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<!DOCTYPE html> <html lang="en"> <head> <meta content="text/html; charset=utf-8" http-equiv="content-type"/> <title>Local Density and its Distributed Approximation</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/2411.12694v2/"/></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/2411.12694v2#S1" title="In Local Density and its Distributed Approximation"><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/2411.12694v2#S1.SS1" title="In 1 Introduction ‣ Local Density and its Distributed Approximation"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">1.1 </span>Local density and results.</span></a> <ol class="ltx_toclist ltx_toclist_subsection"> <li class="ltx_tocentry ltx_tocentry_subparagraph"><a class="ltx_ref" href="https://arxiv.org/html/2411.12694v2#S1.SS1.SSS0.P0.SPx1" title="In 1.1 Local density and results. ‣ 1 Introduction ‣ Local Density and its Distributed Approximation"><span class="ltx_text ltx_ref_title">A: Conceptual results for local density.</span></a></li> <li class="ltx_tocentry ltx_tocentry_subparagraph"><a class="ltx_ref" href="https://arxiv.org/html/2411.12694v2#S1.SS1.SSS0.P0.SPx2" title="In 1.1 Local density and results. ‣ 1 Introduction ‣ Local Density and its Distributed Approximation"><span class="ltx_text ltx_ref_title">B: Results for dynamic algorithms.</span></a></li> <li class="ltx_tocentry ltx_tocentry_subparagraph"><a class="ltx_ref" href="https://arxiv.org/html/2411.12694v2#S1.SS1.SSS0.P0.SPx3" title="In 1.1 Local density and results. ‣ 1 Introduction ‣ Local Density and its Distributed Approximation"><span class="ltx_text ltx_ref_title">C: Results in LOCAL.</span></a></li> <li class="ltx_tocentry ltx_tocentry_subparagraph"><a class="ltx_ref" href="https://arxiv.org/html/2411.12694v2#S1.SS1.SSS0.P0.SPx4" title="In 1.1 Local density and results. ‣ 1 Introduction ‣ Local Density and its Distributed Approximation"><span class="ltx_text ltx_ref_title">D: Results in CONGEST</span></a></li> </ol> </li> </ol> </li> <li class="ltx_tocentry ltx_tocentry_section"> <a class="ltx_ref" href="https://arxiv.org/html/2411.12694v2#S2" title="In Local Density and its Distributed Approximation"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">2 </span>Preliminaries and related work</span></a> <ol class="ltx_toclist ltx_toclist_section"> <li class="ltx_tocentry ltx_tocentry_subparagraph"><a class="ltx_ref" href="https://arxiv.org/html/2411.12694v2#S2.SS0.SSS0.P0.SPx1" title="In 2 Preliminaries and related work ‣ Local Density and its Distributed Approximation"><span class="ltx_text ltx_ref_title">Global graph measures.</span></a></li> <li class="ltx_tocentry ltx_tocentry_subsection"> <a class="ltx_ref" href="https://arxiv.org/html/2411.12694v2#S2.SS1" title="In 2 Preliminaries and related work ‣ Local Density and its Distributed Approximation"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">2.1 </span>Densest subgraph in dynamic algorithms</span></a> <ol class="ltx_toclist ltx_toclist_subsection"> <li class="ltx_tocentry ltx_tocentry_subparagraph"><a class="ltx_ref" href="https://arxiv.org/html/2411.12694v2#S2.SS1.SSS0.P0.SPx1" title="In 2.1 Densest subgraph in dynamic algorithms ‣ 2 Preliminaries and related work ‣ Local Density and its Distributed Approximation"><span class="ltx_text ltx_ref_title">Related work in dynamic algorithms</span></a></li> </ol> </li> <li class="ltx_tocentry ltx_tocentry_subsection"> <a class="ltx_ref" href="https://arxiv.org/html/2411.12694v2#S2.SS2" title="In 2 Preliminaries and related work ‣ Local Density and its Distributed Approximation"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">2.2 </span>Approximate densest subgraph in LOCAL and CONGEST</span></a> <ol class="ltx_toclist ltx_toclist_subsection"> <li class="ltx_tocentry ltx_tocentry_subparagraph"><a class="ltx_ref" href="https://arxiv.org/html/2411.12694v2#S2.SS2.SSS0.P0.SPx1" title="In 2.2 Approximate densest subgraph in LOCAL and CONGEST ‣ 2 Preliminaries and related work ‣ Local Density and its Distributed Approximation"><span class="ltx_text ltx_ref_title">Related work.</span></a></li> </ol> </li> <li class="ltx_tocentry ltx_tocentry_subsection"> <a class="ltx_ref" href="https://arxiv.org/html/2411.12694v2#S2.SS3" title="In 2 Preliminaries and related work ‣ Local Density and its Distributed Approximation"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">2.3 </span>Local density</span></a> <ol class="ltx_toclist ltx_toclist_subsection"> <li class="ltx_tocentry ltx_tocentry_subparagraph"><a class="ltx_ref" href="https://arxiv.org/html/2411.12694v2#S2.SS3.SSS0.P0.SPx1" title="In 2.3 Local density ‣ 2 Preliminaries and related work ‣ Local Density and its Distributed Approximation"><span class="ltx_text ltx_ref_title">The benefit of local measures:</span></a></li> </ol> </li> </ol> </li> <li class="ltx_tocentry ltx_tocentry_section"> <a class="ltx_ref" href="https://arxiv.org/html/2411.12694v2#S3" title="In Local Density and its Distributed Approximation"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">3 </span>Results and organisation</span></a> <ol class="ltx_toclist ltx_toclist_section"> <li class="ltx_tocentry ltx_tocentry_subsubsection"><a class="ltx_ref" href="https://arxiv.org/html/2411.12694v2#S3.SS0.SSS1" title="In 3 Results and organisation ‣ Local Density and its Distributed Approximation"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">3.A </span>Conceptual results for local density</span></a></li> <li class="ltx_tocentry ltx_tocentry_subsubsection"><a class="ltx_ref" href="https://arxiv.org/html/2411.12694v2#S3.SS0.SSS2" title="In 3 Results and organisation ‣ Local Density and its Distributed Approximation"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">3.B </span>Results for dynamic algorithms</span></a></li> <li class="ltx_tocentry ltx_tocentry_subsubsection"><a class="ltx_ref" href="https://arxiv.org/html/2411.12694v2#S3.SS0.SSS3" title="In 3 Results and organisation ‣ Local Density and its Distributed Approximation"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">3.C </span>Results in LOCAL</span></a></li> <li class="ltx_tocentry ltx_tocentry_subsubsection"><a class="ltx_ref" href="https://arxiv.org/html/2411.12694v2#S3.SS0.SSS4" title="In 3 Results and organisation ‣ Local Density and its Distributed Approximation"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">3.D </span>Results in CONGEST</span></a></li> </ol> </li> <li class="ltx_tocentry ltx_tocentry_section"><a class="ltx_ref" href="https://arxiv.org/html/2411.12694v2#S4" title="In Local Density and its Distributed Approximation"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">4 </span>Conceptual results for local density</span></a></li> <li class="ltx_tocentry ltx_tocentry_section"><a class="ltx_ref" href="https://arxiv.org/html/2411.12694v2#S5" title="In Local Density and its Distributed Approximation"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">5 </span>Results for dynamic algorithms</span></a></li> <li class="ltx_tocentry ltx_tocentry_section"><a class="ltx_ref" href="https://arxiv.org/html/2411.12694v2#S6" title="In Local Density and its Distributed Approximation"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">6 </span>Results in LOCAL</span></a></li> <li class="ltx_tocentry ltx_tocentry_section"> <a class="ltx_ref" href="https://arxiv.org/html/2411.12694v2#S7" title="In Local Density and its Distributed Approximation"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">7 </span>Results in CONGEST</span></a> <ol class="ltx_toclist ltx_toclist_section"> <li class="ltx_tocentry ltx_tocentry_subparagraph"><a class="ltx_ref" href="https://arxiv.org/html/2411.12694v2#S7.SS0.SSS0.P0.SPx1" title="In 7 Results in CONGEST ‣ Local Density and its Distributed Approximation"><span class="ltx_text ltx_ref_title">The initialising step.</span></a></li> <li class="ltx_tocentry ltx_tocentry_subsection"> <a class="ltx_ref" href="https://arxiv.org/html/2411.12694v2#S7.SS1" title="In 7 Results in CONGEST ‣ Local Density and its Distributed Approximation"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">7.1 </span>Algorithm overview</span></a> <ol class="ltx_toclist ltx_toclist_subsection"> <li class="ltx_tocentry ltx_tocentry_subparagraph"><a class="ltx_ref" href="https://arxiv.org/html/2411.12694v2#S7.SS1.SSS0.P0.SPx1" title="In 7.1 Algorithm overview ‣ 7 Results in CONGEST ‣ Local Density and its Distributed Approximation"><span class="ltx_text ltx_ref_title">Algorithm (see also Figure <span class="ltx_text ltx_ref_tag">2</span> and Algorithms <span class="ltx_text ltx_ref_tag">2</span> and <span class="ltx_text ltx_ref_tag">3</span> and <span class="ltx_text ltx_ref_tag">4</span> and <span class="ltx_text ltx_ref_tag">5</span>)</span></a></li> </ol> </li> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2411.12694v2#S7.SS2" title="In 7 Results in CONGEST ‣ Local Density and its Distributed Approximation"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">7.2 </span>Formal algorithm definition.</span></a></li> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2411.12694v2#S7.SS3" title="In 7 Results in CONGEST ‣ Local Density and its Distributed Approximation"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">7.3 </span>Sketching our algorithm’s correctness.</span></a></li> <li class="ltx_tocentry ltx_tocentry_subsection"> <a class="ltx_ref" href="https://arxiv.org/html/2411.12694v2#S7.SS4" title="In 7 Results in CONGEST ‣ Local Density and its Distributed Approximation"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">7.4 </span>Formally proving correctness</span></a> <ol class="ltx_toclist ltx_toclist_subsection"> <li class="ltx_tocentry ltx_tocentry_subparagraph"><a class="ltx_ref" href="https://arxiv.org/html/2411.12694v2#S7.SS4.SSS0.P0.SPx1" title="In 7.4 Formally proving correctness ‣ 7 Results in CONGEST ‣ Local Density and its Distributed Approximation"><span class="ltx_text ltx_ref_title">Case 1:</span></a></li> <li class="ltx_tocentry ltx_tocentry_subparagraph"><a class="ltx_ref" href="https://arxiv.org/html/2411.12694v2#S7.SS4.SSS0.P0.SPx2" title="In 7.4 Formally proving correctness ‣ 7 Results in CONGEST ‣ Local Density and its Distributed Approximation"><span class="ltx_text ltx_ref_title">Case 2:</span></a></li> </ol> </li> </ol> </li> <li class="ltx_tocentry ltx_tocentry_appendix"> <a class="ltx_ref" href="https://arxiv.org/html/2411.12694v2#A1" title="In Local Density and its Distributed Approximation"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">A </span>Reporting a densest subgraph</span></a> <ol class="ltx_toclist ltx_toclist_appendix"> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2411.12694v2#A1.SS1" title="In Appendix A Reporting a densest subgraph ‣ Local Density and its Distributed Approximation"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">A.1 </span>Related work: distributed densest subgraph reporting</span></a></li> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2411.12694v2#A1.SS2" title="In Appendix A Reporting a densest subgraph ‣ Local Density and its Distributed Approximation"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">A.2 </span>Reporting a locally densest subgraph</span></a></li> <li class="ltx_tocentry ltx_tocentry_subsection"> <a class="ltx_ref" href="https://arxiv.org/html/2411.12694v2#A1.SS3" title="In Appendix A Reporting a densest subgraph ‣ Local Density and its Distributed Approximation"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">A.3 </span>Our algorithm</span></a> <ol class="ltx_toclist ltx_toclist_subsection"> <li class="ltx_tocentry ltx_tocentry_subparagraph"><a class="ltx_ref" href="https://arxiv.org/html/2411.12694v2#A1.SS3.SSS0.P0.SPx1" title="In A.3 Our algorithm ‣ Appendix A Reporting a densest subgraph ‣ Local Density and its Distributed Approximation"><span class="ltx_text ltx_ref_title">Induction.</span></a></li> <li class="ltx_tocentry ltx_tocentry_subparagraph"><a class="ltx_ref" href="https://arxiv.org/html/2411.12694v2#A1.SS3.SSS0.P0.SPx2" title="In A.3 Our algorithm ‣ Appendix A Reporting a densest subgraph ‣ Local Density and its Distributed Approximation"><span class="ltx_text ltx_ref_title">Analysis:</span></a></li> </ol> </li> </ol> </li> </ol></nav> </nav> <div class="ltx_page_main"> <div class="ltx_page_content"> <article class="ltx_document ltx_authors_1line ltx_fleqn"> <div class="ltx_para" id="p1"> <span class="ltx_ERROR undefined" id="p1.1">\hideLIPIcs</span><span class="ltx_ERROR undefined" id="p1.2">\funding</span> <p class="ltx_p" id="p1.3">This work was supported by the the VILLUM Foundation grant (VIL37507) “Efficient Recomputations for Changeful Problems”, the Independent Research Fund Denmark grant 2020-2023 (9131-00044B) “Dynamic Network Analysis”, and the European Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie grant agreement No 899987. Technical University of Denmark, Kongens Lyngby, Denmarkabgch@dtu.dk Technical University of Denmark, Kongens Lyngby, Denmarkidjva@dtu.dkhttps://orcid.org/0009-0006-2624-0231 Technical University of Denmark, Kongens Lyngby, Denmarkerot@dtu.dkhttps://orcid.org/0000-0001-5853-7909 </p> </div> <div class="ltx_para" id="p2"> <span class="ltx_ERROR undefined" id="p2.1">\ccsdesc</span> <p class="ltx_p" id="p2.2">[500]Theory of computation Graph algorithms analysis</p> </div> <div class="ltx_para" id="p3"> <span class="ltx_ERROR undefined" id="p3.1">\ccsdesc</span> <p class="ltx_p" id="p3.2">[500]Theory of computation Distributed algorithms</p> </div> <h1 class="ltx_title ltx_title_document">Local Density and its Distributed Approximation</h1> <div class="ltx_authors"> <span class="ltx_creator ltx_role_author"> <span class="ltx_personname">Aleksander Bjørn Christiansen </span></span> <span class="ltx_author_before"> </span><span class="ltx_creator ltx_role_author"> <span class="ltx_personname">Ivor van der Hoog </span></span> <span class="ltx_author_before"> </span><span class="ltx_creator ltx_role_author"> <span class="ltx_personname">Eva Rotenberg </span></span> </div> <div class="ltx_abstract"> <h6 class="ltx_title ltx_title_abstract">Abstract</h6> <p class="ltx_p" id="id21.id1">The densest subgraph problem is a classic problem in combinatorial optimisation. Graphs with low maximum subgraph density are often called “uniformly sparse”, leading to algorithms parameterised by this density. However, in reality, the sparsity of a graph is not necessarily uniform. This calls for a formally well-defined, fine-grained notion of density.</p> <p class="ltx_p" id="id8.8">Danisch, Chan, and Sozio propose a definition for <em class="ltx_emph ltx_font_italic" id="id8.8.1">local density</em> that assigns to each vertex <math alttext="v" class="ltx_Math" display="inline" id="id1.1.m1.1"><semantics id="id1.1.m1.1a"><mi id="id1.1.m1.1.1" xref="id1.1.m1.1.1.cmml">v</mi><annotation-xml encoding="MathML-Content" id="id1.1.m1.1b"><ci id="id1.1.m1.1.1.cmml" xref="id1.1.m1.1.1">𝑣</ci></annotation-xml><annotation encoding="application/x-tex" id="id1.1.m1.1c">v</annotation><annotation encoding="application/x-llamapun" id="id1.1.m1.1d">italic_v</annotation></semantics></math> a value <math alttext="\rho^{*}(v)" class="ltx_Math" display="inline" id="id2.2.m2.1"><semantics id="id2.2.m2.1a"><mrow id="id2.2.m2.1.2" xref="id2.2.m2.1.2.cmml"><msup id="id2.2.m2.1.2.2" xref="id2.2.m2.1.2.2.cmml"><mi id="id2.2.m2.1.2.2.2" xref="id2.2.m2.1.2.2.2.cmml">ρ</mi><mo id="id2.2.m2.1.2.2.3" xref="id2.2.m2.1.2.2.3.cmml">∗</mo></msup><mo id="id2.2.m2.1.2.1" xref="id2.2.m2.1.2.1.cmml">⁢</mo><mrow id="id2.2.m2.1.2.3.2" xref="id2.2.m2.1.2.cmml"><mo id="id2.2.m2.1.2.3.2.1" stretchy="false" xref="id2.2.m2.1.2.cmml">(</mo><mi id="id2.2.m2.1.1" xref="id2.2.m2.1.1.cmml">v</mi><mo id="id2.2.m2.1.2.3.2.2" stretchy="false" xref="id2.2.m2.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="id2.2.m2.1b"><apply id="id2.2.m2.1.2.cmml" xref="id2.2.m2.1.2"><times id="id2.2.m2.1.2.1.cmml" xref="id2.2.m2.1.2.1"></times><apply id="id2.2.m2.1.2.2.cmml" xref="id2.2.m2.1.2.2"><csymbol cd="ambiguous" id="id2.2.m2.1.2.2.1.cmml" xref="id2.2.m2.1.2.2">superscript</csymbol><ci id="id2.2.m2.1.2.2.2.cmml" xref="id2.2.m2.1.2.2.2">𝜌</ci><times id="id2.2.m2.1.2.2.3.cmml" xref="id2.2.m2.1.2.2.3"></times></apply><ci id="id2.2.m2.1.1.cmml" xref="id2.2.m2.1.1">𝑣</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="id2.2.m2.1c">\rho^{*}(v)</annotation><annotation encoding="application/x-llamapun" id="id2.2.m2.1d">italic_ρ start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ( italic_v )</annotation></semantics></math>. This local density is a generalisation of the maximum subgraph density of a graph. I.e., if <math alttext="\rho(G)" class="ltx_Math" display="inline" id="id3.3.m3.1"><semantics id="id3.3.m3.1a"><mrow id="id3.3.m3.1.2" xref="id3.3.m3.1.2.cmml"><mi id="id3.3.m3.1.2.2" xref="id3.3.m3.1.2.2.cmml">ρ</mi><mo id="id3.3.m3.1.2.1" xref="id3.3.m3.1.2.1.cmml">⁢</mo><mrow id="id3.3.m3.1.2.3.2" xref="id3.3.m3.1.2.cmml"><mo id="id3.3.m3.1.2.3.2.1" stretchy="false" xref="id3.3.m3.1.2.cmml">(</mo><mi id="id3.3.m3.1.1" xref="id3.3.m3.1.1.cmml">G</mi><mo id="id3.3.m3.1.2.3.2.2" stretchy="false" xref="id3.3.m3.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="id3.3.m3.1b"><apply id="id3.3.m3.1.2.cmml" xref="id3.3.m3.1.2"><times id="id3.3.m3.1.2.1.cmml" xref="id3.3.m3.1.2.1"></times><ci id="id3.3.m3.1.2.2.cmml" xref="id3.3.m3.1.2.2">𝜌</ci><ci id="id3.3.m3.1.1.cmml" xref="id3.3.m3.1.1">𝐺</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="id3.3.m3.1c">\rho(G)</annotation><annotation encoding="application/x-llamapun" id="id3.3.m3.1d">italic_ρ ( italic_G )</annotation></semantics></math> is the subgraph density of a finite graph <math alttext="G" class="ltx_Math" display="inline" id="id4.4.m4.1"><semantics id="id4.4.m4.1a"><mi id="id4.4.m4.1.1" xref="id4.4.m4.1.1.cmml">G</mi><annotation-xml encoding="MathML-Content" id="id4.4.m4.1b"><ci id="id4.4.m4.1.1.cmml" xref="id4.4.m4.1.1">𝐺</ci></annotation-xml><annotation encoding="application/x-tex" id="id4.4.m4.1c">G</annotation><annotation encoding="application/x-llamapun" id="id4.4.m4.1d">italic_G</annotation></semantics></math>, then <math alttext="\rho(G)" class="ltx_Math" display="inline" id="id5.5.m5.1"><semantics id="id5.5.m5.1a"><mrow id="id5.5.m5.1.2" xref="id5.5.m5.1.2.cmml"><mi id="id5.5.m5.1.2.2" xref="id5.5.m5.1.2.2.cmml">ρ</mi><mo id="id5.5.m5.1.2.1" xref="id5.5.m5.1.2.1.cmml">⁢</mo><mrow id="id5.5.m5.1.2.3.2" xref="id5.5.m5.1.2.cmml"><mo id="id5.5.m5.1.2.3.2.1" stretchy="false" xref="id5.5.m5.1.2.cmml">(</mo><mi id="id5.5.m5.1.1" xref="id5.5.m5.1.1.cmml">G</mi><mo id="id5.5.m5.1.2.3.2.2" stretchy="false" xref="id5.5.m5.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="id5.5.m5.1b"><apply id="id5.5.m5.1.2.cmml" xref="id5.5.m5.1.2"><times id="id5.5.m5.1.2.1.cmml" xref="id5.5.m5.1.2.1"></times><ci id="id5.5.m5.1.2.2.cmml" xref="id5.5.m5.1.2.2">𝜌</ci><ci id="id5.5.m5.1.1.cmml" xref="id5.5.m5.1.1">𝐺</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="id5.5.m5.1c">\rho(G)</annotation><annotation encoding="application/x-llamapun" id="id5.5.m5.1d">italic_ρ ( italic_G )</annotation></semantics></math> equals the maximum local density <math alttext="\rho^{*}(v)" class="ltx_Math" display="inline" id="id6.6.m6.1"><semantics id="id6.6.m6.1a"><mrow id="id6.6.m6.1.2" xref="id6.6.m6.1.2.cmml"><msup id="id6.6.m6.1.2.2" xref="id6.6.m6.1.2.2.cmml"><mi id="id6.6.m6.1.2.2.2" xref="id6.6.m6.1.2.2.2.cmml">ρ</mi><mo id="id6.6.m6.1.2.2.3" xref="id6.6.m6.1.2.2.3.cmml">∗</mo></msup><mo id="id6.6.m6.1.2.1" xref="id6.6.m6.1.2.1.cmml">⁢</mo><mrow id="id6.6.m6.1.2.3.2" xref="id6.6.m6.1.2.cmml"><mo id="id6.6.m6.1.2.3.2.1" stretchy="false" xref="id6.6.m6.1.2.cmml">(</mo><mi id="id6.6.m6.1.1" xref="id6.6.m6.1.1.cmml">v</mi><mo id="id6.6.m6.1.2.3.2.2" stretchy="false" xref="id6.6.m6.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="id6.6.m6.1b"><apply id="id6.6.m6.1.2.cmml" xref="id6.6.m6.1.2"><times id="id6.6.m6.1.2.1.cmml" xref="id6.6.m6.1.2.1"></times><apply id="id6.6.m6.1.2.2.cmml" xref="id6.6.m6.1.2.2"><csymbol cd="ambiguous" id="id6.6.m6.1.2.2.1.cmml" xref="id6.6.m6.1.2.2">superscript</csymbol><ci id="id6.6.m6.1.2.2.2.cmml" xref="id6.6.m6.1.2.2.2">𝜌</ci><times id="id6.6.m6.1.2.2.3.cmml" xref="id6.6.m6.1.2.2.3"></times></apply><ci id="id6.6.m6.1.1.cmml" xref="id6.6.m6.1.1">𝑣</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="id6.6.m6.1c">\rho^{*}(v)</annotation><annotation encoding="application/x-llamapun" id="id6.6.m6.1d">italic_ρ start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ( italic_v )</annotation></semantics></math> over vertices <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> in <math alttext="G" class="ltx_Math" display="inline" id="id8.8.m8.1"><semantics id="id8.8.m8.1a"><mi id="id8.8.m8.1.1" xref="id8.8.m8.1.1.cmml">G</mi><annotation-xml encoding="MathML-Content" id="id8.8.m8.1b"><ci id="id8.8.m8.1.1.cmml" xref="id8.8.m8.1.1">𝐺</ci></annotation-xml><annotation encoding="application/x-tex" id="id8.8.m8.1c">G</annotation><annotation encoding="application/x-llamapun" id="id8.8.m8.1d">italic_G</annotation></semantics></math>. They present a Frank-Wolfe-based algorithm to approximate the local density of each vertex with no theoretical (asymptotic) guarantees.</p> <p class="ltx_p" id="id11.11">We provide an extensive study of this local density measure. Just as with (global) maximum subgraph density, we show that there is a dual relation between the local out-degrees and the minimum out-degree orientations of the graph. We introduce the definition of the local out-degree <math alttext="\textsl{g}^{*}(v)" class="ltx_Math" display="inline" id="id9.9.m1.1"><semantics id="id9.9.m1.1a"><mrow id="id9.9.m1.1.2" xref="id9.9.m1.1.2.cmml"><msup id="id9.9.m1.1.2.2" xref="id9.9.m1.1.2.2.cmml"><mtext class="ltx_mathvariant_italic" id="id9.9.m1.1.2.2.2" xref="id9.9.m1.1.2.2.2a.cmml">g</mtext><mo id="id9.9.m1.1.2.2.3" xref="id9.9.m1.1.2.2.3.cmml">∗</mo></msup><mo id="id9.9.m1.1.2.1" xref="id9.9.m1.1.2.1.cmml">⁢</mo><mrow id="id9.9.m1.1.2.3.2" xref="id9.9.m1.1.2.cmml"><mo id="id9.9.m1.1.2.3.2.1" stretchy="false" xref="id9.9.m1.1.2.cmml">(</mo><mi id="id9.9.m1.1.1" xref="id9.9.m1.1.1.cmml">v</mi><mo id="id9.9.m1.1.2.3.2.2" stretchy="false" xref="id9.9.m1.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="id9.9.m1.1b"><apply id="id9.9.m1.1.2.cmml" xref="id9.9.m1.1.2"><times id="id9.9.m1.1.2.1.cmml" xref="id9.9.m1.1.2.1"></times><apply id="id9.9.m1.1.2.2.cmml" xref="id9.9.m1.1.2.2"><csymbol cd="ambiguous" id="id9.9.m1.1.2.2.1.cmml" xref="id9.9.m1.1.2.2">superscript</csymbol><ci id="id9.9.m1.1.2.2.2a.cmml" xref="id9.9.m1.1.2.2.2"><mtext class="ltx_mathvariant_italic" id="id9.9.m1.1.2.2.2.cmml" xref="id9.9.m1.1.2.2.2">g</mtext></ci><times id="id9.9.m1.1.2.2.3.cmml" xref="id9.9.m1.1.2.2.3"></times></apply><ci id="id9.9.m1.1.1.cmml" xref="id9.9.m1.1.1">𝑣</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="id9.9.m1.1c">\textsl{g}^{*}(v)</annotation><annotation encoding="application/x-llamapun" id="id9.9.m1.1d">g start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ( italic_v )</annotation></semantics></math> of a vertex <math alttext="v" class="ltx_Math" display="inline" id="id10.10.m2.1"><semantics id="id10.10.m2.1a"><mi id="id10.10.m2.1.1" xref="id10.10.m2.1.1.cmml">v</mi><annotation-xml encoding="MathML-Content" id="id10.10.m2.1b"><ci id="id10.10.m2.1.1.cmml" xref="id10.10.m2.1.1">𝑣</ci></annotation-xml><annotation encoding="application/x-tex" id="id10.10.m2.1c">v</annotation><annotation encoding="application/x-llamapun" id="id10.10.m2.1d">italic_v</annotation></semantics></math>, and show it to be equal to the local density <math alttext="\rho^{*}(v)" class="ltx_Math" display="inline" id="id11.11.m3.1"><semantics id="id11.11.m3.1a"><mrow id="id11.11.m3.1.2" xref="id11.11.m3.1.2.cmml"><msup id="id11.11.m3.1.2.2" xref="id11.11.m3.1.2.2.cmml"><mi id="id11.11.m3.1.2.2.2" xref="id11.11.m3.1.2.2.2.cmml">ρ</mi><mo id="id11.11.m3.1.2.2.3" xref="id11.11.m3.1.2.2.3.cmml">∗</mo></msup><mo id="id11.11.m3.1.2.1" xref="id11.11.m3.1.2.1.cmml">⁢</mo><mrow id="id11.11.m3.1.2.3.2" xref="id11.11.m3.1.2.cmml"><mo id="id11.11.m3.1.2.3.2.1" stretchy="false" xref="id11.11.m3.1.2.cmml">(</mo><mi id="id11.11.m3.1.1" xref="id11.11.m3.1.1.cmml">v</mi><mo id="id11.11.m3.1.2.3.2.2" stretchy="false" xref="id11.11.m3.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="id11.11.m3.1b"><apply id="id11.11.m3.1.2.cmml" xref="id11.11.m3.1.2"><times id="id11.11.m3.1.2.1.cmml" xref="id11.11.m3.1.2.1"></times><apply id="id11.11.m3.1.2.2.cmml" xref="id11.11.m3.1.2.2"><csymbol cd="ambiguous" id="id11.11.m3.1.2.2.1.cmml" xref="id11.11.m3.1.2.2">superscript</csymbol><ci id="id11.11.m3.1.2.2.2.cmml" xref="id11.11.m3.1.2.2.2">𝜌</ci><times id="id11.11.m3.1.2.2.3.cmml" xref="id11.11.m3.1.2.2.3"></times></apply><ci id="id11.11.m3.1.1.cmml" xref="id11.11.m3.1.1">𝑣</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="id11.11.m3.1c">\rho^{*}(v)</annotation><annotation encoding="application/x-llamapun" id="id11.11.m3.1d">italic_ρ start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ( italic_v )</annotation></semantics></math>. We consider the local out-degree to be conceptually simpler, shorter to define, and easier to compute.</p> <p class="ltx_p" id="id19.19">Using the local out-degree we show a previously unknown fact: that existing algorithms already dynamically approximate the local density for each vertex with polylogarithmic update time. Next, we provide the first distributed algorithms that compute the local density with provable guarantees: given any <math alttext="\varepsilon" class="ltx_Math" display="inline" id="id12.12.m1.1"><semantics id="id12.12.m1.1a"><mi id="id12.12.m1.1.1" xref="id12.12.m1.1.1.cmml">ε</mi><annotation-xml encoding="MathML-Content" id="id12.12.m1.1b"><ci id="id12.12.m1.1.1.cmml" xref="id12.12.m1.1.1">𝜀</ci></annotation-xml><annotation encoding="application/x-tex" id="id12.12.m1.1c">\varepsilon</annotation><annotation encoding="application/x-llamapun" id="id12.12.m1.1d">italic_ε</annotation></semantics></math> such that <math alttext="\varepsilon^{-1}\in O(\operatorname{poly}n)" class="ltx_Math" display="inline" id="id13.13.m2.1"><semantics id="id13.13.m2.1a"><mrow id="id13.13.m2.1.1" xref="id13.13.m2.1.1.cmml"><msup id="id13.13.m2.1.1.3" xref="id13.13.m2.1.1.3.cmml"><mi id="id13.13.m2.1.1.3.2" xref="id13.13.m2.1.1.3.2.cmml">ε</mi><mrow id="id13.13.m2.1.1.3.3" xref="id13.13.m2.1.1.3.3.cmml"><mo id="id13.13.m2.1.1.3.3a" xref="id13.13.m2.1.1.3.3.cmml">−</mo><mn id="id13.13.m2.1.1.3.3.2" xref="id13.13.m2.1.1.3.3.2.cmml">1</mn></mrow></msup><mo id="id13.13.m2.1.1.2" xref="id13.13.m2.1.1.2.cmml">∈</mo><mrow id="id13.13.m2.1.1.1" xref="id13.13.m2.1.1.1.cmml"><mi id="id13.13.m2.1.1.1.3" xref="id13.13.m2.1.1.1.3.cmml">O</mi><mo id="id13.13.m2.1.1.1.2" xref="id13.13.m2.1.1.1.2.cmml">⁢</mo><mrow id="id13.13.m2.1.1.1.1.1" xref="id13.13.m2.1.1.1.1.1.1.cmml"><mo id="id13.13.m2.1.1.1.1.1.2" stretchy="false" xref="id13.13.m2.1.1.1.1.1.1.cmml">(</mo><mrow id="id13.13.m2.1.1.1.1.1.1" xref="id13.13.m2.1.1.1.1.1.1.cmml"><mi id="id13.13.m2.1.1.1.1.1.1.1" xref="id13.13.m2.1.1.1.1.1.1.1.cmml">poly</mi><mo id="id13.13.m2.1.1.1.1.1.1a" lspace="0.167em" xref="id13.13.m2.1.1.1.1.1.1.cmml">⁡</mo><mi id="id13.13.m2.1.1.1.1.1.1.2" xref="id13.13.m2.1.1.1.1.1.1.2.cmml">n</mi></mrow><mo id="id13.13.m2.1.1.1.1.1.3" stretchy="false" xref="id13.13.m2.1.1.1.1.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="id13.13.m2.1b"><apply id="id13.13.m2.1.1.cmml" xref="id13.13.m2.1.1"><in id="id13.13.m2.1.1.2.cmml" xref="id13.13.m2.1.1.2"></in><apply id="id13.13.m2.1.1.3.cmml" xref="id13.13.m2.1.1.3"><csymbol cd="ambiguous" id="id13.13.m2.1.1.3.1.cmml" xref="id13.13.m2.1.1.3">superscript</csymbol><ci id="id13.13.m2.1.1.3.2.cmml" xref="id13.13.m2.1.1.3.2">𝜀</ci><apply id="id13.13.m2.1.1.3.3.cmml" xref="id13.13.m2.1.1.3.3"><minus id="id13.13.m2.1.1.3.3.1.cmml" xref="id13.13.m2.1.1.3.3"></minus><cn id="id13.13.m2.1.1.3.3.2.cmml" type="integer" xref="id13.13.m2.1.1.3.3.2">1</cn></apply></apply><apply id="id13.13.m2.1.1.1.cmml" xref="id13.13.m2.1.1.1"><times id="id13.13.m2.1.1.1.2.cmml" xref="id13.13.m2.1.1.1.2"></times><ci id="id13.13.m2.1.1.1.3.cmml" xref="id13.13.m2.1.1.1.3">𝑂</ci><apply id="id13.13.m2.1.1.1.1.1.1.cmml" xref="id13.13.m2.1.1.1.1.1"><ci id="id13.13.m2.1.1.1.1.1.1.1.cmml" xref="id13.13.m2.1.1.1.1.1.1.1">poly</ci><ci id="id13.13.m2.1.1.1.1.1.1.2.cmml" xref="id13.13.m2.1.1.1.1.1.1.2">𝑛</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="id13.13.m2.1c">\varepsilon^{-1}\in O(\operatorname{poly}n)</annotation><annotation encoding="application/x-llamapun" id="id13.13.m2.1d">italic_ε start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT ∈ italic_O ( roman_poly italic_n )</annotation></semantics></math>, we show a deterministic distributed algorithm in the LOCAL model where, after <math alttext="O(\varepsilon^{-2}\log^{2}n)" class="ltx_Math" display="inline" id="id14.14.m3.1"><semantics id="id14.14.m3.1a"><mrow id="id14.14.m3.1.1" xref="id14.14.m3.1.1.cmml"><mi id="id14.14.m3.1.1.3" xref="id14.14.m3.1.1.3.cmml">O</mi><mo id="id14.14.m3.1.1.2" xref="id14.14.m3.1.1.2.cmml">⁢</mo><mrow id="id14.14.m3.1.1.1.1" xref="id14.14.m3.1.1.1.1.1.cmml"><mo id="id14.14.m3.1.1.1.1.2" stretchy="false" xref="id14.14.m3.1.1.1.1.1.cmml">(</mo><mrow id="id14.14.m3.1.1.1.1.1" xref="id14.14.m3.1.1.1.1.1.cmml"><msup id="id14.14.m3.1.1.1.1.1.2" xref="id14.14.m3.1.1.1.1.1.2.cmml"><mi id="id14.14.m3.1.1.1.1.1.2.2" xref="id14.14.m3.1.1.1.1.1.2.2.cmml">ε</mi><mrow id="id14.14.m3.1.1.1.1.1.2.3" xref="id14.14.m3.1.1.1.1.1.2.3.cmml"><mo id="id14.14.m3.1.1.1.1.1.2.3a" xref="id14.14.m3.1.1.1.1.1.2.3.cmml">−</mo><mn id="id14.14.m3.1.1.1.1.1.2.3.2" xref="id14.14.m3.1.1.1.1.1.2.3.2.cmml">2</mn></mrow></msup><mo id="id14.14.m3.1.1.1.1.1.1" lspace="0.167em" xref="id14.14.m3.1.1.1.1.1.1.cmml">⁢</mo><mrow id="id14.14.m3.1.1.1.1.1.3" xref="id14.14.m3.1.1.1.1.1.3.cmml"><msup id="id14.14.m3.1.1.1.1.1.3.1" xref="id14.14.m3.1.1.1.1.1.3.1.cmml"><mi id="id14.14.m3.1.1.1.1.1.3.1.2" xref="id14.14.m3.1.1.1.1.1.3.1.2.cmml">log</mi><mn id="id14.14.m3.1.1.1.1.1.3.1.3" xref="id14.14.m3.1.1.1.1.1.3.1.3.cmml">2</mn></msup><mo id="id14.14.m3.1.1.1.1.1.3a" lspace="0.167em" xref="id14.14.m3.1.1.1.1.1.3.cmml">⁡</mo><mi id="id14.14.m3.1.1.1.1.1.3.2" xref="id14.14.m3.1.1.1.1.1.3.2.cmml">n</mi></mrow></mrow><mo id="id14.14.m3.1.1.1.1.3" stretchy="false" xref="id14.14.m3.1.1.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="id14.14.m3.1b"><apply id="id14.14.m3.1.1.cmml" xref="id14.14.m3.1.1"><times id="id14.14.m3.1.1.2.cmml" xref="id14.14.m3.1.1.2"></times><ci id="id14.14.m3.1.1.3.cmml" xref="id14.14.m3.1.1.3">𝑂</ci><apply id="id14.14.m3.1.1.1.1.1.cmml" xref="id14.14.m3.1.1.1.1"><times id="id14.14.m3.1.1.1.1.1.1.cmml" xref="id14.14.m3.1.1.1.1.1.1"></times><apply id="id14.14.m3.1.1.1.1.1.2.cmml" xref="id14.14.m3.1.1.1.1.1.2"><csymbol cd="ambiguous" id="id14.14.m3.1.1.1.1.1.2.1.cmml" xref="id14.14.m3.1.1.1.1.1.2">superscript</csymbol><ci id="id14.14.m3.1.1.1.1.1.2.2.cmml" xref="id14.14.m3.1.1.1.1.1.2.2">𝜀</ci><apply id="id14.14.m3.1.1.1.1.1.2.3.cmml" xref="id14.14.m3.1.1.1.1.1.2.3"><minus id="id14.14.m3.1.1.1.1.1.2.3.1.cmml" xref="id14.14.m3.1.1.1.1.1.2.3"></minus><cn id="id14.14.m3.1.1.1.1.1.2.3.2.cmml" type="integer" xref="id14.14.m3.1.1.1.1.1.2.3.2">2</cn></apply></apply><apply id="id14.14.m3.1.1.1.1.1.3.cmml" xref="id14.14.m3.1.1.1.1.1.3"><apply id="id14.14.m3.1.1.1.1.1.3.1.cmml" xref="id14.14.m3.1.1.1.1.1.3.1"><csymbol cd="ambiguous" id="id14.14.m3.1.1.1.1.1.3.1.1.cmml" xref="id14.14.m3.1.1.1.1.1.3.1">superscript</csymbol><log id="id14.14.m3.1.1.1.1.1.3.1.2.cmml" xref="id14.14.m3.1.1.1.1.1.3.1.2"></log><cn id="id14.14.m3.1.1.1.1.1.3.1.3.cmml" type="integer" xref="id14.14.m3.1.1.1.1.1.3.1.3">2</cn></apply><ci id="id14.14.m3.1.1.1.1.1.3.2.cmml" xref="id14.14.m3.1.1.1.1.1.3.2">𝑛</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="id14.14.m3.1c">O(\varepsilon^{-2}\log^{2}n)</annotation><annotation encoding="application/x-llamapun" id="id14.14.m3.1d">italic_O ( italic_ε start_POSTSUPERSCRIPT - 2 end_POSTSUPERSCRIPT roman_log start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_n )</annotation></semantics></math> rounds, every vertex <math alttext="v" class="ltx_Math" display="inline" id="id15.15.m4.1"><semantics id="id15.15.m4.1a"><mi id="id15.15.m4.1.1" xref="id15.15.m4.1.1.cmml">v</mi><annotation-xml encoding="MathML-Content" id="id15.15.m4.1b"><ci id="id15.15.m4.1.1.cmml" xref="id15.15.m4.1.1">𝑣</ci></annotation-xml><annotation encoding="application/x-tex" id="id15.15.m4.1c">v</annotation><annotation encoding="application/x-llamapun" id="id15.15.m4.1d">italic_v</annotation></semantics></math> outputs a <math alttext="(1+\varepsilon)" class="ltx_Math" display="inline" id="id16.16.m5.1"><semantics id="id16.16.m5.1a"><mrow id="id16.16.m5.1.1.1" xref="id16.16.m5.1.1.1.1.cmml"><mo id="id16.16.m5.1.1.1.2" stretchy="false" xref="id16.16.m5.1.1.1.1.cmml">(</mo><mrow id="id16.16.m5.1.1.1.1" xref="id16.16.m5.1.1.1.1.cmml"><mn id="id16.16.m5.1.1.1.1.2" xref="id16.16.m5.1.1.1.1.2.cmml">1</mn><mo id="id16.16.m5.1.1.1.1.1" xref="id16.16.m5.1.1.1.1.1.cmml">+</mo><mi id="id16.16.m5.1.1.1.1.3" xref="id16.16.m5.1.1.1.1.3.cmml">ε</mi></mrow><mo id="id16.16.m5.1.1.1.3" stretchy="false" xref="id16.16.m5.1.1.1.1.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="id16.16.m5.1b"><apply id="id16.16.m5.1.1.1.1.cmml" xref="id16.16.m5.1.1.1"><plus id="id16.16.m5.1.1.1.1.1.cmml" xref="id16.16.m5.1.1.1.1.1"></plus><cn id="id16.16.m5.1.1.1.1.2.cmml" type="integer" xref="id16.16.m5.1.1.1.1.2">1</cn><ci id="id16.16.m5.1.1.1.1.3.cmml" xref="id16.16.m5.1.1.1.1.3">𝜀</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="id16.16.m5.1c">(1+\varepsilon)</annotation><annotation encoding="application/x-llamapun" id="id16.16.m5.1d">( 1 + italic_ε )</annotation></semantics></math>-approximation of their local density <math alttext="\rho^{*}(v)" class="ltx_Math" display="inline" id="id17.17.m6.1"><semantics id="id17.17.m6.1a"><mrow id="id17.17.m6.1.2" xref="id17.17.m6.1.2.cmml"><msup id="id17.17.m6.1.2.2" xref="id17.17.m6.1.2.2.cmml"><mi id="id17.17.m6.1.2.2.2" xref="id17.17.m6.1.2.2.2.cmml">ρ</mi><mo id="id17.17.m6.1.2.2.3" xref="id17.17.m6.1.2.2.3.cmml">∗</mo></msup><mo id="id17.17.m6.1.2.1" xref="id17.17.m6.1.2.1.cmml">⁢</mo><mrow id="id17.17.m6.1.2.3.2" xref="id17.17.m6.1.2.cmml"><mo id="id17.17.m6.1.2.3.2.1" stretchy="false" xref="id17.17.m6.1.2.cmml">(</mo><mi id="id17.17.m6.1.1" xref="id17.17.m6.1.1.cmml">v</mi><mo id="id17.17.m6.1.2.3.2.2" stretchy="false" xref="id17.17.m6.