="algorithm5.9.5.m1.4.4"><ci id="algorithm5.9.5.m1.4.4.3.cmml" xref="algorithm5.9.5.m1.4.4.3">←</ci><ci id="algorithm5.9.5.m1.4.4.4a.cmml" xref="algorithm5.9.5.m1.4.4.4"><mtext id="algorithm5.9.5.m1.4.4.4.cmml" xref="algorithm5.9.5.m1.4.4.4">SOL</mtext></ci><apply id="algorithm5.9.5.m1.4.4.2.cmml" xref="algorithm5.9.5.m1.4.4.2"><union id="algorithm5.9.5.m1.4.4.2.3.cmml" xref="algorithm5.9.5.m1.4.4.2.3"></union><ci id="algorithm5.9.5.m1.4.4.2.4a.cmml" xref="algorithm5.9.5.m1.4.4.2.4"><mtext id="algorithm5.9.5.m1.4.4.2.4.cmml" xref="algorithm5.9.5.m1.4.4.2.4">SOL</mtext></ci><set id="algorithm5.9.5.m1.4.4.2.2.3.cmml" xref="algorithm5.9.5.m1.4.4.2.2.2"><apply id="algorithm5.9.5.m1.3.3.1.1.1.1.cmml" xref="algorithm5.9.5.m1.3.3.1.1.1.1"><times id="algorithm5.9.5.m1.3.3.1.1.1.1.1.cmml" xref="algorithm5.9.5.m1.3.3.1.1.1.1.1"></times><apply id="algorithm5.9.5.m1.3.3.1.1.1.1.2.cmml" xref="algorithm5.9.5.m1.3.3.1.1.1.1.2"><csymbol cd="ambiguous" id="algorithm5.9.5.m1.3.3.1.1.1.1.2.1.cmml" xref="algorithm5.9.5.m1.3.3.1.1.1.1.2">subscript</csymbol><ci id="algorithm5.9.5.m1.3.3.1.1.1.1.2.2.cmml" xref="algorithm5.9.5.m1.3.3.1.1.1.1.2.2">𝐿</ci><ci id="algorithm5.9.5.m1.3.3.1.1.1.1.2.3.cmml" xref="algorithm5.9.5.m1.3.3.1.1.1.1.2.3">𝑢</ci></apply><ci id="algorithm5.9.5.m1.1.1.cmml" xref="algorithm5.9.5.m1.1.1">𝑗</ci></apply><apply id="algorithm5.9.5.m1.4.4.2.2.2.2.cmml" xref="algorithm5.9.5.m1.4.4.2.2.2.2"><times id="algorithm5.9.5.m1.4.4.2.2.2.2.1.cmml" xref="algorithm5.9.5.m1.4.4.2.2.2.2.1"></times><apply id="algorithm5.9.5.m1.4.4.2.2.2.2.2.cmml" xref="algorithm5.9.5.m1.4.4.2.2.2.2.2"><csymbol cd="ambiguous" id="algorithm5.9.5.m1.4.4.2.2.2.2.2.1.cmml" xref="algorithm5.9.5.m1.4.4.2.2.2.2.2">subscript</csymbol><ci id="algorithm5.9.5.m1.4.4.2.2.2.2.2.2.cmml" xref="algorithm5.9.5.m1.4.4.2.2.2.2.2.2">𝐿</ci><ci id="algorithm5.9.5.m1.4.4.2.2.2.2.2.3.cmml" xref="algorithm5.9.5.m1.4.4.2.2.2.2.2.3">𝑣</ci></apply><ci id="algorithm5.9.5.m1.2.2.cmml" xref="algorithm5.9.5.m1.2.2">𝑗</ci></apply></set></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="algorithm5.9.5.m1.4c">\textnormal{SOL}\leftarrow\textnormal{SOL}\cup\{L_{u}(j),L_{v}(j)\}</annotation><annotation encoding="application/x-llamapun" id="algorithm5.9.5.m1.4d">SOL ← SOL ∪ { italic_L start_POSTSUBSCRIPT italic_u end_POSTSUBSCRIPT ( italic_j ) , italic_L start_POSTSUBSCRIPT italic_v end_POSTSUBSCRIPT ( italic_j ) }</annotation></semantics></math> </div> <div class="ltx_listingline" id="algorithm5.10.6"> <span class="ltx_text ltx_font_bold" id="algorithm5.10.6.2">for</span> <em class="ltx_emph ltx_font_italic" id="algorithm5.10.6.1"><math alttext="u\in V" class="ltx_Math" display="inline" id="algorithm5.10.6.1.m1.1"><semantics id="algorithm5.10.6.1.m1.1a"><mrow id="algorithm5.10.6.1.m1.1.1" xref="algorithm5.10.6.1.m1.1.1.cmml"><mi id="algorithm5.10.6.1.m1.1.1.2" xref="algorithm5.10.6.1.m1.1.1.2.cmml">u</mi><mo id="algorithm5.10.6.1.m1.1.1.1" xref="algorithm5.10.6.1.m1.1.1.1.cmml">∈</mo><mi id="algorithm5.10.6.1.m1.1.1.3" xref="algorithm5.10.6.1.m1.1.1.3.cmml">V</mi></mrow><annotation-xml encoding="MathML-Content" id="algorithm5.10.6.1.m1.1b"><apply id="algorithm5.10.6.1.m1.1.1.cmml" xref="algorithm5.10.6.1.m1.1.1"><in id="algorithm5.10.6.1.m1.1.1.1.cmml" xref="algorithm5.10.6.1.m1.1.1.1"></in><ci id="algorithm5.10.6.1.m1.1.1.2.cmml" xref="algorithm5.10.6.1.m1.1.1.2">𝑢</ci><ci id="algorithm5.10.6.1.m1.1.1.3.cmml" xref="algorithm5.10.6.1.m1.1.1.3">𝑉</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="algorithm5.10.6.1.m1.1c">u\in V</annotation><annotation encoding="application/x-llamapun" id="algorithm5.10.6.1.m1.1d">italic_u ∈ italic_V</annotation></semantics></math></em> <span class="ltx_text ltx_font_bold" id="algorithm5.10.6.3">do</span> </div> <div class="ltx_listingline" id="algorithm5.11.7"> <span class="ltx_rule" style="width:1px;height:100%;background:black;display:inline-block;"> </span> <span class="ltx_text ltx_font_bold" id="algorithm5.11.7.2">for</span> <em class="ltx_emph ltx_font_italic" id="algorithm5.11.7.1"><math alttext="v\in C(u)" class="ltx_Math" display="inline" id="algorithm5.11.7.1.m1.1"><semantics id="algorithm5.11.7.1.m1.1a"><mrow id="algorithm5.11.7.1.m1.1.2" xref="algorithm5.11.7.1.m1.1.2.cmml"><mi id="algorithm5.11.7.1.m1.1.2.2" xref="algorithm5.11.7.1.m1.1.2.2.cmml">v</mi><mo id="algorithm5.11.7.1.m1.1.2.1" xref="algorithm5.11.7.1.m1.1.2.1.cmml">∈</mo><mrow id="algorithm5.11.7.1.m1.1.2.3" xref="algorithm5.11.7.1.m1.1.2.3.cmml"><mi id="algorithm5.11.7.1.m1.1.2.3.2" xref="algorithm5.11.7.1.m1.1.2.3.2.cmml">C</mi><mo id="algorithm5.11.7.1.m1.1.2.3.1" xref="algorithm5.11.7.1.m1.1.2.3.1.cmml">⁢</mo><mrow id="algorithm5.11.7.1.m1.1.2.3.3.2" xref="algorithm5.11.7.1.m1.1.2.3.cmml"><mo id="algorithm5.11.7.1.m1.1.2.3.3.2.1" stretchy="false" xref="algorithm5.11.7.1.m1.1.2.3.cmml">(</mo><mi id="algorithm5.11.7.1.m1.1.1" xref="algorithm5.11.7.1.m1.1.1.cmml">u</mi><mo id="algorithm5.11.7.1.m1.1.2.3.3.2.2" stretchy="false" xref="algorithm5.11.7.1.m1.1.2.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="algorithm5.11.7.1.m1.1b"><apply id="algorithm5.11.7.1.m1.1.2.cmml" xref="algorithm5.11.7.1.m1.1.2"><in id="algorithm5.11.7.1.m1.1.2.1.cmml" xref="algorithm5.11.7.1.m1.1.2.1"></in><ci id="algorithm5.11.7.1.m1.1.2.2.cmml" xref="algorithm5.11.7.1.m1.1.2.2">𝑣</ci><apply id="algorithm5.11.7.1.m1.1.2.3.cmml" xref="algorithm5.11.7.1.m1.1.2.3"><times id="algorithm5.11.7.1.m1.1.2.3.1.cmml" xref="algorithm5.11.7.1.m1.1.2.3.1"></times><ci id="algorithm5.11.7.1.m1.1.2.3.2.cmml" xref="algorithm5.11.7.1.m1.1.2.3.2">𝐶</ci><ci id="algorithm5.11.7.1.m1.1.1.cmml" xref="algorithm5.11.7.1.m1.1.1">𝑢</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="algorithm5.11.7.1.m1.1c">v\in C(u)</annotation><annotation encoding="application/x-llamapun" id="algorithm5.11.7.1.m1.1d">italic_v ∈ italic_C ( italic_u )</annotation></semantics></math></em> <span class="ltx_text ltx_font_bold" id="algorithm5.11.7.3">do</span> </div> <div class="ltx_listingline" id="algorithm5.12.8"> <span class="ltx_rule" style="width:1px;height:100%;background:black;display:inline-block;"> </span> <span class="ltx_rule" style="width:1px;height:100%;background:black;display:inline-block;"> </span> <span class="ltx_text ltx_font_bold" id="algorithm5.12.8.2">if</span> <em class="ltx_emph ltx_font_italic" id="algorithm5.12.8.1"><math alttext="\exists e\in\textnormal{OPT}[G_{v},G\setminus G_{u}]" class="ltx_Math" display="inline" id="algorithm5.12.8.1.m1.2"><semantics id="algorithm5.12.8.1.m1.2a"><mrow id="algorithm5.12.8.1.m1.2.2" xref="algorithm5.12.8.1.m1.2.2.cmml"><mrow id="algorithm5.12.8.1.m1.2.2.4" xref="algorithm5.12.8.1.m1.2.2.4.cmml"><mo id="algorithm5.12.8.1.m1.2.2.4.1" rspace="0.167em" xref="algorithm5.12.8.1.m1.2.2.4.1.cmml">∃</mo><mi id="algorithm5.12.8.1.m1.2.2.4.2" xref="algorithm5.12.8.1.m1.2.2.4.2.cmml">e</mi></mrow><mo id="algorithm5.12.8.1.m1.2.2.3" xref="algorithm5.12.8.1.m1.2.2.3.cmml">∈</mo><mrow id="algorithm5.12.8.1.m1.2.2.2" xref="algorithm5.12.8.1.m1.2.2.2.cmml"><mtext id="algorithm5.12.8.1.m1.2.2.2.4" xref="algorithm5.12.8.1.m1.2.2.2.4b.cmml"><em class="ltx_emph ltx_font_upright" id="algorithm5.12.8.1.m1.2.2.2.4.1nest">OPT</em></mtext><mo id="algorithm5.12.8.1.m1.2.2.2.3" xref="algorithm5.12.8.1.m1.2.2.2.3.cmml">⁢</mo><mrow id="algorithm5.12.8.1.m1.2.2.2.2.2" xref="algorithm5.12.8.1.m1.2.2.2.2.3.cmml"><mo id="algorithm5.12.8.1.m1.2.2.2.2.2.3" stretchy="false" xref="algorithm5.12.8.1.m1.2.2.2.2.3.cmml">[</mo><msub id="algorithm5.12.8.1.m1.1.1.1.1.1.1" xref="algorithm5.12.8.1.m1.1.1.1.1.1.1.cmml"><mi id="algorithm5.12.8.1.m1.1.1.1.1.1.1.2" xref="algorithm5.12.8.1.m1.1.1.1.1.1.1.2.cmml">G</mi><mi id="algorithm5.12.8.1.m1.1.1.1.1.1.1.3" xref="algorithm5.12.8.1.m1.1.1.1.1.1.1.3.cmml">v</mi></msub><mo id="algorithm5.12.8.1.m1.2.2.2.2.2.4" xref="algorithm5.12.8.1.m1.2.2.2.2.3.cmml">,</mo><mrow id="algorithm5.12.8.1.m1.2.2.2.2.2.2" xref="algorithm5.12.8.1.m1.2.2.2.2.2.2.cmml"><mi id="algorithm5.12.8.1.m1.2.2.2.2.2.2.2" xref="algorithm5.12.8.1.m1.2.2.2.2.2.2.2.cmml">G</mi><mo id="algorithm5.12.8.1.m1.2.2.2.2.2.2.1" xref="algorithm5.12.8.1.m1.2.2.2.2.2.2.1.cmml">∖</mo><msub id="algorithm5.12.8.1.m1.2.2.2.2.2.2.3" xref="algorithm5.12.8.1.m1.2.2.2.2.2.2.3.cmml"><mi id="algorithm5.12.8.1.m1.2.2.2.2.2.2.3.2" xref="algorithm5.12.8.1.m1.2.2.2.2.2.2.3.2.cmml">G</mi><mi id="algorithm5.12.8.1.m1.2.2.2.2.2.2.3.3" xref="algorithm5.12.8.1.m1.2.2.2.2.2.2.3.3.cmml">u</mi></msub></mrow><mo id="algorithm5.12.8.1.m1.2.2.2.2.2.5" stretchy="false" xref="algorithm5.12.8.1.m1.2.2.2.2.3.cmml">]</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="algorithm5.12.8.1.m1.2b"><apply id="algorithm5.12.8.1.m1.2.2.cmml" xref="algorithm5.12.8.1.m1.2.2"><in id="algorithm5.12.8.1.m1.2.2.3.cmml" xref="algorithm5.12.8.1.m1.2.2.3"></in><apply id="algorithm5.12.8.1.m1.2.2.4.cmml" xref="algorithm5.12.8.1.m1.2.2.4"><exists id="algorithm5.12.8.1.m1.2.2.4.1.cmml" xref="algorithm5.12.8.1.m1.2.2.4.1"></exists><ci id="algorithm5.12.8.1.m1.2.2.4.2.cmml" xref="algorithm5.12.8.1.m1.2.2.4.2">𝑒</ci></apply><apply id="algorithm5.12.8.1.m1.2.2.2.cmml" xref="algorithm5.12.8.1.m1.2.2.2"><times id="algorithm5.12.8.1.m1.2.2.2.3.cmml" xref="algorithm5.12.8.1.m1.2.2.2.3"></times><ci id="algorithm5.12.8.1.m1.2.2.2.4b.cmml" xref="algorithm5.12.8.1.m1.2.2.2.4"><mtext id="algorithm5.12.8.1.m1.2.2.2.4.cmml" xref="algorithm5.12.8.1.m1.2.2.2.4"><em class="ltx_emph ltx_font_upright" id="algorithm5.12.8.1.m1.2.2.2.4.1anest">OPT</em></mtext></ci><interval closure="closed" id="algorithm5.12.8.1.m1.2.2.2.2.3.cmml" xref="algorithm5.12.8.1.m1.2.2.2.2.2"><apply id="algorithm5.12.8.1.m1.1.1.1.1.1.1.cmml" xref="algorithm5.12.8.1.m1.1.1.1.1.1.1"><csymbol cd="ambiguous" id="algorithm5.12.8.1.m1.1.1.1.1.1.1.1.cmml" xref="algorithm5.12.8.1.m1.1.1.1.1.1.1">subscript</csymbol><ci id="algorithm5.12.8.1.m1.1.1.1.1.1.1.2.cmml" xref="algorithm5.12.8.1.m1.1.1.1.1.1.1.2">𝐺</ci><ci id="algorithm5.12.8.1.m1.1.1.1.1.1.1.3.cmml" xref="algorithm5.12.8.1.m1.1.1.1.1.1.1.3">𝑣</ci></apply><apply id="algorithm5.12.8.1.m1.2.2.2.2.2.2.cmml" xref="algorithm5.12.8.1.m1.2.2.2.2.2.2"><setdiff id="algorithm5.12.8.1.m1.2.2.2.2.2.2.1.cmml" xref="algorithm5.12.8.1.m1.2.2.2.2.2.2.1"></setdiff><ci id="algorithm5.12.8.1.m1.2.2.2.2.2.2.2.cmml" xref="algorithm5.12.8.1.m1.2.2.2.2.2.2.2">𝐺</ci><apply id="algorithm5.12.8.1.m1.2.2.2.2.2.2.3.cmml" xref="algorithm5.12.8.1.m1.2.2.2.2.2.2.3"><csymbol cd="ambiguous" id="algorithm5.12.8.1.m1.2.2.2.2.2.2.3.1.cmml" xref="algorithm5.12.8.1.m1.2.2.2.2.2.2.3">subscript</csymbol><ci id="algorithm5.12.8.1.m1.2.2.2.2.2.2.3.2.cmml" xref="algorithm5.12.8.1.m1.2.2.2.2.2.2.3.2">𝐺</ci><ci id="algorithm5.12.8.1.m1.2.2.2.2.2.2.3.3.cmml" xref="algorithm5.12.8.1.m1.2.2.2.2.2.2.3.3">𝑢</ci></apply></apply></interval></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="algorithm5.12.8.1.m1.2c">\exists e\in\textnormal{OPT}[G_{v},G\setminus G_{u}]</annotation><annotation encoding="application/x-llamapun" id="algorithm5.12.8.1.m1.2d">∃ italic_e ∈ OPT [ italic_G start_POSTSUBSCRIPT italic_v end_POSTSUBSCRIPT , italic_G ∖ italic_G start_POSTSUBSCRIPT italic_u end_POSTSUBSCRIPT ]</annotation></semantics></math></em> <span class="ltx_text ltx_font_bold" id="algorithm5.12.8.3">then</span> </div> <div class="ltx_listingline" id="algorithm5.13.9"> <span class="ltx_rule" style="width:1px;height:100%;background:black;display:inline-block;"> </span> <span class="ltx_rule" style="width:1px;height:100%;background:black;display:inline-block;"> </span> <span class="ltx_rule" style="width:1px;height:100%;background:black;display:inline-block;"> </span> Mark <math alttext="v" class="ltx_Math" display="inline" id="algorithm5.13.9.m1.1"><semantics id="algorithm5.13.9.m1.1a"><mi id="algorithm5.13.9.m1.1.1" xref="algorithm5.13.9.m1.1.1.cmml">v</mi><annotation-xml encoding="MathML-Content" id="algorithm5.13.9.m1.1b"><ci id="algorithm5.13.9.m1.1.1.cmml" xref="algorithm5.13.9.m1.1.1">𝑣</ci></annotation-xml><annotation encoding="application/x-tex" id="algorithm5.13.9.m1.1c">v</annotation><annotation encoding="application/x-llamapun" id="algorithm5.13.9.m1.1d">italic_v</annotation></semantics></math> as “good” </div> <div class="ltx_listingline" id="algorithm5.16.12"> <span class="ltx_rule" style="width:1px;height:100%;background:black;display:inline-block;"> </span> Construct <math alttext="C^{\prime\prime}(u)" class="ltx_Math" display="inline" id="algorithm5.14.10.m1.1"><semantics id="algorithm5.14.10.m1.1a"><mrow id="algorithm5.14.10.m1.1.2" xref="algorithm5.14.10.m1.1.2.cmml"><msup id="algorithm5.14.10.m1.1.2.2" xref="algorithm5.14.10.m1.1.2.2.cmml"><mi id="algorithm5.14.10.m1.1.2.2.2" xref="algorithm5.14.10.m1.1.2.2.2.cmml">C</mi><mo id="algorithm5.14.10.m1.1.2.2.3" xref="algorithm5.14.10.m1.1.2.2.3.cmml">′′</mo></msup><mo id="algorithm5.14.10.m1.1.2.1" xref="algorithm5.14.10.m1.1.2.1.cmml">⁢</mo><mrow id="algorithm5.14.10.m1.1.2.3.2" xref="algorithm5.14.10.m1.1.2.cmml"><mo id="algorithm5.14.10.m1.1.2.3.2.1" stretchy="false" xref="algorithm5.14.10.m1.1.2.cmml">(</mo><mi id="algorithm5.14.10.m1.1.1" xref="algorithm5.14.10.m1.1.1.cmml">u</mi><mo id="algorithm5.14.10.m1.1.2.3.2.2" stretchy="false" xref="algorithm5.14.10.m1.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="algorithm5.14.10.m1.1b"><apply id="algorithm5.14.10.m1.1.2.cmml" xref="algorithm5.14.10.m1.1.2"><times id="algorithm5.14.10.m1.1.2.1.cmml" xref="algorithm5.14.10.m1.1.2.1"></times><apply id="algorithm5.14.10.m1.1.2.2.cmml" xref="algorithm5.14.10.m1.1.2.2"><csymbol cd="ambiguous" id="algorithm5.14.10.m1.1.2.2.1.cmml" xref="algorithm5.14.10.m1.1.2.2">superscript</csymbol><ci id="algorithm5.14.10.m1.1.2.2.2.cmml" xref="algorithm5.14.10.m1.1.2.2.2">𝐶</ci><ci id="algorithm5.14.10.m1.1.2.2.3.cmml" xref="algorithm5.14.10.m1.1.2.2.3">′′</ci></apply><ci id="algorithm5.14.10.m1.1.1.cmml" xref="algorithm5.14.10.m1.1.1">𝑢</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="algorithm5.14.10.m1.1c">C^{\prime\prime}(u)</annotation><annotation encoding="application/x-llamapun" id="algorithm5.14.10.m1.1d">italic_C start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT ( italic_u )</annotation></semantics></math> from <math alttext="C^{\prime}(u)" class="ltx_Math" display="inline" id="algorithm5.15.11.m2.1"><semantics id="algorithm5.15.11.m2.1a"><mrow id="algorithm5.15.11.m2.1.2" xref="algorithm5.15.11.m2.1.2.cmml"><msup id="algorithm5.15.11.m2.1.2.2" xref="algorithm5.15.11.m2.1.2.2.cmml"><mi id="algorithm5.15.11.m2.1.2.2.2" xref="algorithm5.15.11.m2.1.2.2.2.cmml">C</mi><mo id="algorithm5.15.11.m2.1.2.2.3" xref="algorithm5.15.11.m2.1.2.2.3.cmml">′</mo></msup><mo id="algorithm5.15.11.m2.1.2.1" xref="algorithm5.15.11.m2.1.2.1.cmml">⁢</mo><mrow id="algorithm5.15.11.m2.1.2.3.2" xref="algorithm5.15.11.m2.1.2.cmml"><mo id="algorithm5.15.11.m2.1.2.3.2.1" stretchy="false" xref="algorithm5.15.11.m2.1.2.cmml">(</mo><mi id="algorithm5.15.11.m2.1.1" xref="algorithm5.15.11.m2.1.1.cmml">u</mi><mo id="algorithm5.15.11.m2.1.2.3.2.2" stretchy="false" xref="algorithm5.15.11.m2.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="algorithm5.15.11.m2.1b"><apply id="algorithm5.15.11.m2.1.2.cmml" xref="algorithm5.15.11.m2.1.2"><times id="algorithm5.15.11.m2.1.2.1.cmml" xref="algorithm5.15.11.m2.1.2.1"></times><apply id="algorithm5.15.11.m2.1.2.2.cmml" xref="algorithm5.15.11.m2.1.2.2"><csymbol cd="ambiguous" id="algorithm5.15.11.m2.1.2.2.1.cmml" xref="algorithm5.15.11.m2.1.2.2">superscript</csymbol><ci id="algorithm5.15.11.m2.1.2.2.2.cmml" xref="algorithm5.15.11.m2.1.2.2.2">𝐶</ci><ci id="algorithm5.15.11.m2.1.2.2.3.cmml" xref="algorithm5.15.11.m2.1.2.2.3">′</ci></apply><ci id="algorithm5.15.11.m2.1.1.cmml" xref="algorithm5.15.11.m2.1.1">𝑢</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="algorithm5.15.11.m2.1c">C^{\prime}(u)</annotation><annotation encoding="application/x-llamapun" id="algorithm5.15.11.m2.1d">italic_C start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ( italic_u )</annotation></semantics></math> (as defined in Algorithm <a class="ltx_ref" href="https://arxiv.org/html/2503.00712v1#algorithm4" title="In 4.1.1 The Streaming Algorithm ‣ 4.1 One-to-Two Augmentation ‣ 4 Vertex Connectivity Augmentation in Link-Arrival Model ‣ Streaming Algorithms for Network Design"><span class="ltx_text ltx_ref_tag">4</span></a>) by contracting all supernodes corresponding to a “good” vertex <math alttext="v\in C(u)" class="ltx_Math" display="inline" id="algorithm5.16.12.m3.1"><semantics id="algorithm5.16.12.m3.1a"><mrow id="algorithm5.16.12.m3.1.2" xref="algorithm5.16.12.m3.1.2.cmml"><mi id="algorithm5.16.12.m3.1.2.2" xref="algorithm5.16.12.m3.1.2.2.cmml">v</mi><mo id="algorithm5.16.12.m3.1.2.1" xref="algorithm5.16.12.m3.1.2.1.cmml">∈</mo><mrow id="algorithm5.16.12.m3.1.2.3" xref="algorithm5.16.12.m3.1.2.3.cmml"><mi id="algorithm5.16.12.m3.1.2.3.2" xref="algorithm5.16.12.m3.1.2.3.2.cmml">C</mi><mo id="algorithm5.16.12.m3.1.2.3.1" xref="algorithm5.16.12.m3.1.2.3.1.cmml">⁢</mo><mrow id="algorithm5.16.12.m3.1.2.3.3.2" xref="algorithm5.16.12.m3.1.2.3.cmml"><mo id="algorithm5.16.12.m3.1.2.3.3.2.1" stretchy="false" xref="algorithm5.16.12.m3.1.2.3.cmml">(</mo><mi id="algorithm5.16.12.m3.1.1" xref="algorithm5.16.12.m3.1.1.cmml">u</mi><mo id="algorithm5.16.12.m3.1.2.3.3.2.2" stretchy="false" xref="algorithm5.16.12.m3.1.2.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="algorithm5.16.12.m3.1b"><apply id="algorithm5.16.12.m3.1.2.cmml" xref="algorithm5.16.12.m3.1.2"><in id="algorithm5.16.12.m3.1.2.1.cmml" xref="algorithm5.16.12.m3.1.2.1"></in><ci id="algorithm5.16.12.m3.1.2.2.cmml" xref="algorithm5.16.12.m3.1.2.2">𝑣</ci><apply id="algorithm5.16.12.m3.1.2.3.cmml" xref="algorithm5.16.12.m3.1.2.3"><times id="algorithm5.16.12.m3.1.2.3.1.cmml" xref="algorithm5.16.12.m3.1.2.3.1"></times><ci id="algorithm5.16.12.m3.1.2.3.2.cmml" xref="algorithm5.16.12.m3.1.2.3.2">𝐶</ci><ci id="algorithm5.16.12.m3.1.1.cmml" xref="algorithm5.16.12.m3.1.1">𝑢</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="algorithm5.16.12.m3.1c">v\in C(u)</annotation><annotation encoding="application/x-llamapun" id="algorithm5.16.12.m3.1d">italic_v ∈ italic_C ( italic_u )</annotation></semantics></math> into one “good” supernode </div> <div class="ltx_listingline" id="algorithm5.18.14"> <span class="ltx_rule" style="width:1px;height:100%;background:black;display:inline-block;"> </span> <math alttext="T^{\prime\prime}_{u}\subseteq T^{\prime}_{u}\leftarrow" class="ltx_Math" display="inline" id="algorithm5.17.13.m1.1"><semantics id="algorithm5.17.13.m1.1a"><mrow id="algorithm5.17.13.m1.1.1" xref="algorithm5.17.13.m1.1.1.cmml"><msubsup id="algorithm5.17.13.m1.1.1.2" xref="algorithm5.17.13.m1.1.1.2.cmml"><mi id="algorithm5.17.13.m1.1.1.2.2.2" xref="algorithm5.17.13.m1.1.1.2.2.2.cmml">T</mi><mi id="algorithm5.17.13.m1.1.1.2.3" xref="algorithm5.17.13.m1.1.1.2.3.cmml">u</mi><mo id="algorithm5.17.13.m1.1.1.2.2.3" xref="algorithm5.17.13.m1.1.1.2.2.3.cmml">′′</mo></msubsup><mo id="algorithm5.17.13.m1.1.1.3" xref="algorithm5.17.13.m1.1.1.3.cmml">⊆</mo><msubsup id="algorithm5.17.13.m1.1.1.4" xref="algorithm5.17.13.m1.1.1.4.cmml"><mi id="algorithm5.17.13.m1.1.1.4.2.2" xref="algorithm5.17.13.m1.1.1.4.2.2.cmml">T</mi><mi id="algorithm5.17.13.m1.1.1.4.3" xref="algorithm5.17.13.m1.1.1.4.3.cmml">u</mi><mo id="algorithm5.17.13.m1.1.1.4.2.3" xref="algorithm5.17.13.m1.1.1.4.2.3.cmml">′</mo></msubsup><mo id="algorithm5.17.13.m1.1.1.5" stretchy="false" xref="algorithm5.17.13.m1.1.1.5.cmml">←</mo><mi id="algorithm5.17.13.m1.1.1.6" xref="algorithm5.17.13.m1.1.1.6.cmml"></mi></mrow><annotation-xml encoding="MathML-Content" id="algorithm5.17.13.m1.1b"><apply id="algorithm5.17.13.m1.1.1.cmml" xref="algorithm5.17.13.m1.1.1"><and id="algorithm5.17.13.m1.1.1a.cmml" xref="algorithm5.17.13.m1.1.1"></and><apply id="algorithm5.17.13.m1.1.1b.cmml" xref="algorithm5.17.13.m1.1.1"><subset id="algorithm5.17.13.m1.1.1.3.cmml" xref="algorithm5.17.13.m1.1.1.3"></subset><apply id="algorithm5.17.13.m1.1.1.2.cmml" xref="algorithm5.17.13.m1.1.1.2"><csymbol cd="ambiguous" id="algorithm5.17.13.m1.1.1.2.1.cmml" xref="algorithm5.17.13.m1.1.1.2">subscript</csymbol><apply id="algorithm5.17.13.m1.1.1.2.2.cmml" xref="algorithm5.17.13.m1.1.1.2"><csymbol cd="ambiguous" id="algorithm5.17.13.m1.1.1.2.2.1.cmml" xref="algorithm5.17.13.m1.1.1.2">superscript</csymbol><ci id="algorithm5.17.13.m1.1.1.2.2.2.cmml" xref="algorithm5.17.13.m1.1.1.2.2.2">𝑇</ci><ci id="algorithm5.17.13.m1.1.1.2.2.3.cmml" xref="algorithm5.17.13.m1.1.1.2.2.3">′′</ci></apply><ci id="algorithm5.17.13.m1.1.1.2.3.cmml" xref="algorithm5.17.13.m1.1.1.2.3">𝑢</ci></apply><apply id="algorithm5.17.13.m1.1.1.4.cmml" xref="algorithm5.17.13.m1.1.1.4"><csymbol cd="ambiguous" id="algorithm5.17.13.m1.1.1.4.1.cmml" xref="algorithm5.17.13.m1.1.1.4">subscript</csymbol><apply id="algorithm5.17.13.m1.1.1.4.2.cmml" xref="algorithm5.17.13.m1.1.1.4"><csymbol cd="ambiguous" id="algorithm5.17.13.m1.1.1.4.2.1.cmml" xref="algorithm5.17.13.m1.1.1.4">superscript</csymbol><ci id="algorithm5.17.13.m1.1.1.4.2.2.cmml" xref="algorithm5.17.13.m1.1.1.4.2.2">𝑇</ci><ci id="algorithm5.17.13.m1.1.1.4.2.3.cmml" xref="algorithm5.17.13.m1.1.1.4.2.3">′</ci></apply><ci id="algorithm5.17.13.m1.1.1.4.3.cmml" xref="algorithm5.17.13.m1.1.1.4.3">𝑢</ci></apply></apply><apply id="algorithm5.17.13.m1.1.1c.cmml" xref="algorithm5.17.13.m1.1.1"><ci id="algorithm5.17.13.m1.1.1.5.cmml" xref="algorithm5.17.13.m1.1.1.5">←</ci><share href="https://arxiv.org/html/2503.00712v1#algorithm5.17.13.m1.1.1.4.cmml" id="algorithm5.17.13.m1.1.1d.cmml" xref="algorithm5.17.13.m1.1.1"></share><csymbol cd="latexml" id="algorithm5.17.13.m1.1.1.6.cmml" xref="algorithm5.17.13.m1.1.1.6">absent</csymbol></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="algorithm5.17.13.m1.1c">T^{\prime\prime}_{u}\subseteq T^{\prime}_{u}\leftarrow</annotation><annotation encoding="application/x-llamapun" id="algorithm5.17.13.m1.1d">italic_T start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_u end_POSTSUBSCRIPT ⊆ italic_T start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_u end_POSTSUBSCRIPT ←</annotation></semantics></math> MST on <math alttext="C^{\prime\prime}(u)" class="ltx_Math" display="inline" id="algorithm5.18.14.m2.1"><semantics id="algorithm5.18.14.m2.1a"><mrow id="algorithm5.18.14.m2.1.2" xref="algorithm5.18.14.m2.1.2.cmml"><msup id="algorithm5.18.14.m2.1.2.2" xref="algorithm5.18.14.m2.1.2.2.cmml"><mi id="algorithm5.18.14.m2.1.2.2.2" xref="algorithm5.18.14.m2.1.2.2.2.cmml">C</mi><mo id="algorithm5.18.14.m2.1.2.2.3" xref="algorithm5.18.14.m2.1.2.2.3.cmml">′′</mo></msup><mo id="algorithm5.18.14.m2.1.2.1" xref="algorithm5.18.14.m2.1.2.1.cmml">⁢</mo><mrow id="algorithm5.18.14.m2.1.2.3.2" xref="algorithm5.18.14.m2.1.2.cmml"><mo id="algorithm5.18.14.m2.1.2.3.2.1" stretchy="false" xref="algorithm5.18.14.m2.1.2.cmml">(</mo><mi id="algorithm5.18.14.m2.1.1" xref="algorithm5.18.14.m2.1.1.cmml">u</mi><mo id="algorithm5.18.14.m2.1.2.3.2.2" stretchy="false" xref="algorithm5.18.14.m2.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="algorithm5.18.14.m2.1b"><apply id="algorithm5.18.14.m2.1.2.cmml" xref="algorithm5.18.14.m2.1.2"><times id="algorithm5.18.14.m2.1.2.1.cmml" xref="algorithm5.18.14.m2.1.2.1"></times><apply id="algorithm5.18.14.m2.1.2.2.cmml" xref="algorithm5.18.14.m2.1.2.2"><csymbol cd="ambiguous" id="algorithm5.18.14.m2.1.2.2.1.cmml" xref="algorithm5.18.14.m2.1.2.2">superscript</csymbol><ci id="algorithm5.18.14.m2.1.2.2.2.cmml" xref="algorithm5.18.14.m2.1.2.2.2">𝐶</ci><ci id="algorithm5.18.14.m2.1.2.2.3.cmml" xref="algorithm5.18.14.m2.1.2.2.3">′′</ci></apply><ci id="algorithm5.18.14.m2.1.1.cmml" xref="algorithm5.18.14.m2.1.1">𝑢</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="algorithm5.18.14.m2.1c">C^{\prime\prime}(u)</annotation><annotation encoding="application/x-llamapun" id="algorithm5.18.14.m2.1d">italic_C start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT ( italic_u )</annotation></semantics></math> </div> <div class="ltx_listingline" id="algorithm5.19.15"> <span class="ltx_rule" style="width:1px;height:100%;background:black;display:inline-block;"> </span> <math alttext="\textnormal{SOL}\leftarrow\textnormal{SOL}\cup T^{\prime\prime}_{u}" class="ltx_Math" display="inline" id="algorithm5.19.15.m1.1"><semantics id="algorithm5.19.15.m1.1a"><mrow id="algorithm5.19.15.m1.1.1" xref="algorithm5.19.15.m1.1.1.cmml"><mtext id="algorithm5.19.15.m1.1.1.2" xref="algorithm5.19.15.m1.1.1.2a.cmml">SOL</mtext><mo id="algorithm5.19.15.m1.1.1.1" stretchy="false" xref="algorithm5.19.15.m1.1.1.1.cmml">←</mo><mrow id="algorithm5.19.15.m1.1.1.3" xref="algorithm5.19.15.m1.1.1.3.cmml"><mtext id="algorithm5.19.15.m1.1.1.3.2" xref="algorithm5.19.15.m1.1.1.3.2a.cmml">SOL</mtext><mo id="algorithm5.19.15.m1.1.1.3.1" xref="algorithm5.19.15.m1.1.1.3.1.cmml">∪</mo><msubsup id="algorithm5.19.15.m1.1.1.3.3" xref="algorithm5.19.15.m1.1.1.3.3.cmml"><mi id="algorithm5.19.15.m1.1.1.3.3.2.2" xref="algorithm5.19.15.m1.1.1.3.3.2.2.cmml">T</mi><mi id="algorithm5.19.15.m1.1.1.3.3.3" xref="algorithm5.19.15.m1.1.1.3.3.3.cmml">u</mi><mo id="algorithm5.19.15.m1.1.1.3.3.2.3" xref="algorithm5.19.15.m1.1.1.3.3.2.3.cmml">′′</mo></msubsup></mrow></mrow><annotation-xml encoding="MathML-Content" id="algorithm5.19.15.m1.1b"><apply id="algorithm5.19.15.m1.1.1.cmml" xref="algorithm5.19.15.m1.1.1"><ci id="algorithm5.19.15.m1.1.1.1.cmml" xref="algorithm5.19.15.m1.1.1.1">←</ci><ci id="algorithm5.19.15.m1.1.1.2a.cmml" xref="algorithm5.19.15.m1.1.1.2"><mtext id="algorithm5.19.15.m1.1.1.2.cmml" xref="algorithm5.19.15.m1.1.1.2">SOL</mtext></ci><apply id="algorithm5.19.15.m1.1.1.3.cmml" xref="algorithm5.19.15.m1.1.1.3"><union id="algorithm5.19.15.m1.1.1.3.1.cmml" xref="algorithm5.19.15.m1.1.1.3.1"></union><ci id="algorithm5.19.15.m1.1.1.3.2a.cmml" xref="algorithm5.19.15.m1.1.1.3.2"><mtext id="algorithm5.19.15.m1.1.1.3.2.cmml" xref="algorithm5.19.15.m1.1.1.3.2">SOL</mtext></ci><apply id="algorithm5.19.15.m1.1.1.3.3.cmml" xref="algorithm5.19.15.m1.1.1.3.3"><csymbol cd="ambiguous" id="algorithm5.19.15.m1.1.1.3.3.1.cmml" xref="algorithm5.19.15.m1.1.1.3.3">subscript</csymbol><apply id="algorithm5.19.15.m1.1.1.3.3.2.cmml" xref="algorithm5.19.15.m1.1.1.3.3"><csymbol cd="ambiguous" id="algorithm5.19.15.m1.1.1.3.3.2.1.cmml" xref="algorithm5.19.15.m1.1.1.3.3">superscript</csymbol><ci id="algorithm5.19.15.m1.1.1.3.3.2.2.cmml" xref="algorithm5.19.15.m1.1.1.3.3.2.2">𝑇</ci><ci id="algorithm5.19.15.m1.1.1.3.3.2.3.cmml" xref="algorithm5.19.15.m1.1.1.3.3.2.3">′′</ci></apply><ci id="algorithm5.19.15.m1.1.1.3.3.3.cmml" xref="algorithm5.19.15.m1.1.1.3.3.3">𝑢</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="algorithm5.19.15.m1.1c">\textnormal{SOL}\leftarrow\textnormal{SOL}\cup T^{\prime\prime}_{u}</annotation><annotation encoding="application/x-llamapun" id="algorithm5.19.15.m1.1d">SOL ← SOL ∪ italic_T start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_u end_POSTSUBSCRIPT</annotation></semantics></math> </div> <div class="ltx_listingline" id="algorithm5.20.16"> <span class="ltx_text ltx_font_bold" id="algorithm5.20.16.1">return</span> <span class="ltx_text ltx_markedasmath" id="algorithm5.20.16.2">SOL</span> </div> </div> <figcaption class="ltx_caption"><span class="ltx_tag ltx_tag_float"><span class="ltx_text ltx_font_bold" id="algorithm5.23.1.1">Algorithm 5</span> </span>Construction of a solution in <math alttext="F" class="ltx_Math" display="inline" id="algorithm5.3.m1.1"><semantics id="algorithm5.3.m1.1b"><mi id="algorithm5.3.m1.1.1" xref="algorithm5.3.m1.1.1.cmml">F</mi><annotation-xml encoding="MathML-Content" id="algorithm5.3.m1.1c"><ci id="algorithm5.3.m1.1.1.cmml" xref="algorithm5.3.m1.1.1">𝐹</ci></annotation-xml><annotation encoding="application/x-tex" id="algorithm5.3.m1.1d">F</annotation><annotation encoding="application/x-llamapun" id="algorithm5.3.m1.1e">italic_F</annotation></semantics></math> from <span class="ltx_text ltx_markedasmath" id="algorithm5.24.2">OPT</span>.</figcaption> </figure> <div class="ltx_theorem ltx_theorem_lemma" id="S4.Thmtheorem5"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="S4.Thmtheorem5.1.1.1">Lemma 4.5</span></span><span class="ltx_text ltx_font_bold" id="S4.Thmtheorem5.2.2">.</span> </h6> <div class="ltx_para" id="S4.Thmtheorem5.p1"> <p class="ltx_p" id="S4.Thmtheorem5.p1.1"><math alttext="(V,E\cup\textnormal{SOL})" class="ltx_Math" display="inline" id="S4.Thmtheorem5.p1.1.m1.2"><semantics id="S4.Thmtheorem5.p1.1.m1.2a"><mrow id="S4.Thmtheorem5.p1.1.m1.2.2.1" xref="S4.Thmtheorem5.p1.1.m1.2.2.2.cmml"><mo id="S4.Thmtheorem5.p1.1.m1.2.2.1.2" stretchy="false" xref="S4.Thmtheorem5.p1.1.m1.2.2.2.cmml">(</mo><mi id="S4.Thmtheorem5.p1.1.m1.1.1" xref="S4.Thmtheorem5.p1.1.m1.1.1.cmml">V</mi><mo id="S4.Thmtheorem5.p1.1.m1.2.2.1.3" xref="S4.Thmtheorem5.p1.1.m1.2.2.2.cmml">,</mo><mrow id="S4.Thmtheorem5.p1.1.m1.2.2.1.1" xref="S4.Thmtheorem5.p1.1.m1.2.2.1.1.cmml"><mi id="S4.Thmtheorem5.p1.1.m1.2.2.1.1.2" xref="S4.Thmtheorem5.p1.1.m1.2.2.1.1.2.cmml">E</mi><mo id="S4.Thmtheorem5.p1.1.m1.2.2.1.1.1" xref="S4.Thmtheorem5.p1.1.m1.2.2.1.1.1.cmml">∪</mo><mtext id="S4.Thmtheorem5.p1.1.m1.2.2.1.1.3" xref="S4.Thmtheorem5.p1.1.m1.2.2.1.1.3a.cmml">SOL</mtext></mrow><mo id="S4.Thmtheorem5.p1.1.m1.2.2.1.4" stretchy="false" xref="S4.Thmtheorem5.p1.1.m1.2.2.2.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem5.p1.1.m1.2b"><interval closure="open" id="S4.Thmtheorem5.p1.1.m1.2.2.2.cmml" xref="S4.Thmtheorem5.p1.1.m1.2.2.1"><ci id="S4.Thmtheorem5.p1.1.m1.1.1.cmml" xref="S4.Thmtheorem5.p1.1.m1.1.1">𝑉</ci><apply id="S4.Thmtheorem5.p1.1.m1.2.2.1.1.cmml" xref="S4.Thmtheorem5.p1.1.m1.2.2.1.1"><union id="S4.Thmtheorem5.p1.1.m1.2.2.1.1.1.cmml" xref="S4.Thmtheorem5.p1.1.m1.2.2.1.1.1"></union><ci id="S4.Thmtheorem5.p1.1.m1.2.2.1.1.2.cmml" xref="S4.Thmtheorem5.p1.1.m1.2.2.1.1.2">𝐸</ci><ci id="S4.Thmtheorem5.p1.1.m1.2.2.1.1.3a.cmml" xref="S4.Thmtheorem5.p1.1.m1.2.2.1.1.3"><mtext id="S4.Thmtheorem5.p1.1.m1.2.2.1.1.3.cmml" xref="S4.Thmtheorem5.p1.1.m1.2.2.1.1.3">SOL</mtext></ci></apply></interval></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem5.p1.1.m1.2c">(V,E\cup\textnormal{SOL})</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem5.p1.1.m1.2d">( italic_V , italic_E ∪ SOL )</annotation></semantics></math> is a 2-connected graph.</p> </div> </div> <div class="ltx_proof" id="S4.SS1.SSS2.3"> <h6 class="ltx_title ltx_runin ltx_font_italic ltx_title_proof">Proof.</h6> <div class="ltx_para" id="S4.SS1.SSS2.1.p1"> <p class="ltx_p" id="S4.SS1.SSS2.1.p1.10">We want to show that for all <math alttext="a\in V" class="ltx_Math" display="inline" id="S4.SS1.SSS2.1.p1.1.m1.1"><semantics id="S4.SS1.SSS2.1.p1.1.m1.1a"><mrow id="S4.SS1.SSS2.1.p1.1.m1.1.1" xref="S4.SS1.SSS2.1.p1.1.m1.1.1.cmml"><mi id="S4.SS1.SSS2.1.p1.1.m1.1.1.2" xref="S4.SS1.SSS2.1.p1.1.m1.1.1.2.cmml">a</mi><mo id="S4.SS1.SSS2.1.p1.1.m1.1.1.1" xref="S4.SS1.SSS2.1.p1.1.m1.1.1.1.cmml">∈</mo><mi id="S4.SS1.SSS2.1.p1.1.m1.1.1.3" xref="S4.SS1.SSS2.1.p1.1.m1.1.1.3.cmml">V</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.SSS2.1.p1.1.m1.1b"><apply id="S4.SS1.SSS2.1.p1.1.m1.1.1.cmml" xref="S4.SS1.SSS2.1.p1.1.m1.1.1"><in id="S4.SS1.SSS2.1.p1.1.m1.1.1.1.cmml" xref="S4.SS1.SSS2.1.p1.1.m1.1.1.1"></in><ci id="S4.SS1.SSS2.1.p1.1.m1.1.1.2.cmml" xref="S4.SS1.SSS2.1.p1.1.m1.1.1.2">𝑎</ci><ci id="S4.SS1.SSS2.1.p1.1.m1.1.1.3.cmml" xref="S4.SS1.SSS2.1.p1.1.m1.1.1.3">𝑉</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.SSS2.1.p1.1.m1.1c">a\in V</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.SSS2.1.p1.1.m1.1d">italic_a ∈ italic_V</annotation></semantics></math>, <math alttext="(V,E\cup\textnormal{SOL})\setminus\{a\}" class="ltx_Math" display="inline" id="S4.SS1.SSS2.1.p1.2.m2.3"><semantics id="S4.SS1.SSS2.1.p1.2.m2.3a"><mrow id="S4.SS1.SSS2.1.p1.2.m2.3.3" xref="S4.SS1.SSS2.1.p1.2.m2.3.3.cmml"><mrow id="S4.SS1.SSS2.1.p1.2.m2.3.3.1.1" xref="S4.SS1.SSS2.1.p1.2.m2.3.3.1.2.cmml"><mo id="S4.SS1.SSS2.1.p1.2.m2.3.3.1.1.2" stretchy="false" xref="S4.SS1.SSS2.1.p1.2.m2.3.3.1.2.cmml">(</mo><mi id="S4.SS1.SSS2.1.p1.2.m2.1.1" xref="S4.SS1.SSS2.1.p1.2.m2.1.1.cmml">V</mi><mo id="S4.SS1.SSS2.1.p1.2.m2.3.3.1.1.3" xref="S4.SS1.SSS2.1.p1.2.m2.3.3.1.2.cmml">,</mo><mrow id="S4.SS1.SSS2.1.p1.2.m2.3.3.1.1.1" xref="S4.SS1.SSS2.1.p1.2.m2.3.3.1.1.1.cmml"><mi id="S4.SS1.SSS2.1.p1.2.m2.3.3.1.1.1.2" xref="S4.SS1.SSS2.1.p1.2.m2.3.3.1.1.1.2.cmml">E</mi><mo id="S4.SS1.SSS2.1.p1.2.m2.3.3.1.1.1.1" xref="S4.SS1.SSS2.1.p1.2.m2.3.3.1.1.1.1.cmml">∪</mo><mtext id="S4.SS1.SSS2.1.p1.2.m2.3.3.1.1.1.3" xref="S4.SS1.SSS2.1.p1.2.m2.3.3.1.1.1.3a.cmml">SOL</mtext></mrow><mo id="S4.SS1.SSS2.1.p1.2.m2.3.3.1.1.4" stretchy="false" xref="S4.SS1.SSS2.1.p1.2.m2.3.3.1.2.cmml">)</mo></mrow><mo id="S4.SS1.SSS2.1.p1.2.m2.3.3.2" xref="S4.SS1.SSS2.1.p1.2.m2.3.3.2.cmml">∖</mo><mrow id="S4.SS1.SSS2.1.p1.2.m2.3.3.3.2" xref="S4.SS1.SSS2.1.p1.2.m2.3.3.3.1.cmml"><mo id="S4.SS1.SSS2.1.p1.2.m2.3.3.3.2.1" stretchy="false" xref="S4.SS1.SSS2.1.p1.2.m2.3.3.3.1.cmml">{</mo><mi id="S4.SS1.SSS2.1.p1.2.m2.2.2" xref="S4.SS1.SSS2.1.p1.2.m2.2.2.cmml">a</mi><mo id="S4.SS1.SSS2.1.p1.2.m2.3.3.3.2.2" stretchy="false" xref="S4.SS1.SSS2.1.p1.2.m2.3.3.3.1.cmml">}</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.SSS2.1.p1.2.m2.3b"><apply id="S4.SS1.SSS2.1.p1.2.m2.3.3.cmml" xref="S4.SS1.SSS2.1.p1.2.m2.3.3"><setdiff id="S4.SS1.SSS2.1.p1.2.m2.3.3.2.cmml" xref="S4.SS1.SSS2.1.p1.2.m2.3.3.2"></setdiff><interval closure="open" id="S4.SS1.SSS2.1.p1.2.m2.3.3.1.2.cmml" xref="S4.SS1.SSS2.1.p1.2.m2.3.3.1.1"><ci id="S4.SS1.SSS2.1.p1.2.m2.1.1.cmml" xref="S4.SS1.SSS2.1.p1.2.m2.1.1">𝑉</ci><apply id="S4.SS1.SSS2.1.p1.2.m2.3.3.1.1.1.cmml" xref="S4.SS1.SSS2.1.p1.2.m2.3.3.1.1.1"><union id="S4.SS1.SSS2.1.p1.2.m2.3.3.1.1.1.1.cmml" xref="S4.SS1.SSS2.1.p1.2.m2.3.3.1.1.1.1"></union><ci id="S4.SS1.SSS2.1.p1.2.m2.3.3.1.1.1.2.cmml" xref="S4.SS1.SSS2.1.p1.2.m2.3.3.1.1.1.2">𝐸</ci><ci id="S4.SS1.SSS2.1.p1.2.m2.3.3.1.1.1.3a.cmml" xref="S4.SS1.SSS2.1.p1.2.m2.3.3.1.1.1.3"><mtext id="S4.SS1.SSS2.1.p1.2.m2.3.3.1.1.1.3.cmml" xref="S4.SS1.SSS2.1.p1.2.m2.3.3.1.1.1.3">SOL</mtext></ci></apply></interval><set id="S4.SS1.SSS2.1.p1.2.m2.3.3.3.1.cmml" xref="S4.SS1.SSS2.1.p1.2.m2.3.3.3.2"><ci id="S4.SS1.SSS2.1.p1.2.m2.2.2.cmml" xref="S4.SS1.SSS2.1.p1.2.m2.2.2">𝑎</ci></set></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.SSS2.1.p1.2.m2.3c">(V,E\cup\textnormal{SOL})\setminus\{a\}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.SSS2.1.p1.2.m2.3d">( italic_V , italic_E ∪ SOL ) ∖ { italic_a }</annotation></semantics></math> is connected. Fix <math alttext="a\in V" class="ltx_Math" display="inline" id="S4.SS1.SSS2.1.p1.3.m3.1"><semantics id="S4.SS1.SSS2.1.p1.3.m3.1a"><mrow id="S4.SS1.SSS2.1.p1.3.m3.1.1" xref="S4.SS1.SSS2.1.p1.3.m3.1.1.cmml"><mi id="S4.SS1.SSS2.1.p1.3.m3.1.1.2" xref="S4.SS1.SSS2.1.p1.3.m3.1.1.2.cmml">a</mi><mo id="S4.SS1.SSS2.1.p1.3.m3.1.1.1" xref="S4.SS1.SSS2.1.p1.3.m3.1.1.1.cmml">∈</mo><mi id="S4.SS1.SSS2.1.p1.3.m3.1.1.3" xref="S4.SS1.SSS2.1.p1.3.m3.1.1.3.cmml">V</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.SSS2.1.p1.3.m3.1b"><apply id="S4.SS1.SSS2.1.p1.3.m3.1.1.cmml" xref="S4.SS1.SSS2.1.p1.3.m3.1.1"><in id="S4.SS1.SSS2.1.p1.3.m3.1.1.1.cmml" xref="S4.SS1.SSS2.1.p1.3.m3.1.1.1"></in><ci id="S4.SS1.SSS2.1.p1.3.m3.1.1.2.cmml" xref="S4.SS1.SSS2.1.p1.3.m3.1.1.2">𝑎</ci><ci id="S4.SS1.SSS2.1.p1.3.m3.1.1.3.cmml" xref="S4.SS1.SSS2.1.p1.3.m3.1.1.3">𝑉</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.SSS2.1.p1.3.m3.1c">a\in V</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.SSS2.1.p1.3.m3.1d">italic_a ∈ italic_V</annotation></semantics></math>. Since <span class="ltx_text ltx_markedasmath" id="S4.SS1.SSS2.1.p1.10.1">OPT</span> is feasible, it suffices to show that for all <math alttext="uv\in\textnormal{OPT}" class="ltx_Math" display="inline" id="S4.SS1.SSS2.1.p1.5.m5.1"><semantics id="S4.SS1.SSS2.1.p1.5.m5.1a"><mrow id="S4.SS1.SSS2.1.p1.5.m5.1.1" xref="S4.SS1.SSS2.1.p1.5.m5.1.1.cmml"><mrow id="S4.SS1.SSS2.1.p1.5.m5.1.1.2" xref="S4.SS1.SSS2.1.p1.5.m5.1.1.2.cmml"><mi id="S4.SS1.SSS2.1.p1.5.m5.1.1.2.2" xref="S4.SS1.SSS2.1.p1.5.m5.1.1.2.2.cmml">u</mi><mo id="S4.SS1.SSS2.1.p1.5.m5.1.1.2.1" xref="S4.SS1.SSS2.1.p1.5.m5.1.1.2.1.cmml">⁢</mo><mi id="S4.SS1.SSS2.1.p1.5.m5.1.1.2.3" xref="S4.SS1.SSS2.1.p1.5.m5.1.1.2.3.cmml">v</mi></mrow><mo id="S4.SS1.SSS2.1.p1.5.m5.1.1.1" xref="S4.SS1.SSS2.1.p1.5.m5.1.1.1.cmml">∈</mo><mtext id="S4.SS1.SSS2.1.p1.5.m5.1.1.3" xref="S4.SS1.SSS2.1.p1.5.m5.1.1.3a.cmml">OPT</mtext></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.SSS2.1.p1.5.m5.1b"><apply id="S4.SS1.SSS2.1.p1.5.m5.1.1.cmml" xref="S4.SS1.SSS2.1.p1.5.m5.1.1"><in id="S4.SS1.SSS2.1.p1.5.m5.1.1.1.cmml" xref="S4.SS1.SSS2.1.p1.5.m5.1.1.1"></in><apply id="S4.SS1.SSS2.1.p1.5.m5.1.1.2.cmml" xref="S4.SS1.SSS2.1.p1.5.m5.1.1.2"><times id="S4.SS1.SSS2.1.p1.5.m5.1.1.2.1.cmml" xref="S4.SS1.SSS2.1.p1.5.m5.1.1.2.1"></times><ci id="S4.SS1.SSS2.1.p1.5.m5.1.1.2.2.cmml" xref="S4.SS1.SSS2.1.p1.5.m5.1.1.2.2">𝑢</ci><ci id="S4.SS1.SSS2.1.p1.5.m5.1.1.2.3.cmml" xref="S4.SS1.SSS2.1.p1.5.m5.1.1.2.3">𝑣</ci></apply><ci id="S4.SS1.SSS2.1.p1.5.m5.1.1.3a.cmml" xref="S4.SS1.SSS2.1.p1.5.m5.1.1.3"><mtext id="S4.SS1.SSS2.1.p1.5.m5.1.1.3.cmml" xref="S4.SS1.SSS2.1.p1.5.m5.1.1.3">OPT</mtext></ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.SSS2.1.p1.5.m5.1c">uv\in\textnormal{OPT}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.SSS2.1.p1.5.m5.1d">italic_u italic_v ∈ OPT</annotation></semantics></math> with <math alttext="u,v\neq a" class="ltx_Math" display="inline" id="S4.SS1.SSS2.1.p1.6.m6.2"><semantics id="S4.SS1.SSS2.1.p1.6.m6.2a"><mrow id="S4.SS1.SSS2.1.p1.6.m6.2.3" xref="S4.SS1.SSS2.1.p1.6.m6.2.3.cmml"><mrow id="S4.SS1.SSS2.1.p1.6.m6.2.3.2.2" xref="S4.SS1.SSS2.1.p1.6.m6.2.3.2.1.cmml"><mi id="S4.SS1.SSS2.1.p1.6.m6.1.1" xref="S4.SS1.SSS2.1.p1.6.m6.1.1.cmml">u</mi><mo id="S4.SS1.SSS2.1.p1.6.m6.2.3.2.2.1" xref="S4.SS1.SSS2.1.p1.6.m6.2.3.2.1.cmml">,</mo><mi id="S4.SS1.SSS2.1.p1.6.m6.2.2" xref="S4.SS1.SSS2.1.p1.6.m6.2.2.cmml">v</mi></mrow><mo id="S4.SS1.SSS2.1.p1.6.m6.2.3.1" xref="S4.SS1.SSS2.1.p1.6.m6.2.3.1.cmml">≠</mo><mi id="S4.SS1.SSS2.1.p1.6.m6.2.3.3" xref="S4.SS1.SSS2.1.p1.6.m6.2.3.3.cmml">a</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.SSS2.1.p1.6.m6.2b"><apply id="S4.SS1.SSS2.1.p1.6.m6.2.3.cmml" xref="S4.SS1.SSS2.1.p1.6.m6.2.3"><neq id="S4.SS1.SSS2.1.p1.6.m6.2.3.1.cmml" xref="S4.SS1.SSS2.1.p1.6.m6.2.3.1"></neq><list id="S4.SS1.SSS2.1.p1.6.m6.2.3.2.1.cmml" xref="S4.SS1.SSS2.1.p1.6.m6.2.3.2.2"><ci id="S4.SS1.SSS2.1.p1.6.m6.1.1.cmml" xref="S4.SS1.SSS2.1.p1.6.m6.1.1">𝑢</ci><ci id="S4.SS1.SSS2.1.p1.6.m6.2.2.cmml" xref="S4.SS1.SSS2.1.p1.6.m6.2.2">𝑣</ci></list><ci id="S4.SS1.SSS2.1.p1.6.m6.2.3.3.cmml" xref="S4.SS1.SSS2.1.p1.6.m6.2.3.3">𝑎</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.SSS2.1.p1.6.m6.2c">u,v\neq a</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.SSS2.1.p1.6.m6.2d">italic_u , italic_v ≠ italic_a</annotation></semantics></math>, there exists a <math alttext="u" class="ltx_Math" display="inline" id="S4.SS1.SSS2.1.p1.7.m7.1"><semantics id="S4.SS1.SSS2.1.p1.7.m7.1a"><mi id="S4.SS1.SSS2.1.p1.7.m7.1.1" xref="S4.SS1.SSS2.1.p1.7.m7.1.1.cmml">u</mi><annotation-xml encoding="MathML-Content" id="S4.SS1.SSS2.1.p1.7.m7.1b"><ci id="S4.SS1.SSS2.1.p1.7.m7.1.1.cmml" xref="S4.SS1.SSS2.1.p1.7.m7.1.1">𝑢</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.SSS2.1.p1.7.m7.1c">u</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.SSS2.1.p1.7.m7.1d">italic_u</annotation></semantics></math>-<math alttext="v" class="ltx_Math" display="inline" id="S4.SS1.SSS2.1.p1.8.m8.1"><semantics id="S4.SS1.SSS2.1.p1.8.m8.1a"><mi id="S4.SS1.SSS2.1.p1.8.m8.1.1" xref="S4.SS1.SSS2.1.p1.8.m8.1.1.cmml">v</mi><annotation-xml encoding="MathML-Content" id="S4.SS1.SSS2.1.p1.8.m8.1b"><ci id="S4.SS1.SSS2.1.p1.8.m8.1.1.cmml" xref="S4.SS1.SSS2.1.p1.8.m8.1.1">𝑣</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.SSS2.1.p1.8.m8.1c">v</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.SSS2.1.p1.8.m8.1d">italic_v</annotation></semantics></math> path in <math alttext="E\cup\textnormal{SOL}" class="ltx_Math" display="inline" id="S4.SS1.SSS2.1.p1.9.m9.1"><semantics id="S4.SS1.SSS2.1.p1.9.m9.1a"><mrow id="S4.SS1.SSS2.1.p1.9.m9.1.1" xref="S4.SS1.SSS2.1.p1.9.m9.1.1.cmml"><mi id="S4.SS1.SSS2.1.p1.9.m9.1.1.2" xref="S4.SS1.SSS2.1.p1.9.m9.1.1.2.cmml">E</mi><mo id="S4.SS1.SSS2.1.p1.9.m9.1.1.1" xref="S4.SS1.SSS2.1.p1.9.m9.1.1.1.cmml">∪</mo><mtext id="S4.SS1.SSS2.1.p1.9.m9.1.1.3" xref="S4.SS1.SSS2.1.p1.9.m9.1.1.3a.cmml">SOL</mtext></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.SSS2.1.p1.9.m9.1b"><apply id="S4.SS1.SSS2.1.p1.9.m9.1.1.cmml" xref="S4.SS1.SSS2.1.p1.9.m9.1.1"><union id="S4.SS1.SSS2.1.p1.9.m9.1.1.1.cmml" xref="S4.SS1.SSS2.1.p1.9.m9.1.1.1"></union><ci id="S4.SS1.SSS2.1.p1.9.m9.1.1.2.cmml" xref="S4.SS1.SSS2.1.p1.9.m9.1.1.2">𝐸</ci><ci id="S4.SS1.SSS2.1.p1.9.m9.1.1.3a.cmml" xref="S4.SS1.SSS2.1.p1.9.m9.1.1.3"><mtext id="S4.SS1.SSS2.1.p1.9.m9.1.1.3.cmml" xref="S4.SS1.SSS2.1.p1.9.m9.1.1.3">SOL</mtext></ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.SSS2.1.p1.9.m9.1c">E\cup\textnormal{SOL}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.SSS2.1.p1.9.m9.1d">italic_E ∪ SOL</annotation></semantics></math> that does not use <math alttext="a" class="ltx_Math" display="inline" id="S4.SS1.SSS2.1.p1.10.m10.1"><semantics id="S4.SS1.SSS2.1.p1.10.m10.1a"><mi id="S4.SS1.SSS2.1.p1.10.m10.1.1" xref="S4.SS1.SSS2.1.p1.10.m10.1.1.cmml">a</mi><annotation-xml encoding="MathML-Content" id="S4.SS1.SSS2.1.p1.10.m10.1b"><ci id="S4.SS1.SSS2.1.p1.10.m10.1.1.cmml" xref="S4.SS1.SSS2.1.p1.10.m10.1.1">𝑎</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.SSS2.1.p1.10.m10.1c">a</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.SSS2.1.p1.10.m10.1d">italic_a</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="S4.SS1.SSS2.2.p2"> <p class="ltx_p" id="S4.SS1.SSS2.2.p2.17">Fix <math alttext="uv\in\textnormal{OPT}" class="ltx_Math" display="inline" id="S4.SS1.SSS2.2.p2.1.m1.1"><semantics id="S4.SS1.SSS2.2.p2.1.m1.1a"><mrow id="S4.SS1.SSS2.2.p2.1.m1.1.1" xref="S4.SS1.SSS2.2.p2.1.m1.1.1.cmml"><mrow id="S4.SS1.SSS2.2.p2.1.m1.1.1.2" xref="S4.SS1.SSS2.2.p2.1.m1.1.1.2.cmml"><mi id="S4.SS1.SSS2.2.p2.1.m1.1.1.2.2" xref="S4.SS1.SSS2.2.p2.1.m1.1.1.2.2.cmml">u</mi><mo id="S4.SS1.SSS2.2.p2.1.m1.1.1.2.1" xref="S4.SS1.SSS2.2.p2.1.m1.1.1.2.1.cmml">⁢</mo><mi id="S4.SS1.SSS2.2.p2.1.m1.1.1.2.3" xref="S4.SS1.SSS2.2.p2.1.m1.1.1.2.3.cmml">v</mi></mrow><mo id="S4.SS1.SSS2.2.p2.1.m1.1.1.1" xref="S4.SS1.SSS2.2.p2.1.m1.1.1.1.cmml">∈</mo><mtext id="S4.SS1.SSS2.2.p2.1.m1.1.1.3" xref="S4.SS1.SSS2.2.p2.1.m1.1.1.3a.cmml">OPT</mtext></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.SSS2.2.p2.1.m1.1b"><apply id="S4.SS1.SSS2.2.p2.1.m1.1.1.cmml" xref="S4.SS1.SSS2.2.p2.1.m1.1.1"><in id="S4.SS1.SSS2.2.p2.1.m1.1.1.1.cmml" xref="S4.SS1.SSS2.2.p2.1.m1.1.1.1"></in><apply id="S4.SS1.SSS2.2.p2.1.m1.1.1.2.cmml" xref="S4.SS1.SSS2.2.p2.1.m1.1.1.2"><times id="S4.SS1.SSS2.2.p2.1.m1.1.1.2.1.cmml" xref="S4.SS1.SSS2.2.p2.1.m1.1.1.2.1"></times><ci id="S4.SS1.SSS2.2.p2.1.m1.1.1.2.2.cmml" xref="S4.SS1.SSS2.2.p2.1.m1.1.1.2.2">𝑢</ci><ci id="S4.SS1.SSS2.2.p2.1.m1.1.1.2.3.cmml" xref="S4.SS1.SSS2.2.p2.1.m1.1.1.2.3">𝑣</ci></apply><ci id="S4.SS1.SSS2.2.p2.1.m1.1.1.3a.cmml" xref="S4.SS1.SSS2.2.p2.1.m1.1.1.3"><mtext id="S4.SS1.SSS2.2.p2.1.m1.1.1.3.cmml" xref="S4.SS1.SSS2.2.p2.1.m1.1.1.3">OPT</mtext></ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.SSS2.2.p2.1.m1.1c">uv\in\textnormal{OPT}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.SSS2.2.p2.1.m1.1d">italic_u italic_v ∈ OPT</annotation></semantics></math>. Consider the unique <math alttext="u" class="ltx_Math" display="inline" id="S4.SS1.SSS2.2.p2.2.m2.1"><semantics id="S4.SS1.SSS2.2.p2.2.m2.1a"><mi id="S4.SS1.SSS2.2.p2.2.m2.1.1" xref="S4.SS1.SSS2.2.p2.2.m2.1.1.cmml">u</mi><annotation-xml encoding="MathML-Content" id="S4.SS1.SSS2.2.p2.2.m2.1b"><ci id="S4.SS1.SSS2.2.p2.2.m2.1.1.cmml" xref="S4.SS1.SSS2.2.p2.2.m2.1.1">𝑢</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.SSS2.2.p2.2.m2.1c">u</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.SSS2.2.p2.2.m2.1d">italic_u</annotation></semantics></math>-<math alttext="v" class="ltx_Math" display="inline" id="S4.SS1.SSS2.2.p2.3.m3.1"><semantics id="S4.SS1.SSS2.2.p2.3.m3.1a"><mi id="S4.SS1.SSS2.2.p2.3.m3.1.1" xref="S4.SS1.SSS2.2.p2.3.m3.1.1.cmml">v</mi><annotation-xml encoding="MathML-Content" id="S4.SS1.SSS2.2.p2.3.m3.1b"><ci id="S4.SS1.SSS2.2.p2.3.m3.1.1.cmml" xref="S4.SS1.SSS2.2.p2.3.m3.1.1">𝑣</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.SSS2.2.p2.3.m3.1c">v</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.SSS2.2.p2.3.m3.1d">italic_v</annotation></semantics></math> path in <math alttext="E" class="ltx_Math" display="inline" id="S4.SS1.SSS2.2.p2.4.m4.1"><semantics id="S4.SS1.SSS2.2.p2.4.m4.1a"><mi id="S4.SS1.SSS2.2.p2.4.m4.1.1" xref="S4.SS1.SSS2.2.p2.4.m4.1.1.cmml">E</mi><annotation-xml encoding="MathML-Content" id="S4.SS1.SSS2.2.p2.4.m4.1b"><ci id="S4.SS1.SSS2.2.p2.4.m4.1.1.cmml" xref="S4.SS1.SSS2.2.p2.4.m4.1.1">𝐸</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.SSS2.2.p2.4.m4.1c">E</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.SSS2.2.p2.4.m4.1d">italic_E</annotation></semantics></math> (recall that <math alttext="G" class="ltx_Math" display="inline" id="S4.SS1.SSS2.2.p2.5.m5.1"><semantics id="S4.SS1.SSS2.2.p2.5.m5.1a"><mi id="S4.SS1.SSS2.2.p2.5.m5.1.1" xref="S4.SS1.SSS2.2.p2.5.m5.1.1.cmml">G</mi><annotation-xml encoding="MathML-Content" id="S4.SS1.SSS2.2.p2.5.m5.1b"><ci id="S4.SS1.SSS2.2.p2.5.m5.1.1.cmml" xref="S4.SS1.SSS2.2.p2.5.m5.1.1">𝐺</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.SSS2.2.p2.5.m5.1c">G</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.SSS2.2.p2.5.m5.1d">italic_G</annotation></semantics></math> is a tree). If this path does not contain <math alttext="a" class="ltx_Math" display="inline" id="S4.SS1.SSS2.2.p2.6.m6.1"><semantics id="S4.SS1.SSS2.2.p2.6.m6.1a"><mi id="S4.SS1.SSS2.2.p2.6.m6.1.1" xref="S4.SS1.SSS2.2.p2.6.m6.1.1.cmml">a</mi><annotation-xml encoding="MathML-Content" id="S4.SS1.SSS2.2.p2.6.m6.1b"><ci id="S4.SS1.SSS2.2.p2.6.m6.1.1.cmml" xref="S4.SS1.SSS2.2.p2.6.m6.1.1">𝑎</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.SSS2.2.p2.6.m6.1c">a</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.SSS2.2.p2.6.m6.1d">italic_a</annotation></semantics></math>, then we are done. Thus we assume <math alttext="a" class="ltx_Math" display="inline" id="S4.SS1.SSS2.2.p2.7.m7.1"><semantics id="S4.SS1.SSS2.2.p2.7.m7.1a"><mi id="S4.SS1.SSS2.2.p2.7.m7.1.1" xref="S4.SS1.SSS2.2.p2.7.m7.1.1.cmml">a</mi><annotation-xml encoding="MathML-Content" id="S4.SS1.SSS2.2.p2.7.m7.1b"><ci id="S4.SS1.SSS2.2.p2.7.m7.1.1.cmml" xref="S4.SS1.SSS2.2.p2.7.m7.1.1">𝑎</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.SSS2.2.p2.7.m7.1c">a</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.SSS2.2.p2.7.m7.1d">italic_a</annotation></semantics></math> is on the <math alttext="u" class="ltx_Math" display="inline" id="S4.SS1.SSS2.2.p2.8.m8.1"><semantics id="S4.SS1.SSS2.2.p2.8.m8.1a"><mi id="S4.SS1.SSS2.2.p2.8.m8.1.1" xref="S4.SS1.SSS2.2.p2.8.m8.1.1.cmml">u</mi><annotation-xml encoding="MathML-Content" id="S4.SS1.SSS2.2.p2.8.m8.1b"><ci id="S4.SS1.SSS2.2.p2.8.m8.1.1.cmml" xref="S4.SS1.SSS2.2.p2.8.m8.1.1">𝑢</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.SSS2.2.p2.8.m8.1c">u</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.SSS2.2.p2.8.m8.1d">italic_u</annotation></semantics></math>-<math alttext="v" class="ltx_Math" display="inline" id="S4.SS1.SSS2.2.p2.9.m9.1"><semantics id="S4.SS1.SSS2.2.p2.9.m9.1a"><mi id="S4.SS1.SSS2.2.p2.9.m9.1.1" xref="S4.SS1.SSS2.2.p2.9.m9.1.1.cmml">v</mi><annotation-xml encoding="MathML-Content" id="S4.SS1.SSS2.2.p2.9.m9.1b"><ci id="S4.SS1.SSS2.2.p2.9.m9.1.1.cmml" xref="S4.SS1.SSS2.2.p2.9.m9.1.1">𝑣</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.SSS2.2.p2.9.m9.1c">v</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.SSS2.2.p2.9.m9.1d">italic_v</annotation></semantics></math> tree path. We case on whether or not <math alttext="a" class="ltx_Math" display="inline" id="S4.SS1.SSS2.2.p2.10.m10.1"><semantics id="S4.SS1.SSS2.2.p2.10.m10.1a"><mi id="S4.SS1.SSS2.2.p2.10.m10.1.1" xref="S4.SS1.SSS2.2.p2.10.m10.1.1.cmml">a</mi><annotation-xml encoding="MathML-Content" id="S4.SS1.SSS2.2.p2.10.m10.1b"><ci id="S4.SS1.SSS2.2.p2.10.m10.1.1.cmml" xref="S4.SS1.SSS2.2.p2.10.m10.1.1">𝑎</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.SSS2.2.p2.10.m10.1c">a</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.SSS2.2.p2.10.m10.1d">italic_a</annotation></semantics></math> is the LCA of <math alttext="u" class="ltx_Math" display="inline" id="S4.SS1.SSS2.2.p2.11.m11.1"><semantics id="S4.SS1.SSS2.2.p2.11.m11.1a"><mi id="S4.SS1.SSS2.2.p2.11.m11.1.1" xref="S4.SS1.SSS2.2.p2.11.m11.1.1.cmml">u</mi><annotation-xml encoding="MathML-Content" id="S4.SS1.SSS2.2.p2.11.m11.1b"><ci id="S4.SS1.SSS2.2.p2.11.m11.1.1.cmml" xref="S4.SS1.SSS2.2.p2.11.m11.1.1">𝑢</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.SSS2.2.p2.11.m11.1c">u</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.SSS2.2.p2.11.m11.1d">italic_u</annotation></semantics></math> and <math alttext="v" class="ltx_Math" display="inline" id="S4.SS1.SSS2.2.p2.12.m12.1"><semantics id="S4.SS1.SSS2.2.p2.12.m12.1a"><mi id="S4.SS1.SSS2.2.p2.12.m12.1.1" xref="S4.SS1.SSS2.2.p2.12.m12.1.1.cmml">v</mi><annotation-xml encoding="MathML-Content" id="S4.SS1.SSS2.2.p2.12.m12.1b"><ci id="S4.SS1.SSS2.2.p2.12.m12.1.1.cmml" xref="S4.SS1.SSS2.2.p2.12.m12.1.1">𝑣</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.SSS2.2.p2.12.m12.1c">v</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.SSS2.2.p2.12.m12.1d">italic_v</annotation></semantics></math>. Let <math alttext="b=\text{LCA}(u,v)" class="ltx_Math" display="inline" id="S4.SS1.SSS2.2.p2.13.m13.2"><semantics id="S4.SS1.SSS2.2.p2.13.m13.2a"><mrow id="S4.SS1.SSS2.2.p2.13.m13.2.3" xref="S4.SS1.SSS2.2.p2.13.m13.2.3.cmml"><mi id="S4.SS1.SSS2.2.p2.13.m13.2.3.2" xref="S4.SS1.SSS2.2.p2.13.m13.2.3.2.cmml">b</mi><mo id="S4.SS1.SSS2.2.p2.13.m13.2.3.1" xref="S4.SS1.SSS2.2.p2.13.m13.2.3.1.cmml">=</mo><mrow id="S4.SS1.SSS2.2.p2.13.m13.2.3.3" xref="S4.SS1.SSS2.2.p2.13.m13.2.3.3.cmml"><mtext id="S4.SS1.SSS2.2.p2.13.m13.2.3.3.2" xref="S4.SS1.SSS2.2.p2.13.m13.2.3.3.2a.cmml">LCA</mtext><mo id="S4.SS1.SSS2.2.p2.13.m13.2.3.3.1" xref="S4.SS1.SSS2.2.p2.13.m13.2.3.3.1.cmml">⁢</mo><mrow id="S4.SS1.SSS2.2.p2.13.m13.2.3.3.3.2" xref="S4.SS1.SSS2.2.p2.13.m13.2.3.3.3.1.cmml"><mo id="S4.SS1.SSS2.2.p2.13.m13.2.3.3.3.2.1" stretchy="false" xref="S4.SS1.SSS2.2.p2.13.m13.2.3.3.3.1.cmml">(</mo><mi id="S4.SS1.SSS2.2.p2.13.m13.1.1" xref="S4.SS1.SSS2.2.p2.13.m13.1.1.cmml">u</mi><mo id="S4.SS1.SSS2.2.p2.13.m13.2.3.3.3.2.2" xref="S4.SS1.SSS2.2.p2.13.m13.2.3.3.3.1.cmml">,</mo><mi id="S4.SS1.SSS2.2.p2.13.m13.2.2" xref="S4.SS1.SSS2.2.p2.13.m13.2.2.cmml">v</mi><mo id="S4.SS1.SSS2.2.p2.13.m13.2.3.3.3.2.3" stretchy="false" xref="S4.SS1.SSS2.2.p2.13.m13.2.3.3.3.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.SSS2.2.p2.13.m13.2b"><apply id="S4.SS1.SSS2.2.p2.13.m13.2.3.cmml" xref="S4.SS1.SSS2.2.p2.13.m13.2.3"><eq id="S4.SS1.SSS2.2.p2.13.m13.2.3.1.cmml" xref="S4.SS1.SSS2.2.p2.13.m13.2.3.1"></eq><ci id="S4.SS1.SSS2.2.p2.13.m13.2.3.2.cmml" xref="S4.SS1.SSS2.2.p2.13.m13.2.3.2">𝑏</ci><apply id="S4.SS1.SSS2.2.p2.13.m13.2.3.3.cmml" xref="S4.SS1.SSS2.2.p2.13.m13.2.3.3"><times id="S4.SS1.SSS2.2.p2.13.m13.2.3.3.1.cmml" xref="S4.SS1.SSS2.2.p2.13.m13.2.3.3.1"></times><ci id="S4.SS1.SSS2.2.p2.13.m13.2.3.3.2a.cmml" xref="S4.SS1.SSS2.2.p2.13.m13.2.3.3.2"><mtext id="S4.SS1.SSS2.2.p2.13.m13.2.3.3.2.cmml" xref="S4.SS1.SSS2.2.p2.13.m13.2.3.3.2">LCA</mtext></ci><interval closure="open" id="S4.SS1.SSS2.2.p2.13.m13.2.3.3.3.1.cmml" xref="S4.SS1.SSS2.2.p2.13.m13.2.3.3.3.2"><ci id="S4.SS1.SSS2.2.p2.13.m13.1.1.cmml" xref="S4.SS1.SSS2.2.p2.13.m13.1.1">𝑢</ci><ci id="S4.SS1.SSS2.2.p2.13.m13.2.2.cmml" xref="S4.SS1.SSS2.2.p2.13.m13.2.2">𝑣</ci></interval></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.SSS2.2.p2.13.m13.2c">b=\text{LCA}(u,v)</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.SSS2.2.p2.13.m13.2d">italic_b = LCA ( italic_u , italic_v )</annotation></semantics></math>. Let <math alttext="u^{\prime},v^{\prime}\in C(b)" class="ltx_Math" display="inline" id="S4.SS1.SSS2.2.p2.14.m14.3"><semantics id="S4.SS1.SSS2.2.p2.14.m14.3a"><mrow id="S4.SS1.SSS2.2.p2.14.m14.3.3" xref="S4.SS1.SSS2.2.p2.14.m14.3.3.cmml"><mrow id="S4.SS1.SSS2.2.p2.14.m14.3.3.2.2" xref="S4.SS1.SSS2.2.p2.14.m14.3.3.2.3.cmml"><msup id="S4.SS1.SSS2.2.p2.14.m14.2.2.1.1.1" xref="S4.SS1.SSS2.2.p2.14.m14.2.2.1.1.1.cmml"><mi id="S4.SS1.SSS2.2.p2.14.m14.2.2.1.1.1.2" xref="S4.SS1.SSS2.2.p2.14.m14.2.2.1.1.1.2.cmml">u</mi><mo id="S4.SS1.SSS2.2.p2.14.m14.2.2.1.1.1.3" xref="S4.SS1.SSS2.2.p2.14.m14.2.2.1.1.1.3.cmml">′</mo></msup><mo id="S4.SS1.SSS2.2.p2.14.m14.3.3.2.2.3" xref="S4.SS1.SSS2.2.p2.14.m14.3.3.2.3.cmml">,</mo><msup id="S4.SS1.SSS2.2.p2.14.m14.3.3.2.2.2" xref="S4.SS1.SSS2.2.p2.14.m14.3.3.2.2.2.cmml"><mi id="S4.SS1.SSS2.2.p2.14.m14.3.3.2.2.2.2" xref="S4.SS1.SSS2.2.p2.14.m14.3.3.2.2.2.2.cmml">v</mi><mo id="S4.SS1.SSS2.2.p2.14.m14.3.3.2.2.2.3" xref="S4.SS1.SSS2.2.p2.14.m14.3.3.2.2.2.3.cmml">′</mo></msup></mrow><mo id="S4.SS1.SSS2.2.p2.14.m14.3.3.3" xref="S4.SS1.SSS2.2.p2.14.m14.3.3.3.cmml">∈</mo><mrow id="S4.SS1.SSS2.2.p2.14.m14.3.3.4" xref="S4.SS1.SSS2.2.p2.14.m14.3.3.4.cmml"><mi id="S4.SS1.SSS2.2.p2.14.m14.3.3.4.2" xref="S4.SS1.SSS2.2.p2.14.m14.3.3.4.2.cmml">C</mi><mo id="S4.SS1.SSS2.2.p2.14.m14.3.3.4.1" xref="S4.SS1.SSS2.2.p2.14.m14.3.3.4.1.cmml">⁢</mo><mrow id="S4.SS1.SSS2.2.p2.14.m14.3.3.4.3.2" xref="S4.SS1.SSS2.2.p2.14.m14.3.3.4.cmml"><mo id="S4.SS1.SSS2.2.p2.14.m14.3.3.4.3.2.1" stretchy="false" xref="S4.SS1.SSS2.2.p2.14.m14.3.3.4.cmml">(</mo><mi id="S4.SS1.SSS2.2.p2.14.m14.1.1" xref="S4.SS1.SSS2.2.p2.14.m14.1.1.cmml">b</mi><mo id="S4.SS1.SSS2.2.p2.14.m14.3.3.4.3.2.2" stretchy="false" xref="S4.SS1.SSS2.2.p2.14.m14.3.3.4.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.SSS2.2.p2.14.m14.3b"><apply id="S4.SS1.SSS2.2.p2.14.m14.3.3.cmml" xref="S4.SS1.SSS2.2.p2.14.m14.3.3"><in id="S4.SS1.SSS2.2.p2.14.m14.3.3.3.cmml" xref="S4.SS1.SSS2.2.p2.14.m14.3.3.3"></in><list id="S4.SS1.SSS2.2.p2.14.m14.3.3.2.3.cmml" xref="S4.SS1.SSS2.2.p2.14.m14.3.3.2.2"><apply id="S4.SS1.SSS2.2.p2.14.m14.2.2.1.1.1.cmml" xref="S4.SS1.SSS2.2.p2.14.m14.2.2.1.1.1"><csymbol cd="ambiguous" id="S4.SS1.SSS2.2.p2.14.m14.2.2.1.1.1.1.cmml" xref="S4.SS1.SSS2.2.p2.14.m14.2.2.1.1.1">superscript</csymbol><ci id="S4.SS1.SSS2.2.p2.14.m14.2.2.1.1.1.2.cmml" xref="S4.SS1.SSS2.2.p2.14.m14.2.2.1.1.1.2">𝑢</ci><ci id="S4.SS1.SSS2.2.p2.14.m14.2.2.1.1.1.3.cmml" xref="S4.SS1.SSS2.2.p2.14.m14.2.2.1.1.1.3">′</ci></apply><apply id="S4.SS1.SSS2.2.p2.14.m14.3.3.2.2.2.cmml" xref="S4.SS1.SSS2.2.p2.14.m14.3.3.2.2.2"><csymbol cd="ambiguous" id="S4.SS1.SSS2.2.p2.14.m14.3.3.2.2.2.1.cmml" xref="S4.SS1.SSS2.2.p2.14.m14.3.3.2.2.2">superscript</csymbol><ci id="S4.SS1.SSS2.2.p2.14.m14.3.3.2.2.2.2.cmml" xref="S4.SS1.SSS2.2.p2.14.m14.3.3.2.2.2.2">𝑣</ci><ci id="S4.SS1.SSS2.2.p2.14.m14.3.3.2.2.2.3.cmml" xref="S4.SS1.SSS2.2.p2.14.m14.3.3.2.2.2.3">′</ci></apply></list><apply id="S4.SS1.SSS2.2.p2.14.m14.3.3.4.cmml" xref="S4.SS1.SSS2.2.p2.14.m14.3.3.4"><times id="S4.SS1.SSS2.2.p2.14.m14.3.3.4.1.cmml" xref="S4.SS1.SSS2.2.p2.14.m14.3.3.4.1"></times><ci id="S4.SS1.SSS2.2.p2.14.m14.3.3.4.2.cmml" xref="S4.SS1.SSS2.2.p2.14.m14.3.3.4.2">𝐶</ci><ci id="S4.SS1.SSS2.2.p2.14.m14.1.1.cmml" xref="S4.SS1.SSS2.2.p2.14.m14.1.1">𝑏</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.SSS2.2.p2.14.m14.3c">u^{\prime},v^{\prime}\in C(b)</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.SSS2.2.p2.14.m14.3d">italic_u start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT , italic_v start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ∈ italic_C ( italic_b )</annotation></semantics></math> be the children of <math alttext="b" class="ltx_Math" display="inline" id="S4.SS1.SSS2.2.p2.15.m15.1"><semantics id="S4.SS1.SSS2.2.p2.15.m15.1a"><mi id="S4.SS1.SSS2.2.p2.15.m15.1.1" xref="S4.SS1.SSS2.2.p2.15.m15.1.1.cmml">b</mi><annotation-xml encoding="MathML-Content" id="S4.SS1.SSS2.2.p2.15.m15.1b"><ci id="S4.SS1.SSS2.2.p2.15.m15.1.1.cmml" xref="S4.SS1.SSS2.2.p2.15.m15.1.1">𝑏</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.SSS2.2.p2.15.m15.1c">b</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.SSS2.2.p2.15.m15.1d">italic_b</annotation></semantics></math> such that <math alttext="u\in G_{u^{\prime}}" class="ltx_Math" display="inline" id="S4.SS1.SSS2.2.p2.16.m16.1"><semantics id="S4.SS1.SSS2.2.p2.16.m16.1a"><mrow id="S4.SS1.SSS2.2.p2.16.m16.1.1" xref="S4.SS1.SSS2.2.p2.16.m16.1.1.cmml"><mi id="S4.SS1.SSS2.2.p2.16.m16.1.1.2" xref="S4.SS1.SSS2.2.p2.16.m16.1.1.2.cmml">u</mi><mo id="S4.SS1.SSS2.2.p2.16.m16.1.1.1" xref="S4.SS1.SSS2.2.p2.16.m16.1.1.1.cmml">∈</mo><msub id="S4.SS1.SSS2.2.p2.16.m16.1.1.3" xref="S4.SS1.SSS2.2.p2.16.m16.1.1.3.cmml"><mi id="S4.SS1.SSS2.2.p2.16.m16.1.1.3.2" xref="S4.SS1.SSS2.2.p2.16.m16.1.1.3.2.cmml">G</mi><msup id="S4.SS1.SSS2.2.p2.16.m16.1.1.3.3" xref="S4.SS1.SSS2.2.p2.16.m16.1.1.3.3.cmml"><mi id="S4.SS1.SSS2.2.p2.16.m16.1.1.3.3.2" xref="S4.SS1.SSS2.2.p2.16.m16.1.1.3.3.2.cmml">u</mi><mo id="S4.SS1.SSS2.2.p2.16.m16.1.1.3.3.3" xref="S4.SS1.SSS2.2.p2.16.m16.1.1.3.3.3.cmml">′</mo></msup></msub></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.SSS2.2.p2.16.m16.1b"><apply id="S4.SS1.SSS2.2.p2.16.m16.1.1.cmml" xref="S4.SS1.SSS2.2.p2.16.m16.1.1"><in id="S4.SS1.SSS2.2.p2.16.m16.1.1.1.cmml" xref="S4.SS1.SSS2.2.p2.16.m16.1.1.1"></in><ci id="S4.SS1.SSS2.2.p2.16.m16.1.1.2.cmml" xref="S4.SS1.SSS2.2.p2.16.m16.1.1.2">𝑢</ci><apply id="S4.SS1.SSS2.2.p2.16.m16.1.1.3.cmml" xref="S4.SS1.SSS2.2.p2.16.m16.1.1.3"><csymbol cd="ambiguous" id="S4.SS1.SSS2.2.p2.16.m16.1.1.3.1.cmml" xref="S4.SS1.SSS2.2.p2.16.m16.1.1.3">subscript</csymbol><ci id="S4.SS1.SSS2.2.p2.16.m16.1.1.3.2.cmml" xref="S4.SS1.SSS2.2.p2.16.m16.1.1.3.2">𝐺</ci><apply id="S4.SS1.SSS2.2.p2.16.m16.1.1.3.3.cmml" xref="S4.SS1.SSS2.2.p2.16.m16.1.1.3.3"><csymbol cd="ambiguous" id="S4.SS1.SSS2.2.p2.16.m16.1.1.3.3.1.cmml" xref="S4.SS1.SSS2.2.p2.16.m16.1.1.3.3">superscript</csymbol><ci id="S4.SS1.SSS2.2.p2.16.m16.1.1.3.3.2.cmml" xref="S4.SS1.SSS2.2.p2.16.m16.1.1.3.3.2">𝑢</ci><ci id="S4.SS1.SSS2.2.p2.16.m16.1.1.3.3.3.cmml" xref="S4.SS1.SSS2.2.p2.16.m16.1.1.3.3.3">′</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.SSS2.2.p2.16.m16.1c">u\in G_{u^{\prime}}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.SSS2.2.p2.16.m16.1d">italic_u ∈ italic_G start_POSTSUBSCRIPT italic_u start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="v\in G_{v^{\prime}}" class="ltx_Math" display="inline" id="S4.SS1.SSS2.2.p2.17.m17.1"><semantics id="S4.SS1.SSS2.2.p2.17.m17.1a"><mrow id="S4.SS1.SSS2.2.p2.17.m17.1.1" xref="S4.SS1.SSS2.2.p2.17.m17.1.1.cmml"><mi id="S4.SS1.SSS2.2.p2.17.m17.1.1.2" xref="S4.SS1.SSS2.2.p2.17.m17.1.1.2.cmml">v</mi><mo id="S4.SS1.SSS2.2.p2.17.m17.1.1.1" xref="S4.SS1.SSS2.2.p2.17.m17.1.1.1.cmml">∈</mo><msub id="S4.SS1.SSS2.2.p2.17.m17.1.1.3" xref="S4.SS1.SSS2.2.p2.17.m17.1.1.3.cmml"><mi id="S4.SS1.SSS2.2.p2.17.m17.1.1.3.2" xref="S4.SS1.SSS2.2.p2.17.m17.1.1.3.2.cmml">G</mi><msup id="S4.SS1.SSS2.2.p2.17.m17.1.1.3.3" xref="S4.SS1.SSS2.2.p2.17.m17.1.1.3.3.cmml"><mi id="S4.SS1.SSS2.2.p2.17.m17.1.1.3.3.2" xref="S4.SS1.SSS2.2.p2.17.m17.1.1.3.3.2.cmml">v</mi><mo id="S4.SS1.SSS2.2.p2.17.m17.1.1.3.3.3" xref="S4.SS1.SSS2.2.p2.17.m17.1.1.3.3.3.cmml">′</mo></msup></msub></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.SSS2.2.p2.17.m17.1b"><apply id="S4.SS1.SSS2.2.p2.17.m17.1.1.cmml" xref="S4.SS1.SSS2.2.p2.17.m17.1.1"><in id="S4.SS1.SSS2.2.p2.17.m17.1.1.1.cmml" xref="S4.SS1.SSS2.2.p2.17.m17.1.1.1"></in><ci id="S4.SS1.SSS2.2.p2.17.m17.1.1.2.cmml" xref="S4.SS1.SSS2.2.p2.17.m17.1.1.2">𝑣</ci><apply id="S4.SS1.SSS2.2.p2.17.m17.1.1.3.cmml" xref="S4.SS1.SSS2.2.p2.17.m17.1.1.3"><csymbol cd="ambiguous" id="S4.SS1.SSS2.2.p2.17.m17.1.1.3.1.cmml" xref="S4.SS1.SSS2.2.p2.17.m17.1.1.3">subscript</csymbol><ci id="S4.SS1.SSS2.2.p2.17.m17.1.1.3.2.cmml" xref="S4.SS1.SSS2.2.p2.17.m17.1.1.3.2">𝐺</ci><apply id="S4.SS1.SSS2.2.p2.17.m17.1.1.3.3.cmml" xref="S4.SS1.SSS2.2.p2.17.m17.1.1.3.3"><csymbol cd="ambiguous" id="S4.SS1.SSS2.2.p2.17.m17.1.1.3.3.1.cmml" xref="S4.SS1.SSS2.2.p2.17.m17.1.1.3.3">superscript</csymbol><ci id="S4.SS1.SSS2.2.p2.17.m17.1.1.3.3.2.cmml" xref="S4.SS1.SSS2.2.p2.17.m17.1.1.3.3.2">𝑣</ci><ci id="S4.SS1.SSS2.2.p2.17.m17.1.1.3.3.3.cmml" xref="S4.SS1.SSS2.2.p2.17.m17.1.1.3.3.3">′</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.SSS2.2.p2.17.m17.1c">v\in G_{v^{\prime}}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.SSS2.2.p2.17.m17.1d">italic_v ∈ italic_G start_POSTSUBSCRIPT italic_v start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="S4.SS1.SSS2.3.p3"> <ul class="ltx_itemize" id="S4.I1"> <li class="ltx_item" id="S4.I1.i1" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">•</span> <div class="ltx_para" id="S4.I1.i1.p1"> <p class="ltx_p" id="S4.I1.i1.p1.18"><span class="ltx_text ltx_font_bold" id="S4.I1.i1.p1.18.1">Case 1:</span> First, suppose <math alttext="a=b" class="ltx_Math" display="inline" id="S4.I1.i1.p1.1.m1.1"><semantics id="S4.I1.i1.p1.1.m1.1a"><mrow id="S4.I1.i1.p1.1.m1.1.1" xref="S4.I1.i1.p1.1.m1.1.1.cmml"><mi id="S4.I1.i1.p1.1.m1.1.1.2" xref="S4.I1.i1.p1.1.m1.1.1.2.cmml">a</mi><mo id="S4.I1.i1.p1.1.m1.1.1.1" xref="S4.I1.i1.p1.1.m1.1.1.1.cmml">=</mo><mi id="S4.I1.i1.p1.1.m1.1.1.3" xref="S4.I1.i1.p1.1.m1.1.1.3.cmml">b</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.I1.i1.p1.1.m1.1b"><apply id="S4.I1.i1.p1.1.m1.1.1.cmml" xref="S4.I1.i1.p1.1.m1.1.1"><eq id="S4.I1.i1.p1.1.m1.1.1.1.cmml" xref="S4.I1.i1.p1.1.m1.1.1.1"></eq><ci id="S4.I1.i1.p1.1.m1.1.1.2.cmml" xref="S4.I1.i1.p1.1.m1.1.1.2">𝑎</ci><ci id="S4.I1.i1.p1.1.m1.1.1.3.cmml" xref="S4.I1.i1.p1.1.m1.1.1.3">𝑏</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.I1.i1.p1.1.m1.1c">a=b</annotation><annotation encoding="application/x-llamapun" id="S4.I1.i1.p1.1.m1.1d">italic_a = italic_b</annotation></semantics></math>. If <math alttext="u^{\prime}" class="ltx_Math" display="inline" id="S4.I1.i1.p1.2.m2.1"><semantics id="S4.I1.i1.p1.2.m2.1a"><msup id="S4.I1.i1.p1.2.m2.1.1" xref="S4.I1.i1.p1.2.m2.1.1.cmml"><mi id="S4.I1.i1.p1.2.m2.1.1.2" xref="S4.I1.i1.p1.2.m2.1.1.2.cmml">u</mi><mo id="S4.I1.i1.p1.2.m2.1.1.3" xref="S4.I1.i1.p1.2.m2.1.1.3.cmml">′</mo></msup><annotation-xml encoding="MathML-Content" id="S4.I1.i1.p1.2.m2.1b"><apply id="S4.I1.i1.p1.2.m2.1.1.cmml" xref="S4.I1.i1.p1.2.m2.1.1"><csymbol cd="ambiguous" id="S4.I1.i1.p1.2.m2.1.1.1.cmml" xref="S4.I1.i1.p1.2.m2.1.1">superscript</csymbol><ci id="S4.I1.i1.p1.2.m2.1.1.2.cmml" xref="S4.I1.i1.p1.2.m2.1.1.2">𝑢</ci><ci id="S4.I1.i1.p1.2.m2.1.1.3.cmml" xref="S4.I1.i1.p1.2.m2.1.1.3">′</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.I1.i1.p1.2.m2.1c">u^{\prime}</annotation><annotation encoding="application/x-llamapun" id="S4.I1.i1.p1.2.m2.1d">italic_u start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math> or <math alttext="v^{\prime}" class="ltx_Math" display="inline" id="S4.I1.i1.p1.3.m3.1"><semantics id="S4.I1.i1.p1.3.m3.1a"><msup id="S4.I1.i1.p1.3.m3.1.1" xref="S4.I1.i1.p1.3.m3.1.1.cmml"><mi id="S4.I1.i1.p1.3.m3.1.1.2" xref="S4.I1.i1.p1.3.m3.1.1.2.cmml">v</mi><mo id="S4.I1.i1.p1.3.m3.1.1.3" xref="S4.I1.i1.p1.3.m3.1.1.3.cmml">′</mo></msup><annotation-xml encoding="MathML-Content" id="S4.I1.i1.p1.3.m3.1b"><apply id="S4.I1.i1.p1.3.m3.1.1.cmml" xref="S4.I1.i1.p1.3.m3.1.1"><csymbol cd="ambiguous" id="S4.I1.i1.p1.3.m3.1.1.1.cmml" xref="S4.I1.i1.p1.3.m3.1.1">superscript</csymbol><ci id="S4.I1.i1.p1.3.m3.1.1.2.cmml" xref="S4.I1.i1.p1.3.m3.1.1.2">𝑣</ci><ci id="S4.I1.i1.p1.3.m3.1.1.3.cmml" xref="S4.I1.i1.p1.3.m3.1.1.3">′</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.I1.i1.p1.3.m3.1c">v^{\prime}</annotation><annotation encoding="application/x-llamapun" id="S4.I1.i1.p1.3.m3.1d">italic_v start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math> are not marked as “good” in Algorithm <a class="ltx_ref" href="https://arxiv.org/html/2503.00712v1#algorithm5" title="In 4.1.2 Bounding Approximation Ratio ‣ 4.1 One-to-Two Augmentation ‣ 4 Vertex Connectivity Augmentation in Link-Arrival Model ‣ Streaming Algorithms for Network Design"><span class="ltx_text ltx_ref_tag">5</span></a>, then <math alttext="u^{\prime}" class="ltx_Math" display="inline" id="S4.I1.i1.p1.4.m4.1"><semantics id="S4.I1.i1.p1.4.m4.1a"><msup id="S4.I1.i1.p1.4.m4.1.1" xref="S4.I1.i1.p1.4.m4.1.1.cmml"><mi id="S4.I1.i1.p1.4.m4.1.1.2" xref="S4.I1.i1.p1.4.m4.1.1.2.cmml">u</mi><mo id="S4.I1.i1.p1.4.m4.1.1.3" xref="S4.I1.i1.p1.4.m4.1.1.3.cmml">′</mo></msup><annotation-xml encoding="MathML-Content" id="S4.I1.i1.p1.4.m4.1b"><apply id="S4.I1.i1.p1.4.m4.1.1.cmml" xref="S4.I1.i1.p1.4.m4.1.1"><csymbol cd="ambiguous" id="S4.I1.i1.p1.4.m4.1.1.1.cmml" xref="S4.I1.i1.p1.4.m4.1.1">superscript</csymbol><ci id="S4.I1.i1.p1.4.m4.1.1.2.cmml" xref="S4.I1.i1.p1.4.m4.1.1.2">𝑢</ci><ci id="S4.I1.i1.p1.4.m4.1.1.3.cmml" xref="S4.I1.i1.p1.4.m4.1.1.3">′</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.I1.i1.p1.4.m4.1c">u^{\prime}</annotation><annotation encoding="application/x-llamapun" id="S4.I1.i1.p1.4.m4.1d">italic_u start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math> and <math alttext="v^{\prime}" class="ltx_Math" display="inline" id="S4.I1.i1.p1.5.m5.1"><semantics id="S4.I1.i1.p1.5.m5.1a"><msup id="S4.I1.i1.p1.5.m5.1.1" xref="S4.I1.i1.p1.5.m5.1.1.cmml"><mi id="S4.I1.i1.p1.5.m5.1.1.2" xref="S4.I1.i1.p1.5.m5.1.1.2.cmml">v</mi><mo id="S4.I1.i1.p1.5.m5.1.1.3" xref="S4.I1.i1.p1.5.m5.1.1.3.cmml">′</mo></msup><annotation-xml encoding="MathML-Content" id="S4.I1.i1.p1.5.m5.1b"><apply id="S4.I1.i1.p1.5.m5.1.1.cmml" xref="S4.I1.i1.p1.5.m5.1.1"><csymbol cd="ambiguous" id="S4.I1.i1.p1.5.m5.1.1.1.cmml" xref="S4.I1.i1.p1.5.m5.1.1">superscript</csymbol><ci id="S4.I1.i1.p1.5.m5.1.1.2.cmml" xref="S4.I1.i1.p1.5.m5.1.1.2">𝑣</ci><ci id="S4.I1.i1.p1.5.m5.1.1.3.cmml" xref="S4.I1.i1.p1.5.m5.1.1.3">′</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.I1.i1.p1.5.m5.1c">v^{\prime}</annotation><annotation encoding="application/x-llamapun" id="S4.I1.i1.p1.5.m5.1d">italic_v start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math> correspond to separate supernodes of <math alttext="C^{\prime\prime}(a)" class="ltx_Math" display="inline" id="S4.I1.i1.p1.6.m6.1"><semantics id="S4.I1.i1.p1.6.m6.1a"><mrow id="S4.I1.i1.p1.6.m6.1.2" xref="S4.I1.i1.p1.6.m6.1.2.cmml"><msup id="S4.I1.i1.p1.6.m6.1.2.2" xref="S4.I1.i1.p1.6.m6.1.2.2.cmml"><mi id="S4.I1.i1.p1.6.m6.1.2.2.2" xref="S4.I1.i1.p1.6.m6.1.2.2.2.cmml">C</mi><mo id="S4.I1.i1.p1.6.m6.1.2.2.3" xref="S4.I1.i1.p1.6.m6.1.2.2.3.cmml">′′</mo></msup><mo id="S4.I1.i1.p1.6.m6.1.2.1" xref="S4.I1.i1.p1.6.m6.1.2.1.cmml">⁢</mo><mrow id="S4.I1.i1.p1.6.m6.1.2.3.2" xref="S4.I1.i1.p1.6.m6.1.2.cmml"><mo id="S4.I1.i1.p1.6.m6.1.2.3.2.1" stretchy="false" xref="S4.I1.i1.p1.6.m6.1.2.cmml">(</mo><mi id="S4.I1.i1.p1.6.m6.1.1" xref="S4.I1.i1.p1.6.m6.1.1.cmml">a</mi><mo id="S4.I1.i1.p1.6.m6.1.2.3.2.2" stretchy="false" xref="S4.I1.i1.p1.6.m6.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.I1.i1.p1.6.m6.1b"><apply id="S4.I1.i1.p1.6.m6.1.2.cmml" xref="S4.I1.i1.p1.6.m6.1.2"><times id="S4.I1.i1.p1.6.m6.1.2.1.cmml" xref="S4.I1.i1.p1.6.m6.1.2.1"></times><apply id="S4.I1.i1.p1.6.m6.1.2.2.cmml" xref="S4.I1.i1.p1.6.m6.1.2.2"><csymbol cd="ambiguous" id="S4.I1.i1.p1.6.m6.1.2.2.1.cmml" xref="S4.I1.i1.p1.6.m6.1.2.2">superscript</csymbol><ci id="S4.I1.i1.p1.6.m6.1.2.2.2.cmml" xref="S4.I1.i1.p1.6.m6.1.2.2.2">𝐶</ci><ci id="S4.I1.i1.p1.6.m6.1.2.2.3.cmml" xref="S4.I1.i1.p1.6.m6.1.2.2.3">′′</ci></apply><ci id="S4.I1.i1.p1.6.m6.1.1.cmml" xref="S4.I1.i1.p1.6.m6.1.1">𝑎</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.I1.i1.p1.6.m6.1c">C^{\prime\prime}(a)</annotation><annotation encoding="application/x-llamapun" id="S4.I1.i1.p1.6.m6.1d">italic_C start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT ( italic_a )</annotation></semantics></math> and thus <math alttext="G_{u^{\prime}}" class="ltx_Math" display="inline" id="S4.I1.i1.p1.7.m7.1"><semantics id="S4.I1.i1.p1.7.m7.1a"><msub id="S4.I1.i1.p1.7.m7.1.1" xref="S4.I1.i1.p1.7.m7.1.1.cmml"><mi id="S4.I1.i1.p1.7.m7.1.1.2" xref="S4.I1.i1.p1.7.m7.1.1.2.cmml">G</mi><msup id="S4.I1.i1.p1.7.m7.1.1.3" xref="S4.I1.i1.p1.7.m7.1.1.3.cmml"><mi id="S4.I1.i1.p1.7.m7.1.1.3.2" xref="S4.I1.i1.p1.7.m7.1.1.3.2.cmml">u</mi><mo id="S4.I1.i1.p1.7.m7.1.1.3.3" xref="S4.I1.i1.p1.7.m7.1.1.3.3.cmml">′</mo></msup></msub><annotation-xml encoding="MathML-Content" id="S4.I1.i1.p1.7.m7.1b"><apply id="S4.I1.i1.p1.7.m7.1.1.cmml" xref="S4.I1.i1.p1.7.m7.1.1"><csymbol cd="ambiguous" id="S4.I1.i1.p1.7.m7.1.1.1.cmml" xref="S4.I1.i1.p1.7.m7.1.1">subscript</csymbol><ci id="S4.I1.i1.p1.7.m7.1.1.2.cmml" xref="S4.I1.i1.p1.7.m7.1.1.2">𝐺</ci><apply id="S4.I1.i1.p1.7.m7.1.1.3.cmml" xref="S4.I1.i1.p1.7.m7.1.1.3"><csymbol cd="ambiguous" id="S4.I1.i1.p1.7.m7.1.1.3.1.cmml" xref="S4.I1.i1.p1.7.m7.1.1.3">superscript</csymbol><ci id="S4.I1.i1.p1.7.m7.1.1.3.2.cmml" xref="S4.I1.i1.p1.7.m7.1.1.3.2">𝑢</ci><ci id="S4.I1.i1.p1.7.m7.1.1.3.3.cmml" xref="S4.I1.i1.p1.7.m7.1.1.3.3">′</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.I1.i1.p1.7.m7.1c">G_{u^{\prime}}</annotation><annotation encoding="application/x-llamapun" id="S4.I1.i1.p1.7.m7.1d">italic_G start_POSTSUBSCRIPT italic_u start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="G_{v^{\prime}}" class="ltx_Math" display="inline" id="S4.I1.i1.p1.8.m8.1"><semantics id="S4.I1.i1.p1.8.m8.1a"><msub id="S4.I1.i1.p1.8.m8.1.1" xref="S4.I1.i1.p1.8.m8.1.1.cmml"><mi id="S4.I1.i1.p1.8.m8.1.1.2" xref="S4.I1.i1.p1.8.m8.1.1.2.cmml">G</mi><msup id="S4.I1.i1.p1.8.m8.1.1.3" xref="S4.I1.i1.p1.8.m8.1.1.3.cmml"><mi id="S4.I1.i1.p1.8.m8.1.1.3.2" xref="S4.I1.i1.p1.8.m8.1.1.3.2.cmml">v</mi><mo id="S4.I1.i1.p1.8.m8.1.1.3.3" xref="S4.I1.i1.p1.8.m8.1.1.3.3.cmml">′</mo></msup></msub><annotation-xml encoding="MathML-Content" id="S4.I1.i1.p1.8.m8.1b"><apply id="S4.I1.i1.p1.8.m8.1.1.cmml" xref="S4.I1.i1.p1.8.m8.1.1"><csymbol cd="ambiguous" id="S4.I1.i1.p1.8.m8.1.1.1.cmml" xref="S4.I1.i1.p1.8.m8.1.1">subscript</csymbol><ci id="S4.I1.i1.p1.8.m8.1.1.2.cmml" xref="S4.I1.i1.p1.8.m8.1.1.2">𝐺</ci><apply id="S4.I1.i1.p1.8.m8.1.1.3.cmml" xref="S4.I1.i1.p1.8.m8.1.1.3"><csymbol cd="ambiguous" id="S4.I1.i1.p1.8.m8.1.1.3.1.cmml" xref="S4.I1.i1.p1.8.m8.1.1.3">superscript</csymbol><ci id="S4.I1.i1.p1.8.m8.1.1.3.2.cmml" xref="S4.I1.i1.p1.8.m8.1.1.3.2">𝑣</ci><ci id="S4.I1.i1.p1.8.m8.1.1.3.3.cmml" xref="S4.I1.i1.p1.8.m8.1.1.3.3">′</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.I1.i1.p1.8.m8.1c">G_{v^{\prime}}</annotation><annotation encoding="application/x-llamapun" id="S4.I1.i1.p1.8.m8.1d">italic_G start_POSTSUBSCRIPT italic_v start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT</annotation></semantics></math> must be connected via <math alttext="T^{\prime\prime}_{a}" class="ltx_Math" display="inline" id="S4.I1.i1.p1.9.m9.1"><semantics id="S4.I1.i1.p1.9.m9.1a"><msubsup id="S4.I1.i1.p1.9.m9.1.1" xref="S4.I1.i1.p1.9.m9.1.1.cmml"><mi id="S4.I1.i1.p1.9.m9.1.1.2.2" xref="S4.I1.i1.p1.9.m9.1.1.2.2.cmml">T</mi><mi id="S4.I1.i1.p1.9.m9.1.1.3" xref="S4.I1.i1.p1.9.m9.1.1.3.cmml">a</mi><mo id="S4.I1.i1.p1.9.m9.1.1.2.3" xref="S4.I1.i1.p1.9.m9.1.1.2.3.cmml">′′</mo></msubsup><annotation-xml encoding="MathML-Content" id="S4.I1.i1.p1.9.m9.1b"><apply id="S4.I1.i1.p1.9.m9.1.1.cmml" xref="S4.I1.i1.p1.9.m9.1.1"><csymbol cd="ambiguous" id="S4.I1.i1.p1.9.m9.1.1.1.cmml" xref="S4.I1.i1.p1.9.m9.1.1">subscript</csymbol><apply id="S4.I1.i1.p1.9.m9.1.1.2.cmml" xref="S4.I1.i1.p1.9.m9.1.1"><csymbol cd="ambiguous" id="S4.I1.i1.p1.9.m9.1.1.2.1.cmml" xref="S4.I1.i1.p1.9.m9.1.1">superscript</csymbol><ci id="S4.I1.i1.p1.9.m9.1.1.2.2.cmml" xref="S4.I1.i1.p1.9.m9.1.1.2.2">𝑇</ci><ci id="S4.I1.i1.p1.9.m9.1.1.2.3.cmml" xref="S4.I1.i1.p1.9.m9.1.1.2.3">′′</ci></apply><ci id="S4.I1.i1.p1.9.m9.1.1.3.cmml" xref="S4.I1.i1.p1.9.m9.1.1.3">𝑎</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.I1.i1.p1.9.m9.1c">T^{\prime\prime}_{a}</annotation><annotation encoding="application/x-llamapun" id="S4.I1.i1.p1.9.m9.1d">italic_T start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT</annotation></semantics></math>. Both <math alttext="G_{u^{\prime}}" class="ltx_Math" display="inline" id="S4.I1.i1.p1.10.m10.1"><semantics id="S4.I1.i1.p1.10.m10.1a"><msub id="S4.I1.i1.p1.10.m10.1.1" xref="S4.I1.i1.p1.10.m10.1.1.cmml"><mi id="S4.I1.i1.p1.10.m10.1.1.2" xref="S4.I1.i1.p1.10.m10.1.1.2.cmml">G</mi><msup id="S4.I1.i1.p1.10.m10.1.1.3" xref="S4.I1.i1.p1.10.m10.1.1.3.cmml"><mi id="S4.I1.i1.p1.10.m10.1.1.3.2" xref="S4.I1.i1.p1.10.m10.1.1.3.2.cmml">u</mi><mo id="S4.I1.i1.p1.10.m10.1.1.3.3" xref="S4.I1.i1.p1.10.m10.1.1.3.3.cmml">′</mo></msup></msub><annotation-xml encoding="MathML-Content" id="S4.I1.i1.p1.10.m10.1b"><apply id="S4.I1.i1.p1.10.m10.1.1.cmml" xref="S4.I1.i1.p1.10.m10.1.1"><csymbol cd="ambiguous" id="S4.I1.i1.p1.10.m10.1.1.1.cmml" xref="S4.I1.i1.p1.10.m10.1.1">subscript</csymbol><ci id="S4.I1.i1.p1.10.m10.1.1.2.cmml" xref="S4.I1.i1.p1.10.m10.1.1.2">𝐺</ci><apply id="S4.I1.i1.p1.10.m10.1.1.3.cmml" xref="S4.I1.i1.p1.10.m10.1.1.3"><csymbol cd="ambiguous" id="S4.I1.i1.p1.10.m10.1.1.3.1.cmml" xref="S4.I1.i1.p1.10.m10.1.1.3">superscript</csymbol><ci id="S4.I1.i1.p1.10.m10.1.1.3.2.cmml" xref="S4.I1.i1.p1.10.m10.1.1.3.2">𝑢</ci><ci id="S4.I1.i1.p1.10.m10.1.1.3.3.cmml" xref="S4.I1.i1.p1.10.m10.1.1.3.3">′</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.I1.i1.p1.10.m10.1c">G_{u^{\prime}}</annotation><annotation encoding="application/x-llamapun" id="S4.I1.i1.p1.10.m10.1d">italic_G start_POSTSUBSCRIPT italic_u start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="G_{v^{\prime}}" class="ltx_Math" display="inline" id="S4.I1.i1.p1.11.m11.1"><semantics id="S4.I1.i1.p1.11.m11.1a"><msub id="S4.I1.i1.p1.11.m11.1.1" xref="S4.I1.i1.p1.11.m11.1.1.cmml"><mi id="S4.I1.i1.p1.11.m11.1.1.2" xref="S4.I1.i1.p1.11.m11.1.1.2.cmml">G</mi><msup id="S4.I1.i1.p1.11.m11.1.1.3" xref="S4.I1.i1.p1.11.m11.1.1.3.cmml"><mi id="S4.I1.i1.p1.11.m11.1.1.3.2" xref="S4.I1.i1.p1.11.m11.1.1.3.2.cmml">v</mi><mo id="S4.I1.i1.p1.11.m11.1.1.3.3" xref="S4.I1.i1.p1.11.m11.1.1.3.3.cmml">′</mo></msup></msub><annotation-xml encoding="MathML-Content" id="S4.I1.i1.p1.11.m11.1b"><apply id="S4.I1.i1.p1.11.m11.1.1.cmml" xref="S4.I1.i1.p1.11.m11.1.1"><csymbol cd="ambiguous" id="S4.I1.i1.p1.11.m11.1.1.1.cmml" xref="S4.I1.i1.p1.11.m11.1.1">subscript</csymbol><ci id="S4.I1.i1.p1.11.m11.1.1.2.cmml" xref="S4.I1.i1.p1.11.m11.1.1.2">𝐺</ci><apply id="S4.I1.i1.p1.11.m11.1.1.3.cmml" xref="S4.I1.i1.p1.11.m11.1.1.3"><csymbol cd="ambiguous" id="S4.I1.i1.p1.11.m11.1.1.3.1.cmml" xref="S4.I1.i1.p1.11.m11.1.1.3">superscript</csymbol><ci id="S4.I1.i1.p1.11.m11.1.1.3.2.cmml" xref="S4.I1.i1.p1.11.m11.1.1.3.2">𝑣</ci><ci id="S4.I1.i1.p1.11.m11.1.1.3.3.cmml" xref="S4.I1.i1.p1.11.m11.1.1.3.3">′</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.I1.i1.p1.11.m11.1c">G_{v^{\prime}}</annotation><annotation encoding="application/x-llamapun" id="S4.I1.i1.p1.11.m11.1d">italic_G start_POSTSUBSCRIPT italic_v start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT</annotation></semantics></math> remain connected despite the deletion of <math alttext="a" class="ltx_Math" display="inline" id="S4.I1.i1.p1.12.m12.1"><semantics id="S4.I1.i1.p1.12.m12.1a"><mi id="S4.I1.i1.p1.12.m12.1.1" xref="S4.I1.i1.p1.12.m12.1.1.cmml">a</mi><annotation-xml encoding="MathML-Content" id="S4.I1.i1.p1.12.m12.1b"><ci id="S4.I1.i1.p1.12.m12.1.1.cmml" xref="S4.I1.i1.p1.12.m12.1.1">𝑎</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.I1.i1.p1.12.m12.1c">a</annotation><annotation encoding="application/x-llamapun" id="S4.I1.i1.p1.12.m12.1d">italic_a</annotation></semantics></math> since <math alttext="u^{\prime},v^{\prime}" class="ltx_Math" display="inline" id="S4.I1.i1.p1.13.m13.2"><semantics id="S4.I1.i1.p1.13.m13.2a"><mrow id="S4.I1.i1.p1.13.m13.2.2.2" xref="S4.I1.i1.p1.13.m13.2.2.3.cmml"><msup id="S4.I1.i1.p1.13.m13.1.1.1.1" xref="S4.I1.i1.p1.13.m13.1.1.1.1.cmml"><mi id="S4.I1.i1.p1.13.m13.1.1.1.1.2" xref="S4.I1.i1.p1.13.m13.1.1.1.1.2.cmml">u</mi><mo id="S4.I1.i1.p1.13.m13.1.1.1.1.3" xref="S4.I1.i1.p1.13.m13.1.1.1.1.3.cmml">′</mo></msup><mo id="S4.I1.i1.p1.13.m13.2.2.2.3" xref="S4.I1.i1.p1.13.m13.2.2.3.cmml">,</mo><msup id="S4.I1.i1.p1.13.m13.2.2.2.2" xref="S4.I1.i1.p1.13.m13.2.2.2.2.cmml"><mi id="S4.I1.i1.p1.13.m13.2.2.2.2.2" xref="S4.I1.i1.p1.13.m13.2.2.2.2.2.cmml">v</mi><mo id="S4.I1.i1.p1.13.m13.2.2.2.2.3" xref="S4.I1.i1.p1.13.m13.2.2.2.2.3.cmml">′</mo></msup></mrow><annotation-xml encoding="MathML-Content" id="S4.I1.i1.p1.13.m13.2b"><list id="S4.I1.i1.p1.13.m13.2.2.3.cmml" xref="S4.I1.i1.p1.13.m13.2.2.2"><apply id="S4.I1.i1.p1.13.m13.1.1.1.1.cmml" xref="S4.I1.i1.p1.13.m13.1.1.1.1"><csymbol cd="ambiguous" id="S4.I1.i1.p1.13.m13.1.1.1.1.1.cmml" xref="S4.I1.i1.p1.13.m13.1.1.1.1">superscript</csymbol><ci id="S4.I1.i1.p1.13.m13.1.1.1.1.2.cmml" xref="S4.I1.i1.p1.13.m13.1.1.1.1.2">𝑢</ci><ci id="S4.I1.i1.p1.13.m13.1.1.1.1.3.cmml" xref="S4.I1.i1.p1.13.m13.1.1.1.1.3">′</ci></apply><apply id="S4.I1.i1.p1.13.m13.2.2.2.2.cmml" xref="S4.I1.i1.p1.13.m13.2.2.2.2"><csymbol cd="ambiguous" id="S4.I1.i1.p1.13.m13.2.2.2.2.1.cmml" xref="S4.I1.i1.p1.13.m13.2.2.2.2">superscript</csymbol><ci id="S4.I1.i1.p1.13.m13.2.2.2.2.2.cmml" xref="S4.I1.i1.p1.13.m13.2.2.2.2.2">𝑣</ci><ci id="S4.I1.i1.p1.13.m13.2.2.2.2.3.cmml" xref="S4.I1.i1.p1.13.m13.2.2.2.2.3">′</ci></apply></list></annotation-xml><annotation encoding="application/x-tex" id="S4.I1.i1.p1.13.m13.2c">u^{\prime},v^{\prime}</annotation><annotation encoding="application/x-llamapun" id="S4.I1.i1.p1.13.m13.2d">italic_u start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT , italic_v start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math> are children of <math alttext="a" class="ltx_Math" display="inline" id="S4.I1.i1.p1.14.m14.1"><semantics id="S4.I1.i1.p1.14.m14.1a"><mi id="S4.I1.i1.p1.14.m14.1.1" xref="S4.I1.i1.p1.14.m14.1.1.cmml">a</mi><annotation-xml encoding="MathML-Content" id="S4.I1.i1.p1.14.m14.1b"><ci id="S4.I1.i1.p1.14.m14.1.1.cmml" xref="S4.I1.i1.p1.14.m14.1.1">𝑎</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.I1.i1.p1.14.m14.1c">a</annotation><annotation encoding="application/x-llamapun" id="S4.I1.i1.p1.14.m14.1d">italic_a</annotation></semantics></math>; thus in this case, <math alttext="u" class="ltx_Math" display="inline" id="S4.I1.i1.p1.15.m15.1"><semantics id="S4.I1.i1.p1.15.m15.1a"><mi id="S4.I1.i1.p1.15.m15.1.1" xref="S4.I1.i1.p1.15.m15.1.1.cmml">u</mi><annotation-xml encoding="MathML-Content" id="S4.I1.i1.p1.15.m15.1b"><ci id="S4.I1.i1.p1.15.m15.1.1.cmml" xref="S4.I1.i1.p1.15.m15.1.1">𝑢</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.I1.i1.p1.15.m15.1c">u</annotation><annotation encoding="application/x-llamapun" id="S4.I1.i1.p1.15.m15.1d">italic_u</annotation></semantics></math> and <math alttext="v" class="ltx_Math" display="inline" id="S4.I1.i1.p1.16.m16.1"><semantics id="S4.I1.i1.p1.16.m16.1a"><mi id="S4.I1.i1.p1.16.m16.1.1" xref="S4.I1.i1.p1.16.m16.1.1.cmml">v</mi><annotation-xml encoding="MathML-Content" id="S4.I1.i1.p1.16.m16.1b"><ci id="S4.I1.i1.p1.16.m16.1.1.cmml" xref="S4.I1.i1.p1.16.m16.1.1">𝑣</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.I1.i1.p1.16.m16.1c">v</annotation><annotation encoding="application/x-llamapun" id="S4.I1.i1.p1.16.m16.1d">italic_v</annotation></semantics></math> are connected via <math alttext="T^{\prime\prime}_{a}" class="ltx_Math" display="inline" id="S4.I1.i1.p1.17.m17.1"><semantics id="S4.I1.i1.p1.17.m17.1a"><msubsup id="S4.I1.i1.p1.17.m17.1.1" xref="S4.I1.i1.p1.17.m17.1.1.cmml"><mi id="S4.I1.i1.p1.17.m17.1.1.2.2" xref="S4.I1.i1.p1.17.m17.1.1.2.2.cmml">T</mi><mi id="S4.I1.i1.p1.17.m17.1.1.3" xref="S4.I1.i1.p1.17.m17.1.1.3.cmml">a</mi><mo id="S4.I1.i1.p1.17.m17.1.1.2.3" xref="S4.I1.i1.p1.17.m17.1.1.2.3.cmml">′′</mo></msubsup><annotation-xml encoding="MathML-Content" id="S4.I1.i1.p1.17.m17.1b"><apply id="S4.I1.i1.p1.17.m17.1.1.cmml" xref="S4.I1.i1.p1.17.m17.1.1"><csymbol cd="ambiguous" id="S4.I1.i1.p1.17.m17.1.1.1.cmml" xref="S4.I1.i1.p1.17.m17.1.1">subscript</csymbol><apply id="S4.I1.i1.p1.17.m17.1.1.2.cmml" xref="S4.I1.i1.p1.17.m17.1.1"><csymbol cd="ambiguous" id="S4.I1.i1.p1.17.m17.1.1.2.1.cmml" xref="S4.I1.i1.p1.17.m17.1.1">superscript</csymbol><ci id="S4.I1.i1.p1.17.m17.1.1.2.2.cmml" xref="S4.I1.i1.p1.17.m17.1.1.2.2">𝑇</ci><ci id="S4.I1.i1.p1.17.m17.1.1.2.3.cmml" xref="S4.I1.i1.p1.17.m17.1.1.2.3">′′</ci></apply><ci id="S4.I1.i1.p1.17.m17.1.1.3.cmml" xref="S4.I1.i1.p1.17.m17.1.1.3">𝑎</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.I1.i1.p1.17.m17.1c">T^{\prime\prime}_{a}</annotation><annotation encoding="application/x-llamapun" id="S4.I1.i1.p1.17.m17.1d">italic_T start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT</annotation></semantics></math>. See Figure <a class="ltx_ref" href="https://arxiv.org/html/2503.00712v1#S4.F3.sf1" title="In Figure 3 ‣ 4.1.2 Bounding Approximation Ratio ‣ 4.1 One-to-Two Augmentation ‣ 4 Vertex Connectivity Augmentation in Link-Arrival Model ‣ Streaming Algorithms for Network Design"><span class="ltx_text ltx_ref_tag">3(a)</span></a> for reference. Note that this case is the only one in which we use the contracted MST <math alttext="T_{a}^{\prime\prime}" class="ltx_Math" display="inline" id="S4.I1.i1.p1.18.m18.1"><semantics id="S4.I1.i1.p1.18.m18.1a"><msubsup id="S4.I1.i1.p1.18.m18.1.1" xref="S4.I1.i1.p1.18.m18.1.1.cmml"><mi id="S4.I1.i1.p1.18.m18.1.1.2.2" xref="S4.I1.i1.p1.18.m18.1.1.2.2.cmml">T</mi><mi id="S4.I1.i1.p1.18.m18.1.1.2.3" xref="S4.I1.i1.p1.18.m18.1.1.2.3.cmml">a</mi><mo id="S4.I1.i1.p1.18.m18.1.1.3" xref="S4.I1.i1.p1.18.m18.1.1.3.cmml">′′</mo></msubsup><annotation-xml encoding="MathML-Content" id="S4.I1.i1.p1.18.m18.1b"><apply id="S4.I1.i1.p1.18.m18.1.1.cmml" xref="S4.I1.i1.p1.18.m18.1.1"><csymbol cd="ambiguous" id="S4.I1.i1.p1.18.m18.1.1.1.cmml" xref="S4.I1.i1.p1.18.m18.1.1">superscript</csymbol><apply id="S4.I1.i1.p1.18.m18.1.1.2.cmml" xref="S4.I1.i1.p1.18.m18.1.1"><csymbol cd="ambiguous" id="S4.I1.i1.p1.18.m18.1.1.2.1.cmml" xref="S4.I1.i1.p1.18.m18.1.1">subscript</csymbol><ci id="S4.I1.i1.p1.18.m18.1.1.2.2.cmml" xref="S4.I1.i1.p1.18.m18.1.1.2.2">𝑇</ci><ci id="S4.I1.i1.p1.18.m18.1.1.2.3.cmml" xref="S4.I1.i1.p1.18.m18.1.1.2.3">𝑎</ci></apply><ci id="S4.I1.i1.p1.18.m18.1.1.3.cmml" xref="S4.I1.i1.p1.18.m18.1.1.3">′′</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.I1.i1.p1.18.m18.1c">T_{a}^{\prime\prime}</annotation><annotation encoding="application/x-llamapun" id="S4.I1.i1.p1.18.m18.1d">italic_T start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="S4.I1.i1.p2"> <p class="ltx_p" id="S4.I1.i1.p2.12">Suppose instead that <em class="ltx_emph ltx_font_italic" id="S4.I1.i1.p2.12.1">both</em> <math alttext="u^{\prime}" class="ltx_Math" display="inline" id="S4.I1.i1.p2.1.m1.1"><semantics id="S4.I1.i1.p2.1.m1.1a"><msup id="S4.I1.i1.p2.1.m1.1.1" xref="S4.I1.i1.p2.1.m1.1.1.cmml"><mi id="S4.I1.i1.p2.1.m1.1.1.2" xref="S4.I1.i1.p2.1.m1.1.1.2.cmml">u</mi><mo id="S4.I1.i1.p2.1.m1.1.1.3" xref="S4.I1.i1.p2.1.m1.1.1.3.cmml">′</mo></msup><annotation-xml encoding="MathML-Content" id="S4.I1.i1.p2.1.m1.1b"><apply id="S4.I1.i1.p2.1.m1.1.1.cmml" xref="S4.I1.i1.p2.1.m1.1.1"><csymbol cd="ambiguous" id="S4.I1.i1.p2.1.m1.1.1.1.cmml" xref="S4.I1.i1.p2.1.m1.1.1">superscript</csymbol><ci id="S4.I1.i1.p2.1.m1.1.1.2.cmml" xref="S4.I1.i1.p2.1.m1.1.1.2">𝑢</ci><ci id="S4.I1.i1.p2.1.m1.1.1.3.cmml" xref="S4.I1.i1.p2.1.m1.1.1.3">′</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.I1.i1.p2.1.m1.1c">u^{\prime}</annotation><annotation encoding="application/x-llamapun" id="S4.I1.i1.p2.1.m1.1d">italic_u start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math> and <math alttext="v^{\prime}" class="ltx_Math" display="inline" id="S4.I1.i1.p2.2.m2.1"><semantics id="S4.I1.i1.p2.2.m2.1a"><msup id="S4.I1.i1.p2.2.m2.1.1" xref="S4.I1.i1.p2.2.m2.1.1.cmml"><mi id="S4.I1.i1.p2.2.m2.1.1.2" xref="S4.I1.i1.p2.2.m2.1.1.2.cmml">v</mi><mo id="S4.I1.i1.p2.2.m2.1.1.3" xref="S4.I1.i1.p2.2.m2.1.1.3.cmml">′</mo></msup><annotation-xml encoding="MathML-Content" id="S4.I1.i1.p2.2.m2.1b"><apply id="S4.I1.i1.p2.2.m2.1.1.cmml" xref="S4.I1.i1.p2.2.m2.1.1"><csymbol cd="ambiguous" id="S4.I1.i1.p2.2.m2.1.1.1.cmml" xref="S4.I1.i1.p2.2.m2.1.1">superscript</csymbol><ci id="S4.I1.i1.p2.2.m2.1.1.2.cmml" xref="S4.I1.i1.p2.2.m2.1.1.2">𝑣</ci><ci id="S4.I1.i1.p2.2.m2.1.1.3.cmml" xref="S4.I1.i1.p2.2.m2.1.1.3">′</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.I1.i1.p2.2.m2.1c">v^{\prime}</annotation><annotation encoding="application/x-llamapun" id="S4.I1.i1.p2.2.m2.1d">italic_v start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math> are marked as “good”, then we will show that they are connected via <math alttext="G\setminus G_{a}" class="ltx_Math" display="inline" id="S4.I1.i1.p2.3.m3.1"><semantics id="S4.I1.i1.p2.3.m3.1a"><mrow id="S4.I1.i1.p2.3.m3.1.1" xref="S4.I1.i1.p2.3.m3.1.1.cmml"><mi id="S4.I1.i1.p2.3.m3.1.1.2" xref="S4.I1.i1.p2.3.m3.1.1.2.cmml">G</mi><mo id="S4.I1.i1.p2.3.m3.1.1.1" xref="S4.I1.i1.p2.3.m3.1.1.1.cmml">∖</mo><msub id="S4.I1.i1.p2.3.m3.1.1.3" xref="S4.I1.i1.p2.3.m3.1.1.3.cmml"><mi id="S4.I1.i1.p2.3.m3.1.1.3.2" xref="S4.I1.i1.p2.3.m3.1.1.3.2.cmml">G</mi><mi id="S4.I1.i1.p2.3.m3.1.1.3.3" xref="S4.I1.i1.p2.3.m3.1.1.3.3.cmml">a</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S4.I1.i1.p2.3.m3.1b"><apply id="S4.I1.i1.p2.3.m3.1.1.cmml" xref="S4.I1.i1.p2.3.m3.1.1"><setdiff id="S4.I1.i1.p2.3.m3.1.1.1.cmml" xref="S4.I1.i1.p2.3.m3.1.1.1"></setdiff><ci id="S4.I1.i1.p2.3.m3.1.1.2.cmml" xref="S4.I1.i1.p2.3.m3.1.1.2">𝐺</ci><apply id="S4.I1.i1.p2.3.m3.1.1.3.cmml" xref="S4.I1.i1.p2.3.m3.1.1.3"><csymbol cd="ambiguous" id="S4.I1.i1.p2.3.m3.1.1.3.1.cmml" xref="S4.I1.i1.p2.3.m3.1.1.3">subscript</csymbol><ci id="S4.I1.i1.p2.3.m3.1.1.3.2.cmml" xref="S4.I1.i1.p2.3.m3.1.1.3.2">𝐺</ci><ci id="S4.I1.i1.p2.3.m3.1.1.3.3.cmml" xref="S4.I1.i1.p2.3.m3.1.1.3.3">𝑎</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.I1.i1.p2.3.m3.1c">G\setminus G_{a}</annotation><annotation encoding="application/x-llamapun" id="S4.I1.i1.p2.3.m3.1d">italic_G ∖ italic_G start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT</annotation></semantics></math> (see Figure <a class="ltx_ref" href="https://arxiv.org/html/2503.00712v1#S4.F3.sf1" title="In Figure 3 ‣ 4.1.2 Bounding Approximation Ratio ‣ 4.1 One-to-Two Augmentation ‣ 4 Vertex Connectivity Augmentation in Link-Arrival Model ‣ Streaming Algorithms for Network Design"><span class="ltx_text ltx_ref_tag">3(a)</span></a>). Since <math alttext="G_{u^{\prime}}" class="ltx_Math" display="inline" id="S4.I1.i1.p2.4.m4.1"><semantics id="S4.I1.i1.p2.4.m4.1a"><msub id="S4.I1.i1.p2.4.m4.1.1" xref="S4.I1.i1.p2.4.m4.1.1.cmml"><mi id="S4.I1.i1.p2.4.m4.1.1.2" xref="S4.I1.i1.p2.4.m4.1.1.2.cmml">G</mi><msup id="S4.I1.i1.p2.4.m4.1.1.3" xref="S4.I1.i1.p2.4.m4.1.1.3.cmml"><mi id="S4.I1.i1.p2.4.m4.1.1.3.2" xref="S4.I1.i1.p2.4.m4.1.1.3.2.cmml">u</mi><mo id="S4.I1.i1.p2.4.m4.1.1.3.3" xref="S4.I1.i1.p2.4.m4.1.1.3.3.cmml">′</mo></msup></msub><annotation-xml encoding="MathML-Content" id="S4.I1.i1.p2.4.m4.1b"><apply id="S4.I1.i1.p2.4.m4.1.1.cmml" xref="S4.I1.i1.p2.4.m4.1.1"><csymbol cd="ambiguous" id="S4.I1.i1.p2.4.m4.1.1.1.cmml" xref="S4.I1.i1.p2.4.m4.1.1">subscript</csymbol><ci id="S4.I1.i1.p2.4.m4.1.1.2.cmml" xref="S4.I1.i1.p2.4.m4.1.1.2">𝐺</ci><apply id="S4.I1.i1.p2.4.m4.1.1.3.cmml" xref="S4.I1.i1.p2.4.m4.1.1.3"><csymbol cd="ambiguous" id="S4.I1.i1.p2.4.m4.1.1.3.1.cmml" xref="S4.I1.i1.p2.4.m4.1.1.3">superscript</csymbol><ci id="S4.I1.i1.p2.4.m4.1.1.3.2.cmml" xref="S4.I1.i1.p2.4.m4.1.1.3.2">𝑢</ci><ci id="S4.I1.i1.p2.4.m4.1.1.3.3.cmml" xref="S4.I1.i1.p2.4.m4.1.1.3.3">′</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.I1.i1.p2.4.m4.1c">G_{u^{\prime}}</annotation><annotation encoding="application/x-llamapun" id="S4.I1.i1.p2.4.m4.1d">italic_G start_POSTSUBSCRIPT italic_u start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT</annotation></semantics></math> is good, there must be some link <math alttext="e_{u}\in\textnormal{OPT}[G_{u^{\prime}},G\setminus G_{a}]" class="ltx_Math" display="inline" id="S4.I1.i1.p2.5.m5.2"><semantics id="S4.I1.i1.p2.5.m5.2a"><mrow id="S4.I1.i1.p2.5.m5.2.2" xref="S4.I1.i1.p2.5.m5.2.2.cmml"><msub id="S4.I1.i1.p2.5.m5.2.2.4" xref="S4.I1.i1.p2.5.m5.2.2.4.cmml"><mi id="S4.I1.i1.p2.5.m5.2.2.4.2" xref="S4.I1.i1.p2.5.m5.2.2.4.2.cmml">e</mi><mi id="S4.I1.i1.p2.5.m5.2.2.4.3" xref="S4.I1.i1.p2.5.m5.2.2.4.3.cmml">u</mi></msub><mo id="S4.I1.i1.p2.5.m5.2.2.3" xref="S4.I1.i1.p2.5.m5.2.2.3.cmml">∈</mo><mrow id="S4.I1.i1.p2.5.m5.2.2.2" xref="S4.I1.i1.p2.5.m5.2.2.2.cmml"><mtext id="S4.I1.i1.p2.5.m5.2.2.2.4" xref="S4.I1.i1.p2.5.m5.2.2.2.4a.cmml">OPT</mtext><mo id="S4.I1.i1.p2.5.m5.2.2.2.3" xref="S4.I1.i1.p2.5.m5.2.2.2.3.cmml">⁢</mo><mrow id="S4.I1.i1.p2.5.m5.2.2.2.2.2" xref="S4.I1.i1.p2.5.m5.2.2.2.2.3.cmml"><mo id="S4.I1.i1.p2.5.m5.2.2.2.2.2.3" stretchy="false" xref="S4.I1.i1.p2.5.m5.2.2.2.2.3.cmml">[</mo><msub id="S4.I1.i1.p2.5.m5.1.1.1.1.1.1" xref="S4.I1.i1.p2.5.m5.1.1.1.1.1.1.cmml"><mi id="S4.I1.i1.p2.5.m5.1.1.1.1.1.1.2" xref="S4.I1.i1.p2.5.m5.1.1.1.1.1.1.2.cmml">G</mi><msup id="S4.I1.i1.p2.5.m5.1.1.1.1.1.1.3" xref="S4.I1.i1.p2.5.m5.1.1.1.1.1.1.3.cmml"><mi id="S4.I1.i1.p2.5.m5.1.1.1.1.1.1.3.2" xref="S4.I1.i1.p2.5.m5.1.1.1.1.1.1.3.2.cmml">u</mi><mo id="S4.I1.i1.p2.5.m5.1.1.1.1.1.1.3.3" xref="S4.I1.i1.p2.5.m5.1.1.1.1.1.1.3.3.cmml">′</mo></msup></msub><mo id="S4.I1.i1.p2.5.m5.2.2.2.2.2.4" xref="S4.I1.i1.p2.5.m5.2.2.2.2.3.cmml">,</mo><mrow id="S4.I1.i1.p2.5.m5.2.2.2.2.2.2" xref="S4.I1.i1.p2.5.m5.2.2.2.2.2.2.cmml"><mi id="S4.I1.i1.p2.5.m5.2.2.2.2.2.2.2" xref="S4.I1.i1.p2.5.m5.2.2.2.2.2.2.2.cmml">G</mi><mo id="S4.I1.i1.p2.5.m5.2.2.2.2.2.2.1" xref="S4.I1.i1.p2.5.m5.2.2.2.2.2.2.1.cmml">∖</mo><msub id="S4.I1.i1.p2.5.m5.2.2.2.2.2.2.3" xref="S4.I1.i1.p2.5.m5.2.2.2.2.2.2.3.cmml"><mi id="S4.I1.i1.p2.5.m5.2.2.2.2.2.2.3.2" xref="S4.I1.i1.p2.5.m5.2.2.2.2.2.2.3.2.cmml">G</mi><mi id="S4.I1.i1.p2.5.m5.2.2.2.2.2.2.3.3" xref="S4.I1.i1.p2.5.m5.2.2.2.2.2.2.3.3.cmml">a</mi></msub></mrow><mo id="S4.I1.i1.p2.5.m5.2.2.2.2.2.5" stretchy="false" xref="S4.I1.i1.p2.5.m5.2.2.2.2.3.cmml">]</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.I1.i1.p2.5.m5.2b"><apply id="S4.I1.i1.p2.5.m5.2.2.cmml" xref="S4.I1.i1.p2.5.m5.2.2"><in id="S4.I1.i1.p2.5.m5.2.2.3.cmml" xref="S4.I1.i1.p2.5.m5.2.2.3"></in><apply id="S4.I1.i1.p2.5.m5.2.2.4.cmml" xref="S4.I1.i1.p2.5.m5.2.2.4"><csymbol cd="ambiguous" id="S4.I1.i1.p2.5.m5.2.2.4.1.cmml" xref="S4.I1.i1.p2.5.m5.2.2.4">subscript</csymbol><ci id="S4.I1.i1.p2.5.m5.2.2.4.2.cmml" xref="S4.I1.i1.p2.5.m5.2.2.4.2">𝑒</ci><ci id="S4.I1.i1.p2.5.m5.2.2.4.3.cmml" xref="S4.I1.i1.p2.5.m5.2.2.4.3">𝑢</ci></apply><apply id="S4.I1.i1.p2.5.m5.2.2.2.cmml" xref="S4.I1.i1.p2.5.m5.2.2.2"><times id="S4.I1.i1.p2.5.m5.2.2.2.3.cmml" xref="S4.I1.i1.p2.5.m5.2.2.2.3"></times><ci id="S4.I1.i1.p2.5.m5.2.2.2.4a.cmml" xref="S4.I1.i1.p2.5.m5.2.2.2.4"><mtext id="S4.I1.i1.p2.5.m5.2.2.2.4.cmml" xref="S4.I1.i1.p2.5.m5.2.2.2.4">OPT</mtext></ci><interval closure="closed" id="S4.I1.i1.p2.5.m5.2.2.2.2.3.cmml" xref="S4.I1.i1.p2.5.m5.2.2.2.2.2"><apply id="S4.I1.i1.p2.5.m5.1.1.1.1.1.1.cmml" xref="S4.I1.i1.p2.5.m5.1.1.1.1.1.1"><csymbol cd="ambiguous" id="S4.I1.i1.p2.5.m5.1.1.1.1.1.1.1.cmml" xref="S4.I1.i1.p2.5.m5.1.1.1.1.1.1">subscript</csymbol><ci id="S4.I1.i1.p2.5.m5.1.1.1.1.1.1.2.cmml" xref="S4.I1.i1.p2.5.m5.1.1.1.1.1.1.2">𝐺</ci><apply id="S4.I1.i1.p2.5.m5.1.1.1.1.1.1.3.cmml" xref="S4.I1.i1.p2.5.m5.1.1.1.1.1.1.3"><csymbol cd="ambiguous" id="S4.I1.i1.p2.5.m5.1.1.1.1.1.1.3.1.cmml" xref="S4.I1.i1.p2.5.m5.1.1.1.1.1.1.3">superscript</csymbol><ci id="S4.I1.i1.p2.5.m5.1.1.1.1.1.1.3.2.cmml" xref="S4.I1.i1.p2.5.m5.1.1.1.1.1.1.3.2">𝑢</ci><ci id="S4.I1.i1.p2.5.m5.1.1.1.1.1.1.3.3.cmml" xref="S4.I1.i1.p2.5.m5.1.1.1.1.1.1.3.3">′</ci></apply></apply><apply id="S4.I1.i1.p2.5.m5.2.2.2.2.2.2.cmml" xref="S4.I1.i1.p2.5.m5.2.2.2.2.2.2"><setdiff id="S4.I1.i1.p2.5.m5.2.2.2.2.2.2.1.cmml" xref="S4.I1.i1.p2.5.m5.2.2.2.2.2.2.1"></setdiff><ci id="S4.I1.i1.p2.5.m5.2.2.2.2.2.2.2.cmml" xref="S4.I1.i1.p2.5.m5.2.2.2.2.2.2.2">𝐺</ci><apply id="S4.I1.i1.p2.5.m5.2.2.2.2.2.2.3.cmml" xref="S4.I1.i1.p2.5.m5.2.2.2.2.2.2.3"><csymbol cd="ambiguous" id="S4.I1.i1.p2.5.m5.2.2.2.2.2.2.3.1.cmml" xref="S4.I1.i1.p2.5.m5.2.2.2.2.2.2.3">subscript</csymbol><ci id="S4.I1.i1.p2.5.m5.2.2.2.2.2.2.3.2.cmml" xref="S4.I1.i1.p2.5.m5.2.2.2.2.2.2.3.2">𝐺</ci><ci id="S4.I1.i1.p2.5.m5.2.2.2.2.2.2.3.3.cmml" xref="S4.I1.i1.p2.5.m5.2.2.2.2.2.2.3.3">𝑎</ci></apply></apply></interval></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.I1.i1.p2.5.m5.2c">e_{u}\in\textnormal{OPT}[G_{u^{\prime}},G\setminus G_{a}]</annotation><annotation encoding="application/x-llamapun" id="S4.I1.i1.p2.5.m5.2d">italic_e start_POSTSUBSCRIPT italic_u end_POSTSUBSCRIPT ∈ OPT [ italic_G start_POSTSUBSCRIPT italic_u start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT , italic_G ∖ italic_G start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT ]</annotation></semantics></math>. Let <math alttext="u^{\prime\prime}" class="ltx_Math" display="inline" id="S4.I1.i1.p2.6.m6.1"><semantics id="S4.I1.i1.p2.6.m6.1a"><msup id="S4.I1.i1.p2.6.m6.1.1" xref="S4.I1.i1.p2.6.m6.1.1.cmml"><mi id="S4.I1.i1.p2.6.m6.1.1.2" xref="S4.I1.i1.p2.6.m6.1.1.2.cmml">u</mi><mo id="S4.I1.i1.p2.6.m6.1.1.3" xref="S4.I1.i1.p2.6.m6.1.1.3.cmml">′′</mo></msup><annotation-xml encoding="MathML-Content" id="S4.I1.i1.p2.6.m6.1b"><apply id="S4.I1.i1.p2.6.m6.1.1.cmml" xref="S4.I1.i1.p2.6.m6.1.1"><csymbol cd="ambiguous" id="S4.I1.i1.p2.6.m6.1.1.1.cmml" xref="S4.I1.i1.p2.6.m6.1.1">superscript</csymbol><ci id="S4.I1.i1.p2.6.m6.1.1.2.cmml" xref="S4.I1.i1.p2.6.m6.1.1.2">𝑢</ci><ci id="S4.I1.i1.p2.6.m6.1.1.3.cmml" xref="S4.I1.i1.p2.6.m6.1.1.3">′′</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.I1.i1.p2.6.m6.1c">u^{\prime\prime}</annotation><annotation encoding="application/x-llamapun" id="S4.I1.i1.p2.6.m6.1d">italic_u start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT</annotation></semantics></math> be the vertex incident to <math alttext="e_{u}" class="ltx_Math" display="inline" id="S4.I1.i1.p2.7.m7.1"><semantics id="S4.I1.i1.p2.7.m7.1a"><msub id="S4.I1.i1.p2.7.m7.1.1" xref="S4.I1.i1.p2.7.m7.1.1.cmml"><mi id="S4.I1.i1.p2.7.m7.1.1.2" xref="S4.I1.i1.p2.7.m7.1.1.2.cmml">e</mi><mi id="S4.I1.i1.p2.7.m7.1.1.3" xref="S4.I1.i1.p2.7.m7.1.1.3.cmml">u</mi></msub><annotation-xml encoding="MathML-Content" id="S4.I1.i1.p2.7.m7.1b"><apply id="S4.I1.i1.p2.7.m7.1.1.cmml" xref="S4.I1.i1.p2.7.m7.1.1"><csymbol cd="ambiguous" id="S4.I1.i1.p2.7.m7.1.1.1.cmml" xref="S4.I1.i1.p2.7.m7.1.1">subscript</csymbol><ci id="S4.I1.i1.p2.7.m7.1.1.2.cmml" xref="S4.I1.i1.p2.7.m7.1.1.2">𝑒</ci><ci id="S4.I1.i1.p2.7.m7.1.1.3.cmml" xref="S4.I1.i1.p2.7.m7.1.1.3">𝑢</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.I1.i1.p2.7.m7.1c">e_{u}</annotation><annotation encoding="application/x-llamapun" id="S4.I1.i1.p2.7.m7.1d">italic_e start_POSTSUBSCRIPT italic_u end_POSTSUBSCRIPT</annotation></semantics></math> in <math alttext="G_{u^{\prime}}" class="ltx_Math" display="inline" id="S4.I1.i1.p2.8.m8.1"><semantics id="S4.I1.i1.p2.8.m8.1a"><msub id="S4.I1.i1.p2.8.m8.1.1" xref="S4.I1.i1.p2.8.m8.1.1.cmml"><mi id="S4.I1.i1.p2.8.m8.1.1.2" xref="S4.I1.i1.p2.8.m8.1.1.2.cmml">G</mi><msup id="S4.I1.i1.p2.8.m8.1.1.3" xref="S4.I1.i1.p2.8.m8.1.1.3.cmml"><mi id="S4.I1.i1.p2.8.m8.1.1.3.2" xref="S4.I1.i1.p2.8.m8.1.1.3.2.cmml">u</mi><mo id="S4.I1.i1.p2.8.m8.1.1.3.3" xref="S4.I1.i1.p2.8.m8.1.1.3.3.cmml">′</mo></msup></msub><annotation-xml encoding="MathML-Content" id="S4.I1.i1.p2.8.m8.1b"><apply id="S4.I1.i1.p2.8.m8.1.1.cmml" xref="S4.I1.i1.p2.8.m8.1.1"><csymbol cd="ambiguous" id="S4.I1.i1.p2.8.m8.1.1.1.cmml" xref="S4.I1.i1.p2.8.m8.1.1">subscript</csymbol><ci id="S4.I1.i1.p2.8.m8.1.1.2.cmml" xref="S4.I1.i1.p2.8.m8.1.1.2">𝐺</ci><apply id="S4.I1.i1.p2.8.m8.1.1.3.cmml" xref="S4.I1.i1.p2.8.m8.1.1.3"><csymbol cd="ambiguous" id="S4.I1.i1.p2.8.m8.1.1.3.1.cmml" xref="S4.I1.i1.p2.8.m8.1.1.3">superscript</csymbol><ci id="S4.I1.i1.p2.8.m8.1.1.3.2.cmml" xref="S4.I1.i1.p2.8.m8.1.1.3.2">𝑢</ci><ci id="S4.I1.i1.p2.8.m8.1.1.3.3.cmml" xref="S4.I1.i1.p2.8.m8.1.1.3.3">′</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.I1.i1.p2.8.m8.1c">G_{u^{\prime}}</annotation><annotation encoding="application/x-llamapun" id="S4.I1.i1.p2.8.m8.1d">italic_G start_POSTSUBSCRIPT italic_u start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT</annotation></semantics></math>, and let <math alttext="j" class="ltx_Math" display="inline" id="S4.I1.i1.p2.9.m9.1"><semantics id="S4.I1.i1.p2.9.m9.1a"><mi id="S4.I1.i1.p2.9.m9.1.1" xref="S4.I1.i1.p2.9.m9.1.1.cmml">j</mi><annotation-xml encoding="MathML-Content" id="S4.I1.i1.p2.9.m9.1b"><ci id="S4.I1.i1.p2.9.m9.1.1.cmml" xref="S4.I1.i1.p2.9.m9.1.1">𝑗</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.I1.i1.p2.9.m9.1c">j</annotation><annotation encoding="application/x-llamapun" id="S4.I1.i1.p2.9.m9.1d">italic_j</annotation></semantics></math> be the weight class of <math alttext="e_{u}" class="ltx_Math" display="inline" id="S4.I1.i1.p2.10.m10.1"><semantics id="S4.I1.i1.p2.10.m10.1a"><msub id="S4.I1.i1.p2.10.m10.1.1" xref="S4.I1.i1.p2.10.m10.1.1.cmml"><mi id="S4.I1.i1.p2.10.m10.1.1.2" xref="S4.I1.i1.p2.10.m10.1.1.2.cmml">e</mi><mi id="S4.I1.i1.p2.10.m10.1.1.3" xref="S4.I1.i1.p2.10.m10.1.1.3.cmml">u</mi></msub><annotation-xml encoding="MathML-Content" id="S4.I1.i1.p2.10.m10.1b"><apply id="S4.I1.i1.p2.10.m10.1.1.cmml" xref="S4.I1.i1.p2.10.m10.1.1"><csymbol cd="ambiguous" id="S4.I1.i1.p2.10.m10.1.1.1.cmml" xref="S4.I1.i1.p2.10.m10.1.1">subscript</csymbol><ci id="S4.I1.i1.p2.10.m10.1.1.2.cmml" xref="S4.I1.i1.p2.10.m10.1.1.2">𝑒</ci><ci id="S4.I1.i1.p2.10.m10.1.1.3.cmml" xref="S4.I1.i1.p2.10.m10.1.1.3">𝑢</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.I1.i1.p2.10.m10.1c">e_{u}</annotation><annotation encoding="application/x-llamapun" id="S4.I1.i1.p2.10.m10.1d">italic_e start_POSTSUBSCRIPT italic_u end_POSTSUBSCRIPT</annotation></semantics></math>. Then <math alttext="L_{u^{\prime\prime}}(j)" class="ltx_Math" display="inline" id="S4.I1.i1.p2.11.m11.1"><semantics id="S4.I1.i1.p2.11.m11.1a"><mrow id="S4.I1.i1.p2.11.m11.1.2" xref="S4.I1.i1.p2.11.m11.1.2.cmml"><msub id="S4.I1.i1.p2.11.m11.1.2.2" xref="S4.I1.i1.p2.11.m11.1.2.2.cmml"><mi id="S4.I1.i1.p2.11.m11.1.2.2.2" xref="S4.I1.i1.p2.11.m11.1.2.2.2.cmml">L</mi><msup id="S4.I1.i1.p2.11.m11.1.2.2.3" xref="S4.I1.i1.p2.11.m11.1.2.2.3.cmml"><mi id="S4.I1.i1.p2.11.m11.1.2.2.3.2" xref="S4.I1.i1.p2.11.m11.1.2.2.3.2.cmml">u</mi><mo id="S4.I1.i1.p2.11.m11.1.2.2.3.3" xref="S4.I1.i1.p2.11.m11.1.2.2.3.3.cmml">′′</mo></msup></msub><mo id="S4.I1.i1.p2.11.m11.1.2.1" xref="S4.I1.i1.p2.11.m11.1.2.1.cmml">⁢</mo><mrow id="S4.I1.i1.p2.11.m11.1.2.3.2" xref="S4.I1.i1.p2.11.m11.1.2.cmml"><mo id="S4.I1.i1.p2.11.m11.1.2.3.2.1" stretchy="false" xref="S4.I1.i1.p2.11.m11.1.2.cmml">(</mo><mi id="S4.I1.i1.p2.11.m11.1.1" xref="S4.I1.i1.p2.11.m11.1.1.cmml">j</mi><mo id="S4.I1.i1.p2.11.m11.1.2.3.2.2" stretchy="false" xref="S4.I1.i1.p2.11.m11.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.I1.i1.p2.11.m11.1b"><apply id="S4.I1.i1.p2.11.m11.1.2.cmml" xref="S4.I1.i1.p2.11.m11.1.2"><times id="S4.I1.i1.p2.11.m11.1.2.1.cmml" xref="S4.I1.i1.p2.11.m11.1.2.1"></times><apply id="S4.I1.i1.p2.11.m11.1.2.2.cmml" xref="S4.I1.i1.p2.11.m11.1.2.2"><csymbol cd="ambiguous" id="S4.I1.i1.p2.11.m11.1.2.2.1.cmml" xref="S4.I1.i1.p2.11.m11.1.2.2">subscript</csymbol><ci id="S4.I1.i1.p2.11.m11.1.2.2.2.cmml" xref="S4.I1.i1.p2.11.m11.1.2.2.2">𝐿</ci><apply id="S4.I1.i1.p2.11.m11.1.2.2.3.cmml" xref="S4.I1.i1.p2.11.m11.1.2.2.3"><csymbol cd="ambiguous" id="S4.I1.i1.p2.11.m11.1.2.2.3.1.cmml" xref="S4.I1.i1.p2.11.m11.1.2.2.3">superscript</csymbol><ci id="S4.I1.i1.p2.11.m11.1.2.2.3.2.cmml" xref="S4.I1.i1.p2.11.m11.1.2.2.3.2">𝑢</ci><ci id="S4.I1.i1.p2.11.m11.1.2.2.3.3.cmml" xref="S4.I1.i1.p2.11.m11.1.2.2.3.3">′′</ci></apply></apply><ci id="S4.I1.i1.p2.11.m11.1.1.cmml" xref="S4.I1.i1.p2.11.m11.1.1">𝑗</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.I1.i1.p2.11.m11.1c">L_{u^{\prime\prime}}(j)</annotation><annotation encoding="application/x-llamapun" id="S4.I1.i1.p2.11.m11.1d">italic_L start_POSTSUBSCRIPT italic_u start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT ( italic_j )</annotation></semantics></math> must have an endpoint in <math alttext="G\setminus G_{a}" class="ltx_Math" display="inline" id="S4.I1.i1.p2.12.m12.1"><semantics id="S4.I1.i1.p2.12.m12.1a"><mrow id="S4.I1.i1.p2.12.m12.1.1" xref="S4.I1.i1.p2.12.m12.1.1.cmml"><mi id="S4.I1.i1.p2.12.m12.1.1.2" xref="S4.I1.i1.p2.12.m12.1.1.2.cmml">G</mi><mo id="S4.I1.i1.p2.12.m12.1.1.1" xref="S4.I1.i1.p2.12.m12.1.1.1.cmml">∖</mo><msub id="S4.I1.i1.p2.12.m12.1.1.3" xref="S4.I1.i1.p2.12.m12.1.1.3.cmml"><mi id="S4.I1.i1.p2.12.m12.1.1.3.2" xref="S4.I1.i1.p2.12.m12.1.1.3.2.cmml">G</mi><mi id="S4.I1.i1.p2.12.m12.1.1.3.3" xref="S4.I1.i1.p2.12.m12.1.1.3.3.cmml">a</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S4.I1.i1.p2.12.m12.1b"><apply id="S4.I1.i1.p2.12.m12.1.1.cmml" xref="S4.I1.i1.p2.12.m12.1.1"><setdiff id="S4.I1.i1.p2.12.m12.1.1.1.cmml" xref="S4.I1.i1.p2.12.m12.1.1.1"></setdiff><ci id="S4.I1.i1.p2.12.m12.1.1.2.cmml" xref="S4.I1.i1.p2.12.m12.1.1.2">𝐺</ci><apply id="S4.I1.i1.p2.12.m12.1.1.3.cmml" xref="S4.I1.i1.p2.12.m12.1.1.3"><csymbol cd="ambiguous" id="S4.I1.i1.p2.12.m12.1.1.3.1.cmml" xref="S4.I1.i1.p2.12.m12.1.1.3">subscript</csymbol><ci id="S4.I1.i1.p2.12.m12.1.1.3.2.cmml" xref="S4.I1.i1.p2.12.m12.1.1.3.2">𝐺</ci><ci id="S4.I1.i1.p2.12.m12.1.1.3.3.cmml" xref="S4.I1.i1.p2.12.m12.1.1.3.3">𝑎</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.I1.i1.p2.12.m12.1c">G\setminus G_{a}</annotation><annotation encoding="application/x-llamapun" id="S4.I1.i1.p2.12.m12.1d">italic_G ∖ italic_G start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT</annotation></semantics></math>, since</p> <table class="ltx_equation ltx_eqn_table" id="S4.Ex9"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="d_{G}(r,\text{LCA}(L_{u^{\prime\prime}}(j)))\leq d_{G}(r,\text{LCA}(e_{u}))<d_% {G}(r,a)." class="ltx_Math" display="block" id="S4.Ex9.m1.6"><semantics id="S4.Ex9.m1.6a"><mrow id="S4.Ex9.m1.6.6.1" xref="S4.Ex9.m1.6.6.1.1.cmml"><mrow id="S4.Ex9.m1.6.6.1.1" xref="S4.Ex9.m1.6.6.1.1.cmml"><mrow id="S4.Ex9.m1.6.6.1.1.1" xref="S4.Ex9.m1.6.6.1.1.1.cmml"><msub id="S4.Ex9.m1.6.6.1.1.1.3" xref="S4.Ex9.m1.6.6.1.1.1.3.cmml"><mi id="S4.Ex9.m1.6.6.1.1.1.3.2" xref="S4.Ex9.m1.6.6.1.1.1.3.2.cmml">d</mi><mi id="S4.Ex9.m1.6.6.1.1.1.3.3" xref="S4.Ex9.m1.6.6.1.1.1.3.3.cmml">G</mi></msub><mo id="S4.Ex9.m1.6.6.1.1.1.2" xref="S4.Ex9.m1.6.6.1.1.1.2.cmml">⁢</mo><mrow id="S4.Ex9.m1.6.6.1.1.1.1.1" xref="S4.Ex9.m1.6.6.1.1.1.1.2.cmml"><mo id="S4.Ex9.m1.6.6.1.1.1.1.1.2" stretchy="false" xref="S4.Ex9.m1.6.6.1.1.1.1.2.cmml">(</mo><mi id="S4.Ex9.m1.2.2" xref="S4.Ex9.m1.2.2.cmml">r</mi><mo id="S4.Ex9.m1.6.6.1.1.1.1.1.3" xref="S4.Ex9.m1.6.6.1.1.1.1.2.cmml">,</mo><mrow id="S4.Ex9.m1.6.6.1.1.1.1.1.1" xref="S4.Ex9.m1.6.6.1.1.1.1.1.1.cmml"><mtext id="S4.Ex9.m1.6.6.1.1.1.1.1.1.3" xref="S4.Ex9.m1.6.6.1.1.1.1.1.1.3a.cmml">LCA</mtext><mo id="S4.Ex9.m1.6.6.1.1.1.1.1.1.2" xref="S4.Ex9.m1.6.6.1.1.1.1.1.1.2.cmml">⁢</mo><mrow id="S4.Ex9.m1.6.6.1.1.1.1.1.1.1.1" xref="S4.Ex9.m1.6.6.1.1.1.1.1.1.1.1.1.cmml"><mo id="S4.Ex9.m1.6.6.1.1.1.1.1.1.1.1.2" stretchy="false" xref="S4.Ex9.m1.6.6.1.1.1.1.1.1.1.1.1.cmml">(</mo><mrow id="S4.Ex9.m1.6.6.1.1.1.1.1.1.1.1.1" xref="S4.Ex9.m1.6.6.1.1.1.1.1.1.1.1.1.cmml"><msub id="S4.Ex9.m1.6.6.1.1.1.1.1.1.1.1.1.2" xref="S4.Ex9.m1.6.6.1.1.1.1.1.1.1.1.1.2.cmml"><mi id="S4.Ex9.m1.6.6.1.1.1.1.1.1.1.1.1.2.2" xref="S4.Ex9.m1.6.6.1.1.1.1.1.1.1.1.1.2.2.cmml">L</mi><msup id="S4.Ex9.m1.6.6.1.1.1.1.1.1.1.1.1.2.3" xref="S4.Ex9.m1.6.6.1.1.1.1.1.1.1.1.1.2.3.cmml"><mi id="S4.Ex9.m1.6.6.1.1.1.1.1.1.1.1.1.2.3.2" xref="S4.Ex9.m1.6.6.1.1.1.1.1.1.1.1.1.2.3.2.cmml">u</mi><mo id="S4.Ex9.m1.6.6.1.1.1.1.1.1.1.1.1.2.3.3" xref="S4.Ex9.m1.6.6.1.1.1.1.1.1.1.1.1.2.3.3.cmml">′′</mo></msup></msub><mo id="S4.Ex9.m1.6.6.1.1.1.1.1.1.1.1.1.1" xref="S4.Ex9.m1.6.6.1.1.1.1.1.1.1.1.1.1.cmml">⁢</mo><mrow id="S4.Ex9.m1.6.6.1.1.1.1.1.1.1.1.1.3.2" xref="S4.Ex9.m1.6.6.1.1.1.1.1.1.1.1.1.cmml"><mo id="S4.Ex9.m1.6.6.1.1.1.1.1.1.1.1.1.3.2.1" stretchy="false" xref="S4.Ex9.m1.6.6.1.1.1.1.1.1.1.1.1.cmml">(</mo><mi id="S4.Ex9.m1.1.1" xref="S4.Ex9.m1.1.1.cmml">j</mi><mo id="S4.Ex9.m1.6.6.1.1.1.1.1.1.1.1.1.3.2.2" stretchy="false" xref="S4.Ex9.m1.6.6.1.1.1.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S4.Ex9.m1.6.6.1.1.1.1.1.1.1.1.3" stretchy="false" xref="S4.Ex9.m1.6.6.1.1.1.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S4.Ex9.m1.6.6.1.1.1.1.1.4" stretchy="false" xref="S4.Ex9.m1.6.6.1.1.1.1.2.cmml">)</mo></mrow></mrow><mo id="S4.Ex9.m1.6.6.1.1.4" xref="S4.Ex9.m1.6.6.1.1.4.cmml">≤</mo><mrow id="S4.Ex9.m1.6.6.1.1.2" xref="S4.Ex9.m1.6.6.1.1.2.cmml"><msub id="S4.Ex9.m1.6.6.1.1.2.3" xref="S4.Ex9.m1.6.6.1.1.2.3.cmml"><mi id="S4.Ex9.m1.6.6.1.1.2.3.2" xref="S4.Ex9.m1.6.6.1.1.2.3.2.cmml">d</mi><mi id="S4.Ex9.m1.6.6.1.1.2.3.3" xref="S4.Ex9.m1.6.6.1.1.2.3.3.cmml">G</mi></msub><mo id="S4.Ex9.m1.6.6.1.1.2.2" xref="S4.Ex9.m1.6.6.1.1.2.2.cmml">⁢</mo><mrow id="S4.Ex9.m1.6.6.1.1.2.1.1" xref="S4.Ex9.m1.6.6.1.1.2.1.2.cmml"><mo id="S4.Ex9.m1.6.6.1.1.2.1.1.2" stretchy="false" xref="S4.Ex9.m1.6.6.1.1.2.1.2.cmml">(</mo><mi id="S4.Ex9.m1.3.3" xref="S4.Ex9.m1.3.3.cmml">r</mi><mo id="S4.Ex9.m1.6.6.1.1.2.1.1.3" xref="S4.Ex9.m1.6.6.1.1.2.1.2.cmml">,</mo><mrow id="S4.Ex9.m1.6.6.1.1.2.1.1.1" xref="S4.Ex9.m1.6.6.1.1.2.1.1.1.cmml"><mtext id="S4.Ex9.m1.6.6.1.1.2.1.1.1.3" xref="S4.Ex9.m1.6.6.1.1.2.1.1.1.3a.cmml">LCA</mtext><mo id="S4.Ex9.m1.6.6.1.1.2.1.1.1.2" xref="S4.Ex9.m1.6.6.1.1.2.1.1.1.2.cmml">⁢</mo><mrow id="S4.Ex9.m1.6.6.1.1.2.1.1.1.1.1" xref="S4.Ex9.m1.6.6.1.1.2.1.1.1.1.1.1.cmml"><mo id="S4.Ex9.m1.6.6.1.1.2.1.1.1.1.1.2" stretchy="false" xref="S4.Ex9.m1.6.6.1.1.2.1.1.1.1.1.1.cmml">(</mo><msub id="S4.Ex9.m1.6.6.1.1.2.1.1.1.1.1.1" xref="S4.Ex9.m1.6.6.1.1.2.1.1.1.1.1.1.cmml"><mi id="S4.Ex9.m1.6.6.1.1.2.1.1.1.1.1.1.2" xref="S4.Ex9.m1.6.6.1.1.2.1.1.1.1.1.1.2.cmml">e</mi><mi id="S4.Ex9.m1.6.6.1.1.2.1.1.1.1.1.1.3" xref="S4.Ex9.m1.6.6.1.1.2.1.1.1.1.1.1.3.cmml">u</mi></msub><mo id="S4.Ex9.m1.6.6.1.1.2.1.1.1.1.1.3" stretchy="false" xref="S4.Ex9.m1.6.6.1.1.2.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S4.Ex9.m1.6.6.1.1.2.1.1.4" stretchy="false" xref="S4.Ex9.m1.6.6.1.1.2.1.2.cmml">)</mo></mrow></mrow><mo id="S4.Ex9.m1.6.6.1.1.5" xref="S4.Ex9.m1.6.6.1.1.5.cmml"><</mo><mrow id="S4.Ex9.m1.6.6.1.1.6" xref="S4.Ex9.m1.6.6.1.1.6.cmml"><msub id="S4.Ex9.m1.6.6.1.1.6.2" xref="S4.Ex9.m1.6.6.1.1.6.2.cmml"><mi id="S4.Ex9.m1.6.6.1.1.6.2.2" 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xref="S4.Ex9.m1.6.6.1.1.2.1.1.1.3"><mtext id="S4.Ex9.m1.6.6.1.1.2.1.1.1.3.cmml" xref="S4.Ex9.m1.6.6.1.1.2.1.1.1.3">LCA</mtext></ci><apply id="S4.Ex9.m1.6.6.1.1.2.1.1.1.1.1.1.cmml" xref="S4.Ex9.m1.6.6.1.1.2.1.1.1.1.1"><csymbol cd="ambiguous" id="S4.Ex9.m1.6.6.1.1.2.1.1.1.1.1.1.1.cmml" xref="S4.Ex9.m1.6.6.1.1.2.1.1.1.1.1">subscript</csymbol><ci id="S4.Ex9.m1.6.6.1.1.2.1.1.1.1.1.1.2.cmml" xref="S4.Ex9.m1.6.6.1.1.2.1.1.1.1.1.1.2">𝑒</ci><ci id="S4.Ex9.m1.6.6.1.1.2.1.1.1.1.1.1.3.cmml" xref="S4.Ex9.m1.6.6.1.1.2.1.1.1.1.1.1.3">𝑢</ci></apply></apply></interval></apply></apply><apply id="S4.Ex9.m1.6.6.1.1c.cmml" xref="S4.Ex9.m1.6.6.1"><lt id="S4.Ex9.m1.6.6.1.1.5.cmml" xref="S4.Ex9.m1.6.6.1.1.5"></lt><share href="https://arxiv.org/html/2503.00712v1#S4.Ex9.m1.6.6.1.1.2.cmml" id="S4.Ex9.m1.6.6.1.1d.cmml" xref="S4.Ex9.m1.6.6.1"></share><apply id="S4.Ex9.m1.6.6.1.1.6.cmml" xref="S4.Ex9.m1.6.6.1.1.6"><times id="S4.Ex9.m1.6.6.1.1.6.1.cmml" xref="S4.Ex9.m1.6.6.1.1.6.1"></times><apply id="S4.Ex9.m1.6.6.1.1.6.2.cmml" xref="S4.Ex9.m1.6.6.1.1.6.2"><csymbol cd="ambiguous" id="S4.Ex9.m1.6.6.1.1.6.2.1.cmml" xref="S4.Ex9.m1.6.6.1.1.6.2">subscript</csymbol><ci id="S4.Ex9.m1.6.6.1.1.6.2.2.cmml" xref="S4.Ex9.m1.6.6.1.1.6.2.2">𝑑</ci><ci id="S4.Ex9.m1.6.6.1.1.6.2.3.cmml" xref="S4.Ex9.m1.6.6.1.1.6.2.3">𝐺</ci></apply><interval closure="open" id="S4.Ex9.m1.6.6.1.1.6.3.1.cmml" xref="S4.Ex9.m1.6.6.1.1.6.3.2"><ci id="S4.Ex9.m1.4.4.cmml" xref="S4.Ex9.m1.4.4">𝑟</ci><ci id="S4.Ex9.m1.5.5.cmml" xref="S4.Ex9.m1.5.5">𝑎</ci></interval></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Ex9.m1.6c">d_{G}(r,\text{LCA}(L_{u^{\prime\prime}}(j)))\leq d_{G}(r,\text{LCA}(e_{u}))<d_% {G}(r,a).</annotation><annotation encoding="application/x-llamapun" id="S4.Ex9.m1.6d">italic_d start_POSTSUBSCRIPT italic_G end_POSTSUBSCRIPT ( italic_r , LCA ( italic_L start_POSTSUBSCRIPT italic_u start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT ( italic_j ) ) ) ≤ italic_d start_POSTSUBSCRIPT italic_G end_POSTSUBSCRIPT ( italic_r , LCA ( italic_e start_POSTSUBSCRIPT italic_u end_POSTSUBSCRIPT ) ) < italic_d start_POSTSUBSCRIPT italic_G end_POSTSUBSCRIPT ( italic_r , italic_a ) .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S4.I1.i1.p2.22">Furthermore, <math alttext="L_{u^{\prime\prime}}(j)" class="ltx_Math" display="inline" id="S4.I1.i1.p2.13.m1.1"><semantics id="S4.I1.i1.p2.13.m1.1a"><mrow id="S4.I1.i1.p2.13.m1.1.2" xref="S4.I1.i1.p2.13.m1.1.2.cmml"><msub id="S4.I1.i1.p2.13.m1.1.2.2" xref="S4.I1.i1.p2.13.m1.1.2.2.cmml"><mi id="S4.I1.i1.p2.13.m1.1.2.2.2" xref="S4.I1.i1.p2.13.m1.1.2.2.2.cmml">L</mi><msup id="S4.I1.i1.p2.13.m1.1.2.2.3" xref="S4.I1.i1.p2.13.m1.1.2.2.3.cmml"><mi id="S4.I1.i1.p2.13.m1.1.2.2.3.2" xref="S4.I1.i1.p2.13.m1.1.2.2.3.2.cmml">u</mi><mo id="S4.I1.i1.p2.13.m1.1.2.2.3.3" xref="S4.I1.i1.p2.13.m1.1.2.2.3.3.cmml">′′</mo></msup></msub><mo id="S4.I1.i1.p2.13.m1.1.2.1" xref="S4.I1.i1.p2.13.m1.1.2.1.cmml">⁢</mo><mrow id="S4.I1.i1.p2.13.m1.1.2.3.2" xref="S4.I1.i1.p2.13.m1.1.2.cmml"><mo id="S4.I1.i1.p2.13.m1.1.2.3.2.1" stretchy="false" xref="S4.I1.i1.p2.13.m1.1.2.cmml">(</mo><mi id="S4.I1.i1.p2.13.m1.1.1" xref="S4.I1.i1.p2.13.m1.1.1.cmml">j</mi><mo id="S4.I1.i1.p2.13.m1.1.2.3.2.2" stretchy="false" xref="S4.I1.i1.p2.13.m1.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.I1.i1.p2.13.m1.1b"><apply id="S4.I1.i1.p2.13.m1.1.2.cmml" xref="S4.I1.i1.p2.13.m1.1.2"><times id="S4.I1.i1.p2.13.m1.1.2.1.cmml" xref="S4.I1.i1.p2.13.m1.1.2.1"></times><apply id="S4.I1.i1.p2.13.m1.1.2.2.cmml" xref="S4.I1.i1.p2.13.m1.1.2.2"><csymbol cd="ambiguous" id="S4.I1.i1.p2.13.m1.1.2.2.1.cmml" xref="S4.I1.i1.p2.13.m1.1.2.2">subscript</csymbol><ci id="S4.I1.i1.p2.13.m1.1.2.2.2.cmml" xref="S4.I1.i1.p2.13.m1.1.2.2.2">𝐿</ci><apply id="S4.I1.i1.p2.13.m1.1.2.2.3.cmml" xref="S4.I1.i1.p2.13.m1.1.2.2.3"><csymbol cd="ambiguous" id="S4.I1.i1.p2.13.m1.1.2.2.3.1.cmml" xref="S4.I1.i1.p2.13.m1.1.2.2.3">superscript</csymbol><ci id="S4.I1.i1.p2.13.m1.1.2.2.3.2.cmml" xref="S4.I1.i1.p2.13.m1.1.2.2.3.2">𝑢</ci><ci id="S4.I1.i1.p2.13.m1.1.2.2.3.3.cmml" xref="S4.I1.i1.p2.13.m1.1.2.2.3.3">′′</ci></apply></apply><ci id="S4.I1.i1.p2.13.m1.1.1.cmml" xref="S4.I1.i1.p2.13.m1.1.1">𝑗</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.I1.i1.p2.13.m1.1c">L_{u^{\prime\prime}}(j)</annotation><annotation encoding="application/x-llamapun" id="S4.I1.i1.p2.13.m1.1d">italic_L start_POSTSUBSCRIPT italic_u start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT ( italic_j )</annotation></semantics></math> is included in <span class="ltx_text ltx_markedasmath" id="S4.I1.i1.p2.22.1">SOL</span> in Algorithm <a class="ltx_ref" href="https://arxiv.org/html/2503.00712v1#algorithm5" title="In 4.1.2 Bounding Approximation Ratio ‣ 4.1 One-to-Two Augmentation ‣ 4 Vertex Connectivity Augmentation in Link-Arrival Model ‣ Streaming Algorithms for Network Design"><span class="ltx_text ltx_ref_tag">5</span></a>, since <math alttext="e_{u}\in\textnormal{OPT}" class="ltx_Math" display="inline" id="S4.I1.i1.p2.15.m3.1"><semantics id="S4.I1.i1.p2.15.m3.1a"><mrow id="S4.I1.i1.p2.15.m3.1.1" xref="S4.I1.i1.p2.15.m3.1.1.cmml"><msub id="S4.I1.i1.p2.15.m3.1.1.2" xref="S4.I1.i1.p2.15.m3.1.1.2.cmml"><mi id="S4.I1.i1.p2.15.m3.1.1.2.2" xref="S4.I1.i1.p2.15.m3.1.1.2.2.cmml">e</mi><mi id="S4.I1.i1.p2.15.m3.1.1.2.3" xref="S4.I1.i1.p2.15.m3.1.1.2.3.cmml">u</mi></msub><mo id="S4.I1.i1.p2.15.m3.1.1.1" xref="S4.I1.i1.p2.15.m3.1.1.1.cmml">∈</mo><mtext id="S4.I1.i1.p2.15.m3.1.1.3" xref="S4.I1.i1.p2.15.m3.1.1.3a.cmml">OPT</mtext></mrow><annotation-xml encoding="MathML-Content" id="S4.I1.i1.p2.15.m3.1b"><apply id="S4.I1.i1.p2.15.m3.1.1.cmml" xref="S4.I1.i1.p2.15.m3.1.1"><in id="S4.I1.i1.p2.15.m3.1.1.1.cmml" xref="S4.I1.i1.p2.15.m3.1.1.1"></in><apply id="S4.I1.i1.p2.15.m3.1.1.2.cmml" xref="S4.I1.i1.p2.15.m3.1.1.2"><csymbol cd="ambiguous" id="S4.I1.i1.p2.15.m3.1.1.2.1.cmml" xref="S4.I1.i1.p2.15.m3.1.1.2">subscript</csymbol><ci id="S4.I1.i1.p2.15.m3.1.1.2.2.cmml" xref="S4.I1.i1.p2.15.m3.1.1.2.2">𝑒</ci><ci id="S4.I1.i1.p2.15.m3.1.1.2.3.cmml" xref="S4.I1.i1.p2.15.m3.1.1.2.3">𝑢</ci></apply><ci id="S4.I1.i1.p2.15.m3.1.1.3a.cmml" xref="S4.I1.i1.p2.15.m3.1.1.3"><mtext id="S4.I1.i1.p2.15.m3.1.1.3.cmml" xref="S4.I1.i1.p2.15.m3.1.1.3">OPT</mtext></ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.I1.i1.p2.15.m3.1c">e_{u}\in\textnormal{OPT}</annotation><annotation encoding="application/x-llamapun" id="S4.I1.i1.p2.15.m3.1d">italic_e start_POSTSUBSCRIPT italic_u end_POSTSUBSCRIPT ∈ OPT</annotation></semantics></math>. Similarly, there must be some <math alttext="v^{\prime\prime}\in G_{v^{\prime}}" class="ltx_Math" display="inline" id="S4.I1.i1.p2.16.m4.1"><semantics id="S4.I1.i1.p2.16.m4.1a"><mrow id="S4.I1.i1.p2.16.m4.1.1" xref="S4.I1.i1.p2.16.m4.1.1.cmml"><msup id="S4.I1.i1.p2.16.m4.1.1.2" xref="S4.I1.i1.p2.16.m4.1.1.2.cmml"><mi id="S4.I1.i1.p2.16.m4.1.1.2.2" xref="S4.I1.i1.p2.16.m4.1.1.2.2.cmml">v</mi><mo id="S4.I1.i1.p2.16.m4.1.1.2.3" xref="S4.I1.i1.p2.16.m4.1.1.2.3.cmml">′′</mo></msup><mo id="S4.I1.i1.p2.16.m4.1.1.1" xref="S4.I1.i1.p2.16.m4.1.1.1.cmml">∈</mo><msub id="S4.I1.i1.p2.16.m4.1.1.3" xref="S4.I1.i1.p2.16.m4.1.1.3.cmml"><mi id="S4.I1.i1.p2.16.m4.1.1.3.2" xref="S4.I1.i1.p2.16.m4.1.1.3.2.cmml">G</mi><msup id="S4.I1.i1.p2.16.m4.1.1.3.3" xref="S4.I1.i1.p2.16.m4.1.1.3.3.cmml"><mi id="S4.I1.i1.p2.16.m4.1.1.3.3.2" xref="S4.I1.i1.p2.16.m4.1.1.3.3.2.cmml">v</mi><mo id="S4.I1.i1.p2.16.m4.1.1.3.3.3" xref="S4.I1.i1.p2.16.m4.1.1.3.3.3.cmml">′</mo></msup></msub></mrow><annotation-xml encoding="MathML-Content" id="S4.I1.i1.p2.16.m4.1b"><apply id="S4.I1.i1.p2.16.m4.1.1.cmml" xref="S4.I1.i1.p2.16.m4.1.1"><in id="S4.I1.i1.p2.16.m4.1.1.1.cmml" xref="S4.I1.i1.p2.16.m4.1.1.1"></in><apply id="S4.I1.i1.p2.16.m4.1.1.2.cmml" xref="S4.I1.i1.p2.16.m4.1.1.2"><csymbol cd="ambiguous" id="S4.I1.i1.p2.16.m4.1.1.2.1.cmml" xref="S4.I1.i1.p2.16.m4.1.1.2">superscript</csymbol><ci id="S4.I1.i1.p2.16.m4.1.1.2.2.cmml" xref="S4.I1.i1.p2.16.m4.1.1.2.2">𝑣</ci><ci id="S4.I1.i1.p2.16.m4.1.1.2.3.cmml" xref="S4.I1.i1.p2.16.m4.1.1.2.3">′′</ci></apply><apply id="S4.I1.i1.p2.16.m4.1.1.3.cmml" xref="S4.I1.i1.p2.16.m4.1.1.3"><csymbol cd="ambiguous" id="S4.I1.i1.p2.16.m4.1.1.3.1.cmml" xref="S4.I1.i1.p2.16.m4.1.1.3">subscript</csymbol><ci id="S4.I1.i1.p2.16.m4.1.1.3.2.cmml" xref="S4.I1.i1.p2.16.m4.1.1.3.2">𝐺</ci><apply id="S4.I1.i1.p2.16.m4.1.1.3.3.cmml" xref="S4.I1.i1.p2.16.m4.1.1.3.3"><csymbol cd="ambiguous" id="S4.I1.i1.p2.16.m4.1.1.3.3.1.cmml" xref="S4.I1.i1.p2.16.m4.1.1.3.3">superscript</csymbol><ci id="S4.I1.i1.p2.16.m4.1.1.3.3.2.cmml" xref="S4.I1.i1.p2.16.m4.1.1.3.3.2">𝑣</ci><ci id="S4.I1.i1.p2.16.m4.1.1.3.3.3.cmml" xref="S4.I1.i1.p2.16.m4.1.1.3.3.3">′</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.I1.i1.p2.16.m4.1c">v^{\prime\prime}\in G_{v^{\prime}}</annotation><annotation encoding="application/x-llamapun" id="S4.I1.i1.p2.16.m4.1d">italic_v start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT ∈ italic_G start_POSTSUBSCRIPT italic_v start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT</annotation></semantics></math> such that <math alttext="L_{v^{\prime\prime}}\cap\textnormal{SOL}" class="ltx_Math" display="inline" id="S4.I1.i1.p2.17.m5.1"><semantics id="S4.I1.i1.p2.17.m5.1a"><mrow id="S4.I1.i1.p2.17.m5.1.1" xref="S4.I1.i1.p2.17.m5.1.1.cmml"><msub id="S4.I1.i1.p2.17.m5.1.1.2" xref="S4.I1.i1.p2.17.m5.1.1.2.cmml"><mi id="S4.I1.i1.p2.17.m5.1.1.2.2" xref="S4.I1.i1.p2.17.m5.1.1.2.2.cmml">L</mi><msup id="S4.I1.i1.p2.17.m5.1.1.2.3" xref="S4.I1.i1.p2.17.m5.1.1.2.3.cmml"><mi id="S4.I1.i1.p2.17.m5.1.1.2.3.2" xref="S4.I1.i1.p2.17.m5.1.1.2.3.2.cmml">v</mi><mo id="S4.I1.i1.p2.17.m5.1.1.2.3.3" xref="S4.I1.i1.p2.17.m5.1.1.2.3.3.cmml">′′</mo></msup></msub><mo id="S4.I1.i1.p2.17.m5.1.1.1" xref="S4.I1.i1.p2.17.m5.1.1.1.cmml">∩</mo><mtext id="S4.I1.i1.p2.17.m5.1.1.3" xref="S4.I1.i1.p2.17.m5.1.1.3a.cmml">SOL</mtext></mrow><annotation-xml encoding="MathML-Content" id="S4.I1.i1.p2.17.m5.1b"><apply id="S4.I1.i1.p2.17.m5.1.1.cmml" xref="S4.I1.i1.p2.17.m5.1.1"><intersect id="S4.I1.i1.p2.17.m5.1.1.1.cmml" xref="S4.I1.i1.p2.17.m5.1.1.1"></intersect><apply id="S4.I1.i1.p2.17.m5.1.1.2.cmml" xref="S4.I1.i1.p2.17.m5.1.1.2"><csymbol cd="ambiguous" id="S4.I1.i1.p2.17.m5.1.1.2.1.cmml" xref="S4.I1.i1.p2.17.m5.1.1.2">subscript</csymbol><ci id="S4.I1.i1.p2.17.m5.1.1.2.2.cmml" xref="S4.I1.i1.p2.17.m5.1.1.2.2">𝐿</ci><apply id="S4.I1.i1.p2.17.m5.1.1.2.3.cmml" xref="S4.I1.i1.p2.17.m5.1.1.2.3"><csymbol cd="ambiguous" id="S4.I1.i1.p2.17.m5.1.1.2.3.1.cmml" xref="S4.I1.i1.p2.17.m5.1.1.2.3">superscript</csymbol><ci id="S4.I1.i1.p2.17.m5.1.1.2.3.2.cmml" xref="S4.I1.i1.p2.17.m5.1.1.2.3.2">𝑣</ci><ci id="S4.I1.i1.p2.17.m5.1.1.2.3.3.cmml" xref="S4.I1.i1.p2.17.m5.1.1.2.3.3">′′</ci></apply></apply><ci id="S4.I1.i1.p2.17.m5.1.1.3a.cmml" xref="S4.I1.i1.p2.17.m5.1.1.3"><mtext id="S4.I1.i1.p2.17.m5.1.1.3.cmml" xref="S4.I1.i1.p2.17.m5.1.1.3">SOL</mtext></ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.I1.i1.p2.17.m5.1c">L_{v^{\prime\prime}}\cap\textnormal{SOL}</annotation><annotation encoding="application/x-llamapun" id="S4.I1.i1.p2.17.m5.1d">italic_L start_POSTSUBSCRIPT italic_v start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT ∩ SOL</annotation></semantics></math> contains a link with one endpoint in <math alttext="G\setminus G_{a}" class="ltx_Math" display="inline" id="S4.I1.i1.p2.18.m6.1"><semantics id="S4.I1.i1.p2.18.m6.1a"><mrow id="S4.I1.i1.p2.18.m6.1.1" xref="S4.I1.i1.p2.18.m6.1.1.cmml"><mi id="S4.I1.i1.p2.18.m6.1.1.2" xref="S4.I1.i1.p2.18.m6.1.1.2.cmml">G</mi><mo id="S4.I1.i1.p2.18.m6.1.1.1" xref="S4.I1.i1.p2.18.m6.1.1.1.cmml">∖</mo><msub id="S4.I1.i1.p2.18.m6.1.1.3" xref="S4.I1.i1.p2.18.m6.1.1.3.cmml"><mi id="S4.I1.i1.p2.18.m6.1.1.3.2" xref="S4.I1.i1.p2.18.m6.1.1.3.2.cmml">G</mi><mi id="S4.I1.i1.p2.18.m6.1.1.3.3" xref="S4.I1.i1.p2.18.m6.1.1.3.3.cmml">a</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S4.I1.i1.p2.18.m6.1b"><apply id="S4.I1.i1.p2.18.m6.1.1.cmml" xref="S4.I1.i1.p2.18.m6.1.1"><setdiff id="S4.I1.i1.p2.18.m6.1.1.1.cmml" xref="S4.I1.i1.p2.18.m6.1.1.1"></setdiff><ci id="S4.I1.i1.p2.18.m6.1.1.2.cmml" xref="S4.I1.i1.p2.18.m6.1.1.2">𝐺</ci><apply id="S4.I1.i1.p2.18.m6.1.1.3.cmml" xref="S4.I1.i1.p2.18.m6.1.1.3"><csymbol cd="ambiguous" id="S4.I1.i1.p2.18.m6.1.1.3.1.cmml" xref="S4.I1.i1.p2.18.m6.1.1.3">subscript</csymbol><ci id="S4.I1.i1.p2.18.m6.1.1.3.2.cmml" xref="S4.I1.i1.p2.18.m6.1.1.3.2">𝐺</ci><ci id="S4.I1.i1.p2.18.m6.1.1.3.3.cmml" xref="S4.I1.i1.p2.18.m6.1.1.3.3">𝑎</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.I1.i1.p2.18.m6.1c">G\setminus G_{a}</annotation><annotation encoding="application/x-llamapun" id="S4.I1.i1.p2.18.m6.1d">italic_G ∖ italic_G start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT</annotation></semantics></math>. Since <math alttext="G\setminus G_{a}" class="ltx_Math" display="inline" id="S4.I1.i1.p2.19.m7.1"><semantics id="S4.I1.i1.p2.19.m7.1a"><mrow id="S4.I1.i1.p2.19.m7.1.1" xref="S4.I1.i1.p2.19.m7.1.1.cmml"><mi id="S4.I1.i1.p2.19.m7.1.1.2" xref="S4.I1.i1.p2.19.m7.1.1.2.cmml">G</mi><mo id="S4.I1.i1.p2.19.m7.1.1.1" xref="S4.I1.i1.p2.19.m7.1.1.1.cmml">∖</mo><msub id="S4.I1.i1.p2.19.m7.1.1.3" xref="S4.I1.i1.p2.19.m7.1.1.3.cmml"><mi id="S4.I1.i1.p2.19.m7.1.1.3.2" xref="S4.I1.i1.p2.19.m7.1.1.3.2.cmml">G</mi><mi id="S4.I1.i1.p2.19.m7.1.1.3.3" xref="S4.I1.i1.p2.19.m7.1.1.3.3.cmml">a</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S4.I1.i1.p2.19.m7.1b"><apply id="S4.I1.i1.p2.19.m7.1.1.cmml" xref="S4.I1.i1.p2.19.m7.1.1"><setdiff id="S4.I1.i1.p2.19.m7.1.1.1.cmml" xref="S4.I1.i1.p2.19.m7.1.1.1"></setdiff><ci id="S4.I1.i1.p2.19.m7.1.1.2.cmml" xref="S4.I1.i1.p2.19.m7.1.1.2">𝐺</ci><apply id="S4.I1.i1.p2.19.m7.1.1.3.cmml" xref="S4.I1.i1.p2.19.m7.1.1.3"><csymbol cd="ambiguous" id="S4.I1.i1.p2.19.m7.1.1.3.1.cmml" xref="S4.I1.i1.p2.19.m7.1.1.3">subscript</csymbol><ci id="S4.I1.i1.p2.19.m7.1.1.3.2.cmml" xref="S4.I1.i1.p2.19.m7.1.1.3.2">𝐺</ci><ci id="S4.I1.i1.p2.19.m7.1.1.3.3.cmml" xref="S4.I1.i1.p2.19.m7.1.1.3.3">𝑎</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.I1.i1.p2.19.m7.1c">G\setminus G_{a}</annotation><annotation encoding="application/x-llamapun" id="S4.I1.i1.p2.19.m7.1d">italic_G ∖ italic_G start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT</annotation></semantics></math> remains connected despite the deletion of <math alttext="a" class="ltx_Math" display="inline" id="S4.I1.i1.p2.20.m8.1"><semantics id="S4.I1.i1.p2.20.m8.1a"><mi id="S4.I1.i1.p2.20.m8.1.1" xref="S4.I1.i1.p2.20.m8.1.1.cmml">a</mi><annotation-xml encoding="MathML-Content" id="S4.I1.i1.p2.20.m8.1b"><ci id="S4.I1.i1.p2.20.m8.1.1.cmml" xref="S4.I1.i1.p2.20.m8.1.1">𝑎</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.I1.i1.p2.20.m8.1c">a</annotation><annotation encoding="application/x-llamapun" id="S4.I1.i1.p2.20.m8.1d">italic_a</annotation></semantics></math>, we obtain our desired <math alttext="u" class="ltx_Math" display="inline" id="S4.I1.i1.p2.21.m9.1"><semantics id="S4.I1.i1.p2.21.m9.1a"><mi id="S4.I1.i1.p2.21.m9.1.1" xref="S4.I1.i1.p2.21.m9.1.1.cmml">u</mi><annotation-xml encoding="MathML-Content" id="S4.I1.i1.p2.21.m9.1b"><ci id="S4.I1.i1.p2.21.m9.1.1.cmml" xref="S4.I1.i1.p2.21.m9.1.1">𝑢</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.I1.i1.p2.21.m9.1c">u</annotation><annotation encoding="application/x-llamapun" id="S4.I1.i1.p2.21.m9.1d">italic_u</annotation></semantics></math>-<math alttext="v" class="ltx_Math" display="inline" id="S4.I1.i1.p2.22.m10.1"><semantics id="S4.I1.i1.p2.22.m10.1a"><mi id="S4.I1.i1.p2.22.m10.1.1" xref="S4.I1.i1.p2.22.m10.1.1.cmml">v</mi><annotation-xml encoding="MathML-Content" id="S4.I1.i1.p2.22.m10.1b"><ci id="S4.I1.i1.p2.22.m10.1.1.cmml" xref="S4.I1.i1.p2.22.m10.1.1">𝑣</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.I1.i1.p2.22.m10.1c">v</annotation><annotation encoding="application/x-llamapun" id="S4.I1.i1.p2.22.m10.1d">italic_v</annotation></semantics></math> path.</p> </div> </li> <li class="ltx_item" id="S4.I1.i2" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">•</span> <div class="ltx_para" id="S4.I1.i2.p1"> <p class="ltx_p" id="S4.I1.i2.p1.7"><span class="ltx_text ltx_font_bold" id="S4.I1.i2.p1.7.1">Case 2:</span> If <math alttext="a\neq b" class="ltx_Math" display="inline" id="S4.I1.i2.p1.1.m1.1"><semantics id="S4.I1.i2.p1.1.m1.1a"><mrow id="S4.I1.i2.p1.1.m1.1.1" xref="S4.I1.i2.p1.1.m1.1.1.cmml"><mi id="S4.I1.i2.p1.1.m1.1.1.2" xref="S4.I1.i2.p1.1.m1.1.1.2.cmml">a</mi><mo id="S4.I1.i2.p1.1.m1.1.1.1" xref="S4.I1.i2.p1.1.m1.1.1.1.cmml">≠</mo><mi id="S4.I1.i2.p1.1.m1.1.1.3" xref="S4.I1.i2.p1.1.m1.1.1.3.cmml">b</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.I1.i2.p1.1.m1.1b"><apply id="S4.I1.i2.p1.1.m1.1.1.cmml" xref="S4.I1.i2.p1.1.m1.1.1"><neq id="S4.I1.i2.p1.1.m1.1.1.1.cmml" xref="S4.I1.i2.p1.1.m1.1.1.1"></neq><ci id="S4.I1.i2.p1.1.m1.1.1.2.cmml" xref="S4.I1.i2.p1.1.m1.1.1.2">𝑎</ci><ci id="S4.I1.i2.p1.1.m1.1.1.3.cmml" xref="S4.I1.i2.p1.1.m1.1.1.3">𝑏</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.I1.i2.p1.1.m1.1c">a\neq b</annotation><annotation encoding="application/x-llamapun" id="S4.I1.i2.p1.1.m1.1d">italic_a ≠ italic_b</annotation></semantics></math>, then <math alttext="a" class="ltx_Math" display="inline" id="S4.I1.i2.p1.2.m2.1"><semantics id="S4.I1.i2.p1.2.m2.1a"><mi id="S4.I1.i2.p1.2.m2.1.1" xref="S4.I1.i2.p1.2.m2.1.1.cmml">a</mi><annotation-xml encoding="MathML-Content" id="S4.I1.i2.p1.2.m2.1b"><ci id="S4.I1.i2.p1.2.m2.1.1.cmml" xref="S4.I1.i2.p1.2.m2.1.1">𝑎</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.I1.i2.p1.2.m2.1c">a</annotation><annotation encoding="application/x-llamapun" id="S4.I1.i2.p1.2.m2.1d">italic_a</annotation></semantics></math> is either in <math alttext="G_{u^{\prime}}" class="ltx_Math" display="inline" id="S4.I1.i2.p1.3.m3.1"><semantics id="S4.I1.i2.p1.3.m3.1a"><msub id="S4.I1.i2.p1.3.m3.1.1" xref="S4.I1.i2.p1.3.m3.1.1.cmml"><mi id="S4.I1.i2.p1.3.m3.1.1.2" xref="S4.I1.i2.p1.3.m3.1.1.2.cmml">G</mi><msup id="S4.I1.i2.p1.3.m3.1.1.3" xref="S4.I1.i2.p1.3.m3.1.1.3.cmml"><mi id="S4.I1.i2.p1.3.m3.1.1.3.2" xref="S4.I1.i2.p1.3.m3.1.1.3.2.cmml">u</mi><mo id="S4.I1.i2.p1.3.m3.1.1.3.3" xref="S4.I1.i2.p1.3.m3.1.1.3.3.cmml">′</mo></msup></msub><annotation-xml encoding="MathML-Content" id="S4.I1.i2.p1.3.m3.1b"><apply id="S4.I1.i2.p1.3.m3.1.1.cmml" xref="S4.I1.i2.p1.3.m3.1.1"><csymbol cd="ambiguous" id="S4.I1.i2.p1.3.m3.1.1.1.cmml" xref="S4.I1.i2.p1.3.m3.1.1">subscript</csymbol><ci id="S4.I1.i2.p1.3.m3.1.1.2.cmml" xref="S4.I1.i2.p1.3.m3.1.1.2">𝐺</ci><apply id="S4.I1.i2.p1.3.m3.1.1.3.cmml" xref="S4.I1.i2.p1.3.m3.1.1.3"><csymbol cd="ambiguous" id="S4.I1.i2.p1.3.m3.1.1.3.1.cmml" xref="S4.I1.i2.p1.3.m3.1.1.3">superscript</csymbol><ci id="S4.I1.i2.p1.3.m3.1.1.3.2.cmml" xref="S4.I1.i2.p1.3.m3.1.1.3.2">𝑢</ci><ci id="S4.I1.i2.p1.3.m3.1.1.3.3.cmml" xref="S4.I1.i2.p1.3.m3.1.1.3.3">′</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.I1.i2.p1.3.m3.1c">G_{u^{\prime}}</annotation><annotation encoding="application/x-llamapun" id="S4.I1.i2.p1.3.m3.1d">italic_G start_POSTSUBSCRIPT italic_u start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT</annotation></semantics></math> or in <math alttext="G_{v^{\prime}}" class="ltx_Math" display="inline" id="S4.I1.i2.p1.4.m4.1"><semantics id="S4.I1.i2.p1.4.m4.1a"><msub id="S4.I1.i2.p1.4.m4.1.1" xref="S4.I1.i2.p1.4.m4.1.1.cmml"><mi id="S4.I1.i2.p1.4.m4.1.1.2" xref="S4.I1.i2.p1.4.m4.1.1.2.cmml">G</mi><msup id="S4.I1.i2.p1.4.m4.1.1.3" xref="S4.I1.i2.p1.4.m4.1.1.3.cmml"><mi id="S4.I1.i2.p1.4.m4.1.1.3.2" xref="S4.I1.i2.p1.4.m4.1.1.3.2.cmml">v</mi><mo id="S4.I1.i2.p1.4.m4.1.1.3.3" xref="S4.I1.i2.p1.4.m4.1.1.3.3.cmml">′</mo></msup></msub><annotation-xml encoding="MathML-Content" id="S4.I1.i2.p1.4.m4.1b"><apply id="S4.I1.i2.p1.4.m4.1.1.cmml" xref="S4.I1.i2.p1.4.m4.1.1"><csymbol cd="ambiguous" id="S4.I1.i2.p1.4.m4.1.1.1.cmml" xref="S4.I1.i2.p1.4.m4.1.1">subscript</csymbol><ci id="S4.I1.i2.p1.4.m4.1.1.2.cmml" xref="S4.I1.i2.p1.4.m4.1.1.2">𝐺</ci><apply id="S4.I1.i2.p1.4.m4.1.1.3.cmml" xref="S4.I1.i2.p1.4.m4.1.1.3"><csymbol cd="ambiguous" id="S4.I1.i2.p1.4.m4.1.1.3.1.cmml" xref="S4.I1.i2.p1.4.m4.1.1.3">superscript</csymbol><ci id="S4.I1.i2.p1.4.m4.1.1.3.2.cmml" xref="S4.I1.i2.p1.4.m4.1.1.3.2">𝑣</ci><ci id="S4.I1.i2.p1.4.m4.1.1.3.3.cmml" xref="S4.I1.i2.p1.4.m4.1.1.3.3">′</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.I1.i2.p1.4.m4.1c">G_{v^{\prime}}</annotation><annotation encoding="application/x-llamapun" id="S4.I1.i2.p1.4.m4.1d">italic_G start_POSTSUBSCRIPT italic_v start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT</annotation></semantics></math>. Suppose without loss of generality that <math alttext="a\in G_{u^{\prime}}" class="ltx_Math" display="inline" id="S4.I1.i2.p1.5.m5.1"><semantics id="S4.I1.i2.p1.5.m5.1a"><mrow id="S4.I1.i2.p1.5.m5.1.1" xref="S4.I1.i2.p1.5.m5.1.1.cmml"><mi id="S4.I1.i2.p1.5.m5.1.1.2" xref="S4.I1.i2.p1.5.m5.1.1.2.cmml">a</mi><mo id="S4.I1.i2.p1.5.m5.1.1.1" xref="S4.I1.i2.p1.5.m5.1.1.1.cmml">∈</mo><msub id="S4.I1.i2.p1.5.m5.1.1.3" xref="S4.I1.i2.p1.5.m5.1.1.3.cmml"><mi id="S4.I1.i2.p1.5.m5.1.1.3.2" xref="S4.I1.i2.p1.5.m5.1.1.3.2.cmml">G</mi><msup id="S4.I1.i2.p1.5.m5.1.1.3.3" xref="S4.I1.i2.p1.5.m5.1.1.3.3.cmml"><mi id="S4.I1.i2.p1.5.m5.1.1.3.3.2" xref="S4.I1.i2.p1.5.m5.1.1.3.3.2.cmml">u</mi><mo id="S4.I1.i2.p1.5.m5.1.1.3.3.3" xref="S4.I1.i2.p1.5.m5.1.1.3.3.3.cmml">′</mo></msup></msub></mrow><annotation-xml encoding="MathML-Content" id="S4.I1.i2.p1.5.m5.1b"><apply id="S4.I1.i2.p1.5.m5.1.1.cmml" xref="S4.I1.i2.p1.5.m5.1.1"><in id="S4.I1.i2.p1.5.m5.1.1.1.cmml" xref="S4.I1.i2.p1.5.m5.1.1.1"></in><ci id="S4.I1.i2.p1.5.m5.1.1.2.cmml" xref="S4.I1.i2.p1.5.m5.1.1.2">𝑎</ci><apply id="S4.I1.i2.p1.5.m5.1.1.3.cmml" xref="S4.I1.i2.p1.5.m5.1.1.3"><csymbol cd="ambiguous" id="S4.I1.i2.p1.5.m5.1.1.3.1.cmml" xref="S4.I1.i2.p1.5.m5.1.1.3">subscript</csymbol><ci id="S4.I1.i2.p1.5.m5.1.1.3.2.cmml" xref="S4.I1.i2.p1.5.m5.1.1.3.2">𝐺</ci><apply id="S4.I1.i2.p1.5.m5.1.1.3.3.cmml" xref="S4.I1.i2.p1.5.m5.1.1.3.3"><csymbol cd="ambiguous" id="S4.I1.i2.p1.5.m5.1.1.3.3.1.cmml" xref="S4.I1.i2.p1.5.m5.1.1.3.3">superscript</csymbol><ci id="S4.I1.i2.p1.5.m5.1.1.3.3.2.cmml" xref="S4.I1.i2.p1.5.m5.1.1.3.3.2">𝑢</ci><ci id="S4.I1.i2.p1.5.m5.1.1.3.3.3.cmml" xref="S4.I1.i2.p1.5.m5.1.1.3.3.3">′</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.I1.i2.p1.5.m5.1c">a\in G_{u^{\prime}}</annotation><annotation encoding="application/x-llamapun" id="S4.I1.i2.p1.5.m5.1d">italic_a ∈ italic_G start_POSTSUBSCRIPT italic_u start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT</annotation></semantics></math>; the other case is analogous. Let <math alttext="j" class="ltx_Math" display="inline" id="S4.I1.i2.p1.6.m6.1"><semantics id="S4.I1.i2.p1.6.m6.1a"><mi id="S4.I1.i2.p1.6.m6.1.1" xref="S4.I1.i2.p1.6.m6.1.1.cmml">j</mi><annotation-xml encoding="MathML-Content" id="S4.I1.i2.p1.6.m6.1b"><ci id="S4.I1.i2.p1.6.m6.1.1.cmml" xref="S4.I1.i2.p1.6.m6.1.1">𝑗</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.I1.i2.p1.6.m6.1c">j</annotation><annotation encoding="application/x-llamapun" id="S4.I1.i2.p1.6.m6.1d">italic_j</annotation></semantics></math> be the weight class of <math alttext="uv" class="ltx_Math" display="inline" id="S4.I1.i2.p1.7.m7.1"><semantics id="S4.I1.i2.p1.7.m7.1a"><mrow id="S4.I1.i2.p1.7.m7.1.1" xref="S4.I1.i2.p1.7.m7.1.1.cmml"><mi id="S4.I1.i2.p1.7.m7.1.1.2" xref="S4.I1.i2.p1.7.m7.1.1.2.cmml">u</mi><mo id="S4.I1.i2.p1.7.m7.1.1.1" xref="S4.I1.i2.p1.7.m7.1.1.1.cmml">⁢</mo><mi id="S4.I1.i2.p1.7.m7.1.1.3" xref="S4.I1.i2.p1.7.m7.1.1.3.cmml">v</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.I1.i2.p1.7.m7.1b"><apply id="S4.I1.i2.p1.7.m7.1.1.cmml" xref="S4.I1.i2.p1.7.m7.1.1"><times id="S4.I1.i2.p1.7.m7.1.1.1.cmml" xref="S4.I1.i2.p1.7.m7.1.1.1"></times><ci id="S4.I1.i2.p1.7.m7.1.1.2.cmml" xref="S4.I1.i2.p1.7.m7.1.1.2">𝑢</ci><ci id="S4.I1.i2.p1.7.m7.1.1.3.cmml" xref="S4.I1.i2.p1.7.m7.1.1.3">𝑣</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.I1.i2.p1.7.m7.1c">uv</annotation><annotation encoding="application/x-llamapun" id="S4.I1.i2.p1.7.m7.1d">italic_u italic_v</annotation></semantics></math>. Then</p> <table class="ltx_equation ltx_eqn_table" id="S4.Ex10"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="d_{G}(r,\text{LCA}(L_{u}(j)))\leq d_{G}(r,b)<d_{G}(r,a);" class="ltx_Math" display="block" id="S4.Ex10.m1.7"><semantics id="S4.Ex10.m1.7a"><mrow id="S4.Ex10.m1.7.7.1" xref="S4.Ex10.m1.7.7.1.1.cmml"><mrow id="S4.Ex10.m1.7.7.1.1" xref="S4.Ex10.m1.7.7.1.1.cmml"><mrow id="S4.Ex10.m1.7.7.1.1.1" xref="S4.Ex10.m1.7.7.1.1.1.cmml"><msub id="S4.Ex10.m1.7.7.1.1.1.3" xref="S4.Ex10.m1.7.7.1.1.1.3.cmml"><mi id="S4.Ex10.m1.7.7.1.1.1.3.2" xref="S4.Ex10.m1.7.7.1.1.1.3.2.cmml">d</mi><mi id="S4.Ex10.m1.7.7.1.1.1.3.3" xref="S4.Ex10.m1.7.7.1.1.1.3.3.cmml">G</mi></msub><mo id="S4.Ex10.m1.7.7.1.1.1.2" xref="S4.Ex10.m1.7.7.1.1.1.2.cmml">⁢</mo><mrow id="S4.Ex10.m1.7.7.1.1.1.1.1" xref="S4.Ex10.m1.7.7.1.1.1.1.2.cmml"><mo id="S4.Ex10.m1.7.7.1.1.1.1.1.2" stretchy="false" xref="S4.Ex10.m1.7.7.1.1.1.1.2.cmml">(</mo><mi id="S4.Ex10.m1.2.2" xref="S4.Ex10.m1.2.2.cmml">r</mi><mo id="S4.Ex10.m1.7.7.1.1.1.1.1.3" xref="S4.Ex10.m1.7.7.1.1.1.1.2.cmml">,</mo><mrow id="S4.Ex10.m1.7.7.1.1.1.1.1.1" xref="S4.Ex10.m1.7.7.1.1.1.1.1.1.cmml"><mtext id="S4.Ex10.m1.7.7.1.1.1.1.1.1.3" xref="S4.Ex10.m1.7.7.1.1.1.1.1.1.3a.cmml">LCA</mtext><mo id="S4.Ex10.m1.7.7.1.1.1.1.1.1.2" xref="S4.Ex10.m1.7.7.1.1.1.1.1.1.2.cmml">⁢</mo><mrow id="S4.Ex10.m1.7.7.1.1.1.1.1.1.1.1" xref="S4.Ex10.m1.7.7.1.1.1.1.1.1.1.1.1.cmml"><mo id="S4.Ex10.m1.7.7.1.1.1.1.1.1.1.1.2" stretchy="false" xref="S4.Ex10.m1.7.7.1.1.1.1.1.1.1.1.1.cmml">(</mo><mrow id="S4.Ex10.m1.7.7.1.1.1.1.1.1.1.1.1" xref="S4.Ex10.m1.7.7.1.1.1.1.1.1.1.1.1.cmml"><msub id="S4.Ex10.m1.7.7.1.1.1.1.1.1.1.1.1.2" xref="S4.Ex10.m1.7.7.1.1.1.1.1.1.1.1.1.2.cmml"><mi id="S4.Ex10.m1.7.7.1.1.1.1.1.1.1.1.1.2.2" xref="S4.Ex10.m1.7.7.1.1.1.1.1.1.1.1.1.2.2.cmml">L</mi><mi id="S4.Ex10.m1.7.7.1.1.1.1.1.1.1.1.1.2.3" xref="S4.Ex10.m1.7.7.1.1.1.1.1.1.1.1.1.2.3.cmml">u</mi></msub><mo id="S4.Ex10.m1.7.7.1.1.1.1.1.1.1.1.1.1" xref="S4.Ex10.m1.7.7.1.1.1.1.1.1.1.1.1.1.cmml">⁢</mo><mrow id="S4.Ex10.m1.7.7.1.1.1.1.1.1.1.1.1.3.2" xref="S4.Ex10.m1.7.7.1.1.1.1.1.1.1.1.1.cmml"><mo id="S4.Ex10.m1.7.7.1.1.1.1.1.1.1.1.1.3.2.1" stretchy="false" xref="S4.Ex10.m1.7.7.1.1.1.1.1.1.1.1.1.cmml">(</mo><mi id="S4.Ex10.m1.1.1" xref="S4.Ex10.m1.1.1.cmml">j</mi><mo id="S4.Ex10.m1.7.7.1.1.1.1.1.1.1.1.1.3.2.2" stretchy="false" xref="S4.Ex10.m1.7.7.1.1.1.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S4.Ex10.m1.7.7.1.1.1.1.1.1.1.1.3" stretchy="false" xref="S4.Ex10.m1.7.7.1.1.1.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S4.Ex10.m1.7.7.1.1.1.1.1.4" stretchy="false" xref="S4.Ex10.m1.7.7.1.1.1.1.2.cmml">)</mo></mrow></mrow><mo id="S4.Ex10.m1.7.7.1.1.3" xref="S4.Ex10.m1.7.7.1.1.3.cmml">≤</mo><mrow id="S4.Ex10.m1.7.7.1.1.4" xref="S4.Ex10.m1.7.7.1.1.4.cmml"><msub id="S4.Ex10.m1.7.7.1.1.4.2" xref="S4.Ex10.m1.7.7.1.1.4.2.cmml"><mi 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id="S4.Ex10.m1.7.7.1.1.6.2.1.cmml" xref="S4.Ex10.m1.7.7.1.1.6.2">subscript</csymbol><ci id="S4.Ex10.m1.7.7.1.1.6.2.2.cmml" xref="S4.Ex10.m1.7.7.1.1.6.2.2">𝑑</ci><ci id="S4.Ex10.m1.7.7.1.1.6.2.3.cmml" xref="S4.Ex10.m1.7.7.1.1.6.2.3">𝐺</ci></apply><interval closure="open" id="S4.Ex10.m1.7.7.1.1.6.3.1.cmml" xref="S4.Ex10.m1.7.7.1.1.6.3.2"><ci id="S4.Ex10.m1.5.5.cmml" xref="S4.Ex10.m1.5.5">𝑟</ci><ci id="S4.Ex10.m1.6.6.cmml" xref="S4.Ex10.m1.6.6">𝑎</ci></interval></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Ex10.m1.7c">d_{G}(r,\text{LCA}(L_{u}(j)))\leq d_{G}(r,b)<d_{G}(r,a);</annotation><annotation encoding="application/x-llamapun" id="S4.Ex10.m1.7d">italic_d start_POSTSUBSCRIPT italic_G end_POSTSUBSCRIPT ( italic_r , LCA ( italic_L start_POSTSUBSCRIPT italic_u end_POSTSUBSCRIPT ( italic_j ) ) ) ≤ italic_d start_POSTSUBSCRIPT italic_G end_POSTSUBSCRIPT ( italic_r , italic_b ) < italic_d start_POSTSUBSCRIPT italic_G end_POSTSUBSCRIPT ( italic_r , italic_a ) ;</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S4.I1.i2.p1.17">the first inequality holds since <math alttext="v\in G\setminus G_{b}" class="ltx_Math" display="inline" id="S4.I1.i2.p1.8.m1.1"><semantics id="S4.I1.i2.p1.8.m1.1a"><mrow id="S4.I1.i2.p1.8.m1.1.1" xref="S4.I1.i2.p1.8.m1.1.1.cmml"><mi id="S4.I1.i2.p1.8.m1.1.1.2" xref="S4.I1.i2.p1.8.m1.1.1.2.cmml">v</mi><mo id="S4.I1.i2.p1.8.m1.1.1.1" xref="S4.I1.i2.p1.8.m1.1.1.1.cmml">∈</mo><mrow id="S4.I1.i2.p1.8.m1.1.1.3" xref="S4.I1.i2.p1.8.m1.1.1.3.cmml"><mi id="S4.I1.i2.p1.8.m1.1.1.3.2" xref="S4.I1.i2.p1.8.m1.1.1.3.2.cmml">G</mi><mo id="S4.I1.i2.p1.8.m1.1.1.3.1" xref="S4.I1.i2.p1.8.m1.1.1.3.1.cmml">∖</mo><msub id="S4.I1.i2.p1.8.m1.1.1.3.3" xref="S4.I1.i2.p1.8.m1.1.1.3.3.cmml"><mi id="S4.I1.i2.p1.8.m1.1.1.3.3.2" xref="S4.I1.i2.p1.8.m1.1.1.3.3.2.cmml">G</mi><mi id="S4.I1.i2.p1.8.m1.1.1.3.3.3" xref="S4.I1.i2.p1.8.m1.1.1.3.3.3.cmml">b</mi></msub></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.I1.i2.p1.8.m1.1b"><apply id="S4.I1.i2.p1.8.m1.1.1.cmml" xref="S4.I1.i2.p1.8.m1.1.1"><in id="S4.I1.i2.p1.8.m1.1.1.1.cmml" xref="S4.I1.i2.p1.8.m1.1.1.1"></in><ci id="S4.I1.i2.p1.8.m1.1.1.2.cmml" xref="S4.I1.i2.p1.8.m1.1.1.2">𝑣</ci><apply id="S4.I1.i2.p1.8.m1.1.1.3.cmml" xref="S4.I1.i2.p1.8.m1.1.1.3"><setdiff id="S4.I1.i2.p1.8.m1.1.1.3.1.cmml" xref="S4.I1.i2.p1.8.m1.1.1.3.1"></setdiff><ci id="S4.I1.i2.p1.8.m1.1.1.3.2.cmml" xref="S4.I1.i2.p1.8.m1.1.1.3.2">𝐺</ci><apply id="S4.I1.i2.p1.8.m1.1.1.3.3.cmml" xref="S4.I1.i2.p1.8.m1.1.1.3.3"><csymbol cd="ambiguous" id="S4.I1.i2.p1.8.m1.1.1.3.3.1.cmml" xref="S4.I1.i2.p1.8.m1.1.1.3.3">subscript</csymbol><ci id="S4.I1.i2.p1.8.m1.1.1.3.3.2.cmml" xref="S4.I1.i2.p1.8.m1.1.1.3.3.2">𝐺</ci><ci id="S4.I1.i2.p1.8.m1.1.1.3.3.3.cmml" xref="S4.I1.i2.p1.8.m1.1.1.3.3.3">𝑏</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.I1.i2.p1.8.m1.1c">v\in G\setminus G_{b}</annotation><annotation encoding="application/x-llamapun" id="S4.I1.i2.p1.8.m1.1d">italic_v ∈ italic_G ∖ italic_G start_POSTSUBSCRIPT italic_b end_POSTSUBSCRIPT</annotation></semantics></math>. Thus <math alttext="L_{u}(j)" class="ltx_Math" display="inline" id="S4.I1.i2.p1.9.m2.1"><semantics id="S4.I1.i2.p1.9.m2.1a"><mrow id="S4.I1.i2.p1.9.m2.1.2" xref="S4.I1.i2.p1.9.m2.1.2.cmml"><msub id="S4.I1.i2.p1.9.m2.1.2.2" xref="S4.I1.i2.p1.9.m2.1.2.2.cmml"><mi id="S4.I1.i2.p1.9.m2.1.2.2.2" xref="S4.I1.i2.p1.9.m2.1.2.2.2.cmml">L</mi><mi id="S4.I1.i2.p1.9.m2.1.2.2.3" xref="S4.I1.i2.p1.9.m2.1.2.2.3.cmml">u</mi></msub><mo id="S4.I1.i2.p1.9.m2.1.2.1" xref="S4.I1.i2.p1.9.m2.1.2.1.cmml">⁢</mo><mrow id="S4.I1.i2.p1.9.m2.1.2.3.2" xref="S4.I1.i2.p1.9.m2.1.2.cmml"><mo id="S4.I1.i2.p1.9.m2.1.2.3.2.1" stretchy="false" xref="S4.I1.i2.p1.9.m2.1.2.cmml">(</mo><mi id="S4.I1.i2.p1.9.m2.1.1" xref="S4.I1.i2.p1.9.m2.1.1.cmml">j</mi><mo id="S4.I1.i2.p1.9.m2.1.2.3.2.2" stretchy="false" xref="S4.I1.i2.p1.9.m2.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.I1.i2.p1.9.m2.1b"><apply id="S4.I1.i2.p1.9.m2.1.2.cmml" xref="S4.I1.i2.p1.9.m2.1.2"><times id="S4.I1.i2.p1.9.m2.1.2.1.cmml" xref="S4.I1.i2.p1.9.m2.1.2.1"></times><apply id="S4.I1.i2.p1.9.m2.1.2.2.cmml" xref="S4.I1.i2.p1.9.m2.1.2.2"><csymbol cd="ambiguous" id="S4.I1.i2.p1.9.m2.1.2.2.1.cmml" xref="S4.I1.i2.p1.9.m2.1.2.2">subscript</csymbol><ci id="S4.I1.i2.p1.9.m2.1.2.2.2.cmml" xref="S4.I1.i2.p1.9.m2.1.2.2.2">𝐿</ci><ci id="S4.I1.i2.p1.9.m2.1.2.2.3.cmml" xref="S4.I1.i2.p1.9.m2.1.2.2.3">𝑢</ci></apply><ci id="S4.I1.i2.p1.9.m2.1.1.cmml" xref="S4.I1.i2.p1.9.m2.1.1">𝑗</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.I1.i2.p1.9.m2.1c">L_{u}(j)</annotation><annotation encoding="application/x-llamapun" id="S4.I1.i2.p1.9.m2.1d">italic_L start_POSTSUBSCRIPT italic_u end_POSTSUBSCRIPT ( italic_j )</annotation></semantics></math> has one endpoint in <math alttext="G\setminus G_{a}" class="ltx_Math" display="inline" id="S4.I1.i2.p1.10.m3.1"><semantics id="S4.I1.i2.p1.10.m3.1a"><mrow id="S4.I1.i2.p1.10.m3.1.1" xref="S4.I1.i2.p1.10.m3.1.1.cmml"><mi id="S4.I1.i2.p1.10.m3.1.1.2" xref="S4.I1.i2.p1.10.m3.1.1.2.cmml">G</mi><mo id="S4.I1.i2.p1.10.m3.1.1.1" xref="S4.I1.i2.p1.10.m3.1.1.1.cmml">∖</mo><msub id="S4.I1.i2.p1.10.m3.1.1.3" xref="S4.I1.i2.p1.10.m3.1.1.3.cmml"><mi id="S4.I1.i2.p1.10.m3.1.1.3.2" xref="S4.I1.i2.p1.10.m3.1.1.3.2.cmml">G</mi><mi id="S4.I1.i2.p1.10.m3.1.1.3.3" xref="S4.I1.i2.p1.10.m3.1.1.3.3.cmml">a</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S4.I1.i2.p1.10.m3.1b"><apply id="S4.I1.i2.p1.10.m3.1.1.cmml" xref="S4.I1.i2.p1.10.m3.1.1"><setdiff id="S4.I1.i2.p1.10.m3.1.1.1.cmml" xref="S4.I1.i2.p1.10.m3.1.1.1"></setdiff><ci id="S4.I1.i2.p1.10.m3.1.1.2.cmml" xref="S4.I1.i2.p1.10.m3.1.1.2">𝐺</ci><apply id="S4.I1.i2.p1.10.m3.1.1.3.cmml" xref="S4.I1.i2.p1.10.m3.1.1.3"><csymbol cd="ambiguous" id="S4.I1.i2.p1.10.m3.1.1.3.1.cmml" xref="S4.I1.i2.p1.10.m3.1.1.3">subscript</csymbol><ci id="S4.I1.i2.p1.10.m3.1.1.3.2.cmml" xref="S4.I1.i2.p1.10.m3.1.1.3.2">𝐺</ci><ci id="S4.I1.i2.p1.10.m3.1.1.3.3.cmml" xref="S4.I1.i2.p1.10.m3.1.1.3.3">𝑎</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.I1.i2.p1.10.m3.1c">G\setminus G_{a}</annotation><annotation encoding="application/x-llamapun" id="S4.I1.i2.p1.10.m3.1d">italic_G ∖ italic_G start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT</annotation></semantics></math>. Furthermore, <math alttext="L_{u}(j)\in\textnormal{SOL}" class="ltx_Math" display="inline" id="S4.I1.i2.p1.11.m4.1"><semantics id="S4.I1.i2.p1.11.m4.1a"><mrow id="S4.I1.i2.p1.11.m4.1.2" xref="S4.I1.i2.p1.11.m4.1.2.cmml"><mrow id="S4.I1.i2.p1.11.m4.1.2.2" xref="S4.I1.i2.p1.11.m4.1.2.2.cmml"><msub id="S4.I1.i2.p1.11.m4.1.2.2.2" xref="S4.I1.i2.p1.11.m4.1.2.2.2.cmml"><mi id="S4.I1.i2.p1.11.m4.1.2.2.2.2" xref="S4.I1.i2.p1.11.m4.1.2.2.2.2.cmml">L</mi><mi id="S4.I1.i2.p1.11.m4.1.2.2.2.3" xref="S4.I1.i2.p1.11.m4.1.2.2.2.3.cmml">u</mi></msub><mo id="S4.I1.i2.p1.11.m4.1.2.2.1" xref="S4.I1.i2.p1.11.m4.1.2.2.1.cmml">⁢</mo><mrow id="S4.I1.i2.p1.11.m4.1.2.2.3.2" xref="S4.I1.i2.p1.11.m4.1.2.2.cmml"><mo id="S4.I1.i2.p1.11.m4.1.2.2.3.2.1" stretchy="false" xref="S4.I1.i2.p1.11.m4.1.2.2.cmml">(</mo><mi id="S4.I1.i2.p1.11.m4.1.1" xref="S4.I1.i2.p1.11.m4.1.1.cmml">j</mi><mo id="S4.I1.i2.p1.11.m4.1.2.2.3.2.2" stretchy="false" xref="S4.I1.i2.p1.11.m4.1.2.2.cmml">)</mo></mrow></mrow><mo id="S4.I1.i2.p1.11.m4.1.2.1" xref="S4.I1.i2.p1.11.m4.1.2.1.cmml">∈</mo><mtext id="S4.I1.i2.p1.11.m4.1.2.3" xref="S4.I1.i2.p1.11.m4.1.2.3a.cmml">SOL</mtext></mrow><annotation-xml encoding="MathML-Content" id="S4.I1.i2.p1.11.m4.1b"><apply id="S4.I1.i2.p1.11.m4.1.2.cmml" xref="S4.I1.i2.p1.11.m4.1.2"><in id="S4.I1.i2.p1.11.m4.1.2.1.cmml" xref="S4.I1.i2.p1.11.m4.1.2.1"></in><apply id="S4.I1.i2.p1.11.m4.1.2.2.cmml" xref="S4.I1.i2.p1.11.m4.1.2.2"><times id="S4.I1.i2.p1.11.m4.1.2.2.1.cmml" xref="S4.I1.i2.p1.11.m4.1.2.2.1"></times><apply id="S4.I1.i2.p1.11.m4.1.2.2.2.cmml" xref="S4.I1.i2.p1.11.m4.1.2.2.2"><csymbol cd="ambiguous" id="S4.I1.i2.p1.11.m4.1.2.2.2.1.cmml" xref="S4.I1.i2.p1.11.m4.1.2.2.2">subscript</csymbol><ci id="S4.I1.i2.p1.11.m4.1.2.2.2.2.cmml" xref="S4.I1.i2.p1.11.m4.1.2.2.2.2">𝐿</ci><ci id="S4.I1.i2.p1.11.m4.1.2.2.2.3.cmml" xref="S4.I1.i2.p1.11.m4.1.2.2.2.3">𝑢</ci></apply><ci id="S4.I1.i2.p1.11.m4.1.1.cmml" xref="S4.I1.i2.p1.11.m4.1.1">𝑗</ci></apply><ci id="S4.I1.i2.p1.11.m4.1.2.3a.cmml" xref="S4.I1.i2.p1.11.m4.1.2.3"><mtext id="S4.I1.i2.p1.11.m4.1.2.3.cmml" xref="S4.I1.i2.p1.11.m4.1.2.3">SOL</mtext></ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.I1.i2.p1.11.m4.1c">L_{u}(j)\in\textnormal{SOL}</annotation><annotation encoding="application/x-llamapun" id="S4.I1.i2.p1.11.m4.1d">italic_L start_POSTSUBSCRIPT italic_u end_POSTSUBSCRIPT ( italic_j ) ∈ SOL</annotation></semantics></math> since <math alttext="uv\in\textnormal{OPT}" class="ltx_Math" display="inline" id="S4.I1.i2.p1.12.m5.1"><semantics id="S4.I1.i2.p1.12.m5.1a"><mrow id="S4.I1.i2.p1.12.m5.1.1" xref="S4.I1.i2.p1.12.m5.1.1.cmml"><mrow id="S4.I1.i2.p1.12.m5.1.1.2" xref="S4.I1.i2.p1.12.m5.1.1.2.cmml"><mi id="S4.I1.i2.p1.12.m5.1.1.2.2" xref="S4.I1.i2.p1.12.m5.1.1.2.2.cmml">u</mi><mo id="S4.I1.i2.p1.12.m5.1.1.2.1" xref="S4.I1.i2.p1.12.m5.1.1.2.1.cmml">⁢</mo><mi id="S4.I1.i2.p1.12.m5.1.1.2.3" xref="S4.I1.i2.p1.12.m5.1.1.2.3.cmml">v</mi></mrow><mo id="S4.I1.i2.p1.12.m5.1.1.1" xref="S4.I1.i2.p1.12.m5.1.1.1.cmml">∈</mo><mtext id="S4.I1.i2.p1.12.m5.1.1.3" xref="S4.I1.i2.p1.12.m5.1.1.3a.cmml">OPT</mtext></mrow><annotation-xml encoding="MathML-Content" id="S4.I1.i2.p1.12.m5.1b"><apply id="S4.I1.i2.p1.12.m5.1.1.cmml" xref="S4.I1.i2.p1.12.m5.1.1"><in id="S4.I1.i2.p1.12.m5.1.1.1.cmml" xref="S4.I1.i2.p1.12.m5.1.1.1"></in><apply id="S4.I1.i2.p1.12.m5.1.1.2.cmml" xref="S4.I1.i2.p1.12.m5.1.1.2"><times id="S4.I1.i2.p1.12.m5.1.1.2.1.cmml" xref="S4.I1.i2.p1.12.m5.1.1.2.1"></times><ci id="S4.I1.i2.p1.12.m5.1.1.2.2.cmml" xref="S4.I1.i2.p1.12.m5.1.1.2.2">𝑢</ci><ci id="S4.I1.i2.p1.12.m5.1.1.2.3.cmml" xref="S4.I1.i2.p1.12.m5.1.1.2.3">𝑣</ci></apply><ci id="S4.I1.i2.p1.12.m5.1.1.3a.cmml" xref="S4.I1.i2.p1.12.m5.1.1.3"><mtext id="S4.I1.i2.p1.12.m5.1.1.3.cmml" xref="S4.I1.i2.p1.12.m5.1.1.3">OPT</mtext></ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.I1.i2.p1.12.m5.1c">uv\in\textnormal{OPT}</annotation><annotation encoding="application/x-llamapun" id="S4.I1.i2.p1.12.m5.1d">italic_u italic_v ∈ OPT</annotation></semantics></math>. Since <math alttext="G\setminus G_{a}" class="ltx_Math" display="inline" id="S4.I1.i2.p1.13.m6.1"><semantics id="S4.I1.i2.p1.13.m6.1a"><mrow id="S4.I1.i2.p1.13.m6.1.1" xref="S4.I1.i2.p1.13.m6.1.1.cmml"><mi id="S4.I1.i2.p1.13.m6.1.1.2" xref="S4.I1.i2.p1.13.m6.1.1.2.cmml">G</mi><mo id="S4.I1.i2.p1.13.m6.1.1.1" xref="S4.I1.i2.p1.13.m6.1.1.1.cmml">∖</mo><msub id="S4.I1.i2.p1.13.m6.1.1.3" xref="S4.I1.i2.p1.13.m6.1.1.3.cmml"><mi id="S4.I1.i2.p1.13.m6.1.1.3.2" xref="S4.I1.i2.p1.13.m6.1.1.3.2.cmml">G</mi><mi id="S4.I1.i2.p1.13.m6.1.1.3.3" xref="S4.I1.i2.p1.13.m6.1.1.3.3.cmml">a</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S4.I1.i2.p1.13.m6.1b"><apply id="S4.I1.i2.p1.13.m6.1.1.cmml" xref="S4.I1.i2.p1.13.m6.1.1"><setdiff id="S4.I1.i2.p1.13.m6.1.1.1.cmml" xref="S4.I1.i2.p1.13.m6.1.1.1"></setdiff><ci id="S4.I1.i2.p1.13.m6.1.1.2.cmml" xref="S4.I1.i2.p1.13.m6.1.1.2">𝐺</ci><apply id="S4.I1.i2.p1.13.m6.1.1.3.cmml" xref="S4.I1.i2.p1.13.m6.1.1.3"><csymbol cd="ambiguous" id="S4.I1.i2.p1.13.m6.1.1.3.1.cmml" xref="S4.I1.i2.p1.13.m6.1.1.3">subscript</csymbol><ci id="S4.I1.i2.p1.13.m6.1.1.3.2.cmml" xref="S4.I1.i2.p1.13.m6.1.1.3.2">𝐺</ci><ci id="S4.I1.i2.p1.13.m6.1.1.3.3.cmml" xref="S4.I1.i2.p1.13.m6.1.1.3.3">𝑎</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.I1.i2.p1.13.m6.1c">G\setminus G_{a}</annotation><annotation encoding="application/x-llamapun" id="S4.I1.i2.p1.13.m6.1d">italic_G ∖ italic_G start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT</annotation></semantics></math> remains connected despite the deletion of <math alttext="a" class="ltx_Math" display="inline" id="S4.I1.i2.p1.14.m7.1"><semantics id="S4.I1.i2.p1.14.m7.1a"><mi id="S4.I1.i2.p1.14.m7.1.1" xref="S4.I1.i2.p1.14.m7.1.1.cmml">a</mi><annotation-xml encoding="MathML-Content" id="S4.I1.i2.p1.14.m7.1b"><ci id="S4.I1.i2.p1.14.m7.1.1.cmml" xref="S4.I1.i2.p1.14.m7.1.1">𝑎</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.I1.i2.p1.14.m7.1c">a</annotation><annotation encoding="application/x-llamapun" id="S4.I1.i2.p1.14.m7.1d">italic_a</annotation></semantics></math>, this gives us our desired <math alttext="u" class="ltx_Math" display="inline" id="S4.I1.i2.p1.15.m8.1"><semantics id="S4.I1.i2.p1.15.m8.1a"><mi id="S4.I1.i2.p1.15.m8.1.1" xref="S4.I1.i2.p1.15.m8.1.1.cmml">u</mi><annotation-xml encoding="MathML-Content" id="S4.I1.i2.p1.15.m8.1b"><ci id="S4.I1.i2.p1.15.m8.1.1.cmml" xref="S4.I1.i2.p1.15.m8.1.1">𝑢</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.I1.i2.p1.15.m8.1c">u</annotation><annotation encoding="application/x-llamapun" id="S4.I1.i2.p1.15.m8.1d">italic_u</annotation></semantics></math>-<math alttext="v" class="ltx_Math" display="inline" id="S4.I1.i2.p1.16.m9.1"><semantics id="S4.I1.i2.p1.16.m9.1a"><mi id="S4.I1.i2.p1.16.m9.1.1" xref="S4.I1.i2.p1.16.m9.1.1.cmml">v</mi><annotation-xml encoding="MathML-Content" id="S4.I1.i2.p1.16.m9.1b"><ci id="S4.I1.i2.p1.16.m9.1.1.cmml" xref="S4.I1.i2.p1.16.m9.1.1">𝑣</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.I1.i2.p1.16.m9.1c">v</annotation><annotation encoding="application/x-llamapun" id="S4.I1.i2.p1.16.m9.1d">italic_v</annotation></semantics></math> path in <math alttext="E\cup\textnormal{SOL}" class="ltx_Math" display="inline" id="S4.I1.i2.p1.17.m10.1"><semantics id="S4.I1.i2.p1.17.m10.1a"><mrow id="S4.I1.i2.p1.17.m10.1.1" xref="S4.I1.i2.p1.17.m10.1.1.cmml"><mi id="S4.I1.i2.p1.17.m10.1.1.2" xref="S4.I1.i2.p1.17.m10.1.1.2.cmml">E</mi><mo id="S4.I1.i2.p1.17.m10.1.1.1" xref="S4.I1.i2.p1.17.m10.1.1.1.cmml">∪</mo><mtext id="S4.I1.i2.p1.17.m10.1.1.3" xref="S4.I1.i2.p1.17.m10.1.1.3a.cmml">SOL</mtext></mrow><annotation-xml encoding="MathML-Content" id="S4.I1.i2.p1.17.m10.1b"><apply id="S4.I1.i2.p1.17.m10.1.1.cmml" xref="S4.I1.i2.p1.17.m10.1.1"><union id="S4.I1.i2.p1.17.m10.1.1.1.cmml" xref="S4.I1.i2.p1.17.m10.1.1.1"></union><ci id="S4.I1.i2.p1.17.m10.1.1.2.cmml" xref="S4.I1.i2.p1.17.m10.1.1.2">𝐸</ci><ci id="S4.I1.i2.p1.17.m10.1.1.3a.cmml" xref="S4.I1.i2.p1.17.m10.1.1.3"><mtext id="S4.I1.i2.p1.17.m10.1.1.3.cmml" xref="S4.I1.i2.p1.17.m10.1.1.3">SOL</mtext></ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.I1.i2.p1.17.m10.1c">E\cup\textnormal{SOL}</annotation><annotation encoding="application/x-llamapun" id="S4.I1.i2.p1.17.m10.1d">italic_E ∪ SOL</annotation></semantics></math>. See Figure <a class="ltx_ref" href="https://arxiv.org/html/2503.00712v1#S4.F3.sf2" title="In Figure 3 ‣ 4.1.2 Bounding Approximation Ratio ‣ 4.1 One-to-Two Augmentation ‣ 4 Vertex Connectivity Augmentation in Link-Arrival Model ‣ Streaming Algorithms for Network Design"><span class="ltx_text ltx_ref_tag">3(b)</span></a> for reference.</p> </div> </li> </ul> <p class="ltx_p" id="S4.SS1.SSS2.3.p3.1">∎</p> </div> </div> <figure class="ltx_figure" id="S4.F3"> <div class="ltx_flex_figure"> <div class="ltx_flex_cell ltx_flex_size_2"> <figure class="ltx_figure ltx_figure_panel ltx_align_center" id="S4.F3.sf1"><img alt="Refer to caption" class="ltx_graphics ltx_centering ltx_img_square" height="213" id="S4.F3.sf1.g1" src="x1.png" width="187"/> <figcaption class="ltx_caption ltx_centering"><span class="ltx_tag ltx_tag_figure"><span class="ltx_text" id="S4.F3.sf1.16.8.1" style="font-size:90%;">(a)</span> </span><span class="ltx_text" id="S4.F3.sf1.14.7" style="font-size:90%;">Case 1: If <math alttext="u^{\prime}" class="ltx_Math" display="inline" id="S4.F3.sf1.8.1.m1.1"><semantics id="S4.F3.sf1.8.1.m1.1b"><msup id="S4.F3.sf1.8.1.m1.1.1" xref="S4.F3.sf1.8.1.m1.1.1.cmml"><mi id="S4.F3.sf1.8.1.m1.1.1.2" xref="S4.F3.sf1.8.1.m1.1.1.2.cmml">u</mi><mo id="S4.F3.sf1.8.1.m1.1.1.3" xref="S4.F3.sf1.8.1.m1.1.1.3.cmml">′</mo></msup><annotation-xml encoding="MathML-Content" id="S4.F3.sf1.8.1.m1.1c"><apply id="S4.F3.sf1.8.1.m1.1.1.cmml" xref="S4.F3.sf1.8.1.m1.1.1"><csymbol cd="ambiguous" id="S4.F3.sf1.8.1.m1.1.1.1.cmml" xref="S4.F3.sf1.8.1.m1.1.1">superscript</csymbol><ci id="S4.F3.sf1.8.1.m1.1.1.2.cmml" xref="S4.F3.sf1.8.1.m1.1.1.2">𝑢</ci><ci id="S4.F3.sf1.8.1.m1.1.1.3.cmml" xref="S4.F3.sf1.8.1.m1.1.1.3">′</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.F3.sf1.8.1.m1.1d">u^{\prime}</annotation><annotation encoding="application/x-llamapun" id="S4.F3.sf1.8.1.m1.1e">italic_u start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math> or <math alttext="v^{\prime}" class="ltx_Math" display="inline" id="S4.F3.sf1.9.2.m2.1"><semantics id="S4.F3.sf1.9.2.m2.1b"><msup id="S4.F3.sf1.9.2.m2.1.1" xref="S4.F3.sf1.9.2.m2.1.1.cmml"><mi id="S4.F3.sf1.9.2.m2.1.1.2" xref="S4.F3.sf1.9.2.m2.1.1.2.cmml">v</mi><mo id="S4.F3.sf1.9.2.m2.1.1.3" xref="S4.F3.sf1.9.2.m2.1.1.3.cmml">′</mo></msup><annotation-xml encoding="MathML-Content" id="S4.F3.sf1.9.2.m2.1c"><apply id="S4.F3.sf1.9.2.m2.1.1.cmml" xref="S4.F3.sf1.9.2.m2.1.1"><csymbol cd="ambiguous" id="S4.F3.sf1.9.2.m2.1.1.1.cmml" xref="S4.F3.sf1.9.2.m2.1.1">superscript</csymbol><ci id="S4.F3.sf1.9.2.m2.1.1.2.cmml" xref="S4.F3.sf1.9.2.m2.1.1.2">𝑣</ci><ci id="S4.F3.sf1.9.2.m2.1.1.3.cmml" xref="S4.F3.sf1.9.2.m2.1.1.3">′</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.F3.sf1.9.2.m2.1d">v^{\prime}</annotation><annotation encoding="application/x-llamapun" id="S4.F3.sf1.9.2.m2.1e">italic_v start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math> are not marked “good”, <math alttext="u" class="ltx_Math" display="inline" id="S4.F3.sf1.10.3.m3.1"><semantics id="S4.F3.sf1.10.3.m3.1b"><mi id="S4.F3.sf1.10.3.m3.1.1" xref="S4.F3.sf1.10.3.m3.1.1.cmml">u</mi><annotation-xml encoding="MathML-Content" id="S4.F3.sf1.10.3.m3.1c"><ci id="S4.F3.sf1.10.3.m3.1.1.cmml" xref="S4.F3.sf1.10.3.m3.1.1">𝑢</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.F3.sf1.10.3.m3.1d">u</annotation><annotation encoding="application/x-llamapun" id="S4.F3.sf1.10.3.m3.1e">italic_u</annotation></semantics></math> and <math alttext="v" class="ltx_Math" display="inline" id="S4.F3.sf1.11.4.m4.1"><semantics id="S4.F3.sf1.11.4.m4.1b"><mi id="S4.F3.sf1.11.4.m4.1.1" xref="S4.F3.sf1.11.4.m4.1.1.cmml">v</mi><annotation-xml encoding="MathML-Content" id="S4.F3.sf1.11.4.m4.1c"><ci id="S4.F3.sf1.11.4.m4.1.1.cmml" xref="S4.F3.sf1.11.4.m4.1.1">𝑣</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.F3.sf1.11.4.m4.1d">v</annotation><annotation encoding="application/x-llamapun" id="S4.F3.sf1.11.4.m4.1e">italic_v</annotation></semantics></math> are connected via the link in <math alttext="T_{a}^{\prime\prime}" class="ltx_Math" display="inline" id="S4.F3.sf1.12.5.m5.1"><semantics id="S4.F3.sf1.12.5.m5.1b"><msubsup id="S4.F3.sf1.12.5.m5.1.1" xref="S4.F3.sf1.12.5.m5.1.1.cmml"><mi id="S4.F3.sf1.12.5.m5.1.1.2.2" xref="S4.F3.sf1.12.5.m5.1.1.2.2.cmml">T</mi><mi id="S4.F3.sf1.12.5.m5.1.1.2.3" xref="S4.F3.sf1.12.5.m5.1.1.2.3.cmml">a</mi><mo id="S4.F3.sf1.12.5.m5.1.1.3" xref="S4.F3.sf1.12.5.m5.1.1.3.cmml">′′</mo></msubsup><annotation-xml encoding="MathML-Content" id="S4.F3.sf1.12.5.m5.1c"><apply id="S4.F3.sf1.12.5.m5.1.1.cmml" xref="S4.F3.sf1.12.5.m5.1.1"><csymbol cd="ambiguous" id="S4.F3.sf1.12.5.m5.1.1.1.cmml" xref="S4.F3.sf1.12.5.m5.1.1">superscript</csymbol><apply id="S4.F3.sf1.12.5.m5.1.1.2.cmml" xref="S4.F3.sf1.12.5.m5.1.1"><csymbol cd="ambiguous" id="S4.F3.sf1.12.5.m5.1.1.2.1.cmml" xref="S4.F3.sf1.12.5.m5.1.1">subscript</csymbol><ci id="S4.F3.sf1.12.5.m5.1.1.2.2.cmml" xref="S4.F3.sf1.12.5.m5.1.1.2.2">𝑇</ci><ci id="S4.F3.sf1.12.5.m5.1.1.2.3.cmml" xref="S4.F3.sf1.12.5.m5.1.1.2.3">𝑎</ci></apply><ci id="S4.F3.sf1.12.5.m5.1.1.3.cmml" xref="S4.F3.sf1.12.5.m5.1.1.3">′′</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.F3.sf1.12.5.m5.1d">T_{a}^{\prime\prime}</annotation><annotation encoding="application/x-llamapun" id="S4.F3.sf1.12.5.m5.1e">italic_T start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT</annotation></semantics></math> shown. Else, they are connected using the links in <math alttext="L_{u^{\prime\prime}}" class="ltx_Math" display="inline" id="S4.F3.sf1.13.6.m6.1"><semantics id="S4.F3.sf1.13.6.m6.1b"><msub id="S4.F3.sf1.13.6.m6.1.1" xref="S4.F3.sf1.13.6.m6.1.1.cmml"><mi id="S4.F3.sf1.13.6.m6.1.1.2" xref="S4.F3.sf1.13.6.m6.1.1.2.cmml">L</mi><msup id="S4.F3.sf1.13.6.m6.1.1.3" xref="S4.F3.sf1.13.6.m6.1.1.3.cmml"><mi id="S4.F3.sf1.13.6.m6.1.1.3.2" xref="S4.F3.sf1.13.6.m6.1.1.3.2.cmml">u</mi><mo id="S4.F3.sf1.13.6.m6.1.1.3.3" xref="S4.F3.sf1.13.6.m6.1.1.3.3.cmml">′′</mo></msup></msub><annotation-xml encoding="MathML-Content" id="S4.F3.sf1.13.6.m6.1c"><apply id="S4.F3.sf1.13.6.m6.1.1.cmml" xref="S4.F3.sf1.13.6.m6.1.1"><csymbol cd="ambiguous" id="S4.F3.sf1.13.6.m6.1.1.1.cmml" xref="S4.F3.sf1.13.6.m6.1.1">subscript</csymbol><ci id="S4.F3.sf1.13.6.m6.1.1.2.cmml" xref="S4.F3.sf1.13.6.m6.1.1.2">𝐿</ci><apply id="S4.F3.sf1.13.6.m6.1.1.3.cmml" xref="S4.F3.sf1.13.6.m6.1.1.3"><csymbol cd="ambiguous" id="S4.F3.sf1.13.6.m6.1.1.3.1.cmml" xref="S4.F3.sf1.13.6.m6.1.1.3">superscript</csymbol><ci id="S4.F3.sf1.13.6.m6.1.1.3.2.cmml" xref="S4.F3.sf1.13.6.m6.1.1.3.2">𝑢</ci><ci id="S4.F3.sf1.13.6.m6.1.1.3.3.cmml" xref="S4.F3.sf1.13.6.m6.1.1.3.3">′′</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.F3.sf1.13.6.m6.1d">L_{u^{\prime\prime}}</annotation><annotation encoding="application/x-llamapun" id="S4.F3.sf1.13.6.m6.1e">italic_L start_POSTSUBSCRIPT italic_u start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT</annotation></semantics></math> and in <math alttext="L_{v^{\prime\prime}}" class="ltx_Math" display="inline" id="S4.F3.sf1.14.7.m7.1"><semantics id="S4.F3.sf1.14.7.m7.1b"><msub id="S4.F3.sf1.14.7.m7.1.1" xref="S4.F3.sf1.14.7.m7.1.1.cmml"><mi id="S4.F3.sf1.14.7.m7.1.1.2" xref="S4.F3.sf1.14.7.m7.1.1.2.cmml">L</mi><msup id="S4.F3.sf1.14.7.m7.1.1.3" xref="S4.F3.sf1.14.7.m7.1.1.3.cmml"><mi id="S4.F3.sf1.14.7.m7.1.1.3.2" xref="S4.F3.sf1.14.7.m7.1.1.3.2.cmml">v</mi><mo id="S4.F3.sf1.14.7.m7.1.1.3.3" xref="S4.F3.sf1.14.7.m7.1.1.3.3.cmml">′′</mo></msup></msub><annotation-xml encoding="MathML-Content" id="S4.F3.sf1.14.7.m7.1c"><apply id="S4.F3.sf1.14.7.m7.1.1.cmml" xref="S4.F3.sf1.14.7.m7.1.1"><csymbol cd="ambiguous" id="S4.F3.sf1.14.7.m7.1.1.1.cmml" xref="S4.F3.sf1.14.7.m7.1.1">subscript</csymbol><ci id="S4.F3.sf1.14.7.m7.1.1.2.cmml" xref="S4.F3.sf1.14.7.m7.1.1.2">𝐿</ci><apply id="S4.F3.sf1.14.7.m7.1.1.3.cmml" xref="S4.F3.sf1.14.7.m7.1.1.3"><csymbol cd="ambiguous" id="S4.F3.sf1.14.7.m7.1.1.3.1.cmml" xref="S4.F3.sf1.14.7.m7.1.1.3">superscript</csymbol><ci id="S4.F3.sf1.14.7.m7.1.1.3.2.cmml" xref="S4.F3.sf1.14.7.m7.1.1.3.2">𝑣</ci><ci id="S4.F3.sf1.14.7.m7.1.1.3.3.cmml" xref="S4.F3.sf1.14.7.m7.1.1.3.3">′′</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.F3.sf1.14.7.m7.1d">L_{v^{\prime\prime}}</annotation><annotation encoding="application/x-llamapun" id="S4.F3.sf1.14.7.m7.1e">italic_L start_POSTSUBSCRIPT italic_v start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT</annotation></semantics></math>.</span></figcaption> </figure> </div> <div class="ltx_flex_cell ltx_flex_size_2"> <figure class="ltx_figure ltx_figure_panel ltx_align_center" id="S4.F3.sf2"><img alt="Refer to caption" class="ltx_graphics ltx_centering ltx_img_square" height="229" id="S4.F3.sf2.g1" src="x2.png" width="222"/> <figcaption class="ltx_caption ltx_centering"><span class="ltx_tag ltx_tag_figure"><span class="ltx_text" id="S4.F3.sf2.8.4.1" style="font-size:90%;">(b)</span> </span><span class="ltx_text" id="S4.F3.sf2.6.3" style="font-size:90%;">Case 2: Here <math alttext="u\in G_{a}" class="ltx_Math" display="inline" id="S4.F3.sf2.4.1.m1.1"><semantics id="S4.F3.sf2.4.1.m1.1b"><mrow id="S4.F3.sf2.4.1.m1.1.1" xref="S4.F3.sf2.4.1.m1.1.1.cmml"><mi id="S4.F3.sf2.4.1.m1.1.1.2" xref="S4.F3.sf2.4.1.m1.1.1.2.cmml">u</mi><mo id="S4.F3.sf2.4.1.m1.1.1.1" xref="S4.F3.sf2.4.1.m1.1.1.1.cmml">∈</mo><msub id="S4.F3.sf2.4.1.m1.1.1.3" xref="S4.F3.sf2.4.1.m1.1.1.3.cmml"><mi id="S4.F3.sf2.4.1.m1.1.1.3.2" xref="S4.F3.sf2.4.1.m1.1.1.3.2.cmml">G</mi><mi id="S4.F3.sf2.4.1.m1.1.1.3.3" xref="S4.F3.sf2.4.1.m1.1.1.3.3.cmml">a</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S4.F3.sf2.4.1.m1.1c"><apply id="S4.F3.sf2.4.1.m1.1.1.cmml" xref="S4.F3.sf2.4.1.m1.1.1"><in id="S4.F3.sf2.4.1.m1.1.1.1.cmml" xref="S4.F3.sf2.4.1.m1.1.1.1"></in><ci id="S4.F3.sf2.4.1.m1.1.1.2.cmml" xref="S4.F3.sf2.4.1.m1.1.1.2">𝑢</ci><apply id="S4.F3.sf2.4.1.m1.1.1.3.cmml" xref="S4.F3.sf2.4.1.m1.1.1.3"><csymbol cd="ambiguous" id="S4.F3.sf2.4.1.m1.1.1.3.1.cmml" xref="S4.F3.sf2.4.1.m1.1.1.3">subscript</csymbol><ci id="S4.F3.sf2.4.1.m1.1.1.3.2.cmml" xref="S4.F3.sf2.4.1.m1.1.1.3.2">𝐺</ci><ci id="S4.F3.sf2.4.1.m1.1.1.3.3.cmml" xref="S4.F3.sf2.4.1.m1.1.1.3.3">𝑎</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.F3.sf2.4.1.m1.1d">u\in G_{a}</annotation><annotation encoding="application/x-llamapun" id="S4.F3.sf2.4.1.m1.1e">italic_u ∈ italic_G start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT</annotation></semantics></math>, <math alttext="v\in G\setminus G_{a}" class="ltx_Math" display="inline" id="S4.F3.sf2.5.2.m2.1"><semantics id="S4.F3.sf2.5.2.m2.1b"><mrow id="S4.F3.sf2.5.2.m2.1.1" xref="S4.F3.sf2.5.2.m2.1.1.cmml"><mi id="S4.F3.sf2.5.2.m2.1.1.2" xref="S4.F3.sf2.5.2.m2.1.1.2.cmml">v</mi><mo id="S4.F3.sf2.5.2.m2.1.1.1" xref="S4.F3.sf2.5.2.m2.1.1.1.cmml">∈</mo><mrow id="S4.F3.sf2.5.2.m2.1.1.3" xref="S4.F3.sf2.5.2.m2.1.1.3.cmml"><mi id="S4.F3.sf2.5.2.m2.1.1.3.2" xref="S4.F3.sf2.5.2.m2.1.1.3.2.cmml">G</mi><mo id="S4.F3.sf2.5.2.m2.1.1.3.1" xref="S4.F3.sf2.5.2.m2.1.1.3.1.cmml">∖</mo><msub id="S4.F3.sf2.5.2.m2.1.1.3.3" xref="S4.F3.sf2.5.2.m2.1.1.3.3.cmml"><mi id="S4.F3.sf2.5.2.m2.1.1.3.3.2" xref="S4.F3.sf2.5.2.m2.1.1.3.3.2.cmml">G</mi><mi id="S4.F3.sf2.5.2.m2.1.1.3.3.3" xref="S4.F3.sf2.5.2.m2.1.1.3.3.3.cmml">a</mi></msub></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.F3.sf2.5.2.m2.1c"><apply id="S4.F3.sf2.5.2.m2.1.1.cmml" xref="S4.F3.sf2.5.2.m2.1.1"><in id="S4.F3.sf2.5.2.m2.1.1.1.cmml" xref="S4.F3.sf2.5.2.m2.1.1.1"></in><ci id="S4.F3.sf2.5.2.m2.1.1.2.cmml" xref="S4.F3.sf2.5.2.m2.1.1.2">𝑣</ci><apply id="S4.F3.sf2.5.2.m2.1.1.3.cmml" xref="S4.F3.sf2.5.2.m2.1.1.3"><setdiff id="S4.F3.sf2.5.2.m2.1.1.3.1.cmml" xref="S4.F3.sf2.5.2.m2.1.1.3.1"></setdiff><ci id="S4.F3.sf2.5.2.m2.1.1.3.2.cmml" xref="S4.F3.sf2.5.2.m2.1.1.3.2">𝐺</ci><apply id="S4.F3.sf2.5.2.m2.1.1.3.3.cmml" xref="S4.F3.sf2.5.2.m2.1.1.3.3"><csymbol cd="ambiguous" id="S4.F3.sf2.5.2.m2.1.1.3.3.1.cmml" xref="S4.F3.sf2.5.2.m2.1.1.3.3">subscript</csymbol><ci id="S4.F3.sf2.5.2.m2.1.1.3.3.2.cmml" xref="S4.F3.sf2.5.2.m2.1.1.3.3.2">𝐺</ci><ci id="S4.F3.sf2.5.2.m2.1.1.3.3.3.cmml" xref="S4.F3.sf2.5.2.m2.1.1.3.3.3">𝑎</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.F3.sf2.5.2.m2.1d">v\in G\setminus G_{a}</annotation><annotation encoding="application/x-llamapun" id="S4.F3.sf2.5.2.m2.1e">italic_v ∈ italic_G ∖ italic_G start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT</annotation></semantics></math>, and <math alttext="a\in G_{u^{\prime}}" class="ltx_Math" display="inline" id="S4.F3.sf2.6.3.m3.1"><semantics id="S4.F3.sf2.6.3.m3.1b"><mrow id="S4.F3.sf2.6.3.m3.1.1" xref="S4.F3.sf2.6.3.m3.1.1.cmml"><mi id="S4.F3.sf2.6.3.m3.1.1.2" xref="S4.F3.sf2.6.3.m3.1.1.2.cmml">a</mi><mo id="S4.F3.sf2.6.3.m3.1.1.1" xref="S4.F3.sf2.6.3.m3.1.1.1.cmml">∈</mo><msub id="S4.F3.sf2.6.3.m3.1.1.3" xref="S4.F3.sf2.6.3.m3.1.1.3.cmml"><mi id="S4.F3.sf2.6.3.m3.1.1.3.2" xref="S4.F3.sf2.6.3.m3.1.1.3.2.cmml">G</mi><msup id="S4.F3.sf2.6.3.m3.1.1.3.3" xref="S4.F3.sf2.6.3.m3.1.1.3.3.cmml"><mi id="S4.F3.sf2.6.3.m3.1.1.3.3.2" xref="S4.F3.sf2.6.3.m3.1.1.3.3.2.cmml">u</mi><mo id="S4.F3.sf2.6.3.m3.1.1.3.3.3" xref="S4.F3.sf2.6.3.m3.1.1.3.3.3.cmml">′</mo></msup></msub></mrow><annotation-xml encoding="MathML-Content" id="S4.F3.sf2.6.3.m3.1c"><apply id="S4.F3.sf2.6.3.m3.1.1.cmml" xref="S4.F3.sf2.6.3.m3.1.1"><in id="S4.F3.sf2.6.3.m3.1.1.1.cmml" xref="S4.F3.sf2.6.3.m3.1.1.1"></in><ci id="S4.F3.sf2.6.3.m3.1.1.2.cmml" xref="S4.F3.sf2.6.3.m3.1.1.2">𝑎</ci><apply id="S4.F3.sf2.6.3.m3.1.1.3.cmml" xref="S4.F3.sf2.6.3.m3.1.1.3"><csymbol cd="ambiguous" id="S4.F3.sf2.6.3.m3.1.1.3.1.cmml" xref="S4.F3.sf2.6.3.m3.1.1.3">subscript</csymbol><ci id="S4.F3.sf2.6.3.m3.1.1.3.2.cmml" xref="S4.F3.sf2.6.3.m3.1.1.3.2">𝐺</ci><apply id="S4.F3.sf2.6.3.m3.1.1.3.3.cmml" xref="S4.F3.sf2.6.3.m3.1.1.3.3"><csymbol cd="ambiguous" id="S4.F3.sf2.6.3.m3.1.1.3.3.1.cmml" xref="S4.F3.sf2.6.3.m3.1.1.3.3">superscript</csymbol><ci id="S4.F3.sf2.6.3.m3.1.1.3.3.2.cmml" xref="S4.F3.sf2.6.3.m3.1.1.3.3.2">𝑢</ci><ci id="S4.F3.sf2.6.3.m3.1.1.3.3.3.cmml" xref="S4.F3.sf2.6.3.m3.1.1.3.3.3">′</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.F3.sf2.6.3.m3.1d">a\in G_{u^{\prime}}</annotation><annotation encoding="application/x-llamapun" id="S4.F3.sf2.6.3.m3.1e">italic_a ∈ italic_G start_POSTSUBSCRIPT italic_u start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT</annotation></semantics></math>.</span></figcaption> </figure> </div> </div> <figcaption class="ltx_caption ltx_centering"><span class="ltx_tag ltx_tag_figure"><span class="ltx_text" id="S4.F3.6.3.1" style="font-size:90%;">Figure 3</span>: </span><span class="ltx_text" id="S4.F3.4.2" style="font-size:90%;">Examples for Lemma <a class="ltx_ref" href="https://arxiv.org/html/2503.00712v1#S4.Thmtheorem5" title="Lemma 4.5. ‣ 4.1.2 Bounding Approximation Ratio ‣ 4.1 One-to-Two Augmentation ‣ 4 Vertex Connectivity Augmentation in Link-Arrival Model ‣ Streaming Algorithms for Network Design"><span class="ltx_text ltx_ref_tag">4.5</span></a>. <math alttext="a" class="ltx_Math" display="inline" id="S4.F3.3.1.m1.1"><semantics id="S4.F3.3.1.m1.1b"><mi id="S4.F3.3.1.m1.1.1" xref="S4.F3.3.1.m1.1.1.cmml">a</mi><annotation-xml encoding="MathML-Content" id="S4.F3.3.1.m1.1c"><ci id="S4.F3.3.1.m1.1.1.cmml" xref="S4.F3.3.1.m1.1.1">𝑎</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.F3.3.1.m1.1d">a</annotation><annotation encoding="application/x-llamapun" id="S4.F3.3.1.m1.1e">italic_a</annotation></semantics></math> represents the deleted vertex and <math alttext="b=\text{LCA}(u,v)" class="ltx_Math" display="inline" id="S4.F3.4.2.m2.2"><semantics id="S4.F3.4.2.m2.2b"><mrow id="S4.F3.4.2.m2.2.3" xref="S4.F3.4.2.m2.2.3.cmml"><mi id="S4.F3.4.2.m2.2.3.2" xref="S4.F3.4.2.m2.2.3.2.cmml">b</mi><mo id="S4.F3.4.2.m2.2.3.1" xref="S4.F3.4.2.m2.2.3.1.cmml">=</mo><mrow id="S4.F3.4.2.m2.2.3.3" xref="S4.F3.4.2.m2.2.3.3.cmml"><mtext id="S4.F3.4.2.m2.2.3.3.2" xref="S4.F3.4.2.m2.2.3.3.2a.cmml">LCA</mtext><mo id="S4.F3.4.2.m2.2.3.3.1" xref="S4.F3.4.2.m2.2.3.3.1.cmml">⁢</mo><mrow id="S4.F3.4.2.m2.2.3.3.3.2" xref="S4.F3.4.2.m2.2.3.3.3.1.cmml"><mo id="S4.F3.4.2.m2.2.3.3.3.2.1" stretchy="false" xref="S4.F3.4.2.m2.2.3.3.3.1.cmml">(</mo><mi id="S4.F3.4.2.m2.1.1" xref="S4.F3.4.2.m2.1.1.cmml">u</mi><mo id="S4.F3.4.2.m2.2.3.3.3.2.2" xref="S4.F3.4.2.m2.2.3.3.3.1.cmml">,</mo><mi id="S4.F3.4.2.m2.2.2" xref="S4.F3.4.2.m2.2.2.cmml">v</mi><mo id="S4.F3.4.2.m2.2.3.3.3.2.3" stretchy="false" xref="S4.F3.4.2.m2.2.3.3.3.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.F3.4.2.m2.2c"><apply id="S4.F3.4.2.m2.2.3.cmml" xref="S4.F3.4.2.m2.2.3"><eq id="S4.F3.4.2.m2.2.3.1.cmml" xref="S4.F3.4.2.m2.2.3.1"></eq><ci id="S4.F3.4.2.m2.2.3.2.cmml" xref="S4.F3.4.2.m2.2.3.2">𝑏</ci><apply id="S4.F3.4.2.m2.2.3.3.cmml" xref="S4.F3.4.2.m2.2.3.3"><times id="S4.F3.4.2.m2.2.3.3.1.cmml" xref="S4.F3.4.2.m2.2.3.3.1"></times><ci id="S4.F3.4.2.m2.2.3.3.2a.cmml" xref="S4.F3.4.2.m2.2.3.3.2"><mtext id="S4.F3.4.2.m2.2.3.3.2.cmml" xref="S4.F3.4.2.m2.2.3.3.2">LCA</mtext></ci><interval closure="open" id="S4.F3.4.2.m2.2.3.3.3.1.cmml" xref="S4.F3.4.2.m2.2.3.3.3.2"><ci id="S4.F3.4.2.m2.1.1.cmml" xref="S4.F3.4.2.m2.1.1">𝑢</ci><ci id="S4.F3.4.2.m2.2.2.cmml" xref="S4.F3.4.2.m2.2.2">𝑣</ci></interval></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.F3.4.2.m2.2d">b=\text{LCA}(u,v)</annotation><annotation encoding="application/x-llamapun" id="S4.F3.4.2.m2.2e">italic_b = LCA ( italic_u , italic_v )</annotation></semantics></math>. Dashed lines represent paths while solid edges represent edges.</span></figcaption> </figure> <div class="ltx_theorem ltx_theorem_lemma" id="S4.Thmtheorem6"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="S4.Thmtheorem6.1.1.1">Lemma 4.6</span></span><span class="ltx_text ltx_font_bold" id="S4.Thmtheorem6.2.2">.</span> </h6> <div class="ltx_para" id="S4.Thmtheorem6.p1"> <p class="ltx_p" id="S4.Thmtheorem6.p1.2">The link set <span class="ltx_text ltx_markedasmath" id="S4.Thmtheorem6.p1.2.1">SOL</span> has weight at most <math alttext="(3+\epsilon)w(\textnormal{OPT})" class="ltx_Math" display="inline" id="S4.Thmtheorem6.p1.2.m2.2"><semantics id="S4.Thmtheorem6.p1.2.m2.2a"><mrow id="S4.Thmtheorem6.p1.2.m2.2.2" xref="S4.Thmtheorem6.p1.2.m2.2.2.cmml"><mrow id="S4.Thmtheorem6.p1.2.m2.2.2.1.1" xref="S4.Thmtheorem6.p1.2.m2.2.2.1.1.1.cmml"><mo id="S4.Thmtheorem6.p1.2.m2.2.2.1.1.2" stretchy="false" xref="S4.Thmtheorem6.p1.2.m2.2.2.1.1.1.cmml">(</mo><mrow id="S4.Thmtheorem6.p1.2.m2.2.2.1.1.1" xref="S4.Thmtheorem6.p1.2.m2.2.2.1.1.1.cmml"><mn id="S4.Thmtheorem6.p1.2.m2.2.2.1.1.1.2" xref="S4.Thmtheorem6.p1.2.m2.2.2.1.1.1.2.cmml">3</mn><mo id="S4.Thmtheorem6.p1.2.m2.2.2.1.1.1.1" xref="S4.Thmtheorem6.p1.2.m2.2.2.1.1.1.1.cmml">+</mo><mi id="S4.Thmtheorem6.p1.2.m2.2.2.1.1.1.3" xref="S4.Thmtheorem6.p1.2.m2.2.2.1.1.1.3.cmml">ϵ</mi></mrow><mo id="S4.Thmtheorem6.p1.2.m2.2.2.1.1.3" stretchy="false" xref="S4.Thmtheorem6.p1.2.m2.2.2.1.1.1.cmml">)</mo></mrow><mo id="S4.Thmtheorem6.p1.2.m2.2.2.2" xref="S4.Thmtheorem6.p1.2.m2.2.2.2.cmml">⁢</mo><mi id="S4.Thmtheorem6.p1.2.m2.2.2.3" xref="S4.Thmtheorem6.p1.2.m2.2.2.3.cmml">w</mi><mo id="S4.Thmtheorem6.p1.2.m2.2.2.2a" xref="S4.Thmtheorem6.p1.2.m2.2.2.2.cmml">⁢</mo><mrow id="S4.Thmtheorem6.p1.2.m2.2.2.4.2" xref="S4.Thmtheorem6.p1.2.m2.1.1a.cmml"><mo id="S4.Thmtheorem6.p1.2.m2.2.2.4.2.1" stretchy="false" xref="S4.Thmtheorem6.p1.2.m2.1.1a.cmml">(</mo><mtext id="S4.Thmtheorem6.p1.2.m2.1.1" xref="S4.Thmtheorem6.p1.2.m2.1.1.cmml">OPT</mtext><mo id="S4.Thmtheorem6.p1.2.m2.2.2.4.2.2" stretchy="false" xref="S4.Thmtheorem6.p1.2.m2.1.1a.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem6.p1.2.m2.2b"><apply id="S4.Thmtheorem6.p1.2.m2.2.2.cmml" xref="S4.Thmtheorem6.p1.2.m2.2.2"><times id="S4.Thmtheorem6.p1.2.m2.2.2.2.cmml" xref="S4.Thmtheorem6.p1.2.m2.2.2.2"></times><apply id="S4.Thmtheorem6.p1.2.m2.2.2.1.1.1.cmml" xref="S4.Thmtheorem6.p1.2.m2.2.2.1.1"><plus id="S4.Thmtheorem6.p1.2.m2.2.2.1.1.1.1.cmml" xref="S4.Thmtheorem6.p1.2.m2.2.2.1.1.1.1"></plus><cn id="S4.Thmtheorem6.p1.2.m2.2.2.1.1.1.2.cmml" type="integer" xref="S4.Thmtheorem6.p1.2.m2.2.2.1.1.1.2">3</cn><ci id="S4.Thmtheorem6.p1.2.m2.2.2.1.1.1.3.cmml" xref="S4.Thmtheorem6.p1.2.m2.2.2.1.1.1.3">italic-ϵ</ci></apply><ci id="S4.Thmtheorem6.p1.2.m2.2.2.3.cmml" xref="S4.Thmtheorem6.p1.2.m2.2.2.3">𝑤</ci><ci id="S4.Thmtheorem6.p1.2.m2.1.1a.cmml" xref="S4.Thmtheorem6.p1.2.m2.2.2.4.2"><mtext id="S4.Thmtheorem6.p1.2.m2.1.1.cmml" xref="S4.Thmtheorem6.p1.2.m2.1.1">OPT</mtext></ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem6.p1.2.m2.2c">(3+\epsilon)w(\textnormal{OPT})</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem6.p1.2.m2.2d">( 3 + italic_ϵ ) italic_w ( OPT )</annotation></semantics></math>.</p> </div> </div> <div class="ltx_proof" id="S4.SS1.SSS2.5"> <h6 class="ltx_title ltx_runin ltx_font_italic ltx_title_proof">Proof.</h6> <div class="ltx_para" id="S4.SS1.SSS2.4.p1"> <p class="ltx_p" id="S4.SS1.SSS2.4.p1.5">In the first part of Algorithm <a class="ltx_ref" href="https://arxiv.org/html/2503.00712v1#algorithm5" title="In 4.1.2 Bounding Approximation Ratio ‣ 4.1 One-to-Two Augmentation ‣ 4 Vertex Connectivity Augmentation in Link-Arrival Model ‣ Streaming Algorithms for Network Design"><span class="ltx_text ltx_ref_tag">5</span></a>, for each link <math alttext="uv\in\textnormal{OPT}" class="ltx_Math" display="inline" id="S4.SS1.SSS2.4.p1.1.m1.1"><semantics id="S4.SS1.SSS2.4.p1.1.m1.1a"><mrow id="S4.SS1.SSS2.4.p1.1.m1.1.1" xref="S4.SS1.SSS2.4.p1.1.m1.1.1.cmml"><mrow id="S4.SS1.SSS2.4.p1.1.m1.1.1.2" xref="S4.SS1.SSS2.4.p1.1.m1.1.1.2.cmml"><mi id="S4.SS1.SSS2.4.p1.1.m1.1.1.2.2" xref="S4.SS1.SSS2.4.p1.1.m1.1.1.2.2.cmml">u</mi><mo id="S4.SS1.SSS2.4.p1.1.m1.1.1.2.1" xref="S4.SS1.SSS2.4.p1.1.m1.1.1.2.1.cmml">⁢</mo><mi id="S4.SS1.SSS2.4.p1.1.m1.1.1.2.3" xref="S4.SS1.SSS2.4.p1.1.m1.1.1.2.3.cmml">v</mi></mrow><mo id="S4.SS1.SSS2.4.p1.1.m1.1.1.1" xref="S4.SS1.SSS2.4.p1.1.m1.1.1.1.cmml">∈</mo><mtext id="S4.SS1.SSS2.4.p1.1.m1.1.1.3" xref="S4.SS1.SSS2.4.p1.1.m1.1.1.3a.cmml">OPT</mtext></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.SSS2.4.p1.1.m1.1b"><apply id="S4.SS1.SSS2.4.p1.1.m1.1.1.cmml" xref="S4.SS1.SSS2.4.p1.1.m1.1.1"><in id="S4.SS1.SSS2.4.p1.1.m1.1.1.1.cmml" xref="S4.SS1.SSS2.4.p1.1.m1.1.1.1"></in><apply id="S4.SS1.SSS2.4.p1.1.m1.1.1.2.cmml" xref="S4.SS1.SSS2.4.p1.1.m1.1.1.2"><times id="S4.SS1.SSS2.4.p1.1.m1.1.1.2.1.cmml" xref="S4.SS1.SSS2.4.p1.1.m1.1.1.2.1"></times><ci id="S4.SS1.SSS2.4.p1.1.m1.1.1.2.2.cmml" xref="S4.SS1.SSS2.4.p1.1.m1.1.1.2.2">𝑢</ci><ci id="S4.SS1.SSS2.4.p1.1.m1.1.1.2.3.cmml" xref="S4.SS1.SSS2.4.p1.1.m1.1.1.2.3">𝑣</ci></apply><ci id="S4.SS1.SSS2.4.p1.1.m1.1.1.3a.cmml" xref="S4.SS1.SSS2.4.p1.1.m1.1.1.3"><mtext id="S4.SS1.SSS2.4.p1.1.m1.1.1.3.cmml" xref="S4.SS1.SSS2.4.p1.1.m1.1.1.3">OPT</mtext></ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.SSS2.4.p1.1.m1.1c">uv\in\textnormal{OPT}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.SSS2.4.p1.1.m1.1d">italic_u italic_v ∈ OPT</annotation></semantics></math>, we add two links to <span class="ltx_text ltx_markedasmath" id="S4.SS1.SSS2.4.p1.5.1">SOL</span>, each of weight at most <math alttext="(1+\epsilon)w(uv)" class="ltx_Math" display="inline" id="S4.SS1.SSS2.4.p1.3.m3.2"><semantics id="S4.SS1.SSS2.4.p1.3.m3.2a"><mrow id="S4.SS1.SSS2.4.p1.3.m3.2.2" xref="S4.SS1.SSS2.4.p1.3.m3.2.2.cmml"><mrow id="S4.SS1.SSS2.4.p1.3.m3.1.1.1.1" xref="S4.SS1.SSS2.4.p1.3.m3.1.1.1.1.1.cmml"><mo id="S4.SS1.SSS2.4.p1.3.m3.1.1.1.1.2" stretchy="false" xref="S4.SS1.SSS2.4.p1.3.m3.1.1.1.1.1.cmml">(</mo><mrow id="S4.SS1.SSS2.4.p1.3.m3.1.1.1.1.1" xref="S4.SS1.SSS2.4.p1.3.m3.1.1.1.1.1.cmml"><mn id="S4.SS1.SSS2.4.p1.3.m3.1.1.1.1.1.2" xref="S4.SS1.SSS2.4.p1.3.m3.1.1.1.1.1.2.cmml">1</mn><mo id="S4.SS1.SSS2.4.p1.3.m3.1.1.1.1.1.1" xref="S4.SS1.SSS2.4.p1.3.m3.1.1.1.1.1.1.cmml">+</mo><mi id="S4.SS1.SSS2.4.p1.3.m3.1.1.1.1.1.3" xref="S4.SS1.SSS2.4.p1.3.m3.1.1.1.1.1.3.cmml">ϵ</mi></mrow><mo id="S4.SS1.SSS2.4.p1.3.m3.1.1.1.1.3" stretchy="false" xref="S4.SS1.SSS2.4.p1.3.m3.1.1.1.1.1.cmml">)</mo></mrow><mo id="S4.SS1.SSS2.4.p1.3.m3.2.2.3" xref="S4.SS1.SSS2.4.p1.3.m3.2.2.3.cmml">⁢</mo><mi id="S4.SS1.SSS2.4.p1.3.m3.2.2.4" xref="S4.SS1.SSS2.4.p1.3.m3.2.2.4.cmml">w</mi><mo id="S4.SS1.SSS2.4.p1.3.m3.2.2.3a" xref="S4.SS1.SSS2.4.p1.3.m3.2.2.3.cmml">⁢</mo><mrow id="S4.SS1.SSS2.4.p1.3.m3.2.2.2.1" xref="S4.SS1.SSS2.4.p1.3.m3.2.2.2.1.1.cmml"><mo id="S4.SS1.SSS2.4.p1.3.m3.2.2.2.1.2" stretchy="false" xref="S4.SS1.SSS2.4.p1.3.m3.2.2.2.1.1.cmml">(</mo><mrow id="S4.SS1.SSS2.4.p1.3.m3.2.2.2.1.1" xref="S4.SS1.SSS2.4.p1.3.m3.2.2.2.1.1.cmml"><mi id="S4.SS1.SSS2.4.p1.3.m3.2.2.2.1.1.2" xref="S4.SS1.SSS2.4.p1.3.m3.2.2.2.1.1.2.cmml">u</mi><mo id="S4.SS1.SSS2.4.p1.3.m3.2.2.2.1.1.1" xref="S4.SS1.SSS2.4.p1.3.m3.2.2.2.1.1.1.cmml">⁢</mo><mi id="S4.SS1.SSS2.4.p1.3.m3.2.2.2.1.1.3" xref="S4.SS1.SSS2.4.p1.3.m3.2.2.2.1.1.3.cmml">v</mi></mrow><mo id="S4.SS1.SSS2.4.p1.3.m3.2.2.2.1.3" stretchy="false" xref="S4.SS1.SSS2.4.p1.3.m3.2.2.2.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.SSS2.4.p1.3.m3.2b"><apply id="S4.SS1.SSS2.4.p1.3.m3.2.2.cmml" xref="S4.SS1.SSS2.4.p1.3.m3.2.2"><times id="S4.SS1.SSS2.4.p1.3.m3.2.2.3.cmml" xref="S4.SS1.SSS2.4.p1.3.m3.2.2.3"></times><apply id="S4.SS1.SSS2.4.p1.3.m3.1.1.1.1.1.cmml" xref="S4.SS1.SSS2.4.p1.3.m3.1.1.1.1"><plus id="S4.SS1.SSS2.4.p1.3.m3.1.1.1.1.1.1.cmml" xref="S4.SS1.SSS2.4.p1.3.m3.1.1.1.1.1.1"></plus><cn id="S4.SS1.SSS2.4.p1.3.m3.1.1.1.1.1.2.cmml" type="integer" xref="S4.SS1.SSS2.4.p1.3.m3.1.1.1.1.1.2">1</cn><ci id="S4.SS1.SSS2.4.p1.3.m3.1.1.1.1.1.3.cmml" xref="S4.SS1.SSS2.4.p1.3.m3.1.1.1.1.1.3">italic-ϵ</ci></apply><ci id="S4.SS1.SSS2.4.p1.3.m3.2.2.4.cmml" xref="S4.SS1.SSS2.4.p1.3.m3.2.2.4">𝑤</ci><apply id="S4.SS1.SSS2.4.p1.3.m3.2.2.2.1.1.cmml" xref="S4.SS1.SSS2.4.p1.3.m3.2.2.2.1"><times id="S4.SS1.SSS2.4.p1.3.m3.2.2.2.1.1.1.cmml" xref="S4.SS1.SSS2.4.p1.3.m3.2.2.2.1.1.1"></times><ci id="S4.SS1.SSS2.4.p1.3.m3.2.2.2.1.1.2.cmml" xref="S4.SS1.SSS2.4.p1.3.m3.2.2.2.1.1.2">𝑢</ci><ci id="S4.SS1.SSS2.4.p1.3.m3.2.2.2.1.1.3.cmml" xref="S4.SS1.SSS2.4.p1.3.m3.2.2.2.1.1.3">𝑣</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.SSS2.4.p1.3.m3.2c">(1+\epsilon)w(uv)</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.SSS2.4.p1.3.m3.2d">( 1 + italic_ϵ ) italic_w ( italic_u italic_v )</annotation></semantics></math>. Thus the total weight of links added to <span class="ltx_text ltx_markedasmath" id="S4.SS1.SSS2.4.p1.5.2">SOL</span> in the first part is at most <math alttext="2(1+\epsilon)w(\textnormal{OPT})" class="ltx_Math" display="inline" id="S4.SS1.SSS2.4.p1.5.m5.2"><semantics id="S4.SS1.SSS2.4.p1.5.m5.2a"><mrow id="S4.SS1.SSS2.4.p1.5.m5.2.2" xref="S4.SS1.SSS2.4.p1.5.m5.2.2.cmml"><mn id="S4.SS1.SSS2.4.p1.5.m5.2.2.3" xref="S4.SS1.SSS2.4.p1.5.m5.2.2.3.cmml">2</mn><mo id="S4.SS1.SSS2.4.p1.5.m5.2.2.2" xref="S4.SS1.SSS2.4.p1.5.m5.2.2.2.cmml">⁢</mo><mrow id="S4.SS1.SSS2.4.p1.5.m5.2.2.1.1" xref="S4.SS1.SSS2.4.p1.5.m5.2.2.1.1.1.cmml"><mo id="S4.SS1.SSS2.4.p1.5.m5.2.2.1.1.2" stretchy="false" xref="S4.SS1.SSS2.4.p1.5.m5.2.2.1.1.1.cmml">(</mo><mrow id="S4.SS1.SSS2.4.p1.5.m5.2.2.1.1.1" xref="S4.SS1.SSS2.4.p1.5.m5.2.2.1.1.1.cmml"><mn id="S4.SS1.SSS2.4.p1.5.m5.2.2.1.1.1.2" xref="S4.SS1.SSS2.4.p1.5.m5.2.2.1.1.1.2.cmml">1</mn><mo id="S4.SS1.SSS2.4.p1.5.m5.2.2.1.1.1.1" xref="S4.SS1.SSS2.4.p1.5.m5.2.2.1.1.1.1.cmml">+</mo><mi id="S4.SS1.SSS2.4.p1.5.m5.2.2.1.1.1.3" xref="S4.SS1.SSS2.4.p1.5.m5.2.2.1.1.1.3.cmml">ϵ</mi></mrow><mo id="S4.SS1.SSS2.4.p1.5.m5.2.2.1.1.3" stretchy="false" xref="S4.SS1.SSS2.4.p1.5.m5.2.2.1.1.1.cmml">)</mo></mrow><mo id="S4.SS1.SSS2.4.p1.5.m5.2.2.2a" xref="S4.SS1.SSS2.4.p1.5.m5.2.2.2.cmml">⁢</mo><mi id="S4.SS1.SSS2.4.p1.5.m5.2.2.4" xref="S4.SS1.SSS2.4.p1.5.m5.2.2.4.cmml">w</mi><mo id="S4.SS1.SSS2.4.p1.5.m5.2.2.2b" xref="S4.SS1.SSS2.4.p1.5.m5.2.2.2.cmml">⁢</mo><mrow id="S4.SS1.SSS2.4.p1.5.m5.2.2.5.2" xref="S4.SS1.SSS2.4.p1.5.m5.1.1a.cmml"><mo id="S4.SS1.SSS2.4.p1.5.m5.2.2.5.2.1" stretchy="false" xref="S4.SS1.SSS2.4.p1.5.m5.1.1a.cmml">(</mo><mtext id="S4.SS1.SSS2.4.p1.5.m5.1.1" xref="S4.SS1.SSS2.4.p1.5.m5.1.1.cmml">OPT</mtext><mo id="S4.SS1.SSS2.4.p1.5.m5.2.2.5.2.2" stretchy="false" xref="S4.SS1.SSS2.4.p1.5.m5.1.1a.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.SSS2.4.p1.5.m5.2b"><apply id="S4.SS1.SSS2.4.p1.5.m5.2.2.cmml" xref="S4.SS1.SSS2.4.p1.5.m5.2.2"><times id="S4.SS1.SSS2.4.p1.5.m5.2.2.2.cmml" xref="S4.SS1.SSS2.4.p1.5.m5.2.2.2"></times><cn id="S4.SS1.SSS2.4.p1.5.m5.2.2.3.cmml" type="integer" xref="S4.SS1.SSS2.4.p1.5.m5.2.2.3">2</cn><apply id="S4.SS1.SSS2.4.p1.5.m5.2.2.1.1.1.cmml" xref="S4.SS1.SSS2.4.p1.5.m5.2.2.1.1"><plus id="S4.SS1.SSS2.4.p1.5.m5.2.2.1.1.1.1.cmml" xref="S4.SS1.SSS2.4.p1.5.m5.2.2.1.1.1.1"></plus><cn id="S4.SS1.SSS2.4.p1.5.m5.2.2.1.1.1.2.cmml" type="integer" xref="S4.SS1.SSS2.4.p1.5.m5.2.2.1.1.1.2">1</cn><ci id="S4.SS1.SSS2.4.p1.5.m5.2.2.1.1.1.3.cmml" xref="S4.SS1.SSS2.4.p1.5.m5.2.2.1.1.1.3">italic-ϵ</ci></apply><ci id="S4.SS1.SSS2.4.p1.5.m5.2.2.4.cmml" xref="S4.SS1.SSS2.4.p1.5.m5.2.2.4">𝑤</ci><ci id="S4.SS1.SSS2.4.p1.5.m5.1.1a.cmml" xref="S4.SS1.SSS2.4.p1.5.m5.2.2.5.2"><mtext id="S4.SS1.SSS2.4.p1.5.m5.1.1.cmml" xref="S4.SS1.SSS2.4.p1.5.m5.1.1">OPT</mtext></ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.SSS2.4.p1.5.m5.2c">2(1+\epsilon)w(\textnormal{OPT})</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.SSS2.4.p1.5.m5.2d">2 ( 1 + italic_ϵ ) italic_w ( OPT )</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="S4.SS1.SSS2.5.p2"> <p class="ltx_p" id="S4.SS1.SSS2.5.p2.23">We claim that the second part of the algorithm costs at most <span class="ltx_text ltx_markedasmath" id="S4.SS1.SSS2.5.p2.23.1">OPT</span>. To show this, consider a partition of <span class="ltx_text ltx_markedasmath" id="S4.SS1.SSS2.5.p2.23.2">OPT</span> based on the <span class="ltx_text" id="S4.SS1.SSS2.5.p2.23.3">LCA</span> of its endpoints: let <math alttext="E^{*}(a)=\{e\in\textnormal{OPT}:\text{LCA}(e)=a\}" class="ltx_Math" display="inline" id="S4.SS1.SSS2.5.p2.3.m3.4"><semantics id="S4.SS1.SSS2.5.p2.3.m3.4a"><mrow id="S4.SS1.SSS2.5.p2.3.m3.4.4" xref="S4.SS1.SSS2.5.p2.3.m3.4.4.cmml"><mrow id="S4.SS1.SSS2.5.p2.3.m3.4.4.4" xref="S4.SS1.SSS2.5.p2.3.m3.4.4.4.cmml"><msup id="S4.SS1.SSS2.5.p2.3.m3.4.4.4.2" xref="S4.SS1.SSS2.5.p2.3.m3.4.4.4.2.cmml"><mi id="S4.SS1.SSS2.5.p2.3.m3.4.4.4.2.2" xref="S4.SS1.SSS2.5.p2.3.m3.4.4.4.2.2.cmml">E</mi><mo id="S4.SS1.SSS2.5.p2.3.m3.4.4.4.2.3" xref="S4.SS1.SSS2.5.p2.3.m3.4.4.4.2.3.cmml">∗</mo></msup><mo id="S4.SS1.SSS2.5.p2.3.m3.4.4.4.1" xref="S4.SS1.SSS2.5.p2.3.m3.4.4.4.1.cmml">⁢</mo><mrow id="S4.SS1.SSS2.5.p2.3.m3.4.4.4.3.2" xref="S4.SS1.SSS2.5.p2.3.m3.4.4.4.cmml"><mo id="S4.SS1.SSS2.5.p2.3.m3.4.4.4.3.2.1" stretchy="false" xref="S4.SS1.SSS2.5.p2.3.m3.4.4.4.cmml">(</mo><mi id="S4.SS1.SSS2.5.p2.3.m3.1.1" xref="S4.SS1.SSS2.5.p2.3.m3.1.1.cmml">a</mi><mo id="S4.SS1.SSS2.5.p2.3.m3.4.4.4.3.2.2" stretchy="false" xref="S4.SS1.SSS2.5.p2.3.m3.4.4.4.cmml">)</mo></mrow></mrow><mo id="S4.SS1.SSS2.5.p2.3.m3.4.4.3" xref="S4.SS1.SSS2.5.p2.3.m3.4.4.3.cmml">=</mo><mrow id="S4.SS1.SSS2.5.p2.3.m3.4.4.2.2" xref="S4.SS1.SSS2.5.p2.3.m3.4.4.2.3.cmml"><mo id="S4.SS1.SSS2.5.p2.3.m3.4.4.2.2.3" stretchy="false" xref="S4.SS1.SSS2.5.p2.3.m3.4.4.2.3.1.cmml">{</mo><mrow id="S4.SS1.SSS2.5.p2.3.m3.3.3.1.1.1" xref="S4.SS1.SSS2.5.p2.3.m3.3.3.1.1.1.cmml"><mi id="S4.SS1.SSS2.5.p2.3.m3.3.3.1.1.1.2" xref="S4.SS1.SSS2.5.p2.3.m3.3.3.1.1.1.2.cmml">e</mi><mo id="S4.SS1.SSS2.5.p2.3.m3.3.3.1.1.1.1" xref="S4.SS1.SSS2.5.p2.3.m3.3.3.1.1.1.1.cmml">∈</mo><mtext id="S4.SS1.SSS2.5.p2.3.m3.3.3.1.1.1.3" xref="S4.SS1.SSS2.5.p2.3.m3.3.3.1.1.1.3a.cmml">OPT</mtext></mrow><mo id="S4.SS1.SSS2.5.p2.3.m3.4.4.2.2.4" lspace="0.278em" rspace="0.278em" xref="S4.SS1.SSS2.5.p2.3.m3.4.4.2.3.1.cmml">:</mo><mrow id="S4.SS1.SSS2.5.p2.3.m3.4.4.2.2.2" xref="S4.SS1.SSS2.5.p2.3.m3.4.4.2.2.2.cmml"><mrow id="S4.SS1.SSS2.5.p2.3.m3.4.4.2.2.2.2" xref="S4.SS1.SSS2.5.p2.3.m3.4.4.2.2.2.2.cmml"><mtext id="S4.SS1.SSS2.5.p2.3.m3.4.4.2.2.2.2.2" xref="S4.SS1.SSS2.5.p2.3.m3.4.4.2.2.2.2.2a.cmml">LCA</mtext><mo id="S4.SS1.SSS2.5.p2.3.m3.4.4.2.2.2.2.1" xref="S4.SS1.SSS2.5.p2.3.m3.4.4.2.2.2.2.1.cmml">⁢</mo><mrow id="S4.SS1.SSS2.5.p2.3.m3.4.4.2.2.2.2.3.2" xref="S4.SS1.SSS2.5.p2.3.m3.4.4.2.2.2.2.cmml"><mo id="S4.SS1.SSS2.5.p2.3.m3.4.4.2.2.2.2.3.2.1" stretchy="false" xref="S4.SS1.SSS2.5.p2.3.m3.4.4.2.2.2.2.cmml">(</mo><mi id="S4.SS1.SSS2.5.p2.3.m3.2.2" xref="S4.SS1.SSS2.5.p2.3.m3.2.2.cmml">e</mi><mo id="S4.SS1.SSS2.5.p2.3.m3.4.4.2.2.2.2.3.2.2" stretchy="false" xref="S4.SS1.SSS2.5.p2.3.m3.4.4.2.2.2.2.cmml">)</mo></mrow></mrow><mo id="S4.SS1.SSS2.5.p2.3.m3.4.4.2.2.2.1" xref="S4.SS1.SSS2.5.p2.3.m3.4.4.2.2.2.1.cmml">=</mo><mi id="S4.SS1.SSS2.5.p2.3.m3.4.4.2.2.2.3" xref="S4.SS1.SSS2.5.p2.3.m3.4.4.2.2.2.3.cmml">a</mi></mrow><mo id="S4.SS1.SSS2.5.p2.3.m3.4.4.2.2.5" stretchy="false" xref="S4.SS1.SSS2.5.p2.3.m3.4.4.2.3.1.cmml">}</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.SSS2.5.p2.3.m3.4b"><apply id="S4.SS1.SSS2.5.p2.3.m3.4.4.cmml" xref="S4.SS1.SSS2.5.p2.3.m3.4.4"><eq id="S4.SS1.SSS2.5.p2.3.m3.4.4.3.cmml" xref="S4.SS1.SSS2.5.p2.3.m3.4.4.3"></eq><apply id="S4.SS1.SSS2.5.p2.3.m3.4.4.4.cmml" xref="S4.SS1.SSS2.5.p2.3.m3.4.4.4"><times id="S4.SS1.SSS2.5.p2.3.m3.4.4.4.1.cmml" xref="S4.SS1.SSS2.5.p2.3.m3.4.4.4.1"></times><apply id="S4.SS1.SSS2.5.p2.3.m3.4.4.4.2.cmml" xref="S4.SS1.SSS2.5.p2.3.m3.4.4.4.2"><csymbol cd="ambiguous" id="S4.SS1.SSS2.5.p2.3.m3.4.4.4.2.1.cmml" xref="S4.SS1.SSS2.5.p2.3.m3.4.4.4.2">superscript</csymbol><ci id="S4.SS1.SSS2.5.p2.3.m3.4.4.4.2.2.cmml" xref="S4.SS1.SSS2.5.p2.3.m3.4.4.4.2.2">𝐸</ci><times id="S4.SS1.SSS2.5.p2.3.m3.4.4.4.2.3.cmml" xref="S4.SS1.SSS2.5.p2.3.m3.4.4.4.2.3"></times></apply><ci id="S4.SS1.SSS2.5.p2.3.m3.1.1.cmml" xref="S4.SS1.SSS2.5.p2.3.m3.1.1">𝑎</ci></apply><apply id="S4.SS1.SSS2.5.p2.3.m3.4.4.2.3.cmml" xref="S4.SS1.SSS2.5.p2.3.m3.4.4.2.2"><csymbol cd="latexml" id="S4.SS1.SSS2.5.p2.3.m3.4.4.2.3.1.cmml" xref="S4.SS1.SSS2.5.p2.3.m3.4.4.2.2.3">conditional-set</csymbol><apply id="S4.SS1.SSS2.5.p2.3.m3.3.3.1.1.1.cmml" xref="S4.SS1.SSS2.5.p2.3.m3.3.3.1.1.1"><in id="S4.SS1.SSS2.5.p2.3.m3.3.3.1.1.1.1.cmml" xref="S4.SS1.SSS2.5.p2.3.m3.3.3.1.1.1.1"></in><ci id="S4.SS1.SSS2.5.p2.3.m3.3.3.1.1.1.2.cmml" xref="S4.SS1.SSS2.5.p2.3.m3.3.3.1.1.1.2">𝑒</ci><ci id="S4.SS1.SSS2.5.p2.3.m3.3.3.1.1.1.3a.cmml" xref="S4.SS1.SSS2.5.p2.3.m3.3.3.1.1.1.3"><mtext id="S4.SS1.SSS2.5.p2.3.m3.3.3.1.1.1.3.cmml" xref="S4.SS1.SSS2.5.p2.3.m3.3.3.1.1.1.3">OPT</mtext></ci></apply><apply id="S4.SS1.SSS2.5.p2.3.m3.4.4.2.2.2.cmml" xref="S4.SS1.SSS2.5.p2.3.m3.4.4.2.2.2"><eq id="S4.SS1.SSS2.5.p2.3.m3.4.4.2.2.2.1.cmml" xref="S4.SS1.SSS2.5.p2.3.m3.4.4.2.2.2.1"></eq><apply id="S4.SS1.SSS2.5.p2.3.m3.4.4.2.2.2.2.cmml" xref="S4.SS1.SSS2.5.p2.3.m3.4.4.2.2.2.2"><times id="S4.SS1.SSS2.5.p2.3.m3.4.4.2.2.2.2.1.cmml" xref="S4.SS1.SSS2.5.p2.3.m3.4.4.2.2.2.2.1"></times><ci id="S4.SS1.SSS2.5.p2.3.m3.4.4.2.2.2.2.2a.cmml" xref="S4.SS1.SSS2.5.p2.3.m3.4.4.2.2.2.2.2"><mtext id="S4.SS1.SSS2.5.p2.3.m3.4.4.2.2.2.2.2.cmml" xref="S4.SS1.SSS2.5.p2.3.m3.4.4.2.2.2.2.2">LCA</mtext></ci><ci id="S4.SS1.SSS2.5.p2.3.m3.2.2.cmml" xref="S4.SS1.SSS2.5.p2.3.m3.2.2">𝑒</ci></apply><ci id="S4.SS1.SSS2.5.p2.3.m3.4.4.2.2.2.3.cmml" xref="S4.SS1.SSS2.5.p2.3.m3.4.4.2.2.2.3">𝑎</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.SSS2.5.p2.3.m3.4c">E^{*}(a)=\{e\in\textnormal{OPT}:\text{LCA}(e)=a\}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.SSS2.5.p2.3.m3.4d">italic_E start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ( italic_a ) = { italic_e ∈ OPT : LCA ( italic_e ) = italic_a }</annotation></semantics></math>. For each <math alttext="v\in C(a)" class="ltx_Math" display="inline" id="S4.SS1.SSS2.5.p2.4.m4.1"><semantics id="S4.SS1.SSS2.5.p2.4.m4.1a"><mrow id="S4.SS1.SSS2.5.p2.4.m4.1.2" xref="S4.SS1.SSS2.5.p2.4.m4.1.2.cmml"><mi id="S4.SS1.SSS2.5.p2.4.m4.1.2.2" xref="S4.SS1.SSS2.5.p2.4.m4.1.2.2.cmml">v</mi><mo id="S4.SS1.SSS2.5.p2.4.m4.1.2.1" xref="S4.SS1.SSS2.5.p2.4.m4.1.2.1.cmml">∈</mo><mrow id="S4.SS1.SSS2.5.p2.4.m4.1.2.3" xref="S4.SS1.SSS2.5.p2.4.m4.1.2.3.cmml"><mi id="S4.SS1.SSS2.5.p2.4.m4.1.2.3.2" xref="S4.SS1.SSS2.5.p2.4.m4.1.2.3.2.cmml">C</mi><mo id="S4.SS1.SSS2.5.p2.4.m4.1.2.3.1" xref="S4.SS1.SSS2.5.p2.4.m4.1.2.3.1.cmml">⁢</mo><mrow id="S4.SS1.SSS2.5.p2.4.m4.1.2.3.3.2" xref="S4.SS1.SSS2.5.p2.4.m4.1.2.3.cmml"><mo id="S4.SS1.SSS2.5.p2.4.m4.1.2.3.3.2.1" stretchy="false" xref="S4.SS1.SSS2.5.p2.4.m4.1.2.3.cmml">(</mo><mi id="S4.SS1.SSS2.5.p2.4.m4.1.1" xref="S4.SS1.SSS2.5.p2.4.m4.1.1.cmml">a</mi><mo id="S4.SS1.SSS2.5.p2.4.m4.1.2.3.3.2.2" stretchy="false" xref="S4.SS1.SSS2.5.p2.4.m4.1.2.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.SSS2.5.p2.4.m4.1b"><apply id="S4.SS1.SSS2.5.p2.4.m4.1.2.cmml" xref="S4.SS1.SSS2.5.p2.4.m4.1.2"><in id="S4.SS1.SSS2.5.p2.4.m4.1.2.1.cmml" xref="S4.SS1.SSS2.5.p2.4.m4.1.2.1"></in><ci id="S4.SS1.SSS2.5.p2.4.m4.1.2.2.cmml" xref="S4.SS1.SSS2.5.p2.4.m4.1.2.2">𝑣</ci><apply id="S4.SS1.SSS2.5.p2.4.m4.1.2.3.cmml" xref="S4.SS1.SSS2.5.p2.4.m4.1.2.3"><times id="S4.SS1.SSS2.5.p2.4.m4.1.2.3.1.cmml" xref="S4.SS1.SSS2.5.p2.4.m4.1.2.3.1"></times><ci id="S4.SS1.SSS2.5.p2.4.m4.1.2.3.2.cmml" xref="S4.SS1.SSS2.5.p2.4.m4.1.2.3.2">𝐶</ci><ci id="S4.SS1.SSS2.5.p2.4.m4.1.1.cmml" xref="S4.SS1.SSS2.5.p2.4.m4.1.1">𝑎</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.SSS2.5.p2.4.m4.1c">v\in C(a)</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.SSS2.5.p2.4.m4.1d">italic_v ∈ italic_C ( italic_a )</annotation></semantics></math>, there must be some path in <math alttext="E\cup\textnormal{OPT}" class="ltx_Math" display="inline" id="S4.SS1.SSS2.5.p2.5.m5.1"><semantics id="S4.SS1.SSS2.5.p2.5.m5.1a"><mrow id="S4.SS1.SSS2.5.p2.5.m5.1.1" xref="S4.SS1.SSS2.5.p2.5.m5.1.1.cmml"><mi id="S4.SS1.SSS2.5.p2.5.m5.1.1.2" xref="S4.SS1.SSS2.5.p2.5.m5.1.1.2.cmml">E</mi><mo id="S4.SS1.SSS2.5.p2.5.m5.1.1.1" xref="S4.SS1.SSS2.5.p2.5.m5.1.1.1.cmml">∪</mo><mtext id="S4.SS1.SSS2.5.p2.5.m5.1.1.3" xref="S4.SS1.SSS2.5.p2.5.m5.1.1.3a.cmml">OPT</mtext></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.SSS2.5.p2.5.m5.1b"><apply id="S4.SS1.SSS2.5.p2.5.m5.1.1.cmml" xref="S4.SS1.SSS2.5.p2.5.m5.1.1"><union id="S4.SS1.SSS2.5.p2.5.m5.1.1.1.cmml" xref="S4.SS1.SSS2.5.p2.5.m5.1.1.1"></union><ci id="S4.SS1.SSS2.5.p2.5.m5.1.1.2.cmml" xref="S4.SS1.SSS2.5.p2.5.m5.1.1.2">𝐸</ci><ci id="S4.SS1.SSS2.5.p2.5.m5.1.1.3a.cmml" xref="S4.SS1.SSS2.5.p2.5.m5.1.1.3"><mtext id="S4.SS1.SSS2.5.p2.5.m5.1.1.3.cmml" xref="S4.SS1.SSS2.5.p2.5.m5.1.1.3">OPT</mtext></ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.SSS2.5.p2.5.m5.1c">E\cup\textnormal{OPT}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.SSS2.5.p2.5.m5.1d">italic_E ∪ OPT</annotation></semantics></math> from <math alttext="G_{v}" class="ltx_Math" display="inline" id="S4.SS1.SSS2.5.p2.6.m6.1"><semantics id="S4.SS1.SSS2.5.p2.6.m6.1a"><msub id="S4.SS1.SSS2.5.p2.6.m6.1.1" xref="S4.SS1.SSS2.5.p2.6.m6.1.1.cmml"><mi id="S4.SS1.SSS2.5.p2.6.m6.1.1.2" xref="S4.SS1.SSS2.5.p2.6.m6.1.1.2.cmml">G</mi><mi id="S4.SS1.SSS2.5.p2.6.m6.1.1.3" xref="S4.SS1.SSS2.5.p2.6.m6.1.1.3.cmml">v</mi></msub><annotation-xml encoding="MathML-Content" id="S4.SS1.SSS2.5.p2.6.m6.1b"><apply id="S4.SS1.SSS2.5.p2.6.m6.1.1.cmml" xref="S4.SS1.SSS2.5.p2.6.m6.1.1"><csymbol cd="ambiguous" id="S4.SS1.SSS2.5.p2.6.m6.1.1.1.cmml" xref="S4.SS1.SSS2.5.p2.6.m6.1.1">subscript</csymbol><ci id="S4.SS1.SSS2.5.p2.6.m6.1.1.2.cmml" xref="S4.SS1.SSS2.5.p2.6.m6.1.1.2">𝐺</ci><ci id="S4.SS1.SSS2.5.p2.6.m6.1.1.3.cmml" xref="S4.SS1.SSS2.5.p2.6.m6.1.1.3">𝑣</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.SSS2.5.p2.6.m6.1c">G_{v}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.SSS2.5.p2.6.m6.1d">italic_G start_POSTSUBSCRIPT italic_v end_POSTSUBSCRIPT</annotation></semantics></math> to <math alttext="G\setminus G_{a}" class="ltx_Math" display="inline" id="S4.SS1.SSS2.5.p2.7.m7.1"><semantics id="S4.SS1.SSS2.5.p2.7.m7.1a"><mrow id="S4.SS1.SSS2.5.p2.7.m7.1.1" xref="S4.SS1.SSS2.5.p2.7.m7.1.1.cmml"><mi id="S4.SS1.SSS2.5.p2.7.m7.1.1.2" xref="S4.SS1.SSS2.5.p2.7.m7.1.1.2.cmml">G</mi><mo id="S4.SS1.SSS2.5.p2.7.m7.1.1.1" xref="S4.SS1.SSS2.5.p2.7.m7.1.1.1.cmml">∖</mo><msub id="S4.SS1.SSS2.5.p2.7.m7.1.1.3" xref="S4.SS1.SSS2.5.p2.7.m7.1.1.3.cmml"><mi id="S4.SS1.SSS2.5.p2.7.m7.1.1.3.2" xref="S4.SS1.SSS2.5.p2.7.m7.1.1.3.2.cmml">G</mi><mi id="S4.SS1.SSS2.5.p2.7.m7.1.1.3.3" xref="S4.SS1.SSS2.5.p2.7.m7.1.1.3.3.cmml">a</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.SSS2.5.p2.7.m7.1b"><apply id="S4.SS1.SSS2.5.p2.7.m7.1.1.cmml" xref="S4.SS1.SSS2.5.p2.7.m7.1.1"><setdiff id="S4.SS1.SSS2.5.p2.7.m7.1.1.1.cmml" xref="S4.SS1.SSS2.5.p2.7.m7.1.1.1"></setdiff><ci id="S4.SS1.SSS2.5.p2.7.m7.1.1.2.cmml" xref="S4.SS1.SSS2.5.p2.7.m7.1.1.2">𝐺</ci><apply id="S4.SS1.SSS2.5.p2.7.m7.1.1.3.cmml" xref="S4.SS1.SSS2.5.p2.7.m7.1.1.3"><csymbol cd="ambiguous" id="S4.SS1.SSS2.5.p2.7.m7.1.1.3.1.cmml" xref="S4.SS1.SSS2.5.p2.7.m7.1.1.3">subscript</csymbol><ci id="S4.SS1.SSS2.5.p2.7.m7.1.1.3.2.cmml" xref="S4.SS1.SSS2.5.p2.7.m7.1.1.3.2">𝐺</ci><ci id="S4.SS1.SSS2.5.p2.7.m7.1.1.3.3.cmml" xref="S4.SS1.SSS2.5.p2.7.m7.1.1.3.3">𝑎</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.SSS2.5.p2.7.m7.1c">G\setminus G_{a}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.SSS2.5.p2.7.m7.1d">italic_G ∖ italic_G start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT</annotation></semantics></math> <em class="ltx_emph ltx_font_italic" id="S4.SS1.SSS2.5.p2.23.4">not</em> containing <math alttext="a" class="ltx_Math" display="inline" id="S4.SS1.SSS2.5.p2.8.m8.1"><semantics id="S4.SS1.SSS2.5.p2.8.m8.1a"><mi id="S4.SS1.SSS2.5.p2.8.m8.1.1" xref="S4.SS1.SSS2.5.p2.8.m8.1.1.cmml">a</mi><annotation-xml encoding="MathML-Content" id="S4.SS1.SSS2.5.p2.8.m8.1b"><ci id="S4.SS1.SSS2.5.p2.8.m8.1.1.cmml" xref="S4.SS1.SSS2.5.p2.8.m8.1.1">𝑎</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.SSS2.5.p2.8.m8.1c">a</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.SSS2.5.p2.8.m8.1d">italic_a</annotation></semantics></math>; else <math alttext="a" class="ltx_Math" display="inline" id="S4.SS1.SSS2.5.p2.9.m9.1"><semantics id="S4.SS1.SSS2.5.p2.9.m9.1a"><mi id="S4.SS1.SSS2.5.p2.9.m9.1.1" xref="S4.SS1.SSS2.5.p2.9.m9.1.1.cmml">a</mi><annotation-xml encoding="MathML-Content" id="S4.SS1.SSS2.5.p2.9.m9.1b"><ci id="S4.SS1.SSS2.5.p2.9.m9.1.1.cmml" xref="S4.SS1.SSS2.5.p2.9.m9.1.1">𝑎</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.SSS2.5.p2.9.m9.1c">a</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.SSS2.5.p2.9.m9.1d">italic_a</annotation></semantics></math> would be a cut-vertex separating <math alttext="G_{v}" class="ltx_Math" display="inline" id="S4.SS1.SSS2.5.p2.10.m10.1"><semantics id="S4.SS1.SSS2.5.p2.10.m10.1a"><msub id="S4.SS1.SSS2.5.p2.10.m10.1.1" xref="S4.SS1.SSS2.5.p2.10.m10.1.1.cmml"><mi id="S4.SS1.SSS2.5.p2.10.m10.1.1.2" xref="S4.SS1.SSS2.5.p2.10.m10.1.1.2.cmml">G</mi><mi id="S4.SS1.SSS2.5.p2.10.m10.1.1.3" xref="S4.SS1.SSS2.5.p2.10.m10.1.1.3.cmml">v</mi></msub><annotation-xml encoding="MathML-Content" id="S4.SS1.SSS2.5.p2.10.m10.1b"><apply id="S4.SS1.SSS2.5.p2.10.m10.1.1.cmml" xref="S4.SS1.SSS2.5.p2.10.m10.1.1"><csymbol cd="ambiguous" id="S4.SS1.SSS2.5.p2.10.m10.1.1.1.cmml" xref="S4.SS1.SSS2.5.p2.10.m10.1.1">subscript</csymbol><ci id="S4.SS1.SSS2.5.p2.10.m10.1.1.2.cmml" xref="S4.SS1.SSS2.5.p2.10.m10.1.1.2">𝐺</ci><ci id="S4.SS1.SSS2.5.p2.10.m10.1.1.3.cmml" xref="S4.SS1.SSS2.5.p2.10.m10.1.1.3">𝑣</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.SSS2.5.p2.10.m10.1c">G_{v}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.SSS2.5.p2.10.m10.1d">italic_G start_POSTSUBSCRIPT italic_v end_POSTSUBSCRIPT</annotation></semantics></math> from the rest of the graph, contradicting feasibility of <span class="ltx_text ltx_markedasmath" id="S4.SS1.SSS2.5.p2.23.5">OPT</span>. Thus this path must contain some edge <math alttext="e\in\textnormal{OPT}[G_{a}\setminus\{a\},G\setminus G_{a}]" class="ltx_Math" display="inline" id="S4.SS1.SSS2.5.p2.12.m12.3"><semantics id="S4.SS1.SSS2.5.p2.12.m12.3a"><mrow id="S4.SS1.SSS2.5.p2.12.m12.3.3" xref="S4.SS1.SSS2.5.p2.12.m12.3.3.cmml"><mi id="S4.SS1.SSS2.5.p2.12.m12.3.3.4" xref="S4.SS1.SSS2.5.p2.12.m12.3.3.4.cmml">e</mi><mo id="S4.SS1.SSS2.5.p2.12.m12.3.3.3" xref="S4.SS1.SSS2.5.p2.12.m12.3.3.3.cmml">∈</mo><mrow id="S4.SS1.SSS2.5.p2.12.m12.3.3.2" xref="S4.SS1.SSS2.5.p2.12.m12.3.3.2.cmml"><mtext id="S4.SS1.SSS2.5.p2.12.m12.3.3.2.4" xref="S4.SS1.SSS2.5.p2.12.m12.3.3.2.4a.cmml">OPT</mtext><mo id="S4.SS1.SSS2.5.p2.12.m12.3.3.2.3" xref="S4.SS1.SSS2.5.p2.12.m12.3.3.2.3.cmml">⁢</mo><mrow id="S4.SS1.SSS2.5.p2.12.m12.3.3.2.2.2" xref="S4.SS1.SSS2.5.p2.12.m12.3.3.2.2.3.cmml"><mo id="S4.SS1.SSS2.5.p2.12.m12.3.3.2.2.2.3" stretchy="false" xref="S4.SS1.SSS2.5.p2.12.m12.3.3.2.2.3.cmml">[</mo><mrow id="S4.SS1.SSS2.5.p2.12.m12.2.2.1.1.1.1" xref="S4.SS1.SSS2.5.p2.12.m12.2.2.1.1.1.1.cmml"><msub id="S4.SS1.SSS2.5.p2.12.m12.2.2.1.1.1.1.2" xref="S4.SS1.SSS2.5.p2.12.m12.2.2.1.1.1.1.2.cmml"><mi id="S4.SS1.SSS2.5.p2.12.m12.2.2.1.1.1.1.2.2" xref="S4.SS1.SSS2.5.p2.12.m12.2.2.1.1.1.1.2.2.cmml">G</mi><mi id="S4.SS1.SSS2.5.p2.12.m12.2.2.1.1.1.1.2.3" xref="S4.SS1.SSS2.5.p2.12.m12.2.2.1.1.1.1.2.3.cmml">a</mi></msub><mo id="S4.SS1.SSS2.5.p2.12.m12.2.2.1.1.1.1.1" xref="S4.SS1.SSS2.5.p2.12.m12.2.2.1.1.1.1.1.cmml">∖</mo><mrow id="S4.SS1.SSS2.5.p2.12.m12.2.2.1.1.1.1.3.2" xref="S4.SS1.SSS2.5.p2.12.m12.2.2.1.1.1.1.3.1.cmml"><mo id="S4.SS1.SSS2.5.p2.12.m12.2.2.1.1.1.1.3.2.1" stretchy="false" xref="S4.SS1.SSS2.5.p2.12.m12.2.2.1.1.1.1.3.1.cmml">{</mo><mi id="S4.SS1.SSS2.5.p2.12.m12.1.1" xref="S4.SS1.SSS2.5.p2.12.m12.1.1.cmml">a</mi><mo id="S4.SS1.SSS2.5.p2.12.m12.2.2.1.1.1.1.3.2.2" stretchy="false" xref="S4.SS1.SSS2.5.p2.12.m12.2.2.1.1.1.1.3.1.cmml">}</mo></mrow></mrow><mo id="S4.SS1.SSS2.5.p2.12.m12.3.3.2.2.2.4" xref="S4.SS1.SSS2.5.p2.12.m12.3.3.2.2.3.cmml">,</mo><mrow id="S4.SS1.SSS2.5.p2.12.m12.3.3.2.2.2.2" xref="S4.SS1.SSS2.5.p2.12.m12.3.3.2.2.2.2.cmml"><mi id="S4.SS1.SSS2.5.p2.12.m12.3.3.2.2.2.2.2" xref="S4.SS1.SSS2.5.p2.12.m12.3.3.2.2.2.2.2.cmml">G</mi><mo id="S4.SS1.SSS2.5.p2.12.m12.3.3.2.2.2.2.1" xref="S4.SS1.SSS2.5.p2.12.m12.3.3.2.2.2.2.1.cmml">∖</mo><msub id="S4.SS1.SSS2.5.p2.12.m12.3.3.2.2.2.2.3" xref="S4.SS1.SSS2.5.p2.12.m12.3.3.2.2.2.2.3.cmml"><mi id="S4.SS1.SSS2.5.p2.12.m12.3.3.2.2.2.2.3.2" xref="S4.SS1.SSS2.5.p2.12.m12.3.3.2.2.2.2.3.2.cmml">G</mi><mi id="S4.SS1.SSS2.5.p2.12.m12.3.3.2.2.2.2.3.3" xref="S4.SS1.SSS2.5.p2.12.m12.3.3.2.2.2.2.3.3.cmml">a</mi></msub></mrow><mo id="S4.SS1.SSS2.5.p2.12.m12.3.3.2.2.2.5" stretchy="false" xref="S4.SS1.SSS2.5.p2.12.m12.3.3.2.2.3.cmml">]</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.SSS2.5.p2.12.m12.3b"><apply id="S4.SS1.SSS2.5.p2.12.m12.3.3.cmml" xref="S4.SS1.SSS2.5.p2.12.m12.3.3"><in id="S4.SS1.SSS2.5.p2.12.m12.3.3.3.cmml" xref="S4.SS1.SSS2.5.p2.12.m12.3.3.3"></in><ci id="S4.SS1.SSS2.5.p2.12.m12.3.3.4.cmml" xref="S4.SS1.SSS2.5.p2.12.m12.3.3.4">𝑒</ci><apply id="S4.SS1.SSS2.5.p2.12.m12.3.3.2.cmml" xref="S4.SS1.SSS2.5.p2.12.m12.3.3.2"><times id="S4.SS1.SSS2.5.p2.12.m12.3.3.2.3.cmml" xref="S4.SS1.SSS2.5.p2.12.m12.3.3.2.3"></times><ci id="S4.SS1.SSS2.5.p2.12.m12.3.3.2.4a.cmml" xref="S4.SS1.SSS2.5.p2.12.m12.3.3.2.4"><mtext id="S4.SS1.SSS2.5.p2.12.m12.3.3.2.4.cmml" xref="S4.SS1.SSS2.5.p2.12.m12.3.3.2.4">OPT</mtext></ci><interval closure="closed" id="S4.SS1.SSS2.5.p2.12.m12.3.3.2.2.3.cmml" xref="S4.SS1.SSS2.5.p2.12.m12.3.3.2.2.2"><apply id="S4.SS1.SSS2.5.p2.12.m12.2.2.1.1.1.1.cmml" xref="S4.SS1.SSS2.5.p2.12.m12.2.2.1.1.1.1"><setdiff id="S4.SS1.SSS2.5.p2.12.m12.2.2.1.1.1.1.1.cmml" xref="S4.SS1.SSS2.5.p2.12.m12.2.2.1.1.1.1.1"></setdiff><apply id="S4.SS1.SSS2.5.p2.12.m12.2.2.1.1.1.1.2.cmml" xref="S4.SS1.SSS2.5.p2.12.m12.2.2.1.1.1.1.2"><csymbol cd="ambiguous" id="S4.SS1.SSS2.5.p2.12.m12.2.2.1.1.1.1.2.1.cmml" xref="S4.SS1.SSS2.5.p2.12.m12.2.2.1.1.1.1.2">subscript</csymbol><ci id="S4.SS1.SSS2.5.p2.12.m12.2.2.1.1.1.1.2.2.cmml" xref="S4.SS1.SSS2.5.p2.12.m12.2.2.1.1.1.1.2.2">𝐺</ci><ci id="S4.SS1.SSS2.5.p2.12.m12.2.2.1.1.1.1.2.3.cmml" xref="S4.SS1.SSS2.5.p2.12.m12.2.2.1.1.1.1.2.3">𝑎</ci></apply><set id="S4.SS1.SSS2.5.p2.12.m12.2.2.1.1.1.1.3.1.cmml" xref="S4.SS1.SSS2.5.p2.12.m12.2.2.1.1.1.1.3.2"><ci id="S4.SS1.SSS2.5.p2.12.m12.1.1.cmml" xref="S4.SS1.SSS2.5.p2.12.m12.1.1">𝑎</ci></set></apply><apply id="S4.SS1.SSS2.5.p2.12.m12.3.3.2.2.2.2.cmml" xref="S4.SS1.SSS2.5.p2.12.m12.3.3.2.2.2.2"><setdiff id="S4.SS1.SSS2.5.p2.12.m12.3.3.2.2.2.2.1.cmml" xref="S4.SS1.SSS2.5.p2.12.m12.3.3.2.2.2.2.1"></setdiff><ci id="S4.SS1.SSS2.5.p2.12.m12.3.3.2.2.2.2.2.cmml" xref="S4.SS1.SSS2.5.p2.12.m12.3.3.2.2.2.2.2">𝐺</ci><apply id="S4.SS1.SSS2.5.p2.12.m12.3.3.2.2.2.2.3.cmml" xref="S4.SS1.SSS2.5.p2.12.m12.3.3.2.2.2.2.3"><csymbol cd="ambiguous" id="S4.SS1.SSS2.5.p2.12.m12.3.3.2.2.2.2.3.1.cmml" xref="S4.SS1.SSS2.5.p2.12.m12.3.3.2.2.2.2.3">subscript</csymbol><ci id="S4.SS1.SSS2.5.p2.12.m12.3.3.2.2.2.2.3.2.cmml" xref="S4.SS1.SSS2.5.p2.12.m12.3.3.2.2.2.2.3.2">𝐺</ci><ci id="S4.SS1.SSS2.5.p2.12.m12.3.3.2.2.2.2.3.3.cmml" xref="S4.SS1.SSS2.5.p2.12.m12.3.3.2.2.2.2.3.3">𝑎</ci></apply></apply></interval></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.SSS2.5.p2.12.m12.3c">e\in\textnormal{OPT}[G_{a}\setminus\{a\},G\setminus G_{a}]</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.SSS2.5.p2.12.m12.3d">italic_e ∈ OPT [ italic_G start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT ∖ { italic_a } , italic_G ∖ italic_G start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT ]</annotation></semantics></math>. By definition, the endpoint of <math alttext="e" class="ltx_Math" display="inline" id="S4.SS1.SSS2.5.p2.13.m13.1"><semantics id="S4.SS1.SSS2.5.p2.13.m13.1a"><mi id="S4.SS1.SSS2.5.p2.13.m13.1.1" xref="S4.SS1.SSS2.5.p2.13.m13.1.1.cmml">e</mi><annotation-xml encoding="MathML-Content" id="S4.SS1.SSS2.5.p2.13.m13.1b"><ci id="S4.SS1.SSS2.5.p2.13.m13.1.1.cmml" xref="S4.SS1.SSS2.5.p2.13.m13.1.1">𝑒</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.SSS2.5.p2.13.m13.1c">e</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.SSS2.5.p2.13.m13.1d">italic_e</annotation></semantics></math> in <math alttext="G_{a}\setminus\{a\}" class="ltx_Math" display="inline" id="S4.SS1.SSS2.5.p2.14.m14.1"><semantics id="S4.SS1.SSS2.5.p2.14.m14.1a"><mrow id="S4.SS1.SSS2.5.p2.14.m14.1.2" xref="S4.SS1.SSS2.5.p2.14.m14.1.2.cmml"><msub id="S4.SS1.SSS2.5.p2.14.m14.1.2.2" xref="S4.SS1.SSS2.5.p2.14.m14.1.2.2.cmml"><mi id="S4.SS1.SSS2.5.p2.14.m14.1.2.2.2" xref="S4.SS1.SSS2.5.p2.14.m14.1.2.2.2.cmml">G</mi><mi id="S4.SS1.SSS2.5.p2.14.m14.1.2.2.3" xref="S4.SS1.SSS2.5.p2.14.m14.1.2.2.3.cmml">a</mi></msub><mo id="S4.SS1.SSS2.5.p2.14.m14.1.2.1" xref="S4.SS1.SSS2.5.p2.14.m14.1.2.1.cmml">∖</mo><mrow id="S4.SS1.SSS2.5.p2.14.m14.1.2.3.2" xref="S4.SS1.SSS2.5.p2.14.m14.1.2.3.1.cmml"><mo id="S4.SS1.SSS2.5.p2.14.m14.1.2.3.2.1" stretchy="false" xref="S4.SS1.SSS2.5.p2.14.m14.1.2.3.1.cmml">{</mo><mi id="S4.SS1.SSS2.5.p2.14.m14.1.1" xref="S4.SS1.SSS2.5.p2.14.m14.1.1.cmml">a</mi><mo id="S4.SS1.SSS2.5.p2.14.m14.1.2.3.2.2" stretchy="false" xref="S4.SS1.SSS2.5.p2.14.m14.1.2.3.1.cmml">}</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.SSS2.5.p2.14.m14.1b"><apply id="S4.SS1.SSS2.5.p2.14.m14.1.2.cmml" xref="S4.SS1.SSS2.5.p2.14.m14.1.2"><setdiff id="S4.SS1.SSS2.5.p2.14.m14.1.2.1.cmml" xref="S4.SS1.SSS2.5.p2.14.m14.1.2.1"></setdiff><apply id="S4.SS1.SSS2.5.p2.14.m14.1.2.2.cmml" xref="S4.SS1.SSS2.5.p2.14.m14.1.2.2"><csymbol cd="ambiguous" id="S4.SS1.SSS2.5.p2.14.m14.1.2.2.1.cmml" xref="S4.SS1.SSS2.5.p2.14.m14.1.2.2">subscript</csymbol><ci id="S4.SS1.SSS2.5.p2.14.m14.1.2.2.2.cmml" xref="S4.SS1.SSS2.5.p2.14.m14.1.2.2.2">𝐺</ci><ci id="S4.SS1.SSS2.5.p2.14.m14.1.2.2.3.cmml" xref="S4.SS1.SSS2.5.p2.14.m14.1.2.2.3">𝑎</ci></apply><set id="S4.SS1.SSS2.5.p2.14.m14.1.2.3.1.cmml" xref="S4.SS1.SSS2.5.p2.14.m14.1.2.3.2"><ci id="S4.SS1.SSS2.5.p2.14.m14.1.1.cmml" xref="S4.SS1.SSS2.5.p2.14.m14.1.1">𝑎</ci></set></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.SSS2.5.p2.14.m14.1c">G_{a}\setminus\{a\}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.SSS2.5.p2.14.m14.1d">italic_G start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT ∖ { italic_a }</annotation></semantics></math> must be in a “good” supernode of <math alttext="C^{\prime}(a)" class="ltx_Math" display="inline" id="S4.SS1.SSS2.5.p2.15.m15.1"><semantics id="S4.SS1.SSS2.5.p2.15.m15.1a"><mrow id="S4.SS1.SSS2.5.p2.15.m15.1.2" xref="S4.SS1.SSS2.5.p2.15.m15.1.2.cmml"><msup id="S4.SS1.SSS2.5.p2.15.m15.1.2.2" xref="S4.SS1.SSS2.5.p2.15.m15.1.2.2.cmml"><mi id="S4.SS1.SSS2.5.p2.15.m15.1.2.2.2" xref="S4.SS1.SSS2.5.p2.15.m15.1.2.2.2.cmml">C</mi><mo id="S4.SS1.SSS2.5.p2.15.m15.1.2.2.3" xref="S4.SS1.SSS2.5.p2.15.m15.1.2.2.3.cmml">′</mo></msup><mo id="S4.SS1.SSS2.5.p2.15.m15.1.2.1" xref="S4.SS1.SSS2.5.p2.15.m15.1.2.1.cmml">⁢</mo><mrow id="S4.SS1.SSS2.5.p2.15.m15.1.2.3.2" xref="S4.SS1.SSS2.5.p2.15.m15.1.2.cmml"><mo id="S4.SS1.SSS2.5.p2.15.m15.1.2.3.2.1" stretchy="false" xref="S4.SS1.SSS2.5.p2.15.m15.1.2.cmml">(</mo><mi id="S4.SS1.SSS2.5.p2.15.m15.1.1" xref="S4.SS1.SSS2.5.p2.15.m15.1.1.cmml">a</mi><mo id="S4.SS1.SSS2.5.p2.15.m15.1.2.3.2.2" stretchy="false" xref="S4.SS1.SSS2.5.p2.15.m15.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.SSS2.5.p2.15.m15.1b"><apply id="S4.SS1.SSS2.5.p2.15.m15.1.2.cmml" xref="S4.SS1.SSS2.5.p2.15.m15.1.2"><times id="S4.SS1.SSS2.5.p2.15.m15.1.2.1.cmml" xref="S4.SS1.SSS2.5.p2.15.m15.1.2.1"></times><apply id="S4.SS1.SSS2.5.p2.15.m15.1.2.2.cmml" xref="S4.SS1.SSS2.5.p2.15.m15.1.2.2"><csymbol cd="ambiguous" id="S4.SS1.SSS2.5.p2.15.m15.1.2.2.1.cmml" xref="S4.SS1.SSS2.5.p2.15.m15.1.2.2">superscript</csymbol><ci id="S4.SS1.SSS2.5.p2.15.m15.1.2.2.2.cmml" xref="S4.SS1.SSS2.5.p2.15.m15.1.2.2.2">𝐶</ci><ci id="S4.SS1.SSS2.5.p2.15.m15.1.2.2.3.cmml" xref="S4.SS1.SSS2.5.p2.15.m15.1.2.2.3">′</ci></apply><ci id="S4.SS1.SSS2.5.p2.15.m15.1.1.cmml" xref="S4.SS1.SSS2.5.p2.15.m15.1.1">𝑎</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.SSS2.5.p2.15.m15.1c">C^{\prime}(a)</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.SSS2.5.p2.15.m15.1d">italic_C start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ( italic_a )</annotation></semantics></math>. Thus all “bad” supernodes of <math alttext="C^{\prime}(a)" class="ltx_Math" display="inline" id="S4.SS1.SSS2.5.p2.16.m16.1"><semantics id="S4.SS1.SSS2.5.p2.16.m16.1a"><mrow id="S4.SS1.SSS2.5.p2.16.m16.1.2" xref="S4.SS1.SSS2.5.p2.16.m16.1.2.cmml"><msup id="S4.SS1.SSS2.5.p2.16.m16.1.2.2" xref="S4.SS1.SSS2.5.p2.16.m16.1.2.2.cmml"><mi id="S4.SS1.SSS2.5.p2.16.m16.1.2.2.2" xref="S4.SS1.SSS2.5.p2.16.m16.1.2.2.2.cmml">C</mi><mo id="S4.SS1.SSS2.5.p2.16.m16.1.2.2.3" xref="S4.SS1.SSS2.5.p2.16.m16.1.2.2.3.cmml">′</mo></msup><mo id="S4.SS1.SSS2.5.p2.16.m16.1.2.1" xref="S4.SS1.SSS2.5.p2.16.m16.1.2.1.cmml">⁢</mo><mrow id="S4.SS1.SSS2.5.p2.16.m16.1.2.3.2" xref="S4.SS1.SSS2.5.p2.16.m16.1.2.cmml"><mo id="S4.SS1.SSS2.5.p2.16.m16.1.2.3.2.1" stretchy="false" xref="S4.SS1.SSS2.5.p2.16.m16.1.2.cmml">(</mo><mi id="S4.SS1.SSS2.5.p2.16.m16.1.1" xref="S4.SS1.SSS2.5.p2.16.m16.1.1.cmml">a</mi><mo id="S4.SS1.SSS2.5.p2.16.m16.1.2.3.2.2" stretchy="false" xref="S4.SS1.SSS2.5.p2.16.m16.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.SSS2.5.p2.16.m16.1b"><apply id="S4.SS1.SSS2.5.p2.16.m16.1.2.cmml" xref="S4.SS1.SSS2.5.p2.16.m16.1.2"><times id="S4.SS1.SSS2.5.p2.16.m16.1.2.1.cmml" xref="S4.SS1.SSS2.5.p2.16.m16.1.2.1"></times><apply id="S4.SS1.SSS2.5.p2.16.m16.1.2.2.cmml" xref="S4.SS1.SSS2.5.p2.16.m16.1.2.2"><csymbol cd="ambiguous" id="S4.SS1.SSS2.5.p2.16.m16.1.2.2.1.cmml" xref="S4.SS1.SSS2.5.p2.16.m16.1.2.2">superscript</csymbol><ci id="S4.SS1.SSS2.5.p2.16.m16.1.2.2.2.cmml" xref="S4.SS1.SSS2.5.p2.16.m16.1.2.2.2">𝐶</ci><ci id="S4.SS1.SSS2.5.p2.16.m16.1.2.2.3.cmml" xref="S4.SS1.SSS2.5.p2.16.m16.1.2.2.3">′</ci></apply><ci id="S4.SS1.SSS2.5.p2.16.m16.1.1.cmml" xref="S4.SS1.SSS2.5.p2.16.m16.1.1">𝑎</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.SSS2.5.p2.16.m16.1c">C^{\prime}(a)</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.SSS2.5.p2.16.m16.1d">italic_C start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ( italic_a )</annotation></semantics></math> are connected to at least one “good” supernode of <math alttext="C^{\prime}(a)" class="ltx_Math" display="inline" id="S4.SS1.SSS2.5.p2.17.m17.1"><semantics id="S4.SS1.SSS2.5.p2.17.m17.1a"><mrow id="S4.SS1.SSS2.5.p2.17.m17.1.2" xref="S4.SS1.SSS2.5.p2.17.m17.1.2.cmml"><msup id="S4.SS1.SSS2.5.p2.17.m17.1.2.2" xref="S4.SS1.SSS2.5.p2.17.m17.1.2.2.cmml"><mi id="S4.SS1.SSS2.5.p2.17.m17.1.2.2.2" xref="S4.SS1.SSS2.5.p2.17.m17.1.2.2.2.cmml">C</mi><mo id="S4.SS1.SSS2.5.p2.17.m17.1.2.2.3" xref="S4.SS1.SSS2.5.p2.17.m17.1.2.2.3.cmml">′</mo></msup><mo id="S4.SS1.SSS2.5.p2.17.m17.1.2.1" xref="S4.SS1.SSS2.5.p2.17.m17.1.2.1.cmml">⁢</mo><mrow id="S4.SS1.SSS2.5.p2.17.m17.1.2.3.2" xref="S4.SS1.SSS2.5.p2.17.m17.1.2.cmml"><mo id="S4.SS1.SSS2.5.p2.17.m17.1.2.3.2.1" stretchy="false" xref="S4.SS1.SSS2.5.p2.17.m17.1.2.cmml">(</mo><mi id="S4.SS1.SSS2.5.p2.17.m17.1.1" xref="S4.SS1.SSS2.5.p2.17.m17.1.1.cmml">a</mi><mo id="S4.SS1.SSS2.5.p2.17.m17.1.2.3.2.2" stretchy="false" xref="S4.SS1.SSS2.5.p2.17.m17.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.SSS2.5.p2.17.m17.1b"><apply id="S4.SS1.SSS2.5.p2.17.m17.1.2.cmml" xref="S4.SS1.SSS2.5.p2.17.m17.1.2"><times id="S4.SS1.SSS2.5.p2.17.m17.1.2.1.cmml" xref="S4.SS1.SSS2.5.p2.17.m17.1.2.1"></times><apply id="S4.SS1.SSS2.5.p2.17.m17.1.2.2.cmml" xref="S4.SS1.SSS2.5.p2.17.m17.1.2.2"><csymbol cd="ambiguous" id="S4.SS1.SSS2.5.p2.17.m17.1.2.2.1.cmml" xref="S4.SS1.SSS2.5.p2.17.m17.1.2.2">superscript</csymbol><ci id="S4.SS1.SSS2.5.p2.17.m17.1.2.2.2.cmml" xref="S4.SS1.SSS2.5.p2.17.m17.1.2.2.2">𝐶</ci><ci id="S4.SS1.SSS2.5.p2.17.m17.1.2.2.3.cmml" xref="S4.SS1.SSS2.5.p2.17.m17.1.2.2.3">′</ci></apply><ci id="S4.SS1.SSS2.5.p2.17.m17.1.1.cmml" xref="S4.SS1.SSS2.5.p2.17.m17.1.1">𝑎</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.SSS2.5.p2.17.m17.1c">C^{\prime}(a)</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.SSS2.5.p2.17.m17.1d">italic_C start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ( italic_a )</annotation></semantics></math> in <span class="ltx_text ltx_markedasmath" id="S4.SS1.SSS2.5.p2.23.6">OPT</span>. Furthermore, a minimal path in the contracted graph from a “bad” supernode to a “good” supernode only uses links in <math alttext="E^{*}(a)" class="ltx_Math" display="inline" id="S4.SS1.SSS2.5.p2.19.m19.1"><semantics id="S4.SS1.SSS2.5.p2.19.m19.1a"><mrow id="S4.SS1.SSS2.5.p2.19.m19.1.2" xref="S4.SS1.SSS2.5.p2.19.m19.1.2.cmml"><msup id="S4.SS1.SSS2.5.p2.19.m19.1.2.2" xref="S4.SS1.SSS2.5.p2.19.m19.1.2.2.cmml"><mi id="S4.SS1.SSS2.5.p2.19.m19.1.2.2.2" xref="S4.SS1.SSS2.5.p2.19.m19.1.2.2.2.cmml">E</mi><mo id="S4.SS1.SSS2.5.p2.19.m19.1.2.2.3" xref="S4.SS1.SSS2.5.p2.19.m19.1.2.2.3.cmml">∗</mo></msup><mo id="S4.SS1.SSS2.5.p2.19.m19.1.2.1" xref="S4.SS1.SSS2.5.p2.19.m19.1.2.1.cmml">⁢</mo><mrow id="S4.SS1.SSS2.5.p2.19.m19.1.2.3.2" xref="S4.SS1.SSS2.5.p2.19.m19.1.2.cmml"><mo id="S4.SS1.SSS2.5.p2.19.m19.1.2.3.2.1" stretchy="false" xref="S4.SS1.SSS2.5.p2.19.m19.1.2.cmml">(</mo><mi id="S4.SS1.SSS2.5.p2.19.m19.1.1" xref="S4.SS1.SSS2.5.p2.19.m19.1.1.cmml">a</mi><mo id="S4.SS1.SSS2.5.p2.19.m19.1.2.3.2.2" stretchy="false" xref="S4.SS1.SSS2.5.p2.19.m19.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.SSS2.5.p2.19.m19.1b"><apply id="S4.SS1.SSS2.5.p2.19.m19.1.2.cmml" xref="S4.SS1.SSS2.5.p2.19.m19.1.2"><times id="S4.SS1.SSS2.5.p2.19.m19.1.2.1.cmml" xref="S4.SS1.SSS2.5.p2.19.m19.1.2.1"></times><apply id="S4.SS1.SSS2.5.p2.19.m19.1.2.2.cmml" xref="S4.SS1.SSS2.5.p2.19.m19.1.2.2"><csymbol cd="ambiguous" id="S4.SS1.SSS2.5.p2.19.m19.1.2.2.1.cmml" xref="S4.SS1.SSS2.5.p2.19.m19.1.2.2">superscript</csymbol><ci id="S4.SS1.SSS2.5.p2.19.m19.1.2.2.2.cmml" xref="S4.SS1.SSS2.5.p2.19.m19.1.2.2.2">𝐸</ci><times id="S4.SS1.SSS2.5.p2.19.m19.1.2.2.3.cmml" xref="S4.SS1.SSS2.5.p2.19.m19.1.2.2.3"></times></apply><ci id="S4.SS1.SSS2.5.p2.19.m19.1.1.cmml" xref="S4.SS1.SSS2.5.p2.19.m19.1.1">𝑎</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.SSS2.5.p2.19.m19.1c">E^{*}(a)</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.SSS2.5.p2.19.m19.1d">italic_E start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ( italic_a )</annotation></semantics></math>, since these links all must go between subtrees of <math alttext="a" class="ltx_Math" display="inline" id="S4.SS1.SSS2.5.p2.20.m20.1"><semantics id="S4.SS1.SSS2.5.p2.20.m20.1a"><mi id="S4.SS1.SSS2.5.p2.20.m20.1.1" xref="S4.SS1.SSS2.5.p2.20.m20.1.1.cmml">a</mi><annotation-xml encoding="MathML-Content" id="S4.SS1.SSS2.5.p2.20.m20.1b"><ci id="S4.SS1.SSS2.5.p2.20.m20.1.1.cmml" xref="S4.SS1.SSS2.5.p2.20.m20.1.1">𝑎</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.SSS2.5.p2.20.m20.1c">a</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.SSS2.5.p2.20.m20.1d">italic_a</annotation></semantics></math>. Thus in <math alttext="E^{*}(a)" class="ltx_Math" display="inline" id="S4.SS1.SSS2.5.p2.21.m21.1"><semantics id="S4.SS1.SSS2.5.p2.21.m21.1a"><mrow id="S4.SS1.SSS2.5.p2.21.m21.1.2" xref="S4.SS1.SSS2.5.p2.21.m21.1.2.cmml"><msup id="S4.SS1.SSS2.5.p2.21.m21.1.2.2" xref="S4.SS1.SSS2.5.p2.21.m21.1.2.2.cmml"><mi id="S4.SS1.SSS2.5.p2.21.m21.1.2.2.2" xref="S4.SS1.SSS2.5.p2.21.m21.1.2.2.2.cmml">E</mi><mo id="S4.SS1.SSS2.5.p2.21.m21.1.2.2.3" xref="S4.SS1.SSS2.5.p2.21.m21.1.2.2.3.cmml">∗</mo></msup><mo id="S4.SS1.SSS2.5.p2.21.m21.1.2.1" xref="S4.SS1.SSS2.5.p2.21.m21.1.2.1.cmml">⁢</mo><mrow id="S4.SS1.SSS2.5.p2.21.m21.1.2.3.2" xref="S4.SS1.SSS2.5.p2.21.m21.1.2.cmml"><mo id="S4.SS1.SSS2.5.p2.21.m21.1.2.3.2.1" stretchy="false" xref="S4.SS1.SSS2.5.p2.21.m21.1.2.cmml">(</mo><mi id="S4.SS1.SSS2.5.p2.21.m21.1.1" xref="S4.SS1.SSS2.5.p2.21.m21.1.1.cmml">a</mi><mo id="S4.SS1.SSS2.5.p2.21.m21.1.2.3.2.2" stretchy="false" xref="S4.SS1.SSS2.5.p2.21.m21.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.SSS2.5.p2.21.m21.1b"><apply id="S4.SS1.SSS2.5.p2.21.m21.1.2.cmml" xref="S4.SS1.SSS2.5.p2.21.m21.1.2"><times id="S4.SS1.SSS2.5.p2.21.m21.1.2.1.cmml" xref="S4.SS1.SSS2.5.p2.21.m21.1.2.1"></times><apply id="S4.SS1.SSS2.5.p2.21.m21.1.2.2.cmml" xref="S4.SS1.SSS2.5.p2.21.m21.1.2.2"><csymbol cd="ambiguous" id="S4.SS1.SSS2.5.p2.21.m21.1.2.2.1.cmml" xref="S4.SS1.SSS2.5.p2.21.m21.1.2.2">superscript</csymbol><ci id="S4.SS1.SSS2.5.p2.21.m21.1.2.2.2.cmml" xref="S4.SS1.SSS2.5.p2.21.m21.1.2.2.2">𝐸</ci><times id="S4.SS1.SSS2.5.p2.21.m21.1.2.2.3.cmml" xref="S4.SS1.SSS2.5.p2.21.m21.1.2.2.3"></times></apply><ci id="S4.SS1.SSS2.5.p2.21.m21.1.1.cmml" xref="S4.SS1.SSS2.5.p2.21.m21.1.1">𝑎</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.SSS2.5.p2.21.m21.1c">E^{*}(a)</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.SSS2.5.p2.21.m21.1d">italic_E start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ( italic_a )</annotation></semantics></math>, all “bad” supernodes are connected to at least one “good” supernode. In particular, <math alttext="E^{*}(a)" class="ltx_Math" display="inline" id="S4.SS1.SSS2.5.p2.22.m22.1"><semantics id="S4.SS1.SSS2.5.p2.22.m22.1a"><mrow id="S4.SS1.SSS2.5.p2.22.m22.1.2" xref="S4.SS1.SSS2.5.p2.22.m22.1.2.cmml"><msup id="S4.SS1.SSS2.5.p2.22.m22.1.2.2" xref="S4.SS1.SSS2.5.p2.22.m22.1.2.2.cmml"><mi id="S4.SS1.SSS2.5.p2.22.m22.1.2.2.2" xref="S4.SS1.SSS2.5.p2.22.m22.1.2.2.2.cmml">E</mi><mo id="S4.SS1.SSS2.5.p2.22.m22.1.2.2.3" xref="S4.SS1.SSS2.5.p2.22.m22.1.2.2.3.cmml">∗</mo></msup><mo id="S4.SS1.SSS2.5.p2.22.m22.1.2.1" xref="S4.SS1.SSS2.5.p2.22.m22.1.2.1.cmml">⁢</mo><mrow id="S4.SS1.SSS2.5.p2.22.m22.1.2.3.2" xref="S4.SS1.SSS2.5.p2.22.m22.1.2.cmml"><mo id="S4.SS1.SSS2.5.p2.22.m22.1.2.3.2.1" stretchy="false" xref="S4.SS1.SSS2.5.p2.22.m22.1.2.cmml">(</mo><mi id="S4.SS1.SSS2.5.p2.22.m22.1.1" xref="S4.SS1.SSS2.5.p2.22.m22.1.1.cmml">a</mi><mo id="S4.SS1.SSS2.5.p2.22.m22.1.2.3.2.2" stretchy="false" xref="S4.SS1.SSS2.5.p2.22.m22.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.SSS2.5.p2.22.m22.1b"><apply id="S4.SS1.SSS2.5.p2.22.m22.1.2.cmml" xref="S4.SS1.SSS2.5.p2.22.m22.1.2"><times id="S4.SS1.SSS2.5.p2.22.m22.1.2.1.cmml" xref="S4.SS1.SSS2.5.p2.22.m22.1.2.1"></times><apply id="S4.SS1.SSS2.5.p2.22.m22.1.2.2.cmml" xref="S4.SS1.SSS2.5.p2.22.m22.1.2.2"><csymbol cd="ambiguous" id="S4.SS1.SSS2.5.p2.22.m22.1.2.2.1.cmml" xref="S4.SS1.SSS2.5.p2.22.m22.1.2.2">superscript</csymbol><ci id="S4.SS1.SSS2.5.p2.22.m22.1.2.2.2.cmml" xref="S4.SS1.SSS2.5.p2.22.m22.1.2.2.2">𝐸</ci><times id="S4.SS1.SSS2.5.p2.22.m22.1.2.2.3.cmml" xref="S4.SS1.SSS2.5.p2.22.m22.1.2.2.3"></times></apply><ci id="S4.SS1.SSS2.5.p2.22.m22.1.1.cmml" xref="S4.SS1.SSS2.5.p2.22.m22.1.1">𝑎</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.SSS2.5.p2.22.m22.1c">E^{*}(a)</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.SSS2.5.p2.22.m22.1d">italic_E start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ( italic_a )</annotation></semantics></math> contains a spanning tree on the contracted graph with vertices <math alttext="C^{\prime\prime}(a)" class="ltx_Math" display="inline" id="S4.SS1.SSS2.5.p2.23.m23.1"><semantics id="S4.SS1.SSS2.5.p2.23.m23.1a"><mrow id="S4.SS1.SSS2.5.p2.23.m23.1.2" xref="S4.SS1.SSS2.5.p2.23.m23.1.2.cmml"><msup id="S4.SS1.SSS2.5.p2.23.m23.1.2.2" xref="S4.SS1.SSS2.5.p2.23.m23.1.2.2.cmml"><mi id="S4.SS1.SSS2.5.p2.23.m23.1.2.2.2" xref="S4.SS1.SSS2.5.p2.23.m23.1.2.2.2.cmml">C</mi><mo id="S4.SS1.SSS2.5.p2.23.m23.1.2.2.3" xref="S4.SS1.SSS2.5.p2.23.m23.1.2.2.3.cmml">′′</mo></msup><mo id="S4.SS1.SSS2.5.p2.23.m23.1.2.1" xref="S4.SS1.SSS2.5.p2.23.m23.1.2.1.cmml">⁢</mo><mrow id="S4.SS1.SSS2.5.p2.23.m23.1.2.3.2" xref="S4.SS1.SSS2.5.p2.23.m23.1.2.cmml"><mo id="S4.SS1.SSS2.5.p2.23.m23.1.2.3.2.1" stretchy="false" xref="S4.SS1.SSS2.5.p2.23.m23.1.2.cmml">(</mo><mi id="S4.SS1.SSS2.5.p2.23.m23.1.1" xref="S4.SS1.SSS2.5.p2.23.m23.1.1.cmml">a</mi><mo id="S4.SS1.SSS2.5.p2.23.m23.1.2.3.2.2" stretchy="false" xref="S4.SS1.SSS2.5.p2.23.m23.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.SSS2.5.p2.23.m23.1b"><apply id="S4.SS1.SSS2.5.p2.23.m23.1.2.cmml" xref="S4.SS1.SSS2.5.p2.23.m23.1.2"><times id="S4.SS1.SSS2.5.p2.23.m23.1.2.1.cmml" xref="S4.SS1.SSS2.5.p2.23.m23.1.2.1"></times><apply id="S4.SS1.SSS2.5.p2.23.m23.1.2.2.cmml" xref="S4.SS1.SSS2.5.p2.23.m23.1.2.2"><csymbol cd="ambiguous" id="S4.SS1.SSS2.5.p2.23.m23.1.2.2.1.cmml" xref="S4.SS1.SSS2.5.p2.23.m23.1.2.2">superscript</csymbol><ci id="S4.SS1.SSS2.5.p2.23.m23.1.2.2.2.cmml" xref="S4.SS1.SSS2.5.p2.23.m23.1.2.2.2">𝐶</ci><ci id="S4.SS1.SSS2.5.p2.23.m23.1.2.2.3.cmml" xref="S4.SS1.SSS2.5.p2.23.m23.1.2.2.3">′′</ci></apply><ci id="S4.SS1.SSS2.5.p2.23.m23.1.1.cmml" xref="S4.SS1.SSS2.5.p2.23.m23.1.1">𝑎</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.SSS2.5.p2.23.m23.1c">C^{\prime\prime}(a)</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.SSS2.5.p2.23.m23.1d">italic_C start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT ( italic_a )</annotation></semantics></math>. Therefore</p> <table class="ltx_equation ltx_eqn_table" id="S4.Ex11"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\sum_{a\in V}w(E(T^{\prime\prime}_{a}))\leq\sum_{a\in V}w(E^{*}(a))\leq w(% \textnormal{OPT})." class="ltx_Math" display="block" id="S4.Ex11.m1.3"><semantics id="S4.Ex11.m1.3a"><mrow id="S4.Ex11.m1.3.3.1" xref="S4.Ex11.m1.3.3.1.1.cmml"><mrow id="S4.Ex11.m1.3.3.1.1" xref="S4.Ex11.m1.3.3.1.1.cmml"><mrow id="S4.Ex11.m1.3.3.1.1.1" xref="S4.Ex11.m1.3.3.1.1.1.cmml"><munder id="S4.Ex11.m1.3.3.1.1.1.2" xref="S4.Ex11.m1.3.3.1.1.1.2.cmml"><mo id="S4.Ex11.m1.3.3.1.1.1.2.2" movablelimits="false" xref="S4.Ex11.m1.3.3.1.1.1.2.2.cmml">∑</mo><mrow id="S4.Ex11.m1.3.3.1.1.1.2.3" xref="S4.Ex11.m1.3.3.1.1.1.2.3.cmml"><mi id="S4.Ex11.m1.3.3.1.1.1.2.3.2" xref="S4.Ex11.m1.3.3.1.1.1.2.3.2.cmml">a</mi><mo id="S4.Ex11.m1.3.3.1.1.1.2.3.1" xref="S4.Ex11.m1.3.3.1.1.1.2.3.1.cmml">∈</mo><mi id="S4.Ex11.m1.3.3.1.1.1.2.3.3" xref="S4.Ex11.m1.3.3.1.1.1.2.3.3.cmml">V</mi></mrow></munder><mrow id="S4.Ex11.m1.3.3.1.1.1.1" xref="S4.Ex11.m1.3.3.1.1.1.1.cmml"><mi id="S4.Ex11.m1.3.3.1.1.1.1.3" xref="S4.Ex11.m1.3.3.1.1.1.1.3.cmml">w</mi><mo id="S4.Ex11.m1.3.3.1.1.1.1.2" xref="S4.Ex11.m1.3.3.1.1.1.1.2.cmml">⁢</mo><mrow id="S4.Ex11.m1.3.3.1.1.1.1.1.1" xref="S4.Ex11.m1.3.3.1.1.1.1.1.1.1.cmml"><mo id="S4.Ex11.m1.3.3.1.1.1.1.1.1.2" stretchy="false" xref="S4.Ex11.m1.3.3.1.1.1.1.1.1.1.cmml">(</mo><mrow id="S4.Ex11.m1.3.3.1.1.1.1.1.1.1" xref="S4.Ex11.m1.3.3.1.1.1.1.1.1.1.cmml"><mi id="S4.Ex11.m1.3.3.1.1.1.1.1.1.1.3" xref="S4.Ex11.m1.3.3.1.1.1.1.1.1.1.3.cmml">E</mi><mo id="S4.Ex11.m1.3.3.1.1.1.1.1.1.1.2" xref="S4.Ex11.m1.3.3.1.1.1.1.1.1.1.2.cmml">⁢</mo><mrow id="S4.Ex11.m1.3.3.1.1.1.1.1.1.1.1.1" xref="S4.Ex11.m1.3.3.1.1.1.1.1.1.1.1.1.1.cmml"><mo id="S4.Ex11.m1.3.3.1.1.1.1.1.1.1.1.1.2" stretchy="false" 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xref="S4.Ex11.m1.2.2">OPT</mtext></ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Ex11.m1.3c">\sum_{a\in V}w(E(T^{\prime\prime}_{a}))\leq\sum_{a\in V}w(E^{*}(a))\leq w(% \textnormal{OPT}).</annotation><annotation encoding="application/x-llamapun" id="S4.Ex11.m1.3d">∑ start_POSTSUBSCRIPT italic_a ∈ italic_V end_POSTSUBSCRIPT italic_w ( italic_E ( italic_T start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT ) ) ≤ ∑ start_POSTSUBSCRIPT italic_a ∈ italic_V end_POSTSUBSCRIPT italic_w ( italic_E start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ( italic_a ) ) ≤ italic_w ( OPT ) .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S4.SS1.SSS2.5.p2.26">Thus the total weight of <span class="ltx_text ltx_markedasmath" id="S4.SS1.SSS2.5.p2.26.1">SOL</span> is at most <math alttext="(3+2\epsilon)w(\textnormal{OPT})" class="ltx_Math" 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xref="S4.SS1.SSS2.5.p2.25.m2.2.2.1.1.1.3.3.cmml">ϵ</mi></mrow></mrow><mo id="S4.SS1.SSS2.5.p2.25.m2.2.2.1.1.3" stretchy="false" xref="S4.SS1.SSS2.5.p2.25.m2.2.2.1.1.1.cmml">)</mo></mrow><mo id="S4.SS1.SSS2.5.p2.25.m2.2.2.2" xref="S4.SS1.SSS2.5.p2.25.m2.2.2.2.cmml">⁢</mo><mi id="S4.SS1.SSS2.5.p2.25.m2.2.2.3" xref="S4.SS1.SSS2.5.p2.25.m2.2.2.3.cmml">w</mi><mo id="S4.SS1.SSS2.5.p2.25.m2.2.2.2a" xref="S4.SS1.SSS2.5.p2.25.m2.2.2.2.cmml">⁢</mo><mrow id="S4.SS1.SSS2.5.p2.25.m2.2.2.4.2" xref="S4.SS1.SSS2.5.p2.25.m2.1.1a.cmml"><mo id="S4.SS1.SSS2.5.p2.25.m2.2.2.4.2.1" stretchy="false" xref="S4.SS1.SSS2.5.p2.25.m2.1.1a.cmml">(</mo><mtext id="S4.SS1.SSS2.5.p2.25.m2.1.1" xref="S4.SS1.SSS2.5.p2.25.m2.1.1.cmml">OPT</mtext><mo id="S4.SS1.SSS2.5.p2.25.m2.2.2.4.2.2" stretchy="false" xref="S4.SS1.SSS2.5.p2.25.m2.1.1a.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.SSS2.5.p2.25.m2.2b"><apply id="S4.SS1.SSS2.5.p2.25.m2.2.2.cmml" xref="S4.SS1.SSS2.5.p2.25.m2.2.2"><times id="S4.SS1.SSS2.5.p2.25.m2.2.2.2.cmml" xref="S4.SS1.SSS2.5.p2.25.m2.2.2.2"></times><apply id="S4.SS1.SSS2.5.p2.25.m2.2.2.1.1.1.cmml" xref="S4.SS1.SSS2.5.p2.25.m2.2.2.1.1"><plus id="S4.SS1.SSS2.5.p2.25.m2.2.2.1.1.1.1.cmml" xref="S4.SS1.SSS2.5.p2.25.m2.2.2.1.1.1.1"></plus><cn id="S4.SS1.SSS2.5.p2.25.m2.2.2.1.1.1.2.cmml" type="integer" xref="S4.SS1.SSS2.5.p2.25.m2.2.2.1.1.1.2">3</cn><apply id="S4.SS1.SSS2.5.p2.25.m2.2.2.1.1.1.3.cmml" xref="S4.SS1.SSS2.5.p2.25.m2.2.2.1.1.1.3"><times id="S4.SS1.SSS2.5.p2.25.m2.2.2.1.1.1.3.1.cmml" xref="S4.SS1.SSS2.5.p2.25.m2.2.2.1.1.1.3.1"></times><cn id="S4.SS1.SSS2.5.p2.25.m2.2.2.1.1.1.3.2.cmml" type="integer" xref="S4.SS1.SSS2.5.p2.25.m2.2.2.1.1.1.3.2">2</cn><ci id="S4.SS1.SSS2.5.p2.25.m2.2.2.1.1.1.3.3.cmml" xref="S4.SS1.SSS2.5.p2.25.m2.2.2.1.1.1.3.3">italic-ϵ</ci></apply></apply><ci id="S4.SS1.SSS2.5.p2.25.m2.2.2.3.cmml" xref="S4.SS1.SSS2.5.p2.25.m2.2.2.3">𝑤</ci><ci id="S4.SS1.SSS2.5.p2.25.m2.1.1a.cmml" xref="S4.SS1.SSS2.5.p2.25.m2.2.2.4.2"><mtext id="S4.SS1.SSS2.5.p2.25.m2.1.1.cmml" xref="S4.SS1.SSS2.5.p2.25.m2.1.1">OPT</mtext></ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.SSS2.5.p2.25.m2.2c">(3+2\epsilon)w(\textnormal{OPT})</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.SSS2.5.p2.25.m2.2d">( 3 + 2 italic_ϵ ) italic_w ( OPT )</annotation></semantics></math>. We can run the algorithm with <math alttext="\epsilon/2" class="ltx_Math" display="inline" id="S4.SS1.SSS2.5.p2.26.m3.1"><semantics id="S4.SS1.SSS2.5.p2.26.m3.1a"><mrow id="S4.SS1.SSS2.5.p2.26.m3.1.1" xref="S4.SS1.SSS2.5.p2.26.m3.1.1.cmml"><mi id="S4.SS1.SSS2.5.p2.26.m3.1.1.2" xref="S4.SS1.SSS2.5.p2.26.m3.1.1.2.cmml">ϵ</mi><mo id="S4.SS1.SSS2.5.p2.26.m3.1.1.1" xref="S4.SS1.SSS2.5.p2.26.m3.1.1.1.cmml">/</mo><mn id="S4.SS1.SSS2.5.p2.26.m3.1.1.3" xref="S4.SS1.SSS2.5.p2.26.m3.1.1.3.cmml">2</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.SSS2.5.p2.26.m3.1b"><apply id="S4.SS1.SSS2.5.p2.26.m3.1.1.cmml" xref="S4.SS1.SSS2.5.p2.26.m3.1.1"><divide id="S4.SS1.SSS2.5.p2.26.m3.1.1.1.cmml" xref="S4.SS1.SSS2.5.p2.26.m3.1.1.1"></divide><ci id="S4.SS1.SSS2.5.p2.26.m3.1.1.2.cmml" xref="S4.SS1.SSS2.5.p2.26.m3.1.1.2">italic-ϵ</ci><cn id="S4.SS1.SSS2.5.p2.26.m3.1.1.3.cmml" type="integer" xref="S4.SS1.SSS2.5.p2.26.m3.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.SSS2.5.p2.26.m3.1c">\epsilon/2</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.SSS2.5.p2.26.m3.1d">italic_ϵ / 2</annotation></semantics></math> to obtain the desired approximation ratio. ∎</p> </div> </div> <div class="ltx_para" id="S4.SS1.SSS2.p3"> <p class="ltx_p" id="S4.SS1.SSS2.p3.3">By Lemmas <a class="ltx_ref" href="https://arxiv.org/html/2503.00712v1#S4.Thmtheorem5" title="Lemma 4.5. ‣ 4.1.2 Bounding Approximation Ratio ‣ 4.1 One-to-Two Augmentation ‣ 4 Vertex Connectivity Augmentation in Link-Arrival Model ‣ Streaming Algorithms for Network Design"><span class="ltx_text ltx_ref_tag">4.5</span></a> and <a class="ltx_ref" href="https://arxiv.org/html/2503.00712v1#S4.Thmtheorem6" title="Lemma 4.6. ‣ 4.1.2 Bounding Approximation Ratio ‣ 4.1 One-to-Two Augmentation ‣ 4 Vertex Connectivity Augmentation in Link-Arrival Model ‣ Streaming Algorithms for Network Design"><span class="ltx_text ltx_ref_tag">4.6</span></a>, Algorithm <a class="ltx_ref" href="https://arxiv.org/html/2503.00712v1#algorithm5" title="In 4.1.2 Bounding Approximation Ratio ‣ 4.1 One-to-Two Augmentation ‣ 4 Vertex Connectivity Augmentation in Link-Arrival Model ‣ Streaming Algorithms for Network Design"><span class="ltx_text ltx_ref_tag">5</span></a> provides a feasible solution to <math alttext="1" class="ltx_Math" display="inline" id="S4.SS1.SSS2.p3.1.m1.1"><semantics id="S4.SS1.SSS2.p3.1.m1.1a"><mn id="S4.SS1.SSS2.p3.1.m1.1.1" xref="S4.SS1.SSS2.p3.1.m1.1.1.cmml">1</mn><annotation-xml encoding="MathML-Content" id="S4.SS1.SSS2.p3.1.m1.1b"><cn id="S4.SS1.SSS2.p3.1.m1.1.1.cmml" type="integer" xref="S4.SS1.SSS2.p3.1.m1.1.1">1</cn></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.SSS2.p3.1.m1.1c">1</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.SSS2.p3.1.m1.1d">1</annotation></semantics></math>-VC-CAP on <math alttext="(V,F)" class="ltx_Math" display="inline" id="S4.SS1.SSS2.p3.2.m2.2"><semantics id="S4.SS1.SSS2.p3.2.m2.2a"><mrow id="S4.SS1.SSS2.p3.2.m2.2.3.2" xref="S4.SS1.SSS2.p3.2.m2.2.3.1.cmml"><mo id="S4.SS1.SSS2.p3.2.m2.2.3.2.1" stretchy="false" xref="S4.SS1.SSS2.p3.2.m2.2.3.1.cmml">(</mo><mi id="S4.SS1.SSS2.p3.2.m2.1.1" xref="S4.SS1.SSS2.p3.2.m2.1.1.cmml">V</mi><mo id="S4.SS1.SSS2.p3.2.m2.2.3.2.2" xref="S4.SS1.SSS2.p3.2.m2.2.3.1.cmml">,</mo><mi id="S4.SS1.SSS2.p3.2.m2.2.2" xref="S4.SS1.SSS2.p3.2.m2.2.2.cmml">F</mi><mo id="S4.SS1.SSS2.p3.2.m2.2.3.2.3" stretchy="false" xref="S4.SS1.SSS2.p3.2.m2.2.3.1.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.SSS2.p3.2.m2.2b"><interval closure="open" id="S4.SS1.SSS2.p3.2.m2.2.3.1.cmml" xref="S4.SS1.SSS2.p3.2.m2.2.3.2"><ci id="S4.SS1.SSS2.p3.2.m2.1.1.cmml" xref="S4.SS1.SSS2.p3.2.m2.1.1">𝑉</ci><ci id="S4.SS1.SSS2.p3.2.m2.2.2.cmml" xref="S4.SS1.SSS2.p3.2.m2.2.2">𝐹</ci></interval></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.SSS2.p3.2.m2.2c">(V,F)</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.SSS2.p3.2.m2.2d">( italic_V , italic_F )</annotation></semantics></math> with weight at most <math alttext="(3+\epsilon)w(\textnormal{OPT})" class="ltx_Math" display="inline" id="S4.SS1.SSS2.p3.3.m3.2"><semantics id="S4.SS1.SSS2.p3.3.m3.2a"><mrow id="S4.SS1.SSS2.p3.3.m3.2.2" xref="S4.SS1.SSS2.p3.3.m3.2.2.cmml"><mrow id="S4.SS1.SSS2.p3.3.m3.2.2.1.1" xref="S4.SS1.SSS2.p3.3.m3.2.2.1.1.1.cmml"><mo id="S4.SS1.SSS2.p3.3.m3.2.2.1.1.2" stretchy="false" xref="S4.SS1.SSS2.p3.3.m3.2.2.1.1.1.cmml">(</mo><mrow id="S4.SS1.SSS2.p3.3.m3.2.2.1.1.1" xref="S4.SS1.SSS2.p3.3.m3.2.2.1.1.1.cmml"><mn id="S4.SS1.SSS2.p3.3.m3.2.2.1.1.1.2" xref="S4.SS1.SSS2.p3.3.m3.2.2.1.1.1.2.cmml">3</mn><mo id="S4.SS1.SSS2.p3.3.m3.2.2.1.1.1.1" xref="S4.SS1.SSS2.p3.3.m3.2.2.1.1.1.1.cmml">+</mo><mi id="S4.SS1.SSS2.p3.3.m3.2.2.1.1.1.3" xref="S4.SS1.SSS2.p3.3.m3.2.2.1.1.1.3.cmml">ϵ</mi></mrow><mo id="S4.SS1.SSS2.p3.3.m3.2.2.1.1.3" stretchy="false" xref="S4.SS1.SSS2.p3.3.m3.2.2.1.1.1.cmml">)</mo></mrow><mo id="S4.SS1.SSS2.p3.3.m3.2.2.2" xref="S4.SS1.SSS2.p3.3.m3.2.2.2.cmml">⁢</mo><mi id="S4.SS1.SSS2.p3.3.m3.2.2.3" xref="S4.SS1.SSS2.p3.3.m3.2.2.3.cmml">w</mi><mo id="S4.SS1.SSS2.p3.3.m3.2.2.2a" xref="S4.SS1.SSS2.p3.3.m3.2.2.2.cmml">⁢</mo><mrow id="S4.SS1.SSS2.p3.3.m3.2.2.4.2" xref="S4.SS1.SSS2.p3.3.m3.1.1a.cmml"><mo id="S4.SS1.SSS2.p3.3.m3.2.2.4.2.1" stretchy="false" xref="S4.SS1.SSS2.p3.3.m3.1.1a.cmml">(</mo><mtext id="S4.SS1.SSS2.p3.3.m3.1.1" xref="S4.SS1.SSS2.p3.3.m3.1.1.cmml">OPT</mtext><mo id="S4.SS1.SSS2.p3.3.m3.2.2.4.2.2" stretchy="false" xref="S4.SS1.SSS2.p3.3.m3.1.1a.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.SSS2.p3.3.m3.2b"><apply id="S4.SS1.SSS2.p3.3.m3.2.2.cmml" xref="S4.SS1.SSS2.p3.3.m3.2.2"><times id="S4.SS1.SSS2.p3.3.m3.2.2.2.cmml" xref="S4.SS1.SSS2.p3.3.m3.2.2.2"></times><apply id="S4.SS1.SSS2.p3.3.m3.2.2.1.1.1.cmml" xref="S4.SS1.SSS2.p3.3.m3.2.2.1.1"><plus id="S4.SS1.SSS2.p3.3.m3.2.2.1.1.1.1.cmml" xref="S4.SS1.SSS2.p3.3.m3.2.2.1.1.1.1"></plus><cn id="S4.SS1.SSS2.p3.3.m3.2.2.1.1.1.2.cmml" type="integer" xref="S4.SS1.SSS2.p3.3.m3.2.2.1.1.1.2">3</cn><ci id="S4.SS1.SSS2.p3.3.m3.2.2.1.1.1.3.cmml" xref="S4.SS1.SSS2.p3.3.m3.2.2.1.1.1.3">italic-ϵ</ci></apply><ci id="S4.SS1.SSS2.p3.3.m3.2.2.3.cmml" xref="S4.SS1.SSS2.p3.3.m3.2.2.3">𝑤</ci><ci id="S4.SS1.SSS2.p3.3.m3.1.1a.cmml" xref="S4.SS1.SSS2.p3.3.m3.2.2.4.2"><mtext id="S4.SS1.SSS2.p3.3.m3.1.1.cmml" xref="S4.SS1.SSS2.p3.3.m3.1.1">OPT</mtext></ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.SSS2.p3.3.m3.2c">(3+\epsilon)w(\textnormal{OPT})</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.SSS2.p3.3.m3.2d">( 3 + italic_ϵ ) italic_w ( OPT )</annotation></semantics></math>; this concludes the proof of Lemma <a class="ltx_ref" href="https://arxiv.org/html/2503.00712v1#S4.Thmtheorem4" title="Lemma 4.4. ‣ 4.1.2 Bounding Approximation Ratio ‣ 4.1 One-to-Two Augmentation ‣ 4 Vertex Connectivity Augmentation in Link-Arrival Model ‣ Streaming Algorithms for Network Design"><span class="ltx_text ltx_ref_tag">4.4</span></a>. This, combined with Lemma <a class="ltx_ref" href="https://arxiv.org/html/2503.00712v1#S4.Thmtheorem3" title="Lemma 4.3. ‣ 4.1.1 The Streaming Algorithm ‣ 4.1 One-to-Two Augmentation ‣ 4 Vertex Connectivity Augmentation in Link-Arrival Model ‣ Streaming Algorithms for Network Design"><span class="ltx_text ltx_ref_tag">4.3</span></a>, concludes the proof of Theorem <a class="ltx_ref" href="https://arxiv.org/html/2503.00712v1#S4.Thmtheorem1" title="Theorem 4.1. ‣ 4.1 One-to-Two Augmentation ‣ 4 Vertex Connectivity Augmentation in Link-Arrival Model ‣ Streaming Algorithms for Network Design"><span class="ltx_text ltx_ref_tag">4.1</span></a>.</p> </div> </section> </section> <section class="ltx_subsection" id="S4.SS2"> <h3 class="ltx_title ltx_title_subsection"> <span class="ltx_tag ltx_tag_subsection">4.2 </span>Two-to-Three Augmentation</h3> <div class="ltx_para" id="S4.SS2.p1"> <p class="ltx_p" id="S4.SS2.p1.1">In this section, we prove the following theorem.</p> </div> <div class="ltx_theorem ltx_theorem_theorem" id="S4.Thmtheorem7"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="S4.Thmtheorem7.1.1.1">Theorem 4.7</span></span><span class="ltx_text ltx_font_bold" id="S4.Thmtheorem7.2.2">.</span> </h6> <div class="ltx_para" id="S4.Thmtheorem7.p1"> <p class="ltx_p" id="S4.Thmtheorem7.p1.4">There exists a streaming algorithm for the <math alttext="2" class="ltx_Math" display="inline" id="S4.Thmtheorem7.p1.1.m1.1"><semantics id="S4.Thmtheorem7.p1.1.m1.1a"><mn id="S4.Thmtheorem7.p1.1.m1.1.1" xref="S4.Thmtheorem7.p1.1.m1.1.1.cmml">2</mn><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem7.p1.1.m1.1b"><cn id="S4.Thmtheorem7.p1.1.m1.1.1.cmml" type="integer" xref="S4.Thmtheorem7.p1.1.m1.1.1">2</cn></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem7.p1.1.m1.1c">2</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem7.p1.1.m1.1d">2</annotation></semantics></math>-VC-CAP problem with edge weights <math alttext="w:E\to[1,W]" class="ltx_Math" display="inline" id="S4.Thmtheorem7.p1.2.m2.2"><semantics id="S4.Thmtheorem7.p1.2.m2.2a"><mrow id="S4.Thmtheorem7.p1.2.m2.2.3" xref="S4.Thmtheorem7.p1.2.m2.2.3.cmml"><mi id="S4.Thmtheorem7.p1.2.m2.2.3.2" xref="S4.Thmtheorem7.p1.2.m2.2.3.2.cmml">w</mi><mo id="S4.Thmtheorem7.p1.2.m2.2.3.1" lspace="0.278em" rspace="0.278em" xref="S4.Thmtheorem7.p1.2.m2.2.3.1.cmml">:</mo><mrow id="S4.Thmtheorem7.p1.2.m2.2.3.3" xref="S4.Thmtheorem7.p1.2.m2.2.3.3.cmml"><mi id="S4.Thmtheorem7.p1.2.m2.2.3.3.2" xref="S4.Thmtheorem7.p1.2.m2.2.3.3.2.cmml">E</mi><mo id="S4.Thmtheorem7.p1.2.m2.2.3.3.1" stretchy="false" xref="S4.Thmtheorem7.p1.2.m2.2.3.3.1.cmml">→</mo><mrow id="S4.Thmtheorem7.p1.2.m2.2.3.3.3.2" xref="S4.Thmtheorem7.p1.2.m2.2.3.3.3.1.cmml"><mo id="S4.Thmtheorem7.p1.2.m2.2.3.3.3.2.1" stretchy="false" xref="S4.Thmtheorem7.p1.2.m2.2.3.3.3.1.cmml">[</mo><mn id="S4.Thmtheorem7.p1.2.m2.1.1" xref="S4.Thmtheorem7.p1.2.m2.1.1.cmml">1</mn><mo id="S4.Thmtheorem7.p1.2.m2.2.3.3.3.2.2" xref="S4.Thmtheorem7.p1.2.m2.2.3.3.3.1.cmml">,</mo><mi id="S4.Thmtheorem7.p1.2.m2.2.2" xref="S4.Thmtheorem7.p1.2.m2.2.2.cmml">W</mi><mo id="S4.Thmtheorem7.p1.2.m2.2.3.3.3.2.3" stretchy="false" xref="S4.Thmtheorem7.p1.2.m2.2.3.3.3.1.cmml">]</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem7.p1.2.m2.2b"><apply id="S4.Thmtheorem7.p1.2.m2.2.3.cmml" xref="S4.Thmtheorem7.p1.2.m2.2.3"><ci id="S4.Thmtheorem7.p1.2.m2.2.3.1.cmml" xref="S4.Thmtheorem7.p1.2.m2.2.3.1">:</ci><ci id="S4.Thmtheorem7.p1.2.m2.2.3.2.cmml" xref="S4.Thmtheorem7.p1.2.m2.2.3.2">𝑤</ci><apply id="S4.Thmtheorem7.p1.2.m2.2.3.3.cmml" xref="S4.Thmtheorem7.p1.2.m2.2.3.3"><ci id="S4.Thmtheorem7.p1.2.m2.2.3.3.1.cmml" xref="S4.Thmtheorem7.p1.2.m2.2.3.3.1">→</ci><ci id="S4.Thmtheorem7.p1.2.m2.2.3.3.2.cmml" xref="S4.Thmtheorem7.p1.2.m2.2.3.3.2">𝐸</ci><interval closure="closed" id="S4.Thmtheorem7.p1.2.m2.2.3.3.3.1.cmml" xref="S4.Thmtheorem7.p1.2.m2.2.3.3.3.2"><cn id="S4.Thmtheorem7.p1.2.m2.1.1.cmml" type="integer" xref="S4.Thmtheorem7.p1.2.m2.1.1">1</cn><ci id="S4.Thmtheorem7.p1.2.m2.2.2.cmml" xref="S4.Thmtheorem7.p1.2.m2.2.2">𝑊</ci></interval></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem7.p1.2.m2.2c">w:E\to[1,W]</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem7.p1.2.m2.2d">italic_w : italic_E → [ 1 , italic_W ]</annotation></semantics></math>, in an insertion-only stream, that uses <math alttext="O(n\epsilon^{-1}\log W)" class="ltx_Math" display="inline" id="S4.Thmtheorem7.p1.3.m3.1"><semantics id="S4.Thmtheorem7.p1.3.m3.1a"><mrow id="S4.Thmtheorem7.p1.3.m3.1.1" xref="S4.Thmtheorem7.p1.3.m3.1.1.cmml"><mi id="S4.Thmtheorem7.p1.3.m3.1.1.3" xref="S4.Thmtheorem7.p1.3.m3.1.1.3.cmml">O</mi><mo id="S4.Thmtheorem7.p1.3.m3.1.1.2" xref="S4.Thmtheorem7.p1.3.m3.1.1.2.cmml">⁢</mo><mrow id="S4.Thmtheorem7.p1.3.m3.1.1.1.1" xref="S4.Thmtheorem7.p1.3.m3.1.1.1.1.1.cmml"><mo id="S4.Thmtheorem7.p1.3.m3.1.1.1.1.2" stretchy="false" xref="S4.Thmtheorem7.p1.3.m3.1.1.1.1.1.cmml">(</mo><mrow id="S4.Thmtheorem7.p1.3.m3.1.1.1.1.1" xref="S4.Thmtheorem7.p1.3.m3.1.1.1.1.1.cmml"><mi id="S4.Thmtheorem7.p1.3.m3.1.1.1.1.1.2" xref="S4.Thmtheorem7.p1.3.m3.1.1.1.1.1.2.cmml">n</mi><mo id="S4.Thmtheorem7.p1.3.m3.1.1.1.1.1.1" xref="S4.Thmtheorem7.p1.3.m3.1.1.1.1.1.1.cmml">⁢</mo><msup id="S4.Thmtheorem7.p1.3.m3.1.1.1.1.1.3" xref="S4.Thmtheorem7.p1.3.m3.1.1.1.1.1.3.cmml"><mi id="S4.Thmtheorem7.p1.3.m3.1.1.1.1.1.3.2" xref="S4.Thmtheorem7.p1.3.m3.1.1.1.1.1.3.2.cmml">ϵ</mi><mrow id="S4.Thmtheorem7.p1.3.m3.1.1.1.1.1.3.3" xref="S4.Thmtheorem7.p1.3.m3.1.1.1.1.1.3.3.cmml"><mo id="S4.Thmtheorem7.p1.3.m3.1.1.1.1.1.3.3a" xref="S4.Thmtheorem7.p1.3.m3.1.1.1.1.1.3.3.cmml">−</mo><mn id="S4.Thmtheorem7.p1.3.m3.1.1.1.1.1.3.3.2" xref="S4.Thmtheorem7.p1.3.m3.1.1.1.1.1.3.3.2.cmml">1</mn></mrow></msup><mo id="S4.Thmtheorem7.p1.3.m3.1.1.1.1.1.1a" lspace="0.167em" xref="S4.Thmtheorem7.p1.3.m3.1.1.1.1.1.1.cmml">⁢</mo><mrow id="S4.Thmtheorem7.p1.3.m3.1.1.1.1.1.4" xref="S4.Thmtheorem7.p1.3.m3.1.1.1.1.1.4.cmml"><mi id="S4.Thmtheorem7.p1.3.m3.1.1.1.1.1.4.1" xref="S4.Thmtheorem7.p1.3.m3.1.1.1.1.1.4.1.cmml">log</mi><mo id="S4.Thmtheorem7.p1.3.m3.1.1.1.1.1.4a" lspace="0.167em" xref="S4.Thmtheorem7.p1.3.m3.1.1.1.1.1.4.cmml">⁡</mo><mi id="S4.Thmtheorem7.p1.3.m3.1.1.1.1.1.4.2" xref="S4.Thmtheorem7.p1.3.m3.1.1.1.1.1.4.2.cmml">W</mi></mrow></mrow><mo id="S4.Thmtheorem7.p1.3.m3.1.1.1.1.3" stretchy="false" xref="S4.Thmtheorem7.p1.3.m3.1.1.