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="id17.17.m6.1b"><apply id="id17.17.m6.1.2.cmml" xref="id17.17.m6.1.2"><times id="id17.17.m6.1.2.1.cmml" xref="id17.17.m6.1.2.1"></times><apply id="id17.17.m6.1.2.2.cmml" xref="id17.17.m6.1.2.2"><csymbol cd="ambiguous" id="id17.17.m6.1.2.2.1.cmml" xref="id17.17.m6.1.2.2">superscript</csymbol><ci id="id17.17.m6.1.2.2.2.cmml" xref="id17.17.m6.1.2.2.2">𝜌</ci><times id="id17.17.m6.1.2.2.3.cmml" xref="id17.17.m6.1.2.2.3"></times></apply><ci id="id17.17.m6.1.1.cmml" xref="id17.17.m6.1.1">𝑣</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="id17.17.m6.1c">\rho^{*}(v)</annotation><annotation encoding="application/x-llamapun" id="id17.17.m6.1d">italic_ρ start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ( italic_v )</annotation></semantics></math>. In CONGEST, we show a deterministic distributed algorithm that requires <math alttext="\text{poly}(\log n,\varepsilon^{-1})\cdot 2^{O(\sqrt{\log n})}" class="ltx_Math" display="inline" id="id18.18.m7.3"><semantics id="id18.18.m7.3a"><mrow id="id18.18.m7.3.3" xref="id18.18.m7.3.3.cmml"><mrow id="id18.18.m7.3.3.2" xref="id18.18.m7.3.3.2.cmml"><mtext id="id18.18.m7.3.3.2.4" xref="id18.18.m7.3.3.2.4a.cmml">poly</mtext><mo id="id18.18.m7.3.3.2.3" xref="id18.18.m7.3.3.2.3.cmml">⁢</mo><mrow id="id18.18.m7.3.3.2.2.2" xref="id18.18.m7.3.3.2.2.3.cmml"><mo id="id18.18.m7.3.3.2.2.2.3" stretchy="false" xref="id18.18.m7.3.3.2.2.3.cmml">(</mo><mrow id="id18.18.m7.2.2.1.1.1.1" xref="id18.18.m7.2.2.1.1.1.1.cmml"><mi id="id18.18.m7.2.2.1.1.1.1.1" xref="id18.18.m7.2.2.1.1.1.1.1.cmml">log</mi><mo id="id18.18.m7.2.2.1.1.1.1a" lspace="0.167em" xref="id18.18.m7.2.2.1.1.1.1.cmml">⁡</mo><mi id="id18.18.m7.2.2.1.1.1.1.2" xref="id18.18.m7.2.2.1.1.1.1.2.cmml">n</mi></mrow><mo id="id18.18.m7.3.3.2.2.2.4" xref="id18.18.m7.3.3.2.2.3.cmml">,</mo><msup id="id18.18.m7.3.3.2.2.2.2" xref="id18.18.m7.3.3.2.2.2.2.cmml"><mi id="id18.18.m7.3.3.2.2.2.2.2" xref="id18.18.m7.3.3.2.2.2.2.2.cmml">ε</mi><mrow id="id18.18.m7.3.3.2.2.2.2.3" xref="id18.18.m7.3.3.2.2.2.2.3.cmml"><mo id="id18.18.m7.3.3.2.2.2.2.3a" xref="id18.18.m7.3.3.2.2.2.2.3.cmml">−</mo><mn id="id18.18.m7.3.3.2.2.2.2.3.2" xref="id18.18.m7.3.3.2.2.2.2.3.2.cmml">1</mn></mrow></msup><mo id="id18.18.m7.3.3.2.2.2.5" rspace="0.055em" stretchy="false" xref="id18.18.m7.3.3.2.2.3.cmml">)</mo></mrow></mrow><mo id="id18.18.m7.3.3.3" rspace="0.222em" xref="id18.18.m7.3.3.3.cmml">⋅</mo><msup id="id18.18.m7.3.3.4" xref="id18.18.m7.3.3.4.cmml"><mn id="id18.18.m7.3.3.4.2" xref="id18.18.m7.3.3.4.2.cmml">2</mn><mrow id="id18.18.m7.1.1.1" xref="id18.18.m7.1.1.1.cmml"><mi id="id18.18.m7.1.1.1.3" xref="id18.18.m7.1.1.1.3.cmml">O</mi><mo id="id18.18.m7.1.1.1.2" xref="id18.18.m7.1.1.1.2.cmml">⁢</mo><mrow id="id18.18.m7.1.1.1.4.2" xref="id18.18.m7.1.1.1.1.cmml"><mo id="id18.18.m7.1.1.1.4.2.1" stretchy="false" xref="id18.18.m7.1.1.1.1.cmml">(</mo><msqrt id="id18.18.m7.1.1.1.1" xref="id18.18.m7.1.1.1.1.cmml"><mrow id="id18.18.m7.1.1.1.1.2" xref="id18.18.m7.1.1.1.1.2.cmml"><mi id="id18.18.m7.1.1.1.1.2.1" xref="id18.18.m7.1.1.1.1.2.1.cmml">log</mi><mo id="id18.18.m7.1.1.1.1.2a" lspace="0.167em" xref="id18.18.m7.1.1.1.1.2.cmml">⁡</mo><mi id="id18.18.m7.1.1.1.1.2.2" xref="id18.18.m7.1.1.1.1.2.2.cmml">n</mi></mrow></msqrt><mo id="id18.18.m7.1.1.1.4.2.2" stretchy="false" xref="id18.18.m7.1.1.1.1.cmml">)</mo></mrow></mrow></msup></mrow><annotation-xml encoding="MathML-Content" id="id18.18.m7.3b"><apply id="id18.18.m7.3.3.cmml" xref="id18.18.m7.3.3"><ci id="id18.18.m7.3.3.3.cmml" xref="id18.18.m7.3.3.3">⋅</ci><apply id="id18.18.m7.3.3.2.cmml" xref="id18.18.m7.3.3.2"><times id="id18.18.m7.3.3.2.3.cmml" xref="id18.18.m7.3.3.2.3"></times><ci id="id18.18.m7.3.3.2.4a.cmml" xref="id18.18.m7.3.3.2.4"><mtext id="id18.18.m7.3.3.2.4.cmml" xref="id18.18.m7.3.3.2.4">poly</mtext></ci><interval closure="open" id="id18.18.m7.3.3.2.2.3.cmml" xref="id18.18.m7.3.3.2.2.2"><apply id="id18.18.m7.2.2.1.1.1.1.cmml" xref="id18.18.m7.2.2.1.1.1.1"><log id="id18.18.m7.2.2.1.1.1.1.1.cmml" xref="id18.18.m7.2.2.1.1.1.1.1"></log><ci id="id18.18.m7.2.2.1.1.1.1.2.cmml" xref="id18.18.m7.2.2.1.1.1.1.2">𝑛</ci></apply><apply id="id18.18.m7.3.3.2.2.2.2.cmml" xref="id18.18.m7.3.3.2.2.2.2"><csymbol cd="ambiguous" id="id18.18.m7.3.3.2.2.2.2.1.cmml" xref="id18.18.m7.3.3.2.2.2.2">superscript</csymbol><ci id="id18.18.m7.3.3.2.2.2.2.2.cmml" xref="id18.18.m7.3.3.2.2.2.2.2">𝜀</ci><apply id="id18.18.m7.3.3.2.2.2.2.3.cmml" xref="id18.18.m7.3.3.2.2.2.2.3"><minus id="id18.18.m7.3.3.2.2.2.2.3.1.cmml" xref="id18.18.m7.3.3.2.2.2.2.3"></minus><cn id="id18.18.m7.3.3.2.2.2.2.3.2.cmml" type="integer" xref="id18.18.m7.3.3.2.2.2.2.3.2">1</cn></apply></apply></interval></apply><apply id="id18.18.m7.3.3.4.cmml" xref="id18.18.m7.3.3.4"><csymbol cd="ambiguous" id="id18.18.m7.3.3.4.1.cmml" xref="id18.18.m7.3.3.4">superscript</csymbol><cn id="id18.18.m7.3.3.4.2.cmml" type="integer" xref="id18.18.m7.3.3.4.2">2</cn><apply id="id18.18.m7.1.1.1.cmml" xref="id18.18.m7.1.1.1"><times id="id18.18.m7.1.1.1.2.cmml" xref="id18.18.m7.1.1.1.2"></times><ci id="id18.18.m7.1.1.1.3.cmml" xref="id18.18.m7.1.1.1.3">𝑂</ci><apply id="id18.18.m7.1.1.1.1.cmml" xref="id18.18.m7.1.1.1.4.2"><root id="id18.18.m7.1.1.1.1a.cmml" xref="id18.18.m7.1.1.1.4.2"></root><apply id="id18.18.m7.1.1.1.1.2.cmml" xref="id18.18.m7.1.1.1.1.2"><log id="id18.18.m7.1.1.1.1.2.1.cmml" xref="id18.18.m7.1.1.1.1.2.1"></log><ci id="id18.18.m7.1.1.1.1.2.2.cmml" xref="id18.18.m7.1.1.1.1.2.2">𝑛</ci></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="id18.18.m7.3c">\text{poly}(\log n,\varepsilon^{-1})\cdot 2^{O(\sqrt{\log n})}</annotation><annotation encoding="application/x-llamapun" id="id18.18.m7.3d">poly ( roman_log italic_n , italic_ε start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT ) ⋅ 2 start_POSTSUPERSCRIPT italic_O ( square-root start_ARG roman_log italic_n end_ARG ) end_POSTSUPERSCRIPT</annotation></semantics></math> rounds, which is sublinear in <math alttext="n" class="ltx_Math" display="inline" id="id19.19.m8.1"><semantics id="id19.19.m8.1a"><mi id="id19.19.m8.1.1" xref="id19.19.m8.1.1.cmml">n</mi><annotation-xml encoding="MathML-Content" id="id19.19.m8.1b"><ci id="id19.19.m8.1.1.cmml" xref="id19.19.m8.1.1">𝑛</ci></annotation-xml><annotation encoding="application/x-tex" id="id19.19.m8.1c">n</annotation><annotation encoding="application/x-llamapun" id="id19.19.m8.1d">italic_n</annotation></semantics></math>.</p> <p class="ltx_p" id="id20.20">As a corollary, we obtain the first deterministic algorithm running in a sublinear number of rounds for <math alttext="(1+\varepsilon)" class="ltx_Math" display="inline" id="id20.20.m1.1"><semantics id="id20.20.m1.1a"><mrow id="id20.20.m1.1.1.1" xref="id20.20.m1.1.1.1.1.cmml"><mo id="id20.20.m1.1.1.1.2" stretchy="false" xref="id20.20.m1.1.1.1.1.cmml">(</mo><mrow id="id20.20.m1.1.1.1.1" xref="id20.20.m1.1.1.1.1.cmml"><mn id="id20.20.m1.1.1.1.1.2" xref="id20.20.m1.1.1.1.1.2.cmml">1</mn><mo id="id20.20.m1.1.1.1.1.1" xref="id20.20.m1.1.1.1.1.1.cmml">+</mo><mi id="id20.20.m1.1.1.1.1.3" xref="id20.20.m1.1.1.1.1.3.cmml">ε</mi></mrow><mo id="id20.20.m1.1.1.1.3" stretchy="false" xref="id20.20.m1.1.1.1.1.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="id20.20.m1.1b"><apply id="id20.20.m1.1.1.1.1.cmml" xref="id20.20.m1.1.1.1"><plus id="id20.20.m1.1.1.1.1.1.cmml" xref="id20.20.m1.1.1.1.1.1"></plus><cn id="id20.20.m1.1.1.1.1.2.cmml" type="integer" xref="id20.20.m1.1.1.1.1.2">1</cn><ci id="id20.20.m1.1.1.1.1.3.cmml" xref="id20.20.m1.1.1.1.1.3">𝜀</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="id20.20.m1.1c">(1+\varepsilon)</annotation><annotation encoding="application/x-llamapun" id="id20.20.m1.1d">( 1 + italic_ε )</annotation></semantics></math>-approximate densest subgraph detection in the CONGEST model.</p> </div> <div class="ltx_classification"> <h6 class="ltx_title ltx_title_classification">keywords: </h6>Distributed graph algorithms, graph density computation, graph density approximation, network analysis theory. </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.p1.8">Density or sparsity measures of graphs are widely studied and have many applications. Examples include the arboricity, the degeneracy, and the maximum subgraph density, all of which are asymptotically related within a factor of <math alttext="2" class="ltx_Math" display="inline" id="S1.p1.1.m1.1"><semantics id="S1.p1.1.m1.1a"><mn id="S1.p1.1.m1.1.1" xref="S1.p1.1.m1.1.1.cmml">2</mn><annotation-xml encoding="MathML-Content" id="S1.p1.1.m1.1b"><cn id="S1.p1.1.m1.1.1.cmml" type="integer" xref="S1.p1.1.m1.1.1">2</cn></annotation-xml><annotation encoding="application/x-tex" id="S1.p1.1.m1.1c">2</annotation><annotation encoding="application/x-llamapun" id="S1.p1.1.m1.1d">2</annotation></semantics></math>. Given a graph or subgraph <math alttext="H" class="ltx_Math" display="inline" id="S1.p1.2.m2.1"><semantics id="S1.p1.2.m2.1a"><mi id="S1.p1.2.m2.1.1" xref="S1.p1.2.m2.1.1.cmml">H</mi><annotation-xml encoding="MathML-Content" id="S1.p1.2.m2.1b"><ci id="S1.p1.2.m2.1.1.cmml" xref="S1.p1.2.m2.1.1">𝐻</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.p1.2.m2.1c">H</annotation><annotation encoding="application/x-llamapun" id="S1.p1.2.m2.1d">italic_H</annotation></semantics></math>, its density, <math alttext="\rho(H)" class="ltx_Math" display="inline" id="S1.p1.3.m3.1"><semantics id="S1.p1.3.m3.1a"><mrow id="S1.p1.3.m3.1.2" xref="S1.p1.3.m3.1.2.cmml"><mi id="S1.p1.3.m3.1.2.2" xref="S1.p1.3.m3.1.2.2.cmml">ρ</mi><mo id="S1.p1.3.m3.1.2.1" xref="S1.p1.3.m3.1.2.1.cmml">⁢</mo><mrow id="S1.p1.3.m3.1.2.3.2" xref="S1.p1.3.m3.1.2.cmml"><mo id="S1.p1.3.m3.1.2.3.2.1" stretchy="false" xref="S1.p1.3.m3.1.2.cmml">(</mo><mi id="S1.p1.3.m3.1.1" xref="S1.p1.3.m3.1.1.cmml">H</mi><mo id="S1.p1.3.m3.1.2.3.2.2" stretchy="false" xref="S1.p1.3.m3.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.p1.3.m3.1b"><apply id="S1.p1.3.m3.1.2.cmml" xref="S1.p1.3.m3.1.2"><times id="S1.p1.3.m3.1.2.1.cmml" xref="S1.p1.3.m3.1.2.1"></times><ci id="S1.p1.3.m3.1.2.2.cmml" xref="S1.p1.3.m3.1.2.2">𝜌</ci><ci id="S1.p1.3.m3.1.1.cmml" xref="S1.p1.3.m3.1.1">𝐻</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p1.3.m3.1c">\rho(H)</annotation><annotation encoding="application/x-llamapun" id="S1.p1.3.m3.1d">italic_ρ ( italic_H )</annotation></semantics></math>, is the average number of edges per vertex in <math alttext="H" class="ltx_Math" display="inline" id="S1.p1.4.m4.1"><semantics id="S1.p1.4.m4.1a"><mi id="S1.p1.4.m4.1.1" xref="S1.p1.4.m4.1.1.cmml">H</mi><annotation-xml encoding="MathML-Content" id="S1.p1.4.m4.1b"><ci id="S1.p1.4.m4.1.1.cmml" xref="S1.p1.4.m4.1.1">𝐻</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.p1.4.m4.1c">H</annotation><annotation encoding="application/x-llamapun" id="S1.p1.4.m4.1d">italic_H</annotation></semantics></math>. The <em class="ltx_emph ltx_font_italic" id="S1.p1.8.1">maximum subgraph density</em> <math alttext="\rho^{\max}(G)" class="ltx_Math" display="inline" id="S1.p1.5.m5.1"><semantics id="S1.p1.5.m5.1a"><mrow id="S1.p1.5.m5.1.2" xref="S1.p1.5.m5.1.2.cmml"><msup id="S1.p1.5.m5.1.2.2" xref="S1.p1.5.m5.1.2.2.cmml"><mi id="S1.p1.5.m5.1.2.2.2" xref="S1.p1.5.m5.1.2.2.2.cmml">ρ</mi><mi id="S1.p1.5.m5.1.2.2.3" xref="S1.p1.5.m5.1.2.2.3.cmml">max</mi></msup><mo id="S1.p1.5.m5.1.2.1" xref="S1.p1.5.m5.1.2.1.cmml">⁢</mo><mrow id="S1.p1.5.m5.1.2.3.2" xref="S1.p1.5.m5.1.2.cmml"><mo id="S1.p1.5.m5.1.2.3.2.1" stretchy="false" xref="S1.p1.5.m5.1.2.cmml">(</mo><mi id="S1.p1.5.m5.1.1" xref="S1.p1.5.m5.1.1.cmml">G</mi><mo id="S1.p1.5.m5.1.2.3.2.2" stretchy="false" xref="S1.p1.5.m5.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.p1.5.m5.1b"><apply id="S1.p1.5.m5.1.2.cmml" xref="S1.p1.5.m5.1.2"><times id="S1.p1.5.m5.1.2.1.cmml" xref="S1.p1.5.m5.1.2.1"></times><apply id="S1.p1.5.m5.1.2.2.cmml" xref="S1.p1.5.m5.1.2.2"><csymbol cd="ambiguous" id="S1.p1.5.m5.1.2.2.1.cmml" xref="S1.p1.5.m5.1.2.2">superscript</csymbol><ci id="S1.p1.5.m5.1.2.2.2.cmml" xref="S1.p1.5.m5.1.2.2.2">𝜌</ci><max id="S1.p1.5.m5.1.2.2.3.cmml" xref="S1.p1.5.m5.1.2.2.3"></max></apply><ci id="S1.p1.5.m5.1.1.cmml" xref="S1.p1.5.m5.1.1">𝐺</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p1.5.m5.1c">\rho^{\max}(G)</annotation><annotation encoding="application/x-llamapun" id="S1.p1.5.m5.1d">italic_ρ start_POSTSUPERSCRIPT roman_max end_POSTSUPERSCRIPT ( italic_G )</annotation></semantics></math> of a graph <math alttext="G" class="ltx_Math" display="inline" id="S1.p1.6.m6.1"><semantics id="S1.p1.6.m6.1a"><mi id="S1.p1.6.m6.1.1" xref="S1.p1.6.m6.1.1.cmml">G</mi><annotation-xml encoding="MathML-Content" id="S1.p1.6.m6.1b"><ci id="S1.p1.6.m6.1.1.cmml" xref="S1.p1.6.m6.1.1">𝐺</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.p1.6.m6.1c">G</annotation><annotation encoding="application/x-llamapun" id="S1.p1.6.m6.1d">italic_G</annotation></semantics></math> is the maximum density <math alttext="\rho(H)" class="ltx_Math" display="inline" id="S1.p1.7.m7.1"><semantics id="S1.p1.7.m7.1a"><mrow id="S1.p1.7.m7.1.2" xref="S1.p1.7.m7.1.2.cmml"><mi id="S1.p1.7.m7.1.2.2" xref="S1.p1.7.m7.1.2.2.cmml">ρ</mi><mo id="S1.p1.7.m7.1.2.1" xref="S1.p1.7.m7.1.2.1.cmml">⁢</mo><mrow id="S1.p1.7.m7.1.2.3.2" xref="S1.p1.7.m7.1.2.cmml"><mo id="S1.p1.7.m7.1.2.3.2.1" stretchy="false" xref="S1.p1.7.m7.1.2.cmml">(</mo><mi id="S1.p1.7.m7.1.1" xref="S1.p1.7.m7.1.1.cmml">H</mi><mo id="S1.p1.7.m7.1.2.3.2.2" stretchy="false" xref="S1.p1.7.m7.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.p1.7.m7.1b"><apply id="S1.p1.7.m7.1.2.cmml" xref="S1.p1.7.m7.1.2"><times id="S1.p1.7.m7.1.2.1.cmml" xref="S1.p1.7.m7.1.2.1"></times><ci id="S1.p1.7.m7.1.2.2.cmml" xref="S1.p1.7.m7.1.2.2">𝜌</ci><ci id="S1.p1.7.m7.1.1.cmml" xref="S1.p1.7.m7.1.1">𝐻</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p1.7.m7.1c">\rho(H)</annotation><annotation encoding="application/x-llamapun" id="S1.p1.7.m7.1d">italic_ρ ( italic_H )</annotation></semantics></math> amongst all subgraphs <math alttext="H\subseteq G" class="ltx_Math" display="inline" id="S1.p1.8.m8.1"><semantics id="S1.p1.8.m8.1a"><mrow id="S1.p1.8.m8.1.1" xref="S1.p1.8.m8.1.1.cmml"><mi id="S1.p1.8.m8.1.1.2" xref="S1.p1.8.m8.1.1.2.cmml">H</mi><mo id="S1.p1.8.m8.1.1.1" xref="S1.p1.8.m8.1.1.1.cmml">⊆</mo><mi id="S1.p1.8.m8.1.1.3" xref="S1.p1.8.m8.1.1.3.cmml">G</mi></mrow><annotation-xml encoding="MathML-Content" id="S1.p1.8.m8.1b"><apply id="S1.p1.8.m8.1.1.cmml" xref="S1.p1.8.m8.1.1"><subset id="S1.p1.8.m8.1.1.1.cmml" xref="S1.p1.8.m8.1.1.1"></subset><ci id="S1.p1.8.m8.1.1.2.cmml" xref="S1.p1.8.m8.1.1.2">𝐻</ci><ci id="S1.p1.8.m8.1.1.3.cmml" xref="S1.p1.8.m8.1.1.3">𝐺</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p1.8.m8.1c">H\subseteq G</annotation><annotation encoding="application/x-llamapun" id="S1.p1.8.m8.1d">italic_H ⊆ italic_G</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="S1.p2"> <p class="ltx_p" id="S1.p2.1">Computing maximum subgraph density has been studied both in the dynamic <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2411.12694v2#bib.bib7" title="">7</a>, <a class="ltx_ref" href="https://arxiv.org/html/2411.12694v2#bib.bib20" title="">20</a>]</cite>, streaming <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2411.12694v2#bib.bib1" title="">1</a>, <a class="ltx_ref" href="https://arxiv.org/html/2411.12694v2#bib.bib4" title="">4</a>]</cite> and distributed <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2411.12694v2#bib.bib11" title="">11</a>, <a class="ltx_ref" href="https://arxiv.org/html/2411.12694v2#bib.bib21" title="">21</a>]</cite> setting. Often these measures are used to parameterise the sparsity of “uniformly sparse graphs” <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2411.12694v2#bib.bib2" title="">2</a>, <a class="ltx_ref" href="https://arxiv.org/html/2411.12694v2#bib.bib5" title="">5</a>, <a class="ltx_ref" href="https://arxiv.org/html/2411.12694v2#bib.bib18" title="">18</a>]</cite>. These measures are global measures in the sense that they measure the sparsity of the most dense part of the graph. In many cases the graph is not equally sparse (or dense) everywhere. Consider for example a lollipop graph: a large clique joined to a long path. The clique is a subgraph of high density, yet the vertices along the path sit in a part of the graph that is significantly less dense. Often, solutions for graph density related problems provide guarantees based on the most dense part of the graph. In some areas of computation more “local” solutions are desirable. Prior works of a global nature often completely disregard certain parts of the graphs, meaning that the output in sparser parts holds little to no information. We give three examples:</p> </div> <div class="ltx_para" id="S1.p3"> <p class="ltx_p" id="S1.p3.1"><math alttext="1)" class="ltx_math_unparsed" display="inline" id="S1.p3.1.m1.1"><semantics id="S1.p3.1.m1.1a"><mrow id="S1.p3.1.m1.1b"><mn id="S1.p3.1.m1.1.1">1</mn><mo id="S1.p3.1.m1.1.2" stretchy="false">)</mo></mrow><annotation encoding="application/x-tex" id="S1.p3.1.m1.1c">1)</annotation><annotation encoding="application/x-llamapun" id="S1.p3.1.m1.1d">1 )</annotation></semantics></math> Many dynamic algorithms for estimating the subgraph density rely on modifying the solution locally. Algorithmic performance is expressed in terms of (global) graph sparsity, and thus fails to exploit the more fine-grained guarantee that local sparse areas yield.</p> </div> <div class="ltx_para" id="S1.p4"> <p class="ltx_p" id="S1.p4.1"><math alttext="2)" class="ltx_math_unparsed" display="inline" id="S1.p4.1.m1.1"><semantics id="S1.p4.1.m1.1a"><mrow id="S1.p4.1.m1.1b"><mn id="S1.p4.1.m1.1.1">2</mn><mo id="S1.p4.1.m1.1.2" stretchy="false">)</mo></mrow><annotation encoding="application/x-tex" id="S1.p4.1.m1.1c">2)</annotation><annotation encoding="application/x-llamapun" id="S1.p4.1.m1.1d">2 )</annotation></semantics></math> In network analysis, one is often interested in determining dense subgraphs as these subgraphs can be interpreted, for instance, as communities within a social network. However, since many classical algorithms are tuned towards only detecting the densest subgraphs, these algorithms might fail to detect communities in sparser parts of the network <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2411.12694v2#bib.bib10" title="">10</a>, <a class="ltx_ref" href="https://arxiv.org/html/2411.12694v2#bib.bib19" title="">19</a>, <a class="ltx_ref" href="https://arxiv.org/html/2411.12694v2#bib.bib22" title="">22</a>]</cite>.</p> </div> <div class="ltx_para" id="S1.p5"> <p class="ltx_p" id="S1.p5.3"><math alttext="3)" class="ltx_math_unparsed" display="inline" id="S1.p5.1.m1.1"><semantics id="S1.p5.1.m1.1a"><mrow id="S1.p5.1.m1.1b"><mn id="S1.p5.1.m1.1.1">3</mn><mo id="S1.p5.1.m1.1.2" stretchy="false">)</mo></mrow><annotation encoding="application/x-tex" id="S1.p5.1.m1.1c">3)</annotation><annotation encoding="application/x-llamapun" id="S1.p5.1.m1.1d">3 )</annotation></semantics></math> Computing the maximum subgraph density is not very local, nor distributed. We consider the models LOCAL and CONGEST, and the lollipop graph. Here, almost instantly, the vertices of the clique realise they are part of a (very dense) clique. The vertices on the path may have to wait for diameter-many rounds before realising the maximum subgraph density of the graph. Distributed algorithms that wish to compute the <em class="ltx_emph ltx_font_italic" id="S1.p5.3.1">value</em> of the subgraph density are thus posed with a choice: either use <math alttext="\Omega(D)" class="ltx_Math" display="inline" id="S1.p5.2.m2.1"><semantics id="S1.p5.2.m2.1a"><mrow id="S1.p5.2.m2.1.2" xref="S1.p5.2.m2.1.2.cmml"><mi id="S1.p5.2.m2.1.2.2" mathvariant="normal" xref="S1.p5.2.m2.1.2.2.cmml">Ω</mi><mo id="S1.p5.2.m2.1.2.1" xref="S1.p5.2.m2.1.2.1.cmml">⁢</mo><mrow id="S1.p5.2.m2.1.2.3.2" xref="S1.p5.2.m2.1.2.cmml"><mo id="S1.p5.2.m2.1.2.3.2.1" stretchy="false" xref="S1.p5.2.m2.1.2.cmml">(</mo><mi id="S1.p5.2.m2.1.1" xref="S1.p5.2.m2.1.1.cmml">D</mi><mo id="S1.p5.2.m2.1.2.3.2.2" stretchy="false" xref="S1.p5.2.m2.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.p5.2.m2.1b"><apply id="S1.p5.2.m2.1.2.cmml" xref="S1.p5.2.m2.1.2"><times id="S1.p5.2.m2.1.2.1.cmml" xref="S1.p5.2.m2.1.2.1"></times><ci id="S1.p5.2.m2.1.2.2.cmml" xref="S1.p5.2.m2.1.2.2">Ω</ci><ci id="S1.p5.2.m2.1.1.cmml" xref="S1.p5.2.m2.1.1">𝐷</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p5.2.m2.1c">\Omega(D)</annotation><annotation encoding="application/x-llamapun" id="S1.p5.2.m2.1d">roman_Ω ( italic_D )</annotation></semantics></math> rounds (where <math alttext="D" class="ltx_Math" display="inline" id="S1.p5.3.m3.1"><semantics id="S1.p5.3.m3.1a"><mi id="S1.p5.3.m3.1.1" xref="S1.p5.3.m3.1.1.cmml">D</mi><annotation-xml encoding="MathML-Content" id="S1.p5.3.m3.1b"><ci id="S1.p5.3.m3.1.1.cmml" xref="S1.p5.3.m3.1.1">𝐷</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.p5.3.m3.1c">D</annotation><annotation encoding="application/x-llamapun" id="S1.p5.3.m3.1d">italic_D</annotation></semantics></math> is the diameter of the graph), or let every vertex output a value that is at most the maximum subgraph density.</p> </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>Local density and results.</h3> <div class="ltx_para" id="S1.SS1.p1"> <p class="ltx_p" id="S1.SS1.p1.2">We consider the definition of <em class="ltx_emph ltx_font_italic" id="S1.SS1.p1.2.1">local density</em> <math alttext="\rho^{*}(v)" class="ltx_Math" display="inline" id="S1.SS1.p1.1.m1.1"><semantics id="S1.SS1.p1.1.m1.1a"><mrow id="S1.SS1.p1.1.m1.1.2" xref="S1.SS1.p1.1.m1.1.2.cmml"><msup id="S1.SS1.p1.1.m1.1.2.2" xref="S1.SS1.p1.1.m1.1.2.2.cmml"><mi id="S1.SS1.p1.1.m1.1.2.2.2" xref="S1.SS1.p1.1.m1.1.2.2.2.cmml">ρ</mi><mo id="S1.SS1.p1.1.m1.1.2.2.3" xref="S1.SS1.p1.1.m1.1.2.2.3.cmml">∗</mo></msup><mo id="S1.SS1.p1.1.m1.1.2.1" xref="S1.SS1.p1.1.m1.1.2.1.cmml">⁢</mo><mrow id="S1.SS1.p1.1.m1.1.2.3.2" xref="S1.SS1.p1.1.m1.1.2.cmml"><mo id="S1.SS1.p1.1.m1.1.2.3.2.1" stretchy="false" xref="S1.SS1.p1.1.m1.1.2.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.m1.1.2.3.2.2" stretchy="false" xref="S1.SS1.p1.1.m1.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.SS1.p1.1.m1.1b"><apply id="S1.SS1.p1.1.m1.1.2.cmml" xref="S1.SS1.p1.1.m1.1.2"><times id="S1.SS1.p1.1.m1.1.2.1.cmml" xref="S1.SS1.p1.1.m1.1.2.1"></times><apply id="S1.SS1.p1.1.m1.1.2.2.cmml" xref="S1.SS1.p1.1.m1.1.2.2"><csymbol cd="ambiguous" id="S1.SS1.p1.1.m1.1.2.2.1.cmml" xref="S1.SS1.p1.1.m1.1.2.2">superscript</csymbol><ci id="S1.SS1.p1.1.m1.1.2.2.2.cmml" xref="S1.SS1.p1.1.m1.1.2.2.2">𝜌</ci><times id="S1.SS1.p1.1.m1.1.2.2.3.cmml" xref="S1.SS1.p1.1.m1.1.2.2.3"></times></apply><ci id="S1.SS1.p1.1.m1.1.1.cmml" xref="S1.SS1.p1.1.m1.1.1">𝑣</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS1.p1.1.m1.1c">\rho^{*}(v)</annotation><annotation encoding="application/x-llamapun" id="S1.SS1.p1.1.m1.1d">italic_ρ start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ( italic_v )</annotation></semantics></math> by Danisch, Chan, and Sozio <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2411.12694v2#bib.bib10" title="">10</a>]</cite>, defined at each node <math alttext="v" class="ltx_Math" display="inline" id="S1.SS1.p1.2.m2.1"><semantics id="S1.SS1.p1.2.m2.1a"><mi id="S1.SS1.p1.2.m2.1.1" xref="S1.SS1.p1.2.m2.1.1.cmml">v</mi><annotation-xml encoding="MathML-Content" id="S1.SS1.p1.2.m2.1b"><ci id="S1.SS1.p1.2.m2.1.1.cmml" xref="S1.SS1.p1.2.m2.1.1">𝑣</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.SS1.p1.2.m2.1c">v</annotation><annotation encoding="application/x-llamapun" id="S1.SS1.p1.2.m2.1d">italic_v</annotation></semantics></math> of the graph (Definition <a class="ltx_ref" href="https://arxiv.org/html/2411.12694v2#S2.Thmtheorem10" title="Definition 2.10 (Definition 2.3 in [10]). ‣ 2.3 Local density ‣ 2 Preliminaries and related work ‣ Local Density and its Distributed Approximation"><span class="ltx_text ltx_ref_tag">2.10</span></a>). Our contributions can be split into four categories, which we present in four sections with corresponding titles.</p> </div> <section class="ltx_subparagraph" id="S1.SS1.SSS0.P0.SPx1"> <h5 class="ltx_title ltx_title_subparagraph">A: Conceptual results for local density. </h5> <div class="ltx_para" id="S1.SS1.SSS0.P0.SPx1.p1"> <p class="ltx_p" id="S1.SS1.SSS0.P0.SPx1.p1.8">Our primary contribution is an extensive overview of the theoretical properties of this local density measure. We show that, just as in the maximum-subgraph density problem, computing local density has a natural dual problem as we define the <em class="ltx_emph ltx_font_italic" id="S1.SS1.SSS0.P0.SPx1.p1.8.1">local out-degree</em>. Consider a (fractional) orientation of the graph that is <em class="ltx_emph ltx_font_italic" id="S1.SS1.SSS0.P0.SPx1.p1.8.2">locally fair</em>. i.e., for each directed edge <math alttext="(u,v)" class="ltx_Math" display="inline" id="S1.SS1.SSS0.P0.SPx1.p1.1.m1.2"><semantics id="S1.SS1.SSS0.P0.SPx1.p1.1.m1.2a"><mrow id="S1.SS1.SSS0.P0.SPx1.p1.1.m1.2.3.2" xref="S1.SS1.SSS0.P0.SPx1.p1.1.m1.2.3.1.cmml"><mo id="S1.SS1.SSS0.P0.SPx1.p1.1.m1.2.3.2.1" stretchy="false" xref="S1.SS1.SSS0.P0.SPx1.p1.1.m1.2.3.1.cmml">(</mo><mi id="S1.SS1.SSS0.P0.SPx1.p1.1.m1.1.1" xref="S1.SS1.SSS0.P0.SPx1.p1.1.m1.1.1.cmml">u</mi><mo id="S1.SS1.SSS0.P0.SPx1.p1.1.m1.2.3.2.2" xref="S1.SS1.SSS0.P0.SPx1.p1.1.m1.2.3.1.cmml">,</mo><mi id="S1.SS1.SSS0.P0.SPx1.p1.1.m1.2.2" xref="S1.SS1.SSS0.P0.SPx1.p1.1.m1.2.2.cmml">v</mi><mo id="S1.SS1.SSS0.P0.SPx1.p1.1.m1.2.3.2.3" stretchy="false" xref="S1.SS1.SSS0.P0.SPx1.p1.1.m1.2.3.1.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S1.SS1.SSS0.P0.SPx1.p1.1.m1.2b"><interval closure="open" id="S1.SS1.SSS0.P0.SPx1.p1.1.m1.2.3.1.cmml" xref="S1.SS1.SSS0.P0.SPx1.p1.1.m1.2.3.2"><ci id="S1.SS1.SSS0.P0.SPx1.p1.1.m1.1.1.cmml" xref="S1.SS1.SSS0.P0.SPx1.p1.1.m1.1.1">𝑢</ci><ci id="S1.SS1.SSS0.P0.SPx1.p1.1.m1.2.2.cmml" xref="S1.SS1.SSS0.P0.SPx1.p1.1.m1.2.2">𝑣</ci></interval></annotation-xml><annotation encoding="application/x-tex" id="S1.SS1.SSS0.P0.SPx1.p1.1.m1.2c">(u,v)</annotation><annotation encoding="application/x-llamapun" id="S1.SS1.SSS0.P0.SPx1.p1.1.m1.2d">( italic_u , italic_v )</annotation></semantics></math>, the out-degree <math alttext="g(u)" class="ltx_Math" display="inline" id="S1.SS1.SSS0.P0.SPx1.p1.2.m2.1"><semantics id="S1.SS1.SSS0.P0.SPx1.p1.2.m2.1a"><mrow id="S1.SS1.SSS0.P0.SPx1.p1.2.m2.1.2" xref="S1.SS1.SSS0.P0.SPx1.p1.2.m2.1.2.cmml"><mi id="S1.SS1.SSS0.P0.SPx1.p1.2.m2.1.2.2" xref="S1.SS1.SSS0.P0.SPx1.p1.2.m2.1.2.2.cmml">g</mi><mo id="S1.SS1.SSS0.P0.SPx1.p1.2.m2.1.2.1" xref="S1.SS1.SSS0.P0.SPx1.p1.2.m2.1.2.1.cmml">⁢</mo><mrow id="S1.SS1.SSS0.P0.SPx1.p1.2.m2.1.2.3.2" xref="S1.SS1.SSS0.P0.SPx1.p1.2.m2.1.2.cmml"><mo id="S1.SS1.SSS0.P0.SPx1.p1.2.m2.1.2.3.2.1" stretchy="false" xref="S1.SS1.SSS0.P0.SPx1.p1.2.m2.1.2.cmml">(</mo><mi id="S1.SS1.SSS0.P0.SPx1.p1.2.m2.1.1" xref="S1.SS1.SSS0.P0.SPx1.p1.2.m2.1.1.cmml">u</mi><mo id="S1.SS1.SSS0.P0.SPx1.p1.2.m2.1.2.3.2.2" stretchy="false" xref="S1.SS1.SSS0.P0.SPx1.p1.2.m2.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.SS1.SSS0.P0.SPx1.p1.2.m2.1b"><apply id="S1.SS1.SSS0.P0.SPx1.p1.2.m2.1.2.cmml" xref="S1.SS1.SSS0.P0.SPx1.p1.2.m2.1.2"><times id="S1.SS1.SSS0.P0.SPx1.p1.2.m2.1.2.1.cmml" xref="S1.SS1.SSS0.P0.SPx1.p1.2.m2.1.2.1"></times><ci id="S1.SS1.SSS0.P0.SPx1.p1.2.m2.1.2.2.cmml" xref="S1.SS1.SSS0.P0.SPx1.p1.2.m2.1.2.2">𝑔</ci><ci id="S1.SS1.SSS0.P0.SPx1.p1.2.m2.1.1.cmml" xref="S1.SS1.SSS0.P0.SPx1.p1.2.m2.1.1">𝑢</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS1.SSS0.P0.SPx1.p1.2.m2.1c">g(u)</annotation><annotation encoding="application/x-llamapun" id="S1.SS1.SSS0.P0.SPx1.p1.2.m2.1d">italic_g ( italic_u )</annotation></semantics></math> is at most <math alttext="g(v)" class="ltx_Math" display="inline" id="S1.SS1.SSS0.P0.SPx1.p1.3.m3.1"><semantics id="S1.SS1.SSS0.P0.SPx1.p1.3.m3.1a"><mrow id="S1.SS1.SSS0.P0.SPx1.p1.3.m3.1.2" xref="S1.SS1.SSS0.P0.SPx1.p1.3.m3.1.2.cmml"><mi id="S1.SS1.SSS0.P0.SPx1.p1.3.m3.1.2.2" xref="S1.SS1.SSS0.P0.SPx1.p1.3.m3.1.2.2.cmml">g</mi><mo id="S1.SS1.SSS0.P0.SPx1.p1.3.m3.1.2.1" xref="S1.SS1.SSS0.P0.SPx1.p1.3.m3.1.2.1.cmml">⁢</mo><mrow id="S1.SS1.SSS0.P0.SPx1.p1.3.m3.1.2.3.2" xref="S1.SS1.SSS0.P0.SPx1.p1.3.m3.1.2.cmml"><mo id="S1.SS1.SSS0.P0.SPx1.p1.3.m3.1.2.3.2.1" stretchy="false" xref="S1.SS1.SSS0.P0.SPx1.p1.3.m3.1.2.cmml">(</mo><mi id="S1.SS1.SSS0.P0.SPx1.p1.3.m3.1.1" xref="S1.SS1.SSS0.P0.SPx1.p1.3.m3.1.1.cmml">v</mi><mo id="S1.SS1.SSS0.P0.SPx1.p1.3.m3.1.2.3.2.2" stretchy="false" xref="S1.SS1.SSS0.P0.SPx1.p1.3.m3.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.SS1.SSS0.P0.SPx1.p1.3.m3.1b"><apply id="S1.SS1.SSS0.P0.SPx1.p1.3.m3.1.2.cmml" xref="S1.SS1.SSS0.P0.SPx1.p1.3.m3.1.2"><times id="S1.SS1.SSS0.P0.SPx1.p1.3.m3.1.2.1.cmml" xref="S1.SS1.SSS0.P0.SPx1.p1.3.m3.1.2.1"></times><ci id="S1.SS1.SSS0.P0.SPx1.p1.3.m3.1.2.2.cmml" xref="S1.SS1.SSS0.P0.SPx1.p1.3.m3.1.2.2">𝑔</ci><ci id="S1.SS1.SSS0.P0.SPx1.p1.3.m3.1.1.cmml" xref="S1.SS1.SSS0.P0.SPx1.p1.3.m3.1.1">𝑣</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS1.SSS0.P0.SPx1.p1.3.m3.1c">g(v)</annotation><annotation encoding="application/x-llamapun" id="S1.SS1.SSS0.P0.SPx1.p1.3.m3.1d">italic_g ( italic_v )</annotation></semantics></math>. We prove that for each vertex <math alttext="v" class="ltx_Math" display="inline" id="S1.SS1.SSS0.P0.SPx1.p1.4.m4.1"><semantics id="S1.SS1.SSS0.P0.SPx1.p1.4.m4.1a"><mi id="S1.SS1.SSS0.P0.SPx1.p1.4.m4.1.1" xref="S1.SS1.SSS0.P0.SPx1.p1.4.m4.1.1.cmml">v</mi><annotation-xml encoding="MathML-Content" id="S1.SS1.SSS0.P0.SPx1.p1.4.m4.1b"><ci id="S1.SS1.SSS0.P0.SPx1.p1.4.m4.1.1.cmml" xref="S1.SS1.SSS0.P0.SPx1.p1.4.m4.1.1">𝑣</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.SS1.SSS0.P0.SPx1.p1.4.m4.1c">v</annotation><annotation encoding="application/x-llamapun" id="S1.SS1.SSS0.P0.SPx1.p1.4.m4.1d">italic_v</annotation></semantics></math>, the out-degree of <math alttext="v" class="ltx_Math" display="inline" id="S1.SS1.SSS0.P0.SPx1.p1.5.m5.1"><semantics id="S1.SS1.SSS0.P0.SPx1.p1.5.m5.1a"><mi id="S1.SS1.SSS0.P0.SPx1.p1.5.m5.1.1" xref="S1.SS1.SSS0.P0.SPx1.p1.5.m5.1.1.cmml">v</mi><annotation-xml encoding="MathML-Content" id="S1.SS1.SSS0.P0.SPx1.p1.5.m5.1b"><ci id="S1.SS1.SSS0.P0.SPx1.p1.5.m5.1.1.cmml" xref="S1.SS1.SSS0.P0.SPx1.p1.5.m5.1.1">𝑣</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.SS1.SSS0.P0.SPx1.p1.5.m5.1c">v</annotation><annotation encoding="application/x-llamapun" id="S1.SS1.SSS0.P0.SPx1.p1.5.m5.1d">italic_v</annotation></semantics></math> has the same value over all locally fair orientations. We define this value <math alttext="\textsl{g}^{*}(v)" class="ltx_Math" display="inline" id="S1.SS1.SSS0.P0.SPx1.p1.6.m6.1"><semantics id="S1.SS1.SSS0.P0.SPx1.p1.6.m6.1a"><mrow id="S1.SS1.SSS0.P0.SPx1.p1.6.m6.1.2" xref="S1.SS1.SSS0.P0.SPx1.p1.6.m6.1.2.cmml"><msup id="S1.SS1.SSS0.P0.SPx1.p1.6.m6.1.2.2" xref="S1.SS1.SSS0.P0.SPx1.p1.6.m6.1.2.2.cmml"><mtext class="ltx_mathvariant_italic" id="S1.SS1.SSS0.P0.SPx1.p1.6.m6.1.2.2.2" xref="S1.SS1.SSS0.P0.SPx1.p1.6.m6.1.2.2.2a.cmml">g</mtext><mo id="S1.SS1.SSS0.P0.SPx1.p1.6.m6.1.2.2.3" xref="S1.SS1.SSS0.P0.SPx1.p1.6.m6.1.2.2.3.cmml">∗</mo></msup><mo id="S1.SS1.SSS0.P0.SPx1.p1.6.m6.1.2.1" xref="S1.SS1.SSS0.P0.SPx1.p1.6.m6.1.2.1.cmml">⁢</mo><mrow id="S1.SS1.SSS0.P0.SPx1.p1.6.m6.1.2.3.2" xref="S1.SS1.SSS0.P0.SPx1.p1.6.m6.1.2.cmml"><mo id="S1.SS1.SSS0.P0.SPx1.p1.6.m6.1.2.3.2.1" stretchy="false" xref="S1.SS1.SSS0.P0.SPx1.p1.6.m6.1.2.cmml">(</mo><mi id="S1.SS1.SSS0.P0.SPx1.p1.6.m6.1.1" xref="S1.SS1.SSS0.P0.SPx1.p1.6.m6.1.1.cmml">v</mi><mo id="S1.SS1.SSS0.P0.SPx1.p1.6.m6.1.2.3.2.2" stretchy="false" xref="S1.SS1.SSS0.P0.SPx1.p1.6.m6.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.SS1.SSS0.P0.SPx1.p1.6.m6.1b"><apply id="S1.SS1.SSS0.P0.SPx1.p1.6.m6.1.2.cmml" xref="S1.SS1.SSS0.P0.SPx1.p1.6.m6.1.2"><times id="S1.SS1.SSS0.P0.SPx1.p1.6.m6.1.2.1.cmml" xref="S1.SS1.SSS0.P0.SPx1.p1.6.m6.1.2.1"></times><apply id="S1.SS1.SSS0.P0.SPx1.p1.6.m6.1.2.2.cmml" xref="S1.SS1.SSS0.P0.SPx1.p1.6.m6.1.2.2"><csymbol cd="ambiguous" id="S1.SS1.SSS0.P0.SPx1.p1.6.m6.1.2.2.1.cmml" xref="S1.SS1.SSS0.P0.SPx1.p1.6.m6.1.2.2">superscript</csymbol><ci id="S1.SS1.SSS0.P0.SPx1.p1.6.m6.1.2.2.2a.cmml" xref="S1.SS1.SSS0.P0.SPx1.p1.6.m6.1.2.2.2"><mtext class="ltx_mathvariant_italic" id="S1.SS1.SSS0.P0.SPx1.p1.6.m6.1.2.2.2.cmml" xref="S1.SS1.SSS0.P0.SPx1.p1.6.m6.1.2.2.2">g</mtext></ci><times id="S1.SS1.SSS0.P0.SPx1.p1.6.m6.1.2.2.3.cmml" xref="S1.SS1.SSS0.P0.SPx1.p1.6.m6.1.2.2.3"></times></apply><ci id="S1.SS1.SSS0.P0.SPx1.p1.6.m6.1.1.cmml" xref="S1.SS1.SSS0.P0.SPx1.p1.6.m6.1.1">𝑣</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS1.SSS0.P0.SPx1.p1.6.m6.1c">\textsl{g}^{*}(v)</annotation><annotation encoding="application/x-llamapun" id="S1.SS1.SSS0.P0.SPx1.p1.6.m6.1d">g start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ( italic_v )</annotation></semantics></math> as the local out-degree of <math alttext="v" class="ltx_Math" display="inline" id="S1.SS1.SSS0.P0.SPx1.p1.7.m7.1"><semantics id="S1.SS1.SSS0.P0.SPx1.p1.7.m7.1a"><mi id="S1.SS1.SSS0.P0.SPx1.p1.7.m7.1.1" xref="S1.SS1.SSS0.P0.SPx1.p1.7.m7.1.1.cmml">v</mi><annotation-xml encoding="MathML-Content" id="S1.SS1.SSS0.P0.SPx1.p1.7.m7.1b"><ci id="S1.SS1.SSS0.P0.SPx1.p1.7.m7.1.1.cmml" xref="S1.SS1.SSS0.P0.SPx1.p1.7.m7.1.1">𝑣</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.SS1.SSS0.P0.SPx1.p1.7.m7.1c">v</annotation><annotation encoding="application/x-llamapun" id="S1.SS1.SSS0.P0.SPx1.p1.7.m7.1d">italic_v</annotation></semantics></math>. We prove that the local density of each vertex is the dual of its local out-degree and thereby <math alttext="\textsl{g}^{*}(v)=\rho^{*}(v)" class="ltx_Math" display="inline" id="S1.SS1.SSS0.P0.SPx1.p1.8.m8.2"><semantics id="S1.SS1.SSS0.P0.SPx1.p1.8.m8.2a"><mrow id="S1.SS1.SSS0.P0.SPx1.p1.8.m8.2.3" xref="S1.SS1.SSS0.P0.SPx1.p1.8.m8.2.3.cmml"><mrow id="S1.SS1.SSS0.P0.SPx1.p1.8.m8.2.3.2" xref="S1.SS1.SSS0.P0.SPx1.p1.8.m8.2.3.2.cmml"><msup id="S1.SS1.SSS0.P0.SPx1.p1.8.m8.2.3.2.2" xref="S1.SS1.SSS0.P0.SPx1.p1.8.m8.2.3.2.2.cmml"><mtext class="ltx_mathvariant_italic" id="S1.SS1.SSS0.P0.SPx1.p1.8.m8.2.3.2.2.2" xref="S1.SS1.SSS0.P0.SPx1.p1.8.m8.2.3.2.2.2a.cmml">g</mtext><mo id="S1.SS1.SSS0.P0.SPx1.p1.8.m8.2.3.2.2.3" xref="S1.SS1.SSS0.P0.SPx1.p1.8.m8.2.3.2.2.3.cmml">∗</mo></msup><mo id="S1.SS1.SSS0.P0.SPx1.p1.8.m8.2.3.2.1" xref="S1.SS1.SSS0.P0.SPx1.p1.8.m8.2.3.2.1.cmml">⁢</mo><mrow id="S1.SS1.SSS0.P0.SPx1.p1.8.m8.2.3.2.3.2" xref="S1.SS1.SSS0.P0.SPx1.p1.8.m8.2.3.2.cmml"><mo id="S1.SS1.SSS0.P0.SPx1.p1.8.m8.2.3.2.3.2.1" stretchy="false" xref="S1.SS1.SSS0.P0.SPx1.p1.8.m8.2.3.2.cmml">(</mo><mi id="S1.SS1.SSS0.P0.SPx1.p1.8.m8.1.1" xref="S1.SS1.SSS0.P0.SPx1.p1.8.m8.1.1.cmml">v</mi><mo id="S1.SS1.SSS0.P0.SPx1.p1.8.m8.2.3.2.3.2.2" stretchy="false" xref="S1.SS1.SSS0.P0.SPx1.p1.8.m8.2.3.2.cmml">)</mo></mrow></mrow><mo id="S1.SS1.SSS0.P0.SPx1.p1.8.m8.2.3.1" xref="S1.SS1.SSS0.P0.SPx1.p1.8.m8.2.3.1.cmml">=</mo><mrow id="S1.SS1.SSS0.P0.SPx1.p1.8.m8.2.3.3" xref="S1.SS1.SSS0.P0.SPx1.p1.8.m8.2.3.3.cmml"><msup id="S1.SS1.SSS0.P0.SPx1.p1.8.m8.2.3.3.2" xref="S1.SS1.SSS0.P0.SPx1.p1.8.m8.2.3.3.2.cmml"><mi id="S1.SS1.SSS0.P0.SPx1.p1.8.m8.2.3.3.2.2" xref="S1.SS1.SSS0.P0.SPx1.p1.8.m8.2.3.3.2.2.cmml">ρ</mi><mo id="S1.SS1.SSS0.P0.SPx1.p1.8.m8.2.3.3.2.3" xref="S1.SS1.SSS0.P0.SPx1.p1.8.m8.2.3.3.2.3.cmml">∗</mo></msup><mo id="S1.SS1.SSS0.P0.SPx1.p1.8.m8.2.3.3.1" xref="S1.SS1.SSS0.P0.SPx1.p1.8.m8.2.3.3.1.cmml">⁢</mo><mrow id="S1.SS1.SSS0.P0.SPx1.p1.8.m8.2.3.3.3.2" xref="S1.SS1.SSS0.P0.SPx1.p1.8.m8.2.3.3.cmml"><mo id="S1.SS1.SSS0.P0.SPx1.p1.8.m8.2.3.3.3.2.1" stretchy="false" xref="S1.SS1.SSS0.P0.SPx1.p1.8.m8.2.3.3.cmml">(</mo><mi id="S1.SS1.SSS0.P0.SPx1.p1.8.m8.2.2" xref="S1.SS1.SSS0.P0.SPx1.p1.8.m8.2.2.cmml">v</mi><mo id="S1.SS1.SSS0.P0.SPx1.p1.8.m8.2.3.3.3.2.2" stretchy="false" xref="S1.SS1.SSS0.P0.SPx1.p1.8.m8.2.3.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.SS1.SSS0.P0.SPx1.p1.8.m8.2b"><apply id="S1.SS1.SSS0.P0.SPx1.p1.8.m8.2.3.cmml" xref="S1.SS1.SSS0.P0.SPx1.p1.8.m8.2.3"><eq id="S1.SS1.SSS0.P0.SPx1.p1.8.m8.2.3.1.cmml" xref="S1.SS1.SSS0.P0.SPx1.p1.8.m8.2.3.1"></eq><apply id="S1.SS1.SSS0.P0.SPx1.p1.8.m8.2.3.2.cmml" xref="S1.SS1.SSS0.P0.SPx1.p1.8.m8.2.3.2"><times id="S1.SS1.SSS0.P0.SPx1.p1.8.m8.2.3.2.1.cmml" xref="S1.SS1.SSS0.P0.SPx1.p1.8.m8.2.3.2.1"></times><apply id="S1.SS1.SSS0.P0.SPx1.p1.8.m8.2.3.2.2.cmml" xref="S1.SS1.SSS0.P0.SPx1.p1.8.m8.2.3.2.2"><csymbol cd="ambiguous" id="S1.SS1.SSS0.P0.SPx1.p1.8.m8.2.3.2.2.1.cmml" xref="S1.SS1.SSS0.P0.SPx1.p1.8.m8.2.3.2.2">superscript</csymbol><ci id="S1.SS1.SSS0.P0.SPx1.p1.8.m8.2.3.2.2.2a.cmml" xref="S1.SS1.SSS0.P0.SPx1.p1.8.m8.2.3.2.2.2"><mtext class="ltx_mathvariant_italic" id="S1.SS1.SSS0.P0.SPx1.p1.8.m8.2.3.2.2.2.cmml" xref="S1.SS1.SSS0.P0.SPx1.p1.8.m8.2.3.2.2.2">g</mtext></ci><times id="S1.SS1.SSS0.P0.SPx1.p1.8.m8.2.3.2.2.3.cmml" xref="S1.SS1.SSS0.P0.SPx1.p1.8.m8.2.3.2.2.3"></times></apply><ci id="S1.SS1.SSS0.P0.SPx1.p1.8.m8.1.1.cmml" xref="S1.SS1.SSS0.P0.SPx1.p1.8.m8.1.1">𝑣</ci></apply><apply id="S1.SS1.SSS0.P0.SPx1.p1.8.m8.2.3.3.cmml" xref="S1.SS1.SSS0.P0.SPx1.p1.8.m8.2.3.3"><times id="S1.SS1.SSS0.P0.SPx1.p1.8.m8.2.3.3.1.cmml" xref="S1.SS1.SSS0.P0.SPx1.p1.8.m8.2.3.3.1"></times><apply id="S1.SS1.SSS0.P0.SPx1.p1.8.m8.2.3.3.2.cmml" xref="S1.SS1.SSS0.P0.SPx1.p1.8.m8.2.3.3.2"><csymbol cd="ambiguous" id="S1.SS1.SSS0.P0.SPx1.p1.8.m8.2.3.3.2.1.cmml" xref="S1.SS1.SSS0.P0.SPx1.p1.8.m8.2.3.3.2">superscript</csymbol><ci id="S1.SS1.SSS0.P0.SPx1.p1.8.m8.2.3.3.2.2.cmml" xref="S1.SS1.SSS0.P0.SPx1.p1.8.m8.2.3.3.2.2">𝜌</ci><times id="S1.SS1.SSS0.P0.SPx1.p1.8.m8.2.3.3.2.3.cmml" xref="S1.SS1.SSS0.P0.SPx1.p1.8.m8.2.3.3.2.3"></times></apply><ci id="S1.SS1.SSS0.P0.SPx1.p1.8.m8.2.2.cmml" xref="S1.SS1.SSS0.P0.SPx1.p1.8.m8.2.2">𝑣</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS1.SSS0.P0.SPx1.p1.8.m8.2c">\textsl{g}^{*}(v)=\rho^{*}(v)</annotation><annotation encoding="application/x-llamapun" id="S1.SS1.SSS0.P0.SPx1.p1.8.m8.2d">g start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ( italic_v ) = italic_ρ start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ( italic_v )</annotation></semantics></math>. This new definition for local out-degree is considerably shorter than the definition for local density. It allows us to show some previously unknown interesting properties of local density:</p> </div> </section> <section class="ltx_subparagraph" id="S1.SS1.SSS0.P0.SPx2"> <h5 class="ltx_title ltx_title_subparagraph">B: Results for dynamic algorithms. </h5> <div class="ltx_para" id="S1.SS1.SSS0.P0.SPx2.p1"> <p class="ltx_p" id="S1.SS1.SSS0.P0.SPx2.p1.6">We prove that in an <em class="ltx_emph ltx_font_italic" id="S1.SS1.SSS0.P0.SPx2.p1.6.1">approximately fair</em> orientation (a definition by Chekuri et al. <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2411.12694v2#bib.bib7" title="">7</a>]</cite>) the out-degree <math alttext="g(v)" class="ltx_Math" display="inline" id="S1.SS1.SSS0.P0.SPx2.p1.1.m1.1"><semantics id="S1.SS1.SSS0.P0.SPx2.p1.1.m1.1a"><mrow id="S1.SS1.SSS0.P0.SPx2.p1.1.m1.1.2" xref="S1.SS1.SSS0.P0.SPx2.p1.1.m1.1.2.cmml"><mi id="S1.SS1.SSS0.P0.SPx2.p1.1.m1.1.2.2" xref="S1.SS1.SSS0.P0.SPx2.p1.1.m1.1.2.2.cmml">g</mi><mo id="S1.SS1.SSS0.P0.SPx2.p1.1.m1.1.2.1" xref="S1.SS1.SSS0.P0.SPx2.p1.1.m1.1.2.1.cmml">⁢</mo><mrow id="S1.SS1.SSS0.P0.SPx2.p1.1.m1.1.2.3.2" xref="S1.SS1.SSS0.P0.SPx2.p1.1.m1.1.2.cmml"><mo id="S1.SS1.SSS0.P0.SPx2.p1.1.m1.1.2.3.2.1" stretchy="false" xref="S1.SS1.SSS0.P0.SPx2.p1.1.m1.1.2.cmml">(</mo><mi id="S1.SS1.SSS0.P0.SPx2.p1.1.m1.1.1" xref="S1.SS1.SSS0.P0.SPx2.p1.1.m1.1.1.cmml">v</mi><mo id="S1.SS1.SSS0.P0.SPx2.p1.1.m1.1.2.3.2.2" stretchy="false" xref="S1.SS1.SSS0.P0.SPx2.p1.1.m1.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.SS1.SSS0.P0.SPx2.p1.1.m1.1b"><apply id="S1.SS1.SSS0.P0.SPx2.p1.1.m1.1.2.cmml" xref="S1.SS1.SSS0.P0.SPx2.p1.1.m1.1.2"><times id="S1.SS1.SSS0.P0.SPx2.p1.1.m1.1.2.1.cmml" xref="S1.SS1.SSS0.P0.SPx2.p1.1.m1.1.2.1"></times><ci id="S1.SS1.SSS0.P0.SPx2.p1.1.m1.1.2.2.cmml" xref="S1.SS1.SSS0.P0.SPx2.p1.1.m1.1.2.2">𝑔</ci><ci id="S1.SS1.SSS0.P0.SPx2.p1.1.m1.1.1.cmml" xref="S1.SS1.SSS0.P0.SPx2.p1.1.m1.1.1">𝑣</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS1.SSS0.P0.SPx2.p1.1.m1.1c">g(v)</annotation><annotation encoding="application/x-llamapun" id="S1.SS1.SSS0.P0.SPx2.p1.1.m1.1d">italic_g ( italic_v )</annotation></semantics></math> of each vertex is a <math alttext="(1+\varepsilon)" class="ltx_Math" display="inline" id="S1.SS1.SSS0.P0.SPx2.p1.2.m2.1"><semantics id="S1.SS1.SSS0.P0.SPx2.p1.2.m2.1a"><mrow id="S1.SS1.SSS0.P0.SPx2.p1.2.m2.1.1.1" xref="S1.SS1.SSS0.P0.SPx2.p1.2.m2.1.1.1.1.cmml"><mo id="S1.SS1.SSS0.P0.SPx2.p1.2.m2.1.1.1.2" stretchy="false" xref="S1.SS1.SSS0.P0.SPx2.p1.2.m2.1.1.1.1.cmml">(</mo><mrow id="S1.SS1.SSS0.P0.SPx2.p1.2.m2.1.1.1.1" xref="S1.SS1.SSS0.P0.SPx2.p1.2.m2.1.1.1.1.cmml"><mn id="S1.SS1.SSS0.P0.SPx2.p1.2.m2.1.1.1.1.2" xref="S1.SS1.SSS0.P0.SPx2.p1.2.m2.1.1.1.1.2.cmml">1</mn><mo id="S1.SS1.SSS0.P0.SPx2.p1.2.m2.1.1.1.1.1" xref="S1.SS1.SSS0.P0.SPx2.p1.2.m2.1.1.1.1.1.cmml">+</mo><mi id="S1.SS1.SSS0.P0.SPx2.p1.2.m2.1.1.1.1.3" xref="S1.SS1.SSS0.P0.SPx2.p1.2.m2.1.1.1.1.3.cmml">ε</mi></mrow><mo id="S1.SS1.SSS0.P0.SPx2.p1.2.m2.1.1.1.3" stretchy="false" xref="S1.SS1.SSS0.P0.SPx2.p1.2.m2.1.1.1.1.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S1.SS1.SSS0.P0.SPx2.p1.2.m2.1b"><apply id="S1.SS1.SSS0.P0.SPx2.p1.2.m2.1.1.1.1.cmml" xref="S1.SS1.SSS0.P0.SPx2.p1.2.m2.1.1.1"><plus id="S1.SS1.SSS0.P0.SPx2.p1.2.m2.1.1.1.1.1.cmml" xref="S1.SS1.SSS0.P0.SPx2.p1.2.m2.1.1.1.1.1"></plus><cn id="S1.SS1.SSS0.P0.SPx2.p1.2.m2.1.1.1.1.2.cmml" type="integer" xref="S1.SS1.SSS0.P0.SPx2.p1.2.m2.1.1.1.1.2">1</cn><ci id="S1.SS1.SSS0.P0.SPx2.p1.2.m2.1.1.1.1.3.cmml" xref="S1.SS1.SSS0.P0.SPx2.p1.2.m2.1.1.1.1.3">𝜀</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS1.SSS0.P0.SPx2.p1.2.m2.1c">(1+\varepsilon)</annotation><annotation encoding="application/x-llamapun" id="S1.SS1.SSS0.P0.SPx2.p1.2.m2.1d">( 1 + italic_ε 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xref="S1.SS1.SSS0.P0.SPx2.p1.3.m3.2.3.2.cmml">(</mo><mi id="S1.SS1.SSS0.P0.SPx2.p1.3.m3.1.1" xref="S1.SS1.SSS0.P0.SPx2.p1.3.m3.1.1.cmml">v</mi><mo id="S1.SS1.SSS0.P0.SPx2.p1.3.m3.2.3.2.3.2.2" stretchy="false" xref="S1.SS1.SSS0.P0.SPx2.p1.3.m3.2.3.2.cmml">)</mo></mrow></mrow><mo id="S1.SS1.SSS0.P0.SPx2.p1.3.m3.2.3.1" xref="S1.SS1.SSS0.P0.SPx2.p1.3.m3.2.3.1.cmml">=</mo><mrow id="S1.SS1.SSS0.P0.SPx2.p1.3.m3.2.3.3" xref="S1.SS1.SSS0.P0.SPx2.p1.3.m3.2.3.3.cmml"><msup id="S1.SS1.SSS0.P0.SPx2.p1.3.m3.2.3.3.2" xref="S1.SS1.SSS0.P0.SPx2.p1.3.m3.2.3.3.2.cmml"><mi id="S1.SS1.SSS0.P0.SPx2.p1.3.m3.2.3.3.2.2" xref="S1.SS1.SSS0.P0.SPx2.p1.3.m3.2.3.3.2.2.cmml">ρ</mi><mo id="S1.SS1.SSS0.P0.SPx2.p1.3.m3.2.3.3.2.3" xref="S1.SS1.SSS0.P0.SPx2.p1.3.m3.2.3.3.2.3.cmml">∗</mo></msup><mo id="S1.SS1.SSS0.P0.SPx2.p1.3.m3.2.3.3.1" xref="S1.SS1.SSS0.P0.SPx2.p1.3.m3.2.3.3.1.cmml">⁢</mo><mrow id="S1.SS1.SSS0.P0.SPx2.p1.3.m3.2.3.3.3.2" xref="S1.SS1.SSS0.P0.SPx2.p1.3.m3.2.3.3.cmml"><mo 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xref="S1.SS1.SSS0.P0.SPx2.p1.3.m3.2.3.2.2">superscript</csymbol><ci id="S1.SS1.SSS0.P0.SPx2.p1.3.m3.2.3.2.2.2a.cmml" xref="S1.SS1.SSS0.P0.SPx2.p1.3.m3.2.3.2.2.2"><mtext class="ltx_mathvariant_italic" id="S1.SS1.SSS0.P0.SPx2.p1.3.m3.2.3.2.2.2.cmml" xref="S1.SS1.SSS0.P0.SPx2.p1.3.m3.2.3.2.2.2">g</mtext></ci><times id="S1.SS1.SSS0.P0.SPx2.p1.3.m3.2.3.2.2.3.cmml" xref="S1.SS1.SSS0.P0.SPx2.p1.3.m3.2.3.2.2.3"></times></apply><ci id="S1.SS1.SSS0.P0.SPx2.p1.3.m3.1.1.cmml" xref="S1.SS1.SSS0.P0.SPx2.p1.3.m3.1.1">𝑣</ci></apply><apply id="S1.SS1.SSS0.P0.SPx2.p1.3.m3.2.3.3.cmml" xref="S1.SS1.SSS0.P0.SPx2.p1.3.m3.2.3.3"><times id="S1.SS1.SSS0.P0.SPx2.p1.3.m3.2.3.3.1.cmml" xref="S1.SS1.SSS0.P0.SPx2.p1.3.m3.2.3.3.1"></times><apply id="S1.SS1.SSS0.P0.SPx2.p1.3.m3.2.3.3.2.cmml" xref="S1.SS1.SSS0.P0.SPx2.p1.3.m3.2.3.3.2"><csymbol cd="ambiguous" id="S1.SS1.SSS0.P0.SPx2.p1.3.m3.2.3.3.2.1.cmml" xref="S1.SS1.SSS0.P0.SPx2.p1.3.m3.2.3.3.2">superscript</csymbol><ci id="S1.SS1.SSS0.P0.SPx2.p1.3.m3.2.3.3.2.2.cmml" xref="S1.SS1.SSS0.P0.SPx2.p1.3.m3.2.3.3.2.2">𝜌</ci><times id="S1.SS1.SSS0.P0.SPx2.p1.3.m3.2.3.3.2.3.cmml" xref="S1.SS1.SSS0.P0.SPx2.p1.3.m3.2.3.3.2.3"></times></apply><ci id="S1.SS1.SSS0.P0.SPx2.p1.3.m3.2.2.cmml" xref="S1.SS1.SSS0.P0.SPx2.p1.3.m3.2.2">𝑣</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS1.SSS0.P0.SPx2.p1.3.m3.2c">\textsl{g}^{*}(v)=\rho^{*}(v)</annotation><annotation encoding="application/x-llamapun" id="S1.SS1.SSS0.P0.SPx2.p1.3.m3.2d">g start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ( italic_v ) = italic_ρ start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ( italic_v )</annotation></semantics></math> (Theorem <a class="ltx_ref" href="https://arxiv.org/html/2411.12694v2#S3.Thmtheorem5" title="Theorem 3.5. ‣ 3.B Results for dynamic algorithms ‣ 3 Results and organisation ‣ Local Density and its Distributed Approximation"><span class="ltx_text ltx_ref_tag">3.5</span></a>). This implies a previously unknown fact: that there exist dynamic polylogarithmic algorithms <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2411.12694v2#bib.bib7" title="">7</a>, <a class="ltx_ref" href="https://arxiv.org/html/2411.12694v2#bib.bib9" title="">9</a>, <a class="ltx_ref" href="https://arxiv.org/html/2411.12694v2#bib.bib8" title="">8</a>]</cite> where each vertex <math alttext="v" class="ltx_Math" display="inline" id="S1.SS1.SSS0.P0.SPx2.p1.4.m4.1"><semantics id="S1.SS1.SSS0.P0.SPx2.p1.4.m4.1a"><mi id="S1.SS1.SSS0.P0.SPx2.p1.4.m4.1.1" xref="S1.SS1.SSS0.P0.SPx2.p1.4.m4.1.1.cmml">v</mi><annotation-xml encoding="MathML-Content" id="S1.SS1.SSS0.P0.SPx2.p1.4.m4.1b"><ci id="S1.SS1.SSS0.P0.SPx2.p1.4.m4.1.1.cmml" xref="S1.SS1.SSS0.P0.SPx2.p1.4.m4.1.1">𝑣</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.SS1.SSS0.P0.SPx2.p1.4.m4.1c">v</annotation><annotation encoding="application/x-llamapun" id="S1.SS1.SSS0.P0.SPx2.p1.4.m4.1d">italic_v</annotation></semantics></math> maintains a <math alttext="(1+\varepsilon)" class="ltx_Math" display="inline" id="S1.SS1.SSS0.P0.SPx2.p1.5.m5.1"><semantics id="S1.SS1.SSS0.P0.SPx2.p1.5.m5.1a"><mrow id="S1.SS1.SSS0.P0.SPx2.p1.5.m5.1.1.1" xref="S1.SS1.SSS0.P0.SPx2.p1.5.m5.1.1.1.1.cmml"><mo id="S1.SS1.SSS0.P0.SPx2.p1.5.m5.1.1.1.2" stretchy="false" xref="S1.SS1.SSS0.P0.SPx2.p1.5.m5.1.1.1.1.cmml">(</mo><mrow id="S1.SS1.SSS0.P0.SPx2.p1.5.m5.1.1.1.1" xref="S1.SS1.SSS0.P0.SPx2.p1.5.m5.1.1.1.1.cmml"><mn id="S1.SS1.SSS0.P0.SPx2.p1.5.m5.1.1.1.1.2" xref="S1.SS1.SSS0.P0.SPx2.p1.5.m5.1.1.1.1.2.cmml">1</mn><mo id="S1.SS1.SSS0.P0.SPx2.p1.5.m5.1.1.1.1.1" xref="S1.SS1.SSS0.P0.SPx2.p1.5.m5.1.1.1.1.1.cmml">+</mo><mi id="S1.SS1.SSS0.P0.SPx2.p1.5.m5.1.1.1.1.3" xref="S1.SS1.SSS0.P0.SPx2.p1.5.m5.1.1.1.1.3.cmml">ε</mi></mrow><mo id="S1.SS1.SSS0.P0.SPx2.p1.5.m5.1.1.1.3" stretchy="false" xref="S1.SS1.SSS0.P0.SPx2.p1.5.m5.1.1.1.1.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S1.SS1.SSS0.P0.SPx2.p1.5.m5.1b"><apply id="S1.SS1.SSS0.P0.SPx2.p1.5.m5.1.1.1.1.cmml" xref="S1.SS1.SSS0.P0.SPx2.p1.5.m5.1.1.1"><plus id="S1.SS1.SSS0.P0.SPx2.p1.5.m5.1.1.1.1.1.cmml" xref="S1.SS1.SSS0.P0.SPx2.p1.5.m5.1.1.1.1.1"></plus><cn id="S1.SS1.SSS0.P0.SPx2.p1.5.m5.1.1.1.1.2.cmml" type="integer" xref="S1.SS1.SSS0.P0.SPx2.p1.5.m5.1.1.1.1.2">1</cn><ci id="S1.SS1.SSS0.P0.SPx2.p1.5.m5.1.1.1.1.3.cmml" xref="S1.SS1.SSS0.P0.SPx2.p1.5.m5.1.1.1.1.3">𝜀</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS1.SSS0.P0.SPx2.p1.5.m5.1c">(1+\varepsilon)</annotation><annotation encoding="application/x-llamapun" id="S1.SS1.SSS0.P0.SPx2.p1.5.m5.1d">( 1 + italic_ε )</annotation></semantics></math>-approximation of its local density <math alttext="\rho^{*}(v)" class="ltx_Math" display="inline" id="S1.SS1.SSS0.P0.SPx2.p1.6.m6.1"><semantics id="S1.SS1.SSS0.P0.SPx2.p1.6.m6.1a"><mrow id="S1.SS1.SSS0.P0.SPx2.p1.6.m6.1.2" xref="S1.SS1.SSS0.P0.SPx2.p1.6.m6.1.2.cmml"><msup id="S1.SS1.SSS0.P0.SPx2.p1.6.m6.1.2.2" xref="S1.SS1.SSS0.P0.SPx2.p1.6.m6.1.2.2.cmml"><mi id="S1.SS1.SSS0.P0.SPx2.p1.6.m6.1.2.2.2" xref="S1.SS1.SSS0.P0.SPx2.p1.6.m6.1.2.2.2.cmml">ρ</mi><mo id="S1.SS1.SSS0.P0.SPx2.p1.6.m6.1.2.2.3" xref="S1.SS1.SSS0.P0.SPx2.p1.6.m6.1.2.2.3.cmml">∗</mo></msup><mo id="S1.SS1.SSS0.P0.SPx2.p1.6.m6.1.2.1" xref="S1.SS1.SSS0.P0.SPx2.p1.6.m6.1.2.1.cmml">⁢</mo><mrow id="S1.SS1.SSS0.P0.SPx2.p1.6.m6.1.2.3.2" xref="S1.SS1.SSS0.P0.SPx2.p1.6.m6.1.2.cmml"><mo id="S1.SS1.SSS0.P0.SPx2.p1.6.m6.1.2.3.2.1" stretchy="false" xref="S1.SS1.SSS0.P0.SPx2.p1.6.m6.1.2.cmml">(</mo><mi id="S1.SS1.SSS0.P0.SPx2.p1.6.m6.1.1" xref="S1.SS1.SSS0.P0.SPx2.p1.6.m6.1.1.cmml">v</mi><mo id="S1.SS1.SSS0.P0.SPx2.p1.6.m6.1.2.3.2.2" stretchy="false" xref="S1.SS1.SSS0.P0.SPx2.p1.6.m6.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.SS1.SSS0.P0.SPx2.p1.6.m6.1b"><apply id="S1.SS1.SSS0.P0.SPx2.p1.6.m6.1.2.cmml" xref="S1.SS1.SSS0.P0.SPx2.p1.6.m6.1.2"><times id="S1.SS1.SSS0.P0.SPx2.p1.6.m6.1.2.1.cmml" xref="S1.SS1.SSS0.P0.SPx2.p1.6.m6.1.2.1"></times><apply id="S1.SS1.SSS0.P0.SPx2.p1.6.m6.1.2.2.cmml" xref="S1.SS1.SSS0.P0.SPx2.p1.6.m6.1.2.2"><csymbol cd="ambiguous" id="S1.SS1.SSS0.P0.SPx2.p1.6.m6.1.2.2.1.cmml" xref="S1.SS1.SSS0.P0.SPx2.p1.6.m6.1.2.2">superscript</csymbol><ci id="S1.SS1.SSS0.P0.SPx2.p1.6.m6.1.2.2.2.cmml" xref="S1.SS1.SSS0.P0.SPx2.p1.6.m6.1.2.2.2">𝜌</ci><times id="S1.SS1.SSS0.P0.SPx2.p1.6.m6.1.2.2.3.cmml" xref="S1.SS1.SSS0.P0.SPx2.p1.6.m6.1.2.2.3"></times></apply><ci id="S1.SS1.SSS0.P0.SPx2.p1.6.m6.1.1.cmml" xref="S1.SS1.SSS0.P0.SPx2.p1.6.m6.1.1">𝑣</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS1.SSS0.P0.SPx2.p1.6.m6.1c">\rho^{*}(v)</annotation><annotation encoding="application/x-llamapun" id="S1.SS1.SSS0.P0.SPx2.p1.6.m6.1d">italic_ρ start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ( italic_v )</annotation></semantics></math> as by Danisch, Chan, and Sozio <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2411.12694v2#bib.bib10" title="">10</a>]</cite>.</p> </div> </section> <section class="ltx_subparagraph" id="S1.SS1.SSS0.P0.SPx3"> <h5 class="ltx_title ltx_title_subparagraph">C: Results in LOCAL. </h5> <div class="ltx_para" id="S1.SS1.SSS0.P0.SPx3.p1"> <p class="ltx_p" id="S1.SS1.SSS0.P0.SPx3.p1.8">We show that each node <math alttext="v" class="ltx_Math" display="inline" id="S1.SS1.SSS0.P0.SPx3.p1.1.m1.1"><semantics id="S1.SS1.SSS0.P0.SPx3.p1.1.m1.1a"><mi id="S1.SS1.SSS0.P0.SPx3.p1.1.m1.1.1" xref="S1.SS1.SSS0.P0.SPx3.p1.1.m1.1.1.cmml">v</mi><annotation-xml encoding="MathML-Content" id="S1.SS1.SSS0.P0.SPx3.p1.1.m1.1b"><ci id="S1.SS1.SSS0.P0.SPx3.p1.1.m1.1.1.cmml" xref="S1.SS1.SSS0.P0.SPx3.p1.1.m1.1.1">𝑣</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.SS1.SSS0.P0.SPx3.p1.1.m1.1c">v</annotation><annotation encoding="application/x-llamapun" id="S1.SS1.SSS0.P0.SPx3.p1.1.m1.1d">italic_v</annotation></semantics></math> can obtain a <math alttext="(1+\varepsilon)" class="ltx_Math" display="inline" id="S1.SS1.SSS0.P0.SPx3.p1.2.m2.1"><semantics id="S1.SS1.SSS0.P0.SPx3.p1.2.m2.1a"><mrow id="S1.SS1.SSS0.P0.SPx3.p1.2.m2.1.1.1" xref="S1.SS1.SSS0.P0.SPx3.p1.2.m2.1.1.1.1.cmml"><mo id="S1.SS1.SSS0.P0.SPx3.p1.2.m2.1.1.1.2" stretchy="false" xref="S1.SS1.SSS0.P0.SPx3.p1.2.m2.1.1.1.1.cmml">(</mo><mrow id="S1.SS1.SSS0.P0.SPx3.p1.2.m2.1.1.1.1" xref="S1.SS1.SSS0.P0.SPx3.p1.2.m2.1.1.1.1.cmml"><mn id="S1.SS1.SSS0.P0.SPx3.p1.2.m2.1.1.1.1.2" xref="S1.SS1.SSS0.P0.SPx3.p1.2.m2.1.1.1.1.2.cmml">1</mn><mo id="S1.SS1.SSS0.P0.SPx3.p1.2.m2.1.1.1.1.1" xref="S1.SS1.SSS0.P0.SPx3.p1.2.m2.1.1.1.1.1.cmml">+</mo><mi id="S1.SS1.SSS0.P0.SPx3.p1.2.m2.1.1.1.1.3" xref="S1.SS1.SSS0.P0.SPx3.p1.2.m2.1.1.1.1.3.cmml">ε</mi></mrow><mo id="S1.SS1.SSS0.P0.SPx3.p1.2.m2.1.1.1.3" stretchy="false" xref="S1.SS1.SSS0.P0.SPx3.p1.2.m2.1.1.1.1.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S1.SS1.SSS0.P0.SPx3.p1.2.m2.1b"><apply id="S1.SS1.SSS0.P0.SPx3.p1.2.m2.1.1.1.1.cmml" xref="S1.SS1.SSS0.P0.SPx3.p1.2.m2.1.1.1"><plus id="S1.SS1.SSS0.P0.SPx3.p1.2.m2.1.1.1.1.1.cmml" xref="S1.SS1.SSS0.P0.SPx3.p1.2.m2.1.1.1.1.1"></plus><cn id="S1.SS1.SSS0.P0.SPx3.p1.2.m2.1.1.1.1.2.cmml" type="integer" xref="S1.SS1.SSS0.P0.SPx3.p1.2.m2.1.1.1.1.2">1</cn><ci id="S1.SS1.SSS0.P0.SPx3.p1.2.m2.1.1.1.1.3.cmml" xref="S1.SS1.SSS0.P0.SPx3.p1.2.m2.1.1.1.1.3">𝜀</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS1.SSS0.P0.SPx3.p1.2.m2.1c">(1+\varepsilon)</annotation><annotation encoding="application/x-llamapun" id="S1.SS1.SSS0.P0.SPx3.p1.2.m2.1d">( 1 + italic_ε )</annotation></semantics></math>-approximation of <math alttext="\rho^{*}(v)" class="ltx_Math" display="inline" id="S1.SS1.SSS0.P0.SPx3.p1.3.m3.1"><semantics id="S1.SS1.SSS0.P0.SPx3.p1.3.m3.1a"><mrow id="S1.SS1.SSS0.P0.SPx3.p1.3.m3.1.2" xref="S1.SS1.SSS0.P0.SPx3.p1.3.m3.1.2.cmml"><msup 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id="S1.SS1.SSS0.P0.SPx3.p1.3.m3.1.2.1.cmml" xref="S1.SS1.SSS0.P0.SPx3.p1.3.m3.1.2.1"></times><apply id="S1.SS1.SSS0.P0.SPx3.p1.3.m3.1.2.2.cmml" xref="S1.SS1.SSS0.P0.SPx3.p1.3.m3.1.2.2"><csymbol cd="ambiguous" id="S1.SS1.SSS0.P0.SPx3.p1.3.m3.1.2.2.1.cmml" xref="S1.SS1.SSS0.P0.SPx3.p1.3.m3.1.2.2">superscript</csymbol><ci id="S1.SS1.SSS0.P0.SPx3.p1.3.m3.1.2.2.2.cmml" xref="S1.SS1.SSS0.P0.SPx3.p1.3.m3.1.2.2.2">𝜌</ci><times id="S1.SS1.SSS0.P0.SPx3.p1.3.m3.1.2.2.3.cmml" xref="S1.SS1.SSS0.P0.SPx3.p1.3.m3.1.2.2.3"></times></apply><ci id="S1.SS1.SSS0.P0.SPx3.p1.3.m3.1.1.cmml" xref="S1.SS1.SSS0.P0.SPx3.p1.3.m3.1.1">𝑣</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS1.SSS0.P0.SPx3.p1.3.m3.1c">\rho^{*}(v)</annotation><annotation encoding="application/x-llamapun" id="S1.SS1.SSS0.P0.SPx3.p1.3.m3.1d">italic_ρ start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ( italic_v )</annotation></semantics></math> by surveying its <math alttext="O(\varepsilon^{-2}\log^{2}n)" class="ltx_Math" display="inline" id="S1.SS1.SSS0.P0.SPx3.p1.4.m4.1"><semantics id="S1.SS1.SSS0.P0.SPx3.p1.4.m4.1a"><mrow id="S1.SS1.SSS0.P0.SPx3.p1.4.m4.1.1" xref="S1.SS1.SSS0.P0.SPx3.p1.4.m4.1.1.cmml"><mi id="S1.SS1.SSS0.P0.SPx3.p1.4.m4.1.1.3" xref="S1.SS1.SSS0.P0.SPx3.p1.4.m4.1.1.3.cmml">O</mi><mo id="S1.SS1.SSS0.P0.SPx3.p1.4.m4.1.1.2" xref="S1.SS1.SSS0.P0.SPx3.p1.4.m4.1.1.2.cmml">⁢</mo><mrow id="S1.SS1.SSS0.P0.SPx3.p1.4.m4.1.1.1.1" xref="S1.SS1.SSS0.P0.SPx3.p1.4.m4.1.1.1.1.1.cmml"><mo id="S1.SS1.SSS0.P0.SPx3.p1.4.m4.1.1.1.1.2" stretchy="false" xref="S1.SS1.SSS0.P0.SPx3.p1.4.m4.1.1.1.1.1.cmml">(</mo><mrow id="S1.SS1.SSS0.P0.SPx3.p1.4.m4.1.1.1.1.1" xref="S1.SS1.SSS0.P0.SPx3.p1.4.m4.1.1.1.1.1.cmml"><msup id="S1.SS1.SSS0.P0.SPx3.p1.4.m4.1.1.1.1.1.2" xref="S1.SS1.SSS0.P0.SPx3.p1.4.m4.1.1.1.1.1.2.cmml"><mi id="S1.SS1.SSS0.P0.SPx3.p1.4.m4.1.1.1.1.1.2.2" xref="S1.SS1.SSS0.P0.SPx3.p1.4.m4.1.1.1.1.1.2.2.cmml">ε</mi><mrow id="S1.SS1.SSS0.P0.SPx3.p1.4.m4.1.1.1.1.1.2.3" xref="S1.SS1.SSS0.P0.SPx3.p1.4.m4.1.1.1.1.1.2.3.cmml"><mo id="S1.SS1.SSS0.P0.SPx3.p1.4.m4.1.1.1.1.1.2.3a" xref="S1.SS1.SSS0.P0.SPx3.p1.4.m4.1.1.1.1.1.2.3.cmml">−</mo><mn id="S1.SS1.SSS0.P0.SPx3.p1.4.m4.1.1.1.1.1.2.3.2" xref="S1.SS1.SSS0.P0.SPx3.p1.4.m4.1.1.1.1.1.2.3.2.cmml">2</mn></mrow></msup><mo id="S1.SS1.SSS0.P0.SPx3.p1.4.m4.1.1.1.1.1.1" lspace="0.167em" xref="S1.SS1.SSS0.P0.SPx3.p1.4.m4.1.1.1.1.1.1.cmml">⁢</mo><mrow id="S1.SS1.SSS0.P0.SPx3.p1.4.m4.1.1.1.1.1.3" xref="S1.SS1.SSS0.P0.SPx3.p1.4.m4.1.1.1.1.1.3.cmml"><msup id="S1.SS1.SSS0.P0.SPx3.p1.4.m4.1.1.1.1.1.3.1" xref="S1.SS1.SSS0.P0.SPx3.p1.4.m4.1.1.1.1.1.3.1.cmml"><mi id="S1.SS1.SSS0.P0.SPx3.p1.4.m4.1.1.1.1.1.3.1.2" xref="S1.SS1.SSS0.P0.SPx3.p1.4.m4.1.1.1.1.1.3.1.2.cmml">log</mi><mn id="S1.SS1.SSS0.P0.SPx3.p1.4.m4.1.1.1.1.1.3.1.3" xref="S1.SS1.SSS0.P0.SPx3.p1.4.m4.1.1.1.1.1.3.1.3.cmml">2</mn></msup><mo id="S1.SS1.SSS0.P0.SPx3.p1.4.m4.1.1.1.1.1.3a" lspace="0.167em" xref="S1.SS1.SSS0.P0.SPx3.p1.4.m4.1.1.1.1.1.3.cmml">⁡</mo><mi 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id="S1.SS1.SSS0.P0.SPx3.p1.4.m4.1.1.1.1.1.2.1.cmml" xref="S1.SS1.SSS0.P0.SPx3.p1.4.m4.1.1.1.1.1.2">superscript</csymbol><ci id="S1.SS1.SSS0.P0.SPx3.p1.4.m4.1.1.1.1.1.2.2.cmml" xref="S1.SS1.SSS0.P0.SPx3.p1.4.m4.1.1.1.1.1.2.2">𝜀</ci><apply id="S1.SS1.SSS0.P0.SPx3.p1.4.m4.1.1.1.1.1.2.3.cmml" xref="S1.SS1.SSS0.P0.SPx3.p1.4.m4.1.1.1.1.1.2.3"><minus id="S1.SS1.SSS0.P0.SPx3.p1.4.m4.1.1.1.1.1.2.3.1.cmml" xref="S1.SS1.SSS0.P0.SPx3.p1.4.m4.1.1.1.1.1.2.3"></minus><cn id="S1.SS1.SSS0.P0.SPx3.p1.4.m4.1.1.1.1.1.2.3.2.cmml" type="integer" xref="S1.SS1.SSS0.P0.SPx3.p1.4.m4.1.1.1.1.1.2.3.2">2</cn></apply></apply><apply id="S1.SS1.SSS0.P0.SPx3.p1.4.m4.1.1.1.1.1.3.cmml" xref="S1.SS1.SSS0.P0.SPx3.p1.4.m4.1.1.1.1.1.3"><apply id="S1.SS1.SSS0.P0.SPx3.p1.4.m4.1.1.1.1.1.3.1.cmml" xref="S1.SS1.SSS0.P0.SPx3.p1.4.m4.1.1.1.1.1.3.1"><csymbol cd="ambiguous" id="S1.SS1.SSS0.P0.SPx3.p1.4.m4.1.1.1.1.1.3.1.1.cmml" xref="S1.SS1.SSS0.P0.SPx3.p1.4.m4.1.1.1.1.1.3.1">superscript</csymbol><log id="S1.SS1.SSS0.P0.SPx3.p1.4.m4.1.1.1.1.1.3.1.2.cmml" xref="S1.SS1.SSS0.P0.SPx3.p1.4.m4.1.1.1.1.1.3.1.2"></log><cn id="S1.SS1.SSS0.P0.SPx3.p1.4.m4.1.1.1.1.1.3.1.3.cmml" type="integer" xref="S1.SS1.SSS0.P0.SPx3.p1.4.m4.1.1.1.1.1.3.1.3">2</cn></apply><ci id="S1.SS1.SSS0.P0.SPx3.p1.4.m4.1.1.1.1.1.3.2.cmml" xref="S1.SS1.SSS0.P0.SPx3.p1.4.m4.1.1.1.1.1.3.2">𝑛</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS1.SSS0.P0.SPx3.p1.4.m4.1c">O(\varepsilon^{-2}\log^{2}n)</annotation><annotation encoding="application/x-llamapun" id="S1.SS1.SSS0.P0.SPx3.p1.4.m4.1d">italic_O ( italic_ε start_POSTSUPERSCRIPT - 2 end_POSTSUPERSCRIPT roman_log start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_n )</annotation></semantics></math>-hop neighbourhood. This induces a LOCAL algorithm where each vertex <math alttext="v\in V" class="ltx_Math" display="inline" id="S1.SS1.SSS0.P0.SPx3.p1.5.m5.1"><semantics id="S1.SS1.SSS0.P0.SPx3.p1.5.m5.1a"><mrow id="S1.SS1.SSS0.P0.SPx3.p1.5.m5.1.1" xref="S1.SS1.SSS0.P0.SPx3.p1.5.m5.1.1.cmml"><mi id="S1.SS1.SSS0.P0.SPx3.p1.5.m5.1.1.2" xref="S1.SS1.SSS0.P0.SPx3.p1.5.m5.1.1.2.cmml">v</mi><mo id="S1.SS1.SSS0.P0.SPx3.p1.5.m5.1.1.1" xref="S1.SS1.SSS0.P0.SPx3.p1.5.m5.1.1.1.cmml">∈</mo><mi id="S1.SS1.SSS0.P0.SPx3.p1.5.m5.1.1.3" xref="S1.SS1.SSS0.P0.SPx3.p1.5.m5.1.1.3.cmml">V</mi></mrow><annotation-xml encoding="MathML-Content" id="S1.SS1.SSS0.P0.SPx3.p1.5.m5.1b"><apply id="S1.SS1.SSS0.P0.SPx3.p1.5.m5.1.1.cmml" xref="S1.SS1.SSS0.P0.SPx3.p1.5.m5.1.1"><in id="S1.SS1.SSS0.P0.SPx3.p1.5.m5.1.1.1.cmml" xref="S1.SS1.SSS0.P0.SPx3.p1.5.m5.1.1.1"></in><ci id="S1.SS1.SSS0.P0.SPx3.p1.5.m5.1.1.2.cmml" xref="S1.SS1.SSS0.P0.SPx3.p1.5.m5.1.1.2">𝑣</ci><ci id="S1.SS1.SSS0.P0.SPx3.p1.5.m5.1.1.3.cmml" xref="S1.SS1.SSS0.P0.SPx3.p1.5.m5.1.1.3">𝑉</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS1.SSS0.P0.SPx3.p1.5.m5.1c">v\in V</annotation><annotation encoding="application/x-llamapun" id="S1.SS1.SSS0.P0.SPx3.p1.5.m5.1d">italic_v ∈ italic_V</annotation></semantics></math> computes a <math alttext="(1+\varepsilon)" class="ltx_Math" display="inline" id="S1.SS1.SSS0.P0.SPx3.p1.6.m6.1"><semantics id="S1.SS1.SSS0.P0.SPx3.p1.6.m6.1a"><mrow id="S1.SS1.SSS0.P0.SPx3.p1.6.m6.1.1.1" xref="S1.SS1.SSS0.P0.SPx3.p1.6.m6.1.1.1.1.cmml"><mo id="S1.SS1.SSS0.P0.SPx3.p1.6.m6.1.1.1.2" stretchy="false" xref="S1.SS1.SSS0.P0.SPx3.p1.6.m6.1.1.1.1.cmml">(</mo><mrow id="S1.SS1.SSS0.P0.SPx3.p1.6.m6.1.1.1.1" xref="S1.SS1.SSS0.P0.SPx3.p1.6.m6.1.1.1.1.cmml"><mn id="S1.SS1.SSS0.P0.SPx3.p1.6.m6.1.1.1.1.2" xref="S1.SS1.SSS0.P0.SPx3.p1.6.m6.1.1.1.1.2.cmml">1</mn><mo id="S1.SS1.SSS0.P0.SPx3.p1.6.m6.1.1.1.1.1" xref="S1.SS1.SSS0.P0.SPx3.p1.6.m6.1.1.1.1.1.cmml">+</mo><mi id="S1.SS1.SSS0.P0.SPx3.p1.6.m6.1.1.1.1.3" xref="S1.SS1.SSS0.P0.SPx3.p1.6.m6.1.1.1.1.3.cmml">ε</mi></mrow><mo id="S1.SS1.SSS0.P0.SPx3.p1.6.m6.1.1.1.3" stretchy="false" xref="S1.SS1.SSS0.P0.SPx3.p1.6.m6.1.1.1.1.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S1.SS1.SSS0.P0.SPx3.p1.6.m6.1b"><apply id="S1.SS1.SSS0.P0.SPx3.p1.6.m6.1.1.1.1.cmml" xref="S1.SS1.SSS0.P0.SPx3.p1.6.m6.1.1.1"><plus id="S1.SS1.SSS0.P0.SPx3.p1.6.m6.1.1.1.1.1.cmml" xref="S1.SS1.SSS0.P0.SPx3.p1.6.m6.1.1.1.1.1"></plus><cn id="S1.SS1.SSS0.P0.SPx3.p1.6.m6.1.1.1.1.2.cmml" type="integer" xref="S1.SS1.SSS0.P0.SPx3.p1.6.m6.1.1.1.1.2">1</cn><ci id="S1.SS1.SSS0.P0.SPx3.p1.6.m6.1.1.1.1.3.cmml" xref="S1.SS1.SSS0.P0.SPx3.p1.6.m6.1.1.1.1.3">𝜀</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS1.SSS0.P0.SPx3.p1.6.m6.1c">(1+\varepsilon)</annotation><annotation encoding="application/x-llamapun" id="S1.SS1.SSS0.P0.SPx3.p1.6.m6.1d">( 1 + italic_ε 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id="S1.SS1.SSS0.P0.SPx3.p1.7.m7.1c">\rho^{*}(v)</annotation><annotation encoding="application/x-llamapun" id="S1.SS1.SSS0.P0.SPx3.p1.7.m7.1d">italic_ρ start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ( italic_v )</annotation></semantics></math> in <math alttext="O(\varepsilon^{-2}\log^{2}n)" class="ltx_Math" display="inline" id="S1.SS1.SSS0.P0.SPx3.p1.8.m8.1"><semantics id="S1.SS1.SSS0.P0.SPx3.p1.8.m8.1a"><mrow id="S1.SS1.SSS0.P0.SPx3.p1.8.m8.1.1" xref="S1.SS1.SSS0.P0.SPx3.p1.8.m8.1.1.cmml"><mi id="S1.SS1.SSS0.P0.SPx3.p1.8.m8.1.1.3" xref="S1.SS1.SSS0.P0.SPx3.p1.8.m8.1.1.3.cmml">O</mi><mo id="S1.SS1.SSS0.P0.SPx3.p1.8.m8.1.1.2" xref="S1.SS1.SSS0.P0.SPx3.p1.8.m8.1.1.2.cmml">⁢</mo><mrow id="S1.SS1.SSS0.P0.SPx3.p1.8.m8.1.1.1.1" xref="S1.SS1.SSS0.P0.SPx3.p1.8.m8.1.1.1.1.1.cmml"><mo id="S1.SS1.SSS0.P0.SPx3.p1.8.m8.1.1.1.1.2" stretchy="false" xref="S1.SS1.SSS0.P0.SPx3.p1.8.m8.1.1.1.1.1.cmml">(</mo><mrow id="S1.SS1.SSS0.P0.SPx3.p1.8.m8.1.1.1.1.1" 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xref="S1.SS1.SSS0.P0.SPx3.p1.8.m8.1.1.1.1.1.3.1.2.cmml">log</mi><mn id="S1.SS1.SSS0.P0.SPx3.p1.8.m8.1.1.1.1.1.3.1.3" xref="S1.SS1.SSS0.P0.SPx3.p1.8.m8.1.1.1.1.1.3.1.3.cmml">2</mn></msup><mo id="S1.SS1.SSS0.P0.SPx3.p1.8.m8.1.1.1.1.1.3a" lspace="0.167em" xref="S1.SS1.SSS0.P0.SPx3.p1.8.m8.1.1.1.1.1.3.cmml">⁡</mo><mi id="S1.SS1.SSS0.P0.SPx3.p1.8.m8.1.1.1.1.1.3.2" xref="S1.SS1.SSS0.P0.SPx3.p1.8.m8.1.1.1.1.1.3.2.cmml">n</mi></mrow></mrow><mo id="S1.SS1.SSS0.P0.SPx3.p1.8.m8.1.1.1.1.3" stretchy="false" xref="S1.SS1.SSS0.P0.SPx3.p1.8.m8.1.1.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.SS1.SSS0.P0.SPx3.p1.8.m8.1b"><apply id="S1.SS1.SSS0.P0.SPx3.p1.8.m8.1.1.cmml" xref="S1.SS1.SSS0.P0.SPx3.p1.8.m8.1.1"><times id="S1.SS1.SSS0.P0.SPx3.p1.8.m8.1.1.2.cmml" xref="S1.SS1.SSS0.P0.SPx3.p1.8.m8.1.1.2"></times><ci id="S1.SS1.SSS0.P0.SPx3.p1.8.m8.1.1.3.cmml" xref="S1.SS1.SSS0.P0.SPx3.p1.8.m8.1.1.3">𝑂</ci><apply id="S1.SS1.SSS0.P0.SPx3.p1.8.m8.1.1.1.1.1.cmml" xref="S1.SS1.SSS0.P0.SPx3.p1.8.m8.1.1.1.1"><times id="S1.SS1.SSS0.P0.SPx3.p1.8.m8.1.1.1.1.1.1.cmml" xref="S1.SS1.SSS0.P0.SPx3.p1.8.m8.1.1.1.1.1.1"></times><apply id="S1.SS1.SSS0.P0.SPx3.p1.8.m8.1.1.1.1.1.2.cmml" xref="S1.SS1.SSS0.P0.SPx3.p1.8.m8.1.1.1.1.1.2"><csymbol cd="ambiguous" id="S1.SS1.SSS0.P0.SPx3.p1.8.m8.1.1.1.1.1.2.1.cmml" xref="S1.SS1.SSS0.P0.SPx3.p1.8.m8.1.1.1.1.1.2">superscript</csymbol><ci id="S1.SS1.SSS0.P0.SPx3.p1.8.m8.1.1.1.1.1.2.2.cmml" xref="S1.SS1.SSS0.P0.SPx3.p1.8.m8.1.1.1.1.1.2.2">𝜀</ci><apply id="S1.SS1.SSS0.P0.SPx3.p1.8.m8.1.1.1.1.1.2.3.cmml" xref="S1.SS1.SSS0.P0.SPx3.p1.8.m8.1.1.1.1.1.2.3"><minus id="S1.SS1.SSS0.P0.SPx3.p1.8.m8.1.1.1.1.1.2.3.1.cmml" xref="S1.SS1.SSS0.P0.SPx3.p1.8.m8.1.1.1.1.1.2.3"></minus><cn id="S1.SS1.SSS0.P0.SPx3.p1.8.m8.1.1.1.1.1.2.3.2.cmml" type="integer" xref="S1.SS1.SSS0.P0.SPx3.p1.8.m8.1.1.1.1.1.2.3.2">2</cn></apply></apply><apply id="S1.SS1.SSS0.P0.SPx3.p1.8.m8.1.1.1.1.1.3.cmml" xref="S1.SS1.SSS0.P0.SPx3.p1.8.m8.1.1.1.1.1.3"><apply id="S1.SS1.SSS0.P0.SPx3.p1.8.m8.1.1.1.1.1.3.1.cmml" xref="S1.SS1.SSS0.P0.SPx3.p1.8.m8.1.1.1.1.1.3.1"><csymbol cd="ambiguous" id="S1.SS1.SSS0.P0.SPx3.p1.8.m8.1.1.1.1.1.3.1.1.cmml" xref="S1.SS1.SSS0.P0.SPx3.p1.8.m8.1.1.1.1.1.3.1">superscript</csymbol><log id="S1.SS1.SSS0.P0.SPx3.p1.8.m8.1.1.1.1.1.3.1.2.cmml" xref="S1.SS1.SSS0.P0.SPx3.p1.8.m8.1.1.1.1.1.3.1.2"></log><cn id="S1.SS1.SSS0.P0.SPx3.p1.8.m8.1.1.1.1.1.3.1.3.cmml" type="integer" xref="S1.SS1.SSS0.P0.SPx3.p1.8.m8.1.1.1.1.1.3.1.3">2</cn></apply><ci id="S1.SS1.SSS0.P0.SPx3.p1.8.m8.1.1.1.1.1.3.2.cmml" xref="S1.SS1.SSS0.P0.SPx3.p1.8.m8.1.1.1.1.1.3.2">𝑛</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS1.SSS0.P0.SPx3.p1.8.m8.1c">O(\varepsilon^{-2}\log^{2}n)</annotation><annotation encoding="application/x-llamapun" id="S1.SS1.SSS0.P0.SPx3.p1.8.m8.1d">italic_O ( italic_ε start_POSTSUPERSCRIPT - 2 end_POSTSUPERSCRIPT roman_log start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_n )</annotation></semantics></math> rounds.</p> </div> <div class="ltx_para" id="S1.SS1.SSS0.P0.SPx3.p2"> <p class="ltx_p" id="S1.SS1.SSS0.P0.SPx3.p2.14"><em class="ltx_emph ltx_font_italic" id="S1.SS1.SSS0.P0.SPx3.p2.14.1">Commentary on runtime:</em> We observe that <math alttext="\rho^{\max}(G)" class="ltx_Math" display="inline" id="S1.SS1.SSS0.P0.SPx3.p2.1.m1.1"><semantics id="S1.SS1.SSS0.P0.SPx3.p2.1.m1.1a"><mrow id="S1.SS1.SSS0.P0.SPx3.p2.1.m1.1.2" xref="S1.SS1.SSS0.P0.SPx3.p2.1.m1.1.2.cmml"><msup id="S1.SS1.SSS0.P0.SPx3.p2.1.m1.1.2.2" xref="S1.SS1.SSS0.P0.SPx3.p2.1.m1.1.2.2.cmml"><mi id="S1.SS1.SSS0.P0.SPx3.p2.1.m1.1.2.2.2" xref="S1.SS1.SSS0.P0.SPx3.p2.1.m1.1.2.2.2.cmml">ρ</mi><mi id="S1.SS1.SSS0.P0.SPx3.p2.1.m1.1.2.2.3" xref="S1.SS1.SSS0.P0.SPx3.p2.1.m1.1.2.2.3.cmml">max</mi></msup><mo id="S1.SS1.SSS0.P0.SPx3.p2.1.m1.1.2.1" xref="S1.SS1.SSS0.P0.SPx3.p2.1.m1.1.2.1.cmml">⁢</mo><mrow id="S1.SS1.SSS0.P0.SPx3.p2.1.m1.1.2.3.2" xref="S1.SS1.SSS0.P0.SPx3.p2.1.m1.1.2.cmml"><mo id="S1.SS1.SSS0.P0.SPx3.p2.1.m1.1.2.3.2.1" stretchy="false" xref="S1.SS1.SSS0.P0.SPx3.p2.1.m1.1.2.cmml">(</mo><mi id="S1.SS1.SSS0.P0.SPx3.p2.1.m1.1.1" xref="S1.SS1.SSS0.P0.SPx3.p2.1.m1.1.1.cmml">G</mi><mo id="S1.SS1.SSS0.P0.SPx3.p2.1.m1.1.2.3.2.2" stretchy="false" xref="S1.SS1.SSS0.P0.SPx3.p2.1.m1.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.SS1.SSS0.P0.SPx3.p2.1.m1.1b"><apply id="S1.SS1.SSS0.P0.SPx3.p2.1.m1.1.2.cmml" xref="S1.SS1.SSS0.P0.SPx3.p2.1.m1.1.2"><times id="S1.SS1.SSS0.P0.SPx3.p2.1.m1.1.2.1.cmml" xref="S1.SS1.SSS0.P0.SPx3.p2.1.m1.1.2.1"></times><apply id="S1.SS1.SSS0.P0.SPx3.p2.1.m1.1.2.2.cmml" xref="S1.SS1.SSS0.P0.SPx3.p2.1.m1.1.2.2"><csymbol cd="ambiguous" id="S1.SS1.SSS0.P0.SPx3.p2.1.m1.1.2.2.1.cmml" xref="S1.SS1.SSS0.P0.SPx3.p2.1.m1.1.2.2">superscript</csymbol><ci id="S1.SS1.SSS0.P0.SPx3.p2.1.m1.1.2.2.2.cmml" xref="S1.SS1.SSS0.P0.SPx3.p2.1.m1.1.2.2.2">𝜌</ci><max id="S1.SS1.SSS0.P0.SPx3.p2.1.m1.1.2.2.3.cmml" xref="S1.SS1.SSS0.P0.SPx3.p2.1.m1.1.2.2.3"></max></apply><ci id="S1.SS1.SSS0.P0.SPx3.p2.1.m1.1.1.cmml" xref="S1.SS1.SSS0.P0.SPx3.p2.1.m1.1.1">𝐺</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS1.SSS0.P0.SPx3.p2.1.m1.1c">\rho^{\max}(G)</annotation><annotation encoding="application/x-llamapun" id="S1.SS1.SSS0.P0.SPx3.p2.1.m1.1d">italic_ρ start_POSTSUPERSCRIPT roman_max end_POSTSUPERSCRIPT ( italic_G )</annotation></semantics></math> can be computed in LOCAL in <math alttext="O(\varepsilon^{-1}\log n)" class="ltx_Math" display="inline" id="S1.SS1.SSS0.P0.SPx3.p2.2.m2.1"><semantics id="S1.SS1.SSS0.P0.SPx3.p2.2.m2.1a"><mrow id="S1.SS1.SSS0.P0.SPx3.p2.2.m2.1.1" xref="S1.SS1.SSS0.P0.SPx3.p2.2.m2.1.1.cmml"><mi id="S1.SS1.SSS0.P0.SPx3.p2.2.m2.1.1.3" xref="S1.SS1.SSS0.P0.SPx3.p2.2.m2.1.1.3.cmml">O</mi><mo id="S1.SS1.SSS0.P0.SPx3.p2.2.m2.1.1.2" xref="S1.SS1.SSS0.P0.SPx3.p2.2.m2.1.1.2.cmml">⁢</mo><mrow id="S1.SS1.SSS0.P0.SPx3.p2.2.m2.1.1.1.1" xref="S1.SS1.SSS0.P0.SPx3.p2.2.m2.1.1.1.1.1.cmml"><mo id="S1.SS1.SSS0.P0.SPx3.p2.2.m2.1.1.1.1.2" stretchy="false" xref="S1.SS1.SSS0.P0.SPx3.p2.2.m2.1.1.1.1.1.cmml">(</mo><mrow id="S1.SS1.SSS0.P0.SPx3.p2.2.m2.1.1.1.1.1" xref="S1.SS1.SSS0.P0.SPx3.p2.2.m2.1.1.1.1.1.cmml"><msup id="S1.SS1.SSS0.P0.SPx3.p2.2.m2.1.1.1.1.1.2" xref="S1.SS1.SSS0.P0.SPx3.p2.2.m2.1.1.1.1.1.2.cmml"><mi id="S1.SS1.SSS0.P0.SPx3.p2.2.m2.1.1.1.1.1.2.2" xref="S1.SS1.SSS0.P0.SPx3.p2.2.m2.1.1.1.1.1.2.2.cmml">ε</mi><mrow id="S1.SS1.SSS0.P0.SPx3.p2.2.m2.1.1.1.1.1.2.3" xref="S1.SS1.SSS0.P0.SPx3.p2.2.m2.1.1.1.1.1.2.3.cmml"><mo id="S1.SS1.SSS0.P0.SPx3.p2.2.m2.1.1.1.1.1.2.3a" xref="S1.SS1.SSS0.P0.SPx3.p2.2.m2.1.1.1.1.1.2.3.cmml">−</mo><mn id="S1.SS1.SSS0.P0.SPx3.p2.2.m2.1.1.1.1.1.2.3.2" xref="S1.SS1.SSS0.P0.SPx3.p2.2.m2.1.1.1.1.1.2.3.2.cmml">1</mn></mrow></msup><mo id="S1.SS1.SSS0.P0.SPx3.p2.2.m2.1.1.1.1.1.1" lspace="0.167em" xref="S1.SS1.SSS0.P0.SPx3.p2.2.m2.1.1.1.1.1.1.cmml">⁢</mo><mrow id="S1.SS1.SSS0.P0.SPx3.p2.2.m2.1.1.1.1.1.3" xref="S1.SS1.SSS0.P0.SPx3.p2.2.m2.1.1.1.1.1.3.cmml"><mi id="S1.SS1.SSS0.P0.SPx3.p2.2.m2.1.1.1.1.1.3.1" xref="S1.SS1.SSS0.P0.SPx3.p2.2.m2.1.1.1.1.1.3.1.cmml">log</mi><mo id="S1.SS1.SSS0.P0.SPx3.p2.2.m2.1.1.1.1.1.3a" lspace="0.167em" xref="S1.SS1.SSS0.P0.SPx3.p2.2.m2.1.1.1.1.1.3.cmml">⁡</mo><mi id="S1.SS1.SSS0.P0.SPx3.p2.2.m2.1.1.1.1.1.3.2" xref="S1.SS1.SSS0.P0.SPx3.p2.2.m2.1.1.1.1.1.3.2.cmml">n</mi></mrow></mrow><mo id="S1.SS1.SSS0.P0.SPx3.p2.2.m2.1.1.1.1.3" stretchy="false" xref="S1.SS1.SSS0.P0.SPx3.p2.2.m2.1.1.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.SS1.SSS0.P0.SPx3.p2.2.m2.1b"><apply id="S1.SS1.SSS0.P0.SPx3.p2.2.m2.1.1.cmml" xref="S1.SS1.SSS0.P0.SPx3.p2.2.m2.1.1"><times id="S1.SS1.SSS0.P0.SPx3.p2.2.m2.1.1.2.cmml" xref="S1.SS1.SSS0.P0.SPx3.p2.2.m2.1.1.2"></times><ci id="S1.SS1.SSS0.P0.SPx3.p2.2.m2.1.1.3.cmml" xref="S1.SS1.SSS0.P0.SPx3.p2.2.m2.1.1.3">𝑂</ci><apply id="S1.SS1.SSS0.P0.SPx3.p2.2.m2.1.1.1.1.1.cmml" xref="S1.SS1.SSS0.P0.SPx3.p2.2.m2.1.1.1.1"><times id="S1.SS1.SSS0.P0.SPx3.p2.2.m2.1.1.1.1.1.1.cmml" xref="S1.SS1.SSS0.P0.SPx3.p2.2.m2.1.1.1.1.1.1"></times><apply id="S1.SS1.SSS0.P0.SPx3.p2.2.m2.1.1.1.1.1.2.cmml" xref="S1.SS1.SSS0.P0.SPx3.p2.2.m2.1.1.1.1.1.2"><csymbol cd="ambiguous" id="S1.SS1.SSS0.P0.SPx3.p2.2.m2.1.1.1.1.1.2.1.cmml" xref="S1.SS1.SSS0.P0.SPx3.p2.2.m2.1.1.1.1.1.2">superscript</csymbol><ci id="S1.SS1.SSS0.P0.SPx3.p2.2.m2.1.1.1.1.1.2.2.cmml" xref="S1.SS1.SSS0.P0.SPx3.p2.2.m2.1.1.1.1.1.2.2">𝜀</ci><apply id="S1.SS1.SSS0.P0.SPx3.p2.2.m2.1.1.1.1.1.2.3.cmml" xref="S1.SS1.SSS0.P0.SPx3.p2.2.m2.1.1.1.1.1.2.3"><minus id="S1.SS1.SSS0.P0.SPx3.p2.2.m2.1.1.1.1.1.2.3.1.cmml" xref="S1.SS1.SSS0.P0.SPx3.p2.2.m2.1.1.1.1.1.2.3"></minus><cn id="S1.SS1.SSS0.P0.SPx3.p2.2.m2.1.1.1.1.1.2.3.2.cmml" type="integer" xref="S1.SS1.SSS0.P0.SPx3.p2.2.m2.1.1.1.1.1.2.3.2">1</cn></apply></apply><apply id="S1.SS1.SSS0.P0.SPx3.p2.2.m2.1.1.1.1.1.3.cmml" xref="S1.SS1.SSS0.P0.SPx3.p2.2.m2.1.1.1.1.1.3"><log id="S1.SS1.SSS0.P0.SPx3.p2.2.m2.1.1.1.1.1.3.1.cmml" xref="S1.SS1.SSS0.P0.SPx3.p2.2.m2.1.1.1.1.1.3.1"></log><ci id="S1.SS1.SSS0.P0.SPx3.p2.2.m2.1.1.1.1.1.3.2.cmml" xref="S1.SS1.SSS0.P0.SPx3.p2.2.m2.1.1.1.1.1.3.2">𝑛</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS1.SSS0.P0.SPx3.p2.2.m2.1c">O(\varepsilon^{-1}\log n)</annotation><annotation encoding="application/x-llamapun" id="S1.SS1.SSS0.P0.SPx3.p2.2.m2.1d">italic_O ( italic_ε start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT roman_log italic_n )</annotation></semantics></math> rounds. In contrast, the stricter local density is computed in <math alttext="O(\varepsilon^{-2}\log^{2}n)" class="ltx_Math" display="inline" id="S1.SS1.SSS0.P0.SPx3.p2.3.m3.1"><semantics id="S1.SS1.SSS0.P0.SPx3.p2.3.m3.1a"><mrow id="S1.SS1.SSS0.P0.SPx3.p2.3.m3.1.1" xref="S1.SS1.SSS0.P0.SPx3.p2.3.m3.1.1.cmml"><mi id="S1.SS1.SSS0.P0.SPx3.p2.3.m3.1.1.3" xref="S1.SS1.SSS0.P0.SPx3.p2.3.m3.1.1.3.cmml">O</mi><mo id="S1.SS1.SSS0.P0.SPx3.p2.3.m3.1.1.2" xref="S1.SS1.SSS0.P0.SPx3.p2.3.m3.1.1.2.cmml">⁢</mo><mrow id="S1.SS1.SSS0.P0.SPx3.p2.3.m3.1.1.1.1" xref="S1.SS1.SSS0.P0.SPx3.p2.3.m3.1.1.1.1.1.cmml"><mo id="S1.SS1.SSS0.P0.SPx3.p2.3.m3.1.1.1.1.2" stretchy="false" xref="S1.SS1.SSS0.P0.SPx3.p2.3.m3.1.1.1.1.1.cmml">(</mo><mrow id="S1.SS1.SSS0.P0.SPx3.p2.3.m3.1.1.1.1.1" xref="S1.SS1.SSS0.P0.SPx3.p2.3.m3.1.1.1.1.1.cmml"><msup id="S1.SS1.SSS0.P0.SPx3.p2.3.m3.1.1.1.1.1.2" xref="S1.SS1.SSS0.P0.SPx3.p2.3.m3.1.1.1.1.1.2.cmml"><mi id="S1.SS1.SSS0.P0.SPx3.p2.3.m3.1.1.1.1.1.2.2" xref="S1.SS1.SSS0.P0.SPx3.p2.3.m3.1.1.1.1.1.2.2.cmml">ε</mi><mrow id="S1.SS1.SSS0.P0.SPx3.p2.3.m3.1.1.1.1.1.2.3" xref="S1.SS1.SSS0.P0.SPx3.p2.3.m3.1.1.1.1.1.2.3.cmml"><mo id="S1.SS1.SSS0.P0.SPx3.p2.3.m3.1.1.1.1.1.2.3a" xref="S1.SS1.SSS0.P0.SPx3.p2.3.m3.1.1.1.1.1.2.3.cmml">−</mo><mn id="S1.SS1.SSS0.P0.SPx3.p2.3.m3.1.1.1.1.1.2.3.2" xref="S1.SS1.SSS0.P0.SPx3.p2.3.m3.1.1.1.1.1.2.3.2.cmml">2</mn></mrow></msup><mo id="S1.SS1.SSS0.P0.SPx3.p2.3.m3.1.1.1.1.1.1" lspace="0.167em" xref="S1.SS1.SSS0.P0.SPx3.p2.3.m3.1.1.1.1.1.1.cmml">⁢</mo><mrow id="S1.SS1.SSS0.P0.SPx3.p2.3.m3.1.1.1.1.1.3" xref="S1.SS1.SSS0.P0.SPx3.p2.3.m3.1.1.1.1.1.3.cmml"><msup id="S1.SS1.SSS0.P0.SPx3.p2.3.m3.1.1.1.1.1.3.1" xref="S1.SS1.SSS0.P0.SPx3.p2.3.m3.1.1.1.1.1.3.1.cmml"><mi id="S1.SS1.SSS0.P0.SPx3.p2.3.m3.1.1.1.1.1.3.1.2" xref="S1.SS1.SSS0.P0.SPx3.p2.3.m3.1.1.1.1.1.3.1.2.cmml">log</mi><mn id="S1.SS1.SSS0.P0.SPx3.p2.3.m3.1.1.1.1.1.3.1.3" xref="S1.SS1.SSS0.P0.SPx3.p2.3.m3.1.1.1.1.1.3.1.3.cmml">2</mn></msup><mo id="S1.SS1.SSS0.P0.SPx3.p2.3.m3.1.1.1.1.1.3a" lspace="0.167em" xref="S1.SS1.SSS0.P0.SPx3.p2.3.m3.1.1.1.1.1.3.cmml">⁡</mo><mi id="S1.SS1.SSS0.P0.SPx3.p2.3.m3.1.1.1.1.1.3.2" xref="S1.SS1.SSS0.P0.SPx3.p2.3.m3.1.1.1.1.1.3.2.cmml">n</mi></mrow></mrow><mo id="S1.SS1.SSS0.P0.SPx3.p2.3.m3.1.1.1.1.3" stretchy="false" xref="S1.SS1.SSS0.P0.SPx3.p2.3.m3.1.1.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.SS1.SSS0.P0.SPx3.p2.3.m3.1b"><apply id="S1.SS1.SSS0.P0.SPx3.p2.3.m3.1.1.cmml" xref="S1.SS1.SSS0.P0.SPx3.p2.3.m3.1.1"><times id="S1.SS1.SSS0.P0.SPx3.p2.3.m3.1.1.2.cmml" xref="S1.SS1.SSS0.P0.SPx3.p2.3.m3.1.1.2"></times><ci id="S1.SS1.SSS0.P0.SPx3.p2.3.m3.1.1.3.cmml" xref="S1.SS1.SSS0.P0.SPx3.p2.3.m3.1.1.3">𝑂</ci><apply id="S1.SS1.SSS0.P0.SPx3.p2.3.m3.1.1.1.1.1.cmml" xref="S1.SS1.SSS0.P0.SPx3.p2.3.m3.1.1.1.1"><times id="S1.SS1.SSS0.P0.SPx3.p2.3.m3.1.1.1.1.1.1.cmml" xref="S1.SS1.SSS0.P0.SPx3.p2.3.m3.1.1.1.1.1.1"></times><apply id="S1.SS1.SSS0.P0.SPx3.p2.3.m3.1.1.1.1.1.2.cmml" xref="S1.SS1.SSS0.P0.SPx3.p2.3.m3.1.1.1.1.1.2"><csymbol cd="ambiguous" id="S1.SS1.SSS0.P0.SPx3.p2.3.m3.1.1.1.1.1.2.1.cmml" xref="S1.SS1.SSS0.P0.SPx3.p2.3.m3.1.1.1.1.1.2">superscript</csymbol><ci id="S1.SS1.SSS0.P0.SPx3.p2.3.m3.1.1.1.1.1.2.2.cmml" xref="S1.SS1.SSS0.P0.SPx3.p2.3.m3.1.1.1.1.1.2.2">𝜀</ci><apply id="S1.SS1.SSS0.P0.SPx3.p2.3.m3.1.1.1.1.1.2.3.cmml" xref="S1.SS1.SSS0.P0.SPx3.p2.3.m3.1.1.1.1.1.2.3"><minus id="S1.SS1.SSS0.P0.SPx3.p2.3.m3.1.1.1.1.1.2.3.1.cmml" xref="S1.SS1.SSS0.P0.SPx3.p2.3.m3.1.1.1.1.1.2.3"></minus><cn id="S1.SS1.SSS0.P0.SPx3.p2.3.m3.1.1.1.1.1.2.3.2.cmml" type="integer" xref="S1.SS1.SSS0.P0.SPx3.p2.3.m3.1.1.1.1.1.2.3.2">2</cn></apply></apply><apply id="S1.SS1.SSS0.P0.SPx3.p2.3.m3.1.1.1.1.1.3.cmml" xref="S1.SS1.SSS0.P0.SPx3.p2.3.m3.1.1.1.1.1.3"><apply id="S1.SS1.SSS0.P0.SPx3.p2.3.m3.1.1.1.1.1.3.1.cmml" xref="S1.SS1.SSS0.P0.SPx3.p2.3.m3.1.1.1.1.1.3.1"><csymbol cd="ambiguous" id="S1.SS1.SSS0.P0.SPx3.p2.3.m3.1.1.1.1.1.3.1.1.cmml" xref="S1.SS1.SSS0.P0.SPx3.p2.3.m3.1.1.1.1.1.3.1">superscript</csymbol><log id="S1.SS1.SSS0.P0.SPx3.p2.3.m3.1.1.1.1.1.3.1.2.cmml" xref="S1.SS1.SSS0.P0.SPx3.p2.3.m3.1.1.1.1.1.3.1.2"></log><cn id="S1.SS1.SSS0.P0.SPx3.p2.3.m3.1.1.1.1.1.3.1.3.cmml" type="integer" xref="S1.SS1.SSS0.P0.SPx3.p2.3.m3.1.1.1.1.1.3.1.3">2</cn></apply><ci id="S1.SS1.SSS0.P0.SPx3.p2.3.m3.1.1.1.1.1.3.2.cmml" xref="S1.SS1.SSS0.P0.SPx3.p2.3.m3.1.1.1.1.1.3.2">𝑛</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS1.SSS0.P0.SPx3.p2.3.m3.1c">O(\varepsilon^{-2}\log^{2}n)</annotation><annotation encoding="application/x-llamapun" id="S1.SS1.SSS0.P0.SPx3.p2.3.m3.1d">italic_O ( italic_ε start_POSTSUPERSCRIPT - 2 end_POSTSUPERSCRIPT roman_log start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_n )</annotation></semantics></math> rounds. This gap may be explained by considering the local density <math alttext="\rho^{*}(v)" class="ltx_Math" display="inline" id="S1.SS1.SSS0.P0.SPx3.p2.4.m4.1"><semantics id="S1.SS1.SSS0.P0.SPx3.p2.4.m4.1a"><mrow id="S1.SS1.SSS0.P0.SPx3.p2.4.m4.1.2" xref="S1.SS1.SSS0.P0.SPx3.p2.4.m4.1.2.cmml"><msup id="S1.SS1.SSS0.P0.SPx3.p2.4.m4.1.2.2" xref="S1.SS1.SSS0.P0.SPx3.p2.4.m4.1.2.2.cmml"><mi id="S1.SS1.SSS0.P0.SPx3.p2.4.m4.1.2.2.2" xref="S1.SS1.SSS0.P0.SPx3.p2.4.m4.1.2.2.2.cmml">ρ</mi><mo id="S1.SS1.SSS0.P0.SPx3.p2.4.m4.1.2.2.3" xref="S1.SS1.SSS0.P0.SPx3.p2.4.m4.1.2.2.3.cmml">∗</mo></msup><mo id="S1.SS1.SSS0.P0.SPx3.p2.4.m4.1.2.1" xref="S1.SS1.SSS0.P0.SPx3.p2.4.m4.1.2.1.cmml">⁢</mo><mrow id="S1.SS1.SSS0.P0.SPx3.p2.4.m4.1.2.3.2" xref="S1.SS1.SSS0.P0.SPx3.p2.4.m4.1.2.cmml"><mo id="S1.SS1.SSS0.P0.SPx3.p2.4.m4.1.2.3.2.1" stretchy="false" xref="S1.SS1.SSS0.P0.SPx3.p2.4.m4.1.2.cmml">(</mo><mi id="S1.SS1.SSS0.P0.SPx3.p2.4.m4.1.1" xref="S1.SS1.SSS0.P0.SPx3.p2.4.m4.1.1.cmml">v</mi><mo id="S1.SS1.SSS0.P0.SPx3.p2.4.m4.1.2.3.2.2" stretchy="false" xref="S1.SS1.SSS0.P0.SPx3.p2.4.m4.