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem7.p1.3.m3.1b"><apply id="S4.Thmtheorem7.p1.3.m3.1.1.cmml" xref="S4.Thmtheorem7.p1.3.m3.1.1"><times id="S4.Thmtheorem7.p1.3.m3.1.1.2.cmml" xref="S4.Thmtheorem7.p1.3.m3.1.1.2"></times><ci id="S4.Thmtheorem7.p1.3.m3.1.1.3.cmml" xref="S4.Thmtheorem7.p1.3.m3.1.1.3">𝑂</ci><apply id="S4.Thmtheorem7.p1.3.m3.1.1.1.1.1.cmml" xref="S4.Thmtheorem7.p1.3.m3.1.1.1.1"><times id="S4.Thmtheorem7.p1.3.m3.1.1.1.1.1.1.cmml" xref="S4.Thmtheorem7.p1.3.m3.1.1.1.1.1.1"></times><ci id="S4.Thmtheorem7.p1.3.m3.1.1.1.1.1.2.cmml" xref="S4.Thmtheorem7.p1.3.m3.1.1.1.1.1.2">𝑛</ci><apply id="S4.Thmtheorem7.p1.3.m3.1.1.1.1.1.3.cmml" xref="S4.Thmtheorem7.p1.3.m3.1.1.1.1.1.3"><csymbol cd="ambiguous" id="S4.Thmtheorem7.p1.3.m3.1.1.1.1.1.3.1.cmml" xref="S4.Thmtheorem7.p1.3.m3.1.1.1.1.1.3">superscript</csymbol><ci id="S4.Thmtheorem7.p1.3.m3.1.1.1.1.1.3.2.cmml" xref="S4.Thmtheorem7.p1.3.m3.1.1.1.1.1.3.2">italic-ϵ</ci><apply id="S4.Thmtheorem7.p1.3.m3.1.1.1.1.1.3.3.cmml" xref="S4.Thmtheorem7.p1.3.m3.1.1.1.1.1.3.3"><minus id="S4.Thmtheorem7.p1.3.m3.1.1.1.1.1.3.3.1.cmml" xref="S4.Thmtheorem7.p1.3.m3.1.1.1.1.1.3.3"></minus><cn id="S4.Thmtheorem7.p1.3.m3.1.1.1.1.1.3.3.2.cmml" type="integer" xref="S4.Thmtheorem7.p1.3.m3.1.1.1.1.1.3.3.2">1</cn></apply></apply><apply id="S4.Thmtheorem7.p1.3.m3.1.1.1.1.1.4.cmml" xref="S4.Thmtheorem7.p1.3.m3.1.1.1.1.1.4"><log id="S4.Thmtheorem7.p1.3.m3.1.1.1.1.1.4.1.cmml" xref="S4.Thmtheorem7.p1.3.m3.1.1.1.1.1.4.1"></log><ci id="S4.Thmtheorem7.p1.3.m3.1.1.1.1.1.4.2.cmml" xref="S4.Thmtheorem7.p1.3.m3.1.1.1.1.1.4.2">𝑊</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem7.p1.3.m3.1c">O(n\epsilon^{-1}\log W)</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem7.p1.3.m3.1d">italic_O ( italic_n italic_ϵ start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT roman_log italic_W )</annotation></semantics></math> space and outputs a <math alttext="(7+\epsilon)" class="ltx_Math" display="inline" id="S4.Thmtheorem7.p1.4.m4.1"><semantics id="S4.Thmtheorem7.p1.4.m4.1a"><mrow id="S4.Thmtheorem7.p1.4.m4.1.1.1" xref="S4.Thmtheorem7.p1.4.m4.1.1.1.1.cmml"><mo id="S4.Thmtheorem7.p1.4.m4.1.1.1.2" stretchy="false" xref="S4.Thmtheorem7.p1.4.m4.1.1.1.1.cmml">(</mo><mrow id="S4.Thmtheorem7.p1.4.m4.1.1.1.1" xref="S4.Thmtheorem7.p1.4.m4.1.1.1.1.cmml"><mn id="S4.Thmtheorem7.p1.4.m4.1.1.1.1.2" xref="S4.Thmtheorem7.p1.4.m4.1.1.1.1.2.cmml">7</mn><mo id="S4.Thmtheorem7.p1.4.m4.1.1.1.1.1" xref="S4.Thmtheorem7.p1.4.m4.1.1.1.1.1.cmml">+</mo><mi id="S4.Thmtheorem7.p1.4.m4.1.1.1.1.3" xref="S4.Thmtheorem7.p1.4.m4.1.1.1.1.3.cmml">ϵ</mi></mrow><mo id="S4.Thmtheorem7.p1.4.m4.1.1.1.3" stretchy="false" xref="S4.Thmtheorem7.p1.4.m4.1.1.1.1.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem7.p1.4.m4.1b"><apply id="S4.Thmtheorem7.p1.4.m4.1.1.1.1.cmml" xref="S4.Thmtheorem7.p1.4.m4.1.1.1"><plus id="S4.Thmtheorem7.p1.4.m4.1.1.1.1.1.cmml" xref="S4.Thmtheorem7.p1.4.m4.1.1.1.1.1"></plus><cn id="S4.Thmtheorem7.p1.4.m4.1.1.1.1.2.cmml" type="integer" xref="S4.Thmtheorem7.p1.4.m4.1.1.1.1.2">7</cn><ci id="S4.Thmtheorem7.p1.4.m4.1.1.1.1.3.cmml" xref="S4.Thmtheorem7.p1.4.m4.1.1.1.1.3">italic-ϵ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem7.p1.4.m4.1c">(7+\epsilon)</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem7.p1.4.m4.1d">( 7 + italic_ϵ )</annotation></semantics></math>-approximate solution.</p> </div> </div> <div class="ltx_para" id="S4.SS2.p2"> <p class="ltx_p" id="S4.SS2.p2.9">Throughout this section, to avoid confusion with two-vertex cuts <math alttext="\{u,v\}" class="ltx_Math" display="inline" id="S4.SS2.p2.1.m1.2"><semantics id="S4.SS2.p2.1.m1.2a"><mrow id="S4.SS2.p2.1.m1.2.3.2" xref="S4.SS2.p2.1.m1.2.3.1.cmml"><mo id="S4.SS2.p2.1.m1.2.3.2.1" stretchy="false" xref="S4.SS2.p2.1.m1.2.3.1.cmml">{</mo><mi id="S4.SS2.p2.1.m1.1.1" xref="S4.SS2.p2.1.m1.1.1.cmml">u</mi><mo id="S4.SS2.p2.1.m1.2.3.2.2" xref="S4.SS2.p2.1.m1.2.3.1.cmml">,</mo><mi id="S4.SS2.p2.1.m1.2.2" xref="S4.SS2.p2.1.m1.2.2.cmml">v</mi><mo id="S4.SS2.p2.1.m1.2.3.2.3" stretchy="false" xref="S4.SS2.p2.1.m1.2.3.1.cmml">}</mo></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.p2.1.m1.2b"><set id="S4.SS2.p2.1.m1.2.3.1.cmml" xref="S4.SS2.p2.1.m1.2.3.2"><ci id="S4.SS2.p2.1.m1.1.1.cmml" xref="S4.SS2.p2.1.m1.1.1">𝑢</ci><ci id="S4.SS2.p2.1.m1.2.2.cmml" xref="S4.SS2.p2.1.m1.2.2">𝑣</ci></set></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p2.1.m1.2c">\{u,v\}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p2.1.m1.2d">{ italic_u , italic_v }</annotation></semantics></math>, we will denote all edges and links as <math alttext="uv" class="ltx_Math" display="inline" id="S4.SS2.p2.2.m2.1"><semantics id="S4.SS2.p2.2.m2.1a"><mrow id="S4.SS2.p2.2.m2.1.1" xref="S4.SS2.p2.2.m2.1.1.cmml"><mi id="S4.SS2.p2.2.m2.1.1.2" xref="S4.SS2.p2.2.m2.1.1.2.cmml">u</mi><mo id="S4.SS2.p2.2.m2.1.1.1" xref="S4.SS2.p2.2.m2.1.1.1.cmml">⁢</mo><mi id="S4.SS2.p2.2.m2.1.1.3" xref="S4.SS2.p2.2.m2.1.1.3.cmml">v</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.p2.2.m2.1b"><apply id="S4.SS2.p2.2.m2.1.1.cmml" xref="S4.SS2.p2.2.m2.1.1"><times id="S4.SS2.p2.2.m2.1.1.1.cmml" xref="S4.SS2.p2.2.m2.1.1.1"></times><ci id="S4.SS2.p2.2.m2.1.1.2.cmml" xref="S4.SS2.p2.2.m2.1.1.2">𝑢</ci><ci id="S4.SS2.p2.2.m2.1.1.3.cmml" xref="S4.SS2.p2.2.m2.1.1.3">𝑣</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p2.2.m2.1c">uv</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p2.2.m2.1d">italic_u italic_v</annotation></semantics></math> or <math alttext="(u,v)" class="ltx_Math" display="inline" id="S4.SS2.p2.3.m3.2"><semantics id="S4.SS2.p2.3.m3.2a"><mrow id="S4.SS2.p2.3.m3.2.3.2" xref="S4.SS2.p2.3.m3.2.3.1.cmml"><mo id="S4.SS2.p2.3.m3.2.3.2.1" stretchy="false" xref="S4.SS2.p2.3.m3.2.3.1.cmml">(</mo><mi id="S4.SS2.p2.3.m3.1.1" xref="S4.SS2.p2.3.m3.1.1.cmml">u</mi><mo id="S4.SS2.p2.3.m3.2.3.2.2" xref="S4.SS2.p2.3.m3.2.3.1.cmml">,</mo><mi id="S4.SS2.p2.3.m3.2.2" xref="S4.SS2.p2.3.m3.2.2.cmml">v</mi><mo id="S4.SS2.p2.3.m3.2.3.2.3" stretchy="false" xref="S4.SS2.p2.3.m3.2.3.1.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.p2.3.m3.2b"><interval closure="open" id="S4.SS2.p2.3.m3.2.3.1.cmml" xref="S4.SS2.p2.3.m3.2.3.2"><ci id="S4.SS2.p2.3.m3.1.1.cmml" xref="S4.SS2.p2.3.m3.1.1">𝑢</ci><ci id="S4.SS2.p2.3.m3.2.2.cmml" xref="S4.SS2.p2.3.m3.2.2">𝑣</ci></interval></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p2.3.m3.2c">(u,v)</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p2.3.m3.2d">( italic_u , italic_v )</annotation></semantics></math>. Following the notation in Section <a class="ltx_ref" href="https://arxiv.org/html/2503.00712v1#S4.SS1" title="4.1 One-to-Two Augmentation ‣ 4 Vertex Connectivity Augmentation in Link-Arrival Model ‣ Streaming Algorithms for Network Design"><span class="ltx_text ltx_ref_tag">4.1</span></a>, for any edge set <math alttext="E^{\prime}\subseteq E\cup L" class="ltx_Math" display="inline" id="S4.SS2.p2.4.m4.1"><semantics id="S4.SS2.p2.4.m4.1a"><mrow id="S4.SS2.p2.4.m4.1.1" xref="S4.SS2.p2.4.m4.1.1.cmml"><msup id="S4.SS2.p2.4.m4.1.1.2" xref="S4.SS2.p2.4.m4.1.1.2.cmml"><mi id="S4.SS2.p2.4.m4.1.1.2.2" xref="S4.SS2.p2.4.m4.1.1.2.2.cmml">E</mi><mo id="S4.SS2.p2.4.m4.1.1.2.3" xref="S4.SS2.p2.4.m4.1.1.2.3.cmml">′</mo></msup><mo id="S4.SS2.p2.4.m4.1.1.1" xref="S4.SS2.p2.4.m4.1.1.1.cmml">⊆</mo><mrow id="S4.SS2.p2.4.m4.1.1.3" xref="S4.SS2.p2.4.m4.1.1.3.cmml"><mi id="S4.SS2.p2.4.m4.1.1.3.2" xref="S4.SS2.p2.4.m4.1.1.3.2.cmml">E</mi><mo id="S4.SS2.p2.4.m4.1.1.3.1" xref="S4.SS2.p2.4.m4.1.1.3.1.cmml">∪</mo><mi id="S4.SS2.p2.4.m4.1.1.3.3" xref="S4.SS2.p2.4.m4.1.1.3.3.cmml">L</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.p2.4.m4.1b"><apply id="S4.SS2.p2.4.m4.1.1.cmml" xref="S4.SS2.p2.4.m4.1.1"><subset id="S4.SS2.p2.4.m4.1.1.1.cmml" xref="S4.SS2.p2.4.m4.1.1.1"></subset><apply id="S4.SS2.p2.4.m4.1.1.2.cmml" xref="S4.SS2.p2.4.m4.1.1.2"><csymbol cd="ambiguous" id="S4.SS2.p2.4.m4.1.1.2.1.cmml" xref="S4.SS2.p2.4.m4.1.1.2">superscript</csymbol><ci id="S4.SS2.p2.4.m4.1.1.2.2.cmml" xref="S4.SS2.p2.4.m4.1.1.2.2">𝐸</ci><ci id="S4.SS2.p2.4.m4.1.1.2.3.cmml" xref="S4.SS2.p2.4.m4.1.1.2.3">′</ci></apply><apply id="S4.SS2.p2.4.m4.1.1.3.cmml" xref="S4.SS2.p2.4.m4.1.1.3"><union id="S4.SS2.p2.4.m4.1.1.3.1.cmml" xref="S4.SS2.p2.4.m4.1.1.3.1"></union><ci id="S4.SS2.p2.4.m4.1.1.3.2.cmml" xref="S4.SS2.p2.4.m4.1.1.3.2">𝐸</ci><ci id="S4.SS2.p2.4.m4.1.1.3.3.cmml" xref="S4.SS2.p2.4.m4.1.1.3.3">𝐿</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p2.4.m4.1c">E^{\prime}\subseteq E\cup L</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p2.4.m4.1d">italic_E start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ⊆ italic_E ∪ italic_L</annotation></semantics></math> and any vertex sets <math alttext="A,B\subseteq V" class="ltx_Math" display="inline" id="S4.SS2.p2.5.m5.2"><semantics id="S4.SS2.p2.5.m5.2a"><mrow id="S4.SS2.p2.5.m5.2.3" xref="S4.SS2.p2.5.m5.2.3.cmml"><mrow id="S4.SS2.p2.5.m5.2.3.2.2" xref="S4.SS2.p2.5.m5.2.3.2.1.cmml"><mi id="S4.SS2.p2.5.m5.1.1" xref="S4.SS2.p2.5.m5.1.1.cmml">A</mi><mo id="S4.SS2.p2.5.m5.2.3.2.2.1" xref="S4.SS2.p2.5.m5.2.3.2.1.cmml">,</mo><mi id="S4.SS2.p2.5.m5.2.2" xref="S4.SS2.p2.5.m5.2.2.cmml">B</mi></mrow><mo id="S4.SS2.p2.5.m5.2.3.1" xref="S4.SS2.p2.5.m5.2.3.1.cmml">⊆</mo><mi id="S4.SS2.p2.5.m5.2.3.3" xref="S4.SS2.p2.5.m5.2.3.3.cmml">V</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.p2.5.m5.2b"><apply id="S4.SS2.p2.5.m5.2.3.cmml" xref="S4.SS2.p2.5.m5.2.3"><subset id="S4.SS2.p2.5.m5.2.3.1.cmml" xref="S4.SS2.p2.5.m5.2.3.1"></subset><list id="S4.SS2.p2.5.m5.2.3.2.1.cmml" xref="S4.SS2.p2.5.m5.2.3.2.2"><ci id="S4.SS2.p2.5.m5.1.1.cmml" xref="S4.SS2.p2.5.m5.1.1">𝐴</ci><ci id="S4.SS2.p2.5.m5.2.2.cmml" xref="S4.SS2.p2.5.m5.2.2">𝐵</ci></list><ci id="S4.SS2.p2.5.m5.2.3.3.cmml" xref="S4.SS2.p2.5.m5.2.3.3">𝑉</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p2.5.m5.2c">A,B\subseteq V</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p2.5.m5.2d">italic_A , italic_B ⊆ italic_V</annotation></semantics></math>, we let <math alttext="E^{\prime}[A,B]" class="ltx_Math" display="inline" id="S4.SS2.p2.6.m6.2"><semantics id="S4.SS2.p2.6.m6.2a"><mrow id="S4.SS2.p2.6.m6.2.3" xref="S4.SS2.p2.6.m6.2.3.cmml"><msup id="S4.SS2.p2.6.m6.2.3.2" xref="S4.SS2.p2.6.m6.2.3.2.cmml"><mi id="S4.SS2.p2.6.m6.2.3.2.2" xref="S4.SS2.p2.6.m6.2.3.2.2.cmml">E</mi><mo id="S4.SS2.p2.6.m6.2.3.2.3" xref="S4.SS2.p2.6.m6.2.3.2.3.cmml">′</mo></msup><mo id="S4.SS2.p2.6.m6.2.3.1" xref="S4.SS2.p2.6.m6.2.3.1.cmml">⁢</mo><mrow id="S4.SS2.p2.6.m6.2.3.3.2" xref="S4.SS2.p2.6.m6.2.3.3.1.cmml"><mo id="S4.SS2.p2.6.m6.2.3.3.2.1" stretchy="false" xref="S4.SS2.p2.6.m6.2.3.3.1.cmml">[</mo><mi id="S4.SS2.p2.6.m6.1.1" xref="S4.SS2.p2.6.m6.1.1.cmml">A</mi><mo id="S4.SS2.p2.6.m6.2.3.3.2.2" xref="S4.SS2.p2.6.m6.2.3.3.1.cmml">,</mo><mi id="S4.SS2.p2.6.m6.2.2" xref="S4.SS2.p2.6.m6.2.2.cmml">B</mi><mo id="S4.SS2.p2.6.m6.2.3.3.2.3" stretchy="false" xref="S4.SS2.p2.6.m6.2.3.3.1.cmml">]</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.p2.6.m6.2b"><apply id="S4.SS2.p2.6.m6.2.3.cmml" xref="S4.SS2.p2.6.m6.2.3"><times id="S4.SS2.p2.6.m6.2.3.1.cmml" xref="S4.SS2.p2.6.m6.2.3.1"></times><apply id="S4.SS2.p2.6.m6.2.3.2.cmml" xref="S4.SS2.p2.6.m6.2.3.2"><csymbol cd="ambiguous" id="S4.SS2.p2.6.m6.2.3.2.1.cmml" xref="S4.SS2.p2.6.m6.2.3.2">superscript</csymbol><ci id="S4.SS2.p2.6.m6.2.3.2.2.cmml" xref="S4.SS2.p2.6.m6.2.3.2.2">𝐸</ci><ci id="S4.SS2.p2.6.m6.2.3.2.3.cmml" xref="S4.SS2.p2.6.m6.2.3.2.3">′</ci></apply><interval closure="closed" id="S4.SS2.p2.6.m6.2.3.3.1.cmml" xref="S4.SS2.p2.6.m6.2.3.3.2"><ci id="S4.SS2.p2.6.m6.1.1.cmml" xref="S4.SS2.p2.6.m6.1.1">𝐴</ci><ci id="S4.SS2.p2.6.m6.2.2.cmml" xref="S4.SS2.p2.6.m6.2.2">𝐵</ci></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p2.6.m6.2c">E^{\prime}[A,B]</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p2.6.m6.2d">italic_E start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT [ italic_A , italic_B ]</annotation></semantics></math> denote the set of edges in <math alttext="E^{\prime}" class="ltx_Math" display="inline" id="S4.SS2.p2.7.m7.1"><semantics id="S4.SS2.p2.7.m7.1a"><msup id="S4.SS2.p2.7.m7.1.1" xref="S4.SS2.p2.7.m7.1.1.cmml"><mi id="S4.SS2.p2.7.m7.1.1.2" xref="S4.SS2.p2.7.m7.1.1.2.cmml">E</mi><mo id="S4.SS2.p2.7.m7.1.1.3" xref="S4.SS2.p2.7.m7.1.1.3.cmml">′</mo></msup><annotation-xml encoding="MathML-Content" id="S4.SS2.p2.7.m7.1b"><apply id="S4.SS2.p2.7.m7.1.1.cmml" xref="S4.SS2.p2.7.m7.1.1"><csymbol cd="ambiguous" id="S4.SS2.p2.7.m7.1.1.1.cmml" xref="S4.SS2.p2.7.m7.1.1">superscript</csymbol><ci id="S4.SS2.p2.7.m7.1.1.2.cmml" xref="S4.SS2.p2.7.m7.1.1.2">𝐸</ci><ci id="S4.SS2.p2.7.m7.1.1.3.cmml" xref="S4.SS2.p2.7.m7.1.1.3">′</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p2.7.m7.1c">E^{\prime}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p2.7.m7.1d">italic_E start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math> with one endpoint in <math alttext="A" class="ltx_Math" display="inline" id="S4.SS2.p2.8.m8.1"><semantics id="S4.SS2.p2.8.m8.1a"><mi id="S4.SS2.p2.8.m8.1.1" xref="S4.SS2.p2.8.m8.1.1.cmml">A</mi><annotation-xml encoding="MathML-Content" id="S4.SS2.p2.8.m8.1b"><ci id="S4.SS2.p2.8.m8.1.1.cmml" xref="S4.SS2.p2.8.m8.1.1">𝐴</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p2.8.m8.1c">A</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p2.8.m8.1d">italic_A</annotation></semantics></math> and the other in <math alttext="B" class="ltx_Math" display="inline" id="S4.SS2.p2.9.m9.1"><semantics id="S4.SS2.p2.9.m9.1a"><mi id="S4.SS2.p2.9.m9.1.1" xref="S4.SS2.p2.9.m9.1.1.cmml">B</mi><annotation-xml encoding="MathML-Content" id="S4.SS2.p2.9.m9.1b"><ci id="S4.SS2.p2.9.m9.1.1.cmml" xref="S4.SS2.p2.9.m9.1.1">𝐵</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p2.9.m9.1c">B</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p2.9.m9.1d">italic_B</annotation></semantics></math>. We start with some background on SPQR trees.</p> </div> <section class="ltx_subsubsection" id="S4.SS2.SSS1"> <h4 class="ltx_title ltx_title_subsubsection"> <span class="ltx_tag ltx_tag_subsubsection">4.2.1 </span>SPQR Trees</h4> <div class="ltx_para" id="S4.SS2.SSS1.p1"> <p class="ltx_p" id="S4.SS2.SSS1.p1.2">An SPQR tree is a data structure that gives a tree-like decomposition of a 2-connected graph into <em class="ltx_emph ltx_font_italic" id="S4.SS2.SSS1.p1.2.1">triconnected</em> components. SPQR trees were first formally defined in <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.00712v1#bib.bibx33" title="">DBT96a</a>]</cite> although they were implicit in prior work <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.00712v1#bib.bibx32" title="">DBT89</a>]</cite>, and the ideas build heavily on work in <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.00712v1#bib.bibx52" title="">HT73</a>]</cite>. We start with some definitions and notation. Note that the following terms are defined on <em class="ltx_emph ltx_font_italic" id="S4.SS2.SSS1.p1.2.2">multigraphs</em> (several parallel edges between a pair of nodes are allowed), even though we assume the input graph <math alttext="G" class="ltx_Math" display="inline" id="S4.SS2.SSS1.p1.1.m1.1"><semantics id="S4.SS2.SSS1.p1.1.m1.1a"><mi id="S4.SS2.SSS1.p1.1.m1.1.1" xref="S4.SS2.SSS1.p1.1.m1.1.1.cmml">G</mi><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS1.p1.1.m1.1b"><ci id="S4.SS2.SSS1.p1.1.m1.1.1.cmml" xref="S4.SS2.SSS1.p1.1.m1.1.1">𝐺</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS1.p1.1.m1.1c">G</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS1.p1.1.m1.1d">italic_G</annotation></semantics></math> is simple. This is because the construction of SPQR trees follows a recursive procedure that may introduce parallel edges. Let <math alttext="G=(V,E)" class="ltx_Math" display="inline" id="S4.SS2.SSS1.p1.2.m2.2"><semantics id="S4.SS2.SSS1.p1.2.m2.2a"><mrow id="S4.SS2.SSS1.p1.2.m2.2.3" xref="S4.SS2.SSS1.p1.2.m2.2.3.cmml"><mi id="S4.SS2.SSS1.p1.2.m2.2.3.2" xref="S4.SS2.SSS1.p1.2.m2.2.3.2.cmml">G</mi><mo id="S4.SS2.SSS1.p1.2.m2.2.3.1" xref="S4.SS2.SSS1.p1.2.m2.2.3.1.cmml">=</mo><mrow id="S4.SS2.SSS1.p1.2.m2.2.3.3.2" xref="S4.SS2.SSS1.p1.2.m2.2.3.3.1.cmml"><mo id="S4.SS2.SSS1.p1.2.m2.2.3.3.2.1" stretchy="false" xref="S4.SS2.SSS1.p1.2.m2.2.3.3.1.cmml">(</mo><mi id="S4.SS2.SSS1.p1.2.m2.1.1" xref="S4.SS2.SSS1.p1.2.m2.1.1.cmml">V</mi><mo id="S4.SS2.SSS1.p1.2.m2.2.3.3.2.2" xref="S4.SS2.SSS1.p1.2.m2.2.3.3.1.cmml">,</mo><mi id="S4.SS2.SSS1.p1.2.m2.2.2" xref="S4.SS2.SSS1.p1.2.m2.2.2.cmml">E</mi><mo id="S4.SS2.SSS1.p1.2.m2.2.3.3.2.3" stretchy="false" xref="S4.SS2.SSS1.p1.2.m2.2.3.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS1.p1.2.m2.2b"><apply id="S4.SS2.SSS1.p1.2.m2.2.3.cmml" xref="S4.SS2.SSS1.p1.2.m2.2.3"><eq id="S4.SS2.SSS1.p1.2.m2.2.3.1.cmml" xref="S4.SS2.SSS1.p1.2.m2.2.3.1"></eq><ci id="S4.SS2.SSS1.p1.2.m2.2.3.2.cmml" xref="S4.SS2.SSS1.p1.2.m2.2.3.2">𝐺</ci><interval closure="open" id="S4.SS2.SSS1.p1.2.m2.2.3.3.1.cmml" xref="S4.SS2.SSS1.p1.2.m2.2.3.3.2"><ci id="S4.SS2.SSS1.p1.2.m2.1.1.cmml" xref="S4.SS2.SSS1.p1.2.m2.1.1">𝑉</ci><ci id="S4.SS2.SSS1.p1.2.m2.2.2.cmml" xref="S4.SS2.SSS1.p1.2.m2.2.2">𝐸</ci></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS1.p1.2.m2.2c">G=(V,E)</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS1.p1.2.m2.2d">italic_G = ( italic_V , italic_E )</annotation></semantics></math> be a 2-connected graph.</p> </div> <div class="ltx_theorem ltx_theorem_definition" id="S4.Thmtheorem8"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="S4.Thmtheorem8.1.1.1">Definition 4.8</span></span><span class="ltx_text ltx_font_bold" id="S4.Thmtheorem8.2.2">.</span> </h6> <div class="ltx_para" id="S4.Thmtheorem8.p1"> <p class="ltx_p" id="S4.Thmtheorem8.p1.2">A <em class="ltx_emph ltx_font_italic" id="S4.Thmtheorem8.p1.2.1">separation pair</em> <math alttext="\{a,b\}" class="ltx_Math" display="inline" id="S4.Thmtheorem8.p1.1.m1.2"><semantics id="S4.Thmtheorem8.p1.1.m1.2a"><mrow id="S4.Thmtheorem8.p1.1.m1.2.3.2" xref="S4.Thmtheorem8.p1.1.m1.2.3.1.cmml"><mo id="S4.Thmtheorem8.p1.1.m1.2.3.2.1" stretchy="false" xref="S4.Thmtheorem8.p1.1.m1.2.3.1.cmml">{</mo><mi id="S4.Thmtheorem8.p1.1.m1.1.1" xref="S4.Thmtheorem8.p1.1.m1.1.1.cmml">a</mi><mo id="S4.Thmtheorem8.p1.1.m1.2.3.2.2" xref="S4.Thmtheorem8.p1.1.m1.2.3.1.cmml">,</mo><mi id="S4.Thmtheorem8.p1.1.m1.2.2" xref="S4.Thmtheorem8.p1.1.m1.2.2.cmml">b</mi><mo id="S4.Thmtheorem8.p1.1.m1.2.3.2.3" stretchy="false" xref="S4.Thmtheorem8.p1.1.m1.2.3.1.cmml">}</mo></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem8.p1.1.m1.2b"><set id="S4.Thmtheorem8.p1.1.m1.2.3.1.cmml" xref="S4.Thmtheorem8.p1.1.m1.2.3.2"><ci id="S4.Thmtheorem8.p1.1.m1.1.1.cmml" xref="S4.Thmtheorem8.p1.1.m1.1.1">𝑎</ci><ci id="S4.Thmtheorem8.p1.1.m1.2.2.cmml" xref="S4.Thmtheorem8.p1.1.m1.2.2">𝑏</ci></set></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem8.p1.1.m1.2c">\{a,b\}</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem8.p1.1.m1.2d">{ italic_a , italic_b }</annotation></semantics></math> is a pair of vertices <math alttext="a,b\in V" class="ltx_Math" display="inline" id="S4.Thmtheorem8.p1.2.m2.2"><semantics id="S4.Thmtheorem8.p1.2.m2.2a"><mrow id="S4.Thmtheorem8.p1.2.m2.2.3" xref="S4.Thmtheorem8.p1.2.m2.2.3.cmml"><mrow id="S4.Thmtheorem8.p1.2.m2.2.3.2.2" xref="S4.Thmtheorem8.p1.2.m2.2.3.2.1.cmml"><mi id="S4.Thmtheorem8.p1.2.m2.1.1" xref="S4.Thmtheorem8.p1.2.m2.1.1.cmml">a</mi><mo id="S4.Thmtheorem8.p1.2.m2.2.3.2.2.1" xref="S4.Thmtheorem8.p1.2.m2.2.3.2.1.cmml">,</mo><mi id="S4.Thmtheorem8.p1.2.m2.2.2" xref="S4.Thmtheorem8.p1.2.m2.2.2.cmml">b</mi></mrow><mo id="S4.Thmtheorem8.p1.2.m2.2.3.1" xref="S4.Thmtheorem8.p1.2.m2.2.3.1.cmml">∈</mo><mi id="S4.Thmtheorem8.p1.2.m2.2.3.3" xref="S4.Thmtheorem8.p1.2.m2.2.3.3.cmml">V</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem8.p1.2.m2.2b"><apply id="S4.Thmtheorem8.p1.2.m2.2.3.cmml" xref="S4.Thmtheorem8.p1.2.m2.2.3"><in id="S4.Thmtheorem8.p1.2.m2.2.3.1.cmml" xref="S4.Thmtheorem8.p1.2.m2.2.3.1"></in><list id="S4.Thmtheorem8.p1.2.m2.2.3.2.1.cmml" xref="S4.Thmtheorem8.p1.2.m2.2.3.2.2"><ci id="S4.Thmtheorem8.p1.2.m2.1.1.cmml" xref="S4.Thmtheorem8.p1.2.m2.1.1">𝑎</ci><ci id="S4.Thmtheorem8.p1.2.m2.2.2.cmml" xref="S4.Thmtheorem8.p1.2.m2.2.2">𝑏</ci></list><ci id="S4.Thmtheorem8.p1.2.m2.2.3.3.cmml" xref="S4.Thmtheorem8.p1.2.m2.2.3.3">𝑉</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem8.p1.2.m2.2c">a,b\in V</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem8.p1.2.m2.2d">italic_a , italic_b ∈ italic_V</annotation></semantics></math> such that at least one of the following hold:</p> <ol class="ltx_enumerate" id="S4.I2"> <li class="ltx_item" id="S4.I2.i1" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">1.</span> <div class="ltx_para" id="S4.I2.i1.p1"> <p class="ltx_p" id="S4.I2.i1.p1.1"><math alttext="G\setminus\{a,b\}" class="ltx_Math" display="inline" id="S4.I2.i1.p1.1.m1.2"><semantics id="S4.I2.i1.p1.1.m1.2a"><mrow id="S4.I2.i1.p1.1.m1.2.3" xref="S4.I2.i1.p1.1.m1.2.3.cmml"><mi id="S4.I2.i1.p1.1.m1.2.3.2" xref="S4.I2.i1.p1.1.m1.2.3.2.cmml">G</mi><mo id="S4.I2.i1.p1.1.m1.2.3.1" xref="S4.I2.i1.p1.1.m1.2.3.1.cmml">∖</mo><mrow id="S4.I2.i1.p1.1.m1.2.3.3.2" xref="S4.I2.i1.p1.1.m1.2.3.3.1.cmml"><mo id="S4.I2.i1.p1.1.m1.2.3.3.2.1" stretchy="false" xref="S4.I2.i1.p1.1.m1.2.3.3.1.cmml">{</mo><mi id="S4.I2.i1.p1.1.m1.1.1" xref="S4.I2.i1.p1.1.m1.1.1.cmml">a</mi><mo id="S4.I2.i1.p1.1.m1.2.3.3.2.2" xref="S4.I2.i1.p1.1.m1.2.3.3.1.cmml">,</mo><mi id="S4.I2.i1.p1.1.m1.2.2" xref="S4.I2.i1.p1.1.m1.2.2.cmml">b</mi><mo id="S4.I2.i1.p1.1.m1.2.3.3.2.3" stretchy="false" xref="S4.I2.i1.p1.1.m1.2.3.3.1.cmml">}</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.I2.i1.p1.1.m1.2b"><apply id="S4.I2.i1.p1.1.m1.2.3.cmml" xref="S4.I2.i1.p1.1.m1.2.3"><setdiff id="S4.I2.i1.p1.1.m1.2.3.1.cmml" xref="S4.I2.i1.p1.1.m1.2.3.1"></setdiff><ci id="S4.I2.i1.p1.1.m1.2.3.2.cmml" xref="S4.I2.i1.p1.1.m1.2.3.2">𝐺</ci><set id="S4.I2.i1.p1.1.m1.2.3.3.1.cmml" xref="S4.I2.i1.p1.1.m1.2.3.3.2"><ci id="S4.I2.i1.p1.1.m1.1.1.cmml" xref="S4.I2.i1.p1.1.m1.1.1">𝑎</ci><ci id="S4.I2.i1.p1.1.m1.2.2.cmml" xref="S4.I2.i1.p1.1.m1.2.2">𝑏</ci></set></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.I2.i1.p1.1.m1.2c">G\setminus\{a,b\}</annotation><annotation encoding="application/x-llamapun" id="S4.I2.i1.p1.1.m1.2d">italic_G ∖ { italic_a , italic_b }</annotation></semantics></math> has at least two connected components.</p> </div> </li> <li class="ltx_item" id="S4.I2.i2" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">2.</span> <div class="ltx_para" id="S4.I2.i2.p1"> <p class="ltx_p" id="S4.I2.i2.p1.3"><math alttext="G" class="ltx_Math" display="inline" id="S4.I2.i2.p1.1.m1.1"><semantics id="S4.I2.i2.p1.1.m1.1a"><mi id="S4.I2.i2.p1.1.m1.1.1" xref="S4.I2.i2.p1.1.m1.1.1.cmml">G</mi><annotation-xml encoding="MathML-Content" id="S4.I2.i2.p1.1.m1.1b"><ci id="S4.I2.i2.p1.1.m1.1.1.cmml" xref="S4.I2.i2.p1.1.m1.1.1">𝐺</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.I2.i2.p1.1.m1.1c">G</annotation><annotation encoding="application/x-llamapun" id="S4.I2.i2.p1.1.m1.1d">italic_G</annotation></semantics></math> contains at least two parallel edges <math alttext="ab" class="ltx_Math" display="inline" id="S4.I2.i2.p1.2.m2.1"><semantics id="S4.I2.i2.p1.2.m2.1a"><mrow id="S4.I2.i2.p1.2.m2.1.1" xref="S4.I2.i2.p1.2.m2.1.1.cmml"><mi id="S4.I2.i2.p1.2.m2.1.1.2" xref="S4.I2.i2.p1.2.m2.1.1.2.cmml">a</mi><mo id="S4.I2.i2.p1.2.m2.1.1.1" xref="S4.I2.i2.p1.2.m2.1.1.1.cmml">⁢</mo><mi id="S4.I2.i2.p1.2.m2.1.1.3" xref="S4.I2.i2.p1.2.m2.1.1.3.cmml">b</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.I2.i2.p1.2.m2.1b"><apply id="S4.I2.i2.p1.2.m2.1.1.cmml" xref="S4.I2.i2.p1.2.m2.1.1"><times id="S4.I2.i2.p1.2.m2.1.1.1.cmml" xref="S4.I2.i2.p1.2.m2.1.1.1"></times><ci id="S4.I2.i2.p1.2.m2.1.1.2.cmml" xref="S4.I2.i2.p1.2.m2.1.1.2">𝑎</ci><ci id="S4.I2.i2.p1.2.m2.1.1.3.cmml" xref="S4.I2.i2.p1.2.m2.1.1.3">𝑏</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.I2.i2.p1.2.m2.1c">ab</annotation><annotation encoding="application/x-llamapun" id="S4.I2.i2.p1.2.m2.1d">italic_a italic_b</annotation></semantics></math> <em class="ltx_emph ltx_font_italic" id="S4.I2.i2.p1.3.1">and</em> <math alttext="G\setminus\{a,b\}\neq\emptyset" class="ltx_Math" display="inline" id="S4.I2.i2.p1.3.m3.2"><semantics id="S4.I2.i2.p1.3.m3.2a"><mrow id="S4.I2.i2.p1.3.m3.2.3" xref="S4.I2.i2.p1.3.m3.2.3.cmml"><mrow id="S4.I2.i2.p1.3.m3.2.3.2" xref="S4.I2.i2.p1.3.m3.2.3.2.cmml"><mi id="S4.I2.i2.p1.3.m3.2.3.2.2" xref="S4.I2.i2.p1.3.m3.2.3.2.2.cmml">G</mi><mo id="S4.I2.i2.p1.3.m3.2.3.2.1" xref="S4.I2.i2.p1.3.m3.2.3.2.1.cmml">∖</mo><mrow id="S4.I2.i2.p1.3.m3.2.3.2.3.2" xref="S4.I2.i2.p1.3.m3.2.3.2.3.1.cmml"><mo id="S4.I2.i2.p1.3.m3.2.3.2.3.2.1" stretchy="false" xref="S4.I2.i2.p1.3.m3.2.3.2.3.1.cmml">{</mo><mi id="S4.I2.i2.p1.3.m3.1.1" xref="S4.I2.i2.p1.3.m3.1.1.cmml">a</mi><mo id="S4.I2.i2.p1.3.m3.2.3.2.3.2.2" xref="S4.I2.i2.p1.3.m3.2.3.2.3.1.cmml">,</mo><mi id="S4.I2.i2.p1.3.m3.2.2" xref="S4.I2.i2.p1.3.m3.2.2.cmml">b</mi><mo id="S4.I2.i2.p1.3.m3.2.3.2.3.2.3" stretchy="false" xref="S4.I2.i2.p1.3.m3.2.3.2.3.1.cmml">}</mo></mrow></mrow><mo id="S4.I2.i2.p1.3.m3.2.3.1" xref="S4.I2.i2.p1.3.m3.2.3.1.cmml">≠</mo><mi id="S4.I2.i2.p1.3.m3.2.3.3" mathvariant="normal" xref="S4.I2.i2.p1.3.m3.2.3.3.cmml">∅</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.I2.i2.p1.3.m3.2b"><apply id="S4.I2.i2.p1.3.m3.2.3.cmml" xref="S4.I2.i2.p1.3.m3.2.3"><neq id="S4.I2.i2.p1.3.m3.2.3.1.cmml" xref="S4.I2.i2.p1.3.m3.2.3.1"></neq><apply id="S4.I2.i2.p1.3.m3.2.3.2.cmml" xref="S4.I2.i2.p1.3.m3.2.3.2"><setdiff id="S4.I2.i2.p1.3.m3.2.3.2.1.cmml" xref="S4.I2.i2.p1.3.m3.2.3.2.1"></setdiff><ci id="S4.I2.i2.p1.3.m3.2.3.2.2.cmml" xref="S4.I2.i2.p1.3.m3.2.3.2.2">𝐺</ci><set id="S4.I2.i2.p1.3.m3.2.3.2.3.1.cmml" xref="S4.I2.i2.p1.3.m3.2.3.2.3.2"><ci id="S4.I2.i2.p1.3.m3.1.1.cmml" xref="S4.I2.i2.p1.3.m3.1.1">𝑎</ci><ci id="S4.I2.i2.p1.3.m3.2.2.cmml" xref="S4.I2.i2.p1.3.m3.2.2">𝑏</ci></set></apply><emptyset id="S4.I2.i2.p1.3.m3.2.3.3.cmml" xref="S4.I2.i2.p1.3.m3.2.3.3"></emptyset></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.I2.i2.p1.3.m3.2c">G\setminus\{a,b\}\neq\emptyset</annotation><annotation encoding="application/x-llamapun" id="S4.I2.i2.p1.3.m3.2d">italic_G ∖ { italic_a , italic_b } ≠ ∅</annotation></semantics></math>.</p> </div> </li> </ol> <p class="ltx_p" id="S4.Thmtheorem8.p1.4">We define corresponding <em class="ltx_emph ltx_font_italic" id="S4.Thmtheorem8.p1.4.1">separation classes</em> as follows:</p> <ol class="ltx_enumerate" id="S4.I3"> <li class="ltx_item" id="S4.I3.i1" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">1.</span> <div class="ltx_para" id="S4.I3.i1.p1"> <p class="ltx_p" id="S4.I3.i1.p1.8">If <math alttext="G\setminus\{a,b\}" class="ltx_Math" display="inline" id="S4.I3.i1.p1.1.m1.2"><semantics id="S4.I3.i1.p1.1.m1.2a"><mrow id="S4.I3.i1.p1.1.m1.2.3" xref="S4.I3.i1.p1.1.m1.2.3.cmml"><mi id="S4.I3.i1.p1.1.m1.2.3.2" xref="S4.I3.i1.p1.1.m1.2.3.2.cmml">G</mi><mo id="S4.I3.i1.p1.1.m1.2.3.1" xref="S4.I3.i1.p1.1.m1.2.3.1.cmml">∖</mo><mrow id="S4.I3.i1.p1.1.m1.2.3.3.2" xref="S4.I3.i1.p1.1.m1.2.3.3.1.cmml"><mo id="S4.I3.i1.p1.1.m1.2.3.3.2.1" stretchy="false" xref="S4.I3.i1.p1.1.m1.2.3.3.1.cmml">{</mo><mi id="S4.I3.i1.p1.1.m1.1.1" xref="S4.I3.i1.p1.1.m1.1.1.cmml">a</mi><mo id="S4.I3.i1.p1.1.m1.2.3.3.2.2" xref="S4.I3.i1.p1.1.m1.2.3.3.1.cmml">,</mo><mi id="S4.I3.i1.p1.1.m1.2.2" xref="S4.I3.i1.p1.1.m1.2.2.cmml">b</mi><mo id="S4.I3.i1.p1.1.m1.2.3.3.2.3" stretchy="false" xref="S4.I3.i1.p1.1.m1.2.3.3.1.cmml">}</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.I3.i1.p1.1.m1.2b"><apply id="S4.I3.i1.p1.1.m1.2.3.cmml" xref="S4.I3.i1.p1.1.m1.2.3"><setdiff id="S4.I3.i1.p1.1.m1.2.3.1.cmml" xref="S4.I3.i1.p1.1.m1.2.3.1"></setdiff><ci id="S4.I3.i1.p1.1.m1.2.3.2.cmml" xref="S4.I3.i1.p1.1.m1.2.3.2">𝐺</ci><set id="S4.I3.i1.p1.1.m1.2.3.3.1.cmml" xref="S4.I3.i1.p1.1.m1.2.3.3.2"><ci id="S4.I3.i1.p1.1.m1.1.1.cmml" xref="S4.I3.i1.p1.1.m1.1.1">𝑎</ci><ci id="S4.I3.i1.p1.1.m1.2.2.cmml" xref="S4.I3.i1.p1.1.m1.2.2">𝑏</ci></set></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.I3.i1.p1.1.m1.2c">G\setminus\{a,b\}</annotation><annotation encoding="application/x-llamapun" id="S4.I3.i1.p1.1.m1.2d">italic_G ∖ { italic_a , italic_b }</annotation></semantics></math> has at least two connected components <math alttext="C_{1},\dots,C_{k}" class="ltx_Math" display="inline" id="S4.I3.i1.p1.2.m2.3"><semantics id="S4.I3.i1.p1.2.m2.3a"><mrow id="S4.I3.i1.p1.2.m2.3.3.2" xref="S4.I3.i1.p1.2.m2.3.3.3.cmml"><msub id="S4.I3.i1.p1.2.m2.2.2.1.1" xref="S4.I3.i1.p1.2.m2.2.2.1.1.cmml"><mi id="S4.I3.i1.p1.2.m2.2.2.1.1.2" xref="S4.I3.i1.p1.2.m2.2.2.1.1.2.cmml">C</mi><mn id="S4.I3.i1.p1.2.m2.2.2.1.1.3" xref="S4.I3.i1.p1.2.m2.2.2.1.1.3.cmml">1</mn></msub><mo id="S4.I3.i1.p1.2.m2.3.3.2.3" xref="S4.I3.i1.p1.2.m2.3.3.3.cmml">,</mo><mi id="S4.I3.i1.p1.2.m2.1.1" mathvariant="normal" xref="S4.I3.i1.p1.2.m2.1.1.cmml">…</mi><mo id="S4.I3.i1.p1.2.m2.3.3.2.4" xref="S4.I3.i1.p1.2.m2.3.3.3.cmml">,</mo><msub id="S4.I3.i1.p1.2.m2.3.3.2.2" xref="S4.I3.i1.p1.2.m2.3.3.2.2.cmml"><mi id="S4.I3.i1.p1.2.m2.3.3.2.2.2" xref="S4.I3.i1.p1.2.m2.3.3.2.2.2.cmml">C</mi><mi id="S4.I3.i1.p1.2.m2.3.3.2.2.3" xref="S4.I3.i1.p1.2.m2.3.3.2.2.3.cmml">k</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S4.I3.i1.p1.2.m2.3b"><list id="S4.I3.i1.p1.2.m2.3.3.3.cmml" xref="S4.I3.i1.p1.2.m2.3.3.2"><apply id="S4.I3.i1.p1.2.m2.2.2.1.1.cmml" xref="S4.I3.i1.p1.2.m2.2.2.1.1"><csymbol cd="ambiguous" id="S4.I3.i1.p1.2.m2.2.2.1.1.1.cmml" xref="S4.I3.i1.p1.2.m2.2.2.1.1">subscript</csymbol><ci id="S4.I3.i1.p1.2.m2.2.2.1.1.2.cmml" xref="S4.I3.i1.p1.2.m2.2.2.1.1.2">𝐶</ci><cn id="S4.I3.i1.p1.2.m2.2.2.1.1.3.cmml" type="integer" xref="S4.I3.i1.p1.2.m2.2.2.1.1.3">1</cn></apply><ci id="S4.I3.i1.p1.2.m2.1.1.cmml" xref="S4.I3.i1.p1.2.m2.1.1">…</ci><apply id="S4.I3.i1.p1.2.m2.3.3.2.2.cmml" xref="S4.I3.i1.p1.2.m2.3.3.2.2"><csymbol cd="ambiguous" id="S4.I3.i1.p1.2.m2.3.3.2.2.1.cmml" xref="S4.I3.i1.p1.2.m2.3.3.2.2">subscript</csymbol><ci id="S4.I3.i1.p1.2.m2.3.3.2.2.2.cmml" xref="S4.I3.i1.p1.2.m2.3.3.2.2.2">𝐶</ci><ci id="S4.I3.i1.p1.2.m2.3.3.2.2.3.cmml" xref="S4.I3.i1.p1.2.m2.3.3.2.2.3">𝑘</ci></apply></list></annotation-xml><annotation encoding="application/x-tex" id="S4.I3.i1.p1.2.m2.3c">C_{1},\dots,C_{k}</annotation><annotation encoding="application/x-llamapun" id="S4.I3.i1.p1.2.m2.3d">italic_C start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , … , italic_C start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT</annotation></semantics></math>, the separation classes are <math alttext="E[C_{i}]\cup E[C_{i},\{a,b\}]" class="ltx_Math" display="inline" id="S4.I3.i1.p1.3.m3.5"><semantics id="S4.I3.i1.p1.3.m3.5a"><mrow id="S4.I3.i1.p1.3.m3.5.5" xref="S4.I3.i1.p1.3.m3.5.5.cmml"><mrow id="S4.I3.i1.p1.3.m3.3.3.1" xref="S4.I3.i1.p1.3.m3.3.3.1.cmml"><mi id="S4.I3.i1.p1.3.m3.3.3.1.3" xref="S4.I3.i1.p1.3.m3.3.3.1.3.cmml">E</mi><mo id="S4.I3.i1.p1.3.m3.3.3.1.2" xref="S4.I3.i1.p1.3.m3.3.3.1.2.cmml">⁢</mo><mrow id="S4.I3.i1.p1.3.m3.3.3.1.1.1" xref="S4.I3.i1.p1.3.m3.3.3.1.1.2.cmml"><mo id="S4.I3.i1.p1.3.m3.3.3.1.1.1.2" stretchy="false" xref="S4.I3.i1.p1.3.m3.3.3.1.1.2.1.cmml">[</mo><msub id="S4.I3.i1.p1.3.m3.3.3.1.1.1.1" xref="S4.I3.i1.p1.3.m3.3.3.1.1.1.1.cmml"><mi id="S4.I3.i1.p1.3.m3.3.3.1.1.1.1.2" xref="S4.I3.i1.p1.3.m3.3.3.1.1.1.1.2.cmml">C</mi><mi id="S4.I3.i1.p1.3.m3.3.3.1.1.1.1.3" xref="S4.I3.i1.p1.3.m3.3.3.1.1.1.1.3.cmml">i</mi></msub><mo id="S4.I3.i1.p1.3.m3.3.3.1.1.1.3" stretchy="false" xref="S4.I3.i1.p1.3.m3.3.3.1.1.2.1.cmml">]</mo></mrow></mrow><mo id="S4.I3.i1.p1.3.m3.5.5.4" xref="S4.I3.i1.p1.3.m3.5.5.4.cmml">∪</mo><mrow id="S4.I3.i1.p1.3.m3.5.5.3" xref="S4.I3.i1.p1.3.m3.5.5.3.cmml"><mi id="S4.I3.i1.p1.3.m3.5.5.3.4" xref="S4.I3.i1.p1.3.m3.5.5.3.4.cmml">E</mi><mo id="S4.I3.i1.p1.3.m3.5.5.3.3" xref="S4.I3.i1.p1.3.m3.5.5.3.3.cmml">⁢</mo><mrow id="S4.I3.i1.p1.3.m3.5.5.3.2.2" xref="S4.I3.i1.p1.3.m3.5.5.3.2.3.cmml"><mo id="S4.I3.i1.p1.3.m3.5.5.3.2.2.3" stretchy="false" xref="S4.I3.i1.p1.3.m3.5.5.3.2.3.cmml">[</mo><msub id="S4.I3.i1.p1.3.m3.4.4.2.1.1.1" xref="S4.I3.i1.p1.3.m3.4.4.2.1.1.1.cmml"><mi id="S4.I3.i1.p1.3.m3.4.4.2.1.1.1.2" xref="S4.I3.i1.p1.3.m3.4.4.2.1.1.1.2.cmml">C</mi><mi id="S4.I3.i1.p1.3.m3.4.4.2.1.1.1.3" xref="S4.I3.i1.p1.3.m3.4.4.2.1.1.1.3.cmml">i</mi></msub><mo id="S4.I3.i1.p1.3.m3.5.5.3.2.2.4" xref="S4.I3.i1.p1.3.m3.5.5.3.2.3.cmml">,</mo><mrow id="S4.I3.i1.p1.3.m3.5.5.3.2.2.2.2" xref="S4.I3.i1.p1.3.m3.5.5.3.2.2.2.1.cmml"><mo id="S4.I3.i1.p1.3.m3.5.5.3.2.2.2.2.1" stretchy="false" xref="S4.I3.i1.p1.3.m3.5.5.3.2.2.2.1.cmml">{</mo><mi id="S4.I3.i1.p1.3.m3.1.1" xref="S4.I3.i1.p1.3.m3.1.1.cmml">a</mi><mo id="S4.I3.i1.p1.3.m3.5.5.3.2.2.2.2.2" xref="S4.I3.i1.p1.3.m3.5.5.3.2.2.2.1.cmml">,</mo><mi id="S4.I3.i1.p1.3.m3.2.2" xref="S4.I3.i1.p1.3.m3.2.2.cmml">b</mi><mo id="S4.I3.i1.p1.3.m3.5.5.3.2.2.2.2.3" stretchy="false" xref="S4.I3.i1.p1.3.m3.5.5.3.2.2.2.1.cmml">}</mo></mrow><mo id="S4.I3.i1.p1.3.m3.5.5.3.2.2.5" stretchy="false" xref="S4.I3.i1.p1.3.m3.5.5.3.2.3.cmml">]</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.I3.i1.p1.3.m3.5b"><apply id="S4.I3.i1.p1.3.m3.5.5.cmml" xref="S4.I3.i1.p1.3.m3.5.5"><union id="S4.I3.i1.p1.3.m3.5.5.4.cmml" xref="S4.I3.i1.p1.3.m3.5.5.4"></union><apply id="S4.I3.i1.p1.3.m3.3.3.1.cmml" xref="S4.I3.i1.p1.3.m3.3.3.1"><times id="S4.I3.i1.p1.3.m3.3.3.1.2.cmml" xref="S4.I3.i1.p1.3.m3.3.3.1.2"></times><ci id="S4.I3.i1.p1.3.m3.3.3.1.3.cmml" xref="S4.I3.i1.p1.3.m3.3.3.1.3">𝐸</ci><apply id="S4.I3.i1.p1.3.m3.3.3.1.1.2.cmml" xref="S4.I3.i1.p1.3.m3.3.3.1.1.1"><csymbol cd="latexml" id="S4.I3.i1.p1.3.m3.3.3.1.1.2.1.cmml" xref="S4.I3.i1.p1.3.m3.3.3.1.1.1.2">delimited-[]</csymbol><apply id="S4.I3.i1.p1.3.m3.3.3.1.1.1.1.cmml" xref="S4.I3.i1.p1.3.m3.3.3.1.1.1.1"><csymbol cd="ambiguous" id="S4.I3.i1.p1.3.m3.3.3.1.1.1.1.1.cmml" xref="S4.I3.i1.p1.3.m3.3.3.1.1.1.1">subscript</csymbol><ci id="S4.I3.i1.p1.3.m3.3.3.1.1.1.1.2.cmml" xref="S4.I3.i1.p1.3.m3.3.3.1.1.1.1.2">𝐶</ci><ci id="S4.I3.i1.p1.3.m3.3.3.1.1.1.1.3.cmml" xref="S4.I3.i1.p1.3.m3.3.3.1.1.1.1.3">𝑖</ci></apply></apply></apply><apply id="S4.I3.i1.p1.3.m3.5.5.3.cmml" xref="S4.I3.i1.p1.3.m3.5.5.3"><times id="S4.I3.i1.p1.3.m3.5.5.3.3.cmml" xref="S4.I3.i1.p1.3.m3.5.5.3.3"></times><ci id="S4.I3.i1.p1.3.m3.5.5.3.4.cmml" xref="S4.I3.i1.p1.3.m3.5.5.3.4">𝐸</ci><interval closure="closed" id="S4.I3.i1.p1.3.m3.5.5.3.2.3.cmml" xref="S4.I3.i1.p1.3.m3.5.5.3.2.2"><apply id="S4.I3.i1.p1.3.m3.4.4.2.1.1.1.cmml" xref="S4.I3.i1.p1.3.m3.4.4.2.1.1.1"><csymbol cd="ambiguous" id="S4.I3.i1.p1.3.m3.4.4.2.1.1.1.1.cmml" xref="S4.I3.i1.p1.3.m3.4.4.2.1.1.1">subscript</csymbol><ci id="S4.I3.i1.p1.3.m3.4.4.2.1.1.1.2.cmml" xref="S4.I3.i1.p1.3.m3.4.4.2.1.1.1.2">𝐶</ci><ci id="S4.I3.i1.p1.3.m3.4.4.2.1.1.1.3.cmml" xref="S4.I3.i1.p1.3.m3.4.4.2.1.1.1.3">𝑖</ci></apply><set id="S4.I3.i1.p1.3.m3.5.5.3.2.2.2.1.cmml" xref="S4.I3.i1.p1.3.m3.5.5.3.2.2.2.2"><ci id="S4.I3.i1.p1.3.m3.1.1.cmml" xref="S4.I3.i1.p1.3.m3.1.1">𝑎</ci><ci id="S4.I3.i1.p1.3.m3.2.2.cmml" xref="S4.I3.i1.p1.3.m3.2.2">𝑏</ci></set></interval></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.I3.i1.p1.3.m3.5c">E[C_{i}]\cup E[C_{i},\{a,b\}]</annotation><annotation encoding="application/x-llamapun" id="S4.I3.i1.p1.3.m3.5d">italic_E [ italic_C start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ] ∪ italic_E [ italic_C start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT , { italic_a , italic_b } ]</annotation></semantics></math> for each <math alttext="i\in[k]" class="ltx_Math" display="inline" id="S4.I3.i1.p1.4.m4.1"><semantics id="S4.I3.i1.p1.4.m4.1a"><mrow id="S4.I3.i1.p1.4.m4.1.2" xref="S4.I3.i1.p1.4.m4.1.2.cmml"><mi id="S4.I3.i1.p1.4.m4.1.2.2" xref="S4.I3.i1.p1.4.m4.1.2.2.cmml">i</mi><mo id="S4.I3.i1.p1.4.m4.1.2.1" xref="S4.I3.i1.p1.4.m4.1.2.1.cmml">∈</mo><mrow id="S4.I3.i1.p1.4.m4.1.2.3.2" xref="S4.I3.i1.p1.4.m4.1.2.3.1.cmml"><mo id="S4.I3.i1.p1.4.m4.1.2.3.2.1" stretchy="false" xref="S4.I3.i1.p1.4.m4.1.2.3.1.1.cmml">[</mo><mi id="S4.I3.i1.p1.4.m4.1.1" xref="S4.I3.i1.p1.4.m4.1.1.cmml">k</mi><mo id="S4.I3.i1.p1.4.m4.1.2.3.2.2" stretchy="false" xref="S4.I3.i1.p1.4.m4.1.2.3.1.1.cmml">]</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.I3.i1.p1.4.m4.1b"><apply id="S4.I3.i1.p1.4.m4.1.2.cmml" xref="S4.I3.i1.p1.4.m4.1.2"><in id="S4.I3.i1.p1.4.m4.1.2.1.cmml" xref="S4.I3.i1.p1.4.m4.1.2.1"></in><ci id="S4.I3.i1.p1.4.m4.1.2.2.cmml" xref="S4.I3.i1.p1.4.m4.1.2.2">𝑖</ci><apply id="S4.I3.i1.p1.4.m4.1.2.3.1.cmml" xref="S4.I3.i1.p1.4.m4.1.2.3.2"><csymbol cd="latexml" id="S4.I3.i1.p1.4.m4.1.2.3.1.1.cmml" xref="S4.I3.i1.p1.4.m4.1.2.3.2.1">delimited-[]</csymbol><ci id="S4.I3.i1.p1.4.m4.1.1.cmml" xref="S4.I3.i1.p1.4.m4.1.1">𝑘</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.I3.i1.p1.4.m4.1c">i\in[k]</annotation><annotation encoding="application/x-llamapun" id="S4.I3.i1.p1.4.m4.1d">italic_i ∈ [ italic_k ]</annotation></semantics></math>. If there are any edges of the form <math alttext="ab" class="ltx_Math" display="inline" id="S4.I3.i1.p1.5.m5.1"><semantics id="S4.I3.i1.p1.5.m5.1a"><mrow id="S4.I3.i1.p1.5.m5.1.1" xref="S4.I3.i1.p1.5.m5.1.1.cmml"><mi id="S4.I3.i1.p1.5.m5.1.1.2" xref="S4.I3.i1.p1.5.m5.1.1.2.cmml">a</mi><mo id="S4.I3.i1.p1.5.m5.1.1.1" xref="S4.I3.i1.p1.5.m5.1.1.1.cmml">⁢</mo><mi id="S4.I3.i1.p1.5.m5.1.1.3" xref="S4.I3.i1.p1.5.m5.1.1.3.cmml">b</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.I3.i1.p1.5.m5.1b"><apply id="S4.I3.i1.p1.5.m5.1.1.cmml" xref="S4.I3.i1.p1.5.m5.1.1"><times id="S4.I3.i1.p1.5.m5.1.1.1.cmml" xref="S4.I3.i1.p1.5.m5.1.1.1"></times><ci id="S4.I3.i1.p1.5.m5.1.1.2.cmml" xref="S4.I3.i1.p1.5.m5.1.1.2">𝑎</ci><ci id="S4.I3.i1.p1.5.m5.1.1.3.cmml" xref="S4.I3.i1.p1.5.m5.1.1.3">𝑏</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.I3.i1.p1.5.m5.1c">ab</annotation><annotation encoding="application/x-llamapun" id="S4.I3.i1.p1.5.m5.1d">italic_a italic_b</annotation></semantics></math>, we add an additional separation class <math alttext="E^{\prime}" class="ltx_Math" display="inline" id="S4.I3.i1.p1.6.m6.1"><semantics id="S4.I3.i1.p1.6.m6.1a"><msup id="S4.I3.i1.p1.6.m6.1.1" xref="S4.I3.i1.p1.6.m6.1.1.cmml"><mi id="S4.I3.i1.p1.6.m6.1.1.2" xref="S4.I3.i1.p1.6.m6.1.1.2.cmml">E</mi><mo id="S4.I3.i1.p1.6.m6.1.1.3" xref="S4.I3.i1.p1.6.m6.1.1.3.cmml">′</mo></msup><annotation-xml encoding="MathML-Content" id="S4.I3.i1.p1.6.m6.1b"><apply id="S4.I3.i1.p1.6.m6.1.1.cmml" xref="S4.I3.i1.p1.6.m6.1.1"><csymbol cd="ambiguous" id="S4.I3.i1.p1.6.m6.1.1.1.cmml" xref="S4.I3.i1.p1.6.m6.1.1">superscript</csymbol><ci id="S4.I3.i1.p1.6.m6.1.1.2.cmml" xref="S4.I3.i1.p1.6.m6.1.1.2">𝐸</ci><ci id="S4.I3.i1.p1.6.m6.1.1.3.cmml" xref="S4.I3.i1.p1.6.m6.1.1.3">′</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.I3.i1.p1.6.m6.1c">E^{\prime}</annotation><annotation encoding="application/x-llamapun" id="S4.I3.i1.p1.6.m6.1d">italic_E start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math> containing all such parallel edges between <math alttext="a" class="ltx_Math" display="inline" id="S4.I3.i1.p1.7.m7.1"><semantics id="S4.I3.i1.p1.7.m7.1a"><mi id="S4.I3.i1.p1.7.m7.1.1" xref="S4.I3.i1.p1.7.m7.1.1.cmml">a</mi><annotation-xml encoding="MathML-Content" id="S4.I3.i1.p1.7.m7.1b"><ci id="S4.I3.i1.p1.7.m7.1.1.cmml" xref="S4.I3.i1.p1.7.m7.1.1">𝑎</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.I3.i1.p1.7.m7.1c">a</annotation><annotation encoding="application/x-llamapun" id="S4.I3.i1.p1.7.m7.1d">italic_a</annotation></semantics></math> and <math alttext="b" class="ltx_Math" display="inline" id="S4.I3.i1.p1.8.m8.1"><semantics id="S4.I3.i1.p1.8.m8.1a"><mi id="S4.I3.i1.p1.8.m8.1.1" xref="S4.I3.i1.p1.8.m8.1.1.cmml">b</mi><annotation-xml encoding="MathML-Content" id="S4.I3.i1.p1.8.m8.1b"><ci id="S4.I3.i1.p1.8.m8.1.1.cmml" xref="S4.I3.i1.p1.8.m8.1.1">𝑏</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.I3.i1.p1.8.m8.1c">b</annotation><annotation encoding="application/x-llamapun" id="S4.I3.i1.p1.8.m8.1d">italic_b</annotation></semantics></math>.</p> </div> </li> <li class="ltx_item" id="S4.I3.i2" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">2.</span> <div class="ltx_para" id="S4.I3.i2.p1"> <p class="ltx_p" id="S4.I3.i2.p1.6">If not, the separation classes are <math alttext="E_{1}=\{e:e=ab\}" class="ltx_Math" display="inline" id="S4.I3.i2.p1.1.m1.2"><semantics id="S4.I3.i2.p1.1.m1.2a"><mrow id="S4.I3.i2.p1.1.m1.2.2" xref="S4.I3.i2.p1.1.m1.2.2.cmml"><msub id="S4.I3.i2.p1.1.m1.2.2.3" xref="S4.I3.i2.p1.1.m1.2.2.3.cmml"><mi id="S4.I3.i2.p1.1.m1.2.2.3.2" xref="S4.I3.i2.p1.1.m1.2.2.3.2.cmml">E</mi><mn id="S4.I3.i2.p1.1.m1.2.2.3.3" xref="S4.I3.i2.p1.1.m1.2.2.3.3.cmml">1</mn></msub><mo id="S4.I3.i2.p1.1.m1.2.2.2" xref="S4.I3.i2.p1.1.m1.2.2.2.cmml">=</mo><mrow id="S4.I3.i2.p1.1.m1.2.2.1.1" xref="S4.I3.i2.p1.1.m1.2.2.1.2.cmml"><mo id="S4.I3.i2.p1.1.m1.2.2.1.1.2" stretchy="false" xref="S4.I3.i2.p1.1.m1.2.2.1.2.1.cmml">{</mo><mi id="S4.I3.i2.p1.1.m1.1.1" xref="S4.I3.i2.p1.1.m1.1.1.cmml">e</mi><mo id="S4.I3.i2.p1.1.m1.2.2.1.1.3" lspace="0.278em" rspace="0.278em" xref="S4.I3.i2.p1.1.m1.2.2.1.2.1.cmml">:</mo><mrow id="S4.I3.i2.p1.1.m1.2.2.1.1.1" xref="S4.I3.i2.p1.1.m1.2.2.1.1.1.cmml"><mi id="S4.I3.i2.p1.1.m1.2.2.1.1.1.2" xref="S4.I3.i2.p1.1.m1.2.2.1.1.1.2.cmml">e</mi><mo id="S4.I3.i2.p1.1.m1.2.2.1.1.1.1" xref="S4.I3.i2.p1.1.m1.2.2.1.1.1.1.cmml">=</mo><mrow id="S4.I3.i2.p1.1.m1.2.2.1.1.1.3" xref="S4.I3.i2.p1.1.m1.2.2.1.1.1.3.cmml"><mi id="S4.I3.i2.p1.1.m1.2.2.1.1.1.3.2" xref="S4.I3.i2.p1.1.m1.2.2.1.1.1.3.2.cmml">a</mi><mo id="S4.I3.i2.p1.1.m1.2.2.1.1.1.3.1" xref="S4.I3.i2.p1.1.m1.2.2.1.1.1.3.1.cmml">⁢</mo><mi id="S4.I3.i2.p1.1.m1.2.2.1.1.1.3.3" xref="S4.I3.i2.p1.1.m1.2.2.1.1.1.3.3.cmml">b</mi></mrow></mrow><mo id="S4.I3.i2.p1.1.m1.2.2.1.1.4" stretchy="false" xref="S4.I3.i2.p1.1.m1.2.2.1.2.1.cmml">}</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.I3.i2.p1.1.m1.2b"><apply id="S4.I3.i2.p1.1.m1.2.2.cmml" xref="S4.I3.i2.p1.1.m1.2.2"><eq id="S4.I3.i2.p1.1.m1.2.2.2.cmml" xref="S4.I3.i2.p1.1.m1.2.2.2"></eq><apply id="S4.I3.i2.p1.1.m1.2.2.3.cmml" xref="S4.I3.i2.p1.1.m1.2.2.3"><csymbol cd="ambiguous" id="S4.I3.i2.p1.1.m1.2.2.3.1.cmml" xref="S4.I3.i2.p1.1.m1.2.2.3">subscript</csymbol><ci id="S4.I3.i2.p1.1.m1.2.2.3.2.cmml" xref="S4.I3.i2.p1.1.m1.2.2.3.2">𝐸</ci><cn id="S4.I3.i2.p1.1.m1.2.2.3.3.cmml" type="integer" xref="S4.I3.i2.p1.1.m1.2.2.3.3">1</cn></apply><apply id="S4.I3.i2.p1.1.m1.2.2.1.2.cmml" xref="S4.I3.i2.p1.1.m1.2.2.1.1"><csymbol cd="latexml" id="S4.I3.i2.p1.1.m1.2.2.1.2.1.cmml" xref="S4.I3.i2.p1.1.m1.2.2.1.1.2">conditional-set</csymbol><ci id="S4.I3.i2.p1.1.m1.1.1.cmml" xref="S4.I3.i2.p1.1.m1.1.1">𝑒</ci><apply id="S4.I3.i2.p1.1.m1.2.2.1.1.1.cmml" xref="S4.I3.i2.p1.1.m1.2.2.1.1.1"><eq id="S4.I3.i2.p1.1.m1.2.2.1.1.1.1.cmml" xref="S4.I3.i2.p1.1.m1.2.2.1.1.1.1"></eq><ci id="S4.I3.i2.p1.1.m1.2.2.1.1.1.2.cmml" xref="S4.I3.i2.p1.1.m1.2.2.1.1.1.2">𝑒</ci><apply id="S4.I3.i2.p1.1.m1.2.2.1.1.1.3.cmml" xref="S4.I3.i2.p1.1.m1.2.2.1.1.1.3"><times id="S4.I3.i2.p1.1.m1.2.2.1.1.1.3.1.cmml" xref="S4.I3.i2.p1.1.m1.2.2.1.1.1.3.1"></times><ci id="S4.I3.i2.p1.1.m1.2.2.1.1.1.3.2.cmml" xref="S4.I3.i2.p1.1.m1.2.2.1.1.1.3.2">𝑎</ci><ci id="S4.I3.i2.p1.1.m1.2.2.1.1.1.3.3.cmml" xref="S4.I3.i2.p1.1.m1.2.2.1.1.1.3.3">𝑏</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.I3.i2.p1.1.m1.2c">E_{1}=\{e:e=ab\}</annotation><annotation encoding="application/x-llamapun" id="S4.I3.i2.p1.1.m1.2d">italic_E start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT = { italic_e : italic_e = italic_a italic_b }</annotation></semantics></math> (all parallel edges between <math alttext="a" class="ltx_Math" display="inline" id="S4.I3.i2.p1.2.m2.1"><semantics id="S4.I3.i2.p1.2.m2.1a"><mi id="S4.I3.i2.p1.2.m2.1.1" xref="S4.I3.i2.p1.2.m2.1.1.cmml">a</mi><annotation-xml encoding="MathML-Content" id="S4.I3.i2.p1.2.m2.1b"><ci id="S4.I3.i2.p1.2.m2.1.1.cmml" xref="S4.I3.i2.p1.2.m2.1.1">𝑎</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.I3.i2.p1.2.m2.1c">a</annotation><annotation encoding="application/x-llamapun" id="S4.I3.i2.p1.2.m2.1d">italic_a</annotation></semantics></math> and <math alttext="b" class="ltx_Math" display="inline" id="S4.I3.i2.p1.3.m3.1"><semantics id="S4.I3.i2.p1.3.m3.1a"><mi id="S4.I3.i2.p1.3.m3.1.1" xref="S4.I3.i2.p1.3.m3.1.1.cmml">b</mi><annotation-xml encoding="MathML-Content" id="S4.I3.i2.p1.3.m3.1b"><ci id="S4.I3.i2.p1.3.m3.1.1.cmml" xref="S4.I3.i2.p1.3.m3.1.1">𝑏</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.I3.i2.p1.3.m3.1c">b</annotation><annotation encoding="application/x-llamapun" id="S4.I3.i2.p1.3.m3.1d">italic_b</annotation></semantics></math>) and <math alttext="E_{2}=E(G)\setminus E_{1}" class="ltx_Math" display="inline" id="S4.I3.i2.p1.4.m4.1"><semantics id="S4.I3.i2.p1.4.m4.1a"><mrow id="S4.I3.i2.p1.4.m4.1.2" xref="S4.I3.i2.p1.4.m4.1.2.cmml"><msub id="S4.I3.i2.p1.4.m4.1.2.2" xref="S4.I3.i2.p1.4.m4.1.2.2.cmml"><mi id="S4.I3.i2.p1.4.m4.1.2.2.2" xref="S4.I3.i2.p1.4.m4.1.2.2.2.cmml">E</mi><mn id="S4.I3.i2.p1.4.m4.1.2.2.3" xref="S4.I3.i2.p1.4.m4.1.2.2.3.cmml">2</mn></msub><mo id="S4.I3.i2.p1.4.m4.1.2.1" xref="S4.I3.i2.p1.4.m4.1.2.1.cmml">=</mo><mrow id="S4.I3.i2.p1.4.m4.1.2.3" xref="S4.I3.i2.p1.4.m4.1.2.3.cmml"><mrow id="S4.I3.i2.p1.4.m4.1.2.3.2" xref="S4.I3.i2.p1.4.m4.1.2.3.2.cmml"><mi id="S4.I3.i2.p1.4.m4.1.2.3.2.2" xref="S4.I3.i2.p1.4.m4.1.2.3.2.2.cmml">E</mi><mo id="S4.I3.i2.p1.4.m4.1.2.3.2.1" xref="S4.I3.i2.p1.4.m4.1.2.3.2.1.cmml">⁢</mo><mrow id="S4.I3.i2.p1.4.m4.1.2.3.2.3.2" xref="S4.I3.i2.p1.4.m4.1.2.3.2.cmml"><mo id="S4.I3.i2.p1.4.m4.1.2.3.2.3.2.1" stretchy="false" xref="S4.I3.i2.p1.4.m4.1.2.3.2.cmml">(</mo><mi id="S4.I3.i2.p1.4.m4.1.1" xref="S4.I3.i2.p1.4.m4.1.1.cmml">G</mi><mo id="S4.I3.i2.p1.4.m4.1.2.3.2.3.2.2" stretchy="false" xref="S4.I3.i2.p1.4.m4.1.2.3.2.cmml">)</mo></mrow></mrow><mo id="S4.I3.i2.p1.4.m4.1.2.3.1" xref="S4.I3.i2.p1.4.m4.1.2.3.1.cmml">∖</mo><msub id="S4.I3.i2.p1.4.m4.1.2.3.3" xref="S4.I3.i2.p1.4.m4.1.2.3.3.cmml"><mi id="S4.I3.i2.p1.4.m4.1.2.3.3.2" xref="S4.I3.i2.p1.4.m4.1.2.3.3.2.cmml">E</mi><mn id="S4.I3.i2.p1.4.m4.1.2.3.3.3" xref="S4.I3.i2.p1.4.m4.1.2.3.3.3.cmml">1</mn></msub></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.I3.i2.p1.4.m4.1b"><apply id="S4.I3.i2.p1.4.m4.1.2.cmml" xref="S4.I3.i2.p1.4.m4.1.2"><eq id="S4.I3.i2.p1.4.m4.1.2.1.cmml" xref="S4.I3.i2.p1.4.m4.1.2.1"></eq><apply id="S4.I3.i2.p1.4.m4.1.2.2.cmml" xref="S4.I3.i2.p1.4.m4.1.2.2"><csymbol cd="ambiguous" id="S4.I3.i2.p1.4.m4.1.2.2.1.cmml" xref="S4.I3.i2.p1.4.m4.1.2.2">subscript</csymbol><ci id="S4.I3.i2.p1.4.m4.1.2.2.2.cmml" xref="S4.I3.i2.p1.4.m4.1.2.2.2">𝐸</ci><cn id="S4.I3.i2.p1.4.m4.1.2.2.3.cmml" type="integer" xref="S4.I3.i2.p1.4.m4.1.2.2.3">2</cn></apply><apply id="S4.I3.i2.p1.4.m4.1.2.3.cmml" xref="S4.I3.i2.p1.4.m4.1.2.3"><setdiff id="S4.I3.i2.p1.4.m4.1.2.3.1.cmml" xref="S4.I3.i2.p1.4.m4.1.2.3.1"></setdiff><apply id="S4.I3.i2.p1.4.m4.1.2.3.2.cmml" xref="S4.I3.i2.p1.4.m4.1.2.3.2"><times id="S4.I3.i2.p1.4.m4.1.2.3.2.1.cmml" xref="S4.I3.i2.p1.4.m4.1.2.3.2.1"></times><ci id="S4.I3.i2.p1.4.m4.1.2.3.2.2.cmml" xref="S4.I3.i2.p1.4.m4.1.2.3.2.2">𝐸</ci><ci id="S4.I3.i2.p1.4.m4.1.1.cmml" xref="S4.I3.i2.p1.4.m4.1.1">𝐺</ci></apply><apply id="S4.I3.i2.p1.4.m4.1.2.3.3.cmml" xref="S4.I3.i2.p1.4.m4.1.2.3.3"><csymbol cd="ambiguous" id="S4.I3.i2.p1.4.m4.1.2.3.3.1.cmml" xref="S4.I3.i2.p1.4.m4.1.2.3.3">subscript</csymbol><ci id="S4.I3.i2.p1.4.m4.1.2.3.3.2.cmml" xref="S4.I3.i2.p1.4.m4.1.2.3.3.2">𝐸</ci><cn id="S4.I3.i2.p1.4.m4.1.2.3.3.3.cmml" type="integer" xref="S4.I3.i2.p1.4.m4.1.2.3.3.3">1</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.I3.i2.p1.4.m4.1c">E_{2}=E(G)\setminus E_{1}</annotation><annotation encoding="application/x-llamapun" id="S4.I3.i2.p1.4.m4.1d">italic_E start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT = italic_E ( italic_G ) ∖ italic_E start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math> (all edges except parallel edges between <math alttext="a" class="ltx_Math" display="inline" id="S4.I3.i2.p1.5.m5.1"><semantics id="S4.I3.i2.p1.5.m5.1a"><mi id="S4.I3.i2.p1.5.m5.1.1" xref="S4.I3.i2.p1.5.m5.1.1.cmml">a</mi><annotation-xml encoding="MathML-Content" id="S4.I3.i2.p1.5.m5.1b"><ci id="S4.I3.i2.p1.5.m5.1.1.cmml" xref="S4.I3.i2.p1.5.m5.1.1">𝑎</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.I3.i2.p1.5.m5.1c">a</annotation><annotation encoding="application/x-llamapun" id="S4.I3.i2.p1.5.m5.1d">italic_a</annotation></semantics></math> and <math alttext="b" class="ltx_Math" display="inline" id="S4.I3.i2.p1.6.m6.1"><semantics id="S4.I3.i2.p1.6.m6.1a"><mi id="S4.I3.i2.p1.6.m6.1.1" xref="S4.I3.i2.p1.6.m6.1.1.cmml">b</mi><annotation-xml encoding="MathML-Content" id="S4.I3.i2.p1.6.m6.1b"><ci id="S4.I3.i2.p1.6.m6.1.1.cmml" xref="S4.I3.i2.p1.6.m6.1.1">𝑏</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.I3.i2.p1.6.m6.1c">b</annotation><annotation encoding="application/x-llamapun" id="S4.I3.i2.p1.6.m6.1d">italic_b</annotation></semantics></math>).</p> </div> </li> </ol> <p class="ltx_p" id="S4.Thmtheorem8.p1.3">Note that the separation classes partition the edge set <math alttext="E(G)" class="ltx_Math" display="inline" id="S4.Thmtheorem8.p1.3.m1.1"><semantics id="S4.Thmtheorem8.p1.3.m1.1a"><mrow id="S4.Thmtheorem8.p1.3.m1.1.2" xref="S4.Thmtheorem8.p1.3.m1.1.2.cmml"><mi id="S4.Thmtheorem8.p1.3.m1.1.2.2" xref="S4.Thmtheorem8.p1.3.m1.1.2.2.cmml">E</mi><mo id="S4.Thmtheorem8.p1.3.m1.1.2.1" xref="S4.Thmtheorem8.p1.3.m1.1.2.1.cmml">⁢</mo><mrow id="S4.Thmtheorem8.p1.3.m1.1.2.3.2" xref="S4.Thmtheorem8.p1.3.m1.1.2.cmml"><mo id="S4.Thmtheorem8.p1.3.m1.1.2.3.2.1" stretchy="false" xref="S4.Thmtheorem8.p1.3.m1.1.2.cmml">(</mo><mi id="S4.Thmtheorem8.p1.3.m1.1.1" xref="S4.Thmtheorem8.p1.3.m1.1.1.cmml">G</mi><mo id="S4.Thmtheorem8.p1.3.m1.1.2.3.2.2" stretchy="false" xref="S4.Thmtheorem8.p1.3.m1.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem8.p1.3.m1.1b"><apply id="S4.Thmtheorem8.p1.3.m1.1.2.cmml" xref="S4.Thmtheorem8.p1.3.m1.1.2"><times id="S4.Thmtheorem8.p1.3.m1.1.2.1.cmml" xref="S4.Thmtheorem8.p1.3.m1.1.2.1"></times><ci id="S4.Thmtheorem8.p1.3.m1.1.2.2.cmml" xref="S4.Thmtheorem8.p1.3.m1.1.2.2">𝐸</ci><ci id="S4.Thmtheorem8.p1.3.m1.1.1.cmml" xref="S4.Thmtheorem8.p1.3.m1.1.1">𝐺</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem8.p1.3.m1.1c">E(G)</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem8.p1.3.m1.1d">italic_E ( italic_G )</annotation></semantics></math>. See Figure <a class="ltx_ref" href="https://arxiv.org/html/2503.00712v1#S4.F4" title="Figure 4 ‣ 4.2.1 SPQR Trees ‣ 4.2 Two-to-Three Augmentation ‣ 4 Vertex Connectivity Augmentation in Link-Arrival Model ‣ Streaming Algorithms for Network Design"><span class="ltx_text ltx_ref_tag">4</span></a> for an example.</p> </div> </div> <figure class="ltx_figure" id="S4.F4"> <div class="ltx_flex_figure"> <div class="ltx_flex_cell ltx_flex_size_2"> <figure class="ltx_figure ltx_figure_panel ltx_align_center" id="S4.F4.sf1"><img alt="Refer to caption" class="ltx_graphics ltx_centering ltx_img_landscape" height="318" id="S4.F4.sf1.g1" src="x3.png" width="581"/> <figcaption class="ltx_caption ltx_centering"><span class="ltx_tag ltx_tag_figure"><span class="ltx_text" id="S4.F4.sf1.20.9.1" style="font-size:90%;">(a)</span> </span><math alttext="G\setminus\{a,b\}" class="ltx_Math" display="inline" id="S4.F4.sf1.10.m1.2"><semantics id="S4.F4.sf1.10.m1.2b"><mrow id="S4.F4.sf1.10.m1.2.3" xref="S4.F4.sf1.10.m1.2.3.cmml"><mi id="S4.F4.sf1.10.m1.2.3.2" mathsize="90%" xref="S4.F4.sf1.10.m1.2.3.2.cmml">G</mi><mo id="S4.F4.sf1.10.m1.2.3.1" mathsize="90%" xref="S4.F4.sf1.10.m1.2.3.1.cmml">∖</mo><mrow id="S4.F4.sf1.10.m1.2.3.3.2" xref="S4.F4.sf1.10.m1.2.3.3.1.cmml"><mo id="S4.F4.sf1.10.m1.2.3.3.2.1" maxsize="90%" minsize="90%" xref="S4.F4.sf1.10.m1.2.3.3.1.cmml">{</mo><mi id="S4.F4.sf1.10.m1.1.1" mathsize="90%" xref="S4.F4.sf1.10.m1.1.1.cmml">a</mi><mo id="S4.F4.sf1.10.m1.2.3.3.2.2" mathsize="90%" xref="S4.F4.sf1.10.m1.2.3.3.1.cmml">,</mo><mi id="S4.F4.sf1.10.m1.2.2" mathsize="90%" xref="S4.F4.sf1.10.m1.2.2.cmml">b</mi><mo id="S4.F4.sf1.10.m1.2.3.3.2.3" maxsize="90%" minsize="90%" xref="S4.F4.sf1.10.m1.2.3.3.1.cmml">}</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.F4.sf1.10.m1.2c"><apply id="S4.F4.sf1.10.m1.2.3.cmml" xref="S4.F4.sf1.10.m1.2.3"><setdiff id="S4.F4.sf1.10.m1.2.3.1.cmml" xref="S4.F4.sf1.10.m1.2.3.1"></setdiff><ci id="S4.F4.sf1.10.m1.2.3.2.cmml" xref="S4.F4.sf1.10.m1.2.3.2">𝐺</ci><set id="S4.F4.sf1.10.m1.2.3.3.1.cmml" xref="S4.F4.sf1.10.m1.2.3.3.2"><ci id="S4.F4.sf1.10.m1.1.1.cmml" xref="S4.F4.sf1.10.m1.1.1">𝑎</ci><ci id="S4.F4.sf1.10.m1.2.2.cmml" xref="S4.F4.sf1.10.m1.2.2">𝑏</ci></set></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.F4.sf1.10.m1.2d">G\setminus\{a,b\}</annotation><annotation encoding="application/x-llamapun" id="S4.F4.sf1.10.m1.2e">italic_G ∖ { italic_a , italic_b }</annotation></semantics></math><span class="ltx_text" id="S4.F4.sf1.18.8" style="font-size:90%;"> has two components: <math alttext="C_{1},C_{2}" class="ltx_Math" display="inline" id="S4.F4.sf1.11.1.m1.2"><semantics id="S4.F4.sf1.11.1.m1.2b"><mrow id="S4.F4.sf1.11.1.m1.2.2.2" xref="S4.F4.sf1.11.1.m1.2.2.3.cmml"><msub id="S4.F4.sf1.11.1.m1.1.1.1.1" xref="S4.F4.sf1.11.1.m1.1.1.1.1.cmml"><mi id="S4.F4.sf1.11.1.m1.1.1.1.1.2" xref="S4.F4.sf1.11.1.m1.1.1.1.1.2.cmml">C</mi><mn id="S4.F4.sf1.11.1.m1.1.1.1.1.3" xref="S4.F4.sf1.11.1.m1.1.1.1.1.3.cmml">1</mn></msub><mo id="S4.F4.sf1.11.1.m1.2.2.2.3" xref="S4.F4.sf1.11.1.m1.2.2.3.cmml">,</mo><msub id="S4.F4.sf1.11.1.m1.2.2.2.2" xref="S4.F4.sf1.11.1.m1.2.2.2.2.cmml"><mi id="S4.F4.sf1.11.1.m1.2.2.2.2.2" xref="S4.F4.sf1.11.1.m1.2.2.2.2.2.cmml">C</mi><mn id="S4.F4.sf1.11.1.m1.2.2.2.2.3" xref="S4.F4.sf1.11.1.m1.2.2.2.2.3.cmml">2</mn></msub></mrow><annotation-xml encoding="MathML-Content" id="S4.F4.sf1.11.1.m1.2c"><list id="S4.F4.sf1.11.1.m1.2.2.3.cmml" xref="S4.F4.sf1.11.1.m1.2.2.2"><apply id="S4.F4.sf1.11.1.m1.1.1.1.1.cmml" xref="S4.F4.sf1.11.1.m1.1.1.1.1"><csymbol cd="ambiguous" id="S4.F4.sf1.11.1.m1.1.1.1.1.1.cmml" xref="S4.F4.sf1.11.1.m1.1.1.1.1">subscript</csymbol><ci id="S4.F4.sf1.11.1.m1.1.1.1.1.2.cmml" xref="S4.F4.sf1.11.1.m1.1.1.1.1.2">𝐶</ci><cn id="S4.F4.sf1.11.1.m1.1.1.1.1.3.cmml" type="integer" xref="S4.F4.sf1.11.1.m1.1.1.1.1.3">1</cn></apply><apply id="S4.F4.sf1.11.1.m1.2.2.2.2.cmml" xref="S4.F4.sf1.11.1.m1.2.2.2.2"><csymbol cd="ambiguous" id="S4.F4.sf1.11.1.m1.2.2.2.2.1.cmml" xref="S4.F4.sf1.11.1.m1.2.2.2.2">subscript</csymbol><ci id="S4.F4.sf1.11.1.m1.2.2.2.2.2.cmml" xref="S4.F4.sf1.11.1.m1.2.2.2.2.2">𝐶</ci><cn id="S4.F4.sf1.11.1.m1.2.2.2.2.3.cmml" type="integer" xref="S4.F4.sf1.11.1.m1.2.2.2.2.3">2</cn></apply></list></annotation-xml><annotation encoding="application/x-tex" id="S4.F4.sf1.11.1.m1.2d">C_{1},C_{2}</annotation><annotation encoding="application/x-llamapun" id="S4.F4.sf1.11.1.m1.2e">italic_C start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_C start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT</annotation></semantics></math>. There are three separation classes: red edges within <math alttext="C_{1}" class="ltx_Math" display="inline" id="S4.F4.sf1.12.2.m2.1"><semantics id="S4.F4.sf1.12.2.m2.1b"><msub id="S4.F4.sf1.12.2.m2.1.1" xref="S4.F4.sf1.12.2.m2.1.1.cmml"><mi id="S4.F4.sf1.12.2.m2.1.1.2" xref="S4.F4.sf1.12.2.m2.1.1.2.cmml">C</mi><mn id="S4.F4.sf1.12.2.m2.1.1.3" xref="S4.F4.sf1.12.2.m2.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="S4.F4.sf1.12.2.m2.1c"><apply id="S4.F4.sf1.12.2.m2.1.1.cmml" xref="S4.F4.sf1.12.2.m2.1.1"><csymbol cd="ambiguous" id="S4.F4.sf1.12.2.m2.1.1.1.cmml" xref="S4.F4.sf1.12.2.m2.1.1">subscript</csymbol><ci id="S4.F4.sf1.12.2.m2.1.1.2.cmml" xref="S4.F4.sf1.12.2.m2.1.1.2">𝐶</ci><cn id="S4.F4.sf1.12.2.m2.1.1.3.cmml" type="integer" xref="S4.F4.sf1.12.2.m2.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.F4.sf1.12.2.m2.1d">C_{1}</annotation><annotation encoding="application/x-llamapun" id="S4.F4.sf1.12.2.m2.1e">italic_C start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math> and between <math alttext="C_{1}" class="ltx_Math" display="inline" id="S4.F4.sf1.13.3.m3.1"><semantics id="S4.F4.sf1.13.3.m3.1b"><msub id="S4.F4.sf1.13.3.m3.1.1" xref="S4.F4.sf1.13.3.m3.1.1.cmml"><mi id="S4.F4.sf1.13.3.m3.1.1.2" xref="S4.F4.sf1.13.3.m3.1.1.2.cmml">C</mi><mn id="S4.F4.sf1.13.3.m3.1.1.3" xref="S4.F4.sf1.13.3.m3.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="S4.F4.sf1.13.3.m3.1c"><apply id="S4.F4.sf1.13.3.m3.1.1.cmml" xref="S4.F4.sf1.13.3.m3.1.1"><csymbol cd="ambiguous" id="S4.F4.sf1.13.3.m3.1.1.1.cmml" xref="S4.F4.sf1.13.3.m3.1.1">subscript</csymbol><ci id="S4.F4.sf1.13.3.m3.1.1.2.cmml" xref="S4.F4.sf1.13.3.m3.1.1.2">𝐶</ci><cn id="S4.F4.sf1.13.3.m3.1.1.3.cmml" type="integer" xref="S4.F4.sf1.13.3.m3.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.F4.sf1.13.3.m3.1d">C_{1}</annotation><annotation encoding="application/x-llamapun" id="S4.F4.sf1.13.3.m3.1e">italic_C start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="\{a,b\}" class="ltx_Math" display="inline" id="S4.F4.sf1.14.4.m4.2"><semantics id="S4.F4.sf1.14.4.m4.2b"><mrow id="S4.F4.sf1.14.4.m4.2.3.2" xref="S4.F4.sf1.14.4.m4.2.3.1.cmml"><mo id="S4.F4.sf1.14.4.m4.2.3.2.1" stretchy="false" xref="S4.F4.sf1.14.4.m4.2.3.1.cmml">{</mo><mi id="S4.F4.sf1.14.4.m4.1.1" xref="S4.F4.sf1.14.4.m4.1.1.cmml">a</mi><mo id="S4.F4.sf1.14.4.m4.2.3.2.2" xref="S4.F4.sf1.14.4.m4.2.3.1.cmml">,</mo><mi id="S4.F4.sf1.14.4.m4.2.2" xref="S4.F4.sf1.14.4.m4.2.2.cmml">b</mi><mo id="S4.F4.sf1.14.4.m4.2.3.2.3" stretchy="false" xref="S4.F4.sf1.14.4.m4.2.3.1.cmml">}</mo></mrow><annotation-xml encoding="MathML-Content" id="S4.F4.sf1.14.4.m4.2c"><set id="S4.F4.sf1.14.4.m4.2.3.1.cmml" xref="S4.F4.sf1.14.4.m4.2.3.2"><ci id="S4.F4.sf1.14.4.m4.1.1.cmml" xref="S4.F4.sf1.14.4.m4.1.1">𝑎</ci><ci id="S4.F4.sf1.14.4.m4.2.2.cmml" xref="S4.F4.sf1.14.4.m4.2.2">𝑏</ci></set></annotation-xml><annotation encoding="application/x-tex" id="S4.F4.sf1.14.4.m4.2d">\{a,b\}</annotation><annotation encoding="application/x-llamapun" id="S4.F4.sf1.14.4.m4.2e">{ italic_a , italic_b }</annotation></semantics></math>, green edges within <math alttext="C_{2}" class="ltx_Math" display="inline" id="S4.F4.sf1.15.5.m5.1"><semantics id="S4.F4.sf1.15.5.m5.1b"><msub id="S4.F4.sf1.15.5.m5.1.1" xref="S4.F4.sf1.15.5.m5.1.1.cmml"><mi id="S4.F4.sf1.15.5.m5.1.1.2" xref="S4.F4.sf1.15.5.m5.1.1.2.cmml">C</mi><mn id="S4.F4.sf1.15.5.m5.1.1.3" xref="S4.F4.sf1.15.5.m5.1.1.3.cmml">2</mn></msub><annotation-xml encoding="MathML-Content" id="S4.F4.sf1.15.5.m5.1c"><apply id="S4.F4.sf1.15.5.m5.1.1.cmml" xref="S4.F4.sf1.15.5.m5.1.1"><csymbol cd="ambiguous" id="S4.F4.sf1.15.5.m5.1.1.1.cmml" xref="S4.F4.sf1.15.5.m5.1.1">subscript</csymbol><ci id="S4.F4.sf1.15.5.m5.1.1.2.cmml" xref="S4.F4.sf1.15.5.m5.1.1.2">𝐶</ci><cn id="S4.F4.sf1.15.5.m5.1.1.3.cmml" type="integer" xref="S4.F4.sf1.15.5.m5.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.F4.sf1.15.5.m5.1d">C_{2}</annotation><annotation encoding="application/x-llamapun" id="S4.F4.sf1.15.5.m5.1e">italic_C start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT</annotation></semantics></math> and between <math alttext="C_{2}" class="ltx_Math" display="inline" id="S4.F4.sf1.16.6.m6.1"><semantics id="S4.F4.sf1.16.6.m6.1b"><msub id="S4.F4.sf1.16.6.m6.1.1" xref="S4.F4.sf1.16.6.m6.1.1.cmml"><mi id="S4.F4.sf1.16.6.m6.1.1.2" xref="S4.F4.sf1.16.6.m6.1.1.2.cmml">C</mi><mn id="S4.F4.sf1.16.6.m6.1.1.3" xref="S4.F4.sf1.16.6.m6.1.1.3.cmml">2</mn></msub><annotation-xml encoding="MathML-Content" id="S4.F4.sf1.16.6.m6.1c"><apply id="S4.F4.sf1.16.6.m6.1.1.cmml" xref="S4.F4.sf1.16.6.m6.1.1"><csymbol cd="ambiguous" id="S4.F4.sf1.16.6.m6.1.1.1.cmml" xref="S4.F4.sf1.16.6.m6.1.1">subscript</csymbol><ci id="S4.F4.sf1.16.6.m6.1.1.2.cmml" xref="S4.F4.sf1.16.6.m6.1.1.2">𝐶</ci><cn id="S4.F4.sf1.16.6.m6.1.1.3.cmml" type="integer" xref="S4.F4.sf1.16.6.m6.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.F4.sf1.16.6.m6.1d">C_{2}</annotation><annotation encoding="application/x-llamapun" id="S4.F4.sf1.16.6.m6.1e">italic_C start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="\{a,b\}" class="ltx_Math" display="inline" id="S4.F4.sf1.17.7.m7.2"><semantics id="S4.F4.sf1.17.7.m7.2b"><mrow id="S4.F4.sf1.17.7.m7.2.3.2" xref="S4.F4.sf1.17.7.m7.2.3.1.cmml"><mo id="S4.F4.sf1.17.7.m7.2.3.2.1" stretchy="false" xref="S4.F4.sf1.17.7.m7.2.3.1.cmml">{</mo><mi id="S4.F4.sf1.17.7.m7.1.1" xref="S4.F4.sf1.17.7.m7.1.1.cmml">a</mi><mo id="S4.F4.sf1.17.7.m7.2.3.2.2" xref="S4.F4.sf1.17.7.m7.2.3.1.cmml">,</mo><mi id="S4.F4.sf1.17.7.m7.2.2" xref="S4.F4.sf1.17.7.m7.2.2.cmml">b</mi><mo id="S4.F4.sf1.17.7.m7.2.3.2.3" stretchy="false" xref="S4.F4.sf1.17.7.m7.2.3.1.cmml">}</mo></mrow><annotation-xml encoding="MathML-Content" id="S4.F4.sf1.17.7.m7.2c"><set id="S4.F4.sf1.17.7.m7.2.3.1.cmml" xref="S4.F4.sf1.17.7.m7.2.3.2"><ci id="S4.F4.sf1.17.7.m7.1.1.cmml" xref="S4.F4.sf1.17.7.m7.1.1">𝑎</ci><ci id="S4.F4.sf1.17.7.m7.2.2.cmml" xref="S4.F4.sf1.17.7.m7.2.2">𝑏</ci></set></annotation-xml><annotation encoding="application/x-tex" id="S4.F4.sf1.17.7.m7.2d">\{a,b\}</annotation><annotation encoding="application/x-llamapun" id="S4.F4.sf1.17.7.m7.2e">{ italic_a , italic_b }</annotation></semantics></math>, and the blue edge <math alttext="ab" class="ltx_Math" display="inline" id="S4.F4.sf1.18.8.m8.1"><semantics id="S4.F4.sf1.18.8.m8.1b"><mrow id="S4.F4.sf1.18.8.m8.1.1" xref="S4.F4.sf1.18.8.m8.1.1.cmml"><mi id="S4.F4.sf1.18.8.m8.1.1.2" xref="S4.F4.sf1.18.8.m8.1.1.2.cmml">a</mi><mo id="S4.F4.sf1.18.8.m8.1.1.1" xref="S4.F4.sf1.18.8.m8.1.1.1.cmml">⁢</mo><mi id="S4.F4.sf1.18.8.m8.1.1.3" xref="S4.F4.sf1.18.8.m8.1.1.3.cmml">b</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.F4.sf1.18.8.m8.1c"><apply id="S4.F4.sf1.18.8.m8.1.1.cmml" xref="S4.F4.sf1.18.8.m8.1.1"><times id="S4.F4.sf1.18.8.m8.1.1.1.cmml" xref="S4.F4.sf1.18.8.m8.1.1.1"></times><ci id="S4.F4.sf1.18.8.m8.1.1.2.cmml" xref="S4.F4.sf1.18.8.m8.1.1.2">𝑎</ci><ci id="S4.F4.sf1.18.8.m8.1.1.3.cmml" xref="S4.F4.sf1.18.8.m8.1.1.3">𝑏</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.F4.sf1.18.8.m8.1d">ab</annotation><annotation encoding="application/x-llamapun" id="S4.F4.sf1.18.8.m8.1e">italic_a italic_b</annotation></semantics></math>.</span></figcaption> </figure> </div> <div class="ltx_flex_cell ltx_flex_size_2"> <figure class="ltx_figure ltx_figure_panel ltx_align_center" id="S4.F4.sf2"><img alt="Refer to caption" class="ltx_graphics ltx_centering ltx_img_square" height="333" id="S4.F4.sf2.g1" src="x4.png" width="373"/> <figcaption class="ltx_caption ltx_centering"><span class="ltx_tag ltx_tag_figure"><span class="ltx_text" id="S4.F4.sf2.14.6.1" style="font-size:90%;">(b)</span> </span><math alttext="G\setminus\{a,b\}" class="ltx_Math" display="inline" id="S4.F4.sf2.7.m1.2"><semantics id="S4.F4.sf2.7.m1.2b"><mrow id="S4.F4.sf2.7.m1.2.3" xref="S4.F4.sf2.7.m1.2.3.cmml"><mi id="S4.F4.sf2.7.m1.2.3.2" mathsize="90%" xref="S4.F4.sf2.7.m1.2.3.2.cmml">G</mi><mo id="S4.F4.sf2.7.m1.2.3.1" mathsize="90%" xref="S4.F4.sf2.7.m1.2.3.1.cmml">∖</mo><mrow id="S4.F4.sf2.7.m1.2.3.3.2" xref="S4.F4.sf2.7.m1.2.3.3.1.cmml"><mo id="S4.F4.sf2.7.m1.2.3.3.2.1" maxsize="90%" minsize="90%" xref="S4.F4.sf2.7.m1.2.3.3.1.cmml">{</mo><mi id="S4.F4.sf2.7.m1.1.1" mathsize="90%" xref="S4.F4.sf2.7.m1.1.1.cmml">a</mi><mo id="S4.F4.sf2.7.m1.2.3.3.2.2" mathsize="90%" xref="S4.F4.sf2.7.m1.2.3.3.1.cmml">,</mo><mi id="S4.F4.sf2.7.m1.2.2" mathsize="90%" xref="S4.F4.sf2.7.m1.2.2.cmml">b</mi><mo id="S4.F4.sf2.7.m1.2.3.3.2.3" maxsize="90%" minsize="90%" xref="S4.F4.sf2.7.m1.2.3.3.1.cmml">}</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.F4.sf2.7.m1.2c"><apply id="S4.F4.sf2.7.m1.2.3.cmml" xref="S4.F4.sf2.7.m1.2.3"><setdiff id="S4.F4.sf2.7.m1.2.3.1.cmml" xref="S4.F4.sf2.7.m1.2.3.1"></setdiff><ci id="S4.F4.sf2.7.m1.2.3.2.cmml" xref="S4.F4.sf2.7.m1.2.3.2">𝐺</ci><set id="S4.F4.sf2.7.m1.2.3.3.1.cmml" xref="S4.F4.sf2.7.m1.2.3.3.2"><ci id="S4.F4.sf2.7.m1.1.1.cmml" xref="S4.F4.sf2.7.m1.1.1">𝑎</ci><ci id="S4.F4.sf2.7.m1.2.2.cmml" xref="S4.F4.sf2.7.m1.2.2">𝑏</ci></set></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.F4.sf2.7.m1.2d">G\setminus\{a,b\}</annotation><annotation encoding="application/x-llamapun" id="S4.F4.sf2.7.m1.2e">italic_G ∖ { italic_a , italic_b }</annotation></semantics></math><span class="ltx_text" id="S4.F4.sf2.12.5" style="font-size:90%;"> has only one component, but there are at least two parallel edges between <math alttext="a" class="ltx_Math" display="inline" id="S4.F4.sf2.8.1.m1.1"><semantics id="S4.F4.sf2.8.1.m1.1b"><mi id="S4.F4.sf2.8.1.m1.1.1" xref="S4.F4.sf2.8.1.m1.1.1.cmml">a</mi><annotation-xml encoding="MathML-Content" id="S4.F4.sf2.8.1.m1.1c"><ci id="S4.F4.sf2.8.1.m1.1.1.cmml" xref="S4.F4.sf2.8.1.m1.1.1">𝑎</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.F4.sf2.8.1.m1.1d">a</annotation><annotation encoding="application/x-llamapun" id="S4.F4.sf2.8.1.m1.1e">italic_a</annotation></semantics></math> and <math alttext="b" class="ltx_Math" display="inline" id="S4.F4.sf2.9.2.m2.1"><semantics id="S4.F4.sf2.9.2.m2.1b"><mi id="S4.F4.sf2.9.2.m2.1.1" xref="S4.F4.sf2.9.2.m2.1.1.cmml">b</mi><annotation-xml encoding="MathML-Content" id="S4.F4.sf2.9.2.m2.1c"><ci id="S4.F4.sf2.9.2.m2.1.1.cmml" xref="S4.F4.sf2.9.2.m2.1.1">𝑏</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.F4.sf2.9.2.m2.1d">b</annotation><annotation encoding="application/x-llamapun" id="S4.F4.sf2.9.2.m2.1e">italic_b</annotation></semantics></math>, and <math alttext="G\setminus\{a,b\}" class="ltx_Math" display="inline" id="S4.F4.sf2.10.3.m3.2"><semantics id="S4.F4.sf2.10.3.m3.2b"><mrow id="S4.F4.sf2.10.3.m3.2.3" xref="S4.F4.sf2.10.3.m3.2.3.cmml"><mi id="S4.F4.sf2.10.3.m3.2.3.2" xref="S4.F4.sf2.10.3.m3.2.3.2.cmml">G</mi><mo id="S4.F4.sf2.10.3.m3.2.3.1" xref="S4.F4.sf2.10.3.m3.2.3.1.cmml">∖</mo><mrow id="S4.F4.sf2.10.3.m3.2.3.3.2" xref="S4.F4.sf2.10.3.m3.2.3.3.1.cmml"><mo id="S4.F4.sf2.10.3.m3.2.3.3.2.1" stretchy="false" xref="S4.F4.sf2.10.3.m3.2.3.3.1.cmml">{</mo><mi id="S4.F4.sf2.10.3.m3.1.1" xref="S4.F4.sf2.10.3.m3.1.1.cmml">a</mi><mo id="S4.F4.sf2.10.3.m3.2.3.3.2.2" xref="S4.F4.sf2.10.3.m3.2.3.3.1.cmml">,</mo><mi id="S4.F4.sf2.10.3.m3.2.2" xref="S4.F4.sf2.10.3.m3.2.2.cmml">b</mi><mo id="S4.F4.sf2.10.3.m3.2.3.3.2.3" stretchy="false" xref="S4.F4.sf2.10.3.m3.2.3.3.1.cmml">}</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.F4.sf2.10.3.m3.2c"><apply id="S4.F4.sf2.10.3.m3.2.3.cmml" xref="S4.F4.sf2.10.3.m3.2.3"><setdiff id="S4.F4.sf2.10.3.m3.2.3.1.cmml" xref="S4.F4.sf2.10.3.m3.2.3.1"></setdiff><ci id="S4.F4.sf2.10.3.m3.2.3.2.cmml" xref="S4.F4.sf2.10.3.m3.2.3.2">𝐺</ci><set id="S4.F4.sf2.10.3.m3.2.3.3.1.cmml" xref="S4.F4.sf2.10.3.m3.2.3.3.2"><ci id="S4.F4.sf2.10.3.m3.1.1.cmml" xref="S4.F4.sf2.10.3.m3.1.1">𝑎</ci><ci id="S4.F4.sf2.10.3.m3.2.2.cmml" xref="S4.F4.sf2.10.3.m3.2.2">𝑏</ci></set></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.F4.sf2.10.3.m3.2d">G\setminus\{a,b\}</annotation><annotation encoding="application/x-llamapun" id="S4.F4.sf2.10.3.m3.2e">italic_G ∖ { italic_a , italic_b }</annotation></semantics></math> is non-empty. There are two separation classes: red parallel edges between <math alttext="a" class="ltx_Math" display="inline" id="S4.F4.sf2.11.4.m4.1"><semantics id="S4.F4.sf2.11.4.m4.1b"><mi id="S4.F4.sf2.11.4.m4.1.1" xref="S4.F4.sf2.11.4.m4.1.1.cmml">a</mi><annotation-xml encoding="MathML-Content" id="S4.F4.sf2.11.4.m4.1c"><ci id="S4.F4.sf2.11.4.m4.1.1.cmml" xref="S4.F4.sf2.11.4.m4.1.1">𝑎</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.F4.sf2.11.4.m4.1d">a</annotation><annotation encoding="application/x-llamapun" id="S4.F4.sf2.11.4.m4.1e">italic_a</annotation></semantics></math> and <math alttext="b" class="ltx_Math" display="inline" id="S4.F4.sf2.12.5.m5.1"><semantics id="S4.F4.sf2.12.5.m5.1b"><mi id="S4.F4.sf2.12.5.m5.1.1" xref="S4.F4.sf2.12.5.m5.1.1.cmml">b</mi><annotation-xml encoding="MathML-Content" id="S4.F4.sf2.12.5.m5.1c"><ci id="S4.F4.sf2.12.5.m5.1.1.cmml" xref="S4.F4.sf2.12.5.m5.1.1">𝑏</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.F4.sf2.12.5.m5.1d">b</annotation><annotation encoding="application/x-llamapun" id="S4.F4.sf2.12.5.m5.1e">italic_b</annotation></semantics></math>, and all other edges in blue.</span></figcaption> </figure> </div> </div> <figcaption class="ltx_caption ltx_centering"><span class="ltx_tag ltx_tag_figure"><span class="ltx_text" id="S4.F4.4.2.1" style="font-size:90%;">Figure 4</span>: </span><span class="ltx_text" id="S4.F4.2.1" style="font-size:90%;">Example of separation pair <math alttext="\{a,b\}" class="ltx_Math" display="inline" id="S4.F4.2.1.m1.2"><semantics id="S4.F4.2.1.m1.2b"><mrow id="S4.F4.2.1.m1.2.3.2" xref="S4.F4.2.1.m1.2.3.1.cmml"><mo id="S4.F4.2.1.m1.2.3.2.1" stretchy="false" xref="S4.F4.2.1.m1.2.3.1.cmml">{</mo><mi id="S4.F4.2.1.m1.1.1" xref="S4.F4.2.1.m1.1.1.cmml">a</mi><mo id="S4.F4.2.1.m1.2.3.2.2" xref="S4.F4.2.1.m1.2.3.1.cmml">,</mo><mi id="S4.F4.2.1.m1.2.2" xref="S4.F4.2.1.m1.2.2.cmml">b</mi><mo id="S4.F4.2.1.m1.2.3.2.3" stretchy="false" xref="S4.F4.2.1.m1.2.3.1.cmml">}</mo></mrow><annotation-xml encoding="MathML-Content" id="S4.F4.2.1.m1.2c"><set id="S4.F4.2.1.m1.2.3.1.cmml" xref="S4.F4.2.1.m1.2.3.2"><ci id="S4.F4.2.1.m1.1.1.cmml" xref="S4.F4.2.1.m1.1.1">𝑎</ci><ci id="S4.F4.2.1.m1.2.2.cmml" xref="S4.F4.2.1.m1.2.2">𝑏</ci></set></annotation-xml><annotation encoding="application/x-tex" id="S4.F4.2.1.m1.2d">\{a,b\}</annotation><annotation encoding="application/x-llamapun" id="S4.F4.2.1.m1.2e">{ italic_a , italic_b }</annotation></semantics></math> and corresponding separation classes shown with different colors.</span></figcaption> </figure> <div class="ltx_para" id="S4.SS2.SSS1.p2"> <p class="ltx_p" id="S4.SS2.SSS1.p2.1">Next, we define two operations: “split” and “merge”.</p> </div> <div class="ltx_theorem ltx_theorem_definition" id="S4.Thmtheorem9"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="S4.Thmtheorem9.1.1.1">Definition 4.9</span></span><span class="ltx_text ltx_font_bold" id="S4.Thmtheorem9.2.2">.</span> </h6> <div class="ltx_para" id="S4.Thmtheorem9.p1"> <p class="ltx_p" id="S4.Thmtheorem9.p1.14">The <em class="ltx_emph ltx_font_italic" id="S4.Thmtheorem9.p1.14.1">split</em> operation on a 2-connected graph <math alttext="G=(V,E)" class="ltx_Math" display="inline" id="S4.Thmtheorem9.p1.1.m1.2"><semantics id="S4.Thmtheorem9.p1.1.m1.2a"><mrow id="S4.Thmtheorem9.p1.1.m1.2.3" xref="S4.Thmtheorem9.p1.1.m1.2.3.cmml"><mi id="S4.Thmtheorem9.p1.1.m1.2.3.2" xref="S4.Thmtheorem9.p1.1.m1.2.3.2.cmml">G</mi><mo id="S4.Thmtheorem9.p1.1.m1.2.3.1" xref="S4.Thmtheorem9.p1.1.m1.2.3.1.cmml">=</mo><mrow id="S4.Thmtheorem9.p1.1.m1.2.3.3.2" xref="S4.Thmtheorem9.p1.1.m1.2.3.3.1.cmml"><mo id="S4.Thmtheorem9.p1.1.m1.2.3.3.2.1" stretchy="false" xref="S4.Thmtheorem9.p1.1.m1.2.3.3.1.cmml">(</mo><mi id="S4.Thmtheorem9.p1.1.m1.1.1" xref="S4.Thmtheorem9.p1.1.m1.1.1.cmml">V</mi><mo id="S4.Thmtheorem9.p1.1.m1.2.3.3.2.2" xref="S4.Thmtheorem9.p1.1.m1.2.3.3.1.cmml">,</mo><mi id="S4.Thmtheorem9.p1.1.m1.2.2" xref="S4.Thmtheorem9.p1.1.m1.2.2.cmml">E</mi><mo id="S4.Thmtheorem9.p1.1.m1.2.3.3.2.3" stretchy="false" xref="S4.Thmtheorem9.p1.1.m1.2.3.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem9.p1.1.m1.2b"><apply id="S4.Thmtheorem9.p1.1.m1.2.3.cmml" xref="S4.Thmtheorem9.p1.1.m1.2.3"><eq id="S4.Thmtheorem9.p1.1.m1.2.3.1.cmml" xref="S4.Thmtheorem9.p1.1.m1.2.3.1"></eq><ci id="S4.Thmtheorem9.p1.1.m1.2.3.2.cmml" xref="S4.Thmtheorem9.p1.1.m1.2.3.2">𝐺</ci><interval closure="open" id="S4.Thmtheorem9.p1.1.m1.2.3.3.1.cmml" xref="S4.Thmtheorem9.p1.1.m1.2.3.3.2"><ci id="S4.Thmtheorem9.p1.1.m1.1.1.cmml" xref="S4.Thmtheorem9.p1.1.m1.1.1">𝑉</ci><ci id="S4.Thmtheorem9.p1.1.m1.2.2.cmml" xref="S4.Thmtheorem9.p1.1.m1.2.2">𝐸</ci></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem9.p1.1.m1.2c">G=(V,E)</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem9.p1.1.m1.2d">italic_G = ( italic_V , italic_E )</annotation></semantics></math> with <math alttext="|E|\geq 4" class="ltx_Math" display="inline" id="S4.Thmtheorem9.p1.2.m2.1"><semantics id="S4.Thmtheorem9.p1.2.m2.1a"><mrow id="S4.Thmtheorem9.p1.2.m2.1.2" xref="S4.Thmtheorem9.p1.2.m2.1.2.cmml"><mrow id="S4.Thmtheorem9.p1.2.m2.1.2.2.2" xref="S4.Thmtheorem9.p1.2.m2.1.2.2.1.cmml"><mo id="S4.Thmtheorem9.p1.2.m2.1.2.2.2.1" stretchy="false" xref="S4.Thmtheorem9.p1.2.m2.1.2.2.1.1.cmml">|</mo><mi id="S4.Thmtheorem9.p1.2.m2.1.1" xref="S4.Thmtheorem9.p1.2.m2.1.1.cmml">E</mi><mo id="S4.Thmtheorem9.p1.2.m2.1.2.2.2.2" stretchy="false" xref="S4.Thmtheorem9.p1.2.m2.1.2.2.1.1.cmml">|</mo></mrow><mo id="S4.Thmtheorem9.p1.2.m2.1.2.1" xref="S4.Thmtheorem9.p1.2.m2.1.2.1.cmml">≥</mo><mn id="S4.Thmtheorem9.p1.2.m2.1.2.3" xref="S4.Thmtheorem9.p1.2.m2.1.2.3.cmml">4</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem9.p1.2.m2.1b"><apply id="S4.Thmtheorem9.p1.2.m2.1.2.cmml" xref="S4.Thmtheorem9.p1.2.m2.1.2"><geq id="S4.Thmtheorem9.p1.2.m2.1.2.1.cmml" xref="S4.Thmtheorem9.p1.2.m2.1.2.1"></geq><apply id="S4.Thmtheorem9.p1.2.m2.1.2.2.1.cmml" xref="S4.Thmtheorem9.p1.2.m2.1.2.2.2"><abs id="S4.Thmtheorem9.p1.2.m2.1.2.2.1.1.cmml" xref="S4.Thmtheorem9.p1.2.m2.1.2.2.2.1"></abs><ci id="S4.Thmtheorem9.p1.2.m2.1.1.cmml" xref="S4.Thmtheorem9.p1.2.m2.1.1">𝐸</ci></apply><cn id="S4.Thmtheorem9.p1.2.m2.1.2.3.cmml" type="integer" xref="S4.Thmtheorem9.p1.2.m2.1.2.3">4</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem9.p1.2.m2.1c">|E|\geq 4</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem9.p1.2.m2.1d">| italic_E | ≥ 4</annotation></semantics></math> and separation pair <math alttext="\{a,b\}" class="ltx_Math" display="inline" id="S4.Thmtheorem9.p1.3.m3.2"><semantics id="S4.Thmtheorem9.p1.3.m3.2a"><mrow id="S4.Thmtheorem9.p1.3.m3.2.3.2" xref="S4.Thmtheorem9.p1.3.m3.2.3.1.cmml"><mo id="S4.Thmtheorem9.p1.3.m3.2.3.2.1" stretchy="false" xref="S4.Thmtheorem9.p1.3.m3.2.3.1.cmml">{</mo><mi id="S4.Thmtheorem9.p1.3.m3.1.1" xref="S4.Thmtheorem9.p1.3.m3.1.1.cmml">a</mi><mo id="S4.Thmtheorem9.p1.3.m3.2.3.2.2" xref="S4.Thmtheorem9.p1.3.m3.2.3.1.cmml">,</mo><mi id="S4.Thmtheorem9.p1.3.m3.2.2" xref="S4.Thmtheorem9.p1.3.m3.2.2.cmml">b</mi><mo id="S4.Thmtheorem9.p1.3.m3.2.3.2.3" stretchy="false" xref="S4.Thmtheorem9.p1.3.m3.2.3.1.cmml">}</mo></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem9.p1.3.m3.2b"><set id="S4.Thmtheorem9.p1.3.m3.2.3.1.cmml" xref="S4.Thmtheorem9.p1.3.m3.2.3.2"><ci id="S4.Thmtheorem9.p1.3.m3.1.1.cmml" xref="S4.Thmtheorem9.p1.3.m3.1.1">𝑎</ci><ci id="S4.Thmtheorem9.p1.3.m3.2.2.cmml" xref="S4.Thmtheorem9.p1.3.m3.2.2">𝑏</ci></set></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem9.p1.3.m3.2c">\{a,b\}</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem9.p1.3.m3.2d">{ italic_a , italic_b }</annotation></semantics></math> is defined as follows. Let <math alttext="E_{1},\dots,E_{k}" class="ltx_Math" display="inline" id="S4.Thmtheorem9.p1.4.m4.3"><semantics id="S4.Thmtheorem9.p1.4.m4.3a"><mrow id="S4.Thmtheorem9.p1.4.m4.3.3.2" xref="S4.Thmtheorem9.p1.4.m4.3.3.3.cmml"><msub id="S4.Thmtheorem9.p1.4.m4.2.2.1.1" xref="S4.Thmtheorem9.p1.4.m4.2.2.1.1.cmml"><mi id="S4.Thmtheorem9.p1.4.m4.2.2.1.1.2" xref="S4.Thmtheorem9.p1.4.m4.2.2.1.1.2.cmml">E</mi><mn id="S4.Thmtheorem9.p1.4.m4.2.2.1.1.3" xref="S4.Thmtheorem9.p1.4.m4.2.2.1.1.3.cmml">1</mn></msub><mo id="S4.Thmtheorem9.p1.4.m4.3.3.2.3" xref="S4.Thmtheorem9.p1.4.m4.3.3.3.cmml">,</mo><mi id="S4.Thmtheorem9.p1.4.m4.1.1" mathvariant="normal" xref="S4.Thmtheorem9.p1.4.m4.1.1.cmml">…</mi><mo id="S4.Thmtheorem9.p1.4.m4.3.3.2.4" xref="S4.Thmtheorem9.p1.4.m4.3.3.3.cmml">,</mo><msub id="S4.Thmtheorem9.p1.4.m4.3.3.2.2" xref="S4.Thmtheorem9.p1.4.m4.3.3.2.2.cmml"><mi id="S4.Thmtheorem9.p1.4.m4.3.3.2.2.2" xref="S4.Thmtheorem9.p1.4.m4.3.3.2.2.2.cmml">E</mi><mi id="S4.Thmtheorem9.p1.4.m4.3.3.2.2.3" xref="S4.Thmtheorem9.p1.4.m4.3.3.2.2.3.cmml">k</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem9.p1.4.m4.3b"><list id="S4.Thmtheorem9.p1.4.m4.3.3.3.cmml" xref="S4.Thmtheorem9.p1.4.m4.3.3.2"><apply id="S4.Thmtheorem9.p1.4.m4.2.2.1.1.cmml" xref="S4.Thmtheorem9.p1.4.m4.2.2.1.1"><csymbol cd="ambiguous" id="S4.Thmtheorem9.p1.4.m4.2.2.1.1.1.cmml" xref="S4.Thmtheorem9.p1.4.m4.2.2.1.1">subscript</csymbol><ci id="S4.Thmtheorem9.p1.4.m4.2.2.1.1.2.cmml" xref="S4.Thmtheorem9.p1.4.m4.2.2.1.1.2">𝐸</ci><cn id="S4.Thmtheorem9.p1.4.m4.2.2.1.1.3.cmml" type="integer" xref="S4.Thmtheorem9.p1.4.m4.2.2.1.1.3">1</cn></apply><ci id="S4.Thmtheorem9.p1.4.m4.1.1.cmml" xref="S4.Thmtheorem9.p1.4.m4.1.1">…</ci><apply id="S4.Thmtheorem9.p1.4.m4.3.3.2.2.cmml" xref="S4.Thmtheorem9.p1.4.m4.3.3.2.2"><csymbol cd="ambiguous" id="S4.Thmtheorem9.p1.4.m4.3.3.2.2.1.cmml" xref="S4.Thmtheorem9.p1.4.m4.3.3.2.2">subscript</csymbol><ci id="S4.Thmtheorem9.p1.4.m4.3.3.2.2.2.cmml" xref="S4.Thmtheorem9.p1.4.m4.3.3.2.2.2">𝐸</ci><ci id="S4.Thmtheorem9.p1.4.m4.3.3.2.2.3.cmml" xref="S4.Thmtheorem9.p1.4.m4.3.3.2.2.3">𝑘</ci></apply></list></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem9.p1.4.m4.3c">E_{1},\dots,E_{k}</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem9.p1.4.m4.3d">italic_E start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , … , italic_E start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT</annotation></semantics></math> be the separation classes of <math alttext="\{a,b\}" class="ltx_Math" display="inline" id="S4.Thmtheorem9.p1.5.m5.2"><semantics id="S4.Thmtheorem9.p1.5.m5.2a"><mrow id="S4.Thmtheorem9.p1.5.m5.2.3.2" xref="S4.Thmtheorem9.p1.5.m5.2.3.1.cmml"><mo id="S4.Thmtheorem9.p1.5.m5.2.3.2.1" stretchy="false" xref="S4.Thmtheorem9.p1.5.m5.2.3.1.cmml">{</mo><mi id="S4.Thmtheorem9.p1.5.m5.1.1" xref="S4.Thmtheorem9.p1.5.m5.1.1.cmml">a</mi><mo id="S4.Thmtheorem9.p1.5.m5.2.3.2.2" xref="S4.Thmtheorem9.p1.5.m5.2.3.1.cmml">,</mo><mi id="S4.Thmtheorem9.p1.5.m5.2.2" xref="S4.Thmtheorem9.p1.5.m5.2.2.cmml">b</mi><mo id="S4.Thmtheorem9.p1.5.m5.2.3.2.3" stretchy="false" xref="S4.Thmtheorem9.p1.5.m5.2.3.1.cmml">}</mo></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem9.p1.5.m5.2b"><set id="S4.Thmtheorem9.p1.5.m5.2.3.1.cmml" xref="S4.Thmtheorem9.p1.5.m5.2.3.2"><ci id="S4.Thmtheorem9.p1.5.m5.1.1.cmml" xref="S4.Thmtheorem9.p1.5.m5.1.1">𝑎</ci><ci id="S4.Thmtheorem9.p1.5.m5.2.2.cmml" xref="S4.Thmtheorem9.p1.5.m5.2.2">𝑏</ci></set></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem9.p1.5.m5.2c">\{a,b\}</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem9.p1.5.m5.2d">{ italic_a , italic_b }</annotation></semantics></math>. For a subset of indices <math alttext="I\subseteq[k]" class="ltx_Math" display="inline" id="S4.Thmtheorem9.p1.6.m6.1"><semantics id="S4.Thmtheorem9.p1.6.m6.1a"><mrow id="S4.Thmtheorem9.p1.6.m6.1.2" xref="S4.Thmtheorem9.p1.6.m6.1.2.cmml"><mi id="S4.Thmtheorem9.p1.6.m6.1.2.2" xref="S4.Thmtheorem9.p1.6.m6.1.2.2.cmml">I</mi><mo id="S4.Thmtheorem9.p1.6.m6.1.2.1" xref="S4.Thmtheorem9.p1.6.m6.1.2.1.cmml">⊆</mo><mrow id="S4.Thmtheorem9.p1.6.m6.1.2.3.2" xref="S4.Thmtheorem9.p1.6.m6.1.2.3.1.cmml"><mo id="S4.Thmtheorem9.p1.6.m6.1.2.3.2.1" stretchy="false" xref="S4.Thmtheorem9.p1.6.m6.1.2.3.1.1.cmml">[</mo><mi id="S4.Thmtheorem9.p1.6.m6.1.1" xref="S4.Thmtheorem9.p1.6.m6.1.1.cmml">k</mi><mo id="S4.Thmtheorem9.p1.6.m6.1.2.3.2.2" stretchy="false" xref="S4.Thmtheorem9.p1.6.m6.1.2.3.1.1.cmml">]</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem9.p1.6.m6.1b"><apply id="S4.Thmtheorem9.p1.6.m6.1.2.cmml" xref="S4.Thmtheorem9.p1.6.m6.1.2"><subset id="S4.Thmtheorem9.p1.6.m6.1.2.1.cmml" xref="S4.Thmtheorem9.p1.6.m6.1.2.1"></subset><ci id="S4.Thmtheorem9.p1.6.m6.1.2.2.cmml" xref="S4.Thmtheorem9.p1.6.m6.1.2.2">𝐼</ci><apply id="S4.Thmtheorem9.p1.6.m6.1.2.3.1.cmml" xref="S4.Thmtheorem9.p1.6.m6.1.2.3.2"><csymbol cd="latexml" id="S4.Thmtheorem9.p1.6.m6.1.2.3.1.1.cmml" xref="S4.Thmtheorem9.p1.6.m6.1.2.3.2.1">delimited-[]</csymbol><ci id="S4.Thmtheorem9.p1.6.m6.1.1.cmml" xref="S4.Thmtheorem9.p1.6.m6.1.1">𝑘</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem9.p1.6.m6.1c">I\subseteq[k]</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem9.p1.6.m6.1d">italic_I ⊆ [ italic_k ]</annotation></semantics></math>, let <math alttext="E_{I}^{\prime}=\cup_{i\in I}E_{i}" class="ltx_Math" display="inline" id="S4.Thmtheorem9.p1.7.m7.1"><semantics id="S4.Thmtheorem9.p1.7.m7.1a"><mrow id="S4.Thmtheorem9.p1.7.m7.1.1" xref="S4.Thmtheorem9.p1.7.m7.1.1.cmml"><msubsup id="S4.Thmtheorem9.p1.7.m7.1.1.2" xref="S4.Thmtheorem9.p1.7.m7.1.1.2.cmml"><mi id="S4.Thmtheorem9.p1.7.m7.1.1.2.2.2" xref="S4.Thmtheorem9.p1.7.m7.1.1.2.2.2.cmml">E</mi><mi id="S4.Thmtheorem9.p1.7.m7.1.1.2.2.3" xref="S4.Thmtheorem9.p1.7.m7.1.1.2.2.3.cmml">I</mi><mo id="S4.Thmtheorem9.p1.7.m7.1.1.2.3" xref="S4.Thmtheorem9.p1.7.m7.1.1.2.3.cmml">′</mo></msubsup><mo id="S4.Thmtheorem9.p1.7.m7.1.1.1" rspace="0em" xref="S4.Thmtheorem9.p1.7.m7.1.1.1.cmml">=</mo><mrow id="S4.Thmtheorem9.p1.7.m7.1.1.3" xref="S4.Thmtheorem9.p1.7.m7.1.1.3.cmml"><msub id="S4.Thmtheorem9.p1.7.m7.1.1.3.1" xref="S4.Thmtheorem9.p1.7.m7.1.1.3.1.cmml"><mo id="S4.Thmtheorem9.p1.7.m7.1.1.3.1.2" lspace="0em" xref="S4.Thmtheorem9.p1.7.m7.1.1.3.1.2.cmml">∪</mo><mrow id="S4.Thmtheorem9.p1.7.m7.1.1.3.1.3" xref="S4.Thmtheorem9.p1.7.m7.1.1.3.1.3.cmml"><mi id="S4.Thmtheorem9.p1.7.m7.1.1.3.1.3.2" xref="S4.Thmtheorem9.p1.7.m7.1.1.3.1.3.2.cmml">i</mi><mo id="S4.Thmtheorem9.p1.7.m7.1.1.3.1.3.1" xref="S4.Thmtheorem9.p1.7.m7.1.1.3.1.3.1.cmml">∈</mo><mi id="S4.Thmtheorem9.p1.7.m7.1.1.3.1.3.3" xref="S4.Thmtheorem9.p1.7.m7.1.1.3.1.3.3.cmml">I</mi></mrow></msub><msub id="S4.Thmtheorem9.p1.7.m7.1.1.3.2" xref="S4.Thmtheorem9.p1.7.m7.1.1.3.2.cmml"><mi id="S4.Thmtheorem9.p1.7.m7.1.1.3.2.2" xref="S4.Thmtheorem9.p1.7.m7.1.1.3.2.2.cmml">E</mi><mi id="S4.Thmtheorem9.p1.7.m7.1.1.3.2.3" xref="S4.Thmtheorem9.p1.7.m7.1.1.3.2.3.cmml">i</mi></msub></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem9.p1.7.m7.1b"><apply id="S4.Thmtheorem9.p1.7.m7.1.1.cmml" xref="S4.Thmtheorem9.p1.7.m7.1.1"><eq id="S4.Thmtheorem9.p1.7.m7.1.1.1.cmml" xref="S4.Thmtheorem9.p1.7.m7.1.1.1"></eq><apply id="S4.Thmtheorem9.p1.7.m7.1.1.2.cmml" xref="S4.Thmtheorem9.p1.7.m7.1.1.2"><csymbol cd="ambiguous" id="S4.Thmtheorem9.p1.7.m7.1.1.2.1.cmml" xref="S4.Thmtheorem9.p1.7.m7.1.1.2">superscript</csymbol><apply id="S4.Thmtheorem9.p1.7.m7.1.1.2.2.cmml" xref="S4.Thmtheorem9.p1.7.m7.1.1.2"><csymbol cd="ambiguous" id="S4.Thmtheorem9.p1.7.m7.1.1.2.2.1.cmml" xref="S4.Thmtheorem9.p1.7.m7.1.1.2">subscript</csymbol><ci id="S4.Thmtheorem9.p1.7.m7.1.1.2.2.2.cmml" xref="S4.Thmtheorem9.p1.7.m7.1.1.2.2.2">𝐸</ci><ci id="S4.Thmtheorem9.p1.7.m7.1.1.2.2.3.cmml" xref="S4.Thmtheorem9.p1.7.m7.1.1.2.2.3">𝐼</ci></apply><ci id="S4.Thmtheorem9.p1.7.m7.1.1.2.3.cmml" xref="S4.Thmtheorem9.p1.7.m7.1.1.2.3">′</ci></apply><apply id="S4.Thmtheorem9.p1.7.m7.1.1.3.cmml" xref="S4.Thmtheorem9.p1.7.m7.1.1.3"><apply id="S4.Thmtheorem9.p1.7.m7.1.1.3.1.cmml" xref="S4.Thmtheorem9.p1.7.m7.1.1.3.1"><csymbol cd="ambiguous" id="S4.Thmtheorem9.p1.7.m7.1.1.3.1.1.cmml" xref="S4.Thmtheorem9.p1.7.m7.1.1.3.1">subscript</csymbol><union id="S4.Thmtheorem9.p1.7.m7.1.1.3.1.2.cmml" xref="S4.Thmtheorem9.p1.7.m7.1.1.3.1.2"></union><apply id="S4.Thmtheorem9.p1.7.m7.1.1.3.1.3.cmml" xref="S4.Thmtheorem9.p1.7.m7.1.1.3.1.3"><in id="S4.Thmtheorem9.p1.7.m7.1.1.3.1.3.1.cmml" xref="S4.Thmtheorem9.p1.7.m7.1.1.3.1.3.1"></in><ci id="S4.Thmtheorem9.p1.7.m7.1.1.3.1.3.2.cmml" xref="S4.Thmtheorem9.p1.7.m7.1.1.3.1.3.2">𝑖</ci><ci id="S4.Thmtheorem9.p1.7.m7.1.1.3.1.3.3.cmml" xref="S4.Thmtheorem9.p1.7.m7.1.1.3.1.3.3">𝐼</ci></apply></apply><apply id="S4.Thmtheorem9.p1.7.m7.1.1.3.2.cmml" xref="S4.Thmtheorem9.p1.7.m7.1.1.3.2"><csymbol cd="ambiguous" id="S4.Thmtheorem9.p1.7.m7.1.1.3.2.1.cmml" xref="S4.Thmtheorem9.p1.7.m7.1.1.3.2">subscript</csymbol><ci id="S4.Thmtheorem9.p1.7.m7.1.1.3.2.2.cmml" xref="S4.Thmtheorem9.p1.7.m7.1.1.3.2.2">𝐸</ci><ci id="S4.Thmtheorem9.p1.7.m7.1.1.3.2.3.cmml" xref="S4.Thmtheorem9.p1.7.m7.1.1.3.2.3">𝑖</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem9.p1.7.m7.1c">E_{I}^{\prime}=\cup_{i\in I}E_{i}</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem9.p1.7.m7.1d">italic_E start_POSTSUBSCRIPT italic_I end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT = ∪ start_POSTSUBSCRIPT italic_i ∈ italic_I end_POSTSUBSCRIPT italic_E start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="E_{I}^{\prime\prime}=\cup_{i\notin I}E_{i}" class="ltx_Math" display="inline" id="S4.Thmtheorem9.p1.8.m8.1"><semantics id="S4.Thmtheorem9.p1.8.m8.1a"><mrow id="S4.Thmtheorem9.p1.8.m8.1.1" xref="S4.Thmtheorem9.p1.8.m8.1.1.cmml"><msubsup id="S4.Thmtheorem9.p1.8.m8.1.1.2" xref="S4.Thmtheorem9.p1.8.m8.1.1.2.cmml"><mi id="S4.Thmtheorem9.p1.8.m8.1.1.2.2.2" xref="S4.Thmtheorem9.p1.8.m8.1.1.2.2.2.cmml">E</mi><mi id="S4.Thmtheorem9.p1.8.m8.1.1.2.2.3" xref="S4.Thmtheorem9.p1.8.m8.1.1.2.2.3.cmml">I</mi><mo id="S4.Thmtheorem9.p1.8.m8.1.1.2.3" xref="S4.Thmtheorem9.p1.8.m8.1.1.2.3.cmml">′′</mo></msubsup><mo id="S4.Thmtheorem9.p1.8.m8.1.1.1" rspace="0em" xref="S4.Thmtheorem9.p1.8.m8.1.1.1.cmml">=</mo><mrow id="S4.Thmtheorem9.p1.8.m8.1.1.3" xref="S4.Thmtheorem9.p1.8.m8.1.1.3.cmml"><msub id="S4.Thmtheorem9.p1.8.m8.1.1.3.1" xref="S4.Thmtheorem9.p1.8.m8.1.1.3.1.cmml"><mo id="S4.Thmtheorem9.p1.8.m8.1.1.3.1.2" lspace="0em" xref="S4.Thmtheorem9.p1.8.m8.1.1.3.1.2.cmml">∪</mo><mrow id="S4.Thmtheorem9.p1.8.m8.1.1.3.1.3" xref="S4.Thmtheorem9.p1.8.m8.1.1.3.1.3.cmml"><mi id="S4.Thmtheorem9.p1.8.m8.1.1.3.1.3.2" xref="S4.Thmtheorem9.p1.8.m8.1.1.3.1.3.2.cmml">i</mi><mo id="S4.Thmtheorem9.p1.8.m8.1.1.3.1.3.1" xref="S4.Thmtheorem9.p1.8.m8.1.1.3.1.3.1.cmml">∉</mo><mi id="S4.Thmtheorem9.p1.8.m8.1.1.3.1.3.3" xref="S4.Thmtheorem9.p1.8.m8.1.1.3.1.3.3.cmml">I</mi></mrow></msub><msub id="S4.Thmtheorem9.p1.8.m8.1.1.3.2" xref="S4.Thmtheorem9.p1.8.m8.1.1.3.2.cmml"><mi id="S4.Thmtheorem9.p1.8.m8.1.1.3.2.2" xref="S4.Thmtheorem9.p1.8.m8.1.1.3.2.2.cmml">E</mi><mi id="S4.Thmtheorem9.p1.8.m8.1.1.3.2.3" xref="S4.Thmtheorem9.p1.8.m8.1.1.3.2.3.cmml">i</mi></msub></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem9.p1.8.m8.1b"><apply id="S4.Thmtheorem9.p1.8.m8.1.1.cmml" xref="S4.Thmtheorem9.p1.8.m8.1.1"><eq id="S4.Thmtheorem9.p1.8.m8.1.1.1.cmml" xref="S4.Thmtheorem9.p1.8.m8.1.1.1"></eq><apply id="S4.Thmtheorem9.p1.8.m8.1.1.2.cmml" xref="S4.Thmtheorem9.p1.8.m8.1.1.2"><csymbol cd="ambiguous" id="S4.Thmtheorem9.p1.8.m8.1.1.2.1.cmml" xref="S4.Thmtheorem9.p1.8.m8.1.1.2">superscript</csymbol><apply id="S4.Thmtheorem9.p1.8.m8.1.1.2.2.cmml" xref="S4.Thmtheorem9.p1.8.m8.1.1.2"><csymbol cd="ambiguous" id="S4.Thmtheorem9.p1.8.m8.1.1.2.2.1.cmml" xref="S4.Thmtheorem9.p1.8.m8.1.1.2">subscript</csymbol><ci id="S4.Thmtheorem9.p1.8.m8.1.1.2.2.2.cmml" xref="S4.Thmtheorem9.p1.8.m8.1.1.2.2.2">𝐸</ci><ci id="S4.Thmtheorem9.p1.8.m8.1.1.2.2.3.cmml" xref="S4.Thmtheorem9.p1.8.m8.1.1.2.2.3">𝐼</ci></apply><ci id="S4.Thmtheorem9.p1.8.m8.1.1.2.3.cmml" xref="S4.Thmtheorem9.p1.8.m8.1.1.2.3">′′</ci></apply><apply id="S4.Thmtheorem9.p1.8.m8.1.1.3.cmml" xref="S4.Thmtheorem9.p1.8.m8.1.1.3"><apply id="S4.Thmtheorem9.p1.8.m8.1.1.3.1.cmml" xref="S4.Thmtheorem9.p1.8.m8.1.1.3.1"><csymbol cd="ambiguous" id="S4.Thmtheorem9.p1.8.m8.1.1.3.1.1.cmml" xref="S4.Thmtheorem9.p1.8.m8.1.1.3.1">subscript</csymbol><union id="S4.Thmtheorem9.p1.8.m8.1.1.3.1.2.cmml" xref="S4.Thmtheorem9.p1.8.m8.1.1.3.1.2"></union><apply id="S4.Thmtheorem9.p1.8.m8.1.1.3.1.3.cmml" xref="S4.Thmtheorem9.p1.8.m8.1.1.3.1.3"><notin id="S4.Thmtheorem9.p1.8.m8.1.1.3.1.3.1.cmml" xref="S4.Thmtheorem9.p1.8.m8.1.1.3.1.3.1"></notin><ci id="S4.Thmtheorem9.p1.8.m8.1.1.3.1.3.2.cmml" xref="S4.Thmtheorem9.p1.8.m8.1.1.3.1.3.2">𝑖</ci><ci id="S4.Thmtheorem9.p1.8.m8.1.1.3.1.3.3.cmml" xref="S4.Thmtheorem9.p1.8.m8.1.1.3.1.3.3">𝐼</ci></apply></apply><apply id="S4.Thmtheorem9.p1.8.m8.1.1.3.2.cmml" xref="S4.Thmtheorem9.p1.8.m8.1.1.3.2"><csymbol cd="ambiguous" id="S4.Thmtheorem9.p1.8.m8.1.1.3.2.1.cmml" xref="S4.Thmtheorem9.p1.8.m8.1.1.3.2">subscript</csymbol><ci id="S4.Thmtheorem9.p1.8.m8.1.1.3.2.2.cmml" xref="S4.Thmtheorem9.p1.8.m8.1.1.3.2.2">𝐸</ci><ci id="S4.Thmtheorem9.p1.8.m8.1.1.3.2.3.cmml" xref="S4.Thmtheorem9.p1.8.m8.1.1.3.2.3">𝑖</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem9.p1.8.m8.1c">E_{I}^{\prime\prime}=\cup_{i\notin I}E_{i}</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem9.p1.8.m8.1d">italic_E start_POSTSUBSCRIPT italic_I end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT = ∪ start_POSTSUBSCRIPT italic_i ∉ italic_I end_POSTSUBSCRIPT italic_E start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math>. Choose <math alttext="I" class="ltx_Math" display="inline" id="S4.Thmtheorem9.p1.9.m9.1"><semantics id="S4.Thmtheorem9.p1.9.m9.1a"><mi id="S4.Thmtheorem9.p1.9.m9.1.1" xref="S4.Thmtheorem9.p1.9.m9.1.1.cmml">I</mi><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem9.p1.9.m9.1b"><ci id="S4.Thmtheorem9.p1.9.m9.1.1.cmml" xref="S4.Thmtheorem9.p1.9.m9.1.1">𝐼</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem9.p1.9.m9.1c">I</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem9.p1.9.m9.1d">italic_I</annotation></semantics></math> arbitrarily such that <math alttext="|E_{I}^{\prime}|\geq 2,|E_{I}^{\prime\prime}|\geq 2" class="ltx_Math" display="inline" id="S4.Thmtheorem9.p1.10.m10.2"><semantics id="S4.Thmtheorem9.p1.10.m10.2a"><mrow id="S4.Thmtheorem9.p1.10.m10.2.2.2" xref="S4.Thmtheorem9.p1.10.m10.2.2.3.cmml"><mrow id="S4.Thmtheorem9.p1.10.m10.1.1.1.1" xref="S4.Thmtheorem9.p1.10.m10.1.1.1.1.cmml"><mrow id="S4.Thmtheorem9.p1.10.m10.1.1.1.1.1.1" xref="S4.Thmtheorem9.p1.10.m10.1.1.1.1.1.2.cmml"><mo id="S4.Thmtheorem9.p1.10.m10.1.1.1.1.1.1.2" stretchy="false" xref="S4.Thmtheorem9.p1.10.m10.1.1.1.1.1.2.1.cmml">|</mo><msubsup id="S4.Thmtheorem9.p1.10.m10.1.1.1.1.1.1.1" xref="S4.Thmtheorem9.p1.10.m10.1.1.1.1.1.1.1.cmml"><mi id="S4.Thmtheorem9.p1.10.m10.1.1.1.1.1.1.1.2.2" xref="S4.Thmtheorem9.p1.10.m10.1.1.1.1.1.1.1.2.2.cmml">E</mi><mi id="S4.Thmtheorem9.p1.10.m10.1.1.1.1.1.1.1.2.3" xref="S4.Thmtheorem9.p1.10.m10.1.1.1.1.1.1.1.2.3.cmml">I</mi><mo id="S4.Thmtheorem9.p1.10.m10.1.1.1.1.1.1.1.3" xref="S4.Thmtheorem9.p1.10.m10.1.1.1.1.1.1.1.3.cmml">′</mo></msubsup><mo id="S4.Thmtheorem9.p1.10.m10.1.1.1.1.1.1.3" stretchy="false" xref="S4.Thmtheorem9.p1.10.m10.1.1.1.1.1.2.1.cmml">|</mo></mrow><mo id="S4.Thmtheorem9.p1.10.m10.1.1.1.1.2" xref="S4.Thmtheorem9.p1.10.m10.1.1.1.1.2.cmml">≥</mo><mn id="S4.Thmtheorem9.p1.10.m10.1.1.1.1.3" xref="S4.Thmtheorem9.p1.10.m10.1.1.1.1.3.cmml">2</mn></mrow><mo id="S4.Thmtheorem9.p1.10.m10.2.2.2.3" xref="S4.Thmtheorem9.p1.10.m10.2.2.3a.cmml">,</mo><mrow id="S4.Thmtheorem9.p1.10.m10.2.2.2.2" xref="S4.Thmtheorem9.p1.10.m10.2.2.2.2.cmml"><mrow id="S4.Thmtheorem9.p1.10.m10.2.2.2.2.1.1" xref="S4.Thmtheorem9.p1.10.m10.2.2.2.2.1.2.cmml"><mo id="S4.Thmtheorem9.p1.10.m10.2.2.2.2.1.1.2" stretchy="false" xref="S4.Thmtheorem9.p1.10.m10.2.2.2.2.1.2.1.cmml">|</mo><msubsup id="S4.Thmtheorem9.p1.10.m10.2.2.2.2.1.1.1" xref="S4.Thmtheorem9.p1.10.m10.2.2.2.2.1.1.1.cmml"><mi id="S4.Thmtheorem9.p1.10.m10.2.2.2.2.1.1.1.2.2" xref="S4.Thmtheorem9.p1.10.m10.2.2.2.2.1.1.1.2.2.cmml">E</mi><mi id="S4.Thmtheorem9.p1.10.m10.2.2.2.2.1.1.1.2.3" xref="S4.Thmtheorem9.p1.10.m10.2.2.2.2.1.1.1.2.3.cmml">I</mi><mo id="S4.Thmtheorem9.p1.10.m10.2.2.2.2.1.1.1.3" xref="S4.Thmtheorem9.p1.10.m10.2.2.2.2.1.1.1.3.cmml">′′</mo></msubsup><mo id="S4.Thmtheorem9.p1.10.m10.2.2.2.2.1.1.3" stretchy="false" xref="S4.Thmtheorem9.p1.10.m10.2.2.2.2.1.2.1.cmml">|</mo></mrow><mo id="S4.Thmtheorem9.p1.10.m10.2.2.2.2.2" xref="S4.Thmtheorem9.p1.10.m10.2.2.2.2.2.cmml">≥</mo><mn id="S4.Thmtheorem9.p1.10.m10.2.2.2.2.3" xref="S4.Thmtheorem9.p1.10.m10.2.2.2.2.3.cmml">2</mn></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem9.p1.10.m10.2b"><apply id="S4.Thmtheorem9.p1.10.m10.2.2.3.cmml" xref="S4.Thmtheorem9.p1.10.m10.2.2.2"><csymbol cd="ambiguous" id="S4.Thmtheorem9.p1.10.m10.2.2.3a.cmml" xref="S4.Thmtheorem9.p1.10.m10.2.2.2.3">formulae-sequence</csymbol><apply id="S4.Thmtheorem9.p1.10.m10.1.1.1.1.cmml" xref="S4.Thmtheorem9.p1.10.m10.1.1.1.1"><geq id="S4.Thmtheorem9.p1.10.m10.1.1.1.1.2.cmml" xref="S4.Thmtheorem9.p1.10.m10.1.1.1.1.2"></geq><apply id="S4.Thmtheorem9.p1.10.m10.1.1.1.1.1.2.cmml" xref="S4.Thmtheorem9.p1.10.m10.1.1.1.1.1.1"><abs id="S4.Thmtheorem9.p1.10.m10.1.1.1.1.1.2.1.cmml" xref="S4.Thmtheorem9.p1.10.m10.1.1.1.1.1.1.2"></abs><apply id="S4.Thmtheorem9.p1.10.m10.1.1.1.1.1.1.1.cmml" xref="S4.Thmtheorem9.p1.10.m10.1.1.1.1.1.1.1"><csymbol cd="ambiguous" id="S4.Thmtheorem9.p1.10.m10.1.1.1.1.1.1.1.1.cmml" xref="S4.Thmtheorem9.p1.10.m10.1.1.1.1.1.1.1">superscript</csymbol><apply id="S4.Thmtheorem9.p1.10.m10.1.1.1.1.1.1.1.2.cmml" xref="S4.Thmtheorem9.p1.10.m10.1.1.1.1.1.1.1"><csymbol cd="ambiguous" id="S4.Thmtheorem9.p1.10.m10.1.1.1.1.1.1.1.2.1.cmml" xref="S4.Thmtheorem9.p1.10.m10.1.1.1.1.1.1.1">subscript</csymbol><ci id="S4.Thmtheorem9.p1.10.m10.1.1.1.1.1.1.1.2.2.cmml" xref="S4.Thmtheorem9.p1.10.m10.1.1.1.1.1.1.1.2.2">𝐸</ci><ci id="S4.Thmtheorem9.p1.10.m10.1.1.1.1.1.1.1.2.3.cmml" xref="S4.Thmtheorem9.p1.10.m10.1.1.1.1.1.1.1.2.3">𝐼</ci></apply><ci id="S4.Thmtheorem9.p1.10.m10.1.1.1.1.1.1.1.3.cmml" xref="S4.Thmtheorem9.p1.10.m10.1.1.1.1.1.1.1.3">′</ci></apply></apply><cn id="S4.Thmtheorem9.p1.10.m10.1.1.1.1.3.cmml" type="integer" xref="S4.Thmtheorem9.p1.10.m10.1.1.1.1.3">2</cn></apply><apply id="S4.Thmtheorem9.p1.10.m10.2.2.2.2.cmml" xref="S4.Thmtheorem9.p1.10.m10.2.2.2.2"><geq id="S4.Thmtheorem9.p1.10.m10.2.2.2.2.2.cmml" xref="S4.Thmtheorem9.p1.10.m10.2.2.2.2.2"></geq><apply id="S4.Thmtheorem9.p1.10.m10.2.2.2.2.1.2.cmml" xref="S4.Thmtheorem9.p1.10.m10.2.2.2.2.1.1"><abs id="S4.Thmtheorem9.p1.10.m10.2.2.2.2.1.2.1.cmml" xref="S4.Thmtheorem9.p1.10.m10.2.2.2.2.1.1.2"></abs><apply id="S4.Thmtheorem9.p1.10.m10.2.2.2.2.1.1.1.cmml" xref="S4.Thmtheorem9.p1.10.m10.2.2.2.2.1.1.1"><csymbol cd="ambiguous" id="S4.Thmtheorem9.p1.10.m10.2.2.2.2.1.1.1.1.cmml" xref="S4.Thmtheorem9.p1.10.m10.2.2.2.2.1.1.1">superscript</csymbol><apply id="S4.Thmtheorem9.p1.10.m10.2.2.2.2.1.1.1.2.cmml" xref="S4.Thmtheorem9.p1.10.m10.2.2.2.2.1.1.1"><csymbol cd="ambiguous" id="S4.Thmtheorem9.p1.10.m10.2.2.2.2.1.1.1.2.1.cmml" xref="S4.Thmtheorem9.p1.10.m10.2.2.2.2.1.1.1">subscript</csymbol><ci id="S4.Thmtheorem9.p1.10.m10.2.2.2.2.1.1.1.2.2.cmml" xref="S4.Thmtheorem9.p1.10.m10.2.2.2.2.1.1.1.2.2">𝐸</ci><ci id="S4.Thmtheorem9.p1.10.m10.2.2.2.2.1.1.1.2.3.cmml" xref="S4.Thmtheorem9.p1.10.m10.2.2.2.2.1.1.1.2.3">𝐼</ci></apply><ci id="S4.Thmtheorem9.p1.10.m10.2.2.2.2.1.1.1.3.cmml" xref="S4.Thmtheorem9.p1.10.m10.2.2.2.2.1.1.1.3">′′</ci></apply></apply><cn id="S4.Thmtheorem9.p1.10.m10.2.2.2.2.3.cmml" type="integer" xref="S4.Thmtheorem9.p1.10.m10.2.2.2.2.3">2</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem9.p1.10.m10.2c">|E_{I}^{\prime}|\geq 2,|E_{I}^{\prime\prime}|\geq 2</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem9.p1.10.m10.2d">| italic_E start_POSTSUBSCRIPT italic_I end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT | ≥ 2 , | italic_E start_POSTSUBSCRIPT italic_I end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT | ≥ 2</annotation></semantics></math>. We introduce a new <em class="ltx_emph ltx_font_italic" id="S4.Thmtheorem9.p1.14.2">virtual edge</em> <math alttext="e=ab" class="ltx_Math" display="inline" id="S4.Thmtheorem9.p1.11.m11.1"><semantics id="S4.Thmtheorem9.p1.11.m11.1a"><mrow id="S4.Thmtheorem9.p1.11.m11.1.1" xref="S4.Thmtheorem9.p1.11.m11.1.1.cmml"><mi id="S4.Thmtheorem9.p1.11.m11.1.1.2" xref="S4.Thmtheorem9.p1.11.m11.1.1.2.cmml">e</mi><mo id="S4.Thmtheorem9.p1.11.m11.1.1.1" xref="S4.Thmtheorem9.p1.11.m11.1.1.1.cmml">=</mo><mrow id="S4.Thmtheorem9.p1.11.m11.1.1.3" xref="S4.Thmtheorem9.p1.11.m11.1.1.3.cmml"><mi id="S4.Thmtheorem9.p1.11.m11.1.1.3.2" xref="S4.Thmtheorem9.p1.11.m11.1.1.3.2.cmml">a</mi><mo id="S4.Thmtheorem9.p1.11.m11.1.1.3.1" xref="S4.Thmtheorem9.p1.11.m11.1.1.3.1.cmml">⁢</mo><mi id="S4.Thmtheorem9.p1.11.m11.1.1.3.3" xref="S4.Thmtheorem9.p1.11.m11.1.1.3.3.cmml">b</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem9.p1.11.m11.1b"><apply id="S4.Thmtheorem9.p1.11.m11.1.1.cmml" xref="S4.Thmtheorem9.p1.11.m11.1.1"><eq id="S4.Thmtheorem9.p1.11.m11.1.1.1.cmml" xref="S4.Thmtheorem9.p1.11.m11.1.1.1"></eq><ci id="S4.Thmtheorem9.p1.11.m11.1.1.2.cmml" xref="S4.Thmtheorem9.p1.11.m11.1.1.2">𝑒</ci><apply id="S4.Thmtheorem9.p1.11.m11.1.1.3.cmml" xref="S4.Thmtheorem9.p1.11.m11.1.1.3"><times id="S4.Thmtheorem9.p1.11.m11.1.1.3.1.cmml" xref="S4.Thmtheorem9.p1.11.m11.1.1.3.1"></times><ci id="S4.Thmtheorem9.p1.11.m11.1.1.3.2.cmml" xref="S4.Thmtheorem9.p1.11.m11.1.1.3.2">𝑎</ci><ci id="S4.Thmtheorem9.p1.11.m11.1.1.3.3.cmml" xref="S4.Thmtheorem9.p1.11.m11.1.1.3.3">𝑏</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem9.p1.11.m11.1c">e=ab</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem9.p1.11.m11.1d">italic_e = italic_a italic_b</annotation></semantics></math>, and define two new multigraphs: <math alttext="G^{\prime}=(V(E_{I}^{\prime}),E_{I}^{\prime}\cup\{e\}),G^{\prime\prime}=(V(E_{% I}^{\prime\prime}),E_{I}^{\prime\prime}\cup\{e\})" class="ltx_Math" display="inline" id="S4.Thmtheorem9.p1.12.m12.4"><semantics id="S4.Thmtheorem9.p1.12.m12.4a"><mrow id="S4.Thmtheorem9.p1.12.m12.4.4.2" xref="S4.Thmtheorem9.p1.12.m12.4.4.3.cmml"><mrow id="S4.Thmtheorem9.p1.12.m12.3.3.1.1" xref="S4.Thmtheorem9.p1.12.m12.3.3.1.1.cmml"><msup id="S4.Thmtheorem9.p1.12.m12.3.3.1.1.4" xref="S4.Thmtheorem9.p1.12.m12.3.3.1.1.4.cmml"><mi id="S4.Thmtheorem9.p1.12.m12.3.3.1.1.4.2" xref="S4.Thmtheorem9.p1.12.m12.3.3.1.1.4.2.cmml">G</mi><mo id="S4.Thmtheorem9.p1.12.m12.3.3.1.1.4.3" xref="S4.Thmtheorem9.p1.12.m12.3.3.1.1.4.3.cmml">′</mo></msup><mo id="S4.Thmtheorem9.p1.12.m12.3.3.1.1.3" xref="S4.Thmtheorem9.p1.12.m12.3.3.1.1.3.cmml">=</mo><mrow id="S4.Thmtheorem9.p1.12.m12.3.3.1.1.2.2" xref="S4.Thmtheorem9.p1.12.m12.3.3.1.1.2.3.cmml"><mo id="S4.Thmtheorem9.p1.12.m12.3.3.1.1.2.2.3" stretchy="false" xref="S4.Thmtheorem9.p1.12.m12.3.3.1.1.2.3.cmml">(</mo><mrow id="S4.Thmtheorem9.p1.12.m12.3.3.1.1.1.1.1" xref="S4.Thmtheorem9.p1.12.m12.3.3.1.1.1.1.1.cmml"><mi id="S4.Thmtheorem9.p1.12.m12.3.3.1.1.1.1.1.3" xref="S4.Thmtheorem9.p1.12.m12.3.3.1.1.1.1.1.3.cmml">V</mi><mo id="S4.Thmtheorem9.p1.12.m12.3.3.1.1.1.1.1.2" xref="S4.Thmtheorem9.p1.12.m12.3.3.1.1.1.1.1.2.cmml">⁢</mo><mrow id="S4.Thmtheorem9.p1.12.m12.3.3.1.1.1.1.1.1.1" xref="S4.Thmtheorem9.p1.12.m12.3.3.1.1.1.1.1.1.1.1.cmml"><mo id="S4.Thmtheorem9.p1.12.m12.3.3.1.1.1.1.1.1.1.2" stretchy="false" xref="S4.Thmtheorem9.p1.12.m12.3.3.1.1.1.1.1.1.1.1.cmml">(</mo><msubsup id="S4.Thmtheorem9.p1.12.m12.3.3.1.1.1.1.1.1.1.1" xref="S4.Thmtheorem9.p1.12.m12.3.3.1.1.1.1.1.1.1.1.cmml"><mi id="S4.Thmtheorem9.p1.12.m12.3.3.1.1.1.1.1.1.1.1.2.2" xref="S4.Thmtheorem9.p1.12.m12.3.3.1.1.1.1.1.1.1.1.2.2.cmml">E</mi><mi id="S4.Thmtheorem9.p1.12.m12.3.3.1.1.1.1.1.1.1.1.2.3" xref="S4.Thmtheorem9.p1.12.m12.3.3.1.1.1.1.1.1.1.1.2.3.cmml">I</mi><mo id="S4.Thmtheorem9.p1.12.m12.3.3.1.1.1.1.1.1.1.1.3" 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xref="S4.Thmtheorem9.p1.12.m12.4.4.2.2.2.2.2.3.2"><ci id="S4.Thmtheorem9.p1.12.m12.2.2.cmml" xref="S4.Thmtheorem9.p1.12.m12.2.2">𝑒</ci></set></apply></interval></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem9.p1.12.m12.4c">G^{\prime}=(V(E_{I}^{\prime}),E_{I}^{\prime}\cup\{e\}),G^{\prime\prime}=(V(E_{% I}^{\prime\prime}),E_{I}^{\prime\prime}\cup\{e\})</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem9.p1.12.m12.4d">italic_G start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT = ( italic_V ( italic_E start_POSTSUBSCRIPT italic_I end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) , italic_E start_POSTSUBSCRIPT italic_I end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ∪ { italic_e } ) , italic_G start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT = ( italic_V ( italic_E start_POSTSUBSCRIPT italic_I end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT ) , italic_E start_POSTSUBSCRIPT italic_I end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT ∪ { italic_e } )</annotation></semantics></math>. The <em class="ltx_emph ltx_font_italic" id="S4.Thmtheorem9.p1.14.3">split</em> operation replaces <math alttext="G" class="ltx_Math" display="inline" id="S4.Thmtheorem9.p1.13.m13.1"><semantics id="S4.Thmtheorem9.p1.13.m13.1a"><mi id="S4.Thmtheorem9.p1.13.m13.1.1" xref="S4.Thmtheorem9.p1.13.m13.1.1.cmml">G</mi><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem9.p1.13.m13.1b"><ci id="S4.Thmtheorem9.p1.13.m13.1.1.cmml" xref="S4.Thmtheorem9.p1.13.m13.1.1">𝐺</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem9.p1.13.m13.1c">G</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem9.p1.13.m13.1d">italic_G</annotation></semantics></math> with <em class="ltx_emph ltx_font_italic" id="S4.Thmtheorem9.p1.14.4">split graphs</em> <math alttext="G^{\prime},G^{\prime\prime}" class="ltx_Math" display="inline" id="S4.Thmtheorem9.p1.14.m14.2"><semantics id="S4.Thmtheorem9.p1.14.m14.2a"><mrow id="S4.Thmtheorem9.p1.14.m14.2.2.2" xref="S4.Thmtheorem9.p1.14.m14.2.2.3.cmml"><msup id="S4.Thmtheorem9.p1.14.m14.1.1.1.1" xref="S4.Thmtheorem9.p1.14.m14.1.1.1.1.cmml"><mi id="S4.Thmtheorem9.p1.14.m14.1.1.1.1.2" xref="S4.Thmtheorem9.p1.14.m14.1.1.1.1.2.cmml">G</mi><mo id="S4.Thmtheorem9.p1.14.m14.1.1.1.1.3" xref="S4.Thmtheorem9.p1.14.m14.1.1.1.1.3.cmml">′</mo></msup><mo id="S4.Thmtheorem9.p1.14.m14.2.2.2.3" xref="S4.Thmtheorem9.p1.14.m14.2.2.3.cmml">,</mo><msup id="S4.Thmtheorem9.p1.14.m14.2.2.2.2" xref="S4.Thmtheorem9.p1.14.m14.2.2.2.2.cmml"><mi id="S4.Thmtheorem9.p1.14.m14.2.2.2.2.2" xref="S4.Thmtheorem9.p1.14.m14.2.2.2.2.2.cmml">G</mi><mo id="S4.Thmtheorem9.p1.14.m14.2.2.2.2.3" xref="S4.Thmtheorem9.p1.14.m14.2.2.2.2.3.cmml">′′</mo></msup></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem9.p1.14.m14.2b"><list id="S4.Thmtheorem9.p1.14.m14.2.2.3.cmml" xref="S4.Thmtheorem9.p1.14.m14.2.2.2"><apply id="S4.Thmtheorem9.p1.14.m14.1.1.1.1.cmml" xref="S4.Thmtheorem9.p1.14.m14.1.1.1.1"><csymbol cd="ambiguous" id="S4.Thmtheorem9.p1.14.m14.1.1.1.1.1.cmml" xref="S4.Thmtheorem9.p1.14.m14.1.1.1.1">superscript</csymbol><ci id="S4.Thmtheorem9.p1.14.m14.1.1.1.1.2.cmml" xref="S4.Thmtheorem9.p1.14.m14.1.1.1.1.2">𝐺</ci><ci id="S4.Thmtheorem9.p1.14.m14.1.1.1.1.3.cmml" xref="S4.Thmtheorem9.p1.14.m14.1.1.1.1.3">′</ci></apply><apply id="S4.Thmtheorem9.p1.14.m14.2.2.2.2.cmml" xref="S4.Thmtheorem9.p1.14.m14.2.2.2.2"><csymbol cd="ambiguous" id="S4.Thmtheorem9.p1.14.m14.2.2.2.2.1.cmml" xref="S4.Thmtheorem9.p1.14.m14.2.2.2.2">superscript</csymbol><ci id="S4.Thmtheorem9.p1.14.m14.2.2.2.2.2.cmml" xref="S4.Thmtheorem9.p1.14.m14.2.2.2.2.2">𝐺</ci><ci id="S4.Thmtheorem9.p1.14.m14.2.2.2.2.3.cmml" xref="S4.Thmtheorem9.p1.14.m14.2.2.2.2.3">′′</ci></apply></list></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem9.p1.14.m14.2c">G^{\prime},G^{\prime\prime}</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem9.p1.14.m14.2d">italic_G start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT , italic_G start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT</annotation></semantics></math>.</p> </div> </div> <div class="ltx_para" id="S4.SS2.SSS1.p3"> <p class="ltx_p" id="S4.SS2.SSS1.p3.3">It is easy to verify that for a 2-connected graph <math alttext="G" class="ltx_Math" display="inline" id="S4.SS2.SSS1.p3.1.m1.1"><semantics id="S4.SS2.SSS1.p3.1.m1.1a"><mi id="S4.SS2.SSS1.p3.1.m1.1.1" xref="S4.SS2.SSS1.p3.1.m1.1.1.cmml">G</mi><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS1.p3.1.m1.1b"><ci id="S4.SS2.SSS1.p3.1.m1.1.1.cmml" xref="S4.SS2.SSS1.p3.1.m1.1.1">𝐺</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS1.p3.1.m1.1c">G</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS1.p3.1.m1.1d">italic_G</annotation></semantics></math>, the split graphs <math alttext="G^{\prime},G^{\prime\prime}" class="ltx_Math" display="inline" id="S4.SS2.SSS1.p3.2.m2.2"><semantics id="S4.SS2.SSS1.p3.2.m2.2a"><mrow id="S4.SS2.SSS1.p3.2.m2.2.2.2" xref="S4.SS2.SSS1.p3.2.m2.2.2.3.cmml"><msup id="S4.SS2.SSS1.p3.2.m2.1.1.1.1" xref="S4.SS2.SSS1.p3.2.m2.1.1.1.1.cmml"><mi id="S4.SS2.SSS1.p3.2.m2.1.1.1.1.2" xref="S4.SS2.SSS1.p3.2.m2.1.1.1.1.2.cmml">G</mi><mo id="S4.SS2.SSS1.p3.2.m2.1.1.1.1.3" xref="S4.SS2.SSS1.p3.2.m2.1.1.1.1.3.cmml">′</mo></msup><mo id="S4.SS2.SSS1.p3.2.m2.2.2.2.3" xref="S4.SS2.SSS1.p3.2.m2.2.2.3.cmml">,</mo><msup id="S4.SS2.SSS1.p3.2.m2.2.2.2.2" xref="S4.SS2.SSS1.p3.2.m2.2.2.2.2.cmml"><mi id="S4.SS2.SSS1.p3.2.m2.2.2.2.2.2" xref="S4.SS2.SSS1.p3.2.m2.2.2.2.2.2.cmml">G</mi><mo id="S4.SS2.SSS1.p3.2.m2.2.2.2.2.3" xref="S4.SS2.SSS1.p3.2.m2.2.2.2.2.3.cmml">′′</mo></msup></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS1.p3.2.m2.2b"><list id="S4.SS2.SSS1.p3.2.m2.2.2.3.cmml" xref="S4.SS2.SSS1.p3.2.m2.2.2.2"><apply id="S4.SS2.SSS1.p3.2.m2.1.1.1.1.cmml" xref="S4.SS2.SSS1.p3.2.m2.1.1.1.1"><csymbol cd="ambiguous" id="S4.SS2.SSS1.p3.2.m2.1.1.1.1.1.cmml" xref="S4.SS2.SSS1.p3.2.m2.1.1.1.1">superscript</csymbol><ci id="S4.SS2.SSS1.p3.2.m2.1.1.1.1.2.cmml" xref="S4.SS2.SSS1.p3.2.m2.1.1.1.1.2">𝐺</ci><ci id="S4.SS2.SSS1.p3.2.m2.1.1.1.1.3.cmml" xref="S4.SS2.SSS1.p3.2.m2.1.1.1.1.3">′</ci></apply><apply id="S4.SS2.SSS1.p3.2.m2.2.2.2.2.cmml" xref="S4.SS2.SSS1.p3.2.m2.2.2.2.2"><csymbol cd="ambiguous" id="S4.SS2.SSS1.p3.2.m2.2.2.2.2.1.cmml" xref="S4.SS2.SSS1.p3.2.m2.2.2.2.2">superscript</csymbol><ci id="S4.SS2.SSS1.p3.2.m2.2.2.2.2.2.cmml" xref="S4.SS2.SSS1.p3.2.m2.2.2.2.2.2">𝐺</ci><ci id="S4.SS2.SSS1.p3.2.m2.2.2.2.2.3.cmml" xref="S4.SS2.SSS1.p3.2.m2.2.2.2.2.3">′′</ci></apply></list></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS1.p3.2.m2.2c">G^{\prime},G^{\prime\prime}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS1.p3.2.m2.2d">italic_G start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT , italic_G start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT</annotation></semantics></math> (constructed using any separation pair) are also <math alttext="2" class="ltx_Math" display="inline" id="S4.SS2.SSS1.p3.3.m3.1"><semantics id="S4.SS2.SSS1.p3.3.m3.1a"><mn id="S4.SS2.SSS1.p3.3.m3.1.1" xref="S4.SS2.SSS1.p3.3.m3.1.1.cmml">2</mn><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS1.p3.3.m3.1b"><cn id="S4.SS2.SSS1.p3.3.m3.1.1.cmml" type="integer" xref="S4.SS2.SSS1.p3.3.m3.1.1">2</cn></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS1.p3.3.m3.1c">2</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS1.p3.3.m3.1d">2</annotation></semantics></math>-connected.</p> </div> <div class="ltx_theorem ltx_theorem_definition" id="S4.Thmtheorem10"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="S4.Thmtheorem10.1.1.1">Definition 4.10</span></span><span class="ltx_text ltx_font_bold" id="S4.Thmtheorem10.2.2">.</span> </h6> <div class="ltx_para" id="S4.Thmtheorem10.p1"> <p class="ltx_p" id="S4.Thmtheorem10.p1.4">The <em class="ltx_emph ltx_font_italic" id="S4.Thmtheorem10.p1.4.1">merge</em> operation merges two split graphs back to their original graph. Given split graphs <math alttext="G^{\prime}=(V^{\prime},E^{\prime})" class="ltx_Math" display="inline" id="S4.Thmtheorem10.p1.1.m1.2"><semantics id="S4.Thmtheorem10.p1.1.m1.2a"><mrow id="S4.Thmtheorem10.p1.1.m1.2.2" xref="S4.Thmtheorem10.p1.1.m1.2.2.cmml"><msup id="S4.Thmtheorem10.p1.1.m1.2.2.4" xref="S4.Thmtheorem10.p1.1.m1.2.2.4.cmml"><mi id="S4.Thmtheorem10.p1.1.m1.2.2.4.2" xref="S4.Thmtheorem10.p1.1.m1.2.2.4.2.cmml">G</mi><mo id="S4.Thmtheorem10.p1.1.m1.2.2.4.3" xref="S4.Thmtheorem10.p1.1.m1.2.2.4.3.cmml">′</mo></msup><mo id="S4.Thmtheorem10.p1.1.m1.2.2.3" xref="S4.Thmtheorem10.p1.1.m1.2.2.3.cmml">=</mo><mrow id="S4.Thmtheorem10.p1.1.m1.2.2.2.2" xref="S4.Thmtheorem10.p1.1.m1.2.2.2.3.cmml"><mo id="S4.Thmtheorem10.p1.1.m1.2.2.2.2.3" stretchy="false" xref="S4.Thmtheorem10.p1.1.m1.2.2.2.3.cmml">(</mo><msup id="S4.Thmtheorem10.p1.1.m1.1.1.1.1.1" xref="S4.Thmtheorem10.p1.1.m1.1.1.1.1.1.cmml"><mi id="S4.Thmtheorem10.p1.1.m1.1.1.1.1.1.2" xref="S4.Thmtheorem10.p1.1.m1.1.1.1.1.1.2.cmml">V</mi><mo id="S4.Thmtheorem10.p1.1.m1.1.1.1.1.1.3" xref="S4.Thmtheorem10.p1.1.m1.1.1.1.1.1.3.cmml">′</mo></msup><mo id="S4.Thmtheorem10.p1.1.m1.2.2.2.2.4" xref="S4.Thmtheorem10.p1.1.m1.2.2.2.3.cmml">,</mo><msup id="S4.Thmtheorem10.p1.1.m1.2.2.2.2.2" xref="S4.Thmtheorem10.p1.1.m1.2.2.2.2.2.cmml"><mi id="S4.Thmtheorem10.p1.1.m1.2.2.2.2.2.2" xref="S4.Thmtheorem10.p1.1.m1.2.2.2.2.2.2.cmml">E</mi><mo id="S4.Thmtheorem10.p1.1.m1.2.2.2.2.2.3" xref="S4.Thmtheorem10.p1.1.m1.2.2.2.2.2.3.cmml">′</mo></msup><mo id="S4.Thmtheorem10.p1.1.m1.2.2.2.2.5" stretchy="false" xref="S4.Thmtheorem10.p1.1.m1.2.2.2.3.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem10.p1.1.m1.2b"><apply id="S4.Thmtheorem10.p1.1.m1.2.2.cmml" xref="S4.Thmtheorem10.p1.1.m1.2.2"><eq id="S4.Thmtheorem10.p1.1.m1.2.2.3.cmml" xref="S4.Thmtheorem10.p1.1.m1.2.2.3"></eq><apply id="S4.Thmtheorem10.p1.1.m1.2.2.4.cmml" xref="S4.Thmtheorem10.p1.1.m1.2.2.4"><csymbol cd="ambiguous" id="S4.Thmtheorem10.p1.1.m1.2.2.4.1.cmml" xref="S4.Thmtheorem10.p1.1.m1.2.2.4">superscript</csymbol><ci id="S4.Thmtheorem10.p1.1.m1.2.2.4.2.cmml" xref="S4.Thmtheorem10.p1.1.m1.2.2.4.2">𝐺</ci><ci id="S4.Thmtheorem10.p1.1.m1.2.2.4.3.cmml" xref="S4.Thmtheorem10.p1.1.m1.2.2.4.3">′</ci></apply><interval closure="open" id="S4.Thmtheorem10.p1.1.m1.2.2.2.3.cmml" xref="S4.Thmtheorem10.p1.1.m1.2.2.2.2"><apply id="S4.Thmtheorem10.p1.1.m1.1.1.1.1.1.cmml" xref="S4.Thmtheorem10.p1.1.m1.1.1.1.1.1"><csymbol cd="ambiguous" id="S4.Thmtheorem10.p1.1.m1.1.1.1.1.1.1.cmml" xref="S4.Thmtheorem10.p1.1.m1.1.1.1.1.1">superscript</csymbol><ci id="S4.Thmtheorem10.p1.1.m1.1.1.1.1.1.2.cmml" xref="S4.Thmtheorem10.p1.1.m1.1.1.1.1.1.2">𝑉</ci><ci id="S4.Thmtheorem10.p1.1.m1.1.1.1.1.1.3.cmml" xref="S4.Thmtheorem10.p1.1.m1.1.1.1.1.1.3">′</ci></apply><apply id="S4.Thmtheorem10.p1.1.m1.2.2.2.2.2.cmml" xref="S4.Thmtheorem10.p1.1.m1.2.2.2.2.2"><csymbol cd="ambiguous" id="S4.Thmtheorem10.p1.1.m1.2.2.2.2.2.1.cmml" xref="S4.Thmtheorem10.p1.1.m1.2.2.2.2.2">superscript</csymbol><ci id="S4.Thmtheorem10.p1.1.m1.2.2.2.2.2.2.cmml" xref="S4.Thmtheorem10.p1.1.m1.2.2.2.2.2.2">𝐸</ci><ci id="S4.Thmtheorem10.p1.1.m1.2.2.2.2.2.3.cmml" xref="S4.Thmtheorem10.p1.1.m1.2.2.2.2.2.3">′</ci></apply></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem10.p1.1.m1.2c">G^{\prime}=(V^{\prime},E^{\prime})</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem10.p1.1.m1.2d">italic_G start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT = ( italic_V start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT , italic_E start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT )</annotation></semantics></math> and <math alttext="G^{\prime\prime}=(V^{\prime\prime},E^{\prime\prime})" class="ltx_Math" display="inline" id="S4.Thmtheorem10.p1.2.m2.2"><semantics id="S4.Thmtheorem10.p1.2.m2.2a"><mrow id="S4.Thmtheorem10.p1.2.m2.2.2" 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id="S4.Thmtheorem10.p1.2.m2.2.2.2.2.2" xref="S4.Thmtheorem10.p1.2.m2.2.2.2.2.2.cmml"><mi id="S4.Thmtheorem10.p1.2.m2.2.2.2.2.2.2" xref="S4.Thmtheorem10.p1.2.m2.2.2.2.2.2.2.cmml">E</mi><mo id="S4.Thmtheorem10.p1.2.m2.2.2.2.2.2.3" xref="S4.Thmtheorem10.p1.2.m2.2.2.2.2.2.3.cmml">′′</mo></msup><mo id="S4.Thmtheorem10.p1.2.m2.2.2.2.2.5" stretchy="false" xref="S4.Thmtheorem10.p1.2.m2.2.2.2.3.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem10.p1.2.m2.2b"><apply id="S4.Thmtheorem10.p1.2.m2.2.2.cmml" xref="S4.Thmtheorem10.p1.2.m2.2.2"><eq id="S4.Thmtheorem10.p1.2.m2.2.2.3.cmml" xref="S4.Thmtheorem10.p1.2.m2.2.2.3"></eq><apply id="S4.Thmtheorem10.p1.2.m2.2.2.4.cmml" xref="S4.Thmtheorem10.p1.2.m2.2.2.4"><csymbol cd="ambiguous" id="S4.Thmtheorem10.p1.2.m2.2.2.4.1.cmml" xref="S4.Thmtheorem10.p1.2.m2.2.2.4">superscript</csymbol><ci id="S4.Thmtheorem10.p1.2.m2.2.2.4.2.cmml" xref="S4.Thmtheorem10.p1.2.m2.2.2.4.2">𝐺</ci><ci id="S4.Thmtheorem10.p1.2.m2.2.2.4.3.cmml" xref="S4.Thmtheorem10.p1.2.m2.2.2.4.3">′′</ci></apply><interval closure="open" id="S4.Thmtheorem10.p1.2.m2.2.2.2.3.cmml" xref="S4.Thmtheorem10.p1.2.m2.2.2.2.2"><apply id="S4.Thmtheorem10.p1.2.m2.1.1.1.1.1.cmml" xref="S4.Thmtheorem10.p1.2.m2.1.1.1.1.1"><csymbol cd="ambiguous" id="S4.Thmtheorem10.p1.2.m2.1.1.1.1.1.1.cmml" xref="S4.Thmtheorem10.p1.2.m2.1.1.1.1.1">superscript</csymbol><ci id="S4.Thmtheorem10.p1.2.m2.1.1.1.1.1.2.cmml" xref="S4.Thmtheorem10.p1.2.m2.1.1.1.1.1.2">𝑉</ci><ci id="S4.Thmtheorem10.p1.2.m2.1.1.1.1.1.3.cmml" xref="S4.Thmtheorem10.p1.2.m2.1.1.1.1.1.3">′′</ci></apply><apply id="S4.Thmtheorem10.p1.2.m2.2.2.2.2.2.cmml" xref="S4.Thmtheorem10.p1.2.m2.2.2.2.2.2"><csymbol cd="ambiguous" id="S4.Thmtheorem10.p1.2.m2.2.2.2.2.2.1.cmml" xref="S4.Thmtheorem10.p1.2.m2.2.2.2.2.2">superscript</csymbol><ci id="S4.Thmtheorem10.p1.2.m2.2.2.2.2.2.2.cmml" xref="S4.Thmtheorem10.p1.2.m2.2.2.2.2.2.2">𝐸</ci><ci id="S4.Thmtheorem10.p1.2.m2.2.2.2.2.2.3.cmml" xref="S4.Thmtheorem10.p1.2.m2.2.2.2.2.2.3">′′</ci></apply></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem10.p1.2.m2.2c">G^{\prime\prime}=(V^{\prime\prime},E^{\prime\prime})</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem10.p1.2.m2.2d">italic_G start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT = ( italic_V start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT , italic_E start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT )</annotation></semantics></math> with the same virtual edge <math alttext="e" class="ltx_Math" display="inline" id="S4.Thmtheorem10.p1.3.m3.1"><semantics id="S4.Thmtheorem10.p1.3.m3.1a"><mi id="S4.Thmtheorem10.p1.3.m3.1.1" xref="S4.Thmtheorem10.p1.3.m3.1.1.cmml">e</mi><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem10.p1.3.m3.1b"><ci id="S4.Thmtheorem10.p1.3.m3.1.1.cmml" xref="S4.Thmtheorem10.p1.3.m3.1.1">𝑒</ci></annotation-xml><annotation encoding="application/x-tex" 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id="S4.Thmtheorem10.p1.4.m4.2.2.1.1.1.2" xref="S4.Thmtheorem10.p1.4.m4.2.2.1.1.1.2.cmml"><mi id="S4.Thmtheorem10.p1.4.m4.2.2.1.1.1.2.2" xref="S4.Thmtheorem10.p1.4.m4.2.2.1.1.1.2.2.cmml">V</mi><mo id="S4.Thmtheorem10.p1.4.m4.2.2.1.1.1.2.3" xref="S4.Thmtheorem10.p1.4.m4.2.2.1.1.1.2.3.cmml">′</mo></msup><mo id="S4.Thmtheorem10.p1.4.m4.2.2.1.1.1.1" xref="S4.Thmtheorem10.p1.4.m4.2.2.1.1.1.1.cmml">∪</mo><msup id="S4.Thmtheorem10.p1.4.m4.2.2.1.1.1.3" xref="S4.Thmtheorem10.p1.4.m4.2.2.1.1.1.3.cmml"><mi id="S4.Thmtheorem10.p1.4.m4.2.2.1.1.1.3.2" xref="S4.Thmtheorem10.p1.4.m4.2.2.1.1.1.3.2.cmml">V</mi><mo id="S4.Thmtheorem10.p1.4.m4.2.2.1.1.1.3.3" xref="S4.Thmtheorem10.p1.4.m4.2.2.1.1.1.3.3.cmml">′′</mo></msup></mrow><mo id="S4.Thmtheorem10.p1.4.m4.3.3.2.2.4" xref="S4.Thmtheorem10.p1.4.m4.3.3.2.3.cmml">,</mo><mrow id="S4.Thmtheorem10.p1.4.m4.3.3.2.2.2" xref="S4.Thmtheorem10.p1.4.m4.3.3.2.2.2.cmml"><mrow id="S4.Thmtheorem10.p1.4.m4.3.3.2.2.2.1.1" 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id="S4.Thmtheorem10.p1.4.m4.3.3.2.2.2.1.1.1.2.1.cmml" xref="S4.Thmtheorem10.p1.4.m4.3.3.2.2.2.1.1.1.2">superscript</csymbol><ci id="S4.Thmtheorem10.p1.4.m4.3.3.2.2.2.1.1.1.2.2.cmml" xref="S4.Thmtheorem10.p1.4.m4.3.3.2.2.2.1.1.1.2.2">𝐸</ci><ci id="S4.Thmtheorem10.p1.4.m4.3.3.2.2.2.1.1.1.2.3.cmml" xref="S4.Thmtheorem10.p1.4.m4.3.3.2.2.2.1.1.1.2.3">′</ci></apply><apply id="S4.Thmtheorem10.p1.4.m4.3.3.2.2.2.1.1.1.3.cmml" xref="S4.Thmtheorem10.p1.4.m4.3.3.2.2.2.1.1.1.3"><csymbol cd="ambiguous" id="S4.Thmtheorem10.p1.4.m4.3.3.2.2.2.1.1.1.3.1.cmml" xref="S4.Thmtheorem10.p1.4.m4.3.3.2.2.2.1.1.1.3">superscript</csymbol><ci id="S4.Thmtheorem10.p1.4.m4.3.3.2.2.2.1.1.1.3.2.cmml" xref="S4.Thmtheorem10.p1.4.m4.3.3.2.2.2.1.1.1.3.2">𝐸</ci><ci id="S4.Thmtheorem10.p1.4.m4.3.3.2.2.2.1.1.1.3.3.cmml" xref="S4.Thmtheorem10.p1.4.m4.3.3.2.2.2.1.1.1.3.3">′′</ci></apply></apply><set id="S4.Thmtheorem10.p1.4.m4.3.3.2.2.2.3.1.cmml" xref="S4.Thmtheorem10.p1.4.m4.3.3.2.2.2.3.2"><ci id="S4.Thmtheorem10.p1.4.m4.1.1.cmml" xref="S4.Thmtheorem10.p1.4.m4.1.1">𝑒</ci></set></apply></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem10.p1.4.m4.3c">G=(V^{\prime}\cup V^{\prime\prime},(E^{\prime}\cup E^{\prime\prime})\setminus% \{e\})</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem10.p1.4.m4.3d">italic_G = ( italic_V start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ∪ italic_V start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT , ( italic_E start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ∪ italic_E start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT ) ∖ { italic_e } )</annotation></semantics></math>.</p> </div> </div> <div class="ltx_para" id="S4.SS2.SSS1.p4"> <p class="ltx_p" id="S4.SS2.SSS1.p4.1">With these operations in hand, we are ready to define SPQR trees.</p> </div> <div class="ltx_theorem ltx_theorem_definition" id="S4.Thmtheorem11"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="S4.Thmtheorem11.1.1.1">Definition 4.11</span></span><span class="ltx_text ltx_font_bold" id="S4.Thmtheorem11.2.2">.</span> </h6> <div class="ltx_para" id="S4.Thmtheorem11.p1"> <p class="ltx_p" id="S4.Thmtheorem11.p1.6">The SPQR tree <math alttext="T" class="ltx_Math" display="inline" id="S4.Thmtheorem11.p1.1.m1.1"><semantics id="S4.Thmtheorem11.p1.1.m1.1a"><mi id="S4.Thmtheorem11.p1.1.m1.1.1" xref="S4.Thmtheorem11.p1.1.m1.1.1.cmml">T</mi><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem11.p1.1.m1.1b"><ci id="S4.Thmtheorem11.p1.1.m1.1.1.cmml" xref="S4.Thmtheorem11.p1.1.m1.1.1">𝑇</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem11.p1.1.m1.1c">T</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem11.p1.1.m1.1d">italic_T</annotation></semantics></math> of a graph <math alttext="G" class="ltx_Math" display="inline" id="S4.Thmtheorem11.p1.2.m2.1"><semantics id="S4.Thmtheorem11.p1.2.m2.1a"><mi id="S4.Thmtheorem11.p1.2.m2.1.1" xref="S4.Thmtheorem11.p1.2.m2.1.1.cmml">G</mi><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem11.p1.2.m2.1b"><ci id="S4.Thmtheorem11.p1.2.m2.1.1.cmml" xref="S4.Thmtheorem11.p1.2.m2.1.1">𝐺</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem11.p1.2.m2.1c">G</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem11.p1.2.m2.1d">italic_G</annotation></semantics></math> is a tree-like data structure in which each node of <math alttext="T" class="ltx_Math" display="inline" id="S4.Thmtheorem11.p1.3.m3.1"><semantics id="S4.Thmtheorem11.p1.3.m3.1a"><mi id="S4.Thmtheorem11.p1.3.m3.1.1" xref="S4.Thmtheorem11.p1.3.m3.1.1.cmml">T</mi><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem11.p1.3.m3.1b"><ci id="S4.Thmtheorem11.p1.3.m3.1.1.cmml" xref="S4.Thmtheorem11.p1.3.m3.1.1">𝑇</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem11.p1.3.m3.1c">T</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem11.p1.3.m3.1d">italic_T</annotation></semantics></math> corresponds to some subgraph of <math alttext="G" class="ltx_Math" display="inline" id="S4.Thmtheorem11.p1.4.m4.1"><semantics id="S4.Thmtheorem11.p1.4.m4.1a"><mi id="S4.Thmtheorem11.p1.4.m4.1.1" xref="S4.Thmtheorem11.p1.4.m4.1.1.cmml">G</mi><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem11.p1.4.m4.1b"><ci id="S4.Thmtheorem11.p1.4.m4.1.1.cmml" xref="S4.Thmtheorem11.p1.4.m4.1.1">𝐺</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem11.p1.4.m4.1c">G</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem11.p1.4.m4.1d">italic_G</annotation></semantics></math>, with additional “virtual” edges. Formally, the SPQR tree <math alttext="T" class="ltx_Math" display="inline" id="S4.Thmtheorem11.p1.5.m5.1"><semantics id="S4.Thmtheorem11.p1.5.m5.1a"><mi id="S4.Thmtheorem11.p1.5.m5.1.1" xref="S4.Thmtheorem11.p1.5.m5.1.1.cmml">T</mi><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem11.p1.5.m5.1b"><ci id="S4.Thmtheorem11.p1.5.m5.1.1.cmml" xref="S4.Thmtheorem11.p1.5.m5.1.1">𝑇</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem11.p1.5.m5.1c">T</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem11.p1.5.m5.1d">italic_T</annotation></semantics></math> is obtained from <math alttext="G" class="ltx_Math" display="inline" id="S4.Thmtheorem11.p1.6.m6.1"><semantics id="S4.Thmtheorem11.p1.6.m6.1a"><mi id="S4.Thmtheorem11.p1.6.m6.1.1" xref="S4.Thmtheorem11.p1.6.m6.1.1.cmml">G</mi><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem11.p1.6.m6.1b"><ci id="S4.Thmtheorem11.p1.6.m6.1.1.cmml" xref="S4.Thmtheorem11.p1.6.m6.1.1">𝐺</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem11.p1.6.m6.1c">G</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem11.p1.6.m6.1d">italic_G</annotation></semantics></math> via the following recursive procedure:</p> <ul class="ltx_itemize" id="S4.I4"> <li class="ltx_item" id="S4.I4.i1" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">•</span> <div class="ltx_para" id="S4.I4.i1.p1"> <p class="ltx_p" id="S4.I4.i1.p1.5">If <math alttext="G" class="ltx_Math" display="inline" id="S4.I4.i1.p1.1.m1.1"><semantics id="S4.I4.i1.p1.1.m1.1a"><mi id="S4.I4.i1.p1.1.m1.1.1" xref="S4.I4.i1.p1.1.m1.1.1.cmml">G</mi><annotation-xml encoding="MathML-Content" id="S4.I4.i1.p1.1.m1.1b"><ci id="S4.I4.i1.p1.1.m1.1.1.cmml" xref="S4.I4.i1.p1.1.m1.1.1">𝐺</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.I4.i1.p1.1.m1.1c">G</annotation><annotation encoding="application/x-llamapun" id="S4.I4.i1.p1.1.m1.1d">italic_G</annotation></semantics></math> has no separation pair, then <math alttext="T" class="ltx_Math" display="inline" id="S4.I4.i1.p1.2.m2.1"><semantics id="S4.I4.i1.p1.2.m2.1a"><mi id="S4.I4.i1.p1.2.m2.1.1" xref="S4.I4.i1.p1.2.m2.1.1.cmml">T</mi><annotation-xml encoding="MathML-Content" id="S4.I4.i1.p1.2.m2.1b"><ci id="S4.I4.i1.p1.2.m2.1.1.cmml" xref="S4.I4.i1.p1.2.m2.1.1">𝑇</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.I4.i1.p1.2.m2.1c">T</annotation><annotation encoding="application/x-llamapun" id="S4.I4.i1.p1.2.m2.1d">italic_T</annotation></semantics></math> is the singleton tree with one node <math alttext="x" class="ltx_Math" display="inline" id="S4.I4.i1.p1.3.m3.1"><semantics id="S4.I4.i1.p1.3.m3.1a"><mi id="S4.I4.i1.p1.3.m3.1.1" xref="S4.I4.i1.p1.3.m3.1.1.cmml">x</mi><annotation-xml encoding="MathML-Content" id="S4.I4.i1.p1.3.m3.1b"><ci id="S4.I4.i1.p1.3.m3.1.1.cmml" xref="S4.I4.i1.p1.3.m3.1.1">𝑥</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.I4.i1.p1.3.m3.1c">x</annotation><annotation encoding="application/x-llamapun" id="S4.I4.i1.p1.3.m3.1d">italic_x</annotation></semantics></math> and no edges. We write <math alttext="G_{x}:=G" class="ltx_Math" display="inline" id="S4.I4.i1.p1.4.m4.1"><semantics id="S4.I4.i1.p1.4.m4.1a"><mrow id="S4.I4.i1.p1.4.m4.1.1" xref="S4.I4.i1.p1.4.m4.1.1.cmml"><msub id="S4.I4.i1.p1.4.m4.1.1.2" xref="S4.I4.i1.p1.4.m4.1.1.2.cmml"><mi id="S4.I4.i1.p1.4.m4.1.1.2.2" xref="S4.I4.i1.p1.4.m4.1.1.2.2.cmml">G</mi><mi id="S4.I4.i1.p1.4.m4.1.1.2.3" xref="S4.I4.i1.p1.4.m4.1.1.2.3.cmml">x</mi></msub><mo id="S4.I4.i1.p1.4.m4.1.1.1" lspace="0.278em" rspace="0.278em" xref="S4.I4.i1.p1.4.m4.1.1.1.cmml">:=</mo><mi id="S4.I4.i1.p1.4.m4.1.1.3" xref="S4.I4.i1.p1.4.m4.1.1.3.cmml">G</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.I4.i1.p1.4.m4.1b"><apply id="S4.I4.i1.p1.4.m4.1.1.cmml" xref="S4.I4.i1.p1.4.m4.1.1"><csymbol cd="latexml" id="S4.I4.i1.p1.4.m4.1.1.1.cmml" xref="S4.I4.i1.p1.4.m4.1.1.1">assign</csymbol><apply id="S4.I4.i1.p1.4.m4.1.1.2.cmml" xref="S4.I4.i1.p1.4.m4.1.1.2"><csymbol cd="ambiguous" id="S4.I4.i1.p1.4.m4.1.1.2.1.cmml" xref="S4.I4.i1.p1.4.m4.1.1.2">subscript</csymbol><ci id="S4.I4.i1.p1.4.m4.1.1.2.2.cmml" xref="S4.I4.i1.p1.4.m4.1.1.2.2">𝐺</ci><ci id="S4.I4.i1.p1.4.m4.1.1.2.3.cmml" xref="S4.I4.i1.p1.4.m4.1.1.2.3">𝑥</ci></apply><ci id="S4.I4.i1.p1.4.m4.1.1.3.cmml" xref="S4.I4.i1.p1.4.m4.1.1.3">𝐺</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.I4.i1.p1.4.m4.1c">G_{x}:=G</annotation><annotation encoding="application/x-llamapun" id="S4.I4.i1.p1.4.m4.1d">italic_G start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT := italic_G</annotation></semantics></math> to denote the graph corresponding to tree node <math alttext="x" class="ltx_Math" display="inline" id="S4.I4.i1.p1.5.m5.1"><semantics id="S4.I4.i1.p1.5.m5.1a"><mi id="S4.I4.i1.p1.5.m5.1.1" xref="S4.I4.i1.p1.5.m5.1.1.cmml">x</mi><annotation-xml encoding="MathML-Content" id="S4.I4.i1.p1.5.m5.1b"><ci id="S4.I4.i1.p1.5.m5.1.1.cmml" xref="S4.I4.i1.p1.5.m5.1.1">𝑥</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.I4.i1.p1.5.m5.1c">x</annotation><annotation encoding="application/x-llamapun" id="S4.I4.i1.p1.5.m5.1d">italic_x</annotation></semantics></math>.</p> </div> </li> <li class="ltx_item" id="S4.I4.i2" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">•</span> <div class="ltx_para" id="S4.I4.i2.p1"> <p class="ltx_p" id="S4.I4.i2.p1.22">If <math alttext="G" class="ltx_Math" display="inline" id="S4.I4.i2.p1.1.m1.1"><semantics id="S4.I4.i2.p1.1.m1.1a"><mi id="S4.I4.i2.p1.1.m1.1.1" xref="S4.I4.i2.p1.1.m1.1.1.cmml">G</mi><annotation-xml encoding="MathML-Content" id="S4.I4.i2.p1.1.m1.1b"><ci id="S4.I4.i2.p1.1.m1.1.1.cmml" xref="S4.I4.i2.p1.1.m1.1.1">𝐺</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.I4.i2.p1.1.m1.1c">G</annotation><annotation encoding="application/x-llamapun" id="S4.I4.i2.p1.1.m1.1d">italic_G</annotation></semantics></math> has a separation pair <math alttext="\{a,b\}" class="ltx_Math" display="inline" id="S4.I4.i2.p1.2.m2.2"><semantics id="S4.I4.i2.p1.2.m2.2a"><mrow id="S4.I4.i2.p1.2.m2.2.3.2" xref="S4.I4.i2.p1.2.m2.2.3.1.cmml"><mo id="S4.I4.i2.p1.2.m2.2.3.2.1" stretchy="false" xref="S4.I4.i2.p1.2.m2.2.3.1.cmml">{</mo><mi id="S4.I4.i2.p1.2.m2.1.1" xref="S4.I4.i2.p1.2.m2.1.1.cmml">a</mi><mo id="S4.I4.i2.p1.2.m2.2.3.2.2" xref="S4.I4.i2.p1.2.m2.2.3.1.cmml">,</mo><mi id="S4.I4.i2.p1.2.m2.2.2" xref="S4.I4.i2.p1.2.m2.2.2.cmml">b</mi><mo id="S4.I4.i2.p1.2.m2.2.3.2.3" stretchy="false" xref="S4.I4.i2.p1.2.m2.2.3.1.cmml">}</mo></mrow><annotation-xml encoding="MathML-Content" id="S4.I4.i2.p1.2.m2.2b"><set id="S4.I4.i2.p1.2.m2.2.3.1.cmml" xref="S4.I4.i2.p1.2.m2.2.3.2"><ci id="S4.I4.i2.p1.2.m2.1.1.cmml" xref="S4.I4.i2.p1.2.m2.1.1">𝑎</ci><ci id="S4.I4.i2.p1.2.m2.2.2.cmml" xref="S4.I4.i2.p1.2.m2.2.2">𝑏</ci></set></annotation-xml><annotation encoding="application/x-tex" id="S4.I4.i2.p1.2.m2.2c">\{a,b\}</annotation><annotation encoding="application/x-llamapun" id="S4.I4.i2.p1.2.m2.2d">{ italic_a , italic_b }</annotation></semantics></math>, we split <math alttext="G" class="ltx_Math" display="inline" id="S4.I4.i2.p1.3.m3.1"><semantics id="S4.I4.i2.p1.3.m3.1a"><mi id="S4.I4.i2.p1.3.m3.1.1" xref="S4.I4.i2.p1.3.m3.1.1.cmml">G</mi><annotation-xml encoding="MathML-Content" id="S4.I4.i2.p1.3.m3.1b"><ci id="S4.I4.i2.p1.3.m3.1.1.cmml" xref="S4.I4.i2.p1.3.m3.1.1">𝐺</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.I4.i2.p1.3.m3.1c">G</annotation><annotation encoding="application/x-llamapun" id="S4.I4.i2.p1.3.m3.1d">italic_G</annotation></semantics></math> on <math alttext="\{a,b\}" class="ltx_Math" display="inline" id="S4.I4.i2.p1.4.m4.2"><semantics id="S4.I4.i2.p1.4.m4.2a"><mrow id="S4.I4.i2.p1.4.m4.2.3.2" xref="S4.I4.i2.p1.4.m4.2.3.1.cmml"><mo id="S4.I4.i2.p1.4.m4.2.3.2.1" stretchy="false" xref="S4.I4.i2.p1.4.m4.2.3.1.cmml">{</mo><mi id="S4.I4.i2.p1.4.m4.1.1" xref="S4.I4.i2.p1.4.m4.1.1.cmml">a</mi><mo id="S4.I4.i2.p1.4.m4.2.3.2.2" xref="S4.I4.i2.p1.4.m4.2.3.1.cmml">,</mo><mi id="S4.I4.i2.p1.4.m4.2.2" xref="S4.I4.i2.p1.4.m4.2.2.cmml">b</mi><mo id="S4.I4.i2.p1.4.m4.2.3.2.3" stretchy="false" xref="S4.I4.i2.p1.4.m4.2.3.1.cmml">}</mo></mrow><annotation-xml encoding="MathML-Content" id="S4.I4.i2.p1.4.m4.2b"><set id="S4.I4.i2.p1.4.m4.2.3.1.cmml" xref="S4.I4.i2.p1.4.m4.2.3.2"><ci id="S4.I4.i2.p1.4.m4.1.1.cmml" xref="S4.I4.i2.p1.4.m4.1.1">𝑎</ci><ci id="S4.I4.i2.p1.4.m4.2.2.cmml" xref="S4.I4.i2.p1.4.m4.2.2">𝑏</ci></set></annotation-xml><annotation encoding="application/x-tex" id="S4.I4.i2.p1.4.m4.2c">\{a,b\}</annotation><annotation encoding="application/x-llamapun" id="S4.I4.i2.p1.4.m4.2d">{ italic_a , italic_b }</annotation></semantics></math> to obtain split graphs <math alttext="G^{\prime},G^{\prime\prime}" class="ltx_Math" display="inline" id="S4.I4.i2.p1.5.m5.2"><semantics id="S4.I4.i2.p1.5.m5.2a"><mrow id="S4.I4.i2.p1.5.m5.2.2.2" xref="S4.I4.i2.p1.5.m5.2.2.3.cmml"><msup id="S4.I4.i2.p1.5.m5.1.1.1.1" xref="S4.I4.i2.p1.5.m5.1.1.1.1.cmml"><mi id="S4.I4.i2.p1.5.m5.1.1.1.1.2" xref="S4.I4.i2.p1.5.m5.1.1.1.1.2.cmml">G</mi><mo id="S4.I4.i2.p1.5.m5.1.1.1.1.3" xref="S4.I4.i2.p1.5.m5.1.1.1.1.3.cmml">′</mo></msup><mo id="S4.I4.i2.p1.5.m5.2.2.2.3" xref="S4.I4.i2.p1.5.m5.2.2.3.cmml">,</mo><msup id="S4.I4.i2.p1.5.m5.2.2.2.2" xref="S4.I4.i2.p1.5.m5.2.2.2.2.cmml"><mi id="S4.I4.i2.p1.5.m5.2.2.2.2.2" xref="S4.I4.i2.p1.5.m5.2.2.2.2.2.cmml">G</mi><mo id="S4.I4.i2.p1.5.m5.2.2.2.2.3" xref="S4.I4.i2.p1.5.m5.2.2.2.2.3.cmml">′′</mo></msup></mrow><annotation-xml encoding="MathML-Content" id="S4.I4.i2.p1.5.m5.2b"><list id="S4.I4.i2.p1.5.m5.2.2.3.cmml" xref="S4.I4.i2.p1.5.m5.2.2.2"><apply id="S4.I4.i2.p1.5.m5.1.1.1.1.cmml" xref="S4.I4.i2.p1.5.m5.1.1.1.1"><csymbol cd="ambiguous" id="S4.I4.i2.p1.5.m5.1.1.1.1.1.cmml" xref="S4.I4.i2.p1.5.m5.1.1.1.1">superscript</csymbol><ci id="S4.I4.i2.p1.5.m5.1.1.1.1.2.cmml" xref="S4.I4.i2.p1.5.m5.1.1.1.1.2">𝐺</ci><ci id="S4.I4.i2.p1.5.m5.1.1.1.1.3.cmml" xref="S4.I4.i2.p1.5.m5.1.1.1.1.3">′</ci></apply><apply id="S4.I4.i2.p1.5.m5.2.2.2.2.cmml" xref="S4.I4.i2.p1.5.m5.2.2.2.2"><csymbol cd="ambiguous" id="S4.I4.i2.p1.5.m5.2.2.2.2.1.cmml" xref="S4.I4.i2.p1.5.m5.2.2.2.2">superscript</csymbol><ci id="S4.I4.i2.p1.5.m5.2.2.2.2.2.cmml" xref="S4.I4.i2.p1.5.m5.2.2.2.2.2">𝐺</ci><ci id="S4.I4.i2.p1.5.m5.2.2.2.2.3.cmml" xref="S4.I4.i2.p1.5.m5.2.2.2.2.3">′′</ci></apply></list></annotation-xml><annotation encoding="application/x-tex" id="S4.I4.i2.p1.5.m5.2c">G^{\prime},G^{\prime\prime}</annotation><annotation encoding="application/x-llamapun" id="S4.I4.i2.p1.5.m5.2d">italic_G start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT , italic_G start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT</annotation></semantics></math>. Let <math alttext="e=ab" class="ltx_Math" display="inline" id="S4.I4.i2.p1.6.m6.1"><semantics id="S4.I4.i2.p1.6.m6.1a"><mrow id="S4.I4.i2.p1.6.m6.1.1" xref="S4.I4.i2.p1.6.m6.1.1.cmml"><mi id="S4.I4.i2.p1.6.m6.1.1.2" xref="S4.I4.i2.p1.6.m6.1.1.2.cmml">e</mi><mo id="S4.I4.i2.p1.6.m6.1.1.1" xref="S4.I4.i2.p1.6.m6.1.1.1.cmml">=</mo><mrow id="S4.I4.i2.p1.6.m6.1.1.3" xref="S4.I4.i2.p1.6.m6.1.1.3.cmml"><mi id="S4.I4.i2.p1.6.m6.1.1.3.2" xref="S4.I4.i2.p1.6.m6.1.1.3.2.cmml">a</mi><mo id="S4.I4.i2.p1.6.m6.1.1.3.1" xref="S4.I4.i2.p1.6.m6.1.1.3.1.cmml">⁢</mo><mi id="S4.I4.i2.p1.6.m6.1.1.3.3" xref="S4.I4.i2.p1.6.m6.1.1.3.3.cmml">b</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.I4.i2.p1.6.m6.1b"><apply id="S4.I4.i2.p1.6.m6.1.1.cmml" xref="S4.I4.i2.p1.6.m6.1.1"><eq id="S4.I4.i2.p1.6.m6.1.1.1.cmml" xref="S4.I4.i2.p1.6.m6.1.1.1"></eq><ci id="S4.I4.i2.p1.6.m6.1.1.2.cmml" xref="S4.I4.i2.p1.6.m6.1.1.2">𝑒</ci><apply id="S4.I4.i2.p1.6.m6.1.1.3.cmml" xref="S4.I4.i2.p1.6.m6.1.1.3"><times id="S4.I4.i2.p1.6.m6.1.1.3.1.cmml" xref="S4.I4.i2.p1.6.m6.1.1.3.1"></times><ci id="S4.I4.i2.p1.6.m6.1.1.3.2.cmml" xref="S4.I4.i2.p1.6.m6.1.1.3.2">𝑎</ci><ci id="S4.I4.i2.p1.6.m6.1.1.3.3.cmml" xref="S4.I4.i2.p1.6.m6.1.1.3.3">𝑏</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.I4.i2.p1.6.m6.1c">e=ab</annotation><annotation encoding="application/x-llamapun" id="S4.I4.i2.p1.6.m6.1d">italic_e = italic_a italic_b</annotation></semantics></math> be the corresponding virtual edge. We recursively construct SPQR trees <math alttext="T^{\prime},T^{\prime\prime}" class="ltx_Math" display="inline" id="S4.I4.i2.p1.7.m7.2"><semantics id="S4.I4.i2.p1.7.m7.2a"><mrow id="S4.I4.i2.p1.7.m7.2.2.2" xref="S4.I4.i2.p1.7.m7.2.2.3.cmml"><msup id="S4.I4.i2.p1.7.m7.1.1.1.1" xref="S4.I4.i2.p1.7.m7.1.1.1.1.cmml"><mi id="S4.I4.i2.p1.7.m7.1.1.1.1.2" xref="S4.I4.i2.p1.7.m7.1.1.1.1.2.cmml">T</mi><mo id="S4.I4.i2.p1.7.m7.1.1.1.1.3" xref="S4.I4.i2.p1.7.m7.1.1.1.1.3.cmml">′</mo></msup><mo id="S4.I4.i2.p1.7.m7.2.2.2.3" xref="S4.I4.i2.p1.7.m7.2.2.3.cmml">,</mo><msup id="S4.I4.i2.p1.7.m7.2.2.2.2" xref="S4.I4.i2.p1.7.m7.2.2.2.2.cmml"><mi id="S4.I4.i2.p1.7.m7.2.2.2.2.2" xref="S4.I4.i2.p1.7.m7.2.2.2.2.2.cmml">T</mi><mo id="S4.I4.i2.p1.7.m7.2.2.2.2.3" xref="S4.I4.i2.p1.7.m7.2.2.2.2.3.cmml">′′</mo></msup></mrow><annotation-xml encoding="MathML-Content" id="S4.I4.i2.p1.7.m7.2b"><list id="S4.I4.i2.p1.7.m7.2.2.3.cmml" xref="S4.I4.i2.p1.7.m7.2.2.2"><apply id="S4.I4.i2.p1.7.m7.1.1.1.1.cmml" xref="S4.I4.i2.p1.7.m7.1.1.1.1"><csymbol cd="ambiguous" id="S4.I4.i2.p1.7.m7.1.1.1.1.1.cmml" xref="S4.I4.i2.p1.7.m7.1.1.1.1">superscript</csymbol><ci id="S4.I4.i2.p1.7.m7.1.1.1.1.2.cmml" xref="S4.I4.i2.p1.7.m7.1.1.1.1.2">𝑇</ci><ci id="S4.I4.i2.p1.7.m7.1.1.1.1.3.cmml" xref="S4.I4.i2.p1.7.m7.1.1.1.1.3">′</ci></apply><apply id="S4.I4.i2.p1.7.m7.2.2.2.2.cmml" xref="S4.I4.i2.p1.7.m7.2.2.2.2"><csymbol cd="ambiguous" id="S4.I4.i2.p1.7.m7.2.2.2.2.1.cmml" xref="S4.I4.i2.p1.7.m7.2.2.2.2">superscript</csymbol><ci id="S4.I4.i2.p1.7.m7.2.2.2.2.2.cmml" xref="S4.I4.i2.p1.7.m7.2.2.2.2.2">𝑇</ci><ci id="S4.I4.i2.p1.7.m7.2.2.2.2.3.cmml" xref="S4.I4.i2.p1.7.m7.2.2.2.2.3">′′</ci></apply></list></annotation-xml><annotation encoding="application/x-tex" id="S4.I4.i2.p1.7.m7.2c">T^{\prime},T^{\prime\prime}</annotation><annotation encoding="application/x-llamapun" id="S4.I4.i2.p1.7.m7.2d">italic_T start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT , italic_T start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT</annotation></semantics></math> on <math alttext="G^{\prime},G^{\prime\prime}" class="ltx_Math" display="inline" id="S4.I4.i2.p1.8.m8.2"><semantics id="S4.I4.i2.p1.8.m8.2a"><mrow id="S4.I4.i2.p1.8.m8.2.2.2" xref="S4.I4.i2.p1.8.m8.2.2.3.cmml"><msup id="S4.I4.i2.p1.8.m8.1.1.1.1" xref="S4.I4.i2.p1.8.m8.1.1.1.1.cmml"><mi id="S4.I4.i2.p1.8.m8.1.1.1.1.2" xref="S4.I4.i2.p1.8.m8.1.1.1.1.2.cmml">G</mi><mo id="S4.I4.i2.p1.8.m8.1.1.1.1.3" xref="S4.I4.i2.p1.8.m8.1.1.1.1.3.cmml">′</mo></msup><mo id="S4.I4.i2.p1.8.m8.2.2.2.3" xref="S4.I4.i2.p1.8.m8.2.2.3.cmml">,</mo><msup id="S4.I4.i2.p1.8.m8.2.2.2.2" xref="S4.I4.i2.p1.8.m8.2.2.2.2.cmml"><mi id="S4.I4.i2.p1.8.m8.2.2.2.2.2" xref="S4.I4.i2.p1.8.m8.2.2.2.2.2.cmml">G</mi><mo id="S4.I4.i2.p1.8.m8.2.2.2.2.3" xref="S4.I4.i2.p1.8.m8.2.2.2.2.3.cmml">′′</mo></msup></mrow><annotation-xml encoding="MathML-Content" id="S4.I4.i2.p1.8.m8.2b"><list id="S4.I4.i2.p1.8.m8.2.2.3.cmml" xref="S4.I4.i2.p1.8.m8.2.2.2"><apply id="S4.I4.i2.p1.8.m8.1.1.1.1.cmml" xref="S4.I4.i2.p1.8.m8.1.1.1.1"><csymbol cd="ambiguous" id="S4.I4.i2.p1.8.m8.1.1.1.1.1.cmml" xref="S4.I4.i2.p1.8.m8.1.1.1.1">superscript</csymbol><ci id="S4.I4.i2.p1.8.m8.1.1.1.1.2.cmml" xref="S4.I4.i2.p1.8.m8.1.1.1.1.2">𝐺</ci><ci id="S4.I4.i2.p1.8.m8.1.1.1.1.3.cmml" xref="S4.I4.i2.p1.8.m8.1.1.1.1.3">′</ci></apply><apply id="S4.I4.i2.p1.8.m8.2.2.2.2.cmml" xref="S4.I4.i2.p1.8.m8.2.2.2.2"><csymbol cd="ambiguous" id="S4.I4.i2.p1.8.m8.2.2.2.2.1.cmml" xref="S4.I4.i2.p1.8.m8.2.2.2.2">superscript</csymbol><ci id="S4.I4.i2.p1.8.m8.2.2.2.2.2.cmml" xref="S4.I4.i2.p1.8.m8.2.2.2.2.2">𝐺</ci><ci id="S4.I4.i2.p1.8.m8.2.2.2.2.3.cmml" xref="S4.I4.i2.p1.8.m8.2.2.2.2.3">′′</ci></apply></list></annotation-xml><annotation encoding="application/x-tex" id="S4.I4.i2.p1.8.m8.2c">G^{\prime},G^{\prime\prime}</annotation><annotation encoding="application/x-llamapun" id="S4.I4.i2.p1.8.m8.2d">italic_G start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT , italic_G start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT</annotation></semantics></math>. Let <math alttext="x^{\prime},x^{\prime\prime}" class="ltx_Math" display="inline" id="S4.I4.i2.p1.9.m9.2"><semantics id="S4.I4.i2.p1.9.m9.2a"><mrow id="S4.I4.i2.p1.9.m9.2.2.2" xref="S4.I4.i2.p1.9.m9.2.2.3.cmml"><msup id="S4.I4.i2.p1.9.m9.1.1.1.1" xref="S4.I4.i2.p1.9.m9.1.1.1.1.cmml"><mi id="S4.I4.i2.p1.9.m9.1.1.1.1.2" xref="S4.I4.i2.p1.9.m9.1.1.1.1.2.cmml">x</mi><mo id="S4.I4.i2.p1.9.m9.1.1.1.1.3" xref="S4.I4.i2.p1.9.m9.1.1.1.1.3.cmml">′</mo></msup><mo id="S4.I4.i2.p1.9.m9.2.2.2.3" xref="S4.I4.i2.p1.9.m9.2.2.3.cmml">,</mo><msup id="S4.I4.i2.p1.9.m9.2.2.2.2" xref="S4.I4.i2.p1.9.m9.2.2.2.2.cmml"><mi id="S4.I4.i2.p1.9.m9.2.2.2.2.2" xref="S4.I4.i2.p1.9.m9.2.2.2.2.2.cmml">x</mi><mo id="S4.I4.i2.p1.9.m9.2.2.2.2.3" xref="S4.I4.i2.p1.9.m9.2.2.2.2.3.cmml">′′</mo></msup></mrow><annotation-xml encoding="MathML-Content" id="S4.I4.i2.p1.9.m9.2b"><list id="S4.I4.i2.p1.9.m9.2.2.3.cmml" xref="S4.I4.i2.p1.9.m9.2.2.2"><apply id="S4.I4.i2.p1.9.m9.1.1.1.1.cmml" xref="S4.I4.i2.p1.9.m9.1.1.1.1"><csymbol cd="ambiguous" id="S4.I4.i2.p1.9.m9.1.1.1.1.1.cmml" xref="S4.I4.i2.p1.9.m9.1.1.1.1">superscript</csymbol><ci id="S4.I4.i2.p1.9.m9.1.1.1.1.2.cmml" xref="S4.I4.i2.p1.9.m9.1.1.1.1.2">𝑥</ci><ci id="S4.I4.i2.p1.9.m9.1.1.1.1.3.cmml" xref="S4.I4.i2.p1.9.m9.1.1.1.1.3">′</ci></apply><apply id="S4.I4.i2.p1.9.m9.2.2.2.2.cmml" xref="S4.I4.i2.p1.9.m9.2.2.2.2"><csymbol cd="ambiguous" id="S4.I4.i2.p1.9.m9.2.2.2.2.1.cmml" xref="S4.I4.i2.p1.9.m9.2.2.2.2">superscript</csymbol><ci id="S4.I4.i2.p1.9.m9.2.2.2.2.2.cmml" xref="S4.I4.i2.p1.9.m9.2.2.2.2.2">𝑥</ci><ci id="S4.I4.i2.p1.9.m9.2.2.2.2.3.cmml" xref="S4.I4.i2.p1.9.m9.2.2.2.2.3">′′</ci></apply></list></annotation-xml><annotation encoding="application/x-tex" id="S4.I4.i2.p1.9.m9.2c">x^{\prime},x^{\prime\prime}</annotation><annotation encoding="application/x-llamapun" id="S4.I4.i2.p1.9.m9.2d">italic_x start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT , italic_x start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT</annotation></semantics></math> be the nodes of <math alttext="T^{\prime},T^{\prime\prime}" class="ltx_Math" display="inline" id="S4.I4.i2.p1.10.m10.2"><semantics id="S4.I4.i2.p1.10.m10.2a"><mrow id="S4.I4.i2.p1.10.m10.2.2.2" xref="S4.I4.i2.p1.10.m10.2.2.3.cmml"><msup id="S4.I4.i2.p1.10.m10.1.1.1.1" xref="S4.I4.i2.p1.10.m10.1.1.1.1.cmml"><mi id="S4.I4.i2.p1.10.m10.1.1.1.1.2" xref="S4.I4.i2.p1.10.m10.1.1.1.1.2.cmml">T</mi><mo id="S4.I4.i2.p1.10.m10.1.1.1.1.3" xref="S4.I4.i2.p1.10.m10.1.1.1.1.3.cmml">′</mo></msup><mo id="S4.I4.i2.p1.10.m10.2.2.2.3" xref="S4.I4.i2.p1.10.m10.2.2.3.cmml">,</mo><msup id="S4.I4.i2.p1.10.m10.2.2.2.2" xref="S4.I4.i2.p1.10.m10.2.2.2.2.cmml"><mi id="S4.I4.i2.p1.10.m10.2.2.2.2.2" xref="S4.I4.i2.p1.10.m10.2.2.2.2.2.cmml">T</mi><mo id="S4.I4.i2.p1.10.m10.2.2.2.2.3" xref="S4.I4.i2.p1.10.m10.2.2.2.2.3.cmml">′′</mo></msup></mrow><annotation-xml encoding="MathML-Content" id="S4.I4.i2.p1.10.m10.2b"><list id="S4.I4.i2.p1.10.m10.2.2.3.cmml" xref="S4.I4.i2.p1.10.m10.2.2.2"><apply id="S4.I4.i2.p1.10.m10.1.1.1.1.cmml" xref="S4.I4.i2.p1.10.m10.1.1.1.1"><csymbol cd="ambiguous" id="S4.I4.i2.p1.10.m10.1.1.1.1.1.cmml" xref="S4.I4.i2.p1.10.m10.1.1.1.1">superscript</csymbol><ci id="S4.I4.i2.p1.10.m10.1.1.1.1.2.cmml" xref="S4.I4.i2.p1.10.m10.1.1.1.1.2">𝑇</ci><ci id="S4.I4.i2.p1.10.m10.1.1.1.1.3.cmml" xref="S4.I4.i2.p1.10.m10.1.1.1.1.3">′</ci></apply><apply id="S4.I4.i2.p1.10.m10.2.2.2.2.cmml" xref="S4.I4.i2.p1.10.m10.2.2.2.2"><csymbol cd="ambiguous" id="S4.I4.i2.p1.10.m10.2.2.2.2.1.cmml" xref="S4.I4.i2.p1.10.m10.2.2.2.2">superscript</csymbol><ci id="S4.I4.i2.p1.10.m10.2.2.2.2.2.cmml" xref="S4.I4.i2.p1.10.m10.2.2.2.2.2">𝑇</ci><ci id="S4.I4.i2.p1.10.m10.2.2.2.2.3.cmml" xref="S4.I4.i2.p1.10.m10.2.2.2.2.3">′′</ci></apply></list></annotation-xml><annotation encoding="application/x-tex" id="S4.I4.i2.p1.10.m10.2c">T^{\prime},T^{\prime\prime}</annotation><annotation encoding="application/x-llamapun" id="S4.I4.i2.p1.10.m10.2d">italic_T start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT , italic_T start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT</annotation></semantics></math> respectively containing edge <math alttext="e" class="ltx_Math" display="inline" id="S4.I4.i2.p1.11.m11.1"><semantics id="S4.I4.i2.p1.11.m11.1a"><mi id="S4.I4.i2.p1.11.m11.1.1" xref="S4.I4.i2.p1.11.m11.1.1.cmml">e</mi><annotation-xml encoding="MathML-Content" id="S4.I4.i2.p1.11.m11.1b"><ci id="S4.I4.i2.p1.11.m11.1.1.cmml" xref="S4.I4.i2.p1.11.m11.1.1">𝑒</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.I4.i2.p1.11.m11.1c">e</annotation><annotation encoding="application/x-llamapun" id="S4.I4.i2.p1.11.m11.1d">italic_e</annotation></semantics></math> – this is well defined, since the split operation partitions the edges of <math alttext="G" class="ltx_Math" display="inline" id="S4.I4.i2.p1.12.m12.1"><semantics id="S4.I4.i2.p1.12.m12.1a"><mi id="S4.I4.i2.p1.12.m12.1.1" xref="S4.I4.i2.p1.12.m12.1.1.cmml">G</mi><annotation-xml encoding="MathML-Content" id="S4.I4.i2.p1.12.m12.1b"><ci id="S4.I4.i2.p1.12.m12.1.1.cmml" xref="S4.I4.i2.p1.12.m12.1.1">𝐺</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.I4.i2.p1.12.m12.1c">G</annotation><annotation encoding="application/x-llamapun" id="S4.I4.i2.p1.12.m12.1d">italic_G</annotation></semantics></math> and <math alttext="e" class="ltx_Math" display="inline" id="S4.I4.i2.p1.13.m13.1"><semantics id="S4.I4.i2.p1.13.m13.1a"><mi id="S4.I4.i2.p1.13.m13.1.1" xref="S4.I4.i2.p1.13.m13.1.1.cmml">e</mi><annotation-xml encoding="MathML-Content" id="S4.I4.i2.p1.13.m13.1b"><ci id="S4.I4.i2.p1.13.m13.1.1.cmml" xref="S4.I4.i2.p1.13.m13.1.1">𝑒</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.I4.i2.p1.13.m13.1c">e</annotation><annotation encoding="application/x-llamapun" id="S4.I4.i2.p1.13.m13.1d">italic_e</annotation></semantics></math> is contained in both <math alttext="G^{\prime}" class="ltx_Math" display="inline" id="S4.I4.i2.p1.14.m14.1"><semantics id="S4.I4.i2.p1.14.m14.1a"><msup id="S4.I4.i2.p1.14.m14.1.1" xref="S4.I4.i2.p1.14.m14.1.1.cmml"><mi id="S4.I4.i2.p1.14.m14.1.1.2" xref="S4.I4.i2.p1.14.m14.1.1.2.cmml">G</mi><mo id="S4.I4.i2.p1.14.m14.1.1.3" xref="S4.I4.i2.p1.14.m14.1.1.3.cmml">′</mo></msup><annotation-xml encoding="MathML-Content" id="S4.I4.i2.p1.14.m14.1b"><apply id="S4.I4.i2.p1.14.m14.1.1.cmml" xref="S4.I4.i2.p1.14.m14.1.1"><csymbol cd="ambiguous" id="S4.I4.i2.p1.14.m14.1.1.1.cmml" xref="S4.I4.i2.p1.14.m14.1.1">superscript</csymbol><ci id="S4.I4.i2.p1.14.m14.1.1.2.cmml" xref="S4.I4.i2.p1.14.m14.1.1.2">𝐺</ci><ci id="S4.I4.i2.p1.14.m14.1.1.3.cmml" xref="S4.I4.i2.p1.14.m14.1.1.3">′</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.I4.i2.p1.14.m14.1c">G^{\prime}</annotation><annotation encoding="application/x-llamapun" id="S4.I4.i2.p1.14.m14.1d">italic_G start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math> and <math alttext="G^{\prime\prime}" class="ltx_Math" display="inline" id="S4.I4.i2.p1.15.m15.1"><semantics id="S4.I4.i2.p1.15.m15.1a"><msup id="S4.I4.i2.p1.15.m15.1.1" xref="S4.I4.i2.p1.15.m15.1.1.cmml"><mi id="S4.I4.i2.p1.15.m15.1.1.2" xref="S4.I4.i2.p1.15.m15.1.1.2.cmml">G</mi><mo id="S4.I4.i2.p1.15.m15.1.1.3" xref="S4.I4.i2.p1.15.m15.1.1.3.cmml">′′</mo></msup><annotation-xml encoding="MathML-Content" id="S4.I4.i2.p1.15.m15.1b"><apply id="S4.I4.i2.p1.15.m15.1.1.cmml" xref="S4.I4.i2.p1.15.m15.1.1"><csymbol cd="ambiguous" id="S4.I4.i2.p1.15.m15.1.1.1.cmml" xref="S4.I4.i2.p1.15.m15.1.1">superscript</csymbol><ci id="S4.I4.i2.p1.15.m15.1.1.2.cmml" xref="S4.I4.i2.p1.15.m15.1.1.2">𝐺</ci><ci id="S4.I4.i2.p1.15.m15.1.1.3.cmml" xref="S4.I4.i2.p1.15.m15.1.1.3">′′</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.I4.i2.p1.15.m15.1c">G^{\prime\prime}</annotation><annotation encoding="application/x-llamapun" id="S4.I4.i2.p1.15.m15.1d">italic_G start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT</annotation></semantics></math>. We let <math alttext="T=T^{\prime}\cup T^{\prime\prime}\cup(x^{\prime},x^{\prime\prime})" class="ltx_Math" display="inline" id="S4.I4.i2.p1.16.m16.2"><semantics id="S4.I4.i2.p1.16.m16.2a"><mrow id="S4.I4.i2.p1.16.m16.2.2" xref="S4.I4.i2.p1.16.m16.2.2.cmml"><mi id="S4.I4.i2.p1.16.m16.2.2.4" xref="S4.I4.i2.p1.16.m16.2.2.4.cmml">T</mi><mo id="S4.I4.i2.p1.16.m16.2.2.3" xref="S4.I4.i2.p1.16.m16.2.2.3.cmml">=</mo><mrow id="S4.I4.i2.p1.16.m16.2.2.2" xref="S4.I4.i2.p1.16.m16.2.2.2.cmml"><msup id="S4.I4.i2.p1.16.m16.2.2.2.4" xref="S4.I4.i2.p1.16.m16.2.2.2.4.cmml"><mi id="S4.I4.i2.p1.16.m16.2.2.2.4.2" xref="S4.I4.i2.p1.16.m16.2.2.2.4.2.cmml">T</mi><mo id="S4.I4.i2.p1.16.m16.2.2.2.4.3" xref="S4.I4.i2.p1.16.m16.2.2.2.4.3.cmml">′</mo></msup><mo id="S4.I4.i2.p1.16.m16.2.2.2.3" xref="S4.I4.i2.p1.16.m16.2.2.2.3.cmml">∪</mo><msup id="S4.I4.i2.p1.16.m16.2.2.2.5" xref="S4.I4.i2.p1.16.m16.2.2.2.5.cmml"><mi id="S4.I4.i2.p1.16.m16.2.2.2.5.2" xref="S4.I4.i2.p1.16.m16.2.2.2.5.2.cmml">T</mi><mo id="S4.I4.i2.p1.16.m16.2.2.2.5.3" xref="S4.I4.i2.p1.16.m16.2.2.2.5.3.cmml">′′</mo></msup><mo id="S4.I4.i2.p1.16.m16.2.2.2.3a" xref="S4.I4.i2.p1.16.m16.2.2.2.3.cmml">∪</mo><mrow id="S4.I4.i2.p1.16.m16.2.2.2.2.2" xref="S4.I4.i2.p1.16.m16.2.2.2.2.3.cmml"><mo id="S4.I4.i2.p1.16.m16.2.2.2.2.2.3" stretchy="false" xref="S4.I4.i2.p1.16.m16.2.2.2.2.3.cmml">(</mo><msup id="S4.I4.i2.p1.16.m16.1.1.1.1.1.1" xref="S4.I4.i2.p1.16.m16.1.1.1.1.1.1.cmml"><mi id="S4.I4.i2.p1.16.m16.1.1.1.1.1.1.2" xref="S4.I4.i2.p1.16.m16.1.1.1.1.1.1.2.cmml">x</mi><mo id="S4.I4.i2.p1.16.m16.1.1.1.1.1.1.3" xref="S4.I4.i2.p1.16.m16.1.1.1.1.1.1.3.cmml">′</mo></msup><mo id="S4.I4.i2.p1.16.m16.2.2.2.2.2.4" xref="S4.I4.i2.p1.16.m16.2.2.2.2.3.cmml">,</mo><msup id="S4.I4.i2.p1.16.m16.2.2.2.2.2.2" xref="S4.I4.i2.p1.16.m16.2.2.2.2.2.2.cmml"><mi id="S4.I4.i2.p1.16.m16.2.2.2.2.2.2.2" xref="S4.I4.i2.p1.16.m16.2.2.2.2.2.2.2.cmml">x</mi><mo id="S4.I4.i2.p1.16.m16.2.2.2.2.2.2.3" xref="S4.I4.i2.p1.16.m16.2.2.2.2.2.2.3.cmml">′′</mo></msup><mo id="S4.I4.i2.p1.16.m16.2.2.2.2.2.5" stretchy="false" xref="S4.I4.i2.p1.16.m16.2.2.2.2.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.I4.i2.p1.16.m16.2b"><apply id="S4.I4.i2.p1.16.m16.2.2.cmml" xref="S4.I4.i2.p1.16.m16.2.2"><eq id="S4.I4.i2.p1.16.m16.2.2.3.cmml" xref="S4.I4.i2.p1.16.m16.2.2.3"></eq><ci id="S4.I4.i2.p1.16.m16.2.2.4.cmml" xref="S4.I4.i2.p1.16.m16.2.2.4">𝑇</ci><apply id="S4.I4.i2.p1.16.m16.2.2.2.cmml" xref="S4.I4.i2.p1.16.m16.2.2.2"><union id="S4.I4.i2.p1.16.m16.2.2.2.3.cmml" xref="S4.I4.i2.p1.16.m16.2.2.2.3"></union><apply id="S4.I4.i2.p1.16.m16.2.2.2.4.cmml" xref="S4.I4.i2.p1.16.m16.2.2.2.4"><csymbol cd="ambiguous" id="S4.I4.i2.p1.16.m16.2.2.2.4.1.cmml" xref="S4.I4.i2.p1.16.m16.2.2.2.4">superscript</csymbol><ci id="S4.I4.i2.p1.16.m16.2.2.2.4.2.cmml" xref="S4.I4.i2.p1.16.m16.2.2.2.4.2">𝑇</ci><ci id="S4.I4.i2.p1.16.m16.2.2.2.4.3.cmml" xref="S4.I4.i2.p1.16.m16.2.2.2.4.3">′</ci></apply><apply id="S4.I4.i2.p1.16.m16.2.2.2.5.cmml" xref="S4.I4.i2.p1.16.m16.2.2.2.5"><csymbol cd="ambiguous" id="S4.I4.i2.p1.16.m16.2.2.2.5.1.cmml" xref="S4.I4.i2.p1.16.m16.2.2.2.5">superscript</csymbol><ci id="S4.I4.i2.p1.16.m16.2.2.2.5.2.cmml" xref="S4.I4.i2.p1.16.m16.2.2.2.5.2">𝑇</ci><ci id="S4.I4.i2.p1.16.m16.2.2.2.5.3.cmml" xref="S4.I4.i2.p1.16.m16.2.2.2.5.3">′′</ci></apply><interval closure="open" id="S4.I4.i2.p1.16.m16.2.2.2.2.3.cmml" xref="S4.I4.i2.p1.16.m16.2.2.2.2.2"><apply id="S4.I4.i2.p1.16.m16.1.1.1.1.1.1.cmml" xref="S4.I4.i2.p1.16.m16.1.1.1.1.1.1"><csymbol cd="ambiguous" id="S4.I4.i2.p1.16.m16.1.1.1.1.1.1.1.cmml" xref="S4.I4.i2.p1.16.m16.1.1.1.1.1.1">superscript</csymbol><ci id="S4.I4.i2.p1.16.m16.1.1.1.1.1.1.2.cmml" xref="S4.I4.i2.p1.16.m16.1.1.1.1.1.1.2">𝑥</ci><ci id="S4.I4.i2.p1.16.m16.1.1.1.1.1.1.3.cmml" xref="S4.I4.i2.p1.16.m16.1.1.1.1.1.1.3">′</ci></apply><apply id="S4.I4.i2.p1.16.m16.2.2.2.2.2.2.cmml" xref="S4.I4.i2.p1.16.m16.2.2.2.2.2.2"><csymbol cd="ambiguous" id="S4.I4.i2.p1.16.m16.2.2.2.2.2.2.1.cmml" xref="S4.I4.i2.p1.16.m16.2.2.2.2.2.2">superscript</csymbol><ci id="S4.I4.i2.p1.16.m16.2.2.2.2.2.2.2.cmml" xref="S4.I4.i2.p1.16.m16.2.2.2.2.2.2.2">𝑥</ci><ci id="S4.I4.i2.p1.16.m16.2.2.2.2.2.2.3.cmml" xref="S4.I4.i2.p1.16.m16.2.2.2.2.2.2.3">′′</ci></apply></interval></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.I4.i2.p1.16.m16.2c">T=T^{\prime}\cup T^{\prime\prime}\cup(x^{\prime},x^{\prime\prime})</annotation><annotation encoding="application/x-llamapun" id="S4.I4.i2.p1.16.m16.2d">italic_T = italic_T start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ∪ italic_T start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT ∪ ( italic_x start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT , italic_x start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT )</annotation></semantics></math>; that is, we combine <math alttext="T^{\prime}" class="ltx_Math" display="inline" id="S4.I4.i2.p1.17.m17.1"><semantics id="S4.I4.i2.p1.17.m17.1a"><msup id="S4.I4.i2.p1.17.m17.1.1" xref="S4.I4.i2.p1.17.m17.1.1.cmml"><mi id="S4.I4.i2.p1.17.m17.1.1.2" xref="S4.I4.i2.p1.17.m17.1.1.2.cmml">T</mi><mo id="S4.I4.i2.p1.17.m17.1.1.3" xref="S4.I4.i2.p1.17.m17.1.1.3.cmml">′</mo></msup><annotation-xml encoding="MathML-Content" id="S4.I4.i2.p1.17.m17.1b"><apply id="S4.I4.i2.p1.17.m17.1.1.cmml" xref="S4.I4.i2.p1.17.m17.1.1"><csymbol cd="ambiguous" id="S4.I4.i2.p1.17.m17.1.1.1.cmml" xref="S4.I4.i2.p1.17.m17.1.1">superscript</csymbol><ci id="S4.I4.i2.p1.17.m17.1.1.2.cmml" xref="S4.I4.i2.p1.17.m17.1.1.2">𝑇</ci><ci id="S4.I4.i2.p1.17.m17.1.1.3.cmml" xref="S4.I4.i2.p1.17.m17.1.1.3">′</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.I4.i2.p1.17.m17.1c">T^{\prime}</annotation><annotation encoding="application/x-llamapun" id="S4.I4.i2.p1.17.m17.1d">italic_T start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math> and <math alttext="T^{\prime\prime}" class="ltx_Math" display="inline" id="S4.I4.i2.p1.18.m18.1"><semantics id="S4.I4.i2.p1.18.m18.1a"><msup id="S4.I4.i2.p1.18.m18.1.1" xref="S4.I4.i2.p1.18.m18.1.1.cmml"><mi id="S4.I4.i2.p1.18.m18.1.1.2" xref="S4.I4.i2.p1.18.m18.1.1.2.cmml">T</mi><mo id="S4.I4.i2.p1.18.m18.1.1.3" xref="S4.I4.i2.p1.18.m18.1.1.3.cmml">′′</mo></msup><annotation-xml encoding="MathML-Content" id="S4.I4.i2.p1.18.m18.1b"><apply id="S4.I4.i2.p1.18.m18.1.1.cmml" xref="S4.I4.i2.p1.18.m18.1.1"><csymbol cd="ambiguous" id="S4.I4.i2.p1.18.m18.1.1.1.cmml" xref="S4.I4.i2.p1.18.m18.1.1">superscript</csymbol><ci id="S4.I4.i2.p1.18.m18.1.1.2.cmml" xref="S4.I4.i2.p1.18.m18.1.1.2">𝑇</ci><ci id="S4.I4.i2.p1.18.m18.1.1.3.cmml" xref="S4.I4.i2.p1.18.m18.1.1.3">′′</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.I4.i2.p1.18.m18.1c">T^{\prime\prime}</annotation><annotation encoding="application/x-llamapun" id="S4.I4.i2.p1.18.m18.1d">italic_T start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT</annotation></semantics></math> via an edge between <math alttext="x^{\prime}" class="ltx_Math" display="inline" id="S4.I4.i2.p1.19.m19.1"><semantics id="S4.I4.i2.p1.19.m19.1a"><msup id="S4.I4.i2.p1.19.m19.1.1" xref="S4.I4.i2.p1.19.m19.1.1.cmml"><mi id="S4.I4.i2.p1.19.m19.1.1.2" xref="S4.I4.i2.p1.19.m19.1.1.2.cmml">x</mi><mo id="S4.I4.i2.p1.19.m19.1.1.3" xref="S4.I4.i2.p1.19.m19.1.1.3.cmml">′</mo></msup><annotation-xml encoding="MathML-Content" id="S4.I4.i2.p1.19.m19.1b"><apply id="S4.I4.i2.p1.19.m19.1.1.cmml" xref="S4.I4.i2.p1.19.m19.1.1"><csymbol cd="ambiguous" id="S4.I4.i2.p1.19.m19.1.1.1.cmml" xref="S4.I4.i2.p1.19.m19.1.1">superscript</csymbol><ci id="S4.I4.i2.p1.19.m19.1.1.2.cmml" xref="S4.I4.i2.p1.19.m19.1.1.2">𝑥</ci><ci id="S4.I4.i2.p1.19.m19.1.1.3.cmml" xref="S4.I4.i2.p1.19.m19.1.1.3">′</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.I4.i2.p1.19.m19.1c">x^{\prime}</annotation><annotation encoding="application/x-llamapun" id="S4.I4.i2.p1.19.m19.1d">italic_x start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math> and <math alttext="x^{\prime\prime}" class="ltx_Math" display="inline" id="S4.I4.i2.p1.20.m20.1"><semantics id="S4.I4.i2.p1.20.m20.1a"><msup id="S4.I4.i2.p1.20.m20.1.1" xref="S4.I4.i2.p1.20.m20.1.1.cmml"><mi id="S4.I4.i2.p1.20.m20.1.1.2" xref="S4.I4.i2.p1.20.m20.1.1.2.cmml">x</mi><mo id="S4.I4.i2.p1.20.m20.1.1.3" xref="S4.I4.i2.p1.20.m20.1.1.3.cmml">′′</mo></msup><annotation-xml encoding="MathML-Content" id="S4.I4.i2.p1.20.m20.1b"><apply id="S4.I4.i2.p1.20.m20.1.1.cmml" xref="S4.I4.i2.p1.20.m20.1.1"><csymbol cd="ambiguous" id="S4.I4.i2.p1.20.m20.1.1.1.cmml" xref="S4.I4.i2.p1.20.m20.1.1">superscript</csymbol><ci id="S4.I4.i2.p1.20.m20.1.1.2.cmml" xref="S4.I4.i2.p1.20.m20.1.1.2">𝑥</ci><ci id="S4.I4.i2.p1.20.m20.1.1.3.cmml" xref="S4.I4.i2.p1.20.m20.1.1.3">′′</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.I4.i2.p1.20.m20.1c">x^{\prime\prime}</annotation><annotation encoding="application/x-llamapun" id="S4.I4.i2.p1.20.m20.1d">italic_x start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT</annotation></semantics></math>. We say the tree edge <math alttext="(x^{\prime},x^{\prime\prime})" class="ltx_Math" display="inline" id="S4.I4.i2.p1.21.m21.2"><semantics id="S4.I4.i2.p1.21.m21.2a"><mrow id="S4.I4.i2.p1.21.m21.2.2.2" xref="S4.I4.i2.p1.21.m21.2.2.3.cmml"><mo id="S4.I4.i2.p1.21.m21.2.2.2.3" stretchy="false" xref="S4.I4.i2.p1.21.m21.2.2.3.cmml">(</mo><msup id="S4.I4.i2.p1.21.m21.1.1.1.1" xref="S4.I4.i2.p1.21.m21.1.1.1.1.cmml"><mi id="S4.I4.i2.p1.21.m21.1.1.1.1.2" xref="S4.I4.i2.p1.21.m21.1.1.1.1.2.cmml">x</mi><mo id="S4.I4.i2.p1.21.m21.1.1.1.1.3" xref="S4.I4.i2.p1.21.m21.1.1.1.1.3.cmml">′</mo></msup><mo id="S4.I4.i2.p1.21.m21.2.2.2.4" xref="S4.I4.i2.p1.21.m21.2.2.3.cmml">,</mo><msup id="S4.I4.i2.p1.21.m21.2.2.2.2" xref="S4.I4.i2.p1.21.m21.2.2.2.2.cmml"><mi id="S4.I4.i2.p1.21.m21.2.2.2.2.2" xref="S4.I4.i2.p1.21.m21.2.2.2.2.2.cmml">x</mi><mo id="S4.I4.i2.p1.21.m21.2.2.2.2.3" xref="S4.I4.i2.p1.21.m21.2.2.2.2.3.cmml">′′</mo></msup><mo id="S4.I4.i2.p1.21.m21.2.2.2.5" stretchy="false" xref="S4.I4.i2.p1.21.m21.2.2.3.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S4.I4.i2.p1.21.m21.2b"><interval closure="open" id="S4.I4.i2.p1.21.m21.2.2.3.cmml" xref="S4.I4.i2.p1.21.m21.2.2.2"><apply id="S4.I4.i2.p1.21.m21.1.1.1.1.cmml" xref="S4.I4.i2.p1.21.m21.1.1.1.1"><csymbol cd="ambiguous" id="S4.I4.i2.p1.21.m21.1.1.1.1.1.cmml" xref="S4.I4.i2.p1.21.m21.1.1.1.1">superscript</csymbol><ci id="S4.I4.i2.p1.21.m21.1.1.1.1.2.cmml" xref="S4.I4.i2.p1.21.m21.1.1.1.1.2">𝑥</ci><ci id="S4.I4.i2.p1.21.m21.1.1.1.1.3.cmml" xref="S4.I4.i2.p1.21.m21.1.1.1.1.3">′</ci></apply><apply id="S4.I4.i2.p1.21.m21.2.2.2.2.cmml" xref="S4.I4.i2.p1.21.m21.2.2.2.2"><csymbol cd="ambiguous" id="S4.I4.i2.p1.21.m21.2.2.2.2.1.cmml" xref="S4.I4.i2.p1.21.m21.2.2.2.2">superscript</csymbol><ci id="S4.I4.i2.p1.21.m21.2.2.2.2.2.cmml" xref="S4.I4.i2.p1.21.m21.2.2.2.2.2">𝑥</ci><ci id="S4.I4.i2.p1.21.m21.2.2.2.2.3.cmml" xref="S4.I4.i2.p1.21.m21.2.2.2.2.3">′′</ci></apply></interval></annotation-xml><annotation encoding="application/x-tex" id="S4.I4.i2.p1.21.m21.2c">(x^{\prime},x^{\prime\prime})</annotation><annotation encoding="application/x-llamapun" id="S4.I4.i2.p1.21.m21.2d">( italic_x start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT , italic_x start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT )</annotation></semantics></math> is associated with the virtual edge <math alttext="e" class="ltx_Math" display="inline" id="S4.I4.i2.p1.22.m22.1"><semantics id="S4.I4.i2.p1.22.m22.1a"><mi id="S4.I4.i2.p1.22.m22.1.1" xref="S4.I4.i2.p1.22.m22.1.1.cmml">e</mi><annotation-xml encoding="MathML-Content" id="S4.I4.i2.p1.22.m22.1b"><ci id="S4.I4.i2.p1.22.m22.1.1.cmml" xref="S4.I4.i2.p1.22.m22.1.1">𝑒</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.I4.i2.p1.22.m22.1c">e</annotation><annotation encoding="application/x-llamapun" id="S4.I4.i2.p1.22.m22.1d">italic_e</annotation></semantics></math>.</p> </div> </li> </ul> <p class="ltx_p" id="S4.Thmtheorem11.p1.17">Then, we postprocess <math alttext="T" class="ltx_Math" display="inline" id="S4.Thmtheorem11.p1.7.m1.1"><semantics id="S4.Thmtheorem11.p1.7.m1.1a"><mi id="S4.Thmtheorem11.p1.7.m1.1.1" xref="S4.Thmtheorem11.p1.7.m1.1.1.cmml">T</mi><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem11.p1.7.m1.1b"><ci id="S4.Thmtheorem11.p1.7.m1.1.1.cmml" xref="S4.Thmtheorem11.p1.7.m1.1.1">𝑇</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem11.p1.7.m1.1c">T</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem11.p1.7.m1.1d">italic_T</annotation></semantics></math> as follows: if there are two adjacent tree nodes <math alttext="x,y" class="ltx_Math" display="inline" id="S4.Thmtheorem11.p1.8.m2.2"><semantics id="S4.Thmtheorem11.p1.8.m2.2a"><mrow id="S4.Thmtheorem11.p1.8.m2.2.3.2" xref="S4.Thmtheorem11.p1.8.m2.2.3.1.cmml"><mi id="S4.Thmtheorem11.p1.8.m2.1.1" xref="S4.Thmtheorem11.p1.8.m2.1.1.cmml">x</mi><mo id="S4.Thmtheorem11.p1.8.m2.2.3.2.1" xref="S4.Thmtheorem11.p1.8.m2.2.3.1.cmml">,</mo><mi id="S4.Thmtheorem11.p1.8.m2.2.2" xref="S4.Thmtheorem11.p1.8.m2.2.2.cmml">y</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem11.p1.8.m2.2b"><list id="S4.Thmtheorem11.p1.8.m2.2.3.1.cmml" xref="S4.Thmtheorem11.p1.8.m2.2.3.2"><ci id="S4.Thmtheorem11.p1.8.m2.1.1.cmml" xref="S4.Thmtheorem11.p1.8.m2.1.1">𝑥</ci><ci id="S4.Thmtheorem11.p1.8.m2.2.2.cmml" xref="S4.Thmtheorem11.p1.8.m2.2.2">𝑦</ci></list></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem11.p1.8.m2.2c">x,y</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem11.p1.8.m2.2d">italic_x , italic_y</annotation></semantics></math> with <math alttext="G_{x},G_{y}" class="ltx_Math" display="inline" id="S4.Thmtheorem11.p1.9.m3.2"><semantics id="S4.Thmtheorem11.p1.9.m3.2a"><mrow id="S4.Thmtheorem11.p1.9.m3.2.2.2" xref="S4.Thmtheorem11.p1.9.m3.2.2.3.cmml"><msub id="S4.Thmtheorem11.p1.9.m3.1.1.1.1" xref="S4.Thmtheorem11.p1.9.m3.1.1.1.1.cmml"><mi id="S4.Thmtheorem11.p1.9.m3.1.1.1.1.2" xref="S4.Thmtheorem11.p1.9.m3.1.1.1.1.2.cmml">G</mi><mi id="S4.Thmtheorem11.p1.9.m3.1.1.1.1.3" xref="S4.Thmtheorem11.p1.9.m3.1.1.1.1.3.cmml">x</mi></msub><mo id="S4.Thmtheorem11.p1.9.m3.2.2.2.3" xref="S4.Thmtheorem11.p1.9.m3.2.2.3.cmml">,</mo><msub id="S4.Thmtheorem11.p1.9.m3.2.2.2.2" xref="S4.Thmtheorem11.p1.9.m3.2.2.2.2.cmml"><mi id="S4.Thmtheorem11.p1.9.m3.2.2.2.2.2" xref="S4.Thmtheorem11.p1.9.m3.2.2.2.2.2.cmml">G</mi><mi id="S4.Thmtheorem11.p1.9.m3.2.2.2.2.3" xref="S4.Thmtheorem11.p1.9.m3.2.2.2.2.3.cmml">y</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem11.p1.9.m3.2b"><list id="S4.Thmtheorem11.p1.9.m3.2.2.3.cmml" xref="S4.Thmtheorem11.p1.9.m3.2.2.2"><apply id="S4.Thmtheorem11.p1.9.m3.1.1.1.1.cmml" xref="S4.Thmtheorem11.p1.9.m3.1.1.1.1"><csymbol cd="ambiguous" id="S4.Thmtheorem11.p1.9.m3.1.1.1.1.1.cmml" xref="S4.Thmtheorem11.p1.9.m3.1.1.1.1">subscript</csymbol><ci id="S4.Thmtheorem11.p1.9.m3.1.1.1.1.2.cmml" xref="S4.Thmtheorem11.p1.9.m3.1.1.1.1.2">𝐺</ci><ci id="S4.Thmtheorem11.p1.9.m3.1.1.1.1.3.cmml" xref="S4.Thmtheorem11.p1.9.m3.1.1.1.1.3">𝑥</ci></apply><apply id="S4.Thmtheorem11.p1.9.m3.2.2.2.2.cmml" xref="S4.Thmtheorem11.p1.9.m3.2.2.2.2"><csymbol cd="ambiguous" id="S4.Thmtheorem11.p1.9.m3.2.2.2.2.1.cmml" xref="S4.Thmtheorem11.p1.9.m3.2.2.2.2">subscript</csymbol><ci id="S4.Thmtheorem11.p1.9.m3.2.2.2.2.2.cmml" xref="S4.Thmtheorem11.p1.9.m3.2.2.2.2.2">𝐺</ci><ci id="S4.Thmtheorem11.p1.9.m3.2.2.2.2.3.cmml" xref="S4.Thmtheorem11.p1.9.m3.2.2.2.2.3">𝑦</ci></apply></list></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem11.p1.9.m3.2c">G_{x},G_{y}</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem11.p1.9.m3.2d">italic_G start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT , italic_G start_POSTSUBSCRIPT italic_y end_POSTSUBSCRIPT</annotation></semantics></math> both dipoles (i.e. <math alttext="|V(G_{x})|=|V(G_{y})|=2" class="ltx_Math" display="inline" id="S4.Thmtheorem11.p1.10.m4.2"><semantics id="S4.Thmtheorem11.p1.10.m4.2a"><mrow id="S4.Thmtheorem11.p1.10.m4.2.2" xref="S4.Thmtheorem11.p1.10.m4.2.2.cmml"><mrow id="S4.Thmtheorem11.p1.10.m4.1.1.1.1" xref="S4.Thmtheorem11.p1.10.m4.1.1.1.2.cmml"><mo id="S4.Thmtheorem11.p1.10.m4.1.1.1.1.2" stretchy="false" xref="S4.Thmtheorem11.p1.10.m4.1.1.1.2.1.cmml">|</mo><mrow id="S4.Thmtheorem11.p1.10.m4.1.1.1.1.1" xref="S4.Thmtheorem11.p1.10.m4.1.1.1.1.1.cmml"><mi id="S4.Thmtheorem11.p1.10.m4.1.1.1.1.1.3" xref="S4.Thmtheorem11.p1.10.m4.1.1.1.1.1.3.cmml">V</mi><mo id="S4.Thmtheorem11.p1.10.m4.1.1.1.1.1.2" xref="S4.Thmtheorem11.p1.10.m4.1.1.1.1.1.2.cmml">⁢</mo><mrow id="S4.Thmtheorem11.p1.10.m4.1.1.1.1.1.1.1" xref="S4.Thmtheorem11.p1.10.m4.1.1.1.1.1.1.1.1.cmml"><mo id="S4.Thmtheorem11.p1.10.m4.1.1.1.1.1.1.1.2" stretchy="false" xref="S4.Thmtheorem11.p1.10.m4.1.1.1.1.1.1.1.1.cmml">(</mo><msub 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xref="S4.Thmtheorem11.p1.10.m4.2.2.2.2.1.cmml">|</mo></mrow><mo id="S4.Thmtheorem11.p1.10.m4.2.2.5" xref="S4.Thmtheorem11.p1.10.m4.2.2.5.cmml">=</mo><mn id="S4.Thmtheorem11.p1.10.m4.2.2.6" xref="S4.Thmtheorem11.p1.10.m4.2.2.6.cmml">2</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem11.p1.10.m4.2b"><apply id="S4.Thmtheorem11.p1.10.m4.2.2.cmml" xref="S4.Thmtheorem11.p1.10.m4.2.2"><and id="S4.Thmtheorem11.p1.10.m4.2.2a.cmml" xref="S4.Thmtheorem11.p1.10.m4.2.2"></and><apply id="S4.Thmtheorem11.p1.10.m4.2.2b.cmml" xref="S4.Thmtheorem11.p1.10.m4.2.2"><eq id="S4.Thmtheorem11.p1.10.m4.2.2.4.cmml" xref="S4.Thmtheorem11.p1.10.m4.2.2.4"></eq><apply id="S4.Thmtheorem11.p1.10.m4.1.1.1.2.cmml" xref="S4.Thmtheorem11.p1.10.m4.1.1.1.1"><abs id="S4.Thmtheorem11.p1.10.m4.1.1.1.2.1.cmml" xref="S4.Thmtheorem11.p1.10.m4.1.1.1.1.2"></abs><apply id="S4.Thmtheorem11.p1.10.m4.1.1.1.1.1.cmml" xref="S4.Thmtheorem11.p1.10.m4.1.1.1.1.1"><times id="S4.Thmtheorem11.p1.10.m4.1.1.1.1.1.2.cmml" xref="S4.Thmtheorem11.p1.10.m4.1.1.1.1.1.2"></times><ci id="S4.Thmtheorem11.p1.10.m4.1.1.1.1.1.3.cmml" xref="S4.Thmtheorem11.p1.10.m4.1.1.1.1.1.3">𝑉</ci><apply id="S4.Thmtheorem11.p1.10.m4.1.1.1.1.1.1.1.1.cmml" xref="S4.Thmtheorem11.p1.10.m4.1.1.1.1.1.1.1"><csymbol cd="ambiguous" id="S4.Thmtheorem11.p1.10.m4.1.1.1.1.1.1.1.1.1.cmml" xref="S4.Thmtheorem11.p1.10.m4.1.1.1.1.1.1.1">subscript</csymbol><ci id="S4.Thmtheorem11.p1.10.m4.1.1.1.1.1.1.1.1.2.cmml" xref="S4.Thmtheorem11.p1.10.m4.1.1.1.1.1.1.1.1.2">𝐺</ci><ci id="S4.Thmtheorem11.p1.10.m4.1.1.1.1.1.1.1.1.3.cmml" xref="S4.Thmtheorem11.p1.10.m4.1.1.1.1.1.1.1.1.3">𝑥</ci></apply></apply></apply><apply id="S4.Thmtheorem11.p1.10.m4.2.2.2.2.cmml" xref="S4.Thmtheorem11.p1.10.m4.2.2.2.1"><abs id="S4.Thmtheorem11.p1.10.m4.2.2.2.2.1.cmml" xref="S4.Thmtheorem11.p1.10.m4.2.2.2.1.2"></abs><apply id="S4.Thmtheorem11.p1.10.m4.2.2.2.1.1.cmml" xref="S4.Thmtheorem11.p1.10.m4.2.2.2.1.1"><times id="S4.Thmtheorem11.p1.10.m4.2.2.2.1.1.2.cmml" xref="S4.Thmtheorem11.p1.10.m4.2.2.2.1.1.2"></times><ci id="S4.Thmtheorem11.p1.10.m4.2.2.2.1.1.3.cmml" xref="S4.Thmtheorem11.p1.10.m4.2.2.2.1.1.3">𝑉</ci><apply id="S4.Thmtheorem11.p1.10.m4.2.2.2.1.1.1.1.1.cmml" xref="S4.Thmtheorem11.p1.10.m4.2.2.2.1.1.1.1"><csymbol cd="ambiguous" id="S4.Thmtheorem11.p1.10.m4.2.2.2.1.1.1.1.1.1.cmml" xref="S4.Thmtheorem11.p1.10.m4.2.2.2.1.1.1.1">subscript</csymbol><ci id="S4.Thmtheorem11.p1.10.m4.2.2.2.1.1.1.1.1.2.cmml" xref="S4.Thmtheorem11.p1.10.m4.2.2.2.1.1.1.1.1.2">𝐺</ci><ci id="S4.Thmtheorem11.p1.10.m4.2.2.2.1.1.1.1.1.3.cmml" xref="S4.Thmtheorem11.p1.10.m4.2.2.2.1.1.1.1.1.3">𝑦</ci></apply></apply></apply></apply><apply id="S4.Thmtheorem11.p1.10.m4.2.2c.cmml" xref="S4.Thmtheorem11.p1.10.m4.2.2"><eq id="S4.Thmtheorem11.p1.10.m4.2.2.5.cmml" xref="S4.Thmtheorem11.p1.10.m4.2.2.5"></eq><share href="https://arxiv.org/html/2503.00712v1#S4.Thmtheorem11.p1.10.m4.2.2.2.cmml" id="S4.Thmtheorem11.p1.10.m4.2.2d.cmml" xref="S4.Thmtheorem11.p1.10.m4.2.2"></share><cn id="S4.Thmtheorem11.p1.10.m4.2.2.6.cmml" type="integer" xref="S4.Thmtheorem11.p1.10.m4.2.2.6">2</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem11.p1.10.m4.2c">|V(G_{x})|=|V(G_{y})|=2</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem11.p1.10.m4.2d">| italic_V ( italic_G start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT ) | = | italic_V ( italic_G start_POSTSUBSCRIPT italic_y end_POSTSUBSCRIPT ) | = 2</annotation></semantics></math>), we merge <math alttext="G_{x}" class="ltx_Math" display="inline" id="S4.Thmtheorem11.p1.11.m5.1"><semantics id="S4.Thmtheorem11.p1.11.m5.1a"><msub id="S4.Thmtheorem11.p1.11.m5.1.1" xref="S4.Thmtheorem11.p1.11.m5.1.1.cmml"><mi id="S4.Thmtheorem11.p1.11.m5.1.1.2" xref="S4.Thmtheorem11.p1.11.m5.1.1.2.cmml">G</mi><mi id="S4.Thmtheorem11.p1.11.m5.1.1.3" xref="S4.Thmtheorem11.p1.11.m5.1.1.3.cmml">x</mi></msub><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem11.p1.11.m5.1b"><apply id="S4.Thmtheorem11.p1.11.m5.1.1.cmml" xref="S4.Thmtheorem11.p1.11.m5.1.1"><csymbol cd="ambiguous" id="S4.Thmtheorem11.p1.11.m5.1.1.1.cmml" xref="S4.Thmtheorem11.p1.11.m5.1.1">subscript</csymbol><ci id="S4.Thmtheorem11.p1.11.m5.1.1.2.cmml" xref="S4.Thmtheorem11.p1.11.m5.1.1.2">𝐺</ci><ci id="S4.Thmtheorem11.p1.11.m5.1.1.3.cmml" xref="S4.Thmtheorem11.p1.11.m5.1.1.3">𝑥</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem11.p1.11.m5.1c">G_{x}</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem11.p1.11.m5.1d">italic_G start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="G_{y}" class="ltx_Math" display="inline" id="S4.Thmtheorem11.p1.12.m6.1"><semantics id="S4.Thmtheorem11.p1.12.m6.1a"><msub id="S4.Thmtheorem11.p1.12.m6.1.1" xref="S4.Thmtheorem11.p1.12.m6.1.1.cmml"><mi id="S4.Thmtheorem11.p1.12.m6.1.1.2" xref="S4.Thmtheorem11.p1.12.m6.1.1.2.cmml">G</mi><mi id="S4.Thmtheorem11.p1.12.m6.1.1.3" xref="S4.Thmtheorem11.p1.12.m6.1.1.3.cmml">y</mi></msub><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem11.p1.12.m6.1b"><apply id="S4.Thmtheorem11.p1.12.m6.1.1.cmml" xref="S4.Thmtheorem11.p1.12.m6.1.1"><csymbol cd="ambiguous" id="S4.Thmtheorem11.p1.12.m6.1.1.1.cmml" xref="S4.Thmtheorem11.p1.12.m6.1.1">subscript</csymbol><ci id="S4.Thmtheorem11.p1.12.m6.1.1.2.cmml" xref="S4.Thmtheorem11.p1.12.m6.1.1.2">𝐺</ci><ci id="S4.Thmtheorem11.p1.12.m6.1.1.3.cmml" xref="S4.Thmtheorem11.p1.12.m6.1.1.3">𝑦</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem11.p1.12.m6.1c">G_{y}</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem11.p1.12.m6.1d">italic_G start_POSTSUBSCRIPT italic_y end_POSTSUBSCRIPT</annotation></semantics></math>. Similarly, if <math alttext="x,y" class="ltx_Math" display="inline" id="S4.Thmtheorem11.p1.13.m7.2"><semantics id="S4.Thmtheorem11.p1.13.m7.2a"><mrow id="S4.Thmtheorem11.p1.13.m7.2.3.2" xref="S4.Thmtheorem11.p1.13.m7.2.3.1.cmml"><mi id="S4.Thmtheorem11.p1.13.m7.1.1" xref="S4.Thmtheorem11.p1.13.m7.1.1.cmml">x</mi><mo id="S4.Thmtheorem11.p1.13.m7.2.3.2.1" xref="S4.Thmtheorem11.p1.13.m7.2.3.1.cmml">,</mo><mi id="S4.Thmtheorem11.p1.13.m7.2.2" xref="S4.Thmtheorem11.p1.13.m7.2.2.cmml">y</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem11.p1.13.m7.2b"><list id="S4.Thmtheorem11.p1.13.m7.2.3.1.cmml" xref="S4.Thmtheorem11.p1.13.m7.2.3.2"><ci id="S4.Thmtheorem11.p1.13.m7.1.1.cmml" xref="S4.Thmtheorem11.p1.13.m7.1.1">𝑥</ci><ci id="S4.Thmtheorem11.p1.13.m7.2.2.cmml" xref="S4.Thmtheorem11.p1.13.m7.2.2">𝑦</ci></list></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem11.p1.13.m7.2c">x,y</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem11.p1.13.m7.2d">italic_x , italic_y</annotation></semantics></math> are adjacent tree nodes and both <math alttext="G_{x}" class="ltx_Math" display="inline" id="S4.Thmtheorem11.p1.14.m8.1"><semantics id="S4.Thmtheorem11.p1.14.m8.1a"><msub id="S4.Thmtheorem11.p1.14.m8.1.1" xref="S4.Thmtheorem11.p1.14.m8.1.1.cmml"><mi id="S4.Thmtheorem11.p1.14.m8.1.1.2" xref="S4.Thmtheorem11.p1.14.m8.1.1.2.cmml">G</mi><mi id="S4.Thmtheorem11.p1.14.m8.1.1.3" xref="S4.Thmtheorem11.p1.14.m8.1.1.3.cmml">x</mi></msub><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem11.p1.14.m8.1b"><apply id="S4.Thmtheorem11.p1.14.m8.1.1.cmml" xref="S4.Thmtheorem11.p1.14.m8.1.1"><csymbol cd="ambiguous" id="S4.Thmtheorem11.p1.14.m8.1.1.1.cmml" xref="S4.Thmtheorem11.p1.14.m8.1.1">subscript</csymbol><ci id="S4.Thmtheorem11.p1.14.m8.1.1.2.cmml" xref="S4.Thmtheorem11.p1.14.m8.1.1.2">𝐺</ci><ci id="S4.Thmtheorem11.p1.14.m8.1.1.3.cmml" xref="S4.Thmtheorem11.p1.14.m8.1.1.3">𝑥</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem11.p1.14.m8.1c">G_{x}</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem11.p1.14.m8.1d">italic_G start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="G_{y}" class="ltx_Math" display="inline" id="S4.Thmtheorem11.p1.15.m9.1"><semantics id="S4.Thmtheorem11.p1.15.m9.1a"><msub id="S4.Thmtheorem11.p1.15.m9.1.1" xref="S4.Thmtheorem11.p1.15.m9.1.1.cmml"><mi id="S4.Thmtheorem11.p1.15.m9.1.1.2" xref="S4.Thmtheorem11.p1.15.m9.1.1.2.cmml">G</mi><mi id="S4.Thmtheorem11.p1.15.m9.1.1.3" xref="S4.Thmtheorem11.p1.15.m9.1.1.3.cmml">y</mi></msub><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem11.p1.15.m9.1b"><apply id="S4.Thmtheorem11.p1.15.m9.1.1.cmml" xref="S4.Thmtheorem11.p1.15.m9.1.1"><csymbol cd="ambiguous" id="S4.Thmtheorem11.p1.15.m9.1.1.1.cmml" xref="S4.Thmtheorem11.p1.15.m9.1.1">subscript</csymbol><ci id="S4.Thmtheorem11.p1.15.m9.1.1.2.cmml" xref="S4.Thmtheorem11.p1.15.m9.1.1.2">𝐺</ci><ci id="S4.Thmtheorem11.p1.15.m9.1.1.3.cmml" xref="S4.Thmtheorem11.p1.15.m9.1.1.3">𝑦</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem11.p1.15.m9.1c">G_{y}</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem11.p1.15.m9.1d">italic_G start_POSTSUBSCRIPT italic_y end_POSTSUBSCRIPT</annotation></semantics></math> are simple cycles, we merge <math alttext="G_{x}" class="ltx_Math" display="inline" id="S4.Thmtheorem11.p1.16.m10.1"><semantics id="S4.Thmtheorem11.p1.16.m10.1a"><msub id="S4.Thmtheorem11.p1.16.m10.1.1" xref="S4.Thmtheorem11.p1.16.m10.1.1.cmml"><mi id="S4.Thmtheorem11.p1.16.m10.1.1.2" xref="S4.Thmtheorem11.p1.16.m10.1.1.2.cmml">G</mi><mi id="S4.Thmtheorem11.p1.16.m10.1.1.3" xref="S4.Thmtheorem11.p1.16.m10.1.1.3.cmml">x</mi></msub><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem11.p1.16.m10.1b"><apply id="S4.Thmtheorem11.p1.16.m10.1.1.cmml" xref="S4.Thmtheorem11.p1.16.m10.1.1"><csymbol cd="ambiguous" id="S4.Thmtheorem11.p1.16.m10.1.1.1.cmml" xref="S4.Thmtheorem11.p1.16.m10.1.1">subscript</csymbol><ci id="S4.Thmtheorem11.p1.16.m10.1.1.2.cmml" xref="S4.Thmtheorem11.p1.16.m10.1.1.2">𝐺</ci><ci id="S4.Thmtheorem11.p1.16.m10.1.1.3.cmml" xref="S4.Thmtheorem11.p1.16.m10.1.1.3">𝑥</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem11.p1.16.m10.1c">G_{x}</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem11.p1.16.m10.1d">italic_G start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="G_{y}" class="ltx_Math" display="inline" id="S4.Thmtheorem11.p1.17.m11.1"><semantics id="S4.Thmtheorem11.p1.17.m11.1a"><msub id="S4.Thmtheorem11.p1.17.m11.1.1" xref="S4.Thmtheorem11.p1.17.m11.1.1.cmml"><mi id="S4.Thmtheorem11.p1.17.m11.1.1.2" xref="S4.Thmtheorem11.p1.17.m11.1.1.2.cmml">G</mi><mi id="S4.Thmtheorem11.p1.17.m11.1.1.3" xref="S4.Thmtheorem11.p1.17.m11.1.1.3.cmml">y</mi></msub><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem11.p1.17.m11.1b"><apply id="S4.Thmtheorem11.p1.17.m11.1.1.cmml" xref="S4.Thmtheorem11.p1.17.m11.1.1"><csymbol cd="ambiguous" id="S4.Thmtheorem11.p1.17.m11.1.1.1.cmml" xref="S4.Thmtheorem11.p1.17.m11.1.1">subscript</csymbol><ci id="S4.Thmtheorem11.p1.17.m11.1.1.2.cmml" xref="S4.Thmtheorem11.p1.17.m11.1.1.2">𝐺</ci><ci id="S4.Thmtheorem11.p1.17.m11.1.1.3.cmml" xref="S4.Thmtheorem11.p1.17.m11.1.1.3">𝑦</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem11.p1.17.m11.1c">G_{y}</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem11.p1.17.m11.1d">italic_G start_POSTSUBSCRIPT italic_y end_POSTSUBSCRIPT</annotation></semantics></math>. See Figure <a class="ltx_ref" href="https://arxiv.org/html/2503.00712v1#S4.F5" title="Figure 5 ‣ 4.2.1 SPQR Trees ‣ 4.2 Two-to-Three Augmentation ‣ 4 Vertex Connectivity Augmentation in Link-Arrival Model ‣ Streaming Algorithms for Network Design"><span class="ltx_text ltx_ref_tag">5</span></a> for an example of the SPQR tree of a 2-connected graph.</p> </div> </div> <figure class="ltx_figure" id="S4.F5"> <div class="ltx_flex_figure"> <div class="ltx_flex_cell ltx_flex_size_2"> <figure class="ltx_figure ltx_figure_panel ltx_align_center" id="S4.F5.sf1"><img alt="Refer to caption" class="ltx_graphics ltx_centering ltx_img_portrait" height="1001" id="S4.F5.sf1.g1" src="x5.png" width="581"/> <figcaption class="ltx_caption ltx_centering"><span class="ltx_tag ltx_tag_figure"><span class="ltx_text" id="S4.F5.sf1.4.2.1" style="font-size:90%;">(a)</span> </span><span class="ltx_text" id="S4.F5.sf1.2.1" style="font-size:90%;">2-connected graph <math alttext="G" class="ltx_Math" display="inline" id="S4.F5.sf1.2.1.m1.1"><semantics id="S4.F5.sf1.2.1.m1.1b"><mi id="S4.F5.sf1.2.1.m1.1.1" xref="S4.F5.sf1.2.1.m1.1.1.cmml">G</mi><annotation-xml encoding="MathML-Content" id="S4.F5.sf1.2.1.m1.1c"><ci id="S4.F5.sf1.2.1.m1.1.1.cmml" xref="S4.F5.sf1.2.1.m1.1.1">𝐺</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.F5.sf1.2.1.m1.1d">G</annotation><annotation encoding="application/x-llamapun" id="S4.F5.sf1.2.1.m1.1e">italic_G</annotation></semantics></math>.</span></figcaption> </figure> </div> <div class="ltx_flex_cell ltx_flex_size_2"> <figure class="ltx_figure ltx_figure_panel ltx_align_center" id="S4.F5.sf2"><img alt="Refer to caption" class="ltx_graphics ltx_centering ltx_img_landscape" height="519" id="S4.F5.sf2.g1" src="x6.png" width="664"/> <figcaption class="ltx_caption ltx_centering"><span class="ltx_tag ltx_tag_figure"><span class="ltx_text" id="S4.F5.sf2.10.5.1" style="font-size:90%;">(b)</span> </span><span class="ltx_text" id="S4.F5.sf2.8.4" style="font-size:90%;">SPQR Tree <math alttext="T" class="ltx_Math" display="inline" id="S4.F5.sf2.5.1.m1.1"><semantics id="S4.F5.sf2.5.1.m1.1b"><mi id="S4.F5.sf2.5.1.m1.1.1" xref="S4.F5.sf2.5.1.m1.1.1.cmml">T</mi><annotation-xml encoding="MathML-Content" id="S4.F5.sf2.5.1.m1.1c"><ci id="S4.F5.sf2.5.1.m1.1.1.cmml" xref="S4.F5.sf2.5.1.m1.1.1">𝑇</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.F5.sf2.5.1.m1.1d">T</annotation><annotation encoding="application/x-llamapun" id="S4.F5.sf2.5.1.m1.1e">italic_T</annotation></semantics></math> with S-nodes <math alttext="A,B,D" class="ltx_Math" display="inline" id="S4.F5.sf2.6.2.m2.3"><semantics id="S4.F5.sf2.6.2.m2.3b"><mrow id="S4.F5.sf2.6.2.m2.3.4.2" xref="S4.F5.sf2.6.2.m2.3.4.1.cmml"><mi id="S4.F5.sf2.6.2.m2.1.1" xref="S4.F5.sf2.6.2.m2.1.1.cmml">A</mi><mo id="S4.F5.sf2.6.2.m2.3.4.2.1" xref="S4.F5.sf2.6.2.m2.3.4.1.cmml">,</mo><mi id="S4.F5.sf2.6.2.m2.2.2" xref="S4.F5.sf2.6.2.m2.2.2.cmml">B</mi><mo id="S4.F5.sf2.6.2.m2.3.4.2.2" xref="S4.F5.sf2.6.2.m2.3.4.1.cmml">,</mo><mi id="S4.F5.sf2.6.2.m2.3.3" xref="S4.F5.sf2.6.2.m2.3.3.cmml">D</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.F5.sf2.6.2.m2.3c"><list id="S4.F5.sf2.6.2.m2.3.4.1.cmml" xref="S4.F5.sf2.6.2.m2.3.4.2"><ci id="S4.F5.sf2.6.2.m2.1.1.cmml" xref="S4.F5.sf2.6.2.m2.1.1">𝐴</ci><ci id="S4.F5.sf2.6.2.m2.2.2.cmml" xref="S4.F5.sf2.6.2.m2.2.2">𝐵</ci><ci id="S4.F5.sf2.6.2.m2.3.3.cmml" xref="S4.F5.sf2.6.2.m2.3.3">𝐷</ci></list></annotation-xml><annotation encoding="application/x-tex" id="S4.F5.sf2.6.2.m2.3d">A,B,D</annotation><annotation encoding="application/x-llamapun" id="S4.F5.sf2.6.2.m2.3e">italic_A , italic_B , italic_D</annotation></semantics></math>, P-node <math alttext="C" class="ltx_Math" display="inline" id="S4.F5.sf2.7.3.m3.1"><semantics id="S4.F5.sf2.7.3.m3.1b"><mi id="S4.F5.sf2.7.3.m3.1.1" xref="S4.F5.sf2.7.3.m3.1.1.cmml">C</mi><annotation-xml encoding="MathML-Content" id="S4.F5.sf2.7.3.m3.1c"><ci id="S4.F5.sf2.7.3.m3.1.1.cmml" xref="S4.F5.sf2.7.3.m3.1.1">𝐶</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.F5.sf2.7.3.m3.1d">C</annotation><annotation encoding="application/x-llamapun" id="S4.F5.sf2.7.3.m3.1e">italic_C</annotation></semantics></math>, R-node <math alttext="E" class="ltx_Math" display="inline" id="S4.F5.sf2.8.4.m4.1"><semantics id="S4.F5.sf2.8.4.m4.1b"><mi id="S4.F5.sf2.8.4.m4.1.1" xref="S4.F5.sf2.8.4.m4.1.1.cmml">E</mi><annotation-xml encoding="MathML-Content" id="S4.F5.sf2.8.4.m4.1c"><ci id="S4.F5.sf2.8.4.m4.1.1.cmml" xref="S4.F5.sf2.8.4.m4.1.1">𝐸</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.F5.sf2.8.4.m4.1d">E</annotation><annotation encoding="application/x-llamapun" id="S4.F5.sf2.8.4.m4.1e">italic_E</annotation></semantics></math>. Virtual edges are dashed.</span></figcaption> </figure> </div> </div> <figcaption class="ltx_caption ltx_centering"><span class="ltx_tag ltx_tag_figure"><span class="ltx_text" id="S4.F5.40.20.1" style="font-size:90%;">Figure 5</span>: </span><span class="ltx_text" id="S4.F5.38.19" style="font-size:90%;">Example of an SPQR tree <math alttext="T" class="ltx_Math" display="inline" id="S4.F5.20.1.m1.1"><semantics id="S4.F5.20.1.m1.1b"><mi id="S4.F5.20.1.m1.1.1" xref="S4.F5.20.1.m1.1.1.cmml">T</mi><annotation-xml encoding="MathML-Content" id="S4.F5.20.1.m1.1c"><ci id="S4.F5.20.1.m1.1.1.cmml" xref="S4.F5.20.1.m1.1.1">𝑇</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.F5.20.1.m1.1d">T</annotation><annotation encoding="application/x-llamapun" id="S4.F5.20.1.m1.1e">italic_T</annotation></semantics></math> constructed from 2-connected graph <math alttext="G" class="ltx_Math" display="inline" id="S4.F5.21.2.m2.1"><semantics id="S4.F5.21.2.m2.1b"><mi id="S4.F5.21.2.m2.1.1" xref="S4.F5.21.2.m2.1.1.cmml">G</mi><annotation-xml encoding="MathML-Content" id="S4.F5.21.2.m2.1c"><ci id="S4.F5.21.2.m2.1.1.cmml" xref="S4.F5.21.2.m2.1.1">𝐺</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.F5.21.2.m2.1d">G</annotation><annotation encoding="application/x-llamapun" id="S4.F5.21.2.m2.1e">italic_G</annotation></semantics></math>. Suppose we root the tree at node <math alttext="C" class="ltx_Math" display="inline" id="S4.F5.22.3.m3.1"><semantics id="S4.F5.22.3.m3.1b"><mi id="S4.F5.22.3.m3.1.1" xref="S4.F5.22.3.m3.1.1.cmml">C</mi><annotation-xml encoding="MathML-Content" id="S4.F5.22.3.m3.1c"><ci id="S4.F5.22.3.m3.1.1.cmml" xref="S4.F5.22.3.m3.1.1">𝐶</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.F5.22.3.m3.1d">C</annotation><annotation encoding="application/x-llamapun" id="S4.F5.22.3.m3.1e">italic_C</annotation></semantics></math>. As defined in Section <a class="ltx_ref" href="https://arxiv.org/html/2503.00712v1#S4.SS2.SSS2" title="4.2.2 The Streaming Algorithm ‣ 4.2 Two-to-Three Augmentation ‣ 4 Vertex Connectivity Augmentation in Link-Arrival Model ‣ Streaming Algorithms for Network Design"><span class="ltx_text ltx_ref_tag">4.2.2</span></a>, <math alttext="\textnormal{parent}(E)=\{5,6\}" class="ltx_Math" display="inline" id="S4.F5.23.4.m4.3"><semantics id="S4.F5.23.4.m4.3b"><mrow id="S4.F5.23.4.m4.3.4" xref="S4.F5.23.4.m4.3.4.cmml"><mrow id="S4.F5.23.4.m4.3.4.2" xref="S4.F5.23.4.m4.3.4.2.cmml"><mtext id="S4.F5.23.4.m4.3.4.2.2" xref="S4.F5.23.4.m4.3.4.2.2a.cmml">parent</mtext><mo id="S4.F5.23.4.m4.3.4.2.1" xref="S4.F5.23.4.m4.3.4.2.1.cmml">⁢</mo><mrow id="S4.F5.23.4.m4.3.4.2.3.2" xref="S4.F5.23.4.m4.3.4.2.cmml"><mo id="S4.F5.23.4.m4.3.4.2.3.2.1" stretchy="false" xref="S4.F5.23.4.m4.3.4.2.cmml">(</mo><mi id="S4.F5.23.4.m4.1.1" xref="S4.F5.23.4.m4.1.1.cmml">E</mi><mo id="S4.F5.23.4.m4.3.4.2.3.2.2" stretchy="false" xref="S4.F5.23.4.m4.3.4.2.cmml">)</mo></mrow></mrow><mo id="S4.F5.23.4.m4.3.4.1" xref="S4.F5.23.4.m4.3.4.1.cmml">=</mo><mrow id="S4.F5.23.4.m4.3.4.3.2" xref="S4.F5.23.4.m4.3.4.3.1.cmml"><mo id="S4.F5.23.4.m4.3.4.3.2.1" stretchy="false" xref="S4.F5.23.4.m4.3.4.3.1.cmml">{</mo><mn id="S4.F5.23.4.m4.2.2" xref="S4.F5.23.4.m4.2.2.cmml">5</mn><mo id="S4.F5.23.4.m4.3.4.3.2.2" xref="S4.F5.23.4.m4.3.4.3.1.cmml">,</mo><mn id="S4.F5.23.4.m4.3.3" xref="S4.F5.23.4.m4.3.3.cmml">6</mn><mo id="S4.F5.23.4.m4.3.4.3.2.3" stretchy="false" xref="S4.F5.23.4.m4.3.4.3.1.cmml">}</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.F5.23.4.m4.3c"><apply id="S4.F5.23.4.m4.3.4.cmml" xref="S4.F5.23.4.m4.3.4"><eq id="S4.F5.23.4.m4.3.4.1.cmml" xref="S4.F5.23.4.m4.3.4.1"></eq><apply id="S4.F5.23.4.m4.3.4.2.cmml" xref="S4.F5.23.4.m4.3.4.2"><times id="S4.F5.23.4.m4.3.4.2.1.cmml" xref="S4.F5.23.4.m4.3.4.2.1"></times><ci id="S4.F5.23.4.m4.3.4.2.2a.cmml" xref="S4.F5.23.4.m4.3.4.2.2"><mtext id="S4.F5.23.4.m4.3.4.2.2.cmml" xref="S4.F5.23.4.m4.3.4.2.2">parent</mtext></ci><ci id="S4.F5.23.4.m4.1.1.cmml" xref="S4.F5.23.4.m4.1.1">𝐸</ci></apply><set id="S4.F5.23.4.m4.3.4.3.1.cmml" xref="S4.F5.23.4.m4.3.4.3.2"><cn id="S4.F5.23.4.m4.2.2.cmml" type="integer" xref="S4.F5.23.4.m4.2.2">5</cn><cn id="S4.F5.23.4.m4.3.3.cmml" type="integer" xref="S4.F5.23.4.m4.3.3">6</cn></set></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.F5.23.4.m4.3d">\textnormal{parent}(E)=\{5,6\}</annotation><annotation encoding="application/x-llamapun" id="S4.F5.23.4.m4.3e">parent ( italic_E ) = { 5 , 6 }</annotation></semantics></math>, and <math alttext="A,B,D" class="ltx_Math" display="inline" id="S4.F5.24.5.m5.3"><semantics id="S4.F5.24.5.m5.3b"><mrow id="S4.F5.24.5.m5.3.4.2" xref="S4.F5.24.5.m5.3.4.1.cmml"><mi id="S4.F5.24.5.m5.1.1" xref="S4.F5.24.5.m5.1.1.cmml">A</mi><mo id="S4.F5.24.5.m5.3.4.2.1" xref="S4.F5.24.5.m5.3.4.1.cmml">,</mo><mi id="S4.F5.24.5.m5.2.2" xref="S4.F5.24.5.m5.2.2.cmml">B</mi><mo id="S4.F5.24.5.m5.3.4.2.2" xref="S4.F5.24.5.m5.3.4.1.cmml">,</mo><mi id="S4.F5.24.5.m5.3.3" xref="S4.F5.24.5.m5.3.3.cmml">D</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.F5.24.5.m5.3c"><list id="S4.F5.24.5.m5.3.4.1.cmml" xref="S4.F5.24.5.m5.3.4.2"><ci id="S4.F5.24.5.m5.1.1.cmml" xref="S4.F5.24.5.m5.1.1">𝐴</ci><ci id="S4.F5.24.5.m5.2.2.cmml" xref="S4.F5.24.5.m5.2.2">𝐵</ci><ci id="S4.F5.24.5.m5.3.3.cmml" xref="S4.F5.24.5.m5.3.3">𝐷</ci></list></annotation-xml><annotation encoding="application/x-tex" id="S4.F5.24.5.m5.3d">A,B,D</annotation><annotation encoding="application/x-llamapun" id="S4.F5.24.5.m5.3e">italic_A , italic_B , italic_D</annotation></semantics></math> all have parent <math alttext="\{1,2\}" class="ltx_Math" display="inline" id="S4.F5.25.6.m6.2"><semantics id="S4.F5.25.6.m6.2b"><mrow id="S4.F5.25.6.m6.2.3.2" xref="S4.F5.25.6.m6.2.3.1.cmml"><mo id="S4.F5.25.6.m6.2.3.2.1" stretchy="false" xref="S4.F5.25.6.m6.2.3.1.cmml">{</mo><mn id="S4.F5.25.6.m6.1.1" xref="S4.F5.25.6.m6.1.1.cmml">1</mn><mo id="S4.F5.25.6.m6.2.3.2.2" xref="S4.F5.25.6.m6.2.3.1.cmml">,</mo><mn id="S4.F5.25.6.m6.2.2" xref="S4.F5.25.6.m6.2.2.cmml">2</mn><mo id="S4.F5.25.6.m6.2.3.2.3" stretchy="false" xref="S4.F5.25.6.m6.2.3.1.cmml">}</mo></mrow><annotation-xml encoding="MathML-Content" id="S4.F5.25.6.m6.2c"><set id="S4.F5.25.6.m6.2.3.1.cmml" xref="S4.F5.25.6.m6.2.3.2"><cn id="S4.F5.25.6.m6.1.1.cmml" type="integer" xref="S4.F5.25.6.m6.1.1">1</cn><cn id="S4.F5.25.6.m6.2.2.cmml" type="integer" xref="S4.F5.25.6.m6.2.2">2</cn></set></annotation-xml><annotation encoding="application/x-tex" id="S4.F5.25.6.m6.2d">\{1,2\}</annotation><annotation encoding="application/x-llamapun" id="S4.F5.25.6.m6.2e">{ 1 , 2 }</annotation></semantics></math> (corresponding to different virtual edges). In this example, <math alttext="h(1)=h(2)=C" class="ltx_Math" display="inline" id="S4.F5.26.7.m7.2"><semantics id="S4.F5.26.7.m7.2b"><mrow id="S4.F5.26.7.m7.2.3" xref="S4.F5.26.7.m7.2.3.cmml"><mrow id="S4.F5.26.7.m7.2.3.2" xref="S4.F5.26.7.m7.2.3.2.cmml"><mi id="S4.F5.26.7.m7.2.3.2.2" xref="S4.F5.26.7.m7.2.3.2.2.cmml">h</mi><mo id="S4.F5.26.7.m7.2.3.2.1" xref="S4.F5.26.7.m7.2.3.2.1.cmml">⁢</mo><mrow id="S4.F5.26.7.m7.2.3.2.3.2" xref="S4.F5.26.7.m7.2.3.2.cmml"><mo id="S4.F5.26.7.m7.2.3.2.3.2.1" stretchy="false" xref="S4.F5.26.7.m7.2.3.2.cmml">(</mo><mn id="S4.F5.26.7.m7.1.1" xref="S4.F5.26.7.m7.1.1.cmml">1</mn><mo id="S4.F5.26.7.m7.2.3.2.3.2.2" stretchy="false" xref="S4.F5.26.7.m7.2.3.2.cmml">)</mo></mrow></mrow><mo id="S4.F5.26.7.m7.2.3.3" xref="S4.F5.26.7.m7.2.3.3.cmml">=</mo><mrow id="S4.F5.26.7.m7.2.3.4" xref="S4.F5.26.7.m7.2.3.4.cmml"><mi id="S4.F5.26.7.m7.2.3.4.2" xref="S4.F5.26.7.m7.2.3.4.2.cmml">h</mi><mo id="S4.F5.26.7.m7.2.3.4.1" xref="S4.F5.26.7.m7.2.3.4.1.cmml">⁢</mo><mrow id="S4.F5.26.7.m7.2.3.4.3.2" xref="S4.F5.26.7.m7.2.3.4.cmml"><mo id="S4.F5.26.7.m7.2.3.4.3.2.1" stretchy="false" xref="S4.F5.26.7.m7.2.3.4.cmml">(</mo><mn id="S4.F5.26.7.m7.2.2" xref="S4.F5.26.7.m7.2.2.cmml">2</mn><mo id="S4.F5.26.7.m7.2.3.4.3.2.2" stretchy="false" xref="S4.F5.26.7.m7.2.3.4.cmml">)</mo></mrow></mrow><mo id="S4.F5.26.7.m7.2.3.5" xref="S4.F5.26.7.m7.2.3.5.cmml">=</mo><mi id="S4.F5.26.7.m7.2.3.6" xref="S4.F5.26.7.m7.2.3.6.cmml">C</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.F5.26.7.m7.2c"><apply id="S4.F5.26.7.m7.2.3.cmml" xref="S4.F5.26.7.m7.2.3"><and id="S4.F5.26.7.m7.2.3a.cmml" xref="S4.F5.26.7.m7.2.3"></and><apply id="S4.F5.26.7.m7.2.3b.cmml" xref="S4.F5.26.7.m7.2.3"><eq id="S4.F5.26.7.m7.2.3.3.cmml" xref="S4.F5.26.7.m7.2.3.3"></eq><apply id="S4.F5.26.7.m7.2.3.2.cmml" xref="S4.F5.26.7.m7.2.3.2"><times id="S4.F5.26.7.m7.2.3.2.1.cmml" xref="S4.F5.26.7.m7.2.3.2.1"></times><ci id="S4.F5.26.7.m7.2.3.2.2.cmml" xref="S4.F5.26.7.m7.2.3.2.2">ℎ</ci><cn id="S4.F5.26.7.m7.1.1.cmml" type="integer" xref="S4.F5.26.7.m7.1.1">1</cn></apply><apply id="S4.F5.26.7.m7.2.3.4.cmml" xref="S4.F5.26.7.m7.2.3.4"><times id="S4.F5.26.7.m7.2.3.4.1.cmml" xref="S4.F5.26.7.m7.2.3.4.1"></times><ci id="S4.F5.26.7.m7.2.3.4.2.cmml" xref="S4.F5.26.7.m7.2.3.4.2">ℎ</ci><cn id="S4.F5.26.7.m7.2.2.cmml" type="integer" xref="S4.F5.26.7.m7.2.2">2</cn></apply></apply><apply id="S4.F5.26.7.m7.2.3c.cmml" xref="S4.F5.26.7.m7.2.3"><eq id="S4.F5.26.7.m7.2.3.5.cmml" xref="S4.F5.26.7.m7.2.3.5"></eq><share href="https://arxiv.org/html/2503.00712v1#S4.F5.26.7.m7.2.3.4.cmml" id="S4.F5.26.7.m7.2.3d.cmml" xref="S4.F5.26.7.m7.2.3"></share><ci id="S4.F5.26.7.m7.2.3.6.cmml" xref="S4.F5.26.7.m7.2.3.6">𝐶</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.F5.26.7.m7.2d">h(1)=h(2)=C</annotation><annotation encoding="application/x-llamapun" id="S4.F5.26.7.m7.2e">italic_h ( 1 ) = italic_h ( 2 ) = italic_C</annotation></semantics></math>, but <math alttext="\ell(1),\ell(2)" class="ltx_Math" display="inline" id="S4.F5.27.8.m8.4"><semantics id="S4.F5.27.8.m8.4b"><mrow id="S4.F5.27.8.m8.4.4.2" xref="S4.F5.27.8.m8.4.4.3.cmml"><mrow id="S4.F5.27.8.m8.3.3.1.1" xref="S4.F5.27.8.m8.3.3.1.1.cmml"><mi id="S4.F5.27.8.m8.3.3.1.1.2" mathvariant="normal" xref="S4.F5.27.8.m8.3.3.1.1.2.cmml">ℓ</mi><mo id="S4.F5.27.8.m8.3.3.1.1.1" xref="S4.F5.27.8.m8.3.3.1.1.1.cmml">⁢</mo><mrow id="S4.F5.27.8.m8.3.3.1.1.3.2" xref="S4.F5.27.8.m8.3.3.1.1.cmml"><mo id="S4.F5.27.8.m8.3.3.1.1.3.2.1" stretchy="false" xref="S4.F5.27.8.m8.3.3.1.1.cmml">(</mo><mn id="S4.F5.27.8.m8.1.1" xref="S4.F5.27.8.m8.1.1.cmml">1</mn><mo id="S4.F5.27.8.m8.3.3.1.1.3.2.2" stretchy="false" xref="S4.F5.27.8.m8.3.3.1.1.cmml">)</mo></mrow></mrow><mo id="S4.F5.27.8.m8.4.4.2.3" xref="S4.F5.27.8.m8.4.4.3.cmml">,</mo><mrow id="S4.F5.27.8.m8.4.4.2.2" xref="S4.F5.27.8.m8.4.4.2.2.cmml"><mi id="S4.F5.27.8.m8.4.4.2.2.2" mathvariant="normal" xref="S4.F5.27.8.m8.4.4.2.2.2.cmml">ℓ</mi><mo id="S4.F5.27.8.m8.4.4.2.2.1" xref="S4.F5.27.8.m8.4.4.2.2.1.cmml">⁢</mo><mrow id="S4.F5.27.8.m8.4.4.2.2.3.2" xref="S4.F5.27.8.m8.4.4.2.2.cmml"><mo id="S4.F5.27.8.m8.4.4.2.2.3.2.1" stretchy="false" xref="S4.F5.27.8.m8.4.4.2.2.cmml">(</mo><mn id="S4.F5.27.8.m8.2.2" xref="S4.F5.27.8.m8.2.2.cmml">2</mn><mo id="S4.F5.27.8.m8.4.4.2.2.3.2.2" stretchy="false" xref="S4.F5.27.8.m8.4.4.2.2.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.F5.27.8.m8.4c"><list id="S4.F5.27.8.m8.4.4.3.cmml" xref="S4.F5.27.8.m8.4.4.2"><apply id="S4.F5.27.8.m8.3.3.1.1.cmml" xref="S4.F5.27.8.m8.3.3.1.1"><times id="S4.F5.27.8.m8.3.3.1.1.1.cmml" xref="S4.F5.27.8.m8.3.3.1.1.1"></times><ci id="S4.F5.27.8.m8.3.3.1.1.2.cmml" xref="S4.F5.27.8.m8.3.3.1.1.2">ℓ</ci><cn id="S4.F5.27.8.m8.1.1.cmml" type="integer" xref="S4.F5.27.8.m8.1.1">1</cn></apply><apply id="S4.F5.27.8.m8.4.4.2.2.cmml" xref="S4.F5.27.8.m8.4.4.2.2"><times id="S4.F5.27.8.m8.4.4.2.2.1.cmml" xref="S4.F5.27.8.m8.4.4.2.2.1"></times><ci id="S4.F5.27.8.m8.4.4.2.2.2.cmml" xref="S4.F5.27.8.m8.4.4.2.2.2">ℓ</ci><cn id="S4.F5.27.8.m8.2.2.cmml" type="integer" xref="S4.F5.27.8.m8.2.2">2</cn></apply></list></annotation-xml><annotation encoding="application/x-tex" id="S4.F5.27.8.m8.4d">\ell(1),\ell(2)</annotation><annotation encoding="application/x-llamapun" id="S4.F5.27.8.m8.4e">roman_ℓ ( 1 ) , roman_ℓ ( 2 )</annotation></semantics></math> can be chosen arbitrarily from <math alttext="\{A,B,D\}" class="ltx_Math" display="inline" id="S4.F5.28.9.m9.3"><semantics id="S4.F5.28.9.m9.3b"><mrow id="S4.F5.28.9.m9.3.4.2" xref="S4.F5.28.9.m9.3.4.1.cmml"><mo id="S4.F5.28.9.m9.3.4.2.1" stretchy="false" xref="S4.F5.28.9.m9.3.4.1.cmml">{</mo><mi id="S4.F5.28.9.m9.1.1" xref="S4.F5.28.9.m9.1.1.cmml">A</mi><mo id="S4.F5.28.9.m9.3.4.2.2" xref="S4.F5.28.9.m9.3.4.1.cmml">,</mo><mi id="S4.F5.28.9.m9.2.2" xref="S4.F5.28.9.m9.2.2.cmml">B</mi><mo id="S4.F5.28.9.m9.3.4.2.3" xref="S4.F5.28.9.m9.3.4.1.cmml">,</mo><mi id="S4.F5.28.9.m9.3.3" xref="S4.F5.28.9.m9.3.3.cmml">D</mi><mo id="S4.F5.28.9.m9.3.4.2.4" stretchy="false" xref="S4.F5.28.9.m9.3.4.1.cmml">}</mo></mrow><annotation-xml encoding="MathML-Content" id="S4.F5.28.9.m9.3c"><set id="S4.F5.28.9.m9.3.4.1.cmml" xref="S4.F5.28.9.m9.3.4.2"><ci id="S4.F5.28.9.m9.1.1.cmml" xref="S4.F5.28.9.m9.1.1">𝐴</ci><ci id="S4.F5.28.9.m9.2.2.cmml" xref="S4.F5.28.9.m9.2.2">𝐵</ci><ci id="S4.F5.28.9.m9.3.3.cmml" xref="S4.F5.28.9.m9.3.3">𝐷</ci></set></annotation-xml><annotation encoding="application/x-tex" id="S4.F5.28.9.m9.3d">\{A,B,D\}</annotation><annotation encoding="application/x-llamapun" id="S4.F5.28.9.m9.3e">{ italic_A , italic_B , italic_D }</annotation></semantics></math>. Vertices <math alttext="3" class="ltx_Math" display="inline" id="S4.F5.29.10.m10.1"><semantics id="S4.F5.29.10.m10.1b"><mn id="S4.F5.29.10.m10.1.1" xref="S4.F5.29.10.m10.1.1.cmml">3</mn><annotation-xml encoding="MathML-Content" id="S4.F5.29.10.m10.1c"><cn id="S4.F5.29.10.m10.1.1.cmml" type="integer" xref="S4.F5.29.10.m10.1.1">3</cn></annotation-xml><annotation encoding="application/x-tex" id="S4.F5.29.10.m10.1d">3</annotation><annotation encoding="application/x-llamapun" id="S4.F5.29.10.m10.1e">3</annotation></semantics></math> and <math alttext="4" class="ltx_Math" display="inline" id="S4.F5.30.11.m11.1"><semantics id="S4.F5.30.11.m11.1b"><mn id="S4.F5.30.11.m11.1.1" xref="S4.F5.30.11.m11.1.1.cmml">4</mn><annotation-xml encoding="MathML-Content" id="S4.F5.30.11.m11.1c"><cn id="S4.F5.30.11.m11.1.1.cmml" type="integer" xref="S4.F5.30.11.m11.1.1">4</cn></annotation-xml><annotation encoding="application/x-tex" id="S4.F5.30.11.m11.1d">4</annotation><annotation encoding="application/x-llamapun" id="S4.F5.30.11.m11.1e">4</annotation></semantics></math> are not included in any virtual edges; thus <math alttext="h(3)=\ell(3)=A" class="ltx_Math" display="inline" id="S4.F5.31.12.m12.2"><semantics id="S4.F5.31.12.m12.2b"><mrow id="S4.F5.31.12.m12.2.3" xref="S4.F5.31.12.m12.2.3.cmml"><mrow id="S4.F5.31.12.m12.2.3.2" xref="S4.F5.31.12.m12.2.3.2.cmml"><mi id="S4.F5.31.12.m12.2.3.2.2" xref="S4.F5.31.12.m12.2.3.2.2.cmml">h</mi><mo id="S4.F5.31.12.m12.2.3.2.1" xref="S4.F5.31.12.m12.2.3.2.1.cmml">⁢</mo><mrow id="S4.F5.31.12.m12.2.3.2.3.2" xref="S4.F5.31.12.m12.2.3.2.cmml"><mo id="S4.F5.31.12.m12.2.3.2.3.2.1" stretchy="false" xref="S4.F5.31.12.m12.2.3.2.cmml">(</mo><mn id="S4.F5.31.12.m12.1.1" xref="S4.F5.31.12.m12.1.1.cmml">3</mn><mo id="S4.F5.31.12.m12.2.3.2.3.2.2" stretchy="false" xref="S4.F5.31.12.m12.2.3.2.cmml">)</mo></mrow></mrow><mo id="S4.F5.31.12.m12.2.3.3" xref="S4.F5.31.12.m12.2.3.3.cmml">=</mo><mrow id="S4.F5.31.12.m12.2.3.4" xref="S4.F5.31.12.m12.2.3.4.cmml"><mi id="S4.F5.31.12.m12.2.3.4.2" mathvariant="normal" xref="S4.F5.31.12.m12.2.3.4.2.cmml">ℓ</mi><mo id="S4.F5.31.12.m12.2.3.4.1" xref="S4.F5.31.12.m12.2.3.4.1.cmml">⁢</mo><mrow id="S4.F5.31.12.m12.2.3.4.3.2" xref="S4.F5.31.12.m12.2.3.4.cmml"><mo id="S4.F5.31.12.m12.2.3.4.3.2.1" stretchy="false" xref="S4.F5.31.12.m12.2.3.4.cmml">(</mo><mn id="S4.F5.31.12.m12.2.2" xref="S4.F5.31.12.m12.2.2.cmml">3</mn><mo id="S4.F5.31.12.m12.2.3.4.3.2.2" stretchy="false" xref="S4.F5.31.12.m12.2.3.4.cmml">)</mo></mrow></mrow><mo id="S4.F5.31.12.m12.2.3.5" xref="S4.F5.31.12.m12.2.3.5.cmml">=</mo><mi id="S4.F5.31.12.m12.2.3.6" xref="S4.F5.31.12.m12.2.3.6.cmml">A</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.F5.31.12.m12.2c"><apply id="S4.F5.31.12.m12.2.3.cmml" xref="S4.F5.31.12.m12.2.3"><and id="S4.F5.31.12.m12.2.3a.cmml" xref="S4.F5.31.12.m12.2.3"></and><apply id="S4.F5.31.12.m12.2.3b.cmml" xref="S4.F5.31.12.m12.2.3"><eq id="S4.F5.31.12.m12.2.3.3.cmml" xref="S4.F5.31.12.m12.2.3.3"></eq><apply id="S4.F5.31.12.m12.2.3.2.cmml" xref="S4.F5.31.12.m12.2.3.2"><times id="S4.F5.31.12.m12.2.3.2.1.cmml" xref="S4.F5.31.12.m12.2.3.2.1"></times><ci id="S4.F5.31.12.m12.2.3.2.2.cmml" xref="S4.F5.31.12.m12.2.3.2.2">ℎ</ci><cn id="S4.F5.31.12.m12.1.1.cmml" type="integer" xref="S4.F5.31.12.m12.1.1">3</cn></apply><apply id="S4.F5.31.12.m12.2.3.4.cmml" xref="S4.F5.31.12.m12.2.3.4"><times id="S4.F5.31.12.m12.2.3.4.1.cmml" xref="S4.F5.31.12.m12.2.3.4.1"></times><ci id="S4.F5.31.12.m12.2.3.4.2.cmml" xref="S4.F5.31.12.m12.2.3.4.2">ℓ</ci><cn id="S4.F5.31.12.m12.2.2.cmml" type="integer" xref="S4.F5.31.12.m12.2.2">3</cn></apply></apply><apply id="S4.F5.31.12.m12.2.3c.cmml" xref="S4.F5.31.12.m12.2.3"><eq id="S4.F5.31.12.m12.2.3.5.cmml" xref="S4.F5.31.12.m12.2.3.5"></eq><share href="https://arxiv.org/html/2503.00712v1#S4.F5.31.12.m12.2.3.4.cmml" id="S4.F5.31.12.m12.2.3d.cmml" xref="S4.F5.31.12.m12.2.3"></share><ci id="S4.F5.31.12.m12.2.3.6.cmml" xref="S4.F5.31.12.m12.2.3.6">𝐴</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.F5.31.12.m12.2d">h(3)=\ell(3)=A</annotation><annotation encoding="application/x-llamapun" id="S4.F5.31.12.m12.2e">italic_h ( 3 ) = roman_ℓ ( 3 ) = italic_A</annotation></semantics></math> and <math alttext="h(4)=\ell(4)=B" class="ltx_Math" display="inline" id="S4.F5.32.13.m13.2"><semantics id="S4.F5.32.13.m13.2b"><mrow id="S4.F5.32.13.m13.2.3" xref="S4.F5.32.13.m13.2.3.cmml"><mrow id="S4.F5.32.13.m13.2.3.2" xref="S4.F5.32.13.m13.2.3.2.cmml"><mi id="S4.F5.32.13.m13.2.3.2.2" xref="S4.F5.32.13.m13.2.3.2.2.cmml">h</mi><mo id="S4.F5.32.13.m13.2.3.2.1" xref="S4.F5.32.13.m13.2.3.2.1.cmml">⁢</mo><mrow id="S4.F5.32.13.m13.2.3.2.3.2" xref="S4.F5.32.13.m13.2.3.2.cmml"><mo id="S4.F5.32.13.m13.2.3.2.3.2.1" stretchy="false" xref="S4.F5.32.13.m13.2.3.2.cmml">(</mo><mn id="S4.F5.32.13.m13.1.1" xref="S4.F5.32.13.m13.1.1.cmml">4</mn><mo id="S4.F5.32.13.m13.2.3.2.3.2.2" stretchy="false" xref="S4.F5.32.13.m13.2.3.2.cmml">)</mo></mrow></mrow><mo id="S4.F5.32.13.m13.2.3.3" xref="S4.F5.32.13.m13.2.3.3.cmml">=</mo><mrow id="S4.F5.32.13.m13.2.3.4" xref="S4.F5.32.13.m13.2.3.4.cmml"><mi id="S4.F5.32.13.m13.2.3.4.2" mathvariant="normal" xref="S4.F5.32.13.m13.2.3.4.2.cmml">ℓ</mi><mo id="S4.F5.32.13.m13.2.3.4.1" xref="S4.F5.32.13.m13.2.3.4.1.cmml">⁢</mo><mrow id="S4.F5.32.13.m13.2.3.4.3.2" xref="S4.F5.32.13.m13.2.3.4.cmml"><mo id="S4.F5.32.13.m13.2.3.4.3.2.1" stretchy="false" xref="S4.F5.32.13.m13.2.3.4.cmml">(</mo><mn id="S4.F5.32.13.m13.2.2" xref="S4.F5.32.13.m13.2.2.cmml">4</mn><mo id="S4.F5.32.13.m13.2.3.4.3.2.2" stretchy="false" xref="S4.F5.32.13.m13.2.3.4.cmml">)</mo></mrow></mrow><mo id="S4.F5.32.13.m13.2.3.5" xref="S4.F5.32.13.m13.2.3.5.cmml">=</mo><mi id="S4.F5.32.13.m13.2.3.6" xref="S4.F5.32.13.m13.2.3.6.cmml">B</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.F5.32.13.m13.2c"><apply id="S4.F5.32.13.m13.2.3.cmml" xref="S4.F5.32.13.m13.2.3"><and id="S4.F5.32.13.m13.2.3a.cmml" xref="S4.F5.32.13.m13.2.3"></and><apply id="S4.F5.32.13.m13.2.3b.cmml" xref="S4.F5.32.13.m13.2.3"><eq id="S4.F5.32.13.m13.2.3.3.cmml" xref="S4.F5.32.13.m13.2.3.3"></eq><apply id="S4.F5.32.13.m13.2.3.2.cmml" xref="S4.F5.32.13.m13.2.3.2"><times id="S4.F5.32.13.m13.2.3.2.1.cmml" xref="S4.F5.32.13.m13.2.3.2.1"></times><ci id="S4.F5.32.13.m13.2.3.2.2.cmml" xref="S4.F5.32.13.m13.2.3.2.2">ℎ</ci><cn id="S4.F5.32.13.m13.1.1.cmml" type="integer" xref="S4.F5.32.13.m13.1.1">4</cn></apply><apply id="S4.F5.32.13.m13.2.3.4.cmml" xref="S4.F5.32.13.m13.2.3.4"><times id="S4.F5.32.13.m13.2.3.4.1.cmml" xref="S4.F5.32.13.m13.2.3.4.1"></times><ci id="S4.F5.32.13.m13.2.3.4.2.cmml" xref="S4.F5.32.13.m13.2.3.4.2">ℓ</ci><cn id="S4.F5.32.13.m13.2.2.cmml" type="integer" xref="S4.F5.32.13.m13.2.2">4</cn></apply></apply><apply id="S4.F5.32.13.m13.2.3c.cmml" xref="S4.F5.32.13.m13.2.3"><eq id="S4.F5.32.13.m13.2.3.5.cmml" xref="S4.F5.32.13.m13.2.3.5"></eq><share href="https://arxiv.org/html/2503.00712v1#S4.F5.32.13.m13.2.3.4.cmml" id="S4.F5.32.13.m13.2.3d.cmml" xref="S4.F5.32.13.m13.2.3"></share><ci id="S4.F5.32.13.m13.2.3.6.cmml" xref="S4.F5.32.13.m13.2.3.6">𝐵</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.F5.32.13.m13.2d">h(4)=\ell(4)=B</annotation><annotation encoding="application/x-llamapun" id="S4.F5.32.13.m13.2e">italic_h ( 4 ) = roman_ℓ ( 4 ) = italic_B</annotation></semantics></math>. The same holds for vertices <math alttext="7,8,9,10" class="ltx_Math" display="inline" id="S4.F5.33.14.m14.4"><semantics id="S4.F5.33.14.m14.4b"><mrow id="S4.F5.33.14.m14.4.5.2" xref="S4.F5.33.14.m14.4.5.1.cmml"><mn id="S4.F5.33.14.m14.1.1" xref="S4.F5.33.14.m14.1.1.cmml">7</mn><mo id="S4.F5.33.14.m14.4.5.2.1" xref="S4.F5.33.14.m14.4.5.1.cmml">,</mo><mn id="S4.F5.33.14.m14.2.2" xref="S4.F5.33.14.m14.2.2.cmml">8</mn><mo id="S4.F5.33.14.m14.4.5.2.2" xref="S4.F5.33.14.m14.4.5.1.cmml">,</mo><mn id="S4.F5.33.14.m14.3.3" xref="S4.F5.33.14.m14.3.3.cmml">9</mn><mo id="S4.F5.33.14.m14.4.5.2.3" xref="S4.F5.33.14.m14.4.5.1.cmml">,</mo><mn id="S4.F5.33.14.m14.4.4" xref="S4.F5.33.14.m14.4.4.cmml">10</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.F5.33.14.m14.4c"><list id="S4.F5.33.14.m14.4.5.1.cmml" xref="S4.F5.33.14.m14.4.5.2"><cn id="S4.F5.33.14.m14.1.1.cmml" type="integer" xref="S4.F5.33.14.m14.1.1">7</cn><cn id="S4.F5.33.14.m14.2.2.cmml" type="integer" xref="S4.F5.33.14.m14.2.2">8</cn><cn id="S4.F5.33.14.m14.3.3.cmml" type="integer" xref="S4.F5.33.14.m14.3.3">9</cn><cn id="S4.F5.33.14.m14.4.4.cmml" type="integer" xref="S4.F5.33.14.m14.4.4">10</cn></list></annotation-xml><annotation encoding="application/x-tex" id="S4.F5.33.14.m14.4d">7,8,9,10</annotation><annotation encoding="application/x-llamapun" id="S4.F5.33.14.m14.4e">7 , 8 , 9 , 10</annotation></semantics></math>; <math alttext="h" class="ltx_Math" display="inline" id="S4.F5.34.15.m15.1"><semantics id="S4.F5.34.15.m15.1b"><mi id="S4.F5.34.15.m15.1.1" xref="S4.F5.34.15.m15.1.1.cmml">h</mi><annotation-xml encoding="MathML-Content" id="S4.F5.34.15.m15.1c"><ci id="S4.F5.34.15.m15.1.1.cmml" xref="S4.F5.34.15.m15.1.1">ℎ</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.F5.34.15.m15.1d">h</annotation><annotation encoding="application/x-llamapun" id="S4.F5.34.15.m15.1e">italic_h</annotation></semantics></math> and <math alttext="\ell" class="ltx_Math" display="inline" id="S4.F5.35.16.m16.1"><semantics id="S4.F5.35.16.m16.1b"><mi id="S4.F5.35.16.m16.1.1" mathvariant="normal" xref="S4.F5.35.16.m16.1.1.cmml">ℓ</mi><annotation-xml encoding="MathML-Content" id="S4.F5.35.16.m16.1c"><ci id="S4.F5.35.16.m16.1.1.cmml" xref="S4.F5.35.16.m16.1.1">ℓ</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.F5.35.16.m16.1d">\ell</annotation><annotation encoding="application/x-llamapun" id="S4.F5.35.16.m16.1e">roman_ℓ</annotation></semantics></math> both map to <math alttext="E" class="ltx_Math" display="inline" id="S4.F5.36.17.m17.1"><semantics id="S4.F5.36.17.m17.1b"><mi id="S4.F5.36.17.m17.1.1" xref="S4.F5.36.17.m17.1.1.cmml">E</mi><annotation-xml encoding="MathML-Content" id="S4.F5.36.17.m17.1c"><ci id="S4.F5.36.17.m17.1.1.cmml" xref="S4.F5.36.17.m17.1.1">𝐸</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.F5.36.17.m17.1d">E</annotation><annotation encoding="application/x-llamapun" id="S4.F5.36.17.m17.1e">italic_E</annotation></semantics></math>. Finally, <math alttext="h(5)=h(6)=D" class="ltx_Math" display="inline" id="S4.F5.37.18.m18.2"><semantics id="S4.F5.37.18.m18.2b"><mrow id="S4.F5.37.18.m18.2.3" xref="S4.F5.37.18.m18.2.3.cmml"><mrow id="S4.F5.37.18.m18.2.3.2" xref="S4.F5.37.18.m18.2.3.2.cmml"><mi id="S4.F5.37.18.m18.2.3.2.2" xref="S4.F5.37.18.m18.2.3.2.2.cmml">h</mi><mo id="S4.F5.37.18.m18.2.3.2.1" xref="S4.F5.37.18.m18.2.3.2.1.cmml">⁢</mo><mrow id="S4.F5.37.18.m18.2.3.2.3.2" xref="S4.F5.37.18.m18.2.3.2.cmml"><mo id="S4.F5.37.18.m18.2.3.2.3.2.1" stretchy="false" xref="S4.F5.37.18.m18.2.3.2.cmml">(</mo><mn id="S4.F5.37.18.m18.1.1" xref="S4.F5.37.18.m18.1.1.cmml">5</mn><mo id="S4.F5.37.18.m18.2.3.2.3.2.2" stretchy="false" xref="S4.F5.37.18.m18.2.3.2.cmml">)</mo></mrow></mrow><mo id="S4.F5.37.18.m18.2.3.3" xref="S4.F5.37.18.m18.2.3.3.cmml">=</mo><mrow id="S4.F5.37.18.m18.2.3.4" xref="S4.F5.37.18.m18.2.3.4.cmml"><mi id="S4.F5.37.18.m18.2.3.4.2" xref="S4.F5.37.18.m18.2.3.4.2.cmml">h</mi><mo id="S4.F5.37.18.m18.2.3.4.1" xref="S4.F5.37.18.m18.2.3.4.1.cmml">⁢</mo><mrow id="S4.F5.37.18.m18.2.3.4.3.2" xref="S4.F5.37.18.m18.2.3.4.cmml"><mo id="S4.F5.37.18.m18.2.3.4.3.2.1" stretchy="false" xref="S4.F5.37.18.m18.2.3.4.cmml">(</mo><mn id="S4.F5.37.18.m18.2.2" xref="S4.F5.37.18.m18.2.2.cmml">6</mn><mo id="S4.F5.37.18.m18.2.3.4.3.2.2" stretchy="false" xref="S4.F5.37.18.m18.2.3.4.cmml">)</mo></mrow></mrow><mo id="S4.F5.37.18.m18.2.3.5" xref="S4.F5.37.18.m18.2.3.5.cmml">=</mo><mi id="S4.F5.37.18.m18.2.3.6" xref="S4.F5.37.18.m18.2.3.6.cmml">D</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.F5.37.18.m18.2c"><apply id="S4.F5.37.18.m18.2.3.cmml" xref="S4.F5.37.18.m18.2.3"><and id="S4.F5.37.18.m18.2.3a.cmml" xref="S4.F5.37.18.m18.2.3"></and><apply id="S4.F5.37.18.m18.2.3b.cmml" xref="S4.F5.37.18.m18.2.3"><eq id="S4.F5.37.18.m18.2.3.3.cmml" xref="S4.F5.37.18.m18.2.3.3"></eq><apply id="S4.F5.37.18.m18.2.3.2.cmml" xref="S4.F5.37.18.m18.2.3.2"><times id="S4.F5.37.18.m18.2.3.2.1.cmml" xref="S4.F5.37.18.m18.2.3.2.1"></times><ci id="S4.F5.37.18.m18.2.3.2.2.cmml" xref="S4.F5.37.18.m18.2.3.2.2">ℎ</ci><cn id="S4.F5.37.18.m18.1.1.cmml" type="integer" xref="S4.F5.37.18.m18.1.1">5</cn></apply><apply id="S4.F5.37.18.m18.2.3.4.cmml" xref="S4.F5.37.18.m18.2.3.4"><times id="S4.F5.37.18.m18.2.3.4.1.cmml" xref="S4.F5.37.18.m18.2.3.4.1"></times><ci id="S4.F5.37.18.m18.2.3.4.2.cmml" xref="S4.F5.37.18.m18.2.3.4.2">ℎ</ci><cn id="S4.F5.37.18.m18.2.2.cmml" type="integer" xref="S4.F5.37.18.m18.2.2">6</cn></apply></apply><apply id="S4.F5.37.18.m18.2.3c.cmml" xref="S4.F5.37.18.m18.2.3"><eq id="S4.F5.37.18.m18.2.3.5.cmml" xref="S4.F5.37.18.m18.2.3.5"></eq><share href="https://arxiv.org/html/2503.00712v1#S4.F5.37.18.m18.2.3.4.cmml" id="S4.F5.37.18.m18.2.3d.cmml" xref="S4.F5.37.18.m18.2.3"></share><ci id="S4.F5.37.18.m18.2.3.6.cmml" xref="S4.F5.37.18.m18.2.3.6">𝐷</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.F5.37.18.m18.2d">h(5)=h(6)=D</annotation><annotation encoding="application/x-llamapun" id="S4.F5.37.18.m18.2e">italic_h ( 5 ) = italic_h ( 6 ) = italic_D</annotation></semantics></math> and <math alttext="\ell(5)=\ell(6)=E" class="ltx_Math" display="inline" id="S4.F5.38.19.m19.2"><semantics id="S4.F5.38.19.m19.2b"><mrow id="S4.F5.38.19.m19.2.3" xref="S4.F5.38.19.m19.2.3.cmml"><mrow id="S4.F5.38.19.m19.2.3.2" xref="S4.F5.38.19.m19.2.3.2.cmml"><mi id="S4.F5.38.19.m19.2.3.2.2" mathvariant="normal" xref="S4.F5.38.19.m19.2.3.2.2.cmml">ℓ</mi><mo id="S4.F5.38.19.m19.2.3.2.1" xref="S4.F5.38.19.m19.2.3.2.1.cmml">⁢</mo><mrow id="S4.F5.38.19.m19.2.3.2.3.2" xref="S4.F5.38.19.m19.2.3.2.cmml"><mo id="S4.F5.38.19.m19.2.3.2.3.2.1" stretchy="false" xref="S4.F5.38.19.m19.2.3.2.cmml">(</mo><mn id="S4.F5.38.19.m19.1.1" xref="S4.F5.38.19.m19.1.1.cmml">5</mn><mo id="S4.F5.38.19.m19.2.3.2.3.2.2" stretchy="false" xref="S4.F5.38.19.m19.2.3.2.cmml">)</mo></mrow></mrow><mo id="S4.F5.38.19.m19.2.3.3" xref="S4.F5.38.19.m19.2.3.3.cmml">=</mo><mrow id="S4.F5.38.19.m19.2.3.4" xref="S4.F5.38.19.m19.2.3.4.cmml"><mi id="S4.F5.38.19.m19.2.3.4.2" mathvariant="normal" xref="S4.F5.38.19.m19.2.3.4.2.cmml">ℓ</mi><mo id="S4.F5.38.19.m19.2.3.4.1" xref="S4.F5.38.19.m19.2.3.4.1.cmml">⁢</mo><mrow id="S4.F5.38.19.m19.2.3.4.3.2" xref="S4.F5.38.19.m19.2.3.4.cmml"><mo id="S4.F5.38.19.m19.2.3.4.3.2.1" stretchy="false" xref="S4.F5.38.19.m19.2.3.4.cmml">(</mo><mn id="S4.F5.38.19.m19.2.2" xref="S4.F5.38.19.m19.2.2.cmml">6</mn><mo id="S4.F5.38.19.m19.2.3.4.3.2.2" stretchy="false" xref="S4.F5.38.19.m19.2.3.4.cmml">)</mo></mrow></mrow><mo id="S4.F5.38.19.m19.2.3.5" xref="S4.F5.38.19.m19.2.3.5.cmml">=</mo><mi id="S4.F5.38.19.m19.2.3.6" xref="S4.F5.38.19.m19.2.3.6.cmml">E</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.F5.38.19.m19.2c"><apply id="S4.F5.38.19.m19.2.3.cmml" xref="S4.F5.38.19.m19.2.3"><and id="S4.F5.38.19.m19.2.3a.cmml" xref="S4.F5.38.19.m19.2.3"></and><apply id="S4.F5.38.19.m19.2.3b.cmml" xref="S4.F5.38.19.m19.2.3"><eq id="S4.F5.38.19.m19.2.3.3.cmml" xref="S4.F5.38.19.m19.2.3.3"></eq><apply id="S4.F5.38.19.m19.2.3.2.cmml" xref="S4.F5.38.19.m19.2.3.2"><times id="S4.F5.38.19.m19.2.3.2.1.cmml" xref="S4.F5.38.19.m19.2.3.2.1"></times><ci id="S4.F5.38.19.m19.2.3.2.2.cmml" xref="S4.F5.38.19.m19.2.3.2.2">ℓ</ci><cn id="S4.F5.38.19.m19.1.1.cmml" type="integer" xref="S4.F5.38.19.m19.1.1">5</cn></apply><apply id="S4.F5.38.19.m19.2.3.4.cmml" xref="S4.F5.38.19.m19.2.3.4"><times id="S4.F5.38.19.m19.2.3.4.1.cmml" xref="S4.F5.38.19.m19.2.3.4.1"></times><ci id="S4.F5.38.19.m19.2.3.4.2.cmml" xref="S4.F5.38.19.m19.2.3.4.2">ℓ</ci><cn id="S4.F5.38.19.m19.2.2.cmml" type="integer" xref="S4.F5.38.19.m19.2.2">6</cn></apply></apply><apply id="S4.F5.38.19.m19.2.3c.cmml" xref="S4.F5.38.19.m19.2.3"><eq id="S4.F5.38.19.m19.2.3.5.cmml" xref="S4.F5.38.19.m19.2.3.5"></eq><share href="https://arxiv.org/html/2503.00712v1#S4.F5.38.19.m19.2.3.4.cmml" id="S4.F5.38.19.m19.2.3d.cmml" xref="S4.F5.38.19.m19.2.3"></share><ci id="S4.F5.38.19.m19.2.3.6.cmml" xref="S4.F5.38.19.m19.2.3.6">𝐸</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.F5.38.19.m19.2d">\ell(5)=\ell(6)=E</annotation><annotation encoding="application/x-llamapun" id="S4.F5.38.19.m19.2e">roman_ℓ ( 5 ) = roman_ℓ ( 6 ) = italic_E</annotation></semantics></math>. </span></figcaption> </figure> <div class="ltx_para" id="S4.SS2.SSS1.p5"> <p class="ltx_p" id="S4.SS2.SSS1.p5.1">We use the following properties of SPQR trees, given mostly by <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.00712v1#bib.bibx52" title="">HT73</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.00712v1#bib.bibx33" title="">DBT96a</a>]</cite>. The linear-time implementation of Lemma <a class="ltx_ref" href="https://arxiv.org/html/2503.00712v1#S4.Thmtheorem14" title="Lemma 4.14. ‣ 4.2.1 SPQR Trees ‣ 4.2 Two-to-Three Augmentation ‣ 4 Vertex Connectivity Augmentation in Link-Arrival Model ‣ Streaming Algorithms for Network Design"><span class="ltx_text ltx_ref_tag">4.14</span></a> was given by <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.00712v1#bib.bibx48" title="">GM00</a>]</cite>.</p> </div> <div class="ltx_theorem ltx_theorem_lemma" id="S4.Thmtheorem12"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="S4.Thmtheorem12.1.1.1">Lemma 4.12</span></span><span class="ltx_text ltx_font_bold" id="S4.Thmtheorem12.2.2">.</span> </h6> <div class="ltx_para" id="S4.Thmtheorem12.p1"> <p class="ltx_p" id="S4.Thmtheorem12.p1.1">Each virtual edge is in exactly two tree nodes and is associated with a unique tree edge. Each edge in <math alttext="E(G)" class="ltx_Math" display="inline" id="S4.Thmtheorem12.p1.1.m1.1"><semantics id="S4.Thmtheorem12.p1.1.m1.1a"><mrow id="S4.Thmtheorem12.p1.1.m1.1.2" xref="S4.Thmtheorem12.p1.1.m1.1.2.cmml"><mi id="S4.Thmtheorem12.p1.1.m1.1.2.2" xref="S4.Thmtheorem12.p1.1.m1.1.2.2.cmml">E</mi><mo id="S4.Thmtheorem12.p1.1.m1.1.2.1" xref="S4.Thmtheorem12.p1.1.m1.1.2.1.cmml">⁢</mo><mrow id="S4.Thmtheorem12.p1.1.m1.1.2.3.2" xref="S4.Thmtheorem12.p1.1.m1.1.2.cmml"><mo id="S4.Thmtheorem12.p1.1.m1.1.2.3.2.1" stretchy="false" xref="S4.Thmtheorem12.p1.1.m1.1.2.cmml">(</mo><mi id="S4.Thmtheorem12.p1.1.m1.1.1" xref="S4.Thmtheorem12.p1.1.m1.1.1.cmml">G</mi><mo id="S4.Thmtheorem12.p1.1.m1.1.2.3.2.2" stretchy="false" xref="S4.Thmtheorem12.p1.1.m1.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem12.p1.1.m1.1b"><apply id="S4.Thmtheorem12.p1.1.m1.1.2.cmml" xref="S4.Thmtheorem12.p1.1.m1.1.2"><times id="S4.Thmtheorem12.p1.1.m1.1.2.1.cmml" xref="S4.Thmtheorem12.p1.1.m1.1.2.1"></times><ci id="S4.Thmtheorem12.p1.1.m1.1.2.2.cmml" xref="S4.Thmtheorem12.p1.1.m1.1.2.2">𝐸</ci><ci id="S4.Thmtheorem12.p1.1.m1.1.1.cmml" xref="S4.Thmtheorem12.p1.1.m1.1.1">𝐺</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem12.p1.1.m1.1c">E(G)</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem12.p1.1.m1.1d">italic_E ( italic_G )</annotation></semantics></math> is in exactly one tree node.</p> </div> </div> <div class="ltx_theorem ltx_theorem_lemma" id="S4.Thmtheorem13"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="S4.Thmtheorem13.1.1.1">Lemma 4.13</span></span><span class="ltx_text ltx_font_bold" id="S4.Thmtheorem13.2.2">.</span> </h6> <div class="ltx_para" id="S4.Thmtheorem13.p1"> <p class="ltx_p" id="S4.Thmtheorem13.p1.3">For any 2-connected graph <math alttext="G" class="ltx_Math" display="inline" id="S4.Thmtheorem13.p1.1.m1.1"><semantics id="S4.Thmtheorem13.p1.1.m1.1a"><mi id="S4.Thmtheorem13.p1.1.m1.1.1" xref="S4.Thmtheorem13.p1.1.m1.1.1.cmml">G</mi><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem13.p1.1.m1.1b"><ci id="S4.Thmtheorem13.p1.1.m1.1.1.cmml" xref="S4.Thmtheorem13.p1.1.m1.1.1">𝐺</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem13.p1.1.m1.1c">G</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem13.p1.1.m1.1d">italic_G</annotation></semantics></math> with SPQR tree <math alttext="T" class="ltx_Math" display="inline" id="S4.Thmtheorem13.p1.2.m2.1"><semantics id="S4.Thmtheorem13.p1.2.m2.1a"><mi id="S4.Thmtheorem13.p1.2.m2.1.1" xref="S4.Thmtheorem13.p1.2.m2.1.1.cmml">T</mi><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem13.p1.2.m2.1b"><ci id="S4.Thmtheorem13.p1.2.m2.1.1.cmml" xref="S4.Thmtheorem13.p1.2.m2.1.1">𝑇</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem13.p1.2.m2.1c">T</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem13.p1.2.m2.1d">italic_T</annotation></semantics></math>, <math alttext="\sum_{x\in V(T)}|E(G_{x})|\leq 3|E(G)|-6" class="ltx_Math" display="inline" id="S4.Thmtheorem13.p1.3.m3.4"><semantics id="S4.Thmtheorem13.p1.3.m3.4a"><mrow id="S4.Thmtheorem13.p1.3.m3.4.4" xref="S4.Thmtheorem13.p1.3.m3.4.4.cmml"><mrow id="S4.Thmtheorem13.p1.3.m3.3.3.1" xref="S4.Thmtheorem13.p1.3.m3.3.3.1.cmml"><msub id="S4.Thmtheorem13.p1.3.m3.3.3.1.2" xref="S4.Thmtheorem13.p1.3.m3.3.3.1.2.cmml"><mo id="S4.Thmtheorem13.p1.3.m3.3.3.1.2.2" xref="S4.Thmtheorem13.p1.3.m3.3.3.1.2.2.cmml">∑</mo><mrow id="S4.Thmtheorem13.p1.3.m3.1.1.1" xref="S4.Thmtheorem13.p1.3.m3.1.1.1.cmml"><mi id="S4.Thmtheorem13.p1.3.m3.1.1.1.3" xref="S4.Thmtheorem13.p1.3.m3.1.1.1.3.cmml">x</mi><mo id="S4.Thmtheorem13.p1.3.m3.1.1.1.2" xref="S4.Thmtheorem13.p1.3.m3.1.1.1.2.cmml">∈</mo><mrow id="S4.Thmtheorem13.p1.3.m3.1.1.1.4" 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xref="S4.Thmtheorem13.p1.3.m3.4.4.2.1.1.1.1.1.cmml">⁢</mo><mrow id="S4.Thmtheorem13.p1.3.m3.4.4.2.1.1.1.1.3.2" xref="S4.Thmtheorem13.p1.3.m3.4.4.2.1.1.1.1.cmml"><mo id="S4.Thmtheorem13.p1.3.m3.4.4.2.1.1.1.1.3.2.1" stretchy="false" xref="S4.Thmtheorem13.p1.3.m3.4.4.2.1.1.1.1.cmml">(</mo><mi id="S4.Thmtheorem13.p1.3.m3.2.2" xref="S4.Thmtheorem13.p1.3.m3.2.2.cmml">G</mi><mo id="S4.Thmtheorem13.p1.3.m3.4.4.2.1.1.1.1.3.2.2" stretchy="false" xref="S4.Thmtheorem13.p1.3.m3.4.4.2.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S4.Thmtheorem13.p1.3.m3.4.4.2.1.1.1.3" stretchy="false" xref="S4.Thmtheorem13.p1.3.m3.4.4.2.1.1.2.1.cmml">|</mo></mrow></mrow><mo id="S4.Thmtheorem13.p1.3.m3.4.4.2.2" xref="S4.Thmtheorem13.p1.3.m3.4.4.2.2.cmml">−</mo><mn id="S4.Thmtheorem13.p1.3.m3.4.4.2.3" xref="S4.Thmtheorem13.p1.3.m3.4.4.2.3.cmml">6</mn></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem13.p1.3.m3.4b"><apply id="S4.Thmtheorem13.p1.3.m3.4.4.cmml" xref="S4.Thmtheorem13.p1.3.m3.4.4"><leq id="S4.Thmtheorem13.p1.3.m3.4.4.3.cmml" xref="S4.Thmtheorem13.p1.3.m3.4.4.3"></leq><apply id="S4.Thmtheorem13.p1.3.m3.3.3.1.cmml" xref="S4.Thmtheorem13.p1.3.m3.3.3.1"><apply id="S4.Thmtheorem13.p1.3.m3.3.3.1.2.cmml" xref="S4.Thmtheorem13.p1.3.m3.3.3.1.2"><csymbol cd="ambiguous" id="S4.Thmtheorem13.p1.3.m3.3.3.1.2.1.cmml" xref="S4.Thmtheorem13.p1.3.m3.3.3.1.2">subscript</csymbol><sum id="S4.Thmtheorem13.p1.3.m3.3.3.1.2.2.cmml" xref="S4.Thmtheorem13.p1.3.m3.3.3.1.2.2"></sum><apply id="S4.Thmtheorem13.p1.3.m3.1.1.1.cmml" xref="S4.Thmtheorem13.p1.3.m3.1.1.1"><in id="S4.Thmtheorem13.p1.3.m3.1.1.1.2.cmml" xref="S4.Thmtheorem13.p1.3.m3.1.1.1.2"></in><ci id="S4.Thmtheorem13.p1.3.m3.1.1.1.3.cmml" xref="S4.Thmtheorem13.p1.3.m3.1.1.1.3">𝑥</ci><apply id="S4.Thmtheorem13.p1.3.m3.1.1.1.4.cmml" xref="S4.Thmtheorem13.p1.3.m3.1.1.1.4"><times id="S4.Thmtheorem13.p1.3.m3.1.1.1.4.1.cmml" xref="S4.Thmtheorem13.p1.3.m3.1.1.1.4.1"></times><ci id="S4.Thmtheorem13.p1.3.m3.1.1.1.4.2.cmml" xref="S4.Thmtheorem13.p1.3.m3.1.1.1.4.2">𝑉</ci><ci id="S4.Thmtheorem13.p1.3.m3.1.1.1.1.cmml" xref="S4.Thmtheorem13.p1.3.m3.1.1.1.1">𝑇</ci></apply></apply></apply><apply id="S4.Thmtheorem13.p1.3.m3.3.3.1.1.2.cmml" xref="S4.Thmtheorem13.p1.3.m3.3.3.1.1.1"><abs id="S4.Thmtheorem13.p1.3.m3.3.3.1.1.2.1.cmml" xref="S4.Thmtheorem13.p1.3.m3.3.3.1.1.1.2"></abs><apply id="S4.Thmtheorem13.p1.3.m3.3.3.1.1.1.1.cmml" xref="S4.Thmtheorem13.p1.3.m3.3.3.1.1.1.1"><times id="S4.Thmtheorem13.p1.3.m3.3.3.1.1.1.1.2.cmml" xref="S4.Thmtheorem13.p1.3.m3.3.3.1.1.1.1.2"></times><ci id="S4.Thmtheorem13.p1.3.m3.3.3.1.1.1.1.3.cmml" xref="S4.Thmtheorem13.p1.3.m3.3.3.1.1.1.1.3">𝐸</ci><apply id="S4.Thmtheorem13.p1.3.m3.3.3.1.1.1.1.1.1.1.cmml" xref="S4.Thmtheorem13.p1.3.m3.3.3.1.1.1.1.1.1"><csymbol cd="ambiguous" id="S4.Thmtheorem13.p1.3.m3.3.3.1.1.1.1.1.1.1.1.cmml" xref="S4.Thmtheorem13.p1.3.m3.3.3.1.1.1.1.1.1">subscript</csymbol><ci id="S4.Thmtheorem13.p1.3.m3.3.3.1.1.1.1.1.1.1.2.cmml" xref="S4.Thmtheorem13.p1.3.m3.3.3.1.1.1.1.1.1.1.2">𝐺</ci><ci id="S4.Thmtheorem13.p1.3.m3.3.3.1.1.1.1.1.1.1.3.cmml" xref="S4.Thmtheorem13.p1.3.m3.3.3.1.1.1.1.1.1.1.3">𝑥</ci></apply></apply></apply></apply><apply id="S4.Thmtheorem13.p1.3.m3.4.4.2.cmml" xref="S4.Thmtheorem13.p1.3.m3.4.4.2"><minus id="S4.Thmtheorem13.p1.3.m3.4.4.2.2.cmml" xref="S4.Thmtheorem13.p1.3.m3.4.4.2.2"></minus><apply id="S4.Thmtheorem13.p1.3.m3.4.4.2.1.cmml" xref="S4.Thmtheorem13.p1.3.m3.4.4.2.1"><times id="S4.Thmtheorem13.p1.3.m3.4.4.2.1.2.cmml" xref="S4.Thmtheorem13.p1.3.m3.4.4.2.1.2"></times><cn id="S4.Thmtheorem13.p1.3.m3.4.4.2.1.3.cmml" type="integer" xref="S4.Thmtheorem13.p1.3.m3.4.4.2.1.3">3</cn><apply id="S4.Thmtheorem13.p1.3.m3.4.4.2.1.1.2.cmml" xref="S4.Thmtheorem13.p1.3.m3.4.4.2.1.1.1"><abs id="S4.Thmtheorem13.p1.3.m3.4.4.2.1.1.2.1.cmml" xref="S4.Thmtheorem13.p1.3.m3.4.4.2.1.1.1.2"></abs><apply id="S4.Thmtheorem13.p1.3.m3.4.4.2.1.1.1.1.cmml" xref="S4.Thmtheorem13.p1.3.m3.4.4.2.1.1.1.1"><times id="S4.Thmtheorem13.p1.3.m3.4.4.2.1.1.1.1.1.cmml" xref="S4.Thmtheorem13.p1.3.m3.4.4.2.1.1.1.1.1"></times><ci id="S4.Thmtheorem13.p1.3.m3.4.4.2.1.1.1.1.2.cmml" xref="S4.Thmtheorem13.p1.3.m3.4.4.2.1.1.1.1.2">𝐸</ci><ci id="S4.Thmtheorem13.p1.3.m3.2.2.cmml" xref="S4.Thmtheorem13.p1.3.m3.2.2">𝐺</ci></apply></apply></apply><cn id="S4.Thmtheorem13.p1.3.m3.4.4.2.3.cmml" type="integer" xref="S4.Thmtheorem13.p1.3.m3.4.4.2.3">6</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem13.p1.3.m3.4c">\sum_{x\in V(T)}|E(G_{x})|\leq 3|E(G)|-6</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem13.p1.3.m3.4d">∑ start_POSTSUBSCRIPT italic_x ∈ italic_V ( italic_T ) end_POSTSUBSCRIPT | italic_E ( italic_G start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT ) | ≤ 3 | italic_E ( italic_G ) | - 6</annotation></semantics></math>.</p> </div> </div> <div class="ltx_theorem ltx_theorem_lemma" id="S4.Thmtheorem14"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="S4.Thmtheorem14.1.1.1">Lemma 4.14</span></span><span class="ltx_text ltx_font_bold" id="S4.Thmtheorem14.2.2">.</span> </h6> <div class="ltx_para" id="S4.Thmtheorem14.p1"> <p class="ltx_p" id="S4.Thmtheorem14.p1.2">For any 2-connected graph <math alttext="G" class="ltx_Math" display="inline" id="S4.Thmtheorem14.p1.1.m1.1"><semantics id="S4.Thmtheorem14.p1.1.m1.1a"><mi id="S4.Thmtheorem14.p1.1.m1.1.1" xref="S4.Thmtheorem14.p1.1.m1.1.1.cmml">G</mi><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem14.p1.1.m1.1b"><ci id="S4.Thmtheorem14.p1.1.m1.1.1.cmml" xref="S4.Thmtheorem14.p1.1.m1.1.1">𝐺</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem14.p1.1.m1.1c">G</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem14.p1.1.m1.1d">italic_G</annotation></semantics></math>, there exists a linear-time algorithm to construct the SPQR tree <math alttext="T" class="ltx_Math" display="inline" id="S4.Thmtheorem14.p1.2.m2.1"><semantics id="S4.Thmtheorem14.p1.2.m2.1a"><mi id="S4.Thmtheorem14.p1.2.m2.1.1" xref="S4.Thmtheorem14.p1.2.m2.1.1.cmml">T</mi><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem14.p1.2.m2.1b"><ci id="S4.Thmtheorem14.p1.2.m2.1.1.cmml" xref="S4.Thmtheorem14.p1.2.m2.1.1">𝑇</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem14.p1.2.m2.1c">T</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem14.p1.2.m2.1d">italic_T</annotation></semantics></math>.</p> </div> </div> <div class="ltx_theorem ltx_theorem_lemma" id="S4.Thmtheorem15"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="S4.Thmtheorem15.1.1.1">Lemma 4.15</span></span><span class="ltx_text ltx_font_bold" id="S4.Thmtheorem15.2.2">.</span> </h6> <div class="ltx_para" id="S4.Thmtheorem15.p1"> <p class="ltx_p" id="S4.Thmtheorem15.p1.2">Let <math alttext="x\in V(T)" class="ltx_Math" display="inline" id="S4.Thmtheorem15.p1.1.m1.1"><semantics id="S4.Thmtheorem15.p1.1.m1.1a"><mrow id="S4.Thmtheorem15.p1.1.m1.1.2" xref="S4.Thmtheorem15.p1.1.m1.1.2.cmml"><mi id="S4.Thmtheorem15.p1.1.m1.1.2.2" xref="S4.Thmtheorem15.p1.1.m1.1.2.2.cmml">x</mi><mo id="S4.Thmtheorem15.p1.1.m1.1.2.1" xref="S4.Thmtheorem15.p1.1.m1.1.2.1.cmml">∈</mo><mrow id="S4.Thmtheorem15.p1.1.m1.1.2.3" xref="S4.Thmtheorem15.p1.1.m1.1.2.3.cmml"><mi id="S4.Thmtheorem15.p1.1.m1.1.2.3.2" xref="S4.Thmtheorem15.p1.1.m1.1.2.3.2.cmml">V</mi><mo id="S4.Thmtheorem15.p1.1.m1.1.2.3.1" xref="S4.Thmtheorem15.p1.1.m1.1.2.3.1.cmml">⁢</mo><mrow id="S4.Thmtheorem15.p1.1.m1.1.2.3.3.2" xref="S4.Thmtheorem15.p1.1.m1.1.2.3.cmml"><mo id="S4.Thmtheorem15.p1.1.m1.1.2.3.3.2.1" stretchy="false" xref="S4.Thmtheorem15.p1.1.m1.1.2.3.cmml">(</mo><mi id="S4.Thmtheorem15.p1.1.m1.1.1" xref="S4.Thmtheorem15.p1.1.m1.1.1.cmml">T</mi><mo id="S4.Thmtheorem15.p1.1.m1.1.2.3.3.2.2" stretchy="false" xref="S4.Thmtheorem15.p1.1.m1.1.2.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem15.p1.1.m1.1b"><apply id="S4.Thmtheorem15.p1.1.m1.1.2.cmml" xref="S4.Thmtheorem15.p1.1.m1.1.2"><in id="S4.Thmtheorem15.p1.1.m1.1.2.1.cmml" xref="S4.Thmtheorem15.p1.1.m1.1.2.1"></in><ci id="S4.Thmtheorem15.p1.1.m1.1.2.2.cmml" xref="S4.Thmtheorem15.p1.1.m1.1.2.2">𝑥</ci><apply id="S4.Thmtheorem15.p1.1.m1.1.2.3.cmml" xref="S4.Thmtheorem15.p1.1.m1.1.2.3"><times id="S4.Thmtheorem15.p1.1.m1.1.2.3.1.cmml" xref="S4.Thmtheorem15.p1.1.m1.1.2.3.1"></times><ci id="S4.Thmtheorem15.p1.1.m1.1.2.3.2.cmml" xref="S4.Thmtheorem15.p1.1.m1.1.2.3.2">𝑉</ci><ci id="S4.Thmtheorem15.p1.1.m1.1.1.cmml" xref="S4.Thmtheorem15.p1.1.m1.1.1">𝑇</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem15.p1.1.m1.1c">x\in V(T)</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem15.p1.1.m1.1d">italic_x ∈ italic_V ( italic_T )</annotation></semantics></math>. Then, <math alttext="G_{x}" class="ltx_Math" display="inline" id="S4.Thmtheorem15.p1.2.m2.1"><semantics id="S4.Thmtheorem15.p1.2.m2.1a"><msub id="S4.Thmtheorem15.p1.2.m2.1.1" xref="S4.Thmtheorem15.p1.2.m2.1.1.cmml"><mi id="S4.Thmtheorem15.p1.2.m2.1.1.2" xref="S4.Thmtheorem15.p1.2.m2.1.1.2.cmml">G</mi><mi id="S4.Thmtheorem15.p1.2.m2.1.1.3" xref="S4.Thmtheorem15.p1.2.m2.1.1.3.cmml">x</mi></msub><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem15.p1.2.m2.1b"><apply id="S4.Thmtheorem15.p1.2.m2.1.1.cmml" xref="S4.Thmtheorem15.p1.2.m2.1.1"><csymbol cd="ambiguous" id="S4.Thmtheorem15.p1.2.m2.1.1.1.cmml" xref="S4.Thmtheorem15.p1.2.m2.1.1">subscript</csymbol><ci id="S4.Thmtheorem15.p1.2.m2.1.1.2.cmml" xref="S4.Thmtheorem15.p1.2.m2.1.1.2">𝐺</ci><ci id="S4.Thmtheorem15.p1.2.m2.1.1.3.cmml" xref="S4.Thmtheorem15.p1.2.m2.1.1.3">𝑥</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem15.p1.2.m2.1c">G_{x}</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem15.p1.2.m2.1d">italic_G start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT</annotation></semantics></math> is exactly one of the following:</p> <ol class="ltx_enumerate" id="S4.I5"> <li class="ltx_item" id="S4.I5.i1" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">1.</span> <div class="ltx_para" id="S4.I5.i1.p1"> <p class="ltx_p" id="S4.I5.i1.p1.1">a graph with exactly two vertices and three or more parallel edges between them;</p> </div> </li> <li class="ltx_item" id="S4.I5.i2" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">2.</span> <div class="ltx_para" id="S4.I5.i2.p1"> <p class="ltx_p" id="S4.I5.i2.p1.1">a simple cycle;</p> </div> </li> <li class="ltx_item" id="S4.I5.i3" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">3.</span> <div class="ltx_para" id="S4.I5.i3.p1"> <p class="ltx_p" id="S4.I5.i3.p1.1">a three-connected graph with at least 4 vertices.</p> </div> </li> </ol> <p class="ltx_p" id="S4.Thmtheorem15.p1.3">We call these P-nodes, S-nodes, and R-nodes respectively.</p> </div> </div> <div class="ltx_theorem ltx_theorem_remark" id="S4.Thmtheorem16"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="S4.Thmtheorem16.1.1.1">Remark 4.16</span></span><span class="ltx_text ltx_font_bold" id="S4.Thmtheorem16.2.2">.</span> </h6> <div class="ltx_para" id="S4.Thmtheorem16.p1"> <p class="ltx_p" id="S4.Thmtheorem16.p1.2">The name “SPQR” tree comes from the different types of graphs that tree nodes can correspond to. A “Q”-node has an associated graph that consists of a single edge: this is necessary for the trivial case where the input graph <math alttext="G" class="ltx_Math" display="inline" id="S4.Thmtheorem16.p1.1.m1.1"><semantics id="S4.Thmtheorem16.p1.1.m1.1a"><mi id="S4.Thmtheorem16.p1.1.m1.1.1" xref="S4.Thmtheorem16.p1.1.m1.1.1.cmml">G</mi><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem16.p1.1.m1.1b"><ci id="S4.Thmtheorem16.p1.1.m1.1.1.cmml" xref="S4.Thmtheorem16.p1.1.m1.1.1">𝐺</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem16.p1.1.m1.1c">G</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem16.p1.1.m1.1d">italic_G</annotation></semantics></math> only has one edge. In some constructions of SPQR trees, there is a separate Q-node for each edge in <math alttext="E(G)" class="ltx_Math" display="inline" id="S4.Thmtheorem16.p1.2.m2.1"><semantics id="S4.Thmtheorem16.p1.2.m2.1a"><mrow id="S4.Thmtheorem16.p1.2.m2.1.2" xref="S4.Thmtheorem16.p1.2.m2.1.2.cmml"><mi id="S4.Thmtheorem16.p1.2.m2.1.2.2" xref="S4.Thmtheorem16.p1.2.m2.1.2.2.cmml">E</mi><mo id="S4.Thmtheorem16.p1.2.m2.1.2.1" xref="S4.Thmtheorem16.p1.2.m2.1.2.1.cmml">⁢</mo><mrow id="S4.Thmtheorem16.p1.2.m2.1.2.3.2" xref="S4.Thmtheorem16.p1.2.m2.1.2.cmml"><mo id="S4.Thmtheorem16.p1.2.m2.1.2.3.2.1" stretchy="false" xref="S4.Thmtheorem16.p1.2.m2.1.2.cmml">(</mo><mi id="S4.Thmtheorem16.p1.2.m2.1.1" xref="S4.Thmtheorem16.p1.2.m2.1.1.cmml">G</mi><mo id="S4.Thmtheorem16.p1.2.m2.1.2.3.2.2" stretchy="false" xref="S4.Thmtheorem16.p1.2.m2.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem16.p1.2.m2.1b"><apply id="S4.Thmtheorem16.p1.2.m2.1.2.cmml" xref="S4.Thmtheorem16.p1.2.m2.1.2"><times id="S4.Thmtheorem16.p1.2.m2.1.2.1.cmml" xref="S4.Thmtheorem16.p1.2.m2.1.2.1"></times><ci id="S4.Thmtheorem16.p1.2.m2.1.2.2.cmml" xref="S4.Thmtheorem16.p1.2.m2.1.2.2">𝐸</ci><ci id="S4.Thmtheorem16.p1.2.m2.1.1.cmml" xref="S4.Thmtheorem16.p1.2.m2.1.1">𝐺</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem16.p1.2.m2.1c">E(G)</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem16.p1.2.m2.1d">italic_E ( italic_G )</annotation></semantics></math>; however, this is unnecessary if one distinguishes between real and virtual edges and thus will not be used in this paper.</p> </div> </div> <div class="ltx_theorem ltx_theorem_lemma" id="S4.Thmtheorem17"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="S4.Thmtheorem17.1.1.1">Lemma 4.17</span></span><span class="ltx_text ltx_font_bold" id="S4.Thmtheorem17.2.2">.</span> </h6> <div class="ltx_para" id="S4.Thmtheorem17.p1"> <p class="ltx_p" id="S4.Thmtheorem17.p1.2">Let <math alttext="\{a,b\}" class="ltx_Math" display="inline" id="S4.Thmtheorem17.p1.1.m1.2"><semantics id="S4.Thmtheorem17.p1.1.m1.2a"><mrow id="S4.Thmtheorem17.p1.1.m1.2.3.2" xref="S4.Thmtheorem17.p1.1.m1.2.3.1.cmml"><mo id="S4.Thmtheorem17.p1.1.m1.2.3.2.1" stretchy="false" xref="S4.Thmtheorem17.p1.1.m1.2.3.1.cmml">{</mo><mi id="S4.Thmtheorem17.p1.1.m1.1.1" xref="S4.Thmtheorem17.p1.1.m1.1.1.cmml">a</mi><mo id="S4.Thmtheorem17.p1.1.m1.2.3.2.2" xref="S4.Thmtheorem17.p1.1.m1.2.3.1.cmml">,</mo><mi id="S4.Thmtheorem17.p1.1.m1.2.2" xref="S4.Thmtheorem17.p1.1.m1.2.2.cmml">b</mi><mo id="S4.Thmtheorem17.p1.1.m1.2.3.2.3" stretchy="false" xref="S4.Thmtheorem17.p1.1.m1.2.3.1.cmml">}</mo></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem17.p1.1.m1.2b"><set id="S4.Thmtheorem17.p1.1.m1.2.3.1.cmml" xref="S4.Thmtheorem17.p1.1.m1.2.3.2"><ci id="S4.Thmtheorem17.p1.1.m1.1.1.cmml" xref="S4.Thmtheorem17.p1.1.m1.1.1">𝑎</ci><ci id="S4.Thmtheorem17.p1.1.m1.2.2.cmml" xref="S4.Thmtheorem17.p1.1.m1.2.2">𝑏</ci></set></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem17.p1.1.m1.2c">\{a,b\}</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem17.p1.1.m1.2d">{ italic_a , italic_b }</annotation></semantics></math> be a 2-cut of <math alttext="G" class="ltx_Math" display="inline" id="S4.Thmtheorem17.p1.2.m2.1"><semantics id="S4.Thmtheorem17.p1.2.m2.1a"><mi id="S4.Thmtheorem17.p1.2.m2.1.1" xref="S4.Thmtheorem17.p1.2.m2.1.1.cmml">G</mi><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem17.p1.2.m2.1b"><ci id="S4.Thmtheorem17.p1.2.m2.1.1.cmml" xref="S4.Thmtheorem17.p1.2.m2.1.1">𝐺</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem17.p1.2.m2.1c">G</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem17.p1.2.m2.1d">italic_G</annotation></semantics></math>. Then, one of the following must be true:</p> <ul class="ltx_itemize" id="S4.I6"> <li class="ltx_item" id="S4.I6.i1" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">•</span> <div class="ltx_para" id="S4.I6.i1.p1"> <p class="ltx_p" id="S4.I6.i1.p1.5"><math alttext="\{a,b\}" class="ltx_Math" display="inline" id="S4.I6.i1.p1.1.m1.2"><semantics id="S4.I6.i1.p1.1.m1.2a"><mrow id="S4.I6.i1.p1.1.m1.2.3.2" xref="S4.I6.i1.p1.1.m1.2.3.1.cmml"><mo id="S4.I6.i1.p1.1.m1.2.3.2.1" stretchy="false" xref="S4.I6.i1.p1.1.m1.2.3.1.cmml">{</mo><mi id="S4.I6.i1.p1.1.m1.1.1" xref="S4.I6.i1.p1.1.m1.1.1.cmml">a</mi><mo id="S4.I6.i1.p1.1.m1.2.3.2.2" xref="S4.I6.i1.p1.1.m1.2.3.1.cmml">,</mo><mi id="S4.I6.i1.p1.1.m1.2.2" xref="S4.I6.i1.p1.1.m1.2.2.cmml">b</mi><mo id="S4.I6.i1.p1.1.m1.2.3.2.3" stretchy="false" xref="S4.I6.i1.p1.1.m1.2.3.1.cmml">}</mo></mrow><annotation-xml encoding="MathML-Content" id="S4.I6.i1.p1.1.m1.2b"><set id="S4.I6.i1.p1.1.m1.2.3.1.cmml" xref="S4.I6.i1.p1.1.m1.2.3.2"><ci id="S4.I6.i1.p1.1.m1.1.1.cmml" xref="S4.I6.i1.p1.1.m1.1.1">𝑎</ci><ci id="S4.I6.i1.p1.1.m1.2.2.cmml" xref="S4.I6.i1.p1.1.m1.2.2">𝑏</ci></set></annotation-xml><annotation encoding="application/x-tex" id="S4.I6.i1.p1.1.m1.2c">\{a,b\}</annotation><annotation encoding="application/x-llamapun" id="S4.I6.i1.p1.1.m1.2d">{ italic_a , italic_b }</annotation></semantics></math> is the vertex set of <math alttext="G_{x}" class="ltx_Math" display="inline" id="S4.I6.i1.p1.2.m2.1"><semantics id="S4.I6.i1.p1.2.m2.1a"><msub id="S4.I6.i1.p1.2.m2.1.1" xref="S4.I6.i1.p1.2.m2.1.1.cmml"><mi id="S4.I6.i1.p1.2.m2.1.1.2" xref="S4.I6.i1.p1.2.m2.1.1.2.cmml">G</mi><mi id="S4.I6.i1.p1.2.m2.1.1.3" xref="S4.I6.i1.p1.2.m2.1.1.3.cmml">x</mi></msub><annotation-xml encoding="MathML-Content" id="S4.I6.i1.p1.2.m2.1b"><apply id="S4.I6.i1.p1.2.m2.1.1.cmml" xref="S4.I6.i1.p1.2.m2.1.1"><csymbol cd="ambiguous" id="S4.I6.i1.p1.2.m2.1.1.1.cmml" xref="S4.I6.i1.p1.2.m2.1.1">subscript</csymbol><ci id="S4.I6.i1.p1.2.m2.1.1.2.cmml" xref="S4.I6.i1.p1.2.m2.1.1.2">𝐺</ci><ci id="S4.I6.i1.p1.2.m2.1.1.3.cmml" xref="S4.I6.i1.p1.2.m2.1.1.3">𝑥</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.I6.i1.p1.2.m2.1c">G_{x}</annotation><annotation encoding="application/x-llamapun" id="S4.I6.i1.p1.2.m2.1d">italic_G start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT</annotation></semantics></math> for a P-node <math alttext="x" class="ltx_Math" display="inline" id="S4.I6.i1.p1.3.m3.1"><semantics id="S4.I6.i1.p1.3.m3.1a"><mi id="S4.I6.i1.p1.3.m3.1.1" xref="S4.I6.i1.p1.3.m3.1.1.cmml">x</mi><annotation-xml encoding="MathML-Content" id="S4.I6.i1.p1.3.m3.1b"><ci id="S4.I6.i1.p1.3.m3.1.1.cmml" xref="S4.I6.i1.p1.3.m3.1.1">𝑥</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.I6.i1.p1.3.m3.1c">x</annotation><annotation encoding="application/x-llamapun" id="S4.I6.i1.p1.3.m3.1d">italic_x</annotation></semantics></math>. The connected components of <math alttext="G\setminus\{a,b\}" class="ltx_Math" display="inline" id="S4.I6.i1.p1.4.m4.2"><semantics id="S4.I6.i1.p1.4.m4.2a"><mrow id="S4.I6.i1.p1.4.m4.2.3" xref="S4.I6.i1.p1.4.m4.2.3.cmml"><mi id="S4.I6.i1.p1.4.m4.2.3.2" xref="S4.I6.i1.p1.4.m4.2.3.2.cmml">G</mi><mo id="S4.I6.i1.p1.4.m4.2.3.1" xref="S4.I6.i1.p1.4.m4.2.3.1.cmml">∖</mo><mrow id="S4.I6.i1.p1.4.m4.2.3.3.2" xref="S4.I6.i1.p1.4.m4.2.3.3.1.cmml"><mo id="S4.I6.i1.p1.4.m4.2.3.3.2.1" stretchy="false" xref="S4.I6.i1.p1.4.m4.2.3.3.1.cmml">{</mo><mi id="S4.I6.i1.p1.4.m4.1.1" xref="S4.I6.i1.p1.4.m4.1.1.cmml">a</mi><mo id="S4.I6.i1.p1.4.m4.2.3.3.2.2" xref="S4.I6.i1.p1.4.m4.2.3.3.1.cmml">,</mo><mi id="S4.I6.i1.p1.4.m4.2.2" xref="S4.I6.i1.p1.4.m4.2.2.cmml">b</mi><mo id="S4.I6.i1.p1.4.m4.2.3.3.2.3" stretchy="false" xref="S4.I6.i1.p1.4.m4.2.3.3.1.cmml">}</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.I6.i1.p1.4.m4.2b"><apply id="S4.I6.i1.p1.4.m4.2.3.cmml" xref="S4.I6.i1.p1.4.m4.2.3"><setdiff id="S4.I6.i1.p1.4.m4.2.3.1.cmml" xref="S4.I6.i1.p1.4.m4.2.3.1"></setdiff><ci id="S4.I6.i1.p1.4.m4.2.3.2.cmml" xref="S4.I6.i1.p1.4.m4.2.3.2">𝐺</ci><set id="S4.I6.i1.p1.4.m4.2.3.3.1.cmml" xref="S4.I6.i1.p1.4.m4.2.3.3.2"><ci id="S4.I6.i1.p1.4.m4.1.1.cmml" xref="S4.I6.i1.p1.4.m4.1.1">𝑎</ci><ci id="S4.I6.i1.p1.4.m4.2.2.cmml" xref="S4.I6.i1.p1.4.m4.2.2">𝑏</ci></set></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.I6.i1.p1.4.m4.2c">G\setminus\{a,b\}</annotation><annotation encoding="application/x-llamapun" id="S4.I6.i1.p1.4.m4.2d">italic_G ∖ { italic_a , italic_b }</annotation></semantics></math> correspond to the subtrees of <math alttext="T\setminus\{x\}" class="ltx_Math" display="inline" id="S4.I6.i1.p1.5.m5.1"><semantics id="S4.I6.i1.p1.5.m5.1a"><mrow id="S4.I6.i1.p1.5.m5.1.2" xref="S4.I6.i1.p1.5.m5.1.2.cmml"><mi id="S4.I6.i1.p1.5.m5.1.2.2" xref="S4.I6.i1.p1.5.m5.1.2.2.cmml">T</mi><mo id="S4.I6.i1.p1.5.m5.1.2.1" xref="S4.I6.i1.p1.5.m5.1.2.1.cmml">∖</mo><mrow id="S4.I6.i1.p1.5.m5.1.2.3.2" xref="S4.I6.i1.p1.5.m5.1.2.3.1.cmml"><mo id="S4.I6.i1.p1.5.m5.1.2.3.2.1" stretchy="false" xref="S4.I6.i1.p1.5.m5.1.2.3.1.cmml">{</mo><mi id="S4.I6.i1.p1.5.m5.1.1" xref="S4.I6.i1.p1.5.m5.1.1.cmml">x</mi><mo id="S4.I6.i1.p1.5.m5.1.2.3.2.2" stretchy="false" xref="S4.I6.i1.p1.5.m5.1.2.3.1.cmml">}</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.I6.i1.p1.5.m5.1b"><apply id="S4.I6.i1.p1.5.m5.1.2.cmml" xref="S4.I6.i1.p1.5.m5.1.2"><setdiff id="S4.I6.i1.p1.5.m5.1.2.1.cmml" xref="S4.I6.i1.p1.5.m5.1.2.1"></setdiff><ci id="S4.I6.i1.p1.5.m5.1.2.2.cmml" xref="S4.I6.i1.p1.5.m5.1.2.2">𝑇</ci><set id="S4.I6.i1.p1.5.m5.1.2.3.1.cmml" xref="S4.I6.i1.p1.5.m5.1.2.3.2"><ci id="S4.I6.i1.p1.5.m5.1.1.cmml" xref="S4.I6.i1.p1.5.m5.1.1">𝑥</ci></set></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.I6.i1.p1.5.m5.1c">T\setminus\{x\}</annotation><annotation encoding="application/x-llamapun" id="S4.I6.i1.p1.5.m5.1d">italic_T ∖ { italic_x }</annotation></semantics></math>.</p> </div> </li> <li class="ltx_item" id="S4.I6.i2" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">•</span> <div class="ltx_para" id="S4.I6.i2.p1"> <p class="ltx_p" id="S4.I6.i2.p1.6">There exists a virtual edge <math alttext="e=ab" class="ltx_Math" display="inline" id="S4.I6.i2.p1.1.m1.1"><semantics id="S4.I6.i2.p1.1.m1.1a"><mrow id="S4.I6.i2.p1.1.m1.1.1" xref="S4.I6.i2.p1.1.m1.1.1.cmml"><mi id="S4.I6.i2.p1.1.m1.1.1.2" xref="S4.I6.i2.p1.1.m1.1.1.2.cmml">e</mi><mo id="S4.I6.i2.p1.1.m1.1.1.1" xref="S4.I6.i2.p1.1.m1.1.1.1.cmml">=</mo><mrow id="S4.I6.i2.p1.1.m1.1.1.3" xref="S4.I6.i2.p1.1.m1.1.1.3.cmml"><mi id="S4.I6.i2.p1.1.m1.1.1.3.2" xref="S4.I6.i2.p1.1.m1.1.1.3.2.cmml">a</mi><mo id="S4.I6.i2.p1.1.m1.1.1.3.1" xref="S4.I6.i2.p1.1.m1.1.1.3.1.cmml">⁢</mo><mi id="S4.I6.i2.p1.1.m1.1.1.3.3" xref="S4.I6.i2.p1.1.m1.1.1.3.3.cmml">b</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.I6.i2.p1.1.m1.1b"><apply id="S4.I6.i2.p1.1.m1.1.1.cmml" xref="S4.I6.i2.p1.1.m1.1.1"><eq id="S4.I6.i2.p1.1.m1.1.1.1.cmml" xref="S4.I6.i2.p1.1.m1.1.1.1"></eq><ci id="S4.I6.i2.p1.1.m1.1.1.2.cmml" xref="S4.I6.i2.p1.1.m1.1.1.2">𝑒</ci><apply id="S4.I6.i2.p1.1.m1.1.1.3.cmml" xref="S4.I6.i2.p1.1.m1.1.1.3"><times id="S4.I6.i2.p1.1.m1.1.1.3.1.cmml" xref="S4.I6.i2.p1.1.m1.1.1.3.1"></times><ci id="S4.I6.i2.p1.1.m1.1.1.3.2.cmml" xref="S4.I6.i2.p1.1.m1.1.1.3.2">𝑎</ci><ci id="S4.I6.i2.p1.1.m1.1.1.3.3.cmml" xref="S4.I6.i2.p1.1.m1.1.1.3.3">𝑏</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.I6.i2.p1.1.m1.1c">e=ab</annotation><annotation encoding="application/x-llamapun" id="S4.I6.i2.p1.1.m1.1d">italic_e = italic_a italic_b</annotation></semantics></math> associated with a tree edge <math alttext="xy" class="ltx_Math" display="inline" id="S4.I6.i2.p1.2.m2.1"><semantics id="S4.I6.i2.p1.2.m2.1a"><mrow id="S4.I6.i2.p1.2.m2.1.1" xref="S4.I6.i2.p1.2.m2.1.1.cmml"><mi id="S4.I6.i2.p1.2.m2.1.1.2" xref="S4.I6.i2.p1.2.m2.1.1.2.cmml">x</mi><mo id="S4.I6.i2.p1.2.m2.1.1.1" xref="S4.I6.i2.p1.2.m2.1.1.1.cmml">⁢</mo><mi id="S4.I6.i2.p1.2.m2.1.1.3" xref="S4.I6.i2.p1.2.m2.1.1.3.cmml">y</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.I6.i2.p1.2.m2.1b"><apply id="S4.I6.i2.p1.2.m2.1.1.cmml" xref="S4.I6.i2.p1.2.m2.1.1"><times id="S4.I6.i2.p1.2.m2.1.1.1.cmml" xref="S4.I6.i2.p1.2.m2.1.1.1"></times><ci id="S4.I6.i2.p1.2.m2.1.1.2.cmml" xref="S4.I6.i2.p1.2.m2.1.1.2">𝑥</ci><ci id="S4.I6.i2.p1.2.m2.1.1.3.cmml" xref="S4.I6.i2.p1.2.m2.1.1.3">𝑦</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.I6.i2.p1.2.m2.1c">xy</annotation><annotation encoding="application/x-llamapun" id="S4.I6.i2.p1.2.m2.1d">italic_x italic_y</annotation></semantics></math> such that at least one of <math alttext="x" class="ltx_Math" display="inline" id="S4.I6.i2.p1.3.m3.1"><semantics id="S4.I6.i2.p1.3.m3.1a"><mi id="S4.I6.i2.p1.3.m3.1.1" xref="S4.I6.i2.p1.3.m3.1.1.cmml">x</mi><annotation-xml encoding="MathML-Content" id="S4.I6.i2.p1.3.m3.1b"><ci id="S4.I6.i2.p1.3.m3.1.1.cmml" xref="S4.I6.i2.p1.3.m3.1.1">𝑥</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.I6.i2.p1.3.m3.1c">x</annotation><annotation encoding="application/x-llamapun" id="S4.I6.i2.p1.3.m3.1d">italic_x</annotation></semantics></math> and <math alttext="y" class="ltx_Math" display="inline" id="S4.I6.i2.p1.4.m4.1"><semantics id="S4.I6.i2.p1.4.m4.1a"><mi id="S4.I6.i2.p1.4.m4.1.1" xref="S4.I6.i2.p1.4.m4.1.1.cmml">y</mi><annotation-xml encoding="MathML-Content" id="S4.I6.i2.p1.4.m4.1b"><ci id="S4.I6.i2.p1.4.m4.1.1.cmml" xref="S4.I6.i2.p1.4.m4.1.1">𝑦</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.I6.i2.p1.4.m4.1c">y</annotation><annotation encoding="application/x-llamapun" id="S4.I6.i2.p1.4.m4.1d">italic_y</annotation></semantics></math> is an R-node; the other is either an R-node or an S-node. The connected components of <math alttext="G\setminus\{a,b\}" class="ltx_Math" display="inline" id="S4.I6.i2.p1.5.m5.2"><semantics id="S4.I6.i2.p1.5.m5.2a"><mrow id="S4.I6.i2.p1.5.m5.2.3" xref="S4.I6.i2.p1.5.m5.2.3.cmml"><mi id="S4.I6.i2.p1.5.m5.2.3.2" xref="S4.I6.i2.p1.5.m5.2.3.2.cmml">G</mi><mo id="S4.I6.i2.p1.5.m5.2.3.1" xref="S4.I6.i2.p1.5.m5.2.3.1.cmml">∖</mo><mrow id="S4.I6.i2.p1.5.m5.2.3.3.2" xref="S4.I6.i2.p1.5.m5.2.3.3.1.cmml"><mo id="S4.I6.i2.p1.5.m5.2.3.3.2.1" stretchy="false" xref="S4.I6.i2.p1.5.m5.2.3.3.1.cmml">{</mo><mi id="S4.I6.i2.p1.5.m5.1.1" xref="S4.I6.i2.p1.5.m5.1.1.cmml">a</mi><mo id="S4.I6.i2.p1.5.m5.2.3.3.2.2" xref="S4.I6.i2.p1.5.m5.2.3.3.1.cmml">,</mo><mi id="S4.I6.i2.p1.5.m5.2.2" xref="S4.I6.i2.p1.5.m5.2.2.cmml">b</mi><mo id="S4.I6.i2.p1.5.m5.2.3.3.2.3" stretchy="false" xref="S4.I6.i2.p1.5.m5.2.3.3.1.cmml">}</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.I6.i2.p1.5.m5.2b"><apply id="S4.I6.i2.p1.5.m5.2.3.cmml" xref="S4.I6.i2.p1.5.m5.2.3"><setdiff id="S4.I6.i2.p1.5.m5.2.3.1.cmml" xref="S4.I6.i2.p1.5.m5.2.3.1"></setdiff><ci id="S4.I6.i2.p1.5.m5.2.3.2.cmml" xref="S4.I6.i2.p1.5.m5.2.3.2">𝐺</ci><set id="S4.I6.i2.p1.5.m5.2.3.3.1.cmml" xref="S4.I6.i2.p1.5.m5.2.3.3.2"><ci id="S4.I6.i2.p1.5.m5.1.1.cmml" xref="S4.I6.i2.p1.5.m5.1.1">𝑎</ci><ci id="S4.I6.i2.p1.5.m5.2.2.cmml" xref="S4.I6.i2.p1.5.m5.2.2">𝑏</ci></set></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.I6.i2.p1.5.m5.2c">G\setminus\{a,b\}</annotation><annotation encoding="application/x-llamapun" id="S4.I6.i2.p1.5.m5.2d">italic_G ∖ { italic_a , italic_b }</annotation></semantics></math> correspond to subtrees of <math alttext="(V(T),E(T)\setminus xy)" class="ltx_Math" display="inline" id="S4.I6.i2.p1.6.m6.4"><semantics id="S4.I6.i2.p1.6.m6.4a"><mrow id="S4.I6.i2.p1.6.m6.4.4.2" xref="S4.I6.i2.p1.6.m6.4.4.3.cmml"><mo id="S4.I6.i2.p1.6.m6.4.4.2.3" stretchy="false" xref="S4.I6.i2.p1.6.m6.4.4.3.cmml">(</mo><mrow id="S4.I6.i2.p1.6.m6.3.3.1.1" xref="S4.I6.i2.p1.6.m6.3.3.1.1.cmml"><mi id="S4.I6.i2.p1.6.m6.3.3.1.1.2" xref="S4.I6.i2.p1.6.m6.3.3.1.1.2.cmml">V</mi><mo id="S4.I6.i2.p1.6.m6.3.3.1.1.1" xref="S4.I6.i2.p1.6.m6.3.3.1.1.1.cmml">⁢</mo><mrow id="S4.I6.i2.p1.6.m6.3.3.1.1.3.2" xref="S4.I6.i2.p1.6.m6.3.3.1.1.cmml"><mo id="S4.I6.i2.p1.6.m6.3.3.1.1.3.2.1" stretchy="false" xref="S4.I6.i2.p1.6.m6.3.3.1.1.cmml">(</mo><mi id="S4.I6.i2.p1.6.m6.1.1" xref="S4.I6.i2.p1.6.m6.1.1.cmml">T</mi><mo id="S4.I6.i2.p1.6.m6.3.3.1.1.3.2.2" stretchy="false" xref="S4.I6.i2.p1.6.m6.3.3.1.1.cmml">)</mo></mrow></mrow><mo id="S4.I6.i2.p1.6.m6.4.4.2.4" xref="S4.I6.i2.p1.6.m6.4.4.3.cmml">,</mo><mrow id="S4.I6.i2.p1.6.m6.4.4.2.2" xref="S4.I6.i2.p1.6.m6.4.4.2.2.cmml"><mrow id="S4.I6.i2.p1.6.m6.4.4.2.2.2" xref="S4.I6.i2.p1.6.m6.4.4.2.2.2.cmml"><mi id="S4.I6.i2.p1.6.m6.4.4.2.2.2.2" xref="S4.I6.i2.p1.6.m6.4.4.2.2.2.2.cmml">E</mi><mo id="S4.I6.i2.p1.6.m6.4.4.2.2.2.1" xref="S4.I6.i2.p1.6.m6.4.4.2.2.2.1.cmml">⁢</mo><mrow id="S4.I6.i2.p1.6.m6.4.4.2.2.2.3.2" xref="S4.I6.i2.p1.6.m6.4.4.2.2.2.cmml"><mo id="S4.I6.i2.p1.6.m6.4.4.2.2.2.3.2.1" stretchy="false" xref="S4.I6.i2.p1.6.m6.4.4.2.2.2.cmml">(</mo><mi id="S4.I6.i2.p1.6.m6.2.2" xref="S4.I6.i2.p1.6.m6.2.2.cmml">T</mi><mo id="S4.I6.i2.p1.6.m6.4.4.2.2.2.3.2.2" stretchy="false" xref="S4.I6.i2.p1.6.m6.4.4.2.2.2.cmml">)</mo></mrow></mrow><mo id="S4.I6.i2.p1.6.m6.4.4.2.2.1" xref="S4.I6.i2.p1.6.m6.4.4.2.2.1.cmml">∖</mo><mrow id="S4.I6.i2.p1.6.m6.4.4.2.2.3" xref="S4.I6.i2.p1.6.m6.4.4.2.2.3.cmml"><mi id="S4.I6.i2.p1.6.m6.4.4.2.2.3.2" xref="S4.I6.i2.p1.6.m6.4.4.2.2.3.2.cmml">x</mi><mo id="S4.I6.i2.p1.6.m6.4.4.2.2.3.1" xref="S4.I6.i2.p1.6.m6.4.4.2.2.3.1.cmml">⁢</mo><mi id="S4.I6.i2.p1.6.m6.4.4.2.2.3.3" xref="S4.I6.i2.p1.6.m6.4.4.2.2.3.3.cmml">y</mi></mrow></mrow><mo id="S4.I6.i2.p1.6.m6.4.4.2.5" stretchy="false" xref="S4.I6.i2.p1.6.m6.4.4.3.