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.SS1.SSS0.P0.SPx3.p2.4.m4.1b"><apply id="S1.SS1.SSS0.P0.SPx3.p2.4.m4.1.2.cmml" xref="S1.SS1.SSS0.P0.SPx3.p2.4.m4.1.2"><times id="S1.SS1.SSS0.P0.SPx3.p2.4.m4.1.2.1.cmml" xref="S1.SS1.SSS0.P0.SPx3.p2.4.m4.1.2.1"></times><apply id="S1.SS1.SSS0.P0.SPx3.p2.4.m4.1.2.2.cmml" xref="S1.SS1.SSS0.P0.SPx3.p2.4.m4.1.2.2"><csymbol cd="ambiguous" id="S1.SS1.SSS0.P0.SPx3.p2.4.m4.1.2.2.1.cmml" xref="S1.SS1.SSS0.P0.SPx3.p2.4.m4.1.2.2">superscript</csymbol><ci id="S1.SS1.SSS0.P0.SPx3.p2.4.m4.1.2.2.2.cmml" xref="S1.SS1.SSS0.P0.SPx3.p2.4.m4.1.2.2.2">𝜌</ci><times id="S1.SS1.SSS0.P0.SPx3.p2.4.m4.1.2.2.3.cmml" xref="S1.SS1.SSS0.P0.SPx3.p2.4.m4.1.2.2.3"></times></apply><ci id="S1.SS1.SSS0.P0.SPx3.p2.4.m4.1.1.cmml" xref="S1.SS1.SSS0.P0.SPx3.p2.4.m4.1.1">𝑣</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS1.SSS0.P0.SPx3.p2.4.m4.1c">\rho^{*}(v)</annotation><annotation encoding="application/x-llamapun" id="S1.SS1.SSS0.P0.SPx3.p2.4.m4.1d">italic_ρ start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ( italic_v )</annotation></semantics></math> for low-local-density vertices <math alttext="v" class="ltx_Math" display="inline" id="S1.SS1.SSS0.P0.SPx3.p2.5.m5.1"><semantics id="S1.SS1.SSS0.P0.SPx3.p2.5.m5.1a"><mi id="S1.SS1.SSS0.P0.SPx3.p2.5.m5.1.1" xref="S1.SS1.SSS0.P0.SPx3.p2.5.m5.1.1.cmml">v</mi><annotation-xml encoding="MathML-Content" id="S1.SS1.SSS0.P0.SPx3.p2.5.m5.1b"><ci id="S1.SS1.SSS0.P0.SPx3.p2.5.m5.1.1.cmml" xref="S1.SS1.SSS0.P0.SPx3.p2.5.m5.1.1">𝑣</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.SS1.SSS0.P0.SPx3.p2.5.m5.1c">v</annotation><annotation encoding="application/x-llamapun" id="S1.SS1.SSS0.P0.SPx3.p2.5.m5.1d">italic_v</annotation></semantics></math> in a graph that has high global density. The local density of <math alttext="v" class="ltx_Math" display="inline" id="S1.SS1.SSS0.P0.SPx3.p2.6.m6.1"><semantics id="S1.SS1.SSS0.P0.SPx3.p2.6.m6.1a"><mi id="S1.SS1.SSS0.P0.SPx3.p2.6.m6.1.1" xref="S1.SS1.SSS0.P0.SPx3.p2.6.m6.1.1.cmml">v</mi><annotation-xml encoding="MathML-Content" id="S1.SS1.SSS0.P0.SPx3.p2.6.m6.1b"><ci id="S1.SS1.SSS0.P0.SPx3.p2.6.m6.1.1.cmml" xref="S1.SS1.SSS0.P0.SPx3.p2.6.m6.1.1">𝑣</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.SS1.SSS0.P0.SPx3.p2.6.m6.1c">v</annotation><annotation encoding="application/x-llamapun" id="S1.SS1.SSS0.P0.SPx3.p2.6.m6.1d">italic_v</annotation></semantics></math> can be affected by a dense subgraph within a hop distance of <math alttext="\Theta(\varepsilon^{-2}\log^{2}n)" class="ltx_Math" display="inline" id="S1.SS1.SSS0.P0.SPx3.p2.7.m7.1"><semantics id="S1.SS1.SSS0.P0.SPx3.p2.7.m7.1a"><mrow id="S1.SS1.SSS0.P0.SPx3.p2.7.m7.1.1" xref="S1.SS1.SSS0.P0.SPx3.p2.7.m7.1.1.cmml"><mi id="S1.SS1.SSS0.P0.SPx3.p2.7.m7.1.1.3" mathvariant="normal" xref="S1.SS1.SSS0.P0.SPx3.p2.7.m7.1.1.3.cmml">Θ</mi><mo id="S1.SS1.SSS0.P0.SPx3.p2.7.m7.1.1.2" xref="S1.SS1.SSS0.P0.SPx3.p2.7.m7.1.1.2.cmml">⁢</mo><mrow id="S1.SS1.SSS0.P0.SPx3.p2.7.m7.1.1.1.1" xref="S1.SS1.SSS0.P0.SPx3.p2.7.m7.1.1.1.1.1.cmml"><mo id="S1.SS1.SSS0.P0.SPx3.p2.7.m7.1.1.1.1.2" stretchy="false" xref="S1.SS1.SSS0.P0.SPx3.p2.7.m7.1.1.1.1.1.cmml">(</mo><mrow id="S1.SS1.SSS0.P0.SPx3.p2.7.m7.1.1.1.1.1" xref="S1.SS1.SSS0.P0.SPx3.p2.7.m7.1.1.1.1.1.cmml"><msup id="S1.SS1.SSS0.P0.SPx3.p2.7.m7.1.1.1.1.1.2" xref="S1.SS1.SSS0.P0.SPx3.p2.7.m7.1.1.1.1.1.2.cmml"><mi id="S1.SS1.SSS0.P0.SPx3.p2.7.m7.1.1.1.1.1.2.2" xref="S1.SS1.SSS0.P0.SPx3.p2.7.m7.1.1.1.1.1.2.2.cmml">ε</mi><mrow id="S1.SS1.SSS0.P0.SPx3.p2.7.m7.1.1.1.1.1.2.3" xref="S1.SS1.SSS0.P0.SPx3.p2.7.m7.1.1.1.1.1.2.3.cmml"><mo id="S1.SS1.SSS0.P0.SPx3.p2.7.m7.1.1.1.1.1.2.3a" xref="S1.SS1.SSS0.P0.SPx3.p2.7.m7.1.1.1.1.1.2.3.cmml">−</mo><mn id="S1.SS1.SSS0.P0.SPx3.p2.7.m7.1.1.1.1.1.2.3.2" xref="S1.SS1.SSS0.P0.SPx3.p2.7.m7.1.1.1.1.1.2.3.2.cmml">2</mn></mrow></msup><mo id="S1.SS1.SSS0.P0.SPx3.p2.7.m7.1.1.1.1.1.1" lspace="0.167em" xref="S1.SS1.SSS0.P0.SPx3.p2.7.m7.1.1.1.1.1.1.cmml">⁢</mo><mrow id="S1.SS1.SSS0.P0.SPx3.p2.7.m7.1.1.1.1.1.3" xref="S1.SS1.SSS0.P0.SPx3.p2.7.m7.1.1.1.1.1.3.cmml"><msup id="S1.SS1.SSS0.P0.SPx3.p2.7.m7.1.1.1.1.1.3.1" xref="S1.SS1.SSS0.P0.SPx3.p2.7.m7.1.1.1.1.1.3.1.cmml"><mi id="S1.SS1.SSS0.P0.SPx3.p2.7.m7.1.1.1.1.1.3.1.2" xref="S1.SS1.SSS0.P0.SPx3.p2.7.m7.1.1.1.1.1.3.1.2.cmml">log</mi><mn id="S1.SS1.SSS0.P0.SPx3.p2.7.m7.1.1.1.1.1.3.1.3" xref="S1.SS1.SSS0.P0.SPx3.p2.7.m7.1.1.1.1.1.3.1.3.cmml">2</mn></msup><mo id="S1.SS1.SSS0.P0.SPx3.p2.7.m7.1.1.1.1.1.3a" lspace="0.167em" xref="S1.SS1.SSS0.P0.SPx3.p2.7.m7.1.1.1.1.1.3.cmml">⁡</mo><mi id="S1.SS1.SSS0.P0.SPx3.p2.7.m7.1.1.1.1.1.3.2" xref="S1.SS1.SSS0.P0.SPx3.p2.7.m7.1.1.1.1.1.3.2.cmml">n</mi></mrow></mrow><mo id="S1.SS1.SSS0.P0.SPx3.p2.7.m7.1.1.1.1.3" stretchy="false" xref="S1.SS1.SSS0.P0.SPx3.p2.7.m7.1.1.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.SS1.SSS0.P0.SPx3.p2.7.m7.1b"><apply id="S1.SS1.SSS0.P0.SPx3.p2.7.m7.1.1.cmml" xref="S1.SS1.SSS0.P0.SPx3.p2.7.m7.1.1"><times id="S1.SS1.SSS0.P0.SPx3.p2.7.m7.1.1.2.cmml" xref="S1.SS1.SSS0.P0.SPx3.p2.7.m7.1.1.2"></times><ci id="S1.SS1.SSS0.P0.SPx3.p2.7.m7.1.1.3.cmml" xref="S1.SS1.SSS0.P0.SPx3.p2.7.m7.1.1.3">Θ</ci><apply id="S1.SS1.SSS0.P0.SPx3.p2.7.m7.1.1.1.1.1.cmml" xref="S1.SS1.SSS0.P0.SPx3.p2.7.m7.1.1.1.1"><times id="S1.SS1.SSS0.P0.SPx3.p2.7.m7.1.1.1.1.1.1.cmml" xref="S1.SS1.SSS0.P0.SPx3.p2.7.m7.1.1.1.1.1.1"></times><apply id="S1.SS1.SSS0.P0.SPx3.p2.7.m7.1.1.1.1.1.2.cmml" xref="S1.SS1.SSS0.P0.SPx3.p2.7.m7.1.1.1.1.1.2"><csymbol cd="ambiguous" id="S1.SS1.SSS0.P0.SPx3.p2.7.m7.1.1.1.1.1.2.1.cmml" xref="S1.SS1.SSS0.P0.SPx3.p2.7.m7.1.1.1.1.1.2">superscript</csymbol><ci id="S1.SS1.SSS0.P0.SPx3.p2.7.m7.1.1.1.1.1.2.2.cmml" xref="S1.SS1.SSS0.P0.SPx3.p2.7.m7.1.1.1.1.1.2.2">𝜀</ci><apply id="S1.SS1.SSS0.P0.SPx3.p2.7.m7.1.1.1.1.1.2.3.cmml" xref="S1.SS1.SSS0.P0.SPx3.p2.7.m7.1.1.1.1.1.2.3"><minus id="S1.SS1.SSS0.P0.SPx3.p2.7.m7.1.1.1.1.1.2.3.1.cmml" xref="S1.SS1.SSS0.P0.SPx3.p2.7.m7.1.1.1.1.1.2.3"></minus><cn id="S1.SS1.SSS0.P0.SPx3.p2.7.m7.1.1.1.1.1.2.3.2.cmml" type="integer" xref="S1.SS1.SSS0.P0.SPx3.p2.7.m7.1.1.1.1.1.2.3.2">2</cn></apply></apply><apply id="S1.SS1.SSS0.P0.SPx3.p2.7.m7.1.1.1.1.1.3.cmml" xref="S1.SS1.SSS0.P0.SPx3.p2.7.m7.1.1.1.1.1.3"><apply id="S1.SS1.SSS0.P0.SPx3.p2.7.m7.1.1.1.1.1.3.1.cmml" xref="S1.SS1.SSS0.P0.SPx3.p2.7.m7.1.1.1.1.1.3.1"><csymbol cd="ambiguous" id="S1.SS1.SSS0.P0.SPx3.p2.7.m7.1.1.1.1.1.3.1.1.cmml" xref="S1.SS1.SSS0.P0.SPx3.p2.7.m7.1.1.1.1.1.3.1">superscript</csymbol><log id="S1.SS1.SSS0.P0.SPx3.p2.7.m7.1.1.1.1.1.3.1.2.cmml" xref="S1.SS1.SSS0.P0.SPx3.p2.7.m7.1.1.1.1.1.3.1.2"></log><cn id="S1.SS1.SSS0.P0.SPx3.p2.7.m7.1.1.1.1.1.3.1.3.cmml" type="integer" xref="S1.SS1.SSS0.P0.SPx3.p2.7.m7.1.1.1.1.1.3.1.3">2</cn></apply><ci id="S1.SS1.SSS0.P0.SPx3.p2.7.m7.1.1.1.1.1.3.2.cmml" xref="S1.SS1.SSS0.P0.SPx3.p2.7.m7.1.1.1.1.1.3.2">𝑛</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS1.SSS0.P0.SPx3.p2.7.m7.1c">\Theta(\varepsilon^{-2}\log^{2}n)</annotation><annotation encoding="application/x-llamapun" id="S1.SS1.SSS0.P0.SPx3.p2.7.m7.1d">roman_Θ ( italic_ε start_POSTSUPERSCRIPT - 2 end_POSTSUPERSCRIPT roman_log start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_n )</annotation></semantics></math> (although it unclear if it can be affected <em class="ltx_emph ltx_font_italic" id="S1.SS1.SSS0.P0.SPx3.p2.14.2">enough</em> to prohibit a <math alttext="(1+\varepsilon)" class="ltx_Math" display="inline" id="S1.SS1.SSS0.P0.SPx3.p2.8.m8.1"><semantics id="S1.SS1.SSS0.P0.SPx3.p2.8.m8.1a"><mrow id="S1.SS1.SSS0.P0.SPx3.p2.8.m8.1.1.1" xref="S1.SS1.SSS0.P0.SPx3.p2.8.m8.1.1.1.1.cmml"><mo id="S1.SS1.SSS0.P0.SPx3.p2.8.m8.1.1.1.2" stretchy="false" xref="S1.SS1.SSS0.P0.SPx3.p2.8.m8.1.1.1.1.cmml">(</mo><mrow id="S1.SS1.SSS0.P0.SPx3.p2.8.m8.1.1.1.1" xref="S1.SS1.SSS0.P0.SPx3.p2.8.m8.1.1.1.1.cmml"><mn id="S1.SS1.SSS0.P0.SPx3.p2.8.m8.1.1.1.1.2" xref="S1.SS1.SSS0.P0.SPx3.p2.8.m8.1.1.1.1.2.cmml">1</mn><mo id="S1.SS1.SSS0.P0.SPx3.p2.8.m8.1.1.1.1.1" xref="S1.SS1.SSS0.P0.SPx3.p2.8.m8.1.1.1.1.1.cmml">+</mo><mi id="S1.SS1.SSS0.P0.SPx3.p2.8.m8.1.1.1.1.3" xref="S1.SS1.SSS0.P0.SPx3.p2.8.m8.1.1.1.1.3.cmml">ε</mi></mrow><mo id="S1.SS1.SSS0.P0.SPx3.p2.8.m8.1.1.1.3" stretchy="false" xref="S1.SS1.SSS0.P0.SPx3.p2.8.m8.1.1.1.1.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S1.SS1.SSS0.P0.SPx3.p2.8.m8.1b"><apply id="S1.SS1.SSS0.P0.SPx3.p2.8.m8.1.1.1.1.cmml" xref="S1.SS1.SSS0.P0.SPx3.p2.8.m8.1.1.1"><plus id="S1.SS1.SSS0.P0.SPx3.p2.8.m8.1.1.1.1.1.cmml" xref="S1.SS1.SSS0.P0.SPx3.p2.8.m8.1.1.1.1.1"></plus><cn id="S1.SS1.SSS0.P0.SPx3.p2.8.m8.1.1.1.1.2.cmml" type="integer" xref="S1.SS1.SSS0.P0.SPx3.p2.8.m8.1.1.1.1.2">1</cn><ci id="S1.SS1.SSS0.P0.SPx3.p2.8.m8.1.1.1.1.3.cmml" xref="S1.SS1.SSS0.P0.SPx3.p2.8.m8.1.1.1.1.3">𝜀</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS1.SSS0.P0.SPx3.p2.8.m8.1c">(1+\varepsilon)</annotation><annotation encoding="application/x-llamapun" id="S1.SS1.SSS0.P0.SPx3.p2.8.m8.1d">( 1 + italic_ε )</annotation></semantics></math>-approximation of <math alttext="\rho^{*}(v)" class="ltx_Math" display="inline" id="S1.SS1.SSS0.P0.SPx3.p2.9.m9.1"><semantics id="S1.SS1.SSS0.P0.SPx3.p2.9.m9.1a"><mrow id="S1.SS1.SSS0.P0.SPx3.p2.9.m9.1.2" xref="S1.SS1.SSS0.P0.SPx3.p2.9.m9.1.2.cmml"><msup id="S1.SS1.SSS0.P0.SPx3.p2.9.m9.1.2.2" xref="S1.SS1.SSS0.P0.SPx3.p2.9.m9.1.2.2.cmml"><mi id="S1.SS1.SSS0.P0.SPx3.p2.9.m9.1.2.2.2" xref="S1.SS1.SSS0.P0.SPx3.p2.9.m9.1.2.2.2.cmml">ρ</mi><mo id="S1.SS1.SSS0.P0.SPx3.p2.9.m9.1.2.2.3" xref="S1.SS1.SSS0.P0.SPx3.p2.9.m9.1.2.2.3.cmml">∗</mo></msup><mo id="S1.SS1.SSS0.P0.SPx3.p2.9.m9.1.2.1" xref="S1.SS1.SSS0.P0.SPx3.p2.9.m9.1.2.1.cmml">⁢</mo><mrow id="S1.SS1.SSS0.P0.SPx3.p2.9.m9.1.2.3.2" xref="S1.SS1.SSS0.P0.SPx3.p2.9.m9.1.2.cmml"><mo id="S1.SS1.SSS0.P0.SPx3.p2.9.m9.1.2.3.2.1" stretchy="false" xref="S1.SS1.SSS0.P0.SPx3.p2.9.m9.1.2.cmml">(</mo><mi id="S1.SS1.SSS0.P0.SPx3.p2.9.m9.1.1" xref="S1.SS1.SSS0.P0.SPx3.p2.9.m9.1.1.cmml">v</mi><mo id="S1.SS1.SSS0.P0.SPx3.p2.9.m9.1.2.3.2.2" stretchy="false" xref="S1.SS1.SSS0.P0.SPx3.p2.9.m9.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.SS1.SSS0.P0.SPx3.p2.9.m9.1b"><apply id="S1.SS1.SSS0.P0.SPx3.p2.9.m9.1.2.cmml" xref="S1.SS1.SSS0.P0.SPx3.p2.9.m9.1.2"><times id="S1.SS1.SSS0.P0.SPx3.p2.9.m9.1.2.1.cmml" xref="S1.SS1.SSS0.P0.SPx3.p2.9.m9.1.2.1"></times><apply id="S1.SS1.SSS0.P0.SPx3.p2.9.m9.1.2.2.cmml" xref="S1.SS1.SSS0.P0.SPx3.p2.9.m9.1.2.2"><csymbol cd="ambiguous" id="S1.SS1.SSS0.P0.SPx3.p2.9.m9.1.2.2.1.cmml" xref="S1.SS1.SSS0.P0.SPx3.p2.9.m9.1.2.2">superscript</csymbol><ci id="S1.SS1.SSS0.P0.SPx3.p2.9.m9.1.2.2.2.cmml" xref="S1.SS1.SSS0.P0.SPx3.p2.9.m9.1.2.2.2">𝜌</ci><times id="S1.SS1.SSS0.P0.SPx3.p2.9.m9.1.2.2.3.cmml" xref="S1.SS1.SSS0.P0.SPx3.p2.9.m9.1.2.2.3"></times></apply><ci id="S1.SS1.SSS0.P0.SPx3.p2.9.m9.1.1.cmml" xref="S1.SS1.SSS0.P0.SPx3.p2.9.m9.1.1">𝑣</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS1.SSS0.P0.SPx3.p2.9.m9.1c">\rho^{*}(v)</annotation><annotation encoding="application/x-llamapun" id="S1.SS1.SSS0.P0.SPx3.p2.9.m9.1d">italic_ρ start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ( italic_v )</annotation></semantics></math> in <math alttext="o(\varepsilon^{-2}\log^{2}n)" class="ltx_Math" display="inline" id="S1.SS1.SSS0.P0.SPx3.p2.10.m10.1"><semantics id="S1.SS1.SSS0.P0.SPx3.p2.10.m10.1a"><mrow id="S1.SS1.SSS0.P0.SPx3.p2.10.m10.1.1" xref="S1.SS1.SSS0.P0.SPx3.p2.10.m10.1.1.cmml"><mi id="S1.SS1.SSS0.P0.SPx3.p2.10.m10.1.1.3" xref="S1.SS1.SSS0.P0.SPx3.p2.10.m10.1.1.3.cmml">o</mi><mo id="S1.SS1.SSS0.P0.SPx3.p2.10.m10.1.1.2" xref="S1.SS1.SSS0.P0.SPx3.p2.10.m10.1.1.2.cmml">⁢</mo><mrow id="S1.SS1.SSS0.P0.SPx3.p2.10.m10.1.1.1.1" xref="S1.SS1.SSS0.P0.SPx3.p2.10.m10.1.1.1.1.1.cmml"><mo id="S1.SS1.SSS0.P0.SPx3.p2.10.m10.1.1.1.1.2" stretchy="false" xref="S1.SS1.SSS0.P0.SPx3.p2.10.m10.1.1.1.1.1.cmml">(</mo><mrow id="S1.SS1.SSS0.P0.SPx3.p2.10.m10.1.1.1.1.1" xref="S1.SS1.SSS0.P0.SPx3.p2.10.m10.1.1.1.1.1.cmml"><msup id="S1.SS1.SSS0.P0.SPx3.p2.10.m10.1.1.1.1.1.2" xref="S1.SS1.SSS0.P0.SPx3.p2.10.m10.1.1.1.1.1.2.cmml"><mi id="S1.SS1.SSS0.P0.SPx3.p2.10.m10.1.1.1.1.1.2.2" xref="S1.SS1.SSS0.P0.SPx3.p2.10.m10.1.1.1.1.1.2.2.cmml">ε</mi><mrow id="S1.SS1.SSS0.P0.SPx3.p2.10.m10.1.1.1.1.1.2.3" xref="S1.SS1.SSS0.P0.SPx3.p2.10.m10.1.1.1.1.1.2.3.cmml"><mo id="S1.SS1.SSS0.P0.SPx3.p2.10.m10.1.1.1.1.1.2.3a" xref="S1.SS1.SSS0.P0.SPx3.p2.10.m10.1.1.1.1.1.2.3.cmml">−</mo><mn id="S1.SS1.SSS0.P0.SPx3.p2.10.m10.1.1.1.1.1.2.3.2" 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type="integer" xref="S1.SS1.SSS0.P0.SPx3.p2.10.m10.1.1.1.1.1.3.1.3">2</cn></apply><ci id="S1.SS1.SSS0.P0.SPx3.p2.10.m10.1.1.1.1.1.3.2.cmml" xref="S1.SS1.SSS0.P0.SPx3.p2.10.m10.1.1.1.1.1.3.2">𝑛</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS1.SSS0.P0.SPx3.p2.10.m10.1c">o(\varepsilon^{-2}\log^{2}n)</annotation><annotation encoding="application/x-llamapun" id="S1.SS1.SSS0.P0.SPx3.p2.10.m10.1d">italic_o ( italic_ε start_POSTSUPERSCRIPT - 2 end_POSTSUPERSCRIPT roman_log start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_n )</annotation></semantics></math> time). The potential for a barrier of <math alttext="O(\varepsilon^{-2}\log^{2}n)" class="ltx_Math" display="inline" id="S1.SS1.SSS0.P0.SPx3.p2.11.m11.1"><semantics id="S1.SS1.SSS0.P0.SPx3.p2.11.m11.1a"><mrow id="S1.SS1.SSS0.P0.SPx3.p2.11.m11.1.1" xref="S1.SS1.SSS0.P0.SPx3.p2.11.m11.1.1.cmml"><mi id="S1.SS1.SSS0.P0.SPx3.p2.11.m11.1.1.3" xref="S1.SS1.SSS0.P0.SPx3.p2.11.m11.1.1.3.cmml">O</mi><mo id="S1.SS1.SSS0.P0.SPx3.p2.11.m11.1.1.2" xref="S1.SS1.SSS0.P0.SPx3.p2.11.m11.1.1.2.cmml">⁢</mo><mrow id="S1.SS1.SSS0.P0.SPx3.p2.11.m11.1.1.1.1" xref="S1.SS1.SSS0.P0.SPx3.p2.11.m11.1.1.1.1.1.cmml"><mo id="S1.SS1.SSS0.P0.SPx3.p2.11.m11.1.1.1.1.2" stretchy="false" xref="S1.SS1.SSS0.P0.SPx3.p2.11.m11.1.1.1.1.1.cmml">(</mo><mrow id="S1.SS1.SSS0.P0.SPx3.p2.11.m11.1.1.1.1.1" xref="S1.SS1.SSS0.P0.SPx3.p2.11.m11.1.1.1.1.1.cmml"><msup id="S1.SS1.SSS0.P0.SPx3.p2.11.m11.1.1.1.1.1.2" xref="S1.SS1.SSS0.P0.SPx3.p2.11.m11.1.1.1.1.1.2.cmml"><mi id="S1.SS1.SSS0.P0.SPx3.p2.11.m11.1.1.1.1.1.2.2" xref="S1.SS1.SSS0.P0.SPx3.p2.11.m11.1.1.1.1.1.2.2.cmml">ε</mi><mrow id="S1.SS1.SSS0.P0.SPx3.p2.11.m11.1.1.1.1.1.2.3" xref="S1.SS1.SSS0.P0.SPx3.p2.11.m11.1.1.1.1.1.2.3.cmml"><mo id="S1.SS1.SSS0.P0.SPx3.p2.11.m11.1.1.1.1.1.2.3a" xref="S1.SS1.SSS0.P0.SPx3.p2.11.m11.1.1.1.1.1.2.3.cmml">−</mo><mn id="S1.SS1.SSS0.P0.SPx3.p2.11.m11.1.1.1.1.1.2.3.2" xref="S1.SS1.SSS0.P0.SPx3.p2.11.m11.1.1.1.1.1.2.3.2.cmml">2</mn></mrow></msup><mo id="S1.SS1.SSS0.P0.SPx3.p2.11.m11.1.1.1.1.1.1" lspace="0.167em" xref="S1.SS1.SSS0.P0.SPx3.p2.11.m11.1.1.1.1.1.1.cmml">⁢</mo><mrow id="S1.SS1.SSS0.P0.SPx3.p2.11.m11.1.1.1.1.1.3" xref="S1.SS1.SSS0.P0.SPx3.p2.11.m11.1.1.1.1.1.3.cmml"><msup id="S1.SS1.SSS0.P0.SPx3.p2.11.m11.1.1.1.1.1.3.1" xref="S1.SS1.SSS0.P0.SPx3.p2.11.m11.1.1.1.1.1.3.1.cmml"><mi id="S1.SS1.SSS0.P0.SPx3.p2.11.m11.1.1.1.1.1.3.1.2" xref="S1.SS1.SSS0.P0.SPx3.p2.11.m11.1.1.1.1.1.3.1.2.cmml">log</mi><mn id="S1.SS1.SSS0.P0.SPx3.p2.11.m11.1.1.1.1.1.3.1.3" xref="S1.SS1.SSS0.P0.SPx3.p2.11.m11.1.1.1.1.1.3.1.3.cmml">2</mn></msup><mo id="S1.SS1.SSS0.P0.SPx3.p2.11.m11.1.1.1.1.1.3a" lspace="0.167em" xref="S1.SS1.SSS0.P0.SPx3.p2.11.m11.1.1.1.1.1.3.cmml">⁡</mo><mi id="S1.SS1.SSS0.P0.SPx3.p2.11.m11.1.1.1.1.1.3.2" xref="S1.SS1.SSS0.P0.SPx3.p2.11.m11.1.1.1.1.1.3.2.cmml">n</mi></mrow></mrow><mo id="S1.SS1.SSS0.P0.SPx3.p2.11.m11.1.1.1.1.3" stretchy="false" xref="S1.SS1.SSS0.P0.SPx3.p2.11.m11.1.1.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.SS1.SSS0.P0.SPx3.p2.11.m11.1b"><apply id="S1.SS1.SSS0.P0.SPx3.p2.11.m11.1.1.cmml" xref="S1.SS1.SSS0.P0.SPx3.p2.11.m11.1.1"><times id="S1.SS1.SSS0.P0.SPx3.p2.11.m11.1.1.2.cmml" xref="S1.SS1.SSS0.P0.SPx3.p2.11.m11.1.1.2"></times><ci id="S1.SS1.SSS0.P0.SPx3.p2.11.m11.1.1.3.cmml" xref="S1.SS1.SSS0.P0.SPx3.p2.11.m11.1.1.3">𝑂</ci><apply id="S1.SS1.SSS0.P0.SPx3.p2.11.m11.1.1.1.1.1.cmml" xref="S1.SS1.SSS0.P0.SPx3.p2.11.m11.1.1.1.1"><times id="S1.SS1.SSS0.P0.SPx3.p2.11.m11.1.1.1.1.1.1.cmml" xref="S1.SS1.SSS0.P0.SPx3.p2.11.m11.1.1.1.1.1.1"></times><apply id="S1.SS1.SSS0.P0.SPx3.p2.11.m11.1.1.1.1.1.2.cmml" xref="S1.SS1.SSS0.P0.SPx3.p2.11.m11.1.1.1.1.1.2"><csymbol cd="ambiguous" id="S1.SS1.SSS0.P0.SPx3.p2.11.m11.1.1.1.1.1.2.1.cmml" xref="S1.SS1.SSS0.P0.SPx3.p2.11.m11.1.1.1.1.1.2">superscript</csymbol><ci id="S1.SS1.SSS0.P0.SPx3.p2.11.m11.1.1.1.1.1.2.2.cmml" xref="S1.SS1.SSS0.P0.SPx3.p2.11.m11.1.1.1.1.1.2.2">𝜀</ci><apply id="S1.SS1.SSS0.P0.SPx3.p2.11.m11.1.1.1.1.1.2.3.cmml" xref="S1.SS1.SSS0.P0.SPx3.p2.11.m11.1.1.1.1.1.2.3"><minus id="S1.SS1.SSS0.P0.SPx3.p2.11.m11.1.1.1.1.1.2.3.1.cmml" xref="S1.SS1.SSS0.P0.SPx3.p2.11.m11.1.1.1.1.1.2.3"></minus><cn id="S1.SS1.SSS0.P0.SPx3.p2.11.m11.1.1.1.1.1.2.3.2.cmml" type="integer" xref="S1.SS1.SSS0.P0.SPx3.p2.11.m11.1.1.1.1.1.2.3.2">2</cn></apply></apply><apply id="S1.SS1.SSS0.P0.SPx3.p2.11.m11.1.1.1.1.1.3.cmml" xref="S1.SS1.SSS0.P0.SPx3.p2.11.m11.1.1.1.1.1.3"><apply id="S1.SS1.SSS0.P0.SPx3.p2.11.m11.1.1.1.1.1.3.1.cmml" xref="S1.SS1.SSS0.P0.SPx3.p2.11.m11.1.1.1.1.1.3.1"><csymbol cd="ambiguous" id="S1.SS1.SSS0.P0.SPx3.p2.11.m11.1.1.1.1.1.3.1.1.cmml" xref="S1.SS1.SSS0.P0.SPx3.p2.11.m11.1.1.1.1.1.3.1">superscript</csymbol><log id="S1.SS1.SSS0.P0.SPx3.p2.11.m11.1.1.1.1.1.3.1.2.cmml" xref="S1.SS1.SSS0.P0.SPx3.p2.11.m11.1.1.1.1.1.3.1.2"></log><cn id="S1.SS1.SSS0.P0.SPx3.p2.11.m11.1.1.1.1.1.3.1.3.cmml" type="integer" xref="S1.SS1.SSS0.P0.SPx3.p2.11.m11.1.1.1.1.1.3.1.3">2</cn></apply><ci id="S1.SS1.SSS0.P0.SPx3.p2.11.m11.1.1.1.1.1.3.2.cmml" xref="S1.SS1.SSS0.P0.SPx3.p2.11.m11.1.1.1.1.1.3.2">𝑛</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS1.SSS0.P0.SPx3.p2.11.m11.1c">O(\varepsilon^{-2}\log^{2}n)</annotation><annotation encoding="application/x-llamapun" id="S1.SS1.SSS0.P0.SPx3.p2.11.m11.1d">italic_O ( italic_ε start_POSTSUPERSCRIPT - 2 end_POSTSUPERSCRIPT roman_log start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_n )</annotation></semantics></math> rounds is also illustrated by existing dynamic algorithms <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2411.12694v2#bib.bib7" title="">7</a>, <a class="ltx_ref" href="https://arxiv.org/html/2411.12694v2#bib.bib21" title="">21</a>]</cite> that maintain <math alttext="\eta" class="ltx_Math" display="inline" id="S1.SS1.SSS0.P0.SPx3.p2.12.m12.1"><semantics id="S1.SS1.SSS0.P0.SPx3.p2.12.m12.1a"><mi id="S1.SS1.SSS0.P0.SPx3.p2.12.m12.1.1" xref="S1.SS1.SSS0.P0.SPx3.p2.12.m12.1.1.cmml">η</mi><annotation-xml encoding="MathML-Content" id="S1.SS1.SSS0.P0.SPx3.p2.12.m12.1b"><ci id="S1.SS1.SSS0.P0.SPx3.p2.12.m12.1.1.cmml" xref="S1.SS1.SSS0.P0.SPx3.p2.12.m12.1.1">𝜂</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.SS1.SSS0.P0.SPx3.p2.12.m12.1c">\eta</annotation><annotation encoding="application/x-llamapun" id="S1.SS1.SSS0.P0.SPx3.p2.12.m12.1d">italic_η</annotation></semantics></math>-fair orientations. In this scenario, these algorithms have a worst-case recourse of <math alttext="\Omega(\varepsilon^{-2}\log^{2}n)" class="ltx_Math" display="inline" id="S1.SS1.SSS0.P0.SPx3.p2.13.m13.1"><semantics id="S1.SS1.SSS0.P0.SPx3.p2.13.m13.1a"><mrow id="S1.SS1.SSS0.P0.SPx3.p2.13.m13.1.1" xref="S1.SS1.SSS0.P0.SPx3.p2.13.m13.1.1.cmml"><mi id="S1.SS1.SSS0.P0.SPx3.p2.13.m13.1.1.3" mathvariant="normal" xref="S1.SS1.SSS0.P0.SPx3.p2.13.m13.1.1.3.cmml">Ω</mi><mo id="S1.SS1.SSS0.P0.SPx3.p2.13.m13.1.1.2" xref="S1.SS1.SSS0.P0.SPx3.p2.13.m13.1.1.2.cmml">⁢</mo><mrow id="S1.SS1.SSS0.P0.SPx3.p2.13.m13.1.1.1.1" xref="S1.SS1.SSS0.P0.SPx3.p2.13.m13.1.1.1.1.1.cmml"><mo id="S1.SS1.SSS0.P0.SPx3.p2.13.m13.1.1.1.1.2" stretchy="false" xref="S1.SS1.SSS0.P0.SPx3.p2.13.m13.1.1.1.1.1.cmml">(</mo><mrow id="S1.SS1.SSS0.P0.SPx3.p2.13.m13.1.1.1.1.1" xref="S1.SS1.SSS0.P0.SPx3.p2.13.m13.1.1.1.1.1.cmml"><msup id="S1.SS1.SSS0.P0.SPx3.p2.13.m13.1.1.1.1.1.2" xref="S1.SS1.SSS0.P0.SPx3.p2.13.m13.1.1.1.1.1.2.cmml"><mi id="S1.SS1.SSS0.P0.SPx3.p2.13.m13.1.1.1.1.1.2.2" xref="S1.SS1.SSS0.P0.SPx3.p2.13.m13.1.1.1.1.1.2.2.cmml">ε</mi><mrow id="S1.SS1.SSS0.P0.SPx3.p2.13.m13.1.1.1.1.1.2.3" xref="S1.SS1.SSS0.P0.SPx3.p2.13.m13.1.1.1.1.1.2.3.cmml"><mo id="S1.SS1.SSS0.P0.SPx3.p2.13.m13.1.1.1.1.1.2.3a" xref="S1.SS1.SSS0.P0.SPx3.p2.13.m13.1.1.1.1.1.2.3.cmml">−</mo><mn id="S1.SS1.SSS0.P0.SPx3.p2.13.m13.1.1.1.1.1.2.3.2" xref="S1.SS1.SSS0.P0.SPx3.p2.13.m13.1.1.1.1.1.2.3.2.cmml">2</mn></mrow></msup><mo id="S1.SS1.SSS0.P0.SPx3.p2.13.m13.1.1.1.1.1.1" lspace="0.167em" xref="S1.SS1.SSS0.P0.SPx3.p2.13.m13.1.1.1.1.1.1.cmml">⁢</mo><mrow id="S1.SS1.SSS0.P0.SPx3.p2.13.m13.1.1.1.1.1.3" xref="S1.SS1.SSS0.P0.SPx3.p2.13.m13.1.1.1.1.1.3.cmml"><msup id="S1.SS1.SSS0.P0.SPx3.p2.13.m13.1.1.1.1.1.3.1" xref="S1.SS1.SSS0.P0.SPx3.p2.13.m13.1.1.1.1.1.3.1.cmml"><mi id="S1.SS1.SSS0.P0.SPx3.p2.13.m13.1.1.1.1.1.3.1.2" xref="S1.SS1.SSS0.P0.SPx3.p2.13.m13.1.1.1.1.1.3.1.2.cmml">log</mi><mn id="S1.SS1.SSS0.P0.SPx3.p2.13.m13.1.1.1.1.1.3.1.3" xref="S1.SS1.SSS0.P0.SPx3.p2.13.m13.1.1.1.1.1.3.1.3.cmml">2</mn></msup><mo id="S1.SS1.SSS0.P0.SPx3.p2.13.m13.1.1.1.1.1.3a" lspace="0.167em" xref="S1.SS1.SSS0.P0.SPx3.p2.13.m13.1.1.1.1.1.3.cmml">⁡</mo><mi id="S1.SS1.SSS0.P0.SPx3.p2.13.m13.1.1.1.1.1.3.2" xref="S1.SS1.SSS0.P0.SPx3.p2.13.m13.1.1.1.1.1.3.2.cmml">n</mi></mrow></mrow><mo id="S1.SS1.SSS0.P0.SPx3.p2.13.m13.1.1.1.1.3" stretchy="false" xref="S1.SS1.SSS0.P0.SPx3.p2.13.m13.1.1.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.SS1.SSS0.P0.SPx3.p2.13.m13.1b"><apply id="S1.SS1.SSS0.P0.SPx3.p2.13.m13.1.1.cmml" xref="S1.SS1.SSS0.P0.SPx3.p2.13.m13.1.1"><times id="S1.SS1.SSS0.P0.SPx3.p2.13.m13.1.1.2.cmml" xref="S1.SS1.SSS0.P0.SPx3.p2.13.m13.1.1.2"></times><ci id="S1.SS1.SSS0.P0.SPx3.p2.13.m13.1.1.3.cmml" xref="S1.SS1.SSS0.P0.SPx3.p2.13.m13.1.1.3">Ω</ci><apply id="S1.SS1.SSS0.P0.SPx3.p2.13.m13.1.1.1.1.1.cmml" xref="S1.SS1.SSS0.P0.SPx3.p2.13.m13.1.1.1.1"><times id="S1.SS1.SSS0.P0.SPx3.p2.13.m13.1.1.1.1.1.1.cmml" xref="S1.SS1.SSS0.P0.SPx3.p2.13.m13.1.1.1.1.1.1"></times><apply id="S1.SS1.SSS0.P0.SPx3.p2.13.m13.1.1.1.1.1.2.cmml" xref="S1.SS1.SSS0.P0.SPx3.p2.13.m13.1.1.1.1.1.2"><csymbol cd="ambiguous" id="S1.SS1.SSS0.P0.SPx3.p2.13.m13.1.1.1.1.1.2.1.cmml" xref="S1.SS1.SSS0.P0.SPx3.p2.13.m13.1.1.1.1.1.2">superscript</csymbol><ci id="S1.SS1.SSS0.P0.SPx3.p2.13.m13.1.1.1.1.1.2.2.cmml" xref="S1.SS1.SSS0.P0.SPx3.p2.13.m13.1.1.1.1.1.2.2">𝜀</ci><apply id="S1.SS1.SSS0.P0.SPx3.p2.13.m13.1.1.1.1.1.2.3.cmml" xref="S1.SS1.SSS0.P0.SPx3.p2.13.m13.1.1.1.1.1.2.3"><minus id="S1.SS1.SSS0.P0.SPx3.p2.13.m13.1.1.1.1.1.2.3.1.cmml" xref="S1.SS1.SSS0.P0.SPx3.p2.13.m13.1.1.1.1.1.2.3"></minus><cn id="S1.SS1.SSS0.P0.SPx3.p2.13.m13.1.1.1.1.1.2.3.2.cmml" type="integer" xref="S1.SS1.SSS0.P0.SPx3.p2.13.m13.1.1.1.1.1.2.3.2">2</cn></apply></apply><apply id="S1.SS1.SSS0.P0.SPx3.p2.13.m13.1.1.1.1.1.3.cmml" xref="S1.SS1.SSS0.P0.SPx3.p2.13.m13.1.1.1.1.1.3"><apply id="S1.SS1.SSS0.P0.SPx3.p2.13.m13.1.1.1.1.1.3.1.cmml" xref="S1.SS1.SSS0.P0.SPx3.p2.13.m13.1.1.1.1.1.3.1"><csymbol cd="ambiguous" id="S1.SS1.SSS0.P0.SPx3.p2.13.m13.1.1.1.1.1.3.1.1.cmml" xref="S1.SS1.SSS0.P0.SPx3.p2.13.m13.1.1.1.1.1.3.1">superscript</csymbol><log id="S1.SS1.SSS0.P0.SPx3.p2.13.m13.1.1.1.1.1.3.1.2.cmml" xref="S1.SS1.SSS0.P0.SPx3.p2.13.m13.1.1.1.1.1.3.1.2"></log><cn id="S1.SS1.SSS0.P0.SPx3.p2.13.m13.1.1.1.1.1.3.1.3.cmml" type="integer" xref="S1.SS1.SSS0.P0.SPx3.p2.13.m13.1.1.1.1.1.3.1.3">2</cn></apply><ci id="S1.SS1.SSS0.P0.SPx3.p2.13.m13.1.1.1.1.1.3.2.cmml" xref="S1.SS1.SSS0.P0.SPx3.p2.13.m13.1.1.1.1.1.3.2">𝑛</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS1.SSS0.P0.SPx3.p2.13.m13.1c">\Omega(\varepsilon^{-2}\log^{2}n)</annotation><annotation encoding="application/x-llamapun" id="S1.SS1.SSS0.P0.SPx3.p2.13.m13.1d">roman_Ω ( italic_ε start_POSTSUPERSCRIPT - 2 end_POSTSUPERSCRIPT roman_log start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_n )</annotation></semantics></math>. We consider it an interesting open problem to either improve our running time in LOCAL, or, show that <math alttext="O(\varepsilon^{-2}\log^{2}n)" class="ltx_Math" display="inline" id="S1.SS1.SSS0.P0.SPx3.p2.14.m14.1"><semantics id="S1.SS1.SSS0.P0.SPx3.p2.14.m14.1a"><mrow id="S1.SS1.SSS0.P0.SPx3.p2.14.m14.1.1" xref="S1.SS1.SSS0.P0.SPx3.p2.14.m14.1.1.cmml"><mi id="S1.SS1.SSS0.P0.SPx3.p2.14.m14.1.1.3" xref="S1.SS1.SSS0.P0.SPx3.p2.14.m14.1.1.3.cmml">O</mi><mo id="S1.SS1.SSS0.P0.SPx3.p2.14.m14.1.1.2" xref="S1.SS1.SSS0.P0.SPx3.p2.14.m14.1.1.2.cmml">⁢</mo><mrow id="S1.SS1.SSS0.P0.SPx3.p2.14.m14.1.1.1.1" xref="S1.SS1.SSS0.P0.SPx3.p2.14.m14.1.1.1.1.1.cmml"><mo id="S1.SS1.SSS0.P0.SPx3.p2.14.m14.1.1.1.1.2" stretchy="false" xref="S1.SS1.SSS0.P0.SPx3.p2.14.m14.1.1.1.1.1.cmml">(</mo><mrow id="S1.SS1.SSS0.P0.SPx3.p2.14.m14.1.1.1.1.1" xref="S1.SS1.SSS0.P0.SPx3.p2.14.m14.1.1.1.1.1.cmml"><msup id="S1.SS1.SSS0.P0.SPx3.p2.14.m14.1.1.1.1.1.2" xref="S1.SS1.SSS0.P0.SPx3.p2.14.m14.1.1.1.1.1.2.cmml"><mi 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italic_n )</annotation></semantics></math> is tight.</p> </div> </section> <section class="ltx_subparagraph" id="S1.SS1.SSS0.P0.SPx4"> <h5 class="ltx_title ltx_title_subparagraph">D: Results in CONGEST</h5> <div class="ltx_para" id="S1.SS1.SSS0.P0.SPx4.p1"> <p class="ltx_p" id="S1.SS1.SSS0.P0.SPx4.p1.7">We show a significantly more involved algorithm in CONGEST, where after <math alttext="O(\operatorname{poly}\{\varepsilon^{-1},\log n\}\cdot 2^{O(\sqrt{\log n})})" class="ltx_Math" display="inline" id="S1.SS1.SSS0.P0.SPx4.p1.1.m1.3"><semantics id="S1.SS1.SSS0.P0.SPx4.p1.1.m1.3a"><mrow id="S1.SS1.SSS0.P0.SPx4.p1.1.m1.3.3" xref="S1.SS1.SSS0.P0.SPx4.p1.1.m1.3.3.cmml"><mi id="S1.SS1.SSS0.P0.SPx4.p1.1.m1.3.3.3" xref="S1.SS1.SSS0.P0.SPx4.p1.1.m1.3.3.3.cmml">O</mi><mo id="S1.SS1.SSS0.P0.SPx4.p1.1.m1.3.3.2" xref="S1.SS1.SSS0.P0.SPx4.p1.1.m1.3.3.2.cmml">⁢</mo><mrow id="S1.SS1.SSS0.P0.SPx4.p1.1.m1.3.3.1.1" xref="S1.SS1.SSS0.P0.SPx4.p1.1.m1.3.3.1.1.1.cmml"><mo 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id="S1.SS1.SSS0.P0.SPx4.p1.1.m1.1.1.1.1.2.1.cmml" xref="S1.SS1.SSS0.P0.SPx4.p1.1.m1.1.1.1.1.2.1"></log><ci id="S1.SS1.SSS0.P0.SPx4.p1.1.m1.1.1.1.1.2.2.cmml" xref="S1.SS1.SSS0.P0.SPx4.p1.1.m1.1.1.1.1.2.2">𝑛</ci></apply></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS1.SSS0.P0.SPx4.p1.1.m1.3c">O(\operatorname{poly}\{\varepsilon^{-1},\log n\}\cdot 2^{O(\sqrt{\log n})})</annotation><annotation encoding="application/x-llamapun" id="S1.SS1.SSS0.P0.SPx4.p1.1.m1.3d">italic_O ( roman_poly { italic_ε start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT , roman_log italic_n } ⋅ 2 start_POSTSUPERSCRIPT italic_O ( square-root start_ARG roman_log italic_n end_ARG ) end_POSTSUPERSCRIPT )</annotation></semantics></math> rounds, each vertex <math alttext="v" class="ltx_Math" display="inline" id="S1.SS1.SSS0.P0.SPx4.p1.2.m2.1"><semantics id="S1.SS1.SSS0.P0.SPx4.p1.2.m2.1a"><mi id="S1.SS1.SSS0.P0.SPx4.p1.2.m2.1.1" xref="S1.SS1.SSS0.P0.SPx4.p1.2.m2.1.1.cmml">v</mi><annotation-xml encoding="MathML-Content" id="S1.SS1.SSS0.P0.SPx4.p1.2.m2.1b"><ci id="S1.SS1.SSS0.P0.SPx4.p1.2.m2.1.1.cmml" xref="S1.SS1.SSS0.P0.SPx4.p1.2.m2.1.1">𝑣</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.SS1.SSS0.P0.SPx4.p1.2.m2.1c">v</annotation><annotation encoding="application/x-llamapun" id="S1.SS1.SSS0.P0.SPx4.p1.2.m2.1d">italic_v</annotation></semantics></math> outputs a <math alttext="(1+\varepsilon)" class="ltx_Math" display="inline" id="S1.SS1.SSS0.P0.SPx4.p1.3.m3.1"><semantics id="S1.SS1.SSS0.P0.SPx4.p1.3.m3.1a"><mrow id="S1.SS1.SSS0.P0.SPx4.p1.3.m3.1.1.1" xref="S1.SS1.SSS0.P0.SPx4.p1.3.m3.1.1.1.1.cmml"><mo id="S1.SS1.SSS0.P0.SPx4.p1.3.m3.1.1.1.2" stretchy="false" xref="S1.SS1.SSS0.P0.SPx4.p1.3.m3.1.1.1.1.cmml">(</mo><mrow id="S1.SS1.SSS0.P0.SPx4.p1.3.m3.1.1.1.1" xref="S1.SS1.SSS0.P0.SPx4.p1.3.m3.1.1.1.1.cmml"><mn id="S1.SS1.SSS0.P0.SPx4.p1.3.m3.1.1.1.1.2" xref="S1.SS1.SSS0.P0.SPx4.p1.3.m3.1.1.1.1.2.cmml">1</mn><mo id="S1.SS1.SSS0.P0.SPx4.p1.3.m3.1.1.1.1.1" xref="S1.SS1.SSS0.P0.SPx4.p1.3.m3.1.1.1.1.1.cmml">+</mo><mi id="S1.SS1.SSS0.P0.SPx4.p1.3.m3.1.1.1.1.3" xref="S1.SS1.SSS0.P0.SPx4.p1.3.m3.1.1.1.1.3.cmml">ε</mi></mrow><mo id="S1.SS1.SSS0.P0.SPx4.p1.3.m3.1.1.1.3" stretchy="false" xref="S1.SS1.SSS0.P0.SPx4.p1.3.m3.1.1.1.1.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S1.SS1.SSS0.P0.SPx4.p1.3.m3.1b"><apply id="S1.SS1.SSS0.P0.SPx4.p1.3.m3.1.1.1.1.cmml" xref="S1.SS1.SSS0.P0.SPx4.p1.3.m3.1.1.1"><plus id="S1.SS1.SSS0.P0.SPx4.p1.3.m3.1.1.1.1.1.cmml" xref="S1.SS1.SSS0.P0.SPx4.p1.3.m3.1.1.1.1.1"></plus><cn id="S1.SS1.SSS0.P0.SPx4.p1.3.m3.1.1.1.1.2.cmml" type="integer" xref="S1.SS1.SSS0.P0.SPx4.p1.3.m3.1.1.1.1.2">1</cn><ci id="S1.SS1.SSS0.P0.SPx4.p1.3.m3.1.1.1.1.3.cmml" xref="S1.SS1.SSS0.P0.SPx4.p1.3.m3.1.1.1.1.3">𝜀</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS1.SSS0.P0.SPx4.p1.3.m3.1c">(1+\varepsilon)</annotation><annotation encoding="application/x-llamapun" id="S1.SS1.SSS0.P0.SPx4.p1.3.m3.1d">( 1 + italic_ε )</annotation></semantics></math>-approximation of <math alttext="\rho^{*}(v)" class="ltx_Math" display="inline" id="S1.SS1.SSS0.P0.SPx4.p1.4.m4.1"><semantics id="S1.SS1.SSS0.P0.SPx4.p1.4.m4.1a"><mrow id="S1.SS1.SSS0.P0.SPx4.p1.4.m4.1.2" xref="S1.SS1.SSS0.P0.SPx4.p1.4.m4.1.2.cmml"><msup id="S1.SS1.SSS0.P0.SPx4.p1.4.m4.1.2.2" xref="S1.SS1.SSS0.P0.SPx4.p1.4.m4.1.2.2.cmml"><mi id="S1.SS1.SSS0.P0.SPx4.p1.4.m4.1.2.2.2" xref="S1.SS1.SSS0.P0.SPx4.p1.4.m4.1.2.2.2.cmml">ρ</mi><mo id="S1.SS1.SSS0.P0.SPx4.p1.4.m4.1.2.2.3" xref="S1.SS1.SSS0.P0.SPx4.p1.4.m4.1.2.2.3.cmml">∗</mo></msup><mo id="S1.SS1.SSS0.P0.SPx4.p1.4.m4.1.2.1" xref="S1.SS1.SSS0.P0.SPx4.p1.4.m4.1.2.1.cmml">⁢</mo><mrow id="S1.SS1.SSS0.P0.SPx4.p1.4.m4.1.2.3.2" xref="S1.SS1.SSS0.P0.SPx4.p1.4.m4.1.2.cmml"><mo id="S1.SS1.SSS0.P0.SPx4.p1.4.m4.1.2.3.2.1" stretchy="false" xref="S1.SS1.SSS0.P0.SPx4.p1.4.m4.1.2.cmml">(</mo><mi id="S1.SS1.SSS0.P0.SPx4.p1.4.m4.1.1" xref="S1.SS1.SSS0.P0.SPx4.p1.4.m4.1.1.cmml">v</mi><mo id="S1.SS1.SSS0.P0.SPx4.p1.4.m4.1.2.3.2.2" stretchy="false" xref="S1.SS1.SSS0.P0.SPx4.p1.4.m4.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.SS1.SSS0.P0.SPx4.p1.4.m4.1b"><apply id="S1.SS1.SSS0.P0.SPx4.p1.4.m4.1.2.cmml" xref="S1.SS1.SSS0.P0.SPx4.p1.4.m4.1.2"><times id="S1.SS1.SSS0.P0.SPx4.p1.4.m4.1.2.1.cmml" xref="S1.SS1.SSS0.P0.SPx4.p1.4.m4.1.2.1"></times><apply id="S1.SS1.SSS0.P0.SPx4.p1.4.m4.1.2.2.cmml" xref="S1.SS1.SSS0.P0.SPx4.p1.4.m4.1.2.2"><csymbol cd="ambiguous" id="S1.SS1.SSS0.P0.SPx4.p1.4.m4.1.2.2.1.cmml" xref="S1.SS1.SSS0.P0.SPx4.p1.4.m4.1.2.2">superscript</csymbol><ci id="S1.SS1.SSS0.P0.SPx4.p1.4.m4.1.2.2.2.cmml" xref="S1.SS1.SSS0.P0.SPx4.p1.4.m4.1.2.2.2">𝜌</ci><times id="S1.SS1.SSS0.P0.SPx4.p1.4.m4.1.2.2.3.cmml" xref="S1.SS1.SSS0.P0.SPx4.p1.4.m4.1.2.2.3"></times></apply><ci id="S1.SS1.SSS0.P0.SPx4.p1.4.m4.1.1.cmml" xref="S1.SS1.SSS0.P0.SPx4.p1.4.m4.1.1">𝑣</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS1.SSS0.P0.SPx4.p1.4.m4.1c">\rho^{*}(v)</annotation><annotation encoding="application/x-llamapun" id="S1.SS1.SSS0.P0.SPx4.p1.4.m4.1d">italic_ρ start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ( italic_v )</annotation></semantics></math>. Since <math alttext="\max\limits_{v\in V}\rho^{*}(v)=\rho^{\max}(G)" class="ltx_Math" display="inline" id="S1.SS1.SSS0.P0.SPx4.p1.5.m5.2"><semantics id="S1.SS1.SSS0.P0.SPx4.p1.5.m5.2a"><mrow id="S1.SS1.SSS0.P0.SPx4.p1.5.m5.2.3" xref="S1.SS1.SSS0.P0.SPx4.p1.5.m5.2.3.cmml"><mrow id="S1.SS1.SSS0.P0.SPx4.p1.5.m5.2.3.2" xref="S1.SS1.SSS0.P0.SPx4.p1.5.m5.2.3.2.cmml"><mrow id="S1.SS1.SSS0.P0.SPx4.p1.5.m5.2.3.2.2" xref="S1.SS1.SSS0.P0.SPx4.p1.5.m5.2.3.2.2.cmml"><munder id="S1.SS1.SSS0.P0.SPx4.p1.5.m5.2.3.2.2.1" xref="S1.SS1.SSS0.P0.SPx4.p1.5.m5.2.3.2.2.1.cmml"><mi id="S1.SS1.SSS0.P0.SPx4.p1.5.m5.2.3.2.2.1.2" xref="S1.SS1.SSS0.P0.SPx4.p1.5.m5.2.3.2.2.1.2.cmml">max</mi><mrow id="S1.SS1.SSS0.P0.SPx4.p1.5.m5.2.3.2.2.1.3" xref="S1.SS1.SSS0.P0.SPx4.p1.5.m5.2.3.2.2.1.3.cmml"><mi id="S1.SS1.SSS0.P0.SPx4.p1.5.m5.2.3.2.2.1.3.2" xref="S1.SS1.SSS0.P0.SPx4.p1.5.m5.2.3.2.2.1.3.2.cmml">v</mi><mo id="S1.SS1.SSS0.P0.SPx4.p1.5.m5.2.3.2.2.1.3.1" xref="S1.SS1.SSS0.P0.SPx4.p1.5.m5.2.3.2.2.1.3.1.cmml">∈</mo><mi id="S1.SS1.SSS0.P0.SPx4.p1.5.m5.2.3.2.2.1.3.3" xref="S1.SS1.SSS0.P0.SPx4.p1.5.m5.2.3.2.2.1.3.3.cmml">V</mi></mrow></munder><mo id="S1.SS1.SSS0.P0.SPx4.p1.5.m5.2.3.2.2a" lspace="0.167em" xref="S1.SS1.SSS0.P0.SPx4.p1.5.m5.2.3.2.2.cmml">⁡</mo><msup id="S1.SS1.SSS0.P0.SPx4.p1.5.m5.2.3.2.2.2" xref="S1.SS1.SSS0.P0.SPx4.p1.5.m5.2.3.2.2.2.cmml"><mi id="S1.SS1.SSS0.P0.SPx4.p1.5.m5.2.3.2.2.2.2" xref="S1.SS1.SSS0.P0.SPx4.p1.5.m5.2.3.2.2.2.2.cmml">ρ</mi><mo id="S1.SS1.SSS0.P0.SPx4.p1.5.m5.2.3.2.2.2.3" xref="S1.SS1.SSS0.P0.SPx4.p1.5.m5.2.3.2.2.2.3.cmml">∗</mo></msup></mrow><mo id="S1.SS1.SSS0.P0.SPx4.p1.5.m5.2.3.2.1" xref="S1.SS1.SSS0.P0.SPx4.p1.5.m5.2.3.2.1.cmml">⁢</mo><mrow id="S1.SS1.SSS0.P0.SPx4.p1.5.m5.2.3.2.3.2" xref="S1.SS1.SSS0.P0.SPx4.p1.5.m5.2.3.2.cmml"><mo id="S1.SS1.SSS0.P0.SPx4.p1.5.m5.2.3.2.3.2.1" stretchy="false" xref="S1.SS1.SSS0.P0.SPx4.p1.5.m5.2.3.2.cmml">(</mo><mi id="S1.SS1.SSS0.P0.SPx4.p1.5.m5.1.1" xref="S1.SS1.SSS0.P0.SPx4.p1.5.m5.1.1.cmml">v</mi><mo id="S1.SS1.SSS0.P0.SPx4.p1.5.m5.2.3.2.3.2.2" stretchy="false" xref="S1.SS1.SSS0.P0.SPx4.p1.5.m5.2.3.2.cmml">)</mo></mrow></mrow><mo id="S1.SS1.SSS0.P0.SPx4.p1.5.m5.2.3.1" xref="S1.SS1.SSS0.P0.SPx4.p1.5.m5.2.3.1.cmml">=</mo><mrow id="S1.SS1.SSS0.P0.SPx4.p1.5.m5.2.3.3" xref="S1.SS1.SSS0.P0.SPx4.p1.5.m5.2.3.3.cmml"><msup id="S1.SS1.SSS0.P0.SPx4.p1.5.m5.2.3.3.2" xref="S1.SS1.SSS0.P0.SPx4.p1.5.m5.2.3.3.2.cmml"><mi id="S1.SS1.SSS0.P0.SPx4.p1.5.m5.2.3.3.2.2" xref="S1.SS1.SSS0.P0.SPx4.p1.5.m5.2.3.3.2.2.cmml">ρ</mi><mi id="S1.SS1.SSS0.P0.SPx4.p1.5.m5.2.3.3.2.3" xref="S1.SS1.SSS0.P0.SPx4.p1.5.m5.2.3.3.2.3.cmml">max</mi></msup><mo id="S1.SS1.SSS0.P0.SPx4.p1.5.m5.2.3.3.1" xref="S1.SS1.SSS0.P0.SPx4.p1.5.m5.2.3.3.1.cmml">⁢</mo><mrow id="S1.SS1.SSS0.P0.SPx4.p1.5.m5.2.3.3.3.2" xref="S1.SS1.SSS0.P0.SPx4.p1.5.m5.2.3.3.cmml"><mo id="S1.SS1.SSS0.P0.SPx4.p1.5.m5.2.3.3.3.2.1" stretchy="false" xref="S1.SS1.SSS0.P0.SPx4.p1.5.m5.2.3.3.cmml">(</mo><mi id="S1.SS1.SSS0.P0.SPx4.p1.5.m5.2.2" xref="S1.SS1.SSS0.P0.SPx4.p1.5.m5.2.2.cmml">G</mi><mo id="S1.SS1.SSS0.P0.SPx4.p1.5.m5.2.3.3.3.2.2" stretchy="false" xref="S1.SS1.SSS0.P0.SPx4.p1.5.m5.2.3.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.SS1.SSS0.P0.SPx4.p1.5.m5.2b"><apply id="S1.SS1.SSS0.P0.SPx4.p1.5.m5.2.3.cmml" xref="S1.SS1.SSS0.P0.SPx4.p1.5.m5.2.3"><eq id="S1.SS1.SSS0.P0.SPx4.p1.5.m5.2.3.1.cmml" xref="S1.SS1.SSS0.P0.SPx4.p1.5.m5.2.3.1"></eq><apply id="S1.SS1.SSS0.P0.SPx4.p1.5.m5.2.3.2.cmml" xref="S1.SS1.SSS0.P0.SPx4.p1.5.m5.2.3.2"><times id="S1.SS1.SSS0.P0.SPx4.p1.5.m5.2.3.2.1.cmml" xref="S1.SS1.SSS0.P0.SPx4.p1.5.m5.2.3.2.1"></times><apply id="S1.SS1.SSS0.P0.SPx4.p1.5.m5.2.3.2.2.cmml" xref="S1.SS1.SSS0.P0.SPx4.p1.5.m5.2.3.2.2"><apply id="S1.SS1.SSS0.P0.SPx4.p1.5.m5.2.3.2.2.1.cmml" xref="S1.SS1.SSS0.P0.SPx4.p1.5.m5.2.3.2.2.1"><csymbol cd="ambiguous" id="S1.SS1.SSS0.P0.SPx4.p1.5.m5.2.3.2.2.1.1.cmml" xref="S1.SS1.SSS0.P0.SPx4.p1.5.m5.2.3.2.2.1">subscript</csymbol><max id="S1.SS1.SSS0.P0.SPx4.p1.5.m5.2.3.2.2.1.2.cmml" xref="S1.SS1.SSS0.P0.SPx4.p1.5.m5.2.3.2.2.1.2"></max><apply id="S1.SS1.SSS0.P0.SPx4.p1.5.m5.2.3.2.2.1.3.cmml" xref="S1.SS1.SSS0.P0.SPx4.p1.5.m5.2.3.2.2.1.3"><in id="S1.SS1.SSS0.P0.SPx4.p1.5.m5.2.3.2.2.1.3.1.cmml" xref="S1.SS1.SSS0.P0.SPx4.p1.5.m5.2.3.2.2.1.3.1"></in><ci id="S1.SS1.SSS0.P0.SPx4.p1.5.m5.2.3.2.2.1.3.2.cmml" xref="S1.SS1.SSS0.P0.SPx4.p1.5.m5.2.3.2.2.1.3.2">𝑣</ci><ci id="S1.SS1.SSS0.P0.SPx4.p1.5.m5.2.3.2.2.1.3.3.cmml" xref="S1.SS1.SSS0.P0.SPx4.p1.5.m5.2.3.2.2.1.3.3">𝑉</ci></apply></apply><apply id="S1.SS1.SSS0.P0.SPx4.p1.5.m5.2.3.2.2.2.cmml" xref="S1.SS1.SSS0.P0.SPx4.p1.5.m5.2.3.2.2.2"><csymbol cd="ambiguous" id="S1.SS1.SSS0.P0.SPx4.p1.5.m5.2.3.2.2.2.1.cmml" xref="S1.SS1.SSS0.P0.SPx4.p1.5.m5.2.3.2.2.2">superscript</csymbol><ci id="S1.SS1.SSS0.P0.SPx4.p1.5.m5.2.3.2.2.2.2.cmml" xref="S1.SS1.SSS0.P0.SPx4.p1.5.m5.2.3.2.2.2.2">𝜌</ci><times id="S1.SS1.SSS0.P0.SPx4.p1.5.m5.2.3.2.2.2.3.cmml" xref="S1.SS1.SSS0.P0.SPx4.p1.5.m5.2.3.2.2.2.3"></times></apply></apply><ci id="S1.SS1.SSS0.P0.SPx4.p1.5.m5.1.1.cmml" xref="S1.SS1.SSS0.P0.SPx4.p1.5.m5.1.1">𝑣</ci></apply><apply id="S1.SS1.SSS0.P0.SPx4.p1.5.m5.2.3.3.cmml" xref="S1.SS1.SSS0.P0.SPx4.p1.5.m5.2.3.3"><times id="S1.SS1.SSS0.P0.SPx4.p1.5.m5.2.3.3.1.cmml" xref="S1.SS1.SSS0.P0.SPx4.p1.5.m5.2.3.3.1"></times><apply id="S1.SS1.SSS0.P0.SPx4.p1.5.m5.2.3.3.2.cmml" xref="S1.SS1.SSS0.P0.SPx4.p1.5.m5.2.3.3.2"><csymbol cd="ambiguous" id="S1.SS1.SSS0.P0.SPx4.p1.5.m5.2.3.3.2.1.cmml" xref="S1.SS1.SSS0.P0.SPx4.p1.5.m5.2.3.3.2">superscript</csymbol><ci id="S1.SS1.SSS0.P0.SPx4.p1.5.m5.2.3.3.2.2.cmml" xref="S1.SS1.SSS0.P0.SPx4.p1.5.m5.2.3.3.2.2">𝜌</ci><max id="S1.SS1.SSS0.P0.SPx4.p1.5.m5.2.3.3.2.3.cmml" xref="S1.SS1.SSS0.P0.SPx4.p1.5.m5.2.3.3.2.3"></max></apply><ci id="S1.SS1.SSS0.P0.SPx4.p1.5.m5.2.2.cmml" xref="S1.SS1.SSS0.P0.SPx4.p1.5.m5.2.2">𝐺</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS1.SSS0.P0.SPx4.p1.5.m5.2c">\max\limits_{v\in V}\rho^{*}(v)=\rho^{\max}(G)</annotation><annotation encoding="application/x-llamapun" id="S1.SS1.SSS0.P0.SPx4.p1.5.m5.2d">roman_max start_POSTSUBSCRIPT italic_v ∈ italic_V end_POSTSUBSCRIPT italic_ρ start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ( italic_v ) = italic_ρ start_POSTSUPERSCRIPT roman_max end_POSTSUPERSCRIPT ( italic_G )</annotation></semantics></math>, this is the first deterministic algorithm for <math alttext="(1+\varepsilon)" class="ltx_Math" display="inline" id="S1.SS1.SSS0.P0.SPx4.p1.6.m6.1"><semantics id="S1.SS1.SSS0.P0.SPx4.p1.6.m6.1a"><mrow id="S1.SS1.SSS0.P0.SPx4.p1.6.m6.1.1.1" xref="S1.SS1.SSS0.P0.SPx4.p1.6.m6.1.1.1.1.cmml"><mo id="S1.SS1.SSS0.P0.SPx4.p1.6.m6.1.1.1.2" stretchy="false" xref="S1.SS1.SSS0.P0.SPx4.p1.6.m6.1.1.1.1.cmml">(</mo><mrow id="S1.SS1.SSS0.P0.SPx4.p1.6.m6.1.1.1.1" xref="S1.SS1.SSS0.P0.SPx4.p1.6.m6.1.1.1.1.cmml"><mn id="S1.SS1.SSS0.P0.SPx4.p1.6.m6.1.1.1.1.2" xref="S1.SS1.SSS0.P0.SPx4.p1.6.m6.1.1.1.1.2.cmml">1</mn><mo id="S1.SS1.SSS0.P0.SPx4.p1.6.m6.1.1.1.1.1" xref="S1.SS1.SSS0.P0.SPx4.p1.6.m6.1.1.1.1.1.cmml">+</mo><mi id="S1.SS1.SSS0.P0.SPx4.p1.6.m6.1.1.1.1.3" xref="S1.SS1.SSS0.P0.SPx4.p1.6.m6.1.1.1.1.3.cmml">ε</mi></mrow><mo id="S1.SS1.SSS0.P0.SPx4.p1.6.m6.1.1.1.3" stretchy="false" xref="S1.SS1.SSS0.P0.SPx4.p1.6.m6.1.1.1.1.