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S4.I6.i2.p1.6.m6.4b"><interval closure="open" id="S4.I6.i2.p1.6.m6.4.4.3.cmml" xref="S4.I6.i2.p1.6.m6.4.4.2"><apply id="S4.I6.i2.p1.6.m6.3.3.1.1.cmml" xref="S4.I6.i2.p1.6.m6.3.3.1.1"><times id="S4.I6.i2.p1.6.m6.3.3.1.1.1.cmml" xref="S4.I6.i2.p1.6.m6.3.3.1.1.1"></times><ci id="S4.I6.i2.p1.6.m6.3.3.1.1.2.cmml" xref="S4.I6.i2.p1.6.m6.3.3.1.1.2">𝑉</ci><ci id="S4.I6.i2.p1.6.m6.1.1.cmml" xref="S4.I6.i2.p1.6.m6.1.1">𝑇</ci></apply><apply id="S4.I6.i2.p1.6.m6.4.4.2.2.cmml" xref="S4.I6.i2.p1.6.m6.4.4.2.2"><setdiff id="S4.I6.i2.p1.6.m6.4.4.2.2.1.cmml" xref="S4.I6.i2.p1.6.m6.4.4.2.2.1"></setdiff><apply id="S4.I6.i2.p1.6.m6.4.4.2.2.2.cmml" xref="S4.I6.i2.p1.6.m6.4.4.2.2.2"><times id="S4.I6.i2.p1.6.m6.4.4.2.2.2.1.cmml" xref="S4.I6.i2.p1.6.m6.4.4.2.2.2.1"></times><ci id="S4.I6.i2.p1.6.m6.4.4.2.2.2.2.cmml" xref="S4.I6.i2.p1.6.m6.4.4.2.2.2.2">𝐸</ci><ci id="S4.I6.i2.p1.6.m6.2.2.cmml" xref="S4.I6.i2.p1.6.m6.2.2">𝑇</ci></apply><apply id="S4.I6.i2.p1.6.m6.4.4.2.2.3.cmml" xref="S4.I6.i2.p1.6.m6.4.4.2.2.3"><times id="S4.I6.i2.p1.6.m6.4.4.2.2.3.1.cmml" xref="S4.I6.i2.p1.6.m6.4.4.2.2.3.1"></times><ci id="S4.I6.i2.p1.6.m6.4.4.2.2.3.2.cmml" xref="S4.I6.i2.p1.6.m6.4.4.2.2.3.2">𝑥</ci><ci id="S4.I6.i2.p1.6.m6.4.4.2.2.3.3.cmml" xref="S4.I6.i2.p1.6.m6.4.4.2.2.3.3">𝑦</ci></apply></apply></interval></annotation-xml><annotation encoding="application/x-tex" id="S4.I6.i2.p1.6.m6.4c">(V(T),E(T)\setminus xy)</annotation><annotation encoding="application/x-llamapun" id="S4.I6.i2.p1.6.m6.4d">( italic_V ( italic_T ) , italic_E ( italic_T ) ∖ italic_x italic_y )</annotation></semantics></math>.</p> </div> </li> <li class="ltx_item" id="S4.I6.i3" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">•</span> <div class="ltx_para" id="S4.I6.i3.p1"> <p class="ltx_p" id="S4.I6.i3.p1.6"><math alttext="a" class="ltx_Math" display="inline" id="S4.I6.i3.p1.1.m1.1"><semantics id="S4.I6.i3.p1.1.m1.1a"><mi id="S4.I6.i3.p1.1.m1.1.1" xref="S4.I6.i3.p1.1.m1.1.1.cmml">a</mi><annotation-xml encoding="MathML-Content" id="S4.I6.i3.p1.1.m1.1b"><ci id="S4.I6.i3.p1.1.m1.1.1.cmml" xref="S4.I6.i3.p1.1.m1.1.1">𝑎</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.I6.i3.p1.1.m1.1c">a</annotation><annotation encoding="application/x-llamapun" id="S4.I6.i3.p1.1.m1.1d">italic_a</annotation></semantics></math> and <math alttext="b" class="ltx_Math" display="inline" id="S4.I6.i3.p1.2.m2.1"><semantics id="S4.I6.i3.p1.2.m2.1a"><mi id="S4.I6.i3.p1.2.m2.1.1" xref="S4.I6.i3.p1.2.m2.1.1.cmml">b</mi><annotation-xml encoding="MathML-Content" id="S4.I6.i3.p1.2.m2.1b"><ci id="S4.I6.i3.p1.2.m2.1.1.cmml" xref="S4.I6.i3.p1.2.m2.1.1">𝑏</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.I6.i3.p1.2.m2.1c">b</annotation><annotation encoding="application/x-llamapun" id="S4.I6.i3.p1.2.m2.1d">italic_b</annotation></semantics></math> are non-adjacent nodes of a cycle <math alttext="G_{x}" class="ltx_Math" display="inline" id="S4.I6.i3.p1.3.m3.1"><semantics id="S4.I6.i3.p1.3.m3.1a"><msub id="S4.I6.i3.p1.3.m3.1.1" xref="S4.I6.i3.p1.3.m3.1.1.cmml"><mi id="S4.I6.i3.p1.3.m3.1.1.2" xref="S4.I6.i3.p1.3.m3.1.1.2.cmml">G</mi><mi id="S4.I6.i3.p1.3.m3.1.1.3" xref="S4.I6.i3.p1.3.m3.1.1.3.cmml">x</mi></msub><annotation-xml encoding="MathML-Content" id="S4.I6.i3.p1.3.m3.1b"><apply id="S4.I6.i3.p1.3.m3.1.1.cmml" xref="S4.I6.i3.p1.3.m3.1.1"><csymbol cd="ambiguous" id="S4.I6.i3.p1.3.m3.1.1.1.cmml" xref="S4.I6.i3.p1.3.m3.1.1">subscript</csymbol><ci id="S4.I6.i3.p1.3.m3.1.1.2.cmml" xref="S4.I6.i3.p1.3.m3.1.1.2">𝐺</ci><ci id="S4.I6.i3.p1.3.m3.1.1.3.cmml" xref="S4.I6.i3.p1.3.m3.1.1.3">𝑥</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.I6.i3.p1.3.m3.1c">G_{x}</annotation><annotation encoding="application/x-llamapun" id="S4.I6.i3.p1.3.m3.1d">italic_G start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT</annotation></semantics></math> for an S-node <math alttext="x" class="ltx_Math" display="inline" id="S4.I6.i3.p1.4.m4.1"><semantics id="S4.I6.i3.p1.4.m4.1a"><mi id="S4.I6.i3.p1.4.m4.1.1" xref="S4.I6.i3.p1.4.m4.1.1.cmml">x</mi><annotation-xml encoding="MathML-Content" id="S4.I6.i3.p1.4.m4.1b"><ci id="S4.I6.i3.p1.4.m4.1.1.cmml" xref="S4.I6.i3.p1.4.m4.1.1">𝑥</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.I6.i3.p1.4.m4.1c">x</annotation><annotation encoding="application/x-llamapun" id="S4.I6.i3.p1.4.m4.1d">italic_x</annotation></semantics></math>. The connected components of <math alttext="G\setminus\{a,b\}" class="ltx_Math" display="inline" id="S4.I6.i3.p1.5.m5.2"><semantics id="S4.I6.i3.p1.5.m5.2a"><mrow id="S4.I6.i3.p1.5.m5.2.3" xref="S4.I6.i3.p1.5.m5.2.3.cmml"><mi id="S4.I6.i3.p1.5.m5.2.3.2" xref="S4.I6.i3.p1.5.m5.2.3.2.cmml">G</mi><mo id="S4.I6.i3.p1.5.m5.2.3.1" xref="S4.I6.i3.p1.5.m5.2.3.1.cmml">∖</mo><mrow id="S4.I6.i3.p1.5.m5.2.3.3.2" xref="S4.I6.i3.p1.5.m5.2.3.3.1.cmml"><mo id="S4.I6.i3.p1.5.m5.2.3.3.2.1" stretchy="false" xref="S4.I6.i3.p1.5.m5.2.3.3.1.cmml">{</mo><mi id="S4.I6.i3.p1.5.m5.1.1" xref="S4.I6.i3.p1.5.m5.1.1.cmml">a</mi><mo id="S4.I6.i3.p1.5.m5.2.3.3.2.2" xref="S4.I6.i3.p1.5.m5.2.3.3.1.cmml">,</mo><mi id="S4.I6.i3.p1.5.m5.2.2" xref="S4.I6.i3.p1.5.m5.2.2.cmml">b</mi><mo id="S4.I6.i3.p1.5.m5.2.3.3.2.3" stretchy="false" xref="S4.I6.i3.p1.5.m5.2.3.3.1.cmml">}</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.I6.i3.p1.5.m5.2b"><apply id="S4.I6.i3.p1.5.m5.2.3.cmml" xref="S4.I6.i3.p1.5.m5.2.3"><setdiff id="S4.I6.i3.p1.5.m5.2.3.1.cmml" xref="S4.I6.i3.p1.5.m5.2.3.1"></setdiff><ci id="S4.I6.i3.p1.5.m5.2.3.2.cmml" xref="S4.I6.i3.p1.5.m5.2.3.2">𝐺</ci><set id="S4.I6.i3.p1.5.m5.2.3.3.1.cmml" xref="S4.I6.i3.p1.5.m5.2.3.3.2"><ci id="S4.I6.i3.p1.5.m5.1.1.cmml" xref="S4.I6.i3.p1.5.m5.1.1">𝑎</ci><ci id="S4.I6.i3.p1.5.m5.2.2.cmml" xref="S4.I6.i3.p1.5.m5.2.2">𝑏</ci></set></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.I6.i3.p1.5.m5.2c">G\setminus\{a,b\}</annotation><annotation encoding="application/x-llamapun" id="S4.I6.i3.p1.5.m5.2d">italic_G ∖ { italic_a , italic_b }</annotation></semantics></math> are the subtrees corresponding to the two components of <math alttext="G_{x}\setminus\{a,b\}" class="ltx_Math" display="inline" id="S4.I6.i3.p1.6.m6.2"><semantics id="S4.I6.i3.p1.6.m6.2a"><mrow id="S4.I6.i3.p1.6.m6.2.3" xref="S4.I6.i3.p1.6.m6.2.3.cmml"><msub id="S4.I6.i3.p1.6.m6.2.3.2" xref="S4.I6.i3.p1.6.m6.2.3.2.cmml"><mi id="S4.I6.i3.p1.6.m6.2.3.2.2" xref="S4.I6.i3.p1.6.m6.2.3.2.2.cmml">G</mi><mi id="S4.I6.i3.p1.6.m6.2.3.2.3" xref="S4.I6.i3.p1.6.m6.2.3.2.3.cmml">x</mi></msub><mo id="S4.I6.i3.p1.6.m6.2.3.1" xref="S4.I6.i3.p1.6.m6.2.3.1.cmml">∖</mo><mrow id="S4.I6.i3.p1.6.m6.2.3.3.2" xref="S4.I6.i3.p1.6.m6.2.3.3.1.cmml"><mo id="S4.I6.i3.p1.6.m6.2.3.3.2.1" stretchy="false" xref="S4.I6.i3.p1.6.m6.2.3.3.1.cmml">{</mo><mi id="S4.I6.i3.p1.6.m6.1.1" xref="S4.I6.i3.p1.6.m6.1.1.cmml">a</mi><mo id="S4.I6.i3.p1.6.m6.2.3.3.2.2" xref="S4.I6.i3.p1.6.m6.2.3.3.1.cmml">,</mo><mi id="S4.I6.i3.p1.6.m6.2.2" xref="S4.I6.i3.p1.6.m6.2.2.cmml">b</mi><mo id="S4.I6.i3.p1.6.m6.2.3.3.2.3" stretchy="false" xref="S4.I6.i3.p1.6.m6.2.3.3.1.cmml">}</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.I6.i3.p1.6.m6.2b"><apply id="S4.I6.i3.p1.6.m6.2.3.cmml" xref="S4.I6.i3.p1.6.m6.2.3"><setdiff id="S4.I6.i3.p1.6.m6.2.3.1.cmml" xref="S4.I6.i3.p1.6.m6.2.3.1"></setdiff><apply id="S4.I6.i3.p1.6.m6.2.3.2.cmml" xref="S4.I6.i3.p1.6.m6.2.3.2"><csymbol cd="ambiguous" id="S4.I6.i3.p1.6.m6.2.3.2.1.cmml" xref="S4.I6.i3.p1.6.m6.2.3.2">subscript</csymbol><ci id="S4.I6.i3.p1.6.m6.2.3.2.2.cmml" xref="S4.I6.i3.p1.6.m6.2.3.2.2">𝐺</ci><ci id="S4.I6.i3.p1.6.m6.2.3.2.3.cmml" xref="S4.I6.i3.p1.6.m6.2.3.2.3">𝑥</ci></apply><set id="S4.I6.i3.p1.6.m6.2.3.3.1.cmml" xref="S4.I6.i3.p1.6.m6.2.3.3.2"><ci id="S4.I6.i3.p1.6.m6.1.1.cmml" xref="S4.I6.i3.p1.6.m6.1.1">𝑎</ci><ci id="S4.I6.i3.p1.6.m6.2.2.cmml" xref="S4.I6.i3.p1.6.m6.2.2">𝑏</ci></set></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.I6.i3.p1.6.m6.2c">G_{x}\setminus\{a,b\}</annotation><annotation encoding="application/x-llamapun" id="S4.I6.i3.p1.6.m6.2d">italic_G start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT ∖ { italic_a , italic_b }</annotation></semantics></math>.</p> </div> </li> </ul> </div> </div> </section> <section class="ltx_subsubsection" id="S4.SS2.SSS2"> <h4 class="ltx_title ltx_title_subsubsection"> <span class="ltx_tag ltx_tag_subsubsection">4.2.2 </span>The Streaming Algorithm</h4> <div class="ltx_para" id="S4.SS2.SSS2.p1"> <p class="ltx_p" id="S4.SS2.SSS2.p1.29">We fix a 2-connected graph <math alttext="G=(V,E)" class="ltx_Math" display="inline" id="S4.SS2.SSS2.p1.1.m1.2"><semantics id="S4.SS2.SSS2.p1.1.m1.2a"><mrow id="S4.SS2.SSS2.p1.1.m1.2.3" xref="S4.SS2.SSS2.p1.1.m1.2.3.cmml"><mi id="S4.SS2.SSS2.p1.1.m1.2.3.2" xref="S4.SS2.SSS2.p1.1.m1.2.3.2.cmml">G</mi><mo id="S4.SS2.SSS2.p1.1.m1.2.3.1" xref="S4.SS2.SSS2.p1.1.m1.2.3.1.cmml">=</mo><mrow id="S4.SS2.SSS2.p1.1.m1.2.3.3.2" xref="S4.SS2.SSS2.p1.1.m1.2.3.3.1.cmml"><mo id="S4.SS2.SSS2.p1.1.m1.2.3.3.2.1" stretchy="false" xref="S4.SS2.SSS2.p1.1.m1.2.3.3.1.cmml">(</mo><mi id="S4.SS2.SSS2.p1.1.m1.1.1" xref="S4.SS2.SSS2.p1.1.m1.1.1.cmml">V</mi><mo id="S4.SS2.SSS2.p1.1.m1.2.3.3.2.2" xref="S4.SS2.SSS2.p1.1.m1.2.3.3.1.cmml">,</mo><mi id="S4.SS2.SSS2.p1.1.m1.2.2" xref="S4.SS2.SSS2.p1.1.m1.2.2.cmml">E</mi><mo id="S4.SS2.SSS2.p1.1.m1.2.3.3.2.3" stretchy="false" xref="S4.SS2.SSS2.p1.1.m1.2.3.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS2.p1.1.m1.2b"><apply id="S4.SS2.SSS2.p1.1.m1.2.3.cmml" xref="S4.SS2.SSS2.p1.1.m1.2.3"><eq id="S4.SS2.SSS2.p1.1.m1.2.3.1.cmml" xref="S4.SS2.SSS2.p1.1.m1.2.3.1"></eq><ci id="S4.SS2.SSS2.p1.1.m1.2.3.2.cmml" xref="S4.SS2.SSS2.p1.1.m1.2.3.2">𝐺</ci><interval closure="open" id="S4.SS2.SSS2.p1.1.m1.2.3.3.1.cmml" xref="S4.SS2.SSS2.p1.1.m1.2.3.3.2"><ci id="S4.SS2.SSS2.p1.1.m1.1.1.cmml" xref="S4.SS2.SSS2.p1.1.m1.1.1">𝑉</ci><ci id="S4.SS2.SSS2.p1.1.m1.2.2.cmml" xref="S4.SS2.SSS2.p1.1.m1.2.2">𝐸</ci></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS2.p1.1.m1.2c">G=(V,E)</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS2.p1.1.m1.2d">italic_G = ( italic_V , italic_E )</annotation></semantics></math>. We can assume without loss of generality that <math alttext="G" class="ltx_Math" display="inline" id="S4.SS2.SSS2.p1.2.m2.1"><semantics id="S4.SS2.SSS2.p1.2.m2.1a"><mi id="S4.SS2.SSS2.p1.2.m2.1.1" xref="S4.SS2.SSS2.p1.2.m2.1.1.cmml">G</mi><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS2.p1.2.m2.1b"><ci id="S4.SS2.SSS2.p1.2.m2.1.1.cmml" xref="S4.SS2.SSS2.p1.2.m2.1.1">𝐺</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS2.p1.2.m2.1c">G</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS2.p1.2.m2.1d">italic_G</annotation></semantics></math> is an edge-minimal 2-vertex-connected graph, using the same reasoning as in the 1-to-2 augmentation setting. We start by constructing the SPQR tree <math alttext="T" class="ltx_Math" display="inline" id="S4.SS2.SSS2.p1.3.m3.1"><semantics id="S4.SS2.SSS2.p1.3.m3.1a"><mi id="S4.SS2.SSS2.p1.3.m3.1.1" xref="S4.SS2.SSS2.p1.3.m3.1.1.cmml">T</mi><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS2.p1.3.m3.1b"><ci id="S4.SS2.SSS2.p1.3.m3.1.1.cmml" xref="S4.SS2.SSS2.p1.3.m3.1.1">𝑇</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS2.p1.3.m3.1c">T</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS2.p1.3.m3.1d">italic_T</annotation></semantics></math> of <math alttext="G" class="ltx_Math" display="inline" id="S4.SS2.SSS2.p1.4.m4.1"><semantics id="S4.SS2.SSS2.p1.4.m4.1a"><mi id="S4.SS2.SSS2.p1.4.m4.1.1" xref="S4.SS2.SSS2.p1.4.m4.1.1.cmml">G</mi><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS2.p1.4.m4.1b"><ci id="S4.SS2.SSS2.p1.4.m4.1.1.cmml" xref="S4.SS2.SSS2.p1.4.m4.1.1">𝐺</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS2.p1.4.m4.1c">G</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS2.p1.4.m4.1d">italic_G</annotation></semantics></math>; this can be done efficiently by Lemma <a class="ltx_ref" href="https://arxiv.org/html/2503.00712v1#S4.Thmtheorem14" title="Lemma 4.14. ‣ 4.2.1 SPQR Trees ‣ 4.2 Two-to-Three Augmentation ‣ 4 Vertex Connectivity Augmentation in Link-Arrival Model ‣ Streaming Algorithms for Network Design"><span class="ltx_text ltx_ref_tag">4.14</span></a>. To avoid confusion, we refer to elements of <math alttext="V(T)" class="ltx_Math" display="inline" id="S4.SS2.SSS2.p1.5.m5.1"><semantics id="S4.SS2.SSS2.p1.5.m5.1a"><mrow id="S4.SS2.SSS2.p1.5.m5.1.2" xref="S4.SS2.SSS2.p1.5.m5.1.2.cmml"><mi id="S4.SS2.SSS2.p1.5.m5.1.2.2" xref="S4.SS2.SSS2.p1.5.m5.1.2.2.cmml">V</mi><mo id="S4.SS2.SSS2.p1.5.m5.1.2.1" xref="S4.SS2.SSS2.p1.5.m5.1.2.1.cmml">⁢</mo><mrow id="S4.SS2.SSS2.p1.5.m5.1.2.3.2" xref="S4.SS2.SSS2.p1.5.m5.1.2.cmml"><mo id="S4.SS2.SSS2.p1.5.m5.1.2.3.2.1" stretchy="false" xref="S4.SS2.SSS2.p1.5.m5.1.2.cmml">(</mo><mi id="S4.SS2.SSS2.p1.5.m5.1.1" xref="S4.SS2.SSS2.p1.5.m5.1.1.cmml">T</mi><mo id="S4.SS2.SSS2.p1.5.m5.1.2.3.2.2" stretchy="false" xref="S4.SS2.SSS2.p1.5.m5.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS2.p1.5.m5.1b"><apply id="S4.SS2.SSS2.p1.5.m5.1.2.cmml" xref="S4.SS2.SSS2.p1.5.m5.1.2"><times id="S4.SS2.SSS2.p1.5.m5.1.2.1.cmml" xref="S4.SS2.SSS2.p1.5.m5.1.2.1"></times><ci id="S4.SS2.SSS2.p1.5.m5.1.2.2.cmml" xref="S4.SS2.SSS2.p1.5.m5.1.2.2">𝑉</ci><ci id="S4.SS2.SSS2.p1.5.m5.1.1.cmml" xref="S4.SS2.SSS2.p1.5.m5.1.1">𝑇</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS2.p1.5.m5.1c">V(T)</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS2.p1.5.m5.1d">italic_V ( italic_T )</annotation></semantics></math> as “nodes” and elements of <math alttext="V(G)" class="ltx_Math" display="inline" id="S4.SS2.SSS2.p1.6.m6.1"><semantics id="S4.SS2.SSS2.p1.6.m6.1a"><mrow id="S4.SS2.SSS2.p1.6.m6.1.2" xref="S4.SS2.SSS2.p1.6.m6.1.2.cmml"><mi id="S4.SS2.SSS2.p1.6.m6.1.2.2" xref="S4.SS2.SSS2.p1.6.m6.1.2.2.cmml">V</mi><mo id="S4.SS2.SSS2.p1.6.m6.1.2.1" xref="S4.SS2.SSS2.p1.6.m6.1.2.1.cmml">⁢</mo><mrow id="S4.SS2.SSS2.p1.6.m6.1.2.3.2" xref="S4.SS2.SSS2.p1.6.m6.1.2.cmml"><mo id="S4.SS2.SSS2.p1.6.m6.1.2.3.2.1" stretchy="false" xref="S4.SS2.SSS2.p1.6.m6.1.2.cmml">(</mo><mi id="S4.SS2.SSS2.p1.6.m6.1.1" xref="S4.SS2.SSS2.p1.6.m6.1.1.cmml">G</mi><mo id="S4.SS2.SSS2.p1.6.m6.1.2.3.2.2" stretchy="false" xref="S4.SS2.SSS2.p1.6.m6.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS2.p1.6.m6.1b"><apply id="S4.SS2.SSS2.p1.6.m6.1.2.cmml" xref="S4.SS2.SSS2.p1.6.m6.1.2"><times id="S4.SS2.SSS2.p1.6.m6.1.2.1.cmml" xref="S4.SS2.SSS2.p1.6.m6.1.2.1"></times><ci id="S4.SS2.SSS2.p1.6.m6.1.2.2.cmml" xref="S4.SS2.SSS2.p1.6.m6.1.2.2">𝑉</ci><ci id="S4.SS2.SSS2.p1.6.m6.1.1.cmml" xref="S4.SS2.SSS2.p1.6.m6.1.1">𝐺</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS2.p1.6.m6.1c">V(G)</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS2.p1.6.m6.1d">italic_V ( italic_G )</annotation></semantics></math> as “vertices”. We choose a root node <math alttext="r" class="ltx_Math" display="inline" id="S4.SS2.SSS2.p1.7.m7.1"><semantics id="S4.SS2.SSS2.p1.7.m7.1a"><mi id="S4.SS2.SSS2.p1.7.m7.1.1" xref="S4.SS2.SSS2.p1.7.m7.1.1.cmml">r</mi><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS2.p1.7.m7.1b"><ci id="S4.SS2.SSS2.p1.7.m7.1.1.cmml" xref="S4.SS2.SSS2.p1.7.m7.1.1">𝑟</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS2.p1.7.m7.1c">r</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS2.p1.7.m7.1d">italic_r</annotation></semantics></math> of <math alttext="T" class="ltx_Math" display="inline" id="S4.SS2.SSS2.p1.8.m8.1"><semantics id="S4.SS2.SSS2.p1.8.m8.1a"><mi id="S4.SS2.SSS2.p1.8.m8.1.1" xref="S4.SS2.SSS2.p1.8.m8.1.1.cmml">T</mi><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS2.p1.8.m8.1b"><ci id="S4.SS2.SSS2.p1.8.m8.1.1.cmml" xref="S4.SS2.SSS2.p1.8.m8.1.1">𝑇</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS2.p1.8.m8.1c">T</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS2.p1.8.m8.1d">italic_T</annotation></semantics></math>. For each tree node <math alttext="x\in V(T)" class="ltx_Math" display="inline" id="S4.SS2.SSS2.p1.9.m9.1"><semantics id="S4.SS2.SSS2.p1.9.m9.1a"><mrow id="S4.SS2.SSS2.p1.9.m9.1.2" xref="S4.SS2.SSS2.p1.9.m9.1.2.cmml"><mi id="S4.SS2.SSS2.p1.9.m9.1.2.2" xref="S4.SS2.SSS2.p1.9.m9.1.2.2.cmml">x</mi><mo id="S4.SS2.SSS2.p1.9.m9.1.2.1" xref="S4.SS2.SSS2.p1.9.m9.1.2.1.cmml">∈</mo><mrow id="S4.SS2.SSS2.p1.9.m9.1.2.3" xref="S4.SS2.SSS2.p1.9.m9.1.2.3.cmml"><mi id="S4.SS2.SSS2.p1.9.m9.1.2.3.2" xref="S4.SS2.SSS2.p1.9.m9.1.2.3.2.cmml">V</mi><mo id="S4.SS2.SSS2.p1.9.m9.1.2.3.1" xref="S4.SS2.SSS2.p1.9.m9.1.2.3.1.cmml">⁢</mo><mrow id="S4.SS2.SSS2.p1.9.m9.1.2.3.3.2" xref="S4.SS2.SSS2.p1.9.m9.1.2.3.cmml"><mo id="S4.SS2.SSS2.p1.9.m9.1.2.3.3.2.1" stretchy="false" xref="S4.SS2.SSS2.p1.9.m9.1.2.3.cmml">(</mo><mi id="S4.SS2.SSS2.p1.9.m9.1.1" xref="S4.SS2.SSS2.p1.9.m9.1.1.cmml">T</mi><mo id="S4.SS2.SSS2.p1.9.m9.1.2.3.3.2.2" stretchy="false" xref="S4.SS2.SSS2.p1.9.m9.1.2.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS2.p1.9.m9.1b"><apply id="S4.SS2.SSS2.p1.9.m9.1.2.cmml" xref="S4.SS2.SSS2.p1.9.m9.1.2"><in id="S4.SS2.SSS2.p1.9.m9.1.2.1.cmml" xref="S4.SS2.SSS2.p1.9.m9.1.2.1"></in><ci id="S4.SS2.SSS2.p1.9.m9.1.2.2.cmml" xref="S4.SS2.SSS2.p1.9.m9.1.2.2">𝑥</ci><apply id="S4.SS2.SSS2.p1.9.m9.1.2.3.cmml" xref="S4.SS2.SSS2.p1.9.m9.1.2.3"><times id="S4.SS2.SSS2.p1.9.m9.1.2.3.1.cmml" xref="S4.SS2.SSS2.p1.9.m9.1.2.3.1"></times><ci id="S4.SS2.SSS2.p1.9.m9.1.2.3.2.cmml" xref="S4.SS2.SSS2.p1.9.m9.1.2.3.2">𝑉</ci><ci id="S4.SS2.SSS2.p1.9.m9.1.1.cmml" xref="S4.SS2.SSS2.p1.9.m9.1.1">𝑇</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS2.p1.9.m9.1c">x\in V(T)</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS2.p1.9.m9.1d">italic_x ∈ italic_V ( italic_T )</annotation></semantics></math>, we let <math alttext="G_{x}" class="ltx_Math" display="inline" id="S4.SS2.SSS2.p1.10.m10.1"><semantics id="S4.SS2.SSS2.p1.10.m10.1a"><msub id="S4.SS2.SSS2.p1.10.m10.1.1" xref="S4.SS2.SSS2.p1.10.m10.1.1.cmml"><mi id="S4.SS2.SSS2.p1.10.m10.1.1.2" xref="S4.SS2.SSS2.p1.10.m10.1.1.2.cmml">G</mi><mi id="S4.SS2.SSS2.p1.10.m10.1.1.3" xref="S4.SS2.SSS2.p1.10.m10.1.1.3.cmml">x</mi></msub><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS2.p1.10.m10.1b"><apply id="S4.SS2.SSS2.p1.10.m10.1.1.cmml" xref="S4.SS2.SSS2.p1.10.m10.1.1"><csymbol cd="ambiguous" id="S4.SS2.SSS2.p1.10.m10.1.1.1.cmml" xref="S4.SS2.SSS2.p1.10.m10.1.1">subscript</csymbol><ci id="S4.SS2.SSS2.p1.10.m10.1.1.2.cmml" xref="S4.SS2.SSS2.p1.10.m10.1.1.2">𝐺</ci><ci id="S4.SS2.SSS2.p1.10.m10.1.1.3.cmml" xref="S4.SS2.SSS2.p1.10.m10.1.1.3">𝑥</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS2.p1.10.m10.1c">G_{x}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS2.p1.10.m10.1d">italic_G start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT</annotation></semantics></math> denote its corresponding graph, we let <math alttext="T_{x}" class="ltx_Math" display="inline" id="S4.SS2.SSS2.p1.11.m11.1"><semantics id="S4.SS2.SSS2.p1.11.m11.1a"><msub id="S4.SS2.SSS2.p1.11.m11.1.1" xref="S4.SS2.SSS2.p1.11.m11.1.1.cmml"><mi id="S4.SS2.SSS2.p1.11.m11.1.1.2" xref="S4.SS2.SSS2.p1.11.m11.1.1.2.cmml">T</mi><mi id="S4.SS2.SSS2.p1.11.m11.1.1.3" xref="S4.SS2.SSS2.p1.11.m11.1.1.3.cmml">x</mi></msub><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS2.p1.11.m11.1b"><apply id="S4.SS2.SSS2.p1.11.m11.1.1.cmml" xref="S4.SS2.SSS2.p1.11.m11.1.1"><csymbol cd="ambiguous" id="S4.SS2.SSS2.p1.11.m11.1.1.1.cmml" xref="S4.SS2.SSS2.p1.11.m11.1.1">subscript</csymbol><ci id="S4.SS2.SSS2.p1.11.m11.1.1.2.cmml" xref="S4.SS2.SSS2.p1.11.m11.1.1.2">𝑇</ci><ci id="S4.SS2.SSS2.p1.11.m11.1.1.3.cmml" xref="S4.SS2.SSS2.p1.11.m11.1.1.3">𝑥</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS2.p1.11.m11.1c">T_{x}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS2.p1.11.m11.1d">italic_T start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT</annotation></semantics></math> denote the subtree of <math alttext="T" class="ltx_Math" display="inline" id="S4.SS2.SSS2.p1.12.m12.1"><semantics id="S4.SS2.SSS2.p1.12.m12.1a"><mi id="S4.SS2.SSS2.p1.12.m12.1.1" xref="S4.SS2.SSS2.p1.12.m12.1.1.cmml">T</mi><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS2.p1.12.m12.1b"><ci id="S4.SS2.SSS2.p1.12.m12.1.1.cmml" xref="S4.SS2.SSS2.p1.12.m12.1.1">𝑇</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS2.p1.12.m12.1c">T</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS2.p1.12.m12.1d">italic_T</annotation></semantics></math> rooted at <math alttext="x" class="ltx_Math" display="inline" id="S4.SS2.SSS2.p1.13.m13.1"><semantics id="S4.SS2.SSS2.p1.13.m13.1a"><mi id="S4.SS2.SSS2.p1.13.m13.1.1" xref="S4.SS2.SSS2.p1.13.m13.1.1.cmml">x</mi><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS2.p1.13.m13.1b"><ci id="S4.SS2.SSS2.p1.13.m13.1.1.cmml" xref="S4.SS2.SSS2.p1.13.m13.1.1">𝑥</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS2.p1.13.m13.1c">x</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS2.p1.13.m13.1d">italic_x</annotation></semantics></math>, and let <math alttext="C(x)" class="ltx_Math" display="inline" id="S4.SS2.SSS2.p1.14.m14.1"><semantics id="S4.SS2.SSS2.p1.14.m14.1a"><mrow id="S4.SS2.SSS2.p1.14.m14.1.2" xref="S4.SS2.SSS2.p1.14.m14.1.2.cmml"><mi id="S4.SS2.SSS2.p1.14.m14.1.2.2" xref="S4.SS2.SSS2.p1.14.m14.1.2.2.cmml">C</mi><mo id="S4.SS2.SSS2.p1.14.m14.1.2.1" xref="S4.SS2.SSS2.p1.14.m14.1.2.1.cmml">⁢</mo><mrow id="S4.SS2.SSS2.p1.14.m14.1.2.3.2" xref="S4.SS2.SSS2.p1.14.m14.1.2.cmml"><mo id="S4.SS2.SSS2.p1.14.m14.1.2.3.2.1" stretchy="false" xref="S4.SS2.SSS2.p1.14.m14.1.2.cmml">(</mo><mi id="S4.SS2.SSS2.p1.14.m14.1.1" xref="S4.SS2.SSS2.p1.14.m14.1.1.cmml">x</mi><mo id="S4.SS2.SSS2.p1.14.m14.1.2.3.2.2" stretchy="false" xref="S4.SS2.SSS2.p1.14.m14.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS2.p1.14.m14.1b"><apply id="S4.SS2.SSS2.p1.14.m14.1.2.cmml" xref="S4.SS2.SSS2.p1.14.m14.1.2"><times id="S4.SS2.SSS2.p1.14.m14.1.2.1.cmml" xref="S4.SS2.SSS2.p1.14.m14.1.2.1"></times><ci id="S4.SS2.SSS2.p1.14.m14.1.2.2.cmml" xref="S4.SS2.SSS2.p1.14.m14.1.2.2">𝐶</ci><ci id="S4.SS2.SSS2.p1.14.m14.1.1.cmml" xref="S4.SS2.SSS2.p1.14.m14.1.1">𝑥</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS2.p1.14.m14.1c">C(x)</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS2.p1.14.m14.1d">italic_C ( italic_x )</annotation></semantics></math> denote the set of children of <math alttext="x" class="ltx_Math" display="inline" id="S4.SS2.SSS2.p1.15.m15.1"><semantics id="S4.SS2.SSS2.p1.15.m15.1a"><mi id="S4.SS2.SSS2.p1.15.m15.1.1" xref="S4.SS2.SSS2.p1.15.m15.1.1.cmml">x</mi><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS2.p1.15.m15.1b"><ci id="S4.SS2.SSS2.p1.15.m15.1.1.cmml" xref="S4.SS2.SSS2.p1.15.m15.1.1">𝑥</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS2.p1.15.m15.1c">x</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS2.p1.15.m15.1d">italic_x</annotation></semantics></math>. We define <math alttext="\textnormal{parent}(x)" class="ltx_Math" display="inline" id="S4.SS2.SSS2.p1.16.m16.1"><semantics id="S4.SS2.SSS2.p1.16.m16.1a"><mrow id="S4.SS2.SSS2.p1.16.m16.1.2" xref="S4.SS2.SSS2.p1.16.m16.1.2.cmml"><mtext id="S4.SS2.SSS2.p1.16.m16.1.2.2" xref="S4.SS2.SSS2.p1.16.m16.1.2.2a.cmml">parent</mtext><mo id="S4.SS2.SSS2.p1.16.m16.1.2.1" xref="S4.SS2.SSS2.p1.16.m16.1.2.1.cmml">⁢</mo><mrow id="S4.SS2.SSS2.p1.16.m16.1.2.3.2" xref="S4.SS2.SSS2.p1.16.m16.1.2.cmml"><mo id="S4.SS2.SSS2.p1.16.m16.1.2.3.2.1" stretchy="false" xref="S4.SS2.SSS2.p1.16.m16.1.2.cmml">(</mo><mi id="S4.SS2.SSS2.p1.16.m16.1.1" xref="S4.SS2.SSS2.p1.16.m16.1.1.cmml">x</mi><mo id="S4.SS2.SSS2.p1.16.m16.1.2.3.2.2" stretchy="false" xref="S4.SS2.SSS2.p1.16.m16.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS2.p1.16.m16.1b"><apply id="S4.SS2.SSS2.p1.16.m16.1.2.cmml" xref="S4.SS2.SSS2.p1.16.m16.1.2"><times id="S4.SS2.SSS2.p1.16.m16.1.2.1.cmml" xref="S4.SS2.SSS2.p1.16.m16.1.2.1"></times><ci id="S4.SS2.SSS2.p1.16.m16.1.2.2a.cmml" xref="S4.SS2.SSS2.p1.16.m16.1.2.2"><mtext id="S4.SS2.SSS2.p1.16.m16.1.2.2.cmml" xref="S4.SS2.SSS2.p1.16.m16.1.2.2">parent</mtext></ci><ci id="S4.SS2.SSS2.p1.16.m16.1.1.cmml" xref="S4.SS2.SSS2.p1.16.m16.1.1">𝑥</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS2.p1.16.m16.1c">\textnormal{parent}(x)</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS2.p1.16.m16.1d">parent ( italic_x )</annotation></semantics></math> as the virtual edge <math alttext="ab" class="ltx_Math" display="inline" id="S4.SS2.SSS2.p1.17.m17.1"><semantics id="S4.SS2.SSS2.p1.17.m17.1a"><mrow id="S4.SS2.SSS2.p1.17.m17.1.1" xref="S4.SS2.SSS2.p1.17.m17.1.1.cmml"><mi id="S4.SS2.SSS2.p1.17.m17.1.1.2" xref="S4.SS2.SSS2.p1.17.m17.1.1.2.cmml">a</mi><mo id="S4.SS2.SSS2.p1.17.m17.1.1.1" xref="S4.SS2.SSS2.p1.17.m17.1.1.1.cmml">⁢</mo><mi id="S4.SS2.SSS2.p1.17.m17.1.1.3" xref="S4.SS2.SSS2.p1.17.m17.1.1.3.cmml">b</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS2.p1.17.m17.1b"><apply id="S4.SS2.SSS2.p1.17.m17.1.1.cmml" xref="S4.SS2.SSS2.p1.17.m17.1.1"><times id="S4.SS2.SSS2.p1.17.m17.1.1.1.cmml" xref="S4.SS2.SSS2.p1.17.m17.1.1.1"></times><ci id="S4.SS2.SSS2.p1.17.m17.1.1.2.cmml" xref="S4.SS2.SSS2.p1.17.m17.1.1.2">𝑎</ci><ci id="S4.SS2.SSS2.p1.17.m17.1.1.3.cmml" xref="S4.SS2.SSS2.p1.17.m17.1.1.3">𝑏</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS2.p1.17.m17.1c">ab</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS2.p1.17.m17.1d">italic_a italic_b</annotation></semantics></math> of <math alttext="G_{x}" class="ltx_Math" display="inline" id="S4.SS2.SSS2.p1.18.m18.1"><semantics id="S4.SS2.SSS2.p1.18.m18.1a"><msub id="S4.SS2.SSS2.p1.18.m18.1.1" xref="S4.SS2.SSS2.p1.18.m18.1.1.cmml"><mi id="S4.SS2.SSS2.p1.18.m18.1.1.2" xref="S4.SS2.SSS2.p1.18.m18.1.1.2.cmml">G</mi><mi id="S4.SS2.SSS2.p1.18.m18.1.1.3" xref="S4.SS2.SSS2.p1.18.m18.1.1.3.cmml">x</mi></msub><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS2.p1.18.m18.1b"><apply id="S4.SS2.SSS2.p1.18.m18.1.1.cmml" xref="S4.SS2.SSS2.p1.18.m18.1.1"><csymbol cd="ambiguous" id="S4.SS2.SSS2.p1.18.m18.1.1.1.cmml" xref="S4.SS2.SSS2.p1.18.m18.1.1">subscript</csymbol><ci id="S4.SS2.SSS2.p1.18.m18.1.1.2.cmml" xref="S4.SS2.SSS2.p1.18.m18.1.1.2">𝐺</ci><ci id="S4.SS2.SSS2.p1.18.m18.1.1.3.cmml" xref="S4.SS2.SSS2.p1.18.m18.1.1.3">𝑥</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS2.p1.18.m18.1c">G_{x}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS2.p1.18.m18.1d">italic_G start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT</annotation></semantics></math> associated with the tree edge on the path from <math alttext="x" class="ltx_Math" display="inline" id="S4.SS2.SSS2.p1.19.m19.1"><semantics id="S4.SS2.SSS2.p1.19.m19.1a"><mi id="S4.SS2.SSS2.p1.19.m19.1.1" xref="S4.SS2.SSS2.p1.19.m19.1.1.cmml">x</mi><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS2.p1.19.m19.1b"><ci id="S4.SS2.SSS2.p1.19.m19.1.1.cmml" xref="S4.SS2.SSS2.p1.19.m19.1.1">𝑥</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS2.p1.19.m19.1c">x</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS2.p1.19.m19.1d">italic_x</annotation></semantics></math> to <math alttext="r" class="ltx_Math" display="inline" id="S4.SS2.SSS2.p1.20.m20.1"><semantics id="S4.SS2.SSS2.p1.20.m20.1a"><mi id="S4.SS2.SSS2.p1.20.m20.1.1" xref="S4.SS2.SSS2.p1.20.m20.1.1.cmml">r</mi><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS2.p1.20.m20.1b"><ci id="S4.SS2.SSS2.p1.20.m20.1.1.cmml" xref="S4.SS2.SSS2.p1.20.m20.1.1">𝑟</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS2.p1.20.m20.1c">r</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS2.p1.20.m20.1d">italic_r</annotation></semantics></math> (see Figure <a class="ltx_ref" href="https://arxiv.org/html/2503.00712v1#S4.F5" title="Figure 5 ‣ 4.2.1 SPQR Trees ‣ 4.2 Two-to-Three Augmentation ‣ 4 Vertex Connectivity Augmentation in Link-Arrival Model ‣ Streaming Algorithms for Network Design"><span class="ltx_text ltx_ref_tag">5</span></a>); this is sometimes overloaded to refer to the set of vertices <math alttext="\{a,b\}" class="ltx_Math" display="inline" id="S4.SS2.SSS2.p1.21.m21.2"><semantics id="S4.SS2.SSS2.p1.21.m21.2a"><mrow id="S4.SS2.SSS2.p1.21.m21.2.3.2" xref="S4.SS2.SSS2.p1.21.m21.2.3.1.cmml"><mo id="S4.SS2.SSS2.p1.21.m21.2.3.2.1" stretchy="false" xref="S4.SS2.SSS2.p1.21.m21.2.3.1.cmml">{</mo><mi id="S4.SS2.SSS2.p1.21.m21.1.1" xref="S4.SS2.SSS2.p1.21.m21.1.1.cmml">a</mi><mo id="S4.SS2.SSS2.p1.21.m21.2.3.2.2" xref="S4.SS2.SSS2.p1.21.m21.2.3.1.cmml">,</mo><mi id="S4.SS2.SSS2.p1.21.m21.2.2" xref="S4.SS2.SSS2.p1.21.m21.2.2.cmml">b</mi><mo id="S4.SS2.SSS2.p1.21.m21.2.3.2.3" stretchy="false" xref="S4.SS2.SSS2.p1.21.m21.2.3.1.cmml">}</mo></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS2.p1.21.m21.2b"><set id="S4.SS2.SSS2.p1.21.m21.2.3.1.cmml" xref="S4.SS2.SSS2.p1.21.m21.2.3.2"><ci id="S4.SS2.SSS2.p1.21.m21.1.1.cmml" xref="S4.SS2.SSS2.p1.21.m21.1.1">𝑎</ci><ci id="S4.SS2.SSS2.p1.21.m21.2.2.cmml" xref="S4.SS2.SSS2.p1.21.m21.2.2">𝑏</ci></set></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS2.p1.21.m21.2c">\{a,b\}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS2.p1.21.m21.2d">{ italic_a , italic_b }</annotation></semantics></math>. Recall from the construction of SPQR trees each vertex <math alttext="u\in V(G)" class="ltx_Math" display="inline" id="S4.SS2.SSS2.p1.22.m22.1"><semantics id="S4.SS2.SSS2.p1.22.m22.1a"><mrow id="S4.SS2.SSS2.p1.22.m22.1.2" xref="S4.SS2.SSS2.p1.22.m22.1.2.cmml"><mi id="S4.SS2.SSS2.p1.22.m22.1.2.2" xref="S4.SS2.SSS2.p1.22.m22.1.2.2.cmml">u</mi><mo id="S4.SS2.SSS2.p1.22.m22.1.2.1" xref="S4.SS2.SSS2.p1.22.m22.1.2.1.cmml">∈</mo><mrow id="S4.SS2.SSS2.p1.22.m22.1.2.3" xref="S4.SS2.SSS2.p1.22.m22.1.2.3.cmml"><mi id="S4.SS2.SSS2.p1.22.m22.1.2.3.2" xref="S4.SS2.SSS2.p1.22.m22.1.2.3.2.cmml">V</mi><mo id="S4.SS2.SSS2.p1.22.m22.1.2.3.1" xref="S4.SS2.SSS2.p1.22.m22.1.2.3.1.cmml">⁢</mo><mrow id="S4.SS2.SSS2.p1.22.m22.1.2.3.3.2" xref="S4.SS2.SSS2.p1.22.m22.1.2.3.cmml"><mo id="S4.SS2.SSS2.p1.22.m22.1.2.3.3.2.1" stretchy="false" xref="S4.SS2.SSS2.p1.22.m22.1.2.3.cmml">(</mo><mi id="S4.SS2.SSS2.p1.22.m22.1.1" xref="S4.SS2.SSS2.p1.22.m22.1.1.cmml">G</mi><mo id="S4.SS2.SSS2.p1.22.m22.1.2.3.3.2.2" stretchy="false" xref="S4.SS2.SSS2.p1.22.m22.1.2.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS2.p1.22.m22.1b"><apply id="S4.SS2.SSS2.p1.22.m22.1.2.cmml" xref="S4.SS2.SSS2.p1.22.m22.1.2"><in id="S4.SS2.SSS2.p1.22.m22.1.2.1.cmml" xref="S4.SS2.SSS2.p1.22.m22.1.2.1"></in><ci id="S4.SS2.SSS2.p1.22.m22.1.2.2.cmml" xref="S4.SS2.SSS2.p1.22.m22.1.2.2">𝑢</ci><apply id="S4.SS2.SSS2.p1.22.m22.1.2.3.cmml" xref="S4.SS2.SSS2.p1.22.m22.1.2.3"><times id="S4.SS2.SSS2.p1.22.m22.1.2.3.1.cmml" xref="S4.SS2.SSS2.p1.22.m22.1.2.3.1"></times><ci id="S4.SS2.SSS2.p1.22.m22.1.2.3.2.cmml" xref="S4.SS2.SSS2.p1.22.m22.1.2.3.2">𝑉</ci><ci id="S4.SS2.SSS2.p1.22.m22.1.1.cmml" xref="S4.SS2.SSS2.p1.22.m22.1.1">𝐺</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS2.p1.22.m22.1c">u\in V(G)</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS2.p1.22.m22.1d">italic_u ∈ italic_V ( italic_G )</annotation></semantics></math> may have several “copies” in <math alttext="T" class="ltx_Math" display="inline" id="S4.SS2.SSS2.p1.23.m23.1"><semantics id="S4.SS2.SSS2.p1.23.m23.1a"><mi id="S4.SS2.SSS2.p1.23.m23.1.1" xref="S4.SS2.SSS2.p1.23.m23.1.1.cmml">T</mi><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS2.p1.23.m23.1b"><ci id="S4.SS2.SSS2.p1.23.m23.1.1.cmml" xref="S4.SS2.SSS2.p1.23.m23.1.1">𝑇</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS2.p1.23.m23.1c">T</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS2.p1.23.m23.1d">italic_T</annotation></semantics></math>; if <math alttext="u" class="ltx_Math" display="inline" id="S4.SS2.SSS2.p1.24.m24.1"><semantics id="S4.SS2.SSS2.p1.24.m24.1a"><mi id="S4.SS2.SSS2.p1.24.m24.1.1" xref="S4.SS2.SSS2.p1.24.m24.1.1.cmml">u</mi><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS2.p1.24.m24.1b"><ci id="S4.SS2.SSS2.p1.24.m24.1.1.cmml" xref="S4.SS2.SSS2.p1.24.m24.1.1">𝑢</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS2.p1.24.m24.1c">u</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS2.p1.24.m24.1d">italic_u</annotation></semantics></math> is part of a separation pair, then it is duplicated in the “split” operation. It is not difficult to see that the set of all tree nodes containing a copy of <math alttext="u" class="ltx_Math" display="inline" id="S4.SS2.SSS2.p1.25.m25.1"><semantics id="S4.SS2.SSS2.p1.25.m25.1a"><mi id="S4.SS2.SSS2.p1.25.m25.1.1" xref="S4.SS2.SSS2.p1.25.m25.1.1.cmml">u</mi><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS2.p1.25.m25.1b"><ci id="S4.SS2.SSS2.p1.25.m25.1.1.cmml" xref="S4.SS2.SSS2.p1.25.m25.1.1">𝑢</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS2.p1.25.m25.1c">u</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS2.p1.25.m25.1d">italic_u</annotation></semantics></math> forms a connected component of <math alttext="T" class="ltx_Math" display="inline" id="S4.SS2.SSS2.p1.26.m26.1"><semantics id="S4.SS2.SSS2.p1.26.m26.1a"><mi id="S4.SS2.SSS2.p1.26.m26.1.1" xref="S4.SS2.SSS2.p1.26.m26.1.1.cmml">T</mi><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS2.p1.26.m26.1b"><ci id="S4.SS2.SSS2.p1.26.m26.1.1.cmml" xref="S4.SS2.SSS2.p1.26.m26.1.1">𝑇</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS2.p1.26.m26.1c">T</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS2.p1.26.m26.1d">italic_T</annotation></semantics></math>. We denote by <math alttext="h(u)" class="ltx_Math" display="inline" id="S4.SS2.SSS2.p1.27.m27.1"><semantics id="S4.SS2.SSS2.p1.27.m27.1a"><mrow id="S4.SS2.SSS2.p1.27.m27.1.2" xref="S4.SS2.SSS2.p1.27.m27.1.2.cmml"><mi id="S4.SS2.SSS2.p1.27.m27.1.2.2" xref="S4.SS2.SSS2.p1.27.m27.1.2.2.cmml">h</mi><mo id="S4.SS2.SSS2.p1.27.m27.1.2.1" xref="S4.SS2.SSS2.p1.27.m27.1.2.1.cmml">⁢</mo><mrow id="S4.SS2.SSS2.p1.27.m27.1.2.3.2" xref="S4.SS2.SSS2.p1.27.m27.1.2.cmml"><mo id="S4.SS2.SSS2.p1.27.m27.1.2.3.2.1" stretchy="false" xref="S4.SS2.SSS2.p1.27.m27.1.2.cmml">(</mo><mi id="S4.SS2.SSS2.p1.27.m27.1.1" xref="S4.SS2.SSS2.p1.27.m27.1.1.cmml">u</mi><mo id="S4.SS2.SSS2.p1.27.m27.1.2.3.2.2" stretchy="false" xref="S4.SS2.SSS2.p1.27.m27.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS2.p1.27.m27.1b"><apply id="S4.SS2.SSS2.p1.27.m27.1.2.cmml" xref="S4.SS2.SSS2.p1.27.m27.1.2"><times id="S4.SS2.SSS2.p1.27.m27.1.2.1.cmml" xref="S4.SS2.SSS2.p1.27.m27.1.2.1"></times><ci id="S4.SS2.SSS2.p1.27.m27.1.2.2.cmml" xref="S4.SS2.SSS2.p1.27.m27.1.2.2">ℎ</ci><ci id="S4.SS2.SSS2.p1.27.m27.1.1.cmml" xref="S4.SS2.SSS2.p1.27.m27.1.1">𝑢</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS2.p1.27.m27.1c">h(u)</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS2.p1.27.m27.1d">italic_h ( italic_u )</annotation></semantics></math> and <math alttext="\ell(u)" class="ltx_Math" display="inline" id="S4.SS2.SSS2.p1.28.m28.1"><semantics id="S4.SS2.SSS2.p1.28.m28.1a"><mrow id="S4.SS2.SSS2.p1.28.m28.1.2" xref="S4.SS2.SSS2.p1.28.m28.1.2.cmml"><mi id="S4.SS2.SSS2.p1.28.m28.1.2.2" mathvariant="normal" xref="S4.SS2.SSS2.p1.28.m28.1.2.2.cmml">ℓ</mi><mo id="S4.SS2.SSS2.p1.28.m28.1.2.1" xref="S4.SS2.SSS2.p1.28.m28.1.2.1.cmml">⁢</mo><mrow id="S4.SS2.SSS2.p1.28.m28.1.2.3.2" xref="S4.SS2.SSS2.p1.28.m28.1.2.cmml"><mo id="S4.SS2.SSS2.p1.28.m28.1.2.3.2.1" stretchy="false" xref="S4.SS2.SSS2.p1.28.m28.1.2.cmml">(</mo><mi id="S4.SS2.SSS2.p1.28.m28.1.1" xref="S4.SS2.SSS2.p1.28.m28.1.1.cmml">u</mi><mo id="S4.SS2.SSS2.p1.28.m28.1.2.3.2.2" stretchy="false" xref="S4.SS2.SSS2.p1.28.m28.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS2.p1.28.m28.1b"><apply id="S4.SS2.SSS2.p1.28.m28.1.2.cmml" xref="S4.SS2.SSS2.p1.28.m28.1.2"><times id="S4.SS2.SSS2.p1.28.m28.1.2.1.cmml" xref="S4.SS2.SSS2.p1.28.m28.1.2.1"></times><ci id="S4.SS2.SSS2.p1.28.m28.1.2.2.cmml" xref="S4.SS2.SSS2.p1.28.m28.1.2.2">ℓ</ci><ci id="S4.SS2.SSS2.p1.28.m28.1.1.cmml" xref="S4.SS2.SSS2.p1.28.m28.1.1">𝑢</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS2.p1.28.m28.1c">\ell(u)</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS2.p1.28.m28.1d">roman_ℓ ( italic_u )</annotation></semantics></math> the tree nodes containing <math alttext="u" class="ltx_Math" display="inline" id="S4.SS2.SSS2.p1.29.m29.1"><semantics id="S4.SS2.SSS2.p1.29.m29.1a"><mi id="S4.SS2.SSS2.p1.29.m29.1.1" xref="S4.SS2.SSS2.p1.29.m29.1.1.cmml">u</mi><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS2.p1.29.m29.1b"><ci id="S4.SS2.SSS2.p1.29.m29.1.1.cmml" xref="S4.SS2.SSS2.p1.29.m29.1.1">𝑢</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS2.p1.29.m29.1c">u</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS2.p1.29.m29.1d">italic_u</annotation></semantics></math> that are the closest to and furthest from the root respectively (breaking ties arbitrarily); see Figure <a class="ltx_ref" href="https://arxiv.org/html/2503.00712v1#S4.F5" title="Figure 5 ‣ 4.2.1 SPQR Trees ‣ 4.2 Two-to-Three Augmentation ‣ 4 Vertex Connectivity Augmentation in Link-Arrival Model ‣ Streaming Algorithms for Network Design"><span class="ltx_text ltx_ref_tag">5</span></a>.</p> </div> <div class="ltx_para" id="S4.SS2.SSS2.p2"> <p class="ltx_p" id="S4.SS2.SSS2.p2.8">We provide a high-level description of the algorithm; this is formalized in Algorithm <a class="ltx_ref" href="https://arxiv.org/html/2503.00712v1#algorithm6" title="In “Cycle” Cuts: ‣ 4.2.2 The Streaming Algorithm ‣ 4.2 Two-to-Three Augmentation ‣ 4 Vertex Connectivity Augmentation in Link-Arrival Model ‣ Streaming Algorithms for Network Design"><span class="ltx_text ltx_ref_tag">6</span></a>. By Lemma <a class="ltx_ref" href="https://arxiv.org/html/2503.00712v1#S4.Thmtheorem17" title="Lemma 4.17. ‣ 4.2.1 SPQR Trees ‣ 4.2 Two-to-Three Augmentation ‣ 4 Vertex Connectivity Augmentation in Link-Arrival Model ‣ Streaming Algorithms for Network Design"><span class="ltx_text ltx_ref_tag">4.17</span></a>, there are three possible type of 2-vertex cuts <math alttext="\{a,b\}" class="ltx_Math" display="inline" id="S4.SS2.SSS2.p2.1.m1.2"><semantics id="S4.SS2.SSS2.p2.1.m1.2a"><mrow id="S4.SS2.SSS2.p2.1.m1.2.3.2" xref="S4.SS2.SSS2.p2.1.m1.2.3.1.cmml"><mo id="S4.SS2.SSS2.p2.1.m1.2.3.2.1" stretchy="false" xref="S4.SS2.SSS2.p2.1.m1.2.3.1.cmml">{</mo><mi id="S4.SS2.SSS2.p2.1.m1.1.1" xref="S4.SS2.SSS2.p2.1.m1.1.1.cmml">a</mi><mo id="S4.SS2.SSS2.p2.1.m1.2.3.2.2" xref="S4.SS2.SSS2.p2.1.m1.2.3.1.cmml">,</mo><mi id="S4.SS2.SSS2.p2.1.m1.2.2" xref="S4.SS2.SSS2.p2.1.m1.2.2.cmml">b</mi><mo id="S4.SS2.SSS2.p2.1.m1.2.3.2.3" stretchy="false" xref="S4.SS2.SSS2.p2.1.m1.2.3.1.cmml">}</mo></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS2.p2.1.m1.2b"><set id="S4.SS2.SSS2.p2.1.m1.2.3.1.cmml" xref="S4.SS2.SSS2.p2.1.m1.2.3.2"><ci id="S4.SS2.SSS2.p2.1.m1.1.1.cmml" xref="S4.SS2.SSS2.p2.1.m1.1.1">𝑎</ci><ci id="S4.SS2.SSS2.p2.1.m1.2.2.cmml" xref="S4.SS2.SSS2.p2.1.m1.2.2">𝑏</ci></set></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS2.p2.1.m1.2c">\{a,b\}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS2.p2.1.m1.2d">{ italic_a , italic_b }</annotation></semantics></math> in <math alttext="G" class="ltx_Math" display="inline" id="S4.SS2.SSS2.p2.2.m2.1"><semantics id="S4.SS2.SSS2.p2.2.m2.1a"><mi id="S4.SS2.SSS2.p2.2.m2.1.1" xref="S4.SS2.SSS2.p2.2.m2.1.1.cmml">G</mi><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS2.p2.2.m2.1b"><ci id="S4.SS2.SSS2.p2.2.m2.1.1.cmml" xref="S4.SS2.SSS2.p2.2.m2.1.1">𝐺</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS2.p2.2.m2.1c">G</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS2.p2.2.m2.1d">italic_G</annotation></semantics></math>: (1) <math alttext="\{a,b\}" class="ltx_Math" display="inline" id="S4.SS2.SSS2.p2.3.m3.2"><semantics id="S4.SS2.SSS2.p2.3.m3.2a"><mrow id="S4.SS2.SSS2.p2.3.m3.2.3.2" xref="S4.SS2.SSS2.p2.3.m3.2.3.1.cmml"><mo id="S4.SS2.SSS2.p2.3.m3.2.3.2.1" stretchy="false" xref="S4.SS2.SSS2.p2.3.m3.2.3.1.cmml">{</mo><mi id="S4.SS2.SSS2.p2.3.m3.1.1" xref="S4.SS2.SSS2.p2.3.m3.1.1.cmml">a</mi><mo id="S4.SS2.SSS2.p2.3.m3.2.3.2.2" xref="S4.SS2.SSS2.p2.3.m3.2.3.1.cmml">,</mo><mi id="S4.SS2.SSS2.p2.3.m3.2.2" xref="S4.SS2.SSS2.p2.3.m3.2.2.cmml">b</mi><mo id="S4.SS2.SSS2.p2.3.m3.2.3.2.3" stretchy="false" xref="S4.SS2.SSS2.p2.3.m3.2.3.1.cmml">}</mo></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS2.p2.3.m3.2b"><set id="S4.SS2.SSS2.p2.3.m3.2.3.1.cmml" xref="S4.SS2.SSS2.p2.3.m3.2.3.2"><ci id="S4.SS2.SSS2.p2.3.m3.1.1.cmml" xref="S4.SS2.SSS2.p2.3.m3.1.1">𝑎</ci><ci id="S4.SS2.SSS2.p2.3.m3.2.2.cmml" xref="S4.SS2.SSS2.p2.3.m3.2.2">𝑏</ci></set></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS2.p2.3.m3.2c">\{a,b\}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS2.p2.3.m3.2d">{ italic_a , italic_b }</annotation></semantics></math> is the graph corresponding to a P-node, (2) <math alttext="ab" class="ltx_Math" display="inline" id="S4.SS2.SSS2.p2.4.m4.1"><semantics id="S4.SS2.SSS2.p2.4.m4.1a"><mrow id="S4.SS2.SSS2.p2.4.m4.1.1" xref="S4.SS2.SSS2.p2.4.m4.1.1.cmml"><mi id="S4.SS2.SSS2.p2.4.m4.1.1.2" xref="S4.SS2.SSS2.p2.4.m4.1.1.2.cmml">a</mi><mo id="S4.SS2.SSS2.p2.4.m4.1.1.1" xref="S4.SS2.SSS2.p2.4.m4.1.1.1.cmml">⁢</mo><mi id="S4.SS2.SSS2.p2.4.m4.1.1.3" xref="S4.SS2.SSS2.p2.4.m4.1.1.3.cmml">b</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS2.p2.4.m4.1b"><apply id="S4.SS2.SSS2.p2.4.m4.1.1.cmml" xref="S4.SS2.SSS2.p2.4.m4.1.1"><times id="S4.SS2.SSS2.p2.4.m4.1.1.1.cmml" xref="S4.SS2.SSS2.p2.4.m4.1.1.1"></times><ci id="S4.SS2.SSS2.p2.4.m4.1.1.2.cmml" xref="S4.SS2.SSS2.p2.4.m4.1.1.2">𝑎</ci><ci id="S4.SS2.SSS2.p2.4.m4.1.1.3.cmml" xref="S4.SS2.SSS2.p2.4.m4.1.1.3">𝑏</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS2.p2.4.m4.1c">ab</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS2.p2.4.m4.1d">italic_a italic_b</annotation></semantics></math> is a virtual edge associated with a tree edge incident to two nodes: one is an R-node and the other is an R-node or S-node, (3) <math alttext="a" class="ltx_Math" display="inline" id="S4.SS2.SSS2.p2.5.m5.1"><semantics id="S4.SS2.SSS2.p2.5.m5.1a"><mi id="S4.SS2.SSS2.p2.5.m5.1.1" xref="S4.SS2.SSS2.p2.5.m5.1.1.cmml">a</mi><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS2.p2.5.m5.1b"><ci id="S4.SS2.SSS2.p2.5.m5.1.1.cmml" xref="S4.SS2.SSS2.p2.5.m5.1.1">𝑎</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS2.p2.5.m5.1c">a</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS2.p2.5.m5.1d">italic_a</annotation></semantics></math> and <math alttext="b" class="ltx_Math" display="inline" id="S4.SS2.SSS2.p2.6.m6.1"><semantics id="S4.SS2.SSS2.p2.6.m6.1a"><mi id="S4.SS2.SSS2.p2.6.m6.1.1" xref="S4.SS2.SSS2.p2.6.m6.1.1.cmml">b</mi><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS2.p2.6.m6.1b"><ci id="S4.SS2.SSS2.p2.6.m6.1.1.cmml" xref="S4.SS2.SSS2.p2.6.m6.1.1">𝑏</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS2.p2.6.m6.1c">b</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS2.p2.6.m6.1d">italic_b</annotation></semantics></math> are non-adjacent nodes of a cycle <math alttext="G_{x}" class="ltx_Math" display="inline" id="S4.SS2.SSS2.p2.7.m7.1"><semantics id="S4.SS2.SSS2.p2.7.m7.1a"><msub id="S4.SS2.SSS2.p2.7.m7.1.1" xref="S4.SS2.SSS2.p2.7.m7.1.1.cmml"><mi id="S4.SS2.SSS2.p2.7.m7.1.1.2" xref="S4.SS2.SSS2.p2.7.m7.1.1.2.cmml">G</mi><mi id="S4.SS2.SSS2.p2.7.m7.1.1.3" xref="S4.SS2.SSS2.p2.7.m7.1.1.3.cmml">x</mi></msub><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS2.p2.7.m7.1b"><apply id="S4.SS2.SSS2.p2.7.m7.1.1.cmml" xref="S4.SS2.SSS2.p2.7.m7.1.1"><csymbol cd="ambiguous" id="S4.SS2.SSS2.p2.7.m7.1.1.1.cmml" xref="S4.SS2.SSS2.p2.7.m7.1.1">subscript</csymbol><ci id="S4.SS2.SSS2.p2.7.m7.1.1.2.cmml" xref="S4.SS2.SSS2.p2.7.m7.1.1.2">𝐺</ci><ci id="S4.SS2.SSS2.p2.7.m7.1.1.3.cmml" xref="S4.SS2.SSS2.p2.7.m7.1.1.3">𝑥</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS2.p2.7.m7.1c">G_{x}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS2.p2.7.m7.1d">italic_G start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT</annotation></semantics></math> for some S-node <math alttext="x" class="ltx_Math" display="inline" id="S4.SS2.SSS2.p2.8.m8.1"><semantics id="S4.SS2.SSS2.p2.8.m8.1a"><mi id="S4.SS2.SSS2.p2.8.m8.1.1" xref="S4.SS2.SSS2.p2.8.m8.1.1.cmml">x</mi><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS2.p2.8.m8.1b"><ci id="S4.SS2.SSS2.p2.8.m8.1.1.cmml" xref="S4.SS2.SSS2.p2.8.m8.1.1">𝑥</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS2.p2.8.m8.1c">x</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS2.p2.8.m8.1d">italic_x</annotation></semantics></math>. Intuitively, one can think of (1) as corresponding to a node being deleted in the tree, (2) corresponding to an edge being deleted in the tree, and (3) to handle connectivity within cycle nodes.</p> </div> <section class="ltx_paragraph" id="S4.SS2.SSS2.Px1"> <h5 class="ltx_title ltx_title_paragraph">“Tree” Cuts:</h5> <div class="ltx_para" id="S4.SS2.SSS2.Px1.p1"> <p class="ltx_p" id="S4.SS2.SSS2.Px1.p1.10">To protect against 2-node cuts of type (1) and (2), we follow a similar approach to the 1-to-2 augmentation described in Section <a class="ltx_ref" href="https://arxiv.org/html/2503.00712v1#S4.SS1" title="4.1 One-to-Two Augmentation ‣ 4 Vertex Connectivity Augmentation in Link-Arrival Model ‣ Streaming Algorithms for Network Design"><span class="ltx_text ltx_ref_tag">4.1</span></a>. The additional difficulty in this setting arises from the fact that <math alttext="T" class="ltx_Math" display="inline" id="S4.SS2.SSS2.Px1.p1.1.m1.1"><semantics id="S4.SS2.SSS2.Px1.p1.1.m1.1a"><mi id="S4.SS2.SSS2.Px1.p1.1.m1.1.1" xref="S4.SS2.SSS2.Px1.p1.1.m1.1.1.cmml">T</mi><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS2.Px1.p1.1.m1.1b"><ci id="S4.SS2.SSS2.Px1.p1.1.m1.1.1.cmml" xref="S4.SS2.SSS2.Px1.p1.1.m1.1.1">𝑇</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS2.Px1.p1.1.m1.1c">T</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS2.Px1.p1.1.m1.1d">italic_T</annotation></semantics></math> may contain multiple copies of each vertex of <math alttext="G" class="ltx_Math" display="inline" id="S4.SS2.SSS2.Px1.p1.2.m2.1"><semantics id="S4.SS2.SSS2.Px1.p1.2.m2.1a"><mi id="S4.SS2.SSS2.Px1.p1.2.m2.1.1" xref="S4.SS2.SSS2.Px1.p1.2.m2.1.1.cmml">G</mi><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS2.Px1.p1.2.m2.1b"><ci id="S4.SS2.SSS2.Px1.p1.2.m2.1.1.cmml" xref="S4.SS2.SSS2.Px1.p1.2.m2.1.1">𝐺</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS2.Px1.p1.2.m2.1c">G</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS2.Px1.p1.2.m2.1d">italic_G</annotation></semantics></math>; thus a link <math alttext="uv" class="ltx_Math" display="inline" id="S4.SS2.SSS2.Px1.p1.3.m3.1"><semantics id="S4.SS2.SSS2.Px1.p1.3.m3.1a"><mrow id="S4.SS2.SSS2.Px1.p1.3.m3.1.1" xref="S4.SS2.SSS2.Px1.p1.3.m3.1.1.cmml"><mi id="S4.SS2.SSS2.Px1.p1.3.m3.1.1.2" xref="S4.SS2.SSS2.Px1.p1.3.m3.1.1.2.cmml">u</mi><mo id="S4.SS2.SSS2.Px1.p1.3.m3.1.1.1" xref="S4.SS2.SSS2.Px1.p1.3.m3.1.1.1.cmml">⁢</mo><mi id="S4.SS2.SSS2.Px1.p1.3.m3.1.1.3" xref="S4.SS2.SSS2.Px1.p1.3.m3.1.1.3.cmml">v</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS2.Px1.p1.3.m3.1b"><apply id="S4.SS2.SSS2.Px1.p1.3.m3.1.1.cmml" xref="S4.SS2.SSS2.Px1.p1.3.m3.1.1"><times id="S4.SS2.SSS2.Px1.p1.3.m3.1.1.1.cmml" xref="S4.SS2.SSS2.Px1.p1.3.m3.1.1.1"></times><ci id="S4.SS2.SSS2.Px1.p1.3.m3.1.1.2.cmml" xref="S4.SS2.SSS2.Px1.p1.3.m3.1.1.2">𝑢</ci><ci id="S4.SS2.SSS2.Px1.p1.3.m3.1.1.3.cmml" xref="S4.SS2.SSS2.Px1.p1.3.m3.1.1.3">𝑣</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS2.Px1.p1.3.m3.1c">uv</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS2.Px1.p1.3.m3.1d">italic_u italic_v</annotation></semantics></math> does not necessarily correspond to a unique “tree link” <math alttext="xy\in V(T)\times V(T)" class="ltx_Math" display="inline" id="S4.SS2.SSS2.Px1.p1.4.m4.2"><semantics id="S4.SS2.SSS2.Px1.p1.4.m4.2a"><mrow id="S4.SS2.SSS2.Px1.p1.4.m4.2.3" xref="S4.SS2.SSS2.Px1.p1.4.m4.2.3.cmml"><mrow id="S4.SS2.SSS2.Px1.p1.4.m4.2.3.2" xref="S4.SS2.SSS2.Px1.p1.4.m4.2.3.2.cmml"><mi id="S4.SS2.SSS2.Px1.p1.4.m4.2.3.2.2" xref="S4.SS2.SSS2.Px1.p1.4.m4.2.3.2.2.cmml">x</mi><mo id="S4.SS2.SSS2.Px1.p1.4.m4.2.3.2.1" xref="S4.SS2.SSS2.Px1.p1.4.m4.2.3.2.1.cmml">⁢</mo><mi id="S4.SS2.SSS2.Px1.p1.4.m4.2.3.2.3" xref="S4.SS2.SSS2.Px1.p1.4.m4.2.3.2.3.cmml">y</mi></mrow><mo id="S4.SS2.SSS2.Px1.p1.4.m4.2.3.1" xref="S4.SS2.SSS2.Px1.p1.4.m4.2.3.1.cmml">∈</mo><mrow id="S4.SS2.SSS2.Px1.p1.4.m4.2.3.3" xref="S4.SS2.SSS2.Px1.p1.4.m4.2.3.3.cmml"><mrow id="S4.SS2.SSS2.Px1.p1.4.m4.2.3.3.2" xref="S4.SS2.SSS2.Px1.p1.4.m4.2.3.3.2.cmml"><mrow id="S4.SS2.SSS2.Px1.p1.4.m4.2.3.3.2.2" xref="S4.SS2.SSS2.Px1.p1.4.m4.2.3.3.2.2.cmml"><mi id="S4.SS2.SSS2.Px1.p1.4.m4.2.3.3.2.2.2" xref="S4.SS2.SSS2.Px1.p1.4.m4.2.3.3.2.2.2.cmml">V</mi><mo id="S4.SS2.SSS2.Px1.p1.4.m4.2.3.3.2.2.1" xref="S4.SS2.SSS2.Px1.p1.4.m4.2.3.3.2.2.1.cmml">⁢</mo><mrow id="S4.SS2.SSS2.Px1.p1.4.m4.2.3.3.2.2.3.2" xref="S4.SS2.SSS2.Px1.p1.4.m4.2.3.3.2.2.cmml"><mo id="S4.SS2.SSS2.Px1.p1.4.m4.2.3.3.2.2.3.2.1" stretchy="false" xref="S4.SS2.SSS2.Px1.p1.4.m4.2.3.3.2.2.cmml">(</mo><mi id="S4.SS2.SSS2.Px1.p1.4.m4.1.1" xref="S4.SS2.SSS2.Px1.p1.4.m4.1.1.cmml">T</mi><mo id="S4.SS2.SSS2.Px1.p1.4.m4.2.3.3.2.2.3.2.2" rspace="0.055em" stretchy="false" xref="S4.SS2.SSS2.Px1.p1.4.m4.2.3.3.2.2.cmml">)</mo></mrow></mrow><mo id="S4.SS2.SSS2.Px1.p1.4.m4.2.3.3.2.1" rspace="0.222em" xref="S4.SS2.SSS2.Px1.p1.4.m4.2.3.3.2.1.cmml">×</mo><mi id="S4.SS2.SSS2.Px1.p1.4.m4.2.3.3.2.3" xref="S4.SS2.SSS2.Px1.p1.4.m4.2.3.3.2.3.cmml">V</mi></mrow><mo id="S4.SS2.SSS2.Px1.p1.4.m4.2.3.3.1" xref="S4.SS2.SSS2.Px1.p1.4.m4.2.3.3.1.cmml">⁢</mo><mrow id="S4.SS2.SSS2.Px1.p1.4.m4.2.3.3.3.2" xref="S4.SS2.SSS2.Px1.p1.4.m4.2.3.3.cmml"><mo id="S4.SS2.SSS2.Px1.p1.4.m4.2.3.3.3.2.1" stretchy="false" xref="S4.SS2.SSS2.Px1.p1.4.m4.2.3.3.cmml">(</mo><mi id="S4.SS2.SSS2.Px1.p1.4.m4.2.2" xref="S4.SS2.SSS2.Px1.p1.4.m4.2.2.cmml">T</mi><mo id="S4.SS2.SSS2.Px1.p1.4.m4.2.3.3.3.2.2" stretchy="false" xref="S4.SS2.SSS2.Px1.p1.4.m4.2.3.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS2.Px1.p1.4.m4.2b"><apply id="S4.SS2.SSS2.Px1.p1.4.m4.2.3.cmml" xref="S4.SS2.SSS2.Px1.p1.4.m4.2.3"><in id="S4.SS2.SSS2.Px1.p1.4.m4.2.3.1.cmml" xref="S4.SS2.SSS2.Px1.p1.4.m4.2.3.1"></in><apply id="S4.SS2.SSS2.Px1.p1.4.m4.2.3.2.cmml" xref="S4.SS2.SSS2.Px1.p1.4.m4.2.3.2"><times id="S4.SS2.SSS2.Px1.p1.4.m4.2.3.2.1.cmml" xref="S4.SS2.SSS2.Px1.p1.4.m4.2.3.2.1"></times><ci id="S4.SS2.SSS2.Px1.p1.4.m4.2.3.2.2.cmml" xref="S4.SS2.SSS2.Px1.p1.4.m4.2.3.2.2">𝑥</ci><ci id="S4.SS2.SSS2.Px1.p1.4.m4.2.3.2.3.cmml" xref="S4.SS2.SSS2.Px1.p1.4.m4.2.3.2.3">𝑦</ci></apply><apply id="S4.SS2.SSS2.Px1.p1.4.m4.2.3.3.cmml" xref="S4.SS2.SSS2.Px1.p1.4.m4.2.3.3"><times id="S4.SS2.SSS2.Px1.p1.4.m4.2.3.3.1.cmml" xref="S4.SS2.SSS2.Px1.p1.4.m4.2.3.3.1"></times><apply id="S4.SS2.SSS2.Px1.p1.4.m4.2.3.3.2.cmml" xref="S4.SS2.SSS2.Px1.p1.4.m4.2.3.3.2"><times id="S4.SS2.SSS2.Px1.p1.4.m4.2.3.3.2.1.cmml" xref="S4.SS2.SSS2.Px1.p1.4.m4.2.3.3.2.1"></times><apply id="S4.SS2.SSS2.Px1.p1.4.m4.2.3.3.2.2.cmml" xref="S4.SS2.SSS2.Px1.p1.4.m4.2.3.3.2.2"><times id="S4.SS2.SSS2.Px1.p1.4.m4.2.3.3.2.2.1.cmml" xref="S4.SS2.SSS2.Px1.p1.4.m4.2.3.3.2.2.1"></times><ci id="S4.SS2.SSS2.Px1.p1.4.m4.2.3.3.2.2.2.cmml" xref="S4.SS2.SSS2.Px1.p1.4.m4.2.3.3.2.2.2">𝑉</ci><ci id="S4.SS2.SSS2.Px1.p1.4.m4.1.1.cmml" xref="S4.SS2.SSS2.Px1.p1.4.m4.1.1">𝑇</ci></apply><ci id="S4.SS2.SSS2.Px1.p1.4.m4.2.3.3.2.3.cmml" xref="S4.SS2.SSS2.Px1.p1.4.m4.2.3.3.2.3">𝑉</ci></apply><ci id="S4.SS2.SSS2.Px1.p1.4.m4.2.2.cmml" xref="S4.SS2.SSS2.Px1.p1.4.m4.2.2">𝑇</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS2.Px1.p1.4.m4.2c">xy\in V(T)\times V(T)</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS2.Px1.p1.4.m4.2d">italic_x italic_y ∈ italic_V ( italic_T ) × italic_V ( italic_T )</annotation></semantics></math>. Furthermore, a 2-vertex cut in <math alttext="G" class="ltx_Math" display="inline" id="S4.SS2.SSS2.Px1.p1.5.m5.1"><semantics id="S4.SS2.SSS2.Px1.p1.5.m5.1a"><mi id="S4.SS2.SSS2.Px1.p1.5.m5.1.1" xref="S4.SS2.SSS2.Px1.p1.5.m5.1.1.cmml">G</mi><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS2.Px1.p1.5.m5.1b"><ci id="S4.SS2.SSS2.Px1.p1.5.m5.1.1.cmml" xref="S4.SS2.SSS2.Px1.p1.5.m5.1.1">𝐺</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS2.Px1.p1.5.m5.1c">G</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS2.Px1.p1.5.m5.1d">italic_G</annotation></semantics></math> may contain copies in several tree nodes. We handle this by strategically choosing to view a link <math alttext="uv\in L" class="ltx_Math" display="inline" id="S4.SS2.SSS2.Px1.p1.6.m6.1"><semantics id="S4.SS2.SSS2.Px1.p1.6.m6.1a"><mrow id="S4.SS2.SSS2.Px1.p1.6.m6.1.1" xref="S4.SS2.SSS2.Px1.p1.6.m6.1.1.cmml"><mrow id="S4.SS2.SSS2.Px1.p1.6.m6.1.1.2" xref="S4.SS2.SSS2.Px1.p1.6.m6.1.1.2.cmml"><mi id="S4.SS2.SSS2.Px1.p1.6.m6.1.1.2.2" xref="S4.SS2.SSS2.Px1.p1.6.m6.1.1.2.2.cmml">u</mi><mo id="S4.SS2.SSS2.Px1.p1.6.m6.1.1.2.1" xref="S4.SS2.SSS2.Px1.p1.6.m6.1.1.2.1.cmml">⁢</mo><mi id="S4.SS2.SSS2.Px1.p1.6.m6.1.1.2.3" xref="S4.SS2.SSS2.Px1.p1.6.m6.1.1.2.3.cmml">v</mi></mrow><mo id="S4.SS2.SSS2.Px1.p1.6.m6.1.1.1" xref="S4.SS2.SSS2.Px1.p1.6.m6.1.1.1.cmml">∈</mo><mi id="S4.SS2.SSS2.Px1.p1.6.m6.1.1.3" xref="S4.SS2.SSS2.Px1.p1.6.m6.1.1.3.cmml">L</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS2.Px1.p1.6.m6.1b"><apply id="S4.SS2.SSS2.Px1.p1.6.m6.1.1.cmml" xref="S4.SS2.SSS2.Px1.p1.6.m6.1.1"><in id="S4.SS2.SSS2.Px1.p1.6.m6.1.1.1.cmml" xref="S4.SS2.SSS2.Px1.p1.6.m6.1.1.1"></in><apply id="S4.SS2.SSS2.Px1.p1.6.m6.1.1.2.cmml" xref="S4.SS2.SSS2.Px1.p1.6.m6.1.1.2"><times id="S4.SS2.SSS2.Px1.p1.6.m6.1.1.2.1.cmml" xref="S4.SS2.SSS2.Px1.p1.6.m6.1.1.2.1"></times><ci id="S4.SS2.SSS2.Px1.p1.6.m6.1.1.2.2.cmml" xref="S4.SS2.SSS2.Px1.p1.6.m6.1.1.2.2">𝑢</ci><ci id="S4.SS2.SSS2.Px1.p1.6.m6.1.1.2.3.cmml" xref="S4.SS2.SSS2.Px1.p1.6.m6.1.1.2.3">𝑣</ci></apply><ci id="S4.SS2.SSS2.Px1.p1.6.m6.1.1.3.cmml" xref="S4.SS2.SSS2.Px1.p1.6.m6.1.1.3">𝐿</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS2.Px1.p1.6.m6.1c">uv\in L</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS2.Px1.p1.6.m6.1d">italic_u italic_v ∈ italic_L</annotation></semantics></math> as a tree link adjacent to either <math alttext="h(u)" class="ltx_Math" display="inline" id="S4.SS2.SSS2.Px1.p1.7.m7.1"><semantics id="S4.SS2.SSS2.Px1.p1.7.m7.1a"><mrow id="S4.SS2.SSS2.Px1.p1.7.m7.1.2" xref="S4.SS2.SSS2.Px1.p1.7.m7.1.2.cmml"><mi id="S4.SS2.SSS2.Px1.p1.7.m7.1.2.2" xref="S4.SS2.SSS2.Px1.p1.7.m7.1.2.2.cmml">h</mi><mo id="S4.SS2.SSS2.Px1.p1.7.m7.1.2.1" xref="S4.SS2.SSS2.Px1.p1.7.m7.1.2.1.cmml">⁢</mo><mrow id="S4.SS2.SSS2.Px1.p1.7.m7.1.2.3.2" xref="S4.SS2.SSS2.Px1.p1.7.m7.1.2.cmml"><mo id="S4.SS2.SSS2.Px1.p1.7.m7.1.2.3.2.1" stretchy="false" xref="S4.SS2.SSS2.Px1.p1.7.m7.1.2.cmml">(</mo><mi id="S4.SS2.SSS2.Px1.p1.7.m7.1.1" xref="S4.SS2.SSS2.Px1.p1.7.m7.1.1.cmml">u</mi><mo id="S4.SS2.SSS2.Px1.p1.7.m7.1.2.3.2.2" stretchy="false" xref="S4.SS2.SSS2.Px1.p1.7.m7.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS2.Px1.p1.7.m7.1b"><apply id="S4.SS2.SSS2.Px1.p1.7.m7.1.2.cmml" xref="S4.SS2.SSS2.Px1.p1.7.m7.1.2"><times id="S4.SS2.SSS2.Px1.p1.7.m7.1.2.1.cmml" xref="S4.SS2.SSS2.Px1.p1.7.m7.1.2.1"></times><ci id="S4.SS2.SSS2.Px1.p1.7.m7.1.2.2.cmml" xref="S4.SS2.SSS2.Px1.p1.7.m7.1.2.2">ℎ</ci><ci id="S4.SS2.SSS2.Px1.p1.7.m7.1.1.cmml" xref="S4.SS2.SSS2.Px1.p1.7.m7.1.1">𝑢</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS2.Px1.p1.7.m7.1c">h(u)</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS2.Px1.p1.7.m7.1d">italic_h ( italic_u )</annotation></semantics></math> or <math alttext="\ell(u)" class="ltx_Math" display="inline" id="S4.SS2.SSS2.Px1.p1.8.m8.1"><semantics id="S4.SS2.SSS2.Px1.p1.8.m8.1a"><mrow id="S4.SS2.SSS2.Px1.p1.8.m8.1.2" xref="S4.SS2.SSS2.Px1.p1.8.m8.1.2.cmml"><mi id="S4.SS2.SSS2.Px1.p1.8.m8.1.2.2" mathvariant="normal" xref="S4.SS2.SSS2.Px1.p1.8.m8.1.2.2.cmml">ℓ</mi><mo id="S4.SS2.SSS2.Px1.p1.8.m8.1.2.1" xref="S4.SS2.SSS2.Px1.p1.8.m8.1.2.1.cmml">⁢</mo><mrow id="S4.SS2.SSS2.Px1.p1.8.m8.1.2.3.2" xref="S4.SS2.SSS2.Px1.p1.8.m8.1.2.cmml"><mo id="S4.SS2.SSS2.Px1.p1.8.m8.1.2.3.2.1" stretchy="false" xref="S4.SS2.SSS2.Px1.p1.8.m8.1.2.cmml">(</mo><mi id="S4.SS2.SSS2.Px1.p1.8.m8.1.1" xref="S4.SS2.SSS2.Px1.p1.8.m8.1.1.cmml">u</mi><mo id="S4.SS2.SSS2.Px1.p1.8.m8.1.2.3.2.2" stretchy="false" xref="S4.SS2.SSS2.Px1.p1.8.m8.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS2.Px1.p1.8.m8.1b"><apply id="S4.SS2.SSS2.Px1.p1.8.m8.1.2.cmml" xref="S4.SS2.SSS2.Px1.p1.8.m8.1.2"><times id="S4.SS2.SSS2.Px1.p1.8.m8.1.2.1.cmml" xref="S4.SS2.SSS2.Px1.p1.8.m8.1.2.1"></times><ci id="S4.SS2.SSS2.Px1.p1.8.m8.1.2.2.cmml" xref="S4.SS2.SSS2.Px1.p1.8.m8.1.2.2">ℓ</ci><ci id="S4.SS2.SSS2.Px1.p1.8.m8.1.1.cmml" xref="S4.SS2.SSS2.Px1.p1.8.m8.1.1">𝑢</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS2.Px1.p1.8.m8.1c">\ell(u)</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS2.Px1.p1.8.m8.1d">roman_ℓ ( italic_u )</annotation></semantics></math> (and <math alttext="h(v)" class="ltx_Math" display="inline" id="S4.SS2.SSS2.Px1.p1.9.m9.1"><semantics id="S4.SS2.SSS2.Px1.p1.9.m9.1a"><mrow id="S4.SS2.SSS2.Px1.p1.9.m9.1.2" xref="S4.SS2.SSS2.Px1.p1.9.m9.1.2.cmml"><mi id="S4.SS2.SSS2.Px1.p1.9.m9.1.2.2" xref="S4.SS2.SSS2.Px1.p1.9.m9.1.2.2.cmml">h</mi><mo id="S4.SS2.SSS2.Px1.p1.9.m9.1.2.1" xref="S4.SS2.SSS2.Px1.p1.9.m9.1.2.1.cmml">⁢</mo><mrow id="S4.SS2.SSS2.Px1.p1.9.m9.1.2.3.2" xref="S4.SS2.SSS2.Px1.p1.9.m9.1.2.cmml"><mo id="S4.SS2.SSS2.Px1.p1.9.m9.1.2.3.2.1" stretchy="false" xref="S4.SS2.SSS2.Px1.p1.9.m9.1.2.cmml">(</mo><mi id="S4.SS2.SSS2.Px1.p1.9.m9.1.1" xref="S4.SS2.SSS2.Px1.p1.9.m9.1.1.cmml">v</mi><mo id="S4.SS2.SSS2.Px1.p1.9.m9.1.2.3.2.2" stretchy="false" xref="S4.SS2.SSS2.Px1.p1.9.m9.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS2.Px1.p1.9.m9.1b"><apply id="S4.SS2.SSS2.Px1.p1.9.m9.1.2.cmml" xref="S4.SS2.SSS2.Px1.p1.9.m9.1.2"><times id="S4.SS2.SSS2.Px1.p1.9.m9.1.2.1.cmml" xref="S4.SS2.SSS2.Px1.p1.9.m9.1.2.1"></times><ci id="S4.SS2.SSS2.Px1.p1.9.m9.1.2.2.cmml" xref="S4.SS2.SSS2.Px1.p1.9.m9.1.2.2">ℎ</ci><ci id="S4.SS2.SSS2.Px1.p1.9.m9.1.1.cmml" xref="S4.SS2.SSS2.Px1.p1.9.m9.1.1">𝑣</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS2.Px1.p1.9.m9.1c">h(v)</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS2.Px1.p1.9.m9.1d">italic_h ( italic_v )</annotation></semantics></math> or <math alttext="\ell(v)" class="ltx_Math" display="inline" id="S4.SS2.SSS2.Px1.p1.10.m10.1"><semantics id="S4.SS2.SSS2.Px1.p1.10.m10.1a"><mrow id="S4.SS2.SSS2.Px1.p1.10.m10.1.2" xref="S4.SS2.SSS2.Px1.p1.10.m10.1.2.cmml"><mi id="S4.SS2.SSS2.Px1.p1.10.m10.1.2.2" mathvariant="normal" xref="S4.SS2.SSS2.Px1.p1.10.m10.1.2.2.cmml">ℓ</mi><mo id="S4.SS2.SSS2.Px1.p1.10.m10.1.2.1" xref="S4.SS2.SSS2.Px1.p1.10.m10.1.2.1.cmml">⁢</mo><mrow id="S4.SS2.SSS2.Px1.p1.10.m10.1.2.3.2" xref="S4.SS2.SSS2.Px1.p1.10.m10.1.2.cmml"><mo id="S4.SS2.SSS2.Px1.p1.10.m10.1.2.3.2.1" stretchy="false" xref="S4.SS2.SSS2.Px1.p1.10.m10.1.2.cmml">(</mo><mi id="S4.SS2.SSS2.Px1.p1.10.m10.1.1" xref="S4.SS2.SSS2.Px1.p1.10.m10.1.1.cmml">v</mi><mo id="S4.SS2.SSS2.Px1.p1.10.m10.1.2.3.2.2" stretchy="false" xref="S4.SS2.SSS2.Px1.p1.10.m10.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS2.Px1.p1.10.m10.1b"><apply id="S4.SS2.SSS2.Px1.p1.10.m10.1.2.cmml" xref="S4.SS2.SSS2.Px1.p1.10.m10.1.2"><times id="S4.SS2.SSS2.Px1.p1.10.m10.1.2.1.cmml" xref="S4.SS2.SSS2.Px1.p1.10.m10.1.2.1"></times><ci id="S4.SS2.SSS2.Px1.p1.10.m10.1.2.2.cmml" xref="S4.SS2.SSS2.Px1.p1.10.m10.1.2.2">ℓ</ci><ci id="S4.SS2.SSS2.Px1.p1.10.m10.1.1.cmml" xref="S4.SS2.SSS2.Px1.p1.10.m10.1.1">𝑣</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS2.Px1.p1.10.m10.1c">\ell(v)</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS2.Px1.p1.10.m10.1d">roman_ℓ ( italic_v )</annotation></semantics></math>) depending on which virtual edge failure we need to protect against. Specifically, we store the following:</p> <ul class="ltx_itemize" id="S4.I7"> <li class="ltx_item" id="S4.I7.i1" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">•</span> <div class="ltx_para" id="S4.I7.i1.p1"> <p class="ltx_p" id="S4.I7.i1.p1.4">for each tree node <math alttext="x" class="ltx_Math" display="inline" id="S4.I7.i1.p1.1.m1.1"><semantics id="S4.I7.i1.p1.1.m1.1a"><mi id="S4.I7.i1.p1.1.m1.1.1" xref="S4.I7.i1.p1.1.m1.1.1.cmml">x</mi><annotation-xml encoding="MathML-Content" id="S4.I7.i1.p1.1.m1.1b"><ci id="S4.I7.i1.p1.1.m1.1.1.cmml" xref="S4.I7.i1.p1.1.m1.1.1">𝑥</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.I7.i1.p1.1.m1.1c">x</annotation><annotation encoding="application/x-llamapun" id="S4.I7.i1.p1.1.m1.1d">italic_x</annotation></semantics></math>, we store the link <math alttext="uv" class="ltx_Math" display="inline" id="S4.I7.i1.p1.2.m2.1"><semantics id="S4.I7.i1.p1.2.m2.1a"><mrow id="S4.I7.i1.p1.2.m2.1.1" xref="S4.I7.i1.p1.2.m2.1.1.cmml"><mi id="S4.I7.i1.p1.2.m2.1.1.2" xref="S4.I7.i1.p1.2.m2.1.1.2.cmml">u</mi><mo id="S4.I7.i1.p1.2.m2.1.1.1" xref="S4.I7.i1.p1.2.m2.1.1.1.cmml">⁢</mo><mi id="S4.I7.i1.p1.2.m2.1.1.3" xref="S4.I7.i1.p1.2.m2.1.1.3.cmml">v</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.I7.i1.p1.2.m2.1b"><apply id="S4.I7.i1.p1.2.m2.1.1.cmml" xref="S4.I7.i1.p1.2.m2.1.1"><times id="S4.I7.i1.p1.2.m2.1.1.1.cmml" xref="S4.I7.i1.p1.2.m2.1.1.1"></times><ci id="S4.I7.i1.p1.2.m2.1.1.2.cmml" xref="S4.I7.i1.p1.2.m2.1.1.2">𝑢</ci><ci id="S4.I7.i1.p1.2.m2.1.1.3.cmml" xref="S4.I7.i1.p1.2.m2.1.1.3">𝑣</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.I7.i1.p1.2.m2.1c">uv</annotation><annotation encoding="application/x-llamapun" id="S4.I7.i1.p1.2.m2.1d">italic_u italic_v</annotation></semantics></math> with <math alttext="x=h(u)" class="ltx_Math" display="inline" id="S4.I7.i1.p1.3.m3.1"><semantics id="S4.I7.i1.p1.3.m3.1a"><mrow id="S4.I7.i1.p1.3.m3.1.2" xref="S4.I7.i1.p1.3.m3.1.2.cmml"><mi id="S4.I7.i1.p1.3.m3.1.2.2" xref="S4.I7.i1.p1.3.m3.1.2.2.cmml">x</mi><mo id="S4.I7.i1.p1.3.m3.1.2.1" xref="S4.I7.i1.p1.3.m3.1.2.1.cmml">=</mo><mrow id="S4.I7.i1.p1.3.m3.1.2.3" xref="S4.I7.i1.p1.3.m3.1.2.3.cmml"><mi id="S4.I7.i1.p1.3.m3.1.2.3.2" xref="S4.I7.i1.p1.3.m3.1.2.3.2.cmml">h</mi><mo id="S4.I7.i1.p1.3.m3.1.2.3.1" xref="S4.I7.i1.p1.3.m3.1.2.3.1.cmml">⁢</mo><mrow id="S4.I7.i1.p1.3.m3.1.2.3.3.2" xref="S4.I7.i1.p1.3.m3.1.2.3.cmml"><mo id="S4.I7.i1.p1.3.m3.1.2.3.3.2.1" stretchy="false" xref="S4.I7.i1.p1.3.m3.1.2.3.cmml">(</mo><mi id="S4.I7.i1.p1.3.m3.1.1" xref="S4.I7.i1.p1.3.m3.1.1.cmml">u</mi><mo id="S4.I7.i1.p1.3.m3.1.2.3.3.2.2" stretchy="false" xref="S4.I7.i1.p1.3.m3.1.2.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.I7.i1.p1.3.m3.1b"><apply id="S4.I7.i1.p1.3.m3.1.2.cmml" xref="S4.I7.i1.p1.3.m3.1.2"><eq id="S4.I7.i1.p1.3.m3.1.2.1.cmml" xref="S4.I7.i1.p1.3.m3.1.2.1"></eq><ci id="S4.I7.i1.p1.3.m3.1.2.2.cmml" xref="S4.I7.i1.p1.3.m3.1.2.2">𝑥</ci><apply id="S4.I7.i1.p1.3.m3.1.2.3.cmml" xref="S4.I7.i1.p1.3.m3.1.2.3"><times id="S4.I7.i1.p1.3.m3.1.2.3.1.cmml" xref="S4.I7.i1.p1.3.m3.1.2.3.1"></times><ci id="S4.I7.i1.p1.3.m3.1.2.3.2.cmml" xref="S4.I7.i1.p1.3.m3.1.2.3.2">ℎ</ci><ci id="S4.I7.i1.p1.3.m3.1.1.cmml" xref="S4.I7.i1.p1.3.m3.1.1">𝑢</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.I7.i1.p1.3.m3.1c">x=h(u)</annotation><annotation encoding="application/x-llamapun" id="S4.I7.i1.p1.3.m3.1d">italic_x = italic_h ( italic_u )</annotation></semantics></math> minimizing <math alttext="d_{T}(r,\text{LCA}(x,\ell(v)))" class="ltx_Math" display="inline" id="S4.I7.i1.p1.4.m4.4"><semantics id="S4.I7.i1.p1.4.m4.4a"><mrow id="S4.I7.i1.p1.4.m4.4.4" xref="S4.I7.i1.p1.4.m4.4.4.cmml"><msub id="S4.I7.i1.p1.4.m4.4.4.3" xref="S4.I7.i1.p1.4.m4.4.4.3.cmml"><mi id="S4.I7.i1.p1.4.m4.4.4.3.2" xref="S4.I7.i1.p1.4.m4.4.4.3.2.cmml">d</mi><mi id="S4.I7.i1.p1.4.m4.4.4.3.3" xref="S4.I7.i1.p1.4.m4.4.4.3.3.cmml">T</mi></msub><mo id="S4.I7.i1.p1.4.m4.4.4.2" xref="S4.I7.i1.p1.4.m4.4.4.2.cmml">⁢</mo><mrow id="S4.I7.i1.p1.4.m4.4.4.1.1" xref="S4.I7.i1.p1.4.m4.4.4.1.2.cmml"><mo id="S4.I7.i1.p1.4.m4.4.4.1.1.2" stretchy="false" xref="S4.I7.i1.p1.4.m4.4.4.1.2.cmml">(</mo><mi id="S4.I7.i1.p1.4.m4.3.3" xref="S4.I7.i1.p1.4.m4.3.3.cmml">r</mi><mo id="S4.I7.i1.p1.4.m4.4.4.1.1.3" xref="S4.I7.i1.p1.4.m4.4.4.1.2.cmml">,</mo><mrow id="S4.I7.i1.p1.4.m4.4.4.1.1.1" xref="S4.I7.i1.p1.4.m4.4.4.1.1.1.cmml"><mtext id="S4.I7.i1.p1.4.m4.4.4.1.1.1.3" xref="S4.I7.i1.p1.4.m4.4.4.1.1.1.3a.cmml">LCA</mtext><mo id="S4.I7.i1.p1.4.m4.4.4.1.1.1.2" xref="S4.I7.i1.p1.4.m4.4.4.1.1.1.2.cmml">⁢</mo><mrow id="S4.I7.i1.p1.4.m4.4.4.1.1.1.1.1" xref="S4.I7.i1.p1.4.m4.4.4.1.1.1.1.2.cmml"><mo id="S4.I7.i1.p1.4.m4.4.4.1.1.1.1.1.2" stretchy="false" xref="S4.I7.i1.p1.4.m4.4.4.1.1.1.1.2.cmml">(</mo><mi id="S4.I7.i1.p1.4.m4.2.2" xref="S4.I7.i1.p1.4.m4.2.2.cmml">x</mi><mo id="S4.I7.i1.p1.4.m4.4.4.1.1.1.1.1.3" xref="S4.I7.i1.p1.4.m4.4.4.1.1.1.1.2.cmml">,</mo><mrow id="S4.I7.i1.p1.4.m4.4.4.1.1.1.1.1.1" xref="S4.I7.i1.p1.4.m4.4.4.1.1.1.1.1.1.cmml"><mi id="S4.I7.i1.p1.4.m4.4.4.1.1.1.1.1.1.2" mathvariant="normal" xref="S4.I7.i1.p1.4.m4.4.4.1.1.1.1.1.1.2.cmml">ℓ</mi><mo id="S4.I7.i1.p1.4.m4.4.4.1.1.1.1.1.1.1" xref="S4.I7.i1.p1.4.m4.4.4.1.1.1.1.1.1.1.cmml">⁢</mo><mrow id="S4.I7.i1.p1.4.m4.4.4.1.1.1.1.1.1.3.2" xref="S4.I7.i1.p1.4.m4.4.4.1.1.1.1.1.1.cmml"><mo id="S4.I7.i1.p1.4.m4.4.4.1.1.1.1.1.1.3.2.1" stretchy="false" xref="S4.I7.i1.p1.4.m4.4.4.1.1.1.1.1.1.cmml">(</mo><mi id="S4.I7.i1.p1.4.m4.1.1" xref="S4.I7.i1.p1.4.m4.1.1.cmml">v</mi><mo id="S4.I7.i1.p1.4.m4.4.4.1.1.1.1.1.1.3.2.2" stretchy="false" xref="S4.I7.i1.p1.4.m4.4.4.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S4.I7.i1.p1.4.m4.4.4.1.1.1.1.1.4" stretchy="false" xref="S4.I7.i1.p1.4.m4.4.4.1.1.1.1.2.cmml">)</mo></mrow></mrow><mo id="S4.I7.i1.p1.4.m4.4.4.1.1.4" stretchy="false" xref="S4.I7.i1.p1.4.m4.4.4.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.I7.i1.p1.4.m4.4b"><apply id="S4.I7.i1.p1.4.m4.4.4.cmml" xref="S4.I7.i1.p1.4.m4.4.4"><times id="S4.I7.i1.p1.4.m4.4.4.2.cmml" xref="S4.I7.i1.p1.4.m4.4.4.2"></times><apply id="S4.I7.i1.p1.4.m4.4.4.3.cmml" xref="S4.I7.i1.p1.4.m4.4.4.3"><csymbol cd="ambiguous" id="S4.I7.i1.p1.4.m4.4.4.3.1.cmml" xref="S4.I7.i1.p1.4.m4.4.4.3">subscript</csymbol><ci id="S4.I7.i1.p1.4.m4.4.4.3.2.cmml" xref="S4.I7.i1.p1.4.m4.4.4.3.2">𝑑</ci><ci id="S4.I7.i1.p1.4.m4.4.4.3.3.cmml" xref="S4.I7.i1.p1.4.m4.4.4.3.3">𝑇</ci></apply><interval closure="open" id="S4.I7.i1.p1.4.m4.4.4.1.2.cmml" xref="S4.I7.i1.p1.4.m4.4.4.1.1"><ci id="S4.I7.i1.p1.4.m4.3.3.cmml" xref="S4.I7.i1.p1.4.m4.3.3">𝑟</ci><apply id="S4.I7.i1.p1.4.m4.4.4.1.1.1.cmml" xref="S4.I7.i1.p1.4.m4.4.4.1.1.1"><times id="S4.I7.i1.p1.4.m4.4.4.1.1.1.2.cmml" xref="S4.I7.i1.p1.4.m4.4.4.1.1.1.2"></times><ci id="S4.I7.i1.p1.4.m4.4.4.1.1.1.3a.cmml" xref="S4.I7.i1.p1.4.m4.4.4.1.1.1.3"><mtext id="S4.I7.i1.p1.4.m4.4.4.1.1.1.3.cmml" xref="S4.I7.i1.p1.4.m4.4.4.1.1.1.3">LCA</mtext></ci><interval closure="open" id="S4.I7.i1.p1.4.m4.4.4.1.1.1.1.2.cmml" xref="S4.I7.i1.p1.4.m4.4.4.1.1.1.1.1"><ci id="S4.I7.i1.p1.4.m4.2.2.cmml" xref="S4.I7.i1.p1.4.m4.2.2">𝑥</ci><apply id="S4.I7.i1.p1.4.m4.4.4.1.1.1.1.1.1.cmml" xref="S4.I7.i1.p1.4.m4.4.4.1.1.1.1.1.1"><times id="S4.I7.i1.p1.4.m4.4.4.1.1.1.1.1.1.1.cmml" xref="S4.I7.i1.p1.4.m4.4.4.1.1.1.1.1.1.1"></times><ci id="S4.I7.i1.p1.4.m4.4.4.1.1.1.1.1.1.2.cmml" xref="S4.I7.i1.p1.4.m4.4.4.1.1.1.1.1.1.2">ℓ</ci><ci id="S4.I7.i1.p1.4.m4.1.1.cmml" xref="S4.I7.i1.p1.4.m4.1.1">𝑣</ci></apply></interval></apply></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.I7.i1.p1.4.m4.4c">d_{T}(r,\text{LCA}(x,\ell(v)))</annotation><annotation encoding="application/x-llamapun" id="S4.I7.i1.p1.4.m4.4d">italic_d start_POSTSUBSCRIPT italic_T end_POSTSUBSCRIPT ( italic_r , LCA ( italic_x , roman_ℓ ( italic_v ) ) )</annotation></semantics></math>;</p> </div> </li> <li class="ltx_item" id="S4.I7.i2" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">•</span> <div class="ltx_para" id="S4.I7.i2.p1"> <p class="ltx_p" id="S4.I7.i2.p1.7">for each P-node <math alttext="x" class="ltx_Math" display="inline" id="S4.I7.i2.p1.1.m1.1"><semantics id="S4.I7.i2.p1.1.m1.1a"><mi id="S4.I7.i2.p1.1.m1.1.1" xref="S4.I7.i2.p1.1.m1.1.1.cmml">x</mi><annotation-xml encoding="MathML-Content" id="S4.I7.i2.p1.1.m1.1b"><ci id="S4.I7.i2.p1.1.m1.1.1.cmml" xref="S4.I7.i2.p1.1.m1.1.1">𝑥</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.I7.i2.p1.1.m1.1c">x</annotation><annotation encoding="application/x-llamapun" id="S4.I7.i2.p1.1.m1.1d">italic_x</annotation></semantics></math>, we construct the following contracted graph: for each child <math alttext="y" class="ltx_Math" display="inline" id="S4.I7.i2.p1.2.m2.1"><semantics id="S4.I7.i2.p1.2.m2.1a"><mi id="S4.I7.i2.p1.2.m2.1.1" xref="S4.I7.i2.p1.2.m2.1.1.cmml">y</mi><annotation-xml encoding="MathML-Content" id="S4.I7.i2.p1.2.m2.1b"><ci id="S4.I7.i2.p1.2.m2.1.1.cmml" xref="S4.I7.i2.p1.2.m2.1.1">𝑦</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.I7.i2.p1.2.m2.1c">y</annotation><annotation encoding="application/x-llamapun" id="S4.I7.i2.p1.2.m2.1d">italic_y</annotation></semantics></math> of <math alttext="x" class="ltx_Math" display="inline" id="S4.I7.i2.p1.3.m3.1"><semantics id="S4.I7.i2.p1.3.m3.1a"><mi id="S4.I7.i2.p1.3.m3.1.1" xref="S4.I7.i2.p1.3.m3.1.1.cmml">x</mi><annotation-xml encoding="MathML-Content" id="S4.I7.i2.p1.3.m3.1b"><ci id="S4.I7.i2.p1.3.m3.1.1.cmml" xref="S4.I7.i2.p1.3.m3.1.1">𝑥</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.I7.i2.p1.3.m3.1c">x</annotation><annotation encoding="application/x-llamapun" id="S4.I7.i2.p1.3.m3.1d">italic_x</annotation></semantics></math>, we contract <math alttext="(\cup_{z\in T_{y}}V(G_{z}))\setminus V(G_{x})" class="ltx_Math" display="inline" id="S4.I7.i2.p1.4.m4.2"><semantics id="S4.I7.i2.p1.4.m4.2a"><mrow id="S4.I7.i2.p1.4.m4.2.2" xref="S4.I7.i2.p1.4.m4.2.2.cmml"><mrow id="S4.I7.i2.p1.4.m4.1.1.1.1" xref="S4.I7.i2.p1.4.m4.1.1.1.1.1.cmml"><mo id="S4.I7.i2.p1.4.m4.1.1.1.1.2" stretchy="false" xref="S4.I7.i2.p1.4.m4.1.1.1.1.1.cmml">(</mo><mrow id="S4.I7.i2.p1.4.m4.1.1.1.1.1" xref="S4.I7.i2.p1.4.m4.1.1.1.1.1.cmml"><msub id="S4.I7.i2.p1.4.m4.1.1.1.1.1.2" xref="S4.I7.i2.p1.4.m4.1.1.1.1.1.2.cmml"><mo id="S4.I7.i2.p1.4.m4.1.1.1.1.1.2.2" lspace="0em" xref="S4.I7.i2.p1.4.m4.1.1.1.1.1.2.2.cmml">∪</mo><mrow id="S4.I7.i2.p1.4.m4.1.1.1.1.1.2.3" xref="S4.I7.i2.p1.4.m4.1.1.1.1.1.2.3.cmml"><mi id="S4.I7.i2.p1.4.m4.1.1.1.1.1.2.3.2" xref="S4.I7.i2.p1.4.m4.1.1.1.1.1.2.3.2.cmml">z</mi><mo id="S4.I7.i2.p1.4.m4.1.1.1.1.1.2.3.1" xref="S4.I7.i2.p1.4.m4.1.1.1.1.1.2.3.1.cmml">∈</mo><msub id="S4.I7.i2.p1.4.m4.1.1.1.1.1.2.3.3" xref="S4.I7.i2.p1.4.m4.1.1.1.1.1.2.3.3.cmml"><mi 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xref="S4.I7.i2.p1.4.m4.1.1.1.1.1.1.1.1.1.2">𝐺</ci><ci id="S4.I7.i2.p1.4.m4.1.1.1.1.1.1.1.1.1.3.cmml" xref="S4.I7.i2.p1.4.m4.1.1.1.1.1.1.1.1.1.3">𝑧</ci></apply></apply></apply><apply id="S4.I7.i2.p1.4.m4.2.2.2.cmml" xref="S4.I7.i2.p1.4.m4.2.2.2"><times id="S4.I7.i2.p1.4.m4.2.2.2.2.cmml" xref="S4.I7.i2.p1.4.m4.2.2.2.2"></times><ci id="S4.I7.i2.p1.4.m4.2.2.2.3.cmml" xref="S4.I7.i2.p1.4.m4.2.2.2.3">𝑉</ci><apply id="S4.I7.i2.p1.4.m4.2.2.2.1.1.1.cmml" xref="S4.I7.i2.p1.4.m4.2.2.2.1.1"><csymbol cd="ambiguous" id="S4.I7.i2.p1.4.m4.2.2.2.1.1.1.1.cmml" xref="S4.I7.i2.p1.4.m4.2.2.2.1.1">subscript</csymbol><ci id="S4.I7.i2.p1.4.m4.2.2.2.1.1.1.2.cmml" xref="S4.I7.i2.p1.4.m4.2.2.2.1.1.1.2">𝐺</ci><ci id="S4.I7.i2.p1.4.m4.2.2.2.1.1.1.3.cmml" xref="S4.I7.i2.p1.4.m4.2.2.2.1.1.1.3">𝑥</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.I7.i2.p1.4.m4.2c">(\cup_{z\in T_{y}}V(G_{z}))\setminus V(G_{x})</annotation><annotation encoding="application/x-llamapun" id="S4.I7.i2.p1.4.m4.2d">( ∪ start_POSTSUBSCRIPT italic_z ∈ italic_T start_POSTSUBSCRIPT italic_y end_POSTSUBSCRIPT end_POSTSUBSCRIPT italic_V ( italic_G start_POSTSUBSCRIPT italic_z end_POSTSUBSCRIPT ) ) ∖ italic_V ( italic_G start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT )</annotation></semantics></math>; this is the set of vertices of <math alttext="G" class="ltx_Math" display="inline" id="S4.I7.i2.p1.5.m5.1"><semantics id="S4.I7.i2.p1.5.m5.1a"><mi id="S4.I7.i2.p1.5.m5.1.1" xref="S4.I7.i2.p1.5.m5.1.1.cmml">G</mi><annotation-xml encoding="MathML-Content" id="S4.I7.i2.p1.5.m5.1b"><ci id="S4.I7.i2.p1.5.m5.1.1.cmml" xref="S4.I7.i2.p1.5.m5.1.1">𝐺</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.I7.i2.p1.5.m5.1c">G</annotation><annotation encoding="application/x-llamapun" id="S4.I7.i2.p1.5.m5.1d">italic_G</annotation></semantics></math> in the subtree <math alttext="T_{y}" class="ltx_Math" display="inline" id="S4.I7.i2.p1.6.m6.1"><semantics id="S4.I7.i2.p1.6.m6.1a"><msub id="S4.I7.i2.p1.6.m6.1.1" xref="S4.I7.i2.p1.6.m6.1.1.cmml"><mi id="S4.I7.i2.p1.6.m6.1.1.2" xref="S4.I7.i2.p1.6.m6.1.1.2.cmml">T</mi><mi id="S4.I7.i2.p1.6.m6.1.1.3" xref="S4.I7.i2.p1.6.m6.1.1.3.cmml">y</mi></msub><annotation-xml encoding="MathML-Content" id="S4.I7.i2.p1.6.m6.1b"><apply id="S4.I7.i2.p1.6.m6.1.1.cmml" xref="S4.I7.i2.p1.6.m6.1.1"><csymbol cd="ambiguous" id="S4.I7.i2.p1.6.m6.1.1.1.cmml" xref="S4.I7.i2.p1.6.m6.1.1">subscript</csymbol><ci id="S4.I7.i2.p1.6.m6.1.1.2.cmml" xref="S4.I7.i2.p1.6.m6.1.1.2">𝑇</ci><ci id="S4.I7.i2.p1.6.m6.1.1.3.cmml" xref="S4.I7.i2.p1.6.m6.1.1.3">𝑦</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.I7.i2.p1.6.m6.1c">T_{y}</annotation><annotation encoding="application/x-llamapun" id="S4.I7.i2.p1.6.m6.1d">italic_T start_POSTSUBSCRIPT italic_y end_POSTSUBSCRIPT</annotation></semantics></math> <em class="ltx_emph ltx_font_italic" id="S4.I7.i2.p1.7.1">not</em> including the two vertices in <math alttext="G_{x}" class="ltx_Math" display="inline" id="S4.I7.i2.p1.7.m7.1"><semantics id="S4.I7.i2.p1.7.m7.1a"><msub id="S4.I7.i2.p1.7.m7.1.1" xref="S4.I7.i2.p1.7.m7.1.1.cmml"><mi id="S4.I7.i2.p1.7.m7.1.1.2" xref="S4.I7.i2.p1.7.m7.1.1.2.cmml">G</mi><mi id="S4.I7.i2.p1.7.m7.1.1.3" xref="S4.I7.i2.p1.7.m7.1.1.3.cmml">x</mi></msub><annotation-xml encoding="MathML-Content" id="S4.I7.i2.p1.7.m7.1b"><apply id="S4.I7.i2.p1.7.m7.1.1.cmml" xref="S4.I7.i2.p1.7.m7.1.1"><csymbol cd="ambiguous" id="S4.I7.i2.p1.7.m7.1.1.1.cmml" xref="S4.I7.i2.p1.7.m7.1.1">subscript</csymbol><ci id="S4.I7.i2.p1.7.m7.1.1.2.cmml" xref="S4.I7.i2.p1.7.m7.1.1.2">𝐺</ci><ci id="S4.I7.i2.p1.7.m7.1.1.3.cmml" xref="S4.I7.i2.p1.7.m7.1.1.3">𝑥</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.I7.i2.p1.7.m7.1c">G_{x}</annotation><annotation encoding="application/x-llamapun" id="S4.I7.i2.p1.7.m7.1d">italic_G start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT</annotation></semantics></math>. We store an MST on this contracted graph.</p> </div> </li> </ul> </div> </section> <section class="ltx_paragraph" id="S4.SS2.SSS2.Px2"> <h5 class="ltx_title ltx_title_paragraph">“Cycle” Cuts:</h5> <div class="ltx_para" id="S4.SS2.SSS2.Px2.p1"> <p class="ltx_p" id="S4.SS2.SSS2.Px2.p1.16">To handle 2-vertex cuts of type (3), we build on ideas developed by <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.00712v1#bib.bibx61" title="">KT93</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.00712v1#bib.bibx62" title="">KV94</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.00712v1#bib.bibx54" title="">JKMV24</a>]</cite> for augmenting a cycle graph to be 3-<em class="ltx_emph ltx_font_italic" id="S4.SS2.SSS2.Px2.p1.16.1">edge</em>-connected. In the unweighted setting, the idea is simple. Suppose we are given some cycle <math alttext="C=\{1,\dots,n\}" class="ltx_Math" display="inline" id="S4.SS2.SSS2.Px2.p1.1.m1.3"><semantics id="S4.SS2.SSS2.Px2.p1.1.m1.3a"><mrow id="S4.SS2.SSS2.Px2.p1.1.m1.3.4" xref="S4.SS2.SSS2.Px2.p1.1.m1.3.4.cmml"><mi id="S4.SS2.SSS2.Px2.p1.1.m1.3.4.2" xref="S4.SS2.SSS2.Px2.p1.1.m1.3.4.2.cmml">C</mi><mo id="S4.SS2.SSS2.Px2.p1.1.m1.3.4.1" xref="S4.SS2.SSS2.Px2.p1.1.m1.3.4.1.cmml">=</mo><mrow id="S4.SS2.SSS2.Px2.p1.1.m1.3.4.3.2" xref="S4.SS2.SSS2.Px2.p1.1.m1.3.4.3.1.cmml"><mo id="S4.SS2.SSS2.Px2.p1.1.m1.3.4.3.2.1" stretchy="false" xref="S4.SS2.SSS2.Px2.p1.1.m1.3.4.3.1.cmml">{</mo><mn id="S4.SS2.SSS2.Px2.p1.1.m1.1.1" xref="S4.SS2.SSS2.Px2.p1.1.m1.1.1.cmml">1</mn><mo id="S4.SS2.SSS2.Px2.p1.1.m1.3.4.3.2.2" xref="S4.SS2.SSS2.Px2.p1.1.m1.3.4.3.1.cmml">,</mo><mi id="S4.SS2.SSS2.Px2.p1.1.m1.2.2" mathvariant="normal" xref="S4.SS2.SSS2.Px2.p1.1.m1.2.2.cmml">…</mi><mo id="S4.SS2.SSS2.Px2.p1.1.m1.3.4.3.2.3" xref="S4.SS2.SSS2.Px2.p1.1.m1.3.4.3.1.cmml">,</mo><mi id="S4.SS2.SSS2.Px2.p1.1.m1.3.3" xref="S4.SS2.SSS2.Px2.p1.1.m1.3.3.cmml">n</mi><mo id="S4.SS2.SSS2.Px2.p1.1.m1.3.4.3.2.4" stretchy="false" xref="S4.SS2.SSS2.Px2.p1.1.m1.3.4.3.1.cmml">}</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS2.Px2.p1.1.m1.3b"><apply id="S4.SS2.SSS2.Px2.p1.1.m1.3.4.cmml" xref="S4.SS2.SSS2.Px2.p1.1.m1.3.4"><eq id="S4.SS2.SSS2.Px2.p1.1.m1.3.4.1.cmml" xref="S4.SS2.SSS2.Px2.p1.1.m1.3.4.1"></eq><ci id="S4.SS2.SSS2.Px2.p1.1.m1.3.4.2.cmml" xref="S4.SS2.SSS2.Px2.p1.1.m1.3.4.2">𝐶</ci><set id="S4.SS2.SSS2.Px2.p1.1.m1.3.4.3.1.cmml" xref="S4.SS2.SSS2.Px2.p1.1.m1.3.4.3.2"><cn id="S4.SS2.SSS2.Px2.p1.1.m1.1.1.cmml" type="integer" xref="S4.SS2.SSS2.Px2.p1.1.m1.1.1">1</cn><ci id="S4.SS2.SSS2.Px2.p1.1.m1.2.2.cmml" xref="S4.SS2.SSS2.Px2.p1.1.m1.2.2">…</ci><ci id="S4.SS2.SSS2.Px2.p1.1.m1.3.3.cmml" xref="S4.SS2.SSS2.Px2.p1.1.m1.3.3">𝑛</ci></set></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS2.Px2.p1.1.m1.3c">C=\{1,\dots,n\}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS2.Px2.p1.1.m1.3d">italic_C = { 1 , … , italic_n }</annotation></semantics></math>. Consider bidirecting all links, that is, each link <math alttext="uv" class="ltx_Math" display="inline" id="S4.SS2.SSS2.Px2.p1.2.m2.1"><semantics id="S4.SS2.SSS2.Px2.p1.2.m2.1a"><mrow id="S4.SS2.SSS2.Px2.p1.2.m2.1.1" xref="S4.SS2.SSS2.Px2.p1.2.m2.1.1.cmml"><mi id="S4.SS2.SSS2.Px2.p1.2.m2.1.1.2" xref="S4.SS2.SSS2.Px2.p1.2.m2.1.1.2.cmml">u</mi><mo id="S4.SS2.SSS2.Px2.p1.2.m2.1.1.1" xref="S4.SS2.SSS2.Px2.p1.2.m2.1.1.1.cmml">⁢</mo><mi id="S4.SS2.SSS2.Px2.p1.2.m2.1.1.3" xref="S4.SS2.SSS2.Px2.p1.2.m2.1.1.3.cmml">v</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS2.Px2.p1.2.m2.1b"><apply id="S4.SS2.SSS2.Px2.p1.2.m2.1.1.cmml" xref="S4.SS2.SSS2.Px2.p1.2.m2.1.1"><times id="S4.SS2.SSS2.Px2.p1.2.m2.1.1.1.cmml" xref="S4.SS2.SSS2.Px2.p1.2.m2.1.1.1"></times><ci id="S4.SS2.SSS2.Px2.p1.2.m2.1.1.2.cmml" xref="S4.SS2.SSS2.Px2.p1.2.m2.1.1.2">𝑢</ci><ci id="S4.SS2.SSS2.Px2.p1.2.m2.1.1.3.cmml" xref="S4.SS2.SSS2.Px2.p1.2.m2.1.1.3">𝑣</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS2.Px2.p1.2.m2.1c">uv</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS2.Px2.p1.2.m2.1d">italic_u italic_v</annotation></semantics></math> is replaced with two <em class="ltx_emph ltx_font_italic" id="S4.SS2.SSS2.Px2.p1.16.2">directed</em> links <math alttext="(u,v)" class="ltx_Math" display="inline" id="S4.SS2.SSS2.Px2.p1.3.m3.2"><semantics id="S4.SS2.SSS2.Px2.p1.3.m3.2a"><mrow id="S4.SS2.SSS2.Px2.p1.3.m3.2.3.2" xref="S4.SS2.SSS2.Px2.p1.3.m3.2.3.1.cmml"><mo id="S4.SS2.SSS2.Px2.p1.3.m3.2.3.2.1" stretchy="false" xref="S4.SS2.SSS2.Px2.p1.3.m3.2.3.1.cmml">(</mo><mi id="S4.SS2.SSS2.Px2.p1.3.m3.1.1" xref="S4.SS2.SSS2.Px2.p1.3.m3.1.1.cmml">u</mi><mo id="S4.SS2.SSS2.Px2.p1.3.m3.2.3.2.2" xref="S4.SS2.SSS2.Px2.p1.3.m3.2.3.1.cmml">,</mo><mi id="S4.SS2.SSS2.Px2.p1.3.m3.2.2" xref="S4.SS2.SSS2.Px2.p1.3.m3.2.2.cmml">v</mi><mo id="S4.SS2.SSS2.Px2.p1.3.m3.2.3.2.3" stretchy="false" xref="S4.SS2.SSS2.Px2.p1.3.m3.2.3.1.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS2.Px2.p1.3.m3.2b"><interval closure="open" id="S4.SS2.SSS2.Px2.p1.3.m3.2.3.1.cmml" xref="S4.SS2.SSS2.Px2.p1.3.m3.2.3.2"><ci id="S4.SS2.SSS2.Px2.p1.3.m3.1.1.cmml" xref="S4.SS2.SSS2.Px2.p1.3.m3.1.1">𝑢</ci><ci id="S4.SS2.SSS2.Px2.p1.3.m3.2.2.cmml" xref="S4.SS2.SSS2.Px2.p1.3.m3.2.2">𝑣</ci></interval></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS2.Px2.p1.3.m3.2c">(u,v)</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS2.Px2.p1.3.m3.2d">( italic_u , italic_v )</annotation></semantics></math> and <math alttext="(v,u)" class="ltx_Math" display="inline" id="S4.SS2.SSS2.Px2.p1.4.m4.2"><semantics id="S4.SS2.SSS2.Px2.p1.4.m4.2a"><mrow id="S4.SS2.SSS2.Px2.p1.4.m4.2.3.2" xref="S4.SS2.SSS2.Px2.p1.4.m4.2.3.1.cmml"><mo id="S4.SS2.SSS2.Px2.p1.4.m4.2.3.2.1" stretchy="false" xref="S4.SS2.SSS2.Px2.p1.4.m4.2.3.1.cmml">(</mo><mi id="S4.SS2.SSS2.Px2.p1.4.m4.1.1" xref="S4.SS2.SSS2.Px2.p1.4.m4.1.1.cmml">v</mi><mo id="S4.SS2.SSS2.Px2.p1.4.m4.2.3.2.2" xref="S4.SS2.SSS2.Px2.p1.4.m4.2.3.1.cmml">,</mo><mi id="S4.SS2.SSS2.Px2.p1.4.m4.2.2" xref="S4.SS2.SSS2.Px2.p1.4.m4.2.2.cmml">u</mi><mo id="S4.SS2.SSS2.Px2.p1.4.m4.2.3.2.3" stretchy="false" xref="S4.SS2.SSS2.Px2.p1.4.m4.2.3.1.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS2.Px2.p1.4.m4.2b"><interval closure="open" id="S4.SS2.SSS2.Px2.p1.4.m4.2.3.1.cmml" xref="S4.SS2.SSS2.Px2.p1.4.m4.2.3.2"><ci id="S4.SS2.SSS2.Px2.p1.4.m4.1.1.cmml" xref="S4.SS2.SSS2.Px2.p1.4.m4.1.1">𝑣</ci><ci id="S4.SS2.SSS2.Px2.p1.4.m4.2.2.cmml" xref="S4.SS2.SSS2.Px2.p1.4.m4.2.2">𝑢</ci></interval></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS2.Px2.p1.4.m4.2c">(v,u)</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS2.Px2.p1.4.m4.2d">( italic_v , italic_u )</annotation></semantics></math>. The goal is, for every interval <math alttext="[i,j]" class="ltx_Math" display="inline" id="S4.SS2.SSS2.Px2.p1.5.m5.2"><semantics id="S4.SS2.SSS2.Px2.p1.5.m5.2a"><mrow id="S4.SS2.SSS2.Px2.p1.5.m5.2.3.2" xref="S4.SS2.SSS2.Px2.p1.5.m5.2.3.1.cmml"><mo id="S4.SS2.SSS2.Px2.p1.5.m5.2.3.2.1" stretchy="false" xref="S4.SS2.SSS2.Px2.p1.5.m5.2.3.1.cmml">[</mo><mi id="S4.SS2.SSS2.Px2.p1.5.m5.1.1" xref="S4.SS2.SSS2.Px2.p1.5.m5.1.1.cmml">i</mi><mo id="S4.SS2.SSS2.Px2.p1.5.m5.2.3.2.2" xref="S4.SS2.SSS2.Px2.p1.5.m5.2.3.1.cmml">,</mo><mi id="S4.SS2.SSS2.Px2.p1.5.m5.2.2" xref="S4.SS2.SSS2.Px2.p1.5.m5.2.2.cmml">j</mi><mo id="S4.SS2.SSS2.Px2.p1.5.m5.2.3.2.3" stretchy="false" xref="S4.SS2.SSS2.Px2.p1.5.m5.2.3.1.cmml">]</mo></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS2.Px2.p1.5.m5.2b"><interval closure="closed" id="S4.SS2.SSS2.Px2.p1.5.m5.2.3.1.cmml" xref="S4.SS2.SSS2.Px2.p1.5.m5.2.3.2"><ci id="S4.SS2.SSS2.Px2.p1.5.m5.1.1.cmml" xref="S4.SS2.SSS2.Px2.p1.5.m5.1.1">𝑖</ci><ci id="S4.SS2.SSS2.Px2.p1.5.m5.2.2.cmml" xref="S4.SS2.SSS2.Px2.p1.5.m5.2.2">𝑗</ci></interval></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS2.Px2.p1.5.m5.2c">[i,j]</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS2.Px2.p1.5.m5.2d">[ italic_i , italic_j ]</annotation></semantics></math>, to include at least one link <math alttext="(u,v)" class="ltx_Math" display="inline" id="S4.SS2.SSS2.Px2.p1.6.m6.2"><semantics id="S4.SS2.SSS2.Px2.p1.6.m6.2a"><mrow id="S4.SS2.SSS2.Px2.p1.6.m6.2.3.2" xref="S4.SS2.SSS2.Px2.p1.6.m6.2.3.1.cmml"><mo id="S4.SS2.SSS2.Px2.p1.6.m6.2.3.2.1" stretchy="false" xref="S4.SS2.SSS2.Px2.p1.6.m6.2.3.1.cmml">(</mo><mi id="S4.SS2.SSS2.Px2.p1.6.m6.1.1" xref="S4.SS2.SSS2.Px2.p1.6.m6.1.1.cmml">u</mi><mo id="S4.SS2.SSS2.Px2.p1.6.m6.2.3.2.2" xref="S4.SS2.SSS2.Px2.p1.6.m6.2.3.1.cmml">,</mo><mi id="S4.SS2.SSS2.Px2.p1.6.m6.2.2" xref="S4.SS2.SSS2.Px2.p1.6.m6.2.2.cmml">v</mi><mo id="S4.SS2.SSS2.Px2.p1.6.m6.2.3.2.3" stretchy="false" xref="S4.SS2.SSS2.Px2.p1.6.m6.2.3.1.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS2.Px2.p1.6.m6.2b"><interval closure="open" id="S4.SS2.SSS2.Px2.p1.6.m6.2.3.1.cmml" xref="S4.SS2.SSS2.Px2.p1.6.m6.2.3.2"><ci id="S4.SS2.SSS2.Px2.p1.6.m6.1.1.cmml" xref="S4.SS2.SSS2.Px2.p1.6.m6.1.1">𝑢</ci><ci id="S4.SS2.SSS2.Px2.p1.6.m6.2.2.cmml" xref="S4.SS2.SSS2.Px2.p1.6.m6.2.2">𝑣</ci></interval></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS2.Px2.p1.6.m6.2c">(u,v)</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS2.Px2.p1.6.m6.2d">( italic_u , italic_v )</annotation></semantics></math> such that <math alttext="u\notin[i,j]" class="ltx_Math" display="inline" id="S4.SS2.SSS2.Px2.p1.7.m7.2"><semantics id="S4.SS2.SSS2.Px2.p1.7.m7.2a"><mrow id="S4.SS2.SSS2.Px2.p1.7.m7.2.3" xref="S4.SS2.SSS2.Px2.p1.7.m7.2.3.cmml"><mi id="S4.SS2.SSS2.Px2.p1.7.m7.2.3.2" xref="S4.SS2.SSS2.Px2.p1.7.m7.2.3.2.cmml">u</mi><mo id="S4.SS2.SSS2.Px2.p1.7.m7.2.3.1" xref="S4.SS2.SSS2.Px2.p1.7.m7.2.3.1.cmml">∉</mo><mrow id="S4.SS2.SSS2.Px2.p1.7.m7.2.3.3.2" xref="S4.SS2.SSS2.Px2.p1.7.m7.2.3.3.1.cmml"><mo id="S4.SS2.SSS2.Px2.p1.7.m7.2.3.3.2.1" stretchy="false" xref="S4.SS2.SSS2.Px2.p1.7.m7.2.3.3.1.cmml">[</mo><mi id="S4.SS2.SSS2.Px2.p1.7.m7.1.1" xref="S4.SS2.SSS2.Px2.p1.7.m7.1.1.cmml">i</mi><mo id="S4.SS2.SSS2.Px2.p1.7.m7.2.3.3.2.2" xref="S4.SS2.SSS2.Px2.p1.7.m7.2.3.3.1.cmml">,</mo><mi id="S4.SS2.SSS2.Px2.p1.7.m7.2.2" xref="S4.SS2.SSS2.Px2.p1.7.m7.2.2.cmml">j</mi><mo id="S4.SS2.SSS2.Px2.p1.7.m7.2.3.3.2.3" stretchy="false" xref="S4.SS2.SSS2.Px2.p1.7.m7.2.3.3.1.cmml">]</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS2.Px2.p1.7.m7.2b"><apply id="S4.SS2.SSS2.Px2.p1.7.m7.2.3.cmml" xref="S4.SS2.SSS2.Px2.p1.7.m7.2.3"><notin id="S4.SS2.SSS2.Px2.p1.7.m7.2.3.1.cmml" xref="S4.SS2.SSS2.Px2.p1.7.m7.2.3.1"></notin><ci id="S4.SS2.SSS2.Px2.p1.7.m7.2.3.2.cmml" xref="S4.SS2.SSS2.Px2.p1.7.m7.2.3.2">𝑢</ci><interval closure="closed" id="S4.SS2.SSS2.Px2.p1.7.m7.2.3.3.1.cmml" xref="S4.SS2.SSS2.Px2.p1.7.m7.2.3.3.2"><ci id="S4.SS2.SSS2.Px2.p1.7.m7.1.1.cmml" xref="S4.SS2.SSS2.Px2.p1.7.m7.1.1">𝑖</ci><ci id="S4.SS2.SSS2.Px2.p1.7.m7.2.2.cmml" xref="S4.SS2.SSS2.Px2.p1.7.m7.2.2">𝑗</ci></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS2.Px2.p1.7.m7.2c">u\notin[i,j]</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS2.Px2.p1.7.m7.2d">italic_u ∉ [ italic_i , italic_j ]</annotation></semantics></math> and <math alttext="v\in[i,j]" class="ltx_Math" display="inline" id="S4.SS2.SSS2.Px2.p1.8.m8.2"><semantics id="S4.SS2.SSS2.Px2.p1.8.m8.2a"><mrow id="S4.SS2.SSS2.Px2.p1.8.m8.2.3" xref="S4.SS2.SSS2.Px2.p1.8.m8.2.3.cmml"><mi id="S4.SS2.SSS2.Px2.p1.8.m8.2.3.2" xref="S4.SS2.SSS2.Px2.p1.8.m8.2.3.2.cmml">v</mi><mo id="S4.SS2.SSS2.Px2.p1.8.m8.2.3.1" xref="S4.SS2.SSS2.Px2.p1.8.m8.2.3.1.cmml">∈</mo><mrow id="S4.SS2.SSS2.Px2.p1.8.m8.2.3.3.2" xref="S4.SS2.SSS2.Px2.p1.8.m8.2.3.3.1.cmml"><mo id="S4.SS2.SSS2.Px2.p1.8.m8.2.3.3.2.1" stretchy="false" xref="S4.SS2.SSS2.Px2.p1.8.m8.2.3.3.1.cmml">[</mo><mi id="S4.SS2.SSS2.Px2.p1.8.m8.1.1" xref="S4.SS2.SSS2.Px2.p1.8.m8.1.1.cmml">i</mi><mo id="S4.SS2.SSS2.Px2.p1.8.m8.2.3.3.2.2" xref="S4.SS2.SSS2.Px2.p1.8.m8.2.3.3.1.cmml">,</mo><mi id="S4.SS2.SSS2.Px2.p1.8.m8.2.2" xref="S4.SS2.SSS2.Px2.p1.8.m8.2.2.cmml">j</mi><mo id="S4.SS2.SSS2.Px2.p1.8.m8.2.3.3.2.3" stretchy="false" xref="S4.SS2.SSS2.Px2.p1.8.m8.2.3.3.1.cmml">]</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS2.Px2.p1.8.m8.2b"><apply id="S4.SS2.SSS2.Px2.p1.8.m8.2.3.cmml" xref="S4.SS2.SSS2.Px2.p1.8.m8.2.3"><in id="S4.SS2.SSS2.Px2.p1.8.m8.2.3.1.cmml" xref="S4.SS2.SSS2.Px2.p1.8.m8.2.3.1"></in><ci id="S4.SS2.SSS2.Px2.p1.8.m8.2.3.2.cmml" xref="S4.SS2.SSS2.Px2.p1.8.m8.2.3.2">𝑣</ci><interval closure="closed" id="S4.SS2.SSS2.Px2.p1.8.m8.2.3.3.1.cmml" xref="S4.SS2.SSS2.Px2.p1.8.m8.2.3.3.2"><ci id="S4.SS2.SSS2.Px2.p1.8.m8.1.1.cmml" xref="S4.SS2.SSS2.Px2.p1.8.m8.1.1">𝑖</ci><ci id="S4.SS2.SSS2.Px2.p1.8.m8.2.2.cmml" xref="S4.SS2.SSS2.Px2.p1.8.m8.2.2">𝑗</ci></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS2.Px2.p1.8.m8.2c">v\in[i,j]</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS2.Px2.p1.8.m8.2d">italic_v ∈ [ italic_i , italic_j ]</annotation></semantics></math>: this corresponds to covering the cut <math alttext="\{(i-1,i),(j,j+1)\}" class="ltx_Math" display="inline" id="S4.SS2.SSS2.Px2.p1.9.m9.4"><semantics id="S4.SS2.SSS2.Px2.p1.9.m9.4a"><mrow id="S4.SS2.SSS2.Px2.p1.9.m9.4.4.2" xref="S4.SS2.SSS2.Px2.p1.9.m9.4.4.3.cmml"><mo id="S4.SS2.SSS2.Px2.p1.9.m9.4.4.2.3" stretchy="false" xref="S4.SS2.SSS2.Px2.p1.9.m9.4.4.3.cmml">{</mo><mrow id="S4.SS2.SSS2.Px2.p1.9.m9.3.3.1.1.1" xref="S4.SS2.SSS2.Px2.p1.9.m9.3.3.1.1.2.cmml"><mo id="S4.SS2.SSS2.Px2.p1.9.m9.3.3.1.1.1.2" stretchy="false" xref="S4.SS2.SSS2.Px2.p1.9.m9.3.3.1.1.2.cmml">(</mo><mrow id="S4.SS2.SSS2.Px2.p1.9.m9.3.3.1.1.1.1" xref="S4.SS2.SSS2.Px2.p1.9.m9.3.3.1.1.1.1.cmml"><mi id="S4.SS2.SSS2.Px2.p1.9.m9.3.3.1.1.1.1.2" xref="S4.SS2.SSS2.Px2.p1.9.m9.3.3.1.1.1.1.2.cmml">i</mi><mo id="S4.SS2.SSS2.Px2.p1.9.m9.3.3.1.1.1.1.1" xref="S4.SS2.SSS2.Px2.p1.9.m9.3.3.1.1.1.1.1.cmml">−</mo><mn id="S4.SS2.SSS2.Px2.p1.9.m9.3.3.1.1.1.1.3" xref="S4.SS2.SSS2.Px2.p1.9.m9.3.3.1.1.1.1.3.cmml">1</mn></mrow><mo id="S4.SS2.SSS2.Px2.p1.9.m9.3.3.1.1.1.3" xref="S4.SS2.SSS2.Px2.p1.9.m9.3.3.1.1.2.cmml">,</mo><mi id="S4.SS2.SSS2.Px2.p1.9.m9.1.1" xref="S4.SS2.SSS2.Px2.p1.9.m9.1.1.cmml">i</mi><mo id="S4.SS2.SSS2.Px2.p1.9.m9.3.3.1.1.1.4" stretchy="false" xref="S4.SS2.SSS2.Px2.p1.9.m9.3.3.1.1.2.cmml">)</mo></mrow><mo id="S4.SS2.SSS2.Px2.p1.9.m9.4.4.2.4" xref="S4.SS2.SSS2.Px2.p1.9.m9.4.4.3.cmml">,</mo><mrow id="S4.SS2.SSS2.Px2.p1.9.m9.4.4.2.2.1" xref="S4.SS2.SSS2.Px2.p1.9.m9.4.4.2.2.2.cmml"><mo id="S4.SS2.SSS2.Px2.p1.9.m9.4.4.2.2.1.2" stretchy="false" xref="S4.SS2.SSS2.Px2.p1.9.m9.4.4.2.2.2.cmml">(</mo><mi id="S4.SS2.SSS2.Px2.p1.9.m9.2.2" xref="S4.SS2.SSS2.Px2.p1.9.m9.2.2.cmml">j</mi><mo id="S4.SS2.SSS2.Px2.p1.9.m9.4.4.2.2.1.3" xref="S4.SS2.SSS2.Px2.p1.9.m9.4.4.2.2.2.cmml">,</mo><mrow id="S4.SS2.SSS2.Px2.p1.9.m9.4.4.2.2.1.1" xref="S4.SS2.SSS2.Px2.p1.9.m9.4.4.2.2.1.1.cmml"><mi id="S4.SS2.SSS2.Px2.p1.9.m9.4.4.2.2.1.1.2" xref="S4.SS2.SSS2.Px2.p1.9.m9.4.4.2.2.1.1.2.cmml">j</mi><mo id="S4.SS2.SSS2.Px2.p1.9.m9.4.4.2.2.1.1.1" xref="S4.SS2.SSS2.Px2.p1.9.m9.4.4.2.2.1.1.1.cmml">+</mo><mn id="S4.SS2.SSS2.Px2.p1.9.m9.4.4.2.2.1.1.3" xref="S4.SS2.SSS2.Px2.p1.9.m9.4.4.2.2.1.1.3.cmml">1</mn></mrow><mo id="S4.SS2.SSS2.Px2.p1.9.m9.4.4.2.2.1.4" stretchy="false" xref="S4.SS2.SSS2.Px2.p1.9.m9.4.4.2.2.2.cmml">)</mo></mrow><mo id="S4.SS2.SSS2.Px2.p1.9.m9.4.4.2.5" stretchy="false" xref="S4.SS2.SSS2.Px2.p1.9.m9.4.4.3.cmml">}</mo></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS2.Px2.p1.9.m9.4b"><set id="S4.SS2.SSS2.Px2.p1.9.m9.4.4.3.cmml" xref="S4.SS2.SSS2.Px2.p1.9.m9.4.4.2"><interval closure="open" id="S4.SS2.SSS2.Px2.p1.9.m9.3.3.1.1.2.cmml" xref="S4.SS2.SSS2.Px2.p1.9.m9.3.3.1.1.1"><apply id="S4.SS2.SSS2.Px2.p1.9.m9.3.3.1.1.1.1.cmml" xref="S4.SS2.SSS2.Px2.p1.9.m9.3.3.1.1.1.1"><minus id="S4.SS2.SSS2.Px2.p1.9.m9.3.3.1.1.1.1.1.cmml" xref="S4.SS2.SSS2.Px2.p1.9.m9.3.3.1.1.1.1.1"></minus><ci id="S4.SS2.SSS2.Px2.p1.9.m9.3.3.1.1.1.1.2.cmml" xref="S4.SS2.SSS2.Px2.p1.9.m9.3.3.1.1.1.1.2">𝑖</ci><cn id="S4.SS2.SSS2.Px2.p1.9.m9.3.3.1.1.1.1.3.cmml" type="integer" xref="S4.SS2.SSS2.Px2.p1.9.m9.3.3.1.1.1.1.3">1</cn></apply><ci id="S4.SS2.SSS2.Px2.p1.9.m9.1.1.cmml" xref="S4.SS2.SSS2.Px2.p1.9.m9.1.1">𝑖</ci></interval><interval closure="open" id="S4.SS2.SSS2.Px2.p1.9.m9.4.4.2.2.2.cmml" xref="S4.SS2.SSS2.Px2.p1.9.m9.4.4.2.2.1"><ci id="S4.SS2.SSS2.Px2.p1.9.m9.2.2.cmml" xref="S4.SS2.SSS2.Px2.p1.9.m9.2.2">𝑗</ci><apply id="S4.SS2.SSS2.Px2.p1.9.m9.4.4.2.2.1.1.cmml" xref="S4.SS2.SSS2.Px2.p1.9.m9.4.4.2.2.1.1"><plus id="S4.SS2.SSS2.Px2.p1.9.m9.4.4.2.2.1.1.1.cmml" xref="S4.SS2.SSS2.Px2.p1.9.m9.4.4.2.2.1.1.1"></plus><ci id="S4.SS2.SSS2.Px2.p1.9.m9.4.4.2.2.1.1.2.cmml" xref="S4.SS2.SSS2.Px2.p1.9.m9.4.4.2.2.1.1.2">𝑗</ci><cn id="S4.SS2.SSS2.Px2.p1.9.m9.4.4.2.2.1.1.3.cmml" type="integer" xref="S4.SS2.SSS2.Px2.p1.9.m9.4.4.2.2.1.1.3">1</cn></apply></interval></set></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS2.Px2.p1.9.m9.4c">\{(i-1,i),(j,j+1)\}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS2.Px2.p1.9.m9.4d">{ ( italic_i - 1 , italic_i ) , ( italic_j , italic_j + 1 ) }</annotation></semantics></math>, and <math alttext="(u,v)" class="ltx_Math" display="inline" id="S4.SS2.SSS2.Px2.p1.10.m10.2"><semantics id="S4.SS2.SSS2.Px2.p1.10.m10.2a"><mrow id="S4.SS2.SSS2.Px2.p1.10.m10.2.3.2" xref="S4.SS2.SSS2.Px2.p1.10.m10.2.3.1.cmml"><mo id="S4.SS2.SSS2.Px2.p1.10.m10.2.3.2.1" stretchy="false" xref="S4.SS2.SSS2.Px2.p1.10.m10.2.3.1.cmml">(</mo><mi id="S4.SS2.SSS2.Px2.p1.10.m10.1.1" xref="S4.SS2.SSS2.Px2.p1.10.m10.1.1.cmml">u</mi><mo id="S4.SS2.SSS2.Px2.p1.10.m10.2.3.2.2" xref="S4.SS2.SSS2.Px2.p1.10.m10.2.3.1.cmml">,</mo><mi id="S4.SS2.SSS2.Px2.p1.10.m10.2.2" xref="S4.SS2.SSS2.Px2.p1.10.m10.2.2.cmml">v</mi><mo id="S4.SS2.SSS2.Px2.p1.10.m10.2.3.2.3" stretchy="false" xref="S4.SS2.SSS2.Px2.p1.10.m10.2.3.1.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS2.Px2.p1.10.m10.2b"><interval closure="open" id="S4.SS2.SSS2.Px2.p1.10.m10.2.3.1.cmml" xref="S4.SS2.SSS2.Px2.p1.10.m10.2.3.2"><ci id="S4.SS2.SSS2.Px2.p1.10.m10.1.1.cmml" xref="S4.SS2.SSS2.Px2.p1.10.m10.1.1">𝑢</ci><ci id="S4.SS2.SSS2.Px2.p1.10.m10.2.2.cmml" xref="S4.SS2.SSS2.Px2.p1.10.m10.2.2">𝑣</ci></interval></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS2.Px2.p1.10.m10.2c">(u,v)</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS2.Px2.p1.10.m10.2d">( italic_u , italic_v )</annotation></semantics></math> goes from the component of <math alttext="C\setminus\{(i-1,i),(j,j+1)\}" class="ltx_Math" display="inline" id="S4.SS2.SSS2.Px2.p1.11.m11.4"><semantics id="S4.SS2.SSS2.Px2.p1.11.m11.4a"><mrow id="S4.SS2.SSS2.Px2.p1.11.m11.4.4" xref="S4.SS2.SSS2.Px2.p1.11.m11.4.4.cmml"><mi id="S4.SS2.SSS2.Px2.p1.11.m11.4.4.4" xref="S4.SS2.SSS2.Px2.p1.11.m11.4.4.4.cmml">C</mi><mo id="S4.SS2.SSS2.Px2.p1.11.m11.4.4.3" xref="S4.SS2.SSS2.Px2.p1.11.m11.4.4.3.cmml">∖</mo><mrow id="S4.SS2.SSS2.Px2.p1.11.m11.4.4.2.2" xref="S4.SS2.SSS2.Px2.p1.11.m11.4.4.2.3.cmml"><mo id="S4.SS2.SSS2.Px2.p1.11.m11.4.4.2.2.3" stretchy="false" xref="S4.SS2.SSS2.Px2.p1.11.m11.4.4.2.3.cmml">{</mo><mrow id="S4.SS2.SSS2.Px2.p1.11.m11.3.3.1.1.1.1" xref="S4.SS2.SSS2.Px2.p1.11.m11.3.3.1.1.1.2.cmml"><mo id="S4.SS2.SSS2.Px2.p1.11.m11.3.3.1.1.1.1.2" stretchy="false" xref="S4.SS2.SSS2.Px2.p1.11.m11.3.3.1.1.1.2.cmml">(</mo><mrow id="S4.SS2.SSS2.Px2.p1.11.m11.3.3.1.1.1.1.1" xref="S4.SS2.SSS2.Px2.p1.11.m11.3.3.1.1.1.1.1.cmml"><mi id="S4.SS2.SSS2.Px2.p1.11.m11.3.3.1.1.1.1.1.2" xref="S4.SS2.SSS2.Px2.p1.11.m11.3.3.1.1.1.1.1.2.cmml">i</mi><mo id="S4.SS2.SSS2.Px2.p1.11.m11.3.3.1.1.1.1.1.1" xref="S4.SS2.SSS2.Px2.p1.11.m11.3.3.1.1.1.1.1.1.cmml">−</mo><mn id="S4.SS2.SSS2.Px2.p1.11.m11.3.3.1.1.1.1.1.3" xref="S4.SS2.SSS2.Px2.p1.11.m11.3.3.1.1.1.1.1.3.cmml">1</mn></mrow><mo id="S4.SS2.SSS2.Px2.p1.11.m11.3.3.1.1.1.1.3" xref="S4.SS2.SSS2.Px2.p1.11.m11.3.3.1.1.1.2.cmml">,</mo><mi id="S4.SS2.SSS2.Px2.p1.11.m11.1.1" xref="S4.SS2.SSS2.Px2.p1.11.m11.1.1.cmml">i</mi><mo id="S4.SS2.SSS2.Px2.p1.11.m11.3.3.1.1.1.1.4" stretchy="false" xref="S4.SS2.SSS2.Px2.p1.11.m11.3.3.1.1.1.2.cmml">)</mo></mrow><mo id="S4.SS2.SSS2.Px2.p1.11.m11.4.4.2.2.4" xref="S4.SS2.SSS2.Px2.p1.11.m11.4.4.2.3.cmml">,</mo><mrow id="S4.SS2.SSS2.Px2.p1.11.m11.4.4.2.2.2.1" xref="S4.SS2.SSS2.Px2.p1.11.m11.4.4.2.2.2.2.cmml"><mo id="S4.SS2.SSS2.Px2.p1.11.m11.4.4.2.2.2.1.2" stretchy="false" xref="S4.SS2.SSS2.Px2.p1.11.m11.4.4.2.2.2.2.cmml">(</mo><mi id="S4.SS2.SSS2.Px2.p1.11.m11.2.2" xref="S4.SS2.SSS2.Px2.p1.11.m11.2.2.cmml">j</mi><mo id="S4.SS2.SSS2.Px2.p1.11.m11.4.4.2.2.2.1.3" xref="S4.SS2.SSS2.Px2.p1.11.m11.4.4.2.2.2.2.cmml">,</mo><mrow id="S4.SS2.SSS2.Px2.p1.11.m11.4.4.2.2.2.1.1" xref="S4.SS2.SSS2.Px2.p1.11.m11.4.4.2.2.2.1.1.cmml"><mi id="S4.SS2.SSS2.Px2.p1.11.m11.4.4.2.2.2.1.1.2" xref="S4.SS2.SSS2.Px2.p1.11.m11.4.4.2.2.2.1.1.2.cmml">j</mi><mo id="S4.SS2.SSS2.Px2.p1.11.m11.4.4.2.2.2.1.1.1" xref="S4.SS2.SSS2.Px2.p1.11.m11.4.4.2.2.2.1.1.1.cmml">+</mo><mn id="S4.SS2.SSS2.Px2.p1.11.m11.4.4.2.2.2.1.1.3" xref="S4.SS2.SSS2.Px2.p1.11.m11.4.4.2.2.2.1.1.3.cmml">1</mn></mrow><mo id="S4.SS2.SSS2.Px2.p1.11.m11.4.4.2.2.2.1.4" stretchy="false" xref="S4.SS2.SSS2.Px2.p1.11.m11.4.4.2.2.2.2.cmml">)</mo></mrow><mo id="S4.SS2.SSS2.Px2.p1.11.m11.4.4.2.2.5" stretchy="false" xref="S4.SS2.SSS2.Px2.p1.11.m11.4.4.2.3.cmml">}</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS2.Px2.p1.11.m11.4b"><apply id="S4.SS2.SSS2.Px2.p1.11.m11.4.4.cmml" xref="S4.SS2.SSS2.Px2.p1.11.m11.4.4"><setdiff id="S4.SS2.SSS2.Px2.p1.11.m11.4.4.3.cmml" xref="S4.SS2.SSS2.Px2.p1.11.m11.4.4.3"></setdiff><ci id="S4.SS2.SSS2.Px2.p1.11.m11.4.4.4.cmml" xref="S4.SS2.SSS2.Px2.p1.11.m11.4.4.4">𝐶</ci><set id="S4.SS2.SSS2.Px2.p1.11.m11.4.4.2.3.cmml" xref="S4.SS2.SSS2.Px2.p1.11.m11.4.4.2.2"><interval closure="open" id="S4.SS2.SSS2.Px2.p1.11.m11.3.3.1.1.1.2.cmml" xref="S4.SS2.SSS2.Px2.p1.11.m11.3.3.1.1.1.1"><apply id="S4.SS2.SSS2.Px2.p1.11.m11.3.3.1.1.1.1.1.cmml" xref="S4.SS2.SSS2.Px2.p1.11.m11.3.3.1.1.1.1.1"><minus id="S4.SS2.SSS2.Px2.p1.11.m11.3.3.1.1.1.1.1.1.cmml" xref="S4.SS2.SSS2.Px2.p1.11.m11.3.3.1.1.1.1.1.1"></minus><ci id="S4.SS2.SSS2.Px2.p1.11.m11.3.3.1.1.1.1.1.2.cmml" xref="S4.SS2.SSS2.Px2.p1.11.m11.3.3.1.1.1.1.1.2">𝑖</ci><cn id="S4.SS2.SSS2.Px2.p1.11.m11.3.3.1.1.1.1.1.3.cmml" type="integer" xref="S4.SS2.SSS2.Px2.p1.11.m11.3.3.1.1.1.1.1.3">1</cn></apply><ci id="S4.SS2.SSS2.Px2.p1.11.m11.1.1.cmml" xref="S4.SS2.SSS2.Px2.p1.11.m11.1.1">𝑖</ci></interval><interval closure="open" id="S4.SS2.SSS2.Px2.p1.11.m11.4.4.2.2.2.2.cmml" xref="S4.SS2.SSS2.Px2.p1.11.m11.4.4.2.2.2.1"><ci id="S4.SS2.SSS2.Px2.p1.11.m11.2.2.cmml" xref="S4.SS2.SSS2.Px2.p1.11.m11.2.2">𝑗</ci><apply id="S4.SS2.SSS2.Px2.p1.11.m11.4.4.2.2.2.1.1.cmml" xref="S4.SS2.SSS2.Px2.p1.11.m11.4.4.2.2.2.1.1"><plus id="S4.SS2.SSS2.Px2.p1.11.m11.4.4.2.2.2.1.1.1.cmml" xref="S4.SS2.SSS2.Px2.p1.11.m11.4.4.2.2.2.1.1.1"></plus><ci id="S4.SS2.SSS2.Px2.p1.11.m11.4.4.2.2.2.1.1.2.cmml" xref="S4.SS2.SSS2.Px2.p1.11.m11.4.4.2.2.2.1.1.2">𝑗</ci><cn id="S4.SS2.SSS2.Px2.p1.11.m11.4.4.2.2.2.1.1.3.cmml" type="integer" xref="S4.SS2.SSS2.Px2.p1.11.m11.4.4.2.2.2.1.1.3">1</cn></apply></interval></set></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS2.Px2.p1.11.m11.4c">C\setminus\{(i-1,i),(j,j+1)\}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS2.Px2.p1.11.m11.4d">italic_C ∖ { ( italic_i - 1 , italic_i ) , ( italic_j , italic_j + 1 ) }</annotation></semantics></math> containing vertex <math alttext="1" class="ltx_Math" display="inline" id="S4.SS2.SSS2.Px2.p1.12.m12.1"><semantics id="S4.SS2.SSS2.Px2.p1.12.m12.1a"><mn id="S4.SS2.SSS2.Px2.p1.12.m12.1.1" xref="S4.SS2.SSS2.Px2.p1.12.m12.1.1.cmml">1</mn><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS2.Px2.p1.12.m12.1b"><cn id="S4.SS2.SSS2.Px2.p1.12.m12.1.1.cmml" type="integer" xref="S4.SS2.SSS2.Px2.p1.12.m12.1.1">1</cn></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS2.Px2.p1.12.m12.1c">1</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS2.Px2.p1.12.m12.1d">1</annotation></semantics></math> to the other component. It is easy to see that for <math alttext="1\leq u\leq u^{\prime}<v" class="ltx_Math" display="inline" id="S4.SS2.SSS2.Px2.p1.13.m13.1"><semantics id="S4.SS2.SSS2.Px2.p1.13.m13.1a"><mrow id="S4.SS2.SSS2.Px2.p1.13.m13.1.1" xref="S4.SS2.SSS2.Px2.p1.13.m13.1.1.cmml"><mn id="S4.SS2.SSS2.Px2.p1.13.m13.1.1.2" xref="S4.SS2.SSS2.Px2.p1.13.m13.1.1.2.cmml">1</mn><mo id="S4.SS2.SSS2.Px2.p1.13.m13.1.1.3" xref="S4.SS2.SSS2.Px2.p1.13.m13.1.1.3.cmml">≤</mo><mi id="S4.SS2.SSS2.Px2.p1.13.m13.1.1.4" xref="S4.SS2.SSS2.Px2.p1.13.m13.1.1.4.cmml">u</mi><mo id="S4.SS2.SSS2.Px2.p1.13.m13.1.1.5" xref="S4.SS2.SSS2.Px2.p1.13.m13.1.1.5.cmml">≤</mo><msup id="S4.SS2.SSS2.Px2.p1.13.m13.1.1.6" xref="S4.SS2.SSS2.Px2.p1.13.m13.1.1.6.cmml"><mi id="S4.SS2.SSS2.Px2.p1.13.m13.1.1.6.2" xref="S4.SS2.SSS2.Px2.p1.13.m13.1.1.6.2.cmml">u</mi><mo id="S4.SS2.SSS2.Px2.p1.13.m13.1.1.6.3" xref="S4.SS2.SSS2.Px2.p1.13.m13.1.1.6.3.cmml">′</mo></msup><mo id="S4.SS2.SSS2.Px2.p1.13.m13.1.1.7" xref="S4.SS2.SSS2.Px2.p1.13.m13.1.1.7.cmml"><</mo><mi id="S4.SS2.SSS2.Px2.p1.13.m13.1.1.8" xref="S4.SS2.SSS2.Px2.p1.13.m13.1.1.8.cmml">v</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS2.Px2.p1.13.m13.1b"><apply id="S4.SS2.SSS2.Px2.p1.13.m13.1.1.cmml" xref="S4.SS2.SSS2.Px2.p1.13.m13.1.1"><and id="S4.SS2.SSS2.Px2.p1.13.m13.1.1a.cmml" xref="S4.SS2.SSS2.Px2.p1.13.m13.1.1"></and><apply id="S4.SS2.SSS2.Px2.p1.13.m13.1.1b.cmml" xref="S4.SS2.SSS2.Px2.p1.13.m13.1.1"><leq id="S4.SS2.SSS2.Px2.p1.13.m13.1.1.3.cmml" xref="S4.SS2.SSS2.Px2.p1.13.m13.1.1.3"></leq><cn id="S4.SS2.SSS2.Px2.p1.13.m13.1.1.2.cmml" type="integer" xref="S4.SS2.SSS2.Px2.p1.13.m13.1.1.2">1</cn><ci id="S4.SS2.SSS2.Px2.p1.13.m13.1.1.4.cmml" xref="S4.SS2.SSS2.Px2.p1.13.m13.1.1.4">𝑢</ci></apply><apply id="S4.SS2.SSS2.Px2.p1.13.m13.1.1c.cmml" xref="S4.SS2.SSS2.Px2.p1.13.m13.1.1"><leq id="S4.SS2.SSS2.Px2.p1.13.m13.1.1.5.cmml" xref="S4.SS2.SSS2.Px2.p1.13.m13.1.1.5"></leq><share href="https://arxiv.org/html/2503.00712v1#S4.SS2.SSS2.Px2.p1.13.m13.1.1.4.cmml" id="S4.SS2.SSS2.Px2.p1.13.m13.1.1d.cmml" xref="S4.SS2.SSS2.Px2.p1.13.m13.1.1"></share><apply id="S4.SS2.SSS2.Px2.p1.13.m13.1.1.6.cmml" xref="S4.SS2.SSS2.Px2.p1.13.m13.1.1.6"><csymbol cd="ambiguous" id="S4.SS2.SSS2.Px2.p1.13.m13.1.1.6.1.cmml" xref="S4.SS2.SSS2.Px2.p1.13.m13.1.1.6">superscript</csymbol><ci id="S4.SS2.SSS2.Px2.p1.13.m13.1.1.6.2.cmml" xref="S4.SS2.SSS2.Px2.p1.13.m13.1.1.6.2">𝑢</ci><ci id="S4.SS2.SSS2.Px2.p1.13.m13.1.1.6.3.cmml" xref="S4.SS2.SSS2.Px2.p1.13.m13.1.1.6.3">′</ci></apply></apply><apply id="S4.SS2.SSS2.Px2.p1.13.m13.1.1e.cmml" xref="S4.SS2.SSS2.Px2.p1.13.m13.1.1"><lt id="S4.SS2.SSS2.Px2.p1.13.m13.1.1.7.cmml" xref="S4.SS2.SSS2.Px2.p1.13.m13.1.1.7"></lt><share href="https://arxiv.org/html/2503.00712v1#S4.SS2.SSS2.Px2.p1.13.m13.1.1.6.cmml" id="S4.SS2.SSS2.Px2.p1.13.m13.1.1f.cmml" xref="S4.SS2.SSS2.Px2.p1.13.m13.1.1"></share><ci id="S4.SS2.SSS2.Px2.p1.13.m13.1.1.8.cmml" xref="S4.SS2.SSS2.Px2.p1.13.m13.1.1.8">𝑣</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS2.Px2.p1.13.m13.1c">1\leq u\leq u^{\prime}<v</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS2.Px2.p1.13.m13.1d">1 ≤ italic_u ≤ italic_u start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT < italic_v</annotation></semantics></math>, the directed link <math alttext="(u,v)" class="ltx_Math" display="inline" id="S4.SS2.SSS2.Px2.p1.14.m14.2"><semantics id="S4.SS2.SSS2.Px2.p1.14.m14.2a"><mrow id="S4.SS2.SSS2.Px2.p1.14.m14.2.3.2" xref="S4.SS2.SSS2.Px2.p1.14.m14.2.3.1.cmml"><mo id="S4.SS2.SSS2.Px2.p1.14.m14.2.3.2.1" stretchy="false" xref="S4.SS2.SSS2.Px2.p1.14.m14.2.3.1.cmml">(</mo><mi id="S4.SS2.SSS2.Px2.p1.14.m14.1.1" xref="S4.SS2.SSS2.Px2.p1.14.m14.1.1.cmml">u</mi><mo id="S4.SS2.SSS2.Px2.p1.14.m14.2.3.2.2" xref="S4.SS2.SSS2.Px2.p1.14.m14.2.3.1.cmml">,</mo><mi id="S4.SS2.SSS2.Px2.p1.14.m14.2.2" xref="S4.SS2.SSS2.Px2.p1.14.m14.2.2.cmml">v</mi><mo id="S4.SS2.SSS2.Px2.p1.14.m14.2.3.2.3" stretchy="false" xref="S4.SS2.SSS2.Px2.p1.14.m14.2.3.1.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS2.Px2.p1.14.m14.2b"><interval closure="open" id="S4.SS2.SSS2.Px2.p1.14.m14.2.3.1.cmml" xref="S4.SS2.SSS2.Px2.p1.14.m14.2.3.2"><ci id="S4.SS2.SSS2.Px2.p1.14.m14.1.1.cmml" xref="S4.SS2.SSS2.Px2.p1.14.m14.1.1">𝑢</ci><ci id="S4.SS2.SSS2.Px2.p1.14.m14.2.2.cmml" xref="S4.SS2.SSS2.Px2.p1.14.m14.2.2">𝑣</ci></interval></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS2.Px2.p1.14.m14.2c">(u,v)</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS2.Px2.p1.14.m14.2d">( italic_u , italic_v )</annotation></semantics></math> covers a superset of the cuts covered by <math alttext="(u^{\prime},v)" class="ltx_Math" display="inline" id="S4.SS2.SSS2.Px2.p1.15.m15.2"><semantics id="S4.SS2.SSS2.Px2.p1.15.m15.2a"><mrow id="S4.SS2.SSS2.Px2.p1.15.m15.2.2.1" xref="S4.SS2.SSS2.Px2.p1.15.m15.2.2.2.cmml"><mo id="S4.SS2.SSS2.Px2.p1.15.m15.2.2.1.2" stretchy="false" xref="S4.SS2.SSS2.Px2.p1.15.m15.2.2.2.cmml">(</mo><msup id="S4.SS2.SSS2.Px2.p1.15.m15.2.2.1.1" xref="S4.SS2.SSS2.Px2.p1.15.m15.2.2.1.1.cmml"><mi id="S4.SS2.SSS2.Px2.p1.15.m15.2.2.1.1.2" xref="S4.SS2.SSS2.Px2.p1.15.m15.2.2.1.1.2.cmml">u</mi><mo id="S4.SS2.SSS2.Px2.p1.15.m15.2.2.1.1.3" xref="S4.SS2.SSS2.Px2.p1.15.m15.2.2.1.1.3.cmml">′</mo></msup><mo id="S4.SS2.SSS2.Px2.p1.15.m15.2.2.1.3" xref="S4.SS2.SSS2.Px2.p1.15.m15.2.2.2.cmml">,</mo><mi id="S4.SS2.SSS2.Px2.p1.15.m15.1.1" xref="S4.SS2.SSS2.Px2.p1.15.m15.1.1.cmml">v</mi><mo id="S4.SS2.SSS2.Px2.p1.15.m15.2.2.1.4" stretchy="false" xref="S4.SS2.SSS2.Px2.p1.15.m15.2.2.2.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS2.Px2.p1.15.m15.2b"><interval closure="open" id="S4.SS2.SSS2.Px2.p1.15.m15.2.2.2.cmml" xref="S4.SS2.SSS2.Px2.p1.15.m15.2.2.1"><apply id="S4.SS2.SSS2.Px2.p1.15.m15.2.2.1.1.cmml" xref="S4.SS2.SSS2.Px2.p1.15.m15.2.2.1.1"><csymbol cd="ambiguous" id="S4.SS2.SSS2.Px2.p1.15.m15.2.2.1.1.1.cmml" xref="S4.SS2.SSS2.Px2.p1.15.m15.2.2.1.1">superscript</csymbol><ci id="S4.SS2.SSS2.Px2.p1.15.m15.2.2.1.1.2.cmml" xref="S4.SS2.SSS2.Px2.p1.15.m15.2.2.1.1.2">𝑢</ci><ci id="S4.SS2.SSS2.Px2.p1.15.m15.2.2.1.1.3.cmml" xref="S4.SS2.SSS2.Px2.p1.15.m15.2.2.1.1.3">′</ci></apply><ci id="S4.SS2.SSS2.Px2.p1.15.m15.1.1.cmml" xref="S4.SS2.SSS2.Px2.p1.15.m15.1.1">𝑣</ci></interval></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS2.Px2.p1.15.m15.2c">(u^{\prime},v)</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS2.Px2.p1.15.m15.2d">( italic_u start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT , italic_v )</annotation></semantics></math>, and the same holds for for <math alttext="v\leq u^{\prime}<u\leq n" class="ltx_Math" display="inline" id="S4.SS2.SSS2.Px2.p1.16.m16.1"><semantics id="S4.SS2.SSS2.Px2.p1.16.m16.1a"><mrow id="S4.SS2.SSS2.Px2.p1.16.m16.1.1" xref="S4.SS2.SSS2.Px2.p1.16.m16.1.1.cmml"><mi id="S4.SS2.SSS2.Px2.p1.16.m16.1.1.2" xref="S4.SS2.SSS2.Px2.p1.16.m16.1.1.2.cmml">v</mi><mo id="S4.SS2.SSS2.Px2.p1.16.m16.1.1.3" xref="S4.SS2.SSS2.Px2.p1.16.m16.1.1.3.cmml">≤</mo><msup id="S4.SS2.SSS2.Px2.p1.16.m16.1.1.4" xref="S4.SS2.SSS2.Px2.p1.16.m16.1.1.4.cmml"><mi id="S4.SS2.SSS2.Px2.p1.16.m16.1.1.4.2" xref="S4.SS2.SSS2.Px2.p1.16.m16.1.1.4.2.cmml">u</mi><mo id="S4.SS2.SSS2.Px2.p1.16.m16.1.1.4.3" xref="S4.SS2.SSS2.Px2.p1.16.m16.1.1.4.3.cmml">′</mo></msup><mo id="S4.SS2.SSS2.Px2.p1.16.m16.1.1.5" xref="S4.SS2.SSS2.Px2.p1.16.m16.1.1.5.cmml"><</mo><mi id="S4.SS2.SSS2.Px2.p1.16.m16.1.1.6" xref="S4.SS2.SSS2.Px2.p1.16.m16.1.1.6.cmml">u</mi><mo id="S4.SS2.SSS2.Px2.p1.16.m16.1.1.7" xref="S4.SS2.SSS2.Px2.p1.16.m16.1.1.7.cmml">≤</mo><mi id="S4.SS2.SSS2.Px2.p1.16.m16.1.1.8" xref="S4.SS2.SSS2.Px2.p1.16.m16.1.1.8.cmml">n</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS2.Px2.p1.16.m16.1b"><apply id="S4.SS2.SSS2.Px2.p1.16.m16.1.1.cmml" xref="S4.SS2.SSS2.Px2.p1.16.m16.1.1"><and id="S4.SS2.SSS2.Px2.p1.16.m16.1.1a.cmml" xref="S4.SS2.SSS2.Px2.p1.16.m16.1.1"></and><apply id="S4.SS2.SSS2.Px2.p1.16.m16.1.1b.cmml" xref="S4.SS2.SSS2.Px2.p1.16.m16.1.1"><leq id="S4.SS2.SSS2.Px2.p1.16.m16.1.1.3.cmml" xref="S4.SS2.SSS2.Px2.p1.16.m16.1.1.3"></leq><ci id="S4.SS2.SSS2.Px2.p1.16.m16.1.1.2.cmml" xref="S4.SS2.SSS2.Px2.p1.16.m16.1.1.2">𝑣</ci><apply id="S4.SS2.SSS2.Px2.p1.16.m16.1.1.4.cmml" xref="S4.SS2.SSS2.Px2.p1.16.m16.1.1.4"><csymbol cd="ambiguous" id="S4.SS2.SSS2.Px2.p1.16.m16.1.1.4.1.cmml" xref="S4.SS2.SSS2.Px2.p1.16.m16.1.1.4">superscript</csymbol><ci id="S4.SS2.SSS2.Px2.p1.16.m16.1.1.4.2.cmml" xref="S4.SS2.SSS2.Px2.p1.16.m16.1.1.4.2">𝑢</ci><ci id="S4.SS2.SSS2.Px2.p1.16.m16.1.1.4.3.cmml" xref="S4.SS2.SSS2.Px2.p1.16.m16.1.1.4.3">′</ci></apply></apply><apply id="S4.SS2.SSS2.Px2.p1.16.m16.1.1c.cmml" xref="S4.SS2.SSS2.Px2.p1.16.m16.1.1"><lt id="S4.SS2.SSS2.Px2.p1.16.m16.1.1.5.cmml" xref="S4.SS2.SSS2.Px2.p1.16.m16.1.1.5"></lt><share href="https://arxiv.org/html/2503.00712v1#S4.SS2.SSS2.Px2.p1.16.m16.1.1.4.cmml" id="S4.SS2.SSS2.Px2.p1.16.m16.1.1d.cmml" xref="S4.SS2.SSS2.Px2.p1.16.m16.1.1"></share><ci id="S4.SS2.SSS2.Px2.p1.16.m16.1.1.6.cmml" xref="S4.SS2.SSS2.Px2.p1.16.m16.1.1.6">𝑢</ci></apply><apply id="S4.SS2.SSS2.Px2.p1.16.m16.1.1e.cmml" xref="S4.SS2.SSS2.Px2.p1.16.m16.1.1"><leq id="S4.SS2.SSS2.Px2.p1.16.m16.1.1.7.cmml" xref="S4.SS2.SSS2.Px2.p1.16.m16.1.1.7"></leq><share href="https://arxiv.org/html/2503.00712v1#S4.SS2.SSS2.Px2.p1.16.m16.1.1.6.cmml" id="S4.SS2.SSS2.Px2.p1.16.m16.1.1f.cmml" xref="S4.SS2.SSS2.Px2.p1.16.m16.1.1"></share><ci id="S4.SS2.SSS2.Px2.p1.16.m16.1.1.8.cmml" xref="S4.SS2.SSS2.Px2.p1.16.m16.1.1.8">𝑛</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS2.Px2.p1.16.m16.1c">v\leq u^{\prime}<u\leq n</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS2.Px2.p1.16.m16.1d">italic_v ≤ italic_u start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT < italic_u ≤ italic_n</annotation></semantics></math>; see Figure <a class="ltx_ref" href="https://arxiv.org/html/2503.00712v1#S4.F7" title="Figure 7 ‣ “Cycle” Cuts: ‣ 4.2.2 The Streaming Algorithm ‣ 4.2 Two-to-Three Augmentation ‣ 4 Vertex Connectivity Augmentation in Link-Arrival Model ‣ Streaming Algorithms for Network Design"><span class="ltx_text ltx_ref_tag">7</span></a> for an example. Thus we can store at most two incoming links per vertex and obtain a 2-approximation in linear space. This can be generalized to weighted graphs using standard weight-bucketing ideas.</p> </div> <div class="ltx_para" id="S4.SS2.SSS2.Px2.p2"> <p class="ltx_p" id="S4.SS2.SSS2.Px2.p2.24">We employ a similar idea for covering 2-vertex cuts of type (3). Fix an S-node <math alttext="x" class="ltx_Math" display="inline" id="S4.SS2.SSS2.Px2.p2.1.m1.1"><semantics id="S4.SS2.SSS2.Px2.p2.1.m1.1a"><mi id="S4.SS2.SSS2.Px2.p2.1.m1.1.1" xref="S4.SS2.SSS2.Px2.p2.1.m1.1.1.cmml">x</mi><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS2.Px2.p2.1.m1.1b"><ci id="S4.SS2.SSS2.Px2.p2.1.m1.1.1.cmml" xref="S4.SS2.SSS2.Px2.p2.1.m1.1.1">𝑥</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS2.Px2.p2.1.m1.1c">x</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS2.Px2.p2.1.m1.1d">italic_x</annotation></semantics></math>. The main difference in this setting is that two components of <math alttext="G_{x}\setminus\{a,b\}" class="ltx_Math" display="inline" id="S4.SS2.SSS2.Px2.p2.2.m2.2"><semantics id="S4.SS2.SSS2.Px2.p2.2.m2.2a"><mrow id="S4.SS2.SSS2.Px2.p2.2.m2.2.3" xref="S4.SS2.SSS2.Px2.p2.2.m2.2.3.cmml"><msub id="S4.SS2.SSS2.Px2.p2.2.m2.2.3.2" xref="S4.SS2.SSS2.Px2.p2.2.m2.2.3.2.cmml"><mi id="S4.SS2.SSS2.Px2.p2.2.m2.2.3.2.2" xref="S4.SS2.SSS2.Px2.p2.2.m2.2.3.2.2.cmml">G</mi><mi id="S4.SS2.SSS2.Px2.p2.2.m2.2.3.2.3" xref="S4.SS2.SSS2.Px2.p2.2.m2.2.3.2.3.cmml">x</mi></msub><mo id="S4.SS2.SSS2.Px2.p2.2.m2.2.3.1" xref="S4.SS2.SSS2.Px2.p2.2.m2.2.3.1.cmml">∖</mo><mrow id="S4.SS2.SSS2.Px2.p2.2.m2.2.3.3.2" xref="S4.SS2.SSS2.Px2.p2.2.m2.2.3.3.1.cmml"><mo id="S4.SS2.SSS2.Px2.p2.2.m2.2.3.3.2.1" stretchy="false" xref="S4.SS2.SSS2.Px2.p2.2.m2.2.3.3.1.cmml">{</mo><mi id="S4.SS2.SSS2.Px2.p2.2.m2.1.1" xref="S4.SS2.SSS2.Px2.p2.2.m2.1.1.cmml">a</mi><mo id="S4.SS2.SSS2.Px2.p2.2.m2.2.3.3.2.2" xref="S4.SS2.SSS2.Px2.p2.2.m2.2.3.3.1.cmml">,</mo><mi id="S4.SS2.SSS2.Px2.p2.2.m2.2.2" xref="S4.SS2.SSS2.Px2.p2.2.m2.2.2.cmml">b</mi><mo id="S4.SS2.SSS2.Px2.p2.2.m2.2.3.3.2.3" stretchy="false" xref="S4.SS2.SSS2.Px2.p2.2.m2.2.3.3.1.cmml">}</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS2.Px2.p2.2.m2.2b"><apply id="S4.SS2.SSS2.Px2.p2.2.m2.2.3.cmml" xref="S4.SS2.SSS2.Px2.p2.2.m2.2.3"><setdiff id="S4.SS2.SSS2.Px2.p2.2.m2.2.3.1.cmml" xref="S4.SS2.SSS2.Px2.p2.2.m2.2.3.1"></setdiff><apply id="S4.SS2.SSS2.Px2.p2.2.m2.2.3.2.cmml" xref="S4.SS2.SSS2.Px2.p2.2.m2.2.3.2"><csymbol cd="ambiguous" id="S4.SS2.SSS2.Px2.p2.2.m2.2.3.2.1.cmml" xref="S4.SS2.SSS2.Px2.p2.2.m2.2.3.2">subscript</csymbol><ci id="S4.SS2.SSS2.Px2.p2.2.m2.2.3.2.2.cmml" xref="S4.SS2.SSS2.Px2.p2.2.m2.2.3.2.2">𝐺</ci><ci id="S4.SS2.SSS2.Px2.p2.2.m2.2.3.2.3.cmml" xref="S4.SS2.SSS2.Px2.p2.2.m2.2.3.2.3">𝑥</ci></apply><set id="S4.SS2.SSS2.Px2.p2.2.m2.2.3.3.1.cmml" xref="S4.SS2.SSS2.Px2.p2.2.m2.2.3.3.2"><ci id="S4.SS2.SSS2.Px2.p2.2.m2.1.1.cmml" xref="S4.SS2.SSS2.Px2.p2.2.m2.1.1">𝑎</ci><ci id="S4.SS2.SSS2.Px2.p2.2.m2.2.2.cmml" xref="S4.SS2.SSS2.Px2.p2.2.m2.2.2">𝑏</ci></set></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS2.Px2.p2.2.m2.2c">G_{x}\setminus\{a,b\}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS2.Px2.p2.2.m2.2d">italic_G start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT ∖ { italic_a , italic_b }</annotation></semantics></math> can be connected via links not in <math alttext="G_{x}" class="ltx_Math" display="inline" id="S4.SS2.SSS2.Px2.p2.3.m3.1"><semantics id="S4.SS2.SSS2.Px2.p2.3.m3.1a"><msub id="S4.SS2.SSS2.Px2.p2.3.m3.1.1" xref="S4.SS2.SSS2.Px2.p2.3.m3.1.1.cmml"><mi id="S4.SS2.SSS2.Px2.p2.3.m3.1.1.2" xref="S4.SS2.SSS2.Px2.p2.3.m3.1.1.2.cmml">G</mi><mi id="S4.SS2.SSS2.Px2.p2.3.m3.1.1.3" xref="S4.SS2.SSS2.Px2.p2.3.m3.1.1.3.cmml">x</mi></msub><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS2.Px2.p2.3.m3.1b"><apply id="S4.SS2.SSS2.Px2.p2.3.m3.1.1.cmml" xref="S4.SS2.SSS2.Px2.p2.3.m3.1.1"><csymbol cd="ambiguous" id="S4.SS2.SSS2.Px2.p2.3.m3.1.1.1.cmml" xref="S4.SS2.SSS2.Px2.p2.3.m3.1.1">subscript</csymbol><ci id="S4.SS2.SSS2.Px2.p2.3.m3.1.1.2.cmml" xref="S4.SS2.SSS2.Px2.p2.3.m3.1.1.2">𝐺</ci><ci id="S4.SS2.SSS2.Px2.p2.3.m3.1.1.3.cmml" xref="S4.SS2.SSS2.Px2.p2.3.m3.1.1.3">𝑥</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS2.Px2.p2.3.m3.1c">G_{x}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS2.Px2.p2.3.m3.1d">italic_G start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT</annotation></semantics></math>: e.g. by a link between the two subtrees of <math alttext="T" class="ltx_Math" display="inline" id="S4.SS2.SSS2.Px2.p2.4.m4.1"><semantics id="S4.SS2.SSS2.Px2.p2.4.m4.1a"><mi id="S4.SS2.SSS2.Px2.p2.4.m4.1.1" xref="S4.SS2.SSS2.Px2.p2.4.m4.1.1.cmml">T</mi><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS2.Px2.p2.4.m4.1b"><ci id="S4.SS2.SSS2.Px2.p2.4.m4.1.1.cmml" xref="S4.SS2.SSS2.Px2.p2.4.m4.1.1">𝑇</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS2.Px2.p2.4.m4.1c">T</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS2.Px2.p2.4.m4.1d">italic_T</annotation></semantics></math> corresponding to the two components of <math alttext="G_{x}\setminus\{a,b\}" class="ltx_Math" display="inline" id="S4.SS2.SSS2.Px2.p2.5.m5.2"><semantics id="S4.SS2.SSS2.Px2.p2.5.m5.2a"><mrow id="S4.SS2.SSS2.Px2.p2.5.m5.2.3" xref="S4.SS2.SSS2.Px2.p2.5.m5.2.3.cmml"><msub id="S4.SS2.SSS2.Px2.p2.5.m5.2.3.2" xref="S4.SS2.SSS2.Px2.p2.5.m5.2.3.2.cmml"><mi id="S4.SS2.SSS2.Px2.p2.5.m5.2.3.2.2" xref="S4.SS2.SSS2.Px2.p2.5.m5.2.3.2.2.cmml">G</mi><mi id="S4.SS2.SSS2.Px2.p2.5.m5.2.3.2.3" xref="S4.SS2.SSS2.Px2.p2.5.m5.2.3.2.3.cmml">x</mi></msub><mo id="S4.SS2.SSS2.Px2.p2.5.m5.2.3.1" xref="S4.SS2.SSS2.Px2.p2.5.m5.2.3.1.cmml">∖</mo><mrow id="S4.SS2.SSS2.Px2.p2.5.m5.2.3.3.2" xref="S4.SS2.SSS2.Px2.p2.5.m5.2.3.3.1.cmml"><mo id="S4.SS2.SSS2.Px2.p2.5.m5.2.3.3.2.1" stretchy="false" xref="S4.SS2.SSS2.Px2.p2.5.m5.2.3.3.1.cmml">{</mo><mi id="S4.SS2.SSS2.Px2.p2.5.m5.1.1" xref="S4.SS2.SSS2.Px2.p2.5.m5.1.1.cmml">a</mi><mo id="S4.SS2.SSS2.Px2.p2.5.m5.2.3.3.2.2" xref="S4.SS2.SSS2.Px2.p2.5.m5.2.3.3.1.cmml">,</mo><mi id="S4.SS2.SSS2.Px2.p2.5.m5.2.2" xref="S4.SS2.SSS2.Px2.p2.5.m5.2.2.cmml">b</mi><mo id="S4.SS2.SSS2.Px2.p2.5.m5.2.3.3.2.3" stretchy="false" xref="S4.SS2.SSS2.Px2.p2.5.m5.2.3.3.1.cmml">}</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS2.Px2.p2.5.m5.2b"><apply id="S4.SS2.SSS2.Px2.p2.5.m5.2.3.cmml" xref="S4.SS2.SSS2.Px2.p2.5.m5.2.3"><setdiff id="S4.SS2.SSS2.Px2.p2.5.m5.2.3.1.cmml" xref="S4.SS2.SSS2.Px2.p2.5.m5.2.3.1"></setdiff><apply id="S4.SS2.SSS2.Px2.p2.5.m5.2.3.2.cmml" xref="S4.SS2.SSS2.Px2.p2.5.m5.2.3.2"><csymbol cd="ambiguous" id="S4.SS2.SSS2.Px2.p2.5.m5.2.3.2.1.cmml" xref="S4.SS2.SSS2.Px2.p2.5.m5.2.3.2">subscript</csymbol><ci id="S4.SS2.SSS2.Px2.p2.5.m5.2.3.2.2.cmml" xref="S4.SS2.SSS2.Px2.p2.5.m5.2.3.2.2">𝐺</ci><ci id="S4.SS2.SSS2.Px2.p2.5.m5.2.3.2.3.cmml" xref="S4.SS2.SSS2.Px2.p2.5.m5.2.3.2.3">𝑥</ci></apply><set id="S4.SS2.SSS2.Px2.p2.5.m5.2.3.3.1.cmml" xref="S4.SS2.SSS2.Px2.p2.5.m5.2.3.3.2"><ci id="S4.SS2.SSS2.Px2.p2.5.m5.1.1.cmml" xref="S4.SS2.SSS2.Px2.p2.5.m5.1.1">𝑎</ci><ci id="S4.SS2.SSS2.Px2.p2.5.m5.2.2.cmml" xref="S4.SS2.SSS2.Px2.p2.5.m5.2.2">𝑏</ci></set></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS2.Px2.p2.5.m5.2c">G_{x}\setminus\{a,b\}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS2.Px2.p2.5.m5.2d">italic_G start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT ∖ { italic_a , italic_b }</annotation></semantics></math>. We label the vertices of <math alttext="G_{x}" class="ltx_Math" display="inline" id="S4.SS2.SSS2.Px2.p2.6.m6.1"><semantics id="S4.SS2.SSS2.Px2.p2.6.m6.1a"><msub id="S4.SS2.SSS2.Px2.p2.6.m6.1.1" xref="S4.SS2.SSS2.Px2.p2.6.m6.1.1.cmml"><mi id="S4.SS2.SSS2.Px2.p2.6.m6.1.1.2" xref="S4.SS2.SSS2.Px2.p2.6.m6.1.1.2.cmml">G</mi><mi id="S4.SS2.SSS2.Px2.p2.6.m6.1.1.3" xref="S4.SS2.SSS2.Px2.p2.6.m6.1.1.3.cmml">x</mi></msub><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS2.Px2.p2.6.m6.1b"><apply id="S4.SS2.SSS2.Px2.p2.6.m6.1.1.cmml" xref="S4.SS2.SSS2.Px2.p2.6.m6.1.1"><csymbol cd="ambiguous" id="S4.SS2.SSS2.Px2.p2.6.m6.1.1.1.cmml" xref="S4.SS2.SSS2.Px2.p2.6.m6.1.1">subscript</csymbol><ci id="S4.SS2.SSS2.Px2.p2.6.m6.1.1.2.cmml" xref="S4.SS2.SSS2.Px2.p2.6.m6.1.1.2">𝐺</ci><ci id="S4.SS2.SSS2.Px2.p2.6.m6.1.1.3.cmml" xref="S4.SS2.SSS2.Px2.p2.6.m6.1.1.3">𝑥</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS2.Px2.p2.6.m6.1c">G_{x}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS2.Px2.p2.6.m6.1d">italic_G start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT</annotation></semantics></math> along the cycle <math alttext="\mu_{0},\dots,\mu_{k}" class="ltx_Math" display="inline" id="S4.SS2.SSS2.Px2.p2.7.m7.3"><semantics id="S4.SS2.SSS2.Px2.p2.7.m7.3a"><mrow id="S4.SS2.SSS2.Px2.p2.7.m7.3.3.2" xref="S4.SS2.SSS2.Px2.p2.7.m7.3.3.3.cmml"><msub id="S4.SS2.SSS2.Px2.p2.7.m7.2.2.1.1" xref="S4.SS2.SSS2.Px2.p2.7.m7.2.2.1.1.cmml"><mi id="S4.SS2.SSS2.Px2.p2.7.m7.2.2.1.1.2" xref="S4.SS2.SSS2.Px2.p2.7.m7.2.2.1.1.2.cmml">μ</mi><mn id="S4.SS2.SSS2.Px2.p2.7.m7.2.2.1.1.3" xref="S4.SS2.SSS2.Px2.p2.7.m7.2.2.1.1.3.cmml">0</mn></msub><mo id="S4.SS2.SSS2.Px2.p2.7.m7.3.3.2.3" xref="S4.SS2.SSS2.Px2.p2.7.m7.3.3.3.cmml">,</mo><mi id="S4.SS2.SSS2.Px2.p2.7.m7.1.1" mathvariant="normal" xref="S4.SS2.SSS2.Px2.p2.7.m7.1.1.cmml">…</mi><mo id="S4.SS2.SSS2.Px2.p2.7.m7.3.3.2.4" xref="S4.SS2.SSS2.Px2.p2.7.m7.3.3.3.cmml">,</mo><msub id="S4.SS2.SSS2.Px2.p2.7.m7.3.3.2.2" xref="S4.SS2.SSS2.Px2.p2.7.m7.3.3.2.2.cmml"><mi id="S4.SS2.SSS2.Px2.p2.7.m7.3.3.2.2.2" xref="S4.SS2.SSS2.Px2.p2.7.m7.3.3.2.2.2.cmml">μ</mi><mi id="S4.SS2.SSS2.Px2.p2.7.m7.3.3.2.2.3" xref="S4.SS2.SSS2.Px2.p2.7.m7.3.3.2.2.3.cmml">k</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS2.Px2.p2.7.m7.3b"><list id="S4.SS2.SSS2.Px2.p2.7.m7.3.3.3.cmml" xref="S4.SS2.SSS2.Px2.p2.7.m7.3.3.2"><apply id="S4.SS2.SSS2.Px2.p2.7.m7.2.2.1.1.cmml" xref="S4.SS2.SSS2.Px2.p2.7.m7.2.2.1.1"><csymbol cd="ambiguous" id="S4.SS2.SSS2.Px2.p2.7.m7.2.2.1.1.1.cmml" xref="S4.SS2.SSS2.Px2.p2.7.m7.2.2.1.1">subscript</csymbol><ci id="S4.SS2.SSS2.Px2.p2.7.m7.2.2.1.1.2.cmml" xref="S4.SS2.SSS2.Px2.p2.7.m7.2.2.1.1.2">𝜇</ci><cn id="S4.SS2.SSS2.Px2.p2.7.m7.2.2.1.1.3.cmml" type="integer" xref="S4.SS2.SSS2.Px2.p2.7.m7.2.2.1.1.3">0</cn></apply><ci id="S4.SS2.SSS2.Px2.p2.7.m7.1.1.cmml" xref="S4.SS2.SSS2.Px2.p2.7.m7.1.1">…</ci><apply id="S4.SS2.SSS2.Px2.p2.7.m7.3.3.2.2.cmml" xref="S4.SS2.SSS2.Px2.p2.7.m7.3.3.2.2"><csymbol cd="ambiguous" id="S4.SS2.SSS2.Px2.p2.7.m7.3.3.2.2.1.cmml" xref="S4.SS2.SSS2.Px2.p2.7.m7.3.3.2.2">subscript</csymbol><ci id="S4.SS2.SSS2.Px2.p2.7.m7.3.3.2.2.2.cmml" xref="S4.SS2.SSS2.Px2.p2.7.m7.3.3.2.2.2">𝜇</ci><ci id="S4.SS2.SSS2.Px2.p2.7.m7.3.3.2.2.3.cmml" xref="S4.SS2.SSS2.Px2.p2.7.m7.3.3.2.2.3">𝑘</ci></apply></list></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS2.Px2.p2.7.m7.3c">\mu_{0},\dots,\mu_{k}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS2.Px2.p2.7.m7.3d">italic_μ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , … , italic_μ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT</annotation></semantics></math>, where <math alttext="\{\mu_{k},\mu_{0}\}=\textnormal{parent}(x)" class="ltx_Math" display="inline" id="S4.SS2.SSS2.Px2.p2.8.m8.3"><semantics id="S4.SS2.SSS2.Px2.p2.8.m8.3a"><mrow id="S4.SS2.SSS2.Px2.p2.8.m8.3.3" xref="S4.SS2.SSS2.Px2.p2.8.m8.3.3.cmml"><mrow id="S4.SS2.SSS2.Px2.p2.8.m8.3.3.2.2" xref="S4.SS2.SSS2.Px2.p2.8.m8.3.3.2.3.cmml"><mo id="S4.SS2.SSS2.Px2.p2.8.m8.3.3.2.2.3" stretchy="false" xref="S4.SS2.SSS2.Px2.p2.8.m8.3.3.2.3.cmml">{</mo><msub id="S4.SS2.SSS2.Px2.p2.8.m8.2.2.1.1.1" xref="S4.SS2.SSS2.Px2.p2.8.m8.2.2.1.1.1.cmml"><mi id="S4.SS2.SSS2.Px2.p2.8.m8.2.2.1.1.1.2" xref="S4.SS2.SSS2.Px2.p2.8.m8.2.2.1.1.1.2.cmml">μ</mi><mi id="S4.SS2.SSS2.Px2.p2.8.m8.2.2.1.1.1.3" xref="S4.SS2.SSS2.Px2.p2.8.m8.2.2.1.1.1.3.cmml">k</mi></msub><mo id="S4.SS2.SSS2.Px2.p2.8.m8.3.3.2.2.4" xref="S4.SS2.SSS2.Px2.p2.8.m8.3.3.2.3.cmml">,</mo><msub id="S4.SS2.SSS2.Px2.p2.8.m8.3.3.2.2.2" xref="S4.SS2.SSS2.Px2.p2.8.m8.3.3.2.2.2.cmml"><mi id="S4.SS2.SSS2.Px2.p2.8.m8.3.3.2.2.2.2" xref="S4.SS2.SSS2.Px2.p2.8.m8.3.3.2.2.2.2.cmml">μ</mi><mn id="S4.SS2.SSS2.Px2.p2.8.m8.3.3.2.2.2.3" xref="S4.SS2.SSS2.Px2.p2.8.m8.3.3.2.2.2.3.cmml">0</mn></msub><mo id="S4.SS2.SSS2.Px2.p2.8.m8.3.3.2.2.5" stretchy="false" xref="S4.SS2.SSS2.Px2.p2.8.m8.3.3.2.3.cmml">}</mo></mrow><mo id="S4.SS2.SSS2.Px2.p2.8.m8.3.3.3" xref="S4.SS2.SSS2.Px2.p2.8.m8.3.3.3.cmml">=</mo><mrow id="S4.SS2.SSS2.Px2.p2.8.m8.3.3.4" xref="S4.SS2.SSS2.Px2.p2.8.m8.3.3.4.cmml"><mtext id="S4.SS2.SSS2.Px2.p2.8.m8.3.3.4.2" xref="S4.SS2.SSS2.Px2.p2.8.m8.3.3.4.2a.cmml">parent</mtext><mo id="S4.SS2.SSS2.Px2.p2.8.m8.3.3.4.1" xref="S4.SS2.SSS2.Px2.p2.8.m8.3.3.4.1.cmml">⁢</mo><mrow id="S4.SS2.SSS2.Px2.p2.8.m8.3.3.4.3.2" xref="S4.SS2.SSS2.Px2.p2.8.m8.3.3.4.cmml"><mo id="S4.SS2.SSS2.Px2.p2.8.m8.3.3.4.3.2.1" stretchy="false" xref="S4.SS2.SSS2.Px2.p2.8.m8.3.3.4.cmml">(</mo><mi id="S4.SS2.SSS2.Px2.p2.8.m8.1.1" xref="S4.SS2.SSS2.Px2.p2.8.m8.1.1.cmml">x</mi><mo id="S4.SS2.SSS2.Px2.p2.8.m8.3.3.4.3.2.2" stretchy="false" xref="S4.SS2.SSS2.Px2.p2.8.m8.3.3.4.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS2.Px2.p2.8.m8.3b"><apply id="S4.SS2.SSS2.Px2.p2.8.m8.3.3.cmml" xref="S4.SS2.SSS2.Px2.p2.8.m8.3.3"><eq id="S4.SS2.SSS2.Px2.p2.8.m8.3.3.3.cmml" xref="S4.SS2.SSS2.Px2.p2.8.m8.3.3.3"></eq><set id="S4.SS2.SSS2.Px2.p2.8.m8.3.3.2.3.cmml" xref="S4.SS2.SSS2.Px2.p2.8.m8.3.3.2.2"><apply id="S4.SS2.SSS2.Px2.p2.8.m8.2.2.1.1.1.cmml" xref="S4.SS2.SSS2.Px2.p2.8.m8.2.2.1.1.1"><csymbol cd="ambiguous" id="S4.SS2.SSS2.Px2.p2.8.m8.2.2.1.1.1.1.cmml" xref="S4.SS2.SSS2.Px2.p2.8.m8.2.2.1.1.1">subscript</csymbol><ci id="S4.SS2.SSS2.Px2.p2.8.m8.2.2.1.1.1.2.cmml" xref="S4.SS2.SSS2.Px2.p2.8.m8.2.2.1.1.1.2">𝜇</ci><ci id="S4.SS2.SSS2.Px2.p2.8.m8.2.2.1.1.1.3.cmml" xref="S4.SS2.SSS2.Px2.p2.8.m8.2.2.1.1.1.3">𝑘</ci></apply><apply id="S4.SS2.SSS2.Px2.p2.8.m8.3.3.2.2.2.cmml" xref="S4.SS2.SSS2.Px2.p2.8.m8.3.3.2.2.2"><csymbol cd="ambiguous" id="S4.SS2.SSS2.Px2.p2.8.m8.3.3.2.2.2.1.cmml" xref="S4.SS2.SSS2.Px2.p2.8.m8.3.3.2.2.2">subscript</csymbol><ci id="S4.SS2.SSS2.Px2.p2.8.m8.3.3.2.2.2.2.cmml" xref="S4.SS2.SSS2.Px2.p2.8.m8.3.3.2.2.2.2">𝜇</ci><cn id="S4.SS2.SSS2.Px2.p2.8.m8.3.3.2.2.2.3.cmml" type="integer" xref="S4.SS2.SSS2.Px2.p2.8.m8.3.3.2.2.2.3">0</cn></apply></set><apply id="S4.SS2.SSS2.Px2.p2.8.m8.3.3.4.cmml" xref="S4.SS2.SSS2.Px2.p2.8.m8.3.3.4"><times id="S4.SS2.SSS2.Px2.p2.8.m8.3.3.4.1.cmml" xref="S4.SS2.SSS2.Px2.p2.8.m8.3.3.4.1"></times><ci id="S4.SS2.SSS2.Px2.p2.8.m8.3.3.4.2a.cmml" xref="S4.SS2.SSS2.Px2.p2.8.m8.3.3.4.2"><mtext id="S4.SS2.SSS2.Px2.p2.8.m8.3.3.4.2.cmml" xref="S4.SS2.SSS2.Px2.p2.8.m8.3.3.4.2">parent</mtext></ci><ci id="S4.SS2.SSS2.Px2.p2.8.m8.1.1.cmml" xref="S4.SS2.SSS2.Px2.p2.8.m8.1.1">𝑥</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS2.Px2.p2.8.m8.3c">\{\mu_{k},\mu_{0}\}=\textnormal{parent}(x)</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS2.Px2.p2.8.m8.3d">{ italic_μ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT , italic_μ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT } = parent ( italic_x )</annotation></semantics></math>. All indices are considered modulo <math alttext="k+1" class="ltx_Math" display="inline" id="S4.SS2.SSS2.Px2.p2.9.m9.1"><semantics id="S4.SS2.SSS2.Px2.p2.9.m9.1a"><mrow id="S4.SS2.SSS2.Px2.p2.9.m9.1.1" xref="S4.SS2.SSS2.Px2.p2.9.m9.1.1.cmml"><mi id="S4.SS2.SSS2.Px2.p2.9.m9.1.1.2" xref="S4.SS2.SSS2.Px2.p2.9.m9.1.1.2.cmml">k</mi><mo id="S4.SS2.SSS2.Px2.p2.9.m9.1.1.1" xref="S4.SS2.SSS2.Px2.p2.9.m9.1.1.1.cmml">+</mo><mn id="S4.SS2.SSS2.Px2.p2.9.m9.1.1.3" xref="S4.SS2.SSS2.Px2.p2.9.m9.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS2.Px2.p2.9.m9.1b"><apply id="S4.SS2.SSS2.Px2.p2.9.m9.1.1.cmml" xref="S4.SS2.SSS2.Px2.p2.9.m9.1.1"><plus id="S4.SS2.SSS2.Px2.p2.9.m9.1.1.1.cmml" xref="S4.SS2.SSS2.Px2.p2.9.m9.1.1.1"></plus><ci id="S4.SS2.SSS2.Px2.p2.9.m9.1.1.2.cmml" xref="S4.SS2.SSS2.Px2.p2.9.m9.1.1.2">𝑘</ci><cn id="S4.SS2.SSS2.Px2.p2.9.m9.1.1.3.cmml" type="integer" xref="S4.SS2.SSS2.Px2.p2.9.m9.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS2.Px2.p2.9.m9.1c">k+1</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS2.Px2.p2.9.m9.1d">italic_k + 1</annotation></semantics></math>; this will sometimes be omitted for the ease of notation. For each <em class="ltx_emph ltx_font_italic" id="S4.SS2.SSS2.Px2.p2.24.1">virtual</em> edge <math alttext="(\mu_{i},\mu_{i+1})" class="ltx_Math" display="inline" id="S4.SS2.SSS2.Px2.p2.10.m10.2"><semantics id="S4.SS2.SSS2.Px2.p2.10.m10.2a"><mrow id="S4.SS2.SSS2.Px2.p2.10.m10.2.2.2" xref="S4.SS2.SSS2.Px2.p2.10.m10.2.2.3.cmml"><mo id="S4.SS2.SSS2.Px2.p2.10.m10.2.2.2.3" stretchy="false" xref="S4.SS2.SSS2.Px2.p2.10.m10.2.2.3.cmml">(</mo><msub id="S4.SS2.SSS2.Px2.p2.10.m10.1.1.1.1" xref="S4.SS2.SSS2.Px2.p2.10.m10.1.1.1.1.cmml"><mi id="S4.SS2.SSS2.Px2.p2.10.m10.1.1.1.1.2" xref="S4.SS2.SSS2.Px2.p2.10.m10.1.1.1.1.2.cmml">μ</mi><mi id="S4.SS2.SSS2.Px2.p2.10.m10.1.1.1.1.3" xref="S4.SS2.SSS2.Px2.p2.10.m10.1.1.1.1.3.cmml">i</mi></msub><mo id="S4.SS2.SSS2.Px2.p2.10.m10.2.2.2.4" xref="S4.SS2.SSS2.Px2.p2.10.m10.2.2.3.cmml">,</mo><msub id="S4.SS2.SSS2.Px2.p2.10.m10.2.2.2.2" xref="S4.SS2.SSS2.Px2.p2.10.m10.2.2.2.2.cmml"><mi id="S4.SS2.SSS2.Px2.p2.10.m10.2.2.2.2.2" xref="S4.SS2.SSS2.Px2.p2.10.m10.2.2.2.2.2.cmml">μ</mi><mrow id="S4.SS2.SSS2.Px2.p2.10.m10.2.2.2.2.3" xref="S4.SS2.SSS2.Px2.p2.10.m10.2.2.2.2.3.cmml"><mi id="S4.SS2.SSS2.Px2.p2.10.m10.2.2.2.2.3.2" xref="S4.SS2.SSS2.Px2.p2.10.m10.2.2.2.2.3.2.cmml">i</mi><mo id="S4.SS2.SSS2.Px2.p2.10.m10.2.2.2.2.3.1" xref="S4.SS2.SSS2.Px2.p2.10.m10.2.2.2.2.3.1.cmml">+</mo><mn id="S4.SS2.SSS2.Px2.p2.10.m10.2.2.2.2.3.3" xref="S4.SS2.SSS2.Px2.p2.10.m10.2.2.2.2.3.3.cmml">1</mn></mrow></msub><mo id="S4.SS2.SSS2.Px2.p2.10.m10.2.2.2.5" stretchy="false" xref="S4.SS2.SSS2.Px2.p2.10.m10.2.2.3.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS2.Px2.p2.10.m10.2b"><interval closure="open" id="S4.SS2.SSS2.Px2.p2.10.m10.2.2.3.cmml" xref="S4.SS2.SSS2.Px2.p2.10.m10.2.2.2"><apply id="S4.SS2.SSS2.Px2.p2.10.m10.1.1.1.1.cmml" xref="S4.SS2.SSS2.Px2.p2.10.m10.1.1.1.1"><csymbol cd="ambiguous" id="S4.SS2.SSS2.Px2.p2.10.m10.1.1.1.1.1.cmml" xref="S4.SS2.SSS2.Px2.p2.10.m10.1.1.1.1">subscript</csymbol><ci id="S4.SS2.SSS2.Px2.p2.10.m10.1.1.1.1.2.cmml" xref="S4.SS2.SSS2.Px2.p2.10.m10.1.1.1.1.2">𝜇</ci><ci id="S4.SS2.SSS2.Px2.p2.10.m10.1.1.1.1.3.cmml" xref="S4.SS2.SSS2.Px2.p2.10.m10.1.1.1.1.3">𝑖</ci></apply><apply id="S4.SS2.SSS2.Px2.p2.10.m10.2.2.2.2.cmml" xref="S4.SS2.SSS2.Px2.p2.10.m10.2.2.2.2"><csymbol cd="ambiguous" id="S4.SS2.SSS2.Px2.p2.10.m10.2.2.2.2.1.cmml" xref="S4.SS2.SSS2.Px2.p2.10.m10.2.2.2.2">subscript</csymbol><ci id="S4.SS2.SSS2.Px2.p2.10.m10.2.2.2.2.2.cmml" xref="S4.SS2.SSS2.Px2.p2.10.m10.2.2.2.2.2">𝜇</ci><apply id="S4.SS2.SSS2.Px2.p2.10.m10.2.2.2.2.3.cmml" xref="S4.SS2.SSS2.Px2.p2.10.m10.2.2.2.2.3"><plus id="S4.SS2.SSS2.Px2.p2.10.m10.2.2.2.2.3.1.cmml" xref="S4.SS2.SSS2.Px2.p2.10.m10.2.2.2.2.3.1"></plus><ci id="S4.SS2.SSS2.Px2.p2.10.m10.2.2.2.2.3.2.cmml" xref="S4.SS2.SSS2.Px2.p2.10.m10.2.2.2.2.3.2">𝑖</ci><cn id="S4.SS2.SSS2.Px2.p2.10.m10.2.2.2.2.3.3.cmml" type="integer" xref="S4.SS2.SSS2.Px2.p2.10.m10.2.2.2.2.3.3">1</cn></apply></apply></interval></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS2.Px2.p2.10.m10.2c">(\mu_{i},\mu_{i+1})</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS2.Px2.p2.10.m10.2d">( italic_μ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT , italic_μ start_POSTSUBSCRIPT italic_i + 1 end_POSTSUBSCRIPT )</annotation></semantics></math>, we subdivide the edge and consider a dummy vertex <math alttext="\mu_{i,i+1}" class="ltx_Math" display="inline" id="S4.SS2.SSS2.Px2.p2.11.m11.2"><semantics id="S4.SS2.SSS2.Px2.p2.11.m11.2a"><msub id="S4.SS2.SSS2.Px2.p2.11.m11.2.3" xref="S4.SS2.SSS2.Px2.p2.11.m11.2.3.cmml"><mi id="S4.SS2.SSS2.Px2.p2.11.m11.2.3.2" xref="S4.SS2.SSS2.Px2.p2.11.m11.2.3.2.cmml">μ</mi><mrow id="S4.SS2.SSS2.Px2.p2.11.m11.2.2.2.2" xref="S4.SS2.SSS2.Px2.p2.11.m11.2.2.2.3.cmml"><mi id="S4.SS2.SSS2.Px2.p2.11.m11.1.1.1.1" xref="S4.SS2.SSS2.Px2.p2.11.m11.1.1.1.1.cmml">i</mi><mo id="S4.SS2.SSS2.Px2.p2.11.m11.2.2.2.2.2" xref="S4.SS2.SSS2.Px2.p2.11.m11.2.2.2.3.cmml">,</mo><mrow id="S4.SS2.SSS2.Px2.p2.11.m11.2.2.2.2.1" xref="S4.SS2.SSS2.Px2.p2.11.m11.2.2.2.2.1.cmml"><mi id="S4.SS2.SSS2.Px2.p2.11.m11.2.2.2.2.1.2" xref="S4.SS2.SSS2.Px2.p2.11.m11.2.2.2.2.1.2.cmml">i</mi><mo id="S4.SS2.SSS2.Px2.p2.11.m11.2.2.2.2.1.1" xref="S4.SS2.SSS2.Px2.p2.11.m11.2.2.2.2.1.1.cmml">+</mo><mn id="S4.SS2.SSS2.Px2.p2.11.m11.2.2.2.2.1.3" xref="S4.SS2.SSS2.Px2.p2.11.m11.2.2.2.2.1.3.cmml">1</mn></mrow></mrow></msub><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS2.Px2.p2.11.m11.2b"><apply id="S4.SS2.SSS2.Px2.p2.11.m11.2.3.cmml" xref="S4.SS2.SSS2.Px2.p2.11.m11.2.3"><csymbol cd="ambiguous" id="S4.SS2.SSS2.Px2.p2.11.m11.2.3.1.cmml" xref="S4.SS2.SSS2.Px2.p2.11.m11.2.3">subscript</csymbol><ci id="S4.SS2.SSS2.Px2.p2.11.m11.2.3.2.cmml" xref="S4.SS2.SSS2.Px2.p2.11.m11.2.3.2">𝜇</ci><list id="S4.SS2.SSS2.Px2.p2.11.m11.2.2.2.3.cmml" xref="S4.SS2.SSS2.Px2.p2.11.m11.2.2.2.2"><ci id="S4.SS2.SSS2.Px2.p2.11.m11.1.1.1.1.cmml" xref="S4.SS2.SSS2.Px2.p2.11.m11.1.1.1.1">𝑖</ci><apply id="S4.SS2.SSS2.Px2.p2.11.m11.2.2.2.2.1.cmml" xref="S4.SS2.SSS2.Px2.p2.11.m11.2.2.2.2.1"><plus id="S4.SS2.SSS2.Px2.p2.11.m11.2.2.2.2.1.1.cmml" xref="S4.SS2.SSS2.Px2.p2.11.m11.2.2.2.2.1.1"></plus><ci id="S4.SS2.SSS2.Px2.p2.11.m11.2.2.2.2.1.2.cmml" xref="S4.SS2.SSS2.Px2.p2.11.m11.2.2.2.2.1.2">𝑖</ci><cn id="S4.SS2.SSS2.Px2.p2.11.m11.2.2.2.2.1.3.cmml" type="integer" xref="S4.SS2.SSS2.Px2.p2.11.m11.2.2.2.2.1.3">1</cn></apply></list></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS2.Px2.p2.11.m11.2c">\mu_{i,i+1}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS2.Px2.p2.11.m11.2d">italic_μ start_POSTSUBSCRIPT italic_i , italic_i + 1 end_POSTSUBSCRIPT</annotation></semantics></math>. For these virtual edges, we let <math alttext="T_{i,i+1}" class="ltx_Math" display="inline" id="S4.SS2.SSS2.Px2.p2.12.m12.2"><semantics id="S4.SS2.SSS2.Px2.p2.12.m12.2a"><msub id="S4.SS2.SSS2.Px2.p2.12.m12.2.3" xref="S4.SS2.SSS2.Px2.p2.12.m12.2.3.cmml"><mi id="S4.SS2.SSS2.Px2.p2.12.m12.2.3.2" xref="S4.SS2.SSS2.Px2.p2.12.m12.2.3.2.cmml">T</mi><mrow id="S4.SS2.SSS2.Px2.p2.12.m12.2.2.2.2" xref="S4.SS2.SSS2.Px2.p2.12.m12.2.2.2.3.cmml"><mi id="S4.SS2.SSS2.Px2.p2.12.m12.1.1.1.1" xref="S4.SS2.SSS2.Px2.p2.12.m12.1.1.1.1.cmml">i</mi><mo id="S4.SS2.SSS2.Px2.p2.12.m12.2.2.2.2.2" xref="S4.SS2.SSS2.Px2.p2.12.m12.2.2.2.3.cmml">,</mo><mrow id="S4.SS2.SSS2.Px2.p2.12.m12.2.2.2.2.1" xref="S4.SS2.SSS2.Px2.p2.12.m12.2.2.2.2.1.cmml"><mi id="S4.SS2.SSS2.Px2.p2.12.m12.2.2.2.2.1.2" xref="S4.SS2.SSS2.Px2.p2.12.m12.2.2.2.2.1.2.cmml">i</mi><mo id="S4.SS2.SSS2.Px2.p2.12.m12.2.2.2.2.1.1" xref="S4.SS2.SSS2.Px2.p2.12.m12.2.2.2.2.1.1.cmml">+</mo><mn id="S4.SS2.SSS2.Px2.p2.12.m12.2.2.2.2.1.3" xref="S4.SS2.SSS2.Px2.p2.12.m12.2.2.2.2.1.3.cmml">1</mn></mrow></mrow></msub><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS2.Px2.p2.12.m12.2b"><apply id="S4.SS2.SSS2.Px2.p2.12.m12.2.3.cmml" xref="S4.SS2.SSS2.Px2.p2.12.m12.2.3"><csymbol cd="ambiguous" id="S4.SS2.SSS2.Px2.p2.12.m12.2.3.1.cmml" xref="S4.SS2.SSS2.Px2.p2.12.m12.2.3">subscript</csymbol><ci id="S4.SS2.SSS2.Px2.p2.12.m12.2.3.2.cmml" xref="S4.SS2.SSS2.Px2.p2.12.m12.2.3.2">𝑇</ci><list id="S4.SS2.SSS2.Px2.p2.12.m12.2.2.2.3.cmml" xref="S4.SS2.SSS2.Px2.p2.12.m12.2.2.2.2"><ci id="S4.SS2.SSS2.Px2.p2.12.m12.1.1.1.1.cmml" xref="S4.SS2.SSS2.Px2.p2.12.m12.1.1.1.1">𝑖</ci><apply id="S4.SS2.SSS2.Px2.p2.12.m12.2.2.2.2.1.cmml" xref="S4.SS2.SSS2.Px2.p2.12.m12.2.2.2.2.1"><plus id="S4.SS2.SSS2.Px2.p2.12.m12.2.2.2.2.1.1.cmml" xref="S4.SS2.SSS2.Px2.p2.12.m12.2.2.2.2.1.1"></plus><ci id="S4.SS2.SSS2.Px2.p2.12.m12.2.2.2.2.1.2.cmml" xref="S4.SS2.SSS2.Px2.p2.12.m12.2.2.2.2.1.2">𝑖</ci><cn id="S4.SS2.SSS2.Px2.p2.12.m12.2.2.2.2.1.3.cmml" type="integer" xref="S4.SS2.SSS2.Px2.p2.12.m12.2.2.2.2.1.3">1</cn></apply></list></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS2.Px2.p2.12.m12.2c">T_{i,i+1}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS2.Px2.p2.12.m12.2d">italic_T start_POSTSUBSCRIPT italic_i , italic_i + 1 end_POSTSUBSCRIPT</annotation></semantics></math> denote the subtree rooted at <math alttext="(\mu_{i},\mu_{i+1})" class="ltx_Math" display="inline" id="S4.SS2.SSS2.Px2.p2.13.m13.2"><semantics id="S4.SS2.SSS2.Px2.p2.13.m13.2a"><mrow id="S4.SS2.SSS2.Px2.p2.13.m13.2.2.2" xref="S4.SS2.SSS2.Px2.p2.13.m13.2.2.3.cmml"><mo id="S4.SS2.SSS2.Px2.p2.13.m13.2.2.2.3" stretchy="false" xref="S4.SS2.SSS2.Px2.p2.13.m13.2.2.3.cmml">(</mo><msub id="S4.SS2.SSS2.Px2.p2.13.m13.1.1.1.1" xref="S4.SS2.SSS2.Px2.p2.13.m13.1.1.1.1.cmml"><mi id="S4.SS2.SSS2.Px2.p2.13.m13.1.1.1.1.2" xref="S4.SS2.SSS2.Px2.p2.13.m13.1.1.1.1.2.cmml">μ</mi><mi id="S4.SS2.SSS2.Px2.p2.13.m13.1.1.1.1.3" xref="S4.SS2.SSS2.Px2.p2.13.m13.1.1.1.1.3.cmml">i</mi></msub><mo id="S4.SS2.SSS2.Px2.p2.13.m13.2.2.2.4" xref="S4.SS2.SSS2.Px2.p2.13.m13.2.2.3.cmml">,</mo><msub id="S4.SS2.SSS2.Px2.p2.13.m13.2.2.2.2" xref="S4.SS2.SSS2.Px2.p2.13.m13.2.2.2.2.cmml"><mi id="S4.SS2.SSS2.Px2.p2.13.m13.2.2.2.2.2" xref="S4.SS2.SSS2.Px2.p2.13.m13.2.2.2.2.2.cmml">μ</mi><mrow id="S4.SS2.SSS2.Px2.p2.13.m13.2.2.2.2.3" xref="S4.SS2.SSS2.Px2.p2.13.m13.2.2.2.2.3.cmml"><mi id="S4.SS2.SSS2.Px2.p2.13.m13.2.2.2.2.3.2" xref="S4.SS2.SSS2.Px2.p2.13.m13.2.2.2.2.3.2.cmml">i</mi><mo id="S4.SS2.SSS2.Px2.p2.13.m13.2.2.2.2.3.1" xref="S4.SS2.SSS2.Px2.p2.13.m13.2.2.2.2.3.1.cmml">+</mo><mn id="S4.SS2.SSS2.Px2.p2.13.m13.2.2.2.2.3.3" xref="S4.SS2.SSS2.Px2.p2.13.m13.2.2.2.2.3.3.cmml">1</mn></mrow></msub><mo id="S4.SS2.SSS2.Px2.p2.13.m13.2.2.2.5" stretchy="false" xref="S4.SS2.SSS2.Px2.p2.13.m13.2.2.3.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS2.Px2.p2.13.m13.2b"><interval closure="open" id="S4.SS2.SSS2.Px2.p2.13.m13.2.2.3.cmml" xref="S4.SS2.SSS2.Px2.p2.13.m13.2.2.2"><apply id="S4.SS2.SSS2.Px2.p2.13.m13.1.1.1.1.cmml" xref="S4.SS2.SSS2.Px2.p2.13.m13.1.1.1.1"><csymbol cd="ambiguous" id="S4.SS2.SSS2.Px2.p2.13.m13.1.1.1.1.1.cmml" xref="S4.SS2.SSS2.Px2.p2.13.m13.1.1.1.1">subscript</csymbol><ci id="S4.SS2.SSS2.Px2.p2.13.m13.1.1.1.1.2.cmml" xref="S4.SS2.SSS2.Px2.p2.13.m13.1.1.1.1.2">𝜇</ci><ci id="S4.SS2.SSS2.Px2.p2.13.m13.1.1.1.1.3.cmml" xref="S4.SS2.SSS2.Px2.p2.13.m13.1.1.1.1.3">𝑖</ci></apply><apply id="S4.SS2.SSS2.Px2.p2.13.m13.2.2.2.2.cmml" xref="S4.SS2.SSS2.Px2.p2.13.m13.2.2.2.2"><csymbol cd="ambiguous" id="S4.SS2.SSS2.Px2.p2.13.m13.2.2.2.2.1.cmml" xref="S4.SS2.SSS2.Px2.p2.13.m13.2.2.2.2">subscript</csymbol><ci id="S4.SS2.SSS2.Px2.p2.13.m13.2.2.2.2.2.cmml" xref="S4.SS2.SSS2.Px2.p2.13.m13.2.2.2.2.2">𝜇</ci><apply id="S4.SS2.SSS2.Px2.p2.13.m13.2.2.2.2.3.cmml" xref="S4.SS2.SSS2.Px2.p2.13.m13.2.2.2.2.3"><plus id="S4.SS2.SSS2.Px2.p2.13.m13.2.2.2.2.3.1.cmml" xref="S4.SS2.SSS2.Px2.p2.13.m13.2.2.2.2.3.1"></plus><ci id="S4.SS2.SSS2.Px2.p2.13.m13.2.2.2.2.3.2.cmml" xref="S4.SS2.SSS2.Px2.p2.13.m13.2.2.2.2.3.2">𝑖</ci><cn id="S4.SS2.SSS2.Px2.p2.13.m13.2.2.2.2.3.3.cmml" type="integer" xref="S4.SS2.SSS2.Px2.p2.13.m13.2.2.2.2.3.3">1</cn></apply></apply></interval></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS2.Px2.p2.13.m13.2c">(\mu_{i},\mu_{i+1})</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS2.Px2.p2.13.m13.2d">( italic_μ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT , italic_μ start_POSTSUBSCRIPT italic_i + 1 end_POSTSUBSCRIPT )</annotation></semantics></math>. We define a function <math alttext="f_{x}" class="ltx_Math" display="inline" id="S4.SS2.SSS2.Px2.p2.14.m14.1"><semantics id="S4.SS2.SSS2.Px2.p2.14.m14.1a"><msub id="S4.SS2.SSS2.Px2.p2.14.m14.1.1" xref="S4.SS2.SSS2.Px2.p2.14.m14.1.1.cmml"><mi id="S4.SS2.SSS2.Px2.p2.14.m14.1.1.2" xref="S4.SS2.SSS2.Px2.p2.14.m14.1.1.2.cmml">f</mi><mi id="S4.SS2.SSS2.Px2.p2.14.m14.1.1.3" xref="S4.SS2.SSS2.Px2.p2.14.m14.1.1.3.cmml">x</mi></msub><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS2.Px2.p2.14.m14.1b"><apply id="S4.SS2.SSS2.Px2.p2.14.m14.1.1.cmml" xref="S4.SS2.SSS2.Px2.p2.14.m14.1.1"><csymbol cd="ambiguous" id="S4.SS2.SSS2.Px2.p2.14.m14.1.1.1.cmml" xref="S4.SS2.SSS2.Px2.p2.14.m14.1.1">subscript</csymbol><ci id="S4.SS2.SSS2.Px2.p2.14.m14.1.1.2.cmml" xref="S4.SS2.SSS2.Px2.p2.14.m14.1.1.2">𝑓</ci><ci id="S4.SS2.SSS2.Px2.p2.14.m14.1.1.3.cmml" xref="S4.SS2.SSS2.Px2.p2.14.m14.1.1.3">𝑥</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS2.Px2.p2.14.m14.1c">f_{x}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS2.Px2.p2.14.m14.1d">italic_f start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT</annotation></semantics></math> on the vertices of <math alttext="G" class="ltx_Math" display="inline" id="S4.SS2.SSS2.Px2.p2.15.m15.1"><semantics id="S4.SS2.SSS2.Px2.p2.15.m15.1a"><mi id="S4.SS2.SSS2.Px2.p2.15.m15.1.1" xref="S4.SS2.SSS2.Px2.p2.15.m15.1.1.cmml">G</mi><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS2.Px2.p2.15.m15.1b"><ci id="S4.SS2.SSS2.Px2.p2.15.m15.1.1.cmml" xref="S4.SS2.SSS2.Px2.p2.15.m15.1.1">𝐺</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS2.Px2.p2.15.m15.1c">G</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS2.Px2.p2.15.m15.1d">italic_G</annotation></semantics></math> that maps each vertex to its corresponding point on the cycle: <math alttext="f_{x}(u)=\mu_{i,i+1}" class="ltx_Math" display="inline" id="S4.SS2.SSS2.Px2.p2.16.m16.3"><semantics id="S4.SS2.SSS2.Px2.p2.16.m16.3a"><mrow id="S4.SS2.SSS2.Px2.p2.16.m16.3.4" xref="S4.SS2.SSS2.Px2.p2.16.m16.3.4.cmml"><mrow id="S4.SS2.SSS2.Px2.p2.16.m16.3.4.2" xref="S4.SS2.SSS2.Px2.p2.16.m16.3.4.2.cmml"><msub id="S4.SS2.SSS2.Px2.p2.16.m16.3.4.2.2" xref="S4.SS2.SSS2.Px2.p2.16.m16.3.4.2.2.cmml"><mi id="S4.SS2.SSS2.Px2.p2.16.m16.3.4.2.2.2" xref="S4.SS2.SSS2.Px2.p2.16.m16.3.4.2.2.2.cmml">f</mi><mi id="S4.SS2.SSS2.Px2.p2.16.m16.3.4.2.2.3" xref="S4.SS2.SSS2.Px2.p2.16.m16.3.4.2.2.3.cmml">x</mi></msub><mo id="S4.SS2.SSS2.Px2.p2.16.m16.3.4.2.1" xref="S4.SS2.SSS2.Px2.p2.16.m16.3.4.2.1.cmml">⁢</mo><mrow id="S4.SS2.SSS2.Px2.p2.16.m16.3.4.2.3.2" xref="S4.SS2.SSS2.Px2.p2.16.m16.3.4.2.cmml"><mo id="S4.SS2.SSS2.Px2.p2.16.m16.3.4.2.3.2.1" stretchy="false" xref="S4.SS2.SSS2.Px2.p2.16.m16.3.4.2.cmml">(</mo><mi id="S4.SS2.SSS2.Px2.p2.16.m16.3.3" xref="S4.SS2.SSS2.Px2.p2.16.m16.3.3.cmml">u</mi><mo id="S4.SS2.SSS2.Px2.p2.16.m16.3.4.2.3.2.2" stretchy="false" xref="S4.SS2.SSS2.Px2.p2.16.m16.3.4.2.cmml">)</mo></mrow></mrow><mo id="S4.SS2.SSS2.Px2.p2.16.m16.3.4.1" xref="S4.SS2.SSS2.Px2.p2.16.m16.3.4.1.cmml">=</mo><msub id="S4.SS2.SSS2.Px2.p2.16.m16.3.4.3" xref="S4.SS2.SSS2.Px2.p2.16.m16.3.4.3.cmml"><mi id="S4.SS2.SSS2.Px2.p2.16.m16.3.4.3.2" xref="S4.SS2.SSS2.Px2.p2.16.m16.3.4.3.2.cmml">μ</mi><mrow id="S4.SS2.SSS2.Px2.p2.16.m16.2.2.2.2" xref="S4.SS2.SSS2.Px2.p2.16.m16.2.2.2.3.cmml"><mi id="S4.SS2.SSS2.Px2.p2.16.m16.1.1.1.1" xref="S4.SS2.SSS2.Px2.p2.16.m16.1.1.1.1.cmml">i</mi><mo id="S4.SS2.SSS2.Px2.p2.16.m16.2.2.2.2.2" xref="S4.SS2.SSS2.Px2.p2.16.m16.2.2.2.3.cmml">,</mo><mrow id="S4.SS2.SSS2.Px2.p2.16.m16.2.2.2.2.1" xref="S4.SS2.SSS2.Px2.p2.16.m16.2.2.2.2.1.cmml"><mi id="S4.SS2.SSS2.Px2.p2.16.m16.2.2.2.2.1.2" xref="S4.SS2.SSS2.Px2.p2.16.m16.2.2.2.2.1.2.cmml">i</mi><mo id="S4.SS2.SSS2.Px2.p2.16.m16.2.2.2.2.1.1" xref="S4.SS2.SSS2.Px2.p2.16.m16.2.2.2.2.1.1.cmml">+</mo><mn id="S4.SS2.SSS2.Px2.p2.16.m16.2.2.2.2.1.3" xref="S4.SS2.SSS2.Px2.p2.16.m16.2.2.2.2.1.3.cmml">1</mn></mrow></mrow></msub></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS2.Px2.p2.16.m16.3b"><apply id="S4.SS2.SSS2.Px2.p2.16.m16.3.4.cmml" xref="S4.SS2.SSS2.Px2.p2.16.m16.3.4"><eq id="S4.SS2.SSS2.Px2.p2.16.m16.3.4.1.cmml" xref="S4.SS2.SSS2.Px2.p2.16.m16.3.4.1"></eq><apply id="S4.SS2.SSS2.Px2.p2.16.m16.3.4.2.cmml" xref="S4.SS2.SSS2.Px2.p2.16.m16.3.4.2"><times id="S4.SS2.SSS2.Px2.p2.16.m16.3.4.2.1.cmml" xref="S4.SS2.SSS2.Px2.p2.16.m16.3.4.2.1"></times><apply id="S4.SS2.SSS2.Px2.p2.16.m16.3.4.2.2.cmml" xref="S4.SS2.SSS2.Px2.p2.16.m16.3.4.2.2"><csymbol cd="ambiguous" id="S4.SS2.SSS2.Px2.p2.16.m16.3.4.2.2.1.cmml" xref="S4.SS2.SSS2.Px2.p2.16.m16.3.4.2.2">subscript</csymbol><ci id="S4.SS2.SSS2.Px2.p2.16.m16.3.4.2.2.2.cmml" xref="S4.SS2.SSS2.Px2.p2.16.m16.3.4.2.2.2">𝑓</ci><ci id="S4.SS2.SSS2.Px2.p2.16.m16.3.4.2.2.3.cmml" xref="S4.SS2.SSS2.Px2.p2.16.m16.3.4.2.2.3">𝑥</ci></apply><ci id="S4.SS2.SSS2.Px2.p2.16.m16.3.3.cmml" xref="S4.SS2.SSS2.Px2.p2.16.m16.3.3">𝑢</ci></apply><apply id="S4.SS2.SSS2.Px2.p2.16.m16.3.4.3.cmml" xref="S4.SS2.SSS2.Px2.p2.16.m16.3.4.3"><csymbol cd="ambiguous" id="S4.SS2.SSS2.Px2.p2.16.m16.3.4.3.1.cmml" xref="S4.SS2.SSS2.Px2.p2.16.m16.3.4.3">subscript</csymbol><ci id="S4.SS2.SSS2.Px2.p2.16.m16.3.4.3.2.cmml" xref="S4.SS2.SSS2.Px2.p2.16.m16.3.4.3.2">𝜇</ci><list id="S4.SS2.SSS2.Px2.p2.16.m16.2.2.2.3.cmml" xref="S4.SS2.SSS2.Px2.p2.16.m16.2.2.2.2"><ci id="S4.SS2.SSS2.Px2.p2.16.m16.1.1.1.1.cmml" xref="S4.SS2.SSS2.Px2.p2.16.m16.1.1.1.1">𝑖</ci><apply id="S4.SS2.SSS2.Px2.p2.16.m16.2.2.2.2.1.cmml" xref="S4.SS2.SSS2.Px2.p2.16.m16.2.2.2.2.1"><plus id="S4.SS2.SSS2.Px2.p2.16.m16.2.2.2.2.1.1.cmml" xref="S4.SS2.SSS2.Px2.p2.16.m16.2.2.2.2.1.1"></plus><ci id="S4.SS2.SSS2.Px2.p2.16.m16.2.2.2.2.1.2.cmml" xref="S4.SS2.SSS2.Px2.p2.16.m16.2.2.2.2.1.2">𝑖</ci><cn id="S4.SS2.SSS2.Px2.p2.16.m16.2.2.2.2.1.3.cmml" type="integer" xref="S4.SS2.SSS2.Px2.p2.16.m16.2.2.2.2.1.3">1</cn></apply></list></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS2.Px2.p2.16.m16.3c">f_{x}(u)=\mu_{i,i+1}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS2.Px2.p2.16.m16.3d">italic_f start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT ( italic_u ) = italic_μ start_POSTSUBSCRIPT italic_i , italic_i + 1 end_POSTSUBSCRIPT</annotation></semantics></math> if <math alttext="u\in T_{i,i+1}\setminus G_{x}" class="ltx_Math" display="inline" id="S4.SS2.SSS2.Px2.p2.17.m17.2"><semantics id="S4.SS2.SSS2.Px2.p2.17.m17.2a"><mrow id="S4.SS2.SSS2.Px2.p2.17.m17.2.3" xref="S4.SS2.SSS2.Px2.p2.17.m17.2.3.cmml"><mi id="S4.SS2.SSS2.Px2.p2.17.m17.2.3.2" xref="S4.SS2.SSS2.Px2.p2.17.m17.2.3.2.cmml">u</mi><mo id="S4.SS2.SSS2.Px2.p2.17.m17.2.3.1" xref="S4.SS2.SSS2.Px2.p2.17.m17.2.3.1.cmml">∈</mo><mrow id="S4.SS2.SSS2.Px2.p2.17.m17.2.3.3" xref="S4.SS2.SSS2.Px2.p2.17.m17.2.3.3.cmml"><msub 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xref="S4.SS2.SSS2.Px2.p2.17.m17.2.3.3.3">subscript</csymbol><ci id="S4.SS2.SSS2.Px2.p2.17.m17.2.3.3.3.2.cmml" xref="S4.SS2.SSS2.Px2.p2.17.m17.2.3.3.3.2">𝐺</ci><ci id="S4.SS2.SSS2.Px2.p2.17.m17.2.3.3.3.3.cmml" xref="S4.SS2.SSS2.Px2.p2.17.m17.2.3.3.3.3">𝑥</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS2.Px2.p2.17.m17.2c">u\in T_{i,i+1}\setminus G_{x}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS2.Px2.p2.17.m17.2d">italic_u ∈ italic_T start_POSTSUBSCRIPT italic_i , italic_i + 1 end_POSTSUBSCRIPT ∖ italic_G start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="f_{x}(\mu_{i})=\mu_{i}" class="ltx_Math" display="inline" id="S4.SS2.SSS2.Px2.p2.18.m18.1"><semantics id="S4.SS2.SSS2.Px2.p2.18.m18.1a"><mrow id="S4.SS2.SSS2.Px2.p2.18.m18.1.1" xref="S4.SS2.SSS2.Px2.p2.18.m18.1.1.cmml"><mrow id="S4.SS2.SSS2.Px2.p2.18.m18.1.1.1" xref="S4.SS2.SSS2.Px2.p2.18.m18.1.1.1.cmml"><msub id="S4.SS2.SSS2.Px2.p2.18.m18.1.1.1.3" xref="S4.SS2.SSS2.Px2.p2.18.m18.1.1.1.3.cmml"><mi id="S4.SS2.SSS2.Px2.p2.18.m18.1.1.1.3.2" xref="S4.SS2.SSS2.Px2.p2.18.m18.1.1.1.3.2.cmml">f</mi><mi id="S4.SS2.SSS2.Px2.p2.18.m18.1.1.1.3.3" xref="S4.SS2.SSS2.Px2.p2.18.m18.1.1.1.3.3.cmml">x</mi></msub><mo id="S4.SS2.SSS2.Px2.p2.18.m18.1.1.1.2" xref="S4.SS2.SSS2.Px2.p2.18.m18.1.1.1.2.cmml">⁢</mo><mrow id="S4.SS2.SSS2.Px2.p2.18.m18.1.1.1.1.1" xref="S4.SS2.SSS2.Px2.p2.18.m18.1.1.1.1.1.1.cmml"><mo id="S4.SS2.SSS2.Px2.p2.18.m18.1.1.1.1.1.2" stretchy="false" xref="S4.SS2.SSS2.Px2.p2.18.m18.1.1.1.1.1.1.cmml">(</mo><msub id="S4.SS2.SSS2.Px2.p2.18.m18.1.1.1.1.1.1" xref="S4.SS2.SSS2.Px2.p2.18.m18.1.1.1.1.1.1.cmml"><mi id="S4.SS2.SSS2.Px2.p2.18.m18.1.1.1.1.1.1.2" xref="S4.SS2.SSS2.Px2.p2.18.m18.1.1.1.1.1.1.2.cmml">μ</mi><mi id="S4.SS2.SSS2.Px2.p2.18.m18.1.1.1.1.1.1.3" xref="S4.SS2.SSS2.Px2.p2.18.m18.1.1.1.1.1.1.3.cmml">i</mi></msub><mo id="S4.SS2.SSS2.Px2.p2.18.m18.1.1.1.1.1.3" stretchy="false" xref="S4.SS2.SSS2.Px2.p2.18.m18.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S4.SS2.SSS2.Px2.p2.18.m18.1.1.2" xref="S4.SS2.SSS2.Px2.p2.18.m18.1.1.2.cmml">=</mo><msub id="S4.SS2.SSS2.Px2.p2.18.m18.1.1.3" xref="S4.SS2.SSS2.Px2.p2.18.m18.1.1.3.cmml"><mi id="S4.SS2.SSS2.Px2.p2.18.m18.1.1.3.2" xref="S4.SS2.SSS2.Px2.p2.18.m18.1.1.3.2.cmml">μ</mi><mi id="S4.SS2.SSS2.Px2.p2.18.m18.1.1.3.3" xref="S4.SS2.SSS2.Px2.p2.18.m18.1.1.3.3.cmml">i</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS2.Px2.p2.18.m18.1b"><apply id="S4.SS2.SSS2.Px2.p2.18.m18.1.1.cmml" xref="S4.SS2.SSS2.Px2.p2.18.m18.1.1"><eq id="S4.SS2.SSS2.Px2.p2.18.m18.1.1.2.cmml" xref="S4.SS2.SSS2.Px2.p2.18.m18.1.1.2"></eq><apply id="S4.SS2.SSS2.Px2.p2.18.m18.1.1.1.cmml" xref="S4.SS2.SSS2.Px2.p2.18.m18.1.1.1"><times id="S4.SS2.SSS2.Px2.p2.18.m18.1.1.1.2.cmml" xref="S4.SS2.SSS2.Px2.p2.18.m18.1.1.1.2"></times><apply id="S4.SS2.SSS2.Px2.p2.18.m18.1.1.1.3.cmml" xref="S4.SS2.SSS2.Px2.p2.18.m18.1.1.1.3"><csymbol cd="ambiguous" id="S4.SS2.SSS2.Px2.p2.18.m18.1.1.1.3.1.cmml" xref="S4.SS2.SSS2.Px2.p2.18.m18.1.1.1.3">subscript</csymbol><ci id="S4.SS2.SSS2.Px2.p2.18.m18.1.1.1.3.2.cmml" xref="S4.SS2.SSS2.Px2.p2.18.m18.1.1.1.3.2">𝑓</ci><ci id="S4.SS2.SSS2.Px2.p2.18.m18.1.1.1.3.3.cmml" xref="S4.SS2.SSS2.Px2.p2.18.m18.1.1.1.3.3">𝑥</ci></apply><apply id="S4.SS2.SSS2.Px2.p2.18.m18.1.1.1.1.1.1.cmml" xref="S4.SS2.SSS2.Px2.p2.18.m18.1.1.1.1.1"><csymbol cd="ambiguous" id="S4.SS2.SSS2.Px2.p2.18.m18.1.1.1.1.1.1.1.cmml" xref="S4.SS2.SSS2.Px2.p2.18.m18.1.1.1.1.1">subscript</csymbol><ci id="S4.SS2.SSS2.Px2.p2.18.m18.1.1.1.1.1.1.2.cmml" xref="S4.SS2.SSS2.Px2.p2.18.m18.1.1.1.1.1.1.2">𝜇</ci><ci id="S4.SS2.SSS2.Px2.p2.18.m18.1.1.1.1.1.1.3.cmml" xref="S4.SS2.SSS2.Px2.p2.18.m18.1.1.1.1.1.1.3">𝑖</ci></apply></apply><apply id="S4.SS2.SSS2.Px2.p2.18.m18.1.1.3.cmml" xref="S4.SS2.SSS2.Px2.p2.18.m18.1.1.3"><csymbol cd="ambiguous" id="S4.SS2.SSS2.Px2.p2.18.m18.1.1.3.1.cmml" xref="S4.SS2.SSS2.Px2.p2.18.m18.1.1.3">subscript</csymbol><ci id="S4.SS2.SSS2.Px2.p2.18.m18.1.1.3.2.cmml" xref="S4.SS2.SSS2.Px2.p2.18.m18.1.1.3.2">𝜇</ci><ci id="S4.SS2.SSS2.Px2.p2.18.m18.1.1.3.3.cmml" xref="S4.SS2.SSS2.Px2.p2.18.m18.1.1.3.3">𝑖</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS2.Px2.p2.18.m18.1c">f_{x}(\mu_{i})=\mu_{i}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS2.Px2.p2.18.m18.1d">italic_f start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT ( italic_μ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ) = italic_μ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math>. See Figure <a class="ltx_ref" href="https://arxiv.org/html/2503.00712v1#S4.F7" title="Figure 7 ‣ “Cycle” Cuts: ‣ 4.2.2 The Streaming Algorithm ‣ 4.2 Two-to-Three Augmentation ‣ 4 Vertex Connectivity Augmentation in Link-Arrival Model ‣ Streaming Algorithms for Network Design"><span class="ltx_text ltx_ref_tag">7</span></a> for an example. We define a natural ordering <math alttext="\prec_{x}" class="ltx_Math" display="inline" id="S4.SS2.SSS2.Px2.p2.19.m19.1"><semantics id="S4.SS2.SSS2.Px2.p2.19.m19.1a"><msub id="S4.SS2.SSS2.Px2.p2.19.m19.1.1" xref="S4.SS2.SSS2.Px2.p2.19.m19.1.1.cmml"><mo id="S4.SS2.SSS2.Px2.p2.19.m19.1.1.2" xref="S4.SS2.SSS2.Px2.p2.19.m19.1.1.2.cmml">≺</mo><mi id="S4.SS2.SSS2.Px2.p2.19.m19.1.1.3" xref="S4.SS2.SSS2.Px2.p2.19.m19.1.1.3.cmml">x</mi></msub><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS2.Px2.p2.19.m19.1b"><apply id="S4.SS2.SSS2.Px2.p2.19.m19.1.1.cmml" xref="S4.SS2.SSS2.Px2.p2.19.m19.1.1"><csymbol cd="ambiguous" id="S4.SS2.SSS2.Px2.p2.19.m19.1.1.1.cmml" xref="S4.SS2.SSS2.Px2.p2.19.m19.1.1">subscript</csymbol><csymbol cd="latexml" id="S4.SS2.SSS2.Px2.p2.19.m19.1.1.2.cmml" xref="S4.SS2.SSS2.Px2.p2.19.m19.1.1.2">precedes</csymbol><ci id="S4.SS2.SSS2.Px2.p2.19.m19.1.1.3.cmml" xref="S4.SS2.SSS2.Px2.p2.19.m19.1.1.3">𝑥</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS2.Px2.p2.19.m19.1c">\prec_{x}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS2.Px2.p2.19.m19.1d">≺ start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT</annotation></semantics></math> on the vertices of the cycle in terms of their appearance on the cycle, so <math alttext="\mu_{0,k}\prec_{x}\mu_{0}\prec_{x}\mu_{0,1}\prec_{x}\mu_{1}\prec_{x}\dots\prec% _{x}\mu_{k}" class="ltx_Math" display="inline" id="S4.SS2.SSS2.Px2.p2.20.m20.4"><semantics id="S4.SS2.SSS2.Px2.p2.20.m20.4a"><mrow id="S4.SS2.SSS2.Px2.p2.20.m20.4.5" xref="S4.SS2.SSS2.Px2.p2.20.m20.4.5.cmml"><msub id="S4.SS2.SSS2.Px2.p2.20.m20.4.5.2" xref="S4.SS2.SSS2.Px2.p2.20.m20.4.5.2.cmml"><mi id="S4.SS2.SSS2.Px2.p2.20.m20.4.5.2.2" xref="S4.SS2.SSS2.Px2.p2.20.m20.4.5.2.2.cmml">μ</mi><mrow id="S4.SS2.SSS2.Px2.p2.20.m20.2.2.2.4" xref="S4.SS2.SSS2.Px2.p2.20.m20.2.2.2.3.cmml"><mn id="S4.SS2.SSS2.Px2.p2.20.m20.1.1.1.1" xref="S4.SS2.SSS2.Px2.p2.20.m20.1.1.1.1.cmml">0</mn><mo 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id="S4.SS2.SSS2.Px2.p2.20.m20.4.5.9.2.cmml" xref="S4.SS2.SSS2.Px2.p2.20.m20.4.5.9.2">precedes</csymbol><ci id="S4.SS2.SSS2.Px2.p2.20.m20.4.5.9.3.cmml" xref="S4.SS2.SSS2.Px2.p2.20.m20.4.5.9.3">𝑥</ci></apply><share href="https://arxiv.org/html/2503.00712v1#S4.SS2.SSS2.Px2.p2.20.m20.4.5.8.cmml" id="S4.SS2.SSS2.Px2.p2.20.m20.4.5h.cmml" xref="S4.SS2.SSS2.Px2.p2.20.m20.4.5"></share><ci id="S4.SS2.SSS2.Px2.p2.20.m20.4.5.10.cmml" xref="S4.SS2.SSS2.Px2.p2.20.m20.4.5.10">⋯</ci></apply><apply id="S4.SS2.SSS2.Px2.p2.20.m20.4.5i.cmml" xref="S4.SS2.SSS2.Px2.p2.20.m20.4.5"><apply id="S4.SS2.SSS2.Px2.p2.20.m20.4.5.11.cmml" xref="S4.SS2.SSS2.Px2.p2.20.m20.4.5.11"><csymbol cd="ambiguous" id="S4.SS2.SSS2.Px2.p2.20.m20.4.5.11.1.cmml" xref="S4.SS2.SSS2.Px2.p2.20.m20.4.5.11">subscript</csymbol><csymbol cd="latexml" id="S4.SS2.SSS2.Px2.p2.20.m20.4.5.11.2.cmml" xref="S4.SS2.SSS2.Px2.p2.20.m20.4.5.11.2">precedes</csymbol><ci id="S4.SS2.SSS2.Px2.p2.20.m20.4.5.11.3.cmml" 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end_POSTSUBSCRIPT ≺ start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT italic_μ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ≺ start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT italic_μ start_POSTSUBSCRIPT 0 , 1 end_POSTSUBSCRIPT ≺ start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT italic_μ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ≺ start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT ⋯ ≺ start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT italic_μ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT</annotation></semantics></math>. For each vertex <math alttext="\mu" class="ltx_Math" display="inline" id="S4.SS2.SSS2.Px2.p2.21.m21.1"><semantics id="S4.SS2.SSS2.Px2.p2.21.m21.1a"><mi id="S4.SS2.SSS2.Px2.p2.21.m21.1.1" xref="S4.SS2.SSS2.Px2.p2.21.m21.1.1.cmml">μ</mi><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS2.Px2.p2.21.m21.1b"><ci id="S4.SS2.SSS2.Px2.p2.21.m21.1.1.cmml" xref="S4.SS2.SSS2.Px2.p2.21.m21.1.1">𝜇</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS2.Px2.p2.21.m21.1c">\mu</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS2.Px2.p2.21.m21.1d">italic_μ</annotation></semantics></math> on the cycle (original or dummy vertex) and each weight class <math alttext="j" class="ltx_Math" display="inline" id="S4.SS2.SSS2.Px2.p2.22.m22.1"><semantics id="S4.SS2.SSS2.Px2.p2.22.m22.1a"><mi id="S4.SS2.SSS2.Px2.p2.22.m22.1.1" xref="S4.SS2.SSS2.Px2.p2.22.m22.1.1.cmml">j</mi><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS2.Px2.p2.22.m22.1b"><ci id="S4.SS2.SSS2.Px2.p2.22.m22.1.1.cmml" xref="S4.SS2.SSS2.Px2.p2.22.m22.1.1">𝑗</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS2.Px2.p2.22.m22.1c">j</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS2.Px2.p2.22.m22.1d">italic_j</annotation></semantics></math>, we store the links <math alttext="\textsc{Min}_{\mu}(j)=\textnormal{argmin}_{(u,v)\in L}\{f_{x}(v):f_{x}(u)=\mu,% w(u,v)\in[(1+\epsilon)^{j},(1+\epsilon)^{j+1})\}" class="ltx_Math" display="inline" id="S4.SS2.SSS2.Px2.p2.23.m23.9"><semantics id="S4.SS2.SSS2.Px2.p2.23.m23.9a"><mrow id="S4.SS2.SSS2.Px2.p2.23.m23.9.9" xref="S4.SS2.SSS2.Px2.p2.23.m23.9.9.cmml"><mrow id="S4.SS2.SSS2.Px2.p2.23.m23.9.9.4" xref="S4.SS2.SSS2.Px2.p2.23.m23.9.9.4.cmml"><msub id="S4.SS2.SSS2.Px2.p2.23.m23.9.9.4.2" xref="S4.SS2.SSS2.Px2.p2.23.m23.9.9.4.2.cmml"><mtext class="ltx_font_smallcaps" id="S4.SS2.SSS2.Px2.p2.23.m23.9.9.4.2.2" xref="S4.SS2.SSS2.Px2.p2.23.m23.9.9.4.2.2a.cmml">Min</mtext><mi 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italic_f start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT ( italic_v ) : italic_f start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT ( italic_u ) = italic_μ , italic_w ( italic_u , italic_v ) ∈ [ ( 1 + italic_ϵ ) start_POSTSUPERSCRIPT italic_j end_POSTSUPERSCRIPT , ( 1 + italic_ϵ ) start_POSTSUPERSCRIPT italic_j + 1 end_POSTSUPERSCRIPT ) }</annotation></semantics></math> and <math alttext="\textsc{Max}_{\mu}(j)=\textnormal{argmax}_{(u,v)\in L}\{f_{x}(v):f_{x}(u)=\mu,% w(u,v)\in[(1+\epsilon)^{j},(1+\epsilon)^{j+1})\}" class="ltx_Math" display="inline" id="S4.SS2.SSS2.Px2.p2.24.m24.9"><semantics id="S4.SS2.SSS2.Px2.p2.24.m24.9a"><mrow id="S4.SS2.SSS2.Px2.p2.24.m24.9.9" xref="S4.SS2.SSS2.Px2.p2.24.m24.9.9.cmml"><mrow id="S4.SS2.SSS2.Px2.p2.24.m24.9.9.4" xref="S4.SS2.SSS2.Px2.p2.24.m24.9.9.4.cmml"><msub id="S4.SS2.SSS2.Px2.p2.24.m24.9.9.4.2" xref="S4.SS2.SSS2.Px2.p2.24.m24.9.9.4.2.cmml"><mtext class="ltx_font_smallcaps" id="S4.SS2.SSS2.Px2.p2.24.m24.9.9.4.2.2" 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italic_f start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT ( italic_v ) : italic_f start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT ( italic_u ) = italic_μ , italic_w ( italic_u , italic_v ) ∈ [ ( 1 + italic_ϵ ) start_POSTSUPERSCRIPT italic_j end_POSTSUPERSCRIPT , ( 1 + italic_ϵ ) start_POSTSUPERSCRIPT italic_j + 1 end_POSTSUPERSCRIPT ) }</annotation></semantics></math>.</p> </div> <figure class="ltx_figure" id="S4.F7"> <div class="ltx_flex_figure"> <div class="ltx_flex_cell ltx_flex_size_2"> <figure class="ltx_figure ltx_figure_panel ltx_minipage ltx_align_center ltx_align_middle" id="S4.F7.1" style="width:173.4pt;"><img alt="Refer to caption" class="ltx_graphics ltx_img_landscape" height="499" id="S4.F7.1.g1" src="x7.png" width="664"/> <figcaption class="ltx_caption ltx_centering"><span class="ltx_tag ltx_tag_figure"><span class="ltx_text" id="S4.F7.1.1.1.1" style="font-size:90%;">Figure 6</span>: </span><span class="ltx_text" id="S4.F7.1.2.2" style="font-size:90%;">Example from <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.00712v1#bib.bibx54" title="">JKMV24</a>]</cite> to augment a cycle to be 3-edge-connected. The blue curved line indicates the cut that needs to be covered.</span></figcaption> </figure> </div> <div class="ltx_flex_cell ltx_flex_size_2"> <figure class="ltx_figure ltx_figure_panel ltx_minipage ltx_align_center ltx_align_middle" id="S4.F7.18" style="width:216.8pt;"><img alt="Refer to caption" class="ltx_graphics ltx_img_square" height="696" id="S4.F7.2.g1" src="x8.png" width="581"/> <figcaption class="ltx_caption ltx_centering"><span class="ltx_tag ltx_tag_figure"><span class="ltx_text" id="S4.F7.18.17.9.1" style="font-size:90%;">Figure 7</span>: </span><span class="ltx_text" id="S4.F7.18.16.8" style="font-size:90%;">Example of S-node <math alttext="x" class="ltx_Math" display="inline" id="S4.F7.11.9.1.m1.1"><semantics id="S4.F7.11.9.1.m1.1b"><mi id="S4.F7.11.9.1.m1.1.1" xref="S4.F7.11.9.1.m1.1.1.cmml">x</mi><annotation-xml encoding="MathML-Content" id="S4.F7.11.9.1.m1.1c"><ci id="S4.F7.11.9.1.m1.1.1.cmml" xref="S4.F7.11.9.1.m1.1.1">𝑥</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.F7.11.9.1.m1.1d">x</annotation><annotation encoding="application/x-llamapun" id="S4.F7.11.9.1.m1.1e">italic_x</annotation></semantics></math> with new dummy vertices. <math alttext="T^{\prime}" class="ltx_Math" display="inline" id="S4.F7.12.10.2.m2.1"><semantics id="S4.F7.12.10.2.m2.1b"><msup id="S4.F7.12.10.2.m2.1.1" xref="S4.F7.12.10.2.m2.1.1.cmml"><mi id="S4.F7.12.10.2.m2.1.1.2" xref="S4.F7.12.10.2.m2.1.1.2.cmml">T</mi><mo id="S4.F7.12.10.2.m2.1.1.3" xref="S4.F7.12.10.2.m2.1.1.3.cmml">′</mo></msup><annotation-xml encoding="MathML-Content" id="S4.F7.12.10.2.m2.1c"><apply id="S4.F7.12.10.2.m2.1.1.cmml" xref="S4.F7.12.10.2.m2.1.1"><csymbol cd="ambiguous" id="S4.F7.12.10.2.m2.1.1.1.cmml" xref="S4.F7.12.10.2.m2.1.1">superscript</csymbol><ci id="S4.F7.12.10.2.m2.1.1.2.cmml" xref="S4.F7.12.10.2.m2.1.1.2">𝑇</ci><ci id="S4.F7.12.10.2.m2.1.1.3.cmml" xref="S4.F7.12.10.2.m2.1.1.3">′</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.F7.12.10.2.m2.1d">T^{\prime}</annotation><annotation encoding="application/x-llamapun" id="S4.F7.12.10.2.m2.1e">italic_T start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math> and <math alttext="T^{\prime\prime}" class="ltx_Math" display="inline" id="S4.F7.13.11.3.m3.1"><semantics id="S4.F7.13.11.3.m3.1b"><msup id="S4.F7.13.11.3.m3.1.1" xref="S4.F7.13.11.3.m3.1.1.cmml"><mi id="S4.F7.13.11.3.m3.1.1.2" xref="S4.F7.13.11.3.m3.1.1.2.cmml">T</mi><mo id="S4.F7.13.11.3.m3.1.1.3" xref="S4.F7.13.11.3.m3.1.1.3.cmml">′′</mo></msup><annotation-xml encoding="MathML-Content" id="S4.F7.13.11.3.m3.1c"><apply id="S4.F7.13.11.3.m3.1.1.cmml" xref="S4.F7.13.11.3.m3.1.1"><csymbol cd="ambiguous" id="S4.F7.13.11.3.m3.1.1.1.cmml" xref="S4.F7.13.11.3.m3.1.1">superscript</csymbol><ci id="S4.F7.13.11.3.m3.1.1.2.cmml" xref="S4.F7.13.11.3.m3.1.1.2">𝑇</ci><ci id="S4.F7.13.11.3.m3.1.1.3.cmml" xref="S4.F7.13.11.3.m3.1.1.3">′′</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.F7.13.11.3.m3.1d">T^{\prime\prime}</annotation><annotation encoding="application/x-llamapun" id="S4.F7.13.11.3.m3.1e">italic_T start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT</annotation></semantics></math> are subtrees of <math alttext="x" class="ltx_Math" display="inline" id="S4.F7.14.12.4.m4.1"><semantics id="S4.F7.14.12.4.m4.1b"><mi id="S4.F7.14.12.4.m4.1.1" xref="S4.F7.14.12.4.m4.1.1.cmml">x</mi><annotation-xml encoding="MathML-Content" id="S4.F7.14.12.4.m4.1c"><ci id="S4.F7.14.12.4.m4.1.1.cmml" xref="S4.F7.14.12.4.m4.1.1">𝑥</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.F7.14.12.4.m4.1d">x</annotation><annotation encoding="application/x-llamapun" id="S4.F7.14.12.4.m4.1e">italic_x</annotation></semantics></math> rooted at virtual edges <math alttext="\{\mu_{2},\mu_{3}\}" class="ltx_Math" display="inline" id="S4.F7.15.13.5.m5.2"><semantics id="S4.F7.15.13.5.m5.2b"><mrow id="S4.F7.15.13.5.m5.2.2.2" xref="S4.F7.15.13.5.m5.2.2.3.cmml"><mo id="S4.F7.15.13.5.m5.2.2.2.3" stretchy="false" xref="S4.F7.15.13.5.m5.2.2.3.cmml">{</mo><msub id="S4.F7.15.13.5.m5.1.1.1.1" xref="S4.F7.15.13.5.m5.1.1.1.1.cmml"><mi id="S4.F7.15.13.5.m5.1.1.1.1.2" xref="S4.F7.15.13.5.m5.1.1.1.1.2.cmml">μ</mi><mn id="S4.F7.15.13.5.m5.1.1.1.1.3" xref="S4.F7.15.13.5.m5.1.1.1.1.3.cmml">2</mn></msub><mo id="S4.F7.15.13.5.m5.2.2.2.4" xref="S4.F7.15.13.5.m5.2.2.3.cmml">,</mo><msub id="S4.F7.15.13.5.m5.2.2.2.2" xref="S4.F7.15.13.5.m5.2.2.2.2.cmml"><mi id="S4.F7.15.13.5.m5.2.2.2.2.2" xref="S4.F7.15.13.5.m5.2.2.2.2.2.cmml">μ</mi><mn id="S4.F7.15.13.5.m5.2.2.2.2.3" xref="S4.F7.15.13.5.m5.2.2.2.2.3.cmml">3</mn></msub><mo id="S4.F7.15.13.5.m5.2.2.2.5" stretchy="false" xref="S4.F7.15.13.5.m5.2.2.3.cmml">}</mo></mrow><annotation-xml encoding="MathML-Content" id="S4.F7.15.13.5.m5.2c"><set id="S4.F7.15.13.5.m5.2.2.3.cmml" xref="S4.F7.15.13.5.m5.2.2.2"><apply id="S4.F7.15.13.5.m5.1.1.1.1.cmml" xref="S4.F7.15.13.5.m5.1.1.1.1"><csymbol cd="ambiguous" id="S4.F7.15.13.5.m5.1.1.1.1.1.cmml" xref="S4.F7.15.13.5.m5.1.1.1.1">subscript</csymbol><ci id="S4.F7.15.13.5.m5.1.1.1.1.2.cmml" xref="S4.F7.15.13.5.m5.1.1.1.1.2">𝜇</ci><cn id="S4.F7.15.13.5.m5.1.1.1.1.3.cmml" type="integer" xref="S4.F7.15.13.5.m5.1.1.1.1.3">2</cn></apply><apply id="S4.F7.15.13.5.m5.2.2.2.2.cmml" xref="S4.F7.15.13.5.m5.2.2.2.2"><csymbol cd="ambiguous" id="S4.F7.15.13.5.m5.2.2.2.2.1.cmml" xref="S4.F7.15.13.5.m5.2.2.2.2">subscript</csymbol><ci id="S4.F7.15.13.5.m5.2.2.2.2.2.cmml" xref="S4.F7.15.13.5.m5.2.2.2.2.2">𝜇</ci><cn id="S4.F7.15.13.5.m5.2.2.2.2.3.cmml" type="integer" xref="S4.F7.15.13.5.m5.2.2.2.2.3">3</cn></apply></set></annotation-xml><annotation encoding="application/x-tex" id="S4.F7.15.13.5.m5.2d">\{\mu_{2},\mu_{3}\}</annotation><annotation encoding="application/x-llamapun" id="S4.F7.15.13.5.m5.2e">{ italic_μ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT , italic_μ start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT }</annotation></semantics></math> and <math alttext="\{\mu_{1},\mu_{2}\}" class="ltx_Math" display="inline" id="S4.F7.16.14.6.m6.2"><semantics id="S4.F7.16.14.6.m6.2b"><mrow id="S4.F7.16.14.6.m6.2.2.2" xref="S4.F7.16.14.6.m6.2.2.3.cmml"><mo id="S4.F7.16.14.6.m6.2.2.2.3" stretchy="false" xref="S4.F7.16.14.6.m6.2.2.3.cmml">{</mo><msub id="S4.F7.16.14.6.m6.1.1.1.1" xref="S4.F7.16.14.6.m6.1.1.1.1.cmml"><mi id="S4.F7.16.14.6.m6.1.1.1.1.2" xref="S4.F7.16.14.6.m6.1.1.1.1.2.cmml">μ</mi><mn id="S4.F7.16.14.6.m6.1.1.1.1.3" xref="S4.F7.16.14.6.m6.1.1.1.1.3.cmml">1</mn></msub><mo id="S4.F7.16.14.6.m6.2.2.2.4" xref="S4.F7.16.14.6.m6.2.2.3.cmml">,</mo><msub id="S4.F7.16.14.6.m6.2.2.2.2" xref="S4.F7.16.14.6.m6.2.2.2.2.cmml"><mi id="S4.F7.16.14.6.m6.2.2.2.2.2" xref="S4.F7.16.14.6.m6.2.2.2.2.2.cmml">μ</mi><mn id="S4.F7.16.14.6.m6.2.2.2.2.3" xref="S4.F7.16.14.6.m6.2.2.2.2.3.cmml">2</mn></msub><mo id="S4.F7.16.14.6.m6.2.2.2.5" stretchy="false" xref="S4.F7.16.14.6.m6.2.2.3.cmml">}</mo></mrow><annotation-xml encoding="MathML-Content" id="S4.F7.16.14.6.m6.2c"><set id="S4.F7.16.14.6.m6.2.2.3.cmml" xref="S4.F7.16.14.6.m6.2.2.2"><apply id="S4.F7.16.14.6.m6.1.1.1.1.cmml" xref="S4.F7.16.14.6.m6.1.1.1.1"><csymbol cd="ambiguous" id="S4.F7.16.14.6.m6.1.1.1.1.1.cmml" xref="S4.F7.16.14.6.m6.1.1.1.1">subscript</csymbol><ci id="S4.F7.16.14.6.m6.1.1.1.1.2.cmml" xref="S4.F7.16.14.6.m6.1.1.1.1.2">𝜇</ci><cn id="S4.F7.16.14.6.m6.1.1.1.1.3.cmml" type="integer" xref="S4.F7.16.14.6.m6.1.1.1.1.3">1</cn></apply><apply id="S4.F7.16.14.6.m6.2.2.2.2.cmml" xref="S4.F7.16.14.6.m6.2.2.2.2"><csymbol cd="ambiguous" id="S4.F7.16.14.6.m6.2.2.2.2.1.cmml" xref="S4.F7.16.14.6.m6.2.2.2.2">subscript</csymbol><ci id="S4.F7.16.14.6.m6.2.2.2.2.2.cmml" xref="S4.F7.16.14.6.m6.2.2.2.2.2">𝜇</ci><cn id="S4.F7.16.14.6.m6.2.2.2.2.3.cmml" type="integer" xref="S4.F7.16.14.6.m6.2.2.2.2.3">2</cn></apply></set></annotation-xml><annotation encoding="application/x-tex" id="S4.F7.16.14.6.m6.2d">\{\mu_{1},\mu_{2}\}</annotation><annotation encoding="application/x-llamapun" id="S4.F7.16.14.6.m6.2e">{ italic_μ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_μ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT }</annotation></semantics></math> respectively. In this example, <math alttext="f(u)=\mu_{2,3}" class="ltx_Math" display="inline" id="S4.F7.17.15.7.m7.3"><semantics id="S4.F7.17.15.7.m7.3b"><mrow id="S4.F7.17.15.7.m7.3.4" xref="S4.F7.17.15.7.m7.3.4.cmml"><mrow id="S4.F7.17.15.7.m7.3.4.2" xref="S4.F7.17.15.7.m7.3.4.2.cmml"><mi id="S4.F7.17.15.7.m7.3.4.2.2" xref="S4.F7.17.15.7.m7.3.4.2.2.cmml">f</mi><mo id="S4.F7.17.15.7.m7.3.4.2.1" xref="S4.F7.17.15.7.m7.3.4.2.1.cmml">⁢</mo><mrow id="S4.F7.17.15.7.m7.3.4.2.3.2" xref="S4.F7.17.15.7.m7.3.4.2.cmml"><mo id="S4.F7.17.15.7.m7.3.4.2.3.2.1" stretchy="false" xref="S4.F7.17.15.7.m7.3.4.2.cmml">(</mo><mi id="S4.F7.17.15.7.m7.3.3" xref="S4.F7.17.15.7.m7.3.3.cmml">u</mi><mo id="S4.F7.17.15.7.m7.3.4.2.3.2.2" stretchy="false" xref="S4.F7.17.15.7.m7.3.4.2.cmml">)</mo></mrow></mrow><mo id="S4.F7.17.15.7.m7.3.4.1" xref="S4.F7.17.15.7.m7.3.4.1.cmml">=</mo><msub id="S4.F7.17.15.7.m7.3.4.3" xref="S4.F7.17.15.7.m7.3.4.3.cmml"><mi id="S4.F7.17.15.7.m7.3.4.3.2" xref="S4.F7.17.15.7.m7.3.4.3.2.cmml">μ</mi><mrow id="S4.F7.17.15.7.m7.2.2.2.4" xref="S4.F7.17.15.7.m7.2.2.2.3.cmml"><mn id="S4.F7.17.15.7.m7.1.1.1.1" xref="S4.F7.17.15.7.m7.1.1.1.1.cmml">2</mn><mo id="S4.F7.17.15.7.m7.2.2.2.4.1" xref="S4.F7.17.15.7.m7.2.2.2.3.cmml">,</mo><mn id="S4.F7.17.15.7.m7.2.2.2.2" xref="S4.F7.17.15.7.m7.2.2.2.2.cmml">3</mn></mrow></msub></mrow><annotation-xml encoding="MathML-Content" id="S4.F7.17.15.7.m7.3c"><apply id="S4.F7.17.15.7.m7.3.4.cmml" xref="S4.F7.17.15.7.m7.3.4"><eq id="S4.F7.17.15.7.m7.3.4.1.cmml" xref="S4.F7.17.15.7.m7.3.4.1"></eq><apply id="S4.F7.17.15.7.m7.3.4.2.cmml" xref="S4.F7.17.15.7.m7.3.4.2"><times id="S4.F7.17.15.7.m7.3.4.2.1.cmml" xref="S4.F7.17.15.7.m7.3.4.2.1"></times><ci id="S4.F7.17.15.7.m7.3.4.2.2.cmml" xref="S4.F7.17.15.7.m7.3.4.2.2">𝑓</ci><ci id="S4.F7.17.15.7.m7.3.3.cmml" xref="S4.F7.17.15.7.m7.3.3">𝑢</ci></apply><apply id="S4.F7.17.15.7.m7.3.4.3.cmml" xref="S4.F7.17.15.7.m7.3.4.3"><csymbol cd="ambiguous" id="S4.F7.17.15.7.m7.3.4.3.1.cmml" xref="S4.F7.17.15.7.m7.3.4.3">subscript</csymbol><ci id="S4.F7.17.15.7.m7.3.4.3.2.cmml" xref="S4.F7.17.15.7.m7.3.4.3.2">𝜇</ci><list id="S4.F7.17.15.7.m7.2.2.2.3.cmml" xref="S4.F7.17.15.7.m7.2.2.2.4"><cn id="S4.F7.17.15.7.m7.1.1.1.1.cmml" type="integer" xref="S4.F7.17.15.7.m7.1.1.1.1">2</cn><cn id="S4.F7.17.15.7.m7.2.2.2.2.cmml" type="integer" xref="S4.F7.17.15.7.m7.2.2.2.2">3</cn></list></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.F7.17.15.7.m7.3d">f(u)=\mu_{2,3}</annotation><annotation encoding="application/x-llamapun" id="S4.F7.17.15.7.m7.3e">italic_f ( italic_u ) = italic_μ start_POSTSUBSCRIPT 2 , 3 end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="f(v)=\mu_{1,2}" class="ltx_Math" display="inline" id="S4.F7.18.16.8.m8.3"><semantics id="S4.F7.18.16.8.m8.3b"><mrow id="S4.F7.18.16.8.m8.3.4" xref="S4.F7.18.16.8.m8.3.4.cmml"><mrow id="S4.F7.18.16.8.m8.3.4.2" xref="S4.F7.18.16.8.m8.3.4.2.cmml"><mi id="S4.F7.18.16.8.m8.3.4.2.2" xref="S4.F7.18.16.8.m8.3.4.2.2.cmml">f</mi><mo id="S4.F7.18.16.8.m8.3.4.2.1" xref="S4.F7.18.16.8.m8.3.4.2.1.cmml">⁢</mo><mrow id="S4.F7.18.16.8.m8.3.4.2.3.2" xref="S4.F7.18.16.8.m8.3.4.2.cmml"><mo id="S4.F7.18.16.8.m8.3.4.2.3.2.1" stretchy="false" xref="S4.F7.18.16.8.m8.3.4.2.cmml">(</mo><mi id="S4.F7.18.16.8.m8.3.3" xref="S4.F7.18.16.8.m8.3.3.cmml">v</mi><mo id="S4.F7.18.16.8.m8.3.4.2.3.2.2" stretchy="false" xref="S4.F7.18.16.8.m8.3.4.2.cmml">)</mo></mrow></mrow><mo id="S4.F7.18.16.8.m8.3.4.1" xref="S4.F7.18.16.8.m8.3.4.1.cmml">=</mo><msub id="S4.F7.18.16.8.m8.3.4.3" xref="S4.F7.18.16.8.m8.3.4.3.cmml"><mi id="S4.F7.18.16.8.m8.3.4.3.2" xref="S4.F7.18.16.8.m8.3.4.3.2.cmml">μ</mi><mrow id="S4.F7.18.16.8.m8.2.2.2.4" xref="S4.F7.18.16.8.m8.2.2.2.3.cmml"><mn id="S4.F7.18.16.8.m8.1.1.1.1" xref="S4.F7.18.16.8.m8.1.1.1.1.cmml">1</mn><mo id="S4.F7.18.16.8.m8.2.2.2.4.1" xref="S4.F7.18.16.8.m8.2.2.2.3.cmml">,</mo><mn id="S4.F7.18.16.8.m8.2.2.2.2" xref="S4.F7.18.16.8.m8.2.2.2.2.cmml">2</mn></mrow></msub></mrow><annotation-xml encoding="MathML-Content" id="S4.F7.18.16.8.m8.3c"><apply id="S4.F7.18.16.8.m8.3.4.cmml" xref="S4.F7.18.16.8.m8.3.4"><eq id="S4.F7.18.16.8.m8.3.4.1.cmml" xref="S4.F7.18.16.8.m8.3.4.1"></eq><apply id="S4.F7.18.16.8.m8.3.4.2.cmml" xref="S4.F7.18.16.8.m8.3.4.2"><times id="S4.F7.18.16.8.m8.3.4.2.1.cmml" xref="S4.F7.18.16.8.m8.3.4.2.1"></times><ci id="S4.F7.18.16.8.m8.3.4.2.2.cmml" xref="S4.F7.18.16.8.m8.3.4.2.2">𝑓</ci><ci id="S4.F7.18.16.8.m8.3.3.cmml" xref="S4.F7.18.16.8.m8.3.3">𝑣</ci></apply><apply id="S4.F7.18.16.8.m8.3.4.3.cmml" xref="S4.F7.18.16.8.m8.3.4.3"><csymbol cd="ambiguous" id="S4.F7.18.16.8.m8.3.4.3.1.cmml" xref="S4.F7.18.16.8.m8.3.4.3">subscript</csymbol><ci id="S4.F7.18.16.8.m8.3.4.3.2.cmml" xref="S4.F7.18.16.8.m8.3.4.3.2">𝜇</ci><list id="S4.F7.18.16.8.m8.2.2.2.3.cmml" xref="S4.F7.18.16.8.m8.2.2.2.4"><cn id="S4.F7.18.16.8.m8.1.1.1.1.cmml" type="integer" xref="S4.F7.18.16.8.m8.1.1.1.1">1</cn><cn id="S4.F7.18.16.8.m8.2.2.2.2.cmml" type="integer" xref="S4.F7.18.16.8.m8.2.2.2.2">2</cn></list></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.F7.18.16.8.m8.3d">f(v)=\mu_{1,2}</annotation><annotation encoding="application/x-llamapun" id="S4.F7.18.16.8.m8.3e">italic_f ( italic_v ) = italic_μ start_POSTSUBSCRIPT 1 , 2 end_POSTSUBSCRIPT</annotation></semantics></math>.</span></figcaption> </figure> </div> </div> </figure> <figure class="ltx_float ltx_algorithm" id="algorithm6"> <div class="ltx_listing ltx_lst_numbers_left ltx_listing" id="algorithm6.45"> <div class="ltx_listingline" id="algorithm6.2.2"> <span class="ltx_text" id="algorithm6.2.2.1"><span class="ltx_text ltx_font_bold" id="algorithm6.2.2.1.1">Input:</span> </span>An edge-minimal 2-vertex-connected graph <math alttext="G=(V,E)" class="ltx_Math" display="inline" id="algorithm6.1.1.m1.2"><semantics id="algorithm6.1.1.m1.2a"><mrow id="algorithm6.1.1.m1.2.3" xref="algorithm6.1.1.m1.2.3.cmml"><mi id="algorithm6.1.1.m1.2.3.2" xref="algorithm6.1.1.m1.2.3.2.cmml">G</mi><mo id="algorithm6.1.1.m1.2.3.1" xref="algorithm6.1.1.m1.2.3.1.cmml">=</mo><mrow id="algorithm6.1.1.m1.2.3.3.2" xref="algorithm6.1.1.m1.2.3.3.1.cmml"><mo id="algorithm6.1.1.m1.2.3.3.2.1" stretchy="false" xref="algorithm6.1.1.m1.2.3.3.1.cmml">(</mo><mi id="algorithm6.1.1.m1.1.1" xref="algorithm6.1.1.m1.1.1.cmml">V</mi><mo id="algorithm6.1.1.m1.2.3.3.2.2" xref="algorithm6.1.1.m1.2.3.3.1.cmml">,</mo><mi id="algorithm6.1.1.m1.2.2" xref="algorithm6.1.1.m1.2.2.cmml">E</mi><mo id="algorithm6.1.1.m1.2.3.3.2.3" stretchy="false" xref="algorithm6.1.1.m1.2.3.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="algorithm6.1.1.m1.2b"><apply id="algorithm6.1.1.m1.2.3.cmml" xref="algorithm6.1.1.m1.2.3"><eq id="algorithm6.1.1.m1.2.3.1.cmml" xref="algorithm6.1.1.m1.2.3.1"></eq><ci id="algorithm6.1.1.m1.2.3.2.cmml" xref="algorithm6.1.1.m1.2.3.2">𝐺</ci><interval closure="open" id="algorithm6.1.1.m1.2.3.3.1.cmml" xref="algorithm6.1.1.m1.2.3.3.2"><ci id="algorithm6.1.1.m1.1.1.cmml" xref="algorithm6.1.1.m1.1.1">𝑉</ci><ci id="algorithm6.1.1.m1.2.2.cmml" xref="algorithm6.1.1.m1.2.2">𝐸</ci></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="algorithm6.1.1.m1.2c">G=(V,E)</annotation><annotation encoding="application/x-llamapun" id="algorithm6.1.1.m1.2d">italic_G = ( italic_V , italic_E )</annotation></semantics></math> with weights <math alttext="w:E\rightarrow[1,W]" class="ltx_Math" display="inline" id="algorithm6.2.2.m2.2"><semantics id="algorithm6.2.2.m2.2a"><mrow id="algorithm6.2.2.m2.2.3" xref="algorithm6.2.2.m2.2.3.cmml"><mi id="algorithm6.2.2.m2.2.3.2" xref="algorithm6.2.2.m2.2.3.2.cmml">w</mi><mo id="algorithm6.2.2.m2.2.3.1" lspace="0.278em" rspace="0.278em" xref="algorithm6.2.2.m2.2.3.1.cmml">:</mo><mrow id="algorithm6.2.2.m2.2.3.3" xref="algorithm6.2.2.m2.2.3.3.cmml"><mi id="algorithm6.2.2.m2.2.3.3.2" xref="algorithm6.2.2.m2.2.3.3.2.cmml">E</mi><mo id="algorithm6.2.2.m2.2.3.3.1" stretchy="false" xref="algorithm6.2.2.m2.2.3.3.1.cmml">→</mo><mrow id="algorithm6.2.2.m2.2.3.3.3.2" xref="algorithm6.2.2.m2.2.3.3.3.1.cmml"><mo id="algorithm6.2.2.m2.2.3.3.3.2.1" stretchy="false" xref="algorithm6.2.2.m2.2.3.3.3.1.cmml">[</mo><mn id="algorithm6.2.2.m2.1.1" xref="algorithm6.2.2.m2.1.1.cmml">1</mn><mo id="algorithm6.2.2.m2.2.3.3.3.2.2" xref="algorithm6.2.2.m2.2.3.3.3.1.cmml">,</mo><mi id="algorithm6.2.2.m2.2.2" xref="algorithm6.2.2.m2.2.2.cmml">W</mi><mo id="algorithm6.2.2.m2.2.3.3.3.2.3" stretchy="false" xref="algorithm6.2.2.m2.2.3.3.3.1.cmml">]</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="algorithm6.2.2.m2.2b"><apply id="algorithm6.2.2.m2.2.3.cmml" xref="algorithm6.2.2.m2.2.3"><ci id="algorithm6.2.2.m2.2.3.1.cmml" xref="algorithm6.2.2.m2.2.3.1">:</ci><ci id="algorithm6.2.2.m2.2.3.2.cmml" xref="algorithm6.2.2.m2.2.3.2">𝑤</ci><apply id="algorithm6.2.2.m2.2.3.3.cmml" xref="algorithm6.2.2.m2.2.3.3"><ci id="algorithm6.2.2.m2.2.3.3.1.cmml" xref="algorithm6.2.2.m2.2.3.3.1">→</ci><ci id="algorithm6.2.2.m2.2.3.3.2.cmml" xref="algorithm6.2.2.m2.2.3.3.2">𝐸</ci><interval closure="closed" id="algorithm6.2.2.m2.2.3.3.3.1.cmml" xref="algorithm6.2.2.m2.2.3.3.3.2"><cn id="algorithm6.2.2.m2.1.1.cmml" type="integer" xref="algorithm6.2.2.m2.1.1">1</cn><ci id="algorithm6.2.2.m2.2.2.cmml" xref="algorithm6.2.2.m2.2.2">𝑊</ci></interval></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="algorithm6.2.2.m2.2c">w:E\rightarrow[1,W]</annotation><annotation encoding="application/x-llamapun" id="algorithm6.2.2.m2.2d">italic_w : italic_E → [ 1 , italic_W ]</annotation></semantics></math>. </div> <div class="ltx_listingline" id="algorithm6.45.46"> <span class="ltx_text" id="algorithm6.45.46.1" style="color:#0000FF;">/* </span><span class="ltx_text ltx_font_smallcaps" id="algorithm6.45.46.2" style="color:#0000FF;">Preprocessing: */</span> </div> <div class="ltx_listingline" id="algorithm6.5.5"> <math alttext="T\leftarrow" class="ltx_Math" display="inline" id="algorithm6.3.3.m1.1"><semantics id="algorithm6.3.3.m1.1a"><mrow id="algorithm6.3.3.m1.1.1" xref="algorithm6.3.3.m1.1.1.cmml"><mi id="algorithm6.3.3.m1.1.1.2" xref="algorithm6.3.3.m1.1.1.2.cmml">T</mi><mo id="algorithm6.3.3.m1.1.1.1" stretchy="false" xref="algorithm6.3.3.m1.1.1.1.cmml">←</mo><mi id="algorithm6.3.3.m1.1.1.3" xref="algorithm6.3.3.m1.1.1.3.cmml"></mi></mrow><annotation-xml encoding="MathML-Content" id="algorithm6.3.3.m1.1b"><apply id="algorithm6.3.3.m1.1.1.cmml" xref="algorithm6.3.3.m1.1.1"><ci id="algorithm6.3.3.m1.1.1.1.cmml" xref="algorithm6.3.3.m1.1.1.1">←</ci><ci id="algorithm6.3.3.m1.1.1.2.cmml" xref="algorithm6.3.3.m1.1.1.2">𝑇</ci><csymbol cd="latexml" id="algorithm6.3.3.m1.1.1.3.cmml" xref="algorithm6.3.3.m1.1.1.3">absent</csymbol></apply></annotation-xml><annotation encoding="application/x-tex" id="algorithm6.3.3.m1.1c">T\leftarrow</annotation><annotation encoding="application/x-llamapun" id="algorithm6.3.3.m1.1d">italic_T ←</annotation></semantics></math> SPQR tree for <math alttext="G" class="ltx_Math" display="inline" id="algorithm6.4.4.m2.1"><semantics id="algorithm6.4.4.m2.1a"><mi id="algorithm6.4.4.m2.1.1" xref="algorithm6.4.4.m2.1.1.cmml">G</mi><annotation-xml encoding="MathML-Content" id="algorithm6.4.4.m2.1b"><ci id="algorithm6.4.4.m2.1.1.cmml" xref="algorithm6.4.4.m2.1.1">𝐺</ci></annotation-xml><annotation encoding="application/x-tex" id="algorithm6.4.4.m2.1c">G</annotation><annotation encoding="application/x-llamapun" id="algorithm6.4.4.m2.1d">italic_G</annotation></semantics></math>; choose root <math alttext="r\in V(T)" class="ltx_Math" display="inline" id="algorithm6.5.5.m3.1"><semantics id="algorithm6.5.5.m3.1a"><mrow id="algorithm6.5.5.m3.1.2" xref="algorithm6.5.5.m3.1.2.cmml"><mi id="algorithm6.5.5.m3.1.2.2" xref="algorithm6.5.5.m3.1.2.2.cmml">r</mi><mo id="algorithm6.5.5.m3.1.2.1" xref="algorithm6.5.5.m3.1.2.1.cmml">∈</mo><mrow id="algorithm6.5.5.m3.1.2.3" xref="algorithm6.5.5.m3.1.2.3.cmml"><mi id="algorithm6.5.5.m3.1.2.3.2" xref="algorithm6.5.5.m3.1.2.3.2.cmml">V</mi><mo id="algorithm6.5.5.m3.1.2.3.1" xref="algorithm6.5.5.m3.1.2.3.1.cmml">⁢</mo><mrow id="algorithm6.5.5.m3.1.2.3.3.2" xref="algorithm6.5.5.m3.1.2.3.cmml"><mo id="algorithm6.5.5.m3.1.2.3.3.2.1" stretchy="false" xref="algorithm6.5.5.m3.1.2.3.cmml">(</mo><mi id="algorithm6.5.5.m3.1.1" xref="algorithm6.5.5.m3.1.1.cmml">T</mi><mo id="algorithm6.5.5.m3.1.2.3.3.2.2" stretchy="false" xref="algorithm6.5.5.m3.1.2.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="algorithm6.5.5.m3.1b"><apply id="algorithm6.5.5.m3.1.2.cmml" xref="algorithm6.5.5.m3.1.2"><in id="algorithm6.5.5.m3.1.2.1.cmml" xref="algorithm6.5.5.m3.1.2.1"></in><ci id="algorithm6.5.5.m3.1.2.2.cmml" xref="algorithm6.5.5.m3.1.2.2">𝑟</ci><apply id="algorithm6.5.5.m3.1.2.3.cmml" xref="algorithm6.5.5.m3.1.2.3"><times id="algorithm6.5.5.m3.1.2.3.1.cmml" xref="algorithm6.5.5.m3.1.2.3.1"></times><ci id="algorithm6.5.5.m3.1.2.3.2.cmml" xref="algorithm6.5.5.m3.1.2.3.2">𝑉</ci><ci id="algorithm6.5.5.m3.1.1.cmml" xref="algorithm6.5.5.m3.1.1">𝑇</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="algorithm6.5.5.m3.1c">r\in V(T)</annotation><annotation encoding="application/x-llamapun" id="algorithm6.5.5.m3.1d">italic_r ∈ italic_V ( italic_T )</annotation></semantics></math> </div> <div class="ltx_listingline" id="algorithm6.6.6"> <span class="ltx_text ltx_font_bold" id="algorithm6.6.6.2">for</span> <em class="ltx_emph ltx_font_italic" id="algorithm6.6.6.1"><math alttext="x\in V(T)" class="ltx_Math" display="inline" id="algorithm6.6.6.1.m1.1"><semantics id="algorithm6.6.6.1.m1.1a"><mrow id="algorithm6.6.6.1.m1.1.2" xref="algorithm6.6.6.1.m1.1.2.cmml"><mi id="algorithm6.6.6.1.m1.1.2.2" xref="algorithm6.6.6.1.m1.1.2.2.cmml">x</mi><mo id="algorithm6.6.6.1.m1.1.2.1" xref="algorithm6.6.6.1.m1.1.2.1.cmml">∈</mo><mrow id="algorithm6.6.6.1.m1.1.2.3" xref="algorithm6.6.6.1.m1.1.2.3.cmml"><mi id="algorithm6.6.6.1.m1.1.2.3.2" xref="algorithm6.6.6.1.m1.1.2.3.2.cmml">V</mi><mo id="algorithm6.6.6.1.m1.1.2.3.1" xref="algorithm6.6.6.1.m1.1.2.3.1.cmml">⁢</mo><mrow id="algorithm6.6.6.1.m1.1.2.3.3.2" xref="algorithm6.6.6.1.m1.1.2.3.cmml"><mo id="algorithm6.6.6.1.m1.1.2.3.3.2.1" stretchy="false" xref="algorithm6.6.6.1.m1.1.2.3.cmml">(</mo><mi id="algorithm6.6.6.1.m1.1.1" xref="algorithm6.6.6.1.m1.1.1.cmml">T</mi><mo id="algorithm6.6.6.1.m1.1.2.3.3.2.2" stretchy="false" xref="algorithm6.6.6.1.m1.1.2.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="algorithm6.6.6.1.m1.1b"><apply id="algorithm6.6.6.1.m1.1.2.cmml" xref="algorithm6.6.6.1.m1.1.2"><in id="algorithm6.6.6.1.m1.1.2.1.cmml" xref="algorithm6.6.6.1.m1.1.2.1"></in><ci id="algorithm6.6.6.1.m1.1.2.2.cmml" xref="algorithm6.6.6.1.m1.1.2.2">𝑥</ci><apply id="algorithm6.6.6.1.m1.1.2.3.cmml" xref="algorithm6.6.6.1.m1.1.2.3"><times id="algorithm6.6.6.1.m1.1.2.3.1.cmml" xref="algorithm6.6.6.1.m1.1.2.3.1"></times><ci id="algorithm6.6.6.1.m1.1.2.3.2.cmml" xref="algorithm6.6.6.1.m1.1.2.3.2">𝑉</ci><ci id="algorithm6.6.6.1.m1.1.1.cmml" xref="algorithm6.6.6.1.m1.1.1">𝑇</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="algorithm6.6.6.1.m1.1c">x\in V(T)</annotation><annotation encoding="application/x-llamapun" id="algorithm6.6.6.1.m1.1d">italic_x ∈ italic_V ( italic_T )</annotation></semantics></math></em> <span class="ltx_text ltx_font_bold" id="algorithm6.6.6.3">do</span> </div> <div class="ltx_listingline" id="algorithm6.7.7"> <span class="ltx_rule" style="width:1px;height:100%;background:black;display:inline-block;"> </span> <span class="ltx_text ltx_font_bold" id="algorithm6.7.7.1">initialize</span> an empty dictionary <math alttext="L_{x}" class="ltx_Math" display="inline" id="algorithm6.7.7.m1.1"><semantics id="algorithm6.7.7.m1.1a"><msub id="algorithm6.7.7.m1.1.1" xref="algorithm6.7.7.m1.1.1.cmml"><mi id="algorithm6.7.7.m1.1.1.2" xref="algorithm6.7.7.m1.1.1.2.cmml">L</mi><mi id="algorithm6.7.7.m1.1.1.3" xref="algorithm6.7.7.m1.1.1.3.cmml">x</mi></msub><annotation-xml encoding="MathML-Content" id="algorithm6.7.7.m1.1b"><apply id="algorithm6.7.7.m1.1.1.cmml" xref="algorithm6.7.7.m1.1.1"><csymbol cd="ambiguous" id="algorithm6.7.7.m1.1.1.1.cmml" xref="algorithm6.7.7.m1.1.1">subscript</csymbol><ci id="algorithm6.7.7.m1.1.1.2.cmml" xref="algorithm6.7.7.m1.1.1.2">𝐿</ci><ci id="algorithm6.7.7.m1.1.1.3.cmml" xref="algorithm6.7.7.m1.1.1.3">𝑥</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="algorithm6.7.7.m1.1c">L_{x}</annotation><annotation encoding="application/x-llamapun" id="algorithm6.7.7.m1.1d">italic_L start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT</annotation></semantics></math> </div> <div class="ltx_listingline" id="algorithm6.45.47"> <span class="ltx_rule" style="width:1px;height:100%;background:black;display:inline-block;"> </span> </div> <div class="ltx_listingline" id="algorithm6.9.9"> <span class="ltx_text ltx_font_bold" id="algorithm6.9.9.3">for</span> <em class="ltx_emph ltx_font_italic" id="algorithm6.9.9.2"><math alttext="x\in V(T)" class="ltx_Math" display="inline" id="algorithm6.8.8.1.m1.1"><semantics id="algorithm6.8.8.1.m1.1a"><mrow id="algorithm6.8.8.1.m1.1.2" xref="algorithm6.8.8.1.m1.1.2.cmml"><mi id="algorithm6.8.8.1.m1.1.2.2" xref="algorithm6.8.8.1.m1.1.2.2.cmml">x</mi><mo id="algorithm6.8.8.1.m1.1.2.1" xref="algorithm6.8.8.1.m1.1.2.1.cmml">∈</mo><mrow id="algorithm6.8.8.1.m1.1.2.3" xref="algorithm6.8.8.1.m1.1.2.3.cmml"><mi id="algorithm6.8.8.1.m1.1.2.3.2" xref="algorithm6.8.8.1.m1.1.2.3.2.cmml">V</mi><mo id="algorithm6.8.8.1.m1.1.2.3.1" xref="algorithm6.8.8.1.m1.1.2.3.1.cmml">⁢</mo><mrow id="algorithm6.8.8.1.m1.1.2.3.3.2" xref="algorithm6.8.8.1.m1.1.2.3.cmml"><mo id="algorithm6.8.8.1.m1.1.2.3.3.2.1" stretchy="false" xref="algorithm6.8.8.1.m1.1.2.3.cmml">(</mo><mi id="algorithm6.8.8.1.m1.1.1" xref="algorithm6.8.8.1.m1.1.1.cmml">T</mi><mo id="algorithm6.8.8.1.m1.1.2.3.3.2.2" stretchy="false" xref="algorithm6.8.8.1.m1.1.2.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="algorithm6.8.8.1.m1.1b"><apply id="algorithm6.8.8.1.m1.1.2.cmml" xref="algorithm6.8.8.1.m1.1.2"><in id="algorithm6.8.8.1.m1.1.2.1.cmml" xref="algorithm6.8.8.1.m1.1.2.1"></in><ci id="algorithm6.8.8.1.m1.1.2.2.cmml" xref="algorithm6.8.8.1.m1.1.2.2">𝑥</ci><apply id="algorithm6.8.8.1.m1.1.2.3.cmml" xref="algorithm6.8.8.1.m1.1.2.3"><times id="algorithm6.8.8.1.m1.1.2.3.1.cmml" xref="algorithm6.8.8.1.m1.1.2.3.1"></times><ci id="algorithm6.8.8.1.m1.1.2.3.2.cmml" xref="algorithm6.8.8.1.m1.1.2.3.2">𝑉</ci><ci id="algorithm6.8.8.1.m1.1.1.cmml" xref="algorithm6.8.8.1.m1.1.1">𝑇</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="algorithm6.8.8.1.m1.1c">x\in V(T)</annotation><annotation encoding="application/x-llamapun" id="algorithm6.8.8.1.m1.1d">italic_x ∈ italic_V ( italic_T )</annotation></semantics></math>; <math alttext="x" class="ltx_Math" display="inline" id="algorithm6.9.9.2.m2.1"><semantics id="algorithm6.9.9.2.m2.1a"><mi id="algorithm6.9.9.2.m2.1.1" xref="algorithm6.9.9.2.m2.1.1.cmml">x</mi><annotation-xml encoding="MathML-Content" id="algorithm6.9.9.2.m2.1b"><ci id="algorithm6.9.9.2.m2.1.1.cmml" xref="algorithm6.9.9.2.m2.1.1">𝑥</ci></annotation-xml><annotation encoding="application/x-tex" id="algorithm6.9.9.2.m2.1c">x</annotation><annotation encoding="application/x-llamapun" id="algorithm6.9.9.2.m2.1d">italic_x</annotation></semantics></math> is a P-node</em> <span class="ltx_text ltx_font_bold" id="algorithm6.9.9.4">do</span> </div> <div class="ltx_listingline" id="algorithm6.15.15"> <span class="ltx_rule" style="width:1px;height:100%;background:black;display:inline-block;"> </span> <span class="ltx_text ltx_font_bold" id="algorithm6.15.15.1">construct</span> <math alttext="C^{\prime}(x)" class="ltx_Math" display="inline" id="algorithm6.10.10.m1.1"><semantics id="algorithm6.10.10.m1.1a"><mrow id="algorithm6.10.10.m1.1.2" xref="algorithm6.10.10.m1.1.2.cmml"><msup id="algorithm6.10.10.m1.1.2.2" xref="algorithm6.10.10.m1.1.2.2.cmml"><mi id="algorithm6.10.10.m1.1.2.2.2" xref="algorithm6.10.10.m1.1.2.2.2.cmml">C</mi><mo id="algorithm6.10.10.m1.1.2.2.3" xref="algorithm6.10.10.m1.1.2.2.3.cmml">′</mo></msup><mo id="algorithm6.10.10.m1.1.2.1" xref="algorithm6.10.10.m1.1.2.1.cmml">⁢</mo><mrow id="algorithm6.10.10.m1.1.2.3.2" xref="algorithm6.10.10.m1.1.2.cmml"><mo id="algorithm6.10.10.m1.1.2.3.2.1" stretchy="false" xref="algorithm6.10.10.m1.1.2.cmml">(</mo><mi id="algorithm6.10.10.m1.1.1" xref="algorithm6.10.10.m1.1.1.cmml">x</mi><mo id="algorithm6.10.10.m1.1.2.3.2.2" stretchy="false" xref="algorithm6.10.10.m1.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="algorithm6.10.10.m1.1b"><apply id="algorithm6.10.10.m1.1.2.cmml" xref="algorithm6.10.10.m1.1.2"><times id="algorithm6.10.10.m1.1.2.1.cmml" xref="algorithm6.10.10.m1.1.2.1"></times><apply id="algorithm6.10.10.m1.1.2.2.cmml" xref="algorithm6.10.10.m1.1.2.2"><csymbol cd="ambiguous" id="algorithm6.10.10.m1.1.2.2.1.cmml" xref="algorithm6.10.10.m1.1.2.2">superscript</csymbol><ci id="algorithm6.10.10.m1.1.2.2.2.cmml" xref="algorithm6.10.10.m1.1.2.2.2">𝐶</ci><ci id="algorithm6.10.10.m1.1.2.2.3.cmml" xref="algorithm6.10.10.m1.1.2.2.3">′</ci></apply><ci id="algorithm6.10.10.m1.1.1.cmml" xref="algorithm6.10.10.m1.1.1">𝑥</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="algorithm6.10.10.m1.1c">C^{\prime}(x)</annotation><annotation encoding="application/x-llamapun" id="algorithm6.10.10.m1.1d">italic_C start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ( italic_x )</annotation></semantics></math>: <span class="ltx_text ltx_font_bold" id="algorithm6.15.15.2">foreach</span> <math alttext="y\in C(x)" class="ltx_Math" display="inline" id="algorithm6.11.11.m2.1"><semantics id="algorithm6.11.11.m2.1a"><mrow id="algorithm6.11.11.m2.1.2" xref="algorithm6.11.11.m2.1.2.cmml"><mi id="algorithm6.11.11.m2.1.2.2" xref="algorithm6.11.11.m2.1.2.2.cmml">y</mi><mo id="algorithm6.11.11.m2.1.2.1" xref="algorithm6.11.11.m2.1.2.1.cmml">∈</mo><mrow id="algorithm6.11.11.m2.1.2.3" xref="algorithm6.11.11.m2.1.2.3.cmml"><mi id="algorithm6.11.11.m2.1.2.3.2" xref="algorithm6.11.11.m2.1.2.3.2.cmml">C</mi><mo id="algorithm6.11.11.m2.1.2.3.1" xref="algorithm6.11.11.m2.1.2.3.1.cmml">⁢</mo><mrow id="algorithm6.11.11.m2.1.2.3.3.2" xref="algorithm6.11.11.m2.1.2.3.cmml"><mo id="algorithm6.11.11.m2.1.2.3.3.2.1" stretchy="false" xref="algorithm6.11.11.m2.1.2.3.cmml">(</mo><mi id="algorithm6.11.11.m2.1.1" xref="algorithm6.11.11.m2.1.1.cmml">x</mi><mo id="algorithm6.11.11.m2.1.2.3.3.2.2" stretchy="false" xref="algorithm6.11.11.m2.1.2.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="algorithm6.11.11.m2.1b"><apply id="algorithm6.11.11.m2.1.2.cmml" xref="algorithm6.11.11.m2.1.2"><in id="algorithm6.11.11.m2.1.2.1.cmml" xref="algorithm6.11.11.m2.1.2.1"></in><ci id="algorithm6.11.11.m2.1.2.2.cmml" xref="algorithm6.11.11.m2.1.2.2">𝑦</ci><apply id="algorithm6.11.11.m2.1.2.3.cmml" xref="algorithm6.11.11.m2.1.2.3"><times id="algorithm6.11.11.m2.1.2.3.1.cmml" xref="algorithm6.11.11.m2.1.2.3.1"></times><ci id="algorithm6.11.11.m2.1.2.3.2.cmml" xref="algorithm6.11.11.m2.1.2.3.2">𝐶</ci><ci id="algorithm6.11.11.m2.1.1.cmml" xref="algorithm6.11.11.m2.1.1">𝑥</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="algorithm6.11.11.m2.1c">y\in C(x)</annotation><annotation encoding="application/x-llamapun" id="algorithm6.11.11.m2.1d">italic_y ∈ italic_C ( italic_x )</annotation></semantics></math>, <span class="ltx_text ltx_font_bold" id="algorithm6.15.15.3">contract</span> <math alttext="(\cup_{z\in T_{y}}V(G_{z}))\setminus V(G_{x})" class="ltx_Math" display="inline" id="algorithm6.12.12.m3.2"><semantics id="algorithm6.12.12.m3.2a"><mrow id="algorithm6.12.12.m3.2.2" xref="algorithm6.12.12.m3.2.2.cmml"><mrow id="algorithm6.12.12.m3.1.1.1.1" xref="algorithm6.12.12.m3.1.1.1.1.1.cmml"><mo id="algorithm6.12.12.m3.1.1.1.1.2" stretchy="false" xref="algorithm6.12.12.m3.1.1.1.1.1.cmml">(</mo><mrow id="algorithm6.12.12.m3.1.1.1.1.1" xref="algorithm6.12.12.m3.1.1.1.1.1.cmml"><msub id="algorithm6.12.12.m3.1.1.1.1.1.2" xref="algorithm6.12.12.m3.1.1.1.1.1.2.cmml"><mo id="algorithm6.12.12.m3.1.1.1.1.1.2.2" lspace="0em" xref="algorithm6.12.12.m3.1.1.1.1.1.2.2.cmml">∪</mo><mrow id="algorithm6.12.12.m3.1.1.1.1.1.2.3" xref="algorithm6.12.12.m3.1.1.1.1.1.2.3.cmml"><mi id="algorithm6.12.12.m3.1.1.1.1.1.2.3.2" xref="algorithm6.12.12.m3.1.1.1.1.1.2.3.2.cmml">z</mi><mo id="algorithm6.12.12.m3.1.1.1.1.1.2.3.1" xref="algorithm6.12.12.m3.1.1.1.1.1.2.3.1.cmml">∈</mo><msub id="algorithm6.12.12.m3.1.1.1.1.1.2.3.3" xref="algorithm6.12.12.m3.1.1.1.1.1.2.3.3.cmml"><mi id="algorithm6.12.12.m3.1.1.1.1.1.2.3.3.2" xref="algorithm6.12.12.m3.1.1.1.1.1.2.3.3.2.cmml">T</mi><mi id="algorithm6.12.12.m3.1.1.1.1.1.2.3.3.3" xref="algorithm6.12.12.m3.1.1.1.1.1.2.3.3.3.cmml">y</mi></msub></mrow></msub><mrow id="algorithm6.12.12.m3.1.1.1.1.1.1" xref="algorithm6.12.12.m3.1.1.1.1.1.1.cmml"><mi id="algorithm6.12.12.m3.1.1.1.1.1.1.3" xref="algorithm6.12.12.m3.1.1.1.1.1.1.3.cmml">V</mi><mo id="algorithm6.12.12.m3.1.1.1.1.1.1.2" xref="algorithm6.12.12.m3.1.1.1.1.1.1.2.cmml">⁢</mo><mrow id="algorithm6.12.12.m3.1.1.1.1.1.1.1.1" xref="algorithm6.12.12.m3.1.1.1.1.1.1.1.1.1.cmml"><mo id="algorithm6.12.12.m3.1.1.1.1.1.1.1.1.2" stretchy="false" xref="algorithm6.12.12.m3.1.1.1.1.1.1.1.1.1.cmml">(</mo><msub id="algorithm6.12.12.m3.1.1.1.1.1.1.1.1.1" xref="algorithm6.12.12.m3.1.1.1.1.1.1.1.1.1.cmml"><mi id="algorithm6.12.12.m3.1.1.1.1.1.1.1.1.1.2" xref="algorithm6.12.12.m3.1.1.1.1.1.1.1.1.1.2.cmml">G</mi><mi id="algorithm6.12.12.m3.1.1.1.1.1.1.1.1.1.3" xref="algorithm6.12.12.m3.1.1.1.1.1.1.1.1.1.3.cmml">z</mi></msub><mo id="algorithm6.12.12.m3.1.1.1.1.1.1.1.1.3" stretchy="false" xref="algorithm6.12.12.m3.1.1.1.1.1.1.1.1.1.cmml">)</mo></mrow></mrow></mrow><mo id="algorithm6.12.12.m3.1.1.1.1.3" stretchy="false" xref="algorithm6.12.12.m3.1.1.1.1.1.cmml">)</mo></mrow><mo id="algorithm6.12.12.m3.2.2.3" xref="algorithm6.12.12.m3.2.2.3.cmml">∖</mo><mrow id="algorithm6.12.12.m3.2.2.2" xref="algorithm6.12.12.m3.2.2.2.cmml"><mi id="algorithm6.12.12.m3.2.2.2.3" xref="algorithm6.12.12.m3.2.2.2.3.cmml">V</mi><mo id="algorithm6.12.12.m3.2.2.2.2" xref="algorithm6.12.12.m3.2.2.2.2.cmml">⁢</mo><mrow id="algorithm6.12.12.m3.2.2.2.1.1" xref="algorithm6.12.12.m3.2.2.2.1.1.1.cmml"><mo id="algorithm6.12.12.m3.2.2.2.1.1.2" stretchy="false" xref="algorithm6.12.12.m3.2.2.2.1.1.1.cmml">(</mo><msub id="algorithm6.12.12.m3.2.2.2.1.1.1" xref="algorithm6.12.12.m3.2.2.2.1.1.1.cmml"><mi id="algorithm6.12.12.m3.2.2.2.1.1.1.2" xref="algorithm6.12.12.m3.2.2.2.1.1.1.2.cmml">G</mi><mi id="algorithm6.12.12.m3.2.2.2.1.1.1.3" xref="algorithm6.12.12.m3.2.2.2.1.1.1.3.cmml">x</mi></msub><mo id="algorithm6.12.12.m3.2.2.2.1.1.3" stretchy="false" xref="algorithm6.12.12.m3.2.2.2.1.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="algorithm6.12.12.m3.2b"><apply id="algorithm6.12.12.m3.2.2.cmml" xref="algorithm6.12.12.m3.2.2"><setdiff id="algorithm6.12.12.m3.2.2.3.cmml" xref="algorithm6.12.12.m3.2.2.3"></setdiff><apply id="algorithm6.12.12.m3.1.1.1.1.1.cmml" xref="algorithm6.12.12.m3.1.1.1.1"><apply id="algorithm6.12.12.m3.1.1.1.1.1.2.cmml" xref="algorithm6.12.12.m3.1.1.1.1.1.2"><csymbol cd="ambiguous" id="algorithm6.12.12.m3.1.1.1.1.1.2.1.cmml" xref="algorithm6.12.12.m3.1.1.1.1.1.2">subscript</csymbol><union id="algorithm6.12.12.m3.1.1.1.1.1.2.2.cmml" xref="algorithm6.12.12.m3.1.1.1.1.1.2.2"></union><apply id="algorithm6.12.12.m3.1.1.1.1.1.2.3.cmml" xref="algorithm6.12.12.m3.1.1.1.1.1.2.3"><in id="algorithm6.12.12.m3.1.1.1.1.1.2.3.1.cmml" xref="algorithm6.12.12.m3.1.1.1.1.1.2.3.1"></in><ci id="algorithm6.12.12.m3.1.1.1.1.1.2.3.2.cmml" xref="algorithm6.12.12.m3.1.1.1.1.1.2.3.2">𝑧</ci><apply id="algorithm6.12.12.m3.1.1.1.1.1.2.3.3.cmml" xref="algorithm6.12.12.m3.1.1.1.1.1.2.3.3"><csymbol cd="ambiguous" id="algorithm6.12.12.m3.1.1.1.1.1.2.3.3.1.cmml" xref="algorithm6.12.12.m3.1.1.1.1.1.2.3.3">subscript</csymbol><ci id="algorithm6.12.12.m3.1.1.1.1.1.2.3.3.2.cmml" xref="algorithm6.12.12.m3.1.1.1.1.1.2.3.3.2">𝑇</ci><ci id="algorithm6.12.12.m3.1.1.1.1.1.2.3.3.3.cmml" xref="algorithm6.12.12.m3.1.1.1.1.1.2.3.3.3">𝑦</ci></apply></apply></apply><apply id="algorithm6.12.12.m3.1.1.1.1.1.1.cmml" xref="algorithm6.12.12.m3.1.1.1.1.1.1"><times id="algorithm6.12.12.m3.1.1.1.1.1.1.2.cmml" xref="algorithm6.12.12.m3.1.1.1.1.1.1.2"></times><ci id="algorithm6.12.12.m3.1.1.1.1.1.1.3.cmml" xref="algorithm6.12.12.m3.1.1.1.1.1.1.3">𝑉</ci><apply id="algorithm6.12.12.m3.1.1.1.1.1.1.1.1.1.cmml" xref="algorithm6.12.12.m3.1.1.1.1.1.1.1.1"><csymbol cd="ambiguous" id="algorithm6.12.12.m3.1.1.1.1.1.1.1.1.1.1.cmml" xref="algorithm6.12.12.m3.1.1.1.1.1.1.1.1">subscript</csymbol><ci id="algorithm6.12.12.m3.1.1.1.1.1.1.1.1.1.2.cmml" xref="algorithm6.12.12.m3.1.1.1.1.1.1.1.1.1.2">𝐺</ci><ci id="algorithm6.12.12.m3.1.1.1.1.1.1.1.1.1.3.cmml" xref="algorithm6.12.12.m3.1.1.1.1.1.1.1.1.1.3">𝑧</ci></apply></apply></apply><apply id="algorithm6.12.12.m3.2.2.2.cmml" xref="algorithm6.12.12.m3.2.2.2"><times id="algorithm6.12.12.m3.2.2.2.2.cmml" xref="algorithm6.12.12.m3.2.2.2.2"></times><ci id="algorithm6.12.12.m3.2.2.2.3.cmml" xref="algorithm6.12.12.m3.2.2.2.3">𝑉</ci><apply id="algorithm6.12.12.m3.2.2.2.1.1.1.cmml" xref="algorithm6.12.12.m3.2.2.2.1.1"><csymbol cd="ambiguous" id="algorithm6.12.12.m3.2.2.2.1.1.1.1.cmml" xref="algorithm6.12.12.m3.2.2.2.1.1">subscript</csymbol><ci id="algorithm6.12.12.m3.2.2.2.1.1.1.2.cmml" xref="algorithm6.12.12.m3.2.2.2.1.1.1.2">𝐺</ci><ci id="algorithm6.12.12.m3.2.2.2.1.1.1.3.cmml" xref="algorithm6.12.12.m3.2.2.2.1.1.1.3">𝑥</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="algorithm6.12.12.m3.2c">(\cup_{z\in T_{y}}V(G_{z}))\setminus V(G_{x})</annotation><annotation encoding="application/x-llamapun" id="algorithm6.12.12.m3.2d">( ∪ start_POSTSUBSCRIPT italic_z ∈ italic_T start_POSTSUBSCRIPT italic_y end_POSTSUBSCRIPT end_POSTSUBSCRIPT italic_V ( italic_G start_POSTSUBSCRIPT italic_z end_POSTSUBSCRIPT ) ) ∖ italic_V ( italic_G start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT )</annotation></semantics></math> into supernode <math alttext="y^{\prime}" class="ltx_Math" display="inline" id="algorithm6.13.13.m4.1"><semantics id="algorithm6.13.13.m4.1a"><msup id="algorithm6.13.13.m4.1.1" xref="algorithm6.13.13.m4.1.1.cmml"><mi id="algorithm6.13.13.m4.1.1.2" xref="algorithm6.13.13.m4.1.1.2.cmml">y</mi><mo id="algorithm6.13.13.m4.1.1.3" xref="algorithm6.13.13.m4.1.1.3.cmml">′</mo></msup><annotation-xml encoding="MathML-Content" id="algorithm6.13.13.m4.1b"><apply id="algorithm6.13.13.m4.1.1.cmml" xref="algorithm6.13.13.m4.1.1"><csymbol cd="ambiguous" id="algorithm6.13.13.m4.1.1.1.cmml" xref="algorithm6.13.13.m4.1.1">superscript</csymbol><ci id="algorithm6.13.13.m4.1.1.2.cmml" xref="algorithm6.13.13.m4.1.1.2">𝑦</ci><ci id="algorithm6.13.13.m4.1.1.3.cmml" xref="algorithm6.13.13.m4.1.1.3">′</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="algorithm6.13.13.m4.1c">y^{\prime}</annotation><annotation encoding="application/x-llamapun" id="algorithm6.13.13.m4.1d">italic_y start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math>, and <span class="ltx_text ltx_font_bold" id="algorithm6.15.15.4">add</span> <math alttext="y^{\prime}" class="ltx_Math" display="inline" id="algorithm6.14.14.m5.1"><semantics id="algorithm6.14.14.m5.1a"><msup id="algorithm6.14.14.m5.1.1" xref="algorithm6.14.14.m5.1.1.cmml"><mi id="algorithm6.14.14.m5.1.1.2" xref="algorithm6.14.14.m5.1.1.2.cmml">y</mi><mo id="algorithm6.14.14.m5.1.1.3" xref="algorithm6.14.14.m5.1.1.3.cmml">′</mo></msup><annotation-xml encoding="MathML-Content" id="algorithm6.14.14.m5.1b"><apply id="algorithm6.14.14.m5.1.1.cmml" xref="algorithm6.14.14.m5.1.1"><csymbol cd="ambiguous" id="algorithm6.14.14.m5.1.1.1.cmml" xref="algorithm6.14.14.m5.1.1">superscript</csymbol><ci id="algorithm6.14.14.m5.1.1.2.cmml" xref="algorithm6.14.14.m5.1.1.2">𝑦</ci><ci id="algorithm6.14.14.m5.1.1.3.cmml" xref="algorithm6.14.14.m5.1.1.3">′</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="algorithm6.14.14.m5.1c">y^{\prime}</annotation><annotation encoding="application/x-llamapun" id="algorithm6.14.14.m5.1d">italic_y start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math> to <math alttext="C^{\prime}(x)" class="ltx_Math" display="inline" id="algorithm6.15.15.m6.1"><semantics id="algorithm6.15.15.m6.1a"><mrow id="algorithm6.15.15.m6.1.2" xref="algorithm6.15.15.m6.1.2.cmml"><msup id="algorithm6.15.15.m6.1.2.2" xref="algorithm6.15.15.m6.1.2.2.cmml"><mi id="algorithm6.15.15.m6.1.2.2.2" xref="algorithm6.15.15.m6.1.2.2.2.cmml">C</mi><mo id="algorithm6.15.15.m6.1.2.2.3" xref="algorithm6.15.15.m6.1.2.2.3.cmml">′</mo></msup><mo id="algorithm6.15.15.m6.1.2.1" xref="algorithm6.15.15.m6.1.2.1.cmml">⁢</mo><mrow id="algorithm6.15.15.m6.1.2.3.2" xref="algorithm6.15.15.m6.1.2.cmml"><mo id="algorithm6.15.15.m6.1.2.3.2.1" stretchy="false" xref="algorithm6.15.15.m6.1.2.cmml">(</mo><mi id="algorithm6.15.15.m6.1.1" xref="algorithm6.15.15.m6.1.1.cmml">x</mi><mo id="algorithm6.15.15.m6.1.2.3.2.2" stretchy="false" xref="algorithm6.15.15.m6.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="algorithm6.15.15.m6.1b"><apply id="algorithm6.15.15.m6.1.2.cmml" xref="algorithm6.15.15.m6.1.2"><times id="algorithm6.15.15.m6.1.2.1.cmml" xref="algorithm6.15.15.m6.1.2.1"></times><apply id="algorithm6.15.15.m6.1.2.2.cmml" xref="algorithm6.15.15.m6.1.2.2"><csymbol cd="ambiguous" id="algorithm6.15.15.m6.1.2.2.1.cmml" xref="algorithm6.15.15.m6.1.2.2">superscript</csymbol><ci id="algorithm6.15.15.m6.1.2.2.2.cmml" xref="algorithm6.15.15.m6.1.2.2.2">𝐶</ci><ci id="algorithm6.15.15.m6.1.2.2.3.cmml" xref="algorithm6.15.15.m6.1.2.2.3">′</ci></apply><ci id="algorithm6.15.15.m6.1.1.cmml" xref="algorithm6.15.15.m6.1.1">𝑥</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="algorithm6.15.15.m6.1c">C^{\prime}(x)</annotation><annotation encoding="application/x-llamapun" id="algorithm6.15.15.m6.1d">italic_C start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ( italic_x )</annotation></semantics></math> </div> <div class="ltx_listingline" id="algorithm6.18.18"> <span class="ltx_rule" style="width:1px;height:100%;background:black;display:inline-block;"> </span> <span class="ltx_text ltx_font_bold" id="algorithm6.18.18.1">define</span> the graph <math alttext="H_{x}" class="ltx_Math" display="inline" id="algorithm6.16.16.m1.1"><semantics id="algorithm6.16.16.m1.1a"><msub id="algorithm6.16.16.m1.1.1" xref="algorithm6.16.16.m1.1.1.cmml"><mi id="algorithm6.16.16.m1.1.1.2" xref="algorithm6.16.16.m1.1.1.2.cmml">H</mi><mi id="algorithm6.16.16.m1.1.1.3" xref="algorithm6.16.16.m1.1.1.3.cmml">x</mi></msub><annotation-xml encoding="MathML-Content" id="algorithm6.16.16.m1.1b"><apply id="algorithm6.16.16.m1.1.1.cmml" xref="algorithm6.16.16.m1.1.1"><csymbol cd="ambiguous" id="algorithm6.16.16.m1.1.1.1.cmml" xref="algorithm6.16.16.m1.1.1">subscript</csymbol><ci id="algorithm6.16.16.m1.1.1.2.cmml" xref="algorithm6.16.16.m1.1.1.2">𝐻</ci><ci id="algorithm6.16.16.m1.1.1.3.cmml" xref="algorithm6.16.16.m1.1.1.3">𝑥</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="algorithm6.16.16.m1.1c">H_{x}</annotation><annotation encoding="application/x-llamapun" id="algorithm6.16.16.m1.1d">italic_H start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT</annotation></semantics></math> on node set <math alttext="C^{\prime}(x)" class="ltx_Math" display="inline" id="algorithm6.17.17.m2.1"><semantics id="algorithm6.17.17.m2.1a"><mrow id="algorithm6.17.17.m2.1.2" xref="algorithm6.17.17.m2.1.2.cmml"><msup id="algorithm6.17.17.m2.1.2.2" xref="algorithm6.17.17.m2.1.2.2.cmml"><mi id="algorithm6.17.17.m2.1.2.2.2" xref="algorithm6.17.17.m2.1.2.2.2.cmml">C</mi><mo id="algorithm6.17.17.m2.1.2.2.3" xref="algorithm6.17.17.m2.1.2.2.3.cmml">′</mo></msup><mo id="algorithm6.17.17.m2.1.2.1" xref="algorithm6.17.17.m2.1.2.1.cmml">⁢</mo><mrow id="algorithm6.17.17.m2.1.2.3.2" xref="algorithm6.17.17.m2.1.2.cmml"><mo id="algorithm6.17.17.m2.1.2.3.2.1" stretchy="false" xref="algorithm6.17.17.m2.1.2.cmml">(</mo><mi id="algorithm6.17.17.m2.1.1" xref="algorithm6.17.17.m2.1.1.cmml">x</mi><mo id="algorithm6.17.17.m2.1.2.3.2.2" stretchy="false" xref="algorithm6.17.17.m2.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="algorithm6.17.17.m2.1b"><apply id="algorithm6.17.17.m2.1.2.cmml" xref="algorithm6.17.17.m2.1.2"><times id="algorithm6.17.17.m2.1.2.1.cmml" xref="algorithm6.17.17.m2.1.2.1"></times><apply id="algorithm6.17.17.m2.1.2.2.cmml" xref="algorithm6.17.17.m2.1.2.2"><csymbol cd="ambiguous" id="algorithm6.17.17.m2.1.2.2.1.cmml" xref="algorithm6.17.17.m2.1.2.2">superscript</csymbol><ci id="algorithm6.17.17.m2.1.2.2.2.cmml" xref="algorithm6.17.17.m2.1.2.2.2">𝐶</ci><ci id="algorithm6.17.17.m2.1.2.2.3.cmml" xref="algorithm6.17.17.m2.1.2.2.3">′</ci></apply><ci id="algorithm6.17.17.m2.1.1.cmml" xref="algorithm6.17.17.m2.1.1">𝑥</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="algorithm6.17.17.m2.1c">C^{\prime}(x)</annotation><annotation encoding="application/x-llamapun" id="algorithm6.17.17.m2.1d">italic_C start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ( italic_x )</annotation></semantics></math> and edge set <math alttext="\emptyset" class="ltx_Math" display="inline" id="algorithm6.18.18.m3.1"><semantics id="algorithm6.18.18.m3.1a"><mi id="algorithm6.18.18.m3.1.1" mathvariant="normal" xref="algorithm6.18.18.m3.1.1.cmml">∅</mi><annotation-xml encoding="MathML-Content" id="algorithm6.18.18.m3.1b"><emptyset id="algorithm6.18.18.m3.1.1.cmml" xref="algorithm6.18.18.m3.1.1"></emptyset></annotation-xml><annotation encoding="application/x-tex" id="algorithm6.18.18.m3.1c">\emptyset</annotation><annotation encoding="application/x-llamapun" id="algorithm6.18.18.m3.1d">∅</annotation></semantics></math> </div> <div class="ltx_listingline" id="algorithm6.20.20"> <span class="ltx_text ltx_font_bold" id="algorithm6.20.20.3">for</span> <em class="ltx_emph ltx_font_italic" id="algorithm6.20.20.2"><math alttext="x\in V(T)" class="ltx_Math" display="inline" id="algorithm6.19.19.1.m1.1"><semantics id="algorithm6.19.19.1.m1.1a"><mrow id="algorithm6.19.19.1.m1.1.2" xref="algorithm6.19.19.1.m1.1.2.cmml"><mi id="algorithm6.19.19.1.m1.1.2.2" xref="algorithm6.19.19.1.m1.1.2.2.cmml">x</mi><mo id="algorithm6.19.19.1.m1.1.2.1" xref="algorithm6.19.19.1.m1.1.2.1.cmml">∈</mo><mrow id="algorithm6.19.19.1.m1.1.2.3" xref="algorithm6.19.19.1.m1.1.2.3.cmml"><mi id="algorithm6.19.19.1.m1.1.2.3.2" xref="algorithm6.19.19.1.m1.1.2.3.2.cmml">V</mi><mo id="algorithm6.19.19.1.m1.1.2.3.1" xref="algorithm6.19.19.1.m1.1.2.3.1.cmml">⁢</mo><mrow id="algorithm6.19.19.1.m1.1.2.3.3.2" xref="algorithm6.19.19.1.m1.1.2.3.cmml"><mo id="algorithm6.19.19.1.m1.1.2.3.3.2.1" stretchy="false" xref="algorithm6.19.19.1.m1.1.2.3.cmml">(</mo><mi id="algorithm6.19.19.1.m1.1.1" xref="algorithm6.19.19.1.m1.1.1.cmml">T</mi><mo id="algorithm6.19.19.1.m1.1.2.3.3.2.2" stretchy="false" xref="algorithm6.19.19.1.m1.1.2.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="algorithm6.19.19.1.m1.1b"><apply id="algorithm6.19.19.1.m1.1.2.cmml" xref="algorithm6.19.19.1.m1.1.2"><in id="algorithm6.19.19.1.m1.1.2.1.cmml" xref="algorithm6.19.19.1.m1.1.2.1"></in><ci id="algorithm6.19.19.1.m1.1.2.2.cmml" xref="algorithm6.19.19.1.m1.1.2.2">𝑥</ci><apply id="algorithm6.19.19.1.m1.1.2.3.cmml" xref="algorithm6.19.19.1.m1.1.2.3"><times id="algorithm6.19.19.1.m1.1.2.3.1.cmml" xref="algorithm6.19.19.1.m1.1.2.3.1"></times><ci id="algorithm6.19.19.1.m1.1.2.3.2.cmml" xref="algorithm6.19.19.1.m1.1.2.3.2">𝑉</ci><ci id="algorithm6.19.19.1.m1.1.1.cmml" xref="algorithm6.19.19.1.m1.1.1">𝑇</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="algorithm6.19.19.1.m1.1c">x\in V(T)</annotation><annotation encoding="application/x-llamapun" id="algorithm6.19.19.1.m1.1d">italic_x ∈ italic_V ( italic_T )</annotation></semantics></math>; <math alttext="x" class="ltx_Math" display="inline" id="algorithm6.20.20.2.m2.1"><semantics id="algorithm6.20.20.2.m2.1a"><mi id="algorithm6.20.20.2.m2.1.1" xref="algorithm6.20.20.2.m2.1.1.cmml">x</mi><annotation-xml encoding="MathML-Content" id="algorithm6.20.20.2.m2.1b"><ci id="algorithm6.20.20.2.m2.1.1.cmml" xref="algorithm6.20.20.2.m2.1.1">𝑥</ci></annotation-xml><annotation encoding="application/x-tex" id="algorithm6.20.20.2.m2.1c">x</annotation><annotation encoding="application/x-llamapun" id="algorithm6.20.20.2.m2.1d">italic_x</annotation></semantics></math> is an S-node</em> <span class="ltx_text ltx_font_bold" id="algorithm6.20.20.4">do</span> </div> <div class="ltx_listingline" id="algorithm6.21.21"> <span class="ltx_rule" style="width:1px;height:100%;background:black;display:inline-block;"> </span> <span class="ltx_text ltx_font_bold" id="algorithm6.21.21.1">create</span> a dummy vertex for each virtual edge in <math alttext="G_{x}" class="ltx_Math" display="inline" id="algorithm6.21.21.m1.1"><semantics id="algorithm6.21.21.m1.1a"><msub id="algorithm6.21.21.m1.1.1" xref="algorithm6.21.21.m1.1.1.cmml"><mi id="algorithm6.21.21.m1.1.1.2" xref="algorithm6.21.21.m1.1.1.2.cmml">G</mi><mi id="algorithm6.21.21.m1.1.1.3" xref="algorithm6.21.21.m1.1.1.3.cmml">x</mi></msub><annotation-xml encoding="MathML-Content" id="algorithm6.21.21.m1.1b"><apply id="algorithm6.21.21.m1.1.1.cmml" xref="algorithm6.21.21.m1.1.1"><csymbol cd="ambiguous" id="algorithm6.21.21.m1.1.1.1.cmml" xref="algorithm6.21.21.m1.1.1">subscript</csymbol><ci id="algorithm6.21.21.m1.1.1.2.cmml" xref="algorithm6.21.21.m1.1.1.2">𝐺</ci><ci id="algorithm6.21.21.m1.1.1.3.cmml" xref="algorithm6.21.21.m1.1.1.3">𝑥</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="algorithm6.21.21.m1.1c">G_{x}</annotation><annotation encoding="application/x-llamapun" id="algorithm6.21.21.m1.1d">italic_G start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT</annotation></semantics></math> </div> <div class="ltx_listingline" id="algorithm6.22.22"> <span class="ltx_rule" style="width:1px;height:100%;background:black;display:inline-block;"> </span> <span class="ltx_text ltx_font_bold" id="algorithm6.22.22.2">for</span> <em class="ltx_emph ltx_font_italic" id="algorithm6.22.22.1"><math alttext="w\in V(G_{x})" class="ltx_Math" display="inline" id="algorithm6.22.22.1.m1.1"><semantics id="algorithm6.22.22.1.m1.1a"><mrow id="algorithm6.22.22.1.m1.1.1" xref="algorithm6.22.22.1.m1.1.1.cmml"><mi id="algorithm6.22.22.1.m1.1.1.3" xref="algorithm6.22.22.1.m1.1.1.3.cmml">w</mi><mo id="algorithm6.22.22.1.m1.1.1.2" xref="algorithm6.22.22.1.m1.1.1.2.cmml">∈</mo><mrow id="algorithm6.22.22.1.m1.1.1.1" xref="algorithm6.22.22.1.m1.1.1.1.cmml"><mi id="algorithm6.22.22.1.m1.1.1.1.3" xref="algorithm6.22.22.1.m1.1.1.1.3.cmml">V</mi><mo id="algorithm6.22.22.1.m1.1.1.1.2" xref="algorithm6.22.22.1.m1.1.1.1.2.cmml">⁢</mo><mrow id="algorithm6.22.22.1.m1.1.1.1.1.1" xref="algorithm6.22.22.1.m1.1.1.1.1.1.1.cmml"><mo id="algorithm6.22.22.1.m1.1.1.1.1.1.2" stretchy="false" xref="algorithm6.22.22.1.m1.1.1.1.1.1.1.cmml">(</mo><msub id="algorithm6.22.22.1.m1.1.1.1.1.1.1" xref="algorithm6.22.22.1.m1.1.1.1.1.1.1.cmml"><mi id="algorithm6.22.22.1.m1.1.1.1.1.1.1.2" xref="algorithm6.22.22.1.m1.1.1.1.1.1.1.2.cmml">G</mi><mi id="algorithm6.22.22.1.m1.1.1.1.1.1.1.3" xref="algorithm6.22.22.1.m1.1.1.1.1.1.1.3.cmml">x</mi></msub><mo id="algorithm6.22.22.1.m1.1.1.1.1.1.3" stretchy="false" xref="algorithm6.22.22.1.m1.1.1.1.1.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="algorithm6.22.22.1.m1.1b"><apply id="algorithm6.22.22.1.m1.1.1.cmml" xref="algorithm6.22.22.1.m1.1.1"><in id="algorithm6.22.22.1.m1.1.1.2.cmml" xref="algorithm6.22.22.1.m1.1.1.2"></in><ci id="algorithm6.22.22.1.m1.1.1.3.cmml" xref="algorithm6.22.22.1.m1.1.1.3">𝑤</ci><apply id="algorithm6.22.22.1.m1.1.1.1.cmml" xref="algorithm6.22.22.1.m1.1.1.1"><times id="algorithm6.22.22.1.m1.1.1.1.2.cmml" xref="algorithm6.22.22.1.m1.1.1.1.2"></times><ci id="algorithm6.22.22.1.m1.1.1.1.3.cmml" xref="algorithm6.22.22.1.m1.1.1.1.3">𝑉</ci><apply id="algorithm6.22.22.1.m1.1.1.1.1.1.1.cmml" xref="algorithm6.22.22.1.m1.1.1.1.1.1"><csymbol cd="ambiguous" id="algorithm6.22.22.1.m1.1.1.1.1.1.1.1.cmml" xref="algorithm6.22.22.1.m1.1.1.1.1.1">subscript</csymbol><ci id="algorithm6.22.22.1.m1.1.1.1.1.1.1.2.cmml" xref="algorithm6.22.22.1.m1.1.1.1.1.1.1.2">𝐺</ci><ci id="algorithm6.22.22.1.m1.1.1.1.1.1.1.3.cmml" xref="algorithm6.22.22.1.m1.1.1.1.1.1.1.3">𝑥</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="algorithm6.22.22.1.m1.1c">w\in V(G_{x})</annotation><annotation encoding="application/x-llamapun" id="algorithm6.22.22.1.m1.1d">italic_w ∈ italic_V ( italic_G start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT )</annotation></semantics></math> (original and dummy)</em> <span class="ltx_text ltx_font_bold" id="algorithm6.22.22.3">do</span> </div> <div class="ltx_listingline" id="algorithm6.24.24"> <span class="ltx_rule" style="width:1px;height:100%;background:black;display:inline-block;"> </span> <span class="ltx_rule" style="width:1px;height:100%;background:black;display:inline-block;"> </span> <span class="ltx_text ltx_font_bold" id="algorithm6.24.24.1">initialize</span> empty dictionaries <math alttext="\textsc{Min}_{w}" class="ltx_Math" display="inline" id="algorithm6.23.23.m1.1"><semantics id="algorithm6.23.23.m1.1a"><msub id="algorithm6.23.23.m1.1.1" xref="algorithm6.23.23.m1.1.1.cmml"><mtext class="ltx_font_smallcaps" id="algorithm6.23.23.m1.1.1.2" xref="algorithm6.23.23.m1.1.1.2a.cmml">Min</mtext><mi id="algorithm6.23.23.m1.1.1.3" xref="algorithm6.23.23.m1.1.1.3.cmml">w</mi></msub><annotation-xml encoding="MathML-Content" id="algorithm6.23.23.m1.1b"><apply id="algorithm6.23.23.m1.1.1.cmml" xref="algorithm6.23.23.m1.1.1"><csymbol cd="ambiguous" id="algorithm6.23.23.m1.1.1.1.cmml" xref="algorithm6.23.23.m1.1.1">subscript</csymbol><ci id="algorithm6.23.23.m1.1.1.2a.cmml" xref="algorithm6.23.23.m1.1.1.2"><mtext class="ltx_font_smallcaps" id="algorithm6.23.23.m1.1.1.2.cmml" xref="algorithm6.23.23.m1.1.1.2">Min</mtext></ci><ci id="algorithm6.23.23.m1.1.1.3.cmml" xref="algorithm6.23.23.m1.1.1.3">𝑤</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="algorithm6.23.23.m1.1c">\textsc{Min}_{w}</annotation><annotation encoding="application/x-llamapun" id="algorithm6.23.23.m1.1d">Min start_POSTSUBSCRIPT italic_w end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="\textsc{Max}_{w}" class="ltx_Math" display="inline" id="algorithm6.24.24.m2.1"><semantics id="algorithm6.24.24.m2.1a"><msub id="algorithm6.24.24.m2.1.1" xref="algorithm6.24.24.m2.1.1.cmml"><mtext class="ltx_font_smallcaps" id="algorithm6.24.24.m2.1.1.2" xref="algorithm6.24.24.m2.1.1.2a.cmml">Max</mtext><mi id="algorithm6.24.24.m2.1.1.3" xref="algorithm6.24.24.m2.1.1.3.cmml">w</mi></msub><annotation-xml encoding="MathML-Content" id="algorithm6.24.24.m2.1b"><apply id="algorithm6.24.24.m2.1.1.cmml" xref="algorithm6.24.24.m2.1.1"><csymbol cd="ambiguous" id="algorithm6.24.24.m2.1.1.1.cmml" xref="algorithm6.24.24.m2.1.1">subscript</csymbol><ci id="algorithm6.24.24.m2.1.1.2a.cmml" xref="algorithm6.24.24.m2.1.1.2"><mtext class="ltx_font_smallcaps" id="algorithm6.24.24.m2.1.1.2.cmml" xref="algorithm6.24.24.m2.1.1.2">Max</mtext></ci><ci id="algorithm6.24.24.m2.1.1.3.cmml" xref="algorithm6.24.24.m2.1.1.3">𝑤</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="algorithm6.24.24.m2.1c">\textsc{Max}_{w}</annotation><annotation encoding="application/x-llamapun" id="algorithm6.24.24.m2.1d">Max start_POSTSUBSCRIPT italic_w end_POSTSUBSCRIPT</annotation></semantics></math> </div> <div class="ltx_listingline" id="algorithm6.45.48"> <span class="ltx_text" id="algorithm6.45.48.1" style="color:#0000FF;">/* </span><span class="ltx_text ltx_font_smallcaps" id="algorithm6.45.48.2" style="color:#0000FF;">In Stream: */</span> </div> <div class="ltx_listingline" id="algorithm6.25.25"> <span class="ltx_text ltx_font_bold" id="algorithm6.25.25.2">for</span> <em class="ltx_emph ltx_font_italic" id="algorithm6.25.25.1"><math alttext="e=uv\in L" class="ltx_Math" display="inline" id="algorithm6.25.25.1.m1.1"><semantics id="algorithm6.25.25.1.m1.1a"><mrow id="algorithm6.25.25.1.m1.1.1" xref="algorithm6.25.25.1.m1.1.1.cmml"><mi id="algorithm6.25.25.1.m1.1.1.2" xref="algorithm6.25.25.1.m1.1.1.2.cmml">e</mi><mo id="algorithm6.25.25.1.m1.1.1.3" xref="algorithm6.25.25.1.m1.1.1.3.cmml">=</mo><mrow id="algorithm6.25.25.1.m1.1.1.4" xref="algorithm6.25.25.1.m1.1.1.4.cmml"><mi id="algorithm6.25.25.1.m1.1.1.4.2" xref="algorithm6.25.25.1.m1.1.1.4.2.cmml">u</mi><mo id="algorithm6.25.25.1.m1.1.1.4.1" xref="algorithm6.25.25.1.m1.1.1.4.1.cmml">⁢</mo><mi id="algorithm6.25.25.1.m1.1.1.4.3" xref="algorithm6.25.25.1.m1.1.1.4.3.cmml">v</mi></mrow><mo id="algorithm6.25.25.1.m1.1.1.5" xref="algorithm6.25.25.1.m1.1.1.5.cmml">∈</mo><mi id="algorithm6.25.25.1.m1.1.1.6" xref="algorithm6.25.25.1.m1.1.1.6.cmml">L</mi></mrow><annotation-xml encoding="MathML-Content" id="algorithm6.25.25.1.m1.1b"><apply id="algorithm6.25.25.1.m1.1.1.cmml" xref="algorithm6.25.25.1.m1.1.1"><and id="algorithm6.25.25.1.m1.1.1a.cmml" xref="algorithm6.25.25.1.m1.1.1"></and><apply id="algorithm6.25.25.1.m1.1.1b.cmml" xref="algorithm6.25.25.1.m1.1.1"><eq id="algorithm6.25.25.1.m1.1.1.3.cmml" xref="algorithm6.25.25.1.m1.1.1.3"></eq><ci id="algorithm6.25.25.1.m1.1.1.2.cmml" xref="algorithm6.25.25.1.m1.1.1.2">𝑒</ci><apply id="algorithm6.25.25.1.m1.1.1.4.cmml" xref="algorithm6.25.25.1.m1.1.1.4"><times id="algorithm6.25.25.1.m1.1.1.4.1.cmml" xref="algorithm6.25.25.1.m1.1.1.4.1"></times><ci id="algorithm6.25.25.1.m1.1.1.4.2.cmml" xref="algorithm6.25.25.1.m1.1.1.4.2">𝑢</ci><ci id="algorithm6.25.25.1.m1.1.1.4.3.cmml" xref="algorithm6.25.25.1.m1.1.1.4.3">𝑣</ci></apply></apply><apply id="algorithm6.25.25.1.m1.1.1c.cmml" xref="algorithm6.25.25.1.m1.1.1"><in id="algorithm6.25.25.1.m1.1.1.5.cmml" xref="algorithm6.25.25.1.m1.1.1.5"></in><share href="https://arxiv.org/html/2503.00712v1#algorithm6.25.25.1.m1.1.1.4.cmml" id="algorithm6.25.25.1.m1.1.1d.cmml" xref="algorithm6.25.25.1.m1.1.1"></share><ci id="algorithm6.25.25.1.m1.1.1.6.cmml" xref="algorithm6.25.25.1.m1.1.1.6">𝐿</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="algorithm6.25.25.1.m1.1c">e=uv\in L</annotation><annotation encoding="application/x-llamapun" id="algorithm6.25.25.1.m1.1d">italic_e = italic_u italic_v ∈ italic_L</annotation></semantics></math> in the stream</em> <span class="ltx_text ltx_font_bold" id="algorithm6.25.25.3">do</span> </div> <div class="ltx_listingline" id="algorithm6.27.27"> <span class="ltx_rule" style="width:1px;height:100%;background:black;display:inline-block;"> </span> <span class="ltx_text ltx_font_bold" id="algorithm6.27.27.1">let</span> <math alttext="j" class="ltx_Math" display="inline" id="algorithm6.26.26.m1.1"><semantics id="algorithm6.26.26.m1.1a"><mi id="algorithm6.26.26.m1.1.1" xref="algorithm6.26.26.m1.1.1.cmml">j</mi><annotation-xml encoding="MathML-Content" id="algorithm6.26.26.m1.1b"><ci id="algorithm6.26.26.m1.1.1.cmml" xref="algorithm6.26.26.m1.1.1">𝑗</ci></annotation-xml><annotation encoding="application/x-tex" id="algorithm6.26.26.m1.1c">j</annotation><annotation encoding="application/x-llamapun" id="algorithm6.26.26.m1.1d">italic_j</annotation></semantics></math> be weight class such that <math alttext="w(e)\in[(1+\epsilon)^{j},(1+\epsilon)^{j+1})" class="ltx_Math" display="inline" id="algorithm6.27.27.m2.3"><semantics id="algorithm6.27.27.m2.3a"><mrow id="algorithm6.27.27.m2.3.3" xref="algorithm6.27.27.m2.3.3.cmml"><mrow id="algorithm6.27.27.m2.3.3.4" xref="algorithm6.27.27.m2.3.3.4.cmml"><mi id="algorithm6.27.27.m2.3.3.4.2" xref="algorithm6.27.27.m2.3.3.4.2.cmml">w</mi><mo id="algorithm6.27.27.m2.3.3.4.1" xref="algorithm6.27.27.m2.3.3.4.1.cmml">⁢</mo><mrow id="algorithm6.27.27.m2.3.3.4.3.2" xref="algorithm6.27.27.m2.3.3.4.cmml"><mo id="algorithm6.27.27.m2.3.3.4.3.2.1" stretchy="false" xref="algorithm6.27.27.m2.3.3.4.cmml">(</mo><mi id="algorithm6.27.27.m2.1.1" xref="algorithm6.27.27.m2.1.1.cmml">e</mi><mo id="algorithm6.27.27.m2.3.3.4.3.2.2" stretchy="false" xref="algorithm6.27.27.m2.3.3.4.cmml">)</mo></mrow></mrow><mo id="algorithm6.27.27.m2.3.3.3" xref="algorithm6.27.27.m2.3.3.3.cmml">∈</mo><mrow id="algorithm6.27.27.m2.3.3.2.2" xref="algorithm6.27.27.m2.3.3.2.3.cmml"><mo id="algorithm6.27.27.m2.3.3.2.2.3" stretchy="false" xref="algorithm6.27.27.m2.3.3.2.3.cmml">[</mo><msup id="algorithm6.27.27.m2.2.2.1.1.1" xref="algorithm6.27.27.m2.2.2.1.1.1.cmml"><mrow id="algorithm6.27.27.m2.2.2.1.1.1.1.1" xref="algorithm6.27.27.m2.2.2.1.1.1.1.1.1.cmml"><mo id="algorithm6.27.27.m2.2.2.1.1.1.1.1.2" stretchy="false" xref="algorithm6.27.27.m2.2.2.1.1.1.1.1.1.cmml">(</mo><mrow id="algorithm6.27.27.m2.2.2.1.1.1.1.1.1" xref="algorithm6.27.27.m2.2.2.1.1.1.1.1.1.cmml"><mn id="algorithm6.27.27.m2.2.2.1.1.1.1.1.1.2" xref="algorithm6.27.27.m2.2.2.1.1.1.1.1.1.2.cmml">1</mn><mo id="algorithm6.27.27.m2.2.2.1.1.1.1.1.1.1" xref="algorithm6.27.27.m2.2.2.1.1.1.1.1.1.1.cmml">+</mo><mi id="algorithm6.27.27.m2.2.2.1.1.1.1.1.1.3" xref="algorithm6.27.27.m2.2.2.1.1.1.1.1.1.3.cmml">ϵ</mi></mrow><mo id="algorithm6.27.27.m2.2.2.1.1.1.1.1.3" stretchy="false" xref="algorithm6.27.27.m2.2.2.1.1.1.1.1.1.cmml">)</mo></mrow><mi id="algorithm6.27.27.m2.2.2.1.1.1.3" xref="algorithm6.27.27.m2.2.2.1.1.1.3.cmml">j</mi></msup><mo id="algorithm6.27.27.m2.3.3.2.2.4" xref="algorithm6.27.27.m2.3.3.2.3.cmml">,</mo><msup id="algorithm6.27.27.m2.3.3.2.2.2" xref="algorithm6.27.27.m2.3.3.2.2.2.cmml"><mrow id="algorithm6.27.27.m2.3.3.2.2.2.1.1" xref="algorithm6.27.27.m2.3.3.2.2.2.1.1.1.cmml"><mo id="algorithm6.27.27.m2.3.3.2.2.2.1.1.2" stretchy="false" xref="algorithm6.27.27.m2.3.3.2.2.2.1.1.1.cmml">(</mo><mrow id="algorithm6.27.27.m2.3.3.2.2.2.1.1.1" xref="algorithm6.27.27.m2.3.3.2.2.2.1.1.1.cmml"><mn id="algorithm6.27.27.m2.3.3.2.2.2.1.1.1.2" xref="algorithm6.27.27.m2.3.3.2.2.2.1.1.1.2.cmml">1</mn><mo id="algorithm6.27.27.m2.3.3.2.2.2.1.1.1.1" xref="algorithm6.27.27.m2.3.3.2.2.2.1.1.1.1.cmml">+</mo><mi id="algorithm6.27.27.m2.3.3.2.2.2.1.1.1.3" xref="algorithm6.27.27.m2.3.3.2.2.2.1.1.1.3.cmml">ϵ</mi></mrow><mo id="algorithm6.27.27.m2.3.3.2.2.2.1.1.3" stretchy="false" xref="algorithm6.27.27.m2.3.3.2.2.2.1.1.1.cmml">)</mo></mrow><mrow id="algorithm6.27.27.m2.3.3.2.2.2.3" xref="algorithm6.27.27.m2.3.3.2.2.2.3.cmml"><mi id="algorithm6.27.27.m2.3.3.2.2.2.3.2" xref="algorithm6.27.27.m2.3.3.2.2.2.3.2.cmml">j</mi><mo id="algorithm6.27.27.m2.3.3.2.2.2.3.1" xref="algorithm6.27.27.m2.3.3.2.2.2.3.1.cmml">+</mo><mn id="algorithm6.27.27.m2.3.3.2.2.2.3.3" xref="algorithm6.27.27.m2.3.3.2.2.2.3.3.cmml">1</mn></mrow></msup><mo id="algorithm6.27.27.m2.3.3.2.2.5" stretchy="false" xref="algorithm6.27.27.m2.3.3.2.3.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="algorithm6.27.27.m2.3b"><apply id="algorithm6.27.27.m2.3.3.cmml" xref="algorithm6.27.27.m2.3.3"><in id="algorithm6.27.27.m2.3.3.3.cmml" xref="algorithm6.27.27.m2.3.3.3"></in><apply id="algorithm6.27.27.m2.3.3.4.cmml" xref="algorithm6.27.27.m2.3.3.4"><times id="algorithm6.27.27.m2.3.3.4.1.cmml" xref="algorithm6.27.27.m2.3.3.4.1"></times><ci id="algorithm6.27.27.m2.3.3.4.2.cmml" xref="algorithm6.27.27.m2.3.3.4.2">𝑤</ci><ci id="algorithm6.27.27.m2.1.1.cmml" xref="algorithm6.27.27.m2.1.1">𝑒</ci></apply><interval closure="closed-open" id="algorithm6.27.27.m2.3.3.2.3.cmml" xref="algorithm6.27.27.m2.3.3.2.2"><apply id="algorithm6.27.27.m2.2.2.1.1.1.cmml" xref="algorithm6.27.27.m2.2.2.1.1.1"><csymbol cd="ambiguous" id="algorithm6.27.27.m2.2.2.1.1.1.2.cmml" xref="algorithm6.27.27.m2.2.2.1.1.1">superscript</csymbol><apply id="algorithm6.27.27.m2.2.2.1.1.1.1.1.1.cmml" xref="algorithm6.27.27.m2.2.2.1.1.1.1.1"><plus id="algorithm6.27.27.m2.2.2.1.1.1.1.1.1.1.cmml" xref="algorithm6.27.27.m2.2.2.1.1.1.1.1.1.1"></plus><cn id="algorithm6.27.27.m2.2.2.1.1.1.1.1.1.2.cmml" type="integer" xref="algorithm6.27.27.m2.2.2.1.1.1.1.1.1.2">1</cn><ci id="algorithm6.27.27.m2.2.2.1.1.1.1.1.1.3.cmml" xref="algorithm6.27.27.m2.2.2.1.1.1.1.1.1.3">italic-ϵ</ci></apply><ci id="algorithm6.27.27.m2.2.2.1.1.1.3.cmml" xref="algorithm6.27.27.m2.2.2.1.1.1.3">𝑗</ci></apply><apply id="algorithm6.27.27.m2.3.3.2.2.2.cmml" xref="algorithm6.27.27.m2.3.3.2.2.2"><csymbol cd="ambiguous" id="algorithm6.27.27.m2.3.3.2.2.2.2.cmml" xref="algorithm6.27.27.m2.3.3.2.2.2">superscript</csymbol><apply id="algorithm6.27.27.m2.3.3.2.2.2.1.1.1.cmml" xref="algorithm6.27.27.m2.3.3.2.2.2.1.1"><plus id="algorithm6.27.27.m2.3.3.2.2.2.1.1.1.1.cmml" xref="algorithm6.27.27.m2.3.3.2.2.2.1.1.1.1"></plus><cn id="algorithm6.27.27.m2.3.3.2.2.2.1.1.1.2.cmml" type="integer" xref="algorithm6.27.27.m2.3.3.2.2.2.1.1.1.2">1</cn><ci id="algorithm6.27.27.m2.3.3.2.2.2.1.1.1.3.cmml" xref="algorithm6.27.27.m2.3.3.2.2.2.1.1.1.3">italic-ϵ</ci></apply><apply id="algorithm6.27.27.m2.3.3.2.2.2.3.cmml" xref="algorithm6.27.27.m2.3.3.2.2.2.3"><plus id="algorithm6.27.27.m2.3.3.2.2.2.3.1.cmml" xref="algorithm6.27.27.m2.3.3.2.2.2.3.1"></plus><ci id="algorithm6.27.27.m2.3.3.2.2.2.3.2.cmml" xref="algorithm6.27.27.m2.3.3.2.2.2.3.2">𝑗</ci><cn id="algorithm6.27.27.m2.3.3.2.2.2.3.3.cmml" type="integer" xref="algorithm6.27.27.m2.3.3.2.2.2.3.3">1</cn></apply></apply></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="algorithm6.27.27.m2.3c">w(e)\in[(1+\epsilon)^{j},(1+\epsilon)^{j+1})</annotation><annotation encoding="application/x-llamapun" id="algorithm6.27.27.m2.3d">italic_w ( italic_e ) ∈ [ ( 1 + italic_ϵ ) start_POSTSUPERSCRIPT italic_j end_POSTSUPERSCRIPT , ( 1 + italic_ϵ ) start_POSTSUPERSCRIPT italic_j + 1 end_POSTSUPERSCRIPT )</annotation></semantics></math> </div> <div class="ltx_listingline" id="algorithm6.29.29"> <span class="ltx_rule" style="width:1px;height:100%;background:black;display:inline-block;"> </span> <span class="ltx_text ltx_font_bold" id="algorithm6.29.29.3">if</span> <em class="ltx_emph ltx_font_italic" id="algorithm6.29.29.2"><math alttext="v" class="ltx_Math" display="inline" id="algorithm6.28.28.1.m1.1"><semantics id="algorithm6.28.28.1.m1.1a"><mi id="algorithm6.28.28.1.m1.1.1" xref="algorithm6.28.28.1.m1.1.1.cmml">v</mi><annotation-xml encoding="MathML-Content" id="algorithm6.28.28.1.m1.1b"><ci id="algorithm6.28.28.1.m1.1.1.cmml" xref="algorithm6.28.28.1.m1.1.1">𝑣</ci></annotation-xml><annotation encoding="application/x-tex" id="algorithm6.28.28.1.m1.1c">v</annotation><annotation encoding="application/x-llamapun" id="algorithm6.28.28.1.m1.1d">italic_v</annotation></semantics></math> minimizes <math alttext="d_{T}(r,\text{LCA}(h(u),\ell(v)))" class="ltx_Math" display="inline" id="algorithm6.29.29.2.m2.4"><semantics id="algorithm6.29.29.2.m2.4a"><mrow id="algorithm6.29.29.2.m2.4.4" xref="algorithm6.29.29.2.m2.4.4.cmml"><msub id="algorithm6.29.29.2.m2.4.4.3" xref="algorithm6.29.29.2.m2.4.4.3.cmml"><mi id="algorithm6.29.29.2.m2.4.4.3.2" xref="algorithm6.29.29.2.m2.4.4.3.2.cmml">d</mi><mi id="algorithm6.29.29.2.m2.4.4.3.3" xref="algorithm6.29.29.2.m2.4.4.3.3.cmml">T</mi></msub><mo id="algorithm6.29.29.2.m2.4.4.2" xref="algorithm6.29.29.2.m2.4.4.2.cmml">⁢</mo><mrow id="algorithm6.29.29.2.m2.4.4.1.1" xref="algorithm6.29.29.2.m2.4.4.1.2.cmml"><mo id="algorithm6.29.29.2.m2.4.4.1.1.2" stretchy="false" xref="algorithm6.29.29.2.m2.4.4.1.2.cmml">(</mo><mi id="algorithm6.29.29.2.m2.3.3" xref="algorithm6.29.29.2.m2.3.3.cmml">r</mi><mo id="algorithm6.29.29.2.m2.4.4.1.1.3" xref="algorithm6.29.29.2.m2.4.4.1.2.cmml">,</mo><mrow id="algorithm6.29.29.2.m2.4.4.1.1.1" xref="algorithm6.29.29.2.m2.4.4.1.1.1.cmml"><mtext class="ltx_mathvariant_italic" id="algorithm6.29.29.2.m2.4.4.1.1.1.4" xref="algorithm6.29.29.2.m2.4.4.1.1.1.4a.cmml">LCA</mtext><mo id="algorithm6.29.29.2.m2.4.4.1.1.1.3" xref="algorithm6.29.29.2.m2.4.4.1.1.1.3.cmml">⁢</mo><mrow id="algorithm6.29.29.2.m2.4.4.1.1.1.2.2" xref="algorithm6.29.29.2.m2.4.4.1.1.1.2.3.cmml"><mo id="algorithm6.29.29.2.m2.4.4.1.1.1.2.2.3" stretchy="false" xref="algorithm6.29.29.2.m2.4.4.1.1.1.2.3.cmml">(</mo><mrow id="algorithm6.29.29.2.m2.4.4.1.1.1.1.1.1" xref="algorithm6.29.29.2.m2.4.4.1.1.1.1.1.1.cmml"><mi id="algorithm6.29.29.2.m2.4.4.1.1.1.1.1.1.2" xref="algorithm6.29.29.2.m2.4.4.1.1.1.1.1.1.2.cmml">h</mi><mo id="algorithm6.29.29.2.m2.4.4.1.1.1.1.1.1.1" xref="algorithm6.29.29.2.m2.4.4.1.1.1.1.1.1.1.cmml">⁢</mo><mrow id="algorithm6.29.29.2.m2.4.4.1.1.1.1.1.1.3.2" xref="algorithm6.29.29.2.m2.4.4.1.1.1.1.1.1.cmml"><mo id="algorithm6.29.29.2.m2.4.4.1.1.1.1.1.1.3.2.1" stretchy="false" xref="algorithm6.29.29.2.m2.4.4.1.1.1.1.1.1.cmml">(</mo><mi id="algorithm6.29.29.2.m2.1.1" xref="algorithm6.29.29.2.m2.1.1.cmml">u</mi><mo id="algorithm6.29.29.2.m2.4.4.1.1.1.1.1.1.3.2.2" stretchy="false" xref="algorithm6.29.29.2.m2.4.4.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="algorithm6.29.29.2.m2.4.4.1.1.1.2.2.4" xref="algorithm6.29.29.2.m2.4.4.1.1.1.2.3.cmml">,</mo><mrow id="algorithm6.29.29.2.m2.4.4.1.1.1.2.2.2" xref="algorithm6.29.29.2.m2.4.4.1.1.1.2.2.2.cmml"><mi id="algorithm6.29.29.2.m2.4.4.1.1.1.2.2.2.2" mathvariant="normal" xref="algorithm6.29.29.2.m2.4.4.1.1.1.2.2.2.2.cmml">ℓ</mi><mo id="algorithm6.29.29.2.m2.4.4.1.1.1.2.2.2.1" xref="algorithm6.29.29.2.m2.4.4.1.1.1.2.2.2.1.cmml">⁢</mo><mrow id="algorithm6.29.29.2.m2.4.4.1.1.1.2.2.2.3.2" xref="algorithm6.29.29.2.m2.4.4.1.1.1.2.2.2.cmml"><mo id="algorithm6.29.29.2.m2.4.4.1.1.1.2.2.2.3.2.1" stretchy="false" xref="algorithm6.29.29.2.m2.4.4.1.1.1.2.2.2.cmml">(</mo><mi id="algorithm6.29.29.2.m2.2.2" xref="algorithm6.29.29.2.m2.2.2.cmml">v</mi><mo id="algorithm6.29.29.2.m2.4.4.1.1.1.2.2.2.3.2.2" stretchy="false" xref="algorithm6.29.29.2.m2.4.4.1.1.1.2.2.2.cmml">)</mo></mrow></mrow><mo id="algorithm6.29.29.2.m2.4.4.1.1.1.2.2.5" stretchy="false" xref="algorithm6.29.29.2.m2.4.4.1.1.1.2.3.cmml">)</mo></mrow></mrow><mo id="algorithm6.29.29.2.m2.4.4.1.1.4" stretchy="false" xref="algorithm6.29.29.2.m2.4.4.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="algorithm6.29.29.2.m2.4b"><apply id="algorithm6.29.29.2.m2.4.4.cmml" xref="algorithm6.29.29.2.m2.4.4"><times id="algorithm6.29.29.2.m2.4.4.2.cmml" xref="algorithm6.29.29.2.m2.4.4.2"></times><apply id="algorithm6.29.29.2.m2.4.4.3.cmml" xref="algorithm6.29.29.2.m2.4.4.3"><csymbol cd="ambiguous" id="algorithm6.29.29.2.m2.4.4.3.1.cmml" xref="algorithm6.29.29.2.m2.4.4.3">subscript</csymbol><ci id="algorithm6.29.29.2.m2.4.4.3.2.cmml" xref="algorithm6.29.29.2.m2.4.4.3.2">𝑑</ci><ci id="algorithm6.29.29.2.m2.4.4.3.3.cmml" xref="algorithm6.29.29.2.m2.4.4.3.3">𝑇</ci></apply><interval closure="open" id="algorithm6.29.29.2.m2.4.4.1.2.cmml" xref="algorithm6.29.29.2.m2.4.4.1.1"><ci id="algorithm6.29.29.2.m2.3.3.cmml" xref="algorithm6.29.29.2.m2.3.3">𝑟</ci><apply id="algorithm6.29.29.2.m2.4.4.1.1.1.cmml" xref="algorithm6.29.29.2.m2.4.4.1.1.1"><times id="algorithm6.29.29.2.m2.4.4.1.1.1.3.cmml" xref="algorithm6.29.29.2.m2.4.4.1.1.1.3"></times><ci id="algorithm6.29.29.2.m2.4.4.1.1.1.4a.cmml" xref="algorithm6.29.29.2.m2.4.4.1.1.1.4"><mtext class="ltx_mathvariant_italic" id="algorithm6.29.29.2.m2.4.4.1.1.1.4.cmml" xref="algorithm6.29.29.2.m2.4.4.1.1.1.4">LCA</mtext></ci><interval closure="open" id="algorithm6.29.29.2.m2.4.4.1.1.1.2.3.cmml" xref="algorithm6.29.29.2.m2.4.4.1.1.1.2.2"><apply id="algorithm6.29.29.2.m2.4.4.1.1.1.1.1.1.cmml" xref="algorithm6.29.29.2.m2.4.4.1.1.1.1.1.1"><times id="algorithm6.29.29.2.m2.4.4.1.1.1.1.1.1.1.cmml" xref="algorithm6.29.29.2.m2.4.4.1.1.1.1.1.1.1"></times><ci id="algorithm6.29.29.2.m2.4.4.1.1.1.1.1.1.2.cmml" xref="algorithm6.29.29.2.m2.4.4.1.1.1.1.1.1.2">ℎ</ci><ci id="algorithm6.29.29.2.m2.1.1.cmml" xref="algorithm6.29.29.2.m2.1.1">𝑢</ci></apply><apply id="algorithm6.29.29.2.m2.4.4.1.1.1.2.2.2.cmml" xref="algorithm6.29.29.2.m2.4.4.1.1.1.2.2.2"><times id="algorithm6.29.29.2.m2.4.4.1.1.1.2.2.2.1.cmml" xref="algorithm6.29.29.2.m2.4.4.1.1.1.2.2.2.1"></times><ci id="algorithm6.29.29.2.m2.4.4.1.1.1.2.2.2.2.cmml" xref="algorithm6.29.29.2.m2.4.4.1.1.1.2.2.2.2">ℓ</ci><ci id="algorithm6.29.29.2.m2.2.2.cmml" xref="algorithm6.29.29.2.m2.2.2">𝑣</ci></apply></interval></apply></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="algorithm6.29.29.2.m2.4c">d_{T}(r,\text{LCA}(h(u),\ell(v)))</annotation><annotation encoding="application/x-llamapun" id="algorithm6.29.29.2.m2.4d">italic_d start_POSTSUBSCRIPT italic_T end_POSTSUBSCRIPT ( italic_r , LCA ( italic_h ( italic_u ) , roman_ℓ ( italic_v ) ) )</annotation></semantics></math></em> <span class="ltx_text ltx_font_bold" id="algorithm6.29.29.4">then</span> </div> <div class="ltx_listingline" id="algorithm6.30.30"> <span class="ltx_rule" style="width:1px;height:100%;background:black;display:inline-block;"> </span> <span class="ltx_rule" style="width:1px;height:100%;background:black;display:inline-block;"> </span> <span class="ltx_text ltx_font_bold" id="algorithm6.30.30.1">update</span> <math alttext="L_{h(u)}(j)\leftarrow e" class="ltx_Math" display="inline" id="algorithm6.30.30.m1.2"><semantics id="algorithm6.30.30.m1.2a"><mrow id="algorithm6.30.30.m1.2.3" xref="algorithm6.30.30.m1.2.3.cmml"><mrow id="algorithm6.30.30.m1.2.3.2" xref="algorithm6.30.30.m1.2.3.2.cmml"><msub id="algorithm6.30.30.m1.2.3.2.2" xref="algorithm6.30.30.m1.2.3.2.2.cmml"><mi id="algorithm6.30.30.m1.2.3.2.2.2" xref="algorithm6.30.30.m1.2.3.2.2.2.cmml">L</mi><mrow id="algorithm6.30.30.m1.1.1.1" xref="algorithm6.30.30.m1.1.1.1.cmml"><mi id="algorithm6.30.30.m1.1.1.1.3" xref="algorithm6.30.30.m1.1.1.1.3.cmml">h</mi><mo id="algorithm6.30.30.m1.1.1.1.2" xref="algorithm6.30.30.m1.1.1.1.2.cmml">⁢</mo><mrow id="algorithm6.30.30.m1.1.1.1.4.2" xref="algorithm6.30.30.m1.1.1.1.cmml"><mo id="algorithm6.30.30.m1.1.1.1.4.2.1" stretchy="false" xref="algorithm6.30.30.m1.1.1.1.cmml">(</mo><mi id="algorithm6.30.30.m1.1.1.1.1" xref="algorithm6.30.30.m1.1.1.1.1.cmml">u</mi><mo id="algorithm6.30.30.m1.1.1.1.4.2.2" stretchy="false" xref="algorithm6.30.30.m1.1.1.1.cmml">)</mo></mrow></mrow></msub><mo id="algorithm6.30.30.m1.2.3.2.1" xref="algorithm6.30.30.m1.2.3.2.1.cmml">⁢</mo><mrow id="algorithm6.30.30.m1.2.3.2.3.2" xref="algorithm6.30.30.m1.2.3.2.cmml"><mo id="algorithm6.30.30.m1.2.3.2.3.2.1" stretchy="false" xref="algorithm6.30.30.m1.2.3.2.cmml">(</mo><mi id="algorithm6.30.30.m1.2.2" xref="algorithm6.30.30.m1.2.2.cmml">j</mi><mo id="algorithm6.30.30.m1.2.3.2.3.2.2" stretchy="false" xref="algorithm6.30.30.m1.2.3.2.cmml">)</mo></mrow></mrow><mo id="algorithm6.30.30.m1.2.3.1" stretchy="false" xref="algorithm6.30.30.m1.2.3.1.cmml">←</mo><mi id="algorithm6.30.30.m1.2.3.3" xref="algorithm6.30.30.m1.2.3.3.cmml">e</mi></mrow><annotation-xml encoding="MathML-Content" id="algorithm6.30.30.m1.2b"><apply id="algorithm6.30.30.m1.2.3.cmml" xref="algorithm6.30.30.m1.2.3"><ci id="algorithm6.30.30.m1.2.3.1.cmml" xref="algorithm6.30.30.m1.2.3.1">←</ci><apply id="algorithm6.30.30.m1.2.3.2.cmml" xref="algorithm6.30.30.m1.2.3.2"><times id="algorithm6.30.30.m1.2.3.2.1.cmml" xref="algorithm6.30.30.m1.2.3.2.1"></times><apply id="algorithm6.30.30.m1.2.3.2.2.cmml" xref="algorithm6.30.30.m1.2.3.2.2"><csymbol cd="ambiguous" id="algorithm6.30.30.m1.2.3.2.2.1.cmml" xref="algorithm6.30.30.m1.2.3.2.2">subscript</csymbol><ci id="algorithm6.30.30.m1.2.3.2.2.2.cmml" xref="algorithm6.30.30.m1.2.3.2.2.2">𝐿</ci><apply id="algorithm6.30.30.m1.1.1.1.cmml" xref="algorithm6.30.30.m1.1.1.1"><times id="algorithm6.30.30.m1.1.1.1.2.cmml" xref="algorithm6.30.30.m1.1.1.1.2"></times><ci id="algorithm6.30.30.m1.1.1.1.3.cmml" xref="algorithm6.30.30.m1.1.1.1.3">ℎ</ci><ci id="algorithm6.30.30.m1.1.1.1.1.cmml" xref="algorithm6.30.30.m1.1.1.1.1">𝑢</ci></apply></apply><ci id="algorithm6.30.30.m1.2.2.cmml" xref="algorithm6.30.30.m1.2.2">𝑗</ci></apply><ci id="algorithm6.30.30.m1.2.3.3.cmml" xref="algorithm6.30.30.m1.2.3.3">𝑒</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="algorithm6.30.30.m1.2c">L_{h(u)}(j)\leftarrow e</annotation><annotation encoding="application/x-llamapun" id="algorithm6.30.30.m1.2d">italic_L start_POSTSUBSCRIPT italic_h ( italic_u ) end_POSTSUBSCRIPT ( italic_j ) ← italic_e</annotation></semantics></math> </div> <div class="ltx_listingline" id="algorithm6.32.32"> <span class="ltx_rule" style="width:1px;height:100%;background:black;display:inline-block;"> </span> <span class="ltx_text ltx_font_bold" id="algorithm6.32.32.3">if</span> <em class="ltx_emph ltx_font_italic" id="algorithm6.32.32.2"><math alttext="u" class="ltx_Math" display="inline" id="algorithm6.31.31.1.m1.1"><semantics id="algorithm6.31.31.1.m1.1a"><mi id="algorithm6.31.31.1.m1.1.1" xref="algorithm6.31.31.1.m1.1.1.cmml">u</mi><annotation-xml encoding="MathML-Content" id="algorithm6.31.31.1.m1.1b"><ci id="algorithm6.31.31.1.m1.1.1.cmml" xref="algorithm6.31.31.1.m1.1.1">𝑢</ci></annotation-xml><annotation encoding="application/x-tex" id="algorithm6.31.31.1.m1.1c">u</annotation><annotation encoding="application/x-llamapun" id="algorithm6.31.31.1.m1.1d">italic_u</annotation></semantics></math> minimizes <math alttext="d_{T}(r,\text{LCA}(h(v),\ell(u)))" class="ltx_Math" display="inline" id="algorithm6.32.32.2.m2.4"><semantics id="algorithm6.32.32.2.m2.4a"><mrow id="algorithm6.32.32.2.m2.4.4" xref="algorithm6.32.32.2.m2.4.4.cmml"><msub id="algorithm6.32.32.2.m2.4.4.3" xref="algorithm6.32.32.2.m2.4.4.3.cmml"><mi id="algorithm6.32.32.2.m2.4.4.3.2" xref="algorithm6.32.32.2.m2.4.4.3.2.cmml">d</mi><mi id="algorithm6.32.32.2.m2.4.4.3.3" xref="algorithm6.32.32.2.m2.4.4.3.3.cmml">T</mi></msub><mo id="algorithm6.32.32.2.m2.4.4.2" xref="algorithm6.32.32.2.m2.4.4.2.cmml">⁢</mo><mrow id="algorithm6.32.32.2.m2.4.4.1.1" xref="algorithm6.32.32.2.m2.4.4.1.2.cmml"><mo id="algorithm6.32.32.2.m2.4.4.1.1.2" stretchy="false" xref="algorithm6.32.32.2.m2.4.4.1.2.cmml">(</mo><mi id="algorithm6.32.32.2.m2.3.3" xref="algorithm6.32.32.2.m2.3.3.cmml">r</mi><mo id="algorithm6.32.32.2.m2.4.4.1.1.3" xref="algorithm6.32.32.2.m2.4.4.1.2.cmml">,</mo><mrow id="algorithm6.32.32.2.m2.4.4.1.1.1" xref="algorithm6.32.32.2.m2.4.4.1.1.1.cmml"><mtext class="ltx_mathvariant_italic" id="algorithm6.32.32.2.m2.4.4.1.1.1.4" xref="algorithm6.32.32.2.m2.4.4.1.1.1.4a.cmml">LCA</mtext><mo id="algorithm6.32.32.2.m2.4.4.1.1.1.3" xref="algorithm6.32.32.2.m2.4.4.1.1.1.3.cmml">⁢</mo><mrow id="algorithm6.32.32.2.m2.4.4.1.1.1.2.2" xref="algorithm6.32.32.2.m2.4.4.1.1.1.2.3.cmml"><mo id="algorithm6.32.32.2.m2.4.4.1.1.1.2.2.3" stretchy="false" xref="algorithm6.32.32.2.m2.4.4.1.1.1.2.3.cmml">(</mo><mrow id="algorithm6.32.32.2.m2.4.4.1.1.1.1.1.1" xref="algorithm6.32.32.2.m2.4.4.1.1.1.1.1.1.cmml"><mi id="algorithm6.32.32.2.m2.4.4.1.1.1.1.1.1.2" xref="algorithm6.32.32.2.m2.4.4.1.1.1.1.1.1.2.cmml">h</mi><mo id="algorithm6.32.32.2.m2.4.4.1.1.1.1.1.1.1" xref="algorithm6.32.32.2.m2.4.4.1.1.1.1.1.1.1.cmml">⁢</mo><mrow id="algorithm6.32.32.2.m2.4.4.1.1.1.1.1.1.3.2" xref="algorithm6.32.32.2.m2.4.4.1.1.1.1.1.1.cmml"><mo id="algorithm6.32.32.2.m2.4.4.1.1.1.1.1.1.3.2.1" stretchy="false" xref="algorithm6.32.32.2.m2.4.4.1.1.1.1.1.1.cmml">(</mo><mi id="algorithm6.32.32.2.m2.1.1" xref="algorithm6.32.32.2.m2.1.1.cmml">v</mi><mo id="algorithm6.32.32.2.m2.4.4.1.1.1.1.1.1.3.2.2" stretchy="false" xref="algorithm6.32.32.2.m2.4.4.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="algorithm6.32.32.2.m2.4.4.1.1.1.2.2.4" xref="algorithm6.32.32.2.m2.4.4.1.1.1.2.3.cmml">,</mo><mrow id="algorithm6.32.32.2.m2.4.4.1.1.1.2.2.2" xref="algorithm6.32.32.2.m2.4.4.1.1.1.2.2.2.cmml"><mi id="algorithm6.32.32.2.m2.4.4.1.1.1.2.2.2.2" mathvariant="normal" xref="algorithm6.32.32.2.m2.4.4.1.1.1.2.2.2.2.cmml">ℓ</mi><mo id="algorithm6.32.32.2.m2.4.4.1.1.1.2.2.2.1" xref="algorithm6.32.32.2.m2.4.4.1.1.1.2.2.2.1.cmml">⁢</mo><mrow id="algorithm6.32.32.2.m2.4.4.1.1.1.2.2.2.3.2" xref="algorithm6.32.32.2.m2.4.4.1.1.1.2.2.2.cmml"><mo id="algorithm6.32.32.2.m2.4.4.1.1.1.2.2.2.3.2.1" stretchy="false" xref="algorithm6.32.32.2.m2.4.4.1.1.1.2.2.2.cmml">(</mo><mi id="algorithm6.32.32.2.m2.2.2" xref="algorithm6.32.32.2.m2.2.2.cmml">u</mi><mo id="algorithm6.32.32.2.m2.4.4.1.1.1.2.2.2.3.2.2" stretchy="false" xref="algorithm6.32.32.2.m2.4.4.1.1.1.2.2.2.cmml">)</mo></mrow></mrow><mo id="algorithm6.32.32.2.m2.4.4.1.1.1.2.2.5" stretchy="false" xref="algorithm6.32.32.2.m2.4.4.1.1.1.2.3.cmml">)</mo></mrow></mrow><mo id="algorithm6.32.32.2.m2.4.4.1.1.4" stretchy="false" xref="algorithm6.32.32.2.m2.4.4.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="algorithm6.32.32.2.m2.4b"><apply id="algorithm6.32.32.2.m2.4.4.cmml" xref="algorithm6.32.32.2.m2.4.4"><times id="algorithm6.32.32.2.m2.4.4.2.cmml" xref="algorithm6.32.32.2.m2.4.4.2"></times><apply id="algorithm6.32.32.2.m2.4.4.3.cmml" xref="algorithm6.32.32.2.m2.4.4.3"><csymbol cd="ambiguous" id="algorithm6.32.32.2.m2.4.4.3.1.cmml" xref="algorithm6.32.32.2.m2.4.4.3">subscript</csymbol><ci id="algorithm6.32.32.2.m2.4.4.3.2.cmml" xref="algorithm6.32.32.2.m2.4.4.3.2">𝑑</ci><ci id="algorithm6.32.32.2.m2.4.4.3.3.cmml" xref="algorithm6.32.32.2.m2.4.4.3.3">𝑇</ci></apply><interval closure="open" id="algorithm6.32.32.2.m2.4.4.1.2.cmml" xref="algorithm6.32.32.2.m2.4.4.1.1"><ci id="algorithm6.32.32.2.m2.3.3.cmml" xref="algorithm6.32.32.2.m2.3.3">𝑟</ci><apply id="algorithm6.32.32.2.m2.4.4.1.1.1.cmml" xref="algorithm6.32.32.2.m2.4.4.1.1.1"><times id="algorithm6.32.32.2.m2.4.4.1.1.1.3.cmml" xref="algorithm6.32.32.2.m2.4.4.1.1.1.3"></times><ci id="algorithm6.32.32.2.m2.4.4.1.1.1.4a.cmml" xref="algorithm6.32.32.2.m2.4.4.1.1.1.4"><mtext class="ltx_mathvariant_italic" id="algorithm6.32.32.2.m2.4.4.1.1.1.4.cmml" xref="algorithm6.32.32.2.m2.4.4.1.1.1.4">LCA</mtext></ci><interval closure="open" id="algorithm6.32.32.2.m2.4.4.1.1.1.2.3.cmml" xref="algorithm6.32.32.2.m2.4.4.1.1.1.2.2"><apply id="algorithm6.32.32.2.m2.4.4.1.1.1.1.1.1.cmml" xref="algorithm6.32.32.2.m2.4.4.1.1.1.1.1.1"><times id="algorithm6.32.32.2.m2.4.4.1.1.1.1.1.1.1.cmml" xref="algorithm6.32.32.2.m2.4.4.1.1.1.1.1.1.1"></times><ci id="algorithm6.32.32.2.m2.4.4.1.1.1.1.1.1.2.cmml" xref="algorithm6.32.32.2.m2.4.4.1.1.1.1.1.1.2">ℎ</ci><ci id="algorithm6.32.32.2.m2.1.1.cmml" xref="algorithm6.32.32.2.m2.1.1">𝑣</ci></apply><apply id="algorithm6.32.32.2.m2.4.4.1.1.1.2.2.2.cmml" xref="algorithm6.32.32.2.m2.4.4.1.1.1.2.2.2"><times id="algorithm6.32.32.2.m2.4.4.1.1.1.2.2.2.1.cmml" xref="algorithm6.32.32.2.m2.4.4.1.1.1.2.2.2.1"></times><ci id="algorithm6.32.32.2.m2.4.4.1.1.1.2.2.2.2.cmml" xref="algorithm6.32.32.2.m2.4.4.1.1.1.2.2.2.2">ℓ</ci><ci id="algorithm6.32.32.2.m2.2.2.cmml" xref="algorithm6.32.32.2.m2.2.2">𝑢</ci></apply></interval></apply></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="algorithm6.32.32.2.m2.4c">d_{T}(r,\text{LCA}(h(v),\ell(u)))</annotation><annotation encoding="application/x-llamapun" id="algorithm6.32.32.2.m2.4d">italic_d start_POSTSUBSCRIPT italic_T end_POSTSUBSCRIPT ( italic_r , LCA ( italic_h ( italic_v ) , roman_ℓ ( italic_u ) ) )</annotation></semantics></math></em> <span class="ltx_text ltx_font_bold" id="algorithm6.32.32.4">then</span> </div> <div class="ltx_listingline" id="algorithm6.33.33"> <span class="ltx_rule" style="width:1px;height:100%;background:black;display:inline-block;"> </span> <span class="ltx_rule" style="width:1px;height:100%;background:black;display:inline-block;"> </span> <span class="ltx_text ltx_font_bold" id="algorithm6.33.33.1">update</span> <math alttext="L_{h(v)}(j)\leftarrow e" class="ltx_Math" display="inline" id="algorithm6.33.33.m1.2"><semantics id="algorithm6.33.33.m1.2a"><mrow id="algorithm6.33.33.m1.2.3" xref="algorithm6.33.33.m1.2.3.cmml"><mrow id="algorithm6.33.33.m1.2.3.2" xref="algorithm6.33.33.m1.2.3.2.cmml"><msub id="algorithm6.33.33.m1.2.3.2.2" xref="algorithm6.33.33.m1.2.3.2.2.cmml"><mi id="algorithm6.33.33.m1.2.3.2.2.2" xref="algorithm6.33.33.m1.2.3.2.2.2.cmml">L</mi><mrow id="algorithm6.33.33.m1.1.1.1" xref="algorithm6.33.33.m1.1.1.1.cmml"><mi id="algorithm6.33.33.m1.1.1.1.3" xref="algorithm6.33.33.m1.1.1.1.3.cmml">h</mi><mo id="algorithm6.33.33.m1.1.1.1.2" xref="algorithm6.33.33.m1.1.1.1.2.cmml">⁢</mo><mrow id="algorithm6.33.33.m1.1.1.1.4.2" xref="algorithm6.33.33.m1.1.1.1.cmml"><mo id="algorithm6.33.33.m1.1.1.1.4.2.1" stretchy="false" xref="algorithm6.33.33.m1.1.1.1.cmml">(</mo><mi id="algorithm6.33.33.m1.1.1.1.1" xref="algorithm6.33.33.m1.1.1.1.1.cmml">v</mi><mo id="algorithm6.33.33.m1.1.1.1.4.2.2" stretchy="false" xref="algorithm6.33.33.m1.1.1.1.cmml">)</mo></mrow></mrow></msub><mo id="algorithm6.33.33.m1.2.3.2.1" xref="algorithm6.33.33.m1.2.3.2.1.cmml">⁢</mo><mrow id="algorithm6.33.33.m1.2.3.2.3.2" xref="algorithm6.33.33.m1.2.3.2.cmml"><mo id="algorithm6.33.33.m1.2.3.2.3.2.1" stretchy="false" xref="algorithm6.33.33.m1.2.3.2.cmml">(</mo><mi id="algorithm6.33.33.m1.2.2" xref="algorithm6.33.33.m1.2.2.cmml">j</mi><mo id="algorithm6.33.33.m1.2.3.2.3.2.2" stretchy="false" xref="algorithm6.33.33.m1.2.3.2.cmml">)</mo></mrow></mrow><mo id="algorithm6.33.33.m1.2.3.1" stretchy="false" xref="algorithm6.33.33.m1.2.3.1.cmml">←</mo><mi id="algorithm6.33.33.m1.2.3.3" xref="algorithm6.33.33.m1.2.3.3.cmml">e</mi></mrow><annotation-xml encoding="MathML-Content" id="algorithm6.33.33.m1.2b"><apply id="algorithm6.33.33.m1.2.3.cmml" xref="algorithm6.33.33.m1.2.3"><ci id="algorithm6.33.33.m1.2.3.1.cmml" xref="algorithm6.33.33.m1.2.3.1">←</ci><apply id="algorithm6.33.33.m1.2.3.2.cmml" xref="algorithm6.33.33.m1.2.3.2"><times id="algorithm6.33.33.m1.2.3.2.1.cmml" xref="algorithm6.33.33.m1.2.3.2.1"></times><apply id="algorithm6.33.33.m1.2.3.2.2.cmml" xref="algorithm6.33.33.m1.2.3.2.2"><csymbol cd="ambiguous" id="algorithm6.33.33.m1.2.3.2.2.1.cmml" xref="algorithm6.33.33.m1.2.3.2.2">subscript</csymbol><ci id="algorithm6.33.33.m1.2.3.2.2.2.cmml" xref="algorithm6.33.33.m1.2.3.2.2.2">𝐿</ci><apply id="algorithm6.33.33.m1.1.1.1.cmml" xref="algorithm6.33.33.m1.1.1.1"><times id="algorithm6.33.33.m1.1.1.1.2.cmml" xref="algorithm6.33.33.m1.1.1.1.2"></times><ci id="algorithm6.33.33.m1.1.1.1.3.cmml" xref="algorithm6.33.33.m1.1.1.1.3">ℎ</ci><ci id="algorithm6.33.33.m1.1.1.1.1.cmml" xref="algorithm6.33.33.m1.1.1.1.1">𝑣</ci></apply></apply><ci id="algorithm6.33.33.m1.2.2.cmml" xref="algorithm6.33.33.m1.2.2">𝑗</ci></apply><ci id="algorithm6.33.33.m1.2.3.3.cmml" xref="algorithm6.33.33.m1.2.3.3">𝑒</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="algorithm6.33.33.m1.2c">L_{h(v)}(j)\leftarrow e</annotation><annotation encoding="application/x-llamapun" id="algorithm6.33.33.m1.2d">italic_L start_POSTSUBSCRIPT italic_h ( italic_v ) end_POSTSUBSCRIPT ( italic_j ) ← italic_e</annotation></semantics></math> </div> <div class="ltx_listingline" id="algorithm6.35.35"> <span class="ltx_rule" style="width:1px;height:100%;background:black;display:inline-block;"> </span> <span class="ltx_text ltx_font_bold" id="algorithm6.35.35.3">for</span> <em class="ltx_emph ltx_font_italic" id="algorithm6.35.35.2"><math alttext="x\in V(T)" class="ltx_Math" display="inline" id="algorithm6.34.34.1.m1.1"><semantics id="algorithm6.34.34.1.m1.1a"><mrow id="algorithm6.34.34.1.m1.1.2" xref="algorithm6.34.34.1.m1.1.2.cmml"><mi id="algorithm6.34.34.1.m1.1.2.2" xref="algorithm6.34.34.1.m1.1.2.2.cmml">x</mi><mo id="algorithm6.34.34.1.m1.1.2.1" xref="algorithm6.34.34.1.m1.1.2.1.cmml">∈</mo><mrow id="algorithm6.34.34.1.m1.1.2.3" xref="algorithm6.34.34.1.m1.1.2.3.cmml"><mi id="algorithm6.34.34.1.m1.1.2.3.2" xref="algorithm6.34.34.1.m1.1.2.3.2.cmml">V</mi><mo id="algorithm6.34.34.1.m1.1.2.3.1" xref="algorithm6.34.34.1.m1.1.2.3.1.cmml">⁢</mo><mrow id="algorithm6.34.34.1.m1.1.2.3.3.2" xref="algorithm6.34.34.1.m1.1.2.3.cmml"><mo id="algorithm6.34.34.1.m1.1.2.3.3.2.1" stretchy="false" xref="algorithm6.34.34.1.m1.1.2.3.cmml">(</mo><mi id="algorithm6.34.34.1.m1.1.1" xref="algorithm6.34.34.1.m1.1.1.cmml">T</mi><mo id="algorithm6.34.34.1.m1.1.2.3.3.2.2" stretchy="false" xref="algorithm6.34.34.1.m1.1.2.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="algorithm6.34.34.1.m1.1b"><apply id="algorithm6.34.34.1.m1.1.2.cmml" xref="algorithm6.34.34.1.m1.1.2"><in id="algorithm6.34.34.1.m1.1.2.1.cmml" xref="algorithm6.34.34.1.m1.1.2.1"></in><ci id="algorithm6.34.34.1.m1.1.2.2.cmml" xref="algorithm6.34.34.1.m1.1.2.2">𝑥</ci><apply id="algorithm6.34.34.1.m1.1.2.3.cmml" xref="algorithm6.34.34.1.m1.1.2.3"><times id="algorithm6.34.34.1.m1.1.2.3.1.cmml" xref="algorithm6.34.34.1.m1.1.2.3.1"></times><ci id="algorithm6.34.34.1.m1.1.2.3.2.cmml" xref="algorithm6.34.34.1.m1.1.2.3.2">𝑉</ci><ci id="algorithm6.34.34.1.m1.1.1.cmml" xref="algorithm6.34.34.1.m1.1.1">𝑇</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="algorithm6.34.34.1.m1.1c">x\in V(T)</annotation><annotation encoding="application/x-llamapun" id="algorithm6.34.34.1.m1.1d">italic_x ∈ italic_V ( italic_T )</annotation></semantics></math>; <math alttext="x" class="ltx_Math" display="inline" id="algorithm6.35.35.2.m2.1"><semantics id="algorithm6.35.35.2.m2.1a"><mi id="algorithm6.35.35.2.m2.1.1" xref="algorithm6.35.35.2.m2.1.1.cmml">x</mi><annotation-xml encoding="MathML-Content" id="algorithm6.35.35.2.m2.1b"><ci id="algorithm6.35.35.2.m2.1.1.cmml" xref="algorithm6.35.35.2.m2.1.1">𝑥</ci></annotation-xml><annotation encoding="application/x-tex" id="algorithm6.35.35.2.m2.1c">x</annotation><annotation encoding="application/x-llamapun" id="algorithm6.35.35.2.m2.1d">italic_x</annotation></semantics></math> is a P-node</em> <span class="ltx_text ltx_font_bold" id="algorithm6.35.35.4">do</span> </div> <div class="ltx_listingline" id="algorithm6.37.37"> <span class="ltx_rule" style="width:1px;height:100%;background:black;display:inline-block;"> </span> <span class="ltx_rule" style="width:1px;height:100%;background:black;display:inline-block;"> </span> <span class="ltx_text ltx_font_bold" id="algorithm6.37.37.1">update</span> MST <math alttext="H_{x}" class="ltx_Math" display="inline" id="algorithm6.36.36.m1.1"><semantics id="algorithm6.36.36.m1.1a"><msub id="algorithm6.36.36.m1.1.1" xref="algorithm6.36.36.m1.1.1.cmml"><mi id="algorithm6.36.36.m1.1.1.2" xref="algorithm6.36.36.m1.1.1.2.cmml">H</mi><mi id="algorithm6.36.36.m1.1.1.3" xref="algorithm6.36.36.m1.1.1.3.cmml">x</mi></msub><annotation-xml encoding="MathML-Content" id="algorithm6.36.36.m1.1b"><apply id="algorithm6.36.36.m1.1.1.cmml" xref="algorithm6.36.36.m1.1.1"><csymbol cd="ambiguous" id="algorithm6.36.36.m1.1.1.1.cmml" xref="algorithm6.36.36.m1.1.1">subscript</csymbol><ci id="algorithm6.36.36.m1.1.1.2.cmml" xref="algorithm6.36.36.m1.1.1.2">𝐻</ci><ci id="algorithm6.36.36.m1.1.1.3.cmml" xref="algorithm6.36.36.m1.1.1.3">𝑥</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="algorithm6.36.36.m1.1c">H_{x}</annotation><annotation encoding="application/x-llamapun" id="algorithm6.36.36.m1.1d">italic_H start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT</annotation></semantics></math> with link <math alttext="uv" class="ltx_Math" display="inline" id="algorithm6.37.37.m2.1"><semantics id="algorithm6.37.37.m2.1a"><mrow id="algorithm6.37.37.m2.1.1" xref="algorithm6.37.37.m2.1.1.cmml"><mi id="algorithm6.37.37.m2.1.1.2" xref="algorithm6.37.37.m2.1.1.2.cmml">u</mi><mo id="algorithm6.37.37.m2.1.1.1" xref="algorithm6.37.37.m2.1.1.1.cmml">⁢</mo><mi id="algorithm6.37.37.m2.1.1.3" xref="algorithm6.37.37.m2.1.1.3.cmml">v</mi></mrow><annotation-xml encoding="MathML-Content" id="algorithm6.37.37.m2.1b"><apply id="algorithm6.37.37.m2.1.1.cmml" xref="algorithm6.37.37.m2.1.1"><times id="algorithm6.37.37.m2.1.1.1.cmml" xref="algorithm6.37.37.m2.1.1.1"></times><ci id="algorithm6.37.37.m2.1.1.2.cmml" xref="algorithm6.37.37.m2.1.1.2">𝑢</ci><ci id="algorithm6.37.37.m2.1.1.3.cmml" xref="algorithm6.37.37.m2.1.1.3">𝑣</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="algorithm6.37.37.m2.1c">uv</annotation><annotation encoding="application/x-llamapun" id="algorithm6.37.37.m2.1d">italic_u italic_v</annotation></semantics></math> using Lemma <a class="ltx_ref" href="https://arxiv.org/html/2503.00712v1#S4.Thmtheorem2" title="Lemma 4.2. ‣ 4.1 One-to-Two Augmentation ‣ 4 Vertex Connectivity Augmentation in Link-Arrival Model ‣ Streaming Algorithms for Network Design"><span class="ltx_text ltx_ref_tag">4.2</span></a> </div> <div class="ltx_listingline" id="algorithm6.39.39"> <span class="ltx_rule" style="width:1px;height:100%;background:black;display:inline-block;"> </span> <span class="ltx_text ltx_font_bold" id="algorithm6.39.39.3">for</span> <em class="ltx_emph ltx_font_italic" id="algorithm6.39.39.2"><math alttext="x\in V(T)" class="ltx_Math" display="inline" id="algorithm6.38.38.1.m1.1"><semantics id="algorithm6.38.38.1.m1.1a"><mrow id="algorithm6.38.38.1.m1.1.2" xref="algorithm6.38.38.1.m1.1.2.cmml"><mi id="algorithm6.38.38.1.m1.1.2.2" xref="algorithm6.38.38.1.m1.1.2.2.cmml">x</mi><mo id="algorithm6.38.38.1.m1.1.2.1" xref="algorithm6.38.38.1.m1.1.2.1.cmml">∈</mo><mrow id="algorithm6.38.38.1.m1.1.2.3" xref="algorithm6.38.38.1.m1.1.2.3.cmml"><mi id="algorithm6.38.38.1.m1.1.2.3.2" xref="algorithm6.38.38.1.m1.1.2.3.2.cmml">V</mi><mo id="algorithm6.38.38.1.m1.1.2.3.1" xref="algorithm6.38.38.1.m1.1.2.3.1.cmml">⁢</mo><mrow id="algorithm6.38.38.1.m1.1.2.3.3.2" xref="algorithm6.38.38.1.m1.1.2.3.cmml"><mo id="algorithm6.38.38.1.m1.1.2.3.3.2.1" stretchy="false" xref="algorithm6.38.38.1.m1.1.2.3.cmml">(</mo><mi id="algorithm6.38.38.1.m1.1.1" xref="algorithm6.38.38.1.m1.1.1.cmml">T</mi><mo id="algorithm6.38.38.1.m1.1.2.3.3.2.2" stretchy="false" xref="algorithm6.38.38.1.m1.1.2.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="algorithm6.38.38.1.m1.1b"><apply id="algorithm6.38.38.1.m1.1.2.cmml" xref="algorithm6.38.38.1.m1.1.2"><in id="algorithm6.38.38.1.m1.1.2.1.cmml" xref="algorithm6.38.38.1.m1.1.2.1"></in><ci id="algorithm6.38.38.1.m1.1.2.2.cmml" xref="algorithm6.38.38.1.m1.1.2.2">𝑥</ci><apply id="algorithm6.38.38.1.m1.1.2.3.cmml" xref="algorithm6.38.38.1.m1.1.2.3"><times id="algorithm6.38.38.1.m1.1.2.3.1.cmml" xref="algorithm6.38.38.1.m1.1.2.3.1"></times><ci id="algorithm6.38.38.1.m1.1.2.3.2.cmml" xref="algorithm6.38.38.1.m1.1.2.3.2">𝑉</ci><ci id="algorithm6.38.38.1.m1.1.1.cmml" xref="algorithm6.38.38.1.m1.1.1">𝑇</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="algorithm6.38.38.1.m1.1c">x\in V(T)</annotation><annotation encoding="application/x-llamapun" id="algorithm6.38.38.1.m1.1d">italic_x ∈ italic_V ( italic_T )</annotation></semantics></math>; <math alttext="x" class="ltx_Math" display="inline" id="algorithm6.39.39.2.m2.1"><semantics id="algorithm6.39.39.2.m2.1a"><mi id="algorithm6.39.39.2.m2.1.1" xref="algorithm6.39.39.2.m2.1.1.cmml">x</mi><annotation-xml encoding="MathML-Content" id="algorithm6.39.39.2.m2.1b"><ci id="algorithm6.39.39.2.m2.1.1.cmml" xref="algorithm6.39.39.2.m2.1.1">𝑥</ci></annotation-xml><annotation encoding="application/x-tex" id="algorithm6.39.39.2.m2.1c">x</annotation><annotation encoding="application/x-llamapun" id="algorithm6.39.39.2.m2.1d">italic_x</annotation></semantics></math> is an S-node</em> <span class="ltx_text ltx_font_bold" id="algorithm6.39.39.4">do</span> </div> <div class="ltx_listingline" id="algorithm6.43.43"> <span class="ltx_rule" style="width:1px;height:100%;background:black;display:inline-block;"> </span> <span class="ltx_rule" style="width:1px;height:100%;background:black;display:inline-block;"> </span> <span class="ltx_text ltx_font_bold" id="algorithm6.43.43.1">update</span> <math alttext="\textsc{Min}_{f_{x}(u)}(j)" class="ltx_Math" display="inline" id="algorithm6.40.40.m1.2"><semantics id="algorithm6.40.40.m1.2a"><mrow id="algorithm6.40.40.m1.2.3" xref="algorithm6.40.40.m1.2.3.cmml"><msub id="algorithm6.40.40.m1.2.3.2" xref="algorithm6.40.40.m1.2.3.2.cmml"><mtext class="ltx_font_smallcaps" id="algorithm6.40.40.m1.2.3.2.2" xref="algorithm6.40.40.m1.2.3.2.2a.cmml">Min</mtext><mrow id="algorithm6.40.40.m1.1.1.1" xref="algorithm6.40.40.m1.1.1.1.cmml"><msub id="algorithm6.40.40.m1.1.1.1.3" xref="algorithm6.40.40.m1.1.1.1.3.cmml"><mi id="algorithm6.40.40.m1.1.1.1.3.2" xref="algorithm6.40.40.m1.1.1.1.3.2.cmml">f</mi><mi id="algorithm6.40.40.m1.1.1.1.3.3" xref="algorithm6.40.40.m1.1.1.1.3.3.cmml">x</mi></msub><mo id="algorithm6.40.40.m1.1.1.1.2" xref="algorithm6.40.40.m1.1.1.1.2.cmml">⁢</mo><mrow id="algorithm6.40.40.m1.1.1.1.4.2" xref="algorithm6.40.40.m1.1.1.1.cmml"><mo id="algorithm6.40.40.m1.1.1.1.4.2.1" stretchy="false" xref="algorithm6.40.40.m1.1.1.1.cmml">(</mo><mi id="algorithm6.40.40.m1.1.1.1.1" xref="algorithm6.40.40.m1.1.1.1.1.cmml">u</mi><mo id="algorithm6.40.40.m1.1.1.1.4.2.2" stretchy="false" xref="algorithm6.40.40.m1.1.1.1.cmml">)</mo></mrow></mrow></msub><mo id="algorithm6.40.40.m1.2.3.1" xref="algorithm6.40.40.m1.2.3.1.cmml">⁢</mo><mrow id="algorithm6.40.40.m1.2.3.3.2" xref="algorithm6.40.40.m1.2.3.cmml"><mo id="algorithm6.40.40.m1.2.3.3.2.1" stretchy="false" xref="algorithm6.40.40.m1.2.3.cmml">(</mo><mi id="algorithm6.40.40.m1.2.2" xref="algorithm6.40.40.m1.2.2.cmml">j</mi><mo id="algorithm6.40.40.m1.2.3.3.2.2" stretchy="false" xref="algorithm6.40.40.m1.2.3.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="algorithm6.40.40.m1.2b"><apply id="algorithm6.40.40.m1.2.3.cmml" xref="algorithm6.40.40.m1.2.3"><times id="algorithm6.40.40.m1.2.3.1.cmml" xref="algorithm6.40.40.m1.2.3.1"></times><apply id="algorithm6.40.40.m1.2.3.2.cmml" xref="algorithm6.40.40.m1.2.3.2"><csymbol cd="ambiguous" id="algorithm6.40.40.m1.2.3.2.1.cmml" xref="algorithm6.40.40.m1.2.3.2">subscript</csymbol><ci id="algorithm6.40.40.m1.2.3.2.2a.cmml" xref="algorithm6.40.40.m1.2.3.2.2"><mtext class="ltx_font_smallcaps" id="algorithm6.40.40.m1.2.3.2.2.cmml" xref="algorithm6.40.40.m1.2.3.2.2">Min</mtext></ci><apply id="algorithm6.40.40.m1.1.1.1.cmml" xref="algorithm6.40.40.m1.1.1.1"><times id="algorithm6.40.40.m1.1.1.1.2.cmml" xref="algorithm6.40.40.m1.1.1.1.2"></times><apply id="algorithm6.40.40.m1.1.1.1.3.cmml" xref="algorithm6.40.40.m1.1.1.1.3"><csymbol cd="ambiguous" id="algorithm6.40.40.m1.1.1.1.3.1.cmml" xref="algorithm6.40.40.m1.1.1.1.3">subscript</csymbol><ci id="algorithm6.40.40.m1.1.1.1.3.2.cmml" xref="algorithm6.40.40.m1.1.1.1.3.2">𝑓</ci><ci id="algorithm6.40.40.m1.1.1.1.3.3.cmml" xref="algorithm6.40.40.m1.1.1.1.3.3">𝑥</ci></apply><ci id="algorithm6.40.40.m1.1.1.1.1.cmml" xref="algorithm6.40.40.m1.1.1.1.1">𝑢</ci></apply></apply><ci id="algorithm6.40.40.m1.2.2.cmml" xref="algorithm6.40.40.m1.2.2">𝑗</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="algorithm6.40.40.m1.2c">\textsc{Min}_{f_{x}(u)}(j)</annotation><annotation encoding="application/x-llamapun" id="algorithm6.40.40.m1.2d">Min start_POSTSUBSCRIPT italic_f start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT ( italic_u ) end_POSTSUBSCRIPT ( italic_j )</annotation></semantics></math>, <math alttext="\textsc{Max}_{f_{x}(u)}(j)" class="ltx_Math" display="inline" id="algorithm6.41.41.m2.2"><semantics id="algorithm6.41.41.m2.2a"><mrow id="algorithm6.41.41.m2.2.3" xref="algorithm6.41.41.m2.2.3.cmml"><msub id="algorithm6.41.41.m2.2.3.2" xref="algorithm6.41.41.m2.2.3.2.cmml"><mtext class="ltx_font_smallcaps" id="algorithm6.41.41.m2.2.3.2.2" xref="algorithm6.41.41.m2.2.3.2.2a.cmml">Max</mtext><mrow id="algorithm6.41.41.m2.1.1.1" xref="algorithm6.41.41.m2.1.1.1.cmml"><msub id="algorithm6.41.41.m2.1.1.1.3" xref="algorithm6.41.41.m2.1.1.1.3.cmml"><mi id="algorithm6.41.41.m2.1.1.1.3.2" xref="algorithm6.41.41.m2.1.1.1.3.2.cmml">f</mi><mi id="algorithm6.41.41.m2.1.1.1.3.3" xref="algorithm6.41.41.m2.1.1.1.3.3.cmml">x</mi></msub><mo id="algorithm6.41.41.m2.1.1.1.2" xref="algorithm6.41.41.m2.1.1.1.2.cmml">⁢</mo><mrow id="algorithm6.41.41.m2.1.1.1.4.2" xref="algorithm6.41.41.m2.1.1.1.cmml"><mo id="algorithm6.41.41.m2.1.1.1.4.2.1" stretchy="false" xref="algorithm6.41.41.m2.1.1.1.cmml">(</mo><mi id="algorithm6.41.41.m2.1.1.1.1" xref="algorithm6.41.41.m2.1.1.1.1.cmml">u</mi><mo id="algorithm6.41.41.m2.1.1.1.4.2.2" stretchy="false" xref="algorithm6.41.41.m2.1.1.1.cmml">)</mo></mrow></mrow></msub><mo id="algorithm6.41.41.m2.2.3.1" xref="algorithm6.41.41.m2.2.3.1.cmml">⁢</mo><mrow id="algorithm6.41.41.m2.2.3.3.2" xref="algorithm6.41.41.m2.2.3.cmml"><mo id="algorithm6.41.41.m2.2.3.3.2.1" stretchy="false" xref="algorithm6.41.41.m2.2.3.cmml">(</mo><mi id="algorithm6.41.41.m2.2.2" xref="algorithm6.41.41.m2.2.2.cmml">j</mi><mo id="algorithm6.41.41.m2.2.3.3.2.2" stretchy="false" xref="algorithm6.41.41.m2.2.3.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="algorithm6.41.41.m2.2b"><apply id="algorithm6.41.41.m2.2.3.cmml" xref="algorithm6.41.41.m2.2.3"><times id="algorithm6.41.41.m2.2.3.1.cmml" xref="algorithm6.41.41.m2.2.3.1"></times><apply id="algorithm6.41.41.m2.2.3.2.cmml" xref="algorithm6.41.41.m2.2.3.2"><csymbol cd="ambiguous" id="algorithm6.41.41.m2.2.3.2.1.cmml" xref="algorithm6.41.41.m2.2.3.2">subscript</csymbol><ci id="algorithm6.41.41.m2.2.3.2.2a.cmml" xref="algorithm6.41.41.m2.2.3.2.2"><mtext class="ltx_font_smallcaps" id="algorithm6.41.41.m2.2.3.2.2.cmml" xref="algorithm6.41.41.m2.2.3.2.2">Max</mtext></ci><apply id="algorithm6.41.41.m2.1.1.1.cmml" xref="algorithm6.41.41.m2.1.1.1"><times id="algorithm6.41.41.m2.1.1.1.2.cmml" xref="algorithm6.41.41.m2.1.1.1.2"></times><apply id="algorithm6.41.41.m2.1.1.1.3.cmml" xref="algorithm6.41.41.m2.1.1.1.3"><csymbol cd="ambiguous" id="algorithm6.41.41.m2.1.1.1.3.1.cmml" xref="algorithm6.41.41.m2.1.1.1.3">subscript</csymbol><ci id="algorithm6.41.41.m2.1.1.1.3.2.cmml" xref="algorithm6.41.41.m2.1.1.1.3.2">𝑓</ci><ci id="algorithm6.41.41.m2.1.1.1.3.3.cmml" xref="algorithm6.41.41.m2.1.1.1.3.3">𝑥</ci></apply><ci id="algorithm6.41.41.m2.1.1.1.1.cmml" xref="algorithm6.41.41.m2.1.1.1.1">𝑢</ci></apply></apply><ci id="algorithm6.41.41.m2.2.2.cmml" xref="algorithm6.41.41.m2.2.2">𝑗</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="algorithm6.41.41.m2.2c">\textsc{Max}_{f_{x}(u)}(j)</annotation><annotation encoding="application/x-llamapun" id="algorithm6.41.41.m2.2d">Max start_POSTSUBSCRIPT italic_f start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT ( italic_u ) end_POSTSUBSCRIPT ( italic_j )</annotation></semantics></math>, <math alttext="\textsc{Min}_{f_{x}(v)}(j)" class="ltx_Math" display="inline" id="algorithm6.42.42.m3.2"><semantics id="algorithm6.42.42.m3.2a"><mrow id="algorithm6.42.42.m3.2.3" xref="algorithm6.42.42.m3.2.3.cmml"><msub id="algorithm6.42.42.m3.2.3.2" xref="algorithm6.42.42.m3.2.3.2.cmml"><mtext class="ltx_font_smallcaps" id="algorithm6.42.42.m3.2.3.2.2" xref="algorithm6.42.42.m3.2.3.2.2a.cmml">Min</mtext><mrow id="algorithm6.42.42.m3.1.1.1" xref="algorithm6.42.42.m3.1.1.1.cmml"><msub id="algorithm6.42.42.m3.1.1.1.3" xref="algorithm6.42.42.m3.1.1.1.3.cmml"><mi id="algorithm6.42.42.m3.1.1.1.3.2" xref="algorithm6.42.42.m3.1.1.1.3.2.cmml">f</mi><mi id="algorithm6.42.42.m3.1.1.1.3.3" xref="algorithm6.42.42.m3.1.1.1.3.3.cmml">x</mi></msub><mo id="algorithm6.42.42.m3.1.1.1.2" xref="algorithm6.42.42.m3.1.1.1.2.cmml">⁢</mo><mrow id="algorithm6.42.42.m3.1.1.1.4.2" xref="algorithm6.42.42.m3.1.1.1.cmml"><mo id="algorithm6.42.42.m3.1.1.1.4.2.1" stretchy="false" xref="algorithm6.42.42.m3.1.1.1.cmml">(</mo><mi id="algorithm6.42.42.m3.1.1.1.1" xref="algorithm6.42.42.m3.1.1.1.1.cmml">v</mi><mo id="algorithm6.42.42.m3.1.1.1.4.2.2" stretchy="false" xref="algorithm6.42.42.m3.1.1.1.cmml">)</mo></mrow></mrow></msub><mo id="algorithm6.42.42.m3.2.3.1" xref="algorithm6.42.42.m3.2.3.1.cmml">⁢</mo><mrow id="algorithm6.42.42.m3.2.3.3.2" xref="algorithm6.42.42.m3.2.3.cmml"><mo id="algorithm6.42.42.m3.2.3.3.2.1" stretchy="false" xref="algorithm6.42.42.m3.2.3.cmml">(</mo><mi id="algorithm6.42.42.m3.2.2" xref="algorithm6.42.42.m3.2.2.cmml">j</mi><mo id="algorithm6.42.42.m3.2.3.3.2.2" stretchy="false" xref="algorithm6.42.42.m3.2.3.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="algorithm6.42.42.m3.2b"><apply id="algorithm6.42.42.m3.2.3.cmml" xref="algorithm6.42.42.m3.2.3"><times id="algorithm6.42.42.m3.2.3.1.cmml" xref="algorithm6.42.42.m3.2.3.1"></times><apply id="algorithm6.42.42.m3.2.3.2.cmml" xref="algorithm6.42.42.m3.2.3.2"><csymbol cd="ambiguous" id="algorithm6.42.42.m3.2.3.2.1.cmml" xref="algorithm6.42.42.m3.2.3.2">subscript</csymbol><ci id="algorithm6.42.42.m3.2.3.2.2a.cmml" xref="algorithm6.42.42.m3.2.3.2.2"><mtext class="ltx_font_smallcaps" id="algorithm6.42.42.m3.2.3.2.2.cmml" xref="algorithm6.42.42.m3.2.3.2.2">Min</mtext></ci><apply id="algorithm6.42.42.m3.1.1.1.cmml" xref="algorithm6.42.42.m3.1.1.1"><times id="algorithm6.42.42.m3.1.1.1.2.cmml" xref="algorithm6.42.42.m3.1.1.1.2"></times><apply id="algorithm6.42.42.m3.1.1.1.3.cmml" xref="algorithm6.42.42.m3.1.1.1.3"><csymbol cd="ambiguous" id="algorithm6.42.42.m3.1.1.1.3.1.cmml" xref="algorithm6.42.42.m3.1.1.1.3">subscript</csymbol><ci id="algorithm6.42.42.m3.1.1.1.3.2.cmml" xref="algorithm6.42.42.m3.1.1.1.3.2">𝑓</ci><ci id="algorithm6.42.42.m3.1.1.1.3.3.cmml" xref="algorithm6.42.42.m3.1.1.1.3.3">𝑥</ci></apply><ci id="algorithm6.42.42.m3.1.1.1.1.cmml" xref="algorithm6.42.42.m3.1.1.1.1">𝑣</ci></apply></apply><ci id="algorithm6.42.42.m3.2.2.cmml" xref="algorithm6.42.42.m3.2.2">𝑗</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="algorithm6.42.42.m3.2c">\textsc{Min}_{f_{x}(v)}(j)</annotation><annotation encoding="application/x-llamapun" id="algorithm6.42.42.m3.2d">Min start_POSTSUBSCRIPT italic_f start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT ( italic_v ) end_POSTSUBSCRIPT ( italic_j )</annotation></semantics></math>, and <math alttext="\textsc{Max}_{f_{x}(v)}(j)" class="ltx_Math" display="inline" id="algorithm6.43.43.m4.2"><semantics id="algorithm6.43.43.m4.2a"><mrow id="algorithm6.43.43.m4.2.3" xref="algorithm6.43.43.m4.2.3.cmml"><msub id="algorithm6.43.43.m4.2.3.2" xref="algorithm6.43.43.m4.2.3.2.cmml"><mtext class="ltx_font_smallcaps" id="algorithm6.43.43.m4.2.3.2.2" xref="algorithm6.43.43.m4.2.3.2.2a.cmml">Max</mtext><mrow id="algorithm6.43.43.m4.1.1.1" xref="algorithm6.43.43.m4.1.1.1.cmml"><msub id="algorithm6.43.43.m4.1.1.1.3" xref="algorithm6.43.43.m4.1.1.1.3.cmml"><mi id="algorithm6.43.43.m4.1.1.1.3.2" xref="algorithm6.43.43.m4.1.1.1.3.2.cmml">f</mi><mi id="algorithm6.43.43.m4.1.1.1.3.3" xref="algorithm6.43.43.m4.1.1.1.3.3.cmml">x</mi></msub><mo id="algorithm6.43.43.m4.1.1.1.2" xref="algorithm6.43.43.m4.1.1.1.2.cmml">⁢</mo><mrow id="algorithm6.43.43.m4.1.1.1.4.2" xref="algorithm6.43.43.m4.1.1.1.cmml"><mo id="algorithm6.43.43.m4.1.1.1.4.2.1" stretchy="false" xref="algorithm6.43.43.m4.1.1.1.cmml">(</mo><mi id="algorithm6.43.43.m4.1.1.1.1" xref="algorithm6.43.43.m4.1.1.1.1.cmml">v</mi><mo id="algorithm6.43.43.m4.1.1.1.4.2.2" stretchy="false" xref="algorithm6.43.43.m4.1.1.1.cmml">)</mo></mrow></mrow></msub><mo id="algorithm6.43.43.m4.2.3.1" xref="algorithm6.43.43.m4.2.3.1.cmml">⁢</mo><mrow id="algorithm6.43.43.m4.2.3.3.2" xref="algorithm6.43.43.m4.2.3.cmml"><mo id="algorithm6.43.43.m4.2.3.3.2.1" stretchy="false" xref="algorithm6.43.43.m4.2.3.cmml">(</mo><mi id="algorithm6.43.43.m4.2.2" xref="algorithm6.43.43.m4.2.2.cmml">j</mi><mo id="algorithm6.43.43.m4.2.3.3.2.2" stretchy="false" xref="algorithm6.43.43.m4.2.3.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="algorithm6.43.43.m4.2b"><apply id="algorithm6.43.43.m4.2.3.cmml" xref="algorithm6.43.43.m4.2.3"><times id="algorithm6.43.43.m4.2.3.1.cmml" xref="algorithm6.43.43.m4.2.3.1"></times><apply id="algorithm6.43.43.m4.2.3.2.cmml" xref="algorithm6.43.43.m4.2.3.2"><csymbol cd="ambiguous" id="algorithm6.43.43.m4.2.3.2.1.cmml" xref="algorithm6.43.43.m4.2.3.2">subscript</csymbol><ci id="algorithm6.43.43.m4.2.3.2.2a.cmml" xref="algorithm6.43.43.m4.2.3.2.2"><mtext class="ltx_font_smallcaps" id="algorithm6.43.43.m4.2.3.2.2.cmml" xref="algorithm6.43.43.m4.2.3.2.2">Max</mtext></ci><apply id="algorithm6.43.43.m4.1.1.1.cmml" xref="algorithm6.43.43.m4.1.1.1"><times id="algorithm6.43.43.m4.1.1.1.2.cmml" xref="algorithm6.43.43.m4.1.1.1.2"></times><apply id="algorithm6.43.43.m4.1.1.1.3.cmml" xref="algorithm6.43.43.m4.1.1.1.3"><csymbol cd="ambiguous" id="algorithm6.43.43.m4.1.1.1.3.1.cmml" xref="algorithm6.43.43.m4.1.1.1.3">subscript</csymbol><ci id="algorithm6.43.43.m4.1.1.1.3.2.cmml" xref="algorithm6.43.43.m4.1.1.1.3.2">𝑓</ci><ci id="algorithm6.43.43.m4.1.1.1.3.3.cmml" xref="algorithm6.43.43.m4.1.1.1.3.3">𝑥</ci></apply><ci id="algorithm6.43.43.m4.1.1.1.1.cmml" xref="algorithm6.43.43.m4.1.1.1.1">𝑣</ci></apply></apply><ci id="algorithm6.43.43.m4.2.2.cmml" xref="algorithm6.43.43.m4.2.2">𝑗</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="algorithm6.43.43.m4.2c">\textsc{Max}_{f_{x}(v)}(j)</annotation><annotation encoding="application/x-llamapun" id="algorithm6.43.43.m4.2d">Max start_POSTSUBSCRIPT italic_f start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT ( italic_v ) end_POSTSUBSCRIPT ( italic_j )</annotation></semantics></math> </div> <div class="ltx_listingline" id="algorithm6.45.49"> <span class="ltx_text" id="algorithm6.45.49.1" style="color:#0000FF;">/* </span><span class="ltx_text ltx_font_smallcaps" id="algorithm6.45.49.2" style="color:#0000FF;">Postprocessing: */</span> </div> <div class="ltx_listingline" id="algorithm6.44.44"> <math alttext="F=(\cup_{x\in V(T)}L_{x})\cup(\cup_{x\in V(T),x\text{ is P-node}}E(H_{x}))\cup% (\cup_{x\in V(T),x\text{ is S-node}}\cup_{\mu}\textsc{Min}_{\mu}\cup\textsc{% Max}_{\mu})" class="ltx_math_unparsed" display="inline" id="algorithm6.44.44.m1.7"><semantics id="algorithm6.44.44.m1.7a"><mrow id="algorithm6.44.44.m1.7b"><mi id="algorithm6.44.44.m1.7.8">F</mi><mo id="algorithm6.44.44.m1.7.9">=</mo><mrow id="algorithm6.44.44.m1.7.10"><mo id="algorithm6.44.44.m1.7.10.1" stretchy="false">(</mo><msub id="algorithm6.44.44.m1.7.10.2"><mo id="algorithm6.44.44.m1.7.10.2.2" lspace="0em">∪</mo><mrow id="algorithm6.44.44.m1.1.1.1"><mi id="algorithm6.44.44.m1.1.1.1.3">x</mi><mo id="algorithm6.44.44.m1.1.1.1.2">∈</mo><mrow id="algorithm6.44.44.m1.1.1.1.4"><mi id="algorithm6.44.44.m1.1.1.1.4.2">V</mi><mo id="algorithm6.44.44.m1.1.1.1.4.1">⁢</mo><mrow id="algorithm6.44.44.m1.1.1.1.4.3.2"><mo id="algorithm6.44.44.m1.1.1.1.4.3.2.1" stretchy="false">(</mo><mi id="algorithm6.44.44.m1.1.1.1.1">T</mi><mo id="algorithm6.44.44.m1.1.1.1.4.3.2.2" stretchy="false">)</mo></mrow></mrow></mrow></msub><msub id="algorithm6.44.44.m1.7.10.3"><mi id="algorithm6.44.44.m1.7.10.3.2">L</mi><mi id="algorithm6.44.44.m1.7.10.3.3">x</mi></msub><mo id="algorithm6.44.44.m1.7.10.4" stretchy="false">)</mo></mrow><mo id="algorithm6.44.44.m1.7.11">∪</mo><mrow id="algorithm6.44.44.m1.7.12"><mo id="algorithm6.44.44.m1.7.12.1" stretchy="false">(</mo><msub id="algorithm6.44.44.m1.7.12.2"><mo id="algorithm6.44.44.m1.7.12.2.2" lspace="0em">∪</mo><mrow id="algorithm6.44.44.m1.4.4.3"><mi id="algorithm6.44.44.m1.4.4.3.5">x</mi><mo id="algorithm6.44.44.m1.4.4.3.4">∈</mo><mrow id="algorithm6.44.44.m1.4.4.3.3.2"><mrow id="algorithm6.44.44.m1.3.3.2.2.1.1"><mi id="algorithm6.44.44.m1.3.3.2.2.1.1.2">V</mi><mo id="algorithm6.44.44.m1.3.3.2.2.1.1.1">⁢</mo><mrow id="algorithm6.44.44.m1.3.3.2.2.1.1.3.2"><mo id="algorithm6.44.44.m1.3.3.2.2.1.1.3.2.1" stretchy="false">(</mo><mi id="algorithm6.44.44.m1.2.2.1.1">T</mi><mo id="algorithm6.44.44.m1.3.3.2.2.1.1.3.2.2" stretchy="false">)</mo></mrow></mrow><mo id="algorithm6.44.44.m1.4.4.3.3.2.3">,</mo><mrow id="algorithm6.44.44.m1.4.4.3.3.2.2"><mi id="algorithm6.44.44.m1.4.4.3.3.2.2.2">x</mi><mo id="algorithm6.44.44.m1.4.4.3.3.2.2.1">⁢</mo><mtext id="algorithm6.44.44.m1.4.4.3.3.2.2.3"> is P-node</mtext></mrow></mrow></mrow></msub><mi id="algorithm6.44.44.m1.7.12.3">E</mi><mrow id="algorithm6.44.44.m1.7.12.4"><mo id="algorithm6.44.44.m1.7.12.4.1" stretchy="false">(</mo><msub id="algorithm6.44.44.m1.7.12.4.2"><mi id="algorithm6.44.44.m1.7.12.4.2.2">H</mi><mi id="algorithm6.44.44.m1.7.12.4.2.3">x</mi></msub><mo id="algorithm6.44.44.m1.7.12.4.3" stretchy="false">)</mo></mrow><mo id="algorithm6.44.44.m1.7.12.5" stretchy="false">)</mo></mrow><mo id="algorithm6.44.44.m1.7.13">∪</mo><mrow id="algorithm6.44.44.m1.7.14"><mo id="algorithm6.44.44.m1.7.14.1" stretchy="false">(</mo><msub id="algorithm6.44.44.m1.7.14.2"><mo id="algorithm6.44.44.m1.7.14.2.2" lspace="0em" rspace="0em">∪</mo><mrow id="algorithm6.44.44.m1.7.7.3"><mi id="algorithm6.44.44.m1.7.7.3.5">x</mi><mo id="algorithm6.44.44.m1.7.7.3.4">∈</mo><mrow id="algorithm6.44.44.m1.7.7.3.3.2"><mrow id="algorithm6.44.44.m1.6.6.2.2.1.1"><mi id="algorithm6.44.44.m1.6.6.2.2.1.1.2">V</mi><mo id="algorithm6.44.44.m1.6.6.2.2.1.1.1">⁢</mo><mrow id="algorithm6.44.44.m1.6.6.2.2.1.1.3.2"><mo id="algorithm6.44.44.m1.6.6.2.2.1.1.3.2.1" stretchy="false">(</mo><mi id="algorithm6.44.44.m1.5.5.1.1">T</mi><mo id="algorithm6.44.44.m1.6.6.2.2.1.1.3.2.2" stretchy="false">)</mo></mrow></mrow><mo id="algorithm6.44.44.m1.7.7.3.3.2.3">,</mo><mrow id="algorithm6.44.44.m1.7.7.3.3.2.2"><mi id="algorithm6.44.44.m1.7.7.3.3.2.2.2">x</mi><mo id="algorithm6.44.44.m1.7.7.3.3.2.2.1">⁢</mo><mtext id="algorithm6.44.44.m1.7.7.3.3.2.2.3"> is S-node</mtext></mrow></mrow></mrow></msub><msub id="algorithm6.44.44.m1.7.14.3"><mo id="algorithm6.44.44.m1.7.14.3.2" lspace="0em">∪</mo><mi id="algorithm6.44.44.m1.7.14.3.3">μ</mi></msub><msub id="algorithm6.44.44.m1.7.14.4"><mtext class="ltx_font_smallcaps" id="algorithm6.44.44.m1.7.14.4.2">Min</mtext><mi id="algorithm6.44.44.m1.7.14.4.3">μ</mi></msub><mo id="algorithm6.44.44.m1.7.14.5">∪</mo><msub id="algorithm6.44.44.m1.7.14.6"><mtext class="ltx_font_smallcaps" id="algorithm6.44.44.m1.7.14.6.2">Max</mtext><mi id="algorithm6.44.44.m1.7.14.6.3">μ</mi></msub><mo id="algorithm6.44.44.m1.7.14.7" stretchy="false">)</mo></mrow></mrow><annotation encoding="application/x-tex" id="algorithm6.44.44.m1.7c">F=(\cup_{x\in V(T)}L_{x})\cup(\cup_{x\in V(T),x\text{ is P-node}}E(H_{x}))\cup% (\cup_{x\in V(T),x\text{ is S-node}}\cup_{\mu}\textsc{Min}_{\mu}\cup\textsc{% Max}_{\mu})</annotation><annotation encoding="application/x-llamapun" id="algorithm6.44.44.m1.7d">italic_F = ( ∪ start_POSTSUBSCRIPT italic_x ∈ italic_V ( italic_T ) end_POSTSUBSCRIPT italic_L start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT ) ∪ ( ∪ start_POSTSUBSCRIPT italic_x ∈ italic_V ( italic_T ) , italic_x is P-node end_POSTSUBSCRIPT italic_E ( italic_H start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT ) ) ∪ ( ∪ start_POSTSUBSCRIPT italic_x ∈ italic_V ( italic_T ) , italic_x is S-node end_POSTSUBSCRIPT ∪ start_POSTSUBSCRIPT italic_μ end_POSTSUBSCRIPT Min start_POSTSUBSCRIPT italic_μ end_POSTSUBSCRIPT ∪ Max start_POSTSUBSCRIPT italic_μ end_POSTSUBSCRIPT )</annotation></semantics></math> </div> <div class="ltx_listingline" id="algorithm6.45.45"> <span class="ltx_text ltx_font_bold" id="algorithm6.45.45.1">return</span> an optimal solution on edge set <math alttext="F" class="ltx_Math" display="inline" id="algorithm6.45.45.m1.1"><semantics id="algorithm6.45.45.m1.1a"><mi id="algorithm6.45.45.m1.1.1" xref="algorithm6.45.45.m1.1.1.cmml">F</mi><annotation-xml encoding="MathML-Content" id="algorithm6.45.45.m1.1b"><ci id="algorithm6.45.45.m1.1.1.cmml" xref="algorithm6.45.45.m1.1.1">𝐹</ci></annotation-xml><annotation encoding="application/x-tex" id="algorithm6.45.45.m1.1c">F</annotation><annotation encoding="application/x-llamapun" id="algorithm6.45.45.m1.1d">italic_F</annotation></semantics></math> </div> </div> <figcaption class="ltx_caption"><span class="ltx_tag ltx_tag_float"><span class="ltx_text ltx_font_bold" id="algorithm6.47.1.1">Algorithm 6</span> </span>The streaming algorithm for 2-to-3 VCSS augmentation</figcaption> </figure> <div class="ltx_para" id="S4.SS2.SSS2.Px2.p3"> <p class="ltx_p" id="S4.SS2.SSS2.Px2.p3.1">In order to bound the space complexity of Algorithm <a class="ltx_ref" href="https://arxiv.org/html/2503.00712v1#algorithm6" title="In “Cycle” Cuts: ‣ 4.2.2 The Streaming Algorithm ‣ 4.2 Two-to-Three Augmentation ‣ 4 Vertex Connectivity Augmentation in Link-Arrival Model ‣ Streaming Algorithms for Network Design"><span class="ltx_text ltx_ref_tag">6</span></a>, we use the following fact; this follows from well-known results including graph sparsification <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.00712v1#bib.bibx69" title="">NI92</a>]</cite> and ear-decompositions of 2-connected graphs.</p> </div> <div class="ltx_theorem ltx_theorem_lemma" id="S4.Thmtheorem18"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="S4.Thmtheorem18.1.1.1">Lemma 4.18</span></span><span class="ltx_text ltx_font_bold" id="S4.Thmtheorem18.2.2">.</span> </h6> <div class="ltx_para" id="S4.Thmtheorem18.p1"> <p class="ltx_p" id="S4.Thmtheorem18.p1.3">Any edge-minimal 2-node-connected graph <math alttext="G=(V,E)" class="ltx_Math" display="inline" id="S4.Thmtheorem18.p1.1.m1.2"><semantics id="S4.Thmtheorem18.p1.1.m1.2a"><mrow id="S4.Thmtheorem18.p1.1.m1.2.3" xref="S4.Thmtheorem18.p1.1.m1.2.3.cmml"><mi id="S4.Thmtheorem18.p1.1.m1.2.3.2" xref="S4.Thmtheorem18.p1.1.m1.2.3.2.cmml">G</mi><mo id="S4.Thmtheorem18.p1.1.m1.2.3.1" xref="S4.Thmtheorem18.p1.1.m1.2.3.1.cmml">=</mo><mrow id="S4.Thmtheorem18.p1.1.m1.2.3.3.2" xref="S4.Thmtheorem18.p1.1.m1.2.3.3.1.cmml"><mo id="S4.Thmtheorem18.p1.1.m1.2.3.3.2.1" stretchy="false" xref="S4.Thmtheorem18.p1.1.m1.2.3.3.1.cmml">(</mo><mi id="S4.Thmtheorem18.p1.1.m1.1.1" xref="S4.Thmtheorem18.p1.1.m1.1.1.cmml">V</mi><mo id="S4.Thmtheorem18.p1.1.m1.2.3.3.2.2" xref="S4.Thmtheorem18.p1.1.m1.2.3.3.1.cmml">,</mo><mi id="S4.Thmtheorem18.p1.1.m1.2.2" xref="S4.Thmtheorem18.p1.1.m1.2.2.cmml">E</mi><mo id="S4.Thmtheorem18.p1.1.m1.2.3.3.2.3" stretchy="false" xref="S4.Thmtheorem18.p1.1.m1.2.3.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem18.p1.1.m1.2b"><apply id="S4.Thmtheorem18.p1.1.m1.2.3.cmml" xref="S4.Thmtheorem18.p1.1.m1.2.3"><eq id="S4.Thmtheorem18.p1.1.m1.2.3.1.cmml" xref="S4.Thmtheorem18.p1.1.m1.2.3.1"></eq><ci id="S4.Thmtheorem18.p1.1.m1.2.3.2.cmml" xref="S4.Thmtheorem18.p1.1.m1.2.3.2">𝐺</ci><interval closure="open" id="S4.Thmtheorem18.p1.1.m1.2.3.3.1.cmml" xref="S4.Thmtheorem18.p1.1.m1.2.3.3.2"><ci id="S4.Thmtheorem18.p1.1.m1.1.1.cmml" xref="S4.Thmtheorem18.p1.1.m1.1.1">𝑉</ci><ci id="S4.Thmtheorem18.p1.1.m1.2.2.cmml" xref="S4.Thmtheorem18.p1.1.m1.2.2">𝐸</ci></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem18.p1.1.m1.2c">G=(V,E)</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem18.p1.1.m1.2d">italic_G = ( italic_V , italic_E )</annotation></semantics></math> has at most <math alttext="2n-2" class="ltx_Math" display="inline" id="S4.Thmtheorem18.p1.2.m2.1"><semantics id="S4.Thmtheorem18.p1.2.m2.1a"><mrow id="S4.Thmtheorem18.p1.2.m2.1.1" xref="S4.Thmtheorem18.p1.2.m2.1.1.cmml"><mrow id="S4.Thmtheorem18.p1.2.m2.1.1.2" xref="S4.Thmtheorem18.p1.2.m2.1.1.2.cmml"><mn id="S4.Thmtheorem18.p1.2.m2.1.1.2.2" xref="S4.Thmtheorem18.p1.2.m2.1.1.2.2.cmml">2</mn><mo id="S4.Thmtheorem18.p1.2.m2.1.1.2.1" xref="S4.Thmtheorem18.p1.2.m2.1.1.2.1.cmml">⁢</mo><mi id="S4.Thmtheorem18.p1.2.m2.1.1.2.3" xref="S4.Thmtheorem18.p1.2.m2.1.1.2.3.cmml">n</mi></mrow><mo id="S4.Thmtheorem18.p1.2.m2.1.1.1" xref="S4.Thmtheorem18.p1.2.m2.1.1.1.cmml">−</mo><mn id="S4.Thmtheorem18.p1.2.m2.1.1.3" xref="S4.Thmtheorem18.p1.2.m2.1.1.3.cmml">2</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem18.p1.2.m2.1b"><apply id="S4.Thmtheorem18.p1.2.m2.1.1.cmml" xref="S4.Thmtheorem18.p1.2.m2.1.1"><minus id="S4.Thmtheorem18.p1.2.m2.1.1.1.cmml" xref="S4.Thmtheorem18.p1.2.m2.1.1.1"></minus><apply id="S4.Thmtheorem18.p1.2.m2.1.1.2.cmml" xref="S4.Thmtheorem18.p1.2.m2.1.1.2"><times id="S4.Thmtheorem18.p1.2.m2.1.1.2.1.cmml" xref="S4.Thmtheorem18.p1.2.m2.1.1.2.1"></times><cn id="S4.Thmtheorem18.p1.2.m2.1.1.2.2.cmml" type="integer" xref="S4.Thmtheorem18.p1.2.m2.1.1.2.2">2</cn><ci id="S4.Thmtheorem18.p1.2.m2.1.1.2.3.cmml" xref="S4.Thmtheorem18.p1.2.m2.1.1.2.3">𝑛</ci></apply><cn id="S4.Thmtheorem18.p1.2.m2.1.1.3.cmml" type="integer" xref="S4.Thmtheorem18.p1.2.m2.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem18.p1.2.m2.1c">2n-2</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem18.p1.2.m2.1d">2 italic_n - 2</annotation></semantics></math> edges, where <math alttext="n=|V|" class="ltx_Math" display="inline" id="S4.Thmtheorem18.p1.3.m3.1"><semantics id="S4.Thmtheorem18.p1.3.m3.1a"><mrow id="S4.Thmtheorem18.p1.3.m3.1.2" xref="S4.Thmtheorem18.p1.3.m3.1.2.cmml"><mi id="S4.Thmtheorem18.p1.3.m3.1.2.2" xref="S4.Thmtheorem18.p1.3.m3.1.2.2.cmml">n</mi><mo id="S4.Thmtheorem18.p1.3.m3.1.2.1" xref="S4.Thmtheorem18.p1.3.m3.1.2.1.cmml">=</mo><mrow id="S4.Thmtheorem18.p1.3.m3.1.2.3.2" xref="S4.Thmtheorem18.p1.3.m3.1.2.3.1.cmml"><mo id="S4.Thmtheorem18.p1.3.m3.1.2.3.2.1" stretchy="false" xref="S4.Thmtheorem18.p1.3.m3.1.2.3.1.1.cmml">|</mo><mi id="S4.Thmtheorem18.p1.3.m3.1.1" xref="S4.Thmtheorem18.p1.3.m3.1.1.cmml">V</mi><mo id="S4.Thmtheorem18.p1.3.m3.1.2.3.2.2" stretchy="false" xref="S4.Thmtheorem18.p1.3.m3.1.2.3.1.1.cmml">|</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem18.p1.3.m3.1b"><apply id="S4.Thmtheorem18.p1.3.m3.1.2.cmml" xref="S4.Thmtheorem18.p1.3.m3.1.2"><eq id="S4.Thmtheorem18.p1.3.m3.1.2.1.cmml" xref="S4.Thmtheorem18.p1.3.m3.1.2.1"></eq><ci id="S4.Thmtheorem18.p1.3.m3.1.2.2.cmml" xref="S4.Thmtheorem18.p1.3.m3.1.2.2">𝑛</ci><apply id="S4.Thmtheorem18.p1.3.m3.1.2.3.1.cmml" xref="S4.Thmtheorem18.p1.3.m3.1.2.3.2"><abs id="S4.Thmtheorem18.p1.3.m3.1.2.3.1.1.cmml" xref="S4.Thmtheorem18.p1.3.m3.1.2.3.2.1"></abs><ci id="S4.Thmtheorem18.p1.3.m3.1.1.cmml" xref="S4.Thmtheorem18.p1.3.m3.1.1">𝑉</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem18.p1.3.m3.1c">n=|V|</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem18.p1.3.m3.1d">italic_n = | italic_V |</annotation></semantics></math>.</p> </div> </div> <div class="ltx_theorem ltx_theorem_lemma" id="S4.Thmtheorem19"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="S4.Thmtheorem19.1.1.1">Lemma 4.19</span></span><span class="ltx_text ltx_font_bold" id="S4.Thmtheorem19.2.2">.</span> </h6> <div class="ltx_para" id="S4.Thmtheorem19.p1"> <p class="ltx_p" id="S4.Thmtheorem19.p1.1">The number of links stored in Algorithm <a class="ltx_ref" href="https://arxiv.org/html/2503.00712v1#algorithm6" title="In “Cycle” Cuts: ‣ 4.2.2 The Streaming Algorithm ‣ 4.2 Two-to-Three Augmentation ‣ 4 Vertex Connectivity Augmentation in Link-Arrival Model ‣ Streaming Algorithms for Network Design"><span class="ltx_text ltx_ref_tag">6</span></a> is <math alttext="O(n\epsilon^{-1}\log W)" class="ltx_Math" display="inline" id="S4.Thmtheorem19.p1.1.m1.1"><semantics id="S4.Thmtheorem19.p1.1.m1.1a"><mrow id="S4.Thmtheorem19.p1.1.m1.1.1" xref="S4.Thmtheorem19.p1.1.m1.1.1.cmml"><mi id="S4.Thmtheorem19.p1.1.m1.1.1.3" xref="S4.Thmtheorem19.p1.1.m1.1.1.3.cmml">O</mi><mo id="S4.Thmtheorem19.p1.1.m1.1.1.2" xref="S4.Thmtheorem19.p1.1.m1.1.1.2.cmml">⁢</mo><mrow id="S4.Thmtheorem19.p1.1.m1.1.1.1.1" xref="S4.Thmtheorem19.p1.1.m1.1.1.1.1.1.cmml"><mo id="S4.Thmtheorem19.p1.1.m1.1.1.1.1.2" stretchy="false" xref="S4.Thmtheorem19.p1.1.m1.1.1.1.1.1.cmml">(</mo><mrow id="S4.Thmtheorem19.p1.1.m1.1.1.1.1.1" xref="S4.Thmtheorem19.p1.1.m1.1.1.1.1.1.cmml"><mi id="S4.Thmtheorem19.p1.1.m1.1.1.1.1.1.2" xref="S4.Thmtheorem19.p1.1.m1.1.1.1.1.1.2.cmml">n</mi><mo id="S4.Thmtheorem19.p1.1.m1.1.1.1.1.1.1" xref="S4.Thmtheorem19.p1.1.m1.1.1.1.1.1.1.cmml">⁢</mo><msup id="S4.Thmtheorem19.p1.1.m1.1.1.1.1.1.3" xref="S4.Thmtheorem19.p1.1.m1.1.1.1.1.1.3.cmml"><mi id="S4.Thmtheorem19.p1.1.m1.1.1.1.1.1.3.2" xref="S4.Thmtheorem19.p1.1.m1.1.1.1.1.1.3.2.cmml">ϵ</mi><mrow id="S4.Thmtheorem19.p1.1.m1.1.1.1.1.1.3.3" xref="S4.Thmtheorem19.p1.1.m1.1.1.1.1.1.3.3.cmml"><mo id="S4.Thmtheorem19.p1.1.m1.1.1.1.1.1.3.3a" xref="S4.Thmtheorem19.p1.1.m1.1.1.1.1.1.3.3.cmml">−</mo><mn id="S4.Thmtheorem19.p1.1.m1.1.1.1.1.1.3.3.2" xref="S4.Thmtheorem19.p1.1.m1.1.1.1.1.1.3.3.2.cmml">1</mn></mrow></msup><mo id="S4.Thmtheorem19.p1.1.m1.1.1.1.1.1.1a" lspace="0.167em" xref="S4.Thmtheorem19.p1.1.m1.1.1.1.1.1.1.cmml">⁢</mo><mrow id="S4.Thmtheorem19.p1.1.m1.1.1.1.1.1.4" xref="S4.Thmtheorem19.p1.1.m1.1.1.1.1.1.4.cmml"><mi id="S4.Thmtheorem19.p1.1.m1.1.1.1.1.1.4.1" xref="S4.Thmtheorem19.p1.1.m1.1.1.1.1.1.4.1.cmml">log</mi><mo id="S4.Thmtheorem19.p1.1.m1.1.1.1.1.1.4a" lspace="0.167em" xref="S4.Thmtheorem19.p1.1.m1.1.1.1.1.1.4.cmml">⁡</mo><mi id="S4.Thmtheorem19.p1.1.m1.1.1.1.1.1.4.2" xref="S4.Thmtheorem19.p1.1.m1.1.1.1.1.1.4.2.cmml">W</mi></mrow></mrow><mo id="S4.Thmtheorem19.p1.1.m1.1.1.1.1.3" stretchy="false" xref="S4.Thmtheorem19.p1.1.m1.1.1.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem19.p1.1.m1.1b"><apply id="S4.Thmtheorem19.p1.1.m1.1.1.cmml" xref="S4.Thmtheorem19.p1.1.m1.1.1"><times id="S4.Thmtheorem19.p1.1.m1.1.1.2.cmml" xref="S4.Thmtheorem19.p1.1.m1.1.1.2"></times><ci id="S4.Thmtheorem19.p1.1.m1.1.1.3.cmml" xref="S4.Thmtheorem19.p1.1.m1.1.1.3">𝑂</ci><apply id="S4.Thmtheorem19.p1.1.m1.1.1.1.1.1.cmml" xref="S4.Thmtheorem19.p1.1.m1.1.1.1.1"><times id="S4.Thmtheorem19.p1.1.m1.1.1.1.1.1.1.cmml" xref="S4.Thmtheorem19.p1.1.m1.1.1.1.1.1.1"></times><ci id="S4.Thmtheorem19.p1.1.m1.1.1.1.1.1.2.cmml" xref="S4.Thmtheorem19.p1.1.m1.1.1.1.1.1.2">𝑛</ci><apply id="S4.Thmtheorem19.p1.1.m1.1.1.1.1.1.3.cmml" xref="S4.Thmtheorem19.p1.1.m1.1.1.1.1.1.3"><csymbol cd="ambiguous" id="S4.Thmtheorem19.p1.1.m1.1.1.1.1.1.3.1.cmml" xref="S4.Thmtheorem19.p1.1.m1.1.1.1.1.1.3">superscript</csymbol><ci id="S4.Thmtheorem19.p1.1.m1.1.1.1.1.1.3.2.cmml" xref="S4.Thmtheorem19.p1.1.m1.1.1.1.1.1.3.2">italic-ϵ</ci><apply id="S4.Thmtheorem19.p1.1.m1.1.1.1.1.1.3.3.cmml" xref="S4.Thmtheorem19.p1.1.m1.1.1.1.1.1.3.3"><minus id="S4.Thmtheorem19.p1.1.m1.1.1.1.1.1.3.3.1.cmml" xref="S4.Thmtheorem19.p1.1.m1.1.1.1.1.1.3.3"></minus><cn id="S4.Thmtheorem19.p1.1.m1.1.1.1.1.1.3.3.2.cmml" type="integer" xref="S4.Thmtheorem19.p1.1.m1.1.1.1.1.1.3.3.2">1</cn></apply></apply><apply id="S4.Thmtheorem19.p1.1.m1.1.1.1.1.1.4.cmml" xref="S4.Thmtheorem19.p1.1.m1.1.1.1.1.1.4"><log id="S4.Thmtheorem19.p1.1.m1.1.1.1.1.1.4.1.cmml" xref="S4.Thmtheorem19.p1.1.m1.1.1.1.1.1.4.1"></log><ci id="S4.Thmtheorem19.p1.1.m1.1.1.1.1.1.4.2.cmml" xref="S4.Thmtheorem19.p1.1.m1.1.1.1.1.1.4.2">𝑊</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem19.p1.1.m1.1c">O(n\epsilon^{-1}\log W)</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem19.p1.1.m1.1d">italic_O ( italic_n italic_ϵ start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT roman_log italic_W )</annotation></semantics></math>.</p> </div> </div> <div class="ltx_proof" id="S4.SS2.SSS2.Px2.1"> <h6 class="ltx_title ltx_runin ltx_font_italic ltx_title_proof">Proof.</h6> <div class="ltx_para" id="S4.SS2.SSS2.Px2.1.p1"> <p class="ltx_p" id="S4.SS2.SSS2.Px2.1.p1.5">Let <math alttext="B=O(\epsilon^{-1}\log W)" class="ltx_Math" display="inline" id="S4.SS2.SSS2.Px2.1.p1.1.m1.1"><semantics id="S4.SS2.SSS2.Px2.1.p1.1.m1.1a"><mrow id="S4.SS2.SSS2.Px2.1.p1.1.m1.1.1" xref="S4.SS2.SSS2.Px2.1.p1.1.m1.1.1.cmml"><mi id="S4.SS2.SSS2.Px2.1.p1.1.m1.1.1.3" xref="S4.SS2.SSS2.Px2.1.p1.1.m1.1.1.3.cmml">B</mi><mo id="S4.SS2.SSS2.Px2.1.p1.1.m1.1.1.2" xref="S4.SS2.SSS2.Px2.1.p1.1.m1.1.1.2.cmml">=</mo><mrow id="S4.SS2.SSS2.Px2.1.p1.1.m1.1.1.1" xref="S4.SS2.SSS2.Px2.1.p1.1.m1.1.1.1.cmml"><mi id="S4.SS2.SSS2.Px2.1.p1.1.m1.1.1.1.3" xref="S4.SS2.SSS2.Px2.1.p1.1.m1.1.1.1.3.cmml">O</mi><mo id="S4.SS2.SSS2.Px2.1.p1.1.m1.1.1.1.2" xref="S4.SS2.SSS2.Px2.1.p1.1.m1.1.1.1.2.cmml">⁢</mo><mrow id="S4.SS2.SSS2.Px2.1.p1.1.m1.1.1.1.1.1" xref="S4.SS2.SSS2.Px2.1.p1.1.m1.1.1.1.1.1.1.cmml"><mo id="S4.SS2.SSS2.Px2.1.p1.1.m1.1.1.1.1.1.2" stretchy="false" xref="S4.SS2.SSS2.Px2.1.p1.1.m1.1.1.1.1.1.1.cmml">(</mo><mrow id="S4.SS2.SSS2.Px2.1.p1.1.m1.1.1.1.1.1.1" xref="S4.SS2.SSS2.Px2.1.p1.1.m1.1.1.1.1.1.1.cmml"><msup id="S4.SS2.SSS2.Px2.1.p1.1.m1.1.1.1.1.1.1.2" xref="S4.SS2.SSS2.Px2.1.p1.1.m1.1.1.1.1.1.1.2.cmml"><mi id="S4.SS2.SSS2.Px2.1.p1.1.m1.1.1.1.1.1.1.2.2" xref="S4.SS2.SSS2.Px2.1.p1.1.m1.1.1.1.1.1.1.2.2.cmml">ϵ</mi><mrow id="S4.SS2.SSS2.Px2.1.p1.1.m1.1.1.1.1.1.1.2.3" xref="S4.SS2.SSS2.Px2.1.p1.1.m1.1.1.1.1.1.1.2.3.cmml"><mo id="S4.SS2.SSS2.Px2.1.p1.1.m1.1.1.1.1.1.1.2.3a" xref="S4.SS2.SSS2.Px2.1.p1.1.m1.1.1.1.1.1.1.2.3.cmml">−</mo><mn id="S4.SS2.SSS2.Px2.1.p1.1.m1.1.1.1.1.1.1.2.3.2" xref="S4.SS2.SSS2.Px2.1.p1.1.m1.1.1.1.1.1.1.2.3.2.cmml">1</mn></mrow></msup><mo id="S4.SS2.SSS2.Px2.1.p1.1.m1.1.1.1.1.1.1.1" lspace="0.167em" xref="S4.SS2.SSS2.Px2.1.p1.1.m1.1.1.1.1.1.1.1.cmml">⁢</mo><mrow id="S4.SS2.SSS2.Px2.1.p1.1.m1.1.1.1.1.1.1.3" xref="S4.SS2.SSS2.Px2.1.p1.1.m1.1.1.1.1.1.1.3.cmml"><mi id="S4.SS2.SSS2.Px2.1.p1.1.m1.1.1.1.1.1.1.3.1" xref="S4.SS2.SSS2.Px2.1.p1.1.m1.1.1.1.1.1.1.3.1.cmml">log</mi><mo id="S4.SS2.SSS2.Px2.1.p1.1.m1.1.1.1.1.1.1.3a" lspace="0.167em" xref="S4.SS2.SSS2.Px2.1.p1.1.m1.1.1.1.1.1.1.3.cmml">⁡</mo><mi id="S4.SS2.SSS2.Px2.1.p1.1.m1.1.1.1.1.1.1.3.2" xref="S4.SS2.SSS2.Px2.1.p1.1.m1.1.1.1.1.1.1.3.2.cmml">W</mi></mrow></mrow><mo id="S4.SS2.SSS2.Px2.1.p1.1.m1.1.1.1.1.1.3" stretchy="false" xref="S4.SS2.SSS2.Px2.1.p1.1.m1.1.1.1.1.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS2.Px2.1.p1.1.m1.1b"><apply id="S4.SS2.SSS2.Px2.1.p1.1.m1.1.1.cmml" xref="S4.SS2.SSS2.Px2.1.p1.1.m1.1.1"><eq id="S4.SS2.SSS2.Px2.1.p1.1.m1.1.1.2.cmml" xref="S4.SS2.SSS2.Px2.1.p1.1.m1.1.1.2"></eq><ci id="S4.SS2.SSS2.Px2.1.p1.1.m1.1.1.3.cmml" xref="S4.SS2.SSS2.Px2.1.p1.1.m1.1.1.3">𝐵</ci><apply id="S4.SS2.SSS2.Px2.1.p1.1.m1.1.1.1.cmml" xref="S4.SS2.SSS2.Px2.1.p1.1.m1.1.1.1"><times id="S4.SS2.SSS2.Px2.1.p1.1.m1.1.1.1.2.cmml" xref="S4.SS2.SSS2.Px2.1.p1.1.m1.1.1.1.2"></times><ci id="S4.SS2.SSS2.Px2.1.p1.1.m1.1.1.1.3.cmml" xref="S4.SS2.SSS2.Px2.1.p1.1.m1.1.1.1.3">𝑂</ci><apply id="S4.SS2.SSS2.Px2.1.p1.1.m1.1.1.1.1.1.1.cmml" xref="S4.SS2.SSS2.Px2.1.p1.1.m1.1.1.1.1.1"><times id="S4.SS2.SSS2.Px2.1.p1.1.m1.1.1.1.1.1.1.1.cmml" xref="S4.SS2.SSS2.Px2.1.p1.1.m1.1.1.1.1.1.1.1"></times><apply id="S4.SS2.SSS2.Px2.1.p1.1.m1.1.1.1.1.1.1.2.cmml" xref="S4.SS2.SSS2.Px2.1.p1.1.m1.1.1.1.1.1.1.2"><csymbol cd="ambiguous" id="S4.SS2.SSS2.Px2.1.p1.1.m1.1.1.1.1.1.1.2.1.cmml" xref="S4.SS2.SSS2.Px2.1.p1.1.m1.1.1.1.1.1.1.2">superscript</csymbol><ci id="S4.SS2.SSS2.Px2.1.p1.1.m1.1.1.1.1.1.1.2.2.cmml" xref="S4.SS2.SSS2.Px2.1.p1.1.m1.1.1.1.1.1.1.2.2">italic-ϵ</ci><apply id="S4.SS2.SSS2.Px2.1.p1.1.m1.1.1.1.1.1.1.2.3.cmml" xref="S4.SS2.SSS2.Px2.1.p1.1.m1.1.1.1.1.1.1.2.3"><minus id="S4.SS2.SSS2.Px2.1.p1.1.m1.1.1.1.1.1.1.2.3.1.cmml" xref="S4.SS2.SSS2.Px2.1.p1.1.m1.1.1.1.1.1.1.2.3"></minus><cn id="S4.SS2.SSS2.Px2.1.p1.1.m1.1.1.1.1.1.1.2.3.2.cmml" type="integer" xref="S4.SS2.SSS2.Px2.1.p1.1.m1.1.1.1.1.1.1.2.3.2">1</cn></apply></apply><apply id="S4.SS2.SSS2.Px2.1.p1.1.m1.1.1.1.1.1.1.3.cmml" xref="S4.SS2.SSS2.Px2.1.p1.1.m1.1.1.1.1.1.1.3"><log id="S4.SS2.SSS2.Px2.1.p1.1.m1.1.1.1.1.1.1.3.1.cmml" xref="S4.SS2.SSS2.Px2.1.p1.1.m1.1.1.1.1.1.1.3.1"></log><ci id="S4.SS2.SSS2.Px2.1.p1.1.m1.1.1.1.1.1.1.3.2.cmml" xref="S4.SS2.SSS2.Px2.1.p1.1.m1.1.1.1.1.1.1.3.2">𝑊</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS2.Px2.1.p1.1.m1.1c">B=O(\epsilon^{-1}\log W)</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS2.Px2.1.p1.1.m1.1d">italic_B = italic_O ( italic_ϵ start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT roman_log italic_W )</annotation></semantics></math> be the number of weight classes. The set of links stored in Algorithm <a class="ltx_ref" href="https://arxiv.org/html/2503.00712v1#algorithm6" title="In “Cycle” Cuts: ‣ 4.2.2 The Streaming Algorithm ‣ 4.2 Two-to-Three Augmentation ‣ 4 Vertex Connectivity Augmentation in Link-Arrival Model ‣ Streaming Algorithms for Network Design"><span class="ltx_text ltx_ref_tag">6</span></a> is exactly <math alttext="F" class="ltx_Math" display="inline" id="S4.SS2.SSS2.Px2.1.p1.2.m2.1"><semantics id="S4.SS2.SSS2.Px2.1.p1.2.m2.1a"><mi id="S4.SS2.SSS2.Px2.1.p1.2.m2.1.1" xref="S4.SS2.SSS2.Px2.1.p1.2.m2.1.1.cmml">F</mi><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS2.Px2.1.p1.2.m2.1b"><ci id="S4.SS2.SSS2.Px2.1.p1.2.m2.1.1.cmml" xref="S4.SS2.SSS2.Px2.1.p1.2.m2.1.1">𝐹</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS2.Px2.1.p1.2.m2.1c">F</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS2.Px2.1.p1.2.m2.1d">italic_F</annotation></semantics></math>. We bound each of the three sets of links separately. First, for each <math alttext="x\in V(T)" class="ltx_Math" display="inline" id="S4.SS2.SSS2.Px2.1.p1.3.m3.1"><semantics id="S4.SS2.SSS2.Px2.1.p1.3.m3.1a"><mrow id="S4.SS2.SSS2.Px2.1.p1.3.m3.1.2" xref="S4.SS2.SSS2.Px2.1.p1.3.m3.1.2.cmml"><mi id="S4.SS2.SSS2.Px2.1.p1.3.m3.1.2.2" xref="S4.SS2.SSS2.Px2.1.p1.3.m3.1.2.2.cmml">x</mi><mo id="S4.SS2.SSS2.Px2.1.p1.3.m3.1.2.1" xref="S4.SS2.SSS2.Px2.1.p1.3.m3.1.2.1.cmml">∈</mo><mrow id="S4.SS2.SSS2.Px2.1.p1.3.m3.1.2.3" xref="S4.SS2.SSS2.Px2.1.p1.3.m3.1.2.3.cmml"><mi id="S4.SS2.SSS2.Px2.1.p1.3.m3.1.2.3.2" xref="S4.SS2.SSS2.Px2.1.p1.3.m3.1.2.3.2.cmml">V</mi><mo id="S4.SS2.SSS2.Px2.1.p1.3.m3.1.2.3.1" xref="S4.SS2.SSS2.Px2.1.p1.3.m3.1.2.3.1.cmml">⁢</mo><mrow id="S4.SS2.SSS2.Px2.1.p1.3.m3.1.2.3.3.2" xref="S4.SS2.SSS2.Px2.1.p1.3.m3.1.2.3.cmml"><mo id="S4.SS2.SSS2.Px2.1.p1.3.m3.1.2.3.3.2.1" stretchy="false" xref="S4.SS2.SSS2.Px2.1.p1.3.m3.1.2.3.cmml">(</mo><mi id="S4.SS2.SSS2.Px2.1.p1.3.m3.1.1" xref="S4.SS2.SSS2.Px2.1.p1.3.m3.1.1.cmml">T</mi><mo id="S4.SS2.SSS2.Px2.1.p1.3.m3.1.2.3.3.2.2" stretchy="false" xref="S4.SS2.SSS2.Px2.1.p1.3.m3.1.2.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS2.Px2.1.p1.3.m3.1b"><apply id="S4.SS2.SSS2.Px2.1.p1.3.m3.1.2.cmml" xref="S4.SS2.SSS2.Px2.1.p1.3.m3.1.2"><in id="S4.SS2.SSS2.Px2.1.p1.3.m3.1.2.1.cmml" xref="S4.SS2.SSS2.Px2.1.p1.3.m3.1.2.1"></in><ci id="S4.SS2.SSS2.Px2.1.p1.3.m3.1.2.2.cmml" xref="S4.SS2.SSS2.Px2.1.p1.3.m3.1.2.2">𝑥</ci><apply id="S4.SS2.SSS2.Px2.1.p1.3.m3.1.2.3.cmml" xref="S4.SS2.SSS2.Px2.1.p1.3.m3.1.2.3"><times id="S4.SS2.SSS2.Px2.1.p1.3.m3.1.2.3.1.cmml" xref="S4.SS2.SSS2.Px2.1.p1.3.m3.1.2.3.1"></times><ci id="S4.SS2.SSS2.Px2.1.p1.3.m3.1.2.3.2.cmml" xref="S4.SS2.SSS2.Px2.1.p1.3.m3.1.2.3.2">𝑉</ci><ci id="S4.SS2.SSS2.Px2.1.p1.3.m3.1.1.cmml" xref="S4.SS2.SSS2.Px2.1.p1.3.m3.1.1">𝑇</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS2.Px2.1.p1.3.m3.1c">x\in V(T)</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS2.Px2.1.p1.3.m3.1d">italic_x ∈ italic_V ( italic_T )</annotation></semantics></math>, we store one link in <math alttext="L_{x}" class="ltx_Math" display="inline" id="S4.SS2.SSS2.Px2.1.p1.4.m4.1"><semantics id="S4.SS2.SSS2.Px2.1.p1.4.m4.1a"><msub id="S4.SS2.SSS2.Px2.1.p1.4.m4.1.1" xref="S4.SS2.SSS2.Px2.1.p1.4.m4.1.1.cmml"><mi id="S4.SS2.SSS2.Px2.1.p1.4.m4.1.1.2" xref="S4.SS2.SSS2.Px2.1.p1.4.m4.1.1.2.cmml">L</mi><mi id="S4.SS2.SSS2.Px2.1.p1.4.m4.1.1.3" xref="S4.SS2.SSS2.Px2.1.p1.4.m4.1.1.3.cmml">x</mi></msub><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS2.Px2.1.p1.4.m4.1b"><apply id="S4.SS2.SSS2.Px2.1.p1.4.m4.1.1.cmml" xref="S4.SS2.SSS2.Px2.1.p1.4.m4.1.1"><csymbol cd="ambiguous" id="S4.SS2.SSS2.Px2.1.p1.4.m4.1.1.1.cmml" xref="S4.SS2.SSS2.Px2.1.p1.4.m4.1.1">subscript</csymbol><ci id="S4.SS2.SSS2.Px2.1.p1.4.m4.1.1.2.cmml" xref="S4.SS2.SSS2.Px2.1.p1.4.m4.1.1.2">𝐿</ci><ci id="S4.SS2.SSS2.Px2.1.p1.4.m4.1.1.3.cmml" xref="S4.SS2.SSS2.Px2.1.p1.4.m4.1.1.3">𝑥</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS2.Px2.1.p1.4.m4.1c">L_{x}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS2.Px2.1.p1.4.m4.1d">italic_L start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT</annotation></semantics></math> per weight class. 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id="S4.SS2.SSS2.Px2.1.p1.5.m5.1.1.1.2" xref="S4.SS2.SSS2.Px2.1.p1.5.m5.1.1.1.2.cmml">∈</mo><mrow id="S4.SS2.SSS2.Px2.1.p1.5.m5.1.1.1.4" xref="S4.SS2.SSS2.Px2.1.p1.5.m5.1.1.1.4.cmml"><mi id="S4.SS2.SSS2.Px2.1.p1.5.m5.1.1.1.4.2" xref="S4.SS2.SSS2.Px2.1.p1.5.m5.1.1.1.4.2.cmml">V</mi><mo id="S4.SS2.SSS2.Px2.1.p1.5.m5.1.1.1.4.1" xref="S4.SS2.SSS2.Px2.1.p1.5.m5.1.1.1.4.1.cmml">⁢</mo><mrow id="S4.SS2.SSS2.Px2.1.p1.5.m5.1.1.1.4.3.2" xref="S4.SS2.SSS2.Px2.1.p1.5.m5.1.1.1.4.cmml"><mo id="S4.SS2.SSS2.Px2.1.p1.5.m5.1.1.1.4.3.2.1" stretchy="false" xref="S4.SS2.SSS2.Px2.1.p1.5.m5.1.1.1.4.cmml">(</mo><mi id="S4.SS2.SSS2.Px2.1.p1.5.m5.1.1.1.1" xref="S4.SS2.SSS2.Px2.1.p1.5.m5.1.1.1.1.cmml">T</mi><mo id="S4.SS2.SSS2.Px2.1.p1.5.m5.1.1.1.4.3.2.2" stretchy="false" xref="S4.SS2.SSS2.Px2.1.p1.5.m5.1.1.1.4.cmml">)</mo></mrow></mrow></mrow></msub><msub id="S4.SS2.SSS2.Px2.1.p1.5.m5.4.4.1.1.1.2" xref="S4.SS2.SSS2.Px2.1.p1.5.m5.4.4.1.1.1.2.cmml"><mi id="S4.SS2.SSS2.Px2.1.p1.5.m5.4.4.1.1.1.2.2" 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Next, we bound the number of links stored in the spanning trees.</p> <table class="ltx_equation ltx_eqn_table" id="S4.Ex12"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\left|\bigcup_{x\in V(T),x\text{ is P-node}}E(H_{x})\right|\leq\sum_{x\in V(T)% ,x\text{ is P-node}}|E(H_{x})|\leq\sum_{x\in V(T)}|C(x)|-1\leq\sum_{x\in V(T)}% \deg_{T}(x)=2|E(T)|." class="ltx_Math" display="block" id="S4.Ex12.m1.12"><semantics id="S4.Ex12.m1.12a"><mrow id="S4.Ex12.m1.12.12.1" xref="S4.Ex12.m1.12.12.1.1.cmml"><mrow id="S4.Ex12.m1.12.12.1.1" xref="S4.Ex12.m1.12.12.1.1.cmml"><mrow id="S4.Ex12.m1.12.12.1.1.1.1" xref="S4.Ex12.m1.12.12.1.1.1.2.cmml"><mo id="S4.Ex12.m1.12.12.1.1.1.1.2" xref="S4.Ex12.m1.12.12.1.1.1.2.1.cmml">|</mo><mrow id="S4.Ex12.m1.12.12.1.1.1.1.1" xref="S4.Ex12.m1.12.12.1.1.1.1.1.cmml"><munder id="S4.Ex12.m1.12.12.1.1.1.1.1.2" xref="S4.Ex12.m1.12.12.1.1.1.1.1.2.cmml"><mo id="S4.Ex12.m1.12.12.1.1.1.1.1.2.2" lspace="0em" movablelimits="false" xref="S4.Ex12.m1.12.12.1.1.1.1.1.2.2.cmml">⋃</mo><mrow id="S4.Ex12.m1.3.3.3" xref="S4.Ex12.m1.3.3.3.cmml"><mi id="S4.Ex12.m1.3.3.3.5" xref="S4.Ex12.m1.3.3.3.5.cmml">x</mi><mo id="S4.Ex12.m1.3.3.3.4" xref="S4.Ex12.m1.3.3.3.4.cmml">∈</mo><mrow id="S4.Ex12.m1.3.3.3.3.2" xref="S4.Ex12.m1.3.3.3.3.3.cmml"><mrow id="S4.Ex12.m1.2.2.2.2.1.1" xref="S4.Ex12.m1.2.2.2.2.1.1.cmml"><mi id="S4.Ex12.m1.2.2.2.2.1.1.2" xref="S4.Ex12.m1.2.2.2.2.1.1.2.cmml">V</mi><mo id="S4.Ex12.m1.2.2.2.2.1.1.1" xref="S4.Ex12.m1.2.2.2.2.1.1.1.cmml">⁢</mo><mrow id="S4.Ex12.m1.2.2.2.2.1.1.3.2" xref="S4.Ex12.m1.2.2.2.2.1.1.cmml"><mo id="S4.Ex12.m1.2.2.2.2.1.1.3.2.1" stretchy="false" xref="S4.Ex12.m1.2.2.2.2.1.1.cmml">(</mo><mi id="S4.Ex12.m1.1.1.1.1" xref="S4.Ex12.m1.1.1.1.1.cmml">T</mi><mo id="S4.Ex12.m1.2.2.2.2.1.1.3.2.2" stretchy="false" xref="S4.Ex12.m1.2.2.2.2.1.1.cmml">)</mo></mrow></mrow><mo id="S4.Ex12.m1.3.3.3.3.2.3" xref="S4.Ex12.m1.3.3.3.3.3.cmml">,</mo><mrow id="S4.Ex12.m1.3.3.3.3.2.2" xref="S4.Ex12.m1.3.3.3.3.2.2.cmml"><mi id="S4.Ex12.m1.3.3.3.3.2.2.2" xref="S4.Ex12.m1.3.3.3.3.2.2.2.cmml">x</mi><mo id="S4.Ex12.m1.3.3.3.3.2.2.1" xref="S4.Ex12.m1.3.3.3.3.2.2.1.cmml">⁢</mo><mtext id="S4.Ex12.m1.3.3.3.3.2.2.3" xref="S4.Ex12.m1.3.3.3.3.2.2.3a.cmml"> is P-node</mtext></mrow></mrow></mrow></munder><mrow id="S4.Ex12.m1.12.12.1.1.1.1.1.1" xref="S4.Ex12.m1.12.12.1.1.1.1.1.1.cmml"><mi id="S4.Ex12.m1.12.12.1.1.1.1.1.1.3" xref="S4.Ex12.m1.12.12.1.1.1.1.1.1.3.cmml">E</mi><mo id="S4.Ex12.m1.12.12.1.1.1.1.1.1.2" xref="S4.Ex12.m1.12.12.1.1.1.1.1.1.2.cmml">⁢</mo><mrow id="S4.Ex12.m1.12.12.1.1.1.1.1.1.1.1" xref="S4.Ex12.m1.12.12.1.1.1.1.1.1.1.1.1.cmml"><mo id="S4.Ex12.m1.12.12.1.1.1.1.1.1.1.1.2" stretchy="false" xref="S4.Ex12.m1.12.12.1.1.1.1.1.1.1.1.1.cmml">(</mo><msub id="S4.Ex12.m1.12.12.1.1.1.1.1.1.1.1.1" xref="S4.Ex12.m1.12.12.1.1.1.1.1.1.1.1.1.cmml"><mi id="S4.Ex12.m1.12.12.1.1.1.1.1.1.1.1.1.2" xref="S4.Ex12.m1.12.12.1.1.1.1.1.1.1.1.1.2.cmml">H</mi><mi id="S4.Ex12.m1.12.12.1.1.1.1.1.1.1.1.1.3" xref="S4.Ex12.m1.12.12.1.1.1.1.1.1.1.1.1.3.cmml">x</mi></msub><mo id="S4.Ex12.m1.12.12.1.1.1.1.1.1.1.1.3" stretchy="false" xref="S4.Ex12.m1.12.12.1.1.1.1.1.1.1.1.1.cmml">)</mo></mrow></mrow></mrow><mo id="S4.Ex12.m1.12.12.1.1.1.1.3" xref="S4.Ex12.m1.12.12.1.1.1.2.1.cmml">|</mo></mrow><mo id="S4.Ex12.m1.12.12.1.1.7" rspace="0.111em" xref="S4.Ex12.m1.12.12.1.1.7.cmml">≤</mo><mrow id="S4.Ex12.m1.12.12.1.1.2" xref="S4.Ex12.m1.12.12.1.1.2.cmml"><munder id="S4.Ex12.m1.12.12.1.1.2.2" xref="S4.Ex12.m1.12.12.1.1.2.2.cmml"><mo id="S4.Ex12.m1.12.12.1.1.2.2.2" movablelimits="false" rspace="0em" xref="S4.Ex12.m1.12.12.1.1.2.2.2.cmml">∑</mo><mrow id="S4.Ex12.m1.6.6.3" xref="S4.Ex12.m1.6.6.3.cmml"><mi id="S4.Ex12.m1.6.6.3.5" xref="S4.Ex12.m1.6.6.3.5.cmml">x</mi><mo id="S4.Ex12.m1.6.6.3.4" xref="S4.Ex12.m1.6.6.3.4.cmml">∈</mo><mrow id="S4.Ex12.m1.6.6.3.3.2" xref="S4.Ex12.m1.6.6.3.3.3.cmml"><mrow id="S4.Ex12.m1.5.5.2.2.1.1" xref="S4.Ex12.m1.5.5.2.2.1.1.cmml"><mi id="S4.Ex12.m1.5.5.2.2.1.1.2" xref="S4.Ex12.m1.5.5.2.2.1.1.2.cmml">V</mi><mo id="S4.Ex12.m1.5.5.2.2.1.1.1" xref="S4.Ex12.m1.5.5.2.2.1.1.1.cmml">⁢</mo><mrow id="S4.Ex12.m1.5.5.2.2.1.1.3.2" xref="S4.Ex12.m1.5.5.2.2.1.1.cmml"><mo id="S4.Ex12.m1.5.5.2.2.1.1.3.2.1" stretchy="false" xref="S4.Ex12.m1.5.5.2.2.1.1.cmml">(</mo><mi id="S4.Ex12.m1.4.4.1.1" xref="S4.Ex12.m1.4.4.1.1.cmml">T</mi><mo id="S4.Ex12.m1.5.5.2.2.1.1.3.2.2" stretchy="false" xref="S4.Ex12.m1.5.5.2.2.1.1.cmml">)</mo></mrow></mrow><mo id="S4.Ex12.m1.6.6.3.3.2.3" xref="S4.Ex12.m1.6.6.3.3.3.cmml">,</mo><mrow id="S4.Ex12.m1.6.6.3.3.2.2" xref="S4.Ex12.m1.6.6.3.3.2.2.cmml"><mi id="S4.Ex12.m1.6.6.3.3.2.2.2" xref="S4.Ex12.m1.6.6.3.3.2.2.2.cmml">x</mi><mo id="S4.Ex12.m1.6.6.3.3.2.2.1" xref="S4.Ex12.m1.6.6.3.3.2.2.1.cmml">⁢</mo><mtext id="S4.Ex12.m1.6.6.3.3.2.2.3" xref="S4.Ex12.m1.6.6.3.3.2.2.3a.cmml"> is P-node</mtext></mrow></mrow></mrow></munder><mrow id="S4.Ex12.m1.12.12.1.1.2.1.1" xref="S4.Ex12.m1.12.12.1.1.2.1.2.cmml"><mo id="S4.Ex12.m1.12.12.1.1.2.1.1.2" stretchy="false" xref="S4.Ex12.m1.12.12.1.1.2.1.2.1.cmml">|</mo><mrow id="S4.Ex12.m1.12.12.1.1.2.1.1.1" xref="S4.Ex12.m1.12.12.1.1.2.1.1.1.cmml"><mi id="S4.Ex12.m1.12.12.1.1.2.1.1.1.3" xref="S4.Ex12.m1.12.12.1.1.2.1.1.1.3.cmml">E</mi><mo id="S4.Ex12.m1.12.12.1.1.2.1.1.1.2" xref="S4.Ex12.m1.12.12.1.1.2.1.1.1.2.cmml">⁢</mo><mrow id="S4.Ex12.m1.12.12.1.1.2.1.1.1.1.1" xref="S4.Ex12.m1.12.12.1.1.2.1.1.1.1.1.1.cmml"><mo id="S4.Ex12.m1.12.12.1.1.2.1.1.1.1.1.2" stretchy="false" xref="S4.Ex12.m1.12.12.1.1.2.1.1.1.1.1.1.cmml">(</mo><msub id="S4.Ex12.m1.12.12.1.1.2.1.1.1.1.1.1" xref="S4.Ex12.m1.12.12.1.1.2.1.1.1.1.1.1.cmml"><mi id="S4.Ex12.m1.12.12.1.1.2.1.1.1.1.1.1.2" xref="S4.Ex12.m1.12.12.1.1.2.1.1.1.1.1.1.2.cmml">H</mi><mi id="S4.Ex12.m1.12.12.1.1.2.1.1.1.1.1.1.3" xref="S4.Ex12.m1.12.12.1.1.2.1.1.1.1.1.1.3.cmml">x</mi></msub><mo id="S4.Ex12.m1.12.12.1.1.2.1.1.1.1.1.3" stretchy="false" xref="S4.Ex12.m1.12.12.1.1.2.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S4.Ex12.m1.12.12.1.1.2.1.1.3" stretchy="false" xref="S4.Ex12.m1.12.12.1.1.2.1.2.1.cmml">|</mo></mrow></mrow><mo id="S4.Ex12.m1.12.12.1.1.8" rspace="0.111em" xref="S4.Ex12.m1.12.12.1.1.8.cmml">≤</mo><mrow id="S4.Ex12.m1.12.12.1.1.3" xref="S4.Ex12.m1.12.12.1.1.3.cmml"><mrow id="S4.Ex12.m1.12.12.1.1.3.1" xref="S4.Ex12.m1.12.12.1.1.3.1.cmml"><munder id="S4.Ex12.m1.12.12.1.1.3.1.2" xref="S4.Ex12.m1.12.12.1.1.3.1.2.cmml"><mo id="S4.Ex12.m1.12.12.1.1.3.1.2.2" movablelimits="false" rspace="0em" xref="S4.Ex12.m1.12.12.1.1.3.1.2.2.cmml">∑</mo><mrow id="S4.Ex12.m1.7.7.1" xref="S4.Ex12.m1.7.7.1.cmml"><mi id="S4.Ex12.m1.7.7.1.3" xref="S4.Ex12.m1.7.7.1.3.cmml">x</mi><mo id="S4.Ex12.m1.7.7.1.2" xref="S4.Ex12.m1.7.7.1.2.cmml">∈</mo><mrow id="S4.Ex12.m1.7.7.1.4" xref="S4.Ex12.m1.7.7.1.4.cmml"><mi id="S4.Ex12.m1.7.7.1.4.2" xref="S4.Ex12.m1.7.7.1.4.2.cmml">V</mi><mo id="S4.Ex12.m1.7.7.1.4.1" xref="S4.Ex12.m1.7.7.1.4.1.cmml">⁢</mo><mrow id="S4.Ex12.m1.7.7.1.4.3.2" xref="S4.Ex12.m1.7.7.1.4.cmml"><mo id="S4.Ex12.m1.7.7.1.4.3.2.1" stretchy="false" xref="S4.Ex12.m1.7.7.1.4.cmml">(</mo><mi id="S4.Ex12.m1.7.7.1.1" xref="S4.Ex12.m1.7.7.1.1.cmml">T</mi><mo id="S4.Ex12.m1.7.7.1.4.3.2.2" stretchy="false" xref="S4.Ex12.m1.7.7.1.4.cmml">)</mo></mrow></mrow></mrow></munder><mrow id="S4.Ex12.m1.12.12.1.1.3.1.1.1" xref="S4.Ex12.m1.12.12.1.1.3.1.1.2.cmml"><mo id="S4.Ex12.m1.12.12.1.1.3.1.1.1.2" stretchy="false" xref="S4.Ex12.m1.12.12.1.1.3.1.1.2.1.cmml">|</mo><mrow id="S4.Ex12.m1.12.12.1.1.3.1.1.1.1" xref="S4.Ex12.m1.12.12.1.1.3.1.1.1.1.cmml"><mi id="S4.Ex12.m1.12.12.1.1.3.1.1.1.1.2" xref="S4.Ex12.m1.12.12.1.1.3.1.1.1.1.2.cmml">C</mi><mo id="S4.Ex12.m1.12.12.1.1.3.1.1.1.1.1" xref="S4.Ex12.m1.12.12.1.1.3.1.1.1.1.1.cmml">⁢</mo><mrow id="S4.Ex12.m1.12.12.1.1.3.1.1.1.1.3.2" xref="S4.Ex12.m1.12.12.1.1.3.1.1.1.1.cmml"><mo id="S4.Ex12.m1.12.12.1.1.3.1.1.1.1.3.2.1" stretchy="false" xref="S4.Ex12.m1.12.12.1.1.3.1.1.1.1.cmml">(</mo><mi id="S4.Ex12.m1.9.9" xref="S4.Ex12.m1.9.9.cmml">x</mi><mo id="S4.Ex12.m1.12.12.1.1.3.1.1.1.1.3.2.2" stretchy="false" xref="S4.Ex12.m1.12.12.1.1.3.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S4.Ex12.m1.12.12.1.1.3.1.1.1.3" stretchy="false" xref="S4.Ex12.m1.12.12.1.1.3.1.1.2.1.cmml">|</mo></mrow></mrow><mo id="S4.Ex12.m1.12.12.1.1.3.2" xref="S4.Ex12.m1.12.12.1.1.3.2.cmml">−</mo><mn id="S4.Ex12.m1.12.12.1.1.3.3" xref="S4.Ex12.m1.12.12.1.1.3.3.cmml">1</mn></mrow><mo id="S4.Ex12.m1.12.12.1.1.9" rspace="0.111em" xref="S4.Ex12.m1.12.12.1.1.9.cmml">≤</mo><mrow id="S4.Ex12.m1.12.12.1.1.4" xref="S4.Ex12.m1.12.12.1.1.4.cmml"><munder id="S4.Ex12.m1.12.12.1.1.4.2" xref="S4.Ex12.m1.12.12.1.1.4.2.cmml"><mo id="S4.Ex12.m1.12.12.1.1.4.2.2" movablelimits="false" xref="S4.Ex12.m1.12.12.1.1.4.2.2.cmml">∑</mo><mrow id="S4.Ex12.m1.8.8.1" xref="S4.Ex12.m1.8.8.1.cmml"><mi id="S4.Ex12.m1.8.8.1.3" xref="S4.Ex12.m1.8.8.1.3.cmml">x</mi><mo 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id="S4.Ex12.m1.12.12.1.1.5.1.1.1.cmml" xref="S4.Ex12.m1.12.12.1.1.5.1.1.1"><times id="S4.Ex12.m1.12.12.1.1.5.1.1.1.1.cmml" xref="S4.Ex12.m1.12.12.1.1.5.1.1.1.1"></times><ci id="S4.Ex12.m1.12.12.1.1.5.1.1.1.2.cmml" xref="S4.Ex12.m1.12.12.1.1.5.1.1.1.2">𝐸</ci><ci id="S4.Ex12.m1.11.11.cmml" xref="S4.Ex12.m1.11.11">𝑇</ci></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Ex12.m1.12c">\left|\bigcup_{x\in V(T),x\text{ is P-node}}E(H_{x})\right|\leq\sum_{x\in V(T)% ,x\text{ is P-node}}|E(H_{x})|\leq\sum_{x\in V(T)}|C(x)|-1\leq\sum_{x\in V(T)}% \deg_{T}(x)=2|E(T)|.</annotation><annotation encoding="application/x-llamapun" id="S4.Ex12.m1.12d">| ⋃ start_POSTSUBSCRIPT italic_x ∈ italic_V ( italic_T ) , italic_x is P-node end_POSTSUBSCRIPT italic_E ( italic_H start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT ) | ≤ ∑ start_POSTSUBSCRIPT italic_x ∈ italic_V ( italic_T ) , italic_x is P-node end_POSTSUBSCRIPT | italic_E ( italic_H start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT ) | ≤ ∑ start_POSTSUBSCRIPT italic_x ∈ italic_V ( italic_T ) end_POSTSUBSCRIPT | italic_C ( italic_x ) | - 1 ≤ ∑ start_POSTSUBSCRIPT italic_x ∈ italic_V ( italic_T ) end_POSTSUBSCRIPT roman_deg start_POSTSUBSCRIPT italic_T end_POSTSUBSCRIPT ( italic_x ) = 2 | italic_E ( italic_T ) | .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S4.SS2.SSS2.Px2.1.p1.14">Finally, for each S-node <math alttext="x" class="ltx_Math" display="inline" id="S4.SS2.SSS2.Px2.1.p1.6.m1.1"><semantics id="S4.SS2.SSS2.Px2.1.p1.6.m1.1a"><mi id="S4.SS2.SSS2.Px2.1.p1.6.m1.1.1" xref="S4.SS2.SSS2.Px2.1.p1.6.m1.1.1.cmml">x</mi><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS2.Px2.1.p1.6.m1.1b"><ci id="S4.SS2.SSS2.Px2.1.p1.6.m1.1.1.cmml" xref="S4.SS2.SSS2.Px2.1.p1.6.m1.1.1">𝑥</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS2.Px2.1.p1.6.m1.1c">x</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS2.Px2.1.p1.6.m1.1d">italic_x</annotation></semantics></math> and each dummy/original node <math alttext="\mu" class="ltx_Math" display="inline" id="S4.SS2.SSS2.Px2.1.p1.7.m2.1"><semantics id="S4.SS2.SSS2.Px2.1.p1.7.m2.1a"><mi id="S4.SS2.SSS2.Px2.1.p1.7.m2.1.1" xref="S4.SS2.SSS2.Px2.1.p1.7.m2.1.1.cmml">μ</mi><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS2.Px2.1.p1.7.m2.1b"><ci id="S4.SS2.SSS2.Px2.1.p1.7.m2.1.1.cmml" xref="S4.SS2.SSS2.Px2.1.p1.7.m2.1.1">𝜇</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS2.Px2.1.p1.7.m2.1c">\mu</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS2.Px2.1.p1.7.m2.1d">italic_μ</annotation></semantics></math>, <math alttext="|\textsc{Min}_{\mu}\cup\textsc{Max}_{\mu}|\leq 2B" class="ltx_Math" display="inline" id="S4.SS2.SSS2.Px2.1.p1.8.m3.1"><semantics id="S4.SS2.SSS2.Px2.1.p1.8.m3.1a"><mrow id="S4.SS2.SSS2.Px2.1.p1.8.m3.1.1" xref="S4.SS2.SSS2.Px2.1.p1.8.m3.1.1.cmml"><mrow id="S4.SS2.SSS2.Px2.1.p1.8.m3.1.1.1.1" xref="S4.SS2.SSS2.Px2.1.p1.8.m3.1.1.1.2.cmml"><mo id="S4.SS2.SSS2.Px2.1.p1.8.m3.1.1.1.1.2" stretchy="false" xref="S4.SS2.SSS2.Px2.1.p1.8.m3.1.1.1.2.1.cmml">|</mo><mrow id="S4.SS2.SSS2.Px2.1.p1.8.m3.1.1.1.1.1" xref="S4.SS2.SSS2.Px2.1.p1.8.m3.1.1.1.1.1.cmml"><msub id="S4.SS2.SSS2.Px2.1.p1.8.m3.1.1.1.1.1.2" xref="S4.SS2.SSS2.Px2.1.p1.8.m3.1.1.1.1.1.2.cmml"><mtext class="ltx_font_smallcaps" id="S4.SS2.SSS2.Px2.1.p1.8.m3.1.1.1.1.1.2.2" xref="S4.SS2.SSS2.Px2.1.p1.8.m3.1.1.1.1.1.2.2a.cmml">Min</mtext><mi id="S4.SS2.SSS2.Px2.1.p1.8.m3.1.1.1.1.1.2.3" xref="S4.SS2.SSS2.Px2.1.p1.8.m3.1.1.1.1.1.2.3.cmml">μ</mi></msub><mo id="S4.SS2.SSS2.Px2.1.p1.8.m3.1.1.1.1.1.1" xref="S4.SS2.SSS2.Px2.1.p1.8.m3.1.1.1.1.1.1.cmml">∪</mo><msub id="S4.SS2.SSS2.Px2.1.p1.8.m3.1.1.1.1.1.3" xref="S4.SS2.SSS2.Px2.1.p1.8.m3.1.1.1.1.1.3.cmml"><mtext class="ltx_font_smallcaps" id="S4.SS2.SSS2.Px2.1.p1.8.m3.1.1.1.1.1.3.2" xref="S4.SS2.SSS2.Px2.1.p1.8.m3.1.1.1.1.1.3.2a.cmml">Max</mtext><mi id="S4.SS2.SSS2.Px2.1.p1.8.m3.1.1.1.1.1.3.3" xref="S4.SS2.SSS2.Px2.1.p1.8.m3.1.1.1.1.1.3.3.cmml">μ</mi></msub></mrow><mo id="S4.SS2.SSS2.Px2.1.p1.8.m3.1.1.1.1.3" stretchy="false" xref="S4.SS2.SSS2.Px2.1.p1.8.m3.1.1.1.2.1.cmml">|</mo></mrow><mo id="S4.SS2.SSS2.Px2.1.p1.8.m3.1.1.2" xref="S4.SS2.SSS2.Px2.1.p1.8.m3.1.1.2.cmml">≤</mo><mrow id="S4.SS2.SSS2.Px2.1.p1.8.m3.1.1.3" xref="S4.SS2.SSS2.Px2.1.p1.8.m3.1.1.3.cmml"><mn id="S4.SS2.SSS2.Px2.1.p1.8.m3.1.1.3.2" xref="S4.SS2.SSS2.Px2.1.p1.8.m3.1.1.3.2.cmml">2</mn><mo id="S4.SS2.SSS2.Px2.1.p1.8.m3.1.1.3.1" xref="S4.SS2.SSS2.Px2.1.p1.8.m3.1.1.3.1.cmml">⁢</mo><mi id="S4.SS2.SSS2.Px2.1.p1.8.m3.1.1.3.3" xref="S4.SS2.SSS2.Px2.1.p1.8.m3.1.1.3.3.cmml">B</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS2.Px2.1.p1.8.m3.1b"><apply id="S4.SS2.SSS2.Px2.1.p1.8.m3.1.1.cmml" xref="S4.SS2.SSS2.Px2.1.p1.8.m3.1.1"><leq id="S4.SS2.SSS2.Px2.1.p1.8.m3.1.1.2.cmml" xref="S4.SS2.SSS2.Px2.1.p1.8.m3.1.1.2"></leq><apply id="S4.SS2.SSS2.Px2.1.p1.8.m3.1.1.1.2.cmml" xref="S4.SS2.SSS2.Px2.1.p1.8.m3.1.1.1.1"><abs id="S4.SS2.SSS2.Px2.1.p1.8.m3.1.1.1.2.1.cmml" xref="S4.SS2.SSS2.Px2.1.p1.8.m3.1.1.1.1.2"></abs><apply id="S4.SS2.SSS2.Px2.1.p1.8.m3.1.1.1.1.1.cmml" xref="S4.SS2.SSS2.Px2.1.p1.8.m3.1.1.1.1.1"><union id="S4.SS2.SSS2.Px2.1.p1.8.m3.1.1.1.1.1.1.cmml" xref="S4.SS2.SSS2.Px2.1.p1.8.m3.1.1.1.1.1.1"></union><apply id="S4.SS2.SSS2.Px2.1.p1.8.m3.1.1.1.1.1.2.cmml" xref="S4.SS2.SSS2.Px2.1.p1.8.m3.1.1.1.1.1.2"><csymbol cd="ambiguous" id="S4.SS2.SSS2.Px2.1.p1.8.m3.1.1.1.1.1.2.1.cmml" xref="S4.SS2.SSS2.Px2.1.p1.8.m3.1.1.1.1.1.2">subscript</csymbol><ci id="S4.SS2.SSS2.Px2.1.p1.8.m3.1.1.1.1.1.2.2a.cmml" xref="S4.SS2.SSS2.Px2.1.p1.8.m3.1.1.1.1.1.2.2"><mtext class="ltx_font_smallcaps" id="S4.SS2.SSS2.Px2.1.p1.8.m3.1.1.1.1.1.2.2.cmml" xref="S4.SS2.SSS2.Px2.1.p1.8.m3.1.1.1.1.1.2.2">Min</mtext></ci><ci id="S4.SS2.SSS2.Px2.1.p1.8.m3.1.1.1.1.1.2.3.cmml" xref="S4.SS2.SSS2.Px2.1.p1.8.m3.1.1.1.1.1.2.3">𝜇</ci></apply><apply id="S4.SS2.SSS2.Px2.1.p1.8.m3.1.1.1.1.1.3.cmml" xref="S4.SS2.SSS2.Px2.1.p1.8.m3.1.1.1.1.1.3"><csymbol cd="ambiguous" id="S4.SS2.SSS2.Px2.1.p1.8.m3.1.1.1.1.1.3.1.cmml" xref="S4.SS2.SSS2.Px2.1.p1.8.m3.1.1.1.1.1.3">subscript</csymbol><ci id="S4.SS2.SSS2.Px2.1.p1.8.m3.1.1.1.1.1.3.2a.cmml" xref="S4.SS2.SSS2.Px2.1.p1.8.m3.1.1.1.1.1.3.2"><mtext class="ltx_font_smallcaps" id="S4.SS2.SSS2.Px2.1.p1.8.m3.1.1.1.1.1.3.2.cmml" xref="S4.SS2.SSS2.Px2.1.p1.8.m3.1.1.1.1.1.3.2">Max</mtext></ci><ci id="S4.SS2.SSS2.Px2.1.p1.8.m3.1.1.1.1.1.3.3.cmml" xref="S4.SS2.SSS2.Px2.1.p1.8.m3.1.1.1.1.1.3.3">𝜇</ci></apply></apply></apply><apply id="S4.SS2.SSS2.Px2.1.p1.8.m3.1.1.3.cmml" xref="S4.SS2.SSS2.Px2.1.p1.8.m3.1.1.3"><times id="S4.SS2.SSS2.Px2.1.p1.8.m3.1.1.3.1.cmml" xref="S4.SS2.SSS2.Px2.1.p1.8.m3.1.1.3.1"></times><cn id="S4.SS2.SSS2.Px2.1.p1.8.m3.1.1.3.2.cmml" type="integer" xref="S4.SS2.SSS2.Px2.1.p1.8.m3.1.1.3.2">2</cn><ci id="S4.SS2.SSS2.Px2.1.p1.8.m3.1.1.3.3.cmml" xref="S4.SS2.SSS2.Px2.1.p1.8.m3.1.1.3.3">𝐵</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS2.Px2.1.p1.8.m3.1c">|\textsc{Min}_{\mu}\cup\textsc{Max}_{\mu}|\leq 2B</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS2.Px2.1.p1.8.m3.1d">| Min start_POSTSUBSCRIPT italic_μ end_POSTSUBSCRIPT ∪ Max start_POSTSUBSCRIPT italic_μ end_POSTSUBSCRIPT | ≤ 2 italic_B</annotation></semantics></math>. We introduce at most one dummy node per edge of <math alttext="G_{x}" class="ltx_Math" display="inline" id="S4.SS2.SSS2.Px2.1.p1.9.m4.1"><semantics id="S4.SS2.SSS2.Px2.1.p1.9.m4.1a"><msub id="S4.SS2.SSS2.Px2.1.p1.9.m4.1.1" xref="S4.SS2.SSS2.Px2.1.p1.9.m4.1.1.cmml"><mi id="S4.SS2.SSS2.Px2.1.p1.9.m4.1.1.2" xref="S4.SS2.SSS2.Px2.1.p1.9.m4.1.1.2.cmml">G</mi><mi id="S4.SS2.SSS2.Px2.1.p1.9.m4.1.1.3" xref="S4.SS2.SSS2.Px2.1.p1.9.m4.1.1.3.cmml">x</mi></msub><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS2.Px2.1.p1.9.m4.1b"><apply id="S4.SS2.SSS2.Px2.1.p1.9.m4.1.1.cmml" xref="S4.SS2.SSS2.Px2.1.p1.9.m4.1.1"><csymbol cd="ambiguous" id="S4.SS2.SSS2.Px2.1.p1.9.m4.1.1.1.cmml" xref="S4.SS2.SSS2.Px2.1.p1.9.m4.1.1">subscript</csymbol><ci id="S4.SS2.SSS2.Px2.1.p1.9.m4.1.1.2.cmml" xref="S4.SS2.SSS2.Px2.1.p1.9.m4.1.1.2">𝐺</ci><ci id="S4.SS2.SSS2.Px2.1.p1.9.m4.1.1.3.cmml" xref="S4.SS2.SSS2.Px2.1.p1.9.m4.1.1.3">𝑥</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS2.Px2.1.p1.9.m4.1c">G_{x}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS2.Px2.1.p1.9.m4.1d">italic_G start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT</annotation></semantics></math>, thus the total number of links stored for <math alttext="x" class="ltx_Math" display="inline" id="S4.SS2.SSS2.Px2.1.p1.10.m5.1"><semantics id="S4.SS2.SSS2.Px2.1.p1.10.m5.1a"><mi id="S4.SS2.SSS2.Px2.1.p1.10.m5.1.1" xref="S4.SS2.SSS2.Px2.1.p1.10.m5.1.1.cmml">x</mi><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS2.Px2.1.p1.10.m5.1b"><ci id="S4.SS2.SSS2.Px2.1.p1.10.m5.1.1.cmml" xref="S4.SS2.SSS2.Px2.1.p1.10.m5.1.1">𝑥</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS2.Px2.1.p1.10.m5.1c">x</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS2.Px2.1.p1.10.m5.1d">italic_x</annotation></semantics></math> is at most <math alttext="2B|V(G_{x})|+2B|E(G_{x})|" class="ltx_Math" display="inline" id="S4.SS2.SSS2.Px2.1.p1.11.m6.2"><semantics id="S4.SS2.SSS2.Px2.1.p1.11.m6.2a"><mrow id="S4.SS2.SSS2.Px2.1.p1.11.m6.2.2" xref="S4.SS2.SSS2.Px2.1.p1.11.m6.2.2.cmml"><mrow id="S4.SS2.SSS2.Px2.1.p1.11.m6.1.1.1" xref="S4.SS2.SSS2.Px2.1.p1.11.m6.1.1.1.cmml"><mn id="S4.SS2.SSS2.Px2.1.p1.11.m6.1.1.1.3" xref="S4.SS2.SSS2.Px2.1.p1.11.m6.1.1.1.3.cmml">2</mn><mo id="S4.SS2.SSS2.Px2.1.p1.11.m6.1.1.1.2" xref="S4.SS2.SSS2.Px2.1.p1.11.m6.1.1.1.2.cmml">⁢</mo><mi id="S4.SS2.SSS2.Px2.1.p1.11.m6.1.1.1.4" xref="S4.SS2.SSS2.Px2.1.p1.11.m6.1.1.1.4.cmml">B</mi><mo id="S4.SS2.SSS2.Px2.1.p1.11.m6.1.1.1.2a" xref="S4.SS2.SSS2.Px2.1.p1.11.m6.1.1.1.2.cmml">⁢</mo><mrow id="S4.SS2.SSS2.Px2.1.p1.11.m6.1.1.1.1.1" xref="S4.SS2.SSS2.Px2.1.p1.11.m6.1.1.1.1.2.cmml"><mo id="S4.SS2.SSS2.Px2.1.p1.11.m6.1.1.1.1.1.2" stretchy="false" xref="S4.SS2.SSS2.Px2.1.p1.11.m6.1.1.1.1.2.1.cmml">|</mo><mrow id="S4.SS2.SSS2.Px2.1.p1.11.m6.1.1.1.1.1.1" xref="S4.SS2.SSS2.Px2.1.p1.11.m6.1.1.1.1.1.1.cmml"><mi id="S4.SS2.SSS2.Px2.1.p1.11.m6.1.1.1.1.1.1.3" xref="S4.SS2.SSS2.Px2.1.p1.11.m6.1.1.1.1.1.1.3.cmml">V</mi><mo id="S4.SS2.SSS2.Px2.1.p1.11.m6.1.1.1.1.1.1.2" xref="S4.SS2.SSS2.Px2.1.p1.11.m6.1.1.1.1.1.1.2.cmml">⁢</mo><mrow id="S4.SS2.SSS2.Px2.1.p1.11.m6.1.1.1.1.1.1.1.1" xref="S4.SS2.SSS2.Px2.1.p1.11.m6.1.1.1.1.1.1.1.1.1.cmml"><mo id="S4.SS2.SSS2.Px2.1.p1.11.m6.1.1.1.1.1.1.1.1.2" stretchy="false" xref="S4.SS2.SSS2.Px2.1.p1.11.m6.1.1.1.1.1.1.1.1.1.cmml">(</mo><msub id="S4.SS2.SSS2.Px2.1.p1.11.m6.1.1.1.1.1.1.1.1.1" xref="S4.SS2.SSS2.Px2.1.p1.11.m6.1.1.1.1.1.1.1.1.1.cmml"><mi id="S4.SS2.SSS2.Px2.1.p1.11.m6.1.1.1.1.1.1.1.1.1.2" xref="S4.SS2.SSS2.Px2.1.p1.11.m6.1.1.1.1.1.1.1.1.1.2.cmml">G</mi><mi id="S4.SS2.SSS2.Px2.1.p1.11.m6.1.1.1.1.1.1.1.1.1.3" xref="S4.SS2.SSS2.Px2.1.p1.11.m6.1.1.1.1.1.1.1.1.1.3.cmml">x</mi></msub><mo id="S4.SS2.SSS2.Px2.1.p1.11.m6.1.1.1.1.1.1.1.1.3" stretchy="false" xref="S4.SS2.SSS2.Px2.1.p1.11.m6.1.1.1.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S4.SS2.SSS2.Px2.1.p1.11.m6.1.1.1.1.1.3" stretchy="false" xref="S4.SS2.SSS2.Px2.1.p1.11.m6.1.1.1.1.2.1.cmml">|</mo></mrow></mrow><mo id="S4.SS2.SSS2.Px2.1.p1.11.m6.2.2.3" xref="S4.SS2.SSS2.Px2.1.p1.11.m6.2.2.3.cmml">+</mo><mrow id="S4.SS2.SSS2.Px2.1.p1.11.m6.2.2.2" xref="S4.SS2.SSS2.Px2.1.p1.11.m6.2.2.2.cmml"><mn id="S4.SS2.SSS2.Px2.1.p1.11.m6.2.2.2.3" xref="S4.SS2.SSS2.Px2.1.p1.11.m6.2.2.2.3.cmml">2</mn><mo id="S4.SS2.SSS2.Px2.1.p1.11.m6.2.2.2.2" xref="S4.SS2.SSS2.Px2.1.p1.11.m6.2.2.2.2.cmml">⁢</mo><mi id="S4.SS2.SSS2.Px2.1.p1.11.m6.2.2.2.4" xref="S4.SS2.SSS2.Px2.1.p1.11.m6.2.2.2.4.cmml">B</mi><mo id="S4.SS2.SSS2.Px2.1.p1.11.m6.2.2.2.2a" xref="S4.SS2.SSS2.Px2.1.p1.11.m6.2.2.2.2.cmml">⁢</mo><mrow id="S4.SS2.SSS2.Px2.1.p1.11.m6.2.2.2.1.1" xref="S4.SS2.SSS2.Px2.1.p1.11.m6.2.2.2.1.2.cmml"><mo id="S4.SS2.SSS2.Px2.1.p1.11.m6.2.2.2.1.1.2" stretchy="false" xref="S4.SS2.SSS2.Px2.1.p1.11.m6.2.2.2.1.2.1.cmml">|</mo><mrow id="S4.SS2.SSS2.Px2.1.p1.11.m6.2.2.2.1.1.1" xref="S4.SS2.SSS2.Px2.1.p1.11.m6.2.2.2.1.1.1.cmml"><mi id="S4.SS2.SSS2.Px2.1.p1.11.m6.2.2.2.1.1.1.3" xref="S4.SS2.SSS2.Px2.1.p1.11.m6.2.2.2.1.1.1.3.cmml">E</mi><mo id="S4.SS2.SSS2.Px2.1.p1.11.m6.2.2.2.1.1.1.2" xref="S4.SS2.SSS2.Px2.1.p1.11.m6.2.2.2.1.1.1.2.cmml">⁢</mo><mrow id="S4.SS2.SSS2.Px2.1.p1.11.m6.2.2.2.1.1.1.1.1" xref="S4.SS2.SSS2.Px2.1.p1.11.m6.2.2.2.1.1.1.1.1.1.cmml"><mo id="S4.SS2.SSS2.Px2.1.p1.11.m6.2.2.2.1.1.1.1.1.2" stretchy="false" xref="S4.SS2.SSS2.Px2.1.p1.11.m6.2.2.2.1.1.1.1.1.1.cmml">(</mo><msub id="S4.SS2.SSS2.Px2.1.p1.11.m6.2.2.2.1.1.1.1.1.1" xref="S4.SS2.SSS2.Px2.1.p1.11.m6.2.2.2.1.1.1.1.1.1.cmml"><mi id="S4.SS2.SSS2.Px2.1.p1.11.m6.2.2.2.1.1.1.1.1.1.2" xref="S4.SS2.SSS2.Px2.1.p1.11.m6.2.2.2.1.1.1.1.1.1.2.cmml">G</mi><mi id="S4.SS2.SSS2.Px2.1.p1.11.m6.2.2.2.1.1.1.1.1.1.3" xref="S4.SS2.SSS2.Px2.1.p1.11.m6.2.2.2.1.1.1.1.1.1.3.cmml">x</mi></msub><mo id="S4.SS2.SSS2.Px2.1.p1.11.m6.2.2.2.1.1.1.1.1.3" stretchy="false" xref="S4.SS2.SSS2.Px2.1.p1.11.m6.2.2.2.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S4.SS2.SSS2.Px2.1.p1.11.m6.2.2.2.1.1.3" stretchy="false" xref="S4.SS2.SSS2.Px2.1.p1.11.m6.2.2.2.1.2.1.cmml">|</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS2.Px2.1.p1.11.m6.2b"><apply id="S4.SS2.SSS2.Px2.1.p1.11.m6.2.2.cmml" xref="S4.SS2.SSS2.Px2.1.p1.11.m6.2.2"><plus id="S4.SS2.SSS2.Px2.1.p1.11.m6.2.2.3.cmml" xref="S4.SS2.SSS2.Px2.1.p1.11.m6.2.2.3"></plus><apply id="S4.SS2.SSS2.Px2.1.p1.11.m6.1.1.1.cmml" xref="S4.SS2.SSS2.Px2.1.p1.11.m6.1.1.1"><times id="S4.SS2.SSS2.Px2.1.p1.11.m6.1.1.1.2.cmml" xref="S4.SS2.SSS2.Px2.1.p1.11.m6.1.1.1.2"></times><cn id="S4.SS2.SSS2.Px2.1.p1.11.m6.1.1.1.3.cmml" type="integer" xref="S4.SS2.SSS2.Px2.1.p1.11.m6.1.1.1.3">2</cn><ci id="S4.SS2.SSS2.Px2.1.p1.11.m6.1.1.1.4.cmml" xref="S4.SS2.SSS2.Px2.1.p1.11.m6.1.1.1.4">𝐵</ci><apply id="S4.SS2.SSS2.Px2.1.p1.11.m6.1.1.1.1.2.cmml" xref="S4.SS2.SSS2.Px2.1.p1.11.m6.1.1.1.1.1"><abs id="S4.SS2.SSS2.Px2.1.p1.11.m6.1.1.1.1.2.1.cmml" xref="S4.SS2.SSS2.Px2.1.p1.11.m6.1.1.1.1.1.2"></abs><apply id="S4.SS2.SSS2.Px2.1.p1.11.m6.1.1.1.1.1.1.cmml" xref="S4.SS2.SSS2.Px2.1.p1.11.m6.1.1.1.1.1.1"><times id="S4.SS2.SSS2.Px2.1.p1.11.m6.1.1.1.1.1.1.2.cmml" xref="S4.SS2.SSS2.Px2.1.p1.11.m6.1.1.1.1.1.1.2"></times><ci id="S4.SS2.SSS2.Px2.1.p1.11.m6.1.1.1.1.1.1.3.cmml" xref="S4.SS2.SSS2.Px2.1.p1.11.m6.1.1.1.1.1.1.3">𝑉</ci><apply id="S4.SS2.SSS2.Px2.1.p1.11.m6.1.1.1.1.1.1.1.1.1.cmml" xref="S4.SS2.SSS2.Px2.1.p1.11.m6.1.1.1.1.1.1.1.1"><csymbol cd="ambiguous" id="S4.SS2.SSS2.Px2.1.p1.11.m6.1.1.1.1.1.1.1.1.1.1.cmml" xref="S4.SS2.SSS2.Px2.1.p1.11.m6.1.1.1.1.1.1.1.1">subscript</csymbol><ci id="S4.SS2.SSS2.Px2.1.p1.11.m6.1.1.1.1.1.1.1.1.1.2.cmml" xref="S4.SS2.SSS2.Px2.1.p1.11.m6.1.1.1.1.1.1.1.1.1.2">𝐺</ci><ci id="S4.SS2.SSS2.Px2.1.p1.11.m6.1.1.1.1.1.1.1.1.1.3.cmml" xref="S4.SS2.SSS2.Px2.1.p1.11.m6.1.1.1.1.1.1.1.1.1.3">𝑥</ci></apply></apply></apply></apply><apply id="S4.SS2.SSS2.Px2.1.p1.11.m6.2.2.2.cmml" xref="S4.SS2.SSS2.Px2.1.p1.11.m6.2.2.2"><times id="S4.SS2.SSS2.Px2.1.p1.11.m6.2.2.2.2.cmml" xref="S4.SS2.SSS2.Px2.1.p1.11.m6.2.2.2.2"></times><cn id="S4.SS2.SSS2.Px2.1.p1.11.m6.2.2.2.3.cmml" type="integer" xref="S4.SS2.SSS2.Px2.1.p1.11.m6.2.2.2.3">2</cn><ci id="S4.SS2.SSS2.Px2.1.p1.11.m6.2.2.2.4.cmml" xref="S4.SS2.SSS2.Px2.1.p1.11.m6.2.2.2.4">𝐵</ci><apply id="S4.SS2.SSS2.Px2.1.p1.11.m6.2.2.2.1.2.cmml" xref="S4.SS2.SSS2.Px2.1.p1.11.m6.2.2.2.1.1"><abs id="S4.SS2.SSS2.Px2.1.p1.11.m6.2.2.2.1.2.1.cmml" xref="S4.SS2.SSS2.Px2.1.p1.11.m6.2.2.2.1.1.2"></abs><apply id="S4.SS2.SSS2.Px2.1.p1.11.m6.2.2.2.1.1.1.cmml" xref="S4.SS2.SSS2.Px2.1.p1.11.m6.2.2.2.1.1.1"><times id="S4.SS2.SSS2.Px2.1.p1.11.m6.2.2.2.1.1.1.2.cmml" xref="S4.SS2.SSS2.Px2.1.p1.11.m6.2.2.2.1.1.1.2"></times><ci id="S4.SS2.SSS2.Px2.1.p1.11.m6.2.2.2.1.1.1.3.cmml" xref="S4.SS2.SSS2.Px2.1.p1.11.m6.2.2.2.1.1.1.3">𝐸</ci><apply id="S4.SS2.SSS2.Px2.1.p1.11.m6.2.2.2.1.1.1.1.1.1.cmml" xref="S4.SS2.SSS2.Px2.1.p1.11.m6.2.2.2.1.1.1.1.1"><csymbol cd="ambiguous" id="S4.SS2.SSS2.Px2.1.p1.11.m6.2.2.2.1.1.1.1.1.1.1.cmml" xref="S4.SS2.SSS2.Px2.1.p1.11.m6.2.2.2.1.1.1.1.1">subscript</csymbol><ci id="S4.SS2.SSS2.Px2.1.p1.11.m6.2.2.2.1.1.1.1.1.1.2.cmml" xref="S4.SS2.SSS2.Px2.1.p1.11.m6.2.2.2.1.1.1.1.1.1.2">𝐺</ci><ci id="S4.SS2.SSS2.Px2.1.p1.11.m6.2.2.2.1.1.1.1.1.1.3.cmml" xref="S4.SS2.SSS2.Px2.1.p1.11.m6.2.2.2.1.1.1.1.1.1.3">𝑥</ci></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS2.Px2.1.p1.11.m6.2c">2B|V(G_{x})|+2B|E(G_{x})|</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS2.Px2.1.p1.11.m6.2d">2 italic_B | italic_V ( italic_G start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT ) | + 2 italic_B | italic_E ( italic_G start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT ) |</annotation></semantics></math>. Since <math alttext="G_{x}" class="ltx_Math" display="inline" id="S4.SS2.SSS2.Px2.1.p1.12.m7.1"><semantics id="S4.SS2.SSS2.Px2.1.p1.12.m7.1a"><msub id="S4.SS2.SSS2.Px2.1.p1.12.m7.1.1" xref="S4.SS2.SSS2.Px2.1.p1.12.m7.1.1.cmml"><mi id="S4.SS2.SSS2.Px2.1.p1.12.m7.1.1.2" xref="S4.SS2.SSS2.Px2.1.p1.12.m7.1.1.2.cmml">G</mi><mi id="S4.SS2.SSS2.Px2.1.p1.12.m7.1.1.3" xref="S4.SS2.SSS2.Px2.1.p1.12.m7.1.1.3.cmml">x</mi></msub><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS2.Px2.1.p1.12.m7.1b"><apply id="S4.SS2.SSS2.Px2.1.p1.12.m7.1.1.cmml" xref="S4.SS2.SSS2.Px2.1.p1.12.m7.1.1"><csymbol cd="ambiguous" id="S4.SS2.SSS2.Px2.1.p1.12.m7.1.1.1.cmml" xref="S4.SS2.SSS2.Px2.1.p1.12.m7.1.1">subscript</csymbol><ci id="S4.SS2.SSS2.Px2.1.p1.12.m7.1.1.2.cmml" xref="S4.SS2.SSS2.Px2.1.p1.12.m7.1.1.2">𝐺</ci><ci id="S4.SS2.SSS2.Px2.1.p1.12.m7.1.1.3.cmml" xref="S4.SS2.SSS2.Px2.1.p1.12.m7.1.1.3">𝑥</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS2.Px2.1.p1.12.m7.1c">G_{x}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS2.Px2.1.p1.12.m7.1d">italic_G start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT</annotation></semantics></math> is a simple cycle, <math alttext="|E(G_{x})|=|V(G_{x})|" class="ltx_Math" display="inline" id="S4.SS2.SSS2.Px2.1.p1.13.m8.2"><semantics id="S4.SS2.SSS2.Px2.1.p1.13.m8.2a"><mrow id="S4.SS2.SSS2.Px2.1.p1.13.m8.2.2" xref="S4.SS2.SSS2.Px2.1.p1.13.m8.2.2.cmml"><mrow id="S4.SS2.SSS2.Px2.1.p1.13.m8.1.1.1.1" xref="S4.SS2.SSS2.Px2.1.p1.13.m8.1.1.1.2.cmml"><mo id="S4.SS2.SSS2.Px2.1.p1.13.m8.1.1.1.1.2" stretchy="false" xref="S4.SS2.SSS2.Px2.1.p1.13.m8.1.1.1.2.1.cmml">|</mo><mrow id="S4.SS2.SSS2.Px2.1.p1.13.m8.1.1.1.1.1" xref="S4.SS2.SSS2.Px2.1.p1.13.m8.1.1.1.1.1.cmml"><mi id="S4.SS2.SSS2.Px2.1.p1.13.m8.1.1.1.1.1.3" xref="S4.SS2.SSS2.Px2.1.p1.13.m8.1.1.1.1.1.3.cmml">E</mi><mo id="S4.SS2.SSS2.Px2.1.p1.13.m8.1.1.1.1.1.2" xref="S4.SS2.SSS2.Px2.1.p1.13.m8.1.1.1.1.1.2.cmml">⁢</mo><mrow id="S4.SS2.SSS2.Px2.1.p1.13.m8.1.1.1.1.1.1.1" xref="S4.SS2.SSS2.Px2.1.p1.13.m8.1.1.1.1.1.1.1.1.cmml"><mo id="S4.SS2.SSS2.Px2.1.p1.13.m8.1.1.1.1.1.1.1.2" stretchy="false" xref="S4.SS2.SSS2.Px2.1.p1.13.m8.1.1.1.1.1.1.1.1.cmml">(</mo><msub id="S4.SS2.SSS2.Px2.1.p1.13.m8.1.1.1.1.1.1.1.1" xref="S4.SS2.SSS2.Px2.1.p1.13.m8.1.1.1.1.1.1.1.1.cmml"><mi id="S4.SS2.SSS2.Px2.1.p1.13.m8.1.1.1.1.1.1.1.1.2" xref="S4.SS2.SSS2.Px2.1.p1.13.m8.1.1.1.1.1.1.1.1.2.cmml">G</mi><mi id="S4.SS2.SSS2.Px2.1.p1.13.m8.1.1.1.1.1.1.1.1.3" xref="S4.SS2.SSS2.Px2.1.p1.13.m8.1.1.1.1.1.1.1.1.3.cmml">x</mi></msub><mo id="S4.SS2.SSS2.Px2.1.p1.13.m8.1.1.1.1.1.1.1.3" stretchy="false" xref="S4.SS2.SSS2.Px2.1.p1.13.m8.1.1.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S4.SS2.SSS2.Px2.1.p1.13.m8.1.1.1.1.3" stretchy="false" xref="S4.SS2.SSS2.Px2.1.p1.13.m8.1.1.1.2.1.cmml">|</mo></mrow><mo id="S4.SS2.SSS2.Px2.1.p1.13.m8.2.2.3" xref="S4.SS2.SSS2.Px2.1.p1.13.m8.2.2.3.cmml">=</mo><mrow id="S4.SS2.SSS2.Px2.1.p1.13.m8.2.2.2.1" xref="S4.SS2.SSS2.Px2.1.p1.13.m8.2.2.2.2.cmml"><mo id="S4.SS2.SSS2.Px2.1.p1.13.m8.2.2.2.1.2" stretchy="false" xref="S4.SS2.SSS2.Px2.1.p1.13.m8.2.2.2.2.1.cmml">|</mo><mrow id="S4.SS2.SSS2.Px2.1.p1.13.m8.2.2.2.1.1" xref="S4.SS2.SSS2.Px2.1.p1.13.m8.2.2.2.1.1.cmml"><mi id="S4.SS2.SSS2.Px2.1.p1.13.m8.2.2.2.1.1.3" xref="S4.SS2.SSS2.Px2.1.p1.13.m8.2.2.2.1.1.3.cmml">V</mi><mo id="S4.SS2.SSS2.Px2.1.p1.13.m8.2.2.2.1.1.2" xref="S4.SS2.SSS2.Px2.1.p1.13.m8.2.2.2.1.1.2.cmml">⁢</mo><mrow id="S4.SS2.SSS2.Px2.1.p1.13.m8.2.2.2.1.1.1.1" xref="S4.SS2.SSS2.Px2.1.p1.13.m8.2.2.2.1.1.1.1.1.cmml"><mo id="S4.SS2.SSS2.Px2.1.p1.13.m8.2.2.2.1.1.1.1.2" stretchy="false" xref="S4.SS2.SSS2.Px2.1.p1.13.m8.2.2.2.1.1.1.1.1.cmml">(</mo><msub id="S4.SS2.SSS2.Px2.1.p1.13.m8.2.2.2.1.1.1.1.1" xref="S4.SS2.SSS2.Px2.1.p1.13.m8.2.2.2.1.1.1.1.1.cmml"><mi id="S4.SS2.SSS2.Px2.1.p1.13.m8.2.2.2.1.1.1.1.1.2" xref="S4.SS2.SSS2.Px2.1.p1.13.m8.2.2.2.1.1.1.1.1.2.cmml">G</mi><mi id="S4.SS2.SSS2.Px2.1.p1.13.m8.2.2.2.1.1.1.1.1.3" xref="S4.SS2.SSS2.Px2.1.p1.13.m8.2.2.2.1.1.1.1.1.3.cmml">x</mi></msub><mo id="S4.SS2.SSS2.Px2.1.p1.13.m8.2.2.2.1.1.1.1.3" stretchy="false" xref="S4.SS2.SSS2.Px2.1.p1.13.m8.2.2.2.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S4.SS2.SSS2.Px2.1.p1.13.m8.2.2.2.1.3" stretchy="false" xref="S4.SS2.SSS2.Px2.1.p1.13.m8.2.2.2.2.1.cmml">|</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS2.Px2.1.p1.13.m8.2b"><apply id="S4.SS2.SSS2.Px2.1.p1.13.m8.2.2.cmml" xref="S4.SS2.SSS2.Px2.1.p1.13.m8.2.2"><eq id="S4.SS2.SSS2.Px2.1.p1.13.m8.2.2.3.cmml" xref="S4.SS2.SSS2.Px2.1.p1.13.m8.2.2.3"></eq><apply id="S4.SS2.SSS2.Px2.1.p1.13.m8.1.1.1.2.cmml" xref="S4.SS2.SSS2.Px2.1.p1.13.m8.1.1.1.1"><abs id="S4.SS2.SSS2.Px2.1.p1.13.m8.1.1.1.2.1.cmml" xref="S4.SS2.SSS2.Px2.1.p1.13.m8.1.1.1.1.2"></abs><apply id="S4.SS2.SSS2.Px2.1.p1.13.m8.1.1.1.1.1.cmml" xref="S4.SS2.SSS2.Px2.1.p1.13.m8.1.1.1.1.1"><times id="S4.SS2.SSS2.Px2.1.p1.13.m8.1.1.1.1.1.2.cmml" xref="S4.SS2.SSS2.Px2.1.p1.13.m8.1.1.1.1.1.2"></times><ci id="S4.SS2.SSS2.Px2.1.p1.13.m8.1.1.1.1.1.3.cmml" xref="S4.SS2.SSS2.Px2.1.p1.13.m8.1.1.1.1.1.3">𝐸</ci><apply id="S4.SS2.SSS2.Px2.1.p1.13.m8.1.1.1.1.1.1.1.1.cmml" xref="S4.SS2.SSS2.Px2.1.p1.13.m8.1.1.1.1.1.1.1"><csymbol cd="ambiguous" id="S4.SS2.SSS2.Px2.1.p1.13.m8.1.1.1.1.1.1.1.1.1.cmml" xref="S4.SS2.SSS2.Px2.1.p1.13.m8.1.1.1.1.1.1.1">subscript</csymbol><ci id="S4.SS2.SSS2.Px2.1.p1.13.m8.1.1.1.1.1.1.1.1.2.cmml" xref="S4.SS2.SSS2.Px2.1.p1.13.m8.1.1.1.1.1.1.1.1.2">𝐺</ci><ci id="S4.SS2.SSS2.Px2.1.p1.13.m8.1.1.1.1.1.1.1.1.3.cmml" xref="S4.SS2.SSS2.Px2.1.p1.13.m8.1.1.1.1.1.1.1.1.3">𝑥</ci></apply></apply></apply><apply id="S4.SS2.SSS2.Px2.1.p1.13.m8.2.2.2.2.cmml" xref="S4.SS2.SSS2.Px2.1.p1.13.m8.2.2.2.1"><abs id="S4.SS2.SSS2.Px2.1.p1.13.m8.2.2.2.2.1.cmml" xref="S4.SS2.SSS2.Px2.1.p1.13.m8.2.2.2.1.2"></abs><apply id="S4.SS2.SSS2.Px2.1.p1.13.m8.2.2.2.1.1.cmml" xref="S4.SS2.SSS2.Px2.1.p1.13.m8.2.2.2.1.1"><times id="S4.SS2.SSS2.Px2.1.p1.13.m8.2.2.2.1.1.2.cmml" xref="S4.SS2.SSS2.Px2.1.p1.13.m8.2.2.2.1.1.2"></times><ci id="S4.SS2.SSS2.Px2.1.p1.13.m8.2.2.2.1.1.3.cmml" xref="S4.SS2.SSS2.Px2.1.p1.13.m8.2.2.2.1.1.3">𝑉</ci><apply id="S4.SS2.SSS2.Px2.1.p1.13.m8.2.2.2.1.1.1.1.1.cmml" xref="S4.SS2.SSS2.Px2.1.p1.13.m8.2.2.2.1.1.1.1"><csymbol cd="ambiguous" id="S4.SS2.SSS2.Px2.1.p1.13.m8.2.2.2.1.1.1.1.1.1.cmml" xref="S4.SS2.SSS2.Px2.1.p1.13.m8.2.2.2.1.1.1.1">subscript</csymbol><ci id="S4.SS2.SSS2.Px2.1.p1.13.m8.2.2.2.1.1.1.1.1.2.cmml" xref="S4.SS2.SSS2.Px2.1.p1.13.m8.2.2.2.1.1.1.1.1.2">𝐺</ci><ci id="S4.SS2.SSS2.Px2.1.p1.13.m8.2.2.2.1.1.1.1.1.3.cmml" xref="S4.SS2.SSS2.Px2.1.p1.13.m8.2.2.2.1.1.1.1.1.3">𝑥</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS2.Px2.1.p1.13.m8.2c">|E(G_{x})|=|V(G_{x})|</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS2.Px2.1.p1.13.m8.2d">| italic_E ( italic_G start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT ) | = | italic_V ( italic_G start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT ) |</annotation></semantics></math>; thus the number of links stored is <math alttext="4B|E(G_{x})|" class="ltx_Math" display="inline" id="S4.SS2.SSS2.Px2.1.p1.14.m9.1"><semantics id="S4.SS2.SSS2.Px2.1.p1.14.m9.1a"><mrow id="S4.SS2.SSS2.Px2.1.p1.14.m9.1.1" xref="S4.SS2.SSS2.Px2.1.p1.14.m9.1.1.cmml"><mn id="S4.SS2.SSS2.Px2.1.p1.14.m9.1.1.3" xref="S4.SS2.SSS2.Px2.1.p1.14.m9.1.1.3.cmml">4</mn><mo id="S4.SS2.SSS2.Px2.1.p1.14.m9.1.1.2" xref="S4.SS2.SSS2.Px2.1.p1.14.m9.1.1.2.cmml">⁢</mo><mi id="S4.SS2.SSS2.Px2.1.p1.14.m9.1.1.4" xref="S4.SS2.SSS2.Px2.1.p1.14.m9.1.1.4.cmml">B</mi><mo id="S4.SS2.SSS2.Px2.1.p1.14.m9.1.1.2a" xref="S4.SS2.SSS2.Px2.1.p1.14.m9.1.1.2.cmml">⁢</mo><mrow id="S4.SS2.SSS2.Px2.1.p1.14.m9.1.1.1.1" xref="S4.SS2.SSS2.Px2.1.p1.14.m9.1.1.1.2.cmml"><mo id="S4.SS2.SSS2.Px2.1.p1.14.m9.1.1.1.1.2" stretchy="false" xref="S4.SS2.SSS2.Px2.1.p1.14.m9.1.1.1.2.1.cmml">|</mo><mrow id="S4.SS2.SSS2.Px2.1.p1.14.m9.1.1.1.1.1" xref="S4.SS2.SSS2.Px2.1.p1.14.m9.1.1.1.1.1.cmml"><mi id="S4.SS2.SSS2.Px2.1.p1.14.m9.1.1.1.1.1.3" xref="S4.SS2.SSS2.Px2.1.p1.14.m9.1.1.1.1.1.3.cmml">E</mi><mo id="S4.SS2.SSS2.Px2.1.p1.14.m9.1.1.1.1.1.2" xref="S4.SS2.SSS2.Px2.1.p1.14.m9.1.1.1.1.1.2.cmml">⁢</mo><mrow id="S4.SS2.SSS2.Px2.1.p1.14.m9.1.1.1.1.1.1.1" xref="S4.SS2.SSS2.Px2.1.p1.14.m9.1.1.1.1.1.1.1.1.cmml"><mo id="S4.SS2.SSS2.Px2.1.p1.14.m9.1.1.1.1.1.1.1.2" stretchy="false" xref="S4.SS2.SSS2.Px2.1.p1.14.m9.1.1.1.1.1.1.1.1.cmml">(</mo><msub id="S4.SS2.SSS2.Px2.1.p1.14.m9.1.1.1.1.1.1.1.1" xref="S4.SS2.SSS2.Px2.1.p1.14.m9.1.1.1.1.1.1.1.1.cmml"><mi id="S4.SS2.SSS2.Px2.1.p1.14.m9.1.1.1.1.1.1.1.1.2" xref="S4.SS2.SSS2.Px2.1.p1.14.m9.1.1.1.1.1.1.1.1.2.cmml">G</mi><mi id="S4.SS2.SSS2.Px2.1.p1.14.m9.1.1.1.1.1.1.1.1.3" xref="S4.SS2.SSS2.Px2.1.p1.14.m9.1.1.1.1.1.1.1.1.3.cmml">x</mi></msub><mo id="S4.SS2.SSS2.Px2.1.p1.14.m9.1.1.1.1.1.1.1.3" stretchy="false" xref="S4.SS2.SSS2.Px2.1.p1.14.m9.1.1.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S4.SS2.SSS2.Px2.1.p1.14.m9.1.1.1.1.3" stretchy="false" xref="S4.SS2.SSS2.Px2.1.p1.14.m9.1.1.1.2.1.cmml">|</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS2.Px2.1.p1.14.m9.1b"><apply id="S4.SS2.SSS2.Px2.1.p1.14.m9.1.1.cmml" xref="S4.SS2.SSS2.Px2.1.p1.14.m9.1.1"><times id="S4.SS2.SSS2.Px2.1.p1.14.m9.1.1.2.cmml" xref="S4.SS2.SSS2.Px2.1.p1.14.m9.1.1.2"></times><cn id="S4.SS2.SSS2.Px2.1.p1.14.m9.1.1.3.cmml" type="integer" xref="S4.SS2.SSS2.Px2.1.p1.14.m9.1.1.3">4</cn><ci id="S4.SS2.SSS2.Px2.1.p1.14.m9.1.1.4.cmml" xref="S4.SS2.SSS2.Px2.1.p1.14.m9.1.1.4">𝐵</ci><apply id="S4.SS2.SSS2.Px2.1.p1.14.m9.1.1.1.2.cmml" xref="S4.SS2.SSS2.Px2.1.p1.14.m9.1.1.1.1"><abs id="S4.SS2.SSS2.Px2.1.p1.14.m9.1.1.1.2.1.cmml" xref="S4.SS2.SSS2.Px2.1.p1.14.m9.1.1.1.1.2"></abs><apply id="S4.SS2.SSS2.Px2.1.p1.14.m9.1.1.1.1.1.cmml" xref="S4.SS2.SSS2.Px2.1.p1.14.m9.1.1.1.1.1"><times id="S4.SS2.SSS2.Px2.1.p1.14.m9.1.1.1.1.1.2.cmml" xref="S4.SS2.SSS2.Px2.1.p1.14.m9.1.1.1.1.1.2"></times><ci id="S4.SS2.SSS2.Px2.1.p1.14.m9.1.1.1.1.1.3.cmml" xref="S4.SS2.SSS2.Px2.1.p1.14.m9.1.1.1.1.1.3">𝐸</ci><apply id="S4.SS2.SSS2.Px2.1.p1.14.m9.1.1.1.1.1.1.1.1.cmml" xref="S4.SS2.SSS2.Px2.1.p1.14.m9.1.1.1.1.1.1.1"><csymbol cd="ambiguous" id="S4.SS2.SSS2.Px2.1.p1.14.m9.1.1.1.1.1.1.1.1.1.cmml" xref="S4.SS2.SSS2.Px2.1.p1.14.m9.1.1.1.1.1.1.1">subscript</csymbol><ci id="S4.SS2.SSS2.Px2.1.p1.14.m9.1.1.1.1.1.1.1.1.2.cmml" xref="S4.SS2.SSS2.Px2.1.p1.14.m9.1.1.1.1.1.1.1.1.2">𝐺</ci><ci id="S4.SS2.SSS2.Px2.1.p1.14.m9.1.1.1.1.1.1.1.1.3.cmml" xref="S4.SS2.SSS2.Px2.1.p1.14.m9.1.1.1.1.1.1.1.1.3">𝑥</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS2.Px2.1.p1.14.m9.1c">4B|E(G_{x})|</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS2.Px2.1.p1.14.m9.1d">4 italic_B | italic_E ( italic_G start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT ) |</annotation></semantics></math>. Combining,</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="Sx1.EGx6"> <tbody id="S4.Ex13"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle|F|\leq B\left[|V(T)|+2|E(T)|+\sum_{x\in V(T)}4|E(G_{x})|\right]% \leq B\left[7\sum_{x\in V(T)}|E(G_{x})|\right]." class="ltx_Math" display="inline" id="S4.Ex13.m1.6"><semantics id="S4.Ex13.m1.6a"><mrow id="S4.Ex13.m1.6.6.1" xref="S4.Ex13.m1.6.6.1.1.cmml"><mrow id="S4.Ex13.m1.6.6.1.1" xref="S4.Ex13.m1.6.6.1.1.cmml"><mrow id="S4.Ex13.m1.6.6.1.1.4.2" xref="S4.Ex13.m1.6.6.1.1.4.1.cmml"><mo id="S4.Ex13.m1.6.6.1.1.4.2.1" stretchy="false" xref="S4.Ex13.m1.6.6.1.1.4.1.1.cmml">|</mo><mi id="S4.Ex13.m1.3.3" xref="S4.Ex13.m1.3.3.cmml">F</mi><mo id="S4.Ex13.m1.6.6.1.1.4.2.2" stretchy="false" xref="S4.Ex13.m1.6.6.1.1.4.1.1.cmml">|</mo></mrow><mo id="S4.Ex13.m1.6.6.1.1.5" xref="S4.Ex13.m1.6.6.1.1.5.cmml">≤</mo><mrow id="S4.Ex13.m1.6.6.1.1.1" xref="S4.Ex13.m1.6.6.1.1.1.cmml"><mi id="S4.Ex13.m1.6.6.1.1.1.3" xref="S4.Ex13.m1.6.6.1.1.1.3.cmml">B</mi><mo id="S4.Ex13.m1.6.6.1.1.1.2" xref="S4.Ex13.m1.6.6.1.1.1.2.cmml">⁢</mo><mrow id="S4.Ex13.m1.6.6.1.1.1.1.1" xref="S4.Ex13.m1.6.6.1.1.1.1.2.cmml"><mo id="S4.Ex13.m1.6.6.1.1.1.1.1.2" xref="S4.Ex13.m1.6.6.1.1.1.1.2.1.cmml">[</mo><mrow id="S4.Ex13.m1.6.6.1.1.1.1.1.1" xref="S4.Ex13.m1.6.6.1.1.1.1.1.1.cmml"><mrow id="S4.Ex13.m1.6.6.1.1.1.1.1.1.1.1" xref="S4.Ex13.m1.6.6.1.1.1.1.1.1.1.2.cmml"><mo id="S4.Ex13.m1.6.6.1.1.1.1.1.1.1.1.2" stretchy="false" xref="S4.Ex13.m1.6.6.1.1.1.1.1.1.1.2.1.cmml">|</mo><mrow id="S4.Ex13.m1.6.6.1.1.1.1.1.1.1.1.1" xref="S4.Ex13.m1.6.6.1.1.1.1.1.1.1.1.1.cmml"><mi id="S4.Ex13.m1.6.6.1.1.1.1.1.1.1.1.1.2" xref="S4.Ex13.m1.6.6.1.1.1.1.1.1.1.1.1.2.cmml">V</mi><mo id="S4.Ex13.m1.6.6.1.1.1.1.1.1.1.1.1.1" xref="S4.Ex13.m1.6.6.1.1.1.1.1.1.1.1.1.1.cmml">⁢</mo><mrow id="S4.Ex13.m1.6.6.1.1.1.1.1.1.1.1.1.3.2" xref="S4.Ex13.m1.6.6.1.1.1.1.1.1.1.1.1.cmml"><mo id="S4.Ex13.m1.6.6.1.1.1.1.1.1.1.1.1.3.2.1" stretchy="false" xref="S4.Ex13.m1.6.6.1.1.1.1.1.1.1.1.1.cmml">(</mo><mi id="S4.Ex13.m1.4.4" xref="S4.Ex13.m1.4.4.cmml">T</mi><mo id="S4.Ex13.m1.6.6.1.1.1.1.1.1.1.1.1.3.2.2" stretchy="false" xref="S4.Ex13.m1.6.6.1.1.1.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S4.Ex13.m1.6.6.1.1.1.1.1.1.1.1.3" stretchy="false" xref="S4.Ex13.m1.6.6.1.1.1.1.1.1.1.2.1.cmml">|</mo></mrow><mo id="S4.Ex13.m1.6.6.1.1.1.1.1.1.4" xref="S4.Ex13.m1.6.6.1.1.1.1.1.1.4.cmml">+</mo><mrow id="S4.Ex13.m1.6.6.1.1.1.1.1.1.2" xref="S4.Ex13.m1.6.6.1.1.1.1.1.1.2.cmml"><mn id="S4.Ex13.m1.6.6.1.1.1.1.1.1.2.3" xref="S4.Ex13.m1.6.6.1.1.1.1.1.1.2.3.cmml">2</mn><mo id="S4.Ex13.m1.6.6.1.1.1.1.1.1.2.2" xref="S4.Ex13.m1.6.6.1.1.1.1.1.1.2.2.cmml">⁢</mo><mrow id="S4.Ex13.m1.6.6.1.1.1.1.1.1.2.1.1" xref="S4.Ex13.m1.6.6.1.1.1.1.1.1.2.1.2.cmml"><mo id="S4.Ex13.m1.6.6.1.1.1.1.1.1.2.1.1.2" stretchy="false" xref="S4.Ex13.m1.6.6.1.1.1.1.1.1.2.1.2.1.cmml">|</mo><mrow id="S4.Ex13.m1.6.6.1.1.1.1.1.1.2.1.1.1" xref="S4.Ex13.m1.6.6.1.1.1.1.1.1.2.1.1.1.cmml"><mi id="S4.Ex13.m1.6.6.1.1.1.1.1.1.2.1.1.1.2" xref="S4.Ex13.m1.6.6.1.1.1.1.1.1.2.1.1.1.2.cmml">E</mi><mo id="S4.Ex13.m1.6.6.1.1.1.1.1.1.2.1.1.1.1" xref="S4.Ex13.m1.6.6.1.1.1.1.1.1.2.1.1.1.1.cmml">⁢</mo><mrow id="S4.Ex13.m1.6.6.1.1.1.1.1.1.2.1.1.1.3.2" xref="S4.Ex13.m1.6.6.1.1.1.1.1.1.2.1.1.1.cmml"><mo id="S4.Ex13.m1.6.6.1.1.1.1.1.1.2.1.1.1.3.2.1" stretchy="false" xref="S4.Ex13.m1.6.6.1.1.1.1.1.1.2.1.1.1.cmml">(</mo><mi id="S4.Ex13.m1.5.5" xref="S4.Ex13.m1.5.5.cmml">T</mi><mo id="S4.Ex13.m1.6.6.1.1.1.1.1.1.2.1.1.1.3.2.2" stretchy="false" xref="S4.Ex13.m1.6.6.1.1.1.1.1.1.2.1.1.1.cmml">)</mo></mrow></mrow><mo id="S4.Ex13.m1.6.6.1.1.1.1.1.1.2.1.1.3" stretchy="false" xref="S4.Ex13.m1.6.6.1.1.1.1.1.1.2.1.2.1.cmml">|</mo></mrow></mrow><mo id="S4.Ex13.m1.6.6.1.1.1.1.1.1.4a" xref="S4.Ex13.m1.6.6.1.1.1.1.1.1.4.cmml">+</mo><mrow id="S4.Ex13.m1.6.6.1.1.1.1.1.1.3" xref="S4.Ex13.m1.6.6.1.1.1.1.1.1.3.cmml"><mstyle displaystyle="true" id="S4.Ex13.m1.6.6.1.1.1.1.1.1.3.2" xref="S4.Ex13.m1.6.6.1.1.1.1.1.1.3.2.cmml"><munder id="S4.Ex13.m1.6.6.1.1.1.1.1.1.3.2a" xref="S4.Ex13.m1.6.6.1.1.1.1.1.1.3.2.cmml"><mo id="S4.Ex13.m1.6.6.1.1.1.1.1.1.3.2.2" movablelimits="false" xref="S4.Ex13.m1.6.6.1.1.1.1.1.1.3.2.2.cmml">∑</mo><mrow id="S4.Ex13.m1.1.1.1" xref="S4.Ex13.m1.1.1.1.cmml"><mi id="S4.Ex13.m1.1.1.1.3" xref="S4.Ex13.m1.1.1.1.3.cmml">x</mi><mo id="S4.Ex13.m1.1.1.1.2" xref="S4.Ex13.m1.1.1.1.2.cmml">∈</mo><mrow id="S4.Ex13.m1.1.1.1.4" xref="S4.Ex13.m1.1.1.1.4.cmml"><mi id="S4.Ex13.m1.1.1.1.4.2" xref="S4.Ex13.m1.1.1.1.4.2.cmml">V</mi><mo id="S4.Ex13.m1.1.1.1.4.1" xref="S4.Ex13.m1.1.1.1.4.1.cmml">⁢</mo><mrow id="S4.Ex13.m1.1.1.1.4.3.2" xref="S4.Ex13.m1.1.1.1.4.cmml"><mo id="S4.Ex13.m1.1.1.1.4.3.2.1" stretchy="false" xref="S4.Ex13.m1.1.1.1.4.cmml">(</mo><mi id="S4.Ex13.m1.1.1.1.1" xref="S4.Ex13.m1.1.1.1.1.cmml">T</mi><mo id="S4.Ex13.m1.1.1.1.4.3.2.2" stretchy="false" xref="S4.Ex13.m1.1.1.1.4.cmml">)</mo></mrow></mrow></mrow></munder></mstyle><mrow id="S4.Ex13.m1.6.6.1.1.1.1.1.1.3.1" xref="S4.Ex13.m1.6.6.1.1.1.1.1.1.3.1.cmml"><mn id="S4.Ex13.m1.6.6.1.1.1.1.1.1.3.1.3" xref="S4.Ex13.m1.6.6.1.1.1.1.1.1.3.1.3.cmml">4</mn><mo id="S4.Ex13.m1.6.6.1.1.1.1.1.1.3.1.2" xref="S4.Ex13.m1.6.6.1.1.1.1.1.1.3.1.2.cmml">⁢</mo><mrow id="S4.Ex13.m1.6.6.1.1.1.1.1.1.3.1.1.1" xref="S4.Ex13.m1.6.6.1.1.1.1.1.1.3.1.1.2.cmml"><mo id="S4.Ex13.m1.6.6.1.1.1.1.1.1.3.1.1.1.2" stretchy="false" xref="S4.Ex13.m1.6.6.1.1.1.1.1.1.3.1.1.2.1.cmml">|</mo><mrow id="S4.Ex13.m1.6.6.1.1.1.1.1.1.3.1.1.1.1" xref="S4.Ex13.m1.6.6.1.1.1.1.1.1.3.1.1.1.1.cmml"><mi id="S4.Ex13.m1.6.6.1.1.1.1.1.1.3.1.1.1.1.3" xref="S4.Ex13.m1.6.6.1.1.1.1.1.1.3.1.1.1.1.3.cmml">E</mi><mo id="S4.Ex13.m1.6.6.1.1.1.1.1.1.3.1.1.1.1.2" xref="S4.Ex13.m1.6.6.1.1.1.1.1.1.3.1.1.1.1.2.cmml">⁢</mo><mrow id="S4.Ex13.m1.6.6.1.1.1.1.1.1.3.1.1.1.1.1.1" xref="S4.Ex13.m1.6.6.1.1.1.1.1.1.3.1.1.1.1.1.1.1.cmml"><mo id="S4.Ex13.m1.6.6.1.1.1.1.1.1.3.1.1.1.1.1.1.2" stretchy="false" xref="S4.Ex13.m1.6.6.1.1.1.1.1.1.3.1.1.1.1.1.1.1.cmml">(</mo><msub id="S4.Ex13.m1.6.6.1.1.1.1.1.1.3.1.1.1.1.1.1.1" xref="S4.Ex13.m1.6.6.1.1.1.1.1.1.3.1.1.1.1.1.1.1.cmml"><mi id="S4.Ex13.m1.6.6.1.1.1.1.1.1.3.1.1.1.1.1.1.1.2" xref="S4.Ex13.m1.6.6.1.1.1.1.1.1.3.1.1.1.1.1.1.1.2.cmml">G</mi><mi id="S4.Ex13.m1.6.6.1.1.1.1.1.1.3.1.1.1.1.1.1.1.3" xref="S4.Ex13.m1.6.6.1.1.1.1.1.1.3.1.1.1.1.1.1.1.3.cmml">x</mi></msub><mo id="S4.Ex13.m1.6.6.1.1.1.1.1.1.3.1.1.1.1.1.1.3" stretchy="false" xref="S4.Ex13.m1.6.6.1.1.1.1.1.1.3.1.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S4.Ex13.m1.6.6.1.1.1.1.1.1.3.1.1.1.3" stretchy="false" xref="S4.Ex13.m1.6.6.1.1.1.1.1.1.3.1.1.2.1.cmml">|</mo></mrow></mrow></mrow></mrow><mo id="S4.Ex13.m1.6.6.1.1.1.1.1.3" xref="S4.Ex13.m1.6.6.1.1.1.1.2.1.cmml">]</mo></mrow></mrow><mo id="S4.Ex13.m1.6.6.1.1.6" xref="S4.Ex13.m1.6.6.1.1.6.cmml">≤</mo><mrow id="S4.Ex13.m1.6.6.1.1.2" xref="S4.Ex13.m1.6.6.1.1.2.cmml"><mi id="S4.Ex13.m1.6.6.1.1.2.3" xref="S4.Ex13.m1.6.6.1.1.2.3.cmml">B</mi><mo id="S4.Ex13.m1.6.6.1.1.2.2" xref="S4.Ex13.m1.6.6.1.1.2.2.cmml">⁢</mo><mrow id="S4.Ex13.m1.6.6.1.1.2.1.1" xref="S4.Ex13.m1.6.6.1.1.2.1.2.cmml"><mo id="S4.Ex13.m1.6.6.1.1.2.1.1.2" xref="S4.Ex13.m1.6.6.1.1.2.1.2.1.cmml">[</mo><mrow id="S4.Ex13.m1.6.6.1.1.2.1.1.1" xref="S4.Ex13.m1.6.6.1.1.2.1.1.1.cmml"><mn id="S4.Ex13.m1.6.6.1.1.2.1.1.1.3" xref="S4.Ex13.m1.6.6.1.1.2.1.1.1.3.cmml">7</mn><mo id="S4.Ex13.m1.6.6.1.1.2.1.1.1.2" xref="S4.Ex13.m1.6.6.1.1.2.1.1.1.2.cmml">⁢</mo><mrow id="S4.Ex13.m1.6.6.1.1.2.1.1.1.1" xref="S4.Ex13.m1.6.6.1.1.2.1.1.1.1.cmml"><mstyle displaystyle="true" id="S4.Ex13.m1.6.6.1.1.2.1.1.1.1.2" xref="S4.Ex13.m1.6.6.1.1.2.1.1.1.1.2.cmml"><munder id="S4.Ex13.m1.6.6.1.1.2.1.1.1.1.2a" xref="S4.Ex13.m1.6.6.1.1.2.1.1.1.1.2.cmml"><mo id="S4.Ex13.m1.6.6.1.1.2.1.1.1.1.2.2" movablelimits="false" 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xref="S4.Ex13.m1.6.6.1.1.2.1.1.1.1.1.1.1.1.1.1.3">𝑥</ci></apply></apply></apply></apply></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Ex13.m1.6c">\displaystyle|F|\leq B\left[|V(T)|+2|E(T)|+\sum_{x\in V(T)}4|E(G_{x})|\right]% \leq B\left[7\sum_{x\in V(T)}|E(G_{x})|\right].</annotation><annotation encoding="application/x-llamapun" id="S4.Ex13.m1.6d">| italic_F | ≤ italic_B [ | italic_V ( italic_T ) | + 2 | italic_E ( italic_T ) | + ∑ start_POSTSUBSCRIPT italic_x ∈ italic_V ( italic_T ) end_POSTSUBSCRIPT 4 | italic_E ( italic_G start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT ) | ] ≤ italic_B [ 7 ∑ start_POSTSUBSCRIPT italic_x ∈ italic_V ( italic_T ) end_POSTSUBSCRIPT | italic_E ( italic_G start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT ) | ] .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S4.SS2.SSS2.Px2.1.p1.16">By Lemma <a class="ltx_ref" href="https://arxiv.org/html/2503.00712v1#S4.Thmtheorem13" title="Lemma 4.13. ‣ 4.2.1 SPQR Trees ‣ 4.2 Two-to-Three Augmentation ‣ 4 Vertex Connectivity Augmentation in Link-Arrival Model ‣ Streaming Algorithms for Network Design"><span class="ltx_text ltx_ref_tag">4.13</span></a>, this is at most <math alttext="O(B\cdot|E|)" class="ltx_Math" display="inline" id="S4.SS2.SSS2.Px2.1.p1.15.m1.2"><semantics id="S4.SS2.SSS2.Px2.1.p1.15.m1.2a"><mrow id="S4.SS2.SSS2.Px2.1.p1.15.m1.2.2" xref="S4.SS2.SSS2.Px2.1.p1.15.m1.2.2.cmml"><mi id="S4.SS2.SSS2.Px2.1.p1.15.m1.2.2.3" xref="S4.SS2.SSS2.Px2.1.p1.15.m1.2.2.3.cmml">O</mi><mo id="S4.SS2.SSS2.Px2.1.p1.15.m1.2.2.2" xref="S4.SS2.SSS2.Px2.1.p1.15.m1.2.2.2.cmml">⁢</mo><mrow id="S4.SS2.SSS2.Px2.1.p1.15.m1.2.2.1.1" xref="S4.SS2.SSS2.Px2.1.p1.15.m1.2.2.1.1.1.cmml"><mo id="S4.SS2.SSS2.Px2.1.p1.15.m1.2.2.1.1.2" stretchy="false" xref="S4.SS2.SSS2.Px2.1.p1.15.m1.2.2.1.1.1.cmml">(</mo><mrow id="S4.SS2.SSS2.Px2.1.p1.15.m1.2.2.1.1.1" xref="S4.SS2.SSS2.Px2.1.p1.15.m1.2.2.1.1.1.cmml"><mi id="S4.SS2.SSS2.Px2.1.p1.15.m1.2.2.1.1.1.2" xref="S4.SS2.SSS2.Px2.1.p1.15.m1.2.2.1.1.1.2.cmml">B</mi><mo id="S4.SS2.SSS2.Px2.1.p1.15.m1.2.2.1.1.1.1" lspace="0.222em" rspace="0.222em" xref="S4.SS2.SSS2.Px2.1.p1.15.m1.2.2.1.1.1.1.cmml">⋅</mo><mrow id="S4.SS2.SSS2.Px2.1.p1.15.m1.2.2.1.1.1.3.2" xref="S4.SS2.SSS2.Px2.1.p1.15.m1.2.2.1.1.1.3.1.cmml"><mo id="S4.SS2.SSS2.Px2.1.p1.15.m1.2.2.1.1.1.3.2.1" stretchy="false" xref="S4.SS2.SSS2.Px2.1.p1.15.m1.2.2.1.1.1.3.1.1.cmml">|</mo><mi id="S4.SS2.SSS2.Px2.1.p1.15.m1.1.1" xref="S4.SS2.SSS2.Px2.1.p1.15.m1.1.1.cmml">E</mi><mo id="S4.SS2.SSS2.Px2.1.p1.15.m1.2.2.1.1.1.3.2.2" stretchy="false" xref="S4.SS2.SSS2.Px2.1.p1.15.m1.2.2.1.1.1.3.1.1.cmml">|</mo></mrow></mrow><mo id="S4.SS2.SSS2.Px2.1.p1.15.m1.2.2.1.1.3" stretchy="false" xref="S4.SS2.SSS2.Px2.1.p1.15.m1.2.2.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS2.Px2.1.p1.15.m1.2b"><apply id="S4.SS2.SSS2.Px2.1.p1.15.m1.2.2.cmml" xref="S4.SS2.SSS2.Px2.1.p1.15.m1.2.2"><times id="S4.SS2.SSS2.Px2.1.p1.15.m1.2.2.2.cmml" xref="S4.SS2.SSS2.Px2.1.p1.15.m1.2.2.2"></times><ci id="S4.SS2.SSS2.Px2.1.p1.15.m1.2.2.3.cmml" xref="S4.SS2.SSS2.Px2.1.p1.15.m1.2.2.3">𝑂</ci><apply id="S4.SS2.SSS2.Px2.1.p1.15.m1.2.2.1.1.1.cmml" xref="S4.SS2.SSS2.Px2.1.p1.15.m1.2.2.1.1"><ci id="S4.SS2.SSS2.Px2.1.p1.15.m1.2.2.1.1.1.1.cmml" xref="S4.SS2.SSS2.Px2.1.p1.15.m1.2.2.1.1.1.1">⋅</ci><ci id="S4.SS2.SSS2.Px2.1.p1.15.m1.2.2.1.1.1.2.cmml" xref="S4.SS2.SSS2.Px2.1.p1.15.m1.2.2.1.1.1.2">𝐵</ci><apply id="S4.SS2.SSS2.Px2.1.p1.15.m1.2.2.1.1.1.3.1.cmml" xref="S4.SS2.SSS2.Px2.1.p1.15.m1.2.2.1.1.1.3.2"><abs id="S4.SS2.SSS2.Px2.1.p1.15.m1.2.2.1.1.1.3.1.1.cmml" xref="S4.SS2.SSS2.Px2.1.p1.15.m1.2.2.1.1.1.3.2.1"></abs><ci id="S4.SS2.SSS2.Px2.1.p1.15.m1.1.1.cmml" xref="S4.SS2.SSS2.Px2.1.p1.15.m1.1.1">𝐸</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS2.Px2.1.p1.15.m1.2c">O(B\cdot|E|)</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS2.Px2.1.p1.15.m1.2d">italic_O ( italic_B ⋅ | italic_E | )</annotation></semantics></math>. Corollary <a class="ltx_ref" href="https://arxiv.org/html/2503.00712v1#S4.Thmtheorem18" title="Lemma 4.18. ‣ “Cycle” Cuts: ‣ 4.2.2 The Streaming Algorithm ‣ 4.2 Two-to-Three Augmentation ‣ 4 Vertex Connectivity Augmentation in Link-Arrival Model ‣ Streaming Algorithms for Network Design"><span class="ltx_text ltx_ref_tag">4.18</span></a> gives us our desired bound of <math alttext="O(n\epsilon^{-1}\log W)" class="ltx_Math" display="inline" id="S4.SS2.SSS2.Px2.1.p1.16.m2.1"><semantics id="S4.SS2.SSS2.Px2.1.p1.16.m2.1a"><mrow id="S4.SS2.SSS2.Px2.1.p1.16.m2.1.1" xref="S4.SS2.SSS2.Px2.1.p1.16.m2.1.1.cmml"><mi id="S4.SS2.SSS2.Px2.1.p1.16.m2.1.1.3" xref="S4.SS2.SSS2.Px2.1.p1.16.m2.1.1.3.cmml">O</mi><mo id="S4.SS2.SSS2.Px2.1.p1.16.m2.1.1.2" xref="S4.SS2.SSS2.Px2.1.p1.16.m2.1.1.2.cmml">⁢</mo><mrow id="S4.SS2.SSS2.Px2.1.p1.16.m2.1.1.1.1" xref="S4.SS2.SSS2.Px2.1.p1.16.m2.1.1.1.1.1.cmml"><mo id="S4.SS2.SSS2.Px2.1.p1.16.m2.1.1.1.1.2" stretchy="false" xref="S4.SS2.SSS2.Px2.1.p1.16.m2.1.1.1.1.1.cmml">(</mo><mrow id="S4.SS2.SSS2.Px2.1.p1.16.m2.1.1.1.1.1" xref="S4.SS2.SSS2.Px2.1.p1.16.m2.1.1.1.1.1.cmml"><mi id="S4.SS2.SSS2.Px2.1.p1.16.m2.1.1.1.1.1.2" xref="S4.SS2.SSS2.Px2.1.p1.16.m2.1.1.1.1.1.2.cmml">n</mi><mo id="S4.SS2.SSS2.Px2.1.p1.16.m2.1.1.1.1.1.1" xref="S4.SS2.SSS2.Px2.1.p1.16.m2.1.1.1.1.1.1.cmml">⁢</mo><msup id="S4.SS2.SSS2.Px2.1.p1.16.m2.1.1.1.1.1.3" xref="S4.SS2.SSS2.Px2.1.p1.16.m2.1.1.1.1.1.3.cmml"><mi id="S4.SS2.SSS2.Px2.1.p1.16.m2.1.1.1.1.1.3.2" xref="S4.SS2.SSS2.Px2.1.p1.16.m2.1.1.1.1.1.3.2.cmml">ϵ</mi><mrow id="S4.SS2.SSS2.Px2.1.p1.16.m2.1.1.1.1.1.3.3" xref="S4.SS2.SSS2.Px2.1.p1.16.m2.1.1.1.1.1.3.3.cmml"><mo id="S4.SS2.SSS2.Px2.1.p1.16.m2.1.1.1.1.1.3.3a" xref="S4.SS2.SSS2.Px2.1.p1.16.m2.1.1.1.1.1.3.3.cmml">−</mo><mn id="S4.SS2.SSS2.Px2.1.p1.16.m2.1.1.1.1.1.3.3.2" xref="S4.SS2.SSS2.Px2.1.p1.16.m2.1.1.1.1.1.3.3.2.cmml">1</mn></mrow></msup><mo id="S4.SS2.SSS2.Px2.1.p1.16.m2.1.1.1.1.1.1a" lspace="0.167em" xref="S4.SS2.SSS2.Px2.1.p1.16.m2.1.1.1.1.1.1.cmml">⁢</mo><mrow id="S4.SS2.SSS2.Px2.1.p1.16.m2.1.1.1.1.1.4" xref="S4.SS2.SSS2.Px2.1.p1.16.m2.1.1.1.1.1.4.cmml"><mi id="S4.SS2.SSS2.Px2.1.p1.16.m2.1.1.1.1.1.4.1" xref="S4.SS2.SSS2.Px2.1.p1.16.m2.1.1.1.1.1.4.1.cmml">log</mi><mo id="S4.SS2.SSS2.Px2.1.p1.16.m2.1.1.1.1.1.4a" lspace="0.167em" xref="S4.SS2.SSS2.Px2.1.p1.16.m2.1.1.1.1.1.4.cmml">⁡</mo><mi id="S4.SS2.SSS2.Px2.1.p1.16.m2.1.1.1.1.1.4.2" xref="S4.SS2.SSS2.Px2.1.p1.16.m2.1.1.1.1.1.4.2.cmml">W</mi></mrow></mrow><mo id="S4.SS2.SSS2.Px2.1.p1.16.m2.1.1.1.1.3" stretchy="false" xref="S4.SS2.SSS2.Px2.1.p1.16.m2.1.1.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS2.Px2.1.p1.16.m2.1b"><apply id="S4.SS2.SSS2.Px2.1.p1.16.m2.1.1.cmml" xref="S4.SS2.SSS2.Px2.1.p1.16.m2.1.1"><times id="S4.SS2.SSS2.Px2.1.p1.16.m2.1.1.2.cmml" xref="S4.SS2.SSS2.Px2.1.p1.16.m2.1.1.2"></times><ci id="S4.SS2.SSS2.Px2.1.p1.16.m2.1.1.3.cmml" xref="S4.SS2.SSS2.Px2.1.p1.16.m2.1.1.3">𝑂</ci><apply id="S4.SS2.SSS2.Px2.1.p1.16.m2.1.1.1.1.1.cmml" xref="S4.SS2.SSS2.Px2.1.p1.16.m2.1.1.1.1"><times id="S4.SS2.SSS2.Px2.1.p1.16.m2.1.1.1.1.1.1.cmml" xref="S4.SS2.SSS2.Px2.1.p1.16.m2.1.1.1.1.1.1"></times><ci id="S4.SS2.SSS2.Px2.1.p1.16.m2.1.1.1.1.1.2.cmml" xref="S4.SS2.SSS2.Px2.1.p1.16.m2.1.1.1.1.1.2">𝑛</ci><apply id="S4.SS2.SSS2.Px2.1.p1.16.m2.1.1.1.1.1.3.cmml" xref="S4.SS2.SSS2.Px2.1.p1.16.m2.1.1.1.1.1.3"><csymbol cd="ambiguous" id="S4.SS2.SSS2.Px2.1.p1.16.m2.1.1.1.1.1.3.1.cmml" xref="S4.SS2.SSS2.Px2.1.p1.16.m2.1.1.1.1.1.3">superscript</csymbol><ci id="S4.SS2.SSS2.Px2.1.p1.16.m2.1.1.1.1.1.3.2.cmml" xref="S4.SS2.SSS2.Px2.1.p1.16.m2.1.1.1.1.1.3.2">italic-ϵ</ci><apply id="S4.SS2.SSS2.Px2.1.p1.16.m2.1.1.1.1.1.3.3.cmml" xref="S4.SS2.SSS2.Px2.1.p1.16.m2.1.1.1.1.1.3.3"><minus id="S4.SS2.SSS2.Px2.1.p1.16.m2.1.1.1.1.1.3.3.1.cmml" xref="S4.SS2.SSS2.Px2.1.p1.16.m2.1.1.1.1.1.3.3"></minus><cn id="S4.SS2.SSS2.Px2.1.p1.16.m2.1.1.1.1.1.3.3.2.cmml" type="integer" xref="S4.SS2.SSS2.Px2.1.p1.16.m2.1.1.1.1.1.3.3.2">1</cn></apply></apply><apply id="S4.SS2.SSS2.Px2.1.p1.16.m2.1.1.1.1.1.4.cmml" xref="S4.SS2.SSS2.Px2.1.p1.16.m2.1.1.1.1.1.4"><log id="S4.SS2.SSS2.Px2.1.p1.16.m2.1.1.1.1.1.4.1.cmml" xref="S4.SS2.SSS2.Px2.1.p1.16.m2.1.1.1.1.1.4.1"></log><ci id="S4.SS2.SSS2.Px2.1.p1.16.m2.1.1.1.1.1.4.2.cmml" xref="S4.SS2.SSS2.Px2.1.p1.16.m2.1.1.1.1.1.4.2">𝑊</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS2.Px2.1.p1.16.m2.1c">O(n\epsilon^{-1}\log W)</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS2.Px2.1.p1.16.m2.1d">italic_O ( italic_n italic_ϵ start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT roman_log italic_W )</annotation></semantics></math>. ∎</p> </div> </div> </section> </section> <section class="ltx_subsubsection" id="S4.SS2.SSS3"> <h4 class="ltx_title ltx_title_subsubsection"> <span class="ltx_tag ltx_tag_subsubsection">4.2.3 </span>Bounding the Approximation Ratio</h4> <div class="ltx_para" id="S4.SS2.SSS3.p1"> <p class="ltx_p" id="S4.SS2.SSS3.p1.1">For the rest of this section, we bound the approximation ratio by proving the following lemma.</p> </div> <div class="ltx_theorem ltx_theorem_lemma" id="S4.Thmtheorem20"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="S4.Thmtheorem20.1.1.1">Lemma 4.20</span></span><span class="ltx_text ltx_font_bold" id="S4.Thmtheorem20.2.2">.</span> </h6> <div class="ltx_para" id="S4.Thmtheorem20.p1"> <p class="ltx_p" id="S4.Thmtheorem20.p1.6">Let <math alttext="F" class="ltx_Math" display="inline" id="S4.Thmtheorem20.p1.1.m1.1"><semantics id="S4.Thmtheorem20.p1.1.m1.1a"><mi id="S4.Thmtheorem20.p1.1.m1.1.1" xref="S4.Thmtheorem20.p1.1.m1.1.1.cmml">F</mi><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem20.p1.1.m1.1b"><ci id="S4.Thmtheorem20.p1.1.m1.1.1.cmml" xref="S4.Thmtheorem20.p1.1.m1.1.1">𝐹</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem20.p1.1.m1.1c">F</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem20.p1.1.m1.1d">italic_F</annotation></semantics></math> be the set of links stored in Algorithm <a class="ltx_ref" href="https://arxiv.org/html/2503.00712v1#algorithm6" title="In “Cycle” Cuts: ‣ 4.2.2 The Streaming Algorithm ‣ 4.2 Two-to-Three Augmentation ‣ 4 Vertex Connectivity Augmentation in Link-Arrival Model ‣ Streaming Algorithms for Network Design"><span class="ltx_text ltx_ref_tag">6</span></a>. The weight of an optimal solution to <math alttext="2" class="ltx_Math" display="inline" id="S4.Thmtheorem20.p1.2.m2.1"><semantics id="S4.Thmtheorem20.p1.2.m2.1a"><mn id="S4.Thmtheorem20.p1.2.m2.1.1" xref="S4.Thmtheorem20.p1.2.m2.1.1.cmml">2</mn><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem20.p1.2.m2.1b"><cn id="S4.Thmtheorem20.p1.2.m2.1.1.cmml" type="integer" xref="S4.Thmtheorem20.p1.2.m2.1.1">2</cn></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem20.p1.2.m2.1c">2</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem20.p1.2.m2.1d">2</annotation></semantics></math>-VC-CAP on <math alttext="(V,F)" class="ltx_Math" display="inline" id="S4.Thmtheorem20.p1.3.m3.2"><semantics id="S4.Thmtheorem20.p1.3.m3.2a"><mrow id="S4.Thmtheorem20.p1.3.m3.2.3.2" xref="S4.Thmtheorem20.p1.3.m3.2.3.1.cmml"><mo id="S4.Thmtheorem20.p1.3.m3.2.3.2.1" stretchy="false" xref="S4.Thmtheorem20.p1.3.m3.2.3.1.cmml">(</mo><mi id="S4.Thmtheorem20.p1.3.m3.1.1" xref="S4.Thmtheorem20.p1.3.m3.1.1.cmml">V</mi><mo id="S4.Thmtheorem20.p1.3.m3.2.3.2.2" xref="S4.Thmtheorem20.p1.3.m3.2.3.1.cmml">,</mo><mi id="S4.Thmtheorem20.p1.3.m3.2.2" xref="S4.Thmtheorem20.p1.3.m3.2.2.cmml">F</mi><mo id="S4.Thmtheorem20.p1.3.m3.2.3.2.3" stretchy="false" xref="S4.Thmtheorem20.p1.3.m3.2.3.1.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem20.p1.3.m3.2b"><interval closure="open" id="S4.Thmtheorem20.p1.3.m3.2.3.1.cmml" xref="S4.Thmtheorem20.p1.3.m3.2.3.2"><ci id="S4.Thmtheorem20.p1.3.m3.1.1.cmml" xref="S4.Thmtheorem20.p1.3.m3.1.1">𝑉</ci><ci id="S4.Thmtheorem20.p1.3.m3.2.2.cmml" xref="S4.Thmtheorem20.p1.3.m3.2.2">𝐹</ci></interval></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem20.p1.3.m3.2c">(V,F)</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem20.p1.3.m3.2d">( italic_V , italic_F )</annotation></semantics></math> is at most <math alttext="(7+\epsilon)" class="ltx_Math" display="inline" id="S4.Thmtheorem20.p1.4.m4.1"><semantics id="S4.Thmtheorem20.p1.4.m4.1a"><mrow id="S4.Thmtheorem20.p1.4.m4.1.1.1" xref="S4.Thmtheorem20.p1.4.m4.1.1.1.1.cmml"><mo id="S4.Thmtheorem20.p1.4.m4.1.1.1.2" stretchy="false" xref="S4.Thmtheorem20.p1.4.m4.1.1.1.1.cmml">(</mo><mrow id="S4.Thmtheorem20.p1.4.m4.1.1.1.1" xref="S4.Thmtheorem20.p1.4.m4.1.1.1.1.cmml"><mn id="S4.Thmtheorem20.p1.4.m4.1.1.1.1.2" xref="S4.Thmtheorem20.p1.4.m4.1.1.1.1.2.cmml">7</mn><mo id="S4.Thmtheorem20.p1.4.m4.1.1.1.1.1" xref="S4.Thmtheorem20.p1.4.m4.1.1.1.1.1.cmml">+</mo><mi id="S4.Thmtheorem20.p1.4.m4.1.1.1.1.3" xref="S4.Thmtheorem20.p1.4.m4.1.1.1.1.3.cmml">ϵ</mi></mrow><mo id="S4.Thmtheorem20.p1.4.m4.1.1.1.3" stretchy="false" xref="S4.Thmtheorem20.p1.4.m4.1.1.1.1.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem20.p1.4.m4.1b"><apply id="S4.Thmtheorem20.p1.4.m4.1.1.1.1.cmml" xref="S4.Thmtheorem20.p1.4.m4.1.1.1"><plus id="S4.Thmtheorem20.p1.4.m4.1.1.1.1.1.cmml" xref="S4.Thmtheorem20.p1.4.m4.1.1.1.1.1"></plus><cn id="S4.Thmtheorem20.p1.4.m4.1.1.1.1.2.cmml" type="integer" xref="S4.Thmtheorem20.p1.4.m4.1.1.1.1.2">7</cn><ci id="S4.Thmtheorem20.p1.4.m4.1.1.1.1.3.cmml" xref="S4.Thmtheorem20.p1.4.m4.1.1.1.1.3">italic-ϵ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem20.p1.4.m4.1c">(7+\epsilon)</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem20.p1.4.m4.1d">( 7 + italic_ϵ )</annotation></semantics></math> times the optimal solution to <math alttext="2" class="ltx_Math" display="inline" id="S4.Thmtheorem20.p1.5.m5.1"><semantics id="S4.Thmtheorem20.p1.5.m5.1a"><mn id="S4.Thmtheorem20.p1.5.m5.1.1" xref="S4.Thmtheorem20.p1.5.m5.1.1.cmml">2</mn><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem20.p1.5.m5.1b"><cn id="S4.Thmtheorem20.p1.5.m5.1.1.cmml" type="integer" xref="S4.Thmtheorem20.p1.5.m5.1.1">2</cn></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem20.p1.5.m5.1c">2</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem20.p1.5.m5.1d">2</annotation></semantics></math>-VC-CAP on <math alttext="G=(V,E)" class="ltx_Math" display="inline" id="S4.Thmtheorem20.p1.6.m6.2"><semantics id="S4.Thmtheorem20.p1.6.m6.2a"><mrow id="S4.Thmtheorem20.p1.6.m6.2.3" xref="S4.Thmtheorem20.p1.6.m6.2.3.cmml"><mi id="S4.Thmtheorem20.p1.6.m6.2.3.2" xref="S4.Thmtheorem20.p1.6.m6.2.3.2.cmml">G</mi><mo id="S4.Thmtheorem20.p1.6.m6.2.3.1" xref="S4.Thmtheorem20.p1.6.m6.2.3.1.cmml">=</mo><mrow id="S4.Thmtheorem20.p1.6.m6.2.3.3.2" xref="S4.Thmtheorem20.p1.6.m6.2.3.3.1.cmml"><mo id="S4.Thmtheorem20.p1.6.m6.2.3.3.2.1" stretchy="false" xref="S4.Thmtheorem20.p1.6.m6.2.3.3.1.cmml">(</mo><mi id="S4.Thmtheorem20.p1.6.m6.1.1" xref="S4.Thmtheorem20.p1.6.m6.1.1.cmml">V</mi><mo id="S4.Thmtheorem20.p1.6.m6.2.3.3.2.2" xref="S4.Thmtheorem20.p1.6.m6.2.3.3.1.cmml">,</mo><mi id="S4.Thmtheorem20.p1.6.m6.2.2" xref="S4.Thmtheorem20.p1.6.m6.2.2.cmml">E</mi><mo id="S4.Thmtheorem20.p1.6.m6.2.3.3.2.3" stretchy="false" xref="S4.Thmtheorem20.p1.6.m6.2.3.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem20.p1.6.m6.2b"><apply id="S4.Thmtheorem20.p1.6.m6.2.3.cmml" xref="S4.Thmtheorem20.p1.6.m6.2.3"><eq id="S4.Thmtheorem20.p1.6.m6.2.3.1.cmml" xref="S4.Thmtheorem20.p1.6.m6.2.3.1"></eq><ci id="S4.Thmtheorem20.p1.6.m6.2.3.2.cmml" xref="S4.Thmtheorem20.p1.6.m6.2.3.2">𝐺</ci><interval closure="open" id="S4.Thmtheorem20.p1.6.m6.2.3.3.1.cmml" xref="S4.Thmtheorem20.p1.6.m6.2.3.3.2"><ci id="S4.Thmtheorem20.p1.6.m6.1.1.cmml" xref="S4.Thmtheorem20.p1.6.m6.1.1">𝑉</ci><ci id="S4.Thmtheorem20.p1.6.m6.2.2.cmml" xref="S4.Thmtheorem20.p1.6.m6.2.2">𝐸</ci></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem20.p1.6.m6.2c">G=(V,E)</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem20.p1.6.m6.2d">italic_G = ( italic_V , italic_E )</annotation></semantics></math>.</p> </div> </div> <div class="ltx_para" id="S4.SS2.SSS3.p2"> <p class="ltx_p" id="S4.SS2.SSS3.p2.16">Fix some optimal solution <span class="ltx_text ltx_markedasmath" id="S4.SS2.SSS3.p2.16.1">OPT</span> on <math alttext="G=(V,E)" class="ltx_Math" display="inline" id="S4.SS2.SSS3.p2.2.m2.2"><semantics id="S4.SS2.SSS3.p2.2.m2.2a"><mrow id="S4.SS2.SSS3.p2.2.m2.2.3" xref="S4.SS2.SSS3.p2.2.m2.2.3.cmml"><mi id="S4.SS2.SSS3.p2.2.m2.2.3.2" xref="S4.SS2.SSS3.p2.2.m2.2.3.2.cmml">G</mi><mo id="S4.SS2.SSS3.p2.2.m2.2.3.1" xref="S4.SS2.SSS3.p2.2.m2.2.3.1.cmml">=</mo><mrow id="S4.SS2.SSS3.p2.2.m2.2.3.3.2" xref="S4.SS2.SSS3.p2.2.m2.2.3.3.1.cmml"><mo id="S4.SS2.SSS3.p2.2.m2.2.3.3.2.1" stretchy="false" xref="S4.SS2.SSS3.p2.2.m2.2.3.3.1.cmml">(</mo><mi id="S4.SS2.SSS3.p2.2.m2.1.1" xref="S4.SS2.SSS3.p2.2.m2.1.1.cmml">V</mi><mo id="S4.SS2.SSS3.p2.2.m2.2.3.3.2.2" xref="S4.SS2.SSS3.p2.2.m2.2.3.3.1.cmml">,</mo><mi id="S4.SS2.SSS3.p2.2.m2.2.2" xref="S4.SS2.SSS3.p2.2.m2.2.2.cmml">E</mi><mo id="S4.SS2.SSS3.p2.2.m2.2.3.3.2.3" stretchy="false" xref="S4.SS2.SSS3.p2.2.m2.2.3.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS3.p2.2.m2.2b"><apply id="S4.SS2.SSS3.p2.2.m2.2.3.cmml" xref="S4.SS2.SSS3.p2.2.m2.2.3"><eq id="S4.SS2.SSS3.p2.2.m2.2.3.1.cmml" xref="S4.SS2.SSS3.p2.2.m2.2.3.1"></eq><ci id="S4.SS2.SSS3.p2.2.m2.2.3.2.cmml" xref="S4.SS2.SSS3.p2.2.m2.2.3.2">𝐺</ci><interval closure="open" id="S4.SS2.SSS3.p2.2.m2.2.3.3.1.cmml" xref="S4.SS2.SSS3.p2.2.m2.2.3.3.2"><ci id="S4.SS2.SSS3.p2.2.m2.1.1.cmml" xref="S4.SS2.SSS3.p2.2.m2.1.1">𝑉</ci><ci id="S4.SS2.SSS3.p2.2.m2.2.2.cmml" xref="S4.SS2.SSS3.p2.2.m2.2.2">𝐸</ci></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS3.p2.2.m2.2c">G=(V,E)</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS3.p2.2.m2.2d">italic_G = ( italic_V , italic_E )</annotation></semantics></math>. To prove Lemma <a class="ltx_ref" href="https://arxiv.org/html/2503.00712v1#S4.Thmtheorem20" title="Lemma 4.20. ‣ 4.2.3 Bounding the Approximation Ratio ‣ 4.2 Two-to-Three Augmentation ‣ 4 Vertex Connectivity Augmentation in Link-Arrival Model ‣ Streaming Algorithms for Network Design"><span class="ltx_text ltx_ref_tag">4.20</span></a>, we provide an algorithm to construct a feasible solution <math alttext="\textnormal{SOL}\subseteq F" class="ltx_Math" display="inline" id="S4.SS2.SSS3.p2.3.m3.1"><semantics id="S4.SS2.SSS3.p2.3.m3.1a"><mrow id="S4.SS2.SSS3.p2.3.m3.1.1" xref="S4.SS2.SSS3.p2.3.m3.1.1.cmml"><mtext id="S4.SS2.SSS3.p2.3.m3.1.1.2" xref="S4.SS2.SSS3.p2.3.m3.1.1.2a.cmml">SOL</mtext><mo id="S4.SS2.SSS3.p2.3.m3.1.1.1" xref="S4.SS2.SSS3.p2.3.m3.1.1.1.cmml">⊆</mo><mi id="S4.SS2.SSS3.p2.3.m3.1.1.3" xref="S4.SS2.SSS3.p2.3.m3.1.1.3.cmml">F</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS3.p2.3.m3.1b"><apply id="S4.SS2.SSS3.p2.3.m3.1.1.cmml" xref="S4.SS2.SSS3.p2.3.m3.1.1"><subset id="S4.SS2.SSS3.p2.3.m3.1.1.1.cmml" xref="S4.SS2.SSS3.p2.3.m3.1.1.1"></subset><ci id="S4.SS2.SSS3.p2.3.m3.1.1.2a.cmml" xref="S4.SS2.SSS3.p2.3.m3.1.1.2"><mtext id="S4.SS2.SSS3.p2.3.m3.1.1.2.cmml" xref="S4.SS2.SSS3.p2.3.m3.1.1.2">SOL</mtext></ci><ci id="S4.SS2.SSS3.p2.3.m3.1.1.3.cmml" xref="S4.SS2.SSS3.p2.3.m3.1.1.3">𝐹</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS3.p2.3.m3.1c">\textnormal{SOL}\subseteq F</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS3.p2.3.m3.1d">SOL ⊆ italic_F</annotation></semantics></math> such that <math alttext="w(\textnormal{SOL})\leq(7+\epsilon)w(\textnormal{OPT})" class="ltx_Math" display="inline" id="S4.SS2.SSS3.p2.4.m4.3"><semantics id="S4.SS2.SSS3.p2.4.m4.3a"><mrow id="S4.SS2.SSS3.p2.4.m4.3.3" xref="S4.SS2.SSS3.p2.4.m4.3.3.cmml"><mrow id="S4.SS2.SSS3.p2.4.m4.3.3.3" xref="S4.SS2.SSS3.p2.4.m4.3.3.3.cmml"><mi id="S4.SS2.SSS3.p2.4.m4.3.3.3.2" xref="S4.SS2.SSS3.p2.4.m4.3.3.3.2.cmml">w</mi><mo id="S4.SS2.SSS3.p2.4.m4.3.3.3.1" xref="S4.SS2.SSS3.p2.4.m4.3.3.3.1.cmml">⁢</mo><mrow id="S4.SS2.SSS3.p2.4.m4.3.3.3.3.2" xref="S4.SS2.SSS3.p2.4.m4.1.1a.cmml"><mo id="S4.SS2.SSS3.p2.4.m4.3.3.3.3.2.1" stretchy="false" xref="S4.SS2.SSS3.p2.4.m4.1.1a.cmml">(</mo><mtext id="S4.SS2.SSS3.p2.4.m4.1.1" xref="S4.SS2.SSS3.p2.4.m4.1.1.cmml">SOL</mtext><mo id="S4.SS2.SSS3.p2.4.m4.3.3.3.3.2.2" stretchy="false" xref="S4.SS2.SSS3.p2.4.m4.1.1a.cmml">)</mo></mrow></mrow><mo id="S4.SS2.SSS3.p2.4.m4.3.3.2" xref="S4.SS2.SSS3.p2.4.m4.3.3.2.cmml">≤</mo><mrow id="S4.SS2.SSS3.p2.4.m4.3.3.1" xref="S4.SS2.SSS3.p2.4.m4.3.3.1.cmml"><mrow id="S4.SS2.SSS3.p2.4.m4.3.3.1.1.1" xref="S4.SS2.SSS3.p2.4.m4.3.3.1.1.1.1.cmml"><mo id="S4.SS2.SSS3.p2.4.m4.3.3.1.1.1.2" stretchy="false" xref="S4.SS2.SSS3.p2.4.m4.3.3.1.1.1.1.cmml">(</mo><mrow id="S4.SS2.SSS3.p2.4.m4.3.3.1.1.1.1" xref="S4.SS2.SSS3.p2.4.m4.3.3.1.1.1.1.cmml"><mn id="S4.SS2.SSS3.p2.4.m4.3.3.1.1.1.1.2" xref="S4.SS2.SSS3.p2.4.m4.3.3.1.1.1.1.2.cmml">7</mn><mo id="S4.SS2.SSS3.p2.4.m4.3.3.1.1.1.1.1" xref="S4.SS2.SSS3.p2.4.m4.3.3.1.1.1.1.1.cmml">+</mo><mi id="S4.SS2.SSS3.p2.4.m4.3.3.1.1.1.1.3" xref="S4.SS2.SSS3.p2.4.m4.3.3.1.1.1.1.3.cmml">ϵ</mi></mrow><mo id="S4.SS2.SSS3.p2.4.m4.3.3.1.1.1.3" stretchy="false" xref="S4.SS2.SSS3.p2.4.m4.3.3.1.1.1.1.cmml">)</mo></mrow><mo id="S4.SS2.SSS3.p2.4.m4.3.3.1.2" xref="S4.SS2.SSS3.p2.4.m4.3.3.1.2.cmml">⁢</mo><mi id="S4.SS2.SSS3.p2.4.m4.3.3.1.3" xref="S4.SS2.SSS3.p2.4.m4.3.3.1.3.cmml">w</mi><mo id="S4.SS2.SSS3.p2.4.m4.3.3.1.2a" xref="S4.SS2.SSS3.p2.4.m4.3.3.1.2.cmml">⁢</mo><mrow id="S4.SS2.SSS3.p2.4.m4.3.3.1.4.2" xref="S4.SS2.SSS3.p2.4.m4.2.2a.cmml"><mo id="S4.SS2.SSS3.p2.4.m4.3.3.1.4.2.1" stretchy="false" xref="S4.SS2.SSS3.p2.4.m4.2.2a.cmml">(</mo><mtext id="S4.SS2.SSS3.p2.4.m4.2.2" xref="S4.SS2.SSS3.p2.4.m4.2.2.cmml">OPT</mtext><mo id="S4.SS2.SSS3.p2.4.m4.3.3.1.4.2.2" stretchy="false" xref="S4.SS2.SSS3.p2.4.m4.2.2a.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS3.p2.4.m4.3b"><apply id="S4.SS2.SSS3.p2.4.m4.3.3.cmml" xref="S4.SS2.SSS3.p2.4.m4.3.3"><leq id="S4.SS2.SSS3.p2.4.m4.3.3.2.cmml" xref="S4.SS2.SSS3.p2.4.m4.3.3.2"></leq><apply id="S4.SS2.SSS3.p2.4.m4.3.3.3.cmml" xref="S4.SS2.SSS3.p2.4.m4.3.3.3"><times id="S4.SS2.SSS3.p2.4.m4.3.3.3.1.cmml" xref="S4.SS2.SSS3.p2.4.m4.3.3.3.1"></times><ci id="S4.SS2.SSS3.p2.4.m4.3.3.3.2.cmml" xref="S4.SS2.SSS3.p2.4.m4.3.3.3.2">𝑤</ci><ci id="S4.SS2.SSS3.p2.4.m4.1.1a.cmml" xref="S4.SS2.SSS3.p2.4.m4.3.3.3.3.2"><mtext id="S4.SS2.SSS3.p2.4.m4.1.1.cmml" xref="S4.SS2.SSS3.p2.4.m4.1.1">SOL</mtext></ci></apply><apply id="S4.SS2.SSS3.p2.4.m4.3.3.1.cmml" xref="S4.SS2.SSS3.p2.4.m4.3.3.1"><times id="S4.SS2.SSS3.p2.4.m4.3.3.1.2.cmml" xref="S4.SS2.SSS3.p2.4.m4.3.3.1.2"></times><apply id="S4.SS2.SSS3.p2.4.m4.3.3.1.1.1.1.cmml" xref="S4.SS2.SSS3.p2.4.m4.3.3.1.1.1"><plus id="S4.SS2.SSS3.p2.4.m4.3.3.1.1.1.1.1.cmml" xref="S4.SS2.SSS3.p2.4.m4.3.3.1.1.1.1.1"></plus><cn id="S4.SS2.SSS3.p2.4.m4.3.3.1.1.1.1.2.cmml" type="integer" xref="S4.SS2.SSS3.p2.4.m4.3.3.1.1.1.1.2">7</cn><ci id="S4.SS2.SSS3.p2.4.m4.3.3.1.1.1.1.3.cmml" xref="S4.SS2.SSS3.p2.4.m4.3.3.1.1.1.1.3">italic-ϵ</ci></apply><ci id="S4.SS2.SSS3.p2.4.m4.3.3.1.3.cmml" xref="S4.SS2.SSS3.p2.4.m4.3.3.1.3">𝑤</ci><ci id="S4.SS2.SSS3.p2.4.m4.2.2a.cmml" xref="S4.SS2.SSS3.p2.4.m4.3.3.1.4.2"><mtext id="S4.SS2.SSS3.p2.4.m4.2.2.cmml" xref="S4.SS2.SSS3.p2.4.m4.2.2">OPT</mtext></ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS3.p2.4.m4.3c">w(\textnormal{SOL})\leq(7+\epsilon)w(\textnormal{OPT})</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS3.p2.4.m4.3d">italic_w ( SOL ) ≤ ( 7 + italic_ϵ ) italic_w ( OPT )</annotation></semantics></math>. Note that this algorithm is for analysis purposes only, as it requires knowledge of an optimal solution <span class="ltx_text ltx_markedasmath" id="S4.SS2.SSS3.p2.16.2">OPT</span> to the instance on <math alttext="G=(V,E)" class="ltx_Math" display="inline" id="S4.SS2.SSS3.p2.6.m6.2"><semantics id="S4.SS2.SSS3.p2.6.m6.2a"><mrow id="S4.SS2.SSS3.p2.6.m6.2.3" xref="S4.SS2.SSS3.p2.6.m6.2.3.cmml"><mi id="S4.SS2.SSS3.p2.6.m6.2.3.2" xref="S4.SS2.SSS3.p2.6.m6.2.3.2.cmml">G</mi><mo id="S4.SS2.SSS3.p2.6.m6.2.3.1" xref="S4.SS2.SSS3.p2.6.m6.2.3.1.cmml">=</mo><mrow id="S4.SS2.SSS3.p2.6.m6.2.3.3.2" xref="S4.SS2.SSS3.p2.6.m6.2.3.3.1.cmml"><mo id="S4.SS2.SSS3.p2.6.m6.2.3.3.2.1" stretchy="false" xref="S4.SS2.SSS3.p2.6.m6.2.3.3.1.cmml">(</mo><mi id="S4.SS2.SSS3.p2.6.m6.1.1" xref="S4.SS2.SSS3.p2.6.m6.1.1.cmml">V</mi><mo id="S4.SS2.SSS3.p2.6.m6.2.3.3.2.2" xref="S4.SS2.SSS3.p2.6.m6.2.3.3.1.cmml">,</mo><mi id="S4.SS2.SSS3.p2.6.m6.2.2" xref="S4.SS2.SSS3.p2.6.m6.2.2.cmml">E</mi><mo id="S4.SS2.SSS3.p2.6.m6.2.3.3.2.3" stretchy="false" xref="S4.SS2.SSS3.p2.6.m6.2.3.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS3.p2.6.m6.2b"><apply id="S4.SS2.SSS3.p2.6.m6.2.3.cmml" xref="S4.SS2.SSS3.p2.6.m6.2.3"><eq id="S4.SS2.SSS3.p2.6.m6.2.3.1.cmml" xref="S4.SS2.SSS3.p2.6.m6.2.3.1"></eq><ci id="S4.SS2.SSS3.p2.6.m6.2.3.2.cmml" xref="S4.SS2.SSS3.p2.6.m6.2.3.2">𝐺</ci><interval closure="open" id="S4.SS2.SSS3.p2.6.m6.2.3.3.1.cmml" xref="S4.SS2.SSS3.p2.6.m6.2.3.3.2"><ci id="S4.SS2.SSS3.p2.6.m6.1.1.cmml" xref="S4.SS2.SSS3.p2.6.m6.1.1">𝑉</ci><ci id="S4.SS2.SSS3.p2.6.m6.2.2.cmml" xref="S4.SS2.SSS3.p2.6.m6.2.2">𝐸</ci></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS3.p2.6.m6.2c">G=(V,E)</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS3.p2.6.m6.2d">italic_G = ( italic_V , italic_E )</annotation></semantics></math>. The construction of <span class="ltx_text ltx_markedasmath" id="S4.SS2.SSS3.p2.16.3">SOL</span> from the links in <math alttext="\cup_{x}L_{x}" class="ltx_Math" display="inline" id="S4.SS2.SSS3.p2.8.m8.1"><semantics id="S4.SS2.SSS3.p2.8.m8.1a"><mrow id="S4.SS2.SSS3.p2.8.m8.1.1" xref="S4.SS2.SSS3.p2.8.m8.1.1.cmml"><msub id="S4.SS2.SSS3.p2.8.m8.1.1.1" xref="S4.SS2.SSS3.p2.8.m8.1.1.1.cmml"><mo id="S4.SS2.SSS3.p2.8.m8.1.1.1.2" xref="S4.SS2.SSS3.p2.8.m8.1.1.1.2.cmml">∪</mo><mi id="S4.SS2.SSS3.p2.8.m8.1.1.1.3" xref="S4.SS2.SSS3.p2.8.m8.1.1.1.3.cmml">x</mi></msub><msub id="S4.SS2.SSS3.p2.8.m8.1.1.2" xref="S4.SS2.SSS3.p2.8.m8.1.1.2.cmml"><mi id="S4.SS2.SSS3.p2.8.m8.1.1.2.2" xref="S4.SS2.SSS3.p2.8.m8.1.1.2.2.cmml">L</mi><mi id="S4.SS2.SSS3.p2.8.m8.1.1.2.3" xref="S4.SS2.SSS3.p2.8.m8.1.1.2.3.cmml">x</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS3.p2.8.m8.1b"><apply id="S4.SS2.SSS3.p2.8.m8.1.1.cmml" xref="S4.SS2.SSS3.p2.8.m8.1.1"><apply id="S4.SS2.SSS3.p2.8.m8.1.1.1.cmml" xref="S4.SS2.SSS3.p2.8.m8.1.1.1"><csymbol cd="ambiguous" id="S4.SS2.SSS3.p2.8.m8.1.1.1.1.cmml" xref="S4.SS2.SSS3.p2.8.m8.1.1.1">subscript</csymbol><union id="S4.SS2.SSS3.p2.8.m8.1.1.1.2.cmml" xref="S4.SS2.SSS3.p2.8.m8.1.1.1.2"></union><ci id="S4.SS2.SSS3.p2.8.m8.1.1.1.3.cmml" xref="S4.SS2.SSS3.p2.8.m8.1.1.1.3">𝑥</ci></apply><apply id="S4.SS2.SSS3.p2.8.m8.1.1.2.cmml" xref="S4.SS2.SSS3.p2.8.m8.1.1.2"><csymbol cd="ambiguous" id="S4.SS2.SSS3.p2.8.m8.1.1.2.1.cmml" xref="S4.SS2.SSS3.p2.8.m8.1.1.2">subscript</csymbol><ci id="S4.SS2.SSS3.p2.8.m8.1.1.2.2.cmml" xref="S4.SS2.SSS3.p2.8.m8.1.1.2.2">𝐿</ci><ci id="S4.SS2.SSS3.p2.8.m8.1.1.2.3.cmml" xref="S4.SS2.SSS3.p2.8.m8.1.1.2.3">𝑥</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS3.p2.8.m8.1c">\cup_{x}L_{x}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS3.p2.8.m8.1d">∪ start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT italic_L start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT</annotation></semantics></math> (links that minimize LCA distance to root) and <math alttext="\cup_{x}E(H_{x})" class="ltx_Math" display="inline" id="S4.SS2.SSS3.p2.9.m9.1"><semantics id="S4.SS2.SSS3.p2.9.m9.1a"><mrow id="S4.SS2.SSS3.p2.9.m9.1.1" xref="S4.SS2.SSS3.p2.9.m9.1.1.cmml"><msub id="S4.SS2.SSS3.p2.9.m9.1.1.2" xref="S4.SS2.SSS3.p2.9.m9.1.1.2.cmml"><mo id="S4.SS2.SSS3.p2.9.m9.1.1.2.2" xref="S4.SS2.SSS3.p2.9.m9.1.1.2.2.cmml">∪</mo><mi id="S4.SS2.SSS3.p2.9.m9.1.1.2.3" xref="S4.SS2.SSS3.p2.9.m9.1.1.2.3.cmml">x</mi></msub><mrow id="S4.SS2.SSS3.p2.9.m9.1.1.1" xref="S4.SS2.SSS3.p2.9.m9.1.1.1.cmml"><mi id="S4.SS2.SSS3.p2.9.m9.1.1.1.3" xref="S4.SS2.SSS3.p2.9.m9.1.1.1.3.cmml">E</mi><mo id="S4.SS2.SSS3.p2.9.m9.1.1.1.2" xref="S4.SS2.SSS3.p2.9.m9.1.1.1.2.cmml">⁢</mo><mrow id="S4.SS2.SSS3.p2.9.m9.1.1.1.1.1" xref="S4.SS2.SSS3.p2.9.m9.1.1.1.1.1.1.cmml"><mo id="S4.SS2.SSS3.p2.9.m9.1.1.1.1.1.2" stretchy="false" xref="S4.SS2.SSS3.p2.9.m9.1.1.1.1.1.1.cmml">(</mo><msub id="S4.SS2.SSS3.p2.9.m9.1.1.1.1.1.1" xref="S4.SS2.SSS3.p2.9.m9.1.1.1.1.1.1.cmml"><mi id="S4.SS2.SSS3.p2.9.m9.1.1.1.1.1.1.2" xref="S4.SS2.SSS3.p2.9.m9.1.1.1.1.1.1.2.cmml">H</mi><mi id="S4.SS2.SSS3.p2.9.m9.1.1.1.1.1.1.3" xref="S4.SS2.SSS3.p2.9.m9.1.1.1.1.1.1.3.cmml">x</mi></msub><mo id="S4.SS2.SSS3.p2.9.m9.1.1.1.1.1.3" stretchy="false" xref="S4.SS2.SSS3.p2.9.m9.1.1.1.1.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS3.p2.9.m9.1b"><apply id="S4.SS2.SSS3.p2.9.m9.1.1.cmml" xref="S4.SS2.SSS3.p2.9.m9.1.1"><apply id="S4.SS2.SSS3.p2.9.m9.1.1.2.cmml" xref="S4.SS2.SSS3.p2.9.m9.1.1.2"><csymbol cd="ambiguous" id="S4.SS2.SSS3.p2.9.m9.1.1.2.1.cmml" xref="S4.SS2.SSS3.p2.9.m9.1.1.2">subscript</csymbol><union id="S4.SS2.SSS3.p2.9.m9.1.1.2.2.cmml" xref="S4.SS2.SSS3.p2.9.m9.1.1.2.2"></union><ci id="S4.SS2.SSS3.p2.9.m9.1.1.2.3.cmml" xref="S4.SS2.SSS3.p2.9.m9.1.1.2.3">𝑥</ci></apply><apply id="S4.SS2.SSS3.p2.9.m9.1.1.1.cmml" xref="S4.SS2.SSS3.p2.9.m9.1.1.1"><times id="S4.SS2.SSS3.p2.9.m9.1.1.1.2.cmml" xref="S4.SS2.SSS3.p2.9.m9.1.1.1.2"></times><ci id="S4.SS2.SSS3.p2.9.m9.1.1.1.3.cmml" xref="S4.SS2.SSS3.p2.9.m9.1.1.1.3">𝐸</ci><apply id="S4.SS2.SSS3.p2.9.m9.1.1.1.1.1.1.cmml" xref="S4.SS2.SSS3.p2.9.m9.1.1.1.1.1"><csymbol cd="ambiguous" id="S4.SS2.SSS3.p2.9.m9.1.1.1.1.1.1.1.cmml" xref="S4.SS2.SSS3.p2.9.m9.1.1.1.1.1">subscript</csymbol><ci id="S4.SS2.SSS3.p2.9.m9.1.1.1.1.1.1.2.cmml" xref="S4.SS2.SSS3.p2.9.m9.1.1.1.1.1.1.2">𝐻</ci><ci id="S4.SS2.SSS3.p2.9.m9.1.1.1.1.1.1.3.cmml" xref="S4.SS2.SSS3.p2.9.m9.1.1.1.1.1.1.3">𝑥</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS3.p2.9.m9.1c">\cup_{x}E(H_{x})</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS3.p2.9.m9.1d">∪ start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT italic_E ( italic_H start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT )</annotation></semantics></math> (MST links) follows a similar approach to the <math alttext="1" class="ltx_Math" display="inline" id="S4.SS2.SSS3.p2.10.m10.1"><semantics id="S4.SS2.SSS3.p2.10.m10.1a"><mn id="S4.SS2.SSS3.p2.10.m10.1.1" xref="S4.SS2.SSS3.p2.10.m10.1.1.cmml">1</mn><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS3.p2.10.m10.1b"><cn id="S4.SS2.SSS3.p2.10.m10.1.1.cmml" type="integer" xref="S4.SS2.SSS3.p2.10.m10.1.1">1</cn></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS3.p2.10.m10.1c">1</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS3.p2.10.m10.1d">1</annotation></semantics></math>-VC-CAP setting. The key technical challenge lies in the cycle case. This is because a single link <math alttext="uv\in L" class="ltx_Math" display="inline" id="S4.SS2.SSS3.p2.11.m11.1"><semantics id="S4.SS2.SSS3.p2.11.m11.1a"><mrow id="S4.SS2.SSS3.p2.11.m11.1.1" xref="S4.SS2.SSS3.p2.11.m11.1.1.cmml"><mrow id="S4.SS2.SSS3.p2.11.m11.1.1.2" xref="S4.SS2.SSS3.p2.11.m11.1.1.2.cmml"><mi id="S4.SS2.SSS3.p2.11.m11.1.1.2.2" xref="S4.SS2.SSS3.p2.11.m11.1.1.2.2.cmml">u</mi><mo id="S4.SS2.SSS3.p2.11.m11.1.1.2.1" xref="S4.SS2.SSS3.p2.11.m11.1.1.2.1.cmml">⁢</mo><mi id="S4.SS2.SSS3.p2.11.m11.1.1.2.3" xref="S4.SS2.SSS3.p2.11.m11.1.1.2.3.cmml">v</mi></mrow><mo id="S4.SS2.SSS3.p2.11.m11.1.1.1" xref="S4.SS2.SSS3.p2.11.m11.1.1.1.cmml">∈</mo><mi id="S4.SS2.SSS3.p2.11.m11.1.1.3" xref="S4.SS2.SSS3.p2.11.m11.1.1.3.cmml">L</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS3.p2.11.m11.1b"><apply id="S4.SS2.SSS3.p2.11.m11.1.1.cmml" xref="S4.SS2.SSS3.p2.11.m11.1.1"><in id="S4.SS2.SSS3.p2.11.m11.1.1.1.cmml" xref="S4.SS2.SSS3.p2.11.m11.1.1.1"></in><apply id="S4.SS2.SSS3.p2.11.m11.1.1.2.cmml" xref="S4.SS2.SSS3.p2.11.m11.1.1.2"><times id="S4.SS2.SSS3.p2.11.m11.1.1.2.1.cmml" xref="S4.SS2.SSS3.p2.11.m11.1.1.2.1"></times><ci id="S4.SS2.SSS3.p2.11.m11.1.1.2.2.cmml" xref="S4.SS2.SSS3.p2.11.m11.1.1.2.2">𝑢</ci><ci id="S4.SS2.SSS3.p2.11.m11.1.1.2.3.cmml" xref="S4.SS2.SSS3.p2.11.m11.1.1.2.3">𝑣</ci></apply><ci id="S4.SS2.SSS3.p2.11.m11.1.1.3.cmml" xref="S4.SS2.SSS3.p2.11.m11.1.1.3">𝐿</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS3.p2.11.m11.1c">uv\in L</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS3.p2.11.m11.1d">italic_u italic_v ∈ italic_L</annotation></semantics></math> may be responsible for increasing the connectivity of multiple S-nodes in the tree, leading to a large blow-up in the approximation factor. To circumvent this issue, we show that for each <math alttext="uv\in L" class="ltx_Math" display="inline" id="S4.SS2.SSS3.p2.12.m12.1"><semantics id="S4.SS2.SSS3.p2.12.m12.1a"><mrow id="S4.SS2.SSS3.p2.12.m12.1.1" xref="S4.SS2.SSS3.p2.12.m12.1.1.cmml"><mrow id="S4.SS2.SSS3.p2.12.m12.1.1.2" xref="S4.SS2.SSS3.p2.12.m12.1.1.2.cmml"><mi id="S4.SS2.SSS3.p2.12.m12.1.1.2.2" xref="S4.SS2.SSS3.p2.12.m12.1.1.2.2.cmml">u</mi><mo id="S4.SS2.SSS3.p2.12.m12.1.1.2.1" xref="S4.SS2.SSS3.p2.12.m12.1.1.2.1.cmml">⁢</mo><mi id="S4.SS2.SSS3.p2.12.m12.1.1.2.3" xref="S4.SS2.SSS3.p2.12.m12.1.1.2.3.cmml">v</mi></mrow><mo id="S4.SS2.SSS3.p2.12.m12.1.1.1" xref="S4.SS2.SSS3.p2.12.m12.1.1.1.cmml">∈</mo><mi id="S4.SS2.SSS3.p2.12.m12.1.1.3" xref="S4.SS2.SSS3.p2.12.m12.1.1.3.cmml">L</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS3.p2.12.m12.1b"><apply id="S4.SS2.SSS3.p2.12.m12.1.1.cmml" xref="S4.SS2.SSS3.p2.12.m12.1.1"><in id="S4.SS2.SSS3.p2.12.m12.1.1.1.cmml" xref="S4.SS2.SSS3.p2.12.m12.1.1.1"></in><apply id="S4.SS2.SSS3.p2.12.m12.1.1.2.cmml" xref="S4.SS2.SSS3.p2.12.m12.1.1.2"><times id="S4.SS2.SSS3.p2.12.m12.1.1.2.1.cmml" xref="S4.SS2.SSS3.p2.12.m12.1.1.2.1"></times><ci id="S4.SS2.SSS3.p2.12.m12.1.1.2.2.cmml" xref="S4.SS2.SSS3.p2.12.m12.1.1.2.2">𝑢</ci><ci id="S4.SS2.SSS3.p2.12.m12.1.1.2.3.cmml" xref="S4.SS2.SSS3.p2.12.m12.1.1.2.3">𝑣</ci></apply><ci id="S4.SS2.SSS3.p2.12.m12.1.1.3.cmml" xref="S4.SS2.SSS3.p2.12.m12.1.1.3">𝐿</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS3.p2.12.m12.1c">uv\in L</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS3.p2.12.m12.1d">italic_u italic_v ∈ italic_L</annotation></semantics></math>, we can restrict attention to at most three S-nodes in the tree: the LCA of <math alttext="\ell(u)" class="ltx_Math" display="inline" id="S4.SS2.SSS3.p2.13.m13.1"><semantics id="S4.SS2.SSS3.p2.13.m13.1a"><mrow id="S4.SS2.SSS3.p2.13.m13.1.2" xref="S4.SS2.SSS3.p2.13.m13.1.2.cmml"><mi id="S4.SS2.SSS3.p2.13.m13.1.2.2" mathvariant="normal" xref="S4.SS2.SSS3.p2.13.m13.1.2.2.cmml">ℓ</mi><mo id="S4.SS2.SSS3.p2.13.m13.1.2.1" xref="S4.SS2.SSS3.p2.13.m13.1.2.1.cmml">⁢</mo><mrow id="S4.SS2.SSS3.p2.13.m13.1.2.3.2" xref="S4.SS2.SSS3.p2.13.m13.1.2.cmml"><mo id="S4.SS2.SSS3.p2.13.m13.1.2.3.2.1" stretchy="false" xref="S4.SS2.SSS3.p2.13.m13.1.2.cmml">(</mo><mi id="S4.SS2.SSS3.p2.13.m13.1.1" xref="S4.SS2.SSS3.p2.13.m13.1.1.cmml">u</mi><mo id="S4.SS2.SSS3.p2.13.m13.1.2.3.2.2" stretchy="false" xref="S4.SS2.SSS3.p2.13.m13.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS3.p2.13.m13.1b"><apply id="S4.SS2.SSS3.p2.13.m13.1.2.cmml" xref="S4.SS2.SSS3.p2.13.m13.1.2"><times id="S4.SS2.SSS3.p2.13.m13.1.2.1.cmml" xref="S4.SS2.SSS3.p2.13.m13.1.2.1"></times><ci id="S4.SS2.SSS3.p2.13.m13.1.2.2.cmml" xref="S4.SS2.SSS3.p2.13.m13.1.2.2">ℓ</ci><ci id="S4.SS2.SSS3.p2.13.m13.1.1.cmml" xref="S4.SS2.SSS3.p2.13.m13.1.1">𝑢</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS3.p2.13.m13.1c">\ell(u)</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS3.p2.13.m13.1d">roman_ℓ ( italic_u )</annotation></semantics></math> and <math alttext="\ell(v)" class="ltx_Math" display="inline" id="S4.SS2.SSS3.p2.14.m14.1"><semantics id="S4.SS2.SSS3.p2.14.m14.1a"><mrow id="S4.SS2.SSS3.p2.14.m14.1.2" xref="S4.SS2.SSS3.p2.14.m14.1.2.cmml"><mi id="S4.SS2.SSS3.p2.14.m14.1.2.2" mathvariant="normal" xref="S4.SS2.SSS3.p2.14.m14.1.2.2.cmml">ℓ</mi><mo id="S4.SS2.SSS3.p2.14.m14.1.2.1" xref="S4.SS2.SSS3.p2.14.m14.1.2.1.cmml">⁢</mo><mrow id="S4.SS2.SSS3.p2.14.m14.1.2.3.2" xref="S4.SS2.SSS3.p2.14.m14.1.2.cmml"><mo id="S4.SS2.SSS3.p2.14.m14.1.2.3.2.1" stretchy="false" xref="S4.SS2.SSS3.p2.14.m14.1.2.cmml">(</mo><mi id="S4.SS2.SSS3.p2.14.m14.1.1" xref="S4.SS2.SSS3.p2.14.m14.1.1.cmml">v</mi><mo id="S4.SS2.SSS3.p2.14.m14.1.2.3.2.2" stretchy="false" xref="S4.SS2.SSS3.p2.14.m14.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS3.p2.14.m14.1b"><apply id="S4.SS2.SSS3.p2.14.m14.1.2.cmml" xref="S4.SS2.SSS3.p2.14.m14.1.2"><times id="S4.SS2.SSS3.p2.14.m14.1.2.1.cmml" xref="S4.SS2.SSS3.p2.14.m14.1.2.1"></times><ci id="S4.SS2.SSS3.p2.14.m14.1.2.2.cmml" xref="S4.SS2.SSS3.p2.14.m14.1.2.2">ℓ</ci><ci id="S4.SS2.SSS3.p2.14.m14.1.1.cmml" xref="S4.SS2.SSS3.p2.14.m14.1.1">𝑣</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS3.p2.14.m14.1c">\ell(v)</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS3.p2.14.m14.1d">roman_ℓ ( italic_v )</annotation></semantics></math>, an S-node containing <math alttext="u" class="ltx_Math" display="inline" id="S4.SS2.SSS3.p2.15.m15.1"><semantics id="S4.SS2.SSS3.p2.15.m15.1a"><mi id="S4.SS2.SSS3.p2.15.m15.1.1" xref="S4.SS2.SSS3.p2.15.m15.1.1.cmml">u</mi><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS3.p2.15.m15.1b"><ci id="S4.SS2.SSS3.p2.15.m15.1.1.cmml" xref="S4.SS2.SSS3.p2.15.m15.1.1">𝑢</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS3.p2.15.m15.1c">u</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS3.p2.15.m15.1d">italic_u</annotation></semantics></math>, and an S-node containing <math alttext="v" class="ltx_Math" display="inline" id="S4.SS2.SSS3.p2.16.m16.1"><semantics id="S4.SS2.SSS3.p2.16.m16.1a"><mi id="S4.SS2.SSS3.p2.16.m16.1.1" xref="S4.SS2.SSS3.p2.16.m16.1.1.cmml">v</mi><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS3.p2.16.m16.1b"><ci id="S4.SS2.SSS3.p2.16.m16.1.1.cmml" xref="S4.SS2.SSS3.p2.16.m16.1.1">𝑣</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS3.p2.16.m16.1c">v</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS3.p2.16.m16.1d">italic_v</annotation></semantics></math>. Details are given in Algorithm <a class="ltx_ref" href="https://arxiv.org/html/2503.00712v1#algorithm7" title="In 4.2.3 Bounding the Approximation Ratio ‣ 4.2 Two-to-Three Augmentation ‣ 4 Vertex Connectivity Augmentation in Link-Arrival Model ‣ Streaming Algorithms for Network Design"><span class="ltx_text ltx_ref_tag">7</span></a> and the following analysis.</p> </div> <figure class="ltx_float ltx_algorithm" id="algorithm7"> <div class="ltx_listing ltx_lst_numbers_left ltx_listing" id="algorithm7.38"> <div class="ltx_listingline" id="algorithm7.5.1"> <math alttext="\textnormal{SOL}\leftarrow\emptyset" class="ltx_Math" display="inline" id="algorithm7.5.1.m1.1"><semantics id="algorithm7.5.1.m1.1a"><mrow id="algorithm7.5.1.m1.1.1" xref="algorithm7.5.1.m1.1.1.cmml"><mtext id="algorithm7.5.1.m1.1.1.2" xref="algorithm7.5.1.m1.1.1.2a.cmml">SOL</mtext><mo id="algorithm7.5.1.m1.1.1.1" stretchy="false" xref="algorithm7.5.1.m1.1.1.1.cmml">←</mo><mi id="algorithm7.5.1.m1.1.1.3" mathvariant="normal" xref="algorithm7.5.1.m1.1.1.3.cmml">∅</mi></mrow><annotation-xml encoding="MathML-Content" id="algorithm7.5.1.m1.1b"><apply id="algorithm7.5.1.m1.1.1.cmml" xref="algorithm7.5.1.m1.1.1"><ci id="algorithm7.5.1.m1.1.1.1.cmml" xref="algorithm7.5.1.m1.1.1.1">←</ci><ci id="algorithm7.5.1.m1.1.1.2a.cmml" xref="algorithm7.5.1.m1.1.1.2"><mtext id="algorithm7.5.1.m1.1.1.2.cmml" xref="algorithm7.5.1.m1.1.1.2">SOL</mtext></ci><emptyset id="algorithm7.5.1.m1.1.1.3.cmml" xref="algorithm7.5.1.m1.1.1.3"></emptyset></apply></annotation-xml><annotation encoding="application/x-tex" id="algorithm7.5.1.m1.1c">\textnormal{SOL}\leftarrow\emptyset</annotation><annotation encoding="application/x-llamapun" id="algorithm7.5.1.m1.1d">SOL ← ∅</annotation></semantics></math> </div> <div class="ltx_listingline" id="algorithm7.6.2"> <span class="ltx_text ltx_font_bold" id="algorithm7.6.2.2">for</span> <em class="ltx_emph ltx_font_italic" id="algorithm7.6.2.1"><math alttext="e=(u,v)\in\textnormal{OPT}" class="ltx_Math" display="inline" id="algorithm7.6.2.1.m1.2"><semantics id="algorithm7.6.2.1.m1.2a"><mrow id="algorithm7.6.2.1.m1.2.3" xref="algorithm7.6.2.1.m1.2.3.cmml"><mi id="algorithm7.6.2.1.m1.2.3.2" xref="algorithm7.6.2.1.m1.2.3.2.cmml">e</mi><mo id="algorithm7.6.2.1.m1.2.3.3" xref="algorithm7.6.2.1.m1.2.3.3.cmml">=</mo><mrow id="algorithm7.6.2.1.m1.2.3.4.2" xref="algorithm7.6.2.1.m1.2.3.4.1.cmml"><mo id="algorithm7.6.2.1.m1.2.3.4.2.1" stretchy="false" xref="algorithm7.6.2.1.m1.2.3.4.1.cmml">(</mo><mi id="algorithm7.6.2.1.m1.1.1" xref="algorithm7.6.2.1.m1.1.1.cmml">u</mi><mo id="algorithm7.6.2.1.m1.2.3.4.2.2" xref="algorithm7.6.2.1.m1.2.3.4.1.cmml">,</mo><mi id="algorithm7.6.2.1.m1.2.2" xref="algorithm7.6.2.1.m1.2.2.cmml">v</mi><mo id="algorithm7.6.2.1.m1.2.3.4.2.3" stretchy="false" xref="algorithm7.6.2.1.m1.2.3.4.1.cmml">)</mo></mrow><mo id="algorithm7.6.2.1.m1.2.3.5" xref="algorithm7.6.2.1.m1.2.3.5.cmml">∈</mo><mtext id="algorithm7.6.2.1.m1.2.3.6" xref="algorithm7.6.2.1.m1.2.3.6b.cmml"><em class="ltx_emph ltx_font_upright" id="algorithm7.6.2.1.m1.2.3.6.1nest">OPT</em></mtext></mrow><annotation-xml encoding="MathML-Content" id="algorithm7.6.2.1.m1.2b"><apply id="algorithm7.6.2.1.m1.2.3.cmml" xref="algorithm7.6.2.1.m1.2.3"><and id="algorithm7.6.2.1.m1.2.3a.cmml" xref="algorithm7.6.2.1.m1.2.3"></and><apply id="algorithm7.6.2.1.m1.2.3b.cmml" xref="algorithm7.6.2.1.m1.2.3"><eq id="algorithm7.6.2.1.m1.2.3.3.cmml" xref="algorithm7.6.2.1.m1.2.3.3"></eq><ci id="algorithm7.6.2.1.m1.2.3.2.cmml" xref="algorithm7.6.2.1.m1.2.3.2">𝑒</ci><interval closure="open" id="algorithm7.6.2.1.m1.2.3.4.1.cmml" xref="algorithm7.6.2.1.m1.2.3.4.2"><ci id="algorithm7.6.2.1.m1.1.1.cmml" xref="algorithm7.6.2.1.m1.1.1">𝑢</ci><ci id="algorithm7.6.2.1.m1.2.2.cmml" xref="algorithm7.6.2.1.m1.2.2">𝑣</ci></interval></apply><apply id="algorithm7.6.2.1.m1.2.3c.cmml" xref="algorithm7.6.2.1.m1.2.3"><in id="algorithm7.6.2.1.m1.2.3.5.cmml" xref="algorithm7.6.2.1.m1.2.3.5"></in><share href="https://arxiv.org/html/2503.00712v1#algorithm7.6.2.1.m1.2.3.4.cmml" id="algorithm7.6.2.1.m1.2.3d.cmml" xref="algorithm7.6.2.1.m1.2.3"></share><ci id="algorithm7.6.2.1.m1.2.3.6b.cmml" xref="algorithm7.6.2.1.m1.2.3.6"><mtext id="algorithm7.6.2.1.m1.2.3.6.cmml" xref="algorithm7.6.2.1.m1.2.3.6"><em class="ltx_emph ltx_font_upright" id="algorithm7.6.2.1.m1.2.3.6.1anest">OPT</em></mtext></ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="algorithm7.6.2.1.m1.2c">e=(u,v)\in\textnormal{OPT}</annotation><annotation encoding="application/x-llamapun" id="algorithm7.6.2.1.m1.2d">italic_e = ( italic_u , italic_v ) ∈ OPT</annotation></semantics></math></em> <span class="ltx_text ltx_font_bold" id="algorithm7.6.2.3">do</span> </div> <div class="ltx_listingline" id="algorithm7.8.4"> <span class="ltx_rule" style="width:1px;height:100%;background:black;display:inline-block;"> </span> <span class="ltx_text ltx_font_bold" id="algorithm7.8.4.1">let</span> <math alttext="j" class="ltx_Math" display="inline" id="algorithm7.7.3.m1.1"><semantics id="algorithm7.7.3.m1.1a"><mi id="algorithm7.7.3.m1.1.1" xref="algorithm7.7.3.m1.1.1.cmml">j</mi><annotation-xml encoding="MathML-Content" id="algorithm7.7.3.m1.1b"><ci id="algorithm7.7.3.m1.1.1.cmml" xref="algorithm7.7.3.m1.1.1">𝑗</ci></annotation-xml><annotation encoding="application/x-tex" id="algorithm7.7.3.m1.1c">j</annotation><annotation encoding="application/x-llamapun" id="algorithm7.7.3.m1.1d">italic_j</annotation></semantics></math> be weight class such that <math alttext="w(e)\in[(1+\epsilon)^{j},(1+\epsilon)^{j+1})" class="ltx_Math" display="inline" id="algorithm7.8.4.m2.3"><semantics id="algorithm7.8.4.m2.3a"><mrow id="algorithm7.8.4.m2.3.3" xref="algorithm7.8.4.m2.3.3.cmml"><mrow id="algorithm7.8.4.m2.3.3.4" xref="algorithm7.8.4.m2.3.3.4.cmml"><mi id="algorithm7.8.4.m2.3.3.4.2" xref="algorithm7.8.4.m2.3.3.4.2.cmml">w</mi><mo id="algorithm7.8.4.m2.3.3.4.1" xref="algorithm7.8.4.m2.3.3.4.1.cmml">⁢</mo><mrow id="algorithm7.8.4.m2.3.3.4.3.2" xref="algorithm7.8.4.m2.3.3.4.cmml"><mo id="algorithm7.8.4.m2.3.3.4.3.2.1" stretchy="false" xref="algorithm7.8.4.m2.3.3.4.cmml">(</mo><mi id="algorithm7.8.4.m2.1.1" xref="algorithm7.8.4.m2.1.1.cmml">e</mi><mo id="algorithm7.8.4.m2.3.3.4.3.2.2" stretchy="false" xref="algorithm7.8.4.m2.3.3.4.cmml">)</mo></mrow></mrow><mo id="algorithm7.8.4.m2.3.3.3" xref="algorithm7.8.4.m2.3.3.3.cmml">∈</mo><mrow id="algorithm7.8.4.m2.3.3.2.2" xref="algorithm7.8.4.m2.3.3.2.3.cmml"><mo id="algorithm7.8.4.m2.3.3.2.2.3" stretchy="false" xref="algorithm7.8.4.m2.3.3.2.3.cmml">[</mo><msup id="algorithm7.8.4.m2.2.2.1.1.1" xref="algorithm7.8.4.m2.2.2.1.1.1.cmml"><mrow id="algorithm7.8.4.m2.2.2.1.1.1.1.1" xref="algorithm7.8.4.m2.2.2.1.1.1.1.1.1.cmml"><mo id="algorithm7.8.4.m2.2.2.1.1.1.1.1.2" stretchy="false" xref="algorithm7.8.4.m2.2.2.1.1.1.1.1.1.cmml">(</mo><mrow id="algorithm7.8.4.m2.2.2.1.1.1.1.1.1" xref="algorithm7.8.4.m2.2.2.1.1.1.1.1.1.cmml"><mn id="algorithm7.8.4.m2.2.2.1.1.1.1.1.1.2" xref="algorithm7.8.4.m2.2.2.1.1.1.1.1.1.2.cmml">1</mn><mo id="algorithm7.8.4.m2.2.2.1.1.1.1.1.1.1" xref="algorithm7.8.4.m2.2.2.1.1.1.1.1.1.1.cmml">+</mo><mi id="algorithm7.8.4.m2.2.2.1.1.1.1.1.1.3" xref="algorithm7.8.4.m2.2.2.1.1.1.1.1.1.3.cmml">ϵ</mi></mrow><mo id="algorithm7.8.4.m2.2.2.1.1.1.1.1.3" stretchy="false" xref="algorithm7.8.4.m2.2.2.1.1.1.1.1.1.cmml">)</mo></mrow><mi id="algorithm7.8.4.m2.2.2.1.1.1.3" xref="algorithm7.8.4.m2.2.2.1.1.1.3.cmml">j</mi></msup><mo id="algorithm7.8.4.m2.3.3.2.2.4" xref="algorithm7.8.4.m2.3.3.2.3.cmml">,</mo><msup id="algorithm7.8.4.m2.3.3.2.2.2" xref="algorithm7.8.4.m2.3.3.2.2.2.cmml"><mrow id="algorithm7.8.4.m2.3.3.2.2.2.1.1" xref="algorithm7.8.4.m2.3.3.2.2.2.1.1.1.cmml"><mo id="algorithm7.8.4.m2.3.3.2.2.2.1.1.2" stretchy="false" xref="algorithm7.8.4.m2.3.3.2.2.2.1.1.1.cmml">(</mo><mrow id="algorithm7.8.4.m2.3.3.2.2.2.1.1.1" xref="algorithm7.8.4.m2.3.3.2.2.2.1.1.1.cmml"><mn id="algorithm7.8.4.m2.3.3.2.2.2.1.1.1.2" xref="algorithm7.8.4.m2.3.3.2.2.2.1.1.1.2.cmml">1</mn><mo id="algorithm7.8.4.m2.3.3.2.2.2.1.1.1.1" xref="algorithm7.8.4.m2.3.3.2.2.2.1.1.1.1.cmml">+</mo><mi id="algorithm7.8.4.m2.3.3.2.2.2.1.1.1.3" xref="algorithm7.8.4.m2.3.3.2.2.2.1.1.1.3.cmml">ϵ</mi></mrow><mo id="algorithm7.8.4.m2.3.3.2.2.2.1.1.3" stretchy="false" xref="algorithm7.8.4.m2.3.3.2.2.2.1.1.1.cmml">)</mo></mrow><mrow id="algorithm7.8.4.m2.3.3.2.2.2.3" xref="algorithm7.8.4.m2.3.3.2.2.2.3.cmml"><mi id="algorithm7.8.4.m2.3.3.2.2.2.3.2" xref="algorithm7.8.4.m2.3.3.2.2.2.3.2.cmml">j</mi><mo id="algorithm7.8.4.m2.3.3.2.2.2.3.1" xref="algorithm7.8.4.m2.3.3.2.2.2.3.1.cmml">+</mo><mn id="algorithm7.8.4.m2.3.3.2.2.2.3.3" xref="algorithm7.8.4.m2.3.3.2.2.2.3.3.cmml">1</mn></mrow></msup><mo id="algorithm7.8.4.m2.3.3.2.2.5" stretchy="false" xref="algorithm7.8.4.m2.3.3.2.3.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="algorithm7.8.4.m2.3b"><apply id="algorithm7.8.4.m2.3.3.cmml" xref="algorithm7.8.4.m2.3.3"><in id="algorithm7.8.4.m2.3.3.3.cmml" xref="algorithm7.8.4.m2.3.3.3"></in><apply id="algorithm7.8.4.m2.3.3.4.cmml" xref="algorithm7.8.4.m2.3.3.4"><times id="algorithm7.8.4.m2.3.3.4.1.cmml" xref="algorithm7.8.4.m2.3.3.4.1"></times><ci id="algorithm7.8.4.m2.3.3.4.2.cmml" xref="algorithm7.8.4.m2.3.3.4.2">𝑤</ci><ci id="algorithm7.8.4.m2.1.1.cmml" xref="algorithm7.8.4.m2.1.1">𝑒</ci></apply><interval closure="closed-open" id="algorithm7.8.4.m2.3.3.2.3.cmml" xref="algorithm7.8.4.m2.3.3.2.2"><apply id="algorithm7.8.4.m2.2.2.1.1.1.cmml" xref="algorithm7.8.4.m2.2.2.1.1.1"><csymbol cd="ambiguous" id="algorithm7.8.4.m2.2.2.1.1.1.2.cmml" xref="algorithm7.8.4.m2.2.2.1.1.1">superscript</csymbol><apply id="algorithm7.8.4.m2.2.2.1.1.1.1.1.1.cmml" xref="algorithm7.8.4.m2.2.2.1.1.1.1.1"><plus id="algorithm7.8.4.m2.2.2.1.1.1.1.1.1.1.cmml" xref="algorithm7.8.4.m2.2.2.1.1.1.1.1.1.1"></plus><cn id="algorithm7.8.4.m2.2.2.1.1.1.1.1.1.2.cmml" type="integer" xref="algorithm7.8.4.m2.2.2.1.1.1.1.1.1.2">1</cn><ci id="algorithm7.8.4.m2.2.2.1.1.1.1.1.1.3.cmml" xref="algorithm7.8.4.m2.2.2.1.1.1.1.1.1.3">italic-ϵ</ci></apply><ci id="algorithm7.8.4.m2.2.2.1.1.1.3.cmml" xref="algorithm7.8.4.m2.2.2.1.1.1.3">𝑗</ci></apply><apply id="algorithm7.8.4.m2.3.3.2.2.2.cmml" xref="algorithm7.8.4.m2.3.3.2.2.2"><csymbol cd="ambiguous" id="algorithm7.8.4.m2.3.3.2.2.2.2.cmml" xref="algorithm7.8.4.m2.3.3.2.2.2">superscript</csymbol><apply id="algorithm7.8.4.m2.3.3.2.2.2.1.1.1.cmml" xref="algorithm7.8.4.m2.3.3.2.2.2.1.1"><plus id="algorithm7.8.4.m2.3.3.2.2.2.1.1.1.1.cmml" xref="algorithm7.8.4.m2.3.3.2.2.2.1.1.1.1"></plus><cn id="algorithm7.8.4.m2.3.3.2.2.2.1.1.1.2.cmml" type="integer" xref="algorithm7.8.4.m2.3.3.2.2.2.1.1.1.2">1</cn><ci id="algorithm7.8.4.m2.3.3.2.2.2.1.1.1.3.cmml" xref="algorithm7.8.4.m2.3.3.2.2.2.1.1.1.3">italic-ϵ</ci></apply><apply id="algorithm7.8.4.m2.3.3.2.2.2.3.cmml" xref="algorithm7.8.4.m2.3.3.2.2.2.3"><plus id="algorithm7.8.4.m2.3.3.2.2.2.3.1.cmml" xref="algorithm7.8.4.m2.3.3.2.2.2.3.1"></plus><ci id="algorithm7.8.4.m2.3.3.2.2.2.3.2.cmml" xref="algorithm7.8.4.m2.3.3.2.2.2.3.2">𝑗</ci><cn id="algorithm7.8.4.m2.3.3.2.2.2.3.3.cmml" type="integer" xref="algorithm7.8.4.m2.3.3.2.2.2.3.3">1</cn></apply></apply></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="algorithm7.8.4.m2.3c">w(e)\in[(1+\epsilon)^{j},(1+\epsilon)^{j+1})</annotation><annotation encoding="application/x-llamapun" id="algorithm7.8.4.m2.3d">italic_w ( italic_e ) ∈ [ ( 1 + italic_ϵ ) start_POSTSUPERSCRIPT italic_j end_POSTSUPERSCRIPT , ( 1 + italic_ϵ ) start_POSTSUPERSCRIPT italic_j + 1 end_POSTSUPERSCRIPT )</annotation></semantics></math> </div> <div class="ltx_listingline" id="algorithm7.9.5"> <span class="ltx_rule" style="width:1px;height:100%;background:black;display:inline-block;"> </span> <math alttext="\textnormal{SOL}\leftarrow\textnormal{SOL}\cup\{L_{h(u)}(j),L_{h(v)}(j)\}" class="ltx_Math" display="inline" id="algorithm7.9.5.m1.6"><semantics id="algorithm7.9.5.m1.6a"><mrow id="algorithm7.9.5.m1.6.6" xref="algorithm7.9.5.m1.6.6.cmml"><mtext id="algorithm7.9.5.m1.6.6.4" xref="algorithm7.9.5.m1.6.6.4a.cmml">SOL</mtext><mo id="algorithm7.9.5.m1.6.6.3" stretchy="false" xref="algorithm7.9.5.m1.6.6.3.cmml">←</mo><mrow id="algorithm7.9.5.m1.6.6.2" xref="algorithm7.9.5.m1.6.6.2.cmml"><mtext id="algorithm7.9.5.m1.6.6.2.4" xref="algorithm7.9.5.m1.6.6.2.4a.cmml">SOL</mtext><mo id="algorithm7.9.5.m1.6.6.2.3" xref="algorithm7.9.5.m1.6.6.2.3.cmml">∪</mo><mrow id="algorithm7.9.5.m1.6.6.2.2.2" xref="algorithm7.9.5.m1.6.6.2.2.3.cmml"><mo id="algorithm7.9.5.m1.6.6.2.2.2.3" stretchy="false" xref="algorithm7.9.5.m1.6.6.2.2.3.cmml">{</mo><mrow id="algorithm7.9.5.m1.5.5.1.1.1.1" xref="algorithm7.9.5.m1.5.5.1.1.1.1.cmml"><msub id="algorithm7.9.5.m1.5.5.1.1.1.1.2" xref="algorithm7.9.5.m1.5.5.1.1.1.1.2.cmml"><mi id="algorithm7.9.5.m1.5.5.1.1.1.1.2.2" xref="algorithm7.9.5.m1.5.5.1.1.1.1.2.2.cmml">L</mi><mrow id="algorithm7.9.5.m1.1.1.1" xref="algorithm7.9.5.m1.1.1.1.cmml"><mi id="algorithm7.9.5.m1.1.1.1.3" xref="algorithm7.9.5.m1.1.1.1.3.cmml">h</mi><mo id="algorithm7.9.5.m1.1.1.1.2" xref="algorithm7.9.5.m1.1.1.1.2.cmml">⁢</mo><mrow id="algorithm7.9.5.m1.1.1.1.4.2" xref="algorithm7.9.5.m1.1.1.1.cmml"><mo id="algorithm7.9.5.m1.1.1.1.4.2.1" stretchy="false" xref="algorithm7.9.5.m1.1.1.1.cmml">(</mo><mi id="algorithm7.9.5.m1.1.1.1.1" xref="algorithm7.9.5.m1.1.1.1.1.cmml">u</mi><mo id="algorithm7.9.5.m1.1.1.1.4.2.2" stretchy="false" xref="algorithm7.9.5.m1.1.1.1.cmml">)</mo></mrow></mrow></msub><mo id="algorithm7.9.5.m1.5.5.1.1.1.1.1" xref="algorithm7.9.5.m1.5.5.1.1.1.1.1.cmml">⁢</mo><mrow id="algorithm7.9.5.m1.5.5.1.1.1.1.3.2" xref="algorithm7.9.5.m1.5.5.1.1.1.1.cmml"><mo id="algorithm7.9.5.m1.5.5.1.1.1.1.3.2.1" stretchy="false" xref="algorithm7.9.5.m1.5.5.1.1.1.1.cmml">(</mo><mi id="algorithm7.9.5.m1.3.3" xref="algorithm7.9.5.m1.3.3.cmml">j</mi><mo id="algorithm7.9.5.m1.5.5.1.1.1.1.3.2.2" stretchy="false" xref="algorithm7.9.5.m1.5.5.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="algorithm7.9.5.m1.6.6.2.2.2.4" xref="algorithm7.9.5.m1.6.6.2.2.3.cmml">,</mo><mrow id="algorithm7.9.5.m1.6.6.2.2.2.2" xref="algorithm7.9.5.m1.6.6.2.2.2.2.cmml"><msub id="algorithm7.9.5.m1.6.6.2.2.2.2.2" xref="algorithm7.9.5.m1.6.6.2.2.2.2.2.cmml"><mi id="algorithm7.9.5.m1.6.6.2.2.2.2.2.2" xref="algorithm7.9.5.m1.6.6.2.2.2.2.2.2.cmml">L</mi><mrow id="algorithm7.9.5.m1.2.2.1" xref="algorithm7.9.5.m1.2.2.1.cmml"><mi id="algorithm7.9.5.m1.2.2.1.3" xref="algorithm7.9.5.m1.2.2.1.3.cmml">h</mi><mo id="algorithm7.9.5.m1.2.2.1.2" xref="algorithm7.9.5.m1.2.2.1.2.cmml">⁢</mo><mrow id="algorithm7.9.5.m1.2.2.1.4.2" xref="algorithm7.9.5.m1.2.2.1.cmml"><mo id="algorithm7.9.5.m1.2.2.1.4.2.1" stretchy="false" xref="algorithm7.9.5.m1.2.2.1.cmml">(</mo><mi id="algorithm7.9.5.m1.2.2.1.1" xref="algorithm7.9.5.m1.2.2.1.1.cmml">v</mi><mo id="algorithm7.9.5.m1.2.2.1.4.2.2" stretchy="false" xref="algorithm7.9.5.m1.2.2.1.cmml">)</mo></mrow></mrow></msub><mo id="algorithm7.9.5.m1.6.6.2.2.2.2.1" xref="algorithm7.9.5.m1.6.6.2.2.2.2.1.cmml">⁢</mo><mrow id="algorithm7.9.5.m1.6.6.2.2.2.2.3.2" xref="algorithm7.9.5.m1.6.6.2.2.2.2.cmml"><mo id="algorithm7.9.5.m1.6.6.2.2.2.2.3.2.1" stretchy="false" xref="algorithm7.9.5.m1.6.6.2.2.2.2.cmml">(</mo><mi id="algorithm7.9.5.m1.4.4" xref="algorithm7.9.5.m1.4.4.cmml">j</mi><mo id="algorithm7.9.5.m1.6.6.2.2.2.2.3.2.2" stretchy="false" xref="algorithm7.9.5.m1.6.6.2.2.2.2.cmml">)</mo></mrow></mrow><mo id="algorithm7.9.5.m1.6.6.2.2.2.5" stretchy="false" xref="algorithm7.9.5.m1.6.6.2.2.3.cmml">}</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="algorithm7.9.5.m1.6b"><apply id="algorithm7.9.5.m1.6.6.cmml" xref="algorithm7.9.5.m1.6.6"><ci id="algorithm7.9.5.m1.6.6.3.cmml" xref="algorithm7.9.5.m1.6.6.3">←</ci><ci id="algorithm7.9.5.m1.6.6.4a.cmml" xref="algorithm7.9.5.m1.6.6.4"><mtext id="algorithm7.9.5.m1.6.6.4.cmml" xref="algorithm7.9.5.m1.6.6.4">SOL</mtext></ci><apply id="algorithm7.9.5.m1.6.6.2.cmml" xref="algorithm7.9.5.m1.6.6.2"><union id="algorithm7.9.5.m1.6.6.2.3.cmml" xref="algorithm7.9.5.m1.6.6.2.3"></union><ci id="algorithm7.9.5.m1.6.6.2.4a.cmml" xref="algorithm7.9.5.m1.6.6.2.4"><mtext id="algorithm7.9.5.m1.6.6.2.4.cmml" xref="algorithm7.9.5.m1.6.6.2.4">SOL</mtext></ci><set id="algorithm7.9.5.m1.6.6.2.2.3.cmml" xref="algorithm7.9.5.m1.6.6.2.2.2"><apply id="algorithm7.9.5.m1.5.5.1.1.1.1.cmml" xref="algorithm7.9.5.m1.5.5.1.1.1.1"><times id="algorithm7.9.5.m1.5.5.1.1.1.1.1.cmml" xref="algorithm7.9.5.m1.5.5.1.1.1.1.1"></times><apply id="algorithm7.9.5.m1.5.5.1.1.1.1.2.cmml" xref="algorithm7.9.5.m1.5.5.1.1.1.1.2"><csymbol cd="ambiguous" id="algorithm7.9.5.m1.5.5.1.1.1.1.2.1.cmml" xref="algorithm7.9.5.m1.5.5.1.1.1.1.2">subscript</csymbol><ci id="algorithm7.9.5.m1.5.5.1.1.1.1.2.2.cmml" xref="algorithm7.9.5.m1.5.5.1.1.1.1.2.2">𝐿</ci><apply id="algorithm7.9.5.m1.1.1.1.cmml" xref="algorithm7.9.5.m1.1.1.1"><times id="algorithm7.9.5.m1.1.1.1.2.cmml" xref="algorithm7.9.5.m1.1.1.1.2"></times><ci id="algorithm7.9.5.m1.1.1.1.3.cmml" xref="algorithm7.9.5.m1.1.1.1.3">ℎ</ci><ci id="algorithm7.9.5.m1.1.1.1.1.cmml" xref="algorithm7.9.5.m1.1.1.1.1">𝑢</ci></apply></apply><ci id="algorithm7.9.5.m1.3.3.cmml" xref="algorithm7.9.5.m1.3.3">𝑗</ci></apply><apply id="algorithm7.9.5.m1.6.6.2.2.2.2.cmml" xref="algorithm7.9.5.m1.6.6.2.2.2.2"><times id="algorithm7.9.5.m1.6.6.2.2.2.2.1.cmml" xref="algorithm7.9.5.m1.6.6.2.2.2.2.1"></times><apply id="algorithm7.9.5.m1.6.6.2.2.2.2.2.cmml" xref="algorithm7.9.5.m1.6.6.2.2.2.2.2"><csymbol cd="ambiguous" id="algorithm7.9.5.m1.6.6.2.2.2.2.2.1.cmml" xref="algorithm7.9.5.m1.6.6.2.2.2.2.2">subscript</csymbol><ci id="algorithm7.9.5.m1.6.6.2.2.2.2.2.2.cmml" xref="algorithm7.9.5.m1.6.6.2.2.2.2.2.2">𝐿</ci><apply id="algorithm7.9.5.m1.2.2.1.cmml" xref="algorithm7.9.5.m1.2.2.1"><times id="algorithm7.9.5.m1.2.2.1.2.cmml" xref="algorithm7.9.5.m1.2.2.1.2"></times><ci id="algorithm7.9.5.m1.2.2.1.3.cmml" xref="algorithm7.9.5.m1.2.2.1.3">ℎ</ci><ci id="algorithm7.9.5.m1.2.2.1.1.cmml" xref="algorithm7.9.5.m1.2.2.1.1">𝑣</ci></apply></apply><ci id="algorithm7.9.5.m1.4.4.cmml" xref="algorithm7.9.5.m1.4.4">𝑗</ci></apply></set></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="algorithm7.9.5.m1.6c">\textnormal{SOL}\leftarrow\textnormal{SOL}\cup\{L_{h(u)}(j),L_{h(v)}(j)\}</annotation><annotation encoding="application/x-llamapun" id="algorithm7.9.5.m1.6d">SOL ← SOL ∪ { italic_L start_POSTSUBSCRIPT italic_h ( italic_u ) end_POSTSUBSCRIPT ( italic_j ) , italic_L start_POSTSUBSCRIPT italic_h ( italic_v ) end_POSTSUBSCRIPT ( italic_j ) }</annotation></semantics></math> </div> <div class="ltx_listingline" id="algorithm7.10.6"> <span class="ltx_rule" style="width:1px;height:100%;background:black;display:inline-block;"> </span> <span class="ltx_text ltx_font_bold" id="algorithm7.10.6.2">if</span> <em class="ltx_emph ltx_font_italic" id="algorithm7.10.6.1"><math alttext="x=\text{LCA}(\ell(u),\ell(v))" class="ltx_Math" display="inline" id="algorithm7.10.6.1.m1.4"><semantics id="algorithm7.10.6.1.m1.4a"><mrow id="algorithm7.10.6.1.m1.4.4" xref="algorithm7.10.6.1.m1.4.4.cmml"><mi id="algorithm7.10.6.1.m1.4.4.4" xref="algorithm7.10.6.1.m1.4.4.4.cmml">x</mi><mo id="algorithm7.10.6.1.m1.4.4.3" xref="algorithm7.10.6.1.m1.4.4.3.cmml">=</mo><mrow id="algorithm7.10.6.1.m1.4.4.2" xref="algorithm7.10.6.1.m1.4.4.2.cmml"><mtext class="ltx_mathvariant_italic" id="algorithm7.10.6.1.m1.4.4.2.4" xref="algorithm7.10.6.1.m1.4.4.2.4a.cmml">LCA</mtext><mo id="algorithm7.10.6.1.m1.4.4.2.3" xref="algorithm7.10.6.1.m1.4.4.2.3.cmml">⁢</mo><mrow id="algorithm7.10.6.1.m1.4.4.2.2.2" xref="algorithm7.10.6.1.m1.4.4.2.2.3.cmml"><mo id="algorithm7.10.6.1.m1.4.4.2.2.2.3" stretchy="false" xref="algorithm7.10.6.1.m1.4.4.2.2.3.cmml">(</mo><mrow id="algorithm7.10.6.1.m1.3.3.1.1.1.1" xref="algorithm7.10.6.1.m1.3.3.1.1.1.1.cmml"><mi id="algorithm7.10.6.1.m1.3.3.1.1.1.1.2" mathvariant="normal" xref="algorithm7.10.6.1.m1.3.3.1.1.1.1.2.cmml">ℓ</mi><mo id="algorithm7.10.6.1.m1.3.3.1.1.1.1.1" xref="algorithm7.10.6.1.m1.3.3.1.1.1.1.1.cmml">⁢</mo><mrow id="algorithm7.10.6.1.m1.3.3.1.1.1.1.3.2" xref="algorithm7.10.6.1.m1.3.3.1.1.1.1.cmml"><mo id="algorithm7.10.6.1.m1.3.3.1.1.1.1.3.2.1" stretchy="false" xref="algorithm7.10.6.1.m1.3.3.1.1.1.1.cmml">(</mo><mi id="algorithm7.10.6.1.m1.1.1" xref="algorithm7.10.6.1.m1.1.1.cmml">u</mi><mo id="algorithm7.10.6.1.m1.3.3.1.1.1.1.3.2.2" stretchy="false" xref="algorithm7.10.6.1.m1.3.3.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="algorithm7.10.6.1.m1.4.4.2.2.2.4" xref="algorithm7.10.6.1.m1.4.4.2.2.3.cmml">,</mo><mrow id="algorithm7.10.6.1.m1.4.4.2.2.2.2" xref="algorithm7.10.6.1.m1.4.4.2.2.2.2.cmml"><mi id="algorithm7.10.6.1.m1.4.4.2.2.2.2.2" mathvariant="normal" xref="algorithm7.10.6.1.m1.4.4.2.2.2.2.2.cmml">ℓ</mi><mo id="algorithm7.10.6.1.m1.4.4.2.2.2.2.1" xref="algorithm7.10.6.1.m1.4.4.2.2.2.2.1.cmml">⁢</mo><mrow id="algorithm7.10.6.1.m1.4.4.2.2.2.2.3.2" xref="algorithm7.10.6.1.m1.4.4.2.2.2.2.cmml"><mo id="algorithm7.10.6.1.m1.4.4.2.2.2.2.3.2.1" stretchy="false" xref="algorithm7.10.6.1.m1.4.4.2.2.2.2.cmml">(</mo><mi id="algorithm7.10.6.1.m1.2.2" xref="algorithm7.10.6.1.m1.2.2.cmml">v</mi><mo id="algorithm7.10.6.1.m1.4.4.2.2.2.2.3.2.2" stretchy="false" xref="algorithm7.10.6.1.m1.4.4.2.2.2.2.cmml">)</mo></mrow></mrow><mo id="algorithm7.10.6.1.m1.4.4.2.2.2.5" stretchy="false" xref="algorithm7.10.6.1.m1.4.4.2.2.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="algorithm7.10.6.1.m1.4b"><apply id="algorithm7.10.6.1.m1.4.4.cmml" xref="algorithm7.10.6.1.m1.4.4"><eq id="algorithm7.10.6.1.m1.4.4.3.cmml" xref="algorithm7.10.6.1.m1.4.4.3"></eq><ci id="algorithm7.10.6.1.m1.4.4.4.cmml" xref="algorithm7.10.6.1.m1.4.4.4">𝑥</ci><apply id="algorithm7.10.6.1.m1.4.4.2.cmml" xref="algorithm7.10.6.1.m1.4.4.2"><times id="algorithm7.10.6.1.m1.4.4.2.3.cmml" xref="algorithm7.10.6.1.m1.4.4.2.3"></times><ci id="algorithm7.10.6.1.m1.4.4.2.4a.cmml" xref="algorithm7.10.6.1.m1.4.4.2.4"><mtext class="ltx_mathvariant_italic" id="algorithm7.10.6.1.m1.4.4.2.4.cmml" xref="algorithm7.10.6.1.m1.4.4.2.4">LCA</mtext></ci><interval closure="open" id="algorithm7.10.6.1.m1.4.4.2.2.3.cmml" xref="algorithm7.10.6.1.m1.4.4.2.2.2"><apply id="algorithm7.10.6.1.m1.3.3.1.1.1.1.cmml" xref="algorithm7.10.6.1.m1.3.3.1.1.1.1"><times id="algorithm7.10.6.1.m1.3.3.1.1.1.1.1.cmml" xref="algorithm7.10.6.1.m1.3.3.1.1.1.1.1"></times><ci id="algorithm7.10.6.1.m1.3.3.1.1.1.1.2.cmml" xref="algorithm7.10.6.1.m1.3.3.1.1.1.1.2">ℓ</ci><ci id="algorithm7.10.6.1.m1.1.1.cmml" xref="algorithm7.10.6.1.m1.1.1">𝑢</ci></apply><apply id="algorithm7.10.6.1.m1.4.4.2.2.2.2.cmml" xref="algorithm7.10.6.1.m1.4.4.2.2.2.2"><times id="algorithm7.10.6.1.m1.4.4.2.2.2.2.1.cmml" xref="algorithm7.10.6.1.m1.4.4.2.2.2.2.1"></times><ci id="algorithm7.10.6.1.m1.4.4.2.2.2.2.2.cmml" xref="algorithm7.10.6.1.m1.4.4.2.2.2.2.2">ℓ</ci><ci id="algorithm7.10.6.1.m1.2.2.cmml" xref="algorithm7.10.6.1.m1.2.2">𝑣</ci></apply></interval></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="algorithm7.10.6.1.m1.4c">x=\text{LCA}(\ell(u),\ell(v))</annotation><annotation encoding="application/x-llamapun" id="algorithm7.10.6.1.m1.4d">italic_x = LCA ( roman_ℓ ( italic_u ) , roman_ℓ ( italic_v ) )</annotation></semantics></math> is S-node</em> <span class="ltx_text ltx_font_bold" id="algorithm7.10.6.3">then</span> </div> <div class="ltx_listingline" id="algorithm7.11.7"> <span class="ltx_rule" style="width:1px;height:100%;background:black;display:inline-block;"> </span> <span class="ltx_rule" style="width:1px;height:100%;background:black;display:inline-block;"> </span> <span class="ltx_text ltx_font_bold" id="algorithm7.11.7.2">if</span> <em class="ltx_emph ltx_font_italic" id="algorithm7.11.7.1"><math alttext="f_{x}(u)\prec_{x}f_{x}(v)" class="ltx_Math" display="inline" id="algorithm7.11.7.1.m1.2"><semantics id="algorithm7.11.7.1.m1.2a"><mrow id="algorithm7.11.7.1.m1.2.3" xref="algorithm7.11.7.1.m1.2.3.cmml"><mrow id="algorithm7.11.7.1.m1.2.3.2" xref="algorithm7.11.7.1.m1.2.3.2.cmml"><msub id="algorithm7.11.7.1.m1.2.3.2.2" xref="algorithm7.11.7.1.m1.2.3.2.2.cmml"><mi id="algorithm7.11.7.1.m1.2.3.2.2.2" xref="algorithm7.11.7.1.m1.2.3.2.2.2.cmml">f</mi><mi id="algorithm7.11.7.1.m1.2.3.2.2.3" xref="algorithm7.11.7.1.m1.2.3.2.2.3.cmml">x</mi></msub><mo id="algorithm7.11.7.1.m1.2.3.2.1" xref="algorithm7.11.7.1.m1.2.3.2.1.cmml">⁢</mo><mrow id="algorithm7.11.7.1.m1.2.3.2.3.2" xref="algorithm7.11.7.1.m1.2.3.2.cmml"><mo id="algorithm7.11.7.1.m1.2.3.2.3.2.1" stretchy="false" xref="algorithm7.11.7.1.m1.2.3.2.cmml">(</mo><mi id="algorithm7.11.7.1.m1.1.1" xref="algorithm7.11.7.1.m1.1.1.cmml">u</mi><mo id="algorithm7.11.7.1.m1.2.3.2.3.2.2" stretchy="false" xref="algorithm7.11.7.1.m1.2.3.2.cmml">)</mo></mrow></mrow><msub id="algorithm7.11.7.1.m1.2.3.1" xref="algorithm7.11.7.1.m1.2.3.1.cmml"><mo id="algorithm7.11.7.1.m1.2.3.1.2" xref="algorithm7.11.7.1.m1.2.3.1.2.cmml">≺</mo><mi id="algorithm7.11.7.1.m1.2.3.1.3" xref="algorithm7.11.7.1.m1.2.3.1.3.cmml">x</mi></msub><mrow id="algorithm7.11.7.1.m1.2.3.3" xref="algorithm7.11.7.1.m1.2.3.3.cmml"><msub id="algorithm7.11.7.1.m1.2.3.3.2" xref="algorithm7.11.7.1.m1.2.3.3.2.cmml"><mi id="algorithm7.11.7.1.m1.2.3.3.2.2" xref="algorithm7.11.7.1.m1.2.3.3.2.2.cmml">f</mi><mi id="algorithm7.11.7.1.m1.2.3.3.2.3" xref="algorithm7.11.7.1.m1.2.3.3.2.3.cmml">x</mi></msub><mo id="algorithm7.11.7.1.m1.2.3.3.1" xref="algorithm7.11.7.1.m1.2.3.3.1.cmml">⁢</mo><mrow id="algorithm7.11.7.1.m1.2.3.3.3.2" xref="algorithm7.11.7.1.m1.2.3.3.cmml"><mo id="algorithm7.11.7.1.m1.2.3.3.3.2.1" stretchy="false" xref="algorithm7.11.7.1.m1.2.3.3.cmml">(</mo><mi id="algorithm7.11.7.1.m1.2.2" xref="algorithm7.11.7.1.m1.2.2.cmml">v</mi><mo id="algorithm7.11.7.1.m1.2.3.3.3.2.2" stretchy="false" xref="algorithm7.11.7.1.m1.2.3.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="algorithm7.11.7.1.m1.2b"><apply id="algorithm7.11.7.1.m1.2.3.cmml" xref="algorithm7.11.7.1.m1.2.3"><apply id="algorithm7.11.7.1.m1.2.3.1.cmml" xref="algorithm7.11.7.1.m1.2.3.1"><csymbol cd="ambiguous" id="algorithm7.11.7.1.m1.2.3.1.1.cmml" xref="algorithm7.11.7.1.m1.2.3.1">subscript</csymbol><csymbol cd="latexml" id="algorithm7.11.7.1.m1.2.3.1.2.cmml" xref="algorithm7.11.7.1.m1.2.3.1.2">precedes</csymbol><ci id="algorithm7.11.7.1.m1.2.3.1.3.cmml" xref="algorithm7.11.7.1.m1.2.3.1.3">𝑥</ci></apply><apply id="algorithm7.11.7.1.m1.2.3.2.cmml" xref="algorithm7.11.7.1.m1.2.3.2"><times id="algorithm7.11.7.1.m1.2.3.2.1.cmml" xref="algorithm7.11.7.1.m1.2.3.2.1"></times><apply id="algorithm7.11.7.1.m1.2.3.2.2.cmml" xref="algorithm7.11.7.1.m1.2.3.2.2"><csymbol cd="ambiguous" id="algorithm7.11.7.1.m1.2.3.2.2.1.cmml" xref="algorithm7.11.7.1.m1.2.3.2.2">subscript</csymbol><ci id="algorithm7.11.7.1.m1.2.3.2.2.2.cmml" xref="algorithm7.11.7.1.m1.2.3.2.2.2">𝑓</ci><ci id="algorithm7.11.7.1.m1.2.3.2.2.3.cmml" xref="algorithm7.11.7.1.m1.2.3.2.2.3">𝑥</ci></apply><ci id="algorithm7.11.7.1.m1.1.1.cmml" xref="algorithm7.11.7.1.m1.1.1">𝑢</ci></apply><apply id="algorithm7.11.7.1.m1.2.3.3.cmml" xref="algorithm7.11.7.1.m1.2.3.3"><times id="algorithm7.11.7.1.m1.2.3.3.1.cmml" xref="algorithm7.11.7.1.m1.2.3.3.1"></times><apply id="algorithm7.11.7.1.m1.2.3.3.2.cmml" xref="algorithm7.11.7.1.m1.2.3.3.2"><csymbol cd="ambiguous" id="algorithm7.11.7.1.m1.2.3.3.2.1.cmml" xref="algorithm7.11.7.1.m1.2.3.3.2">subscript</csymbol><ci id="algorithm7.11.7.1.m1.2.3.3.2.2.cmml" xref="algorithm7.11.7.1.m1.2.3.3.2.2">𝑓</ci><ci id="algorithm7.11.7.1.m1.2.3.3.2.3.cmml" xref="algorithm7.11.7.1.m1.2.3.3.2.3">𝑥</ci></apply><ci id="algorithm7.11.7.1.m1.2.2.cmml" xref="algorithm7.11.7.1.m1.2.2">𝑣</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="algorithm7.11.7.1.m1.2c">f_{x}(u)\prec_{x}f_{x}(v)</annotation><annotation encoding="application/x-llamapun" id="algorithm7.11.7.1.m1.2d">italic_f start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT ( italic_u ) ≺ start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT italic_f start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT ( italic_v )</annotation></semantics></math></em> <span class="ltx_text ltx_font_bold" id="algorithm7.11.7.3">then</span> </div> <div class="ltx_listingline" id="algorithm7.12.8"> <span class="ltx_rule" style="width:1px;height:100%;background:black;display:inline-block;"> </span> <span class="ltx_rule" style="width:1px;height:100%;background:black;display:inline-block;"> </span> <span class="ltx_rule" style="width:1px;height:100%;background:black;display:inline-block;"> </span> <math alttext="\textnormal{SOL}\leftarrow\textnormal{SOL}\cup\{\textsc{Min}_{f_{x}(v)}(j),% \textsc{Max}_{f_{x}(u)}(j)\}" class="ltx_Math" display="inline" id="algorithm7.12.8.m1.6"><semantics id="algorithm7.12.8.m1.6a"><mrow id="algorithm7.12.8.m1.6.6" xref="algorithm7.12.8.m1.6.6.cmml"><mtext id="algorithm7.12.8.m1.6.6.4" xref="algorithm7.12.8.m1.6.6.4a.cmml">SOL</mtext><mo id="algorithm7.12.8.m1.6.6.3" stretchy="false" xref="algorithm7.12.8.m1.6.6.3.cmml">←</mo><mrow id="algorithm7.12.8.m1.6.6.2" xref="algorithm7.12.8.m1.6.6.2.cmml"><mtext id="algorithm7.12.8.m1.6.6.2.4" xref="algorithm7.12.8.m1.6.6.2.4a.cmml">SOL</mtext><mo id="algorithm7.12.8.m1.6.6.2.3" xref="algorithm7.12.8.m1.6.6.2.3.cmml">∪</mo><mrow id="algorithm7.12.8.m1.6.6.2.2.2" xref="algorithm7.12.8.m1.6.6.2.2.3.cmml"><mo id="algorithm7.12.8.m1.6.6.2.2.2.3" stretchy="false" xref="algorithm7.12.8.m1.6.6.2.2.3.cmml">{</mo><mrow id="algorithm7.12.8.m1.5.5.1.1.1.1" xref="algorithm7.12.8.m1.5.5.1.1.1.1.cmml"><msub id="algorithm7.12.8.m1.5.5.1.1.1.1.2" xref="algorithm7.12.8.m1.5.5.1.1.1.1.2.cmml"><mtext class="ltx_font_smallcaps" id="algorithm7.12.8.m1.5.5.1.1.1.1.2.2" xref="algorithm7.12.8.m1.5.5.1.1.1.1.2.2a.cmml">Min</mtext><mrow id="algorithm7.12.8.m1.1.1.1" xref="algorithm7.12.8.m1.1.1.1.cmml"><msub id="algorithm7.12.8.m1.1.1.1.3" xref="algorithm7.12.8.m1.1.1.1.3.cmml"><mi id="algorithm7.12.8.m1.1.1.1.3.2" xref="algorithm7.12.8.m1.1.1.1.3.2.cmml">f</mi><mi id="algorithm7.12.8.m1.1.1.1.3.3" xref="algorithm7.12.8.m1.1.1.1.3.3.cmml">x</mi></msub><mo id="algorithm7.12.8.m1.1.1.1.2" xref="algorithm7.12.8.m1.1.1.1.2.cmml">⁢</mo><mrow id="algorithm7.12.8.m1.1.1.1.4.2" xref="algorithm7.12.8.m1.1.1.1.cmml"><mo id="algorithm7.12.8.m1.1.1.1.4.2.1" stretchy="false" xref="algorithm7.12.8.m1.1.1.1.cmml">(</mo><mi id="algorithm7.12.8.m1.1.1.1.1" xref="algorithm7.12.8.m1.1.1.1.1.cmml">v</mi><mo id="algorithm7.12.8.m1.1.1.1.4.2.2" stretchy="false" xref="algorithm7.12.8.m1.1.1.1.cmml">)</mo></mrow></mrow></msub><mo id="algorithm7.12.8.m1.5.5.1.1.1.1.1" xref="algorithm7.12.8.m1.5.5.1.1.1.1.1.cmml">⁢</mo><mrow id="algorithm7.12.8.m1.5.5.1.1.1.1.3.2" xref="algorithm7.12.8.m1.5.5.1.1.1.1.cmml"><mo id="algorithm7.12.8.m1.5.5.1.1.1.1.3.2.1" stretchy="false" xref="algorithm7.12.8.m1.5.5.1.1.1.1.cmml">(</mo><mi id="algorithm7.12.8.m1.3.3" xref="algorithm7.12.8.m1.3.3.cmml">j</mi><mo id="algorithm7.12.8.m1.5.5.1.1.1.1.3.2.2" stretchy="false" xref="algorithm7.12.8.m1.5.5.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="algorithm7.12.8.m1.6.6.2.2.2.4" xref="algorithm7.12.8.m1.6.6.2.2.3.cmml">,</mo><mrow id="algorithm7.12.8.m1.6.6.2.2.2.2" xref="algorithm7.12.8.m1.6.6.2.2.2.2.cmml"><msub id="algorithm7.12.8.m1.6.6.2.2.2.2.2" xref="algorithm7.12.8.m1.6.6.2.2.2.2.2.cmml"><mtext class="ltx_font_smallcaps" id="algorithm7.12.8.m1.6.6.2.2.2.2.2.2" xref="algorithm7.12.8.m1.6.6.2.2.2.2.2.2a.cmml">Max</mtext><mrow id="algorithm7.12.8.m1.2.2.1" xref="algorithm7.12.8.m1.2.2.1.cmml"><msub id="algorithm7.12.8.m1.2.2.1.3" xref="algorithm7.12.8.m1.2.2.1.3.cmml"><mi id="algorithm7.12.8.m1.2.2.1.3.2" xref="algorithm7.12.8.m1.2.2.1.3.2.cmml">f</mi><mi id="algorithm7.12.8.m1.2.2.1.3.3" xref="algorithm7.12.8.m1.2.2.1.3.3.cmml">x</mi></msub><mo id="algorithm7.12.8.m1.2.2.1.2" xref="algorithm7.12.8.m1.2.2.1.2.cmml">⁢</mo><mrow id="algorithm7.12.8.m1.2.2.1.4.2" xref="algorithm7.12.8.m1.2.2.1.cmml"><mo id="algorithm7.12.8.m1.2.2.1.4.2.1" stretchy="false" xref="algorithm7.12.8.m1.2.2.1.cmml">(</mo><mi id="algorithm7.12.8.m1.2.2.1.1" xref="algorithm7.12.8.m1.2.2.1.1.cmml">u</mi><mo id="algorithm7.12.8.m1.2.2.1.4.2.2" stretchy="false" xref="algorithm7.12.8.m1.2.2.1.cmml">)</mo></mrow></mrow></msub><mo id="algorithm7.12.8.m1.6.6.2.2.2.2.1" xref="algorithm7.12.8.m1.6.6.2.2.2.2.1.cmml">⁢</mo><mrow id="algorithm7.12.8.m1.6.6.2.2.2.2.3.2" xref="algorithm7.12.8.m1.6.6.2.2.2.2.cmml"><mo id="algorithm7.12.8.m1.6.6.2.2.2.2.3.2.1" stretchy="false" xref="algorithm7.12.8.m1.6.6.2.2.2.2.cmml">(</mo><mi id="algorithm7.12.8.m1.4.4" xref="algorithm7.12.8.m1.4.4.cmml">j</mi><mo id="algorithm7.12.8.m1.6.6.2.2.2.2.3.2.2" stretchy="false" xref="algorithm7.12.8.m1.6.6.2.2.2.2.cmml">)</mo></mrow></mrow><mo id="algorithm7.12.8.m1.6.6.2.2.2.5" stretchy="false" xref="algorithm7.12.8.m1.6.6.2.2.3.cmml">}</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="algorithm7.12.8.m1.6b"><apply id="algorithm7.12.8.m1.6.6.cmml" xref="algorithm7.12.8.m1.6.6"><ci id="algorithm7.12.8.m1.6.6.3.cmml" xref="algorithm7.12.8.m1.6.6.3">←</ci><ci id="algorithm7.12.8.m1.6.6.4a.cmml" xref="algorithm7.12.8.m1.6.6.4"><mtext id="algorithm7.12.8.m1.6.6.4.cmml" xref="algorithm7.12.8.m1.6.6.4">SOL</mtext></ci><apply id="algorithm7.12.8.m1.6.6.2.cmml" xref="algorithm7.12.8.m1.6.6.2"><union id="algorithm7.12.8.m1.6.6.2.3.cmml" xref="algorithm7.12.8.m1.6.6.2.3"></union><ci id="algorithm7.12.8.m1.6.6.2.4a.cmml" xref="algorithm7.12.8.m1.6.6.2.4"><mtext id="algorithm7.12.8.m1.6.6.2.4.cmml" xref="algorithm7.12.8.m1.6.6.2.4">SOL</mtext></ci><set id="algorithm7.12.8.m1.6.6.2.2.3.cmml" xref="algorithm7.12.8.m1.6.6.2.2.2"><apply id="algorithm7.12.8.m1.5.5.1.1.1.1.cmml" xref="algorithm7.12.8.m1.5.5.1.1.1.1"><times id="algorithm7.12.8.m1.5.5.1.1.1.1.1.cmml" xref="algorithm7.12.8.m1.5.5.1.1.1.1.1"></times><apply id="algorithm7.12.8.m1.5.5.1.1.1.1.2.cmml" xref="algorithm7.12.8.m1.5.5.1.1.1.1.2"><csymbol cd="ambiguous" id="algorithm7.12.8.m1.5.5.1.1.1.1.2.1.cmml" xref="algorithm7.12.8.m1.5.5.1.1.1.1.2">subscript</csymbol><ci id="algorithm7.12.8.m1.5.5.1.1.1.1.2.2a.cmml" xref="algorithm7.12.8.m1.5.5.1.1.1.1.2.2"><mtext class="ltx_font_smallcaps" id="algorithm7.12.8.m1.5.5.1.1.1.1.2.2.cmml" xref="algorithm7.12.8.m1.5.5.1.1.1.1.2.2">Min</mtext></ci><apply id="algorithm7.12.8.m1.1.1.1.cmml" xref="algorithm7.12.8.m1.1.1.1"><times id="algorithm7.12.8.m1.1.1.1.2.cmml" xref="algorithm7.12.8.m1.1.1.1.2"></times><apply id="algorithm7.12.8.m1.1.1.1.3.cmml" xref="algorithm7.12.8.m1.1.1.1.3"><csymbol cd="ambiguous" id="algorithm7.12.8.m1.1.1.1.3.1.cmml" xref="algorithm7.12.8.m1.1.1.1.3">subscript</csymbol><ci id="algorithm7.12.8.m1.1.1.1.3.2.cmml" xref="algorithm7.12.8.m1.1.1.1.3.2">𝑓</ci><ci id="algorithm7.12.8.m1.1.1.1.3.3.cmml" xref="algorithm7.12.8.m1.1.1.1.3.3">𝑥</ci></apply><ci id="algorithm7.12.8.m1.1.1.1.1.cmml" xref="algorithm7.12.8.m1.1.1.1.1">𝑣</ci></apply></apply><ci id="algorithm7.12.8.m1.3.3.cmml" xref="algorithm7.12.8.m1.3.3">𝑗</ci></apply><apply id="algorithm7.12.8.m1.6.6.2.2.2.2.cmml" xref="algorithm7.12.8.m1.6.6.2.2.2.2"><times id="algorithm7.12.8.m1.6.6.2.2.2.2.1.cmml" xref="algorithm7.12.8.m1.6.6.2.2.2.2.1"></times><apply id="algorithm7.12.8.m1.6.6.2.2.2.2.2.cmml" xref="algorithm7.12.8.m1.6.6.2.2.2.2.2"><csymbol cd="ambiguous" id="algorithm7.12.8.m1.6.6.2.2.2.2.2.1.cmml" xref="algorithm7.12.8.m1.6.6.2.2.2.2.2">subscript</csymbol><ci id="algorithm7.12.8.m1.6.6.2.2.2.2.2.2a.cmml" xref="algorithm7.12.8.m1.6.6.2.2.2.2.2.2"><mtext class="ltx_font_smallcaps" id="algorithm7.12.8.m1.6.6.2.2.2.2.2.2.cmml" xref="algorithm7.12.8.m1.6.6.2.2.2.2.2.2">Max</mtext></ci><apply id="algorithm7.12.8.m1.2.2.1.cmml" xref="algorithm7.12.8.m1.2.2.1"><times id="algorithm7.12.8.m1.2.2.1.2.cmml" xref="algorithm7.12.8.m1.2.2.1.2"></times><apply id="algorithm7.12.8.m1.2.2.1.3.cmml" xref="algorithm7.12.8.m1.2.2.1.3"><csymbol cd="ambiguous" id="algorithm7.12.8.m1.2.2.1.3.1.cmml" xref="algorithm7.12.8.m1.2.2.1.3">subscript</csymbol><ci id="algorithm7.12.8.m1.2.2.1.3.2.cmml" xref="algorithm7.12.8.m1.2.2.1.3.2">𝑓</ci><ci id="algorithm7.12.8.m1.2.2.1.3.3.cmml" xref="algorithm7.12.8.m1.2.2.1.3.3">𝑥</ci></apply><ci id="algorithm7.12.8.m1.2.2.1.1.cmml" xref="algorithm7.12.8.m1.2.2.1.1">𝑢</ci></apply></apply><ci id="algorithm7.12.8.m1.4.4.cmml" xref="algorithm7.12.8.m1.4.4">𝑗</ci></apply></set></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="algorithm7.12.8.m1.6c">\textnormal{SOL}\leftarrow\textnormal{SOL}\cup\{\textsc{Min}_{f_{x}(v)}(j),% \textsc{Max}_{f_{x}(u)}(j)\}</annotation><annotation encoding="application/x-llamapun" id="algorithm7.12.8.m1.6d">SOL ← SOL ∪ { Min start_POSTSUBSCRIPT italic_f start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT ( italic_v ) end_POSTSUBSCRIPT ( italic_j ) , Max start_POSTSUBSCRIPT italic_f start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT ( italic_u ) end_POSTSUBSCRIPT ( italic_j ) }</annotation></semantics></math> </div> <div class="ltx_listingline" id="algorithm7.13.9"> <span class="ltx_rule" style="width:1px;height:100%;background:black;display:inline-block;"> </span> <span class="ltx_rule" style="width:1px;height:100%;background:black;display:inline-block;"> </span> <span class="ltx_text ltx_font_bold" id="algorithm7.13.9.2">if</span> <em class="ltx_emph ltx_font_italic" id="algorithm7.13.9.1"><math alttext="f_{x}(u)\prec_{x}f_{x}(v)" class="ltx_Math" display="inline" id="algorithm7.13.9.1.m1.2"><semantics id="algorithm7.13.9.1.m1.2a"><mrow id="algorithm7.13.9.1.m1.2.3" xref="algorithm7.13.9.1.m1.2.3.cmml"><mrow id="algorithm7.13.9.1.m1.2.3.2" xref="algorithm7.13.9.1.m1.2.3.2.cmml"><msub id="algorithm7.13.9.1.m1.2.3.2.2" xref="algorithm7.13.9.1.m1.2.3.2.2.cmml"><mi id="algorithm7.13.9.1.m1.2.3.2.2.2" xref="algorithm7.13.9.1.m1.2.3.2.2.2.cmml">f</mi><mi id="algorithm7.13.9.1.m1.2.3.2.2.3" xref="algorithm7.13.9.1.m1.2.3.2.2.3.cmml">x</mi></msub><mo id="algorithm7.13.9.1.m1.2.3.2.1" xref="algorithm7.13.9.1.m1.2.3.2.1.cmml">⁢</mo><mrow id="algorithm7.13.9.1.m1.2.3.2.3.2" xref="algorithm7.13.9.1.m1.2.3.2.cmml"><mo id="algorithm7.13.9.1.m1.2.3.2.3.2.1" stretchy="false" xref="algorithm7.13.9.1.m1.2.3.2.cmml">(</mo><mi id="algorithm7.13.9.1.m1.1.1" xref="algorithm7.13.9.1.m1.1.1.cmml">u</mi><mo id="algorithm7.13.9.1.m1.2.3.2.3.2.2" stretchy="false" xref="algorithm7.13.9.1.m1.2.3.2.cmml">)</mo></mrow></mrow><msub id="algorithm7.13.9.1.m1.2.3.1" xref="algorithm7.13.9.1.m1.2.3.1.cmml"><mo id="algorithm7.13.9.1.m1.2.3.1.2" xref="algorithm7.13.9.1.m1.2.3.1.2.cmml">≺</mo><mi id="algorithm7.13.9.1.m1.2.3.1.3" xref="algorithm7.13.9.1.m1.2.3.1.3.cmml">x</mi></msub><mrow id="algorithm7.13.9.1.m1.2.3.3" xref="algorithm7.13.9.1.m1.2.3.3.cmml"><msub id="algorithm7.13.9.1.m1.2.3.3.2" xref="algorithm7.13.9.1.m1.2.3.3.2.cmml"><mi id="algorithm7.13.9.1.m1.2.3.3.2.2" xref="algorithm7.13.9.1.m1.2.3.3.2.2.cmml">f</mi><mi id="algorithm7.13.9.1.m1.2.3.3.2.3" xref="algorithm7.13.9.1.m1.2.3.3.2.3.cmml">x</mi></msub><mo id="algorithm7.13.9.1.m1.2.3.3.1" xref="algorithm7.13.9.1.m1.2.3.3.1.cmml">⁢</mo><mrow id="algorithm7.13.9.1.m1.2.3.3.3.2" xref="algorithm7.13.9.1.m1.2.3.3.cmml"><mo id="algorithm7.13.9.1.m1.2.3.3.3.2.1" stretchy="false" xref="algorithm7.13.9.1.m1.2.3.3.cmml">(</mo><mi id="algorithm7.13.9.1.m1.2.2" xref="algorithm7.13.9.1.m1.2.2.cmml">v</mi><mo id="algorithm7.13.9.1.m1.2.3.3.3.2.2" stretchy="false" xref="algorithm7.13.9.1.m1.2.3.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="algorithm7.13.9.1.m1.2b"><apply id="algorithm7.13.9.1.m1.2.3.cmml" xref="algorithm7.13.9.1.m1.2.3"><apply id="algorithm7.13.9.1.m1.2.3.1.cmml" xref="algorithm7.13.9.1.m1.2.3.1"><csymbol cd="ambiguous" id="algorithm7.13.9.1.m1.2.3.1.1.cmml" xref="algorithm7.13.9.1.m1.2.3.1">subscript</csymbol><csymbol cd="latexml" id="algorithm7.13.9.1.m1.2.3.1.2.cmml" xref="algorithm7.13.9.1.m1.2.3.1.2">precedes</csymbol><ci id="algorithm7.13.9.1.m1.2.3.1.3.cmml" xref="algorithm7.13.9.1.m1.2.3.1.3">𝑥</ci></apply><apply id="algorithm7.13.9.1.m1.2.3.2.cmml" xref="algorithm7.13.9.1.m1.2.3.2"><times id="algorithm7.13.9.1.m1.2.3.2.1.cmml" xref="algorithm7.13.9.1.m1.2.3.2.1"></times><apply id="algorithm7.13.9.1.m1.2.3.2.2.cmml" xref="algorithm7.13.9.1.m1.2.3.2.2"><csymbol cd="ambiguous" id="algorithm7.13.9.1.m1.2.3.2.2.1.cmml" xref="algorithm7.13.9.1.m1.2.3.2.2">subscript</csymbol><ci id="algorithm7.13.9.1.m1.2.3.2.2.2.cmml" xref="algorithm7.13.9.1.m1.2.3.2.2.2">𝑓</ci><ci id="algorithm7.13.9.1.m1.2.3.2.2.3.cmml" xref="algorithm7.13.9.1.m1.2.3.2.2.3">𝑥</ci></apply><ci id="algorithm7.13.9.1.m1.1.1.cmml" xref="algorithm7.13.9.1.m1.1.1">𝑢</ci></apply><apply id="algorithm7.13.9.1.m1.2.3.3.cmml" xref="algorithm7.13.9.1.m1.2.3.3"><times id="algorithm7.13.9.1.m1.2.3.3.1.cmml" xref="algorithm7.13.9.1.m1.2.3.3.1"></times><apply id="algorithm7.13.9.1.m1.2.3.3.2.cmml" xref="algorithm7.13.9.1.m1.2.3.3.2"><csymbol cd="ambiguous" id="algorithm7.13.9.1.m1.2.3.3.2.1.cmml" xref="algorithm7.13.9.1.m1.2.3.3.2">subscript</csymbol><ci id="algorithm7.13.9.1.m1.2.3.3.2.2.cmml" xref="algorithm7.13.9.1.m1.2.3.3.2.2">𝑓</ci><ci id="algorithm7.13.9.1.m1.2.3.3.2.3.cmml" xref="algorithm7.13.9.1.m1.2.3.3.2.3">𝑥</ci></apply><ci id="algorithm7.13.9.1.m1.2.2.cmml" xref="algorithm7.13.9.1.m1.2.2">𝑣</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="algorithm7.13.9.1.m1.2c">f_{x}(u)\prec_{x}f_{x}(v)</annotation><annotation encoding="application/x-llamapun" id="algorithm7.13.9.1.m1.2d">italic_f start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT ( italic_u ) ≺ start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT italic_f start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT ( italic_v )</annotation></semantics></math></em> <span class="ltx_text ltx_font_bold" id="algorithm7.13.9.3">then</span> </div> <div class="ltx_listingline" id="algorithm7.14.10"> <span class="ltx_rule" style="width:1px;height:100%;background:black;display:inline-block;"> </span> <span class="ltx_rule" style="width:1px;height:100%;background:black;display:inline-block;"> </span> <span class="ltx_rule" style="width:1px;height:100%;background:black;display:inline-block;"> </span> <math alttext="\textnormal{SOL}\leftarrow\textnormal{SOL}\cup\{\textsc{Min}_{f_{x}(u)}(j),% \textsc{Max}_{f_{x}(v)}(j)\}" class="ltx_Math" display="inline" id="algorithm7.14.10.m1.6"><semantics id="algorithm7.14.10.m1.6a"><mrow id="algorithm7.14.10.m1.6.6" xref="algorithm7.14.10.m1.6.6.cmml"><mtext id="algorithm7.14.10.m1.6.6.4" xref="algorithm7.14.10.m1.6.6.4a.cmml">SOL</mtext><mo id="algorithm7.14.10.m1.6.6.3" stretchy="false" xref="algorithm7.14.10.m1.6.6.3.cmml">←</mo><mrow id="algorithm7.14.10.m1.6.6.2" xref="algorithm7.14.10.m1.6.6.2.cmml"><mtext id="algorithm7.14.10.m1.6.6.2.4" xref="algorithm7.14.10.m1.6.6.2.4a.cmml">SOL</mtext><mo id="algorithm7.14.10.m1.6.6.2.3" xref="algorithm7.14.10.m1.6.6.2.3.cmml">∪</mo><mrow id="algorithm7.14.10.m1.6.6.2.2.2" xref="algorithm7.14.10.m1.6.6.2.2.3.cmml"><mo id="algorithm7.14.10.m1.6.6.2.2.2.3" stretchy="false" xref="algorithm7.14.10.m1.6.6.2.2.3.cmml">{</mo><mrow id="algorithm7.14.10.m1.5.5.1.1.1.1" xref="algorithm7.14.10.m1.5.5.1.1.1.1.cmml"><msub id="algorithm7.14.10.m1.5.5.1.1.1.1.2" xref="algorithm7.14.10.m1.5.5.1.1.1.1.2.cmml"><mtext class="ltx_font_smallcaps" id="algorithm7.14.10.m1.5.5.1.1.1.1.2.2" xref="algorithm7.14.10.m1.5.5.1.1.1.1.2.2a.cmml">Min</mtext><mrow id="algorithm7.14.10.m1.1.1.1" xref="algorithm7.14.10.m1.1.1.1.cmml"><msub id="algorithm7.14.10.m1.1.1.1.3" xref="algorithm7.14.10.m1.1.1.1.3.cmml"><mi id="algorithm7.14.10.m1.1.1.1.3.2" xref="algorithm7.14.10.m1.1.1.1.3.2.cmml">f</mi><mi id="algorithm7.14.10.m1.1.1.1.3.3" xref="algorithm7.14.10.m1.1.1.1.3.3.cmml">x</mi></msub><mo id="algorithm7.14.10.m1.1.1.1.2" xref="algorithm7.14.10.m1.1.1.1.2.cmml">⁢</mo><mrow id="algorithm7.14.10.m1.1.1.1.4.2" xref="algorithm7.14.10.m1.1.1.1.cmml"><mo id="algorithm7.14.10.m1.1.1.1.4.2.1" stretchy="false" xref="algorithm7.14.10.m1.1.1.1.cmml">(</mo><mi id="algorithm7.14.10.m1.1.1.1.1" xref="algorithm7.14.10.m1.1.1.1.1.cmml">u</mi><mo id="algorithm7.14.10.m1.1.1.1.4.2.2" stretchy="false" xref="algorithm7.14.10.m1.1.1.1.cmml">)</mo></mrow></mrow></msub><mo id="algorithm7.14.10.m1.5.5.1.1.1.1.1" xref="algorithm7.14.10.m1.5.5.1.1.1.1.1.cmml">⁢</mo><mrow id="algorithm7.14.10.m1.5.5.1.1.1.1.3.2" xref="algorithm7.14.10.m1.5.5.1.1.1.1.cmml"><mo id="algorithm7.14.10.m1.5.5.1.1.1.1.3.2.1" stretchy="false" xref="algorithm7.14.10.m1.5.5.1.1.1.1.cmml">(</mo><mi id="algorithm7.14.10.m1.3.3" xref="algorithm7.14.10.m1.3.3.cmml">j</mi><mo id="algorithm7.14.10.m1.5.5.1.1.1.1.3.2.2" stretchy="false" xref="algorithm7.14.10.m1.5.5.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="algorithm7.14.10.m1.6.6.2.2.2.4" xref="algorithm7.14.10.m1.6.6.2.2.3.cmml">,</mo><mrow id="algorithm7.14.10.m1.6.6.2.2.2.2" xref="algorithm7.14.10.m1.6.6.2.2.2.2.cmml"><msub id="algorithm7.14.10.m1.6.6.2.2.2.2.2" xref="algorithm7.14.10.m1.6.6.2.2.2.2.2.cmml"><mtext class="ltx_font_smallcaps" id="algorithm7.14.10.m1.6.6.2.2.2.2.2.2" xref="algorithm7.14.10.m1.6.6.2.2.2.2.2.2a.cmml">Max</mtext><mrow id="algorithm7.14.10.m1.2.2.1" xref="algorithm7.14.10.m1.2.2.1.cmml"><msub id="algorithm7.14.10.m1.2.2.1.3" xref="algorithm7.14.10.m1.2.2.1.3.cmml"><mi id="algorithm7.14.10.m1.2.2.1.3.2" xref="algorithm7.14.10.m1.2.2.1.3.2.cmml">f</mi><mi id="algorithm7.14.10.m1.2.2.1.3.3" xref="algorithm7.14.10.m1.2.2.1.3.3.cmml">x</mi></msub><mo id="algorithm7.14.10.m1.2.2.1.2" xref="algorithm7.14.10.m1.2.2.1.2.cmml">⁢</mo><mrow id="algorithm7.14.10.m1.2.2.1.4.2" xref="algorithm7.14.10.m1.2.2.1.cmml"><mo id="algorithm7.14.10.m1.2.2.1.4.2.1" stretchy="false" xref="algorithm7.14.10.m1.2.2.1.cmml">(</mo><mi id="algorithm7.14.10.m1.2.2.1.1" xref="algorithm7.14.10.m1.2.2.1.1.cmml">v</mi><mo id="algorithm7.14.10.m1.2.2.1.4.2.2" stretchy="false" xref="algorithm7.14.10.m1.2.2.1.cmml">)</mo></mrow></mrow></msub><mo id="algorithm7.14.10.m1.6.6.2.2.2.2.1" xref="algorithm7.14.10.m1.6.6.2.2.2.2.1.cmml">⁢</mo><mrow id="algorithm7.14.10.m1.6.6.2.2.2.2.3.2" xref="algorithm7.14.10.m1.6.6.2.2.2.2.cmml"><mo id="algorithm7.14.10.m1.6.6.2.2.2.2.3.2.1" stretchy="false" xref="algorithm7.14.10.m1.6.6.2.2.2.2.cmml">(</mo><mi id="algorithm7.14.10.m1.4.4" xref="algorithm7.14.10.m1.4.4.cmml">j</mi><mo id="algorithm7.14.10.m1.6.6.2.2.2.2.3.2.2" stretchy="false" xref="algorithm7.14.10.m1.6.6.2.2.2.2.cmml">)</mo></mrow></mrow><mo id="algorithm7.14.10.m1.6.6.2.2.2.5" stretchy="false" xref="algorithm7.14.10.m1.6.6.2.2.3.cmml">}</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="algorithm7.14.10.m1.6b"><apply id="algorithm7.14.10.m1.6.6.cmml" xref="algorithm7.14.10.m1.6.6"><ci id="algorithm7.14.10.m1.6.6.3.cmml" xref="algorithm7.14.10.m1.6.6.3">←</ci><ci id="algorithm7.14.10.m1.6.6.4a.cmml" xref="algorithm7.14.10.m1.6.6.4"><mtext id="algorithm7.14.10.m1.6.6.4.cmml" xref="algorithm7.14.10.m1.6.6.4">SOL</mtext></ci><apply id="algorithm7.14.10.m1.6.6.2.cmml" xref="algorithm7.14.10.m1.6.6.2"><union id="algorithm7.14.10.m1.6.6.2.3.cmml" xref="algorithm7.14.10.m1.6.6.2.3"></union><ci id="algorithm7.14.10.m1.6.6.2.4a.cmml" xref="algorithm7.14.10.m1.6.6.2.4"><mtext id="algorithm7.14.10.m1.6.6.2.4.cmml" xref="algorithm7.14.10.m1.6.6.2.4">SOL</mtext></ci><set id="algorithm7.14.10.m1.6.6.2.2.3.cmml" xref="algorithm7.14.10.m1.6.6.2.2.2"><apply id="algorithm7.14.10.m1.5.5.1.1.1.1.cmml" xref="algorithm7.14.10.m1.5.5.1.1.1.1"><times id="algorithm7.14.10.m1.5.5.1.1.1.1.1.cmml" xref="algorithm7.14.10.m1.5.5.1.1.1.1.1"></times><apply id="algorithm7.14.10.m1.5.5.1.1.1.1.2.cmml" xref="algorithm7.14.10.m1.5.5.1.1.1.1.2"><csymbol cd="ambiguous" id="algorithm7.14.10.m1.5.5.1.1.1.1.2.1.cmml" xref="algorithm7.14.10.m1.5.5.1.1.1.1.2">subscript</csymbol><ci id="algorithm7.14.10.m1.5.5.1.1.1.1.2.2a.cmml" xref="algorithm7.14.10.m1.5.5.1.1.1.1.2.2"><mtext class="ltx_font_smallcaps" id="algorithm7.14.10.m1.5.5.1.1.1.1.2.2.cmml" xref="algorithm7.14.10.m1.5.5.1.1.1.1.2.2">Min</mtext></ci><apply id="algorithm7.14.10.m1.1.1.1.cmml" xref="algorithm7.14.10.m1.1.1.1"><times id="algorithm7.14.10.m1.1.1.1.2.cmml" xref="algorithm7.14.10.m1.1.1.1.2"></times><apply id="algorithm7.14.10.m1.1.1.1.3.cmml" xref="algorithm7.14.10.m1.1.1.1.3"><csymbol cd="ambiguous" id="algorithm7.14.10.m1.1.1.1.3.1.cmml" xref="algorithm7.14.10.m1.1.1.1.3">subscript</csymbol><ci id="algorithm7.14.10.m1.1.1.1.3.2.cmml" xref="algorithm7.14.10.m1.1.1.1.3.2">𝑓</ci><ci id="algorithm7.14.10.m1.1.1.1.3.3.cmml" xref="algorithm7.14.10.m1.1.1.1.3.3">𝑥</ci></apply><ci id="algorithm7.14.10.m1.1.1.1.1.cmml" xref="algorithm7.14.10.m1.1.1.1.1">𝑢</ci></apply></apply><ci id="algorithm7.14.10.m1.3.3.cmml" xref="algorithm7.14.10.m1.3.3">𝑗</ci></apply><apply id="algorithm7.14.10.m1.6.6.2.2.2.2.cmml" xref="algorithm7.14.10.m1.6.6.2.2.2.2"><times id="algorithm7.14.10.m1.6.6.2.2.2.2.1.cmml" xref="algorithm7.14.10.m1.6.6.2.2.2.2.1"></times><apply id="algorithm7.14.10.m1.6.6.2.2.2.2.2.cmml" xref="algorithm7.14.10.m1.6.6.2.2.2.2.2"><csymbol cd="ambiguous" id="algorithm7.14.10.m1.6.6.2.2.2.2.2.1.cmml" xref="algorithm7.14.10.m1.6.6.2.2.2.2.2">subscript</csymbol><ci id="algorithm7.14.10.m1.6.6.2.2.2.2.2.2a.cmml" xref="algorithm7.14.10.m1.6.6.2.2.2.2.2.2"><mtext class="ltx_font_smallcaps" id="algorithm7.14.10.m1.6.6.2.2.2.2.2.2.cmml" xref="algorithm7.14.10.m1.6.6.2.2.2.2.2.2">Max</mtext></ci><apply id="algorithm7.14.10.m1.2.2.1.cmml" xref="algorithm7.14.10.m1.2.2.1"><times id="algorithm7.14.10.m1.2.2.1.2.cmml" xref="algorithm7.14.10.m1.2.2.1.2"></times><apply id="algorithm7.14.10.m1.2.2.1.3.cmml" xref="algorithm7.14.10.m1.2.2.1.3"><csymbol cd="ambiguous" id="algorithm7.14.10.m1.2.2.1.3.1.cmml" xref="algorithm7.14.10.m1.2.2.1.3">subscript</csymbol><ci id="algorithm7.14.10.m1.2.2.1.3.2.cmml" xref="algorithm7.14.10.m1.2.2.1.3.2">𝑓</ci><ci id="algorithm7.14.10.m1.2.2.1.3.3.cmml" xref="algorithm7.14.10.m1.2.2.1.3.3">𝑥</ci></apply><ci id="algorithm7.14.10.m1.2.2.1.1.cmml" xref="algorithm7.14.10.m1.2.2.1.1">𝑣</ci></apply></apply><ci id="algorithm7.14.10.m1.4.4.cmml" xref="algorithm7.14.10.m1.4.4">𝑗</ci></apply></set></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="algorithm7.14.10.m1.6c">\textnormal{SOL}\leftarrow\textnormal{SOL}\cup\{\textsc{Min}_{f_{x}(u)}(j),% \textsc{Max}_{f_{x}(v)}(j)\}</annotation><annotation encoding="application/x-llamapun" id="algorithm7.14.10.m1.6d">SOL ← SOL ∪ { Min start_POSTSUBSCRIPT italic_f start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT ( italic_u ) end_POSTSUBSCRIPT ( italic_j ) , Max start_POSTSUBSCRIPT italic_f start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT ( italic_v ) end_POSTSUBSCRIPT ( italic_j ) }</annotation></semantics></math> </div> <div class="ltx_listingline" id="algorithm7.18.14"> <span class="ltx_rule" style="width:1px;height:100%;background:black;display:inline-block;"> </span> <span class="ltx_text ltx_font_bold" id="algorithm7.18.14.5">if</span> <em class="ltx_emph ltx_font_italic" id="algorithm7.18.14.4"><math alttext="\exists" class="ltx_Math" display="inline" id="algorithm7.15.11.1.m1.1"><semantics id="algorithm7.15.11.1.m1.1a"><mo id="algorithm7.15.11.1.m1.1.1" xref="algorithm7.15.11.1.m1.1.1.cmml">∃</mo><annotation-xml encoding="MathML-Content" id="algorithm7.15.11.1.m1.1b"><exists id="algorithm7.15.11.1.m1.1.1.cmml" xref="algorithm7.15.11.1.m1.1.1"></exists></annotation-xml><annotation encoding="application/x-tex" id="algorithm7.15.11.1.m1.1c">\exists</annotation><annotation encoding="application/x-llamapun" id="algorithm7.15.11.1.m1.1d">∃</annotation></semantics></math> S-node <math alttext="x" class="ltx_Math" display="inline" id="algorithm7.16.12.2.m2.1"><semantics id="algorithm7.16.12.2.m2.1a"><mi id="algorithm7.16.12.2.m2.1.1" xref="algorithm7.16.12.2.m2.1.1.cmml">x</mi><annotation-xml encoding="MathML-Content" id="algorithm7.16.12.2.m2.1b"><ci id="algorithm7.16.12.2.m2.1.1.cmml" xref="algorithm7.16.12.2.m2.1.1">𝑥</ci></annotation-xml><annotation encoding="application/x-tex" id="algorithm7.16.12.2.m2.1c">x</annotation><annotation encoding="application/x-llamapun" id="algorithm7.16.12.2.m2.1d">italic_x</annotation></semantics></math> such that <math alttext="u\in G_{x}\setminus\textnormal{parent}(x)" class="ltx_Math" display="inline" id="algorithm7.17.13.3.m3.1"><semantics id="algorithm7.17.13.3.m3.1a"><mrow id="algorithm7.17.13.3.m3.1.2" xref="algorithm7.17.13.3.m3.1.2.cmml"><mi id="algorithm7.17.13.3.m3.1.2.2" xref="algorithm7.17.13.3.m3.1.2.2.cmml">u</mi><mo id="algorithm7.17.13.3.m3.1.2.1" xref="algorithm7.17.13.3.m3.1.2.1.cmml">∈</mo><mrow id="algorithm7.17.13.3.m3.1.2.3" xref="algorithm7.17.13.3.m3.1.2.3.cmml"><msub id="algorithm7.17.13.3.m3.1.2.3.2" xref="algorithm7.17.13.3.m3.1.2.3.2.cmml"><mi id="algorithm7.17.13.3.m3.1.2.3.2.2" xref="algorithm7.17.13.3.m3.1.2.3.2.2.cmml">G</mi><mi id="algorithm7.17.13.3.m3.1.2.3.2.3" xref="algorithm7.17.13.3.m3.1.2.3.2.3.cmml">x</mi></msub><mo id="algorithm7.17.13.3.m3.1.2.3.1" xref="algorithm7.17.13.3.m3.1.2.3.1.cmml">∖</mo><mrow id="algorithm7.17.13.3.m3.1.2.3.3" xref="algorithm7.17.13.3.m3.1.2.3.3.cmml"><mtext id="algorithm7.17.13.3.m3.1.2.3.3.2" xref="algorithm7.17.13.3.m3.1.2.3.3.2b.cmml"><em class="ltx_emph ltx_font_upright" id="algorithm7.17.13.3.m3.1.2.3.3.2.1nest">parent</em></mtext><mo id="algorithm7.17.13.3.m3.1.2.3.3.1" xref="algorithm7.17.13.3.m3.1.2.3.3.1.cmml">⁢</mo><mrow id="algorithm7.17.13.3.m3.1.2.3.3.3.2" xref="algorithm7.17.13.3.m3.1.2.3.3.cmml"><mo id="algorithm7.17.13.3.m3.1.2.3.3.3.2.1" stretchy="false" xref="algorithm7.17.13.3.m3.1.2.3.3.cmml">(</mo><mi id="algorithm7.17.13.3.m3.1.1" xref="algorithm7.17.13.3.m3.1.1.cmml">x</mi><mo id="algorithm7.17.13.3.m3.1.2.3.3.3.2.2" stretchy="false" xref="algorithm7.17.13.3.m3.1.2.3.3.cmml">)</mo></mrow></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="algorithm7.17.13.3.m3.1b"><apply id="algorithm7.17.13.3.m3.1.2.cmml" xref="algorithm7.17.13.3.m3.1.2"><in id="algorithm7.17.13.3.m3.1.2.1.cmml" xref="algorithm7.17.13.3.m3.1.2.1"></in><ci id="algorithm7.17.13.3.m3.1.2.2.cmml" xref="algorithm7.17.13.3.m3.1.2.2">𝑢</ci><apply id="algorithm7.17.13.3.m3.1.2.3.cmml" xref="algorithm7.17.13.3.m3.1.2.3"><setdiff id="algorithm7.17.13.3.m3.1.2.3.1.cmml" xref="algorithm7.17.13.3.m3.1.2.3.1"></setdiff><apply id="algorithm7.17.13.3.m3.1.2.3.2.cmml" xref="algorithm7.17.13.3.m3.1.2.3.2"><csymbol cd="ambiguous" id="algorithm7.17.13.3.m3.1.2.3.2.1.cmml" xref="algorithm7.17.13.3.m3.1.2.3.2">subscript</csymbol><ci id="algorithm7.17.13.3.m3.1.2.3.2.2.cmml" xref="algorithm7.17.13.3.m3.1.2.3.2.2">𝐺</ci><ci id="algorithm7.17.13.3.m3.1.2.3.2.3.cmml" xref="algorithm7.17.13.3.m3.1.2.3.2.3">𝑥</ci></apply><apply id="algorithm7.17.13.3.m3.1.2.3.3.cmml" xref="algorithm7.17.13.3.m3.1.2.3.3"><times id="algorithm7.17.13.3.m3.1.2.3.3.1.cmml" xref="algorithm7.17.13.3.m3.1.2.3.3.1"></times><ci id="algorithm7.17.13.3.m3.1.2.3.3.2b.cmml" xref="algorithm7.17.13.3.m3.1.2.3.3.2"><mtext id="algorithm7.17.13.3.m3.1.2.3.3.2.cmml" xref="algorithm7.17.13.3.m3.1.2.3.3.2"><em class="ltx_emph ltx_font_upright" id="algorithm7.17.13.3.m3.1.2.3.3.2.1anest">parent</em></mtext></ci><ci id="algorithm7.17.13.3.m3.1.1.cmml" xref="algorithm7.17.13.3.m3.1.1">𝑥</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="algorithm7.17.13.3.m3.1c">u\in G_{x}\setminus\textnormal{parent}(x)</annotation><annotation encoding="application/x-llamapun" id="algorithm7.17.13.3.m3.1d">italic_u ∈ italic_G start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT ∖ parent ( italic_x )</annotation></semantics></math>, <math alttext="\ell(v)\in T\setminus T_{x}" class="ltx_Math" display="inline" id="algorithm7.18.14.4.m4.1"><semantics id="algorithm7.18.14.4.m4.1a"><mrow id="algorithm7.18.14.4.m4.1.2" xref="algorithm7.18.14.4.m4.1.2.cmml"><mrow id="algorithm7.18.14.4.m4.1.2.2" xref="algorithm7.18.14.4.m4.1.2.2.cmml"><mi id="algorithm7.18.14.4.m4.1.2.2.2" mathvariant="normal" xref="algorithm7.18.14.4.m4.1.2.2.2.cmml">ℓ</mi><mo id="algorithm7.18.14.4.m4.1.2.2.1" xref="algorithm7.18.14.4.m4.1.2.2.1.cmml">⁢</mo><mrow id="algorithm7.18.14.4.m4.1.2.2.3.2" xref="algorithm7.18.14.4.m4.1.2.2.cmml"><mo id="algorithm7.18.14.4.m4.1.2.2.3.2.1" stretchy="false" xref="algorithm7.18.14.4.m4.1.2.2.cmml">(</mo><mi id="algorithm7.18.14.4.m4.1.1" xref="algorithm7.18.14.4.m4.1.1.cmml">v</mi><mo id="algorithm7.18.14.4.m4.1.2.2.3.2.2" stretchy="false" xref="algorithm7.18.14.4.m4.1.2.2.cmml">)</mo></mrow></mrow><mo id="algorithm7.18.14.4.m4.1.2.1" xref="algorithm7.18.14.4.m4.1.2.1.cmml">∈</mo><mrow id="algorithm7.18.14.4.m4.1.2.3" xref="algorithm7.18.14.4.m4.1.2.3.cmml"><mi id="algorithm7.18.14.4.m4.1.2.3.2" xref="algorithm7.18.14.4.m4.1.2.3.2.cmml">T</mi><mo id="algorithm7.18.14.4.m4.1.2.3.1" xref="algorithm7.18.14.4.m4.1.2.3.1.cmml">∖</mo><msub id="algorithm7.18.14.4.m4.1.2.3.3" xref="algorithm7.18.14.4.m4.1.2.3.3.cmml"><mi id="algorithm7.18.14.4.m4.1.2.3.3.2" xref="algorithm7.18.14.4.m4.1.2.3.3.2.cmml">T</mi><mi id="algorithm7.18.14.4.m4.1.2.3.3.3" xref="algorithm7.18.14.4.m4.1.2.3.3.3.cmml">x</mi></msub></mrow></mrow><annotation-xml encoding="MathML-Content" id="algorithm7.18.14.4.m4.1b"><apply id="algorithm7.18.14.4.m4.1.2.cmml" xref="algorithm7.18.14.4.m4.1.2"><in id="algorithm7.18.14.4.m4.1.2.1.cmml" xref="algorithm7.18.14.4.m4.1.2.1"></in><apply id="algorithm7.18.14.4.m4.1.2.2.cmml" xref="algorithm7.18.14.4.m4.1.2.2"><times id="algorithm7.18.14.4.m4.1.2.2.1.cmml" xref="algorithm7.18.14.4.m4.1.2.2.1"></times><ci id="algorithm7.18.14.4.m4.1.2.2.2.cmml" xref="algorithm7.18.14.4.m4.1.2.2.2">ℓ</ci><ci id="algorithm7.18.14.4.m4.1.1.cmml" xref="algorithm7.18.14.4.m4.1.1">𝑣</ci></apply><apply id="algorithm7.18.14.4.m4.1.2.3.cmml" xref="algorithm7.18.14.4.m4.1.2.3"><setdiff id="algorithm7.18.14.4.m4.1.2.3.1.cmml" xref="algorithm7.18.14.4.m4.1.2.3.1"></setdiff><ci id="algorithm7.18.14.4.m4.1.2.3.2.cmml" xref="algorithm7.18.14.4.m4.1.2.3.2">𝑇</ci><apply id="algorithm7.18.14.4.m4.1.2.3.3.cmml" xref="algorithm7.18.14.4.m4.1.2.3.3"><csymbol cd="ambiguous" id="algorithm7.18.14.4.m4.1.2.3.3.1.cmml" xref="algorithm7.18.14.4.m4.1.2.3.3">subscript</csymbol><ci id="algorithm7.18.14.4.m4.1.2.3.3.2.cmml" xref="algorithm7.18.14.4.m4.1.2.3.3.2">𝑇</ci><ci id="algorithm7.18.14.4.m4.1.2.3.3.3.cmml" xref="algorithm7.18.14.4.m4.1.2.3.3.3">𝑥</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="algorithm7.18.14.4.m4.1c">\ell(v)\in T\setminus T_{x}</annotation><annotation encoding="application/x-llamapun" id="algorithm7.18.14.4.m4.1d">roman_ℓ ( italic_v ) ∈ italic_T ∖ italic_T start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT</annotation></semantics></math></em> <span class="ltx_text ltx_font_bold" id="algorithm7.18.14.6">then</span> </div> <div class="ltx_listingline" id="algorithm7.19.15"> <span class="ltx_rule" style="width:1px;height:100%;background:black;display:inline-block;"> </span> <span class="ltx_rule" style="width:1px;height:100%;background:black;display:inline-block;"> </span> <math alttext="\textnormal{SOL}\leftarrow\textnormal{SOL}\cup\{\textsc{Min}_{u}(j)\}" class="ltx_Math" display="inline" id="algorithm7.19.15.m1.2"><semantics id="algorithm7.19.15.m1.2a"><mrow id="algorithm7.19.15.m1.2.2" xref="algorithm7.19.15.m1.2.2.cmml"><mtext id="algorithm7.19.15.m1.2.2.3" xref="algorithm7.19.15.m1.2.2.3a.cmml">SOL</mtext><mo id="algorithm7.19.15.m1.2.2.2" stretchy="false" xref="algorithm7.19.15.m1.2.2.2.cmml">←</mo><mrow id="algorithm7.19.15.m1.2.2.1" xref="algorithm7.19.15.m1.2.2.1.cmml"><mtext id="algorithm7.19.15.m1.2.2.1.3" xref="algorithm7.19.15.m1.2.2.1.3a.cmml">SOL</mtext><mo id="algorithm7.19.15.m1.2.2.1.2" xref="algorithm7.19.15.m1.2.2.1.2.cmml">∪</mo><mrow id="algorithm7.19.15.m1.2.2.1.1.1" xref="algorithm7.19.15.m1.2.2.1.1.2.cmml"><mo id="algorithm7.19.15.m1.2.2.1.1.1.2" stretchy="false" xref="algorithm7.19.15.m1.2.2.1.1.2.cmml">{</mo><mrow id="algorithm7.19.15.m1.2.2.1.1.1.1" xref="algorithm7.19.15.m1.2.2.1.1.1.1.cmml"><msub id="algorithm7.19.15.m1.2.2.1.1.1.1.2" xref="algorithm7.19.15.m1.2.2.1.1.1.1.2.cmml"><mtext class="ltx_font_smallcaps" id="algorithm7.19.15.m1.2.2.1.1.1.1.2.2" xref="algorithm7.19.15.m1.2.2.1.1.1.1.2.2a.cmml">Min</mtext><mi id="algorithm7.19.15.m1.2.2.1.1.1.1.2.3" xref="algorithm7.19.15.m1.2.2.1.1.1.1.2.3.cmml">u</mi></msub><mo id="algorithm7.19.15.m1.2.2.1.1.1.1.1" xref="algorithm7.19.15.m1.2.2.1.1.1.1.1.cmml">⁢</mo><mrow id="algorithm7.19.15.m1.2.2.1.1.1.1.3.2" xref="algorithm7.19.15.m1.2.2.1.1.1.1.cmml"><mo id="algorithm7.19.15.m1.2.2.1.1.1.1.3.2.1" stretchy="false" xref="algorithm7.19.15.m1.2.2.1.1.1.1.cmml">(</mo><mi id="algorithm7.19.15.m1.1.1" xref="algorithm7.19.15.m1.1.1.cmml">j</mi><mo id="algorithm7.19.15.m1.2.2.1.1.1.1.3.2.2" stretchy="false" xref="algorithm7.19.15.m1.2.2.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="algorithm7.19.15.m1.2.2.1.1.1.3" stretchy="false" xref="algorithm7.19.15.m1.2.2.1.1.2.cmml">}</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="algorithm7.19.15.m1.2b"><apply id="algorithm7.19.15.m1.2.2.cmml" xref="algorithm7.19.15.m1.2.2"><ci id="algorithm7.19.15.m1.2.2.2.cmml" xref="algorithm7.19.15.m1.2.2.2">←</ci><ci id="algorithm7.19.15.m1.2.2.3a.cmml" xref="algorithm7.19.15.m1.2.2.3"><mtext id="algorithm7.19.15.m1.2.2.3.cmml" xref="algorithm7.19.15.m1.2.2.3">SOL</mtext></ci><apply id="algorithm7.19.15.m1.2.2.1.cmml" xref="algorithm7.19.15.m1.2.2.1"><union id="algorithm7.19.15.m1.2.2.1.2.cmml" xref="algorithm7.19.15.m1.2.2.1.2"></union><ci id="algorithm7.19.15.m1.2.2.1.3a.cmml" xref="algorithm7.19.15.m1.2.2.1.3"><mtext id="algorithm7.19.15.m1.2.2.1.3.cmml" xref="algorithm7.19.15.m1.2.2.1.3">SOL</mtext></ci><set id="algorithm7.19.15.m1.2.2.1.1.2.cmml" xref="algorithm7.19.15.m1.2.2.1.1.1"><apply id="algorithm7.19.15.m1.2.2.1.1.1.1.cmml" xref="algorithm7.19.15.m1.2.2.1.1.1.1"><times id="algorithm7.19.15.m1.2.2.1.1.1.1.1.cmml" xref="algorithm7.19.15.m1.2.2.1.1.1.1.1"></times><apply id="algorithm7.19.15.m1.2.2.1.1.1.1.2.cmml" xref="algorithm7.19.15.m1.2.2.1.1.1.1.2"><csymbol cd="ambiguous" id="algorithm7.19.15.m1.2.2.1.1.1.1.2.1.cmml" xref="algorithm7.19.15.m1.2.2.1.1.1.1.2">subscript</csymbol><ci id="algorithm7.19.15.m1.2.2.1.1.1.1.2.2a.cmml" xref="algorithm7.19.15.m1.2.2.1.1.1.1.2.2"><mtext class="ltx_font_smallcaps" id="algorithm7.19.15.m1.2.2.1.1.1.1.2.2.cmml" xref="algorithm7.19.15.m1.2.2.1.1.1.1.2.2">Min</mtext></ci><ci id="algorithm7.19.15.m1.2.2.1.1.1.1.2.3.cmml" xref="algorithm7.19.15.m1.2.2.1.1.1.1.2.3">𝑢</ci></apply><ci id="algorithm7.19.15.m1.1.1.cmml" xref="algorithm7.19.15.m1.1.1">𝑗</ci></apply></set></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="algorithm7.19.15.m1.2c">\textnormal{SOL}\leftarrow\textnormal{SOL}\cup\{\textsc{Min}_{u}(j)\}</annotation><annotation encoding="application/x-llamapun" id="algorithm7.19.15.m1.2d">SOL ← SOL ∪ { Min start_POSTSUBSCRIPT italic_u end_POSTSUBSCRIPT ( italic_j ) }</annotation></semantics></math> </div> <div class="ltx_listingline" id="algorithm7.23.19"> <span class="ltx_rule" style="width:1px;height:100%;background:black;display:inline-block;"> </span> <span class="ltx_text ltx_font_bold" id="algorithm7.23.19.5">if</span> <em class="ltx_emph ltx_font_italic" id="algorithm7.23.19.4"><math alttext="\exists" class="ltx_Math" display="inline" id="algorithm7.20.16.1.m1.1"><semantics id="algorithm7.20.16.1.m1.1a"><mo id="algorithm7.20.16.1.m1.1.1" xref="algorithm7.20.16.1.m1.1.1.cmml">∃</mo><annotation-xml encoding="MathML-Content" id="algorithm7.20.16.1.m1.1b"><exists id="algorithm7.20.16.1.m1.1.1.cmml" xref="algorithm7.20.16.1.m1.1.1"></exists></annotation-xml><annotation encoding="application/x-tex" id="algorithm7.20.16.1.m1.1c">\exists</annotation><annotation encoding="application/x-llamapun" id="algorithm7.20.16.1.m1.1d">∃</annotation></semantics></math> S-node <math alttext="x" class="ltx_Math" display="inline" id="algorithm7.21.17.2.m2.1"><semantics id="algorithm7.21.17.2.m2.1a"><mi id="algorithm7.21.17.2.m2.1.1" xref="algorithm7.21.17.2.m2.1.1.cmml">x</mi><annotation-xml encoding="MathML-Content" id="algorithm7.21.17.2.m2.1b"><ci id="algorithm7.21.17.2.m2.1.1.cmml" xref="algorithm7.21.17.2.m2.1.1">𝑥</ci></annotation-xml><annotation encoding="application/x-tex" id="algorithm7.21.17.2.m2.1c">x</annotation><annotation encoding="application/x-llamapun" id="algorithm7.21.17.2.m2.1d">italic_x</annotation></semantics></math> such that <math alttext="v\in G_{x}\setminus\textnormal{parent}(x)" class="ltx_Math" display="inline" id="algorithm7.22.18.3.m3.1"><semantics id="algorithm7.22.18.3.m3.1a"><mrow id="algorithm7.22.18.3.m3.1.2" xref="algorithm7.22.18.3.m3.1.2.cmml"><mi id="algorithm7.22.18.3.m3.1.2.2" xref="algorithm7.22.18.3.m3.1.2.2.cmml">v</mi><mo id="algorithm7.22.18.3.m3.1.2.1" xref="algorithm7.22.18.3.m3.1.2.1.cmml">∈</mo><mrow id="algorithm7.22.18.3.m3.1.2.3" xref="algorithm7.22.18.3.m3.1.2.3.cmml"><msub id="algorithm7.22.18.3.m3.1.2.3.2" xref="algorithm7.22.18.3.m3.1.2.3.2.cmml"><mi id="algorithm7.22.18.3.m3.1.2.3.2.2" xref="algorithm7.22.18.3.m3.1.2.3.2.2.cmml">G</mi><mi id="algorithm7.22.18.3.m3.1.2.3.2.3" xref="algorithm7.22.18.3.m3.1.2.3.2.3.cmml">x</mi></msub><mo id="algorithm7.22.18.3.m3.1.2.3.1" xref="algorithm7.22.18.3.m3.1.2.3.1.cmml">∖</mo><mrow id="algorithm7.22.18.3.m3.1.2.3.3" xref="algorithm7.22.18.3.m3.1.2.3.3.cmml"><mtext id="algorithm7.22.18.3.m3.1.2.3.3.2" xref="algorithm7.22.18.3.m3.1.2.3.3.2b.cmml"><em class="ltx_emph ltx_font_upright" id="algorithm7.22.18.3.m3.1.2.3.3.2.1nest">parent</em></mtext><mo id="algorithm7.22.18.3.m3.1.2.3.3.1" xref="algorithm7.22.18.3.m3.1.2.3.3.1.cmml">⁢</mo><mrow id="algorithm7.22.18.3.m3.1.2.3.3.3.2" xref="algorithm7.22.18.3.m3.1.2.3.3.cmml"><mo id="algorithm7.22.18.3.m3.1.2.3.3.3.2.1" stretchy="false" xref="algorithm7.22.18.3.m3.1.2.3.3.cmml">(</mo><mi id="algorithm7.22.18.3.m3.1.1" xref="algorithm7.22.18.3.m3.1.1.cmml">x</mi><mo id="algorithm7.22.18.3.m3.1.2.3.3.3.2.2" stretchy="false" xref="algorithm7.22.18.3.m3.1.2.3.3.cmml">)</mo></mrow></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="algorithm7.22.18.3.m3.1b"><apply id="algorithm7.22.18.3.m3.1.2.cmml" xref="algorithm7.22.18.3.m3.1.2"><in id="algorithm7.22.18.3.m3.1.2.1.cmml" xref="algorithm7.22.18.3.m3.1.2.1"></in><ci id="algorithm7.22.18.3.m3.1.2.2.cmml" xref="algorithm7.22.18.3.m3.1.2.2">𝑣</ci><apply id="algorithm7.22.18.3.m3.1.2.3.cmml" xref="algorithm7.22.18.3.m3.1.2.3"><setdiff id="algorithm7.22.18.3.m3.1.2.3.1.cmml" xref="algorithm7.22.18.3.m3.1.2.3.1"></setdiff><apply id="algorithm7.22.18.3.m3.1.2.3.2.cmml" xref="algorithm7.22.18.3.m3.1.2.3.2"><csymbol cd="ambiguous" id="algorithm7.22.18.3.m3.1.2.3.2.1.cmml" xref="algorithm7.22.18.3.m3.1.2.3.2">subscript</csymbol><ci id="algorithm7.22.18.3.m3.1.2.3.2.2.cmml" xref="algorithm7.22.18.3.m3.1.2.3.2.2">𝐺</ci><ci id="algorithm7.22.18.3.m3.1.2.3.2.3.cmml" xref="algorithm7.22.18.3.m3.1.2.3.2.3">𝑥</ci></apply><apply id="algorithm7.22.18.3.m3.1.2.3.3.cmml" xref="algorithm7.22.18.3.m3.1.2.3.3"><times id="algorithm7.22.18.3.m3.1.2.3.3.1.cmml" xref="algorithm7.22.18.3.m3.1.2.3.3.1"></times><ci id="algorithm7.22.18.3.m3.1.2.3.3.2b.cmml" xref="algorithm7.22.18.3.m3.1.2.3.3.2"><mtext id="algorithm7.22.18.3.m3.1.2.3.3.2.cmml" xref="algorithm7.22.18.3.m3.1.2.3.3.2"><em class="ltx_emph ltx_font_upright" id="algorithm7.22.18.3.m3.1.2.3.3.2.1anest">parent</em></mtext></ci><ci id="algorithm7.22.18.3.m3.1.1.cmml" xref="algorithm7.22.18.3.m3.1.1">𝑥</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="algorithm7.22.18.3.m3.1c">v\in G_{x}\setminus\textnormal{parent}(x)</annotation><annotation encoding="application/x-llamapun" id="algorithm7.22.18.3.m3.1d">italic_v ∈ italic_G start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT ∖ parent ( italic_x )</annotation></semantics></math>, <math alttext="\ell(u)\in T\setminus T_{x}" class="ltx_Math" display="inline" id="algorithm7.23.19.4.m4.1"><semantics id="algorithm7.23.19.4.m4.1a"><mrow id="algorithm7.23.19.4.m4.1.2" xref="algorithm7.23.19.4.m4.1.2.cmml"><mrow id="algorithm7.23.19.4.m4.1.2.2" xref="algorithm7.23.19.4.m4.1.2.2.cmml"><mi id="algorithm7.23.19.4.m4.1.2.2.2" mathvariant="normal" xref="algorithm7.23.19.4.m4.1.2.2.2.cmml">ℓ</mi><mo id="algorithm7.23.19.4.m4.1.2.2.1" xref="algorithm7.23.19.4.m4.1.2.2.1.cmml">⁢</mo><mrow id="algorithm7.23.19.4.m4.1.2.2.3.2" xref="algorithm7.23.19.4.m4.1.2.2.cmml"><mo id="algorithm7.23.19.4.m4.1.2.2.3.2.1" stretchy="false" xref="algorithm7.23.19.4.m4.1.2.2.cmml">(</mo><mi id="algorithm7.23.19.4.m4.1.1" xref="algorithm7.23.19.4.m4.1.1.cmml">u</mi><mo id="algorithm7.23.19.4.m4.1.2.2.3.2.2" stretchy="false" xref="algorithm7.23.19.4.m4.1.2.2.cmml">)</mo></mrow></mrow><mo id="algorithm7.23.19.4.m4.1.2.1" xref="algorithm7.23.19.4.m4.1.2.1.cmml">∈</mo><mrow id="algorithm7.23.19.4.m4.1.2.3" xref="algorithm7.23.19.4.m4.1.2.3.cmml"><mi id="algorithm7.23.19.4.m4.1.2.3.2" xref="algorithm7.23.19.4.m4.1.2.3.2.cmml">T</mi><mo id="algorithm7.23.19.4.m4.1.2.3.1" xref="algorithm7.23.19.4.m4.1.2.3.1.cmml">∖</mo><msub id="algorithm7.23.19.4.m4.1.2.3.3" xref="algorithm7.23.19.4.m4.1.2.3.3.cmml"><mi id="algorithm7.23.19.4.m4.1.2.3.3.2" xref="algorithm7.23.19.4.m4.1.2.3.3.2.cmml">T</mi><mi id="algorithm7.23.19.4.m4.1.2.3.3.3" xref="algorithm7.23.19.4.m4.1.2.3.3.3.cmml">x</mi></msub></mrow></mrow><annotation-xml encoding="MathML-Content" id="algorithm7.23.19.4.m4.1b"><apply id="algorithm7.23.19.4.m4.1.2.cmml" xref="algorithm7.23.19.4.m4.1.2"><in id="algorithm7.23.19.4.m4.1.2.1.cmml" xref="algorithm7.23.19.4.m4.1.2.1"></in><apply id="algorithm7.23.19.4.m4.1.2.2.cmml" xref="algorithm7.23.19.4.m4.1.2.2"><times id="algorithm7.23.19.4.m4.1.2.2.1.cmml" xref="algorithm7.23.19.4.m4.1.2.2.1"></times><ci id="algorithm7.23.19.4.m4.1.2.2.2.cmml" xref="algorithm7.23.19.4.m4.1.2.2.2">ℓ</ci><ci id="algorithm7.23.19.4.m4.1.1.cmml" xref="algorithm7.23.19.4.m4.1.1">𝑢</ci></apply><apply id="algorithm7.23.19.4.m4.1.2.3.cmml" xref="algorithm7.23.19.4.m4.1.2.3"><setdiff id="algorithm7.23.19.4.m4.1.2.3.1.cmml" xref="algorithm7.23.19.4.m4.1.2.3.1"></setdiff><ci id="algorithm7.23.19.4.m4.1.2.3.2.cmml" xref="algorithm7.23.19.4.m4.1.2.3.2">𝑇</ci><apply id="algorithm7.23.19.4.m4.1.2.3.3.cmml" xref="algorithm7.23.19.4.m4.1.2.3.3"><csymbol cd="ambiguous" id="algorithm7.23.19.4.m4.1.2.3.3.1.cmml" xref="algorithm7.23.19.4.m4.1.2.3.3">subscript</csymbol><ci id="algorithm7.23.19.4.m4.1.2.3.3.2.cmml" xref="algorithm7.23.19.4.m4.1.2.3.3.2">𝑇</ci><ci id="algorithm7.23.19.4.m4.1.2.3.3.3.cmml" xref="algorithm7.23.19.4.m4.1.2.3.3.3">𝑥</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="algorithm7.23.19.4.m4.1c">\ell(u)\in T\setminus T_{x}</annotation><annotation encoding="application/x-llamapun" id="algorithm7.23.19.4.m4.1d">roman_ℓ ( italic_u ) ∈ italic_T ∖ italic_T start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT</annotation></semantics></math></em> <span class="ltx_text ltx_font_bold" id="algorithm7.23.19.6">then</span> </div> <div class="ltx_listingline" id="algorithm7.24.20"> <span class="ltx_rule" style="width:1px;height:100%;background:black;display:inline-block;"> </span> <span class="ltx_rule" style="width:1px;height:100%;background:black;display:inline-block;"> </span> <math alttext="\textnormal{SOL}\leftarrow\textnormal{SOL}\cup\{\textsc{Min}_{v}(j)\}" class="ltx_Math" display="inline" id="algorithm7.24.20.m1.2"><semantics id="algorithm7.24.20.m1.2a"><mrow id="algorithm7.24.20.m1.2.2" xref="algorithm7.24.20.m1.2.2.cmml"><mtext id="algorithm7.24.20.m1.2.2.3" xref="algorithm7.24.20.m1.2.2.3a.cmml">SOL</mtext><mo id="algorithm7.24.20.m1.2.2.2" stretchy="false" xref="algorithm7.24.20.m1.2.2.2.cmml">←</mo><mrow id="algorithm7.24.20.m1.2.2.1" xref="algorithm7.24.20.m1.2.2.1.cmml"><mtext id="algorithm7.24.20.m1.2.2.1.3" xref="algorithm7.24.20.m1.2.2.1.3a.cmml">SOL</mtext><mo id="algorithm7.24.20.m1.2.2.1.2" xref="algorithm7.24.20.m1.2.2.1.2.cmml">∪</mo><mrow id="algorithm7.24.20.m1.2.2.1.1.1" xref="algorithm7.24.20.m1.2.2.1.1.2.cmml"><mo id="algorithm7.24.20.m1.2.2.1.1.1.2" stretchy="false" xref="algorithm7.24.20.m1.2.2.1.1.2.cmml">{</mo><mrow id="algorithm7.24.20.m1.2.2.1.1.1.1" xref="algorithm7.24.20.m1.2.2.1.1.1.1.cmml"><msub id="algorithm7.24.20.m1.2.2.1.1.1.1.2" xref="algorithm7.24.20.m1.2.2.1.1.1.1.2.cmml"><mtext class="ltx_font_smallcaps" id="algorithm7.24.20.m1.2.2.1.1.1.1.2.2" xref="algorithm7.24.20.m1.2.2.1.1.1.1.2.2a.cmml">Min</mtext><mi id="algorithm7.24.20.m1.2.2.1.1.1.1.2.3" xref="algorithm7.24.20.m1.2.2.1.1.1.1.2.3.cmml">v</mi></msub><mo id="algorithm7.24.20.m1.2.2.1.1.1.1.1" xref="algorithm7.24.20.m1.2.2.1.1.1.1.1.cmml">⁢</mo><mrow id="algorithm7.24.20.m1.2.2.1.1.1.1.3.2" xref="algorithm7.24.20.m1.2.2.1.1.1.1.cmml"><mo id="algorithm7.24.20.m1.2.2.1.1.1.1.3.2.1" stretchy="false" xref="algorithm7.24.20.m1.2.2.1.1.1.1.cmml">(</mo><mi id="algorithm7.24.20.m1.1.1" xref="algorithm7.24.20.m1.1.1.cmml">j</mi><mo id="algorithm7.24.20.m1.2.2.1.1.1.1.3.2.2" stretchy="false" xref="algorithm7.24.20.m1.2.2.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="algorithm7.24.20.m1.2.2.1.1.1.3" stretchy="false" xref="algorithm7.24.20.m1.2.2.1.1.2.cmml">}</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="algorithm7.24.20.m1.2b"><apply id="algorithm7.24.20.m1.2.2.cmml" xref="algorithm7.24.20.m1.2.2"><ci id="algorithm7.24.20.m1.2.2.2.cmml" xref="algorithm7.24.20.m1.2.2.2">←</ci><ci id="algorithm7.24.20.m1.2.2.3a.cmml" xref="algorithm7.24.20.m1.2.2.3"><mtext id="algorithm7.24.20.m1.2.2.3.cmml" xref="algorithm7.24.20.m1.2.2.3">SOL</mtext></ci><apply id="algorithm7.24.20.m1.2.2.1.cmml" xref="algorithm7.24.20.m1.2.2.1"><union id="algorithm7.24.20.m1.2.2.1.2.cmml" xref="algorithm7.24.20.m1.2.2.1.2"></union><ci id="algorithm7.24.20.m1.2.2.1.3a.cmml" xref="algorithm7.24.20.m1.2.2.1.3"><mtext id="algorithm7.24.20.m1.2.2.1.3.cmml" xref="algorithm7.24.20.m1.2.2.1.3">SOL</mtext></ci><set id="algorithm7.24.20.m1.2.2.1.1.2.cmml" xref="algorithm7.24.20.m1.2.2.1.1.1"><apply id="algorithm7.24.20.m1.2.2.1.1.1.1.cmml" xref="algorithm7.24.20.m1.2.2.1.1.1.1"><times id="algorithm7.24.20.m1.2.2.1.1.1.1.1.cmml" xref="algorithm7.24.20.m1.2.2.1.1.1.1.1"></times><apply id="algorithm7.24.20.m1.2.2.1.1.1.1.2.cmml" xref="algorithm7.24.20.m1.2.2.1.1.1.1.2"><csymbol cd="ambiguous" id="algorithm7.24.20.m1.2.2.1.1.1.1.2.1.cmml" xref="algorithm7.24.20.m1.2.2.1.1.1.1.2">subscript</csymbol><ci id="algorithm7.24.20.m1.2.2.1.1.1.1.2.2a.cmml" xref="algorithm7.24.20.m1.2.2.1.1.1.1.2.2"><mtext class="ltx_font_smallcaps" id="algorithm7.24.20.m1.2.2.1.1.1.1.2.2.cmml" xref="algorithm7.24.20.m1.2.2.1.1.1.1.2.2">Min</mtext></ci><ci id="algorithm7.24.20.m1.2.2.1.1.1.1.2.3.cmml" xref="algorithm7.24.20.m1.2.2.1.1.1.1.2.3">𝑣</ci></apply><ci id="algorithm7.24.20.m1.1.1.cmml" xref="algorithm7.24.20.m1.1.1">𝑗</ci></apply></set></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="algorithm7.24.20.m1.2c">\textnormal{SOL}\leftarrow\textnormal{SOL}\cup\{\textsc{Min}_{v}(j)\}</annotation><annotation encoding="application/x-llamapun" id="algorithm7.24.20.m1.2d">SOL ← SOL ∪ { Min start_POSTSUBSCRIPT italic_v end_POSTSUBSCRIPT ( italic_j ) }</annotation></semantics></math> </div> <div class="ltx_listingline" id="algorithm7.26.22"> <span class="ltx_text ltx_font_bold" id="algorithm7.26.22.3">for</span> <em class="ltx_emph ltx_font_italic" id="algorithm7.26.22.2"><math alttext="x\in V(T)" class="ltx_Math" display="inline" id="algorithm7.25.21.1.m1.1"><semantics id="algorithm7.25.21.1.m1.1a"><mrow id="algorithm7.25.21.1.m1.1.2" xref="algorithm7.25.21.1.m1.1.2.cmml"><mi id="algorithm7.25.21.1.m1.1.2.2" xref="algorithm7.25.21.1.m1.1.2.2.cmml">x</mi><mo id="algorithm7.25.21.1.m1.1.2.1" xref="algorithm7.25.21.1.m1.1.2.1.cmml">∈</mo><mrow id="algorithm7.25.21.1.m1.1.2.3" xref="algorithm7.25.21.1.m1.1.2.3.cmml"><mi id="algorithm7.25.21.1.m1.1.2.3.2" xref="algorithm7.25.21.1.m1.1.2.3.2.cmml">V</mi><mo id="algorithm7.25.21.1.m1.1.2.3.1" xref="algorithm7.25.21.1.m1.1.2.3.1.cmml">⁢</mo><mrow id="algorithm7.25.21.1.m1.1.2.3.3.2" xref="algorithm7.25.21.1.m1.1.2.3.cmml"><mo id="algorithm7.25.21.1.m1.1.2.3.3.2.1" stretchy="false" xref="algorithm7.25.21.1.m1.1.2.3.cmml">(</mo><mi id="algorithm7.25.21.1.m1.1.1" xref="algorithm7.25.21.1.m1.1.1.cmml">T</mi><mo id="algorithm7.25.21.1.m1.1.2.3.3.2.2" stretchy="false" xref="algorithm7.25.21.1.m1.1.2.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="algorithm7.25.21.1.m1.1b"><apply id="algorithm7.25.21.1.m1.1.2.cmml" xref="algorithm7.25.21.1.m1.1.2"><in id="algorithm7.25.21.1.m1.1.2.1.cmml" xref="algorithm7.25.21.1.m1.1.2.1"></in><ci id="algorithm7.25.21.1.m1.1.2.2.cmml" xref="algorithm7.25.21.1.m1.1.2.2">𝑥</ci><apply id="algorithm7.25.21.1.m1.1.2.3.cmml" xref="algorithm7.25.21.1.m1.1.2.3"><times id="algorithm7.25.21.1.m1.1.2.3.1.cmml" xref="algorithm7.25.21.1.m1.1.2.3.1"></times><ci id="algorithm7.25.21.1.m1.1.2.3.2.cmml" xref="algorithm7.25.21.1.m1.1.2.3.2">𝑉</ci><ci id="algorithm7.25.21.1.m1.1.1.cmml" xref="algorithm7.25.21.1.m1.1.1">𝑇</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="algorithm7.25.21.1.m1.1c">x\in V(T)</annotation><annotation encoding="application/x-llamapun" id="algorithm7.25.21.1.m1.1d">italic_x ∈ italic_V ( italic_T )</annotation></semantics></math>, <math alttext="x" class="ltx_Math" display="inline" id="algorithm7.26.22.2.m2.1"><semantics id="algorithm7.26.22.2.m2.1a"><mi id="algorithm7.26.22.2.m2.1.1" xref="algorithm7.26.22.2.m2.1.1.cmml">x</mi><annotation-xml encoding="MathML-Content" id="algorithm7.26.22.2.m2.1b"><ci id="algorithm7.26.22.2.m2.1.1.cmml" xref="algorithm7.26.22.2.m2.1.1">𝑥</ci></annotation-xml><annotation encoding="application/x-tex" id="algorithm7.26.22.2.m2.1c">x</annotation><annotation encoding="application/x-llamapun" id="algorithm7.26.22.2.m2.1d">italic_x</annotation></semantics></math> is P-node</em> <span class="ltx_text ltx_font_bold" id="algorithm7.26.22.4">do</span> </div> <div class="ltx_listingline" id="algorithm7.27.23"> <span class="ltx_rule" style="width:1px;height:100%;background:black;display:inline-block;"> </span> <span class="ltx_text ltx_font_bold" id="algorithm7.27.23.2">for</span> <em class="ltx_emph ltx_font_italic" id="algorithm7.27.23.1"><math alttext="y\in C(x)" class="ltx_Math" display="inline" id="algorithm7.27.23.1.m1.1"><semantics id="algorithm7.27.23.1.m1.1a"><mrow id="algorithm7.27.23.1.m1.1.2" xref="algorithm7.27.23.1.m1.1.2.cmml"><mi id="algorithm7.27.23.1.m1.1.2.2" xref="algorithm7.27.23.1.m1.1.2.2.cmml">y</mi><mo id="algorithm7.27.23.1.m1.1.2.1" xref="algorithm7.27.23.1.m1.1.2.1.cmml">∈</mo><mrow id="algorithm7.27.23.1.m1.1.2.3" xref="algorithm7.27.23.1.m1.1.2.3.cmml"><mi id="algorithm7.27.23.1.m1.1.2.3.2" xref="algorithm7.27.23.1.m1.1.2.3.2.cmml">C</mi><mo id="algorithm7.27.23.1.m1.1.2.3.1" xref="algorithm7.27.23.1.m1.1.2.3.1.cmml">⁢</mo><mrow id="algorithm7.27.23.1.m1.1.2.3.3.2" xref="algorithm7.27.23.1.m1.1.2.3.cmml"><mo id="algorithm7.27.23.1.m1.1.2.3.3.2.1" stretchy="false" xref="algorithm7.27.23.1.m1.1.2.3.cmml">(</mo><mi id="algorithm7.27.23.1.m1.1.1" xref="algorithm7.27.23.1.m1.1.1.cmml">x</mi><mo id="algorithm7.27.23.1.m1.1.2.3.3.2.2" stretchy="false" xref="algorithm7.27.23.1.m1.1.2.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="algorithm7.27.23.1.m1.1b"><apply id="algorithm7.27.23.1.m1.1.2.cmml" xref="algorithm7.27.23.1.m1.1.2"><in id="algorithm7.27.23.1.m1.1.2.1.cmml" xref="algorithm7.27.23.1.m1.1.2.1"></in><ci id="algorithm7.27.23.1.m1.1.2.2.cmml" xref="algorithm7.27.23.1.m1.1.2.2">𝑦</ci><apply id="algorithm7.27.23.1.m1.1.2.3.cmml" xref="algorithm7.27.23.1.m1.1.2.3"><times id="algorithm7.27.23.1.m1.1.2.3.1.cmml" xref="algorithm7.27.23.1.m1.1.2.3.1"></times><ci id="algorithm7.27.23.1.m1.1.2.3.2.cmml" xref="algorithm7.27.23.1.m1.1.2.3.2">𝐶</ci><ci id="algorithm7.27.23.1.m1.1.1.cmml" xref="algorithm7.27.23.1.m1.1.1">𝑥</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="algorithm7.27.23.1.m1.1c">y\in C(x)</annotation><annotation encoding="application/x-llamapun" id="algorithm7.27.23.1.m1.1d">italic_y ∈ italic_C ( italic_x )</annotation></semantics></math></em> <span class="ltx_text ltx_font_bold" id="algorithm7.27.23.3">do</span> </div> <div class="ltx_listingline" id="algorithm7.30.26"> <span class="ltx_rule" style="width:1px;height:100%;background:black;display:inline-block;"> </span> <span class="ltx_rule" style="width:1px;height:100%;background:black;display:inline-block;"> </span> <span class="ltx_text ltx_font_bold" id="algorithm7.30.26.4">if</span> <em class="ltx_emph ltx_font_italic" id="algorithm7.30.26.3"><math alttext="\exists e=(u,v)\in\textnormal{OPT}" class="ltx_Math" display="inline" id="algorithm7.28.24.1.m1.2"><semantics id="algorithm7.28.24.1.m1.2a"><mrow id="algorithm7.28.24.1.m1.2.3" xref="algorithm7.28.24.1.m1.2.3.cmml"><mrow id="algorithm7.28.24.1.m1.2.3.2" xref="algorithm7.28.24.1.m1.2.3.2.cmml"><mo id="algorithm7.28.24.1.m1.2.3.2.1" rspace="0.167em" xref="algorithm7.28.24.1.m1.2.3.2.1.cmml">∃</mo><mi id="algorithm7.28.24.1.m1.2.3.2.2" xref="algorithm7.28.24.1.m1.2.3.2.2.cmml">e</mi></mrow><mo id="algorithm7.28.24.1.m1.2.3.3" xref="algorithm7.28.24.1.m1.2.3.3.cmml">=</mo><mrow id="algorithm7.28.24.1.m1.2.3.4.2" xref="algorithm7.28.24.1.m1.2.3.4.1.cmml"><mo id="algorithm7.28.24.1.m1.2.3.4.2.1" stretchy="false" xref="algorithm7.28.24.1.m1.2.3.4.1.cmml">(</mo><mi id="algorithm7.28.24.1.m1.1.1" xref="algorithm7.28.24.1.m1.1.1.cmml">u</mi><mo id="algorithm7.28.24.1.m1.2.3.4.2.2" xref="algorithm7.28.24.1.m1.2.3.4.1.cmml">,</mo><mi id="algorithm7.28.24.1.m1.2.2" xref="algorithm7.28.24.1.m1.2.2.cmml">v</mi><mo id="algorithm7.28.24.1.m1.2.3.4.2.3" stretchy="false" xref="algorithm7.28.24.1.m1.2.3.4.1.cmml">)</mo></mrow><mo id="algorithm7.28.24.1.m1.2.3.5" xref="algorithm7.28.24.1.m1.2.3.5.cmml">∈</mo><mtext id="algorithm7.28.24.1.m1.2.3.6" xref="algorithm7.28.24.1.m1.2.3.6b.cmml"><em class="ltx_emph ltx_font_upright" id="algorithm7.28.24.1.m1.2.3.6.1nest">OPT</em></mtext></mrow><annotation-xml encoding="MathML-Content" id="algorithm7.28.24.1.m1.2b"><apply id="algorithm7.28.24.1.m1.2.3.cmml" xref="algorithm7.28.24.1.m1.2.3"><and id="algorithm7.28.24.1.m1.2.3a.cmml" xref="algorithm7.28.24.1.m1.2.3"></and><apply id="algorithm7.28.24.1.m1.2.3b.cmml" xref="algorithm7.28.24.1.m1.2.3"><eq id="algorithm7.28.24.1.m1.2.3.3.cmml" xref="algorithm7.28.24.1.m1.2.3.3"></eq><apply id="algorithm7.28.24.1.m1.2.3.2.cmml" xref="algorithm7.28.24.1.m1.2.3.2"><exists id="algorithm7.28.24.1.m1.2.3.2.1.cmml" xref="algorithm7.28.24.1.m1.2.3.2.1"></exists><ci id="algorithm7.28.24.1.m1.2.3.2.2.cmml" xref="algorithm7.28.24.1.m1.2.3.2.2">𝑒</ci></apply><interval closure="open" id="algorithm7.28.24.1.m1.2.3.4.1.cmml" xref="algorithm7.28.24.1.m1.2.3.4.2"><ci id="algorithm7.28.24.1.m1.1.1.cmml" xref="algorithm7.28.24.1.m1.1.1">𝑢</ci><ci id="algorithm7.28.24.1.m1.2.2.cmml" xref="algorithm7.28.24.1.m1.2.2">𝑣</ci></interval></apply><apply id="algorithm7.28.24.1.m1.2.3c.cmml" xref="algorithm7.28.24.1.m1.2.3"><in id="algorithm7.28.24.1.m1.2.3.5.cmml" xref="algorithm7.28.24.1.m1.2.3.5"></in><share href="https://arxiv.org/html/2503.00712v1#algorithm7.28.24.1.m1.2.3.4.cmml" id="algorithm7.28.24.1.m1.2.3d.cmml" xref="algorithm7.28.24.1.m1.2.3"></share><ci id="algorithm7.28.24.1.m1.2.3.6b.cmml" xref="algorithm7.28.24.1.m1.2.3.6"><mtext id="algorithm7.28.24.1.m1.2.3.6.cmml" xref="algorithm7.28.24.1.m1.2.3.6"><em class="ltx_emph ltx_font_upright" id="algorithm7.28.24.1.m1.2.3.6.1anest">OPT</em></mtext></ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="algorithm7.28.24.1.m1.2c">\exists e=(u,v)\in\textnormal{OPT}</annotation><annotation encoding="application/x-llamapun" id="algorithm7.28.24.1.m1.2d">∃ italic_e = ( italic_u , italic_v ) ∈ OPT</annotation></semantics></math> such that <math alttext="h(u)\subseteq V(T_{y})" class="ltx_Math" display="inline" id="algorithm7.29.25.2.m2.2"><semantics id="algorithm7.29.25.2.m2.2a"><mrow id="algorithm7.29.25.2.m2.2.2" xref="algorithm7.29.25.2.m2.2.2.cmml"><mrow id="algorithm7.29.25.2.m2.2.2.3" xref="algorithm7.29.25.2.m2.2.2.3.cmml"><mi id="algorithm7.29.25.2.m2.2.2.3.2" xref="algorithm7.29.25.2.m2.2.2.3.2.cmml">h</mi><mo id="algorithm7.29.25.2.m2.2.2.3.1" xref="algorithm7.29.25.2.m2.2.2.3.1.cmml">⁢</mo><mrow id="algorithm7.29.25.2.m2.2.2.3.3.2" xref="algorithm7.29.25.2.m2.2.2.3.cmml"><mo id="algorithm7.29.25.2.m2.2.2.3.3.2.1" stretchy="false" xref="algorithm7.29.25.2.m2.2.2.3.cmml">(</mo><mi id="algorithm7.29.25.2.m2.1.1" xref="algorithm7.29.25.2.m2.1.1.cmml">u</mi><mo id="algorithm7.29.25.2.m2.2.2.3.3.2.2" stretchy="false" xref="algorithm7.29.25.2.m2.2.2.3.cmml">)</mo></mrow></mrow><mo id="algorithm7.29.25.2.m2.2.2.2" xref="algorithm7.29.25.2.m2.2.2.2.cmml">⊆</mo><mrow id="algorithm7.29.25.2.m2.2.2.1" xref="algorithm7.29.25.2.m2.2.2.1.cmml"><mi id="algorithm7.29.25.2.m2.2.2.1.3" xref="algorithm7.29.25.2.m2.2.2.1.3.cmml">V</mi><mo id="algorithm7.29.25.2.m2.2.2.1.2" xref="algorithm7.29.25.2.m2.2.2.1.2.cmml">⁢</mo><mrow id="algorithm7.29.25.2.m2.2.2.1.1.1" xref="algorithm7.29.25.2.m2.2.2.1.1.1.1.cmml"><mo id="algorithm7.29.25.2.m2.2.2.1.1.1.2" stretchy="false" xref="algorithm7.29.25.2.m2.2.2.1.1.1.1.cmml">(</mo><msub id="algorithm7.29.25.2.m2.2.2.1.1.1.1" xref="algorithm7.29.25.2.m2.2.2.1.1.1.1.cmml"><mi id="algorithm7.29.25.2.m2.2.2.1.1.1.1.2" xref="algorithm7.29.25.2.m2.2.2.1.1.1.1.2.cmml">T</mi><mi id="algorithm7.29.25.2.m2.2.2.1.1.1.1.3" xref="algorithm7.29.25.2.m2.2.2.1.1.1.1.3.cmml">y</mi></msub><mo id="algorithm7.29.25.2.m2.2.2.1.1.1.3" stretchy="false" xref="algorithm7.29.25.2.m2.2.2.1.1.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="algorithm7.29.25.2.m2.2b"><apply id="algorithm7.29.25.2.m2.2.2.cmml" xref="algorithm7.29.25.2.m2.2.2"><subset id="algorithm7.29.25.2.m2.2.2.2.cmml" xref="algorithm7.29.25.2.m2.2.2.2"></subset><apply id="algorithm7.29.25.2.m2.2.2.3.cmml" xref="algorithm7.29.25.2.m2.2.2.3"><times id="algorithm7.29.25.2.m2.2.2.3.1.cmml" xref="algorithm7.29.25.2.m2.2.2.3.1"></times><ci id="algorithm7.29.25.2.m2.2.2.3.2.cmml" xref="algorithm7.29.25.2.m2.2.2.3.2">ℎ</ci><ci id="algorithm7.29.25.2.m2.1.1.cmml" xref="algorithm7.29.25.2.m2.1.1">𝑢</ci></apply><apply id="algorithm7.29.25.2.m2.2.2.1.cmml" xref="algorithm7.29.25.2.m2.2.2.1"><times id="algorithm7.29.25.2.m2.2.2.1.2.cmml" xref="algorithm7.29.25.2.m2.2.2.1.2"></times><ci id="algorithm7.29.25.2.m2.2.2.1.3.cmml" xref="algorithm7.29.25.2.m2.2.2.1.3">𝑉</ci><apply id="algorithm7.29.25.2.m2.2.2.1.1.1.1.cmml" xref="algorithm7.29.25.2.m2.2.2.1.1.1"><csymbol cd="ambiguous" id="algorithm7.29.25.2.m2.2.2.1.1.1.1.1.cmml" xref="algorithm7.29.25.2.m2.2.2.1.1.1">subscript</csymbol><ci id="algorithm7.29.25.2.m2.2.2.1.1.1.1.2.cmml" xref="algorithm7.29.25.2.m2.2.2.1.1.1.1.2">𝑇</ci><ci id="algorithm7.29.25.2.m2.2.2.1.1.1.1.3.cmml" xref="algorithm7.29.25.2.m2.2.2.1.1.1.1.3">𝑦</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="algorithm7.29.25.2.m2.2c">h(u)\subseteq V(T_{y})</annotation><annotation encoding="application/x-llamapun" id="algorithm7.29.25.2.m2.2d">italic_h ( italic_u ) ⊆ italic_V ( italic_T start_POSTSUBSCRIPT italic_y end_POSTSUBSCRIPT )</annotation></semantics></math> and <math alttext="\ell(v)\subseteq V(T\setminus T_{x})" class="ltx_Math" display="inline" id="algorithm7.30.26.3.m3.2"><semantics id="algorithm7.30.26.3.m3.2a"><mrow id="algorithm7.30.26.3.m3.2.2" xref="algorithm7.30.26.3.m3.2.2.cmml"><mrow id="algorithm7.30.26.3.m3.2.2.3" xref="algorithm7.30.26.3.m3.2.2.3.cmml"><mi id="algorithm7.30.26.3.m3.2.2.3.2" mathvariant="normal" xref="algorithm7.30.26.3.m3.2.2.3.2.cmml">ℓ</mi><mo id="algorithm7.30.26.3.m3.2.2.3.1" xref="algorithm7.30.26.3.m3.2.2.3.1.cmml">⁢</mo><mrow id="algorithm7.30.26.3.m3.2.2.3.3.2" xref="algorithm7.30.26.3.m3.2.2.3.cmml"><mo id="algorithm7.30.26.3.m3.2.2.3.3.2.1" stretchy="false" xref="algorithm7.30.26.3.m3.2.2.3.cmml">(</mo><mi id="algorithm7.30.26.3.m3.1.1" xref="algorithm7.30.26.3.m3.1.1.cmml">v</mi><mo id="algorithm7.30.26.3.m3.2.2.3.3.2.2" stretchy="false" xref="algorithm7.30.26.3.m3.2.2.3.cmml">)</mo></mrow></mrow><mo id="algorithm7.30.26.3.m3.2.2.2" xref="algorithm7.30.26.3.m3.2.2.2.cmml">⊆</mo><mrow id="algorithm7.30.26.3.m3.2.2.1" xref="algorithm7.30.26.3.m3.2.2.1.cmml"><mi id="algorithm7.30.26.3.m3.2.2.1.3" xref="algorithm7.30.26.3.m3.2.2.1.3.cmml">V</mi><mo id="algorithm7.30.26.3.m3.2.2.1.2" xref="algorithm7.30.26.3.m3.2.2.1.2.cmml">⁢</mo><mrow id="algorithm7.30.26.3.m3.2.2.1.1.1" xref="algorithm7.30.26.3.m3.2.2.1.1.1.1.cmml"><mo id="algorithm7.30.26.3.m3.2.2.1.1.1.2" stretchy="false" xref="algorithm7.30.26.3.m3.2.2.1.1.1.1.cmml">(</mo><mrow id="algorithm7.30.26.3.m3.2.2.1.1.1.1" xref="algorithm7.30.26.3.m3.2.2.1.1.1.1.cmml"><mi id="algorithm7.30.26.3.m3.2.2.1.1.1.1.2" xref="algorithm7.30.26.3.m3.2.2.1.1.1.1.2.cmml">T</mi><mo id="algorithm7.30.26.3.m3.2.2.1.1.1.1.1" xref="algorithm7.30.26.3.m3.2.2.1.1.1.1.1.cmml">∖</mo><msub id="algorithm7.30.26.3.m3.2.2.1.1.1.1.3" xref="algorithm7.30.26.3.m3.2.2.1.1.1.1.3.cmml"><mi id="algorithm7.30.26.3.m3.2.2.1.1.1.1.3.2" xref="algorithm7.30.26.3.m3.2.2.1.1.1.1.3.2.cmml">T</mi><mi id="algorithm7.30.26.3.m3.2.2.1.1.1.1.3.3" xref="algorithm7.30.26.3.m3.2.2.1.1.1.1.3.3.cmml">x</mi></msub></mrow><mo id="algorithm7.30.26.3.m3.2.2.1.1.1.3" stretchy="false" xref="algorithm7.30.26.3.m3.2.2.1.1.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="algorithm7.30.26.3.m3.2b"><apply id="algorithm7.30.26.3.m3.2.2.cmml" xref="algorithm7.30.26.3.m3.2.2"><subset id="algorithm7.30.26.3.m3.2.2.2.cmml" xref="algorithm7.30.26.3.m3.2.2.2"></subset><apply id="algorithm7.30.26.3.m3.2.2.3.cmml" xref="algorithm7.30.26.3.m3.2.2.3"><times id="algorithm7.30.26.3.m3.2.2.3.1.cmml" xref="algorithm7.30.26.3.m3.2.2.3.1"></times><ci id="algorithm7.30.26.3.m3.2.2.3.2.cmml" xref="algorithm7.30.26.3.m3.2.2.3.2">ℓ</ci><ci id="algorithm7.30.26.3.m3.1.1.cmml" xref="algorithm7.30.26.3.m3.1.1">𝑣</ci></apply><apply id="algorithm7.30.26.3.m3.2.2.1.cmml" xref="algorithm7.30.26.3.m3.2.2.1"><times id="algorithm7.30.26.3.m3.2.2.1.2.cmml" xref="algorithm7.30.26.3.m3.2.2.1.2"></times><ci id="algorithm7.30.26.3.m3.2.2.1.3.cmml" xref="algorithm7.30.26.3.m3.2.2.1.3">𝑉</ci><apply id="algorithm7.30.26.3.m3.2.2.1.1.1.1.cmml" xref="algorithm7.30.26.3.m3.2.2.1.1.1"><setdiff id="algorithm7.30.26.3.m3.2.2.1.1.1.1.1.cmml" xref="algorithm7.30.26.3.m3.2.2.1.1.1.1.1"></setdiff><ci id="algorithm7.30.26.3.m3.2.2.1.1.1.1.2.cmml" xref="algorithm7.30.26.3.m3.2.2.1.1.1.1.2">𝑇</ci><apply id="algorithm7.30.26.3.m3.2.2.1.1.1.1.3.cmml" xref="algorithm7.30.26.3.m3.2.2.1.1.1.1.3"><csymbol cd="ambiguous" id="algorithm7.30.26.3.m3.2.2.1.1.1.1.3.1.cmml" xref="algorithm7.30.26.3.m3.2.2.1.1.1.1.3">subscript</csymbol><ci id="algorithm7.30.26.3.m3.2.2.1.1.1.1.3.2.cmml" xref="algorithm7.30.26.3.m3.2.2.1.1.1.1.3.2">𝑇</ci><ci id="algorithm7.30.26.3.m3.2.2.1.1.1.1.3.3.cmml" xref="algorithm7.30.26.3.m3.2.2.1.1.1.1.3.3">𝑥</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="algorithm7.30.26.3.m3.2c">\ell(v)\subseteq V(T\setminus T_{x})</annotation><annotation encoding="application/x-llamapun" id="algorithm7.30.26.3.m3.2d">roman_ℓ ( italic_v ) ⊆ italic_V ( italic_T ∖ italic_T start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT )</annotation></semantics></math></em> <span class="ltx_text ltx_font_bold" id="algorithm7.30.26.5">then</span> </div> <div class="ltx_listingline" id="algorithm7.31.27"> <span class="ltx_rule" style="width:1px;height:100%;background:black;display:inline-block;"> </span> <span class="ltx_rule" style="width:1px;height:100%;background:black;display:inline-block;"> </span> <span class="ltx_rule" style="width:1px;height:100%;background:black;display:inline-block;"> </span> <span class="ltx_text ltx_font_bold" id="algorithm7.31.27.1">mark</span> <math alttext="y" class="ltx_Math" display="inline" id="algorithm7.31.27.m1.1"><semantics id="algorithm7.31.27.m1.1a"><mi id="algorithm7.31.27.m1.1.1" xref="algorithm7.31.27.m1.1.1.cmml">y</mi><annotation-xml encoding="MathML-Content" id="algorithm7.31.27.m1.1b"><ci id="algorithm7.31.27.m1.1.1.cmml" xref="algorithm7.31.27.m1.1.1">𝑦</ci></annotation-xml><annotation encoding="application/x-tex" id="algorithm7.31.27.m1.1c">y</annotation><annotation encoding="application/x-llamapun" id="algorithm7.31.27.m1.1d">italic_y</annotation></semantics></math> as “good” </div> <div class="ltx_listingline" id="algorithm7.34.30"> <span class="ltx_rule" style="width:1px;height:100%;background:black;display:inline-block;"> </span> <span class="ltx_text ltx_font_bold" id="algorithm7.34.30.1">construct</span> <math alttext="C^{\prime\prime}(x)" class="ltx_Math" display="inline" id="algorithm7.32.28.m1.1"><semantics id="algorithm7.32.28.m1.1a"><mrow id="algorithm7.32.28.m1.1.2" xref="algorithm7.32.28.m1.1.2.cmml"><msup id="algorithm7.32.28.m1.1.2.2" xref="algorithm7.32.28.m1.1.2.2.cmml"><mi id="algorithm7.32.28.m1.1.2.2.2" xref="algorithm7.32.28.m1.1.2.2.2.cmml">C</mi><mo id="algorithm7.32.28.m1.1.2.2.3" xref="algorithm7.32.28.m1.1.2.2.3.cmml">′′</mo></msup><mo id="algorithm7.32.28.m1.1.2.1" xref="algorithm7.32.28.m1.1.2.1.cmml">⁢</mo><mrow id="algorithm7.32.28.m1.1.2.3.2" xref="algorithm7.32.28.m1.1.2.cmml"><mo id="algorithm7.32.28.m1.1.2.3.2.1" stretchy="false" xref="algorithm7.32.28.m1.1.2.cmml">(</mo><mi id="algorithm7.32.28.m1.1.1" xref="algorithm7.32.28.m1.1.1.cmml">x</mi><mo id="algorithm7.32.28.m1.1.2.3.2.2" stretchy="false" xref="algorithm7.32.28.m1.