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S1.SS1.SSS0.P0.SPx4.p1.6.m6.1b"><apply id="S1.SS1.SSS0.P0.SPx4.p1.6.m6.1.1.1.1.cmml" xref="S1.SS1.SSS0.P0.SPx4.p1.6.m6.1.1.1"><plus id="S1.SS1.SSS0.P0.SPx4.p1.6.m6.1.1.1.1.1.cmml" xref="S1.SS1.SSS0.P0.SPx4.p1.6.m6.1.1.1.1.1"></plus><cn id="S1.SS1.SSS0.P0.SPx4.p1.6.m6.1.1.1.1.2.cmml" type="integer" xref="S1.SS1.SSS0.P0.SPx4.p1.6.m6.1.1.1.1.2">1</cn><ci id="S1.SS1.SSS0.P0.SPx4.p1.6.m6.1.1.1.1.3.cmml" xref="S1.SS1.SSS0.P0.SPx4.p1.6.m6.1.1.1.1.3">𝜀</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS1.SSS0.P0.SPx4.p1.6.m6.1c">(1+\varepsilon)</annotation><annotation encoding="application/x-llamapun" id="S1.SS1.SSS0.P0.SPx4.p1.6.m6.1d">( 1 + italic_ε )</annotation></semantics></math>-approximating of the global subgraph density <math alttext="\rho^{\max}(G)" class="ltx_Math" display="inline" id="S1.SS1.SSS0.P0.SPx4.p1.7.m7.1"><semantics id="S1.SS1.SSS0.P0.SPx4.p1.7.m7.1a"><mrow id="S1.SS1.SSS0.P0.SPx4.p1.7.m7.1.2" xref="S1.SS1.SSS0.P0.SPx4.p1.7.m7.1.2.cmml"><msup id="S1.SS1.SSS0.P0.SPx4.p1.7.m7.1.2.2" xref="S1.SS1.SSS0.P0.SPx4.p1.7.m7.1.2.2.cmml"><mi id="S1.SS1.SSS0.P0.SPx4.p1.7.m7.1.2.2.2" xref="S1.SS1.SSS0.P0.SPx4.p1.7.m7.1.2.2.2.cmml">ρ</mi><mi id="S1.SS1.SSS0.P0.SPx4.p1.7.m7.1.2.2.3" xref="S1.SS1.SSS0.P0.SPx4.p1.7.m7.1.2.2.3.cmml">max</mi></msup><mo id="S1.SS1.SSS0.P0.SPx4.p1.7.m7.1.2.1" xref="S1.SS1.SSS0.P0.SPx4.p1.7.m7.1.2.1.cmml">⁢</mo><mrow id="S1.SS1.SSS0.P0.SPx4.p1.7.m7.1.2.3.2" xref="S1.SS1.SSS0.P0.SPx4.p1.7.m7.1.2.cmml"><mo id="S1.SS1.SSS0.P0.SPx4.p1.7.m7.1.2.3.2.1" stretchy="false" xref="S1.SS1.SSS0.P0.SPx4.p1.7.m7.1.2.cmml">(</mo><mi id="S1.SS1.SSS0.P0.SPx4.p1.7.m7.1.1" xref="S1.SS1.SSS0.P0.SPx4.p1.7.m7.1.1.cmml">G</mi><mo id="S1.SS1.SSS0.P0.SPx4.p1.7.m7.1.2.3.2.2" stretchy="false" xref="S1.SS1.SSS0.P0.SPx4.p1.7.m7.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.SS1.SSS0.P0.SPx4.p1.7.m7.1b"><apply id="S1.SS1.SSS0.P0.SPx4.p1.7.m7.1.2.cmml" xref="S1.SS1.SSS0.P0.SPx4.p1.7.m7.1.2"><times id="S1.SS1.SSS0.P0.SPx4.p1.7.m7.1.2.1.cmml" xref="S1.SS1.SSS0.P0.SPx4.p1.7.m7.1.2.1"></times><apply id="S1.SS1.SSS0.P0.SPx4.p1.7.m7.1.2.2.cmml" xref="S1.SS1.SSS0.P0.SPx4.p1.7.m7.1.2.2"><csymbol cd="ambiguous" id="S1.SS1.SSS0.P0.SPx4.p1.7.m7.1.2.2.1.cmml" xref="S1.SS1.SSS0.P0.SPx4.p1.7.m7.1.2.2">superscript</csymbol><ci id="S1.SS1.SSS0.P0.SPx4.p1.7.m7.1.2.2.2.cmml" xref="S1.SS1.SSS0.P0.SPx4.p1.7.m7.1.2.2.2">𝜌</ci><max id="S1.SS1.SSS0.P0.SPx4.p1.7.m7.1.2.2.3.cmml" xref="S1.SS1.SSS0.P0.SPx4.p1.7.m7.1.2.2.3"></max></apply><ci id="S1.SS1.SSS0.P0.SPx4.p1.7.m7.1.1.cmml" xref="S1.SS1.SSS0.P0.SPx4.p1.7.m7.1.1">𝐺</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS1.SSS0.P0.SPx4.p1.7.m7.1c">\rho^{\max}(G)</annotation><annotation encoding="application/x-llamapun" id="S1.SS1.SSS0.P0.SPx4.p1.7.m7.1d">italic_ρ start_POSTSUPERSCRIPT roman_max end_POSTSUPERSCRIPT ( italic_G )</annotation></semantics></math> in CONGEST, that runs in a number of rounds that is sublinear in the diameter of the graph.</p> </div> <div class="ltx_para" id="S1.SS1.SSS0.P0.SPx4.p2"> <p class="ltx_p" id="S1.SS1.SSS0.P0.SPx4.p2.7">In the main body, we focus on the value variant where we want each vertex <math alttext="v" class="ltx_Math" display="inline" id="S1.SS1.SSS0.P0.SPx4.p2.1.m1.1"><semantics id="S1.SS1.SSS0.P0.SPx4.p2.1.m1.1a"><mi id="S1.SS1.SSS0.P0.SPx4.p2.1.m1.1.1" xref="S1.SS1.SSS0.P0.SPx4.p2.1.m1.1.1.cmml">v</mi><annotation-xml encoding="MathML-Content" id="S1.SS1.SSS0.P0.SPx4.p2.1.m1.1b"><ci id="S1.SS1.SSS0.P0.SPx4.p2.1.m1.1.1.cmml" xref="S1.SS1.SSS0.P0.SPx4.p2.1.m1.1.1">𝑣</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.SS1.SSS0.P0.SPx4.p2.1.m1.1c">v</annotation><annotation encoding="application/x-llamapun" id="S1.SS1.SSS0.P0.SPx4.p2.1.m1.1d">italic_v</annotation></semantics></math> to output an approximation of <math alttext="\rho^{*}(v)" class="ltx_Math" display="inline" id="S1.SS1.SSS0.P0.SPx4.p2.2.m2.1"><semantics id="S1.SS1.SSS0.P0.SPx4.p2.2.m2.1a"><mrow id="S1.SS1.SSS0.P0.SPx4.p2.2.m2.1.2" xref="S1.SS1.SSS0.P0.SPx4.p2.2.m2.1.2.cmml"><msup id="S1.SS1.SSS0.P0.SPx4.p2.2.m2.1.2.2" xref="S1.SS1.SSS0.P0.SPx4.p2.2.m2.1.2.2.cmml"><mi id="S1.SS1.SSS0.P0.SPx4.p2.2.m2.1.2.2.2" xref="S1.SS1.SSS0.P0.SPx4.p2.2.m2.1.2.2.2.cmml">ρ</mi><mo id="S1.SS1.SSS0.P0.SPx4.p2.2.m2.1.2.2.3" xref="S1.SS1.SSS0.P0.SPx4.p2.2.m2.1.2.2.3.cmml">∗</mo></msup><mo id="S1.SS1.SSS0.P0.SPx4.p2.2.m2.1.2.1" xref="S1.SS1.SSS0.P0.SPx4.p2.2.m2.1.2.1.cmml">⁢</mo><mrow id="S1.SS1.SSS0.P0.SPx4.p2.2.m2.1.2.3.2" xref="S1.SS1.SSS0.P0.SPx4.p2.2.m2.1.2.cmml"><mo id="S1.SS1.SSS0.P0.SPx4.p2.2.m2.1.2.3.2.1" stretchy="false" xref="S1.SS1.SSS0.P0.SPx4.p2.2.m2.1.2.cmml">(</mo><mi id="S1.SS1.SSS0.P0.SPx4.p2.2.m2.1.1" xref="S1.SS1.SSS0.P0.SPx4.p2.2.m2.1.1.cmml">v</mi><mo id="S1.SS1.SSS0.P0.SPx4.p2.2.m2.1.2.3.2.2" stretchy="false" xref="S1.SS1.SSS0.P0.SPx4.p2.2.m2.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.SS1.SSS0.P0.SPx4.p2.2.m2.1b"><apply id="S1.SS1.SSS0.P0.SPx4.p2.2.m2.1.2.cmml" xref="S1.SS1.SSS0.P0.SPx4.p2.2.m2.1.2"><times id="S1.SS1.SSS0.P0.SPx4.p2.2.m2.1.2.1.cmml" xref="S1.SS1.SSS0.P0.SPx4.p2.2.m2.1.2.1"></times><apply id="S1.SS1.SSS0.P0.SPx4.p2.2.m2.1.2.2.cmml" xref="S1.SS1.SSS0.P0.SPx4.p2.2.m2.1.2.2"><csymbol cd="ambiguous" id="S1.SS1.SSS0.P0.SPx4.p2.2.m2.1.2.2.1.cmml" xref="S1.SS1.SSS0.P0.SPx4.p2.2.m2.1.2.2">superscript</csymbol><ci id="S1.SS1.SSS0.P0.SPx4.p2.2.m2.1.2.2.2.cmml" xref="S1.SS1.SSS0.P0.SPx4.p2.2.m2.1.2.2.2">𝜌</ci><times id="S1.SS1.SSS0.P0.SPx4.p2.2.m2.1.2.2.3.cmml" xref="S1.SS1.SSS0.P0.SPx4.p2.2.m2.1.2.2.3"></times></apply><ci id="S1.SS1.SSS0.P0.SPx4.p2.2.m2.1.1.cmml" xref="S1.SS1.SSS0.P0.SPx4.p2.2.m2.1.1">𝑣</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS1.SSS0.P0.SPx4.p2.2.m2.1c">\rho^{*}(v)</annotation><annotation encoding="application/x-llamapun" id="S1.SS1.SSS0.P0.SPx4.p2.2.m2.1d">italic_ρ start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ( italic_v )</annotation></semantics></math>. In Appendix <a class="ltx_ref" href="https://arxiv.org/html/2411.12694v2#A1" title="Appendix A Reporting a densest subgraph ‣ Local Density and its Distributed Approximation"><span class="ltx_text ltx_ref_tag">A</span></a> we extend our analysis so that each vertex <math alttext="v" class="ltx_Math" display="inline" id="S1.SS1.SSS0.P0.SPx4.p2.3.m3.1"><semantics id="S1.SS1.SSS0.P0.SPx4.p2.3.m3.1a"><mi id="S1.SS1.SSS0.P0.SPx4.p2.3.m3.1.1" xref="S1.SS1.SSS0.P0.SPx4.p2.3.m3.1.1.cmml">v</mi><annotation-xml encoding="MathML-Content" id="S1.SS1.SSS0.P0.SPx4.p2.3.m3.1b"><ci id="S1.SS1.SSS0.P0.SPx4.p2.3.m3.1.1.cmml" xref="S1.SS1.SSS0.P0.SPx4.p2.3.m3.1.1">𝑣</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.SS1.SSS0.P0.SPx4.p2.3.m3.1c">v</annotation><annotation encoding="application/x-llamapun" id="S1.SS1.SSS0.P0.SPx4.p2.3.m3.1d">italic_v</annotation></semantics></math> can output a subgraph <math alttext="H" class="ltx_Math" display="inline" id="S1.SS1.SSS0.P0.SPx4.p2.4.m4.1"><semantics id="S1.SS1.SSS0.P0.SPx4.p2.4.m4.1a"><mi id="S1.SS1.SSS0.P0.SPx4.p2.4.m4.1.1" xref="S1.SS1.SSS0.P0.SPx4.p2.4.m4.1.1.cmml">H</mi><annotation-xml encoding="MathML-Content" id="S1.SS1.SSS0.P0.SPx4.p2.4.m4.1b"><ci id="S1.SS1.SSS0.P0.SPx4.p2.4.m4.1.1.cmml" xref="S1.SS1.SSS0.P0.SPx4.p2.4.m4.1.1">𝐻</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.SS1.SSS0.P0.SPx4.p2.4.m4.1c">H</annotation><annotation encoding="application/x-llamapun" id="S1.SS1.SSS0.P0.SPx4.p2.4.m4.1d">italic_H</annotation></semantics></math> with <math alttext="v\in H" class="ltx_Math" display="inline" id="S1.SS1.SSS0.P0.SPx4.p2.5.m5.1"><semantics id="S1.SS1.SSS0.P0.SPx4.p2.5.m5.1a"><mrow id="S1.SS1.SSS0.P0.SPx4.p2.5.m5.1.1" xref="S1.SS1.SSS0.P0.SPx4.p2.5.m5.1.1.cmml"><mi id="S1.SS1.SSS0.P0.SPx4.p2.5.m5.1.1.2" xref="S1.SS1.SSS0.P0.SPx4.p2.5.m5.1.1.2.cmml">v</mi><mo id="S1.SS1.SSS0.P0.SPx4.p2.5.m5.1.1.1" xref="S1.SS1.SSS0.P0.SPx4.p2.5.m5.1.1.1.cmml">∈</mo><mi id="S1.SS1.SSS0.P0.SPx4.p2.5.m5.1.1.3" xref="S1.SS1.SSS0.P0.SPx4.p2.5.m5.1.1.3.cmml">H</mi></mrow><annotation-xml encoding="MathML-Content" id="S1.SS1.SSS0.P0.SPx4.p2.5.m5.1b"><apply id="S1.SS1.SSS0.P0.SPx4.p2.5.m5.1.1.cmml" xref="S1.SS1.SSS0.P0.SPx4.p2.5.m5.1.1"><in id="S1.SS1.SSS0.P0.SPx4.p2.5.m5.1.1.1.cmml" xref="S1.SS1.SSS0.P0.SPx4.p2.5.m5.1.1.1"></in><ci id="S1.SS1.SSS0.P0.SPx4.p2.5.m5.1.1.2.cmml" xref="S1.SS1.SSS0.P0.SPx4.p2.5.m5.1.1.2">𝑣</ci><ci id="S1.SS1.SSS0.P0.SPx4.p2.5.m5.1.1.3.cmml" xref="S1.SS1.SSS0.P0.SPx4.p2.5.m5.1.1.3">𝐻</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS1.SSS0.P0.SPx4.p2.5.m5.1c">v\in H</annotation><annotation encoding="application/x-llamapun" id="S1.SS1.SSS0.P0.SPx4.p2.5.m5.1d">italic_v ∈ italic_H</annotation></semantics></math> where <math alttext="\rho(H)" class="ltx_Math" display="inline" id="S1.SS1.SSS0.P0.SPx4.p2.6.m6.1"><semantics id="S1.SS1.SSS0.P0.SPx4.p2.6.m6.1a"><mrow id="S1.SS1.SSS0.P0.SPx4.p2.6.m6.1.2" xref="S1.SS1.SSS0.P0.SPx4.p2.6.m6.1.2.cmml"><mi id="S1.SS1.SSS0.P0.SPx4.p2.6.m6.1.2.2" xref="S1.SS1.SSS0.P0.SPx4.p2.6.m6.1.2.2.cmml">ρ</mi><mo id="S1.SS1.SSS0.P0.SPx4.p2.6.m6.1.2.1" xref="S1.SS1.SSS0.P0.SPx4.p2.6.m6.1.2.1.cmml">⁢</mo><mrow id="S1.SS1.SSS0.P0.SPx4.p2.6.m6.1.2.3.2" xref="S1.SS1.SSS0.P0.SPx4.p2.6.m6.1.2.cmml"><mo id="S1.SS1.SSS0.P0.SPx4.p2.6.m6.1.2.3.2.1" stretchy="false" xref="S1.SS1.SSS0.P0.SPx4.p2.6.m6.1.2.cmml">(</mo><mi id="S1.SS1.SSS0.P0.SPx4.p2.6.m6.1.1" xref="S1.SS1.SSS0.P0.SPx4.p2.6.m6.1.1.cmml">H</mi><mo id="S1.SS1.SSS0.P0.SPx4.p2.6.m6.1.2.3.2.2" stretchy="false" xref="S1.SS1.SSS0.P0.SPx4.p2.6.m6.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.SS1.SSS0.P0.SPx4.p2.6.m6.1b"><apply id="S1.SS1.SSS0.P0.SPx4.p2.6.m6.1.2.cmml" xref="S1.SS1.SSS0.P0.SPx4.p2.6.m6.1.2"><times id="S1.SS1.SSS0.P0.SPx4.p2.6.m6.1.2.1.cmml" xref="S1.SS1.SSS0.P0.SPx4.p2.6.m6.1.2.1"></times><ci id="S1.SS1.SSS0.P0.SPx4.p2.6.m6.1.2.2.cmml" xref="S1.SS1.SSS0.P0.SPx4.p2.6.m6.1.2.2">𝜌</ci><ci id="S1.SS1.SSS0.P0.SPx4.p2.6.m6.1.1.cmml" xref="S1.SS1.SSS0.P0.SPx4.p2.6.m6.1.1">𝐻</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS1.SSS0.P0.SPx4.p2.6.m6.1c">\rho(H)</annotation><annotation encoding="application/x-llamapun" id="S1.SS1.SSS0.P0.SPx4.p2.6.m6.1d">italic_ρ ( italic_H )</annotation></semantics></math> approximates <math alttext="\rho^{*}(v)" class="ltx_Math" display="inline" id="S1.SS1.SSS0.P0.SPx4.p2.7.m7.1"><semantics id="S1.SS1.SSS0.P0.SPx4.p2.7.m7.1a"><mrow id="S1.SS1.SSS0.P0.SPx4.p2.7.m7.1.2" xref="S1.SS1.SSS0.P0.SPx4.p2.7.m7.1.2.cmml"><msup id="S1.SS1.SSS0.P0.SPx4.p2.7.m7.1.2.2" xref="S1.SS1.SSS0.P0.SPx4.p2.7.m7.1.2.2.cmml"><mi id="S1.SS1.SSS0.P0.SPx4.p2.7.m7.1.2.2.2" xref="S1.SS1.SSS0.P0.SPx4.p2.7.m7.1.2.2.2.cmml">ρ</mi><mo id="S1.SS1.SSS0.P0.SPx4.p2.7.m7.1.2.2.3" xref="S1.SS1.SSS0.P0.SPx4.p2.7.m7.1.2.2.3.cmml">∗</mo></msup><mo id="S1.SS1.SSS0.P0.SPx4.p2.7.m7.1.2.1" xref="S1.SS1.SSS0.P0.SPx4.p2.7.m7.1.2.1.cmml">⁢</mo><mrow id="S1.SS1.SSS0.P0.SPx4.p2.7.m7.1.2.3.2" xref="S1.SS1.SSS0.P0.SPx4.p2.7.m7.1.2.cmml"><mo id="S1.SS1.SSS0.P0.SPx4.p2.7.m7.1.2.3.2.1" stretchy="false" xref="S1.SS1.SSS0.P0.SPx4.p2.7.m7.1.2.cmml">(</mo><mi id="S1.SS1.SSS0.P0.SPx4.p2.7.m7.1.1" xref="S1.SS1.SSS0.P0.SPx4.p2.7.m7.1.1.cmml">v</mi><mo id="S1.SS1.SSS0.P0.SPx4.p2.7.m7.1.2.3.2.2" stretchy="false" xref="S1.SS1.SSS0.P0.SPx4.p2.7.m7.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.SS1.SSS0.P0.SPx4.p2.7.m7.1b"><apply id="S1.SS1.SSS0.P0.SPx4.p2.7.m7.1.2.cmml" xref="S1.SS1.SSS0.P0.SPx4.p2.7.m7.1.2"><times id="S1.SS1.SSS0.P0.SPx4.p2.7.m7.1.2.1.cmml" xref="S1.SS1.SSS0.P0.SPx4.p2.7.m7.1.2.1"></times><apply id="S1.SS1.SSS0.P0.SPx4.p2.7.m7.1.2.2.cmml" xref="S1.SS1.SSS0.P0.SPx4.p2.7.m7.1.2.2"><csymbol cd="ambiguous" id="S1.SS1.SSS0.P0.SPx4.p2.7.m7.1.2.2.1.cmml" xref="S1.SS1.SSS0.P0.SPx4.p2.7.m7.1.2.2">superscript</csymbol><ci id="S1.SS1.SSS0.P0.SPx4.p2.7.m7.1.2.2.2.cmml" xref="S1.SS1.SSS0.P0.SPx4.p2.7.m7.1.2.2.2">𝜌</ci><times id="S1.SS1.SSS0.P0.SPx4.p2.7.m7.1.2.2.3.cmml" xref="S1.SS1.SSS0.P0.SPx4.p2.7.m7.1.2.2.3"></times></apply><ci id="S1.SS1.SSS0.P0.SPx4.p2.7.m7.1.1.cmml" xref="S1.SS1.SSS0.P0.SPx4.p2.7.m7.1.1">𝑣</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS1.SSS0.P0.SPx4.p2.7.m7.1c">\rho^{*}(v)</annotation><annotation encoding="application/x-llamapun" id="S1.SS1.SSS0.P0.SPx4.p2.7.m7.1d">italic_ρ start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ( italic_v )</annotation></semantics></math>. See also Tables <a class="ltx_ref" href="https://arxiv.org/html/2411.12694v2#S1.T1" title="Table 1 ‣ D: Results in CONGEST ‣ 1.1 Local density and results. ‣ 1 Introduction ‣ Local Density and its Distributed Approximation"><span class="ltx_text ltx_ref_tag">1</span></a> and <a class="ltx_ref" href="https://arxiv.org/html/2411.12694v2#A1.T2" title="Table 2 ‣ A.2 Reporting a locally densest subgraph ‣ Appendix A Reporting a densest subgraph ‣ Local Density and its Distributed Approximation"><span class="ltx_text ltx_ref_tag">2</span></a>.</p> </div> <figure class="ltx_table" id="S1.T1"> <p class="ltx_p ltx_align_center" id="S1.T1.20"><span class="ltx_text ltx_inline-block" id="S1.T1.20.20" style="width:433.6pt;"> <span class="ltx_inline-block ltx_transformed_outer" id="S1.T1.20.20.20.20.20" style="width:488.1pt;height:181pt;vertical-align:-1.0pt;"><span class="ltx_transformed_inner" style="transform:translate(0.0pt,0.0pt) scale(1,1) ;"> <span class="ltx_p" id="S1.T1.20.20.20.20.20.20"><span class="ltx_text" id="S1.T1.20.20.20.20.20.20.20"> <span class="ltx_tabular ltx_align_middle" id="S1.T1.20.20.20.20.20.20.20.20"> <span class="ltx_tr" id="S1.T1.2.2.2.2.2.2.2.2.2"> <span class="ltx_td ltx_align_center ltx_border_r" id="S1.T1.2.2.2.2.2.2.2.2.2.3">Model</span> <span class="ltx_td ltx_align_center ltx_border_r" id="S1.T1.2.2.2.2.2.2.2.2.2.4">Problem</span> <span class="ltx_td ltx_align_left ltx_border_r" id="S1.T1.2.2.2.2.2.2.2.2.2.2">Each <math alttext="v" class="ltx_Math" display="inline" id="S1.T1.1.1.1.1.1.1.1.1.1.1.m1.1"><semantics id="S1.T1.1.1.1.1.1.1.1.1.1.1.m1.1a"><mi id="S1.T1.1.1.1.1.1.1.1.1.1.1.m1.1.1" xref="S1.T1.1.1.1.1.1.1.1.1.1.1.m1.1.1.cmml">v</mi><annotation-xml encoding="MathML-Content" id="S1.T1.1.1.1.1.1.1.1.1.1.1.m1.1b"><ci id="S1.T1.1.1.1.1.1.1.1.1.1.1.m1.1.1.cmml" xref="S1.T1.1.1.1.1.1.1.1.1.1.1.m1.1.1">𝑣</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.T1.1.1.1.1.1.1.1.1.1.1.m1.1c">v</annotation><annotation encoding="application/x-llamapun" id="S1.T1.1.1.1.1.1.1.1.1.1.1.m1.1d">italic_v</annotation></semantics></math> outputs <math alttext="\rho_{v}" class="ltx_Math" display="inline" id="S1.T1.2.2.2.2.2.2.2.2.2.2.m2.1"><semantics id="S1.T1.2.2.2.2.2.2.2.2.2.2.m2.1a"><msub id="S1.T1.2.2.2.2.2.2.2.2.2.2.m2.1.1" xref="S1.T1.2.2.2.2.2.2.2.2.2.2.m2.1.1.cmml"><mi id="S1.T1.2.2.2.2.2.2.2.2.2.2.m2.1.1.2" xref="S1.T1.2.2.2.2.2.2.2.2.2.2.m2.1.1.2.cmml">ρ</mi><mi id="S1.T1.2.2.2.2.2.2.2.2.2.2.m2.1.1.3" xref="S1.T1.2.2.2.2.2.2.2.2.2.2.m2.1.1.3.cmml">v</mi></msub><annotation-xml encoding="MathML-Content" id="S1.T1.2.2.2.2.2.2.2.2.2.2.m2.1b"><apply id="S1.T1.2.2.2.2.2.2.2.2.2.2.m2.1.1.cmml" xref="S1.T1.2.2.2.2.2.2.2.2.2.2.m2.1.1"><csymbol cd="ambiguous" id="S1.T1.2.2.2.2.2.2.2.2.2.2.m2.1.1.1.cmml" xref="S1.T1.2.2.2.2.2.2.2.2.2.2.m2.1.1">subscript</csymbol><ci id="S1.T1.2.2.2.2.2.2.2.2.2.2.m2.1.1.2.cmml" xref="S1.T1.2.2.2.2.2.2.2.2.2.2.m2.1.1.2">𝜌</ci><ci id="S1.T1.2.2.2.2.2.2.2.2.2.2.m2.1.1.3.cmml" xref="S1.T1.2.2.2.2.2.2.2.2.2.2.m2.1.1.3">𝑣</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.T1.2.2.2.2.2.2.2.2.2.2.m2.1c">\rho_{v}</annotation><annotation encoding="application/x-llamapun" id="S1.T1.2.2.2.2.2.2.2.2.2.2.m2.1d">italic_ρ start_POSTSUBSCRIPT italic_v end_POSTSUBSCRIPT</annotation></semantics></math> with</span> <span class="ltx_td ltx_align_left ltx_border_r" id="S1.T1.2.2.2.2.2.2.2.2.2.5">Rounds</span> <span class="ltx_td ltx_align_left" id="S1.T1.2.2.2.2.2.2.2.2.2.6">Source</span></span> <span class="ltx_tr" id="S1.T1.4.4.4.4.4.4.4.4.4"> <span class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S1.T1.4.4.4.4.4.4.4.4.4.3">L</span> <span class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S1.T1.4.4.4.4.4.4.4.4.4.4"><a class="ltx_ref" href="https://arxiv.org/html/2411.12694v2#S2.Thmtheorem7" title="Problem 2.7. ‣ 2.2 Approximate densest subgraph in LOCAL and CONGEST ‣ 2 Preliminaries and related work ‣ Local Density and its Distributed Approximation"><span class="ltx_text ltx_ref_tag">2.7</span></a>.1</span> <span class="ltx_td ltx_align_left ltx_border_r ltx_border_t" id="S1.T1.3.3.3.3.3.3.3.3.3.1"><math alttext="\rho_{v}\in[(1+\varepsilon)^{-1}\rho^{\max}(G),(1+\varepsilon)\rho^{\max}(G)]" class="ltx_Math" display="inline" id="S1.T1.3.3.3.3.3.3.3.3.3.1.m1.4"><semantics id="S1.T1.3.3.3.3.3.3.3.3.3.1.m1.4a"><mrow id="S1.T1.3.3.3.3.3.3.3.3.3.1.m1.4.4" xref="S1.T1.3.3.3.3.3.3.3.3.3.1.m1.4.4.cmml"><msub id="S1.T1.3.3.3.3.3.3.3.3.3.1.m1.4.4.4" xref="S1.T1.3.3.3.3.3.3.3.3.3.1.m1.4.4.4.cmml"><mi id="S1.T1.3.3.3.3.3.3.3.3.3.1.m1.4.4.4.2" xref="S1.T1.3.3.3.3.3.3.3.3.3.1.m1.4.4.4.2.cmml">ρ</mi><mi id="S1.T1.3.3.3.3.3.3.3.3.3.1.m1.4.4.4.3" xref="S1.T1.3.3.3.3.3.3.3.3.3.1.m1.4.4.4.3.cmml">v</mi></msub><mo id="S1.T1.3.3.3.3.3.3.3.3.3.1.m1.4.4.3" xref="S1.T1.3.3.3.3.3.3.3.3.3.1.m1.4.4.3.cmml">∈</mo><mrow id="S1.T1.3.3.3.3.3.3.3.3.3.1.m1.4.4.2.2" xref="S1.T1.3.3.3.3.3.3.3.3.3.1.m1.4.4.2.3.cmml"><mo id="S1.T1.3.3.3.3.3.3.3.3.3.1.m1.4.4.2.2.3" stretchy="false" xref="S1.T1.3.3.3.3.3.3.3.3.3.1.m1.4.4.2.3.cmml">[</mo><mrow id="S1.T1.3.3.3.3.3.3.3.3.3.1.m1.3.3.1.1.1" xref="S1.T1.3.3.3.3.3.3.3.3.3.1.m1.3.3.1.1.1.cmml"><msup id="S1.T1.3.3.3.3.3.3.3.3.3.1.m1.3.3.1.1.1.1" xref="S1.T1.3.3.3.3.3.3.3.3.3.1.m1.3.3.1.1.1.1.cmml"><mrow id="S1.T1.3.3.3.3.3.3.3.3.3.1.m1.3.3.1.1.1.1.1.1" xref="S1.T1.3.3.3.3.3.3.3.3.3.1.m1.3.3.1.1.1.1.1.1.1.cmml"><mo id="S1.T1.3.3.3.3.3.3.3.3.3.1.m1.3.3.1.1.1.1.1.1.2" stretchy="false" xref="S1.T1.3.3.3.3.3.3.3.3.3.1.m1.3.3.1.1.1.1.1.1.1.cmml">(</mo><mrow id="S1.T1.3.3.3.3.3.3.3.3.3.1.m1.3.3.1.1.1.1.1.1.1" xref="S1.T1.3.3.3.3.3.3.3.3.3.1.m1.3.3.1.1.1.1.1.1.1.cmml"><mn id="S1.T1.3.3.3.3.3.3.3.3.3.1.m1.3.3.1.1.1.1.1.1.1.2" xref="S1.T1.3.3.3.3.3.3.3.3.3.1.m1.3.3.1.1.1.1.1.1.1.2.cmml">1</mn><mo id="S1.T1.3.3.3.3.3.3.3.3.3.1.m1.3.3.1.1.1.1.1.1.1.1" 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id="S1.T1.3.3.3.3.3.3.3.3.3.1.m1.4.4.2.2.2.3.3.cmml" xref="S1.T1.3.3.3.3.3.3.3.3.3.1.m1.4.4.2.2.2.3.3"></max></apply><ci id="S1.T1.3.3.3.3.3.3.3.3.3.1.m1.2.2.cmml" xref="S1.T1.3.3.3.3.3.3.3.3.3.1.m1.2.2">𝐺</ci></apply></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.T1.3.3.3.3.3.3.3.3.3.1.m1.4c">\rho_{v}\in[(1+\varepsilon)^{-1}\rho^{\max}(G),(1+\varepsilon)\rho^{\max}(G)]</annotation><annotation encoding="application/x-llamapun" id="S1.T1.3.3.3.3.3.3.3.3.3.1.m1.4d">italic_ρ start_POSTSUBSCRIPT italic_v end_POSTSUBSCRIPT ∈ [ ( 1 + italic_ε ) start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT italic_ρ start_POSTSUPERSCRIPT roman_max end_POSTSUPERSCRIPT ( italic_G ) , ( 1 + italic_ε ) italic_ρ start_POSTSUPERSCRIPT roman_max end_POSTSUPERSCRIPT ( italic_G ) ]</annotation></semantics></math></span> <span class="ltx_td ltx_align_left ltx_border_r ltx_border_t" id="S1.T1.4.4.4.4.4.4.4.4.4.2"><math alttext="\Theta(D)" class="ltx_Math" display="inline" id="S1.T1.4.4.4.4.4.4.4.4.4.2.m1.1"><semantics id="S1.T1.4.4.4.4.4.4.4.4.4.2.m1.1a"><mrow id="S1.T1.4.4.4.4.4.4.4.4.4.2.m1.1.2" xref="S1.T1.4.4.4.4.4.4.4.4.4.2.m1.1.2.cmml"><mi id="S1.T1.4.4.4.4.4.4.4.4.4.2.m1.1.2.2" mathvariant="normal" xref="S1.T1.4.4.4.4.4.4.4.4.4.2.m1.1.2.2.cmml">Θ</mi><mo id="S1.T1.4.4.4.4.4.4.4.4.4.2.m1.1.2.1" xref="S1.T1.4.4.4.4.4.4.4.4.4.2.m1.1.2.1.cmml">⁢</mo><mrow id="S1.T1.4.4.4.4.4.4.4.4.4.2.m1.1.2.3.2" xref="S1.T1.4.4.4.4.4.4.4.4.4.2.m1.1.2.cmml"><mo id="S1.T1.4.4.4.4.4.4.4.4.4.2.m1.1.2.3.2.1" stretchy="false" xref="S1.T1.4.4.4.4.4.4.4.4.4.2.m1.1.2.cmml">(</mo><mi id="S1.T1.4.4.4.4.4.4.4.4.4.2.m1.1.1" xref="S1.T1.4.4.4.4.4.4.4.4.4.2.m1.1.1.cmml">D</mi><mo id="S1.T1.4.4.4.4.4.4.4.4.4.2.m1.1.2.3.2.2" stretchy="false" xref="S1.T1.4.4.4.4.4.4.4.4.4.2.m1.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.T1.4.4.4.4.4.4.4.4.4.2.m1.1b"><apply id="S1.T1.4.4.4.4.4.4.4.4.4.2.m1.1.2.cmml" xref="S1.T1.4.4.4.4.4.4.4.4.4.2.m1.1.2"><times id="S1.T1.4.4.4.4.4.4.4.4.4.2.m1.1.2.1.cmml" xref="S1.T1.4.4.4.4.4.4.4.4.4.2.m1.1.2.1"></times><ci id="S1.T1.4.4.4.4.4.4.4.4.4.2.m1.1.2.2.cmml" xref="S1.T1.4.4.4.4.4.4.4.4.4.2.m1.1.2.2">Θ</ci><ci id="S1.T1.4.4.4.4.4.4.4.4.4.2.m1.1.1.cmml" xref="S1.T1.4.4.4.4.4.4.4.4.4.2.m1.1.1">𝐷</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.T1.4.4.4.4.4.4.4.4.4.2.m1.1c">\Theta(D)</annotation><annotation encoding="application/x-llamapun" id="S1.T1.4.4.4.4.4.4.4.4.4.2.m1.1d">roman_Θ ( italic_D )</annotation></semantics></math></span> <span class="ltx_td ltx_align_left ltx_border_t" id="S1.T1.4.4.4.4.4.4.4.4.4.5"><cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2411.12694v2#bib.bib21" title="">21</a>]</cite></span></span> <span class="ltx_tr" id="S1.T1.6.6.6.6.6.6.6.6.6"> <span class="ltx_td ltx_border_r" id="S1.T1.6.6.6.6.6.6.6.6.6.3"></span> <span class="ltx_td ltx_align_center ltx_border_r" id="S1.T1.6.6.6.6.6.6.6.6.6.4"><a 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id="S1.T1.6.6.6.6.6.6.6.6.6.2.m1.1.1.1.1.1.3.2.cmml" xref="S1.T1.6.6.6.6.6.6.6.6.6.2.m1.1.1.1.1.1.3.2">𝑛</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.T1.6.6.6.6.6.6.6.6.6.2.m1.1c">O(\varepsilon^{-1}\log n)</annotation><annotation encoding="application/x-llamapun" id="S1.T1.6.6.6.6.6.6.6.6.6.2.m1.1d">italic_O ( italic_ε start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT roman_log italic_n )</annotation></semantics></math></span> <span class="ltx_td ltx_align_left" id="S1.T1.6.6.6.6.6.6.6.6.6.5"><cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2411.12694v2#bib.bib21" title="">21</a>]</cite></span></span> <span class="ltx_tr" id="S1.T1.8.8.8.8.8.8.8.8.8"> <span class="ltx_td ltx_border_r" id="S1.T1.8.8.8.8.8.8.8.8.8.3"></span> <span class="ltx_td ltx_align_center ltx_border_r" id="S1.T1.8.8.8.8.8.8.8.8.8.4"><a class="ltx_ref ltx_font_bold" href="https://arxiv.org/html/2411.12694v2#S2.Thmtheorem11" title="Problem 2.11. ‣ The benefit of local measures: ‣ 2.3 Local density ‣ 2 Preliminaries and related work ‣ Local Density and its Distributed Approximation"><span class="ltx_text ltx_ref_tag">2.11</span></a></span> <span class="ltx_td ltx_align_left ltx_border_r" id="S1.T1.7.7.7.7.7.7.7.7.7.1"><math alttext="\boldsymbol{\rho_{v}\in[(1+\varepsilon)^{-1}\rho^{*}(v),(1+\varepsilon)\rho^{*% }(v)]}" class="ltx_Math" display="inline" id="S1.T1.7.7.7.7.7.7.7.7.7.1.m1.4"><semantics id="S1.T1.7.7.7.7.7.7.7.7.7.1.m1.4a"><mrow id="S1.T1.7.7.7.7.7.7.7.7.7.1.m1.4.4" xref="S1.T1.7.7.7.7.7.7.7.7.7.1.m1.4.4.cmml"><msub id="S1.T1.7.7.7.7.7.7.7.7.7.1.m1.4.4.4" xref="S1.T1.7.7.7.7.7.7.7.7.7.1.m1.4.4.4.cmml"><mi id="S1.T1.7.7.7.7.7.7.7.7.7.1.m1.4.4.4.2" xref="S1.T1.7.7.7.7.7.7.7.7.7.1.m1.4.4.4.2.cmml">𝝆</mi><mi id="S1.T1.7.7.7.7.7.7.7.7.7.1.m1.4.4.4.3" xref="S1.T1.7.7.7.7.7.7.7.7.7.1.m1.4.4.4.3.cmml">𝒗</mi></msub><mo class="ltx_mathvariant_bold" id="S1.T1.7.7.7.7.7.7.7.7.7.1.m1.4.4.3" 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id="S1.T1.8.8.8.8.8.8.8.8.8.2.m1.1.1.1.1.1.3a" lspace="0.167em" xref="S1.T1.8.8.8.8.8.8.8.8.8.2.m1.1.1.1.1.1.3.cmml">⁡</mo><mi id="S1.T1.8.8.8.8.8.8.8.8.8.2.m1.1.1.1.1.1.3.2" xref="S1.T1.8.8.8.8.8.8.8.8.8.2.m1.1.1.1.1.1.3.2.cmml">𝒏</mi></mrow></mrow><mo class="ltx_mathvariant_bold" id="S1.T1.8.8.8.8.8.8.8.8.8.2.m1.1.1.1.1.3" mathvariant="bold" stretchy="false" xref="S1.T1.8.8.8.8.8.8.8.8.8.2.m1.1.1.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.T1.8.8.8.8.8.8.8.8.8.2.m1.1b"><apply id="S1.T1.8.8.8.8.8.8.8.8.8.2.m1.1.1.cmml" xref="S1.T1.8.8.8.8.8.8.8.8.8.2.m1.1.1"><times id="S1.T1.8.8.8.8.8.8.8.8.8.2.m1.1.1.2.cmml" xref="S1.T1.8.8.8.8.8.8.8.8.8.2.m1.1.1.2"></times><ci id="S1.T1.8.8.8.8.8.8.8.8.8.2.m1.1.1.3.cmml" xref="S1.T1.8.8.8.8.8.8.8.8.8.2.m1.1.1.3">𝑶</ci><apply id="S1.T1.8.8.8.8.8.8.8.8.8.2.m1.1.1.1.1.1.cmml" xref="S1.T1.8.8.8.8.8.8.8.8.8.2.m1.1.1.1.1"><times id="S1.T1.8.8.8.8.8.8.8.8.8.2.m1.1.1.1.1.1.1.cmml" 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xref="S1.T1.8.8.8.8.8.8.8.8.8.2.m1.1.1.1.1.1.3.1"><csymbol cd="ambiguous" id="S1.T1.8.8.8.8.8.8.8.8.8.2.m1.1.1.1.1.1.3.1.1.cmml" xref="S1.T1.8.8.8.8.8.8.8.8.8.2.m1.1.1.1.1.1.3.1">superscript</csymbol><log id="S1.T1.8.8.8.8.8.8.8.8.8.2.m1.1.1.1.1.1.3.1.2.cmml" xref="S1.T1.8.8.8.8.8.8.8.8.8.2.m1.1.1.1.1.1.3.1.2"></log><cn id="S1.T1.8.8.8.8.8.8.8.8.8.2.m1.1.1.1.1.1.3.1.3.cmml" type="integer" xref="S1.T1.8.8.8.8.8.8.8.8.8.2.m1.1.1.1.1.1.3.1.3">2</cn></apply><ci id="S1.T1.8.8.8.8.8.8.8.8.8.2.m1.1.1.1.1.1.3.2.cmml" xref="S1.T1.8.8.8.8.8.8.8.8.8.2.m1.1.1.1.1.1.3.2">𝒏</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.T1.8.8.8.8.8.8.8.8.8.2.m1.1c">\boldsymbol{O(\varepsilon^{-2}\log^{2}n)}</annotation><annotation encoding="application/x-llamapun" id="S1.T1.8.8.8.8.8.8.8.8.8.2.m1.1d">bold_italic_O bold_( bold_italic_ε start_POSTSUPERSCRIPT bold_- bold_2 end_POSTSUPERSCRIPT bold_log start_POSTSUPERSCRIPT bold_2 end_POSTSUPERSCRIPT bold_italic_n bold_)</annotation></semantics></math></span> <span class="ltx_td ltx_align_left" id="S1.T1.8.8.8.8.8.8.8.8.8.5"><span class="ltx_text ltx_font_bold" id="S1.T1.8.8.8.8.8.8.8.8.8.5.1">Cor. <a class="ltx_ref" href="https://arxiv.org/html/2411.12694v2#S3.Thmtheorem8" title="Corollary 3.8. ‣ 3.C Results in LOCAL ‣ 3 Results and organisation ‣ Local Density and its Distributed Approximation"><span class="ltx_text ltx_ref_tag">3.8</span></a></span></span></span> <span class="ltx_tr" id="S1.T1.10.10.10.10.10.10.10.10.10"> <span class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S1.T1.10.10.10.10.10.10.10.10.10.3">C</span> <span class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S1.T1.10.10.10.10.10.10.10.10.10.4"><a class="ltx_ref" href="https://arxiv.org/html/2411.12694v2#S2.Thmtheorem7" title="Problem 2.7. ‣ 2.2 Approximate densest subgraph in LOCAL and CONGEST ‣ 2 Preliminaries and related work ‣ Local Density and its Distributed Approximation"><span class="ltx_text ltx_ref_tag">2.7</span></a>.1</span> <span class="ltx_td ltx_align_left ltx_border_r ltx_border_t" id="S1.T1.9.9.9.9.9.9.9.9.9.1"><math alttext="\rho_{v}\in[(2+\varepsilon)^{-1}\rho^{\max}(G),(2+\varepsilon)\rho^{\max}(G)]" class="ltx_Math" display="inline" id="S1.T1.9.9.9.9.9.9.9.9.9.1.m1.4"><semantics id="S1.T1.9.9.9.9.9.9.9.9.9.1.m1.4a"><mrow id="S1.T1.9.9.9.9.9.9.9.9.9.1.m1.4.4" xref="S1.T1.9.9.9.9.9.9.9.9.9.1.m1.4.4.cmml"><msub id="S1.T1.9.9.9.9.9.9.9.9.9.1.m1.4.4.4" xref="S1.T1.9.9.9.9.9.9.9.9.9.1.m1.4.4.4.cmml"><mi id="S1.T1.9.9.9.9.9.9.9.9.9.1.m1.4.4.4.2" xref="S1.T1.9.9.9.9.9.9.9.9.9.1.m1.4.4.4.2.cmml">ρ</mi><mi id="S1.T1.9.9.9.9.9.9.9.9.9.1.m1.4.4.4.3" xref="S1.T1.9.9.9.9.9.9.9.9.9.1.m1.4.4.4.3.cmml">v</mi></msub><mo id="S1.T1.9.9.9.9.9.9.9.9.9.1.m1.4.4.3" xref="S1.T1.9.9.9.9.9.9.9.9.9.1.m1.4.4.3.cmml">∈</mo><mrow id="S1.T1.9.9.9.9.9.9.9.9.9.1.m1.4.4.2.2" xref="S1.T1.9.9.9.9.9.9.9.9.9.1.m1.4.4.2.3.cmml"><mo id="S1.T1.9.9.9.9.9.9.9.9.9.1.m1.4.4.2.2.3" stretchy="false" 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xref="S1.T1.10.10.10.10.10.10.10.10.10.2.m1.1.1.1.1.1.3.1"></log><ci id="S1.T1.10.10.10.10.10.10.10.10.10.2.m1.1.1.1.1.1.3.2.cmml" xref="S1.T1.10.10.10.10.10.10.10.10.10.2.m1.1.1.1.1.1.3.2">𝑛</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.T1.10.10.10.10.10.10.10.10.10.2.m1.1c">O(D\cdot\varepsilon^{-1}\log n)</annotation><annotation encoding="application/x-llamapun" id="S1.T1.10.10.10.10.10.10.10.10.10.2.m1.1d">italic_O ( italic_D ⋅ italic_ε start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT roman_log italic_n )</annotation></semantics></math></span> <span class="ltx_td ltx_align_left ltx_border_t" id="S1.T1.10.10.10.10.10.10.10.10.10.5"><cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2411.12694v2#bib.bib11" title="">11</a>]</cite></span></span> <span class="ltx_tr" id="S1.T1.12.12.12.12.12.12.12.12.12"> <span class="ltx_td ltx_border_r" id="S1.T1.12.12.12.12.12.12.12.12.12.3"></span> <span class="ltx_td 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xref="S1.T1.11.11.11.11.11.11.11.11.11.1.m1.2.2">𝐺</ci></apply></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.T1.11.11.11.11.11.11.11.11.11.1.m1.4c">\rho_{v}\in[(1+\varepsilon)^{-1}\rho^{\max}(G),(1+\varepsilon)\rho^{\max}(G)]</annotation><annotation encoding="application/x-llamapun" id="S1.T1.11.11.11.11.11.11.11.11.11.1.m1.4d">italic_ρ start_POSTSUBSCRIPT italic_v end_POSTSUBSCRIPT ∈ [ ( 1 + italic_ε ) start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT italic_ρ start_POSTSUPERSCRIPT roman_max end_POSTSUPERSCRIPT ( italic_G ) , ( 1 + italic_ε ) italic_ρ start_POSTSUPERSCRIPT roman_max end_POSTSUPERSCRIPT ( italic_G ) ]</annotation></semantics></math></span> <span class="ltx_td ltx_align_left ltx_border_r" id="S1.T1.12.12.12.12.12.12.12.12.12.2"><span class="ltx_text" id="S1.T1.12.12.12.12.12.12.12.12.12.2.1" style="color:#FF8000;"><math alttext="O(\varepsilon^{-4}\log^{4}n+D)" class="ltx_Math" display="inline" 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href="https://arxiv.org/html/2411.12694v2#bib.bib21" title="">21</a>]</cite></span></span> <span class="ltx_tr" id="S1.T1.14.14.14.14.14.14.14.14.14"> <span class="ltx_td ltx_border_r" id="S1.T1.14.14.14.14.14.14.14.14.14.3"></span> <span class="ltx_td ltx_align_center ltx_border_r" id="S1.T1.14.14.14.14.14.14.14.14.14.4"><a class="ltx_ref" href="https://arxiv.org/html/2411.12694v2#S2.Thmtheorem7" title="Problem 2.7. ‣ 2.2 Approximate densest subgraph in LOCAL and CONGEST ‣ 2 Preliminaries and related work ‣ Local Density and its Distributed Approximation"><span class="ltx_text ltx_ref_tag">2.7</span></a>.2</span> <span class="ltx_td ltx_align_left ltx_border_r" id="S1.T1.13.13.13.13.13.13.13.13.13.1"><math alttext="\max_{v}\rho_{v}\in[(1+\varepsilon)^{-1}\rho^{\max}(G),(1+\varepsilon)\rho^{% \max}(G)]" class="ltx_Math" display="inline" id="S1.T1.13.13.13.13.13.13.13.13.13.1.m1.4"><semantics id="S1.T1.13.13.13.13.13.13.13.13.13.1.m1.4a"><mrow 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ltx_font_bold" href="https://arxiv.org/html/2411.12694v2#S2.Thmtheorem7" title="Problem 2.7. ‣ 2.2 Approximate densest subgraph in LOCAL and CONGEST ‣ 2 Preliminaries and related work ‣ Local Density and its Distributed Approximation"><span class="ltx_text ltx_ref_tag">2.7</span></a><span class="ltx_text ltx_font_bold" id="S1.T1.16.16.16.16.16.16.16.16.16.4.1">.2</span></span> <span class="ltx_td ltx_align_left ltx_border_r" id="S1.T1.15.15.15.15.15.15.15.15.15.1"><math alttext="\boldsymbol{\rho_{v}\in[(1+\varepsilon)^{-1}\rho^{*}(v),(1+\varepsilon)\rho^{*% }(v)]}" class="ltx_Math" display="inline" id="S1.T1.15.15.15.15.15.15.15.15.15.1.m1.4"><semantics id="S1.T1.15.15.15.15.15.15.15.15.15.1.m1.4a"><mrow id="S1.T1.15.15.15.15.15.15.15.15.15.1.m1.4.4" xref="S1.T1.15.15.15.15.15.15.15.15.15.1.m1.4.4.cmml"><msub id="S1.T1.15.15.15.15.15.15.15.15.15.1.m1.4.4.4" xref="S1.T1.15.15.15.15.15.15.15.15.15.1.m1.4.4.4.cmml"><mi id="S1.T1.15.15.15.15.15.15.15.15.15.1.m1.4.4.4.2" 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id="S1.T1.18.18.18.18.18.18.18.18.18.2.m1.3c">\boldsymbol{O(\operatorname{poly}\{\log n,\varepsilon^{-1}\})\cdot 2^{O(\sqrt{% \log n})}}</annotation><annotation encoding="application/x-llamapun" id="S1.T1.18.18.18.18.18.18.18.18.18.2.m1.3d">bold_italic_O bold_( bold_poly bold_{ bold_log bold_italic_n bold_, bold_italic_ε start_POSTSUPERSCRIPT bold_- bold_1 end_POSTSUPERSCRIPT bold_} bold_) bold_⋅ bold_2 start_POSTSUPERSCRIPT bold_italic_O bold_( square-root start_ARG bold_log bold_italic_n end_ARG bold_) end_POSTSUPERSCRIPT</annotation></semantics></math></span> <span class="ltx_td ltx_align_left" id="S1.T1.18.18.18.18.18.18.18.18.18.5"><span class="ltx_text ltx_font_bold" id="S1.T1.18.18.18.18.18.18.18.18.18.5.1">Thm. <a class="ltx_ref" href="https://arxiv.org/html/2411.12694v2#S3.Thmtheorem9" title="Theorem 3.9. ‣ 3.D Results in CONGEST ‣ 3 Results and organisation ‣ Local Density and its Distributed Approximation"><span class="ltx_text 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xref="S1.T1.20.20.20.20.20.20.20.20.20.2.m1.2.2.1.1.1.2.2.2">superscript</csymbol><ci id="S1.T1.20.20.20.20.20.20.20.20.20.2.m1.2.2.1.1.1.2.2.2.2.cmml" xref="S1.T1.20.20.20.20.20.20.20.20.20.2.m1.2.2.1.1.1.2.2.2.2">𝜺</ci><apply id="S1.T1.20.20.20.20.20.20.20.20.20.2.m1.2.2.1.1.1.2.2.2.3.cmml" xref="S1.T1.20.20.20.20.20.20.20.20.20.2.m1.2.2.1.1.1.2.2.2.3"><minus id="S1.T1.20.20.20.20.20.20.20.20.20.2.m1.2.2.1.1.1.2.2.2.3.1.cmml" xref="S1.T1.20.20.20.20.20.20.20.20.20.2.m1.2.2.1.1.1.2.2.2.3"></minus><cn id="S1.T1.20.20.20.20.20.20.20.20.20.2.m1.2.2.1.1.1.2.2.2.3.2.cmml" type="integer" xref="S1.T1.20.20.20.20.20.20.20.20.20.2.m1.2.2.1.1.1.2.2.2.3.2">1</cn></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.T1.20.20.20.20.20.20.20.20.20.2.m1.2c">\boldsymbol{O(\operatorname{poly}\{\log n,\varepsilon^{-1}\})}</annotation><annotation encoding="application/x-llamapun" id="S1.T1.20.20.20.20.20.20.20.20.20.2.m1.2d">bold_italic_O bold_( bold_poly bold_{ bold_log bold_italic_n bold_, bold_italic_ε start_POSTSUPERSCRIPT bold_- bold_1 end_POSTSUPERSCRIPT bold_} bold_)</annotation></semantics></math><span class="ltx_text" id="S1.T1.20.20.20.20.20.20.20.20.20.2.1" style="color:#FF8000;"> whp.</span></span> <span class="ltx_td ltx_align_left" id="S1.T1.20.20.20.20.20.20.20.20.20.5"><span class="ltx_text ltx_font_bold" id="S1.T1.20.20.20.20.20.20.20.20.20.5.1">Thm. <a class="ltx_ref" href="https://arxiv.org/html/2411.12694v2#S3.Thmtheorem9" title="Theorem 3.9. ‣ 3.D Results in CONGEST ‣ 3 Results and organisation ‣ Local Density and its Distributed Approximation"><span class="ltx_text ltx_ref_tag">3.9</span></a></span></span></span> </span></span></span> </span></span></span></p> <figcaption class="ltx_caption ltx_centering"><span class="ltx_tag ltx_tag_table"><span class="ltx_text" id="S1.T1.22.1.1" style="font-size:90%;">Table 1</span>: </span><span class="ltx_text" id="S1.T1.23.2" style="font-size:90%;">Results in LOCAL (L) or CONGEST (C) where prior work for computing the global subgraph density is compared to our running time for the local subgraph density. D denotes the diameter. Orange running times are not deterministic and occur with high probability. </span></figcaption> </figure> <div class="ltx_pagination ltx_role_newpage"></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 and related work</h2> <div class="ltx_para" id="S2.p1"> <p class="ltx_p" id="S2.p1.13">Let <math alttext="G=(V,E)" class="ltx_Math" display="inline" id="S2.p1.1.m1.2"><semantics id="S2.p1.1.m1.2a"><mrow id="S2.p1.1.m1.2.3" xref="S2.p1.1.m1.2.3.cmml"><mi id="S2.p1.1.m1.2.3.2" xref="S2.p1.1.m1.2.3.2.cmml">G</mi><mo id="S2.p1.1.m1.2.3.1" xref="S2.p1.1.m1.2.3.1.cmml">=</mo><mrow id="S2.p1.1.m1.2.3.3.2" xref="S2.p1.1.m1.2.3.3.1.cmml"><mo id="S2.p1.1.m1.2.3.3.2.1" stretchy="false" xref="S2.p1.1.m1.2.3.3.1.cmml">(</mo><mi id="S2.p1.1.m1.1.1" xref="S2.p1.1.m1.1.1.cmml">V</mi><mo id="S2.p1.1.m1.2.3.3.2.2" xref="S2.p1.1.m1.2.3.3.1.cmml">,</mo><mi id="S2.p1.1.m1.2.2" xref="S2.p1.1.m1.2.2.cmml">E</mi><mo id="S2.p1.1.m1.2.3.3.2.3" stretchy="false" xref="S2.p1.1.m1.2.3.