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="algorithm7.32.28.m1.1b"><apply id="algorithm7.32.28.m1.1.2.cmml" xref="algorithm7.32.28.m1.1.2"><times id="algorithm7.32.28.m1.1.2.1.cmml" xref="algorithm7.32.28.m1.1.2.1"></times><apply id="algorithm7.32.28.m1.1.2.2.cmml" xref="algorithm7.32.28.m1.1.2.2"><csymbol cd="ambiguous" id="algorithm7.32.28.m1.1.2.2.1.cmml" xref="algorithm7.32.28.m1.1.2.2">superscript</csymbol><ci id="algorithm7.32.28.m1.1.2.2.2.cmml" xref="algorithm7.32.28.m1.1.2.2.2">𝐶</ci><ci id="algorithm7.32.28.m1.1.2.2.3.cmml" xref="algorithm7.32.28.m1.1.2.2.3">′′</ci></apply><ci id="algorithm7.32.28.m1.1.1.cmml" xref="algorithm7.32.28.m1.1.1">𝑥</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="algorithm7.32.28.m1.1c">C^{\prime\prime}(x)</annotation><annotation encoding="application/x-llamapun" id="algorithm7.32.28.m1.1d">italic_C start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT ( italic_x )</annotation></semantics></math> from <math alttext="C^{\prime}(x)" class="ltx_Math" display="inline" id="algorithm7.33.29.m2.1"><semantics id="algorithm7.33.29.m2.1a"><mrow id="algorithm7.33.29.m2.1.2" xref="algorithm7.33.29.m2.1.2.cmml"><msup id="algorithm7.33.29.m2.1.2.2" xref="algorithm7.33.29.m2.1.2.2.cmml"><mi id="algorithm7.33.29.m2.1.2.2.2" xref="algorithm7.33.29.m2.1.2.2.2.cmml">C</mi><mo id="algorithm7.33.29.m2.1.2.2.3" xref="algorithm7.33.29.m2.1.2.2.3.cmml">′</mo></msup><mo id="algorithm7.33.29.m2.1.2.1" xref="algorithm7.33.29.m2.1.2.1.cmml">⁢</mo><mrow id="algorithm7.33.29.m2.1.2.3.2" xref="algorithm7.33.29.m2.1.2.cmml"><mo id="algorithm7.33.29.m2.1.2.3.2.1" stretchy="false" xref="algorithm7.33.29.m2.1.2.cmml">(</mo><mi id="algorithm7.33.29.m2.1.1" xref="algorithm7.33.29.m2.1.1.cmml">x</mi><mo id="algorithm7.33.29.m2.1.2.3.2.2" stretchy="false" xref="algorithm7.33.29.m2.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="algorithm7.33.29.m2.1b"><apply id="algorithm7.33.29.m2.1.2.cmml" xref="algorithm7.33.29.m2.1.2"><times id="algorithm7.33.29.m2.1.2.1.cmml" xref="algorithm7.33.29.m2.1.2.1"></times><apply id="algorithm7.33.29.m2.1.2.2.cmml" xref="algorithm7.33.29.m2.1.2.2"><csymbol cd="ambiguous" id="algorithm7.33.29.m2.1.2.2.1.cmml" xref="algorithm7.33.29.m2.1.2.2">superscript</csymbol><ci id="algorithm7.33.29.m2.1.2.2.2.cmml" xref="algorithm7.33.29.m2.1.2.2.2">𝐶</ci><ci id="algorithm7.33.29.m2.1.2.2.3.cmml" xref="algorithm7.33.29.m2.1.2.2.3">′</ci></apply><ci id="algorithm7.33.29.m2.1.1.cmml" xref="algorithm7.33.29.m2.1.1">𝑥</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="algorithm7.33.29.m2.1c">C^{\prime}(x)</annotation><annotation encoding="application/x-llamapun" id="algorithm7.33.29.m2.1d">italic_C start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ( italic_x )</annotation></semantics></math> (as in Algorithm <a class="ltx_ref" href="https://arxiv.org/html/2503.00712v1#algorithm6" title="In “Cycle” Cuts: ‣ 4.2.2 The Streaming Algorithm ‣ 4.2 Two-to-Three Augmentation ‣ 4 Vertex Connectivity Augmentation in Link-Arrival Model ‣ Streaming Algorithms for Network Design"><span class="ltx_text ltx_ref_tag">6</span></a>) by contracting all supernodes corresponding to a “good” node <math alttext="y\in C(x)" class="ltx_Math" display="inline" id="algorithm7.34.30.m3.1"><semantics id="algorithm7.34.30.m3.1a"><mrow id="algorithm7.34.30.m3.1.2" xref="algorithm7.34.30.m3.1.2.cmml"><mi id="algorithm7.34.30.m3.1.2.2" xref="algorithm7.34.30.m3.1.2.2.cmml">y</mi><mo id="algorithm7.34.30.m3.1.2.1" xref="algorithm7.34.30.m3.1.2.1.cmml">∈</mo><mrow id="algorithm7.34.30.m3.1.2.3" xref="algorithm7.34.30.m3.1.2.3.cmml"><mi id="algorithm7.34.30.m3.1.2.3.2" xref="algorithm7.34.30.m3.1.2.3.2.cmml">C</mi><mo id="algorithm7.34.30.m3.1.2.3.1" xref="algorithm7.34.30.m3.1.2.3.1.cmml">⁢</mo><mrow id="algorithm7.34.30.m3.1.2.3.3.2" xref="algorithm7.34.30.m3.1.2.3.cmml"><mo id="algorithm7.34.30.m3.1.2.3.3.2.1" stretchy="false" xref="algorithm7.34.30.m3.1.2.3.cmml">(</mo><mi id="algorithm7.34.30.m3.1.1" xref="algorithm7.34.30.m3.1.1.cmml">x</mi><mo id="algorithm7.34.30.m3.1.2.3.3.2.2" stretchy="false" xref="algorithm7.34.30.m3.1.2.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="algorithm7.34.30.m3.1b"><apply id="algorithm7.34.30.m3.1.2.cmml" xref="algorithm7.34.30.m3.1.2"><in id="algorithm7.34.30.m3.1.2.1.cmml" xref="algorithm7.34.30.m3.1.2.1"></in><ci id="algorithm7.34.30.m3.1.2.2.cmml" xref="algorithm7.34.30.m3.1.2.2">𝑦</ci><apply id="algorithm7.34.30.m3.1.2.3.cmml" xref="algorithm7.34.30.m3.1.2.3"><times id="algorithm7.34.30.m3.1.2.3.1.cmml" xref="algorithm7.34.30.m3.1.2.3.1"></times><ci id="algorithm7.34.30.m3.1.2.3.2.cmml" xref="algorithm7.34.30.m3.1.2.3.2">𝐶</ci><ci id="algorithm7.34.30.m3.1.1.cmml" xref="algorithm7.34.30.m3.1.1">𝑥</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="algorithm7.34.30.m3.1c">y\in C(x)</annotation><annotation encoding="application/x-llamapun" id="algorithm7.34.30.m3.1d">italic_y ∈ italic_C ( italic_x )</annotation></semantics></math> into one “good” supernode </div> <div class="ltx_listingline" id="algorithm7.36.32"> <span class="ltx_rule" style="width:1px;height:100%;background:black;display:inline-block;"> </span> <math alttext="H_{x}^{\prime}\subseteq H_{x}\leftarrow" class="ltx_Math" display="inline" id="algorithm7.35.31.m1.1"><semantics id="algorithm7.35.31.m1.1a"><mrow id="algorithm7.35.31.m1.1.1" xref="algorithm7.35.31.m1.1.1.cmml"><msubsup id="algorithm7.35.31.m1.1.1.2" xref="algorithm7.35.31.m1.1.1.2.cmml"><mi id="algorithm7.35.31.m1.1.1.2.2.2" xref="algorithm7.35.31.m1.1.1.2.2.2.cmml">H</mi><mi id="algorithm7.35.31.m1.1.1.2.2.3" xref="algorithm7.35.31.m1.1.1.2.2.3.cmml">x</mi><mo id="algorithm7.35.31.m1.1.1.2.3" xref="algorithm7.35.31.m1.1.1.2.3.cmml">′</mo></msubsup><mo id="algorithm7.35.31.m1.1.1.3" xref="algorithm7.35.31.m1.1.1.3.cmml">⊆</mo><msub id="algorithm7.35.31.m1.1.1.4" xref="algorithm7.35.31.m1.1.1.4.cmml"><mi id="algorithm7.35.31.m1.1.1.4.2" xref="algorithm7.35.31.m1.1.1.4.2.cmml">H</mi><mi id="algorithm7.35.31.m1.1.1.4.3" xref="algorithm7.35.31.m1.1.1.4.3.cmml">x</mi></msub><mo id="algorithm7.35.31.m1.1.1.5" stretchy="false" xref="algorithm7.35.31.m1.1.1.5.cmml">←</mo><mi id="algorithm7.35.31.m1.1.1.6" xref="algorithm7.35.31.m1.1.1.6.cmml"></mi></mrow><annotation-xml encoding="MathML-Content" id="algorithm7.35.31.m1.1b"><apply id="algorithm7.35.31.m1.1.1.cmml" xref="algorithm7.35.31.m1.1.1"><and id="algorithm7.35.31.m1.1.1a.cmml" xref="algorithm7.35.31.m1.1.1"></and><apply id="algorithm7.35.31.m1.1.1b.cmml" xref="algorithm7.35.31.m1.1.1"><subset id="algorithm7.35.31.m1.1.1.3.cmml" xref="algorithm7.35.31.m1.1.1.3"></subset><apply id="algorithm7.35.31.m1.1.1.2.cmml" xref="algorithm7.35.31.m1.1.1.2"><csymbol cd="ambiguous" id="algorithm7.35.31.m1.1.1.2.1.cmml" xref="algorithm7.35.31.m1.1.1.2">superscript</csymbol><apply id="algorithm7.35.31.m1.1.1.2.2.cmml" xref="algorithm7.35.31.m1.1.1.2"><csymbol cd="ambiguous" id="algorithm7.35.31.m1.1.1.2.2.1.cmml" xref="algorithm7.35.31.m1.1.1.2">subscript</csymbol><ci id="algorithm7.35.31.m1.1.1.2.2.2.cmml" xref="algorithm7.35.31.m1.1.1.2.2.2">𝐻</ci><ci id="algorithm7.35.31.m1.1.1.2.2.3.cmml" xref="algorithm7.35.31.m1.1.1.2.2.3">𝑥</ci></apply><ci id="algorithm7.35.31.m1.1.1.2.3.cmml" xref="algorithm7.35.31.m1.1.1.2.3">′</ci></apply><apply id="algorithm7.35.31.m1.1.1.4.cmml" xref="algorithm7.35.31.m1.1.1.4"><csymbol cd="ambiguous" id="algorithm7.35.31.m1.1.1.4.1.cmml" xref="algorithm7.35.31.m1.1.1.4">subscript</csymbol><ci id="algorithm7.35.31.m1.1.1.4.2.cmml" xref="algorithm7.35.31.m1.1.1.4.2">𝐻</ci><ci id="algorithm7.35.31.m1.1.1.4.3.cmml" xref="algorithm7.35.31.m1.1.1.4.3">𝑥</ci></apply></apply><apply id="algorithm7.35.31.m1.1.1c.cmml" xref="algorithm7.35.31.m1.1.1"><ci id="algorithm7.35.31.m1.1.1.5.cmml" xref="algorithm7.35.31.m1.1.1.5">←</ci><share href="https://arxiv.org/html/2503.00712v1#algorithm7.35.31.m1.1.1.4.cmml" id="algorithm7.35.31.m1.1.1d.cmml" xref="algorithm7.35.31.m1.1.1"></share><csymbol cd="latexml" id="algorithm7.35.31.m1.1.1.6.cmml" xref="algorithm7.35.31.m1.1.1.6">absent</csymbol></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="algorithm7.35.31.m1.1c">H_{x}^{\prime}\subseteq H_{x}\leftarrow</annotation><annotation encoding="application/x-llamapun" id="algorithm7.35.31.m1.1d">italic_H start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ⊆ italic_H start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT ←</annotation></semantics></math> MST on <math alttext="C^{\prime\prime}(x)" class="ltx_Math" display="inline" id="algorithm7.36.32.m2.1"><semantics id="algorithm7.36.32.m2.1a"><mrow id="algorithm7.36.32.m2.1.2" xref="algorithm7.36.32.m2.1.2.cmml"><msup id="algorithm7.36.32.m2.1.2.2" xref="algorithm7.36.32.m2.1.2.2.cmml"><mi id="algorithm7.36.32.m2.1.2.2.2" xref="algorithm7.36.32.m2.1.2.2.2.cmml">C</mi><mo id="algorithm7.36.32.m2.1.2.2.3" xref="algorithm7.36.32.m2.1.2.2.3.cmml">′′</mo></msup><mo id="algorithm7.36.32.m2.1.2.1" xref="algorithm7.36.32.m2.1.2.1.cmml">⁢</mo><mrow id="algorithm7.36.32.m2.1.2.3.2" xref="algorithm7.36.32.m2.1.2.cmml"><mo id="algorithm7.36.32.m2.1.2.3.2.1" stretchy="false" xref="algorithm7.36.32.m2.1.2.cmml">(</mo><mi id="algorithm7.36.32.m2.1.1" xref="algorithm7.36.32.m2.1.1.cmml">x</mi><mo id="algorithm7.36.32.m2.1.2.3.2.2" stretchy="false" xref="algorithm7.36.32.m2.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="algorithm7.36.32.m2.1b"><apply id="algorithm7.36.32.m2.1.2.cmml" xref="algorithm7.36.32.m2.1.2"><times id="algorithm7.36.32.m2.1.2.1.cmml" xref="algorithm7.36.32.m2.1.2.1"></times><apply id="algorithm7.36.32.m2.1.2.2.cmml" xref="algorithm7.36.32.m2.1.2.2"><csymbol cd="ambiguous" id="algorithm7.36.32.m2.1.2.2.1.cmml" xref="algorithm7.36.32.m2.1.2.2">superscript</csymbol><ci id="algorithm7.36.32.m2.1.2.2.2.cmml" xref="algorithm7.36.32.m2.1.2.2.2">𝐶</ci><ci id="algorithm7.36.32.m2.1.2.2.3.cmml" xref="algorithm7.36.32.m2.1.2.2.3">′′</ci></apply><ci id="algorithm7.36.32.m2.1.1.cmml" xref="algorithm7.36.32.m2.1.1">𝑥</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="algorithm7.36.32.m2.1c">C^{\prime\prime}(x)</annotation><annotation encoding="application/x-llamapun" id="algorithm7.36.32.m2.1d">italic_C start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT ( italic_x )</annotation></semantics></math> </div> <div class="ltx_listingline" id="algorithm7.37.33"> <span class="ltx_rule" style="width:1px;height:100%;background:black;display:inline-block;"> </span> <math alttext="\textnormal{SOL}\leftarrow\textnormal{SOL}\cup E(H_{x}^{\prime})" class="ltx_Math" display="inline" id="algorithm7.37.33.m1.1"><semantics id="algorithm7.37.33.m1.1a"><mrow id="algorithm7.37.33.m1.1.1" xref="algorithm7.37.33.m1.1.1.cmml"><mtext id="algorithm7.37.33.m1.1.1.3" xref="algorithm7.37.33.m1.1.1.3a.cmml">SOL</mtext><mo id="algorithm7.37.33.m1.1.1.2" stretchy="false" xref="algorithm7.37.33.m1.1.1.2.cmml">←</mo><mrow id="algorithm7.37.33.m1.1.1.1" xref="algorithm7.37.33.m1.1.1.1.cmml"><mtext id="algorithm7.37.33.m1.1.1.1.3" xref="algorithm7.37.33.m1.1.1.1.3a.cmml">SOL</mtext><mo id="algorithm7.37.33.m1.1.1.1.2" xref="algorithm7.37.33.m1.1.1.1.2.cmml">∪</mo><mrow id="algorithm7.37.33.m1.1.1.1.1" xref="algorithm7.37.33.m1.1.1.1.1.cmml"><mi id="algorithm7.37.33.m1.1.1.1.1.3" xref="algorithm7.37.33.m1.1.1.1.1.3.cmml">E</mi><mo id="algorithm7.37.33.m1.1.1.1.1.2" xref="algorithm7.37.33.m1.1.1.1.1.2.cmml">⁢</mo><mrow id="algorithm7.37.33.m1.1.1.1.1.1.1" xref="algorithm7.37.33.m1.1.1.1.1.1.1.1.cmml"><mo id="algorithm7.37.33.m1.1.1.1.1.1.1.2" stretchy="false" xref="algorithm7.37.33.m1.1.1.1.1.1.1.1.cmml">(</mo><msubsup id="algorithm7.37.33.m1.1.1.1.1.1.1.1" xref="algorithm7.37.33.m1.1.1.1.1.1.1.1.cmml"><mi id="algorithm7.37.33.m1.1.1.1.1.1.1.1.2.2" xref="algorithm7.37.33.m1.1.1.1.1.1.1.1.2.2.cmml">H</mi><mi id="algorithm7.37.33.m1.1.1.1.1.1.1.1.2.3" xref="algorithm7.37.33.m1.1.1.1.1.1.1.1.2.3.cmml">x</mi><mo id="algorithm7.37.33.m1.1.1.1.1.1.1.1.3" xref="algorithm7.37.33.m1.1.1.1.1.1.1.1.3.cmml">′</mo></msubsup><mo id="algorithm7.37.33.m1.1.1.1.1.1.1.3" stretchy="false" xref="algorithm7.37.33.m1.1.1.1.1.1.1.1.cmml">)</mo></mrow></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="algorithm7.37.33.m1.1b"><apply id="algorithm7.37.33.m1.1.1.cmml" xref="algorithm7.37.33.m1.1.1"><ci id="algorithm7.37.33.m1.1.1.2.cmml" xref="algorithm7.37.33.m1.1.1.2">←</ci><ci id="algorithm7.37.33.m1.1.1.3a.cmml" xref="algorithm7.37.33.m1.1.1.3"><mtext id="algorithm7.37.33.m1.1.1.3.cmml" xref="algorithm7.37.33.m1.1.1.3">SOL</mtext></ci><apply id="algorithm7.37.33.m1.1.1.1.cmml" xref="algorithm7.37.33.m1.1.1.1"><union id="algorithm7.37.33.m1.1.1.1.2.cmml" xref="algorithm7.37.33.m1.1.1.1.2"></union><ci id="algorithm7.37.33.m1.1.1.1.3a.cmml" xref="algorithm7.37.33.m1.1.1.1.3"><mtext id="algorithm7.37.33.m1.1.1.1.3.cmml" xref="algorithm7.37.33.m1.1.1.1.3">SOL</mtext></ci><apply id="algorithm7.37.33.m1.1.1.1.1.cmml" xref="algorithm7.37.33.m1.1.1.1.1"><times id="algorithm7.37.33.m1.1.1.1.1.2.cmml" xref="algorithm7.37.33.m1.1.1.1.1.2"></times><ci id="algorithm7.37.33.m1.1.1.1.1.3.cmml" xref="algorithm7.37.33.m1.1.1.1.1.3">𝐸</ci><apply id="algorithm7.37.33.m1.1.1.1.1.1.1.1.cmml" xref="algorithm7.37.33.m1.1.1.1.1.1.1"><csymbol cd="ambiguous" id="algorithm7.37.33.m1.1.1.1.1.1.1.1.1.cmml" xref="algorithm7.37.33.m1.1.1.1.1.1.1">superscript</csymbol><apply id="algorithm7.37.33.m1.1.1.1.1.1.1.1.2.cmml" xref="algorithm7.37.33.m1.1.1.1.1.1.1"><csymbol cd="ambiguous" id="algorithm7.37.33.m1.1.1.1.1.1.1.1.2.1.cmml" xref="algorithm7.37.33.m1.1.1.1.1.1.1">subscript</csymbol><ci id="algorithm7.37.33.m1.1.1.1.1.1.1.1.2.2.cmml" xref="algorithm7.37.33.m1.1.1.1.1.1.1.1.2.2">𝐻</ci><ci id="algorithm7.37.33.m1.1.1.1.1.1.1.1.2.3.cmml" xref="algorithm7.37.33.m1.1.1.1.1.1.1.1.2.3">𝑥</ci></apply><ci id="algorithm7.37.33.m1.1.1.1.1.1.1.1.3.cmml" xref="algorithm7.37.33.m1.1.1.1.1.1.1.1.3">′</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="algorithm7.37.33.m1.1c">\textnormal{SOL}\leftarrow\textnormal{SOL}\cup E(H_{x}^{\prime})</annotation><annotation encoding="application/x-llamapun" id="algorithm7.37.33.m1.1d">SOL ← SOL ∪ italic_E ( italic_H start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT )</annotation></semantics></math> </div> <div class="ltx_listingline" id="algorithm7.38.34"> <span class="ltx_text ltx_font_bold" id="algorithm7.38.34.1">return</span> <span class="ltx_text ltx_markedasmath" id="algorithm7.38.34.2">SOL</span> </div> </div> <figcaption class="ltx_caption"><span class="ltx_tag ltx_tag_float"><span class="ltx_text ltx_font_bold" id="algorithm7.41.1.1">Algorithm 7</span> </span>Construction of a solution in <math alttext="F" class="ltx_Math" display="inline" id="algorithm7.3.m1.1"><semantics id="algorithm7.3.m1.1b"><mi id="algorithm7.3.m1.1.1" xref="algorithm7.3.m1.1.1.cmml">F</mi><annotation-xml encoding="MathML-Content" id="algorithm7.3.m1.1c"><ci id="algorithm7.3.m1.1.1.cmml" xref="algorithm7.3.m1.1.1">𝐹</ci></annotation-xml><annotation encoding="application/x-tex" id="algorithm7.3.m1.1d">F</annotation><annotation encoding="application/x-llamapun" id="algorithm7.3.m1.1e">italic_F</annotation></semantics></math> from <span class="ltx_text ltx_markedasmath" id="algorithm7.42.2">OPT</span>.</figcaption> </figure> <div class="ltx_para" id="S4.SS2.SSS3.p3"> <p class="ltx_p" id="S4.SS2.SSS3.p3.1">We fix <span class="ltx_text ltx_markedasmath" id="S4.SS2.SSS3.p3.1.1">SOL</span> to be the set of links given by Algorithm <a class="ltx_ref" href="https://arxiv.org/html/2503.00712v1#algorithm7" title="In 4.2.3 Bounding the Approximation Ratio ‣ 4.2 Two-to-Three Augmentation ‣ 4 Vertex Connectivity Augmentation in Link-Arrival Model ‣ Streaming Algorithms for Network Design"><span class="ltx_text ltx_ref_tag">7</span></a>.</p> </div> <div class="ltx_theorem ltx_theorem_lemma" id="S4.Thmtheorem21"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="S4.Thmtheorem21.1.1.1">Lemma 4.21</span></span><span class="ltx_text ltx_font_bold" id="S4.Thmtheorem21.2.2">.</span> </h6> <div class="ltx_para" id="S4.Thmtheorem21.p1"> <p class="ltx_p" id="S4.Thmtheorem21.p1.2">The edge set <span class="ltx_text ltx_markedasmath" id="S4.Thmtheorem21.p1.2.1">SOL</span> has weight at most <math alttext="(7+\epsilon)w(\textnormal{OPT})" class="ltx_Math" display="inline" id="S4.Thmtheorem21.p1.2.m2.2"><semantics id="S4.Thmtheorem21.p1.2.m2.2a"><mrow id="S4.Thmtheorem21.p1.2.m2.2.2" xref="S4.Thmtheorem21.p1.2.m2.2.2.cmml"><mrow id="S4.Thmtheorem21.p1.2.m2.2.2.1.1" xref="S4.Thmtheorem21.p1.2.m2.2.2.1.1.1.cmml"><mo id="S4.Thmtheorem21.p1.2.m2.2.2.1.1.2" stretchy="false" xref="S4.Thmtheorem21.p1.2.m2.2.2.1.1.1.cmml">(</mo><mrow id="S4.Thmtheorem21.p1.2.m2.2.2.1.1.1" xref="S4.Thmtheorem21.p1.2.m2.2.2.1.1.1.cmml"><mn id="S4.Thmtheorem21.p1.2.m2.2.2.1.1.1.2" xref="S4.Thmtheorem21.p1.2.m2.2.2.1.1.1.2.cmml">7</mn><mo id="S4.Thmtheorem21.p1.2.m2.2.2.1.1.1.1" xref="S4.Thmtheorem21.p1.2.m2.2.2.1.1.1.1.cmml">+</mo><mi id="S4.Thmtheorem21.p1.2.m2.2.2.1.1.1.3" xref="S4.Thmtheorem21.p1.2.m2.2.2.1.1.1.3.cmml">ϵ</mi></mrow><mo id="S4.Thmtheorem21.p1.2.m2.2.2.1.1.3" stretchy="false" xref="S4.Thmtheorem21.p1.2.m2.2.2.1.1.1.cmml">)</mo></mrow><mo id="S4.Thmtheorem21.p1.2.m2.2.2.2" xref="S4.Thmtheorem21.p1.2.m2.2.2.2.cmml">⁢</mo><mi id="S4.Thmtheorem21.p1.2.m2.2.2.3" xref="S4.Thmtheorem21.p1.2.m2.2.2.3.cmml">w</mi><mo id="S4.Thmtheorem21.p1.2.m2.2.2.2a" xref="S4.Thmtheorem21.p1.2.m2.2.2.2.cmml">⁢</mo><mrow id="S4.Thmtheorem21.p1.2.m2.2.2.4.2" xref="S4.Thmtheorem21.p1.2.m2.1.1a.cmml"><mo id="S4.Thmtheorem21.p1.2.m2.2.2.4.2.1" stretchy="false" xref="S4.Thmtheorem21.p1.2.m2.1.1a.cmml">(</mo><mtext id="S4.Thmtheorem21.p1.2.m2.1.1" xref="S4.Thmtheorem21.p1.2.m2.1.1.cmml">OPT</mtext><mo id="S4.Thmtheorem21.p1.2.m2.2.2.4.2.2" stretchy="false" xref="S4.Thmtheorem21.p1.2.m2.1.1a.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem21.p1.2.m2.2b"><apply id="S4.Thmtheorem21.p1.2.m2.2.2.cmml" xref="S4.Thmtheorem21.p1.2.m2.2.2"><times id="S4.Thmtheorem21.p1.2.m2.2.2.2.cmml" xref="S4.Thmtheorem21.p1.2.m2.2.2.2"></times><apply id="S4.Thmtheorem21.p1.2.m2.2.2.1.1.1.cmml" xref="S4.Thmtheorem21.p1.2.m2.2.2.1.1"><plus id="S4.Thmtheorem21.p1.2.m2.2.2.1.1.1.1.cmml" xref="S4.Thmtheorem21.p1.2.m2.2.2.1.1.1.1"></plus><cn id="S4.Thmtheorem21.p1.2.m2.2.2.1.1.1.2.cmml" type="integer" xref="S4.Thmtheorem21.p1.2.m2.2.2.1.1.1.2">7</cn><ci id="S4.Thmtheorem21.p1.2.m2.2.2.1.1.1.3.cmml" xref="S4.Thmtheorem21.p1.2.m2.2.2.1.1.1.3">italic-ϵ</ci></apply><ci id="S4.Thmtheorem21.p1.2.m2.2.2.3.cmml" xref="S4.Thmtheorem21.p1.2.m2.2.2.3">𝑤</ci><ci id="S4.Thmtheorem21.p1.2.m2.1.1a.cmml" xref="S4.Thmtheorem21.p1.2.m2.2.2.4.2"><mtext id="S4.Thmtheorem21.p1.2.m2.1.1.cmml" xref="S4.Thmtheorem21.p1.2.m2.1.1">OPT</mtext></ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem21.p1.2.m2.2c">(7+\epsilon)w(\textnormal{OPT})</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem21.p1.2.m2.2d">( 7 + italic_ϵ ) italic_w ( OPT )</annotation></semantics></math>.</p> </div> </div> <div class="ltx_proof" id="S4.SS2.SSS3.3"> <h6 class="ltx_title ltx_runin ltx_font_italic ltx_title_proof">Proof.</h6> <div class="ltx_para" id="S4.SS2.SSS3.1.p1"> <p class="ltx_p" id="S4.SS2.SSS3.1.p1.5">We first consider <math alttext="\textnormal{SOL}\cap\cup_{x}L_{x}" class="ltx_math_unparsed" display="inline" id="S4.SS2.SSS3.1.p1.1.m1.1"><semantics id="S4.SS2.SSS3.1.p1.1.m1.1a"><mrow id="S4.SS2.SSS3.1.p1.1.m1.1b"><mtext id="S4.SS2.SSS3.1.p1.1.m1.1.1">SOL</mtext><mo id="S4.SS2.SSS3.1.p1.1.m1.1.2" rspace="0em">∩</mo><msub id="S4.SS2.SSS3.1.p1.1.m1.1.3"><mo id="S4.SS2.SSS3.1.p1.1.m1.1.3.2" lspace="0em">∪</mo><mi id="S4.SS2.SSS3.1.p1.1.m1.1.3.3">x</mi></msub><msub id="S4.SS2.SSS3.1.p1.1.m1.1.4"><mi id="S4.SS2.SSS3.1.p1.1.m1.1.4.2">L</mi><mi id="S4.SS2.SSS3.1.p1.1.m1.1.4.3">x</mi></msub></mrow><annotation encoding="application/x-tex" id="S4.SS2.SSS3.1.p1.1.m1.1c">\textnormal{SOL}\cap\cup_{x}L_{x}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS3.1.p1.1.m1.1d">SOL ∩ ∪ start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT italic_L start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT</annotation></semantics></math>. In the first part of Algorithm <a class="ltx_ref" href="https://arxiv.org/html/2503.00712v1#algorithm7" title="In 4.2.3 Bounding the Approximation Ratio ‣ 4.2 Two-to-Three Augmentation ‣ 4 Vertex Connectivity Augmentation in Link-Arrival Model ‣ Streaming Algorithms for Network Design"><span class="ltx_text ltx_ref_tag">7</span></a>, for each link <math alttext="uv\in\textnormal{OPT}" class="ltx_Math" display="inline" id="S4.SS2.SSS3.1.p1.2.m2.1"><semantics id="S4.SS2.SSS3.1.p1.2.m2.1a"><mrow id="S4.SS2.SSS3.1.p1.2.m2.1.1" xref="S4.SS2.SSS3.1.p1.2.m2.1.1.cmml"><mrow id="S4.SS2.SSS3.1.p1.2.m2.1.1.2" xref="S4.SS2.SSS3.1.p1.2.m2.1.1.2.cmml"><mi id="S4.SS2.SSS3.1.p1.2.m2.1.1.2.2" xref="S4.SS2.SSS3.1.p1.2.m2.1.1.2.2.cmml">u</mi><mo id="S4.SS2.SSS3.1.p1.2.m2.1.1.2.1" xref="S4.SS2.SSS3.1.p1.2.m2.1.1.2.1.cmml">⁢</mo><mi id="S4.SS2.SSS3.1.p1.2.m2.1.1.2.3" xref="S4.SS2.SSS3.1.p1.2.m2.1.1.2.3.cmml">v</mi></mrow><mo id="S4.SS2.SSS3.1.p1.2.m2.1.1.1" xref="S4.SS2.SSS3.1.p1.2.m2.1.1.1.cmml">∈</mo><mtext id="S4.SS2.SSS3.1.p1.2.m2.1.1.3" xref="S4.SS2.SSS3.1.p1.2.m2.1.1.3a.cmml">OPT</mtext></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS3.1.p1.2.m2.1b"><apply id="S4.SS2.SSS3.1.p1.2.m2.1.1.cmml" xref="S4.SS2.SSS3.1.p1.2.m2.1.1"><in id="S4.SS2.SSS3.1.p1.2.m2.1.1.1.cmml" xref="S4.SS2.SSS3.1.p1.2.m2.1.1.1"></in><apply id="S4.SS2.SSS3.1.p1.2.m2.1.1.2.cmml" xref="S4.SS2.SSS3.1.p1.2.m2.1.1.2"><times id="S4.SS2.SSS3.1.p1.2.m2.1.1.2.1.cmml" xref="S4.SS2.SSS3.1.p1.2.m2.1.1.2.1"></times><ci id="S4.SS2.SSS3.1.p1.2.m2.1.1.2.2.cmml" xref="S4.SS2.SSS3.1.p1.2.m2.1.1.2.2">𝑢</ci><ci id="S4.SS2.SSS3.1.p1.2.m2.1.1.2.3.cmml" xref="S4.SS2.SSS3.1.p1.2.m2.1.1.2.3">𝑣</ci></apply><ci id="S4.SS2.SSS3.1.p1.2.m2.1.1.3a.cmml" xref="S4.SS2.SSS3.1.p1.2.m2.1.1.3"><mtext id="S4.SS2.SSS3.1.p1.2.m2.1.1.3.cmml" xref="S4.SS2.SSS3.1.p1.2.m2.1.1.3">OPT</mtext></ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS3.1.p1.2.m2.1c">uv\in\textnormal{OPT}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS3.1.p1.2.m2.1d">italic_u italic_v ∈ OPT</annotation></semantics></math>, we add two links to <span class="ltx_text ltx_markedasmath" id="S4.SS2.SSS3.1.p1.5.1">SOL</span>, each of weight at most <math alttext="(1+\epsilon)w(uv)" class="ltx_Math" display="inline" id="S4.SS2.SSS3.1.p1.4.m4.2"><semantics id="S4.SS2.SSS3.1.p1.4.m4.2a"><mrow id="S4.SS2.SSS3.1.p1.4.m4.2.2" xref="S4.SS2.SSS3.1.p1.4.m4.2.2.cmml"><mrow id="S4.SS2.SSS3.1.p1.4.m4.1.1.1.1" xref="S4.SS2.SSS3.1.p1.4.m4.1.1.1.1.1.cmml"><mo id="S4.SS2.SSS3.1.p1.4.m4.1.1.1.1.2" stretchy="false" xref="S4.SS2.SSS3.1.p1.4.m4.1.1.1.1.1.cmml">(</mo><mrow id="S4.SS2.SSS3.1.p1.4.m4.1.1.1.1.1" xref="S4.SS2.SSS3.1.p1.4.m4.1.1.1.1.1.cmml"><mn id="S4.SS2.SSS3.1.p1.4.m4.1.1.1.1.1.2" xref="S4.SS2.SSS3.1.p1.4.m4.1.1.1.1.1.2.cmml">1</mn><mo id="S4.SS2.SSS3.1.p1.4.m4.1.1.1.1.1.1" xref="S4.SS2.SSS3.1.p1.4.m4.1.1.1.1.1.1.cmml">+</mo><mi id="S4.SS2.SSS3.1.p1.4.m4.1.1.1.1.1.3" xref="S4.SS2.SSS3.1.p1.4.m4.1.1.1.1.1.3.cmml">ϵ</mi></mrow><mo id="S4.SS2.SSS3.1.p1.4.m4.1.1.1.1.3" stretchy="false" xref="S4.SS2.SSS3.1.p1.4.m4.1.1.1.1.1.cmml">)</mo></mrow><mo id="S4.SS2.SSS3.1.p1.4.m4.2.2.3" xref="S4.SS2.SSS3.1.p1.4.m4.2.2.3.cmml">⁢</mo><mi id="S4.SS2.SSS3.1.p1.4.m4.2.2.4" xref="S4.SS2.SSS3.1.p1.4.m4.2.2.4.cmml">w</mi><mo id="S4.SS2.SSS3.1.p1.4.m4.2.2.3a" xref="S4.SS2.SSS3.1.p1.4.m4.2.2.3.cmml">⁢</mo><mrow id="S4.SS2.SSS3.1.p1.4.m4.2.2.2.1" xref="S4.SS2.SSS3.1.p1.4.m4.2.2.2.1.1.cmml"><mo id="S4.SS2.SSS3.1.p1.4.m4.2.2.2.1.2" stretchy="false" xref="S4.SS2.SSS3.1.p1.4.m4.2.2.2.1.1.cmml">(</mo><mrow id="S4.SS2.SSS3.1.p1.4.m4.2.2.2.1.1" xref="S4.SS2.SSS3.1.p1.4.m4.2.2.2.1.1.cmml"><mi id="S4.SS2.SSS3.1.p1.4.m4.2.2.2.1.1.2" xref="S4.SS2.SSS3.1.p1.4.m4.2.2.2.1.1.2.cmml">u</mi><mo id="S4.SS2.SSS3.1.p1.4.m4.2.2.2.1.1.1" xref="S4.SS2.SSS3.1.p1.4.m4.2.2.2.1.1.1.cmml">⁢</mo><mi id="S4.SS2.SSS3.1.p1.4.m4.2.2.2.1.1.3" xref="S4.SS2.SSS3.1.p1.4.m4.2.2.2.1.1.3.cmml">v</mi></mrow><mo id="S4.SS2.SSS3.1.p1.4.m4.2.2.2.1.3" stretchy="false" xref="S4.SS2.SSS3.1.p1.4.m4.2.2.2.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS3.1.p1.4.m4.2b"><apply id="S4.SS2.SSS3.1.p1.4.m4.2.2.cmml" xref="S4.SS2.SSS3.1.p1.4.m4.2.2"><times id="S4.SS2.SSS3.1.p1.4.m4.2.2.3.cmml" xref="S4.SS2.SSS3.1.p1.4.m4.2.2.3"></times><apply id="S4.SS2.SSS3.1.p1.4.m4.1.1.1.1.1.cmml" xref="S4.SS2.SSS3.1.p1.4.m4.1.1.1.1"><plus id="S4.SS2.SSS3.1.p1.4.m4.1.1.1.1.1.1.cmml" xref="S4.SS2.SSS3.1.p1.4.m4.1.1.1.1.1.1"></plus><cn id="S4.SS2.SSS3.1.p1.4.m4.1.1.1.1.1.2.cmml" type="integer" xref="S4.SS2.SSS3.1.p1.4.m4.1.1.1.1.1.2">1</cn><ci id="S4.SS2.SSS3.1.p1.4.m4.1.1.1.1.1.3.cmml" xref="S4.SS2.SSS3.1.p1.4.m4.1.1.1.1.1.3">italic-ϵ</ci></apply><ci id="S4.SS2.SSS3.1.p1.4.m4.2.2.4.cmml" xref="S4.SS2.SSS3.1.p1.4.m4.2.2.4">𝑤</ci><apply id="S4.SS2.SSS3.1.p1.4.m4.2.2.2.1.1.cmml" xref="S4.SS2.SSS3.1.p1.4.m4.2.2.2.1"><times id="S4.SS2.SSS3.1.p1.4.m4.2.2.2.1.1.1.cmml" xref="S4.SS2.SSS3.1.p1.4.m4.2.2.2.1.1.1"></times><ci id="S4.SS2.SSS3.1.p1.4.m4.2.2.2.1.1.2.cmml" xref="S4.SS2.SSS3.1.p1.4.m4.2.2.2.1.1.2">𝑢</ci><ci id="S4.SS2.SSS3.1.p1.4.m4.2.2.2.1.1.3.cmml" xref="S4.SS2.SSS3.1.p1.4.m4.2.2.2.1.1.3">𝑣</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS3.1.p1.4.m4.2c">(1+\epsilon)w(uv)</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS3.1.p1.4.m4.2d">( 1 + italic_ϵ ) italic_w ( italic_u italic_v )</annotation></semantics></math>. Thus <math alttext="w(\textnormal{SOL}\cap\cup_{x}L_{x})\leq 2(1+\epsilon)w(\textnormal{OPT})" class="ltx_math_unparsed" display="inline" id="S4.SS2.SSS3.1.p1.5.m5.1"><semantics id="S4.SS2.SSS3.1.p1.5.m5.1a"><mrow id="S4.SS2.SSS3.1.p1.5.m5.1b"><mi id="S4.SS2.SSS3.1.p1.5.m5.1.1">w</mi><mrow id="S4.SS2.SSS3.1.p1.5.m5.1.2"><mo id="S4.SS2.SSS3.1.p1.5.m5.1.2.1" stretchy="false">(</mo><mtext id="S4.SS2.SSS3.1.p1.5.m5.1.2.2">SOL</mtext><mo id="S4.SS2.SSS3.1.p1.5.m5.1.2.3" rspace="0em">∩</mo><msub id="S4.SS2.SSS3.1.p1.5.m5.1.2.4"><mo id="S4.SS2.SSS3.1.p1.5.m5.1.2.4.2" lspace="0em">∪</mo><mi id="S4.SS2.SSS3.1.p1.5.m5.1.2.4.3">x</mi></msub><msub id="S4.SS2.SSS3.1.p1.5.m5.1.2.5"><mi id="S4.SS2.SSS3.1.p1.5.m5.1.2.5.2">L</mi><mi id="S4.SS2.SSS3.1.p1.5.m5.1.2.5.3">x</mi></msub><mo id="S4.SS2.SSS3.1.p1.5.m5.1.2.6" stretchy="false">)</mo></mrow><mo id="S4.SS2.SSS3.1.p1.5.m5.1.3">≤</mo><mn id="S4.SS2.SSS3.1.p1.5.m5.1.4">2</mn><mrow id="S4.SS2.SSS3.1.p1.5.m5.1.5"><mo id="S4.SS2.SSS3.1.p1.5.m5.1.5.1" stretchy="false">(</mo><mn id="S4.SS2.SSS3.1.p1.5.m5.1.5.2">1</mn><mo id="S4.SS2.SSS3.1.p1.5.m5.1.5.3">+</mo><mi id="S4.SS2.SSS3.1.p1.5.m5.1.5.4">ϵ</mi><mo id="S4.SS2.SSS3.1.p1.5.m5.1.5.5" stretchy="false">)</mo></mrow><mi id="S4.SS2.SSS3.1.p1.5.m5.1.6">w</mi><mrow id="S4.SS2.SSS3.1.p1.5.m5.1.7"><mo id="S4.SS2.SSS3.1.p1.5.m5.1.7.1" stretchy="false">(</mo><mtext id="S4.SS2.SSS3.1.p1.5.m5.1.7.2">OPT</mtext><mo id="S4.SS2.SSS3.1.p1.5.m5.1.7.3" stretchy="false">)</mo></mrow></mrow><annotation encoding="application/x-tex" id="S4.SS2.SSS3.1.p1.5.m5.1c">w(\textnormal{SOL}\cap\cup_{x}L_{x})\leq 2(1+\epsilon)w(\textnormal{OPT})</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS3.1.p1.5.m5.1d">italic_w ( SOL ∩ ∪ start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT italic_L start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT ) ≤ 2 ( 1 + italic_ϵ ) italic_w ( OPT )</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="S4.SS2.SSS3.2.p2"> <p class="ltx_p" id="S4.SS2.SSS3.2.p2.31">Next, we consider <math alttext="\textnormal{SOL}\cap\cup_{x}E(H_{x})" class="ltx_math_unparsed" display="inline" id="S4.SS2.SSS3.2.p2.1.m1.1"><semantics id="S4.SS2.SSS3.2.p2.1.m1.1a"><mrow id="S4.SS2.SSS3.2.p2.1.m1.1b"><mtext id="S4.SS2.SSS3.2.p2.1.m1.1.1">SOL</mtext><mo id="S4.SS2.SSS3.2.p2.1.m1.1.2" rspace="0em">∩</mo><msub id="S4.SS2.SSS3.2.p2.1.m1.1.3"><mo id="S4.SS2.SSS3.2.p2.1.m1.1.3.2" lspace="0em">∪</mo><mi id="S4.SS2.SSS3.2.p2.1.m1.1.3.3">x</mi></msub><mi id="S4.SS2.SSS3.2.p2.1.m1.1.4">E</mi><mrow id="S4.SS2.SSS3.2.p2.1.m1.1.5"><mo id="S4.SS2.SSS3.2.p2.1.m1.1.5.1" stretchy="false">(</mo><msub id="S4.SS2.SSS3.2.p2.1.m1.1.5.2"><mi id="S4.SS2.SSS3.2.p2.1.m1.1.5.2.2">H</mi><mi id="S4.SS2.SSS3.2.p2.1.m1.1.5.2.3">x</mi></msub><mo id="S4.SS2.SSS3.2.p2.1.m1.1.5.3" stretchy="false">)</mo></mrow></mrow><annotation encoding="application/x-tex" id="S4.SS2.SSS3.2.p2.1.m1.1c">\textnormal{SOL}\cap\cup_{x}E(H_{x})</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS3.2.p2.1.m1.1d">SOL ∩ ∪ start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT italic_E ( italic_H start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT )</annotation></semantics></math> and show that this has weight at most <span class="ltx_text ltx_markedasmath" id="S4.SS2.SSS3.2.p2.31.1">OPT</span>; the reasoning is similar to the case of <math alttext="1" class="ltx_Math" display="inline" id="S4.SS2.SSS3.2.p2.3.m3.1"><semantics id="S4.SS2.SSS3.2.p2.3.m3.1a"><mn id="S4.SS2.SSS3.2.p2.3.m3.1.1" xref="S4.SS2.SSS3.2.p2.3.m3.1.1.cmml">1</mn><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS3.2.p2.3.m3.1b"><cn id="S4.SS2.SSS3.2.p2.3.m3.1.1.cmml" type="integer" xref="S4.SS2.SSS3.2.p2.3.m3.1.1">1</cn></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS3.2.p2.3.m3.1c">1</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS3.2.p2.3.m3.1d">1</annotation></semantics></math>-VC-CAP and we rewrite it here for completeness. Consider a partition of <span class="ltx_text ltx_markedasmath" id="S4.SS2.SSS3.2.p2.31.2">OPT</span> based on the LCA of its endpoints: for each P-node <math alttext="x" class="ltx_Math" display="inline" id="S4.SS2.SSS3.2.p2.5.m5.1"><semantics id="S4.SS2.SSS3.2.p2.5.m5.1a"><mi id="S4.SS2.SSS3.2.p2.5.m5.1.1" xref="S4.SS2.SSS3.2.p2.5.m5.1.1.cmml">x</mi><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS3.2.p2.5.m5.1b"><ci id="S4.SS2.SSS3.2.p2.5.m5.1.1.cmml" xref="S4.SS2.SSS3.2.p2.5.m5.1.1">𝑥</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS3.2.p2.5.m5.1c">x</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS3.2.p2.5.m5.1d">italic_x</annotation></semantics></math>, let <math alttext="E^{*}(x)=\{uv\in\textnormal{OPT}:\text{LCA}(\ell(u),\ell(v))=x\}" class="ltx_Math" display="inline" id="S4.SS2.SSS3.2.p2.6.m6.5"><semantics id="S4.SS2.SSS3.2.p2.6.m6.5a"><mrow id="S4.SS2.SSS3.2.p2.6.m6.5.5" xref="S4.SS2.SSS3.2.p2.6.m6.5.5.cmml"><mrow id="S4.SS2.SSS3.2.p2.6.m6.5.5.4" xref="S4.SS2.SSS3.2.p2.6.m6.5.5.4.cmml"><msup id="S4.SS2.SSS3.2.p2.6.m6.5.5.4.2" xref="S4.SS2.SSS3.2.p2.6.m6.5.5.4.2.cmml"><mi id="S4.SS2.SSS3.2.p2.6.m6.5.5.4.2.2" xref="S4.SS2.SSS3.2.p2.6.m6.5.5.4.2.2.cmml">E</mi><mo id="S4.SS2.SSS3.2.p2.6.m6.5.5.4.2.3" xref="S4.SS2.SSS3.2.p2.6.m6.5.5.4.2.3.cmml">∗</mo></msup><mo id="S4.SS2.SSS3.2.p2.6.m6.5.5.4.1" xref="S4.SS2.SSS3.2.p2.6.m6.5.5.4.1.cmml">⁢</mo><mrow id="S4.SS2.SSS3.2.p2.6.m6.5.5.4.3.2" xref="S4.SS2.SSS3.2.p2.6.m6.5.5.4.cmml"><mo id="S4.SS2.SSS3.2.p2.6.m6.5.5.4.3.2.1" stretchy="false" xref="S4.SS2.SSS3.2.p2.6.m6.5.5.4.cmml">(</mo><mi id="S4.SS2.SSS3.2.p2.6.m6.1.1" xref="S4.SS2.SSS3.2.p2.6.m6.1.1.cmml">x</mi><mo id="S4.SS2.SSS3.2.p2.6.m6.5.5.4.3.2.2" stretchy="false" xref="S4.SS2.SSS3.2.p2.6.m6.5.5.4.cmml">)</mo></mrow></mrow><mo id="S4.SS2.SSS3.2.p2.6.m6.5.5.3" xref="S4.SS2.SSS3.2.p2.6.m6.5.5.3.cmml">=</mo><mrow id="S4.SS2.SSS3.2.p2.6.m6.5.5.2.2" xref="S4.SS2.SSS3.2.p2.6.m6.5.5.2.3.cmml"><mo id="S4.SS2.SSS3.2.p2.6.m6.5.5.2.2.3" stretchy="false" xref="S4.SS2.SSS3.2.p2.6.m6.5.5.2.3.1.cmml">{</mo><mrow id="S4.SS2.SSS3.2.p2.6.m6.4.4.1.1.1" xref="S4.SS2.SSS3.2.p2.6.m6.4.4.1.1.1.cmml"><mrow id="S4.SS2.SSS3.2.p2.6.m6.4.4.1.1.1.2" xref="S4.SS2.SSS3.2.p2.6.m6.4.4.1.1.1.2.cmml"><mi id="S4.SS2.SSS3.2.p2.6.m6.4.4.1.1.1.2.2" xref="S4.SS2.SSS3.2.p2.6.m6.4.4.1.1.1.2.2.cmml">u</mi><mo id="S4.SS2.SSS3.2.p2.6.m6.4.4.1.1.1.2.1" xref="S4.SS2.SSS3.2.p2.6.m6.4.4.1.1.1.2.1.cmml">⁢</mo><mi id="S4.SS2.SSS3.2.p2.6.m6.4.4.1.1.1.2.3" xref="S4.SS2.SSS3.2.p2.6.m6.4.4.1.1.1.2.3.cmml">v</mi></mrow><mo id="S4.SS2.SSS3.2.p2.6.m6.4.4.1.1.1.1" xref="S4.SS2.SSS3.2.p2.6.m6.4.4.1.1.1.1.cmml">∈</mo><mtext id="S4.SS2.SSS3.2.p2.6.m6.4.4.1.1.1.3" xref="S4.SS2.SSS3.2.p2.6.m6.4.4.1.1.1.3a.cmml">OPT</mtext></mrow><mo id="S4.SS2.SSS3.2.p2.6.m6.5.5.2.2.4" lspace="0.278em" rspace="0.278em" xref="S4.SS2.SSS3.2.p2.6.m6.5.5.2.3.1.cmml">:</mo><mrow id="S4.SS2.SSS3.2.p2.6.m6.5.5.2.2.2" xref="S4.SS2.SSS3.2.p2.6.m6.5.5.2.2.2.cmml"><mrow id="S4.SS2.SSS3.2.p2.6.m6.5.5.2.2.2.2" xref="S4.SS2.SSS3.2.p2.6.m6.5.5.2.2.2.2.cmml"><mtext id="S4.SS2.SSS3.2.p2.6.m6.5.5.2.2.2.2.4" xref="S4.SS2.SSS3.2.p2.6.m6.5.5.2.2.2.2.4a.cmml">LCA</mtext><mo id="S4.SS2.SSS3.2.p2.6.m6.5.5.2.2.2.2.3" xref="S4.SS2.SSS3.2.p2.6.m6.5.5.2.2.2.2.3.cmml">⁢</mo><mrow id="S4.SS2.SSS3.2.p2.6.m6.5.5.2.2.2.2.2.2" xref="S4.SS2.SSS3.2.p2.6.m6.5.5.2.2.2.2.2.3.cmml"><mo id="S4.SS2.SSS3.2.p2.6.m6.5.5.2.2.2.2.2.2.3" stretchy="false" xref="S4.SS2.SSS3.2.p2.6.m6.5.5.2.2.2.2.2.3.cmml">(</mo><mrow id="S4.SS2.SSS3.2.p2.6.m6.5.5.2.2.2.1.1.1.1" xref="S4.SS2.SSS3.2.p2.6.m6.5.5.2.2.2.1.1.1.1.cmml"><mi id="S4.SS2.SSS3.2.p2.6.m6.5.5.2.2.2.1.1.1.1.2" mathvariant="normal" xref="S4.SS2.SSS3.2.p2.6.m6.5.5.2.2.2.1.1.1.1.2.cmml">ℓ</mi><mo id="S4.SS2.SSS3.2.p2.6.m6.5.5.2.2.2.1.1.1.1.1" xref="S4.SS2.SSS3.2.p2.6.m6.5.5.2.2.2.1.1.1.1.1.cmml">⁢</mo><mrow id="S4.SS2.SSS3.2.p2.6.m6.5.5.2.2.2.1.1.1.1.3.2" xref="S4.SS2.SSS3.2.p2.6.m6.5.5.2.2.2.1.1.1.1.cmml"><mo id="S4.SS2.SSS3.2.p2.6.m6.5.5.2.2.2.1.1.1.1.3.2.1" stretchy="false" xref="S4.SS2.SSS3.2.p2.6.m6.5.5.2.2.2.1.1.1.1.cmml">(</mo><mi id="S4.SS2.SSS3.2.p2.6.m6.2.2" xref="S4.SS2.SSS3.2.p2.6.m6.2.2.cmml">u</mi><mo id="S4.SS2.SSS3.2.p2.6.m6.5.5.2.2.2.1.1.1.1.3.2.2" stretchy="false" xref="S4.SS2.SSS3.2.p2.6.m6.5.5.2.2.2.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S4.SS2.SSS3.2.p2.6.m6.5.5.2.2.2.2.2.2.4" xref="S4.SS2.SSS3.2.p2.6.m6.5.5.2.2.2.2.2.3.cmml">,</mo><mrow id="S4.SS2.SSS3.2.p2.6.m6.5.5.2.2.2.2.2.2.2" xref="S4.SS2.SSS3.2.p2.6.m6.5.5.2.2.2.2.2.2.2.cmml"><mi id="S4.SS2.SSS3.2.p2.6.m6.5.5.2.2.2.2.2.2.2.2" mathvariant="normal" xref="S4.SS2.SSS3.2.p2.6.m6.5.5.2.2.2.2.2.2.2.2.cmml">ℓ</mi><mo id="S4.SS2.SSS3.2.p2.6.m6.5.5.2.2.2.2.2.2.2.1" xref="S4.SS2.SSS3.2.p2.6.m6.5.5.2.2.2.2.2.2.2.1.cmml">⁢</mo><mrow id="S4.SS2.SSS3.2.p2.6.m6.5.5.2.2.2.2.2.2.2.3.2" xref="S4.SS2.SSS3.2.p2.6.m6.5.5.2.2.2.2.2.2.2.cmml"><mo 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xref="S4.SS2.SSS3.2.p2.6.m6.5.5.3"></eq><apply id="S4.SS2.SSS3.2.p2.6.m6.5.5.4.cmml" xref="S4.SS2.SSS3.2.p2.6.m6.5.5.4"><times id="S4.SS2.SSS3.2.p2.6.m6.5.5.4.1.cmml" xref="S4.SS2.SSS3.2.p2.6.m6.5.5.4.1"></times><apply id="S4.SS2.SSS3.2.p2.6.m6.5.5.4.2.cmml" xref="S4.SS2.SSS3.2.p2.6.m6.5.5.4.2"><csymbol cd="ambiguous" id="S4.SS2.SSS3.2.p2.6.m6.5.5.4.2.1.cmml" xref="S4.SS2.SSS3.2.p2.6.m6.5.5.4.2">superscript</csymbol><ci id="S4.SS2.SSS3.2.p2.6.m6.5.5.4.2.2.cmml" xref="S4.SS2.SSS3.2.p2.6.m6.5.5.4.2.2">𝐸</ci><times id="S4.SS2.SSS3.2.p2.6.m6.5.5.4.2.3.cmml" xref="S4.SS2.SSS3.2.p2.6.m6.5.5.4.2.3"></times></apply><ci id="S4.SS2.SSS3.2.p2.6.m6.1.1.cmml" xref="S4.SS2.SSS3.2.p2.6.m6.1.1">𝑥</ci></apply><apply id="S4.SS2.SSS3.2.p2.6.m6.5.5.2.3.cmml" xref="S4.SS2.SSS3.2.p2.6.m6.5.5.2.2"><csymbol cd="latexml" id="S4.SS2.SSS3.2.p2.6.m6.5.5.2.3.1.cmml" xref="S4.SS2.SSS3.2.p2.6.m6.5.5.2.2.3">conditional-set</csymbol><apply id="S4.SS2.SSS3.2.p2.6.m6.4.4.1.1.1.cmml" xref="S4.SS2.SSS3.2.p2.6.m6.4.4.1.1.1"><in id="S4.SS2.SSS3.2.p2.6.m6.4.4.1.1.1.1.cmml" xref="S4.SS2.SSS3.2.p2.6.m6.4.4.1.1.1.1"></in><apply id="S4.SS2.SSS3.2.p2.6.m6.4.4.1.1.1.2.cmml" xref="S4.SS2.SSS3.2.p2.6.m6.4.4.1.1.1.2"><times id="S4.SS2.SSS3.2.p2.6.m6.4.4.1.1.1.2.1.cmml" xref="S4.SS2.SSS3.2.p2.6.m6.4.4.1.1.1.2.1"></times><ci id="S4.SS2.SSS3.2.p2.6.m6.4.4.1.1.1.2.2.cmml" xref="S4.SS2.SSS3.2.p2.6.m6.4.4.1.1.1.2.2">𝑢</ci><ci id="S4.SS2.SSS3.2.p2.6.m6.4.4.1.1.1.2.3.cmml" xref="S4.SS2.SSS3.2.p2.6.m6.4.4.1.1.1.2.3">𝑣</ci></apply><ci id="S4.SS2.SSS3.2.p2.6.m6.4.4.1.1.1.3a.cmml" xref="S4.SS2.SSS3.2.p2.6.m6.4.4.1.1.1.3"><mtext id="S4.SS2.SSS3.2.p2.6.m6.4.4.1.1.1.3.cmml" xref="S4.SS2.SSS3.2.p2.6.m6.4.4.1.1.1.3">OPT</mtext></ci></apply><apply id="S4.SS2.SSS3.2.p2.6.m6.5.5.2.2.2.cmml" xref="S4.SS2.SSS3.2.p2.6.m6.5.5.2.2.2"><eq id="S4.SS2.SSS3.2.p2.6.m6.5.5.2.2.2.3.cmml" xref="S4.SS2.SSS3.2.p2.6.m6.5.5.2.2.2.3"></eq><apply id="S4.SS2.SSS3.2.p2.6.m6.5.5.2.2.2.2.cmml" xref="S4.SS2.SSS3.2.p2.6.m6.5.5.2.2.2.2"><times id="S4.SS2.SSS3.2.p2.6.m6.5.5.2.2.2.2.3.cmml" xref="S4.SS2.SSS3.2.p2.6.m6.5.5.2.2.2.2.3"></times><ci id="S4.SS2.SSS3.2.p2.6.m6.5.5.2.2.2.2.4a.cmml" xref="S4.SS2.SSS3.2.p2.6.m6.5.5.2.2.2.2.4"><mtext id="S4.SS2.SSS3.2.p2.6.m6.5.5.2.2.2.2.4.cmml" xref="S4.SS2.SSS3.2.p2.6.m6.5.5.2.2.2.2.4">LCA</mtext></ci><interval closure="open" id="S4.SS2.SSS3.2.p2.6.m6.5.5.2.2.2.2.2.3.cmml" xref="S4.SS2.SSS3.2.p2.6.m6.5.5.2.2.2.2.2.2"><apply id="S4.SS2.SSS3.2.p2.6.m6.5.5.2.2.2.1.1.1.1.cmml" xref="S4.SS2.SSS3.2.p2.6.m6.5.5.2.2.2.1.1.1.1"><times id="S4.SS2.SSS3.2.p2.6.m6.5.5.2.2.2.1.1.1.1.1.cmml" xref="S4.SS2.SSS3.2.p2.6.m6.5.5.2.2.2.1.1.1.1.1"></times><ci id="S4.SS2.SSS3.2.p2.6.m6.5.5.2.2.2.1.1.1.1.2.cmml" xref="S4.SS2.SSS3.2.p2.6.m6.5.5.2.2.2.1.1.1.1.2">ℓ</ci><ci id="S4.SS2.SSS3.2.p2.6.m6.2.2.cmml" xref="S4.SS2.SSS3.2.p2.6.m6.2.2">𝑢</ci></apply><apply id="S4.SS2.SSS3.2.p2.6.m6.5.5.2.2.2.2.2.2.2.cmml" xref="S4.SS2.SSS3.2.p2.6.m6.5.5.2.2.2.2.2.2.2"><times id="S4.SS2.SSS3.2.p2.6.m6.5.5.2.2.2.2.2.2.2.1.cmml" xref="S4.SS2.SSS3.2.p2.6.m6.5.5.2.2.2.2.2.2.2.1"></times><ci id="S4.SS2.SSS3.2.p2.6.m6.5.5.2.2.2.2.2.2.2.2.cmml" xref="S4.SS2.SSS3.2.p2.6.m6.5.5.2.2.2.2.2.2.2.2">ℓ</ci><ci id="S4.SS2.SSS3.2.p2.6.m6.3.3.cmml" xref="S4.SS2.SSS3.2.p2.6.m6.3.3">𝑣</ci></apply></interval></apply><ci id="S4.SS2.SSS3.2.p2.6.m6.5.5.2.2.2.4.cmml" xref="S4.SS2.SSS3.2.p2.6.m6.5.5.2.2.2.4">𝑥</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS3.2.p2.6.m6.5c">E^{*}(x)=\{uv\in\textnormal{OPT}:\text{LCA}(\ell(u),\ell(v))=x\}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS3.2.p2.6.m6.5d">italic_E start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ( italic_x ) = { italic_u italic_v ∈ OPT : LCA ( roman_ℓ ( italic_u ) , roman_ℓ ( italic_v ) ) = italic_x }</annotation></semantics></math>. For each node <math alttext="y\in C(x)" class="ltx_Math" display="inline" id="S4.SS2.SSS3.2.p2.7.m7.1"><semantics id="S4.SS2.SSS3.2.p2.7.m7.1a"><mrow id="S4.SS2.SSS3.2.p2.7.m7.1.2" xref="S4.SS2.SSS3.2.p2.7.m7.1.2.cmml"><mi id="S4.SS2.SSS3.2.p2.7.m7.1.2.2" xref="S4.SS2.SSS3.2.p2.7.m7.1.2.2.cmml">y</mi><mo id="S4.SS2.SSS3.2.p2.7.m7.1.2.1" xref="S4.SS2.SSS3.2.p2.7.m7.1.2.1.cmml">∈</mo><mrow id="S4.SS2.SSS3.2.p2.7.m7.1.2.3" xref="S4.SS2.SSS3.2.p2.7.m7.1.2.3.cmml"><mi id="S4.SS2.SSS3.2.p2.7.m7.1.2.3.2" xref="S4.SS2.SSS3.2.p2.7.m7.1.2.3.2.cmml">C</mi><mo id="S4.SS2.SSS3.2.p2.7.m7.1.2.3.1" xref="S4.SS2.SSS3.2.p2.7.m7.1.2.3.1.cmml">⁢</mo><mrow id="S4.SS2.SSS3.2.p2.7.m7.1.2.3.3.2" xref="S4.SS2.SSS3.2.p2.7.m7.1.2.3.cmml"><mo id="S4.SS2.SSS3.2.p2.7.m7.1.2.3.3.2.1" stretchy="false" xref="S4.SS2.SSS3.2.p2.7.m7.1.2.3.cmml">(</mo><mi id="S4.SS2.SSS3.2.p2.7.m7.1.1" xref="S4.SS2.SSS3.2.p2.7.m7.1.1.cmml">x</mi><mo id="S4.SS2.SSS3.2.p2.7.m7.1.2.3.3.2.2" stretchy="false" xref="S4.SS2.SSS3.2.p2.7.m7.1.2.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS3.2.p2.7.m7.1b"><apply id="S4.SS2.SSS3.2.p2.7.m7.1.2.cmml" xref="S4.SS2.SSS3.2.p2.7.m7.1.2"><in id="S4.SS2.SSS3.2.p2.7.m7.1.2.1.cmml" xref="S4.SS2.SSS3.2.p2.7.m7.1.2.1"></in><ci id="S4.SS2.SSS3.2.p2.7.m7.1.2.2.cmml" xref="S4.SS2.SSS3.2.p2.7.m7.1.2.2">𝑦</ci><apply id="S4.SS2.SSS3.2.p2.7.m7.1.2.3.cmml" xref="S4.SS2.SSS3.2.p2.7.m7.1.2.3"><times id="S4.SS2.SSS3.2.p2.7.m7.1.2.3.1.cmml" xref="S4.SS2.SSS3.2.p2.7.m7.1.2.3.1"></times><ci id="S4.SS2.SSS3.2.p2.7.m7.1.2.3.2.cmml" xref="S4.SS2.SSS3.2.p2.7.m7.1.2.3.2">𝐶</ci><ci id="S4.SS2.SSS3.2.p2.7.m7.1.1.cmml" xref="S4.SS2.SSS3.2.p2.7.m7.1.1">𝑥</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS3.2.p2.7.m7.1c">y\in C(x)</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS3.2.p2.7.m7.1d">italic_y ∈ italic_C ( italic_x )</annotation></semantics></math>, there must be some path in <math alttext="E\cup\textnormal{OPT}" class="ltx_Math" display="inline" id="S4.SS2.SSS3.2.p2.8.m8.1"><semantics id="S4.SS2.SSS3.2.p2.8.m8.1a"><mrow id="S4.SS2.SSS3.2.p2.8.m8.1.1" xref="S4.SS2.SSS3.2.p2.8.m8.1.1.cmml"><mi id="S4.SS2.SSS3.2.p2.8.m8.1.1.2" xref="S4.SS2.SSS3.2.p2.8.m8.1.1.2.cmml">E</mi><mo id="S4.SS2.SSS3.2.p2.8.m8.1.1.1" xref="S4.SS2.SSS3.2.p2.8.m8.1.1.1.cmml">∪</mo><mtext id="S4.SS2.SSS3.2.p2.8.m8.1.1.3" xref="S4.SS2.SSS3.2.p2.8.m8.1.1.3a.cmml">OPT</mtext></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS3.2.p2.8.m8.1b"><apply id="S4.SS2.SSS3.2.p2.8.m8.1.1.cmml" xref="S4.SS2.SSS3.2.p2.8.m8.1.1"><union id="S4.SS2.SSS3.2.p2.8.m8.1.1.1.cmml" xref="S4.SS2.SSS3.2.p2.8.m8.1.1.1"></union><ci id="S4.SS2.SSS3.2.p2.8.m8.1.1.2.cmml" xref="S4.SS2.SSS3.2.p2.8.m8.1.1.2">𝐸</ci><ci id="S4.SS2.SSS3.2.p2.8.m8.1.1.3a.cmml" xref="S4.SS2.SSS3.2.p2.8.m8.1.1.3"><mtext id="S4.SS2.SSS3.2.p2.8.m8.1.1.3.cmml" xref="S4.SS2.SSS3.2.p2.8.m8.1.1.3">OPT</mtext></ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS3.2.p2.8.m8.1c">E\cup\textnormal{OPT}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS3.2.p2.8.m8.1d">italic_E ∪ OPT</annotation></semantics></math> from the vertices of <math alttext="G" class="ltx_Math" display="inline" id="S4.SS2.SSS3.2.p2.9.m9.1"><semantics id="S4.SS2.SSS3.2.p2.9.m9.1a"><mi id="S4.SS2.SSS3.2.p2.9.m9.1.1" xref="S4.SS2.SSS3.2.p2.9.m9.1.1.cmml">G</mi><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS3.2.p2.9.m9.1b"><ci id="S4.SS2.SSS3.2.p2.9.m9.1.1.cmml" xref="S4.SS2.SSS3.2.p2.9.m9.1.1">𝐺</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS3.2.p2.9.m9.1c">G</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS3.2.p2.9.m9.1d">italic_G</annotation></semantics></math> corresponding to <math alttext="T_{y}" class="ltx_Math" display="inline" id="S4.SS2.SSS3.2.p2.10.m10.1"><semantics id="S4.SS2.SSS3.2.p2.10.m10.1a"><msub id="S4.SS2.SSS3.2.p2.10.m10.1.1" xref="S4.SS2.SSS3.2.p2.10.m10.1.1.cmml"><mi id="S4.SS2.SSS3.2.p2.10.m10.1.1.2" xref="S4.SS2.SSS3.2.p2.10.m10.1.1.2.cmml">T</mi><mi id="S4.SS2.SSS3.2.p2.10.m10.1.1.3" xref="S4.SS2.SSS3.2.p2.10.m10.1.1.3.cmml">y</mi></msub><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS3.2.p2.10.m10.1b"><apply id="S4.SS2.SSS3.2.p2.10.m10.1.1.cmml" xref="S4.SS2.SSS3.2.p2.10.m10.1.1"><csymbol cd="ambiguous" id="S4.SS2.SSS3.2.p2.10.m10.1.1.1.cmml" xref="S4.SS2.SSS3.2.p2.10.m10.1.1">subscript</csymbol><ci id="S4.SS2.SSS3.2.p2.10.m10.1.1.2.cmml" xref="S4.SS2.SSS3.2.p2.10.m10.1.1.2">𝑇</ci><ci id="S4.SS2.SSS3.2.p2.10.m10.1.1.3.cmml" xref="S4.SS2.SSS3.2.p2.10.m10.1.1.3">𝑦</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS3.2.p2.10.m10.1c">T_{y}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS3.2.p2.10.m10.1d">italic_T start_POSTSUBSCRIPT italic_y end_POSTSUBSCRIPT</annotation></semantics></math> to the vertices of <math alttext="G" class="ltx_Math" display="inline" id="S4.SS2.SSS3.2.p2.11.m11.1"><semantics id="S4.SS2.SSS3.2.p2.11.m11.1a"><mi id="S4.SS2.SSS3.2.p2.11.m11.1.1" xref="S4.SS2.SSS3.2.p2.11.m11.1.1.cmml">G</mi><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS3.2.p2.11.m11.1b"><ci id="S4.SS2.SSS3.2.p2.11.m11.1.1.cmml" xref="S4.SS2.SSS3.2.p2.11.m11.1.1">𝐺</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS3.2.p2.11.m11.1c">G</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS3.2.p2.11.m11.1d">italic_G</annotation></semantics></math> corresponding to <math alttext="T\setminus T_{x}" class="ltx_Math" display="inline" id="S4.SS2.SSS3.2.p2.12.m12.1"><semantics id="S4.SS2.SSS3.2.p2.12.m12.1a"><mrow id="S4.SS2.SSS3.2.p2.12.m12.1.1" xref="S4.SS2.SSS3.2.p2.12.m12.1.1.cmml"><mi id="S4.SS2.SSS3.2.p2.12.m12.1.1.2" xref="S4.SS2.SSS3.2.p2.12.m12.1.1.2.cmml">T</mi><mo id="S4.SS2.SSS3.2.p2.12.m12.1.1.1" xref="S4.SS2.SSS3.2.p2.12.m12.1.1.1.cmml">∖</mo><msub id="S4.SS2.SSS3.2.p2.12.m12.1.1.3" xref="S4.SS2.SSS3.2.p2.12.m12.1.1.3.cmml"><mi id="S4.SS2.SSS3.2.p2.12.m12.1.1.3.2" xref="S4.SS2.SSS3.2.p2.12.m12.1.1.3.2.cmml">T</mi><mi id="S4.SS2.SSS3.2.p2.12.m12.1.1.3.3" xref="S4.SS2.SSS3.2.p2.12.m12.1.1.3.3.cmml">x</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS3.2.p2.12.m12.1b"><apply id="S4.SS2.SSS3.2.p2.12.m12.1.1.cmml" xref="S4.SS2.SSS3.2.p2.12.m12.1.1"><setdiff id="S4.SS2.SSS3.2.p2.12.m12.1.1.1.cmml" xref="S4.SS2.SSS3.2.p2.12.m12.1.1.1"></setdiff><ci id="S4.SS2.SSS3.2.p2.12.m12.1.1.2.cmml" xref="S4.SS2.SSS3.2.p2.12.m12.1.1.2">𝑇</ci><apply id="S4.SS2.SSS3.2.p2.12.m12.1.1.3.cmml" xref="S4.SS2.SSS3.2.p2.12.m12.1.1.3"><csymbol cd="ambiguous" id="S4.SS2.SSS3.2.p2.12.m12.1.1.3.1.cmml" xref="S4.SS2.SSS3.2.p2.12.m12.1.1.3">subscript</csymbol><ci id="S4.SS2.SSS3.2.p2.12.m12.1.1.3.2.cmml" xref="S4.SS2.SSS3.2.p2.12.m12.1.1.3.2">𝑇</ci><ci id="S4.SS2.SSS3.2.p2.12.m12.1.1.3.3.cmml" xref="S4.SS2.SSS3.2.p2.12.m12.1.1.3.3">𝑥</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS3.2.p2.12.m12.1c">T\setminus T_{x}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS3.2.p2.12.m12.1d">italic_T ∖ italic_T start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT</annotation></semantics></math> <em class="ltx_emph ltx_font_italic" id="S4.SS2.SSS3.2.p2.31.3">without</em> using the two nodes in <math alttext="V(G_{x})" class="ltx_Math" display="inline" id="S4.SS2.SSS3.2.p2.13.m13.1"><semantics id="S4.SS2.SSS3.2.p2.13.m13.1a"><mrow id="S4.SS2.SSS3.2.p2.13.m13.1.1" xref="S4.SS2.SSS3.2.p2.13.m13.1.1.cmml"><mi id="S4.SS2.SSS3.2.p2.13.m13.1.1.3" xref="S4.SS2.SSS3.2.p2.13.m13.1.1.3.cmml">V</mi><mo id="S4.SS2.SSS3.2.p2.13.m13.1.1.2" xref="S4.SS2.SSS3.2.p2.13.m13.1.1.2.cmml">⁢</mo><mrow id="S4.SS2.SSS3.2.p2.13.m13.1.1.1.1" xref="S4.SS2.SSS3.2.p2.13.m13.1.1.1.1.1.cmml"><mo id="S4.SS2.SSS3.2.p2.13.m13.1.1.1.1.2" stretchy="false" xref="S4.SS2.SSS3.2.p2.13.m13.1.1.1.1.1.cmml">(</mo><msub id="S4.SS2.SSS3.2.p2.13.m13.1.1.1.1.1" xref="S4.SS2.SSS3.2.p2.13.m13.1.1.1.1.1.cmml"><mi id="S4.SS2.SSS3.2.p2.13.m13.1.1.1.1.1.2" xref="S4.SS2.SSS3.2.p2.13.m13.1.1.1.1.1.2.cmml">G</mi><mi id="S4.SS2.SSS3.2.p2.13.m13.1.1.1.1.1.3" xref="S4.SS2.SSS3.2.p2.13.m13.1.1.1.1.1.3.cmml">x</mi></msub><mo id="S4.SS2.SSS3.2.p2.13.m13.1.1.1.1.3" stretchy="false" xref="S4.SS2.SSS3.2.p2.13.m13.1.1.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS3.2.p2.13.m13.1b"><apply id="S4.SS2.SSS3.2.p2.13.m13.1.1.cmml" xref="S4.SS2.SSS3.2.p2.13.m13.1.1"><times id="S4.SS2.SSS3.2.p2.13.m13.1.1.2.cmml" xref="S4.SS2.SSS3.2.p2.13.m13.1.1.2"></times><ci id="S4.SS2.SSS3.2.p2.13.m13.1.1.3.cmml" xref="S4.SS2.SSS3.2.p2.13.m13.1.1.3">𝑉</ci><apply id="S4.SS2.SSS3.2.p2.13.m13.1.1.1.1.1.cmml" xref="S4.SS2.SSS3.2.p2.13.m13.1.1.1.1"><csymbol cd="ambiguous" id="S4.SS2.SSS3.2.p2.13.m13.1.1.1.1.1.1.cmml" xref="S4.SS2.SSS3.2.p2.13.m13.1.1.1.1">subscript</csymbol><ci id="S4.SS2.SSS3.2.p2.13.m13.1.1.1.1.1.2.cmml" xref="S4.SS2.SSS3.2.p2.13.m13.1.1.1.1.1.2">𝐺</ci><ci id="S4.SS2.SSS3.2.p2.13.m13.1.1.1.1.1.3.cmml" xref="S4.SS2.SSS3.2.p2.13.m13.1.1.1.1.1.3">𝑥</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS3.2.p2.13.m13.1c">V(G_{x})</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS3.2.p2.13.m13.1d">italic_V ( italic_G start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT )</annotation></semantics></math>; else <math alttext="V(G_{x})" class="ltx_Math" display="inline" id="S4.SS2.SSS3.2.p2.14.m14.1"><semantics id="S4.SS2.SSS3.2.p2.14.m14.1a"><mrow id="S4.SS2.SSS3.2.p2.14.m14.1.1" xref="S4.SS2.SSS3.2.p2.14.m14.1.1.cmml"><mi id="S4.SS2.SSS3.2.p2.14.m14.1.1.3" xref="S4.SS2.SSS3.2.p2.14.m14.1.1.3.cmml">V</mi><mo id="S4.SS2.SSS3.2.p2.14.m14.1.1.2" xref="S4.SS2.SSS3.2.p2.14.m14.1.1.2.cmml">⁢</mo><mrow id="S4.SS2.SSS3.2.p2.14.m14.1.1.1.1" xref="S4.SS2.SSS3.2.p2.14.m14.1.1.1.1.1.cmml"><mo id="S4.SS2.SSS3.2.p2.14.m14.1.1.1.1.2" stretchy="false" xref="S4.SS2.SSS3.2.p2.14.m14.1.1.1.1.1.cmml">(</mo><msub id="S4.SS2.SSS3.2.p2.14.m14.1.1.1.1.1" xref="S4.SS2.SSS3.2.p2.14.m14.1.1.1.1.1.cmml"><mi id="S4.SS2.SSS3.2.p2.14.m14.1.1.1.1.1.2" xref="S4.SS2.SSS3.2.p2.14.m14.1.1.1.1.1.2.cmml">G</mi><mi id="S4.SS2.SSS3.2.p2.14.m14.1.1.1.1.1.3" xref="S4.SS2.SSS3.2.p2.14.m14.1.1.1.1.1.3.cmml">x</mi></msub><mo id="S4.SS2.SSS3.2.p2.14.m14.1.1.1.1.3" stretchy="false" xref="S4.SS2.SSS3.2.p2.14.m14.1.1.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS3.2.p2.14.m14.1b"><apply id="S4.SS2.SSS3.2.p2.14.m14.1.1.cmml" xref="S4.SS2.SSS3.2.p2.14.m14.1.1"><times id="S4.SS2.SSS3.2.p2.14.m14.1.1.2.cmml" xref="S4.SS2.SSS3.2.p2.14.m14.1.1.2"></times><ci id="S4.SS2.SSS3.2.p2.14.m14.1.1.3.cmml" xref="S4.SS2.SSS3.2.p2.14.m14.1.1.3">𝑉</ci><apply id="S4.SS2.SSS3.2.p2.14.m14.1.1.1.1.1.cmml" xref="S4.SS2.SSS3.2.p2.14.m14.1.1.1.1"><csymbol cd="ambiguous" id="S4.SS2.SSS3.2.p2.14.m14.1.1.1.1.1.1.cmml" xref="S4.SS2.SSS3.2.p2.14.m14.1.1.1.1">subscript</csymbol><ci id="S4.SS2.SSS3.2.p2.14.m14.1.1.1.1.1.2.cmml" xref="S4.SS2.SSS3.2.p2.14.m14.1.1.1.1.1.2">𝐺</ci><ci id="S4.SS2.SSS3.2.p2.14.m14.1.1.1.1.1.3.cmml" xref="S4.SS2.SSS3.2.p2.14.m14.1.1.1.1.1.3">𝑥</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS3.2.p2.14.m14.1c">V(G_{x})</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS3.2.p2.14.m14.1d">italic_V ( italic_G start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT )</annotation></semantics></math> would be a 2-cut separating the vertices corresponding to <math alttext="T_{y}" class="ltx_Math" display="inline" id="S4.SS2.SSS3.2.p2.15.m15.1"><semantics id="S4.SS2.SSS3.2.p2.15.m15.1a"><msub id="S4.SS2.SSS3.2.p2.15.m15.1.1" xref="S4.SS2.SSS3.2.p2.15.m15.1.1.cmml"><mi id="S4.SS2.SSS3.2.p2.15.m15.1.1.2" xref="S4.SS2.SSS3.2.p2.15.m15.1.1.2.cmml">T</mi><mi id="S4.SS2.SSS3.2.p2.15.m15.1.1.3" xref="S4.SS2.SSS3.2.p2.15.m15.1.1.3.cmml">y</mi></msub><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS3.2.p2.15.m15.1b"><apply id="S4.SS2.SSS3.2.p2.15.m15.1.1.cmml" xref="S4.SS2.SSS3.2.p2.15.m15.1.1"><csymbol cd="ambiguous" id="S4.SS2.SSS3.2.p2.15.m15.1.1.1.cmml" xref="S4.SS2.SSS3.2.p2.15.m15.1.1">subscript</csymbol><ci id="S4.SS2.SSS3.2.p2.15.m15.1.1.2.cmml" xref="S4.SS2.SSS3.2.p2.15.m15.1.1.2">𝑇</ci><ci id="S4.SS2.SSS3.2.p2.15.m15.1.1.3.cmml" xref="S4.SS2.SSS3.2.p2.15.m15.1.1.3">𝑦</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS3.2.p2.15.m15.1c">T_{y}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS3.2.p2.15.m15.1d">italic_T start_POSTSUBSCRIPT italic_y end_POSTSUBSCRIPT</annotation></semantics></math> from the rest of the graph, contradicting feasibility of <span class="ltx_text ltx_markedasmath" id="S4.SS2.SSS3.2.p2.31.4">OPT</span>. These paths must each have a link “leaving” <math alttext="T_{x}" class="ltx_Math" display="inline" id="S4.SS2.SSS3.2.p2.17.m17.1"><semantics id="S4.SS2.SSS3.2.p2.17.m17.1a"><msub id="S4.SS2.SSS3.2.p2.17.m17.1.1" xref="S4.SS2.SSS3.2.p2.17.m17.1.1.cmml"><mi id="S4.SS2.SSS3.2.p2.17.m17.1.1.2" xref="S4.SS2.SSS3.2.p2.17.m17.1.1.2.cmml">T</mi><mi id="S4.SS2.SSS3.2.p2.17.m17.1.1.3" xref="S4.SS2.SSS3.2.p2.17.m17.1.1.3.cmml">x</mi></msub><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS3.2.p2.17.m17.1b"><apply id="S4.SS2.SSS3.2.p2.17.m17.1.1.cmml" xref="S4.SS2.SSS3.2.p2.17.m17.1.1"><csymbol cd="ambiguous" id="S4.SS2.SSS3.2.p2.17.m17.1.1.1.cmml" xref="S4.SS2.SSS3.2.p2.17.m17.1.1">subscript</csymbol><ci id="S4.SS2.SSS3.2.p2.17.m17.1.1.2.cmml" xref="S4.SS2.SSS3.2.p2.17.m17.1.1.2">𝑇</ci><ci id="S4.SS2.SSS3.2.p2.17.m17.1.1.3.cmml" xref="S4.SS2.SSS3.2.p2.17.m17.1.1.3">𝑥</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS3.2.p2.17.m17.1c">T_{x}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS3.2.p2.17.m17.1d">italic_T start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT</annotation></semantics></math>; that is, a link <math alttext="e" class="ltx_Math" display="inline" id="S4.SS2.SSS3.2.p2.18.m18.1"><semantics id="S4.SS2.SSS3.2.p2.18.m18.1a"><mi id="S4.SS2.SSS3.2.p2.18.m18.1.1" xref="S4.SS2.SSS3.2.p2.18.m18.1.1.cmml">e</mi><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS3.2.p2.18.m18.1b"><ci id="S4.SS2.SSS3.2.p2.18.m18.1.1.cmml" xref="S4.SS2.SSS3.2.p2.18.m18.1.1">𝑒</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS3.2.p2.18.m18.1c">e</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS3.2.p2.18.m18.1d">italic_e</annotation></semantics></math> with one endpoint in the vertex set of <math alttext="G" class="ltx_Math" display="inline" id="S4.SS2.SSS3.2.p2.19.m19.1"><semantics id="S4.SS2.SSS3.2.p2.19.m19.1a"><mi id="S4.SS2.SSS3.2.p2.19.m19.1.1" xref="S4.SS2.SSS3.2.p2.19.m19.1.1.cmml">G</mi><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS3.2.p2.19.m19.1b"><ci id="S4.SS2.SSS3.2.p2.19.m19.1.1.cmml" xref="S4.SS2.SSS3.2.p2.19.m19.1.1">𝐺</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS3.2.p2.19.m19.1c">G</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS3.2.p2.19.m19.1d">italic_G</annotation></semantics></math> corresponding to <math alttext="T_{x}" class="ltx_Math" display="inline" id="S4.SS2.SSS3.2.p2.20.m20.1"><semantics id="S4.SS2.SSS3.2.p2.20.m20.1a"><msub id="S4.SS2.SSS3.2.p2.20.m20.1.1" xref="S4.SS2.SSS3.2.p2.20.m20.1.1.cmml"><mi id="S4.SS2.SSS3.2.p2.20.m20.1.1.2" xref="S4.SS2.SSS3.2.p2.20.m20.1.1.2.cmml">T</mi><mi id="S4.SS2.SSS3.2.p2.20.m20.1.1.3" xref="S4.SS2.SSS3.2.p2.20.m20.1.1.3.cmml">x</mi></msub><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS3.2.p2.20.m20.1b"><apply id="S4.SS2.SSS3.2.p2.20.m20.1.1.cmml" xref="S4.SS2.SSS3.2.p2.20.m20.1.1"><csymbol cd="ambiguous" id="S4.SS2.SSS3.2.p2.20.m20.1.1.1.cmml" xref="S4.SS2.SSS3.2.p2.20.m20.1.1">subscript</csymbol><ci id="S4.SS2.SSS3.2.p2.20.m20.1.1.2.cmml" xref="S4.SS2.SSS3.2.p2.20.m20.1.1.2">𝑇</ci><ci id="S4.SS2.SSS3.2.p2.20.m20.1.1.3.cmml" xref="S4.SS2.SSS3.2.p2.20.m20.1.1.3">𝑥</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS3.2.p2.20.m20.1c">T_{x}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS3.2.p2.20.m20.1d">italic_T start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT</annotation></semantics></math> and the other in the vertex set of <math alttext="G" class="ltx_Math" display="inline" id="S4.SS2.SSS3.2.p2.21.m21.1"><semantics id="S4.SS2.SSS3.2.p2.21.m21.1a"><mi id="S4.SS2.SSS3.2.p2.21.m21.1.1" xref="S4.SS2.SSS3.2.p2.21.m21.1.1.cmml">G</mi><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS3.2.p2.21.m21.1b"><ci id="S4.SS2.SSS3.2.p2.21.m21.1.1.cmml" xref="S4.SS2.SSS3.2.p2.21.m21.1.1">𝐺</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS3.2.p2.21.m21.1c">G</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS3.2.p2.21.m21.1d">italic_G</annotation></semantics></math> corresponding to <math alttext="T\setminus T_{x}" class="ltx_Math" display="inline" id="S4.SS2.SSS3.2.p2.22.m22.1"><semantics id="S4.SS2.SSS3.2.p2.22.m22.1a"><mrow id="S4.SS2.SSS3.2.p2.22.m22.1.1" xref="S4.SS2.SSS3.2.p2.22.m22.1.1.cmml"><mi id="S4.SS2.SSS3.2.p2.22.m22.1.1.2" xref="S4.SS2.SSS3.2.p2.22.m22.1.1.2.cmml">T</mi><mo id="S4.SS2.SSS3.2.p2.22.m22.1.1.1" xref="S4.SS2.SSS3.2.p2.22.m22.1.1.1.cmml">∖</mo><msub id="S4.SS2.SSS3.2.p2.22.m22.1.1.3" xref="S4.SS2.SSS3.2.p2.22.m22.1.1.3.cmml"><mi id="S4.SS2.SSS3.2.p2.22.m22.1.1.3.2" xref="S4.SS2.SSS3.2.p2.22.m22.1.1.3.2.cmml">T</mi><mi id="S4.SS2.SSS3.2.p2.22.m22.1.1.3.3" xref="S4.SS2.SSS3.2.p2.22.m22.1.1.3.3.cmml">x</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS3.2.p2.22.m22.1b"><apply id="S4.SS2.SSS3.2.p2.22.m22.1.1.cmml" xref="S4.SS2.SSS3.2.p2.22.m22.1.1"><setdiff id="S4.SS2.SSS3.2.p2.22.m22.1.1.1.cmml" xref="S4.SS2.SSS3.2.p2.22.m22.1.1.1"></setdiff><ci id="S4.SS2.SSS3.2.p2.22.m22.1.1.2.cmml" xref="S4.SS2.SSS3.2.p2.22.m22.1.1.2">𝑇</ci><apply id="S4.SS2.SSS3.2.p2.22.m22.1.1.3.cmml" xref="S4.SS2.SSS3.2.p2.22.m22.1.1.3"><csymbol cd="ambiguous" id="S4.SS2.SSS3.2.p2.22.m22.1.1.3.1.cmml" xref="S4.SS2.SSS3.2.p2.22.m22.1.1.3">subscript</csymbol><ci id="S4.SS2.SSS3.2.p2.22.m22.1.1.3.2.cmml" xref="S4.SS2.SSS3.2.p2.22.m22.1.1.3.2">𝑇</ci><ci id="S4.SS2.SSS3.2.p2.22.m22.1.1.3.3.cmml" xref="S4.SS2.SSS3.2.p2.22.m22.1.1.3.3">𝑥</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS3.2.p2.22.m22.1c">T\setminus T_{x}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS3.2.p2.22.m22.1d">italic_T ∖ italic_T start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT</annotation></semantics></math>. By construction, endpoint of <math alttext="e" class="ltx_Math" display="inline" id="S4.SS2.SSS3.2.p2.23.m23.1"><semantics id="S4.SS2.SSS3.2.p2.23.m23.1a"><mi id="S4.SS2.SSS3.2.p2.23.m23.1.1" xref="S4.SS2.SSS3.2.p2.23.m23.1.1.cmml">e</mi><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS3.2.p2.23.m23.1b"><ci id="S4.SS2.SSS3.2.p2.23.m23.1.1.cmml" xref="S4.SS2.SSS3.2.p2.23.m23.1.1">𝑒</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS3.2.p2.23.m23.1c">e</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS3.2.p2.23.m23.1d">italic_e</annotation></semantics></math> in <math alttext="T_{x}" class="ltx_Math" display="inline" id="S4.SS2.SSS3.2.p2.24.m24.1"><semantics id="S4.SS2.SSS3.2.p2.24.m24.1a"><msub id="S4.SS2.SSS3.2.p2.24.m24.1.1" xref="S4.SS2.SSS3.2.p2.24.m24.1.1.cmml"><mi id="S4.SS2.SSS3.2.p2.24.m24.1.1.2" xref="S4.SS2.SSS3.2.p2.24.m24.1.1.2.cmml">T</mi><mi id="S4.SS2.SSS3.2.p2.24.m24.1.1.3" xref="S4.SS2.SSS3.2.p2.24.m24.1.1.3.cmml">x</mi></msub><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS3.2.p2.24.m24.1b"><apply id="S4.SS2.SSS3.2.p2.24.m24.1.1.cmml" xref="S4.SS2.SSS3.2.p2.24.m24.1.1"><csymbol cd="ambiguous" id="S4.SS2.SSS3.2.p2.24.m24.1.1.1.cmml" xref="S4.SS2.SSS3.2.p2.24.m24.1.1">subscript</csymbol><ci id="S4.SS2.SSS3.2.p2.24.m24.1.1.2.cmml" xref="S4.SS2.SSS3.2.p2.24.m24.1.1.2">𝑇</ci><ci id="S4.SS2.SSS3.2.p2.24.m24.1.1.3.cmml" xref="S4.SS2.SSS3.2.p2.24.m24.1.1.3">𝑥</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS3.2.p2.24.m24.1c">T_{x}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS3.2.p2.24.m24.1d">italic_T start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT</annotation></semantics></math> must be in a “good” supernode. Thus all “bad” supernodes are connected to at least one “good” supernode in <span class="ltx_text ltx_markedasmath" id="S4.SS2.SSS3.2.p2.31.5">OPT</span>. Furthermore, a minimal path in the contracted graph from a “bad” supernode to a “good” supernode only uses links in <math alttext="E^{*}" class="ltx_Math" display="inline" id="S4.SS2.SSS3.2.p2.26.m26.1"><semantics id="S4.SS2.SSS3.2.p2.26.m26.1a"><msup id="S4.SS2.SSS3.2.p2.26.m26.1.1" xref="S4.SS2.SSS3.2.p2.26.m26.1.1.cmml"><mi id="S4.SS2.SSS3.2.p2.26.m26.1.1.2" xref="S4.SS2.SSS3.2.p2.26.m26.1.1.2.cmml">E</mi><mo id="S4.SS2.SSS3.2.p2.26.m26.1.1.3" xref="S4.SS2.SSS3.2.p2.26.m26.1.1.3.cmml">∗</mo></msup><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS3.2.p2.26.m26.1b"><apply id="S4.SS2.SSS3.2.p2.26.m26.1.1.cmml" xref="S4.SS2.SSS3.2.p2.26.m26.1.1"><csymbol cd="ambiguous" id="S4.SS2.SSS3.2.p2.26.m26.1.1.1.cmml" xref="S4.SS2.SSS3.2.p2.26.m26.1.1">superscript</csymbol><ci id="S4.SS2.SSS3.2.p2.26.m26.1.1.2.cmml" xref="S4.SS2.SSS3.2.p2.26.m26.1.1.2">𝐸</ci><times id="S4.SS2.SSS3.2.p2.26.m26.1.1.3.cmml" xref="S4.SS2.SSS3.2.p2.26.m26.1.1.3"></times></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS3.2.p2.26.m26.1c">E^{*}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS3.2.p2.26.m26.1d">italic_E start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT</annotation></semantics></math>, since these links all must go between subtrees of <math alttext="x" class="ltx_Math" display="inline" id="S4.SS2.SSS3.2.p2.27.m27.1"><semantics id="S4.SS2.SSS3.2.p2.27.m27.1a"><mi id="S4.SS2.SSS3.2.p2.27.m27.1.1" xref="S4.SS2.SSS3.2.p2.27.m27.1.1.cmml">x</mi><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS3.2.p2.27.m27.1b"><ci id="S4.SS2.SSS3.2.p2.27.m27.1.1.cmml" xref="S4.SS2.SSS3.2.p2.27.m27.1.1">𝑥</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS3.2.p2.27.m27.1c">x</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS3.2.p2.27.m27.1d">italic_x</annotation></semantics></math> and avoid <math alttext="V(G_{x})" class="ltx_Math" display="inline" id="S4.SS2.SSS3.2.p2.28.m28.1"><semantics id="S4.SS2.SSS3.2.p2.28.m28.1a"><mrow id="S4.SS2.SSS3.2.p2.28.m28.1.1" xref="S4.SS2.SSS3.2.p2.28.m28.1.1.cmml"><mi id="S4.SS2.SSS3.2.p2.28.m28.1.1.3" xref="S4.SS2.SSS3.2.p2.28.m28.1.1.3.cmml">V</mi><mo id="S4.SS2.SSS3.2.p2.28.m28.1.1.2" xref="S4.SS2.SSS3.2.p2.28.m28.1.1.2.cmml">⁢</mo><mrow id="S4.SS2.SSS3.2.p2.28.m28.1.1.1.1" xref="S4.SS2.SSS3.2.p2.28.m28.1.1.1.1.1.cmml"><mo id="S4.SS2.SSS3.2.p2.28.m28.1.1.1.1.2" stretchy="false" xref="S4.SS2.SSS3.2.p2.28.m28.1.1.1.1.1.cmml">(</mo><msub id="S4.SS2.SSS3.2.p2.28.m28.1.1.1.1.1" xref="S4.SS2.SSS3.2.p2.28.m28.1.1.1.1.1.cmml"><mi id="S4.SS2.SSS3.2.p2.28.m28.1.1.1.1.1.2" xref="S4.SS2.SSS3.2.p2.28.m28.1.1.1.1.1.2.cmml">G</mi><mi id="S4.SS2.SSS3.2.p2.28.m28.1.1.1.1.1.3" xref="S4.SS2.SSS3.2.p2.28.m28.1.1.1.1.1.3.cmml">x</mi></msub><mo id="S4.SS2.SSS3.2.p2.28.m28.1.1.1.1.3" stretchy="false" xref="S4.SS2.SSS3.2.p2.28.m28.1.1.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS3.2.p2.28.m28.1b"><apply id="S4.SS2.SSS3.2.p2.28.m28.1.1.cmml" xref="S4.SS2.SSS3.2.p2.28.m28.1.1"><times id="S4.SS2.SSS3.2.p2.28.m28.1.1.2.cmml" xref="S4.SS2.SSS3.2.p2.28.m28.1.1.2"></times><ci id="S4.SS2.SSS3.2.p2.28.m28.1.1.3.cmml" xref="S4.SS2.SSS3.2.p2.28.m28.1.1.3">𝑉</ci><apply id="S4.SS2.SSS3.2.p2.28.m28.1.1.1.1.1.cmml" xref="S4.SS2.SSS3.2.p2.28.m28.1.1.1.1"><csymbol cd="ambiguous" id="S4.SS2.SSS3.2.p2.28.m28.1.1.1.1.1.1.cmml" xref="S4.SS2.SSS3.2.p2.28.m28.1.1.1.1">subscript</csymbol><ci id="S4.SS2.SSS3.2.p2.28.m28.1.1.1.1.1.2.cmml" xref="S4.SS2.SSS3.2.p2.28.m28.1.1.1.1.1.2">𝐺</ci><ci id="S4.SS2.SSS3.2.p2.28.m28.1.1.1.1.1.3.cmml" xref="S4.SS2.SSS3.2.p2.28.m28.1.1.1.1.1.3">𝑥</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS3.2.p2.28.m28.1c">V(G_{x})</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS3.2.p2.28.m28.1d">italic_V ( italic_G start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT )</annotation></semantics></math>. Thus in <math alttext="E^{*}(x)" class="ltx_Math" display="inline" id="S4.SS2.SSS3.2.p2.29.m29.1"><semantics id="S4.SS2.SSS3.2.p2.29.m29.1a"><mrow id="S4.SS2.SSS3.2.p2.29.m29.1.2" xref="S4.SS2.SSS3.2.p2.29.m29.1.2.cmml"><msup id="S4.SS2.SSS3.2.p2.29.m29.1.2.2" xref="S4.SS2.SSS3.2.p2.29.m29.1.2.2.cmml"><mi id="S4.SS2.SSS3.2.p2.29.m29.1.2.2.2" xref="S4.SS2.SSS3.2.p2.29.m29.1.2.2.2.cmml">E</mi><mo id="S4.SS2.SSS3.2.p2.29.m29.1.2.2.3" xref="S4.SS2.SSS3.2.p2.29.m29.1.2.2.3.cmml">∗</mo></msup><mo id="S4.SS2.SSS3.2.p2.29.m29.1.2.1" xref="S4.SS2.SSS3.2.p2.29.m29.1.2.1.cmml">⁢</mo><mrow id="S4.SS2.SSS3.2.p2.29.m29.1.2.3.2" xref="S4.SS2.SSS3.2.p2.29.m29.1.2.cmml"><mo id="S4.SS2.SSS3.2.p2.29.m29.1.2.3.2.1" stretchy="false" xref="S4.SS2.SSS3.2.p2.29.m29.1.2.cmml">(</mo><mi id="S4.SS2.SSS3.2.p2.29.m29.1.1" xref="S4.SS2.SSS3.2.p2.29.m29.1.1.cmml">x</mi><mo id="S4.SS2.SSS3.2.p2.29.m29.1.2.3.2.2" stretchy="false" xref="S4.SS2.SSS3.2.p2.29.m29.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS3.2.p2.29.m29.1b"><apply id="S4.SS2.SSS3.2.p2.29.m29.1.2.cmml" xref="S4.SS2.SSS3.2.p2.29.m29.1.2"><times id="S4.SS2.SSS3.2.p2.29.m29.1.2.1.cmml" xref="S4.SS2.SSS3.2.p2.29.m29.1.2.1"></times><apply id="S4.SS2.SSS3.2.p2.29.m29.1.2.2.cmml" xref="S4.SS2.SSS3.2.p2.29.m29.1.2.2"><csymbol cd="ambiguous" id="S4.SS2.SSS3.2.p2.29.m29.1.2.2.1.cmml" xref="S4.SS2.SSS3.2.p2.29.m29.1.2.2">superscript</csymbol><ci id="S4.SS2.SSS3.2.p2.29.m29.1.2.2.2.cmml" xref="S4.SS2.SSS3.2.p2.29.m29.1.2.2.2">𝐸</ci><times id="S4.SS2.SSS3.2.p2.29.m29.1.2.2.3.cmml" xref="S4.SS2.SSS3.2.p2.29.m29.1.2.2.3"></times></apply><ci id="S4.SS2.SSS3.2.p2.29.m29.1.1.cmml" xref="S4.SS2.SSS3.2.p2.29.m29.1.1">𝑥</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS3.2.p2.29.m29.1c">E^{*}(x)</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS3.2.p2.29.m29.1d">italic_E start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ( italic_x )</annotation></semantics></math>, all “bad” supernodes are connected to at least one “good” supernode. In particular, <math alttext="E^{*}(x)" class="ltx_Math" display="inline" id="S4.SS2.SSS3.2.p2.30.m30.1"><semantics id="S4.SS2.SSS3.2.p2.30.m30.1a"><mrow id="S4.SS2.SSS3.2.p2.30.m30.1.2" xref="S4.SS2.SSS3.2.p2.30.m30.1.2.cmml"><msup id="S4.SS2.SSS3.2.p2.30.m30.1.2.2" xref="S4.SS2.SSS3.2.p2.30.m30.1.2.2.cmml"><mi id="S4.SS2.SSS3.2.p2.30.m30.1.2.2.2" xref="S4.SS2.SSS3.2.p2.30.m30.1.2.2.2.cmml">E</mi><mo id="S4.SS2.SSS3.2.p2.30.m30.1.2.2.3" xref="S4.SS2.SSS3.2.p2.30.m30.1.2.2.3.cmml">∗</mo></msup><mo id="S4.SS2.SSS3.2.p2.30.m30.1.2.1" xref="S4.SS2.SSS3.2.p2.30.m30.1.2.1.cmml">⁢</mo><mrow id="S4.SS2.SSS3.2.p2.30.m30.1.2.3.2" xref="S4.SS2.SSS3.2.p2.30.m30.1.2.cmml"><mo id="S4.SS2.SSS3.2.p2.30.m30.1.2.3.2.1" stretchy="false" xref="S4.SS2.SSS3.2.p2.30.m30.1.2.cmml">(</mo><mi id="S4.SS2.SSS3.2.p2.30.m30.1.1" xref="S4.SS2.SSS3.2.p2.30.m30.1.1.cmml">x</mi><mo id="S4.SS2.SSS3.2.p2.30.m30.1.2.3.2.2" stretchy="false" xref="S4.SS2.SSS3.2.p2.30.m30.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS3.2.p2.30.m30.1b"><apply id="S4.SS2.SSS3.2.p2.30.m30.1.2.cmml" xref="S4.SS2.SSS3.2.p2.30.m30.1.2"><times id="S4.SS2.SSS3.2.p2.30.m30.1.2.1.cmml" xref="S4.SS2.SSS3.2.p2.30.m30.1.2.1"></times><apply id="S4.SS2.SSS3.2.p2.30.m30.1.2.2.cmml" xref="S4.SS2.SSS3.2.p2.30.m30.1.2.2"><csymbol cd="ambiguous" id="S4.SS2.SSS3.2.p2.30.m30.1.2.2.1.cmml" xref="S4.SS2.SSS3.2.p2.30.m30.1.2.2">superscript</csymbol><ci id="S4.SS2.SSS3.2.p2.30.m30.1.2.2.2.cmml" xref="S4.SS2.SSS3.2.p2.30.m30.1.2.2.2">𝐸</ci><times id="S4.SS2.SSS3.2.p2.30.m30.1.2.2.3.cmml" xref="S4.SS2.SSS3.2.p2.30.m30.1.2.2.3"></times></apply><ci id="S4.SS2.SSS3.2.p2.30.m30.1.1.cmml" xref="S4.SS2.SSS3.2.p2.30.m30.1.1">𝑥</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS3.2.p2.30.m30.1c">E^{*}(x)</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS3.2.p2.30.m30.1d">italic_E start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ( italic_x )</annotation></semantics></math> contains a spanning tree on the <math alttext="C^{\prime\prime}(x)" class="ltx_Math" display="inline" id="S4.SS2.SSS3.2.p2.31.m31.1"><semantics id="S4.SS2.SSS3.2.p2.31.m31.1a"><mrow id="S4.SS2.SSS3.2.p2.31.m31.1.2" xref="S4.SS2.SSS3.2.p2.31.m31.1.2.cmml"><msup id="S4.SS2.SSS3.2.p2.31.m31.1.2.2" xref="S4.SS2.SSS3.2.p2.31.m31.1.2.2.cmml"><mi id="S4.SS2.SSS3.2.p2.31.m31.1.2.2.2" xref="S4.SS2.SSS3.2.p2.31.m31.1.2.2.2.cmml">C</mi><mo id="S4.SS2.SSS3.2.p2.31.m31.1.2.2.3" xref="S4.SS2.SSS3.2.p2.31.m31.1.2.2.3.cmml">′′</mo></msup><mo id="S4.SS2.SSS3.2.p2.31.m31.1.2.1" xref="S4.SS2.SSS3.2.p2.31.m31.1.2.1.cmml">⁢</mo><mrow id="S4.SS2.SSS3.2.p2.31.m31.1.2.3.2" xref="S4.SS2.SSS3.2.p2.31.m31.1.2.cmml"><mo id="S4.SS2.SSS3.2.p2.31.m31.1.2.3.2.1" stretchy="false" xref="S4.SS2.SSS3.2.p2.31.m31.1.2.cmml">(</mo><mi id="S4.SS2.SSS3.2.p2.31.m31.1.1" xref="S4.SS2.SSS3.2.p2.31.m31.1.1.cmml">x</mi><mo id="S4.SS2.SSS3.2.p2.31.m31.1.2.3.2.2" stretchy="false" xref="S4.SS2.SSS3.2.p2.31.m31.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS3.2.p2.31.m31.1b"><apply id="S4.SS2.SSS3.2.p2.31.m31.1.2.cmml" xref="S4.SS2.SSS3.2.p2.31.m31.1.2"><times id="S4.SS2.SSS3.2.p2.31.m31.1.2.1.cmml" xref="S4.SS2.SSS3.2.p2.31.m31.1.2.1"></times><apply id="S4.SS2.SSS3.2.p2.31.m31.1.2.2.cmml" xref="S4.SS2.SSS3.2.p2.31.m31.1.2.2"><csymbol cd="ambiguous" id="S4.SS2.SSS3.2.p2.31.m31.1.2.2.1.cmml" xref="S4.SS2.SSS3.2.p2.31.m31.1.2.2">superscript</csymbol><ci id="S4.SS2.SSS3.2.p2.31.m31.1.2.2.2.cmml" xref="S4.SS2.SSS3.2.p2.31.m31.1.2.2.2">𝐶</ci><ci id="S4.SS2.SSS3.2.p2.31.m31.1.2.2.3.cmml" xref="S4.SS2.SSS3.2.p2.31.m31.1.2.2.3">′′</ci></apply><ci id="S4.SS2.SSS3.2.p2.31.m31.1.1.cmml" xref="S4.SS2.SSS3.2.p2.31.m31.1.1">𝑥</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS3.2.p2.31.m31.1c">C^{\prime\prime}(x)</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS3.2.p2.31.m31.1d">italic_C start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT ( italic_x )</annotation></semantics></math>. Therefore</p> <table class="ltx_equation ltx_eqn_table" id="S4.Ex14"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="w(\textnormal{SOL}\cap\cup_{x}E(H_{x}))=\sum_{x\in V(T):x\text{ is P-node}}w(E% (H^{\prime}_{x}))\leq\sum_{x\in V(T),x\text{ is P-node}}w(E^{*}(x))\leq w(% \textnormal{OPT})." class="ltx_math_unparsed" display="block" id="S4.Ex14.m1.4"><semantics id="S4.Ex14.m1.4a"><mrow id="S4.Ex14.m1.4b"><mi id="S4.Ex14.m1.4.5">w</mi><mrow id="S4.Ex14.m1.4.6"><mo id="S4.Ex14.m1.4.6.1" stretchy="false">(</mo><mtext id="S4.Ex14.m1.4.6.2">SOL</mtext><mo id="S4.Ex14.m1.4.6.3" rspace="0em">∩</mo><msub id="S4.Ex14.m1.4.6.4"><mo id="S4.Ex14.m1.4.6.4.2" lspace="0em">∪</mo><mi id="S4.Ex14.m1.4.6.4.3">x</mi></msub><mi id="S4.Ex14.m1.4.6.5">E</mi><mrow id="S4.Ex14.m1.4.6.6"><mo id="S4.Ex14.m1.4.6.6.1" stretchy="false">(</mo><msub id="S4.Ex14.m1.4.6.6.2"><mi id="S4.Ex14.m1.4.6.6.2.2">H</mi><mi id="S4.Ex14.m1.4.6.6.2.3">x</mi></msub><mo id="S4.Ex14.m1.4.6.6.3" stretchy="false">)</mo></mrow><mo id="S4.Ex14.m1.4.6.7" stretchy="false">)</mo></mrow><mo id="S4.Ex14.m1.4.7" rspace="0.111em">=</mo><munder id="S4.Ex14.m1.4.8"><mo id="S4.Ex14.m1.4.8.2" movablelimits="false">∑</mo><mrow id="S4.Ex14.m1.1.1.1"><mrow id="S4.Ex14.m1.1.1.1.3"><mi id="S4.Ex14.m1.1.1.1.3.2">x</mi><mo id="S4.Ex14.m1.1.1.1.3.1">∈</mo><mrow id="S4.Ex14.m1.1.1.1.3.3"><mi id="S4.Ex14.m1.1.1.1.3.3.2">V</mi><mo id="S4.Ex14.m1.1.1.1.3.3.1">⁢</mo><mrow id="S4.Ex14.m1.1.1.1.3.3.3.2"><mo id="S4.Ex14.m1.1.1.1.3.3.3.2.1" stretchy="false">(</mo><mi id="S4.Ex14.m1.1.1.1.1">T</mi><mo id="S4.Ex14.m1.1.1.1.3.3.3.2.2" rspace="0.278em" stretchy="false">)</mo></mrow></mrow></mrow><mo id="S4.Ex14.m1.1.1.1.2" rspace="0.278em">:</mo><mrow id="S4.Ex14.m1.1.1.1.4"><mi id="S4.Ex14.m1.1.1.1.4.2">x</mi><mo id="S4.Ex14.m1.1.1.1.4.1">⁢</mo><mtext id="S4.Ex14.m1.1.1.1.4.3"> is P-node</mtext></mrow></mrow></munder><mi id="S4.Ex14.m1.4.9">w</mi><mrow id="S4.Ex14.m1.4.10"><mo id="S4.Ex14.m1.4.10.1" stretchy="false">(</mo><mi id="S4.Ex14.m1.4.10.2">E</mi><mrow id="S4.Ex14.m1.4.10.3"><mo id="S4.Ex14.m1.4.10.3.1" stretchy="false">(</mo><msubsup id="S4.Ex14.m1.4.10.3.2"><mi id="S4.Ex14.m1.4.10.3.2.2.2">H</mi><mi id="S4.Ex14.m1.4.10.3.2.3">x</mi><mo id="S4.Ex14.m1.4.10.3.2.2.3">′</mo></msubsup><mo id="S4.Ex14.m1.4.10.3.3" stretchy="false">)</mo></mrow><mo id="S4.Ex14.m1.4.10.4" stretchy="false">)</mo></mrow><mo id="S4.Ex14.m1.4.11" rspace="0.111em">≤</mo><munder id="S4.Ex14.m1.4.12"><mo id="S4.Ex14.m1.4.12.2" movablelimits="false">∑</mo><mrow id="S4.Ex14.m1.4.4.3"><mi id="S4.Ex14.m1.4.4.3.5">x</mi><mo id="S4.Ex14.m1.4.4.3.4">∈</mo><mrow id="S4.Ex14.m1.4.4.3.3.2"><mrow id="S4.Ex14.m1.3.3.2.2.1.1"><mi id="S4.Ex14.m1.3.3.2.2.1.1.2">V</mi><mo id="S4.Ex14.m1.3.3.2.2.1.1.1">⁢</mo><mrow id="S4.Ex14.m1.3.3.2.2.1.1.3.2"><mo id="S4.Ex14.m1.3.3.2.2.1.1.3.2.1" stretchy="false">(</mo><mi id="S4.Ex14.m1.2.2.1.1">T</mi><mo id="S4.Ex14.m1.3.3.2.2.1.1.3.2.2" stretchy="false">)</mo></mrow></mrow><mo id="S4.Ex14.m1.4.4.3.3.2.3">,</mo><mrow id="S4.Ex14.m1.4.4.3.3.2.2"><mi id="S4.Ex14.m1.4.4.3.3.2.2.2">x</mi><mo id="S4.Ex14.m1.4.4.3.3.2.2.1">⁢</mo><mtext id="S4.Ex14.m1.4.4.3.3.2.2.3"> is P-node</mtext></mrow></mrow></mrow></munder><mi id="S4.Ex14.m1.4.13">w</mi><mrow id="S4.Ex14.m1.4.14"><mo id="S4.Ex14.m1.4.14.1" stretchy="false">(</mo><msup id="S4.Ex14.m1.4.14.2"><mi id="S4.Ex14.m1.4.14.2.2">E</mi><mo id="S4.Ex14.m1.4.14.2.3">∗</mo></msup><mrow id="S4.Ex14.m1.4.14.3"><mo id="S4.Ex14.m1.4.14.3.1" stretchy="false">(</mo><mi id="S4.Ex14.m1.4.14.3.2">x</mi><mo id="S4.Ex14.m1.4.14.3.3" stretchy="false">)</mo></mrow><mo id="S4.Ex14.m1.4.14.4" stretchy="false">)</mo></mrow><mo id="S4.Ex14.m1.4.15">≤</mo><mi id="S4.Ex14.m1.4.16">w</mi><mrow id="S4.Ex14.m1.4.17"><mo id="S4.Ex14.m1.4.17.1" stretchy="false">(</mo><mtext id="S4.Ex14.m1.4.17.2">OPT</mtext><mo id="S4.Ex14.m1.4.17.3" stretchy="false">)</mo></mrow><mo id="S4.Ex14.m1.4.18" lspace="0em">.</mo></mrow><annotation encoding="application/x-tex" id="S4.Ex14.m1.4c">w(\textnormal{SOL}\cap\cup_{x}E(H_{x}))=\sum_{x\in V(T):x\text{ is P-node}}w(E% (H^{\prime}_{x}))\leq\sum_{x\in V(T),x\text{ is P-node}}w(E^{*}(x))\leq w(% \textnormal{OPT}).</annotation><annotation encoding="application/x-llamapun" id="S4.Ex14.m1.4d">italic_w ( SOL ∩ ∪ start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT italic_E ( italic_H start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT ) ) = ∑ start_POSTSUBSCRIPT italic_x ∈ italic_V ( italic_T ) : italic_x is P-node end_POSTSUBSCRIPT italic_w ( italic_E ( italic_H start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT ) ) ≤ ∑ start_POSTSUBSCRIPT italic_x ∈ italic_V ( italic_T ) , italic_x is P-node end_POSTSUBSCRIPT italic_w ( italic_E start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ( italic_x ) ) ≤ italic_w ( OPT ) .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> </div> <div class="ltx_para" id="S4.SS2.SSS3.3.p3"> <p class="ltx_p" id="S4.SS2.SSS3.3.p3.2">Finally, we bound the weight of <math alttext="\textnormal{SOL}\cap(\cup_{x}(\cup_{\mu}\textsc{Min}_{u}\cup\textsc{Max}_{u}))" class="ltx_Math" display="inline" id="S4.SS2.SSS3.3.p3.1.m1.1"><semantics id="S4.SS2.SSS3.3.p3.1.m1.1a"><mrow id="S4.SS2.SSS3.3.p3.1.m1.1.1" xref="S4.SS2.SSS3.3.p3.1.m1.1.1.cmml"><mtext id="S4.SS2.SSS3.3.p3.1.m1.1.1.3" xref="S4.SS2.SSS3.3.p3.1.m1.1.1.3a.cmml">SOL</mtext><mo id="S4.SS2.SSS3.3.p3.1.m1.1.1.2" xref="S4.SS2.SSS3.3.p3.1.m1.1.1.2.cmml">∩</mo><mrow id="S4.SS2.SSS3.3.p3.1.m1.1.1.1.1" xref="S4.SS2.SSS3.3.p3.1.m1.1.1.1.1.1.cmml"><mo id="S4.SS2.SSS3.3.p3.1.m1.1.1.1.1.2" stretchy="false" xref="S4.SS2.SSS3.3.p3.1.m1.1.1.1.1.1.cmml">(</mo><mrow id="S4.SS2.SSS3.3.p3.1.m1.1.1.1.1.1" xref="S4.SS2.SSS3.3.p3.1.m1.1.1.1.1.1.cmml"><msub id="S4.SS2.SSS3.3.p3.1.m1.1.1.1.1.1.2" xref="S4.SS2.SSS3.3.p3.1.m1.1.1.1.1.1.2.cmml"><mo id="S4.SS2.SSS3.3.p3.1.m1.1.1.1.1.1.2.2" lspace="0em" xref="S4.SS2.SSS3.3.p3.1.m1.1.1.1.1.1.2.2.cmml">∪</mo><mi id="S4.SS2.SSS3.3.p3.1.m1.1.1.1.1.1.2.3" xref="S4.SS2.SSS3.3.p3.1.m1.1.1.1.1.1.2.3.cmml">x</mi></msub><mrow id="S4.SS2.SSS3.3.p3.1.m1.1.1.1.1.1.1.1" xref="S4.SS2.SSS3.3.p3.1.m1.1.1.1.1.1.1.1.1.cmml"><mo id="S4.SS2.SSS3.3.p3.1.m1.1.1.1.1.1.1.1.2" stretchy="false" xref="S4.SS2.SSS3.3.p3.1.m1.1.1.1.1.1.1.1.1.cmml">(</mo><mrow id="S4.SS2.SSS3.3.p3.1.m1.1.1.1.1.1.1.1.1" xref="S4.SS2.SSS3.3.p3.1.m1.1.1.1.1.1.1.1.1.cmml"><mrow id="S4.SS2.SSS3.3.p3.1.m1.1.1.1.1.1.1.1.1.2" xref="S4.SS2.SSS3.3.p3.1.m1.1.1.1.1.1.1.1.1.2.cmml"><msub id="S4.SS2.SSS3.3.p3.1.m1.1.1.1.1.1.1.1.1.2.1" xref="S4.SS2.SSS3.3.p3.1.m1.1.1.1.1.1.1.1.1.2.1.cmml"><mo id="S4.SS2.SSS3.3.p3.1.m1.1.1.1.1.1.1.1.1.2.1.2" lspace="0em" xref="S4.SS2.SSS3.3.p3.1.m1.1.1.1.1.1.1.1.1.2.1.2.cmml">∪</mo><mi id="S4.SS2.SSS3.3.p3.1.m1.1.1.1.1.1.1.1.1.2.1.3" xref="S4.SS2.SSS3.3.p3.1.m1.1.1.1.1.1.1.1.1.2.1.3.cmml">μ</mi></msub><msub id="S4.SS2.SSS3.3.p3.1.m1.1.1.1.1.1.1.1.1.2.2" xref="S4.SS2.SSS3.3.p3.1.m1.1.1.1.1.1.1.1.1.2.2.cmml"><mtext class="ltx_font_smallcaps" id="S4.SS2.SSS3.3.p3.1.m1.1.1.1.1.1.1.1.1.2.2.2" xref="S4.SS2.SSS3.3.p3.1.m1.1.1.1.1.1.1.1.1.2.2.2a.cmml">Min</mtext><mi id="S4.SS2.SSS3.3.p3.1.m1.1.1.1.1.1.1.1.1.2.2.3" xref="S4.SS2.SSS3.3.p3.1.m1.1.1.1.1.1.1.1.1.2.2.3.cmml">u</mi></msub></mrow><mo id="S4.SS2.SSS3.3.p3.1.m1.1.1.1.1.1.1.1.1.1" xref="S4.SS2.SSS3.3.p3.1.m1.1.1.1.1.1.1.1.1.1.cmml">∪</mo><msub id="S4.SS2.SSS3.3.p3.1.m1.1.1.1.1.1.1.1.1.3" xref="S4.SS2.SSS3.3.p3.1.m1.1.1.1.1.1.1.1.1.3.cmml"><mtext class="ltx_font_smallcaps" id="S4.SS2.SSS3.3.p3.1.m1.1.1.1.1.1.1.1.1.3.2" xref="S4.SS2.SSS3.3.p3.1.m1.1.1.1.1.1.1.1.1.3.2a.cmml">Max</mtext><mi id="S4.SS2.SSS3.3.p3.1.m1.1.1.1.1.1.1.1.1.3.3" xref="S4.SS2.SSS3.3.p3.1.m1.1.1.1.1.1.1.1.1.3.3.cmml">u</mi></msub></mrow><mo id="S4.SS2.SSS3.3.p3.1.m1.1.1.1.1.1.1.1.3" stretchy="false" xref="S4.SS2.SSS3.3.p3.1.m1.1.1.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S4.SS2.SSS3.3.p3.1.m1.1.1.1.1.3" stretchy="false" xref="S4.SS2.SSS3.3.p3.1.m1.1.1.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS3.3.p3.1.m1.1b"><apply id="S4.SS2.SSS3.3.p3.1.m1.1.1.cmml" xref="S4.SS2.SSS3.3.p3.1.m1.1.1"><intersect id="S4.SS2.SSS3.3.p3.1.m1.1.1.2.cmml" xref="S4.SS2.SSS3.3.p3.1.m1.1.1.2"></intersect><ci id="S4.SS2.SSS3.3.p3.1.m1.1.1.3a.cmml" xref="S4.SS2.SSS3.3.p3.1.m1.1.1.3"><mtext id="S4.SS2.SSS3.3.p3.1.m1.1.1.3.cmml" xref="S4.SS2.SSS3.3.p3.1.m1.1.1.3">SOL</mtext></ci><apply id="S4.SS2.SSS3.3.p3.1.m1.1.1.1.1.1.cmml" xref="S4.SS2.SSS3.3.p3.1.m1.1.1.1.1"><apply id="S4.SS2.SSS3.3.p3.1.m1.1.1.1.1.1.2.cmml" xref="S4.SS2.SSS3.3.p3.1.m1.1.1.1.1.1.2"><csymbol cd="ambiguous" id="S4.SS2.SSS3.3.p3.1.m1.1.1.1.1.1.2.1.cmml" xref="S4.SS2.SSS3.3.p3.1.m1.1.1.1.1.1.2">subscript</csymbol><union 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xref="S4.SS2.SSS3.3.p3.1.m1.1.1.1.1.1.1.1.1.2.1.3">𝜇</ci></apply><apply id="S4.SS2.SSS3.3.p3.1.m1.1.1.1.1.1.1.1.1.2.2.cmml" xref="S4.SS2.SSS3.3.p3.1.m1.1.1.1.1.1.1.1.1.2.2"><csymbol cd="ambiguous" id="S4.SS2.SSS3.3.p3.1.m1.1.1.1.1.1.1.1.1.2.2.1.cmml" xref="S4.SS2.SSS3.3.p3.1.m1.1.1.1.1.1.1.1.1.2.2">subscript</csymbol><ci id="S4.SS2.SSS3.3.p3.1.m1.1.1.1.1.1.1.1.1.2.2.2a.cmml" xref="S4.SS2.SSS3.3.p3.1.m1.1.1.1.1.1.1.1.1.2.2.2"><mtext class="ltx_font_smallcaps" id="S4.SS2.SSS3.3.p3.1.m1.1.1.1.1.1.1.1.1.2.2.2.cmml" xref="S4.SS2.SSS3.3.p3.1.m1.1.1.1.1.1.1.1.1.2.2.2">Min</mtext></ci><ci id="S4.SS2.SSS3.3.p3.1.m1.1.1.1.1.1.1.1.1.2.2.3.cmml" xref="S4.SS2.SSS3.3.p3.1.m1.1.1.1.1.1.1.1.1.2.2.3">𝑢</ci></apply></apply><apply id="S4.SS2.SSS3.3.p3.1.m1.1.1.1.1.1.1.1.1.3.cmml" xref="S4.SS2.SSS3.3.p3.1.m1.1.1.1.1.1.1.1.1.3"><csymbol cd="ambiguous" id="S4.SS2.SSS3.3.p3.1.m1.1.1.1.1.1.1.1.1.3.1.cmml" xref="S4.SS2.SSS3.3.p3.1.m1.1.1.1.1.1.1.1.1.3">subscript</csymbol><ci id="S4.SS2.SSS3.3.p3.1.m1.1.1.1.1.1.1.1.1.3.2a.cmml" xref="S4.SS2.SSS3.3.p3.1.m1.1.1.1.1.1.1.1.1.3.2"><mtext class="ltx_font_smallcaps" id="S4.SS2.SSS3.3.p3.1.m1.1.1.1.1.1.1.1.1.3.2.cmml" xref="S4.SS2.SSS3.3.p3.1.m1.1.1.1.1.1.1.1.1.3.2">Max</mtext></ci><ci id="S4.SS2.SSS3.3.p3.1.m1.1.1.1.1.1.1.1.1.3.3.cmml" xref="S4.SS2.SSS3.3.p3.1.m1.1.1.1.1.1.1.1.1.3.3">𝑢</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS3.3.p3.1.m1.1c">\textnormal{SOL}\cap(\cup_{x}(\cup_{\mu}\textsc{Min}_{u}\cup\textsc{Max}_{u}))</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS3.3.p3.1.m1.1d">SOL ∩ ( ∪ start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT ( ∪ start_POSTSUBSCRIPT italic_μ end_POSTSUBSCRIPT Min start_POSTSUBSCRIPT italic_u end_POSTSUBSCRIPT ∪ Max start_POSTSUBSCRIPT italic_u end_POSTSUBSCRIPT ) )</annotation></semantics></math>. For each <math alttext="uv\in\textnormal{OPT}" class="ltx_Math" display="inline" id="S4.SS2.SSS3.3.p3.2.m2.1"><semantics id="S4.SS2.SSS3.3.p3.2.m2.1a"><mrow id="S4.SS2.SSS3.3.p3.2.m2.1.1" xref="S4.SS2.SSS3.3.p3.2.m2.1.1.cmml"><mrow id="S4.SS2.SSS3.3.p3.2.m2.1.1.2" xref="S4.SS2.SSS3.3.p3.2.m2.1.1.2.cmml"><mi id="S4.SS2.SSS3.3.p3.2.m2.1.1.2.2" xref="S4.SS2.SSS3.3.p3.2.m2.1.1.2.2.cmml">u</mi><mo id="S4.SS2.SSS3.3.p3.2.m2.1.1.2.1" xref="S4.SS2.SSS3.3.p3.2.m2.1.1.2.1.cmml">⁢</mo><mi id="S4.SS2.SSS3.3.p3.2.m2.1.1.2.3" xref="S4.SS2.SSS3.3.p3.2.m2.1.1.2.3.cmml">v</mi></mrow><mo id="S4.SS2.SSS3.3.p3.2.m2.1.1.1" xref="S4.SS2.SSS3.3.p3.2.m2.1.1.1.cmml">∈</mo><mtext id="S4.SS2.SSS3.3.p3.2.m2.1.1.3" xref="S4.SS2.SSS3.3.p3.2.m2.1.1.3a.cmml">OPT</mtext></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS3.3.p3.2.m2.1b"><apply id="S4.SS2.SSS3.3.p3.2.m2.1.1.cmml" xref="S4.SS2.SSS3.3.p3.2.m2.1.1"><in id="S4.SS2.SSS3.3.p3.2.m2.1.1.1.cmml" xref="S4.SS2.SSS3.3.p3.2.m2.1.1.1"></in><apply id="S4.SS2.SSS3.3.p3.2.m2.1.1.2.cmml" xref="S4.SS2.SSS3.3.p3.2.m2.1.1.2"><times id="S4.SS2.SSS3.3.p3.2.m2.1.1.2.1.cmml" xref="S4.SS2.SSS3.3.p3.2.m2.1.1.2.1"></times><ci id="S4.SS2.SSS3.3.p3.2.m2.1.1.2.2.cmml" xref="S4.SS2.SSS3.3.p3.2.m2.1.1.2.2">𝑢</ci><ci id="S4.SS2.SSS3.3.p3.2.m2.1.1.2.3.cmml" xref="S4.SS2.SSS3.3.p3.2.m2.1.1.2.3">𝑣</ci></apply><ci id="S4.SS2.SSS3.3.p3.2.m2.1.1.3a.cmml" xref="S4.SS2.SSS3.3.p3.2.m2.1.1.3"><mtext id="S4.SS2.SSS3.3.p3.2.m2.1.1.3.cmml" xref="S4.SS2.SSS3.3.p3.2.m2.1.1.3">OPT</mtext></ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS3.3.p3.2.m2.1c">uv\in\textnormal{OPT}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS3.3.p3.2.m2.1d">italic_u italic_v ∈ OPT</annotation></semantics></math>, we consider at most 3 S-nodes</p> <ul class="ltx_itemize" id="S4.I8"> <li class="ltx_item" id="S4.I8.i1" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">•</span> <div class="ltx_para" id="S4.I8.i1.p1"> <p class="ltx_p" id="S4.I8.i1.p1.3">If <math alttext="\text{LCA}(\ell(u),\ell(v))" class="ltx_Math" display="inline" id="S4.I8.i1.p1.1.m1.4"><semantics id="S4.I8.i1.p1.1.m1.4a"><mrow id="S4.I8.i1.p1.1.m1.4.4" xref="S4.I8.i1.p1.1.m1.4.4.cmml"><mtext id="S4.I8.i1.p1.1.m1.4.4.4" xref="S4.I8.i1.p1.1.m1.4.4.4a.cmml">LCA</mtext><mo id="S4.I8.i1.p1.1.m1.4.4.3" xref="S4.I8.i1.p1.1.m1.4.4.3.cmml">⁢</mo><mrow id="S4.I8.i1.p1.1.m1.4.4.2.2" xref="S4.I8.i1.p1.1.m1.4.4.2.3.cmml"><mo id="S4.I8.i1.p1.1.m1.4.4.2.2.3" stretchy="false" xref="S4.I8.i1.p1.1.m1.4.4.2.3.cmml">(</mo><mrow id="S4.I8.i1.p1.1.m1.3.3.1.1.1" xref="S4.I8.i1.p1.1.m1.3.3.1.1.1.cmml"><mi id="S4.I8.i1.p1.1.m1.3.3.1.1.1.2" mathvariant="normal" xref="S4.I8.i1.p1.1.m1.3.3.1.1.1.2.cmml">ℓ</mi><mo id="S4.I8.i1.p1.1.m1.3.3.1.1.1.1" xref="S4.I8.i1.p1.1.m1.3.3.1.1.1.1.cmml">⁢</mo><mrow id="S4.I8.i1.p1.1.m1.3.3.1.1.1.3.2" xref="S4.I8.i1.p1.1.m1.3.3.1.1.1.cmml"><mo id="S4.I8.i1.p1.1.m1.3.3.1.1.1.3.2.1" stretchy="false" xref="S4.I8.i1.p1.1.m1.3.3.1.1.1.cmml">(</mo><mi id="S4.I8.i1.p1.1.m1.1.1" xref="S4.I8.i1.p1.1.m1.1.1.cmml">u</mi><mo id="S4.I8.i1.p1.1.m1.3.3.1.1.1.3.2.2" stretchy="false" xref="S4.I8.i1.p1.1.m1.3.3.1.1.1.cmml">)</mo></mrow></mrow><mo id="S4.I8.i1.p1.1.m1.4.4.2.2.4" xref="S4.I8.i1.p1.1.m1.4.4.2.3.cmml">,</mo><mrow id="S4.I8.i1.p1.1.m1.4.4.2.2.2" xref="S4.I8.i1.p1.1.m1.4.4.2.2.2.cmml"><mi id="S4.I8.i1.p1.1.m1.4.4.2.2.2.2" mathvariant="normal" xref="S4.I8.i1.p1.1.m1.4.4.2.2.2.2.cmml">ℓ</mi><mo id="S4.I8.i1.p1.1.m1.4.4.2.2.2.1" xref="S4.I8.i1.p1.1.m1.4.4.2.2.2.1.cmml">⁢</mo><mrow id="S4.I8.i1.p1.1.m1.4.4.2.2.2.3.2" xref="S4.I8.i1.p1.1.m1.4.4.2.2.2.cmml"><mo id="S4.I8.i1.p1.1.m1.4.4.2.2.2.3.2.1" stretchy="false" xref="S4.I8.i1.p1.1.m1.4.4.2.2.2.cmml">(</mo><mi id="S4.I8.i1.p1.1.m1.2.2" xref="S4.I8.i1.p1.1.m1.2.2.cmml">v</mi><mo id="S4.I8.i1.p1.1.m1.4.4.2.2.2.3.2.2" stretchy="false" xref="S4.I8.i1.p1.1.m1.4.4.2.2.2.cmml">)</mo></mrow></mrow><mo id="S4.I8.i1.p1.1.m1.4.4.2.2.5" stretchy="false" xref="S4.I8.i1.p1.1.m1.4.4.2.3.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.I8.i1.p1.1.m1.4b"><apply id="S4.I8.i1.p1.1.m1.4.4.cmml" xref="S4.I8.i1.p1.1.m1.4.4"><times id="S4.I8.i1.p1.1.m1.4.4.3.cmml" xref="S4.I8.i1.p1.1.m1.4.4.3"></times><ci id="S4.I8.i1.p1.1.m1.4.4.4a.cmml" xref="S4.I8.i1.p1.1.m1.4.4.4"><mtext id="S4.I8.i1.p1.1.m1.4.4.4.cmml" xref="S4.I8.i1.p1.1.m1.4.4.4">LCA</mtext></ci><interval closure="open" id="S4.I8.i1.p1.1.m1.4.4.2.3.cmml" xref="S4.I8.i1.p1.1.m1.4.4.2.2"><apply id="S4.I8.i1.p1.1.m1.3.3.1.1.1.cmml" xref="S4.I8.i1.p1.1.m1.3.3.1.1.1"><times id="S4.I8.i1.p1.1.m1.3.3.1.1.1.1.cmml" xref="S4.I8.i1.p1.1.m1.3.3.1.1.1.1"></times><ci id="S4.I8.i1.p1.1.m1.3.3.1.1.1.2.cmml" xref="S4.I8.i1.p1.1.m1.3.3.1.1.1.2">ℓ</ci><ci id="S4.I8.i1.p1.1.m1.1.1.cmml" xref="S4.I8.i1.p1.1.m1.1.1">𝑢</ci></apply><apply id="S4.I8.i1.p1.1.m1.4.4.2.2.2.cmml" xref="S4.I8.i1.p1.1.m1.4.4.2.2.2"><times id="S4.I8.i1.p1.1.m1.4.4.2.2.2.1.cmml" xref="S4.I8.i1.p1.1.m1.4.4.2.2.2.1"></times><ci id="S4.I8.i1.p1.1.m1.4.4.2.2.2.2.cmml" xref="S4.I8.i1.p1.1.m1.4.4.2.2.2.2">ℓ</ci><ci id="S4.I8.i1.p1.1.m1.2.2.cmml" xref="S4.I8.i1.p1.1.m1.2.2">𝑣</ci></apply></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.I8.i1.p1.1.m1.4c">\text{LCA}(\ell(u),\ell(v))</annotation><annotation encoding="application/x-llamapun" id="S4.I8.i1.p1.1.m1.4d">LCA ( roman_ℓ ( italic_u ) , roman_ℓ ( italic_v ) )</annotation></semantics></math> is an S-node, we add 2 links to <span class="ltx_text ltx_markedasmath" id="S4.I8.i1.p1.3.1">SOL</span> of weight at most <math alttext="(1+\epsilon)w(uv)" class="ltx_Math" display="inline" id="S4.I8.i1.p1.3.m3.2"><semantics id="S4.I8.i1.p1.3.m3.2a"><mrow id="S4.I8.i1.p1.3.m3.2.2" xref="S4.I8.i1.p1.3.m3.2.2.cmml"><mrow id="S4.I8.i1.p1.3.m3.1.1.1.1" xref="S4.I8.i1.p1.3.m3.1.1.1.1.1.cmml"><mo id="S4.I8.i1.p1.3.m3.1.1.1.1.2" stretchy="false" xref="S4.I8.i1.p1.3.m3.1.1.1.1.1.cmml">(</mo><mrow id="S4.I8.i1.p1.3.m3.1.1.1.1.1" xref="S4.I8.i1.p1.3.m3.1.1.1.1.1.cmml"><mn id="S4.I8.i1.p1.3.m3.1.1.1.1.1.2" xref="S4.I8.i1.p1.3.m3.1.1.1.1.1.2.cmml">1</mn><mo id="S4.I8.i1.p1.3.m3.1.1.1.1.1.1" xref="S4.I8.i1.p1.3.m3.1.1.1.1.1.1.cmml">+</mo><mi id="S4.I8.i1.p1.3.m3.1.1.1.1.1.3" xref="S4.I8.i1.p1.3.m3.1.1.1.1.1.3.cmml">ϵ</mi></mrow><mo id="S4.I8.i1.p1.3.m3.1.1.1.1.3" stretchy="false" xref="S4.I8.i1.p1.3.m3.1.1.1.1.1.cmml">)</mo></mrow><mo id="S4.I8.i1.p1.3.m3.2.2.3" xref="S4.I8.i1.p1.3.m3.2.2.3.cmml">⁢</mo><mi id="S4.I8.i1.p1.3.m3.2.2.4" xref="S4.I8.i1.p1.3.m3.2.2.4.cmml">w</mi><mo id="S4.I8.i1.p1.3.m3.2.2.3a" xref="S4.I8.i1.p1.3.m3.2.2.3.cmml">⁢</mo><mrow id="S4.I8.i1.p1.3.m3.2.2.2.1" xref="S4.I8.i1.p1.3.m3.2.2.2.1.1.cmml"><mo id="S4.I8.i1.p1.3.m3.2.2.2.1.2" stretchy="false" xref="S4.I8.i1.p1.3.m3.2.2.2.1.1.cmml">(</mo><mrow id="S4.I8.i1.p1.3.m3.2.2.2.1.1" xref="S4.I8.i1.p1.3.m3.2.2.2.1.1.cmml"><mi id="S4.I8.i1.p1.3.m3.2.2.2.1.1.2" xref="S4.I8.i1.p1.3.m3.2.2.2.1.1.2.cmml">u</mi><mo id="S4.I8.i1.p1.3.m3.2.2.2.1.1.1" xref="S4.I8.i1.p1.3.m3.2.2.2.1.1.1.cmml">⁢</mo><mi id="S4.I8.i1.p1.3.m3.2.2.2.1.1.3" xref="S4.I8.i1.p1.3.m3.2.2.2.1.1.3.cmml">v</mi></mrow><mo id="S4.I8.i1.p1.3.m3.2.2.2.1.3" stretchy="false" xref="S4.I8.i1.p1.3.m3.2.2.2.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.I8.i1.p1.3.m3.2b"><apply id="S4.I8.i1.p1.3.m3.2.2.cmml" xref="S4.I8.i1.p1.3.m3.2.2"><times id="S4.I8.i1.p1.3.m3.2.2.3.cmml" xref="S4.I8.i1.p1.3.m3.2.2.3"></times><apply id="S4.I8.i1.p1.3.m3.1.1.1.1.1.cmml" xref="S4.I8.i1.p1.3.m3.1.1.1.1"><plus id="S4.I8.i1.p1.3.m3.1.1.1.1.1.1.cmml" xref="S4.I8.i1.p1.3.m3.1.1.1.1.1.1"></plus><cn id="S4.I8.i1.p1.3.m3.1.1.1.1.1.2.cmml" type="integer" xref="S4.I8.i1.p1.3.m3.1.1.1.1.1.2">1</cn><ci id="S4.I8.i1.p1.3.m3.1.1.1.1.1.3.cmml" xref="S4.I8.i1.p1.3.m3.1.1.1.1.1.3">italic-ϵ</ci></apply><ci id="S4.I8.i1.p1.3.m3.2.2.4.cmml" xref="S4.I8.i1.p1.3.m3.2.2.4">𝑤</ci><apply id="S4.I8.i1.p1.3.m3.2.2.2.1.1.cmml" xref="S4.I8.i1.p1.3.m3.2.2.2.1"><times id="S4.I8.i1.p1.3.m3.2.2.2.1.1.1.cmml" xref="S4.I8.i1.p1.3.m3.2.2.2.1.1.1"></times><ci id="S4.I8.i1.p1.3.m3.2.2.2.1.1.2.cmml" xref="S4.I8.i1.p1.3.m3.2.2.2.1.1.2">𝑢</ci><ci id="S4.I8.i1.p1.3.m3.2.2.2.1.1.3.cmml" xref="S4.I8.i1.p1.3.m3.2.2.2.1.1.3">𝑣</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.I8.i1.p1.3.m3.2c">(1+\epsilon)w(uv)</annotation><annotation encoding="application/x-llamapun" id="S4.I8.i1.p1.3.m3.2d">( 1 + italic_ϵ ) italic_w ( italic_u italic_v )</annotation></semantics></math> each;</p> </div> </li> <li class="ltx_item" id="S4.I8.i2" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">•</span> <div class="ltx_para" id="S4.I8.i2.p1"> <p class="ltx_p" id="S4.I8.i2.p1.7">If <math alttext="u\in G_{x}\setminus\textnormal{parent}(x)" class="ltx_Math" display="inline" id="S4.I8.i2.p1.1.m1.1"><semantics id="S4.I8.i2.p1.1.m1.1a"><mrow id="S4.I8.i2.p1.1.m1.1.2" xref="S4.I8.i2.p1.1.m1.1.2.cmml"><mi id="S4.I8.i2.p1.1.m1.1.2.2" xref="S4.I8.i2.p1.1.m1.1.2.2.cmml">u</mi><mo id="S4.I8.i2.p1.1.m1.1.2.1" xref="S4.I8.i2.p1.1.m1.1.2.1.cmml">∈</mo><mrow id="S4.I8.i2.p1.1.m1.1.2.3" xref="S4.I8.i2.p1.1.m1.1.2.3.cmml"><msub id="S4.I8.i2.p1.1.m1.1.2.3.2" xref="S4.I8.i2.p1.1.m1.1.2.3.2.cmml"><mi id="S4.I8.i2.p1.1.m1.1.2.3.2.2" xref="S4.I8.i2.p1.1.m1.1.2.3.2.2.cmml">G</mi><mi id="S4.I8.i2.p1.1.m1.1.2.3.2.3" xref="S4.I8.i2.p1.1.m1.1.2.3.2.3.cmml">x</mi></msub><mo id="S4.I8.i2.p1.1.m1.1.2.3.1" xref="S4.I8.i2.p1.1.m1.1.2.3.1.cmml">∖</mo><mrow id="S4.I8.i2.p1.1.m1.1.2.3.3" xref="S4.I8.i2.p1.1.m1.1.2.3.3.cmml"><mtext id="S4.I8.i2.p1.1.m1.1.2.3.3.2" xref="S4.I8.i2.p1.1.m1.1.2.3.3.2a.cmml">parent</mtext><mo id="S4.I8.i2.p1.1.m1.1.2.3.3.1" xref="S4.I8.i2.p1.1.m1.1.2.3.3.1.cmml">⁢</mo><mrow id="S4.I8.i2.p1.1.m1.1.2.3.3.3.2" xref="S4.I8.i2.p1.1.m1.1.2.3.3.cmml"><mo id="S4.I8.i2.p1.1.m1.1.2.3.3.3.2.1" stretchy="false" xref="S4.I8.i2.p1.1.m1.1.2.3.3.cmml">(</mo><mi id="S4.I8.i2.p1.1.m1.1.1" xref="S4.I8.i2.p1.1.m1.1.1.cmml">x</mi><mo id="S4.I8.i2.p1.1.m1.1.2.3.3.3.2.2" stretchy="false" xref="S4.I8.i2.p1.1.m1.1.2.3.3.cmml">)</mo></mrow></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.I8.i2.p1.1.m1.1b"><apply id="S4.I8.i2.p1.1.m1.1.2.cmml" xref="S4.I8.i2.p1.1.m1.1.2"><in id="S4.I8.i2.p1.1.m1.1.2.1.cmml" xref="S4.I8.i2.p1.1.m1.1.2.1"></in><ci id="S4.I8.i2.p1.1.m1.1.2.2.cmml" xref="S4.I8.i2.p1.1.m1.1.2.2">𝑢</ci><apply id="S4.I8.i2.p1.1.m1.1.2.3.cmml" xref="S4.I8.i2.p1.1.m1.1.2.3"><setdiff id="S4.I8.i2.p1.1.m1.1.2.3.1.cmml" xref="S4.I8.i2.p1.1.m1.1.2.3.1"></setdiff><apply id="S4.I8.i2.p1.1.m1.1.2.3.2.cmml" xref="S4.I8.i2.p1.1.m1.1.2.3.2"><csymbol cd="ambiguous" id="S4.I8.i2.p1.1.m1.1.2.3.2.1.cmml" xref="S4.I8.i2.p1.1.m1.1.2.3.2">subscript</csymbol><ci id="S4.I8.i2.p1.1.m1.1.2.3.2.2.cmml" xref="S4.I8.i2.p1.1.m1.1.2.3.2.2">𝐺</ci><ci id="S4.I8.i2.p1.1.m1.1.2.3.2.3.cmml" xref="S4.I8.i2.p1.1.m1.1.2.3.2.3">𝑥</ci></apply><apply id="S4.I8.i2.p1.1.m1.1.2.3.3.cmml" xref="S4.I8.i2.p1.1.m1.1.2.3.3"><times id="S4.I8.i2.p1.1.m1.1.2.3.3.1.cmml" xref="S4.I8.i2.p1.1.m1.1.2.3.3.1"></times><ci id="S4.I8.i2.p1.1.m1.1.2.3.3.2a.cmml" xref="S4.I8.i2.p1.1.m1.1.2.3.3.2"><mtext id="S4.I8.i2.p1.1.m1.1.2.3.3.2.cmml" xref="S4.I8.i2.p1.1.m1.1.2.3.3.2">parent</mtext></ci><ci id="S4.I8.i2.p1.1.m1.1.1.cmml" xref="S4.I8.i2.p1.1.m1.1.1">𝑥</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.I8.i2.p1.1.m1.1c">u\in G_{x}\setminus\textnormal{parent}(x)</annotation><annotation encoding="application/x-llamapun" id="S4.I8.i2.p1.1.m1.1d">italic_u ∈ italic_G start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT ∖ parent ( italic_x )</annotation></semantics></math> for some S-node <math alttext="x" class="ltx_Math" display="inline" id="S4.I8.i2.p1.2.m2.1"><semantics id="S4.I8.i2.p1.2.m2.1a"><mi id="S4.I8.i2.p1.2.m2.1.1" xref="S4.I8.i2.p1.2.m2.1.1.cmml">x</mi><annotation-xml encoding="MathML-Content" id="S4.I8.i2.p1.2.m2.1b"><ci id="S4.I8.i2.p1.2.m2.1.1.cmml" xref="S4.I8.i2.p1.2.m2.1.1">𝑥</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.I8.i2.p1.2.m2.1c">x</annotation><annotation encoding="application/x-llamapun" id="S4.I8.i2.p1.2.m2.1d">italic_x</annotation></semantics></math> and <math alttext="v" class="ltx_Math" display="inline" id="S4.I8.i2.p1.3.m3.1"><semantics id="S4.I8.i2.p1.3.m3.1a"><mi id="S4.I8.i2.p1.3.m3.1.1" xref="S4.I8.i2.p1.3.m3.1.1.cmml">v</mi><annotation-xml encoding="MathML-Content" id="S4.I8.i2.p1.3.m3.1b"><ci id="S4.I8.i2.p1.3.m3.1.1.cmml" xref="S4.I8.i2.p1.3.m3.1.1">𝑣</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.I8.i2.p1.3.m3.1c">v</annotation><annotation encoding="application/x-llamapun" id="S4.I8.i2.p1.3.m3.1d">italic_v</annotation></semantics></math> is outside the subtree rooted at <math alttext="x" class="ltx_Math" display="inline" id="S4.I8.i2.p1.4.m4.1"><semantics id="S4.I8.i2.p1.4.m4.1a"><mi id="S4.I8.i2.p1.4.m4.1.1" xref="S4.I8.i2.p1.4.m4.1.1.cmml">x</mi><annotation-xml encoding="MathML-Content" id="S4.I8.i2.p1.4.m4.1b"><ci id="S4.I8.i2.p1.4.m4.1.1.cmml" xref="S4.I8.i2.p1.4.m4.1.1">𝑥</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.I8.i2.p1.4.m4.1c">x</annotation><annotation encoding="application/x-llamapun" id="S4.I8.i2.p1.4.m4.1d">italic_x</annotation></semantics></math>, then we add one link to <span class="ltx_text ltx_markedasmath" id="S4.I8.i2.p1.7.1">SOL</span> of weight at most <math alttext="(1+\epsilon)w(uv)" class="ltx_Math" display="inline" id="S4.I8.i2.p1.6.m6.2"><semantics id="S4.I8.i2.p1.6.m6.2a"><mrow id="S4.I8.i2.p1.6.m6.2.2" xref="S4.I8.i2.p1.6.m6.2.2.cmml"><mrow id="S4.I8.i2.p1.6.m6.1.1.1.1" xref="S4.I8.i2.p1.6.m6.1.1.1.1.1.cmml"><mo id="S4.I8.i2.p1.6.m6.1.1.1.1.2" stretchy="false" xref="S4.I8.i2.p1.6.m6.1.1.1.1.1.cmml">(</mo><mrow id="S4.I8.i2.p1.6.m6.1.1.1.1.1" xref="S4.I8.i2.p1.6.m6.1.1.1.1.1.cmml"><mn id="S4.I8.i2.p1.6.m6.1.1.1.1.1.2" xref="S4.I8.i2.p1.6.m6.1.1.1.1.1.2.cmml">1</mn><mo id="S4.I8.i2.p1.6.m6.1.1.1.1.1.1" xref="S4.I8.i2.p1.6.m6.1.1.1.1.1.1.cmml">+</mo><mi id="S4.I8.i2.p1.6.m6.1.1.1.1.1.3" xref="S4.I8.i2.p1.6.m6.1.1.1.1.1.3.cmml">ϵ</mi></mrow><mo id="S4.I8.i2.p1.6.m6.1.1.1.1.3" stretchy="false" xref="S4.I8.i2.p1.6.m6.1.1.1.1.1.cmml">)</mo></mrow><mo id="S4.I8.i2.p1.6.m6.2.2.3" xref="S4.I8.i2.p1.6.m6.2.2.3.cmml">⁢</mo><mi id="S4.I8.i2.p1.6.m6.2.2.4" xref="S4.I8.i2.p1.6.m6.2.2.4.cmml">w</mi><mo id="S4.I8.i2.p1.6.m6.2.2.3a" xref="S4.I8.i2.p1.6.m6.2.2.3.cmml">⁢</mo><mrow id="S4.I8.i2.p1.6.m6.2.2.2.1" xref="S4.I8.i2.p1.6.m6.2.2.2.1.1.cmml"><mo id="S4.I8.i2.p1.6.m6.2.2.2.1.2" stretchy="false" xref="S4.I8.i2.p1.6.m6.2.2.2.1.1.cmml">(</mo><mrow id="S4.I8.i2.p1.6.m6.2.2.2.1.1" xref="S4.I8.i2.p1.6.m6.2.2.2.1.1.cmml"><mi id="S4.I8.i2.p1.6.m6.2.2.2.1.1.2" xref="S4.I8.i2.p1.6.m6.2.2.2.1.1.2.cmml">u</mi><mo id="S4.I8.i2.p1.6.m6.2.2.2.1.1.1" xref="S4.I8.i2.p1.6.m6.2.2.2.1.1.1.cmml">⁢</mo><mi id="S4.I8.i2.p1.6.m6.2.2.2.1.1.3" xref="S4.I8.i2.p1.6.m6.2.2.2.1.1.3.cmml">v</mi></mrow><mo id="S4.I8.i2.p1.6.m6.2.2.2.1.3" stretchy="false" xref="S4.I8.i2.p1.6.m6.2.2.2.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.I8.i2.p1.6.m6.2b"><apply id="S4.I8.i2.p1.6.m6.2.2.cmml" xref="S4.I8.i2.p1.6.m6.2.2"><times id="S4.I8.i2.p1.6.m6.2.2.3.cmml" xref="S4.I8.i2.p1.6.m6.2.2.3"></times><apply id="S4.I8.i2.p1.6.m6.1.1.1.1.1.cmml" xref="S4.I8.i2.p1.6.m6.1.1.1.1"><plus id="S4.I8.i2.p1.6.m6.1.1.1.1.1.1.cmml" xref="S4.I8.i2.p1.6.m6.1.1.1.1.1.1"></plus><cn id="S4.I8.i2.p1.6.m6.1.1.1.1.1.2.cmml" type="integer" xref="S4.I8.i2.p1.6.m6.1.1.1.1.1.2">1</cn><ci id="S4.I8.i2.p1.6.m6.1.1.1.1.1.3.cmml" xref="S4.I8.i2.p1.6.m6.1.1.1.1.1.3">italic-ϵ</ci></apply><ci id="S4.I8.i2.p1.6.m6.2.2.4.cmml" xref="S4.I8.i2.p1.6.m6.2.2.4">𝑤</ci><apply id="S4.I8.i2.p1.6.m6.2.2.2.1.1.cmml" xref="S4.I8.i2.p1.6.m6.2.2.2.1"><times id="S4.I8.i2.p1.6.m6.2.2.2.1.1.1.cmml" xref="S4.I8.i2.p1.6.m6.2.2.2.1.1.1"></times><ci id="S4.I8.i2.p1.6.m6.2.2.2.1.1.2.cmml" xref="S4.I8.i2.p1.6.m6.2.2.2.1.1.2">𝑢</ci><ci id="S4.I8.i2.p1.6.m6.2.2.2.1.1.3.cmml" xref="S4.I8.i2.p1.6.m6.2.2.2.1.1.3">𝑣</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.I8.i2.p1.6.m6.2c">(1+\epsilon)w(uv)</annotation><annotation encoding="application/x-llamapun" id="S4.I8.i2.p1.6.m6.2d">( 1 + italic_ϵ ) italic_w ( italic_u italic_v )</annotation></semantics></math> – there can be at most one such S-node, since if <math alttext="u" class="ltx_Math" display="inline" id="S4.I8.i2.p1.7.m7.1"><semantics id="S4.I8.i2.p1.7.m7.1a"><mi id="S4.I8.i2.p1.7.m7.1.1" xref="S4.I8.i2.p1.7.m7.1.1.cmml">u</mi><annotation-xml encoding="MathML-Content" id="S4.I8.i2.p1.7.m7.1b"><ci id="S4.I8.i2.p1.7.m7.1.1.cmml" xref="S4.I8.i2.p1.7.m7.1.1">𝑢</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.I8.i2.p1.7.m7.1c">u</annotation><annotation encoding="application/x-llamapun" id="S4.I8.i2.p1.7.m7.1d">italic_u</annotation></semantics></math> has a a copy in multiple S-nodes, then it must be in the parent of all but one of them;</p> </div> </li> <li class="ltx_item" id="S4.I8.i3" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">•</span> <div class="ltx_para" id="S4.I8.i3.p1"> <p class="ltx_p" id="S4.I8.i3.p1.6">If <math alttext="v\in G_{x}\setminus\textnormal{parent}(x)" class="ltx_Math" display="inline" id="S4.I8.i3.p1.1.m1.1"><semantics id="S4.I8.i3.p1.1.m1.1a"><mrow id="S4.I8.i3.p1.1.m1.1.2" xref="S4.I8.i3.p1.1.m1.1.2.cmml"><mi id="S4.I8.i3.p1.1.m1.1.2.2" xref="S4.I8.i3.p1.1.m1.1.2.2.cmml">v</mi><mo id="S4.I8.i3.p1.1.m1.1.2.1" xref="S4.I8.i3.p1.1.m1.1.2.1.cmml">∈</mo><mrow id="S4.I8.i3.p1.1.m1.1.2.3" xref="S4.I8.i3.p1.1.m1.1.2.3.cmml"><msub id="S4.I8.i3.p1.1.m1.1.2.3.2" xref="S4.I8.i3.p1.1.m1.1.2.3.2.cmml"><mi id="S4.I8.i3.p1.1.m1.1.2.3.2.2" xref="S4.I8.i3.p1.1.m1.1.2.3.2.2.cmml">G</mi><mi id="S4.I8.i3.p1.1.m1.1.2.3.2.3" xref="S4.I8.i3.p1.1.m1.1.2.3.2.3.cmml">x</mi></msub><mo id="S4.I8.i3.p1.1.m1.1.2.3.1" xref="S4.I8.i3.p1.1.m1.1.2.3.1.cmml">∖</mo><mrow id="S4.I8.i3.p1.1.m1.1.2.3.3" xref="S4.I8.i3.p1.1.m1.1.2.3.3.cmml"><mtext id="S4.I8.i3.p1.1.m1.1.2.3.3.2" xref="S4.I8.i3.p1.1.m1.1.2.3.3.2a.cmml">parent</mtext><mo id="S4.I8.i3.p1.1.m1.1.2.3.3.1" xref="S4.I8.i3.p1.1.m1.1.2.3.3.1.cmml">⁢</mo><mrow id="S4.I8.i3.p1.1.m1.1.2.3.3.3.2" xref="S4.I8.i3.p1.1.m1.1.2.3.3.cmml"><mo id="S4.I8.i3.p1.1.m1.1.2.3.3.3.2.1" stretchy="false" xref="S4.I8.i3.p1.1.m1.1.2.3.3.cmml">(</mo><mi id="S4.I8.i3.p1.1.m1.1.1" xref="S4.I8.i3.p1.1.m1.1.1.cmml">x</mi><mo id="S4.I8.i3.p1.1.m1.1.2.3.3.3.2.2" stretchy="false" xref="S4.I8.i3.p1.1.m1.1.2.3.3.cmml">)</mo></mrow></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.I8.i3.p1.1.m1.1b"><apply id="S4.I8.i3.p1.1.m1.1.2.cmml" xref="S4.I8.i3.p1.1.m1.1.2"><in id="S4.I8.i3.p1.1.m1.1.2.1.cmml" xref="S4.I8.i3.p1.1.m1.1.2.1"></in><ci id="S4.I8.i3.p1.1.m1.1.2.2.cmml" xref="S4.I8.i3.p1.1.m1.1.2.2">𝑣</ci><apply id="S4.I8.i3.p1.1.m1.1.2.3.cmml" xref="S4.I8.i3.p1.1.m1.1.2.3"><setdiff id="S4.I8.i3.p1.1.m1.1.2.3.1.cmml" xref="S4.I8.i3.p1.1.m1.1.2.3.1"></setdiff><apply id="S4.I8.i3.p1.1.m1.1.2.3.2.cmml" xref="S4.I8.i3.p1.1.m1.1.2.3.2"><csymbol cd="ambiguous" id="S4.I8.i3.p1.1.m1.1.2.3.2.1.cmml" xref="S4.I8.i3.p1.1.m1.1.2.3.2">subscript</csymbol><ci id="S4.I8.i3.p1.1.m1.1.2.3.2.2.cmml" xref="S4.I8.i3.p1.1.m1.1.2.3.2.2">𝐺</ci><ci id="S4.I8.i3.p1.1.m1.1.2.3.2.3.cmml" xref="S4.I8.i3.p1.1.m1.1.2.3.2.3">𝑥</ci></apply><apply id="S4.I8.i3.p1.1.m1.1.2.3.3.cmml" xref="S4.I8.i3.p1.1.m1.1.2.3.3"><times id="S4.I8.i3.p1.1.m1.1.2.3.3.1.cmml" xref="S4.I8.i3.p1.1.m1.1.2.3.3.1"></times><ci id="S4.I8.i3.p1.1.m1.1.2.3.3.2a.cmml" xref="S4.I8.i3.p1.1.m1.1.2.3.3.2"><mtext id="S4.I8.i3.p1.1.m1.1.2.3.3.2.cmml" xref="S4.I8.i3.p1.1.m1.1.2.3.3.2">parent</mtext></ci><ci id="S4.I8.i3.p1.1.m1.1.1.cmml" xref="S4.I8.i3.p1.1.m1.1.1">𝑥</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.I8.i3.p1.1.m1.1c">v\in G_{x}\setminus\textnormal{parent}(x)</annotation><annotation encoding="application/x-llamapun" id="S4.I8.i3.p1.1.m1.1d">italic_v ∈ italic_G start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT ∖ parent ( italic_x )</annotation></semantics></math> for some S-node <math alttext="x" class="ltx_Math" display="inline" id="S4.I8.i3.p1.2.m2.1"><semantics id="S4.I8.i3.p1.2.m2.1a"><mi id="S4.I8.i3.p1.2.m2.1.1" xref="S4.I8.i3.p1.2.m2.1.1.cmml">x</mi><annotation-xml encoding="MathML-Content" id="S4.I8.i3.p1.2.m2.1b"><ci id="S4.I8.i3.p1.2.m2.1.1.cmml" xref="S4.I8.i3.p1.2.m2.1.1">𝑥</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.I8.i3.p1.2.m2.1c">x</annotation><annotation encoding="application/x-llamapun" id="S4.I8.i3.p1.2.m2.1d">italic_x</annotation></semantics></math> and <math alttext="u" class="ltx_Math" display="inline" id="S4.I8.i3.p1.3.m3.1"><semantics id="S4.I8.i3.p1.3.m3.1a"><mi id="S4.I8.i3.p1.3.m3.1.1" xref="S4.I8.i3.p1.3.m3.1.1.cmml">u</mi><annotation-xml encoding="MathML-Content" id="S4.I8.i3.p1.3.m3.1b"><ci id="S4.I8.i3.p1.3.m3.1.1.cmml" xref="S4.I8.i3.p1.3.m3.1.1">𝑢</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.I8.i3.p1.3.m3.1c">u</annotation><annotation encoding="application/x-llamapun" id="S4.I8.i3.p1.3.m3.1d">italic_u</annotation></semantics></math> is outside the subtree rooted at <math alttext="x" class="ltx_Math" display="inline" id="S4.I8.i3.p1.4.m4.1"><semantics id="S4.I8.i3.p1.4.m4.1a"><mi id="S4.I8.i3.p1.4.m4.1.1" xref="S4.I8.i3.p1.4.m4.1.1.cmml">x</mi><annotation-xml encoding="MathML-Content" id="S4.I8.i3.p1.4.m4.1b"><ci id="S4.I8.i3.p1.4.m4.1.1.cmml" xref="S4.I8.i3.p1.4.m4.1.1">𝑥</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.I8.i3.p1.4.m4.1c">x</annotation><annotation encoding="application/x-llamapun" id="S4.I8.i3.p1.4.m4.1d">italic_x</annotation></semantics></math>, then we add one link to <span class="ltx_text ltx_markedasmath" id="S4.I8.i3.p1.6.1">SOL</span> of weight at most <math alttext="(1+\epsilon)w(uv)" class="ltx_Math" display="inline" id="S4.I8.i3.p1.6.m6.2"><semantics id="S4.I8.i3.p1.6.m6.2a"><mrow id="S4.I8.i3.p1.6.m6.2.2" xref="S4.I8.i3.p1.6.m6.2.2.cmml"><mrow id="S4.I8.i3.p1.6.m6.1.1.1.1" xref="S4.I8.i3.p1.6.m6.1.1.1.1.1.cmml"><mo id="S4.I8.i3.p1.6.m6.1.1.1.1.2" stretchy="false" xref="S4.I8.i3.p1.6.m6.1.1.1.1.1.cmml">(</mo><mrow id="S4.I8.i3.p1.6.m6.1.1.1.1.1" xref="S4.I8.i3.p1.6.m6.1.1.1.1.1.cmml"><mn id="S4.I8.i3.p1.6.m6.1.1.1.1.1.2" xref="S4.I8.i3.p1.6.m6.1.1.1.1.1.2.cmml">1</mn><mo id="S4.I8.i3.p1.6.m6.1.1.1.1.1.1" xref="S4.I8.i3.p1.6.m6.1.1.1.1.1.1.cmml">+</mo><mi id="S4.I8.i3.p1.6.m6.1.1.1.1.1.3" xref="S4.I8.i3.p1.6.m6.1.1.1.1.1.3.cmml">ϵ</mi></mrow><mo id="S4.I8.i3.p1.6.m6.1.1.1.1.3" stretchy="false" xref="S4.I8.i3.p1.6.m6.1.1.1.1.1.cmml">)</mo></mrow><mo id="S4.I8.i3.p1.6.m6.2.2.3" xref="S4.I8.i3.p1.6.m6.2.2.3.cmml">⁢</mo><mi id="S4.I8.i3.p1.6.m6.2.2.4" xref="S4.I8.i3.p1.6.m6.2.2.4.cmml">w</mi><mo id="S4.I8.i3.p1.6.m6.2.2.3a" xref="S4.I8.i3.p1.6.m6.2.2.3.cmml">⁢</mo><mrow id="S4.I8.i3.p1.6.m6.2.2.2.1" xref="S4.I8.i3.p1.6.m6.2.2.2.1.1.cmml"><mo id="S4.I8.i3.p1.6.m6.2.2.2.1.2" stretchy="false" xref="S4.I8.i3.p1.6.m6.2.2.2.1.1.cmml">(</mo><mrow id="S4.I8.i3.p1.6.m6.2.2.2.1.1" xref="S4.I8.i3.p1.6.m6.2.2.2.1.1.cmml"><mi id="S4.I8.i3.p1.6.m6.2.2.2.1.1.2" xref="S4.I8.i3.p1.6.m6.2.2.2.1.1.2.cmml">u</mi><mo id="S4.I8.i3.p1.6.m6.2.2.2.1.1.1" xref="S4.I8.i3.p1.6.m6.2.2.2.1.1.1.cmml">⁢</mo><mi id="S4.I8.i3.p1.6.m6.2.2.2.1.1.3" xref="S4.I8.i3.p1.6.m6.2.2.2.1.1.3.cmml">v</mi></mrow><mo id="S4.I8.i3.p1.6.m6.2.2.2.1.3" stretchy="false" xref="S4.I8.i3.p1.6.m6.2.2.2.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.I8.i3.p1.6.m6.2b"><apply id="S4.I8.i3.p1.6.m6.2.2.cmml" xref="S4.I8.i3.p1.6.m6.2.2"><times id="S4.I8.i3.p1.6.m6.2.2.3.cmml" xref="S4.I8.i3.p1.6.m6.2.2.3"></times><apply id="S4.I8.i3.p1.6.m6.1.1.1.1.1.cmml" xref="S4.I8.i3.p1.6.m6.1.1.1.1"><plus id="S4.I8.i3.p1.6.m6.1.1.1.1.1.1.cmml" xref="S4.I8.i3.p1.6.m6.1.1.1.1.1.1"></plus><cn id="S4.I8.i3.p1.6.m6.1.1.1.1.1.2.cmml" type="integer" xref="S4.I8.i3.p1.6.m6.1.1.1.1.1.2">1</cn><ci id="S4.I8.i3.p1.6.m6.1.1.1.1.1.3.cmml" xref="S4.I8.i3.p1.6.m6.1.1.1.1.1.3">italic-ϵ</ci></apply><ci id="S4.I8.i3.p1.6.m6.2.2.4.cmml" xref="S4.I8.i3.p1.6.m6.2.2.4">𝑤</ci><apply id="S4.I8.i3.p1.6.m6.2.2.2.1.1.cmml" xref="S4.I8.i3.p1.6.m6.2.2.2.1"><times id="S4.I8.i3.p1.6.m6.2.2.2.1.1.1.cmml" xref="S4.I8.i3.p1.6.m6.2.2.2.1.1.1"></times><ci id="S4.I8.i3.p1.6.m6.2.2.2.1.1.2.cmml" xref="S4.I8.i3.p1.6.m6.2.2.2.1.1.2">𝑢</ci><ci id="S4.I8.i3.p1.6.m6.2.2.2.1.1.3.cmml" xref="S4.I8.i3.p1.6.m6.2.2.2.1.1.3">𝑣</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.I8.i3.p1.6.m6.2c">(1+\epsilon)w(uv)</annotation><annotation encoding="application/x-llamapun" id="S4.I8.i3.p1.6.m6.2d">( 1 + italic_ϵ ) italic_w ( italic_u italic_v )</annotation></semantics></math> – there is at most one such S-node by the same argument as above.</p> </div> </li> </ul> <p class="ltx_p" id="S4.SS2.SSS3.3.p3.7">Thus for each link <math alttext="uv\in\textnormal{OPT}" class="ltx_Math" display="inline" id="S4.SS2.SSS3.3.p3.3.m1.1"><semantics id="S4.SS2.SSS3.3.p3.3.m1.1a"><mrow id="S4.SS2.SSS3.3.p3.3.m1.1.1" xref="S4.SS2.SSS3.3.p3.3.m1.1.1.cmml"><mrow id="S4.SS2.SSS3.3.p3.3.m1.1.1.2" xref="S4.SS2.SSS3.3.p3.3.m1.1.1.2.cmml"><mi id="S4.SS2.SSS3.3.p3.3.m1.1.1.2.2" xref="S4.SS2.SSS3.3.p3.3.m1.1.1.2.2.cmml">u</mi><mo id="S4.SS2.SSS3.3.p3.3.m1.1.1.2.1" xref="S4.SS2.SSS3.3.p3.3.m1.1.1.2.1.cmml">⁢</mo><mi id="S4.SS2.SSS3.3.p3.3.m1.1.1.2.3" xref="S4.SS2.SSS3.3.p3.3.m1.1.1.2.3.cmml">v</mi></mrow><mo id="S4.SS2.SSS3.3.p3.3.m1.1.1.1" xref="S4.SS2.SSS3.3.p3.3.m1.1.1.1.cmml">∈</mo><mtext id="S4.SS2.SSS3.3.p3.3.m1.1.1.3" xref="S4.SS2.SSS3.3.p3.3.m1.1.1.3a.cmml">OPT</mtext></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS3.3.p3.3.m1.1b"><apply id="S4.SS2.SSS3.3.p3.3.m1.1.1.cmml" xref="S4.SS2.SSS3.3.p3.3.m1.1.1"><in id="S4.SS2.SSS3.3.p3.3.m1.1.1.1.cmml" xref="S4.SS2.SSS3.3.p3.3.m1.1.1.1"></in><apply id="S4.SS2.SSS3.3.p3.3.m1.1.1.2.cmml" xref="S4.SS2.SSS3.3.p3.3.m1.1.1.2"><times id="S4.SS2.SSS3.3.p3.3.m1.1.1.2.1.cmml" xref="S4.SS2.SSS3.3.p3.3.m1.1.1.2.1"></times><ci id="S4.SS2.SSS3.3.p3.3.m1.1.1.2.2.cmml" xref="S4.SS2.SSS3.3.p3.3.m1.1.1.2.2">𝑢</ci><ci id="S4.SS2.SSS3.3.p3.3.m1.1.1.2.3.cmml" xref="S4.SS2.SSS3.3.p3.3.m1.1.1.2.3">𝑣</ci></apply><ci id="S4.SS2.SSS3.3.p3.3.m1.1.1.3a.cmml" xref="S4.SS2.SSS3.3.p3.3.m1.1.1.3"><mtext id="S4.SS2.SSS3.3.p3.3.m1.1.1.3.cmml" xref="S4.SS2.SSS3.3.p3.3.m1.1.1.3">OPT</mtext></ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS3.3.p3.3.m1.1c">uv\in\textnormal{OPT}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS3.3.p3.3.m1.1d">italic_u italic_v ∈ OPT</annotation></semantics></math>, we add a total weight of at most <math alttext="4(1+\epsilon)" class="ltx_Math" display="inline" id="S4.SS2.SSS3.3.p3.4.m2.1"><semantics id="S4.SS2.SSS3.3.p3.4.m2.1a"><mrow id="S4.SS2.SSS3.3.p3.4.m2.1.1" xref="S4.SS2.SSS3.3.p3.4.m2.1.1.cmml"><mn id="S4.SS2.SSS3.3.p3.4.m2.1.1.3" xref="S4.SS2.SSS3.3.p3.4.m2.1.1.3.cmml">4</mn><mo id="S4.SS2.SSS3.3.p3.4.m2.1.1.2" xref="S4.SS2.SSS3.3.p3.4.m2.1.1.2.cmml">⁢</mo><mrow id="S4.SS2.SSS3.3.p3.4.m2.1.1.1.1" xref="S4.SS2.SSS3.3.p3.4.m2.1.1.1.1.1.cmml"><mo id="S4.SS2.SSS3.3.p3.4.m2.1.1.1.1.2" stretchy="false" xref="S4.SS2.SSS3.3.p3.4.m2.1.1.1.1.1.cmml">(</mo><mrow id="S4.SS2.SSS3.3.p3.4.m2.1.1.1.1.1" xref="S4.SS2.SSS3.3.p3.4.m2.1.1.1.1.1.cmml"><mn id="S4.SS2.SSS3.3.p3.4.m2.1.1.1.1.1.2" xref="S4.SS2.SSS3.3.p3.4.m2.1.1.1.1.1.2.cmml">1</mn><mo id="S4.SS2.SSS3.3.p3.4.m2.1.1.1.1.1.1" xref="S4.SS2.SSS3.3.p3.4.m2.1.1.1.1.1.1.cmml">+</mo><mi id="S4.SS2.SSS3.3.p3.4.m2.1.1.1.1.1.3" xref="S4.SS2.SSS3.3.p3.4.m2.1.1.1.1.1.3.cmml">ϵ</mi></mrow><mo id="S4.SS2.SSS3.3.p3.4.m2.1.1.1.1.3" stretchy="false" xref="S4.SS2.SSS3.3.p3.4.m2.1.1.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS3.3.p3.4.m2.1b"><apply id="S4.SS2.SSS3.3.p3.4.m2.1.1.cmml" xref="S4.SS2.SSS3.3.p3.4.m2.1.1"><times id="S4.SS2.SSS3.3.p3.4.m2.1.1.2.cmml" xref="S4.SS2.SSS3.3.p3.4.m2.1.1.2"></times><cn id="S4.SS2.SSS3.3.p3.4.m2.1.1.3.cmml" type="integer" xref="S4.SS2.SSS3.3.p3.4.m2.1.1.3">4</cn><apply id="S4.SS2.SSS3.3.p3.4.m2.1.1.1.1.1.cmml" xref="S4.SS2.SSS3.3.p3.4.m2.1.1.1.1"><plus id="S4.SS2.SSS3.3.p3.4.m2.1.1.1.1.1.1.cmml" xref="S4.SS2.SSS3.3.p3.4.m2.1.1.1.1.1.1"></plus><cn id="S4.SS2.SSS3.3.p3.4.m2.1.1.1.1.1.2.cmml" type="integer" xref="S4.SS2.SSS3.3.p3.4.m2.1.1.1.1.1.2">1</cn><ci id="S4.SS2.SSS3.3.p3.4.m2.1.1.1.1.1.3.cmml" xref="S4.SS2.SSS3.3.p3.4.m2.1.1.1.1.1.3">italic-ϵ</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS3.3.p3.4.m2.1c">4(1+\epsilon)</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS3.3.p3.4.m2.1d">4 ( 1 + italic_ϵ )</annotation></semantics></math> to <span class="ltx_text ltx_markedasmath" id="S4.SS2.SSS3.3.p3.7.1">SOL</span>. Combining all of the above, <math alttext="w(\textnormal{SOL})\leq[2(1+\epsilon)+1+4(1+\epsilon)]w(\textnormal{OPT})=(7+6% \epsilon)w(\textnormal{OPT})" class="ltx_Math" display="inline" id="S4.SS2.SSS3.3.p3.6.m4.5"><semantics id="S4.SS2.SSS3.3.p3.6.m4.5a"><mrow id="S4.SS2.SSS3.3.p3.6.m4.5.5" xref="S4.SS2.SSS3.3.p3.6.m4.5.5.cmml"><mrow id="S4.SS2.SSS3.3.p3.6.m4.5.5.4" xref="S4.SS2.SSS3.3.p3.6.m4.5.5.4.cmml"><mi id="S4.SS2.SSS3.3.p3.6.m4.5.5.4.2" xref="S4.SS2.SSS3.3.p3.6.m4.5.5.4.2.cmml">w</mi><mo id="S4.SS2.SSS3.3.p3.6.m4.5.5.4.1" xref="S4.SS2.SSS3.3.p3.6.m4.5.5.4.1.cmml">⁢</mo><mrow id="S4.SS2.SSS3.3.p3.6.m4.5.5.4.3.2" xref="S4.SS2.SSS3.3.p3.6.m4.1.1a.cmml"><mo id="S4.SS2.SSS3.3.p3.6.m4.5.5.4.3.2.1" stretchy="false" xref="S4.SS2.SSS3.3.p3.6.m4.1.1a.cmml">(</mo><mtext id="S4.SS2.SSS3.3.p3.6.m4.1.1" xref="S4.SS2.SSS3.3.p3.6.m4.1.1.cmml">SOL</mtext><mo id="S4.SS2.SSS3.3.p3.6.m4.5.5.4.3.2.2" stretchy="false" xref="S4.SS2.SSS3.3.p3.6.m4.1.1a.cmml">)</mo></mrow></mrow><mo id="S4.SS2.SSS3.3.p3.6.m4.5.5.5" xref="S4.SS2.SSS3.3.p3.6.m4.5.5.5.cmml">≤</mo><mrow id="S4.SS2.SSS3.3.p3.6.m4.4.4.1" xref="S4.SS2.SSS3.3.p3.6.m4.4.4.1.cmml"><mrow id="S4.SS2.SSS3.3.p3.6.m4.4.4.1.1.1" xref="S4.SS2.SSS3.3.p3.6.m4.4.4.1.1.2.cmml"><mo id="S4.SS2.SSS3.3.p3.6.m4.4.4.1.1.1.2" stretchy="false" xref="S4.SS2.SSS3.3.p3.6.m4.4.4.1.1.2.1.cmml">[</mo><mrow id="S4.SS2.SSS3.3.p3.6.m4.4.4.1.1.1.1" xref="S4.SS2.SSS3.3.p3.6.m4.4.4.1.1.1.1.cmml"><mrow id="S4.SS2.SSS3.3.p3.6.m4.4.4.1.1.1.1.1" xref="S4.SS2.SSS3.3.p3.6.m4.4.4.1.1.1.1.1.cmml"><mn id="S4.SS2.SSS3.3.p3.6.m4.4.4.1.1.1.1.1.3" xref="S4.SS2.SSS3.3.p3.6.m4.4.4.1.1.1.1.1.3.cmml">2</mn><mo id="S4.SS2.SSS3.3.p3.6.m4.4.4.1.1.1.1.1.2" xref="S4.SS2.SSS3.3.p3.6.m4.4.4.1.1.1.1.1.2.cmml">⁢</mo><mrow id="S4.SS2.SSS3.3.p3.6.m4.4.4.1.1.1.1.1.1.1" xref="S4.SS2.SSS3.3.p3.6.m4.4.4.1.1.1.1.1.1.1.1.cmml"><mo id="S4.SS2.SSS3.3.p3.6.m4.4.4.1.1.1.1.1.1.1.2" stretchy="false" xref="S4.SS2.SSS3.3.p3.6.m4.4.4.1.1.1.1.1.1.1.1.cmml">(</mo><mrow id="S4.SS2.SSS3.3.p3.6.m4.4.4.1.1.1.1.1.1.1.1" xref="S4.SS2.SSS3.3.p3.6.m4.4.4.1.1.1.1.1.1.1.1.cmml"><mn id="S4.SS2.SSS3.3.p3.6.m4.4.4.1.1.1.1.1.1.1.1.2" xref="S4.SS2.SSS3.3.p3.6.m4.4.4.1.1.1.1.1.1.1.1.2.cmml">1</mn><mo id="S4.SS2.SSS3.3.p3.6.m4.4.4.1.1.1.1.1.1.1.1.1" xref="S4.SS2.SSS3.3.p3.6.m4.4.4.1.1.1.1.1.1.1.1.1.cmml">+</mo><mi id="S4.SS2.SSS3.3.p3.6.m4.4.4.1.1.1.1.1.1.1.1.3" xref="S4.SS2.SSS3.3.p3.6.m4.4.4.1.1.1.1.1.1.1.1.3.cmml">ϵ</mi></mrow><mo id="S4.SS2.SSS3.3.p3.6.m4.4.4.1.1.1.1.1.1.1.3" stretchy="false" xref="S4.SS2.SSS3.3.p3.6.m4.4.4.1.1.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S4.SS2.SSS3.3.p3.6.m4.4.4.1.1.1.1.3" xref="S4.SS2.SSS3.3.p3.6.m4.4.4.1.1.1.1.3.cmml">+</mo><mn id="S4.SS2.SSS3.3.p3.6.m4.4.4.1.1.1.1.4" xref="S4.SS2.SSS3.3.p3.6.m4.4.4.1.1.1.1.4.cmml">1</mn><mo id="S4.SS2.SSS3.3.p3.6.m4.4.4.1.1.1.1.3a" xref="S4.SS2.SSS3.3.p3.6.m4.4.4.1.1.1.1.3.cmml">+</mo><mrow id="S4.SS2.SSS3.3.p3.6.m4.4.4.1.1.1.1.2" xref="S4.SS2.SSS3.3.p3.6.m4.4.4.1.1.1.1.2.cmml"><mn 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xref="S4.SS2.SSS3.3.p3.6.m4.4.4.1.1.1.1.2.1.1.1.cmml">)</mo></mrow></mrow></mrow><mo id="S4.SS2.SSS3.3.p3.6.m4.4.4.1.1.1.3" stretchy="false" xref="S4.SS2.SSS3.3.p3.6.m4.4.4.1.1.2.1.cmml">]</mo></mrow><mo id="S4.SS2.SSS3.3.p3.6.m4.4.4.1.2" xref="S4.SS2.SSS3.3.p3.6.m4.4.4.1.2.cmml">⁢</mo><mi id="S4.SS2.SSS3.3.p3.6.m4.4.4.1.3" xref="S4.SS2.SSS3.3.p3.6.m4.4.4.1.3.cmml">w</mi><mo id="S4.SS2.SSS3.3.p3.6.m4.4.4.1.2a" xref="S4.SS2.SSS3.3.p3.6.m4.4.4.1.2.cmml">⁢</mo><mrow id="S4.SS2.SSS3.3.p3.6.m4.4.4.1.4.2" xref="S4.SS2.SSS3.3.p3.6.m4.2.2a.cmml"><mo id="S4.SS2.SSS3.3.p3.6.m4.4.4.1.4.2.1" stretchy="false" xref="S4.SS2.SSS3.3.p3.6.m4.2.2a.cmml">(</mo><mtext id="S4.SS2.SSS3.3.p3.6.m4.2.2" xref="S4.SS2.SSS3.3.p3.6.m4.2.2.cmml">OPT</mtext><mo id="S4.SS2.SSS3.3.p3.6.m4.4.4.1.4.2.2" stretchy="false" xref="S4.SS2.SSS3.3.p3.6.m4.2.2a.cmml">)</mo></mrow></mrow><mo id="S4.SS2.SSS3.3.p3.6.m4.5.5.6" xref="S4.SS2.SSS3.3.p3.6.m4.5.5.6.cmml">=</mo><mrow id="S4.SS2.SSS3.3.p3.6.m4.5.5.2" xref="S4.SS2.SSS3.3.p3.6.m4.5.5.2.cmml"><mrow id="S4.SS2.SSS3.3.p3.6.m4.5.5.2.1.1" xref="S4.SS2.SSS3.3.p3.6.m4.5.5.2.1.1.1.cmml"><mo id="S4.SS2.SSS3.3.p3.6.m4.5.5.2.1.1.2" stretchy="false" xref="S4.SS2.SSS3.3.p3.6.m4.5.5.2.1.1.1.cmml">(</mo><mrow id="S4.SS2.SSS3.3.p3.6.m4.5.5.2.1.1.1" xref="S4.SS2.SSS3.3.p3.6.m4.5.5.2.1.1.1.cmml"><mn id="S4.SS2.SSS3.3.p3.6.m4.5.5.2.1.1.1.2" xref="S4.SS2.SSS3.3.p3.6.m4.5.5.2.1.1.1.2.cmml">7</mn><mo id="S4.SS2.SSS3.3.p3.6.m4.5.5.2.1.1.1.1" xref="S4.SS2.SSS3.3.p3.6.m4.5.5.2.1.1.1.1.cmml">+</mo><mrow id="S4.SS2.SSS3.3.p3.6.m4.5.5.2.1.1.1.3" xref="S4.SS2.SSS3.3.p3.6.m4.5.5.2.1.1.1.3.cmml"><mn id="S4.SS2.SSS3.3.p3.6.m4.5.5.2.1.1.1.3.2" xref="S4.SS2.SSS3.3.p3.6.m4.5.5.2.1.1.1.3.2.cmml">6</mn><mo id="S4.SS2.SSS3.3.p3.6.m4.5.5.2.1.1.1.3.1" xref="S4.SS2.SSS3.3.p3.6.m4.5.5.2.1.1.1.3.1.cmml">⁢</mo><mi id="S4.SS2.SSS3.3.p3.6.m4.5.5.2.1.1.1.3.3" xref="S4.SS2.SSS3.3.p3.6.m4.5.5.2.1.1.1.3.3.cmml">ϵ</mi></mrow></mrow><mo id="S4.SS2.SSS3.3.p3.6.m4.5.5.2.1.1.3" stretchy="false" xref="S4.SS2.SSS3.3.p3.6.m4.5.5.2.1.1.1.cmml">)</mo></mrow><mo id="S4.SS2.SSS3.3.p3.6.m4.5.5.2.2" xref="S4.SS2.SSS3.3.p3.6.m4.5.5.2.2.cmml">⁢</mo><mi id="S4.SS2.SSS3.3.p3.6.m4.5.5.2.3" xref="S4.SS2.SSS3.3.p3.6.m4.5.5.2.3.cmml">w</mi><mo id="S4.SS2.SSS3.3.p3.6.m4.5.5.2.2a" xref="S4.SS2.SSS3.3.p3.6.m4.5.5.2.2.cmml">⁢</mo><mrow id="S4.SS2.SSS3.3.p3.6.m4.5.5.2.4.2" xref="S4.SS2.SSS3.3.p3.6.m4.3.3a.cmml"><mo id="S4.SS2.SSS3.3.p3.6.m4.5.5.2.4.2.1" stretchy="false" xref="S4.SS2.SSS3.3.p3.6.m4.3.3a.cmml">(</mo><mtext id="S4.SS2.SSS3.3.p3.6.m4.3.3" xref="S4.SS2.SSS3.3.p3.6.m4.3.3.cmml">OPT</mtext><mo id="S4.SS2.SSS3.3.p3.6.m4.5.5.2.4.2.2" stretchy="false" xref="S4.SS2.SSS3.3.p3.6.m4.3.3a.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS3.3.p3.6.m4.5b"><apply id="S4.SS2.SSS3.3.p3.6.m4.5.5.cmml" xref="S4.SS2.SSS3.3.p3.6.m4.5.5"><and id="S4.SS2.SSS3.3.p3.6.m4.5.5a.cmml" xref="S4.SS2.SSS3.3.p3.6.m4.5.5"></and><apply 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id="S4.SS2.SSS3.3.p3.6.m4.5c">w(\textnormal{SOL})\leq[2(1+\epsilon)+1+4(1+\epsilon)]w(\textnormal{OPT})=(7+6% \epsilon)w(\textnormal{OPT})</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS3.3.p3.6.m4.5d">italic_w ( SOL ) ≤ [ 2 ( 1 + italic_ϵ ) + 1 + 4 ( 1 + italic_ϵ ) ] italic_w ( OPT ) = ( 7 + 6 italic_ϵ ) italic_w ( OPT )</annotation></semantics></math>. We run the algorithm with <math alttext="\epsilon/6" class="ltx_Math" display="inline" id="S4.SS2.SSS3.3.p3.7.m5.1"><semantics id="S4.SS2.SSS3.3.p3.7.m5.1a"><mrow id="S4.SS2.SSS3.3.p3.7.m5.1.1" xref="S4.SS2.SSS3.3.p3.7.m5.1.1.cmml"><mi id="S4.SS2.SSS3.3.p3.7.m5.1.1.2" xref="S4.SS2.SSS3.3.p3.7.m5.1.1.2.cmml">ϵ</mi><mo id="S4.SS2.SSS3.3.p3.7.m5.1.1.1" xref="S4.SS2.SSS3.3.p3.7.m5.1.1.1.cmml">/</mo><mn id="S4.SS2.SSS3.3.p3.7.m5.1.1.3" xref="S4.SS2.SSS3.3.p3.7.m5.1.1.3.cmml">6</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS3.3.p3.7.m5.1b"><apply id="S4.SS2.SSS3.3.p3.7.m5.1.1.cmml" xref="S4.SS2.SSS3.3.p3.7.m5.1.1"><divide id="S4.SS2.SSS3.3.p3.7.m5.1.1.1.cmml" xref="S4.SS2.SSS3.3.p3.7.m5.1.1.1"></divide><ci id="S4.SS2.SSS3.3.p3.7.m5.1.1.2.cmml" xref="S4.SS2.SSS3.3.p3.7.m5.1.1.2">italic-ϵ</ci><cn id="S4.SS2.SSS3.3.p3.7.m5.1.1.3.cmml" type="integer" xref="S4.SS2.SSS3.3.p3.7.m5.1.1.3">6</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS3.3.p3.7.m5.1c">\epsilon/6</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS3.3.p3.7.m5.1d">italic_ϵ / 6</annotation></semantics></math> to obtain the desired approximation ratio. ∎</p> </div> </div> <div class="ltx_theorem ltx_theorem_lemma" id="S4.Thmtheorem22"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="S4.Thmtheorem22.1.1.1">Lemma 4.22</span></span><span class="ltx_text ltx_font_bold" id="S4.Thmtheorem22.2.2">.</span> </h6> <div class="ltx_para" id="S4.Thmtheorem22.p1"> <p class="ltx_p" id="S4.Thmtheorem22.p1.1"><math alttext="(V,E\cup\textnormal{SOL})" class="ltx_Math" display="inline" id="S4.Thmtheorem22.p1.1.m1.2"><semantics id="S4.Thmtheorem22.p1.1.m1.2a"><mrow id="S4.Thmtheorem22.p1.1.m1.2.2.1" xref="S4.Thmtheorem22.p1.1.m1.2.2.2.cmml"><mo id="S4.Thmtheorem22.p1.1.m1.2.2.1.2" stretchy="false" xref="S4.Thmtheorem22.p1.1.m1.2.2.2.cmml">(</mo><mi id="S4.Thmtheorem22.p1.1.m1.1.1" xref="S4.Thmtheorem22.p1.1.m1.1.1.cmml">V</mi><mo id="S4.Thmtheorem22.p1.1.m1.2.2.1.3" xref="S4.Thmtheorem22.p1.1.m1.2.2.2.cmml">,</mo><mrow id="S4.Thmtheorem22.p1.1.m1.2.2.1.1" xref="S4.Thmtheorem22.p1.1.m1.2.2.1.1.cmml"><mi id="S4.Thmtheorem22.p1.1.m1.2.2.1.1.2" xref="S4.Thmtheorem22.p1.1.m1.2.2.1.1.2.cmml">E</mi><mo id="S4.Thmtheorem22.p1.1.m1.2.2.1.1.1" xref="S4.Thmtheorem22.p1.1.m1.2.2.1.1.1.cmml">∪</mo><mtext id="S4.Thmtheorem22.p1.1.m1.2.2.1.1.3" xref="S4.Thmtheorem22.p1.1.m1.2.2.1.1.3a.cmml">SOL</mtext></mrow><mo id="S4.Thmtheorem22.p1.1.m1.2.2.1.4" stretchy="false" xref="S4.Thmtheorem22.p1.1.m1.2.2.2.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem22.p1.1.m1.2b"><interval closure="open" id="S4.Thmtheorem22.p1.1.m1.2.2.2.cmml" xref="S4.Thmtheorem22.p1.1.m1.2.2.1"><ci id="S4.Thmtheorem22.p1.1.m1.1.1.cmml" xref="S4.Thmtheorem22.p1.1.m1.1.1">𝑉</ci><apply id="S4.Thmtheorem22.p1.1.m1.2.2.1.1.cmml" xref="S4.Thmtheorem22.p1.1.m1.2.2.1.1"><union id="S4.Thmtheorem22.p1.1.m1.2.2.1.1.1.cmml" xref="S4.Thmtheorem22.p1.1.m1.2.2.1.1.1"></union><ci id="S4.Thmtheorem22.p1.1.m1.2.2.1.1.2.cmml" xref="S4.Thmtheorem22.p1.1.m1.2.2.1.1.2">𝐸</ci><ci id="S4.Thmtheorem22.p1.1.m1.2.2.1.1.3a.cmml" xref="S4.Thmtheorem22.p1.1.m1.2.2.1.1.3"><mtext id="S4.Thmtheorem22.p1.1.m1.2.2.1.1.3.cmml" xref="S4.Thmtheorem22.p1.1.m1.2.2.1.1.3">SOL</mtext></ci></apply></interval></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem22.p1.1.m1.2c">(V,E\cup\textnormal{SOL})</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem22.p1.1.m1.2d">( italic_V , italic_E ∪ SOL )</annotation></semantics></math> is a 3-vertex-connected graph.</p> </div> </div> <div class="ltx_proof" id="S4.SS2.SSS3.5"> <h6 class="ltx_title ltx_runin ltx_font_italic ltx_title_proof">Proof.</h6> <div class="ltx_para" id="S4.SS2.SSS3.4.p1"> <p class="ltx_p" id="S4.SS2.SSS3.4.p1.10">We want to show that for 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id="S4.SS2.SSS3.4.p1.1.m1.2b"><apply id="S4.SS2.SSS3.4.p1.1.m1.2.3.cmml" xref="S4.SS2.SSS3.4.p1.1.m1.2.3"><in id="S4.SS2.SSS3.4.p1.1.m1.2.3.1.cmml" xref="S4.SS2.SSS3.4.p1.1.m1.2.3.1"></in><set id="S4.SS2.SSS3.4.p1.1.m1.2.3.2.1.cmml" xref="S4.SS2.SSS3.4.p1.1.m1.2.3.2.2"><ci id="S4.SS2.SSS3.4.p1.1.m1.1.1.cmml" xref="S4.SS2.SSS3.4.p1.1.m1.1.1">𝑎</ci><ci id="S4.SS2.SSS3.4.p1.1.m1.2.2.cmml" xref="S4.SS2.SSS3.4.p1.1.m1.2.2">𝑏</ci></set><ci id="S4.SS2.SSS3.4.p1.1.m1.2.3.3.cmml" xref="S4.SS2.SSS3.4.p1.1.m1.2.3.3">𝑉</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS3.4.p1.1.m1.2c">\{a,b\}\in V</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS3.4.p1.1.m1.2d">{ italic_a , italic_b } ∈ italic_V</annotation></semantics></math>, <math alttext="(V,E\cup\textnormal{SOL})\setminus\{a,b\}" class="ltx_Math" display="inline" id="S4.SS2.SSS3.4.p1.2.m2.4"><semantics id="S4.SS2.SSS3.4.p1.2.m2.4a"><mrow id="S4.SS2.SSS3.4.p1.2.m2.4.4" xref="S4.SS2.SSS3.4.p1.2.m2.4.4.cmml"><mrow id="S4.SS2.SSS3.4.p1.2.m2.4.4.1.1" xref="S4.SS2.SSS3.4.p1.2.m2.4.4.1.2.cmml"><mo id="S4.SS2.SSS3.4.p1.2.m2.4.4.1.1.2" stretchy="false" xref="S4.SS2.SSS3.4.p1.2.m2.4.4.1.2.cmml">(</mo><mi id="S4.SS2.SSS3.4.p1.2.m2.1.1" xref="S4.SS2.SSS3.4.p1.2.m2.1.1.cmml">V</mi><mo id="S4.SS2.SSS3.4.p1.2.m2.4.4.1.1.3" xref="S4.SS2.SSS3.4.p1.2.m2.4.4.1.2.cmml">,</mo><mrow id="S4.SS2.SSS3.4.p1.2.m2.4.4.1.1.1" xref="S4.SS2.SSS3.4.p1.2.m2.4.4.1.1.1.cmml"><mi id="S4.SS2.SSS3.4.p1.2.m2.4.4.1.1.1.2" xref="S4.SS2.SSS3.4.p1.2.m2.4.4.1.1.1.2.cmml">E</mi><mo id="S4.SS2.SSS3.4.p1.2.m2.4.4.1.1.1.1" xref="S4.SS2.SSS3.4.p1.2.m2.4.4.1.1.1.1.cmml">∪</mo><mtext id="S4.SS2.SSS3.4.p1.2.m2.4.4.1.1.1.3" xref="S4.SS2.SSS3.4.p1.2.m2.4.4.1.1.1.3a.cmml">SOL</mtext></mrow><mo id="S4.SS2.SSS3.4.p1.2.m2.4.4.1.1.4" stretchy="false" xref="S4.SS2.SSS3.4.p1.2.m2.4.4.1.2.cmml">)</mo></mrow><mo id="S4.SS2.SSS3.4.p1.2.m2.4.4.2" xref="S4.SS2.SSS3.4.p1.2.m2.4.4.2.cmml">∖</mo><mrow id="S4.SS2.SSS3.4.p1.2.m2.4.4.3.2" xref="S4.SS2.SSS3.4.p1.2.m2.4.4.3.1.cmml"><mo id="S4.SS2.SSS3.4.p1.2.m2.4.4.3.2.1" stretchy="false" xref="S4.SS2.SSS3.4.p1.2.m2.4.4.3.1.cmml">{</mo><mi id="S4.SS2.SSS3.4.p1.2.m2.2.2" xref="S4.SS2.SSS3.4.p1.2.m2.2.2.cmml">a</mi><mo id="S4.SS2.SSS3.4.p1.2.m2.4.4.3.2.2" xref="S4.SS2.SSS3.4.p1.2.m2.4.4.3.1.cmml">,</mo><mi id="S4.SS2.SSS3.4.p1.2.m2.3.3" xref="S4.SS2.SSS3.4.p1.2.m2.3.3.cmml">b</mi><mo id="S4.SS2.SSS3.4.p1.2.m2.4.4.3.2.3" stretchy="false" xref="S4.SS2.SSS3.4.p1.2.m2.4.4.3.1.cmml">}</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS3.4.p1.2.m2.4b"><apply id="S4.SS2.SSS3.4.p1.2.m2.4.4.cmml" xref="S4.SS2.SSS3.4.p1.2.m2.4.4"><setdiff id="S4.SS2.SSS3.4.p1.2.m2.4.4.2.cmml" xref="S4.SS2.SSS3.4.p1.2.m2.4.4.2"></setdiff><interval closure="open" id="S4.SS2.SSS3.4.p1.2.m2.4.4.1.2.cmml" xref="S4.SS2.SSS3.4.p1.2.m2.4.4.1.1"><ci id="S4.SS2.SSS3.4.p1.2.m2.1.1.cmml" xref="S4.SS2.SSS3.4.p1.2.m2.1.1">𝑉</ci><apply id="S4.SS2.SSS3.4.p1.2.m2.4.4.1.1.1.cmml" xref="S4.SS2.SSS3.4.p1.2.m2.4.4.1.1.1"><union id="S4.SS2.SSS3.4.p1.2.m2.4.4.1.1.1.1.cmml" xref="S4.SS2.SSS3.4.p1.2.m2.4.4.1.1.1.1"></union><ci id="S4.SS2.SSS3.4.p1.2.m2.4.4.1.1.1.2.cmml" xref="S4.SS2.SSS3.4.p1.2.m2.4.4.1.1.1.2">𝐸</ci><ci id="S4.SS2.SSS3.4.p1.2.m2.4.4.1.1.1.3a.cmml" xref="S4.SS2.SSS3.4.p1.2.m2.4.4.1.1.1.3"><mtext id="S4.SS2.SSS3.4.p1.2.m2.4.4.1.1.1.3.cmml" xref="S4.SS2.SSS3.4.p1.2.m2.4.4.1.1.1.3">SOL</mtext></ci></apply></interval><set id="S4.SS2.SSS3.4.p1.2.m2.4.4.3.1.cmml" xref="S4.SS2.SSS3.4.p1.2.m2.4.4.3.2"><ci id="S4.SS2.SSS3.4.p1.2.m2.2.2.cmml" xref="S4.SS2.SSS3.4.p1.2.m2.2.2">𝑎</ci><ci id="S4.SS2.SSS3.4.p1.2.m2.3.3.cmml" xref="S4.SS2.SSS3.4.p1.2.m2.3.3">𝑏</ci></set></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS3.4.p1.2.m2.4c">(V,E\cup\textnormal{SOL})\setminus\{a,b\}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS3.4.p1.2.m2.4d">( italic_V , italic_E ∪ SOL ) ∖ { italic_a , italic_b }</annotation></semantics></math> is connected. Fix <math alttext="\{a,b\}\in V" class="ltx_Math" display="inline" id="S4.SS2.SSS3.4.p1.3.m3.2"><semantics id="S4.SS2.SSS3.4.p1.3.m3.2a"><mrow id="S4.SS2.SSS3.4.p1.3.m3.2.3" xref="S4.SS2.SSS3.4.p1.3.m3.2.3.cmml"><mrow id="S4.SS2.SSS3.4.p1.3.m3.2.3.2.2" xref="S4.SS2.SSS3.4.p1.3.m3.2.3.2.1.cmml"><mo id="S4.SS2.SSS3.4.p1.3.m3.2.3.2.2.1" stretchy="false" xref="S4.SS2.SSS3.4.p1.3.m3.2.3.2.1.cmml">{</mo><mi id="S4.SS2.SSS3.4.p1.3.m3.1.1" xref="S4.SS2.SSS3.4.p1.3.m3.1.1.cmml">a</mi><mo id="S4.SS2.SSS3.4.p1.3.m3.2.3.2.2.2" xref="S4.SS2.SSS3.4.p1.3.m3.2.3.2.1.cmml">,</mo><mi id="S4.SS2.SSS3.4.p1.3.m3.2.2" xref="S4.SS2.SSS3.4.p1.3.m3.2.2.cmml">b</mi><mo id="S4.SS2.SSS3.4.p1.3.m3.2.3.2.2.3" stretchy="false" xref="S4.SS2.SSS3.4.p1.3.m3.2.3.2.1.cmml">}</mo></mrow><mo id="S4.SS2.SSS3.4.p1.3.m3.2.3.1" xref="S4.SS2.SSS3.4.p1.3.m3.2.3.1.cmml">∈</mo><mi id="S4.SS2.SSS3.4.p1.3.m3.2.3.3" xref="S4.SS2.SSS3.4.p1.3.m3.2.3.3.cmml">V</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS3.4.p1.3.m3.2b"><apply id="S4.SS2.SSS3.4.p1.3.m3.2.3.cmml" xref="S4.SS2.SSS3.4.p1.3.m3.2.3"><in id="S4.SS2.SSS3.4.p1.3.m3.2.3.1.cmml" xref="S4.SS2.SSS3.4.p1.3.m3.2.3.1"></in><set id="S4.SS2.SSS3.4.p1.3.m3.2.3.2.1.cmml" xref="S4.SS2.SSS3.4.p1.3.m3.2.3.2.2"><ci id="S4.SS2.SSS3.4.p1.3.m3.1.1.cmml" xref="S4.SS2.SSS3.4.p1.3.m3.1.1">𝑎</ci><ci id="S4.SS2.SSS3.4.p1.3.m3.2.2.cmml" xref="S4.SS2.SSS3.4.p1.3.m3.2.2">𝑏</ci></set><ci id="S4.SS2.SSS3.4.p1.3.m3.2.3.3.cmml" xref="S4.SS2.SSS3.4.p1.3.m3.2.3.3">𝑉</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS3.4.p1.3.m3.2c">\{a,b\}\in V</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS3.4.p1.3.m3.2d">{ italic_a , italic_b } ∈ italic_V</annotation></semantics></math>. Since <span class="ltx_text ltx_markedasmath" id="S4.SS2.SSS3.4.p1.10.1">OPT</span> is feasible, it suffices to show that for all <math alttext="uv\in\textnormal{OPT}" class="ltx_Math" display="inline" id="S4.SS2.SSS3.4.p1.5.m5.1"><semantics id="S4.SS2.SSS3.4.p1.5.m5.1a"><mrow id="S4.SS2.SSS3.4.p1.5.m5.1.1" xref="S4.SS2.SSS3.4.p1.5.m5.1.1.cmml"><mrow id="S4.SS2.SSS3.4.p1.5.m5.1.1.2" xref="S4.SS2.SSS3.4.p1.5.m5.1.1.2.cmml"><mi id="S4.SS2.SSS3.4.p1.5.m5.1.1.2.2" xref="S4.SS2.SSS3.4.p1.5.m5.1.1.2.2.cmml">u</mi><mo id="S4.SS2.SSS3.4.p1.5.m5.1.1.2.1" xref="S4.SS2.SSS3.4.p1.5.m5.1.1.2.1.cmml">⁢</mo><mi id="S4.SS2.SSS3.4.p1.5.m5.1.1.2.3" xref="S4.SS2.SSS3.4.p1.5.m5.1.1.2.3.cmml">v</mi></mrow><mo id="S4.SS2.SSS3.4.p1.5.m5.1.1.1" xref="S4.SS2.SSS3.4.p1.5.m5.1.1.1.cmml">∈</mo><mtext id="S4.SS2.SSS3.4.p1.5.m5.1.1.3" xref="S4.SS2.SSS3.4.p1.5.m5.1.1.3a.cmml">OPT</mtext></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS3.4.p1.5.m5.1b"><apply id="S4.SS2.SSS3.4.p1.5.m5.1.1.cmml" xref="S4.SS2.SSS3.4.p1.5.m5.1.1"><in id="S4.SS2.SSS3.4.p1.5.m5.1.1.1.cmml" xref="S4.SS2.SSS3.4.p1.5.m5.1.1.1"></in><apply id="S4.SS2.SSS3.4.p1.5.m5.1.1.2.cmml" xref="S4.SS2.SSS3.4.p1.5.m5.1.1.2"><times id="S4.SS2.SSS3.4.p1.5.m5.1.1.2.1.cmml" xref="S4.SS2.SSS3.4.p1.5.m5.1.1.2.1"></times><ci id="S4.SS2.SSS3.4.p1.5.m5.1.1.2.2.cmml" xref="S4.SS2.SSS3.4.p1.5.m5.1.1.2.2">𝑢</ci><ci id="S4.SS2.SSS3.4.p1.5.m5.1.1.2.3.cmml" xref="S4.SS2.SSS3.4.p1.5.m5.1.1.2.3">𝑣</ci></apply><ci id="S4.SS2.SSS3.4.p1.5.m5.1.1.3a.cmml" xref="S4.SS2.SSS3.4.p1.5.m5.1.1.3"><mtext id="S4.SS2.SSS3.4.p1.5.m5.1.1.3.cmml" xref="S4.SS2.SSS3.4.p1.5.m5.1.1.3">OPT</mtext></ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS3.4.p1.5.m5.1c">uv\in\textnormal{OPT}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS3.4.p1.5.m5.1d">italic_u italic_v ∈ OPT</annotation></semantics></math> with <math alttext="u,v\notin\{a,b\}" class="ltx_Math" display="inline" id="S4.SS2.SSS3.4.p1.6.m6.4"><semantics id="S4.SS2.SSS3.4.p1.6.m6.4a"><mrow id="S4.SS2.SSS3.4.p1.6.m6.4.5" xref="S4.SS2.SSS3.4.p1.6.m6.4.5.cmml"><mrow id="S4.SS2.SSS3.4.p1.6.m6.4.5.2.2" xref="S4.SS2.SSS3.4.p1.6.m6.4.5.2.1.cmml"><mi id="S4.SS2.SSS3.4.p1.6.m6.3.3" xref="S4.SS2.SSS3.4.p1.6.m6.3.3.cmml">u</mi><mo id="S4.SS2.SSS3.4.p1.6.m6.4.5.2.2.1" xref="S4.SS2.SSS3.4.p1.6.m6.4.5.2.1.cmml">,</mo><mi id="S4.SS2.SSS3.4.p1.6.m6.4.4" xref="S4.SS2.SSS3.4.p1.6.m6.4.4.cmml">v</mi></mrow><mo id="S4.SS2.SSS3.4.p1.6.m6.4.5.1" xref="S4.SS2.SSS3.4.p1.6.m6.4.5.1.cmml">∉</mo><mrow id="S4.SS2.SSS3.4.p1.6.m6.4.5.3.2" xref="S4.SS2.SSS3.4.p1.6.m6.4.5.3.1.cmml"><mo id="S4.SS2.SSS3.4.p1.6.m6.4.5.3.2.1" stretchy="false" xref="S4.SS2.SSS3.4.p1.6.m6.4.5.3.1.cmml">{</mo><mi id="S4.SS2.SSS3.4.p1.6.m6.1.1" xref="S4.SS2.SSS3.4.p1.6.m6.1.1.cmml">a</mi><mo id="S4.SS2.SSS3.4.p1.6.m6.4.5.3.2.2" xref="S4.SS2.SSS3.4.p1.6.m6.4.5.3.1.cmml">,</mo><mi id="S4.SS2.SSS3.4.p1.6.m6.2.2" xref="S4.SS2.SSS3.4.p1.6.m6.2.2.cmml">b</mi><mo id="S4.SS2.SSS3.4.p1.6.m6.4.5.3.2.3" stretchy="false" xref="S4.SS2.SSS3.4.p1.6.m6.4.5.3.1.cmml">}</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS3.4.p1.6.m6.4b"><apply id="S4.SS2.SSS3.4.p1.6.m6.4.5.cmml" xref="S4.SS2.SSS3.4.p1.6.m6.4.5"><notin id="S4.SS2.SSS3.4.p1.6.m6.4.5.1.cmml" xref="S4.SS2.SSS3.4.p1.6.m6.4.5.1"></notin><list id="S4.SS2.SSS3.4.p1.6.m6.4.5.2.1.cmml" xref="S4.SS2.SSS3.4.p1.6.m6.4.5.2.2"><ci id="S4.SS2.SSS3.4.p1.6.m6.3.3.cmml" xref="S4.SS2.SSS3.4.p1.6.m6.3.3">𝑢</ci><ci id="S4.SS2.SSS3.4.p1.6.m6.4.4.cmml" xref="S4.SS2.SSS3.4.p1.6.m6.4.4">𝑣</ci></list><set id="S4.SS2.SSS3.4.p1.6.m6.4.5.3.1.cmml" xref="S4.SS2.SSS3.4.p1.6.m6.4.5.3.2"><ci id="S4.SS2.SSS3.4.p1.6.m6.1.1.cmml" xref="S4.SS2.SSS3.4.p1.6.m6.1.1">𝑎</ci><ci id="S4.SS2.SSS3.4.p1.6.m6.2.2.cmml" xref="S4.SS2.SSS3.4.p1.6.m6.2.2">𝑏</ci></set></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS3.4.p1.6.m6.4c">u,v\notin\{a,b\}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS3.4.p1.6.m6.4d">italic_u , italic_v ∉ { italic_a , italic_b }</annotation></semantics></math>, there exists a <math alttext="u" class="ltx_Math" display="inline" id="S4.SS2.SSS3.4.p1.7.m7.1"><semantics id="S4.SS2.SSS3.4.p1.7.m7.1a"><mi id="S4.SS2.SSS3.4.p1.7.m7.1.1" xref="S4.SS2.SSS3.4.p1.7.m7.1.1.cmml">u</mi><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS3.4.p1.7.m7.1b"><ci id="S4.SS2.SSS3.4.p1.7.m7.1.1.cmml" xref="S4.SS2.SSS3.4.p1.7.m7.1.1">𝑢</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS3.4.p1.7.m7.1c">u</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS3.4.p1.7.m7.1d">italic_u</annotation></semantics></math>-<math alttext="v" class="ltx_Math" display="inline" id="S4.SS2.SSS3.4.p1.8.m8.1"><semantics id="S4.SS2.SSS3.4.p1.8.m8.1a"><mi id="S4.SS2.SSS3.4.p1.8.m8.1.1" xref="S4.SS2.SSS3.4.p1.8.m8.1.1.cmml">v</mi><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS3.4.p1.8.m8.1b"><ci id="S4.SS2.SSS3.4.p1.8.m8.1.1.cmml" xref="S4.SS2.SSS3.4.p1.8.m8.1.1">𝑣</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS3.4.p1.8.m8.1c">v</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS3.4.p1.8.m8.1d">italic_v</annotation></semantics></math> path in <math alttext="E\cup\textnormal{SOL}" class="ltx_Math" display="inline" id="S4.SS2.SSS3.4.p1.9.m9.1"><semantics id="S4.SS2.SSS3.4.p1.9.m9.1a"><mrow id="S4.SS2.SSS3.4.p1.9.m9.1.1" xref="S4.SS2.SSS3.4.p1.9.m9.1.1.cmml"><mi id="S4.SS2.SSS3.4.p1.9.m9.1.1.2" xref="S4.SS2.SSS3.4.p1.9.m9.1.1.2.cmml">E</mi><mo id="S4.SS2.SSS3.4.p1.9.m9.1.1.1" xref="S4.SS2.SSS3.4.p1.9.m9.1.1.1.cmml">∪</mo><mtext id="S4.SS2.SSS3.4.p1.9.m9.1.1.3" xref="S4.SS2.SSS3.4.p1.9.m9.1.1.3a.cmml">SOL</mtext></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS3.4.p1.9.m9.1b"><apply id="S4.SS2.SSS3.4.p1.9.m9.1.1.cmml" xref="S4.SS2.SSS3.4.p1.9.m9.1.1"><union id="S4.SS2.SSS3.4.p1.9.m9.1.1.1.cmml" xref="S4.SS2.SSS3.4.p1.9.m9.1.1.1"></union><ci id="S4.SS2.SSS3.4.p1.9.m9.1.1.2.cmml" xref="S4.SS2.SSS3.4.p1.9.m9.1.1.2">𝐸</ci><ci id="S4.SS2.SSS3.4.p1.9.m9.1.1.3a.cmml" xref="S4.SS2.SSS3.4.p1.9.m9.1.1.3"><mtext id="S4.SS2.SSS3.4.p1.9.m9.1.1.3.cmml" xref="S4.SS2.SSS3.4.p1.9.m9.1.1.3">SOL</mtext></ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS3.4.p1.9.m9.1c">E\cup\textnormal{SOL}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS3.4.p1.9.m9.1d">italic_E ∪ SOL</annotation></semantics></math> that does not use <math alttext="\{a,b\}" class="ltx_Math" display="inline" id="S4.SS2.SSS3.4.p1.10.m10.2"><semantics id="S4.SS2.SSS3.4.p1.10.m10.2a"><mrow id="S4.SS2.SSS3.4.p1.10.m10.2.3.2" xref="S4.SS2.SSS3.4.p1.10.m10.2.3.1.cmml"><mo id="S4.SS2.SSS3.4.p1.10.m10.2.3.2.1" stretchy="false" xref="S4.SS2.SSS3.4.p1.10.m10.2.3.1.cmml">{</mo><mi id="S4.SS2.SSS3.4.p1.10.m10.1.1" xref="S4.SS2.SSS3.4.p1.10.m10.1.1.cmml">a</mi><mo id="S4.SS2.SSS3.4.p1.10.m10.2.3.2.2" xref="S4.SS2.SSS3.4.p1.10.m10.2.3.1.cmml">,</mo><mi id="S4.SS2.SSS3.4.p1.10.m10.2.2" xref="S4.SS2.SSS3.4.p1.10.m10.2.2.cmml">b</mi><mo id="S4.SS2.SSS3.4.p1.10.m10.2.3.2.3" stretchy="false" xref="S4.SS2.SSS3.4.p1.10.m10.2.3.1.cmml">}</mo></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS3.4.p1.10.m10.2b"><set id="S4.SS2.SSS3.4.p1.10.m10.2.3.1.cmml" xref="S4.SS2.SSS3.4.p1.10.m10.2.3.2"><ci id="S4.SS2.SSS3.4.p1.10.m10.1.1.cmml" xref="S4.SS2.SSS3.4.p1.10.m10.1.1">𝑎</ci><ci id="S4.SS2.SSS3.4.p1.10.m10.2.2.cmml" xref="S4.SS2.SSS3.4.p1.10.m10.2.2">𝑏</ci></set></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS3.4.p1.10.m10.2c">\{a,b\}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS3.4.p1.10.m10.2d">{ italic_a , italic_b }</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="S4.SS2.SSS3.5.p2"> <p class="ltx_p" id="S4.SS2.SSS3.5.p2.8">Fix <math alttext="uv\in\textnormal{OPT}" class="ltx_Math" display="inline" id="S4.SS2.SSS3.5.p2.1.m1.1"><semantics id="S4.SS2.SSS3.5.p2.1.m1.1a"><mrow id="S4.SS2.SSS3.5.p2.1.m1.1.1" xref="S4.SS2.SSS3.5.p2.1.m1.1.1.cmml"><mrow id="S4.SS2.SSS3.5.p2.1.m1.1.1.2" xref="S4.SS2.SSS3.5.p2.1.m1.1.1.2.cmml"><mi id="S4.SS2.SSS3.5.p2.1.m1.1.1.2.2" xref="S4.SS2.SSS3.5.p2.1.m1.1.1.2.2.cmml">u</mi><mo id="S4.SS2.SSS3.5.p2.1.m1.1.1.2.1" xref="S4.SS2.SSS3.5.p2.1.m1.1.1.2.1.cmml">⁢</mo><mi id="S4.SS2.SSS3.5.p2.1.m1.1.1.2.3" xref="S4.SS2.SSS3.5.p2.1.m1.1.1.2.3.cmml">v</mi></mrow><mo id="S4.SS2.SSS3.5.p2.1.m1.1.1.1" xref="S4.SS2.SSS3.5.p2.1.m1.1.1.1.cmml">∈</mo><mtext id="S4.SS2.SSS3.5.p2.1.m1.1.1.3" xref="S4.SS2.SSS3.5.p2.1.m1.1.1.3a.cmml">OPT</mtext></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS3.5.p2.1.m1.1b"><apply id="S4.SS2.SSS3.5.p2.1.m1.1.1.cmml" xref="S4.SS2.SSS3.5.p2.1.m1.1.1"><in id="S4.SS2.SSS3.5.p2.1.m1.1.1.1.cmml" xref="S4.SS2.SSS3.5.p2.1.m1.1.1.1"></in><apply id="S4.SS2.SSS3.5.p2.1.m1.1.1.2.cmml" xref="S4.SS2.SSS3.5.p2.1.m1.1.1.2"><times id="S4.SS2.SSS3.5.p2.1.m1.1.1.2.1.cmml" xref="S4.SS2.SSS3.5.p2.1.m1.1.1.2.1"></times><ci id="S4.SS2.SSS3.5.p2.1.m1.1.1.2.2.cmml" xref="S4.SS2.SSS3.5.p2.1.m1.1.1.2.2">𝑢</ci><ci id="S4.SS2.SSS3.5.p2.1.m1.1.1.2.3.cmml" xref="S4.SS2.SSS3.5.p2.1.m1.1.1.2.3">𝑣</ci></apply><ci id="S4.SS2.SSS3.5.p2.1.m1.1.1.3a.cmml" xref="S4.SS2.SSS3.5.p2.1.m1.1.1.3"><mtext id="S4.SS2.SSS3.5.p2.1.m1.1.1.3.cmml" xref="S4.SS2.SSS3.5.p2.1.m1.1.1.3">OPT</mtext></ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS3.5.p2.1.m1.1c">uv\in\textnormal{OPT}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS3.5.p2.1.m1.1d">italic_u italic_v ∈ OPT</annotation></semantics></math> and let <math alttext="j" class="ltx_Math" display="inline" id="S4.SS2.SSS3.5.p2.2.m2.1"><semantics id="S4.SS2.SSS3.5.p2.2.m2.1a"><mi id="S4.SS2.SSS3.5.p2.2.m2.1.1" xref="S4.SS2.SSS3.5.p2.2.m2.1.1.cmml">j</mi><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS3.5.p2.2.m2.1b"><ci id="S4.SS2.SSS3.5.p2.2.m2.1.1.cmml" xref="S4.SS2.SSS3.5.p2.2.m2.1.1">𝑗</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS3.5.p2.2.m2.1c">j</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS3.5.p2.2.m2.1d">italic_j</annotation></semantics></math> be the weight class such that <math alttext="w(uv)\in[(1+\epsilon)^{j},(1+\epsilon)^{j+1})" class="ltx_Math" display="inline" id="S4.SS2.SSS3.5.p2.3.m3.3"><semantics id="S4.SS2.SSS3.5.p2.3.m3.3a"><mrow id="S4.SS2.SSS3.5.p2.3.m3.3.3" xref="S4.SS2.SSS3.5.p2.3.m3.3.3.cmml"><mrow id="S4.SS2.SSS3.5.p2.3.m3.1.1.1" xref="S4.SS2.SSS3.5.p2.3.m3.1.1.1.cmml"><mi id="S4.SS2.SSS3.5.p2.3.m3.1.1.1.3" xref="S4.SS2.SSS3.5.p2.3.m3.1.1.1.3.cmml">w</mi><mo id="S4.SS2.SSS3.5.p2.3.m3.1.1.1.2" xref="S4.SS2.SSS3.5.p2.3.m3.1.1.1.2.cmml">⁢</mo><mrow id="S4.SS2.SSS3.5.p2.3.m3.1.1.1.1.1" xref="S4.SS2.SSS3.5.p2.3.m3.1.1.1.1.1.1.cmml"><mo id="S4.SS2.SSS3.5.p2.3.m3.1.1.1.1.1.2" stretchy="false" xref="S4.SS2.SSS3.5.p2.3.m3.1.1.1.1.1.1.cmml">(</mo><mrow id="S4.SS2.SSS3.5.p2.3.m3.1.1.1.1.1.1" xref="S4.SS2.SSS3.5.p2.3.m3.1.1.1.1.1.1.cmml"><mi id="S4.SS2.SSS3.5.p2.3.m3.1.1.1.1.1.1.2" xref="S4.SS2.SSS3.5.p2.3.m3.1.1.1.1.1.1.2.cmml">u</mi><mo id="S4.SS2.SSS3.5.p2.3.m3.1.1.1.1.1.1.1" xref="S4.SS2.SSS3.5.p2.3.m3.1.1.1.1.1.1.1.cmml">⁢</mo><mi id="S4.SS2.SSS3.5.p2.3.m3.1.1.1.1.1.1.3" xref="S4.SS2.SSS3.5.p2.3.m3.1.1.1.1.1.1.3.cmml">v</mi></mrow><mo id="S4.SS2.SSS3.5.p2.3.m3.1.1.1.1.1.3" stretchy="false" xref="S4.SS2.SSS3.5.p2.3.m3.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S4.SS2.SSS3.5.p2.3.m3.3.3.4" xref="S4.SS2.SSS3.5.p2.3.m3.3.3.4.cmml">∈</mo><mrow id="S4.SS2.SSS3.5.p2.3.m3.3.3.3.2" xref="S4.SS2.SSS3.5.p2.3.m3.3.3.3.3.cmml"><mo id="S4.SS2.SSS3.5.p2.3.m3.3.3.3.2.3" stretchy="false" xref="S4.SS2.SSS3.5.p2.3.m3.3.3.3.3.cmml">[</mo><msup id="S4.SS2.SSS3.5.p2.3.m3.2.2.2.1.1" xref="S4.SS2.SSS3.5.p2.3.m3.2.2.2.1.1.cmml"><mrow id="S4.SS2.SSS3.5.p2.3.m3.2.2.2.1.1.1.1" xref="S4.SS2.SSS3.5.p2.3.m3.2.2.2.1.1.1.1.1.cmml"><mo id="S4.SS2.SSS3.5.p2.3.m3.2.2.2.1.1.1.1.2" stretchy="false" xref="S4.SS2.SSS3.5.p2.3.m3.2.2.2.1.1.1.1.1.cmml">(</mo><mrow id="S4.SS2.SSS3.5.p2.3.m3.2.2.2.1.1.1.1.1" xref="S4.SS2.SSS3.5.p2.3.m3.2.2.2.1.1.1.1.1.cmml"><mn id="S4.SS2.SSS3.5.p2.3.m3.2.2.2.1.1.1.1.1.2" xref="S4.SS2.SSS3.5.p2.3.m3.2.2.2.1.1.1.1.1.2.cmml">1</mn><mo id="S4.SS2.SSS3.5.p2.3.m3.2.2.2.1.1.1.1.1.1" xref="S4.SS2.SSS3.5.p2.3.m3.2.2.2.1.1.1.1.1.1.cmml">+</mo><mi id="S4.SS2.SSS3.5.p2.3.m3.2.2.2.1.1.1.1.1.3" xref="S4.SS2.SSS3.5.p2.3.m3.2.2.2.1.1.1.1.1.3.cmml">ϵ</mi></mrow><mo id="S4.SS2.SSS3.5.p2.3.m3.2.2.2.1.1.1.1.3" stretchy="false" xref="S4.SS2.SSS3.5.p2.3.m3.2.2.2.1.1.1.1.1.cmml">)</mo></mrow><mi id="S4.SS2.SSS3.5.p2.3.m3.2.2.2.1.1.3" xref="S4.SS2.SSS3.5.p2.3.m3.2.2.2.1.1.3.cmml">j</mi></msup><mo id="S4.SS2.SSS3.5.p2.3.m3.3.3.3.2.4" xref="S4.SS2.SSS3.5.p2.3.m3.3.3.3.3.cmml">,</mo><msup id="S4.SS2.SSS3.5.p2.3.m3.3.3.3.2.2" xref="S4.SS2.SSS3.5.p2.3.m3.3.3.3.2.2.cmml"><mrow id="S4.SS2.SSS3.5.p2.3.m3.3.3.3.2.2.1.1" xref="S4.SS2.SSS3.5.p2.3.m3.3.3.3.2.2.1.1.1.cmml"><mo id="S4.SS2.SSS3.5.p2.3.m3.3.3.3.2.2.1.1.2" stretchy="false" xref="S4.SS2.SSS3.5.p2.3.m3.3.3.3.2.2.1.1.1.cmml">(</mo><mrow id="S4.SS2.SSS3.5.p2.3.m3.3.3.3.2.2.1.1.1" xref="S4.SS2.SSS3.5.p2.3.m3.3.3.3.2.2.1.1.1.cmml"><mn id="S4.SS2.SSS3.5.p2.3.m3.3.3.3.2.2.1.1.1.2" xref="S4.SS2.SSS3.5.p2.3.m3.3.3.3.2.2.1.1.1.2.cmml">1</mn><mo id="S4.SS2.SSS3.5.p2.3.m3.3.3.3.2.2.1.1.1.1" xref="S4.SS2.SSS3.5.p2.3.m3.3.3.3.2.2.1.1.1.1.cmml">+</mo><mi id="S4.SS2.SSS3.5.p2.3.m3.3.3.3.2.2.1.1.1.3" xref="S4.SS2.SSS3.5.p2.3.m3.3.3.3.2.2.1.1.1.3.cmml">ϵ</mi></mrow><mo id="S4.SS2.SSS3.5.p2.3.m3.3.3.3.2.2.1.1.3" stretchy="false" xref="S4.SS2.SSS3.5.p2.3.m3.3.3.3.2.2.1.1.1.cmml">)</mo></mrow><mrow id="S4.SS2.SSS3.5.p2.3.m3.3.3.3.2.2.3" xref="S4.SS2.SSS3.5.p2.3.m3.3.3.3.2.2.3.cmml"><mi id="S4.SS2.SSS3.5.p2.3.m3.3.3.3.2.2.3.2" xref="S4.SS2.SSS3.5.p2.3.m3.3.3.3.2.2.3.2.cmml">j</mi><mo id="S4.SS2.SSS3.5.p2.3.m3.3.3.3.2.2.3.1" xref="S4.SS2.SSS3.5.p2.3.m3.3.3.3.2.2.3.1.cmml">+</mo><mn id="S4.SS2.SSS3.5.p2.3.m3.3.3.3.2.2.3.3" xref="S4.SS2.SSS3.5.p2.3.m3.3.3.3.2.2.3.3.cmml">1</mn></mrow></msup><mo id="S4.SS2.SSS3.5.p2.3.m3.3.3.3.2.5" stretchy="false" xref="S4.SS2.SSS3.5.p2.3.m3.3.3.3.3.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS3.5.p2.3.m3.3b"><apply id="S4.SS2.SSS3.5.p2.3.m3.3.3.cmml" xref="S4.SS2.SSS3.5.p2.3.m3.3.3"><in id="S4.SS2.SSS3.5.p2.3.m3.3.3.4.cmml" xref="S4.SS2.SSS3.5.p2.3.m3.3.3.4"></in><apply id="S4.SS2.SSS3.5.p2.3.m3.1.1.1.cmml" xref="S4.SS2.SSS3.5.p2.3.m3.1.1.1"><times id="S4.SS2.SSS3.5.p2.3.m3.1.1.1.2.cmml" xref="S4.SS2.SSS3.5.p2.3.m3.1.1.1.2"></times><ci 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xref="S4.SS2.SSS3.5.p2.3.m3.3.3.3.2.2.1.1.1.3">italic-ϵ</ci></apply><apply id="S4.SS2.SSS3.5.p2.3.m3.3.3.3.2.2.3.cmml" xref="S4.SS2.SSS3.5.p2.3.m3.3.3.3.2.2.3"><plus id="S4.SS2.SSS3.5.p2.3.m3.3.3.3.2.2.3.1.cmml" xref="S4.SS2.SSS3.5.p2.3.m3.3.3.3.2.2.3.1"></plus><ci id="S4.SS2.SSS3.5.p2.3.m3.3.3.3.2.2.3.2.cmml" xref="S4.SS2.SSS3.5.p2.3.m3.3.3.3.2.2.3.2">𝑗</ci><cn id="S4.SS2.SSS3.5.p2.3.m3.3.3.3.2.2.3.3.cmml" type="integer" xref="S4.SS2.SSS3.5.p2.3.m3.3.3.3.2.2.3.3">1</cn></apply></apply></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS3.5.p2.3.m3.3c">w(uv)\in[(1+\epsilon)^{j},(1+\epsilon)^{j+1})</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS3.5.p2.3.m3.3d">italic_w ( italic_u italic_v ) ∈ [ ( 1 + italic_ϵ ) start_POSTSUPERSCRIPT italic_j end_POSTSUPERSCRIPT , ( 1 + italic_ϵ ) start_POSTSUPERSCRIPT italic_j + 1 end_POSTSUPERSCRIPT )</annotation></semantics></math>. By Lemma <a class="ltx_ref" href="https://arxiv.org/html/2503.00712v1#S4.Thmtheorem17" title="Lemma 4.17. ‣ 4.2.1 SPQR Trees ‣ 4.2 Two-to-Three Augmentation ‣ 4 Vertex Connectivity Augmentation in Link-Arrival Model ‣ Streaming Algorithms for Network Design"><span class="ltx_text ltx_ref_tag">4.17</span></a>, there are three cases in which <math alttext="G\setminus\{a,b\}" class="ltx_Math" display="inline" id="S4.SS2.SSS3.5.p2.4.m4.2"><semantics id="S4.SS2.SSS3.5.p2.4.m4.2a"><mrow id="S4.SS2.SSS3.5.p2.4.m4.2.3" xref="S4.SS2.SSS3.5.p2.4.m4.2.3.cmml"><mi id="S4.SS2.SSS3.5.p2.4.m4.2.3.2" xref="S4.SS2.SSS3.5.p2.4.m4.2.3.2.cmml">G</mi><mo id="S4.SS2.SSS3.5.p2.4.m4.2.3.1" xref="S4.SS2.SSS3.5.p2.4.m4.2.3.1.cmml">∖</mo><mrow id="S4.SS2.SSS3.5.p2.4.m4.2.3.3.2" xref="S4.SS2.SSS3.5.p2.4.m4.2.3.3.1.cmml"><mo id="S4.SS2.SSS3.5.p2.4.m4.2.3.3.2.1" stretchy="false" xref="S4.SS2.SSS3.5.p2.4.m4.2.3.3.1.cmml">{</mo><mi id="S4.SS2.SSS3.5.p2.4.m4.1.1" xref="S4.SS2.SSS3.5.p2.4.m4.1.1.cmml">a</mi><mo id="S4.SS2.SSS3.5.p2.4.m4.2.3.3.2.2" xref="S4.SS2.SSS3.5.p2.4.m4.2.3.3.1.cmml">,</mo><mi id="S4.SS2.SSS3.5.p2.4.m4.2.2" xref="S4.SS2.SSS3.5.p2.4.m4.2.2.cmml">b</mi><mo id="S4.SS2.SSS3.5.p2.4.m4.2.3.3.2.3" stretchy="false" xref="S4.SS2.SSS3.5.p2.4.m4.2.3.3.1.cmml">}</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS3.5.p2.4.m4.2b"><apply id="S4.SS2.SSS3.5.p2.4.m4.2.3.cmml" xref="S4.SS2.SSS3.5.p2.4.m4.2.3"><setdiff id="S4.SS2.SSS3.5.p2.4.m4.2.3.1.cmml" xref="S4.SS2.SSS3.5.p2.4.m4.2.3.1"></setdiff><ci id="S4.SS2.SSS3.5.p2.4.m4.2.3.2.cmml" xref="S4.SS2.SSS3.5.p2.4.m4.2.3.2">𝐺</ci><set id="S4.SS2.SSS3.5.p2.4.m4.2.3.3.1.cmml" xref="S4.SS2.SSS3.5.p2.4.m4.2.3.3.2"><ci id="S4.SS2.SSS3.5.p2.4.m4.1.1.cmml" xref="S4.SS2.SSS3.5.p2.4.m4.1.1">𝑎</ci><ci id="S4.SS2.SSS3.5.p2.4.m4.2.2.cmml" xref="S4.SS2.SSS3.5.p2.4.m4.2.2">𝑏</ci></set></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS3.5.p2.4.m4.2c">G\setminus\{a,b\}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS3.5.p2.4.m4.2d">italic_G ∖ { italic_a , italic_b }</annotation></semantics></math> is disconnected: (1) <math alttext="ab" class="ltx_Math" display="inline" id="S4.SS2.SSS3.5.p2.5.m5.1"><semantics id="S4.SS2.SSS3.5.p2.5.m5.1a"><mrow id="S4.SS2.SSS3.5.p2.5.m5.1.1" xref="S4.SS2.SSS3.5.p2.5.m5.1.1.cmml"><mi id="S4.SS2.SSS3.5.p2.5.m5.1.1.2" xref="S4.SS2.SSS3.5.p2.5.m5.1.1.2.cmml">a</mi><mo id="S4.SS2.SSS3.5.p2.5.m5.1.1.1" xref="S4.SS2.SSS3.5.p2.5.m5.1.1.1.cmml">⁢</mo><mi id="S4.SS2.SSS3.5.p2.5.m5.1.1.3" xref="S4.SS2.SSS3.5.p2.5.m5.1.1.3.cmml">b</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS3.5.p2.5.m5.1b"><apply id="S4.SS2.SSS3.5.p2.5.m5.1.1.cmml" xref="S4.SS2.SSS3.5.p2.5.m5.1.1"><times id="S4.SS2.SSS3.5.p2.5.m5.1.1.1.cmml" xref="S4.SS2.SSS3.5.p2.5.m5.1.1.1"></times><ci id="S4.SS2.SSS3.5.p2.5.m5.1.1.2.cmml" xref="S4.SS2.SSS3.5.p2.5.m5.1.1.2">𝑎</ci><ci id="S4.SS2.SSS3.5.p2.5.m5.1.1.3.cmml" xref="S4.SS2.SSS3.5.p2.5.m5.1.1.3">𝑏</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS3.5.p2.5.m5.1c">ab</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS3.5.p2.5.m5.1d">italic_a italic_b</annotation></semantics></math> is a virtual edge incident to either two R-nodes or one R-node and one S-node, (2) <math alttext="\{a,b\}" class="ltx_Math" display="inline" id="S4.SS2.SSS3.5.p2.6.m6.2"><semantics id="S4.SS2.SSS3.5.p2.6.m6.2a"><mrow id="S4.SS2.SSS3.5.p2.6.m6.2.3.2" xref="S4.SS2.SSS3.5.p2.6.m6.2.3.1.cmml"><mo id="S4.SS2.SSS3.5.p2.6.m6.2.3.2.1" stretchy="false" xref="S4.SS2.SSS3.5.p2.6.m6.2.3.1.cmml">{</mo><mi id="S4.SS2.SSS3.5.p2.6.m6.1.1" xref="S4.SS2.SSS3.5.p2.6.m6.1.1.cmml">a</mi><mo id="S4.SS2.SSS3.5.p2.6.m6.2.3.2.2" xref="S4.SS2.SSS3.5.p2.6.m6.2.3.1.cmml">,</mo><mi id="S4.SS2.SSS3.5.p2.6.m6.2.2" xref="S4.SS2.SSS3.5.p2.6.m6.2.2.cmml">b</mi><mo id="S4.SS2.SSS3.5.p2.6.m6.2.3.2.3" stretchy="false" xref="S4.SS2.SSS3.5.p2.6.m6.2.3.1.cmml">}</mo></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS3.5.p2.6.m6.2b"><set id="S4.SS2.SSS3.5.p2.6.m6.2.3.1.cmml" xref="S4.SS2.SSS3.5.p2.6.m6.2.3.2"><ci id="S4.SS2.SSS3.5.p2.6.m6.1.1.cmml" xref="S4.SS2.SSS3.5.p2.6.m6.1.1">𝑎</ci><ci id="S4.SS2.SSS3.5.p2.6.m6.2.2.cmml" xref="S4.SS2.SSS3.5.p2.6.m6.2.2">𝑏</ci></set></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS3.5.p2.6.m6.2c">\{a,b\}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS3.5.p2.6.m6.2d">{ italic_a , italic_b }</annotation></semantics></math> is the vertex set associated with a P-node, and (3) <math alttext="a" class="ltx_Math" display="inline" id="S4.SS2.SSS3.5.p2.7.m7.1"><semantics id="S4.SS2.SSS3.5.p2.7.m7.1a"><mi id="S4.SS2.SSS3.5.p2.7.m7.1.1" xref="S4.SS2.SSS3.5.p2.7.m7.1.1.cmml">a</mi><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS3.5.p2.7.m7.1b"><ci id="S4.SS2.SSS3.5.p2.7.m7.1.1.cmml" xref="S4.SS2.SSS3.5.p2.7.m7.1.1">𝑎</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS3.5.p2.7.m7.1c">a</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS3.5.p2.7.m7.1d">italic_a</annotation></semantics></math> and <math alttext="b" class="ltx_Math" display="inline" id="S4.SS2.SSS3.5.p2.8.m8.1"><semantics id="S4.SS2.SSS3.5.p2.8.m8.1a"><mi id="S4.SS2.SSS3.5.p2.8.m8.1.1" xref="S4.SS2.SSS3.5.p2.8.m8.1.1.cmml">b</mi><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS3.5.p2.8.m8.1b"><ci id="S4.SS2.SSS3.5.p2.8.m8.1.1.cmml" xref="S4.SS2.SSS3.5.p2.8.m8.1.1">𝑏</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS3.5.p2.8.m8.1c">b</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS3.5.p2.8.m8.1d">italic_b</annotation></semantics></math> are non-adjacent nodes of an S-node. We consider each of these cases separately.</p> </div> </div> <section class="ltx_paragraph" id="S4.SS2.SSS3.Px1"> <h5 class="ltx_title ltx_title_paragraph">Case 1: <math alttext="\boldsymbol{e=ab}" class="ltx_Math" display="inline" id="S4.SS2.SSS3.Px1.1.m1.1"><semantics id="S4.SS2.SSS3.Px1.1.m1.1b"><mrow id="S4.SS2.SSS3.Px1.1.m1.1.1" xref="S4.SS2.SSS3.Px1.1.m1.1.1.cmml"><mi id="S4.SS2.SSS3.Px1.1.m1.1.1.2" xref="S4.SS2.SSS3.Px1.1.m1.1.1.2.cmml">𝒆</mi><mo class="ltx_mathvariant_bold" id="S4.SS2.SSS3.Px1.1.m1.1.1.1" mathvariant="bold" xref="S4.SS2.SSS3.Px1.1.m1.1.1.1.cmml">=</mo><mrow id="S4.SS2.SSS3.Px1.1.m1.1.1.3" xref="S4.SS2.SSS3.Px1.1.m1.1.1.3.cmml"><mi id="S4.SS2.SSS3.Px1.1.m1.1.1.3.2" xref="S4.SS2.SSS3.Px1.1.m1.1.1.3.2.cmml">𝒂</mi><mo id="S4.SS2.SSS3.Px1.1.m1.1.1.3.1" xref="S4.SS2.SSS3.Px1.1.m1.1.1.3.1.cmml">⁢</mo><mi id="S4.SS2.SSS3.Px1.1.m1.1.1.3.3" xref="S4.SS2.SSS3.Px1.1.m1.1.1.3.3.cmml">𝒃</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS3.Px1.1.m1.1c"><apply id="S4.SS2.SSS3.Px1.1.m1.1.1.cmml" xref="S4.SS2.SSS3.Px1.1.m1.1.1"><eq id="S4.SS2.SSS3.Px1.1.m1.1.1.1.cmml" xref="S4.SS2.SSS3.Px1.1.m1.1.1.1"></eq><ci id="S4.SS2.SSS3.Px1.1.m1.1.1.2.cmml" xref="S4.SS2.SSS3.Px1.1.m1.1.1.2">𝒆</ci><apply id="S4.SS2.SSS3.Px1.1.m1.1.1.3.cmml" xref="S4.SS2.SSS3.Px1.1.m1.1.1.3"><times id="S4.SS2.SSS3.Px1.1.m1.1.1.3.1.cmml" xref="S4.SS2.SSS3.Px1.1.m1.1.1.3.1"></times><ci id="S4.SS2.SSS3.Px1.1.m1.1.1.3.2.cmml" xref="S4.SS2.SSS3.Px1.1.m1.1.1.3.2">𝒂</ci><ci id="S4.SS2.SSS3.Px1.1.m1.1.1.3.3.cmml" xref="S4.SS2.SSS3.Px1.1.m1.1.1.3.3">𝒃</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS3.Px1.1.m1.1d">\boldsymbol{e=ab}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS3.Px1.1.m1.1e">bold_italic_e bold_= bold_italic_a bold_italic_b</annotation></semantics></math> is a virtual edge of an R-node and an R or S-node:</h5> <div class="ltx_para" id="S4.SS2.SSS3.Px1.p1"> <p class="ltx_p" id="S4.SS2.SSS3.Px1.p1.11">Let <math alttext="e^{\prime}=xy\in E(T)" class="ltx_Math" display="inline" id="S4.SS2.SSS3.Px1.p1.1.m1.1"><semantics id="S4.SS2.SSS3.Px1.p1.1.m1.1a"><mrow id="S4.SS2.SSS3.Px1.p1.1.m1.1.2" xref="S4.SS2.SSS3.Px1.p1.1.m1.1.2.cmml"><msup id="S4.SS2.SSS3.Px1.p1.1.m1.1.2.2" xref="S4.SS2.SSS3.Px1.p1.1.m1.1.2.2.cmml"><mi id="S4.SS2.SSS3.Px1.p1.1.m1.1.2.2.2" xref="S4.SS2.SSS3.Px1.p1.1.m1.1.2.2.2.cmml">e</mi><mo id="S4.SS2.SSS3.Px1.p1.1.m1.1.2.2.3" xref="S4.SS2.SSS3.Px1.p1.1.m1.1.2.2.3.cmml">′</mo></msup><mo id="S4.SS2.SSS3.Px1.p1.1.m1.1.2.3" xref="S4.SS2.SSS3.Px1.p1.1.m1.1.2.3.cmml">=</mo><mrow id="S4.SS2.SSS3.Px1.p1.1.m1.1.2.4" xref="S4.SS2.SSS3.Px1.p1.1.m1.1.2.4.cmml"><mi id="S4.SS2.SSS3.Px1.p1.1.m1.1.2.4.2" xref="S4.SS2.SSS3.Px1.p1.1.m1.1.2.4.2.cmml">x</mi><mo id="S4.SS2.SSS3.Px1.p1.1.m1.1.2.4.1" xref="S4.SS2.SSS3.Px1.p1.1.m1.1.2.4.1.cmml">⁢</mo><mi id="S4.SS2.SSS3.Px1.p1.1.m1.1.2.4.3" xref="S4.SS2.SSS3.Px1.p1.1.m1.1.2.4.3.cmml">y</mi></mrow><mo id="S4.SS2.SSS3.Px1.p1.1.m1.1.2.5" xref="S4.SS2.SSS3.Px1.p1.1.m1.1.2.5.cmml">∈</mo><mrow id="S4.SS2.SSS3.Px1.p1.1.m1.1.2.6" xref="S4.SS2.SSS3.Px1.p1.1.m1.1.2.6.cmml"><mi id="S4.SS2.SSS3.Px1.p1.1.m1.1.2.6.2" xref="S4.SS2.SSS3.Px1.p1.1.m1.1.2.6.2.cmml">E</mi><mo id="S4.SS2.SSS3.Px1.p1.1.m1.1.2.6.1" xref="S4.SS2.SSS3.Px1.p1.1.m1.1.2.6.1.cmml">⁢</mo><mrow id="S4.SS2.SSS3.Px1.p1.1.m1.1.2.6.3.2" xref="S4.SS2.SSS3.Px1.p1.1.m1.1.2.6.cmml"><mo id="S4.SS2.SSS3.Px1.p1.1.m1.1.2.6.3.2.1" stretchy="false" xref="S4.SS2.SSS3.Px1.p1.1.m1.1.2.6.cmml">(</mo><mi id="S4.SS2.SSS3.Px1.p1.1.m1.1.1" xref="S4.SS2.SSS3.Px1.p1.1.m1.1.1.cmml">T</mi><mo id="S4.SS2.SSS3.Px1.p1.1.m1.1.2.6.3.2.2" stretchy="false" xref="S4.SS2.SSS3.Px1.p1.1.m1.1.2.6.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS3.Px1.p1.1.m1.1b"><apply id="S4.SS2.SSS3.Px1.p1.1.m1.1.2.cmml" xref="S4.SS2.SSS3.Px1.p1.1.m1.1.2"><and id="S4.SS2.SSS3.Px1.p1.1.m1.1.2a.cmml" xref="S4.SS2.SSS3.Px1.p1.1.m1.1.2"></and><apply id="S4.SS2.SSS3.Px1.p1.1.m1.1.2b.cmml" xref="S4.SS2.SSS3.Px1.p1.1.m1.1.2"><eq id="S4.SS2.SSS3.Px1.p1.1.m1.1.2.3.cmml" xref="S4.SS2.SSS3.Px1.p1.1.m1.1.2.3"></eq><apply id="S4.SS2.SSS3.Px1.p1.1.m1.1.2.2.cmml" xref="S4.SS2.SSS3.Px1.p1.1.m1.1.2.2"><csymbol cd="ambiguous" id="S4.SS2.SSS3.Px1.p1.1.m1.1.2.2.1.cmml" xref="S4.SS2.SSS3.Px1.p1.1.m1.1.2.2">superscript</csymbol><ci id="S4.SS2.SSS3.Px1.p1.1.m1.1.2.2.2.cmml" xref="S4.SS2.SSS3.Px1.p1.1.m1.1.2.2.2">𝑒</ci><ci id="S4.SS2.SSS3.Px1.p1.1.m1.1.2.2.3.cmml" xref="S4.SS2.SSS3.Px1.p1.1.m1.1.2.2.3">′</ci></apply><apply id="S4.SS2.SSS3.Px1.p1.1.m1.1.2.4.cmml" xref="S4.SS2.SSS3.Px1.p1.1.m1.1.2.4"><times id="S4.SS2.SSS3.Px1.p1.1.m1.1.2.4.1.cmml" xref="S4.SS2.SSS3.Px1.p1.1.m1.1.2.4.1"></times><ci id="S4.SS2.SSS3.Px1.p1.1.m1.1.2.4.2.cmml" xref="S4.SS2.SSS3.Px1.p1.1.m1.1.2.4.2">𝑥</ci><ci id="S4.SS2.SSS3.Px1.p1.1.m1.1.2.4.3.cmml" xref="S4.SS2.SSS3.Px1.p1.1.m1.1.2.4.3">𝑦</ci></apply></apply><apply id="S4.SS2.SSS3.Px1.p1.1.m1.1.2c.cmml" xref="S4.SS2.SSS3.Px1.p1.1.m1.1.2"><in id="S4.SS2.SSS3.Px1.p1.1.m1.1.2.5.cmml" xref="S4.SS2.SSS3.Px1.p1.1.m1.1.2.5"></in><share href="https://arxiv.org/html/2503.00712v1#S4.SS2.SSS3.Px1.p1.1.m1.1.2.4.cmml" id="S4.SS2.SSS3.Px1.p1.1.m1.1.2d.cmml" xref="S4.SS2.SSS3.Px1.p1.1.m1.1.2"></share><apply id="S4.SS2.SSS3.Px1.p1.1.m1.1.2.6.cmml" xref="S4.SS2.SSS3.Px1.p1.1.m1.1.2.6"><times id="S4.SS2.SSS3.Px1.p1.1.m1.1.2.6.1.cmml" xref="S4.SS2.SSS3.Px1.p1.1.m1.1.2.6.1"></times><ci id="S4.SS2.SSS3.Px1.p1.1.m1.1.2.6.2.cmml" xref="S4.SS2.SSS3.Px1.p1.1.m1.1.2.6.2">𝐸</ci><ci id="S4.SS2.SSS3.Px1.p1.1.m1.1.1.cmml" xref="S4.SS2.SSS3.Px1.p1.1.m1.1.1">𝑇</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS3.Px1.p1.1.m1.1c">e^{\prime}=xy\in E(T)</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS3.Px1.p1.1.m1.1d">italic_e start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT = italic_x italic_y ∈ italic_E ( italic_T )</annotation></semantics></math> be the tree edge associated with virtual edge <math alttext="e" class="ltx_Math" display="inline" id="S4.SS2.SSS3.Px1.p1.2.m2.1"><semantics id="S4.SS2.SSS3.Px1.p1.2.m2.1a"><mi id="S4.SS2.SSS3.Px1.p1.2.m2.1.1" xref="S4.SS2.SSS3.Px1.p1.2.m2.1.1.cmml">e</mi><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS3.Px1.p1.2.m2.1b"><ci id="S4.SS2.SSS3.Px1.p1.2.m2.1.1.cmml" xref="S4.SS2.SSS3.Px1.p1.2.m2.1.1">𝑒</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS3.Px1.p1.2.m2.1c">e</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS3.Px1.p1.2.m2.1d">italic_e</annotation></semantics></math>. We let <math alttext="x" class="ltx_Math" display="inline" id="S4.SS2.SSS3.Px1.p1.3.m3.1"><semantics id="S4.SS2.SSS3.Px1.p1.3.m3.1a"><mi id="S4.SS2.SSS3.Px1.p1.3.m3.1.1" xref="S4.SS2.SSS3.Px1.p1.3.m3.1.1.cmml">x</mi><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS3.Px1.p1.3.m3.1b"><ci id="S4.SS2.SSS3.Px1.p1.3.m3.1.1.cmml" xref="S4.SS2.SSS3.Px1.p1.3.m3.1.1">𝑥</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS3.Px1.p1.3.m3.1c">x</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS3.Px1.p1.3.m3.1d">italic_x</annotation></semantics></math> denote the child node and <math alttext="y" class="ltx_Math" display="inline" id="S4.SS2.SSS3.Px1.p1.4.m4.1"><semantics id="S4.SS2.SSS3.Px1.p1.4.m4.1a"><mi id="S4.SS2.SSS3.Px1.p1.4.m4.1.1" xref="S4.SS2.SSS3.Px1.p1.4.m4.1.1.cmml">y</mi><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS3.Px1.p1.4.m4.1b"><ci id="S4.SS2.SSS3.Px1.p1.4.m4.1.1.cmml" xref="S4.SS2.SSS3.Px1.p1.4.m4.1.1">𝑦</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS3.Px1.p1.4.m4.1c">y</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS3.Px1.p1.4.m4.1d">italic_y</annotation></semantics></math> the parent; note that this implies that <math alttext="\{a,b\}=\textnormal{parent}(x)" class="ltx_Math" display="inline" id="S4.SS2.SSS3.Px1.p1.5.m5.3"><semantics id="S4.SS2.SSS3.Px1.p1.5.m5.3a"><mrow id="S4.SS2.SSS3.Px1.p1.5.m5.3.4" xref="S4.SS2.SSS3.Px1.p1.5.m5.3.4.cmml"><mrow id="S4.SS2.SSS3.Px1.p1.5.m5.3.4.2.2" xref="S4.SS2.SSS3.Px1.p1.5.m5.3.4.2.1.cmml"><mo id="S4.SS2.SSS3.Px1.p1.5.m5.3.4.2.2.1" stretchy="false" xref="S4.SS2.SSS3.Px1.p1.5.m5.3.4.2.1.cmml">{</mo><mi id="S4.SS2.SSS3.Px1.p1.5.m5.1.1" xref="S4.SS2.SSS3.Px1.p1.5.m5.1.1.cmml">a</mi><mo id="S4.SS2.SSS3.Px1.p1.5.m5.3.4.2.2.2" xref="S4.SS2.SSS3.Px1.p1.5.m5.3.4.2.1.cmml">,</mo><mi id="S4.SS2.SSS3.Px1.p1.5.m5.2.2" xref="S4.SS2.SSS3.Px1.p1.5.m5.2.2.cmml">b</mi><mo id="S4.SS2.SSS3.Px1.p1.5.m5.3.4.2.2.3" stretchy="false" xref="S4.SS2.SSS3.Px1.p1.5.m5.3.4.2.1.cmml">}</mo></mrow><mo id="S4.SS2.SSS3.Px1.p1.5.m5.3.4.1" xref="S4.SS2.SSS3.Px1.p1.5.m5.3.4.1.cmml">=</mo><mrow id="S4.SS2.SSS3.Px1.p1.5.m5.3.4.3" xref="S4.SS2.SSS3.Px1.p1.5.m5.3.4.3.cmml"><mtext id="S4.SS2.SSS3.Px1.p1.5.m5.3.4.3.2" xref="S4.SS2.SSS3.Px1.p1.5.m5.3.4.3.2a.cmml">parent</mtext><mo id="S4.SS2.SSS3.Px1.p1.5.m5.3.4.3.1" xref="S4.SS2.SSS3.Px1.p1.5.m5.3.4.3.1.cmml">⁢</mo><mrow id="S4.SS2.SSS3.Px1.p1.5.m5.3.4.3.3.2" xref="S4.SS2.SSS3.Px1.p1.5.m5.3.4.3.cmml"><mo id="S4.SS2.SSS3.Px1.p1.5.m5.3.4.3.3.2.1" stretchy="false" xref="S4.SS2.SSS3.Px1.p1.5.m5.3.4.3.cmml">(</mo><mi id="S4.SS2.SSS3.Px1.p1.5.m5.3.3" xref="S4.SS2.SSS3.Px1.p1.5.m5.3.3.cmml">x</mi><mo id="S4.SS2.SSS3.Px1.p1.5.m5.3.4.3.3.2.2" stretchy="false" xref="S4.SS2.SSS3.Px1.p1.5.m5.3.4.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS3.Px1.p1.5.m5.3b"><apply id="S4.SS2.SSS3.Px1.p1.5.m5.3.4.cmml" xref="S4.SS2.SSS3.Px1.p1.5.m5.3.4"><eq id="S4.SS2.SSS3.Px1.p1.5.m5.3.4.1.cmml" xref="S4.SS2.SSS3.Px1.p1.5.m5.3.4.1"></eq><set id="S4.SS2.SSS3.Px1.p1.5.m5.3.4.2.1.cmml" xref="S4.SS2.SSS3.Px1.p1.5.m5.3.4.2.2"><ci id="S4.SS2.SSS3.Px1.p1.5.m5.1.1.cmml" xref="S4.SS2.SSS3.Px1.p1.5.m5.1.1">𝑎</ci><ci id="S4.SS2.SSS3.Px1.p1.5.m5.2.2.cmml" xref="S4.SS2.SSS3.Px1.p1.5.m5.2.2">𝑏</ci></set><apply id="S4.SS2.SSS3.Px1.p1.5.m5.3.4.3.cmml" xref="S4.SS2.SSS3.Px1.p1.5.m5.3.4.3"><times id="S4.SS2.SSS3.Px1.p1.5.m5.3.4.3.1.cmml" xref="S4.SS2.SSS3.Px1.p1.5.m5.3.4.3.1"></times><ci id="S4.SS2.SSS3.Px1.p1.5.m5.3.4.3.2a.cmml" xref="S4.SS2.SSS3.Px1.p1.5.m5.3.4.3.2"><mtext id="S4.SS2.SSS3.Px1.p1.5.m5.3.4.3.2.cmml" xref="S4.SS2.SSS3.Px1.p1.5.m5.3.4.3.2">parent</mtext></ci><ci id="S4.SS2.SSS3.Px1.p1.5.m5.3.3.cmml" xref="S4.SS2.SSS3.Px1.p1.5.m5.3.3">𝑥</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS3.Px1.p1.5.m5.3c">\{a,b\}=\textnormal{parent}(x)</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS3.Px1.p1.5.m5.3d">{ italic_a , italic_b } = parent ( italic_x )</annotation></semantics></math>. Since <math alttext="x" class="ltx_Math" display="inline" id="S4.SS2.SSS3.Px1.p1.6.m6.1"><semantics id="S4.SS2.SSS3.Px1.p1.6.m6.1a"><mi id="S4.SS2.SSS3.Px1.p1.6.m6.1.1" xref="S4.SS2.SSS3.Px1.p1.6.m6.1.1.cmml">x</mi><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS3.Px1.p1.6.m6.1b"><ci id="S4.SS2.SSS3.Px1.p1.6.m6.1.1.cmml" xref="S4.SS2.SSS3.Px1.p1.6.m6.1.1">𝑥</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS3.Px1.p1.6.m6.1c">x</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS3.Px1.p1.6.m6.1d">italic_x</annotation></semantics></math> and <math alttext="y" class="ltx_Math" display="inline" id="S4.SS2.SSS3.Px1.p1.7.m7.1"><semantics id="S4.SS2.SSS3.Px1.p1.7.m7.1a"><mi id="S4.SS2.SSS3.Px1.p1.7.m7.1.1" xref="S4.SS2.SSS3.Px1.p1.7.m7.1.1.cmml">y</mi><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS3.Px1.p1.7.m7.1b"><ci id="S4.SS2.SSS3.Px1.p1.7.m7.1.1.cmml" xref="S4.SS2.SSS3.Px1.p1.7.m7.1.1">𝑦</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS3.Px1.p1.7.m7.1c">y</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS3.Px1.p1.7.m7.1d">italic_y</annotation></semantics></math> are both R or S nodes, <math alttext="G_{x}" class="ltx_Math" display="inline" id="S4.SS2.SSS3.Px1.p1.8.m8.1"><semantics id="S4.SS2.SSS3.Px1.p1.8.m8.1a"><msub id="S4.SS2.SSS3.Px1.p1.8.m8.1.1" xref="S4.SS2.SSS3.Px1.p1.8.m8.1.1.cmml"><mi id="S4.SS2.SSS3.Px1.p1.8.m8.1.1.2" xref="S4.SS2.SSS3.Px1.p1.8.m8.1.1.2.cmml">G</mi><mi id="S4.SS2.SSS3.Px1.p1.8.m8.1.1.3" xref="S4.SS2.SSS3.Px1.p1.8.m8.1.1.3.cmml">x</mi></msub><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS3.Px1.p1.8.m8.1b"><apply id="S4.SS2.SSS3.Px1.p1.8.m8.1.1.cmml" xref="S4.SS2.SSS3.Px1.p1.8.m8.1.1"><csymbol cd="ambiguous" id="S4.SS2.SSS3.Px1.p1.8.m8.1.1.1.cmml" xref="S4.SS2.SSS3.Px1.p1.8.m8.1.1">subscript</csymbol><ci id="S4.SS2.SSS3.Px1.p1.8.m8.1.1.2.cmml" xref="S4.SS2.SSS3.Px1.p1.8.m8.1.1.2">𝐺</ci><ci id="S4.SS2.SSS3.Px1.p1.8.m8.1.1.3.cmml" xref="S4.SS2.SSS3.Px1.p1.8.m8.1.1.3">𝑥</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS3.Px1.p1.8.m8.1c">G_{x}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS3.Px1.p1.8.m8.1d">italic_G start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="G_{y}" class="ltx_Math" display="inline" id="S4.SS2.SSS3.Px1.p1.9.m9.1"><semantics id="S4.SS2.SSS3.Px1.p1.9.m9.1a"><msub id="S4.SS2.SSS3.Px1.p1.9.m9.1.1" xref="S4.SS2.SSS3.Px1.p1.9.m9.1.1.cmml"><mi id="S4.SS2.SSS3.Px1.p1.9.m9.1.1.2" xref="S4.SS2.SSS3.Px1.p1.9.m9.1.1.2.cmml">G</mi><mi id="S4.SS2.SSS3.Px1.p1.9.m9.1.1.3" xref="S4.SS2.SSS3.Px1.p1.9.m9.1.1.3.cmml">y</mi></msub><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS3.Px1.p1.9.m9.1b"><apply id="S4.SS2.SSS3.Px1.p1.9.m9.1.1.cmml" xref="S4.SS2.SSS3.Px1.p1.9.m9.1.1"><csymbol cd="ambiguous" id="S4.SS2.SSS3.Px1.p1.9.m9.1.1.1.cmml" xref="S4.SS2.SSS3.Px1.p1.9.m9.1.1">subscript</csymbol><ci id="S4.SS2.SSS3.Px1.p1.9.m9.1.1.2.cmml" xref="S4.SS2.SSS3.Px1.p1.9.m9.1.1.2">𝐺</ci><ci id="S4.SS2.SSS3.Px1.p1.9.m9.1.1.3.cmml" xref="S4.SS2.SSS3.Px1.p1.9.m9.1.1.3">𝑦</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS3.Px1.p1.9.m9.1c">G_{y}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS3.Px1.p1.9.m9.1d">italic_G start_POSTSUBSCRIPT italic_y end_POSTSUBSCRIPT</annotation></semantics></math> each have at least 3 vertices. Furthermore, <math alttext="G_{x}\setminus\{a,b\}" class="ltx_Math" display="inline" id="S4.SS2.SSS3.Px1.p1.10.m10.2"><semantics id="S4.SS2.SSS3.Px1.p1.10.m10.2a"><mrow id="S4.SS2.SSS3.Px1.p1.10.m10.2.3" xref="S4.SS2.SSS3.Px1.p1.10.m10.2.3.cmml"><msub id="S4.SS2.SSS3.Px1.p1.10.m10.2.3.2" xref="S4.SS2.SSS3.Px1.p1.10.m10.2.3.2.cmml"><mi id="S4.SS2.SSS3.Px1.p1.10.m10.2.3.2.2" xref="S4.SS2.SSS3.Px1.p1.10.m10.2.3.2.2.cmml">G</mi><mi id="S4.SS2.SSS3.Px1.p1.10.m10.2.3.2.3" xref="S4.SS2.SSS3.Px1.p1.10.m10.2.3.2.3.cmml">x</mi></msub><mo id="S4.SS2.SSS3.Px1.p1.10.m10.2.3.1" xref="S4.SS2.SSS3.Px1.p1.10.m10.2.3.1.cmml">∖</mo><mrow id="S4.SS2.SSS3.Px1.p1.10.m10.2.3.3.2" xref="S4.SS2.SSS3.Px1.p1.10.m10.2.3.3.1.cmml"><mo id="S4.SS2.SSS3.Px1.p1.10.m10.2.3.3.2.1" stretchy="false" xref="S4.SS2.SSS3.Px1.p1.10.m10.2.3.3.1.cmml">{</mo><mi id="S4.SS2.SSS3.Px1.p1.10.m10.1.1" xref="S4.SS2.SSS3.Px1.p1.10.m10.1.1.cmml">a</mi><mo id="S4.SS2.SSS3.Px1.p1.10.m10.2.3.3.2.2" xref="S4.SS2.SSS3.Px1.p1.10.m10.2.3.3.1.cmml">,</mo><mi id="S4.SS2.SSS3.Px1.p1.10.m10.2.2" xref="S4.SS2.SSS3.Px1.p1.10.m10.2.2.cmml">b</mi><mo id="S4.SS2.SSS3.Px1.p1.10.m10.2.3.3.2.3" stretchy="false" xref="S4.SS2.SSS3.Px1.p1.10.m10.2.3.3.1.cmml">}</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS3.Px1.p1.10.m10.2b"><apply id="S4.SS2.SSS3.Px1.p1.10.m10.2.3.cmml" xref="S4.SS2.SSS3.Px1.p1.10.m10.2.3"><setdiff id="S4.SS2.SSS3.Px1.p1.10.m10.2.3.1.cmml" xref="S4.SS2.SSS3.Px1.p1.10.m10.2.3.1"></setdiff><apply id="S4.SS2.SSS3.Px1.p1.10.m10.2.3.2.cmml" xref="S4.SS2.SSS3.Px1.p1.10.m10.2.3.2"><csymbol cd="ambiguous" id="S4.SS2.SSS3.Px1.p1.10.m10.2.3.2.1.cmml" xref="S4.SS2.SSS3.Px1.p1.10.m10.2.3.2">subscript</csymbol><ci id="S4.SS2.SSS3.Px1.p1.10.m10.2.3.2.2.cmml" xref="S4.SS2.SSS3.Px1.p1.10.m10.2.3.2.2">𝐺</ci><ci id="S4.SS2.SSS3.Px1.p1.10.m10.2.3.2.3.cmml" xref="S4.SS2.SSS3.Px1.p1.10.m10.2.3.2.3">𝑥</ci></apply><set id="S4.SS2.SSS3.Px1.p1.10.m10.2.3.3.1.cmml" xref="S4.SS2.SSS3.Px1.p1.10.m10.2.3.3.2"><ci id="S4.SS2.SSS3.Px1.p1.10.m10.1.1.cmml" xref="S4.SS2.SSS3.Px1.p1.10.m10.1.1">𝑎</ci><ci id="S4.SS2.SSS3.Px1.p1.10.m10.2.2.cmml" xref="S4.SS2.SSS3.Px1.p1.10.m10.2.2">𝑏</ci></set></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS3.Px1.p1.10.m10.2c">G_{x}\setminus\{a,b\}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS3.Px1.p1.10.m10.2d">italic_G start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT ∖ { italic_a , italic_b }</annotation></semantics></math> and <math alttext="G_{y}\setminus\{a,b\}" class="ltx_Math" display="inline" id="S4.SS2.SSS3.Px1.p1.11.m11.2"><semantics id="S4.SS2.SSS3.Px1.p1.11.m11.2a"><mrow id="S4.SS2.SSS3.Px1.p1.11.m11.2.3" xref="S4.SS2.SSS3.Px1.p1.11.m11.2.3.cmml"><msub id="S4.SS2.SSS3.Px1.p1.11.m11.2.3.2" xref="S4.SS2.SSS3.Px1.p1.11.m11.2.3.2.cmml"><mi id="S4.SS2.SSS3.Px1.p1.11.m11.2.3.2.2" xref="S4.SS2.SSS3.Px1.p1.11.m11.2.3.2.2.cmml">G</mi><mi id="S4.SS2.SSS3.Px1.p1.11.m11.2.3.2.3" xref="S4.SS2.SSS3.Px1.p1.11.m11.2.3.2.3.cmml">y</mi></msub><mo id="S4.SS2.SSS3.Px1.p1.11.m11.2.3.1" xref="S4.SS2.SSS3.Px1.p1.11.m11.2.3.1.cmml">∖</mo><mrow id="S4.SS2.SSS3.Px1.p1.11.m11.2.3.3.2" xref="S4.SS2.SSS3.Px1.p1.11.m11.2.3.3.1.cmml"><mo id="S4.SS2.SSS3.Px1.p1.11.m11.2.3.3.2.1" stretchy="false" xref="S4.SS2.SSS3.Px1.p1.11.m11.2.3.3.1.cmml">{</mo><mi id="S4.SS2.SSS3.Px1.p1.11.m11.1.1" xref="S4.SS2.SSS3.Px1.p1.11.m11.1.1.cmml">a</mi><mo id="S4.SS2.SSS3.Px1.p1.11.m11.2.3.3.2.2" xref="S4.SS2.SSS3.Px1.p1.11.m11.2.3.3.1.cmml">,</mo><mi id="S4.SS2.SSS3.Px1.p1.11.m11.2.2" xref="S4.SS2.SSS3.Px1.p1.11.m11.2.2.cmml">b</mi><mo id="S4.SS2.SSS3.Px1.p1.11.m11.2.3.3.2.3" stretchy="false" xref="S4.SS2.SSS3.Px1.p1.11.m11.2.3.3.1.cmml">}</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS3.Px1.p1.11.m11.2b"><apply id="S4.SS2.SSS3.Px1.p1.11.m11.2.3.cmml" xref="S4.SS2.SSS3.Px1.p1.11.m11.2.3"><setdiff id="S4.SS2.SSS3.Px1.p1.11.m11.2.3.1.cmml" xref="S4.SS2.SSS3.Px1.p1.11.m11.2.3.1"></setdiff><apply id="S4.SS2.SSS3.Px1.p1.11.m11.2.3.2.cmml" xref="S4.SS2.SSS3.Px1.p1.11.m11.2.3.2"><csymbol cd="ambiguous" id="S4.SS2.SSS3.Px1.p1.11.m11.2.3.2.1.cmml" xref="S4.SS2.SSS3.Px1.p1.11.m11.2.3.2">subscript</csymbol><ci id="S4.SS2.SSS3.Px1.p1.11.m11.2.3.2.2.cmml" xref="S4.SS2.SSS3.Px1.p1.11.m11.2.3.2.2">𝐺</ci><ci id="S4.SS2.SSS3.Px1.p1.11.m11.2.3.2.3.cmml" xref="S4.SS2.SSS3.Px1.p1.11.m11.2.3.2.3">𝑦</ci></apply><set id="S4.SS2.SSS3.Px1.p1.11.m11.2.3.3.1.cmml" xref="S4.SS2.SSS3.Px1.p1.11.m11.2.3.3.2"><ci id="S4.SS2.SSS3.Px1.p1.11.m11.1.1.cmml" xref="S4.SS2.SSS3.Px1.p1.11.m11.1.1">𝑎</ci><ci id="S4.SS2.SSS3.Px1.p1.11.m11.2.2.cmml" xref="S4.SS2.SSS3.Px1.p1.11.m11.2.2">𝑏</ci></set></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS3.Px1.p1.11.m11.2c">G_{y}\setminus\{a,b\}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS3.Px1.p1.11.m11.2d">italic_G start_POSTSUBSCRIPT italic_y end_POSTSUBSCRIPT ∖ { italic_a , italic_b }</annotation></semantics></math> are both connected: the R-node case is clear by definition, and cycles remain connected after the deletion of two adjacent nodes.</p> </div> <div class="ltx_para" id="S4.SS2.SSS3.Px1.p2"> <p class="ltx_p" id="S4.SS2.SSS3.Px1.p2.20">First, suppose <math alttext="u,v" class="ltx_Math" display="inline" id="S4.SS2.SSS3.Px1.p2.1.m1.2"><semantics id="S4.SS2.SSS3.Px1.p2.1.m1.2a"><mrow id="S4.SS2.SSS3.Px1.p2.1.m1.2.3.2" xref="S4.SS2.SSS3.Px1.p2.1.m1.2.3.1.cmml"><mi id="S4.SS2.SSS3.Px1.p2.1.m1.1.1" xref="S4.SS2.SSS3.Px1.p2.1.m1.1.1.cmml">u</mi><mo id="S4.SS2.SSS3.Px1.p2.1.m1.2.3.2.1" xref="S4.SS2.SSS3.Px1.p2.1.m1.2.3.1.cmml">,</mo><mi id="S4.SS2.SSS3.Px1.p2.1.m1.2.2" xref="S4.SS2.SSS3.Px1.p2.1.m1.2.2.cmml">v</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS3.Px1.p2.1.m1.2b"><list id="S4.SS2.SSS3.Px1.p2.1.m1.2.3.1.cmml" xref="S4.SS2.SSS3.Px1.p2.1.m1.2.3.2"><ci id="S4.SS2.SSS3.Px1.p2.1.m1.1.1.cmml" xref="S4.SS2.SSS3.Px1.p2.1.m1.1.1">𝑢</ci><ci id="S4.SS2.SSS3.Px1.p2.1.m1.2.2.cmml" xref="S4.SS2.SSS3.Px1.p2.1.m1.2.2">𝑣</ci></list></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS3.Px1.p2.1.m1.2c">u,v</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS3.Px1.p2.1.m1.2d">italic_u , italic_v</annotation></semantics></math> are both in the set of vertices of <math alttext="G" class="ltx_Math" display="inline" id="S4.SS2.SSS3.Px1.p2.2.m2.1"><semantics id="S4.SS2.SSS3.Px1.p2.2.m2.1a"><mi id="S4.SS2.SSS3.Px1.p2.2.m2.1.1" xref="S4.SS2.SSS3.Px1.p2.2.m2.1.1.cmml">G</mi><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS3.Px1.p2.2.m2.1b"><ci id="S4.SS2.SSS3.Px1.p2.2.m2.1.1.cmml" xref="S4.SS2.SSS3.Px1.p2.2.m2.1.1">𝐺</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS3.Px1.p2.2.m2.1c">G</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS3.Px1.p2.2.m2.1d">italic_G</annotation></semantics></math> associated with <math alttext="T_{x}" class="ltx_Math" display="inline" id="S4.SS2.SSS3.Px1.p2.3.m3.1"><semantics id="S4.SS2.SSS3.Px1.p2.3.m3.1a"><msub id="S4.SS2.SSS3.Px1.p2.3.m3.1.1" xref="S4.SS2.SSS3.Px1.p2.3.m3.1.1.cmml"><mi id="S4.SS2.SSS3.Px1.p2.3.m3.1.1.2" xref="S4.SS2.SSS3.Px1.p2.3.m3.1.1.2.cmml">T</mi><mi id="S4.SS2.SSS3.Px1.p2.3.m3.1.1.3" xref="S4.SS2.SSS3.Px1.p2.3.m3.1.1.3.cmml">x</mi></msub><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS3.Px1.p2.3.m3.1b"><apply id="S4.SS2.SSS3.Px1.p2.3.m3.1.1.cmml" xref="S4.SS2.SSS3.Px1.p2.3.m3.1.1"><csymbol cd="ambiguous" id="S4.SS2.SSS3.Px1.p2.3.m3.1.1.1.cmml" xref="S4.SS2.SSS3.Px1.p2.3.m3.1.1">subscript</csymbol><ci id="S4.SS2.SSS3.Px1.p2.3.m3.1.1.2.cmml" xref="S4.SS2.SSS3.Px1.p2.3.m3.1.1.2">𝑇</ci><ci id="S4.SS2.SSS3.Px1.p2.3.m3.1.1.3.cmml" xref="S4.SS2.SSS3.Px1.p2.3.m3.1.1.3">𝑥</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS3.Px1.p2.3.m3.1c">T_{x}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS3.Px1.p2.3.m3.1d">italic_T start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT</annotation></semantics></math>. Then, since <math alttext="T_{x}" class="ltx_Math" display="inline" id="S4.SS2.SSS3.Px1.p2.4.m4.1"><semantics id="S4.SS2.SSS3.Px1.p2.4.m4.1a"><msub id="S4.SS2.SSS3.Px1.p2.4.m4.1.1" xref="S4.SS2.SSS3.Px1.p2.4.m4.1.1.cmml"><mi id="S4.SS2.SSS3.Px1.p2.4.m4.1.1.2" xref="S4.SS2.SSS3.Px1.p2.4.m4.1.1.2.cmml">T</mi><mi id="S4.SS2.SSS3.Px1.p2.4.m4.1.1.3" xref="S4.SS2.SSS3.Px1.p2.4.m4.1.1.3.cmml">x</mi></msub><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS3.Px1.p2.4.m4.1b"><apply id="S4.SS2.SSS3.Px1.p2.4.m4.1.1.cmml" xref="S4.SS2.SSS3.Px1.p2.4.m4.1.1"><csymbol cd="ambiguous" id="S4.SS2.SSS3.Px1.p2.4.m4.1.1.1.cmml" xref="S4.SS2.SSS3.Px1.p2.4.m4.1.1">subscript</csymbol><ci id="S4.SS2.SSS3.Px1.p2.4.m4.1.1.2.cmml" xref="S4.SS2.SSS3.Px1.p2.4.m4.1.1.2">𝑇</ci><ci id="S4.SS2.SSS3.Px1.p2.4.m4.1.1.3.cmml" xref="S4.SS2.SSS3.Px1.p2.4.m4.1.1.3">𝑥</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS3.Px1.p2.4.m4.1c">T_{x}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS3.Px1.p2.4.m4.1d">italic_T start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT</annotation></semantics></math> remains connected despite the deletion of <math alttext="e^{\prime}" class="ltx_Math" display="inline" id="S4.SS2.SSS3.Px1.p2.5.m5.1"><semantics id="S4.SS2.SSS3.Px1.p2.5.m5.1a"><msup id="S4.SS2.SSS3.Px1.p2.5.m5.1.1" xref="S4.SS2.SSS3.Px1.p2.5.m5.1.1.cmml"><mi id="S4.SS2.SSS3.Px1.p2.5.m5.1.1.2" xref="S4.SS2.SSS3.Px1.p2.5.m5.1.1.2.cmml">e</mi><mo id="S4.SS2.SSS3.Px1.p2.5.m5.1.1.3" xref="S4.SS2.SSS3.Px1.p2.5.m5.1.1.3.cmml">′</mo></msup><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS3.Px1.p2.5.m5.1b"><apply id="S4.SS2.SSS3.Px1.p2.5.m5.1.1.cmml" xref="S4.SS2.SSS3.Px1.p2.5.m5.1.1"><csymbol cd="ambiguous" id="S4.SS2.SSS3.Px1.p2.5.m5.1.1.1.cmml" xref="S4.SS2.SSS3.Px1.p2.5.m5.1.1">superscript</csymbol><ci id="S4.SS2.SSS3.Px1.p2.5.m5.1.1.2.cmml" xref="S4.SS2.SSS3.Px1.p2.5.m5.1.1.2">𝑒</ci><ci id="S4.SS2.SSS3.Px1.p2.5.m5.1.1.3.cmml" xref="S4.SS2.SSS3.Px1.p2.5.m5.1.1.3">′</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS3.Px1.p2.5.m5.1c">e^{\prime}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS3.Px1.p2.5.m5.1d">italic_e start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math> and <math alttext="G_{x}\setminus\{a,b\}" class="ltx_Math" display="inline" id="S4.SS2.SSS3.Px1.p2.6.m6.2"><semantics id="S4.SS2.SSS3.Px1.p2.6.m6.2a"><mrow id="S4.SS2.SSS3.Px1.p2.6.m6.2.3" xref="S4.SS2.SSS3.Px1.p2.6.m6.2.3.cmml"><msub id="S4.SS2.SSS3.Px1.p2.6.m6.2.3.2" xref="S4.SS2.SSS3.Px1.p2.6.m6.2.3.2.cmml"><mi id="S4.SS2.SSS3.Px1.p2.6.m6.2.3.2.2" xref="S4.SS2.SSS3.Px1.p2.6.m6.2.3.2.2.cmml">G</mi><mi id="S4.SS2.SSS3.Px1.p2.6.m6.2.3.2.3" xref="S4.SS2.SSS3.Px1.p2.6.m6.2.3.2.3.cmml">x</mi></msub><mo id="S4.SS2.SSS3.Px1.p2.6.m6.2.3.1" xref="S4.SS2.SSS3.Px1.p2.6.m6.2.3.1.cmml">∖</mo><mrow id="S4.SS2.SSS3.Px1.p2.6.m6.2.3.3.2" xref="S4.SS2.SSS3.Px1.p2.6.m6.2.3.3.1.cmml"><mo id="S4.SS2.SSS3.Px1.p2.6.m6.2.3.3.2.1" stretchy="false" xref="S4.SS2.SSS3.Px1.p2.6.m6.2.3.3.1.cmml">{</mo><mi id="S4.SS2.SSS3.Px1.p2.6.m6.1.1" xref="S4.SS2.SSS3.Px1.p2.6.m6.1.1.cmml">a</mi><mo id="S4.SS2.SSS3.Px1.p2.6.m6.2.3.3.2.2" xref="S4.SS2.SSS3.Px1.p2.6.m6.2.3.3.1.cmml">,</mo><mi id="S4.SS2.SSS3.Px1.p2.6.m6.2.2" xref="S4.SS2.SSS3.Px1.p2.6.m6.2.2.cmml">b</mi><mo id="S4.SS2.SSS3.Px1.p2.6.m6.2.3.3.2.3" stretchy="false" xref="S4.SS2.SSS3.Px1.p2.6.m6.2.3.3.1.cmml">}</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS3.Px1.p2.6.m6.2b"><apply id="S4.SS2.SSS3.Px1.p2.6.m6.2.3.cmml" xref="S4.SS2.SSS3.Px1.p2.6.m6.2.3"><setdiff id="S4.SS2.SSS3.Px1.p2.6.m6.2.3.1.cmml" xref="S4.SS2.SSS3.Px1.p2.6.m6.2.3.1"></setdiff><apply id="S4.SS2.SSS3.Px1.p2.6.m6.2.3.2.cmml" xref="S4.SS2.SSS3.Px1.p2.6.m6.2.3.2"><csymbol cd="ambiguous" id="S4.SS2.SSS3.Px1.p2.6.m6.2.3.2.1.cmml" xref="S4.SS2.SSS3.Px1.p2.6.m6.2.3.2">subscript</csymbol><ci id="S4.SS2.SSS3.Px1.p2.6.m6.2.3.2.2.cmml" xref="S4.SS2.SSS3.Px1.p2.6.m6.2.3.2.2">𝐺</ci><ci id="S4.SS2.SSS3.Px1.p2.6.m6.2.3.2.3.cmml" xref="S4.SS2.SSS3.Px1.p2.6.m6.2.3.2.3">𝑥</ci></apply><set id="S4.SS2.SSS3.Px1.p2.6.m6.2.3.3.1.cmml" xref="S4.SS2.SSS3.Px1.p2.6.m6.2.3.3.2"><ci id="S4.SS2.SSS3.Px1.p2.6.m6.1.1.cmml" xref="S4.SS2.SSS3.Px1.p2.6.m6.1.1">𝑎</ci><ci id="S4.SS2.SSS3.Px1.p2.6.m6.2.2.cmml" xref="S4.SS2.SSS3.Px1.p2.6.m6.2.2">𝑏</ci></set></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS3.Px1.p2.6.m6.2c">G_{x}\setminus\{a,b\}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS3.Px1.p2.6.m6.2d">italic_G start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT ∖ { italic_a , italic_b }</annotation></semantics></math> is connected, there exists a <math alttext="u" class="ltx_Math" display="inline" id="S4.SS2.SSS3.Px1.p2.7.m7.1"><semantics id="S4.SS2.SSS3.Px1.p2.7.m7.1a"><mi id="S4.SS2.SSS3.Px1.p2.7.m7.1.1" xref="S4.SS2.SSS3.Px1.p2.7.m7.1.1.cmml">u</mi><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS3.Px1.p2.7.m7.1b"><ci id="S4.SS2.SSS3.Px1.p2.7.m7.1.1.cmml" xref="S4.SS2.SSS3.Px1.p2.7.m7.1.1">𝑢</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS3.Px1.p2.7.m7.1c">u</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS3.Px1.p2.7.m7.1d">italic_u</annotation></semantics></math>-<math alttext="v" class="ltx_Math" display="inline" id="S4.SS2.SSS3.Px1.p2.8.m8.1"><semantics id="S4.SS2.SSS3.Px1.p2.8.m8.1a"><mi id="S4.SS2.SSS3.Px1.p2.8.m8.1.1" xref="S4.SS2.SSS3.Px1.p2.8.m8.1.1.cmml">v</mi><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS3.Px1.p2.8.m8.1b"><ci id="S4.SS2.SSS3.Px1.p2.8.m8.1.1.cmml" xref="S4.SS2.SSS3.Px1.p2.8.m8.1.1">𝑣</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS3.Px1.p2.8.m8.1c">v</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS3.Px1.p2.8.m8.1d">italic_v</annotation></semantics></math> path in <math alttext="E" class="ltx_Math" display="inline" id="S4.SS2.SSS3.Px1.p2.9.m9.1"><semantics id="S4.SS2.SSS3.Px1.p2.9.m9.1a"><mi id="S4.SS2.SSS3.Px1.p2.9.m9.1.1" xref="S4.SS2.SSS3.Px1.p2.9.m9.1.1.cmml">E</mi><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS3.Px1.p2.9.m9.1b"><ci id="S4.SS2.SSS3.Px1.p2.9.m9.1.1.cmml" xref="S4.SS2.SSS3.Px1.p2.9.m9.1.1">𝐸</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS3.Px1.p2.9.m9.1c">E</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS3.Px1.p2.9.m9.1d">italic_E</annotation></semantics></math> avoiding <math alttext="\{a,b\}" class="ltx_Math" display="inline" id="S4.SS2.SSS3.Px1.p2.10.m10.2"><semantics id="S4.SS2.SSS3.Px1.p2.10.m10.2a"><mrow id="S4.SS2.SSS3.Px1.p2.10.m10.2.3.2" xref="S4.SS2.SSS3.Px1.p2.10.m10.2.3.1.cmml"><mo id="S4.SS2.SSS3.Px1.p2.10.m10.2.3.2.1" stretchy="false" xref="S4.SS2.SSS3.Px1.p2.10.m10.2.3.1.cmml">{</mo><mi id="S4.SS2.SSS3.Px1.p2.10.m10.1.1" xref="S4.SS2.SSS3.Px1.p2.10.m10.1.1.cmml">a</mi><mo id="S4.SS2.SSS3.Px1.p2.10.m10.2.3.2.2" xref="S4.SS2.SSS3.Px1.p2.10.m10.2.3.1.cmml">,</mo><mi id="S4.SS2.SSS3.Px1.p2.10.m10.2.2" xref="S4.SS2.SSS3.Px1.p2.10.m10.2.2.cmml">b</mi><mo id="S4.SS2.SSS3.Px1.p2.10.m10.2.3.2.3" stretchy="false" xref="S4.SS2.SSS3.Px1.p2.10.m10.2.3.1.cmml">}</mo></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS3.Px1.p2.10.m10.2b"><set id="S4.SS2.SSS3.Px1.p2.10.m10.2.3.1.cmml" xref="S4.SS2.SSS3.Px1.p2.10.m10.2.3.2"><ci id="S4.SS2.SSS3.Px1.p2.10.m10.1.1.cmml" xref="S4.SS2.SSS3.Px1.p2.10.m10.1.1">𝑎</ci><ci id="S4.SS2.SSS3.Px1.p2.10.m10.2.2.cmml" xref="S4.SS2.SSS3.Px1.p2.10.m10.2.2">𝑏</ci></set></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS3.Px1.p2.10.m10.2c">\{a,b\}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS3.Px1.p2.10.m10.2d">{ italic_a , italic_b }</annotation></semantics></math>. Similarly, if <math alttext="u,v" class="ltx_Math" display="inline" id="S4.SS2.SSS3.Px1.p2.11.m11.2"><semantics id="S4.SS2.SSS3.Px1.p2.11.m11.2a"><mrow id="S4.SS2.SSS3.Px1.p2.11.m11.2.3.2" xref="S4.SS2.SSS3.Px1.p2.11.m11.2.3.1.cmml"><mi id="S4.SS2.SSS3.Px1.p2.11.m11.1.1" xref="S4.SS2.SSS3.Px1.p2.11.m11.1.1.cmml">u</mi><mo id="S4.SS2.SSS3.Px1.p2.11.m11.2.3.2.1" xref="S4.SS2.SSS3.Px1.p2.11.m11.2.3.1.cmml">,</mo><mi id="S4.SS2.SSS3.Px1.p2.11.m11.2.2" xref="S4.SS2.SSS3.Px1.p2.11.m11.2.2.cmml">v</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS3.Px1.p2.11.m11.2b"><list id="S4.SS2.SSS3.Px1.p2.11.m11.2.3.1.cmml" xref="S4.SS2.SSS3.Px1.p2.11.m11.2.3.2"><ci id="S4.SS2.SSS3.Px1.p2.11.m11.1.1.cmml" xref="S4.SS2.SSS3.Px1.p2.11.m11.1.1">𝑢</ci><ci id="S4.SS2.SSS3.Px1.p2.11.m11.2.2.cmml" xref="S4.SS2.SSS3.Px1.p2.11.m11.2.2">𝑣</ci></list></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS3.Px1.p2.11.m11.2c">u,v</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS3.Px1.p2.11.m11.2d">italic_u , italic_v</annotation></semantics></math> are both in the set of vertices of <math alttext="G" class="ltx_Math" display="inline" id="S4.SS2.SSS3.Px1.p2.12.m12.1"><semantics id="S4.SS2.SSS3.Px1.p2.12.m12.1a"><mi id="S4.SS2.SSS3.Px1.p2.12.m12.1.1" xref="S4.SS2.SSS3.Px1.p2.12.m12.1.1.cmml">G</mi><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS3.Px1.p2.12.m12.1b"><ci id="S4.SS2.SSS3.Px1.p2.12.m12.1.1.cmml" xref="S4.SS2.SSS3.Px1.p2.12.m12.1.1">𝐺</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS3.Px1.p2.12.m12.1c">G</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS3.Px1.p2.12.m12.1d">italic_G</annotation></semantics></math> associated with <math alttext="T\setminus T_{x}" class="ltx_Math" display="inline" id="S4.SS2.SSS3.Px1.p2.13.m13.1"><semantics id="S4.SS2.SSS3.Px1.p2.13.m13.1a"><mrow id="S4.SS2.SSS3.Px1.p2.13.m13.1.1" xref="S4.SS2.SSS3.Px1.p2.13.m13.1.1.cmml"><mi id="S4.SS2.SSS3.Px1.p2.13.m13.1.1.2" xref="S4.SS2.SSS3.Px1.p2.13.m13.1.1.2.cmml">T</mi><mo id="S4.SS2.SSS3.Px1.p2.13.m13.1.1.1" xref="S4.SS2.SSS3.Px1.p2.13.m13.1.1.1.cmml">∖</mo><msub id="S4.SS2.SSS3.Px1.p2.13.m13.1.1.3" xref="S4.SS2.SSS3.Px1.p2.13.m13.1.1.3.cmml"><mi id="S4.SS2.SSS3.Px1.p2.13.m13.1.1.3.2" xref="S4.SS2.SSS3.Px1.p2.13.m13.1.1.3.2.cmml">T</mi><mi id="S4.SS2.SSS3.Px1.p2.13.m13.1.1.3.3" xref="S4.SS2.SSS3.Px1.p2.13.m13.1.1.3.3.cmml">x</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS3.Px1.p2.13.m13.1b"><apply id="S4.SS2.SSS3.Px1.p2.13.m13.1.1.cmml" xref="S4.SS2.SSS3.Px1.p2.13.m13.1.1"><setdiff id="S4.SS2.SSS3.Px1.p2.13.m13.1.1.1.cmml" xref="S4.SS2.SSS3.Px1.p2.13.m13.1.1.1"></setdiff><ci id="S4.SS2.SSS3.Px1.p2.13.m13.1.1.2.cmml" xref="S4.SS2.SSS3.Px1.p2.13.m13.1.1.2">𝑇</ci><apply id="S4.SS2.SSS3.Px1.p2.13.m13.1.1.3.cmml" xref="S4.SS2.SSS3.Px1.p2.13.m13.1.1.3"><csymbol cd="ambiguous" id="S4.SS2.SSS3.Px1.p2.13.m13.1.1.3.1.cmml" xref="S4.SS2.SSS3.Px1.p2.13.m13.1.1.3">subscript</csymbol><ci id="S4.SS2.SSS3.Px1.p2.13.m13.1.1.3.2.cmml" xref="S4.SS2.SSS3.Px1.p2.13.m13.1.1.3.2">𝑇</ci><ci id="S4.SS2.SSS3.Px1.p2.13.m13.1.1.3.3.cmml" xref="S4.SS2.SSS3.Px1.p2.13.m13.1.1.3.3">𝑥</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS3.Px1.p2.13.m13.1c">T\setminus T_{x}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS3.Px1.p2.13.m13.1d">italic_T ∖ italic_T start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT</annotation></semantics></math>, then since <math alttext="T\setminus T_{x}" class="ltx_Math" display="inline" id="S4.SS2.SSS3.Px1.p2.14.m14.1"><semantics id="S4.SS2.SSS3.Px1.p2.14.m14.1a"><mrow id="S4.SS2.SSS3.Px1.p2.14.m14.1.1" xref="S4.SS2.SSS3.Px1.p2.14.m14.1.1.cmml"><mi id="S4.SS2.SSS3.Px1.p2.14.m14.1.1.2" xref="S4.SS2.SSS3.Px1.p2.14.m14.1.1.2.cmml">T</mi><mo id="S4.SS2.SSS3.Px1.p2.14.m14.1.1.1" xref="S4.SS2.SSS3.Px1.p2.14.m14.1.1.1.cmml">∖</mo><msub id="S4.SS2.SSS3.Px1.p2.14.m14.1.1.3" xref="S4.SS2.SSS3.Px1.p2.14.m14.1.1.3.cmml"><mi id="S4.SS2.SSS3.Px1.p2.14.m14.1.1.3.2" xref="S4.SS2.SSS3.Px1.p2.14.m14.1.1.3.2.cmml">T</mi><mi id="S4.SS2.SSS3.Px1.p2.14.m14.1.1.3.3" xref="S4.SS2.SSS3.Px1.p2.14.m14.1.1.3.3.cmml">x</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS3.Px1.p2.14.m14.1b"><apply id="S4.SS2.SSS3.Px1.p2.14.m14.1.1.cmml" xref="S4.SS2.SSS3.Px1.p2.14.m14.1.1"><setdiff id="S4.SS2.SSS3.Px1.p2.14.m14.1.1.1.cmml" xref="S4.SS2.SSS3.Px1.p2.14.m14.1.1.1"></setdiff><ci id="S4.SS2.SSS3.Px1.p2.14.m14.1.1.2.cmml" xref="S4.SS2.SSS3.Px1.p2.14.m14.1.1.2">𝑇</ci><apply id="S4.SS2.SSS3.Px1.p2.14.m14.1.1.3.cmml" xref="S4.SS2.SSS3.Px1.p2.14.m14.1.1.3"><csymbol cd="ambiguous" id="S4.SS2.SSS3.Px1.p2.14.m14.1.1.3.1.cmml" xref="S4.SS2.SSS3.Px1.p2.14.m14.1.1.3">subscript</csymbol><ci id="S4.SS2.SSS3.Px1.p2.14.m14.1.1.3.2.cmml" xref="S4.SS2.SSS3.Px1.p2.14.m14.1.1.3.2">𝑇</ci><ci id="S4.SS2.SSS3.Px1.p2.14.m14.1.1.3.3.cmml" xref="S4.SS2.SSS3.Px1.p2.14.m14.1.1.3.3">𝑥</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS3.Px1.p2.14.m14.1c">T\setminus T_{x}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS3.Px1.p2.14.m14.1d">italic_T ∖ italic_T start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT</annotation></semantics></math> remains connected despite the deletion of <math alttext="e^{\prime}" class="ltx_Math" display="inline" id="S4.SS2.SSS3.Px1.p2.15.m15.1"><semantics id="S4.SS2.SSS3.Px1.p2.15.m15.1a"><msup id="S4.SS2.SSS3.Px1.p2.15.m15.1.1" xref="S4.SS2.SSS3.Px1.p2.15.m15.1.1.cmml"><mi id="S4.SS2.SSS3.Px1.p2.15.m15.1.1.2" xref="S4.SS2.SSS3.Px1.p2.15.m15.1.1.2.cmml">e</mi><mo id="S4.SS2.SSS3.Px1.p2.15.m15.1.1.3" xref="S4.SS2.SSS3.Px1.p2.15.m15.1.1.3.cmml">′</mo></msup><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS3.Px1.p2.15.m15.1b"><apply id="S4.SS2.SSS3.Px1.p2.15.m15.1.1.cmml" xref="S4.SS2.SSS3.Px1.p2.15.m15.1.1"><csymbol cd="ambiguous" id="S4.SS2.SSS3.Px1.p2.15.m15.1.1.1.cmml" xref="S4.SS2.SSS3.Px1.p2.15.m15.1.1">superscript</csymbol><ci id="S4.SS2.SSS3.Px1.p2.15.m15.1.1.2.cmml" xref="S4.SS2.SSS3.Px1.p2.15.m15.1.1.2">𝑒</ci><ci id="S4.SS2.SSS3.Px1.p2.15.m15.1.1.3.cmml" xref="S4.SS2.SSS3.Px1.p2.15.m15.1.1.3">′</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS3.Px1.p2.15.m15.1c">e^{\prime}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS3.Px1.p2.15.m15.1d">italic_e start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math> and <math alttext="G_{y}\setminus\{a,b\}" class="ltx_Math" display="inline" id="S4.SS2.SSS3.Px1.p2.16.m16.2"><semantics id="S4.SS2.SSS3.Px1.p2.16.m16.2a"><mrow id="S4.SS2.SSS3.Px1.p2.16.m16.2.3" xref="S4.SS2.SSS3.Px1.p2.16.m16.2.3.cmml"><msub id="S4.SS2.SSS3.Px1.p2.16.m16.2.3.2" xref="S4.SS2.SSS3.Px1.p2.16.m16.2.3.2.cmml"><mi id="S4.SS2.SSS3.Px1.p2.16.m16.2.3.2.2" xref="S4.SS2.SSS3.Px1.p2.16.m16.2.3.2.2.cmml">G</mi><mi id="S4.SS2.SSS3.Px1.p2.16.m16.2.3.2.3" xref="S4.SS2.SSS3.Px1.p2.16.m16.2.3.2.3.cmml">y</mi></msub><mo id="S4.SS2.SSS3.Px1.p2.16.m16.2.3.1" xref="S4.SS2.SSS3.Px1.p2.16.m16.2.3.1.cmml">∖</mo><mrow id="S4.SS2.SSS3.Px1.p2.16.m16.2.3.3.2" xref="S4.SS2.SSS3.Px1.p2.16.m16.2.3.3.1.cmml"><mo id="S4.SS2.SSS3.Px1.p2.16.m16.2.3.3.2.1" stretchy="false" xref="S4.SS2.SSS3.Px1.p2.16.m16.2.3.3.1.cmml">{</mo><mi id="S4.SS2.SSS3.Px1.p2.16.m16.1.1" xref="S4.SS2.SSS3.Px1.p2.16.m16.1.1.cmml">a</mi><mo id="S4.SS2.SSS3.Px1.p2.16.m16.2.3.3.2.2" xref="S4.SS2.SSS3.Px1.p2.16.m16.2.3.3.1.cmml">,</mo><mi id="S4.SS2.SSS3.Px1.p2.16.m16.2.2" xref="S4.SS2.SSS3.Px1.p2.16.m16.2.2.cmml">b</mi><mo id="S4.SS2.SSS3.Px1.p2.16.m16.2.3.3.2.3" stretchy="false" xref="S4.SS2.SSS3.Px1.p2.16.m16.2.3.3.1.cmml">}</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS3.Px1.p2.16.m16.2b"><apply id="S4.SS2.SSS3.Px1.p2.16.m16.2.3.cmml" xref="S4.SS2.SSS3.Px1.p2.16.m16.2.3"><setdiff id="S4.SS2.SSS3.Px1.p2.16.m16.2.3.1.cmml" xref="S4.SS2.SSS3.Px1.p2.16.m16.2.3.1"></setdiff><apply id="S4.SS2.SSS3.Px1.p2.16.m16.2.3.2.cmml" xref="S4.SS2.SSS3.Px1.p2.16.m16.2.3.2"><csymbol cd="ambiguous" id="S4.SS2.SSS3.Px1.p2.16.m16.2.3.2.1.cmml" xref="S4.SS2.SSS3.Px1.p2.16.m16.2.3.2">subscript</csymbol><ci id="S4.SS2.SSS3.Px1.p2.16.m16.2.3.2.2.cmml" xref="S4.SS2.SSS3.Px1.p2.16.m16.2.3.2.2">𝐺</ci><ci id="S4.SS2.SSS3.Px1.p2.16.m16.2.3.2.3.cmml" xref="S4.SS2.SSS3.Px1.p2.16.m16.2.3.2.3">𝑦</ci></apply><set id="S4.SS2.SSS3.Px1.p2.16.m16.2.3.3.1.cmml" xref="S4.SS2.SSS3.Px1.p2.16.m16.2.3.3.2"><ci id="S4.SS2.SSS3.Px1.p2.16.m16.1.1.cmml" xref="S4.SS2.SSS3.Px1.p2.16.m16.1.1">𝑎</ci><ci id="S4.SS2.SSS3.Px1.p2.16.m16.2.2.cmml" xref="S4.SS2.SSS3.Px1.p2.16.m16.2.2">𝑏</ci></set></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS3.Px1.p2.16.m16.2c">G_{y}\setminus\{a,b\}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS3.Px1.p2.16.m16.2d">italic_G start_POSTSUBSCRIPT italic_y end_POSTSUBSCRIPT ∖ { italic_a , italic_b }</annotation></semantics></math> is connected, there exists a <math alttext="u" class="ltx_Math" display="inline" id="S4.SS2.SSS3.Px1.p2.17.m17.1"><semantics id="S4.SS2.SSS3.Px1.p2.17.m17.1a"><mi id="S4.SS2.SSS3.Px1.p2.17.m17.1.1" xref="S4.SS2.SSS3.Px1.p2.17.m17.1.1.cmml">u</mi><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS3.Px1.p2.17.m17.1b"><ci id="S4.SS2.SSS3.Px1.p2.17.m17.1.1.cmml" xref="S4.SS2.SSS3.Px1.p2.17.m17.1.1">𝑢</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS3.Px1.p2.17.m17.1c">u</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS3.Px1.p2.17.m17.1d">italic_u</annotation></semantics></math>-<math alttext="v" class="ltx_Math" display="inline" id="S4.SS2.SSS3.Px1.p2.18.m18.1"><semantics id="S4.SS2.SSS3.Px1.p2.18.m18.1a"><mi id="S4.SS2.SSS3.Px1.p2.18.m18.1.1" xref="S4.SS2.SSS3.Px1.p2.18.m18.1.1.cmml">v</mi><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS3.Px1.p2.18.m18.1b"><ci id="S4.SS2.SSS3.Px1.p2.18.m18.1.1.cmml" xref="S4.SS2.SSS3.Px1.p2.18.m18.1.1">𝑣</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS3.Px1.p2.18.m18.1c">v</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS3.Px1.p2.18.m18.1d">italic_v</annotation></semantics></math> path in <math alttext="E" class="ltx_Math" display="inline" id="S4.SS2.SSS3.Px1.p2.19.m19.1"><semantics id="S4.SS2.SSS3.Px1.p2.19.m19.1a"><mi id="S4.SS2.SSS3.Px1.p2.19.m19.1.1" xref="S4.SS2.SSS3.Px1.p2.19.m19.1.1.cmml">E</mi><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS3.Px1.p2.19.m19.1b"><ci id="S4.SS2.SSS3.Px1.p2.19.m19.1.1.cmml" xref="S4.SS2.SSS3.Px1.p2.19.m19.1.1">𝐸</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS3.Px1.p2.19.m19.1c">E</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS3.Px1.p2.19.m19.1d">italic_E</annotation></semantics></math> avoiding <math alttext="\{a,b\}" class="ltx_Math" display="inline" id="S4.SS2.SSS3.Px1.p2.20.m20.2"><semantics id="S4.SS2.SSS3.Px1.p2.20.m20.2a"><mrow id="S4.SS2.SSS3.Px1.p2.20.m20.2.3.2" xref="S4.SS2.SSS3.Px1.p2.20.m20.2.3.1.cmml"><mo id="S4.SS2.SSS3.Px1.p2.20.m20.2.3.2.1" stretchy="false" xref="S4.SS2.SSS3.Px1.p2.20.m20.2.3.1.cmml">{</mo><mi id="S4.SS2.SSS3.Px1.p2.20.m20.1.1" xref="S4.SS2.SSS3.Px1.p2.20.m20.1.1.cmml">a</mi><mo id="S4.SS2.SSS3.Px1.p2.20.m20.2.3.2.2" xref="S4.SS2.SSS3.Px1.p2.20.m20.2.3.1.cmml">,</mo><mi id="S4.SS2.SSS3.Px1.p2.20.m20.2.2" xref="S4.SS2.SSS3.Px1.p2.20.m20.2.2.cmml">b</mi><mo id="S4.SS2.SSS3.Px1.p2.20.m20.2.3.2.3" stretchy="false" xref="S4.SS2.SSS3.Px1.p2.20.m20.2.3.1.cmml">}</mo></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS3.Px1.p2.20.m20.2b"><set id="S4.SS2.SSS3.Px1.p2.20.m20.2.3.1.cmml" xref="S4.SS2.SSS3.Px1.p2.20.m20.2.3.2"><ci id="S4.SS2.SSS3.Px1.p2.20.m20.1.1.cmml" xref="S4.SS2.SSS3.Px1.p2.20.m20.1.1">𝑎</ci><ci id="S4.SS2.SSS3.Px1.p2.20.m20.2.2.cmml" xref="S4.SS2.SSS3.Px1.p2.20.m20.2.2">𝑏</ci></set></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS3.Px1.p2.20.m20.2c">\{a,b\}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS3.Px1.p2.20.m20.2d">{ italic_a , italic_b }</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="S4.SS2.SSS3.Px1.p3"> <p class="ltx_p" id="S4.SS2.SSS3.Px1.p3.16">Thus, one of <math alttext="u" class="ltx_Math" display="inline" id="S4.SS2.SSS3.Px1.p3.1.m1.1"><semantics id="S4.SS2.SSS3.Px1.p3.1.m1.1a"><mi id="S4.SS2.SSS3.Px1.p3.1.m1.1.1" xref="S4.SS2.SSS3.Px1.p3.1.m1.1.1.cmml">u</mi><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS3.Px1.p3.1.m1.1b"><ci id="S4.SS2.SSS3.Px1.p3.1.m1.1.1.cmml" xref="S4.SS2.SSS3.Px1.p3.1.m1.1.1">𝑢</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS3.Px1.p3.1.m1.1c">u</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS3.Px1.p3.1.m1.1d">italic_u</annotation></semantics></math> and <math alttext="v" class="ltx_Math" display="inline" id="S4.SS2.SSS3.Px1.p3.2.m2.1"><semantics id="S4.SS2.SSS3.Px1.p3.2.m2.1a"><mi id="S4.SS2.SSS3.Px1.p3.2.m2.1.1" xref="S4.SS2.SSS3.Px1.p3.2.m2.1.1.cmml">v</mi><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS3.Px1.p3.2.m2.1b"><ci id="S4.SS2.SSS3.Px1.p3.2.m2.1.1.cmml" xref="S4.SS2.SSS3.Px1.p3.2.m2.1.1">𝑣</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS3.Px1.p3.2.m2.1c">v</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS3.Px1.p3.2.m2.1d">italic_v</annotation></semantics></math> must be in <math alttext="T_{x}" class="ltx_Math" display="inline" id="S4.SS2.SSS3.Px1.p3.3.m3.1"><semantics id="S4.SS2.SSS3.Px1.p3.3.m3.1a"><msub id="S4.SS2.SSS3.Px1.p3.3.m3.1.1" xref="S4.SS2.SSS3.Px1.p3.3.m3.1.1.cmml"><mi id="S4.SS2.SSS3.Px1.p3.3.m3.1.1.2" xref="S4.SS2.SSS3.Px1.p3.3.m3.1.1.2.cmml">T</mi><mi id="S4.SS2.SSS3.Px1.p3.3.m3.1.1.3" xref="S4.SS2.SSS3.Px1.p3.3.m3.1.1.3.cmml">x</mi></msub><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS3.Px1.p3.3.m3.1b"><apply id="S4.SS2.SSS3.Px1.p3.3.m3.1.1.cmml" xref="S4.SS2.SSS3.Px1.p3.3.m3.1.1"><csymbol cd="ambiguous" id="S4.SS2.SSS3.Px1.p3.3.m3.1.1.1.cmml" xref="S4.SS2.SSS3.Px1.p3.3.m3.1.1">subscript</csymbol><ci id="S4.SS2.SSS3.Px1.p3.3.m3.1.1.2.cmml" xref="S4.SS2.SSS3.Px1.p3.3.m3.1.1.2">𝑇</ci><ci id="S4.SS2.SSS3.Px1.p3.3.m3.1.1.3.cmml" xref="S4.SS2.SSS3.Px1.p3.3.m3.1.1.3">𝑥</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS3.Px1.p3.3.m3.1c">T_{x}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS3.Px1.p3.3.m3.1d">italic_T start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT</annotation></semantics></math> and the other outside <math alttext="T_{x}" class="ltx_Math" display="inline" id="S4.SS2.SSS3.Px1.p3.4.m4.1"><semantics id="S4.SS2.SSS3.Px1.p3.4.m4.1a"><msub id="S4.SS2.SSS3.Px1.p3.4.m4.1.1" xref="S4.SS2.SSS3.Px1.p3.4.m4.1.1.cmml"><mi id="S4.SS2.SSS3.Px1.p3.4.m4.1.1.2" xref="S4.SS2.SSS3.Px1.p3.4.m4.1.1.2.cmml">T</mi><mi id="S4.SS2.SSS3.Px1.p3.4.m4.1.1.3" xref="S4.SS2.SSS3.Px1.p3.4.m4.1.1.3.cmml">x</mi></msub><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS3.Px1.p3.4.m4.1b"><apply id="S4.SS2.SSS3.Px1.p3.4.m4.1.1.cmml" xref="S4.SS2.SSS3.Px1.p3.4.m4.1.1"><csymbol cd="ambiguous" id="S4.SS2.SSS3.Px1.p3.4.m4.1.1.1.cmml" xref="S4.SS2.SSS3.Px1.p3.4.m4.1.1">subscript</csymbol><ci id="S4.SS2.SSS3.Px1.p3.4.m4.1.1.2.cmml" xref="S4.SS2.SSS3.Px1.p3.4.m4.1.1.2">𝑇</ci><ci id="S4.SS2.SSS3.Px1.p3.4.m4.1.1.3.cmml" xref="S4.SS2.SSS3.Px1.p3.4.m4.1.1.3">𝑥</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS3.Px1.p3.4.m4.1c">T_{x}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS3.Px1.p3.4.m4.1d">italic_T start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT</annotation></semantics></math>. We assume without loss of generality that <math alttext="u" class="ltx_Math" display="inline" id="S4.SS2.SSS3.Px1.p3.5.m5.1"><semantics id="S4.SS2.SSS3.Px1.p3.5.m5.1a"><mi id="S4.SS2.SSS3.Px1.p3.5.m5.1.1" xref="S4.SS2.SSS3.Px1.p3.5.m5.1.1.cmml">u</mi><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS3.Px1.p3.5.m5.1b"><ci id="S4.SS2.SSS3.Px1.p3.5.m5.1.1.cmml" xref="S4.SS2.SSS3.Px1.p3.5.m5.1.1">𝑢</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS3.Px1.p3.5.m5.1c">u</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS3.Px1.p3.5.m5.1d">italic_u</annotation></semantics></math> is associated with <math alttext="T_{x}" class="ltx_Math" display="inline" id="S4.SS2.SSS3.Px1.p3.6.m6.1"><semantics id="S4.SS2.SSS3.Px1.p3.6.m6.1a"><msub id="S4.SS2.SSS3.Px1.p3.6.m6.1.1" xref="S4.SS2.SSS3.Px1.p3.6.m6.1.1.cmml"><mi id="S4.SS2.SSS3.Px1.p3.6.m6.1.1.2" xref="S4.SS2.SSS3.Px1.p3.6.m6.1.1.2.cmml">T</mi><mi id="S4.SS2.SSS3.Px1.p3.6.m6.1.1.3" xref="S4.SS2.SSS3.Px1.p3.6.m6.1.1.3.cmml">x</mi></msub><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS3.Px1.p3.6.m6.1b"><apply id="S4.SS2.SSS3.Px1.p3.6.m6.1.1.cmml" xref="S4.SS2.SSS3.Px1.p3.6.m6.1.1"><csymbol cd="ambiguous" id="S4.SS2.SSS3.Px1.p3.6.m6.1.1.1.cmml" xref="S4.SS2.SSS3.Px1.p3.6.m6.1.1">subscript</csymbol><ci id="S4.SS2.SSS3.Px1.p3.6.m6.1.1.2.cmml" xref="S4.SS2.SSS3.Px1.p3.6.m6.1.1.2">𝑇</ci><ci id="S4.SS2.SSS3.Px1.p3.6.m6.1.1.3.cmml" xref="S4.SS2.SSS3.Px1.p3.6.m6.1.1.3">𝑥</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS3.Px1.p3.6.m6.1c">T_{x}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS3.Px1.p3.6.m6.1d">italic_T start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT</annotation></semantics></math> (i.e. <math alttext="h(u)\in V(T_{x})" class="ltx_Math" display="inline" id="S4.SS2.SSS3.Px1.p3.7.m7.2"><semantics id="S4.SS2.SSS3.Px1.p3.7.m7.2a"><mrow id="S4.SS2.SSS3.Px1.p3.7.m7.2.2" xref="S4.SS2.SSS3.Px1.p3.7.m7.2.2.cmml"><mrow id="S4.SS2.SSS3.Px1.p3.7.m7.2.2.3" xref="S4.SS2.SSS3.Px1.p3.7.m7.2.2.3.cmml"><mi id="S4.SS2.SSS3.Px1.p3.7.m7.2.2.3.2" xref="S4.SS2.SSS3.Px1.p3.7.m7.2.2.3.2.cmml">h</mi><mo id="S4.SS2.SSS3.Px1.p3.7.m7.2.2.3.1" xref="S4.SS2.SSS3.Px1.p3.7.m7.2.2.3.1.cmml">⁢</mo><mrow id="S4.SS2.SSS3.Px1.p3.7.m7.2.2.3.3.2" xref="S4.SS2.SSS3.Px1.p3.7.m7.2.2.3.cmml"><mo id="S4.SS2.SSS3.Px1.p3.7.m7.2.2.3.3.2.1" stretchy="false" xref="S4.SS2.SSS3.Px1.p3.7.m7.2.2.3.cmml">(</mo><mi id="S4.SS2.SSS3.Px1.p3.7.m7.1.1" xref="S4.SS2.SSS3.Px1.p3.7.m7.1.1.cmml">u</mi><mo id="S4.SS2.SSS3.Px1.p3.7.m7.2.2.3.3.2.2" stretchy="false" xref="S4.SS2.SSS3.Px1.p3.7.m7.2.2.3.cmml">)</mo></mrow></mrow><mo id="S4.SS2.SSS3.Px1.p3.7.m7.2.2.2" xref="S4.SS2.SSS3.Px1.p3.7.m7.2.2.2.cmml">∈</mo><mrow id="S4.SS2.SSS3.Px1.p3.7.m7.2.2.1" xref="S4.SS2.SSS3.Px1.p3.7.m7.2.2.1.cmml"><mi id="S4.SS2.SSS3.Px1.p3.7.m7.2.2.1.3" xref="S4.SS2.SSS3.Px1.p3.7.m7.2.2.1.3.cmml">V</mi><mo id="S4.SS2.SSS3.Px1.p3.7.m7.2.2.1.2" xref="S4.SS2.SSS3.Px1.p3.7.m7.2.2.1.2.cmml">⁢</mo><mrow id="S4.SS2.SSS3.Px1.p3.7.m7.2.2.1.1.1" xref="S4.SS2.SSS3.Px1.p3.7.m7.2.2.1.1.1.1.cmml"><mo id="S4.SS2.SSS3.Px1.p3.7.m7.2.2.1.1.1.2" stretchy="false" xref="S4.SS2.SSS3.Px1.p3.7.m7.2.2.1.1.1.1.cmml">(</mo><msub id="S4.SS2.SSS3.Px1.p3.7.m7.2.2.1.1.1.1" xref="S4.SS2.SSS3.Px1.p3.7.m7.2.2.1.1.1.1.cmml"><mi id="S4.SS2.SSS3.Px1.p3.7.m7.2.2.1.1.1.1.2" xref="S4.SS2.SSS3.Px1.p3.7.m7.2.2.1.1.1.1.2.cmml">T</mi><mi id="S4.SS2.SSS3.Px1.p3.7.m7.2.2.1.1.1.1.3" xref="S4.SS2.SSS3.Px1.p3.7.m7.2.2.1.1.1.1.3.cmml">x</mi></msub><mo id="S4.SS2.SSS3.Px1.p3.7.m7.2.2.1.1.1.3" stretchy="false" xref="S4.SS2.SSS3.Px1.p3.7.m7.2.2.1.1.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS3.Px1.p3.7.m7.2b"><apply id="S4.SS2.SSS3.Px1.p3.7.m7.2.2.cmml" xref="S4.SS2.SSS3.Px1.p3.7.m7.2.2"><in id="S4.SS2.SSS3.Px1.p3.7.m7.2.2.2.cmml" xref="S4.SS2.SSS3.Px1.p3.7.m7.2.2.2"></in><apply id="S4.SS2.SSS3.Px1.p3.7.m7.2.2.3.cmml" xref="S4.SS2.SSS3.Px1.p3.7.m7.2.2.3"><times id="S4.SS2.SSS3.Px1.p3.7.m7.2.2.3.1.cmml" xref="S4.SS2.SSS3.Px1.p3.7.m7.2.2.3.1"></times><ci id="S4.SS2.SSS3.Px1.p3.7.m7.2.2.3.2.cmml" xref="S4.SS2.SSS3.Px1.p3.7.m7.2.2.3.2">ℎ</ci><ci id="S4.SS2.SSS3.Px1.p3.7.m7.1.1.cmml" xref="S4.SS2.SSS3.Px1.p3.7.m7.1.1">𝑢</ci></apply><apply id="S4.SS2.SSS3.Px1.p3.7.m7.2.2.1.cmml" xref="S4.SS2.SSS3.Px1.p3.7.m7.2.2.1"><times id="S4.SS2.SSS3.Px1.p3.7.m7.2.2.1.2.cmml" xref="S4.SS2.SSS3.Px1.p3.7.m7.2.2.1.2"></times><ci id="S4.SS2.SSS3.Px1.p3.7.m7.2.2.1.3.cmml" xref="S4.SS2.SSS3.Px1.p3.7.m7.2.2.1.3">𝑉</ci><apply id="S4.SS2.SSS3.Px1.p3.7.m7.2.2.1.1.1.1.cmml" xref="S4.SS2.SSS3.Px1.p3.7.m7.2.2.1.1.1"><csymbol cd="ambiguous" id="S4.SS2.SSS3.Px1.p3.7.m7.2.2.1.1.1.1.1.cmml" xref="S4.SS2.SSS3.Px1.p3.7.m7.2.2.1.1.1">subscript</csymbol><ci id="S4.SS2.SSS3.Px1.p3.7.m7.2.2.1.1.1.1.2.cmml" xref="S4.SS2.SSS3.Px1.p3.7.m7.2.2.1.1.1.1.2">𝑇</ci><ci id="S4.SS2.SSS3.Px1.p3.7.m7.2.2.1.1.1.1.3.cmml" xref="S4.SS2.SSS3.Px1.p3.7.m7.2.2.1.1.1.1.3">𝑥</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS3.Px1.p3.7.m7.2c">h(u)\in V(T_{x})</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS3.Px1.p3.7.m7.2d">italic_h ( italic_u ) ∈ italic_V ( italic_T start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT )</annotation></semantics></math>) and <math alttext="v" class="ltx_Math" display="inline" id="S4.SS2.SSS3.Px1.p3.8.m8.1"><semantics id="S4.SS2.SSS3.Px1.p3.8.m8.1a"><mi id="S4.SS2.SSS3.Px1.p3.8.m8.1.1" xref="S4.SS2.SSS3.Px1.p3.8.m8.1.1.cmml">v</mi><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS3.Px1.p3.8.m8.1b"><ci id="S4.SS2.SSS3.Px1.p3.8.m8.1.1.cmml" xref="S4.SS2.SSS3.Px1.p3.8.m8.1.1">𝑣</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS3.Px1.p3.8.m8.1c">v</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS3.Px1.p3.8.m8.1d">italic_v</annotation></semantics></math> with <math alttext="T\setminus T_{x}" class="ltx_Math" display="inline" id="S4.SS2.SSS3.Px1.p3.9.m9.1"><semantics id="S4.SS2.SSS3.Px1.p3.9.m9.1a"><mrow id="S4.SS2.SSS3.Px1.p3.9.m9.1.1" xref="S4.SS2.SSS3.Px1.p3.9.m9.1.1.cmml"><mi id="S4.SS2.SSS3.Px1.p3.9.m9.1.1.2" xref="S4.SS2.SSS3.Px1.p3.9.m9.1.1.2.cmml">T</mi><mo id="S4.SS2.SSS3.Px1.p3.9.m9.1.1.1" xref="S4.SS2.SSS3.Px1.p3.9.m9.1.1.1.cmml">∖</mo><msub id="S4.SS2.SSS3.Px1.p3.9.m9.1.1.3" xref="S4.SS2.SSS3.Px1.p3.9.m9.1.1.3.cmml"><mi id="S4.SS2.SSS3.Px1.p3.9.m9.1.1.3.2" xref="S4.SS2.SSS3.Px1.p3.9.m9.1.1.3.2.cmml">T</mi><mi id="S4.SS2.SSS3.Px1.p3.9.m9.1.1.3.3" xref="S4.SS2.SSS3.Px1.p3.9.m9.1.1.3.3.cmml">x</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS3.Px1.p3.9.m9.1b"><apply id="S4.SS2.SSS3.Px1.p3.9.m9.1.1.cmml" xref="S4.SS2.SSS3.Px1.p3.9.m9.1.1"><setdiff id="S4.SS2.SSS3.Px1.p3.9.m9.1.1.1.cmml" xref="S4.SS2.SSS3.Px1.p3.9.m9.1.1.1"></setdiff><ci id="S4.SS2.SSS3.Px1.p3.9.m9.1.1.2.cmml" xref="S4.SS2.SSS3.Px1.p3.9.m9.1.1.2">𝑇</ci><apply id="S4.SS2.SSS3.Px1.p3.9.m9.1.1.3.cmml" xref="S4.SS2.SSS3.Px1.p3.9.m9.1.1.3"><csymbol cd="ambiguous" id="S4.SS2.SSS3.Px1.p3.9.m9.1.1.3.1.cmml" xref="S4.SS2.SSS3.Px1.p3.9.m9.1.1.3">subscript</csymbol><ci id="S4.SS2.SSS3.Px1.p3.9.m9.1.1.3.2.cmml" xref="S4.SS2.SSS3.Px1.p3.9.m9.1.1.3.2">𝑇</ci><ci id="S4.SS2.SSS3.Px1.p3.9.m9.1.1.3.3.cmml" xref="S4.SS2.SSS3.Px1.p3.9.m9.1.1.3.3">𝑥</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS3.Px1.p3.9.m9.1c">T\setminus T_{x}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS3.Px1.p3.9.m9.1d">italic_T ∖ italic_T start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT</annotation></semantics></math> (i.e. <math alttext="\ell(v)\in V(T\setminus T_{x})" class="ltx_Math" display="inline" id="S4.SS2.SSS3.Px1.p3.10.m10.2"><semantics id="S4.SS2.SSS3.Px1.p3.10.m10.2a"><mrow id="S4.SS2.SSS3.Px1.p3.10.m10.2.2" xref="S4.SS2.SSS3.Px1.p3.10.m10.2.2.cmml"><mrow id="S4.SS2.SSS3.Px1.p3.10.m10.2.2.3" xref="S4.SS2.SSS3.Px1.p3.10.m10.2.2.3.cmml"><mi id="S4.SS2.SSS3.Px1.p3.10.m10.2.2.3.2" mathvariant="normal" xref="S4.SS2.SSS3.Px1.p3.10.m10.2.2.3.2.cmml">ℓ</mi><mo id="S4.SS2.SSS3.Px1.p3.10.m10.2.2.3.1" xref="S4.SS2.SSS3.Px1.p3.10.m10.2.2.3.1.cmml">⁢</mo><mrow id="S4.SS2.SSS3.Px1.p3.10.m10.2.2.3.3.2" xref="S4.SS2.SSS3.Px1.p3.10.m10.2.2.3.cmml"><mo id="S4.SS2.SSS3.Px1.p3.10.m10.2.2.3.3.2.1" stretchy="false" xref="S4.SS2.SSS3.Px1.p3.10.m10.2.2.3.cmml">(</mo><mi id="S4.SS2.SSS3.Px1.p3.10.m10.1.1" xref="S4.SS2.SSS3.Px1.p3.10.m10.1.1.cmml">v</mi><mo id="S4.SS2.SSS3.Px1.p3.10.m10.2.2.3.3.2.2" stretchy="false" xref="S4.SS2.SSS3.Px1.p3.10.m10.2.2.3.cmml">)</mo></mrow></mrow><mo id="S4.SS2.SSS3.Px1.p3.10.m10.2.2.2" xref="S4.SS2.SSS3.Px1.p3.10.m10.2.2.2.cmml">∈</mo><mrow id="S4.SS2.SSS3.Px1.p3.10.m10.2.2.1" xref="S4.SS2.SSS3.Px1.p3.10.m10.2.2.1.cmml"><mi id="S4.SS2.SSS3.Px1.p3.10.m10.2.2.1.3" xref="S4.SS2.SSS3.Px1.p3.10.m10.2.2.1.3.cmml">V</mi><mo id="S4.SS2.SSS3.Px1.p3.10.m10.2.2.1.2" xref="S4.SS2.SSS3.Px1.p3.10.m10.2.2.1.2.cmml">⁢</mo><mrow id="S4.SS2.SSS3.Px1.p3.10.m10.2.2.1.1.1" xref="S4.SS2.SSS3.Px1.p3.10.m10.2.2.1.1.1.1.cmml"><mo id="S4.SS2.SSS3.Px1.p3.10.m10.2.2.1.1.1.2" stretchy="false" xref="S4.SS2.SSS3.Px1.p3.10.m10.2.2.1.1.1.1.cmml">(</mo><mrow id="S4.SS2.SSS3.Px1.p3.10.m10.2.2.1.1.1.1" xref="S4.SS2.SSS3.Px1.p3.10.m10.2.2.1.1.1.1.cmml"><mi id="S4.SS2.SSS3.Px1.p3.10.m10.2.2.1.1.1.1.2" xref="S4.SS2.SSS3.Px1.p3.10.m10.2.2.1.1.1.1.2.cmml">T</mi><mo id="S4.SS2.SSS3.Px1.p3.10.m10.2.2.1.1.1.1.1" xref="S4.SS2.SSS3.Px1.p3.10.m10.2.2.1.1.1.1.1.cmml">∖</mo><msub id="S4.SS2.SSS3.Px1.p3.10.m10.2.2.1.1.1.1.3" xref="S4.SS2.SSS3.Px1.p3.10.m10.2.2.1.1.1.1.3.cmml"><mi id="S4.SS2.SSS3.Px1.p3.10.m10.2.2.1.1.1.1.3.2" xref="S4.SS2.SSS3.Px1.p3.10.m10.2.2.1.1.1.1.3.2.cmml">T</mi><mi id="S4.SS2.SSS3.Px1.p3.10.m10.2.2.1.1.1.1.3.3" xref="S4.SS2.SSS3.Px1.p3.10.m10.2.2.1.1.1.1.3.3.cmml">x</mi></msub></mrow><mo id="S4.SS2.SSS3.Px1.p3.10.m10.2.2.1.1.1.3" stretchy="false" xref="S4.SS2.SSS3.Px1.p3.10.m10.2.2.1.1.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS3.Px1.p3.10.m10.2b"><apply id="S4.SS2.SSS3.Px1.p3.10.m10.2.2.cmml" xref="S4.SS2.SSS3.Px1.p3.10.m10.2.2"><in id="S4.SS2.SSS3.Px1.p3.10.m10.2.2.2.cmml" xref="S4.SS2.SSS3.Px1.p3.10.m10.2.2.2"></in><apply id="S4.SS2.SSS3.Px1.p3.10.m10.2.2.3.cmml" xref="S4.SS2.SSS3.Px1.p3.10.m10.2.2.3"><times id="S4.SS2.SSS3.Px1.p3.10.m10.2.2.3.1.cmml" xref="S4.SS2.SSS3.Px1.p3.10.m10.2.2.3.1"></times><ci id="S4.SS2.SSS3.Px1.p3.10.m10.2.2.3.2.cmml" xref="S4.SS2.SSS3.Px1.p3.10.m10.2.2.3.2">ℓ</ci><ci id="S4.SS2.SSS3.Px1.p3.10.m10.1.1.cmml" xref="S4.SS2.SSS3.Px1.p3.10.m10.1.1">𝑣</ci></apply><apply id="S4.SS2.SSS3.Px1.p3.10.m10.2.2.1.cmml" xref="S4.SS2.SSS3.Px1.p3.10.m10.2.2.1"><times id="S4.SS2.SSS3.Px1.p3.10.m10.2.2.1.2.cmml" xref="S4.SS2.SSS3.Px1.p3.10.m10.2.2.1.2"></times><ci id="S4.SS2.SSS3.Px1.p3.10.m10.2.2.1.3.cmml" xref="S4.SS2.SSS3.Px1.p3.10.m10.2.2.1.3">𝑉</ci><apply id="S4.SS2.SSS3.Px1.p3.10.m10.2.2.1.1.1.1.cmml" xref="S4.SS2.SSS3.Px1.p3.10.m10.2.2.1.1.1"><setdiff id="S4.SS2.SSS3.Px1.p3.10.m10.2.2.1.1.1.1.1.cmml" xref="S4.SS2.SSS3.Px1.p3.10.m10.2.2.1.1.1.1.1"></setdiff><ci id="S4.SS2.SSS3.Px1.p3.10.m10.2.2.1.1.1.1.2.cmml" xref="S4.SS2.SSS3.Px1.p3.10.m10.2.2.1.1.1.1.2">𝑇</ci><apply id="S4.SS2.SSS3.Px1.p3.10.m10.2.2.1.1.1.1.3.cmml" xref="S4.SS2.SSS3.Px1.p3.10.m10.2.2.1.1.1.1.3"><csymbol cd="ambiguous" id="S4.SS2.SSS3.Px1.p3.10.m10.2.2.1.1.1.1.3.1.cmml" xref="S4.SS2.SSS3.Px1.p3.10.m10.2.2.1.1.1.1.3">subscript</csymbol><ci id="S4.SS2.SSS3.Px1.p3.10.m10.2.2.1.1.1.1.3.2.cmml" xref="S4.SS2.SSS3.Px1.p3.10.m10.2.2.1.1.1.1.3.2">𝑇</ci><ci id="S4.SS2.SSS3.Px1.p3.10.m10.2.2.1.1.1.1.3.3.cmml" xref="S4.SS2.SSS3.Px1.p3.10.m10.2.2.1.1.1.1.3.3">𝑥</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS3.Px1.p3.10.m10.2c">\ell(v)\in V(T\setminus T_{x})</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS3.Px1.p3.10.m10.2d">roman_ℓ ( italic_v ) ∈ italic_V ( italic_T ∖ italic_T start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT )</annotation></semantics></math>). Let <math alttext="(u^{\prime},u^{\prime\prime})=L_{h(u)}(j)" class="ltx_Math" display="inline" id="S4.SS2.SSS3.Px1.p3.11.m11.4"><semantics id="S4.SS2.SSS3.Px1.p3.11.m11.4a"><mrow id="S4.SS2.SSS3.Px1.p3.11.m11.4.4" xref="S4.SS2.SSS3.Px1.p3.11.m11.4.4.cmml"><mrow id="S4.SS2.SSS3.Px1.p3.11.m11.4.4.2.2" xref="S4.SS2.SSS3.Px1.p3.11.m11.4.4.2.3.cmml"><mo id="S4.SS2.SSS3.Px1.p3.11.m11.4.4.2.2.3" stretchy="false" xref="S4.SS2.SSS3.Px1.p3.11.m11.4.4.2.3.cmml">(</mo><msup id="S4.SS2.SSS3.Px1.p3.11.m11.3.3.1.1.1" xref="S4.SS2.SSS3.Px1.p3.11.m11.3.3.1.1.1.cmml"><mi id="S4.SS2.SSS3.Px1.p3.11.m11.3.3.1.1.1.2" xref="S4.SS2.SSS3.Px1.p3.11.m11.3.3.1.1.1.2.cmml">u</mi><mo id="S4.SS2.SSS3.Px1.p3.11.m11.3.3.1.1.1.3" xref="S4.SS2.SSS3.Px1.p3.11.m11.3.3.1.1.1.3.cmml">′</mo></msup><mo id="S4.SS2.SSS3.Px1.p3.11.m11.4.4.2.2.4" xref="S4.SS2.SSS3.Px1.p3.11.m11.4.4.2.3.cmml">,</mo><msup id="S4.SS2.SSS3.Px1.p3.11.m11.4.4.2.2.2" xref="S4.SS2.SSS3.Px1.p3.11.m11.4.4.2.2.2.cmml"><mi id="S4.SS2.SSS3.Px1.p3.11.m11.4.4.2.2.2.2" xref="S4.SS2.SSS3.Px1.p3.11.m11.4.4.2.2.2.2.cmml">u</mi><mo id="S4.SS2.SSS3.Px1.p3.11.m11.4.4.2.2.2.3" xref="S4.SS2.SSS3.Px1.p3.11.m11.4.4.2.2.2.3.cmml">′′</mo></msup><mo id="S4.SS2.SSS3.Px1.p3.11.m11.4.4.2.2.5" stretchy="false" xref="S4.SS2.SSS3.Px1.p3.11.m11.4.4.2.3.cmml">)</mo></mrow><mo id="S4.SS2.SSS3.Px1.p3.11.m11.4.4.3" xref="S4.SS2.SSS3.Px1.p3.11.m11.4.4.3.cmml">=</mo><mrow id="S4.SS2.SSS3.Px1.p3.11.m11.4.4.4" xref="S4.SS2.SSS3.Px1.p3.11.m11.4.4.4.cmml"><msub id="S4.SS2.SSS3.Px1.p3.11.m11.4.4.4.2" xref="S4.SS2.SSS3.Px1.p3.11.m11.4.4.4.2.cmml"><mi id="S4.SS2.SSS3.Px1.p3.11.m11.4.4.4.2.2" xref="S4.SS2.SSS3.Px1.p3.11.m11.4.4.4.2.2.cmml">L</mi><mrow id="S4.SS2.SSS3.Px1.p3.11.m11.1.1.1" xref="S4.SS2.SSS3.Px1.p3.11.m11.1.1.1.cmml"><mi id="S4.SS2.SSS3.Px1.p3.11.m11.1.1.1.3" xref="S4.SS2.SSS3.Px1.p3.11.m11.1.1.1.3.cmml">h</mi><mo id="S4.SS2.SSS3.Px1.p3.11.m11.1.1.1.2" xref="S4.SS2.SSS3.Px1.p3.11.m11.1.1.1.2.cmml">⁢</mo><mrow id="S4.SS2.SSS3.Px1.p3.11.m11.1.1.1.4.2" xref="S4.SS2.SSS3.Px1.p3.11.m11.1.1.1.cmml"><mo id="S4.SS2.SSS3.Px1.p3.11.m11.1.1.1.4.2.1" stretchy="false" xref="S4.SS2.SSS3.Px1.p3.11.m11.1.1.1.cmml">(</mo><mi id="S4.SS2.SSS3.Px1.p3.11.m11.1.1.1.1" xref="S4.SS2.SSS3.Px1.p3.11.m11.1.1.1.1.cmml">u</mi><mo id="S4.SS2.SSS3.Px1.p3.11.m11.1.1.1.4.2.2" stretchy="false" xref="S4.SS2.SSS3.Px1.p3.11.m11.1.1.1.cmml">)</mo></mrow></mrow></msub><mo id="S4.SS2.SSS3.Px1.p3.11.m11.4.4.4.1" xref="S4.SS2.SSS3.Px1.p3.11.m11.4.4.4.1.cmml">⁢</mo><mrow id="S4.SS2.SSS3.Px1.p3.11.m11.4.4.4.3.2" xref="S4.SS2.SSS3.Px1.p3.11.m11.4.4.4.cmml"><mo id="S4.SS2.SSS3.Px1.p3.11.m11.4.4.4.3.2.1" stretchy="false" xref="S4.SS2.SSS3.Px1.p3.11.m11.4.4.4.cmml">(</mo><mi id="S4.SS2.SSS3.Px1.p3.11.m11.2.2" xref="S4.SS2.SSS3.Px1.p3.11.m11.2.2.cmml">j</mi><mo id="S4.SS2.SSS3.Px1.p3.11.m11.4.4.4.3.2.2" stretchy="false" xref="S4.SS2.SSS3.Px1.p3.11.m11.4.4.4.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS3.Px1.p3.11.m11.4b"><apply id="S4.SS2.SSS3.Px1.p3.11.m11.4.4.cmml" xref="S4.SS2.SSS3.Px1.p3.11.m11.4.4"><eq id="S4.SS2.SSS3.Px1.p3.11.m11.4.4.3.cmml" xref="S4.SS2.SSS3.Px1.p3.11.m11.4.4.3"></eq><interval closure="open" id="S4.SS2.SSS3.Px1.p3.11.m11.4.4.2.3.cmml" xref="S4.SS2.SSS3.Px1.p3.11.m11.4.4.2.2"><apply id="S4.SS2.SSS3.Px1.p3.11.m11.3.3.1.1.1.cmml" xref="S4.SS2.SSS3.Px1.p3.11.m11.3.3.1.1.1"><csymbol cd="ambiguous" id="S4.SS2.SSS3.Px1.p3.11.m11.3.3.1.1.1.1.cmml" xref="S4.SS2.SSS3.Px1.p3.11.m11.3.3.1.1.1">superscript</csymbol><ci id="S4.SS2.SSS3.Px1.p3.11.m11.3.3.1.1.1.2.cmml" xref="S4.SS2.SSS3.Px1.p3.11.m11.3.3.1.1.1.2">𝑢</ci><ci id="S4.SS2.SSS3.Px1.p3.11.m11.3.3.1.1.1.3.cmml" xref="S4.SS2.SSS3.Px1.p3.11.m11.3.3.1.1.1.3">′</ci></apply><apply id="S4.SS2.SSS3.Px1.p3.11.m11.4.4.2.2.2.cmml" xref="S4.SS2.SSS3.Px1.p3.11.m11.4.4.2.2.2"><csymbol cd="ambiguous" id="S4.SS2.SSS3.Px1.p3.11.m11.4.4.2.2.2.1.cmml" xref="S4.SS2.SSS3.Px1.p3.11.m11.4.4.2.2.2">superscript</csymbol><ci id="S4.SS2.SSS3.Px1.p3.11.m11.4.4.2.2.2.2.cmml" xref="S4.SS2.SSS3.Px1.p3.11.m11.4.4.2.2.2.2">𝑢</ci><ci id="S4.SS2.SSS3.Px1.p3.11.m11.4.4.2.2.2.3.cmml" xref="S4.SS2.SSS3.Px1.p3.11.m11.4.4.2.2.2.3">′′</ci></apply></interval><apply id="S4.SS2.SSS3.Px1.p3.11.m11.4.4.4.cmml" xref="S4.SS2.SSS3.Px1.p3.11.m11.4.4.4"><times id="S4.SS2.SSS3.Px1.p3.11.m11.4.4.4.1.cmml" xref="S4.SS2.SSS3.Px1.p3.11.m11.4.4.4.1"></times><apply id="S4.SS2.SSS3.Px1.p3.11.m11.4.4.4.2.cmml" xref="S4.SS2.SSS3.Px1.p3.11.m11.4.4.4.2"><csymbol cd="ambiguous" id="S4.SS2.SSS3.Px1.p3.11.m11.4.4.4.2.1.cmml" xref="S4.SS2.SSS3.Px1.p3.11.m11.4.4.4.2">subscript</csymbol><ci id="S4.SS2.SSS3.Px1.p3.11.m11.4.4.4.2.2.cmml" xref="S4.SS2.SSS3.Px1.p3.11.m11.4.4.4.2.2">𝐿</ci><apply id="S4.SS2.SSS3.Px1.p3.11.m11.1.1.1.cmml" xref="S4.SS2.SSS3.Px1.p3.11.m11.1.1.1"><times id="S4.SS2.SSS3.Px1.p3.11.m11.1.1.1.2.cmml" xref="S4.SS2.SSS3.Px1.p3.11.m11.1.1.1.2"></times><ci id="S4.SS2.SSS3.Px1.p3.11.m11.1.1.1.3.cmml" xref="S4.SS2.SSS3.Px1.p3.11.m11.1.1.1.3">ℎ</ci><ci id="S4.SS2.SSS3.Px1.p3.11.m11.1.1.1.1.cmml" xref="S4.SS2.SSS3.Px1.p3.11.m11.1.1.1.1">𝑢</ci></apply></apply><ci id="S4.SS2.SSS3.Px1.p3.11.m11.2.2.cmml" xref="S4.SS2.SSS3.Px1.p3.11.m11.2.2">𝑗</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS3.Px1.p3.11.m11.4c">(u^{\prime},u^{\prime\prime})=L_{h(u)}(j)</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS3.Px1.p3.11.m11.4d">( italic_u start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT , italic_u start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT ) = italic_L start_POSTSUBSCRIPT italic_h ( italic_u ) end_POSTSUBSCRIPT ( italic_j )</annotation></semantics></math> where <math alttext="h(u^{\prime})=h(u)" class="ltx_Math" display="inline" id="S4.SS2.SSS3.Px1.p3.12.m12.2"><semantics id="S4.SS2.SSS3.Px1.p3.12.m12.2a"><mrow id="S4.SS2.SSS3.Px1.p3.12.m12.2.2" xref="S4.SS2.SSS3.Px1.p3.12.m12.2.2.cmml"><mrow id="S4.SS2.SSS3.Px1.p3.12.m12.2.2.1" xref="S4.SS2.SSS3.Px1.p3.12.m12.2.2.1.cmml"><mi id="S4.SS2.SSS3.Px1.p3.12.m12.2.2.1.3" xref="S4.SS2.SSS3.Px1.p3.12.m12.2.2.1.3.cmml">h</mi><mo id="S4.SS2.SSS3.Px1.p3.12.m12.2.2.1.2" xref="S4.SS2.SSS3.Px1.p3.12.m12.2.2.1.2.cmml">⁢</mo><mrow id="S4.SS2.SSS3.Px1.p3.12.m12.2.2.1.1.1" xref="S4.SS2.SSS3.Px1.p3.12.m12.2.2.1.1.1.1.cmml"><mo id="S4.SS2.SSS3.Px1.p3.12.m12.2.2.1.1.1.2" stretchy="false" xref="S4.SS2.SSS3.Px1.p3.12.m12.2.2.1.1.1.1.cmml">(</mo><msup id="S4.SS2.SSS3.Px1.p3.12.m12.2.2.1.1.1.1" xref="S4.SS2.SSS3.Px1.p3.12.m12.2.2.1.1.1.1.cmml"><mi id="S4.SS2.SSS3.Px1.p3.12.m12.2.2.1.1.1.1.2" xref="S4.SS2.SSS3.Px1.p3.12.m12.2.2.1.1.1.1.2.cmml">u</mi><mo id="S4.SS2.SSS3.Px1.p3.12.m12.2.2.1.1.1.1.3" xref="S4.SS2.SSS3.Px1.p3.12.m12.2.2.1.1.1.1.3.cmml">′</mo></msup><mo id="S4.SS2.SSS3.Px1.p3.12.m12.2.2.1.1.1.3" stretchy="false" xref="S4.SS2.SSS3.Px1.p3.12.m12.2.2.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S4.SS2.SSS3.Px1.p3.12.m12.2.2.2" xref="S4.SS2.SSS3.Px1.p3.12.m12.2.2.2.cmml">=</mo><mrow id="S4.SS2.SSS3.Px1.p3.12.m12.2.2.3" xref="S4.SS2.SSS3.Px1.p3.12.m12.2.2.3.cmml"><mi id="S4.SS2.SSS3.Px1.p3.12.m12.2.2.3.2" xref="S4.SS2.SSS3.Px1.p3.12.m12.2.2.3.2.cmml">h</mi><mo id="S4.SS2.SSS3.Px1.p3.12.m12.2.2.3.1" xref="S4.SS2.SSS3.Px1.p3.12.m12.2.2.3.1.cmml">⁢</mo><mrow id="S4.SS2.SSS3.Px1.p3.12.m12.2.2.3.3.2" xref="S4.SS2.SSS3.Px1.p3.12.m12.2.2.3.cmml"><mo id="S4.SS2.SSS3.Px1.p3.12.m12.2.2.3.3.2.1" stretchy="false" xref="S4.SS2.SSS3.Px1.p3.12.m12.2.2.3.cmml">(</mo><mi id="S4.SS2.SSS3.Px1.p3.12.m12.1.1" xref="S4.SS2.SSS3.Px1.p3.12.m12.1.1.cmml">u</mi><mo id="S4.SS2.SSS3.Px1.p3.12.m12.2.2.3.3.2.2" stretchy="false" xref="S4.SS2.SSS3.Px1.p3.12.m12.2.2.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS3.Px1.p3.12.m12.2b"><apply id="S4.SS2.SSS3.Px1.p3.12.m12.2.2.cmml" xref="S4.SS2.SSS3.Px1.p3.12.m12.2.2"><eq id="S4.SS2.SSS3.Px1.p3.12.m12.2.2.2.cmml" xref="S4.SS2.SSS3.Px1.p3.12.m12.2.2.2"></eq><apply id="S4.SS2.SSS3.Px1.p3.12.m12.2.2.1.cmml" xref="S4.SS2.SSS3.Px1.p3.12.m12.2.2.1"><times id="S4.SS2.SSS3.Px1.p3.12.m12.2.2.1.2.cmml" xref="S4.SS2.SSS3.Px1.p3.12.m12.2.2.1.2"></times><ci id="S4.SS2.SSS3.Px1.p3.12.m12.2.2.1.3.cmml" xref="S4.SS2.SSS3.Px1.p3.12.m12.2.2.1.3">ℎ</ci><apply id="S4.SS2.SSS3.Px1.p3.12.m12.2.2.1.1.1.1.cmml" xref="S4.SS2.SSS3.Px1.p3.12.m12.2.2.1.1.1"><csymbol cd="ambiguous" id="S4.SS2.SSS3.Px1.p3.12.m12.2.2.1.1.1.1.1.cmml" xref="S4.SS2.SSS3.Px1.p3.12.m12.2.2.1.1.1">superscript</csymbol><ci id="S4.SS2.SSS3.Px1.p3.12.m12.2.2.1.1.1.1.2.cmml" xref="S4.SS2.SSS3.Px1.p3.12.m12.2.2.1.1.1.1.2">𝑢</ci><ci id="S4.SS2.SSS3.Px1.p3.12.m12.2.2.1.1.1.1.3.cmml" xref="S4.SS2.SSS3.Px1.p3.12.m12.2.2.1.1.1.1.3">′</ci></apply></apply><apply id="S4.SS2.SSS3.Px1.p3.12.m12.2.2.3.cmml" xref="S4.SS2.SSS3.Px1.p3.12.m12.2.2.3"><times id="S4.SS2.SSS3.Px1.p3.12.m12.2.2.3.1.cmml" xref="S4.SS2.SSS3.Px1.p3.12.m12.2.2.3.1"></times><ci id="S4.SS2.SSS3.Px1.p3.12.m12.2.2.3.2.cmml" xref="S4.SS2.SSS3.Px1.p3.12.m12.2.2.3.2">ℎ</ci><ci id="S4.SS2.SSS3.Px1.p3.12.m12.1.1.cmml" xref="S4.SS2.SSS3.Px1.p3.12.m12.1.1">𝑢</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS3.Px1.p3.12.m12.2c">h(u^{\prime})=h(u)</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS3.Px1.p3.12.m12.2d">italic_h ( italic_u start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) = italic_h ( italic_u )</annotation></semantics></math>; this must be in <span class="ltx_text ltx_markedasmath" id="S4.SS2.SSS3.Px1.p3.16.1">SOL</span> since <math alttext="uv\in\textnormal{OPT}" class="ltx_Math" display="inline" id="S4.SS2.SSS3.Px1.p3.14.m14.1"><semantics id="S4.SS2.SSS3.Px1.p3.14.m14.1a"><mrow id="S4.SS2.SSS3.Px1.p3.14.m14.1.1" xref="S4.SS2.SSS3.Px1.p3.14.m14.1.1.cmml"><mrow id="S4.SS2.SSS3.Px1.p3.14.m14.1.1.2" xref="S4.SS2.SSS3.Px1.p3.14.m14.1.1.2.cmml"><mi id="S4.SS2.SSS3.Px1.p3.14.m14.1.1.2.2" xref="S4.SS2.SSS3.Px1.p3.14.m14.1.1.2.2.cmml">u</mi><mo id="S4.SS2.SSS3.Px1.p3.14.m14.1.1.2.1" xref="S4.SS2.SSS3.Px1.p3.14.m14.1.1.2.1.cmml">⁢</mo><mi id="S4.SS2.SSS3.Px1.p3.14.m14.1.1.2.3" xref="S4.SS2.SSS3.Px1.p3.14.m14.1.1.2.3.cmml">v</mi></mrow><mo id="S4.SS2.SSS3.Px1.p3.14.m14.1.1.1" xref="S4.SS2.SSS3.Px1.p3.14.m14.1.1.1.cmml">∈</mo><mtext id="S4.SS2.SSS3.Px1.p3.14.m14.1.1.3" xref="S4.SS2.SSS3.Px1.p3.14.m14.1.1.3a.cmml">OPT</mtext></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS3.Px1.p3.14.m14.1b"><apply id="S4.SS2.SSS3.Px1.p3.14.m14.1.1.cmml" xref="S4.SS2.SSS3.Px1.p3.14.m14.1.1"><in id="S4.SS2.SSS3.Px1.p3.14.m14.1.1.1.cmml" xref="S4.SS2.SSS3.Px1.p3.14.m14.1.1.1"></in><apply id="S4.SS2.SSS3.Px1.p3.14.m14.1.1.2.cmml" xref="S4.SS2.SSS3.Px1.p3.14.m14.1.1.2"><times id="S4.SS2.SSS3.Px1.p3.14.m14.1.1.2.1.cmml" xref="S4.SS2.SSS3.Px1.p3.14.m14.1.1.2.1"></times><ci id="S4.SS2.SSS3.Px1.p3.14.m14.1.1.2.2.cmml" xref="S4.SS2.SSS3.Px1.p3.14.m14.1.1.2.2">𝑢</ci><ci id="S4.SS2.SSS3.Px1.p3.14.m14.1.1.2.3.cmml" xref="S4.SS2.SSS3.Px1.p3.14.m14.1.1.2.3">𝑣</ci></apply><ci id="S4.SS2.SSS3.Px1.p3.14.m14.1.1.3a.cmml" xref="S4.SS2.SSS3.Px1.p3.14.m14.1.1.3"><mtext id="S4.SS2.SSS3.Px1.p3.14.m14.1.1.3.cmml" xref="S4.SS2.SSS3.Px1.p3.14.m14.1.1.3">OPT</mtext></ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS3.Px1.p3.14.m14.1c">uv\in\textnormal{OPT}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS3.Px1.p3.14.m14.1d">italic_u italic_v ∈ OPT</annotation></semantics></math>. By construction, <math alttext="d_{T}(r,\text{LCA}(h(u),\ell(u^{\prime\prime})))\leq d_{T}(r,\text{LCA}(h(u),% \ell(v)))" class="ltx_Math" display="inline" id="S4.SS2.SSS3.Px1.p3.15.m15.7"><semantics id="S4.SS2.SSS3.Px1.p3.15.m15.7a"><mrow id="S4.SS2.SSS3.Px1.p3.15.m15.7.7" xref="S4.SS2.SSS3.Px1.p3.15.m15.7.7.cmml"><mrow id="S4.SS2.SSS3.Px1.p3.15.m15.6.6.1" xref="S4.SS2.SSS3.Px1.p3.15.m15.6.6.1.cmml"><msub id="S4.SS2.SSS3.Px1.p3.15.m15.6.6.1.3" xref="S4.SS2.SSS3.Px1.p3.15.m15.6.6.1.3.cmml"><mi id="S4.SS2.SSS3.Px1.p3.15.m15.6.6.1.3.2" xref="S4.SS2.SSS3.Px1.p3.15.m15.6.6.1.3.2.cmml">d</mi><mi id="S4.SS2.SSS3.Px1.p3.15.m15.6.6.1.3.3" xref="S4.SS2.SSS3.Px1.p3.15.m15.6.6.1.3.3.cmml">T</mi></msub><mo id="S4.SS2.SSS3.Px1.p3.15.m15.6.6.1.2" xref="S4.SS2.SSS3.Px1.p3.15.m15.6.6.1.2.cmml">⁢</mo><mrow id="S4.SS2.SSS3.Px1.p3.15.m15.6.6.1.1.1" xref="S4.SS2.SSS3.Px1.p3.15.m15.6.6.1.1.2.cmml"><mo id="S4.SS2.SSS3.Px1.p3.15.m15.6.6.1.1.1.2" stretchy="false" xref="S4.SS2.SSS3.Px1.p3.15.m15.6.6.1.1.2.cmml">(</mo><mi id="S4.SS2.SSS3.Px1.p3.15.m15.2.2" xref="S4.SS2.SSS3.Px1.p3.15.m15.2.2.cmml">r</mi><mo id="S4.SS2.SSS3.Px1.p3.15.m15.6.6.1.1.1.3" xref="S4.SS2.SSS3.Px1.p3.15.m15.6.6.1.1.2.cmml">,</mo><mrow id="S4.SS2.SSS3.Px1.p3.15.m15.6.6.1.1.1.1" xref="S4.SS2.SSS3.Px1.p3.15.m15.6.6.1.1.1.1.cmml"><mtext id="S4.SS2.SSS3.Px1.p3.15.m15.6.6.1.1.1.1.4" xref="S4.SS2.SSS3.Px1.p3.15.m15.6.6.1.1.1.1.4a.cmml">LCA</mtext><mo id="S4.SS2.SSS3.Px1.p3.15.m15.6.6.1.1.1.1.3" xref="S4.SS2.SSS3.Px1.p3.15.m15.6.6.1.1.1.1.3.cmml">⁢</mo><mrow id="S4.SS2.SSS3.Px1.p3.15.m15.6.6.1.1.1.1.2.2" xref="S4.SS2.SSS3.Px1.p3.15.m15.6.6.1.1.1.1.2.3.cmml"><mo id="S4.SS2.SSS3.Px1.p3.15.m15.6.6.1.1.1.1.2.2.3" stretchy="false" xref="S4.SS2.SSS3.Px1.p3.15.m15.6.6.1.1.1.1.2.3.cmml">(</mo><mrow id="S4.SS2.SSS3.Px1.p3.15.m15.6.6.1.1.1.1.1.1.1" xref="S4.SS2.SSS3.Px1.p3.15.m15.6.6.1.1.1.1.1.1.1.cmml"><mi id="S4.SS2.SSS3.Px1.p3.15.m15.6.6.1.1.1.1.1.1.1.2" xref="S4.SS2.SSS3.Px1.p3.15.m15.6.6.1.1.1.1.1.1.1.2.cmml">h</mi><mo id="S4.SS2.SSS3.Px1.p3.15.m15.6.6.1.1.1.1.1.1.1.1" xref="S4.SS2.SSS3.Px1.p3.15.m15.6.6.1.1.1.1.1.1.1.1.cmml">⁢</mo><mrow id="S4.SS2.SSS3.Px1.p3.15.m15.6.6.1.1.1.1.1.1.1.3.2" xref="S4.SS2.SSS3.Px1.p3.15.m15.6.6.1.1.1.1.1.1.1.cmml"><mo id="S4.SS2.SSS3.Px1.p3.15.m15.6.6.1.1.1.1.1.1.1.3.2.1" stretchy="false" xref="S4.SS2.SSS3.Px1.p3.15.m15.6.6.1.1.1.1.1.1.1.cmml">(</mo><mi id="S4.SS2.SSS3.Px1.p3.15.m15.1.1" xref="S4.SS2.SSS3.Px1.p3.15.m15.1.1.cmml">u</mi><mo id="S4.SS2.SSS3.Px1.p3.15.m15.6.6.1.1.1.1.1.1.1.3.2.2" stretchy="false" xref="S4.SS2.SSS3.Px1.p3.15.m15.6.6.1.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S4.SS2.SSS3.Px1.p3.15.m15.6.6.1.1.1.1.2.2.4" xref="S4.SS2.SSS3.Px1.p3.15.m15.6.6.1.1.1.1.2.3.cmml">,</mo><mrow id="S4.SS2.SSS3.Px1.p3.15.m15.6.6.1.1.1.1.2.2.2" xref="S4.SS2.SSS3.Px1.p3.15.m15.6.6.1.1.1.1.2.2.2.cmml"><mi id="S4.SS2.SSS3.Px1.p3.15.m15.6.6.1.1.1.1.2.2.2.3" mathvariant="normal" xref="S4.SS2.SSS3.Px1.p3.15.m15.6.6.1.1.1.1.2.2.2.3.cmml">ℓ</mi><mo id="S4.SS2.SSS3.Px1.p3.15.m15.6.6.1.1.1.1.2.2.2.2" xref="S4.SS2.SSS3.Px1.p3.15.m15.6.6.1.1.1.1.2.2.2.2.cmml">⁢</mo><mrow id="S4.SS2.SSS3.Px1.p3.15.m15.6.6.1.1.1.1.2.2.2.1.1" xref="S4.SS2.SSS3.Px1.p3.15.m15.6.6.1.1.1.1.2.2.2.1.1.1.cmml"><mo id="S4.SS2.SSS3.Px1.p3.15.m15.6.6.1.1.1.1.2.2.2.1.1.2" stretchy="false" xref="S4.SS2.SSS3.Px1.p3.15.m15.6.6.1.1.1.1.2.2.2.1.1.1.cmml">(</mo><msup id="S4.SS2.SSS3.Px1.p3.15.m15.6.6.1.1.1.1.2.2.2.1.1.1" xref="S4.SS2.SSS3.Px1.p3.15.m15.6.6.1.1.1.1.2.2.2.1.1.1.cmml"><mi id="S4.SS2.SSS3.Px1.p3.15.m15.6.6.1.1.1.1.2.2.2.1.1.1.2" xref="S4.SS2.SSS3.Px1.p3.15.m15.6.6.1.1.1.1.2.2.2.1.1.1.2.cmml">u</mi><mo id="S4.SS2.SSS3.Px1.p3.15.m15.6.6.1.1.1.1.2.2.2.1.1.1.3" xref="S4.SS2.SSS3.Px1.p3.15.m15.6.6.1.1.1.1.2.2.2.1.1.1.3.cmml">′′</mo></msup><mo id="S4.SS2.SSS3.Px1.p3.15.m15.6.6.1.1.1.1.2.2.2.1.1.3" stretchy="false" xref="S4.SS2.SSS3.Px1.p3.15.m15.6.6.1.1.1.1.2.2.2.1.1.1.cmml">)</mo></mrow></mrow><mo id="S4.SS2.SSS3.Px1.p3.15.m15.6.6.1.1.1.1.2.2.5" stretchy="false" xref="S4.SS2.SSS3.Px1.p3.15.m15.6.6.1.1.1.1.2.3.cmml">)</mo></mrow></mrow><mo id="S4.SS2.SSS3.Px1.p3.15.m15.6.6.1.1.1.4" stretchy="false" xref="S4.SS2.SSS3.Px1.p3.15.m15.6.6.1.1.2.cmml">)</mo></mrow></mrow><mo id="S4.SS2.SSS3.Px1.p3.15.m15.7.7.3" xref="S4.SS2.SSS3.Px1.p3.15.m15.7.7.3.cmml">≤</mo><mrow id="S4.SS2.SSS3.Px1.p3.15.m15.7.7.2" xref="S4.SS2.SSS3.Px1.p3.15.m15.7.7.2.cmml"><msub id="S4.SS2.SSS3.Px1.p3.15.m15.7.7.2.3" xref="S4.SS2.SSS3.Px1.p3.15.m15.7.7.2.3.cmml"><mi id="S4.SS2.SSS3.Px1.p3.15.m15.7.7.2.3.2" xref="S4.SS2.SSS3.Px1.p3.15.m15.7.7.2.3.2.cmml">d</mi><mi id="S4.SS2.SSS3.Px1.p3.15.m15.7.7.2.3.3" xref="S4.SS2.SSS3.Px1.p3.15.m15.7.7.2.3.3.cmml">T</mi></msub><mo id="S4.SS2.SSS3.Px1.p3.15.m15.7.7.2.2" xref="S4.SS2.SSS3.Px1.p3.15.m15.7.7.2.2.cmml">⁢</mo><mrow id="S4.SS2.SSS3.Px1.p3.15.m15.7.7.2.1.1" xref="S4.SS2.SSS3.Px1.p3.15.m15.7.7.2.1.2.cmml"><mo id="S4.SS2.SSS3.Px1.p3.15.m15.7.7.2.1.1.2" stretchy="false" xref="S4.SS2.SSS3.Px1.p3.15.m15.7.7.2.1.2.cmml">(</mo><mi id="S4.SS2.SSS3.Px1.p3.15.m15.5.5" xref="S4.SS2.SSS3.Px1.p3.15.m15.5.5.cmml">r</mi><mo id="S4.SS2.SSS3.Px1.p3.15.m15.7.7.2.1.1.3" xref="S4.SS2.SSS3.Px1.p3.15.m15.7.7.2.1.2.cmml">,</mo><mrow id="S4.SS2.SSS3.Px1.p3.15.m15.7.7.2.1.1.1" xref="S4.SS2.SSS3.Px1.p3.15.m15.7.7.2.1.1.1.cmml"><mtext id="S4.SS2.SSS3.Px1.p3.15.m15.7.7.2.1.1.1.4" xref="S4.SS2.SSS3.Px1.p3.15.m15.7.7.2.1.1.1.4a.cmml">LCA</mtext><mo id="S4.SS2.SSS3.Px1.p3.15.m15.7.7.2.1.1.1.3" xref="S4.SS2.SSS3.Px1.p3.15.m15.7.7.2.1.1.1.3.cmml">⁢</mo><mrow id="S4.SS2.SSS3.Px1.p3.15.m15.7.7.2.1.1.1.2.2" xref="S4.SS2.SSS3.Px1.p3.15.m15.7.7.2.1.1.1.2.3.cmml"><mo id="S4.SS2.SSS3.Px1.p3.15.m15.7.7.2.1.1.1.2.2.3" stretchy="false" xref="S4.SS2.SSS3.Px1.p3.15.m15.7.7.2.1.1.1.2.3.cmml">(</mo><mrow id="S4.SS2.SSS3.Px1.p3.15.m15.7.7.2.1.1.1.1.1.1" xref="S4.SS2.SSS3.Px1.p3.15.m15.7.7.2.1.1.1.1.1.1.cmml"><mi id="S4.SS2.SSS3.Px1.p3.15.m15.7.7.2.1.1.1.1.1.1.2" xref="S4.SS2.SSS3.Px1.p3.15.m15.7.7.2.1.1.1.1.1.1.2.cmml">h</mi><mo id="S4.SS2.SSS3.Px1.p3.15.m15.7.7.2.1.1.1.1.1.1.1" xref="S4.SS2.SSS3.Px1.p3.15.m15.7.7.2.1.1.1.1.1.1.1.cmml">⁢</mo><mrow id="S4.SS2.SSS3.Px1.p3.15.m15.7.7.2.1.1.1.1.1.1.3.2" xref="S4.SS2.SSS3.Px1.p3.15.m15.7.7.2.1.1.1.1.1.1.cmml"><mo id="S4.SS2.SSS3.Px1.p3.15.m15.7.7.2.1.1.1.1.1.1.3.2.1" stretchy="false" xref="S4.SS2.SSS3.Px1.p3.15.m15.7.7.2.1.1.1.1.1.1.cmml">(</mo><mi id="S4.SS2.SSS3.Px1.p3.15.m15.3.3" xref="S4.SS2.SSS3.Px1.p3.15.m15.3.3.cmml">u</mi><mo id="S4.SS2.SSS3.Px1.p3.15.m15.7.7.2.1.1.1.1.1.1.3.2.2" stretchy="false" xref="S4.SS2.SSS3.Px1.p3.15.m15.7.7.2.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S4.SS2.SSS3.Px1.p3.15.m15.7.7.2.1.1.1.2.2.4" xref="S4.SS2.SSS3.Px1.p3.15.m15.7.7.2.1.1.1.2.3.cmml">,</mo><mrow id="S4.SS2.SSS3.Px1.p3.15.m15.7.7.2.1.1.1.2.2.2" 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stretchy="false" xref="S4.SS2.SSS3.Px1.p3.15.m15.7.7.2.1.2.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS3.Px1.p3.15.m15.7b"><apply id="S4.SS2.SSS3.Px1.p3.15.m15.7.7.cmml" xref="S4.SS2.SSS3.Px1.p3.15.m15.7.7"><leq id="S4.SS2.SSS3.Px1.p3.15.m15.7.7.3.cmml" xref="S4.SS2.SSS3.Px1.p3.15.m15.7.7.3"></leq><apply id="S4.SS2.SSS3.Px1.p3.15.m15.6.6.1.cmml" xref="S4.SS2.SSS3.Px1.p3.15.m15.6.6.1"><times id="S4.SS2.SSS3.Px1.p3.15.m15.6.6.1.2.cmml" xref="S4.SS2.SSS3.Px1.p3.15.m15.6.6.1.2"></times><apply id="S4.SS2.SSS3.Px1.p3.15.m15.6.6.1.3.cmml" xref="S4.SS2.SSS3.Px1.p3.15.m15.6.6.1.3"><csymbol cd="ambiguous" id="S4.SS2.SSS3.Px1.p3.15.m15.6.6.1.3.1.cmml" xref="S4.SS2.SSS3.Px1.p3.15.m15.6.6.1.3">subscript</csymbol><ci id="S4.SS2.SSS3.Px1.p3.15.m15.6.6.1.3.2.cmml" xref="S4.SS2.SSS3.Px1.p3.15.m15.6.6.1.3.2">𝑑</ci><ci id="S4.SS2.SSS3.Px1.p3.15.m15.6.6.1.3.3.cmml" xref="S4.SS2.SSS3.Px1.p3.15.m15.6.6.1.3.3">𝑇</ci></apply><interval closure="open" 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xref="S4.SS2.SSS3.Px1.p3.15.m15.5.5">𝑟</ci><apply id="S4.SS2.SSS3.Px1.p3.15.m15.7.7.2.1.1.1.cmml" xref="S4.SS2.SSS3.Px1.p3.15.m15.7.7.2.1.1.1"><times id="S4.SS2.SSS3.Px1.p3.15.m15.7.7.2.1.1.1.3.cmml" xref="S4.SS2.SSS3.Px1.p3.15.m15.7.7.2.1.1.1.3"></times><ci id="S4.SS2.SSS3.Px1.p3.15.m15.7.7.2.1.1.1.4a.cmml" xref="S4.SS2.SSS3.Px1.p3.15.m15.7.7.2.1.1.1.4"><mtext id="S4.SS2.SSS3.Px1.p3.15.m15.7.7.2.1.1.1.4.cmml" xref="S4.SS2.SSS3.Px1.p3.15.m15.7.7.2.1.1.1.4">LCA</mtext></ci><interval closure="open" id="S4.SS2.SSS3.Px1.p3.15.m15.7.7.2.1.1.1.2.3.cmml" xref="S4.SS2.SSS3.Px1.p3.15.m15.7.7.2.1.1.1.2.2"><apply id="S4.SS2.SSS3.Px1.p3.15.m15.7.7.2.1.1.1.1.1.1.cmml" xref="S4.SS2.SSS3.Px1.p3.15.m15.7.7.2.1.1.1.1.1.1"><times id="S4.SS2.SSS3.Px1.p3.15.m15.7.7.2.1.1.1.1.1.1.1.cmml" xref="S4.SS2.SSS3.Px1.p3.15.m15.7.7.2.1.1.1.1.1.1.1"></times><ci id="S4.SS2.SSS3.Px1.p3.15.m15.7.7.2.1.1.1.1.1.1.2.cmml" xref="S4.SS2.SSS3.Px1.p3.15.m15.7.7.2.1.1.1.1.1.1.2">ℎ</ci><ci id="S4.SS2.SSS3.Px1.p3.15.m15.3.3.cmml" xref="S4.SS2.SSS3.Px1.p3.15.m15.3.3">𝑢</ci></apply><apply id="S4.SS2.SSS3.Px1.p3.15.m15.7.7.2.1.1.1.2.2.2.cmml" xref="S4.SS2.SSS3.Px1.p3.15.m15.7.7.2.1.1.1.2.2.2"><times id="S4.SS2.SSS3.Px1.p3.15.m15.7.7.2.1.1.1.2.2.2.1.cmml" xref="S4.SS2.SSS3.Px1.p3.15.m15.7.7.2.1.1.1.2.2.2.1"></times><ci id="S4.SS2.SSS3.Px1.p3.15.m15.7.7.2.1.1.1.2.2.2.2.cmml" xref="S4.SS2.SSS3.Px1.p3.15.m15.7.7.2.1.1.1.2.2.2.2">ℓ</ci><ci id="S4.SS2.SSS3.Px1.p3.15.m15.4.4.cmml" xref="S4.SS2.SSS3.Px1.p3.15.m15.4.4">𝑣</ci></apply></interval></apply></interval></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS3.Px1.p3.15.m15.7c">d_{T}(r,\text{LCA}(h(u),\ell(u^{\prime\prime})))\leq d_{T}(r,\text{LCA}(h(u),% \ell(v)))</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS3.Px1.p3.15.m15.7d">italic_d start_POSTSUBSCRIPT italic_T end_POSTSUBSCRIPT ( italic_r , LCA ( italic_h ( italic_u ) , roman_ℓ ( italic_u start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT ) ) ) ≤ italic_d start_POSTSUBSCRIPT italic_T end_POSTSUBSCRIPT ( italic_r , LCA ( italic_h ( italic_u ) , roman_ℓ ( italic_v ) ) )</annotation></semantics></math>. In particular, <math alttext="\text{LCA}(h(u),\ell(u^{\prime\prime}))\in T\setminus T_{x}" class="ltx_Math" display="inline" id="S4.SS2.SSS3.Px1.p3.16.m16.3"><semantics id="S4.SS2.SSS3.Px1.p3.16.m16.3a"><mrow id="S4.SS2.SSS3.Px1.p3.16.m16.3.3" xref="S4.SS2.SSS3.Px1.p3.16.m16.3.3.cmml"><mrow id="S4.SS2.SSS3.Px1.p3.16.m16.3.3.2" xref="S4.SS2.SSS3.Px1.p3.16.m16.3.3.2.cmml"><mtext id="S4.SS2.SSS3.Px1.p3.16.m16.3.3.2.4" xref="S4.SS2.SSS3.Px1.p3.16.m16.3.3.2.4a.cmml">LCA</mtext><mo id="S4.SS2.SSS3.Px1.p3.16.m16.3.3.2.3" xref="S4.SS2.SSS3.Px1.p3.16.m16.3.3.2.3.cmml">⁢</mo><mrow id="S4.SS2.SSS3.Px1.p3.16.m16.3.3.2.2.2" xref="S4.SS2.SSS3.Px1.p3.16.m16.3.3.2.2.3.cmml"><mo id="S4.SS2.SSS3.Px1.p3.16.m16.3.3.2.2.2.3" stretchy="false" xref="S4.SS2.SSS3.Px1.p3.16.m16.3.3.2.2.3.cmml">(</mo><mrow id="S4.SS2.SSS3.Px1.p3.16.m16.2.2.1.1.1.1" xref="S4.SS2.SSS3.Px1.p3.16.m16.2.2.1.1.1.1.cmml"><mi id="S4.SS2.SSS3.Px1.p3.16.m16.2.2.1.1.1.1.2" xref="S4.SS2.SSS3.Px1.p3.16.m16.2.2.1.1.1.1.2.cmml">h</mi><mo 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xref="S4.SS2.SSS3.Px1.p3.16.m16.3.3.2.3"></times><ci id="S4.SS2.SSS3.Px1.p3.16.m16.3.3.2.4a.cmml" xref="S4.SS2.SSS3.Px1.p3.16.m16.3.3.2.4"><mtext id="S4.SS2.SSS3.Px1.p3.16.m16.3.3.2.4.cmml" xref="S4.SS2.SSS3.Px1.p3.16.m16.3.3.2.4">LCA</mtext></ci><interval closure="open" id="S4.SS2.SSS3.Px1.p3.16.m16.3.3.2.2.3.cmml" xref="S4.SS2.SSS3.Px1.p3.16.m16.3.3.2.2.2"><apply id="S4.SS2.SSS3.Px1.p3.16.m16.2.2.1.1.1.1.cmml" xref="S4.SS2.SSS3.Px1.p3.16.m16.2.2.1.1.1.1"><times id="S4.SS2.SSS3.Px1.p3.16.m16.2.2.1.1.1.1.1.cmml" xref="S4.SS2.SSS3.Px1.p3.16.m16.2.2.1.1.1.1.1"></times><ci id="S4.SS2.SSS3.Px1.p3.16.m16.2.2.1.1.1.1.2.cmml" xref="S4.SS2.SSS3.Px1.p3.16.m16.2.2.1.1.1.1.2">ℎ</ci><ci id="S4.SS2.SSS3.Px1.p3.16.m16.1.1.cmml" xref="S4.SS2.SSS3.Px1.p3.16.m16.1.1">𝑢</ci></apply><apply id="S4.SS2.SSS3.Px1.p3.16.m16.3.3.2.2.2.2.cmml" xref="S4.SS2.SSS3.Px1.p3.16.m16.3.3.2.2.2.2"><times id="S4.SS2.SSS3.Px1.p3.16.m16.3.3.2.2.2.2.2.cmml" xref="S4.SS2.SSS3.Px1.p3.16.m16.3.3.2.2.2.2.2"></times><ci id="S4.SS2.SSS3.Px1.p3.16.m16.3.3.2.2.2.2.3.cmml" xref="S4.SS2.SSS3.Px1.p3.16.m16.3.3.2.2.2.2.3">ℓ</ci><apply id="S4.SS2.SSS3.Px1.p3.16.m16.3.3.2.2.2.2.1.1.1.cmml" xref="S4.SS2.SSS3.Px1.p3.16.m16.3.3.2.2.2.2.1.1"><csymbol cd="ambiguous" id="S4.SS2.SSS3.Px1.p3.16.m16.3.3.2.2.2.2.1.1.1.1.cmml" xref="S4.SS2.SSS3.Px1.p3.16.m16.3.3.2.2.2.2.1.1">superscript</csymbol><ci id="S4.SS2.SSS3.Px1.p3.16.m16.3.3.2.2.2.2.1.1.1.2.cmml" xref="S4.SS2.SSS3.Px1.p3.16.m16.3.3.2.2.2.2.1.1.1.2">𝑢</ci><ci id="S4.SS2.SSS3.Px1.p3.16.m16.3.3.2.2.2.2.1.1.1.3.cmml" xref="S4.SS2.SSS3.Px1.p3.16.m16.3.3.2.2.2.2.1.1.1.3">′′</ci></apply></apply></interval></apply><apply id="S4.SS2.SSS3.Px1.p3.16.m16.3.3.4.cmml" xref="S4.SS2.SSS3.Px1.p3.16.m16.3.3.4"><setdiff id="S4.SS2.SSS3.Px1.p3.16.m16.3.3.4.1.cmml" xref="S4.SS2.SSS3.Px1.p3.16.m16.3.3.4.1"></setdiff><ci id="S4.SS2.SSS3.Px1.p3.16.m16.3.3.4.2.cmml" xref="S4.SS2.SSS3.Px1.p3.16.m16.3.3.4.2">𝑇</ci><apply id="S4.SS2.SSS3.Px1.p3.16.m16.3.3.4.3.cmml" xref="S4.SS2.SSS3.Px1.p3.16.m16.3.3.4.3"><csymbol cd="ambiguous" id="S4.SS2.SSS3.Px1.p3.16.m16.3.3.4.3.1.cmml" xref="S4.SS2.SSS3.Px1.p3.16.m16.3.3.4.3">subscript</csymbol><ci id="S4.SS2.SSS3.Px1.p3.16.m16.3.3.4.3.2.cmml" xref="S4.SS2.SSS3.Px1.p3.16.m16.3.3.4.3.2">𝑇</ci><ci id="S4.SS2.SSS3.Px1.p3.16.m16.3.3.4.3.3.cmml" xref="S4.SS2.SSS3.Px1.p3.16.m16.3.3.4.3.3">𝑥</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS3.Px1.p3.16.m16.3c">\text{LCA}(h(u),\ell(u^{\prime\prime}))\in T\setminus T_{x}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS3.Px1.p3.16.m16.3d">LCA ( italic_h ( italic_u ) , roman_ℓ ( italic_u start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT ) ) ∈ italic_T ∖ italic_T start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT</annotation></semantics></math>. See Figure <a class="ltx_ref" href="https://arxiv.org/html/2503.00712v1#S4.F8" title="Figure 8 ‣ Case 1: 𝒆=𝒂⁢𝒃 is a virtual edge of an R-node and an R or S-node: ‣ 4.2.3 Bounding the Approximation Ratio ‣ 4.2 Two-to-Three Augmentation ‣ 4 Vertex Connectivity Augmentation in Link-Arrival Model ‣ Streaming Algorithms for Network Design"><span class="ltx_text ltx_ref_tag">8</span></a> for reference.</p> </div> <div class="ltx_para" id="S4.SS2.SSS3.Px1.p4"> <p class="ltx_p" id="S4.SS2.SSS3.Px1.p4.4">This gives us the following <math alttext="u" class="ltx_Math" display="inline" id="S4.SS2.SSS3.Px1.p4.1.m1.1"><semantics id="S4.SS2.SSS3.Px1.p4.1.m1.1a"><mi id="S4.SS2.SSS3.Px1.p4.1.m1.1.1" xref="S4.SS2.SSS3.Px1.p4.1.m1.1.1.cmml">u</mi><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS3.Px1.p4.1.m1.1b"><ci id="S4.SS2.SSS3.Px1.p4.1.m1.1.1.cmml" xref="S4.SS2.SSS3.Px1.p4.1.m1.1.1">𝑢</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS3.Px1.p4.1.m1.1c">u</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS3.Px1.p4.1.m1.1d">italic_u</annotation></semantics></math>-<math alttext="v" class="ltx_Math" display="inline" id="S4.SS2.SSS3.Px1.p4.2.m2.1"><semantics id="S4.SS2.SSS3.Px1.p4.2.m2.1a"><mi id="S4.SS2.SSS3.Px1.p4.2.m2.1.1" xref="S4.SS2.SSS3.Px1.p4.2.m2.1.1.cmml">v</mi><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS3.Px1.p4.2.m2.1b"><ci id="S4.SS2.SSS3.Px1.p4.2.m2.1.1.cmml" xref="S4.SS2.SSS3.Px1.p4.2.m2.1.1">𝑣</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS3.Px1.p4.2.m2.1c">v</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS3.Px1.p4.2.m2.1d">italic_v</annotation></semantics></math> path in <math alttext="\textnormal{SOL}\cup E" class="ltx_Math" display="inline" id="S4.SS2.SSS3.Px1.p4.3.m3.1"><semantics id="S4.SS2.SSS3.Px1.p4.3.m3.1a"><mrow id="S4.SS2.SSS3.Px1.p4.3.m3.1.1" xref="S4.SS2.SSS3.Px1.p4.3.m3.1.1.cmml"><mtext id="S4.SS2.SSS3.Px1.p4.3.m3.1.1.2" xref="S4.SS2.SSS3.Px1.p4.3.m3.1.1.2a.cmml">SOL</mtext><mo id="S4.SS2.SSS3.Px1.p4.3.m3.1.1.1" xref="S4.SS2.SSS3.Px1.p4.3.m3.1.1.1.cmml">∪</mo><mi id="S4.SS2.SSS3.Px1.p4.3.m3.1.1.3" xref="S4.SS2.SSS3.Px1.p4.3.m3.1.1.3.cmml">E</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS3.Px1.p4.3.m3.1b"><apply id="S4.SS2.SSS3.Px1.p4.3.m3.1.1.cmml" xref="S4.SS2.SSS3.Px1.p4.3.m3.1.1"><union id="S4.SS2.SSS3.Px1.p4.3.m3.1.1.1.cmml" xref="S4.SS2.SSS3.Px1.p4.3.m3.1.1.1"></union><ci id="S4.SS2.SSS3.Px1.p4.3.m3.1.1.2a.cmml" xref="S4.SS2.SSS3.Px1.p4.3.m3.1.1.2"><mtext id="S4.SS2.SSS3.Px1.p4.3.m3.1.1.2.cmml" xref="S4.SS2.SSS3.Px1.p4.3.m3.1.1.2">SOL</mtext></ci><ci id="S4.SS2.SSS3.Px1.p4.3.m3.1.1.3.cmml" xref="S4.SS2.SSS3.Px1.p4.3.m3.1.1.3">𝐸</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS3.Px1.p4.3.m3.1c">\textnormal{SOL}\cup E</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS3.Px1.p4.3.m3.1d">SOL ∪ italic_E</annotation></semantics></math> without using <math alttext="\{a,b\}" class="ltx_Math" display="inline" id="S4.SS2.SSS3.Px1.p4.4.m4.2"><semantics id="S4.SS2.SSS3.Px1.p4.4.m4.2a"><mrow id="S4.SS2.SSS3.Px1.p4.4.m4.2.3.2" xref="S4.SS2.SSS3.Px1.p4.4.m4.2.3.1.cmml"><mo id="S4.SS2.SSS3.Px1.p4.4.m4.2.3.2.1" stretchy="false" xref="S4.SS2.SSS3.Px1.p4.4.m4.2.3.1.cmml">{</mo><mi id="S4.SS2.SSS3.Px1.p4.4.m4.1.1" xref="S4.SS2.SSS3.Px1.p4.4.m4.1.1.cmml">a</mi><mo id="S4.SS2.SSS3.Px1.p4.4.m4.2.3.2.2" xref="S4.SS2.SSS3.Px1.p4.4.m4.2.3.1.cmml">,</mo><mi id="S4.SS2.SSS3.Px1.p4.4.m4.2.2" xref="S4.SS2.SSS3.Px1.p4.4.m4.2.2.cmml">b</mi><mo id="S4.SS2.SSS3.Px1.p4.4.m4.2.3.2.3" stretchy="false" xref="S4.SS2.SSS3.Px1.p4.4.m4.2.3.1.cmml">}</mo></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS3.Px1.p4.4.m4.2b"><set id="S4.SS2.SSS3.Px1.p4.4.m4.2.3.1.cmml" xref="S4.SS2.SSS3.Px1.p4.4.m4.2.3.2"><ci id="S4.SS2.SSS3.Px1.p4.4.m4.1.1.cmml" xref="S4.SS2.SSS3.Px1.p4.4.m4.1.1">𝑎</ci><ci id="S4.SS2.SSS3.Px1.p4.4.m4.2.2.cmml" xref="S4.SS2.SSS3.Px1.p4.4.m4.2.2">𝑏</ci></set></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS3.Px1.p4.4.m4.2c">\{a,b\}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS3.Px1.p4.4.m4.2d">{ italic_a , italic_b }</annotation></semantics></math>.</p> <ul class="ltx_itemize" id="S4.I9"> <li class="ltx_item" id="S4.I9.i1" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">•</span> <div class="ltx_para" id="S4.I9.i1.p1"> <p class="ltx_p" id="S4.I9.i1.p1.12"><math alttext="u\to u^{\prime}" class="ltx_Math" display="inline" id="S4.I9.i1.p1.1.m1.1"><semantics id="S4.I9.i1.p1.1.m1.1a"><mrow id="S4.I9.i1.p1.1.m1.1.1" xref="S4.I9.i1.p1.1.m1.1.1.cmml"><mi id="S4.I9.i1.p1.1.m1.1.1.2" xref="S4.I9.i1.p1.1.m1.1.1.2.cmml">u</mi><mo id="S4.I9.i1.p1.1.m1.1.1.1" stretchy="false" xref="S4.I9.i1.p1.1.m1.1.1.1.cmml">→</mo><msup id="S4.I9.i1.p1.1.m1.1.1.3" xref="S4.I9.i1.p1.1.m1.1.1.3.cmml"><mi id="S4.I9.i1.p1.1.m1.1.1.3.2" xref="S4.I9.i1.p1.1.m1.1.1.3.2.cmml">u</mi><mo id="S4.I9.i1.p1.1.m1.1.1.3.3" xref="S4.I9.i1.p1.1.m1.1.1.3.3.cmml">′</mo></msup></mrow><annotation-xml encoding="MathML-Content" id="S4.I9.i1.p1.1.m1.1b"><apply id="S4.I9.i1.p1.1.m1.1.1.cmml" xref="S4.I9.i1.p1.1.m1.1.1"><ci id="S4.I9.i1.p1.1.m1.1.1.1.cmml" xref="S4.I9.i1.p1.1.m1.1.1.1">→</ci><ci id="S4.I9.i1.p1.1.m1.1.1.2.cmml" xref="S4.I9.i1.p1.1.m1.1.1.2">𝑢</ci><apply id="S4.I9.i1.p1.1.m1.1.1.3.cmml" xref="S4.I9.i1.p1.1.m1.1.1.3"><csymbol cd="ambiguous" id="S4.I9.i1.p1.1.m1.1.1.3.1.cmml" xref="S4.I9.i1.p1.1.m1.1.1.3">superscript</csymbol><ci id="S4.I9.i1.p1.1.m1.1.1.3.2.cmml" xref="S4.I9.i1.p1.1.m1.1.1.3.2">𝑢</ci><ci id="S4.I9.i1.p1.1.m1.1.1.3.3.cmml" xref="S4.I9.i1.p1.1.m1.1.1.3.3">′</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.I9.i1.p1.1.m1.1c">u\to u^{\prime}</annotation><annotation encoding="application/x-llamapun" id="S4.I9.i1.p1.1.m1.1d">italic_u → italic_u start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math>: since <math alttext="u" class="ltx_Math" display="inline" id="S4.I9.i1.p1.2.m2.1"><semantics id="S4.I9.i1.p1.2.m2.1a"><mi id="S4.I9.i1.p1.2.m2.1.1" xref="S4.I9.i1.p1.2.m2.1.1.cmml">u</mi><annotation-xml encoding="MathML-Content" id="S4.I9.i1.p1.2.m2.1b"><ci id="S4.I9.i1.p1.2.m2.1.1.cmml" xref="S4.I9.i1.p1.2.m2.1.1">𝑢</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.I9.i1.p1.2.m2.1c">u</annotation><annotation encoding="application/x-llamapun" id="S4.I9.i1.p1.2.m2.1d">italic_u</annotation></semantics></math> and <math alttext="u^{\prime}" class="ltx_Math" display="inline" id="S4.I9.i1.p1.3.m3.1"><semantics id="S4.I9.i1.p1.3.m3.1a"><msup id="S4.I9.i1.p1.3.m3.1.1" xref="S4.I9.i1.p1.3.m3.1.1.cmml"><mi id="S4.I9.i1.p1.3.m3.1.1.2" xref="S4.I9.i1.p1.3.m3.1.1.2.cmml">u</mi><mo id="S4.I9.i1.p1.3.m3.1.1.3" xref="S4.I9.i1.p1.3.m3.1.1.3.cmml">′</mo></msup><annotation-xml encoding="MathML-Content" id="S4.I9.i1.p1.3.m3.1b"><apply id="S4.I9.i1.p1.3.m3.1.1.cmml" xref="S4.I9.i1.p1.3.m3.1.1"><csymbol cd="ambiguous" id="S4.I9.i1.p1.3.m3.1.1.1.cmml" xref="S4.I9.i1.p1.3.m3.1.1">superscript</csymbol><ci id="S4.I9.i1.p1.3.m3.1.1.2.cmml" xref="S4.I9.i1.p1.3.m3.1.1.2">𝑢</ci><ci id="S4.I9.i1.p1.3.m3.1.1.3.cmml" xref="S4.I9.i1.p1.3.m3.1.1.3">′</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.I9.i1.p1.3.m3.1c">u^{\prime}</annotation><annotation encoding="application/x-llamapun" id="S4.I9.i1.p1.3.m3.1d">italic_u start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math> are both in the vertex set of <math alttext="G_{h(u)}" class="ltx_Math" display="inline" id="S4.I9.i1.p1.4.m4.1"><semantics id="S4.I9.i1.p1.4.m4.1a"><msub id="S4.I9.i1.p1.4.m4.1.2" xref="S4.I9.i1.p1.4.m4.1.2.cmml"><mi id="S4.I9.i1.p1.4.m4.1.2.2" xref="S4.I9.i1.p1.4.m4.1.2.2.cmml">G</mi><mrow id="S4.I9.i1.p1.4.m4.1.1.1" xref="S4.I9.i1.p1.4.m4.1.1.1.cmml"><mi id="S4.I9.i1.p1.4.m4.1.1.1.3" xref="S4.I9.i1.p1.4.m4.1.1.1.3.cmml">h</mi><mo id="S4.I9.i1.p1.4.m4.1.1.1.2" xref="S4.I9.i1.p1.4.m4.1.1.1.2.cmml">⁢</mo><mrow id="S4.I9.i1.p1.4.m4.1.1.1.4.2" xref="S4.I9.i1.p1.4.m4.1.1.1.cmml"><mo id="S4.I9.i1.p1.4.m4.1.1.1.4.2.1" stretchy="false" xref="S4.I9.i1.p1.4.m4.1.1.1.cmml">(</mo><mi id="S4.I9.i1.p1.4.m4.1.1.1.1" xref="S4.I9.i1.p1.4.m4.1.1.1.1.cmml">u</mi><mo id="S4.I9.i1.p1.4.m4.1.1.1.4.2.2" stretchy="false" xref="S4.I9.i1.p1.4.m4.1.1.1.cmml">)</mo></mrow></mrow></msub><annotation-xml encoding="MathML-Content" id="S4.I9.i1.p1.4.m4.1b"><apply id="S4.I9.i1.p1.4.m4.1.2.cmml" xref="S4.I9.i1.p1.4.m4.1.2"><csymbol cd="ambiguous" id="S4.I9.i1.p1.4.m4.1.2.1.cmml" xref="S4.I9.i1.p1.4.m4.1.2">subscript</csymbol><ci id="S4.I9.i1.p1.4.m4.1.2.2.cmml" xref="S4.I9.i1.p1.4.m4.1.2.2">𝐺</ci><apply id="S4.I9.i1.p1.4.m4.1.1.1.cmml" xref="S4.I9.i1.p1.4.m4.1.1.1"><times id="S4.I9.i1.p1.4.m4.1.1.1.2.cmml" xref="S4.I9.i1.p1.4.m4.1.1.1.2"></times><ci id="S4.I9.i1.p1.4.m4.1.1.1.3.cmml" xref="S4.I9.i1.p1.4.m4.1.1.1.3">ℎ</ci><ci id="S4.I9.i1.p1.4.m4.1.1.1.1.cmml" xref="S4.I9.i1.p1.4.m4.1.1.1.1">𝑢</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.I9.i1.p1.4.m4.1c">G_{h(u)}</annotation><annotation encoding="application/x-llamapun" id="S4.I9.i1.p1.4.m4.1d">italic_G start_POSTSUBSCRIPT italic_h ( italic_u ) end_POSTSUBSCRIPT</annotation></semantics></math>, and all tree nodes stay connected despite the deletion of <math alttext="\{a,b\}" class="ltx_Math" display="inline" id="S4.I9.i1.p1.5.m5.2"><semantics id="S4.I9.i1.p1.5.m5.2a"><mrow id="S4.I9.i1.p1.5.m5.2.3.2" xref="S4.I9.i1.p1.5.m5.2.3.1.cmml"><mo id="S4.I9.i1.p1.5.m5.2.3.2.1" stretchy="false" xref="S4.I9.i1.p1.5.m5.2.3.1.cmml">{</mo><mi id="S4.I9.i1.p1.5.m5.1.1" xref="S4.I9.i1.p1.5.m5.1.1.cmml">a</mi><mo id="S4.I9.i1.p1.5.m5.2.3.2.2" xref="S4.I9.i1.p1.5.m5.2.3.1.cmml">,</mo><mi id="S4.I9.i1.p1.5.m5.2.2" xref="S4.I9.i1.p1.5.m5.2.2.cmml">b</mi><mo id="S4.I9.i1.p1.5.m5.2.3.2.3" stretchy="false" xref="S4.I9.i1.p1.5.m5.2.3.1.cmml">}</mo></mrow><annotation-xml encoding="MathML-Content" id="S4.I9.i1.p1.5.m5.2b"><set id="S4.I9.i1.p1.5.m5.2.3.1.cmml" xref="S4.I9.i1.p1.5.m5.2.3.2"><ci id="S4.I9.i1.p1.5.m5.1.1.cmml" xref="S4.I9.i1.p1.5.m5.1.1">𝑎</ci><ci id="S4.I9.i1.p1.5.m5.2.2.cmml" xref="S4.I9.i1.p1.5.m5.2.2">𝑏</ci></set></annotation-xml><annotation encoding="application/x-tex" id="S4.I9.i1.p1.5.m5.2c">\{a,b\}</annotation><annotation encoding="application/x-llamapun" id="S4.I9.i1.p1.5.m5.2d">{ italic_a , italic_b }</annotation></semantics></math>, there is a path from <math alttext="u" class="ltx_Math" display="inline" id="S4.I9.i1.p1.6.m6.1"><semantics id="S4.I9.i1.p1.6.m6.1a"><mi id="S4.I9.i1.p1.6.m6.1.1" xref="S4.I9.i1.p1.6.m6.1.1.cmml">u</mi><annotation-xml encoding="MathML-Content" id="S4.I9.i1.p1.6.m6.1b"><ci id="S4.I9.i1.p1.6.m6.1.1.cmml" xref="S4.I9.i1.p1.6.m6.1.1">𝑢</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.I9.i1.p1.6.m6.1c">u</annotation><annotation encoding="application/x-llamapun" id="S4.I9.i1.p1.6.m6.1d">italic_u</annotation></semantics></math> to <math alttext="u^{\prime}" class="ltx_Math" display="inline" id="S4.I9.i1.p1.7.m7.1"><semantics id="S4.I9.i1.p1.7.m7.1a"><msup id="S4.I9.i1.p1.7.m7.1.1" xref="S4.I9.i1.p1.7.m7.1.1.cmml"><mi id="S4.I9.i1.p1.7.m7.1.1.2" xref="S4.I9.i1.p1.7.m7.1.1.2.cmml">u</mi><mo id="S4.I9.i1.p1.7.m7.1.1.3" xref="S4.I9.i1.p1.7.m7.1.1.3.cmml">′</mo></msup><annotation-xml encoding="MathML-Content" id="S4.I9.i1.p1.7.m7.1b"><apply id="S4.I9.i1.p1.7.m7.1.1.cmml" xref="S4.I9.i1.p1.7.m7.1.1"><csymbol cd="ambiguous" id="S4.I9.i1.p1.7.m7.1.1.1.cmml" xref="S4.I9.i1.p1.7.m7.1.1">superscript</csymbol><ci id="S4.I9.i1.p1.7.m7.1.1.2.cmml" xref="S4.I9.i1.p1.7.m7.1.1.2">𝑢</ci><ci id="S4.I9.i1.p1.7.m7.1.1.3.cmml" xref="S4.I9.i1.p1.7.m7.1.1.3">′</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.I9.i1.p1.7.m7.1c">u^{\prime}</annotation><annotation encoding="application/x-llamapun" id="S4.I9.i1.p1.7.m7.1d">italic_u start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math> in <math alttext="E" class="ltx_Math" display="inline" id="S4.I9.i1.p1.8.m8.1"><semantics id="S4.I9.i1.p1.8.m8.1a"><mi id="S4.I9.i1.p1.8.m8.1.1" xref="S4.I9.i1.p1.8.m8.1.1.cmml">E</mi><annotation-xml encoding="MathML-Content" id="S4.I9.i1.p1.8.m8.1b"><ci id="S4.I9.i1.p1.8.m8.1.1.cmml" xref="S4.I9.i1.p1.8.m8.1.1">𝐸</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.I9.i1.p1.8.m8.1c">E</annotation><annotation encoding="application/x-llamapun" id="S4.I9.i1.p1.8.m8.1d">italic_E</annotation></semantics></math> without <math alttext="\{a,b\}" class="ltx_Math" display="inline" id="S4.I9.i1.p1.9.m9.2"><semantics id="S4.I9.i1.p1.9.m9.2a"><mrow id="S4.I9.i1.p1.9.m9.2.3.2" xref="S4.I9.i1.p1.9.m9.2.3.1.cmml"><mo id="S4.I9.i1.p1.9.m9.2.3.2.1" stretchy="false" xref="S4.I9.i1.p1.9.m9.2.3.1.cmml">{</mo><mi id="S4.I9.i1.p1.9.m9.1.1" xref="S4.I9.i1.p1.9.m9.1.1.cmml">a</mi><mo id="S4.I9.i1.p1.9.m9.2.3.2.2" xref="S4.I9.i1.p1.9.m9.2.3.1.cmml">,</mo><mi id="S4.I9.i1.p1.9.m9.2.2" xref="S4.I9.i1.p1.9.m9.2.2.cmml">b</mi><mo id="S4.I9.i1.p1.9.m9.2.3.2.3" stretchy="false" xref="S4.I9.i1.p1.9.m9.2.3.1.cmml">}</mo></mrow><annotation-xml encoding="MathML-Content" id="S4.I9.i1.p1.9.m9.2b"><set id="S4.I9.i1.p1.9.m9.2.3.1.cmml" xref="S4.I9.i1.p1.9.m9.2.3.2"><ci id="S4.I9.i1.p1.9.m9.1.1.cmml" xref="S4.I9.i1.p1.9.m9.1.1">𝑎</ci><ci id="S4.I9.i1.p1.9.m9.2.2.cmml" xref="S4.I9.i1.p1.9.m9.2.2">𝑏</ci></set></annotation-xml><annotation encoding="application/x-tex" id="S4.I9.i1.p1.9.m9.2c">\{a,b\}</annotation><annotation encoding="application/x-llamapun" id="S4.I9.i1.p1.9.m9.2d">{ italic_a , italic_b }</annotation></semantics></math>. Note that <math alttext="u^{\prime}\notin\{a,b\}" class="ltx_Math" display="inline" id="S4.I9.i1.p1.10.m10.2"><semantics id="S4.I9.i1.p1.10.m10.2a"><mrow id="S4.I9.i1.p1.10.m10.2.3" xref="S4.I9.i1.p1.10.m10.2.3.cmml"><msup id="S4.I9.i1.p1.10.m10.2.3.2" xref="S4.I9.i1.p1.10.m10.2.3.2.cmml"><mi id="S4.I9.i1.p1.10.m10.2.3.2.2" xref="S4.I9.i1.p1.10.m10.2.3.2.2.cmml">u</mi><mo id="S4.I9.i1.p1.10.m10.2.3.2.3" xref="S4.I9.i1.p1.10.m10.2.3.2.3.cmml">′</mo></msup><mo id="S4.I9.i1.p1.10.m10.2.3.1" xref="S4.I9.i1.p1.10.m10.2.3.1.cmml">∉</mo><mrow id="S4.I9.i1.p1.10.m10.2.3.3.2" xref="S4.I9.i1.p1.10.m10.2.3.3.1.cmml"><mo id="S4.I9.i1.p1.10.m10.2.3.3.2.1" stretchy="false" xref="S4.I9.i1.p1.10.m10.2.3.3.1.cmml">{</mo><mi id="S4.I9.i1.p1.10.m10.1.1" xref="S4.I9.i1.p1.10.m10.1.1.cmml">a</mi><mo id="S4.I9.i1.p1.10.m10.2.3.3.2.2" xref="S4.I9.i1.p1.10.m10.2.3.3.1.cmml">,</mo><mi id="S4.I9.i1.p1.10.m10.2.2" xref="S4.I9.i1.p1.10.m10.2.2.cmml">b</mi><mo id="S4.I9.i1.p1.10.m10.2.3.3.2.3" stretchy="false" xref="S4.I9.i1.p1.10.m10.2.3.3.1.cmml">}</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.I9.i1.p1.10.m10.2b"><apply id="S4.I9.i1.p1.10.m10.2.3.cmml" xref="S4.I9.i1.p1.10.m10.2.3"><notin id="S4.I9.i1.p1.10.m10.2.3.1.cmml" xref="S4.I9.i1.p1.10.m10.2.3.1"></notin><apply id="S4.I9.i1.p1.10.m10.2.3.2.cmml" xref="S4.I9.i1.p1.10.m10.2.3.2"><csymbol cd="ambiguous" id="S4.I9.i1.p1.10.m10.2.3.2.1.cmml" xref="S4.I9.i1.p1.10.m10.2.3.2">superscript</csymbol><ci id="S4.I9.i1.p1.10.m10.2.3.2.2.cmml" xref="S4.I9.i1.p1.10.m10.2.3.2.2">𝑢</ci><ci id="S4.I9.i1.p1.10.m10.2.3.2.3.cmml" xref="S4.I9.i1.p1.10.m10.2.3.2.3">′</ci></apply><set id="S4.I9.i1.p1.10.m10.2.3.3.1.cmml" xref="S4.I9.i1.p1.10.m10.2.3.3.2"><ci id="S4.I9.i1.p1.10.m10.1.1.cmml" xref="S4.I9.i1.p1.10.m10.1.1">𝑎</ci><ci id="S4.I9.i1.p1.10.m10.2.2.cmml" xref="S4.I9.i1.p1.10.m10.2.2">𝑏</ci></set></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.I9.i1.p1.10.m10.2c">u^{\prime}\notin\{a,b\}</annotation><annotation encoding="application/x-llamapun" id="S4.I9.i1.p1.10.m10.2d">italic_u start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ∉ { italic_a , italic_b }</annotation></semantics></math>, since <math alttext="h(u)=h(u^{\prime})\in T_{x}" class="ltx_Math" display="inline" id="S4.I9.i1.p1.11.m11.2"><semantics id="S4.I9.i1.p1.11.m11.2a"><mrow id="S4.I9.i1.p1.11.m11.2.2" xref="S4.I9.i1.p1.11.m11.2.2.cmml"><mrow id="S4.I9.i1.p1.11.m11.2.2.3" xref="S4.I9.i1.p1.11.m11.2.2.3.cmml"><mi id="S4.I9.i1.p1.11.m11.2.2.3.2" xref="S4.I9.i1.p1.11.m11.2.2.3.2.cmml">h</mi><mo id="S4.I9.i1.p1.11.m11.2.2.3.1" xref="S4.I9.i1.p1.11.m11.2.2.3.1.cmml">⁢</mo><mrow id="S4.I9.i1.p1.11.m11.2.2.3.3.2" xref="S4.I9.i1.p1.11.m11.2.2.3.cmml"><mo id="S4.I9.i1.p1.11.m11.2.2.3.3.2.1" stretchy="false" xref="S4.I9.i1.p1.11.m11.2.2.3.cmml">(</mo><mi id="S4.I9.i1.p1.11.m11.1.1" xref="S4.I9.i1.p1.11.m11.1.1.cmml">u</mi><mo id="S4.I9.i1.p1.11.m11.2.2.3.3.2.2" stretchy="false" xref="S4.I9.i1.p1.11.m11.2.2.3.cmml">)</mo></mrow></mrow><mo id="S4.I9.i1.p1.11.m11.2.2.4" xref="S4.I9.i1.p1.11.m11.2.2.4.cmml">=</mo><mrow id="S4.I9.i1.p1.11.m11.2.2.1" xref="S4.I9.i1.p1.11.m11.2.2.1.cmml"><mi id="S4.I9.i1.p1.11.m11.2.2.1.3" xref="S4.I9.i1.p1.11.m11.2.2.1.3.cmml">h</mi><mo id="S4.I9.i1.p1.11.m11.2.2.1.2" xref="S4.I9.i1.p1.11.m11.2.2.1.2.cmml">⁢</mo><mrow id="S4.I9.i1.p1.11.m11.2.2.1.1.1" xref="S4.I9.i1.p1.11.m11.2.2.1.1.1.1.cmml"><mo id="S4.I9.i1.p1.11.m11.2.2.1.1.1.2" stretchy="false" xref="S4.I9.i1.p1.11.m11.2.2.1.1.1.1.cmml">(</mo><msup id="S4.I9.i1.p1.11.m11.2.2.1.1.1.1" xref="S4.I9.i1.p1.11.m11.2.2.1.1.1.1.cmml"><mi id="S4.I9.i1.p1.11.m11.2.2.1.1.1.1.2" xref="S4.I9.i1.p1.11.m11.2.2.1.1.1.1.2.cmml">u</mi><mo id="S4.I9.i1.p1.11.m11.2.2.1.1.1.1.3" xref="S4.I9.i1.p1.11.m11.2.2.1.1.1.1.3.cmml">′</mo></msup><mo id="S4.I9.i1.p1.11.m11.2.2.1.1.1.3" stretchy="false" xref="S4.I9.i1.p1.11.m11.2.2.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S4.I9.i1.p1.11.m11.2.2.5" xref="S4.I9.i1.p1.11.m11.2.2.5.cmml">∈</mo><msub id="S4.I9.i1.p1.11.m11.2.2.6" xref="S4.I9.i1.p1.11.m11.2.2.6.cmml"><mi id="S4.I9.i1.p1.11.m11.2.2.6.2" xref="S4.I9.i1.p1.11.m11.2.2.6.2.cmml">T</mi><mi id="S4.I9.i1.p1.11.m11.2.2.6.3" xref="S4.I9.i1.p1.11.m11.2.2.6.3.cmml">x</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S4.I9.i1.p1.11.m11.2b"><apply id="S4.I9.i1.p1.11.m11.2.2.cmml" xref="S4.I9.i1.p1.11.m11.2.2"><and id="S4.I9.i1.p1.11.m11.2.2a.cmml" xref="S4.I9.i1.p1.11.m11.2.2"></and><apply id="S4.I9.i1.p1.11.m11.2.2b.cmml" xref="S4.I9.i1.p1.11.m11.2.2"><eq id="S4.I9.i1.p1.11.m11.2.2.4.cmml" xref="S4.I9.i1.p1.11.m11.2.2.4"></eq><apply id="S4.I9.i1.p1.11.m11.2.2.3.cmml" xref="S4.I9.i1.p1.11.m11.2.2.3"><times id="S4.I9.i1.p1.11.m11.2.2.3.1.cmml" xref="S4.I9.i1.p1.11.m11.2.2.3.1"></times><ci id="S4.I9.i1.p1.11.m11.2.2.3.2.cmml" xref="S4.I9.i1.p1.11.m11.2.2.3.2">ℎ</ci><ci id="S4.I9.i1.p1.11.m11.1.1.cmml" xref="S4.I9.i1.p1.11.m11.1.1">𝑢</ci></apply><apply id="S4.I9.i1.p1.11.m11.2.2.1.cmml" xref="S4.I9.i1.p1.11.m11.2.2.1"><times id="S4.I9.i1.p1.11.m11.2.2.1.2.cmml" xref="S4.I9.i1.p1.11.m11.2.2.1.2"></times><ci id="S4.I9.i1.p1.11.m11.2.2.1.3.cmml" xref="S4.I9.i1.p1.11.m11.2.2.1.3">ℎ</ci><apply id="S4.I9.i1.p1.11.m11.2.2.1.1.1.1.cmml" xref="S4.I9.i1.p1.11.m11.2.2.1.1.1"><csymbol cd="ambiguous" id="S4.I9.i1.p1.11.m11.2.2.1.1.1.1.1.cmml" xref="S4.I9.i1.p1.11.m11.2.2.1.1.1">superscript</csymbol><ci id="S4.I9.i1.p1.11.m11.2.2.1.1.1.1.2.cmml" xref="S4.I9.i1.p1.11.m11.2.2.1.1.1.1.2">𝑢</ci><ci id="S4.I9.i1.p1.11.m11.2.2.1.1.1.1.3.cmml" xref="S4.I9.i1.p1.11.m11.2.2.1.1.1.1.3">′</ci></apply></apply></apply><apply id="S4.I9.i1.p1.11.m11.2.2c.cmml" xref="S4.I9.i1.p1.11.m11.2.2"><in id="S4.I9.i1.p1.11.m11.2.2.5.cmml" xref="S4.I9.i1.p1.11.m11.2.2.5"></in><share href="https://arxiv.org/html/2503.00712v1#S4.I9.i1.p1.11.m11.2.2.1.cmml" id="S4.I9.i1.p1.11.m11.2.2d.cmml" xref="S4.I9.i1.p1.11.m11.2.2"></share><apply id="S4.I9.i1.p1.11.m11.2.2.6.cmml" xref="S4.I9.i1.p1.11.m11.2.2.6"><csymbol cd="ambiguous" id="S4.I9.i1.p1.11.m11.2.2.6.1.cmml" xref="S4.I9.i1.p1.11.m11.2.2.6">subscript</csymbol><ci id="S4.I9.i1.p1.11.m11.2.2.6.2.cmml" xref="S4.I9.i1.p1.11.m11.2.2.6.2">𝑇</ci><ci id="S4.I9.i1.p1.11.m11.2.2.6.3.cmml" xref="S4.I9.i1.p1.11.m11.2.2.6.3">𝑥</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.I9.i1.p1.11.m11.2c">h(u)=h(u^{\prime})\in T_{x}</annotation><annotation encoding="application/x-llamapun" id="S4.I9.i1.p1.11.m11.2d">italic_h ( italic_u ) = italic_h ( italic_u start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) ∈ italic_T start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT</annotation></semantics></math>, and <math alttext="h(a)=h(b)=y" class="ltx_Math" display="inline" id="S4.I9.i1.p1.12.m12.2"><semantics id="S4.I9.i1.p1.12.m12.2a"><mrow id="S4.I9.i1.p1.12.m12.2.3" xref="S4.I9.i1.p1.12.m12.2.3.cmml"><mrow id="S4.I9.i1.p1.12.m12.2.3.2" xref="S4.I9.i1.p1.12.m12.2.3.2.cmml"><mi id="S4.I9.i1.p1.12.m12.2.3.2.2" xref="S4.I9.i1.p1.12.m12.2.3.2.2.cmml">h</mi><mo id="S4.I9.i1.p1.12.m12.2.3.2.1" xref="S4.I9.i1.p1.12.m12.2.3.2.1.cmml">⁢</mo><mrow id="S4.I9.i1.p1.12.m12.2.3.2.3.2" xref="S4.I9.i1.p1.12.m12.2.3.2.cmml"><mo id="S4.I9.i1.p1.12.m12.2.3.2.3.2.1" stretchy="false" xref="S4.I9.i1.p1.12.m12.2.3.2.cmml">(</mo><mi id="S4.I9.i1.p1.12.m12.1.1" xref="S4.I9.i1.p1.12.m12.1.1.cmml">a</mi><mo id="S4.I9.i1.p1.12.m12.2.3.2.3.2.2" stretchy="false" xref="S4.I9.i1.p1.12.m12.2.3.2.cmml">)</mo></mrow></mrow><mo id="S4.I9.i1.p1.12.m12.2.3.3" xref="S4.I9.i1.p1.12.m12.2.3.3.cmml">=</mo><mrow id="S4.I9.i1.p1.12.m12.2.3.4" xref="S4.I9.i1.p1.12.m12.2.3.4.cmml"><mi id="S4.I9.i1.p1.12.m12.2.3.4.2" xref="S4.I9.i1.p1.12.m12.2.3.4.2.cmml">h</mi><mo id="S4.I9.i1.p1.12.m12.2.3.4.1" xref="S4.I9.i1.p1.12.m12.2.3.4.1.cmml">⁢</mo><mrow id="S4.I9.i1.p1.12.m12.2.3.4.3.2" xref="S4.I9.i1.p1.12.m12.2.3.4.cmml"><mo id="S4.I9.i1.p1.12.m12.2.3.4.3.2.1" stretchy="false" xref="S4.I9.i1.p1.12.m12.2.3.4.cmml">(</mo><mi id="S4.I9.i1.p1.12.m12.2.2" xref="S4.I9.i1.p1.12.m12.2.2.cmml">b</mi><mo id="S4.I9.i1.p1.12.m12.2.3.4.3.2.2" stretchy="false" xref="S4.I9.i1.p1.12.m12.2.3.4.cmml">)</mo></mrow></mrow><mo id="S4.I9.i1.p1.12.m12.2.3.5" xref="S4.I9.i1.p1.12.m12.2.3.5.cmml">=</mo><mi id="S4.I9.i1.p1.12.m12.2.3.6" xref="S4.I9.i1.p1.12.m12.2.3.6.cmml">y</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.I9.i1.p1.12.m12.2b"><apply id="S4.I9.i1.p1.12.m12.2.3.cmml" xref="S4.I9.i1.p1.12.m12.2.3"><and id="S4.I9.i1.p1.12.m12.2.3a.cmml" xref="S4.I9.i1.p1.12.m12.2.3"></and><apply id="S4.I9.i1.p1.12.m12.2.3b.cmml" xref="S4.I9.i1.p1.12.m12.2.3"><eq id="S4.I9.i1.p1.12.m12.2.3.3.cmml" xref="S4.I9.i1.p1.12.m12.2.3.3"></eq><apply id="S4.I9.i1.p1.12.m12.2.3.2.cmml" xref="S4.I9.i1.p1.12.m12.2.3.2"><times id="S4.I9.i1.p1.12.m12.2.3.2.1.cmml" xref="S4.I9.i1.p1.12.m12.2.3.2.1"></times><ci id="S4.I9.i1.p1.12.m12.2.3.2.2.cmml" xref="S4.I9.i1.p1.12.m12.2.3.2.2">ℎ</ci><ci id="S4.I9.i1.p1.12.m12.1.1.cmml" xref="S4.I9.i1.p1.12.m12.1.1">𝑎</ci></apply><apply id="S4.I9.i1.p1.12.m12.2.3.4.cmml" xref="S4.I9.i1.p1.12.m12.2.3.4"><times id="S4.I9.i1.p1.12.m12.2.3.4.1.cmml" xref="S4.I9.i1.p1.12.m12.2.3.4.1"></times><ci id="S4.I9.i1.p1.12.m12.2.3.4.2.cmml" xref="S4.I9.i1.p1.12.m12.2.3.4.2">ℎ</ci><ci id="S4.I9.i1.p1.12.m12.2.2.cmml" xref="S4.I9.i1.p1.12.m12.2.2">𝑏</ci></apply></apply><apply id="S4.I9.i1.p1.12.m12.2.3c.cmml" xref="S4.I9.i1.p1.12.m12.2.3"><eq id="S4.I9.i1.p1.12.m12.2.3.5.cmml" xref="S4.I9.i1.p1.12.m12.2.3.5"></eq><share href="https://arxiv.org/html/2503.00712v1#S4.I9.i1.p1.12.m12.2.3.4.cmml" id="S4.I9.i1.p1.12.m12.2.3d.cmml" xref="S4.I9.i1.p1.12.m12.2.3"></share><ci id="S4.I9.i1.p1.12.m12.2.3.6.cmml" xref="S4.I9.i1.p1.12.m12.2.3.6">𝑦</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.I9.i1.p1.12.m12.2c">h(a)=h(b)=y</annotation><annotation encoding="application/x-llamapun" id="S4.I9.i1.p1.12.m12.2d">italic_h ( italic_a ) = italic_h ( italic_b ) = italic_y</annotation></semantics></math>.</p> </div> </li> <li class="ltx_item" id="S4.I9.i2" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">•</span> <div class="ltx_para" id="S4.I9.i2.p1"> <p class="ltx_p" id="S4.I9.i2.p1.7"><math alttext="u^{\prime}\to u^{\prime\prime}" class="ltx_Math" display="inline" id="S4.I9.i2.p1.1.m1.1"><semantics id="S4.I9.i2.p1.1.m1.1a"><mrow id="S4.I9.i2.p1.1.m1.1.1" xref="S4.I9.i2.p1.1.m1.1.1.cmml"><msup id="S4.I9.i2.p1.1.m1.1.1.2" xref="S4.I9.i2.p1.1.m1.1.1.2.cmml"><mi id="S4.I9.i2.p1.1.m1.1.1.2.2" xref="S4.I9.i2.p1.1.m1.1.1.2.2.cmml">u</mi><mo id="S4.I9.i2.p1.1.m1.1.1.2.3" xref="S4.I9.i2.p1.1.m1.1.1.2.3.cmml">′</mo></msup><mo id="S4.I9.i2.p1.1.m1.1.1.1" stretchy="false" xref="S4.I9.i2.p1.1.m1.1.1.1.cmml">→</mo><msup id="S4.I9.i2.p1.1.m1.1.1.3" xref="S4.I9.i2.p1.1.m1.1.1.3.cmml"><mi id="S4.I9.i2.p1.1.m1.1.1.3.2" xref="S4.I9.i2.p1.1.m1.1.1.3.2.cmml">u</mi><mo id="S4.I9.i2.p1.1.m1.1.1.3.3" xref="S4.I9.i2.p1.1.m1.1.1.3.3.cmml">′′</mo></msup></mrow><annotation-xml encoding="MathML-Content" id="S4.I9.i2.p1.1.m1.1b"><apply id="S4.I9.i2.p1.1.m1.1.1.cmml" xref="S4.I9.i2.p1.1.m1.1.1"><ci id="S4.I9.i2.p1.1.m1.1.1.1.cmml" xref="S4.I9.i2.p1.1.m1.1.1.1">→</ci><apply id="S4.I9.i2.p1.1.m1.1.1.2.cmml" xref="S4.I9.i2.p1.1.m1.1.1.2"><csymbol cd="ambiguous" id="S4.I9.i2.p1.1.m1.1.1.2.1.cmml" xref="S4.I9.i2.p1.1.m1.1.1.2">superscript</csymbol><ci id="S4.I9.i2.p1.1.m1.1.1.2.2.cmml" xref="S4.I9.i2.p1.1.m1.1.1.2.2">𝑢</ci><ci id="S4.I9.i2.p1.1.m1.1.1.2.3.cmml" xref="S4.I9.i2.p1.1.m1.1.1.2.3">′</ci></apply><apply id="S4.I9.i2.p1.1.m1.1.1.3.cmml" xref="S4.I9.i2.p1.1.m1.1.1.3"><csymbol cd="ambiguous" id="S4.I9.i2.p1.1.m1.1.1.3.1.cmml" xref="S4.I9.i2.p1.1.m1.1.1.3">superscript</csymbol><ci id="S4.I9.i2.p1.1.m1.1.1.3.2.cmml" xref="S4.I9.i2.p1.1.m1.1.1.3.2">𝑢</ci><ci id="S4.I9.i2.p1.1.m1.1.1.3.3.cmml" xref="S4.I9.i2.p1.1.m1.1.1.3.3">′′</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.I9.i2.p1.1.m1.1c">u^{\prime}\to u^{\prime\prime}</annotation><annotation encoding="application/x-llamapun" id="S4.I9.i2.p1.1.m1.1d">italic_u start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT → italic_u start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT</annotation></semantics></math>: as described above, this link is contained in <span class="ltx_text ltx_markedasmath" id="S4.I9.i2.p1.7.1">SOL</span>. Note that <math alttext="u^{\prime\prime}\notin\{a,b\}" class="ltx_Math" display="inline" id="S4.I9.i2.p1.3.m3.2"><semantics id="S4.I9.i2.p1.3.m3.2a"><mrow id="S4.I9.i2.p1.3.m3.2.3" xref="S4.I9.i2.p1.3.m3.2.3.cmml"><msup id="S4.I9.i2.p1.3.m3.2.3.2" xref="S4.I9.i2.p1.3.m3.2.3.2.cmml"><mi id="S4.I9.i2.p1.3.m3.2.3.2.2" xref="S4.I9.i2.p1.3.m3.2.3.2.2.cmml">u</mi><mo id="S4.I9.i2.p1.3.m3.2.3.2.3" xref="S4.I9.i2.p1.3.m3.2.3.2.3.cmml">′′</mo></msup><mo id="S4.I9.i2.p1.3.m3.2.3.1" xref="S4.I9.i2.p1.3.m3.2.3.1.cmml">∉</mo><mrow id="S4.I9.i2.p1.3.m3.2.3.3.2" xref="S4.I9.i2.p1.3.m3.2.3.3.1.cmml"><mo id="S4.I9.i2.p1.3.m3.2.3.3.2.1" stretchy="false" xref="S4.I9.i2.p1.3.m3.2.3.3.1.cmml">{</mo><mi id="S4.I9.i2.p1.3.m3.1.1" xref="S4.I9.i2.p1.3.m3.1.1.cmml">a</mi><mo id="S4.I9.i2.p1.3.m3.2.3.3.2.2" xref="S4.I9.i2.p1.3.m3.2.3.3.1.cmml">,</mo><mi id="S4.I9.i2.p1.3.m3.2.2" xref="S4.I9.i2.p1.3.m3.2.2.cmml">b</mi><mo id="S4.I9.i2.p1.3.m3.2.3.3.2.3" stretchy="false" xref="S4.I9.i2.p1.3.m3.2.3.3.1.cmml">}</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.I9.i2.p1.3.m3.2b"><apply id="S4.I9.i2.p1.3.m3.2.3.cmml" xref="S4.I9.i2.p1.3.m3.2.3"><notin id="S4.I9.i2.p1.3.m3.2.3.1.cmml" xref="S4.I9.i2.p1.3.m3.2.3.1"></notin><apply id="S4.I9.i2.p1.3.m3.2.3.2.cmml" xref="S4.I9.i2.p1.3.m3.2.3.2"><csymbol cd="ambiguous" id="S4.I9.i2.p1.3.m3.2.3.2.1.cmml" xref="S4.I9.i2.p1.3.m3.2.3.2">superscript</csymbol><ci id="S4.I9.i2.p1.3.m3.2.3.2.2.cmml" xref="S4.I9.i2.p1.3.m3.2.3.2.2">𝑢</ci><ci id="S4.I9.i2.p1.3.m3.2.3.2.3.cmml" xref="S4.I9.i2.p1.3.m3.2.3.2.3">′′</ci></apply><set id="S4.I9.i2.p1.3.m3.2.3.3.1.cmml" xref="S4.I9.i2.p1.3.m3.2.3.3.2"><ci id="S4.I9.i2.p1.3.m3.1.1.cmml" xref="S4.I9.i2.p1.3.m3.1.1">𝑎</ci><ci id="S4.I9.i2.p1.3.m3.2.2.cmml" xref="S4.I9.i2.p1.3.m3.2.2">𝑏</ci></set></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.I9.i2.p1.3.m3.2c">u^{\prime\prime}\notin\{a,b\}</annotation><annotation encoding="application/x-llamapun" id="S4.I9.i2.p1.3.m3.2d">italic_u start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT ∉ { italic_a , italic_b }</annotation></semantics></math>: <math alttext="\text{LCA}(h(u),\ell(u^{\prime\prime}))\in T\setminus T_{x}" class="ltx_Math" display="inline" id="S4.I9.i2.p1.4.m4.3"><semantics id="S4.I9.i2.p1.4.m4.3a"><mrow id="S4.I9.i2.p1.4.m4.3.3" xref="S4.I9.i2.p1.4.m4.3.3.cmml"><mrow id="S4.I9.i2.p1.4.m4.3.3.2" xref="S4.I9.i2.p1.4.m4.3.3.2.cmml"><mtext id="S4.I9.i2.p1.4.m4.3.3.2.4" xref="S4.I9.i2.p1.4.m4.3.3.2.4a.cmml">LCA</mtext><mo id="S4.I9.i2.p1.4.m4.3.3.2.3" xref="S4.I9.i2.p1.4.m4.3.3.2.3.cmml">⁢</mo><mrow id="S4.I9.i2.p1.4.m4.3.3.2.2.2" xref="S4.I9.i2.p1.4.m4.3.3.2.2.3.cmml"><mo id="S4.I9.i2.p1.4.m4.3.3.2.2.2.3" stretchy="false" xref="S4.I9.i2.p1.4.m4.3.3.2.2.3.cmml">(</mo><mrow id="S4.I9.i2.p1.4.m4.2.2.1.1.1.1" xref="S4.I9.i2.p1.4.m4.2.2.1.1.1.1.cmml"><mi id="S4.I9.i2.p1.4.m4.2.2.1.1.1.1.2" xref="S4.I9.i2.p1.4.m4.2.2.1.1.1.1.2.cmml">h</mi><mo id="S4.I9.i2.p1.4.m4.2.2.1.1.1.1.1" xref="S4.I9.i2.p1.4.m4.2.2.1.1.1.1.1.cmml">⁢</mo><mrow id="S4.I9.i2.p1.4.m4.2.2.1.1.1.1.3.2" xref="S4.I9.i2.p1.4.m4.2.2.1.1.1.1.cmml"><mo id="S4.I9.i2.p1.4.m4.2.2.1.1.1.1.3.2.1" stretchy="false" xref="S4.I9.i2.p1.4.m4.2.2.1.1.1.1.cmml">(</mo><mi id="S4.I9.i2.p1.4.m4.1.1" xref="S4.I9.i2.p1.4.m4.1.1.cmml">u</mi><mo id="S4.I9.i2.p1.4.m4.2.2.1.1.1.1.3.2.2" stretchy="false" xref="S4.I9.i2.p1.4.m4.2.2.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S4.I9.i2.p1.4.m4.3.3.2.2.2.4" xref="S4.I9.i2.p1.4.m4.3.3.2.2.3.cmml">,</mo><mrow id="S4.I9.i2.p1.4.m4.3.3.2.2.2.2" xref="S4.I9.i2.p1.4.m4.3.3.2.2.2.2.cmml"><mi id="S4.I9.i2.p1.4.m4.3.3.2.2.2.2.3" mathvariant="normal" xref="S4.I9.i2.p1.4.m4.3.3.2.2.2.2.3.cmml">ℓ</mi><mo id="S4.I9.i2.p1.4.m4.3.3.2.2.2.2.2" xref="S4.I9.i2.p1.4.m4.3.3.2.2.2.2.2.cmml">⁢</mo><mrow id="S4.I9.i2.p1.4.m4.3.3.2.2.2.2.1.1" xref="S4.I9.i2.p1.4.m4.3.3.2.2.2.2.1.1.1.cmml"><mo id="S4.I9.i2.p1.4.m4.3.3.2.2.2.2.1.1.2" stretchy="false" xref="S4.I9.i2.p1.4.m4.3.3.2.2.2.2.1.1.1.cmml">(</mo><msup id="S4.I9.i2.p1.4.m4.3.3.2.2.2.2.1.1.1" xref="S4.I9.i2.p1.4.m4.3.3.2.2.2.2.1.1.1.cmml"><mi id="S4.I9.i2.p1.4.m4.3.3.2.2.2.2.1.1.1.2" xref="S4.I9.i2.p1.4.m4.3.3.2.2.2.2.1.1.1.2.cmml">u</mi><mo id="S4.I9.i2.p1.4.m4.3.3.2.2.2.2.1.1.1.3" xref="S4.I9.i2.p1.4.m4.3.3.2.2.2.2.1.1.1.3.cmml">′′</mo></msup><mo id="S4.I9.i2.p1.4.m4.3.3.2.2.2.2.1.1.3" stretchy="false" xref="S4.I9.i2.p1.4.m4.3.3.2.2.2.2.1.1.1.cmml">)</mo></mrow></mrow><mo id="S4.I9.i2.p1.4.m4.3.3.2.2.2.5" stretchy="false" xref="S4.I9.i2.p1.4.m4.3.3.2.2.3.cmml">)</mo></mrow></mrow><mo id="S4.I9.i2.p1.4.m4.3.3.3" xref="S4.I9.i2.p1.4.m4.3.3.3.cmml">∈</mo><mrow id="S4.I9.i2.p1.4.m4.3.3.4" xref="S4.I9.i2.p1.4.m4.3.3.4.cmml"><mi id="S4.I9.i2.p1.4.m4.3.3.4.2" xref="S4.I9.i2.p1.4.m4.3.3.4.2.cmml">T</mi><mo id="S4.I9.i2.p1.4.m4.3.3.4.1" xref="S4.I9.i2.p1.4.m4.3.3.4.1.cmml">∖</mo><msub id="S4.I9.i2.p1.4.m4.3.3.4.3" xref="S4.I9.i2.p1.4.m4.3.3.4.3.cmml"><mi id="S4.I9.i2.p1.4.m4.3.3.4.3.2" xref="S4.I9.i2.p1.4.m4.3.3.4.3.2.cmml">T</mi><mi id="S4.I9.i2.p1.4.m4.3.3.4.3.3" xref="S4.I9.i2.p1.4.m4.3.3.4.3.3.cmml">x</mi></msub></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.I9.i2.p1.4.m4.3b"><apply id="S4.I9.i2.p1.4.m4.3.3.cmml" xref="S4.I9.i2.p1.4.m4.3.3"><in id="S4.I9.i2.p1.4.m4.3.3.3.cmml" xref="S4.I9.i2.p1.4.m4.3.3.3"></in><apply id="S4.I9.i2.p1.4.m4.3.3.2.cmml" xref="S4.I9.i2.p1.4.m4.3.3.2"><times id="S4.I9.i2.p1.4.m4.3.3.2.3.cmml" xref="S4.I9.i2.p1.4.m4.3.3.2.3"></times><ci id="S4.I9.i2.p1.4.m4.3.3.2.4a.cmml" xref="S4.I9.i2.p1.4.m4.3.3.2.4"><mtext id="S4.I9.i2.p1.4.m4.3.3.2.4.cmml" xref="S4.I9.i2.p1.4.m4.3.3.2.4">LCA</mtext></ci><interval closure="open" id="S4.I9.i2.p1.4.m4.3.3.2.2.3.cmml" xref="S4.I9.i2.p1.4.m4.3.3.2.2.2"><apply id="S4.I9.i2.p1.4.m4.2.2.1.1.1.1.cmml" xref="S4.I9.i2.p1.4.m4.2.2.1.1.1.1"><times id="S4.I9.i2.p1.4.m4.2.2.1.1.1.1.1.cmml" xref="S4.I9.i2.p1.4.m4.2.2.1.1.1.1.1"></times><ci id="S4.I9.i2.p1.4.m4.2.2.1.1.1.1.2.cmml" xref="S4.I9.i2.p1.4.m4.2.2.1.1.1.1.2">ℎ</ci><ci id="S4.I9.i2.p1.4.m4.1.1.cmml" xref="S4.I9.i2.p1.4.m4.1.1">𝑢</ci></apply><apply id="S4.I9.i2.p1.4.m4.3.3.2.2.2.2.cmml" xref="S4.I9.i2.p1.4.m4.3.3.2.2.2.2"><times id="S4.I9.i2.p1.4.m4.3.3.2.2.2.2.2.cmml" xref="S4.I9.i2.p1.4.m4.3.3.2.2.2.2.2"></times><ci id="S4.I9.i2.p1.4.m4.3.3.2.2.2.2.3.cmml" xref="S4.I9.i2.p1.4.m4.3.3.2.2.2.2.3">ℓ</ci><apply id="S4.I9.i2.p1.4.m4.3.3.2.2.2.2.1.1.1.cmml" xref="S4.I9.i2.p1.4.m4.3.3.2.2.2.2.1.1"><csymbol cd="ambiguous" id="S4.I9.i2.p1.4.m4.3.3.2.2.2.2.1.1.1.1.cmml" xref="S4.I9.i2.p1.4.m4.3.3.2.2.2.2.1.1">superscript</csymbol><ci id="S4.I9.i2.p1.4.m4.3.3.2.2.2.2.1.1.1.2.cmml" xref="S4.I9.i2.p1.4.m4.3.3.2.2.2.2.1.1.1.2">𝑢</ci><ci id="S4.I9.i2.p1.4.m4.3.3.2.2.2.2.1.1.1.3.cmml" xref="S4.I9.i2.p1.4.m4.3.3.2.2.2.2.1.1.1.3">′′</ci></apply></apply></interval></apply><apply id="S4.I9.i2.p1.4.m4.3.3.4.cmml" xref="S4.I9.i2.p1.4.m4.3.3.4"><setdiff id="S4.I9.i2.p1.4.m4.3.3.4.1.cmml" xref="S4.I9.i2.p1.4.m4.3.3.4.1"></setdiff><ci id="S4.I9.i2.p1.4.m4.3.3.4.2.cmml" xref="S4.I9.i2.p1.4.m4.3.3.4.2">𝑇</ci><apply id="S4.I9.i2.p1.4.m4.3.3.4.3.cmml" xref="S4.I9.i2.p1.4.m4.3.3.4.3"><csymbol cd="ambiguous" id="S4.I9.i2.p1.4.m4.3.3.4.3.1.cmml" xref="S4.I9.i2.p1.4.m4.3.3.4.3">subscript</csymbol><ci id="S4.I9.i2.p1.4.m4.3.3.4.3.2.cmml" xref="S4.I9.i2.p1.4.m4.3.3.4.3.2">𝑇</ci><ci id="S4.I9.i2.p1.4.m4.3.3.4.3.3.cmml" xref="S4.I9.i2.p1.4.m4.3.3.4.3.3">𝑥</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.I9.i2.p1.4.m4.3c">\text{LCA}(h(u),\ell(u^{\prime\prime}))\in T\setminus T_{x}</annotation><annotation encoding="application/x-llamapun" id="S4.I9.i2.p1.4.m4.3d">LCA ( italic_h ( italic_u ) , roman_ℓ ( italic_u start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT ) ) ∈ italic_T ∖ italic_T start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT</annotation></semantics></math> implies that <math alttext="\ell(u^{\prime\prime})\in T\setminus T_{x}" class="ltx_Math" display="inline" id="S4.I9.i2.p1.5.m5.1"><semantics id="S4.I9.i2.p1.5.m5.1a"><mrow id="S4.I9.i2.p1.5.m5.1.1" xref="S4.I9.i2.p1.5.m5.1.1.cmml"><mrow id="S4.I9.i2.p1.5.m5.1.1.1" xref="S4.I9.i2.p1.5.m5.1.1.1.cmml"><mi id="S4.I9.i2.p1.5.m5.1.1.1.3" mathvariant="normal" xref="S4.I9.i2.p1.5.m5.1.1.1.3.cmml">ℓ</mi><mo id="S4.I9.i2.p1.5.m5.1.1.1.2" xref="S4.I9.i2.p1.5.m5.1.1.1.2.cmml">⁢</mo><mrow id="S4.I9.i2.p1.5.m5.1.1.1.1.1" xref="S4.I9.i2.p1.5.m5.1.1.1.1.1.1.cmml"><mo id="S4.I9.i2.p1.5.m5.1.1.1.1.1.2" stretchy="false" xref="S4.I9.i2.p1.5.m5.1.1.1.1.1.1.cmml">(</mo><msup id="S4.I9.i2.p1.5.m5.1.1.1.1.1.1" xref="S4.I9.i2.p1.5.m5.1.1.1.1.1.1.cmml"><mi id="S4.I9.i2.p1.5.m5.1.1.1.1.1.1.2" xref="S4.I9.i2.p1.5.m5.1.1.1.1.1.1.2.cmml">u</mi><mo id="S4.I9.i2.p1.5.m5.1.1.1.1.1.1.3" xref="S4.I9.i2.p1.5.m5.1.1.1.1.1.1.3.cmml">′′</mo></msup><mo id="S4.I9.i2.p1.5.m5.1.1.1.1.1.3" stretchy="false" xref="S4.I9.i2.p1.5.m5.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S4.I9.i2.p1.5.m5.1.1.2" xref="S4.I9.i2.p1.5.m5.1.1.2.cmml">∈</mo><mrow id="S4.I9.i2.p1.5.m5.1.1.3" xref="S4.I9.i2.p1.5.m5.1.1.3.cmml"><mi id="S4.I9.i2.p1.5.m5.1.1.3.2" xref="S4.I9.i2.p1.5.m5.1.1.3.2.cmml">T</mi><mo id="S4.I9.i2.p1.5.m5.1.1.3.1" xref="S4.I9.i2.p1.5.m5.1.1.3.1.cmml">∖</mo><msub id="S4.I9.i2.p1.5.m5.1.1.3.3" xref="S4.I9.i2.p1.5.m5.1.1.3.3.cmml"><mi id="S4.I9.i2.p1.5.m5.1.1.3.3.2" xref="S4.I9.i2.p1.5.m5.1.1.3.3.2.cmml">T</mi><mi id="S4.I9.i2.p1.5.m5.1.1.3.3.3" xref="S4.I9.i2.p1.5.m5.1.1.3.3.3.cmml">x</mi></msub></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.I9.i2.p1.5.m5.1b"><apply id="S4.I9.i2.p1.5.m5.1.1.cmml" xref="S4.I9.i2.p1.5.m5.1.1"><in id="S4.I9.i2.p1.5.m5.1.1.2.cmml" xref="S4.I9.i2.p1.5.m5.1.1.2"></in><apply id="S4.I9.i2.p1.5.m5.1.1.1.cmml" xref="S4.I9.i2.p1.5.m5.1.1.1"><times id="S4.I9.i2.p1.5.m5.1.1.1.2.cmml" xref="S4.I9.i2.p1.5.m5.1.1.1.2"></times><ci id="S4.I9.i2.p1.5.m5.1.1.1.3.cmml" xref="S4.I9.i2.p1.5.m5.1.1.1.3">ℓ</ci><apply id="S4.I9.i2.p1.5.m5.1.1.1.1.1.1.cmml" xref="S4.I9.i2.p1.5.m5.1.1.1.1.1"><csymbol cd="ambiguous" id="S4.I9.i2.p1.5.m5.1.1.1.1.1.1.1.cmml" xref="S4.I9.i2.p1.5.m5.1.1.1.1.1">superscript</csymbol><ci id="S4.I9.i2.p1.5.m5.1.1.1.1.1.1.2.cmml" xref="S4.I9.i2.p1.5.m5.1.1.1.1.1.1.2">𝑢</ci><ci id="S4.I9.i2.p1.5.m5.1.1.1.1.1.1.3.cmml" xref="S4.I9.i2.p1.5.m5.1.1.1.1.1.1.3">′′</ci></apply></apply><apply id="S4.I9.i2.p1.5.m5.1.1.3.cmml" xref="S4.I9.i2.p1.5.m5.1.1.3"><setdiff id="S4.I9.i2.p1.5.m5.1.1.3.1.cmml" xref="S4.I9.i2.p1.5.m5.1.1.3.1"></setdiff><ci id="S4.I9.i2.p1.5.m5.1.1.3.2.cmml" xref="S4.I9.i2.p1.5.m5.1.1.3.2">𝑇</ci><apply id="S4.I9.i2.p1.5.m5.1.1.3.3.cmml" xref="S4.I9.i2.p1.5.m5.1.1.3.3"><csymbol cd="ambiguous" id="S4.I9.i2.p1.5.m5.1.1.3.3.1.cmml" xref="S4.I9.i2.p1.5.m5.1.1.3.3">subscript</csymbol><ci id="S4.I9.i2.p1.5.m5.1.1.3.3.2.cmml" xref="S4.I9.i2.p1.5.m5.1.1.3.3.2">𝑇</ci><ci id="S4.I9.i2.p1.5.m5.1.1.3.3.3.cmml" xref="S4.I9.i2.p1.5.m5.1.1.3.3.3">𝑥</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.I9.i2.p1.5.m5.1c">\ell(u^{\prime\prime})\in T\setminus T_{x}</annotation><annotation encoding="application/x-llamapun" id="S4.I9.i2.p1.5.m5.1d">roman_ℓ ( italic_u start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT ) ∈ italic_T ∖ italic_T start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT</annotation></semantics></math> (since <math alttext="h(u)\in T_{x}" class="ltx_Math" display="inline" id="S4.I9.i2.p1.6.m6.1"><semantics id="S4.I9.i2.p1.6.m6.1a"><mrow id="S4.I9.i2.p1.6.m6.1.2" xref="S4.I9.i2.p1.6.m6.1.2.cmml"><mrow id="S4.I9.i2.p1.6.m6.1.2.2" xref="S4.I9.i2.p1.6.m6.1.2.2.cmml"><mi id="S4.I9.i2.p1.6.m6.1.2.2.2" xref="S4.I9.i2.p1.6.m6.1.2.2.2.cmml">h</mi><mo id="S4.I9.i2.p1.6.m6.1.2.2.1" xref="S4.I9.i2.p1.6.m6.1.2.2.1.cmml">⁢</mo><mrow id="S4.I9.i2.p1.6.m6.1.2.2.3.2" xref="S4.I9.i2.p1.6.m6.1.2.2.cmml"><mo id="S4.I9.i2.p1.6.m6.1.2.2.3.2.1" stretchy="false" xref="S4.I9.i2.p1.6.m6.1.2.2.cmml">(</mo><mi id="S4.I9.i2.p1.6.m6.1.1" xref="S4.I9.i2.p1.6.m6.1.1.cmml">u</mi><mo id="S4.I9.i2.p1.6.m6.1.2.2.3.2.2" stretchy="false" xref="S4.I9.i2.p1.6.m6.1.2.2.cmml">)</mo></mrow></mrow><mo id="S4.I9.i2.p1.6.m6.1.2.1" xref="S4.I9.i2.p1.6.m6.1.2.1.cmml">∈</mo><msub id="S4.I9.i2.p1.6.m6.1.2.3" xref="S4.I9.i2.p1.6.m6.1.2.3.cmml"><mi id="S4.I9.i2.p1.6.m6.1.2.3.2" xref="S4.I9.i2.p1.6.m6.1.2.3.2.cmml">T</mi><mi id="S4.I9.i2.p1.6.m6.1.2.3.3" xref="S4.I9.i2.p1.6.m6.1.2.3.3.cmml">x</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S4.I9.i2.p1.6.m6.1b"><apply id="S4.I9.i2.p1.6.m6.1.2.cmml" xref="S4.I9.i2.p1.6.m6.1.2"><in id="S4.I9.i2.p1.6.m6.1.2.1.cmml" xref="S4.I9.i2.p1.6.m6.1.2.1"></in><apply id="S4.I9.i2.p1.6.m6.1.2.2.cmml" xref="S4.I9.i2.p1.6.m6.1.2.2"><times id="S4.I9.i2.p1.6.m6.1.2.2.1.cmml" xref="S4.I9.i2.p1.6.m6.1.2.2.1"></times><ci id="S4.I9.i2.p1.6.m6.1.2.2.2.cmml" xref="S4.I9.i2.p1.6.m6.1.2.2.2">ℎ</ci><ci id="S4.I9.i2.p1.6.m6.1.1.cmml" xref="S4.I9.i2.p1.6.m6.1.1">𝑢</ci></apply><apply id="S4.I9.i2.p1.6.m6.1.2.3.cmml" xref="S4.I9.i2.p1.6.m6.1.2.3"><csymbol cd="ambiguous" id="S4.I9.i2.p1.6.m6.1.2.3.1.cmml" xref="S4.I9.i2.p1.6.m6.1.2.3">subscript</csymbol><ci id="S4.I9.i2.p1.6.m6.1.2.3.2.cmml" xref="S4.I9.i2.p1.6.m6.1.2.3.2">𝑇</ci><ci id="S4.I9.i2.p1.6.m6.1.2.3.3.cmml" xref="S4.I9.i2.p1.6.m6.1.2.3.3">𝑥</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.I9.i2.p1.6.m6.1c">h(u)\in T_{x}</annotation><annotation encoding="application/x-llamapun" id="S4.I9.i2.p1.6.m6.1d">italic_h ( italic_u ) ∈ italic_T start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT</annotation></semantics></math>), and <math alttext="\ell(a)=\ell(b)=x" class="ltx_Math" display="inline" id="S4.I9.i2.p1.7.m7.2"><semantics id="S4.I9.i2.p1.7.m7.2a"><mrow id="S4.I9.i2.p1.7.m7.2.3" xref="S4.I9.i2.p1.7.m7.2.3.cmml"><mrow id="S4.I9.i2.p1.7.m7.2.3.2" xref="S4.I9.i2.p1.7.m7.2.3.2.cmml"><mi id="S4.I9.i2.p1.7.m7.2.3.2.2" mathvariant="normal" xref="S4.I9.i2.p1.7.m7.2.3.2.2.cmml">ℓ</mi><mo id="S4.I9.i2.p1.7.m7.2.3.2.1" xref="S4.I9.i2.p1.7.m7.2.3.2.1.cmml">⁢</mo><mrow id="S4.I9.i2.p1.7.m7.2.3.2.3.2" xref="S4.I9.i2.p1.7.m7.2.3.2.cmml"><mo id="S4.I9.i2.p1.7.m7.2.3.2.3.2.1" stretchy="false" xref="S4.I9.i2.p1.7.m7.2.3.2.cmml">(</mo><mi id="S4.I9.i2.p1.7.m7.1.1" xref="S4.I9.i2.p1.7.m7.1.1.cmml">a</mi><mo id="S4.I9.i2.p1.7.m7.2.3.2.3.2.2" stretchy="false" xref="S4.I9.i2.p1.7.m7.2.3.2.cmml">)</mo></mrow></mrow><mo id="S4.I9.i2.p1.7.m7.2.3.3" xref="S4.I9.i2.p1.7.m7.2.3.3.cmml">=</mo><mrow id="S4.I9.i2.p1.7.m7.2.3.4" xref="S4.I9.i2.p1.7.m7.2.3.4.cmml"><mi id="S4.I9.i2.p1.7.m7.2.3.4.2" mathvariant="normal" xref="S4.I9.i2.p1.7.m7.2.3.4.2.cmml">ℓ</mi><mo id="S4.I9.i2.p1.7.m7.2.3.4.1" xref="S4.I9.i2.p1.7.m7.2.3.4.1.cmml">⁢</mo><mrow id="S4.I9.i2.p1.7.m7.2.3.4.3.2" xref="S4.I9.i2.p1.7.m7.2.3.4.cmml"><mo id="S4.I9.i2.p1.7.m7.2.3.4.3.2.1" stretchy="false" xref="S4.I9.i2.p1.7.m7.2.3.4.cmml">(</mo><mi id="S4.I9.i2.p1.7.m7.2.2" xref="S4.I9.i2.p1.7.m7.2.2.cmml">b</mi><mo id="S4.I9.i2.p1.7.m7.2.3.4.3.2.2" stretchy="false" xref="S4.I9.i2.p1.7.m7.2.3.4.cmml">)</mo></mrow></mrow><mo id="S4.I9.i2.p1.7.m7.2.3.5" xref="S4.I9.i2.p1.7.m7.2.3.5.cmml">=</mo><mi id="S4.I9.i2.p1.7.m7.2.3.6" xref="S4.I9.i2.p1.7.m7.2.3.6.cmml">x</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.I9.i2.p1.7.m7.2b"><apply id="S4.I9.i2.p1.7.m7.2.3.cmml" xref="S4.I9.i2.p1.7.m7.2.3"><and id="S4.I9.i2.p1.7.m7.2.3a.cmml" xref="S4.I9.i2.p1.7.m7.2.3"></and><apply id="S4.I9.i2.p1.7.m7.2.3b.cmml" xref="S4.I9.i2.p1.7.m7.2.3"><eq id="S4.I9.i2.p1.7.m7.2.3.3.cmml" xref="S4.I9.i2.p1.7.m7.2.3.3"></eq><apply id="S4.I9.i2.p1.7.m7.2.3.2.cmml" xref="S4.I9.i2.p1.7.m7.2.3.2"><times id="S4.I9.i2.p1.7.m7.2.3.2.1.cmml" xref="S4.I9.i2.p1.7.m7.2.3.2.1"></times><ci id="S4.I9.i2.p1.7.m7.2.3.2.2.cmml" xref="S4.I9.i2.p1.7.m7.2.3.2.2">ℓ</ci><ci id="S4.I9.i2.p1.7.m7.1.1.cmml" xref="S4.I9.i2.p1.7.m7.1.1">𝑎</ci></apply><apply id="S4.I9.i2.p1.7.m7.2.3.4.cmml" xref="S4.I9.i2.p1.7.m7.2.3.4"><times id="S4.I9.i2.p1.7.m7.2.3.4.1.cmml" xref="S4.I9.i2.p1.7.m7.2.3.4.1"></times><ci id="S4.I9.i2.p1.7.m7.2.3.4.2.cmml" xref="S4.I9.i2.p1.7.m7.2.3.4.2">ℓ</ci><ci id="S4.I9.i2.p1.7.m7.2.2.cmml" xref="S4.I9.i2.p1.7.m7.2.2">𝑏</ci></apply></apply><apply id="S4.I9.i2.p1.7.m7.2.3c.cmml" xref="S4.I9.i2.p1.7.m7.2.3"><eq id="S4.I9.i2.p1.7.m7.2.3.5.cmml" xref="S4.I9.i2.p1.7.m7.2.3.5"></eq><share href="https://arxiv.org/html/2503.00712v1#S4.I9.i2.p1.7.m7.2.3.4.cmml" id="S4.I9.i2.p1.7.m7.2.3d.cmml" xref="S4.I9.i2.p1.7.m7.2.3"></share><ci id="S4.I9.i2.p1.7.m7.2.3.6.cmml" xref="S4.I9.i2.p1.7.m7.2.3.6">𝑥</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.I9.i2.p1.7.m7.2c">\ell(a)=\ell(b)=x</annotation><annotation encoding="application/x-llamapun" id="S4.I9.i2.p1.7.m7.2d">roman_ℓ ( italic_a ) = roman_ℓ ( italic_b ) = italic_x</annotation></semantics></math>.</p> </div> </li> <li class="ltx_item" id="S4.I9.i3" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">•</span> <div class="ltx_para" id="S4.I9.i3.p1"> <p class="ltx_p" id="S4.I9.i3.p1.8"><math alttext="u^{\prime\prime}\to v" class="ltx_Math" display="inline" id="S4.I9.i3.p1.1.m1.1"><semantics id="S4.I9.i3.p1.1.m1.1a"><mrow id="S4.I9.i3.p1.1.m1.1.1" xref="S4.I9.i3.p1.1.m1.1.1.cmml"><msup id="S4.I9.i3.p1.1.m1.1.1.2" xref="S4.I9.i3.p1.1.m1.1.1.2.cmml"><mi id="S4.I9.i3.p1.1.m1.1.1.2.2" xref="S4.I9.i3.p1.1.m1.1.1.2.2.cmml">u</mi><mo id="S4.I9.i3.p1.1.m1.1.1.2.3" xref="S4.I9.i3.p1.1.m1.1.1.2.3.cmml">′′</mo></msup><mo id="S4.I9.i3.p1.1.m1.1.1.1" stretchy="false" xref="S4.I9.i3.p1.1.m1.1.1.1.cmml">→</mo><mi id="S4.I9.i3.p1.1.m1.1.1.3" xref="S4.I9.i3.p1.1.m1.1.1.3.cmml">v</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.I9.i3.p1.1.m1.1b"><apply id="S4.I9.i3.p1.1.m1.1.1.cmml" xref="S4.I9.i3.p1.1.m1.1.1"><ci id="S4.I9.i3.p1.1.m1.1.1.1.cmml" xref="S4.I9.i3.p1.1.m1.1.1.1">→</ci><apply id="S4.I9.i3.p1.1.m1.1.1.2.cmml" xref="S4.I9.i3.p1.1.m1.1.1.2"><csymbol cd="ambiguous" id="S4.I9.i3.p1.1.m1.1.1.2.1.cmml" xref="S4.I9.i3.p1.1.m1.1.1.2">superscript</csymbol><ci id="S4.I9.i3.p1.1.m1.1.1.2.2.cmml" xref="S4.I9.i3.p1.1.m1.1.1.2.2">𝑢</ci><ci id="S4.I9.i3.p1.1.m1.1.1.2.3.cmml" xref="S4.I9.i3.p1.1.m1.1.1.2.3">′′</ci></apply><ci id="S4.I9.i3.p1.1.m1.1.1.3.cmml" xref="S4.I9.i3.p1.1.m1.1.1.3">𝑣</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.I9.i3.p1.1.m1.1c">u^{\prime\prime}\to v</annotation><annotation encoding="application/x-llamapun" id="S4.I9.i3.p1.1.m1.1d">italic_u start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT → italic_v</annotation></semantics></math>: since <math alttext="G_{y}\setminus\{a,b\}" class="ltx_Math" display="inline" id="S4.I9.i3.p1.2.m2.2"><semantics id="S4.I9.i3.p1.2.m2.2a"><mrow id="S4.I9.i3.p1.2.m2.2.3" xref="S4.I9.i3.p1.2.m2.2.3.cmml"><msub id="S4.I9.i3.p1.2.m2.2.3.2" xref="S4.I9.i3.p1.2.m2.2.3.2.cmml"><mi id="S4.I9.i3.p1.2.m2.2.3.2.2" xref="S4.I9.i3.p1.2.m2.2.3.2.2.cmml">G</mi><mi id="S4.I9.i3.p1.2.m2.2.3.2.3" xref="S4.I9.i3.p1.2.m2.2.3.2.3.cmml">y</mi></msub><mo id="S4.I9.i3.p1.2.m2.2.3.1" xref="S4.I9.i3.p1.2.m2.2.3.1.cmml">∖</mo><mrow id="S4.I9.i3.p1.2.m2.2.3.3.2" xref="S4.I9.i3.p1.2.m2.2.3.3.1.cmml"><mo id="S4.I9.i3.p1.2.m2.2.3.3.2.1" stretchy="false" xref="S4.I9.i3.p1.2.m2.2.3.3.1.cmml">{</mo><mi id="S4.I9.i3.p1.2.m2.1.1" xref="S4.I9.i3.p1.2.m2.1.1.cmml">a</mi><mo id="S4.I9.i3.p1.2.m2.2.3.3.2.2" xref="S4.I9.i3.p1.2.m2.2.3.3.1.cmml">,</mo><mi id="S4.I9.i3.p1.2.m2.2.2" xref="S4.I9.i3.p1.2.m2.2.2.cmml">b</mi><mo id="S4.I9.i3.p1.2.m2.2.3.3.2.3" stretchy="false" xref="S4.I9.i3.p1.2.m2.2.3.3.1.cmml">}</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.I9.i3.p1.2.m2.2b"><apply id="S4.I9.i3.p1.2.m2.2.3.cmml" xref="S4.I9.i3.p1.2.m2.2.3"><setdiff id="S4.I9.i3.p1.2.m2.2.3.1.cmml" xref="S4.I9.i3.p1.2.m2.2.3.1"></setdiff><apply id="S4.I9.i3.p1.2.m2.2.3.2.cmml" xref="S4.I9.i3.p1.2.m2.2.3.2"><csymbol cd="ambiguous" id="S4.I9.i3.p1.2.m2.2.3.2.1.cmml" xref="S4.I9.i3.p1.2.m2.2.3.2">subscript</csymbol><ci id="S4.I9.i3.p1.2.m2.2.3.2.2.cmml" xref="S4.I9.i3.p1.2.m2.2.3.2.2">𝐺</ci><ci id="S4.I9.i3.p1.2.m2.2.3.2.3.cmml" xref="S4.I9.i3.p1.2.m2.2.3.2.3">𝑦</ci></apply><set id="S4.I9.i3.p1.2.m2.2.3.3.1.cmml" xref="S4.I9.i3.p1.2.m2.2.3.3.2"><ci id="S4.I9.i3.p1.2.m2.1.1.cmml" xref="S4.I9.i3.p1.2.m2.1.1">𝑎</ci><ci id="S4.I9.i3.p1.2.m2.2.2.cmml" xref="S4.I9.i3.p1.2.m2.2.2">𝑏</ci></set></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.I9.i3.p1.2.m2.2c">G_{y}\setminus\{a,b\}</annotation><annotation encoding="application/x-llamapun" id="S4.I9.i3.p1.2.m2.2d">italic_G start_POSTSUBSCRIPT italic_y end_POSTSUBSCRIPT ∖ { italic_a , italic_b }</annotation></semantics></math> is connected and <math alttext="T\setminus T_{x}" class="ltx_Math" display="inline" id="S4.I9.i3.p1.3.m3.1"><semantics id="S4.I9.i3.p1.3.m3.1a"><mrow id="S4.I9.i3.p1.3.m3.1.1" xref="S4.I9.i3.p1.3.m3.1.1.cmml"><mi id="S4.I9.i3.p1.3.m3.1.1.2" xref="S4.I9.i3.p1.3.m3.1.1.2.cmml">T</mi><mo id="S4.I9.i3.p1.3.m3.1.1.1" xref="S4.I9.i3.p1.3.m3.1.1.1.cmml">∖</mo><msub id="S4.I9.i3.p1.3.m3.1.1.3" xref="S4.I9.i3.p1.3.m3.1.1.3.cmml"><mi id="S4.I9.i3.p1.3.m3.1.1.3.2" xref="S4.I9.i3.p1.3.m3.1.1.3.2.cmml">T</mi><mi id="S4.I9.i3.p1.3.m3.1.1.3.3" xref="S4.I9.i3.p1.3.m3.1.1.3.3.cmml">x</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S4.I9.i3.p1.3.m3.1b"><apply id="S4.I9.i3.p1.3.m3.1.1.cmml" xref="S4.I9.i3.p1.3.m3.1.1"><setdiff id="S4.I9.i3.p1.3.m3.1.1.1.cmml" xref="S4.I9.i3.p1.3.m3.1.1.1"></setdiff><ci id="S4.I9.i3.p1.3.m3.1.1.2.cmml" xref="S4.I9.i3.p1.3.m3.1.1.2">𝑇</ci><apply id="S4.I9.i3.p1.3.m3.1.1.3.cmml" xref="S4.I9.i3.p1.3.m3.1.1.3"><csymbol cd="ambiguous" id="S4.I9.i3.p1.3.m3.1.1.3.1.cmml" xref="S4.I9.i3.p1.3.m3.1.1.3">subscript</csymbol><ci id="S4.I9.i3.p1.3.m3.1.1.3.2.cmml" xref="S4.I9.i3.p1.3.m3.1.1.3.2">𝑇</ci><ci id="S4.I9.i3.p1.3.m3.1.1.3.3.cmml" xref="S4.I9.i3.p1.3.m3.1.1.3.3">𝑥</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.I9.i3.p1.3.m3.1c">T\setminus T_{x}</annotation><annotation encoding="application/x-llamapun" id="S4.I9.i3.p1.3.m3.1d">italic_T ∖ italic_T start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT</annotation></semantics></math> remains connected despite the deletion of <math alttext="e^{\prime}" class="ltx_Math" display="inline" id="S4.I9.i3.p1.4.m4.1"><semantics id="S4.I9.i3.p1.4.m4.1a"><msup id="S4.I9.i3.p1.4.m4.1.1" xref="S4.I9.i3.p1.4.m4.1.1.cmml"><mi id="S4.I9.i3.p1.4.m4.1.1.2" xref="S4.I9.i3.p1.4.m4.1.1.2.cmml">e</mi><mo id="S4.I9.i3.p1.4.m4.1.1.3" xref="S4.I9.i3.p1.4.m4.1.1.3.cmml">′</mo></msup><annotation-xml encoding="MathML-Content" id="S4.I9.i3.p1.4.m4.1b"><apply id="S4.I9.i3.p1.4.m4.1.1.cmml" xref="S4.I9.i3.p1.4.m4.1.1"><csymbol cd="ambiguous" id="S4.I9.i3.p1.4.m4.1.1.1.cmml" xref="S4.I9.i3.p1.4.m4.1.1">superscript</csymbol><ci id="S4.I9.i3.p1.4.m4.1.1.2.cmml" xref="S4.I9.i3.p1.4.m4.1.1.2">𝑒</ci><ci id="S4.I9.i3.p1.4.m4.1.1.3.cmml" xref="S4.I9.i3.p1.4.m4.1.1.3">′</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.I9.i3.p1.4.m4.1c">e^{\prime}</annotation><annotation encoding="application/x-llamapun" id="S4.I9.i3.p1.4.m4.1d">italic_e start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math>, there is a <math alttext="u^{\prime\prime}" class="ltx_Math" display="inline" id="S4.I9.i3.p1.5.m5.1"><semantics id="S4.I9.i3.p1.5.m5.1a"><msup id="S4.I9.i3.p1.5.m5.1.1" xref="S4.I9.i3.p1.5.m5.1.1.cmml"><mi id="S4.I9.i3.p1.5.m5.1.1.2" xref="S4.I9.i3.p1.5.m5.1.1.2.cmml">u</mi><mo id="S4.I9.i3.p1.5.m5.1.1.3" xref="S4.I9.i3.p1.5.m5.1.1.3.cmml">′′</mo></msup><annotation-xml encoding="MathML-Content" id="S4.I9.i3.p1.5.m5.1b"><apply id="S4.I9.i3.p1.5.m5.1.1.cmml" xref="S4.I9.i3.p1.5.m5.1.1"><csymbol cd="ambiguous" id="S4.I9.i3.p1.5.m5.1.1.1.cmml" xref="S4.I9.i3.p1.5.m5.1.1">superscript</csymbol><ci id="S4.I9.i3.p1.5.m5.1.1.2.cmml" xref="S4.I9.i3.p1.5.m5.1.1.2">𝑢</ci><ci id="S4.I9.i3.p1.5.m5.1.1.3.cmml" xref="S4.I9.i3.p1.5.m5.1.1.3">′′</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.I9.i3.p1.5.m5.1c">u^{\prime\prime}</annotation><annotation encoding="application/x-llamapun" id="S4.I9.i3.p1.5.m5.1d">italic_u start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT</annotation></semantics></math>-<math alttext="v" class="ltx_Math" display="inline" id="S4.I9.i3.p1.6.m6.1"><semantics id="S4.I9.i3.p1.6.m6.1a"><mi id="S4.I9.i3.p1.6.m6.1.1" xref="S4.I9.i3.p1.6.m6.1.1.cmml">v</mi><annotation-xml encoding="MathML-Content" id="S4.I9.i3.p1.6.m6.1b"><ci id="S4.I9.i3.p1.6.m6.1.1.cmml" xref="S4.I9.i3.p1.6.m6.1.1">𝑣</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.I9.i3.p1.6.m6.1c">v</annotation><annotation encoding="application/x-llamapun" id="S4.I9.i3.p1.6.m6.1d">italic_v</annotation></semantics></math> path in <math alttext="E" class="ltx_Math" display="inline" id="S4.I9.i3.p1.7.m7.1"><semantics id="S4.I9.i3.p1.7.m7.1a"><mi id="S4.I9.i3.p1.7.m7.1.1" xref="S4.I9.i3.p1.7.m7.1.1.cmml">E</mi><annotation-xml encoding="MathML-Content" id="S4.I9.i3.p1.7.m7.1b"><ci id="S4.I9.i3.p1.7.m7.1.1.cmml" xref="S4.I9.i3.p1.7.m7.1.1">𝐸</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.I9.i3.p1.7.m7.1c">E</annotation><annotation encoding="application/x-llamapun" id="S4.I9.i3.p1.7.m7.1d">italic_E</annotation></semantics></math> avoiding <math alttext="\{a,b\}" class="ltx_Math" display="inline" id="S4.I9.i3.p1.8.m8.2"><semantics id="S4.I9.i3.p1.8.m8.2a"><mrow id="S4.I9.i3.p1.8.m8.2.3.2" xref="S4.I9.i3.p1.8.m8.2.3.1.cmml"><mo id="S4.I9.i3.p1.8.m8.2.3.2.1" stretchy="false" xref="S4.I9.i3.p1.8.m8.2.3.1.cmml">{</mo><mi id="S4.I9.i3.p1.8.m8.1.1" xref="S4.I9.i3.p1.8.m8.1.1.cmml">a</mi><mo id="S4.I9.i3.p1.8.m8.2.3.2.2" xref="S4.I9.i3.p1.8.m8.2.3.1.cmml">,</mo><mi id="S4.I9.i3.p1.8.m8.2.2" xref="S4.I9.i3.p1.8.m8.2.2.cmml">b</mi><mo id="S4.I9.i3.p1.8.m8.2.3.2.3" stretchy="false" xref="S4.I9.i3.p1.8.m8.2.3.1.cmml">}</mo></mrow><annotation-xml encoding="MathML-Content" id="S4.I9.i3.p1.8.m8.2b"><set id="S4.I9.i3.p1.8.m8.2.3.1.cmml" xref="S4.I9.i3.p1.8.m8.2.3.2"><ci id="S4.I9.i3.p1.8.m8.1.1.cmml" xref="S4.I9.i3.p1.8.m8.1.1">𝑎</ci><ci id="S4.I9.i3.p1.8.m8.2.2.cmml" xref="S4.I9.i3.p1.8.m8.2.2">𝑏</ci></set></annotation-xml><annotation encoding="application/x-tex" id="S4.I9.i3.p1.8.m8.2c">\{a,b\}</annotation><annotation encoding="application/x-llamapun" id="S4.I9.i3.p1.8.m8.2d">{ italic_a , italic_b }</annotation></semantics></math>.</p> </div> </li> </ul> </div> <figure class="ltx_figure" id="S4.F8"><img alt="Refer to caption" class="ltx_graphics ltx_centering ltx_img_square" height="252" id="S4.F8.g1" src="x9.png" width="270"/> <figcaption class="ltx_caption ltx_centering"><span class="ltx_tag ltx_tag_figure"><span class="ltx_text" id="S4.F8.12.6.1" style="font-size:90%;">Figure 8</span>: </span><span class="ltx_text" id="S4.F8.10.5" style="font-size:90%;">Case 1 example. Tree nodes are shown with boxes, while graph nodes are given by dots. Note that while the figure shows <math alttext="h(u)" class="ltx_Math" display="inline" id="S4.F8.6.1.m1.1"><semantics id="S4.F8.6.1.m1.1b"><mrow id="S4.F8.6.1.m1.1.2" xref="S4.F8.6.1.m1.1.2.cmml"><mi id="S4.F8.6.1.m1.1.2.2" xref="S4.F8.6.1.m1.1.2.2.cmml">h</mi><mo id="S4.F8.6.1.m1.1.2.1" xref="S4.F8.6.1.m1.1.2.1.cmml">⁢</mo><mrow id="S4.F8.6.1.m1.1.2.3.2" xref="S4.F8.6.1.m1.1.2.cmml"><mo id="S4.F8.6.1.m1.1.2.3.2.1" stretchy="false" xref="S4.F8.6.1.m1.1.2.cmml">(</mo><mi id="S4.F8.6.1.m1.1.1" xref="S4.F8.6.1.m1.1.1.cmml">u</mi><mo id="S4.F8.6.1.m1.1.2.3.2.2" stretchy="false" xref="S4.F8.6.1.m1.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.F8.6.1.m1.1c"><apply id="S4.F8.6.1.m1.1.2.cmml" xref="S4.F8.6.1.m1.1.2"><times id="S4.F8.6.1.m1.1.2.1.cmml" xref="S4.F8.6.1.m1.1.2.1"></times><ci id="S4.F8.6.1.m1.1.2.2.cmml" xref="S4.F8.6.1.m1.1.2.2">ℎ</ci><ci id="S4.F8.6.1.m1.1.1.cmml" xref="S4.F8.6.1.m1.1.1">𝑢</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.F8.6.1.m1.1d">h(u)</annotation><annotation encoding="application/x-llamapun" id="S4.F8.6.1.m1.1e">italic_h ( italic_u )</annotation></semantics></math> and <math alttext="x" class="ltx_Math" display="inline" id="S4.F8.7.2.m2.1"><semantics id="S4.F8.7.2.m2.1b"><mi id="S4.F8.7.2.m2.1.1" xref="S4.F8.7.2.m2.1.1.cmml">x</mi><annotation-xml encoding="MathML-Content" id="S4.F8.7.2.m2.1c"><ci id="S4.F8.7.2.m2.1.1.cmml" xref="S4.F8.7.2.m2.1.1">𝑥</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.F8.7.2.m2.1d">x</annotation><annotation encoding="application/x-llamapun" id="S4.F8.7.2.m2.1e">italic_x</annotation></semantics></math> to be separate, it is possible that <math alttext="h(u)=x" class="ltx_Math" display="inline" id="S4.F8.8.3.m3.1"><semantics id="S4.F8.8.3.m3.1b"><mrow id="S4.F8.8.3.m3.1.2" xref="S4.F8.8.3.m3.1.2.cmml"><mrow id="S4.F8.8.3.m3.1.2.2" xref="S4.F8.8.3.m3.1.2.2.cmml"><mi id="S4.F8.8.3.m3.1.2.2.2" xref="S4.F8.8.3.m3.1.2.2.2.cmml">h</mi><mo id="S4.F8.8.3.m3.1.2.2.1" xref="S4.F8.8.3.m3.1.2.2.1.cmml">⁢</mo><mrow id="S4.F8.8.3.m3.1.2.2.3.2" xref="S4.F8.8.3.m3.1.2.2.cmml"><mo id="S4.F8.8.3.m3.1.2.2.3.2.1" stretchy="false" xref="S4.F8.8.3.m3.1.2.2.cmml">(</mo><mi id="S4.F8.8.3.m3.1.1" xref="S4.F8.8.3.m3.1.1.cmml">u</mi><mo id="S4.F8.8.3.m3.1.2.2.3.2.2" stretchy="false" xref="S4.F8.8.3.m3.1.2.2.cmml">)</mo></mrow></mrow><mo id="S4.F8.8.3.m3.1.2.1" xref="S4.F8.8.3.m3.1.2.1.cmml">=</mo><mi id="S4.F8.8.3.m3.1.2.3" xref="S4.F8.8.3.m3.1.2.3.cmml">x</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.F8.8.3.m3.1c"><apply id="S4.F8.8.3.m3.1.2.cmml" xref="S4.F8.8.3.m3.1.2"><eq id="S4.F8.8.3.m3.1.2.1.cmml" xref="S4.F8.8.3.m3.1.2.1"></eq><apply id="S4.F8.8.3.m3.1.2.2.cmml" xref="S4.F8.8.3.m3.1.2.2"><times id="S4.F8.8.3.m3.1.2.2.1.cmml" xref="S4.F8.8.3.m3.1.2.2.1"></times><ci id="S4.F8.8.3.m3.1.2.2.2.cmml" xref="S4.F8.8.3.m3.1.2.2.2">ℎ</ci><ci id="S4.F8.8.3.m3.1.1.cmml" xref="S4.F8.8.3.m3.1.1">𝑢</ci></apply><ci id="S4.F8.8.3.m3.1.2.3.cmml" xref="S4.F8.8.3.m3.1.2.3">𝑥</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.F8.8.3.m3.1d">h(u)=x</annotation><annotation encoding="application/x-llamapun" id="S4.F8.8.3.m3.1e">italic_h ( italic_u ) = italic_x</annotation></semantics></math>. Similarly, it is possible that <math alttext="v,u^{\prime\prime}\in G_{y}" class="ltx_Math" display="inline" id="S4.F8.9.4.m4.2"><semantics id="S4.F8.9.4.m4.2b"><mrow id="S4.F8.9.4.m4.2.2" xref="S4.F8.9.4.m4.2.2.cmml"><mrow id="S4.F8.9.4.m4.2.2.1.1" xref="S4.F8.9.4.m4.2.2.1.2.cmml"><mi id="S4.F8.9.4.m4.1.1" xref="S4.F8.9.4.m4.1.1.cmml">v</mi><mo id="S4.F8.9.4.m4.2.2.1.1.2" xref="S4.F8.9.4.m4.2.2.1.2.cmml">,</mo><msup id="S4.F8.9.4.m4.2.2.1.1.1" xref="S4.F8.9.4.m4.2.2.1.1.1.cmml"><mi id="S4.F8.9.4.m4.2.2.1.1.1.2" xref="S4.F8.9.4.m4.2.2.1.1.1.2.cmml">u</mi><mo id="S4.F8.9.4.m4.2.2.1.1.1.3" xref="S4.F8.9.4.m4.2.2.1.1.1.3.cmml">′′</mo></msup></mrow><mo id="S4.F8.9.4.m4.2.2.2" xref="S4.F8.9.4.m4.2.2.2.cmml">∈</mo><msub id="S4.F8.9.4.m4.2.2.3" xref="S4.F8.9.4.m4.2.2.3.cmml"><mi id="S4.F8.9.4.m4.2.2.3.2" xref="S4.F8.9.4.m4.2.2.3.2.cmml">G</mi><mi id="S4.F8.9.4.m4.2.2.3.3" xref="S4.F8.9.4.m4.2.2.3.3.cmml">y</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S4.F8.9.4.m4.2c"><apply id="S4.F8.9.4.m4.2.2.cmml" xref="S4.F8.9.4.m4.2.2"><in id="S4.F8.9.4.m4.2.2.2.cmml" xref="S4.F8.9.4.m4.2.2.2"></in><list id="S4.F8.9.4.m4.2.2.1.2.cmml" xref="S4.F8.9.4.m4.2.2.1.1"><ci id="S4.F8.9.4.m4.1.1.cmml" xref="S4.F8.9.4.m4.1.1">𝑣</ci><apply id="S4.F8.9.4.m4.2.2.1.1.1.cmml" xref="S4.F8.9.4.m4.2.2.1.1.1"><csymbol cd="ambiguous" id="S4.F8.9.4.m4.2.2.1.1.1.1.cmml" xref="S4.F8.9.4.m4.2.2.1.1.1">superscript</csymbol><ci id="S4.F8.9.4.m4.2.2.1.1.1.2.cmml" xref="S4.F8.9.4.m4.2.2.1.1.1.2">𝑢</ci><ci id="S4.F8.9.4.m4.2.2.1.1.1.3.cmml" xref="S4.F8.9.4.m4.2.2.1.1.1.3">′′</ci></apply></list><apply id="S4.F8.9.4.m4.2.2.3.cmml" xref="S4.F8.9.4.m4.2.2.3"><csymbol cd="ambiguous" id="S4.F8.9.4.m4.2.2.3.1.cmml" xref="S4.F8.9.4.m4.2.2.3">subscript</csymbol><ci id="S4.F8.9.4.m4.2.2.3.2.cmml" xref="S4.F8.9.4.m4.2.2.3.2">𝐺</ci><ci id="S4.F8.9.4.m4.2.2.3.3.cmml" xref="S4.F8.9.4.m4.2.2.3.3">𝑦</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.F8.9.4.m4.2d">v,u^{\prime\prime}\in G_{y}</annotation><annotation encoding="application/x-llamapun" id="S4.F8.9.4.m4.2e">italic_v , italic_u start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT ∈ italic_G start_POSTSUBSCRIPT italic_y end_POSTSUBSCRIPT</annotation></semantics></math>, and possible that <math alttext="y=r" class="ltx_Math" display="inline" id="S4.F8.10.5.m5.1"><semantics id="S4.F8.10.5.m5.1b"><mrow id="S4.F8.10.5.m5.1.1" xref="S4.F8.10.5.m5.1.1.cmml"><mi id="S4.F8.10.5.m5.1.1.2" xref="S4.F8.10.5.m5.1.1.2.cmml">y</mi><mo id="S4.F8.10.5.m5.1.1.1" xref="S4.F8.10.5.m5.1.1.1.cmml">=</mo><mi id="S4.F8.10.5.m5.1.1.3" xref="S4.F8.10.5.m5.1.1.3.cmml">r</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.F8.10.5.m5.1c"><apply id="S4.F8.10.5.m5.1.1.cmml" xref="S4.F8.10.5.m5.1.1"><eq id="S4.F8.10.5.m5.1.1.1.cmml" xref="S4.F8.10.5.m5.1.1.1"></eq><ci id="S4.F8.10.5.m5.1.1.2.cmml" xref="S4.F8.10.5.m5.1.1.2">𝑦</ci><ci id="S4.F8.10.5.m5.1.1.3.cmml" xref="S4.F8.10.5.m5.1.1.3">𝑟</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.F8.10.5.m5.1d">y=r</annotation><annotation encoding="application/x-llamapun" id="S4.F8.10.5.m5.1e">italic_y = italic_r</annotation></semantics></math>.</span></figcaption> </figure> </section> <section class="ltx_paragraph" id="S4.SS2.SSS3.Px2"> <h5 class="ltx_title ltx_title_paragraph">Case 2: <math alttext="\boldsymbol{V(G_{x})=\{a,b\}}" class="ltx_Math" display="inline" id="S4.SS2.SSS3.Px2.1.m1.3"><semantics id="S4.SS2.SSS3.Px2.1.m1.3b"><mrow id="S4.SS2.SSS3.Px2.1.m1.3.3" xref="S4.SS2.SSS3.Px2.1.m1.3.3.cmml"><mrow id="S4.SS2.SSS3.Px2.1.m1.3.3.1" xref="S4.SS2.SSS3.Px2.1.m1.3.3.1.cmml"><mi id="S4.SS2.SSS3.Px2.1.m1.3.3.1.3" xref="S4.SS2.SSS3.Px2.1.m1.3.3.1.3.cmml">𝑽</mi><mo id="S4.SS2.SSS3.Px2.1.m1.3.3.1.2" xref="S4.SS2.SSS3.Px2.1.m1.3.3.1.2.cmml">⁢</mo><mrow id="S4.SS2.SSS3.Px2.1.m1.3.3.1.1.1" xref="S4.SS2.SSS3.Px2.1.m1.3.3.1.1.1.1.cmml"><mo class="ltx_mathvariant_bold" id="S4.SS2.SSS3.Px2.1.m1.3.3.1.1.1.2" mathvariant="bold" stretchy="false" xref="S4.SS2.SSS3.Px2.1.m1.3.3.1.1.1.1.cmml">(</mo><msub id="S4.SS2.SSS3.Px2.1.m1.3.3.1.1.1.1" xref="S4.SS2.SSS3.Px2.1.m1.3.3.1.1.1.1.cmml"><mi id="S4.SS2.SSS3.Px2.1.m1.3.3.1.1.1.1.2" xref="S4.SS2.SSS3.Px2.1.m1.3.3.1.1.1.1.2.cmml">𝑮</mi><mi id="S4.SS2.SSS3.Px2.1.m1.3.3.1.1.1.1.3" xref="S4.SS2.SSS3.Px2.1.m1.3.3.1.1.1.1.3.cmml">𝒙</mi></msub><mo class="ltx_mathvariant_bold" id="S4.SS2.SSS3.Px2.1.m1.3.3.1.1.1.3" mathvariant="bold" stretchy="false" xref="S4.SS2.SSS3.Px2.1.m1.3.3.1.1.1.1.cmml">)</mo></mrow></mrow><mo class="ltx_mathvariant_bold" id="S4.SS2.SSS3.Px2.1.m1.3.3.2" mathvariant="bold" xref="S4.SS2.SSS3.Px2.1.m1.3.3.2.cmml">=</mo><mrow id="S4.SS2.SSS3.Px2.1.m1.3.3.3.2" xref="S4.SS2.SSS3.Px2.1.m1.3.3.3.1.cmml"><mo class="ltx_mathvariant_bold" id="S4.SS2.SSS3.Px2.1.m1.3.3.3.2.1" mathvariant="bold" stretchy="false" xref="S4.SS2.SSS3.Px2.1.m1.3.3.3.1.cmml">{</mo><mi id="S4.SS2.SSS3.Px2.1.m1.1.1" xref="S4.SS2.SSS3.Px2.1.m1.1.1.cmml">𝒂</mi><mo class="ltx_mathvariant_bold" id="S4.SS2.SSS3.Px2.1.m1.3.3.3.2.2" mathvariant="bold" xref="S4.SS2.SSS3.Px2.1.m1.3.3.3.1.cmml">,</mo><mi id="S4.SS2.SSS3.Px2.1.m1.2.2" xref="S4.SS2.SSS3.Px2.1.m1.2.2.cmml">𝒃</mi><mo class="ltx_mathvariant_bold" id="S4.SS2.SSS3.Px2.1.m1.3.3.3.2.3" mathvariant="bold" stretchy="false" xref="S4.SS2.SSS3.Px2.1.m1.3.3.3.1.cmml">}</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS3.Px2.1.m1.3c"><apply id="S4.SS2.SSS3.Px2.1.m1.3.3.cmml" xref="S4.SS2.SSS3.Px2.1.m1.3.3"><eq id="S4.SS2.SSS3.Px2.1.m1.3.3.2.cmml" xref="S4.SS2.SSS3.Px2.1.m1.3.3.2"></eq><apply id="S4.SS2.SSS3.Px2.1.m1.3.3.1.cmml" xref="S4.SS2.SSS3.Px2.1.m1.3.3.1"><times id="S4.SS2.SSS3.Px2.1.m1.3.3.1.2.cmml" xref="S4.SS2.SSS3.Px2.1.m1.3.3.1.2"></times><ci id="S4.SS2.SSS3.Px2.1.m1.3.3.1.3.cmml" xref="S4.SS2.SSS3.Px2.1.m1.3.3.1.3">𝑽</ci><apply id="S4.SS2.SSS3.Px2.1.m1.3.3.1.1.1.1.cmml" xref="S4.SS2.SSS3.Px2.1.m1.3.3.1.1.1"><csymbol cd="ambiguous" id="S4.SS2.SSS3.Px2.1.m1.3.3.1.1.1.1.1.cmml" xref="S4.SS2.SSS3.Px2.1.m1.3.3.1.1.1">subscript</csymbol><ci id="S4.SS2.SSS3.Px2.1.m1.3.3.1.1.1.1.2.cmml" xref="S4.SS2.SSS3.Px2.1.m1.3.3.1.1.1.1.2">𝑮</ci><ci id="S4.SS2.SSS3.Px2.1.m1.3.3.1.1.1.1.3.cmml" xref="S4.SS2.SSS3.Px2.1.m1.3.3.1.1.1.1.3">𝒙</ci></apply></apply><set id="S4.SS2.SSS3.Px2.1.m1.3.3.3.1.cmml" xref="S4.SS2.SSS3.Px2.1.m1.3.3.3.2"><ci id="S4.SS2.SSS3.Px2.1.m1.1.1.cmml" xref="S4.SS2.SSS3.Px2.1.m1.1.1">𝒂</ci><ci id="S4.SS2.SSS3.Px2.1.m1.2.2.cmml" xref="S4.SS2.SSS3.Px2.1.m1.2.2">𝒃</ci></set></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS3.Px2.1.m1.3d">\boldsymbol{V(G_{x})=\{a,b\}}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS3.Px2.1.m1.3e">bold_italic_V bold_( bold_italic_G start_POSTSUBSCRIPT bold_italic_x end_POSTSUBSCRIPT bold_) bold_= bold_{ bold_italic_a bold_, bold_italic_b bold_}</annotation></semantics></math> for a P-node <math alttext="\boldsymbol{x}" class="ltx_Math" display="inline" id="S4.SS2.SSS3.Px2.2.m2.1"><semantics id="S4.SS2.SSS3.Px2.2.m2.1b"><mi id="S4.SS2.SSS3.Px2.2.m2.1.1" xref="S4.SS2.SSS3.Px2.2.m2.1.1.cmml">𝒙</mi><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS3.Px2.2.m2.1c"><ci id="S4.SS2.SSS3.Px2.2.m2.1.1.cmml" xref="S4.SS2.SSS3.Px2.2.m2.1.1">𝒙</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS3.Px2.2.m2.1d">\boldsymbol{x}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS3.Px2.2.m2.1e">bold_italic_x</annotation></semantics></math>:</h5> <div class="ltx_para" id="S4.SS2.SSS3.Px2.p1"> <p class="ltx_p" id="S4.SS2.SSS3.Px2.p1.12">Since no two P-nodes are adjacent to each other, all neighbors of <math alttext="x" class="ltx_Math" display="inline" id="S4.SS2.SSS3.Px2.p1.1.m1.1"><semantics id="S4.SS2.SSS3.Px2.p1.1.m1.1a"><mi id="S4.SS2.SSS3.Px2.p1.1.m1.1.1" xref="S4.SS2.SSS3.Px2.p1.1.m1.1.1.cmml">x</mi><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS3.Px2.p1.1.m1.1b"><ci id="S4.SS2.SSS3.Px2.p1.1.m1.1.1.cmml" xref="S4.SS2.SSS3.Px2.p1.1.m1.1.1">𝑥</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS3.Px2.p1.1.m1.1c">x</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS3.Px2.p1.1.m1.1d">italic_x</annotation></semantics></math> in <math alttext="T" class="ltx_Math" display="inline" id="S4.SS2.SSS3.Px2.p1.2.m2.1"><semantics id="S4.SS2.SSS3.Px2.p1.2.m2.1a"><mi id="S4.SS2.SSS3.Px2.p1.2.m2.1.1" xref="S4.SS2.SSS3.Px2.p1.2.m2.1.1.cmml">T</mi><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS3.Px2.p1.2.m2.1b"><ci id="S4.SS2.SSS3.Px2.p1.2.m2.1.1.cmml" xref="S4.SS2.SSS3.Px2.p1.2.m2.1.1">𝑇</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS3.Px2.p1.2.m2.1c">T</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS3.Px2.p1.2.m2.1d">italic_T</annotation></semantics></math> are R or S nodes. In particular, this means that for all <math alttext="y\in V(T)\setminus\{x\}" class="ltx_Math" display="inline" id="S4.SS2.SSS3.Px2.p1.3.m3.2"><semantics id="S4.SS2.SSS3.Px2.p1.3.m3.2a"><mrow id="S4.SS2.SSS3.Px2.p1.3.m3.2.3" xref="S4.SS2.SSS3.Px2.p1.3.m3.2.3.cmml"><mi id="S4.SS2.SSS3.Px2.p1.3.m3.2.3.2" xref="S4.SS2.SSS3.Px2.p1.3.m3.2.3.2.cmml">y</mi><mo id="S4.SS2.SSS3.Px2.p1.3.m3.2.3.1" xref="S4.SS2.SSS3.Px2.p1.3.m3.2.3.1.cmml">∈</mo><mrow id="S4.SS2.SSS3.Px2.p1.3.m3.2.3.3" xref="S4.SS2.SSS3.Px2.p1.3.m3.2.3.3.cmml"><mrow id="S4.SS2.SSS3.Px2.p1.3.m3.2.3.3.2" xref="S4.SS2.SSS3.Px2.p1.3.m3.2.3.3.2.cmml"><mi id="S4.SS2.SSS3.Px2.p1.3.m3.2.3.3.2.2" xref="S4.SS2.SSS3.Px2.p1.3.m3.2.3.3.2.2.cmml">V</mi><mo id="S4.SS2.SSS3.Px2.p1.3.m3.2.3.3.2.1" xref="S4.SS2.SSS3.Px2.p1.3.m3.2.3.3.2.1.cmml">⁢</mo><mrow id="S4.SS2.SSS3.Px2.p1.3.m3.2.3.3.2.3.2" xref="S4.SS2.SSS3.Px2.p1.3.m3.2.3.3.2.cmml"><mo id="S4.SS2.SSS3.Px2.p1.3.m3.2.3.3.2.3.2.1" stretchy="false" xref="S4.SS2.SSS3.Px2.p1.3.m3.2.3.3.2.cmml">(</mo><mi id="S4.SS2.SSS3.Px2.p1.3.m3.1.1" xref="S4.SS2.SSS3.Px2.p1.3.m3.1.1.cmml">T</mi><mo id="S4.SS2.SSS3.Px2.p1.3.m3.2.3.3.2.3.2.2" stretchy="false" xref="S4.SS2.SSS3.Px2.p1.3.m3.2.3.3.2.cmml">)</mo></mrow></mrow><mo id="S4.SS2.SSS3.Px2.p1.3.m3.2.3.3.1" xref="S4.SS2.SSS3.Px2.p1.3.m3.2.3.3.1.cmml">∖</mo><mrow id="S4.SS2.SSS3.Px2.p1.3.m3.2.3.3.3.2" xref="S4.SS2.SSS3.Px2.p1.3.m3.2.3.3.3.1.cmml"><mo id="S4.SS2.SSS3.Px2.p1.3.m3.2.3.3.3.2.1" stretchy="false" xref="S4.SS2.SSS3.Px2.p1.3.m3.2.3.3.3.1.cmml">{</mo><mi id="S4.SS2.SSS3.Px2.p1.3.m3.2.2" xref="S4.SS2.SSS3.Px2.p1.3.m3.2.2.cmml">x</mi><mo id="S4.SS2.SSS3.Px2.p1.3.m3.2.3.3.3.2.2" stretchy="false" xref="S4.SS2.SSS3.Px2.p1.3.m3.2.3.3.3.1.cmml">}</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS3.Px2.p1.3.m3.2b"><apply id="S4.SS2.SSS3.Px2.p1.3.m3.2.3.cmml" xref="S4.SS2.SSS3.Px2.p1.3.m3.2.3"><in id="S4.SS2.SSS3.Px2.p1.3.m3.2.3.1.cmml" xref="S4.SS2.SSS3.Px2.p1.3.m3.2.3.1"></in><ci id="S4.SS2.SSS3.Px2.p1.3.m3.2.3.2.cmml" xref="S4.SS2.SSS3.Px2.p1.3.m3.2.3.2">𝑦</ci><apply id="S4.SS2.SSS3.Px2.p1.3.m3.2.3.3.cmml" xref="S4.SS2.SSS3.Px2.p1.3.m3.2.3.3"><setdiff id="S4.SS2.SSS3.Px2.p1.3.m3.2.3.3.1.cmml" xref="S4.SS2.SSS3.Px2.p1.3.m3.2.3.3.1"></setdiff><apply id="S4.SS2.SSS3.Px2.p1.3.m3.2.3.3.2.cmml" xref="S4.SS2.SSS3.Px2.p1.3.m3.2.3.3.2"><times id="S4.SS2.SSS3.Px2.p1.3.m3.2.3.3.2.1.cmml" xref="S4.SS2.SSS3.Px2.p1.3.m3.2.3.3.2.1"></times><ci id="S4.SS2.SSS3.Px2.p1.3.m3.2.3.3.2.2.cmml" xref="S4.SS2.SSS3.Px2.p1.3.m3.2.3.3.2.2">𝑉</ci><ci id="S4.SS2.SSS3.Px2.p1.3.m3.1.1.cmml" xref="S4.SS2.SSS3.Px2.p1.3.m3.1.1">𝑇</ci></apply><set id="S4.SS2.SSS3.Px2.p1.3.m3.2.3.3.3.1.cmml" xref="S4.SS2.SSS3.Px2.p1.3.m3.2.3.3.3.2"><ci id="S4.SS2.SSS3.Px2.p1.3.m3.2.2.cmml" xref="S4.SS2.SSS3.Px2.p1.3.m3.2.2">𝑥</ci></set></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS3.Px2.p1.3.m3.2c">y\in V(T)\setminus\{x\}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS3.Px2.p1.3.m3.2d">italic_y ∈ italic_V ( italic_T ) ∖ { italic_x }</annotation></semantics></math>, <math alttext="G_{y}\setminus\{a,b\}" class="ltx_Math" display="inline" id="S4.SS2.SSS3.Px2.p1.4.m4.2"><semantics id="S4.SS2.SSS3.Px2.p1.4.m4.2a"><mrow id="S4.SS2.SSS3.Px2.p1.4.m4.2.3" xref="S4.SS2.SSS3.Px2.p1.4.m4.2.3.cmml"><msub id="S4.SS2.SSS3.Px2.p1.4.m4.2.3.2" xref="S4.SS2.SSS3.Px2.p1.4.m4.2.3.2.cmml"><mi id="S4.SS2.SSS3.Px2.p1.4.m4.2.3.2.2" xref="S4.SS2.SSS3.Px2.p1.4.m4.2.3.2.2.cmml">G</mi><mi id="S4.SS2.SSS3.Px2.p1.4.m4.2.3.2.3" xref="S4.SS2.SSS3.Px2.p1.4.m4.2.3.2.3.cmml">y</mi></msub><mo id="S4.SS2.SSS3.Px2.p1.4.m4.2.3.1" xref="S4.SS2.SSS3.Px2.p1.4.m4.2.3.1.cmml">∖</mo><mrow id="S4.SS2.SSS3.Px2.p1.4.m4.2.3.3.2" xref="S4.SS2.SSS3.Px2.p1.4.m4.2.3.3.1.cmml"><mo id="S4.SS2.SSS3.Px2.p1.4.m4.2.3.3.2.1" stretchy="false" xref="S4.SS2.SSS3.Px2.p1.4.m4.2.3.3.1.cmml">{</mo><mi id="S4.SS2.SSS3.Px2.p1.4.m4.1.1" xref="S4.SS2.SSS3.Px2.p1.4.m4.1.1.cmml">a</mi><mo id="S4.SS2.SSS3.Px2.p1.4.m4.2.3.3.2.2" xref="S4.SS2.SSS3.Px2.p1.4.m4.2.3.3.1.cmml">,</mo><mi id="S4.SS2.SSS3.Px2.p1.4.m4.2.2" xref="S4.SS2.SSS3.Px2.p1.4.m4.2.2.cmml">b</mi><mo id="S4.SS2.SSS3.Px2.p1.4.m4.2.3.3.2.3" stretchy="false" xref="S4.SS2.SSS3.Px2.p1.4.m4.2.3.3.1.cmml">}</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS3.Px2.p1.4.m4.2b"><apply id="S4.SS2.SSS3.Px2.p1.4.m4.2.3.cmml" xref="S4.SS2.SSS3.Px2.p1.4.m4.2.3"><setdiff id="S4.SS2.SSS3.Px2.p1.4.m4.2.3.1.cmml" xref="S4.SS2.SSS3.Px2.p1.4.m4.2.3.1"></setdiff><apply id="S4.SS2.SSS3.Px2.p1.4.m4.2.3.2.cmml" xref="S4.SS2.SSS3.Px2.p1.4.m4.2.3.2"><csymbol cd="ambiguous" id="S4.SS2.SSS3.Px2.p1.4.m4.2.3.2.1.cmml" xref="S4.SS2.SSS3.Px2.p1.4.m4.2.3.2">subscript</csymbol><ci id="S4.SS2.SSS3.Px2.p1.4.m4.2.3.2.2.cmml" xref="S4.SS2.SSS3.Px2.p1.4.m4.2.3.2.2">𝐺</ci><ci id="S4.SS2.SSS3.Px2.p1.4.m4.2.3.2.3.cmml" xref="S4.SS2.SSS3.Px2.p1.4.m4.2.3.2.3">𝑦</ci></apply><set id="S4.SS2.SSS3.Px2.p1.4.m4.2.3.3.1.cmml" xref="S4.SS2.SSS3.Px2.p1.4.m4.2.3.3.2"><ci id="S4.SS2.SSS3.Px2.p1.4.m4.1.1.cmml" xref="S4.SS2.SSS3.Px2.p1.4.m4.1.1">𝑎</ci><ci id="S4.SS2.SSS3.Px2.p1.4.m4.2.2.cmml" xref="S4.SS2.SSS3.Px2.p1.4.m4.2.2">𝑏</ci></set></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS3.Px2.p1.4.m4.2c">G_{y}\setminus\{a,b\}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS3.Px2.p1.4.m4.2d">italic_G start_POSTSUBSCRIPT italic_y end_POSTSUBSCRIPT ∖ { italic_a , italic_b }</annotation></semantics></math> is connected. Therefore, if <math alttext="u,v" class="ltx_Math" display="inline" id="S4.SS2.SSS3.Px2.p1.5.m5.2"><semantics id="S4.SS2.SSS3.Px2.p1.5.m5.2a"><mrow id="S4.SS2.SSS3.Px2.p1.5.m5.2.3.2" xref="S4.SS2.SSS3.Px2.p1.5.m5.2.3.1.cmml"><mi id="S4.SS2.SSS3.Px2.p1.5.m5.1.1" xref="S4.SS2.SSS3.Px2.p1.5.m5.1.1.cmml">u</mi><mo id="S4.SS2.SSS3.Px2.p1.5.m5.2.3.2.1" xref="S4.SS2.SSS3.Px2.p1.5.m5.2.3.1.cmml">,</mo><mi id="S4.SS2.SSS3.Px2.p1.5.m5.2.2" xref="S4.SS2.SSS3.Px2.p1.5.m5.2.2.cmml">v</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS3.Px2.p1.5.m5.2b"><list id="S4.SS2.SSS3.Px2.p1.5.m5.2.3.1.cmml" xref="S4.SS2.SSS3.Px2.p1.5.m5.2.3.2"><ci id="S4.SS2.SSS3.Px2.p1.5.m5.1.1.cmml" xref="S4.SS2.SSS3.Px2.p1.5.m5.1.1">𝑢</ci><ci id="S4.SS2.SSS3.Px2.p1.5.m5.2.2.cmml" xref="S4.SS2.SSS3.Px2.p1.5.m5.2.2">𝑣</ci></list></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS3.Px2.p1.5.m5.2c">u,v</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS3.Px2.p1.5.m5.2d">italic_u , italic_v</annotation></semantics></math> are in the same component of <math alttext="T\setminus\{x\}" class="ltx_Math" display="inline" id="S4.SS2.SSS3.Px2.p1.6.m6.1"><semantics id="S4.SS2.SSS3.Px2.p1.6.m6.1a"><mrow id="S4.SS2.SSS3.Px2.p1.6.m6.1.2" xref="S4.SS2.SSS3.Px2.p1.6.m6.1.2.cmml"><mi id="S4.SS2.SSS3.Px2.p1.6.m6.1.2.2" xref="S4.SS2.SSS3.Px2.p1.6.m6.1.2.2.cmml">T</mi><mo id="S4.SS2.SSS3.Px2.p1.6.m6.1.2.1" xref="S4.SS2.SSS3.Px2.p1.6.m6.1.2.1.cmml">∖</mo><mrow id="S4.SS2.SSS3.Px2.p1.6.m6.1.2.3.2" xref="S4.SS2.SSS3.Px2.p1.6.m6.1.2.3.1.cmml"><mo id="S4.SS2.SSS3.Px2.p1.6.m6.1.2.3.2.1" stretchy="false" xref="S4.SS2.SSS3.Px2.p1.6.m6.1.2.3.1.cmml">{</mo><mi id="S4.SS2.SSS3.Px2.p1.6.m6.1.1" xref="S4.SS2.SSS3.Px2.p1.6.m6.1.1.cmml">x</mi><mo id="S4.SS2.SSS3.Px2.p1.6.m6.1.2.3.2.2" stretchy="false" xref="S4.SS2.SSS3.Px2.p1.6.m6.1.2.3.1.cmml">}</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS3.Px2.p1.6.m6.1b"><apply id="S4.SS2.SSS3.Px2.p1.6.m6.1.2.cmml" xref="S4.SS2.SSS3.Px2.p1.6.m6.1.2"><setdiff id="S4.SS2.SSS3.Px2.p1.6.m6.1.2.1.cmml" xref="S4.SS2.SSS3.Px2.p1.6.m6.1.2.1"></setdiff><ci id="S4.SS2.SSS3.Px2.p1.6.m6.1.2.2.cmml" xref="S4.SS2.SSS3.Px2.p1.6.m6.1.2.2">𝑇</ci><set id="S4.SS2.SSS3.Px2.p1.6.m6.1.2.3.1.cmml" xref="S4.SS2.SSS3.Px2.p1.6.m6.1.2.3.2"><ci id="S4.SS2.SSS3.Px2.p1.6.m6.1.1.cmml" xref="S4.SS2.SSS3.Px2.p1.6.m6.1.1">𝑥</ci></set></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS3.Px2.p1.6.m6.1c">T\setminus\{x\}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS3.Px2.p1.6.m6.1d">italic_T ∖ { italic_x }</annotation></semantics></math>, then there is a <math alttext="u" class="ltx_Math" display="inline" id="S4.SS2.SSS3.Px2.p1.7.m7.1"><semantics id="S4.SS2.SSS3.Px2.p1.7.m7.1a"><mi id="S4.SS2.SSS3.Px2.p1.7.m7.1.1" xref="S4.SS2.SSS3.Px2.p1.7.m7.1.1.cmml">u</mi><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS3.Px2.p1.7.m7.1b"><ci id="S4.SS2.SSS3.Px2.p1.7.m7.1.1.cmml" xref="S4.SS2.SSS3.Px2.p1.7.m7.1.1">𝑢</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS3.Px2.p1.7.m7.1c">u</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS3.Px2.p1.7.m7.1d">italic_u</annotation></semantics></math>-<math alttext="v" class="ltx_Math" display="inline" id="S4.SS2.SSS3.Px2.p1.8.m8.1"><semantics id="S4.SS2.SSS3.Px2.p1.8.m8.1a"><mi id="S4.SS2.SSS3.Px2.p1.8.m8.1.1" xref="S4.SS2.SSS3.Px2.p1.8.m8.1.1.cmml">v</mi><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS3.Px2.p1.8.m8.1b"><ci id="S4.SS2.SSS3.Px2.p1.8.m8.1.1.cmml" xref="S4.SS2.SSS3.Px2.p1.8.m8.1.1">𝑣</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS3.Px2.p1.8.m8.1c">v</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS3.Px2.p1.8.m8.1d">italic_v</annotation></semantics></math> path in <math alttext="E\setminus\{a,b\}" class="ltx_Math" display="inline" id="S4.SS2.SSS3.Px2.p1.9.m9.2"><semantics id="S4.SS2.SSS3.Px2.p1.9.m9.2a"><mrow id="S4.SS2.SSS3.Px2.p1.9.m9.2.3" xref="S4.SS2.SSS3.Px2.p1.9.m9.2.3.cmml"><mi id="S4.SS2.SSS3.Px2.p1.9.m9.2.3.2" xref="S4.SS2.SSS3.Px2.p1.9.m9.2.3.2.cmml">E</mi><mo id="S4.SS2.SSS3.Px2.p1.9.m9.2.3.1" xref="S4.SS2.SSS3.Px2.p1.9.m9.2.3.1.cmml">∖</mo><mrow id="S4.SS2.SSS3.Px2.p1.9.m9.2.3.3.2" xref="S4.SS2.SSS3.Px2.p1.9.m9.2.3.3.1.cmml"><mo id="S4.SS2.SSS3.Px2.p1.9.m9.2.3.3.2.1" stretchy="false" xref="S4.SS2.SSS3.Px2.p1.9.m9.2.3.3.1.cmml">{</mo><mi id="S4.SS2.SSS3.Px2.p1.9.m9.1.1" xref="S4.SS2.SSS3.Px2.p1.9.m9.1.1.cmml">a</mi><mo id="S4.SS2.SSS3.Px2.p1.9.m9.2.3.3.2.2" xref="S4.SS2.SSS3.Px2.p1.9.m9.2.3.3.1.cmml">,</mo><mi id="S4.SS2.SSS3.Px2.p1.9.m9.2.2" xref="S4.SS2.SSS3.Px2.p1.9.m9.2.2.cmml">b</mi><mo id="S4.SS2.SSS3.Px2.p1.9.m9.2.3.3.2.3" stretchy="false" xref="S4.SS2.SSS3.Px2.p1.9.m9.2.3.3.1.cmml">}</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS3.Px2.p1.9.m9.2b"><apply id="S4.SS2.SSS3.Px2.p1.9.m9.2.3.cmml" xref="S4.SS2.SSS3.Px2.p1.9.m9.2.3"><setdiff id="S4.SS2.SSS3.Px2.p1.9.m9.2.3.1.cmml" xref="S4.SS2.SSS3.Px2.p1.9.m9.2.3.1"></setdiff><ci id="S4.SS2.SSS3.Px2.p1.9.m9.2.3.2.cmml" xref="S4.SS2.SSS3.Px2.p1.9.m9.2.3.2">𝐸</ci><set id="S4.SS2.SSS3.Px2.p1.9.m9.2.3.3.1.cmml" xref="S4.SS2.SSS3.Px2.p1.9.m9.2.3.3.2"><ci id="S4.SS2.SSS3.Px2.p1.9.m9.1.1.cmml" xref="S4.SS2.SSS3.Px2.p1.9.m9.1.1">𝑎</ci><ci id="S4.SS2.SSS3.Px2.p1.9.m9.2.2.cmml" xref="S4.SS2.SSS3.Px2.p1.9.m9.2.2">𝑏</ci></set></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS3.Px2.p1.9.m9.2c">E\setminus\{a,b\}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS3.Px2.p1.9.m9.2d">italic_E ∖ { italic_a , italic_b }</annotation></semantics></math>. Thus we assume <math alttext="u" class="ltx_Math" display="inline" id="S4.SS2.SSS3.Px2.p1.10.m10.1"><semantics id="S4.SS2.SSS3.Px2.p1.10.m10.1a"><mi id="S4.SS2.SSS3.Px2.p1.10.m10.1.1" xref="S4.SS2.SSS3.Px2.p1.10.m10.1.1.cmml">u</mi><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS3.Px2.p1.10.m10.1b"><ci id="S4.SS2.SSS3.Px2.p1.10.m10.1.1.cmml" xref="S4.SS2.SSS3.Px2.p1.10.m10.1.1">𝑢</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS3.Px2.p1.10.m10.1c">u</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS3.Px2.p1.10.m10.1d">italic_u</annotation></semantics></math> and <math alttext="v" class="ltx_Math" display="inline" id="S4.SS2.SSS3.Px2.p1.11.m11.1"><semantics id="S4.SS2.SSS3.Px2.p1.11.m11.1a"><mi id="S4.SS2.SSS3.Px2.p1.11.m11.1.1" xref="S4.SS2.SSS3.Px2.p1.11.m11.1.1.cmml">v</mi><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS3.Px2.p1.11.m11.1b"><ci id="S4.SS2.SSS3.Px2.p1.11.m11.1.1.cmml" xref="S4.SS2.SSS3.Px2.p1.11.m11.1.1">𝑣</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS3.Px2.p1.11.m11.1c">v</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS3.Px2.p1.11.m11.1d">italic_v</annotation></semantics></math> are in distinct components of <math alttext="T\setminus\{x\}" class="ltx_Math" display="inline" id="S4.SS2.SSS3.Px2.p1.12.m12.1"><semantics id="S4.SS2.SSS3.Px2.p1.12.m12.1a"><mrow id="S4.SS2.SSS3.Px2.p1.12.m12.1.2" xref="S4.SS2.SSS3.Px2.p1.12.m12.1.2.cmml"><mi id="S4.SS2.SSS3.Px2.p1.12.m12.1.2.2" xref="S4.SS2.SSS3.Px2.p1.12.m12.1.2.2.cmml">T</mi><mo id="S4.SS2.SSS3.Px2.p1.12.m12.1.2.1" xref="S4.SS2.SSS3.Px2.p1.12.m12.1.2.1.cmml">∖</mo><mrow id="S4.SS2.SSS3.Px2.p1.12.m12.1.2.3.2" xref="S4.SS2.SSS3.Px2.p1.12.m12.1.2.3.1.cmml"><mo id="S4.SS2.SSS3.Px2.p1.12.m12.1.2.3.2.1" stretchy="false" xref="S4.SS2.SSS3.Px2.p1.12.m12.1.2.3.1.cmml">{</mo><mi id="S4.SS2.SSS3.Px2.p1.12.m12.1.1" xref="S4.SS2.SSS3.Px2.p1.12.m12.1.1.cmml">x</mi><mo id="S4.SS2.SSS3.Px2.p1.12.m12.1.2.3.2.2" stretchy="false" xref="S4.SS2.SSS3.Px2.p1.12.m12.1.2.3.1.cmml">}</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS3.Px2.p1.12.m12.1b"><apply id="S4.SS2.SSS3.Px2.p1.12.m12.1.2.cmml" xref="S4.SS2.SSS3.Px2.p1.12.m12.1.2"><setdiff id="S4.SS2.SSS3.Px2.p1.12.m12.1.2.1.cmml" xref="S4.SS2.SSS3.Px2.p1.12.m12.1.2.1"></setdiff><ci id="S4.SS2.SSS3.Px2.p1.12.m12.1.2.2.cmml" xref="S4.SS2.SSS3.Px2.p1.12.m12.1.2.2">𝑇</ci><set id="S4.SS2.SSS3.Px2.p1.12.m12.1.2.3.1.cmml" xref="S4.SS2.SSS3.Px2.p1.12.m12.1.2.3.2"><ci id="S4.SS2.SSS3.Px2.p1.12.m12.1.1.cmml" xref="S4.SS2.SSS3.Px2.p1.12.m12.1.1">𝑥</ci></set></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS3.Px2.p1.12.m12.1c">T\setminus\{x\}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS3.Px2.p1.12.m12.1d">italic_T ∖ { italic_x }</annotation></semantics></math>.</p> <ul class="ltx_itemize" id="S4.I10"> <li class="ltx_item" id="S4.I10.i1" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">•</span> <div class="ltx_para" id="S4.I10.i1.p1"> <p class="ltx_p" id="S4.I10.i1.p1.16"><span class="ltx_text ltx_font_bold" id="S4.I10.i1.p1.2.2">Case 2a: <math alttext="\boldsymbol{h(u),h(v)}" class="ltx_Math" display="inline" id="S4.I10.i1.p1.1.1.m1.4"><semantics id="S4.I10.i1.p1.1.1.m1.4a"><mrow id="S4.I10.i1.p1.1.1.m1.4.4.2" xref="S4.I10.i1.p1.1.1.m1.4.4.3.cmml"><mrow id="S4.I10.i1.p1.1.1.m1.3.3.1.1" xref="S4.I10.i1.p1.1.1.m1.3.3.1.1.cmml"><mi id="S4.I10.i1.p1.1.1.m1.3.3.1.1.2" xref="S4.I10.i1.p1.1.1.m1.3.3.1.1.2.cmml">h</mi><mo id="S4.I10.i1.p1.1.1.m1.3.3.1.1.1" xref="S4.I10.i1.p1.1.1.m1.3.3.1.1.1.cmml">⁢</mo><mrow id="S4.I10.i1.p1.1.1.m1.3.3.1.1.3.2" xref="S4.I10.i1.p1.1.1.m1.3.3.1.1.cmml"><mo class="ltx_mathvariant_bold" id="S4.I10.i1.p1.1.1.m1.3.3.1.1.3.2.1" mathvariant="bold" stretchy="false" xref="S4.I10.i1.p1.1.1.m1.3.3.1.1.cmml">(</mo><mi id="S4.I10.i1.p1.1.1.m1.1.1" xref="S4.I10.i1.p1.1.1.m1.1.1.cmml">u</mi><mo class="ltx_mathvariant_bold" id="S4.I10.i1.p1.1.1.m1.3.3.1.1.3.2.2" mathvariant="bold" stretchy="false" xref="S4.I10.i1.p1.1.1.m1.3.3.1.1.cmml">)</mo></mrow></mrow><mo class="ltx_mathvariant_bold" id="S4.I10.i1.p1.1.1.m1.4.4.2.3" mathvariant="bold" xref="S4.I10.i1.p1.1.1.m1.4.4.3.cmml">,</mo><mrow id="S4.I10.i1.p1.1.1.m1.4.4.2.2" xref="S4.I10.i1.p1.1.1.m1.4.4.2.2.cmml"><mi id="S4.I10.i1.p1.1.1.m1.4.4.2.2.2" xref="S4.I10.i1.p1.1.1.m1.4.4.2.2.2.cmml">h</mi><mo id="S4.I10.i1.p1.1.1.m1.4.4.2.2.1" xref="S4.I10.i1.p1.1.1.m1.4.4.2.2.1.cmml">⁢</mo><mrow id="S4.I10.i1.p1.1.1.m1.4.4.2.2.3.2" xref="S4.I10.i1.p1.1.1.m1.4.4.2.2.cmml"><mo class="ltx_mathvariant_bold" id="S4.I10.i1.p1.1.1.m1.4.4.2.2.3.2.1" mathvariant="bold" stretchy="false" xref="S4.I10.i1.p1.1.1.m1.4.4.2.2.cmml">(</mo><mi id="S4.I10.i1.p1.1.1.m1.2.2" xref="S4.I10.i1.p1.1.1.m1.2.2.cmml">v</mi><mo class="ltx_mathvariant_bold" id="S4.I10.i1.p1.1.1.m1.4.4.2.2.3.2.2" mathvariant="bold" stretchy="false" xref="S4.I10.i1.p1.1.1.m1.4.4.2.2.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.I10.i1.p1.1.1.m1.4b"><list id="S4.I10.i1.p1.1.1.m1.4.4.3.cmml" xref="S4.I10.i1.p1.1.1.m1.4.4.2"><apply id="S4.I10.i1.p1.1.1.m1.3.3.1.1.cmml" xref="S4.I10.i1.p1.1.1.m1.3.3.1.1"><times id="S4.I10.i1.p1.1.1.m1.3.3.1.1.1.cmml" xref="S4.I10.i1.p1.1.1.m1.3.3.1.1.1"></times><ci id="S4.I10.i1.p1.1.1.m1.3.3.1.1.2.cmml" xref="S4.I10.i1.p1.1.1.m1.3.3.1.1.2">ℎ</ci><ci id="S4.I10.i1.p1.1.1.m1.1.1.cmml" xref="S4.I10.i1.p1.1.1.m1.1.1">𝑢</ci></apply><apply id="S4.I10.i1.p1.1.1.m1.4.4.2.2.cmml" xref="S4.I10.i1.p1.1.1.m1.4.4.2.2"><times id="S4.I10.i1.p1.1.1.m1.4.4.2.2.1.cmml" xref="S4.I10.i1.p1.1.1.m1.4.4.2.2.1"></times><ci id="S4.I10.i1.p1.1.1.m1.4.4.2.2.2.cmml" xref="S4.I10.i1.p1.1.1.m1.4.4.2.2.2">ℎ</ci><ci id="S4.I10.i1.p1.1.1.m1.2.2.cmml" xref="S4.I10.i1.p1.1.1.m1.2.2">𝑣</ci></apply></list></annotation-xml><annotation encoding="application/x-tex" id="S4.I10.i1.p1.1.1.m1.4c">\boldsymbol{h(u),h(v)}</annotation><annotation encoding="application/x-llamapun" id="S4.I10.i1.p1.1.1.m1.4d">bold_italic_h bold_( bold_italic_u bold_) bold_, bold_italic_h bold_( bold_italic_v bold_)</annotation></semantics></math> are both in <math alttext="\boldsymbol{T_{x}}" class="ltx_Math" display="inline" id="S4.I10.i1.p1.2.2.m2.1"><semantics id="S4.I10.i1.p1.2.2.m2.1a"><msub id="S4.I10.i1.p1.2.2.m2.1.1" xref="S4.I10.i1.p1.2.2.m2.1.1.cmml"><mi id="S4.I10.i1.p1.2.2.m2.1.1.2" xref="S4.I10.i1.p1.2.2.m2.1.1.2.cmml">T</mi><mi id="S4.I10.i1.p1.2.2.m2.1.1.3" xref="S4.I10.i1.p1.2.2.m2.1.1.3.cmml">x</mi></msub><annotation-xml encoding="MathML-Content" id="S4.I10.i1.p1.2.2.m2.1b"><apply id="S4.I10.i1.p1.2.2.m2.1.1.cmml" xref="S4.I10.i1.p1.2.2.m2.1.1"><csymbol cd="ambiguous" id="S4.I10.i1.p1.2.2.m2.1.1.1.cmml" xref="S4.I10.i1.p1.2.2.m2.1.1">subscript</csymbol><ci id="S4.I10.i1.p1.2.2.m2.1.1.2.cmml" xref="S4.I10.i1.p1.2.2.m2.1.1.2">𝑇</ci><ci id="S4.I10.i1.p1.2.2.m2.1.1.3.cmml" xref="S4.I10.i1.p1.2.2.m2.1.1.3">𝑥</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.I10.i1.p1.2.2.m2.1c">\boldsymbol{T_{x}}</annotation><annotation encoding="application/x-llamapun" id="S4.I10.i1.p1.2.2.m2.1d">bold_italic_T start_POSTSUBSCRIPT bold_italic_x end_POSTSUBSCRIPT</annotation></semantics></math>:</span> Let <math alttext="T(u)" class="ltx_Math" display="inline" id="S4.I10.i1.p1.3.m1.1"><semantics id="S4.I10.i1.p1.3.m1.1a"><mrow id="S4.I10.i1.p1.3.m1.1.2" xref="S4.I10.i1.p1.3.m1.1.2.cmml"><mi id="S4.I10.i1.p1.3.m1.1.2.2" xref="S4.I10.i1.p1.3.m1.1.2.2.cmml">T</mi><mo id="S4.I10.i1.p1.3.m1.1.2.1" xref="S4.I10.i1.p1.3.m1.1.2.1.cmml">⁢</mo><mrow id="S4.I10.i1.p1.3.m1.1.2.3.2" xref="S4.I10.i1.p1.3.m1.1.2.cmml"><mo id="S4.I10.i1.p1.3.m1.1.2.3.2.1" stretchy="false" xref="S4.I10.i1.p1.3.m1.1.2.cmml">(</mo><mi id="S4.I10.i1.p1.3.m1.1.1" xref="S4.I10.i1.p1.3.m1.1.1.cmml">u</mi><mo id="S4.I10.i1.p1.3.m1.1.2.3.2.2" stretchy="false" xref="S4.I10.i1.p1.3.m1.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.I10.i1.p1.3.m1.1b"><apply id="S4.I10.i1.p1.3.m1.1.2.cmml" xref="S4.I10.i1.p1.3.m1.1.2"><times id="S4.I10.i1.p1.3.m1.1.2.1.cmml" xref="S4.I10.i1.p1.3.m1.1.2.1"></times><ci id="S4.I10.i1.p1.3.m1.1.2.2.cmml" xref="S4.I10.i1.p1.3.m1.1.2.2">𝑇</ci><ci id="S4.I10.i1.p1.3.m1.1.1.cmml" xref="S4.I10.i1.p1.3.m1.1.1">𝑢</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.I10.i1.p1.3.m1.1c">T(u)</annotation><annotation encoding="application/x-llamapun" id="S4.I10.i1.p1.3.m1.1d">italic_T ( italic_u )</annotation></semantics></math> and <math alttext="T(v)" class="ltx_Math" display="inline" id="S4.I10.i1.p1.4.m2.1"><semantics id="S4.I10.i1.p1.4.m2.1a"><mrow id="S4.I10.i1.p1.4.m2.1.2" xref="S4.I10.i1.p1.4.m2.1.2.cmml"><mi id="S4.I10.i1.p1.4.m2.1.2.2" xref="S4.I10.i1.p1.4.m2.1.2.2.cmml">T</mi><mo id="S4.I10.i1.p1.4.m2.1.2.1" xref="S4.I10.i1.p1.4.m2.1.2.1.cmml">⁢</mo><mrow id="S4.I10.i1.p1.4.m2.1.2.3.2" xref="S4.I10.i1.p1.4.m2.1.2.cmml"><mo id="S4.I10.i1.p1.4.m2.1.2.3.2.1" stretchy="false" xref="S4.I10.i1.p1.4.m2.1.2.cmml">(</mo><mi id="S4.I10.i1.p1.4.m2.1.1" xref="S4.I10.i1.p1.4.m2.1.1.cmml">v</mi><mo id="S4.I10.i1.p1.4.m2.1.2.3.2.2" stretchy="false" xref="S4.I10.i1.p1.4.m2.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.I10.i1.p1.4.m2.1b"><apply id="S4.I10.i1.p1.4.m2.1.2.cmml" xref="S4.I10.i1.p1.4.m2.1.2"><times id="S4.I10.i1.p1.4.m2.1.2.1.cmml" xref="S4.I10.i1.p1.4.m2.1.2.1"></times><ci id="S4.I10.i1.p1.4.m2.1.2.2.cmml" xref="S4.I10.i1.p1.4.m2.1.2.2">𝑇</ci><ci id="S4.I10.i1.p1.4.m2.1.1.cmml" xref="S4.I10.i1.p1.4.m2.1.1">𝑣</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.I10.i1.p1.4.m2.1c">T(v)</annotation><annotation encoding="application/x-llamapun" id="S4.I10.i1.p1.4.m2.1d">italic_T ( italic_v )</annotation></semantics></math> be the subtrees of <math alttext="T_{x}" class="ltx_Math" display="inline" id="S4.I10.i1.p1.5.m3.1"><semantics id="S4.I10.i1.p1.5.m3.1a"><msub id="S4.I10.i1.p1.5.m3.1.1" xref="S4.I10.i1.p1.5.m3.1.1.cmml"><mi id="S4.I10.i1.p1.5.m3.1.1.2" xref="S4.I10.i1.p1.5.m3.1.1.2.cmml">T</mi><mi id="S4.I10.i1.p1.5.m3.1.1.3" xref="S4.I10.i1.p1.5.m3.1.1.3.cmml">x</mi></msub><annotation-xml encoding="MathML-Content" id="S4.I10.i1.p1.5.m3.1b"><apply id="S4.I10.i1.p1.5.m3.1.1.cmml" xref="S4.I10.i1.p1.5.m3.1.1"><csymbol cd="ambiguous" id="S4.I10.i1.p1.5.m3.1.1.1.cmml" xref="S4.I10.i1.p1.5.m3.1.1">subscript</csymbol><ci id="S4.I10.i1.p1.5.m3.1.1.2.cmml" xref="S4.I10.i1.p1.5.m3.1.1.2">𝑇</ci><ci id="S4.I10.i1.p1.5.m3.1.1.3.cmml" xref="S4.I10.i1.p1.5.m3.1.1.3">𝑥</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.I10.i1.p1.5.m3.1c">T_{x}</annotation><annotation encoding="application/x-llamapun" id="S4.I10.i1.p1.5.m3.1d">italic_T start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT</annotation></semantics></math> containing <math alttext="h(u)" class="ltx_Math" display="inline" id="S4.I10.i1.p1.6.m4.1"><semantics id="S4.I10.i1.p1.6.m4.1a"><mrow id="S4.I10.i1.p1.6.m4.1.2" xref="S4.I10.i1.p1.6.m4.1.2.cmml"><mi id="S4.I10.i1.p1.6.m4.1.2.2" xref="S4.I10.i1.p1.6.m4.1.2.2.cmml">h</mi><mo id="S4.I10.i1.p1.6.m4.1.2.1" xref="S4.I10.i1.p1.6.m4.1.2.1.cmml">⁢</mo><mrow id="S4.I10.i1.p1.6.m4.1.2.3.2" xref="S4.I10.i1.p1.6.m4.1.2.cmml"><mo id="S4.I10.i1.p1.6.m4.1.2.3.2.1" stretchy="false" xref="S4.I10.i1.p1.6.m4.1.2.cmml">(</mo><mi id="S4.I10.i1.p1.6.m4.1.1" xref="S4.I10.i1.p1.6.m4.1.1.cmml">u</mi><mo id="S4.I10.i1.p1.6.m4.1.2.3.2.2" stretchy="false" xref="S4.I10.i1.p1.6.m4.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.I10.i1.p1.6.m4.1b"><apply id="S4.I10.i1.p1.6.m4.1.2.cmml" xref="S4.I10.i1.p1.6.m4.1.2"><times id="S4.I10.i1.p1.6.m4.1.2.1.cmml" xref="S4.I10.i1.p1.6.m4.1.2.1"></times><ci id="S4.I10.i1.p1.6.m4.1.2.2.cmml" xref="S4.I10.i1.p1.6.m4.1.2.2">ℎ</ci><ci id="S4.I10.i1.p1.6.m4.1.1.cmml" xref="S4.I10.i1.p1.6.m4.1.1">𝑢</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.I10.i1.p1.6.m4.1c">h(u)</annotation><annotation encoding="application/x-llamapun" id="S4.I10.i1.p1.6.m4.1d">italic_h ( italic_u )</annotation></semantics></math> and <math alttext="h(v)" class="ltx_Math" display="inline" id="S4.I10.i1.p1.7.m5.1"><semantics id="S4.I10.i1.p1.7.m5.1a"><mrow id="S4.I10.i1.p1.7.m5.1.2" xref="S4.I10.i1.p1.7.m5.1.2.cmml"><mi id="S4.I10.i1.p1.7.m5.1.2.2" xref="S4.I10.i1.p1.7.m5.1.2.2.cmml">h</mi><mo id="S4.I10.i1.p1.7.m5.1.2.1" xref="S4.I10.i1.p1.7.m5.1.2.1.cmml">⁢</mo><mrow id="S4.I10.i1.p1.7.m5.1.2.3.2" xref="S4.I10.i1.p1.7.m5.1.2.cmml"><mo id="S4.I10.i1.p1.7.m5.1.2.3.2.1" stretchy="false" xref="S4.I10.i1.p1.7.m5.1.2.cmml">(</mo><mi id="S4.I10.i1.p1.7.m5.1.1" xref="S4.I10.i1.p1.7.m5.1.1.cmml">v</mi><mo id="S4.I10.i1.p1.7.m5.1.2.3.2.2" stretchy="false" xref="S4.I10.i1.p1.7.m5.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.I10.i1.p1.7.m5.1b"><apply id="S4.I10.i1.p1.7.m5.1.2.cmml" xref="S4.I10.i1.p1.7.m5.1.2"><times id="S4.I10.i1.p1.7.m5.1.2.1.cmml" xref="S4.I10.i1.p1.7.m5.1.2.1"></times><ci id="S4.I10.i1.p1.7.m5.1.2.2.cmml" xref="S4.I10.i1.p1.7.m5.1.2.2">ℎ</ci><ci id="S4.I10.i1.p1.7.m5.1.1.cmml" xref="S4.I10.i1.p1.7.m5.1.1">𝑣</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.I10.i1.p1.7.m5.1c">h(v)</annotation><annotation encoding="application/x-llamapun" id="S4.I10.i1.p1.7.m5.1d">italic_h ( italic_v )</annotation></semantics></math> respectively. Suppose either of the subtrees are rooted at nodes that are not “good” (as defined in Algorithm <a class="ltx_ref" href="https://arxiv.org/html/2503.00712v1#algorithm7" title="In 4.2.3 Bounding the Approximation Ratio ‣ 4.2 Two-to-Three Augmentation ‣ 4 Vertex Connectivity Augmentation in Link-Arrival Model ‣ Streaming Algorithms for Network Design"><span class="ltx_text ltx_ref_tag">7</span></a>). Then, there must be two distinct supernodes <math alttext="z,z^{\prime}" class="ltx_Math" display="inline" id="S4.I10.i1.p1.8.m6.2"><semantics id="S4.I10.i1.p1.8.m6.2a"><mrow id="S4.I10.i1.p1.8.m6.2.2.1" xref="S4.I10.i1.p1.8.m6.2.2.2.cmml"><mi id="S4.I10.i1.p1.8.m6.1.1" xref="S4.I10.i1.p1.8.m6.1.1.cmml">z</mi><mo id="S4.I10.i1.p1.8.m6.2.2.1.2" xref="S4.I10.i1.p1.8.m6.2.2.2.cmml">,</mo><msup id="S4.I10.i1.p1.8.m6.2.2.1.1" xref="S4.I10.i1.p1.8.m6.2.2.1.1.cmml"><mi id="S4.I10.i1.p1.8.m6.2.2.1.1.2" xref="S4.I10.i1.p1.8.m6.2.2.1.1.2.cmml">z</mi><mo id="S4.I10.i1.p1.8.m6.2.2.1.1.3" xref="S4.I10.i1.p1.8.m6.2.2.1.1.3.cmml">′</mo></msup></mrow><annotation-xml encoding="MathML-Content" id="S4.I10.i1.p1.8.m6.2b"><list id="S4.I10.i1.p1.8.m6.2.2.2.cmml" xref="S4.I10.i1.p1.8.m6.2.2.1"><ci id="S4.I10.i1.p1.8.m6.1.1.cmml" xref="S4.I10.i1.p1.8.m6.1.1">𝑧</ci><apply id="S4.I10.i1.p1.8.m6.2.2.1.1.cmml" xref="S4.I10.i1.p1.8.m6.2.2.1.1"><csymbol cd="ambiguous" id="S4.I10.i1.p1.8.m6.2.2.1.1.1.cmml" xref="S4.I10.i1.p1.8.m6.2.2.1.1">superscript</csymbol><ci id="S4.I10.i1.p1.8.m6.2.2.1.1.2.cmml" xref="S4.I10.i1.p1.8.m6.2.2.1.1.2">𝑧</ci><ci id="S4.I10.i1.p1.8.m6.2.2.1.1.3.cmml" xref="S4.I10.i1.p1.8.m6.2.2.1.1.3">′</ci></apply></list></annotation-xml><annotation encoding="application/x-tex" id="S4.I10.i1.p1.8.m6.2c">z,z^{\prime}</annotation><annotation encoding="application/x-llamapun" id="S4.I10.i1.p1.8.m6.2d">italic_z , italic_z start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math> in <math alttext="C^{\prime\prime}(x)" class="ltx_Math" display="inline" id="S4.I10.i1.p1.9.m7.1"><semantics id="S4.I10.i1.p1.9.m7.1a"><mrow id="S4.I10.i1.p1.9.m7.1.2" xref="S4.I10.i1.p1.9.m7.1.2.cmml"><msup id="S4.I10.i1.p1.9.m7.1.2.2" xref="S4.I10.i1.p1.9.m7.1.2.2.cmml"><mi id="S4.I10.i1.p1.9.m7.1.2.2.2" xref="S4.I10.i1.p1.9.m7.1.2.2.2.cmml">C</mi><mo id="S4.I10.i1.p1.9.m7.1.2.2.3" xref="S4.I10.i1.p1.9.m7.1.2.2.3.cmml">′′</mo></msup><mo id="S4.I10.i1.p1.9.m7.1.2.1" xref="S4.I10.i1.p1.9.m7.1.2.1.cmml">⁢</mo><mrow id="S4.I10.i1.p1.9.m7.1.2.3.2" xref="S4.I10.i1.p1.9.m7.1.2.cmml"><mo id="S4.I10.i1.p1.9.m7.1.2.3.2.1" stretchy="false" xref="S4.I10.i1.p1.9.m7.1.2.cmml">(</mo><mi id="S4.I10.i1.p1.9.m7.1.1" xref="S4.I10.i1.p1.9.m7.1.1.cmml">x</mi><mo id="S4.I10.i1.p1.9.m7.1.2.3.2.2" stretchy="false" xref="S4.I10.i1.p1.9.m7.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.I10.i1.p1.9.m7.1b"><apply id="S4.I10.i1.p1.9.m7.1.2.cmml" xref="S4.I10.i1.p1.9.m7.1.2"><times id="S4.I10.i1.p1.9.m7.1.2.1.cmml" xref="S4.I10.i1.p1.9.m7.1.2.1"></times><apply id="S4.I10.i1.p1.9.m7.1.2.2.cmml" xref="S4.I10.i1.p1.9.m7.1.2.2"><csymbol cd="ambiguous" id="S4.I10.i1.p1.9.m7.1.2.2.1.cmml" xref="S4.I10.i1.p1.9.m7.1.2.2">superscript</csymbol><ci id="S4.I10.i1.p1.9.m7.1.2.2.2.cmml" xref="S4.I10.i1.p1.9.m7.1.2.2.2">𝐶</ci><ci id="S4.I10.i1.p1.9.m7.1.2.2.3.cmml" xref="S4.I10.i1.p1.9.m7.1.2.2.3">′′</ci></apply><ci id="S4.I10.i1.p1.9.m7.1.1.cmml" xref="S4.I10.i1.p1.9.m7.1.1">𝑥</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.I10.i1.p1.9.m7.1c">C^{\prime\prime}(x)</annotation><annotation encoding="application/x-llamapun" id="S4.I10.i1.p1.9.m7.1d">italic_C start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT ( italic_x )</annotation></semantics></math> such that all vertices associated with <math alttext="T(u)" class="ltx_Math" display="inline" id="S4.I10.i1.p1.10.m8.1"><semantics id="S4.I10.i1.p1.10.m8.1a"><mrow id="S4.I10.i1.p1.10.m8.1.2" xref="S4.I10.i1.p1.10.m8.1.2.cmml"><mi id="S4.I10.i1.p1.10.m8.1.2.2" xref="S4.I10.i1.p1.10.m8.1.2.2.cmml">T</mi><mo id="S4.I10.i1.p1.10.m8.1.2.1" xref="S4.I10.i1.p1.10.m8.1.2.1.cmml">⁢</mo><mrow id="S4.I10.i1.p1.10.m8.1.2.3.2" xref="S4.I10.i1.p1.10.m8.1.2.cmml"><mo id="S4.I10.i1.p1.10.m8.1.2.3.2.1" stretchy="false" xref="S4.I10.i1.p1.10.m8.1.2.cmml">(</mo><mi id="S4.I10.i1.p1.10.m8.1.1" xref="S4.I10.i1.p1.10.m8.1.1.cmml">u</mi><mo id="S4.I10.i1.p1.10.m8.1.2.3.2.2" stretchy="false" xref="S4.I10.i1.p1.10.m8.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.I10.i1.p1.10.m8.1b"><apply id="S4.I10.i1.p1.10.m8.1.2.cmml" xref="S4.I10.i1.p1.10.m8.1.2"><times id="S4.I10.i1.p1.10.m8.1.2.1.cmml" xref="S4.I10.i1.p1.10.m8.1.2.1"></times><ci id="S4.I10.i1.p1.10.m8.1.2.2.cmml" xref="S4.I10.i1.p1.10.m8.1.2.2">𝑇</ci><ci id="S4.I10.i1.p1.10.m8.1.1.cmml" xref="S4.I10.i1.p1.10.m8.1.1">𝑢</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.I10.i1.p1.10.m8.1c">T(u)</annotation><annotation encoding="application/x-llamapun" id="S4.I10.i1.p1.10.m8.1d">italic_T ( italic_u )</annotation></semantics></math> are contracted into <math alttext="z" class="ltx_Math" display="inline" id="S4.I10.i1.p1.11.m9.1"><semantics id="S4.I10.i1.p1.11.m9.1a"><mi id="S4.I10.i1.p1.11.m9.1.1" xref="S4.I10.i1.p1.11.m9.1.1.cmml">z</mi><annotation-xml encoding="MathML-Content" id="S4.I10.i1.p1.11.m9.1b"><ci id="S4.I10.i1.p1.11.m9.1.1.cmml" xref="S4.I10.i1.p1.11.m9.1.1">𝑧</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.I10.i1.p1.11.m9.1c">z</annotation><annotation encoding="application/x-llamapun" id="S4.I10.i1.p1.11.m9.1d">italic_z</annotation></semantics></math> and all vertices associated with <math alttext="T(v)" class="ltx_Math" display="inline" id="S4.I10.i1.p1.12.m10.1"><semantics id="S4.I10.i1.p1.12.m10.1a"><mrow id="S4.I10.i1.p1.12.m10.1.2" xref="S4.I10.i1.p1.12.m10.1.2.cmml"><mi id="S4.I10.i1.p1.12.m10.1.2.2" xref="S4.I10.i1.p1.12.m10.1.2.2.cmml">T</mi><mo id="S4.I10.i1.p1.12.m10.1.2.1" xref="S4.I10.i1.p1.12.m10.1.2.1.cmml">⁢</mo><mrow id="S4.I10.i1.p1.12.m10.1.2.3.2" xref="S4.I10.i1.p1.12.m10.1.2.cmml"><mo id="S4.I10.i1.p1.12.m10.1.2.3.2.1" stretchy="false" xref="S4.I10.i1.p1.12.m10.1.2.cmml">(</mo><mi id="S4.I10.i1.p1.12.m10.1.1" xref="S4.I10.i1.p1.12.m10.1.1.cmml">v</mi><mo id="S4.I10.i1.p1.12.m10.1.2.3.2.2" stretchy="false" xref="S4.I10.i1.p1.12.m10.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.I10.i1.p1.12.m10.1b"><apply id="S4.I10.i1.p1.12.m10.1.2.cmml" xref="S4.I10.i1.p1.12.m10.1.2"><times id="S4.I10.i1.p1.12.m10.1.2.1.cmml" xref="S4.I10.i1.p1.12.m10.1.2.1"></times><ci id="S4.I10.i1.p1.12.m10.1.2.2.cmml" xref="S4.I10.i1.p1.12.m10.1.2.2">𝑇</ci><ci id="S4.I10.i1.p1.12.m10.1.1.cmml" xref="S4.I10.i1.p1.12.m10.1.1">𝑣</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.I10.i1.p1.12.m10.1c">T(v)</annotation><annotation encoding="application/x-llamapun" id="S4.I10.i1.p1.12.m10.1d">italic_T ( italic_v )</annotation></semantics></math> are contracted into <math alttext="z^{\prime}" class="ltx_Math" display="inline" id="S4.I10.i1.p1.13.m11.1"><semantics id="S4.I10.i1.p1.13.m11.1a"><msup id="S4.I10.i1.p1.13.m11.1.1" xref="S4.I10.i1.p1.13.m11.1.1.cmml"><mi id="S4.I10.i1.p1.13.m11.1.1.2" xref="S4.I10.i1.p1.13.m11.1.1.2.cmml">z</mi><mo id="S4.I10.i1.p1.13.m11.1.1.3" xref="S4.I10.i1.p1.13.m11.1.1.3.cmml">′</mo></msup><annotation-xml encoding="MathML-Content" id="S4.I10.i1.p1.13.m11.1b"><apply id="S4.I10.i1.p1.13.m11.1.1.cmml" xref="S4.I10.i1.p1.13.m11.1.1"><csymbol cd="ambiguous" id="S4.I10.i1.p1.13.m11.1.1.1.cmml" xref="S4.I10.i1.p1.13.m11.1.1">superscript</csymbol><ci id="S4.I10.i1.p1.13.m11.1.1.2.cmml" xref="S4.I10.i1.p1.13.m11.1.1.2">𝑧</ci><ci id="S4.I10.i1.p1.13.m11.1.1.3.cmml" xref="S4.I10.i1.p1.13.m11.1.1.3">′</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.I10.i1.p1.13.m11.1c">z^{\prime}</annotation><annotation encoding="application/x-llamapun" id="S4.I10.i1.p1.13.m11.1d">italic_z start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math>. Thus <math alttext="T(u)" class="ltx_Math" display="inline" id="S4.I10.i1.p1.14.m12.1"><semantics id="S4.I10.i1.p1.14.m12.1a"><mrow id="S4.I10.i1.p1.14.m12.1.2" xref="S4.I10.i1.p1.14.m12.1.2.cmml"><mi id="S4.I10.i1.p1.14.m12.1.2.2" xref="S4.I10.i1.p1.14.m12.1.2.2.cmml">T</mi><mo id="S4.I10.i1.p1.14.m12.1.2.1" xref="S4.I10.i1.p1.14.m12.1.2.1.cmml">⁢</mo><mrow id="S4.I10.i1.p1.14.m12.1.2.3.2" xref="S4.I10.i1.p1.14.m12.1.2.cmml"><mo id="S4.I10.i1.p1.14.m12.1.2.3.2.1" stretchy="false" xref="S4.I10.i1.p1.14.m12.1.2.cmml">(</mo><mi id="S4.I10.i1.p1.14.m12.1.1" xref="S4.I10.i1.p1.14.m12.1.1.cmml">u</mi><mo id="S4.I10.i1.p1.14.m12.1.2.3.2.2" stretchy="false" xref="S4.I10.i1.p1.14.m12.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.I10.i1.p1.14.m12.1b"><apply id="S4.I10.i1.p1.14.m12.1.2.cmml" xref="S4.I10.i1.p1.14.m12.1.2"><times id="S4.I10.i1.p1.14.m12.1.2.1.cmml" xref="S4.I10.i1.p1.14.m12.1.2.1"></times><ci id="S4.I10.i1.p1.14.m12.1.2.2.cmml" xref="S4.I10.i1.p1.14.m12.1.2.2">𝑇</ci><ci id="S4.I10.i1.p1.14.m12.1.1.cmml" xref="S4.I10.i1.p1.14.m12.1.1">𝑢</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.I10.i1.p1.14.m12.1c">T(u)</annotation><annotation encoding="application/x-llamapun" id="S4.I10.i1.p1.14.m12.1d">italic_T ( italic_u )</annotation></semantics></math> and <math alttext="T(v)" class="ltx_Math" display="inline" id="S4.I10.i1.p1.15.m13.1"><semantics id="S4.I10.i1.p1.15.m13.1a"><mrow id="S4.I10.i1.p1.15.m13.1.2" xref="S4.I10.i1.p1.15.m13.1.2.cmml"><mi id="S4.I10.i1.p1.15.m13.1.2.2" xref="S4.I10.i1.p1.15.m13.1.2.2.cmml">T</mi><mo id="S4.I10.i1.p1.15.m13.1.2.1" xref="S4.I10.i1.p1.15.m13.1.2.1.cmml">⁢</mo><mrow id="S4.I10.i1.p1.15.m13.1.2.3.2" xref="S4.I10.i1.p1.15.m13.1.2.cmml"><mo id="S4.I10.i1.p1.15.m13.1.2.3.2.1" stretchy="false" xref="S4.I10.i1.p1.15.m13.1.2.cmml">(</mo><mi id="S4.I10.i1.p1.15.m13.1.1" xref="S4.I10.i1.p1.15.m13.1.1.cmml">v</mi><mo id="S4.I10.i1.p1.15.m13.1.2.3.2.2" stretchy="false" xref="S4.I10.i1.p1.15.m13.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.I10.i1.p1.15.m13.1b"><apply id="S4.I10.i1.p1.15.m13.1.2.cmml" xref="S4.I10.i1.p1.15.m13.1.2"><times id="S4.I10.i1.p1.15.m13.1.2.1.cmml" xref="S4.I10.i1.p1.15.m13.1.2.1"></times><ci id="S4.I10.i1.p1.15.m13.1.2.2.cmml" xref="S4.I10.i1.p1.15.m13.1.2.2">𝑇</ci><ci id="S4.I10.i1.p1.15.m13.1.1.cmml" xref="S4.I10.i1.p1.15.m13.1.1">𝑣</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.I10.i1.p1.15.m13.1c">T(v)</annotation><annotation encoding="application/x-llamapun" id="S4.I10.i1.p1.15.m13.1d">italic_T ( italic_v )</annotation></semantics></math> must be connected by <math alttext="H^{\prime}_{x}" class="ltx_Math" display="inline" id="S4.I10.i1.p1.16.m14.1"><semantics id="S4.I10.i1.p1.16.m14.1a"><msubsup id="S4.I10.i1.p1.16.m14.1.1" xref="S4.I10.i1.p1.16.m14.1.1.cmml"><mi id="S4.I10.i1.p1.16.m14.1.1.2.2" xref="S4.I10.i1.p1.16.m14.1.1.2.2.cmml">H</mi><mi id="S4.I10.i1.p1.16.m14.1.1.3" xref="S4.I10.i1.p1.16.m14.1.1.3.cmml">x</mi><mo id="S4.I10.i1.p1.16.m14.1.1.2.3" xref="S4.I10.i1.p1.16.m14.1.1.2.3.cmml">′</mo></msubsup><annotation-xml encoding="MathML-Content" id="S4.I10.i1.p1.16.m14.1b"><apply id="S4.I10.i1.p1.16.m14.1.1.cmml" xref="S4.I10.i1.p1.16.m14.1.1"><csymbol cd="ambiguous" id="S4.I10.i1.p1.16.m14.1.1.1.cmml" xref="S4.I10.i1.p1.16.m14.1.1">subscript</csymbol><apply id="S4.I10.i1.p1.16.m14.1.1.2.cmml" xref="S4.I10.i1.p1.16.m14.1.1"><csymbol cd="ambiguous" id="S4.I10.i1.p1.16.m14.1.1.2.1.cmml" xref="S4.I10.i1.p1.16.m14.1.1">superscript</csymbol><ci id="S4.I10.i1.p1.16.m14.1.1.2.2.cmml" xref="S4.I10.i1.p1.16.m14.1.1.2.2">𝐻</ci><ci id="S4.I10.i1.p1.16.m14.1.1.2.3.cmml" xref="S4.I10.i1.p1.16.m14.1.1.2.3">′</ci></apply><ci id="S4.I10.i1.p1.16.m14.1.1.3.cmml" xref="S4.I10.i1.p1.16.m14.1.1.3">𝑥</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.I10.i1.p1.16.m14.1c">H^{\prime}_{x}</annotation><annotation encoding="application/x-llamapun" id="S4.I10.i1.p1.16.m14.1d">italic_H start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="S4.I10.i1.p2"> <p class="ltx_p" id="S4.I10.i1.p2.17">Therefore, we assume both subtrees are good. We will show that <span class="ltx_text ltx_markedasmath" id="S4.I10.i1.p2.17.1">SOL</span> contains a link <math alttext="e_{u}" class="ltx_Math" display="inline" id="S4.I10.i1.p2.2.m2.1"><semantics id="S4.I10.i1.p2.2.m2.1a"><msub id="S4.I10.i1.p2.2.m2.1.1" xref="S4.I10.i1.p2.2.m2.1.1.cmml"><mi id="S4.I10.i1.p2.2.m2.1.1.2" xref="S4.I10.i1.p2.2.m2.1.1.2.cmml">e</mi><mi id="S4.I10.i1.p2.2.m2.1.1.3" xref="S4.I10.i1.p2.2.m2.1.1.3.cmml">u</mi></msub><annotation-xml encoding="MathML-Content" id="S4.I10.i1.p2.2.m2.1b"><apply id="S4.I10.i1.p2.2.m2.1.1.cmml" xref="S4.I10.i1.p2.2.m2.1.1"><csymbol cd="ambiguous" id="S4.I10.i1.p2.2.m2.1.1.1.cmml" xref="S4.I10.i1.p2.2.m2.1.1">subscript</csymbol><ci id="S4.I10.i1.p2.2.m2.1.1.2.cmml" xref="S4.I10.i1.p2.2.m2.1.1.2">𝑒</ci><ci id="S4.I10.i1.p2.2.m2.1.1.3.cmml" xref="S4.I10.i1.p2.2.m2.1.1.3">𝑢</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.I10.i1.p2.2.m2.1c">e_{u}</annotation><annotation encoding="application/x-llamapun" id="S4.I10.i1.p2.2.m2.1d">italic_e start_POSTSUBSCRIPT italic_u end_POSTSUBSCRIPT</annotation></semantics></math> between <math alttext="T(u)" class="ltx_Math" display="inline" id="S4.I10.i1.p2.3.m3.1"><semantics id="S4.I10.i1.p2.3.m3.1a"><mrow id="S4.I10.i1.p2.3.m3.1.2" xref="S4.I10.i1.p2.3.m3.1.2.cmml"><mi id="S4.I10.i1.p2.3.m3.1.2.2" xref="S4.I10.i1.p2.3.m3.1.2.2.cmml">T</mi><mo id="S4.I10.i1.p2.3.m3.1.2.1" xref="S4.I10.i1.p2.3.m3.1.2.1.cmml">⁢</mo><mrow id="S4.I10.i1.p2.3.m3.1.2.3.2" xref="S4.I10.i1.p2.3.m3.1.2.cmml"><mo id="S4.I10.i1.p2.3.m3.1.2.3.2.1" stretchy="false" xref="S4.I10.i1.p2.3.m3.1.2.cmml">(</mo><mi id="S4.I10.i1.p2.3.m3.1.1" xref="S4.I10.i1.p2.3.m3.1.1.cmml">u</mi><mo id="S4.I10.i1.p2.3.m3.1.2.3.2.2" stretchy="false" xref="S4.I10.i1.p2.3.m3.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.I10.i1.p2.3.m3.1b"><apply id="S4.I10.i1.p2.3.m3.1.2.cmml" xref="S4.I10.i1.p2.3.m3.1.2"><times id="S4.I10.i1.p2.3.m3.1.2.1.cmml" xref="S4.I10.i1.p2.3.m3.1.2.1"></times><ci id="S4.I10.i1.p2.3.m3.1.2.2.cmml" xref="S4.I10.i1.p2.3.m3.1.2.2">𝑇</ci><ci id="S4.I10.i1.p2.3.m3.1.1.cmml" xref="S4.I10.i1.p2.3.m3.1.1">𝑢</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.I10.i1.p2.3.m3.1c">T(u)</annotation><annotation encoding="application/x-llamapun" id="S4.I10.i1.p2.3.m3.1d">italic_T ( italic_u )</annotation></semantics></math> and <math alttext="T\setminus T_{x}" class="ltx_Math" display="inline" id="S4.I10.i1.p2.4.m4.1"><semantics id="S4.I10.i1.p2.4.m4.1a"><mrow id="S4.I10.i1.p2.4.m4.1.1" xref="S4.I10.i1.p2.4.m4.1.1.cmml"><mi id="S4.I10.i1.p2.4.m4.1.1.2" xref="S4.I10.i1.p2.4.m4.1.1.2.cmml">T</mi><mo id="S4.I10.i1.p2.4.m4.1.1.1" xref="S4.I10.i1.p2.4.m4.1.1.1.cmml">∖</mo><msub id="S4.I10.i1.p2.4.m4.1.1.3" xref="S4.I10.i1.p2.4.m4.1.1.3.cmml"><mi id="S4.I10.i1.p2.4.m4.1.1.3.2" xref="S4.I10.i1.p2.4.m4.1.1.3.2.cmml">T</mi><mi id="S4.I10.i1.p2.4.m4.1.1.3.3" xref="S4.I10.i1.p2.4.m4.1.1.3.3.cmml">x</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S4.I10.i1.p2.4.m4.1b"><apply id="S4.I10.i1.p2.4.m4.1.1.cmml" xref="S4.I10.i1.p2.4.m4.1.1"><setdiff id="S4.I10.i1.p2.4.m4.1.1.1.cmml" xref="S4.I10.i1.p2.4.m4.1.1.1"></setdiff><ci id="S4.I10.i1.p2.4.m4.1.1.2.cmml" xref="S4.I10.i1.p2.4.m4.1.1.2">𝑇</ci><apply id="S4.I10.i1.p2.4.m4.1.1.3.cmml" xref="S4.I10.i1.p2.4.m4.1.1.3"><csymbol cd="ambiguous" id="S4.I10.i1.p2.4.m4.1.1.3.1.cmml" xref="S4.I10.i1.p2.4.m4.1.1.3">subscript</csymbol><ci id="S4.I10.i1.p2.4.m4.1.1.3.2.cmml" xref="S4.I10.i1.p2.4.m4.1.1.3.2">𝑇</ci><ci id="S4.I10.i1.p2.4.m4.1.1.3.3.cmml" xref="S4.I10.i1.p2.4.m4.1.1.3.3">𝑥</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.I10.i1.p2.4.m4.1c">T\setminus T_{x}</annotation><annotation encoding="application/x-llamapun" id="S4.I10.i1.p2.4.m4.1d">italic_T ∖ italic_T start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT</annotation></semantics></math> and a link <math alttext="e_{v}" class="ltx_Math" display="inline" id="S4.I10.i1.p2.5.m5.1"><semantics id="S4.I10.i1.p2.5.m5.1a"><msub id="S4.I10.i1.p2.5.m5.1.1" xref="S4.I10.i1.p2.5.m5.1.1.cmml"><mi id="S4.I10.i1.p2.5.m5.1.1.2" xref="S4.I10.i1.p2.5.m5.1.1.2.cmml">e</mi><mi id="S4.I10.i1.p2.5.m5.1.1.3" xref="S4.I10.i1.p2.5.m5.1.1.3.cmml">v</mi></msub><annotation-xml encoding="MathML-Content" id="S4.I10.i1.p2.5.m5.1b"><apply id="S4.I10.i1.p2.5.m5.1.1.cmml" xref="S4.I10.i1.p2.5.m5.1.1"><csymbol cd="ambiguous" id="S4.I10.i1.p2.5.m5.1.1.1.cmml" xref="S4.I10.i1.p2.5.m5.1.1">subscript</csymbol><ci id="S4.I10.i1.p2.5.m5.1.1.2.cmml" xref="S4.I10.i1.p2.5.m5.1.1.2">𝑒</ci><ci id="S4.I10.i1.p2.5.m5.1.1.3.cmml" xref="S4.I10.i1.p2.5.m5.1.1.3">𝑣</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.I10.i1.p2.5.m5.1c">e_{v}</annotation><annotation encoding="application/x-llamapun" id="S4.I10.i1.p2.5.m5.1d">italic_e start_POSTSUBSCRIPT italic_v end_POSTSUBSCRIPT</annotation></semantics></math> between <math alttext="T(v)" class="ltx_Math" display="inline" id="S4.I10.i1.p2.6.m6.1"><semantics id="S4.I10.i1.p2.6.m6.1a"><mrow id="S4.I10.i1.p2.6.m6.1.2" xref="S4.I10.i1.p2.6.m6.1.2.cmml"><mi id="S4.I10.i1.p2.6.m6.1.2.2" xref="S4.I10.i1.p2.6.m6.1.2.2.cmml">T</mi><mo id="S4.I10.i1.p2.6.m6.1.2.1" xref="S4.I10.i1.p2.6.m6.1.2.1.cmml">⁢</mo><mrow id="S4.I10.i1.p2.6.m6.1.2.3.2" xref="S4.I10.i1.p2.6.m6.1.2.cmml"><mo id="S4.I10.i1.p2.6.m6.1.2.3.2.1" stretchy="false" xref="S4.I10.i1.p2.6.m6.1.2.cmml">(</mo><mi id="S4.I10.i1.p2.6.m6.1.1" xref="S4.I10.i1.p2.6.m6.1.1.cmml">v</mi><mo id="S4.I10.i1.p2.6.m6.1.2.3.2.2" stretchy="false" xref="S4.I10.i1.p2.6.m6.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.I10.i1.p2.6.m6.1b"><apply id="S4.I10.i1.p2.6.m6.1.2.cmml" xref="S4.I10.i1.p2.6.m6.1.2"><times id="S4.I10.i1.p2.6.m6.1.2.1.cmml" xref="S4.I10.i1.p2.6.m6.1.2.1"></times><ci id="S4.I10.i1.p2.6.m6.1.2.2.cmml" xref="S4.I10.i1.p2.6.m6.1.2.2">𝑇</ci><ci id="S4.I10.i1.p2.6.m6.1.1.cmml" xref="S4.I10.i1.p2.6.m6.1.1">𝑣</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.I10.i1.p2.6.m6.1c">T(v)</annotation><annotation encoding="application/x-llamapun" id="S4.I10.i1.p2.6.m6.1d">italic_T ( italic_v )</annotation></semantics></math> and <math alttext="T\setminus T_{x}" class="ltx_Math" display="inline" id="S4.I10.i1.p2.7.m7.1"><semantics id="S4.I10.i1.p2.7.m7.1a"><mrow id="S4.I10.i1.p2.7.m7.1.1" xref="S4.I10.i1.p2.7.m7.1.1.cmml"><mi id="S4.I10.i1.p2.7.m7.1.1.2" xref="S4.I10.i1.p2.7.m7.1.1.2.cmml">T</mi><mo id="S4.I10.i1.p2.7.m7.1.1.1" xref="S4.I10.i1.p2.7.m7.1.1.1.cmml">∖</mo><msub id="S4.I10.i1.p2.7.m7.1.1.3" xref="S4.I10.i1.p2.7.m7.1.1.3.cmml"><mi id="S4.I10.i1.p2.7.m7.1.1.3.2" xref="S4.I10.i1.p2.7.m7.1.1.3.2.cmml">T</mi><mi id="S4.I10.i1.p2.7.m7.1.1.3.3" xref="S4.I10.i1.p2.7.m7.1.1.3.3.cmml">x</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S4.I10.i1.p2.7.m7.1b"><apply id="S4.I10.i1.p2.7.m7.1.1.cmml" xref="S4.I10.i1.p2.7.m7.1.1"><setdiff id="S4.I10.i1.p2.7.m7.1.1.1.cmml" xref="S4.I10.i1.p2.7.m7.1.1.1"></setdiff><ci id="S4.I10.i1.p2.7.m7.1.1.2.cmml" xref="S4.I10.i1.p2.7.m7.1.1.2">𝑇</ci><apply id="S4.I10.i1.p2.7.m7.1.1.3.cmml" xref="S4.I10.i1.p2.7.m7.1.1.3"><csymbol cd="ambiguous" id="S4.I10.i1.p2.7.m7.1.1.3.1.cmml" xref="S4.I10.i1.p2.7.m7.1.1.3">subscript</csymbol><ci id="S4.I10.i1.p2.7.m7.1.1.3.2.cmml" xref="S4.I10.i1.p2.7.m7.1.1.3.2">𝑇</ci><ci id="S4.I10.i1.p2.7.m7.1.1.3.3.cmml" xref="S4.I10.i1.p2.7.m7.1.1.3.3">𝑥</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.I10.i1.p2.7.m7.1c">T\setminus T_{x}</annotation><annotation encoding="application/x-llamapun" id="S4.I10.i1.p2.7.m7.1d">italic_T ∖ italic_T start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT</annotation></semantics></math> such that neither <math alttext="e_{u}" class="ltx_Math" display="inline" id="S4.I10.i1.p2.8.m8.1"><semantics id="S4.I10.i1.p2.8.m8.1a"><msub id="S4.I10.i1.p2.8.m8.1.1" xref="S4.I10.i1.p2.8.m8.1.1.cmml"><mi id="S4.I10.i1.p2.8.m8.1.1.2" xref="S4.I10.i1.p2.8.m8.1.1.2.cmml">e</mi><mi id="S4.I10.i1.p2.8.m8.1.1.3" xref="S4.I10.i1.p2.8.m8.1.1.3.cmml">u</mi></msub><annotation-xml encoding="MathML-Content" id="S4.I10.i1.p2.8.m8.1b"><apply id="S4.I10.i1.p2.8.m8.1.1.cmml" xref="S4.I10.i1.p2.8.m8.1.1"><csymbol cd="ambiguous" id="S4.I10.i1.p2.8.m8.1.1.1.cmml" xref="S4.I10.i1.p2.8.m8.1.1">subscript</csymbol><ci id="S4.I10.i1.p2.8.m8.1.1.2.cmml" xref="S4.I10.i1.p2.8.m8.1.1.2">𝑒</ci><ci id="S4.I10.i1.p2.8.m8.1.1.3.cmml" xref="S4.I10.i1.p2.8.m8.1.1.3">𝑢</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.I10.i1.p2.8.m8.1c">e_{u}</annotation><annotation encoding="application/x-llamapun" id="S4.I10.i1.p2.8.m8.1d">italic_e start_POSTSUBSCRIPT italic_u end_POSTSUBSCRIPT</annotation></semantics></math> nor <math alttext="e_{v}" class="ltx_Math" display="inline" id="S4.I10.i1.p2.9.m9.1"><semantics id="S4.I10.i1.p2.9.m9.1a"><msub id="S4.I10.i1.p2.9.m9.1.1" xref="S4.I10.i1.p2.9.m9.1.1.cmml"><mi id="S4.I10.i1.p2.9.m9.1.1.2" xref="S4.I10.i1.p2.9.m9.1.1.2.cmml">e</mi><mi id="S4.I10.i1.p2.9.m9.1.1.3" xref="S4.I10.i1.p2.9.m9.1.1.3.cmml">v</mi></msub><annotation-xml encoding="MathML-Content" id="S4.I10.i1.p2.9.m9.1b"><apply id="S4.I10.i1.p2.9.m9.1.1.cmml" xref="S4.I10.i1.p2.9.m9.1.1"><csymbol cd="ambiguous" id="S4.I10.i1.p2.9.m9.1.1.1.cmml" xref="S4.I10.i1.p2.9.m9.1.1">subscript</csymbol><ci id="S4.I10.i1.p2.9.m9.1.1.2.cmml" xref="S4.I10.i1.p2.9.m9.1.1.2">𝑒</ci><ci id="S4.I10.i1.p2.9.m9.1.1.3.cmml" xref="S4.I10.i1.p2.9.m9.1.1.3">𝑣</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.I10.i1.p2.9.m9.1c">e_{v}</annotation><annotation encoding="application/x-llamapun" id="S4.I10.i1.p2.9.m9.1d">italic_e start_POSTSUBSCRIPT italic_v end_POSTSUBSCRIPT</annotation></semantics></math> are incident to <math alttext="\{a,b\}" class="ltx_Math" display="inline" id="S4.I10.i1.p2.10.m10.2"><semantics id="S4.I10.i1.p2.10.m10.2a"><mrow id="S4.I10.i1.p2.10.m10.2.3.2" xref="S4.I10.i1.p2.10.m10.2.3.1.cmml"><mo id="S4.I10.i1.p2.10.m10.2.3.2.1" stretchy="false" xref="S4.I10.i1.p2.10.m10.2.3.1.cmml">{</mo><mi id="S4.I10.i1.p2.10.m10.1.1" xref="S4.I10.i1.p2.10.m10.1.1.cmml">a</mi><mo id="S4.I10.i1.p2.10.m10.2.3.2.2" xref="S4.I10.i1.p2.10.m10.2.3.1.cmml">,</mo><mi id="S4.I10.i1.p2.10.m10.2.2" xref="S4.I10.i1.p2.10.m10.2.2.cmml">b</mi><mo id="S4.I10.i1.p2.10.m10.2.3.2.3" stretchy="false" xref="S4.I10.i1.p2.10.m10.2.3.1.cmml">}</mo></mrow><annotation-xml encoding="MathML-Content" id="S4.I10.i1.p2.10.m10.2b"><set id="S4.I10.i1.p2.10.m10.2.3.1.cmml" xref="S4.I10.i1.p2.10.m10.2.3.2"><ci id="S4.I10.i1.p2.10.m10.1.1.cmml" xref="S4.I10.i1.p2.10.m10.1.1">𝑎</ci><ci id="S4.I10.i1.p2.10.m10.2.2.cmml" xref="S4.I10.i1.p2.10.m10.2.2">𝑏</ci></set></annotation-xml><annotation encoding="application/x-tex" id="S4.I10.i1.p2.10.m10.2c">\{a,b\}</annotation><annotation encoding="application/x-llamapun" id="S4.I10.i1.p2.10.m10.2d">{ italic_a , italic_b }</annotation></semantics></math>. By the above discussion, since all components of <math alttext="T\setminus x" class="ltx_Math" display="inline" id="S4.I10.i1.p2.11.m11.1"><semantics id="S4.I10.i1.p2.11.m11.1a"><mrow id="S4.I10.i1.p2.11.m11.1.1" xref="S4.I10.i1.p2.11.m11.1.1.cmml"><mi id="S4.I10.i1.p2.11.m11.1.1.2" xref="S4.I10.i1.p2.11.m11.1.1.2.cmml">T</mi><mo id="S4.I10.i1.p2.11.m11.1.1.1" xref="S4.I10.i1.p2.11.m11.1.1.1.cmml">∖</mo><mi id="S4.I10.i1.p2.11.m11.1.1.3" xref="S4.I10.i1.p2.11.m11.1.1.3.cmml">x</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.I10.i1.p2.11.m11.1b"><apply id="S4.I10.i1.p2.11.m11.1.1.cmml" xref="S4.I10.i1.p2.11.m11.1.1"><setdiff id="S4.I10.i1.p2.11.m11.1.1.1.cmml" xref="S4.I10.i1.p2.11.m11.1.1.1"></setdiff><ci id="S4.I10.i1.p2.11.m11.1.1.2.cmml" xref="S4.I10.i1.p2.11.m11.1.1.2">𝑇</ci><ci id="S4.I10.i1.p2.11.m11.1.1.3.cmml" xref="S4.I10.i1.p2.11.m11.1.1.3">𝑥</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.I10.i1.p2.11.m11.1c">T\setminus x</annotation><annotation encoding="application/x-llamapun" id="S4.I10.i1.p2.11.m11.1d">italic_T ∖ italic_x</annotation></semantics></math> (and thus their corresponding vertices in <math alttext="G" class="ltx_Math" display="inline" id="S4.I10.i1.p2.12.m12.1"><semantics id="S4.I10.i1.p2.12.m12.1a"><mi id="S4.I10.i1.p2.12.m12.1.1" xref="S4.I10.i1.p2.12.m12.1.1.cmml">G</mi><annotation-xml encoding="MathML-Content" id="S4.I10.i1.p2.12.m12.1b"><ci id="S4.I10.i1.p2.12.m12.1.1.cmml" xref="S4.I10.i1.p2.12.m12.1.1">𝐺</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.I10.i1.p2.12.m12.1c">G</annotation><annotation encoding="application/x-llamapun" id="S4.I10.i1.p2.12.m12.1d">italic_G</annotation></semantics></math>) remain connected despite the removal of <math alttext="\{a,b\}" class="ltx_Math" display="inline" id="S4.I10.i1.p2.13.m13.2"><semantics id="S4.I10.i1.p2.13.m13.2a"><mrow id="S4.I10.i1.p2.13.m13.2.3.2" xref="S4.I10.i1.p2.13.m13.2.3.1.cmml"><mo id="S4.I10.i1.p2.13.m13.2.3.2.1" stretchy="false" xref="S4.I10.i1.p2.13.m13.2.3.1.cmml">{</mo><mi id="S4.I10.i1.p2.13.m13.1.1" xref="S4.I10.i1.p2.13.m13.1.1.cmml">a</mi><mo id="S4.I10.i1.p2.13.m13.2.3.2.2" xref="S4.I10.i1.p2.13.m13.2.3.1.cmml">,</mo><mi id="S4.I10.i1.p2.13.m13.2.2" xref="S4.I10.i1.p2.13.m13.2.2.cmml">b</mi><mo id="S4.I10.i1.p2.13.m13.2.3.2.3" stretchy="false" xref="S4.I10.i1.p2.13.m13.2.3.1.cmml">}</mo></mrow><annotation-xml encoding="MathML-Content" id="S4.I10.i1.p2.13.m13.2b"><set id="S4.I10.i1.p2.13.m13.2.3.1.cmml" xref="S4.I10.i1.p2.13.m13.2.3.2"><ci id="S4.I10.i1.p2.13.m13.1.1.cmml" xref="S4.I10.i1.p2.13.m13.1.1">𝑎</ci><ci id="S4.I10.i1.p2.13.m13.2.2.cmml" xref="S4.I10.i1.p2.13.m13.2.2">𝑏</ci></set></annotation-xml><annotation encoding="application/x-tex" id="S4.I10.i1.p2.13.m13.2c">\{a,b\}</annotation><annotation encoding="application/x-llamapun" id="S4.I10.i1.p2.13.m13.2d">{ italic_a , italic_b }</annotation></semantics></math>, this suffices to show that there exists a <math alttext="u" class="ltx_Math" display="inline" id="S4.I10.i1.p2.14.m14.1"><semantics id="S4.I10.i1.p2.14.m14.1a"><mi id="S4.I10.i1.p2.14.m14.1.1" xref="S4.I10.i1.p2.14.m14.1.1.cmml">u</mi><annotation-xml encoding="MathML-Content" id="S4.I10.i1.p2.14.m14.1b"><ci id="S4.I10.i1.p2.14.m14.1.1.cmml" xref="S4.I10.i1.p2.14.m14.1.1">𝑢</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.I10.i1.p2.14.m14.1c">u</annotation><annotation encoding="application/x-llamapun" id="S4.I10.i1.p2.14.m14.1d">italic_u</annotation></semantics></math>-<math alttext="v" class="ltx_Math" display="inline" id="S4.I10.i1.p2.15.m15.1"><semantics id="S4.I10.i1.p2.15.m15.1a"><mi id="S4.I10.i1.p2.15.m15.1.1" xref="S4.I10.i1.p2.15.m15.1.1.cmml">v</mi><annotation-xml encoding="MathML-Content" id="S4.I10.i1.p2.15.m15.1b"><ci id="S4.I10.i1.p2.15.m15.1.1.cmml" xref="S4.I10.i1.p2.15.m15.1.1">𝑣</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.I10.i1.p2.15.m15.1c">v</annotation><annotation encoding="application/x-llamapun" id="S4.I10.i1.p2.15.m15.1d">italic_v</annotation></semantics></math> path in <math alttext="E\cup\textnormal{SOL}" class="ltx_Math" display="inline" id="S4.I10.i1.p2.16.m16.1"><semantics id="S4.I10.i1.p2.16.m16.1a"><mrow id="S4.I10.i1.p2.16.m16.1.1" xref="S4.I10.i1.p2.16.m16.1.1.cmml"><mi id="S4.I10.i1.p2.16.m16.1.1.2" xref="S4.I10.i1.p2.16.m16.1.1.2.cmml">E</mi><mo id="S4.I10.i1.p2.16.m16.1.1.1" xref="S4.I10.i1.p2.16.m16.1.1.1.cmml">∪</mo><mtext id="S4.I10.i1.p2.16.m16.1.1.3" xref="S4.I10.i1.p2.16.m16.1.1.3a.cmml">SOL</mtext></mrow><annotation-xml encoding="MathML-Content" id="S4.I10.i1.p2.16.m16.1b"><apply id="S4.I10.i1.p2.16.m16.1.1.cmml" xref="S4.I10.i1.p2.16.m16.1.1"><union id="S4.I10.i1.p2.16.m16.1.1.1.cmml" xref="S4.I10.i1.p2.16.m16.1.1.1"></union><ci id="S4.I10.i1.p2.16.m16.1.1.2.cmml" xref="S4.I10.i1.p2.16.m16.1.1.2">𝐸</ci><ci id="S4.I10.i1.p2.16.m16.1.1.3a.cmml" xref="S4.I10.i1.p2.16.m16.1.1.3"><mtext id="S4.I10.i1.p2.16.m16.1.1.3.cmml" xref="S4.I10.i1.p2.16.m16.1.1.3">SOL</mtext></ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.I10.i1.p2.16.m16.1c">E\cup\textnormal{SOL}</annotation><annotation encoding="application/x-llamapun" id="S4.I10.i1.p2.16.m16.1d">italic_E ∪ SOL</annotation></semantics></math> avoiding <math alttext="\{a,b\}" class="ltx_Math" display="inline" id="S4.I10.i1.p2.17.m17.2"><semantics id="S4.I10.i1.p2.17.m17.2a"><mrow id="S4.I10.i1.p2.17.m17.2.3.2" xref="S4.I10.i1.p2.17.m17.2.3.1.cmml"><mo id="S4.I10.i1.p2.17.m17.2.3.2.1" stretchy="false" xref="S4.I10.i1.p2.17.m17.2.3.1.cmml">{</mo><mi id="S4.I10.i1.p2.17.m17.1.1" xref="S4.I10.i1.p2.17.m17.1.1.cmml">a</mi><mo id="S4.I10.i1.p2.17.m17.2.3.2.2" xref="S4.I10.i1.p2.17.m17.2.3.1.cmml">,</mo><mi id="S4.I10.i1.p2.17.m17.2.2" xref="S4.I10.i1.p2.17.m17.2.2.cmml">b</mi><mo id="S4.I10.i1.p2.17.m17.2.3.2.3" stretchy="false" xref="S4.I10.i1.p2.17.m17.2.3.1.cmml">}</mo></mrow><annotation-xml encoding="MathML-Content" id="S4.I10.i1.p2.17.m17.2b"><set id="S4.I10.i1.p2.17.m17.2.3.1.cmml" xref="S4.I10.i1.p2.17.m17.2.3.2"><ci id="S4.I10.i1.p2.17.m17.1.1.cmml" xref="S4.I10.i1.p2.17.m17.1.1">𝑎</ci><ci id="S4.I10.i1.p2.17.m17.2.2.cmml" xref="S4.I10.i1.p2.17.m17.2.2">𝑏</ci></set></annotation-xml><annotation encoding="application/x-tex" id="S4.I10.i1.p2.17.m17.2c">\{a,b\}</annotation><annotation encoding="application/x-llamapun" id="S4.I10.i1.p2.17.m17.2d">{ italic_a , italic_b }</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="S4.I10.i1.p3"> <p class="ltx_p" id="S4.I10.i1.p3.27">Since <math alttext="T(u)" class="ltx_Math" display="inline" id="S4.I10.i1.p3.1.m1.1"><semantics id="S4.I10.i1.p3.1.m1.1a"><mrow id="S4.I10.i1.p3.1.m1.1.2" xref="S4.I10.i1.p3.1.m1.1.2.cmml"><mi id="S4.I10.i1.p3.1.m1.1.2.2" xref="S4.I10.i1.p3.1.m1.1.2.2.cmml">T</mi><mo id="S4.I10.i1.p3.1.m1.1.2.1" xref="S4.I10.i1.p3.1.m1.1.2.1.cmml">⁢</mo><mrow id="S4.I10.i1.p3.1.m1.1.2.3.2" xref="S4.I10.i1.p3.1.m1.1.2.cmml"><mo id="S4.I10.i1.p3.1.m1.1.2.3.2.1" stretchy="false" xref="S4.I10.i1.p3.1.m1.1.2.cmml">(</mo><mi id="S4.I10.i1.p3.1.m1.1.1" xref="S4.I10.i1.p3.1.m1.1.1.cmml">u</mi><mo id="S4.I10.i1.p3.1.m1.1.2.3.2.2" stretchy="false" xref="S4.I10.i1.p3.1.m1.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.I10.i1.p3.1.m1.1b"><apply id="S4.I10.i1.p3.1.m1.1.2.cmml" xref="S4.I10.i1.p3.1.m1.1.2"><times id="S4.I10.i1.p3.1.m1.1.2.1.cmml" xref="S4.I10.i1.p3.1.m1.1.2.1"></times><ci id="S4.I10.i1.p3.1.m1.1.2.2.cmml" xref="S4.I10.i1.p3.1.m1.1.2.2">𝑇</ci><ci id="S4.I10.i1.p3.1.m1.1.1.cmml" xref="S4.I10.i1.p3.1.m1.1.1">𝑢</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.I10.i1.p3.1.m1.1c">T(u)</annotation><annotation encoding="application/x-llamapun" id="S4.I10.i1.p3.1.m1.1d">italic_T ( italic_u )</annotation></semantics></math> is good, there exists a link <math alttext="(u_{1},u_{2})\in\textnormal{OPT}" class="ltx_Math" display="inline" id="S4.I10.i1.p3.2.m2.2"><semantics id="S4.I10.i1.p3.2.m2.2a"><mrow id="S4.I10.i1.p3.2.m2.2.2" xref="S4.I10.i1.p3.2.m2.2.2.cmml"><mrow id="S4.I10.i1.p3.2.m2.2.2.2.2" xref="S4.I10.i1.p3.2.m2.2.2.2.3.cmml"><mo id="S4.I10.i1.p3.2.m2.2.2.2.2.3" stretchy="false" xref="S4.I10.i1.p3.2.m2.2.2.2.3.cmml">(</mo><msub id="S4.I10.i1.p3.2.m2.1.1.1.1.1" xref="S4.I10.i1.p3.2.m2.1.1.1.1.1.cmml"><mi id="S4.I10.i1.p3.2.m2.1.1.1.1.1.2" xref="S4.I10.i1.p3.2.m2.1.1.1.1.1.2.cmml">u</mi><mn id="S4.I10.i1.p3.2.m2.1.1.1.1.1.3" xref="S4.I10.i1.p3.2.m2.1.1.1.1.1.3.cmml">1</mn></msub><mo id="S4.I10.i1.p3.2.m2.2.2.2.2.4" xref="S4.I10.i1.p3.2.m2.2.2.2.3.cmml">,</mo><msub id="S4.I10.i1.p3.2.m2.2.2.2.2.2" xref="S4.I10.i1.p3.2.m2.2.2.2.2.2.cmml"><mi id="S4.I10.i1.p3.2.m2.2.2.2.2.2.2" xref="S4.I10.i1.p3.2.m2.2.2.2.2.2.2.cmml">u</mi><mn id="S4.I10.i1.p3.2.m2.2.2.2.2.2.3" xref="S4.I10.i1.p3.2.m2.2.2.2.2.2.3.cmml">2</mn></msub><mo id="S4.I10.i1.p3.2.m2.2.2.2.2.5" stretchy="false" xref="S4.I10.i1.p3.2.m2.2.2.2.3.cmml">)</mo></mrow><mo id="S4.I10.i1.p3.2.m2.2.2.3" xref="S4.I10.i1.p3.2.m2.2.2.3.cmml">∈</mo><mtext id="S4.I10.i1.p3.2.m2.2.2.4" xref="S4.I10.i1.p3.2.m2.2.2.4a.cmml">OPT</mtext></mrow><annotation-xml encoding="MathML-Content" id="S4.I10.i1.p3.2.m2.2b"><apply id="S4.I10.i1.p3.2.m2.2.2.cmml" xref="S4.I10.i1.p3.2.m2.2.2"><in id="S4.I10.i1.p3.2.m2.2.2.3.cmml" xref="S4.I10.i1.p3.2.m2.2.2.3"></in><interval closure="open" id="S4.I10.i1.p3.2.m2.2.2.2.3.cmml" xref="S4.I10.i1.p3.2.m2.2.2.2.2"><apply id="S4.I10.i1.p3.2.m2.1.1.1.1.1.cmml" xref="S4.I10.i1.p3.2.m2.1.1.1.1.1"><csymbol cd="ambiguous" id="S4.I10.i1.p3.2.m2.1.1.1.1.1.1.cmml" xref="S4.I10.i1.p3.2.m2.1.1.1.1.1">subscript</csymbol><ci id="S4.I10.i1.p3.2.m2.1.1.1.1.1.2.cmml" xref="S4.I10.i1.p3.2.m2.1.1.1.1.1.2">𝑢</ci><cn id="S4.I10.i1.p3.2.m2.1.1.1.1.1.3.cmml" type="integer" xref="S4.I10.i1.p3.2.m2.1.1.1.1.1.3">1</cn></apply><apply id="S4.I10.i1.p3.2.m2.2.2.2.2.2.cmml" xref="S4.I10.i1.p3.2.m2.2.2.2.2.2"><csymbol cd="ambiguous" id="S4.I10.i1.p3.2.m2.2.2.2.2.2.1.cmml" xref="S4.I10.i1.p3.2.m2.2.2.2.2.2">subscript</csymbol><ci id="S4.I10.i1.p3.2.m2.2.2.2.2.2.2.cmml" xref="S4.I10.i1.p3.2.m2.2.2.2.2.2.2">𝑢</ci><cn id="S4.I10.i1.p3.2.m2.2.2.2.2.2.3.cmml" type="integer" xref="S4.I10.i1.p3.2.m2.2.2.2.2.2.3">2</cn></apply></interval><ci id="S4.I10.i1.p3.2.m2.2.2.4a.cmml" xref="S4.I10.i1.p3.2.m2.2.2.4"><mtext id="S4.I10.i1.p3.2.m2.2.2.4.cmml" xref="S4.I10.i1.p3.2.m2.2.2.4">OPT</mtext></ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.I10.i1.p3.2.m2.2c">(u_{1},u_{2})\in\textnormal{OPT}</annotation><annotation encoding="application/x-llamapun" id="S4.I10.i1.p3.2.m2.2d">( italic_u start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_u start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) ∈ OPT</annotation></semantics></math> with <math alttext="h(u_{1})\subseteq T(u)" class="ltx_Math" display="inline" id="S4.I10.i1.p3.3.m3.2"><semantics id="S4.I10.i1.p3.3.m3.2a"><mrow id="S4.I10.i1.p3.3.m3.2.2" xref="S4.I10.i1.p3.3.m3.2.2.cmml"><mrow id="S4.I10.i1.p3.3.m3.2.2.1" xref="S4.I10.i1.p3.3.m3.2.2.1.cmml"><mi id="S4.I10.i1.p3.3.m3.2.2.1.3" xref="S4.I10.i1.p3.3.m3.2.2.1.3.cmml">h</mi><mo id="S4.I10.i1.p3.3.m3.2.2.1.2" xref="S4.I10.i1.p3.3.m3.2.2.1.2.cmml">⁢</mo><mrow id="S4.I10.i1.p3.3.m3.2.2.1.1.1" xref="S4.I10.i1.p3.3.m3.2.2.1.1.1.1.cmml"><mo id="S4.I10.i1.p3.3.m3.2.2.1.1.1.2" stretchy="false" xref="S4.I10.i1.p3.3.m3.2.2.1.1.1.1.cmml">(</mo><msub id="S4.I10.i1.p3.3.m3.2.2.1.1.1.1" xref="S4.I10.i1.p3.3.m3.2.2.1.1.1.1.cmml"><mi id="S4.I10.i1.p3.3.m3.2.2.1.1.1.1.2" xref="S4.I10.i1.p3.3.m3.2.2.1.1.1.1.2.cmml">u</mi><mn id="S4.I10.i1.p3.3.m3.2.2.1.1.1.1.3" xref="S4.I10.i1.p3.3.m3.2.2.1.1.1.1.3.cmml">1</mn></msub><mo id="S4.I10.i1.p3.3.m3.2.2.1.1.1.3" stretchy="false" xref="S4.I10.i1.p3.3.m3.2.2.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S4.I10.i1.p3.3.m3.2.2.2" xref="S4.I10.i1.p3.3.m3.2.2.2.cmml">⊆</mo><mrow id="S4.I10.i1.p3.3.m3.2.2.3" xref="S4.I10.i1.p3.3.m3.2.2.3.cmml"><mi id="S4.I10.i1.p3.3.m3.2.2.3.2" xref="S4.I10.i1.p3.3.m3.2.2.3.2.cmml">T</mi><mo id="S4.I10.i1.p3.3.m3.2.2.3.1" xref="S4.I10.i1.p3.3.m3.2.2.3.1.cmml">⁢</mo><mrow id="S4.I10.i1.p3.3.m3.2.2.3.3.2" xref="S4.I10.i1.p3.3.m3.2.2.3.cmml"><mo id="S4.I10.i1.p3.3.m3.2.2.3.3.2.1" stretchy="false" xref="S4.I10.i1.p3.3.m3.2.2.3.cmml">(</mo><mi id="S4.I10.i1.p3.3.m3.1.1" xref="S4.I10.i1.p3.3.m3.1.1.cmml">u</mi><mo id="S4.I10.i1.p3.3.m3.2.2.3.3.2.2" stretchy="false" xref="S4.I10.i1.p3.3.m3.2.2.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.I10.i1.p3.3.m3.2b"><apply id="S4.I10.i1.p3.3.m3.2.2.cmml" xref="S4.I10.i1.p3.3.m3.2.2"><subset id="S4.I10.i1.p3.3.m3.2.2.2.cmml" xref="S4.I10.i1.p3.3.m3.2.2.2"></subset><apply id="S4.I10.i1.p3.3.m3.2.2.1.cmml" xref="S4.I10.i1.p3.3.m3.2.2.1"><times id="S4.I10.i1.p3.3.m3.2.2.1.2.cmml" xref="S4.I10.i1.p3.3.m3.2.2.1.2"></times><ci id="S4.I10.i1.p3.3.m3.2.2.1.3.cmml" xref="S4.I10.i1.p3.3.m3.2.2.1.3">ℎ</ci><apply id="S4.I10.i1.p3.3.m3.2.2.1.1.1.1.cmml" xref="S4.I10.i1.p3.3.m3.2.2.1.1.1"><csymbol cd="ambiguous" id="S4.I10.i1.p3.3.m3.2.2.1.1.1.1.1.cmml" xref="S4.I10.i1.p3.3.m3.2.2.1.1.1">subscript</csymbol><ci id="S4.I10.i1.p3.3.m3.2.2.1.1.1.1.2.cmml" xref="S4.I10.i1.p3.3.m3.2.2.1.1.1.1.2">𝑢</ci><cn id="S4.I10.i1.p3.3.m3.2.2.1.1.1.1.3.cmml" type="integer" xref="S4.I10.i1.p3.3.m3.2.2.1.1.1.1.3">1</cn></apply></apply><apply id="S4.I10.i1.p3.3.m3.2.2.3.cmml" xref="S4.I10.i1.p3.3.m3.2.2.3"><times id="S4.I10.i1.p3.3.m3.2.2.3.1.cmml" xref="S4.I10.i1.p3.3.m3.2.2.3.1"></times><ci id="S4.I10.i1.p3.3.m3.2.2.3.2.cmml" xref="S4.I10.i1.p3.3.m3.2.2.3.2">𝑇</ci><ci id="S4.I10.i1.p3.3.m3.1.1.cmml" xref="S4.I10.i1.p3.3.m3.1.1">𝑢</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.I10.i1.p3.3.m3.2c">h(u_{1})\subseteq T(u)</annotation><annotation encoding="application/x-llamapun" id="S4.I10.i1.p3.3.m3.2d">italic_h ( italic_u start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) ⊆ italic_T ( italic_u )</annotation></semantics></math> and <math alttext="\ell(u_{2})\subseteq T\setminus T_{x}" class="ltx_Math" display="inline" id="S4.I10.i1.p3.4.m4.1"><semantics id="S4.I10.i1.p3.4.m4.1a"><mrow id="S4.I10.i1.p3.4.m4.1.1" xref="S4.I10.i1.p3.4.m4.1.1.cmml"><mrow id="S4.I10.i1.p3.4.m4.1.1.1" xref="S4.I10.i1.p3.4.m4.1.1.1.cmml"><mi id="S4.I10.i1.p3.4.m4.1.1.1.3" mathvariant="normal" xref="S4.I10.i1.p3.4.m4.1.1.1.3.cmml">ℓ</mi><mo id="S4.I10.i1.p3.4.m4.1.1.1.2" xref="S4.I10.i1.p3.4.m4.1.1.1.2.cmml">⁢</mo><mrow id="S4.I10.i1.p3.4.m4.1.1.1.1.1" xref="S4.I10.i1.p3.4.m4.1.1.1.1.1.1.cmml"><mo id="S4.I10.i1.p3.4.m4.1.1.1.1.1.2" stretchy="false" xref="S4.I10.i1.p3.4.m4.1.1.1.1.1.1.cmml">(</mo><msub id="S4.I10.i1.p3.4.m4.1.1.1.1.1.1" xref="S4.I10.i1.p3.4.m4.1.1.1.1.1.1.cmml"><mi id="S4.I10.i1.p3.4.m4.1.1.1.1.1.1.2" xref="S4.I10.i1.p3.4.m4.1.1.1.1.1.1.2.cmml">u</mi><mn id="S4.I10.i1.p3.4.m4.1.1.1.1.1.1.3" xref="S4.I10.i1.p3.4.m4.1.1.1.1.1.1.3.cmml">2</mn></msub><mo id="S4.I10.i1.p3.4.m4.1.1.1.1.1.3" stretchy="false" xref="S4.I10.i1.p3.4.m4.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S4.I10.i1.p3.4.m4.1.1.2" xref="S4.I10.i1.p3.4.m4.1.1.2.cmml">⊆</mo><mrow id="S4.I10.i1.p3.4.m4.1.1.3" xref="S4.I10.i1.p3.4.m4.1.1.3.cmml"><mi id="S4.I10.i1.p3.4.m4.1.1.3.2" xref="S4.I10.i1.p3.4.m4.1.1.3.2.cmml">T</mi><mo id="S4.I10.i1.p3.4.m4.1.1.3.1" xref="S4.I10.i1.p3.4.m4.1.1.3.1.cmml">∖</mo><msub id="S4.I10.i1.p3.4.m4.1.1.3.3" xref="S4.I10.i1.p3.4.m4.1.1.3.3.cmml"><mi id="S4.I10.i1.p3.4.m4.1.1.3.3.2" xref="S4.I10.i1.p3.4.m4.1.1.3.3.2.cmml">T</mi><mi id="S4.I10.i1.p3.4.m4.1.1.3.3.3" xref="S4.I10.i1.p3.4.m4.1.1.3.3.3.cmml">x</mi></msub></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.I10.i1.p3.4.m4.1b"><apply id="S4.I10.i1.p3.4.m4.1.1.cmml" xref="S4.I10.i1.p3.4.m4.1.1"><subset id="S4.I10.i1.p3.4.m4.1.1.2.cmml" xref="S4.I10.i1.p3.4.m4.1.1.2"></subset><apply id="S4.I10.i1.p3.4.m4.1.1.1.cmml" xref="S4.I10.i1.p3.4.m4.1.1.1"><times id="S4.I10.i1.p3.4.m4.1.1.1.2.cmml" xref="S4.I10.i1.p3.4.m4.1.1.1.2"></times><ci id="S4.I10.i1.p3.4.m4.1.1.1.3.cmml" xref="S4.I10.i1.p3.4.m4.1.1.1.3">ℓ</ci><apply id="S4.I10.i1.p3.4.m4.1.1.1.1.1.1.cmml" xref="S4.I10.i1.p3.4.m4.1.1.1.1.1"><csymbol cd="ambiguous" id="S4.I10.i1.p3.4.m4.1.1.1.1.1.1.1.cmml" xref="S4.I10.i1.p3.4.m4.1.1.1.1.1">subscript</csymbol><ci id="S4.I10.i1.p3.4.m4.1.1.1.1.1.1.2.cmml" xref="S4.I10.i1.p3.4.m4.1.1.1.1.1.1.2">𝑢</ci><cn id="S4.I10.i1.p3.4.m4.1.1.1.1.1.1.3.cmml" type="integer" xref="S4.I10.i1.p3.4.m4.1.1.1.1.1.1.3">2</cn></apply></apply><apply id="S4.I10.i1.p3.4.m4.1.1.3.cmml" xref="S4.I10.i1.p3.4.m4.1.1.3"><setdiff id="S4.I10.i1.p3.4.m4.1.1.3.1.cmml" xref="S4.I10.i1.p3.4.m4.1.1.3.1"></setdiff><ci id="S4.I10.i1.p3.4.m4.1.1.3.2.cmml" xref="S4.I10.i1.p3.4.m4.1.1.3.2">𝑇</ci><apply id="S4.I10.i1.p3.4.m4.1.1.3.3.cmml" xref="S4.I10.i1.p3.4.m4.1.1.3.3"><csymbol cd="ambiguous" id="S4.I10.i1.p3.4.m4.1.1.3.3.1.cmml" xref="S4.I10.i1.p3.4.m4.1.1.3.3">subscript</csymbol><ci id="S4.I10.i1.p3.4.m4.1.1.3.3.2.cmml" xref="S4.I10.i1.p3.4.m4.1.1.3.3.2">𝑇</ci><ci id="S4.I10.i1.p3.4.m4.1.1.3.3.3.cmml" xref="S4.I10.i1.p3.4.m4.1.1.3.3.3">𝑥</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.I10.i1.p3.4.m4.1c">\ell(u_{2})\subseteq T\setminus T_{x}</annotation><annotation encoding="application/x-llamapun" id="S4.I10.i1.p3.4.m4.1d">roman_ℓ ( italic_u start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) ⊆ italic_T ∖ italic_T start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT</annotation></semantics></math>. In particular, this means that <math alttext="\text{LCA}(h(u_{1}),\ell(u_{2}))\in T\setminus T_{x}" class="ltx_Math" display="inline" id="S4.I10.i1.p3.5.m5.2"><semantics id="S4.I10.i1.p3.5.m5.2a"><mrow id="S4.I10.i1.p3.5.m5.2.2" xref="S4.I10.i1.p3.5.m5.2.2.cmml"><mrow id="S4.I10.i1.p3.5.m5.2.2.2" xref="S4.I10.i1.p3.5.m5.2.2.2.cmml"><mtext id="S4.I10.i1.p3.5.m5.2.2.2.4" xref="S4.I10.i1.p3.5.m5.2.2.2.4a.cmml">LCA</mtext><mo id="S4.I10.i1.p3.5.m5.2.2.2.3" xref="S4.I10.i1.p3.5.m5.2.2.2.3.cmml">⁢</mo><mrow id="S4.I10.i1.p3.5.m5.2.2.2.2.2" xref="S4.I10.i1.p3.5.m5.2.2.2.2.3.cmml"><mo id="S4.I10.i1.p3.5.m5.2.2.2.2.2.3" stretchy="false" xref="S4.I10.i1.p3.5.m5.2.2.2.2.3.cmml">(</mo><mrow id="S4.I10.i1.p3.5.m5.1.1.1.1.1.1" xref="S4.I10.i1.p3.5.m5.1.1.1.1.1.1.cmml"><mi id="S4.I10.i1.p3.5.m5.1.1.1.1.1.1.3" xref="S4.I10.i1.p3.5.m5.1.1.1.1.1.1.3.cmml">h</mi><mo id="S4.I10.i1.p3.5.m5.1.1.1.1.1.1.2" xref="S4.I10.i1.p3.5.m5.1.1.1.1.1.1.2.cmml">⁢</mo><mrow id="S4.I10.i1.p3.5.m5.1.1.1.1.1.1.1.1" xref="S4.I10.i1.p3.5.m5.1.1.1.1.1.1.1.1.1.cmml"><mo id="S4.I10.i1.p3.5.m5.1.1.1.1.1.1.1.1.2" stretchy="false" xref="S4.I10.i1.p3.5.m5.1.1.1.1.1.1.1.1.1.cmml">(</mo><msub id="S4.I10.i1.p3.5.m5.1.1.1.1.1.1.1.1.1" xref="S4.I10.i1.p3.5.m5.1.1.1.1.1.1.1.1.1.cmml"><mi id="S4.I10.i1.p3.5.m5.1.1.1.1.1.1.1.1.1.2" xref="S4.I10.i1.p3.5.m5.1.1.1.1.1.1.1.1.1.2.cmml">u</mi><mn id="S4.I10.i1.p3.5.m5.1.1.1.1.1.1.1.1.1.3" xref="S4.I10.i1.p3.5.m5.1.1.1.1.1.1.1.1.1.3.cmml">1</mn></msub><mo id="S4.I10.i1.p3.5.m5.1.1.1.1.1.1.1.1.3" stretchy="false" xref="S4.I10.i1.p3.5.m5.1.1.1.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S4.I10.i1.p3.5.m5.2.2.2.2.2.4" xref="S4.I10.i1.p3.5.m5.2.2.2.2.3.cmml">,</mo><mrow id="S4.I10.i1.p3.5.m5.2.2.2.2.2.2" xref="S4.I10.i1.p3.5.m5.2.2.2.2.2.2.cmml"><mi id="S4.I10.i1.p3.5.m5.2.2.2.2.2.2.3" mathvariant="normal" xref="S4.I10.i1.p3.5.m5.2.2.2.2.2.2.3.cmml">ℓ</mi><mo id="S4.I10.i1.p3.5.m5.2.2.2.2.2.2.2" xref="S4.I10.i1.p3.5.m5.2.2.2.2.2.2.2.cmml">⁢</mo><mrow id="S4.I10.i1.p3.5.m5.2.2.2.2.2.2.1.1" xref="S4.I10.i1.p3.5.m5.2.2.2.2.2.2.1.1.1.cmml"><mo id="S4.I10.i1.p3.5.m5.2.2.2.2.2.2.1.1.2" stretchy="false" xref="S4.I10.i1.p3.5.m5.2.2.2.2.2.2.1.1.1.cmml">(</mo><msub id="S4.I10.i1.p3.5.m5.2.2.2.2.2.2.1.1.1" xref="S4.I10.i1.p3.5.m5.2.2.2.2.2.2.1.1.1.cmml"><mi id="S4.I10.i1.p3.5.m5.2.2.2.2.2.2.1.1.1.2" xref="S4.I10.i1.p3.5.m5.2.2.2.2.2.2.1.1.1.2.cmml">u</mi><mn id="S4.I10.i1.p3.5.m5.2.2.2.2.2.2.1.1.1.3" xref="S4.I10.i1.p3.5.m5.2.2.2.2.2.2.1.1.1.3.cmml">2</mn></msub><mo id="S4.I10.i1.p3.5.m5.2.2.2.2.2.2.1.1.3" stretchy="false" xref="S4.I10.i1.p3.5.m5.2.2.2.2.2.2.1.1.1.cmml">)</mo></mrow></mrow><mo id="S4.I10.i1.p3.5.m5.2.2.2.2.2.5" stretchy="false" xref="S4.I10.i1.p3.5.m5.2.2.2.2.3.cmml">)</mo></mrow></mrow><mo id="S4.I10.i1.p3.5.m5.2.2.3" xref="S4.I10.i1.p3.5.m5.2.2.3.cmml">∈</mo><mrow id="S4.I10.i1.p3.5.m5.2.2.4" xref="S4.I10.i1.p3.5.m5.2.2.4.cmml"><mi id="S4.I10.i1.p3.5.m5.2.2.4.2" xref="S4.I10.i1.p3.5.m5.2.2.4.2.cmml">T</mi><mo id="S4.I10.i1.p3.5.m5.2.2.4.1" xref="S4.I10.i1.p3.5.m5.2.2.4.1.cmml">∖</mo><msub id="S4.I10.i1.p3.5.m5.2.2.4.3" xref="S4.I10.i1.p3.5.m5.2.2.4.3.cmml"><mi id="S4.I10.i1.p3.5.m5.2.2.4.3.2" xref="S4.I10.i1.p3.5.m5.2.2.4.3.2.cmml">T</mi><mi id="S4.I10.i1.p3.5.m5.2.2.4.3.3" xref="S4.I10.i1.p3.5.m5.2.2.4.3.3.cmml">x</mi></msub></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.I10.i1.p3.5.m5.2b"><apply id="S4.I10.i1.p3.5.m5.2.2.cmml" xref="S4.I10.i1.p3.5.m5.2.2"><in id="S4.I10.i1.p3.5.m5.2.2.3.cmml" xref="S4.I10.i1.p3.5.m5.2.2.3"></in><apply id="S4.I10.i1.p3.5.m5.2.2.2.cmml" xref="S4.I10.i1.p3.5.m5.2.2.2"><times id="S4.I10.i1.p3.5.m5.2.2.2.3.cmml" xref="S4.I10.i1.p3.5.m5.2.2.2.3"></times><ci id="S4.I10.i1.p3.5.m5.2.2.2.4a.cmml" xref="S4.I10.i1.p3.5.m5.2.2.2.4"><mtext id="S4.I10.i1.p3.5.m5.2.2.2.4.cmml" xref="S4.I10.i1.p3.5.m5.2.2.2.4">LCA</mtext></ci><interval closure="open" id="S4.I10.i1.p3.5.m5.2.2.2.2.3.cmml" xref="S4.I10.i1.p3.5.m5.2.2.2.2.2"><apply id="S4.I10.i1.p3.5.m5.1.1.1.1.1.1.cmml" xref="S4.I10.i1.p3.5.m5.1.1.1.1.1.1"><times id="S4.I10.i1.p3.5.m5.1.1.1.1.1.1.2.cmml" xref="S4.I10.i1.p3.5.m5.1.1.1.1.1.1.2"></times><ci id="S4.I10.i1.p3.5.m5.1.1.1.1.1.1.3.cmml" xref="S4.I10.i1.p3.5.m5.1.1.1.1.1.1.3">ℎ</ci><apply id="S4.I10.i1.p3.5.m5.1.1.1.1.1.1.1.1.1.cmml" xref="S4.I10.i1.p3.5.m5.1.1.1.1.1.1.1.1"><csymbol cd="ambiguous" id="S4.I10.i1.p3.5.m5.1.1.1.1.1.1.1.1.1.1.cmml" xref="S4.I10.i1.p3.5.m5.1.1.1.1.1.1.1.1">subscript</csymbol><ci id="S4.I10.i1.p3.5.m5.1.1.1.1.1.1.1.1.1.2.cmml" xref="S4.I10.i1.p3.5.m5.1.1.1.1.1.1.1.1.1.2">𝑢</ci><cn id="S4.I10.i1.p3.5.m5.1.1.1.1.1.1.1.1.1.3.cmml" type="integer" xref="S4.I10.i1.p3.5.m5.1.1.1.1.1.1.1.1.1.3">1</cn></apply></apply><apply id="S4.I10.i1.p3.5.m5.2.2.2.2.2.2.cmml" xref="S4.I10.i1.p3.5.m5.2.2.2.2.2.2"><times id="S4.I10.i1.p3.5.m5.2.2.2.2.2.2.2.cmml" xref="S4.I10.i1.p3.5.m5.2.2.2.2.2.2.2"></times><ci id="S4.I10.i1.p3.5.m5.2.2.2.2.2.2.3.cmml" xref="S4.I10.i1.p3.5.m5.2.2.2.2.2.2.3">ℓ</ci><apply id="S4.I10.i1.p3.5.m5.2.2.2.2.2.2.1.1.1.cmml" xref="S4.I10.i1.p3.5.m5.2.2.2.2.2.2.1.1"><csymbol cd="ambiguous" id="S4.I10.i1.p3.5.m5.2.2.2.2.2.2.1.1.1.1.cmml" xref="S4.I10.i1.p3.5.m5.2.2.2.2.2.2.1.1">subscript</csymbol><ci id="S4.I10.i1.p3.5.m5.2.2.2.2.2.2.1.1.1.2.cmml" xref="S4.I10.i1.p3.5.m5.2.2.2.2.2.2.1.1.1.2">𝑢</ci><cn id="S4.I10.i1.p3.5.m5.2.2.2.2.2.2.1.1.1.3.cmml" type="integer" xref="S4.I10.i1.p3.5.m5.2.2.2.2.2.2.1.1.1.3">2</cn></apply></apply></interval></apply><apply id="S4.I10.i1.p3.5.m5.2.2.4.cmml" xref="S4.I10.i1.p3.5.m5.2.2.4"><setdiff id="S4.I10.i1.p3.5.m5.2.2.4.1.cmml" xref="S4.I10.i1.p3.5.m5.2.2.4.1"></setdiff><ci id="S4.I10.i1.p3.5.m5.2.2.4.2.cmml" xref="S4.I10.i1.p3.5.m5.2.2.4.2">𝑇</ci><apply id="S4.I10.i1.p3.5.m5.2.2.4.3.cmml" xref="S4.I10.i1.p3.5.m5.2.2.4.3"><csymbol cd="ambiguous" id="S4.I10.i1.p3.5.m5.2.2.4.3.1.cmml" xref="S4.I10.i1.p3.5.m5.2.2.4.3">subscript</csymbol><ci id="S4.I10.i1.p3.5.m5.2.2.4.3.2.cmml" xref="S4.I10.i1.p3.5.m5.2.2.4.3.2">𝑇</ci><ci id="S4.I10.i1.p3.5.m5.2.2.4.3.3.cmml" xref="S4.I10.i1.p3.5.m5.2.2.4.3.3">𝑥</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.I10.i1.p3.5.m5.2c">\text{LCA}(h(u_{1}),\ell(u_{2}))\in T\setminus T_{x}</annotation><annotation encoding="application/x-llamapun" id="S4.I10.i1.p3.5.m5.2d">LCA ( italic_h ( italic_u start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) , roman_ℓ ( italic_u start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) ) ∈ italic_T ∖ italic_T start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT</annotation></semantics></math>. Let <math alttext="j^{\prime}" class="ltx_Math" display="inline" id="S4.I10.i1.p3.6.m6.1"><semantics id="S4.I10.i1.p3.6.m6.1a"><msup id="S4.I10.i1.p3.6.m6.1.1" xref="S4.I10.i1.p3.6.m6.1.1.cmml"><mi id="S4.I10.i1.p3.6.m6.1.1.2" xref="S4.I10.i1.p3.6.m6.1.1.2.cmml">j</mi><mo id="S4.I10.i1.p3.6.m6.1.1.3" xref="S4.I10.i1.p3.6.m6.1.1.3.cmml">′</mo></msup><annotation-xml encoding="MathML-Content" id="S4.I10.i1.p3.6.m6.1b"><apply id="S4.I10.i1.p3.6.m6.1.1.cmml" xref="S4.I10.i1.p3.6.m6.1.1"><csymbol cd="ambiguous" id="S4.I10.i1.p3.6.m6.1.1.1.cmml" xref="S4.I10.i1.p3.6.m6.1.1">superscript</csymbol><ci id="S4.I10.i1.p3.6.m6.1.1.2.cmml" xref="S4.I10.i1.p3.6.m6.1.1.2">𝑗</ci><ci id="S4.I10.i1.p3.6.m6.1.1.3.cmml" xref="S4.I10.i1.p3.6.m6.1.1.3">′</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.I10.i1.p3.6.m6.1c">j^{\prime}</annotation><annotation encoding="application/x-llamapun" id="S4.I10.i1.p3.6.m6.1d">italic_j start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math> be the weight class of <math alttext="(u_{1},u_{2})" class="ltx_Math" display="inline" id="S4.I10.i1.p3.7.m7.2"><semantics id="S4.I10.i1.p3.7.m7.2a"><mrow id="S4.I10.i1.p3.7.m7.2.2.2" xref="S4.I10.i1.p3.7.m7.2.2.3.cmml"><mo id="S4.I10.i1.p3.7.m7.2.2.2.3" stretchy="false" xref="S4.I10.i1.p3.7.m7.2.2.3.cmml">(</mo><msub id="S4.I10.i1.p3.7.m7.1.1.1.1" xref="S4.I10.i1.p3.7.m7.1.1.1.1.cmml"><mi id="S4.I10.i1.p3.7.m7.1.1.1.1.2" xref="S4.I10.i1.p3.7.m7.1.1.1.1.2.cmml">u</mi><mn id="S4.I10.i1.p3.7.m7.1.1.1.1.3" xref="S4.I10.i1.p3.7.m7.1.1.1.1.3.cmml">1</mn></msub><mo id="S4.I10.i1.p3.7.m7.2.2.2.4" xref="S4.I10.i1.p3.7.m7.2.2.3.cmml">,</mo><msub id="S4.I10.i1.p3.7.m7.2.2.2.2" xref="S4.I10.i1.p3.7.m7.2.2.2.2.cmml"><mi id="S4.I10.i1.p3.7.m7.2.2.2.2.2" xref="S4.I10.i1.p3.7.m7.2.2.2.2.2.cmml">u</mi><mn id="S4.I10.i1.p3.7.m7.2.2.2.2.3" xref="S4.I10.i1.p3.7.m7.2.2.2.2.3.cmml">2</mn></msub><mo id="S4.I10.i1.p3.7.m7.2.2.2.5" stretchy="false" xref="S4.I10.i1.p3.7.m7.2.2.3.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S4.I10.i1.p3.7.m7.2b"><interval closure="open" id="S4.I10.i1.p3.7.m7.2.2.3.cmml" xref="S4.I10.i1.p3.7.m7.2.2.2"><apply id="S4.I10.i1.p3.7.m7.1.1.1.1.cmml" xref="S4.I10.i1.p3.7.m7.1.1.1.1"><csymbol cd="ambiguous" id="S4.I10.i1.p3.7.m7.1.1.1.1.1.cmml" xref="S4.I10.i1.p3.7.m7.1.1.1.1">subscript</csymbol><ci id="S4.I10.i1.p3.7.m7.1.1.1.1.2.cmml" xref="S4.I10.i1.p3.7.m7.1.1.1.1.2">𝑢</ci><cn id="S4.I10.i1.p3.7.m7.1.1.1.1.3.cmml" type="integer" xref="S4.I10.i1.p3.7.m7.1.1.1.1.3">1</cn></apply><apply id="S4.I10.i1.p3.7.m7.2.2.2.2.cmml" xref="S4.I10.i1.p3.7.m7.2.2.2.2"><csymbol cd="ambiguous" id="S4.I10.i1.p3.7.m7.2.2.2.2.1.cmml" xref="S4.I10.i1.p3.7.m7.2.2.2.2">subscript</csymbol><ci id="S4.I10.i1.p3.7.m7.2.2.2.2.2.cmml" xref="S4.I10.i1.p3.7.m7.2.2.2.2.2">𝑢</ci><cn id="S4.I10.i1.p3.7.m7.2.2.2.2.3.cmml" type="integer" xref="S4.I10.i1.p3.7.m7.2.2.2.2.3">2</cn></apply></interval></annotation-xml><annotation encoding="application/x-tex" id="S4.I10.i1.p3.7.m7.2c">(u_{1},u_{2})</annotation><annotation encoding="application/x-llamapun" id="S4.I10.i1.p3.7.m7.2d">( italic_u start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_u start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT )</annotation></semantics></math>, and let <math alttext="(u_{3},u_{4})=L_{h(u_{1})}(j^{\prime})" class="ltx_Math" display="inline" id="S4.I10.i1.p3.8.m8.4"><semantics id="S4.I10.i1.p3.8.m8.4a"><mrow id="S4.I10.i1.p3.8.m8.4.4" xref="S4.I10.i1.p3.8.m8.4.4.cmml"><mrow id="S4.I10.i1.p3.8.m8.3.3.2.2" xref="S4.I10.i1.p3.8.m8.3.3.2.3.cmml"><mo id="S4.I10.i1.p3.8.m8.3.3.2.2.3" stretchy="false" xref="S4.I10.i1.p3.8.m8.3.3.2.3.cmml">(</mo><msub id="S4.I10.i1.p3.8.m8.2.2.1.1.1" xref="S4.I10.i1.p3.8.m8.2.2.1.1.1.cmml"><mi id="S4.I10.i1.p3.8.m8.2.2.1.1.1.2" xref="S4.I10.i1.p3.8.m8.2.2.1.1.1.2.cmml">u</mi><mn id="S4.I10.i1.p3.8.m8.2.2.1.1.1.3" xref="S4.I10.i1.p3.8.m8.2.2.1.1.1.3.cmml">3</mn></msub><mo id="S4.I10.i1.p3.8.m8.3.3.2.2.4" xref="S4.I10.i1.p3.8.m8.3.3.2.3.cmml">,</mo><msub id="S4.I10.i1.p3.8.m8.3.3.2.2.2" xref="S4.I10.i1.p3.8.m8.3.3.2.2.2.cmml"><mi id="S4.I10.i1.p3.8.m8.3.3.2.2.2.2" xref="S4.I10.i1.p3.8.m8.3.3.2.2.2.2.cmml">u</mi><mn id="S4.I10.i1.p3.8.m8.3.3.2.2.2.3" xref="S4.I10.i1.p3.8.m8.3.3.2.2.2.3.cmml">4</mn></msub><mo id="S4.I10.i1.p3.8.m8.3.3.2.2.5" stretchy="false" xref="S4.I10.i1.p3.8.m8.3.3.2.3.cmml">)</mo></mrow><mo id="S4.I10.i1.p3.8.m8.4.4.4" xref="S4.I10.i1.p3.8.m8.4.4.4.cmml">=</mo><mrow id="S4.I10.i1.p3.8.m8.4.4.3" xref="S4.I10.i1.p3.8.m8.4.4.3.cmml"><msub id="S4.I10.i1.p3.8.m8.4.4.3.3" xref="S4.I10.i1.p3.8.m8.4.4.3.3.cmml"><mi id="S4.I10.i1.p3.8.m8.4.4.3.3.2" xref="S4.I10.i1.p3.8.m8.4.4.3.3.2.cmml">L</mi><mrow id="S4.I10.i1.p3.8.m8.1.1.1" xref="S4.I10.i1.p3.8.m8.1.1.1.cmml"><mi id="S4.I10.i1.p3.8.m8.1.1.1.3" xref="S4.I10.i1.p3.8.m8.1.1.1.3.cmml">h</mi><mo id="S4.I10.i1.p3.8.m8.1.1.1.2" xref="S4.I10.i1.p3.8.m8.1.1.1.2.cmml">⁢</mo><mrow id="S4.I10.i1.p3.8.m8.1.1.1.1.1" xref="S4.I10.i1.p3.8.m8.1.1.1.1.1.1.cmml"><mo id="S4.I10.i1.p3.8.m8.1.1.1.1.1.2" stretchy="false" xref="S4.I10.i1.p3.8.m8.1.1.1.1.1.1.cmml">(</mo><msub id="S4.I10.i1.p3.8.m8.1.1.1.1.1.1" xref="S4.I10.i1.p3.8.m8.1.1.1.1.1.1.cmml"><mi id="S4.I10.i1.p3.8.m8.1.1.1.1.1.1.2" xref="S4.I10.i1.p3.8.m8.1.1.1.1.1.1.2.cmml">u</mi><mn id="S4.I10.i1.p3.8.m8.1.1.1.1.1.1.3" xref="S4.I10.i1.p3.8.m8.1.1.1.1.1.1.3.cmml">1</mn></msub><mo id="S4.I10.i1.p3.8.m8.1.1.1.1.1.3" stretchy="false" xref="S4.I10.i1.p3.8.m8.1.1.1.1.1.1.cmml">)</mo></mrow></mrow></msub><mo id="S4.I10.i1.p3.8.m8.4.4.3.2" xref="S4.I10.i1.p3.8.m8.4.4.3.2.cmml">⁢</mo><mrow id="S4.I10.i1.p3.8.m8.4.4.3.1.1" xref="S4.I10.i1.p3.8.m8.4.4.3.1.1.1.cmml"><mo id="S4.I10.i1.p3.8.m8.4.4.3.1.1.2" stretchy="false" xref="S4.I10.i1.p3.8.m8.4.4.3.1.1.1.cmml">(</mo><msup id="S4.I10.i1.p3.8.m8.4.4.3.1.1.1" xref="S4.I10.i1.p3.8.m8.4.4.3.1.1.1.cmml"><mi id="S4.I10.i1.p3.8.m8.4.4.3.1.1.1.2" xref="S4.I10.i1.p3.8.m8.4.4.3.1.1.1.2.cmml">j</mi><mo id="S4.I10.i1.p3.8.m8.4.4.3.1.1.1.3" xref="S4.I10.i1.p3.8.m8.4.4.3.1.1.1.3.cmml">′</mo></msup><mo id="S4.I10.i1.p3.8.m8.4.4.3.1.1.3" stretchy="false" xref="S4.I10.i1.p3.8.m8.4.4.3.1.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.I10.i1.p3.8.m8.4b"><apply id="S4.I10.i1.p3.8.m8.4.4.cmml" xref="S4.I10.i1.p3.8.m8.4.4"><eq id="S4.I10.i1.p3.8.m8.4.4.4.cmml" xref="S4.I10.i1.p3.8.m8.4.4.4"></eq><interval closure="open" id="S4.I10.i1.p3.8.m8.3.3.2.3.cmml" xref="S4.I10.i1.p3.8.m8.3.3.2.2"><apply id="S4.I10.i1.p3.8.m8.2.2.1.1.1.cmml" xref="S4.I10.i1.p3.8.m8.2.2.1.1.1"><csymbol cd="ambiguous" id="S4.I10.i1.p3.8.m8.2.2.1.1.1.1.cmml" xref="S4.I10.i1.p3.8.m8.2.2.1.1.1">subscript</csymbol><ci id="S4.I10.i1.p3.8.m8.2.2.1.1.1.2.cmml" xref="S4.I10.i1.p3.8.m8.2.2.1.1.1.2">𝑢</ci><cn id="S4.I10.i1.p3.8.m8.2.2.1.1.1.3.cmml" type="integer" xref="S4.I10.i1.p3.8.m8.2.2.1.1.1.3">3</cn></apply><apply id="S4.I10.i1.p3.8.m8.3.3.2.2.2.cmml" xref="S4.I10.i1.p3.8.m8.3.3.2.2.2"><csymbol cd="ambiguous" id="S4.I10.i1.p3.8.m8.3.3.2.2.2.1.cmml" xref="S4.I10.i1.p3.8.m8.3.3.2.2.2">subscript</csymbol><ci id="S4.I10.i1.p3.8.m8.3.3.2.2.2.2.cmml" xref="S4.I10.i1.p3.8.m8.3.3.2.2.2.2">𝑢</ci><cn id="S4.I10.i1.p3.8.m8.3.3.2.2.2.3.cmml" type="integer" xref="S4.I10.i1.p3.8.m8.3.3.2.2.2.3">4</cn></apply></interval><apply id="S4.I10.i1.p3.8.m8.4.4.3.cmml" xref="S4.I10.i1.p3.8.m8.4.4.3"><times id="S4.I10.i1.p3.8.m8.4.4.3.2.cmml" xref="S4.I10.i1.p3.8.m8.4.4.3.2"></times><apply id="S4.I10.i1.p3.8.m8.4.4.3.3.cmml" xref="S4.I10.i1.p3.8.m8.4.4.3.3"><csymbol cd="ambiguous" id="S4.I10.i1.p3.8.m8.4.4.3.3.1.cmml" xref="S4.I10.i1.p3.8.m8.4.4.3.3">subscript</csymbol><ci id="S4.I10.i1.p3.8.m8.4.4.3.3.2.cmml" xref="S4.I10.i1.p3.8.m8.4.4.3.3.2">𝐿</ci><apply id="S4.I10.i1.p3.8.m8.1.1.1.cmml" xref="S4.I10.i1.p3.8.m8.1.1.1"><times id="S4.I10.i1.p3.8.m8.1.1.1.2.cmml" xref="S4.I10.i1.p3.8.m8.1.1.1.2"></times><ci id="S4.I10.i1.p3.8.m8.1.1.1.3.cmml" xref="S4.I10.i1.p3.8.m8.1.1.1.3">ℎ</ci><apply id="S4.I10.i1.p3.8.m8.1.1.1.1.1.1.cmml" xref="S4.I10.i1.p3.8.m8.1.1.1.1.1"><csymbol cd="ambiguous" id="S4.I10.i1.p3.8.m8.1.1.1.1.1.1.1.cmml" xref="S4.I10.i1.p3.8.m8.1.1.1.1.1">subscript</csymbol><ci id="S4.I10.i1.p3.8.m8.1.1.1.1.1.1.2.cmml" xref="S4.I10.i1.p3.8.m8.1.1.1.1.1.1.2">𝑢</ci><cn id="S4.I10.i1.p3.8.m8.1.1.1.1.1.1.3.cmml" type="integer" xref="S4.I10.i1.p3.8.m8.1.1.1.1.1.1.3">1</cn></apply></apply></apply><apply id="S4.I10.i1.p3.8.m8.4.4.3.1.1.1.cmml" xref="S4.I10.i1.p3.8.m8.4.4.3.1.1"><csymbol cd="ambiguous" id="S4.I10.i1.p3.8.m8.4.4.3.1.1.1.1.cmml" xref="S4.I10.i1.p3.8.m8.4.4.3.1.1">superscript</csymbol><ci id="S4.I10.i1.p3.8.m8.4.4.3.1.1.1.2.cmml" xref="S4.I10.i1.p3.8.m8.4.4.3.1.1.1.2">𝑗</ci><ci id="S4.I10.i1.p3.8.m8.4.4.3.1.1.1.3.cmml" xref="S4.I10.i1.p3.8.m8.4.4.3.1.1.1.3">′</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.I10.i1.p3.8.m8.4c">(u_{3},u_{4})=L_{h(u_{1})}(j^{\prime})</annotation><annotation encoding="application/x-llamapun" id="S4.I10.i1.p3.8.m8.4d">( italic_u start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT , italic_u start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ) = italic_L start_POSTSUBSCRIPT italic_h ( italic_u start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) end_POSTSUBSCRIPT ( italic_j start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT )</annotation></semantics></math> with <math alttext="h(u_{1})=h(u_{3})" class="ltx_Math" display="inline" id="S4.I10.i1.p3.9.m9.2"><semantics id="S4.I10.i1.p3.9.m9.2a"><mrow id="S4.I10.i1.p3.9.m9.2.2" xref="S4.I10.i1.p3.9.m9.2.2.cmml"><mrow id="S4.I10.i1.p3.9.m9.1.1.1" xref="S4.I10.i1.p3.9.m9.1.1.1.cmml"><mi id="S4.I10.i1.p3.9.m9.1.1.1.3" xref="S4.I10.i1.p3.9.m9.1.1.1.3.cmml">h</mi><mo id="S4.I10.i1.p3.9.m9.1.1.1.2" xref="S4.I10.i1.p3.9.m9.1.1.1.2.cmml">⁢</mo><mrow id="S4.I10.i1.p3.9.m9.1.1.1.1.1" xref="S4.I10.i1.p3.9.m9.1.1.1.1.1.1.cmml"><mo id="S4.I10.i1.p3.9.m9.1.1.1.1.1.2" stretchy="false" xref="S4.I10.i1.p3.9.m9.1.1.1.1.1.1.cmml">(</mo><msub id="S4.I10.i1.p3.9.m9.1.1.1.1.1.1" xref="S4.I10.i1.p3.9.m9.1.1.1.1.1.1.cmml"><mi id="S4.I10.i1.p3.9.m9.1.1.1.1.1.1.2" xref="S4.I10.i1.p3.9.m9.1.1.1.1.1.1.2.cmml">u</mi><mn id="S4.I10.i1.p3.9.m9.1.1.1.1.1.1.3" xref="S4.I10.i1.p3.9.m9.1.1.1.1.1.1.3.cmml">1</mn></msub><mo id="S4.I10.i1.p3.9.m9.1.1.1.1.1.3" stretchy="false" xref="S4.I10.i1.p3.9.m9.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S4.I10.i1.p3.9.m9.2.2.3" xref="S4.I10.i1.p3.9.m9.2.2.3.cmml">=</mo><mrow id="S4.I10.i1.p3.9.m9.2.2.2" xref="S4.I10.i1.p3.9.m9.2.2.2.cmml"><mi id="S4.I10.i1.p3.9.m9.2.2.2.3" xref="S4.I10.i1.p3.9.m9.2.2.2.3.cmml">h</mi><mo id="S4.I10.i1.p3.9.m9.2.2.2.2" xref="S4.I10.i1.p3.9.m9.2.2.2.2.cmml">⁢</mo><mrow id="S4.I10.i1.p3.9.m9.2.2.2.1.1" xref="S4.I10.i1.p3.9.m9.2.2.2.1.1.1.cmml"><mo id="S4.I10.i1.p3.9.m9.2.2.2.1.1.2" stretchy="false" xref="S4.I10.i1.p3.9.m9.2.2.2.1.1.1.cmml">(</mo><msub id="S4.I10.i1.p3.9.m9.2.2.2.1.1.1" xref="S4.I10.i1.p3.9.m9.2.2.2.1.1.1.cmml"><mi id="S4.I10.i1.p3.9.m9.2.2.2.1.1.1.2" xref="S4.I10.i1.p3.9.m9.2.2.2.1.1.1.2.cmml">u</mi><mn id="S4.I10.i1.p3.9.m9.2.2.2.1.1.1.3" xref="S4.I10.i1.p3.9.m9.2.2.2.1.1.1.3.cmml">3</mn></msub><mo id="S4.I10.i1.p3.9.m9.2.2.2.1.1.3" stretchy="false" xref="S4.I10.i1.p3.9.m9.2.2.2.1.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.I10.i1.p3.9.m9.2b"><apply id="S4.I10.i1.p3.9.m9.2.2.cmml" xref="S4.I10.i1.p3.9.m9.2.2"><eq id="S4.I10.i1.p3.9.m9.2.2.3.cmml" xref="S4.I10.i1.p3.9.m9.2.2.3"></eq><apply id="S4.I10.i1.p3.9.m9.1.1.1.cmml" xref="S4.I10.i1.p3.9.m9.1.1.1"><times id="S4.I10.i1.p3.9.m9.1.1.1.2.cmml" xref="S4.I10.i1.p3.9.m9.1.1.1.2"></times><ci id="S4.I10.i1.p3.9.m9.1.1.1.3.cmml" xref="S4.I10.i1.p3.9.m9.1.1.1.3">ℎ</ci><apply id="S4.I10.i1.p3.9.m9.1.1.1.1.1.1.cmml" xref="S4.I10.i1.p3.9.m9.1.1.1.1.1"><csymbol cd="ambiguous" id="S4.I10.i1.p3.9.m9.1.1.1.1.1.1.1.cmml" xref="S4.I10.i1.p3.9.m9.1.1.1.1.1">subscript</csymbol><ci id="S4.I10.i1.p3.9.m9.1.1.1.1.1.1.2.cmml" xref="S4.I10.i1.p3.9.m9.1.1.1.1.1.1.2">𝑢</ci><cn id="S4.I10.i1.p3.9.m9.1.1.1.1.1.1.3.cmml" type="integer" xref="S4.I10.i1.p3.9.m9.1.1.1.1.1.1.3">1</cn></apply></apply><apply id="S4.I10.i1.p3.9.m9.2.2.2.cmml" xref="S4.I10.i1.p3.9.m9.2.2.2"><times id="S4.I10.i1.p3.9.m9.2.2.2.2.cmml" xref="S4.I10.i1.p3.9.m9.2.2.2.2"></times><ci id="S4.I10.i1.p3.9.m9.2.2.2.3.cmml" xref="S4.I10.i1.p3.9.m9.2.2.2.3">ℎ</ci><apply id="S4.I10.i1.p3.9.m9.2.2.2.1.1.1.cmml" xref="S4.I10.i1.p3.9.m9.2.2.2.1.1"><csymbol cd="ambiguous" id="S4.I10.i1.p3.9.m9.2.2.2.1.1.1.1.cmml" xref="S4.I10.i1.p3.9.m9.2.2.2.1.1">subscript</csymbol><ci id="S4.I10.i1.p3.9.m9.2.2.2.1.1.1.2.cmml" xref="S4.I10.i1.p3.9.m9.2.2.2.1.1.1.2">𝑢</ci><cn id="S4.I10.i1.p3.9.m9.2.2.2.1.1.1.3.cmml" type="integer" xref="S4.I10.i1.p3.9.m9.2.2.2.1.1.1.3">3</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.I10.i1.p3.9.m9.2c">h(u_{1})=h(u_{3})</annotation><annotation encoding="application/x-llamapun" id="S4.I10.i1.p3.9.m9.2d">italic_h ( italic_u start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) = italic_h ( italic_u start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT )</annotation></semantics></math>. This must be in <span class="ltx_text ltx_markedasmath" id="S4.I10.i1.p3.27.1">SOL</span> since <math alttext="(u_{1},u_{2})\in\textnormal{OPT}" class="ltx_Math" display="inline" id="S4.I10.i1.p3.11.m11.2"><semantics id="S4.I10.i1.p3.11.m11.2a"><mrow id="S4.I10.i1.p3.11.m11.2.2" xref="S4.I10.i1.p3.11.m11.2.2.cmml"><mrow id="S4.I10.i1.p3.11.m11.2.2.2.2" xref="S4.I10.i1.p3.11.m11.2.2.2.3.cmml"><mo id="S4.I10.i1.p3.11.m11.2.2.2.2.3" stretchy="false" xref="S4.I10.i1.p3.11.m11.2.2.2.3.cmml">(</mo><msub id="S4.I10.i1.p3.11.m11.1.1.1.1.1" xref="S4.I10.i1.p3.11.m11.1.1.1.1.1.cmml"><mi id="S4.I10.i1.p3.11.m11.1.1.1.1.1.2" xref="S4.I10.i1.p3.11.m11.1.1.1.1.1.2.cmml">u</mi><mn id="S4.I10.i1.p3.11.m11.1.1.1.1.1.3" xref="S4.I10.i1.p3.11.m11.1.1.1.1.1.3.cmml">1</mn></msub><mo id="S4.I10.i1.p3.11.m11.2.2.2.2.4" xref="S4.I10.i1.p3.11.m11.2.2.2.3.cmml">,</mo><msub id="S4.I10.i1.p3.11.m11.2.2.2.2.2" xref="S4.I10.i1.p3.11.m11.2.2.2.2.2.cmml"><mi id="S4.I10.i1.p3.11.m11.2.2.2.2.2.2" xref="S4.I10.i1.p3.11.m11.2.2.2.2.2.2.cmml">u</mi><mn id="S4.I10.i1.p3.11.m11.2.2.2.2.2.3" xref="S4.I10.i1.p3.11.m11.2.2.2.2.2.3.cmml">2</mn></msub><mo id="S4.I10.i1.p3.11.m11.2.2.2.2.5" stretchy="false" xref="S4.I10.i1.p3.11.m11.2.2.2.3.cmml">)</mo></mrow><mo id="S4.I10.i1.p3.11.m11.2.2.3" xref="S4.I10.i1.p3.11.m11.2.2.3.cmml">∈</mo><mtext id="S4.I10.i1.p3.11.m11.2.2.4" xref="S4.I10.i1.p3.11.m11.2.2.4a.cmml">OPT</mtext></mrow><annotation-xml encoding="MathML-Content" id="S4.I10.i1.p3.11.m11.2b"><apply id="S4.I10.i1.p3.11.m11.2.2.cmml" xref="S4.I10.i1.p3.11.m11.2.2"><in id="S4.I10.i1.p3.11.m11.2.2.3.cmml" xref="S4.I10.i1.p3.11.m11.2.2.3"></in><interval closure="open" id="S4.I10.i1.p3.11.m11.2.2.2.3.cmml" xref="S4.I10.i1.p3.11.m11.2.2.2.2"><apply id="S4.I10.i1.p3.11.m11.1.1.1.1.1.cmml" xref="S4.I10.i1.p3.11.m11.1.1.1.1.1"><csymbol cd="ambiguous" id="S4.I10.i1.p3.11.m11.1.1.1.1.1.1.cmml" xref="S4.I10.i1.p3.11.m11.1.1.1.1.1">subscript</csymbol><ci id="S4.I10.i1.p3.11.m11.1.1.1.1.1.2.cmml" xref="S4.I10.i1.p3.11.m11.1.1.1.1.1.2">𝑢</ci><cn id="S4.I10.i1.p3.11.m11.1.1.1.1.1.3.cmml" type="integer" xref="S4.I10.i1.p3.11.m11.1.1.1.1.1.3">1</cn></apply><apply id="S4.I10.i1.p3.11.m11.2.2.2.2.2.cmml" xref="S4.I10.i1.p3.11.m11.2.2.2.2.2"><csymbol cd="ambiguous" id="S4.I10.i1.p3.11.m11.2.2.2.2.2.1.cmml" xref="S4.I10.i1.p3.11.m11.2.2.2.2.2">subscript</csymbol><ci id="S4.I10.i1.p3.11.m11.2.2.2.2.2.2.cmml" xref="S4.I10.i1.p3.11.m11.2.2.2.2.2.2">𝑢</ci><cn id="S4.I10.i1.p3.11.m11.2.2.2.2.2.3.cmml" type="integer" xref="S4.I10.i1.p3.11.m11.2.2.2.2.2.3">2</cn></apply></interval><ci id="S4.I10.i1.p3.11.m11.2.2.4a.cmml" xref="S4.I10.i1.p3.11.m11.2.2.4"><mtext id="S4.I10.i1.p3.11.m11.2.2.4.cmml" xref="S4.I10.i1.p3.11.m11.2.2.4">OPT</mtext></ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.I10.i1.p3.11.m11.2c">(u_{1},u_{2})\in\textnormal{OPT}</annotation><annotation encoding="application/x-llamapun" id="S4.I10.i1.p3.11.m11.2d">( italic_u start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_u start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) ∈ OPT</annotation></semantics></math>. By construction, <math alttext="d_{T}(r,\text{LCA}(h(u_{1}),\ell(u_{2})))\leq d_{T}(r,\text{LCA}(h(u_{1}),\ell% (u_{4})))" class="ltx_Math" display="inline" id="S4.I10.i1.p3.12.m12.4"><semantics id="S4.I10.i1.p3.12.m12.4a"><mrow id="S4.I10.i1.p3.12.m12.4.4" xref="S4.I10.i1.p3.12.m12.4.4.cmml"><mrow id="S4.I10.i1.p3.12.m12.3.3.1" xref="S4.I10.i1.p3.12.m12.3.3.1.cmml"><msub id="S4.I10.i1.p3.12.m12.3.3.1.3" xref="S4.I10.i1.p3.12.m12.3.3.1.3.cmml"><mi id="S4.I10.i1.p3.12.m12.3.3.1.3.2" xref="S4.I10.i1.p3.12.m12.3.3.1.3.2.cmml">d</mi><mi id="S4.I10.i1.p3.12.m12.3.3.1.3.3" xref="S4.I10.i1.p3.12.m12.3.3.1.3.3.cmml">T</mi></msub><mo id="S4.I10.i1.p3.12.m12.3.3.1.2" xref="S4.I10.i1.p3.12.m12.3.3.1.2.cmml">⁢</mo><mrow id="S4.I10.i1.p3.12.m12.3.3.1.1.1" xref="S4.I10.i1.p3.12.m12.3.3.1.1.2.cmml"><mo id="S4.I10.i1.p3.12.m12.3.3.1.1.1.2" stretchy="false" xref="S4.I10.i1.p3.12.m12.3.3.1.1.2.cmml">(</mo><mi id="S4.I10.i1.p3.12.m12.1.1" xref="S4.I10.i1.p3.12.m12.1.1.cmml">r</mi><mo 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xref="S4.I10.i1.p3.12.m12.4.4.2.1.1"><ci id="S4.I10.i1.p3.12.m12.2.2.cmml" xref="S4.I10.i1.p3.12.m12.2.2">𝑟</ci><apply id="S4.I10.i1.p3.12.m12.4.4.2.1.1.1.cmml" xref="S4.I10.i1.p3.12.m12.4.4.2.1.1.1"><times id="S4.I10.i1.p3.12.m12.4.4.2.1.1.1.3.cmml" xref="S4.I10.i1.p3.12.m12.4.4.2.1.1.1.3"></times><ci id="S4.I10.i1.p3.12.m12.4.4.2.1.1.1.4a.cmml" xref="S4.I10.i1.p3.12.m12.4.4.2.1.1.1.4"><mtext id="S4.I10.i1.p3.12.m12.4.4.2.1.1.1.4.cmml" xref="S4.I10.i1.p3.12.m12.4.4.2.1.1.1.4">LCA</mtext></ci><interval closure="open" id="S4.I10.i1.p3.12.m12.4.4.2.1.1.1.2.3.cmml" xref="S4.I10.i1.p3.12.m12.4.4.2.1.1.1.2.2"><apply id="S4.I10.i1.p3.12.m12.4.4.2.1.1.1.1.1.1.cmml" xref="S4.I10.i1.p3.12.m12.4.4.2.1.1.1.1.1.1"><times id="S4.I10.i1.p3.12.m12.4.4.2.1.1.1.1.1.1.2.cmml" xref="S4.I10.i1.p3.12.m12.4.4.2.1.1.1.1.1.1.2"></times><ci id="S4.I10.i1.p3.12.m12.4.4.2.1.1.1.1.1.1.3.cmml" xref="S4.I10.i1.p3.12.m12.4.4.2.1.1.1.1.1.1.3">ℎ</ci><apply id="S4.I10.i1.p3.12.m12.4.4.2.1.1.1.1.1.1.1.1.1.cmml" xref="S4.I10.i1.p3.12.m12.4.4.2.1.1.1.1.1.1.1.1"><csymbol cd="ambiguous" id="S4.I10.i1.p3.12.m12.4.4.2.1.1.1.1.1.1.1.1.1.1.cmml" xref="S4.I10.i1.p3.12.m12.4.4.2.1.1.1.1.1.1.1.1">subscript</csymbol><ci id="S4.I10.i1.p3.12.m12.4.4.2.1.1.1.1.1.1.1.1.1.2.cmml" xref="S4.I10.i1.p3.12.m12.4.4.2.1.1.1.1.1.1.1.1.1.2">𝑢</ci><cn id="S4.I10.i1.p3.12.m12.4.4.2.1.1.1.1.1.1.1.1.1.3.cmml" type="integer" xref="S4.I10.i1.p3.12.m12.4.4.2.1.1.1.1.1.1.1.1.1.3">1</cn></apply></apply><apply id="S4.I10.i1.p3.12.m12.4.4.2.1.1.1.2.2.2.cmml" xref="S4.I10.i1.p3.12.m12.4.4.2.1.1.1.2.2.2"><times id="S4.I10.i1.p3.12.m12.4.4.2.1.1.1.2.2.2.2.cmml" xref="S4.I10.i1.p3.12.m12.4.4.2.1.1.1.2.2.2.2"></times><ci id="S4.I10.i1.p3.12.m12.4.4.2.1.1.1.2.2.2.3.cmml" xref="S4.I10.i1.p3.12.m12.4.4.2.1.1.1.2.2.2.3">ℓ</ci><apply id="S4.I10.i1.p3.12.m12.4.4.2.1.1.1.2.2.2.1.1.1.cmml" xref="S4.I10.i1.p3.12.m12.4.4.2.1.1.1.2.2.2.1.1"><csymbol cd="ambiguous" id="S4.I10.i1.p3.12.m12.4.4.2.1.1.1.2.2.2.1.1.1.1.cmml" xref="S4.I10.i1.p3.12.m12.4.4.2.1.1.1.2.2.2.1.1">subscript</csymbol><ci id="S4.I10.i1.p3.12.m12.4.4.2.1.1.1.2.2.2.1.1.1.2.cmml" xref="S4.I10.i1.p3.12.m12.4.4.2.1.1.1.2.2.2.1.1.1.2">𝑢</ci><cn id="S4.I10.i1.p3.12.m12.4.4.2.1.1.1.2.2.2.1.1.1.3.cmml" type="integer" xref="S4.I10.i1.p3.12.m12.4.4.2.1.1.1.2.2.2.1.1.1.3">4</cn></apply></apply></interval></apply></interval></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.I10.i1.p3.12.m12.4c">d_{T}(r,\text{LCA}(h(u_{1}),\ell(u_{2})))\leq d_{T}(r,\text{LCA}(h(u_{1}),\ell% (u_{4})))</annotation><annotation encoding="application/x-llamapun" id="S4.I10.i1.p3.12.m12.4d">italic_d start_POSTSUBSCRIPT italic_T end_POSTSUBSCRIPT ( italic_r , LCA ( italic_h ( italic_u start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) , roman_ℓ ( italic_u start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) ) ) ≤ italic_d start_POSTSUBSCRIPT italic_T end_POSTSUBSCRIPT ( italic_r , LCA ( italic_h ( italic_u start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) , roman_ℓ ( italic_u start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ) ) )</annotation></semantics></math>. In particular, <math alttext="\text{LCA}(h(u_{1}),\ell(u_{4}))\in T\setminus T_{x}" class="ltx_Math" display="inline" id="S4.I10.i1.p3.13.m13.2"><semantics id="S4.I10.i1.p3.13.m13.2a"><mrow id="S4.I10.i1.p3.13.m13.2.2" xref="S4.I10.i1.p3.13.m13.2.2.cmml"><mrow id="S4.I10.i1.p3.13.m13.2.2.2" xref="S4.I10.i1.p3.13.m13.2.2.2.cmml"><mtext id="S4.I10.i1.p3.13.m13.2.2.2.4" xref="S4.I10.i1.p3.13.m13.2.2.2.4a.cmml">LCA</mtext><mo id="S4.I10.i1.p3.13.m13.2.2.2.3" xref="S4.I10.i1.p3.13.m13.2.2.2.3.cmml">⁢</mo><mrow id="S4.I10.i1.p3.13.m13.2.2.2.2.2" xref="S4.I10.i1.p3.13.m13.2.2.2.2.3.cmml"><mo id="S4.I10.i1.p3.13.m13.2.2.2.2.2.3" stretchy="false" xref="S4.I10.i1.p3.13.m13.2.2.2.2.3.cmml">(</mo><mrow id="S4.I10.i1.p3.13.m13.1.1.1.1.1.1" xref="S4.I10.i1.p3.13.m13.1.1.1.1.1.1.cmml"><mi id="S4.I10.i1.p3.13.m13.1.1.1.1.1.1.3" xref="S4.I10.i1.p3.13.m13.1.1.1.1.1.1.3.cmml">h</mi><mo id="S4.I10.i1.p3.13.m13.1.1.1.1.1.1.2" xref="S4.I10.i1.p3.13.m13.1.1.1.1.1.1.2.cmml">⁢</mo><mrow 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xref="S4.I10.i1.p3.13.m13.2.2.2.2.2.2.2.cmml">⁢</mo><mrow id="S4.I10.i1.p3.13.m13.2.2.2.2.2.2.1.1" xref="S4.I10.i1.p3.13.m13.2.2.2.2.2.2.1.1.1.cmml"><mo id="S4.I10.i1.p3.13.m13.2.2.2.2.2.2.1.1.2" stretchy="false" xref="S4.I10.i1.p3.13.m13.2.2.2.2.2.2.1.1.1.cmml">(</mo><msub id="S4.I10.i1.p3.13.m13.2.2.2.2.2.2.1.1.1" xref="S4.I10.i1.p3.13.m13.2.2.2.2.2.2.1.1.1.cmml"><mi id="S4.I10.i1.p3.13.m13.2.2.2.2.2.2.1.1.1.2" xref="S4.I10.i1.p3.13.m13.2.2.2.2.2.2.1.1.1.2.cmml">u</mi><mn id="S4.I10.i1.p3.13.m13.2.2.2.2.2.2.1.1.1.3" xref="S4.I10.i1.p3.13.m13.2.2.2.2.2.2.1.1.1.3.cmml">4</mn></msub><mo id="S4.I10.i1.p3.13.m13.2.2.2.2.2.2.1.1.3" stretchy="false" xref="S4.I10.i1.p3.13.m13.2.2.2.2.2.2.1.1.1.cmml">)</mo></mrow></mrow><mo id="S4.I10.i1.p3.13.m13.2.2.2.2.2.5" stretchy="false" xref="S4.I10.i1.p3.13.m13.2.2.2.2.3.cmml">)</mo></mrow></mrow><mo id="S4.I10.i1.p3.13.m13.2.2.3" xref="S4.I10.i1.p3.13.m13.2.2.3.cmml">∈</mo><mrow id="S4.I10.i1.p3.13.m13.2.2.4" xref="S4.I10.i1.p3.13.m13.2.2.4.cmml"><mi id="S4.I10.i1.p3.13.m13.2.2.4.2" xref="S4.I10.i1.p3.13.m13.2.2.4.2.cmml">T</mi><mo id="S4.I10.i1.p3.13.m13.2.2.4.1" xref="S4.I10.i1.p3.13.m13.2.2.4.1.cmml">∖</mo><msub id="S4.I10.i1.p3.13.m13.2.2.4.3" xref="S4.I10.i1.p3.13.m13.2.2.4.3.cmml"><mi id="S4.I10.i1.p3.13.m13.2.2.4.3.2" xref="S4.I10.i1.p3.13.m13.2.2.4.3.2.cmml">T</mi><mi id="S4.I10.i1.p3.13.m13.2.2.4.3.3" xref="S4.I10.i1.p3.13.m13.2.2.4.3.3.cmml">x</mi></msub></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.I10.i1.p3.13.m13.2b"><apply id="S4.I10.i1.p3.13.m13.2.2.cmml" xref="S4.I10.i1.p3.13.m13.2.2"><in id="S4.I10.i1.p3.13.m13.2.2.3.cmml" xref="S4.I10.i1.p3.13.m13.2.2.3"></in><apply id="S4.I10.i1.p3.13.m13.2.2.2.cmml" xref="S4.I10.i1.p3.13.m13.2.2.2"><times id="S4.I10.i1.p3.13.m13.2.2.2.3.cmml" xref="S4.I10.i1.p3.13.m13.2.2.2.3"></times><ci id="S4.I10.i1.p3.13.m13.2.2.2.4a.cmml" xref="S4.I10.i1.p3.13.m13.2.2.2.4"><mtext id="S4.I10.i1.p3.13.m13.2.2.2.4.cmml" xref="S4.I10.i1.p3.13.m13.2.2.2.4">LCA</mtext></ci><interval closure="open" id="S4.I10.i1.p3.13.m13.2.2.2.2.3.cmml" xref="S4.I10.i1.p3.13.m13.2.2.2.2.2"><apply id="S4.I10.i1.p3.13.m13.1.1.1.1.1.1.cmml" xref="S4.I10.i1.p3.13.m13.1.1.1.1.1.1"><times id="S4.I10.i1.p3.13.m13.1.1.1.1.1.1.2.cmml" xref="S4.I10.i1.p3.13.m13.1.1.1.1.1.1.2"></times><ci id="S4.I10.i1.p3.13.m13.1.1.1.1.1.1.3.cmml" xref="S4.I10.i1.p3.13.m13.1.1.1.1.1.1.3">ℎ</ci><apply id="S4.I10.i1.p3.13.m13.1.1.1.1.1.1.1.1.1.cmml" xref="S4.I10.i1.p3.13.m13.1.1.1.1.1.1.1.1"><csymbol cd="ambiguous" id="S4.I10.i1.p3.13.m13.1.1.1.1.1.1.1.1.1.1.cmml" xref="S4.I10.i1.p3.13.m13.1.1.1.1.1.1.1.1">subscript</csymbol><ci id="S4.I10.i1.p3.13.m13.1.1.1.1.1.1.1.1.1.2.cmml" xref="S4.I10.i1.p3.13.m13.1.1.1.1.1.1.1.1.1.2">𝑢</ci><cn id="S4.I10.i1.p3.13.m13.1.1.1.1.1.1.1.1.1.3.cmml" type="integer" xref="S4.I10.i1.p3.13.m13.1.1.1.1.1.1.1.1.1.3">1</cn></apply></apply><apply id="S4.I10.i1.p3.13.m13.2.2.2.2.2.2.cmml" xref="S4.I10.i1.p3.13.m13.2.2.2.2.2.2"><times id="S4.I10.i1.p3.13.m13.2.2.2.2.2.2.2.cmml" xref="S4.I10.i1.p3.13.m13.2.2.2.2.2.2.2"></times><ci id="S4.I10.i1.p3.13.m13.2.2.2.2.2.2.3.cmml" xref="S4.I10.i1.p3.13.m13.2.2.2.2.2.2.3">ℓ</ci><apply id="S4.I10.i1.p3.13.m13.2.2.2.2.2.2.1.1.1.cmml" xref="S4.I10.i1.p3.13.m13.2.2.2.2.2.2.1.1"><csymbol cd="ambiguous" id="S4.I10.i1.p3.13.m13.2.2.2.2.2.2.1.1.1.1.cmml" xref="S4.I10.i1.p3.13.m13.2.2.2.2.2.2.1.1">subscript</csymbol><ci id="S4.I10.i1.p3.13.m13.2.2.2.2.2.2.1.1.1.2.cmml" xref="S4.I10.i1.p3.13.m13.2.2.2.2.2.2.1.1.1.2">𝑢</ci><cn id="S4.I10.i1.p3.13.m13.2.2.2.2.2.2.1.1.1.3.cmml" type="integer" xref="S4.I10.i1.p3.13.m13.2.2.2.2.2.2.1.1.1.3">4</cn></apply></apply></interval></apply><apply id="S4.I10.i1.p3.13.m13.2.2.4.cmml" xref="S4.I10.i1.p3.13.m13.2.2.4"><setdiff id="S4.I10.i1.p3.13.m13.2.2.4.1.cmml" xref="S4.I10.i1.p3.13.m13.2.2.4.1"></setdiff><ci id="S4.I10.i1.p3.13.m13.2.2.4.2.cmml" xref="S4.I10.i1.p3.13.m13.2.2.4.2">𝑇</ci><apply id="S4.I10.i1.p3.13.m13.2.2.4.3.cmml" xref="S4.I10.i1.p3.13.m13.2.2.4.3"><csymbol cd="ambiguous" id="S4.I10.i1.p3.13.m13.2.2.4.3.1.cmml" xref="S4.I10.i1.p3.13.m13.2.2.4.3">subscript</csymbol><ci id="S4.I10.i1.p3.13.m13.2.2.4.3.2.cmml" xref="S4.I10.i1.p3.13.m13.2.2.4.3.2">𝑇</ci><ci id="S4.I10.i1.p3.13.m13.2.2.4.3.3.cmml" xref="S4.I10.i1.p3.13.m13.2.2.4.3.3">𝑥</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.I10.i1.p3.13.m13.2c">\text{LCA}(h(u_{1}),\ell(u_{4}))\in T\setminus T_{x}</annotation><annotation encoding="application/x-llamapun" id="S4.I10.i1.p3.13.m13.2d">LCA ( italic_h ( italic_u start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) , roman_ℓ ( italic_u start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ) ) ∈ italic_T ∖ italic_T start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT</annotation></semantics></math>, so <math alttext="\ell(u_{4})\in T\setminus T_{x}" class="ltx_Math" display="inline" id="S4.I10.i1.p3.14.m14.1"><semantics id="S4.I10.i1.p3.14.m14.1a"><mrow id="S4.I10.i1.p3.14.m14.1.1" xref="S4.I10.i1.p3.14.m14.1.1.cmml"><mrow id="S4.I10.i1.p3.14.m14.1.1.1" xref="S4.I10.i1.p3.14.m14.1.1.1.cmml"><mi id="S4.I10.i1.p3.14.m14.1.1.1.3" mathvariant="normal" xref="S4.I10.i1.p3.14.m14.1.1.1.3.cmml">ℓ</mi><mo id="S4.I10.i1.p3.14.m14.1.1.1.2" xref="S4.I10.i1.p3.14.m14.1.1.1.2.cmml">⁢</mo><mrow id="S4.I10.i1.p3.14.m14.1.1.1.1.1" xref="S4.I10.i1.p3.14.m14.1.1.1.1.1.1.cmml"><mo id="S4.I10.i1.p3.14.m14.1.1.1.1.1.2" stretchy="false" xref="S4.I10.i1.p3.14.m14.1.1.1.1.1.1.cmml">(</mo><msub id="S4.I10.i1.p3.14.m14.1.1.1.1.1.1" xref="S4.I10.i1.p3.14.m14.1.1.1.1.1.1.cmml"><mi id="S4.I10.i1.p3.14.m14.1.1.1.1.1.1.2" xref="S4.I10.i1.p3.14.m14.1.1.1.1.1.1.2.cmml">u</mi><mn id="S4.I10.i1.p3.14.m14.1.1.1.1.1.1.3" xref="S4.I10.i1.p3.14.m14.1.1.1.1.1.1.3.cmml">4</mn></msub><mo id="S4.I10.i1.p3.14.m14.1.1.1.1.1.3" stretchy="false" xref="S4.I10.i1.p3.14.m14.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S4.I10.i1.p3.14.m14.1.1.2" xref="S4.I10.i1.p3.14.m14.1.1.2.cmml">∈</mo><mrow id="S4.I10.i1.p3.14.m14.1.1.3" xref="S4.I10.i1.p3.14.m14.1.1.3.cmml"><mi id="S4.I10.i1.p3.14.m14.1.1.3.2" xref="S4.I10.i1.p3.14.m14.1.1.3.2.cmml">T</mi><mo id="S4.I10.i1.p3.14.m14.1.1.3.1" xref="S4.I10.i1.p3.14.m14.1.1.3.1.cmml">∖</mo><msub id="S4.I10.i1.p3.14.m14.1.1.3.3" xref="S4.I10.i1.p3.14.m14.1.1.3.3.cmml"><mi id="S4.I10.i1.p3.14.m14.1.1.3.3.2" xref="S4.I10.i1.p3.14.m14.1.1.3.3.2.cmml">T</mi><mi id="S4.I10.i1.p3.14.m14.1.1.3.3.3" xref="S4.I10.i1.p3.14.m14.1.1.3.3.3.cmml">x</mi></msub></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.I10.i1.p3.14.m14.1b"><apply id="S4.I10.i1.p3.14.m14.1.1.cmml" xref="S4.I10.i1.p3.14.m14.1.1"><in id="S4.I10.i1.p3.14.m14.1.1.2.cmml" xref="S4.I10.i1.p3.14.m14.1.1.2"></in><apply id="S4.I10.i1.p3.14.m14.1.1.1.cmml" xref="S4.I10.i1.p3.14.m14.1.1.1"><times id="S4.I10.i1.p3.14.m14.1.1.1.2.cmml" xref="S4.I10.i1.p3.14.m14.1.1.1.2"></times><ci id="S4.I10.i1.p3.14.m14.1.1.1.3.cmml" xref="S4.I10.i1.p3.14.m14.1.1.1.3">ℓ</ci><apply id="S4.I10.i1.p3.14.m14.1.1.1.1.1.1.cmml" xref="S4.I10.i1.p3.14.m14.1.1.1.1.1"><csymbol cd="ambiguous" id="S4.I10.i1.p3.14.m14.1.1.1.1.1.1.1.cmml" xref="S4.I10.i1.p3.14.m14.1.1.1.1.1">subscript</csymbol><ci id="S4.I10.i1.p3.14.m14.1.1.1.1.1.1.2.cmml" xref="S4.I10.i1.p3.14.m14.1.1.1.1.1.1.2">𝑢</ci><cn id="S4.I10.i1.p3.14.m14.1.1.1.1.1.1.3.cmml" type="integer" xref="S4.I10.i1.p3.14.m14.1.1.1.1.1.1.3">4</cn></apply></apply><apply id="S4.I10.i1.p3.14.m14.1.1.3.cmml" xref="S4.I10.i1.p3.14.m14.1.1.3"><setdiff id="S4.I10.i1.p3.14.m14.1.1.3.1.cmml" xref="S4.I10.i1.p3.14.m14.1.1.3.1"></setdiff><ci id="S4.I10.i1.p3.14.m14.1.1.3.2.cmml" xref="S4.I10.i1.p3.14.m14.1.1.3.2">𝑇</ci><apply id="S4.I10.i1.p3.14.m14.1.1.3.3.cmml" xref="S4.I10.i1.p3.14.m14.1.1.3.3"><csymbol cd="ambiguous" id="S4.I10.i1.p3.14.m14.1.1.3.3.1.cmml" xref="S4.I10.i1.p3.14.m14.1.1.3.3">subscript</csymbol><ci id="S4.I10.i1.p3.14.m14.1.1.3.3.2.cmml" xref="S4.I10.i1.p3.14.m14.1.1.3.3.2">𝑇</ci><ci id="S4.I10.i1.p3.14.m14.1.1.3.3.3.cmml" xref="S4.I10.i1.p3.14.m14.1.1.3.3.3">𝑥</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.I10.i1.p3.14.m14.1c">\ell(u_{4})\in T\setminus T_{x}</annotation><annotation encoding="application/x-llamapun" id="S4.I10.i1.p3.14.m14.1d">roman_ℓ ( italic_u start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ) ∈ italic_T ∖ italic_T start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT</annotation></semantics></math>. Note that <math alttext="u_{3}\neq\{a,b\}" class="ltx_Math" display="inline" id="S4.I10.i1.p3.15.m15.2"><semantics id="S4.I10.i1.p3.15.m15.2a"><mrow id="S4.I10.i1.p3.15.m15.2.3" xref="S4.I10.i1.p3.15.m15.2.3.cmml"><msub id="S4.I10.i1.p3.15.m15.2.3.2" xref="S4.I10.i1.p3.15.m15.2.3.2.cmml"><mi id="S4.I10.i1.p3.15.m15.2.3.2.2" xref="S4.I10.i1.p3.15.m15.2.3.2.2.cmml">u</mi><mn id="S4.I10.i1.p3.15.m15.2.3.2.3" xref="S4.I10.i1.p3.15.m15.2.3.2.3.cmml">3</mn></msub><mo id="S4.I10.i1.p3.15.m15.2.3.1" xref="S4.I10.i1.p3.15.m15.2.3.1.cmml">≠</mo><mrow id="S4.I10.i1.p3.15.m15.2.3.3.2" xref="S4.I10.i1.p3.15.m15.2.3.3.1.cmml"><mo id="S4.I10.i1.p3.15.m15.2.3.3.2.1" stretchy="false" xref="S4.I10.i1.p3.15.m15.2.3.3.1.cmml">{</mo><mi id="S4.I10.i1.p3.15.m15.1.1" xref="S4.I10.i1.p3.15.m15.1.1.cmml">a</mi><mo id="S4.I10.i1.p3.15.m15.2.3.3.2.2" xref="S4.I10.i1.p3.15.m15.2.3.3.1.cmml">,</mo><mi id="S4.I10.i1.p3.15.m15.2.2" xref="S4.I10.i1.p3.15.m15.2.2.cmml">b</mi><mo id="S4.I10.i1.p3.15.m15.2.3.3.2.3" stretchy="false" xref="S4.I10.i1.p3.15.m15.2.3.3.1.cmml">}</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.I10.i1.p3.15.m15.2b"><apply id="S4.I10.i1.p3.15.m15.2.3.cmml" xref="S4.I10.i1.p3.15.m15.2.3"><neq id="S4.I10.i1.p3.15.m15.2.3.1.cmml" xref="S4.I10.i1.p3.15.m15.2.3.1"></neq><apply id="S4.I10.i1.p3.15.m15.2.3.2.cmml" xref="S4.I10.i1.p3.15.m15.2.3.2"><csymbol cd="ambiguous" id="S4.I10.i1.p3.15.m15.2.3.2.1.cmml" xref="S4.I10.i1.p3.15.m15.2.3.2">subscript</csymbol><ci id="S4.I10.i1.p3.15.m15.2.3.2.2.cmml" xref="S4.I10.i1.p3.15.m15.2.3.2.2">𝑢</ci><cn id="S4.I10.i1.p3.15.m15.2.3.2.3.cmml" type="integer" xref="S4.I10.i1.p3.15.m15.2.3.2.3">3</cn></apply><set id="S4.I10.i1.p3.15.m15.2.3.3.1.cmml" xref="S4.I10.i1.p3.15.m15.2.3.3.2"><ci id="S4.I10.i1.p3.15.m15.1.1.cmml" xref="S4.I10.i1.p3.15.m15.1.1">𝑎</ci><ci id="S4.I10.i1.p3.15.m15.2.2.cmml" xref="S4.I10.i1.p3.15.m15.2.2">𝑏</ci></set></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.I10.i1.p3.15.m15.2c">u_{3}\neq\{a,b\}</annotation><annotation encoding="application/x-llamapun" id="S4.I10.i1.p3.15.m15.2d">italic_u start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ≠ { italic_a , italic_b }</annotation></semantics></math> since <math alttext="h(u_{3})=h(u_{1})\in T(u)" class="ltx_Math" display="inline" id="S4.I10.i1.p3.16.m16.3"><semantics id="S4.I10.i1.p3.16.m16.3a"><mrow id="S4.I10.i1.p3.16.m16.3.3" xref="S4.I10.i1.p3.16.m16.3.3.cmml"><mrow id="S4.I10.i1.p3.16.m16.2.2.1" xref="S4.I10.i1.p3.16.m16.2.2.1.cmml"><mi id="S4.I10.i1.p3.16.m16.2.2.1.3" xref="S4.I10.i1.p3.16.m16.2.2.1.3.cmml">h</mi><mo id="S4.I10.i1.p3.16.m16.2.2.1.2" xref="S4.I10.i1.p3.16.m16.2.2.1.2.cmml">⁢</mo><mrow id="S4.I10.i1.p3.16.m16.2.2.1.1.1" xref="S4.I10.i1.p3.16.m16.2.2.1.1.1.1.cmml"><mo id="S4.I10.i1.p3.16.m16.2.2.1.1.1.2" stretchy="false" xref="S4.I10.i1.p3.16.m16.2.2.1.1.1.1.cmml">(</mo><msub id="S4.I10.i1.p3.16.m16.2.2.1.1.1.1" xref="S4.I10.i1.p3.16.m16.2.2.1.1.1.1.cmml"><mi id="S4.I10.i1.p3.16.m16.2.2.1.1.1.1.2" xref="S4.I10.i1.p3.16.m16.2.2.1.1.1.1.2.cmml">u</mi><mn id="S4.I10.i1.p3.16.m16.2.2.1.1.1.1.3" xref="S4.I10.i1.p3.16.m16.2.2.1.1.1.1.3.cmml">3</mn></msub><mo id="S4.I10.i1.p3.16.m16.2.2.1.1.1.3" stretchy="false" xref="S4.I10.i1.p3.16.m16.2.2.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S4.I10.i1.p3.16.m16.3.3.4" xref="S4.I10.i1.p3.16.m16.3.3.4.cmml">=</mo><mrow id="S4.I10.i1.p3.16.m16.3.3.2" xref="S4.I10.i1.p3.16.m16.3.3.2.cmml"><mi id="S4.I10.i1.p3.16.m16.3.3.2.3" xref="S4.I10.i1.p3.16.m16.3.3.2.3.cmml">h</mi><mo id="S4.I10.i1.p3.16.m16.3.3.2.2" xref="S4.I10.i1.p3.16.m16.3.3.2.2.cmml">⁢</mo><mrow id="S4.I10.i1.p3.16.m16.3.3.2.1.1" xref="S4.I10.i1.p3.16.m16.3.3.2.1.1.1.cmml"><mo id="S4.I10.i1.p3.16.m16.3.3.2.1.1.2" stretchy="false" xref="S4.I10.i1.p3.16.m16.3.3.2.1.1.1.cmml">(</mo><msub id="S4.I10.i1.p3.16.m16.3.3.2.1.1.1" xref="S4.I10.i1.p3.16.m16.3.3.2.1.1.1.cmml"><mi id="S4.I10.i1.p3.16.m16.3.3.2.1.1.1.2" xref="S4.I10.i1.p3.16.m16.3.3.2.1.1.1.2.cmml">u</mi><mn id="S4.I10.i1.p3.16.m16.3.3.2.1.1.1.3" xref="S4.I10.i1.p3.16.m16.3.3.2.1.1.1.3.cmml">1</mn></msub><mo id="S4.I10.i1.p3.16.m16.3.3.2.1.1.3" stretchy="false" xref="S4.I10.i1.p3.16.m16.3.3.2.1.1.1.cmml">)</mo></mrow></mrow><mo id="S4.I10.i1.p3.16.m16.3.3.5" xref="S4.I10.i1.p3.16.m16.3.3.5.cmml">∈</mo><mrow id="S4.I10.i1.p3.16.m16.3.3.6" xref="S4.I10.i1.p3.16.m16.3.3.6.cmml"><mi id="S4.I10.i1.p3.16.m16.3.3.6.2" xref="S4.I10.i1.p3.16.m16.3.3.6.2.cmml">T</mi><mo id="S4.I10.i1.p3.16.m16.3.3.6.1" xref="S4.I10.i1.p3.16.m16.3.3.6.1.cmml">⁢</mo><mrow id="S4.I10.i1.p3.16.m16.3.3.6.3.2" xref="S4.I10.i1.p3.16.m16.3.3.6.cmml"><mo id="S4.I10.i1.p3.16.m16.3.3.6.3.2.1" stretchy="false" xref="S4.I10.i1.p3.16.m16.3.3.6.cmml">(</mo><mi id="S4.I10.i1.p3.16.m16.1.1" xref="S4.I10.i1.p3.16.m16.1.1.cmml">u</mi><mo id="S4.I10.i1.p3.16.m16.3.3.6.3.2.2" stretchy="false" xref="S4.I10.i1.p3.16.m16.3.3.6.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.I10.i1.p3.16.m16.3b"><apply id="S4.I10.i1.p3.16.m16.3.3.cmml" xref="S4.I10.i1.p3.16.m16.3.3"><and id="S4.I10.i1.p3.16.m16.3.3a.cmml" xref="S4.I10.i1.p3.16.m16.3.3"></and><apply id="S4.I10.i1.p3.16.m16.3.3b.cmml" xref="S4.I10.i1.p3.16.m16.3.3"><eq id="S4.I10.i1.p3.16.m16.3.3.4.cmml" xref="S4.I10.i1.p3.16.m16.3.3.4"></eq><apply id="S4.I10.i1.p3.16.m16.2.2.1.cmml" xref="S4.I10.i1.p3.16.m16.2.2.1"><times id="S4.I10.i1.p3.16.m16.2.2.1.2.cmml" xref="S4.I10.i1.p3.16.m16.2.2.1.2"></times><ci id="S4.I10.i1.p3.16.m16.2.2.1.3.cmml" xref="S4.I10.i1.p3.16.m16.2.2.1.3">ℎ</ci><apply id="S4.I10.i1.p3.16.m16.2.2.1.1.1.1.cmml" xref="S4.I10.i1.p3.16.m16.2.2.1.1.1"><csymbol cd="ambiguous" id="S4.I10.i1.p3.16.m16.2.2.1.1.1.1.1.cmml" xref="S4.I10.i1.p3.16.m16.2.2.1.1.1">subscript</csymbol><ci id="S4.I10.i1.p3.16.m16.2.2.1.1.1.1.2.cmml" xref="S4.I10.i1.p3.16.m16.2.2.1.1.1.1.2">𝑢</ci><cn id="S4.I10.i1.p3.16.m16.2.2.1.1.1.1.3.cmml" type="integer" xref="S4.I10.i1.p3.16.m16.2.2.1.1.1.1.3">3</cn></apply></apply><apply id="S4.I10.i1.p3.16.m16.3.3.2.cmml" xref="S4.I10.i1.p3.16.m16.3.3.2"><times id="S4.I10.i1.p3.16.m16.3.3.2.2.cmml" xref="S4.I10.i1.p3.16.m16.3.3.2.2"></times><ci id="S4.I10.i1.p3.16.m16.3.3.2.3.cmml" xref="S4.I10.i1.p3.16.m16.3.3.2.3">ℎ</ci><apply id="S4.I10.i1.p3.16.m16.3.3.2.1.1.1.cmml" xref="S4.I10.i1.p3.16.m16.3.3.2.1.1"><csymbol cd="ambiguous" id="S4.I10.i1.p3.16.m16.3.3.2.1.1.1.1.cmml" xref="S4.I10.i1.p3.16.m16.3.3.2.1.1">subscript</csymbol><ci id="S4.I10.i1.p3.16.m16.3.3.2.1.1.1.2.cmml" xref="S4.I10.i1.p3.16.m16.3.3.2.1.1.1.2">𝑢</ci><cn id="S4.I10.i1.p3.16.m16.3.3.2.1.1.1.3.cmml" type="integer" xref="S4.I10.i1.p3.16.m16.3.3.2.1.1.1.3">1</cn></apply></apply></apply><apply id="S4.I10.i1.p3.16.m16.3.3c.cmml" xref="S4.I10.i1.p3.16.m16.3.3"><in id="S4.I10.i1.p3.16.m16.3.3.5.cmml" xref="S4.I10.i1.p3.16.m16.3.3.5"></in><share href="https://arxiv.org/html/2503.00712v1#S4.I10.i1.p3.16.m16.3.3.2.cmml" id="S4.I10.i1.p3.16.m16.3.3d.cmml" xref="S4.I10.i1.p3.16.m16.3.3"></share><apply id="S4.I10.i1.p3.16.m16.3.3.6.cmml" xref="S4.I10.i1.p3.16.m16.3.3.6"><times id="S4.I10.i1.p3.16.m16.3.3.6.1.cmml" xref="S4.I10.i1.p3.16.m16.3.3.6.1"></times><ci id="S4.I10.i1.p3.16.m16.3.3.6.2.cmml" xref="S4.I10.i1.p3.16.m16.3.3.6.2">𝑇</ci><ci id="S4.I10.i1.p3.16.m16.1.1.cmml" xref="S4.I10.i1.p3.16.m16.1.1">𝑢</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.I10.i1.p3.16.m16.3c">h(u_{3})=h(u_{1})\in T(u)</annotation><annotation encoding="application/x-llamapun" id="S4.I10.i1.p3.16.m16.3d">italic_h ( italic_u start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ) = italic_h ( italic_u start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) ∈ italic_T ( italic_u )</annotation></semantics></math>, and <math alttext="h(a)=h(b)\in T\setminus T_{x}" class="ltx_Math" display="inline" id="S4.I10.i1.p3.17.m17.2"><semantics id="S4.I10.i1.p3.17.m17.2a"><mrow id="S4.I10.i1.p3.17.m17.2.3" xref="S4.I10.i1.p3.17.m17.2.3.cmml"><mrow id="S4.I10.i1.p3.17.m17.2.3.2" xref="S4.I10.i1.p3.17.m17.2.3.2.cmml"><mi id="S4.I10.i1.p3.17.m17.2.3.2.2" xref="S4.I10.i1.p3.17.m17.2.3.2.2.cmml">h</mi><mo id="S4.I10.i1.p3.17.m17.2.3.2.1" xref="S4.I10.i1.p3.17.m17.2.3.2.1.cmml">⁢</mo><mrow id="S4.I10.i1.p3.17.m17.2.3.2.3.2" xref="S4.I10.i1.p3.17.m17.2.3.2.cmml"><mo id="S4.I10.i1.p3.17.m17.2.3.2.3.2.1" stretchy="false" xref="S4.I10.i1.p3.17.m17.2.3.2.cmml">(</mo><mi id="S4.I10.i1.p3.17.m17.1.1" xref="S4.I10.i1.p3.17.m17.1.1.cmml">a</mi><mo id="S4.I10.i1.p3.17.m17.2.3.2.3.2.2" stretchy="false" xref="S4.I10.i1.p3.17.m17.2.3.2.cmml">)</mo></mrow></mrow><mo id="S4.I10.i1.p3.17.m17.2.3.3" xref="S4.I10.i1.p3.17.m17.2.3.3.cmml">=</mo><mrow id="S4.I10.i1.p3.17.m17.2.3.4" xref="S4.I10.i1.p3.17.m17.2.3.4.cmml"><mi id="S4.I10.i1.p3.17.m17.2.3.4.2" xref="S4.I10.i1.p3.17.m17.2.3.4.2.cmml">h</mi><mo id="S4.I10.i1.p3.17.m17.2.3.4.1" xref="S4.I10.i1.p3.17.m17.2.3.4.1.cmml">⁢</mo><mrow id="S4.I10.i1.p3.17.m17.2.3.4.3.2" xref="S4.I10.i1.p3.17.m17.2.3.4.cmml"><mo id="S4.I10.i1.p3.17.m17.2.3.4.3.2.1" stretchy="false" xref="S4.I10.i1.p3.17.m17.2.3.4.cmml">(</mo><mi id="S4.I10.i1.p3.17.m17.2.2" xref="S4.I10.i1.p3.17.m17.2.2.cmml">b</mi><mo id="S4.I10.i1.p3.17.m17.2.3.4.3.2.2" stretchy="false" xref="S4.I10.i1.p3.17.m17.2.3.4.cmml">)</mo></mrow></mrow><mo id="S4.I10.i1.p3.17.m17.2.3.5" xref="S4.I10.i1.p3.17.m17.2.3.5.cmml">∈</mo><mrow id="S4.I10.i1.p3.17.m17.2.3.6" xref="S4.I10.i1.p3.17.m17.2.3.6.cmml"><mi id="S4.I10.i1.p3.17.m17.2.3.6.2" xref="S4.I10.i1.p3.17.m17.2.3.6.2.cmml">T</mi><mo id="S4.I10.i1.p3.17.m17.2.3.6.1" xref="S4.I10.i1.p3.17.m17.2.3.6.1.cmml">∖</mo><msub id="S4.I10.i1.p3.17.m17.2.3.6.3" xref="S4.I10.i1.p3.17.m17.2.3.6.3.cmml"><mi id="S4.I10.i1.p3.17.m17.2.3.6.3.2" xref="S4.I10.i1.p3.17.m17.2.3.6.3.2.cmml">T</mi><mi id="S4.I10.i1.p3.17.m17.2.3.6.3.3" xref="S4.I10.i1.p3.17.m17.2.3.6.3.3.cmml">x</mi></msub></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.I10.i1.p3.17.m17.2b"><apply id="S4.I10.i1.p3.17.m17.2.3.cmml" xref="S4.I10.i1.p3.17.m17.2.3"><and id="S4.I10.i1.p3.17.m17.2.3a.cmml" xref="S4.I10.i1.p3.17.m17.2.3"></and><apply id="S4.I10.i1.p3.17.m17.2.3b.cmml" xref="S4.I10.i1.p3.17.m17.2.3"><eq id="S4.I10.i1.p3.17.m17.2.3.3.cmml" xref="S4.I10.i1.p3.17.m17.2.3.3"></eq><apply id="S4.I10.i1.p3.17.m17.2.3.2.cmml" xref="S4.I10.i1.p3.17.m17.2.3.2"><times id="S4.I10.i1.p3.17.m17.2.3.2.1.cmml" xref="S4.I10.i1.p3.17.m17.2.3.2.1"></times><ci id="S4.I10.i1.p3.17.m17.2.3.2.2.cmml" xref="S4.I10.i1.p3.17.m17.2.3.2.2">ℎ</ci><ci id="S4.I10.i1.p3.17.m17.1.1.cmml" xref="S4.I10.i1.p3.17.m17.1.1">𝑎</ci></apply><apply id="S4.I10.i1.p3.17.m17.2.3.4.cmml" xref="S4.I10.i1.p3.17.m17.2.3.4"><times id="S4.I10.i1.p3.17.m17.2.3.4.1.cmml" xref="S4.I10.i1.p3.17.m17.2.3.4.1"></times><ci id="S4.I10.i1.p3.17.m17.2.3.4.2.cmml" xref="S4.I10.i1.p3.17.m17.2.3.4.2">ℎ</ci><ci id="S4.I10.i1.p3.17.m17.2.2.cmml" xref="S4.I10.i1.p3.17.m17.2.2">𝑏</ci></apply></apply><apply id="S4.I10.i1.p3.17.m17.2.3c.cmml" xref="S4.I10.i1.p3.17.m17.2.3"><in id="S4.I10.i1.p3.17.m17.2.3.5.cmml" xref="S4.I10.i1.p3.17.m17.2.3.5"></in><share href="https://arxiv.org/html/2503.00712v1#S4.I10.i1.p3.17.m17.2.3.4.cmml" id="S4.I10.i1.p3.17.m17.2.3d.cmml" xref="S4.I10.i1.p3.17.m17.2.3"></share><apply id="S4.I10.i1.p3.17.m17.2.3.6.cmml" xref="S4.I10.i1.p3.17.m17.2.3.6"><setdiff id="S4.I10.i1.p3.17.m17.2.3.6.1.cmml" xref="S4.I10.i1.p3.17.m17.2.3.6.1"></setdiff><ci id="S4.I10.i1.p3.17.m17.2.3.6.2.cmml" xref="S4.I10.i1.p3.17.m17.2.3.6.2">𝑇</ci><apply id="S4.I10.i1.p3.17.m17.2.3.6.3.cmml" xref="S4.I10.i1.p3.17.m17.2.3.6.3"><csymbol cd="ambiguous" id="S4.I10.i1.p3.17.m17.2.3.6.3.1.cmml" xref="S4.I10.i1.p3.17.m17.2.3.6.3">subscript</csymbol><ci id="S4.I10.i1.p3.17.m17.2.3.6.3.2.cmml" xref="S4.I10.i1.p3.17.m17.2.3.6.3.2">𝑇</ci><ci id="S4.I10.i1.p3.17.m17.2.3.6.3.3.cmml" xref="S4.I10.i1.p3.17.m17.2.3.6.3.3">𝑥</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.I10.i1.p3.17.m17.2c">h(a)=h(b)\in T\setminus T_{x}</annotation><annotation encoding="application/x-llamapun" id="S4.I10.i1.p3.17.m17.2d">italic_h ( italic_a ) = italic_h ( italic_b ) ∈ italic_T ∖ italic_T start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT</annotation></semantics></math>. Furthermore, <math alttext="u_{4}\neq\{a,b\}" class="ltx_Math" display="inline" id="S4.I10.i1.p3.18.m18.2"><semantics id="S4.I10.i1.p3.18.m18.2a"><mrow id="S4.I10.i1.p3.18.m18.2.3" xref="S4.I10.i1.p3.18.m18.2.3.cmml"><msub id="S4.I10.i1.p3.18.m18.2.3.2" xref="S4.I10.i1.p3.18.m18.2.3.2.cmml"><mi id="S4.I10.i1.p3.18.m18.2.3.2.2" xref="S4.I10.i1.p3.18.m18.2.3.2.2.cmml">u</mi><mn id="S4.I10.i1.p3.18.m18.2.3.2.3" xref="S4.I10.i1.p3.18.m18.2.3.2.3.cmml">4</mn></msub><mo id="S4.I10.i1.p3.18.m18.2.3.1" xref="S4.I10.i1.p3.18.m18.2.3.1.cmml">≠</mo><mrow id="S4.I10.i1.p3.18.m18.2.3.3.2" xref="S4.I10.i1.p3.18.m18.2.3.3.1.cmml"><mo id="S4.I10.i1.p3.18.m18.2.3.3.2.1" stretchy="false" xref="S4.I10.i1.p3.18.m18.2.3.3.1.cmml">{</mo><mi id="S4.I10.i1.p3.18.m18.1.1" xref="S4.I10.i1.p3.18.m18.1.1.cmml">a</mi><mo id="S4.I10.i1.p3.18.m18.2.3.3.2.2" xref="S4.I10.i1.p3.18.m18.2.3.3.1.cmml">,</mo><mi id="S4.I10.i1.p3.18.m18.2.2" xref="S4.I10.i1.p3.18.m18.2.2.cmml">b</mi><mo id="S4.I10.i1.p3.18.m18.2.3.3.2.3" stretchy="false" xref="S4.I10.i1.p3.18.m18.2.3.3.1.cmml">}</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.I10.i1.p3.18.m18.2b"><apply id="S4.I10.i1.p3.18.m18.2.3.cmml" xref="S4.I10.i1.p3.18.m18.2.3"><neq id="S4.I10.i1.p3.18.m18.2.3.1.cmml" xref="S4.I10.i1.p3.18.m18.2.3.1"></neq><apply id="S4.I10.i1.p3.18.m18.2.3.2.cmml" xref="S4.I10.i1.p3.18.m18.2.3.2"><csymbol cd="ambiguous" id="S4.I10.i1.p3.18.m18.2.3.2.1.cmml" xref="S4.I10.i1.p3.18.m18.2.3.2">subscript</csymbol><ci id="S4.I10.i1.p3.18.m18.2.3.2.2.cmml" xref="S4.I10.i1.p3.18.m18.2.3.2.2">𝑢</ci><cn id="S4.I10.i1.p3.18.m18.2.3.2.3.cmml" type="integer" xref="S4.I10.i1.p3.18.m18.2.3.2.3">4</cn></apply><set id="S4.I10.i1.p3.18.m18.2.3.3.1.cmml" xref="S4.I10.i1.p3.18.m18.2.3.3.2"><ci id="S4.I10.i1.p3.18.m18.1.1.cmml" xref="S4.I10.i1.p3.18.m18.1.1">𝑎</ci><ci id="S4.I10.i1.p3.18.m18.2.2.cmml" xref="S4.I10.i1.p3.18.m18.2.2">𝑏</ci></set></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.I10.i1.p3.18.m18.2c">u_{4}\neq\{a,b\}</annotation><annotation encoding="application/x-llamapun" id="S4.I10.i1.p3.18.m18.2d">italic_u start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ≠ { italic_a , italic_b }</annotation></semantics></math>, since <math alttext="\ell(u_{4})\in T\setminus T_{x}" class="ltx_Math" display="inline" id="S4.I10.i1.p3.19.m19.1"><semantics id="S4.I10.i1.p3.19.m19.1a"><mrow id="S4.I10.i1.p3.19.m19.1.1" xref="S4.I10.i1.p3.19.m19.1.1.cmml"><mrow id="S4.I10.i1.p3.19.m19.1.1.1" xref="S4.I10.i1.p3.19.m19.1.1.1.cmml"><mi id="S4.I10.i1.p3.19.m19.1.1.1.3" mathvariant="normal" xref="S4.I10.i1.p3.19.m19.1.1.1.3.cmml">ℓ</mi><mo id="S4.I10.i1.p3.19.m19.1.1.1.2" xref="S4.I10.i1.p3.19.m19.1.1.1.2.cmml">⁢</mo><mrow id="S4.I10.i1.p3.19.m19.1.1.1.1.1" xref="S4.I10.i1.p3.19.m19.1.1.1.1.1.1.cmml"><mo id="S4.I10.i1.p3.19.m19.1.1.1.1.1.2" stretchy="false" xref="S4.I10.i1.p3.19.m19.1.1.1.1.1.1.cmml">(</mo><msub id="S4.I10.i1.p3.19.m19.1.1.1.1.1.1" xref="S4.I10.i1.p3.19.m19.1.1.1.1.1.1.cmml"><mi id="S4.I10.i1.p3.19.m19.1.1.1.1.1.1.2" xref="S4.I10.i1.p3.19.m19.1.1.1.1.1.1.2.cmml">u</mi><mn id="S4.I10.i1.p3.19.m19.1.1.1.1.1.1.3" xref="S4.I10.i1.p3.19.m19.1.1.1.1.1.1.3.cmml">4</mn></msub><mo id="S4.I10.i1.p3.19.m19.1.1.1.1.1.3" stretchy="false" xref="S4.I10.i1.p3.19.m19.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S4.I10.i1.p3.19.m19.1.1.2" xref="S4.I10.i1.p3.19.m19.1.1.2.cmml">∈</mo><mrow id="S4.I10.i1.p3.19.m19.1.1.3" xref="S4.I10.i1.p3.19.m19.1.1.3.cmml"><mi id="S4.I10.i1.p3.19.m19.1.1.3.2" xref="S4.I10.i1.p3.19.m19.1.1.3.2.cmml">T</mi><mo id="S4.I10.i1.p3.19.m19.1.1.3.1" xref="S4.I10.i1.p3.19.m19.1.1.3.1.cmml">∖</mo><msub id="S4.I10.i1.p3.19.m19.1.1.3.3" xref="S4.I10.i1.p3.19.m19.1.1.3.3.cmml"><mi id="S4.I10.i1.p3.19.m19.1.1.3.3.2" xref="S4.I10.i1.p3.19.m19.1.1.3.3.2.cmml">T</mi><mi id="S4.I10.i1.p3.19.m19.1.1.3.3.3" xref="S4.I10.i1.p3.19.m19.1.1.3.3.3.cmml">x</mi></msub></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.I10.i1.p3.19.m19.1b"><apply id="S4.I10.i1.p3.19.m19.1.1.cmml" xref="S4.I10.i1.p3.19.m19.1.1"><in id="S4.I10.i1.p3.19.m19.1.1.2.cmml" xref="S4.I10.i1.p3.19.m19.1.1.2"></in><apply id="S4.I10.i1.p3.19.m19.1.1.1.cmml" xref="S4.I10.i1.p3.19.m19.1.1.1"><times id="S4.I10.i1.p3.19.m19.1.1.1.2.cmml" xref="S4.I10.i1.p3.19.m19.1.1.1.2"></times><ci id="S4.I10.i1.p3.19.m19.1.1.1.3.cmml" xref="S4.I10.i1.p3.19.m19.1.1.1.3">ℓ</ci><apply id="S4.I10.i1.p3.19.m19.1.1.1.1.1.1.cmml" xref="S4.I10.i1.p3.19.m19.1.1.1.1.1"><csymbol cd="ambiguous" id="S4.I10.i1.p3.19.m19.1.1.1.1.1.1.1.cmml" xref="S4.I10.i1.p3.19.m19.1.1.1.1.1">subscript</csymbol><ci id="S4.I10.i1.p3.19.m19.1.1.1.1.1.1.2.cmml" xref="S4.I10.i1.p3.19.m19.1.1.1.1.1.1.2">𝑢</ci><cn id="S4.I10.i1.p3.19.m19.1.1.1.1.1.1.3.cmml" type="integer" xref="S4.I10.i1.p3.19.m19.1.1.1.1.1.1.3">4</cn></apply></apply><apply id="S4.I10.i1.p3.19.m19.1.1.3.cmml" xref="S4.I10.i1.p3.19.m19.1.1.3"><setdiff id="S4.I10.i1.p3.19.m19.1.1.3.1.cmml" xref="S4.I10.i1.p3.19.m19.1.1.3.1"></setdiff><ci id="S4.I10.i1.p3.19.m19.1.1.3.2.cmml" xref="S4.I10.i1.p3.19.m19.1.1.3.2">𝑇</ci><apply id="S4.I10.i1.p3.19.m19.1.1.3.3.cmml" xref="S4.I10.i1.p3.19.m19.1.1.3.3"><csymbol cd="ambiguous" id="S4.I10.i1.p3.19.m19.1.1.3.3.1.cmml" xref="S4.I10.i1.p3.19.m19.1.1.3.3">subscript</csymbol><ci id="S4.I10.i1.p3.19.m19.1.1.3.3.2.cmml" xref="S4.I10.i1.p3.19.m19.1.1.3.3.2">𝑇</ci><ci id="S4.I10.i1.p3.19.m19.1.1.3.3.3.cmml" xref="S4.I10.i1.p3.19.m19.1.1.3.3.3">𝑥</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.I10.i1.p3.19.m19.1c">\ell(u_{4})\in T\setminus T_{x}</annotation><annotation encoding="application/x-llamapun" id="S4.I10.i1.p3.19.m19.1d">roman_ℓ ( italic_u start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ) ∈ italic_T ∖ italic_T start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT</annotation></semantics></math>, but <math alttext="\ell(a),\ell(b)\in T_{x}" class="ltx_Math" display="inline" id="S4.I10.i1.p3.20.m20.4"><semantics id="S4.I10.i1.p3.20.m20.4a"><mrow id="S4.I10.i1.p3.20.m20.4.4" xref="S4.I10.i1.p3.20.m20.4.4.cmml"><mrow id="S4.I10.i1.p3.20.m20.4.4.2.2" xref="S4.I10.i1.p3.20.m20.4.4.2.3.cmml"><mrow id="S4.I10.i1.p3.20.m20.3.3.1.1.1" xref="S4.I10.i1.p3.20.m20.3.3.1.1.1.cmml"><mi id="S4.I10.i1.p3.20.m20.3.3.1.1.1.2" mathvariant="normal" xref="S4.I10.i1.p3.20.m20.3.3.1.1.1.2.cmml">ℓ</mi><mo id="S4.I10.i1.p3.20.m20.3.3.1.1.1.1" xref="S4.I10.i1.p3.20.m20.3.3.1.1.1.1.cmml">⁢</mo><mrow id="S4.I10.i1.p3.20.m20.3.3.1.1.1.3.2" xref="S4.I10.i1.p3.20.m20.3.3.1.1.1.cmml"><mo id="S4.I10.i1.p3.20.m20.3.3.1.1.1.3.2.1" stretchy="false" xref="S4.I10.i1.p3.20.m20.3.3.1.1.1.cmml">(</mo><mi id="S4.I10.i1.p3.20.m20.1.1" xref="S4.I10.i1.p3.20.m20.1.1.cmml">a</mi><mo id="S4.I10.i1.p3.20.m20.3.3.1.1.1.3.2.2" stretchy="false" xref="S4.I10.i1.p3.20.m20.3.3.1.1.1.cmml">)</mo></mrow></mrow><mo id="S4.I10.i1.p3.20.m20.4.4.2.2.3" xref="S4.I10.i1.p3.20.m20.4.4.2.3.cmml">,</mo><mrow id="S4.I10.i1.p3.20.m20.4.4.2.2.2" xref="S4.I10.i1.p3.20.m20.4.4.2.2.2.cmml"><mi id="S4.I10.i1.p3.20.m20.4.4.2.2.2.2" mathvariant="normal" xref="S4.I10.i1.p3.20.m20.4.4.2.2.2.2.cmml">ℓ</mi><mo id="S4.I10.i1.p3.20.m20.4.4.2.2.2.1" xref="S4.I10.i1.p3.20.m20.4.4.2.2.2.1.cmml">⁢</mo><mrow id="S4.I10.i1.p3.20.m20.4.4.2.2.2.3.2" xref="S4.I10.i1.p3.20.m20.4.4.2.2.2.cmml"><mo id="S4.I10.i1.p3.20.m20.4.4.2.2.2.3.2.1" stretchy="false" xref="S4.I10.i1.p3.20.m20.4.4.2.2.2.cmml">(</mo><mi id="S4.I10.i1.p3.20.m20.2.2" xref="S4.I10.i1.p3.20.m20.2.2.cmml">b</mi><mo id="S4.I10.i1.p3.20.m20.4.4.2.2.2.3.2.2" stretchy="false" xref="S4.I10.i1.p3.20.m20.4.4.2.2.2.cmml">)</mo></mrow></mrow></mrow><mo id="S4.I10.i1.p3.20.m20.4.4.3" xref="S4.I10.i1.p3.20.m20.4.4.3.cmml">∈</mo><msub id="S4.I10.i1.p3.20.m20.4.4.4" xref="S4.I10.i1.p3.20.m20.4.4.4.cmml"><mi id="S4.I10.i1.p3.20.m20.4.4.4.2" xref="S4.I10.i1.p3.20.m20.4.4.4.2.cmml">T</mi><mi id="S4.I10.i1.p3.20.m20.4.4.4.3" xref="S4.I10.i1.p3.20.m20.4.4.4.3.cmml">x</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S4.I10.i1.p3.20.m20.4b"><apply id="S4.I10.i1.p3.20.m20.4.4.cmml" xref="S4.I10.i1.p3.20.m20.4.4"><in id="S4.I10.i1.p3.20.m20.4.4.3.cmml" xref="S4.I10.i1.p3.20.m20.4.4.3"></in><list id="S4.I10.i1.p3.20.m20.4.4.2.3.cmml" xref="S4.I10.i1.p3.20.m20.4.4.2.2"><apply id="S4.I10.i1.p3.20.m20.3.3.1.1.1.cmml" xref="S4.I10.i1.p3.20.m20.3.3.1.1.1"><times id="S4.I10.i1.p3.20.m20.3.3.1.1.1.1.cmml" xref="S4.I10.i1.p3.20.m20.3.3.1.1.1.1"></times><ci id="S4.I10.i1.p3.20.m20.3.3.1.1.1.2.cmml" xref="S4.I10.i1.p3.20.m20.3.3.1.1.1.2">ℓ</ci><ci id="S4.I10.i1.p3.20.m20.1.1.cmml" xref="S4.I10.i1.p3.20.m20.1.1">𝑎</ci></apply><apply id="S4.I10.i1.p3.20.m20.4.4.2.2.2.cmml" xref="S4.I10.i1.p3.20.m20.4.4.2.2.2"><times id="S4.I10.i1.p3.20.m20.4.4.2.2.2.1.cmml" xref="S4.I10.i1.p3.20.m20.4.4.2.2.2.1"></times><ci id="S4.I10.i1.p3.20.m20.4.4.2.2.2.2.cmml" xref="S4.I10.i1.p3.20.m20.4.4.2.2.2.2">ℓ</ci><ci id="S4.I10.i1.p3.20.m20.2.2.cmml" xref="S4.I10.i1.p3.20.m20.2.2">𝑏</ci></apply></list><apply id="S4.I10.i1.p3.20.m20.4.4.4.cmml" xref="S4.I10.i1.p3.20.m20.4.4.4"><csymbol cd="ambiguous" id="S4.I10.i1.p3.20.m20.4.4.4.1.cmml" xref="S4.I10.i1.p3.20.m20.4.4.4">subscript</csymbol><ci id="S4.I10.i1.p3.20.m20.4.4.4.2.cmml" xref="S4.I10.i1.p3.20.m20.4.4.4.2">𝑇</ci><ci id="S4.I10.i1.p3.20.m20.4.4.4.3.cmml" xref="S4.I10.i1.p3.20.m20.4.4.4.3">𝑥</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.I10.i1.p3.20.m20.4c">\ell(a),\ell(b)\in T_{x}</annotation><annotation encoding="application/x-llamapun" id="S4.I10.i1.p3.20.m20.4d">roman_ℓ ( italic_a ) , roman_ℓ ( italic_b ) ∈ italic_T start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT</annotation></semantics></math>. Thus <math alttext="(u_{3},u_{4})" class="ltx_Math" display="inline" id="S4.I10.i1.p3.21.m21.2"><semantics id="S4.I10.i1.p3.21.m21.2a"><mrow id="S4.I10.i1.p3.21.m21.2.2.2" xref="S4.I10.i1.p3.21.m21.2.2.3.cmml"><mo id="S4.I10.i1.p3.21.m21.2.2.2.3" stretchy="false" xref="S4.I10.i1.p3.21.m21.2.2.3.cmml">(</mo><msub id="S4.I10.i1.p3.21.m21.1.1.1.1" xref="S4.I10.i1.p3.21.m21.1.1.1.1.cmml"><mi id="S4.I10.i1.p3.21.m21.1.1.1.1.2" xref="S4.I10.i1.p3.21.m21.1.1.1.1.2.cmml">u</mi><mn id="S4.I10.i1.p3.21.m21.1.1.1.1.3" xref="S4.I10.i1.p3.21.m21.1.1.1.1.3.cmml">3</mn></msub><mo id="S4.I10.i1.p3.21.m21.2.2.2.4" xref="S4.I10.i1.p3.21.m21.2.2.3.cmml">,</mo><msub id="S4.I10.i1.p3.21.m21.2.2.2.2" xref="S4.I10.i1.p3.21.m21.2.2.2.2.cmml"><mi id="S4.I10.i1.p3.21.m21.2.2.2.2.2" xref="S4.I10.i1.p3.21.m21.2.2.2.2.2.cmml">u</mi><mn id="S4.I10.i1.p3.21.m21.2.2.2.2.3" xref="S4.I10.i1.p3.21.m21.2.2.2.2.3.cmml">4</mn></msub><mo id="S4.I10.i1.p3.21.m21.2.2.2.5" stretchy="false" xref="S4.I10.i1.p3.21.m21.2.2.3.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S4.I10.i1.p3.21.m21.2b"><interval closure="open" id="S4.I10.i1.p3.21.m21.2.2.3.cmml" xref="S4.I10.i1.p3.21.m21.2.2.2"><apply id="S4.I10.i1.p3.21.m21.1.1.1.1.cmml" xref="S4.I10.i1.p3.21.m21.1.1.1.1"><csymbol cd="ambiguous" id="S4.I10.i1.p3.21.m21.1.1.1.1.1.cmml" xref="S4.I10.i1.p3.21.m21.1.1.1.1">subscript</csymbol><ci id="S4.I10.i1.p3.21.m21.1.1.1.1.2.cmml" xref="S4.I10.i1.p3.21.m21.1.1.1.1.2">𝑢</ci><cn id="S4.I10.i1.p3.21.m21.1.1.1.1.3.cmml" type="integer" xref="S4.I10.i1.p3.21.m21.1.1.1.1.3">3</cn></apply><apply id="S4.I10.i1.p3.21.m21.2.2.2.2.cmml" xref="S4.I10.i1.p3.21.m21.2.2.2.2"><csymbol cd="ambiguous" id="S4.I10.i1.p3.21.m21.2.2.2.2.1.cmml" xref="S4.I10.i1.p3.21.m21.2.2.2.2">subscript</csymbol><ci id="S4.I10.i1.p3.21.m21.2.2.2.2.2.cmml" xref="S4.I10.i1.p3.21.m21.2.2.2.2.2">𝑢</ci><cn id="S4.I10.i1.p3.21.m21.2.2.2.2.3.cmml" type="integer" xref="S4.I10.i1.p3.21.m21.2.2.2.2.3">4</cn></apply></interval></annotation-xml><annotation encoding="application/x-tex" id="S4.I10.i1.p3.21.m21.2c">(u_{3},u_{4})</annotation><annotation encoding="application/x-llamapun" id="S4.I10.i1.p3.21.m21.2d">( italic_u start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT , italic_u start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT )</annotation></semantics></math> is a link in <span class="ltx_text ltx_markedasmath" id="S4.I10.i1.p3.27.2">SOL</span> between <math alttext="T(u)" class="ltx_Math" display="inline" id="S4.I10.i1.p3.23.m23.1"><semantics id="S4.I10.i1.p3.23.m23.1a"><mrow id="S4.I10.i1.p3.23.m23.1.2" xref="S4.I10.i1.p3.23.m23.1.2.cmml"><mi id="S4.I10.i1.p3.23.m23.1.2.2" xref="S4.I10.i1.p3.23.m23.1.2.2.cmml">T</mi><mo id="S4.I10.i1.p3.23.m23.1.2.1" xref="S4.I10.i1.p3.23.m23.1.2.1.cmml">⁢</mo><mrow id="S4.I10.i1.p3.23.m23.1.2.3.2" xref="S4.I10.i1.p3.23.m23.1.2.cmml"><mo id="S4.I10.i1.p3.23.m23.1.2.3.2.1" stretchy="false" xref="S4.I10.i1.p3.23.m23.1.2.cmml">(</mo><mi id="S4.I10.i1.p3.23.m23.1.1" xref="S4.I10.i1.p3.23.m23.1.1.cmml">u</mi><mo id="S4.I10.i1.p3.23.m23.1.2.3.2.2" stretchy="false" xref="S4.I10.i1.p3.23.m23.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.I10.i1.p3.23.m23.1b"><apply id="S4.I10.i1.p3.23.m23.1.2.cmml" xref="S4.I10.i1.p3.23.m23.1.2"><times id="S4.I10.i1.p3.23.m23.1.2.1.cmml" xref="S4.I10.i1.p3.23.m23.1.2.1"></times><ci id="S4.I10.i1.p3.23.m23.1.2.2.cmml" xref="S4.I10.i1.p3.23.m23.1.2.2">𝑇</ci><ci id="S4.I10.i1.p3.23.m23.1.1.cmml" xref="S4.I10.i1.p3.23.m23.1.1">𝑢</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.I10.i1.p3.23.m23.1c">T(u)</annotation><annotation encoding="application/x-llamapun" id="S4.I10.i1.p3.23.m23.1d">italic_T ( italic_u )</annotation></semantics></math> and <math alttext="T\setminus T_{x}" class="ltx_Math" display="inline" id="S4.I10.i1.p3.24.m24.1"><semantics id="S4.I10.i1.p3.24.m24.1a"><mrow id="S4.I10.i1.p3.24.m24.1.1" xref="S4.I10.i1.p3.24.m24.1.1.cmml"><mi id="S4.I10.i1.p3.24.m24.1.1.2" xref="S4.I10.i1.p3.24.m24.1.1.2.cmml">T</mi><mo id="S4.I10.i1.p3.24.m24.1.1.1" xref="S4.I10.i1.p3.24.m24.1.1.1.cmml">∖</mo><msub id="S4.I10.i1.p3.24.m24.1.1.3" xref="S4.I10.i1.p3.24.m24.1.1.3.cmml"><mi id="S4.I10.i1.p3.24.m24.1.1.3.2" xref="S4.I10.i1.p3.24.m24.1.1.3.2.cmml">T</mi><mi id="S4.I10.i1.p3.24.m24.1.1.3.3" xref="S4.I10.i1.p3.24.m24.1.1.3.3.cmml">x</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S4.I10.i1.p3.24.m24.1b"><apply id="S4.I10.i1.p3.24.m24.1.1.cmml" xref="S4.I10.i1.p3.24.m24.1.1"><setdiff id="S4.I10.i1.p3.24.m24.1.1.1.cmml" xref="S4.I10.i1.p3.24.m24.1.1.1"></setdiff><ci id="S4.I10.i1.p3.24.m24.1.1.2.cmml" xref="S4.I10.i1.p3.24.m24.1.1.2">𝑇</ci><apply id="S4.I10.i1.p3.24.m24.1.1.3.cmml" xref="S4.I10.i1.p3.24.m24.1.1.3"><csymbol cd="ambiguous" id="S4.I10.i1.p3.24.m24.1.1.3.1.cmml" xref="S4.I10.i1.p3.24.m24.1.1.3">subscript</csymbol><ci id="S4.I10.i1.p3.24.m24.1.1.3.2.cmml" xref="S4.I10.i1.p3.24.m24.1.1.3.2">𝑇</ci><ci id="S4.I10.i1.p3.24.m24.1.1.3.3.cmml" xref="S4.I10.i1.p3.24.m24.1.1.3.3">𝑥</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.I10.i1.p3.24.m24.1c">T\setminus T_{x}</annotation><annotation encoding="application/x-llamapun" id="S4.I10.i1.p3.24.m24.1d">italic_T ∖ italic_T start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT</annotation></semantics></math> that is not incident to <math alttext="a" class="ltx_Math" display="inline" id="S4.I10.i1.p3.25.m25.1"><semantics id="S4.I10.i1.p3.25.m25.1a"><mi id="S4.I10.i1.p3.25.m25.1.1" xref="S4.I10.i1.p3.25.m25.1.1.cmml">a</mi><annotation-xml encoding="MathML-Content" id="S4.I10.i1.p3.25.m25.1b"><ci id="S4.I10.i1.p3.25.m25.1.1.cmml" xref="S4.I10.i1.p3.25.m25.1.1">𝑎</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.I10.i1.p3.25.m25.1c">a</annotation><annotation encoding="application/x-llamapun" id="S4.I10.i1.p3.25.m25.1d">italic_a</annotation></semantics></math> or <math alttext="b" class="ltx_Math" display="inline" id="S4.I10.i1.p3.26.m26.1"><semantics id="S4.I10.i1.p3.26.m26.1a"><mi id="S4.I10.i1.p3.26.m26.1.1" xref="S4.I10.i1.p3.26.m26.1.1.cmml">b</mi><annotation-xml encoding="MathML-Content" id="S4.I10.i1.p3.26.m26.1b"><ci id="S4.I10.i1.p3.26.m26.1.1.cmml" xref="S4.I10.i1.p3.26.m26.1.1">𝑏</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.I10.i1.p3.26.m26.1c">b</annotation><annotation encoding="application/x-llamapun" id="S4.I10.i1.p3.26.m26.1d">italic_b</annotation></semantics></math>, as desired. The analysis for <math alttext="v" class="ltx_Math" display="inline" id="S4.I10.i1.p3.27.m27.1"><semantics id="S4.I10.i1.p3.27.m27.1a"><mi id="S4.I10.i1.p3.27.m27.1.1" xref="S4.I10.i1.p3.27.m27.1.1.cmml">v</mi><annotation-xml encoding="MathML-Content" id="S4.I10.i1.p3.27.m27.1b"><ci id="S4.I10.i1.p3.27.m27.1.1.cmml" xref="S4.I10.i1.p3.27.m27.1.1">𝑣</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.I10.i1.p3.27.m27.1c">v</annotation><annotation encoding="application/x-llamapun" id="S4.I10.i1.p3.27.m27.1d">italic_v</annotation></semantics></math> is analogous. See Figure <a class="ltx_ref" href="https://arxiv.org/html/2503.00712v1#S4.F9" title="Figure 9 ‣ Case 2: 𝑽⁢(𝑮_𝒙)={𝒂,𝒃} for a P-node 𝒙: ‣ 4.2.3 Bounding the Approximation Ratio ‣ 4.2 Two-to-Three Augmentation ‣ 4 Vertex Connectivity Augmentation in Link-Arrival Model ‣ Streaming Algorithms for Network Design"><span class="ltx_text ltx_ref_tag">9</span></a> for reference.</p> </div> </li> <li class="ltx_item" id="S4.I10.i2" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">•</span> <div class="ltx_para" id="S4.I10.i2.p1"> <p class="ltx_p" id="S4.I10.i2.p1.21"><span class="ltx_text ltx_font_bold" id="S4.I10.i2.p1.4.4">Case 2b: One of <math alttext="\boldsymbol{u}" class="ltx_Math" display="inline" id="S4.I10.i2.p1.1.1.m1.1"><semantics id="S4.I10.i2.p1.1.1.m1.1a"><mi id="S4.I10.i2.p1.1.1.m1.1.1" xref="S4.I10.i2.p1.1.1.m1.1.1.cmml">u</mi><annotation-xml encoding="MathML-Content" id="S4.I10.i2.p1.1.1.m1.1b"><ci id="S4.I10.i2.p1.1.1.m1.1.1.cmml" xref="S4.I10.i2.p1.1.1.m1.1.1">𝑢</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.I10.i2.p1.1.1.m1.1c">\boldsymbol{u}</annotation><annotation encoding="application/x-llamapun" id="S4.I10.i2.p1.1.1.m1.1d">bold_italic_u</annotation></semantics></math> and <math alttext="\boldsymbol{v}" class="ltx_Math" display="inline" id="S4.I10.i2.p1.2.2.m2.1"><semantics id="S4.I10.i2.p1.2.2.m2.1a"><mi id="S4.I10.i2.p1.2.2.m2.1.1" xref="S4.I10.i2.p1.2.2.m2.1.1.cmml">v</mi><annotation-xml encoding="MathML-Content" id="S4.I10.i2.p1.2.2.m2.1b"><ci id="S4.I10.i2.p1.2.2.m2.1.1.cmml" xref="S4.I10.i2.p1.2.2.m2.1.1">𝑣</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.I10.i2.p1.2.2.m2.1c">\boldsymbol{v}</annotation><annotation encoding="application/x-llamapun" id="S4.I10.i2.p1.2.2.m2.1d">bold_italic_v</annotation></semantics></math> is in <math alttext="\boldsymbol{T_{x}}" class="ltx_Math" display="inline" id="S4.I10.i2.p1.3.3.m3.1"><semantics id="S4.I10.i2.p1.3.3.m3.1a"><msub id="S4.I10.i2.p1.3.3.m3.1.1" xref="S4.I10.i2.p1.3.3.m3.1.1.cmml"><mi id="S4.I10.i2.p1.3.3.m3.1.1.2" xref="S4.I10.i2.p1.3.3.m3.1.1.2.cmml">T</mi><mi id="S4.I10.i2.p1.3.3.m3.1.1.3" xref="S4.I10.i2.p1.3.3.m3.1.1.3.cmml">x</mi></msub><annotation-xml encoding="MathML-Content" id="S4.I10.i2.p1.3.3.m3.1b"><apply id="S4.I10.i2.p1.3.3.m3.1.1.cmml" xref="S4.I10.i2.p1.3.3.m3.1.1"><csymbol cd="ambiguous" id="S4.I10.i2.p1.3.3.m3.1.1.1.cmml" xref="S4.I10.i2.p1.3.3.m3.1.1">subscript</csymbol><ci id="S4.I10.i2.p1.3.3.m3.1.1.2.cmml" xref="S4.I10.i2.p1.3.3.m3.1.1.2">𝑇</ci><ci id="S4.I10.i2.p1.3.3.m3.1.1.3.cmml" xref="S4.I10.i2.p1.3.3.m3.1.1.3">𝑥</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.I10.i2.p1.3.3.m3.1c">\boldsymbol{T_{x}}</annotation><annotation encoding="application/x-llamapun" id="S4.I10.i2.p1.3.3.m3.1d">bold_italic_T start_POSTSUBSCRIPT bold_italic_x end_POSTSUBSCRIPT</annotation></semantics></math> and the other is in <math alttext="\boldsymbol{T\setminus T_{x}}" class="ltx_Math" display="inline" id="S4.I10.i2.p1.4.4.m4.1"><semantics id="S4.I10.i2.p1.4.4.m4.1a"><mrow id="S4.I10.i2.p1.4.4.m4.1.1" xref="S4.I10.i2.p1.4.4.m4.1.1.cmml"><mi id="S4.I10.i2.p1.4.4.m4.1.1.2" xref="S4.I10.i2.p1.4.4.m4.1.1.2.cmml">T</mi><mo class="ltx_mathvariant_bold" id="S4.I10.i2.p1.4.4.m4.1.1.1" mathvariant="bold" xref="S4.I10.i2.p1.4.4.m4.1.1.1.cmml">∖</mo><msub id="S4.I10.i2.p1.4.4.m4.1.1.3" xref="S4.I10.i2.p1.4.4.m4.1.1.3.cmml"><mi id="S4.I10.i2.p1.4.4.m4.1.1.3.2" xref="S4.I10.i2.p1.4.4.m4.1.1.3.2.cmml">T</mi><mi id="S4.I10.i2.p1.4.4.m4.1.1.3.3" xref="S4.I10.i2.p1.4.4.m4.1.1.3.3.cmml">x</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S4.I10.i2.p1.4.4.m4.1b"><apply id="S4.I10.i2.p1.4.4.m4.1.1.cmml" xref="S4.I10.i2.p1.4.4.m4.1.1"><setdiff id="S4.I10.i2.p1.4.4.m4.1.1.1.cmml" xref="S4.I10.i2.p1.4.4.m4.1.1.1"></setdiff><ci id="S4.I10.i2.p1.4.4.m4.1.1.2.cmml" xref="S4.I10.i2.p1.4.4.m4.1.1.2">𝑇</ci><apply id="S4.I10.i2.p1.4.4.m4.1.1.3.cmml" xref="S4.I10.i2.p1.4.4.m4.1.1.3"><csymbol cd="ambiguous" id="S4.I10.i2.p1.4.4.m4.1.1.3.1.cmml" xref="S4.I10.i2.p1.4.4.m4.1.1.3">subscript</csymbol><ci id="S4.I10.i2.p1.4.4.m4.1.1.3.2.cmml" xref="S4.I10.i2.p1.4.4.m4.1.1.3.2">𝑇</ci><ci id="S4.I10.i2.p1.4.4.m4.1.1.3.3.cmml" xref="S4.I10.i2.p1.4.4.m4.1.1.3.3">𝑥</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.I10.i2.p1.4.4.m4.1c">\boldsymbol{T\setminus T_{x}}</annotation><annotation encoding="application/x-llamapun" id="S4.I10.i2.p1.4.4.m4.1d">bold_italic_T bold_∖ bold_italic_T start_POSTSUBSCRIPT bold_italic_x end_POSTSUBSCRIPT</annotation></semantics></math>:</span> Without loss of generality, suppose <math alttext="h(u)\in T_{x}" class="ltx_Math" display="inline" id="S4.I10.i2.p1.5.m1.1"><semantics id="S4.I10.i2.p1.5.m1.1a"><mrow id="S4.I10.i2.p1.5.m1.1.2" xref="S4.I10.i2.p1.5.m1.1.2.cmml"><mrow id="S4.I10.i2.p1.5.m1.1.2.2" xref="S4.I10.i2.p1.5.m1.1.2.2.cmml"><mi id="S4.I10.i2.p1.5.m1.1.2.2.2" xref="S4.I10.i2.p1.5.m1.1.2.2.2.cmml">h</mi><mo id="S4.I10.i2.p1.5.m1.1.2.2.1" xref="S4.I10.i2.p1.5.m1.1.2.2.1.cmml">⁢</mo><mrow id="S4.I10.i2.p1.5.m1.1.2.2.3.2" xref="S4.I10.i2.p1.5.m1.1.2.2.cmml"><mo id="S4.I10.i2.p1.5.m1.1.2.2.3.2.1" stretchy="false" xref="S4.I10.i2.p1.5.m1.1.2.2.cmml">(</mo><mi id="S4.I10.i2.p1.5.m1.1.1" xref="S4.I10.i2.p1.5.m1.1.1.cmml">u</mi><mo id="S4.I10.i2.p1.5.m1.1.2.2.3.2.2" stretchy="false" xref="S4.I10.i2.p1.5.m1.1.2.2.cmml">)</mo></mrow></mrow><mo id="S4.I10.i2.p1.5.m1.1.2.1" xref="S4.I10.i2.p1.5.m1.1.2.1.cmml">∈</mo><msub id="S4.I10.i2.p1.5.m1.1.2.3" xref="S4.I10.i2.p1.5.m1.1.2.3.cmml"><mi id="S4.I10.i2.p1.5.m1.1.2.3.2" xref="S4.I10.i2.p1.5.m1.1.2.3.2.cmml">T</mi><mi id="S4.I10.i2.p1.5.m1.1.2.3.3" xref="S4.I10.i2.p1.5.m1.1.2.3.3.cmml">x</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S4.I10.i2.p1.5.m1.1b"><apply id="S4.I10.i2.p1.5.m1.1.2.cmml" xref="S4.I10.i2.p1.5.m1.1.2"><in id="S4.I10.i2.p1.5.m1.1.2.1.cmml" xref="S4.I10.i2.p1.5.m1.1.2.1"></in><apply id="S4.I10.i2.p1.5.m1.1.2.2.cmml" xref="S4.I10.i2.p1.5.m1.1.2.2"><times id="S4.I10.i2.p1.5.m1.1.2.2.1.cmml" xref="S4.I10.i2.p1.5.m1.1.2.2.1"></times><ci id="S4.I10.i2.p1.5.m1.1.2.2.2.cmml" xref="S4.I10.i2.p1.5.m1.1.2.2.2">ℎ</ci><ci id="S4.I10.i2.p1.5.m1.1.1.cmml" xref="S4.I10.i2.p1.5.m1.1.1">𝑢</ci></apply><apply id="S4.I10.i2.p1.5.m1.1.2.3.cmml" xref="S4.I10.i2.p1.5.m1.1.2.3"><csymbol cd="ambiguous" id="S4.I10.i2.p1.5.m1.1.2.3.1.cmml" xref="S4.I10.i2.p1.5.m1.1.2.3">subscript</csymbol><ci id="S4.I10.i2.p1.5.m1.1.2.3.2.cmml" xref="S4.I10.i2.p1.5.m1.1.2.3.2">𝑇</ci><ci id="S4.I10.i2.p1.5.m1.1.2.3.3.cmml" xref="S4.I10.i2.p1.5.m1.1.2.3.3">𝑥</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.I10.i2.p1.5.m1.1c">h(u)\in T_{x}</annotation><annotation encoding="application/x-llamapun" id="S4.I10.i2.p1.5.m1.1d">italic_h ( italic_u ) ∈ italic_T start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="\ell(v)\in T\setminus T_{x}" class="ltx_Math" display="inline" id="S4.I10.i2.p1.6.m2.1"><semantics id="S4.I10.i2.p1.6.m2.1a"><mrow id="S4.I10.i2.p1.6.m2.1.2" xref="S4.I10.i2.p1.6.m2.1.2.cmml"><mrow id="S4.I10.i2.p1.6.m2.1.2.2" xref="S4.I10.i2.p1.6.m2.1.2.2.cmml"><mi id="S4.I10.i2.p1.6.m2.1.2.2.2" mathvariant="normal" xref="S4.I10.i2.p1.6.m2.1.2.2.2.cmml">ℓ</mi><mo id="S4.I10.i2.p1.6.m2.1.2.2.1" xref="S4.I10.i2.p1.6.m2.1.2.2.1.cmml">⁢</mo><mrow id="S4.I10.i2.p1.6.m2.1.2.2.3.2" xref="S4.I10.i2.p1.6.m2.1.2.2.cmml"><mo id="S4.I10.i2.p1.6.m2.1.2.2.3.2.1" stretchy="false" xref="S4.I10.i2.p1.6.m2.1.2.2.cmml">(</mo><mi id="S4.I10.i2.p1.6.m2.1.1" xref="S4.I10.i2.p1.6.m2.1.1.cmml">v</mi><mo id="S4.I10.i2.p1.6.m2.1.2.2.3.2.2" stretchy="false" xref="S4.I10.i2.p1.6.m2.1.2.2.cmml">)</mo></mrow></mrow><mo id="S4.I10.i2.p1.6.m2.1.2.1" xref="S4.I10.i2.p1.6.m2.1.2.1.cmml">∈</mo><mrow id="S4.I10.i2.p1.6.m2.1.2.3" xref="S4.I10.i2.p1.6.m2.1.2.3.cmml"><mi id="S4.I10.i2.p1.6.m2.1.2.3.2" xref="S4.I10.i2.p1.6.m2.1.2.3.2.cmml">T</mi><mo id="S4.I10.i2.p1.6.m2.1.2.3.1" xref="S4.I10.i2.p1.6.m2.1.2.3.1.cmml">∖</mo><msub id="S4.I10.i2.p1.6.m2.1.2.3.3" xref="S4.I10.i2.p1.6.m2.1.2.3.3.cmml"><mi id="S4.I10.i2.p1.6.m2.1.2.3.3.2" xref="S4.I10.i2.p1.6.m2.1.2.3.3.2.cmml">T</mi><mi id="S4.I10.i2.p1.6.m2.1.2.3.3.3" xref="S4.I10.i2.p1.6.m2.1.2.3.3.3.cmml">x</mi></msub></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.I10.i2.p1.6.m2.1b"><apply id="S4.I10.i2.p1.6.m2.1.2.cmml" xref="S4.I10.i2.p1.6.m2.1.2"><in id="S4.I10.i2.p1.6.m2.1.2.1.cmml" xref="S4.I10.i2.p1.6.m2.1.2.1"></in><apply id="S4.I10.i2.p1.6.m2.1.2.2.cmml" xref="S4.I10.i2.p1.6.m2.1.2.2"><times id="S4.I10.i2.p1.6.m2.1.2.2.1.cmml" xref="S4.I10.i2.p1.6.m2.1.2.2.1"></times><ci id="S4.I10.i2.p1.6.m2.1.2.2.2.cmml" xref="S4.I10.i2.p1.6.m2.1.2.2.2">ℓ</ci><ci id="S4.I10.i2.p1.6.m2.1.1.cmml" xref="S4.I10.i2.p1.6.m2.1.1">𝑣</ci></apply><apply id="S4.I10.i2.p1.6.m2.1.2.3.cmml" xref="S4.I10.i2.p1.6.m2.1.2.3"><setdiff id="S4.I10.i2.p1.6.m2.1.2.3.1.cmml" xref="S4.I10.i2.p1.6.m2.1.2.3.1"></setdiff><ci id="S4.I10.i2.p1.6.m2.1.2.3.2.cmml" xref="S4.I10.i2.p1.6.m2.1.2.3.2">𝑇</ci><apply id="S4.I10.i2.p1.6.m2.1.2.3.3.cmml" xref="S4.I10.i2.p1.6.m2.1.2.3.3"><csymbol cd="ambiguous" id="S4.I10.i2.p1.6.m2.1.2.3.3.1.cmml" xref="S4.I10.i2.p1.6.m2.1.2.3.3">subscript</csymbol><ci id="S4.I10.i2.p1.6.m2.1.2.3.3.2.cmml" xref="S4.I10.i2.p1.6.m2.1.2.3.3.2">𝑇</ci><ci id="S4.I10.i2.p1.6.m2.1.2.3.3.3.cmml" xref="S4.I10.i2.p1.6.m2.1.2.3.3.3">𝑥</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.I10.i2.p1.6.m2.1c">\ell(v)\in T\setminus T_{x}</annotation><annotation encoding="application/x-llamapun" id="S4.I10.i2.p1.6.m2.1d">roman_ℓ ( italic_v ) ∈ italic_T ∖ italic_T start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT</annotation></semantics></math>. In this case, we use the same argument as Case 1: let <math alttext="(u^{\prime},u^{\prime\prime})=L_{h(u)}(j)" class="ltx_Math" display="inline" id="S4.I10.i2.p1.7.m3.4"><semantics id="S4.I10.i2.p1.7.m3.4a"><mrow id="S4.I10.i2.p1.7.m3.4.4" xref="S4.I10.i2.p1.7.m3.4.4.cmml"><mrow id="S4.I10.i2.p1.7.m3.4.4.2.2" xref="S4.I10.i2.p1.7.m3.4.4.2.3.cmml"><mo id="S4.I10.i2.p1.7.m3.4.4.2.2.3" stretchy="false" xref="S4.I10.i2.p1.7.m3.4.4.2.3.cmml">(</mo><msup id="S4.I10.i2.p1.7.m3.3.3.1.1.1" xref="S4.I10.i2.p1.7.m3.3.3.1.1.1.cmml"><mi id="S4.I10.i2.p1.7.m3.3.3.1.1.1.2" xref="S4.I10.i2.p1.7.m3.3.3.1.1.1.2.cmml">u</mi><mo id="S4.I10.i2.p1.7.m3.3.3.1.1.1.3" xref="S4.I10.i2.p1.7.m3.3.3.1.1.1.3.cmml">′</mo></msup><mo id="S4.I10.i2.p1.7.m3.4.4.2.2.4" xref="S4.I10.i2.p1.7.m3.4.4.2.3.cmml">,</mo><msup id="S4.I10.i2.p1.7.m3.4.4.2.2.2" xref="S4.I10.i2.p1.7.m3.4.4.2.2.2.cmml"><mi id="S4.I10.i2.p1.7.m3.4.4.2.2.2.2" xref="S4.I10.i2.p1.7.m3.4.4.2.2.2.2.cmml">u</mi><mo id="S4.I10.i2.p1.7.m3.4.4.2.2.2.3" xref="S4.I10.i2.p1.7.m3.4.4.2.2.2.3.cmml">′′</mo></msup><mo id="S4.I10.i2.p1.7.m3.4.4.2.2.5" stretchy="false" xref="S4.I10.i2.p1.7.m3.4.4.2.3.cmml">)</mo></mrow><mo id="S4.I10.i2.p1.7.m3.4.4.3" xref="S4.I10.i2.p1.7.m3.4.4.3.cmml">=</mo><mrow id="S4.I10.i2.p1.7.m3.4.4.4" xref="S4.I10.i2.p1.7.m3.4.4.4.cmml"><msub id="S4.I10.i2.p1.7.m3.4.4.4.2" xref="S4.I10.i2.p1.7.m3.4.4.4.2.cmml"><mi id="S4.I10.i2.p1.7.m3.4.4.4.2.2" xref="S4.I10.i2.p1.7.m3.4.4.4.2.2.cmml">L</mi><mrow id="S4.I10.i2.p1.7.m3.1.1.1" xref="S4.I10.i2.p1.7.m3.1.1.1.cmml"><mi id="S4.I10.i2.p1.7.m3.1.1.1.3" xref="S4.I10.i2.p1.7.m3.1.1.1.3.cmml">h</mi><mo id="S4.I10.i2.p1.7.m3.1.1.1.2" xref="S4.I10.i2.p1.7.m3.1.1.1.2.cmml">⁢</mo><mrow id="S4.I10.i2.p1.7.m3.1.1.1.4.2" xref="S4.I10.i2.p1.7.m3.1.1.1.cmml"><mo id="S4.I10.i2.p1.7.m3.1.1.1.4.2.1" stretchy="false" xref="S4.I10.i2.p1.7.m3.1.1.1.cmml">(</mo><mi id="S4.I10.i2.p1.7.m3.1.1.1.1" xref="S4.I10.i2.p1.7.m3.1.1.1.1.cmml">u</mi><mo id="S4.I10.i2.p1.7.m3.1.1.1.4.2.2" stretchy="false" xref="S4.I10.i2.p1.7.m3.1.1.1.cmml">)</mo></mrow></mrow></msub><mo id="S4.I10.i2.p1.7.m3.4.4.4.1" xref="S4.I10.i2.p1.7.m3.4.4.4.1.cmml">⁢</mo><mrow id="S4.I10.i2.p1.7.m3.4.4.4.3.2" xref="S4.I10.i2.p1.7.m3.4.4.4.cmml"><mo id="S4.I10.i2.p1.7.m3.4.4.4.3.2.1" stretchy="false" xref="S4.I10.i2.p1.7.m3.4.4.4.cmml">(</mo><mi id="S4.I10.i2.p1.7.m3.2.2" xref="S4.I10.i2.p1.7.m3.2.2.cmml">j</mi><mo id="S4.I10.i2.p1.7.m3.4.4.4.3.2.2" stretchy="false" xref="S4.I10.i2.p1.7.m3.4.4.4.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.I10.i2.p1.7.m3.4b"><apply id="S4.I10.i2.p1.7.m3.4.4.cmml" xref="S4.I10.i2.p1.7.m3.4.4"><eq id="S4.I10.i2.p1.7.m3.4.4.3.cmml" xref="S4.I10.i2.p1.7.m3.4.4.3"></eq><interval closure="open" id="S4.I10.i2.p1.7.m3.4.4.2.3.cmml" xref="S4.I10.i2.p1.7.m3.4.4.2.2"><apply id="S4.I10.i2.p1.7.m3.3.3.1.1.1.cmml" xref="S4.I10.i2.p1.7.m3.3.3.1.1.1"><csymbol cd="ambiguous" id="S4.I10.i2.p1.7.m3.3.3.1.1.1.1.cmml" xref="S4.I10.i2.p1.7.m3.3.3.1.1.1">superscript</csymbol><ci id="S4.I10.i2.p1.7.m3.3.3.1.1.1.2.cmml" xref="S4.I10.i2.p1.7.m3.3.3.1.1.1.2">𝑢</ci><ci id="S4.I10.i2.p1.7.m3.3.3.1.1.1.3.cmml" xref="S4.I10.i2.p1.7.m3.3.3.1.1.1.3">′</ci></apply><apply id="S4.I10.i2.p1.7.m3.4.4.2.2.2.cmml" xref="S4.I10.i2.p1.7.m3.4.4.2.2.2"><csymbol cd="ambiguous" id="S4.I10.i2.p1.7.m3.4.4.2.2.2.1.cmml" xref="S4.I10.i2.p1.7.m3.4.4.2.2.2">superscript</csymbol><ci id="S4.I10.i2.p1.7.m3.4.4.2.2.2.2.cmml" xref="S4.I10.i2.p1.7.m3.4.4.2.2.2.2">𝑢</ci><ci id="S4.I10.i2.p1.7.m3.4.4.2.2.2.3.cmml" xref="S4.I10.i2.p1.7.m3.4.4.2.2.2.3">′′</ci></apply></interval><apply id="S4.I10.i2.p1.7.m3.4.4.4.cmml" xref="S4.I10.i2.p1.7.m3.4.4.4"><times id="S4.I10.i2.p1.7.m3.4.4.4.1.cmml" xref="S4.I10.i2.p1.7.m3.4.4.4.1"></times><apply id="S4.I10.i2.p1.7.m3.4.4.4.2.cmml" xref="S4.I10.i2.p1.7.m3.4.4.4.2"><csymbol cd="ambiguous" id="S4.I10.i2.p1.7.m3.4.4.4.2.1.cmml" xref="S4.I10.i2.p1.7.m3.4.4.4.2">subscript</csymbol><ci id="S4.I10.i2.p1.7.m3.4.4.4.2.2.cmml" xref="S4.I10.i2.p1.7.m3.4.4.4.2.2">𝐿</ci><apply id="S4.I10.i2.p1.7.m3.1.1.1.cmml" xref="S4.I10.i2.p1.7.m3.1.1.1"><times id="S4.I10.i2.p1.7.m3.1.1.1.2.cmml" xref="S4.I10.i2.p1.7.m3.1.1.1.2"></times><ci id="S4.I10.i2.p1.7.m3.1.1.1.3.cmml" xref="S4.I10.i2.p1.7.m3.1.1.1.3">ℎ</ci><ci id="S4.I10.i2.p1.7.m3.1.1.1.1.cmml" xref="S4.I10.i2.p1.7.m3.1.1.1.1">𝑢</ci></apply></apply><ci id="S4.I10.i2.p1.7.m3.2.2.cmml" xref="S4.I10.i2.p1.7.m3.2.2">𝑗</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.I10.i2.p1.7.m3.4c">(u^{\prime},u^{\prime\prime})=L_{h(u)}(j)</annotation><annotation encoding="application/x-llamapun" id="S4.I10.i2.p1.7.m3.4d">( italic_u start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT , italic_u start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT ) = italic_L start_POSTSUBSCRIPT italic_h ( italic_u ) end_POSTSUBSCRIPT ( italic_j )</annotation></semantics></math> where <math alttext="h(u^{\prime})=h(u)" class="ltx_Math" display="inline" id="S4.I10.i2.p1.8.m4.2"><semantics id="S4.I10.i2.p1.8.m4.2a"><mrow id="S4.I10.i2.p1.8.m4.2.2" xref="S4.I10.i2.p1.8.m4.2.2.cmml"><mrow id="S4.I10.i2.p1.8.m4.2.2.1" xref="S4.I10.i2.p1.8.m4.2.2.1.cmml"><mi id="S4.I10.i2.p1.8.m4.2.2.1.3" xref="S4.I10.i2.p1.8.m4.2.2.1.3.cmml">h</mi><mo id="S4.I10.i2.p1.8.m4.2.2.1.2" xref="S4.I10.i2.p1.8.m4.2.2.1.2.cmml">⁢</mo><mrow id="S4.I10.i2.p1.8.m4.2.2.1.1.1" xref="S4.I10.i2.p1.8.m4.2.2.1.1.1.1.cmml"><mo id="S4.I10.i2.p1.8.m4.2.2.1.1.1.2" stretchy="false" xref="S4.I10.i2.p1.8.m4.2.2.1.1.1.1.cmml">(</mo><msup id="S4.I10.i2.p1.8.m4.2.2.1.1.1.1" xref="S4.I10.i2.p1.8.m4.2.2.1.1.1.1.cmml"><mi id="S4.I10.i2.p1.8.m4.2.2.1.1.1.1.2" xref="S4.I10.i2.p1.8.m4.2.2.1.1.1.1.2.cmml">u</mi><mo id="S4.I10.i2.p1.8.m4.2.2.1.1.1.1.3" xref="S4.I10.i2.p1.8.m4.2.2.1.1.1.1.3.cmml">′</mo></msup><mo id="S4.I10.i2.p1.8.m4.2.2.1.1.1.3" stretchy="false" xref="S4.I10.i2.p1.8.m4.2.2.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S4.I10.i2.p1.8.m4.2.2.2" xref="S4.I10.i2.p1.8.m4.2.2.2.cmml">=</mo><mrow id="S4.I10.i2.p1.8.m4.2.2.3" xref="S4.I10.i2.p1.8.m4.2.2.3.cmml"><mi id="S4.I10.i2.p1.8.m4.2.2.3.2" xref="S4.I10.i2.p1.8.m4.2.2.3.2.cmml">h</mi><mo id="S4.I10.i2.p1.8.m4.2.2.3.1" xref="S4.I10.i2.p1.8.m4.2.2.3.1.cmml">⁢</mo><mrow id="S4.I10.i2.p1.8.m4.2.2.3.3.2" xref="S4.I10.i2.p1.8.m4.2.2.3.cmml"><mo id="S4.I10.i2.p1.8.m4.2.2.3.3.2.1" stretchy="false" xref="S4.I10.i2.p1.8.m4.2.2.3.cmml">(</mo><mi id="S4.I10.i2.p1.8.m4.1.1" xref="S4.I10.i2.p1.8.m4.1.1.cmml">u</mi><mo id="S4.I10.i2.p1.8.m4.2.2.3.3.2.2" stretchy="false" xref="S4.I10.i2.p1.8.m4.2.2.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.I10.i2.p1.8.m4.2b"><apply id="S4.I10.i2.p1.8.m4.2.2.cmml" xref="S4.I10.i2.p1.8.m4.2.2"><eq id="S4.I10.i2.p1.8.m4.2.2.2.cmml" xref="S4.I10.i2.p1.8.m4.2.2.2"></eq><apply id="S4.I10.i2.p1.8.m4.2.2.1.cmml" xref="S4.I10.i2.p1.8.m4.2.2.1"><times id="S4.I10.i2.p1.8.m4.2.2.1.2.cmml" xref="S4.I10.i2.p1.8.m4.2.2.1.2"></times><ci id="S4.I10.i2.p1.8.m4.2.2.1.3.cmml" xref="S4.I10.i2.p1.8.m4.2.2.1.3">ℎ</ci><apply id="S4.I10.i2.p1.8.m4.2.2.1.1.1.1.cmml" xref="S4.I10.i2.p1.8.m4.2.2.1.1.1"><csymbol cd="ambiguous" id="S4.I10.i2.p1.8.m4.2.2.1.1.1.1.1.cmml" xref="S4.I10.i2.p1.8.m4.2.2.1.1.1">superscript</csymbol><ci id="S4.I10.i2.p1.8.m4.2.2.1.1.1.1.2.cmml" xref="S4.I10.i2.p1.8.m4.2.2.1.1.1.1.2">𝑢</ci><ci id="S4.I10.i2.p1.8.m4.2.2.1.1.1.1.3.cmml" xref="S4.I10.i2.p1.8.m4.2.2.1.1.1.1.3">′</ci></apply></apply><apply id="S4.I10.i2.p1.8.m4.2.2.3.cmml" xref="S4.I10.i2.p1.8.m4.2.2.3"><times id="S4.I10.i2.p1.8.m4.2.2.3.1.cmml" xref="S4.I10.i2.p1.8.m4.2.2.3.1"></times><ci id="S4.I10.i2.p1.8.m4.2.2.3.2.cmml" xref="S4.I10.i2.p1.8.m4.2.2.3.2">ℎ</ci><ci id="S4.I10.i2.p1.8.m4.1.1.cmml" xref="S4.I10.i2.p1.8.m4.1.1">𝑢</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.I10.i2.p1.8.m4.2c">h(u^{\prime})=h(u)</annotation><annotation encoding="application/x-llamapun" id="S4.I10.i2.p1.8.m4.2d">italic_h ( italic_u start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) = italic_h ( italic_u )</annotation></semantics></math>; this must be in <span class="ltx_text ltx_markedasmath" id="S4.I10.i2.p1.21.5">SOL</span> since <math alttext="uv\in\textnormal{OPT}" class="ltx_Math" display="inline" id="S4.I10.i2.p1.10.m6.1"><semantics id="S4.I10.i2.p1.10.m6.1a"><mrow id="S4.I10.i2.p1.10.m6.1.1" xref="S4.I10.i2.p1.10.m6.1.1.cmml"><mrow id="S4.I10.i2.p1.10.m6.1.1.2" xref="S4.I10.i2.p1.10.m6.1.1.2.cmml"><mi id="S4.I10.i2.p1.10.m6.1.1.2.2" xref="S4.I10.i2.p1.10.m6.1.1.2.2.cmml">u</mi><mo id="S4.I10.i2.p1.10.m6.1.1.2.1" xref="S4.I10.i2.p1.10.m6.1.1.2.1.cmml">⁢</mo><mi id="S4.I10.i2.p1.10.m6.1.1.2.3" xref="S4.I10.i2.p1.10.m6.1.1.2.3.cmml">v</mi></mrow><mo id="S4.I10.i2.p1.10.m6.1.1.1" xref="S4.I10.i2.p1.10.m6.1.1.1.cmml">∈</mo><mtext id="S4.I10.i2.p1.10.m6.1.1.3" xref="S4.I10.i2.p1.10.m6.1.1.3a.cmml">OPT</mtext></mrow><annotation-xml encoding="MathML-Content" id="S4.I10.i2.p1.10.m6.1b"><apply id="S4.I10.i2.p1.10.m6.1.1.cmml" xref="S4.I10.i2.p1.10.m6.1.1"><in id="S4.I10.i2.p1.10.m6.1.1.1.cmml" xref="S4.I10.i2.p1.10.m6.1.1.1"></in><apply id="S4.I10.i2.p1.10.m6.1.1.2.cmml" xref="S4.I10.i2.p1.10.m6.1.1.2"><times id="S4.I10.i2.p1.10.m6.1.1.2.1.cmml" xref="S4.I10.i2.p1.10.m6.1.1.2.1"></times><ci id="S4.I10.i2.p1.10.m6.1.1.2.2.cmml" xref="S4.I10.i2.p1.10.m6.1.1.2.2">𝑢</ci><ci id="S4.I10.i2.p1.10.m6.1.1.2.3.cmml" xref="S4.I10.i2.p1.10.m6.1.1.2.3">𝑣</ci></apply><ci id="S4.I10.i2.p1.10.m6.1.1.3a.cmml" xref="S4.I10.i2.p1.10.m6.1.1.3"><mtext id="S4.I10.i2.p1.10.m6.1.1.3.cmml" xref="S4.I10.i2.p1.10.m6.1.1.3">OPT</mtext></ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.I10.i2.p1.10.m6.1c">uv\in\textnormal{OPT}</annotation><annotation encoding="application/x-llamapun" id="S4.I10.i2.p1.10.m6.1d">italic_u italic_v ∈ OPT</annotation></semantics></math>. Clearly <math alttext="u^{\prime}\notin\{a,b\}" class="ltx_Math" display="inline" id="S4.I10.i2.p1.11.m7.2"><semantics id="S4.I10.i2.p1.11.m7.2a"><mrow id="S4.I10.i2.p1.11.m7.2.3" xref="S4.I10.i2.p1.11.m7.2.3.cmml"><msup id="S4.I10.i2.p1.11.m7.2.3.2" xref="S4.I10.i2.p1.11.m7.2.3.2.cmml"><mi id="S4.I10.i2.p1.11.m7.2.3.2.2" xref="S4.I10.i2.p1.11.m7.2.3.2.2.cmml">u</mi><mo id="S4.I10.i2.p1.11.m7.2.3.2.3" xref="S4.I10.i2.p1.11.m7.2.3.2.3.cmml">′</mo></msup><mo id="S4.I10.i2.p1.11.m7.2.3.1" xref="S4.I10.i2.p1.11.m7.2.3.1.cmml">∉</mo><mrow id="S4.I10.i2.p1.11.m7.2.3.3.2" xref="S4.I10.i2.p1.11.m7.2.3.3.1.cmml"><mo id="S4.I10.i2.p1.11.m7.2.3.3.2.1" stretchy="false" xref="S4.I10.i2.p1.11.m7.2.3.3.1.cmml">{</mo><mi id="S4.I10.i2.p1.11.m7.1.1" xref="S4.I10.i2.p1.11.m7.1.1.cmml">a</mi><mo id="S4.I10.i2.p1.11.m7.2.3.3.2.2" xref="S4.I10.i2.p1.11.m7.2.3.3.1.cmml">,</mo><mi id="S4.I10.i2.p1.11.m7.2.2" xref="S4.I10.i2.p1.11.m7.2.2.cmml">b</mi><mo id="S4.I10.i2.p1.11.m7.2.3.3.2.3" stretchy="false" xref="S4.I10.i2.p1.11.m7.2.3.3.1.cmml">}</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.I10.i2.p1.11.m7.2b"><apply id="S4.I10.i2.p1.11.m7.2.3.cmml" xref="S4.I10.i2.p1.11.m7.2.3"><notin id="S4.I10.i2.p1.11.m7.2.3.1.cmml" xref="S4.I10.i2.p1.11.m7.2.3.1"></notin><apply id="S4.I10.i2.p1.11.m7.2.3.2.cmml" xref="S4.I10.i2.p1.11.m7.2.3.2"><csymbol cd="ambiguous" id="S4.I10.i2.p1.11.m7.2.3.2.1.cmml" xref="S4.I10.i2.p1.11.m7.2.3.2">superscript</csymbol><ci id="S4.I10.i2.p1.11.m7.2.3.2.2.cmml" xref="S4.I10.i2.p1.11.m7.2.3.2.2">𝑢</ci><ci id="S4.I10.i2.p1.11.m7.2.3.2.3.cmml" xref="S4.I10.i2.p1.11.m7.2.3.2.3">′</ci></apply><set id="S4.I10.i2.p1.11.m7.2.3.3.1.cmml" xref="S4.I10.i2.p1.11.m7.2.3.3.2"><ci id="S4.I10.i2.p1.11.m7.1.1.cmml" xref="S4.I10.i2.p1.11.m7.1.1">𝑎</ci><ci id="S4.I10.i2.p1.11.m7.2.2.cmml" xref="S4.I10.i2.p1.11.m7.2.2">𝑏</ci></set></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.I10.i2.p1.11.m7.2c">u^{\prime}\notin\{a,b\}</annotation><annotation encoding="application/x-llamapun" id="S4.I10.i2.p1.11.m7.2d">italic_u start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ∉ { italic_a , italic_b }</annotation></semantics></math>, since <math alttext="h(u^{\prime})=h(u)\in T(u)" class="ltx_Math" display="inline" id="S4.I10.i2.p1.12.m8.3"><semantics id="S4.I10.i2.p1.12.m8.3a"><mrow id="S4.I10.i2.p1.12.m8.3.3" xref="S4.I10.i2.p1.12.m8.3.3.cmml"><mrow id="S4.I10.i2.p1.12.m8.3.3.1" xref="S4.I10.i2.p1.12.m8.3.3.1.cmml"><mi id="S4.I10.i2.p1.12.m8.3.3.1.3" xref="S4.I10.i2.p1.12.m8.3.3.1.3.cmml">h</mi><mo id="S4.I10.i2.p1.12.m8.3.3.1.2" xref="S4.I10.i2.p1.12.m8.3.3.1.2.cmml">⁢</mo><mrow id="S4.I10.i2.p1.12.m8.3.3.1.1.1" xref="S4.I10.i2.p1.12.m8.3.3.1.1.1.1.cmml"><mo id="S4.I10.i2.p1.12.m8.3.3.1.1.1.2" stretchy="false" xref="S4.I10.i2.p1.12.m8.3.3.1.1.1.1.cmml">(</mo><msup id="S4.I10.i2.p1.12.m8.3.3.1.1.1.1" xref="S4.I10.i2.p1.12.m8.3.3.1.1.1.1.cmml"><mi id="S4.I10.i2.p1.12.m8.3.3.1.1.1.1.2" xref="S4.I10.i2.p1.12.m8.3.3.1.1.1.1.2.cmml">u</mi><mo id="S4.I10.i2.p1.12.m8.3.3.1.1.1.1.3" xref="S4.I10.i2.p1.12.m8.3.3.1.1.1.1.3.cmml">′</mo></msup><mo id="S4.I10.i2.p1.12.m8.3.3.1.1.1.3" stretchy="false" xref="S4.I10.i2.p1.12.m8.3.3.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S4.I10.i2.p1.12.m8.3.3.3" xref="S4.I10.i2.p1.12.m8.3.3.3.cmml">=</mo><mrow id="S4.I10.i2.p1.12.m8.3.3.4" xref="S4.I10.i2.p1.12.m8.3.3.4.cmml"><mi id="S4.I10.i2.p1.12.m8.3.3.4.2" xref="S4.I10.i2.p1.12.m8.3.3.4.2.cmml">h</mi><mo id="S4.I10.i2.p1.12.m8.3.3.4.1" xref="S4.I10.i2.p1.12.m8.3.3.4.1.cmml">⁢</mo><mrow id="S4.I10.i2.p1.12.m8.3.3.4.3.2" xref="S4.I10.i2.p1.12.m8.3.3.4.cmml"><mo id="S4.I10.i2.p1.12.m8.3.3.4.3.2.1" stretchy="false" xref="S4.I10.i2.p1.12.m8.3.3.4.cmml">(</mo><mi id="S4.I10.i2.p1.12.m8.1.1" xref="S4.I10.i2.p1.12.m8.1.1.cmml">u</mi><mo id="S4.I10.i2.p1.12.m8.3.3.4.3.2.2" stretchy="false" xref="S4.I10.i2.p1.12.m8.3.3.4.cmml">)</mo></mrow></mrow><mo id="S4.I10.i2.p1.12.m8.3.3.5" xref="S4.I10.i2.p1.12.m8.3.3.5.cmml">∈</mo><mrow id="S4.I10.i2.p1.12.m8.3.3.6" xref="S4.I10.i2.p1.12.m8.3.3.6.cmml"><mi id="S4.I10.i2.p1.12.m8.3.3.6.2" xref="S4.I10.i2.p1.12.m8.3.3.6.2.cmml">T</mi><mo id="S4.I10.i2.p1.12.m8.3.3.6.1" xref="S4.I10.i2.p1.12.m8.3.3.6.1.cmml">⁢</mo><mrow id="S4.I10.i2.p1.12.m8.3.3.6.3.2" xref="S4.I10.i2.p1.12.m8.3.3.6.cmml"><mo id="S4.I10.i2.p1.12.m8.3.3.6.3.2.1" stretchy="false" xref="S4.I10.i2.p1.12.m8.3.3.6.cmml">(</mo><mi id="S4.I10.i2.p1.12.m8.2.2" xref="S4.I10.i2.p1.12.m8.2.2.cmml">u</mi><mo id="S4.I10.i2.p1.12.m8.3.3.6.3.2.2" stretchy="false" xref="S4.I10.i2.p1.12.m8.3.3.6.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.I10.i2.p1.12.m8.3b"><apply id="S4.I10.i2.p1.12.m8.3.3.cmml" xref="S4.I10.i2.p1.12.m8.3.3"><and id="S4.I10.i2.p1.12.m8.3.3a.cmml" xref="S4.I10.i2.p1.12.m8.3.3"></and><apply id="S4.I10.i2.p1.12.m8.3.3b.cmml" xref="S4.I10.i2.p1.12.m8.3.3"><eq id="S4.I10.i2.p1.12.m8.3.3.3.cmml" xref="S4.I10.i2.p1.12.m8.3.3.3"></eq><apply id="S4.I10.i2.p1.12.m8.3.3.1.cmml" xref="S4.I10.i2.p1.12.m8.3.3.1"><times id="S4.I10.i2.p1.12.m8.3.3.1.2.cmml" xref="S4.I10.i2.p1.12.m8.3.3.1.2"></times><ci id="S4.I10.i2.p1.12.m8.3.3.1.3.cmml" xref="S4.I10.i2.p1.12.m8.3.3.1.3">ℎ</ci><apply id="S4.I10.i2.p1.12.m8.3.3.1.1.1.1.cmml" xref="S4.I10.i2.p1.12.m8.3.3.1.1.1"><csymbol cd="ambiguous" id="S4.I10.i2.p1.12.m8.3.3.1.1.1.1.1.cmml" xref="S4.I10.i2.p1.12.m8.3.3.1.1.1">superscript</csymbol><ci id="S4.I10.i2.p1.12.m8.3.3.1.1.1.1.2.cmml" xref="S4.I10.i2.p1.12.m8.3.3.1.1.1.1.2">𝑢</ci><ci id="S4.I10.i2.p1.12.m8.3.3.1.1.1.1.3.cmml" xref="S4.I10.i2.p1.12.m8.3.3.1.1.1.1.3">′</ci></apply></apply><apply id="S4.I10.i2.p1.12.m8.3.3.4.cmml" xref="S4.I10.i2.p1.12.m8.3.3.4"><times id="S4.I10.i2.p1.12.m8.3.3.4.1.cmml" xref="S4.I10.i2.p1.12.m8.3.3.4.1"></times><ci id="S4.I10.i2.p1.12.m8.3.3.4.2.cmml" xref="S4.I10.i2.p1.12.m8.3.3.4.2">ℎ</ci><ci id="S4.I10.i2.p1.12.m8.1.1.cmml" xref="S4.I10.i2.p1.12.m8.1.1">𝑢</ci></apply></apply><apply id="S4.I10.i2.p1.12.m8.3.3c.cmml" xref="S4.I10.i2.p1.12.m8.3.3"><in id="S4.I10.i2.p1.12.m8.3.3.5.cmml" xref="S4.I10.i2.p1.12.m8.3.3.5"></in><share href="https://arxiv.org/html/2503.00712v1#S4.I10.i2.p1.12.m8.3.3.4.cmml" id="S4.I10.i2.p1.12.m8.3.3d.cmml" xref="S4.I10.i2.p1.12.m8.3.3"></share><apply id="S4.I10.i2.p1.12.m8.3.3.6.cmml" xref="S4.I10.i2.p1.12.m8.3.3.6"><times id="S4.I10.i2.p1.12.m8.3.3.6.1.cmml" xref="S4.I10.i2.p1.12.m8.3.3.6.1"></times><ci id="S4.I10.i2.p1.12.m8.3.3.6.2.cmml" xref="S4.I10.i2.p1.12.m8.3.3.6.2">𝑇</ci><ci id="S4.I10.i2.p1.12.m8.2.2.cmml" xref="S4.I10.i2.p1.12.m8.2.2">𝑢</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.I10.i2.p1.12.m8.3c">h(u^{\prime})=h(u)\in T(u)</annotation><annotation encoding="application/x-llamapun" id="S4.I10.i2.p1.12.m8.3d">italic_h ( italic_u start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) = italic_h ( italic_u ) ∈ italic_T ( italic_u )</annotation></semantics></math> and <math alttext="h(a),h(b)\in T\setminus T_{x}" class="ltx_Math" display="inline" id="S4.I10.i2.p1.13.m9.4"><semantics id="S4.I10.i2.p1.13.m9.4a"><mrow id="S4.I10.i2.p1.13.m9.4.4" xref="S4.I10.i2.p1.13.m9.4.4.cmml"><mrow id="S4.I10.i2.p1.13.m9.4.4.2.2" xref="S4.I10.i2.p1.13.m9.4.4.2.3.cmml"><mrow id="S4.I10.i2.p1.13.m9.3.3.1.1.1" xref="S4.I10.i2.p1.13.m9.3.3.1.1.1.cmml"><mi id="S4.I10.i2.p1.13.m9.3.3.1.1.1.2" xref="S4.I10.i2.p1.13.m9.3.3.1.1.1.2.cmml">h</mi><mo id="S4.I10.i2.p1.13.m9.3.3.1.1.1.1" xref="S4.I10.i2.p1.13.m9.3.3.1.1.1.1.cmml">⁢</mo><mrow id="S4.I10.i2.p1.13.m9.3.3.1.1.1.3.2" xref="S4.I10.i2.p1.13.m9.3.3.1.1.1.cmml"><mo id="S4.I10.i2.p1.13.m9.3.3.1.1.1.3.2.1" stretchy="false" xref="S4.I10.i2.p1.13.m9.3.3.1.1.1.cmml">(</mo><mi id="S4.I10.i2.p1.13.m9.1.1" xref="S4.I10.i2.p1.13.m9.1.1.cmml">a</mi><mo id="S4.I10.i2.p1.13.m9.3.3.1.1.1.3.2.2" stretchy="false" xref="S4.I10.i2.p1.13.m9.3.3.1.1.1.cmml">)</mo></mrow></mrow><mo id="S4.I10.i2.p1.13.m9.4.4.2.2.3" xref="S4.I10.i2.p1.13.m9.4.4.2.3.cmml">,</mo><mrow id="S4.I10.i2.p1.13.m9.4.4.2.2.2" xref="S4.I10.i2.p1.13.m9.4.4.2.2.2.cmml"><mi id="S4.I10.i2.p1.13.m9.4.4.2.2.2.2" xref="S4.I10.i2.p1.13.m9.4.4.2.2.2.2.cmml">h</mi><mo id="S4.I10.i2.p1.13.m9.4.4.2.2.2.1" xref="S4.I10.i2.p1.13.m9.4.4.2.2.2.1.cmml">⁢</mo><mrow id="S4.I10.i2.p1.13.m9.4.4.2.2.2.3.2" xref="S4.I10.i2.p1.13.m9.4.4.2.2.2.cmml"><mo id="S4.I10.i2.p1.13.m9.4.4.2.2.2.3.2.1" stretchy="false" xref="S4.I10.i2.p1.13.m9.4.4.2.2.2.cmml">(</mo><mi id="S4.I10.i2.p1.13.m9.2.2" xref="S4.I10.i2.p1.13.m9.2.2.cmml">b</mi><mo id="S4.I10.i2.p1.13.m9.4.4.2.2.2.3.2.2" stretchy="false" xref="S4.I10.i2.p1.13.m9.4.4.2.2.2.cmml">)</mo></mrow></mrow></mrow><mo id="S4.I10.i2.p1.13.m9.4.4.3" xref="S4.I10.i2.p1.13.m9.4.4.3.cmml">∈</mo><mrow id="S4.I10.i2.p1.13.m9.4.4.4" xref="S4.I10.i2.p1.13.m9.4.4.4.cmml"><mi id="S4.I10.i2.p1.13.m9.4.4.4.2" xref="S4.I10.i2.p1.13.m9.4.4.4.2.cmml">T</mi><mo id="S4.I10.i2.p1.13.m9.4.4.4.1" xref="S4.I10.i2.p1.13.m9.4.4.4.1.cmml">∖</mo><msub id="S4.I10.i2.p1.13.m9.4.4.4.3" xref="S4.I10.i2.p1.13.m9.4.4.4.3.cmml"><mi id="S4.I10.i2.p1.13.m9.4.4.4.3.2" xref="S4.I10.i2.p1.13.m9.4.4.4.3.2.cmml">T</mi><mi id="S4.I10.i2.p1.13.m9.4.4.4.3.3" xref="S4.I10.i2.p1.13.m9.4.4.4.3.3.cmml">x</mi></msub></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.I10.i2.p1.13.m9.4b"><apply id="S4.I10.i2.p1.13.m9.4.4.cmml" xref="S4.I10.i2.p1.13.m9.4.4"><in id="S4.I10.i2.p1.13.m9.4.4.3.cmml" xref="S4.I10.i2.p1.13.m9.4.4.3"></in><list id="S4.I10.i2.p1.13.m9.4.4.2.3.cmml" xref="S4.I10.i2.p1.13.m9.4.4.2.2"><apply id="S4.I10.i2.p1.13.m9.3.3.1.1.1.cmml" xref="S4.I10.i2.p1.13.m9.3.3.1.1.1"><times id="S4.I10.i2.p1.13.m9.3.3.1.1.1.1.cmml" xref="S4.I10.i2.p1.13.m9.3.3.1.1.1.1"></times><ci id="S4.I10.i2.p1.13.m9.3.3.1.1.1.2.cmml" xref="S4.I10.i2.p1.13.m9.3.3.1.1.1.2">ℎ</ci><ci id="S4.I10.i2.p1.13.m9.1.1.cmml" xref="S4.I10.i2.p1.13.m9.1.1">𝑎</ci></apply><apply id="S4.I10.i2.p1.13.m9.4.4.2.2.2.cmml" xref="S4.I10.i2.p1.13.m9.4.4.2.2.2"><times id="S4.I10.i2.p1.13.m9.4.4.2.2.2.1.cmml" xref="S4.I10.i2.p1.13.m9.4.4.2.2.2.1"></times><ci id="S4.I10.i2.p1.13.m9.4.4.2.2.2.2.cmml" xref="S4.I10.i2.p1.13.m9.4.4.2.2.2.2">ℎ</ci><ci id="S4.I10.i2.p1.13.m9.2.2.cmml" xref="S4.I10.i2.p1.13.m9.2.2">𝑏</ci></apply></list><apply id="S4.I10.i2.p1.13.m9.4.4.4.cmml" xref="S4.I10.i2.p1.13.m9.4.4.4"><setdiff id="S4.I10.i2.p1.13.m9.4.4.4.1.cmml" xref="S4.I10.i2.p1.13.m9.4.4.4.1"></setdiff><ci id="S4.I10.i2.p1.13.m9.4.4.4.2.cmml" xref="S4.I10.i2.p1.13.m9.4.4.4.2">𝑇</ci><apply id="S4.I10.i2.p1.13.m9.4.4.4.3.cmml" xref="S4.I10.i2.p1.13.m9.4.4.4.3"><csymbol cd="ambiguous" id="S4.I10.i2.p1.13.m9.4.4.4.3.1.cmml" xref="S4.I10.i2.p1.13.m9.4.4.4.3">subscript</csymbol><ci id="S4.I10.i2.p1.13.m9.4.4.4.3.2.cmml" xref="S4.I10.i2.p1.13.m9.4.4.4.3.2">𝑇</ci><ci id="S4.I10.i2.p1.13.m9.4.4.4.3.3.cmml" xref="S4.I10.i2.p1.13.m9.4.4.4.3.3">𝑥</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.I10.i2.p1.13.m9.4c">h(a),h(b)\in T\setminus T_{x}</annotation><annotation encoding="application/x-llamapun" id="S4.I10.i2.p1.13.m9.4d">italic_h ( italic_a ) , italic_h ( italic_b ) ∈ italic_T ∖ italic_T start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT</annotation></semantics></math>. By the same reasoning as in Case 1, <math alttext="\ell(u^{\prime\prime})\in T\setminus T_{x}" class="ltx_Math" display="inline" id="S4.I10.i2.p1.14.m10.1"><semantics id="S4.I10.i2.p1.14.m10.1a"><mrow id="S4.I10.i2.p1.14.m10.1.1" xref="S4.I10.i2.p1.14.m10.1.1.cmml"><mrow id="S4.I10.i2.p1.14.m10.1.1.1" xref="S4.I10.i2.p1.14.m10.1.1.1.cmml"><mi id="S4.I10.i2.p1.14.m10.1.1.1.3" mathvariant="normal" xref="S4.I10.i2.p1.14.m10.1.1.1.3.cmml">ℓ</mi><mo id="S4.I10.i2.p1.14.m10.1.1.1.2" xref="S4.I10.i2.p1.14.m10.1.1.1.2.cmml">⁢</mo><mrow id="S4.I10.i2.p1.14.m10.1.1.1.1.1" xref="S4.I10.i2.p1.14.m10.1.1.1.1.1.1.cmml"><mo id="S4.I10.i2.p1.14.m10.1.1.1.1.1.2" stretchy="false" xref="S4.I10.i2.p1.14.m10.1.1.1.1.1.1.cmml">(</mo><msup id="S4.I10.i2.p1.14.m10.1.1.1.1.1.1" xref="S4.I10.i2.p1.14.m10.1.1.1.1.1.1.cmml"><mi id="S4.I10.i2.p1.14.m10.1.1.1.1.1.1.2" xref="S4.I10.i2.p1.14.m10.1.1.1.1.1.1.2.cmml">u</mi><mo id="S4.I10.i2.p1.14.m10.1.1.1.1.1.1.3" xref="S4.I10.i2.p1.14.m10.1.1.1.1.1.1.3.cmml">′′</mo></msup><mo id="S4.I10.i2.p1.14.m10.1.1.1.1.1.3" stretchy="false" xref="S4.I10.i2.p1.14.m10.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S4.I10.i2.p1.14.m10.1.1.2" xref="S4.I10.i2.p1.14.m10.1.1.2.cmml">∈</mo><mrow id="S4.I10.i2.p1.14.m10.1.1.3" xref="S4.I10.i2.p1.14.m10.1.1.3.cmml"><mi id="S4.I10.i2.p1.14.m10.1.1.3.2" xref="S4.I10.i2.p1.14.m10.1.1.3.2.cmml">T</mi><mo id="S4.I10.i2.p1.14.m10.1.1.3.1" xref="S4.I10.i2.p1.14.m10.1.1.3.1.cmml">∖</mo><msub id="S4.I10.i2.p1.14.m10.1.1.3.3" xref="S4.I10.i2.p1.14.m10.1.1.3.3.cmml"><mi id="S4.I10.i2.p1.14.m10.1.1.3.3.2" xref="S4.I10.i2.p1.14.m10.1.1.3.3.2.cmml">T</mi><mi id="S4.I10.i2.p1.14.m10.1.1.3.3.3" xref="S4.I10.i2.p1.14.m10.1.1.3.3.3.cmml">x</mi></msub></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.I10.i2.p1.14.m10.1b"><apply id="S4.I10.i2.p1.14.m10.1.1.cmml" xref="S4.I10.i2.p1.14.m10.1.1"><in id="S4.I10.i2.p1.14.m10.1.1.2.cmml" xref="S4.I10.i2.p1.14.m10.1.1.2"></in><apply id="S4.I10.i2.p1.14.m10.1.1.1.cmml" xref="S4.I10.i2.p1.14.m10.1.1.1"><times id="S4.I10.i2.p1.14.m10.1.1.1.2.cmml" xref="S4.I10.i2.p1.14.m10.1.1.1.2"></times><ci id="S4.I10.i2.p1.14.m10.1.1.1.3.cmml" xref="S4.I10.i2.p1.14.m10.1.1.1.3">ℓ</ci><apply id="S4.I10.i2.p1.14.m10.1.1.1.1.1.1.cmml" xref="S4.I10.i2.p1.14.m10.1.1.1.1.1"><csymbol cd="ambiguous" id="S4.I10.i2.p1.14.m10.1.1.1.1.1.1.1.cmml" xref="S4.I10.i2.p1.14.m10.1.1.1.1.1">superscript</csymbol><ci id="S4.I10.i2.p1.14.m10.1.1.1.1.1.1.2.cmml" xref="S4.I10.i2.p1.14.m10.1.1.1.1.1.1.2">𝑢</ci><ci id="S4.I10.i2.p1.14.m10.1.1.1.1.1.1.3.cmml" xref="S4.I10.i2.p1.14.m10.1.1.1.1.1.1.3">′′</ci></apply></apply><apply id="S4.I10.i2.p1.14.m10.1.1.3.cmml" xref="S4.I10.i2.p1.14.m10.1.1.3"><setdiff id="S4.I10.i2.p1.14.m10.1.1.3.1.cmml" xref="S4.I10.i2.p1.14.m10.1.1.3.1"></setdiff><ci id="S4.I10.i2.p1.14.m10.1.1.3.2.cmml" xref="S4.I10.i2.p1.14.m10.1.1.3.2">𝑇</ci><apply id="S4.I10.i2.p1.14.m10.1.1.3.3.cmml" xref="S4.I10.i2.p1.14.m10.1.1.3.3"><csymbol cd="ambiguous" id="S4.I10.i2.p1.14.m10.1.1.3.3.1.cmml" xref="S4.I10.i2.p1.14.m10.1.1.3.3">subscript</csymbol><ci id="S4.I10.i2.p1.14.m10.1.1.3.3.2.cmml" xref="S4.I10.i2.p1.14.m10.1.1.3.3.2">𝑇</ci><ci id="S4.I10.i2.p1.14.m10.1.1.3.3.3.cmml" xref="S4.I10.i2.p1.14.m10.1.1.3.3.3">𝑥</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.I10.i2.p1.14.m10.1c">\ell(u^{\prime\prime})\in T\setminus T_{x}</annotation><annotation encoding="application/x-llamapun" id="S4.I10.i2.p1.14.m10.1d">roman_ℓ ( italic_u start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT ) ∈ italic_T ∖ italic_T start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT</annotation></semantics></math>, so <math alttext="u^{\prime\prime}\notin\{a,b\}" class="ltx_Math" display="inline" id="S4.I10.i2.p1.15.m11.2"><semantics id="S4.I10.i2.p1.15.m11.2a"><mrow id="S4.I10.i2.p1.15.m11.2.3" xref="S4.I10.i2.p1.15.m11.2.3.cmml"><msup id="S4.I10.i2.p1.15.m11.2.3.2" xref="S4.I10.i2.p1.15.m11.2.3.2.cmml"><mi id="S4.I10.i2.p1.15.m11.2.3.2.2" xref="S4.I10.i2.p1.15.m11.2.3.2.2.cmml">u</mi><mo id="S4.I10.i2.p1.15.m11.2.3.2.3" xref="S4.I10.i2.p1.15.m11.2.3.2.3.cmml">′′</mo></msup><mo id="S4.I10.i2.p1.15.m11.2.3.1" xref="S4.I10.i2.p1.15.m11.2.3.1.cmml">∉</mo><mrow id="S4.I10.i2.p1.15.m11.2.3.3.2" xref="S4.I10.i2.p1.15.m11.2.3.3.1.cmml"><mo id="S4.I10.i2.p1.15.m11.2.3.3.2.1" stretchy="false" xref="S4.I10.i2.p1.15.m11.2.3.3.1.cmml">{</mo><mi id="S4.I10.i2.p1.15.m11.1.1" xref="S4.I10.i2.p1.15.m11.1.1.cmml">a</mi><mo id="S4.I10.i2.p1.15.m11.2.3.3.2.2" xref="S4.I10.i2.p1.15.m11.2.3.3.1.cmml">,</mo><mi id="S4.I10.i2.p1.15.m11.2.2" xref="S4.I10.i2.p1.15.m11.2.2.cmml">b</mi><mo id="S4.I10.i2.p1.15.m11.2.3.3.2.3" stretchy="false" xref="S4.I10.i2.p1.15.m11.2.3.3.1.cmml">}</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.I10.i2.p1.15.m11.2b"><apply id="S4.I10.i2.p1.15.m11.2.3.cmml" xref="S4.I10.i2.p1.15.m11.2.3"><notin id="S4.I10.i2.p1.15.m11.2.3.1.cmml" xref="S4.I10.i2.p1.15.m11.2.3.1"></notin><apply id="S4.I10.i2.p1.15.m11.2.3.2.cmml" xref="S4.I10.i2.p1.15.m11.2.3.2"><csymbol cd="ambiguous" id="S4.I10.i2.p1.15.m11.2.3.2.1.cmml" xref="S4.I10.i2.p1.15.m11.2.3.2">superscript</csymbol><ci id="S4.I10.i2.p1.15.m11.2.3.2.2.cmml" xref="S4.I10.i2.p1.15.m11.2.3.2.2">𝑢</ci><ci id="S4.I10.i2.p1.15.m11.2.3.2.3.cmml" xref="S4.I10.i2.p1.15.m11.2.3.2.3">′′</ci></apply><set id="S4.I10.i2.p1.15.m11.2.3.3.1.cmml" xref="S4.I10.i2.p1.15.m11.2.3.3.2"><ci id="S4.I10.i2.p1.15.m11.1.1.cmml" xref="S4.I10.i2.p1.15.m11.1.1">𝑎</ci><ci id="S4.I10.i2.p1.15.m11.2.2.cmml" xref="S4.I10.i2.p1.15.m11.2.2">𝑏</ci></set></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.I10.i2.p1.15.m11.2c">u^{\prime\prime}\notin\{a,b\}</annotation><annotation encoding="application/x-llamapun" id="S4.I10.i2.p1.15.m11.2d">italic_u start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT ∉ { italic_a , italic_b }</annotation></semantics></math>, since <math alttext="\ell(a),\ell(b)\in T_{x}" class="ltx_Math" display="inline" id="S4.I10.i2.p1.16.m12.4"><semantics id="S4.I10.i2.p1.16.m12.4a"><mrow id="S4.I10.i2.p1.16.m12.4.4" xref="S4.I10.i2.p1.16.m12.4.4.cmml"><mrow id="S4.I10.i2.p1.16.m12.4.4.2.2" xref="S4.I10.i2.p1.16.m12.4.4.2.3.cmml"><mrow id="S4.I10.i2.p1.16.m12.3.3.1.1.1" xref="S4.I10.i2.p1.16.m12.3.3.1.1.1.cmml"><mi id="S4.I10.i2.p1.16.m12.3.3.1.1.1.2" mathvariant="normal" xref="S4.I10.i2.p1.16.m12.3.3.1.1.1.2.cmml">ℓ</mi><mo id="S4.I10.i2.p1.16.m12.3.3.1.1.1.1" xref="S4.I10.i2.p1.16.m12.3.3.1.1.1.1.cmml">⁢</mo><mrow id="S4.I10.i2.p1.16.m12.3.3.1.1.1.3.2" xref="S4.I10.i2.p1.16.m12.3.3.1.1.1.cmml"><mo id="S4.I10.i2.p1.16.m12.3.3.1.1.1.3.2.1" stretchy="false" xref="S4.I10.i2.p1.16.m12.3.3.1.1.1.cmml">(</mo><mi id="S4.I10.i2.p1.16.m12.1.1" xref="S4.I10.i2.p1.16.m12.1.1.cmml">a</mi><mo id="S4.I10.i2.p1.16.m12.3.3.1.1.1.3.2.2" stretchy="false" xref="S4.I10.i2.p1.16.m12.3.3.1.1.1.cmml">)</mo></mrow></mrow><mo id="S4.I10.i2.p1.16.m12.4.4.2.2.3" xref="S4.I10.i2.p1.16.m12.4.4.2.3.cmml">,</mo><mrow id="S4.I10.i2.p1.16.m12.4.4.2.2.2" xref="S4.I10.i2.p1.16.m12.4.4.2.2.2.cmml"><mi id="S4.I10.i2.p1.16.m12.4.4.2.2.2.2" mathvariant="normal" xref="S4.I10.i2.p1.16.m12.4.4.2.2.2.2.cmml">ℓ</mi><mo id="S4.I10.i2.p1.16.m12.4.4.2.2.2.1" xref="S4.I10.i2.p1.16.m12.4.4.2.2.2.1.cmml">⁢</mo><mrow id="S4.I10.i2.p1.16.m12.4.4.2.2.2.3.2" xref="S4.I10.i2.p1.16.m12.4.4.2.2.2.cmml"><mo id="S4.I10.i2.p1.16.m12.4.4.2.2.2.3.2.1" stretchy="false" xref="S4.I10.i2.p1.16.m12.4.4.2.2.2.cmml">(</mo><mi id="S4.I10.i2.p1.16.m12.2.2" xref="S4.I10.i2.p1.16.m12.2.2.cmml">b</mi><mo id="S4.I10.i2.p1.16.m12.4.4.2.2.2.3.2.2" stretchy="false" xref="S4.I10.i2.p1.16.m12.4.4.2.2.2.cmml">)</mo></mrow></mrow></mrow><mo id="S4.I10.i2.p1.16.m12.4.4.3" xref="S4.I10.i2.p1.16.m12.4.4.3.cmml">∈</mo><msub id="S4.I10.i2.p1.16.m12.4.4.4" xref="S4.I10.i2.p1.16.m12.4.4.4.cmml"><mi id="S4.I10.i2.p1.16.m12.4.4.4.2" xref="S4.I10.i2.p1.16.m12.4.4.4.2.cmml">T</mi><mi id="S4.I10.i2.p1.16.m12.4.4.4.3" xref="S4.I10.i2.p1.16.m12.4.4.4.3.cmml">x</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S4.I10.i2.p1.16.m12.4b"><apply id="S4.I10.i2.p1.16.m12.4.4.cmml" xref="S4.I10.i2.p1.16.m12.4.4"><in id="S4.I10.i2.p1.16.m12.4.4.3.cmml" xref="S4.I10.i2.p1.16.m12.4.4.3"></in><list id="S4.I10.i2.p1.16.m12.4.4.2.3.cmml" xref="S4.I10.i2.p1.16.m12.4.4.2.2"><apply id="S4.I10.i2.p1.16.m12.3.3.1.1.1.cmml" xref="S4.I10.i2.p1.16.m12.3.3.1.1.1"><times id="S4.I10.i2.p1.16.m12.3.3.1.1.1.1.cmml" xref="S4.I10.i2.p1.16.m12.3.3.1.1.1.1"></times><ci id="S4.I10.i2.p1.16.m12.3.3.1.1.1.2.cmml" xref="S4.I10.i2.p1.16.m12.3.3.1.1.1.2">ℓ</ci><ci id="S4.I10.i2.p1.16.m12.1.1.cmml" xref="S4.I10.i2.p1.16.m12.1.1">𝑎</ci></apply><apply id="S4.I10.i2.p1.16.m12.4.4.2.2.2.cmml" xref="S4.I10.i2.p1.16.m12.4.4.2.2.2"><times id="S4.I10.i2.p1.16.m12.4.4.2.2.2.1.cmml" xref="S4.I10.i2.p1.16.m12.4.4.2.2.2.1"></times><ci id="S4.I10.i2.p1.16.m12.4.4.2.2.2.2.cmml" xref="S4.I10.i2.p1.16.m12.4.4.2.2.2.2">ℓ</ci><ci id="S4.I10.i2.p1.16.m12.2.2.cmml" xref="S4.I10.i2.p1.16.m12.2.2">𝑏</ci></apply></list><apply id="S4.I10.i2.p1.16.m12.4.4.4.cmml" xref="S4.I10.i2.p1.16.m12.4.4.4"><csymbol cd="ambiguous" id="S4.I10.i2.p1.16.m12.4.4.4.1.cmml" xref="S4.I10.i2.p1.16.m12.4.4.4">subscript</csymbol><ci id="S4.I10.i2.p1.16.m12.4.4.4.2.cmml" xref="S4.I10.i2.p1.16.m12.4.4.4.2">𝑇</ci><ci id="S4.I10.i2.p1.16.m12.4.4.4.3.cmml" xref="S4.I10.i2.p1.16.m12.4.4.4.3">𝑥</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.I10.i2.p1.16.m12.4c">\ell(a),\ell(b)\in T_{x}</annotation><annotation encoding="application/x-llamapun" id="S4.I10.i2.p1.16.m12.4d">roman_ℓ ( italic_a ) , roman_ℓ ( italic_b ) ∈ italic_T start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT</annotation></semantics></math>. Thus <math alttext="E\cup\textnormal{OPT}" class="ltx_Math" display="inline" id="S4.I10.i2.p1.17.m13.1"><semantics id="S4.I10.i2.p1.17.m13.1a"><mrow id="S4.I10.i2.p1.17.m13.1.1" xref="S4.I10.i2.p1.17.m13.1.1.cmml"><mi id="S4.I10.i2.p1.17.m13.1.1.2" xref="S4.I10.i2.p1.17.m13.1.1.2.cmml">E</mi><mo id="S4.I10.i2.p1.17.m13.1.1.1" xref="S4.I10.i2.p1.17.m13.1.1.1.cmml">∪</mo><mtext id="S4.I10.i2.p1.17.m13.1.1.3" xref="S4.I10.i2.p1.17.m13.1.1.3a.cmml">OPT</mtext></mrow><annotation-xml encoding="MathML-Content" id="S4.I10.i2.p1.17.m13.1b"><apply id="S4.I10.i2.p1.17.m13.1.1.cmml" xref="S4.I10.i2.p1.17.m13.1.1"><union id="S4.I10.i2.p1.17.m13.1.1.1.cmml" xref="S4.I10.i2.p1.17.m13.1.1.1"></union><ci id="S4.I10.i2.p1.17.m13.1.1.2.cmml" xref="S4.I10.i2.p1.17.m13.1.1.2">𝐸</ci><ci id="S4.I10.i2.p1.17.m13.1.1.3a.cmml" xref="S4.I10.i2.p1.17.m13.1.1.3"><mtext id="S4.I10.i2.p1.17.m13.1.1.3.cmml" xref="S4.I10.i2.p1.17.m13.1.1.3">OPT</mtext></ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.I10.i2.p1.17.m13.1c">E\cup\textnormal{OPT}</annotation><annotation encoding="application/x-llamapun" id="S4.I10.i2.p1.17.m13.1d">italic_E ∪ OPT</annotation></semantics></math> contains a <math alttext="u" class="ltx_Math" display="inline" id="S4.I10.i2.p1.18.m14.1"><semantics id="S4.I10.i2.p1.18.m14.1a"><mi id="S4.I10.i2.p1.18.m14.1.1" xref="S4.I10.i2.p1.18.m14.1.1.cmml">u</mi><annotation-xml encoding="MathML-Content" id="S4.I10.i2.p1.18.m14.1b"><ci id="S4.I10.i2.p1.18.m14.1.1.cmml" xref="S4.I10.i2.p1.18.m14.1.1">𝑢</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.I10.i2.p1.18.m14.1c">u</annotation><annotation encoding="application/x-llamapun" id="S4.I10.i2.p1.18.m14.1d">italic_u</annotation></semantics></math>-<math alttext="v" class="ltx_Math" display="inline" id="S4.I10.i2.p1.19.m15.1"><semantics id="S4.I10.i2.p1.19.m15.1a"><mi id="S4.I10.i2.p1.19.m15.1.1" xref="S4.I10.i2.p1.19.m15.1.1.cmml">v</mi><annotation-xml encoding="MathML-Content" id="S4.I10.i2.p1.19.m15.1b"><ci id="S4.I10.i2.p1.19.m15.1.1.cmml" xref="S4.I10.i2.p1.19.m15.1.1">𝑣</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.I10.i2.p1.19.m15.1c">v</annotation><annotation encoding="application/x-llamapun" id="S4.I10.i2.p1.19.m15.1d">italic_v</annotation></semantics></math> path avoiding <math alttext="\{a,b\}" class="ltx_Math" display="inline" id="S4.I10.i2.p1.20.m16.2"><semantics id="S4.I10.i2.p1.20.m16.2a"><mrow id="S4.I10.i2.p1.20.m16.2.3.2" xref="S4.I10.i2.p1.20.m16.2.3.1.cmml"><mo id="S4.I10.i2.p1.20.m16.2.3.2.1" stretchy="false" xref="S4.I10.i2.p1.20.m16.2.3.1.cmml">{</mo><mi id="S4.I10.i2.p1.20.m16.1.1" xref="S4.I10.i2.p1.20.m16.1.1.cmml">a</mi><mo id="S4.I10.i2.p1.20.m16.2.3.2.2" xref="S4.I10.i2.p1.20.m16.2.3.1.cmml">,</mo><mi id="S4.I10.i2.p1.20.m16.2.2" xref="S4.I10.i2.p1.20.m16.2.2.cmml">b</mi><mo id="S4.I10.i2.p1.20.m16.2.3.2.3" stretchy="false" xref="S4.I10.i2.p1.20.m16.2.3.1.cmml">}</mo></mrow><annotation-xml encoding="MathML-Content" id="S4.I10.i2.p1.20.m16.2b"><set id="S4.I10.i2.p1.20.m16.2.3.1.cmml" xref="S4.I10.i2.p1.20.m16.2.3.2"><ci id="S4.I10.i2.p1.20.m16.1.1.cmml" xref="S4.I10.i2.p1.20.m16.1.1">𝑎</ci><ci id="S4.I10.i2.p1.20.m16.2.2.cmml" xref="S4.I10.i2.p1.20.m16.2.2">𝑏</ci></set></annotation-xml><annotation encoding="application/x-tex" id="S4.I10.i2.p1.20.m16.2c">\{a,b\}</annotation><annotation encoding="application/x-llamapun" id="S4.I10.i2.p1.20.m16.2d">{ italic_a , italic_b }</annotation></semantics></math> via the link <math alttext="(u^{\prime},u^{\prime\prime})" class="ltx_Math" display="inline" id="S4.I10.i2.p1.21.m17.2"><semantics id="S4.I10.i2.p1.21.m17.2a"><mrow id="S4.I10.i2.p1.21.m17.2.2.2" xref="S4.I10.i2.p1.21.m17.2.2.3.cmml"><mo id="S4.I10.i2.p1.21.m17.2.2.2.3" stretchy="false" xref="S4.I10.i2.p1.21.m17.2.2.3.cmml">(</mo><msup id="S4.I10.i2.p1.21.m17.1.1.1.1" xref="S4.I10.i2.p1.21.m17.1.1.1.1.cmml"><mi id="S4.I10.i2.p1.21.m17.1.1.1.1.2" xref="S4.I10.i2.p1.21.m17.1.1.1.1.2.cmml">u</mi><mo id="S4.I10.i2.p1.21.m17.1.1.1.1.3" xref="S4.I10.i2.p1.21.m17.1.1.1.1.3.cmml">′</mo></msup><mo id="S4.I10.i2.p1.21.m17.2.2.2.4" xref="S4.I10.i2.p1.21.m17.2.2.3.cmml">,</mo><msup id="S4.I10.i2.p1.21.m17.2.2.2.2" xref="S4.I10.i2.p1.21.m17.2.2.2.2.cmml"><mi id="S4.I10.i2.p1.21.m17.2.2.2.2.2" xref="S4.I10.i2.p1.21.m17.2.2.2.2.2.cmml">u</mi><mo id="S4.I10.i2.p1.21.m17.2.2.2.2.3" xref="S4.I10.i2.p1.21.m17.2.2.2.2.3.cmml">′′</mo></msup><mo id="S4.I10.i2.p1.21.m17.2.2.2.5" stretchy="false" xref="S4.I10.i2.p1.21.m17.2.2.3.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S4.I10.i2.p1.21.m17.2b"><interval closure="open" id="S4.I10.i2.p1.21.m17.2.2.3.cmml" xref="S4.I10.i2.p1.21.m17.2.2.2"><apply id="S4.I10.i2.p1.21.m17.1.1.1.1.cmml" xref="S4.I10.i2.p1.21.m17.1.1.1.1"><csymbol cd="ambiguous" id="S4.I10.i2.p1.21.m17.1.1.1.1.1.cmml" xref="S4.I10.i2.p1.21.m17.1.1.1.1">superscript</csymbol><ci id="S4.I10.i2.p1.21.m17.1.1.1.1.2.cmml" xref="S4.I10.i2.p1.21.m17.1.1.1.1.2">𝑢</ci><ci id="S4.I10.i2.p1.21.m17.1.1.1.1.3.cmml" xref="S4.I10.i2.p1.21.m17.1.1.1.1.3">′</ci></apply><apply id="S4.I10.i2.p1.21.m17.2.2.2.2.cmml" xref="S4.I10.i2.p1.21.m17.2.2.2.2"><csymbol cd="ambiguous" id="S4.I10.i2.p1.21.m17.2.2.2.2.1.cmml" xref="S4.I10.i2.p1.21.m17.2.2.2.2">superscript</csymbol><ci id="S4.I10.i2.p1.21.m17.2.2.2.2.2.cmml" xref="S4.I10.i2.p1.21.m17.2.2.2.2.2">𝑢</ci><ci id="S4.I10.i2.p1.21.m17.2.2.2.2.3.cmml" xref="S4.I10.i2.p1.21.m17.2.2.2.2.3">′′</ci></apply></interval></annotation-xml><annotation encoding="application/x-tex" id="S4.I10.i2.p1.21.m17.2c">(u^{\prime},u^{\prime\prime})</annotation><annotation encoding="application/x-llamapun" id="S4.I10.i2.p1.21.m17.2d">( italic_u start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT , italic_u start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT )</annotation></semantics></math>.</p> </div> </li> </ul> </div> <figure class="ltx_figure" id="S4.F9"> <div class="ltx_flex_figure"> <div class="ltx_flex_cell ltx_flex_size_2"> <figure class="ltx_figure ltx_figure_panel ltx_align_center" id="S4.F9.sf1"><img alt="Refer to caption" class="ltx_graphics ltx_img_square" height="208" id="S4.F9.sf1.g1" src="x10.png" width="193"/> <figcaption class="ltx_caption"><span class="ltx_tag ltx_tag_figure"><span class="ltx_text" id="S4.F9.sf1.8.4.1" style="font-size:90%;">(a)</span> </span><span class="ltx_text" id="S4.F9.sf1.6.3" style="font-size:90%;">Either <math alttext="T(v)" class="ltx_Math" display="inline" id="S4.F9.sf1.4.1.m1.1"><semantics id="S4.F9.sf1.4.1.m1.1b"><mrow id="S4.F9.sf1.4.1.m1.1.2" xref="S4.F9.sf1.4.1.m1.1.2.cmml"><mi id="S4.F9.sf1.4.1.m1.1.2.2" xref="S4.F9.sf1.4.1.m1.1.2.2.cmml">T</mi><mo id="S4.F9.sf1.4.1.m1.1.2.1" xref="S4.F9.sf1.4.1.m1.1.2.1.cmml">⁢</mo><mrow id="S4.F9.sf1.4.1.m1.1.2.3.2" xref="S4.F9.sf1.4.1.m1.1.2.cmml"><mo id="S4.F9.sf1.4.1.m1.1.2.3.2.1" stretchy="false" xref="S4.F9.sf1.4.1.m1.1.2.cmml">(</mo><mi id="S4.F9.sf1.4.1.m1.1.1" xref="S4.F9.sf1.4.1.m1.1.1.cmml">v</mi><mo id="S4.F9.sf1.4.1.m1.1.2.3.2.2" stretchy="false" xref="S4.F9.sf1.4.1.m1.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.F9.sf1.4.1.m1.1c"><apply id="S4.F9.sf1.4.1.m1.1.2.cmml" xref="S4.F9.sf1.4.1.m1.1.2"><times id="S4.F9.sf1.4.1.m1.1.2.1.cmml" xref="S4.F9.sf1.4.1.m1.1.2.1"></times><ci id="S4.F9.sf1.4.1.m1.1.2.2.cmml" xref="S4.F9.sf1.4.1.m1.1.2.2">𝑇</ci><ci id="S4.F9.sf1.4.1.m1.1.1.cmml" xref="S4.F9.sf1.4.1.m1.1.1">𝑣</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.F9.sf1.4.1.m1.1d">T(v)</annotation><annotation encoding="application/x-llamapun" id="S4.F9.sf1.4.1.m1.1e">italic_T ( italic_v )</annotation></semantics></math> or <math alttext="T(u)" class="ltx_Math" display="inline" id="S4.F9.sf1.5.2.m2.1"><semantics id="S4.F9.sf1.5.2.m2.1b"><mrow id="S4.F9.sf1.5.2.m2.1.2" xref="S4.F9.sf1.5.2.m2.1.2.cmml"><mi id="S4.F9.sf1.5.2.m2.1.2.2" xref="S4.F9.sf1.5.2.m2.1.2.2.cmml">T</mi><mo id="S4.F9.sf1.5.2.m2.1.2.1" xref="S4.F9.sf1.5.2.m2.1.2.1.cmml">⁢</mo><mrow id="S4.F9.sf1.5.2.m2.1.2.3.2" xref="S4.F9.sf1.5.2.m2.1.2.cmml"><mo id="S4.F9.sf1.5.2.m2.1.2.3.2.1" stretchy="false" xref="S4.F9.sf1.5.2.m2.1.2.cmml">(</mo><mi id="S4.F9.sf1.5.2.m2.1.1" xref="S4.F9.sf1.5.2.m2.1.1.cmml">u</mi><mo id="S4.F9.sf1.5.2.m2.1.2.3.2.2" stretchy="false" xref="S4.F9.sf1.5.2.m2.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.F9.sf1.5.2.m2.1c"><apply id="S4.F9.sf1.5.2.m2.1.2.cmml" xref="S4.F9.sf1.5.2.m2.1.2"><times id="S4.F9.sf1.5.2.m2.1.2.1.cmml" xref="S4.F9.sf1.5.2.m2.1.2.1"></times><ci id="S4.F9.sf1.5.2.m2.1.2.2.cmml" xref="S4.F9.sf1.5.2.m2.1.2.2">𝑇</ci><ci id="S4.F9.sf1.5.2.m2.1.1.cmml" xref="S4.F9.sf1.5.2.m2.1.1">𝑢</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.F9.sf1.5.2.m2.1d">T(u)</annotation><annotation encoding="application/x-llamapun" id="S4.F9.sf1.5.2.m2.1e">italic_T ( italic_u )</annotation></semantics></math> are not good; then they are connected via the MST <math alttext="H_{x}^{\prime}" class="ltx_Math" display="inline" id="S4.F9.sf1.6.3.m3.1"><semantics id="S4.F9.sf1.6.3.m3.1b"><msubsup id="S4.F9.sf1.6.3.m3.1.1" xref="S4.F9.sf1.6.3.m3.1.1.cmml"><mi id="S4.F9.sf1.6.3.m3.1.1.2.2" xref="S4.F9.sf1.6.3.m3.1.1.2.2.cmml">H</mi><mi id="S4.F9.sf1.6.3.m3.1.1.2.3" xref="S4.F9.sf1.6.3.m3.1.1.2.3.cmml">x</mi><mo id="S4.F9.sf1.6.3.m3.1.1.3" xref="S4.F9.sf1.6.3.m3.1.1.3.cmml">′</mo></msubsup><annotation-xml encoding="MathML-Content" id="S4.F9.sf1.6.3.m3.1c"><apply id="S4.F9.sf1.6.3.m3.1.1.cmml" xref="S4.F9.sf1.6.3.m3.1.1"><csymbol cd="ambiguous" id="S4.F9.sf1.6.3.m3.1.1.1.cmml" xref="S4.F9.sf1.6.3.m3.1.1">superscript</csymbol><apply id="S4.F9.sf1.6.3.m3.1.1.2.cmml" xref="S4.F9.sf1.6.3.m3.1.1"><csymbol cd="ambiguous" id="S4.F9.sf1.6.3.m3.1.1.2.1.cmml" xref="S4.F9.sf1.6.3.m3.1.1">subscript</csymbol><ci id="S4.F9.sf1.6.3.m3.1.1.2.2.cmml" xref="S4.F9.sf1.6.3.m3.1.1.2.2">𝐻</ci><ci id="S4.F9.sf1.6.3.m3.1.1.2.3.cmml" xref="S4.F9.sf1.6.3.m3.1.1.2.3">𝑥</ci></apply><ci id="S4.F9.sf1.6.3.m3.1.1.3.cmml" xref="S4.F9.sf1.6.3.m3.1.1.3">′</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.F9.sf1.6.3.m3.1d">H_{x}^{\prime}</annotation><annotation encoding="application/x-llamapun" id="S4.F9.sf1.6.3.m3.1e">italic_H start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math>.</span></figcaption> </figure> </div> <div class="ltx_flex_cell ltx_flex_size_2"> <figure class="ltx_figure ltx_figure_panel ltx_align_center" id="S4.F9.sf2"><img alt="Refer to caption" class="ltx_graphics ltx_img_landscape" height="228" id="S4.F9.sf2.g1" src="x11.png" width="375"/> <figcaption class="ltx_caption"><span class="ltx_tag ltx_tag_figure"><span class="ltx_text" id="S4.F9.sf2.10.5.1" style="font-size:90%;">(b)</span> </span><span class="ltx_text" id="S4.F9.sf2.8.4" style="font-size:90%;">Both <math alttext="T(u)" class="ltx_Math" display="inline" id="S4.F9.sf2.5.1.m1.1"><semantics id="S4.F9.sf2.5.1.m1.1b"><mrow id="S4.F9.sf2.5.1.m1.1.2" xref="S4.F9.sf2.5.1.m1.1.2.cmml"><mi id="S4.F9.sf2.5.1.m1.1.2.2" xref="S4.F9.sf2.5.1.m1.1.2.2.cmml">T</mi><mo id="S4.F9.sf2.5.1.m1.1.2.1" xref="S4.F9.sf2.5.1.m1.1.2.1.cmml">⁢</mo><mrow id="S4.F9.sf2.5.1.m1.1.2.3.2" xref="S4.F9.sf2.5.1.m1.1.2.cmml"><mo id="S4.F9.sf2.5.1.m1.1.2.3.2.1" stretchy="false" xref="S4.F9.sf2.5.1.m1.1.2.cmml">(</mo><mi id="S4.F9.sf2.5.1.m1.1.1" xref="S4.F9.sf2.5.1.m1.1.1.cmml">u</mi><mo id="S4.F9.sf2.5.1.m1.1.2.3.2.2" stretchy="false" xref="S4.F9.sf2.5.1.m1.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.F9.sf2.5.1.m1.1c"><apply id="S4.F9.sf2.5.1.m1.1.2.cmml" xref="S4.F9.sf2.5.1.m1.1.2"><times id="S4.F9.sf2.5.1.m1.1.2.1.cmml" xref="S4.F9.sf2.5.1.m1.1.2.1"></times><ci id="S4.F9.sf2.5.1.m1.1.2.2.cmml" xref="S4.F9.sf2.5.1.m1.1.2.2">𝑇</ci><ci id="S4.F9.sf2.5.1.m1.1.1.cmml" xref="S4.F9.sf2.5.1.m1.1.1">𝑢</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.F9.sf2.5.1.m1.1d">T(u)</annotation><annotation encoding="application/x-llamapun" id="S4.F9.sf2.5.1.m1.1e">italic_T ( italic_u )</annotation></semantics></math> and <math alttext="T(v)" class="ltx_Math" display="inline" id="S4.F9.sf2.6.2.m2.1"><semantics id="S4.F9.sf2.6.2.m2.1b"><mrow id="S4.F9.sf2.6.2.m2.1.2" xref="S4.F9.sf2.6.2.m2.1.2.cmml"><mi id="S4.F9.sf2.6.2.m2.1.2.2" xref="S4.F9.sf2.6.2.m2.1.2.2.cmml">T</mi><mo id="S4.F9.sf2.6.2.m2.1.2.1" xref="S4.F9.sf2.6.2.m2.1.2.1.cmml">⁢</mo><mrow id="S4.F9.sf2.6.2.m2.1.2.3.2" xref="S4.F9.sf2.6.2.m2.1.2.cmml"><mo id="S4.F9.sf2.6.2.m2.1.2.3.2.1" stretchy="false" xref="S4.F9.sf2.6.2.m2.1.2.cmml">(</mo><mi id="S4.F9.sf2.6.2.m2.1.1" xref="S4.F9.sf2.6.2.m2.1.1.cmml">v</mi><mo id="S4.F9.sf2.6.2.m2.1.2.3.2.2" stretchy="false" xref="S4.F9.sf2.6.2.m2.