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.p1.1.m1.2b"><apply id="S2.p1.1.m1.2.3.cmml" xref="S2.p1.1.m1.2.3"><eq id="S2.p1.1.m1.2.3.1.cmml" xref="S2.p1.1.m1.2.3.1"></eq><ci id="S2.p1.1.m1.2.3.2.cmml" xref="S2.p1.1.m1.2.3.2">𝐺</ci><interval closure="open" id="S2.p1.1.m1.2.3.3.1.cmml" xref="S2.p1.1.m1.2.3.3.2"><ci id="S2.p1.1.m1.1.1.cmml" xref="S2.p1.1.m1.1.1">𝑉</ci><ci id="S2.p1.1.m1.2.2.cmml" xref="S2.p1.1.m1.2.2">𝐸</ci></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p1.1.m1.2c">G=(V,E)</annotation><annotation encoding="application/x-llamapun" id="S2.p1.1.m1.2d">italic_G = ( italic_V , italic_E )</annotation></semantics></math> be an undirected weighted graph with <math alttext="n" 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">n</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">𝑛</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.p1.2.m2.1c">n</annotation><annotation encoding="application/x-llamapun" id="S2.p1.2.m2.1d">italic_n</annotation></semantics></math> vertices and <math alttext="m" 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">m</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">𝑚</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.p1.3.m3.1c">m</annotation><annotation encoding="application/x-llamapun" id="S2.p1.3.m3.1d">italic_m</annotation></semantics></math> edges. For any <math alttext="v\in V" class="ltx_Math" display="inline" id="S2.p1.4.m4.1"><semantics id="S2.p1.4.m4.1a"><mrow id="S2.p1.4.m4.1.1" xref="S2.p1.4.m4.1.1.cmml"><mi id="S2.p1.4.m4.1.1.2" xref="S2.p1.4.m4.1.1.2.cmml">v</mi><mo id="S2.p1.4.m4.1.1.1" xref="S2.p1.4.m4.1.1.1.cmml">∈</mo><mi id="S2.p1.4.m4.1.1.3" xref="S2.p1.4.m4.1.1.3.cmml">V</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.p1.4.m4.1b"><apply id="S2.p1.4.m4.1.1.cmml" xref="S2.p1.4.m4.1.1"><in id="S2.p1.4.m4.1.1.1.cmml" xref="S2.p1.4.m4.1.1.1"></in><ci id="S2.p1.4.m4.1.1.2.cmml" xref="S2.p1.4.m4.1.1.2">𝑣</ci><ci id="S2.p1.4.m4.1.1.3.cmml" xref="S2.p1.4.m4.1.1.3">𝑉</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p1.4.m4.1c">v\in V</annotation><annotation encoding="application/x-llamapun" id="S2.p1.4.m4.1d">italic_v ∈ italic_V</annotation></semantics></math> and any integer <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>, we denote by <math alttext="H_{k}(v)" class="ltx_Math" display="inline" id="S2.p1.6.m6.1"><semantics id="S2.p1.6.m6.1a"><mrow id="S2.p1.6.m6.1.2" xref="S2.p1.6.m6.1.2.cmml"><msub id="S2.p1.6.m6.1.2.2" xref="S2.p1.6.m6.1.2.2.cmml"><mi id="S2.p1.6.m6.1.2.2.2" xref="S2.p1.6.m6.1.2.2.2.cmml">H</mi><mi id="S2.p1.6.m6.1.2.2.3" xref="S2.p1.6.m6.1.2.2.3.cmml">k</mi></msub><mo id="S2.p1.6.m6.1.2.1" xref="S2.p1.6.m6.1.2.1.cmml">⁢</mo><mrow id="S2.p1.6.m6.1.2.3.2" xref="S2.p1.6.m6.1.2.cmml"><mo id="S2.p1.6.m6.1.2.3.2.1" stretchy="false" xref="S2.p1.6.m6.1.2.cmml">(</mo><mi id="S2.p1.6.m6.1.1" xref="S2.p1.6.m6.1.1.cmml">v</mi><mo id="S2.p1.6.m6.1.2.3.2.2" stretchy="false" xref="S2.p1.6.m6.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.p1.6.m6.1b"><apply id="S2.p1.6.m6.1.2.cmml" xref="S2.p1.6.m6.1.2"><times id="S2.p1.6.m6.1.2.1.cmml" xref="S2.p1.6.m6.1.2.1"></times><apply id="S2.p1.6.m6.1.2.2.cmml" xref="S2.p1.6.m6.1.2.2"><csymbol cd="ambiguous" id="S2.p1.6.m6.1.2.2.1.cmml" xref="S2.p1.6.m6.1.2.2">subscript</csymbol><ci id="S2.p1.6.m6.1.2.2.2.cmml" xref="S2.p1.6.m6.1.2.2.2">𝐻</ci><ci id="S2.p1.6.m6.1.2.2.3.cmml" xref="S2.p1.6.m6.1.2.2.3">𝑘</ci></apply><ci id="S2.p1.6.m6.1.1.cmml" xref="S2.p1.6.m6.1.1">𝑣</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p1.6.m6.1c">H_{k}(v)</annotation><annotation encoding="application/x-llamapun" id="S2.p1.6.m6.1d">italic_H start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ( italic_v )</annotation></semantics></math> the <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>-hop neighborhood of <math alttext="v" class="ltx_Math" display="inline" id="S2.p1.8.m8.1"><semantics id="S2.p1.8.m8.1a"><mi id="S2.p1.8.m8.1.1" xref="S2.p1.8.m8.1.1.cmml">v</mi><annotation-xml encoding="MathML-Content" id="S2.p1.8.m8.1b"><ci id="S2.p1.8.m8.1.1.cmml" xref="S2.p1.8.m8.1.1">𝑣</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.p1.8.m8.1c">v</annotation><annotation encoding="application/x-llamapun" id="S2.p1.8.m8.1d">italic_v</annotation></semantics></math>. For each edge <math alttext="e\in E" class="ltx_Math" display="inline" id="S2.p1.9.m9.1"><semantics id="S2.p1.9.m9.1a"><mrow id="S2.p1.9.m9.1.1" xref="S2.p1.9.m9.1.1.cmml"><mi id="S2.p1.9.m9.1.1.2" xref="S2.p1.9.m9.1.1.2.cmml">e</mi><mo id="S2.p1.9.m9.1.1.1" xref="S2.p1.9.m9.1.1.1.cmml">∈</mo><mi id="S2.p1.9.m9.1.1.3" xref="S2.p1.9.m9.1.1.3.cmml">E</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.p1.9.m9.1b"><apply id="S2.p1.9.m9.1.1.cmml" xref="S2.p1.9.m9.1.1"><in id="S2.p1.9.m9.1.1.1.cmml" xref="S2.p1.9.m9.1.1.1"></in><ci id="S2.p1.9.m9.1.1.2.cmml" xref="S2.p1.9.m9.1.1.2">𝑒</ci><ci id="S2.p1.9.m9.1.1.3.cmml" xref="S2.p1.9.m9.1.1.3">𝐸</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p1.9.m9.1c">e\in E</annotation><annotation encoding="application/x-llamapun" id="S2.p1.9.m9.1d">italic_e ∈ italic_E</annotation></semantics></math> we denote by <math alttext="\textsl{g}(e)" class="ltx_Math" display="inline" id="S2.p1.10.m10.1"><semantics id="S2.p1.10.m10.1a"><mrow id="S2.p1.10.m10.1.2" xref="S2.p1.10.m10.1.2.cmml"><mtext class="ltx_mathvariant_italic" id="S2.p1.10.m10.1.2.2" xref="S2.p1.10.m10.1.2.2a.cmml">g</mtext><mo id="S2.p1.10.m10.1.2.1" xref="S2.p1.10.m10.1.2.1.cmml">⁢</mo><mrow id="S2.p1.10.m10.1.2.3.2" xref="S2.p1.10.m10.1.2.cmml"><mo id="S2.p1.10.m10.1.2.3.2.1" stretchy="false" xref="S2.p1.10.m10.1.2.cmml">(</mo><mi id="S2.p1.10.m10.1.1" xref="S2.p1.10.m10.1.1.cmml">e</mi><mo id="S2.p1.10.m10.1.2.3.2.2" stretchy="false" xref="S2.p1.10.m10.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.p1.10.m10.1b"><apply id="S2.p1.10.m10.1.2.cmml" xref="S2.p1.10.m10.1.2"><times id="S2.p1.10.m10.1.2.1.cmml" xref="S2.p1.10.m10.1.2.1"></times><ci id="S2.p1.10.m10.1.2.2a.cmml" xref="S2.p1.10.m10.1.2.2"><mtext class="ltx_mathvariant_italic" id="S2.p1.10.m10.1.2.2.cmml" xref="S2.p1.10.m10.1.2.2">g</mtext></ci><ci id="S2.p1.10.m10.1.1.cmml" xref="S2.p1.10.m10.1.1">𝑒</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p1.10.m10.1c">\textsl{g}(e)</annotation><annotation encoding="application/x-llamapun" id="S2.p1.10.m10.1d">g ( italic_e )</annotation></semantics></math> the <em class="ltx_emph ltx_font_italic" id="S2.p1.13.1">weight</em> of <math alttext="e" 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">e</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">𝑒</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.p1.11.m11.1c">e</annotation><annotation encoding="application/x-llamapun" id="S2.p1.11.m11.1d">italic_e</annotation></semantics></math>. Any edge with endpoints <math alttext="u,v" class="ltx_Math" display="inline" id="S2.p1.12.m12.2"><semantics id="S2.p1.12.m12.2a"><mrow id="S2.p1.12.m12.2.3.2" xref="S2.p1.12.m12.2.3.1.cmml"><mi id="S2.p1.12.m12.1.1" xref="S2.p1.12.m12.1.1.cmml">u</mi><mo id="S2.p1.12.m12.2.3.2.1" xref="S2.p1.12.m12.2.3.1.cmml">,</mo><mi id="S2.p1.12.m12.2.2" xref="S2.p1.12.m12.2.2.cmml">v</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.p1.12.m12.2b"><list id="S2.p1.12.m12.2.3.1.cmml" xref="S2.p1.12.m12.2.3.2"><ci id="S2.p1.12.m12.1.1.cmml" xref="S2.p1.12.m12.1.1">𝑢</ci><ci id="S2.p1.12.m12.2.2.cmml" xref="S2.p1.12.m12.2.2">𝑣</ci></list></annotation-xml><annotation encoding="application/x-tex" id="S2.p1.12.m12.2c">u,v</annotation><annotation encoding="application/x-llamapun" id="S2.p1.12.m12.2d">italic_u , italic_v</annotation></semantics></math> may be denoted as <math alttext="\overline{uv}" class="ltx_Math" display="inline" id="S2.p1.13.m13.1"><semantics id="S2.p1.13.m13.1a"><mover accent="true" id="S2.p1.13.m13.1.1" xref="S2.p1.13.m13.1.1.cmml"><mrow id="S2.p1.13.m13.1.1.2" xref="S2.p1.13.m13.1.1.2.cmml"><mi id="S2.p1.13.m13.1.1.2.2" xref="S2.p1.13.m13.1.1.2.2.cmml">u</mi><mo id="S2.p1.13.m13.1.1.2.1" xref="S2.p1.13.m13.1.1.2.1.cmml">⁢</mo><mi id="S2.p1.13.m13.1.1.2.3" xref="S2.p1.13.m13.1.1.2.3.cmml">v</mi></mrow><mo id="S2.p1.13.m13.1.1.1" xref="S2.p1.13.m13.1.1.1.cmml">¯</mo></mover><annotation-xml encoding="MathML-Content" id="S2.p1.13.m13.1b"><apply id="S2.p1.13.m13.1.1.cmml" xref="S2.p1.13.m13.1.1"><ci id="S2.p1.13.m13.1.1.1.cmml" xref="S2.p1.13.m13.1.1.1">¯</ci><apply id="S2.p1.13.m13.1.1.2.cmml" xref="S2.p1.13.m13.1.1.2"><times id="S2.p1.13.m13.1.1.2.1.cmml" xref="S2.p1.13.m13.1.1.2.1"></times><ci id="S2.p1.13.m13.1.1.2.2.cmml" xref="S2.p1.13.m13.1.1.2.2">𝑢</ci><ci id="S2.p1.13.m13.1.1.2.3.cmml" xref="S2.p1.13.m13.1.1.2.3">𝑣</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p1.13.m13.1c">\overline{uv}</annotation><annotation encoding="application/x-llamapun" id="S2.p1.13.m13.1d">over¯ start_ARG italic_u italic_v end_ARG</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="S2.p2"> <p class="ltx_p" id="S2.p2.18">We can augment any weighted graph <math alttext="G" class="ltx_Math" display="inline" id="S2.p2.1.m1.1"><semantics id="S2.p2.1.m1.1a"><mi id="S2.p2.1.m1.1.1" xref="S2.p2.1.m1.1.1.cmml">G</mi><annotation-xml encoding="MathML-Content" id="S2.p2.1.m1.1b"><ci id="S2.p2.1.m1.1.1.cmml" xref="S2.p2.1.m1.1.1">𝐺</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.p2.1.m1.1c">G</annotation><annotation encoding="application/x-llamapun" id="S2.p2.1.m1.1d">italic_G</annotation></semantics></math> with a <em class="ltx_emph ltx_font_italic" id="S2.p2.18.1">(fractional) orientation</em>. An orientation <math alttext="\overrightarrow{G}" class="ltx_Math" display="inline" id="S2.p2.2.m2.1"><semantics id="S2.p2.2.m2.1a"><mover accent="true" 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">G</mi><mo id="S2.p2.2.m2.1.1.1" stretchy="false" xref="S2.p2.2.m2.1.1.1.cmml">→</mo></mover><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"><ci id="S2.p2.2.m2.1.1.1.cmml" xref="S2.p2.2.m2.1.1.1">→</ci><ci id="S2.p2.2.m2.1.1.2.cmml" xref="S2.p2.2.m2.1.1.2">𝐺</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p2.2.m2.1c">\overrightarrow{G}</annotation><annotation encoding="application/x-llamapun" id="S2.p2.2.m2.1d">over→ start_ARG italic_G end_ARG</annotation></semantics></math> assigns to each edge <math alttext="\overline{uv}" 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"><mrow id="S2.p2.3.m3.1.1.2" xref="S2.p2.3.m3.1.1.2.cmml"><mi id="S2.p2.3.m3.1.1.2.2" xref="S2.p2.3.m3.1.1.2.2.cmml">u</mi><mo id="S2.p2.3.m3.1.1.2.1" xref="S2.p2.3.m3.1.1.2.1.cmml">⁢</mo><mi id="S2.p2.3.m3.1.1.2.3" xref="S2.p2.3.m3.1.1.2.3.cmml">v</mi></mrow><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><apply id="S2.p2.3.m3.1.1.2.cmml" xref="S2.p2.3.m3.1.1.2"><times id="S2.p2.3.m3.1.1.2.1.cmml" xref="S2.p2.3.m3.1.1.2.1"></times><ci id="S2.p2.3.m3.1.1.2.2.cmml" xref="S2.p2.3.m3.1.1.2.2">𝑢</ci><ci id="S2.p2.3.m3.1.1.2.3.cmml" xref="S2.p2.3.m3.1.1.2.3">𝑣</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p2.3.m3.1c">\overline{uv}</annotation><annotation encoding="application/x-llamapun" id="S2.p2.3.m3.1d">over¯ start_ARG italic_u italic_v end_ARG</annotation></semantics></math> two positive real values: <math alttext="\textsl{g}(u\!\to\!v)" class="ltx_Math" display="inline" id="S2.p2.4.m4.1"><semantics id="S2.p2.4.m4.1a"><mrow id="S2.p2.4.m4.1.1" xref="S2.p2.4.m4.1.1.cmml"><mtext class="ltx_mathvariant_italic" id="S2.p2.4.m4.1.1.3" xref="S2.p2.4.m4.1.1.3a.cmml">g</mtext><mo id="S2.p2.4.m4.1.1.2" xref="S2.p2.4.m4.1.1.2.cmml">⁢</mo><mrow id="S2.p2.4.m4.1.1.1.1" xref="S2.p2.4.m4.1.1.1.1.1.cmml"><mo id="S2.p2.4.m4.1.1.1.1.2" stretchy="false" xref="S2.p2.4.m4.1.1.1.1.1.cmml">(</mo><mrow id="S2.p2.4.m4.1.1.1.1.1" xref="S2.p2.4.m4.1.1.1.1.1.cmml"><mi id="S2.p2.4.m4.1.1.1.1.1.2" xref="S2.p2.4.m4.1.1.1.1.1.2.cmml">u</mi><mo id="S2.p2.4.m4.1.1.1.1.1.1" lspace="0.108em" rspace="0.108em" stretchy="false" xref="S2.p2.4.m4.1.1.1.1.1.1.cmml">→</mo><mi id="S2.p2.4.m4.1.1.1.1.1.3" xref="S2.p2.4.m4.1.1.1.1.1.3.cmml">v</mi></mrow><mo id="S2.p2.4.m4.1.1.1.1.3" stretchy="false" xref="S2.p2.4.m4.1.1.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.p2.4.m4.1b"><apply id="S2.p2.4.m4.1.1.cmml" xref="S2.p2.4.m4.1.1"><times id="S2.p2.4.m4.1.1.2.cmml" xref="S2.p2.4.m4.1.1.2"></times><ci id="S2.p2.4.m4.1.1.3a.cmml" xref="S2.p2.4.m4.1.1.3"><mtext class="ltx_mathvariant_italic" id="S2.p2.4.m4.1.1.3.cmml" xref="S2.p2.4.m4.1.1.3">g</mtext></ci><apply id="S2.p2.4.m4.1.1.1.1.1.cmml" xref="S2.p2.4.m4.1.1.1.1"><ci id="S2.p2.4.m4.1.1.1.1.1.1.cmml" xref="S2.p2.4.m4.1.1.1.1.1.1">→</ci><ci id="S2.p2.4.m4.1.1.1.1.1.2.cmml" xref="S2.p2.4.m4.1.1.1.1.1.2">𝑢</ci><ci id="S2.p2.4.m4.1.1.1.1.1.3.cmml" xref="S2.p2.4.m4.1.1.1.1.1.3">𝑣</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p2.4.m4.1c">\textsl{g}(u\!\to\!v)</annotation><annotation encoding="application/x-llamapun" id="S2.p2.4.m4.1d">g ( italic_u → italic_v )</annotation></semantics></math> and <math alttext="\textsl{g}(v\!\to\!u)" 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"><mtext class="ltx_mathvariant_italic" id="S2.p2.5.m5.1.1.3" xref="S2.p2.5.m5.1.1.3a.cmml">g</mtext><mo id="S2.p2.5.m5.1.1.2" xref="S2.p2.5.m5.1.1.2.cmml">⁢</mo><mrow id="S2.p2.5.m5.1.1.1.1" xref="S2.p2.5.m5.1.1.1.1.1.cmml"><mo id="S2.p2.5.m5.1.1.1.1.2" stretchy="false" xref="S2.p2.5.m5.1.1.1.1.1.cmml">(</mo><mrow id="S2.p2.5.m5.1.1.1.1.1" xref="S2.p2.5.m5.1.1.1.1.1.cmml"><mi id="S2.p2.5.m5.1.1.1.1.1.2" xref="S2.p2.5.m5.1.1.1.1.1.2.cmml">v</mi><mo id="S2.p2.5.m5.1.1.1.1.1.1" lspace="0.108em" rspace="0.108em" stretchy="false" xref="S2.p2.5.m5.1.1.1.1.1.1.cmml">→</mo><mi id="S2.p2.5.m5.1.1.1.1.1.3" xref="S2.p2.5.m5.1.1.1.1.1.3.cmml">u</mi></mrow><mo id="S2.p2.5.m5.1.1.1.1.3" stretchy="false" xref="S2.p2.5.m5.1.1.1.1.1.cmml">)</mo></mrow></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"><times id="S2.p2.5.m5.1.1.2.cmml" xref="S2.p2.5.m5.1.1.2"></times><ci id="S2.p2.5.m5.1.1.3a.cmml" xref="S2.p2.5.m5.1.1.3"><mtext class="ltx_mathvariant_italic" id="S2.p2.5.m5.1.1.3.cmml" xref="S2.p2.5.m5.1.1.3">g</mtext></ci><apply id="S2.p2.5.m5.1.1.1.1.1.cmml" xref="S2.p2.5.m5.1.1.1.1"><ci id="S2.p2.5.m5.1.1.1.1.1.1.cmml" xref="S2.p2.5.m5.1.1.1.1.1.1">→</ci><ci id="S2.p2.5.m5.1.1.1.1.1.2.cmml" xref="S2.p2.5.m5.1.1.1.1.1.2">𝑣</ci><ci id="S2.p2.5.m5.1.1.1.1.1.3.cmml" xref="S2.p2.5.m5.1.1.1.1.1.3">𝑢</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p2.5.m5.1c">\textsl{g}(v\!\to\!u)</annotation><annotation encoding="application/x-llamapun" id="S2.p2.5.m5.1d">g ( italic_v → italic_u )</annotation></semantics></math> such that <math alttext="\textsl{g}(\overline{uv})=\textsl{g}(u\!\to\!v)+\textsl{g}(v\!\to\!u)" class="ltx_Math" display="inline" id="S2.p2.6.m6.3"><semantics id="S2.p2.6.m6.3a"><mrow id="S2.p2.6.m6.3.3" xref="S2.p2.6.m6.3.3.cmml"><mrow id="S2.p2.6.m6.3.3.4" xref="S2.p2.6.m6.3.3.4.cmml"><mtext class="ltx_mathvariant_italic" id="S2.p2.6.m6.3.3.4.2" xref="S2.p2.6.m6.3.3.4.2a.cmml">g</mtext><mo id="S2.p2.6.m6.3.3.4.1" xref="S2.p2.6.m6.3.3.4.1.cmml">⁢</mo><mrow id="S2.p2.6.m6.3.3.4.3.2" xref="S2.p2.6.m6.1.1.cmml"><mo id="S2.p2.6.m6.3.3.4.3.2.1" 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"><mrow id="S2.p2.6.m6.1.1.2" xref="S2.p2.6.m6.1.1.2.cmml"><mi id="S2.p2.6.m6.1.1.2.2" xref="S2.p2.6.m6.1.1.2.2.cmml">u</mi><mo id="S2.p2.6.m6.1.1.2.1" xref="S2.p2.6.m6.1.1.2.1.cmml">⁢</mo><mi id="S2.p2.6.m6.1.1.2.3" xref="S2.p2.6.m6.1.1.2.3.cmml">v</mi></mrow><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.m6.3.3.4.3.2.2" stretchy="false" xref="S2.p2.6.m6.1.1.cmml">)</mo></mrow></mrow><mo id="S2.p2.6.m6.3.3.3" xref="S2.p2.6.m6.3.3.3.cmml">=</mo><mrow id="S2.p2.6.m6.3.3.2" xref="S2.p2.6.m6.3.3.2.cmml"><mrow id="S2.p2.6.m6.2.2.1.1" xref="S2.p2.6.m6.2.2.1.1.cmml"><mtext class="ltx_mathvariant_italic" id="S2.p2.6.m6.2.2.1.1.3" xref="S2.p2.6.m6.2.2.1.1.3a.cmml">g</mtext><mo id="S2.p2.6.m6.2.2.1.1.2" xref="S2.p2.6.m6.2.2.1.1.2.cmml">⁢</mo><mrow id="S2.p2.6.m6.2.2.1.1.1.1" xref="S2.p2.6.m6.2.2.1.1.1.1.1.cmml"><mo id="S2.p2.6.m6.2.2.1.1.1.1.2" stretchy="false" xref="S2.p2.6.m6.2.2.1.1.1.1.1.cmml">(</mo><mrow id="S2.p2.6.m6.2.2.1.1.1.1.1" xref="S2.p2.6.m6.2.2.1.1.1.1.1.cmml"><mi id="S2.p2.6.m6.2.2.1.1.1.1.1.2" xref="S2.p2.6.m6.2.2.1.1.1.1.1.2.cmml">u</mi><mo id="S2.p2.6.m6.2.2.1.1.1.1.1.1" lspace="0.108em" rspace="0.108em" stretchy="false" xref="S2.p2.6.m6.2.2.1.1.1.1.1.1.cmml">→</mo><mi id="S2.p2.6.m6.2.2.1.1.1.1.1.3" xref="S2.p2.6.m6.2.2.1.1.1.1.1.3.cmml">v</mi></mrow><mo id="S2.p2.6.m6.2.2.1.1.1.1.3" stretchy="false" xref="S2.p2.6.m6.2.2.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S2.p2.6.m6.3.3.2.3" xref="S2.p2.6.m6.3.3.2.3.cmml">+</mo><mrow id="S2.p2.6.m6.3.3.2.2" xref="S2.p2.6.m6.3.3.2.2.cmml"><mtext class="ltx_mathvariant_italic" id="S2.p2.6.m6.3.3.2.2.3" xref="S2.p2.6.m6.3.3.2.2.3a.cmml">g</mtext><mo id="S2.p2.6.m6.3.3.2.2.2" xref="S2.p2.6.m6.3.3.2.2.2.cmml">⁢</mo><mrow id="S2.p2.6.m6.3.3.2.2.1.1" xref="S2.p2.6.m6.3.3.2.2.1.1.1.cmml"><mo id="S2.p2.6.m6.3.3.2.2.1.1.2" stretchy="false" xref="S2.p2.6.m6.3.3.2.2.1.1.1.cmml">(</mo><mrow id="S2.p2.6.m6.3.3.2.2.1.1.1" xref="S2.p2.6.m6.3.3.2.2.1.1.1.cmml"><mi id="S2.p2.6.m6.3.3.2.2.1.1.1.2" xref="S2.p2.6.m6.3.3.2.2.1.1.1.2.cmml">v</mi><mo id="S2.p2.6.m6.3.3.2.2.1.1.1.1" lspace="0.108em" rspace="0.108em" stretchy="false" xref="S2.p2.6.m6.3.3.2.2.1.1.1.1.cmml">→</mo><mi id="S2.p2.6.m6.3.3.2.2.1.1.1.3" xref="S2.p2.6.m6.3.3.2.2.1.1.1.3.cmml">u</mi></mrow><mo id="S2.p2.6.m6.3.3.2.2.1.1.3" stretchy="false" xref="S2.p2.6.m6.3.3.2.2.1.1.1.cmml">)</mo></mrow></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.p2.6.m6.3b"><apply id="S2.p2.6.m6.3.3.cmml" xref="S2.p2.6.m6.3.3"><eq id="S2.p2.6.m6.3.3.3.cmml" xref="S2.p2.6.m6.3.3.3"></eq><apply id="S2.p2.6.m6.3.3.4.cmml" xref="S2.p2.6.m6.3.3.4"><times id="S2.p2.6.m6.3.3.4.1.cmml" xref="S2.p2.6.m6.3.3.4.1"></times><ci id="S2.p2.6.m6.3.3.4.2a.cmml" xref="S2.p2.6.m6.3.3.4.2"><mtext class="ltx_mathvariant_italic" id="S2.p2.6.m6.3.3.4.2.cmml" xref="S2.p2.6.m6.3.3.4.2">g</mtext></ci><apply id="S2.p2.6.m6.1.1.cmml" xref="S2.p2.6.m6.3.3.4.3.2"><ci id="S2.p2.6.m6.1.1.1.cmml" xref="S2.p2.6.m6.1.1.1">¯</ci><apply id="S2.p2.6.m6.1.1.2.cmml" xref="S2.p2.6.m6.1.1.2"><times id="S2.p2.6.m6.1.1.2.1.cmml" xref="S2.p2.6.m6.1.1.2.1"></times><ci id="S2.p2.6.m6.1.1.2.2.cmml" xref="S2.p2.6.m6.1.1.2.2">𝑢</ci><ci id="S2.p2.6.m6.1.1.2.3.cmml" xref="S2.p2.6.m6.1.1.2.3">𝑣</ci></apply></apply></apply><apply id="S2.p2.6.m6.3.3.2.cmml" xref="S2.p2.6.m6.3.3.2"><plus id="S2.p2.6.m6.3.3.2.3.cmml" xref="S2.p2.6.m6.3.3.2.3"></plus><apply id="S2.p2.6.m6.2.2.1.1.cmml" xref="S2.p2.6.m6.2.2.1.1"><times id="S2.p2.6.m6.2.2.1.1.2.cmml" xref="S2.p2.6.m6.2.2.1.1.2"></times><ci id="S2.p2.6.m6.2.2.1.1.3a.cmml" xref="S2.p2.6.m6.2.2.1.1.3"><mtext class="ltx_mathvariant_italic" id="S2.p2.6.m6.2.2.1.1.3.cmml" xref="S2.p2.6.m6.2.2.1.1.3">g</mtext></ci><apply id="S2.p2.6.m6.2.2.1.1.1.1.1.cmml" xref="S2.p2.6.m6.2.2.1.1.1.1"><ci id="S2.p2.6.m6.2.2.1.1.1.1.1.1.cmml" xref="S2.p2.6.m6.2.2.1.1.1.1.1.1">→</ci><ci id="S2.p2.6.m6.2.2.1.1.1.1.1.2.cmml" xref="S2.p2.6.m6.2.2.1.1.1.1.1.2">𝑢</ci><ci id="S2.p2.6.m6.2.2.1.1.1.1.1.3.cmml" xref="S2.p2.6.m6.2.2.1.1.1.1.1.3">𝑣</ci></apply></apply><apply id="S2.p2.6.m6.3.3.2.2.cmml" xref="S2.p2.6.m6.3.3.2.2"><times id="S2.p2.6.m6.3.3.2.2.2.cmml" xref="S2.p2.6.m6.3.3.2.2.2"></times><ci id="S2.p2.6.m6.3.3.2.2.3a.cmml" xref="S2.p2.6.m6.3.3.2.2.3"><mtext class="ltx_mathvariant_italic" id="S2.p2.6.m6.3.3.2.2.3.cmml" xref="S2.p2.6.m6.3.3.2.2.3">g</mtext></ci><apply id="S2.p2.6.m6.3.3.2.2.1.1.1.cmml" xref="S2.p2.6.m6.3.3.2.2.1.1"><ci id="S2.p2.6.m6.3.3.2.2.1.1.1.1.cmml" xref="S2.p2.6.m6.3.3.2.2.1.1.1.1">→</ci><ci id="S2.p2.6.m6.3.3.2.2.1.1.1.2.cmml" xref="S2.p2.6.m6.3.3.2.2.1.1.1.2">𝑣</ci><ci id="S2.p2.6.m6.3.3.2.2.1.1.1.3.cmml" xref="S2.p2.6.m6.3.3.2.2.1.1.1.3">𝑢</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p2.6.m6.3c">\textsl{g}(\overline{uv})=\textsl{g}(u\!\to\!v)+\textsl{g}(v\!\to\!u)</annotation><annotation encoding="application/x-llamapun" id="S2.p2.6.m6.3d">g ( over¯ start_ARG italic_u italic_v end_ARG ) = g ( italic_u → italic_v ) + g ( italic_v → italic_u )</annotation></semantics></math>. These values may be interpreted as pointing a fraction of the edge <math alttext="\overline{uv}" class="ltx_Math" display="inline" id="S2.p2.7.m7.1"><semantics id="S2.p2.7.m7.1a"><mover accent="true" id="S2.p2.7.m7.1.1" xref="S2.p2.7.m7.1.1.cmml"><mrow id="S2.p2.7.m7.1.1.2" xref="S2.p2.7.m7.1.1.2.cmml"><mi id="S2.p2.7.m7.1.1.2.2" xref="S2.p2.7.m7.1.1.2.2.cmml">u</mi><mo id="S2.p2.7.m7.1.1.2.1" xref="S2.p2.7.m7.1.1.2.1.cmml">⁢</mo><mi id="S2.p2.7.m7.1.1.2.3" xref="S2.p2.7.m7.1.1.2.3.cmml">v</mi></mrow><mo id="S2.p2.7.m7.1.1.1" xref="S2.p2.7.m7.1.1.1.cmml">¯</mo></mover><annotation-xml encoding="MathML-Content" id="S2.p2.7.m7.1b"><apply id="S2.p2.7.m7.1.1.cmml" xref="S2.p2.7.m7.1.1"><ci id="S2.p2.7.m7.1.1.1.cmml" xref="S2.p2.7.m7.1.1.1">¯</ci><apply id="S2.p2.7.m7.1.1.2.cmml" xref="S2.p2.7.m7.1.1.2"><times id="S2.p2.7.m7.1.1.2.1.cmml" xref="S2.p2.7.m7.1.1.2.1"></times><ci id="S2.p2.7.m7.1.1.2.2.cmml" xref="S2.p2.7.m7.1.1.2.2">𝑢</ci><ci id="S2.p2.7.m7.1.1.2.3.cmml" xref="S2.p2.7.m7.1.1.2.3">𝑣</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p2.7.m7.1c">\overline{uv}</annotation><annotation encoding="application/x-llamapun" id="S2.p2.7.m7.1d">over¯ start_ARG italic_u italic_v end_ARG</annotation></semantics></math> from <math alttext="u" class="ltx_Math" display="inline" id="S2.p2.8.m8.1"><semantics id="S2.p2.8.m8.1a"><mi id="S2.p2.8.m8.1.1" xref="S2.p2.8.m8.1.1.cmml">u</mi><annotation-xml encoding="MathML-Content" id="S2.p2.8.m8.1b"><ci id="S2.p2.8.m8.1.1.cmml" xref="S2.p2.8.m8.1.1">𝑢</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.p2.8.m8.1c">u</annotation><annotation encoding="application/x-llamapun" id="S2.p2.8.m8.1d">italic_u</annotation></semantics></math> to <math alttext="v" class="ltx_Math" display="inline" id="S2.p2.9.m9.1"><semantics id="S2.p2.9.m9.1a"><mi id="S2.p2.9.m9.1.1" xref="S2.p2.9.m9.1.1.cmml">v</mi><annotation-xml encoding="MathML-Content" id="S2.p2.9.m9.1b"><ci id="S2.p2.9.m9.1.1.cmml" xref="S2.p2.9.m9.1.1">𝑣</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.p2.9.m9.1c">v</annotation><annotation encoding="application/x-llamapun" id="S2.p2.9.m9.1d">italic_v</annotation></semantics></math>, and the other fraction from <math alttext="v" class="ltx_Math" display="inline" id="S2.p2.10.m10.1"><semantics id="S2.p2.10.m10.1a"><mi id="S2.p2.10.m10.1.1" xref="S2.p2.10.m10.1.1.cmml">v</mi><annotation-xml encoding="MathML-Content" id="S2.p2.10.m10.1b"><ci id="S2.p2.10.m10.1.1.cmml" xref="S2.p2.10.m10.1.1">𝑣</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.p2.10.m10.1c">v</annotation><annotation encoding="application/x-llamapun" id="S2.p2.10.m10.1d">italic_v</annotation></semantics></math> to <math alttext="u" class="ltx_Math" display="inline" id="S2.p2.11.m11.1"><semantics id="S2.p2.11.m11.1a"><mi id="S2.p2.11.m11.1.1" xref="S2.p2.11.m11.1.1.cmml">u</mi><annotation-xml encoding="MathML-Content" id="S2.p2.11.m11.1b"><ci id="S2.p2.11.m11.1.1.cmml" xref="S2.p2.11.m11.1.1">𝑢</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.p2.11.m11.1c">u</annotation><annotation encoding="application/x-llamapun" id="S2.p2.11.m11.1d">italic_u</annotation></semantics></math>. Given an orientation <math alttext="\overrightarrow{G}" class="ltx_Math" display="inline" id="S2.p2.12.m12.1"><semantics id="S2.p2.12.m12.1a"><mover accent="true" id="S2.p2.12.m12.1.1" xref="S2.p2.12.m12.1.1.cmml"><mi id="S2.p2.12.m12.1.1.2" xref="S2.p2.12.m12.1.1.2.cmml">G</mi><mo id="S2.p2.12.m12.1.1.1" stretchy="false" xref="S2.p2.12.m12.1.1.1.cmml">→</mo></mover><annotation-xml encoding="MathML-Content" id="S2.p2.12.m12.1b"><apply id="S2.p2.12.m12.1.1.cmml" xref="S2.p2.12.m12.1.1"><ci id="S2.p2.12.m12.1.1.1.cmml" xref="S2.p2.12.m12.1.1.1">→</ci><ci id="S2.p2.12.m12.1.1.2.cmml" xref="S2.p2.12.m12.1.1.2">𝐺</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p2.12.m12.1c">\overrightarrow{G}</annotation><annotation encoding="application/x-llamapun" id="S2.p2.12.m12.1d">over→ start_ARG italic_G end_ARG</annotation></semantics></math>, we denote by <math alttext="\textsl{g}(u)=\sum\limits_{v\in V}\textsl{g}(u\!\to\!v)" class="ltx_Math" display="inline" id="S2.p2.13.m13.2"><semantics id="S2.p2.13.m13.2a"><mrow id="S2.p2.13.m13.2.2" xref="S2.p2.13.m13.2.2.cmml"><mrow id="S2.p2.13.m13.2.2.3" xref="S2.p2.13.m13.2.2.3.cmml"><mtext class="ltx_mathvariant_italic" id="S2.p2.13.m13.2.2.3.2" xref="S2.p2.13.m13.2.2.3.2a.cmml">g</mtext><mo id="S2.p2.13.m13.2.2.3.1" xref="S2.p2.13.m13.2.2.3.1.cmml">⁢</mo><mrow id="S2.p2.13.m13.2.2.3.3.2" xref="S2.p2.13.m13.2.2.3.cmml"><mo id="S2.p2.13.m13.2.2.3.3.2.1" stretchy="false" xref="S2.p2.13.m13.2.2.3.cmml">(</mo><mi id="S2.p2.13.m13.1.1" xref="S2.p2.13.m13.1.1.cmml">u</mi><mo id="S2.p2.13.m13.2.2.3.3.2.2" stretchy="false" xref="S2.p2.13.m13.2.2.3.cmml">)</mo></mrow></mrow><mo id="S2.p2.13.m13.2.2.2" rspace="0.111em" xref="S2.p2.13.m13.2.2.2.cmml">=</mo><mrow id="S2.p2.13.m13.2.2.1" xref="S2.p2.13.m13.2.2.1.cmml"><munder id="S2.p2.13.m13.2.2.1.2" xref="S2.p2.13.m13.2.2.1.2.cmml"><mo id="S2.p2.13.m13.2.2.1.2.2" movablelimits="false" xref="S2.p2.13.m13.2.2.1.2.2.cmml">∑</mo><mrow id="S2.p2.13.m13.2.2.1.2.3" xref="S2.p2.13.m13.2.2.1.2.3.cmml"><mi id="S2.p2.13.m13.2.2.1.2.3.2" xref="S2.p2.13.m13.2.2.1.2.3.2.cmml">v</mi><mo id="S2.p2.13.m13.2.2.1.2.3.1" xref="S2.p2.13.m13.2.2.1.2.3.1.cmml">∈</mo><mi id="S2.p2.13.m13.2.2.1.2.3.3" xref="S2.p2.13.m13.2.2.1.2.3.3.cmml">V</mi></mrow></munder><mrow id="S2.p2.13.m13.2.2.1.1" xref="S2.p2.13.m13.2.2.1.1.cmml"><mtext class="ltx_mathvariant_italic" id="S2.p2.13.m13.2.2.1.1.3" xref="S2.p2.13.m13.2.2.1.1.3a.cmml">g</mtext><mo id="S2.p2.13.m13.2.2.1.1.2" xref="S2.p2.13.m13.2.2.1.1.2.cmml">⁢</mo><mrow id="S2.p2.13.m13.2.2.1.1.1.1" xref="S2.p2.13.m13.2.2.1.1.1.1.1.cmml"><mo id="S2.p2.13.m13.2.2.1.1.1.1.2" stretchy="false" xref="S2.p2.13.m13.2.2.1.1.1.1.1.cmml">(</mo><mrow id="S2.p2.13.m13.2.2.1.1.1.1.1" xref="S2.p2.13.m13.2.2.1.1.1.1.1.cmml"><mi id="S2.p2.13.m13.2.2.1.1.1.1.1.2" xref="S2.p2.13.m13.2.2.1.1.1.1.1.2.cmml">u</mi><mo id="S2.p2.13.m13.2.2.1.1.1.1.1.1" lspace="0.108em" rspace="0.108em" stretchy="false" xref="S2.p2.13.m13.2.2.1.1.1.1.1.1.cmml">→</mo><mi id="S2.p2.13.m13.2.2.1.1.1.1.1.3" xref="S2.p2.13.m13.2.2.1.1.1.1.1.3.cmml">v</mi></mrow><mo id="S2.p2.13.m13.2.2.1.1.1.1.3" stretchy="false" xref="S2.p2.13.m13.2.2.1.1.1.1.1.cmml">)</mo></mrow></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.p2.13.m13.2b"><apply id="S2.p2.13.m13.2.2.cmml" xref="S2.p2.13.m13.2.2"><eq id="S2.p2.13.m13.2.2.2.cmml" xref="S2.p2.13.m13.2.2.2"></eq><apply id="S2.p2.13.m13.2.2.3.cmml" xref="S2.p2.13.m13.2.2.3"><times id="S2.p2.13.m13.2.2.3.1.cmml" xref="S2.p2.13.m13.2.2.3.1"></times><ci id="S2.p2.13.m13.2.2.3.2a.cmml" xref="S2.p2.13.m13.2.2.3.2"><mtext class="ltx_mathvariant_italic" id="S2.p2.13.m13.2.2.3.2.cmml" xref="S2.p2.13.m13.2.2.3.2">g</mtext></ci><ci id="S2.p2.13.m13.1.1.cmml" xref="S2.p2.13.m13.1.1">𝑢</ci></apply><apply id="S2.p2.13.m13.2.2.1.cmml" xref="S2.p2.13.m13.2.2.1"><apply id="S2.p2.13.m13.2.2.1.2.cmml" xref="S2.p2.13.m13.2.2.1.2"><csymbol cd="ambiguous" id="S2.p2.13.m13.2.2.1.2.1.cmml" xref="S2.p2.13.m13.2.2.1.2">subscript</csymbol><sum id="S2.p2.13.m13.2.2.1.2.2.cmml" xref="S2.p2.13.m13.2.2.1.2.2"></sum><apply id="S2.p2.13.m13.2.2.1.2.3.cmml" xref="S2.p2.13.m13.2.2.1.2.3"><in id="S2.p2.13.m13.2.2.1.2.3.1.cmml" xref="S2.p2.13.m13.2.2.1.2.3.1"></in><ci id="S2.p2.13.m13.2.2.1.2.3.2.cmml" xref="S2.p2.13.m13.2.2.1.2.3.2">𝑣</ci><ci id="S2.p2.13.m13.2.2.1.2.3.3.cmml" xref="S2.p2.13.m13.2.2.1.2.3.3">𝑉</ci></apply></apply><apply id="S2.p2.13.m13.2.2.1.1.cmml" xref="S2.p2.13.m13.2.2.1.1"><times id="S2.p2.13.m13.2.2.1.1.2.cmml" xref="S2.p2.13.m13.2.2.1.1.2"></times><ci id="S2.p2.13.m13.2.2.1.1.3a.cmml" xref="S2.p2.13.m13.2.2.1.1.3"><mtext class="ltx_mathvariant_italic" id="S2.p2.13.m13.2.2.1.1.3.cmml" xref="S2.p2.13.m13.2.2.1.1.3">g</mtext></ci><apply id="S2.p2.13.m13.2.2.1.1.1.1.1.cmml" xref="S2.p2.13.m13.2.2.1.1.1.1"><ci id="S2.p2.13.m13.2.2.1.1.1.1.1.1.cmml" xref="S2.p2.13.m13.2.2.1.1.1.1.1.1">→</ci><ci id="S2.p2.13.m13.2.2.1.1.1.1.1.2.cmml" xref="S2.p2.13.m13.2.2.1.1.1.1.1.2">𝑢</ci><ci id="S2.p2.13.m13.2.2.1.1.1.1.1.3.cmml" xref="S2.p2.13.m13.2.2.1.1.1.1.1.3">𝑣</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p2.13.m13.2c">\textsl{g}(u)=\sum\limits_{v\in V}\textsl{g}(u\!\to\!v)</annotation><annotation encoding="application/x-llamapun" id="S2.p2.13.m13.2d">g ( italic_u ) = ∑ start_POSTSUBSCRIPT italic_v ∈ italic_V end_POSTSUBSCRIPT g ( italic_u → italic_v )</annotation></semantics></math> the <em class="ltx_emph ltx_font_italic" id="S2.p2.18.2">out-degree</em> of <math alttext="u" class="ltx_Math" display="inline" id="S2.p2.14.m14.1"><semantics id="S2.p2.14.m14.1a"><mi id="S2.p2.14.m14.1.1" xref="S2.p2.14.m14.1.1.cmml">u</mi><annotation-xml encoding="MathML-Content" id="S2.p2.14.m14.1b"><ci id="S2.p2.14.m14.1.1.cmml" xref="S2.p2.14.m14.1.1">𝑢</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.p2.14.m14.1c">u</annotation><annotation encoding="application/x-llamapun" id="S2.p2.14.m14.1d">italic_u</annotation></semantics></math> (i.e., how much fractional edges point outwards from <math alttext="u" class="ltx_Math" display="inline" id="S2.p2.15.m15.1"><semantics id="S2.p2.15.m15.1a"><mi id="S2.p2.15.m15.1.1" xref="S2.p2.15.m15.1.1.cmml">u</mi><annotation-xml encoding="MathML-Content" id="S2.p2.15.m15.1b"><ci id="S2.p2.15.m15.1.1.cmml" xref="S2.p2.15.m15.1.1">𝑢</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.p2.15.m15.1c">u</annotation><annotation encoding="application/x-llamapun" id="S2.p2.15.m15.1d">italic_u</annotation></semantics></math> in <math alttext="\overrightarrow{G}" class="ltx_Math" display="inline" id="S2.p2.16.m16.1"><semantics id="S2.p2.16.m16.1a"><mover accent="true" id="S2.p2.16.m16.1.1" xref="S2.p2.16.m16.1.1.cmml"><mi id="S2.p2.16.m16.1.1.2" xref="S2.p2.16.m16.1.1.2.cmml">G</mi><mo id="S2.p2.16.m16.1.1.1" stretchy="false" xref="S2.p2.16.m16.1.1.1.cmml">→</mo></mover><annotation-xml encoding="MathML-Content" id="S2.p2.16.m16.1b"><apply id="S2.p2.16.m16.1.1.cmml" xref="S2.p2.16.m16.1.1"><ci id="S2.p2.16.m16.1.1.1.cmml" xref="S2.p2.16.m16.1.1.1">→</ci><ci id="S2.p2.16.m16.1.1.2.cmml" xref="S2.p2.16.m16.1.1.2">𝐺</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p2.16.m16.1c">\overrightarrow{G}</annotation><annotation encoding="application/x-llamapun" id="S2.p2.16.m16.1d">over→ start_ARG italic_G end_ARG</annotation></semantics></math>). Given these definitions, we can consider two graph measures of <math alttext="G" class="ltx_Math" display="inline" id="S2.p2.17.m17.1"><semantics id="S2.p2.17.m17.1a"><mi id="S2.p2.17.m17.1.1" xref="S2.p2.17.m17.1.1.cmml">G</mi><annotation-xml encoding="MathML-Content" id="S2.p2.17.m17.1b"><ci id="S2.p2.17.m17.1.1.cmml" xref="S2.p2.17.m17.1.1">𝐺</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.p2.17.m17.1c">G</annotation><annotation encoding="application/x-llamapun" id="S2.p2.17.m17.1d">italic_G</annotation></semantics></math>: the maximum subgraph density and the minimum orientation of <math alttext="G" class="ltx_Math" display="inline" id="S2.p2.18.m18.1"><semantics id="S2.p2.18.m18.1a"><mi id="S2.p2.18.m18.1.1" xref="S2.p2.18.m18.1.1.cmml">G</mi><annotation-xml encoding="MathML-Content" id="S2.p2.18.m18.1b"><ci id="S2.p2.18.m18.1.1.cmml" xref="S2.p2.18.m18.1.1">𝐺</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.p2.18.m18.1c">G</annotation><annotation encoding="application/x-llamapun" id="S2.p2.18.m18.1d">italic_G</annotation></semantics></math>.</p> </div> <section class="ltx_subparagraph" id="S2.SS0.SSS0.P0.SPx1"> <h5 class="ltx_title ltx_title_subparagraph">Global graph measures.</h5> <div class="ltx_para" id="S2.SS0.SSS0.P0.SPx1.p1"> <p class="ltx_p" id="S2.SS0.SSS0.P0.SPx1.p1.19">For any subgraph <math alttext="H\subseteq G" class="ltx_Math" display="inline" id="S2.SS0.SSS0.P0.SPx1.p1.1.m1.1"><semantics id="S2.SS0.SSS0.P0.SPx1.p1.1.m1.1a"><mrow id="S2.SS0.SSS0.P0.SPx1.p1.1.m1.1.1" xref="S2.SS0.SSS0.P0.SPx1.p1.1.m1.1.1.cmml"><mi id="S2.SS0.SSS0.P0.SPx1.p1.1.m1.1.1.2" xref="S2.SS0.SSS0.P0.SPx1.p1.1.m1.1.1.2.cmml">H</mi><mo id="S2.SS0.SSS0.P0.SPx1.p1.1.m1.1.1.1" xref="S2.SS0.SSS0.P0.SPx1.p1.1.m1.1.1.1.cmml">⊆</mo><mi id="S2.SS0.SSS0.P0.SPx1.p1.1.m1.1.1.3" xref="S2.SS0.SSS0.P0.SPx1.p1.1.m1.1.1.3.cmml">G</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.P0.SPx1.p1.1.m1.1b"><apply id="S2.SS0.SSS0.P0.SPx1.p1.1.m1.1.1.cmml" xref="S2.SS0.SSS0.P0.SPx1.p1.1.m1.1.1"><subset id="S2.SS0.SSS0.P0.SPx1.p1.1.m1.1.1.1.cmml" xref="S2.SS0.SSS0.P0.SPx1.p1.1.m1.1.1.1"></subset><ci id="S2.SS0.SSS0.P0.SPx1.p1.1.m1.1.1.2.cmml" xref="S2.SS0.SSS0.P0.SPx1.p1.1.m1.1.1.2">𝐻</ci><ci id="S2.SS0.SSS0.P0.SPx1.p1.1.m1.1.1.3.cmml" xref="S2.SS0.SSS0.P0.SPx1.p1.1.m1.1.1.3">𝐺</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.P0.SPx1.p1.1.m1.1c">H\subseteq G</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.P0.SPx1.p1.1.m1.1d">italic_H ⊆ italic_G</annotation></semantics></math>, its <em class="ltx_emph ltx_font_italic" id="S2.SS0.SSS0.P0.SPx1.p1.19.1">density</em> <math alttext="\rho(H)" class="ltx_Math" display="inline" id="S2.SS0.SSS0.P0.SPx1.p1.2.m2.1"><semantics id="S2.SS0.SSS0.P0.SPx1.p1.2.m2.1a"><mrow id="S2.SS0.SSS0.P0.SPx1.p1.2.m2.1.2" xref="S2.SS0.SSS0.P0.SPx1.p1.2.m2.1.2.cmml"><mi id="S2.SS0.SSS0.P0.SPx1.p1.2.m2.1.2.2" xref="S2.SS0.SSS0.P0.SPx1.p1.2.m2.1.2.2.cmml">ρ</mi><mo id="S2.SS0.SSS0.P0.SPx1.p1.2.m2.1.2.1" xref="S2.SS0.SSS0.P0.SPx1.p1.2.m2.1.2.1.cmml">⁢</mo><mrow id="S2.SS0.SSS0.P0.SPx1.p1.2.m2.1.2.3.2" xref="S2.SS0.SSS0.P0.SPx1.p1.2.m2.1.2.cmml"><mo id="S2.SS0.SSS0.P0.SPx1.p1.2.m2.1.2.3.2.1" stretchy="false" xref="S2.SS0.SSS0.P0.SPx1.p1.2.m2.1.2.cmml">(</mo><mi id="S2.SS0.SSS0.P0.SPx1.p1.2.m2.1.1" xref="S2.SS0.SSS0.P0.SPx1.p1.2.m2.1.1.cmml">H</mi><mo id="S2.SS0.SSS0.P0.SPx1.p1.2.m2.1.2.3.2.2" stretchy="false" xref="S2.SS0.SSS0.P0.SPx1.p1.2.m2.