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.F9.sf2.6.2.m2.1c"><apply id="S4.F9.sf2.6.2.m2.1.2.cmml" xref="S4.F9.sf2.6.2.m2.1.2"><times id="S4.F9.sf2.6.2.m2.1.2.1.cmml" xref="S4.F9.sf2.6.2.m2.1.2.1"></times><ci id="S4.F9.sf2.6.2.m2.1.2.2.cmml" xref="S4.F9.sf2.6.2.m2.1.2.2">𝑇</ci><ci id="S4.F9.sf2.6.2.m2.1.1.cmml" xref="S4.F9.sf2.6.2.m2.1.1">𝑣</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.F9.sf2.6.2.m2.1d">T(v)</annotation><annotation encoding="application/x-llamapun" id="S4.F9.sf2.6.2.m2.1e">italic_T ( italic_v )</annotation></semantics></math> are good. Note that <math alttext="u" class="ltx_Math" display="inline" id="S4.F9.sf2.7.3.m3.1"><semantics id="S4.F9.sf2.7.3.m3.1b"><mi id="S4.F9.sf2.7.3.m3.1.1" xref="S4.F9.sf2.7.3.m3.1.1.cmml">u</mi><annotation-xml encoding="MathML-Content" id="S4.F9.sf2.7.3.m3.1c"><ci id="S4.F9.sf2.7.3.m3.1.1.cmml" xref="S4.F9.sf2.7.3.m3.1.1">𝑢</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.F9.sf2.7.3.m3.1d">u</annotation><annotation encoding="application/x-llamapun" id="S4.F9.sf2.7.3.m3.1e">italic_u</annotation></semantics></math> may be in <math alttext="h(u_{1})" class="ltx_Math" display="inline" id="S4.F9.sf2.8.4.m4.1"><semantics id="S4.F9.sf2.8.4.m4.1b"><mrow id="S4.F9.sf2.8.4.m4.1.1" xref="S4.F9.sf2.8.4.m4.1.1.cmml"><mi id="S4.F9.sf2.8.4.m4.1.1.3" xref="S4.F9.sf2.8.4.m4.1.1.3.cmml">h</mi><mo id="S4.F9.sf2.8.4.m4.1.1.2" xref="S4.F9.sf2.8.4.m4.1.1.2.cmml">⁢</mo><mrow id="S4.F9.sf2.8.4.m4.1.1.1.1" xref="S4.F9.sf2.8.4.m4.1.1.1.1.1.cmml"><mo id="S4.F9.sf2.8.4.m4.1.1.1.1.2" stretchy="false" xref="S4.F9.sf2.8.4.m4.1.1.1.1.1.cmml">(</mo><msub id="S4.F9.sf2.8.4.m4.1.1.1.1.1" xref="S4.F9.sf2.8.4.m4.1.1.1.1.1.cmml"><mi id="S4.F9.sf2.8.4.m4.1.1.1.1.1.2" xref="S4.F9.sf2.8.4.m4.1.1.1.1.1.2.cmml">u</mi><mn id="S4.F9.sf2.8.4.m4.1.1.1.1.1.3" xref="S4.F9.sf2.8.4.m4.1.1.1.1.1.3.cmml">1</mn></msub><mo id="S4.F9.sf2.8.4.m4.1.1.1.1.3" stretchy="false" xref="S4.F9.sf2.8.4.m4.1.1.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.F9.sf2.8.4.m4.1c"><apply id="S4.F9.sf2.8.4.m4.1.1.cmml" xref="S4.F9.sf2.8.4.m4.1.1"><times id="S4.F9.sf2.8.4.m4.1.1.2.cmml" xref="S4.F9.sf2.8.4.m4.1.1.2"></times><ci id="S4.F9.sf2.8.4.m4.1.1.3.cmml" xref="S4.F9.sf2.8.4.m4.1.1.3">ℎ</ci><apply id="S4.F9.sf2.8.4.m4.1.1.1.1.1.cmml" xref="S4.F9.sf2.8.4.m4.1.1.1.1"><csymbol cd="ambiguous" id="S4.F9.sf2.8.4.m4.1.1.1.1.1.1.cmml" xref="S4.F9.sf2.8.4.m4.1.1.1.1">subscript</csymbol><ci id="S4.F9.sf2.8.4.m4.1.1.1.1.1.2.cmml" xref="S4.F9.sf2.8.4.m4.1.1.1.1.1.2">𝑢</ci><cn id="S4.F9.sf2.8.4.m4.1.1.1.1.1.3.cmml" type="integer" xref="S4.F9.sf2.8.4.m4.1.1.1.1.1.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.F9.sf2.8.4.m4.1d">h(u_{1})</annotation><annotation encoding="application/x-llamapun" id="S4.F9.sf2.8.4.m4.1e">italic_h ( italic_u start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT )</annotation></semantics></math>, though this is not necessary as demonstrated in the figure.</span></figcaption> </figure> </div> </div> <figcaption class="ltx_caption ltx_centering"><span class="ltx_tag ltx_tag_figure"><span class="ltx_text" id="S4.F9.8.4.1" style="font-size:90%;">Figure 9</span>: </span><span class="ltx_text" id="S4.F9.6.3" style="font-size:90%;">Case 2a example. Tree nodes are shown with boxes, while graph vertices are given by dots. Here <math alttext="x" class="ltx_Math" display="inline" id="S4.F9.4.1.m1.1"><semantics id="S4.F9.4.1.m1.1b"><mi id="S4.F9.4.1.m1.1.1" xref="S4.F9.4.1.m1.1.1.cmml">x</mi><annotation-xml encoding="MathML-Content" id="S4.F9.4.1.m1.1c"><ci id="S4.F9.4.1.m1.1.1.cmml" xref="S4.F9.4.1.m1.1.1">𝑥</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.F9.4.1.m1.1d">x</annotation><annotation encoding="application/x-llamapun" id="S4.F9.4.1.m1.1e">italic_x</annotation></semantics></math> has two subtrees <math alttext="T(u)" class="ltx_Math" display="inline" id="S4.F9.5.2.m2.1"><semantics id="S4.F9.5.2.m2.1b"><mrow id="S4.F9.5.2.m2.1.2" xref="S4.F9.5.2.m2.1.2.cmml"><mi id="S4.F9.5.2.m2.1.2.2" xref="S4.F9.5.2.m2.1.2.2.cmml">T</mi><mo id="S4.F9.5.2.m2.1.2.1" xref="S4.F9.5.2.m2.1.2.1.cmml">⁢</mo><mrow id="S4.F9.5.2.m2.1.2.3.2" xref="S4.F9.5.2.m2.1.2.cmml"><mo id="S4.F9.5.2.m2.1.2.3.2.1" stretchy="false" xref="S4.F9.5.2.m2.1.2.cmml">(</mo><mi id="S4.F9.5.2.m2.1.1" xref="S4.F9.5.2.m2.1.1.cmml">u</mi><mo id="S4.F9.5.2.m2.1.2.3.2.2" stretchy="false" xref="S4.F9.5.2.m2.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.F9.5.2.m2.1c"><apply id="S4.F9.5.2.m2.1.2.cmml" xref="S4.F9.5.2.m2.1.2"><times id="S4.F9.5.2.m2.1.2.1.cmml" xref="S4.F9.5.2.m2.1.2.1"></times><ci id="S4.F9.5.2.m2.1.2.2.cmml" xref="S4.F9.5.2.m2.1.2.2">𝑇</ci><ci id="S4.F9.5.2.m2.1.1.cmml" xref="S4.F9.5.2.m2.1.1">𝑢</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.F9.5.2.m2.1d">T(u)</annotation><annotation encoding="application/x-llamapun" id="S4.F9.5.2.m2.1e">italic_T ( italic_u )</annotation></semantics></math> and <math alttext="T(v)" class="ltx_Math" display="inline" id="S4.F9.6.3.m3.1"><semantics id="S4.F9.6.3.m3.1b"><mrow id="S4.F9.6.3.m3.1.2" xref="S4.F9.6.3.m3.1.2.cmml"><mi id="S4.F9.6.3.m3.1.2.2" xref="S4.F9.6.3.m3.1.2.2.cmml">T</mi><mo id="S4.F9.6.3.m3.1.2.1" xref="S4.F9.6.3.m3.1.2.1.cmml">⁢</mo><mrow id="S4.F9.6.3.m3.1.2.3.2" xref="S4.F9.6.3.m3.1.2.cmml"><mo id="S4.F9.6.3.m3.1.2.3.2.1" stretchy="false" xref="S4.F9.6.3.m3.1.2.cmml">(</mo><mi id="S4.F9.6.3.m3.1.1" xref="S4.F9.6.3.m3.1.1.cmml">v</mi><mo id="S4.F9.6.3.m3.1.2.3.2.2" stretchy="false" xref="S4.F9.6.3.m3.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.F9.6.3.m3.1c"><apply id="S4.F9.6.3.m3.1.2.cmml" xref="S4.F9.6.3.m3.1.2"><times id="S4.F9.6.3.m3.1.2.1.cmml" xref="S4.F9.6.3.m3.1.2.1"></times><ci id="S4.F9.6.3.m3.1.2.2.cmml" xref="S4.F9.6.3.m3.1.2.2">𝑇</ci><ci id="S4.F9.6.3.m3.1.1.cmml" xref="S4.F9.6.3.m3.1.1">𝑣</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.F9.6.3.m3.1d">T(v)</annotation><annotation encoding="application/x-llamapun" id="S4.F9.6.3.m3.1e">italic_T ( italic_v )</annotation></semantics></math>.</span></figcaption> </figure> </section> <section class="ltx_paragraph" id="S4.SS2.SSS3.Px3"> <h5 class="ltx_title ltx_title_paragraph">Case 3: <math alttext="\boldsymbol{a}" class="ltx_Math" display="inline" id="S4.SS2.SSS3.Px3.1.m1.1"><semantics id="S4.SS2.SSS3.Px3.1.m1.1b"><mi id="S4.SS2.SSS3.Px3.1.m1.1.1" xref="S4.SS2.SSS3.Px3.1.m1.1.1.cmml">𝒂</mi><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS3.Px3.1.m1.1c"><ci id="S4.SS2.SSS3.Px3.1.m1.1.1.cmml" xref="S4.SS2.SSS3.Px3.1.m1.1.1">𝒂</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS3.Px3.1.m1.1d">\boldsymbol{a}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS3.Px3.1.m1.1e">bold_italic_a</annotation></semantics></math> and <math alttext="\boldsymbol{b}" class="ltx_Math" display="inline" id="S4.SS2.SSS3.Px3.2.m2.1"><semantics id="S4.SS2.SSS3.Px3.2.m2.1b"><mi id="S4.SS2.SSS3.Px3.2.m2.1.1" xref="S4.SS2.SSS3.Px3.2.m2.1.1.cmml">𝒃</mi><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS3.Px3.2.m2.1c"><ci id="S4.SS2.SSS3.Px3.2.m2.1.1.cmml" xref="S4.SS2.SSS3.Px3.2.m2.1.1">𝒃</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS3.Px3.2.m2.1d">\boldsymbol{b}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS3.Px3.2.m2.1e">bold_italic_b</annotation></semantics></math> are non-adjacent nodes of <math alttext="\boldsymbol{G_{x}}" class="ltx_Math" display="inline" id="S4.SS2.SSS3.Px3.3.m3.1"><semantics id="S4.SS2.SSS3.Px3.3.m3.1b"><msub id="S4.SS2.SSS3.Px3.3.m3.1.1" xref="S4.SS2.SSS3.Px3.3.m3.1.1.cmml"><mi id="S4.SS2.SSS3.Px3.3.m3.1.1.2" xref="S4.SS2.SSS3.Px3.3.m3.1.1.2.cmml">𝑮</mi><mi id="S4.SS2.SSS3.Px3.3.m3.1.1.3" xref="S4.SS2.SSS3.Px3.3.m3.1.1.3.cmml">𝒙</mi></msub><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS3.Px3.3.m3.1c"><apply id="S4.SS2.SSS3.Px3.3.m3.1.1.cmml" xref="S4.SS2.SSS3.Px3.3.m3.1.1"><csymbol cd="ambiguous" id="S4.SS2.SSS3.Px3.3.m3.1.1.1.cmml" xref="S4.SS2.SSS3.Px3.3.m3.1.1">subscript</csymbol><ci id="S4.SS2.SSS3.Px3.3.m3.1.1.2.cmml" xref="S4.SS2.SSS3.Px3.3.m3.1.1.2">𝑮</ci><ci id="S4.SS2.SSS3.Px3.3.m3.1.1.3.cmml" xref="S4.SS2.SSS3.Px3.3.m3.1.1.3">𝒙</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS3.Px3.3.m3.1d">\boldsymbol{G_{x}}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS3.Px3.3.m3.1e">bold_italic_G start_POSTSUBSCRIPT bold_italic_x end_POSTSUBSCRIPT</annotation></semantics></math> for an S-node <math alttext="\boldsymbol{x}" class="ltx_Math" display="inline" id="S4.SS2.SSS3.Px3.4.m4.1"><semantics id="S4.SS2.SSS3.Px3.4.m4.1b"><mi id="S4.SS2.SSS3.Px3.4.m4.1.1" xref="S4.SS2.SSS3.Px3.4.m4.1.1.cmml">𝒙</mi><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS3.Px3.4.m4.1c"><ci id="S4.SS2.SSS3.Px3.4.m4.1.1.cmml" xref="S4.SS2.SSS3.Px3.4.m4.1.1">𝒙</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS3.Px3.4.m4.1d">\boldsymbol{x}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS3.Px3.4.m4.1e">bold_italic_x</annotation></semantics></math>:</h5> <div class="ltx_para" id="S4.SS2.SSS3.Px3.p1"> <p class="ltx_p" id="S4.SS2.SSS3.Px3.p1.20">Let <math alttext="C_{1}" class="ltx_Math" display="inline" id="S4.SS2.SSS3.Px3.p1.1.m1.1"><semantics id="S4.SS2.SSS3.Px3.p1.1.m1.1a"><msub id="S4.SS2.SSS3.Px3.p1.1.m1.1.1" xref="S4.SS2.SSS3.Px3.p1.1.m1.1.1.cmml"><mi id="S4.SS2.SSS3.Px3.p1.1.m1.1.1.2" xref="S4.SS2.SSS3.Px3.p1.1.m1.1.1.2.cmml">C</mi><mn id="S4.SS2.SSS3.Px3.p1.1.m1.1.1.3" xref="S4.SS2.SSS3.Px3.p1.1.m1.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS3.Px3.p1.1.m1.1b"><apply id="S4.SS2.SSS3.Px3.p1.1.m1.1.1.cmml" xref="S4.SS2.SSS3.Px3.p1.1.m1.1.1"><csymbol cd="ambiguous" id="S4.SS2.SSS3.Px3.p1.1.m1.1.1.1.cmml" xref="S4.SS2.SSS3.Px3.p1.1.m1.1.1">subscript</csymbol><ci id="S4.SS2.SSS3.Px3.p1.1.m1.1.1.2.cmml" xref="S4.SS2.SSS3.Px3.p1.1.m1.1.1.2">𝐶</ci><cn id="S4.SS2.SSS3.Px3.p1.1.m1.1.1.3.cmml" type="integer" xref="S4.SS2.SSS3.Px3.p1.1.m1.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS3.Px3.p1.1.m1.1c">C_{1}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS3.Px3.p1.1.m1.1d">italic_C start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="C_{2}" class="ltx_Math" display="inline" id="S4.SS2.SSS3.Px3.p1.2.m2.1"><semantics id="S4.SS2.SSS3.Px3.p1.2.m2.1a"><msub id="S4.SS2.SSS3.Px3.p1.2.m2.1.1" xref="S4.SS2.SSS3.Px3.p1.2.m2.1.1.cmml"><mi id="S4.SS2.SSS3.Px3.p1.2.m2.1.1.2" xref="S4.SS2.SSS3.Px3.p1.2.m2.1.1.2.cmml">C</mi><mn id="S4.SS2.SSS3.Px3.p1.2.m2.1.1.3" xref="S4.SS2.SSS3.Px3.p1.2.m2.1.1.3.cmml">2</mn></msub><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS3.Px3.p1.2.m2.1b"><apply id="S4.SS2.SSS3.Px3.p1.2.m2.1.1.cmml" xref="S4.SS2.SSS3.Px3.p1.2.m2.1.1"><csymbol cd="ambiguous" id="S4.SS2.SSS3.Px3.p1.2.m2.1.1.1.cmml" xref="S4.SS2.SSS3.Px3.p1.2.m2.1.1">subscript</csymbol><ci id="S4.SS2.SSS3.Px3.p1.2.m2.1.1.2.cmml" xref="S4.SS2.SSS3.Px3.p1.2.m2.1.1.2">𝐶</ci><cn id="S4.SS2.SSS3.Px3.p1.2.m2.1.1.3.cmml" type="integer" xref="S4.SS2.SSS3.Px3.p1.2.m2.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS3.Px3.p1.2.m2.1c">C_{2}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS3.Px3.p1.2.m2.1d">italic_C start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT</annotation></semantics></math> be the two components of the cycle <math alttext="G_{x}" class="ltx_Math" display="inline" id="S4.SS2.SSS3.Px3.p1.3.m3.1"><semantics id="S4.SS2.SSS3.Px3.p1.3.m3.1a"><msub id="S4.SS2.SSS3.Px3.p1.3.m3.1.1" xref="S4.SS2.SSS3.Px3.p1.3.m3.1.1.cmml"><mi id="S4.SS2.SSS3.Px3.p1.3.m3.1.1.2" xref="S4.SS2.SSS3.Px3.p1.3.m3.1.1.2.cmml">G</mi><mi id="S4.SS2.SSS3.Px3.p1.3.m3.1.1.3" xref="S4.SS2.SSS3.Px3.p1.3.m3.1.1.3.cmml">x</mi></msub><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS3.Px3.p1.3.m3.1b"><apply id="S4.SS2.SSS3.Px3.p1.3.m3.1.1.cmml" xref="S4.SS2.SSS3.Px3.p1.3.m3.1.1"><csymbol cd="ambiguous" id="S4.SS2.SSS3.Px3.p1.3.m3.1.1.1.cmml" xref="S4.SS2.SSS3.Px3.p1.3.m3.1.1">subscript</csymbol><ci id="S4.SS2.SSS3.Px3.p1.3.m3.1.1.2.cmml" xref="S4.SS2.SSS3.Px3.p1.3.m3.1.1.2">𝐺</ci><ci id="S4.SS2.SSS3.Px3.p1.3.m3.1.1.3.cmml" xref="S4.SS2.SSS3.Px3.p1.3.m3.1.1.3">𝑥</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS3.Px3.p1.3.m3.1c">G_{x}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS3.Px3.p1.3.m3.1d">italic_G start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT</annotation></semantics></math> formed by the removal of <math alttext="a" class="ltx_Math" display="inline" id="S4.SS2.SSS3.Px3.p1.4.m4.1"><semantics id="S4.SS2.SSS3.Px3.p1.4.m4.1a"><mi id="S4.SS2.SSS3.Px3.p1.4.m4.1.1" xref="S4.SS2.SSS3.Px3.p1.4.m4.1.1.cmml">a</mi><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS3.Px3.p1.4.m4.1b"><ci id="S4.SS2.SSS3.Px3.p1.4.m4.1.1.cmml" xref="S4.SS2.SSS3.Px3.p1.4.m4.1.1">𝑎</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS3.Px3.p1.4.m4.1c">a</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS3.Px3.p1.4.m4.1d">italic_a</annotation></semantics></math> and <math alttext="b" class="ltx_Math" display="inline" id="S4.SS2.SSS3.Px3.p1.5.m5.1"><semantics id="S4.SS2.SSS3.Px3.p1.5.m5.1a"><mi id="S4.SS2.SSS3.Px3.p1.5.m5.1.1" xref="S4.SS2.SSS3.Px3.p1.5.m5.1.1.cmml">b</mi><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS3.Px3.p1.5.m5.1b"><ci id="S4.SS2.SSS3.Px3.p1.5.m5.1.1.cmml" xref="S4.SS2.SSS3.Px3.p1.5.m5.1.1">𝑏</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS3.Px3.p1.5.m5.1c">b</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS3.Px3.p1.5.m5.1d">italic_b</annotation></semantics></math>. We assume that <math alttext="u" class="ltx_Math" display="inline" id="S4.SS2.SSS3.Px3.p1.6.m6.1"><semantics id="S4.SS2.SSS3.Px3.p1.6.m6.1a"><mi id="S4.SS2.SSS3.Px3.p1.6.m6.1.1" xref="S4.SS2.SSS3.Px3.p1.6.m6.1.1.cmml">u</mi><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS3.Px3.p1.6.m6.1b"><ci id="S4.SS2.SSS3.Px3.p1.6.m6.1.1.cmml" xref="S4.SS2.SSS3.Px3.p1.6.m6.1.1">𝑢</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS3.Px3.p1.6.m6.1c">u</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS3.Px3.p1.6.m6.1d">italic_u</annotation></semantics></math> and <math alttext="v" class="ltx_Math" display="inline" id="S4.SS2.SSS3.Px3.p1.7.m7.1"><semantics id="S4.SS2.SSS3.Px3.p1.7.m7.1a"><mi id="S4.SS2.SSS3.Px3.p1.7.m7.1.1" xref="S4.SS2.SSS3.Px3.p1.7.m7.1.1.cmml">v</mi><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS3.Px3.p1.7.m7.1b"><ci id="S4.SS2.SSS3.Px3.p1.7.m7.1.1.cmml" xref="S4.SS2.SSS3.Px3.p1.7.m7.1.1">𝑣</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS3.Px3.p1.7.m7.1c">v</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS3.Px3.p1.7.m7.1d">italic_v</annotation></semantics></math> are separated in <math alttext="G" class="ltx_Math" display="inline" id="S4.SS2.SSS3.Px3.p1.8.m8.1"><semantics id="S4.SS2.SSS3.Px3.p1.8.m8.1a"><mi id="S4.SS2.SSS3.Px3.p1.8.m8.1.1" xref="S4.SS2.SSS3.Px3.p1.8.m8.1.1.cmml">G</mi><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS3.Px3.p1.8.m8.1b"><ci id="S4.SS2.SSS3.Px3.p1.8.m8.1.1.cmml" xref="S4.SS2.SSS3.Px3.p1.8.m8.1.1">𝐺</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS3.Px3.p1.8.m8.1c">G</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS3.Px3.p1.8.m8.1d">italic_G</annotation></semantics></math> by the removal of <math alttext="\{a,b\}" class="ltx_Math" display="inline" id="S4.SS2.SSS3.Px3.p1.9.m9.2"><semantics id="S4.SS2.SSS3.Px3.p1.9.m9.2a"><mrow id="S4.SS2.SSS3.Px3.p1.9.m9.2.3.2" xref="S4.SS2.SSS3.Px3.p1.9.m9.2.3.1.cmml"><mo id="S4.SS2.SSS3.Px3.p1.9.m9.2.3.2.1" stretchy="false" xref="S4.SS2.SSS3.Px3.p1.9.m9.2.3.1.cmml">{</mo><mi id="S4.SS2.SSS3.Px3.p1.9.m9.1.1" xref="S4.SS2.SSS3.Px3.p1.9.m9.1.1.cmml">a</mi><mo id="S4.SS2.SSS3.Px3.p1.9.m9.2.3.2.2" xref="S4.SS2.SSS3.Px3.p1.9.m9.2.3.1.cmml">,</mo><mi id="S4.SS2.SSS3.Px3.p1.9.m9.2.2" xref="S4.SS2.SSS3.Px3.p1.9.m9.2.2.cmml">b</mi><mo id="S4.SS2.SSS3.Px3.p1.9.m9.2.3.2.3" stretchy="false" xref="S4.SS2.SSS3.Px3.p1.9.m9.2.3.1.cmml">}</mo></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS3.Px3.p1.9.m9.2b"><set id="S4.SS2.SSS3.Px3.p1.9.m9.2.3.1.cmml" xref="S4.SS2.SSS3.Px3.p1.9.m9.2.3.2"><ci id="S4.SS2.SSS3.Px3.p1.9.m9.1.1.cmml" xref="S4.SS2.SSS3.Px3.p1.9.m9.1.1">𝑎</ci><ci id="S4.SS2.SSS3.Px3.p1.9.m9.2.2.cmml" xref="S4.SS2.SSS3.Px3.p1.9.m9.2.2">𝑏</ci></set></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS3.Px3.p1.9.m9.2c">\{a,b\}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS3.Px3.p1.9.m9.2d">{ italic_a , italic_b }</annotation></semantics></math>. This implies that <math alttext="x" class="ltx_Math" display="inline" id="S4.SS2.SSS3.Px3.p1.10.m10.1"><semantics id="S4.SS2.SSS3.Px3.p1.10.m10.1a"><mi id="S4.SS2.SSS3.Px3.p1.10.m10.1.1" xref="S4.SS2.SSS3.Px3.p1.10.m10.1.1.cmml">x</mi><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS3.Px3.p1.10.m10.1b"><ci id="S4.SS2.SSS3.Px3.p1.10.m10.1.1.cmml" xref="S4.SS2.SSS3.Px3.p1.10.m10.1.1">𝑥</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS3.Px3.p1.10.m10.1c">x</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS3.Px3.p1.10.m10.1d">italic_x</annotation></semantics></math> must be on the tree path from <math alttext="u" class="ltx_Math" display="inline" id="S4.SS2.SSS3.Px3.p1.11.m11.1"><semantics id="S4.SS2.SSS3.Px3.p1.11.m11.1a"><mi id="S4.SS2.SSS3.Px3.p1.11.m11.1.1" xref="S4.SS2.SSS3.Px3.p1.11.m11.1.1.cmml">u</mi><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS3.Px3.p1.11.m11.1b"><ci id="S4.SS2.SSS3.Px3.p1.11.m11.1.1.cmml" xref="S4.SS2.SSS3.Px3.p1.11.m11.1.1">𝑢</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS3.Px3.p1.11.m11.1c">u</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS3.Px3.p1.11.m11.1d">italic_u</annotation></semantics></math> to <math alttext="v" class="ltx_Math" display="inline" id="S4.SS2.SSS3.Px3.p1.12.m12.1"><semantics id="S4.SS2.SSS3.Px3.p1.12.m12.1a"><mi id="S4.SS2.SSS3.Px3.p1.12.m12.1.1" xref="S4.SS2.SSS3.Px3.p1.12.m12.1.1.cmml">v</mi><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS3.Px3.p1.12.m12.1b"><ci id="S4.SS2.SSS3.Px3.p1.12.m12.1.1.cmml" xref="S4.SS2.SSS3.Px3.p1.12.m12.1.1">𝑣</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS3.Px3.p1.12.m12.1c">v</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS3.Px3.p1.12.m12.1d">italic_v</annotation></semantics></math> (this includes the case that <math alttext="u" class="ltx_Math" display="inline" id="S4.SS2.SSS3.Px3.p1.13.m13.1"><semantics id="S4.SS2.SSS3.Px3.p1.13.m13.1a"><mi id="S4.SS2.SSS3.Px3.p1.13.m13.1.1" xref="S4.SS2.SSS3.Px3.p1.13.m13.1.1.cmml">u</mi><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS3.Px3.p1.13.m13.1b"><ci id="S4.SS2.SSS3.Px3.p1.13.m13.1.1.cmml" xref="S4.SS2.SSS3.Px3.p1.13.m13.1.1">𝑢</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS3.Px3.p1.13.m13.1c">u</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS3.Px3.p1.13.m13.1d">italic_u</annotation></semantics></math> and <math alttext="v" class="ltx_Math" display="inline" id="S4.SS2.SSS3.Px3.p1.14.m14.1"><semantics id="S4.SS2.SSS3.Px3.p1.14.m14.1a"><mi id="S4.SS2.SSS3.Px3.p1.14.m14.1.1" xref="S4.SS2.SSS3.Px3.p1.14.m14.1.1.cmml">v</mi><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS3.Px3.p1.14.m14.1b"><ci id="S4.SS2.SSS3.Px3.p1.14.m14.1.1.cmml" xref="S4.SS2.SSS3.Px3.p1.14.m14.1.1">𝑣</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS3.Px3.p1.14.m14.1c">v</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS3.Px3.p1.14.m14.1d">italic_v</annotation></semantics></math> are in the cycle <math alttext="G_{x}" class="ltx_Math" display="inline" id="S4.SS2.SSS3.Px3.p1.15.m15.1"><semantics id="S4.SS2.SSS3.Px3.p1.15.m15.1a"><msub id="S4.SS2.SSS3.Px3.p1.15.m15.1.1" xref="S4.SS2.SSS3.Px3.p1.15.m15.1.1.cmml"><mi id="S4.SS2.SSS3.Px3.p1.15.m15.1.1.2" xref="S4.SS2.SSS3.Px3.p1.15.m15.1.1.2.cmml">G</mi><mi id="S4.SS2.SSS3.Px3.p1.15.m15.1.1.3" xref="S4.SS2.SSS3.Px3.p1.15.m15.1.1.3.cmml">x</mi></msub><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS3.Px3.p1.15.m15.1b"><apply id="S4.SS2.SSS3.Px3.p1.15.m15.1.1.cmml" xref="S4.SS2.SSS3.Px3.p1.15.m15.1.1"><csymbol cd="ambiguous" id="S4.SS2.SSS3.Px3.p1.15.m15.1.1.1.cmml" xref="S4.SS2.SSS3.Px3.p1.15.m15.1.1">subscript</csymbol><ci id="S4.SS2.SSS3.Px3.p1.15.m15.1.1.2.cmml" xref="S4.SS2.SSS3.Px3.p1.15.m15.1.1.2">𝐺</ci><ci id="S4.SS2.SSS3.Px3.p1.15.m15.1.1.3.cmml" xref="S4.SS2.SSS3.Px3.p1.15.m15.1.1.3">𝑥</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS3.Px3.p1.15.m15.1c">G_{x}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS3.Px3.p1.15.m15.1d">italic_G start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT</annotation></semantics></math>). Note that since <math alttext="a" class="ltx_Math" display="inline" id="S4.SS2.SSS3.Px3.p1.16.m16.1"><semantics id="S4.SS2.SSS3.Px3.p1.16.m16.1a"><mi id="S4.SS2.SSS3.Px3.p1.16.m16.1.1" xref="S4.SS2.SSS3.Px3.p1.16.m16.1.1.cmml">a</mi><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS3.Px3.p1.16.m16.1b"><ci id="S4.SS2.SSS3.Px3.p1.16.m16.1.1.cmml" xref="S4.SS2.SSS3.Px3.p1.16.m16.1.1">𝑎</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS3.Px3.p1.16.m16.1c">a</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS3.Px3.p1.16.m16.1d">italic_a</annotation></semantics></math> and <math alttext="b" class="ltx_Math" display="inline" id="S4.SS2.SSS3.Px3.p1.17.m17.1"><semantics id="S4.SS2.SSS3.Px3.p1.17.m17.1a"><mi id="S4.SS2.SSS3.Px3.p1.17.m17.1.1" xref="S4.SS2.SSS3.Px3.p1.17.m17.1.1.cmml">b</mi><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS3.Px3.p1.17.m17.1b"><ci id="S4.SS2.SSS3.Px3.p1.17.m17.1.1.cmml" xref="S4.SS2.SSS3.Px3.p1.17.m17.1.1">𝑏</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS3.Px3.p1.17.m17.1c">b</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS3.Px3.p1.17.m17.1d">italic_b</annotation></semantics></math> are non-adjacent, all dummy nodes remain intact. We overload notation and write <math alttext="G_{x}" class="ltx_Math" display="inline" id="S4.SS2.SSS3.Px3.p1.18.m18.1"><semantics id="S4.SS2.SSS3.Px3.p1.18.m18.1a"><msub id="S4.SS2.SSS3.Px3.p1.18.m18.1.1" xref="S4.SS2.SSS3.Px3.p1.18.m18.1.1.cmml"><mi id="S4.SS2.SSS3.Px3.p1.18.m18.1.1.2" xref="S4.SS2.SSS3.Px3.p1.18.m18.1.1.2.cmml">G</mi><mi id="S4.SS2.SSS3.Px3.p1.18.m18.1.1.3" xref="S4.SS2.SSS3.Px3.p1.18.m18.1.1.3.cmml">x</mi></msub><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS3.Px3.p1.18.m18.1b"><apply id="S4.SS2.SSS3.Px3.p1.18.m18.1.1.cmml" xref="S4.SS2.SSS3.Px3.p1.18.m18.1.1"><csymbol cd="ambiguous" id="S4.SS2.SSS3.Px3.p1.18.m18.1.1.1.cmml" xref="S4.SS2.SSS3.Px3.p1.18.m18.1.1">subscript</csymbol><ci id="S4.SS2.SSS3.Px3.p1.18.m18.1.1.2.cmml" xref="S4.SS2.SSS3.Px3.p1.18.m18.1.1.2">𝐺</ci><ci id="S4.SS2.SSS3.Px3.p1.18.m18.1.1.3.cmml" xref="S4.SS2.SSS3.Px3.p1.18.m18.1.1.3">𝑥</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS3.Px3.p1.18.m18.1c">G_{x}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS3.Px3.p1.18.m18.1d">italic_G start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT</annotation></semantics></math> to include dummy nodes. For ease of notation we write <math alttext="f:=f_{x}" class="ltx_Math" display="inline" id="S4.SS2.SSS3.Px3.p1.19.m19.1"><semantics id="S4.SS2.SSS3.Px3.p1.19.m19.1a"><mrow id="S4.SS2.SSS3.Px3.p1.19.m19.1.1" xref="S4.SS2.SSS3.Px3.p1.19.m19.1.1.cmml"><mi id="S4.SS2.SSS3.Px3.p1.19.m19.1.1.2" xref="S4.SS2.SSS3.Px3.p1.19.m19.1.1.2.cmml">f</mi><mo id="S4.SS2.SSS3.Px3.p1.19.m19.1.1.1" lspace="0.278em" rspace="0.278em" xref="S4.SS2.SSS3.Px3.p1.19.m19.1.1.1.cmml">:=</mo><msub id="S4.SS2.SSS3.Px3.p1.19.m19.1.1.3" xref="S4.SS2.SSS3.Px3.p1.19.m19.1.1.3.cmml"><mi id="S4.SS2.SSS3.Px3.p1.19.m19.1.1.3.2" xref="S4.SS2.SSS3.Px3.p1.19.m19.1.1.3.2.cmml">f</mi><mi id="S4.SS2.SSS3.Px3.p1.19.m19.1.1.3.3" xref="S4.SS2.SSS3.Px3.p1.19.m19.1.1.3.3.cmml">x</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS3.Px3.p1.19.m19.1b"><apply id="S4.SS2.SSS3.Px3.p1.19.m19.1.1.cmml" xref="S4.SS2.SSS3.Px3.p1.19.m19.1.1"><csymbol cd="latexml" id="S4.SS2.SSS3.Px3.p1.19.m19.1.1.1.cmml" xref="S4.SS2.SSS3.Px3.p1.19.m19.1.1.1">assign</csymbol><ci id="S4.SS2.SSS3.Px3.p1.19.m19.1.1.2.cmml" xref="S4.SS2.SSS3.Px3.p1.19.m19.1.1.2">𝑓</ci><apply id="S4.SS2.SSS3.Px3.p1.19.m19.1.1.3.cmml" xref="S4.SS2.SSS3.Px3.p1.19.m19.1.1.3"><csymbol cd="ambiguous" id="S4.SS2.SSS3.Px3.p1.19.m19.1.1.3.1.cmml" xref="S4.SS2.SSS3.Px3.p1.19.m19.1.1.3">subscript</csymbol><ci id="S4.SS2.SSS3.Px3.p1.19.m19.1.1.3.2.cmml" xref="S4.SS2.SSS3.Px3.p1.19.m19.1.1.3.2">𝑓</ci><ci id="S4.SS2.SSS3.Px3.p1.19.m19.1.1.3.3.cmml" xref="S4.SS2.SSS3.Px3.p1.19.m19.1.1.3.3">𝑥</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS3.Px3.p1.19.m19.1c">f:=f_{x}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS3.Px3.p1.19.m19.1d">italic_f := italic_f start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT</annotation></semantics></math>, <math alttext="\prec:=\prec_{x}" class="ltx_Math" display="inline" id="S4.SS2.SSS3.Px3.p1.20.m20.3"><semantics id="S4.SS2.SSS3.Px3.p1.20.m20.3a"><mrow id="S4.SS2.SSS3.Px3.p1.20.m20.3.3.1" xref="S4.SS2.SSS3.Px3.p1.20.m20.3.3.2.cmml"><mo id="S4.SS2.SSS3.Px3.p1.20.m20.1.1" xref="S4.SS2.SSS3.Px3.p1.20.m20.1.1.cmml">≺</mo><mo id="S4.SS2.SSS3.Px3.p1.20.m20.3.3.1.2" lspace="0em" xref="S4.SS2.SSS3.Px3.p1.20.m20.3.3.2.cmml">⁣</mo><mo id="S4.SS2.SSS3.Px3.p1.20.m20.2.2" xref="S4.SS2.SSS3.Px3.p1.20.m20.2.2.cmml">:=</mo><mo id="S4.SS2.SSS3.Px3.p1.20.m20.3.3.1.3" lspace="0em" xref="S4.SS2.SSS3.Px3.p1.20.m20.3.3.2.cmml">⁣</mo><msub id="S4.SS2.SSS3.Px3.p1.20.m20.3.3.1.1" xref="S4.SS2.SSS3.Px3.p1.20.m20.3.3.1.1.cmml"><mo id="S4.SS2.SSS3.Px3.p1.20.m20.3.3.1.1.2" xref="S4.SS2.SSS3.Px3.p1.20.m20.3.3.1.1.2.cmml">≺</mo><mi id="S4.SS2.SSS3.Px3.p1.20.m20.3.3.1.1.3" xref="S4.SS2.SSS3.Px3.p1.20.m20.3.3.1.1.3.cmml">x</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS3.Px3.p1.20.m20.3b"><list id="S4.SS2.SSS3.Px3.p1.20.m20.3.3.2.cmml" xref="S4.SS2.SSS3.Px3.p1.20.m20.3.3.1"><csymbol cd="latexml" id="S4.SS2.SSS3.Px3.p1.20.m20.1.1.cmml" xref="S4.SS2.SSS3.Px3.p1.20.m20.1.1">precedes</csymbol><csymbol cd="latexml" id="S4.SS2.SSS3.Px3.p1.20.m20.2.2.cmml" xref="S4.SS2.SSS3.Px3.p1.20.m20.2.2">assign</csymbol><apply id="S4.SS2.SSS3.Px3.p1.20.m20.3.3.1.1.cmml" xref="S4.SS2.SSS3.Px3.p1.20.m20.3.3.1.1"><csymbol cd="ambiguous" id="S4.SS2.SSS3.Px3.p1.20.m20.3.3.1.1.1.cmml" xref="S4.SS2.SSS3.Px3.p1.20.m20.3.3.1.1">subscript</csymbol><csymbol cd="latexml" id="S4.SS2.SSS3.Px3.p1.20.m20.3.3.1.1.2.cmml" xref="S4.SS2.SSS3.Px3.p1.20.m20.3.3.1.1.2">precedes</csymbol><ci id="S4.SS2.SSS3.Px3.p1.20.m20.3.3.1.1.3.cmml" xref="S4.SS2.SSS3.Px3.p1.20.m20.3.3.1.1.3">𝑥</ci></apply></list></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS3.Px3.p1.20.m20.3c">\prec:=\prec_{x}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS3.Px3.p1.20.m20.3d">≺ := ≺ start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT</annotation></semantics></math> for this section.</p> <ul class="ltx_itemize" id="S4.I11"> <li class="ltx_item" id="S4.I11.i1" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">•</span> <div class="ltx_para" id="S4.I11.i1.p1"> <p class="ltx_p" id="S4.I11.i1.p1.11"><span class="ltx_text ltx_font_bold" id="S4.I11.i1.p1.1.1">Case 3a: <math alttext="\boldsymbol{\text{LCA}(\ell(u),\ell(v))=x}" class="ltx_Math" display="inline" id="S4.I11.i1.p1.1.1.m1.4"><semantics id="S4.I11.i1.p1.1.1.m1.4a"><mrow id="S4.I11.i1.p1.1.1.m1.4.4" xref="S4.I11.i1.p1.1.1.m1.4.4.cmml"><mrow id="S4.I11.i1.p1.1.1.m1.4.4.2" xref="S4.I11.i1.p1.1.1.m1.4.4.2.cmml"><mtext class="ltx_mathvariant_bold" id="S4.I11.i1.p1.1.1.m1.4.4.2.4" xref="S4.I11.i1.p1.1.1.m1.4.4.2.4a.cmml">LCA</mtext><mo id="S4.I11.i1.p1.1.1.m1.4.4.2.3" xref="S4.I11.i1.p1.1.1.m1.4.4.2.3.cmml">⁢</mo><mrow id="S4.I11.i1.p1.1.1.m1.4.4.2.2.2" xref="S4.I11.i1.p1.1.1.m1.4.4.2.2.3.cmml"><mo class="ltx_mathvariant_bold" id="S4.I11.i1.p1.1.1.m1.4.4.2.2.2.3" mathvariant="bold" stretchy="false" xref="S4.I11.i1.p1.1.1.m1.4.4.2.2.3.cmml">(</mo><mrow id="S4.I11.i1.p1.1.1.m1.3.3.1.1.1.1" xref="S4.I11.i1.p1.1.1.m1.3.3.1.1.1.1.cmml"><mi class="ltx_mathvariant_bold" id="S4.I11.i1.p1.1.1.m1.3.3.1.1.1.1.2" mathvariant="bold" xref="S4.I11.i1.p1.1.1.m1.3.3.1.1.1.1.2.cmml">ℓ</mi><mo id="S4.I11.i1.p1.1.1.m1.3.3.1.1.1.1.1" xref="S4.I11.i1.p1.1.1.m1.3.3.1.1.1.1.1.cmml">⁢</mo><mrow id="S4.I11.i1.p1.1.1.m1.3.3.1.1.1.1.3.2" xref="S4.I11.i1.p1.1.1.m1.3.3.1.1.1.1.cmml"><mo class="ltx_mathvariant_bold" id="S4.I11.i1.p1.1.1.m1.3.3.1.1.1.1.3.2.1" mathvariant="bold" stretchy="false" xref="S4.I11.i1.p1.1.1.m1.3.3.1.1.1.1.cmml">(</mo><mi id="S4.I11.i1.p1.1.1.m1.1.1" xref="S4.I11.i1.p1.1.1.m1.1.1.cmml">u</mi><mo class="ltx_mathvariant_bold" id="S4.I11.i1.p1.1.1.m1.3.3.1.1.1.1.3.2.2" mathvariant="bold" stretchy="false" xref="S4.I11.i1.p1.1.1.m1.3.3.1.1.1.1.cmml">)</mo></mrow></mrow><mo class="ltx_mathvariant_bold" id="S4.I11.i1.p1.1.1.m1.4.4.2.2.2.4" mathvariant="bold" xref="S4.I11.i1.p1.1.1.m1.4.4.2.2.3.cmml">,</mo><mrow id="S4.I11.i1.p1.1.1.m1.4.4.2.2.2.2" xref="S4.I11.i1.p1.1.1.m1.4.4.2.2.2.2.cmml"><mi class="ltx_mathvariant_bold" id="S4.I11.i1.p1.1.1.m1.4.4.2.2.2.2.2" mathvariant="bold" xref="S4.I11.i1.p1.1.1.m1.4.4.2.2.2.2.2.cmml">ℓ</mi><mo id="S4.I11.i1.p1.1.1.m1.4.4.2.2.2.2.1" xref="S4.I11.i1.p1.1.1.m1.4.4.2.2.2.2.1.cmml">⁢</mo><mrow id="S4.I11.i1.p1.1.1.m1.4.4.2.2.2.2.3.2" xref="S4.I11.i1.p1.1.1.m1.4.4.2.2.2.2.cmml"><mo class="ltx_mathvariant_bold" id="S4.I11.i1.p1.1.1.m1.4.4.2.2.2.2.3.2.1" mathvariant="bold" stretchy="false" xref="S4.I11.i1.p1.1.1.m1.4.4.2.2.2.2.cmml">(</mo><mi id="S4.I11.i1.p1.1.1.m1.2.2" xref="S4.I11.i1.p1.1.1.m1.2.2.cmml">v</mi><mo class="ltx_mathvariant_bold" id="S4.I11.i1.p1.1.1.m1.4.4.2.2.2.2.3.2.2" mathvariant="bold" stretchy="false" xref="S4.I11.i1.p1.1.1.m1.4.4.2.2.2.2.cmml">)</mo></mrow></mrow><mo class="ltx_mathvariant_bold" id="S4.I11.i1.p1.1.1.m1.4.4.2.2.2.5" mathvariant="bold" stretchy="false" xref="S4.I11.i1.p1.1.1.m1.4.4.2.2.3.cmml">)</mo></mrow></mrow><mo class="ltx_mathvariant_bold" id="S4.I11.i1.p1.1.1.m1.4.4.3" mathvariant="bold" xref="S4.I11.i1.p1.1.1.m1.4.4.3.cmml">=</mo><mi id="S4.I11.i1.p1.1.1.m1.4.4.4" xref="S4.I11.i1.p1.1.1.m1.4.4.4.cmml">x</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.I11.i1.p1.1.1.m1.4b"><apply id="S4.I11.i1.p1.1.1.m1.4.4.cmml" xref="S4.I11.i1.p1.1.1.m1.4.4"><eq id="S4.I11.i1.p1.1.1.m1.4.4.3.cmml" xref="S4.I11.i1.p1.1.1.m1.4.4.3"></eq><apply id="S4.I11.i1.p1.1.1.m1.4.4.2.cmml" xref="S4.I11.i1.p1.1.1.m1.4.4.2"><times id="S4.I11.i1.p1.1.1.m1.4.4.2.3.cmml" xref="S4.I11.i1.p1.1.1.m1.4.4.2.3"></times><ci id="S4.I11.i1.p1.1.1.m1.4.4.2.4a.cmml" xref="S4.I11.i1.p1.1.1.m1.4.4.2.4"><mtext class="ltx_mathvariant_bold" id="S4.I11.i1.p1.1.1.m1.4.4.2.4.cmml" xref="S4.I11.i1.p1.1.1.m1.4.4.2.4">LCA</mtext></ci><interval closure="open" id="S4.I11.i1.p1.1.1.m1.4.4.2.2.3.cmml" xref="S4.I11.i1.p1.1.1.m1.4.4.2.2.2"><apply id="S4.I11.i1.p1.1.1.m1.3.3.1.1.1.1.cmml" xref="S4.I11.i1.p1.1.1.m1.3.3.1.1.1.1"><times id="S4.I11.i1.p1.1.1.m1.3.3.1.1.1.1.1.cmml" xref="S4.I11.i1.p1.1.1.m1.3.3.1.1.1.1.1"></times><ci id="S4.I11.i1.p1.1.1.m1.3.3.1.1.1.1.2.cmml" xref="S4.I11.i1.p1.1.1.m1.3.3.1.1.1.1.2">bold-ℓ</ci><ci id="S4.I11.i1.p1.1.1.m1.1.1.cmml" xref="S4.I11.i1.p1.1.1.m1.1.1">𝑢</ci></apply><apply id="S4.I11.i1.p1.1.1.m1.4.4.2.2.2.2.cmml" xref="S4.I11.i1.p1.1.1.m1.4.4.2.2.2.2"><times id="S4.I11.i1.p1.1.1.m1.4.4.2.2.2.2.1.cmml" xref="S4.I11.i1.p1.1.1.m1.4.4.2.2.2.2.1"></times><ci id="S4.I11.i1.p1.1.1.m1.4.4.2.2.2.2.2.cmml" xref="S4.I11.i1.p1.1.1.m1.4.4.2.2.2.2.2">bold-ℓ</ci><ci id="S4.I11.i1.p1.1.1.m1.2.2.cmml" xref="S4.I11.i1.p1.1.1.m1.2.2">𝑣</ci></apply></interval></apply><ci id="S4.I11.i1.p1.1.1.m1.4.4.4.cmml" xref="S4.I11.i1.p1.1.1.m1.4.4.4">𝑥</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.I11.i1.p1.1.1.m1.4c">\boldsymbol{\text{LCA}(\ell(u),\ell(v))=x}</annotation><annotation encoding="application/x-llamapun" id="S4.I11.i1.p1.1.1.m1.4d">LCA bold_( bold_ℓ bold_( bold_italic_u bold_) bold_, bold_ℓ bold_( bold_italic_v bold_) bold_) bold_= bold_italic_x</annotation></semantics></math>:</span> Notice that <math alttext="u" class="ltx_Math" display="inline" id="S4.I11.i1.p1.2.m1.1"><semantics id="S4.I11.i1.p1.2.m1.1a"><mi id="S4.I11.i1.p1.2.m1.1.1" xref="S4.I11.i1.p1.2.m1.1.1.cmml">u</mi><annotation-xml encoding="MathML-Content" id="S4.I11.i1.p1.2.m1.1b"><ci id="S4.I11.i1.p1.2.m1.1.1.cmml" xref="S4.I11.i1.p1.2.m1.1.1">𝑢</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.I11.i1.p1.2.m1.1c">u</annotation><annotation encoding="application/x-llamapun" id="S4.I11.i1.p1.2.m1.1d">italic_u</annotation></semantics></math> and <math alttext="f(u)" class="ltx_Math" display="inline" id="S4.I11.i1.p1.3.m2.1"><semantics id="S4.I11.i1.p1.3.m2.1a"><mrow id="S4.I11.i1.p1.3.m2.1.2" xref="S4.I11.i1.p1.3.m2.1.2.cmml"><mi id="S4.I11.i1.p1.3.m2.1.2.2" xref="S4.I11.i1.p1.3.m2.1.2.2.cmml">f</mi><mo id="S4.I11.i1.p1.3.m2.1.2.1" xref="S4.I11.i1.p1.3.m2.1.2.1.cmml">⁢</mo><mrow id="S4.I11.i1.p1.3.m2.1.2.3.2" xref="S4.I11.i1.p1.3.m2.1.2.cmml"><mo id="S4.I11.i1.p1.3.m2.1.2.3.2.1" stretchy="false" xref="S4.I11.i1.p1.3.m2.1.2.cmml">(</mo><mi id="S4.I11.i1.p1.3.m2.1.1" xref="S4.I11.i1.p1.3.m2.1.1.cmml">u</mi><mo id="S4.I11.i1.p1.3.m2.1.2.3.2.2" stretchy="false" xref="S4.I11.i1.p1.3.m2.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.I11.i1.p1.3.m2.1b"><apply id="S4.I11.i1.p1.3.m2.1.2.cmml" xref="S4.I11.i1.p1.3.m2.1.2"><times id="S4.I11.i1.p1.3.m2.1.2.1.cmml" xref="S4.I11.i1.p1.3.m2.1.2.1"></times><ci id="S4.I11.i1.p1.3.m2.1.2.2.cmml" xref="S4.I11.i1.p1.3.m2.1.2.2">𝑓</ci><ci id="S4.I11.i1.p1.3.m2.1.1.cmml" xref="S4.I11.i1.p1.3.m2.1.1">𝑢</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.I11.i1.p1.3.m2.1c">f(u)</annotation><annotation encoding="application/x-llamapun" id="S4.I11.i1.p1.3.m2.1d">italic_f ( italic_u )</annotation></semantics></math> (as well as <math alttext="v" class="ltx_Math" display="inline" id="S4.I11.i1.p1.4.m3.1"><semantics id="S4.I11.i1.p1.4.m3.1a"><mi id="S4.I11.i1.p1.4.m3.1.1" xref="S4.I11.i1.p1.4.m3.1.1.cmml">v</mi><annotation-xml encoding="MathML-Content" id="S4.I11.i1.p1.4.m3.1b"><ci id="S4.I11.i1.p1.4.m3.1.1.cmml" xref="S4.I11.i1.p1.4.m3.1.1">𝑣</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.I11.i1.p1.4.m3.1c">v</annotation><annotation encoding="application/x-llamapun" id="S4.I11.i1.p1.4.m3.1d">italic_v</annotation></semantics></math> and <math alttext="f(v)" class="ltx_Math" display="inline" id="S4.I11.i1.p1.5.m4.1"><semantics id="S4.I11.i1.p1.5.m4.1a"><mrow id="S4.I11.i1.p1.5.m4.1.2" xref="S4.I11.i1.p1.5.m4.1.2.cmml"><mi id="S4.I11.i1.p1.5.m4.1.2.2" xref="S4.I11.i1.p1.5.m4.1.2.2.cmml">f</mi><mo id="S4.I11.i1.p1.5.m4.1.2.1" xref="S4.I11.i1.p1.5.m4.1.2.1.cmml">⁢</mo><mrow id="S4.I11.i1.p1.5.m4.1.2.3.2" xref="S4.I11.i1.p1.5.m4.1.2.cmml"><mo id="S4.I11.i1.p1.5.m4.1.2.3.2.1" stretchy="false" xref="S4.I11.i1.p1.5.m4.1.2.cmml">(</mo><mi id="S4.I11.i1.p1.5.m4.1.1" xref="S4.I11.i1.p1.5.m4.1.1.cmml">v</mi><mo id="S4.I11.i1.p1.5.m4.1.2.3.2.2" stretchy="false" xref="S4.I11.i1.p1.5.m4.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.I11.i1.p1.5.m4.1b"><apply id="S4.I11.i1.p1.5.m4.1.2.cmml" xref="S4.I11.i1.p1.5.m4.1.2"><times id="S4.I11.i1.p1.5.m4.1.2.1.cmml" xref="S4.I11.i1.p1.5.m4.1.2.1"></times><ci id="S4.I11.i1.p1.5.m4.1.2.2.cmml" xref="S4.I11.i1.p1.5.m4.1.2.2">𝑓</ci><ci id="S4.I11.i1.p1.5.m4.1.1.cmml" xref="S4.I11.i1.p1.5.m4.1.1">𝑣</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.I11.i1.p1.5.m4.1c">f(v)</annotation><annotation encoding="application/x-llamapun" id="S4.I11.i1.p1.5.m4.1d">italic_f ( italic_v )</annotation></semantics></math>) remain connected despite the deletion of <math alttext="\{a,b\}" class="ltx_Math" display="inline" id="S4.I11.i1.p1.6.m5.2"><semantics id="S4.I11.i1.p1.6.m5.2a"><mrow id="S4.I11.i1.p1.6.m5.2.3.2" xref="S4.I11.i1.p1.6.m5.2.3.1.cmml"><mo id="S4.I11.i1.p1.6.m5.2.3.2.1" stretchy="false" xref="S4.I11.i1.p1.6.m5.2.3.1.cmml">{</mo><mi id="S4.I11.i1.p1.6.m5.1.1" xref="S4.I11.i1.p1.6.m5.1.1.cmml">a</mi><mo id="S4.I11.i1.p1.6.m5.2.3.2.2" xref="S4.I11.i1.p1.6.m5.2.3.1.cmml">,</mo><mi id="S4.I11.i1.p1.6.m5.2.2" xref="S4.I11.i1.p1.6.m5.2.2.cmml">b</mi><mo id="S4.I11.i1.p1.6.m5.2.3.2.3" stretchy="false" xref="S4.I11.i1.p1.6.m5.2.3.1.cmml">}</mo></mrow><annotation-xml encoding="MathML-Content" id="S4.I11.i1.p1.6.m5.2b"><set id="S4.I11.i1.p1.6.m5.2.3.1.cmml" xref="S4.I11.i1.p1.6.m5.2.3.2"><ci id="S4.I11.i1.p1.6.m5.1.1.cmml" xref="S4.I11.i1.p1.6.m5.1.1">𝑎</ci><ci id="S4.I11.i1.p1.6.m5.2.2.cmml" xref="S4.I11.i1.p1.6.m5.2.2">𝑏</ci></set></annotation-xml><annotation encoding="application/x-tex" id="S4.I11.i1.p1.6.m5.2c">\{a,b\}</annotation><annotation encoding="application/x-llamapun" id="S4.I11.i1.p1.6.m5.2d">{ italic_a , italic_b }</annotation></semantics></math>. Thus <math alttext="f(u)" class="ltx_Math" display="inline" id="S4.I11.i1.p1.7.m6.1"><semantics id="S4.I11.i1.p1.7.m6.1a"><mrow id="S4.I11.i1.p1.7.m6.1.2" xref="S4.I11.i1.p1.7.m6.1.2.cmml"><mi id="S4.I11.i1.p1.7.m6.1.2.2" xref="S4.I11.i1.p1.7.m6.1.2.2.cmml">f</mi><mo id="S4.I11.i1.p1.7.m6.1.2.1" xref="S4.I11.i1.p1.7.m6.1.2.1.cmml">⁢</mo><mrow id="S4.I11.i1.p1.7.m6.1.2.3.2" xref="S4.I11.i1.p1.7.m6.1.2.cmml"><mo id="S4.I11.i1.p1.7.m6.1.2.3.2.1" stretchy="false" xref="S4.I11.i1.p1.7.m6.1.2.cmml">(</mo><mi id="S4.I11.i1.p1.7.m6.1.1" xref="S4.I11.i1.p1.7.m6.1.1.cmml">u</mi><mo id="S4.I11.i1.p1.7.m6.1.2.3.2.2" stretchy="false" xref="S4.I11.i1.p1.7.m6.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.I11.i1.p1.7.m6.1b"><apply id="S4.I11.i1.p1.7.m6.1.2.cmml" xref="S4.I11.i1.p1.7.m6.1.2"><times id="S4.I11.i1.p1.7.m6.1.2.1.cmml" xref="S4.I11.i1.p1.7.m6.1.2.1"></times><ci id="S4.I11.i1.p1.7.m6.1.2.2.cmml" xref="S4.I11.i1.p1.7.m6.1.2.2">𝑓</ci><ci id="S4.I11.i1.p1.7.m6.1.1.cmml" xref="S4.I11.i1.p1.7.m6.1.1">𝑢</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.I11.i1.p1.7.m6.1c">f(u)</annotation><annotation encoding="application/x-llamapun" id="S4.I11.i1.p1.7.m6.1d">italic_f ( italic_u )</annotation></semantics></math> and <math alttext="f(v)" class="ltx_Math" display="inline" id="S4.I11.i1.p1.8.m7.1"><semantics id="S4.I11.i1.p1.8.m7.1a"><mrow id="S4.I11.i1.p1.8.m7.1.2" xref="S4.I11.i1.p1.8.m7.1.2.cmml"><mi id="S4.I11.i1.p1.8.m7.1.2.2" xref="S4.I11.i1.p1.8.m7.1.2.2.cmml">f</mi><mo id="S4.I11.i1.p1.8.m7.1.2.1" xref="S4.I11.i1.p1.8.m7.1.2.1.cmml">⁢</mo><mrow id="S4.I11.i1.p1.8.m7.1.2.3.2" xref="S4.I11.i1.p1.8.m7.1.2.cmml"><mo id="S4.I11.i1.p1.8.m7.1.2.3.2.1" stretchy="false" xref="S4.I11.i1.p1.8.m7.1.2.cmml">(</mo><mi id="S4.I11.i1.p1.8.m7.1.1" xref="S4.I11.i1.p1.8.m7.1.1.cmml">v</mi><mo id="S4.I11.i1.p1.8.m7.1.2.3.2.2" stretchy="false" xref="S4.I11.i1.p1.8.m7.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.I11.i1.p1.8.m7.1b"><apply id="S4.I11.i1.p1.8.m7.1.2.cmml" xref="S4.I11.i1.p1.8.m7.1.2"><times id="S4.I11.i1.p1.8.m7.1.2.1.cmml" xref="S4.I11.i1.p1.8.m7.1.2.1"></times><ci id="S4.I11.i1.p1.8.m7.1.2.2.cmml" xref="S4.I11.i1.p1.8.m7.1.2.2">𝑓</ci><ci id="S4.I11.i1.p1.8.m7.1.1.cmml" xref="S4.I11.i1.p1.8.m7.1.1">𝑣</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.I11.i1.p1.8.m7.1c">f(v)</annotation><annotation encoding="application/x-llamapun" id="S4.I11.i1.p1.8.m7.1d">italic_f ( italic_v )</annotation></semantics></math> must be in distinct components of <math alttext="G_{x}\setminus\{a,b\}" class="ltx_Math" display="inline" id="S4.I11.i1.p1.9.m8.2"><semantics id="S4.I11.i1.p1.9.m8.2a"><mrow id="S4.I11.i1.p1.9.m8.2.3" xref="S4.I11.i1.p1.9.m8.2.3.cmml"><msub id="S4.I11.i1.p1.9.m8.2.3.2" xref="S4.I11.i1.p1.9.m8.2.3.2.cmml"><mi id="S4.I11.i1.p1.9.m8.2.3.2.2" xref="S4.I11.i1.p1.9.m8.2.3.2.2.cmml">G</mi><mi id="S4.I11.i1.p1.9.m8.2.3.2.3" xref="S4.I11.i1.p1.9.m8.2.3.2.3.cmml">x</mi></msub><mo id="S4.I11.i1.p1.9.m8.2.3.1" xref="S4.I11.i1.p1.9.m8.2.3.1.cmml">∖</mo><mrow id="S4.I11.i1.p1.9.m8.2.3.3.2" xref="S4.I11.i1.p1.9.m8.2.3.3.1.cmml"><mo id="S4.I11.i1.p1.9.m8.2.3.3.2.1" stretchy="false" xref="S4.I11.i1.p1.9.m8.2.3.3.1.cmml">{</mo><mi id="S4.I11.i1.p1.9.m8.1.1" xref="S4.I11.i1.p1.9.m8.1.1.cmml">a</mi><mo id="S4.I11.i1.p1.9.m8.2.3.3.2.2" xref="S4.I11.i1.p1.9.m8.2.3.3.1.cmml">,</mo><mi id="S4.I11.i1.p1.9.m8.2.2" xref="S4.I11.i1.p1.9.m8.2.2.cmml">b</mi><mo id="S4.I11.i1.p1.9.m8.2.3.3.2.3" stretchy="false" xref="S4.I11.i1.p1.9.m8.2.3.3.1.cmml">}</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.I11.i1.p1.9.m8.2b"><apply id="S4.I11.i1.p1.9.m8.2.3.cmml" xref="S4.I11.i1.p1.9.m8.2.3"><setdiff id="S4.I11.i1.p1.9.m8.2.3.1.cmml" xref="S4.I11.i1.p1.9.m8.2.3.1"></setdiff><apply id="S4.I11.i1.p1.9.m8.2.3.2.cmml" xref="S4.I11.i1.p1.9.m8.2.3.2"><csymbol cd="ambiguous" id="S4.I11.i1.p1.9.m8.2.3.2.1.cmml" xref="S4.I11.i1.p1.9.m8.2.3.2">subscript</csymbol><ci id="S4.I11.i1.p1.9.m8.2.3.2.2.cmml" xref="S4.I11.i1.p1.9.m8.2.3.2.2">𝐺</ci><ci id="S4.I11.i1.p1.9.m8.2.3.2.3.cmml" xref="S4.I11.i1.p1.9.m8.2.3.2.3">𝑥</ci></apply><set id="S4.I11.i1.p1.9.m8.2.3.3.1.cmml" xref="S4.I11.i1.p1.9.m8.2.3.3.2"><ci id="S4.I11.i1.p1.9.m8.1.1.cmml" xref="S4.I11.i1.p1.9.m8.1.1">𝑎</ci><ci id="S4.I11.i1.p1.9.m8.2.2.cmml" xref="S4.I11.i1.p1.9.m8.2.2">𝑏</ci></set></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.I11.i1.p1.9.m8.2c">G_{x}\setminus\{a,b\}</annotation><annotation encoding="application/x-llamapun" id="S4.I11.i1.p1.9.m8.2d">italic_G start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT ∖ { italic_a , italic_b }</annotation></semantics></math>; without loss of generality suppose <math alttext="f(u)\in C_{1}" class="ltx_Math" display="inline" id="S4.I11.i1.p1.10.m9.1"><semantics id="S4.I11.i1.p1.10.m9.1a"><mrow id="S4.I11.i1.p1.10.m9.1.2" xref="S4.I11.i1.p1.10.m9.1.2.cmml"><mrow id="S4.I11.i1.p1.10.m9.1.2.2" xref="S4.I11.i1.p1.10.m9.1.2.2.cmml"><mi id="S4.I11.i1.p1.10.m9.1.2.2.2" xref="S4.I11.i1.p1.10.m9.1.2.2.2.cmml">f</mi><mo id="S4.I11.i1.p1.10.m9.1.2.2.1" xref="S4.I11.i1.p1.10.m9.1.2.2.1.cmml">⁢</mo><mrow id="S4.I11.i1.p1.10.m9.1.2.2.3.2" xref="S4.I11.i1.p1.10.m9.1.2.2.cmml"><mo id="S4.I11.i1.p1.10.m9.1.2.2.3.2.1" stretchy="false" xref="S4.I11.i1.p1.10.m9.1.2.2.cmml">(</mo><mi id="S4.I11.i1.p1.10.m9.1.1" xref="S4.I11.i1.p1.10.m9.1.1.cmml">u</mi><mo id="S4.I11.i1.p1.10.m9.1.2.2.3.2.2" stretchy="false" xref="S4.I11.i1.p1.10.m9.1.2.2.cmml">)</mo></mrow></mrow><mo id="S4.I11.i1.p1.10.m9.1.2.1" xref="S4.I11.i1.p1.10.m9.1.2.1.cmml">∈</mo><msub id="S4.I11.i1.p1.10.m9.1.2.3" xref="S4.I11.i1.p1.10.m9.1.2.3.cmml"><mi id="S4.I11.i1.p1.10.m9.1.2.3.2" xref="S4.I11.i1.p1.10.m9.1.2.3.2.cmml">C</mi><mn id="S4.I11.i1.p1.10.m9.1.2.3.3" xref="S4.I11.i1.p1.10.m9.1.2.3.3.cmml">1</mn></msub></mrow><annotation-xml encoding="MathML-Content" id="S4.I11.i1.p1.10.m9.1b"><apply id="S4.I11.i1.p1.10.m9.1.2.cmml" xref="S4.I11.i1.p1.10.m9.1.2"><in id="S4.I11.i1.p1.10.m9.1.2.1.cmml" xref="S4.I11.i1.p1.10.m9.1.2.1"></in><apply id="S4.I11.i1.p1.10.m9.1.2.2.cmml" xref="S4.I11.i1.p1.10.m9.1.2.2"><times id="S4.I11.i1.p1.10.m9.1.2.2.1.cmml" xref="S4.I11.i1.p1.10.m9.1.2.2.1"></times><ci id="S4.I11.i1.p1.10.m9.1.2.2.2.cmml" xref="S4.I11.i1.p1.10.m9.1.2.2.2">𝑓</ci><ci id="S4.I11.i1.p1.10.m9.1.1.cmml" xref="S4.I11.i1.p1.10.m9.1.1">𝑢</ci></apply><apply id="S4.I11.i1.p1.10.m9.1.2.3.cmml" xref="S4.I11.i1.p1.10.m9.1.2.3"><csymbol cd="ambiguous" id="S4.I11.i1.p1.10.m9.1.2.3.1.cmml" xref="S4.I11.i1.p1.10.m9.1.2.3">subscript</csymbol><ci id="S4.I11.i1.p1.10.m9.1.2.3.2.cmml" xref="S4.I11.i1.p1.10.m9.1.2.3.2">𝐶</ci><cn id="S4.I11.i1.p1.10.m9.1.2.3.3.cmml" type="integer" xref="S4.I11.i1.p1.10.m9.1.2.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.I11.i1.p1.10.m9.1c">f(u)\in C_{1}</annotation><annotation encoding="application/x-llamapun" id="S4.I11.i1.p1.10.m9.1d">italic_f ( italic_u ) ∈ italic_C start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="f(v)\in C_{2}" class="ltx_Math" display="inline" id="S4.I11.i1.p1.11.m10.1"><semantics id="S4.I11.i1.p1.11.m10.1a"><mrow id="S4.I11.i1.p1.11.m10.1.2" xref="S4.I11.i1.p1.11.m10.1.2.cmml"><mrow id="S4.I11.i1.p1.11.m10.1.2.2" xref="S4.I11.i1.p1.11.m10.1.2.2.cmml"><mi id="S4.I11.i1.p1.11.m10.1.2.2.2" xref="S4.I11.i1.p1.11.m10.1.2.2.2.cmml">f</mi><mo id="S4.I11.i1.p1.11.m10.1.2.2.1" xref="S4.I11.i1.p1.11.m10.1.2.2.1.cmml">⁢</mo><mrow id="S4.I11.i1.p1.11.m10.1.2.2.3.2" xref="S4.I11.i1.p1.11.m10.1.2.2.cmml"><mo id="S4.I11.i1.p1.11.m10.1.2.2.3.2.1" stretchy="false" xref="S4.I11.i1.p1.11.m10.1.2.2.cmml">(</mo><mi id="S4.I11.i1.p1.11.m10.1.1" xref="S4.I11.i1.p1.11.m10.1.1.cmml">v</mi><mo id="S4.I11.i1.p1.11.m10.1.2.2.3.2.2" stretchy="false" xref="S4.I11.i1.p1.11.m10.1.2.2.cmml">)</mo></mrow></mrow><mo id="S4.I11.i1.p1.11.m10.1.2.1" xref="S4.I11.i1.p1.11.m10.1.2.1.cmml">∈</mo><msub id="S4.I11.i1.p1.11.m10.1.2.3" xref="S4.I11.i1.p1.11.m10.1.2.3.cmml"><mi id="S4.I11.i1.p1.11.m10.1.2.3.2" xref="S4.I11.i1.p1.11.m10.1.2.3.2.cmml">C</mi><mn id="S4.I11.i1.p1.11.m10.1.2.3.3" xref="S4.I11.i1.p1.11.m10.1.2.3.3.cmml">2</mn></msub></mrow><annotation-xml encoding="MathML-Content" id="S4.I11.i1.p1.11.m10.1b"><apply id="S4.I11.i1.p1.11.m10.1.2.cmml" xref="S4.I11.i1.p1.11.m10.1.2"><in id="S4.I11.i1.p1.11.m10.1.2.1.cmml" xref="S4.I11.i1.p1.11.m10.1.2.1"></in><apply id="S4.I11.i1.p1.11.m10.1.2.2.cmml" xref="S4.I11.i1.p1.11.m10.1.2.2"><times id="S4.I11.i1.p1.11.m10.1.2.2.1.cmml" xref="S4.I11.i1.p1.11.m10.1.2.2.1"></times><ci id="S4.I11.i1.p1.11.m10.1.2.2.2.cmml" xref="S4.I11.i1.p1.11.m10.1.2.2.2">𝑓</ci><ci id="S4.I11.i1.p1.11.m10.1.1.cmml" xref="S4.I11.i1.p1.11.m10.1.1">𝑣</ci></apply><apply id="S4.I11.i1.p1.11.m10.1.2.3.cmml" xref="S4.I11.i1.p1.11.m10.1.2.3"><csymbol cd="ambiguous" id="S4.I11.i1.p1.11.m10.1.2.3.1.cmml" xref="S4.I11.i1.p1.11.m10.1.2.3">subscript</csymbol><ci id="S4.I11.i1.p1.11.m10.1.2.3.2.cmml" xref="S4.I11.i1.p1.11.m10.1.2.3.2">𝐶</ci><cn id="S4.I11.i1.p1.11.m10.1.2.3.3.cmml" type="integer" xref="S4.I11.i1.p1.11.m10.1.2.3.3">2</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.I11.i1.p1.11.m10.1c">f(v)\in C_{2}</annotation><annotation encoding="application/x-llamapun" id="S4.I11.i1.p1.11.m10.1d">italic_f ( italic_v ) ∈ italic_C start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT</annotation></semantics></math>.</p> <ul class="ltx_itemize" id="S4.I11.i1.I1"> <li class="ltx_item" id="S4.I11.i1.I1.i1" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item"><span class="ltx_text ltx_font_bold" id="S4.I11.i1.I1.i1.1.1.1">–</span></span> <div class="ltx_para" id="S4.I11.i1.I1.i1.p1"> <p class="ltx_p" id="S4.I11.i1.I1.i1.p1.26">If <math alttext="f(u)\prec f(v)" class="ltx_Math" display="inline" id="S4.I11.i1.I1.i1.p1.1.m1.2"><semantics id="S4.I11.i1.I1.i1.p1.1.m1.2a"><mrow id="S4.I11.i1.I1.i1.p1.1.m1.2.3" xref="S4.I11.i1.I1.i1.p1.1.m1.2.3.cmml"><mrow id="S4.I11.i1.I1.i1.p1.1.m1.2.3.2" xref="S4.I11.i1.I1.i1.p1.1.m1.2.3.2.cmml"><mi id="S4.I11.i1.I1.i1.p1.1.m1.2.3.2.2" xref="S4.I11.i1.I1.i1.p1.1.m1.2.3.2.2.cmml">f</mi><mo id="S4.I11.i1.I1.i1.p1.1.m1.2.3.2.1" xref="S4.I11.i1.I1.i1.p1.1.m1.2.3.2.1.cmml">⁢</mo><mrow id="S4.I11.i1.I1.i1.p1.1.m1.2.3.2.3.2" xref="S4.I11.i1.I1.i1.p1.1.m1.2.3.2.cmml"><mo id="S4.I11.i1.I1.i1.p1.1.m1.2.3.2.3.2.1" stretchy="false" xref="S4.I11.i1.I1.i1.p1.1.m1.2.3.2.cmml">(</mo><mi id="S4.I11.i1.I1.i1.p1.1.m1.1.1" xref="S4.I11.i1.I1.i1.p1.1.m1.1.1.cmml">u</mi><mo id="S4.I11.i1.I1.i1.p1.1.m1.2.3.2.3.2.2" stretchy="false" xref="S4.I11.i1.I1.i1.p1.1.m1.2.3.2.cmml">)</mo></mrow></mrow><mo id="S4.I11.i1.I1.i1.p1.1.m1.2.3.1" xref="S4.I11.i1.I1.i1.p1.1.m1.2.3.1.cmml">≺</mo><mrow id="S4.I11.i1.I1.i1.p1.1.m1.2.3.3" xref="S4.I11.i1.I1.i1.p1.1.m1.2.3.3.cmml"><mi id="S4.I11.i1.I1.i1.p1.1.m1.2.3.3.2" xref="S4.I11.i1.I1.i1.p1.1.m1.2.3.3.2.cmml">f</mi><mo id="S4.I11.i1.I1.i1.p1.1.m1.2.3.3.1" xref="S4.I11.i1.I1.i1.p1.1.m1.2.3.3.1.cmml">⁢</mo><mrow id="S4.I11.i1.I1.i1.p1.1.m1.2.3.3.3.2" xref="S4.I11.i1.I1.i1.p1.1.m1.2.3.3.cmml"><mo id="S4.I11.i1.I1.i1.p1.1.m1.2.3.3.3.2.1" stretchy="false" xref="S4.I11.i1.I1.i1.p1.1.m1.2.3.3.cmml">(</mo><mi id="S4.I11.i1.I1.i1.p1.1.m1.2.2" xref="S4.I11.i1.I1.i1.p1.1.m1.2.2.cmml">v</mi><mo id="S4.I11.i1.I1.i1.p1.1.m1.2.3.3.3.2.2" stretchy="false" xref="S4.I11.i1.I1.i1.p1.1.m1.2.3.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.I11.i1.I1.i1.p1.1.m1.2b"><apply id="S4.I11.i1.I1.i1.p1.1.m1.2.3.cmml" xref="S4.I11.i1.I1.i1.p1.1.m1.2.3"><csymbol cd="latexml" id="S4.I11.i1.I1.i1.p1.1.m1.2.3.1.cmml" xref="S4.I11.i1.I1.i1.p1.1.m1.2.3.1">precedes</csymbol><apply id="S4.I11.i1.I1.i1.p1.1.m1.2.3.2.cmml" xref="S4.I11.i1.I1.i1.p1.1.m1.2.3.2"><times id="S4.I11.i1.I1.i1.p1.1.m1.2.3.2.1.cmml" xref="S4.I11.i1.I1.i1.p1.1.m1.2.3.2.1"></times><ci id="S4.I11.i1.I1.i1.p1.1.m1.2.3.2.2.cmml" xref="S4.I11.i1.I1.i1.p1.1.m1.2.3.2.2">𝑓</ci><ci id="S4.I11.i1.I1.i1.p1.1.m1.1.1.cmml" xref="S4.I11.i1.I1.i1.p1.1.m1.1.1">𝑢</ci></apply><apply id="S4.I11.i1.I1.i1.p1.1.m1.2.3.3.cmml" xref="S4.I11.i1.I1.i1.p1.1.m1.2.3.3"><times id="S4.I11.i1.I1.i1.p1.1.m1.2.3.3.1.cmml" xref="S4.I11.i1.I1.i1.p1.1.m1.2.3.3.1"></times><ci id="S4.I11.i1.I1.i1.p1.1.m1.2.3.3.2.cmml" xref="S4.I11.i1.I1.i1.p1.1.m1.2.3.3.2">𝑓</ci><ci id="S4.I11.i1.I1.i1.p1.1.m1.2.2.cmml" xref="S4.I11.i1.I1.i1.p1.1.m1.2.2">𝑣</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.I11.i1.I1.i1.p1.1.m1.2c">f(u)\prec f(v)</annotation><annotation encoding="application/x-llamapun" id="S4.I11.i1.I1.i1.p1.1.m1.2d">italic_f ( italic_u ) ≺ italic_f ( italic_v )</annotation></semantics></math>, then let <math alttext="(v^{\prime},v^{\prime\prime})=\textsc{Min}_{f(v)}(j)" class="ltx_Math" display="inline" id="S4.I11.i1.I1.i1.p1.2.m2.4"><semantics id="S4.I11.i1.I1.i1.p1.2.m2.4a"><mrow id="S4.I11.i1.I1.i1.p1.2.m2.4.4" xref="S4.I11.i1.I1.i1.p1.2.m2.4.4.cmml"><mrow id="S4.I11.i1.I1.i1.p1.2.m2.4.4.2.2" xref="S4.I11.i1.I1.i1.p1.2.m2.4.4.2.3.cmml"><mo id="S4.I11.i1.I1.i1.p1.2.m2.4.4.2.2.3" stretchy="false" xref="S4.I11.i1.I1.i1.p1.2.m2.4.4.2.3.cmml">(</mo><msup id="S4.I11.i1.I1.i1.p1.2.m2.3.3.1.1.1" xref="S4.I11.i1.I1.i1.p1.2.m2.3.3.1.1.1.cmml"><mi id="S4.I11.i1.I1.i1.p1.2.m2.3.3.1.1.1.2" xref="S4.I11.i1.I1.i1.p1.2.m2.3.3.1.1.1.2.cmml">v</mi><mo id="S4.I11.i1.I1.i1.p1.2.m2.3.3.1.1.1.3" xref="S4.I11.i1.I1.i1.p1.2.m2.3.3.1.1.1.3.cmml">′</mo></msup><mo id="S4.I11.i1.I1.i1.p1.2.m2.4.4.2.2.4" xref="S4.I11.i1.I1.i1.p1.2.m2.4.4.2.3.cmml">,</mo><msup id="S4.I11.i1.I1.i1.p1.2.m2.4.4.2.2.2" xref="S4.I11.i1.I1.i1.p1.2.m2.4.4.2.2.2.cmml"><mi id="S4.I11.i1.I1.i1.p1.2.m2.4.4.2.2.2.2" xref="S4.I11.i1.I1.i1.p1.2.m2.4.4.2.2.2.2.cmml">v</mi><mo id="S4.I11.i1.I1.i1.p1.2.m2.4.4.2.2.2.3" xref="S4.I11.i1.I1.i1.p1.2.m2.4.4.2.2.2.3.cmml">′′</mo></msup><mo id="S4.I11.i1.I1.i1.p1.2.m2.4.4.2.2.5" stretchy="false" xref="S4.I11.i1.I1.i1.p1.2.m2.4.4.2.3.cmml">)</mo></mrow><mo id="S4.I11.i1.I1.i1.p1.2.m2.4.4.3" xref="S4.I11.i1.I1.i1.p1.2.m2.4.4.3.cmml">=</mo><mrow id="S4.I11.i1.I1.i1.p1.2.m2.4.4.4" xref="S4.I11.i1.I1.i1.p1.2.m2.4.4.4.cmml"><msub id="S4.I11.i1.I1.i1.p1.2.m2.4.4.4.2" xref="S4.I11.i1.I1.i1.p1.2.m2.4.4.4.2.cmml"><mtext class="ltx_font_smallcaps" id="S4.I11.i1.I1.i1.p1.2.m2.4.4.4.2.2" xref="S4.I11.i1.I1.i1.p1.2.m2.4.4.4.2.2a.cmml">Min</mtext><mrow id="S4.I11.i1.I1.i1.p1.2.m2.1.1.1" xref="S4.I11.i1.I1.i1.p1.2.m2.1.1.1.cmml"><mi id="S4.I11.i1.I1.i1.p1.2.m2.1.1.1.3" xref="S4.I11.i1.I1.i1.p1.2.m2.1.1.1.3.cmml">f</mi><mo id="S4.I11.i1.I1.i1.p1.2.m2.1.1.1.2" xref="S4.I11.i1.I1.i1.p1.2.m2.1.1.1.2.cmml">⁢</mo><mrow id="S4.I11.i1.I1.i1.p1.2.m2.1.1.1.4.2" xref="S4.I11.i1.I1.i1.p1.2.m2.1.1.1.cmml"><mo id="S4.I11.i1.I1.i1.p1.2.m2.1.1.1.4.2.1" stretchy="false" xref="S4.I11.i1.I1.i1.p1.2.m2.1.1.1.cmml">(</mo><mi id="S4.I11.i1.I1.i1.p1.2.m2.1.1.1.1" xref="S4.I11.i1.I1.i1.p1.2.m2.1.1.1.1.cmml">v</mi><mo id="S4.I11.i1.I1.i1.p1.2.m2.1.1.1.4.2.2" stretchy="false" xref="S4.I11.i1.I1.i1.p1.2.m2.1.1.1.cmml">)</mo></mrow></mrow></msub><mo id="S4.I11.i1.I1.i1.p1.2.m2.4.4.4.1" xref="S4.I11.i1.I1.i1.p1.2.m2.4.4.4.1.cmml">⁢</mo><mrow id="S4.I11.i1.I1.i1.p1.2.m2.4.4.4.3.2" xref="S4.I11.i1.I1.i1.p1.2.m2.4.4.4.cmml"><mo id="S4.I11.i1.I1.i1.p1.2.m2.4.4.4.3.2.1" stretchy="false" xref="S4.I11.i1.I1.i1.p1.2.m2.4.4.4.cmml">(</mo><mi id="S4.I11.i1.I1.i1.p1.2.m2.2.2" xref="S4.I11.i1.I1.i1.p1.2.m2.2.2.cmml">j</mi><mo id="S4.I11.i1.I1.i1.p1.2.m2.4.4.4.3.2.2" stretchy="false" xref="S4.I11.i1.I1.i1.p1.2.m2.4.4.4.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.I11.i1.I1.i1.p1.2.m2.4b"><apply id="S4.I11.i1.I1.i1.p1.2.m2.4.4.cmml" xref="S4.I11.i1.I1.i1.p1.2.m2.4.4"><eq id="S4.I11.i1.I1.i1.p1.2.m2.4.4.3.cmml" xref="S4.I11.i1.I1.i1.p1.2.m2.4.4.3"></eq><interval closure="open" id="S4.I11.i1.I1.i1.p1.2.m2.4.4.2.3.cmml" xref="S4.I11.i1.I1.i1.p1.2.m2.4.4.2.2"><apply id="S4.I11.i1.I1.i1.p1.2.m2.3.3.1.1.1.cmml" xref="S4.I11.i1.I1.i1.p1.2.m2.3.3.1.1.1"><csymbol cd="ambiguous" id="S4.I11.i1.I1.i1.p1.2.m2.3.3.1.1.1.1.cmml" xref="S4.I11.i1.I1.i1.p1.2.m2.3.3.1.1.1">superscript</csymbol><ci id="S4.I11.i1.I1.i1.p1.2.m2.3.3.1.1.1.2.cmml" xref="S4.I11.i1.I1.i1.p1.2.m2.3.3.1.1.1.2">𝑣</ci><ci id="S4.I11.i1.I1.i1.p1.2.m2.3.3.1.1.1.3.cmml" xref="S4.I11.i1.I1.i1.p1.2.m2.3.3.1.1.1.3">′</ci></apply><apply id="S4.I11.i1.I1.i1.p1.2.m2.4.4.2.2.2.cmml" xref="S4.I11.i1.I1.i1.p1.2.m2.4.4.2.2.2"><csymbol cd="ambiguous" id="S4.I11.i1.I1.i1.p1.2.m2.4.4.2.2.2.1.cmml" xref="S4.I11.i1.I1.i1.p1.2.m2.4.4.2.2.2">superscript</csymbol><ci id="S4.I11.i1.I1.i1.p1.2.m2.4.4.2.2.2.2.cmml" xref="S4.I11.i1.I1.i1.p1.2.m2.4.4.2.2.2.2">𝑣</ci><ci id="S4.I11.i1.I1.i1.p1.2.m2.4.4.2.2.2.3.cmml" xref="S4.I11.i1.I1.i1.p1.2.m2.4.4.2.2.2.3">′′</ci></apply></interval><apply id="S4.I11.i1.I1.i1.p1.2.m2.4.4.4.cmml" xref="S4.I11.i1.I1.i1.p1.2.m2.4.4.4"><times id="S4.I11.i1.I1.i1.p1.2.m2.4.4.4.1.cmml" xref="S4.I11.i1.I1.i1.p1.2.m2.4.4.4.1"></times><apply id="S4.I11.i1.I1.i1.p1.2.m2.4.4.4.2.cmml" xref="S4.I11.i1.I1.i1.p1.2.m2.4.4.4.2"><csymbol cd="ambiguous" id="S4.I11.i1.I1.i1.p1.2.m2.4.4.4.2.1.cmml" xref="S4.I11.i1.I1.i1.p1.2.m2.4.4.4.2">subscript</csymbol><ci id="S4.I11.i1.I1.i1.p1.2.m2.4.4.4.2.2a.cmml" xref="S4.I11.i1.I1.i1.p1.2.m2.4.4.4.2.2"><mtext class="ltx_font_smallcaps" id="S4.I11.i1.I1.i1.p1.2.m2.4.4.4.2.2.cmml" xref="S4.I11.i1.I1.i1.p1.2.m2.4.4.4.2.2">Min</mtext></ci><apply id="S4.I11.i1.I1.i1.p1.2.m2.1.1.1.cmml" xref="S4.I11.i1.I1.i1.p1.2.m2.1.1.1"><times id="S4.I11.i1.I1.i1.p1.2.m2.1.1.1.2.cmml" xref="S4.I11.i1.I1.i1.p1.2.m2.1.1.1.2"></times><ci id="S4.I11.i1.I1.i1.p1.2.m2.1.1.1.3.cmml" xref="S4.I11.i1.I1.i1.p1.2.m2.1.1.1.3">𝑓</ci><ci id="S4.I11.i1.I1.i1.p1.2.m2.1.1.1.1.cmml" xref="S4.I11.i1.I1.i1.p1.2.m2.1.1.1.1">𝑣</ci></apply></apply><ci id="S4.I11.i1.I1.i1.p1.2.m2.2.2.cmml" xref="S4.I11.i1.I1.i1.p1.2.m2.2.2">𝑗</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.I11.i1.I1.i1.p1.2.m2.4c">(v^{\prime},v^{\prime\prime})=\textsc{Min}_{f(v)}(j)</annotation><annotation encoding="application/x-llamapun" id="S4.I11.i1.I1.i1.p1.2.m2.4d">( italic_v start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT , italic_v start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT ) = Min start_POSTSUBSCRIPT italic_f ( italic_v ) end_POSTSUBSCRIPT ( italic_j )</annotation></semantics></math> with <math alttext="f(v^{\prime})=f(v)" class="ltx_Math" display="inline" id="S4.I11.i1.I1.i1.p1.3.m3.2"><semantics id="S4.I11.i1.I1.i1.p1.3.m3.2a"><mrow id="S4.I11.i1.I1.i1.p1.3.m3.2.2" xref="S4.I11.i1.I1.i1.p1.3.m3.2.2.cmml"><mrow id="S4.I11.i1.I1.i1.p1.3.m3.2.2.1" xref="S4.I11.i1.I1.i1.p1.3.m3.2.2.1.cmml"><mi id="S4.I11.i1.I1.i1.p1.3.m3.2.2.1.3" xref="S4.I11.i1.I1.i1.p1.3.m3.2.2.1.3.cmml">f</mi><mo id="S4.I11.i1.I1.i1.p1.3.m3.2.2.1.2" xref="S4.I11.i1.I1.i1.p1.3.m3.2.2.1.2.cmml">⁢</mo><mrow id="S4.I11.i1.I1.i1.p1.3.m3.2.2.1.1.1" xref="S4.I11.i1.I1.i1.p1.3.m3.2.2.1.1.1.1.cmml"><mo id="S4.I11.i1.I1.i1.p1.3.m3.2.2.1.1.1.2" stretchy="false" xref="S4.I11.i1.I1.i1.p1.3.m3.2.2.1.1.1.1.cmml">(</mo><msup id="S4.I11.i1.I1.i1.p1.3.m3.2.2.1.1.1.1" xref="S4.I11.i1.I1.i1.p1.3.m3.2.2.1.1.1.1.cmml"><mi id="S4.I11.i1.I1.i1.p1.3.m3.2.2.1.1.1.1.2" xref="S4.I11.i1.I1.i1.p1.3.m3.2.2.1.1.1.1.2.cmml">v</mi><mo id="S4.I11.i1.I1.i1.p1.3.m3.2.2.1.1.1.1.3" xref="S4.I11.i1.I1.i1.p1.3.m3.2.2.1.1.1.1.3.cmml">′</mo></msup><mo id="S4.I11.i1.I1.i1.p1.3.m3.2.2.1.1.1.3" stretchy="false" xref="S4.I11.i1.I1.i1.p1.3.m3.2.2.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S4.I11.i1.I1.i1.p1.3.m3.2.2.2" xref="S4.I11.i1.I1.i1.p1.3.m3.2.2.2.cmml">=</mo><mrow id="S4.I11.i1.I1.i1.p1.3.m3.2.2.3" xref="S4.I11.i1.I1.i1.p1.3.m3.2.2.3.cmml"><mi id="S4.I11.i1.I1.i1.p1.3.m3.2.2.3.2" xref="S4.I11.i1.I1.i1.p1.3.m3.2.2.3.2.cmml">f</mi><mo id="S4.I11.i1.I1.i1.p1.3.m3.2.2.3.1" xref="S4.I11.i1.I1.i1.p1.3.m3.2.2.3.1.cmml">⁢</mo><mrow id="S4.I11.i1.I1.i1.p1.3.m3.2.2.3.3.2" xref="S4.I11.i1.I1.i1.p1.3.m3.2.2.3.cmml"><mo id="S4.I11.i1.I1.i1.p1.3.m3.2.2.3.3.2.1" stretchy="false" xref="S4.I11.i1.I1.i1.p1.3.m3.2.2.3.cmml">(</mo><mi id="S4.I11.i1.I1.i1.p1.3.m3.1.1" xref="S4.I11.i1.I1.i1.p1.3.m3.1.1.cmml">v</mi><mo id="S4.I11.i1.I1.i1.p1.3.m3.2.2.3.3.2.2" stretchy="false" xref="S4.I11.i1.I1.i1.p1.3.m3.2.2.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.I11.i1.I1.i1.p1.3.m3.2b"><apply id="S4.I11.i1.I1.i1.p1.3.m3.2.2.cmml" xref="S4.I11.i1.I1.i1.p1.3.m3.2.2"><eq id="S4.I11.i1.I1.i1.p1.3.m3.2.2.2.cmml" xref="S4.I11.i1.I1.i1.p1.3.m3.2.2.2"></eq><apply id="S4.I11.i1.I1.i1.p1.3.m3.2.2.1.cmml" xref="S4.I11.i1.I1.i1.p1.3.m3.2.2.1"><times id="S4.I11.i1.I1.i1.p1.3.m3.2.2.1.2.cmml" xref="S4.I11.i1.I1.i1.p1.3.m3.2.2.1.2"></times><ci id="S4.I11.i1.I1.i1.p1.3.m3.2.2.1.3.cmml" xref="S4.I11.i1.I1.i1.p1.3.m3.2.2.1.3">𝑓</ci><apply id="S4.I11.i1.I1.i1.p1.3.m3.2.2.1.1.1.1.cmml" xref="S4.I11.i1.I1.i1.p1.3.m3.2.2.1.1.1"><csymbol cd="ambiguous" id="S4.I11.i1.I1.i1.p1.3.m3.2.2.1.1.1.1.1.cmml" xref="S4.I11.i1.I1.i1.p1.3.m3.2.2.1.1.1">superscript</csymbol><ci id="S4.I11.i1.I1.i1.p1.3.m3.2.2.1.1.1.1.2.cmml" xref="S4.I11.i1.I1.i1.p1.3.m3.2.2.1.1.1.1.2">𝑣</ci><ci id="S4.I11.i1.I1.i1.p1.3.m3.2.2.1.1.1.1.3.cmml" xref="S4.I11.i1.I1.i1.p1.3.m3.2.2.1.1.1.1.3">′</ci></apply></apply><apply id="S4.I11.i1.I1.i1.p1.3.m3.2.2.3.cmml" xref="S4.I11.i1.I1.i1.p1.3.m3.2.2.3"><times id="S4.I11.i1.I1.i1.p1.3.m3.2.2.3.1.cmml" xref="S4.I11.i1.I1.i1.p1.3.m3.2.2.3.1"></times><ci id="S4.I11.i1.I1.i1.p1.3.m3.2.2.3.2.cmml" xref="S4.I11.i1.I1.i1.p1.3.m3.2.2.3.2">𝑓</ci><ci id="S4.I11.i1.I1.i1.p1.3.m3.1.1.cmml" xref="S4.I11.i1.I1.i1.p1.3.m3.1.1">𝑣</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.I11.i1.I1.i1.p1.3.m3.2c">f(v^{\prime})=f(v)</annotation><annotation encoding="application/x-llamapun" id="S4.I11.i1.I1.i1.p1.3.m3.2d">italic_f ( italic_v start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) = italic_f ( italic_v )</annotation></semantics></math>. Clearly, <math alttext="v^{\prime},f(v^{\prime})\notin\{a,b\}" class="ltx_Math" display="inline" id="S4.I11.i1.I1.i1.p1.4.m4.4"><semantics id="S4.I11.i1.I1.i1.p1.4.m4.4a"><mrow id="S4.I11.i1.I1.i1.p1.4.m4.4.4" xref="S4.I11.i1.I1.i1.p1.4.m4.4.4.cmml"><mrow id="S4.I11.i1.I1.i1.p1.4.m4.4.4.2.2" xref="S4.I11.i1.I1.i1.p1.4.m4.4.4.2.3.cmml"><msup id="S4.I11.i1.I1.i1.p1.4.m4.3.3.1.1.1" xref="S4.I11.i1.I1.i1.p1.4.m4.3.3.1.1.1.cmml"><mi id="S4.I11.i1.I1.i1.p1.4.m4.3.3.1.1.1.2" xref="S4.I11.i1.I1.i1.p1.4.m4.3.3.1.1.1.2.cmml">v</mi><mo id="S4.I11.i1.I1.i1.p1.4.m4.3.3.1.1.1.3" xref="S4.I11.i1.I1.i1.p1.4.m4.3.3.1.1.1.3.cmml">′</mo></msup><mo id="S4.I11.i1.I1.i1.p1.4.m4.4.4.2.2.3" xref="S4.I11.i1.I1.i1.p1.4.m4.4.4.2.3.cmml">,</mo><mrow id="S4.I11.i1.I1.i1.p1.4.m4.4.4.2.2.2" xref="S4.I11.i1.I1.i1.p1.4.m4.4.4.2.2.2.cmml"><mi id="S4.I11.i1.I1.i1.p1.4.m4.4.4.2.2.2.3" xref="S4.I11.i1.I1.i1.p1.4.m4.4.4.2.2.2.3.cmml">f</mi><mo id="S4.I11.i1.I1.i1.p1.4.m4.4.4.2.2.2.2" xref="S4.I11.i1.I1.i1.p1.4.m4.4.4.2.2.2.2.cmml">⁢</mo><mrow id="S4.I11.i1.I1.i1.p1.4.m4.4.4.2.2.2.1.1" xref="S4.I11.i1.I1.i1.p1.4.m4.4.4.2.2.2.1.1.1.cmml"><mo id="S4.I11.i1.I1.i1.p1.4.m4.4.4.2.2.2.1.1.2" stretchy="false" xref="S4.I11.i1.I1.i1.p1.4.m4.4.4.2.2.2.1.1.1.cmml">(</mo><msup id="S4.I11.i1.I1.i1.p1.4.m4.4.4.2.2.2.1.1.1" xref="S4.I11.i1.I1.i1.p1.4.m4.4.4.2.2.2.1.1.1.cmml"><mi id="S4.I11.i1.I1.i1.p1.4.m4.4.4.2.2.2.1.1.1.2" xref="S4.I11.i1.I1.i1.p1.4.m4.4.4.2.2.2.1.1.1.2.cmml">v</mi><mo id="S4.I11.i1.I1.i1.p1.4.m4.4.4.2.2.2.1.1.1.3" xref="S4.I11.i1.I1.i1.p1.4.m4.4.4.2.2.2.1.1.1.3.cmml">′</mo></msup><mo id="S4.I11.i1.I1.i1.p1.4.m4.4.4.2.2.2.1.1.3" stretchy="false" xref="S4.I11.i1.I1.i1.p1.4.m4.4.4.2.2.2.1.1.1.cmml">)</mo></mrow></mrow></mrow><mo id="S4.I11.i1.I1.i1.p1.4.m4.4.4.3" xref="S4.I11.i1.I1.i1.p1.4.m4.4.4.3.cmml">∉</mo><mrow id="S4.I11.i1.I1.i1.p1.4.m4.4.4.4.2" xref="S4.I11.i1.I1.i1.p1.4.m4.4.4.4.1.cmml"><mo id="S4.I11.i1.I1.i1.p1.4.m4.4.4.4.2.1" stretchy="false" xref="S4.I11.i1.I1.i1.p1.4.m4.4.4.4.1.cmml">{</mo><mi id="S4.I11.i1.I1.i1.p1.4.m4.1.1" xref="S4.I11.i1.I1.i1.p1.4.m4.1.1.cmml">a</mi><mo id="S4.I11.i1.I1.i1.p1.4.m4.4.4.4.2.2" xref="S4.I11.i1.I1.i1.p1.4.m4.4.4.4.1.cmml">,</mo><mi id="S4.I11.i1.I1.i1.p1.4.m4.2.2" xref="S4.I11.i1.I1.i1.p1.4.m4.2.2.cmml">b</mi><mo id="S4.I11.i1.I1.i1.p1.4.m4.4.4.4.2.3" stretchy="false" xref="S4.I11.i1.I1.i1.p1.4.m4.4.4.4.1.cmml">}</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.I11.i1.I1.i1.p1.4.m4.4b"><apply id="S4.I11.i1.I1.i1.p1.4.m4.4.4.cmml" xref="S4.I11.i1.I1.i1.p1.4.m4.4.4"><notin id="S4.I11.i1.I1.i1.p1.4.m4.4.4.3.cmml" xref="S4.I11.i1.I1.i1.p1.4.m4.4.4.3"></notin><list id="S4.I11.i1.I1.i1.p1.4.m4.4.4.2.3.cmml" xref="S4.I11.i1.I1.i1.p1.4.m4.4.4.2.2"><apply id="S4.I11.i1.I1.i1.p1.4.m4.3.3.1.1.1.cmml" xref="S4.I11.i1.I1.i1.p1.4.m4.3.3.1.1.1"><csymbol cd="ambiguous" id="S4.I11.i1.I1.i1.p1.4.m4.3.3.1.1.1.1.cmml" xref="S4.I11.i1.I1.i1.p1.4.m4.3.3.1.1.1">superscript</csymbol><ci id="S4.I11.i1.I1.i1.p1.4.m4.3.3.1.1.1.2.cmml" xref="S4.I11.i1.I1.i1.p1.4.m4.3.3.1.1.1.2">𝑣</ci><ci id="S4.I11.i1.I1.i1.p1.4.m4.3.3.1.1.1.3.cmml" xref="S4.I11.i1.I1.i1.p1.4.m4.3.3.1.1.1.3">′</ci></apply><apply id="S4.I11.i1.I1.i1.p1.4.m4.4.4.2.2.2.cmml" xref="S4.I11.i1.I1.i1.p1.4.m4.4.4.2.2.2"><times id="S4.I11.i1.I1.i1.p1.4.m4.4.4.2.2.2.2.cmml" xref="S4.I11.i1.I1.i1.p1.4.m4.4.4.2.2.2.2"></times><ci id="S4.I11.i1.I1.i1.p1.4.m4.4.4.2.2.2.3.cmml" xref="S4.I11.i1.I1.i1.p1.4.m4.4.4.2.2.2.3">𝑓</ci><apply id="S4.I11.i1.I1.i1.p1.4.m4.4.4.2.2.2.1.1.1.cmml" xref="S4.I11.i1.I1.i1.p1.4.m4.4.4.2.2.2.1.1"><csymbol cd="ambiguous" id="S4.I11.i1.I1.i1.p1.4.m4.4.4.2.2.2.1.1.1.1.cmml" xref="S4.I11.i1.I1.i1.p1.4.m4.4.4.2.2.2.1.1">superscript</csymbol><ci id="S4.I11.i1.I1.i1.p1.4.m4.4.4.2.2.2.1.1.1.2.cmml" xref="S4.I11.i1.I1.i1.p1.4.m4.4.4.2.2.2.1.1.1.2">𝑣</ci><ci id="S4.I11.i1.I1.i1.p1.4.m4.4.4.2.2.2.1.1.1.3.cmml" xref="S4.I11.i1.I1.i1.p1.4.m4.4.4.2.2.2.1.1.1.3">′</ci></apply></apply></list><set id="S4.I11.i1.I1.i1.p1.4.m4.4.4.4.1.cmml" xref="S4.I11.i1.I1.i1.p1.4.m4.4.4.4.2"><ci id="S4.I11.i1.I1.i1.p1.4.m4.1.1.cmml" xref="S4.I11.i1.I1.i1.p1.4.m4.1.1">𝑎</ci><ci id="S4.I11.i1.I1.i1.p1.4.m4.2.2.cmml" xref="S4.I11.i1.I1.i1.p1.4.m4.2.2">𝑏</ci></set></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.I11.i1.I1.i1.p1.4.m4.4c">v^{\prime},f(v^{\prime})\notin\{a,b\}</annotation><annotation encoding="application/x-llamapun" id="S4.I11.i1.I1.i1.p1.4.m4.4d">italic_v start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT , italic_f ( italic_v start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) ∉ { italic_a , italic_b }</annotation></semantics></math>, since <math alttext="f(v)=f(v^{\prime})" class="ltx_Math" display="inline" id="S4.I11.i1.I1.i1.p1.5.m5.2"><semantics id="S4.I11.i1.I1.i1.p1.5.m5.2a"><mrow id="S4.I11.i1.I1.i1.p1.5.m5.2.2" xref="S4.I11.i1.I1.i1.p1.5.m5.2.2.cmml"><mrow id="S4.I11.i1.I1.i1.p1.5.m5.2.2.3" xref="S4.I11.i1.I1.i1.p1.5.m5.2.2.3.cmml"><mi id="S4.I11.i1.I1.i1.p1.5.m5.2.2.3.2" xref="S4.I11.i1.I1.i1.p1.5.m5.2.2.3.2.cmml">f</mi><mo id="S4.I11.i1.I1.i1.p1.5.m5.2.2.3.1" xref="S4.I11.i1.I1.i1.p1.5.m5.2.2.3.1.cmml">⁢</mo><mrow id="S4.I11.i1.I1.i1.p1.5.m5.2.2.3.3.2" xref="S4.I11.i1.I1.i1.p1.5.m5.2.2.3.cmml"><mo id="S4.I11.i1.I1.i1.p1.5.m5.2.2.3.3.2.1" stretchy="false" xref="S4.I11.i1.I1.i1.p1.5.m5.2.2.3.cmml">(</mo><mi id="S4.I11.i1.I1.i1.p1.5.m5.1.1" xref="S4.I11.i1.I1.i1.p1.5.m5.1.1.cmml">v</mi><mo id="S4.I11.i1.I1.i1.p1.5.m5.2.2.3.3.2.2" stretchy="false" xref="S4.I11.i1.I1.i1.p1.5.m5.2.2.3.cmml">)</mo></mrow></mrow><mo id="S4.I11.i1.I1.i1.p1.5.m5.2.2.2" xref="S4.I11.i1.I1.i1.p1.5.m5.2.2.2.cmml">=</mo><mrow id="S4.I11.i1.I1.i1.p1.5.m5.2.2.1" xref="S4.I11.i1.I1.i1.p1.5.m5.2.2.1.cmml"><mi id="S4.I11.i1.I1.i1.p1.5.m5.2.2.1.3" xref="S4.I11.i1.I1.i1.p1.5.m5.2.2.1.3.cmml">f</mi><mo id="S4.I11.i1.I1.i1.p1.5.m5.2.2.1.2" xref="S4.I11.i1.I1.i1.p1.5.m5.2.2.1.2.cmml">⁢</mo><mrow id="S4.I11.i1.I1.i1.p1.5.m5.2.2.1.1.1" xref="S4.I11.i1.I1.i1.p1.5.m5.2.2.1.1.1.1.cmml"><mo id="S4.I11.i1.I1.i1.p1.5.m5.2.2.1.1.1.2" stretchy="false" xref="S4.I11.i1.I1.i1.p1.5.m5.2.2.1.1.1.1.cmml">(</mo><msup id="S4.I11.i1.I1.i1.p1.5.m5.2.2.1.1.1.1" xref="S4.I11.i1.I1.i1.p1.5.m5.2.2.1.1.1.1.cmml"><mi id="S4.I11.i1.I1.i1.p1.5.m5.2.2.1.1.1.1.2" xref="S4.I11.i1.I1.i1.p1.5.m5.2.2.1.1.1.1.2.cmml">v</mi><mo id="S4.I11.i1.I1.i1.p1.5.m5.2.2.1.1.1.1.3" xref="S4.I11.i1.I1.i1.p1.5.m5.2.2.1.1.1.1.3.cmml">′</mo></msup><mo id="S4.I11.i1.I1.i1.p1.5.m5.2.2.1.1.1.3" stretchy="false" xref="S4.I11.i1.I1.i1.p1.5.m5.2.2.1.1.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.I11.i1.I1.i1.p1.5.m5.2b"><apply id="S4.I11.i1.I1.i1.p1.5.m5.2.2.cmml" xref="S4.I11.i1.I1.i1.p1.5.m5.2.2"><eq id="S4.I11.i1.I1.i1.p1.5.m5.2.2.2.cmml" xref="S4.I11.i1.I1.i1.p1.5.m5.2.2.2"></eq><apply id="S4.I11.i1.I1.i1.p1.5.m5.2.2.3.cmml" xref="S4.I11.i1.I1.i1.p1.5.m5.2.2.3"><times id="S4.I11.i1.I1.i1.p1.5.m5.2.2.3.1.cmml" xref="S4.I11.i1.I1.i1.p1.5.m5.2.2.3.1"></times><ci id="S4.I11.i1.I1.i1.p1.5.m5.2.2.3.2.cmml" xref="S4.I11.i1.I1.i1.p1.5.m5.2.2.3.2">𝑓</ci><ci id="S4.I11.i1.I1.i1.p1.5.m5.1.1.cmml" xref="S4.I11.i1.I1.i1.p1.5.m5.1.1">𝑣</ci></apply><apply id="S4.I11.i1.I1.i1.p1.5.m5.2.2.1.cmml" xref="S4.I11.i1.I1.i1.p1.5.m5.2.2.1"><times id="S4.I11.i1.I1.i1.p1.5.m5.2.2.1.2.cmml" xref="S4.I11.i1.I1.i1.p1.5.m5.2.2.1.2"></times><ci id="S4.I11.i1.I1.i1.p1.5.m5.2.2.1.3.cmml" xref="S4.I11.i1.I1.i1.p1.5.m5.2.2.1.3">𝑓</ci><apply id="S4.I11.i1.I1.i1.p1.5.m5.2.2.1.1.1.1.cmml" xref="S4.I11.i1.I1.i1.p1.5.m5.2.2.1.1.1"><csymbol cd="ambiguous" id="S4.I11.i1.I1.i1.p1.5.m5.2.2.1.1.1.1.1.cmml" xref="S4.I11.i1.I1.i1.p1.5.m5.2.2.1.1.1">superscript</csymbol><ci id="S4.I11.i1.I1.i1.p1.5.m5.2.2.1.1.1.1.2.cmml" xref="S4.I11.i1.I1.i1.p1.5.m5.2.2.1.1.1.1.2">𝑣</ci><ci id="S4.I11.i1.I1.i1.p1.5.m5.2.2.1.1.1.1.3.cmml" xref="S4.I11.i1.I1.i1.p1.5.m5.2.2.1.1.1.1.3">′</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.I11.i1.I1.i1.p1.5.m5.2c">f(v)=f(v^{\prime})</annotation><annotation encoding="application/x-llamapun" id="S4.I11.i1.I1.i1.p1.5.m5.2d">italic_f ( italic_v ) = italic_f ( italic_v start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT )</annotation></semantics></math>, so either <math alttext="v^{\prime}=v=f(v)" class="ltx_Math" display="inline" id="S4.I11.i1.I1.i1.p1.6.m6.1"><semantics id="S4.I11.i1.I1.i1.p1.6.m6.1a"><mrow id="S4.I11.i1.I1.i1.p1.6.m6.1.2" xref="S4.I11.i1.I1.i1.p1.6.m6.1.2.cmml"><msup id="S4.I11.i1.I1.i1.p1.6.m6.1.2.2" xref="S4.I11.i1.I1.i1.p1.6.m6.1.2.2.cmml"><mi id="S4.I11.i1.I1.i1.p1.6.m6.1.2.2.2" xref="S4.I11.i1.I1.i1.p1.6.m6.1.2.2.2.cmml">v</mi><mo id="S4.I11.i1.I1.i1.p1.6.m6.1.2.2.3" xref="S4.I11.i1.I1.i1.p1.6.m6.1.2.2.3.cmml">′</mo></msup><mo id="S4.I11.i1.I1.i1.p1.6.m6.1.2.3" xref="S4.I11.i1.I1.i1.p1.6.m6.1.2.3.cmml">=</mo><mi id="S4.I11.i1.I1.i1.p1.6.m6.1.2.4" xref="S4.I11.i1.I1.i1.p1.6.m6.1.2.4.cmml">v</mi><mo id="S4.I11.i1.I1.i1.p1.6.m6.1.2.5" xref="S4.I11.i1.I1.i1.p1.6.m6.1.2.5.cmml">=</mo><mrow id="S4.I11.i1.I1.i1.p1.6.m6.1.2.6" xref="S4.I11.i1.I1.i1.p1.6.m6.1.2.6.cmml"><mi id="S4.I11.i1.I1.i1.p1.6.m6.1.2.6.2" xref="S4.I11.i1.I1.i1.p1.6.m6.1.2.6.2.cmml">f</mi><mo id="S4.I11.i1.I1.i1.p1.6.m6.1.2.6.1" xref="S4.I11.i1.I1.i1.p1.6.m6.1.2.6.1.cmml">⁢</mo><mrow id="S4.I11.i1.I1.i1.p1.6.m6.1.2.6.3.2" xref="S4.I11.i1.I1.i1.p1.6.m6.1.2.6.cmml"><mo id="S4.I11.i1.I1.i1.p1.6.m6.1.2.6.3.2.1" stretchy="false" xref="S4.I11.i1.I1.i1.p1.6.m6.1.2.6.cmml">(</mo><mi id="S4.I11.i1.I1.i1.p1.6.m6.1.1" xref="S4.I11.i1.I1.i1.p1.6.m6.1.1.cmml">v</mi><mo id="S4.I11.i1.I1.i1.p1.6.m6.1.2.6.3.2.2" stretchy="false" xref="S4.I11.i1.I1.i1.p1.6.m6.1.2.6.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.I11.i1.I1.i1.p1.6.m6.1b"><apply id="S4.I11.i1.I1.i1.p1.6.m6.1.2.cmml" xref="S4.I11.i1.I1.i1.p1.6.m6.1.2"><and id="S4.I11.i1.I1.i1.p1.6.m6.1.2a.cmml" xref="S4.I11.i1.I1.i1.p1.6.m6.1.2"></and><apply id="S4.I11.i1.I1.i1.p1.6.m6.1.2b.cmml" xref="S4.I11.i1.I1.i1.p1.6.m6.1.2"><eq id="S4.I11.i1.I1.i1.p1.6.m6.1.2.3.cmml" xref="S4.I11.i1.I1.i1.p1.6.m6.1.2.3"></eq><apply id="S4.I11.i1.I1.i1.p1.6.m6.1.2.2.cmml" xref="S4.I11.i1.I1.i1.p1.6.m6.1.2.2"><csymbol cd="ambiguous" id="S4.I11.i1.I1.i1.p1.6.m6.1.2.2.1.cmml" xref="S4.I11.i1.I1.i1.p1.6.m6.1.2.2">superscript</csymbol><ci id="S4.I11.i1.I1.i1.p1.6.m6.1.2.2.2.cmml" xref="S4.I11.i1.I1.i1.p1.6.m6.1.2.2.2">𝑣</ci><ci id="S4.I11.i1.I1.i1.p1.6.m6.1.2.2.3.cmml" xref="S4.I11.i1.I1.i1.p1.6.m6.1.2.2.3">′</ci></apply><ci id="S4.I11.i1.I1.i1.p1.6.m6.1.2.4.cmml" xref="S4.I11.i1.I1.i1.p1.6.m6.1.2.4">𝑣</ci></apply><apply id="S4.I11.i1.I1.i1.p1.6.m6.1.2c.cmml" xref="S4.I11.i1.I1.i1.p1.6.m6.1.2"><eq id="S4.I11.i1.I1.i1.p1.6.m6.1.2.5.cmml" xref="S4.I11.i1.I1.i1.p1.6.m6.1.2.5"></eq><share href="https://arxiv.org/html/2503.00712v1#S4.I11.i1.I1.i1.p1.6.m6.1.2.4.cmml" id="S4.I11.i1.I1.i1.p1.6.m6.1.2d.cmml" xref="S4.I11.i1.I1.i1.p1.6.m6.1.2"></share><apply id="S4.I11.i1.I1.i1.p1.6.m6.1.2.6.cmml" xref="S4.I11.i1.I1.i1.p1.6.m6.1.2.6"><times id="S4.I11.i1.I1.i1.p1.6.m6.1.2.6.1.cmml" xref="S4.I11.i1.I1.i1.p1.6.m6.1.2.6.1"></times><ci id="S4.I11.i1.I1.i1.p1.6.m6.1.2.6.2.cmml" xref="S4.I11.i1.I1.i1.p1.6.m6.1.2.6.2">𝑓</ci><ci id="S4.I11.i1.I1.i1.p1.6.m6.1.1.cmml" xref="S4.I11.i1.I1.i1.p1.6.m6.1.1">𝑣</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.I11.i1.I1.i1.p1.6.m6.1c">v^{\prime}=v=f(v)</annotation><annotation encoding="application/x-llamapun" id="S4.I11.i1.I1.i1.p1.6.m6.1d">italic_v start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT = italic_v = italic_f ( italic_v )</annotation></semantics></math> or <math alttext="v" class="ltx_Math" display="inline" id="S4.I11.i1.I1.i1.p1.7.m7.1"><semantics id="S4.I11.i1.I1.i1.p1.7.m7.1a"><mi id="S4.I11.i1.I1.i1.p1.7.m7.1.1" xref="S4.I11.i1.I1.i1.p1.7.m7.1.1.cmml">v</mi><annotation-xml encoding="MathML-Content" id="S4.I11.i1.I1.i1.p1.7.m7.1b"><ci id="S4.I11.i1.I1.i1.p1.7.m7.1.1.cmml" xref="S4.I11.i1.I1.i1.p1.7.m7.1.1">𝑣</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.I11.i1.I1.i1.p1.7.m7.1c">v</annotation><annotation encoding="application/x-llamapun" id="S4.I11.i1.I1.i1.p1.7.m7.1d">italic_v</annotation></semantics></math> and <math alttext="v^{\prime}" class="ltx_Math" display="inline" id="S4.I11.i1.I1.i1.p1.8.m8.1"><semantics id="S4.I11.i1.I1.i1.p1.8.m8.1a"><msup id="S4.I11.i1.I1.i1.p1.8.m8.1.1" xref="S4.I11.i1.I1.i1.p1.8.m8.1.1.cmml"><mi id="S4.I11.i1.I1.i1.p1.8.m8.1.1.2" xref="S4.I11.i1.I1.i1.p1.8.m8.1.1.2.cmml">v</mi><mo id="S4.I11.i1.I1.i1.p1.8.m8.1.1.3" xref="S4.I11.i1.I1.i1.p1.8.m8.1.1.3.cmml">′</mo></msup><annotation-xml encoding="MathML-Content" id="S4.I11.i1.I1.i1.p1.8.m8.1b"><apply id="S4.I11.i1.I1.i1.p1.8.m8.1.1.cmml" xref="S4.I11.i1.I1.i1.p1.8.m8.1.1"><csymbol cd="ambiguous" id="S4.I11.i1.I1.i1.p1.8.m8.1.1.1.cmml" xref="S4.I11.i1.I1.i1.p1.8.m8.1.1">superscript</csymbol><ci id="S4.I11.i1.I1.i1.p1.8.m8.1.1.2.cmml" xref="S4.I11.i1.I1.i1.p1.8.m8.1.1.2">𝑣</ci><ci id="S4.I11.i1.I1.i1.p1.8.m8.1.1.3.cmml" xref="S4.I11.i1.I1.i1.p1.8.m8.1.1.3">′</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.I11.i1.I1.i1.p1.8.m8.1c">v^{\prime}</annotation><annotation encoding="application/x-llamapun" id="S4.I11.i1.I1.i1.p1.8.m8.1d">italic_v start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math> are in the same subtree rooted at <math alttext="x" class="ltx_Math" display="inline" id="S4.I11.i1.I1.i1.p1.9.m9.1"><semantics id="S4.I11.i1.I1.i1.p1.9.m9.1a"><mi id="S4.I11.i1.I1.i1.p1.9.m9.1.1" xref="S4.I11.i1.I1.i1.p1.9.m9.1.1.cmml">x</mi><annotation-xml encoding="MathML-Content" id="S4.I11.i1.I1.i1.p1.9.m9.1b"><ci id="S4.I11.i1.I1.i1.p1.9.m9.1.1.cmml" xref="S4.I11.i1.I1.i1.p1.9.m9.1.1">𝑥</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.I11.i1.I1.i1.p1.9.m9.1c">x</annotation><annotation encoding="application/x-llamapun" id="S4.I11.i1.I1.i1.p1.9.m9.1d">italic_x</annotation></semantics></math> and <math alttext="f(v^{\prime})" class="ltx_Math" display="inline" id="S4.I11.i1.I1.i1.p1.10.m10.1"><semantics id="S4.I11.i1.I1.i1.p1.10.m10.1a"><mrow id="S4.I11.i1.I1.i1.p1.10.m10.1.1" xref="S4.I11.i1.I1.i1.p1.10.m10.1.1.cmml"><mi id="S4.I11.i1.I1.i1.p1.10.m10.1.1.3" xref="S4.I11.i1.I1.i1.p1.10.m10.1.1.3.cmml">f</mi><mo id="S4.I11.i1.I1.i1.p1.10.m10.1.1.2" xref="S4.I11.i1.I1.i1.p1.10.m10.1.1.2.cmml">⁢</mo><mrow id="S4.I11.i1.I1.i1.p1.10.m10.1.1.1.1" xref="S4.I11.i1.I1.i1.p1.10.m10.1.1.1.1.1.cmml"><mo id="S4.I11.i1.I1.i1.p1.10.m10.1.1.1.1.2" stretchy="false" xref="S4.I11.i1.I1.i1.p1.10.m10.1.1.1.1.1.cmml">(</mo><msup id="S4.I11.i1.I1.i1.p1.10.m10.1.1.1.1.1" xref="S4.I11.i1.I1.i1.p1.10.m10.1.1.1.1.1.cmml"><mi id="S4.I11.i1.I1.i1.p1.10.m10.1.1.1.1.1.2" xref="S4.I11.i1.I1.i1.p1.10.m10.1.1.1.1.1.2.cmml">v</mi><mo id="S4.I11.i1.I1.i1.p1.10.m10.1.1.1.1.1.3" xref="S4.I11.i1.I1.i1.p1.10.m10.1.1.1.1.1.3.cmml">′</mo></msup><mo id="S4.I11.i1.I1.i1.p1.10.m10.1.1.1.1.3" stretchy="false" xref="S4.I11.i1.I1.i1.p1.10.m10.1.1.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.I11.i1.I1.i1.p1.10.m10.1b"><apply id="S4.I11.i1.I1.i1.p1.10.m10.1.1.cmml" xref="S4.I11.i1.I1.i1.p1.10.m10.1.1"><times id="S4.I11.i1.I1.i1.p1.10.m10.1.1.2.cmml" xref="S4.I11.i1.I1.i1.p1.10.m10.1.1.2"></times><ci id="S4.I11.i1.I1.i1.p1.10.m10.1.1.3.cmml" xref="S4.I11.i1.I1.i1.p1.10.m10.1.1.3">𝑓</ci><apply id="S4.I11.i1.I1.i1.p1.10.m10.1.1.1.1.1.cmml" xref="S4.I11.i1.I1.i1.p1.10.m10.1.1.1.1"><csymbol cd="ambiguous" id="S4.I11.i1.I1.i1.p1.10.m10.1.1.1.1.1.1.cmml" xref="S4.I11.i1.I1.i1.p1.10.m10.1.1.1.1">superscript</csymbol><ci id="S4.I11.i1.I1.i1.p1.10.m10.1.1.1.1.1.2.cmml" xref="S4.I11.i1.I1.i1.p1.10.m10.1.1.1.1.1.2">𝑣</ci><ci id="S4.I11.i1.I1.i1.p1.10.m10.1.1.1.1.1.3.cmml" xref="S4.I11.i1.I1.i1.p1.10.m10.1.1.1.1.1.3">′</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.I11.i1.I1.i1.p1.10.m10.1c">f(v^{\prime})</annotation><annotation encoding="application/x-llamapun" id="S4.I11.i1.I1.i1.p1.10.m10.1d">italic_f ( italic_v start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT )</annotation></semantics></math> is a dummy node. Furthermore, note that <math alttext="(v^{\prime},v^{\prime\prime})\in\textnormal{SOL}" class="ltx_Math" display="inline" id="S4.I11.i1.I1.i1.p1.11.m11.2"><semantics id="S4.I11.i1.I1.i1.p1.11.m11.2a"><mrow id="S4.I11.i1.I1.i1.p1.11.m11.2.2" xref="S4.I11.i1.I1.i1.p1.11.m11.2.2.cmml"><mrow id="S4.I11.i1.I1.i1.p1.11.m11.2.2.2.2" xref="S4.I11.i1.I1.i1.p1.11.m11.2.2.2.3.cmml"><mo id="S4.I11.i1.I1.i1.p1.11.m11.2.2.2.2.3" stretchy="false" xref="S4.I11.i1.I1.i1.p1.11.m11.2.2.2.3.cmml">(</mo><msup id="S4.I11.i1.I1.i1.p1.11.m11.1.1.1.1.1" xref="S4.I11.i1.I1.i1.p1.11.m11.1.1.1.1.1.cmml"><mi id="S4.I11.i1.I1.i1.p1.11.m11.1.1.1.1.1.2" xref="S4.I11.i1.I1.i1.p1.11.m11.1.1.1.1.1.2.cmml">v</mi><mo id="S4.I11.i1.I1.i1.p1.11.m11.1.1.1.1.1.3" xref="S4.I11.i1.I1.i1.p1.11.m11.1.1.1.1.1.3.cmml">′</mo></msup><mo id="S4.I11.i1.I1.i1.p1.11.m11.2.2.2.2.4" xref="S4.I11.i1.I1.i1.p1.11.m11.2.2.2.3.cmml">,</mo><msup id="S4.I11.i1.I1.i1.p1.11.m11.2.2.2.2.2" xref="S4.I11.i1.I1.i1.p1.11.m11.2.2.2.2.2.cmml"><mi id="S4.I11.i1.I1.i1.p1.11.m11.2.2.2.2.2.2" xref="S4.I11.i1.I1.i1.p1.11.m11.2.2.2.2.2.2.cmml">v</mi><mo id="S4.I11.i1.I1.i1.p1.11.m11.2.2.2.2.2.3" xref="S4.I11.i1.I1.i1.p1.11.m11.2.2.2.2.2.3.cmml">′′</mo></msup><mo id="S4.I11.i1.I1.i1.p1.11.m11.2.2.2.2.5" stretchy="false" xref="S4.I11.i1.I1.i1.p1.11.m11.2.2.2.3.cmml">)</mo></mrow><mo id="S4.I11.i1.I1.i1.p1.11.m11.2.2.3" xref="S4.I11.i1.I1.i1.p1.11.m11.2.2.3.cmml">∈</mo><mtext id="S4.I11.i1.I1.i1.p1.11.m11.2.2.4" xref="S4.I11.i1.I1.i1.p1.11.m11.2.2.4a.cmml">SOL</mtext></mrow><annotation-xml encoding="MathML-Content" id="S4.I11.i1.I1.i1.p1.11.m11.2b"><apply id="S4.I11.i1.I1.i1.p1.11.m11.2.2.cmml" xref="S4.I11.i1.I1.i1.p1.11.m11.2.2"><in id="S4.I11.i1.I1.i1.p1.11.m11.2.2.3.cmml" xref="S4.I11.i1.I1.i1.p1.11.m11.2.2.3"></in><interval closure="open" id="S4.I11.i1.I1.i1.p1.11.m11.2.2.2.3.cmml" xref="S4.I11.i1.I1.i1.p1.11.m11.2.2.2.2"><apply id="S4.I11.i1.I1.i1.p1.11.m11.1.1.1.1.1.cmml" xref="S4.I11.i1.I1.i1.p1.11.m11.1.1.1.1.1"><csymbol cd="ambiguous" id="S4.I11.i1.I1.i1.p1.11.m11.1.1.1.1.1.1.cmml" xref="S4.I11.i1.I1.i1.p1.11.m11.1.1.1.1.1">superscript</csymbol><ci id="S4.I11.i1.I1.i1.p1.11.m11.1.1.1.1.1.2.cmml" xref="S4.I11.i1.I1.i1.p1.11.m11.1.1.1.1.1.2">𝑣</ci><ci id="S4.I11.i1.I1.i1.p1.11.m11.1.1.1.1.1.3.cmml" xref="S4.I11.i1.I1.i1.p1.11.m11.1.1.1.1.1.3">′</ci></apply><apply id="S4.I11.i1.I1.i1.p1.11.m11.2.2.2.2.2.cmml" xref="S4.I11.i1.I1.i1.p1.11.m11.2.2.2.2.2"><csymbol cd="ambiguous" id="S4.I11.i1.I1.i1.p1.11.m11.2.2.2.2.2.1.cmml" xref="S4.I11.i1.I1.i1.p1.11.m11.2.2.2.2.2">superscript</csymbol><ci id="S4.I11.i1.I1.i1.p1.11.m11.2.2.2.2.2.2.cmml" xref="S4.I11.i1.I1.i1.p1.11.m11.2.2.2.2.2.2">𝑣</ci><ci id="S4.I11.i1.I1.i1.p1.11.m11.2.2.2.2.2.3.cmml" xref="S4.I11.i1.I1.i1.p1.11.m11.2.2.2.2.2.3">′′</ci></apply></interval><ci id="S4.I11.i1.I1.i1.p1.11.m11.2.2.4a.cmml" xref="S4.I11.i1.I1.i1.p1.11.m11.2.2.4"><mtext id="S4.I11.i1.I1.i1.p1.11.m11.2.2.4.cmml" xref="S4.I11.i1.I1.i1.p1.11.m11.2.2.4">SOL</mtext></ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.I11.i1.I1.i1.p1.11.m11.2c">(v^{\prime},v^{\prime\prime})\in\textnormal{SOL}</annotation><annotation encoding="application/x-llamapun" id="S4.I11.i1.I1.i1.p1.11.m11.2d">( italic_v start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT , italic_v start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT ) ∈ SOL</annotation></semantics></math> since <math alttext="uv\in\textnormal{OPT}" class="ltx_Math" display="inline" id="S4.I11.i1.I1.i1.p1.12.m12.1"><semantics id="S4.I11.i1.I1.i1.p1.12.m12.1a"><mrow id="S4.I11.i1.I1.i1.p1.12.m12.1.1" xref="S4.I11.i1.I1.i1.p1.12.m12.1.1.cmml"><mrow id="S4.I11.i1.I1.i1.p1.12.m12.1.1.2" xref="S4.I11.i1.I1.i1.p1.12.m12.1.1.2.cmml"><mi id="S4.I11.i1.I1.i1.p1.12.m12.1.1.2.2" xref="S4.I11.i1.I1.i1.p1.12.m12.1.1.2.2.cmml">u</mi><mo id="S4.I11.i1.I1.i1.p1.12.m12.1.1.2.1" xref="S4.I11.i1.I1.i1.p1.12.m12.1.1.2.1.cmml">⁢</mo><mi id="S4.I11.i1.I1.i1.p1.12.m12.1.1.2.3" xref="S4.I11.i1.I1.i1.p1.12.m12.1.1.2.3.cmml">v</mi></mrow><mo id="S4.I11.i1.I1.i1.p1.12.m12.1.1.1" xref="S4.I11.i1.I1.i1.p1.12.m12.1.1.1.cmml">∈</mo><mtext id="S4.I11.i1.I1.i1.p1.12.m12.1.1.3" xref="S4.I11.i1.I1.i1.p1.12.m12.1.1.3a.cmml">OPT</mtext></mrow><annotation-xml encoding="MathML-Content" id="S4.I11.i1.I1.i1.p1.12.m12.1b"><apply id="S4.I11.i1.I1.i1.p1.12.m12.1.1.cmml" xref="S4.I11.i1.I1.i1.p1.12.m12.1.1"><in id="S4.I11.i1.I1.i1.p1.12.m12.1.1.1.cmml" xref="S4.I11.i1.I1.i1.p1.12.m12.1.1.1"></in><apply id="S4.I11.i1.I1.i1.p1.12.m12.1.1.2.cmml" xref="S4.I11.i1.I1.i1.p1.12.m12.1.1.2"><times id="S4.I11.i1.I1.i1.p1.12.m12.1.1.2.1.cmml" xref="S4.I11.i1.I1.i1.p1.12.m12.1.1.2.1"></times><ci id="S4.I11.i1.I1.i1.p1.12.m12.1.1.2.2.cmml" xref="S4.I11.i1.I1.i1.p1.12.m12.1.1.2.2">𝑢</ci><ci id="S4.I11.i1.I1.i1.p1.12.m12.1.1.2.3.cmml" xref="S4.I11.i1.I1.i1.p1.12.m12.1.1.2.3">𝑣</ci></apply><ci id="S4.I11.i1.I1.i1.p1.12.m12.1.1.3a.cmml" xref="S4.I11.i1.I1.i1.p1.12.m12.1.1.3"><mtext id="S4.I11.i1.I1.i1.p1.12.m12.1.1.3.cmml" xref="S4.I11.i1.I1.i1.p1.12.m12.1.1.3">OPT</mtext></ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.I11.i1.I1.i1.p1.12.m12.1c">uv\in\textnormal{OPT}</annotation><annotation encoding="application/x-llamapun" id="S4.I11.i1.I1.i1.p1.12.m12.1d">italic_u italic_v ∈ OPT</annotation></semantics></math> and <math alttext="\text{LCA}(\ell(u),\ell(v))=x" class="ltx_Math" display="inline" id="S4.I11.i1.I1.i1.p1.13.m13.4"><semantics id="S4.I11.i1.I1.i1.p1.13.m13.4a"><mrow id="S4.I11.i1.I1.i1.p1.13.m13.4.4" xref="S4.I11.i1.I1.i1.p1.13.m13.4.4.cmml"><mrow id="S4.I11.i1.I1.i1.p1.13.m13.4.4.2" xref="S4.I11.i1.I1.i1.p1.13.m13.4.4.2.cmml"><mtext id="S4.I11.i1.I1.i1.p1.13.m13.4.4.2.4" xref="S4.I11.i1.I1.i1.p1.13.m13.4.4.2.4a.cmml">LCA</mtext><mo id="S4.I11.i1.I1.i1.p1.13.m13.4.4.2.3" xref="S4.I11.i1.I1.i1.p1.13.m13.4.4.2.3.cmml">⁢</mo><mrow id="S4.I11.i1.I1.i1.p1.13.m13.4.4.2.2.2" xref="S4.I11.i1.I1.i1.p1.13.m13.4.4.2.2.3.cmml"><mo id="S4.I11.i1.I1.i1.p1.13.m13.4.4.2.2.2.3" stretchy="false" xref="S4.I11.i1.I1.i1.p1.13.m13.4.4.2.2.3.cmml">(</mo><mrow id="S4.I11.i1.I1.i1.p1.13.m13.3.3.1.1.1.1" xref="S4.I11.i1.I1.i1.p1.13.m13.3.3.1.1.1.1.cmml"><mi id="S4.I11.i1.I1.i1.p1.13.m13.3.3.1.1.1.1.2" mathvariant="normal" xref="S4.I11.i1.I1.i1.p1.13.m13.3.3.1.1.1.1.2.cmml">ℓ</mi><mo id="S4.I11.i1.I1.i1.p1.13.m13.3.3.1.1.1.1.1" xref="S4.I11.i1.I1.i1.p1.13.m13.3.3.1.1.1.1.1.cmml">⁢</mo><mrow id="S4.I11.i1.I1.i1.p1.13.m13.3.3.1.1.1.1.3.2" xref="S4.I11.i1.I1.i1.p1.13.m13.3.3.1.1.1.1.cmml"><mo id="S4.I11.i1.I1.i1.p1.13.m13.3.3.1.1.1.1.3.2.1" stretchy="false" xref="S4.I11.i1.I1.i1.p1.13.m13.3.3.1.1.1.1.cmml">(</mo><mi id="S4.I11.i1.I1.i1.p1.13.m13.1.1" xref="S4.I11.i1.I1.i1.p1.13.m13.1.1.cmml">u</mi><mo id="S4.I11.i1.I1.i1.p1.13.m13.3.3.1.1.1.1.3.2.2" stretchy="false" xref="S4.I11.i1.I1.i1.p1.13.m13.3.3.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S4.I11.i1.I1.i1.p1.13.m13.4.4.2.2.2.4" xref="S4.I11.i1.I1.i1.p1.13.m13.4.4.2.2.3.cmml">,</mo><mrow id="S4.I11.i1.I1.i1.p1.13.m13.4.4.2.2.2.2" xref="S4.I11.i1.I1.i1.p1.13.m13.4.4.2.2.2.2.cmml"><mi id="S4.I11.i1.I1.i1.p1.13.m13.4.4.2.2.2.2.2" mathvariant="normal" xref="S4.I11.i1.I1.i1.p1.13.m13.4.4.2.2.2.2.2.cmml">ℓ</mi><mo id="S4.I11.i1.I1.i1.p1.13.m13.4.4.2.2.2.2.1" xref="S4.I11.i1.I1.i1.p1.13.m13.4.4.2.2.2.2.1.cmml">⁢</mo><mrow id="S4.I11.i1.I1.i1.p1.13.m13.4.4.2.2.2.2.3.2" xref="S4.I11.i1.I1.i1.p1.13.m13.4.4.2.2.2.2.cmml"><mo id="S4.I11.i1.I1.i1.p1.13.m13.4.4.2.2.2.2.3.2.1" stretchy="false" xref="S4.I11.i1.I1.i1.p1.13.m13.4.4.2.2.2.2.cmml">(</mo><mi id="S4.I11.i1.I1.i1.p1.13.m13.2.2" xref="S4.I11.i1.I1.i1.p1.13.m13.2.2.cmml">v</mi><mo id="S4.I11.i1.I1.i1.p1.13.m13.4.4.2.2.2.2.3.2.2" stretchy="false" xref="S4.I11.i1.I1.i1.p1.13.m13.4.4.2.2.2.2.cmml">)</mo></mrow></mrow><mo id="S4.I11.i1.I1.i1.p1.13.m13.4.4.2.2.2.5" stretchy="false" xref="S4.I11.i1.I1.i1.p1.13.m13.4.4.2.2.3.cmml">)</mo></mrow></mrow><mo id="S4.I11.i1.I1.i1.p1.13.m13.4.4.3" xref="S4.I11.i1.I1.i1.p1.13.m13.4.4.3.cmml">=</mo><mi id="S4.I11.i1.I1.i1.p1.13.m13.4.4.4" xref="S4.I11.i1.I1.i1.p1.13.m13.4.4.4.cmml">x</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.I11.i1.I1.i1.p1.13.m13.4b"><apply id="S4.I11.i1.I1.i1.p1.13.m13.4.4.cmml" xref="S4.I11.i1.I1.i1.p1.13.m13.4.4"><eq id="S4.I11.i1.I1.i1.p1.13.m13.4.4.3.cmml" xref="S4.I11.i1.I1.i1.p1.13.m13.4.4.3"></eq><apply id="S4.I11.i1.I1.i1.p1.13.m13.4.4.2.cmml" xref="S4.I11.i1.I1.i1.p1.13.m13.4.4.2"><times id="S4.I11.i1.I1.i1.p1.13.m13.4.4.2.3.cmml" xref="S4.I11.i1.I1.i1.p1.13.m13.4.4.2.3"></times><ci id="S4.I11.i1.I1.i1.p1.13.m13.4.4.2.4a.cmml" xref="S4.I11.i1.I1.i1.p1.13.m13.4.4.2.4"><mtext id="S4.I11.i1.I1.i1.p1.13.m13.4.4.2.4.cmml" xref="S4.I11.i1.I1.i1.p1.13.m13.4.4.2.4">LCA</mtext></ci><interval closure="open" id="S4.I11.i1.I1.i1.p1.13.m13.4.4.2.2.3.cmml" xref="S4.I11.i1.I1.i1.p1.13.m13.4.4.2.2.2"><apply id="S4.I11.i1.I1.i1.p1.13.m13.3.3.1.1.1.1.cmml" xref="S4.I11.i1.I1.i1.p1.13.m13.3.3.1.1.1.1"><times id="S4.I11.i1.I1.i1.p1.13.m13.3.3.1.1.1.1.1.cmml" xref="S4.I11.i1.I1.i1.p1.13.m13.3.3.1.1.1.1.1"></times><ci id="S4.I11.i1.I1.i1.p1.13.m13.3.3.1.1.1.1.2.cmml" xref="S4.I11.i1.I1.i1.p1.13.m13.3.3.1.1.1.1.2">ℓ</ci><ci id="S4.I11.i1.I1.i1.p1.13.m13.1.1.cmml" xref="S4.I11.i1.I1.i1.p1.13.m13.1.1">𝑢</ci></apply><apply id="S4.I11.i1.I1.i1.p1.13.m13.4.4.2.2.2.2.cmml" xref="S4.I11.i1.I1.i1.p1.13.m13.4.4.2.2.2.2"><times id="S4.I11.i1.I1.i1.p1.13.m13.4.4.2.2.2.2.1.cmml" xref="S4.I11.i1.I1.i1.p1.13.m13.4.4.2.2.2.2.1"></times><ci id="S4.I11.i1.I1.i1.p1.13.m13.4.4.2.2.2.2.2.cmml" xref="S4.I11.i1.I1.i1.p1.13.m13.4.4.2.2.2.2.2">ℓ</ci><ci id="S4.I11.i1.I1.i1.p1.13.m13.2.2.cmml" xref="S4.I11.i1.I1.i1.p1.13.m13.2.2">𝑣</ci></apply></interval></apply><ci id="S4.I11.i1.I1.i1.p1.13.m13.4.4.4.cmml" xref="S4.I11.i1.I1.i1.p1.13.m13.4.4.4">𝑥</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.I11.i1.I1.i1.p1.13.m13.4c">\text{LCA}(\ell(u),\ell(v))=x</annotation><annotation encoding="application/x-llamapun" id="S4.I11.i1.I1.i1.p1.13.m13.4d">LCA ( roman_ℓ ( italic_u ) , roman_ℓ ( italic_v ) ) = italic_x</annotation></semantics></math>. By construction, <math alttext="f(v^{\prime\prime})\preceq f(u)" class="ltx_Math" display="inline" id="S4.I11.i1.I1.i1.p1.14.m14.2"><semantics id="S4.I11.i1.I1.i1.p1.14.m14.2a"><mrow id="S4.I11.i1.I1.i1.p1.14.m14.2.2" xref="S4.I11.i1.I1.i1.p1.14.m14.2.2.cmml"><mrow id="S4.I11.i1.I1.i1.p1.14.m14.2.2.1" xref="S4.I11.i1.I1.i1.p1.14.m14.2.2.1.cmml"><mi id="S4.I11.i1.I1.i1.p1.14.m14.2.2.1.3" xref="S4.I11.i1.I1.i1.p1.14.m14.2.2.1.3.cmml">f</mi><mo id="S4.I11.i1.I1.i1.p1.14.m14.2.2.1.2" xref="S4.I11.i1.I1.i1.p1.14.m14.2.2.1.2.cmml">⁢</mo><mrow id="S4.I11.i1.I1.i1.p1.14.m14.2.2.1.1.1" xref="S4.I11.i1.I1.i1.p1.14.m14.2.2.1.1.1.1.cmml"><mo id="S4.I11.i1.I1.i1.p1.14.m14.2.2.1.1.1.2" stretchy="false" xref="S4.I11.i1.I1.i1.p1.14.m14.2.2.1.1.1.1.cmml">(</mo><msup id="S4.I11.i1.I1.i1.p1.14.m14.2.2.1.1.1.1" xref="S4.I11.i1.I1.i1.p1.14.m14.2.2.1.1.1.1.cmml"><mi id="S4.I11.i1.I1.i1.p1.14.m14.2.2.1.1.1.1.2" xref="S4.I11.i1.I1.i1.p1.14.m14.2.2.1.1.1.1.2.cmml">v</mi><mo id="S4.I11.i1.I1.i1.p1.14.m14.2.2.1.1.1.1.3" xref="S4.I11.i1.I1.i1.p1.14.m14.2.2.1.1.1.1.3.cmml">′′</mo></msup><mo id="S4.I11.i1.I1.i1.p1.14.m14.2.2.1.1.1.3" stretchy="false" xref="S4.I11.i1.I1.i1.p1.14.m14.2.2.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S4.I11.i1.I1.i1.p1.14.m14.2.2.2" xref="S4.I11.i1.I1.i1.p1.14.m14.2.2.2.cmml">⪯</mo><mrow id="S4.I11.i1.I1.i1.p1.14.m14.2.2.3" xref="S4.I11.i1.I1.i1.p1.14.m14.2.2.3.cmml"><mi id="S4.I11.i1.I1.i1.p1.14.m14.2.2.3.2" xref="S4.I11.i1.I1.i1.p1.14.m14.2.2.3.2.cmml">f</mi><mo id="S4.I11.i1.I1.i1.p1.14.m14.2.2.3.1" xref="S4.I11.i1.I1.i1.p1.14.m14.2.2.3.1.cmml">⁢</mo><mrow id="S4.I11.i1.I1.i1.p1.14.m14.2.2.3.3.2" xref="S4.I11.i1.I1.i1.p1.14.m14.2.2.3.cmml"><mo id="S4.I11.i1.I1.i1.p1.14.m14.2.2.3.3.2.1" stretchy="false" xref="S4.I11.i1.I1.i1.p1.14.m14.2.2.3.cmml">(</mo><mi id="S4.I11.i1.I1.i1.p1.14.m14.1.1" xref="S4.I11.i1.I1.i1.p1.14.m14.1.1.cmml">u</mi><mo id="S4.I11.i1.I1.i1.p1.14.m14.2.2.3.3.2.2" stretchy="false" xref="S4.I11.i1.I1.i1.p1.14.m14.2.2.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.I11.i1.I1.i1.p1.14.m14.2b"><apply id="S4.I11.i1.I1.i1.p1.14.m14.2.2.cmml" xref="S4.I11.i1.I1.i1.p1.14.m14.2.2"><csymbol cd="latexml" id="S4.I11.i1.I1.i1.p1.14.m14.2.2.2.cmml" xref="S4.I11.i1.I1.i1.p1.14.m14.2.2.2">precedes-or-equals</csymbol><apply id="S4.I11.i1.I1.i1.p1.14.m14.2.2.1.cmml" xref="S4.I11.i1.I1.i1.p1.14.m14.2.2.1"><times id="S4.I11.i1.I1.i1.p1.14.m14.2.2.1.2.cmml" xref="S4.I11.i1.I1.i1.p1.14.m14.2.2.1.2"></times><ci id="S4.I11.i1.I1.i1.p1.14.m14.2.2.1.3.cmml" xref="S4.I11.i1.I1.i1.p1.14.m14.2.2.1.3">𝑓</ci><apply id="S4.I11.i1.I1.i1.p1.14.m14.2.2.1.1.1.1.cmml" xref="S4.I11.i1.I1.i1.p1.14.m14.2.2.1.1.1"><csymbol cd="ambiguous" id="S4.I11.i1.I1.i1.p1.14.m14.2.2.1.1.1.1.1.cmml" xref="S4.I11.i1.I1.i1.p1.14.m14.2.2.1.1.1">superscript</csymbol><ci id="S4.I11.i1.I1.i1.p1.14.m14.2.2.1.1.1.1.2.cmml" xref="S4.I11.i1.I1.i1.p1.14.m14.2.2.1.1.1.1.2">𝑣</ci><ci id="S4.I11.i1.I1.i1.p1.14.m14.2.2.1.1.1.1.3.cmml" xref="S4.I11.i1.I1.i1.p1.14.m14.2.2.1.1.1.1.3">′′</ci></apply></apply><apply id="S4.I11.i1.I1.i1.p1.14.m14.2.2.3.cmml" xref="S4.I11.i1.I1.i1.p1.14.m14.2.2.3"><times id="S4.I11.i1.I1.i1.p1.14.m14.2.2.3.1.cmml" xref="S4.I11.i1.I1.i1.p1.14.m14.2.2.3.1"></times><ci id="S4.I11.i1.I1.i1.p1.14.m14.2.2.3.2.cmml" xref="S4.I11.i1.I1.i1.p1.14.m14.2.2.3.2">𝑓</ci><ci id="S4.I11.i1.I1.i1.p1.14.m14.1.1.cmml" xref="S4.I11.i1.I1.i1.p1.14.m14.1.1">𝑢</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.I11.i1.I1.i1.p1.14.m14.2c">f(v^{\prime\prime})\preceq f(u)</annotation><annotation encoding="application/x-llamapun" id="S4.I11.i1.I1.i1.p1.14.m14.2d">italic_f ( italic_v start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT ) ⪯ italic_f ( italic_u )</annotation></semantics></math> and is therefore also in <math alttext="C_{1}" class="ltx_Math" display="inline" id="S4.I11.i1.I1.i1.p1.15.m15.1"><semantics id="S4.I11.i1.I1.i1.p1.15.m15.1a"><msub id="S4.I11.i1.I1.i1.p1.15.m15.1.1" xref="S4.I11.i1.I1.i1.p1.15.m15.1.1.cmml"><mi id="S4.I11.i1.I1.i1.p1.15.m15.1.1.2" xref="S4.I11.i1.I1.i1.p1.15.m15.1.1.2.cmml">C</mi><mn id="S4.I11.i1.I1.i1.p1.15.m15.1.1.3" xref="S4.I11.i1.I1.i1.p1.15.m15.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="S4.I11.i1.I1.i1.p1.15.m15.1b"><apply id="S4.I11.i1.I1.i1.p1.15.m15.1.1.cmml" xref="S4.I11.i1.I1.i1.p1.15.m15.1.1"><csymbol cd="ambiguous" id="S4.I11.i1.I1.i1.p1.15.m15.1.1.1.cmml" xref="S4.I11.i1.I1.i1.p1.15.m15.1.1">subscript</csymbol><ci id="S4.I11.i1.I1.i1.p1.15.m15.1.1.2.cmml" xref="S4.I11.i1.I1.i1.p1.15.m15.1.1.2">𝐶</ci><cn id="S4.I11.i1.I1.i1.p1.15.m15.1.1.3.cmml" type="integer" xref="S4.I11.i1.I1.i1.p1.15.m15.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.I11.i1.I1.i1.p1.15.m15.1c">C_{1}</annotation><annotation encoding="application/x-llamapun" id="S4.I11.i1.I1.i1.p1.15.m15.1d">italic_C start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math>. Since <math alttext="f(u)\in C_{1}" class="ltx_Math" display="inline" id="S4.I11.i1.I1.i1.p1.16.m16.1"><semantics id="S4.I11.i1.I1.i1.p1.16.m16.1a"><mrow id="S4.I11.i1.I1.i1.p1.16.m16.1.2" xref="S4.I11.i1.I1.i1.p1.16.m16.1.2.cmml"><mrow id="S4.I11.i1.I1.i1.p1.16.m16.1.2.2" xref="S4.I11.i1.I1.i1.p1.16.m16.1.2.2.cmml"><mi id="S4.I11.i1.I1.i1.p1.16.m16.1.2.2.2" xref="S4.I11.i1.I1.i1.p1.16.m16.1.2.2.2.cmml">f</mi><mo id="S4.I11.i1.I1.i1.p1.16.m16.1.2.2.1" xref="S4.I11.i1.I1.i1.p1.16.m16.1.2.2.1.cmml">⁢</mo><mrow id="S4.I11.i1.I1.i1.p1.16.m16.1.2.2.3.2" xref="S4.I11.i1.I1.i1.p1.16.m16.1.2.2.cmml"><mo id="S4.I11.i1.I1.i1.p1.16.m16.1.2.2.3.2.1" stretchy="false" xref="S4.I11.i1.I1.i1.p1.16.m16.1.2.2.cmml">(</mo><mi id="S4.I11.i1.I1.i1.p1.16.m16.1.1" xref="S4.I11.i1.I1.i1.p1.16.m16.1.1.cmml">u</mi><mo id="S4.I11.i1.I1.i1.p1.16.m16.1.2.2.3.2.2" stretchy="false" xref="S4.I11.i1.I1.i1.p1.16.m16.1.2.2.cmml">)</mo></mrow></mrow><mo id="S4.I11.i1.I1.i1.p1.16.m16.1.2.1" xref="S4.I11.i1.I1.i1.p1.16.m16.1.2.1.cmml">∈</mo><msub id="S4.I11.i1.I1.i1.p1.16.m16.1.2.3" xref="S4.I11.i1.I1.i1.p1.16.m16.1.2.3.cmml"><mi id="S4.I11.i1.I1.i1.p1.16.m16.1.2.3.2" xref="S4.I11.i1.I1.i1.p1.16.m16.1.2.3.2.cmml">C</mi><mn id="S4.I11.i1.I1.i1.p1.16.m16.1.2.3.3" xref="S4.I11.i1.I1.i1.p1.16.m16.1.2.3.3.cmml">1</mn></msub></mrow><annotation-xml encoding="MathML-Content" id="S4.I11.i1.I1.i1.p1.16.m16.1b"><apply id="S4.I11.i1.I1.i1.p1.16.m16.1.2.cmml" xref="S4.I11.i1.I1.i1.p1.16.m16.1.2"><in id="S4.I11.i1.I1.i1.p1.16.m16.1.2.1.cmml" xref="S4.I11.i1.I1.i1.p1.16.m16.1.2.1"></in><apply id="S4.I11.i1.I1.i1.p1.16.m16.1.2.2.cmml" xref="S4.I11.i1.I1.i1.p1.16.m16.1.2.2"><times id="S4.I11.i1.I1.i1.p1.16.m16.1.2.2.1.cmml" xref="S4.I11.i1.I1.i1.p1.16.m16.1.2.2.1"></times><ci id="S4.I11.i1.I1.i1.p1.16.m16.1.2.2.2.cmml" xref="S4.I11.i1.I1.i1.p1.16.m16.1.2.2.2">𝑓</ci><ci id="S4.I11.i1.I1.i1.p1.16.m16.1.1.cmml" xref="S4.I11.i1.I1.i1.p1.16.m16.1.1">𝑢</ci></apply><apply id="S4.I11.i1.I1.i1.p1.16.m16.1.2.3.cmml" xref="S4.I11.i1.I1.i1.p1.16.m16.1.2.3"><csymbol cd="ambiguous" id="S4.I11.i1.I1.i1.p1.16.m16.1.2.3.1.cmml" xref="S4.I11.i1.I1.i1.p1.16.m16.1.2.3">subscript</csymbol><ci id="S4.I11.i1.I1.i1.p1.16.m16.1.2.3.2.cmml" xref="S4.I11.i1.I1.i1.p1.16.m16.1.2.3.2">𝐶</ci><cn id="S4.I11.i1.I1.i1.p1.16.m16.1.2.3.3.cmml" type="integer" xref="S4.I11.i1.I1.i1.p1.16.m16.1.2.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.I11.i1.I1.i1.p1.16.m16.1c">f(u)\in C_{1}</annotation><annotation encoding="application/x-llamapun" id="S4.I11.i1.I1.i1.p1.16.m16.1d">italic_f ( italic_u ) ∈ italic_C start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math>, <math alttext="f(u)\preceq\min(a,b)" class="ltx_Math" display="inline" id="S4.I11.i1.I1.i1.p1.17.m17.4"><semantics id="S4.I11.i1.I1.i1.p1.17.m17.4a"><mrow id="S4.I11.i1.I1.i1.p1.17.m17.4.5" xref="S4.I11.i1.I1.i1.p1.17.m17.4.5.cmml"><mrow id="S4.I11.i1.I1.i1.p1.17.m17.4.5.2" xref="S4.I11.i1.I1.i1.p1.17.m17.4.5.2.cmml"><mi id="S4.I11.i1.I1.i1.p1.17.m17.4.5.2.2" xref="S4.I11.i1.I1.i1.p1.17.m17.4.5.2.2.cmml">f</mi><mo id="S4.I11.i1.I1.i1.p1.17.m17.4.5.2.1" xref="S4.I11.i1.I1.i1.p1.17.m17.4.5.2.1.cmml">⁢</mo><mrow id="S4.I11.i1.I1.i1.p1.17.m17.4.5.2.3.2" xref="S4.I11.i1.I1.i1.p1.17.m17.4.5.2.cmml"><mo id="S4.I11.i1.I1.i1.p1.17.m17.4.5.2.3.2.1" stretchy="false" xref="S4.I11.i1.I1.i1.p1.17.m17.4.5.2.cmml">(</mo><mi id="S4.I11.i1.I1.i1.p1.17.m17.1.1" xref="S4.I11.i1.I1.i1.p1.17.m17.1.1.cmml">u</mi><mo id="S4.I11.i1.I1.i1.p1.17.m17.4.5.2.3.2.2" stretchy="false" xref="S4.I11.i1.I1.i1.p1.17.m17.4.5.2.cmml">)</mo></mrow></mrow><mo id="S4.I11.i1.I1.i1.p1.17.m17.4.5.1" xref="S4.I11.i1.I1.i1.p1.17.m17.4.5.1.cmml">⪯</mo><mrow id="S4.I11.i1.I1.i1.p1.17.m17.4.5.3.2" xref="S4.I11.i1.I1.i1.p1.17.m17.4.5.3.1.cmml"><mi id="S4.I11.i1.I1.i1.p1.17.m17.2.2" xref="S4.I11.i1.I1.i1.p1.17.m17.2.2.cmml">min</mi><mo id="S4.I11.i1.I1.i1.p1.17.m17.4.5.3.2a" xref="S4.I11.i1.I1.i1.p1.17.m17.4.5.3.1.cmml">⁡</mo><mrow id="S4.I11.i1.I1.i1.p1.17.m17.4.5.3.2.1" xref="S4.I11.i1.I1.i1.p1.17.m17.4.5.3.1.cmml"><mo id="S4.I11.i1.I1.i1.p1.17.m17.4.5.3.2.1.1" stretchy="false" xref="S4.I11.i1.I1.i1.p1.17.m17.4.5.3.1.cmml">(</mo><mi id="S4.I11.i1.I1.i1.p1.17.m17.3.3" xref="S4.I11.i1.I1.i1.p1.17.m17.3.3.cmml">a</mi><mo id="S4.I11.i1.I1.i1.p1.17.m17.4.5.3.2.1.2" xref="S4.I11.i1.I1.i1.p1.17.m17.4.5.3.1.cmml">,</mo><mi id="S4.I11.i1.I1.i1.p1.17.m17.4.4" xref="S4.I11.i1.I1.i1.p1.17.m17.4.4.cmml">b</mi><mo id="S4.I11.i1.I1.i1.p1.17.m17.4.5.3.2.1.3" stretchy="false" xref="S4.I11.i1.I1.i1.p1.17.m17.4.5.3.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.I11.i1.I1.i1.p1.17.m17.4b"><apply id="S4.I11.i1.I1.i1.p1.17.m17.4.5.cmml" xref="S4.I11.i1.I1.i1.p1.17.m17.4.5"><csymbol cd="latexml" id="S4.I11.i1.I1.i1.p1.17.m17.4.5.1.cmml" xref="S4.I11.i1.I1.i1.p1.17.m17.4.5.1">precedes-or-equals</csymbol><apply id="S4.I11.i1.I1.i1.p1.17.m17.4.5.2.cmml" xref="S4.I11.i1.I1.i1.p1.17.m17.4.5.2"><times id="S4.I11.i1.I1.i1.p1.17.m17.4.5.2.1.cmml" xref="S4.I11.i1.I1.i1.p1.17.m17.4.5.2.1"></times><ci id="S4.I11.i1.I1.i1.p1.17.m17.4.5.2.2.cmml" xref="S4.I11.i1.I1.i1.p1.17.m17.4.5.2.2">𝑓</ci><ci id="S4.I11.i1.I1.i1.p1.17.m17.1.1.cmml" xref="S4.I11.i1.I1.i1.p1.17.m17.1.1">𝑢</ci></apply><apply id="S4.I11.i1.I1.i1.p1.17.m17.4.5.3.1.cmml" xref="S4.I11.i1.I1.i1.p1.17.m17.4.5.3.2"><min id="S4.I11.i1.I1.i1.p1.17.m17.2.2.cmml" xref="S4.I11.i1.I1.i1.p1.17.m17.2.2"></min><ci id="S4.I11.i1.I1.i1.p1.17.m17.3.3.cmml" xref="S4.I11.i1.I1.i1.p1.17.m17.3.3">𝑎</ci><ci id="S4.I11.i1.I1.i1.p1.17.m17.4.4.cmml" xref="S4.I11.i1.I1.i1.p1.17.m17.4.4">𝑏</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.I11.i1.I1.i1.p1.17.m17.4c">f(u)\preceq\min(a,b)</annotation><annotation encoding="application/x-llamapun" id="S4.I11.i1.I1.i1.p1.17.m17.4d">italic_f ( italic_u ) ⪯ roman_min ( italic_a , italic_b )</annotation></semantics></math> (where <math alttext="\min" class="ltx_Math" display="inline" id="S4.I11.i1.I1.i1.p1.18.m18.1"><semantics id="S4.I11.i1.I1.i1.p1.18.m18.1a"><mi id="S4.I11.i1.I1.i1.p1.18.m18.1.1" xref="S4.I11.i1.I1.i1.p1.18.m18.1.1.cmml">min</mi><annotation-xml encoding="MathML-Content" id="S4.I11.i1.I1.i1.p1.18.m18.1b"><min id="S4.I11.i1.I1.i1.p1.18.m18.1.1.cmml" xref="S4.I11.i1.I1.i1.p1.18.m18.1.1"></min></annotation-xml><annotation encoding="application/x-tex" id="S4.I11.i1.I1.i1.p1.18.m18.1c">\min</annotation><annotation encoding="application/x-llamapun" id="S4.I11.i1.I1.i1.p1.18.m18.1d">roman_min</annotation></semantics></math> is with respect to <math alttext="\prec" class="ltx_Math" display="inline" id="S4.I11.i1.I1.i1.p1.19.m19.1"><semantics id="S4.I11.i1.I1.i1.p1.19.m19.1a"><mo id="S4.I11.i1.I1.i1.p1.19.m19.1.1" xref="S4.I11.i1.I1.i1.p1.19.m19.1.1.cmml">≺</mo><annotation-xml encoding="MathML-Content" id="S4.I11.i1.I1.i1.p1.19.m19.1b"><csymbol cd="latexml" id="S4.I11.i1.I1.i1.p1.19.m19.1.1.cmml" xref="S4.I11.i1.I1.i1.p1.19.m19.1.1">precedes</csymbol></annotation-xml><annotation encoding="application/x-tex" id="S4.I11.i1.I1.i1.p1.19.m19.1c">\prec</annotation><annotation encoding="application/x-llamapun" id="S4.I11.i1.I1.i1.p1.19.m19.1d">≺</annotation></semantics></math>), thus <math alttext="f(v^{\prime\prime})\notin\{a,b\}" class="ltx_Math" display="inline" id="S4.I11.i1.I1.i1.p1.20.m20.3"><semantics id="S4.I11.i1.I1.i1.p1.20.m20.3a"><mrow id="S4.I11.i1.I1.i1.p1.20.m20.3.3" xref="S4.I11.i1.I1.i1.p1.20.m20.3.3.cmml"><mrow id="S4.I11.i1.I1.i1.p1.20.m20.3.3.1" xref="S4.I11.i1.I1.i1.p1.20.m20.3.3.1.cmml"><mi id="S4.I11.i1.I1.i1.p1.20.m20.3.3.1.3" xref="S4.I11.i1.I1.i1.p1.20.m20.3.3.1.3.cmml">f</mi><mo id="S4.I11.i1.I1.i1.p1.20.m20.3.3.1.2" xref="S4.I11.i1.I1.i1.p1.20.m20.3.3.1.2.cmml">⁢</mo><mrow id="S4.I11.i1.I1.i1.p1.20.m20.3.3.1.1.1" xref="S4.I11.i1.I1.i1.p1.20.m20.3.3.1.1.1.1.cmml"><mo id="S4.I11.i1.I1.i1.p1.20.m20.3.3.1.1.1.2" stretchy="false" xref="S4.I11.i1.I1.i1.p1.20.m20.3.3.1.1.1.1.cmml">(</mo><msup id="S4.I11.i1.I1.i1.p1.20.m20.3.3.1.1.1.1" xref="S4.I11.i1.I1.i1.p1.20.m20.3.3.1.1.1.1.cmml"><mi id="S4.I11.i1.I1.i1.p1.20.m20.3.3.1.1.1.1.2" xref="S4.I11.i1.I1.i1.p1.20.m20.3.3.1.1.1.1.2.cmml">v</mi><mo id="S4.I11.i1.I1.i1.p1.20.m20.3.3.1.1.1.1.3" xref="S4.I11.i1.I1.i1.p1.20.m20.3.3.1.1.1.1.3.cmml">′′</mo></msup><mo id="S4.I11.i1.I1.i1.p1.20.m20.3.3.1.1.1.3" stretchy="false" xref="S4.I11.i1.I1.i1.p1.20.m20.3.3.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S4.I11.i1.I1.i1.p1.20.m20.3.3.2" xref="S4.I11.i1.I1.i1.p1.20.m20.3.3.2.cmml">∉</mo><mrow id="S4.I11.i1.I1.i1.p1.20.m20.3.3.3.2" xref="S4.I11.i1.I1.i1.p1.20.m20.3.3.3.1.cmml"><mo id="S4.I11.i1.I1.i1.p1.20.m20.3.3.3.2.1" stretchy="false" xref="S4.I11.i1.I1.i1.p1.20.m20.3.3.3.1.cmml">{</mo><mi id="S4.I11.i1.I1.i1.p1.20.m20.1.1" xref="S4.I11.i1.I1.i1.p1.20.m20.1.1.cmml">a</mi><mo id="S4.I11.i1.I1.i1.p1.20.m20.3.3.3.2.2" xref="S4.I11.i1.I1.i1.p1.20.m20.3.3.3.1.cmml">,</mo><mi id="S4.I11.i1.I1.i1.p1.20.m20.2.2" xref="S4.I11.i1.I1.i1.p1.20.m20.2.2.cmml">b</mi><mo id="S4.I11.i1.I1.i1.p1.20.m20.3.3.3.2.3" stretchy="false" xref="S4.I11.i1.I1.i1.p1.20.m20.3.3.3.1.cmml">}</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.I11.i1.I1.i1.p1.20.m20.3b"><apply id="S4.I11.i1.I1.i1.p1.20.m20.3.3.cmml" xref="S4.I11.i1.I1.i1.p1.20.m20.3.3"><notin id="S4.I11.i1.I1.i1.p1.20.m20.3.3.2.cmml" xref="S4.I11.i1.I1.i1.p1.20.m20.3.3.2"></notin><apply id="S4.I11.i1.I1.i1.p1.20.m20.3.3.1.cmml" xref="S4.I11.i1.I1.i1.p1.20.m20.3.3.1"><times id="S4.I11.i1.I1.i1.p1.20.m20.3.3.1.2.cmml" xref="S4.I11.i1.I1.i1.p1.20.m20.3.3.1.2"></times><ci id="S4.I11.i1.I1.i1.p1.20.m20.3.3.1.3.cmml" xref="S4.I11.i1.I1.i1.p1.20.m20.3.3.1.3">𝑓</ci><apply id="S4.I11.i1.I1.i1.p1.20.m20.3.3.1.1.1.1.cmml" xref="S4.I11.i1.I1.i1.p1.20.m20.3.3.1.1.1"><csymbol cd="ambiguous" id="S4.I11.i1.I1.i1.p1.20.m20.3.3.1.1.1.1.1.cmml" xref="S4.I11.i1.I1.i1.p1.20.m20.3.3.1.1.1">superscript</csymbol><ci id="S4.I11.i1.I1.i1.p1.20.m20.3.3.1.1.1.1.2.cmml" xref="S4.I11.i1.I1.i1.p1.20.m20.3.3.1.1.1.1.2">𝑣</ci><ci id="S4.I11.i1.I1.i1.p1.20.m20.3.3.1.1.1.1.3.cmml" xref="S4.I11.i1.I1.i1.p1.20.m20.3.3.1.1.1.1.3">′′</ci></apply></apply><set id="S4.I11.i1.I1.i1.p1.20.m20.3.3.3.1.cmml" xref="S4.I11.i1.I1.i1.p1.20.m20.3.3.3.2"><ci id="S4.I11.i1.I1.i1.p1.20.m20.1.1.cmml" xref="S4.I11.i1.I1.i1.p1.20.m20.1.1">𝑎</ci><ci id="S4.I11.i1.I1.i1.p1.20.m20.2.2.cmml" xref="S4.I11.i1.I1.i1.p1.20.m20.2.2">𝑏</ci></set></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.I11.i1.I1.i1.p1.20.m20.3c">f(v^{\prime\prime})\notin\{a,b\}</annotation><annotation encoding="application/x-llamapun" id="S4.I11.i1.I1.i1.p1.20.m20.3d">italic_f ( italic_v start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT ) ∉ { italic_a , italic_b }</annotation></semantics></math> and by extension <math alttext="v^{\prime\prime}\notin\{a,b\}" class="ltx_Math" display="inline" id="S4.I11.i1.I1.i1.p1.21.m21.2"><semantics id="S4.I11.i1.I1.i1.p1.21.m21.2a"><mrow id="S4.I11.i1.I1.i1.p1.21.m21.2.3" xref="S4.I11.i1.I1.i1.p1.21.m21.2.3.cmml"><msup id="S4.I11.i1.I1.i1.p1.21.m21.2.3.2" xref="S4.I11.i1.I1.i1.p1.21.m21.2.3.2.cmml"><mi id="S4.I11.i1.I1.i1.p1.21.m21.2.3.2.2" xref="S4.I11.i1.I1.i1.p1.21.m21.2.3.2.2.cmml">v</mi><mo id="S4.I11.i1.I1.i1.p1.21.m21.2.3.2.3" xref="S4.I11.i1.I1.i1.p1.21.m21.2.3.2.3.cmml">′′</mo></msup><mo id="S4.I11.i1.I1.i1.p1.21.m21.2.3.1" xref="S4.I11.i1.I1.i1.p1.21.m21.2.3.1.cmml">∉</mo><mrow id="S4.I11.i1.I1.i1.p1.21.m21.2.3.3.2" xref="S4.I11.i1.I1.i1.p1.21.m21.2.3.3.1.cmml"><mo id="S4.I11.i1.I1.i1.p1.21.m21.2.3.3.2.1" stretchy="false" xref="S4.I11.i1.I1.i1.p1.21.m21.2.3.3.1.cmml">{</mo><mi id="S4.I11.i1.I1.i1.p1.21.m21.1.1" xref="S4.I11.i1.I1.i1.p1.21.m21.1.1.cmml">a</mi><mo id="S4.I11.i1.I1.i1.p1.21.m21.2.3.3.2.2" xref="S4.I11.i1.I1.i1.p1.21.m21.2.3.3.1.cmml">,</mo><mi id="S4.I11.i1.I1.i1.p1.21.m21.2.2" xref="S4.I11.i1.I1.i1.p1.21.m21.2.2.cmml">b</mi><mo id="S4.I11.i1.I1.i1.p1.21.m21.2.3.3.2.3" stretchy="false" xref="S4.I11.i1.I1.i1.p1.21.m21.2.3.3.1.cmml">}</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.I11.i1.I1.i1.p1.21.m21.2b"><apply id="S4.I11.i1.I1.i1.p1.21.m21.2.3.cmml" xref="S4.I11.i1.I1.i1.p1.21.m21.2.3"><notin id="S4.I11.i1.I1.i1.p1.21.m21.2.3.1.cmml" xref="S4.I11.i1.I1.i1.p1.21.m21.2.3.1"></notin><apply id="S4.I11.i1.I1.i1.p1.21.m21.2.3.2.cmml" xref="S4.I11.i1.I1.i1.p1.21.m21.2.3.2"><csymbol cd="ambiguous" id="S4.I11.i1.I1.i1.p1.21.m21.2.3.2.1.cmml" xref="S4.I11.i1.I1.i1.p1.21.m21.2.3.2">superscript</csymbol><ci id="S4.I11.i1.I1.i1.p1.21.m21.2.3.2.2.cmml" xref="S4.I11.i1.I1.i1.p1.21.m21.2.3.2.2">𝑣</ci><ci id="S4.I11.i1.I1.i1.p1.21.m21.2.3.2.3.cmml" xref="S4.I11.i1.I1.i1.p1.21.m21.2.3.2.3">′′</ci></apply><set id="S4.I11.i1.I1.i1.p1.21.m21.2.3.3.1.cmml" xref="S4.I11.i1.I1.i1.p1.21.m21.2.3.3.2"><ci id="S4.I11.i1.I1.i1.p1.21.m21.1.1.cmml" xref="S4.I11.i1.I1.i1.p1.21.m21.1.1">𝑎</ci><ci id="S4.I11.i1.I1.i1.p1.21.m21.2.2.cmml" xref="S4.I11.i1.I1.i1.p1.21.m21.2.2">𝑏</ci></set></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.I11.i1.I1.i1.p1.21.m21.2c">v^{\prime\prime}\notin\{a,b\}</annotation><annotation encoding="application/x-llamapun" id="S4.I11.i1.I1.i1.p1.21.m21.2d">italic_v start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT ∉ { italic_a , italic_b }</annotation></semantics></math>. Thus there exists a <math alttext="u" class="ltx_Math" display="inline" id="S4.I11.i1.I1.i1.p1.22.m22.1"><semantics id="S4.I11.i1.I1.i1.p1.22.m22.1a"><mi id="S4.I11.i1.I1.i1.p1.22.m22.1.1" xref="S4.I11.i1.I1.i1.p1.22.m22.1.1.cmml">u</mi><annotation-xml encoding="MathML-Content" id="S4.I11.i1.I1.i1.p1.22.m22.1b"><ci id="S4.I11.i1.I1.i1.p1.22.m22.1.1.cmml" xref="S4.I11.i1.I1.i1.p1.22.m22.1.1">𝑢</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.I11.i1.I1.i1.p1.22.m22.1c">u</annotation><annotation encoding="application/x-llamapun" id="S4.I11.i1.I1.i1.p1.22.m22.1d">italic_u</annotation></semantics></math>-<math alttext="v" class="ltx_Math" display="inline" id="S4.I11.i1.I1.i1.p1.23.m23.1"><semantics id="S4.I11.i1.I1.i1.p1.23.m23.1a"><mi id="S4.I11.i1.I1.i1.p1.23.m23.1.1" xref="S4.I11.i1.I1.i1.p1.23.m23.1.1.cmml">v</mi><annotation-xml encoding="MathML-Content" id="S4.I11.i1.I1.i1.p1.23.m23.1b"><ci id="S4.I11.i1.I1.i1.p1.23.m23.1.1.cmml" xref="S4.I11.i1.I1.i1.p1.23.m23.1.1">𝑣</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.I11.i1.I1.i1.p1.23.m23.1c">v</annotation><annotation encoding="application/x-llamapun" id="S4.I11.i1.I1.i1.p1.23.m23.1d">italic_v</annotation></semantics></math> path: <math alttext="u\to f(u)\to f(v^{\prime\prime})" class="ltx_Math" display="inline" id="S4.I11.i1.I1.i1.p1.24.m24.2"><semantics id="S4.I11.i1.I1.i1.p1.24.m24.2a"><mrow id="S4.I11.i1.I1.i1.p1.24.m24.2.2" xref="S4.I11.i1.I1.i1.p1.24.m24.2.2.cmml"><mi id="S4.I11.i1.I1.i1.p1.24.m24.2.2.3" xref="S4.I11.i1.I1.i1.p1.24.m24.2.2.3.cmml">u</mi><mo id="S4.I11.i1.I1.i1.p1.24.m24.2.2.4" stretchy="false" xref="S4.I11.i1.I1.i1.p1.24.m24.2.2.4.cmml">→</mo><mrow id="S4.I11.i1.I1.i1.p1.24.m24.2.2.5" xref="S4.I11.i1.I1.i1.p1.24.m24.2.2.5.cmml"><mi id="S4.I11.i1.I1.i1.p1.24.m24.2.2.5.2" xref="S4.I11.i1.I1.i1.p1.24.m24.2.2.5.2.cmml">f</mi><mo id="S4.I11.i1.I1.i1.p1.24.m24.2.2.5.1" xref="S4.I11.i1.I1.i1.p1.24.m24.2.2.5.1.cmml">⁢</mo><mrow id="S4.I11.i1.I1.i1.p1.24.m24.2.2.5.3.2" xref="S4.I11.i1.I1.i1.p1.24.m24.2.2.5.cmml"><mo id="S4.I11.i1.I1.i1.p1.24.m24.2.2.5.3.2.1" stretchy="false" xref="S4.I11.i1.I1.i1.p1.24.m24.2.2.5.cmml">(</mo><mi id="S4.I11.i1.I1.i1.p1.24.m24.1.1" xref="S4.I11.i1.I1.i1.p1.24.m24.1.1.cmml">u</mi><mo id="S4.I11.i1.I1.i1.p1.24.m24.2.2.5.3.2.2" stretchy="false" xref="S4.I11.i1.I1.i1.p1.24.m24.2.2.5.cmml">)</mo></mrow></mrow><mo id="S4.I11.i1.I1.i1.p1.24.m24.2.2.6" stretchy="false" xref="S4.I11.i1.I1.i1.p1.24.m24.2.2.6.cmml">→</mo><mrow id="S4.I11.i1.I1.i1.p1.24.m24.2.2.1" xref="S4.I11.i1.I1.i1.p1.24.m24.2.2.1.cmml"><mi id="S4.I11.i1.I1.i1.p1.24.m24.2.2.1.3" xref="S4.I11.i1.I1.i1.p1.24.m24.2.2.1.3.cmml">f</mi><mo id="S4.I11.i1.I1.i1.p1.24.m24.2.2.1.2" xref="S4.I11.i1.I1.i1.p1.24.m24.2.2.1.2.cmml">⁢</mo><mrow id="S4.I11.i1.I1.i1.p1.24.m24.2.2.1.1.1" xref="S4.I11.i1.I1.i1.p1.24.m24.2.2.1.1.1.1.cmml"><mo id="S4.I11.i1.I1.i1.p1.24.m24.2.2.1.1.1.2" stretchy="false" xref="S4.I11.i1.I1.i1.p1.24.m24.2.2.1.1.1.1.cmml">(</mo><msup id="S4.I11.i1.I1.i1.p1.24.m24.2.2.1.1.1.1" xref="S4.I11.i1.I1.i1.p1.24.m24.2.2.1.1.1.1.cmml"><mi id="S4.I11.i1.I1.i1.p1.24.m24.2.2.1.1.1.1.2" xref="S4.I11.i1.I1.i1.p1.24.m24.2.2.1.1.1.1.2.cmml">v</mi><mo id="S4.I11.i1.I1.i1.p1.24.m24.2.2.1.1.1.1.3" xref="S4.I11.i1.I1.i1.p1.24.m24.2.2.1.1.1.1.3.cmml">′′</mo></msup><mo id="S4.I11.i1.I1.i1.p1.24.m24.2.2.1.1.1.3" stretchy="false" xref="S4.I11.i1.I1.i1.p1.24.m24.2.2.1.1.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.I11.i1.I1.i1.p1.24.m24.2b"><apply id="S4.I11.i1.I1.i1.p1.24.m24.2.2.cmml" xref="S4.I11.i1.I1.i1.p1.24.m24.2.2"><and id="S4.I11.i1.I1.i1.p1.24.m24.2.2a.cmml" xref="S4.I11.i1.I1.i1.p1.24.m24.2.2"></and><apply id="S4.I11.i1.I1.i1.p1.24.m24.2.2b.cmml" xref="S4.I11.i1.I1.i1.p1.24.m24.2.2"><ci id="S4.I11.i1.I1.i1.p1.24.m24.2.2.4.cmml" xref="S4.I11.i1.I1.i1.p1.24.m24.2.2.4">→</ci><ci id="S4.I11.i1.I1.i1.p1.24.m24.2.2.3.cmml" xref="S4.I11.i1.I1.i1.p1.24.m24.2.2.3">𝑢</ci><apply id="S4.I11.i1.I1.i1.p1.24.m24.2.2.5.cmml" xref="S4.I11.i1.I1.i1.p1.24.m24.2.2.5"><times id="S4.I11.i1.I1.i1.p1.24.m24.2.2.5.1.cmml" xref="S4.I11.i1.I1.i1.p1.24.m24.2.2.5.1"></times><ci id="S4.I11.i1.I1.i1.p1.24.m24.2.2.5.2.cmml" xref="S4.I11.i1.I1.i1.p1.24.m24.2.2.5.2">𝑓</ci><ci id="S4.I11.i1.I1.i1.p1.24.m24.1.1.cmml" xref="S4.I11.i1.I1.i1.p1.24.m24.1.1">𝑢</ci></apply></apply><apply id="S4.I11.i1.I1.i1.p1.24.m24.2.2c.cmml" xref="S4.I11.i1.I1.i1.p1.24.m24.2.2"><ci id="S4.I11.i1.I1.i1.p1.24.m24.2.2.6.cmml" xref="S4.I11.i1.I1.i1.p1.24.m24.2.2.6">→</ci><share href="https://arxiv.org/html/2503.00712v1#S4.I11.i1.I1.i1.p1.24.m24.2.2.5.cmml" id="S4.I11.i1.I1.i1.p1.24.m24.2.2d.cmml" xref="S4.I11.i1.I1.i1.p1.24.m24.2.2"></share><apply id="S4.I11.i1.I1.i1.p1.24.m24.2.2.1.cmml" xref="S4.I11.i1.I1.i1.p1.24.m24.2.2.1"><times id="S4.I11.i1.I1.i1.p1.24.m24.2.2.1.2.cmml" xref="S4.I11.i1.I1.i1.p1.24.m24.2.2.1.2"></times><ci id="S4.I11.i1.I1.i1.p1.24.m24.2.2.1.3.cmml" xref="S4.I11.i1.I1.i1.p1.24.m24.2.2.1.3">𝑓</ci><apply id="S4.I11.i1.I1.i1.p1.24.m24.2.2.1.1.1.1.cmml" xref="S4.I11.i1.I1.i1.p1.24.m24.2.2.1.1.1"><csymbol cd="ambiguous" id="S4.I11.i1.I1.i1.p1.24.m24.2.2.1.1.1.1.1.cmml" xref="S4.I11.i1.I1.i1.p1.24.m24.2.2.1.1.1">superscript</csymbol><ci id="S4.I11.i1.I1.i1.p1.24.m24.2.2.1.1.1.1.2.cmml" xref="S4.I11.i1.I1.i1.p1.24.m24.2.2.1.1.1.1.2">𝑣</ci><ci id="S4.I11.i1.I1.i1.p1.24.m24.2.2.1.1.1.1.3.cmml" xref="S4.I11.i1.I1.i1.p1.24.m24.2.2.1.1.1.1.3">′′</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.I11.i1.I1.i1.p1.24.m24.2c">u\to f(u)\to f(v^{\prime\prime})</annotation><annotation encoding="application/x-llamapun" id="S4.I11.i1.I1.i1.p1.24.m24.2d">italic_u → italic_f ( italic_u ) → italic_f ( italic_v start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT )</annotation></semantics></math> (since <math alttext="C_{1}" class="ltx_Math" display="inline" id="S4.I11.i1.I1.i1.p1.25.m25.1"><semantics id="S4.I11.i1.I1.i1.p1.25.m25.1a"><msub id="S4.I11.i1.I1.i1.p1.25.m25.1.1" xref="S4.I11.i1.I1.i1.p1.25.m25.1.1.cmml"><mi id="S4.I11.i1.I1.i1.p1.25.m25.1.1.2" xref="S4.I11.i1.I1.i1.p1.25.m25.1.1.2.cmml">C</mi><mn id="S4.I11.i1.I1.i1.p1.25.m25.1.1.3" xref="S4.I11.i1.I1.i1.p1.25.m25.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="S4.I11.i1.I1.i1.p1.25.m25.1b"><apply id="S4.I11.i1.I1.i1.p1.25.m25.1.1.cmml" xref="S4.I11.i1.I1.i1.p1.25.m25.1.1"><csymbol cd="ambiguous" id="S4.I11.i1.I1.i1.p1.25.m25.1.1.1.cmml" xref="S4.I11.i1.I1.i1.p1.25.m25.1.1">subscript</csymbol><ci id="S4.I11.i1.I1.i1.p1.25.m25.1.1.2.cmml" xref="S4.I11.i1.I1.i1.p1.25.m25.1.1.2">𝐶</ci><cn id="S4.I11.i1.I1.i1.p1.25.m25.1.1.3.cmml" type="integer" xref="S4.I11.i1.I1.i1.p1.25.m25.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.I11.i1.I1.i1.p1.25.m25.1c">C_{1}</annotation><annotation encoding="application/x-llamapun" id="S4.I11.i1.I1.i1.p1.25.m25.1d">italic_C start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math> remains connected), <math alttext="f(v^{\prime\prime})\to v^{\prime\prime}\to v^{\prime}\to f(v)\to v" class="ltx_Math" display="inline" id="S4.I11.i1.I1.i1.p1.26.m26.2"><semantics id="S4.I11.i1.I1.i1.p1.26.m26.2a"><mrow id="S4.I11.i1.I1.i1.p1.26.m26.2.2" xref="S4.I11.i1.I1.i1.p1.26.m26.2.2.cmml"><mrow id="S4.I11.i1.I1.i1.p1.26.m26.2.2.1" xref="S4.I11.i1.I1.i1.p1.26.m26.2.2.1.cmml"><mi id="S4.I11.i1.I1.i1.p1.26.m26.2.2.1.3" xref="S4.I11.i1.I1.i1.p1.26.m26.2.2.1.3.cmml">f</mi><mo id="S4.I11.i1.I1.i1.p1.26.m26.2.2.1.2" xref="S4.I11.i1.I1.i1.p1.26.m26.2.2.1.2.cmml">⁢</mo><mrow id="S4.I11.i1.I1.i1.p1.26.m26.2.2.1.1.1" xref="S4.I11.i1.I1.i1.p1.26.m26.2.2.1.1.1.1.cmml"><mo id="S4.I11.i1.I1.i1.p1.26.m26.2.2.1.1.1.2" stretchy="false" xref="S4.I11.i1.I1.i1.p1.26.m26.2.2.1.1.1.1.cmml">(</mo><msup id="S4.I11.i1.I1.i1.p1.26.m26.2.2.1.1.1.1" xref="S4.I11.i1.I1.i1.p1.26.m26.2.2.1.1.1.1.cmml"><mi id="S4.I11.i1.I1.i1.p1.26.m26.2.2.1.1.1.1.2" xref="S4.I11.i1.I1.i1.p1.26.m26.2.2.1.1.1.1.2.cmml">v</mi><mo id="S4.I11.i1.I1.i1.p1.26.m26.2.2.1.1.1.1.3" xref="S4.I11.i1.I1.i1.p1.26.m26.2.2.1.1.1.1.3.cmml">′′</mo></msup><mo id="S4.I11.i1.I1.i1.p1.26.m26.2.2.1.1.1.3" stretchy="false" xref="S4.I11.i1.I1.i1.p1.26.m26.2.2.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S4.I11.i1.I1.i1.p1.26.m26.2.2.3" stretchy="false" xref="S4.I11.i1.I1.i1.p1.26.m26.2.2.3.cmml">→</mo><msup id="S4.I11.i1.I1.i1.p1.26.m26.2.2.4" xref="S4.I11.i1.I1.i1.p1.26.m26.2.2.4.cmml"><mi id="S4.I11.i1.I1.i1.p1.26.m26.2.2.4.2" xref="S4.I11.i1.I1.i1.p1.26.m26.2.2.4.2.cmml">v</mi><mo id="S4.I11.i1.I1.i1.p1.26.m26.2.2.4.3" xref="S4.I11.i1.I1.i1.p1.26.m26.2.2.4.3.cmml">′′</mo></msup><mo id="S4.I11.i1.I1.i1.p1.26.m26.2.2.5" stretchy="false" xref="S4.I11.i1.I1.i1.p1.26.m26.2.2.5.cmml">→</mo><msup id="S4.I11.i1.I1.i1.p1.26.m26.2.2.6" xref="S4.I11.i1.I1.i1.p1.26.m26.2.2.6.cmml"><mi id="S4.I11.i1.I1.i1.p1.26.m26.2.2.6.2" xref="S4.I11.i1.I1.i1.p1.26.m26.2.2.6.2.cmml">v</mi><mo id="S4.I11.i1.I1.i1.p1.26.m26.2.2.6.3" xref="S4.I11.i1.I1.i1.p1.26.m26.2.2.6.3.cmml">′</mo></msup><mo id="S4.I11.i1.I1.i1.p1.26.m26.2.2.7" stretchy="false" xref="S4.I11.i1.I1.i1.p1.26.m26.2.2.7.cmml">→</mo><mrow id="S4.I11.i1.I1.i1.p1.26.m26.2.2.8" xref="S4.I11.i1.I1.i1.p1.26.m26.2.2.8.cmml"><mi id="S4.I11.i1.I1.i1.p1.26.m26.2.2.8.2" xref="S4.I11.i1.I1.i1.p1.26.m26.2.2.8.2.cmml">f</mi><mo id="S4.I11.i1.I1.i1.p1.26.m26.2.2.8.1" xref="S4.I11.i1.I1.i1.p1.26.m26.2.2.8.1.cmml">⁢</mo><mrow id="S4.I11.i1.I1.i1.p1.26.m26.2.2.8.3.2" xref="S4.I11.i1.I1.i1.p1.26.m26.2.2.8.cmml"><mo id="S4.I11.i1.I1.i1.p1.26.m26.2.2.8.3.2.1" stretchy="false" xref="S4.I11.i1.I1.i1.p1.26.m26.2.2.8.cmml">(</mo><mi id="S4.I11.i1.I1.i1.p1.26.m26.1.1" xref="S4.I11.i1.I1.i1.p1.26.m26.1.1.cmml">v</mi><mo id="S4.I11.i1.I1.i1.p1.26.m26.2.2.8.3.2.2" stretchy="false" xref="S4.I11.i1.I1.i1.p1.26.m26.2.2.8.cmml">)</mo></mrow></mrow><mo id="S4.I11.i1.I1.i1.p1.26.m26.2.2.9" stretchy="false" xref="S4.I11.i1.I1.i1.p1.26.m26.2.2.9.cmml">→</mo><mi id="S4.I11.i1.I1.i1.p1.26.m26.2.2.10" xref="S4.I11.i1.I1.i1.p1.26.m26.2.2.10.cmml">v</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.I11.i1.I1.i1.p1.26.m26.2b"><apply id="S4.I11.i1.I1.i1.p1.26.m26.2.2.cmml" xref="S4.I11.i1.I1.i1.p1.26.m26.2.2"><and id="S4.I11.i1.I1.i1.p1.26.m26.2.2a.cmml" xref="S4.I11.i1.I1.i1.p1.26.m26.2.2"></and><apply id="S4.I11.i1.I1.i1.p1.26.m26.2.2b.cmml" xref="S4.I11.i1.I1.i1.p1.26.m26.2.2"><ci id="S4.I11.i1.I1.i1.p1.26.m26.2.2.3.cmml" xref="S4.I11.i1.I1.i1.p1.26.m26.2.2.3">→</ci><apply id="S4.I11.i1.I1.i1.p1.26.m26.2.2.1.cmml" xref="S4.I11.i1.I1.i1.p1.26.m26.2.2.1"><times id="S4.I11.i1.I1.i1.p1.26.m26.2.2.1.2.cmml" xref="S4.I11.i1.I1.i1.p1.26.m26.2.2.1.2"></times><ci id="S4.I11.i1.I1.i1.p1.26.m26.2.2.1.3.cmml" xref="S4.I11.i1.I1.i1.p1.26.m26.2.2.1.3">𝑓</ci><apply id="S4.I11.i1.I1.i1.p1.26.m26.2.2.1.1.1.1.cmml" xref="S4.I11.i1.I1.i1.p1.26.m26.2.2.1.1.1"><csymbol cd="ambiguous" id="S4.I11.i1.I1.i1.p1.26.m26.2.2.1.1.1.1.1.cmml" xref="S4.I11.i1.I1.i1.p1.26.m26.2.2.1.1.1">superscript</csymbol><ci id="S4.I11.i1.I1.i1.p1.26.m26.2.2.1.1.1.1.2.cmml" xref="S4.I11.i1.I1.i1.p1.26.m26.2.2.1.1.1.1.2">𝑣</ci><ci id="S4.I11.i1.I1.i1.p1.26.m26.2.2.1.1.1.1.3.cmml" xref="S4.I11.i1.I1.i1.p1.26.m26.2.2.1.1.1.1.3">′′</ci></apply></apply><apply id="S4.I11.i1.I1.i1.p1.26.m26.2.2.4.cmml" xref="S4.I11.i1.I1.i1.p1.26.m26.2.2.4"><csymbol cd="ambiguous" id="S4.I11.i1.I1.i1.p1.26.m26.2.2.4.1.cmml" xref="S4.I11.i1.I1.i1.p1.26.m26.2.2.4">superscript</csymbol><ci id="S4.I11.i1.I1.i1.p1.26.m26.2.2.4.2.cmml" xref="S4.I11.i1.I1.i1.p1.26.m26.2.2.4.2">𝑣</ci><ci id="S4.I11.i1.I1.i1.p1.26.m26.2.2.4.3.cmml" xref="S4.I11.i1.I1.i1.p1.26.m26.2.2.4.3">′′</ci></apply></apply><apply id="S4.I11.i1.I1.i1.p1.26.m26.2.2c.cmml" xref="S4.I11.i1.I1.i1.p1.26.m26.2.2"><ci id="S4.I11.i1.I1.i1.p1.26.m26.2.2.5.cmml" xref="S4.I11.i1.I1.i1.p1.26.m26.2.2.5">→</ci><share href="https://arxiv.org/html/2503.00712v1#S4.I11.i1.I1.i1.p1.26.m26.2.2.4.cmml" id="S4.I11.i1.I1.i1.p1.26.m26.2.2d.cmml" xref="S4.I11.i1.I1.i1.p1.26.m26.2.2"></share><apply id="S4.I11.i1.I1.i1.p1.26.m26.2.2.6.cmml" xref="S4.I11.i1.I1.i1.p1.26.m26.2.2.6"><csymbol cd="ambiguous" id="S4.I11.i1.I1.i1.p1.26.m26.2.2.6.1.cmml" xref="S4.I11.i1.I1.i1.p1.26.m26.2.2.6">superscript</csymbol><ci id="S4.I11.i1.I1.i1.p1.26.m26.2.2.6.2.cmml" xref="S4.I11.i1.I1.i1.p1.26.m26.2.2.6.2">𝑣</ci><ci id="S4.I11.i1.I1.i1.p1.26.m26.2.2.6.3.cmml" xref="S4.I11.i1.I1.i1.p1.26.m26.2.2.6.3">′</ci></apply></apply><apply id="S4.I11.i1.I1.i1.p1.26.m26.2.2e.cmml" xref="S4.I11.i1.I1.i1.p1.26.m26.2.2"><ci id="S4.I11.i1.I1.i1.p1.26.m26.2.2.7.cmml" xref="S4.I11.i1.I1.i1.p1.26.m26.2.2.7">→</ci><share href="https://arxiv.org/html/2503.00712v1#S4.I11.i1.I1.i1.p1.26.m26.2.2.6.cmml" id="S4.I11.i1.I1.i1.p1.26.m26.2.2f.cmml" xref="S4.I11.i1.I1.i1.p1.26.m26.2.2"></share><apply id="S4.I11.i1.I1.i1.p1.26.m26.2.2.8.cmml" xref="S4.I11.i1.I1.i1.p1.26.m26.2.2.8"><times id="S4.I11.i1.I1.i1.p1.26.m26.2.2.8.1.cmml" xref="S4.I11.i1.I1.i1.p1.26.m26.2.2.8.1"></times><ci id="S4.I11.i1.I1.i1.p1.26.m26.2.2.8.2.cmml" xref="S4.I11.i1.I1.i1.p1.26.m26.2.2.8.2">𝑓</ci><ci id="S4.I11.i1.I1.i1.p1.26.m26.1.1.cmml" xref="S4.I11.i1.I1.i1.p1.26.m26.1.1">𝑣</ci></apply></apply><apply id="S4.I11.i1.I1.i1.p1.26.m26.2.2g.cmml" xref="S4.I11.i1.I1.i1.p1.26.m26.2.2"><ci id="S4.I11.i1.I1.i1.p1.26.m26.2.2.9.cmml" xref="S4.I11.i1.I1.i1.p1.26.m26.2.2.9">→</ci><share href="https://arxiv.org/html/2503.00712v1#S4.I11.i1.I1.i1.p1.26.m26.2.2.8.cmml" id="S4.I11.i1.I1.i1.p1.26.m26.2.2h.cmml" xref="S4.I11.i1.I1.i1.p1.26.m26.2.2"></share><ci id="S4.I11.i1.I1.i1.p1.26.m26.2.2.10.cmml" xref="S4.I11.i1.I1.i1.p1.26.m26.2.2.10">𝑣</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.I11.i1.I1.i1.p1.26.m26.2c">f(v^{\prime\prime})\to v^{\prime\prime}\to v^{\prime}\to f(v)\to v</annotation><annotation encoding="application/x-llamapun" id="S4.I11.i1.I1.i1.p1.26.m26.2d">italic_f ( italic_v start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT ) → italic_v start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT → italic_v start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT → italic_f ( italic_v ) → italic_v</annotation></semantics></math>. See Figure <a class="ltx_ref" href="https://arxiv.org/html/2503.00712v1#S4.F10.sf1" title="In Figure 10 ‣ Case 3: 𝒂 and 𝒃 are non-adjacent nodes of 𝑮_𝒙 for an S-node 𝒙: ‣ 4.2.3 Bounding the Approximation Ratio ‣ 4.2 Two-to-Three Augmentation ‣ 4 Vertex Connectivity Augmentation in Link-Arrival Model ‣ Streaming Algorithms for Network Design"><span class="ltx_text ltx_ref_tag">10(a)</span></a> for reference.</p> </div> </li> <li class="ltx_item" id="S4.I11.i1.I1.i2" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item"><span class="ltx_text ltx_font_bold" id="S4.I11.i1.I1.i2.1.1.1">–</span></span> <div class="ltx_para" id="S4.I11.i1.I1.i2.p1"> <p class="ltx_p" id="S4.I11.i1.I1.i2.p1.8">The case with <math alttext="f(v)\prec f(u)" class="ltx_Math" display="inline" id="S4.I11.i1.I1.i2.p1.1.m1.2"><semantics id="S4.I11.i1.I1.i2.p1.1.m1.2a"><mrow id="S4.I11.i1.I1.i2.p1.1.m1.2.3" xref="S4.I11.i1.I1.i2.p1.1.m1.2.3.cmml"><mrow id="S4.I11.i1.I1.i2.p1.1.m1.2.3.2" xref="S4.I11.i1.I1.i2.p1.1.m1.2.3.2.cmml"><mi id="S4.I11.i1.I1.i2.p1.1.m1.2.3.2.2" xref="S4.I11.i1.I1.i2.p1.1.m1.2.3.2.2.cmml">f</mi><mo id="S4.I11.i1.I1.i2.p1.1.m1.2.3.2.1" xref="S4.I11.i1.I1.i2.p1.1.m1.2.3.2.1.cmml">⁢</mo><mrow id="S4.I11.i1.I1.i2.p1.1.m1.2.3.2.3.2" xref="S4.I11.i1.I1.i2.p1.1.m1.2.3.2.cmml"><mo id="S4.I11.i1.I1.i2.p1.1.m1.2.3.2.3.2.1" stretchy="false" xref="S4.I11.i1.I1.i2.p1.1.m1.2.3.2.cmml">(</mo><mi id="S4.I11.i1.I1.i2.p1.1.m1.1.1" xref="S4.I11.i1.I1.i2.p1.1.m1.1.1.cmml">v</mi><mo id="S4.I11.i1.I1.i2.p1.1.m1.2.3.2.3.2.2" stretchy="false" xref="S4.I11.i1.I1.i2.p1.1.m1.2.3.2.cmml">)</mo></mrow></mrow><mo id="S4.I11.i1.I1.i2.p1.1.m1.2.3.1" xref="S4.I11.i1.I1.i2.p1.1.m1.2.3.1.cmml">≺</mo><mrow id="S4.I11.i1.I1.i2.p1.1.m1.2.3.3" xref="S4.I11.i1.I1.i2.p1.1.m1.2.3.3.cmml"><mi id="S4.I11.i1.I1.i2.p1.1.m1.2.3.3.2" xref="S4.I11.i1.I1.i2.p1.1.m1.2.3.3.2.cmml">f</mi><mo id="S4.I11.i1.I1.i2.p1.1.m1.2.3.3.1" xref="S4.I11.i1.I1.i2.p1.1.m1.2.3.3.1.cmml">⁢</mo><mrow id="S4.I11.i1.I1.i2.p1.1.m1.2.3.3.3.2" xref="S4.I11.i1.I1.i2.p1.1.m1.2.3.3.cmml"><mo id="S4.I11.i1.I1.i2.p1.1.m1.2.3.3.3.2.1" stretchy="false" xref="S4.I11.i1.I1.i2.p1.1.m1.2.3.3.cmml">(</mo><mi id="S4.I11.i1.I1.i2.p1.1.m1.2.2" xref="S4.I11.i1.I1.i2.p1.1.m1.2.2.cmml">u</mi><mo id="S4.I11.i1.I1.i2.p1.1.m1.2.3.3.3.2.2" stretchy="false" xref="S4.I11.i1.I1.i2.p1.1.m1.2.3.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.I11.i1.I1.i2.p1.1.m1.2b"><apply id="S4.I11.i1.I1.i2.p1.1.m1.2.3.cmml" xref="S4.I11.i1.I1.i2.p1.1.m1.2.3"><csymbol cd="latexml" id="S4.I11.i1.I1.i2.p1.1.m1.2.3.1.cmml" xref="S4.I11.i1.I1.i2.p1.1.m1.2.3.1">precedes</csymbol><apply id="S4.I11.i1.I1.i2.p1.1.m1.2.3.2.cmml" xref="S4.I11.i1.I1.i2.p1.1.m1.2.3.2"><times id="S4.I11.i1.I1.i2.p1.1.m1.2.3.2.1.cmml" xref="S4.I11.i1.I1.i2.p1.1.m1.2.3.2.1"></times><ci id="S4.I11.i1.I1.i2.p1.1.m1.2.3.2.2.cmml" xref="S4.I11.i1.I1.i2.p1.1.m1.2.3.2.2">𝑓</ci><ci id="S4.I11.i1.I1.i2.p1.1.m1.1.1.cmml" xref="S4.I11.i1.I1.i2.p1.1.m1.1.1">𝑣</ci></apply><apply id="S4.I11.i1.I1.i2.p1.1.m1.2.3.3.cmml" xref="S4.I11.i1.I1.i2.p1.1.m1.2.3.3"><times id="S4.I11.i1.I1.i2.p1.1.m1.2.3.3.1.cmml" xref="S4.I11.i1.I1.i2.p1.1.m1.2.3.3.1"></times><ci id="S4.I11.i1.I1.i2.p1.1.m1.2.3.3.2.cmml" xref="S4.I11.i1.I1.i2.p1.1.m1.2.3.3.2">𝑓</ci><ci id="S4.I11.i1.I1.i2.p1.1.m1.2.2.cmml" xref="S4.I11.i1.I1.i2.p1.1.m1.2.2">𝑢</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.I11.i1.I1.i2.p1.1.m1.2c">f(v)\prec f(u)</annotation><annotation encoding="application/x-llamapun" id="S4.I11.i1.I1.i2.p1.1.m1.2d">italic_f ( italic_v ) ≺ italic_f ( italic_u )</annotation></semantics></math> is similar; in this case we let <math alttext="(v^{\prime},v^{\prime\prime})=\textsc{Max}_{f(v)}(j)" class="ltx_Math" display="inline" id="S4.I11.i1.I1.i2.p1.2.m2.4"><semantics id="S4.I11.i1.I1.i2.p1.2.m2.4a"><mrow id="S4.I11.i1.I1.i2.p1.2.m2.4.4" xref="S4.I11.i1.I1.i2.p1.2.m2.4.4.cmml"><mrow id="S4.I11.i1.I1.i2.p1.2.m2.4.4.2.2" xref="S4.I11.i1.I1.i2.p1.2.m2.4.4.2.3.cmml"><mo id="S4.I11.i1.I1.i2.p1.2.m2.4.4.2.2.3" stretchy="false" xref="S4.I11.i1.I1.i2.p1.2.m2.4.4.2.3.cmml">(</mo><msup id="S4.I11.i1.I1.i2.p1.2.m2.3.3.1.1.1" xref="S4.I11.i1.I1.i2.p1.2.m2.3.3.1.1.1.cmml"><mi id="S4.I11.i1.I1.i2.p1.2.m2.3.3.1.1.1.2" xref="S4.I11.i1.I1.i2.p1.2.m2.3.3.1.1.1.2.cmml">v</mi><mo id="S4.I11.i1.I1.i2.p1.2.m2.3.3.1.1.1.3" xref="S4.I11.i1.I1.i2.p1.2.m2.3.3.1.1.1.3.cmml">′</mo></msup><mo id="S4.I11.i1.I1.i2.p1.2.m2.4.4.2.2.4" xref="S4.I11.i1.I1.i2.p1.2.m2.4.4.2.3.cmml">,</mo><msup id="S4.I11.i1.I1.i2.p1.2.m2.4.4.2.2.2" xref="S4.I11.i1.I1.i2.p1.2.m2.4.4.2.2.2.cmml"><mi id="S4.I11.i1.I1.i2.p1.2.m2.4.4.2.2.2.2" xref="S4.I11.i1.I1.i2.p1.2.m2.4.4.2.2.2.2.cmml">v</mi><mo id="S4.I11.i1.I1.i2.p1.2.m2.4.4.2.2.2.3" xref="S4.I11.i1.I1.i2.p1.2.m2.4.4.2.2.2.3.cmml">′′</mo></msup><mo id="S4.I11.i1.I1.i2.p1.2.m2.4.4.2.2.5" stretchy="false" xref="S4.I11.i1.I1.i2.p1.2.m2.4.4.2.3.cmml">)</mo></mrow><mo id="S4.I11.i1.I1.i2.p1.2.m2.4.4.3" xref="S4.I11.i1.I1.i2.p1.2.m2.4.4.3.cmml">=</mo><mrow id="S4.I11.i1.I1.i2.p1.2.m2.4.4.4" xref="S4.I11.i1.I1.i2.p1.2.m2.4.4.4.cmml"><msub id="S4.I11.i1.I1.i2.p1.2.m2.4.4.4.2" xref="S4.I11.i1.I1.i2.p1.2.m2.4.4.4.2.cmml"><mtext class="ltx_font_smallcaps" id="S4.I11.i1.I1.i2.p1.2.m2.4.4.4.2.2" xref="S4.I11.i1.I1.i2.p1.2.m2.4.4.4.2.2a.cmml">Max</mtext><mrow id="S4.I11.i1.I1.i2.p1.2.m2.1.1.1" xref="S4.I11.i1.I1.i2.p1.2.m2.1.1.1.cmml"><mi id="S4.I11.i1.I1.i2.p1.2.m2.1.1.1.3" xref="S4.I11.i1.I1.i2.p1.2.m2.1.1.1.3.cmml">f</mi><mo id="S4.I11.i1.I1.i2.p1.2.m2.1.1.1.2" xref="S4.I11.i1.I1.i2.p1.2.m2.1.1.1.2.cmml">⁢</mo><mrow id="S4.I11.i1.I1.i2.p1.2.m2.1.1.1.4.2" xref="S4.I11.i1.I1.i2.p1.2.m2.1.1.1.cmml"><mo id="S4.I11.i1.I1.i2.p1.2.m2.1.1.1.4.2.1" stretchy="false" xref="S4.I11.i1.I1.i2.p1.2.m2.1.1.1.cmml">(</mo><mi id="S4.I11.i1.I1.i2.p1.2.m2.1.1.1.1" xref="S4.I11.i1.I1.i2.p1.2.m2.1.1.1.1.cmml">v</mi><mo id="S4.I11.i1.I1.i2.p1.2.m2.1.1.1.4.2.2" stretchy="false" xref="S4.I11.i1.I1.i2.p1.2.m2.1.1.1.cmml">)</mo></mrow></mrow></msub><mo id="S4.I11.i1.I1.i2.p1.2.m2.4.4.4.1" xref="S4.I11.i1.I1.i2.p1.2.m2.4.4.4.1.cmml">⁢</mo><mrow id="S4.I11.i1.I1.i2.p1.2.m2.4.4.4.3.2" xref="S4.I11.i1.I1.i2.p1.2.m2.4.4.4.cmml"><mo id="S4.I11.i1.I1.i2.p1.2.m2.4.4.4.3.2.1" stretchy="false" xref="S4.I11.i1.I1.i2.p1.2.m2.4.4.4.cmml">(</mo><mi id="S4.I11.i1.I1.i2.p1.2.m2.2.2" xref="S4.I11.i1.I1.i2.p1.2.m2.2.2.cmml">j</mi><mo id="S4.I11.i1.I1.i2.p1.2.m2.4.4.4.3.2.2" stretchy="false" xref="S4.I11.i1.I1.i2.p1.2.m2.4.4.4.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.I11.i1.I1.i2.p1.2.m2.4b"><apply id="S4.I11.i1.I1.i2.p1.2.m2.4.4.cmml" xref="S4.I11.i1.I1.i2.p1.2.m2.4.4"><eq id="S4.I11.i1.I1.i2.p1.2.m2.4.4.3.cmml" xref="S4.I11.i1.I1.i2.p1.2.m2.4.4.3"></eq><interval closure="open" id="S4.I11.i1.I1.i2.p1.2.m2.4.4.2.3.cmml" xref="S4.I11.i1.I1.i2.p1.2.m2.4.4.2.2"><apply id="S4.I11.i1.I1.i2.p1.2.m2.3.3.1.1.1.cmml" xref="S4.I11.i1.I1.i2.p1.2.m2.3.3.1.1.1"><csymbol cd="ambiguous" id="S4.I11.i1.I1.i2.p1.2.m2.3.3.1.1.1.1.cmml" xref="S4.I11.i1.I1.i2.p1.2.m2.3.3.1.1.1">superscript</csymbol><ci id="S4.I11.i1.I1.i2.p1.2.m2.3.3.1.1.1.2.cmml" xref="S4.I11.i1.I1.i2.p1.2.m2.3.3.1.1.1.2">𝑣</ci><ci id="S4.I11.i1.I1.i2.p1.2.m2.3.3.1.1.1.3.cmml" xref="S4.I11.i1.I1.i2.p1.2.m2.3.3.1.1.1.3">′</ci></apply><apply id="S4.I11.i1.I1.i2.p1.2.m2.4.4.2.2.2.cmml" xref="S4.I11.i1.I1.i2.p1.2.m2.4.4.2.2.2"><csymbol cd="ambiguous" id="S4.I11.i1.I1.i2.p1.2.m2.4.4.2.2.2.1.cmml" xref="S4.I11.i1.I1.i2.p1.2.m2.4.4.2.2.2">superscript</csymbol><ci id="S4.I11.i1.I1.i2.p1.2.m2.4.4.2.2.2.2.cmml" xref="S4.I11.i1.I1.i2.p1.2.m2.4.4.2.2.2.2">𝑣</ci><ci id="S4.I11.i1.I1.i2.p1.2.m2.4.4.2.2.2.3.cmml" xref="S4.I11.i1.I1.i2.p1.2.m2.4.4.2.2.2.3">′′</ci></apply></interval><apply id="S4.I11.i1.I1.i2.p1.2.m2.4.4.4.cmml" xref="S4.I11.i1.I1.i2.p1.2.m2.4.4.4"><times id="S4.I11.i1.I1.i2.p1.2.m2.4.4.4.1.cmml" xref="S4.I11.i1.I1.i2.p1.2.m2.4.4.4.1"></times><apply id="S4.I11.i1.I1.i2.p1.2.m2.4.4.4.2.cmml" xref="S4.I11.i1.I1.i2.p1.2.m2.4.4.4.2"><csymbol cd="ambiguous" id="S4.I11.i1.I1.i2.p1.2.m2.4.4.4.2.1.cmml" xref="S4.I11.i1.I1.i2.p1.2.m2.4.4.4.2">subscript</csymbol><ci id="S4.I11.i1.I1.i2.p1.2.m2.4.4.4.2.2a.cmml" xref="S4.I11.i1.I1.i2.p1.2.m2.4.4.4.2.2"><mtext class="ltx_font_smallcaps" id="S4.I11.i1.I1.i2.p1.2.m2.4.4.4.2.2.cmml" xref="S4.I11.i1.I1.i2.p1.2.m2.4.4.4.2.2">Max</mtext></ci><apply id="S4.I11.i1.I1.i2.p1.2.m2.1.1.1.cmml" xref="S4.I11.i1.I1.i2.p1.2.m2.1.1.1"><times id="S4.I11.i1.I1.i2.p1.2.m2.1.1.1.2.cmml" xref="S4.I11.i1.I1.i2.p1.2.m2.1.1.1.2"></times><ci id="S4.I11.i1.I1.i2.p1.2.m2.1.1.1.3.cmml" xref="S4.I11.i1.I1.i2.p1.2.m2.1.1.1.3">𝑓</ci><ci id="S4.I11.i1.I1.i2.p1.2.m2.1.1.1.1.cmml" xref="S4.I11.i1.I1.i2.p1.2.m2.1.1.1.1">𝑣</ci></apply></apply><ci id="S4.I11.i1.I1.i2.p1.2.m2.2.2.cmml" xref="S4.I11.i1.I1.i2.p1.2.m2.2.2">𝑗</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.I11.i1.I1.i2.p1.2.m2.4c">(v^{\prime},v^{\prime\prime})=\textsc{Max}_{f(v)}(j)</annotation><annotation encoding="application/x-llamapun" id="S4.I11.i1.I1.i2.p1.2.m2.4d">( italic_v start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT , italic_v start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT ) = Max start_POSTSUBSCRIPT italic_f ( italic_v ) end_POSTSUBSCRIPT ( italic_j )</annotation></semantics></math>. It is easy to see that by the same argument as above, <math alttext="v^{\prime},v^{\prime\prime},f(v^{\prime\prime})" class="ltx_Math" display="inline" id="S4.I11.i1.I1.i2.p1.3.m3.3"><semantics id="S4.I11.i1.I1.i2.p1.3.m3.3a"><mrow id="S4.I11.i1.I1.i2.p1.3.m3.3.3.3" xref="S4.I11.i1.I1.i2.p1.3.m3.3.3.4.cmml"><msup id="S4.I11.i1.I1.i2.p1.3.m3.1.1.1.1" xref="S4.I11.i1.I1.i2.p1.3.m3.1.1.1.1.cmml"><mi id="S4.I11.i1.I1.i2.p1.3.m3.1.1.1.1.2" xref="S4.I11.i1.I1.i2.p1.3.m3.1.1.1.1.2.cmml">v</mi><mo id="S4.I11.i1.I1.i2.p1.3.m3.1.1.1.1.3" xref="S4.I11.i1.I1.i2.p1.3.m3.1.1.1.1.3.cmml">′</mo></msup><mo id="S4.I11.i1.I1.i2.p1.3.m3.3.3.3.4" xref="S4.I11.i1.I1.i2.p1.3.m3.3.3.4.cmml">,</mo><msup id="S4.I11.i1.I1.i2.p1.3.m3.2.2.2.2" xref="S4.I11.i1.I1.i2.p1.3.m3.2.2.2.2.cmml"><mi id="S4.I11.i1.I1.i2.p1.3.m3.2.2.2.2.2" xref="S4.I11.i1.I1.i2.p1.3.m3.2.2.2.2.2.cmml">v</mi><mo id="S4.I11.i1.I1.i2.p1.3.m3.2.2.2.2.3" xref="S4.I11.i1.I1.i2.p1.3.m3.2.2.2.2.3.cmml">′′</mo></msup><mo id="S4.I11.i1.I1.i2.p1.3.m3.3.3.3.5" xref="S4.I11.i1.I1.i2.p1.3.m3.3.3.4.cmml">,</mo><mrow id="S4.I11.i1.I1.i2.p1.3.m3.3.3.3.3" xref="S4.I11.i1.I1.i2.p1.3.m3.3.3.3.3.cmml"><mi id="S4.I11.i1.I1.i2.p1.3.m3.3.3.3.3.3" xref="S4.I11.i1.I1.i2.p1.3.m3.3.3.3.3.3.cmml">f</mi><mo id="S4.I11.i1.I1.i2.p1.3.m3.3.3.3.3.2" xref="S4.I11.i1.I1.i2.p1.3.m3.3.3.3.3.2.cmml">⁢</mo><mrow id="S4.I11.i1.I1.i2.p1.3.m3.3.3.3.3.1.1" xref="S4.I11.i1.I1.i2.p1.3.m3.3.3.3.3.1.1.1.cmml"><mo id="S4.I11.i1.I1.i2.p1.3.m3.3.3.3.3.1.1.2" stretchy="false" xref="S4.I11.i1.I1.i2.p1.3.m3.3.3.3.3.1.1.1.cmml">(</mo><msup id="S4.I11.i1.I1.i2.p1.3.m3.3.3.3.3.1.1.1" xref="S4.I11.i1.I1.i2.p1.3.m3.3.3.3.3.1.1.1.cmml"><mi id="S4.I11.i1.I1.i2.p1.3.m3.3.3.3.3.1.1.1.2" xref="S4.I11.i1.I1.i2.p1.3.m3.3.3.3.3.1.1.1.2.cmml">v</mi><mo id="S4.I11.i1.I1.i2.p1.3.m3.3.3.3.3.1.1.1.3" xref="S4.I11.i1.I1.i2.p1.3.m3.3.3.3.3.1.1.1.3.cmml">′′</mo></msup><mo id="S4.I11.i1.I1.i2.p1.3.m3.3.3.3.3.1.1.3" stretchy="false" xref="S4.I11.i1.I1.i2.p1.3.m3.3.3.3.3.1.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.I11.i1.I1.i2.p1.3.m3.3b"><list id="S4.I11.i1.I1.i2.p1.3.m3.3.3.4.cmml" xref="S4.I11.i1.I1.i2.p1.3.m3.3.3.3"><apply id="S4.I11.i1.I1.i2.p1.3.m3.1.1.1.1.cmml" xref="S4.I11.i1.I1.i2.p1.3.m3.1.1.1.1"><csymbol cd="ambiguous" id="S4.I11.i1.I1.i2.p1.3.m3.1.1.1.1.1.cmml" xref="S4.I11.i1.I1.i2.p1.3.m3.1.1.1.1">superscript</csymbol><ci id="S4.I11.i1.I1.i2.p1.3.m3.1.1.1.1.2.cmml" xref="S4.I11.i1.I1.i2.p1.3.m3.1.1.1.1.2">𝑣</ci><ci id="S4.I11.i1.I1.i2.p1.3.m3.1.1.1.1.3.cmml" xref="S4.I11.i1.I1.i2.p1.3.m3.1.1.1.1.3">′</ci></apply><apply id="S4.I11.i1.I1.i2.p1.3.m3.2.2.2.2.cmml" xref="S4.I11.i1.I1.i2.p1.3.m3.2.2.2.2"><csymbol cd="ambiguous" id="S4.I11.i1.I1.i2.p1.3.m3.2.2.2.2.1.cmml" xref="S4.I11.i1.I1.i2.p1.3.m3.2.2.2.2">superscript</csymbol><ci id="S4.I11.i1.I1.i2.p1.3.m3.2.2.2.2.2.cmml" xref="S4.I11.i1.I1.i2.p1.3.m3.2.2.2.2.2">𝑣</ci><ci id="S4.I11.i1.I1.i2.p1.3.m3.2.2.2.2.3.cmml" xref="S4.I11.i1.I1.i2.p1.3.m3.2.2.2.2.3">′′</ci></apply><apply id="S4.I11.i1.I1.i2.p1.3.m3.3.3.3.3.cmml" xref="S4.I11.i1.I1.i2.p1.3.m3.3.3.3.3"><times id="S4.I11.i1.I1.i2.p1.3.m3.3.3.3.3.2.cmml" xref="S4.I11.i1.I1.i2.p1.3.m3.3.3.3.3.2"></times><ci id="S4.I11.i1.I1.i2.p1.3.m3.3.3.3.3.3.cmml" xref="S4.I11.i1.I1.i2.p1.3.m3.3.3.3.3.3">𝑓</ci><apply id="S4.I11.i1.I1.i2.p1.3.m3.3.3.3.3.1.1.1.cmml" xref="S4.I11.i1.I1.i2.p1.3.m3.3.3.3.3.1.1"><csymbol cd="ambiguous" id="S4.I11.i1.I1.i2.p1.3.m3.3.3.3.3.1.1.1.1.cmml" xref="S4.I11.i1.I1.i2.p1.3.m3.3.3.3.3.1.1">superscript</csymbol><ci id="S4.I11.i1.I1.i2.p1.3.m3.3.3.3.3.1.1.1.2.cmml" xref="S4.I11.i1.I1.i2.p1.3.m3.3.3.3.3.1.1.1.2">𝑣</ci><ci id="S4.I11.i1.I1.i2.p1.3.m3.3.3.3.3.1.1.1.3.cmml" xref="S4.I11.i1.I1.i2.p1.3.m3.3.3.3.3.1.1.1.3">′′</ci></apply></apply></list></annotation-xml><annotation encoding="application/x-tex" id="S4.I11.i1.I1.i2.p1.3.m3.3c">v^{\prime},v^{\prime\prime},f(v^{\prime\prime})</annotation><annotation encoding="application/x-llamapun" id="S4.I11.i1.I1.i2.p1.3.m3.3d">italic_v start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT , italic_v start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT , italic_f ( italic_v start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT )</annotation></semantics></math> are all not contained in <math alttext="\{a,b\}" class="ltx_Math" display="inline" id="S4.I11.i1.I1.i2.p1.4.m4.2"><semantics id="S4.I11.i1.I1.i2.p1.4.m4.2a"><mrow id="S4.I11.i1.I1.i2.p1.4.m4.2.3.2" xref="S4.I11.i1.I1.i2.p1.4.m4.2.3.1.cmml"><mo id="S4.I11.i1.I1.i2.p1.4.m4.2.3.2.1" stretchy="false" xref="S4.I11.i1.I1.i2.p1.4.m4.2.3.1.cmml">{</mo><mi id="S4.I11.i1.I1.i2.p1.4.m4.1.1" xref="S4.I11.i1.I1.i2.p1.4.m4.1.1.cmml">a</mi><mo id="S4.I11.i1.I1.i2.p1.4.m4.2.3.2.2" xref="S4.I11.i1.I1.i2.p1.4.m4.2.3.1.cmml">,</mo><mi id="S4.I11.i1.I1.i2.p1.4.m4.2.2" xref="S4.I11.i1.I1.i2.p1.4.m4.2.2.cmml">b</mi><mo id="S4.I11.i1.I1.i2.p1.4.m4.2.3.2.3" stretchy="false" xref="S4.I11.i1.I1.i2.p1.4.m4.2.3.1.cmml">}</mo></mrow><annotation-xml encoding="MathML-Content" id="S4.I11.i1.I1.i2.p1.4.m4.2b"><set id="S4.I11.i1.I1.i2.p1.4.m4.2.3.1.cmml" xref="S4.I11.i1.I1.i2.p1.4.m4.2.3.2"><ci id="S4.I11.i1.I1.i2.p1.4.m4.1.1.cmml" xref="S4.I11.i1.I1.i2.p1.4.m4.1.1">𝑎</ci><ci id="S4.I11.i1.I1.i2.p1.4.m4.2.2.cmml" xref="S4.I11.i1.I1.i2.p1.4.m4.2.2">𝑏</ci></set></annotation-xml><annotation encoding="application/x-tex" id="S4.I11.i1.I1.i2.p1.4.m4.2c">\{a,b\}</annotation><annotation encoding="application/x-llamapun" id="S4.I11.i1.I1.i2.p1.4.m4.2d">{ italic_a , italic_b }</annotation></semantics></math>, and thus there exists a <math alttext="u" class="ltx_Math" display="inline" id="S4.I11.i1.I1.i2.p1.5.m5.1"><semantics id="S4.I11.i1.I1.i2.p1.5.m5.1a"><mi id="S4.I11.i1.I1.i2.p1.5.m5.1.1" xref="S4.I11.i1.I1.i2.p1.5.m5.1.1.cmml">u</mi><annotation-xml encoding="MathML-Content" id="S4.I11.i1.I1.i2.p1.5.m5.1b"><ci id="S4.I11.i1.I1.i2.p1.5.m5.1.1.cmml" xref="S4.I11.i1.I1.i2.p1.5.m5.1.1">𝑢</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.I11.i1.I1.i2.p1.5.m5.1c">u</annotation><annotation encoding="application/x-llamapun" id="S4.I11.i1.I1.i2.p1.5.m5.1d">italic_u</annotation></semantics></math>-<math alttext="v" class="ltx_Math" display="inline" id="S4.I11.i1.I1.i2.p1.6.m6.1"><semantics id="S4.I11.i1.I1.i2.p1.6.m6.1a"><mi id="S4.I11.i1.I1.i2.p1.6.m6.1.1" xref="S4.I11.i1.I1.i2.p1.6.m6.1.1.cmml">v</mi><annotation-xml encoding="MathML-Content" id="S4.I11.i1.I1.i2.p1.6.m6.1b"><ci id="S4.I11.i1.I1.i2.p1.6.m6.1.1.cmml" xref="S4.I11.i1.I1.i2.p1.6.m6.1.1">𝑣</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.I11.i1.I1.i2.p1.6.m6.1c">v</annotation><annotation encoding="application/x-llamapun" id="S4.I11.i1.I1.i2.p1.6.m6.1d">italic_v</annotation></semantics></math> path via the <math alttext="(v^{\prime},v^{\prime\prime})" class="ltx_Math" display="inline" id="S4.I11.i1.I1.i2.p1.7.m7.2"><semantics id="S4.I11.i1.I1.i2.p1.7.m7.2a"><mrow id="S4.I11.i1.I1.i2.p1.7.m7.2.2.2" xref="S4.I11.i1.I1.i2.p1.7.m7.2.2.3.cmml"><mo id="S4.I11.i1.I1.i2.p1.7.m7.2.2.2.3" stretchy="false" xref="S4.I11.i1.I1.i2.p1.7.m7.2.2.3.cmml">(</mo><msup id="S4.I11.i1.I1.i2.p1.7.m7.1.1.1.1" xref="S4.I11.i1.I1.i2.p1.7.m7.1.1.1.1.cmml"><mi id="S4.I11.i1.I1.i2.p1.7.m7.1.1.1.1.2" xref="S4.I11.i1.I1.i2.p1.7.m7.1.1.1.1.2.cmml">v</mi><mo id="S4.I11.i1.I1.i2.p1.7.m7.1.1.1.1.3" xref="S4.I11.i1.I1.i2.p1.7.m7.1.1.1.1.3.cmml">′</mo></msup><mo id="S4.I11.i1.I1.i2.p1.7.m7.2.2.2.4" xref="S4.I11.i1.I1.i2.p1.7.m7.2.2.3.cmml">,</mo><msup id="S4.I11.i1.I1.i2.p1.7.m7.2.2.2.2" xref="S4.I11.i1.I1.i2.p1.7.m7.2.2.2.2.cmml"><mi id="S4.I11.i1.I1.i2.p1.7.m7.2.2.2.2.2" xref="S4.I11.i1.I1.i2.p1.7.m7.2.2.2.2.2.cmml">v</mi><mo id="S4.I11.i1.I1.i2.p1.7.m7.2.2.2.2.3" xref="S4.I11.i1.I1.i2.p1.7.m7.2.2.2.2.3.cmml">′′</mo></msup><mo id="S4.I11.i1.I1.i2.p1.7.m7.2.2.2.5" stretchy="false" xref="S4.I11.i1.I1.i2.p1.7.m7.2.2.3.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S4.I11.i1.I1.i2.p1.7.m7.2b"><interval closure="open" id="S4.I11.i1.I1.i2.p1.7.m7.2.2.3.cmml" xref="S4.I11.i1.I1.i2.p1.7.m7.2.2.2"><apply id="S4.I11.i1.I1.i2.p1.7.m7.1.1.1.1.cmml" xref="S4.I11.i1.I1.i2.p1.7.m7.1.1.1.1"><csymbol cd="ambiguous" id="S4.I11.i1.I1.i2.p1.7.m7.1.1.1.1.1.cmml" xref="S4.I11.i1.I1.i2.p1.7.m7.1.1.1.1">superscript</csymbol><ci id="S4.I11.i1.I1.i2.p1.7.m7.1.1.1.1.2.cmml" xref="S4.I11.i1.I1.i2.p1.7.m7.1.1.1.1.2">𝑣</ci><ci id="S4.I11.i1.I1.i2.p1.7.m7.1.1.1.1.3.cmml" xref="S4.I11.i1.I1.i2.p1.7.m7.1.1.1.1.3">′</ci></apply><apply id="S4.I11.i1.I1.i2.p1.7.m7.2.2.2.2.cmml" xref="S4.I11.i1.I1.i2.p1.7.m7.2.2.2.2"><csymbol cd="ambiguous" id="S4.I11.i1.I1.i2.p1.7.m7.2.2.2.2.1.cmml" xref="S4.I11.i1.I1.i2.p1.7.m7.2.2.2.2">superscript</csymbol><ci id="S4.I11.i1.I1.i2.p1.7.m7.2.2.2.2.2.cmml" xref="S4.I11.i1.I1.i2.p1.7.m7.2.2.2.2.2">𝑣</ci><ci id="S4.I11.i1.I1.i2.p1.7.m7.2.2.2.2.3.cmml" xref="S4.I11.i1.I1.i2.p1.7.m7.2.2.2.2.3">′′</ci></apply></interval></annotation-xml><annotation encoding="application/x-tex" id="S4.I11.i1.I1.i2.p1.7.m7.2c">(v^{\prime},v^{\prime\prime})</annotation><annotation encoding="application/x-llamapun" id="S4.I11.i1.I1.i2.p1.7.m7.2d">( italic_v start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT , italic_v start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT )</annotation></semantics></math> link in <span class="ltx_text ltx_markedasmath" id="S4.I11.i1.I1.i2.p1.8.1">SOL</span>.</p> </div> </li> </ul> </div> </li> <li class="ltx_item" id="S4.I11.i2" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">•</span> <div class="ltx_para" id="S4.I11.i2.p1"> <p class="ltx_p" id="S4.I11.i2.p1.9"><span class="ltx_text ltx_font_bold" id="S4.I11.i2.p1.1.1">Case 3b: <math alttext="\boldsymbol{\text{LCA}(\ell(u),\ell(v))\neq x}" class="ltx_Math" display="inline" id="S4.I11.i2.p1.1.1.m1.4"><semantics id="S4.I11.i2.p1.1.1.m1.4a"><mrow id="S4.I11.i2.p1.1.1.m1.4.4" xref="S4.I11.i2.p1.1.1.m1.4.4.cmml"><mrow id="S4.I11.i2.p1.1.1.m1.4.4.2" xref="S4.I11.i2.p1.1.1.m1.4.4.2.cmml"><mtext class="ltx_mathvariant_bold" id="S4.I11.i2.p1.1.1.m1.4.4.2.4" xref="S4.I11.i2.p1.1.1.m1.4.4.2.4a.cmml">LCA</mtext><mo id="S4.I11.i2.p1.1.1.m1.4.4.2.3" xref="S4.I11.i2.p1.1.1.m1.4.4.2.3.cmml">⁢</mo><mrow id="S4.I11.i2.p1.1.1.m1.4.4.2.2.2" xref="S4.I11.i2.p1.1.1.m1.4.4.2.2.3.cmml"><mo class="ltx_mathvariant_bold" id="S4.I11.i2.p1.1.1.m1.4.4.2.2.2.3" mathvariant="bold" stretchy="false" xref="S4.I11.i2.p1.1.1.m1.4.4.2.2.3.cmml">(</mo><mrow id="S4.I11.i2.p1.1.1.m1.3.3.1.1.1.1" xref="S4.I11.i2.p1.1.1.m1.3.3.1.1.1.1.cmml"><mi class="ltx_mathvariant_bold" id="S4.I11.i2.p1.1.1.m1.3.3.1.1.1.1.2" mathvariant="bold" xref="S4.I11.i2.p1.1.1.m1.3.3.1.1.1.1.2.cmml">ℓ</mi><mo id="S4.I11.i2.p1.1.1.m1.3.3.1.1.1.1.1" xref="S4.I11.i2.p1.1.1.m1.3.3.1.1.1.1.1.cmml">⁢</mo><mrow id="S4.I11.i2.p1.1.1.m1.3.3.1.1.1.1.3.2" xref="S4.I11.i2.p1.1.1.m1.3.3.1.1.1.1.cmml"><mo class="ltx_mathvariant_bold" id="S4.I11.i2.p1.1.1.m1.3.3.1.1.1.1.3.2.1" mathvariant="bold" stretchy="false" xref="S4.I11.i2.p1.1.1.m1.3.3.1.1.1.1.cmml">(</mo><mi id="S4.I11.i2.p1.1.1.m1.1.1" xref="S4.I11.i2.p1.1.1.m1.1.1.cmml">u</mi><mo class="ltx_mathvariant_bold" id="S4.I11.i2.p1.1.1.m1.3.3.1.1.1.1.3.2.2" mathvariant="bold" stretchy="false" xref="S4.I11.i2.p1.1.1.m1.3.3.1.1.1.1.cmml">)</mo></mrow></mrow><mo class="ltx_mathvariant_bold" id="S4.I11.i2.p1.1.1.m1.4.4.2.2.2.4" mathvariant="bold" xref="S4.I11.i2.p1.1.1.m1.4.4.2.2.3.cmml">,</mo><mrow id="S4.I11.i2.p1.1.1.m1.4.4.2.2.2.2" xref="S4.I11.i2.p1.1.1.m1.4.4.2.2.2.2.cmml"><mi class="ltx_mathvariant_bold" id="S4.I11.i2.p1.1.1.m1.4.4.2.2.2.2.2" mathvariant="bold" xref="S4.I11.i2.p1.1.1.m1.4.4.2.2.2.2.2.cmml">ℓ</mi><mo id="S4.I11.i2.p1.1.1.m1.4.4.2.2.2.2.1" xref="S4.I11.i2.p1.1.1.m1.4.4.2.2.2.2.1.cmml">⁢</mo><mrow id="S4.I11.i2.p1.1.1.m1.4.4.2.2.2.2.3.2" xref="S4.I11.i2.p1.1.1.m1.4.4.2.2.2.2.cmml"><mo class="ltx_mathvariant_bold" id="S4.I11.i2.p1.1.1.m1.4.4.2.2.2.2.3.2.1" mathvariant="bold" stretchy="false" xref="S4.I11.i2.p1.1.1.m1.4.4.2.2.2.2.cmml">(</mo><mi id="S4.I11.i2.p1.1.1.m1.2.2" xref="S4.I11.i2.p1.1.1.m1.2.2.cmml">v</mi><mo class="ltx_mathvariant_bold" id="S4.I11.i2.p1.1.1.m1.4.4.2.2.2.2.3.2.2" mathvariant="bold" stretchy="false" xref="S4.I11.i2.p1.1.1.m1.4.4.2.2.2.2.cmml">)</mo></mrow></mrow><mo class="ltx_mathvariant_bold" id="S4.I11.i2.p1.1.1.m1.4.4.2.2.2.5" mathvariant="bold" stretchy="false" xref="S4.I11.i2.p1.1.1.m1.4.4.2.2.3.cmml">)</mo></mrow></mrow><mo class="ltx_mathvariant_bold" id="S4.I11.i2.p1.1.1.m1.4.4.3" mathvariant="bold" xref="S4.I11.i2.p1.1.1.m1.4.4.3.cmml">≠</mo><mi id="S4.I11.i2.p1.1.1.m1.4.4.4" xref="S4.I11.i2.p1.1.1.m1.4.4.4.cmml">x</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.I11.i2.p1.1.1.m1.4b"><apply id="S4.I11.i2.p1.1.1.m1.4.4.cmml" xref="S4.I11.i2.p1.1.1.m1.4.4"><neq id="S4.I11.i2.p1.1.1.m1.4.4.3.cmml" xref="S4.I11.i2.p1.1.1.m1.4.4.3"></neq><apply id="S4.I11.i2.p1.1.1.m1.4.4.2.cmml" xref="S4.I11.i2.p1.1.1.m1.4.4.2"><times id="S4.I11.i2.p1.1.1.m1.4.4.2.3.cmml" xref="S4.I11.i2.p1.1.1.m1.4.4.2.3"></times><ci id="S4.I11.i2.p1.1.1.m1.4.4.2.4a.cmml" xref="S4.I11.i2.p1.1.1.m1.4.4.2.4"><mtext class="ltx_mathvariant_bold" id="S4.I11.i2.p1.1.1.m1.4.4.2.4.cmml" xref="S4.I11.i2.p1.1.1.m1.4.4.2.4">LCA</mtext></ci><interval closure="open" id="S4.I11.i2.p1.1.1.m1.4.4.2.2.3.cmml" xref="S4.I11.i2.p1.1.1.m1.4.4.2.2.2"><apply id="S4.I11.i2.p1.1.1.m1.3.3.1.1.1.1.cmml" xref="S4.I11.i2.p1.1.1.m1.3.3.1.1.1.1"><times id="S4.I11.i2.p1.1.1.m1.3.3.1.1.1.1.1.cmml" xref="S4.I11.i2.p1.1.1.m1.3.3.1.1.1.1.1"></times><ci id="S4.I11.i2.p1.1.1.m1.3.3.1.1.1.1.2.cmml" xref="S4.I11.i2.p1.1.1.m1.3.3.1.1.1.1.2">bold-ℓ</ci><ci id="S4.I11.i2.p1.1.1.m1.1.1.cmml" xref="S4.I11.i2.p1.1.1.m1.1.1">𝑢</ci></apply><apply id="S4.I11.i2.p1.1.1.m1.4.4.2.2.2.2.cmml" xref="S4.I11.i2.p1.1.1.m1.4.4.2.2.2.2"><times id="S4.I11.i2.p1.1.1.m1.4.4.2.2.2.2.1.cmml" xref="S4.I11.i2.p1.1.1.m1.4.4.2.2.2.2.1"></times><ci id="S4.I11.i2.p1.1.1.m1.4.4.2.2.2.2.2.cmml" xref="S4.I11.i2.p1.1.1.m1.4.4.2.2.2.2.2">bold-ℓ</ci><ci id="S4.I11.i2.p1.1.1.m1.2.2.cmml" xref="S4.I11.i2.p1.1.1.m1.2.2">𝑣</ci></apply></interval></apply><ci id="S4.I11.i2.p1.1.1.m1.4.4.4.cmml" xref="S4.I11.i2.p1.1.1.m1.4.4.4">𝑥</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.I11.i2.p1.1.1.m1.4c">\boldsymbol{\text{LCA}(\ell(u),\ell(v))\neq x}</annotation><annotation encoding="application/x-llamapun" id="S4.I11.i2.p1.1.1.m1.4d">LCA bold_( bold_ℓ bold_( bold_italic_u bold_) bold_, bold_ℓ bold_( bold_italic_v bold_) bold_) bold_≠ bold_italic_x</annotation></semantics></math>: </span> Since we assume that <math alttext="x" class="ltx_Math" display="inline" id="S4.I11.i2.p1.2.m1.1"><semantics id="S4.I11.i2.p1.2.m1.1a"><mi id="S4.I11.i2.p1.2.m1.1.1" xref="S4.I11.i2.p1.2.m1.1.1.cmml">x</mi><annotation-xml encoding="MathML-Content" id="S4.I11.i2.p1.2.m1.1b"><ci id="S4.I11.i2.p1.2.m1.1.1.cmml" xref="S4.I11.i2.p1.2.m1.1.1">𝑥</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.I11.i2.p1.2.m1.1c">x</annotation><annotation encoding="application/x-llamapun" id="S4.I11.i2.p1.2.m1.1d">italic_x</annotation></semantics></math> is on the tree path between <math alttext="u" class="ltx_Math" display="inline" id="S4.I11.i2.p1.3.m2.1"><semantics id="S4.I11.i2.p1.3.m2.1a"><mi id="S4.I11.i2.p1.3.m2.1.1" xref="S4.I11.i2.p1.3.m2.1.1.cmml">u</mi><annotation-xml encoding="MathML-Content" id="S4.I11.i2.p1.3.m2.1b"><ci id="S4.I11.i2.p1.3.m2.1.1.cmml" xref="S4.I11.i2.p1.3.m2.1.1">𝑢</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.I11.i2.p1.3.m2.1c">u</annotation><annotation encoding="application/x-llamapun" id="S4.I11.i2.p1.3.m2.1d">italic_u</annotation></semantics></math> and <math alttext="v" class="ltx_Math" display="inline" id="S4.I11.i2.p1.4.m3.1"><semantics id="S4.I11.i2.p1.4.m3.1a"><mi id="S4.I11.i2.p1.4.m3.1.1" xref="S4.I11.i2.p1.4.m3.1.1.cmml">v</mi><annotation-xml encoding="MathML-Content" id="S4.I11.i2.p1.4.m3.1b"><ci id="S4.I11.i2.p1.4.m3.1.1.cmml" xref="S4.I11.i2.p1.4.m3.1.1">𝑣</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.I11.i2.p1.4.m3.1c">v</annotation><annotation encoding="application/x-llamapun" id="S4.I11.i2.p1.4.m3.1d">italic_v</annotation></semantics></math>, we can assume without loss of generality that <math alttext="\ell(u)\in T_{x}" class="ltx_Math" display="inline" id="S4.I11.i2.p1.5.m4.1"><semantics id="S4.I11.i2.p1.5.m4.1a"><mrow id="S4.I11.i2.p1.5.m4.1.2" xref="S4.I11.i2.p1.5.m4.1.2.cmml"><mrow id="S4.I11.i2.p1.5.m4.1.2.2" xref="S4.I11.i2.p1.5.m4.1.2.2.cmml"><mi id="S4.I11.i2.p1.5.m4.1.2.2.2" mathvariant="normal" xref="S4.I11.i2.p1.5.m4.1.2.2.2.cmml">ℓ</mi><mo id="S4.I11.i2.p1.5.m4.1.2.2.1" xref="S4.I11.i2.p1.5.m4.1.2.2.1.cmml">⁢</mo><mrow id="S4.I11.i2.p1.5.m4.1.2.2.3.2" xref="S4.I11.i2.p1.5.m4.1.2.2.cmml"><mo id="S4.I11.i2.p1.5.m4.1.2.2.3.2.1" stretchy="false" xref="S4.I11.i2.p1.5.m4.1.2.2.cmml">(</mo><mi id="S4.I11.i2.p1.5.m4.1.1" xref="S4.I11.i2.p1.5.m4.1.1.cmml">u</mi><mo id="S4.I11.i2.p1.5.m4.1.2.2.3.2.2" stretchy="false" xref="S4.I11.i2.p1.5.m4.1.2.2.cmml">)</mo></mrow></mrow><mo id="S4.I11.i2.p1.5.m4.1.2.1" xref="S4.I11.i2.p1.5.m4.1.2.1.cmml">∈</mo><msub id="S4.I11.i2.p1.5.m4.1.2.3" xref="S4.I11.i2.p1.5.m4.1.2.3.cmml"><mi id="S4.I11.i2.p1.5.m4.1.2.3.2" xref="S4.I11.i2.p1.5.m4.1.2.3.2.cmml">T</mi><mi id="S4.I11.i2.p1.5.m4.1.2.3.3" xref="S4.I11.i2.p1.5.m4.1.2.3.3.cmml">x</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S4.I11.i2.p1.5.m4.1b"><apply id="S4.I11.i2.p1.5.m4.1.2.cmml" xref="S4.I11.i2.p1.5.m4.1.2"><in id="S4.I11.i2.p1.5.m4.1.2.1.cmml" xref="S4.I11.i2.p1.5.m4.1.2.1"></in><apply id="S4.I11.i2.p1.5.m4.1.2.2.cmml" xref="S4.I11.i2.p1.5.m4.1.2.2"><times id="S4.I11.i2.p1.5.m4.1.2.2.1.cmml" xref="S4.I11.i2.p1.5.m4.1.2.2.1"></times><ci id="S4.I11.i2.p1.5.m4.1.2.2.2.cmml" xref="S4.I11.i2.p1.5.m4.1.2.2.2">ℓ</ci><ci id="S4.I11.i2.p1.5.m4.1.1.cmml" xref="S4.I11.i2.p1.5.m4.1.1">𝑢</ci></apply><apply id="S4.I11.i2.p1.5.m4.1.2.3.cmml" xref="S4.I11.i2.p1.5.m4.1.2.3"><csymbol cd="ambiguous" id="S4.I11.i2.p1.5.m4.1.2.3.1.cmml" xref="S4.I11.i2.p1.5.m4.1.2.3">subscript</csymbol><ci id="S4.I11.i2.p1.5.m4.1.2.3.2.cmml" xref="S4.I11.i2.p1.5.m4.1.2.3.2">𝑇</ci><ci id="S4.I11.i2.p1.5.m4.1.2.3.3.cmml" xref="S4.I11.i2.p1.5.m4.1.2.3.3">𝑥</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.I11.i2.p1.5.m4.1c">\ell(u)\in T_{x}</annotation><annotation encoding="application/x-llamapun" id="S4.I11.i2.p1.5.m4.1d">roman_ℓ ( italic_u ) ∈ italic_T start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="\ell(v)\in T\setminus T_{x}" class="ltx_Math" display="inline" id="S4.I11.i2.p1.6.m5.1"><semantics id="S4.I11.i2.p1.6.m5.1a"><mrow id="S4.I11.i2.p1.6.m5.1.2" xref="S4.I11.i2.p1.6.m5.1.2.cmml"><mrow id="S4.I11.i2.p1.6.m5.1.2.2" xref="S4.I11.i2.p1.6.m5.1.2.2.cmml"><mi id="S4.I11.i2.p1.6.m5.1.2.2.2" mathvariant="normal" xref="S4.I11.i2.p1.6.m5.1.2.2.2.cmml">ℓ</mi><mo id="S4.I11.i2.p1.6.m5.1.2.2.1" xref="S4.I11.i2.p1.6.m5.1.2.2.1.cmml">⁢</mo><mrow id="S4.I11.i2.p1.6.m5.1.2.2.3.2" xref="S4.I11.i2.p1.6.m5.1.2.2.cmml"><mo id="S4.I11.i2.p1.6.m5.1.2.2.3.2.1" stretchy="false" xref="S4.I11.i2.p1.6.m5.1.2.2.cmml">(</mo><mi id="S4.I11.i2.p1.6.m5.1.1" xref="S4.I11.i2.p1.6.m5.1.1.cmml">v</mi><mo id="S4.I11.i2.p1.6.m5.1.2.2.3.2.2" stretchy="false" xref="S4.I11.i2.p1.6.m5.1.2.2.cmml">)</mo></mrow></mrow><mo id="S4.I11.i2.p1.6.m5.1.2.1" xref="S4.I11.i2.p1.6.m5.1.2.1.cmml">∈</mo><mrow id="S4.I11.i2.p1.6.m5.1.2.3" xref="S4.I11.i2.p1.6.m5.1.2.3.cmml"><mi id="S4.I11.i2.p1.6.m5.1.2.3.2" xref="S4.I11.i2.p1.6.m5.1.2.3.2.cmml">T</mi><mo id="S4.I11.i2.p1.6.m5.1.2.3.1" xref="S4.I11.i2.p1.6.m5.1.2.3.1.cmml">∖</mo><msub id="S4.I11.i2.p1.6.m5.1.2.3.3" xref="S4.I11.i2.p1.6.m5.1.2.3.3.cmml"><mi id="S4.I11.i2.p1.6.m5.1.2.3.3.2" xref="S4.I11.i2.p1.6.m5.1.2.3.3.2.cmml">T</mi><mi id="S4.I11.i2.p1.6.m5.1.2.3.3.3" xref="S4.I11.i2.p1.6.m5.1.2.3.3.3.cmml">x</mi></msub></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.I11.i2.p1.6.m5.1b"><apply id="S4.I11.i2.p1.6.m5.1.2.cmml" xref="S4.I11.i2.p1.6.m5.1.2"><in id="S4.I11.i2.p1.6.m5.1.2.1.cmml" xref="S4.I11.i2.p1.6.m5.1.2.1"></in><apply id="S4.I11.i2.p1.6.m5.1.2.2.cmml" xref="S4.I11.i2.p1.6.m5.1.2.2"><times id="S4.I11.i2.p1.6.m5.1.2.2.1.cmml" xref="S4.I11.i2.p1.6.m5.1.2.2.1"></times><ci id="S4.I11.i2.p1.6.m5.1.2.2.2.cmml" xref="S4.I11.i2.p1.6.m5.1.2.2.2">ℓ</ci><ci id="S4.I11.i2.p1.6.m5.1.1.cmml" xref="S4.I11.i2.p1.6.m5.1.1">𝑣</ci></apply><apply id="S4.I11.i2.p1.6.m5.1.2.3.cmml" xref="S4.I11.i2.p1.6.m5.1.2.3"><setdiff id="S4.I11.i2.p1.6.m5.1.2.3.1.cmml" xref="S4.I11.i2.p1.6.m5.1.2.3.1"></setdiff><ci id="S4.I11.i2.p1.6.m5.1.2.3.2.cmml" xref="S4.I11.i2.p1.6.m5.1.2.3.2">𝑇</ci><apply id="S4.I11.i2.p1.6.m5.1.2.3.3.cmml" xref="S4.I11.i2.p1.6.m5.1.2.3.3"><csymbol cd="ambiguous" id="S4.I11.i2.p1.6.m5.1.2.3.3.1.cmml" xref="S4.I11.i2.p1.6.m5.1.2.3.3">subscript</csymbol><ci id="S4.I11.i2.p1.6.m5.1.2.3.3.2.cmml" xref="S4.I11.i2.p1.6.m5.1.2.3.3.2">𝑇</ci><ci id="S4.I11.i2.p1.6.m5.1.2.3.3.3.cmml" xref="S4.I11.i2.p1.6.m5.1.2.3.3.3">𝑥</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.I11.i2.p1.6.m5.1c">\ell(v)\in T\setminus T_{x}</annotation><annotation encoding="application/x-llamapun" id="S4.I11.i2.p1.6.m5.1d">roman_ℓ ( italic_v ) ∈ italic_T ∖ italic_T start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT</annotation></semantics></math>. Note that this implies that <math alttext="v\notin G_{x}" class="ltx_Math" display="inline" id="S4.I11.i2.p1.7.m6.1"><semantics id="S4.I11.i2.p1.7.m6.1a"><mrow id="S4.I11.i2.p1.7.m6.1.1" xref="S4.I11.i2.p1.7.m6.1.1.cmml"><mi id="S4.I11.i2.p1.7.m6.1.1.2" xref="S4.I11.i2.p1.7.m6.1.1.2.cmml">v</mi><mo id="S4.I11.i2.p1.7.m6.1.1.1" xref="S4.I11.i2.p1.7.m6.1.1.1.cmml">∉</mo><msub id="S4.I11.i2.p1.7.m6.1.1.3" xref="S4.I11.i2.p1.7.m6.1.1.3.cmml"><mi id="S4.I11.i2.p1.7.m6.1.1.3.2" xref="S4.I11.i2.p1.7.m6.1.1.3.2.cmml">G</mi><mi id="S4.I11.i2.p1.7.m6.1.1.3.3" xref="S4.I11.i2.p1.7.m6.1.1.3.3.cmml">x</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S4.I11.i2.p1.7.m6.1b"><apply id="S4.I11.i2.p1.7.m6.1.1.cmml" xref="S4.I11.i2.p1.7.m6.1.1"><notin id="S4.I11.i2.p1.7.m6.1.1.1.cmml" xref="S4.I11.i2.p1.7.m6.1.1.1"></notin><ci id="S4.I11.i2.p1.7.m6.1.1.2.cmml" xref="S4.I11.i2.p1.7.m6.1.1.2">𝑣</ci><apply id="S4.I11.i2.p1.7.m6.1.1.3.cmml" xref="S4.I11.i2.p1.7.m6.1.1.3"><csymbol cd="ambiguous" id="S4.I11.i2.p1.7.m6.1.1.3.1.cmml" xref="S4.I11.i2.p1.7.m6.1.1.3">subscript</csymbol><ci id="S4.I11.i2.p1.7.m6.1.1.3.2.cmml" xref="S4.I11.i2.p1.7.m6.1.1.3.2">𝐺</ci><ci id="S4.I11.i2.p1.7.m6.1.1.3.3.cmml" xref="S4.I11.i2.p1.7.m6.1.1.3.3">𝑥</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.I11.i2.p1.7.m6.1c">v\notin G_{x}</annotation><annotation encoding="application/x-llamapun" id="S4.I11.i2.p1.7.m6.1d">italic_v ∉ italic_G start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="f(v)" class="ltx_Math" display="inline" id="S4.I11.i2.p1.8.m7.1"><semantics id="S4.I11.i2.p1.8.m7.1a"><mrow id="S4.I11.i2.p1.8.m7.1.2" xref="S4.I11.i2.p1.8.m7.1.2.cmml"><mi id="S4.I11.i2.p1.8.m7.1.2.2" xref="S4.I11.i2.p1.8.m7.1.2.2.cmml">f</mi><mo id="S4.I11.i2.p1.8.m7.1.2.1" xref="S4.I11.i2.p1.8.m7.1.2.1.cmml">⁢</mo><mrow id="S4.I11.i2.p1.8.m7.1.2.3.2" xref="S4.I11.i2.p1.8.m7.1.2.cmml"><mo id="S4.I11.i2.p1.8.m7.1.2.3.2.1" stretchy="false" xref="S4.I11.i2.p1.8.m7.1.2.cmml">(</mo><mi id="S4.I11.i2.p1.8.m7.1.1" xref="S4.I11.i2.p1.8.m7.1.1.cmml">v</mi><mo id="S4.I11.i2.p1.8.m7.1.2.3.2.2" stretchy="false" xref="S4.I11.i2.p1.8.m7.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.I11.i2.p1.8.m7.1b"><apply id="S4.I11.i2.p1.8.m7.1.2.cmml" xref="S4.I11.i2.p1.8.m7.1.2"><times id="S4.I11.i2.p1.8.m7.1.2.1.cmml" xref="S4.I11.i2.p1.8.m7.1.2.1"></times><ci id="S4.I11.i2.p1.8.m7.1.2.2.cmml" xref="S4.I11.i2.p1.8.m7.1.2.2">𝑓</ci><ci id="S4.I11.i2.p1.8.m7.1.1.cmml" xref="S4.I11.i2.p1.8.m7.1.1">𝑣</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.I11.i2.p1.8.m7.1c">f(v)</annotation><annotation encoding="application/x-llamapun" id="S4.I11.i2.p1.8.m7.1d">italic_f ( italic_v )</annotation></semantics></math> is the dummy node corresponding to the parent edge of <math alttext="x" class="ltx_Math" display="inline" id="S4.I11.i2.p1.9.m8.1"><semantics id="S4.I11.i2.p1.9.m8.1a"><mi id="S4.I11.i2.p1.9.m8.1.1" xref="S4.I11.i2.p1.9.m8.1.1.cmml">x</mi><annotation-xml encoding="MathML-Content" id="S4.I11.i2.p1.9.m8.1b"><ci id="S4.I11.i2.p1.9.m8.1.1.cmml" xref="S4.I11.i2.p1.9.m8.1.1">𝑥</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.I11.i2.p1.9.m8.1c">x</annotation><annotation encoding="application/x-llamapun" id="S4.I11.i2.p1.9.m8.1d">italic_x</annotation></semantics></math>.</p> <ul class="ltx_itemize" id="S4.I11.i2.I1"> <li class="ltx_item" id="S4.I11.i2.I1.i1" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item"><span class="ltx_text ltx_font_bold" id="S4.I11.i2.I1.i1.1.1.1">–</span></span> <div class="ltx_para" id="S4.I11.i2.I1.i1.p1"> <p class="ltx_p" id="S4.I11.i2.I1.i1.p1.17">Suppose <math alttext="u\in G_{x}\setminus\textnormal{parent}(x)" class="ltx_Math" display="inline" id="S4.I11.i2.I1.i1.p1.1.m1.1"><semantics id="S4.I11.i2.I1.i1.p1.1.m1.1a"><mrow id="S4.I11.i2.I1.i1.p1.1.m1.1.2" xref="S4.I11.i2.I1.i1.p1.1.m1.1.2.cmml"><mi id="S4.I11.i2.I1.i1.p1.1.m1.1.2.2" xref="S4.I11.i2.I1.i1.p1.1.m1.1.2.2.cmml">u</mi><mo id="S4.I11.i2.I1.i1.p1.1.m1.1.2.1" xref="S4.I11.i2.I1.i1.p1.1.m1.1.2.1.cmml">∈</mo><mrow id="S4.I11.i2.I1.i1.p1.1.m1.1.2.3" xref="S4.I11.i2.I1.i1.p1.1.m1.1.2.3.cmml"><msub id="S4.I11.i2.I1.i1.p1.1.m1.1.2.3.2" xref="S4.I11.i2.I1.i1.p1.1.m1.1.2.3.2.cmml"><mi id="S4.I11.i2.I1.i1.p1.1.m1.1.2.3.2.2" xref="S4.I11.i2.I1.i1.p1.1.m1.1.2.3.2.2.cmml">G</mi><mi id="S4.I11.i2.I1.i1.p1.1.m1.1.2.3.2.3" xref="S4.I11.i2.I1.i1.p1.1.m1.1.2.3.2.3.cmml">x</mi></msub><mo id="S4.I11.i2.I1.i1.p1.1.m1.1.2.3.1" xref="S4.I11.i2.I1.i1.p1.1.m1.1.2.3.1.cmml">∖</mo><mrow id="S4.I11.i2.I1.i1.p1.1.m1.1.2.3.3" xref="S4.I11.i2.I1.i1.p1.1.m1.1.2.3.3.cmml"><mtext id="S4.I11.i2.I1.i1.p1.1.m1.1.2.3.3.2" xref="S4.I11.i2.I1.i1.p1.1.m1.1.2.3.3.2a.cmml">parent</mtext><mo id="S4.I11.i2.I1.i1.p1.1.m1.1.2.3.3.1" xref="S4.I11.i2.I1.i1.p1.1.m1.1.2.3.3.1.cmml">⁢</mo><mrow id="S4.I11.i2.I1.i1.p1.1.m1.1.2.3.3.3.2" xref="S4.I11.i2.I1.i1.p1.1.m1.1.2.3.3.cmml"><mo id="S4.I11.i2.I1.i1.p1.1.m1.1.2.3.3.3.2.1" stretchy="false" xref="S4.I11.i2.I1.i1.p1.1.m1.1.2.3.3.cmml">(</mo><mi id="S4.I11.i2.I1.i1.p1.1.m1.1.1" xref="S4.I11.i2.I1.i1.p1.1.m1.1.1.cmml">x</mi><mo id="S4.I11.i2.I1.i1.p1.1.m1.1.2.3.3.3.2.2" stretchy="false" xref="S4.I11.i2.I1.i1.p1.1.m1.1.2.3.3.cmml">)</mo></mrow></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.I11.i2.I1.i1.p1.1.m1.1b"><apply id="S4.I11.i2.I1.i1.p1.1.m1.1.2.cmml" xref="S4.I11.i2.I1.i1.p1.1.m1.1.2"><in id="S4.I11.i2.I1.i1.p1.1.m1.1.2.1.cmml" xref="S4.I11.i2.I1.i1.p1.1.m1.1.2.1"></in><ci id="S4.I11.i2.I1.i1.p1.1.m1.1.2.2.cmml" xref="S4.I11.i2.I1.i1.p1.1.m1.1.2.2">𝑢</ci><apply id="S4.I11.i2.I1.i1.p1.1.m1.1.2.3.cmml" xref="S4.I11.i2.I1.i1.p1.1.m1.1.2.3"><setdiff id="S4.I11.i2.I1.i1.p1.1.m1.1.2.3.1.cmml" xref="S4.I11.i2.I1.i1.p1.1.m1.1.2.3.1"></setdiff><apply id="S4.I11.i2.I1.i1.p1.1.m1.1.2.3.2.cmml" xref="S4.I11.i2.I1.i1.p1.1.m1.1.2.3.2"><csymbol cd="ambiguous" id="S4.I11.i2.I1.i1.p1.1.m1.1.2.3.2.1.cmml" xref="S4.I11.i2.I1.i1.p1.1.m1.1.2.3.2">subscript</csymbol><ci id="S4.I11.i2.I1.i1.p1.1.m1.1.2.3.2.2.cmml" xref="S4.I11.i2.I1.i1.p1.1.m1.1.2.3.2.2">𝐺</ci><ci id="S4.I11.i2.I1.i1.p1.1.m1.1.2.3.2.3.cmml" xref="S4.I11.i2.I1.i1.p1.1.m1.1.2.3.2.3">𝑥</ci></apply><apply id="S4.I11.i2.I1.i1.p1.1.m1.1.2.3.3.cmml" xref="S4.I11.i2.I1.i1.p1.1.m1.1.2.3.3"><times id="S4.I11.i2.I1.i1.p1.1.m1.1.2.3.3.1.cmml" xref="S4.I11.i2.I1.i1.p1.1.m1.1.2.3.3.1"></times><ci id="S4.I11.i2.I1.i1.p1.1.m1.1.2.3.3.2a.cmml" xref="S4.I11.i2.I1.i1.p1.1.m1.1.2.3.3.2"><mtext id="S4.I11.i2.I1.i1.p1.1.m1.1.2.3.3.2.cmml" xref="S4.I11.i2.I1.i1.p1.1.m1.1.2.3.3.2">parent</mtext></ci><ci id="S4.I11.i2.I1.i1.p1.1.m1.1.1.cmml" xref="S4.I11.i2.I1.i1.p1.1.m1.1.1">𝑥</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.I11.i2.I1.i1.p1.1.m1.1c">u\in G_{x}\setminus\textnormal{parent}(x)</annotation><annotation encoding="application/x-llamapun" id="S4.I11.i2.I1.i1.p1.1.m1.1d">italic_u ∈ italic_G start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT ∖ parent ( italic_x )</annotation></semantics></math>. Then <math alttext="\textsc{Min}_{u}(j)=(u,v^{\prime})" class="ltx_Math" display="inline" id="S4.I11.i2.I1.i1.p1.2.m2.3"><semantics id="S4.I11.i2.I1.i1.p1.2.m2.3a"><mrow id="S4.I11.i2.I1.i1.p1.2.m2.3.3" xref="S4.I11.i2.I1.i1.p1.2.m2.3.3.cmml"><mrow id="S4.I11.i2.I1.i1.p1.2.m2.3.3.3" xref="S4.I11.i2.I1.i1.p1.2.m2.3.3.3.cmml"><msub id="S4.I11.i2.I1.i1.p1.2.m2.3.3.3.2" xref="S4.I11.i2.I1.i1.p1.2.m2.3.3.3.2.cmml"><mtext class="ltx_font_smallcaps" id="S4.I11.i2.I1.i1.p1.2.m2.3.3.3.2.2" xref="S4.I11.i2.I1.i1.p1.2.m2.3.3.3.2.2a.cmml">Min</mtext><mi id="S4.I11.i2.I1.i1.p1.2.m2.3.3.3.2.3" xref="S4.I11.i2.I1.i1.p1.2.m2.3.3.3.2.3.cmml">u</mi></msub><mo id="S4.I11.i2.I1.i1.p1.2.m2.3.3.3.1" xref="S4.I11.i2.I1.i1.p1.2.m2.3.3.3.1.cmml">⁢</mo><mrow id="S4.I11.i2.I1.i1.p1.2.m2.3.3.3.3.2" xref="S4.I11.i2.I1.i1.p1.2.m2.3.3.3.cmml"><mo id="S4.I11.i2.I1.i1.p1.2.m2.3.3.3.3.2.1" stretchy="false" xref="S4.I11.i2.I1.i1.p1.2.m2.3.3.3.cmml">(</mo><mi id="S4.I11.i2.I1.i1.p1.2.m2.1.1" xref="S4.I11.i2.I1.i1.p1.2.m2.1.1.cmml">j</mi><mo id="S4.I11.i2.I1.i1.p1.2.m2.3.3.3.3.2.2" stretchy="false" xref="S4.I11.i2.I1.i1.p1.2.m2.3.3.3.cmml">)</mo></mrow></mrow><mo id="S4.I11.i2.I1.i1.p1.2.m2.3.3.2" xref="S4.I11.i2.I1.i1.p1.2.m2.3.3.2.cmml">=</mo><mrow id="S4.I11.i2.I1.i1.p1.2.m2.3.3.1.1" xref="S4.I11.i2.I1.i1.p1.2.m2.3.3.1.2.cmml"><mo id="S4.I11.i2.I1.i1.p1.2.m2.3.3.1.1.2" stretchy="false" xref="S4.I11.i2.I1.i1.p1.2.m2.3.3.1.2.cmml">(</mo><mi id="S4.I11.i2.I1.i1.p1.2.m2.2.2" xref="S4.I11.i2.I1.i1.p1.2.m2.2.2.cmml">u</mi><mo id="S4.I11.i2.I1.i1.p1.2.m2.3.3.1.1.3" xref="S4.I11.i2.I1.i1.p1.2.m2.3.3.1.2.cmml">,</mo><msup id="S4.I11.i2.I1.i1.p1.2.m2.3.3.1.1.1" xref="S4.I11.i2.I1.i1.p1.2.m2.3.3.1.1.1.cmml"><mi id="S4.I11.i2.I1.i1.p1.2.m2.3.3.1.1.1.2" xref="S4.I11.i2.I1.i1.p1.2.m2.3.3.1.1.1.2.cmml">v</mi><mo id="S4.I11.i2.I1.i1.p1.2.m2.3.3.1.1.1.3" xref="S4.I11.i2.I1.i1.p1.2.m2.3.3.1.1.1.3.cmml">′</mo></msup><mo id="S4.I11.i2.I1.i1.p1.2.m2.3.3.1.1.4" stretchy="false" xref="S4.I11.i2.I1.i1.p1.2.m2.3.3.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.I11.i2.I1.i1.p1.2.m2.3b"><apply id="S4.I11.i2.I1.i1.p1.2.m2.3.3.cmml" xref="S4.I11.i2.I1.i1.p1.2.m2.3.3"><eq id="S4.I11.i2.I1.i1.p1.2.m2.3.3.2.cmml" xref="S4.I11.i2.I1.i1.p1.2.m2.3.3.2"></eq><apply id="S4.I11.i2.I1.i1.p1.2.m2.3.3.3.cmml" xref="S4.I11.i2.I1.i1.p1.2.m2.3.3.3"><times id="S4.I11.i2.I1.i1.p1.2.m2.3.3.3.1.cmml" xref="S4.I11.i2.I1.i1.p1.2.m2.3.3.3.1"></times><apply id="S4.I11.i2.I1.i1.p1.2.m2.3.3.3.2.cmml" xref="S4.I11.i2.I1.i1.p1.2.m2.3.3.3.2"><csymbol cd="ambiguous" id="S4.I11.i2.I1.i1.p1.2.m2.3.3.3.2.1.cmml" xref="S4.I11.i2.I1.i1.p1.2.m2.3.3.3.2">subscript</csymbol><ci id="S4.I11.i2.I1.i1.p1.2.m2.3.3.3.2.2a.cmml" xref="S4.I11.i2.I1.i1.p1.2.m2.3.3.3.2.2"><mtext class="ltx_font_smallcaps" id="S4.I11.i2.I1.i1.p1.2.m2.3.3.3.2.2.cmml" xref="S4.I11.i2.I1.i1.p1.2.m2.3.3.3.2.2">Min</mtext></ci><ci id="S4.I11.i2.I1.i1.p1.2.m2.3.3.3.2.3.cmml" xref="S4.I11.i2.I1.i1.p1.2.m2.3.3.3.2.3">𝑢</ci></apply><ci id="S4.I11.i2.I1.i1.p1.2.m2.1.1.cmml" xref="S4.I11.i2.I1.i1.p1.2.m2.1.1">𝑗</ci></apply><interval closure="open" id="S4.I11.i2.I1.i1.p1.2.m2.3.3.1.2.cmml" xref="S4.I11.i2.I1.i1.p1.2.m2.3.3.1.1"><ci id="S4.I11.i2.I1.i1.p1.2.m2.2.2.cmml" xref="S4.I11.i2.I1.i1.p1.2.m2.2.2">𝑢</ci><apply id="S4.I11.i2.I1.i1.p1.2.m2.3.3.1.1.1.cmml" xref="S4.I11.i2.I1.i1.p1.2.m2.3.3.1.1.1"><csymbol cd="ambiguous" id="S4.I11.i2.I1.i1.p1.2.m2.3.3.1.1.1.1.cmml" xref="S4.I11.i2.I1.i1.p1.2.m2.3.3.1.1.1">superscript</csymbol><ci id="S4.I11.i2.I1.i1.p1.2.m2.3.3.1.1.1.2.cmml" xref="S4.I11.i2.I1.i1.p1.2.m2.3.3.1.1.1.2">𝑣</ci><ci id="S4.I11.i2.I1.i1.p1.2.m2.3.3.1.1.1.3.cmml" xref="S4.I11.i2.I1.i1.p1.2.m2.3.3.1.1.1.3">′</ci></apply></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.I11.i2.I1.i1.p1.2.m2.3c">\textsc{Min}_{u}(j)=(u,v^{\prime})</annotation><annotation encoding="application/x-llamapun" id="S4.I11.i2.I1.i1.p1.2.m2.3d">Min start_POSTSUBSCRIPT italic_u end_POSTSUBSCRIPT ( italic_j ) = ( italic_u , italic_v start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT )</annotation></semantics></math> where <math alttext="f(v^{\prime})=f(v)" class="ltx_Math" display="inline" id="S4.I11.i2.I1.i1.p1.3.m3.2"><semantics id="S4.I11.i2.I1.i1.p1.3.m3.2a"><mrow id="S4.I11.i2.I1.i1.p1.3.m3.2.2" xref="S4.I11.i2.I1.i1.p1.3.m3.2.2.cmml"><mrow id="S4.I11.i2.I1.i1.p1.3.m3.2.2.1" xref="S4.I11.i2.I1.i1.p1.3.m3.2.2.1.cmml"><mi id="S4.I11.i2.I1.i1.p1.3.m3.2.2.1.3" xref="S4.I11.i2.I1.i1.p1.3.m3.2.2.1.3.cmml">f</mi><mo id="S4.I11.i2.I1.i1.p1.3.m3.2.2.1.2" xref="S4.I11.i2.I1.i1.p1.3.m3.2.2.1.2.cmml">⁢</mo><mrow id="S4.I11.i2.I1.i1.p1.3.m3.2.2.1.1.1" xref="S4.I11.i2.I1.i1.p1.3.m3.2.2.1.1.1.1.cmml"><mo id="S4.I11.i2.I1.i1.p1.3.m3.2.2.1.1.1.2" stretchy="false" xref="S4.I11.i2.I1.i1.p1.3.m3.2.2.1.1.1.1.cmml">(</mo><msup id="S4.I11.i2.I1.i1.p1.3.m3.2.2.1.1.1.1" xref="S4.I11.i2.I1.i1.p1.3.m3.2.2.1.1.1.1.cmml"><mi id="S4.I11.i2.I1.i1.p1.3.m3.2.2.1.1.1.1.2" xref="S4.I11.i2.I1.i1.p1.3.m3.2.2.1.1.1.1.2.cmml">v</mi><mo id="S4.I11.i2.I1.i1.p1.3.m3.2.2.1.1.1.1.3" xref="S4.I11.i2.I1.i1.p1.3.m3.2.2.1.1.1.1.3.cmml">′</mo></msup><mo id="S4.I11.i2.I1.i1.p1.3.m3.2.2.1.1.1.3" stretchy="false" xref="S4.I11.i2.I1.i1.p1.3.m3.2.2.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S4.I11.i2.I1.i1.p1.3.m3.2.2.2" xref="S4.I11.i2.I1.i1.p1.3.m3.2.2.2.cmml">=</mo><mrow id="S4.I11.i2.I1.i1.p1.3.m3.2.2.3" xref="S4.I11.i2.I1.i1.p1.3.m3.2.2.3.cmml"><mi id="S4.I11.i2.I1.i1.p1.3.m3.2.2.3.2" xref="S4.I11.i2.I1.i1.p1.3.m3.2.2.3.2.cmml">f</mi><mo id="S4.I11.i2.I1.i1.p1.3.m3.2.2.3.1" xref="S4.I11.i2.I1.i1.p1.3.m3.2.2.3.1.cmml">⁢</mo><mrow id="S4.I11.i2.I1.i1.p1.3.m3.2.2.3.3.2" xref="S4.I11.i2.I1.i1.p1.3.m3.2.2.3.cmml"><mo id="S4.I11.i2.I1.i1.p1.3.m3.2.2.3.3.2.1" stretchy="false" xref="S4.I11.i2.I1.i1.p1.3.m3.2.2.3.cmml">(</mo><mi id="S4.I11.i2.I1.i1.p1.3.m3.1.1" xref="S4.I11.i2.I1.i1.p1.3.m3.1.1.cmml">v</mi><mo id="S4.I11.i2.I1.i1.p1.3.m3.2.2.3.3.2.2" stretchy="false" xref="S4.I11.i2.I1.i1.p1.3.m3.2.2.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.I11.i2.I1.i1.p1.3.m3.2b"><apply id="S4.I11.i2.I1.i1.p1.3.m3.2.2.cmml" xref="S4.I11.i2.I1.i1.p1.3.m3.2.2"><eq id="S4.I11.i2.I1.i1.p1.3.m3.2.2.2.cmml" xref="S4.I11.i2.I1.i1.p1.3.m3.2.2.2"></eq><apply id="S4.I11.i2.I1.i1.p1.3.m3.2.2.1.cmml" xref="S4.I11.i2.I1.i1.p1.3.m3.2.2.1"><times id="S4.I11.i2.I1.i1.p1.3.m3.2.2.1.2.cmml" xref="S4.I11.i2.I1.i1.p1.3.m3.2.2.1.2"></times><ci id="S4.I11.i2.I1.i1.p1.3.m3.2.2.1.3.cmml" xref="S4.I11.i2.I1.i1.p1.3.m3.2.2.1.3">𝑓</ci><apply id="S4.I11.i2.I1.i1.p1.3.m3.2.2.1.1.1.1.cmml" xref="S4.I11.i2.I1.i1.p1.3.m3.2.2.1.1.1"><csymbol cd="ambiguous" id="S4.I11.i2.I1.i1.p1.3.m3.2.2.1.1.1.1.1.cmml" xref="S4.I11.i2.I1.i1.p1.3.m3.2.2.1.1.1">superscript</csymbol><ci id="S4.I11.i2.I1.i1.p1.3.m3.2.2.1.1.1.1.2.cmml" xref="S4.I11.i2.I1.i1.p1.3.m3.2.2.1.1.1.1.2">𝑣</ci><ci id="S4.I11.i2.I1.i1.p1.3.m3.2.2.1.1.1.1.3.cmml" xref="S4.I11.i2.I1.i1.p1.3.m3.2.2.1.1.1.1.3">′</ci></apply></apply><apply id="S4.I11.i2.I1.i1.p1.3.m3.2.2.3.cmml" xref="S4.I11.i2.I1.i1.p1.3.m3.2.2.3"><times id="S4.I11.i2.I1.i1.p1.3.m3.2.2.3.1.cmml" xref="S4.I11.i2.I1.i1.p1.3.m3.2.2.3.1"></times><ci id="S4.I11.i2.I1.i1.p1.3.m3.2.2.3.2.cmml" xref="S4.I11.i2.I1.i1.p1.3.m3.2.2.3.2">𝑓</ci><ci id="S4.I11.i2.I1.i1.p1.3.m3.1.1.cmml" xref="S4.I11.i2.I1.i1.p1.3.m3.1.1">𝑣</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.I11.i2.I1.i1.p1.3.m3.2c">f(v^{\prime})=f(v)</annotation><annotation encoding="application/x-llamapun" id="S4.I11.i2.I1.i1.p1.3.m3.2d">italic_f ( italic_v start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) = italic_f ( italic_v )</annotation></semantics></math>, so <math alttext="\ell(v^{\prime})\in T\setminus T_{x}" class="ltx_Math" display="inline" id="S4.I11.i2.I1.i1.p1.4.m4.1"><semantics id="S4.I11.i2.I1.i1.p1.4.m4.1a"><mrow id="S4.I11.i2.I1.i1.p1.4.m4.1.1" xref="S4.I11.i2.I1.i1.p1.4.m4.1.1.cmml"><mrow id="S4.I11.i2.I1.i1.p1.4.m4.1.1.1" xref="S4.I11.i2.I1.i1.p1.4.m4.1.1.1.cmml"><mi id="S4.I11.i2.I1.i1.p1.4.m4.1.1.1.3" mathvariant="normal" xref="S4.I11.i2.I1.i1.p1.4.m4.1.1.1.3.cmml">ℓ</mi><mo id="S4.I11.i2.I1.i1.p1.4.m4.1.1.1.2" xref="S4.I11.i2.I1.i1.p1.4.m4.1.1.1.2.cmml">⁢</mo><mrow id="S4.I11.i2.I1.i1.p1.4.m4.1.1.1.1.1" xref="S4.I11.i2.I1.i1.p1.4.m4.1.1.1.1.1.1.cmml"><mo id="S4.I11.i2.I1.i1.p1.4.m4.1.1.1.1.1.2" stretchy="false" xref="S4.I11.i2.I1.i1.p1.4.m4.1.1.1.1.1.1.cmml">(</mo><msup id="S4.I11.i2.I1.i1.p1.4.m4.1.1.1.1.1.1" xref="S4.I11.i2.I1.i1.p1.4.m4.1.1.1.1.1.1.cmml"><mi id="S4.I11.i2.I1.i1.p1.4.m4.1.1.1.1.1.1.2" xref="S4.I11.i2.I1.i1.p1.4.m4.1.1.1.1.1.1.2.cmml">v</mi><mo id="S4.I11.i2.I1.i1.p1.4.m4.1.1.1.1.1.1.3" xref="S4.I11.i2.I1.i1.p1.4.m4.1.1.1.1.1.1.3.cmml">′</mo></msup><mo id="S4.I11.i2.I1.i1.p1.4.m4.1.1.1.1.1.3" stretchy="false" xref="S4.I11.i2.I1.i1.p1.4.m4.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S4.I11.i2.I1.i1.p1.4.m4.1.1.2" xref="S4.I11.i2.I1.i1.p1.4.m4.1.1.2.cmml">∈</mo><mrow id="S4.I11.i2.I1.i1.p1.4.m4.1.1.3" xref="S4.I11.i2.I1.i1.p1.4.m4.1.1.3.cmml"><mi id="S4.I11.i2.I1.i1.p1.4.m4.1.1.3.2" xref="S4.I11.i2.I1.i1.p1.4.m4.1.1.3.2.cmml">T</mi><mo id="S4.I11.i2.I1.i1.p1.4.m4.1.1.3.1" xref="S4.I11.i2.I1.i1.p1.4.m4.1.1.3.1.cmml">∖</mo><msub id="S4.I11.i2.I1.i1.p1.4.m4.1.1.3.3" xref="S4.I11.i2.I1.i1.p1.4.m4.1.1.3.3.cmml"><mi id="S4.I11.i2.I1.i1.p1.4.m4.1.1.3.3.2" xref="S4.I11.i2.I1.i1.p1.4.m4.1.1.3.3.2.cmml">T</mi><mi id="S4.I11.i2.I1.i1.p1.4.m4.1.1.3.3.3" xref="S4.I11.i2.I1.i1.p1.4.m4.1.1.3.3.3.cmml">x</mi></msub></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.I11.i2.I1.i1.p1.4.m4.1b"><apply id="S4.I11.i2.I1.i1.p1.4.m4.1.1.cmml" xref="S4.I11.i2.I1.i1.p1.4.m4.1.1"><in id="S4.I11.i2.I1.i1.p1.4.m4.1.1.2.cmml" xref="S4.I11.i2.I1.i1.p1.4.m4.1.1.2"></in><apply id="S4.I11.i2.I1.i1.p1.4.m4.1.1.1.cmml" xref="S4.I11.i2.I1.i1.p1.4.m4.1.1.1"><times id="S4.I11.i2.I1.i1.p1.4.m4.1.1.1.2.cmml" xref="S4.I11.i2.I1.i1.p1.4.m4.1.1.1.2"></times><ci id="S4.I11.i2.I1.i1.p1.4.m4.1.1.1.3.cmml" xref="S4.I11.i2.I1.i1.p1.4.m4.1.1.1.3">ℓ</ci><apply id="S4.I11.i2.I1.i1.p1.4.m4.1.1.1.1.1.1.cmml" xref="S4.I11.i2.I1.i1.p1.4.m4.1.1.1.1.1"><csymbol cd="ambiguous" id="S4.I11.i2.I1.i1.p1.4.m4.1.1.1.1.1.1.1.cmml" xref="S4.I11.i2.I1.i1.p1.4.m4.1.1.1.1.1">superscript</csymbol><ci id="S4.I11.i2.I1.i1.p1.4.m4.1.1.1.1.1.1.2.cmml" xref="S4.I11.i2.I1.i1.p1.4.m4.1.1.1.1.1.1.2">𝑣</ci><ci id="S4.I11.i2.I1.i1.p1.4.m4.1.1.1.1.1.1.3.cmml" xref="S4.I11.i2.I1.i1.p1.4.m4.1.1.1.1.1.1.3">′</ci></apply></apply><apply id="S4.I11.i2.I1.i1.p1.4.m4.1.1.3.cmml" xref="S4.I11.i2.I1.i1.p1.4.m4.1.1.3"><setdiff id="S4.I11.i2.I1.i1.p1.4.m4.1.1.3.1.cmml" xref="S4.I11.i2.I1.i1.p1.4.m4.1.1.3.1"></setdiff><ci id="S4.I11.i2.I1.i1.p1.4.m4.1.1.3.2.cmml" xref="S4.I11.i2.I1.i1.p1.4.m4.1.1.3.2">𝑇</ci><apply id="S4.I11.i2.I1.i1.p1.4.m4.1.1.3.3.cmml" xref="S4.I11.i2.I1.i1.p1.4.m4.1.1.3.3"><csymbol cd="ambiguous" id="S4.I11.i2.I1.i1.p1.4.m4.1.1.3.3.1.cmml" xref="S4.I11.i2.I1.i1.p1.4.m4.1.1.3.3">subscript</csymbol><ci id="S4.I11.i2.I1.i1.p1.4.m4.1.1.3.3.2.cmml" xref="S4.I11.i2.I1.i1.p1.4.m4.1.1.3.3.2">𝑇</ci><ci id="S4.I11.i2.I1.i1.p1.4.m4.1.1.3.3.3.cmml" xref="S4.I11.i2.I1.i1.p1.4.m4.1.1.3.3.3">𝑥</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.I11.i2.I1.i1.p1.4.m4.1c">\ell(v^{\prime})\in T\setminus T_{x}</annotation><annotation encoding="application/x-llamapun" id="S4.I11.i2.I1.i1.p1.4.m4.1d">roman_ℓ ( italic_v start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) ∈ italic_T ∖ italic_T start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT</annotation></semantics></math>. It is easy to see that <math alttext="T\setminus T_{x}" class="ltx_Math" display="inline" id="S4.I11.i2.I1.i1.p1.5.m5.1"><semantics id="S4.I11.i2.I1.i1.p1.5.m5.1a"><mrow id="S4.I11.i2.I1.i1.p1.5.m5.1.1" xref="S4.I11.i2.I1.i1.p1.5.m5.1.1.cmml"><mi id="S4.I11.i2.I1.i1.p1.5.m5.1.1.2" xref="S4.I11.i2.I1.i1.p1.5.m5.1.1.2.cmml">T</mi><mo id="S4.I11.i2.I1.i1.p1.5.m5.1.1.1" xref="S4.I11.i2.I1.i1.p1.5.m5.1.1.1.cmml">∖</mo><msub id="S4.I11.i2.I1.i1.p1.5.m5.1.1.3" xref="S4.I11.i2.I1.i1.p1.5.m5.1.1.3.cmml"><mi id="S4.I11.i2.I1.i1.p1.5.m5.1.1.3.2" xref="S4.I11.i2.I1.i1.p1.5.m5.1.1.3.2.cmml">T</mi><mi id="S4.I11.i2.I1.i1.p1.5.m5.1.1.3.3" xref="S4.I11.i2.I1.i1.p1.5.m5.1.1.3.3.cmml">x</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S4.I11.i2.I1.i1.p1.5.m5.1b"><apply id="S4.I11.i2.I1.i1.p1.5.m5.1.1.cmml" xref="S4.I11.i2.I1.i1.p1.5.m5.1.1"><setdiff id="S4.I11.i2.I1.i1.p1.5.m5.1.1.1.cmml" xref="S4.I11.i2.I1.i1.p1.5.m5.1.1.1"></setdiff><ci id="S4.I11.i2.I1.i1.p1.5.m5.1.1.2.cmml" xref="S4.I11.i2.I1.i1.p1.5.m5.1.1.2">𝑇</ci><apply id="S4.I11.i2.I1.i1.p1.5.m5.1.1.3.cmml" xref="S4.I11.i2.I1.i1.p1.5.m5.1.1.3"><csymbol cd="ambiguous" id="S4.I11.i2.I1.i1.p1.5.m5.1.1.3.1.cmml" xref="S4.I11.i2.I1.i1.p1.5.m5.1.1.3">subscript</csymbol><ci id="S4.I11.i2.I1.i1.p1.5.m5.1.1.3.2.cmml" xref="S4.I11.i2.I1.i1.p1.5.m5.1.1.3.2">𝑇</ci><ci id="S4.I11.i2.I1.i1.p1.5.m5.1.1.3.3.cmml" xref="S4.I11.i2.I1.i1.p1.5.m5.1.1.3.3">𝑥</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.I11.i2.I1.i1.p1.5.m5.1c">T\setminus T_{x}</annotation><annotation encoding="application/x-llamapun" id="S4.I11.i2.I1.i1.p1.5.m5.1d">italic_T ∖ italic_T start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT</annotation></semantics></math> remains connected despite the removal of <math alttext="\{a,b\}" class="ltx_Math" display="inline" id="S4.I11.i2.I1.i1.p1.6.m6.2"><semantics id="S4.I11.i2.I1.i1.p1.6.m6.2a"><mrow id="S4.I11.i2.I1.i1.p1.6.m6.2.3.2" xref="S4.I11.i2.I1.i1.p1.6.m6.2.3.1.cmml"><mo id="S4.I11.i2.I1.i1.p1.6.m6.2.3.2.1" stretchy="false" xref="S4.I11.i2.I1.i1.p1.6.m6.2.3.1.cmml">{</mo><mi id="S4.I11.i2.I1.i1.p1.6.m6.1.1" xref="S4.I11.i2.I1.i1.p1.6.m6.1.1.cmml">a</mi><mo id="S4.I11.i2.I1.i1.p1.6.m6.2.3.2.2" xref="S4.I11.i2.I1.i1.p1.6.m6.2.3.1.cmml">,</mo><mi id="S4.I11.i2.I1.i1.p1.6.m6.2.2" xref="S4.I11.i2.I1.i1.p1.6.m6.2.2.cmml">b</mi><mo id="S4.I11.i2.I1.i1.p1.6.m6.2.3.2.3" stretchy="false" xref="S4.I11.i2.I1.i1.p1.6.m6.2.3.1.cmml">}</mo></mrow><annotation-xml encoding="MathML-Content" id="S4.I11.i2.I1.i1.p1.6.m6.2b"><set id="S4.I11.i2.I1.i1.p1.6.m6.2.3.1.cmml" xref="S4.I11.i2.I1.i1.p1.6.m6.2.3.2"><ci id="S4.I11.i2.I1.i1.p1.6.m6.1.1.cmml" xref="S4.I11.i2.I1.i1.p1.6.m6.1.1">𝑎</ci><ci id="S4.I11.i2.I1.i1.p1.6.m6.2.2.cmml" xref="S4.I11.i2.I1.i1.p1.6.m6.2.2">𝑏</ci></set></annotation-xml><annotation encoding="application/x-tex" id="S4.I11.i2.I1.i1.p1.6.m6.2c">\{a,b\}</annotation><annotation encoding="application/x-llamapun" id="S4.I11.i2.I1.i1.p1.6.m6.2d">{ italic_a , italic_b }</annotation></semantics></math>. Furthermore, <math alttext="(u,v^{\prime})\in\textnormal{SOL}" class="ltx_Math" display="inline" id="S4.I11.i2.I1.i1.p1.7.m7.2"><semantics id="S4.I11.i2.I1.i1.p1.7.m7.2a"><mrow id="S4.I11.i2.I1.i1.p1.7.m7.2.2" xref="S4.I11.i2.I1.i1.p1.7.m7.2.2.cmml"><mrow id="S4.I11.i2.I1.i1.p1.7.m7.2.2.1.1" xref="S4.I11.i2.I1.i1.p1.7.m7.2.2.1.2.cmml"><mo id="S4.I11.i2.I1.i1.p1.7.m7.2.2.1.1.2" stretchy="false" xref="S4.I11.i2.I1.i1.p1.7.m7.2.2.1.2.cmml">(</mo><mi id="S4.I11.i2.I1.i1.p1.7.m7.1.1" xref="S4.I11.i2.I1.i1.p1.7.m7.1.1.cmml">u</mi><mo id="S4.I11.i2.I1.i1.p1.7.m7.2.2.1.1.3" xref="S4.I11.i2.I1.i1.p1.7.m7.2.2.1.2.cmml">,</mo><msup id="S4.I11.i2.I1.i1.p1.7.m7.2.2.1.1.1" xref="S4.I11.i2.I1.i1.p1.7.m7.2.2.1.1.1.cmml"><mi id="S4.I11.i2.I1.i1.p1.7.m7.2.2.1.1.1.2" xref="S4.I11.i2.I1.i1.p1.7.m7.2.2.1.1.1.2.cmml">v</mi><mo id="S4.I11.i2.I1.i1.p1.7.m7.2.2.1.1.1.3" xref="S4.I11.i2.I1.i1.p1.7.m7.2.2.1.1.1.3.cmml">′</mo></msup><mo id="S4.I11.i2.I1.i1.p1.7.m7.2.2.1.1.4" stretchy="false" xref="S4.I11.i2.I1.i1.p1.7.m7.2.2.1.2.cmml">)</mo></mrow><mo id="S4.I11.i2.I1.i1.p1.7.m7.2.2.2" xref="S4.I11.i2.I1.i1.p1.7.m7.2.2.2.cmml">∈</mo><mtext id="S4.I11.i2.I1.i1.p1.7.m7.2.2.3" xref="S4.I11.i2.I1.i1.p1.7.m7.2.2.3a.cmml">SOL</mtext></mrow><annotation-xml encoding="MathML-Content" id="S4.I11.i2.I1.i1.p1.7.m7.2b"><apply id="S4.I11.i2.I1.i1.p1.7.m7.2.2.cmml" xref="S4.I11.i2.I1.i1.p1.7.m7.2.2"><in id="S4.I11.i2.I1.i1.p1.7.m7.2.2.2.cmml" xref="S4.I11.i2.I1.i1.p1.7.m7.2.2.2"></in><interval closure="open" id="S4.I11.i2.I1.i1.p1.7.m7.2.2.1.2.cmml" xref="S4.I11.i2.I1.i1.p1.7.m7.2.2.1.1"><ci id="S4.I11.i2.I1.i1.p1.7.m7.1.1.cmml" xref="S4.I11.i2.I1.i1.p1.7.m7.1.1">𝑢</ci><apply id="S4.I11.i2.I1.i1.p1.7.m7.2.2.1.1.1.cmml" xref="S4.I11.i2.I1.i1.p1.7.m7.2.2.1.1.1"><csymbol cd="ambiguous" id="S4.I11.i2.I1.i1.p1.7.m7.2.2.1.1.1.1.cmml" xref="S4.I11.i2.I1.i1.p1.7.m7.2.2.1.1.1">superscript</csymbol><ci id="S4.I11.i2.I1.i1.p1.7.m7.2.2.1.1.1.2.cmml" xref="S4.I11.i2.I1.i1.p1.7.m7.2.2.1.1.1.2">𝑣</ci><ci id="S4.I11.i2.I1.i1.p1.7.m7.2.2.1.1.1.3.cmml" xref="S4.I11.i2.I1.i1.p1.7.m7.2.2.1.1.1.3">′</ci></apply></interval><ci id="S4.I11.i2.I1.i1.p1.7.m7.2.2.3a.cmml" xref="S4.I11.i2.I1.i1.p1.7.m7.2.2.3"><mtext id="S4.I11.i2.I1.i1.p1.7.m7.2.2.3.cmml" xref="S4.I11.i2.I1.i1.p1.7.m7.2.2.3">SOL</mtext></ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.I11.i2.I1.i1.p1.7.m7.2c">(u,v^{\prime})\in\textnormal{SOL}</annotation><annotation encoding="application/x-llamapun" id="S4.I11.i2.I1.i1.p1.7.m7.2d">( italic_u , italic_v start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) ∈ SOL</annotation></semantics></math>, since <math alttext="(u,v)\in\textnormal{OPT}" class="ltx_Math" display="inline" id="S4.I11.i2.I1.i1.p1.8.m8.2"><semantics id="S4.I11.i2.I1.i1.p1.8.m8.2a"><mrow id="S4.I11.i2.I1.i1.p1.8.m8.2.3" xref="S4.I11.i2.I1.i1.p1.8.m8.2.3.cmml"><mrow id="S4.I11.i2.I1.i1.p1.8.m8.2.3.2.2" xref="S4.I11.i2.I1.i1.p1.8.m8.2.3.2.1.cmml"><mo id="S4.I11.i2.I1.i1.p1.8.m8.2.3.2.2.1" stretchy="false" xref="S4.I11.i2.I1.i1.p1.8.m8.2.3.2.1.cmml">(</mo><mi id="S4.I11.i2.I1.i1.p1.8.m8.1.1" xref="S4.I11.i2.I1.i1.p1.8.m8.1.1.cmml">u</mi><mo id="S4.I11.i2.I1.i1.p1.8.m8.2.3.2.2.2" xref="S4.I11.i2.I1.i1.p1.8.m8.2.3.2.1.cmml">,</mo><mi id="S4.I11.i2.I1.i1.p1.8.m8.2.2" xref="S4.I11.i2.I1.i1.p1.8.m8.2.2.cmml">v</mi><mo id="S4.I11.i2.I1.i1.p1.8.m8.2.3.2.2.3" stretchy="false" xref="S4.I11.i2.I1.i1.p1.8.m8.2.3.2.1.cmml">)</mo></mrow><mo id="S4.I11.i2.I1.i1.p1.8.m8.2.3.1" xref="S4.I11.i2.I1.i1.p1.8.m8.2.3.1.cmml">∈</mo><mtext id="S4.I11.i2.I1.i1.p1.8.m8.2.3.3" xref="S4.I11.i2.I1.i1.p1.8.m8.2.3.3a.cmml">OPT</mtext></mrow><annotation-xml encoding="MathML-Content" id="S4.I11.i2.I1.i1.p1.8.m8.2b"><apply id="S4.I11.i2.I1.i1.p1.8.m8.2.3.cmml" xref="S4.I11.i2.I1.i1.p1.8.m8.2.3"><in id="S4.I11.i2.I1.i1.p1.8.m8.2.3.1.cmml" xref="S4.I11.i2.I1.i1.p1.8.m8.2.3.1"></in><interval closure="open" id="S4.I11.i2.I1.i1.p1.8.m8.2.3.2.1.cmml" xref="S4.I11.i2.I1.i1.p1.8.m8.2.3.2.2"><ci id="S4.I11.i2.I1.i1.p1.8.m8.1.1.cmml" xref="S4.I11.i2.I1.i1.p1.8.m8.1.1">𝑢</ci><ci id="S4.I11.i2.I1.i1.p1.8.m8.2.2.cmml" xref="S4.I11.i2.I1.i1.p1.8.m8.2.2">𝑣</ci></interval><ci id="S4.I11.i2.I1.i1.p1.8.m8.2.3.3a.cmml" xref="S4.I11.i2.I1.i1.p1.8.m8.2.3.3"><mtext id="S4.I11.i2.I1.i1.p1.8.m8.2.3.3.cmml" xref="S4.I11.i2.I1.i1.p1.8.m8.2.3.3">OPT</mtext></ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.I11.i2.I1.i1.p1.8.m8.2c">(u,v)\in\textnormal{OPT}</annotation><annotation encoding="application/x-llamapun" id="S4.I11.i2.I1.i1.p1.8.m8.2d">( italic_u , italic_v ) ∈ OPT</annotation></semantics></math>, <math alttext="u\in G_{x}\setminus\textnormal{parent}(x)" class="ltx_Math" display="inline" id="S4.I11.i2.I1.i1.p1.9.m9.1"><semantics id="S4.I11.i2.I1.i1.p1.9.m9.1a"><mrow id="S4.I11.i2.I1.i1.p1.9.m9.1.2" xref="S4.I11.i2.I1.i1.p1.9.m9.1.2.cmml"><mi id="S4.I11.i2.I1.i1.p1.9.m9.1.2.2" xref="S4.I11.i2.I1.i1.p1.9.m9.1.2.2.cmml">u</mi><mo id="S4.I11.i2.I1.i1.p1.9.m9.1.2.1" xref="S4.I11.i2.I1.i1.p1.9.m9.1.2.1.cmml">∈</mo><mrow id="S4.I11.i2.I1.i1.p1.9.m9.1.2.3" xref="S4.I11.i2.I1.i1.p1.9.m9.1.2.3.cmml"><msub id="S4.I11.i2.I1.i1.p1.9.m9.1.2.3.2" xref="S4.I11.i2.I1.i1.p1.9.m9.1.2.3.2.cmml"><mi id="S4.I11.i2.I1.i1.p1.9.m9.1.2.3.2.2" xref="S4.I11.i2.I1.i1.p1.9.m9.1.2.3.2.2.cmml">G</mi><mi id="S4.I11.i2.I1.i1.p1.9.m9.1.2.3.2.3" xref="S4.I11.i2.I1.i1.p1.9.m9.1.2.3.2.3.cmml">x</mi></msub><mo id="S4.I11.i2.I1.i1.p1.9.m9.1.2.3.1" xref="S4.I11.i2.I1.i1.p1.9.m9.1.2.3.1.cmml">∖</mo><mrow id="S4.I11.i2.I1.i1.p1.9.m9.1.2.3.3" xref="S4.I11.i2.I1.i1.p1.9.m9.1.2.3.3.cmml"><mtext id="S4.I11.i2.I1.i1.p1.9.m9.1.2.3.3.2" xref="S4.I11.i2.I1.i1.p1.9.m9.1.2.3.3.2a.cmml">parent</mtext><mo id="S4.I11.i2.I1.i1.p1.9.m9.1.2.3.3.1" xref="S4.I11.i2.I1.i1.p1.9.m9.1.2.3.3.1.cmml">⁢</mo><mrow id="S4.I11.i2.I1.i1.p1.9.m9.1.2.3.3.3.2" xref="S4.I11.i2.I1.i1.p1.9.m9.1.2.3.3.cmml"><mo id="S4.I11.i2.I1.i1.p1.9.m9.1.2.3.3.3.2.1" stretchy="false" xref="S4.I11.i2.I1.i1.p1.9.m9.1.2.3.3.cmml">(</mo><mi id="S4.I11.i2.I1.i1.p1.9.m9.1.1" xref="S4.I11.i2.I1.i1.p1.9.m9.1.1.cmml">x</mi><mo id="S4.I11.i2.I1.i1.p1.9.m9.1.2.3.3.3.2.2" stretchy="false" xref="S4.I11.i2.I1.i1.p1.9.m9.1.2.3.3.cmml">)</mo></mrow></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.I11.i2.I1.i1.p1.9.m9.1b"><apply id="S4.I11.i2.I1.i1.p1.9.m9.1.2.cmml" xref="S4.I11.i2.I1.i1.p1.9.m9.1.2"><in id="S4.I11.i2.I1.i1.p1.9.m9.1.2.1.cmml" xref="S4.I11.i2.I1.i1.p1.9.m9.1.2.1"></in><ci id="S4.I11.i2.I1.i1.p1.9.m9.1.2.2.cmml" xref="S4.I11.i2.I1.i1.p1.9.m9.1.2.2">𝑢</ci><apply id="S4.I11.i2.I1.i1.p1.9.m9.1.2.3.cmml" xref="S4.I11.i2.I1.i1.p1.9.m9.1.2.3"><setdiff id="S4.I11.i2.I1.i1.p1.9.m9.1.2.3.1.cmml" xref="S4.I11.i2.I1.i1.p1.9.m9.1.2.3.1"></setdiff><apply id="S4.I11.i2.I1.i1.p1.9.m9.1.2.3.2.cmml" xref="S4.I11.i2.I1.i1.p1.9.m9.1.2.3.2"><csymbol cd="ambiguous" id="S4.I11.i2.I1.i1.p1.9.m9.1.2.3.2.1.cmml" xref="S4.I11.i2.I1.i1.p1.9.m9.1.2.3.2">subscript</csymbol><ci id="S4.I11.i2.I1.i1.p1.9.m9.1.2.3.2.2.cmml" xref="S4.I11.i2.I1.i1.p1.9.m9.1.2.3.2.2">𝐺</ci><ci id="S4.I11.i2.I1.i1.p1.9.m9.1.2.3.2.3.cmml" xref="S4.I11.i2.I1.i1.p1.9.m9.1.2.3.2.3">𝑥</ci></apply><apply id="S4.I11.i2.I1.i1.p1.9.m9.1.2.3.3.cmml" xref="S4.I11.i2.I1.i1.p1.9.m9.1.2.3.3"><times id="S4.I11.i2.I1.i1.p1.9.m9.1.2.3.3.1.cmml" xref="S4.I11.i2.I1.i1.p1.9.m9.1.2.3.3.1"></times><ci id="S4.I11.i2.I1.i1.p1.9.m9.1.2.3.3.2a.cmml" xref="S4.I11.i2.I1.i1.p1.9.m9.1.2.3.3.2"><mtext id="S4.I11.i2.I1.i1.p1.9.m9.1.2.3.3.2.cmml" xref="S4.I11.i2.I1.i1.p1.9.m9.1.2.3.3.2">parent</mtext></ci><ci id="S4.I11.i2.I1.i1.p1.9.m9.1.1.cmml" xref="S4.I11.i2.I1.i1.p1.9.m9.1.1">𝑥</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.I11.i2.I1.i1.p1.9.m9.1c">u\in G_{x}\setminus\textnormal{parent}(x)</annotation><annotation encoding="application/x-llamapun" id="S4.I11.i2.I1.i1.p1.9.m9.1d">italic_u ∈ italic_G start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT ∖ parent ( italic_x )</annotation></semantics></math>, and <math alttext="\ell(v)\in T\setminus T_{x}" class="ltx_Math" display="inline" id="S4.I11.i2.I1.i1.p1.10.m10.1"><semantics id="S4.I11.i2.I1.i1.p1.10.m10.1a"><mrow id="S4.I11.i2.I1.i1.p1.10.m10.1.2" xref="S4.I11.i2.I1.i1.p1.10.m10.1.2.cmml"><mrow id="S4.I11.i2.I1.i1.p1.10.m10.1.2.2" xref="S4.I11.i2.I1.i1.p1.10.m10.1.2.2.cmml"><mi id="S4.I11.i2.I1.i1.p1.10.m10.1.2.2.2" mathvariant="normal" xref="S4.I11.i2.I1.i1.p1.10.m10.1.2.2.2.cmml">ℓ</mi><mo id="S4.I11.i2.I1.i1.p1.10.m10.1.2.2.1" xref="S4.I11.i2.I1.i1.p1.10.m10.1.2.2.1.cmml">⁢</mo><mrow id="S4.I11.i2.I1.i1.p1.10.m10.1.2.2.3.2" xref="S4.I11.i2.I1.i1.p1.10.m10.1.2.2.cmml"><mo id="S4.I11.i2.I1.i1.p1.10.m10.1.2.2.3.2.1" stretchy="false" xref="S4.I11.i2.I1.i1.p1.10.m10.1.2.2.cmml">(</mo><mi id="S4.I11.i2.I1.i1.p1.10.m10.1.1" xref="S4.I11.i2.I1.i1.p1.10.m10.1.1.cmml">v</mi><mo id="S4.I11.i2.I1.i1.p1.10.m10.1.2.2.3.2.2" stretchy="false" xref="S4.I11.i2.I1.i1.p1.10.m10.1.2.2.cmml">)</mo></mrow></mrow><mo id="S4.I11.i2.I1.i1.p1.10.m10.1.2.1" xref="S4.I11.i2.I1.i1.p1.10.m10.1.2.1.cmml">∈</mo><mrow id="S4.I11.i2.I1.i1.p1.10.m10.1.2.3" xref="S4.I11.i2.I1.i1.p1.10.m10.1.2.3.cmml"><mi id="S4.I11.i2.I1.i1.p1.10.m10.1.2.3.2" xref="S4.I11.i2.I1.i1.p1.10.m10.1.2.3.2.cmml">T</mi><mo id="S4.I11.i2.I1.i1.p1.10.m10.1.2.3.1" xref="S4.I11.i2.I1.i1.p1.10.m10.1.2.3.1.cmml">∖</mo><msub id="S4.I11.i2.I1.i1.p1.10.m10.1.2.3.3" xref="S4.I11.i2.I1.i1.p1.10.m10.1.2.3.3.cmml"><mi id="S4.I11.i2.I1.i1.p1.10.m10.1.2.3.3.2" xref="S4.I11.i2.I1.i1.p1.10.m10.1.2.3.3.2.cmml">T</mi><mi id="S4.I11.i2.I1.i1.p1.10.m10.1.2.3.3.3" xref="S4.I11.i2.I1.i1.p1.10.m10.1.2.3.3.3.cmml">x</mi></msub></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.I11.i2.I1.i1.p1.10.m10.1b"><apply id="S4.I11.i2.I1.i1.p1.10.m10.1.2.cmml" xref="S4.I11.i2.I1.i1.p1.10.m10.1.2"><in id="S4.I11.i2.I1.i1.p1.10.m10.1.2.1.cmml" xref="S4.I11.i2.I1.i1.p1.10.m10.1.2.1"></in><apply id="S4.I11.i2.I1.i1.p1.10.m10.1.2.2.cmml" xref="S4.I11.i2.I1.i1.p1.10.m10.1.2.2"><times id="S4.I11.i2.I1.i1.p1.10.m10.1.2.2.1.cmml" xref="S4.I11.i2.I1.i1.p1.10.m10.1.2.2.1"></times><ci id="S4.I11.i2.I1.i1.p1.10.m10.1.2.2.2.cmml" xref="S4.I11.i2.I1.i1.p1.10.m10.1.2.2.2">ℓ</ci><ci id="S4.I11.i2.I1.i1.p1.10.m10.1.1.cmml" xref="S4.I11.i2.I1.i1.p1.10.m10.1.1">𝑣</ci></apply><apply id="S4.I11.i2.I1.i1.p1.10.m10.1.2.3.cmml" xref="S4.I11.i2.I1.i1.p1.10.m10.1.2.3"><setdiff id="S4.I11.i2.I1.i1.p1.10.m10.1.2.3.1.cmml" xref="S4.I11.i2.I1.i1.p1.10.m10.1.2.3.1"></setdiff><ci id="S4.I11.i2.I1.i1.p1.10.m10.1.2.3.2.cmml" xref="S4.I11.i2.I1.i1.p1.10.m10.1.2.3.2">𝑇</ci><apply id="S4.I11.i2.I1.i1.p1.10.m10.1.2.3.3.cmml" xref="S4.I11.i2.I1.i1.p1.10.m10.1.2.3.3"><csymbol cd="ambiguous" id="S4.I11.i2.I1.i1.p1.10.m10.1.2.3.3.1.cmml" xref="S4.I11.i2.I1.i1.p1.10.m10.1.2.3.3">subscript</csymbol><ci id="S4.I11.i2.I1.i1.p1.10.m10.1.2.3.3.2.cmml" xref="S4.I11.i2.I1.i1.p1.10.m10.1.2.3.3.2">𝑇</ci><ci id="S4.I11.i2.I1.i1.p1.10.m10.1.2.3.3.3.cmml" xref="S4.I11.i2.I1.i1.p1.10.m10.1.2.3.3.3">𝑥</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.I11.i2.I1.i1.p1.10.m10.1c">\ell(v)\in T\setminus T_{x}</annotation><annotation encoding="application/x-llamapun" id="S4.I11.i2.I1.i1.p1.10.m10.1d">roman_ℓ ( italic_v ) ∈ italic_T ∖ italic_T start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT</annotation></semantics></math>. Finally, <math alttext="v^{\prime}\notin\{a,b\}" class="ltx_Math" display="inline" id="S4.I11.i2.I1.i1.p1.11.m11.2"><semantics id="S4.I11.i2.I1.i1.p1.11.m11.2a"><mrow id="S4.I11.i2.I1.i1.p1.11.m11.2.3" xref="S4.I11.i2.I1.i1.p1.11.m11.2.3.cmml"><msup id="S4.I11.i2.I1.i1.p1.11.m11.2.3.2" xref="S4.I11.i2.I1.i1.p1.11.m11.2.3.2.cmml"><mi id="S4.I11.i2.I1.i1.p1.11.m11.2.3.2.2" xref="S4.I11.i2.I1.i1.p1.11.m11.2.3.2.2.cmml">v</mi><mo id="S4.I11.i2.I1.i1.p1.11.m11.2.3.2.3" xref="S4.I11.i2.I1.i1.p1.11.m11.2.3.2.3.cmml">′</mo></msup><mo id="S4.I11.i2.I1.i1.p1.11.m11.2.3.1" xref="S4.I11.i2.I1.i1.p1.11.m11.2.3.1.cmml">∉</mo><mrow id="S4.I11.i2.I1.i1.p1.11.m11.2.3.3.2" xref="S4.I11.i2.I1.i1.p1.11.m11.2.3.3.1.cmml"><mo id="S4.I11.i2.I1.i1.p1.11.m11.2.3.3.2.1" stretchy="false" xref="S4.I11.i2.I1.i1.p1.11.m11.2.3.3.1.cmml">{</mo><mi id="S4.I11.i2.I1.i1.p1.11.m11.1.1" xref="S4.I11.i2.I1.i1.p1.11.m11.1.1.cmml">a</mi><mo id="S4.I11.i2.I1.i1.p1.11.m11.2.3.3.2.2" xref="S4.I11.i2.I1.i1.p1.11.m11.2.3.3.1.cmml">,</mo><mi id="S4.I11.i2.I1.i1.p1.11.m11.2.2" xref="S4.I11.i2.I1.i1.p1.11.m11.2.2.cmml">b</mi><mo id="S4.I11.i2.I1.i1.p1.11.m11.2.3.3.2.3" stretchy="false" xref="S4.I11.i2.I1.i1.p1.11.m11.2.3.3.1.cmml">}</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.I11.i2.I1.i1.p1.11.m11.2b"><apply id="S4.I11.i2.I1.i1.p1.11.m11.2.3.cmml" xref="S4.I11.i2.I1.i1.p1.11.m11.2.3"><notin id="S4.I11.i2.I1.i1.p1.11.m11.2.3.1.cmml" xref="S4.I11.i2.I1.i1.p1.11.m11.2.3.1"></notin><apply id="S4.I11.i2.I1.i1.p1.11.m11.2.3.2.cmml" xref="S4.I11.i2.I1.i1.p1.11.m11.2.3.2"><csymbol cd="ambiguous" id="S4.I11.i2.I1.i1.p1.11.m11.2.3.2.1.cmml" xref="S4.I11.i2.I1.i1.p1.11.m11.2.3.2">superscript</csymbol><ci id="S4.I11.i2.I1.i1.p1.11.m11.2.3.2.2.cmml" xref="S4.I11.i2.I1.i1.p1.11.m11.2.3.2.2">𝑣</ci><ci id="S4.I11.i2.I1.i1.p1.11.m11.2.3.2.3.cmml" xref="S4.I11.i2.I1.i1.p1.11.m11.2.3.2.3">′</ci></apply><set id="S4.I11.i2.I1.i1.p1.11.m11.2.3.3.1.cmml" xref="S4.I11.i2.I1.i1.p1.11.m11.2.3.3.2"><ci id="S4.I11.i2.I1.i1.p1.11.m11.1.1.cmml" xref="S4.I11.i2.I1.i1.p1.11.m11.1.1">𝑎</ci><ci id="S4.I11.i2.I1.i1.p1.11.m11.2.2.cmml" xref="S4.I11.i2.I1.i1.p1.11.m11.2.2">𝑏</ci></set></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.I11.i2.I1.i1.p1.11.m11.2c">v^{\prime}\notin\{a,b\}</annotation><annotation encoding="application/x-llamapun" id="S4.I11.i2.I1.i1.p1.11.m11.2d">italic_v start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ∉ { italic_a , italic_b }</annotation></semantics></math> since <math alttext="v^{\prime}\notin G_{x}" class="ltx_Math" display="inline" id="S4.I11.i2.I1.i1.p1.12.m12.1"><semantics id="S4.I11.i2.I1.i1.p1.12.m12.1a"><mrow id="S4.I11.i2.I1.i1.p1.12.m12.1.1" xref="S4.I11.i2.I1.i1.p1.12.m12.1.1.cmml"><msup id="S4.I11.i2.I1.i1.p1.12.m12.1.1.2" xref="S4.I11.i2.I1.i1.p1.12.m12.1.1.2.cmml"><mi id="S4.I11.i2.I1.i1.p1.12.m12.1.1.2.2" xref="S4.I11.i2.I1.i1.p1.12.m12.1.1.2.2.cmml">v</mi><mo id="S4.I11.i2.I1.i1.p1.12.m12.1.1.2.3" xref="S4.I11.i2.I1.i1.p1.12.m12.1.1.2.3.cmml">′</mo></msup><mo id="S4.I11.i2.I1.i1.p1.12.m12.1.1.1" xref="S4.I11.i2.I1.i1.p1.12.m12.1.1.1.cmml">∉</mo><msub id="S4.I11.i2.I1.i1.p1.12.m12.1.1.3" xref="S4.I11.i2.I1.i1.p1.12.m12.1.1.3.cmml"><mi id="S4.I11.i2.I1.i1.p1.12.m12.1.1.3.2" xref="S4.I11.i2.I1.i1.p1.12.m12.1.1.3.2.cmml">G</mi><mi id="S4.I11.i2.I1.i1.p1.12.m12.1.1.3.3" xref="S4.I11.i2.I1.i1.p1.12.m12.1.1.3.3.cmml">x</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S4.I11.i2.I1.i1.p1.12.m12.1b"><apply id="S4.I11.i2.I1.i1.p1.12.m12.1.1.cmml" xref="S4.I11.i2.I1.i1.p1.12.m12.1.1"><notin id="S4.I11.i2.I1.i1.p1.12.m12.1.1.1.cmml" xref="S4.I11.i2.I1.i1.p1.12.m12.1.1.1"></notin><apply id="S4.I11.i2.I1.i1.p1.12.m12.1.1.2.cmml" xref="S4.I11.i2.I1.i1.p1.12.m12.1.1.2"><csymbol cd="ambiguous" id="S4.I11.i2.I1.i1.p1.12.m12.1.1.2.1.cmml" xref="S4.I11.i2.I1.i1.p1.12.m12.1.1.2">superscript</csymbol><ci id="S4.I11.i2.I1.i1.p1.12.m12.1.1.2.2.cmml" xref="S4.I11.i2.I1.i1.p1.12.m12.1.1.2.2">𝑣</ci><ci id="S4.I11.i2.I1.i1.p1.12.m12.1.1.2.3.cmml" xref="S4.I11.i2.I1.i1.p1.12.m12.1.1.2.3">′</ci></apply><apply id="S4.I11.i2.I1.i1.p1.12.m12.1.1.3.cmml" xref="S4.I11.i2.I1.i1.p1.12.m12.1.1.3"><csymbol cd="ambiguous" id="S4.I11.i2.I1.i1.p1.12.m12.1.1.3.1.cmml" xref="S4.I11.i2.I1.i1.p1.12.m12.1.1.3">subscript</csymbol><ci id="S4.I11.i2.I1.i1.p1.12.m12.1.1.3.2.cmml" xref="S4.I11.i2.I1.i1.p1.12.m12.1.1.3.2">𝐺</ci><ci id="S4.I11.i2.I1.i1.p1.12.m12.1.1.3.3.cmml" xref="S4.I11.i2.I1.i1.p1.12.m12.1.1.3.3">𝑥</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.I11.i2.I1.i1.p1.12.m12.1c">v^{\prime}\notin G_{x}</annotation><annotation encoding="application/x-llamapun" id="S4.I11.i2.I1.i1.p1.12.m12.1d">italic_v start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ∉ italic_G start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT</annotation></semantics></math>. Thus there exists a <math alttext="u" class="ltx_Math" display="inline" id="S4.I11.i2.I1.i1.p1.13.m13.1"><semantics id="S4.I11.i2.I1.i1.p1.13.m13.1a"><mi id="S4.I11.i2.I1.i1.p1.13.m13.1.1" xref="S4.I11.i2.I1.i1.p1.13.m13.1.1.cmml">u</mi><annotation-xml encoding="MathML-Content" id="S4.I11.i2.I1.i1.p1.13.m13.1b"><ci id="S4.I11.i2.I1.i1.p1.13.m13.1.1.cmml" xref="S4.I11.i2.I1.i1.p1.13.m13.1.1">𝑢</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.I11.i2.I1.i1.p1.13.m13.1c">u</annotation><annotation encoding="application/x-llamapun" id="S4.I11.i2.I1.i1.p1.13.m13.1d">italic_u</annotation></semantics></math>-<math alttext="v" class="ltx_Math" display="inline" id="S4.I11.i2.I1.i1.p1.14.m14.1"><semantics id="S4.I11.i2.I1.i1.p1.14.m14.1a"><mi id="S4.I11.i2.I1.i1.p1.14.m14.1.1" xref="S4.I11.i2.I1.i1.p1.14.m14.1.1.cmml">v</mi><annotation-xml encoding="MathML-Content" id="S4.I11.i2.I1.i1.p1.14.m14.1b"><ci id="S4.I11.i2.I1.i1.p1.14.m14.1.1.cmml" xref="S4.I11.i2.I1.i1.p1.14.m14.1.1">𝑣</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.I11.i2.I1.i1.p1.14.m14.1c">v</annotation><annotation encoding="application/x-llamapun" id="S4.I11.i2.I1.i1.p1.14.m14.1d">italic_v</annotation></semantics></math> path in <math alttext="E\cup\textnormal{SOL}" class="ltx_Math" display="inline" id="S4.I11.i2.I1.i1.p1.15.m15.1"><semantics id="S4.I11.i2.I1.i1.p1.15.m15.1a"><mrow id="S4.I11.i2.I1.i1.p1.15.m15.1.1" xref="S4.I11.i2.I1.i1.p1.15.m15.1.1.cmml"><mi id="S4.I11.i2.I1.i1.p1.15.m15.1.1.2" xref="S4.I11.i2.I1.i1.p1.15.m15.1.1.2.cmml">E</mi><mo id="S4.I11.i2.I1.i1.p1.15.m15.1.1.1" xref="S4.I11.i2.I1.i1.p1.15.m15.1.1.1.cmml">∪</mo><mtext id="S4.I11.i2.I1.i1.p1.15.m15.1.1.3" xref="S4.I11.i2.I1.i1.p1.15.m15.1.1.3a.cmml">SOL</mtext></mrow><annotation-xml encoding="MathML-Content" id="S4.I11.i2.I1.i1.p1.15.m15.1b"><apply id="S4.I11.i2.I1.i1.p1.15.m15.1.1.cmml" xref="S4.I11.i2.I1.i1.p1.15.m15.1.1"><union id="S4.I11.i2.I1.i1.p1.15.m15.1.1.1.cmml" xref="S4.I11.i2.I1.i1.p1.15.m15.1.1.1"></union><ci id="S4.I11.i2.I1.i1.p1.15.m15.1.1.2.cmml" xref="S4.I11.i2.I1.i1.p1.15.m15.1.1.2">𝐸</ci><ci id="S4.I11.i2.I1.i1.p1.15.m15.1.1.3a.cmml" xref="S4.I11.i2.I1.i1.p1.15.m15.1.1.3"><mtext id="S4.I11.i2.I1.i1.p1.15.m15.1.1.3.cmml" xref="S4.I11.i2.I1.i1.p1.15.m15.1.1.3">SOL</mtext></ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.I11.i2.I1.i1.p1.15.m15.1c">E\cup\textnormal{SOL}</annotation><annotation encoding="application/x-llamapun" id="S4.I11.i2.I1.i1.p1.15.m15.1d">italic_E ∪ SOL</annotation></semantics></math> avoiding <math alttext="\{a,b\}" class="ltx_Math" display="inline" id="S4.I11.i2.I1.i1.p1.16.m16.2"><semantics id="S4.I11.i2.I1.i1.p1.16.m16.2a"><mrow id="S4.I11.i2.I1.i1.p1.16.m16.2.3.2" xref="S4.I11.i2.I1.i1.p1.16.m16.2.3.1.cmml"><mo id="S4.I11.i2.I1.i1.p1.16.m16.2.3.2.1" stretchy="false" xref="S4.I11.i2.I1.i1.p1.16.m16.2.3.1.cmml">{</mo><mi id="S4.I11.i2.I1.i1.p1.16.m16.1.1" xref="S4.I11.i2.I1.i1.p1.16.m16.1.1.cmml">a</mi><mo id="S4.I11.i2.I1.i1.p1.16.m16.2.3.2.2" xref="S4.I11.i2.I1.i1.p1.16.m16.2.3.1.cmml">,</mo><mi id="S4.I11.i2.I1.i1.p1.16.m16.2.2" xref="S4.I11.i2.I1.i1.p1.16.m16.2.2.cmml">b</mi><mo id="S4.I11.i2.I1.i1.p1.16.m16.2.3.2.3" stretchy="false" xref="S4.I11.i2.I1.i1.p1.16.m16.2.3.1.cmml">}</mo></mrow><annotation-xml encoding="MathML-Content" id="S4.I11.i2.I1.i1.p1.16.m16.2b"><set id="S4.I11.i2.I1.i1.p1.16.m16.2.3.1.cmml" xref="S4.I11.i2.I1.i1.p1.16.m16.2.3.2"><ci id="S4.I11.i2.I1.i1.p1.16.m16.1.1.cmml" xref="S4.I11.i2.I1.i1.p1.16.m16.1.1">𝑎</ci><ci id="S4.I11.i2.I1.i1.p1.16.m16.2.2.cmml" xref="S4.I11.i2.I1.i1.p1.16.m16.2.2">𝑏</ci></set></annotation-xml><annotation encoding="application/x-tex" id="S4.I11.i2.I1.i1.p1.16.m16.2c">\{a,b\}</annotation><annotation encoding="application/x-llamapun" id="S4.I11.i2.I1.i1.p1.16.m16.2d">{ italic_a , italic_b }</annotation></semantics></math> via the link <math alttext="\{u,v^{\prime}\}" class="ltx_Math" display="inline" id="S4.I11.i2.I1.i1.p1.17.m17.2"><semantics id="S4.I11.i2.I1.i1.p1.17.m17.2a"><mrow id="S4.I11.i2.I1.i1.p1.17.m17.2.2.1" xref="S4.I11.i2.I1.i1.p1.17.m17.2.2.2.cmml"><mo id="S4.I11.i2.I1.i1.p1.17.m17.2.2.1.2" stretchy="false" xref="S4.I11.i2.I1.i1.p1.17.m17.2.2.2.cmml">{</mo><mi id="S4.I11.i2.I1.i1.p1.17.m17.1.1" xref="S4.I11.i2.I1.i1.p1.17.m17.1.1.cmml">u</mi><mo id="S4.I11.i2.I1.i1.p1.17.m17.2.2.1.3" xref="S4.I11.i2.I1.i1.p1.17.m17.2.2.2.cmml">,</mo><msup id="S4.I11.i2.I1.i1.p1.17.m17.2.2.1.1" xref="S4.I11.i2.I1.i1.p1.17.m17.2.2.1.1.cmml"><mi id="S4.I11.i2.I1.i1.p1.17.m17.2.2.1.1.2" xref="S4.I11.i2.I1.i1.p1.17.m17.2.2.1.1.2.cmml">v</mi><mo id="S4.I11.i2.I1.i1.p1.17.m17.2.2.1.1.3" xref="S4.I11.i2.I1.i1.p1.17.m17.2.2.1.1.3.cmml">′</mo></msup><mo id="S4.I11.i2.I1.i1.p1.17.m17.2.2.1.4" stretchy="false" xref="S4.I11.i2.I1.i1.p1.17.m17.2.2.2.cmml">}</mo></mrow><annotation-xml encoding="MathML-Content" id="S4.I11.i2.I1.i1.p1.17.m17.2b"><set id="S4.I11.i2.I1.i1.p1.17.m17.2.2.2.cmml" xref="S4.I11.i2.I1.i1.p1.17.m17.2.2.1"><ci id="S4.I11.i2.I1.i1.p1.17.m17.1.1.cmml" xref="S4.I11.i2.I1.i1.p1.17.m17.1.1">𝑢</ci><apply id="S4.I11.i2.I1.i1.p1.17.m17.2.2.1.1.cmml" xref="S4.I11.i2.I1.i1.p1.17.m17.2.2.1.1"><csymbol cd="ambiguous" id="S4.I11.i2.I1.i1.p1.17.m17.2.2.1.1.1.cmml" xref="S4.I11.i2.I1.i1.p1.17.m17.2.2.1.1">superscript</csymbol><ci id="S4.I11.i2.I1.i1.p1.17.m17.2.2.1.1.2.cmml" xref="S4.I11.i2.I1.i1.p1.17.m17.2.2.1.1.2">𝑣</ci><ci id="S4.I11.i2.I1.i1.p1.17.m17.2.2.1.1.3.cmml" xref="S4.I11.i2.I1.i1.p1.17.m17.2.2.1.1.3">′</ci></apply></set></annotation-xml><annotation encoding="application/x-tex" id="S4.I11.i2.I1.i1.p1.17.m17.2c">\{u,v^{\prime}\}</annotation><annotation encoding="application/x-llamapun" id="S4.I11.i2.I1.i1.p1.17.m17.2d">{ italic_u , italic_v start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT }</annotation></semantics></math>. See Figure <a class="ltx_ref" href="https://arxiv.org/html/2503.00712v1#S4.F10.sf2" title="In Figure 10 ‣ Case 3: 𝒂 and 𝒃 are non-adjacent nodes of 𝑮_𝒙 for an S-node 𝒙: ‣ 4.2.3 Bounding the Approximation Ratio ‣ 4.2 Two-to-Three Augmentation ‣ 4 Vertex Connectivity Augmentation in Link-Arrival Model ‣ Streaming Algorithms for Network Design"><span class="ltx_text ltx_ref_tag">10(b)</span></a> for reference.</p> </div> </li> <li class="ltx_item" id="S4.I11.i2.I1.i2" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item"><span class="ltx_text ltx_font_bold" id="S4.I11.i2.I1.i2.1.1.1">–</span></span> <div class="ltx_para" id="S4.I11.i2.I1.i2.p1"> <p class="ltx_p" id="S4.I11.i2.I1.i2.p1.11">Suppose <math alttext="u\in\textnormal{parent}(x)" class="ltx_Math" display="inline" id="S4.I11.i2.I1.i2.p1.1.m1.1"><semantics id="S4.I11.i2.I1.i2.p1.1.m1.1a"><mrow id="S4.I11.i2.I1.i2.p1.1.m1.1.2" xref="S4.I11.i2.I1.i2.p1.1.m1.1.2.cmml"><mi id="S4.I11.i2.I1.i2.p1.1.m1.1.2.2" xref="S4.I11.i2.I1.i2.p1.1.m1.1.2.2.cmml">u</mi><mo id="S4.I11.i2.I1.i2.p1.1.m1.1.2.1" xref="S4.I11.i2.I1.i2.p1.1.m1.1.2.1.cmml">∈</mo><mrow id="S4.I11.i2.I1.i2.p1.1.m1.1.2.3" xref="S4.I11.i2.I1.i2.p1.1.m1.1.2.3.cmml"><mtext id="S4.I11.i2.I1.i2.p1.1.m1.1.2.3.2" xref="S4.I11.i2.I1.i2.p1.1.m1.1.2.3.2a.cmml">parent</mtext><mo id="S4.I11.i2.I1.i2.p1.1.m1.1.2.3.1" xref="S4.I11.i2.I1.i2.p1.1.m1.1.2.3.1.cmml">⁢</mo><mrow id="S4.I11.i2.I1.i2.p1.1.m1.1.2.3.3.2" xref="S4.I11.i2.I1.i2.p1.1.m1.1.2.3.cmml"><mo id="S4.I11.i2.I1.i2.p1.1.m1.1.2.3.3.2.1" stretchy="false" xref="S4.I11.i2.I1.i2.p1.1.m1.1.2.3.cmml">(</mo><mi id="S4.I11.i2.I1.i2.p1.1.m1.1.1" xref="S4.I11.i2.I1.i2.p1.1.m1.1.1.cmml">x</mi><mo id="S4.I11.i2.I1.i2.p1.1.m1.1.2.3.3.2.2" stretchy="false" xref="S4.I11.i2.I1.i2.p1.1.m1.1.2.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.I11.i2.I1.i2.p1.1.m1.1b"><apply id="S4.I11.i2.I1.i2.p1.1.m1.1.2.cmml" xref="S4.I11.i2.I1.i2.p1.1.m1.1.2"><in id="S4.I11.i2.I1.i2.p1.1.m1.1.2.1.cmml" xref="S4.I11.i2.I1.i2.p1.1.m1.1.2.1"></in><ci id="S4.I11.i2.I1.i2.p1.1.m1.1.2.2.cmml" xref="S4.I11.i2.I1.i2.p1.1.m1.1.2.2">𝑢</ci><apply id="S4.I11.i2.I1.i2.p1.1.m1.1.2.3.cmml" xref="S4.I11.i2.I1.i2.p1.1.m1.1.2.3"><times id="S4.I11.i2.I1.i2.p1.1.m1.1.2.3.1.cmml" xref="S4.I11.i2.I1.i2.p1.1.m1.1.2.3.1"></times><ci id="S4.I11.i2.I1.i2.p1.1.m1.1.2.3.2a.cmml" xref="S4.I11.i2.I1.i2.p1.1.m1.1.2.3.2"><mtext id="S4.I11.i2.I1.i2.p1.1.m1.1.2.3.2.cmml" xref="S4.I11.i2.I1.i2.p1.1.m1.1.2.3.2">parent</mtext></ci><ci id="S4.I11.i2.I1.i2.p1.1.m1.1.1.cmml" xref="S4.I11.i2.I1.i2.p1.1.m1.1.1">𝑥</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.I11.i2.I1.i2.p1.1.m1.1c">u\in\textnormal{parent}(x)</annotation><annotation encoding="application/x-llamapun" id="S4.I11.i2.I1.i2.p1.1.m1.1d">italic_u ∈ parent ( italic_x )</annotation></semantics></math>. Then, since <math alttext="a" class="ltx_Math" display="inline" id="S4.I11.i2.I1.i2.p1.2.m2.1"><semantics id="S4.I11.i2.I1.i2.p1.2.m2.1a"><mi id="S4.I11.i2.I1.i2.p1.2.m2.1.1" xref="S4.I11.i2.I1.i2.p1.2.m2.1.1.cmml">a</mi><annotation-xml encoding="MathML-Content" id="S4.I11.i2.I1.i2.p1.2.m2.1b"><ci id="S4.I11.i2.I1.i2.p1.2.m2.1.1.cmml" xref="S4.I11.i2.I1.i2.p1.2.m2.1.1">𝑎</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.I11.i2.I1.i2.p1.2.m2.1c">a</annotation><annotation encoding="application/x-llamapun" id="S4.I11.i2.I1.i2.p1.2.m2.1d">italic_a</annotation></semantics></math> and <math alttext="b" class="ltx_Math" display="inline" id="S4.I11.i2.I1.i2.p1.3.m3.1"><semantics id="S4.I11.i2.I1.i2.p1.3.m3.1a"><mi id="S4.I11.i2.I1.i2.p1.3.m3.1.1" xref="S4.I11.i2.I1.i2.p1.3.m3.1.1.cmml">b</mi><annotation-xml encoding="MathML-Content" id="S4.I11.i2.I1.i2.p1.3.m3.1b"><ci id="S4.I11.i2.I1.i2.p1.3.m3.1.1.cmml" xref="S4.I11.i2.I1.i2.p1.3.m3.1.1">𝑏</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.I11.i2.I1.i2.p1.3.m3.1c">b</annotation><annotation encoding="application/x-llamapun" id="S4.I11.i2.I1.i2.p1.3.m3.1d">italic_b</annotation></semantics></math> are non-adjacent nodes of <math alttext="G_{x}" class="ltx_Math" display="inline" id="S4.I11.i2.I1.i2.p1.4.m4.1"><semantics id="S4.I11.i2.I1.i2.p1.4.m4.1a"><msub id="S4.I11.i2.I1.i2.p1.4.m4.1.1" xref="S4.I11.i2.I1.i2.p1.4.m4.1.1.cmml"><mi id="S4.I11.i2.I1.i2.p1.4.m4.1.1.2" xref="S4.I11.i2.I1.i2.p1.4.m4.1.1.2.cmml">G</mi><mi id="S4.I11.i2.I1.i2.p1.4.m4.1.1.3" xref="S4.I11.i2.I1.i2.p1.4.m4.1.1.3.cmml">x</mi></msub><annotation-xml encoding="MathML-Content" id="S4.I11.i2.I1.i2.p1.4.m4.1b"><apply id="S4.I11.i2.I1.i2.p1.4.m4.1.1.cmml" xref="S4.I11.i2.I1.i2.p1.4.m4.1.1"><csymbol cd="ambiguous" id="S4.I11.i2.I1.i2.p1.4.m4.1.1.1.cmml" xref="S4.I11.i2.I1.i2.p1.4.m4.1.1">subscript</csymbol><ci id="S4.I11.i2.I1.i2.p1.4.m4.1.1.2.cmml" xref="S4.I11.i2.I1.i2.p1.4.m4.1.1.2">𝐺</ci><ci id="S4.I11.i2.I1.i2.p1.4.m4.1.1.3.cmml" xref="S4.I11.i2.I1.i2.p1.4.m4.1.1.3">𝑥</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.I11.i2.I1.i2.p1.4.m4.1c">G_{x}</annotation><annotation encoding="application/x-llamapun" id="S4.I11.i2.I1.i2.p1.4.m4.1d">italic_G start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT</annotation></semantics></math>, <math alttext="u" class="ltx_Math" display="inline" id="S4.I11.i2.I1.i2.p1.5.m5.1"><semantics id="S4.I11.i2.I1.i2.p1.5.m5.1a"><mi id="S4.I11.i2.I1.i2.p1.5.m5.1.1" xref="S4.I11.i2.I1.i2.p1.5.m5.1.1.cmml">u</mi><annotation-xml encoding="MathML-Content" id="S4.I11.i2.I1.i2.p1.5.m5.1b"><ci id="S4.I11.i2.I1.i2.p1.5.m5.1.1.cmml" xref="S4.I11.i2.I1.i2.p1.5.m5.1.1">𝑢</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.I11.i2.I1.i2.p1.5.m5.1c">u</annotation><annotation encoding="application/x-llamapun" id="S4.I11.i2.I1.i2.p1.5.m5.1d">italic_u</annotation></semantics></math> is connected to <math alttext="T\setminus T_{x}" class="ltx_Math" display="inline" id="S4.I11.i2.I1.i2.p1.6.m6.1"><semantics id="S4.I11.i2.I1.i2.p1.6.m6.1a"><mrow id="S4.I11.i2.I1.i2.p1.6.m6.1.1" xref="S4.I11.i2.I1.i2.p1.6.m6.1.1.cmml"><mi id="S4.I11.i2.I1.i2.p1.6.m6.1.1.2" xref="S4.I11.i2.I1.i2.p1.6.m6.1.1.2.cmml">T</mi><mo id="S4.I11.i2.I1.i2.p1.6.m6.1.1.1" xref="S4.I11.i2.I1.i2.p1.6.m6.1.1.1.cmml">∖</mo><msub id="S4.I11.i2.I1.i2.p1.6.m6.1.1.3" xref="S4.I11.i2.I1.i2.p1.6.m6.1.1.3.cmml"><mi id="S4.I11.i2.I1.i2.p1.6.m6.1.1.3.2" xref="S4.I11.i2.I1.i2.p1.6.m6.1.1.3.2.cmml">T</mi><mi id="S4.I11.i2.I1.i2.p1.6.m6.1.1.3.3" xref="S4.I11.i2.I1.i2.p1.6.m6.1.1.3.3.cmml">x</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S4.I11.i2.I1.i2.p1.6.m6.1b"><apply id="S4.I11.i2.I1.i2.p1.6.m6.1.1.cmml" xref="S4.I11.i2.I1.i2.p1.6.m6.1.1"><setdiff id="S4.I11.i2.I1.i2.p1.6.m6.1.1.1.cmml" xref="S4.I11.i2.I1.i2.p1.6.m6.1.1.1"></setdiff><ci id="S4.I11.i2.I1.i2.p1.6.m6.1.1.2.cmml" xref="S4.I11.i2.I1.i2.p1.6.m6.1.1.2">𝑇</ci><apply id="S4.I11.i2.I1.i2.p1.6.m6.1.1.3.cmml" xref="S4.I11.i2.I1.i2.p1.6.m6.1.1.3"><csymbol cd="ambiguous" id="S4.I11.i2.I1.i2.p1.6.m6.1.1.3.1.cmml" xref="S4.I11.i2.I1.i2.p1.6.m6.1.1.3">subscript</csymbol><ci id="S4.I11.i2.I1.i2.p1.6.m6.1.1.3.2.cmml" xref="S4.I11.i2.I1.i2.p1.6.m6.1.1.3.2">𝑇</ci><ci id="S4.I11.i2.I1.i2.p1.6.m6.1.1.3.3.cmml" xref="S4.I11.i2.I1.i2.p1.6.m6.1.1.3.3">𝑥</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.I11.i2.I1.i2.p1.6.m6.1c">T\setminus T_{x}</annotation><annotation encoding="application/x-llamapun" id="S4.I11.i2.I1.i2.p1.6.m6.1d">italic_T ∖ italic_T start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT</annotation></semantics></math> despite the removal of <math alttext="\{a,b\}" class="ltx_Math" display="inline" id="S4.I11.i2.I1.i2.p1.7.m7.2"><semantics id="S4.I11.i2.I1.i2.p1.7.m7.2a"><mrow id="S4.I11.i2.I1.i2.p1.7.m7.2.3.2" xref="S4.I11.i2.I1.i2.p1.7.m7.2.3.1.cmml"><mo id="S4.I11.i2.I1.i2.p1.7.m7.2.3.2.1" stretchy="false" xref="S4.I11.i2.I1.i2.p1.7.m7.2.3.1.cmml">{</mo><mi id="S4.I11.i2.I1.i2.p1.7.m7.1.1" xref="S4.I11.i2.I1.i2.p1.7.m7.1.1.cmml">a</mi><mo id="S4.I11.i2.I1.i2.p1.7.m7.2.3.2.2" xref="S4.I11.i2.I1.i2.p1.7.m7.2.3.1.cmml">,</mo><mi id="S4.I11.i2.I1.i2.p1.7.m7.2.2" xref="S4.I11.i2.I1.i2.p1.7.m7.2.2.cmml">b</mi><mo id="S4.I11.i2.I1.i2.p1.7.m7.2.3.2.3" stretchy="false" xref="S4.I11.i2.I1.i2.p1.7.m7.2.3.1.cmml">}</mo></mrow><annotation-xml encoding="MathML-Content" id="S4.I11.i2.I1.i2.p1.7.m7.2b"><set id="S4.I11.i2.I1.i2.p1.7.m7.2.3.1.cmml" xref="S4.I11.i2.I1.i2.p1.7.m7.2.3.2"><ci id="S4.I11.i2.I1.i2.p1.7.m7.1.1.cmml" xref="S4.I11.i2.I1.i2.p1.7.m7.1.1">𝑎</ci><ci id="S4.I11.i2.I1.i2.p1.7.m7.2.2.cmml" xref="S4.I11.i2.I1.i2.p1.7.m7.2.2">𝑏</ci></set></annotation-xml><annotation encoding="application/x-tex" id="S4.I11.i2.I1.i2.p1.7.m7.2c">\{a,b\}</annotation><annotation encoding="application/x-llamapun" id="S4.I11.i2.I1.i2.p1.7.m7.2d">{ italic_a , italic_b }</annotation></semantics></math>, so <math alttext="E" class="ltx_Math" display="inline" id="S4.I11.i2.I1.i2.p1.8.m8.1"><semantics id="S4.I11.i2.I1.i2.p1.8.m8.1a"><mi id="S4.I11.i2.I1.i2.p1.8.m8.1.1" xref="S4.I11.i2.I1.i2.p1.8.m8.1.1.cmml">E</mi><annotation-xml encoding="MathML-Content" id="S4.I11.i2.I1.i2.p1.8.m8.1b"><ci id="S4.I11.i2.I1.i2.p1.8.m8.1.1.cmml" xref="S4.I11.i2.I1.i2.p1.8.m8.1.1">𝐸</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.I11.i2.I1.i2.p1.8.m8.1c">E</annotation><annotation encoding="application/x-llamapun" id="S4.I11.i2.I1.i2.p1.8.m8.1d">italic_E</annotation></semantics></math> contains a <math alttext="u" class="ltx_Math" display="inline" id="S4.I11.i2.I1.i2.p1.9.m9.1"><semantics id="S4.I11.i2.I1.i2.p1.9.m9.1a"><mi id="S4.I11.i2.I1.i2.p1.9.m9.1.1" xref="S4.I11.i2.I1.i2.p1.9.m9.1.1.cmml">u</mi><annotation-xml encoding="MathML-Content" id="S4.I11.i2.I1.i2.p1.9.m9.1b"><ci id="S4.I11.i2.I1.i2.p1.9.m9.1.1.cmml" xref="S4.I11.i2.I1.i2.p1.9.m9.1.1">𝑢</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.I11.i2.I1.i2.p1.9.m9.1c">u</annotation><annotation encoding="application/x-llamapun" id="S4.I11.i2.I1.i2.p1.9.m9.1d">italic_u</annotation></semantics></math>-<math alttext="v" class="ltx_Math" display="inline" id="S4.I11.i2.I1.i2.p1.10.m10.1"><semantics id="S4.I11.i2.I1.i2.p1.10.m10.1a"><mi id="S4.I11.i2.I1.i2.p1.10.m10.1.1" xref="S4.I11.i2.I1.i2.p1.10.m10.1.1.cmml">v</mi><annotation-xml encoding="MathML-Content" id="S4.I11.i2.I1.i2.p1.10.m10.1b"><ci id="S4.I11.i2.I1.i2.p1.10.m10.1.1.cmml" xref="S4.I11.i2.I1.i2.p1.10.m10.1.1">𝑣</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.I11.i2.I1.i2.p1.10.m10.1c">v</annotation><annotation encoding="application/x-llamapun" id="S4.I11.i2.I1.i2.p1.10.m10.1d">italic_v</annotation></semantics></math> path avoiding <math alttext="\{a,b\}" class="ltx_Math" display="inline" id="S4.I11.i2.I1.i2.p1.11.m11.2"><semantics id="S4.I11.i2.I1.i2.p1.11.m11.2a"><mrow id="S4.I11.i2.I1.i2.p1.11.m11.2.3.2" xref="S4.I11.i2.I1.i2.p1.11.m11.2.3.1.cmml"><mo id="S4.I11.i2.I1.i2.p1.11.m11.2.3.2.1" stretchy="false" xref="S4.I11.i2.I1.i2.p1.11.m11.2.3.1.cmml">{</mo><mi id="S4.I11.i2.I1.i2.p1.11.m11.1.1" xref="S4.I11.i2.I1.i2.p1.11.m11.1.1.cmml">a</mi><mo id="S4.I11.i2.I1.i2.p1.11.m11.2.3.2.2" xref="S4.I11.i2.I1.i2.p1.11.m11.2.3.1.cmml">,</mo><mi id="S4.I11.i2.I1.i2.p1.11.m11.2.2" xref="S4.I11.i2.I1.i2.p1.11.m11.2.2.cmml">b</mi><mo id="S4.I11.i2.I1.i2.p1.11.m11.2.3.2.3" stretchy="false" xref="S4.I11.i2.I1.i2.p1.11.m11.2.3.1.cmml">}</mo></mrow><annotation-xml encoding="MathML-Content" id="S4.I11.i2.I1.i2.p1.11.m11.2b"><set id="S4.I11.i2.I1.i2.p1.11.m11.2.3.1.cmml" xref="S4.I11.i2.I1.i2.p1.11.m11.2.3.2"><ci id="S4.I11.i2.I1.i2.p1.11.m11.1.1.cmml" xref="S4.I11.i2.I1.i2.p1.11.m11.1.1">𝑎</ci><ci id="S4.I11.i2.I1.i2.p1.11.m11.2.2.cmml" xref="S4.I11.i2.I1.i2.p1.11.m11.2.2">𝑏</ci></set></annotation-xml><annotation encoding="application/x-tex" id="S4.I11.i2.I1.i2.p1.11.m11.2c">\{a,b\}</annotation><annotation encoding="application/x-llamapun" id="S4.I11.i2.I1.i2.p1.11.m11.2d">{ italic_a , italic_b }</annotation></semantics></math>.</p> </div> </li> <li class="ltx_item" id="S4.I11.i2.I1.i3" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item"><span class="ltx_text ltx_font_bold" id="S4.I11.i2.I1.i3.1.1.1">–</span></span> <div class="ltx_para" id="S4.I11.i2.I1.i3.p1"> <p class="ltx_p" id="S4.I11.i2.I1.i3.p1.18">Suppose <math alttext="u\notin G_{x}" class="ltx_Math" display="inline" id="S4.I11.i2.I1.i3.p1.1.m1.1"><semantics id="S4.I11.i2.I1.i3.p1.1.m1.1a"><mrow id="S4.I11.i2.I1.i3.p1.1.m1.1.1" xref="S4.I11.i2.I1.i3.p1.1.m1.1.1.cmml"><mi id="S4.I11.i2.I1.i3.p1.1.m1.1.1.2" xref="S4.I11.i2.I1.i3.p1.1.m1.1.1.2.cmml">u</mi><mo id="S4.I11.i2.I1.i3.p1.1.m1.1.1.1" xref="S4.I11.i2.I1.i3.p1.1.m1.1.1.1.cmml">∉</mo><msub id="S4.I11.i2.I1.i3.p1.1.m1.1.1.3" xref="S4.I11.i2.I1.i3.p1.1.m1.1.1.3.cmml"><mi id="S4.I11.i2.I1.i3.p1.1.m1.1.1.3.2" xref="S4.I11.i2.I1.i3.p1.1.m1.1.1.3.2.cmml">G</mi><mi id="S4.I11.i2.I1.i3.p1.1.m1.1.1.3.3" xref="S4.I11.i2.I1.i3.p1.1.m1.1.1.3.3.cmml">x</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S4.I11.i2.I1.i3.p1.1.m1.1b"><apply id="S4.I11.i2.I1.i3.p1.1.m1.1.1.cmml" xref="S4.I11.i2.I1.i3.p1.1.m1.1.1"><notin id="S4.I11.i2.I1.i3.p1.1.m1.1.1.1.cmml" xref="S4.I11.i2.I1.i3.p1.1.m1.1.1.1"></notin><ci id="S4.I11.i2.I1.i3.p1.1.m1.1.1.2.cmml" xref="S4.I11.i2.I1.i3.p1.1.m1.1.1.2">𝑢</ci><apply id="S4.I11.i2.I1.i3.p1.1.m1.1.1.3.cmml" xref="S4.I11.i2.I1.i3.p1.1.m1.1.1.3"><csymbol cd="ambiguous" id="S4.I11.i2.I1.i3.p1.1.m1.1.1.3.1.cmml" xref="S4.I11.i2.I1.i3.p1.1.m1.1.1.3">subscript</csymbol><ci id="S4.I11.i2.I1.i3.p1.1.m1.1.1.3.2.cmml" xref="S4.I11.i2.I1.i3.p1.1.m1.1.1.3.2">𝐺</ci><ci id="S4.I11.i2.I1.i3.p1.1.m1.1.1.3.3.cmml" xref="S4.I11.i2.I1.i3.p1.1.m1.1.1.3.3">𝑥</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.I11.i2.I1.i3.p1.1.m1.1c">u\notin G_{x}</annotation><annotation encoding="application/x-llamapun" id="S4.I11.i2.I1.i3.p1.1.m1.1d">italic_u ∉ italic_G start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT</annotation></semantics></math>. Then we can use the same argument as in Case 1. let <math alttext="(u^{\prime},u^{\prime\prime})=L_{h(u)}(j)" class="ltx_Math" display="inline" id="S4.I11.i2.I1.i3.p1.2.m2.4"><semantics id="S4.I11.i2.I1.i3.p1.2.m2.4a"><mrow id="S4.I11.i2.I1.i3.p1.2.m2.4.4" xref="S4.I11.i2.I1.i3.p1.2.m2.4.4.cmml"><mrow id="S4.I11.i2.I1.i3.p1.2.m2.4.4.2.2" xref="S4.I11.i2.I1.i3.p1.2.m2.4.4.2.3.cmml"><mo id="S4.I11.i2.I1.i3.p1.2.m2.4.4.2.2.3" stretchy="false" xref="S4.I11.i2.I1.i3.p1.2.m2.4.4.2.3.cmml">(</mo><msup id="S4.I11.i2.I1.i3.p1.2.m2.3.3.1.1.1" xref="S4.I11.i2.I1.i3.p1.2.m2.3.3.1.1.1.cmml"><mi id="S4.I11.i2.I1.i3.p1.2.m2.3.3.1.1.1.2" xref="S4.I11.i2.I1.i3.p1.2.m2.3.3.1.1.1.2.cmml">u</mi><mo id="S4.I11.i2.I1.i3.p1.2.m2.3.3.1.1.1.3" xref="S4.I11.i2.I1.i3.p1.2.m2.3.3.1.1.1.3.cmml">′</mo></msup><mo id="S4.I11.i2.I1.i3.p1.2.m2.4.4.2.2.4" xref="S4.I11.i2.I1.i3.p1.2.m2.4.4.2.3.cmml">,</mo><msup id="S4.I11.i2.I1.i3.p1.2.m2.4.4.2.2.2" xref="S4.I11.i2.I1.i3.p1.2.m2.4.4.2.2.2.cmml"><mi id="S4.I11.i2.I1.i3.p1.2.m2.4.4.2.2.2.2" xref="S4.I11.i2.I1.i3.p1.2.m2.4.4.2.2.2.2.cmml">u</mi><mo id="S4.I11.i2.I1.i3.p1.2.m2.4.4.2.2.2.3" xref="S4.I11.i2.I1.i3.p1.2.m2.4.4.2.2.2.3.cmml">′′</mo></msup><mo id="S4.I11.i2.I1.i3.p1.2.m2.4.4.2.2.5" stretchy="false" xref="S4.I11.i2.I1.i3.p1.2.m2.4.4.2.3.cmml">)</mo></mrow><mo id="S4.I11.i2.I1.i3.p1.2.m2.4.4.3" xref="S4.I11.i2.I1.i3.p1.2.m2.4.4.3.cmml">=</mo><mrow id="S4.I11.i2.I1.i3.p1.2.m2.4.4.4" xref="S4.I11.i2.I1.i3.p1.2.m2.4.4.4.cmml"><msub id="S4.I11.i2.I1.i3.p1.2.m2.4.4.4.2" xref="S4.I11.i2.I1.i3.p1.2.m2.4.4.4.2.cmml"><mi id="S4.I11.i2.I1.i3.p1.2.m2.4.4.4.2.2" xref="S4.I11.i2.I1.i3.p1.2.m2.4.4.4.2.2.cmml">L</mi><mrow id="S4.I11.i2.I1.i3.p1.2.m2.1.1.1" xref="S4.I11.i2.I1.i3.p1.2.m2.1.1.1.cmml"><mi id="S4.I11.i2.I1.i3.p1.2.m2.1.1.1.3" xref="S4.I11.i2.I1.i3.p1.2.m2.1.1.1.3.cmml">h</mi><mo id="S4.I11.i2.I1.i3.p1.2.m2.1.1.1.2" xref="S4.I11.i2.I1.i3.p1.2.m2.1.1.1.2.cmml">⁢</mo><mrow id="S4.I11.i2.I1.i3.p1.2.m2.1.1.1.4.2" xref="S4.I11.i2.I1.i3.p1.2.m2.1.1.1.cmml"><mo id="S4.I11.i2.I1.i3.p1.2.m2.1.1.1.4.2.1" stretchy="false" xref="S4.I11.i2.I1.i3.p1.2.m2.1.1.1.cmml">(</mo><mi id="S4.I11.i2.I1.i3.p1.2.m2.1.1.1.1" xref="S4.I11.i2.I1.i3.p1.2.m2.1.1.1.1.cmml">u</mi><mo id="S4.I11.i2.I1.i3.p1.2.m2.1.1.1.4.2.2" stretchy="false" xref="S4.I11.i2.I1.i3.p1.2.m2.1.1.1.cmml">)</mo></mrow></mrow></msub><mo id="S4.I11.i2.I1.i3.p1.2.m2.4.4.4.1" xref="S4.I11.i2.I1.i3.p1.2.m2.4.4.4.1.cmml">⁢</mo><mrow id="S4.I11.i2.I1.i3.p1.2.m2.4.4.4.3.2" xref="S4.I11.i2.I1.i3.p1.2.m2.4.4.4.cmml"><mo id="S4.I11.i2.I1.i3.p1.2.m2.4.4.4.3.2.1" stretchy="false" xref="S4.I11.i2.I1.i3.p1.2.m2.4.4.4.cmml">(</mo><mi id="S4.I11.i2.I1.i3.p1.2.m2.2.2" xref="S4.I11.i2.I1.i3.p1.2.m2.2.2.cmml">j</mi><mo id="S4.I11.i2.I1.i3.p1.2.m2.4.4.4.3.2.2" stretchy="false" xref="S4.I11.i2.I1.i3.p1.2.m2.4.4.4.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.I11.i2.I1.i3.p1.2.m2.4b"><apply id="S4.I11.i2.I1.i3.p1.2.m2.4.4.cmml" xref="S4.I11.i2.I1.i3.p1.2.m2.4.4"><eq id="S4.I11.i2.I1.i3.p1.2.m2.4.4.3.cmml" xref="S4.I11.i2.I1.i3.p1.2.m2.4.4.3"></eq><interval closure="open" id="S4.I11.i2.I1.i3.p1.2.m2.4.4.2.3.cmml" xref="S4.I11.i2.I1.i3.p1.2.m2.4.4.2.2"><apply id="S4.I11.i2.I1.i3.p1.2.m2.3.3.1.1.1.cmml" xref="S4.I11.i2.I1.i3.p1.2.m2.3.3.1.1.1"><csymbol cd="ambiguous" id="S4.I11.i2.I1.i3.p1.2.m2.3.3.1.1.1.1.cmml" xref="S4.I11.i2.I1.i3.p1.2.m2.3.3.1.1.1">superscript</csymbol><ci id="S4.I11.i2.I1.i3.p1.2.m2.3.3.1.1.1.2.cmml" xref="S4.I11.i2.I1.i3.p1.2.m2.3.3.1.1.1.2">𝑢</ci><ci id="S4.I11.i2.I1.i3.p1.2.m2.3.3.1.1.1.3.cmml" xref="S4.I11.i2.I1.i3.p1.2.m2.3.3.1.1.1.3">′</ci></apply><apply id="S4.I11.i2.I1.i3.p1.2.m2.4.4.2.2.2.cmml" xref="S4.I11.i2.I1.i3.p1.2.m2.4.4.2.2.2"><csymbol cd="ambiguous" id="S4.I11.i2.I1.i3.p1.2.m2.4.4.2.2.2.1.cmml" xref="S4.I11.i2.I1.i3.p1.2.m2.4.4.2.2.2">superscript</csymbol><ci id="S4.I11.i2.I1.i3.p1.2.m2.4.4.2.2.2.2.cmml" xref="S4.I11.i2.I1.i3.p1.2.m2.4.4.2.2.2.2">𝑢</ci><ci id="S4.I11.i2.I1.i3.p1.2.m2.4.4.2.2.2.3.cmml" xref="S4.I11.i2.I1.i3.p1.2.m2.4.4.2.2.2.3">′′</ci></apply></interval><apply id="S4.I11.i2.I1.i3.p1.2.m2.4.4.4.cmml" xref="S4.I11.i2.I1.i3.p1.2.m2.4.4.4"><times id="S4.I11.i2.I1.i3.p1.2.m2.4.4.4.1.cmml" xref="S4.I11.i2.I1.i3.p1.2.m2.4.4.4.1"></times><apply id="S4.I11.i2.I1.i3.p1.2.m2.4.4.4.2.cmml" xref="S4.I11.i2.I1.i3.p1.2.m2.4.4.4.2"><csymbol cd="ambiguous" id="S4.I11.i2.I1.i3.p1.2.m2.4.4.4.2.1.cmml" xref="S4.I11.i2.I1.i3.p1.2.m2.4.4.4.2">subscript</csymbol><ci id="S4.I11.i2.I1.i3.p1.2.m2.4.4.4.2.2.cmml" xref="S4.I11.i2.I1.i3.p1.2.m2.4.4.4.2.2">𝐿</ci><apply id="S4.I11.i2.I1.i3.p1.2.m2.1.1.1.cmml" xref="S4.I11.i2.I1.i3.p1.2.m2.1.1.1"><times id="S4.I11.i2.I1.i3.p1.2.m2.1.1.1.2.cmml" xref="S4.I11.i2.I1.i3.p1.2.m2.1.1.1.2"></times><ci id="S4.I11.i2.I1.i3.p1.2.m2.1.1.1.3.cmml" xref="S4.I11.i2.I1.i3.p1.2.m2.1.1.1.3">ℎ</ci><ci id="S4.I11.i2.I1.i3.p1.2.m2.1.1.1.1.cmml" xref="S4.I11.i2.I1.i3.p1.2.m2.1.1.1.1">𝑢</ci></apply></apply><ci id="S4.I11.i2.I1.i3.p1.2.m2.2.2.cmml" xref="S4.I11.i2.I1.i3.p1.2.m2.2.2">𝑗</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.I11.i2.I1.i3.p1.2.m2.4c">(u^{\prime},u^{\prime\prime})=L_{h(u)}(j)</annotation><annotation encoding="application/x-llamapun" id="S4.I11.i2.I1.i3.p1.2.m2.4d">( italic_u start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT , italic_u start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT ) = italic_L start_POSTSUBSCRIPT italic_h ( italic_u ) end_POSTSUBSCRIPT ( italic_j )</annotation></semantics></math> where <math alttext="h(u^{\prime})=h(u)" class="ltx_Math" display="inline" id="S4.I11.i2.I1.i3.p1.3.m3.2"><semantics id="S4.I11.i2.I1.i3.p1.3.m3.2a"><mrow id="S4.I11.i2.I1.i3.p1.3.m3.2.2" xref="S4.I11.i2.I1.i3.p1.3.m3.2.2.cmml"><mrow id="S4.I11.i2.I1.i3.p1.3.m3.2.2.1" xref="S4.I11.i2.I1.i3.p1.3.m3.2.2.1.cmml"><mi id="S4.I11.i2.I1.i3.p1.3.m3.2.2.1.3" xref="S4.I11.i2.I1.i3.p1.3.m3.2.2.1.3.cmml">h</mi><mo id="S4.I11.i2.I1.i3.p1.3.m3.2.2.1.2" xref="S4.I11.i2.I1.i3.p1.3.m3.2.2.1.2.cmml">⁢</mo><mrow id="S4.I11.i2.I1.i3.p1.3.m3.2.2.1.1.1" xref="S4.I11.i2.I1.i3.p1.3.m3.2.2.1.1.1.1.cmml"><mo id="S4.I11.i2.I1.i3.p1.3.m3.2.2.1.1.1.2" stretchy="false" xref="S4.I11.i2.I1.i3.p1.3.m3.2.2.1.1.1.1.cmml">(</mo><msup id="S4.I11.i2.I1.i3.p1.3.m3.2.2.1.1.1.1" xref="S4.I11.i2.I1.i3.p1.3.m3.2.2.1.1.1.1.cmml"><mi id="S4.I11.i2.I1.i3.p1.3.m3.2.2.1.1.1.1.2" xref="S4.I11.i2.I1.i3.p1.3.m3.2.2.1.1.1.1.2.cmml">u</mi><mo id="S4.I11.i2.I1.i3.p1.3.m3.2.2.1.1.1.1.3" xref="S4.I11.i2.I1.i3.p1.3.m3.2.2.1.1.1.1.3.cmml">′</mo></msup><mo id="S4.I11.i2.I1.i3.p1.3.m3.2.2.1.1.1.3" stretchy="false" xref="S4.I11.i2.I1.i3.p1.3.m3.2.2.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S4.I11.i2.I1.i3.p1.3.m3.2.2.2" xref="S4.I11.i2.I1.i3.p1.3.m3.2.2.2.cmml">=</mo><mrow id="S4.I11.i2.I1.i3.p1.3.m3.2.2.3" xref="S4.I11.i2.I1.i3.p1.3.m3.2.2.3.cmml"><mi id="S4.I11.i2.I1.i3.p1.3.m3.2.2.3.2" xref="S4.I11.i2.I1.i3.p1.3.m3.2.2.3.2.cmml">h</mi><mo id="S4.I11.i2.I1.i3.p1.3.m3.2.2.3.1" xref="S4.I11.i2.I1.i3.p1.3.m3.2.2.3.1.cmml">⁢</mo><mrow id="S4.I11.i2.I1.i3.p1.3.m3.2.2.3.3.2" xref="S4.I11.i2.I1.i3.p1.3.m3.2.2.3.cmml"><mo id="S4.I11.i2.I1.i3.p1.3.m3.2.2.3.3.2.1" stretchy="false" xref="S4.I11.i2.I1.i3.p1.3.m3.2.2.3.cmml">(</mo><mi id="S4.I11.i2.I1.i3.p1.3.m3.1.1" xref="S4.I11.i2.I1.i3.p1.3.m3.1.1.cmml">u</mi><mo id="S4.I11.i2.I1.i3.p1.3.m3.2.2.3.3.2.2" stretchy="false" xref="S4.I11.i2.I1.i3.p1.3.m3.2.2.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.I11.i2.I1.i3.p1.3.m3.2b"><apply id="S4.I11.i2.I1.i3.p1.3.m3.2.2.cmml" xref="S4.I11.i2.I1.i3.p1.3.m3.2.2"><eq id="S4.I11.i2.I1.i3.p1.3.m3.2.2.2.cmml" xref="S4.I11.i2.I1.i3.p1.3.m3.2.2.2"></eq><apply id="S4.I11.i2.I1.i3.p1.3.m3.2.2.1.cmml" xref="S4.I11.i2.I1.i3.p1.3.m3.2.2.1"><times id="S4.I11.i2.I1.i3.p1.3.m3.2.2.1.2.cmml" xref="S4.I11.i2.I1.i3.p1.3.m3.2.2.1.2"></times><ci id="S4.I11.i2.I1.i3.p1.3.m3.2.2.1.3.cmml" xref="S4.I11.i2.I1.i3.p1.3.m3.2.2.1.3">ℎ</ci><apply id="S4.I11.i2.I1.i3.p1.3.m3.2.2.1.1.1.1.cmml" xref="S4.I11.i2.I1.i3.p1.3.m3.2.2.1.1.1"><csymbol cd="ambiguous" id="S4.I11.i2.I1.i3.p1.3.m3.2.2.1.1.1.1.1.cmml" xref="S4.I11.i2.I1.i3.p1.3.m3.2.2.1.1.1">superscript</csymbol><ci id="S4.I11.i2.I1.i3.p1.3.m3.2.2.1.1.1.1.2.cmml" xref="S4.I11.i2.I1.i3.p1.3.m3.2.2.1.1.1.1.2">𝑢</ci><ci id="S4.I11.i2.I1.i3.p1.3.m3.2.2.1.1.1.1.3.cmml" xref="S4.I11.i2.I1.i3.p1.3.m3.2.2.1.1.1.1.3">′</ci></apply></apply><apply id="S4.I11.i2.I1.i3.p1.3.m3.2.2.3.cmml" xref="S4.I11.i2.I1.i3.p1.3.m3.2.2.3"><times id="S4.I11.i2.I1.i3.p1.3.m3.2.2.3.1.cmml" xref="S4.I11.i2.I1.i3.p1.3.m3.2.2.3.1"></times><ci id="S4.I11.i2.I1.i3.p1.3.m3.2.2.3.2.cmml" xref="S4.I11.i2.I1.i3.p1.3.m3.2.2.3.2">ℎ</ci><ci id="S4.I11.i2.I1.i3.p1.3.m3.1.1.cmml" xref="S4.I11.i2.I1.i3.p1.3.m3.1.1">𝑢</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.I11.i2.I1.i3.p1.3.m3.2c">h(u^{\prime})=h(u)</annotation><annotation encoding="application/x-llamapun" id="S4.I11.i2.I1.i3.p1.3.m3.2d">italic_h ( italic_u start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) = italic_h ( italic_u )</annotation></semantics></math>; this must be in <span class="ltx_text ltx_markedasmath" id="S4.I11.i2.I1.i3.p1.18.1">SOL</span> since <math alttext="(u,v)\in\textnormal{OPT}" class="ltx_Math" display="inline" id="S4.I11.i2.I1.i3.p1.5.m5.2"><semantics id="S4.I11.i2.I1.i3.p1.5.m5.2a"><mrow id="S4.I11.i2.I1.i3.p1.5.m5.2.3" xref="S4.I11.i2.I1.i3.p1.5.m5.2.3.cmml"><mrow id="S4.I11.i2.I1.i3.p1.5.m5.2.3.2.2" xref="S4.I11.i2.I1.i3.p1.5.m5.2.3.2.1.cmml"><mo id="S4.I11.i2.I1.i3.p1.5.m5.2.3.2.2.1" stretchy="false" xref="S4.I11.i2.I1.i3.p1.5.m5.2.3.2.1.cmml">(</mo><mi id="S4.I11.i2.I1.i3.p1.5.m5.1.1" xref="S4.I11.i2.I1.i3.p1.5.m5.1.1.cmml">u</mi><mo id="S4.I11.i2.I1.i3.p1.5.m5.2.3.2.2.2" xref="S4.I11.i2.I1.i3.p1.5.m5.2.3.2.1.cmml">,</mo><mi id="S4.I11.i2.I1.i3.p1.5.m5.2.2" xref="S4.I11.i2.I1.i3.p1.5.m5.2.2.cmml">v</mi><mo id="S4.I11.i2.I1.i3.p1.5.m5.2.3.2.2.3" stretchy="false" xref="S4.I11.i2.I1.i3.p1.5.m5.2.3.2.1.cmml">)</mo></mrow><mo id="S4.I11.i2.I1.i3.p1.5.m5.2.3.1" xref="S4.I11.i2.I1.i3.p1.5.m5.2.3.1.cmml">∈</mo><mtext id="S4.I11.i2.I1.i3.p1.5.m5.2.3.3" xref="S4.I11.i2.I1.i3.p1.5.m5.2.3.3a.cmml">OPT</mtext></mrow><annotation-xml encoding="MathML-Content" id="S4.I11.i2.I1.i3.p1.5.m5.2b"><apply id="S4.I11.i2.I1.i3.p1.5.m5.2.3.cmml" xref="S4.I11.i2.I1.i3.p1.5.m5.2.3"><in id="S4.I11.i2.I1.i3.p1.5.m5.2.3.1.cmml" xref="S4.I11.i2.I1.i3.p1.5.m5.2.3.1"></in><interval closure="open" id="S4.I11.i2.I1.i3.p1.5.m5.2.3.2.1.cmml" xref="S4.I11.i2.I1.i3.p1.5.m5.2.3.2.2"><ci id="S4.I11.i2.I1.i3.p1.5.m5.1.1.cmml" xref="S4.I11.i2.I1.i3.p1.5.m5.1.1">𝑢</ci><ci id="S4.I11.i2.I1.i3.p1.5.m5.2.2.cmml" xref="S4.I11.i2.I1.i3.p1.5.m5.2.2">𝑣</ci></interval><ci id="S4.I11.i2.I1.i3.p1.5.m5.2.3.3a.cmml" xref="S4.I11.i2.I1.i3.p1.5.m5.2.3.3"><mtext id="S4.I11.i2.I1.i3.p1.5.m5.2.3.3.cmml" xref="S4.I11.i2.I1.i3.p1.5.m5.2.3.3">OPT</mtext></ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.I11.i2.I1.i3.p1.5.m5.2c">(u,v)\in\textnormal{OPT}</annotation><annotation encoding="application/x-llamapun" id="S4.I11.i2.I1.i3.p1.5.m5.2d">( italic_u , italic_v ) ∈ OPT</annotation></semantics></math>. Furthermore, <math alttext="u^{\prime}\notin\{a,b\}" class="ltx_Math" display="inline" id="S4.I11.i2.I1.i3.p1.6.m6.2"><semantics id="S4.I11.i2.I1.i3.p1.6.m6.2a"><mrow id="S4.I11.i2.I1.i3.p1.6.m6.2.3" xref="S4.I11.i2.I1.i3.p1.6.m6.2.3.cmml"><msup id="S4.I11.i2.I1.i3.p1.6.m6.2.3.2" xref="S4.I11.i2.I1.i3.p1.6.m6.2.3.2.cmml"><mi id="S4.I11.i2.I1.i3.p1.6.m6.2.3.2.2" xref="S4.I11.i2.I1.i3.p1.6.m6.2.3.2.2.cmml">u</mi><mo id="S4.I11.i2.I1.i3.p1.6.m6.2.3.2.3" xref="S4.I11.i2.I1.i3.p1.6.m6.2.3.2.3.cmml">′</mo></msup><mo id="S4.I11.i2.I1.i3.p1.6.m6.2.3.1" xref="S4.I11.i2.I1.i3.p1.6.m6.2.3.1.cmml">∉</mo><mrow id="S4.I11.i2.I1.i3.p1.6.m6.2.3.3.2" xref="S4.I11.i2.I1.i3.p1.6.m6.2.3.3.1.cmml"><mo id="S4.I11.i2.I1.i3.p1.6.m6.2.3.3.2.1" stretchy="false" xref="S4.I11.i2.I1.i3.p1.6.m6.2.3.3.1.cmml">{</mo><mi id="S4.I11.i2.I1.i3.p1.6.m6.1.1" xref="S4.I11.i2.I1.i3.p1.6.m6.1.1.cmml">a</mi><mo id="S4.I11.i2.I1.i3.p1.6.m6.2.3.3.2.2" xref="S4.I11.i2.I1.i3.p1.6.m6.2.3.3.1.cmml">,</mo><mi id="S4.I11.i2.I1.i3.p1.6.m6.2.2" xref="S4.I11.i2.I1.i3.p1.6.m6.2.2.cmml">b</mi><mo id="S4.I11.i2.I1.i3.p1.6.m6.2.3.3.2.3" stretchy="false" xref="S4.I11.i2.I1.i3.p1.6.m6.2.3.3.1.cmml">}</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.I11.i2.I1.i3.p1.6.m6.2b"><apply id="S4.I11.i2.I1.i3.p1.6.m6.2.3.cmml" xref="S4.I11.i2.I1.i3.p1.6.m6.2.3"><notin id="S4.I11.i2.I1.i3.p1.6.m6.2.3.1.cmml" xref="S4.I11.i2.I1.i3.p1.6.m6.2.3.1"></notin><apply id="S4.I11.i2.I1.i3.p1.6.m6.2.3.2.cmml" xref="S4.I11.i2.I1.i3.p1.6.m6.2.3.2"><csymbol cd="ambiguous" id="S4.I11.i2.I1.i3.p1.6.m6.2.3.2.1.cmml" xref="S4.I11.i2.I1.i3.p1.6.m6.2.3.2">superscript</csymbol><ci id="S4.I11.i2.I1.i3.p1.6.m6.2.3.2.2.cmml" xref="S4.I11.i2.I1.i3.p1.6.m6.2.3.2.2">𝑢</ci><ci id="S4.I11.i2.I1.i3.p1.6.m6.2.3.2.3.cmml" xref="S4.I11.i2.I1.i3.p1.6.m6.2.3.2.3">′</ci></apply><set id="S4.I11.i2.I1.i3.p1.6.m6.2.3.3.1.cmml" xref="S4.I11.i2.I1.i3.p1.6.m6.2.3.3.2"><ci id="S4.I11.i2.I1.i3.p1.6.m6.1.1.cmml" xref="S4.I11.i2.I1.i3.p1.6.m6.1.1">𝑎</ci><ci id="S4.I11.i2.I1.i3.p1.6.m6.2.2.cmml" xref="S4.I11.i2.I1.i3.p1.6.m6.2.2">𝑏</ci></set></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.I11.i2.I1.i3.p1.6.m6.2c">u^{\prime}\notin\{a,b\}</annotation><annotation encoding="application/x-llamapun" id="S4.I11.i2.I1.i3.p1.6.m6.2d">italic_u start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ∉ { italic_a , italic_b }</annotation></semantics></math>, since <math alttext="h(u^{\prime})=h(u)" class="ltx_Math" display="inline" id="S4.I11.i2.I1.i3.p1.7.m7.2"><semantics id="S4.I11.i2.I1.i3.p1.7.m7.2a"><mrow id="S4.I11.i2.I1.i3.p1.7.m7.2.2" xref="S4.I11.i2.I1.i3.p1.7.m7.2.2.cmml"><mrow id="S4.I11.i2.I1.i3.p1.7.m7.2.2.1" xref="S4.I11.i2.I1.i3.p1.7.m7.2.2.1.cmml"><mi id="S4.I11.i2.I1.i3.p1.7.m7.2.2.1.3" xref="S4.I11.i2.I1.i3.p1.7.m7.2.2.1.3.cmml">h</mi><mo id="S4.I11.i2.I1.i3.p1.7.m7.2.2.1.2" xref="S4.I11.i2.I1.i3.p1.7.m7.2.2.1.2.cmml">⁢</mo><mrow id="S4.I11.i2.I1.i3.p1.7.m7.2.2.1.1.1" xref="S4.I11.i2.I1.i3.p1.7.m7.2.2.1.1.1.1.cmml"><mo id="S4.I11.i2.I1.i3.p1.7.m7.2.2.1.1.1.2" stretchy="false" xref="S4.I11.i2.I1.i3.p1.7.m7.2.2.1.1.1.1.cmml">(</mo><msup id="S4.I11.i2.I1.i3.p1.7.m7.2.2.1.1.1.1" xref="S4.I11.i2.I1.i3.p1.7.m7.2.2.1.1.1.1.cmml"><mi id="S4.I11.i2.I1.i3.p1.7.m7.2.2.1.1.1.1.2" xref="S4.I11.i2.I1.i3.p1.7.m7.2.2.1.1.1.1.2.cmml">u</mi><mo id="S4.I11.i2.I1.i3.p1.7.m7.2.2.1.1.1.1.3" xref="S4.I11.i2.I1.i3.p1.7.m7.2.2.1.1.1.1.3.cmml">′</mo></msup><mo id="S4.I11.i2.I1.i3.p1.7.m7.2.2.1.1.1.3" stretchy="false" xref="S4.I11.i2.I1.i3.p1.7.m7.2.2.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S4.I11.i2.I1.i3.p1.7.m7.2.2.2" xref="S4.I11.i2.I1.i3.p1.7.m7.2.2.2.cmml">=</mo><mrow id="S4.I11.i2.I1.i3.p1.7.m7.2.2.3" xref="S4.I11.i2.I1.i3.p1.7.m7.2.2.3.cmml"><mi id="S4.I11.i2.I1.i3.p1.7.m7.2.2.3.2" xref="S4.I11.i2.I1.i3.p1.7.m7.2.2.3.2.cmml">h</mi><mo id="S4.I11.i2.I1.i3.p1.7.m7.2.2.3.1" xref="S4.I11.i2.I1.i3.p1.7.m7.2.2.3.1.cmml">⁢</mo><mrow id="S4.I11.i2.I1.i3.p1.7.m7.2.2.3.3.2" xref="S4.I11.i2.I1.i3.p1.7.m7.2.2.3.cmml"><mo id="S4.I11.i2.I1.i3.p1.7.m7.2.2.3.3.2.1" stretchy="false" xref="S4.I11.i2.I1.i3.p1.7.m7.2.2.3.cmml">(</mo><mi id="S4.I11.i2.I1.i3.p1.7.m7.1.1" xref="S4.I11.i2.I1.i3.p1.7.m7.1.1.cmml">u</mi><mo id="S4.I11.i2.I1.i3.p1.7.m7.2.2.3.3.2.2" stretchy="false" xref="S4.I11.i2.I1.i3.p1.7.m7.2.2.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.I11.i2.I1.i3.p1.7.m7.2b"><apply id="S4.I11.i2.I1.i3.p1.7.m7.2.2.cmml" xref="S4.I11.i2.I1.i3.p1.7.m7.2.2"><eq id="S4.I11.i2.I1.i3.p1.7.m7.2.2.2.cmml" xref="S4.I11.i2.I1.i3.p1.7.m7.2.2.2"></eq><apply id="S4.I11.i2.I1.i3.p1.7.m7.2.2.1.cmml" xref="S4.I11.i2.I1.i3.p1.7.m7.2.2.1"><times id="S4.I11.i2.I1.i3.p1.7.m7.2.2.1.2.cmml" xref="S4.I11.i2.I1.i3.p1.7.m7.2.2.1.2"></times><ci id="S4.I11.i2.I1.i3.p1.7.m7.2.2.1.3.cmml" xref="S4.I11.i2.I1.i3.p1.7.m7.2.2.1.3">ℎ</ci><apply id="S4.I11.i2.I1.i3.p1.7.m7.2.2.1.1.1.1.cmml" xref="S4.I11.i2.I1.i3.p1.7.m7.2.2.1.1.1"><csymbol cd="ambiguous" id="S4.I11.i2.I1.i3.p1.7.m7.2.2.1.1.1.1.1.cmml" xref="S4.I11.i2.I1.i3.p1.7.m7.2.2.1.1.1">superscript</csymbol><ci id="S4.I11.i2.I1.i3.p1.7.m7.2.2.1.1.1.1.2.cmml" xref="S4.I11.i2.I1.i3.p1.7.m7.2.2.1.1.1.1.2">𝑢</ci><ci id="S4.I11.i2.I1.i3.p1.7.m7.2.2.1.1.1.1.3.cmml" xref="S4.I11.i2.I1.i3.p1.7.m7.2.2.1.1.1.1.3">′</ci></apply></apply><apply id="S4.I11.i2.I1.i3.p1.7.m7.2.2.3.cmml" xref="S4.I11.i2.I1.i3.p1.7.m7.2.2.3"><times id="S4.I11.i2.I1.i3.p1.7.m7.2.2.3.1.cmml" xref="S4.I11.i2.I1.i3.p1.7.m7.2.2.3.1"></times><ci id="S4.I11.i2.I1.i3.p1.7.m7.2.2.3.2.cmml" xref="S4.I11.i2.I1.i3.p1.7.m7.2.2.3.2">ℎ</ci><ci id="S4.I11.i2.I1.i3.p1.7.m7.1.1.cmml" xref="S4.I11.i2.I1.i3.p1.7.m7.1.1">𝑢</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.I11.i2.I1.i3.p1.7.m7.2c">h(u^{\prime})=h(u)</annotation><annotation encoding="application/x-llamapun" id="S4.I11.i2.I1.i3.p1.7.m7.2d">italic_h ( italic_u start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) = italic_h ( italic_u )</annotation></semantics></math> which is in a strict subtree of <math alttext="T_{x}" class="ltx_Math" display="inline" id="S4.I11.i2.I1.i3.p1.8.m8.1"><semantics id="S4.I11.i2.I1.i3.p1.8.m8.1a"><msub id="S4.I11.i2.I1.i3.p1.8.m8.1.1" xref="S4.I11.i2.I1.i3.p1.8.m8.1.1.cmml"><mi id="S4.I11.i2.I1.i3.p1.8.m8.1.1.2" xref="S4.I11.i2.I1.i3.p1.8.m8.1.1.2.cmml">T</mi><mi id="S4.I11.i2.I1.i3.p1.8.m8.1.1.3" xref="S4.I11.i2.I1.i3.p1.8.m8.1.1.3.cmml">x</mi></msub><annotation-xml encoding="MathML-Content" id="S4.I11.i2.I1.i3.p1.8.m8.1b"><apply id="S4.I11.i2.I1.i3.p1.8.m8.1.1.cmml" xref="S4.I11.i2.I1.i3.p1.8.m8.1.1"><csymbol cd="ambiguous" id="S4.I11.i2.I1.i3.p1.8.m8.1.1.1.cmml" xref="S4.I11.i2.I1.i3.p1.8.m8.1.1">subscript</csymbol><ci id="S4.I11.i2.I1.i3.p1.8.m8.1.1.2.cmml" xref="S4.I11.i2.I1.i3.p1.8.m8.1.1.2">𝑇</ci><ci id="S4.I11.i2.I1.i3.p1.8.m8.1.1.3.cmml" xref="S4.I11.i2.I1.i3.p1.8.m8.1.1.3">𝑥</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.I11.i2.I1.i3.p1.8.m8.1c">T_{x}</annotation><annotation encoding="application/x-llamapun" id="S4.I11.i2.I1.i3.p1.8.m8.1d">italic_T start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT</annotation></semantics></math>, but <math alttext="h(a),h(b)" class="ltx_Math" display="inline" id="S4.I11.i2.I1.i3.p1.9.m9.4"><semantics id="S4.I11.i2.I1.i3.p1.9.m9.4a"><mrow id="S4.I11.i2.I1.i3.p1.9.m9.4.4.2" xref="S4.I11.i2.I1.i3.p1.9.m9.4.4.3.cmml"><mrow id="S4.I11.i2.I1.i3.p1.9.m9.3.3.1.1" xref="S4.I11.i2.I1.i3.p1.9.m9.3.3.1.1.cmml"><mi id="S4.I11.i2.I1.i3.p1.9.m9.3.3.1.1.2" xref="S4.I11.i2.I1.i3.p1.9.m9.3.3.1.1.2.cmml">h</mi><mo id="S4.I11.i2.I1.i3.p1.9.m9.3.3.1.1.1" xref="S4.I11.i2.I1.i3.p1.9.m9.3.3.1.1.1.cmml">⁢</mo><mrow id="S4.I11.i2.I1.i3.p1.9.m9.3.3.1.1.3.2" xref="S4.I11.i2.I1.i3.p1.9.m9.3.3.1.1.cmml"><mo id="S4.I11.i2.I1.i3.p1.9.m9.3.3.1.1.3.2.1" stretchy="false" xref="S4.I11.i2.I1.i3.p1.9.m9.3.3.1.1.cmml">(</mo><mi id="S4.I11.i2.I1.i3.p1.9.m9.1.1" xref="S4.I11.i2.I1.i3.p1.9.m9.1.1.cmml">a</mi><mo id="S4.I11.i2.I1.i3.p1.9.m9.3.3.1.1.3.2.2" stretchy="false" xref="S4.I11.i2.I1.i3.p1.9.m9.3.3.1.1.cmml">)</mo></mrow></mrow><mo id="S4.I11.i2.I1.i3.p1.9.m9.4.4.2.3" xref="S4.I11.i2.I1.i3.p1.9.m9.4.4.3.cmml">,</mo><mrow id="S4.I11.i2.I1.i3.p1.9.m9.4.4.2.2" xref="S4.I11.i2.I1.i3.p1.9.m9.4.4.2.2.cmml"><mi id="S4.I11.i2.I1.i3.p1.9.m9.4.4.2.2.2" xref="S4.I11.i2.I1.i3.p1.9.m9.4.4.2.2.2.cmml">h</mi><mo id="S4.I11.i2.I1.i3.p1.9.m9.4.4.2.2.1" xref="S4.I11.i2.I1.i3.p1.9.m9.4.4.2.2.1.cmml">⁢</mo><mrow id="S4.I11.i2.I1.i3.p1.9.m9.4.4.2.2.3.2" xref="S4.I11.i2.I1.i3.p1.9.m9.4.4.2.2.cmml"><mo id="S4.I11.i2.I1.i3.p1.9.m9.4.4.2.2.3.2.1" stretchy="false" xref="S4.I11.i2.I1.i3.p1.9.m9.4.4.2.2.cmml">(</mo><mi id="S4.I11.i2.I1.i3.p1.9.m9.2.2" xref="S4.I11.i2.I1.i3.p1.9.m9.2.2.cmml">b</mi><mo id="S4.I11.i2.I1.i3.p1.9.m9.4.4.2.2.3.2.2" stretchy="false" xref="S4.I11.i2.I1.i3.p1.9.m9.4.4.2.2.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.I11.i2.I1.i3.p1.9.m9.4b"><list id="S4.I11.i2.I1.i3.p1.9.m9.4.4.3.cmml" xref="S4.I11.i2.I1.i3.p1.9.m9.4.4.2"><apply id="S4.I11.i2.I1.i3.p1.9.m9.3.3.1.1.cmml" xref="S4.I11.i2.I1.i3.p1.9.m9.3.3.1.1"><times id="S4.I11.i2.I1.i3.p1.9.m9.3.3.1.1.1.cmml" xref="S4.I11.i2.I1.i3.p1.9.m9.3.3.1.1.1"></times><ci id="S4.I11.i2.I1.i3.p1.9.m9.3.3.1.1.2.cmml" xref="S4.I11.i2.I1.i3.p1.9.m9.3.3.1.1.2">ℎ</ci><ci id="S4.I11.i2.I1.i3.p1.9.m9.1.1.cmml" xref="S4.I11.i2.I1.i3.p1.9.m9.1.1">𝑎</ci></apply><apply id="S4.I11.i2.I1.i3.p1.9.m9.4.4.2.2.cmml" xref="S4.I11.i2.I1.i3.p1.9.m9.4.4.2.2"><times id="S4.I11.i2.I1.i3.p1.9.m9.4.4.2.2.1.cmml" xref="S4.I11.i2.I1.i3.p1.9.m9.4.4.2.2.1"></times><ci id="S4.I11.i2.I1.i3.p1.9.m9.4.4.2.2.2.cmml" xref="S4.I11.i2.I1.i3.p1.9.m9.4.4.2.2.2">ℎ</ci><ci id="S4.I11.i2.I1.i3.p1.9.m9.2.2.cmml" xref="S4.I11.i2.I1.i3.p1.9.m9.2.2">𝑏</ci></apply></list></annotation-xml><annotation encoding="application/x-tex" id="S4.I11.i2.I1.i3.p1.9.m9.4c">h(a),h(b)</annotation><annotation encoding="application/x-llamapun" id="S4.I11.i2.I1.i3.p1.9.m9.4d">italic_h ( italic_a ) , italic_h ( italic_b )</annotation></semantics></math> are at least as close to the root as <math alttext="x" class="ltx_Math" display="inline" id="S4.I11.i2.I1.i3.p1.10.m10.1"><semantics id="S4.I11.i2.I1.i3.p1.10.m10.1a"><mi id="S4.I11.i2.I1.i3.p1.10.m10.1.1" xref="S4.I11.i2.I1.i3.p1.10.m10.1.1.cmml">x</mi><annotation-xml encoding="MathML-Content" id="S4.I11.i2.I1.i3.p1.10.m10.1b"><ci id="S4.I11.i2.I1.i3.p1.10.m10.1.1.cmml" xref="S4.I11.i2.I1.i3.p1.10.m10.1.1">𝑥</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.I11.i2.I1.i3.p1.10.m10.1c">x</annotation><annotation encoding="application/x-llamapun" id="S4.I11.i2.I1.i3.p1.10.m10.1d">italic_x</annotation></semantics></math>. By the same reasoning as Case 1, <math alttext="\ell(u^{\prime\prime})\in T\setminus T_{x}" class="ltx_Math" display="inline" id="S4.I11.i2.I1.i3.p1.11.m11.1"><semantics id="S4.I11.i2.I1.i3.p1.11.m11.1a"><mrow id="S4.I11.i2.I1.i3.p1.11.m11.1.1" xref="S4.I11.i2.I1.i3.p1.11.m11.1.1.cmml"><mrow id="S4.I11.i2.I1.i3.p1.11.m11.1.1.1" xref="S4.I11.i2.I1.i3.p1.11.m11.1.1.1.cmml"><mi id="S4.I11.i2.I1.i3.p1.11.m11.1.1.1.3" mathvariant="normal" xref="S4.I11.i2.I1.i3.p1.11.m11.1.1.1.3.cmml">ℓ</mi><mo id="S4.I11.i2.I1.i3.p1.11.m11.1.1.1.2" xref="S4.I11.i2.I1.i3.p1.11.m11.1.1.1.2.cmml">⁢</mo><mrow id="S4.I11.i2.I1.i3.p1.11.m11.1.1.1.1.1" xref="S4.I11.i2.I1.i3.p1.11.m11.1.1.1.1.1.1.cmml"><mo id="S4.I11.i2.I1.i3.p1.11.m11.1.1.1.1.1.2" stretchy="false" xref="S4.I11.i2.I1.i3.p1.11.m11.1.1.1.1.1.1.cmml">(</mo><msup id="S4.I11.i2.I1.i3.p1.11.m11.1.1.1.1.1.1" xref="S4.I11.i2.I1.i3.p1.11.m11.1.1.1.1.1.1.cmml"><mi id="S4.I11.i2.I1.i3.p1.11.m11.1.1.1.1.1.1.2" xref="S4.I11.i2.I1.i3.p1.11.m11.1.1.1.1.1.1.2.cmml">u</mi><mo id="S4.I11.i2.I1.i3.p1.11.m11.1.1.1.1.1.1.3" xref="S4.I11.i2.I1.i3.p1.11.m11.1.1.1.1.1.1.3.cmml">′′</mo></msup><mo id="S4.I11.i2.I1.i3.p1.11.m11.1.1.1.1.1.3" stretchy="false" xref="S4.I11.i2.I1.i3.p1.11.m11.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S4.I11.i2.I1.i3.p1.11.m11.1.1.2" xref="S4.I11.i2.I1.i3.p1.11.m11.1.1.2.cmml">∈</mo><mrow id="S4.I11.i2.I1.i3.p1.11.m11.1.1.3" xref="S4.I11.i2.I1.i3.p1.11.m11.1.1.3.cmml"><mi id="S4.I11.i2.I1.i3.p1.11.m11.1.1.3.2" xref="S4.I11.i2.I1.i3.p1.11.m11.1.1.3.2.cmml">T</mi><mo id="S4.I11.i2.I1.i3.p1.11.m11.1.1.3.1" xref="S4.I11.i2.I1.i3.p1.11.m11.1.1.3.1.cmml">∖</mo><msub id="S4.I11.i2.I1.i3.p1.11.m11.1.1.3.3" xref="S4.I11.i2.I1.i3.p1.11.m11.1.1.3.3.cmml"><mi id="S4.I11.i2.I1.i3.p1.11.m11.1.1.3.3.2" xref="S4.I11.i2.I1.i3.p1.11.m11.1.1.3.3.2.cmml">T</mi><mi id="S4.I11.i2.I1.i3.p1.11.m11.1.1.3.3.3" xref="S4.I11.i2.I1.i3.p1.11.m11.1.1.3.3.3.cmml">x</mi></msub></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.I11.i2.I1.i3.p1.11.m11.1b"><apply id="S4.I11.i2.I1.i3.p1.11.m11.1.1.cmml" xref="S4.I11.i2.I1.i3.p1.11.m11.1.1"><in id="S4.I11.i2.I1.i3.p1.11.m11.1.1.2.cmml" xref="S4.I11.i2.I1.i3.p1.11.m11.1.1.2"></in><apply id="S4.I11.i2.I1.i3.p1.11.m11.1.1.1.cmml" xref="S4.I11.i2.I1.i3.p1.11.m11.1.1.1"><times id="S4.I11.i2.I1.i3.p1.11.m11.1.1.1.2.cmml" xref="S4.I11.i2.I1.i3.p1.11.m11.1.1.1.2"></times><ci id="S4.I11.i2.I1.i3.p1.11.m11.1.1.1.3.cmml" xref="S4.I11.i2.I1.i3.p1.11.m11.1.1.1.3">ℓ</ci><apply id="S4.I11.i2.I1.i3.p1.11.m11.1.1.1.1.1.1.cmml" xref="S4.I11.i2.I1.i3.p1.11.m11.1.1.1.1.1"><csymbol cd="ambiguous" id="S4.I11.i2.I1.i3.p1.11.m11.1.1.1.1.1.1.1.cmml" xref="S4.I11.i2.I1.i3.p1.11.m11.1.1.1.1.1">superscript</csymbol><ci id="S4.I11.i2.I1.i3.p1.11.m11.1.1.1.1.1.1.2.cmml" xref="S4.I11.i2.I1.i3.p1.11.m11.1.1.1.1.1.1.2">𝑢</ci><ci id="S4.I11.i2.I1.i3.p1.11.m11.1.1.1.1.1.1.3.cmml" xref="S4.I11.i2.I1.i3.p1.11.m11.1.1.1.1.1.1.3">′′</ci></apply></apply><apply id="S4.I11.i2.I1.i3.p1.11.m11.1.1.3.cmml" xref="S4.I11.i2.I1.i3.p1.11.m11.1.1.3"><setdiff id="S4.I11.i2.I1.i3.p1.11.m11.1.1.3.1.cmml" xref="S4.I11.i2.I1.i3.p1.11.m11.1.1.3.1"></setdiff><ci id="S4.I11.i2.I1.i3.p1.11.m11.1.1.3.2.cmml" xref="S4.I11.i2.I1.i3.p1.11.m11.1.1.3.2">𝑇</ci><apply id="S4.I11.i2.I1.i3.p1.11.m11.1.1.3.3.cmml" xref="S4.I11.i2.I1.i3.p1.11.m11.1.1.3.3"><csymbol cd="ambiguous" id="S4.I11.i2.I1.i3.p1.11.m11.1.1.3.3.1.cmml" xref="S4.I11.i2.I1.i3.p1.11.m11.1.1.3.3">subscript</csymbol><ci id="S4.I11.i2.I1.i3.p1.11.m11.1.1.3.3.2.cmml" xref="S4.I11.i2.I1.i3.p1.11.m11.1.1.3.3.2">𝑇</ci><ci id="S4.I11.i2.I1.i3.p1.11.m11.1.1.3.3.3.cmml" xref="S4.I11.i2.I1.i3.p1.11.m11.1.1.3.3.3">𝑥</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.I11.i2.I1.i3.p1.11.m11.1c">\ell(u^{\prime\prime})\in T\setminus T_{x}</annotation><annotation encoding="application/x-llamapun" id="S4.I11.i2.I1.i3.p1.11.m11.1d">roman_ℓ ( italic_u start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT ) ∈ italic_T ∖ italic_T start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT</annotation></semantics></math>, so <math alttext="u^{\prime\prime}\notin\{a,b\}" class="ltx_Math" display="inline" id="S4.I11.i2.I1.i3.p1.12.m12.2"><semantics id="S4.I11.i2.I1.i3.p1.12.m12.2a"><mrow id="S4.I11.i2.I1.i3.p1.12.m12.2.3" xref="S4.I11.i2.I1.i3.p1.12.m12.2.3.cmml"><msup id="S4.I11.i2.I1.i3.p1.12.m12.2.3.2" xref="S4.I11.i2.I1.i3.p1.12.m12.2.3.2.cmml"><mi id="S4.I11.i2.I1.i3.p1.12.m12.2.3.2.2" xref="S4.I11.i2.I1.i3.p1.12.m12.2.3.2.2.cmml">u</mi><mo id="S4.I11.i2.I1.i3.p1.12.m12.2.3.2.3" xref="S4.I11.i2.I1.i3.p1.12.m12.2.3.2.3.cmml">′′</mo></msup><mo id="S4.I11.i2.I1.i3.p1.12.m12.2.3.1" xref="S4.I11.i2.I1.i3.p1.12.m12.2.3.1.cmml">∉</mo><mrow id="S4.I11.i2.I1.i3.p1.12.m12.2.3.3.2" xref="S4.I11.i2.I1.i3.p1.12.m12.2.3.3.1.cmml"><mo id="S4.I11.i2.I1.i3.p1.12.m12.2.3.3.2.1" stretchy="false" xref="S4.I11.i2.I1.i3.p1.12.m12.2.3.3.1.cmml">{</mo><mi id="S4.I11.i2.I1.i3.p1.12.m12.1.1" xref="S4.I11.i2.I1.i3.p1.12.m12.1.1.cmml">a</mi><mo id="S4.I11.i2.I1.i3.p1.12.m12.2.3.3.2.2" xref="S4.I11.i2.I1.i3.p1.12.m12.2.3.3.1.cmml">,</mo><mi id="S4.I11.i2.I1.i3.p1.12.m12.2.2" xref="S4.I11.i2.I1.i3.p1.12.m12.2.2.cmml">b</mi><mo id="S4.I11.i2.I1.i3.p1.12.m12.2.3.3.2.3" stretchy="false" xref="S4.I11.i2.I1.i3.p1.12.m12.2.3.3.1.cmml">}</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.I11.i2.I1.i3.p1.12.m12.2b"><apply id="S4.I11.i2.I1.i3.p1.12.m12.2.3.cmml" xref="S4.I11.i2.I1.i3.p1.12.m12.2.3"><notin id="S4.I11.i2.I1.i3.p1.12.m12.2.3.1.cmml" xref="S4.I11.i2.I1.i3.p1.12.m12.2.3.1"></notin><apply id="S4.I11.i2.I1.i3.p1.12.m12.2.3.2.cmml" xref="S4.I11.i2.I1.i3.p1.12.m12.2.3.2"><csymbol cd="ambiguous" id="S4.I11.i2.I1.i3.p1.12.m12.2.3.2.1.cmml" xref="S4.I11.i2.I1.i3.p1.12.m12.2.3.2">superscript</csymbol><ci id="S4.I11.i2.I1.i3.p1.12.m12.2.3.2.2.cmml" xref="S4.I11.i2.I1.i3.p1.12.m12.2.3.2.2">𝑢</ci><ci id="S4.I11.i2.I1.i3.p1.12.m12.2.3.2.3.cmml" xref="S4.I11.i2.I1.i3.p1.12.m12.2.3.2.3">′′</ci></apply><set id="S4.I11.i2.I1.i3.p1.12.m12.2.3.3.1.cmml" xref="S4.I11.i2.I1.i3.p1.12.m12.2.3.3.2"><ci id="S4.I11.i2.I1.i3.p1.12.m12.1.1.cmml" xref="S4.I11.i2.I1.i3.p1.12.m12.1.1">𝑎</ci><ci id="S4.I11.i2.I1.i3.p1.12.m12.2.2.cmml" xref="S4.I11.i2.I1.i3.p1.12.m12.2.2">𝑏</ci></set></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.I11.i2.I1.i3.p1.12.m12.2c">u^{\prime\prime}\notin\{a,b\}</annotation><annotation encoding="application/x-llamapun" id="S4.I11.i2.I1.i3.p1.12.m12.2d">italic_u start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT ∉ { italic_a , italic_b }</annotation></semantics></math>, since <math alttext="\ell(a),\ell(b)\in T_{x}" class="ltx_Math" display="inline" id="S4.I11.i2.I1.i3.p1.13.m13.4"><semantics id="S4.I11.i2.I1.i3.p1.13.m13.4a"><mrow id="S4.I11.i2.I1.i3.p1.13.m13.4.4" xref="S4.I11.i2.I1.i3.p1.13.m13.4.4.cmml"><mrow id="S4.I11.i2.I1.i3.p1.13.m13.4.4.2.2" xref="S4.I11.i2.I1.i3.p1.13.m13.4.4.2.3.cmml"><mrow id="S4.I11.i2.I1.i3.p1.13.m13.3.3.1.1.1" xref="S4.I11.i2.I1.i3.p1.13.m13.3.3.1.1.1.cmml"><mi id="S4.I11.i2.I1.i3.p1.13.m13.3.3.1.1.1.2" mathvariant="normal" xref="S4.I11.i2.I1.i3.p1.13.m13.3.3.1.1.1.2.cmml">ℓ</mi><mo id="S4.I11.i2.I1.i3.p1.13.m13.3.3.1.1.1.1" xref="S4.I11.i2.I1.i3.p1.13.m13.3.3.1.1.1.1.cmml">⁢</mo><mrow id="S4.I11.i2.I1.i3.p1.13.m13.3.3.1.1.1.3.2" xref="S4.I11.i2.I1.i3.p1.13.m13.3.3.1.1.1.cmml"><mo id="S4.I11.i2.I1.i3.p1.13.m13.3.3.1.1.1.3.2.1" stretchy="false" xref="S4.I11.i2.I1.i3.p1.13.m13.3.3.1.1.1.cmml">(</mo><mi id="S4.I11.i2.I1.i3.p1.13.m13.1.1" xref="S4.I11.i2.I1.i3.p1.13.m13.1.1.cmml">a</mi><mo id="S4.I11.i2.I1.i3.p1.13.m13.3.3.1.1.1.3.2.2" stretchy="false" xref="S4.I11.i2.I1.i3.p1.13.m13.3.3.1.1.1.cmml">)</mo></mrow></mrow><mo id="S4.I11.i2.I1.i3.p1.13.m13.4.4.2.2.3" xref="S4.I11.i2.I1.i3.p1.13.m13.4.4.2.3.cmml">,</mo><mrow id="S4.I11.i2.I1.i3.p1.13.m13.4.4.2.2.2" xref="S4.I11.i2.I1.i3.p1.13.m13.4.4.2.2.2.cmml"><mi id="S4.I11.i2.I1.i3.p1.13.m13.4.4.2.2.2.2" mathvariant="normal" xref="S4.I11.i2.I1.i3.p1.13.m13.4.4.2.2.2.2.cmml">ℓ</mi><mo id="S4.I11.i2.I1.i3.p1.13.m13.4.4.2.2.2.1" xref="S4.I11.i2.I1.i3.p1.13.m13.4.4.2.2.2.1.cmml">⁢</mo><mrow id="S4.I11.i2.I1.i3.p1.13.m13.4.4.2.2.2.3.2" xref="S4.I11.i2.I1.i3.p1.13.m13.4.4.2.2.2.cmml"><mo id="S4.I11.i2.I1.i3.p1.13.m13.4.4.2.2.2.3.2.1" stretchy="false" xref="S4.I11.i2.I1.i3.p1.13.m13.4.4.2.2.2.cmml">(</mo><mi id="S4.I11.i2.I1.i3.p1.13.m13.2.2" xref="S4.I11.i2.I1.i3.p1.13.m13.2.2.cmml">b</mi><mo id="S4.I11.i2.I1.i3.p1.13.m13.4.4.2.2.2.3.2.2" stretchy="false" xref="S4.I11.i2.I1.i3.p1.13.m13.4.4.2.2.2.cmml">)</mo></mrow></mrow></mrow><mo id="S4.I11.i2.I1.i3.p1.13.m13.4.4.3" xref="S4.I11.i2.I1.i3.p1.13.m13.4.4.3.cmml">∈</mo><msub id="S4.I11.i2.I1.i3.p1.13.m13.4.4.4" xref="S4.I11.i2.I1.i3.p1.13.m13.4.4.4.cmml"><mi id="S4.I11.i2.I1.i3.p1.13.m13.4.4.4.2" xref="S4.I11.i2.I1.i3.p1.13.m13.4.4.4.2.cmml">T</mi><mi id="S4.I11.i2.I1.i3.p1.13.m13.4.4.4.3" xref="S4.I11.i2.I1.i3.p1.13.m13.4.4.4.3.cmml">x</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S4.I11.i2.I1.i3.p1.13.m13.4b"><apply id="S4.I11.i2.I1.i3.p1.13.m13.4.4.cmml" xref="S4.I11.i2.I1.i3.p1.13.m13.4.4"><in id="S4.I11.i2.I1.i3.p1.13.m13.4.4.3.cmml" xref="S4.I11.i2.I1.i3.p1.13.m13.4.4.3"></in><list id="S4.I11.i2.I1.i3.p1.13.m13.4.4.2.3.cmml" xref="S4.I11.i2.I1.i3.p1.13.m13.4.4.2.2"><apply id="S4.I11.i2.I1.i3.p1.13.m13.3.3.1.1.1.cmml" xref="S4.I11.i2.I1.i3.p1.13.m13.3.3.1.1.1"><times id="S4.I11.i2.I1.i3.p1.13.m13.3.3.1.1.1.1.cmml" xref="S4.I11.i2.I1.i3.p1.13.m13.3.3.1.1.1.1"></times><ci id="S4.I11.i2.I1.i3.p1.13.m13.3.3.1.1.1.2.cmml" xref="S4.I11.i2.I1.i3.p1.13.m13.3.3.1.1.1.2">ℓ</ci><ci id="S4.I11.i2.I1.i3.p1.13.m13.1.1.cmml" xref="S4.I11.i2.I1.i3.p1.13.m13.1.1">𝑎</ci></apply><apply id="S4.I11.i2.I1.i3.p1.13.m13.4.4.2.2.2.cmml" xref="S4.I11.i2.I1.i3.p1.13.m13.4.4.2.2.2"><times id="S4.I11.i2.I1.i3.p1.13.m13.4.4.2.2.2.1.cmml" xref="S4.I11.i2.I1.i3.p1.13.m13.4.4.2.2.2.1"></times><ci id="S4.I11.i2.I1.i3.p1.13.m13.4.4.2.2.2.2.cmml" xref="S4.I11.i2.I1.i3.p1.13.m13.4.4.2.2.2.2">ℓ</ci><ci id="S4.I11.i2.I1.i3.p1.13.m13.2.2.cmml" xref="S4.I11.i2.I1.i3.p1.13.m13.2.2">𝑏</ci></apply></list><apply id="S4.I11.i2.I1.i3.p1.13.m13.4.4.4.cmml" xref="S4.I11.i2.I1.i3.p1.13.m13.4.4.4"><csymbol cd="ambiguous" id="S4.I11.i2.I1.i3.p1.13.m13.4.4.4.1.cmml" xref="S4.I11.i2.I1.i3.p1.13.m13.4.4.4">subscript</csymbol><ci id="S4.I11.i2.I1.i3.p1.13.m13.4.4.4.2.cmml" xref="S4.I11.i2.I1.i3.p1.13.m13.4.4.4.2">𝑇</ci><ci id="S4.I11.i2.I1.i3.p1.13.m13.4.4.4.3.cmml" xref="S4.I11.i2.I1.i3.p1.13.m13.4.4.4.3">𝑥</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.I11.i2.I1.i3.p1.13.m13.4c">\ell(a),\ell(b)\in T_{x}</annotation><annotation encoding="application/x-llamapun" id="S4.I11.i2.I1.i3.p1.13.m13.4d">roman_ℓ ( italic_a ) , roman_ℓ ( italic_b ) ∈ italic_T start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT</annotation></semantics></math>. Thus <math alttext="E\cup\textnormal{OPT}" class="ltx_Math" display="inline" id="S4.I11.i2.I1.i3.p1.14.m14.1"><semantics id="S4.I11.i2.I1.i3.p1.14.m14.1a"><mrow id="S4.I11.i2.I1.i3.p1.14.m14.1.1" xref="S4.I11.i2.I1.i3.p1.14.m14.1.1.cmml"><mi id="S4.I11.i2.I1.i3.p1.14.m14.1.1.2" xref="S4.I11.i2.I1.i3.p1.14.m14.1.1.2.cmml">E</mi><mo id="S4.I11.i2.I1.i3.p1.14.m14.1.1.1" xref="S4.I11.i2.I1.i3.p1.14.m14.1.1.1.cmml">∪</mo><mtext id="S4.I11.i2.I1.i3.p1.14.m14.1.1.3" xref="S4.I11.i2.I1.i3.p1.14.m14.1.1.3a.cmml">OPT</mtext></mrow><annotation-xml encoding="MathML-Content" id="S4.I11.i2.I1.i3.p1.14.m14.1b"><apply id="S4.I11.i2.I1.i3.p1.14.m14.1.1.cmml" xref="S4.I11.i2.I1.i3.p1.14.m14.1.1"><union id="S4.I11.i2.I1.i3.p1.14.m14.1.1.1.cmml" xref="S4.I11.i2.I1.i3.p1.14.m14.1.1.1"></union><ci id="S4.I11.i2.I1.i3.p1.14.m14.1.1.2.cmml" xref="S4.I11.i2.I1.i3.p1.14.m14.1.1.2">𝐸</ci><ci id="S4.I11.i2.I1.i3.p1.14.m14.1.1.3a.cmml" xref="S4.I11.i2.I1.i3.p1.14.m14.1.1.3"><mtext id="S4.I11.i2.I1.i3.p1.14.m14.1.1.3.cmml" xref="S4.I11.i2.I1.i3.p1.14.m14.1.1.3">OPT</mtext></ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.I11.i2.I1.i3.p1.14.m14.1c">E\cup\textnormal{OPT}</annotation><annotation encoding="application/x-llamapun" id="S4.I11.i2.I1.i3.p1.14.m14.1d">italic_E ∪ OPT</annotation></semantics></math> contains a <math alttext="u" class="ltx_Math" display="inline" id="S4.I11.i2.I1.i3.p1.15.m15.1"><semantics id="S4.I11.i2.I1.i3.p1.15.m15.1a"><mi id="S4.I11.i2.I1.i3.p1.15.m15.1.1" xref="S4.I11.i2.I1.i3.p1.15.m15.1.1.cmml">u</mi><annotation-xml encoding="MathML-Content" id="S4.I11.i2.I1.i3.p1.15.m15.1b"><ci id="S4.I11.i2.I1.i3.p1.15.m15.1.1.cmml" xref="S4.I11.i2.I1.i3.p1.15.m15.1.1">𝑢</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.I11.i2.I1.i3.p1.15.m15.1c">u</annotation><annotation encoding="application/x-llamapun" id="S4.I11.i2.I1.i3.p1.15.m15.1d">italic_u</annotation></semantics></math>-<math alttext="v" class="ltx_Math" display="inline" id="S4.I11.i2.I1.i3.p1.16.m16.1"><semantics id="S4.I11.i2.I1.i3.p1.16.m16.1a"><mi id="S4.I11.i2.I1.i3.p1.16.m16.1.1" xref="S4.I11.i2.I1.i3.p1.16.m16.1.1.cmml">v</mi><annotation-xml encoding="MathML-Content" id="S4.I11.i2.I1.i3.p1.16.m16.1b"><ci id="S4.I11.i2.I1.i3.p1.16.m16.1.1.cmml" xref="S4.I11.i2.I1.i3.p1.16.m16.1.1">𝑣</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.I11.i2.I1.i3.p1.16.m16.1c">v</annotation><annotation encoding="application/x-llamapun" id="S4.I11.i2.I1.i3.p1.16.m16.1d">italic_v</annotation></semantics></math> path avoiding <math alttext="\{a,b\}" class="ltx_Math" display="inline" id="S4.I11.i2.I1.i3.p1.17.m17.2"><semantics id="S4.I11.i2.I1.i3.p1.17.m17.2a"><mrow id="S4.I11.i2.I1.i3.p1.17.m17.2.3.2" xref="S4.I11.i2.I1.i3.p1.17.m17.2.3.1.cmml"><mo id="S4.I11.i2.I1.i3.p1.17.m17.2.3.2.1" stretchy="false" xref="S4.I11.i2.I1.i3.p1.17.m17.2.3.1.cmml">{</mo><mi id="S4.I11.i2.I1.i3.p1.17.m17.1.1" xref="S4.I11.i2.I1.i3.p1.17.m17.1.1.cmml">a</mi><mo id="S4.I11.i2.I1.i3.p1.17.m17.2.3.2.2" xref="S4.I11.i2.I1.i3.p1.17.m17.2.3.1.cmml">,</mo><mi id="S4.I11.i2.I1.i3.p1.17.m17.2.2" xref="S4.I11.i2.I1.i3.p1.17.m17.2.2.cmml">b</mi><mo id="S4.I11.i2.I1.i3.p1.17.m17.2.3.2.3" stretchy="false" xref="S4.I11.i2.I1.i3.p1.17.m17.2.3.1.cmml">}</mo></mrow><annotation-xml encoding="MathML-Content" id="S4.I11.i2.I1.i3.p1.17.m17.2b"><set id="S4.I11.i2.I1.i3.p1.17.m17.2.3.1.cmml" xref="S4.I11.i2.I1.i3.p1.17.m17.2.3.2"><ci id="S4.I11.i2.I1.i3.p1.17.m17.1.1.cmml" xref="S4.I11.i2.I1.i3.p1.17.m17.1.1">𝑎</ci><ci id="S4.I11.i2.I1.i3.p1.17.m17.2.2.cmml" xref="S4.I11.i2.I1.i3.p1.17.m17.2.2">𝑏</ci></set></annotation-xml><annotation encoding="application/x-tex" id="S4.I11.i2.I1.i3.p1.17.m17.2c">\{a,b\}</annotation><annotation encoding="application/x-llamapun" id="S4.I11.i2.I1.i3.p1.17.m17.2d">{ italic_a , italic_b }</annotation></semantics></math> via the link <math alttext="(u^{\prime},u^{\prime\prime})" class="ltx_Math" display="inline" id="S4.I11.i2.I1.i3.p1.18.m18.2"><semantics id="S4.I11.i2.I1.i3.p1.18.m18.2a"><mrow id="S4.I11.i2.I1.i3.p1.18.m18.2.2.2" xref="S4.I11.i2.I1.i3.p1.18.m18.2.2.3.cmml"><mo id="S4.I11.i2.I1.i3.p1.18.m18.2.2.2.3" stretchy="false" xref="S4.I11.i2.I1.i3.p1.18.m18.2.2.3.cmml">(</mo><msup id="S4.I11.i2.I1.i3.p1.18.m18.1.1.1.1" xref="S4.I11.i2.I1.i3.p1.18.m18.1.1.1.1.cmml"><mi id="S4.I11.i2.I1.i3.p1.18.m18.1.1.1.1.2" xref="S4.I11.i2.I1.i3.p1.18.m18.1.1.1.1.2.cmml">u</mi><mo id="S4.I11.i2.I1.i3.p1.18.m18.1.1.1.1.3" xref="S4.I11.i2.I1.i3.p1.18.m18.1.1.1.1.3.cmml">′</mo></msup><mo id="S4.I11.i2.I1.i3.p1.18.m18.2.2.2.4" xref="S4.I11.i2.I1.i3.p1.18.m18.2.2.3.cmml">,</mo><msup id="S4.I11.i2.I1.i3.p1.18.m18.2.2.2.2" xref="S4.I11.i2.I1.i3.p1.18.m18.2.2.2.2.cmml"><mi id="S4.I11.i2.I1.i3.p1.18.m18.2.2.2.2.2" xref="S4.I11.i2.I1.i3.p1.18.m18.2.2.2.2.2.cmml">u</mi><mo id="S4.I11.i2.I1.i3.p1.18.m18.2.2.2.2.3" xref="S4.I11.i2.I1.i3.p1.18.m18.2.2.2.2.3.cmml">′′</mo></msup><mo id="S4.I11.i2.I1.i3.p1.18.m18.2.2.2.5" stretchy="false" xref="S4.I11.i2.I1.i3.p1.18.m18.2.2.3.