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.P0.SPx1.p1.2.m2.1b"><apply id="S2.SS0.SSS0.P0.SPx1.p1.2.m2.1.2.cmml" xref="S2.SS0.SSS0.P0.SPx1.p1.2.m2.1.2"><times id="S2.SS0.SSS0.P0.SPx1.p1.2.m2.1.2.1.cmml" xref="S2.SS0.SSS0.P0.SPx1.p1.2.m2.1.2.1"></times><ci id="S2.SS0.SSS0.P0.SPx1.p1.2.m2.1.2.2.cmml" xref="S2.SS0.SSS0.P0.SPx1.p1.2.m2.1.2.2">𝜌</ci><ci id="S2.SS0.SSS0.P0.SPx1.p1.2.m2.1.1.cmml" xref="S2.SS0.SSS0.P0.SPx1.p1.2.m2.1.1">𝐻</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.P0.SPx1.p1.2.m2.1c">\rho(H)</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.P0.SPx1.p1.2.m2.1d">italic_ρ ( italic_H )</annotation></semantics></math> is defined as <math alttext="\rho(H)=\frac{1}{|V(H)|}\sum\limits_{e\in E(H)}\textsl{g}(e)" class="ltx_Math" display="inline" id="S2.SS0.SSS0.P0.SPx1.p1.3.m3.5"><semantics id="S2.SS0.SSS0.P0.SPx1.p1.3.m3.5a"><mrow id="S2.SS0.SSS0.P0.SPx1.p1.3.m3.5.6" xref="S2.SS0.SSS0.P0.SPx1.p1.3.m3.5.6.cmml"><mrow id="S2.SS0.SSS0.P0.SPx1.p1.3.m3.5.6.2" xref="S2.SS0.SSS0.P0.SPx1.p1.3.m3.5.6.2.cmml"><mi id="S2.SS0.SSS0.P0.SPx1.p1.3.m3.5.6.2.2" xref="S2.SS0.SSS0.P0.SPx1.p1.3.m3.5.6.2.2.cmml">ρ</mi><mo id="S2.SS0.SSS0.P0.SPx1.p1.3.m3.5.6.2.1" xref="S2.SS0.SSS0.P0.SPx1.p1.3.m3.5.6.2.1.cmml">⁢</mo><mrow id="S2.SS0.SSS0.P0.SPx1.p1.3.m3.5.6.2.3.2" xref="S2.SS0.SSS0.P0.SPx1.p1.3.m3.5.6.2.cmml"><mo id="S2.SS0.SSS0.P0.SPx1.p1.3.m3.5.6.2.3.2.1" stretchy="false" xref="S2.SS0.SSS0.P0.SPx1.p1.3.m3.5.6.2.cmml">(</mo><mi id="S2.SS0.SSS0.P0.SPx1.p1.3.m3.4.4" xref="S2.SS0.SSS0.P0.SPx1.p1.3.m3.4.4.cmml">H</mi><mo id="S2.SS0.SSS0.P0.SPx1.p1.3.m3.5.6.2.3.2.2" stretchy="false" xref="S2.SS0.SSS0.P0.SPx1.p1.3.m3.5.6.2.cmml">)</mo></mrow></mrow><mo id="S2.SS0.SSS0.P0.SPx1.p1.3.m3.5.6.1" xref="S2.SS0.SSS0.P0.SPx1.p1.3.m3.5.6.1.cmml">=</mo><mrow id="S2.SS0.SSS0.P0.SPx1.p1.3.m3.5.6.3" xref="S2.SS0.SSS0.P0.SPx1.p1.3.m3.5.6.3.cmml"><mfrac id="S2.SS0.SSS0.P0.SPx1.p1.3.m3.2.2" xref="S2.SS0.SSS0.P0.SPx1.p1.3.m3.2.2.cmml"><mn id="S2.SS0.SSS0.P0.SPx1.p1.3.m3.2.2.4" xref="S2.SS0.SSS0.P0.SPx1.p1.3.m3.2.2.4.cmml">1</mn><mrow id="S2.SS0.SSS0.P0.SPx1.p1.3.m3.2.2.2.2" xref="S2.SS0.SSS0.P0.SPx1.p1.3.m3.2.2.2.3.cmml"><mo id="S2.SS0.SSS0.P0.SPx1.p1.3.m3.2.2.2.2.2" stretchy="false" xref="S2.SS0.SSS0.P0.SPx1.p1.3.m3.2.2.2.3.1.cmml">|</mo><mrow id="S2.SS0.SSS0.P0.SPx1.p1.3.m3.2.2.2.2.1" xref="S2.SS0.SSS0.P0.SPx1.p1.3.m3.2.2.2.2.1.cmml"><mi id="S2.SS0.SSS0.P0.SPx1.p1.3.m3.2.2.2.2.1.2" xref="S2.SS0.SSS0.P0.SPx1.p1.3.m3.2.2.2.2.1.2.cmml">V</mi><mo id="S2.SS0.SSS0.P0.SPx1.p1.3.m3.2.2.2.2.1.1" xref="S2.SS0.SSS0.P0.SPx1.p1.3.m3.2.2.2.2.1.1.cmml">⁢</mo><mrow id="S2.SS0.SSS0.P0.SPx1.p1.3.m3.2.2.2.2.1.3.2" xref="S2.SS0.SSS0.P0.SPx1.p1.3.m3.2.2.2.2.1.cmml"><mo id="S2.SS0.SSS0.P0.SPx1.p1.3.m3.2.2.2.2.1.3.2.1" stretchy="false" xref="S2.SS0.SSS0.P0.SPx1.p1.3.m3.2.2.2.2.1.cmml">(</mo><mi id="S2.SS0.SSS0.P0.SPx1.p1.3.m3.1.1.1.1" xref="S2.SS0.SSS0.P0.SPx1.p1.3.m3.1.1.1.1.cmml">H</mi><mo id="S2.SS0.SSS0.P0.SPx1.p1.3.m3.2.2.2.2.1.3.2.2" stretchy="false" xref="S2.SS0.SSS0.P0.SPx1.p1.3.m3.2.2.2.2.1.cmml">)</mo></mrow></mrow><mo id="S2.SS0.SSS0.P0.SPx1.p1.3.m3.2.2.2.2.3" stretchy="false" xref="S2.SS0.SSS0.P0.SPx1.p1.3.m3.2.2.2.3.1.cmml">|</mo></mrow></mfrac><mo id="S2.SS0.SSS0.P0.SPx1.p1.3.m3.5.6.3.1" xref="S2.SS0.SSS0.P0.SPx1.p1.3.m3.5.6.3.1.cmml">⁢</mo><mrow id="S2.SS0.SSS0.P0.SPx1.p1.3.m3.5.6.3.2" xref="S2.SS0.SSS0.P0.SPx1.p1.3.m3.5.6.3.2.cmml"><munder id="S2.SS0.SSS0.P0.SPx1.p1.3.m3.5.6.3.2.1" xref="S2.SS0.SSS0.P0.SPx1.p1.3.m3.5.6.3.2.1.cmml"><mo id="S2.SS0.SSS0.P0.SPx1.p1.3.m3.5.6.3.2.1.2" movablelimits="false" xref="S2.SS0.SSS0.P0.SPx1.p1.3.m3.5.6.3.2.1.2.cmml">∑</mo><mrow id="S2.SS0.SSS0.P0.SPx1.p1.3.m3.3.3.1" xref="S2.SS0.SSS0.P0.SPx1.p1.3.m3.3.3.1.cmml"><mi id="S2.SS0.SSS0.P0.SPx1.p1.3.m3.3.3.1.3" xref="S2.SS0.SSS0.P0.SPx1.p1.3.m3.3.3.1.3.cmml">e</mi><mo id="S2.SS0.SSS0.P0.SPx1.p1.3.m3.3.3.1.2" xref="S2.SS0.SSS0.P0.SPx1.p1.3.m3.3.3.1.2.cmml">∈</mo><mrow id="S2.SS0.SSS0.P0.SPx1.p1.3.m3.3.3.1.4" xref="S2.SS0.SSS0.P0.SPx1.p1.3.m3.3.3.1.4.cmml"><mi id="S2.SS0.SSS0.P0.SPx1.p1.3.m3.3.3.1.4.2" xref="S2.SS0.SSS0.P0.SPx1.p1.3.m3.3.3.1.4.2.cmml">E</mi><mo id="S2.SS0.SSS0.P0.SPx1.p1.3.m3.3.3.1.4.1" xref="S2.SS0.SSS0.P0.SPx1.p1.3.m3.3.3.1.4.1.cmml">⁢</mo><mrow 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xref="S2.SS0.SSS0.P0.SPx1.p1.3.m3.5.6.3.2.2.cmml">(</mo><mi id="S2.SS0.SSS0.P0.SPx1.p1.3.m3.5.5" xref="S2.SS0.SSS0.P0.SPx1.p1.3.m3.5.5.cmml">e</mi><mo id="S2.SS0.SSS0.P0.SPx1.p1.3.m3.5.6.3.2.2.3.2.2" stretchy="false" xref="S2.SS0.SSS0.P0.SPx1.p1.3.m3.5.6.3.2.2.cmml">)</mo></mrow></mrow></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.P0.SPx1.p1.3.m3.5b"><apply id="S2.SS0.SSS0.P0.SPx1.p1.3.m3.5.6.cmml" xref="S2.SS0.SSS0.P0.SPx1.p1.3.m3.5.6"><eq id="S2.SS0.SSS0.P0.SPx1.p1.3.m3.5.6.1.cmml" xref="S2.SS0.SSS0.P0.SPx1.p1.3.m3.5.6.1"></eq><apply id="S2.SS0.SSS0.P0.SPx1.p1.3.m3.5.6.2.cmml" xref="S2.SS0.SSS0.P0.SPx1.p1.3.m3.5.6.2"><times id="S2.SS0.SSS0.P0.SPx1.p1.3.m3.5.6.2.1.cmml" xref="S2.SS0.SSS0.P0.SPx1.p1.3.m3.5.6.2.1"></times><ci id="S2.SS0.SSS0.P0.SPx1.p1.3.m3.5.6.2.2.cmml" xref="S2.SS0.SSS0.P0.SPx1.p1.3.m3.5.6.2.2">𝜌</ci><ci id="S2.SS0.SSS0.P0.SPx1.p1.3.m3.4.4.cmml" xref="S2.SS0.SSS0.P0.SPx1.p1.3.m3.4.4">𝐻</ci></apply><apply id="S2.SS0.SSS0.P0.SPx1.p1.3.m3.5.6.3.cmml" xref="S2.SS0.SSS0.P0.SPx1.p1.3.m3.5.6.3"><times id="S2.SS0.SSS0.P0.SPx1.p1.3.m3.5.6.3.1.cmml" xref="S2.SS0.SSS0.P0.SPx1.p1.3.m3.5.6.3.1"></times><apply id="S2.SS0.SSS0.P0.SPx1.p1.3.m3.2.2.cmml" xref="S2.SS0.SSS0.P0.SPx1.p1.3.m3.2.2"><divide id="S2.SS0.SSS0.P0.SPx1.p1.3.m3.2.2.3.cmml" xref="S2.SS0.SSS0.P0.SPx1.p1.3.m3.2.2"></divide><cn id="S2.SS0.SSS0.P0.SPx1.p1.3.m3.2.2.4.cmml" type="integer" xref="S2.SS0.SSS0.P0.SPx1.p1.3.m3.2.2.4">1</cn><apply id="S2.SS0.SSS0.P0.SPx1.p1.3.m3.2.2.2.3.cmml" xref="S2.SS0.SSS0.P0.SPx1.p1.3.m3.2.2.2.2"><abs id="S2.SS0.SSS0.P0.SPx1.p1.3.m3.2.2.2.3.1.cmml" xref="S2.SS0.SSS0.P0.SPx1.p1.3.m3.2.2.2.2.2"></abs><apply id="S2.SS0.SSS0.P0.SPx1.p1.3.m3.2.2.2.2.1.cmml" xref="S2.SS0.SSS0.P0.SPx1.p1.3.m3.2.2.2.2.1"><times id="S2.SS0.SSS0.P0.SPx1.p1.3.m3.2.2.2.2.1.1.cmml" xref="S2.SS0.SSS0.P0.SPx1.p1.3.m3.2.2.2.2.1.1"></times><ci id="S2.SS0.SSS0.P0.SPx1.p1.3.m3.2.2.2.2.1.2.cmml" xref="S2.SS0.SSS0.P0.SPx1.p1.3.m3.2.2.2.2.1.2">𝑉</ci><ci id="S2.SS0.SSS0.P0.SPx1.p1.3.m3.1.1.1.1.cmml" xref="S2.SS0.SSS0.P0.SPx1.p1.3.m3.1.1.1.1">𝐻</ci></apply></apply></apply><apply id="S2.SS0.SSS0.P0.SPx1.p1.3.m3.5.6.3.2.cmml" xref="S2.SS0.SSS0.P0.SPx1.p1.3.m3.5.6.3.2"><apply id="S2.SS0.SSS0.P0.SPx1.p1.3.m3.5.6.3.2.1.cmml" xref="S2.SS0.SSS0.P0.SPx1.p1.3.m3.5.6.3.2.1"><csymbol cd="ambiguous" id="S2.SS0.SSS0.P0.SPx1.p1.3.m3.5.6.3.2.1.1.cmml" xref="S2.SS0.SSS0.P0.SPx1.p1.3.m3.5.6.3.2.1">subscript</csymbol><sum id="S2.SS0.SSS0.P0.SPx1.p1.3.m3.5.6.3.2.1.2.cmml" xref="S2.SS0.SSS0.P0.SPx1.p1.3.m3.5.6.3.2.1.2"></sum><apply id="S2.SS0.SSS0.P0.SPx1.p1.3.m3.3.3.1.cmml" xref="S2.SS0.SSS0.P0.SPx1.p1.3.m3.3.3.1"><in id="S2.SS0.SSS0.P0.SPx1.p1.3.m3.3.3.1.2.cmml" xref="S2.SS0.SSS0.P0.SPx1.p1.3.m3.3.3.1.2"></in><ci id="S2.SS0.SSS0.P0.SPx1.p1.3.m3.3.3.1.3.cmml" xref="S2.SS0.SSS0.P0.SPx1.p1.3.m3.3.3.1.3">𝑒</ci><apply id="S2.SS0.SSS0.P0.SPx1.p1.3.m3.3.3.1.4.cmml" xref="S2.SS0.SSS0.P0.SPx1.p1.3.m3.3.3.1.4"><times id="S2.SS0.SSS0.P0.SPx1.p1.3.m3.3.3.1.4.1.cmml" xref="S2.SS0.SSS0.P0.SPx1.p1.3.m3.3.3.1.4.1"></times><ci id="S2.SS0.SSS0.P0.SPx1.p1.3.m3.3.3.1.4.2.cmml" xref="S2.SS0.SSS0.P0.SPx1.p1.3.m3.3.3.1.4.2">𝐸</ci><ci id="S2.SS0.SSS0.P0.SPx1.p1.3.m3.3.3.1.1.cmml" xref="S2.SS0.SSS0.P0.SPx1.p1.3.m3.3.3.1.1">𝐻</ci></apply></apply></apply><apply id="S2.SS0.SSS0.P0.SPx1.p1.3.m3.5.6.3.2.2.cmml" xref="S2.SS0.SSS0.P0.SPx1.p1.3.m3.5.6.3.2.2"><times id="S2.SS0.SSS0.P0.SPx1.p1.3.m3.5.6.3.2.2.1.cmml" xref="S2.SS0.SSS0.P0.SPx1.p1.3.m3.5.6.3.2.2.1"></times><ci id="S2.SS0.SSS0.P0.SPx1.p1.3.m3.5.6.3.2.2.2a.cmml" xref="S2.SS0.SSS0.P0.SPx1.p1.3.m3.5.6.3.2.2.2"><mtext class="ltx_mathvariant_italic" id="S2.SS0.SSS0.P0.SPx1.p1.3.m3.5.6.3.2.2.2.cmml" xref="S2.SS0.SSS0.P0.SPx1.p1.3.m3.5.6.3.2.2.2">g</mtext></ci><ci id="S2.SS0.SSS0.P0.SPx1.p1.3.m3.5.5.cmml" xref="S2.SS0.SSS0.P0.SPx1.p1.3.m3.5.5">𝑒</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.P0.SPx1.p1.3.m3.5c">\rho(H)=\frac{1}{|V(H)|}\sum\limits_{e\in E(H)}\textsl{g}(e)</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.P0.SPx1.p1.3.m3.5d">italic_ρ ( italic_H ) = divide start_ARG 1 end_ARG start_ARG | italic_V ( italic_H ) | end_ARG ∑ start_POSTSUBSCRIPT italic_e ∈ italic_E ( italic_H ) end_POSTSUBSCRIPT g ( italic_e )</annotation></semantics></math>. The <em class="ltx_emph ltx_font_italic" id="S2.SS0.SSS0.P0.SPx1.p1.19.2">maximum subgraph density</em> <math alttext="\rho^{\max}(G)" class="ltx_Math" display="inline" id="S2.SS0.SSS0.P0.SPx1.p1.4.m4.1"><semantics id="S2.SS0.SSS0.P0.SPx1.p1.4.m4.1a"><mrow id="S2.SS0.SSS0.P0.SPx1.p1.4.m4.1.2" xref="S2.SS0.SSS0.P0.SPx1.p1.4.m4.1.2.cmml"><msup id="S2.SS0.SSS0.P0.SPx1.p1.4.m4.1.2.2" xref="S2.SS0.SSS0.P0.SPx1.p1.4.m4.1.2.2.cmml"><mi id="S2.SS0.SSS0.P0.SPx1.p1.4.m4.1.2.2.2" xref="S2.SS0.SSS0.P0.SPx1.p1.4.m4.1.2.2.2.cmml">ρ</mi><mi id="S2.SS0.SSS0.P0.SPx1.p1.4.m4.1.2.2.3" xref="S2.SS0.SSS0.P0.SPx1.p1.4.m4.1.2.2.3.cmml">max</mi></msup><mo id="S2.SS0.SSS0.P0.SPx1.p1.4.m4.1.2.1" xref="S2.SS0.SSS0.P0.SPx1.p1.4.m4.1.2.1.cmml">⁢</mo><mrow id="S2.SS0.SSS0.P0.SPx1.p1.4.m4.1.2.3.2" xref="S2.SS0.SSS0.P0.SPx1.p1.4.m4.1.2.cmml"><mo id="S2.SS0.SSS0.P0.SPx1.p1.4.m4.1.2.3.2.1" stretchy="false" xref="S2.SS0.SSS0.P0.SPx1.p1.4.m4.1.2.cmml">(</mo><mi id="S2.SS0.SSS0.P0.SPx1.p1.4.m4.1.1" xref="S2.SS0.SSS0.P0.SPx1.p1.4.m4.1.1.cmml">G</mi><mo id="S2.SS0.SSS0.P0.SPx1.p1.4.m4.1.2.3.2.2" stretchy="false" xref="S2.SS0.SSS0.P0.SPx1.p1.4.m4.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.P0.SPx1.p1.4.m4.1b"><apply id="S2.SS0.SSS0.P0.SPx1.p1.4.m4.1.2.cmml" xref="S2.SS0.SSS0.P0.SPx1.p1.4.m4.1.2"><times id="S2.SS0.SSS0.P0.SPx1.p1.4.m4.1.2.1.cmml" xref="S2.SS0.SSS0.P0.SPx1.p1.4.m4.1.2.1"></times><apply id="S2.SS0.SSS0.P0.SPx1.p1.4.m4.1.2.2.cmml" xref="S2.SS0.SSS0.P0.SPx1.p1.4.m4.1.2.2"><csymbol cd="ambiguous" id="S2.SS0.SSS0.P0.SPx1.p1.4.m4.1.2.2.1.cmml" xref="S2.SS0.SSS0.P0.SPx1.p1.4.m4.1.2.2">superscript</csymbol><ci id="S2.SS0.SSS0.P0.SPx1.p1.4.m4.1.2.2.2.cmml" xref="S2.SS0.SSS0.P0.SPx1.p1.4.m4.1.2.2.2">𝜌</ci><max id="S2.SS0.SSS0.P0.SPx1.p1.4.m4.1.2.2.3.cmml" xref="S2.SS0.SSS0.P0.SPx1.p1.4.m4.1.2.2.3"></max></apply><ci id="S2.SS0.SSS0.P0.SPx1.p1.4.m4.1.1.cmml" xref="S2.SS0.SSS0.P0.SPx1.p1.4.m4.1.1">𝐺</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.P0.SPx1.p1.4.m4.1c">\rho^{\max}(G)</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.P0.SPx1.p1.4.m4.1d">italic_ρ start_POSTSUPERSCRIPT roman_max end_POSTSUPERSCRIPT ( italic_G )</annotation></semantics></math> is then the maximum over all <math alttext="H\subseteq G" class="ltx_Math" display="inline" id="S2.SS0.SSS0.P0.SPx1.p1.5.m5.1"><semantics id="S2.SS0.SSS0.P0.SPx1.p1.5.m5.1a"><mrow id="S2.SS0.SSS0.P0.SPx1.p1.5.m5.1.1" xref="S2.SS0.SSS0.P0.SPx1.p1.5.m5.1.1.cmml"><mi id="S2.SS0.SSS0.P0.SPx1.p1.5.m5.1.1.2" xref="S2.SS0.SSS0.P0.SPx1.p1.5.m5.1.1.2.cmml">H</mi><mo id="S2.SS0.SSS0.P0.SPx1.p1.5.m5.1.1.1" xref="S2.SS0.SSS0.P0.SPx1.p1.5.m5.1.1.1.cmml">⊆</mo><mi id="S2.SS0.SSS0.P0.SPx1.p1.5.m5.1.1.3" xref="S2.SS0.SSS0.P0.SPx1.p1.5.m5.1.1.3.cmml">G</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.P0.SPx1.p1.5.m5.1b"><apply id="S2.SS0.SSS0.P0.SPx1.p1.5.m5.1.1.cmml" xref="S2.SS0.SSS0.P0.SPx1.p1.5.m5.1.1"><subset id="S2.SS0.SSS0.P0.SPx1.p1.5.m5.1.1.1.cmml" xref="S2.SS0.SSS0.P0.SPx1.p1.5.m5.1.1.1"></subset><ci id="S2.SS0.SSS0.P0.SPx1.p1.5.m5.1.1.2.cmml" xref="S2.SS0.SSS0.P0.SPx1.p1.5.m5.1.1.2">𝐻</ci><ci id="S2.SS0.SSS0.P0.SPx1.p1.5.m5.1.1.3.cmml" xref="S2.SS0.SSS0.P0.SPx1.p1.5.m5.1.1.3">𝐺</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.P0.SPx1.p1.5.m5.1c">H\subseteq G</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.P0.SPx1.p1.5.m5.1d">italic_H ⊆ italic_G</annotation></semantics></math> of <math alttext="\rho(H)" class="ltx_Math" display="inline" id="S2.SS0.SSS0.P0.SPx1.p1.6.m6.1"><semantics id="S2.SS0.SSS0.P0.SPx1.p1.6.m6.1a"><mrow id="S2.SS0.SSS0.P0.SPx1.p1.6.m6.1.2" xref="S2.SS0.SSS0.P0.SPx1.p1.6.m6.1.2.cmml"><mi id="S2.SS0.SSS0.P0.SPx1.p1.6.m6.1.2.2" xref="S2.SS0.SSS0.P0.SPx1.p1.6.m6.1.2.2.cmml">ρ</mi><mo id="S2.SS0.SSS0.P0.SPx1.p1.6.m6.1.2.1" xref="S2.SS0.SSS0.P0.SPx1.p1.6.m6.1.2.1.cmml">⁢</mo><mrow id="S2.SS0.SSS0.P0.SPx1.p1.6.m6.1.2.3.2" xref="S2.SS0.SSS0.P0.SPx1.p1.6.m6.1.2.cmml"><mo id="S2.SS0.SSS0.P0.SPx1.p1.6.m6.1.2.3.2.1" stretchy="false" xref="S2.SS0.SSS0.P0.SPx1.p1.6.m6.1.2.cmml">(</mo><mi id="S2.SS0.SSS0.P0.SPx1.p1.6.m6.1.1" xref="S2.SS0.SSS0.P0.SPx1.p1.6.m6.1.1.cmml">H</mi><mo id="S2.SS0.SSS0.P0.SPx1.p1.6.m6.1.2.3.2.2" stretchy="false" xref="S2.SS0.SSS0.P0.SPx1.p1.6.m6.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.P0.SPx1.p1.6.m6.1b"><apply id="S2.SS0.SSS0.P0.SPx1.p1.6.m6.1.2.cmml" xref="S2.SS0.SSS0.P0.SPx1.p1.6.m6.1.2"><times id="S2.SS0.SSS0.P0.SPx1.p1.6.m6.1.2.1.cmml" xref="S2.SS0.SSS0.P0.SPx1.p1.6.m6.1.2.1"></times><ci id="S2.SS0.SSS0.P0.SPx1.p1.6.m6.1.2.2.cmml" xref="S2.SS0.SSS0.P0.SPx1.p1.6.m6.1.2.2">𝜌</ci><ci id="S2.SS0.SSS0.P0.SPx1.p1.6.m6.1.1.cmml" xref="S2.SS0.SSS0.P0.SPx1.p1.6.m6.1.1">𝐻</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.P0.SPx1.p1.6.m6.1c">\rho(H)</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.P0.SPx1.p1.6.m6.1d">italic_ρ ( italic_H )</annotation></semantics></math>. A subgraph <math alttext="H\subseteq G" class="ltx_Math" display="inline" id="S2.SS0.SSS0.P0.SPx1.p1.7.m7.1"><semantics id="S2.SS0.SSS0.P0.SPx1.p1.7.m7.1a"><mrow id="S2.SS0.SSS0.P0.SPx1.p1.7.m7.1.1" xref="S2.SS0.SSS0.P0.SPx1.p1.7.m7.1.1.cmml"><mi id="S2.SS0.SSS0.P0.SPx1.p1.7.m7.1.1.2" xref="S2.SS0.SSS0.P0.SPx1.p1.7.m7.1.1.2.cmml">H</mi><mo id="S2.SS0.SSS0.P0.SPx1.p1.7.m7.1.1.1" xref="S2.SS0.SSS0.P0.SPx1.p1.7.m7.1.1.1.cmml">⊆</mo><mi id="S2.SS0.SSS0.P0.SPx1.p1.7.m7.1.1.3" xref="S2.SS0.SSS0.P0.SPx1.p1.7.m7.1.1.3.cmml">G</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.P0.SPx1.p1.7.m7.1b"><apply id="S2.SS0.SSS0.P0.SPx1.p1.7.m7.1.1.cmml" xref="S2.SS0.SSS0.P0.SPx1.p1.7.m7.1.1"><subset id="S2.SS0.SSS0.P0.SPx1.p1.7.m7.1.1.1.cmml" xref="S2.SS0.SSS0.P0.SPx1.p1.7.m7.1.1.1"></subset><ci id="S2.SS0.SSS0.P0.SPx1.p1.7.m7.1.1.2.cmml" xref="S2.SS0.SSS0.P0.SPx1.p1.7.m7.1.1.2">𝐻</ci><ci id="S2.SS0.SSS0.P0.SPx1.p1.7.m7.1.1.3.cmml" xref="S2.SS0.SSS0.P0.SPx1.p1.7.m7.1.1.3">𝐺</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.P0.SPx1.p1.7.m7.1c">H\subseteq G</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.P0.SPx1.p1.7.m7.1d">italic_H ⊆ italic_G</annotation></semantics></math> is <em class="ltx_emph ltx_font_italic" id="S2.SS0.SSS0.P0.SPx1.p1.19.3">densest</em> whenever <math alttext="\rho(H)=\rho^{\max}(G)" class="ltx_Math" display="inline" id="S2.SS0.SSS0.P0.SPx1.p1.8.m8.2"><semantics id="S2.SS0.SSS0.P0.SPx1.p1.8.m8.2a"><mrow id="S2.SS0.SSS0.P0.SPx1.p1.8.m8.2.3" xref="S2.SS0.SSS0.P0.SPx1.p1.8.m8.2.3.cmml"><mrow id="S2.SS0.SSS0.P0.SPx1.p1.8.m8.2.3.2" xref="S2.SS0.SSS0.P0.SPx1.p1.8.m8.2.3.2.cmml"><mi id="S2.SS0.SSS0.P0.SPx1.p1.8.m8.2.3.2.2" xref="S2.SS0.SSS0.P0.SPx1.p1.8.m8.2.3.2.2.cmml">ρ</mi><mo id="S2.SS0.SSS0.P0.SPx1.p1.8.m8.2.3.2.1" xref="S2.SS0.SSS0.P0.SPx1.p1.8.m8.2.3.2.1.cmml">⁢</mo><mrow id="S2.SS0.SSS0.P0.SPx1.p1.8.m8.2.3.2.3.2" xref="S2.SS0.SSS0.P0.SPx1.p1.8.m8.2.3.2.cmml"><mo id="S2.SS0.SSS0.P0.SPx1.p1.8.m8.2.3.2.3.2.1" stretchy="false" xref="S2.SS0.SSS0.P0.SPx1.p1.8.m8.2.3.2.cmml">(</mo><mi id="S2.SS0.SSS0.P0.SPx1.p1.8.m8.1.1" xref="S2.SS0.SSS0.P0.SPx1.p1.8.m8.1.1.cmml">H</mi><mo id="S2.SS0.SSS0.P0.SPx1.p1.8.m8.2.3.2.3.2.2" stretchy="false" xref="S2.SS0.SSS0.P0.SPx1.p1.8.m8.2.3.2.cmml">)</mo></mrow></mrow><mo id="S2.SS0.SSS0.P0.SPx1.p1.8.m8.2.3.1" xref="S2.SS0.SSS0.P0.SPx1.p1.8.m8.2.3.1.cmml">=</mo><mrow id="S2.SS0.SSS0.P0.SPx1.p1.8.m8.2.3.3" xref="S2.SS0.SSS0.P0.SPx1.p1.8.m8.2.3.3.cmml"><msup id="S2.SS0.SSS0.P0.SPx1.p1.8.m8.2.3.3.2" xref="S2.SS0.SSS0.P0.SPx1.p1.8.m8.2.3.3.2.cmml"><mi id="S2.SS0.SSS0.P0.SPx1.p1.8.m8.2.3.3.2.2" xref="S2.SS0.SSS0.P0.SPx1.p1.8.m8.2.3.3.2.2.cmml">ρ</mi><mi id="S2.SS0.SSS0.P0.SPx1.p1.8.m8.2.3.3.2.3" xref="S2.SS0.SSS0.P0.SPx1.p1.8.m8.2.3.3.2.3.cmml">max</mi></msup><mo id="S2.SS0.SSS0.P0.SPx1.p1.8.m8.2.3.3.1" xref="S2.SS0.SSS0.P0.SPx1.p1.8.m8.2.3.3.1.cmml">⁢</mo><mrow id="S2.SS0.SSS0.P0.SPx1.p1.8.m8.2.3.3.3.2" xref="S2.SS0.SSS0.P0.SPx1.p1.8.m8.2.3.3.cmml"><mo id="S2.SS0.SSS0.P0.SPx1.p1.8.m8.2.3.3.3.2.1" stretchy="false" xref="S2.SS0.SSS0.P0.SPx1.p1.8.m8.2.3.3.cmml">(</mo><mi id="S2.SS0.SSS0.P0.SPx1.p1.8.m8.2.2" xref="S2.SS0.SSS0.P0.SPx1.p1.8.m8.2.2.cmml">G</mi><mo id="S2.SS0.SSS0.P0.SPx1.p1.8.m8.2.3.3.3.2.2" stretchy="false" xref="S2.SS0.SSS0.P0.SPx1.p1.8.m8.2.3.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.P0.SPx1.p1.8.m8.2b"><apply id="S2.SS0.SSS0.P0.SPx1.p1.8.m8.2.3.cmml" xref="S2.SS0.SSS0.P0.SPx1.p1.8.m8.2.3"><eq id="S2.SS0.SSS0.P0.SPx1.p1.8.m8.2.3.1.cmml" xref="S2.SS0.SSS0.P0.SPx1.p1.8.m8.2.3.1"></eq><apply id="S2.SS0.SSS0.P0.SPx1.p1.8.m8.2.3.2.cmml" xref="S2.SS0.SSS0.P0.SPx1.p1.8.m8.2.3.2"><times id="S2.SS0.SSS0.P0.SPx1.p1.8.m8.2.3.2.1.cmml" xref="S2.SS0.SSS0.P0.SPx1.p1.8.m8.2.3.2.1"></times><ci id="S2.SS0.SSS0.P0.SPx1.p1.8.m8.2.3.2.2.cmml" xref="S2.SS0.SSS0.P0.SPx1.p1.8.m8.2.3.2.2">𝜌</ci><ci id="S2.SS0.SSS0.P0.SPx1.p1.8.m8.1.1.cmml" xref="S2.SS0.SSS0.P0.SPx1.p1.8.m8.1.1">𝐻</ci></apply><apply id="S2.SS0.SSS0.P0.SPx1.p1.8.m8.2.3.3.cmml" xref="S2.SS0.SSS0.P0.SPx1.p1.8.m8.2.3.3"><times id="S2.SS0.SSS0.P0.SPx1.p1.8.m8.2.3.3.1.cmml" xref="S2.SS0.SSS0.P0.SPx1.p1.8.m8.2.3.3.1"></times><apply id="S2.SS0.SSS0.P0.SPx1.p1.8.m8.2.3.3.2.cmml" xref="S2.SS0.SSS0.P0.SPx1.p1.8.m8.2.3.3.2"><csymbol cd="ambiguous" id="S2.SS0.SSS0.P0.SPx1.p1.8.m8.2.3.3.2.1.cmml" xref="S2.SS0.SSS0.P0.SPx1.p1.8.m8.2.3.3.2">superscript</csymbol><ci id="S2.SS0.SSS0.P0.SPx1.p1.8.m8.2.3.3.2.2.cmml" xref="S2.SS0.SSS0.P0.SPx1.p1.8.m8.2.3.3.2.2">𝜌</ci><max id="S2.SS0.SSS0.P0.SPx1.p1.8.m8.2.3.3.2.3.cmml" xref="S2.SS0.SSS0.P0.SPx1.p1.8.m8.2.3.3.2.3"></max></apply><ci id="S2.SS0.SSS0.P0.SPx1.p1.8.m8.2.2.cmml" xref="S2.SS0.SSS0.P0.SPx1.p1.8.m8.2.2">𝐺</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.P0.SPx1.p1.8.m8.2c">\rho(H)=\rho^{\max}(G)</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.P0.SPx1.p1.8.m8.2d">italic_ρ ( italic_H ) = italic_ρ start_POSTSUPERSCRIPT roman_max end_POSTSUPERSCRIPT ( italic_G )</annotation></semantics></math>. For any orientation <math alttext="\overrightarrow{G}" class="ltx_Math" display="inline" id="S2.SS0.SSS0.P0.SPx1.p1.9.m9.1"><semantics id="S2.SS0.SSS0.P0.SPx1.p1.9.m9.1a"><mover accent="true" id="S2.SS0.SSS0.P0.SPx1.p1.9.m9.1.1" xref="S2.SS0.SSS0.P0.SPx1.p1.9.m9.1.1.cmml"><mi id="S2.SS0.SSS0.P0.SPx1.p1.9.m9.1.1.2" xref="S2.SS0.SSS0.P0.SPx1.p1.9.m9.1.1.2.cmml">G</mi><mo id="S2.SS0.SSS0.P0.SPx1.p1.9.m9.1.1.1" stretchy="false" xref="S2.SS0.SSS0.P0.SPx1.p1.9.m9.1.1.1.cmml">→</mo></mover><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.P0.SPx1.p1.9.m9.1b"><apply id="S2.SS0.SSS0.P0.SPx1.p1.9.m9.1.1.cmml" xref="S2.SS0.SSS0.P0.SPx1.p1.9.m9.1.1"><ci id="S2.SS0.SSS0.P0.SPx1.p1.9.m9.1.1.1.cmml" xref="S2.SS0.SSS0.P0.SPx1.p1.9.m9.1.1.1">→</ci><ci id="S2.SS0.SSS0.P0.SPx1.p1.9.m9.1.1.2.cmml" xref="S2.SS0.SSS0.P0.SPx1.p1.9.m9.1.1.2">𝐺</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.P0.SPx1.p1.9.m9.1c">\overrightarrow{G}</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.P0.SPx1.p1.9.m9.1d">over→ start_ARG italic_G end_ARG</annotation></semantics></math> of <math alttext="G" class="ltx_Math" display="inline" id="S2.SS0.SSS0.P0.SPx1.p1.10.m10.1"><semantics id="S2.SS0.SSS0.P0.SPx1.p1.10.m10.1a"><mi id="S2.SS0.SSS0.P0.SPx1.p1.10.m10.1.1" xref="S2.SS0.SSS0.P0.SPx1.p1.10.m10.1.1.cmml">G</mi><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.P0.SPx1.p1.10.m10.1b"><ci id="S2.SS0.SSS0.P0.SPx1.p1.10.m10.1.1.cmml" xref="S2.SS0.SSS0.P0.SPx1.p1.10.m10.1.1">𝐺</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.P0.SPx1.p1.10.m10.1c">G</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.P0.SPx1.p1.10.m10.1d">italic_G</annotation></semantics></math>, its <em class="ltx_emph ltx_font_italic" id="S2.SS0.SSS0.P0.SPx1.p1.19.4">maximum out-degree</em> <math alttext="\Delta(\overrightarrow{G})" class="ltx_Math" display="inline" id="S2.SS0.SSS0.P0.SPx1.p1.11.m11.1"><semantics id="S2.SS0.SSS0.P0.SPx1.p1.11.m11.1a"><mrow id="S2.SS0.SSS0.P0.SPx1.p1.11.m11.1.2" xref="S2.SS0.SSS0.P0.SPx1.p1.11.m11.1.2.cmml"><mi id="S2.SS0.SSS0.P0.SPx1.p1.11.m11.1.2.2" mathvariant="normal" xref="S2.SS0.SSS0.P0.SPx1.p1.11.m11.1.2.2.cmml">Δ</mi><mo id="S2.SS0.SSS0.P0.SPx1.p1.11.m11.1.2.1" xref="S2.SS0.SSS0.P0.SPx1.p1.11.m11.1.2.1.cmml">⁢</mo><mrow id="S2.SS0.SSS0.P0.SPx1.p1.11.m11.1.2.3.2" xref="S2.SS0.SSS0.P0.SPx1.p1.11.m11.1.1.cmml"><mo id="S2.SS0.SSS0.P0.SPx1.p1.11.m11.1.2.3.2.1" stretchy="false" xref="S2.SS0.SSS0.P0.SPx1.p1.11.m11.1.1.cmml">(</mo><mover accent="true" id="S2.SS0.SSS0.P0.SPx1.p1.11.m11.1.1" xref="S2.SS0.SSS0.P0.SPx1.p1.11.m11.1.1.cmml"><mi id="S2.SS0.SSS0.P0.SPx1.p1.11.m11.1.1.2" xref="S2.SS0.SSS0.P0.SPx1.p1.11.m11.1.1.2.cmml">G</mi><mo id="S2.SS0.SSS0.P0.SPx1.p1.11.m11.1.1.1" stretchy="false" xref="S2.SS0.SSS0.P0.SPx1.p1.11.m11.1.1.1.cmml">→</mo></mover><mo id="S2.SS0.SSS0.P0.SPx1.p1.11.m11.1.2.3.2.2" stretchy="false" xref="S2.SS0.SSS0.P0.SPx1.p1.11.m11.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.P0.SPx1.p1.11.m11.1b"><apply id="S2.SS0.SSS0.P0.SPx1.p1.11.m11.1.2.cmml" xref="S2.SS0.SSS0.P0.SPx1.p1.11.m11.1.2"><times id="S2.SS0.SSS0.P0.SPx1.p1.11.m11.1.2.1.cmml" xref="S2.SS0.SSS0.P0.SPx1.p1.11.m11.1.2.1"></times><ci id="S2.SS0.SSS0.P0.SPx1.p1.11.m11.1.2.2.cmml" xref="S2.SS0.SSS0.P0.SPx1.p1.11.m11.1.2.2">Δ</ci><apply id="S2.SS0.SSS0.P0.SPx1.p1.11.m11.1.1.cmml" xref="S2.SS0.SSS0.P0.SPx1.p1.11.m11.1.2.3.2"><ci id="S2.SS0.SSS0.P0.SPx1.p1.11.m11.1.1.1.cmml" xref="S2.SS0.SSS0.P0.SPx1.p1.11.m11.1.1.1">→</ci><ci id="S2.SS0.SSS0.P0.SPx1.p1.11.m11.1.1.2.cmml" xref="S2.SS0.SSS0.P0.SPx1.p1.11.m11.1.1.2">𝐺</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.P0.SPx1.p1.11.m11.1c">\Delta(\overrightarrow{G})</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.P0.SPx1.p1.11.m11.1d">roman_Δ ( over→ start_ARG italic_G end_ARG )</annotation></semantics></math> is the maximum over all <math alttext="u" class="ltx_Math" display="inline" id="S2.SS0.SSS0.P0.SPx1.p1.12.m12.1"><semantics id="S2.SS0.SSS0.P0.SPx1.p1.12.m12.1a"><mi id="S2.SS0.SSS0.P0.SPx1.p1.12.m12.1.1" xref="S2.SS0.SSS0.P0.SPx1.p1.12.m12.1.1.cmml">u</mi><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.P0.SPx1.p1.12.m12.1b"><ci id="S2.SS0.SSS0.P0.SPx1.p1.12.m12.1.1.cmml" xref="S2.SS0.SSS0.P0.SPx1.p1.12.m12.1.1">𝑢</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.P0.SPx1.p1.12.m12.1c">u</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.P0.SPx1.p1.12.m12.1d">italic_u</annotation></semantics></math> of the out-degree <math alttext="\textsl{g}(u)" class="ltx_Math" display="inline" id="S2.SS0.SSS0.P0.SPx1.p1.13.m13.1"><semantics id="S2.SS0.SSS0.P0.SPx1.p1.13.m13.1a"><mrow id="S2.SS0.SSS0.P0.SPx1.p1.13.m13.1.2" xref="S2.SS0.SSS0.P0.SPx1.p1.13.m13.1.2.cmml"><mtext class="ltx_mathvariant_italic" id="S2.SS0.SSS0.P0.SPx1.p1.13.m13.1.2.2" xref="S2.SS0.SSS0.P0.SPx1.p1.13.m13.1.2.2a.cmml">g</mtext><mo id="S2.SS0.SSS0.P0.SPx1.p1.13.m13.1.2.1" xref="S2.SS0.SSS0.P0.SPx1.p1.13.m13.1.2.1.cmml">⁢</mo><mrow id="S2.SS0.SSS0.P0.SPx1.p1.13.m13.1.2.3.2" xref="S2.SS0.SSS0.P0.SPx1.p1.13.m13.1.2.cmml"><mo id="S2.SS0.SSS0.P0.SPx1.p1.13.m13.1.2.3.2.1" stretchy="false" xref="S2.SS0.SSS0.P0.SPx1.p1.13.m13.1.2.cmml">(</mo><mi id="S2.SS0.SSS0.P0.SPx1.p1.13.m13.1.1" xref="S2.SS0.SSS0.P0.SPx1.p1.13.m13.1.1.cmml">u</mi><mo id="S2.SS0.SSS0.P0.SPx1.p1.13.m13.1.2.3.2.2" stretchy="false" xref="S2.SS0.SSS0.P0.SPx1.p1.13.m13.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.P0.SPx1.p1.13.m13.1b"><apply id="S2.SS0.SSS0.P0.SPx1.p1.13.m13.1.2.cmml" xref="S2.SS0.SSS0.P0.SPx1.p1.13.m13.1.2"><times id="S2.SS0.SSS0.P0.SPx1.p1.13.m13.1.2.1.cmml" xref="S2.SS0.SSS0.P0.SPx1.p1.13.m13.1.2.1"></times><ci id="S2.SS0.SSS0.P0.SPx1.p1.13.m13.1.2.2a.cmml" xref="S2.SS0.SSS0.P0.SPx1.p1.13.m13.1.2.2"><mtext class="ltx_mathvariant_italic" id="S2.SS0.SSS0.P0.SPx1.p1.13.m13.1.2.2.cmml" xref="S2.SS0.SSS0.P0.SPx1.p1.13.m13.1.2.2">g</mtext></ci><ci id="S2.SS0.SSS0.P0.SPx1.p1.13.m13.1.1.cmml" xref="S2.SS0.SSS0.P0.SPx1.p1.13.m13.1.1">𝑢</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.P0.SPx1.p1.13.m13.1c">\textsl{g}(u)</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.P0.SPx1.p1.13.m13.1d">g ( italic_u )</annotation></semantics></math>. The <em class="ltx_emph ltx_font_italic" id="S2.SS0.SSS0.P0.SPx1.p1.19.5">optimal out-degree</em> of <math alttext="G" class="ltx_Math" display="inline" id="S2.SS0.SSS0.P0.SPx1.p1.14.m14.1"><semantics id="S2.SS0.SSS0.P0.SPx1.p1.14.m14.1a"><mi id="S2.SS0.SSS0.P0.SPx1.p1.14.m14.1.1" xref="S2.SS0.SSS0.P0.SPx1.p1.14.m14.1.1.cmml">G</mi><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.P0.SPx1.p1.14.m14.1b"><ci id="S2.SS0.SSS0.P0.SPx1.p1.14.m14.1.1.cmml" xref="S2.SS0.SSS0.P0.SPx1.p1.14.m14.1.1">𝐺</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.P0.SPx1.p1.14.m14.1c">G</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.P0.SPx1.p1.14.m14.1d">italic_G</annotation></semantics></math>, denoted by <math alttext="\Delta^{\min}(G)" class="ltx_Math" display="inline" id="S2.SS0.SSS0.P0.SPx1.p1.15.m15.1"><semantics id="S2.SS0.SSS0.P0.SPx1.p1.15.m15.1a"><mrow id="S2.SS0.SSS0.P0.SPx1.p1.15.m15.1.2" xref="S2.SS0.SSS0.P0.SPx1.p1.15.m15.1.2.cmml"><msup id="S2.SS0.SSS0.P0.SPx1.p1.15.m15.1.2.2" xref="S2.SS0.SSS0.P0.SPx1.p1.15.m15.1.2.2.cmml"><mi id="S2.SS0.SSS0.P0.SPx1.p1.15.m15.1.2.2.2" mathvariant="normal" xref="S2.SS0.SSS0.P0.SPx1.p1.15.m15.1.2.2.2.cmml">Δ</mi><mi id="S2.SS0.SSS0.P0.SPx1.p1.15.m15.1.2.2.3" xref="S2.SS0.SSS0.P0.SPx1.p1.15.m15.1.2.2.3.cmml">min</mi></msup><mo id="S2.SS0.SSS0.P0.SPx1.p1.15.m15.1.2.1" xref="S2.SS0.SSS0.P0.SPx1.p1.15.m15.1.2.1.cmml">⁢</mo><mrow id="S2.SS0.SSS0.P0.SPx1.p1.15.m15.1.2.3.2" xref="S2.SS0.SSS0.P0.SPx1.p1.15.m15.1.2.cmml"><mo id="S2.SS0.SSS0.P0.SPx1.p1.15.m15.1.2.3.2.1" stretchy="false" xref="S2.SS0.SSS0.P0.SPx1.p1.15.m15.1.2.cmml">(</mo><mi id="S2.SS0.SSS0.P0.SPx1.p1.15.m15.1.1" xref="S2.SS0.SSS0.P0.SPx1.p1.15.m15.1.1.cmml">G</mi><mo id="S2.SS0.SSS0.P0.SPx1.p1.15.m15.1.2.3.2.2" stretchy="false" xref="S2.SS0.SSS0.P0.SPx1.p1.15.m15.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.P0.SPx1.p1.15.m15.1b"><apply id="S2.SS0.SSS0.P0.SPx1.p1.15.m15.1.2.cmml" xref="S2.SS0.SSS0.P0.SPx1.p1.15.m15.1.2"><times id="S2.SS0.SSS0.P0.SPx1.p1.15.m15.1.2.1.cmml" xref="S2.SS0.SSS0.P0.SPx1.p1.15.m15.1.2.1"></times><apply id="S2.SS0.SSS0.P0.SPx1.p1.15.m15.1.2.2.cmml" xref="S2.SS0.SSS0.P0.SPx1.p1.15.m15.1.2.2"><csymbol cd="ambiguous" id="S2.SS0.SSS0.P0.SPx1.p1.15.m15.1.2.2.1.cmml" xref="S2.SS0.SSS0.P0.SPx1.p1.15.m15.1.2.2">superscript</csymbol><ci id="S2.SS0.SSS0.P0.SPx1.p1.15.m15.1.2.2.2.cmml" xref="S2.SS0.SSS0.P0.SPx1.p1.15.m15.1.2.2.2">Δ</ci><min id="S2.SS0.SSS0.P0.SPx1.p1.15.m15.1.2.2.3.cmml" xref="S2.SS0.SSS0.P0.SPx1.p1.15.m15.1.2.2.3"></min></apply><ci id="S2.SS0.SSS0.P0.SPx1.p1.15.m15.1.1.cmml" xref="S2.SS0.SSS0.P0.SPx1.p1.15.m15.1.1">𝐺</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.P0.SPx1.p1.15.m15.1c">\Delta^{\min}(G)</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.P0.SPx1.p1.15.m15.1d">roman_Δ start_POSTSUPERSCRIPT roman_min end_POSTSUPERSCRIPT ( italic_G )</annotation></semantics></math>, is subsequently the minimum over all <math alttext="\overrightarrow{G}" class="ltx_Math" display="inline" id="S2.SS0.SSS0.P0.SPx1.p1.16.m16.1"><semantics id="S2.SS0.SSS0.P0.SPx1.p1.16.m16.1a"><mover accent="true" id="S2.SS0.SSS0.P0.SPx1.p1.16.m16.1.1" xref="S2.SS0.SSS0.P0.SPx1.p1.16.m16.1.1.cmml"><mi id="S2.SS0.SSS0.P0.SPx1.p1.16.m16.1.1.2" xref="S2.SS0.SSS0.P0.SPx1.p1.16.m16.1.1.2.cmml">G</mi><mo id="S2.SS0.SSS0.P0.SPx1.p1.16.m16.1.1.1" stretchy="false" xref="S2.SS0.SSS0.P0.SPx1.p1.16.m16.1.1.1.cmml">→</mo></mover><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.P0.SPx1.p1.16.m16.1b"><apply id="S2.SS0.SSS0.P0.SPx1.p1.16.m16.1.1.cmml" xref="S2.SS0.SSS0.P0.SPx1.p1.16.m16.1.1"><ci id="S2.SS0.SSS0.P0.SPx1.p1.16.m16.1.1.1.cmml" xref="S2.SS0.SSS0.P0.SPx1.p1.16.m16.1.1.1">→</ci><ci id="S2.SS0.SSS0.P0.SPx1.p1.16.m16.1.1.2.cmml" xref="S2.SS0.SSS0.P0.SPx1.p1.16.m16.1.1.2">𝐺</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.P0.SPx1.p1.16.m16.1c">\overrightarrow{G}</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.P0.SPx1.p1.16.m16.1d">over→ start_ARG italic_G end_ARG</annotation></semantics></math> of <math alttext="\Delta(\overrightarrow{G})" class="ltx_Math" display="inline" id="S2.SS0.SSS0.P0.SPx1.p1.17.m17.1"><semantics id="S2.SS0.SSS0.P0.SPx1.p1.17.m17.1a"><mrow id="S2.SS0.SSS0.P0.SPx1.p1.17.m17.1.2" xref="S2.SS0.SSS0.P0.SPx1.p1.17.m17.1.2.cmml"><mi id="S2.SS0.SSS0.P0.SPx1.p1.17.m17.1.2.2" mathvariant="normal" xref="S2.SS0.SSS0.P0.SPx1.p1.17.m17.1.2.2.cmml">Δ</mi><mo id="S2.SS0.SSS0.P0.SPx1.p1.17.m17.1.2.1" xref="S2.SS0.SSS0.P0.SPx1.p1.17.m17.1.2.1.cmml">⁢</mo><mrow id="S2.SS0.SSS0.P0.SPx1.p1.17.m17.1.2.3.2" xref="S2.SS0.SSS0.P0.SPx1.p1.17.m17.1.1.cmml"><mo id="S2.SS0.SSS0.P0.SPx1.p1.17.m17.1.2.3.2.1" stretchy="false" xref="S2.SS0.SSS0.P0.SPx1.p1.17.m17.1.1.cmml">(</mo><mover accent="true" id="S2.SS0.SSS0.P0.SPx1.p1.17.m17.1.1" xref="S2.SS0.SSS0.P0.SPx1.p1.17.m17.1.1.cmml"><mi id="S2.SS0.SSS0.P0.SPx1.p1.17.m17.1.1.2" xref="S2.SS0.SSS0.P0.SPx1.p1.17.m17.1.1.2.cmml">G</mi><mo id="S2.SS0.SSS0.P0.SPx1.p1.17.m17.1.1.1" stretchy="false" xref="S2.SS0.SSS0.P0.SPx1.p1.17.m17.1.1.1.cmml">→</mo></mover><mo id="S2.SS0.SSS0.P0.SPx1.p1.17.m17.1.2.3.2.2" stretchy="false" xref="S2.SS0.SSS0.P0.SPx1.p1.17.m17.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.P0.SPx1.p1.17.m17.1b"><apply id="S2.SS0.SSS0.P0.SPx1.p1.17.m17.1.2.cmml" xref="S2.SS0.SSS0.P0.SPx1.p1.17.m17.1.2"><times id="S2.SS0.SSS0.P0.SPx1.p1.17.m17.1.2.1.cmml" xref="S2.SS0.SSS0.P0.SPx1.p1.17.m17.1.2.1"></times><ci id="S2.SS0.SSS0.P0.SPx1.p1.17.m17.1.2.2.cmml" xref="S2.SS0.SSS0.P0.SPx1.p1.17.m17.1.2.2">Δ</ci><apply id="S2.SS0.SSS0.P0.SPx1.p1.17.m17.1.1.cmml" xref="S2.SS0.SSS0.P0.SPx1.p1.17.m17.1.2.3.2"><ci id="S2.SS0.SSS0.P0.SPx1.p1.17.m17.1.1.1.cmml" xref="S2.SS0.SSS0.P0.SPx1.p1.17.m17.1.1.1">→</ci><ci id="S2.SS0.SSS0.P0.SPx1.p1.17.m17.1.1.2.cmml" xref="S2.SS0.SSS0.P0.SPx1.p1.17.m17.1.1.2">𝐺</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.P0.SPx1.p1.17.m17.1c">\Delta(\overrightarrow{G})</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.P0.SPx1.p1.17.m17.1d">roman_Δ ( over→ start_ARG italic_G end_ARG )</annotation></semantics></math>. An orientation <math alttext="\overrightarrow{G}" class="ltx_Math" display="inline" id="S2.SS0.SSS0.P0.SPx1.p1.18.m18.1"><semantics id="S2.SS0.SSS0.P0.SPx1.p1.18.m18.1a"><mover accent="true" id="S2.SS0.SSS0.P0.SPx1.p1.18.m18.1.1" xref="S2.SS0.SSS0.P0.SPx1.p1.18.m18.1.1.cmml"><mi id="S2.SS0.SSS0.P0.SPx1.p1.18.m18.1.1.2" xref="S2.SS0.SSS0.P0.SPx1.p1.18.m18.1.1.2.cmml">G</mi><mo id="S2.SS0.SSS0.P0.SPx1.p1.18.m18.1.1.1" stretchy="false" xref="S2.SS0.SSS0.P0.SPx1.p1.18.m18.1.1.1.cmml">→</mo></mover><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.P0.SPx1.p1.18.m18.1b"><apply id="S2.SS0.SSS0.P0.SPx1.p1.18.m18.1.1.cmml" xref="S2.SS0.SSS0.P0.SPx1.p1.18.m18.1.1"><ci id="S2.SS0.SSS0.P0.SPx1.p1.18.m18.1.1.1.cmml" xref="S2.SS0.SSS0.P0.SPx1.p1.18.m18.1.1.1">→</ci><ci id="S2.SS0.SSS0.P0.SPx1.p1.18.m18.1.1.2.cmml" xref="S2.SS0.SSS0.P0.SPx1.p1.18.m18.1.1.2">𝐺</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.P0.SPx1.p1.18.m18.1c">\overrightarrow{G}</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.P0.SPx1.p1.18.m18.1d">over→ start_ARG italic_G end_ARG</annotation></semantics></math> itself is <em class="ltx_emph ltx_font_italic" id="S2.SS0.SSS0.P0.SPx1.p1.19.6">minimum</em> whenever <math alttext="\Delta(\overrightarrow{G})=\Delta^{\min}(G)" class="ltx_Math" display="inline" id="S2.SS0.SSS0.P0.SPx1.p1.19.m19.2"><semantics id="S2.SS0.SSS0.P0.SPx1.p1.19.m19.2a"><mrow id="S2.SS0.SSS0.P0.SPx1.p1.19.m19.2.3" xref="S2.SS0.SSS0.P0.SPx1.p1.19.m19.2.3.cmml"><mrow id="S2.SS0.SSS0.P0.SPx1.p1.19.m19.2.3.2" xref="S2.SS0.SSS0.P0.SPx1.p1.19.m19.2.3.2.cmml"><mi id="S2.SS0.SSS0.P0.SPx1.p1.19.m19.2.3.2.2" mathvariant="normal" xref="S2.SS0.SSS0.P0.SPx1.p1.19.m19.2.3.2.2.cmml">Δ</mi><mo id="S2.SS0.SSS0.P0.SPx1.p1.19.m19.2.3.2.1" xref="S2.SS0.SSS0.P0.SPx1.p1.19.m19.2.3.2.1.cmml">⁢</mo><mrow id="S2.SS0.SSS0.P0.SPx1.p1.19.m19.2.3.2.3.2" xref="S2.SS0.SSS0.P0.SPx1.p1.19.m19.1.1.cmml"><mo id="S2.SS0.SSS0.P0.SPx1.p1.19.m19.2.3.2.3.2.1" stretchy="false" xref="S2.SS0.SSS0.P0.SPx1.p1.19.m19.1.1.cmml">(</mo><mover accent="true" id="S2.SS0.SSS0.P0.SPx1.p1.19.m19.1.1" xref="S2.SS0.SSS0.P0.SPx1.p1.19.m19.1.1.cmml"><mi id="S2.SS0.SSS0.P0.SPx1.p1.19.m19.1.1.2" xref="S2.SS0.SSS0.P0.SPx1.p1.19.m19.1.1.2.cmml">G</mi><mo id="S2.SS0.SSS0.P0.SPx1.p1.19.m19.1.1.1" stretchy="false" xref="S2.SS0.SSS0.P0.SPx1.p1.19.m19.1.1.1.cmml">→</mo></mover><mo id="S2.SS0.SSS0.P0.SPx1.p1.19.m19.2.3.2.3.2.2" stretchy="false" xref="S2.SS0.SSS0.P0.SPx1.p1.19.m19.1.1.cmml">)</mo></mrow></mrow><mo id="S2.SS0.SSS0.P0.SPx1.p1.19.m19.2.3.1" xref="S2.SS0.SSS0.P0.SPx1.p1.19.m19.2.3.1.cmml">=</mo><mrow id="S2.SS0.SSS0.P0.SPx1.p1.19.m19.2.3.3" xref="S2.SS0.SSS0.P0.SPx1.p1.19.m19.2.3.3.cmml"><msup id="S2.SS0.SSS0.P0.SPx1.p1.19.m19.2.3.3.2" xref="S2.SS0.SSS0.P0.SPx1.p1.19.m19.2.3.3.2.cmml"><mi id="S2.SS0.SSS0.P0.SPx1.p1.19.m19.2.3.3.2.2" mathvariant="normal" xref="S2.SS0.SSS0.P0.SPx1.p1.19.m19.2.3.3.2.2.cmml">Δ</mi><mi id="S2.SS0.SSS0.P0.SPx1.p1.19.m19.2.3.3.2.3" xref="S2.SS0.SSS0.P0.SPx1.p1.19.m19.2.3.3.2.3.cmml">min</mi></msup><mo id="S2.SS0.SSS0.P0.SPx1.p1.19.m19.2.3.3.1" xref="S2.SS0.SSS0.P0.SPx1.p1.19.m19.2.3.3.1.cmml">⁢</mo><mrow id="S2.SS0.SSS0.P0.SPx1.p1.19.m19.2.3.3.3.2" xref="S2.SS0.SSS0.P0.SPx1.p1.19.m19.2.3.3.cmml"><mo id="S2.SS0.SSS0.P0.SPx1.p1.19.m19.2.3.3.3.2.1" stretchy="false" xref="S2.SS0.SSS0.P0.SPx1.p1.19.m19.2.3.3.cmml">(</mo><mi id="S2.SS0.SSS0.P0.SPx1.p1.19.m19.2.2" xref="S2.SS0.SSS0.P0.SPx1.p1.19.m19.2.2.cmml">G</mi><mo id="S2.SS0.SSS0.P0.SPx1.p1.19.m19.2.3.3.3.2.2" stretchy="false" xref="S2.SS0.SSS0.P0.SPx1.p1.19.m19.2.3.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.P0.SPx1.p1.19.m19.2b"><apply id="S2.SS0.SSS0.P0.SPx1.p1.19.m19.2.3.cmml" xref="S2.SS0.SSS0.P0.SPx1.p1.19.m19.2.3"><eq id="S2.SS0.SSS0.P0.SPx1.p1.19.m19.2.3.1.cmml" xref="S2.SS0.SSS0.P0.SPx1.p1.19.m19.2.3.1"></eq><apply id="S2.SS0.SSS0.P0.SPx1.p1.19.m19.2.3.2.cmml" xref="S2.SS0.SSS0.P0.SPx1.p1.19.m19.2.3.2"><times id="S2.SS0.SSS0.P0.SPx1.p1.19.m19.2.3.2.1.cmml" xref="S2.SS0.SSS0.P0.SPx1.p1.19.m19.2.3.2.1"></times><ci id="S2.SS0.SSS0.P0.SPx1.p1.19.m19.2.3.2.2.cmml" xref="S2.SS0.SSS0.P0.SPx1.p1.19.m19.2.3.2.2">Δ</ci><apply id="S2.SS0.SSS0.P0.SPx1.p1.19.m19.1.1.cmml" xref="S2.SS0.SSS0.P0.SPx1.p1.19.m19.2.3.2.3.2"><ci id="S2.SS0.SSS0.P0.SPx1.p1.19.m19.1.1.1.cmml" xref="S2.SS0.SSS0.P0.SPx1.p1.19.m19.1.1.1">→</ci><ci id="S2.SS0.SSS0.P0.SPx1.p1.19.m19.1.1.2.cmml" xref="S2.SS0.SSS0.P0.SPx1.p1.19.m19.1.1.2">𝐺</ci></apply></apply><apply id="S2.SS0.SSS0.P0.SPx1.p1.19.m19.2.3.3.cmml" xref="S2.SS0.SSS0.P0.SPx1.p1.19.m19.2.3.3"><times id="S2.SS0.SSS0.P0.SPx1.p1.19.m19.2.3.3.1.cmml" xref="S2.SS0.SSS0.P0.SPx1.p1.19.m19.2.3.3.1"></times><apply id="S2.SS0.SSS0.P0.SPx1.p1.19.m19.2.3.3.2.cmml" xref="S2.SS0.SSS0.P0.SPx1.p1.19.m19.2.3.3.2"><csymbol cd="ambiguous" id="S2.SS0.SSS0.P0.SPx1.p1.19.m19.2.3.3.2.1.cmml" xref="S2.SS0.SSS0.P0.SPx1.p1.19.m19.2.3.3.2">superscript</csymbol><ci id="S2.SS0.SSS0.P0.SPx1.p1.19.m19.2.3.3.2.2.cmml" xref="S2.SS0.SSS0.P0.SPx1.p1.19.m19.2.3.3.2.2">Δ</ci><min id="S2.SS0.SSS0.P0.SPx1.p1.19.m19.2.3.3.2.3.cmml" xref="S2.SS0.SSS0.P0.SPx1.p1.19.m19.2.3.3.2.3"></min></apply><ci id="S2.SS0.SSS0.P0.SPx1.p1.19.m19.2.2.cmml" xref="S2.SS0.SSS0.P0.SPx1.p1.19.m19.2.2">𝐺</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.P0.SPx1.p1.19.m19.2c">\Delta(\overrightarrow{G})=\Delta^{\min}(G)</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.P0.SPx1.p1.19.m19.2d">roman_Δ ( over→ start_ARG italic_G end_ARG ) = roman_Δ start_POSTSUPERSCRIPT roman_min end_POSTSUPERSCRIPT ( italic_G )</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="S2.SS0.SSS0.P0.SPx1.p2"> <p class="ltx_p" id="S2.SS0.SSS0.P0.SPx1.p2.1">The density of <math alttext="G" class="ltx_Math" display="inline" id="S2.SS0.SSS0.P0.SPx1.p2.1.m1.1"><semantics id="S2.SS0.SSS0.P0.SPx1.p2.1.m1.1a"><mi id="S2.SS0.SSS0.P0.SPx1.p2.1.m1.1.1" xref="S2.SS0.SSS0.P0.SPx1.p2.1.m1.1.1.cmml">G</mi><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.P0.SPx1.p2.1.m1.1b"><ci id="S2.SS0.SSS0.P0.SPx1.p2.1.m1.1.1.cmml" xref="S2.SS0.SSS0.P0.SPx1.p2.1.m1.1.1">𝐺</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.P0.SPx1.p2.1.m1.1c">G</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.P0.SPx1.p2.1.m1.1d">italic_G</annotation></semantics></math> and the optimal out-degree are closely related. One way to illustrate this is through the following dual linear programs:</p> </div> <div class="ltx_para" id="S2.SS0.SSS0.P0.SPx1.p3"> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="A1.EGx1"> <tbody id="S2.Ex1"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_left_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><span class="ltx_text ltx_markedasmath ltx_font_bold" id="S2.Ex1.5.2.1.1">DS (Densest Subgraph)</span></td> <td class="ltx_td ltx_eqn_cell"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle\|" class="ltx_Math" display="inline" id="S2.Ex1.m2.1"><semantics id="S2.Ex1.m2.1a"><mo id="S2.Ex1.m2.1.1" xref="S2.Ex1.m2.1.1.cmml">∥</mo><annotation-xml encoding="MathML-Content" id="S2.Ex1.m2.1b"><ci id="S2.Ex1.m2.1.1.cmml" xref="S2.Ex1.m2.1.1">∥</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.Ex1.m2.1c">\displaystyle\|</annotation><annotation encoding="application/x-llamapun" id="S2.Ex1.m2.1d">∥</annotation></semantics></math></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><span class="ltx_text ltx_markedasmath ltx_font_bold" id="S2.Ex1.8.2.2.1">FO (Fractional Orientation)</span></td> <td class="ltx_eqn_cell ltx_eqn_left_padright" colspan="2"></td> </tr></tbody> <tbody id="S2.Ex2"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_left_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle\max\sum_{\overline{uv}\in E}\textsl{g}(\overline{uv})\cdot y_{u,v}" class="ltx_Math" display="inline" id="S2.Ex2.m1.3"><semantics id="S2.Ex2.m1.3a"><mrow id="S2.Ex2.m1.3.4" xref="S2.Ex2.m1.3.4.cmml"><mi id="S2.Ex2.m1.3.4.2" xref="S2.Ex2.m1.3.4.2.cmml">max</mi><mo id="S2.Ex2.m1.3.4.1" lspace="0.167em" xref="S2.Ex2.m1.3.4.1.cmml">⁢</mo><mrow id="S2.Ex2.m1.3.4.3" xref="S2.Ex2.m1.3.4.3.cmml"><mstyle displaystyle="true" id="S2.Ex2.m1.3.4.3.1" 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id="S2.Ex2.m1.3.4.3.2.2.2.cmml" xref="S2.Ex2.m1.3.4.3.2.2.2">g</mtext></ci><apply id="S2.Ex2.m1.3.3.cmml" xref="S2.Ex2.m1.3.4.3.2.2.3.2"><ci id="S2.Ex2.m1.3.3.1.cmml" xref="S2.Ex2.m1.3.3.1">¯</ci><apply id="S2.Ex2.m1.3.3.2.cmml" xref="S2.Ex2.m1.3.3.2"><times id="S2.Ex2.m1.3.3.2.1.cmml" xref="S2.Ex2.m1.3.3.2.1"></times><ci id="S2.Ex2.m1.3.3.2.2.cmml" xref="S2.Ex2.m1.3.3.2.2">𝑢</ci><ci id="S2.Ex2.m1.3.3.2.3.cmml" xref="S2.Ex2.m1.3.3.2.3">𝑣</ci></apply></apply></apply><apply id="S2.Ex2.m1.3.4.3.2.3.cmml" xref="S2.Ex2.m1.3.4.3.2.3"><csymbol cd="ambiguous" id="S2.Ex2.m1.3.4.3.2.3.1.cmml" xref="S2.Ex2.m1.3.4.3.2.3">subscript</csymbol><ci id="S2.Ex2.m1.3.4.3.2.3.2.cmml" xref="S2.Ex2.m1.3.4.3.2.3.2">𝑦</ci><list id="S2.Ex2.m1.2.2.2.3.cmml" xref="S2.Ex2.m1.2.2.2.4"><ci id="S2.Ex2.m1.1.1.1.1.cmml" xref="S2.Ex2.m1.1.1.1.1">𝑢</ci><ci id="S2.Ex2.m1.2.2.2.2.cmml" xref="S2.Ex2.m1.2.2.2.2">𝑣</ci></list></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Ex2.m1.3c">\displaystyle\max\sum_{\overline{uv}\in E}\textsl{g}(\overline{uv})\cdot y_{u,v}</annotation><annotation encoding="application/x-llamapun" id="S2.Ex2.m1.3d">roman_max ∑ start_POSTSUBSCRIPT over¯ start_ARG italic_u italic_v end_ARG ∈ italic_E end_POSTSUBSCRIPT g ( over¯ start_ARG italic_u italic_v end_ARG ) ⋅ italic_y start_POSTSUBSCRIPT italic_u , italic_v end_POSTSUBSCRIPT</annotation></semantics></math></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><span class="ltx_text ltx_markedasmath" id="S2.Ex2.5.2.2.1">s.t.