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S4.I11.i2.I1.i3.p1.18.m18.2b"><interval closure="open" id="S4.I11.i2.I1.i3.p1.18.m18.2.2.3.cmml" xref="S4.I11.i2.I1.i3.p1.18.m18.2.2.2"><apply id="S4.I11.i2.I1.i3.p1.18.m18.1.1.1.1.cmml" xref="S4.I11.i2.I1.i3.p1.18.m18.1.1.1.1"><csymbol cd="ambiguous" id="S4.I11.i2.I1.i3.p1.18.m18.1.1.1.1.1.cmml" xref="S4.I11.i2.I1.i3.p1.18.m18.1.1.1.1">superscript</csymbol><ci id="S4.I11.i2.I1.i3.p1.18.m18.1.1.1.1.2.cmml" xref="S4.I11.i2.I1.i3.p1.18.m18.1.1.1.1.2">𝑢</ci><ci id="S4.I11.i2.I1.i3.p1.18.m18.1.1.1.1.3.cmml" xref="S4.I11.i2.I1.i3.p1.18.m18.1.1.1.1.3">′</ci></apply><apply id="S4.I11.i2.I1.i3.p1.18.m18.2.2.2.2.cmml" xref="S4.I11.i2.I1.i3.p1.18.m18.2.2.2.2"><csymbol cd="ambiguous" id="S4.I11.i2.I1.i3.p1.18.m18.2.2.2.2.1.cmml" xref="S4.I11.i2.I1.i3.p1.18.m18.2.2.2.2">superscript</csymbol><ci id="S4.I11.i2.I1.i3.p1.18.m18.2.2.2.2.2.cmml" xref="S4.I11.i2.I1.i3.p1.18.m18.2.2.2.2.2">𝑢</ci><ci id="S4.I11.i2.I1.i3.p1.18.m18.2.2.2.2.3.cmml" xref="S4.I11.i2.I1.i3.p1.18.m18.2.2.2.2.3">′′</ci></apply></interval></annotation-xml><annotation encoding="application/x-tex" id="S4.I11.i2.I1.i3.p1.18.m18.2c">(u^{\prime},u^{\prime\prime})</annotation><annotation encoding="application/x-llamapun" id="S4.I11.i2.I1.i3.p1.18.m18.2d">( italic_u start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT , italic_u start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT )</annotation></semantics></math>.</p> </div> </li> </ul> </div> </li> </ul> </div> <figure class="ltx_figure" id="S4.F10"> <div class="ltx_flex_figure"> <div class="ltx_flex_cell ltx_flex_size_2"> <figure class="ltx_figure ltx_figure_panel ltx_align_center" id="S4.F10.sf1"><img alt="Refer to caption" class="ltx_graphics ltx_centering ltx_img_square" height="241" id="S4.F10.sf1.g1" src="x12.png" width="219"/> <figcaption class="ltx_caption ltx_centering"><span class="ltx_tag ltx_tag_figure"><span class="ltx_text" id="S4.F10.sf1.12.6.1" style="font-size:90%;">(a)</span> </span><span class="ltx_text" id="S4.F10.sf1.10.5" style="font-size:90%;">Case 3a: Here <math alttext="f(u)\prec f(v)" class="ltx_Math" display="inline" id="S4.F10.sf1.6.1.m1.2"><semantics id="S4.F10.sf1.6.1.m1.2b"><mrow id="S4.F10.sf1.6.1.m1.2.3" xref="S4.F10.sf1.6.1.m1.2.3.cmml"><mrow id="S4.F10.sf1.6.1.m1.2.3.2" xref="S4.F10.sf1.6.1.m1.2.3.2.cmml"><mi id="S4.F10.sf1.6.1.m1.2.3.2.2" xref="S4.F10.sf1.6.1.m1.2.3.2.2.cmml">f</mi><mo id="S4.F10.sf1.6.1.m1.2.3.2.1" xref="S4.F10.sf1.6.1.m1.2.3.2.1.cmml">⁢</mo><mrow id="S4.F10.sf1.6.1.m1.2.3.2.3.2" xref="S4.F10.sf1.6.1.m1.2.3.2.cmml"><mo id="S4.F10.sf1.6.1.m1.2.3.2.3.2.1" stretchy="false" xref="S4.F10.sf1.6.1.m1.2.3.2.cmml">(</mo><mi id="S4.F10.sf1.6.1.m1.1.1" xref="S4.F10.sf1.6.1.m1.1.1.cmml">u</mi><mo id="S4.F10.sf1.6.1.m1.2.3.2.3.2.2" stretchy="false" xref="S4.F10.sf1.6.1.m1.2.3.2.cmml">)</mo></mrow></mrow><mo id="S4.F10.sf1.6.1.m1.2.3.1" xref="S4.F10.sf1.6.1.m1.2.3.1.cmml">≺</mo><mrow id="S4.F10.sf1.6.1.m1.2.3.3" xref="S4.F10.sf1.6.1.m1.2.3.3.cmml"><mi id="S4.F10.sf1.6.1.m1.2.3.3.2" xref="S4.F10.sf1.6.1.m1.2.3.3.2.cmml">f</mi><mo id="S4.F10.sf1.6.1.m1.2.3.3.1" xref="S4.F10.sf1.6.1.m1.2.3.3.1.cmml">⁢</mo><mrow id="S4.F10.sf1.6.1.m1.2.3.3.3.2" xref="S4.F10.sf1.6.1.m1.2.3.3.cmml"><mo id="S4.F10.sf1.6.1.m1.2.3.3.3.2.1" stretchy="false" xref="S4.F10.sf1.6.1.m1.2.3.3.cmml">(</mo><mi id="S4.F10.sf1.6.1.m1.2.2" xref="S4.F10.sf1.6.1.m1.2.2.cmml">v</mi><mo id="S4.F10.sf1.6.1.m1.2.3.3.3.2.2" stretchy="false" xref="S4.F10.sf1.6.1.m1.2.3.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.F10.sf1.6.1.m1.2c"><apply id="S4.F10.sf1.6.1.m1.2.3.cmml" xref="S4.F10.sf1.6.1.m1.2.3"><csymbol cd="latexml" id="S4.F10.sf1.6.1.m1.2.3.1.cmml" xref="S4.F10.sf1.6.1.m1.2.3.1">precedes</csymbol><apply id="S4.F10.sf1.6.1.m1.2.3.2.cmml" xref="S4.F10.sf1.6.1.m1.2.3.2"><times id="S4.F10.sf1.6.1.m1.2.3.2.1.cmml" xref="S4.F10.sf1.6.1.m1.2.3.2.1"></times><ci id="S4.F10.sf1.6.1.m1.2.3.2.2.cmml" xref="S4.F10.sf1.6.1.m1.2.3.2.2">𝑓</ci><ci id="S4.F10.sf1.6.1.m1.1.1.cmml" xref="S4.F10.sf1.6.1.m1.1.1">𝑢</ci></apply><apply id="S4.F10.sf1.6.1.m1.2.3.3.cmml" xref="S4.F10.sf1.6.1.m1.2.3.3"><times id="S4.F10.sf1.6.1.m1.2.3.3.1.cmml" xref="S4.F10.sf1.6.1.m1.2.3.3.1"></times><ci id="S4.F10.sf1.6.1.m1.2.3.3.2.cmml" xref="S4.F10.sf1.6.1.m1.2.3.3.2">𝑓</ci><ci id="S4.F10.sf1.6.1.m1.2.2.cmml" xref="S4.F10.sf1.6.1.m1.2.2">𝑣</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.F10.sf1.6.1.m1.2d">f(u)\prec f(v)</annotation><annotation encoding="application/x-llamapun" id="S4.F10.sf1.6.1.m1.2e">italic_f ( italic_u ) ≺ italic_f ( italic_v )</annotation></semantics></math>. In this example, <math alttext="f(v^{\prime\prime})=v^{\prime\prime}" class="ltx_Math" display="inline" id="S4.F10.sf1.7.2.m2.1"><semantics id="S4.F10.sf1.7.2.m2.1b"><mrow id="S4.F10.sf1.7.2.m2.1.1" xref="S4.F10.sf1.7.2.m2.1.1.cmml"><mrow id="S4.F10.sf1.7.2.m2.1.1.1" xref="S4.F10.sf1.7.2.m2.1.1.1.cmml"><mi id="S4.F10.sf1.7.2.m2.1.1.1.3" xref="S4.F10.sf1.7.2.m2.1.1.1.3.cmml">f</mi><mo id="S4.F10.sf1.7.2.m2.1.1.1.2" xref="S4.F10.sf1.7.2.m2.1.1.1.2.cmml">⁢</mo><mrow id="S4.F10.sf1.7.2.m2.1.1.1.1.1" xref="S4.F10.sf1.7.2.m2.1.1.1.1.1.1.cmml"><mo id="S4.F10.sf1.7.2.m2.1.1.1.1.1.2" stretchy="false" xref="S4.F10.sf1.7.2.m2.1.1.1.1.1.1.cmml">(</mo><msup id="S4.F10.sf1.7.2.m2.1.1.1.1.1.1" xref="S4.F10.sf1.7.2.m2.1.1.1.1.1.1.cmml"><mi id="S4.F10.sf1.7.2.m2.1.1.1.1.1.1.2" xref="S4.F10.sf1.7.2.m2.1.1.1.1.1.1.2.cmml">v</mi><mo id="S4.F10.sf1.7.2.m2.1.1.1.1.1.1.3" xref="S4.F10.sf1.7.2.m2.1.1.1.1.1.1.3.cmml">′′</mo></msup><mo id="S4.F10.sf1.7.2.m2.1.1.1.1.1.3" stretchy="false" xref="S4.F10.sf1.7.2.m2.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S4.F10.sf1.7.2.m2.1.1.2" xref="S4.F10.sf1.7.2.m2.1.1.2.cmml">=</mo><msup id="S4.F10.sf1.7.2.m2.1.1.3" xref="S4.F10.sf1.7.2.m2.1.1.3.cmml"><mi id="S4.F10.sf1.7.2.m2.1.1.3.2" xref="S4.F10.sf1.7.2.m2.1.1.3.2.cmml">v</mi><mo id="S4.F10.sf1.7.2.m2.1.1.3.3" xref="S4.F10.sf1.7.2.m2.1.1.3.3.cmml">′′</mo></msup></mrow><annotation-xml encoding="MathML-Content" id="S4.F10.sf1.7.2.m2.1c"><apply id="S4.F10.sf1.7.2.m2.1.1.cmml" xref="S4.F10.sf1.7.2.m2.1.1"><eq id="S4.F10.sf1.7.2.m2.1.1.2.cmml" xref="S4.F10.sf1.7.2.m2.1.1.2"></eq><apply id="S4.F10.sf1.7.2.m2.1.1.1.cmml" xref="S4.F10.sf1.7.2.m2.1.1.1"><times id="S4.F10.sf1.7.2.m2.1.1.1.2.cmml" xref="S4.F10.sf1.7.2.m2.1.1.1.2"></times><ci id="S4.F10.sf1.7.2.m2.1.1.1.3.cmml" xref="S4.F10.sf1.7.2.m2.1.1.1.3">𝑓</ci><apply id="S4.F10.sf1.7.2.m2.1.1.1.1.1.1.cmml" xref="S4.F10.sf1.7.2.m2.1.1.1.1.1"><csymbol cd="ambiguous" id="S4.F10.sf1.7.2.m2.1.1.1.1.1.1.1.cmml" xref="S4.F10.sf1.7.2.m2.1.1.1.1.1">superscript</csymbol><ci id="S4.F10.sf1.7.2.m2.1.1.1.1.1.1.2.cmml" xref="S4.F10.sf1.7.2.m2.1.1.1.1.1.1.2">𝑣</ci><ci id="S4.F10.sf1.7.2.m2.1.1.1.1.1.1.3.cmml" xref="S4.F10.sf1.7.2.m2.1.1.1.1.1.1.3">′′</ci></apply></apply><apply id="S4.F10.sf1.7.2.m2.1.1.3.cmml" xref="S4.F10.sf1.7.2.m2.1.1.3"><csymbol cd="ambiguous" id="S4.F10.sf1.7.2.m2.1.1.3.1.cmml" xref="S4.F10.sf1.7.2.m2.1.1.3">superscript</csymbol><ci id="S4.F10.sf1.7.2.m2.1.1.3.2.cmml" xref="S4.F10.sf1.7.2.m2.1.1.3.2">𝑣</ci><ci id="S4.F10.sf1.7.2.m2.1.1.3.3.cmml" xref="S4.F10.sf1.7.2.m2.1.1.3.3">′′</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.F10.sf1.7.2.m2.1d">f(v^{\prime\prime})=v^{\prime\prime}</annotation><annotation encoding="application/x-llamapun" id="S4.F10.sf1.7.2.m2.1e">italic_f ( italic_v start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT ) = italic_v start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT</annotation></semantics></math> and <math alttext="f(u)=u" class="ltx_Math" display="inline" id="S4.F10.sf1.8.3.m3.1"><semantics id="S4.F10.sf1.8.3.m3.1b"><mrow id="S4.F10.sf1.8.3.m3.1.2" xref="S4.F10.sf1.8.3.m3.1.2.cmml"><mrow id="S4.F10.sf1.8.3.m3.1.2.2" xref="S4.F10.sf1.8.3.m3.1.2.2.cmml"><mi id="S4.F10.sf1.8.3.m3.1.2.2.2" xref="S4.F10.sf1.8.3.m3.1.2.2.2.cmml">f</mi><mo id="S4.F10.sf1.8.3.m3.1.2.2.1" xref="S4.F10.sf1.8.3.m3.1.2.2.1.cmml">⁢</mo><mrow id="S4.F10.sf1.8.3.m3.1.2.2.3.2" xref="S4.F10.sf1.8.3.m3.1.2.2.cmml"><mo id="S4.F10.sf1.8.3.m3.1.2.2.3.2.1" stretchy="false" xref="S4.F10.sf1.8.3.m3.1.2.2.cmml">(</mo><mi id="S4.F10.sf1.8.3.m3.1.1" xref="S4.F10.sf1.8.3.m3.1.1.cmml">u</mi><mo id="S4.F10.sf1.8.3.m3.1.2.2.3.2.2" stretchy="false" xref="S4.F10.sf1.8.3.m3.1.2.2.cmml">)</mo></mrow></mrow><mo id="S4.F10.sf1.8.3.m3.1.2.1" xref="S4.F10.sf1.8.3.m3.1.2.1.cmml">=</mo><mi id="S4.F10.sf1.8.3.m3.1.2.3" xref="S4.F10.sf1.8.3.m3.1.2.3.cmml">u</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.F10.sf1.8.3.m3.1c"><apply id="S4.F10.sf1.8.3.m3.1.2.cmml" xref="S4.F10.sf1.8.3.m3.1.2"><eq id="S4.F10.sf1.8.3.m3.1.2.1.cmml" xref="S4.F10.sf1.8.3.m3.1.2.1"></eq><apply id="S4.F10.sf1.8.3.m3.1.2.2.cmml" xref="S4.F10.sf1.8.3.m3.1.2.2"><times id="S4.F10.sf1.8.3.m3.1.2.2.1.cmml" xref="S4.F10.sf1.8.3.m3.1.2.2.1"></times><ci id="S4.F10.sf1.8.3.m3.1.2.2.2.cmml" xref="S4.F10.sf1.8.3.m3.1.2.2.2">𝑓</ci><ci id="S4.F10.sf1.8.3.m3.1.1.cmml" xref="S4.F10.sf1.8.3.m3.1.1">𝑢</ci></apply><ci id="S4.F10.sf1.8.3.m3.1.2.3.cmml" xref="S4.F10.sf1.8.3.m3.1.2.3">𝑢</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.F10.sf1.8.3.m3.1d">f(u)=u</annotation><annotation encoding="application/x-llamapun" id="S4.F10.sf1.8.3.m3.1e">italic_f ( italic_u ) = italic_u</annotation></semantics></math>, but <math alttext="v,v^{\prime}" class="ltx_Math" display="inline" id="S4.F10.sf1.9.4.m4.2"><semantics id="S4.F10.sf1.9.4.m4.2b"><mrow id="S4.F10.sf1.9.4.m4.2.2.1" xref="S4.F10.sf1.9.4.m4.2.2.2.cmml"><mi id="S4.F10.sf1.9.4.m4.1.1" xref="S4.F10.sf1.9.4.m4.1.1.cmml">v</mi><mo id="S4.F10.sf1.9.4.m4.2.2.1.2" xref="S4.F10.sf1.9.4.m4.2.2.2.cmml">,</mo><msup id="S4.F10.sf1.9.4.m4.2.2.1.1" xref="S4.F10.sf1.9.4.m4.2.2.1.1.cmml"><mi id="S4.F10.sf1.9.4.m4.2.2.1.1.2" xref="S4.F10.sf1.9.4.m4.2.2.1.1.2.cmml">v</mi><mo id="S4.F10.sf1.9.4.m4.2.2.1.1.3" xref="S4.F10.sf1.9.4.m4.2.2.1.1.3.cmml">′</mo></msup></mrow><annotation-xml encoding="MathML-Content" id="S4.F10.sf1.9.4.m4.2c"><list id="S4.F10.sf1.9.4.m4.2.2.2.cmml" xref="S4.F10.sf1.9.4.m4.2.2.1"><ci id="S4.F10.sf1.9.4.m4.1.1.cmml" xref="S4.F10.sf1.9.4.m4.1.1">𝑣</ci><apply id="S4.F10.sf1.9.4.m4.2.2.1.1.cmml" xref="S4.F10.sf1.9.4.m4.2.2.1.1"><csymbol cd="ambiguous" id="S4.F10.sf1.9.4.m4.2.2.1.1.1.cmml" xref="S4.F10.sf1.9.4.m4.2.2.1.1">superscript</csymbol><ci id="S4.F10.sf1.9.4.m4.2.2.1.1.2.cmml" xref="S4.F10.sf1.9.4.m4.2.2.1.1.2">𝑣</ci><ci id="S4.F10.sf1.9.4.m4.2.2.1.1.3.cmml" xref="S4.F10.sf1.9.4.m4.2.2.1.1.3">′</ci></apply></list></annotation-xml><annotation encoding="application/x-tex" id="S4.F10.sf1.9.4.m4.2d">v,v^{\prime}</annotation><annotation encoding="application/x-llamapun" id="S4.F10.sf1.9.4.m4.2e">italic_v , italic_v start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math> are not on the cycle and thus <math alttext="f(v)" class="ltx_Math" display="inline" id="S4.F10.sf1.10.5.m5.1"><semantics id="S4.F10.sf1.10.5.m5.1b"><mrow id="S4.F10.sf1.10.5.m5.1.2" xref="S4.F10.sf1.10.5.m5.1.2.cmml"><mi id="S4.F10.sf1.10.5.m5.1.2.2" xref="S4.F10.sf1.10.5.m5.1.2.2.cmml">f</mi><mo id="S4.F10.sf1.10.5.m5.1.2.1" xref="S4.F10.sf1.10.5.m5.1.2.1.cmml">⁢</mo><mrow id="S4.F10.sf1.10.5.m5.1.2.3.2" xref="S4.F10.sf1.10.5.m5.1.2.cmml"><mo id="S4.F10.sf1.10.5.m5.1.2.3.2.1" stretchy="false" xref="S4.F10.sf1.10.5.m5.1.2.cmml">(</mo><mi id="S4.F10.sf1.10.5.m5.1.1" xref="S4.F10.sf1.10.5.m5.1.1.cmml">v</mi><mo id="S4.F10.sf1.10.5.m5.1.2.3.2.2" stretchy="false" xref="S4.F10.sf1.10.5.m5.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.F10.sf1.10.5.m5.1c"><apply id="S4.F10.sf1.10.5.m5.1.2.cmml" xref="S4.F10.sf1.10.5.m5.1.2"><times id="S4.F10.sf1.10.5.m5.1.2.1.cmml" xref="S4.F10.sf1.10.5.m5.1.2.1"></times><ci id="S4.F10.sf1.10.5.m5.1.2.2.cmml" xref="S4.F10.sf1.10.5.m5.1.2.2">𝑓</ci><ci id="S4.F10.sf1.10.5.m5.1.1.cmml" xref="S4.F10.sf1.10.5.m5.1.1">𝑣</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.F10.sf1.10.5.m5.1d">f(v)</annotation><annotation encoding="application/x-llamapun" id="S4.F10.sf1.10.5.m5.1e">italic_f ( italic_v )</annotation></semantics></math> is a dummy node.</span></figcaption> </figure> </div> <div class="ltx_flex_cell ltx_flex_size_2"> <figure class="ltx_figure ltx_figure_panel ltx_align_center" id="S4.F10.sf2"><img alt="Refer to caption" class="ltx_graphics ltx_centering ltx_img_square" height="227" id="S4.F10.sf2.g1" src="x13.png" width="205"/> <figcaption class="ltx_caption ltx_centering"><span class="ltx_tag ltx_tag_figure"><span class="ltx_text" id="S4.F10.sf2.4.2.1" style="font-size:90%;">(b)</span> </span><span class="ltx_text" id="S4.F10.sf2.2.1" style="font-size:90%;">Case 3b: Here <math alttext="u\in G_{x}\setminus\textnormal{parent}(x)" class="ltx_Math" display="inline" id="S4.F10.sf2.2.1.m1.1"><semantics id="S4.F10.sf2.2.1.m1.1b"><mrow id="S4.F10.sf2.2.1.m1.1.2" xref="S4.F10.sf2.2.1.m1.1.2.cmml"><mi id="S4.F10.sf2.2.1.m1.1.2.2" xref="S4.F10.sf2.2.1.m1.1.2.2.cmml">u</mi><mo id="S4.F10.sf2.2.1.m1.1.2.1" xref="S4.F10.sf2.2.1.m1.1.2.1.cmml">∈</mo><mrow id="S4.F10.sf2.2.1.m1.1.2.3" xref="S4.F10.sf2.2.1.m1.1.2.3.cmml"><msub id="S4.F10.sf2.2.1.m1.1.2.3.2" xref="S4.F10.sf2.2.1.m1.1.2.3.2.cmml"><mi id="S4.F10.sf2.2.1.m1.1.2.3.2.2" xref="S4.F10.sf2.2.1.m1.1.2.3.2.2.cmml">G</mi><mi id="S4.F10.sf2.2.1.m1.1.2.3.2.3" xref="S4.F10.sf2.2.1.m1.1.2.3.2.3.cmml">x</mi></msub><mo id="S4.F10.sf2.2.1.m1.1.2.3.1" xref="S4.F10.sf2.2.1.m1.1.2.3.1.cmml">∖</mo><mrow id="S4.F10.sf2.2.1.m1.1.2.3.3" xref="S4.F10.sf2.2.1.m1.1.2.3.3.cmml"><mtext id="S4.F10.sf2.2.1.m1.1.2.3.3.2" xref="S4.F10.sf2.2.1.m1.1.2.3.3.2a.cmml">parent</mtext><mo id="S4.F10.sf2.2.1.m1.1.2.3.3.1" xref="S4.F10.sf2.2.1.m1.1.2.3.3.1.cmml">⁢</mo><mrow id="S4.F10.sf2.2.1.m1.1.2.3.3.3.2" xref="S4.F10.sf2.2.1.m1.1.2.3.3.cmml"><mo id="S4.F10.sf2.2.1.m1.1.2.3.3.3.2.1" stretchy="false" xref="S4.F10.sf2.2.1.m1.1.2.3.3.cmml">(</mo><mi id="S4.F10.sf2.2.1.m1.1.1" xref="S4.F10.sf2.2.1.m1.1.1.cmml">x</mi><mo id="S4.F10.sf2.2.1.m1.1.2.3.3.3.2.2" stretchy="false" xref="S4.F10.sf2.2.1.m1.1.2.3.3.cmml">)</mo></mrow></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.F10.sf2.2.1.m1.1c"><apply id="S4.F10.sf2.2.1.m1.1.2.cmml" xref="S4.F10.sf2.2.1.m1.1.2"><in id="S4.F10.sf2.2.1.m1.1.2.1.cmml" xref="S4.F10.sf2.2.1.m1.1.2.1"></in><ci id="S4.F10.sf2.2.1.m1.1.2.2.cmml" xref="S4.F10.sf2.2.1.m1.1.2.2">𝑢</ci><apply id="S4.F10.sf2.2.1.m1.1.2.3.cmml" xref="S4.F10.sf2.2.1.m1.1.2.3"><setdiff id="S4.F10.sf2.2.1.m1.1.2.3.1.cmml" xref="S4.F10.sf2.2.1.m1.1.2.3.1"></setdiff><apply id="S4.F10.sf2.2.1.m1.1.2.3.2.cmml" xref="S4.F10.sf2.2.1.m1.1.2.3.2"><csymbol cd="ambiguous" id="S4.F10.sf2.2.1.m1.1.2.3.2.1.cmml" xref="S4.F10.sf2.2.1.m1.1.2.3.2">subscript</csymbol><ci id="S4.F10.sf2.2.1.m1.1.2.3.2.2.cmml" xref="S4.F10.sf2.2.1.m1.1.2.3.2.2">𝐺</ci><ci id="S4.F10.sf2.2.1.m1.1.2.3.2.3.cmml" xref="S4.F10.sf2.2.1.m1.1.2.3.2.3">𝑥</ci></apply><apply id="S4.F10.sf2.2.1.m1.1.2.3.3.cmml" xref="S4.F10.sf2.2.1.m1.1.2.3.3"><times id="S4.F10.sf2.2.1.m1.1.2.3.3.1.cmml" xref="S4.F10.sf2.2.1.m1.1.2.3.3.1"></times><ci id="S4.F10.sf2.2.1.m1.1.2.3.3.2a.cmml" xref="S4.F10.sf2.2.1.m1.1.2.3.3.2"><mtext id="S4.F10.sf2.2.1.m1.1.2.3.3.2.cmml" xref="S4.F10.sf2.2.1.m1.1.2.3.3.2">parent</mtext></ci><ci id="S4.F10.sf2.2.1.m1.1.1.cmml" xref="S4.F10.sf2.2.1.m1.1.1">𝑥</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.F10.sf2.2.1.m1.1d">u\in G_{x}\setminus\textnormal{parent}(x)</annotation><annotation encoding="application/x-llamapun" id="S4.F10.sf2.2.1.m1.1e">italic_u ∈ italic_G start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT ∖ parent ( italic_x )</annotation></semantics></math>.</span></figcaption> </figure> </div> </div> <figcaption class="ltx_caption ltx_centering"><span class="ltx_tag ltx_tag_figure"><span class="ltx_text" id="S4.F10.2.1.1" style="font-size:90%;">Figure 10</span>: </span><span class="ltx_text" id="S4.F10.3.2" style="font-size:90%;">Case 3 example. Tree nodes are shown with boxes, while graph vertices are given by dots.</span></figcaption> </figure> <div class="ltx_para" id="S4.SS2.SSS3.Px3.p2"> <p class="ltx_p" id="S4.SS2.SSS3.Px3.p2.1">∎</p> </div> <div class="ltx_para" id="S4.SS2.SSS3.Px3.p3"> <p class="ltx_p" id="S4.SS2.SSS3.Px3.p3.3">By Lemmas <a class="ltx_ref" href="https://arxiv.org/html/2503.00712v1#S4.Thmtheorem22" title="Lemma 4.22. ‣ 4.2.3 Bounding the Approximation Ratio ‣ 4.2 Two-to-Three Augmentation ‣ 4 Vertex Connectivity Augmentation in Link-Arrival Model ‣ Streaming Algorithms for Network Design"><span class="ltx_text ltx_ref_tag">4.22</span></a> and <a class="ltx_ref" href="https://arxiv.org/html/2503.00712v1#S4.Thmtheorem21" title="Lemma 4.21. ‣ 4.2.3 Bounding the Approximation Ratio ‣ 4.2 Two-to-Three Augmentation ‣ 4 Vertex Connectivity Augmentation in Link-Arrival Model ‣ Streaming Algorithms for Network Design"><span class="ltx_text ltx_ref_tag">4.21</span></a>, Algorithm <a class="ltx_ref" href="https://arxiv.org/html/2503.00712v1#algorithm7" title="In 4.2.3 Bounding the Approximation Ratio ‣ 4.2 Two-to-Three Augmentation ‣ 4 Vertex Connectivity Augmentation in Link-Arrival Model ‣ Streaming Algorithms for Network Design"><span class="ltx_text ltx_ref_tag">7</span></a> provides a feasible solution to <math alttext="2" class="ltx_Math" display="inline" id="S4.SS2.SSS3.Px3.p3.1.m1.1"><semantics id="S4.SS2.SSS3.Px3.p3.1.m1.1a"><mn id="S4.SS2.SSS3.Px3.p3.1.m1.1.1" xref="S4.SS2.SSS3.Px3.p3.1.m1.1.1.cmml">2</mn><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS3.Px3.p3.1.m1.1b"><cn id="S4.SS2.SSS3.Px3.p3.1.m1.1.1.cmml" type="integer" xref="S4.SS2.SSS3.Px3.p3.1.m1.1.1">2</cn></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS3.Px3.p3.1.m1.1c">2</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS3.Px3.p3.1.m1.1d">2</annotation></semantics></math>-VC-CAP on <math alttext="(V,F)" class="ltx_Math" display="inline" id="S4.SS2.SSS3.Px3.p3.2.m2.2"><semantics id="S4.SS2.SSS3.Px3.p3.2.m2.2a"><mrow id="S4.SS2.SSS3.Px3.p3.2.m2.2.3.2" xref="S4.SS2.SSS3.Px3.p3.2.m2.2.3.1.cmml"><mo id="S4.SS2.SSS3.Px3.p3.2.m2.2.3.2.1" stretchy="false" xref="S4.SS2.SSS3.Px3.p3.2.m2.2.3.1.cmml">(</mo><mi id="S4.SS2.SSS3.Px3.p3.2.m2.1.1" xref="S4.SS2.SSS3.Px3.p3.2.m2.1.1.cmml">V</mi><mo id="S4.SS2.SSS3.Px3.p3.2.m2.2.3.2.2" xref="S4.SS2.SSS3.Px3.p3.2.m2.2.3.1.cmml">,</mo><mi id="S4.SS2.SSS3.Px3.p3.2.m2.2.2" xref="S4.SS2.SSS3.Px3.p3.2.m2.2.2.cmml">F</mi><mo id="S4.SS2.SSS3.Px3.p3.2.m2.2.3.2.3" stretchy="false" xref="S4.SS2.SSS3.Px3.p3.2.m2.2.3.1.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS3.Px3.p3.2.m2.2b"><interval closure="open" id="S4.SS2.SSS3.Px3.p3.2.m2.2.3.1.cmml" xref="S4.SS2.SSS3.Px3.p3.2.m2.2.3.2"><ci id="S4.SS2.SSS3.Px3.p3.2.m2.1.1.cmml" xref="S4.SS2.SSS3.Px3.p3.2.m2.1.1">𝑉</ci><ci id="S4.SS2.SSS3.Px3.p3.2.m2.2.2.cmml" xref="S4.SS2.SSS3.Px3.p3.2.m2.2.2">𝐹</ci></interval></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS3.Px3.p3.2.m2.2c">(V,F)</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS3.Px3.p3.2.m2.2d">( italic_V , italic_F )</annotation></semantics></math> with weight at most <math alttext="(7+\epsilon)w(\textnormal{OPT})" class="ltx_Math" display="inline" id="S4.SS2.SSS3.Px3.p3.3.m3.2"><semantics id="S4.SS2.SSS3.Px3.p3.3.m3.2a"><mrow id="S4.SS2.SSS3.Px3.p3.3.m3.2.2" xref="S4.SS2.SSS3.Px3.p3.3.m3.2.2.cmml"><mrow id="S4.SS2.SSS3.Px3.p3.3.m3.2.2.1.1" xref="S4.SS2.SSS3.Px3.p3.3.m3.2.2.1.1.1.cmml"><mo id="S4.SS2.SSS3.Px3.p3.3.m3.2.2.1.1.2" stretchy="false" xref="S4.SS2.SSS3.Px3.p3.3.m3.2.2.1.1.1.cmml">(</mo><mrow id="S4.SS2.SSS3.Px3.p3.3.m3.2.2.1.1.1" xref="S4.SS2.SSS3.Px3.p3.3.m3.2.2.1.1.1.cmml"><mn id="S4.SS2.SSS3.Px3.p3.3.m3.2.2.1.1.1.2" xref="S4.SS2.SSS3.Px3.p3.3.m3.2.2.1.1.1.2.cmml">7</mn><mo id="S4.SS2.SSS3.Px3.p3.3.m3.2.2.1.1.1.1" xref="S4.SS2.SSS3.Px3.p3.3.m3.2.2.1.1.1.1.cmml">+</mo><mi id="S4.SS2.SSS3.Px3.p3.3.m3.2.2.1.1.1.3" xref="S4.SS2.SSS3.Px3.p3.3.m3.2.2.1.1.1.3.cmml">ϵ</mi></mrow><mo id="S4.SS2.SSS3.Px3.p3.3.m3.2.2.1.1.3" stretchy="false" xref="S4.SS2.SSS3.Px3.p3.3.m3.2.2.1.1.1.cmml">)</mo></mrow><mo id="S4.SS2.SSS3.Px3.p3.3.m3.2.2.2" xref="S4.SS2.SSS3.Px3.p3.3.m3.2.2.2.cmml">⁢</mo><mi id="S4.SS2.SSS3.Px3.p3.3.m3.2.2.3" xref="S4.SS2.SSS3.Px3.p3.3.m3.2.2.3.cmml">w</mi><mo id="S4.SS2.SSS3.Px3.p3.3.m3.2.2.2a" xref="S4.SS2.SSS3.Px3.p3.3.m3.2.2.2.cmml">⁢</mo><mrow id="S4.SS2.SSS3.Px3.p3.3.m3.2.2.4.2" xref="S4.SS2.SSS3.Px3.p3.3.m3.1.1a.cmml"><mo id="S4.SS2.SSS3.Px3.p3.3.m3.2.2.4.2.1" stretchy="false" xref="S4.SS2.SSS3.Px3.p3.3.m3.1.1a.cmml">(</mo><mtext id="S4.SS2.SSS3.Px3.p3.3.m3.1.1" xref="S4.SS2.SSS3.Px3.p3.3.m3.1.1.cmml">OPT</mtext><mo id="S4.SS2.SSS3.Px3.p3.3.m3.2.2.4.2.2" stretchy="false" xref="S4.SS2.SSS3.Px3.p3.3.m3.1.1a.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS3.Px3.p3.3.m3.2b"><apply id="S4.SS2.SSS3.Px3.p3.3.m3.2.2.cmml" xref="S4.SS2.SSS3.Px3.p3.3.m3.2.2"><times id="S4.SS2.SSS3.Px3.p3.3.m3.2.2.2.cmml" xref="S4.SS2.SSS3.Px3.p3.3.m3.2.2.2"></times><apply id="S4.SS2.SSS3.Px3.p3.3.m3.2.2.1.1.1.cmml" xref="S4.SS2.SSS3.Px3.p3.3.m3.2.2.1.1"><plus id="S4.SS2.SSS3.Px3.p3.3.m3.2.2.1.1.1.1.cmml" xref="S4.SS2.SSS3.Px3.p3.3.m3.2.2.1.1.1.1"></plus><cn id="S4.SS2.SSS3.Px3.p3.3.m3.2.2.1.1.1.2.cmml" type="integer" xref="S4.SS2.SSS3.Px3.p3.3.m3.2.2.1.1.1.2">7</cn><ci id="S4.SS2.SSS3.Px3.p3.3.m3.2.2.1.1.1.3.cmml" xref="S4.SS2.SSS3.Px3.p3.3.m3.2.2.1.1.1.3">italic-ϵ</ci></apply><ci id="S4.SS2.SSS3.Px3.p3.3.m3.2.2.3.cmml" xref="S4.SS2.SSS3.Px3.p3.3.m3.2.2.3">𝑤</ci><ci id="S4.SS2.SSS3.Px3.p3.3.m3.1.1a.cmml" xref="S4.SS2.SSS3.Px3.p3.3.m3.2.2.4.2"><mtext id="S4.SS2.SSS3.Px3.p3.3.m3.1.1.cmml" xref="S4.SS2.SSS3.Px3.p3.3.m3.1.1">OPT</mtext></ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS3.Px3.p3.3.m3.2c">(7+\epsilon)w(\textnormal{OPT})</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS3.Px3.p3.3.m3.2d">( 7 + italic_ϵ ) italic_w ( OPT )</annotation></semantics></math>; this concludes the proof of Lemma <a class="ltx_ref" href="https://arxiv.org/html/2503.00712v1#S4.Thmtheorem20" title="Lemma 4.20. ‣ 4.2.3 Bounding the Approximation Ratio ‣ 4.2 Two-to-Three Augmentation ‣ 4 Vertex Connectivity Augmentation in Link-Arrival Model ‣ Streaming Algorithms for Network Design"><span class="ltx_text ltx_ref_tag">4.20</span></a>. This, combined with Lemma <a class="ltx_ref" href="https://arxiv.org/html/2503.00712v1#S4.Thmtheorem19" title="Lemma 4.19. ‣ “Cycle” Cuts: ‣ 4.2.2 The Streaming Algorithm ‣ 4.2 Two-to-Three Augmentation ‣ 4 Vertex Connectivity Augmentation in Link-Arrival Model ‣ Streaming Algorithms for Network Design"><span class="ltx_text ltx_ref_tag">4.19</span></a>, concludes the proof of Theorem <a class="ltx_ref" href="https://arxiv.org/html/2503.00712v1#S4.Thmtheorem7" title="Theorem 4.7. ‣ 4.2 Two-to-Three Augmentation ‣ 4 Vertex Connectivity Augmentation in Link-Arrival Model ‣ Streaming Algorithms for Network Design"><span class="ltx_text ltx_ref_tag">4.7</span></a>.</p> </div> </section> </section> </section> </section> <section class="ltx_section" id="S5"> <h2 class="ltx_title ltx_title_section"> <span class="ltx_tag ltx_tag_section">5 </span>Lower Bounds for Streaming Network Design</h2> <div class="ltx_para" id="S5.p1"> <p class="ltx_p" id="S5.p1.1">We describe a lower bound for the vertex-connectivity tree-augmentation problem (VC-TAP), i.e., <math alttext="1" class="ltx_Math" display="inline" id="S5.p1.1.m1.1"><semantics id="S5.p1.1.m1.1a"><mn id="S5.p1.1.m1.1.1" xref="S5.p1.1.m1.1.1.cmml">1</mn><annotation-xml encoding="MathML-Content" id="S5.p1.1.m1.1b"><cn id="S5.p1.1.m1.1.1.cmml" type="integer" xref="S5.p1.1.m1.1.1">1</cn></annotation-xml><annotation encoding="application/x-tex" id="S5.p1.1.m1.1c">1</annotation><annotation encoding="application/x-llamapun" id="S5.p1.1.m1.1d">1</annotation></semantics></math>-VC-CAP.</p> </div> <div class="ltx_theorem ltx_theorem_theorem" id="S5.Thmtheorem1"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="S5.Thmtheorem1.1.1.1">Theorem 5.1</span></span><span class="ltx_text ltx_font_bold" id="S5.Thmtheorem1.2.2">.</span> </h6> <div class="ltx_para" id="S5.Thmtheorem1.p1"> <p class="ltx_p" id="S5.Thmtheorem1.p1.4">Consider the unweighted VC-TAP where both <math alttext="E" class="ltx_Math" display="inline" id="S5.Thmtheorem1.p1.1.m1.1"><semantics id="S5.Thmtheorem1.p1.1.m1.1a"><mi id="S5.Thmtheorem1.p1.1.m1.1.1" xref="S5.Thmtheorem1.p1.1.m1.1.1.cmml">E</mi><annotation-xml encoding="MathML-Content" id="S5.Thmtheorem1.p1.1.m1.1b"><ci id="S5.Thmtheorem1.p1.1.m1.1.1.cmml" xref="S5.Thmtheorem1.p1.1.m1.1.1">𝐸</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmtheorem1.p1.1.m1.1c">E</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem1.p1.1.m1.1d">italic_E</annotation></semantics></math> and <math alttext="L" class="ltx_Math" display="inline" id="S5.Thmtheorem1.p1.2.m2.1"><semantics id="S5.Thmtheorem1.p1.2.m2.1a"><mi id="S5.Thmtheorem1.p1.2.m2.1.1" xref="S5.Thmtheorem1.p1.2.m2.1.1.cmml">L</mi><annotation-xml encoding="MathML-Content" id="S5.Thmtheorem1.p1.2.m2.1b"><ci id="S5.Thmtheorem1.p1.2.m2.1.1.cmml" xref="S5.Thmtheorem1.p1.2.m2.1.1">𝐿</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmtheorem1.p1.2.m2.1c">L</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem1.p1.2.m2.1d">italic_L</annotation></semantics></math> arrives as a stream in an arbitrary order. Any single-pass algorithm returning a better than <math alttext="(2t+1)" class="ltx_Math" display="inline" id="S5.Thmtheorem1.p1.3.m3.1"><semantics id="S5.Thmtheorem1.p1.3.m3.1a"><mrow id="S5.Thmtheorem1.p1.3.m3.1.1.1" xref="S5.Thmtheorem1.p1.3.m3.1.1.1.1.cmml"><mo id="S5.Thmtheorem1.p1.3.m3.1.1.1.2" stretchy="false" xref="S5.Thmtheorem1.p1.3.m3.1.1.1.1.cmml">(</mo><mrow id="S5.Thmtheorem1.p1.3.m3.1.1.1.1" xref="S5.Thmtheorem1.p1.3.m3.1.1.1.1.cmml"><mrow id="S5.Thmtheorem1.p1.3.m3.1.1.1.1.2" xref="S5.Thmtheorem1.p1.3.m3.1.1.1.1.2.cmml"><mn id="S5.Thmtheorem1.p1.3.m3.1.1.1.1.2.2" xref="S5.Thmtheorem1.p1.3.m3.1.1.1.1.2.2.cmml">2</mn><mo id="S5.Thmtheorem1.p1.3.m3.1.1.1.1.2.1" xref="S5.Thmtheorem1.p1.3.m3.1.1.1.1.2.1.cmml">⁢</mo><mi id="S5.Thmtheorem1.p1.3.m3.1.1.1.1.2.3" xref="S5.Thmtheorem1.p1.3.m3.1.1.1.1.2.3.cmml">t</mi></mrow><mo id="S5.Thmtheorem1.p1.3.m3.1.1.1.1.1" xref="S5.Thmtheorem1.p1.3.m3.1.1.1.1.1.cmml">+</mo><mn id="S5.Thmtheorem1.p1.3.m3.1.1.1.1.3" xref="S5.Thmtheorem1.p1.3.m3.1.1.1.1.3.cmml">1</mn></mrow><mo id="S5.Thmtheorem1.p1.3.m3.1.1.1.3" stretchy="false" xref="S5.Thmtheorem1.p1.3.m3.1.1.1.1.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S5.Thmtheorem1.p1.3.m3.1b"><apply id="S5.Thmtheorem1.p1.3.m3.1.1.1.1.cmml" xref="S5.Thmtheorem1.p1.3.m3.1.1.1"><plus id="S5.Thmtheorem1.p1.3.m3.1.1.1.1.1.cmml" xref="S5.Thmtheorem1.p1.3.m3.1.1.1.1.1"></plus><apply id="S5.Thmtheorem1.p1.3.m3.1.1.1.1.2.cmml" xref="S5.Thmtheorem1.p1.3.m3.1.1.1.1.2"><times id="S5.Thmtheorem1.p1.3.m3.1.1.1.1.2.1.cmml" xref="S5.Thmtheorem1.p1.3.m3.1.1.1.1.2.1"></times><cn id="S5.Thmtheorem1.p1.3.m3.1.1.1.1.2.2.cmml" type="integer" xref="S5.Thmtheorem1.p1.3.m3.1.1.1.1.2.2">2</cn><ci id="S5.Thmtheorem1.p1.3.m3.1.1.1.1.2.3.cmml" xref="S5.Thmtheorem1.p1.3.m3.1.1.1.1.2.3">𝑡</ci></apply><cn id="S5.Thmtheorem1.p1.3.m3.1.1.1.1.3.cmml" type="integer" xref="S5.Thmtheorem1.p1.3.m3.1.1.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmtheorem1.p1.3.m3.1c">(2t+1)</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem1.p1.3.m3.1d">( 2 italic_t + 1 )</annotation></semantics></math>-approximation requires <math alttext="\Omega(n^{1+1/t})" class="ltx_Math" display="inline" id="S5.Thmtheorem1.p1.4.m4.1"><semantics id="S5.Thmtheorem1.p1.4.m4.1a"><mrow id="S5.Thmtheorem1.p1.4.m4.1.1" xref="S5.Thmtheorem1.p1.4.m4.1.1.cmml"><mi id="S5.Thmtheorem1.p1.4.m4.1.1.3" mathvariant="normal" xref="S5.Thmtheorem1.p1.4.m4.1.1.3.cmml">Ω</mi><mo id="S5.Thmtheorem1.p1.4.m4.1.1.2" xref="S5.Thmtheorem1.p1.4.m4.1.1.2.cmml">⁢</mo><mrow id="S5.Thmtheorem1.p1.4.m4.1.1.1.1" xref="S5.Thmtheorem1.p1.4.m4.1.1.1.1.1.cmml"><mo id="S5.Thmtheorem1.p1.4.m4.1.1.1.1.2" stretchy="false" xref="S5.Thmtheorem1.p1.4.m4.1.1.1.1.1.cmml">(</mo><msup id="S5.Thmtheorem1.p1.4.m4.1.1.1.1.1" xref="S5.Thmtheorem1.p1.4.m4.1.1.1.1.1.cmml"><mi id="S5.Thmtheorem1.p1.4.m4.1.1.1.1.1.2" xref="S5.Thmtheorem1.p1.4.m4.1.1.1.1.1.2.cmml">n</mi><mrow id="S5.Thmtheorem1.p1.4.m4.1.1.1.1.1.3" xref="S5.Thmtheorem1.p1.4.m4.1.1.1.1.1.3.cmml"><mn id="S5.Thmtheorem1.p1.4.m4.1.1.1.1.1.3.2" xref="S5.Thmtheorem1.p1.4.m4.1.1.1.1.1.3.2.cmml">1</mn><mo id="S5.Thmtheorem1.p1.4.m4.1.1.1.1.1.3.1" xref="S5.Thmtheorem1.p1.4.m4.1.1.1.1.1.3.1.cmml">+</mo><mrow id="S5.Thmtheorem1.p1.4.m4.1.1.1.1.1.3.3" xref="S5.Thmtheorem1.p1.4.m4.1.1.1.1.1.3.3.cmml"><mn id="S5.Thmtheorem1.p1.4.m4.1.1.1.1.1.3.3.2" xref="S5.Thmtheorem1.p1.4.m4.1.1.1.1.1.3.3.2.cmml">1</mn><mo id="S5.Thmtheorem1.p1.4.m4.1.1.1.1.1.3.3.1" xref="S5.Thmtheorem1.p1.4.m4.1.1.1.1.1.3.3.1.cmml">/</mo><mi id="S5.Thmtheorem1.p1.4.m4.1.1.1.1.1.3.3.3" xref="S5.Thmtheorem1.p1.4.m4.1.1.1.1.1.3.3.3.cmml">t</mi></mrow></mrow></msup><mo id="S5.Thmtheorem1.p1.4.m4.1.1.1.1.3" stretchy="false" xref="S5.Thmtheorem1.p1.4.m4.1.1.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.Thmtheorem1.p1.4.m4.1b"><apply id="S5.Thmtheorem1.p1.4.m4.1.1.cmml" xref="S5.Thmtheorem1.p1.4.m4.1.1"><times id="S5.Thmtheorem1.p1.4.m4.1.1.2.cmml" xref="S5.Thmtheorem1.p1.4.m4.1.1.2"></times><ci id="S5.Thmtheorem1.p1.4.m4.1.1.3.cmml" xref="S5.Thmtheorem1.p1.4.m4.1.1.3">Ω</ci><apply id="S5.Thmtheorem1.p1.4.m4.1.1.1.1.1.cmml" xref="S5.Thmtheorem1.p1.4.m4.1.1.1.1"><csymbol cd="ambiguous" id="S5.Thmtheorem1.p1.4.m4.1.1.1.1.1.1.cmml" xref="S5.Thmtheorem1.p1.4.m4.1.1.1.1">superscript</csymbol><ci id="S5.Thmtheorem1.p1.4.m4.1.1.1.1.1.2.cmml" xref="S5.Thmtheorem1.p1.4.m4.1.1.1.1.1.2">𝑛</ci><apply id="S5.Thmtheorem1.p1.4.m4.1.1.1.1.1.3.cmml" xref="S5.Thmtheorem1.p1.4.m4.1.1.1.1.1.3"><plus id="S5.Thmtheorem1.p1.4.m4.1.1.1.1.1.3.1.cmml" xref="S5.Thmtheorem1.p1.4.m4.1.1.1.1.1.3.1"></plus><cn id="S5.Thmtheorem1.p1.4.m4.1.1.1.1.1.3.2.cmml" type="integer" xref="S5.Thmtheorem1.p1.4.m4.1.1.1.1.1.3.2">1</cn><apply id="S5.Thmtheorem1.p1.4.m4.1.1.1.1.1.3.3.cmml" xref="S5.Thmtheorem1.p1.4.m4.1.1.1.1.1.3.3"><divide id="S5.Thmtheorem1.p1.4.m4.1.1.1.1.1.3.3.1.cmml" xref="S5.Thmtheorem1.p1.4.m4.1.1.1.1.1.3.3.1"></divide><cn id="S5.Thmtheorem1.p1.4.m4.1.1.1.1.1.3.3.2.cmml" type="integer" xref="S5.Thmtheorem1.p1.4.m4.1.1.1.1.1.3.3.2">1</cn><ci id="S5.Thmtheorem1.p1.4.m4.1.1.1.1.1.3.3.3.cmml" xref="S5.Thmtheorem1.p1.4.m4.1.1.1.1.1.3.3.3">𝑡</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmtheorem1.p1.4.m4.1c">\Omega(n^{1+1/t})</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem1.p1.4.m4.1d">roman_Ω ( italic_n start_POSTSUPERSCRIPT 1 + 1 / italic_t end_POSTSUPERSCRIPT )</annotation></semantics></math> space.</p> </div> </div> <div class="ltx_proof" id="S5.3"> <h6 class="ltx_title ltx_runin ltx_font_italic ltx_title_proof">Proof.</h6> <div class="ltx_para" id="S5.1.p1"> <p class="ltx_p" id="S5.1.p1.11">Let <math alttext="G=(V,E)" class="ltx_Math" display="inline" id="S5.1.p1.1.m1.2"><semantics id="S5.1.p1.1.m1.2a"><mrow id="S5.1.p1.1.m1.2.3" xref="S5.1.p1.1.m1.2.3.cmml"><mi id="S5.1.p1.1.m1.2.3.2" xref="S5.1.p1.1.m1.2.3.2.cmml">G</mi><mo id="S5.1.p1.1.m1.2.3.1" xref="S5.1.p1.1.m1.2.3.1.cmml">=</mo><mrow id="S5.1.p1.1.m1.2.3.3.2" xref="S5.1.p1.1.m1.2.3.3.1.cmml"><mo id="S5.1.p1.1.m1.2.3.3.2.1" stretchy="false" xref="S5.1.p1.1.m1.2.3.3.1.cmml">(</mo><mi id="S5.1.p1.1.m1.1.1" xref="S5.1.p1.1.m1.1.1.cmml">V</mi><mo id="S5.1.p1.1.m1.2.3.3.2.2" xref="S5.1.p1.1.m1.2.3.3.1.cmml">,</mo><mi id="S5.1.p1.1.m1.2.2" xref="S5.1.p1.1.m1.2.2.cmml">E</mi><mo id="S5.1.p1.1.m1.2.3.3.2.3" stretchy="false" xref="S5.1.p1.1.m1.2.3.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.1.p1.1.m1.2b"><apply id="S5.1.p1.1.m1.2.3.cmml" xref="S5.1.p1.1.m1.2.3"><eq id="S5.1.p1.1.m1.2.3.1.cmml" xref="S5.1.p1.1.m1.2.3.1"></eq><ci id="S5.1.p1.1.m1.2.3.2.cmml" xref="S5.1.p1.1.m1.2.3.2">𝐺</ci><interval closure="open" id="S5.1.p1.1.m1.2.3.3.1.cmml" xref="S5.1.p1.1.m1.2.3.3.2"><ci id="S5.1.p1.1.m1.1.1.cmml" xref="S5.1.p1.1.m1.1.1">𝑉</ci><ci id="S5.1.p1.1.m1.2.2.cmml" xref="S5.1.p1.1.m1.2.2">𝐸</ci></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.1.p1.1.m1.2c">G=(V,E)</annotation><annotation encoding="application/x-llamapun" id="S5.1.p1.1.m1.2d">italic_G = ( italic_V , italic_E )</annotation></semantics></math> be a fixed graph on <math alttext="n" class="ltx_Math" display="inline" id="S5.1.p1.2.m2.1"><semantics id="S5.1.p1.2.m2.1a"><mi id="S5.1.p1.2.m2.1.1" xref="S5.1.p1.2.m2.1.1.cmml">n</mi><annotation-xml encoding="MathML-Content" id="S5.1.p1.2.m2.1b"><ci id="S5.1.p1.2.m2.1.1.cmml" xref="S5.1.p1.2.m2.1.1">𝑛</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.1.p1.2.m2.1c">n</annotation><annotation encoding="application/x-llamapun" id="S5.1.p1.2.m2.1d">italic_n</annotation></semantics></math> vertices with girth strictly larger than <math alttext="2t+1" class="ltx_Math" display="inline" id="S5.1.p1.3.m3.1"><semantics id="S5.1.p1.3.m3.1a"><mrow id="S5.1.p1.3.m3.1.1" xref="S5.1.p1.3.m3.1.1.cmml"><mrow id="S5.1.p1.3.m3.1.1.2" xref="S5.1.p1.3.m3.1.1.2.cmml"><mn id="S5.1.p1.3.m3.1.1.2.2" xref="S5.1.p1.3.m3.1.1.2.2.cmml">2</mn><mo id="S5.1.p1.3.m3.1.1.2.1" xref="S5.1.p1.3.m3.1.1.2.1.cmml">⁢</mo><mi id="S5.1.p1.3.m3.1.1.2.3" xref="S5.1.p1.3.m3.1.1.2.3.cmml">t</mi></mrow><mo id="S5.1.p1.3.m3.1.1.1" xref="S5.1.p1.3.m3.1.1.1.cmml">+</mo><mn id="S5.1.p1.3.m3.1.1.3" xref="S5.1.p1.3.m3.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S5.1.p1.3.m3.1b"><apply id="S5.1.p1.3.m3.1.1.cmml" xref="S5.1.p1.3.m3.1.1"><plus id="S5.1.p1.3.m3.1.1.1.cmml" xref="S5.1.p1.3.m3.1.1.1"></plus><apply id="S5.1.p1.3.m3.1.1.2.cmml" xref="S5.1.p1.3.m3.1.1.2"><times id="S5.1.p1.3.m3.1.1.2.1.cmml" xref="S5.1.p1.3.m3.1.1.2.1"></times><cn id="S5.1.p1.3.m3.1.1.2.2.cmml" type="integer" xref="S5.1.p1.3.m3.1.1.2.2">2</cn><ci id="S5.1.p1.3.m3.1.1.2.3.cmml" xref="S5.1.p1.3.m3.1.1.2.3">𝑡</ci></apply><cn id="S5.1.p1.3.m3.1.1.3.cmml" type="integer" xref="S5.1.p1.3.m3.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.1.p1.3.m3.1c">2t+1</annotation><annotation encoding="application/x-llamapun" id="S5.1.p1.3.m3.1d">2 italic_t + 1</annotation></semantics></math>. Consider the INDEX problem: Alice has a bit string from <math alttext="\{0,1\}^{E(G)}" class="ltx_Math" display="inline" id="S5.1.p1.4.m4.3"><semantics id="S5.1.p1.4.m4.3a"><msup id="S5.1.p1.4.m4.3.4" xref="S5.1.p1.4.m4.3.4.cmml"><mrow id="S5.1.p1.4.m4.3.4.2.2" xref="S5.1.p1.4.m4.3.4.2.1.cmml"><mo id="S5.1.p1.4.m4.3.4.2.2.1" stretchy="false" xref="S5.1.p1.4.m4.3.4.2.1.cmml">{</mo><mn id="S5.1.p1.4.m4.2.2" xref="S5.1.p1.4.m4.2.2.cmml">0</mn><mo id="S5.1.p1.4.m4.3.4.2.2.2" xref="S5.1.p1.4.m4.3.4.2.1.cmml">,</mo><mn id="S5.1.p1.4.m4.3.3" xref="S5.1.p1.4.m4.3.3.cmml">1</mn><mo id="S5.1.p1.4.m4.3.4.2.2.3" stretchy="false" xref="S5.1.p1.4.m4.3.4.2.1.cmml">}</mo></mrow><mrow id="S5.1.p1.4.m4.1.1.1" xref="S5.1.p1.4.m4.1.1.1.cmml"><mi id="S5.1.p1.4.m4.1.1.1.3" xref="S5.1.p1.4.m4.1.1.1.3.cmml">E</mi><mo id="S5.1.p1.4.m4.1.1.1.2" xref="S5.1.p1.4.m4.1.1.1.2.cmml">⁢</mo><mrow id="S5.1.p1.4.m4.1.1.1.4.2" xref="S5.1.p1.4.m4.1.1.1.cmml"><mo id="S5.1.p1.4.m4.1.1.1.4.2.1" stretchy="false" xref="S5.1.p1.4.m4.1.1.1.cmml">(</mo><mi id="S5.1.p1.4.m4.1.1.1.1" xref="S5.1.p1.4.m4.1.1.1.1.cmml">G</mi><mo id="S5.1.p1.4.m4.1.1.1.4.2.2" stretchy="false" xref="S5.1.p1.4.m4.1.1.1.cmml">)</mo></mrow></mrow></msup><annotation-xml encoding="MathML-Content" id="S5.1.p1.4.m4.3b"><apply id="S5.1.p1.4.m4.3.4.cmml" xref="S5.1.p1.4.m4.3.4"><csymbol cd="ambiguous" id="S5.1.p1.4.m4.3.4.1.cmml" xref="S5.1.p1.4.m4.3.4">superscript</csymbol><set id="S5.1.p1.4.m4.3.4.2.1.cmml" xref="S5.1.p1.4.m4.3.4.2.2"><cn id="S5.1.p1.4.m4.2.2.cmml" type="integer" xref="S5.1.p1.4.m4.2.2">0</cn><cn id="S5.1.p1.4.m4.3.3.cmml" type="integer" xref="S5.1.p1.4.m4.3.3">1</cn></set><apply id="S5.1.p1.4.m4.1.1.1.cmml" xref="S5.1.p1.4.m4.1.1.1"><times id="S5.1.p1.4.m4.1.1.1.2.cmml" xref="S5.1.p1.4.m4.1.1.1.2"></times><ci id="S5.1.p1.4.m4.1.1.1.3.cmml" xref="S5.1.p1.4.m4.1.1.1.3">𝐸</ci><ci id="S5.1.p1.4.m4.1.1.1.1.cmml" xref="S5.1.p1.4.m4.1.1.1.1">𝐺</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.1.p1.4.m4.3c">\{0,1\}^{E(G)}</annotation><annotation encoding="application/x-llamapun" id="S5.1.p1.4.m4.3d">{ 0 , 1 } start_POSTSUPERSCRIPT italic_E ( italic_G ) end_POSTSUPERSCRIPT</annotation></semantics></math>, representing a subgraph <math alttext="G^{\prime}\subseteq G" class="ltx_Math" display="inline" id="S5.1.p1.5.m5.1"><semantics id="S5.1.p1.5.m5.1a"><mrow id="S5.1.p1.5.m5.1.1" xref="S5.1.p1.5.m5.1.1.cmml"><msup id="S5.1.p1.5.m5.1.1.2" xref="S5.1.p1.5.m5.1.1.2.cmml"><mi id="S5.1.p1.5.m5.1.1.2.2" xref="S5.1.p1.5.m5.1.1.2.2.cmml">G</mi><mo id="S5.1.p1.5.m5.1.1.2.3" xref="S5.1.p1.5.m5.1.1.2.3.cmml">′</mo></msup><mo id="S5.1.p1.5.m5.1.1.1" xref="S5.1.p1.5.m5.1.1.1.cmml">⊆</mo><mi id="S5.1.p1.5.m5.1.1.3" xref="S5.1.p1.5.m5.1.1.3.cmml">G</mi></mrow><annotation-xml encoding="MathML-Content" id="S5.1.p1.5.m5.1b"><apply id="S5.1.p1.5.m5.1.1.cmml" xref="S5.1.p1.5.m5.1.1"><subset id="S5.1.p1.5.m5.1.1.1.cmml" xref="S5.1.p1.5.m5.1.1.1"></subset><apply id="S5.1.p1.5.m5.1.1.2.cmml" xref="S5.1.p1.5.m5.1.1.2"><csymbol cd="ambiguous" id="S5.1.p1.5.m5.1.1.2.1.cmml" xref="S5.1.p1.5.m5.1.1.2">superscript</csymbol><ci id="S5.1.p1.5.m5.1.1.2.2.cmml" xref="S5.1.p1.5.m5.1.1.2.2">𝐺</ci><ci id="S5.1.p1.5.m5.1.1.2.3.cmml" xref="S5.1.p1.5.m5.1.1.2.3">′</ci></apply><ci id="S5.1.p1.5.m5.1.1.3.cmml" xref="S5.1.p1.5.m5.1.1.3">𝐺</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.1.p1.5.m5.1c">G^{\prime}\subseteq G</annotation><annotation encoding="application/x-llamapun" id="S5.1.p1.5.m5.1d">italic_G start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ⊆ italic_G</annotation></semantics></math>. Alice then sends a message to Bob, who must recover the <math alttext="i" class="ltx_Math" display="inline" id="S5.1.p1.6.m6.1"><semantics id="S5.1.p1.6.m6.1a"><mi id="S5.1.p1.6.m6.1.1" xref="S5.1.p1.6.m6.1.1.cmml">i</mi><annotation-xml encoding="MathML-Content" id="S5.1.p1.6.m6.1b"><ci id="S5.1.p1.6.m6.1.1.cmml" xref="S5.1.p1.6.m6.1.1">𝑖</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.1.p1.6.m6.1c">i</annotation><annotation encoding="application/x-llamapun" id="S5.1.p1.6.m6.1d">italic_i</annotation></semantics></math>-th bit of the string for a given index <math alttext="i" class="ltx_Math" display="inline" id="S5.1.p1.7.m7.1"><semantics id="S5.1.p1.7.m7.1a"><mi id="S5.1.p1.7.m7.1.1" xref="S5.1.p1.7.m7.1.1.cmml">i</mi><annotation-xml encoding="MathML-Content" id="S5.1.p1.7.m7.1b"><ci id="S5.1.p1.7.m7.1.1.cmml" xref="S5.1.p1.7.m7.1.1">𝑖</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.1.p1.7.m7.1c">i</annotation><annotation encoding="application/x-llamapun" id="S5.1.p1.7.m7.1d">italic_i</annotation></semantics></math>, or equivalently, determine whether <math alttext="(u,v)\in G^{\prime}" class="ltx_Math" display="inline" id="S5.1.p1.8.m8.2"><semantics id="S5.1.p1.8.m8.2a"><mrow id="S5.1.p1.8.m8.2.3" xref="S5.1.p1.8.m8.2.3.cmml"><mrow id="S5.1.p1.8.m8.2.3.2.2" xref="S5.1.p1.8.m8.2.3.2.1.cmml"><mo id="S5.1.p1.8.m8.2.3.2.2.1" stretchy="false" xref="S5.1.p1.8.m8.2.3.2.1.cmml">(</mo><mi id="S5.1.p1.8.m8.1.1" xref="S5.1.p1.8.m8.1.1.cmml">u</mi><mo id="S5.1.p1.8.m8.2.3.2.2.2" xref="S5.1.p1.8.m8.2.3.2.1.cmml">,</mo><mi id="S5.1.p1.8.m8.2.2" xref="S5.1.p1.8.m8.2.2.cmml">v</mi><mo id="S5.1.p1.8.m8.2.3.2.2.3" stretchy="false" xref="S5.1.p1.8.m8.2.3.2.1.cmml">)</mo></mrow><mo id="S5.1.p1.8.m8.2.3.1" xref="S5.1.p1.8.m8.2.3.1.cmml">∈</mo><msup id="S5.1.p1.8.m8.2.3.3" xref="S5.1.p1.8.m8.2.3.3.cmml"><mi id="S5.1.p1.8.m8.2.3.3.2" xref="S5.1.p1.8.m8.2.3.3.2.cmml">G</mi><mo id="S5.1.p1.8.m8.2.3.3.3" xref="S5.1.p1.8.m8.2.3.3.3.cmml">′</mo></msup></mrow><annotation-xml encoding="MathML-Content" id="S5.1.p1.8.m8.2b"><apply id="S5.1.p1.8.m8.2.3.cmml" xref="S5.1.p1.8.m8.2.3"><in id="S5.1.p1.8.m8.2.3.1.cmml" xref="S5.1.p1.8.m8.2.3.1"></in><interval closure="open" id="S5.1.p1.8.m8.2.3.2.1.cmml" xref="S5.1.p1.8.m8.2.3.2.2"><ci id="S5.1.p1.8.m8.1.1.cmml" xref="S5.1.p1.8.m8.1.1">𝑢</ci><ci id="S5.1.p1.8.m8.2.2.cmml" xref="S5.1.p1.8.m8.2.2">𝑣</ci></interval><apply id="S5.1.p1.8.m8.2.3.3.cmml" xref="S5.1.p1.8.m8.2.3.3"><csymbol cd="ambiguous" id="S5.1.p1.8.m8.2.3.3.1.cmml" xref="S5.1.p1.8.m8.2.3.3">superscript</csymbol><ci id="S5.1.p1.8.m8.2.3.3.2.cmml" xref="S5.1.p1.8.m8.2.3.3.2">𝐺</ci><ci id="S5.1.p1.8.m8.2.3.3.3.cmml" xref="S5.1.p1.8.m8.2.3.3.3">′</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.1.p1.8.m8.2c">(u,v)\in G^{\prime}</annotation><annotation encoding="application/x-llamapun" id="S5.1.p1.8.m8.2d">( italic_u , italic_v ) ∈ italic_G start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math> for a specified edge <math alttext="(u,v)\in G" class="ltx_Math" display="inline" id="S5.1.p1.9.m9.2"><semantics id="S5.1.p1.9.m9.2a"><mrow id="S5.1.p1.9.m9.2.3" xref="S5.1.p1.9.m9.2.3.cmml"><mrow id="S5.1.p1.9.m9.2.3.2.2" xref="S5.1.p1.9.m9.2.3.2.1.cmml"><mo id="S5.1.p1.9.m9.2.3.2.2.1" stretchy="false" xref="S5.1.p1.9.m9.2.3.2.1.cmml">(</mo><mi id="S5.1.p1.9.m9.1.1" xref="S5.1.p1.9.m9.1.1.cmml">u</mi><mo id="S5.1.p1.9.m9.2.3.2.2.2" xref="S5.1.p1.9.m9.2.3.2.1.cmml">,</mo><mi id="S5.1.p1.9.m9.2.2" xref="S5.1.p1.9.m9.2.2.cmml">v</mi><mo id="S5.1.p1.9.m9.2.3.2.2.3" stretchy="false" xref="S5.1.p1.9.m9.2.3.2.1.cmml">)</mo></mrow><mo id="S5.1.p1.9.m9.2.3.1" xref="S5.1.p1.9.m9.2.3.1.cmml">∈</mo><mi id="S5.1.p1.9.m9.2.3.3" xref="S5.1.p1.9.m9.2.3.3.cmml">G</mi></mrow><annotation-xml encoding="MathML-Content" id="S5.1.p1.9.m9.2b"><apply id="S5.1.p1.9.m9.2.3.cmml" xref="S5.1.p1.9.m9.2.3"><in id="S5.1.p1.9.m9.2.3.1.cmml" xref="S5.1.p1.9.m9.2.3.1"></in><interval closure="open" id="S5.1.p1.9.m9.2.3.2.1.cmml" xref="S5.1.p1.9.m9.2.3.2.2"><ci id="S5.1.p1.9.m9.1.1.cmml" xref="S5.1.p1.9.m9.1.1">𝑢</ci><ci id="S5.1.p1.9.m9.2.2.cmml" xref="S5.1.p1.9.m9.2.2">𝑣</ci></interval><ci id="S5.1.p1.9.m9.2.3.3.cmml" xref="S5.1.p1.9.m9.2.3.3">𝐺</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.1.p1.9.m9.2c">(u,v)\in G</annotation><annotation encoding="application/x-llamapun" id="S5.1.p1.9.m9.2d">( italic_u , italic_v ) ∈ italic_G</annotation></semantics></math>. It was shown by Miltersen et al. <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.00712v1#bib.bibx68" title="">MNSW98</a>]</cite> that any bounded-error randomized protocol for INDEX requires a message of size <math alttext="\Omega(|E(G)|)" class="ltx_Math" display="inline" id="S5.1.p1.10.m10.2"><semantics id="S5.1.p1.10.m10.2a"><mrow id="S5.1.p1.10.m10.2.2" xref="S5.1.p1.10.m10.2.2.cmml"><mi id="S5.1.p1.10.m10.2.2.3" mathvariant="normal" xref="S5.1.p1.10.m10.2.2.3.cmml">Ω</mi><mo id="S5.1.p1.10.m10.2.2.2" xref="S5.1.p1.10.m10.2.2.2.cmml">⁢</mo><mrow id="S5.1.p1.10.m10.2.2.1.1" xref="S5.1.p1.10.m10.2.2.cmml"><mo id="S5.1.p1.10.m10.2.2.1.1.2" stretchy="false" xref="S5.1.p1.10.m10.2.2.cmml">(</mo><mrow id="S5.1.p1.10.m10.2.2.1.1.1.1" xref="S5.1.p1.10.m10.2.2.1.1.1.2.cmml"><mo id="S5.1.p1.10.m10.2.2.1.1.1.1.2" stretchy="false" xref="S5.1.p1.10.m10.2.2.1.1.1.2.1.cmml">|</mo><mrow id="S5.1.p1.10.m10.2.2.1.1.1.1.1" xref="S5.1.p1.10.m10.2.2.1.1.1.1.1.cmml"><mi id="S5.1.p1.10.m10.2.2.1.1.1.1.1.2" xref="S5.1.p1.10.m10.2.2.1.1.1.1.1.2.cmml">E</mi><mo id="S5.1.p1.10.m10.2.2.1.1.1.1.1.1" xref="S5.1.p1.10.m10.2.2.1.1.1.1.1.1.cmml">⁢</mo><mrow id="S5.1.p1.10.m10.2.2.1.1.1.1.1.3.2" xref="S5.1.p1.10.m10.2.2.1.1.1.1.1.cmml"><mo id="S5.1.p1.10.m10.2.2.1.1.1.1.1.3.2.1" stretchy="false" xref="S5.1.p1.10.m10.2.2.1.1.1.1.1.cmml">(</mo><mi id="S5.1.p1.10.m10.1.1" xref="S5.1.p1.10.m10.1.1.cmml">G</mi><mo id="S5.1.p1.10.m10.2.2.1.1.1.1.1.3.2.2" stretchy="false" xref="S5.1.p1.10.m10.2.2.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S5.1.p1.10.m10.2.2.1.1.1.1.3" stretchy="false" xref="S5.1.p1.10.m10.2.2.1.1.1.2.1.cmml">|</mo></mrow><mo id="S5.1.p1.10.m10.2.2.1.1.3" stretchy="false" xref="S5.1.p1.10.m10.2.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.1.p1.10.m10.2b"><apply id="S5.1.p1.10.m10.2.2.cmml" xref="S5.1.p1.10.m10.2.2"><times id="S5.1.p1.10.m10.2.2.2.cmml" xref="S5.1.p1.10.m10.2.2.2"></times><ci id="S5.1.p1.10.m10.2.2.3.cmml" xref="S5.1.p1.10.m10.2.2.3">Ω</ci><apply id="S5.1.p1.10.m10.2.2.1.1.1.2.cmml" xref="S5.1.p1.10.m10.2.2.1.1.1.1"><abs id="S5.1.p1.10.m10.2.2.1.1.1.2.1.cmml" xref="S5.1.p1.10.m10.2.2.1.1.1.1.2"></abs><apply id="S5.1.p1.10.m10.2.2.1.1.1.1.1.cmml" xref="S5.1.p1.10.m10.2.2.1.1.1.1.1"><times id="S5.1.p1.10.m10.2.2.1.1.1.1.1.1.cmml" xref="S5.1.p1.10.m10.2.2.1.1.1.1.1.1"></times><ci id="S5.1.p1.10.m10.2.2.1.1.1.1.1.2.cmml" xref="S5.1.p1.10.m10.2.2.1.1.1.1.1.2">𝐸</ci><ci id="S5.1.p1.10.m10.1.1.cmml" xref="S5.1.p1.10.m10.1.1">𝐺</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.1.p1.10.m10.2c">\Omega(|E(G)|)</annotation><annotation encoding="application/x-llamapun" id="S5.1.p1.10.m10.2d">roman_Ω ( | italic_E ( italic_G ) | )</annotation></semantics></math> bits. Note that the graph <math alttext="G" class="ltx_Math" display="inline" id="S5.1.p1.11.m11.1"><semantics id="S5.1.p1.11.m11.1a"><mi id="S5.1.p1.11.m11.1.1" xref="S5.1.p1.11.m11.1.1.cmml">G</mi><annotation-xml encoding="MathML-Content" id="S5.1.p1.11.m11.1b"><ci id="S5.1.p1.11.m11.1.1.cmml" xref="S5.1.p1.11.m11.1.1">𝐺</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.1.p1.11.m11.1c">G</annotation><annotation encoding="application/x-llamapun" id="S5.1.p1.11.m11.1d">italic_G</annotation></semantics></math> and its edges are known to both Alice and Bob.</p> </div> <div class="ltx_para" id="S5.2.p2"> <p class="ltx_p" id="S5.2.p2.3">We now use the streaming algorithm <math alttext="\mathcal{A}" class="ltx_Math" display="inline" id="S5.2.p2.1.m1.1"><semantics id="S5.2.p2.1.m1.1a"><mi class="ltx_font_mathcaligraphic" id="S5.2.p2.1.m1.1.1" xref="S5.2.p2.1.m1.1.1.cmml">𝒜</mi><annotation-xml encoding="MathML-Content" id="S5.2.p2.1.m1.1b"><ci id="S5.2.p2.1.m1.1.1.cmml" xref="S5.2.p2.1.m1.1.1">𝒜</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.2.p2.1.m1.1c">\mathcal{A}</annotation><annotation encoding="application/x-llamapun" id="S5.2.p2.1.m1.1d">caligraphic_A</annotation></semantics></math> for VC-TAP to design a protocol for the INDEX problem. Alice and Bob jointly construct a TAP instance <math alttext="(E,L)" class="ltx_Math" display="inline" id="S5.2.p2.2.m2.2"><semantics id="S5.2.p2.2.m2.2a"><mrow id="S5.2.p2.2.m2.2.3.2" xref="S5.2.p2.2.m2.2.3.1.cmml"><mo id="S5.2.p2.2.m2.2.3.2.1" stretchy="false" xref="S5.2.p2.2.m2.2.3.1.cmml">(</mo><mi id="S5.2.p2.2.m2.1.1" xref="S5.2.p2.2.m2.1.1.cmml">E</mi><mo id="S5.2.p2.2.m2.2.3.2.2" xref="S5.2.p2.2.m2.2.3.1.cmml">,</mo><mi id="S5.2.p2.2.m2.2.2" xref="S5.2.p2.2.m2.2.2.cmml">L</mi><mo id="S5.2.p2.2.m2.2.3.2.3" stretchy="false" xref="S5.2.p2.2.m2.2.3.1.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S5.2.p2.2.m2.2b"><interval closure="open" id="S5.2.p2.2.m2.2.3.1.cmml" xref="S5.2.p2.2.m2.2.3.2"><ci id="S5.2.p2.2.m2.1.1.cmml" xref="S5.2.p2.2.m2.1.1">𝐸</ci><ci id="S5.2.p2.2.m2.2.2.cmml" xref="S5.2.p2.2.m2.2.2">𝐿</ci></interval></annotation-xml><annotation encoding="application/x-tex" id="S5.2.p2.2.m2.2c">(E,L)</annotation><annotation encoding="application/x-llamapun" id="S5.2.p2.2.m2.2d">( italic_E , italic_L )</annotation></semantics></math> for <math alttext="\mathcal{A}" class="ltx_Math" display="inline" id="S5.2.p2.3.m3.1"><semantics id="S5.2.p2.3.m3.1a"><mi class="ltx_font_mathcaligraphic" id="S5.2.p2.3.m3.1.1" xref="S5.2.p2.3.m3.1.1.cmml">𝒜</mi><annotation-xml encoding="MathML-Content" id="S5.2.p2.3.m3.1b"><ci id="S5.2.p2.3.m3.1.1.cmml" xref="S5.2.p2.3.m3.1.1">𝒜</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.2.p2.3.m3.1c">\mathcal{A}</annotation><annotation encoding="application/x-llamapun" id="S5.2.p2.3.m3.1d">caligraphic_A</annotation></semantics></math> as follows:</p> <ul class="ltx_itemize" id="S5.I1"> <li class="ltx_item" id="S5.I1.i1" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">•</span> <div class="ltx_para" id="S5.I1.i1.p1"> <p class="ltx_p" id="S5.I1.i1.p1.15">First, Alice sets <math alttext="L:=E(G^{\prime})" class="ltx_Math" display="inline" id="S5.I1.i1.p1.1.m1.1"><semantics id="S5.I1.i1.p1.1.m1.1a"><mrow id="S5.I1.i1.p1.1.m1.1.1" xref="S5.I1.i1.p1.1.m1.1.1.cmml"><mi id="S5.I1.i1.p1.1.m1.1.1.3" xref="S5.I1.i1.p1.1.m1.1.1.3.cmml">L</mi><mo id="S5.I1.i1.p1.1.m1.1.1.2" lspace="0.278em" rspace="0.278em" xref="S5.I1.i1.p1.1.m1.1.1.2.cmml">:=</mo><mrow id="S5.I1.i1.p1.1.m1.1.1.1" xref="S5.I1.i1.p1.1.m1.1.1.1.cmml"><mi id="S5.I1.i1.p1.1.m1.1.1.1.3" xref="S5.I1.i1.p1.1.m1.1.1.1.3.cmml">E</mi><mo id="S5.I1.i1.p1.1.m1.1.1.1.2" xref="S5.I1.i1.p1.1.m1.1.1.1.2.cmml">⁢</mo><mrow id="S5.I1.i1.p1.1.m1.1.1.1.1.1" xref="S5.I1.i1.p1.1.m1.1.1.1.1.1.1.cmml"><mo id="S5.I1.i1.p1.1.m1.1.1.1.1.1.2" stretchy="false" xref="S5.I1.i1.p1.1.m1.1.1.1.1.1.1.cmml">(</mo><msup id="S5.I1.i1.p1.1.m1.1.1.1.1.1.1" xref="S5.I1.i1.p1.1.m1.1.1.1.1.1.1.cmml"><mi id="S5.I1.i1.p1.1.m1.1.1.1.1.1.1.2" xref="S5.I1.i1.p1.1.m1.1.1.1.1.1.1.2.cmml">G</mi><mo id="S5.I1.i1.p1.1.m1.1.1.1.1.1.1.3" xref="S5.I1.i1.p1.1.m1.1.1.1.1.1.1.3.cmml">′</mo></msup><mo id="S5.I1.i1.p1.1.m1.1.1.1.1.1.3" stretchy="false" xref="S5.I1.i1.p1.1.m1.1.1.1.1.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.I1.i1.p1.1.m1.1b"><apply id="S5.I1.i1.p1.1.m1.1.1.cmml" xref="S5.I1.i1.p1.1.m1.1.1"><csymbol cd="latexml" id="S5.I1.i1.p1.1.m1.1.1.2.cmml" xref="S5.I1.i1.p1.1.m1.1.1.2">assign</csymbol><ci id="S5.I1.i1.p1.1.m1.1.1.3.cmml" xref="S5.I1.i1.p1.1.m1.1.1.3">𝐿</ci><apply id="S5.I1.i1.p1.1.m1.1.1.1.cmml" xref="S5.I1.i1.p1.1.m1.1.1.1"><times id="S5.I1.i1.p1.1.m1.1.1.1.2.cmml" xref="S5.I1.i1.p1.1.m1.1.1.1.2"></times><ci id="S5.I1.i1.p1.1.m1.1.1.1.3.cmml" xref="S5.I1.i1.p1.1.m1.1.1.1.3">𝐸</ci><apply id="S5.I1.i1.p1.1.m1.1.1.1.1.1.1.cmml" xref="S5.I1.i1.p1.1.m1.1.1.1.1.1"><csymbol cd="ambiguous" id="S5.I1.i1.p1.1.m1.1.1.1.1.1.1.1.cmml" xref="S5.I1.i1.p1.1.m1.1.1.1.1.1">superscript</csymbol><ci id="S5.I1.i1.p1.1.m1.1.1.1.1.1.1.2.cmml" xref="S5.I1.i1.p1.1.m1.1.1.1.1.1.1.2">𝐺</ci><ci id="S5.I1.i1.p1.1.m1.1.1.1.1.1.1.3.cmml" xref="S5.I1.i1.p1.1.m1.1.1.1.1.1.1.3">′</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.I1.i1.p1.1.m1.1c">L:=E(G^{\prime})</annotation><annotation encoding="application/x-llamapun" id="S5.I1.i1.p1.1.m1.1d">italic_L := italic_E ( italic_G start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT )</annotation></semantics></math> and feeds it into <math alttext="\mathcal{A}" class="ltx_Math" display="inline" id="S5.I1.i1.p1.2.m2.1"><semantics id="S5.I1.i1.p1.2.m2.1a"><mi class="ltx_font_mathcaligraphic" id="S5.I1.i1.p1.2.m2.1.1" xref="S5.I1.i1.p1.2.m2.1.1.cmml">𝒜</mi><annotation-xml encoding="MathML-Content" id="S5.I1.i1.p1.2.m2.1b"><ci id="S5.I1.i1.p1.2.m2.1.1.cmml" xref="S5.I1.i1.p1.2.m2.1.1">𝒜</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.I1.i1.p1.2.m2.1c">\mathcal{A}</annotation><annotation encoding="application/x-llamapun" id="S5.I1.i1.p1.2.m2.1d">caligraphic_A</annotation></semantics></math>. She then sends the memory contents of <math alttext="\mathcal{A}" class="ltx_Math" display="inline" id="S5.I1.i1.p1.3.m3.1"><semantics id="S5.I1.i1.p1.3.m3.1a"><mi class="ltx_font_mathcaligraphic" id="S5.I1.i1.p1.3.m3.1.1" xref="S5.I1.i1.p1.3.m3.1.1.cmml">𝒜</mi><annotation-xml encoding="MathML-Content" id="S5.I1.i1.p1.3.m3.1b"><ci id="S5.I1.i1.p1.3.m3.1.1.cmml" xref="S5.I1.i1.p1.3.m3.1.1">𝒜</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.I1.i1.p1.3.m3.1c">\mathcal{A}</annotation><annotation encoding="application/x-llamapun" id="S5.I1.i1.p1.3.m3.1d">caligraphic_A</annotation></semantics></math> to Bob. To determine whether <math alttext="(u,v)\in G^{\prime}" class="ltx_Math" display="inline" id="S5.I1.i1.p1.4.m4.2"><semantics id="S5.I1.i1.p1.4.m4.2a"><mrow id="S5.I1.i1.p1.4.m4.2.3" xref="S5.I1.i1.p1.4.m4.2.3.cmml"><mrow id="S5.I1.i1.p1.4.m4.2.3.2.2" xref="S5.I1.i1.p1.4.m4.2.3.2.1.cmml"><mo id="S5.I1.i1.p1.4.m4.2.3.2.2.1" stretchy="false" xref="S5.I1.i1.p1.4.m4.2.3.2.1.cmml">(</mo><mi id="S5.I1.i1.p1.4.m4.1.1" xref="S5.I1.i1.p1.4.m4.1.1.cmml">u</mi><mo id="S5.I1.i1.p1.4.m4.2.3.2.2.2" xref="S5.I1.i1.p1.4.m4.2.3.2.1.cmml">,</mo><mi id="S5.I1.i1.p1.4.m4.2.2" xref="S5.I1.i1.p1.4.m4.2.2.cmml">v</mi><mo id="S5.I1.i1.p1.4.m4.2.3.2.2.3" stretchy="false" xref="S5.I1.i1.p1.4.m4.2.3.2.1.cmml">)</mo></mrow><mo id="S5.I1.i1.p1.4.m4.2.3.1" xref="S5.I1.i1.p1.4.m4.2.3.1.cmml">∈</mo><msup id="S5.I1.i1.p1.4.m4.2.3.3" xref="S5.I1.i1.p1.4.m4.2.3.3.cmml"><mi id="S5.I1.i1.p1.4.m4.2.3.3.2" xref="S5.I1.i1.p1.4.m4.2.3.3.2.cmml">G</mi><mo id="S5.I1.i1.p1.4.m4.2.3.3.3" xref="S5.I1.i1.p1.4.m4.2.3.3.3.cmml">′</mo></msup></mrow><annotation-xml encoding="MathML-Content" id="S5.I1.i1.p1.4.m4.2b"><apply id="S5.I1.i1.p1.4.m4.2.3.cmml" xref="S5.I1.i1.p1.4.m4.2.3"><in id="S5.I1.i1.p1.4.m4.2.3.1.cmml" xref="S5.I1.i1.p1.4.m4.2.3.1"></in><interval closure="open" id="S5.I1.i1.p1.4.m4.2.3.2.1.cmml" xref="S5.I1.i1.p1.4.m4.2.3.2.2"><ci id="S5.I1.i1.p1.4.m4.1.1.cmml" xref="S5.I1.i1.p1.4.m4.1.1">𝑢</ci><ci id="S5.I1.i1.p1.4.m4.2.2.cmml" xref="S5.I1.i1.p1.4.m4.2.2">𝑣</ci></interval><apply id="S5.I1.i1.p1.4.m4.2.3.3.cmml" xref="S5.I1.i1.p1.4.m4.2.3.3"><csymbol cd="ambiguous" id="S5.I1.i1.p1.4.m4.2.3.3.1.cmml" xref="S5.I1.i1.p1.4.m4.2.3.3">superscript</csymbol><ci id="S5.I1.i1.p1.4.m4.2.3.3.2.cmml" xref="S5.I1.i1.p1.4.m4.2.3.3.2">𝐺</ci><ci id="S5.I1.i1.p1.4.m4.2.3.3.3.cmml" xref="S5.I1.i1.p1.4.m4.2.3.3.3">′</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.I1.i1.p1.4.m4.2c">(u,v)\in G^{\prime}</annotation><annotation 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id="S5.I1.i1.p1.5.m5.6c">E:=\{(x_{1},x_{2}),(x_{2},x_{3}),\dots,(x_{|V|-1},x_{|V|})\}</annotation><annotation encoding="application/x-llamapun" id="S5.I1.i1.p1.5.m5.6d">italic_E := { ( italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_x start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) , ( italic_x start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT , italic_x start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ) , … , ( italic_x start_POSTSUBSCRIPT | italic_V | - 1 end_POSTSUBSCRIPT , italic_x start_POSTSUBSCRIPT | italic_V | end_POSTSUBSCRIPT ) }</annotation></semantics></math> and feeds <math alttext="E" class="ltx_Math" display="inline" id="S5.I1.i1.p1.6.m6.1"><semantics id="S5.I1.i1.p1.6.m6.1a"><mi id="S5.I1.i1.p1.6.m6.1.1" xref="S5.I1.i1.p1.6.m6.1.1.cmml">E</mi><annotation-xml encoding="MathML-Content" id="S5.I1.i1.p1.6.m6.1b"><ci id="S5.I1.i1.p1.6.m6.1.1.cmml" xref="S5.I1.i1.p1.6.m6.1.1">𝐸</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.I1.i1.p1.6.m6.1c">E</annotation><annotation encoding="application/x-llamapun" id="S5.I1.i1.p1.6.m6.1d">italic_E</annotation></semantics></math> to <math alttext="\mathcal{A}" class="ltx_Math" display="inline" id="S5.I1.i1.p1.7.m7.1"><semantics id="S5.I1.i1.p1.7.m7.1a"><mi class="ltx_font_mathcaligraphic" id="S5.I1.i1.p1.7.m7.1.1" xref="S5.I1.i1.p1.7.m7.1.1.cmml">𝒜</mi><annotation-xml encoding="MathML-Content" id="S5.I1.i1.p1.7.m7.1b"><ci id="S5.I1.i1.p1.7.m7.1.1.cmml" xref="S5.I1.i1.p1.7.m7.1.1">𝒜</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.I1.i1.p1.7.m7.1c">\mathcal{A}</annotation><annotation encoding="application/x-llamapun" id="S5.I1.i1.p1.7.m7.1d">caligraphic_A</annotation></semantics></math>, where <math alttext="x_{1}:=u" class="ltx_Math" display="inline" id="S5.I1.i1.p1.8.m8.1"><semantics id="S5.I1.i1.p1.8.m8.1a"><mrow id="S5.I1.i1.p1.8.m8.1.1" xref="S5.I1.i1.p1.8.m8.1.1.cmml"><msub id="S5.I1.i1.p1.8.m8.1.1.2" xref="S5.I1.i1.p1.8.m8.1.1.2.cmml"><mi id="S5.I1.i1.p1.8.m8.1.1.2.2" xref="S5.I1.i1.p1.8.m8.1.1.2.2.cmml">x</mi><mn id="S5.I1.i1.p1.8.m8.1.1.2.3" xref="S5.I1.i1.p1.8.m8.1.1.2.3.cmml">1</mn></msub><mo id="S5.I1.i1.p1.8.m8.1.1.1" lspace="0.278em" rspace="0.278em" xref="S5.I1.i1.p1.8.m8.1.1.1.cmml">:=</mo><mi id="S5.I1.i1.p1.8.m8.1.1.3" xref="S5.I1.i1.p1.8.m8.1.1.3.cmml">u</mi></mrow><annotation-xml encoding="MathML-Content" id="S5.I1.i1.p1.8.m8.1b"><apply id="S5.I1.i1.p1.8.m8.1.1.cmml" xref="S5.I1.i1.p1.8.m8.1.1"><csymbol cd="latexml" id="S5.I1.i1.p1.8.m8.1.1.1.cmml" xref="S5.I1.i1.p1.8.m8.1.1.1">assign</csymbol><apply id="S5.I1.i1.p1.8.m8.1.1.2.cmml" xref="S5.I1.i1.p1.8.m8.1.1.2"><csymbol cd="ambiguous" id="S5.I1.i1.p1.8.m8.1.1.2.1.cmml" xref="S5.I1.i1.p1.8.m8.1.1.2">subscript</csymbol><ci id="S5.I1.i1.p1.8.m8.1.1.2.2.cmml" xref="S5.I1.i1.p1.8.m8.1.1.2.2">𝑥</ci><cn id="S5.I1.i1.p1.8.m8.1.1.2.3.cmml" type="integer" xref="S5.I1.i1.p1.8.m8.1.1.2.3">1</cn></apply><ci id="S5.I1.i1.p1.8.m8.1.1.3.cmml" xref="S5.I1.i1.p1.8.m8.1.1.3">𝑢</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.I1.i1.p1.8.m8.1c">x_{1}:=u</annotation><annotation encoding="application/x-llamapun" id="S5.I1.i1.p1.8.m8.1d">italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT := italic_u</annotation></semantics></math>, <math alttext="x_{|V|}:=v" class="ltx_Math" display="inline" id="S5.I1.i1.p1.9.m9.1"><semantics id="S5.I1.i1.p1.9.m9.1a"><mrow id="S5.I1.i1.p1.9.m9.1.2" xref="S5.I1.i1.p1.9.m9.1.2.cmml"><msub id="S5.I1.i1.p1.9.m9.1.2.2" xref="S5.I1.i1.p1.9.m9.1.2.2.cmml"><mi id="S5.I1.i1.p1.9.m9.1.2.2.2" xref="S5.I1.i1.p1.9.m9.1.2.2.2.cmml">x</mi><mrow id="S5.I1.i1.p1.9.m9.1.1.1.3" xref="S5.I1.i1.p1.9.m9.1.1.1.2.cmml"><mo id="S5.I1.i1.p1.9.m9.1.1.1.3.1" stretchy="false" xref="S5.I1.i1.p1.9.m9.1.1.1.2.1.cmml">|</mo><mi id="S5.I1.i1.p1.9.m9.1.1.1.1" xref="S5.I1.i1.p1.9.m9.1.1.1.1.cmml">V</mi><mo id="S5.I1.i1.p1.9.m9.1.1.1.3.2" stretchy="false" xref="S5.I1.i1.p1.9.m9.1.1.1.2.1.cmml">|</mo></mrow></msub><mo id="S5.I1.i1.p1.9.m9.1.2.1" lspace="0.278em" rspace="0.278em" xref="S5.I1.i1.p1.9.m9.1.2.1.cmml">:=</mo><mi id="S5.I1.i1.p1.9.m9.1.2.3" xref="S5.I1.i1.p1.9.m9.1.2.3.cmml">v</mi></mrow><annotation-xml encoding="MathML-Content" id="S5.I1.i1.p1.9.m9.1b"><apply id="S5.I1.i1.p1.9.m9.1.2.cmml" xref="S5.I1.i1.p1.9.m9.1.2"><csymbol cd="latexml" id="S5.I1.i1.p1.9.m9.1.2.1.cmml" xref="S5.I1.i1.p1.9.m9.1.2.1">assign</csymbol><apply id="S5.I1.i1.p1.9.m9.1.2.2.cmml" xref="S5.I1.i1.p1.9.m9.1.2.2"><csymbol cd="ambiguous" id="S5.I1.i1.p1.9.m9.1.2.2.1.cmml" xref="S5.I1.i1.p1.9.m9.1.2.2">subscript</csymbol><ci id="S5.I1.i1.p1.9.m9.1.2.2.2.cmml" xref="S5.I1.i1.p1.9.m9.1.2.2.2">𝑥</ci><apply id="S5.I1.i1.p1.9.m9.1.1.1.2.cmml" xref="S5.I1.i1.p1.9.m9.1.1.1.3"><abs id="S5.I1.i1.p1.9.m9.1.1.1.2.1.cmml" xref="S5.I1.i1.p1.9.m9.1.1.1.3.1"></abs><ci id="S5.I1.i1.p1.9.m9.1.1.1.1.cmml" xref="S5.I1.i1.p1.9.m9.1.1.1.1">𝑉</ci></apply></apply><ci id="S5.I1.i1.p1.9.m9.1.2.3.cmml" xref="S5.I1.i1.p1.9.m9.1.2.3">𝑣</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.I1.i1.p1.9.m9.1c">x_{|V|}:=v</annotation><annotation encoding="application/x-llamapun" id="S5.I1.i1.p1.9.m9.1d">italic_x start_POSTSUBSCRIPT | italic_V | end_POSTSUBSCRIPT := italic_v</annotation></semantics></math>, and the intermediate vertices <math alttext="\{x_{2},\dots,x_{|V|-1}\}=V\setminus\{u,v\}" class="ltx_Math" display="inline" id="S5.I1.i1.p1.10.m10.6"><semantics id="S5.I1.i1.p1.10.m10.6a"><mrow id="S5.I1.i1.p1.10.m10.6.6" xref="S5.I1.i1.p1.10.m10.6.6.cmml"><mrow id="S5.I1.i1.p1.10.m10.6.6.2.2" xref="S5.I1.i1.p1.10.m10.6.6.2.3.cmml"><mo id="S5.I1.i1.p1.10.m10.6.6.2.2.3" stretchy="false" xref="S5.I1.i1.p1.10.m10.6.6.2.3.cmml">{</mo><msub id="S5.I1.i1.p1.10.m10.5.5.1.1.1" xref="S5.I1.i1.p1.10.m10.5.5.1.1.1.cmml"><mi id="S5.I1.i1.p1.10.m10.5.5.1.1.1.2" xref="S5.I1.i1.p1.10.m10.5.5.1.1.1.2.cmml">x</mi><mn id="S5.I1.i1.p1.10.m10.5.5.1.1.1.3" xref="S5.I1.i1.p1.10.m10.5.5.1.1.1.3.cmml">2</mn></msub><mo id="S5.I1.i1.p1.10.m10.6.6.2.2.4" xref="S5.I1.i1.p1.10.m10.6.6.2.3.cmml">,</mo><mi id="S5.I1.i1.p1.10.m10.2.2" mathvariant="normal" xref="S5.I1.i1.p1.10.m10.2.2.cmml">…</mi><mo id="S5.I1.i1.p1.10.m10.6.6.2.2.5" xref="S5.I1.i1.p1.10.m10.6.6.2.3.cmml">,</mo><msub id="S5.I1.i1.p1.10.m10.6.6.2.2.2" xref="S5.I1.i1.p1.10.m10.6.6.2.2.2.cmml"><mi id="S5.I1.i1.p1.10.m10.6.6.2.2.2.2" xref="S5.I1.i1.p1.10.m10.6.6.2.2.2.2.cmml">x</mi><mrow id="S5.I1.i1.p1.10.m10.1.1.1" xref="S5.I1.i1.p1.10.m10.1.1.1.cmml"><mrow id="S5.I1.i1.p1.10.m10.1.1.1.3.2" xref="S5.I1.i1.p1.10.m10.1.1.1.3.1.cmml"><mo id="S5.I1.i1.p1.10.m10.1.1.1.3.2.1" stretchy="false" xref="S5.I1.i1.p1.10.m10.1.1.1.3.1.1.cmml">|</mo><mi id="S5.I1.i1.p1.10.m10.1.1.1.1" xref="S5.I1.i1.p1.10.m10.1.1.1.1.cmml">V</mi><mo id="S5.I1.i1.p1.10.m10.1.1.1.3.2.2" stretchy="false" xref="S5.I1.i1.p1.10.m10.1.1.1.3.1.1.cmml">|</mo></mrow><mo id="S5.I1.i1.p1.10.m10.1.1.1.2" xref="S5.I1.i1.p1.10.m10.1.1.1.2.cmml">−</mo><mn id="S5.I1.i1.p1.10.m10.1.1.1.4" xref="S5.I1.i1.p1.10.m10.1.1.1.4.cmml">1</mn></mrow></msub><mo id="S5.I1.i1.p1.10.m10.6.6.2.2.6" stretchy="false" xref="S5.I1.i1.p1.10.m10.6.6.2.3.cmml">}</mo></mrow><mo id="S5.I1.i1.p1.10.m10.6.6.3" xref="S5.I1.i1.p1.10.m10.6.6.3.cmml">=</mo><mrow id="S5.I1.i1.p1.10.m10.6.6.4" xref="S5.I1.i1.p1.10.m10.6.6.4.cmml"><mi id="S5.I1.i1.p1.10.m10.6.6.4.2" xref="S5.I1.i1.p1.10.m10.6.6.4.2.cmml">V</mi><mo id="S5.I1.i1.p1.10.m10.6.6.4.1" xref="S5.I1.i1.p1.10.m10.6.6.4.1.cmml">∖</mo><mrow id="S5.I1.i1.p1.10.m10.6.6.4.3.2" xref="S5.I1.i1.p1.10.m10.6.6.4.3.1.cmml"><mo id="S5.I1.i1.p1.10.m10.6.6.4.3.2.1" stretchy="false" xref="S5.I1.i1.p1.10.m10.6.6.4.3.1.cmml">{</mo><mi id="S5.I1.i1.p1.10.m10.3.3" xref="S5.I1.i1.p1.10.m10.3.3.cmml">u</mi><mo id="S5.I1.i1.p1.10.m10.6.6.4.3.2.2" xref="S5.I1.i1.p1.10.m10.6.6.4.3.1.cmml">,</mo><mi id="S5.I1.i1.p1.10.m10.4.4" xref="S5.I1.i1.p1.10.m10.4.4.cmml">v</mi><mo id="S5.I1.i1.p1.10.m10.6.6.4.3.2.3" stretchy="false" xref="S5.I1.i1.p1.10.m10.6.6.4.3.1.cmml">}</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.I1.i1.p1.10.m10.6b"><apply id="S5.I1.i1.p1.10.m10.6.6.cmml" xref="S5.I1.i1.p1.10.m10.6.6"><eq id="S5.I1.i1.p1.10.m10.6.6.3.cmml" xref="S5.I1.i1.p1.10.m10.6.6.3"></eq><set id="S5.I1.i1.p1.10.m10.6.6.2.3.cmml" xref="S5.I1.i1.p1.10.m10.6.6.2.2"><apply id="S5.I1.i1.p1.10.m10.5.5.1.1.1.cmml" xref="S5.I1.i1.p1.10.m10.5.5.1.1.1"><csymbol cd="ambiguous" id="S5.I1.i1.p1.10.m10.5.5.1.1.1.1.cmml" xref="S5.I1.i1.p1.10.m10.5.5.1.1.1">subscript</csymbol><ci id="S5.I1.i1.p1.10.m10.5.5.1.1.1.2.cmml" xref="S5.I1.i1.p1.10.m10.5.5.1.1.1.2">𝑥</ci><cn id="S5.I1.i1.p1.10.m10.5.5.1.1.1.3.cmml" type="integer" xref="S5.I1.i1.p1.10.m10.5.5.1.1.1.3">2</cn></apply><ci id="S5.I1.i1.p1.10.m10.2.2.cmml" xref="S5.I1.i1.p1.10.m10.2.2">…</ci><apply id="S5.I1.i1.p1.10.m10.6.6.2.2.2.cmml" xref="S5.I1.i1.p1.10.m10.6.6.2.2.2"><csymbol cd="ambiguous" id="S5.I1.i1.p1.10.m10.6.6.2.2.2.1.cmml" xref="S5.I1.i1.p1.10.m10.6.6.2.2.2">subscript</csymbol><ci id="S5.I1.i1.p1.10.m10.6.6.2.2.2.2.cmml" xref="S5.I1.i1.p1.10.m10.6.6.2.2.2.2">𝑥</ci><apply id="S5.I1.i1.p1.10.m10.1.1.1.cmml" xref="S5.I1.i1.p1.10.m10.1.1.1"><minus id="S5.I1.i1.p1.10.m10.1.1.1.2.cmml" xref="S5.I1.i1.p1.10.m10.1.1.1.2"></minus><apply id="S5.I1.i1.p1.10.m10.1.1.1.3.1.cmml" xref="S5.I1.i1.p1.10.m10.1.1.1.3.2"><abs id="S5.I1.i1.p1.10.m10.1.1.1.3.1.1.cmml" xref="S5.I1.i1.p1.10.m10.1.1.1.3.2.1"></abs><ci id="S5.I1.i1.p1.10.m10.1.1.1.1.cmml" xref="S5.I1.i1.p1.10.m10.1.1.1.1">𝑉</ci></apply><cn id="S5.I1.i1.p1.10.m10.1.1.1.4.cmml" type="integer" xref="S5.I1.i1.p1.10.m10.1.1.1.4">1</cn></apply></apply></set><apply id="S5.I1.i1.p1.10.m10.6.6.4.cmml" xref="S5.I1.i1.p1.10.m10.6.6.4"><setdiff id="S5.I1.i1.p1.10.m10.6.6.4.1.cmml" xref="S5.I1.i1.p1.10.m10.6.6.4.1"></setdiff><ci id="S5.I1.i1.p1.10.m10.6.6.4.2.cmml" xref="S5.I1.i1.p1.10.m10.6.6.4.2">𝑉</ci><set id="S5.I1.i1.p1.10.m10.6.6.4.3.1.cmml" xref="S5.I1.i1.p1.10.m10.6.6.4.3.2"><ci id="S5.I1.i1.p1.10.m10.3.3.cmml" xref="S5.I1.i1.p1.10.m10.3.3">𝑢</ci><ci id="S5.I1.i1.p1.10.m10.4.4.cmml" xref="S5.I1.i1.p1.10.m10.4.4">𝑣</ci></set></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.I1.i1.p1.10.m10.6c">\{x_{2},\dots,x_{|V|-1}\}=V\setminus\{u,v\}</annotation><annotation encoding="application/x-llamapun" id="S5.I1.i1.p1.10.m10.6d">{ italic_x start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT , … , italic_x start_POSTSUBSCRIPT | italic_V | - 1 end_POSTSUBSCRIPT } = italic_V ∖ { italic_u , italic_v }</annotation></semantics></math> are ordered so that <math alttext="d_{H}(u,x_{j-1})\leq d_{H}(u,x_{j})" class="ltx_Math" display="inline" id="S5.I1.i1.p1.11.m11.4"><semantics id="S5.I1.i1.p1.11.m11.4a"><mrow id="S5.I1.i1.p1.11.m11.4.4" xref="S5.I1.i1.p1.11.m11.4.4.cmml"><mrow id="S5.I1.i1.p1.11.m11.3.3.1" xref="S5.I1.i1.p1.11.m11.3.3.1.cmml"><msub id="S5.I1.i1.p1.11.m11.3.3.1.3" xref="S5.I1.i1.p1.11.m11.3.3.1.3.cmml"><mi id="S5.I1.i1.p1.11.m11.3.3.1.3.2" xref="S5.I1.i1.p1.11.m11.3.3.1.3.2.cmml">d</mi><mi id="S5.I1.i1.p1.11.m11.3.3.1.3.3" xref="S5.I1.i1.p1.11.m11.3.3.1.3.3.cmml">H</mi></msub><mo id="S5.I1.i1.p1.11.m11.3.3.1.2" xref="S5.I1.i1.p1.11.m11.3.3.1.2.cmml">⁢</mo><mrow id="S5.I1.i1.p1.11.m11.3.3.1.1.1" xref="S5.I1.i1.p1.11.m11.3.3.1.1.2.cmml"><mo id="S5.I1.i1.p1.11.m11.3.3.1.1.1.2" stretchy="false" xref="S5.I1.i1.p1.11.m11.3.3.1.1.2.cmml">(</mo><mi id="S5.I1.i1.p1.11.m11.1.1" xref="S5.I1.i1.p1.11.m11.1.1.cmml">u</mi><mo id="S5.I1.i1.p1.11.m11.3.3.1.1.1.3" xref="S5.I1.i1.p1.11.m11.3.3.1.1.2.cmml">,</mo><msub id="S5.I1.i1.p1.11.m11.3.3.1.1.1.1" xref="S5.I1.i1.p1.11.m11.3.3.1.1.1.1.cmml"><mi id="S5.I1.i1.p1.11.m11.3.3.1.1.1.1.2" xref="S5.I1.i1.p1.11.m11.3.3.1.1.1.1.2.cmml">x</mi><mrow id="S5.I1.i1.p1.11.m11.3.3.1.1.1.1.3" xref="S5.I1.i1.p1.11.m11.3.3.1.1.1.1.3.cmml"><mi id="S5.I1.i1.p1.11.m11.3.3.1.1.1.1.3.2" xref="S5.I1.i1.p1.11.m11.3.3.1.1.1.1.3.2.cmml">j</mi><mo id="S5.I1.i1.p1.11.m11.3.3.1.1.1.1.3.1" xref="S5.I1.i1.p1.11.m11.3.3.1.1.1.1.3.1.cmml">−</mo><mn id="S5.I1.i1.p1.11.m11.3.3.1.1.1.1.3.3" xref="S5.I1.i1.p1.11.m11.3.3.1.1.1.1.3.3.cmml">1</mn></mrow></msub><mo id="S5.I1.i1.p1.11.m11.3.3.1.1.1.4" stretchy="false" xref="S5.I1.i1.p1.11.m11.3.3.1.1.2.cmml">)</mo></mrow></mrow><mo id="S5.I1.i1.p1.11.m11.4.4.3" xref="S5.I1.i1.p1.11.m11.4.4.3.cmml">≤</mo><mrow id="S5.I1.i1.p1.11.m11.4.4.2" xref="S5.I1.i1.p1.11.m11.4.4.2.cmml"><msub id="S5.I1.i1.p1.11.m11.4.4.2.3" xref="S5.I1.i1.p1.11.m11.4.4.2.3.cmml"><mi id="S5.I1.i1.p1.11.m11.4.4.2.3.2" xref="S5.I1.i1.p1.11.m11.4.4.2.3.2.cmml">d</mi><mi id="S5.I1.i1.p1.11.m11.4.4.2.3.3" xref="S5.I1.i1.p1.11.m11.4.4.2.3.3.cmml">H</mi></msub><mo id="S5.I1.i1.p1.11.m11.4.4.2.2" xref="S5.I1.i1.p1.11.m11.4.4.2.2.cmml">⁢</mo><mrow id="S5.I1.i1.p1.11.m11.4.4.2.1.1" xref="S5.I1.i1.p1.11.m11.4.4.2.1.2.cmml"><mo id="S5.I1.i1.p1.11.m11.4.4.2.1.1.2" stretchy="false" xref="S5.I1.i1.p1.11.m11.4.4.2.1.2.cmml">(</mo><mi id="S5.I1.i1.p1.11.m11.2.2" xref="S5.I1.i1.p1.11.m11.2.2.cmml">u</mi><mo id="S5.I1.i1.p1.11.m11.4.4.2.1.1.3" xref="S5.I1.i1.p1.11.m11.4.4.2.1.2.cmml">,</mo><msub id="S5.I1.i1.p1.11.m11.4.4.2.1.1.1" xref="S5.I1.i1.p1.11.m11.4.4.2.1.1.1.cmml"><mi id="S5.I1.i1.p1.11.m11.4.4.2.1.1.1.2" xref="S5.I1.i1.p1.11.m11.4.4.2.1.1.1.2.cmml">x</mi><mi id="S5.I1.i1.p1.11.m11.4.4.2.1.1.1.3" xref="S5.I1.i1.p1.11.m11.4.4.2.1.1.1.3.cmml">j</mi></msub><mo id="S5.I1.i1.p1.11.m11.4.4.2.1.1.4" stretchy="false" xref="S5.I1.i1.p1.11.m11.4.4.2.1.2.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.I1.i1.p1.11.m11.4b"><apply id="S5.I1.i1.p1.11.m11.4.4.cmml" xref="S5.I1.i1.p1.11.m11.4.4"><leq id="S5.I1.i1.p1.11.m11.4.4.3.cmml" xref="S5.I1.i1.p1.11.m11.4.4.3"></leq><apply id="S5.I1.i1.p1.11.m11.3.3.1.cmml" xref="S5.I1.i1.p1.11.m11.3.3.1"><times id="S5.I1.i1.p1.11.m11.3.3.1.2.cmml" xref="S5.I1.i1.p1.11.m11.3.3.1.2"></times><apply id="S5.I1.i1.p1.11.m11.3.3.1.3.cmml" xref="S5.I1.i1.p1.11.m11.3.3.1.3"><csymbol cd="ambiguous" id="S5.I1.i1.p1.11.m11.3.3.1.3.1.cmml" xref="S5.I1.i1.p1.11.m11.3.3.1.3">subscript</csymbol><ci id="S5.I1.i1.p1.11.m11.3.3.1.3.2.cmml" xref="S5.I1.i1.p1.11.m11.3.3.1.3.2">𝑑</ci><ci id="S5.I1.i1.p1.11.m11.3.3.1.3.3.cmml" xref="S5.I1.i1.p1.11.m11.3.3.1.3.3">𝐻</ci></apply><interval closure="open" id="S5.I1.i1.p1.11.m11.3.3.1.1.2.cmml" xref="S5.I1.i1.p1.11.m11.3.3.1.1.1"><ci id="S5.I1.i1.p1.11.m11.1.1.cmml" xref="S5.I1.i1.p1.11.m11.1.1">𝑢</ci><apply id="S5.I1.i1.p1.11.m11.3.3.1.1.1.1.cmml" xref="S5.I1.i1.p1.11.m11.3.3.1.1.1.1"><csymbol cd="ambiguous" id="S5.I1.i1.p1.11.m11.3.3.1.1.1.1.1.cmml" xref="S5.I1.i1.p1.11.m11.3.3.1.1.1.1">subscript</csymbol><ci id="S5.I1.i1.p1.11.m11.3.3.1.1.1.1.2.cmml" xref="S5.I1.i1.p1.11.m11.3.3.1.1.1.1.2">𝑥</ci><apply id="S5.I1.i1.p1.11.m11.3.3.1.1.1.1.3.cmml" xref="S5.I1.i1.p1.11.m11.3.3.1.1.1.1.3"><minus id="S5.I1.i1.p1.11.m11.3.3.1.1.1.1.3.1.cmml" xref="S5.I1.i1.p1.11.m11.3.3.1.1.1.1.3.1"></minus><ci id="S5.I1.i1.p1.11.m11.3.3.1.1.1.1.3.2.cmml" xref="S5.I1.i1.p1.11.m11.3.3.1.1.1.1.3.2">𝑗</ci><cn id="S5.I1.i1.p1.11.m11.3.3.1.1.1.1.3.3.cmml" type="integer" xref="S5.I1.i1.p1.11.m11.3.3.1.1.1.1.3.3">1</cn></apply></apply></interval></apply><apply id="S5.I1.i1.p1.11.m11.4.4.2.cmml" xref="S5.I1.i1.p1.11.m11.4.4.2"><times id="S5.I1.i1.p1.11.m11.4.4.2.2.cmml" xref="S5.I1.i1.p1.11.m11.4.4.2.2"></times><apply id="S5.I1.i1.p1.11.m11.4.4.2.3.cmml" xref="S5.I1.i1.p1.11.m11.4.4.2.3"><csymbol cd="ambiguous" id="S5.I1.i1.p1.11.m11.4.4.2.3.1.cmml" xref="S5.I1.i1.p1.11.m11.4.4.2.3">subscript</csymbol><ci id="S5.I1.i1.p1.11.m11.4.4.2.3.2.cmml" xref="S5.I1.i1.p1.11.m11.4.4.2.3.2">𝑑</ci><ci id="S5.I1.i1.p1.11.m11.4.4.2.3.3.cmml" xref="S5.I1.i1.p1.11.m11.4.4.2.3.3">𝐻</ci></apply><interval closure="open" id="S5.I1.i1.p1.11.m11.4.4.2.1.2.cmml" xref="S5.I1.i1.p1.11.m11.4.4.2.1.1"><ci id="S5.I1.i1.p1.11.m11.2.2.cmml" xref="S5.I1.i1.p1.11.m11.2.2">𝑢</ci><apply id="S5.I1.i1.p1.11.m11.4.4.2.1.1.1.cmml" xref="S5.I1.i1.p1.11.m11.4.4.2.1.1.1"><csymbol cd="ambiguous" id="S5.I1.i1.p1.11.m11.4.4.2.1.1.1.1.cmml" xref="S5.I1.i1.p1.11.m11.4.4.2.1.1.1">subscript</csymbol><ci id="S5.I1.i1.p1.11.m11.4.4.2.1.1.1.2.cmml" xref="S5.I1.i1.p1.11.m11.4.4.2.1.1.1.2">𝑥</ci><ci id="S5.I1.i1.p1.11.m11.4.4.2.1.1.1.3.cmml" xref="S5.I1.i1.p1.11.m11.4.4.2.1.1.1.3">𝑗</ci></apply></interval></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.I1.i1.p1.11.m11.4c">d_{H}(u,x_{j-1})\leq d_{H}(u,x_{j})</annotation><annotation encoding="application/x-llamapun" id="S5.I1.i1.p1.11.m11.4d">italic_d start_POSTSUBSCRIPT italic_H end_POSTSUBSCRIPT ( italic_u , italic_x start_POSTSUBSCRIPT italic_j - 1 end_POSTSUBSCRIPT ) ≤ italic_d start_POSTSUBSCRIPT italic_H end_POSTSUBSCRIPT ( italic_u , italic_x start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT )</annotation></semantics></math> for <math alttext="2\leq j\leq|V|-1" class="ltx_Math" display="inline" id="S5.I1.i1.p1.12.m12.1"><semantics id="S5.I1.i1.p1.12.m12.1a"><mrow id="S5.I1.i1.p1.12.m12.1.2" xref="S5.I1.i1.p1.12.m12.1.2.cmml"><mn id="S5.I1.i1.p1.12.m12.1.2.2" xref="S5.I1.i1.p1.12.m12.1.2.2.cmml">2</mn><mo id="S5.I1.i1.p1.12.m12.1.2.3" xref="S5.I1.i1.p1.12.m12.1.2.3.cmml">≤</mo><mi id="S5.I1.i1.p1.12.m12.1.2.4" xref="S5.I1.i1.p1.12.m12.1.2.4.cmml">j</mi><mo id="S5.I1.i1.p1.12.m12.1.2.5" xref="S5.I1.i1.p1.12.m12.1.2.5.cmml">≤</mo><mrow id="S5.I1.i1.p1.12.m12.1.2.6" xref="S5.I1.i1.p1.12.m12.1.2.6.cmml"><mrow id="S5.I1.i1.p1.12.m12.1.2.6.2.2" xref="S5.I1.i1.p1.12.m12.1.2.6.2.1.cmml"><mo id="S5.I1.i1.p1.12.m12.1.2.6.2.2.1" stretchy="false" xref="S5.I1.i1.p1.12.m12.1.2.6.2.1.1.cmml">|</mo><mi id="S5.I1.i1.p1.12.m12.1.1" xref="S5.I1.i1.p1.12.m12.1.1.cmml">V</mi><mo id="S5.I1.i1.p1.12.m12.1.2.6.2.2.2" stretchy="false" xref="S5.I1.i1.p1.12.m12.1.2.6.2.1.1.cmml">|</mo></mrow><mo id="S5.I1.i1.p1.12.m12.1.2.6.1" xref="S5.I1.i1.p1.12.m12.1.2.6.1.cmml">−</mo><mn id="S5.I1.i1.p1.12.m12.1.2.6.3" xref="S5.I1.i1.p1.12.m12.1.2.6.3.cmml">1</mn></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.I1.i1.p1.12.m12.1b"><apply id="S5.I1.i1.p1.12.m12.1.2.cmml" xref="S5.I1.i1.p1.12.m12.1.2"><and id="S5.I1.i1.p1.12.m12.1.2a.cmml" xref="S5.I1.i1.p1.12.m12.1.2"></and><apply id="S5.I1.i1.p1.12.m12.1.2b.cmml" xref="S5.I1.i1.p1.12.m12.1.2"><leq id="S5.I1.i1.p1.12.m12.1.2.3.cmml" xref="S5.I1.i1.p1.12.m12.1.2.3"></leq><cn id="S5.I1.i1.p1.12.m12.1.2.2.cmml" type="integer" xref="S5.I1.i1.p1.12.m12.1.2.2">2</cn><ci id="S5.I1.i1.p1.12.m12.1.2.4.cmml" xref="S5.I1.i1.p1.12.m12.1.2.4">𝑗</ci></apply><apply id="S5.I1.i1.p1.12.m12.1.2c.cmml" xref="S5.I1.i1.p1.12.m12.1.2"><leq id="S5.I1.i1.p1.12.m12.1.2.5.cmml" xref="S5.I1.i1.p1.12.m12.1.2.5"></leq><share href="https://arxiv.org/html/2503.00712v1#S5.I1.i1.p1.12.m12.1.2.4.cmml" id="S5.I1.i1.p1.12.m12.1.2d.cmml" xref="S5.I1.i1.p1.12.m12.1.2"></share><apply id="S5.I1.i1.p1.12.m12.1.2.6.cmml" xref="S5.I1.i1.p1.12.m12.1.2.6"><minus id="S5.I1.i1.p1.12.m12.1.2.6.1.cmml" xref="S5.I1.i1.p1.12.m12.1.2.6.1"></minus><apply id="S5.I1.i1.p1.12.m12.1.2.6.2.1.cmml" xref="S5.I1.i1.p1.12.m12.1.2.6.2.2"><abs id="S5.I1.i1.p1.12.m12.1.2.6.2.1.1.cmml" xref="S5.I1.i1.p1.12.m12.1.2.6.2.2.1"></abs><ci id="S5.I1.i1.p1.12.m12.1.1.cmml" xref="S5.I1.i1.p1.12.m12.1.1">𝑉</ci></apply><cn id="S5.I1.i1.p1.12.m12.1.2.6.3.cmml" type="integer" xref="S5.I1.i1.p1.12.m12.1.2.6.3">1</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.I1.i1.p1.12.m12.1c">2\leq j\leq|V|-1</annotation><annotation encoding="application/x-llamapun" id="S5.I1.i1.p1.12.m12.1d">2 ≤ italic_j ≤ | italic_V | - 1</annotation></semantics></math>. Here, the graph <math alttext="H" class="ltx_Math" display="inline" id="S5.I1.i1.p1.13.m13.1"><semantics id="S5.I1.i1.p1.13.m13.1a"><mi id="S5.I1.i1.p1.13.m13.1.1" xref="S5.I1.i1.p1.13.m13.1.1.cmml">H</mi><annotation-xml encoding="MathML-Content" id="S5.I1.i1.p1.13.m13.1b"><ci id="S5.I1.i1.p1.13.m13.1.1.cmml" xref="S5.I1.i1.p1.13.m13.1.1">𝐻</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.I1.i1.p1.13.m13.1c">H</annotation><annotation encoding="application/x-llamapun" id="S5.I1.i1.p1.13.m13.1d">italic_H</annotation></semantics></math> is defined as <math alttext="G" class="ltx_Math" display="inline" id="S5.I1.i1.p1.14.m14.1"><semantics id="S5.I1.i1.p1.14.m14.1a"><mi id="S5.I1.i1.p1.14.m14.1.1" xref="S5.I1.i1.p1.14.m14.1.1.cmml">G</mi><annotation-xml encoding="MathML-Content" id="S5.I1.i1.p1.14.m14.1b"><ci id="S5.I1.i1.p1.14.m14.1.1.cmml" xref="S5.I1.i1.p1.14.m14.1.1">𝐺</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.I1.i1.p1.14.m14.1c">G</annotation><annotation encoding="application/x-llamapun" id="S5.I1.i1.p1.14.m14.1d">italic_G</annotation></semantics></math> with the edge <math alttext="(u,v)" class="ltx_Math" display="inline" id="S5.I1.i1.p1.15.m15.2"><semantics id="S5.I1.i1.p1.15.m15.2a"><mrow id="S5.I1.i1.p1.15.m15.2.3.2" xref="S5.I1.i1.p1.15.m15.2.3.1.cmml"><mo id="S5.I1.i1.p1.15.m15.2.3.2.1" stretchy="false" xref="S5.I1.i1.p1.15.m15.2.3.1.cmml">(</mo><mi id="S5.I1.i1.p1.15.m15.1.1" xref="S5.I1.i1.p1.15.m15.1.1.cmml">u</mi><mo id="S5.I1.i1.p1.15.m15.2.3.2.2" xref="S5.I1.i1.p1.15.m15.2.3.1.cmml">,</mo><mi id="S5.I1.i1.p1.15.m15.2.2" xref="S5.I1.i1.p1.15.m15.2.2.cmml">v</mi><mo id="S5.I1.i1.p1.15.m15.2.3.2.3" stretchy="false" xref="S5.I1.i1.p1.15.m15.2.3.1.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S5.I1.i1.p1.15.m15.2b"><interval closure="open" id="S5.I1.i1.p1.15.m15.2.3.1.cmml" xref="S5.I1.i1.p1.15.m15.2.3.2"><ci id="S5.I1.i1.p1.15.m15.1.1.cmml" xref="S5.I1.i1.p1.15.m15.1.1">𝑢</ci><ci id="S5.I1.i1.p1.15.m15.2.2.cmml" xref="S5.I1.i1.p1.15.m15.2.2">𝑣</ci></interval></annotation-xml><annotation encoding="application/x-tex" id="S5.I1.i1.p1.15.m15.2c">(u,v)</annotation><annotation encoding="application/x-llamapun" id="S5.I1.i1.p1.15.m15.2d">( italic_u , italic_v )</annotation></semantics></math> removed.</p> </div> </li> <li class="ltx_item" id="S5.I1.i2" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">•</span> <div class="ltx_para" id="S5.I1.i2.p1"> <p class="ltx_p" id="S5.I1.i2.p1.4">Bob then determines that <math alttext="(u,v)\in G^{\prime}" class="ltx_Math" display="inline" id="S5.I1.i2.p1.1.m1.2"><semantics id="S5.I1.i2.p1.1.m1.2a"><mrow id="S5.I1.i2.p1.1.m1.2.3" xref="S5.I1.i2.p1.1.m1.2.3.cmml"><mrow id="S5.I1.i2.p1.1.m1.2.3.2.2" xref="S5.I1.i2.p1.1.m1.2.3.2.1.cmml"><mo id="S5.I1.i2.p1.1.m1.2.3.2.2.1" stretchy="false" xref="S5.I1.i2.p1.1.m1.2.3.2.1.cmml">(</mo><mi id="S5.I1.i2.p1.1.m1.1.1" xref="S5.I1.i2.p1.1.m1.1.1.cmml">u</mi><mo id="S5.I1.i2.p1.1.m1.2.3.2.2.2" xref="S5.I1.i2.p1.1.m1.2.3.2.1.cmml">,</mo><mi id="S5.I1.i2.p1.1.m1.2.2" xref="S5.I1.i2.p1.1.m1.2.2.cmml">v</mi><mo id="S5.I1.i2.p1.1.m1.2.3.2.2.3" stretchy="false" xref="S5.I1.i2.p1.1.m1.2.3.2.1.cmml">)</mo></mrow><mo id="S5.I1.i2.p1.1.m1.2.3.1" xref="S5.I1.i2.p1.1.m1.2.3.1.cmml">∈</mo><msup id="S5.I1.i2.p1.1.m1.2.3.3" xref="S5.I1.i2.p1.1.m1.2.3.3.cmml"><mi id="S5.I1.i2.p1.1.m1.2.3.3.2" xref="S5.I1.i2.p1.1.m1.2.3.3.2.cmml">G</mi><mo id="S5.I1.i2.p1.1.m1.2.3.3.3" xref="S5.I1.i2.p1.1.m1.2.3.3.3.cmml">′</mo></msup></mrow><annotation-xml encoding="MathML-Content" id="S5.I1.i2.p1.1.m1.2b"><apply id="S5.I1.i2.p1.1.m1.2.3.cmml" xref="S5.I1.i2.p1.1.m1.2.3"><in id="S5.I1.i2.p1.1.m1.2.3.1.cmml" xref="S5.I1.i2.p1.1.m1.2.3.1"></in><interval closure="open" id="S5.I1.i2.p1.1.m1.2.3.2.1.cmml" xref="S5.I1.i2.p1.1.m1.2.3.2.2"><ci id="S5.I1.i2.p1.1.m1.1.1.cmml" xref="S5.I1.i2.p1.1.m1.1.1">𝑢</ci><ci id="S5.I1.i2.p1.1.m1.2.2.cmml" xref="S5.I1.i2.p1.1.m1.2.2">𝑣</ci></interval><apply id="S5.I1.i2.p1.1.m1.2.3.3.cmml" xref="S5.I1.i2.p1.1.m1.2.3.3"><csymbol cd="ambiguous" id="S5.I1.i2.p1.1.m1.2.3.3.1.cmml" xref="S5.I1.i2.p1.1.m1.2.3.3">superscript</csymbol><ci id="S5.I1.i2.p1.1.m1.2.3.3.2.cmml" xref="S5.I1.i2.p1.1.m1.2.3.3.2">𝐺</ci><ci id="S5.I1.i2.p1.1.m1.2.3.3.3.cmml" xref="S5.I1.i2.p1.1.m1.2.3.3.3">′</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.I1.i2.p1.1.m1.2c">(u,v)\in G^{\prime}</annotation><annotation encoding="application/x-llamapun" id="S5.I1.i2.p1.1.m1.2d">( italic_u , italic_v ) ∈ italic_G start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math> if and only if <math alttext="\mathcal{A}" class="ltx_Math" display="inline" id="S5.I1.i2.p1.2.m2.1"><semantics id="S5.I1.i2.p1.2.m2.1a"><mi class="ltx_font_mathcaligraphic" id="S5.I1.i2.p1.2.m2.1.1" xref="S5.I1.i2.p1.2.m2.1.1.cmml">𝒜</mi><annotation-xml encoding="MathML-Content" id="S5.I1.i2.p1.2.m2.1b"><ci id="S5.I1.i2.p1.2.m2.1.1.cmml" xref="S5.I1.i2.p1.2.m2.1.1">𝒜</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.I1.i2.p1.2.m2.1c">\mathcal{A}</annotation><annotation encoding="application/x-llamapun" id="S5.I1.i2.p1.2.m2.1d">caligraphic_A</annotation></semantics></math> returns an approximate solution with cost less than <math alttext="2t+1" class="ltx_Math" display="inline" id="S5.I1.i2.p1.3.m3.1"><semantics id="S5.I1.i2.p1.3.m3.1a"><mrow id="S5.I1.i2.p1.3.m3.1.1" xref="S5.I1.i2.p1.3.m3.1.1.cmml"><mrow id="S5.I1.i2.p1.3.m3.1.1.2" xref="S5.I1.i2.p1.3.m3.1.1.2.cmml"><mn id="S5.I1.i2.p1.3.m3.1.1.2.2" xref="S5.I1.i2.p1.3.m3.1.1.2.2.cmml">2</mn><mo id="S5.I1.i2.p1.3.m3.1.1.2.1" xref="S5.I1.i2.p1.3.m3.1.1.2.1.cmml">⁢</mo><mi id="S5.I1.i2.p1.3.m3.1.1.2.3" xref="S5.I1.i2.p1.3.m3.1.1.2.3.cmml">t</mi></mrow><mo id="S5.I1.i2.p1.3.m3.1.1.1" xref="S5.I1.i2.p1.3.m3.1.1.1.cmml">+</mo><mn id="S5.I1.i2.p1.3.m3.1.1.3" xref="S5.I1.i2.p1.3.m3.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S5.I1.i2.p1.3.m3.1b"><apply id="S5.I1.i2.p1.3.m3.1.1.cmml" xref="S5.I1.i2.p1.3.m3.1.1"><plus id="S5.I1.i2.p1.3.m3.1.1.1.cmml" xref="S5.I1.i2.p1.3.m3.1.1.1"></plus><apply id="S5.I1.i2.p1.3.m3.1.1.2.cmml" xref="S5.I1.i2.p1.3.m3.1.1.2"><times id="S5.I1.i2.p1.3.m3.1.1.2.1.cmml" xref="S5.I1.i2.p1.3.m3.1.1.2.1"></times><cn id="S5.I1.i2.p1.3.m3.1.1.2.2.cmml" type="integer" xref="S5.I1.i2.p1.3.m3.1.1.2.2">2</cn><ci id="S5.I1.i2.p1.3.m3.1.1.2.3.cmml" xref="S5.I1.i2.p1.3.m3.1.1.2.3">𝑡</ci></apply><cn id="S5.I1.i2.p1.3.m3.1.1.3.cmml" type="integer" xref="S5.I1.i2.p1.3.m3.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.I1.i2.p1.3.m3.1c">2t+1</annotation><annotation encoding="application/x-llamapun" id="S5.I1.i2.p1.3.m3.1d">2 italic_t + 1</annotation></semantics></math> for the instance <math alttext="(E,L)" class="ltx_Math" display="inline" id="S5.I1.i2.p1.4.m4.2"><semantics id="S5.I1.i2.p1.4.m4.2a"><mrow id="S5.I1.i2.p1.4.m4.2.3.2" xref="S5.I1.i2.p1.4.m4.2.3.1.cmml"><mo id="S5.I1.i2.p1.4.m4.2.3.2.1" stretchy="false" xref="S5.I1.i2.p1.4.m4.2.3.1.cmml">(</mo><mi id="S5.I1.i2.p1.4.m4.1.1" xref="S5.I1.i2.p1.4.m4.1.1.cmml">E</mi><mo id="S5.I1.i2.p1.4.m4.2.3.2.2" xref="S5.I1.i2.p1.4.m4.2.3.1.cmml">,</mo><mi id="S5.I1.i2.p1.4.m4.2.2" xref="S5.I1.i2.p1.4.m4.2.2.cmml">L</mi><mo id="S5.I1.i2.p1.4.m4.2.3.2.3" stretchy="false" xref="S5.I1.i2.p1.4.m4.2.3.1.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S5.I1.i2.p1.4.m4.2b"><interval closure="open" id="S5.I1.i2.p1.4.m4.2.3.1.cmml" xref="S5.I1.i2.p1.4.m4.2.3.2"><ci id="S5.I1.i2.p1.4.m4.1.1.cmml" xref="S5.I1.i2.p1.4.m4.1.1">𝐸</ci><ci id="S5.I1.i2.p1.4.m4.2.2.cmml" xref="S5.I1.i2.p1.4.m4.2.2">𝐿</ci></interval></annotation-xml><annotation encoding="application/x-tex" id="S5.I1.i2.p1.4.m4.2c">(E,L)</annotation><annotation encoding="application/x-llamapun" id="S5.I1.i2.p1.4.m4.2d">( italic_E , italic_L )</annotation></semantics></math>.</p> </div> </li> </ul> <p class="ltx_p" id="S5.2.p2.4">Next, we prove the correctness of the described protocol that uses <math alttext="\mathcal{A}" class="ltx_Math" display="inline" id="S5.2.p2.4.m1.1"><semantics id="S5.2.p2.4.m1.1a"><mi class="ltx_font_mathcaligraphic" id="S5.2.p2.4.m1.1.1" xref="S5.2.p2.4.m1.1.1.cmml">𝒜</mi><annotation-xml encoding="MathML-Content" id="S5.2.p2.4.m1.1b"><ci id="S5.2.p2.4.m1.1.1.cmml" xref="S5.2.p2.4.m1.1.1">𝒜</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.2.p2.4.m1.1c">\mathcal{A}</annotation><annotation encoding="application/x-llamapun" id="S5.2.p2.4.m1.1d">caligraphic_A</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="S5.3.p3"> <ul class="ltx_itemize" id="S5.I2"> <li class="ltx_item" id="S5.I2.i1" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">•</span> <div class="ltx_para" id="S5.I2.i1.p1"> <p class="ltx_p" id="S5.I2.i1.p1.6"><span class="ltx_text ltx_font_bold" id="S5.I2.i1.p1.1.1">If <math alttext="(u,v)\in E(G^{\prime})" class="ltx_Math" display="inline" id="S5.I2.i1.p1.1.1.m1.3"><semantics id="S5.I2.i1.p1.1.1.m1.3a"><mrow id="S5.I2.i1.p1.1.1.m1.3.3" xref="S5.I2.i1.p1.1.1.m1.3.3.cmml"><mrow id="S5.I2.i1.p1.1.1.m1.3.3.3.2" xref="S5.I2.i1.p1.1.1.m1.3.3.3.1.cmml"><mo id="S5.I2.i1.p1.1.1.m1.3.3.3.2.1" stretchy="false" xref="S5.I2.i1.p1.1.1.m1.3.3.3.1.cmml">(</mo><mi id="S5.I2.i1.p1.1.1.m1.1.1" xref="S5.I2.i1.p1.1.1.m1.1.1.cmml">u</mi><mo id="S5.I2.i1.p1.1.1.m1.3.3.3.2.2" xref="S5.I2.i1.p1.1.1.m1.3.3.3.1.cmml">,</mo><mi id="S5.I2.i1.p1.1.1.m1.2.2" xref="S5.I2.i1.p1.1.1.m1.2.2.cmml">v</mi><mo id="S5.I2.i1.p1.1.1.m1.3.3.3.2.3" stretchy="false" xref="S5.I2.i1.p1.1.1.m1.3.3.3.1.cmml">)</mo></mrow><mo id="S5.I2.i1.p1.1.1.m1.3.3.2" xref="S5.I2.i1.p1.1.1.m1.3.3.2.cmml">∈</mo><mrow id="S5.I2.i1.p1.1.1.m1.3.3.1" xref="S5.I2.i1.p1.1.1.m1.3.3.1.cmml"><mi id="S5.I2.i1.p1.1.1.m1.3.3.1.3" xref="S5.I2.i1.p1.1.1.m1.3.3.1.3.cmml">E</mi><mo id="S5.I2.i1.p1.1.1.m1.3.3.1.2" xref="S5.I2.i1.p1.1.1.m1.3.3.1.2.cmml">⁢</mo><mrow id="S5.I2.i1.p1.1.1.m1.3.3.1.1.1" xref="S5.I2.i1.p1.1.1.m1.3.3.1.1.1.1.cmml"><mo id="S5.I2.i1.p1.1.1.m1.3.3.1.1.1.2" stretchy="false" xref="S5.I2.i1.p1.1.1.m1.3.3.1.1.1.1.cmml">(</mo><msup id="S5.I2.i1.p1.1.1.m1.3.3.1.1.1.1" xref="S5.I2.i1.p1.1.1.m1.3.3.1.1.1.1.cmml"><mi id="S5.I2.i1.p1.1.1.m1.3.3.1.1.1.1.2" xref="S5.I2.i1.p1.1.1.m1.3.3.1.1.1.1.2.cmml">G</mi><mo id="S5.I2.i1.p1.1.1.m1.3.3.1.1.1.1.3" xref="S5.I2.i1.p1.1.1.m1.3.3.1.1.1.1.3.cmml">′</mo></msup><mo id="S5.I2.i1.p1.1.1.m1.3.3.1.1.1.3" stretchy="false" xref="S5.I2.i1.p1.1.1.m1.3.3.1.1.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.I2.i1.p1.1.1.m1.3b"><apply id="S5.I2.i1.p1.1.1.m1.3.3.cmml" xref="S5.I2.i1.p1.1.1.m1.3.3"><in id="S5.I2.i1.p1.1.1.m1.3.3.2.cmml" xref="S5.I2.i1.p1.1.1.m1.3.3.2"></in><interval closure="open" id="S5.I2.i1.p1.1.1.m1.3.3.3.1.cmml" xref="S5.I2.i1.p1.1.1.m1.3.3.3.2"><ci id="S5.I2.i1.p1.1.1.m1.1.1.cmml" xref="S5.I2.i1.p1.1.1.m1.1.1">𝑢</ci><ci id="S5.I2.i1.p1.1.1.m1.2.2.cmml" xref="S5.I2.i1.p1.1.1.m1.2.2">𝑣</ci></interval><apply id="S5.I2.i1.p1.1.1.m1.3.3.1.cmml" xref="S5.I2.i1.p1.1.1.m1.3.3.1"><times id="S5.I2.i1.p1.1.1.m1.3.3.1.2.cmml" xref="S5.I2.i1.p1.1.1.m1.3.3.1.2"></times><ci id="S5.I2.i1.p1.1.1.m1.3.3.1.3.cmml" xref="S5.I2.i1.p1.1.1.m1.3.3.1.3">𝐸</ci><apply id="S5.I2.i1.p1.1.1.m1.3.3.1.1.1.1.cmml" xref="S5.I2.i1.p1.1.1.m1.3.3.1.1.1"><csymbol cd="ambiguous" id="S5.I2.i1.p1.1.1.m1.3.3.1.1.1.1.1.cmml" xref="S5.I2.i1.p1.1.1.m1.3.3.1.1.1">superscript</csymbol><ci id="S5.I2.i1.p1.1.1.m1.3.3.1.1.1.1.2.cmml" xref="S5.I2.i1.p1.1.1.m1.3.3.1.1.1.1.2">𝐺</ci><ci id="S5.I2.i1.p1.1.1.m1.3.3.1.1.1.1.3.cmml" xref="S5.I2.i1.p1.1.1.m1.3.3.1.1.1.1.3">′</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.I2.i1.p1.1.1.m1.3c">(u,v)\in E(G^{\prime})</annotation><annotation encoding="application/x-llamapun" id="S5.I2.i1.p1.1.1.m1.3d">( italic_u , italic_v ) ∈ italic_E ( italic_G start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT )</annotation></semantics></math>:</span> In this case, the optimal solution for augmenting the chain <math alttext="E" class="ltx_Math" display="inline" id="S5.I2.i1.p1.2.m1.1"><semantics id="S5.I2.i1.p1.2.m1.1a"><mi id="S5.I2.i1.p1.2.m1.1.1" xref="S5.I2.i1.p1.2.m1.1.1.cmml">E</mi><annotation-xml encoding="MathML-Content" id="S5.I2.i1.p1.2.m1.1b"><ci id="S5.I2.i1.p1.2.m1.1.1.cmml" xref="S5.I2.i1.p1.2.m1.1.1">𝐸</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.I2.i1.p1.2.m1.1c">E</annotation><annotation encoding="application/x-llamapun" id="S5.I2.i1.p1.2.m1.1d">italic_E</annotation></semantics></math> is to add the single edge <math alttext="(u,v)\in L=E(G^{\prime})" class="ltx_Math" display="inline" id="S5.I2.i1.p1.3.m2.3"><semantics id="S5.I2.i1.p1.3.m2.3a"><mrow id="S5.I2.i1.p1.3.m2.3.3" 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xref="S5.I2.i1.p1.3.m2.3.3.3.2"><ci id="S5.I2.i1.p1.3.m2.1.1.cmml" xref="S5.I2.i1.p1.3.m2.1.1">𝑢</ci><ci id="S5.I2.i1.p1.3.m2.2.2.cmml" xref="S5.I2.i1.p1.3.m2.2.2">𝑣</ci></interval><ci id="S5.I2.i1.p1.3.m2.3.3.5.cmml" xref="S5.I2.i1.p1.3.m2.3.3.5">𝐿</ci></apply><apply id="S5.I2.i1.p1.3.m2.3.3c.cmml" xref="S5.I2.i1.p1.3.m2.3.3"><eq id="S5.I2.i1.p1.3.m2.3.3.6.cmml" xref="S5.I2.i1.p1.3.m2.3.3.6"></eq><share href="https://arxiv.org/html/2503.00712v1#S5.I2.i1.p1.3.m2.3.3.5.cmml" id="S5.I2.i1.p1.3.m2.3.3d.cmml" xref="S5.I2.i1.p1.3.m2.3.3"></share><apply id="S5.I2.i1.p1.3.m2.3.3.1.cmml" xref="S5.I2.i1.p1.3.m2.3.3.1"><times id="S5.I2.i1.p1.3.m2.3.3.1.2.cmml" xref="S5.I2.i1.p1.3.m2.3.3.1.2"></times><ci id="S5.I2.i1.p1.3.m2.3.3.1.3.cmml" xref="S5.I2.i1.p1.3.m2.3.3.1.3">𝐸</ci><apply id="S5.I2.i1.p1.3.m2.3.3.1.1.1.1.cmml" xref="S5.I2.i1.p1.3.m2.3.3.1.1.1"><csymbol cd="ambiguous" id="S5.I2.i1.p1.3.m2.3.3.1.1.1.1.1.cmml" xref="S5.I2.i1.p1.3.m2.3.3.1.1.1">superscript</csymbol><ci id="S5.I2.i1.p1.3.m2.3.3.1.1.1.1.2.cmml" xref="S5.I2.i1.p1.3.m2.3.3.1.1.1.1.2">𝐺</ci><ci id="S5.I2.i1.p1.3.m2.3.3.1.1.1.1.3.cmml" xref="S5.I2.i1.p1.3.m2.3.3.1.1.1.1.3">′</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.I2.i1.p1.3.m2.3c">(u,v)\in L=E(G^{\prime})</annotation><annotation encoding="application/x-llamapun" id="S5.I2.i1.p1.3.m2.3d">( italic_u , italic_v ) ∈ italic_L = italic_E ( italic_G start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT )</annotation></semantics></math>, completing a spanning cycle. Hence, <math alttext="\mathcal{A}" class="ltx_Math" display="inline" id="S5.I2.i1.p1.4.m3.1"><semantics id="S5.I2.i1.p1.4.m3.1a"><mi class="ltx_font_mathcaligraphic" id="S5.I2.i1.p1.4.m3.1.1" xref="S5.I2.i1.p1.4.m3.1.1.cmml">𝒜</mi><annotation-xml encoding="MathML-Content" id="S5.I2.i1.p1.4.m3.1b"><ci id="S5.I2.i1.p1.4.m3.1.1.cmml" xref="S5.I2.i1.p1.4.m3.1.1">𝒜</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.I2.i1.p1.4.m3.1c">\mathcal{A}</annotation><annotation encoding="application/x-llamapun" id="S5.I2.i1.p1.4.m3.1d">caligraphic_A</annotation></semantics></math> will report an approximate solution of cost at most <math alttext="2t+1" class="ltx_Math" display="inline" id="S5.I2.i1.p1.5.m4.1"><semantics id="S5.I2.i1.p1.5.m4.1a"><mrow id="S5.I2.i1.p1.5.m4.1.1" xref="S5.I2.i1.p1.5.m4.1.1.cmml"><mrow id="S5.I2.i1.p1.5.m4.1.1.2" xref="S5.I2.i1.p1.5.m4.1.1.2.cmml"><mn id="S5.I2.i1.p1.5.m4.1.1.2.2" xref="S5.I2.i1.p1.5.m4.1.1.2.2.cmml">2</mn><mo id="S5.I2.i1.p1.5.m4.1.1.2.1" xref="S5.I2.i1.p1.5.m4.1.1.2.1.cmml">⁢</mo><mi id="S5.I2.i1.p1.5.m4.1.1.2.3" xref="S5.I2.i1.p1.5.m4.1.1.2.3.cmml">t</mi></mrow><mo id="S5.I2.i1.p1.5.m4.1.1.1" xref="S5.I2.i1.p1.5.m4.1.1.1.cmml">+</mo><mn id="S5.I2.i1.p1.5.m4.1.1.3" xref="S5.I2.i1.p1.5.m4.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S5.I2.i1.p1.5.m4.1b"><apply id="S5.I2.i1.p1.5.m4.1.1.cmml" xref="S5.I2.i1.p1.5.m4.1.1"><plus id="S5.I2.i1.p1.5.m4.1.1.1.cmml" xref="S5.I2.i1.p1.5.m4.1.1.1"></plus><apply id="S5.I2.i1.p1.5.m4.1.1.2.cmml" xref="S5.I2.i1.p1.5.m4.1.1.2"><times id="S5.I2.i1.p1.5.m4.1.1.2.1.cmml" xref="S5.I2.i1.p1.5.m4.1.1.2.1"></times><cn id="S5.I2.i1.p1.5.m4.1.1.2.2.cmml" type="integer" xref="S5.I2.i1.p1.5.m4.1.1.2.2">2</cn><ci id="S5.I2.i1.p1.5.m4.1.1.2.3.cmml" xref="S5.I2.i1.p1.5.m4.1.1.2.3">𝑡</ci></apply><cn id="S5.I2.i1.p1.5.m4.1.1.3.cmml" type="integer" xref="S5.I2.i1.p1.5.m4.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.I2.i1.p1.5.m4.1c">2t+1</annotation><annotation encoding="application/x-llamapun" id="S5.I2.i1.p1.5.m4.1d">2 italic_t + 1</annotation></semantics></math> and Bob can correctly decide <math alttext="(u,v)\in E(G^{\prime})" class="ltx_Math" display="inline" id="S5.I2.i1.p1.6.m5.3"><semantics id="S5.I2.i1.p1.6.m5.3a"><mrow id="S5.I2.i1.p1.6.m5.3.3" xref="S5.I2.i1.p1.6.m5.3.3.cmml"><mrow id="S5.I2.i1.p1.6.m5.3.3.3.2" xref="S5.I2.i1.p1.6.m5.3.3.3.1.cmml"><mo id="S5.I2.i1.p1.6.m5.3.3.3.2.1" stretchy="false" xref="S5.I2.i1.p1.6.m5.3.3.3.1.cmml">(</mo><mi id="S5.I2.i1.p1.6.m5.1.1" xref="S5.I2.i1.p1.6.m5.1.1.cmml">u</mi><mo id="S5.I2.i1.p1.6.m5.3.3.3.2.2" xref="S5.I2.i1.p1.6.m5.3.3.3.1.cmml">,</mo><mi id="S5.I2.i1.p1.6.m5.2.2" xref="S5.I2.i1.p1.6.m5.2.2.cmml">v</mi><mo id="S5.I2.i1.p1.6.m5.3.3.3.2.3" stretchy="false" xref="S5.I2.i1.p1.6.m5.3.3.3.1.cmml">)</mo></mrow><mo id="S5.I2.i1.p1.6.m5.3.3.2" xref="S5.I2.i1.p1.6.m5.3.3.2.cmml">∈</mo><mrow id="S5.I2.i1.p1.6.m5.3.3.1" xref="S5.I2.i1.p1.6.m5.3.3.1.cmml"><mi id="S5.I2.i1.p1.6.m5.3.3.1.3" xref="S5.I2.i1.p1.6.m5.3.3.1.3.cmml">E</mi><mo id="S5.I2.i1.p1.6.m5.3.3.1.2" xref="S5.I2.i1.p1.6.m5.3.3.1.2.cmml">⁢</mo><mrow id="S5.I2.i1.p1.6.m5.3.3.1.1.1" xref="S5.I2.i1.p1.6.m5.3.3.1.1.1.1.cmml"><mo id="S5.I2.i1.p1.6.m5.3.3.1.1.1.2" stretchy="false" xref="S5.I2.i1.p1.6.m5.3.3.1.1.1.1.cmml">(</mo><msup id="S5.I2.i1.p1.6.m5.3.3.1.1.1.1" xref="S5.I2.i1.p1.6.m5.3.3.1.1.1.1.cmml"><mi id="S5.I2.i1.p1.6.m5.3.3.1.1.1.1.2" xref="S5.I2.i1.p1.6.m5.3.3.1.1.1.1.2.cmml">G</mi><mo id="S5.I2.i1.p1.6.m5.3.3.1.1.1.1.3" xref="S5.I2.i1.p1.6.m5.3.3.1.1.1.1.3.cmml">′</mo></msup><mo id="S5.I2.i1.p1.6.m5.3.3.1.1.1.3" stretchy="false" xref="S5.I2.i1.p1.6.m5.3.3.1.1.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.I2.i1.p1.6.m5.3b"><apply id="S5.I2.i1.p1.6.m5.3.3.cmml" xref="S5.I2.i1.p1.6.m5.3.3"><in id="S5.I2.i1.p1.6.m5.3.3.2.cmml" xref="S5.I2.i1.p1.6.m5.3.3.2"></in><interval closure="open" id="S5.I2.i1.p1.6.m5.3.3.3.1.cmml" xref="S5.I2.i1.p1.6.m5.3.3.3.2"><ci id="S5.I2.i1.p1.6.m5.1.1.cmml" xref="S5.I2.i1.p1.6.m5.1.1">𝑢</ci><ci id="S5.I2.i1.p1.6.m5.2.2.cmml" xref="S5.I2.i1.p1.6.m5.2.2">𝑣</ci></interval><apply id="S5.I2.i1.p1.6.m5.3.3.1.cmml" xref="S5.I2.i1.p1.6.m5.3.3.1"><times id="S5.I2.i1.p1.6.m5.3.3.1.2.cmml" xref="S5.I2.i1.p1.6.m5.3.3.1.2"></times><ci id="S5.I2.i1.p1.6.m5.3.3.1.3.cmml" xref="S5.I2.i1.p1.6.m5.3.3.1.3">𝐸</ci><apply id="S5.I2.i1.p1.6.m5.3.3.1.1.1.1.cmml" xref="S5.I2.i1.p1.6.m5.3.3.1.1.1"><csymbol cd="ambiguous" id="S5.I2.i1.p1.6.m5.3.3.1.1.1.1.1.cmml" xref="S5.I2.i1.p1.6.m5.3.3.1.1.1">superscript</csymbol><ci id="S5.I2.i1.p1.6.m5.3.3.1.1.1.1.2.cmml" xref="S5.I2.i1.p1.6.m5.3.3.1.1.1.1.2">𝐺</ci><ci id="S5.I2.i1.p1.6.m5.3.3.1.1.1.1.3.cmml" xref="S5.I2.i1.p1.6.m5.3.3.1.1.1.1.3">′</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.I2.i1.p1.6.m5.3c">(u,v)\in E(G^{\prime})</annotation><annotation encoding="application/x-llamapun" id="S5.I2.i1.p1.6.m5.3d">( italic_u , italic_v ) ∈ italic_E ( italic_G start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT )</annotation></semantics></math>.</p> </div> </li> <li class="ltx_item" id="S5.I2.i2" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">•</span> <div class="ltx_para" id="S5.I2.i2.p1"> <p class="ltx_p" id="S5.I2.i2.p1.5"><span class="ltx_text ltx_font_bold" id="S5.I2.i2.p1.1.1">If <math alttext="(u,v)\notin E(G^{\prime})" class="ltx_Math" display="inline" id="S5.I2.i2.p1.1.1.m1.3"><semantics id="S5.I2.i2.p1.1.1.m1.3a"><mrow id="S5.I2.i2.p1.1.1.m1.3.3" xref="S5.I2.i2.p1.1.1.m1.3.3.cmml"><mrow id="S5.I2.i2.p1.1.1.m1.3.3.3.2" xref="S5.I2.i2.p1.1.1.m1.3.3.3.1.cmml"><mo id="S5.I2.i2.p1.1.1.m1.3.3.3.2.1" stretchy="false" xref="S5.I2.i2.p1.1.1.m1.3.3.3.1.cmml">(</mo><mi id="S5.I2.i2.p1.1.1.m1.1.1" xref="S5.I2.i2.p1.1.1.m1.1.1.cmml">u</mi><mo id="S5.I2.i2.p1.1.1.m1.3.3.3.2.2" xref="S5.I2.i2.p1.1.1.m1.3.3.3.1.cmml">,</mo><mi id="S5.I2.i2.p1.1.1.m1.2.2" xref="S5.I2.i2.p1.1.1.m1.2.2.cmml">v</mi><mo id="S5.I2.i2.p1.1.1.m1.3.3.3.2.3" stretchy="false" xref="S5.I2.i2.p1.1.1.m1.3.3.3.1.cmml">)</mo></mrow><mo id="S5.I2.i2.p1.1.1.m1.3.3.2" xref="S5.I2.i2.p1.1.1.m1.3.3.2.cmml">∉</mo><mrow id="S5.I2.i2.p1.1.1.m1.3.3.1" xref="S5.I2.i2.p1.1.1.m1.3.3.1.cmml"><mi id="S5.I2.i2.p1.1.1.m1.3.3.1.3" xref="S5.I2.i2.p1.1.1.m1.3.3.1.3.cmml">E</mi><mo id="S5.I2.i2.p1.1.1.m1.3.3.1.2" xref="S5.I2.i2.p1.1.1.m1.3.3.1.2.cmml">⁢</mo><mrow id="S5.I2.i2.p1.1.1.m1.3.3.1.1.1" xref="S5.I2.i2.p1.1.1.m1.3.3.1.1.1.1.cmml"><mo id="S5.I2.i2.p1.1.1.m1.3.3.1.1.1.2" stretchy="false" xref="S5.I2.i2.p1.1.1.m1.3.3.1.1.1.1.cmml">(</mo><msup id="S5.I2.i2.p1.1.1.m1.3.3.1.1.1.1" xref="S5.I2.i2.p1.1.1.m1.3.3.1.1.1.1.cmml"><mi id="S5.I2.i2.p1.1.1.m1.3.3.1.1.1.1.2" xref="S5.I2.i2.p1.1.1.m1.3.3.1.1.1.1.2.cmml">G</mi><mo id="S5.I2.i2.p1.1.1.m1.3.3.1.1.1.1.3" xref="S5.I2.i2.p1.1.1.m1.3.3.1.1.1.1.3.cmml">′</mo></msup><mo id="S5.I2.i2.p1.1.1.m1.3.3.1.1.1.3" stretchy="false" xref="S5.I2.i2.p1.1.1.m1.3.3.1.1.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.I2.i2.p1.1.1.m1.3b"><apply id="S5.I2.i2.p1.1.1.m1.3.3.cmml" xref="S5.I2.i2.p1.1.1.m1.3.3"><notin id="S5.I2.i2.p1.1.1.m1.3.3.2.cmml" xref="S5.I2.i2.p1.1.1.m1.3.3.2"></notin><interval closure="open" id="S5.I2.i2.p1.1.1.m1.3.3.3.1.cmml" xref="S5.I2.i2.p1.1.1.m1.3.3.3.2"><ci id="S5.I2.i2.p1.1.1.m1.1.1.cmml" xref="S5.I2.i2.p1.1.1.m1.1.1">𝑢</ci><ci id="S5.I2.i2.p1.1.1.m1.2.2.cmml" xref="S5.I2.i2.p1.1.1.m1.2.2">𝑣</ci></interval><apply id="S5.I2.i2.p1.1.1.m1.3.3.1.cmml" xref="S5.I2.i2.p1.1.1.m1.3.3.1"><times id="S5.I2.i2.p1.1.1.m1.3.3.1.2.cmml" xref="S5.I2.i2.p1.1.1.m1.3.3.1.2"></times><ci id="S5.I2.i2.p1.1.1.m1.3.3.1.3.cmml" xref="S5.I2.i2.p1.1.1.m1.3.3.1.3">𝐸</ci><apply id="S5.I2.i2.p1.1.1.m1.3.3.1.1.1.1.cmml" xref="S5.I2.i2.p1.1.1.m1.3.3.1.1.1"><csymbol cd="ambiguous" id="S5.I2.i2.p1.1.1.m1.3.3.1.1.1.1.1.cmml" xref="S5.I2.i2.p1.1.1.m1.3.3.1.1.1">superscript</csymbol><ci id="S5.I2.i2.p1.1.1.m1.3.3.1.1.1.1.2.cmml" xref="S5.I2.i2.p1.1.1.m1.3.3.1.1.1.1.2">𝐺</ci><ci id="S5.I2.i2.p1.1.1.m1.3.3.1.1.1.1.3.cmml" xref="S5.I2.i2.p1.1.1.m1.3.3.1.1.1.1.3">′</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.I2.i2.p1.1.1.m1.3c">(u,v)\notin E(G^{\prime})</annotation><annotation encoding="application/x-llamapun" id="S5.I2.i2.p1.1.1.m1.3d">( italic_u , italic_v ) ∉ italic_E ( italic_G start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT )</annotation></semantics></math>:</span> To show that Bob can correctly decide <math alttext="(u,v)\notin E(G^{\prime})" class="ltx_Math" display="inline" id="S5.I2.i2.p1.2.m1.3"><semantics id="S5.I2.i2.p1.2.m1.3a"><mrow id="S5.I2.i2.p1.2.m1.3.3" xref="S5.I2.i2.p1.2.m1.3.3.cmml"><mrow id="S5.I2.i2.p1.2.m1.3.3.3.2" xref="S5.I2.i2.p1.2.m1.3.3.3.1.cmml"><mo id="S5.I2.i2.p1.2.m1.3.3.3.2.1" stretchy="false" xref="S5.I2.i2.p1.2.m1.3.3.3.1.cmml">(</mo><mi id="S5.I2.i2.p1.2.m1.1.1" xref="S5.I2.i2.p1.2.m1.1.1.cmml">u</mi><mo id="S5.I2.i2.p1.2.m1.3.3.3.2.2" xref="S5.I2.i2.p1.2.m1.3.3.3.1.cmml">,</mo><mi id="S5.I2.i2.p1.2.m1.2.2" xref="S5.I2.i2.p1.2.m1.2.2.cmml">v</mi><mo id="S5.I2.i2.p1.2.m1.3.3.3.2.3" stretchy="false" xref="S5.I2.i2.p1.2.m1.3.3.3.1.cmml">)</mo></mrow><mo id="S5.I2.i2.p1.2.m1.3.3.2" xref="S5.I2.i2.p1.2.m1.3.3.2.cmml">∉</mo><mrow id="S5.I2.i2.p1.2.m1.3.3.1" xref="S5.I2.i2.p1.2.m1.3.3.1.cmml"><mi id="S5.I2.i2.p1.2.m1.3.3.1.3" xref="S5.I2.i2.p1.2.m1.3.3.1.3.cmml">E</mi><mo id="S5.I2.i2.p1.2.m1.3.3.1.2" xref="S5.I2.i2.p1.2.m1.3.3.1.2.cmml">⁢</mo><mrow id="S5.I2.i2.p1.2.m1.3.3.1.1.1" xref="S5.I2.i2.p1.2.m1.3.3.1.1.1.1.cmml"><mo id="S5.I2.i2.p1.2.m1.3.3.1.1.1.2" stretchy="false" xref="S5.I2.i2.p1.2.m1.3.3.1.1.1.1.cmml">(</mo><msup id="S5.I2.i2.p1.2.m1.3.3.1.1.1.1" xref="S5.I2.i2.p1.2.m1.3.3.1.1.1.1.cmml"><mi id="S5.I2.i2.p1.2.m1.3.3.1.1.1.1.2" xref="S5.I2.i2.p1.2.m1.3.3.1.1.1.1.2.cmml">G</mi><mo id="S5.I2.i2.p1.2.m1.3.3.1.1.1.1.3" xref="S5.I2.i2.p1.2.m1.3.3.1.1.1.1.3.cmml">′</mo></msup><mo id="S5.I2.i2.p1.2.m1.3.3.1.1.1.3" stretchy="false" xref="S5.I2.i2.p1.2.m1.3.3.1.1.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.I2.i2.p1.2.m1.3b"><apply id="S5.I2.i2.p1.2.m1.3.3.cmml" xref="S5.I2.i2.p1.2.m1.3.3"><notin id="S5.I2.i2.p1.2.m1.3.3.2.cmml" xref="S5.I2.i2.p1.2.m1.3.3.2"></notin><interval closure="open" id="S5.I2.i2.p1.2.m1.3.3.3.1.cmml" xref="S5.I2.i2.p1.2.m1.3.3.3.2"><ci id="S5.I2.i2.p1.2.m1.1.1.cmml" xref="S5.I2.i2.p1.2.m1.1.1">𝑢</ci><ci id="S5.I2.i2.p1.2.m1.2.2.cmml" xref="S5.I2.i2.p1.2.m1.2.2">𝑣</ci></interval><apply id="S5.I2.i2.p1.2.m1.3.3.1.cmml" xref="S5.I2.i2.p1.2.m1.3.3.1"><times id="S5.I2.i2.p1.2.m1.3.3.1.2.cmml" xref="S5.I2.i2.p1.2.m1.3.3.1.2"></times><ci id="S5.I2.i2.p1.2.m1.3.3.1.3.cmml" xref="S5.I2.i2.p1.2.m1.3.3.1.3">𝐸</ci><apply id="S5.I2.i2.p1.2.m1.3.3.1.1.1.1.cmml" xref="S5.I2.i2.p1.2.m1.3.3.1.1.1"><csymbol cd="ambiguous" id="S5.I2.i2.p1.2.m1.3.3.1.1.1.1.1.cmml" xref="S5.I2.i2.p1.2.m1.3.3.1.1.1">superscript</csymbol><ci id="S5.I2.i2.p1.2.m1.3.3.1.1.1.1.2.cmml" xref="S5.I2.i2.p1.2.m1.3.3.1.1.1.1.2">𝐺</ci><ci id="S5.I2.i2.p1.2.m1.3.3.1.1.1.1.3.cmml" xref="S5.I2.i2.p1.2.m1.3.3.1.1.1.1.3">′</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.I2.i2.p1.2.m1.3c">(u,v)\notin E(G^{\prime})</annotation><annotation encoding="application/x-llamapun" id="S5.I2.i2.p1.2.m1.3d">( italic_u , italic_v ) ∉ italic_E ( italic_G start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT )</annotation></semantics></math>, it suffices to show that any feasible augmentation set <math alttext="S=\{(x_{i_{1}},x_{j_{1}}),(x_{i_{2}},x_{j_{2}}),\dots," class="ltx_math_unparsed" display="inline" id="S5.I2.i2.p1.3.m2.1"><semantics id="S5.I2.i2.p1.3.m2.1a"><mrow id="S5.I2.i2.p1.3.m2.1b"><mi id="S5.I2.i2.p1.3.m2.1.2">S</mi><mo id="S5.I2.i2.p1.3.m2.1.3">=</mo><mrow id="S5.I2.i2.p1.3.m2.1.4"><mo id="S5.I2.i2.p1.3.m2.1.4.1" stretchy="false">{</mo><mrow id="S5.I2.i2.p1.3.m2.1.4.2"><mo id="S5.I2.i2.p1.3.m2.1.4.2.1" stretchy="false">(</mo><msub id="S5.I2.i2.p1.3.m2.1.4.2.2"><mi id="S5.I2.i2.p1.3.m2.1.4.2.2.2">x</mi><msub id="S5.I2.i2.p1.3.m2.1.4.2.2.3"><mi id="S5.I2.i2.p1.3.m2.1.4.2.2.3.2">i</mi><mn id="S5.I2.i2.p1.3.m2.1.4.2.2.3.3">1</mn></msub></msub><mo id="S5.I2.i2.p1.3.m2.1.4.2.3">,</mo><msub id="S5.I2.i2.p1.3.m2.1.4.2.4"><mi id="S5.I2.i2.p1.3.m2.1.4.2.4.2">x</mi><msub id="S5.I2.i2.p1.3.m2.1.4.2.4.3"><mi id="S5.I2.i2.p1.3.m2.1.4.2.4.3.2">j</mi><mn id="S5.I2.i2.p1.3.m2.1.4.2.4.3.3">1</mn></msub></msub><mo id="S5.I2.i2.p1.3.m2.1.4.2.5" stretchy="false">)</mo></mrow><mo id="S5.I2.i2.p1.3.m2.1.4.3">,</mo><mrow id="S5.I2.i2.p1.3.m2.1.4.4"><mo id="S5.I2.i2.p1.3.m2.1.4.4.1" stretchy="false">(</mo><msub id="S5.I2.i2.p1.3.m2.1.4.4.2"><mi id="S5.I2.i2.p1.3.m2.1.4.4.2.2">x</mi><msub id="S5.I2.i2.p1.3.m2.1.4.4.2.3"><mi id="S5.I2.i2.p1.3.m2.1.4.4.2.3.2">i</mi><mn id="S5.I2.i2.p1.3.m2.1.4.4.2.3.3">2</mn></msub></msub><mo id="S5.I2.i2.p1.3.m2.1.4.4.3">,</mo><msub id="S5.I2.i2.p1.3.m2.1.4.4.4"><mi id="S5.I2.i2.p1.3.m2.1.4.4.4.2">x</mi><msub id="S5.I2.i2.p1.3.m2.1.4.4.4.3"><mi id="S5.I2.i2.p1.3.m2.1.4.4.4.3.2">j</mi><mn id="S5.I2.i2.p1.3.m2.1.4.4.4.3.3">2</mn></msub></msub><mo id="S5.I2.i2.p1.3.m2.1.4.4.5" stretchy="false">)</mo></mrow><mo id="S5.I2.i2.p1.3.m2.1.4.5">,</mo><mi id="S5.I2.i2.p1.3.m2.1.1" mathvariant="normal">…</mi><mo id="S5.I2.i2.p1.3.m2.1.4.6">,</mo></mrow></mrow><annotation encoding="application/x-tex" id="S5.I2.i2.p1.3.m2.1c">S=\{(x_{i_{1}},x_{j_{1}}),(x_{i_{2}},x_{j_{2}}),\dots,</annotation><annotation encoding="application/x-llamapun" id="S5.I2.i2.p1.3.m2.1d">italic_S = { ( italic_x start_POSTSUBSCRIPT italic_i start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT , italic_x start_POSTSUBSCRIPT italic_j start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ) , ( italic_x start_POSTSUBSCRIPT italic_i start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_POSTSUBSCRIPT , italic_x start_POSTSUBSCRIPT italic_j start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ) , … ,</annotation></semantics></math> <math alttext="(x_{i_{s}},x_{j_{s}})\}\subseteq L=E(G^{\prime})" class="ltx_math_unparsed" display="inline" id="S5.I2.i2.p1.4.m3.1"><semantics id="S5.I2.i2.p1.4.m3.1a"><mrow id="S5.I2.i2.p1.4.m3.1b"><mrow id="S5.I2.i2.p1.4.m3.1.1"><mo id="S5.I2.i2.p1.4.m3.1.1.1" stretchy="false">(</mo><msub id="S5.I2.i2.p1.4.m3.1.1.2"><mi id="S5.I2.i2.p1.4.m3.1.1.2.2">x</mi><msub id="S5.I2.i2.p1.4.m3.1.1.2.3"><mi id="S5.I2.i2.p1.4.m3.1.1.2.3.2">i</mi><mi id="S5.I2.i2.p1.4.m3.1.1.2.3.3">s</mi></msub></msub><mo id="S5.I2.i2.p1.4.m3.1.1.3">,</mo><msub id="S5.I2.i2.p1.4.m3.1.1.4"><mi id="S5.I2.i2.p1.4.m3.1.1.4.2">x</mi><msub id="S5.I2.i2.p1.4.m3.1.1.4.3"><mi id="S5.I2.i2.p1.4.m3.1.1.4.3.2">j</mi><mi id="S5.I2.i2.p1.4.m3.1.1.4.3.3">s</mi></msub></msub><mo id="S5.I2.i2.p1.4.m3.1.1.5" stretchy="false">)</mo></mrow><mo id="S5.I2.i2.p1.4.m3.1.2" stretchy="false">}</mo><mo id="S5.I2.i2.p1.4.m3.1.3">⊆</mo><mi id="S5.I2.i2.p1.4.m3.1.4">L</mi><mo id="S5.I2.i2.p1.4.m3.1.5">=</mo><mi id="S5.I2.i2.p1.4.m3.1.6">E</mi><mo id="S5.I2.i2.p1.4.m3.1.7" stretchy="false">(</mo><msup id="S5.I2.i2.p1.4.m3.1.8"><mi id="S5.I2.i2.p1.4.m3.1.8.2">G</mi><mo id="S5.I2.i2.p1.4.m3.1.8.3">′</mo></msup><mo id="S5.I2.i2.p1.4.m3.1.9" stretchy="false">)</mo></mrow><annotation encoding="application/x-tex" id="S5.I2.i2.p1.4.m3.1c">(x_{i_{s}},x_{j_{s}})\}\subseteq L=E(G^{\prime})</annotation><annotation encoding="application/x-llamapun" id="S5.I2.i2.p1.4.m3.1d">( italic_x start_POSTSUBSCRIPT italic_i start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT end_POSTSUBSCRIPT , italic_x start_POSTSUBSCRIPT italic_j start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT end_POSTSUBSCRIPT ) } ⊆ italic_L = italic_E ( italic_G start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT )</annotation></semantics></math> has size at least <math alttext="2t+1" class="ltx_Math" display="inline" id="S5.I2.i2.p1.5.m4.1"><semantics id="S5.I2.i2.p1.5.m4.1a"><mrow id="S5.I2.i2.p1.5.m4.1.1" xref="S5.I2.i2.p1.5.m4.1.1.cmml"><mrow id="S5.I2.i2.p1.5.m4.1.1.2" xref="S5.I2.i2.p1.5.m4.1.1.2.cmml"><mn id="S5.I2.i2.p1.5.m4.1.1.2.2" xref="S5.I2.i2.p1.5.m4.1.1.2.2.cmml">2</mn><mo id="S5.I2.i2.p1.5.m4.1.1.2.1" xref="S5.I2.i2.p1.5.m4.1.1.2.1.cmml">⁢</mo><mi id="S5.I2.i2.p1.5.m4.1.1.2.3" xref="S5.I2.i2.p1.5.m4.1.1.2.3.cmml">t</mi></mrow><mo id="S5.I2.i2.p1.5.m4.1.1.1" xref="S5.I2.i2.p1.5.m4.1.1.1.cmml">+</mo><mn id="S5.I2.i2.p1.5.m4.1.1.3" xref="S5.I2.i2.p1.5.m4.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S5.I2.i2.p1.5.m4.1b"><apply id="S5.I2.i2.p1.5.m4.1.1.cmml" xref="S5.I2.i2.p1.5.m4.1.1"><plus id="S5.I2.i2.p1.5.m4.1.1.1.cmml" xref="S5.I2.i2.p1.5.m4.1.1.1"></plus><apply id="S5.I2.i2.p1.5.m4.1.1.2.cmml" xref="S5.I2.i2.p1.5.m4.1.1.2"><times id="S5.I2.i2.p1.5.m4.1.1.2.1.cmml" xref="S5.I2.i2.p1.5.m4.1.1.2.1"></times><cn id="S5.I2.i2.p1.5.m4.1.1.2.2.cmml" type="integer" xref="S5.I2.i2.p1.5.m4.1.1.2.2">2</cn><ci id="S5.I2.i2.p1.5.m4.1.1.2.3.cmml" xref="S5.I2.i2.p1.5.m4.1.1.2.3">𝑡</ci></apply><cn id="S5.I2.i2.p1.5.m4.1.1.3.cmml" type="integer" xref="S5.I2.i2.p1.5.m4.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.I2.i2.p1.5.m4.1c">2t+1</annotation><annotation encoding="application/x-llamapun" id="S5.I2.i2.p1.5.m4.1d">2 italic_t + 1</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="S5.I2.i2.p2"> <p class="ltx_p" id="S5.I2.i2.p2.21">We assume <math alttext="i_{k}<j_{k}" class="ltx_Math" display="inline" id="S5.I2.i2.p2.1.m1.1"><semantics id="S5.I2.i2.p2.1.m1.1a"><mrow id="S5.I2.i2.p2.1.m1.1.1" xref="S5.I2.i2.p2.1.m1.1.1.cmml"><msub id="S5.I2.i2.p2.1.m1.1.1.2" xref="S5.I2.i2.p2.1.m1.1.1.2.cmml"><mi id="S5.I2.i2.p2.1.m1.1.1.2.2" xref="S5.I2.i2.p2.1.m1.1.1.2.2.cmml">i</mi><mi id="S5.I2.i2.p2.1.m1.1.1.2.3" xref="S5.I2.i2.p2.1.m1.1.1.2.3.cmml">k</mi></msub><mo id="S5.I2.i2.p2.1.m1.1.1.1" xref="S5.I2.i2.p2.1.m1.1.1.1.cmml"><</mo><msub id="S5.I2.i2.p2.1.m1.1.1.3" xref="S5.I2.i2.p2.1.m1.1.1.3.cmml"><mi id="S5.I2.i2.p2.1.m1.1.1.3.2" xref="S5.I2.i2.p2.1.m1.1.1.3.2.cmml">j</mi><mi id="S5.I2.i2.p2.1.m1.1.1.3.3" xref="S5.I2.i2.p2.1.m1.1.1.3.3.cmml">k</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S5.I2.i2.p2.1.m1.1b"><apply id="S5.I2.i2.p2.1.m1.1.1.cmml" xref="S5.I2.i2.p2.1.m1.1.1"><lt id="S5.I2.i2.p2.1.m1.1.1.1.cmml" xref="S5.I2.i2.p2.1.m1.1.1.1"></lt><apply id="S5.I2.i2.p2.1.m1.1.1.2.cmml" xref="S5.I2.i2.p2.1.m1.1.1.2"><csymbol cd="ambiguous" id="S5.I2.i2.p2.1.m1.1.1.2.1.cmml" xref="S5.I2.i2.p2.1.m1.1.1.2">subscript</csymbol><ci id="S5.I2.i2.p2.1.m1.1.1.2.2.cmml" xref="S5.I2.i2.p2.1.m1.1.1.2.2">𝑖</ci><ci id="S5.I2.i2.p2.1.m1.1.1.2.3.cmml" xref="S5.I2.i2.p2.1.m1.1.1.2.3">𝑘</ci></apply><apply id="S5.I2.i2.p2.1.m1.1.1.3.cmml" xref="S5.I2.i2.p2.1.m1.1.1.3"><csymbol cd="ambiguous" id="S5.I2.i2.p2.1.m1.1.1.3.1.cmml" xref="S5.I2.i2.p2.1.m1.1.1.3">subscript</csymbol><ci id="S5.I2.i2.p2.1.m1.1.1.3.2.cmml" xref="S5.I2.i2.p2.1.m1.1.1.3.2">𝑗</ci><ci id="S5.I2.i2.p2.1.m1.1.1.3.3.cmml" xref="S5.I2.i2.p2.1.m1.1.1.3.3">𝑘</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.I2.i2.p2.1.m1.1c">i_{k}<j_{k}</annotation><annotation encoding="application/x-llamapun" id="S5.I2.i2.p2.1.m1.1d">italic_i start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT < italic_j start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT</annotation></semantics></math> for all <math alttext="1\leq k\leq s" class="ltx_Math" display="inline" id="S5.I2.i2.p2.2.m2.1"><semantics id="S5.I2.i2.p2.2.m2.1a"><mrow id="S5.I2.i2.p2.2.m2.1.1" xref="S5.I2.i2.p2.2.m2.1.1.cmml"><mn id="S5.I2.i2.p2.2.m2.1.1.2" xref="S5.I2.i2.p2.2.m2.1.1.2.cmml">1</mn><mo id="S5.I2.i2.p2.2.m2.1.1.3" xref="S5.I2.i2.p2.2.m2.1.1.3.cmml">≤</mo><mi id="S5.I2.i2.p2.2.m2.1.1.4" xref="S5.I2.i2.p2.2.m2.1.1.4.cmml">k</mi><mo id="S5.I2.i2.p2.2.m2.1.1.5" xref="S5.I2.i2.p2.2.m2.1.1.5.cmml">≤</mo><mi id="S5.I2.i2.p2.2.m2.1.1.6" xref="S5.I2.i2.p2.2.m2.1.1.6.cmml">s</mi></mrow><annotation-xml encoding="MathML-Content" id="S5.I2.i2.p2.2.m2.1b"><apply id="S5.I2.i2.p2.2.m2.1.1.cmml" xref="S5.I2.i2.p2.2.m2.1.1"><and id="S5.I2.i2.p2.2.m2.1.1a.cmml" xref="S5.I2.i2.p2.2.m2.1.1"></and><apply id="S5.I2.i2.p2.2.m2.1.1b.cmml" xref="S5.I2.i2.p2.2.m2.1.1"><leq id="S5.I2.i2.p2.2.m2.1.1.3.cmml" xref="S5.I2.i2.p2.2.m2.1.1.3"></leq><cn id="S5.I2.i2.p2.2.m2.1.1.2.cmml" type="integer" xref="S5.I2.i2.p2.2.m2.1.1.2">1</cn><ci id="S5.I2.i2.p2.2.m2.1.1.4.cmml" xref="S5.I2.i2.p2.2.m2.1.1.4">𝑘</ci></apply><apply id="S5.I2.i2.p2.2.m2.1.1c.cmml" xref="S5.I2.i2.p2.2.m2.1.1"><leq id="S5.I2.i2.p2.2.m2.1.1.5.cmml" xref="S5.I2.i2.p2.2.m2.1.1.5"></leq><share href="https://arxiv.org/html/2503.00712v1#S5.I2.i2.p2.2.m2.1.1.4.cmml" id="S5.I2.i2.p2.2.m2.1.1d.cmml" xref="S5.I2.i2.p2.2.m2.1.1"></share><ci id="S5.I2.i2.p2.2.m2.1.1.6.cmml" xref="S5.I2.i2.p2.2.m2.1.1.6">𝑠</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.I2.i2.p2.2.m2.1c">1\leq k\leq s</annotation><annotation encoding="application/x-llamapun" id="S5.I2.i2.p2.2.m2.1d">1 ≤ italic_k ≤ italic_s</annotation></semantics></math>. Since <math alttext="\{(x_{1},x_{2}),(x_{2},x_{3}),\dots," class="ltx_math_unparsed" display="inline" id="S5.I2.i2.p2.3.m3.1"><semantics id="S5.I2.i2.p2.3.m3.1a"><mrow id="S5.I2.i2.p2.3.m3.1b"><mo id="S5.I2.i2.p2.3.m3.1.2" stretchy="false">{</mo><mrow id="S5.I2.i2.p2.3.m3.1.3"><mo id="S5.I2.i2.p2.3.m3.1.3.1" stretchy="false">(</mo><msub id="S5.I2.i2.p2.3.m3.1.3.2"><mi id="S5.I2.i2.p2.3.m3.1.3.2.2">x</mi><mn id="S5.I2.i2.p2.3.m3.1.3.2.3">1</mn></msub><mo id="S5.I2.i2.p2.3.m3.1.3.3">,</mo><msub id="S5.I2.i2.p2.3.m3.1.3.4"><mi id="S5.I2.i2.p2.3.m3.1.3.4.2">x</mi><mn id="S5.I2.i2.p2.3.m3.1.3.4.3">2</mn></msub><mo id="S5.I2.i2.p2.3.m3.1.3.5" stretchy="false">)</mo></mrow><mo id="S5.I2.i2.p2.3.m3.1.4">,</mo><mrow id="S5.I2.i2.p2.3.m3.1.5"><mo id="S5.I2.i2.p2.3.m3.1.5.1" stretchy="false">(</mo><msub id="S5.I2.i2.p2.3.m3.1.5.2"><mi id="S5.I2.i2.p2.3.m3.1.5.2.2">x</mi><mn id="S5.I2.i2.p2.3.m3.1.5.2.3">2</mn></msub><mo id="S5.I2.i2.p2.3.m3.1.5.3">,</mo><msub id="S5.I2.i2.p2.3.m3.1.5.4"><mi id="S5.I2.i2.p2.3.m3.1.5.4.2">x</mi><mn id="S5.I2.i2.p2.3.m3.1.5.4.3">3</mn></msub><mo id="S5.I2.i2.p2.3.m3.1.5.5" stretchy="false">)</mo></mrow><mo id="S5.I2.i2.p2.3.m3.1.6">,</mo><mi id="S5.I2.i2.p2.3.m3.1.1" mathvariant="normal">…</mi><mo id="S5.I2.i2.p2.3.m3.1.7">,</mo></mrow><annotation encoding="application/x-tex" id="S5.I2.i2.p2.3.m3.1c">\{(x_{1},x_{2}),(x_{2},x_{3}),\dots,</annotation><annotation encoding="application/x-llamapun" id="S5.I2.i2.p2.3.m3.1d">{ ( italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_x start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) , ( italic_x start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT , italic_x start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ) , … ,</annotation></semantics></math> <math alttext="(x_{|V|-1},x_{|V|})\}\cup S" class="ltx_math_unparsed" display="inline" id="S5.I2.i2.p2.4.m4.2"><semantics id="S5.I2.i2.p2.4.m4.2a"><mrow id="S5.I2.i2.p2.4.m4.2b"><mrow id="S5.I2.i2.p2.4.m4.2.3"><mo id="S5.I2.i2.p2.4.m4.2.3.1" stretchy="false">(</mo><msub id="S5.I2.i2.p2.4.m4.2.3.2"><mi id="S5.I2.i2.p2.4.m4.2.3.2.2">x</mi><mrow id="S5.I2.i2.p2.4.m4.1.1.1"><mrow id="S5.I2.i2.p2.4.m4.1.1.1.3.2"><mo id="S5.I2.i2.p2.4.m4.1.1.1.3.2.1" stretchy="false">|</mo><mi id="S5.I2.i2.p2.4.m4.1.1.1.1">V</mi><mo id="S5.I2.i2.p2.4.m4.1.1.1.3.2.2" stretchy="false">|</mo></mrow><mo id="S5.I2.i2.p2.4.m4.1.1.1.2">−</mo><mn id="S5.I2.i2.p2.4.m4.1.1.1.4">1</mn></mrow></msub><mo id="S5.I2.i2.p2.4.m4.2.3.3">,</mo><msub id="S5.I2.i2.p2.4.m4.2.3.4"><mi id="S5.I2.i2.p2.4.m4.2.3.4.2">x</mi><mrow id="S5.I2.i2.p2.4.m4.2.2.1.3"><mo id="S5.I2.i2.p2.4.m4.2.2.1.3.1" stretchy="false">|</mo><mi id="S5.I2.i2.p2.4.m4.2.2.1.1">V</mi><mo id="S5.I2.i2.p2.4.m4.2.2.1.3.2" stretchy="false">|</mo></mrow></msub><mo id="S5.I2.i2.p2.4.m4.2.3.5" stretchy="false">)</mo></mrow><mo id="S5.I2.i2.p2.4.m4.2.4" stretchy="false">}</mo><mo id="S5.I2.i2.p2.4.m4.2.5">∪</mo><mi id="S5.I2.i2.p2.4.m4.2.6">S</mi></mrow><annotation encoding="application/x-tex" 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xref="S5.I2.i2.p2.6.m6.3.3.2.2.2.2"><minus id="S5.I2.i2.p2.6.m6.3.3.2.2.2.2.1.cmml" xref="S5.I2.i2.p2.6.m6.3.3.2.2.2.2.1"></minus><apply id="S5.I2.i2.p2.6.m6.3.3.2.2.2.2.2.cmml" xref="S5.I2.i2.p2.6.m6.3.3.2.2.2.2.2"><csymbol cd="ambiguous" id="S5.I2.i2.p2.6.m6.3.3.2.2.2.2.2.1.cmml" xref="S5.I2.i2.p2.6.m6.3.3.2.2.2.2.2">subscript</csymbol><ci id="S5.I2.i2.p2.6.m6.3.3.2.2.2.2.2.2.cmml" xref="S5.I2.i2.p2.6.m6.3.3.2.2.2.2.2.2">𝑗</ci><ci id="S5.I2.i2.p2.6.m6.3.3.2.2.2.2.2.3.cmml" xref="S5.I2.i2.p2.6.m6.3.3.2.2.2.2.2.3">𝑠</ci></apply><cn id="S5.I2.i2.p2.6.m6.3.3.2.2.2.2.3.cmml" type="integer" xref="S5.I2.i2.p2.6.m6.3.3.2.2.2.2.3">1</cn></apply></interval></list></annotation-xml><annotation encoding="application/x-tex" id="S5.I2.i2.p2.6.m6.3c">[i_{1}+1,j_{1}-1],\dots,[i_{s}+1,j_{s}-1]</annotation><annotation encoding="application/x-llamapun" id="S5.I2.i2.p2.6.m6.3d">[ italic_i start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT + 1 , italic_j start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT - 1 ] , … , [ italic_i start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT + 1 , italic_j start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT - 1 ]</annotation></semantics></math> covers all <math alttext="\{2,\dots,|V|-1\}" class="ltx_Math" display="inline" id="S5.I2.i2.p2.7.m7.4"><semantics id="S5.I2.i2.p2.7.m7.4a"><mrow id="S5.I2.i2.p2.7.m7.4.4.1" xref="S5.I2.i2.p2.7.m7.4.4.2.cmml"><mo id="S5.I2.i2.p2.7.m7.4.4.1.2" stretchy="false" xref="S5.I2.i2.p2.7.m7.4.4.2.cmml">{</mo><mn id="S5.I2.i2.p2.7.m7.1.1" xref="S5.I2.i2.p2.7.m7.1.1.cmml">2</mn><mo id="S5.I2.i2.p2.7.m7.4.4.1.3" xref="S5.I2.i2.p2.7.m7.4.4.2.cmml">,</mo><mi id="S5.I2.i2.p2.7.m7.2.2" mathvariant="normal" xref="S5.I2.i2.p2.7.m7.2.2.cmml">…</mi><mo id="S5.I2.i2.p2.7.m7.4.4.1.4" xref="S5.I2.i2.p2.7.m7.4.4.2.cmml">,</mo><mrow id="S5.I2.i2.p2.7.m7.4.4.1.1" xref="S5.I2.i2.p2.7.m7.4.4.1.1.cmml"><mrow id="S5.I2.i2.p2.7.m7.4.4.1.1.2.2" xref="S5.I2.i2.p2.7.m7.4.4.1.1.2.1.cmml"><mo id="S5.I2.i2.p2.7.m7.4.4.1.1.2.2.1" stretchy="false" xref="S5.I2.i2.p2.7.m7.4.4.1.1.2.1.1.cmml">|</mo><mi id="S5.I2.i2.p2.7.m7.3.3" xref="S5.I2.i2.p2.7.m7.3.3.cmml">V</mi><mo id="S5.I2.i2.p2.7.m7.4.4.1.1.2.2.2" stretchy="false" xref="S5.I2.i2.p2.7.m7.4.4.1.1.2.1.1.cmml">|</mo></mrow><mo id="S5.I2.i2.p2.7.m7.4.4.1.1.1" xref="S5.I2.i2.p2.7.m7.4.4.1.1.1.cmml">−</mo><mn id="S5.I2.i2.p2.7.m7.4.4.1.1.3" xref="S5.I2.i2.p2.7.m7.4.4.1.1.3.cmml">1</mn></mrow><mo id="S5.I2.i2.p2.7.m7.4.4.1.5" stretchy="false" xref="S5.I2.i2.p2.7.m7.4.4.2.cmml">}</mo></mrow><annotation-xml encoding="MathML-Content" id="S5.I2.i2.p2.7.m7.4b"><set id="S5.I2.i2.p2.7.m7.4.4.2.cmml" xref="S5.I2.i2.p2.7.m7.4.4.1"><cn id="S5.I2.i2.p2.7.m7.1.1.cmml" type="integer" xref="S5.I2.i2.p2.7.m7.1.1">2</cn><ci id="S5.I2.i2.p2.7.m7.2.2.cmml" xref="S5.I2.i2.p2.7.m7.2.2">…</ci><apply id="S5.I2.i2.p2.7.m7.4.4.1.1.cmml" xref="S5.I2.i2.p2.7.m7.4.4.1.1"><minus id="S5.I2.i2.p2.7.m7.4.4.1.1.1.cmml" xref="S5.I2.i2.p2.7.m7.4.4.1.1.1"></minus><apply id="S5.I2.i2.p2.7.m7.4.4.1.1.2.1.cmml" xref="S5.I2.i2.p2.7.m7.4.4.1.1.2.2"><abs id="S5.I2.i2.p2.7.m7.4.4.1.1.2.1.1.cmml" xref="S5.I2.i2.p2.7.m7.4.4.1.1.2.2.1"></abs><ci id="S5.I2.i2.p2.7.m7.3.3.cmml" xref="S5.I2.i2.p2.7.m7.3.3">𝑉</ci></apply><cn id="S5.I2.i2.p2.7.m7.4.4.1.1.3.cmml" type="integer" xref="S5.I2.i2.p2.7.m7.4.4.1.1.3">1</cn></apply></set></annotation-xml><annotation encoding="application/x-tex" id="S5.I2.i2.p2.7.m7.4c">\{2,\dots,|V|-1\}</annotation><annotation encoding="application/x-llamapun" id="S5.I2.i2.p2.7.m7.4d">{ 2 , … , | italic_V | - 1 }</annotation></semantics></math>. This is because, once we add an edge <math alttext="(i_{k},j_{k})" class="ltx_Math" display="inline" id="S5.I2.i2.p2.8.m8.2"><semantics id="S5.I2.i2.p2.8.m8.2a"><mrow id="S5.I2.i2.p2.8.m8.2.2.2" xref="S5.I2.i2.p2.8.m8.2.2.3.cmml"><mo id="S5.I2.i2.p2.8.m8.2.2.2.3" stretchy="false" xref="S5.I2.i2.p2.8.m8.2.2.3.cmml">(</mo><msub id="S5.I2.i2.p2.8.m8.1.1.1.1" xref="S5.I2.i2.p2.8.m8.1.1.1.1.cmml"><mi id="S5.I2.i2.p2.8.m8.1.1.1.1.2" xref="S5.I2.i2.p2.8.m8.1.1.1.1.2.cmml">i</mi><mi id="S5.I2.i2.p2.8.m8.1.1.1.1.3" xref="S5.I2.i2.p2.8.m8.1.1.1.1.3.cmml">k</mi></msub><mo id="S5.I2.i2.p2.8.m8.2.2.2.4" xref="S5.I2.i2.p2.8.m8.2.2.3.cmml">,</mo><msub id="S5.I2.i2.p2.8.m8.2.2.2.2" xref="S5.I2.i2.p2.8.m8.2.2.2.2.cmml"><mi id="S5.I2.i2.p2.8.m8.2.2.2.2.2" xref="S5.I2.i2.p2.8.m8.2.2.2.2.2.cmml">j</mi><mi id="S5.I2.i2.p2.8.m8.2.2.2.2.3" xref="S5.I2.i2.p2.8.m8.2.2.2.2.3.cmml">k</mi></msub><mo id="S5.I2.i2.p2.8.m8.2.2.2.5" stretchy="false" xref="S5.I2.i2.p2.8.m8.2.2.3.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S5.I2.i2.p2.8.m8.2b"><interval closure="open" id="S5.I2.i2.p2.8.m8.2.2.3.cmml" xref="S5.I2.i2.p2.8.m8.2.2.2"><apply id="S5.I2.i2.p2.8.m8.1.1.1.1.cmml" xref="S5.I2.i2.p2.8.m8.1.1.1.1"><csymbol cd="ambiguous" id="S5.I2.i2.p2.8.m8.1.1.1.1.1.cmml" xref="S5.I2.i2.p2.8.m8.1.1.1.1">subscript</csymbol><ci id="S5.I2.i2.p2.8.m8.1.1.1.1.2.cmml" xref="S5.I2.i2.p2.8.m8.1.1.1.1.2">𝑖</ci><ci id="S5.I2.i2.p2.8.m8.1.1.1.1.3.cmml" xref="S5.I2.i2.p2.8.m8.1.1.1.1.3">𝑘</ci></apply><apply id="S5.I2.i2.p2.8.m8.2.2.2.2.cmml" xref="S5.I2.i2.p2.8.m8.2.2.2.2"><csymbol cd="ambiguous" id="S5.I2.i2.p2.8.m8.2.2.2.2.1.cmml" xref="S5.I2.i2.p2.8.m8.2.2.2.2">subscript</csymbol><ci id="S5.I2.i2.p2.8.m8.2.2.2.2.2.cmml" xref="S5.I2.i2.p2.8.m8.2.2.2.2.2">𝑗</ci><ci id="S5.I2.i2.p2.8.m8.2.2.2.2.3.cmml" xref="S5.I2.i2.p2.8.m8.2.2.2.2.3">𝑘</ci></apply></interval></annotation-xml><annotation encoding="application/x-tex" id="S5.I2.i2.p2.8.m8.2c">(i_{k},j_{k})</annotation><annotation encoding="application/x-llamapun" id="S5.I2.i2.p2.8.m8.2d">( italic_i start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT , italic_j start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT )</annotation></semantics></math>, removing any vertex in <math alttext="[i_{k}+1,j_{k}-1]" class="ltx_Math" display="inline" id="S5.I2.i2.p2.9.m9.2"><semantics id="S5.I2.i2.p2.9.m9.2a"><mrow id="S5.I2.i2.p2.9.m9.2.2.2" xref="S5.I2.i2.p2.9.m9.2.2.3.cmml"><mo id="S5.I2.i2.p2.9.m9.2.2.2.3" stretchy="false" xref="S5.I2.i2.p2.9.m9.2.2.3.cmml">[</mo><mrow id="S5.I2.i2.p2.9.m9.1.1.1.1" xref="S5.I2.i2.p2.9.m9.1.1.1.1.cmml"><msub id="S5.I2.i2.p2.9.m9.1.1.1.1.2" xref="S5.I2.i2.p2.9.m9.1.1.1.1.2.cmml"><mi id="S5.I2.i2.p2.9.m9.1.1.1.1.2.2" xref="S5.I2.i2.p2.9.m9.1.1.1.1.2.2.cmml">i</mi><mi id="S5.I2.i2.p2.9.m9.1.1.1.1.2.3" xref="S5.I2.i2.p2.9.m9.1.1.1.1.2.3.cmml">k</mi></msub><mo id="S5.I2.i2.p2.9.m9.1.1.1.1.1" xref="S5.I2.i2.p2.9.m9.1.1.1.1.1.cmml">+</mo><mn id="S5.I2.i2.p2.9.m9.1.1.1.1.3" xref="S5.I2.i2.p2.9.m9.1.1.1.1.3.cmml">1</mn></mrow><mo id="S5.I2.i2.p2.9.m9.2.2.2.4" xref="S5.I2.i2.p2.9.m9.2.2.3.cmml">,</mo><mrow id="S5.I2.i2.p2.9.m9.2.2.2.2" xref="S5.I2.i2.p2.9.m9.2.2.2.2.cmml"><msub id="S5.I2.i2.p2.9.m9.2.2.2.2.2" xref="S5.I2.i2.p2.9.m9.2.2.2.2.2.cmml"><mi id="S5.I2.i2.p2.9.m9.2.2.2.2.2.2" xref="S5.I2.i2.p2.9.m9.2.2.2.2.2.2.cmml">j</mi><mi id="S5.I2.i2.p2.9.m9.2.2.2.2.2.3" xref="S5.I2.i2.p2.9.m9.2.2.2.2.2.3.cmml">k</mi></msub><mo id="S5.I2.i2.p2.9.m9.2.2.2.2.1" xref="S5.I2.i2.p2.9.m9.2.2.2.2.1.cmml">−</mo><mn id="S5.I2.i2.p2.9.m9.2.2.2.2.3" xref="S5.I2.i2.p2.9.m9.2.2.2.2.3.cmml">1</mn></mrow><mo id="S5.I2.i2.p2.9.m9.2.2.2.5" stretchy="false" xref="S5.I2.i2.p2.9.m9.2.2.3.cmml">]</mo></mrow><annotation-xml encoding="MathML-Content" id="S5.I2.i2.p2.9.m9.2b"><interval closure="closed" id="S5.I2.i2.p2.9.m9.2.2.3.cmml" xref="S5.I2.i2.p2.9.m9.2.2.2"><apply id="S5.I2.i2.p2.9.m9.1.1.1.1.cmml" xref="S5.I2.i2.p2.9.m9.1.1.1.1"><plus id="S5.I2.i2.p2.9.m9.1.1.1.1.1.cmml" xref="S5.I2.i2.p2.9.m9.1.1.1.1.1"></plus><apply id="S5.I2.i2.p2.9.m9.1.1.1.1.2.cmml" xref="S5.I2.i2.p2.9.m9.1.1.1.1.2"><csymbol cd="ambiguous" id="S5.I2.i2.p2.9.m9.1.1.1.1.2.1.cmml" xref="S5.I2.i2.p2.9.m9.1.1.1.1.2">subscript</csymbol><ci id="S5.I2.i2.p2.9.m9.1.1.1.1.2.2.cmml" xref="S5.I2.i2.p2.9.m9.1.1.1.1.2.2">𝑖</ci><ci id="S5.I2.i2.p2.9.m9.1.1.1.1.2.3.cmml" xref="S5.I2.i2.p2.9.m9.1.1.1.1.2.3">𝑘</ci></apply><cn id="S5.I2.i2.p2.9.m9.1.1.1.1.3.cmml" type="integer" xref="S5.I2.i2.p2.9.m9.1.1.1.1.3">1</cn></apply><apply id="S5.I2.i2.p2.9.m9.2.2.2.2.cmml" xref="S5.I2.i2.p2.9.m9.2.2.2.2"><minus id="S5.I2.i2.p2.9.m9.2.2.2.2.1.cmml" xref="S5.I2.i2.p2.9.m9.2.2.2.2.1"></minus><apply id="S5.I2.i2.p2.9.m9.2.2.2.2.2.cmml" xref="S5.I2.i2.p2.9.m9.2.2.2.2.2"><csymbol cd="ambiguous" id="S5.I2.i2.p2.9.m9.2.2.2.2.2.1.cmml" xref="S5.I2.i2.p2.9.m9.2.2.2.2.2">subscript</csymbol><ci id="S5.I2.i2.p2.9.m9.2.2.2.2.2.2.cmml" xref="S5.I2.i2.p2.9.m9.2.2.2.2.2.2">𝑗</ci><ci id="S5.I2.i2.p2.9.m9.2.2.2.2.2.3.cmml" xref="S5.I2.i2.p2.9.m9.2.2.2.2.2.3">𝑘</ci></apply><cn id="S5.I2.i2.p2.9.m9.2.2.2.2.3.cmml" type="integer" xref="S5.I2.i2.p2.9.m9.2.2.2.2.3">1</cn></apply></interval></annotation-xml><annotation encoding="application/x-tex" id="S5.I2.i2.p2.9.m9.2c">[i_{k}+1,j_{k}-1]</annotation><annotation encoding="application/x-llamapun" id="S5.I2.i2.p2.9.m9.2d">[ italic_i start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT + 1 , italic_j start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT - 1 ]</annotation></semantics></math> does not disconnect the graph, as the connectivity is preserved by the edge <math alttext="(i_{k},j_{k})" class="ltx_Math" display="inline" id="S5.I2.i2.p2.10.m10.2"><semantics id="S5.I2.i2.p2.10.m10.2a"><mrow id="S5.I2.i2.p2.10.m10.2.2.2" xref="S5.I2.i2.p2.10.m10.2.2.3.cmml"><mo id="S5.I2.i2.p2.10.m10.2.2.2.3" stretchy="false" xref="S5.I2.i2.p2.10.m10.2.2.3.cmml">(</mo><msub id="S5.I2.i2.p2.10.m10.1.1.1.1" xref="S5.I2.i2.p2.10.m10.1.1.1.1.cmml"><mi id="S5.I2.i2.p2.10.m10.1.1.1.1.2" xref="S5.I2.i2.p2.10.m10.1.1.1.1.2.cmml">i</mi><mi id="S5.I2.i2.p2.10.m10.1.1.1.1.3" xref="S5.I2.i2.p2.10.m10.1.1.1.1.3.cmml">k</mi></msub><mo id="S5.I2.i2.p2.10.m10.2.2.2.4" xref="S5.I2.i2.p2.10.m10.2.2.3.cmml">,</mo><msub id="S5.I2.i2.p2.10.m10.2.2.2.2" xref="S5.I2.i2.p2.10.m10.2.2.2.2.cmml"><mi id="S5.I2.i2.p2.10.m10.2.2.2.2.2" xref="S5.I2.i2.p2.10.m10.2.2.2.2.2.cmml">j</mi><mi id="S5.I2.i2.p2.10.m10.2.2.2.2.3" xref="S5.I2.i2.p2.10.m10.2.2.2.2.3.cmml">k</mi></msub><mo id="S5.I2.i2.p2.10.m10.2.2.2.5" stretchy="false" xref="S5.I2.i2.p2.10.m10.2.2.3.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S5.I2.i2.p2.10.m10.2b"><interval closure="open" id="S5.I2.i2.p2.10.m10.2.2.3.cmml" xref="S5.I2.i2.p2.10.m10.2.2.2"><apply id="S5.I2.i2.p2.10.m10.1.1.1.1.cmml" xref="S5.I2.i2.p2.10.m10.1.1.1.1"><csymbol cd="ambiguous" id="S5.I2.i2.p2.10.m10.1.1.1.1.1.cmml" xref="S5.I2.i2.p2.10.m10.1.1.1.1">subscript</csymbol><ci id="S5.I2.i2.p2.10.m10.1.1.1.1.2.cmml" xref="S5.I2.i2.p2.10.m10.1.1.1.1.2">𝑖</ci><ci id="S5.I2.i2.p2.10.m10.1.1.1.1.3.cmml" xref="S5.I2.i2.p2.10.m10.1.1.1.1.3">𝑘</ci></apply><apply id="S5.I2.i2.p2.10.m10.2.2.2.2.cmml" xref="S5.I2.i2.p2.10.m10.2.2.2.2"><csymbol cd="ambiguous" id="S5.I2.i2.p2.10.m10.2.2.2.2.1.cmml" xref="S5.I2.i2.p2.10.m10.2.2.2.2">subscript</csymbol><ci id="S5.I2.i2.p2.10.m10.2.2.2.2.2.cmml" xref="S5.I2.i2.p2.10.m10.2.2.2.2.2">𝑗</ci><ci id="S5.I2.i2.p2.10.m10.2.2.2.2.3.cmml" xref="S5.I2.i2.p2.10.m10.2.2.2.2.3">𝑘</ci></apply></interval></annotation-xml><annotation encoding="application/x-tex" id="S5.I2.i2.p2.10.m10.2c">(i_{k},j_{k})</annotation><annotation encoding="application/x-llamapun" id="S5.I2.i2.p2.10.m10.2d">( italic_i start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT , italic_j start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT )</annotation></semantics></math>. We can assume, without loss of generality, that <math alttext="1=i_{1}<i_{2}<\dots<i_{s}" class="ltx_Math" display="inline" id="S5.I2.i2.p2.11.m11.1"><semantics id="S5.I2.i2.p2.11.m11.1a"><mrow id="S5.I2.i2.p2.11.m11.1.1" xref="S5.I2.i2.p2.11.m11.1.1.cmml"><mn id="S5.I2.i2.p2.11.m11.1.1.2" xref="S5.I2.i2.p2.11.m11.1.1.2.cmml">1</mn><mo id="S5.I2.i2.p2.11.m11.1.1.3" xref="S5.I2.i2.p2.11.m11.1.1.3.cmml">=</mo><msub id="S5.I2.i2.p2.11.m11.1.1.4" xref="S5.I2.i2.p2.11.m11.1.1.4.cmml"><mi id="S5.I2.i2.p2.11.m11.1.1.4.2" xref="S5.I2.i2.p2.11.m11.1.1.4.2.cmml">i</mi><mn id="S5.I2.i2.p2.11.m11.1.1.4.3" xref="S5.I2.i2.p2.11.m11.1.1.4.3.cmml">1</mn></msub><mo id="S5.I2.i2.p2.11.m11.1.1.5" xref="S5.I2.i2.p2.11.m11.1.1.5.cmml"><</mo><msub id="S5.I2.i2.p2.11.m11.1.1.6" xref="S5.I2.i2.p2.11.m11.1.1.6.cmml"><mi id="S5.I2.i2.p2.11.m11.1.1.6.2" xref="S5.I2.i2.p2.11.m11.1.1.6.2.cmml">i</mi><mn id="S5.I2.i2.p2.11.m11.1.1.6.3" xref="S5.I2.i2.p2.11.m11.1.1.6.3.cmml">2</mn></msub><mo id="S5.I2.i2.p2.11.m11.1.1.7" xref="S5.I2.i2.p2.11.m11.1.1.7.cmml"><</mo><mi id="S5.I2.i2.p2.11.m11.1.1.8" mathvariant="normal" xref="S5.I2.i2.p2.11.m11.1.1.8.cmml">⋯</mi><mo id="S5.I2.i2.p2.11.m11.1.1.9" xref="S5.I2.i2.p2.11.m11.1.1.9.cmml"><</mo><msub id="S5.I2.i2.p2.11.m11.1.1.10" xref="S5.I2.i2.p2.11.m11.1.1.10.cmml"><mi id="S5.I2.i2.p2.11.m11.1.1.10.2" xref="S5.I2.i2.p2.11.m11.1.1.10.2.cmml">i</mi><mi id="S5.I2.i2.p2.11.m11.1.1.10.3" xref="S5.I2.i2.p2.11.m11.1.1.10.3.cmml">s</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S5.I2.i2.p2.11.m11.1b"><apply id="S5.I2.i2.p2.11.m11.1.1.cmml" xref="S5.I2.i2.p2.11.m11.1.1"><and id="S5.I2.i2.p2.11.m11.1.1a.cmml" xref="S5.I2.i2.p2.11.m11.1.1"></and><apply id="S5.I2.i2.p2.11.m11.1.1b.cmml" xref="S5.I2.i2.p2.11.m11.1.1"><eq id="S5.I2.i2.p2.11.m11.1.1.3.cmml" xref="S5.I2.i2.p2.11.m11.1.1.3"></eq><cn id="S5.I2.i2.p2.11.m11.1.1.2.cmml" type="integer" xref="S5.I2.i2.p2.11.m11.1.1.2">1</cn><apply id="S5.I2.i2.p2.11.m11.1.1.4.cmml" xref="S5.I2.i2.p2.11.m11.1.1.4"><csymbol cd="ambiguous" id="S5.I2.i2.p2.11.m11.1.1.4.1.cmml" xref="S5.I2.i2.p2.11.m11.1.1.4">subscript</csymbol><ci id="S5.I2.i2.p2.11.m11.1.1.4.2.cmml" xref="S5.I2.i2.p2.11.m11.1.1.4.2">𝑖</ci><cn id="S5.I2.i2.p2.11.m11.1.1.4.3.cmml" type="integer" xref="S5.I2.i2.p2.11.m11.1.1.4.3">1</cn></apply></apply><apply id="S5.I2.i2.p2.11.m11.1.1c.cmml" xref="S5.I2.i2.p2.11.m11.1.1"><lt id="S5.I2.i2.p2.11.m11.1.1.5.cmml" xref="S5.I2.i2.p2.11.m11.1.1.5"></lt><share href="https://arxiv.org/html/2503.00712v1#S5.I2.i2.p2.11.m11.1.1.4.cmml" id="S5.I2.i2.p2.11.m11.1.1d.cmml" xref="S5.I2.i2.p2.11.m11.1.1"></share><apply id="S5.I2.i2.p2.11.m11.1.1.6.cmml" xref="S5.I2.i2.p2.11.m11.1.1.6"><csymbol cd="ambiguous" id="S5.I2.i2.p2.11.m11.1.1.6.1.cmml" xref="S5.I2.i2.p2.11.m11.1.1.6">subscript</csymbol><ci id="S5.I2.i2.p2.11.m11.1.1.6.2.cmml" xref="S5.I2.i2.p2.11.m11.1.1.6.2">𝑖</ci><cn id="S5.I2.i2.p2.11.m11.1.1.6.3.cmml" type="integer" xref="S5.I2.i2.p2.11.m11.1.1.6.3">2</cn></apply></apply><apply id="S5.I2.i2.p2.11.m11.1.1e.cmml" xref="S5.I2.i2.p2.11.m11.1.1"><lt id="S5.I2.i2.p2.11.m11.1.1.7.cmml" xref="S5.I2.i2.p2.11.m11.1.1.7"></lt><share href="https://arxiv.org/html/2503.00712v1#S5.I2.i2.p2.11.m11.1.1.6.cmml" id="S5.I2.i2.p2.11.m11.1.1f.cmml" xref="S5.I2.i2.p2.11.m11.1.1"></share><ci id="S5.I2.i2.p2.11.m11.1.1.8.cmml" xref="S5.I2.i2.p2.11.m11.1.1.8">⋯</ci></apply><apply id="S5.I2.i2.p2.11.m11.1.1g.cmml" xref="S5.I2.i2.p2.11.m11.1.1"><lt id="S5.I2.i2.p2.11.m11.1.1.9.cmml" xref="S5.I2.i2.p2.11.m11.1.1.9"></lt><share href="https://arxiv.org/html/2503.00712v1#S5.I2.i2.p2.11.m11.1.1.8.cmml" id="S5.I2.i2.p2.11.m11.1.1h.cmml" xref="S5.I2.i2.p2.11.m11.1.1"></share><apply id="S5.I2.i2.p2.11.m11.1.1.10.cmml" xref="S5.I2.i2.p2.11.m11.1.1.10"><csymbol cd="ambiguous" id="S5.I2.i2.p2.11.m11.1.1.10.1.cmml" xref="S5.I2.i2.p2.11.m11.1.1.10">subscript</csymbol><ci id="S5.I2.i2.p2.11.m11.1.1.10.2.cmml" xref="S5.I2.i2.p2.11.m11.1.1.10.2">𝑖</ci><ci id="S5.I2.i2.p2.11.m11.1.1.10.3.cmml" xref="S5.I2.i2.p2.11.m11.1.1.10.3">𝑠</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.I2.i2.p2.11.m11.1c">1=i_{1}<i_{2}<\dots<i_{s}</annotation><annotation encoding="application/x-llamapun" id="S5.I2.i2.p2.11.m11.1d">1 = italic_i start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT < italic_i start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT < ⋯ < italic_i start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="j_{1}<\dots<j_{s}=|V|" class="ltx_Math" display="inline" id="S5.I2.i2.p2.12.m12.1"><semantics id="S5.I2.i2.p2.12.m12.1a"><mrow id="S5.I2.i2.p2.12.m12.1.2" xref="S5.I2.i2.p2.12.m12.1.2.cmml"><msub id="S5.I2.i2.p2.12.m12.1.2.2" xref="S5.I2.i2.p2.12.m12.1.2.2.cmml"><mi id="S5.I2.i2.p2.12.m12.1.2.2.2" xref="S5.I2.i2.p2.12.m12.1.2.2.2.cmml">j</mi><mn id="S5.I2.i2.p2.12.m12.1.2.2.3" xref="S5.I2.i2.p2.12.m12.1.2.2.3.cmml">1</mn></msub><mo id="S5.I2.i2.p2.12.m12.1.2.3" xref="S5.I2.i2.p2.12.m12.1.2.3.cmml"><</mo><mi id="S5.I2.i2.p2.12.m12.1.2.4" mathvariant="normal" xref="S5.I2.i2.p2.12.m12.1.2.4.cmml">⋯</mi><mo id="S5.I2.i2.p2.12.m12.1.2.5" xref="S5.I2.i2.p2.12.m12.1.2.5.cmml"><</mo><msub id="S5.I2.i2.p2.12.m12.1.2.6" xref="S5.I2.i2.p2.12.m12.1.2.6.cmml"><mi id="S5.I2.i2.p2.12.m12.1.2.6.2" xref="S5.I2.i2.p2.12.m12.1.2.6.2.cmml">j</mi><mi id="S5.I2.i2.p2.12.m12.1.2.6.3" xref="S5.I2.i2.p2.12.m12.1.2.6.3.cmml">s</mi></msub><mo id="S5.I2.i2.p2.12.m12.1.2.7" xref="S5.I2.i2.p2.12.m12.1.2.7.cmml">=</mo><mrow id="S5.I2.i2.p2.12.m12.1.2.8.2" xref="S5.I2.i2.p2.12.m12.1.2.8.1.cmml"><mo id="S5.I2.i2.p2.12.m12.1.2.8.2.1" stretchy="false" xref="S5.I2.i2.p2.12.m12.1.2.8.1.1.cmml">|</mo><mi id="S5.I2.i2.p2.12.m12.1.1" xref="S5.I2.i2.p2.12.m12.1.1.cmml">V</mi><mo id="S5.I2.i2.p2.12.m12.1.2.8.2.2" stretchy="false" xref="S5.I2.i2.p2.12.m12.1.2.8.1.1.cmml">|</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.I2.i2.p2.12.m12.1b"><apply id="S5.I2.i2.p2.12.m12.1.2.cmml" xref="S5.I2.i2.p2.12.m12.1.2"><and id="S5.I2.i2.p2.12.m12.1.2a.cmml" xref="S5.I2.i2.p2.12.m12.1.2"></and><apply id="S5.I2.i2.p2.12.m12.1.2b.cmml" xref="S5.I2.i2.p2.12.m12.1.2"><lt id="S5.I2.i2.p2.12.m12.1.2.3.cmml" xref="S5.I2.i2.p2.12.m12.1.2.3"></lt><apply id="S5.I2.i2.p2.12.m12.1.2.2.cmml" xref="S5.I2.i2.p2.12.m12.1.2.2"><csymbol cd="ambiguous" id="S5.I2.i2.p2.12.m12.1.2.2.1.cmml" xref="S5.I2.i2.p2.12.m12.1.2.2">subscript</csymbol><ci id="S5.I2.i2.p2.12.m12.1.2.2.2.cmml" xref="S5.I2.i2.p2.12.m12.1.2.2.2">𝑗</ci><cn id="S5.I2.i2.p2.12.m12.1.2.2.3.cmml" type="integer" xref="S5.I2.i2.p2.12.m12.1.2.2.3">1</cn></apply><ci id="S5.I2.i2.p2.12.m12.1.2.4.cmml" xref="S5.I2.i2.p2.12.m12.1.2.4">⋯</ci></apply><apply id="S5.I2.i2.p2.12.m12.1.2c.cmml" xref="S5.I2.i2.p2.12.m12.1.2"><lt id="S5.I2.i2.p2.12.m12.1.2.5.cmml" xref="S5.I2.i2.p2.12.m12.1.2.5"></lt><share href="https://arxiv.org/html/2503.00712v1#S5.I2.i2.p2.12.m12.1.2.4.cmml" id="S5.I2.i2.p2.12.m12.1.2d.cmml" xref="S5.I2.i2.p2.12.m12.1.2"></share><apply id="S5.I2.i2.p2.12.m12.1.2.6.cmml" xref="S5.I2.i2.p2.12.m12.1.2.6"><csymbol cd="ambiguous" id="S5.I2.i2.p2.12.m12.1.2.6.1.cmml" xref="S5.I2.i2.p2.12.m12.1.2.6">subscript</csymbol><ci id="S5.I2.i2.p2.12.m12.1.2.6.2.cmml" xref="S5.I2.i2.p2.12.m12.1.2.6.2">𝑗</ci><ci id="S5.I2.i2.p2.12.m12.1.2.6.3.cmml" xref="S5.I2.i2.p2.12.m12.1.2.6.3">𝑠</ci></apply></apply><apply id="S5.I2.i2.p2.12.m12.1.2e.cmml" xref="S5.I2.i2.p2.12.m12.1.2"><eq id="S5.I2.i2.p2.12.m12.1.2.7.cmml" xref="S5.I2.i2.p2.12.m12.1.2.7"></eq><share href="https://arxiv.org/html/2503.00712v1#S5.I2.i2.p2.12.m12.1.2.6.cmml" id="S5.I2.i2.p2.12.m12.1.2f.cmml" xref="S5.I2.i2.p2.12.m12.1.2"></share><apply id="S5.I2.i2.p2.12.m12.1.2.8.1.cmml" xref="S5.I2.i2.p2.12.m12.1.2.8.2"><abs id="S5.I2.i2.p2.12.m12.1.2.8.1.1.cmml" xref="S5.I2.i2.p2.12.m12.1.2.8.2.1"></abs><ci id="S5.I2.i2.p2.12.m12.1.1.cmml" xref="S5.I2.i2.p2.12.m12.1.1">𝑉</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.I2.i2.p2.12.m12.1c">j_{1}<\dots<j_{s}=|V|</annotation><annotation encoding="application/x-llamapun" id="S5.I2.i2.p2.12.m12.1d">italic_j start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT < ⋯ < italic_j start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT = | italic_V |</annotation></semantics></math>, with <math alttext="j_{k-1}>i_{k}" class="ltx_Math" display="inline" id="S5.I2.i2.p2.13.m13.1"><semantics id="S5.I2.i2.p2.13.m13.1a"><mrow id="S5.I2.i2.p2.13.m13.1.1" xref="S5.I2.i2.p2.13.m13.1.1.cmml"><msub id="S5.I2.i2.p2.13.m13.1.1.2" xref="S5.I2.i2.p2.13.m13.1.1.2.cmml"><mi id="S5.I2.i2.p2.13.m13.1.1.2.2" xref="S5.I2.i2.p2.13.m13.1.1.2.2.cmml">j</mi><mrow id="S5.I2.i2.p2.13.m13.1.1.2.3" xref="S5.I2.i2.p2.13.m13.1.1.2.3.cmml"><mi id="S5.I2.i2.p2.13.m13.1.1.2.3.2" xref="S5.I2.i2.p2.13.m13.1.1.2.3.2.cmml">k</mi><mo id="S5.I2.i2.p2.13.m13.1.1.2.3.1" xref="S5.I2.i2.p2.13.m13.1.1.2.3.1.cmml">−</mo><mn id="S5.I2.i2.p2.13.m13.1.1.2.3.3" xref="S5.I2.i2.p2.13.m13.1.1.2.3.3.cmml">1</mn></mrow></msub><mo id="S5.I2.i2.p2.13.m13.1.1.1" xref="S5.I2.i2.p2.13.m13.1.1.1.cmml">></mo><msub id="S5.I2.i2.p2.13.m13.1.1.3" xref="S5.I2.i2.p2.13.m13.1.1.3.cmml"><mi id="S5.I2.i2.p2.13.m13.1.1.3.2" xref="S5.I2.i2.p2.13.m13.1.1.3.2.cmml">i</mi><mi id="S5.I2.i2.p2.13.m13.1.1.3.3" xref="S5.I2.i2.p2.13.m13.1.1.3.3.cmml">k</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S5.I2.i2.p2.13.m13.1b"><apply id="S5.I2.i2.p2.13.m13.1.1.cmml" xref="S5.I2.i2.p2.13.m13.1.1"><gt id="S5.I2.i2.p2.13.m13.1.1.1.cmml" xref="S5.I2.i2.p2.13.m13.1.1.1"></gt><apply id="S5.I2.i2.p2.13.m13.1.1.2.cmml" xref="S5.I2.i2.p2.13.m13.1.1.2"><csymbol cd="ambiguous" id="S5.I2.i2.p2.13.m13.1.1.2.1.cmml" xref="S5.I2.i2.p2.13.m13.1.1.2">subscript</csymbol><ci id="S5.I2.i2.p2.13.m13.1.1.2.2.cmml" xref="S5.I2.i2.p2.13.m13.1.1.2.2">𝑗</ci><apply id="S5.I2.i2.p2.13.m13.1.1.2.3.cmml" xref="S5.I2.i2.p2.13.m13.1.1.2.3"><minus id="S5.I2.i2.p2.13.m13.1.1.2.3.1.cmml" xref="S5.I2.i2.p2.13.m13.1.1.2.3.1"></minus><ci id="S5.I2.i2.p2.13.m13.1.1.2.3.2.cmml" xref="S5.I2.i2.p2.13.m13.1.1.2.3.2">𝑘</ci><cn id="S5.I2.i2.p2.13.m13.1.1.2.3.3.cmml" type="integer" xref="S5.I2.i2.p2.13.m13.1.1.2.3.3">1</cn></apply></apply><apply id="S5.I2.i2.p2.13.m13.1.1.3.cmml" xref="S5.I2.i2.p2.13.m13.1.1.3"><csymbol cd="ambiguous" id="S5.I2.i2.p2.13.m13.1.1.3.1.cmml" xref="S5.I2.i2.p2.13.m13.1.1.3">subscript</csymbol><ci id="S5.I2.i2.p2.13.m13.1.1.3.2.cmml" xref="S5.I2.i2.p2.13.m13.1.1.3.2">𝑖</ci><ci id="S5.I2.i2.p2.13.m13.1.1.3.3.cmml" xref="S5.I2.i2.p2.13.m13.1.1.3.3">𝑘</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.I2.i2.p2.13.m13.1c">j_{k-1}>i_{k}</annotation><annotation encoding="application/x-llamapun" id="S5.I2.i2.p2.13.m13.1d">italic_j start_POSTSUBSCRIPT italic_k - 1 end_POSTSUBSCRIPT > italic_i start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT</annotation></semantics></math>, by keeping a minimal feasible subset of <math alttext="S" class="ltx_Math" display="inline" id="S5.I2.i2.p2.14.m14.1"><semantics id="S5.I2.i2.p2.14.m14.1a"><mi id="S5.I2.i2.p2.14.m14.1.1" xref="S5.I2.i2.p2.14.m14.1.1.cmml">S</mi><annotation-xml encoding="MathML-Content" id="S5.I2.i2.p2.14.m14.1b"><ci id="S5.I2.i2.p2.14.m14.1.1.cmml" xref="S5.I2.i2.p2.14.m14.1.1">𝑆</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.I2.i2.p2.14.m14.1c">S</annotation><annotation encoding="application/x-llamapun" id="S5.I2.i2.p2.14.m14.1d">italic_S</annotation></semantics></math>. This is because if <math alttext="i_{k-1}<i_{k}" class="ltx_Math" display="inline" id="S5.I2.i2.p2.15.m15.1"><semantics id="S5.I2.i2.p2.15.m15.1a"><mrow id="S5.I2.i2.p2.15.m15.1.1" xref="S5.I2.i2.p2.15.m15.1.1.cmml"><msub id="S5.I2.i2.p2.15.m15.1.1.2" xref="S5.I2.i2.p2.15.m15.1.1.2.cmml"><mi id="S5.I2.i2.p2.15.m15.1.1.2.2" xref="S5.I2.i2.p2.15.m15.1.1.2.2.cmml">i</mi><mrow id="S5.I2.i2.p2.15.m15.1.1.2.3" xref="S5.I2.i2.p2.15.m15.1.1.2.3.cmml"><mi id="S5.I2.i2.p2.15.m15.1.1.2.3.2" xref="S5.I2.i2.p2.15.m15.1.1.2.3.2.cmml">k</mi><mo id="S5.I2.i2.p2.15.m15.1.1.2.3.1" xref="S5.I2.i2.p2.15.m15.1.1.2.3.1.cmml">−</mo><mn id="S5.I2.i2.p2.15.m15.1.1.2.3.3" xref="S5.I2.i2.p2.15.m15.1.1.2.3.3.cmml">1</mn></mrow></msub><mo id="S5.I2.i2.p2.15.m15.1.1.1" xref="S5.I2.i2.p2.15.m15.1.1.1.cmml"><</mo><msub id="S5.I2.i2.p2.15.m15.1.1.3" xref="S5.I2.i2.p2.15.m15.1.1.3.cmml"><mi id="S5.I2.i2.p2.15.m15.1.1.3.2" xref="S5.I2.i2.p2.15.m15.1.1.3.2.cmml">i</mi><mi id="S5.I2.i2.p2.15.m15.1.1.3.3" xref="S5.I2.i2.p2.15.m15.1.1.3.3.cmml">k</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S5.I2.i2.p2.15.m15.1b"><apply id="S5.I2.i2.p2.15.m15.1.1.cmml" xref="S5.I2.i2.p2.15.m15.1.1"><lt id="S5.I2.i2.p2.15.m15.1.1.1.cmml" xref="S5.I2.i2.p2.15.m15.1.1.1"></lt><apply id="S5.I2.i2.p2.15.m15.1.1.2.cmml" xref="S5.I2.i2.p2.15.m15.1.1.2"><csymbol cd="ambiguous" id="S5.I2.i2.p2.15.m15.1.1.2.1.cmml" xref="S5.I2.i2.p2.15.m15.1.1.2">subscript</csymbol><ci id="S5.I2.i2.p2.15.m15.1.1.2.2.cmml" xref="S5.I2.i2.p2.15.m15.1.1.2.2">𝑖</ci><apply id="S5.I2.i2.p2.15.m15.1.1.2.3.cmml" xref="S5.I2.i2.p2.15.m15.1.1.2.3"><minus id="S5.I2.i2.p2.15.m15.1.1.2.3.1.cmml" xref="S5.I2.i2.p2.15.m15.1.1.2.3.1"></minus><ci id="S5.I2.i2.p2.15.m15.1.1.2.3.2.cmml" xref="S5.I2.i2.p2.15.m15.1.1.2.3.2">𝑘</ci><cn id="S5.I2.i2.p2.15.m15.1.1.2.3.3.cmml" type="integer" xref="S5.I2.i2.p2.15.m15.1.1.2.3.3">1</cn></apply></apply><apply id="S5.I2.i2.p2.15.m15.1.1.3.cmml" xref="S5.I2.i2.p2.15.m15.1.1.3"><csymbol cd="ambiguous" id="S5.I2.i2.p2.15.m15.1.1.3.1.cmml" xref="S5.I2.i2.p2.15.m15.1.1.3">subscript</csymbol><ci id="S5.I2.i2.p2.15.m15.1.1.3.2.cmml" xref="S5.I2.i2.p2.15.m15.1.1.3.2">𝑖</ci><ci id="S5.I2.i2.p2.15.m15.1.1.3.3.cmml" xref="S5.I2.i2.p2.15.m15.1.1.3.3">𝑘</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.I2.i2.p2.15.m15.1c">i_{k-1}<i_{k}</annotation><annotation encoding="application/x-llamapun" id="S5.I2.i2.p2.15.m15.1d">italic_i start_POSTSUBSCRIPT italic_k - 1 end_POSTSUBSCRIPT < italic_i start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="j_{k-1}>j_{k}" class="ltx_Math" display="inline" id="S5.I2.i2.p2.16.m16.1"><semantics id="S5.I2.i2.p2.16.m16.1a"><mrow id="S5.I2.i2.p2.16.m16.1.1" xref="S5.I2.i2.p2.16.m16.1.1.cmml"><msub id="S5.I2.i2.p2.16.m16.1.1.2" xref="S5.I2.i2.p2.16.m16.1.1.2.cmml"><mi id="S5.I2.i2.p2.16.m16.1.1.2.2" xref="S5.I2.i2.p2.16.m16.1.1.2.2.cmml">j</mi><mrow id="S5.I2.i2.p2.16.m16.1.1.2.3" xref="S5.I2.i2.p2.16.m16.1.1.2.3.cmml"><mi id="S5.I2.i2.p2.16.m16.1.1.2.3.2" xref="S5.I2.i2.p2.16.m16.1.1.2.3.2.cmml">k</mi><mo id="S5.I2.i2.p2.16.m16.1.1.2.3.1" xref="S5.I2.i2.p2.16.m16.1.1.2.3.1.cmml">−</mo><mn id="S5.I2.i2.p2.16.m16.1.1.2.3.3" xref="S5.I2.i2.p2.16.m16.1.1.2.3.3.cmml">1</mn></mrow></msub><mo id="S5.I2.i2.p2.16.m16.1.1.1" xref="S5.I2.i2.p2.16.m16.1.1.1.cmml">></mo><msub id="S5.I2.i2.p2.16.m16.1.1.3" xref="S5.I2.i2.p2.16.m16.1.1.3.cmml"><mi id="S5.I2.i2.p2.16.m16.1.1.3.2" xref="S5.I2.i2.p2.16.m16.1.1.3.2.cmml">j</mi><mi id="S5.I2.i2.p2.16.m16.1.1.3.3" xref="S5.I2.i2.p2.16.m16.1.1.3.3.cmml">k</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S5.I2.i2.p2.16.m16.1b"><apply id="S5.I2.i2.p2.16.m16.1.1.cmml" xref="S5.I2.i2.p2.16.m16.1.1"><gt id="S5.I2.i2.p2.16.m16.1.1.1.cmml" xref="S5.I2.i2.p2.16.m16.1.1.1"></gt><apply id="S5.I2.i2.p2.16.m16.1.1.2.cmml" xref="S5.I2.i2.p2.16.m16.1.1.2"><csymbol cd="ambiguous" id="S5.I2.i2.p2.16.m16.1.1.2.1.cmml" xref="S5.I2.i2.p2.16.m16.1.1.2">subscript</csymbol><ci id="S5.I2.i2.p2.16.m16.1.1.2.2.cmml" xref="S5.I2.i2.p2.16.m16.1.1.2.2">𝑗</ci><apply id="S5.I2.i2.p2.16.m16.1.1.2.3.cmml" xref="S5.I2.i2.p2.16.m16.1.1.2.3"><minus id="S5.I2.i2.p2.16.m16.1.1.2.3.1.cmml" xref="S5.I2.i2.p2.16.m16.1.1.2.3.1"></minus><ci id="S5.I2.i2.p2.16.m16.1.1.2.3.2.cmml" xref="S5.I2.i2.p2.16.m16.1.1.2.3.2">𝑘</ci><cn id="S5.I2.i2.p2.16.m16.1.1.2.3.3.cmml" type="integer" xref="S5.I2.i2.p2.16.m16.1.1.2.3.3">1</cn></apply></apply><apply id="S5.I2.i2.p2.16.m16.1.1.3.cmml" xref="S5.I2.i2.p2.16.m16.1.1.3"><csymbol cd="ambiguous" id="S5.I2.i2.p2.16.m16.1.1.3.1.cmml" xref="S5.I2.i2.p2.16.m16.1.1.3">subscript</csymbol><ci id="S5.I2.i2.p2.16.m16.1.1.3.2.cmml" xref="S5.I2.i2.p2.16.m16.1.1.3.2">𝑗</ci><ci id="S5.I2.i2.p2.16.m16.1.1.3.3.cmml" xref="S5.I2.i2.p2.16.m16.1.1.3.3">𝑘</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.I2.i2.p2.16.m16.1c">j_{k-1}>j_{k}</annotation><annotation encoding="application/x-llamapun" id="S5.I2.i2.p2.16.m16.1d">italic_j start_POSTSUBSCRIPT italic_k - 1 end_POSTSUBSCRIPT > italic_j start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT</annotation></semantics></math>, then removing <math alttext="(i_{k},j_{k})" class="ltx_Math" display="inline" id="S5.I2.i2.p2.17.m17.2"><semantics id="S5.I2.i2.p2.17.m17.2a"><mrow id="S5.I2.i2.p2.17.m17.2.2.2" xref="S5.I2.i2.p2.17.m17.2.2.3.cmml"><mo id="S5.I2.i2.p2.17.m17.2.2.2.3" stretchy="false" xref="S5.I2.i2.p2.17.m17.2.2.3.cmml">(</mo><msub id="S5.I2.i2.p2.17.m17.1.1.1.1" xref="S5.I2.i2.p2.17.m17.1.1.1.1.cmml"><mi id="S5.I2.i2.p2.17.m17.1.1.1.1.2" xref="S5.I2.i2.p2.17.m17.1.1.1.1.2.cmml">i</mi><mi id="S5.I2.i2.p2.17.m17.1.1.1.1.3" xref="S5.I2.i2.p2.17.m17.1.1.1.1.3.cmml">k</mi></msub><mo id="S5.I2.i2.p2.17.m17.2.2.2.4" xref="S5.I2.i2.p2.17.m17.2.2.3.cmml">,</mo><msub id="S5.I2.i2.p2.17.m17.2.2.2.2" xref="S5.I2.i2.p2.17.m17.2.2.2.2.cmml"><mi id="S5.I2.i2.p2.17.m17.2.2.2.2.2" xref="S5.I2.i2.p2.17.m17.2.2.2.2.2.cmml">j</mi><mi id="S5.I2.i2.p2.17.m17.2.2.2.2.3" xref="S5.I2.i2.p2.17.m17.2.2.2.2.3.cmml">k</mi></msub><mo id="S5.I2.i2.p2.17.m17.2.2.2.5" stretchy="false" xref="S5.I2.i2.p2.17.m17.2.2.3.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S5.I2.i2.p2.17.m17.2b"><interval closure="open" id="S5.I2.i2.p2.17.m17.2.2.3.cmml" xref="S5.I2.i2.p2.17.m17.2.2.2"><apply id="S5.I2.i2.p2.17.m17.1.1.1.1.cmml" xref="S5.I2.i2.p2.17.m17.1.1.1.1"><csymbol cd="ambiguous" id="S5.I2.i2.p2.17.m17.1.1.1.1.1.cmml" xref="S5.I2.i2.p2.17.m17.1.1.1.1">subscript</csymbol><ci id="S5.I2.i2.p2.17.m17.1.1.1.1.2.cmml" xref="S5.I2.i2.p2.17.m17.1.1.1.1.2">𝑖</ci><ci id="S5.I2.i2.p2.17.m17.1.1.1.1.3.cmml" xref="S5.I2.i2.p2.17.m17.1.1.1.1.3">𝑘</ci></apply><apply id="S5.I2.i2.p2.17.m17.2.2.2.2.cmml" xref="S5.I2.i2.p2.17.m17.2.2.2.2"><csymbol cd="ambiguous" id="S5.I2.i2.p2.17.m17.2.2.2.2.1.cmml" xref="S5.I2.i2.p2.17.m17.2.2.2.2">subscript</csymbol><ci id="S5.I2.i2.p2.17.m17.2.2.2.2.2.cmml" xref="S5.I2.i2.p2.17.m17.2.2.2.2.2">𝑗</ci><ci id="S5.I2.i2.p2.17.m17.2.2.2.2.3.cmml" xref="S5.I2.i2.p2.17.m17.2.2.2.2.3">𝑘</ci></apply></interval></annotation-xml><annotation encoding="application/x-tex" id="S5.I2.i2.p2.17.m17.2c">(i_{k},j_{k})</annotation><annotation encoding="application/x-llamapun" id="S5.I2.i2.p2.17.m17.2d">( italic_i start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT , italic_j start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT )</annotation></semantics></math> still leaves all cuts covered by <math alttext="(i_{k-1},j_{k-1})" class="ltx_Math" display="inline" id="S5.I2.i2.p2.18.m18.2"><semantics id="S5.I2.i2.p2.18.m18.2a"><mrow id="S5.I2.i2.p2.18.m18.2.2.2" xref="S5.I2.i2.p2.18.m18.2.2.3.cmml"><mo id="S5.I2.i2.p2.18.m18.2.2.2.3" stretchy="false" xref="S5.I2.i2.p2.18.m18.2.2.3.cmml">(</mo><msub id="S5.I2.i2.p2.18.m18.1.1.1.1" xref="S5.I2.i2.p2.18.m18.1.1.1.1.cmml"><mi id="S5.I2.i2.p2.18.m18.1.1.1.1.2" xref="S5.I2.i2.p2.18.m18.1.1.1.1.2.cmml">i</mi><mrow id="S5.I2.i2.p2.18.m18.1.1.1.1.3" xref="S5.I2.i2.p2.18.m18.1.1.1.1.3.cmml"><mi id="S5.I2.i2.p2.18.m18.1.1.1.1.3.2" xref="S5.I2.i2.p2.18.m18.1.1.1.1.3.2.cmml">k</mi><mo id="S5.I2.i2.p2.18.m18.1.1.1.1.3.1" xref="S5.I2.i2.p2.18.m18.1.1.1.1.3.1.cmml">−</mo><mn id="S5.I2.i2.p2.18.m18.1.1.1.1.3.3" xref="S5.I2.i2.p2.18.m18.1.1.1.1.3.3.cmml">1</mn></mrow></msub><mo id="S5.I2.i2.p2.18.m18.2.2.2.4" xref="S5.I2.i2.p2.18.m18.2.2.3.cmml">,</mo><msub id="S5.I2.i2.p2.18.m18.2.2.2.2" xref="S5.I2.i2.p2.18.m18.2.2.2.2.cmml"><mi id="S5.I2.i2.p2.18.m18.2.2.2.2.2" xref="S5.I2.i2.p2.18.m18.2.2.2.2.2.cmml">j</mi><mrow id="S5.I2.i2.p2.18.m18.2.2.2.2.3" xref="S5.I2.i2.p2.18.m18.2.2.2.2.3.cmml"><mi id="S5.I2.i2.p2.18.m18.2.2.2.2.3.2" xref="S5.I2.i2.p2.18.m18.2.2.2.2.3.2.cmml">k</mi><mo id="S5.I2.i2.p2.18.m18.2.2.2.2.3.1" xref="S5.I2.i2.p2.18.m18.2.2.2.2.3.1.cmml">−</mo><mn id="S5.I2.i2.p2.18.m18.2.2.2.2.3.3" xref="S5.I2.i2.p2.18.m18.2.2.2.2.3.3.cmml">1</mn></mrow></msub><mo id="S5.I2.i2.p2.18.m18.2.2.2.5" stretchy="false" xref="S5.I2.i2.p2.18.m18.2.2.3.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S5.I2.i2.p2.18.m18.2b"><interval closure="open" id="S5.I2.i2.p2.18.m18.2.2.3.cmml" xref="S5.I2.i2.p2.18.m18.2.2.2"><apply id="S5.I2.i2.p2.18.m18.1.1.1.1.cmml" xref="S5.I2.i2.p2.18.m18.1.1.1.1"><csymbol cd="ambiguous" id="S5.I2.i2.p2.18.m18.1.1.1.1.1.cmml" xref="S5.I2.i2.p2.18.m18.1.1.1.1">subscript</csymbol><ci id="S5.I2.i2.p2.18.m18.1.1.1.1.2.cmml" xref="S5.I2.i2.p2.18.m18.1.1.1.1.2">𝑖</ci><apply id="S5.I2.i2.p2.18.m18.1.1.1.1.3.cmml" xref="S5.I2.i2.p2.18.m18.1.1.1.1.3"><minus id="S5.I2.i2.p2.18.m18.1.1.1.1.3.1.cmml" xref="S5.I2.i2.p2.18.m18.1.1.1.1.3.1"></minus><ci id="S5.I2.i2.p2.18.m18.1.1.1.1.3.2.cmml" xref="S5.I2.i2.p2.18.m18.1.1.1.1.3.2">𝑘</ci><cn id="S5.I2.i2.p2.18.m18.1.1.1.1.3.3.cmml" type="integer" xref="S5.I2.i2.p2.18.m18.1.1.1.1.3.3">1</cn></apply></apply><apply id="S5.I2.i2.p2.18.m18.2.2.2.2.cmml" xref="S5.I2.i2.p2.18.m18.2.2.2.2"><csymbol cd="ambiguous" id="S5.I2.i2.p2.18.m18.2.2.2.2.1.cmml" xref="S5.I2.i2.p2.18.m18.2.2.2.2">subscript</csymbol><ci id="S5.I2.i2.p2.18.m18.2.2.2.2.2.cmml" xref="S5.I2.i2.p2.18.m18.2.2.2.2.2">𝑗</ci><apply id="S5.I2.i2.p2.18.m18.2.2.2.2.3.cmml" xref="S5.I2.i2.p2.18.m18.2.2.2.2.3"><minus id="S5.I2.i2.p2.18.m18.2.2.2.2.3.1.cmml" xref="S5.I2.i2.p2.18.m18.2.2.2.2.3.1"></minus><ci id="S5.I2.i2.p2.18.m18.2.2.2.2.3.2.cmml" xref="S5.I2.i2.p2.18.m18.2.2.2.2.3.2">𝑘</ci><cn id="S5.I2.i2.p2.18.m18.2.2.2.2.3.3.cmml" type="integer" xref="S5.I2.i2.p2.18.m18.2.2.2.2.3.3">1</cn></apply></apply></interval></annotation-xml><annotation encoding="application/x-tex" id="S5.I2.i2.p2.18.m18.2c">(i_{k-1},j_{k-1})</annotation><annotation encoding="application/x-llamapun" id="S5.I2.i2.p2.18.m18.2d">( italic_i start_POSTSUBSCRIPT italic_k - 1 end_POSTSUBSCRIPT , italic_j start_POSTSUBSCRIPT italic_k - 1 end_POSTSUBSCRIPT )</annotation></semantics></math>. Moreover, if <math alttext="j_{k-1}\leq i_{k}" class="ltx_Math" display="inline" id="S5.I2.i2.p2.19.m19.1"><semantics id="S5.I2.i2.p2.19.m19.1a"><mrow id="S5.I2.i2.p2.19.m19.1.1" xref="S5.I2.i2.p2.19.m19.1.1.cmml"><msub id="S5.I2.i2.p2.19.m19.1.1.2" xref="S5.I2.i2.p2.19.m19.1.1.2.cmml"><mi id="S5.I2.i2.p2.19.m19.1.1.2.2" xref="S5.I2.i2.p2.19.m19.1.1.2.2.cmml">j</mi><mrow id="S5.I2.i2.p2.19.m19.1.1.2.3" xref="S5.I2.i2.p2.19.m19.1.1.2.3.cmml"><mi id="S5.I2.i2.p2.19.m19.1.1.2.3.2" xref="S5.I2.i2.p2.19.m19.1.1.2.3.2.cmml">k</mi><mo id="S5.I2.i2.p2.19.m19.1.1.2.3.1" xref="S5.I2.i2.p2.19.m19.1.1.2.3.1.cmml">−</mo><mn id="S5.I2.i2.p2.19.m19.1.1.2.3.3" xref="S5.I2.i2.p2.19.m19.1.1.2.3.3.cmml">1</mn></mrow></msub><mo id="S5.I2.i2.p2.19.m19.1.1.1" xref="S5.I2.i2.p2.19.m19.1.1.1.cmml">≤</mo><msub id="S5.I2.i2.p2.19.m19.1.1.3" xref="S5.I2.i2.p2.19.m19.1.1.3.cmml"><mi id="S5.I2.i2.p2.19.m19.1.1.3.2" xref="S5.I2.i2.p2.19.m19.1.1.3.2.cmml">i</mi><mi id="S5.I2.i2.p2.19.m19.1.1.3.3" xref="S5.I2.i2.p2.19.m19.1.1.3.3.cmml">k</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S5.I2.i2.p2.19.m19.1b"><apply id="S5.I2.i2.p2.19.m19.1.1.cmml" xref="S5.I2.i2.p2.19.m19.1.1"><leq id="S5.I2.i2.p2.19.m19.1.1.1.cmml" xref="S5.I2.i2.p2.19.m19.1.1.1"></leq><apply id="S5.I2.i2.p2.19.m19.1.1.2.cmml" xref="S5.I2.i2.p2.19.m19.1.1.2"><csymbol cd="ambiguous" id="S5.I2.i2.p2.19.m19.1.1.2.1.cmml" xref="S5.I2.i2.p2.19.m19.1.1.2">subscript</csymbol><ci id="S5.I2.i2.p2.19.m19.1.1.2.2.cmml" xref="S5.I2.i2.p2.19.m19.1.1.2.2">𝑗</ci><apply id="S5.I2.i2.p2.19.m19.1.1.2.3.cmml" xref="S5.I2.i2.p2.19.m19.1.1.2.3"><minus id="S5.I2.i2.p2.19.m19.1.1.2.3.1.cmml" xref="S5.I2.i2.p2.19.m19.1.1.2.3.1"></minus><ci id="S5.I2.i2.p2.19.m19.1.1.2.3.2.cmml" xref="S5.I2.i2.p2.19.m19.1.1.2.3.2">𝑘</ci><cn id="S5.I2.i2.p2.19.m19.1.1.2.3.3.cmml" type="integer" xref="S5.I2.i2.p2.19.m19.1.1.2.3.3">1</cn></apply></apply><apply id="S5.I2.i2.p2.19.m19.1.1.3.cmml" xref="S5.I2.i2.p2.19.m19.1.1.3"><csymbol cd="ambiguous" id="S5.I2.i2.p2.19.m19.1.1.3.1.cmml" xref="S5.I2.i2.p2.19.m19.1.1.3">subscript</csymbol><ci id="S5.I2.i2.p2.19.m19.1.1.3.2.cmml" xref="S5.I2.i2.p2.19.m19.1.1.3.2">𝑖</ci><ci id="S5.I2.i2.p2.19.m19.1.1.3.3.cmml" xref="S5.I2.i2.p2.19.m19.1.1.3.3">𝑘</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.I2.i2.p2.19.m19.1c">j_{k-1}\leq i_{k}</annotation><annotation encoding="application/x-llamapun" id="S5.I2.i2.p2.19.m19.1d">italic_j start_POSTSUBSCRIPT italic_k - 1 end_POSTSUBSCRIPT ≤ italic_i start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT</annotation></semantics></math>, the removing any nodes in <math alttext="[j_{k-1},i_{k}]" class="ltx_Math" display="inline" id="S5.I2.i2.p2.20.m20.2"><semantics id="S5.I2.i2.p2.20.m20.2a"><mrow id="S5.I2.i2.p2.20.m20.2.2.2" xref="S5.I2.i2.p2.20.m20.2.2.3.cmml"><mo id="S5.I2.i2.p2.20.m20.2.2.2.3" stretchy="false" xref="S5.I2.i2.p2.20.m20.2.2.3.cmml">[</mo><msub id="S5.I2.i2.p2.20.m20.1.1.1.1" xref="S5.I2.i2.p2.20.m20.1.1.1.1.cmml"><mi id="S5.I2.i2.p2.20.m20.1.1.1.1.2" xref="S5.I2.i2.p2.20.m20.1.1.1.1.2.cmml">j</mi><mrow id="S5.I2.i2.p2.20.m20.1.1.1.1.3" xref="S5.I2.i2.p2.20.m20.1.1.1.1.3.cmml"><mi id="S5.I2.i2.p2.20.m20.1.1.1.1.3.2" xref="S5.I2.i2.p2.20.m20.1.1.1.1.3.2.cmml">k</mi><mo id="S5.I2.i2.p2.20.m20.1.1.1.1.3.1" xref="S5.I2.i2.p2.20.m20.1.1.1.1.3.1.cmml">−</mo><mn id="S5.I2.i2.p2.20.m20.1.1.1.1.3.3" xref="S5.I2.i2.p2.20.m20.1.1.1.1.3.3.cmml">1</mn></mrow></msub><mo id="S5.I2.i2.p2.20.m20.2.2.2.4" xref="S5.I2.i2.p2.20.m20.2.2.3.cmml">,</mo><msub id="S5.I2.i2.p2.20.m20.2.2.2.2" xref="S5.I2.i2.p2.20.m20.2.2.2.2.cmml"><mi id="S5.I2.i2.p2.20.m20.2.2.2.2.2" xref="S5.I2.i2.p2.20.m20.2.2.2.2.2.cmml">i</mi><mi id="S5.I2.i2.p2.20.m20.2.2.2.2.3" xref="S5.I2.i2.p2.20.m20.2.2.2.2.3.cmml">k</mi></msub><mo id="S5.I2.i2.p2.20.m20.2.2.2.5" stretchy="false" xref="S5.I2.i2.p2.20.m20.2.2.3.cmml">]</mo></mrow><annotation-xml encoding="MathML-Content" id="S5.I2.i2.p2.20.m20.2b"><interval closure="closed" id="S5.I2.i2.p2.20.m20.2.2.3.cmml" xref="S5.I2.i2.p2.20.m20.2.2.2"><apply id="S5.I2.i2.p2.20.m20.1.1.1.1.cmml" xref="S5.I2.i2.p2.20.m20.1.1.1.1"><csymbol cd="ambiguous" id="S5.I2.i2.p2.20.m20.1.1.1.1.1.cmml" xref="S5.I2.i2.p2.20.m20.1.1.1.1">subscript</csymbol><ci id="S5.I2.i2.p2.20.m20.1.1.1.1.2.cmml" xref="S5.I2.i2.p2.20.m20.1.1.1.1.2">𝑗</ci><apply id="S5.I2.i2.p2.20.m20.1.1.1.1.3.cmml" xref="S5.I2.i2.p2.20.m20.1.1.1.1.3"><minus id="S5.I2.i2.p2.20.m20.1.1.1.1.3.1.cmml" xref="S5.I2.i2.p2.20.m20.1.1.1.1.3.1"></minus><ci id="S5.I2.i2.p2.20.m20.1.1.1.1.3.2.cmml" xref="S5.I2.i2.p2.20.m20.1.1.1.1.3.2">𝑘</ci><cn id="S5.I2.i2.p2.20.m20.1.1.1.1.3.3.cmml" type="integer" xref="S5.I2.i2.p2.20.m20.1.1.1.1.3.3">1</cn></apply></apply><apply id="S5.I2.i2.p2.20.m20.2.2.2.2.cmml" xref="S5.I2.i2.p2.20.m20.2.2.2.2"><csymbol cd="ambiguous" id="S5.I2.i2.p2.20.m20.2.2.2.2.1.cmml" xref="S5.I2.i2.p2.20.m20.2.2.2.2">subscript</csymbol><ci id="S5.I2.i2.p2.20.m20.2.2.2.2.2.cmml" xref="S5.I2.i2.p2.20.m20.2.2.2.2.2">𝑖</ci><ci id="S5.I2.i2.p2.20.m20.2.2.2.2.3.cmml" xref="S5.I2.i2.p2.20.m20.2.2.2.2.3">𝑘</ci></apply></interval></annotation-xml><annotation encoding="application/x-tex" id="S5.I2.i2.p2.20.m20.2c">[j_{k-1},i_{k}]</annotation><annotation encoding="application/x-llamapun" id="S5.I2.i2.p2.20.m20.2d">[ italic_j start_POSTSUBSCRIPT italic_k - 1 end_POSTSUBSCRIPT , italic_i start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ]</annotation></semantics></math> will leave the graph <math alttext="\{(x_{1},x_{2}),(x_{2},x_{3}),\dots,(x_{|V|-1},x_{|V|})\}\cup S" class="ltx_Math" display="inline" id="S5.I2.i2.p2.21.m21.6"><semantics id="S5.I2.i2.p2.21.m21.6a"><mrow id="S5.I2.i2.p2.21.m21.6.6" xref="S5.I2.i2.p2.21.m21.6.6.cmml"><mrow id="S5.I2.i2.p2.21.m21.6.6.3.3" xref="S5.I2.i2.p2.21.m21.6.6.3.4.cmml"><mo id="S5.I2.i2.p2.21.m21.6.6.3.3.4" stretchy="false" xref="S5.I2.i2.p2.21.m21.6.6.3.4.cmml">{</mo><mrow id="S5.I2.i2.p2.21.m21.4.4.1.1.1.2" xref="S5.I2.i2.p2.21.m21.4.4.1.1.1.3.cmml"><mo id="S5.I2.i2.p2.21.m21.4.4.1.1.1.2.3" stretchy="false" xref="S5.I2.i2.p2.21.m21.4.4.1.1.1.3.cmml">(</mo><msub id="S5.I2.i2.p2.21.m21.4.4.1.1.1.1.1" xref="S5.I2.i2.p2.21.m21.4.4.1.1.1.1.1.cmml"><mi id="S5.I2.i2.p2.21.m21.4.4.1.1.1.1.1.2" xref="S5.I2.i2.p2.21.m21.4.4.1.1.1.1.1.2.cmml">x</mi><mn id="S5.I2.i2.p2.21.m21.4.4.1.1.1.1.1.3" xref="S5.I2.i2.p2.21.m21.4.4.1.1.1.1.1.3.cmml">1</mn></msub><mo id="S5.I2.i2.p2.21.m21.4.4.1.1.1.2.4" xref="S5.I2.i2.p2.21.m21.4.4.1.1.1.3.cmml">,</mo><msub id="S5.I2.i2.p2.21.m21.4.4.1.1.1.2.2" xref="S5.I2.i2.p2.21.m21.4.4.1.1.1.2.2.cmml"><mi id="S5.I2.i2.p2.21.m21.4.4.1.1.1.2.2.2" xref="S5.I2.i2.p2.21.m21.4.4.1.1.1.2.2.2.cmml">x</mi><mn id="S5.I2.i2.p2.21.m21.4.4.1.1.1.2.2.3" xref="S5.I2.i2.p2.21.m21.4.4.1.1.1.2.2.3.cmml">2</mn></msub><mo id="S5.I2.i2.p2.21.m21.4.4.1.1.1.2.5" stretchy="false" xref="S5.I2.i2.p2.21.m21.4.4.1.1.1.3.cmml">)</mo></mrow><mo id="S5.I2.i2.p2.21.m21.6.6.3.3.5" xref="S5.I2.i2.p2.21.m21.6.6.3.4.cmml">,</mo><mrow id="S5.I2.i2.p2.21.m21.5.5.2.2.2.2" xref="S5.I2.i2.p2.21.m21.5.5.2.2.2.3.cmml"><mo id="S5.I2.i2.p2.21.m21.5.5.2.2.2.2.3" stretchy="false" xref="S5.I2.i2.p2.21.m21.5.5.2.2.2.3.cmml">(</mo><msub id="S5.I2.i2.p2.21.m21.5.5.2.2.2.1.1" xref="S5.I2.i2.p2.21.m21.5.5.2.2.2.1.1.cmml"><mi id="S5.I2.i2.p2.21.m21.5.5.2.2.2.1.1.2" xref="S5.I2.i2.p2.21.m21.5.5.2.2.2.1.1.2.cmml">x</mi><mn id="S5.I2.i2.p2.21.m21.5.5.2.2.2.1.1.3" xref="S5.I2.i2.p2.21.m21.5.5.2.2.2.1.1.3.cmml">2</mn></msub><mo id="S5.I2.i2.p2.21.m21.5.5.2.2.2.2.4" xref="S5.I2.i2.p2.21.m21.5.5.2.2.2.3.cmml">,</mo><msub id="S5.I2.i2.p2.21.m21.5.5.2.2.2.2.2" xref="S5.I2.i2.p2.21.m21.5.5.2.2.2.2.2.cmml"><mi id="S5.I2.i2.p2.21.m21.5.5.2.2.2.2.2.2" xref="S5.I2.i2.p2.21.m21.5.5.2.2.2.2.2.2.cmml">x</mi><mn id="S5.I2.i2.p2.21.m21.5.5.2.2.2.2.2.3" xref="S5.I2.i2.p2.21.m21.5.5.2.2.2.2.2.3.cmml">3</mn></msub><mo id="S5.I2.i2.p2.21.m21.5.5.2.2.2.2.5" stretchy="false" 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xref="S5.I2.i2.p2.21.m21.1.1.1.1.cmml">V</mi><mo id="S5.I2.i2.p2.21.m21.1.1.1.3.2.2" stretchy="false" xref="S5.I2.i2.p2.21.m21.1.1.1.3.1.1.cmml">|</mo></mrow><mo id="S5.I2.i2.p2.21.m21.1.1.1.2" xref="S5.I2.i2.p2.21.m21.1.1.1.2.cmml">−</mo><mn id="S5.I2.i2.p2.21.m21.1.1.1.4" xref="S5.I2.i2.p2.21.m21.1.1.1.4.cmml">1</mn></mrow></msub><mo id="S5.I2.i2.p2.21.m21.6.6.3.3.3.2.4" xref="S5.I2.i2.p2.21.m21.6.6.3.3.3.3.cmml">,</mo><msub id="S5.I2.i2.p2.21.m21.6.6.3.3.3.2.2" xref="S5.I2.i2.p2.21.m21.6.6.3.3.3.2.2.cmml"><mi id="S5.I2.i2.p2.21.m21.6.6.3.3.3.2.2.2" xref="S5.I2.i2.p2.21.m21.6.6.3.3.3.2.2.2.cmml">x</mi><mrow id="S5.I2.i2.p2.21.m21.2.2.1.3" xref="S5.I2.i2.p2.21.m21.2.2.1.2.cmml"><mo id="S5.I2.i2.p2.21.m21.2.2.1.3.1" stretchy="false" xref="S5.I2.i2.p2.21.m21.2.2.1.2.1.cmml">|</mo><mi id="S5.I2.i2.p2.21.m21.2.2.1.1" xref="S5.I2.i2.p2.21.m21.2.2.1.1.cmml">V</mi><mo id="S5.I2.i2.p2.21.m21.2.2.1.3.2" stretchy="false" xref="S5.I2.i2.p2.21.m21.2.2.1.2.1.cmml">|</mo></mrow></msub><mo id="S5.I2.i2.p2.21.m21.6.6.3.3.3.2.5" stretchy="false" xref="S5.I2.i2.p2.21.m21.6.6.3.3.3.3.cmml">)</mo></mrow><mo id="S5.I2.i2.p2.21.m21.6.6.3.3.8" stretchy="false" xref="S5.I2.i2.p2.21.m21.6.6.3.4.cmml">}</mo></mrow><mo id="S5.I2.i2.p2.21.m21.6.6.4" xref="S5.I2.i2.p2.21.m21.6.6.4.cmml">∪</mo><mi id="S5.I2.i2.p2.21.m21.6.6.5" xref="S5.I2.i2.p2.21.m21.6.6.5.cmml">S</mi></mrow><annotation-xml encoding="MathML-Content" id="S5.I2.i2.p2.21.m21.6b"><apply id="S5.I2.i2.p2.21.m21.6.6.cmml" xref="S5.I2.i2.p2.21.m21.6.6"><union id="S5.I2.i2.p2.21.m21.6.6.4.cmml" xref="S5.I2.i2.p2.21.m21.6.6.4"></union><set id="S5.I2.i2.p2.21.m21.6.6.3.4.cmml" xref="S5.I2.i2.p2.21.m21.6.6.3.3"><interval closure="open" id="S5.I2.i2.p2.21.m21.4.4.1.1.1.3.cmml" xref="S5.I2.i2.p2.21.m21.4.4.1.1.1.2"><apply id="S5.I2.i2.p2.21.m21.4.4.1.1.1.1.1.cmml" xref="S5.I2.i2.p2.21.m21.4.4.1.1.1.1.1"><csymbol cd="ambiguous" id="S5.I2.i2.p2.21.m21.4.4.1.1.1.1.1.1.cmml" xref="S5.I2.i2.p2.21.m21.4.4.1.1.1.1.1">subscript</csymbol><ci id="S5.I2.i2.p2.21.m21.4.4.1.1.1.1.1.2.cmml" xref="S5.I2.i2.p2.21.m21.4.4.1.1.1.1.1.2">𝑥</ci><cn id="S5.I2.i2.p2.21.m21.4.4.1.1.1.1.1.3.cmml" type="integer" xref="S5.I2.i2.p2.21.m21.4.4.1.1.1.1.1.3">1</cn></apply><apply id="S5.I2.i2.p2.21.m21.4.4.1.1.1.2.2.cmml" xref="S5.I2.i2.p2.21.m21.4.4.1.1.1.2.2"><csymbol cd="ambiguous" id="S5.I2.i2.p2.21.m21.4.4.1.1.1.2.2.1.cmml" xref="S5.I2.i2.p2.21.m21.4.4.1.1.1.2.2">subscript</csymbol><ci id="S5.I2.i2.p2.21.m21.4.4.1.1.1.2.2.2.cmml" xref="S5.I2.i2.p2.21.m21.4.4.1.1.1.2.2.2">𝑥</ci><cn id="S5.I2.i2.p2.21.m21.4.4.1.1.1.2.2.3.cmml" type="integer" xref="S5.I2.i2.p2.21.m21.4.4.1.1.1.2.2.3">2</cn></apply></interval><interval closure="open" id="S5.I2.i2.p2.21.m21.5.5.2.2.2.3.cmml" xref="S5.I2.i2.p2.21.m21.5.5.2.2.2.2"><apply id="S5.I2.i2.p2.21.m21.5.5.2.2.2.1.1.cmml" xref="S5.I2.i2.p2.21.m21.5.5.2.2.2.1.1"><csymbol cd="ambiguous" id="S5.I2.i2.p2.21.m21.5.5.2.2.2.1.1.1.cmml" 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cd="ambiguous" id="S5.I2.i2.p2.21.m21.6.6.3.3.3.1.1.1.cmml" xref="S5.I2.i2.p2.21.m21.6.6.3.3.3.1.1">subscript</csymbol><ci id="S5.I2.i2.p2.21.m21.6.6.3.3.3.1.1.2.cmml" xref="S5.I2.i2.p2.21.m21.6.6.3.3.3.1.1.2">𝑥</ci><apply id="S5.I2.i2.p2.21.m21.1.1.1.cmml" xref="S5.I2.i2.p2.21.m21.1.1.1"><minus id="S5.I2.i2.p2.21.m21.1.1.1.2.cmml" xref="S5.I2.i2.p2.21.m21.1.1.1.2"></minus><apply id="S5.I2.i2.p2.21.m21.1.1.1.3.1.cmml" xref="S5.I2.i2.p2.21.m21.1.1.1.3.2"><abs id="S5.I2.i2.p2.21.m21.1.1.1.3.1.1.cmml" xref="S5.I2.i2.p2.21.m21.1.1.1.3.2.1"></abs><ci id="S5.I2.i2.p2.21.m21.1.1.1.1.cmml" xref="S5.I2.i2.p2.21.m21.1.1.1.1">𝑉</ci></apply><cn id="S5.I2.i2.p2.21.m21.1.1.1.4.cmml" type="integer" xref="S5.I2.i2.p2.21.m21.1.1.1.4">1</cn></apply></apply><apply id="S5.I2.i2.p2.21.m21.6.6.3.3.3.2.2.cmml" xref="S5.I2.i2.p2.21.m21.6.6.3.3.3.2.2"><csymbol cd="ambiguous" id="S5.I2.i2.p2.21.m21.6.6.3.3.3.2.2.1.cmml" xref="S5.I2.i2.p2.21.m21.6.6.3.3.3.2.2">subscript</csymbol><ci id="S5.I2.i2.p2.21.m21.6.6.3.3.3.2.2.2.cmml" xref="S5.I2.i2.p2.21.m21.6.6.3.3.3.2.2.2">𝑥</ci><apply id="S5.I2.i2.p2.21.m21.2.2.1.2.cmml" xref="S5.I2.i2.p2.21.m21.2.2.1.3"><abs id="S5.I2.i2.p2.21.m21.2.2.1.2.1.cmml" xref="S5.I2.i2.p2.21.m21.2.2.1.3.1"></abs><ci id="S5.I2.i2.p2.21.m21.2.2.1.1.cmml" xref="S5.I2.i2.p2.21.m21.2.2.1.1">𝑉</ci></apply></apply></interval></set><ci id="S5.I2.i2.p2.21.m21.6.6.5.cmml" xref="S5.I2.i2.p2.21.m21.6.6.5">𝑆</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.I2.i2.p2.21.m21.6c">\{(x_{1},x_{2}),(x_{2},x_{3}),\dots,(x_{|V|-1},x_{|V|})\}\cup S</annotation><annotation encoding="application/x-llamapun" id="S5.I2.i2.p2.21.m21.6d">{ ( italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_x start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) , ( italic_x start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT , italic_x start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ) , … , ( italic_x start_POSTSUBSCRIPT | italic_V | - 1 end_POSTSUBSCRIPT , italic_x start_POSTSUBSCRIPT | italic_V | end_POSTSUBSCRIPT ) } ∪ italic_S</annotation></semantics></math> disconnected.</p> </div> <div class="ltx_para" id="S5.I2.i2.p3"> <p class="ltx_p" id="S5.I2.i2.p3.6">Note that <math alttext="S\subseteq E(G^{\prime})\subseteq E(G)\setminus\{(u,v)\}=E(H)" class="ltx_Math" display="inline" id="S5.I2.i2.p3.1.m1.6"><semantics id="S5.I2.i2.p3.1.m1.6a"><mrow id="S5.I2.i2.p3.1.m1.6.6" xref="S5.I2.i2.p3.1.m1.6.6.cmml"><mi id="S5.I2.i2.p3.1.m1.6.6.4" xref="S5.I2.i2.p3.1.m1.6.6.4.cmml">S</mi><mo id="S5.I2.i2.p3.1.m1.6.6.5" xref="S5.I2.i2.p3.1.m1.6.6.5.cmml">⊆</mo><mrow id="S5.I2.i2.p3.1.m1.5.5.1" xref="S5.I2.i2.p3.1.m1.5.5.1.cmml"><mi id="S5.I2.i2.p3.1.m1.5.5.1.3" xref="S5.I2.i2.p3.1.m1.5.5.1.3.cmml">E</mi><mo id="S5.I2.i2.p3.1.m1.5.5.1.2" xref="S5.I2.i2.p3.1.m1.5.5.1.2.cmml">⁢</mo><mrow id="S5.I2.i2.p3.1.m1.5.5.1.1.1" xref="S5.I2.i2.p3.1.m1.5.5.1.1.1.1.cmml"><mo id="S5.I2.i2.p3.1.m1.5.5.1.1.1.2" stretchy="false" xref="S5.I2.i2.p3.1.m1.5.5.1.1.1.1.cmml">(</mo><msup id="S5.I2.i2.p3.1.m1.5.5.1.1.1.1" xref="S5.I2.i2.p3.1.m1.5.5.1.1.1.1.cmml"><mi id="S5.I2.i2.p3.1.m1.5.5.1.1.1.1.2" xref="S5.I2.i2.p3.1.m1.5.5.1.1.1.1.2.cmml">G</mi><mo id="S5.I2.i2.p3.1.m1.5.5.1.1.1.1.3" xref="S5.I2.i2.p3.1.m1.5.5.1.1.1.1.3.cmml">′</mo></msup><mo id="S5.I2.i2.p3.1.m1.5.5.1.1.1.3" stretchy="false" xref="S5.I2.i2.p3.1.m1.5.5.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S5.I2.i2.p3.1.m1.6.6.6" xref="S5.I2.i2.p3.1.m1.6.6.6.cmml">⊆</mo><mrow id="S5.I2.i2.p3.1.m1.6.6.2" xref="S5.I2.i2.p3.1.m1.6.6.2.cmml"><mrow id="S5.I2.i2.p3.1.m1.6.6.2.3" xref="S5.I2.i2.p3.1.m1.6.6.2.3.cmml"><mi id="S5.I2.i2.p3.1.m1.6.6.2.3.2" xref="S5.I2.i2.p3.1.m1.6.6.2.3.2.cmml">E</mi><mo id="S5.I2.i2.p3.1.m1.6.6.2.3.1" xref="S5.I2.i2.p3.1.m1.6.6.2.3.1.cmml">⁢</mo><mrow id="S5.I2.i2.p3.1.m1.6.6.2.3.3.2" xref="S5.I2.i2.p3.1.m1.6.6.2.3.cmml"><mo id="S5.I2.i2.p3.1.m1.6.6.2.3.3.2.1" stretchy="false" xref="S5.I2.i2.p3.1.m1.6.6.2.3.cmml">(</mo><mi id="S5.I2.i2.p3.1.m1.1.1" xref="S5.I2.i2.p3.1.m1.1.1.cmml">G</mi><mo id="S5.I2.i2.p3.1.m1.6.6.2.3.3.2.2" stretchy="false" xref="S5.I2.i2.p3.1.m1.6.6.2.3.cmml">)</mo></mrow></mrow><mo id="S5.I2.i2.p3.1.m1.6.6.2.2" xref="S5.I2.i2.p3.1.m1.6.6.2.2.cmml">∖</mo><mrow id="S5.I2.i2.p3.1.m1.6.6.2.1.1" xref="S5.I2.i2.p3.1.m1.6.6.2.1.2.cmml"><mo id="S5.I2.i2.p3.1.m1.6.6.2.1.1.2" stretchy="false" xref="S5.I2.i2.p3.1.m1.6.6.2.1.2.cmml">{</mo><mrow id="S5.I2.i2.p3.1.m1.6.6.2.1.1.1.2" xref="S5.I2.i2.p3.1.m1.6.6.2.1.1.1.1.cmml"><mo id="S5.I2.i2.p3.1.m1.6.6.2.1.1.1.2.1" stretchy="false" xref="S5.I2.i2.p3.1.m1.6.6.2.1.1.1.1.cmml">(</mo><mi id="S5.I2.i2.p3.1.m1.2.2" xref="S5.I2.i2.p3.1.m1.2.2.cmml">u</mi><mo id="S5.I2.i2.p3.1.m1.6.6.2.1.1.1.2.2" xref="S5.I2.i2.p3.1.m1.6.6.2.1.1.1.1.cmml">,</mo><mi id="S5.I2.i2.p3.1.m1.3.3" xref="S5.I2.i2.p3.1.m1.3.3.cmml">v</mi><mo id="S5.I2.i2.p3.1.m1.6.6.2.1.1.1.2.3" stretchy="false" xref="S5.I2.i2.p3.1.m1.6.6.2.1.1.1.1.cmml">)</mo></mrow><mo id="S5.I2.i2.p3.1.m1.6.6.2.1.1.3" stretchy="false" xref="S5.I2.i2.p3.1.m1.6.6.2.1.2.cmml">}</mo></mrow></mrow><mo id="S5.I2.i2.p3.1.m1.6.6.7" xref="S5.I2.i2.p3.1.m1.6.6.7.cmml">=</mo><mrow id="S5.I2.i2.p3.1.m1.6.6.8" xref="S5.I2.i2.p3.1.m1.6.6.8.cmml"><mi id="S5.I2.i2.p3.1.m1.6.6.8.2" xref="S5.I2.i2.p3.1.m1.6.6.8.2.cmml">E</mi><mo id="S5.I2.i2.p3.1.m1.6.6.8.1" xref="S5.I2.i2.p3.1.m1.6.6.8.1.cmml">⁢</mo><mrow id="S5.I2.i2.p3.1.m1.6.6.8.3.2" xref="S5.I2.i2.p3.1.m1.6.6.8.cmml"><mo id="S5.I2.i2.p3.1.m1.6.6.8.3.2.1" stretchy="false" xref="S5.I2.i2.p3.1.m1.6.6.8.cmml">(</mo><mi id="S5.I2.i2.p3.1.m1.4.4" xref="S5.I2.i2.p3.1.m1.4.4.cmml">H</mi><mo id="S5.I2.i2.p3.1.m1.6.6.8.3.2.2" stretchy="false" xref="S5.I2.i2.p3.1.m1.6.6.8.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.I2.i2.p3.1.m1.6b"><apply id="S5.I2.i2.p3.1.m1.6.6.cmml" xref="S5.I2.i2.p3.1.m1.6.6"><and id="S5.I2.i2.p3.1.m1.6.6a.cmml" xref="S5.I2.i2.p3.1.m1.6.6"></and><apply id="S5.I2.i2.p3.1.m1.6.6b.cmml" xref="S5.I2.i2.p3.1.m1.6.6"><subset id="S5.I2.i2.p3.1.m1.6.6.5.cmml" xref="S5.I2.i2.p3.1.m1.6.6.5"></subset><ci id="S5.I2.i2.p3.1.m1.6.6.4.cmml" xref="S5.I2.i2.p3.1.m1.6.6.4">𝑆</ci><apply id="S5.I2.i2.p3.1.m1.5.5.1.cmml" xref="S5.I2.i2.p3.1.m1.5.5.1"><times id="S5.I2.i2.p3.1.m1.5.5.1.2.cmml" xref="S5.I2.i2.p3.1.m1.5.5.1.2"></times><ci id="S5.I2.i2.p3.1.m1.5.5.1.3.cmml" xref="S5.I2.i2.p3.1.m1.5.5.1.3">𝐸</ci><apply id="S5.I2.i2.p3.1.m1.5.5.1.1.1.1.cmml" xref="S5.I2.i2.p3.1.m1.5.5.1.1.1"><csymbol cd="ambiguous" id="S5.I2.i2.p3.1.m1.5.5.1.1.1.1.1.cmml" xref="S5.I2.i2.p3.1.m1.5.5.1.1.1">superscript</csymbol><ci id="S5.I2.i2.p3.1.m1.5.5.1.1.1.1.2.cmml" xref="S5.I2.i2.p3.1.m1.5.5.1.1.1.1.2">𝐺</ci><ci id="S5.I2.i2.p3.1.m1.5.5.1.1.1.1.3.cmml" xref="S5.I2.i2.p3.1.m1.5.5.1.1.1.1.3">′</ci></apply></apply></apply><apply id="S5.I2.i2.p3.1.m1.6.6c.cmml" xref="S5.I2.i2.p3.1.m1.6.6"><subset id="S5.I2.i2.p3.1.m1.6.6.6.cmml" xref="S5.I2.i2.p3.1.m1.6.6.6"></subset><share href="https://arxiv.org/html/2503.00712v1#S5.I2.i2.p3.1.m1.5.5.1.cmml" id="S5.I2.i2.p3.1.m1.6.6d.cmml" xref="S5.I2.i2.p3.1.m1.6.6"></share><apply id="S5.I2.i2.p3.1.m1.6.6.2.cmml" xref="S5.I2.i2.p3.1.m1.6.6.2"><setdiff id="S5.I2.i2.p3.1.m1.6.6.2.2.cmml" xref="S5.I2.i2.p3.1.m1.6.6.2.2"></setdiff><apply id="S5.I2.i2.p3.1.m1.6.6.2.3.cmml" xref="S5.I2.i2.p3.1.m1.6.6.2.3"><times id="S5.I2.i2.p3.1.m1.6.6.2.3.1.cmml" xref="S5.I2.i2.p3.1.m1.6.6.2.3.1"></times><ci id="S5.I2.i2.p3.1.m1.6.6.2.3.2.cmml" xref="S5.I2.i2.p3.1.m1.6.6.2.3.2">𝐸</ci><ci id="S5.I2.i2.p3.1.m1.1.1.cmml" xref="S5.I2.i2.p3.1.m1.1.1">𝐺</ci></apply><set id="S5.I2.i2.p3.1.m1.6.6.2.1.2.cmml" xref="S5.I2.i2.p3.1.m1.6.6.2.1.1"><interval closure="open" id="S5.I2.i2.p3.1.m1.6.6.2.1.1.1.1.cmml" xref="S5.I2.i2.p3.1.m1.6.6.2.1.1.1.2"><ci id="S5.I2.i2.p3.1.m1.2.2.cmml" xref="S5.I2.i2.p3.1.m1.2.2">𝑢</ci><ci id="S5.I2.i2.p3.1.m1.3.3.cmml" xref="S5.I2.i2.p3.1.m1.3.3">𝑣</ci></interval></set></apply></apply><apply id="S5.I2.i2.p3.1.m1.6.6e.cmml" xref="S5.I2.i2.p3.1.m1.6.6"><eq id="S5.I2.i2.p3.1.m1.6.6.7.cmml" xref="S5.I2.i2.p3.1.m1.6.6.7"></eq><share href="https://arxiv.org/html/2503.00712v1#S5.I2.i2.p3.1.m1.6.6.2.cmml" id="S5.I2.i2.p3.1.m1.6.6f.cmml" xref="S5.I2.i2.p3.1.m1.6.6"></share><apply id="S5.I2.i2.p3.1.m1.6.6.8.cmml" xref="S5.I2.i2.p3.1.m1.6.6.8"><times id="S5.I2.i2.p3.1.m1.6.6.8.1.cmml" xref="S5.I2.i2.p3.1.m1.6.6.8.1"></times><ci id="S5.I2.i2.p3.1.m1.6.6.8.2.cmml" xref="S5.I2.i2.p3.1.m1.6.6.8.2">𝐸</ci><ci id="S5.I2.i2.p3.1.m1.4.4.cmml" xref="S5.I2.i2.p3.1.m1.4.4">𝐻</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.I2.i2.p3.1.m1.6c">S\subseteq E(G^{\prime})\subseteq E(G)\setminus\{(u,v)\}=E(H)</annotation><annotation encoding="application/x-llamapun" id="S5.I2.i2.p3.1.m1.6d">italic_S ⊆ italic_E ( italic_G start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) ⊆ italic_E ( italic_G ) ∖ { ( italic_u , italic_v ) } = italic_E ( italic_H )</annotation></semantics></math>. Now we inductively prove for every <math alttext="1\leq k\leq s" class="ltx_Math" display="inline" id="S5.I2.i2.p3.2.m2.1"><semantics id="S5.I2.i2.p3.2.m2.1a"><mrow id="S5.I2.i2.p3.2.m2.1.1" xref="S5.I2.i2.p3.2.m2.1.1.cmml"><mn id="S5.I2.i2.p3.2.m2.1.1.2" xref="S5.I2.i2.p3.2.m2.1.1.2.cmml">1</mn><mo id="S5.I2.i2.p3.2.m2.1.1.3" xref="S5.I2.i2.p3.2.m2.1.1.3.cmml">≤</mo><mi id="S5.I2.i2.p3.2.m2.1.1.4" xref="S5.I2.i2.p3.2.m2.1.1.4.cmml">k</mi><mo id="S5.I2.i2.p3.2.m2.1.1.5" xref="S5.I2.i2.p3.2.m2.1.1.5.cmml">≤</mo><mi id="S5.I2.i2.p3.2.m2.1.1.6" xref="S5.I2.i2.p3.2.m2.1.1.6.cmml">s</mi></mrow><annotation-xml encoding="MathML-Content" id="S5.I2.i2.p3.2.m2.1b"><apply id="S5.I2.i2.p3.2.m2.1.1.cmml" xref="S5.I2.i2.p3.2.m2.1.1"><and id="S5.I2.i2.p3.2.m2.1.1a.cmml" xref="S5.I2.i2.p3.2.m2.1.1"></and><apply id="S5.I2.i2.p3.2.m2.1.1b.cmml" xref="S5.I2.i2.p3.2.m2.1.1"><leq id="S5.I2.i2.p3.2.m2.1.1.3.cmml" xref="S5.I2.i2.p3.2.m2.1.1.3"></leq><cn id="S5.I2.i2.p3.2.m2.1.1.2.cmml" type="integer" xref="S5.I2.i2.p3.2.m2.1.1.2">1</cn><ci id="S5.I2.i2.p3.2.m2.1.1.4.cmml" xref="S5.I2.i2.p3.2.m2.1.1.4">𝑘</ci></apply><apply id="S5.I2.i2.p3.2.m2.1.1c.cmml" xref="S5.I2.i2.p3.2.m2.1.1"><leq id="S5.I2.i2.p3.2.m2.1.1.5.cmml" xref="S5.I2.i2.p3.2.m2.1.1.5"></leq><share href="https://arxiv.org/html/2503.00712v1#S5.I2.i2.p3.2.m2.1.1.4.cmml" id="S5.I2.i2.p3.2.m2.1.1d.cmml" xref="S5.I2.i2.p3.2.m2.1.1"></share><ci id="S5.I2.i2.p3.2.m2.1.1.6.cmml" xref="S5.I2.i2.p3.2.m2.1.1.6">𝑠</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.I2.i2.p3.2.m2.1c">1\leq k\leq s</annotation><annotation encoding="application/x-llamapun" id="S5.I2.i2.p3.2.m2.1d">1 ≤ italic_k ≤ italic_s</annotation></semantics></math> that <math alttext="d_{H}(u,x_{j_{k}})\leq k" class="ltx_Math" display="inline" id="S5.I2.i2.p3.3.m3.2"><semantics id="S5.I2.i2.p3.3.m3.2a"><mrow id="S5.I2.i2.p3.3.m3.2.2" xref="S5.I2.i2.p3.3.m3.2.2.cmml"><mrow id="S5.I2.i2.p3.3.m3.2.2.1" xref="S5.I2.i2.p3.3.m3.2.2.1.cmml"><msub id="S5.I2.i2.p3.3.m3.2.2.1.3" xref="S5.I2.i2.p3.3.m3.2.2.1.3.cmml"><mi id="S5.I2.i2.p3.3.m3.2.2.1.3.2" xref="S5.I2.i2.p3.3.m3.2.2.1.3.2.cmml">d</mi><mi id="S5.I2.i2.p3.3.m3.2.2.1.3.3" xref="S5.I2.i2.p3.3.m3.2.2.1.3.3.cmml">H</mi></msub><mo id="S5.I2.i2.p3.3.m3.2.2.1.2" xref="S5.I2.i2.p3.3.m3.2.2.1.2.cmml">⁢</mo><mrow id="S5.I2.i2.p3.3.m3.2.2.1.1.1" xref="S5.I2.i2.p3.3.m3.2.2.1.1.2.cmml"><mo id="S5.I2.i2.p3.3.m3.2.2.1.1.1.2" stretchy="false" xref="S5.I2.i2.p3.3.m3.2.2.1.1.2.cmml">(</mo><mi id="S5.I2.i2.p3.3.m3.1.1" xref="S5.I2.i2.p3.3.m3.1.1.cmml">u</mi><mo id="S5.I2.i2.p3.3.m3.2.2.1.1.1.3" xref="S5.I2.i2.p3.3.m3.2.2.1.1.2.cmml">,</mo><msub id="S5.I2.i2.p3.3.m3.2.2.1.1.1.1" xref="S5.I2.i2.p3.3.m3.2.2.1.1.1.1.cmml"><mi id="S5.I2.i2.p3.3.m3.2.2.1.1.1.1.2" xref="S5.I2.i2.p3.3.m3.2.2.1.1.1.1.2.cmml">x</mi><msub id="S5.I2.i2.p3.3.m3.2.2.1.1.1.1.3" xref="S5.I2.i2.p3.3.m3.2.2.1.1.1.1.3.cmml"><mi id="S5.I2.i2.p3.3.m3.2.2.1.1.1.1.3.2" xref="S5.I2.i2.p3.3.m3.2.2.1.1.1.1.3.2.cmml">j</mi><mi id="S5.I2.i2.p3.3.m3.2.2.1.1.1.1.3.3" xref="S5.I2.i2.p3.3.m3.2.2.1.1.1.1.3.3.cmml">k</mi></msub></msub><mo id="S5.I2.i2.p3.3.m3.2.2.1.1.1.4" stretchy="false" xref="S5.I2.i2.p3.3.m3.2.2.1.1.2.cmml">)</mo></mrow></mrow><mo id="S5.I2.i2.p3.3.m3.2.2.2" xref="S5.I2.i2.p3.3.m3.2.2.2.cmml">≤</mo><mi id="S5.I2.i2.p3.3.m3.2.2.3" xref="S5.I2.i2.p3.3.m3.2.2.3.cmml">k</mi></mrow><annotation-xml encoding="MathML-Content" id="S5.I2.i2.p3.3.m3.2b"><apply id="S5.I2.i2.p3.3.m3.2.2.cmml" xref="S5.I2.i2.p3.3.m3.2.2"><leq id="S5.I2.i2.p3.3.m3.2.2.2.cmml" xref="S5.I2.i2.p3.3.m3.2.2.2"></leq><apply id="S5.I2.i2.p3.3.m3.2.2.1.cmml" xref="S5.I2.i2.p3.3.m3.2.2.1"><times id="S5.I2.i2.p3.3.m3.2.2.1.2.cmml" xref="S5.I2.i2.p3.3.m3.2.2.1.2"></times><apply id="S5.I2.i2.p3.3.m3.2.2.1.3.cmml" xref="S5.I2.i2.p3.3.m3.2.2.1.3"><csymbol cd="ambiguous" id="S5.I2.i2.p3.3.m3.2.2.1.3.1.cmml" xref="S5.I2.i2.p3.3.m3.2.2.1.3">subscript</csymbol><ci id="S5.I2.i2.p3.3.m3.2.2.1.3.2.cmml" xref="S5.I2.i2.p3.3.m3.2.2.1.3.2">𝑑</ci><ci id="S5.I2.i2.p3.3.m3.2.2.1.3.3.cmml" xref="S5.I2.i2.p3.3.m3.2.2.1.3.3">𝐻</ci></apply><interval closure="open" id="S5.I2.i2.p3.3.m3.2.2.1.1.2.cmml" xref="S5.I2.i2.p3.3.m3.2.2.1.1.1"><ci id="S5.I2.i2.p3.3.m3.1.1.cmml" xref="S5.I2.i2.p3.3.m3.1.1">𝑢</ci><apply id="S5.I2.i2.p3.3.m3.2.2.1.1.1.1.cmml" xref="S5.I2.i2.p3.3.m3.2.2.1.1.1.1"><csymbol cd="ambiguous" id="S5.I2.i2.p3.3.m3.2.2.1.1.1.1.1.cmml" xref="S5.I2.i2.p3.3.m3.2.2.1.1.1.1">subscript</csymbol><ci id="S5.I2.i2.p3.3.m3.2.2.1.1.1.1.2.cmml" xref="S5.I2.i2.p3.3.m3.2.2.1.1.1.1.2">𝑥</ci><apply id="S5.I2.i2.p3.3.m3.2.2.1.1.1.1.3.cmml" xref="S5.I2.i2.p3.3.m3.2.2.1.1.1.1.3"><csymbol cd="ambiguous" id="S5.I2.i2.p3.3.m3.2.2.1.1.1.1.3.1.cmml" xref="S5.I2.i2.p3.3.m3.2.2.1.1.1.1.3">subscript</csymbol><ci id="S5.I2.i2.p3.3.m3.2.2.1.1.1.1.3.2.cmml" xref="S5.I2.i2.p3.3.m3.2.2.1.1.1.1.3.2">𝑗</ci><ci id="S5.I2.i2.p3.3.m3.2.2.1.1.1.1.3.3.cmml" xref="S5.I2.i2.p3.3.m3.2.2.1.1.1.1.3.3">𝑘</ci></apply></apply></interval></apply><ci id="S5.I2.i2.p3.3.m3.2.2.3.cmml" xref="S5.I2.i2.p3.3.m3.2.2.3">𝑘</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.I2.i2.p3.3.m3.2c">d_{H}(u,x_{j_{k}})\leq k</annotation><annotation encoding="application/x-llamapun" id="S5.I2.i2.p3.3.m3.2d">italic_d start_POSTSUBSCRIPT italic_H end_POSTSUBSCRIPT ( italic_u , italic_x start_POSTSUBSCRIPT italic_j start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT end_POSTSUBSCRIPT ) ≤ italic_k</annotation></semantics></math>. The base case <math alttext="k=1" class="ltx_Math" display="inline" id="S5.I2.i2.p3.4.m4.1"><semantics id="S5.I2.i2.p3.4.m4.1a"><mrow id="S5.I2.i2.p3.4.m4.1.1" xref="S5.I2.i2.p3.4.m4.1.1.cmml"><mi id="S5.I2.i2.p3.4.m4.1.1.2" xref="S5.I2.i2.p3.4.m4.1.1.2.cmml">k</mi><mo id="S5.I2.i2.p3.4.m4.1.1.1" xref="S5.I2.i2.p3.4.m4.1.1.1.cmml">=</mo><mn id="S5.I2.i2.p3.4.m4.1.1.3" xref="S5.I2.i2.p3.4.m4.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S5.I2.i2.p3.4.m4.1b"><apply id="S5.I2.i2.p3.4.m4.1.1.cmml" xref="S5.I2.i2.p3.4.m4.1.1"><eq id="S5.I2.i2.p3.4.m4.1.1.1.cmml" xref="S5.I2.i2.p3.4.m4.1.1.1"></eq><ci id="S5.I2.i2.p3.4.m4.1.1.2.cmml" xref="S5.I2.i2.p3.4.m4.1.1.2">𝑘</ci><cn id="S5.I2.i2.p3.4.m4.1.1.3.cmml" type="integer" xref="S5.I2.i2.p3.4.m4.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.I2.i2.p3.4.m4.1c">k=1</annotation><annotation encoding="application/x-llamapun" id="S5.I2.i2.p3.4.m4.1d">italic_k = 1</annotation></semantics></math> is immediate: <math alttext="d_{H}(u,x_{j_{1}})=d_{H}(x_{i_{1}},x_{j_{1}})=1" class="ltx_Math" display="inline" id="S5.I2.i2.p3.5.m5.4"><semantics id="S5.I2.i2.p3.5.m5.4a"><mrow id="S5.I2.i2.p3.5.m5.4.4" xref="S5.I2.i2.p3.5.m5.4.4.cmml"><mrow id="S5.I2.i2.p3.5.m5.2.2.1" xref="S5.I2.i2.p3.5.m5.2.2.1.cmml"><msub id="S5.I2.i2.p3.5.m5.2.2.1.3" xref="S5.I2.i2.p3.5.m5.2.2.1.3.cmml"><mi id="S5.I2.i2.p3.5.m5.2.2.1.3.2" xref="S5.I2.i2.p3.5.m5.2.2.1.3.2.cmml">d</mi><mi id="S5.I2.i2.p3.5.m5.2.2.1.3.3" xref="S5.I2.i2.p3.5.m5.2.2.1.3.3.cmml">H</mi></msub><mo id="S5.I2.i2.p3.5.m5.2.2.1.2" xref="S5.I2.i2.p3.5.m5.2.2.1.2.cmml">⁢</mo><mrow id="S5.I2.i2.p3.5.m5.2.2.1.1.1" xref="S5.I2.i2.p3.5.m5.2.2.1.1.2.cmml"><mo id="S5.I2.i2.p3.5.m5.2.2.1.1.1.2" stretchy="false" xref="S5.I2.i2.p3.5.m5.2.2.1.1.2.cmml">(</mo><mi id="S5.I2.i2.p3.5.m5.1.1" xref="S5.I2.i2.p3.5.m5.1.1.cmml">u</mi><mo id="S5.I2.i2.p3.5.m5.2.2.1.1.1.3" xref="S5.I2.i2.p3.5.m5.2.2.1.1.2.cmml">,</mo><msub id="S5.I2.i2.p3.5.m5.2.2.1.1.1.1" xref="S5.I2.i2.p3.5.m5.2.2.1.1.1.1.cmml"><mi id="S5.I2.i2.p3.5.m5.2.2.1.1.1.1.2" xref="S5.I2.i2.p3.5.m5.2.2.1.1.1.1.2.cmml">x</mi><msub id="S5.I2.i2.p3.5.m5.2.2.1.1.1.1.3" xref="S5.I2.i2.p3.5.m5.2.2.1.1.1.1.3.cmml"><mi id="S5.I2.i2.p3.5.m5.2.2.1.1.1.1.3.2" xref="S5.I2.i2.p3.5.m5.2.2.1.1.1.1.3.2.cmml">j</mi><mn id="S5.I2.i2.p3.5.m5.2.2.1.1.1.1.3.3" xref="S5.I2.i2.p3.5.m5.2.2.1.1.1.1.3.3.cmml">1</mn></msub></msub><mo id="S5.I2.i2.p3.5.m5.2.2.1.1.1.4" stretchy="false" xref="S5.I2.i2.p3.5.m5.2.2.1.1.2.cmml">)</mo></mrow></mrow><mo id="S5.I2.i2.p3.5.m5.4.4.5" xref="S5.I2.i2.p3.5.m5.4.4.5.cmml">=</mo><mrow id="S5.I2.i2.p3.5.m5.4.4.3" xref="S5.I2.i2.p3.5.m5.4.4.3.cmml"><msub id="S5.I2.i2.p3.5.m5.4.4.3.4" xref="S5.I2.i2.p3.5.m5.4.4.3.4.cmml"><mi id="S5.I2.i2.p3.5.m5.4.4.3.4.2" xref="S5.I2.i2.p3.5.m5.4.4.3.4.2.cmml">d</mi><mi id="S5.I2.i2.p3.5.m5.4.4.3.4.3" xref="S5.I2.i2.p3.5.m5.4.4.3.4.3.cmml">H</mi></msub><mo id="S5.I2.i2.p3.5.m5.4.4.3.3" xref="S5.I2.i2.p3.5.m5.4.4.3.3.cmml">⁢</mo><mrow id="S5.I2.i2.p3.5.m5.4.4.3.2.2" xref="S5.I2.i2.p3.5.m5.4.4.3.2.3.cmml"><mo id="S5.I2.i2.p3.5.m5.4.4.3.2.2.3" stretchy="false" xref="S5.I2.i2.p3.5.m5.4.4.3.2.3.cmml">(</mo><msub id="S5.I2.i2.p3.5.m5.3.3.2.1.1.1" xref="S5.I2.i2.p3.5.m5.3.3.2.1.1.1.cmml"><mi id="S5.I2.i2.p3.5.m5.3.3.2.1.1.1.2" xref="S5.I2.i2.p3.5.m5.3.3.2.1.1.1.2.cmml">x</mi><msub id="S5.I2.i2.p3.5.m5.3.3.2.1.1.1.3" xref="S5.I2.i2.p3.5.m5.3.3.2.1.1.1.3.cmml"><mi id="S5.I2.i2.p3.5.m5.3.3.2.1.1.1.3.2" xref="S5.I2.i2.p3.5.m5.3.3.2.1.1.1.3.2.cmml">i</mi><mn id="S5.I2.i2.p3.5.m5.3.3.2.1.1.1.3.3" xref="S5.I2.i2.p3.5.m5.3.3.2.1.1.1.3.3.cmml">1</mn></msub></msub><mo id="S5.I2.i2.p3.5.m5.4.4.3.2.2.4" xref="S5.I2.i2.p3.5.m5.4.4.3.2.3.cmml">,</mo><msub id="S5.I2.i2.p3.5.m5.4.4.3.2.2.2" xref="S5.I2.i2.p3.5.m5.4.4.3.2.2.2.cmml"><mi id="S5.I2.i2.p3.5.m5.4.4.3.2.2.2.2" xref="S5.I2.i2.p3.5.m5.4.4.3.2.2.2.2.cmml">x</mi><msub id="S5.I2.i2.p3.5.m5.4.4.3.2.2.2.3" xref="S5.I2.i2.p3.5.m5.4.4.3.2.2.2.3.cmml"><mi id="S5.I2.i2.p3.5.m5.4.4.3.2.2.2.3.2" xref="S5.I2.i2.p3.5.m5.4.4.3.2.2.2.3.2.cmml">j</mi><mn id="S5.I2.i2.p3.5.m5.4.4.3.2.2.2.3.3" xref="S5.I2.i2.p3.5.m5.4.4.3.2.2.2.3.3.cmml">1</mn></msub></msub><mo id="S5.I2.i2.p3.5.m5.4.4.3.2.2.5" stretchy="false" xref="S5.I2.i2.p3.5.m5.4.4.3.2.3.cmml">)</mo></mrow></mrow><mo id="S5.I2.i2.p3.5.m5.4.4.6" xref="S5.I2.i2.p3.5.m5.4.4.6.cmml">=</mo><mn id="S5.I2.i2.p3.5.m5.4.4.7" xref="S5.I2.i2.p3.5.m5.4.4.7.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S5.I2.i2.p3.5.m5.4b"><apply id="S5.I2.i2.p3.5.m5.4.4.cmml" xref="S5.I2.i2.p3.5.m5.4.4"><and id="S5.I2.i2.p3.5.m5.4.4a.cmml" xref="S5.I2.i2.p3.5.m5.4.4"></and><apply id="S5.I2.i2.p3.5.m5.4.4b.cmml" xref="S5.I2.i2.p3.5.m5.4.4"><eq id="S5.I2.i2.p3.5.m5.4.4.5.cmml" xref="S5.I2.i2.p3.5.m5.4.4.5"></eq><apply id="S5.I2.i2.p3.5.m5.2.2.1.cmml" xref="S5.I2.i2.p3.5.m5.2.2.1"><times id="S5.I2.i2.p3.5.m5.2.2.1.2.cmml" xref="S5.I2.i2.p3.5.m5.2.2.1.2"></times><apply id="S5.I2.i2.p3.5.m5.2.2.1.3.cmml" xref="S5.I2.i2.p3.5.m5.2.2.1.3"><csymbol cd="ambiguous" id="S5.I2.i2.p3.5.m5.2.2.1.3.1.cmml" xref="S5.I2.i2.p3.5.m5.2.2.1.3">subscript</csymbol><ci id="S5.I2.i2.p3.5.m5.2.2.1.3.2.cmml" xref="S5.I2.i2.p3.5.m5.2.2.1.3.2">𝑑</ci><ci id="S5.I2.i2.p3.5.m5.2.2.1.3.3.cmml" xref="S5.I2.i2.p3.5.m5.2.2.1.3.3">𝐻</ci></apply><interval closure="open" id="S5.I2.i2.p3.5.m5.2.2.1.1.2.cmml" xref="S5.I2.i2.p3.5.m5.2.2.1.1.1"><ci id="S5.I2.i2.p3.5.m5.1.1.cmml" xref="S5.I2.i2.p3.5.m5.1.1">𝑢</ci><apply id="S5.I2.i2.p3.5.m5.2.2.1.1.1.1.cmml" xref="S5.I2.i2.p3.5.m5.2.2.1.1.1.1"><csymbol cd="ambiguous" id="S5.I2.i2.p3.5.m5.2.2.1.1.1.1.1.cmml" xref="S5.I2.i2.p3.5.m5.2.2.1.1.1.1">subscript</csymbol><ci id="S5.I2.i2.p3.5.m5.2.2.1.1.1.1.2.cmml" xref="S5.I2.i2.p3.5.m5.2.2.1.1.1.1.2">𝑥</ci><apply id="S5.I2.i2.p3.5.m5.2.2.1.1.1.1.3.cmml" xref="S5.I2.i2.p3.5.m5.2.2.1.1.1.1.3"><csymbol cd="ambiguous" 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xref="S5.I2.i2.p3.5.m5.3.3.2.1.1.1"><csymbol cd="ambiguous" id="S5.I2.i2.p3.5.m5.3.3.2.1.1.1.1.cmml" xref="S5.I2.i2.p3.5.m5.3.3.2.1.1.1">subscript</csymbol><ci id="S5.I2.i2.p3.5.m5.3.3.2.1.1.1.2.cmml" xref="S5.I2.i2.p3.5.m5.3.3.2.1.1.1.2">𝑥</ci><apply id="S5.I2.i2.p3.5.m5.3.3.2.1.1.1.3.cmml" xref="S5.I2.i2.p3.5.m5.3.3.2.1.1.1.3"><csymbol cd="ambiguous" id="S5.I2.i2.p3.5.m5.3.3.2.1.1.1.3.1.cmml" xref="S5.I2.i2.p3.5.m5.3.3.2.1.1.1.3">subscript</csymbol><ci id="S5.I2.i2.p3.5.m5.3.3.2.1.1.1.3.2.cmml" xref="S5.I2.i2.p3.5.m5.3.3.2.1.1.1.3.2">𝑖</ci><cn id="S5.I2.i2.p3.5.m5.3.3.2.1.1.1.3.3.cmml" type="integer" xref="S5.I2.i2.p3.5.m5.3.3.2.1.1.1.3.3">1</cn></apply></apply><apply id="S5.I2.i2.p3.5.m5.4.4.3.2.2.2.cmml" xref="S5.I2.i2.p3.5.m5.4.4.3.2.2.2"><csymbol cd="ambiguous" id="S5.I2.i2.p3.5.m5.4.4.3.2.2.2.1.cmml" xref="S5.I2.i2.p3.5.m5.4.4.3.2.2.2">subscript</csymbol><ci id="S5.I2.i2.p3.5.m5.4.4.3.2.2.2.2.cmml" xref="S5.I2.i2.p3.5.m5.4.4.3.2.2.2.2">𝑥</ci><apply id="S5.I2.i2.p3.5.m5.4.4.3.2.2.2.3.cmml" xref="S5.I2.i2.p3.5.m5.4.4.3.2.2.2.3"><csymbol cd="ambiguous" id="S5.I2.i2.p3.5.m5.4.4.3.2.2.2.3.1.cmml" xref="S5.I2.i2.p3.5.m5.4.4.3.2.2.2.3">subscript</csymbol><ci id="S5.I2.i2.p3.5.m5.4.4.3.2.2.2.3.2.cmml" xref="S5.I2.i2.p3.5.m5.4.4.3.2.2.2.3.2">𝑗</ci><cn id="S5.I2.i2.p3.5.m5.4.4.3.2.2.2.3.3.cmml" type="integer" xref="S5.I2.i2.p3.5.m5.4.4.3.2.2.2.3.3">1</cn></apply></apply></interval></apply></apply><apply id="S5.I2.i2.p3.5.m5.4.4c.cmml" xref="S5.I2.i2.p3.5.m5.4.4"><eq id="S5.I2.i2.p3.5.m5.4.4.6.cmml" xref="S5.I2.i2.p3.5.m5.4.4.6"></eq><share href="https://arxiv.org/html/2503.00712v1#S5.I2.i2.p3.5.m5.4.4.3.cmml" id="S5.I2.i2.p3.5.m5.4.4d.cmml" xref="S5.I2.i2.p3.5.m5.4.4"></share><cn id="S5.I2.i2.p3.5.m5.4.4.7.cmml" type="integer" xref="S5.I2.i2.p3.5.m5.4.4.7">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.I2.i2.p3.5.m5.4c">d_{H}(u,x_{j_{1}})=d_{H}(x_{i_{1}},x_{j_{1}})=1</annotation><annotation encoding="application/x-llamapun" id="S5.I2.i2.p3.5.m5.4d">italic_d start_POSTSUBSCRIPT italic_H end_POSTSUBSCRIPT ( italic_u , italic_x start_POSTSUBSCRIPT italic_j start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ) = italic_d start_POSTSUBSCRIPT italic_H end_POSTSUBSCRIPT ( italic_x start_POSTSUBSCRIPT italic_i start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT , italic_x start_POSTSUBSCRIPT italic_j start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ) = 1</annotation></semantics></math>. For the inductive step <math alttext="2\leq k\leq s" class="ltx_Math" display="inline" id="S5.I2.i2.p3.6.m6.1"><semantics id="S5.I2.i2.p3.6.m6.1a"><mrow id="S5.I2.i2.p3.6.m6.1.1" xref="S5.I2.i2.p3.6.m6.1.1.cmml"><mn id="S5.I2.i2.p3.6.m6.1.1.2" xref="S5.I2.i2.p3.6.m6.1.1.2.cmml">2</mn><mo id="S5.I2.i2.p3.6.m6.1.1.3" xref="S5.I2.i2.p3.6.m6.1.1.3.cmml">≤</mo><mi id="S5.I2.i2.p3.6.m6.1.1.4" xref="S5.I2.i2.p3.6.m6.1.1.4.cmml">k</mi><mo id="S5.I2.i2.p3.6.m6.1.1.5" xref="S5.I2.i2.p3.6.m6.1.1.5.cmml">≤</mo><mi id="S5.I2.i2.p3.6.m6.1.1.6" xref="S5.I2.i2.p3.6.m6.1.1.6.cmml">s</mi></mrow><annotation-xml encoding="MathML-Content" id="S5.I2.i2.p3.6.m6.1b"><apply id="S5.I2.i2.p3.6.m6.1.1.cmml" xref="S5.I2.i2.p3.6.m6.1.1"><and id="S5.I2.i2.p3.6.m6.1.1a.cmml" xref="S5.I2.i2.p3.6.m6.1.1"></and><apply id="S5.I2.i2.p3.6.m6.1.1b.cmml" xref="S5.I2.i2.p3.6.m6.1.1"><leq id="S5.I2.i2.p3.6.m6.1.1.3.cmml" xref="S5.I2.i2.p3.6.m6.1.1.3"></leq><cn id="S5.I2.i2.p3.6.m6.1.1.2.cmml" type="integer" xref="S5.I2.i2.p3.6.m6.1.1.2">2</cn><ci id="S5.I2.i2.p3.6.m6.1.1.4.cmml" xref="S5.I2.i2.p3.6.m6.1.1.4">𝑘</ci></apply><apply id="S5.I2.i2.p3.6.m6.1.1c.cmml" xref="S5.I2.i2.p3.6.m6.1.1"><leq id="S5.I2.i2.p3.6.m6.1.1.5.cmml" xref="S5.I2.i2.p3.6.m6.1.1.5"></leq><share href="https://arxiv.org/html/2503.00712v1#S5.I2.i2.p3.6.m6.1.1.4.cmml" id="S5.I2.i2.p3.6.m6.1.1d.cmml" xref="S5.I2.i2.p3.6.m6.1.1"></share><ci id="S5.I2.i2.p3.6.m6.1.1.6.cmml" xref="S5.I2.i2.p3.6.m6.1.1.6">𝑠</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.I2.i2.p3.6.m6.1c">2\leq k\leq s</annotation><annotation encoding="application/x-llamapun" id="S5.I2.i2.p3.6.m6.1d">2 ≤ italic_k ≤ italic_s</annotation></semantics></math>, we have</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="Sx1.EGx7"> <tbody id="S5.Ex15"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle d_{H}(u,x_{j_{k}})" class="ltx_Math" display="inline" id="S5.Ex15.m1.2"><semantics id="S5.Ex15.m1.2a"><mrow id="S5.Ex15.m1.2.2" xref="S5.Ex15.m1.2.2.cmml"><msub id="S5.Ex15.m1.2.2.3" xref="S5.Ex15.m1.2.2.3.cmml"><mi id="S5.Ex15.m1.2.2.3.2" xref="S5.Ex15.m1.2.2.3.2.cmml">d</mi><mi id="S5.Ex15.m1.2.2.3.3" xref="S5.Ex15.m1.2.2.3.3.cmml">H</mi></msub><mo id="S5.Ex15.m1.2.2.2" xref="S5.Ex15.m1.2.2.2.cmml">⁢</mo><mrow id="S5.Ex15.m1.2.2.1.1" xref="S5.Ex15.m1.2.2.1.2.cmml"><mo id="S5.Ex15.m1.2.2.1.1.2" stretchy="false" xref="S5.Ex15.m1.2.2.1.2.cmml">(</mo><mi id="S5.Ex15.m1.1.1" xref="S5.Ex15.m1.1.1.cmml">u</mi><mo id="S5.Ex15.m1.2.2.1.1.3" xref="S5.Ex15.m1.2.2.1.2.cmml">,</mo><msub id="S5.Ex15.m1.2.2.1.1.1" xref="S5.Ex15.m1.2.2.1.1.1.cmml"><mi id="S5.Ex15.m1.2.2.1.1.1.2" xref="S5.Ex15.m1.2.2.1.1.1.2.cmml">x</mi><msub id="S5.Ex15.m1.2.2.1.1.1.3" xref="S5.Ex15.m1.2.2.1.1.1.3.cmml"><mi id="S5.Ex15.m1.2.2.1.1.1.3.2" xref="S5.Ex15.m1.2.2.1.1.1.3.2.cmml">j</mi><mi id="S5.Ex15.m1.2.2.1.1.1.3.3" xref="S5.Ex15.m1.2.2.1.1.1.3.3.cmml">k</mi></msub></msub><mo id="S5.Ex15.m1.2.2.1.1.4" stretchy="false" xref="S5.Ex15.m1.2.2.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.Ex15.m1.2b"><apply id="S5.Ex15.m1.2.2.cmml" xref="S5.Ex15.m1.2.2"><times id="S5.Ex15.m1.2.2.2.cmml" xref="S5.Ex15.m1.2.2.2"></times><apply id="S5.Ex15.m1.2.2.3.cmml" xref="S5.Ex15.m1.2.2.3"><csymbol cd="ambiguous" id="S5.Ex15.m1.2.2.3.1.cmml" xref="S5.Ex15.m1.2.2.3">subscript</csymbol><ci id="S5.Ex15.m1.2.2.3.2.cmml" xref="S5.Ex15.m1.2.2.3.2">𝑑</ci><ci id="S5.Ex15.m1.2.2.3.3.cmml" xref="S5.Ex15.m1.2.2.3.3">𝐻</ci></apply><interval closure="open" id="S5.Ex15.m1.2.2.1.2.cmml" xref="S5.Ex15.m1.2.2.1.1"><ci id="S5.Ex15.m1.1.1.cmml" xref="S5.Ex15.m1.1.1">𝑢</ci><apply id="S5.Ex15.m1.2.2.1.1.1.cmml" xref="S5.Ex15.m1.2.2.1.1.1"><csymbol cd="ambiguous" id="S5.Ex15.m1.2.2.1.1.1.1.cmml" xref="S5.Ex15.m1.2.2.1.1.1">subscript</csymbol><ci id="S5.Ex15.m1.2.2.1.1.1.2.cmml" xref="S5.Ex15.m1.2.2.1.1.1.2">𝑥</ci><apply id="S5.Ex15.m1.2.2.1.1.1.3.cmml" xref="S5.Ex15.m1.2.2.1.1.1.3"><csymbol cd="ambiguous" id="S5.Ex15.m1.2.2.1.1.1.3.1.cmml" xref="S5.Ex15.m1.2.2.1.1.1.3">subscript</csymbol><ci id="S5.Ex15.m1.2.2.1.1.1.3.2.cmml" xref="S5.Ex15.m1.2.2.1.1.1.3.2">𝑗</ci><ci id="S5.Ex15.m1.2.2.1.1.1.3.3.cmml" xref="S5.Ex15.m1.2.2.1.1.1.3.3">𝑘</ci></apply></apply></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Ex15.m1.2c">\displaystyle d_{H}(u,x_{j_{k}})</annotation><annotation encoding="application/x-llamapun" id="S5.Ex15.m1.2d">italic_d start_POSTSUBSCRIPT italic_H end_POSTSUBSCRIPT ( italic_u , italic_x start_POSTSUBSCRIPT italic_j start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT end_POSTSUBSCRIPT )</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle\leq d_{H}(u,x_{i_{k}})+d_{H}(x_{i_{k}},x_{j_{k}})" class="ltx_Math" display="inline" id="S5.Ex15.m2.4"><semantics id="S5.Ex15.m2.4a"><mrow id="S5.Ex15.m2.4.4" xref="S5.Ex15.m2.4.4.cmml"><mi id="S5.Ex15.m2.4.4.5" xref="S5.Ex15.m2.4.4.5.cmml"></mi><mo id="S5.Ex15.m2.4.4.4" xref="S5.Ex15.m2.4.4.4.cmml">≤</mo><mrow id="S5.Ex15.m2.4.4.3" xref="S5.Ex15.m2.4.4.3.cmml"><mrow id="S5.Ex15.m2.2.2.1.1" xref="S5.Ex15.m2.2.2.1.1.cmml"><msub id="S5.Ex15.m2.2.2.1.1.3" xref="S5.Ex15.m2.2.2.1.1.3.cmml"><mi id="S5.Ex15.m2.2.2.1.1.3.2" xref="S5.Ex15.m2.2.2.1.1.3.2.cmml">d</mi><mi id="S5.Ex15.m2.2.2.1.1.3.3" xref="S5.Ex15.m2.2.2.1.1.3.3.cmml">H</mi></msub><mo id="S5.Ex15.m2.2.2.1.1.2" xref="S5.Ex15.m2.2.2.1.1.2.cmml">⁢</mo><mrow id="S5.Ex15.m2.2.2.1.1.1.1" xref="S5.Ex15.m2.2.2.1.1.1.2.cmml"><mo id="S5.Ex15.m2.2.2.1.1.1.1.2" stretchy="false" xref="S5.Ex15.m2.2.2.1.1.1.2.cmml">(</mo><mi id="S5.Ex15.m2.1.1" xref="S5.Ex15.m2.1.1.cmml">u</mi><mo id="S5.Ex15.m2.2.2.1.1.1.1.3" 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id="S5.Ex16.m1.2.2.1.1.1.1.1.3.3.cmml" xref="S5.Ex16.m1.2.2.1.1.1.1.1.3.3">𝑘</ci></apply></apply></interval></apply><cn id="S5.Ex16.m1.2.2.1.3.cmml" type="integer" xref="S5.Ex16.m1.2.2.1.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Ex16.m1.2c">\displaystyle=d_{H}(u,x_{i_{k}})+1</annotation><annotation encoding="application/x-llamapun" id="S5.Ex16.m1.2d">= italic_d start_POSTSUBSCRIPT italic_H end_POSTSUBSCRIPT ( italic_u , italic_x start_POSTSUBSCRIPT italic_i start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT end_POSTSUBSCRIPT ) + 1</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright" colspan="2"></td> </tr></tbody> <tbody id="S5.Ex17"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_eqn_cell"></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle\leq d_{H}(u,x_{j_{k-1}})+1" class="ltx_Math" 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encoding="application/x-tex" id="S5.Ex17.m1.2c">\displaystyle\leq d_{H}(u,x_{j_{k-1}})+1</annotation><annotation encoding="application/x-llamapun" id="S5.Ex17.m1.2d">≤ italic_d start_POSTSUBSCRIPT italic_H end_POSTSUBSCRIPT ( italic_u , italic_x start_POSTSUBSCRIPT italic_j start_POSTSUBSCRIPT italic_k - 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ) + 1</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle\rhd\text{by $i_{k}\leq j_{k-1}<|V|$ and monotonicity of $d_{H}(u% ,x_{*})$}" class="ltx_Math" display="inline" id="S5.Ex17.m2.2"><semantics id="S5.Ex17.m2.2a"><mrow id="S5.Ex17.m2.2.3" xref="S5.Ex17.m2.2.3.cmml"><mo id="S5.Ex17.m2.2.3a" rspace="0em" xref="S5.Ex17.m2.2.3.cmml">⊳</mo><mrow id="S5.Ex17.m2.2.2.2a" xref="S5.Ex17.m2.2.2.2ac.cmml"><mtext id="S5.Ex17.m2.2.2.2aa" xref="S5.Ex17.m2.2.2.2ac.cmml">by </mtext><mrow id="S5.Ex17.m2.1.1.1.m1.1.2" xref="S5.Ex17.m2.1.1.1.m1.1.2.cmml"><msub id="S5.Ex17.m2.1.1.1.m1.1.2.2" xref="S5.Ex17.m2.1.1.1.m1.1.2.2.cmml"><mi id="S5.Ex17.m2.1.1.1.m1.1.2.2.2" xref="S5.Ex17.m2.1.1.1.m1.1.2.2.2.cmml">i</mi><mi id="S5.Ex17.m2.1.1.1.m1.1.2.2.3" xref="S5.Ex17.m2.1.1.1.m1.1.2.2.3.cmml">k</mi></msub><mo id="S5.Ex17.m2.1.1.1.m1.1.2.3" xref="S5.Ex17.m2.1.1.1.m1.1.2.3.cmml">≤</mo><msub id="S5.Ex17.m2.1.1.1.m1.1.2.4" xref="S5.Ex17.m2.1.1.1.m1.1.2.4.cmml"><mi id="S5.Ex17.m2.1.1.1.m1.1.2.4.2" xref="S5.Ex17.m2.1.1.1.m1.1.2.4.2.cmml">j</mi><mrow id="S5.Ex17.m2.1.1.1.m1.1.2.4.3" xref="S5.Ex17.m2.1.1.1.m1.1.2.4.3.cmml"><mi id="S5.Ex17.m2.1.1.1.m1.1.2.4.3.2" xref="S5.Ex17.m2.1.1.1.m1.1.2.4.3.2.cmml">k</mi><mo id="S5.Ex17.m2.1.1.1.m1.1.2.4.3.1" xref="S5.Ex17.m2.1.1.1.m1.1.2.4.3.1.cmml">−</mo><mn id="S5.Ex17.m2.1.1.1.m1.1.2.4.3.3" xref="S5.Ex17.m2.1.1.1.m1.1.2.4.3.3.cmml">1</mn></mrow></msub><mo id="S5.Ex17.m2.1.1.1.m1.1.2.5" xref="S5.Ex17.m2.1.1.1.m1.1.2.5.cmml"><</mo><mrow id="S5.Ex17.m2.1.1.1.m1.1.2.6.2" xref="S5.Ex17.m2.1.1.1.m1.1.2.6.2.cmml"><mo id="S5.Ex17.m2.1.1.1.m1.1.2.6.2.1" stretchy="false" xref="S5.Ex17.m2.1.1.1.m1.1.2.6.2.cmml">|</mo><mi id="S5.Ex17.m2.1.1.1.m1.1.1" xref="S5.Ex17.m2.1.1.1.m1.1.1.cmml">V</mi><mo id="S5.Ex17.m2.1.1.1.m1.1.2.6.2.2" stretchy="false" xref="S5.Ex17.m2.1.1.1.m1.1.2.6.2.cmml">|</mo></mrow></mrow><mtext id="S5.Ex17.m2.2.2.2ab" xref="S5.Ex17.m2.2.2.2ac.cmml"> and monotonicity of </mtext><mrow id="S5.Ex17.m2.2.2.2.m2.2.2" xref="S5.Ex17.m2.2.2.2.m2.2.2.cmml"><msub id="S5.Ex17.m2.2.2.2.m2.2.2.3" xref="S5.Ex17.m2.2.2.2.m2.2.2.3.cmml"><mi id="S5.Ex17.m2.2.2.2.m2.2.2.3.2" xref="S5.Ex17.m2.2.2.2.m2.2.2.3.2.cmml">d</mi><mi id="S5.Ex17.m2.2.2.2.m2.2.2.3.3" xref="S5.Ex17.m2.2.2.2.m2.2.2.3.3.cmml">H</mi></msub><mo id="S5.Ex17.m2.2.2.2.m2.2.2.2" xref="S5.Ex17.m2.2.2.2.m2.2.2.2.cmml">⁢</mo><mrow id="S5.Ex17.m2.2.2.2.m2.2.2.1.1" xref="S5.Ex17.m2.2.2.2.m2.2.2.1.1.cmml"><mo id="S5.Ex17.m2.2.2.2.m2.2.2.1.1.2" stretchy="false" xref="S5.Ex17.m2.2.2.2.m2.2.2.1.1.cmml">(</mo><mi id="S5.Ex17.m2.2.2.2.m2.1.1" xref="S5.Ex17.m2.2.2.2.m2.1.1.cmml">u</mi><mo id="S5.Ex17.m2.2.2.2.m2.2.2.1.1.3" xref="S5.Ex17.m2.2.2.2.m2.2.2.1.1.cmml">,</mo><msub id="S5.Ex17.m2.2.2.2.m2.2.2.1.1.1" xref="S5.Ex17.m2.2.2.2.m2.2.2.1.1.1.cmml"><mi id="S5.Ex17.m2.2.2.2.m2.2.2.1.1.1.2" xref="S5.Ex17.m2.2.2.2.m2.2.2.1.1.1.2.cmml">x</mi><mo id="S5.Ex17.m2.2.2.2.m2.2.2.1.1.1.3" xref="S5.Ex17.m2.2.2.2.m2.2.2.1.1.1.3.cmml">∗</mo></msub><mo id="S5.Ex17.m2.2.2.2.m2.2.2.1.1.4" stretchy="false" xref="S5.Ex17.m2.2.2.2.m2.2.2.1.1.cmml">)</mo></mrow></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.Ex17.m2.2b"><apply id="S5.Ex17.m2.2.3.cmml" xref="S5.Ex17.m2.2.3"><csymbol cd="latexml" id="S5.Ex17.m2.2.3.1.cmml" xref="S5.Ex17.m2.2.3">contains-as-subgroup</csymbol><ci id="S5.Ex17.m2.2.2.2ac.cmml" xref="S5.Ex17.m2.2.2.2a"><mrow id="S5.Ex17.m2.2.2.2a.cmml" xref="S5.Ex17.m2.2.2.2a"><mtext id="S5.Ex17.m2.2.2.2aa.cmml" xref="S5.Ex17.m2.2.2.2a">by </mtext><mrow id="S5.Ex17.m2.1.1.1.m1.1.2.cmml" xref="S5.Ex17.m2.1.1.1.m1.1.2"><msub id="S5.Ex17.m2.1.1.1.m1.1.2.2.cmml" xref="S5.Ex17.m2.1.1.1.m1.1.2.2"><mi id="S5.Ex17.m2.1.1.1.m1.1.2.2.2.cmml" xref="S5.Ex17.m2.1.1.1.m1.1.2.2.2">i</mi><mi id="S5.Ex17.m2.1.1.1.m1.1.2.2.3.cmml" xref="S5.Ex17.m2.1.1.1.m1.1.2.2.3">k</mi></msub><mo id="S5.Ex17.m2.1.1.1.m1.1.2.3.cmml" xref="S5.Ex17.m2.1.1.1.m1.1.2.3">≤</mo><msub id="S5.Ex17.m2.1.1.1.m1.1.2.4.cmml" xref="S5.Ex17.m2.1.1.1.m1.1.2.4"><mi id="S5.Ex17.m2.1.1.1.m1.1.2.4.2.cmml" xref="S5.Ex17.m2.1.1.1.m1.1.2.4.2">j</mi><mrow id="S5.Ex17.m2.1.1.1.m1.1.2.4.3.cmml" xref="S5.Ex17.m2.1.1.1.m1.1.2.4.3"><mi id="S5.Ex17.m2.1.1.1.m1.1.2.4.3.2.cmml" xref="S5.Ex17.m2.1.1.1.m1.1.2.4.3.2">k</mi><mo id="S5.Ex17.m2.1.1.1.m1.1.2.4.3.1.cmml" xref="S5.Ex17.m2.1.1.1.m1.1.2.4.3.1">−</mo><mn id="S5.Ex17.m2.1.1.1.m1.1.2.4.3.3.cmml" xref="S5.Ex17.m2.1.1.1.m1.1.2.4.3.3">1</mn></mrow></msub><mo id="S5.Ex17.m2.1.1.1.m1.1.2.5.cmml" xref="S5.Ex17.m2.1.1.1.m1.1.2.5"><</mo><mrow id="S5.Ex17.m2.1.1.1.m1.1.2.6.2.cmml" xref="S5.Ex17.m2.1.1.1.m1.1.2.6.2"><mo id="S5.Ex17.m2.1.1.1.m1.1.2.6.2.1.cmml" stretchy="false" xref="S5.Ex17.m2.1.1.1.m1.1.2.6.2">|</mo><mi id="S5.Ex17.m2.1.1.1.m1.1.1.cmml" xref="S5.Ex17.m2.1.1.1.m1.1.1">V</mi><mo id="S5.Ex17.m2.1.1.1.m1.1.2.6.2.2.cmml" stretchy="false" xref="S5.Ex17.m2.1.1.1.m1.1.2.6.2">|</mo></mrow></mrow><mtext id="S5.Ex17.m2.2.2.2ab.cmml" xref="S5.Ex17.m2.2.2.2a"> and monotonicity of </mtext><mrow id="S5.Ex17.m2.2.2.2.m2.2.2.cmml" xref="S5.Ex17.m2.2.2.2.m2.2.2"><msub id="S5.Ex17.m2.2.2.2.m2.2.2.3.cmml" xref="S5.Ex17.m2.2.2.2.m2.2.2.3"><mi id="S5.Ex17.m2.2.2.2.m2.2.2.3.2.cmml" xref="S5.Ex17.m2.2.2.2.m2.2.2.3.2">d</mi><mi id="S5.Ex17.m2.2.2.2.m2.2.2.3.3.cmml" xref="S5.Ex17.m2.2.2.2.m2.2.2.3.3">H</mi></msub><mo id="S5.Ex17.m2.2.2.2.m2.2.2.2.cmml" xref="S5.Ex17.m2.2.2.2.m2.2.2.2">⁢</mo><mrow id="S5.Ex17.m2.2.2.2.m2.2.2.1.1.cmml" xref="S5.Ex17.m2.2.2.2.m2.2.2.1.1"><mo id="S5.Ex17.m2.2.2.2.m2.2.2.1.1.2.cmml" stretchy="false" xref="S5.Ex17.m2.2.2.2.m2.2.2.1.1">(</mo><mi id="S5.Ex17.m2.2.2.2.m2.1.1.cmml" xref="S5.Ex17.m2.2.2.2.m2.1.1">u</mi><mo id="S5.Ex17.m2.2.2.2.m2.2.2.1.1.3.cmml" xref="S5.Ex17.m2.2.2.2.m2.2.2.1.1">,</mo><msub id="S5.Ex17.m2.2.2.2.m2.2.2.1.1.1.cmml" xref="S5.Ex17.m2.2.2.2.m2.2.2.1.1.1"><mi id="S5.Ex17.m2.2.2.2.m2.2.2.1.1.1.2.cmml" xref="S5.Ex17.m2.2.2.2.m2.2.2.1.1.1.2">x</mi><mo id="S5.Ex17.m2.2.2.2.m2.2.2.1.1.1.3.cmml" xref="S5.Ex17.m2.2.2.2.m2.2.2.1.1.1.3">∗</mo></msub><mo id="S5.Ex17.m2.2.2.2.m2.2.2.1.1.4.cmml" stretchy="false" xref="S5.Ex17.m2.2.2.2.m2.2.2.1.1">)</mo></mrow></mrow></mrow></ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Ex17.m2.2c">\displaystyle\rhd\text{by $i_{k}\leq j_{k-1}<|V|$ and monotonicity of $d_{H}(u% ,x_{*})$}</annotation><annotation encoding="application/x-llamapun" id="S5.Ex17.m2.2d">⊳ by italic_i start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ≤ italic_j start_POSTSUBSCRIPT italic_k - 1 end_POSTSUBSCRIPT < | italic_V | and monotonicity of italic_d start_POSTSUBSCRIPT italic_H end_POSTSUBSCRIPT ( italic_u , italic_x start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT )</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> <tbody id="S5.Ex18"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_eqn_cell"></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle\leq k" class="ltx_Math" display="inline" id="S5.Ex18.m1.1"><semantics id="S5.Ex18.m1.1a"><mrow id="S5.Ex18.m1.1.1" xref="S5.Ex18.m1.1.1.cmml"><mi id="S5.Ex18.m1.1.1.2" xref="S5.Ex18.m1.1.1.2.cmml"></mi><mo id="S5.Ex18.m1.1.1.1" xref="S5.Ex18.m1.1.1.1.cmml">≤</mo><mi id="S5.Ex18.m1.1.1.3" xref="S5.Ex18.m1.1.1.3.cmml">k</mi></mrow><annotation-xml encoding="MathML-Content" id="S5.Ex18.m1.1b"><apply id="S5.Ex18.m1.1.1.cmml" xref="S5.Ex18.m1.1.1"><leq id="S5.Ex18.m1.1.1.1.cmml" xref="S5.Ex18.m1.1.1.1"></leq><csymbol cd="latexml" id="S5.Ex18.m1.1.1.2.cmml" xref="S5.Ex18.m1.1.1.2">absent</csymbol><ci id="S5.Ex18.m1.1.1.3.cmml" xref="S5.Ex18.m1.1.1.3">𝑘</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Ex18.m1.1c">\displaystyle\leq k</annotation><annotation encoding="application/x-llamapun" id="S5.Ex18.m1.1d">≤ italic_k</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle\rhd\text{by induction hypothesis}" class="ltx_Math" display="inline" id="S5.Ex18.m2.1"><semantics id="S5.Ex18.m2.1a"><mrow id="S5.Ex18.m2.1.1" xref="S5.Ex18.m2.1.1.cmml"><mo id="S5.Ex18.m2.1.1a" rspace="0em" xref="S5.Ex18.m2.1.1.cmml">⊳</mo><mtext id="S5.Ex18.m2.1.1.2" xref="S5.Ex18.m2.1.1.2a.cmml">by induction hypothesis</mtext></mrow><annotation-xml encoding="MathML-Content" id="S5.Ex18.m2.1b"><apply id="S5.Ex18.m2.1.1.cmml" xref="S5.Ex18.m2.1.1"><csymbol cd="latexml" id="S5.Ex18.m2.1.1.1.cmml" xref="S5.Ex18.m2.1.1">contains-as-subgroup</csymbol><ci id="S5.Ex18.m2.1.1.2a.cmml" xref="S5.Ex18.m2.1.1.2"><mtext id="S5.Ex18.m2.1.1.2.cmml" xref="S5.Ex18.m2.1.1.2">by induction hypothesis</mtext></ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Ex18.m2.1c">\displaystyle\rhd\text{by induction hypothesis}</annotation><annotation encoding="application/x-llamapun" id="S5.Ex18.m2.1d">⊳ by induction hypothesis</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S5.I2.i2.p3.12">Hence, this shows that <math alttext="d_{H}(u,v)=d_{H}(u,x_{j_{s}})\leq s" class="ltx_Math" display="inline" id="S5.I2.i2.p3.7.m1.4"><semantics id="S5.I2.i2.p3.7.m1.4a"><mrow id="S5.I2.i2.p3.7.m1.4.4" xref="S5.I2.i2.p3.7.m1.4.4.cmml"><mrow id="S5.I2.i2.p3.7.m1.4.4.3" xref="S5.I2.i2.p3.7.m1.4.4.3.cmml"><msub id="S5.I2.i2.p3.7.m1.4.4.3.2" xref="S5.I2.i2.p3.7.m1.4.4.3.2.cmml"><mi id="S5.I2.i2.p3.7.m1.4.4.3.2.2" xref="S5.I2.i2.p3.7.m1.4.4.3.2.2.cmml">d</mi><mi id="S5.I2.i2.p3.7.m1.4.4.3.2.3" 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xref="S5.I2.i2.p3.7.m1.3.3">𝑢</ci><apply id="S5.I2.i2.p3.7.m1.4.4.1.1.1.1.cmml" xref="S5.I2.i2.p3.7.m1.4.4.1.1.1.1"><csymbol cd="ambiguous" id="S5.I2.i2.p3.7.m1.4.4.1.1.1.1.1.cmml" xref="S5.I2.i2.p3.7.m1.4.4.1.1.1.1">subscript</csymbol><ci id="S5.I2.i2.p3.7.m1.4.4.1.1.1.1.2.cmml" xref="S5.I2.i2.p3.7.m1.4.4.1.1.1.1.2">𝑥</ci><apply id="S5.I2.i2.p3.7.m1.4.4.1.1.1.1.3.cmml" xref="S5.I2.i2.p3.7.m1.4.4.1.1.1.1.3"><csymbol cd="ambiguous" id="S5.I2.i2.p3.7.m1.4.4.1.1.1.1.3.1.cmml" xref="S5.I2.i2.p3.7.m1.4.4.1.1.1.1.3">subscript</csymbol><ci id="S5.I2.i2.p3.7.m1.4.4.1.1.1.1.3.2.cmml" xref="S5.I2.i2.p3.7.m1.4.4.1.1.1.1.3.2">𝑗</ci><ci id="S5.I2.i2.p3.7.m1.4.4.1.1.1.1.3.3.cmml" xref="S5.I2.i2.p3.7.m1.4.4.1.1.1.1.3.3">𝑠</ci></apply></apply></interval></apply></apply><apply id="S5.I2.i2.p3.7.m1.4.4c.cmml" xref="S5.I2.i2.p3.7.m1.4.4"><leq id="S5.I2.i2.p3.7.m1.4.4.5.cmml" xref="S5.I2.i2.p3.7.m1.4.4.5"></leq><share href="https://arxiv.org/html/2503.00712v1#S5.I2.i2.p3.7.m1.4.4.1.cmml" id="S5.I2.i2.p3.7.m1.4.4d.cmml" xref="S5.I2.i2.p3.7.m1.4.4"></share><ci id="S5.I2.i2.p3.7.m1.4.4.6.cmml" xref="S5.I2.i2.p3.7.m1.4.4.6">𝑠</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.I2.i2.p3.7.m1.4c">d_{H}(u,v)=d_{H}(u,x_{j_{s}})\leq s</annotation><annotation encoding="application/x-llamapun" id="S5.I2.i2.p3.7.m1.4d">italic_d start_POSTSUBSCRIPT italic_H end_POSTSUBSCRIPT ( italic_u , italic_v ) = italic_d start_POSTSUBSCRIPT italic_H end_POSTSUBSCRIPT ( italic_u , italic_x start_POSTSUBSCRIPT italic_j start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT end_POSTSUBSCRIPT ) ≤ italic_s</annotation></semantics></math>. So <math alttext="G=H\cup\{(u,v)\}" class="ltx_Math" display="inline" id="S5.I2.i2.p3.8.m2.3"><semantics id="S5.I2.i2.p3.8.m2.3a"><mrow id="S5.I2.i2.p3.8.m2.3.3" xref="S5.I2.i2.p3.8.m2.3.3.cmml"><mi id="S5.I2.i2.p3.8.m2.3.3.3" xref="S5.I2.i2.p3.8.m2.3.3.3.cmml">G</mi><mo id="S5.I2.i2.p3.8.m2.3.3.2" xref="S5.I2.i2.p3.8.m2.3.3.2.cmml">=</mo><mrow id="S5.I2.i2.p3.8.m2.3.3.1" xref="S5.I2.i2.p3.8.m2.3.3.1.cmml"><mi id="S5.I2.i2.p3.8.m2.3.3.1.3" xref="S5.I2.i2.p3.8.m2.3.3.1.3.cmml">H</mi><mo id="S5.I2.i2.p3.8.m2.3.3.1.2" xref="S5.I2.i2.p3.8.m2.3.3.1.2.cmml">∪</mo><mrow id="S5.I2.i2.p3.8.m2.3.3.1.1.1" xref="S5.I2.i2.p3.8.m2.3.3.1.1.2.cmml"><mo id="S5.I2.i2.p3.8.m2.3.3.1.1.1.2" stretchy="false" xref="S5.I2.i2.p3.8.m2.3.3.1.1.2.cmml">{</mo><mrow id="S5.I2.i2.p3.8.m2.3.3.1.1.1.1.2" xref="S5.I2.i2.p3.8.m2.3.3.1.1.1.1.1.cmml"><mo id="S5.I2.i2.p3.8.m2.3.3.1.1.1.1.2.1" stretchy="false" xref="S5.I2.i2.p3.8.m2.3.3.1.1.1.1.1.cmml">(</mo><mi id="S5.I2.i2.p3.8.m2.1.1" xref="S5.I2.i2.p3.8.m2.1.1.cmml">u</mi><mo id="S5.I2.i2.p3.8.m2.3.3.1.1.1.1.2.2" xref="S5.I2.i2.p3.8.m2.3.3.1.1.1.1.1.cmml">,</mo><mi id="S5.I2.i2.p3.8.m2.2.2" xref="S5.I2.i2.p3.8.m2.2.2.cmml">v</mi><mo id="S5.I2.i2.p3.8.m2.3.3.1.1.1.1.2.3" stretchy="false" xref="S5.I2.i2.p3.8.m2.3.3.1.1.1.1.1.cmml">)</mo></mrow><mo id="S5.I2.i2.p3.8.m2.3.3.1.1.1.3" stretchy="false" xref="S5.I2.i2.p3.8.m2.3.3.1.1.2.cmml">}</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.I2.i2.p3.8.m2.3b"><apply id="S5.I2.i2.p3.8.m2.3.3.cmml" xref="S5.I2.i2.p3.8.m2.3.3"><eq id="S5.I2.i2.p3.8.m2.3.3.2.cmml" xref="S5.I2.i2.p3.8.m2.3.3.2"></eq><ci id="S5.I2.i2.p3.8.m2.3.3.3.cmml" xref="S5.I2.i2.p3.8.m2.3.3.3">𝐺</ci><apply id="S5.I2.i2.p3.8.m2.3.3.1.cmml" xref="S5.I2.i2.p3.8.m2.3.3.1"><union id="S5.I2.i2.p3.8.m2.3.3.1.2.cmml" xref="S5.I2.i2.p3.8.m2.3.3.1.2"></union><ci id="S5.I2.i2.p3.8.m2.3.3.1.3.cmml" xref="S5.I2.i2.p3.8.m2.3.3.1.3">𝐻</ci><set id="S5.I2.i2.p3.8.m2.3.3.1.1.2.cmml" xref="S5.I2.i2.p3.8.m2.3.3.1.1.1"><interval closure="open" id="S5.I2.i2.p3.8.m2.3.3.1.1.1.1.1.cmml" xref="S5.I2.i2.p3.8.m2.3.3.1.1.1.1.2"><ci id="S5.I2.i2.p3.8.m2.1.1.cmml" xref="S5.I2.i2.p3.8.m2.1.1">𝑢</ci><ci id="S5.I2.i2.p3.8.m2.2.2.cmml" xref="S5.I2.i2.p3.8.m2.2.2">𝑣</ci></interval></set></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.I2.i2.p3.8.m2.3c">G=H\cup\{(u,v)\}</annotation><annotation encoding="application/x-llamapun" id="S5.I2.i2.p3.8.m2.3d">italic_G = italic_H ∪ { ( italic_u , italic_v ) }</annotation></semantics></math> contains a cycle of length at most <math alttext="s+1" class="ltx_Math" display="inline" id="S5.I2.i2.p3.9.m3.1"><semantics id="S5.I2.i2.p3.9.m3.1a"><mrow id="S5.I2.i2.p3.9.m3.1.1" xref="S5.I2.i2.p3.9.m3.1.1.cmml"><mi id="S5.I2.i2.p3.9.m3.1.1.2" xref="S5.I2.i2.p3.9.m3.1.1.2.cmml">s</mi><mo id="S5.I2.i2.p3.9.m3.1.1.1" xref="S5.I2.i2.p3.9.m3.1.1.1.cmml">+</mo><mn id="S5.I2.i2.p3.9.m3.1.1.3" xref="S5.I2.i2.p3.9.m3.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S5.I2.i2.p3.9.m3.1b"><apply id="S5.I2.i2.p3.9.m3.1.1.cmml" xref="S5.I2.i2.p3.9.m3.1.1"><plus id="S5.I2.i2.p3.9.m3.1.1.1.cmml" xref="S5.I2.i2.p3.9.m3.1.1.1"></plus><ci id="S5.I2.i2.p3.9.m3.1.1.2.cmml" xref="S5.I2.i2.p3.9.m3.1.1.2">𝑠</ci><cn id="S5.I2.i2.p3.9.m3.1.1.3.cmml" type="integer" xref="S5.I2.i2.p3.9.m3.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.I2.i2.p3.9.m3.1c">s+1</annotation><annotation encoding="application/x-llamapun" id="S5.I2.i2.p3.9.m3.1d">italic_s + 1</annotation></semantics></math>. Since <math alttext="G" class="ltx_Math" display="inline" id="S5.I2.i2.p3.10.m4.1"><semantics id="S5.I2.i2.p3.10.m4.1a"><mi id="S5.I2.i2.p3.10.m4.1.1" xref="S5.I2.i2.p3.10.m4.1.1.cmml">G</mi><annotation-xml encoding="MathML-Content" id="S5.I2.i2.p3.10.m4.1b"><ci id="S5.I2.i2.p3.10.m4.1.1.cmml" xref="S5.I2.i2.p3.10.m4.1.1">𝐺</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.I2.i2.p3.10.m4.1c">G</annotation><annotation encoding="application/x-llamapun" id="S5.I2.i2.p3.10.m4.1d">italic_G</annotation></semantics></math> has girth at least <math alttext="2t+2" class="ltx_Math" display="inline" id="S5.I2.i2.p3.11.m5.1"><semantics id="S5.I2.i2.p3.11.m5.1a"><mrow id="S5.I2.i2.p3.11.m5.1.1" xref="S5.I2.i2.p3.11.m5.1.1.cmml"><mrow id="S5.I2.i2.p3.11.m5.1.1.2" xref="S5.I2.i2.p3.11.m5.1.1.2.cmml"><mn id="S5.I2.i2.p3.11.m5.1.1.2.2" xref="S5.I2.i2.p3.11.m5.1.1.2.2.cmml">2</mn><mo id="S5.I2.i2.p3.11.m5.1.1.2.1" xref="S5.I2.i2.p3.11.m5.1.1.2.1.cmml">⁢</mo><mi id="S5.I2.i2.p3.11.m5.1.1.2.3" xref="S5.I2.i2.p3.11.m5.1.1.2.3.cmml">t</mi></mrow><mo id="S5.I2.i2.p3.11.m5.1.1.1" xref="S5.I2.i2.p3.11.m5.1.1.1.cmml">+</mo><mn id="S5.I2.i2.p3.11.m5.1.1.3" xref="S5.I2.i2.p3.11.m5.1.1.3.cmml">2</mn></mrow><annotation-xml encoding="MathML-Content" id="S5.I2.i2.p3.11.m5.1b"><apply id="S5.I2.i2.p3.11.m5.1.1.cmml" xref="S5.I2.i2.p3.11.m5.1.1"><plus id="S5.I2.i2.p3.11.m5.1.1.1.cmml" xref="S5.I2.i2.p3.11.m5.1.1.1"></plus><apply id="S5.I2.i2.p3.11.m5.1.1.2.cmml" xref="S5.I2.i2.p3.11.m5.1.1.2"><times id="S5.I2.i2.p3.11.m5.1.1.2.1.cmml" xref="S5.I2.i2.p3.11.m5.1.1.2.1"></times><cn id="S5.I2.i2.p3.11.m5.1.1.2.2.cmml" type="integer" xref="S5.I2.i2.p3.11.m5.1.1.2.2">2</cn><ci id="S5.I2.i2.p3.11.m5.1.1.2.3.cmml" xref="S5.I2.i2.p3.11.m5.1.1.2.3">𝑡</ci></apply><cn id="S5.I2.i2.p3.11.m5.1.1.3.cmml" type="integer" xref="S5.I2.i2.p3.11.m5.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.I2.i2.p3.11.m5.1c">2t+2</annotation><annotation encoding="application/x-llamapun" id="S5.I2.i2.p3.11.m5.1d">2 italic_t + 2</annotation></semantics></math>, we conclude <math alttext="s\geq 2t+1" class="ltx_Math" display="inline" id="S5.I2.i2.p3.12.m6.1"><semantics id="S5.I2.i2.p3.12.m6.1a"><mrow id="S5.I2.i2.p3.12.m6.1.1" xref="S5.I2.i2.p3.12.m6.1.1.cmml"><mi id="S5.I2.i2.p3.12.m6.1.1.2" xref="S5.I2.i2.p3.12.m6.1.1.2.cmml">s</mi><mo id="S5.I2.i2.p3.12.m6.1.1.1" xref="S5.I2.i2.p3.12.m6.1.1.1.cmml">≥</mo><mrow id="S5.I2.i2.p3.12.m6.1.1.3" xref="S5.I2.i2.p3.12.m6.1.1.3.cmml"><mrow id="S5.I2.i2.p3.12.m6.1.1.3.2" xref="S5.I2.i2.p3.12.m6.1.1.3.2.cmml"><mn id="S5.I2.i2.p3.12.m6.1.1.3.2.2" xref="S5.I2.i2.p3.12.m6.1.1.3.2.2.cmml">2</mn><mo id="S5.I2.i2.p3.12.m6.1.1.3.2.1" xref="S5.I2.i2.p3.12.m6.1.1.3.2.1.cmml">⁢</mo><mi id="S5.I2.i2.p3.12.m6.1.1.3.2.3" xref="S5.I2.i2.p3.12.m6.1.1.3.2.3.cmml">t</mi></mrow><mo id="S5.I2.i2.p3.12.m6.1.1.3.1" xref="S5.I2.i2.p3.12.m6.1.1.3.1.cmml">+</mo><mn id="S5.I2.i2.p3.12.m6.1.1.3.3" xref="S5.I2.i2.p3.12.m6.1.1.3.3.cmml">1</mn></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.I2.i2.p3.12.m6.1b"><apply id="S5.I2.i2.p3.12.m6.1.1.cmml" xref="S5.I2.i2.p3.12.m6.1.1"><geq id="S5.I2.i2.p3.12.m6.1.1.1.cmml" xref="S5.I2.i2.p3.12.m6.1.1.1"></geq><ci id="S5.I2.i2.p3.12.m6.1.1.2.cmml" xref="S5.I2.i2.p3.12.m6.1.1.2">𝑠</ci><apply id="S5.I2.i2.p3.12.m6.1.1.3.cmml" xref="S5.I2.i2.p3.12.m6.1.1.3"><plus id="S5.I2.i2.p3.12.m6.1.1.3.1.cmml" xref="S5.I2.i2.p3.12.m6.1.1.3.1"></plus><apply id="S5.I2.i2.p3.12.m6.1.1.3.2.cmml" xref="S5.I2.i2.p3.12.m6.1.1.3.2"><times id="S5.I2.i2.p3.12.m6.1.1.3.2.1.cmml" xref="S5.I2.i2.p3.12.m6.1.1.3.2.1"></times><cn id="S5.I2.i2.p3.12.m6.1.1.3.2.2.cmml" type="integer" xref="S5.I2.i2.p3.12.m6.1.1.3.2.2">2</cn><ci id="S5.I2.i2.p3.12.m6.1.1.3.2.3.cmml" xref="S5.I2.i2.p3.12.m6.1.1.3.2.3">𝑡</ci></apply><cn id="S5.I2.i2.p3.12.m6.1.1.3.3.cmml" type="integer" xref="S5.I2.i2.p3.12.m6.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.I2.i2.p3.12.m6.1c">s\geq 2t+1</annotation><annotation encoding="application/x-llamapun" id="S5.I2.i2.p3.12.m6.1d">italic_s ≥ 2 italic_t + 1</annotation></semantics></math>.</p> </div> </li> </ul> <p class="ltx_p" id="S5.3.p3.1">∎</p> </div> </div> <div class="ltx_para" id="S5.p2"> <p class="ltx_p" id="S5.p2.1">Theorem <a class="ltx_ref" href="https://arxiv.org/html/2503.00712v1#S5.Thmtheorem1" title="Theorem 5.1. ‣ 5 Lower Bounds for Streaming Network Design ‣ Streaming Algorithms for Network Design"><span class="ltx_text ltx_ref_tag">5.1</span></a> immediately implies the following for <math alttext="k" class="ltx_Math" display="inline" id="S5.p2.1.m1.1"><semantics id="S5.p2.1.m1.1a"><mi id="S5.p2.1.m1.1.1" xref="S5.p2.1.m1.1.1.cmml">k</mi><annotation-xml encoding="MathML-Content" id="S5.p2.1.m1.1b"><ci id="S5.p2.1.m1.1.1.cmml" xref="S5.p2.1.m1.1.1">𝑘</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.p2.1.m1.1c">k</annotation><annotation encoding="application/x-llamapun" id="S5.p2.1.m1.1d">italic_k</annotation></semantics></math>-VC-CAP.</p> </div> <div class="ltx_theorem ltx_theorem_corollary" id="S5.Thmtheorem2"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="S5.Thmtheorem2.1.1.1">Corollary 5.2</span></span><span class="ltx_text ltx_font_bold" id="S5.Thmtheorem2.2.2">.</span> </h6> <div class="ltx_para" id="S5.Thmtheorem2.p1"> <p class="ltx_p" id="S5.Thmtheorem2.p1.3">Any algorithm approximating <math alttext="k" class="ltx_Math" display="inline" id="S5.Thmtheorem2.p1.1.m1.1"><semantics id="S5.Thmtheorem2.p1.1.m1.1a"><mi id="S5.Thmtheorem2.p1.1.m1.1.1" xref="S5.Thmtheorem2.p1.1.m1.1.1.cmml">k</mi><annotation-xml encoding="MathML-Content" id="S5.Thmtheorem2.p1.1.m1.1b"><ci id="S5.Thmtheorem2.p1.1.m1.1.1.cmml" xref="S5.Thmtheorem2.p1.1.m1.1.1">𝑘</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmtheorem2.p1.1.m1.1c">k</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem2.p1.1.m1.1d">italic_k</annotation></semantics></math>-VC-CAP with a factor better than <math alttext="2t+1" class="ltx_Math" display="inline" id="S5.Thmtheorem2.p1.2.m2.1"><semantics id="S5.Thmtheorem2.p1.2.m2.1a"><mrow id="S5.Thmtheorem2.p1.2.m2.1.1" xref="S5.Thmtheorem2.p1.2.m2.1.1.cmml"><mrow id="S5.Thmtheorem2.p1.2.m2.1.1.2" xref="S5.Thmtheorem2.p1.2.m2.1.1.2.cmml"><mn id="S5.Thmtheorem2.p1.2.m2.1.1.2.2" xref="S5.Thmtheorem2.p1.2.m2.1.1.2.2.cmml">2</mn><mo id="S5.Thmtheorem2.p1.2.m2.1.1.2.1" xref="S5.Thmtheorem2.p1.2.m2.1.1.2.1.cmml">⁢</mo><mi id="S5.Thmtheorem2.p1.2.m2.1.1.2.3" xref="S5.Thmtheorem2.p1.2.m2.1.1.2.3.cmml">t</mi></mrow><mo id="S5.Thmtheorem2.p1.2.m2.1.1.1" xref="S5.Thmtheorem2.p1.2.m2.1.1.1.cmml">+</mo><mn id="S5.Thmtheorem2.p1.2.m2.1.1.3" xref="S5.Thmtheorem2.p1.2.m2.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S5.Thmtheorem2.p1.2.m2.1b"><apply id="S5.Thmtheorem2.p1.2.m2.1.1.cmml" xref="S5.Thmtheorem2.p1.2.m2.1.1"><plus id="S5.Thmtheorem2.p1.2.m2.1.1.1.cmml" xref="S5.Thmtheorem2.p1.2.m2.1.1.1"></plus><apply id="S5.Thmtheorem2.p1.2.m2.1.1.2.cmml" xref="S5.Thmtheorem2.p1.2.m2.1.1.2"><times id="S5.Thmtheorem2.p1.2.m2.1.1.2.1.cmml" xref="S5.Thmtheorem2.p1.2.m2.1.1.2.1"></times><cn id="S5.Thmtheorem2.p1.2.m2.1.1.2.2.cmml" type="integer" xref="S5.Thmtheorem2.p1.2.m2.1.1.2.2">2</cn><ci id="S5.Thmtheorem2.p1.2.m2.1.1.2.3.cmml" xref="S5.Thmtheorem2.p1.2.m2.1.1.2.3">𝑡</ci></apply><cn id="S5.Thmtheorem2.p1.2.m2.1.1.3.cmml" type="integer" xref="S5.Thmtheorem2.p1.2.m2.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmtheorem2.p1.2.m2.1c">2t+1</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem2.p1.2.m2.1d">2 italic_t + 1</annotation></semantics></math> in fully streaming requires <math alttext="\Omega(n^{1+1/t})" class="ltx_Math" display="inline" id="S5.Thmtheorem2.p1.3.m3.1"><semantics id="S5.Thmtheorem2.p1.3.m3.1a"><mrow id="S5.Thmtheorem2.p1.3.m3.1.1" xref="S5.Thmtheorem2.p1.3.m3.1.1.cmml"><mi id="S5.Thmtheorem2.p1.3.m3.1.1.3" mathvariant="normal" xref="S5.Thmtheorem2.p1.3.m3.1.1.3.cmml">Ω</mi><mo id="S5.Thmtheorem2.p1.3.m3.1.1.2" xref="S5.Thmtheorem2.p1.3.m3.1.1.2.cmml">⁢</mo><mrow id="S5.Thmtheorem2.p1.3.m3.1.1.1.1" xref="S5.Thmtheorem2.p1.3.m3.1.1.1.1.1.cmml"><mo id="S5.Thmtheorem2.p1.3.m3.1.1.1.1.2" stretchy="false" xref="S5.Thmtheorem2.p1.3.m3.1.1.1.1.1.cmml">(</mo><msup id="S5.Thmtheorem2.p1.3.m3.1.1.1.1.1" xref="S5.Thmtheorem2.p1.3.m3.1.1.1.1.1.cmml"><mi id="S5.Thmtheorem2.p1.3.m3.1.1.1.1.1.2" xref="S5.Thmtheorem2.p1.3.m3.1.1.1.1.1.2.cmml">n</mi><mrow id="S5.Thmtheorem2.p1.3.m3.1.1.1.1.1.3" xref="S5.Thmtheorem2.p1.3.m3.1.1.1.1.1.3.cmml"><mn id="S5.Thmtheorem2.p1.3.m3.1.1.1.1.1.3.2" xref="S5.Thmtheorem2.p1.3.m3.1.1.1.1.1.3.2.cmml">1</mn><mo id="S5.Thmtheorem2.p1.3.m3.1.1.1.1.1.3.1" xref="S5.Thmtheorem2.p1.3.m3.1.1.1.1.1.3.1.cmml">+</mo><mrow id="S5.Thmtheorem2.p1.3.m3.1.1.1.1.1.3.3" xref="S5.Thmtheorem2.p1.3.m3.1.1.1.1.1.3.3.cmml"><mn id="S5.Thmtheorem2.p1.3.m3.1.1.1.1.1.3.3.2" xref="S5.Thmtheorem2.p1.3.m3.1.1.1.1.1.3.3.2.cmml">1</mn><mo id="S5.Thmtheorem2.p1.3.m3.1.1.1.1.1.3.3.1" xref="S5.Thmtheorem2.p1.3.m3.1.1.1.1.1.3.3.1.cmml">/</mo><mi id="S5.Thmtheorem2.p1.3.m3.1.1.1.1.1.3.3.3" xref="S5.Thmtheorem2.p1.3.m3.1.1.1.1.1.3.3.3.cmml">t</mi></mrow></mrow></msup><mo id="S5.Thmtheorem2.p1.3.m3.1.1.1.1.3" stretchy="false" xref="S5.Thmtheorem2.p1.3.m3.1.1.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.Thmtheorem2.p1.3.m3.1b"><apply id="S5.Thmtheorem2.p1.3.m3.1.1.cmml" xref="S5.Thmtheorem2.p1.3.m3.1.1"><times id="S5.Thmtheorem2.p1.3.m3.1.1.2.cmml" xref="S5.Thmtheorem2.p1.3.m3.1.1.2"></times><ci id="S5.Thmtheorem2.p1.3.m3.1.1.3.cmml" xref="S5.Thmtheorem2.p1.3.m3.1.1.3">Ω</ci><apply id="S5.Thmtheorem2.p1.3.m3.1.1.1.1.1.cmml" xref="S5.Thmtheorem2.p1.3.m3.1.1.1.1"><csymbol cd="ambiguous" id="S5.Thmtheorem2.p1.3.m3.1.1.1.1.1.1.cmml" xref="S5.Thmtheorem2.p1.3.m3.1.1.1.1">superscript</csymbol><ci id="S5.Thmtheorem2.p1.3.m3.1.1.1.1.1.2.cmml" xref="S5.Thmtheorem2.p1.3.m3.1.1.1.1.1.2">𝑛</ci><apply id="S5.Thmtheorem2.p1.3.m3.1.1.1.1.1.3.cmml" xref="S5.Thmtheorem2.p1.3.m3.1.1.1.1.1.3"><plus id="S5.Thmtheorem2.p1.3.m3.1.1.1.1.1.3.1.cmml" xref="S5.Thmtheorem2.p1.3.m3.1.1.1.1.1.3.1"></plus><cn id="S5.Thmtheorem2.p1.3.m3.1.1.1.1.1.3.2.cmml" type="integer" xref="S5.Thmtheorem2.p1.3.m3.1.1.1.1.1.3.2">1</cn><apply id="S5.Thmtheorem2.p1.3.m3.1.1.1.1.1.3.3.cmml" xref="S5.Thmtheorem2.p1.3.m3.1.1.1.1.1.3.3"><divide id="S5.Thmtheorem2.p1.3.m3.1.1.1.1.1.3.3.1.cmml" xref="S5.Thmtheorem2.p1.3.m3.1.1.1.1.1.3.3.1"></divide><cn id="S5.Thmtheorem2.p1.3.m3.1.1.1.1.1.3.3.2.cmml" type="integer" xref="S5.Thmtheorem2.p1.3.m3.1.1.1.1.1.3.3.2">1</cn><ci id="S5.Thmtheorem2.p1.3.m3.1.1.1.1.1.3.3.3.cmml" xref="S5.Thmtheorem2.p1.3.m3.1.1.1.1.1.3.3.3">𝑡</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmtheorem2.p1.3.m3.1c">\Omega(n^{1+1/t})</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem2.p1.3.m3.1d">roman_Ω ( italic_n start_POSTSUPERSCRIPT 1 + 1 / italic_t end_POSTSUPERSCRIPT )</annotation></semantics></math> space.</p> </div> </div> <div class="ltx_para" id="S5.p3"> <p class="ltx_p" id="S5.p3.2">Moreover, by Theorem <a class="ltx_ref" href="https://arxiv.org/html/2503.00712v1#S5.Thmtheorem1" title="Theorem 5.1. ‣ 5 Lower Bounds for Streaming Network Design ‣ Streaming Algorithms for Network Design"><span class="ltx_text ltx_ref_tag">5.1</span></a> and the fact that any feasible solution for <math alttext="k" class="ltx_Math" display="inline" id="S5.p3.1.m1.1"><semantics id="S5.p3.1.m1.1a"><mi id="S5.p3.1.m1.1.1" xref="S5.p3.1.m1.1.1.cmml">k</mi><annotation-xml encoding="MathML-Content" id="S5.p3.1.m1.1b"><ci id="S5.p3.1.m1.1.1.cmml" xref="S5.p3.1.m1.1.1">𝑘</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.p3.1.m1.1c">k</annotation><annotation encoding="application/x-llamapun" id="S5.p3.1.m1.1d">italic_k</annotation></semantics></math>-VCSS and VC-SNDP (in general) has size <math alttext="\Omega(nk)" class="ltx_Math" display="inline" id="S5.p3.2.m2.1"><semantics id="S5.p3.2.m2.1a"><mrow id="S5.p3.2.m2.1.1" xref="S5.p3.2.m2.1.1.cmml"><mi id="S5.p3.2.m2.1.1.3" mathvariant="normal" xref="S5.p3.2.m2.1.1.3.cmml">Ω</mi><mo id="S5.p3.2.m2.1.1.2" xref="S5.p3.2.m2.1.1.2.cmml">⁢</mo><mrow id="S5.p3.2.m2.1.1.1.1" xref="S5.p3.2.m2.1.1.1.1.1.cmml"><mo id="S5.p3.2.m2.1.1.1.1.2" stretchy="false" xref="S5.p3.2.m2.1.1.1.1.1.cmml">(</mo><mrow id="S5.p3.2.m2.1.1.1.1.1" xref="S5.p3.2.m2.1.1.1.1.1.cmml"><mi id="S5.p3.2.m2.1.1.1.1.1.2" xref="S5.p3.2.m2.1.1.1.1.1.2.cmml">n</mi><mo id="S5.p3.2.m2.1.1.1.1.1.1" xref="S5.p3.2.m2.1.1.1.1.1.1.cmml">⁢</mo><mi id="S5.p3.2.m2.1.1.1.1.1.3" xref="S5.p3.2.m2.1.1.1.1.1.3.cmml">k</mi></mrow><mo id="S5.p3.2.m2.1.1.1.1.3" stretchy="false" xref="S5.p3.2.m2.1.1.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.p3.2.m2.1b"><apply id="S5.p3.2.m2.1.1.cmml" xref="S5.p3.2.m2.1.1"><times id="S5.p3.2.m2.1.1.2.cmml" xref="S5.p3.2.m2.1.1.2"></times><ci id="S5.p3.2.m2.1.1.3.cmml" xref="S5.p3.2.m2.1.1.3">Ω</ci><apply id="S5.p3.2.m2.1.1.1.1.1.cmml" xref="S5.p3.2.m2.1.1.1.1"><times id="S5.p3.2.m2.1.1.1.1.1.1.cmml" xref="S5.p3.2.m2.1.1.1.1.1.1"></times><ci id="S5.p3.2.m2.1.1.1.1.1.2.cmml" xref="S5.p3.2.m2.1.1.1.1.1.2">𝑛</ci><ci id="S5.p3.2.m2.1.1.1.1.1.3.cmml" xref="S5.p3.2.m2.1.1.1.1.1.3">𝑘</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.p3.2.m2.1c">\Omega(nk)</annotation><annotation encoding="application/x-llamapun" id="S5.p3.2.m2.1d">roman_Ω ( italic_n italic_k )</annotation></semantics></math>, the following corollaries hold.</p> </div> <div class="ltx_theorem ltx_theorem_corollary" id="S5.Thmtheorem3"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="S5.Thmtheorem3.1.1.1">Corollary 5.3</span></span><span class="ltx_text ltx_font_bold" id="S5.Thmtheorem3.2.2">.</span> </h6> <div class="ltx_para" id="S5.Thmtheorem3.p1"> <p class="ltx_p" id="S5.Thmtheorem3.p1.3">Any algorithm approximating <math alttext="k" class="ltx_Math" display="inline" id="S5.Thmtheorem3.p1.1.m1.1"><semantics id="S5.Thmtheorem3.p1.1.m1.1a"><mi id="S5.Thmtheorem3.p1.1.m1.1.1" xref="S5.Thmtheorem3.p1.1.m1.1.1.cmml">k</mi><annotation-xml encoding="MathML-Content" id="S5.Thmtheorem3.p1.1.m1.1b"><ci id="S5.Thmtheorem3.p1.1.m1.1.1.cmml" xref="S5.Thmtheorem3.p1.1.m1.1.1">𝑘</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmtheorem3.p1.1.m1.1c">k</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem3.p1.1.m1.1d">italic_k</annotation></semantics></math>-VCSS with a factor better than <math alttext="2t+1" class="ltx_Math" display="inline" id="S5.Thmtheorem3.p1.2.m2.1"><semantics id="S5.Thmtheorem3.p1.2.m2.1a"><mrow id="S5.Thmtheorem3.p1.2.m2.1.1" xref="S5.Thmtheorem3.p1.2.m2.1.1.cmml"><mrow id="S5.Thmtheorem3.p1.2.m2.1.1.2" xref="S5.Thmtheorem3.p1.2.m2.1.1.2.cmml"><mn id="S5.Thmtheorem3.p1.2.m2.1.1.2.2" xref="S5.Thmtheorem3.p1.2.m2.1.1.2.2.cmml">2</mn><mo id="S5.Thmtheorem3.p1.2.m2.1.1.2.1" xref="S5.Thmtheorem3.p1.2.m2.1.1.2.1.cmml">⁢</mo><mi id="S5.Thmtheorem3.p1.2.m2.1.1.2.3" xref="S5.Thmtheorem3.p1.2.m2.1.1.2.3.cmml">t</mi></mrow><mo id="S5.Thmtheorem3.p1.2.m2.1.1.1" xref="S5.Thmtheorem3.p1.2.m2.1.1.1.cmml">+</mo><mn id="S5.Thmtheorem3.p1.2.m2.1.1.3" xref="S5.Thmtheorem3.p1.2.m2.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S5.Thmtheorem3.p1.2.m2.1b"><apply id="S5.Thmtheorem3.p1.2.m2.1.1.cmml" xref="S5.Thmtheorem3.p1.2.m2.1.1"><plus id="S5.Thmtheorem3.p1.2.m2.1.1.1.cmml" xref="S5.Thmtheorem3.p1.2.m2.1.1.1"></plus><apply id="S5.Thmtheorem3.p1.2.m2.1.1.2.cmml" xref="S5.Thmtheorem3.p1.2.m2.1.1.2"><times id="S5.Thmtheorem3.p1.2.m2.1.1.2.1.cmml" xref="S5.Thmtheorem3.p1.2.m2.1.1.2.1"></times><cn id="S5.Thmtheorem3.p1.2.m2.1.1.2.2.cmml" type="integer" xref="S5.Thmtheorem3.p1.2.m2.1.1.2.2">2</cn><ci id="S5.Thmtheorem3.p1.2.m2.1.1.2.3.cmml" xref="S5.Thmtheorem3.p1.2.m2.1.1.2.3">𝑡</ci></apply><cn id="S5.Thmtheorem3.p1.2.m2.1.1.3.cmml" type="integer" xref="S5.Thmtheorem3.p1.2.m2.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmtheorem3.p1.2.m2.1c">2t+1</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem3.p1.2.m2.1d">2 italic_t + 1</annotation></semantics></math>, in insertion-only streams, requires <math alttext="\Omega(nk+n^{1+1/t})" class="ltx_Math" display="inline" id="S5.Thmtheorem3.p1.3.m3.1"><semantics id="S5.Thmtheorem3.p1.3.m3.1a"><mrow id="S5.Thmtheorem3.p1.3.m3.1.1" xref="S5.Thmtheorem3.p1.3.m3.1.1.cmml"><mi id="S5.Thmtheorem3.p1.3.m3.1.1.3" mathvariant="normal" xref="S5.Thmtheorem3.p1.3.m3.1.1.3.cmml">Ω</mi><mo id="S5.Thmtheorem3.p1.3.m3.1.1.2" xref="S5.Thmtheorem3.p1.3.m3.1.1.2.cmml">⁢</mo><mrow id="S5.Thmtheorem3.p1.3.m3.1.1.1.1" xref="S5.Thmtheorem3.p1.3.m3.1.1.1.1.1.cmml"><mo id="S5.Thmtheorem3.p1.3.m3.1.1.1.1.2" stretchy="false" xref="S5.Thmtheorem3.p1.3.m3.1.1.1.1.1.cmml">(</mo><mrow id="S5.Thmtheorem3.p1.3.m3.1.1.1.1.1" xref="S5.Thmtheorem3.p1.3.m3.1.1.1.1.1.cmml"><mrow id="S5.Thmtheorem3.p1.3.m3.1.1.1.1.1.2" xref="S5.Thmtheorem3.p1.3.m3.1.1.1.1.1.2.cmml"><mi id="S5.Thmtheorem3.p1.3.m3.1.1.1.1.1.2.2" xref="S5.Thmtheorem3.p1.3.m3.1.1.1.1.1.2.2.cmml">n</mi><mo id="S5.Thmtheorem3.p1.3.m3.1.1.1.1.1.2.1" xref="S5.Thmtheorem3.p1.3.m3.1.1.1.1.1.2.1.cmml">⁢</mo><mi id="S5.Thmtheorem3.p1.3.m3.1.1.1.1.1.2.3" xref="S5.Thmtheorem3.p1.3.m3.1.1.1.1.1.2.3.cmml">k</mi></mrow><mo id="S5.Thmtheorem3.p1.3.m3.1.1.1.1.1.1" xref="S5.Thmtheorem3.p1.3.m3.1.1.1.1.1.1.cmml">+</mo><msup id="S5.Thmtheorem3.p1.3.m3.1.1.1.1.1.3" xref="S5.Thmtheorem3.p1.3.m3.1.1.1.1.1.3.cmml"><mi id="S5.Thmtheorem3.p1.3.m3.1.1.1.1.1.3.2" xref="S5.Thmtheorem3.p1.3.m3.1.1.1.1.1.3.2.cmml">n</mi><mrow id="S5.Thmtheorem3.p1.3.m3.1.1.1.1.1.3.3" xref="S5.Thmtheorem3.p1.3.m3.1.1.1.1.1.3.3.cmml"><mn id="S5.Thmtheorem3.p1.3.m3.1.1.1.1.1.3.3.2" xref="S5.Thmtheorem3.p1.3.m3.1.1.1.1.1.3.3.2.cmml">1</mn><mo id="S5.Thmtheorem3.p1.3.m3.1.1.1.1.1.3.3.1" xref="S5.Thmtheorem3.p1.3.m3.1.1.1.1.1.3.3.1.cmml">+</mo><mrow id="S5.Thmtheorem3.p1.3.m3.1.1.1.1.1.3.3.3" xref="S5.Thmtheorem3.p1.3.m3.1.1.1.1.1.3.3.3.cmml"><mn id="S5.Thmtheorem3.p1.3.m3.1.1.1.1.1.3.3.3.2" xref="S5.Thmtheorem3.p1.3.m3.1.1.1.1.1.3.3.3.2.cmml">1</mn><mo id="S5.Thmtheorem3.p1.3.m3.1.1.1.1.1.3.3.3.1" xref="S5.Thmtheorem3.p1.3.m3.1.1.1.1.1.3.3.3.1.cmml">/</mo><mi id="S5.Thmtheorem3.p1.3.m3.1.1.1.1.1.3.3.3.3" xref="S5.Thmtheorem3.p1.3.m3.1.1.1.1.1.3.3.3.3.cmml">t</mi></mrow></mrow></msup></mrow><mo id="S5.Thmtheorem3.p1.3.m3.1.1.1.1.3" stretchy="false" xref="S5.Thmtheorem3.p1.3.m3.1.1.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.Thmtheorem3.p1.3.m3.1b"><apply id="S5.Thmtheorem3.p1.3.m3.1.1.cmml" xref="S5.Thmtheorem3.p1.3.m3.1.1"><times id="S5.Thmtheorem3.p1.3.m3.1.1.2.cmml" xref="S5.Thmtheorem3.p1.3.m3.1.1.2"></times><ci id="S5.Thmtheorem3.p1.3.m3.1.1.3.cmml" xref="S5.Thmtheorem3.p1.3.m3.1.1.3">Ω</ci><apply id="S5.Thmtheorem3.p1.3.m3.1.1.1.1.1.cmml" xref="S5.Thmtheorem3.p1.3.m3.1.1.1.1"><plus id="S5.Thmtheorem3.p1.3.m3.1.1.1.1.1.1.cmml" xref="S5.Thmtheorem3.p1.3.m3.1.1.1.1.1.1"></plus><apply id="S5.Thmtheorem3.p1.3.m3.1.1.1.1.1.2.cmml" xref="S5.Thmtheorem3.p1.3.m3.1.1.1.1.1.2"><times id="S5.Thmtheorem3.p1.3.m3.1.1.1.1.1.2.1.cmml" xref="S5.Thmtheorem3.p1.3.m3.1.1.1.1.1.2.1"></times><ci id="S5.Thmtheorem3.p1.3.m3.1.1.1.1.1.2.2.cmml" xref="S5.Thmtheorem3.p1.3.m3.1.1.1.1.1.2.2">𝑛</ci><ci id="S5.Thmtheorem3.p1.3.m3.1.1.1.1.1.2.3.cmml" xref="S5.Thmtheorem3.p1.3.m3.1.1.1.1.1.2.3">𝑘</ci></apply><apply id="S5.Thmtheorem3.p1.3.m3.1.1.1.1.1.3.cmml" xref="S5.Thmtheorem3.p1.3.m3.1.1.1.1.1.3"><csymbol cd="ambiguous" id="S5.Thmtheorem3.p1.3.m3.1.1.1.1.1.3.1.cmml" xref="S5.Thmtheorem3.p1.3.m3.1.1.1.1.1.3">superscript</csymbol><ci id="S5.Thmtheorem3.p1.3.m3.1.1.1.1.1.3.2.cmml" xref="S5.Thmtheorem3.p1.3.m3.1.1.1.1.1.3.2">𝑛</ci><apply id="S5.Thmtheorem3.p1.3.m3.1.1.1.1.1.3.3.cmml" xref="S5.Thmtheorem3.p1.3.m3.1.1.1.1.1.3.3"><plus id="S5.Thmtheorem3.p1.3.m3.1.1.1.1.1.3.3.1.cmml" xref="S5.Thmtheorem3.p1.3.m3.1.1.1.1.1.3.3.1"></plus><cn id="S5.Thmtheorem3.p1.3.m3.1.1.1.1.1.3.3.2.cmml" type="integer" xref="S5.Thmtheorem3.p1.3.m3.1.1.1.1.1.3.3.2">1</cn><apply id="S5.Thmtheorem3.p1.3.m3.1.1.1.1.1.3.3.3.cmml" xref="S5.Thmtheorem3.p1.3.m3.1.1.1.1.1.3.3.3"><divide id="S5.Thmtheorem3.p1.3.m3.1.1.1.1.1.3.3.3.1.cmml" xref="S5.Thmtheorem3.p1.3.m3.1.1.1.1.1.3.3.3.1"></divide><cn id="S5.Thmtheorem3.p1.3.m3.1.1.1.1.1.3.3.3.2.cmml" type="integer" xref="S5.Thmtheorem3.p1.3.m3.1.1.1.1.1.3.3.3.2">1</cn><ci id="S5.Thmtheorem3.p1.3.m3.1.1.1.1.1.3.3.3.3.cmml" xref="S5.Thmtheorem3.p1.3.m3.1.1.1.1.1.3.3.3.3">𝑡</ci></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmtheorem3.p1.3.m3.1c">\Omega(nk+n^{1+1/t})</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem3.p1.3.m3.1d">roman_Ω ( italic_n italic_k + italic_n start_POSTSUPERSCRIPT 1 + 1 / italic_t end_POSTSUPERSCRIPT )</annotation></semantics></math> space.</p> </div> </div> <div class="ltx_theorem ltx_theorem_corollary" id="S5.Thmtheorem4"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="S5.Thmtheorem4.1.1.1">Corollary 5.4</span></span><span class="ltx_text ltx_font_bold" id="S5.Thmtheorem4.2.2">.</span> </h6> <div class="ltx_para" id="S5.Thmtheorem4.p1"> <p class="ltx_p" id="S5.Thmtheorem4.p1.3">Any algorithm approximating VC-SNDP with maximum connectivity requirement <math alttext="k" class="ltx_Math" display="inline" id="S5.Thmtheorem4.p1.1.m1.1"><semantics id="S5.Thmtheorem4.p1.1.m1.1a"><mi id="S5.Thmtheorem4.p1.1.m1.1.1" xref="S5.Thmtheorem4.p1.1.m1.1.1.cmml">k</mi><annotation-xml encoding="MathML-Content" id="S5.Thmtheorem4.p1.1.m1.1b"><ci id="S5.Thmtheorem4.p1.1.m1.1.1.cmml" xref="S5.Thmtheorem4.p1.1.m1.1.1">𝑘</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmtheorem4.p1.1.m1.1c">k</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem4.p1.1.m1.1d">italic_k</annotation></semantics></math> with a factor better than <math alttext="2t+1" class="ltx_Math" display="inline" id="S5.Thmtheorem4.p1.2.m2.1"><semantics id="S5.Thmtheorem4.p1.2.m2.1a"><mrow id="S5.Thmtheorem4.p1.2.m2.1.1" xref="S5.Thmtheorem4.p1.2.m2.1.1.cmml"><mrow id="S5.Thmtheorem4.p1.2.m2.1.1.2" xref="S5.Thmtheorem4.p1.2.m2.1.1.2.cmml"><mn id="S5.Thmtheorem4.p1.2.m2.1.1.2.2" xref="S5.Thmtheorem4.p1.2.m2.1.1.2.2.cmml">2</mn><mo id="S5.Thmtheorem4.p1.2.m2.1.1.2.1" xref="S5.Thmtheorem4.p1.2.m2.1.1.2.1.cmml">⁢</mo><mi id="S5.Thmtheorem4.p1.2.m2.1.1.2.3" xref="S5.Thmtheorem4.p1.2.m2.1.1.2.3.cmml">t</mi></mrow><mo id="S5.Thmtheorem4.p1.2.m2.1.1.1" xref="S5.Thmtheorem4.p1.2.m2.1.1.1.cmml">+</mo><mn id="S5.Thmtheorem4.p1.2.m2.1.1.3" xref="S5.Thmtheorem4.p1.2.m2.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S5.Thmtheorem4.p1.2.m2.1b"><apply id="S5.Thmtheorem4.p1.2.m2.1.1.cmml" xref="S5.Thmtheorem4.p1.2.m2.1.1"><plus id="S5.Thmtheorem4.p1.2.m2.1.1.1.cmml" xref="S5.Thmtheorem4.p1.2.m2.1.1.1"></plus><apply id="S5.Thmtheorem4.p1.2.m2.1.1.2.cmml" xref="S5.Thmtheorem4.p1.2.m2.1.1.2"><times id="S5.Thmtheorem4.p1.2.m2.1.1.2.1.cmml" xref="S5.Thmtheorem4.p1.2.m2.1.1.2.1"></times><cn id="S5.Thmtheorem4.p1.2.m2.1.1.2.2.cmml" type="integer" xref="S5.Thmtheorem4.p1.2.m2.1.1.2.2">2</cn><ci id="S5.Thmtheorem4.p1.2.m2.1.1.2.3.cmml" xref="S5.Thmtheorem4.p1.2.m2.1.1.2.3">𝑡</ci></apply><cn id="S5.Thmtheorem4.p1.2.m2.1.1.3.cmml" type="integer" xref="S5.Thmtheorem4.p1.2.m2.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmtheorem4.p1.2.m2.1c">2t+1</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem4.p1.2.m2.1d">2 italic_t + 1</annotation></semantics></math>, in insertion-only streams, requires <math alttext="\Omega(nk+n^{1+1/t})" class="ltx_Math" display="inline" id="S5.Thmtheorem4.p1.3.m3.1"><semantics id="S5.Thmtheorem4.p1.3.m3.1a"><mrow id="S5.Thmtheorem4.p1.3.m3.1.1" xref="S5.Thmtheorem4.p1.3.m3.1.1.cmml"><mi id="S5.Thmtheorem4.p1.3.m3.1.1.3" mathvariant="normal" xref="S5.Thmtheorem4.p1.3.m3.1.1.3.cmml">Ω</mi><mo id="S5.Thmtheorem4.p1.3.m3.1.1.2" xref="S5.Thmtheorem4.p1.3.m3.1.1.2.cmml">⁢</mo><mrow id="S5.Thmtheorem4.p1.3.m3.1.1.1.1" xref="S5.Thmtheorem4.p1.3.m3.1.1.1.1.1.cmml"><mo id="S5.Thmtheorem4.p1.3.m3.1.1.1.1.2" stretchy="false" xref="S5.Thmtheorem4.p1.3.m3.1.1.1.1.1.cmml">(</mo><mrow id="S5.Thmtheorem4.p1.3.m3.1.1.1.1.1" xref="S5.Thmtheorem4.p1.3.m3.1.1.1.1.1.cmml"><mrow id="S5.Thmtheorem4.p1.3.m3.1.1.1.1.1.2" xref="S5.Thmtheorem4.p1.3.m3.1.1.1.1.1.2.cmml"><mi id="S5.Thmtheorem4.p1.3.m3.1.1.1.1.1.2.2" xref="S5.Thmtheorem4.p1.3.m3.1.1.1.1.1.2.2.cmml">n</mi><mo id="S5.Thmtheorem4.p1.3.m3.1.1.1.1.1.2.1" xref="S5.Thmtheorem4.p1.3.m3.1.1.1.1.1.2.1.cmml">⁢</mo><mi id="S5.Thmtheorem4.p1.3.m3.1.1.1.1.1.2.3" xref="S5.Thmtheorem4.p1.3.m3.1.1.1.1.1.2.3.cmml">k</mi></mrow><mo id="S5.Thmtheorem4.p1.3.m3.1.1.1.1.1.1" xref="S5.Thmtheorem4.p1.3.m3.1.1.1.1.1.1.cmml">+</mo><msup id="S5.Thmtheorem4.p1.3.m3.1.1.1.1.1.3" xref="S5.Thmtheorem4.p1.3.m3.1.1.1.1.1.3.cmml"><mi id="S5.Thmtheorem4.p1.3.m3.1.1.1.1.1.3.2" xref="S5.Thmtheorem4.p1.3.m3.1.1.1.1.1.3.2.cmml">n</mi><mrow id="S5.Thmtheorem4.p1.3.m3.1.1.1.1.1.3.3" xref="S5.Thmtheorem4.p1.3.m3.1.1.1.1.1.3.3.cmml"><mn id="S5.Thmtheorem4.p1.3.m3.1.1.1.1.1.3.3.2" xref="S5.Thmtheorem4.p1.3.m3.1.1.1.1.1.3.3.2.cmml">1</mn><mo id="S5.Thmtheorem4.p1.3.m3.1.1.1.1.1.3.3.1" xref="S5.Thmtheorem4.p1.3.m3.1.1.1.1.1.3.3.1.cmml">+</mo><mrow id="S5.Thmtheorem4.p1.3.m3.1.1.1.1.1.3.3.3" xref="S5.Thmtheorem4.p1.3.m3.1.1.1.1.1.3.3.3.cmml"><mn id="S5.Thmtheorem4.p1.3.m3.1.1.1.1.1.3.3.3.2" xref="S5.Thmtheorem4.p1.3.m3.1.1.1.1.1.3.3.3.2.cmml">1</mn><mo id="S5.Thmtheorem4.p1.3.m3.1.1.1.1.1.3.3.3.1" xref="S5.Thmtheorem4.p1.3.m3.1.1.1.1.1.3.3.3.1.cmml">/</mo><mi id="S5.Thmtheorem4.p1.3.m3.1.1.1.1.1.3.3.3.3" xref="S5.Thmtheorem4.p1.3.m3.1.1.1.1.1.3.3.3.3.cmml">t</mi></mrow></mrow></msup></mrow><mo id="S5.Thmtheorem4.p1.3.m3.1.1.1.1.3" stretchy="false" xref="S5.Thmtheorem4.p1.3.m3.1.1.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.Thmtheorem4.p1.3.m3.1b"><apply id="S5.Thmtheorem4.p1.3.m3.1.1.cmml" xref="S5.Thmtheorem4.p1.3.m3.1.1"><times id="S5.Thmtheorem4.p1.3.m3.1.1.2.cmml" xref="S5.Thmtheorem4.p1.3.m3.1.1.2"></times><ci id="S5.Thmtheorem4.p1.3.m3.1.1.3.cmml" xref="S5.Thmtheorem4.p1.3.m3.1.1.3">Ω</ci><apply id="S5.Thmtheorem4.p1.3.m3.1.1.1.1.1.cmml" xref="S5.Thmtheorem4.p1.3.m3.1.1.1.1"><plus id="S5.Thmtheorem4.p1.3.m3.1.1.1.1.1.1.cmml" xref="S5.Thmtheorem4.p1.3.m3.1.1.1.1.1.1"></plus><apply id="S5.Thmtheorem4.p1.3.m3.1.1.1.1.1.2.cmml" xref="S5.Thmtheorem4.p1.3.m3.1.1.1.1.1.2"><times id="S5.Thmtheorem4.p1.3.m3.1.1.1.1.1.2.1.cmml" xref="S5.Thmtheorem4.p1.3.m3.1.1.1.1.1.2.1"></times><ci id="S5.Thmtheorem4.p1.3.m3.1.1.1.1.1.2.2.cmml" xref="S5.Thmtheorem4.p1.3.m3.1.1.1.1.1.2.2">𝑛</ci><ci id="S5.Thmtheorem4.p1.3.m3.1.1.1.1.1.2.3.cmml" xref="S5.Thmtheorem4.p1.3.m3.1.1.1.1.1.2.3">𝑘</ci></apply><apply id="S5.Thmtheorem4.p1.3.m3.1.1.1.1.1.3.cmml" xref="S5.Thmtheorem4.p1.3.m3.1.1.1.1.1.3"><csymbol cd="ambiguous" id="S5.Thmtheorem4.p1.3.m3.1.1.1.1.1.3.1.cmml" xref="S5.Thmtheorem4.p1.3.m3.1.1.1.1.1.3">superscript</csymbol><ci id="S5.Thmtheorem4.p1.3.m3.1.1.1.1.1.3.2.cmml" xref="S5.Thmtheorem4.p1.3.m3.1.1.1.1.1.3.2">𝑛</ci><apply id="S5.Thmtheorem4.p1.3.m3.1.1.1.1.1.3.3.cmml" xref="S5.Thmtheorem4.p1.3.m3.1.1.1.1.1.3.3"><plus id="S5.Thmtheorem4.p1.3.m3.1.1.1.1.1.3.3.1.cmml" xref="S5.Thmtheorem4.p1.3.m3.1.1.1.1.1.3.3.1"></plus><cn id="S5.Thmtheorem4.p1.3.m3.1.1.1.1.1.3.3.2.cmml" type="integer" xref="S5.Thmtheorem4.p1.3.m3.1.1.1.1.1.3.3.2">1</cn><apply id="S5.Thmtheorem4.p1.3.m3.1.1.1.1.1.3.3.3.cmml" xref="S5.Thmtheorem4.p1.3.m3.1.1.1.1.1.3.3.3"><divide id="S5.Thmtheorem4.p1.3.m3.1.1.1.1.1.3.3.3.1.cmml" xref="S5.Thmtheorem4.p1.3.m3.1.1.1.1.1.3.3.3.1"></divide><cn id="S5.Thmtheorem4.p1.3.m3.1.1.1.1.1.3.3.3.2.cmml" type="integer" xref="S5.Thmtheorem4.p1.3.m3.1.1.1.1.1.3.3.3.2">1</cn><ci id="S5.Thmtheorem4.p1.3.m3.1.1.1.1.1.3.3.3.3.cmml" xref="S5.Thmtheorem4.p1.3.m3.1.1.1.1.1.3.3.3.3">𝑡</ci></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmtheorem4.p1.3.m3.1c">\Omega(nk+n^{1+1/t})</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem4.p1.3.m3.1d">roman_Ω ( italic_n italic_k + italic_n start_POSTSUPERSCRIPT 1 + 1 / italic_t end_POSTSUPERSCRIPT )</annotation></semantics></math> space.</p> </div> </div> </section> <section class="ltx_section" id="Sx1"> <h2 class="ltx_title ltx_title_section">Acknowledgment</h2> <div class="ltx_para" id="Sx1.p1"> <p class="ltx_p" id="Sx1.p1.1">This work was conducted in part while Chandra Chekuri, Sepideh Mahabadi and Ali Vakilian were visitors at the Simons Institute for the Theory of Computing as part of the Data Structures and Optimization for Fast Algorithms program. The authors thank Ce Jin for his contributions during the early stage of the project.</p> </div> </section> <section class="ltx_bibliography" id="bib"> <h2 class="ltx_title ltx_title_bibliography">References</h2> <ul class="ltx_biblist"> <li class="ltx_bibitem" id="bib.bibx1"> <span class="ltx_tag ltx_tag_bibitem">[AD21]</span> <span class="ltx_bibblock"> Sepehr Assadi and Aditi Dudeja. </span> <span class="ltx_bibblock">A simple semi-streaming algorithm for global minimum cuts. </span> <span class="ltx_bibblock">In <span class="ltx_text ltx_font_italic" id="bib.bibx1.1.1">Symposium on Simplicity in Algorithms (SOSA)</span>, pages 172–180. 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