</span></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle\|" class="ltx_Math" display="inline" id="S2.Ex2.m3.1"><semantics id="S2.Ex2.m3.1a"><mo id="S2.Ex2.m3.1.1" xref="S2.Ex2.m3.1.1.cmml">∥</mo><annotation-xml encoding="MathML-Content" id="S2.Ex2.m3.1b"><ci id="S2.Ex2.m3.1.1.cmml" xref="S2.Ex2.m3.1.1">∥</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.Ex2.m3.1c">\displaystyle\|</annotation><annotation encoding="application/x-llamapun" id="S2.Ex2.m3.1d">∥</annotation></semantics></math></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle\min\rho" class="ltx_Math" display="inline" id="S2.Ex2.m4.1"><semantics id="S2.Ex2.m4.1a"><mrow id="S2.Ex2.m4.1.1" xref="S2.Ex2.m4.1.1.cmml"><mi id="S2.Ex2.m4.1.1.1" xref="S2.Ex2.m4.1.1.1.cmml">min</mi><mo id="S2.Ex2.m4.1.1a" lspace="0.167em" xref="S2.Ex2.m4.1.1.cmml">⁡</mo><mi id="S2.Ex2.m4.1.1.2" xref="S2.Ex2.m4.1.1.2.cmml">ρ</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.Ex2.m4.1b"><apply id="S2.Ex2.m4.1.1.cmml" xref="S2.Ex2.m4.1.1"><min id="S2.Ex2.m4.1.1.1.cmml" xref="S2.Ex2.m4.1.1.1"></min><ci id="S2.Ex2.m4.1.1.2.cmml" xref="S2.Ex2.m4.1.1.2">𝜌</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Ex2.m4.1c">\displaystyle\min\rho</annotation><annotation encoding="application/x-llamapun" id="S2.Ex2.m4.1d">roman_min italic_ρ</annotation></semantics></math></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><span class="ltx_text ltx_markedasmath" id="S2.Ex2.9.2.1.1">s.t.</span></td> <td class="ltx_eqn_cell ltx_eqn_left_padright"></td> </tr></tbody> <tbody id="S2.Ex3"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_left_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle x_{u},x_{v}\geq y_{u,v}" class="ltx_Math" display="inline" id="S2.Ex3.m1.4"><semantics id="S2.Ex3.m1.4a"><mrow id="S2.Ex3.m1.4.4" xref="S2.Ex3.m1.4.4.cmml"><mrow id="S2.Ex3.m1.4.4.2.2" xref="S2.Ex3.m1.4.4.2.3.cmml"><msub id="S2.Ex3.m1.3.3.1.1.1" xref="S2.Ex3.m1.3.3.1.1.1.cmml"><mi id="S2.Ex3.m1.3.3.1.1.1.2" xref="S2.Ex3.m1.3.3.1.1.1.2.cmml">x</mi><mi id="S2.Ex3.m1.3.3.1.1.1.3" xref="S2.Ex3.m1.3.3.1.1.1.3.cmml">u</mi></msub><mo id="S2.Ex3.m1.4.4.2.2.3" xref="S2.Ex3.m1.4.4.2.3.cmml">,</mo><msub id="S2.Ex3.m1.4.4.2.2.2" xref="S2.Ex3.m1.4.4.2.2.2.cmml"><mi id="S2.Ex3.m1.4.4.2.2.2.2" xref="S2.Ex3.m1.4.4.2.2.2.2.cmml">x</mi><mi id="S2.Ex3.m1.4.4.2.2.2.3" xref="S2.Ex3.m1.4.4.2.2.2.3.cmml">v</mi></msub></mrow><mo id="S2.Ex3.m1.4.4.3" xref="S2.Ex3.m1.4.4.3.cmml">≥</mo><msub id="S2.Ex3.m1.4.4.4" xref="S2.Ex3.m1.4.4.4.cmml"><mi id="S2.Ex3.m1.4.4.4.2" xref="S2.Ex3.m1.4.4.4.2.cmml">y</mi><mrow id="S2.Ex3.m1.2.2.2.4" xref="S2.Ex3.m1.2.2.2.3.cmml"><mi id="S2.Ex3.m1.1.1.1.1" xref="S2.Ex3.m1.1.1.1.1.cmml">u</mi><mo id="S2.Ex3.m1.2.2.2.4.1" xref="S2.Ex3.m1.2.2.2.3.cmml">,</mo><mi id="S2.Ex3.m1.2.2.2.2" xref="S2.Ex3.m1.2.2.2.2.cmml">v</mi></mrow></msub></mrow><annotation-xml encoding="MathML-Content" id="S2.Ex3.m1.4b"><apply id="S2.Ex3.m1.4.4.cmml" xref="S2.Ex3.m1.4.4"><geq id="S2.Ex3.m1.4.4.3.cmml" xref="S2.Ex3.m1.4.4.3"></geq><list id="S2.Ex3.m1.4.4.2.3.cmml" xref="S2.Ex3.m1.4.4.2.2"><apply id="S2.Ex3.m1.3.3.1.1.1.cmml" xref="S2.Ex3.m1.3.3.1.1.1"><csymbol cd="ambiguous" id="S2.Ex3.m1.3.3.1.1.1.1.cmml" xref="S2.Ex3.m1.3.3.1.1.1">subscript</csymbol><ci id="S2.Ex3.m1.3.3.1.1.1.2.cmml" xref="S2.Ex3.m1.3.3.1.1.1.2">𝑥</ci><ci id="S2.Ex3.m1.3.3.1.1.1.3.cmml" xref="S2.Ex3.m1.3.3.1.1.1.3">𝑢</ci></apply><apply id="S2.Ex3.m1.4.4.2.2.2.cmml" xref="S2.Ex3.m1.4.4.2.2.2"><csymbol cd="ambiguous" id="S2.Ex3.m1.4.4.2.2.2.1.cmml" xref="S2.Ex3.m1.4.4.2.2.2">subscript</csymbol><ci id="S2.Ex3.m1.4.4.2.2.2.2.cmml" xref="S2.Ex3.m1.4.4.2.2.2.2">𝑥</ci><ci id="S2.Ex3.m1.4.4.2.2.2.3.cmml" xref="S2.Ex3.m1.4.4.2.2.2.3">𝑣</ci></apply></list><apply id="S2.Ex3.m1.4.4.4.cmml" xref="S2.Ex3.m1.4.4.4"><csymbol cd="ambiguous" id="S2.Ex3.m1.4.4.4.1.cmml" xref="S2.Ex3.m1.4.4.4">subscript</csymbol><ci id="S2.Ex3.m1.4.4.4.2.cmml" xref="S2.Ex3.m1.4.4.4.2">𝑦</ci><list id="S2.Ex3.m1.2.2.2.3.cmml" xref="S2.Ex3.m1.2.2.2.4"><ci id="S2.Ex3.m1.1.1.1.1.cmml" xref="S2.Ex3.m1.1.1.1.1">𝑢</ci><ci id="S2.Ex3.m1.2.2.2.2.cmml" xref="S2.Ex3.m1.2.2.2.2">𝑣</ci></list></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Ex3.m1.4c">\displaystyle x_{u},x_{v}\geq y_{u,v}</annotation><annotation encoding="application/x-llamapun" id="S2.Ex3.m1.4d">italic_x start_POSTSUBSCRIPT italic_u end_POSTSUBSCRIPT , italic_x start_POSTSUBSCRIPT italic_v end_POSTSUBSCRIPT ≥ italic_y start_POSTSUBSCRIPT italic_u , italic_v end_POSTSUBSCRIPT</annotation></semantics></math></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle\forall\overline{uv}\in E" class="ltx_Math" display="inline" id="S2.Ex3.m2.1"><semantics id="S2.Ex3.m2.1a"><mrow id="S2.Ex3.m2.1.1" xref="S2.Ex3.m2.1.1.cmml"><mrow id="S2.Ex3.m2.1.1.2" xref="S2.Ex3.m2.1.1.2.cmml"><mo id="S2.Ex3.m2.1.1.2.1" rspace="0.167em" xref="S2.Ex3.m2.1.1.2.1.cmml">∀</mo><mover accent="true" id="S2.Ex3.m2.1.1.2.2" xref="S2.Ex3.m2.1.1.2.2.cmml"><mrow id="S2.Ex3.m2.1.1.2.2.2" xref="S2.Ex3.m2.1.1.2.2.2.cmml"><mi id="S2.Ex3.m2.1.1.2.2.2.2" xref="S2.Ex3.m2.1.1.2.2.2.2.cmml">u</mi><mo id="S2.Ex3.m2.1.1.2.2.2.1" 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id="S2.Ex3.m2.1.1.2.2.2.3.cmml" xref="S2.Ex3.m2.1.1.2.2.2.3">𝑣</ci></apply></apply></apply><ci id="S2.Ex3.m2.1.1.3.cmml" xref="S2.Ex3.m2.1.1.3">𝐸</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Ex3.m2.1c">\displaystyle\forall\overline{uv}\in E</annotation><annotation encoding="application/x-llamapun" id="S2.Ex3.m2.1d">∀ over¯ start_ARG italic_u italic_v end_ARG ∈ italic_E</annotation></semantics></math></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle\|" class="ltx_Math" display="inline" id="S2.Ex3.m3.1"><semantics id="S2.Ex3.m3.1a"><mo id="S2.Ex3.m3.1.1" xref="S2.Ex3.m3.1.1.cmml">∥</mo><annotation-xml encoding="MathML-Content" id="S2.Ex3.m3.1b"><ci id="S2.Ex3.m3.1.1.cmml" xref="S2.Ex3.m3.1.1">∥</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.Ex3.m3.1c">\displaystyle\|</annotation><annotation encoding="application/x-llamapun" id="S2.Ex3.m3.1d">∥</annotation></semantics></math></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle\textsl{g}(u\!\to\!v)+\textsl{g}(v\!\to\!u)\geq\textsl{g}(% \overline{uv})" class="ltx_Math" display="inline" id="S2.Ex3.m4.3"><semantics id="S2.Ex3.m4.3a"><mrow id="S2.Ex3.m4.3.3" xref="S2.Ex3.m4.3.3.cmml"><mrow id="S2.Ex3.m4.3.3.2" xref="S2.Ex3.m4.3.3.2.cmml"><mrow id="S2.Ex3.m4.2.2.1.1" xref="S2.Ex3.m4.2.2.1.1.cmml"><mtext class="ltx_mathvariant_italic" id="S2.Ex3.m4.2.2.1.1.3" xref="S2.Ex3.m4.2.2.1.1.3a.cmml">g</mtext><mo id="S2.Ex3.m4.2.2.1.1.2" xref="S2.Ex3.m4.2.2.1.1.2.cmml">⁢</mo><mrow id="S2.Ex3.m4.2.2.1.1.1.1" xref="S2.Ex3.m4.2.2.1.1.1.1.1.cmml"><mo id="S2.Ex3.m4.2.2.1.1.1.1.2" stretchy="false" xref="S2.Ex3.m4.2.2.1.1.1.1.1.cmml">(</mo><mrow id="S2.Ex3.m4.2.2.1.1.1.1.1" xref="S2.Ex3.m4.2.2.1.1.1.1.1.cmml"><mi id="S2.Ex3.m4.2.2.1.1.1.1.1.2" xref="S2.Ex3.m4.2.2.1.1.1.1.1.2.cmml">u</mi><mo id="S2.Ex3.m4.2.2.1.1.1.1.1.1" lspace="0.108em" rspace="0.108em" stretchy="false" xref="S2.Ex3.m4.2.2.1.1.1.1.1.1.cmml">→</mo><mi id="S2.Ex3.m4.2.2.1.1.1.1.1.3" xref="S2.Ex3.m4.2.2.1.1.1.1.1.3.cmml">v</mi></mrow><mo id="S2.Ex3.m4.2.2.1.1.1.1.3" stretchy="false" xref="S2.Ex3.m4.2.2.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S2.Ex3.m4.3.3.2.3" xref="S2.Ex3.m4.3.3.2.3.cmml">+</mo><mrow id="S2.Ex3.m4.3.3.2.2" xref="S2.Ex3.m4.3.3.2.2.cmml"><mtext class="ltx_mathvariant_italic" id="S2.Ex3.m4.3.3.2.2.3" xref="S2.Ex3.m4.3.3.2.2.3a.cmml">g</mtext><mo id="S2.Ex3.m4.3.3.2.2.2" xref="S2.Ex3.m4.3.3.2.2.2.cmml">⁢</mo><mrow id="S2.Ex3.m4.3.3.2.2.1.1" xref="S2.Ex3.m4.3.3.2.2.1.1.1.cmml"><mo id="S2.Ex3.m4.3.3.2.2.1.1.2" stretchy="false" xref="S2.Ex3.m4.3.3.2.2.1.1.1.cmml">(</mo><mrow id="S2.Ex3.m4.3.3.2.2.1.1.1" xref="S2.Ex3.m4.3.3.2.2.1.1.1.cmml"><mi id="S2.Ex3.m4.3.3.2.2.1.1.1.2" xref="S2.Ex3.m4.3.3.2.2.1.1.1.2.cmml">v</mi><mo id="S2.Ex3.m4.3.3.2.2.1.1.1.1" lspace="0.108em" rspace="0.108em" stretchy="false" xref="S2.Ex3.m4.3.3.2.2.1.1.1.1.cmml">→</mo><mi id="S2.Ex3.m4.3.3.2.2.1.1.1.3" xref="S2.Ex3.m4.3.3.2.2.1.1.1.3.cmml">u</mi></mrow><mo id="S2.Ex3.m4.3.3.2.2.1.1.3" stretchy="false" xref="S2.Ex3.m4.3.3.2.2.1.1.1.cmml">)</mo></mrow></mrow></mrow><mo id="S2.Ex3.m4.3.3.3" xref="S2.Ex3.m4.3.3.3.cmml">≥</mo><mrow id="S2.Ex3.m4.3.3.4" xref="S2.Ex3.m4.3.3.4.cmml"><mtext class="ltx_mathvariant_italic" id="S2.Ex3.m4.3.3.4.2" xref="S2.Ex3.m4.3.3.4.2a.cmml">g</mtext><mo id="S2.Ex3.m4.3.3.4.1" xref="S2.Ex3.m4.3.3.4.1.cmml">⁢</mo><mrow id="S2.Ex3.m4.3.3.4.3.2" xref="S2.Ex3.m4.1.1.cmml"><mo id="S2.Ex3.m4.3.3.4.3.2.1" stretchy="false" xref="S2.Ex3.m4.1.1.cmml">(</mo><mover accent="true" id="S2.Ex3.m4.1.1" xref="S2.Ex3.m4.1.1.cmml"><mrow id="S2.Ex3.m4.1.1.2" xref="S2.Ex3.m4.1.1.2.cmml"><mi id="S2.Ex3.m4.1.1.2.2" xref="S2.Ex3.m4.1.1.2.2.cmml">u</mi><mo id="S2.Ex3.m4.1.1.2.1" xref="S2.Ex3.m4.1.1.2.1.cmml">⁢</mo><mi id="S2.Ex3.m4.1.1.2.3" xref="S2.Ex3.m4.1.1.2.3.cmml">v</mi></mrow><mo id="S2.Ex3.m4.1.1.1" xref="S2.Ex3.m4.1.1.1.cmml">¯</mo></mover><mo id="S2.Ex3.m4.3.3.4.3.2.2" stretchy="false" xref="S2.Ex3.m4.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.Ex3.m4.3b"><apply id="S2.Ex3.m4.3.3.cmml" xref="S2.Ex3.m4.3.3"><geq id="S2.Ex3.m4.3.3.3.cmml" xref="S2.Ex3.m4.3.3.3"></geq><apply id="S2.Ex3.m4.3.3.2.cmml" xref="S2.Ex3.m4.3.3.2"><plus id="S2.Ex3.m4.3.3.2.3.cmml" xref="S2.Ex3.m4.3.3.2.3"></plus><apply id="S2.Ex3.m4.2.2.1.1.cmml" xref="S2.Ex3.m4.2.2.1.1"><times id="S2.Ex3.m4.2.2.1.1.2.cmml" xref="S2.Ex3.m4.2.2.1.1.2"></times><ci id="S2.Ex3.m4.2.2.1.1.3a.cmml" xref="S2.Ex3.m4.2.2.1.1.3"><mtext class="ltx_mathvariant_italic" id="S2.Ex3.m4.2.2.1.1.3.cmml" xref="S2.Ex3.m4.2.2.1.1.3">g</mtext></ci><apply id="S2.Ex3.m4.2.2.1.1.1.1.1.cmml" xref="S2.Ex3.m4.2.2.1.1.1.1"><ci id="S2.Ex3.m4.2.2.1.1.1.1.1.1.cmml" xref="S2.Ex3.m4.2.2.1.1.1.1.1.1">→</ci><ci id="S2.Ex3.m4.2.2.1.1.1.1.1.2.cmml" xref="S2.Ex3.m4.2.2.1.1.1.1.1.2">𝑢</ci><ci id="S2.Ex3.m4.2.2.1.1.1.1.1.3.cmml" xref="S2.Ex3.m4.2.2.1.1.1.1.1.3">𝑣</ci></apply></apply><apply id="S2.Ex3.m4.3.3.2.2.cmml" xref="S2.Ex3.m4.3.3.2.2"><times id="S2.Ex3.m4.3.3.2.2.2.cmml" xref="S2.Ex3.m4.3.3.2.2.2"></times><ci id="S2.Ex3.m4.3.3.2.2.3a.cmml" xref="S2.Ex3.m4.3.3.2.2.3"><mtext class="ltx_mathvariant_italic" id="S2.Ex3.m4.3.3.2.2.3.cmml" xref="S2.Ex3.m4.3.3.2.2.3">g</mtext></ci><apply id="S2.Ex3.m4.3.3.2.2.1.1.1.cmml" xref="S2.Ex3.m4.3.3.2.2.1.1"><ci id="S2.Ex3.m4.3.3.2.2.1.1.1.1.cmml" xref="S2.Ex3.m4.3.3.2.2.1.1.1.1">→</ci><ci id="S2.Ex3.m4.3.3.2.2.1.1.1.2.cmml" xref="S2.Ex3.m4.3.3.2.2.1.1.1.2">𝑣</ci><ci id="S2.Ex3.m4.3.3.2.2.1.1.1.3.cmml" xref="S2.Ex3.m4.3.3.2.2.1.1.1.3">𝑢</ci></apply></apply></apply><apply id="S2.Ex3.m4.3.3.4.cmml" xref="S2.Ex3.m4.3.3.4"><times id="S2.Ex3.m4.3.3.4.1.cmml" xref="S2.Ex3.m4.3.3.4.1"></times><ci id="S2.Ex3.m4.3.3.4.2a.cmml" xref="S2.Ex3.m4.3.3.4.2"><mtext class="ltx_mathvariant_italic" id="S2.Ex3.m4.3.3.4.2.cmml" xref="S2.Ex3.m4.3.3.4.2">g</mtext></ci><apply id="S2.Ex3.m4.1.1.cmml" xref="S2.Ex3.m4.3.3.4.3.2"><ci id="S2.Ex3.m4.1.1.1.cmml" xref="S2.Ex3.m4.1.1.1">¯</ci><apply id="S2.Ex3.m4.1.1.2.cmml" xref="S2.Ex3.m4.1.1.2"><times id="S2.Ex3.m4.1.1.2.1.cmml" xref="S2.Ex3.m4.1.1.2.1"></times><ci id="S2.Ex3.m4.1.1.2.2.cmml" xref="S2.Ex3.m4.1.1.2.2">𝑢</ci><ci id="S2.Ex3.m4.1.1.2.3.cmml" xref="S2.Ex3.m4.1.1.2.3">𝑣</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Ex3.m4.3c">\displaystyle\textsl{g}(u\!\to\!v)+\textsl{g}(v\!\to\!u)\geq\textsl{g}(% \overline{uv})</annotation><annotation encoding="application/x-llamapun" id="S2.Ex3.m4.3d">g ( italic_u → italic_v ) + g ( italic_v → italic_u ) ≥ g ( over¯ start_ARG italic_u italic_v end_ARG )</annotation></semantics></math></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle\forall\overline{uv}\in E" class="ltx_Math" display="inline" id="S2.Ex3.m5.1"><semantics id="S2.Ex3.m5.1a"><mrow id="S2.Ex3.m5.1.1" xref="S2.Ex3.m5.1.1.cmml"><mrow id="S2.Ex3.m5.1.1.2" 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id="S2.Ex3.m5.1.1.2.2.cmml" xref="S2.Ex3.m5.1.1.2.2"><ci id="S2.Ex3.m5.1.1.2.2.1.cmml" xref="S2.Ex3.m5.1.1.2.2.1">¯</ci><apply id="S2.Ex3.m5.1.1.2.2.2.cmml" xref="S2.Ex3.m5.1.1.2.2.2"><times id="S2.Ex3.m5.1.1.2.2.2.1.cmml" xref="S2.Ex3.m5.1.1.2.2.2.1"></times><ci id="S2.Ex3.m5.1.1.2.2.2.2.cmml" xref="S2.Ex3.m5.1.1.2.2.2.2">𝑢</ci><ci id="S2.Ex3.m5.1.1.2.2.2.3.cmml" xref="S2.Ex3.m5.1.1.2.2.2.3">𝑣</ci></apply></apply></apply><ci id="S2.Ex3.m5.1.1.3.cmml" xref="S2.Ex3.m5.1.1.3">𝐸</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Ex3.m5.1c">\displaystyle\forall\overline{uv}\in E</annotation><annotation encoding="application/x-llamapun" id="S2.Ex3.m5.1d">∀ over¯ start_ARG italic_u italic_v end_ARG ∈ italic_E</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_left_padright"></td> </tr></tbody> <tbody id="S2.Ex4"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_left_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle\sum_{v\in V}x_{v}\leq 1" class="ltx_Math" display="inline" id="S2.Ex4.m1.1"><semantics id="S2.Ex4.m1.1a"><mrow id="S2.Ex4.m1.1.1" xref="S2.Ex4.m1.1.1.cmml"><mrow id="S2.Ex4.m1.1.1.2" xref="S2.Ex4.m1.1.1.2.cmml"><mstyle displaystyle="true" id="S2.Ex4.m1.1.1.2.1" xref="S2.Ex4.m1.1.1.2.1.cmml"><munder id="S2.Ex4.m1.1.1.2.1a" xref="S2.Ex4.m1.1.1.2.1.cmml"><mo id="S2.Ex4.m1.1.1.2.1.2" movablelimits="false" xref="S2.Ex4.m1.1.1.2.1.2.cmml">∑</mo><mrow id="S2.Ex4.m1.1.1.2.1.3" xref="S2.Ex4.m1.1.1.2.1.3.cmml"><mi id="S2.Ex4.m1.1.1.2.1.3.2" xref="S2.Ex4.m1.1.1.2.1.3.2.cmml">v</mi><mo id="S2.Ex4.m1.1.1.2.1.3.1" xref="S2.Ex4.m1.1.1.2.1.3.1.cmml">∈</mo><mi id="S2.Ex4.m1.1.1.2.1.3.3" xref="S2.Ex4.m1.1.1.2.1.3.3.cmml">V</mi></mrow></munder></mstyle><msub id="S2.Ex4.m1.1.1.2.2" xref="S2.Ex4.m1.1.1.2.2.cmml"><mi id="S2.Ex4.m1.1.1.2.2.2" xref="S2.Ex4.m1.1.1.2.2.2.cmml">x</mi><mi id="S2.Ex4.m1.1.1.2.2.3" xref="S2.Ex4.m1.1.1.2.2.3.cmml">v</mi></msub></mrow><mo id="S2.Ex4.m1.1.1.1" xref="S2.Ex4.m1.1.1.1.cmml">≤</mo><mn id="S2.Ex4.m1.1.1.3" xref="S2.Ex4.m1.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S2.Ex4.m1.1b"><apply id="S2.Ex4.m1.1.1.cmml" xref="S2.Ex4.m1.1.1"><leq id="S2.Ex4.m1.1.1.1.cmml" xref="S2.Ex4.m1.1.1.1"></leq><apply id="S2.Ex4.m1.1.1.2.cmml" xref="S2.Ex4.m1.1.1.2"><apply id="S2.Ex4.m1.1.1.2.1.cmml" xref="S2.Ex4.m1.1.1.2.1"><csymbol cd="ambiguous" id="S2.Ex4.m1.1.1.2.1.1.cmml" xref="S2.Ex4.m1.1.1.2.1">subscript</csymbol><sum id="S2.Ex4.m1.1.1.2.1.2.cmml" xref="S2.Ex4.m1.1.1.2.1.2"></sum><apply id="S2.Ex4.m1.1.1.2.1.3.cmml" xref="S2.Ex4.m1.1.1.2.1.3"><in id="S2.Ex4.m1.1.1.2.1.3.1.cmml" xref="S2.Ex4.m1.1.1.2.1.3.1"></in><ci id="S2.Ex4.m1.1.1.2.1.3.2.cmml" xref="S2.Ex4.m1.1.1.2.1.3.2">𝑣</ci><ci id="S2.Ex4.m1.1.1.2.1.3.3.cmml" xref="S2.Ex4.m1.1.1.2.1.3.3">𝑉</ci></apply></apply><apply id="S2.Ex4.m1.1.1.2.2.cmml" xref="S2.Ex4.m1.1.1.2.2"><csymbol cd="ambiguous" id="S2.Ex4.m1.1.1.2.2.1.cmml" xref="S2.Ex4.m1.1.1.2.2">subscript</csymbol><ci id="S2.Ex4.m1.1.1.2.2.2.cmml" xref="S2.Ex4.m1.1.1.2.2.2">𝑥</ci><ci id="S2.Ex4.m1.1.1.2.2.3.cmml" xref="S2.Ex4.m1.1.1.2.2.3">𝑣</ci></apply></apply><cn id="S2.Ex4.m1.1.1.3.cmml" type="integer" xref="S2.Ex4.m1.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Ex4.m1.1c">\displaystyle\sum_{v\in V}x_{v}\leq 1</annotation><annotation encoding="application/x-llamapun" id="S2.Ex4.m1.1d">∑ start_POSTSUBSCRIPT italic_v ∈ italic_V end_POSTSUBSCRIPT italic_x start_POSTSUBSCRIPT italic_v end_POSTSUBSCRIPT ≤ 1</annotation></semantics></math></td> <td class="ltx_td ltx_eqn_cell"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle\|" class="ltx_Math" display="inline" id="S2.Ex4.m2.1"><semantics id="S2.Ex4.m2.1a"><mo id="S2.Ex4.m2.1.1" xref="S2.Ex4.m2.1.1.cmml">∥</mo><annotation-xml encoding="MathML-Content" id="S2.Ex4.m2.1b"><ci id="S2.Ex4.m2.1.1.cmml" xref="S2.Ex4.m2.1.1">∥</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.Ex4.m2.1c">\displaystyle\|</annotation><annotation encoding="application/x-llamapun" id="S2.Ex4.m2.1d">∥</annotation></semantics></math></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle\rho\geq\sum_{v\in V}\textsl{g}(u\!\to\!v)" class="ltx_Math" display="inline" id="S2.Ex4.m3.1"><semantics id="S2.Ex4.m3.1a"><mrow id="S2.Ex4.m3.1.1" xref="S2.Ex4.m3.1.1.cmml"><mi id="S2.Ex4.m3.1.1.3" xref="S2.Ex4.m3.1.1.3.cmml">ρ</mi><mo id="S2.Ex4.m3.1.1.2" xref="S2.Ex4.m3.1.1.2.cmml">≥</mo><mrow id="S2.Ex4.m3.1.1.1" xref="S2.Ex4.m3.1.1.1.cmml"><mstyle displaystyle="true" id="S2.Ex4.m3.1.1.1.2" xref="S2.Ex4.m3.1.1.1.2.cmml"><munder id="S2.Ex4.m3.1.1.1.2a" xref="S2.Ex4.m3.1.1.1.2.cmml"><mo id="S2.Ex4.m3.1.1.1.2.2" movablelimits="false" xref="S2.Ex4.m3.1.1.1.2.2.cmml">∑</mo><mrow id="S2.Ex4.m3.1.1.1.2.3" xref="S2.Ex4.m3.1.1.1.2.3.cmml"><mi id="S2.Ex4.m3.1.1.1.2.3.2" xref="S2.Ex4.m3.1.1.1.2.3.2.cmml">v</mi><mo id="S2.Ex4.m3.1.1.1.2.3.1" xref="S2.Ex4.m3.1.1.1.2.3.1.cmml">∈</mo><mi id="S2.Ex4.m3.1.1.1.2.3.3" xref="S2.Ex4.m3.1.1.1.2.3.3.cmml">V</mi></mrow></munder></mstyle><mrow id="S2.Ex4.m3.1.1.1.1" xref="S2.Ex4.m3.1.1.1.1.cmml"><mtext class="ltx_mathvariant_italic" id="S2.Ex4.m3.1.1.1.1.3" xref="S2.Ex4.m3.1.1.1.1.3a.cmml">g</mtext><mo id="S2.Ex4.m3.1.1.1.1.2" xref="S2.Ex4.m3.1.1.1.1.2.cmml">⁢</mo><mrow id="S2.Ex4.m3.1.1.1.1.1.1" xref="S2.Ex4.m3.1.1.1.1.1.1.1.cmml"><mo id="S2.Ex4.m3.1.1.1.1.1.1.2" stretchy="false" xref="S2.Ex4.m3.1.1.1.1.1.1.1.cmml">(</mo><mrow id="S2.Ex4.m3.1.1.1.1.1.1.1" xref="S2.Ex4.m3.1.1.1.1.1.1.1.cmml"><mi id="S2.Ex4.m3.1.1.1.1.1.1.1.2" xref="S2.Ex4.m3.1.1.1.1.1.1.1.2.cmml">u</mi><mo id="S2.Ex4.m3.1.1.1.1.1.1.1.1" lspace="0.108em" rspace="0.108em" stretchy="false" xref="S2.Ex4.m3.1.1.1.1.1.1.1.1.cmml">→</mo><mi id="S2.Ex4.m3.1.1.1.1.1.1.1.3" xref="S2.Ex4.m3.1.1.1.1.1.1.1.3.cmml">v</mi></mrow><mo id="S2.Ex4.m3.1.1.1.1.1.1.3" stretchy="false" xref="S2.Ex4.m3.1.1.1.1.1.1.1.cmml">)</mo></mrow></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.Ex4.m3.1b"><apply id="S2.Ex4.m3.1.1.cmml" xref="S2.Ex4.m3.1.1"><geq id="S2.Ex4.m3.1.1.2.cmml" xref="S2.Ex4.m3.1.1.2"></geq><ci id="S2.Ex4.m3.1.1.3.cmml" xref="S2.Ex4.m3.1.1.3">𝜌</ci><apply id="S2.Ex4.m3.1.1.1.cmml" xref="S2.Ex4.m3.1.1.1"><apply id="S2.Ex4.m3.1.1.1.2.cmml" xref="S2.Ex4.m3.1.1.1.2"><csymbol cd="ambiguous" id="S2.Ex4.m3.1.1.1.2.1.cmml" xref="S2.Ex4.m3.1.1.1.2">subscript</csymbol><sum id="S2.Ex4.m3.1.1.1.2.2.cmml" xref="S2.Ex4.m3.1.1.1.2.2"></sum><apply id="S2.Ex4.m3.1.1.1.2.3.cmml" xref="S2.Ex4.m3.1.1.1.2.3"><in id="S2.Ex4.m3.1.1.1.2.3.1.cmml" xref="S2.Ex4.m3.1.1.1.2.3.1"></in><ci id="S2.Ex4.m3.1.1.1.2.3.2.cmml" xref="S2.Ex4.m3.1.1.1.2.3.2">𝑣</ci><ci id="S2.Ex4.m3.1.1.1.2.3.3.cmml" xref="S2.Ex4.m3.1.1.1.2.3.3">𝑉</ci></apply></apply><apply id="S2.Ex4.m3.1.1.1.1.cmml" xref="S2.Ex4.m3.1.1.1.1"><times id="S2.Ex4.m3.1.1.1.1.2.cmml" xref="S2.Ex4.m3.1.1.1.1.2"></times><ci id="S2.Ex4.m3.1.1.1.1.3a.cmml" xref="S2.Ex4.m3.1.1.1.1.3"><mtext class="ltx_mathvariant_italic" id="S2.Ex4.m3.1.1.1.1.3.cmml" xref="S2.Ex4.m3.1.1.1.1.3">g</mtext></ci><apply id="S2.Ex4.m3.1.1.1.1.1.1.1.cmml" xref="S2.Ex4.m3.1.1.1.1.1.1"><ci id="S2.Ex4.m3.1.1.1.1.1.1.1.1.cmml" xref="S2.Ex4.m3.1.1.1.1.1.1.1.1">→</ci><ci id="S2.Ex4.m3.1.1.1.1.1.1.1.2.cmml" xref="S2.Ex4.m3.1.1.1.1.1.1.1.2">𝑢</ci><ci id="S2.Ex4.m3.1.1.1.1.1.1.1.3.cmml" xref="S2.Ex4.m3.1.1.1.1.1.1.1.3">𝑣</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Ex4.m3.1c">\displaystyle\rho\geq\sum_{v\in V}\textsl{g}(u\!\to\!v)</annotation><annotation encoding="application/x-llamapun" id="S2.Ex4.m3.1d">italic_ρ ≥ ∑ start_POSTSUBSCRIPT italic_v ∈ italic_V end_POSTSUBSCRIPT g ( italic_u → italic_v )</annotation></semantics></math></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle\forall u\in V" class="ltx_Math" display="inline" id="S2.Ex4.m4.1"><semantics id="S2.Ex4.m4.1a"><mrow id="S2.Ex4.m4.1.1" xref="S2.Ex4.m4.1.1.cmml"><mrow id="S2.Ex4.m4.1.1.2" xref="S2.Ex4.m4.1.1.2.cmml"><mo id="S2.Ex4.m4.1.1.2.1" rspace="0.167em" xref="S2.Ex4.m4.1.1.2.1.cmml">∀</mo><mi id="S2.Ex4.m4.1.1.2.2" xref="S2.Ex4.m4.1.1.2.2.cmml">u</mi></mrow><mo id="S2.Ex4.m4.1.1.1" xref="S2.Ex4.m4.1.1.1.cmml">∈</mo><mi id="S2.Ex4.m4.1.1.3" xref="S2.Ex4.m4.1.1.3.cmml">V</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.Ex4.m4.1b"><apply id="S2.Ex4.m4.1.1.cmml" xref="S2.Ex4.m4.1.1"><in id="S2.Ex4.m4.1.1.1.cmml" xref="S2.Ex4.m4.1.1.1"></in><apply id="S2.Ex4.m4.1.1.2.cmml" xref="S2.Ex4.m4.1.1.2"><csymbol cd="latexml" id="S2.Ex4.m4.1.1.2.1.cmml" xref="S2.Ex4.m4.1.1.2.1">for-all</csymbol><ci id="S2.Ex4.m4.1.1.2.2.cmml" xref="S2.Ex4.m4.1.1.2.2">𝑢</ci></apply><ci id="S2.Ex4.m4.1.1.3.cmml" xref="S2.Ex4.m4.1.1.3">𝑉</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Ex4.m4.1c">\displaystyle\forall u\in V</annotation><annotation encoding="application/x-llamapun" id="S2.Ex4.m4.1d">∀ italic_u ∈ italic_V</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_left_padright"></td> </tr></tbody> <tbody id="S2.Ex5"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_left_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle\quad x_{v},y_{u,v}\geq 0" class="ltx_Math" display="inline" id="S2.Ex5.m1.4"><semantics id="S2.Ex5.m1.4a"><mrow id="S2.Ex5.m1.4.4" xref="S2.Ex5.m1.4.4.cmml"><mrow id="S2.Ex5.m1.4.4.2.2" xref="S2.Ex5.m1.4.4.2.3.cmml"><msub id="S2.Ex5.m1.3.3.1.1.1" xref="S2.Ex5.m1.3.3.1.1.1.cmml"><mi id="S2.Ex5.m1.3.3.1.1.1.2" xref="S2.Ex5.m1.3.3.1.1.1.2.cmml">x</mi><mi id="S2.Ex5.m1.3.3.1.1.1.3" xref="S2.Ex5.m1.3.3.1.1.1.3.cmml">v</mi></msub><mo id="S2.Ex5.m1.4.4.2.2.3" xref="S2.Ex5.m1.4.4.2.3.cmml">,</mo><msub id="S2.Ex5.m1.4.4.2.2.2" xref="S2.Ex5.m1.4.4.2.2.2.cmml"><mi id="S2.Ex5.m1.4.4.2.2.2.2" xref="S2.Ex5.m1.4.4.2.2.2.2.cmml">y</mi><mrow id="S2.Ex5.m1.2.2.2.4" xref="S2.Ex5.m1.2.2.2.3.cmml"><mi id="S2.Ex5.m1.1.1.1.1" xref="S2.Ex5.m1.1.1.1.1.cmml">u</mi><mo id="S2.Ex5.m1.2.2.2.4.1" xref="S2.Ex5.m1.2.2.2.3.cmml">,</mo><mi id="S2.Ex5.m1.2.2.2.2" xref="S2.Ex5.m1.2.2.2.2.cmml">v</mi></mrow></msub></mrow><mo id="S2.Ex5.m1.4.4.3" xref="S2.Ex5.m1.4.4.3.cmml">≥</mo><mn id="S2.Ex5.m1.4.4.4" xref="S2.Ex5.m1.4.4.4.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S2.Ex5.m1.4b"><apply id="S2.Ex5.m1.4.4.cmml" xref="S2.Ex5.m1.4.4"><geq id="S2.Ex5.m1.4.4.3.cmml" xref="S2.Ex5.m1.4.4.3"></geq><list id="S2.Ex5.m1.4.4.2.3.cmml" xref="S2.Ex5.m1.4.4.2.2"><apply id="S2.Ex5.m1.3.3.1.1.1.cmml" xref="S2.Ex5.m1.3.3.1.1.1"><csymbol cd="ambiguous" id="S2.Ex5.m1.3.3.1.1.1.1.cmml" xref="S2.Ex5.m1.3.3.1.1.1">subscript</csymbol><ci id="S2.Ex5.m1.3.3.1.1.1.2.cmml" xref="S2.Ex5.m1.3.3.1.1.1.2">𝑥</ci><ci id="S2.Ex5.m1.3.3.1.1.1.3.cmml" xref="S2.Ex5.m1.3.3.1.1.1.3">𝑣</ci></apply><apply id="S2.Ex5.m1.4.4.2.2.2.cmml" xref="S2.Ex5.m1.4.4.2.2.2"><csymbol cd="ambiguous" id="S2.Ex5.m1.4.4.2.2.2.1.cmml" xref="S2.Ex5.m1.4.4.2.2.2">subscript</csymbol><ci id="S2.Ex5.m1.4.4.2.2.2.2.cmml" xref="S2.Ex5.m1.4.4.2.2.2.2">𝑦</ci><list id="S2.Ex5.m1.2.2.2.3.cmml" xref="S2.Ex5.m1.2.2.2.4"><ci id="S2.Ex5.m1.1.1.1.1.cmml" xref="S2.Ex5.m1.1.1.1.1">𝑢</ci><ci id="S2.Ex5.m1.2.2.2.2.cmml" xref="S2.Ex5.m1.2.2.2.2">𝑣</ci></list></apply></list><cn id="S2.Ex5.m1.4.4.4.cmml" type="integer" xref="S2.Ex5.m1.4.4.4">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Ex5.m1.4c">\displaystyle\quad x_{v},y_{u,v}\geq 0</annotation><annotation encoding="application/x-llamapun" id="S2.Ex5.m1.4d">italic_x start_POSTSUBSCRIPT italic_v end_POSTSUBSCRIPT , italic_y start_POSTSUBSCRIPT italic_u , italic_v end_POSTSUBSCRIPT ≥ 0</annotation></semantics></math></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle\forall u,v\in V" class="ltx_Math" display="inline" id="S2.Ex5.m2.2"><semantics id="S2.Ex5.m2.2a"><mrow id="S2.Ex5.m2.2.2" xref="S2.Ex5.m2.2.2.cmml"><mrow id="S2.Ex5.m2.2.2.1.1" xref="S2.Ex5.m2.2.2.1.2.cmml"><mrow id="S2.Ex5.m2.2.2.1.1.1" xref="S2.Ex5.m2.2.2.1.1.1.cmml"><mo id="S2.Ex5.m2.2.2.1.1.1.1" rspace="0.167em" xref="S2.Ex5.m2.2.2.1.1.1.1.cmml">∀</mo><mi id="S2.Ex5.m2.2.2.1.1.1.2" xref="S2.Ex5.m2.2.2.1.1.1.2.cmml">u</mi></mrow><mo id="S2.Ex5.m2.2.2.1.1.2" xref="S2.Ex5.m2.2.2.1.2.cmml">,</mo><mi id="S2.Ex5.m2.1.1" xref="S2.Ex5.m2.1.1.cmml">v</mi></mrow><mo id="S2.Ex5.m2.2.2.2" xref="S2.Ex5.m2.2.2.2.cmml">∈</mo><mi id="S2.Ex5.m2.2.2.3" xref="S2.Ex5.m2.2.2.3.cmml">V</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.Ex5.m2.2b"><apply id="S2.Ex5.m2.2.2.cmml" xref="S2.Ex5.m2.2.2"><in id="S2.Ex5.m2.2.2.2.cmml" xref="S2.Ex5.m2.2.2.2"></in><list id="S2.Ex5.m2.2.2.1.2.cmml" xref="S2.Ex5.m2.2.2.1.1"><apply id="S2.Ex5.m2.2.2.1.1.1.cmml" xref="S2.Ex5.m2.2.2.1.1.1"><csymbol cd="latexml" id="S2.Ex5.m2.2.2.1.1.1.1.cmml" xref="S2.Ex5.m2.2.2.1.1.1.1">for-all</csymbol><ci id="S2.Ex5.m2.2.2.1.1.1.2.cmml" xref="S2.Ex5.m2.2.2.1.1.1.2">𝑢</ci></apply><ci id="S2.Ex5.m2.1.1.cmml" xref="S2.Ex5.m2.1.1">𝑣</ci></list><ci id="S2.Ex5.m2.2.2.3.cmml" xref="S2.Ex5.m2.2.2.3">𝑉</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Ex5.m2.2c">\displaystyle\forall u,v\in V</annotation><annotation encoding="application/x-llamapun" id="S2.Ex5.m2.2d">∀ italic_u , italic_v ∈ italic_V</annotation></semantics></math></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle\|" class="ltx_Math" display="inline" id="S2.Ex5.m3.1"><semantics id="S2.Ex5.m3.1a"><mo id="S2.Ex5.m3.1.1" xref="S2.Ex5.m3.1.1.cmml">∥</mo><annotation-xml encoding="MathML-Content" id="S2.Ex5.m3.1b"><ci id="S2.Ex5.m3.1.1.cmml" xref="S2.Ex5.m3.1.1">∥</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.Ex5.m3.1c">\displaystyle\|</annotation><annotation encoding="application/x-llamapun" id="S2.Ex5.m3.1d">∥</annotation></semantics></math></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle\textsl{g}(u\!\to\!v),\textsl{g}(v\!\to\!u)\geq 0" class="ltx_Math" display="inline" id="S2.Ex5.m4.2"><semantics id="S2.Ex5.m4.2a"><mrow id="S2.Ex5.m4.2.2" xref="S2.Ex5.m4.2.2.cmml"><mrow id="S2.Ex5.m4.2.2.2.2" xref="S2.Ex5.m4.2.2.2.3.cmml"><mrow id="S2.Ex5.m4.1.1.1.1.1" xref="S2.Ex5.m4.1.1.1.1.1.cmml"><mtext class="ltx_mathvariant_italic" id="S2.Ex5.m4.1.1.1.1.1.3" xref="S2.Ex5.m4.1.1.1.1.1.3a.cmml">g</mtext><mo id="S2.Ex5.m4.1.1.1.1.1.2" xref="S2.Ex5.m4.1.1.1.1.1.2.cmml">⁢</mo><mrow id="S2.Ex5.m4.1.1.1.1.1.1.1" xref="S2.Ex5.m4.1.1.1.1.1.1.1.1.cmml"><mo id="S2.Ex5.m4.1.1.1.1.1.1.1.2" stretchy="false" xref="S2.Ex5.m4.1.1.1.1.1.1.1.1.cmml">(</mo><mrow id="S2.Ex5.m4.1.1.1.1.1.1.1.1" xref="S2.Ex5.m4.1.1.1.1.1.1.1.1.cmml"><mi id="S2.Ex5.m4.1.1.1.1.1.1.1.1.2" xref="S2.Ex5.m4.1.1.1.1.1.1.1.1.2.cmml">u</mi><mo id="S2.Ex5.m4.1.1.1.1.1.1.1.1.1" lspace="0.108em" rspace="0.108em" stretchy="false" xref="S2.Ex5.m4.1.1.1.1.1.1.1.1.1.cmml">→</mo><mi id="S2.Ex5.m4.1.1.1.1.1.1.1.1.3" xref="S2.Ex5.m4.1.1.1.1.1.1.1.1.3.cmml">v</mi></mrow><mo id="S2.Ex5.m4.1.1.1.1.1.1.1.3" stretchy="false" xref="S2.Ex5.m4.1.1.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S2.Ex5.m4.2.2.2.2.3" xref="S2.Ex5.m4.2.2.2.3.cmml">,</mo><mrow id="S2.Ex5.m4.2.2.2.2.2" xref="S2.Ex5.m4.2.2.2.2.2.cmml"><mtext class="ltx_mathvariant_italic" id="S2.Ex5.m4.2.2.2.2.2.3" xref="S2.Ex5.m4.2.2.2.2.2.3a.cmml">g</mtext><mo id="S2.Ex5.m4.2.2.2.2.2.2" xref="S2.Ex5.m4.2.2.2.2.2.2.cmml">⁢</mo><mrow id="S2.Ex5.m4.2.2.2.2.2.1.1" xref="S2.Ex5.m4.2.2.2.2.2.1.1.1.cmml"><mo id="S2.Ex5.m4.2.2.2.2.2.1.1.2" stretchy="false" xref="S2.Ex5.m4.2.2.2.2.2.1.1.1.cmml">(</mo><mrow id="S2.Ex5.m4.2.2.2.2.2.1.1.1" xref="S2.Ex5.m4.2.2.2.2.2.1.1.1.cmml"><mi id="S2.Ex5.m4.2.2.2.2.2.1.1.1.2" xref="S2.Ex5.m4.2.2.2.2.2.1.1.1.2.cmml">v</mi><mo id="S2.Ex5.m4.2.2.2.2.2.1.1.1.1" lspace="0.108em" rspace="0.108em" stretchy="false" xref="S2.Ex5.m4.2.2.2.2.2.1.1.1.1.cmml">→</mo><mi id="S2.Ex5.m4.2.2.2.2.2.1.1.1.3" xref="S2.Ex5.m4.2.2.2.2.2.1.1.1.3.cmml">u</mi></mrow><mo id="S2.Ex5.m4.2.2.2.2.2.1.1.3" stretchy="false" xref="S2.Ex5.m4.2.2.2.2.2.1.1.1.cmml">)</mo></mrow></mrow></mrow><mo id="S2.Ex5.m4.2.2.3" xref="S2.Ex5.m4.2.2.3.cmml">≥</mo><mn id="S2.Ex5.m4.2.2.4" xref="S2.Ex5.m4.2.2.4.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S2.Ex5.m4.2b"><apply id="S2.Ex5.m4.2.2.cmml" xref="S2.Ex5.m4.2.2"><geq id="S2.Ex5.m4.2.2.3.cmml" xref="S2.Ex5.m4.2.2.3"></geq><list id="S2.Ex5.m4.2.2.2.3.cmml" xref="S2.Ex5.m4.2.2.2.2"><apply id="S2.Ex5.m4.1.1.1.1.1.cmml" xref="S2.Ex5.m4.1.1.1.1.1"><times id="S2.Ex5.m4.1.1.1.1.1.2.cmml" xref="S2.Ex5.m4.1.1.1.1.1.2"></times><ci id="S2.Ex5.m4.1.1.1.1.1.3a.cmml" xref="S2.Ex5.m4.1.1.1.1.1.3"><mtext class="ltx_mathvariant_italic" id="S2.Ex5.m4.1.1.1.1.1.3.cmml" xref="S2.Ex5.m4.1.1.1.1.1.3">g</mtext></ci><apply id="S2.Ex5.m4.1.1.1.1.1.1.1.1.cmml" xref="S2.Ex5.m4.1.1.1.1.1.1.1"><ci id="S2.Ex5.m4.1.1.1.1.1.1.1.1.1.cmml" xref="S2.Ex5.m4.1.1.1.1.1.1.1.1.1">→</ci><ci id="S2.Ex5.m4.1.1.1.1.1.1.1.1.2.cmml" xref="S2.Ex5.m4.1.1.1.1.1.1.1.1.2">𝑢</ci><ci id="S2.Ex5.m4.1.1.1.1.1.1.1.1.3.cmml" xref="S2.Ex5.m4.1.1.1.1.1.1.1.1.3">𝑣</ci></apply></apply><apply id="S2.Ex5.m4.2.2.2.2.2.cmml" xref="S2.Ex5.m4.2.2.2.2.2"><times id="S2.Ex5.m4.2.2.2.2.2.2.cmml" xref="S2.Ex5.m4.2.2.2.2.2.2"></times><ci id="S2.Ex5.m4.2.2.2.2.2.3a.cmml" xref="S2.Ex5.m4.2.2.2.2.2.3"><mtext class="ltx_mathvariant_italic" id="S2.Ex5.m4.2.2.2.2.2.3.cmml" xref="S2.Ex5.m4.2.2.2.2.2.3">g</mtext></ci><apply id="S2.Ex5.m4.2.2.2.2.2.1.1.1.cmml" xref="S2.Ex5.m4.2.2.2.2.2.1.1"><ci id="S2.Ex5.m4.2.2.2.2.2.1.1.1.1.cmml" xref="S2.Ex5.m4.2.2.2.2.2.1.1.1.1">→</ci><ci id="S2.Ex5.m4.2.2.2.2.2.1.1.1.2.cmml" xref="S2.Ex5.m4.2.2.2.2.2.1.1.1.2">𝑣</ci><ci id="S2.Ex5.m4.2.2.2.2.2.1.1.1.3.cmml" xref="S2.Ex5.m4.2.2.2.2.2.1.1.1.3">𝑢</ci></apply></apply></list><cn id="S2.Ex5.m4.2.2.4.cmml" type="integer" xref="S2.Ex5.m4.2.2.4">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Ex5.m4.2c">\displaystyle\textsl{g}(u\!\to\!v),\textsl{g}(v\!\to\!u)\geq 0</annotation><annotation encoding="application/x-llamapun" id="S2.Ex5.m4.2d">g ( italic_u → italic_v ) , g ( italic_v → italic_u ) ≥ 0</annotation></semantics></math></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle\forall u,v\in V" class="ltx_Math" display="inline" id="S2.Ex5.m5.2"><semantics id="S2.Ex5.m5.2a"><mrow id="S2.Ex5.m5.2.2" xref="S2.Ex5.m5.2.2.cmml"><mrow id="S2.Ex5.m5.2.2.1.1" xref="S2.Ex5.m5.2.2.1.2.cmml"><mrow id="S2.Ex5.m5.2.2.1.1.1" xref="S2.Ex5.m5.2.2.1.1.1.cmml"><mo id="S2.Ex5.m5.2.2.1.1.1.1" rspace="0.167em" xref="S2.Ex5.m5.2.2.1.1.1.1.cmml">∀</mo><mi id="S2.Ex5.m5.2.2.1.1.1.2" xref="S2.Ex5.m5.2.2.1.1.1.2.cmml">u</mi></mrow><mo id="S2.Ex5.m5.2.2.1.1.2" xref="S2.Ex5.m5.2.2.1.2.cmml">,</mo><mi id="S2.Ex5.m5.1.1" xref="S2.Ex5.m5.1.1.cmml">v</mi></mrow><mo id="S2.Ex5.m5.2.2.2" xref="S2.Ex5.m5.2.2.2.cmml">∈</mo><mi id="S2.Ex5.m5.2.2.3" xref="S2.Ex5.m5.2.2.3.cmml">V</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.Ex5.m5.2b"><apply id="S2.Ex5.m5.2.2.cmml" xref="S2.Ex5.m5.2.2"><in id="S2.Ex5.m5.2.2.2.cmml" xref="S2.Ex5.m5.2.2.2"></in><list id="S2.Ex5.m5.2.2.1.2.cmml" xref="S2.Ex5.m5.2.2.1.1"><apply id="S2.Ex5.m5.2.2.1.1.1.cmml" xref="S2.Ex5.m5.2.2.1.1.1"><csymbol cd="latexml" id="S2.Ex5.m5.2.2.1.1.1.1.cmml" xref="S2.Ex5.m5.2.2.1.1.1.1">for-all</csymbol><ci id="S2.Ex5.m5.2.2.1.1.1.2.cmml" xref="S2.Ex5.m5.2.2.1.1.1.2">𝑢</ci></apply><ci id="S2.Ex5.m5.1.1.cmml" xref="S2.Ex5.m5.1.1">𝑣</ci></list><ci id="S2.Ex5.m5.2.2.3.cmml" xref="S2.Ex5.m5.2.2.3">𝑉</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Ex5.m5.2c">\displaystyle\forall u,v\in V</annotation><annotation encoding="application/x-llamapun" id="S2.Ex5.m5.2d">∀ italic_u , italic_v ∈ italic_V</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_left_padright"></td> </tr></tbody> </table> </div> <div class="ltx_para ltx_noindent" id="S2.SS0.SSS0.P0.SPx1.p4"> <p class="ltx_p" id="S2.SS0.SSS0.P0.SPx1.p4.3">Denote by <math alttext="R" class="ltx_Math" display="inline" id="S2.SS0.SSS0.P0.SPx1.p4.1.m1.1"><semantics id="S2.SS0.SSS0.P0.SPx1.p4.1.m1.1a"><mi id="S2.SS0.SSS0.P0.SPx1.p4.1.m1.1.1" xref="S2.SS0.SSS0.P0.SPx1.p4.1.m1.1.1.cmml">R</mi><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.P0.SPx1.p4.1.m1.1b"><ci id="S2.SS0.SSS0.P0.SPx1.p4.1.m1.1.1.cmml" xref="S2.SS0.SSS0.P0.SPx1.p4.1.m1.1.1">𝑅</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.P0.SPx1.p4.1.m1.1c">R</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.P0.SPx1.p4.1.m1.1d">italic_R</annotation></semantics></math> the optimal value of DS and by <math alttext="\Delta" class="ltx_Math" display="inline" id="S2.SS0.SSS0.P0.SPx1.p4.2.m2.1"><semantics id="S2.SS0.SSS0.P0.SPx1.p4.2.m2.1a"><mi id="S2.SS0.SSS0.P0.SPx1.p4.2.m2.1.1" mathvariant="normal" xref="S2.SS0.SSS0.P0.SPx1.p4.2.m2.1.1.cmml">Δ</mi><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.P0.SPx1.p4.2.m2.1b"><ci id="S2.SS0.SSS0.P0.SPx1.p4.2.m2.1.1.cmml" xref="S2.SS0.SSS0.P0.SPx1.p4.2.m2.1.1">Δ</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.P0.SPx1.p4.2.m2.1c">\Delta</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.P0.SPx1.p4.2.m2.1d">roman_Δ</annotation></semantics></math> the optimal value of FO. By duality, <math alttext="R=\Delta" class="ltx_Math" display="inline" id="S2.SS0.SSS0.P0.SPx1.p4.3.m3.1"><semantics id="S2.SS0.SSS0.P0.SPx1.p4.3.m3.1a"><mrow id="S2.SS0.SSS0.P0.SPx1.p4.3.m3.1.1" xref="S2.SS0.SSS0.P0.SPx1.p4.3.m3.1.1.cmml"><mi id="S2.SS0.SSS0.P0.SPx1.p4.3.m3.1.1.2" xref="S2.SS0.SSS0.P0.SPx1.p4.3.m3.1.1.2.cmml">R</mi><mo id="S2.SS0.SSS0.P0.SPx1.p4.3.m3.1.1.1" xref="S2.SS0.SSS0.P0.SPx1.p4.3.m3.1.1.1.cmml">=</mo><mi id="S2.SS0.SSS0.P0.SPx1.p4.3.m3.1.1.3" mathvariant="normal" xref="S2.SS0.SSS0.P0.SPx1.p4.3.m3.1.1.3.cmml">Δ</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.P0.SPx1.p4.3.m3.1b"><apply id="S2.SS0.SSS0.P0.SPx1.p4.3.m3.1.1.cmml" xref="S2.SS0.SSS0.P0.SPx1.p4.3.m3.1.1"><eq id="S2.SS0.SSS0.P0.SPx1.p4.3.m3.1.1.1.cmml" xref="S2.SS0.SSS0.P0.SPx1.p4.3.m3.1.1.1"></eq><ci id="S2.SS0.SSS0.P0.SPx1.p4.3.m3.1.1.2.cmml" xref="S2.SS0.SSS0.P0.SPx1.p4.3.m3.1.1.2">𝑅</ci><ci id="S2.SS0.SSS0.P0.SPx1.p4.3.m3.1.1.3.cmml" xref="S2.SS0.SSS0.P0.SPx1.p4.3.m3.1.1.3">Δ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.P0.SPx1.p4.3.m3.1c">R=\Delta</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.P0.SPx1.p4.3.m3.1d">italic_R = roman_Δ</annotation></semantics></math>. Moreover, Charikar <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2411.12694v2#bib.bib6" title="">6</a>]</cite> relates these two linear programs to the densest subgraph problem:</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>(Theorem 1 in <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2411.12694v2#bib.bib6" title="">6</a>]</cite>)<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.p1.5"><span class="ltx_text ltx_font_italic" id="S2.Thmtheorem1.p1.5.5">Let <math alttext="G" class="ltx_Math" display="inline" id="S2.Thmtheorem1.p1.1.1.m1.1"><semantics id="S2.Thmtheorem1.p1.1.1.m1.1a"><mi id="S2.Thmtheorem1.p1.1.1.m1.1.1" xref="S2.Thmtheorem1.p1.1.1.m1.1.1.cmml">G</mi><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem1.p1.1.1.m1.1b"><ci id="S2.Thmtheorem1.p1.1.1.m1.1.1.cmml" xref="S2.Thmtheorem1.p1.1.1.m1.1.1">𝐺</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem1.p1.1.1.m1.1c">G</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem1.p1.1.1.m1.1d">italic_G</annotation></semantics></math> be a unit weight graph. Denote by <math alttext="R" class="ltx_Math" display="inline" id="S2.Thmtheorem1.p1.2.2.m2.1"><semantics id="S2.Thmtheorem1.p1.2.2.m2.1a"><mi id="S2.Thmtheorem1.p1.2.2.m2.1.1" xref="S2.Thmtheorem1.p1.2.2.m2.1.1.cmml">R</mi><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem1.p1.2.2.m2.1b"><ci id="S2.Thmtheorem1.p1.2.2.m2.1.1.cmml" xref="S2.Thmtheorem1.p1.2.2.m2.1.1">𝑅</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem1.p1.2.2.m2.1c">R</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem1.p1.2.2.m2.1d">italic_R</annotation></semantics></math> the optimal solution of DS and by <math alttext="D" class="ltx_Math" display="inline" id="S2.Thmtheorem1.p1.3.3.m3.1"><semantics id="S2.Thmtheorem1.p1.3.3.m3.1a"><mi id="S2.Thmtheorem1.p1.3.3.m3.1.1" xref="S2.Thmtheorem1.p1.3.3.m3.1.1.cmml">D</mi><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem1.p1.3.3.m3.1b"><ci id="S2.Thmtheorem1.p1.3.3.m3.1.1.cmml" xref="S2.Thmtheorem1.p1.3.3.m3.1.1">𝐷</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem1.p1.3.3.m3.1c">D</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem1.p1.3.3.m3.1d">italic_D</annotation></semantics></math> the optimal solution of <math alttext="FO" class="ltx_Math" display="inline" id="S2.Thmtheorem1.p1.4.4.m4.1"><semantics id="S2.Thmtheorem1.p1.4.4.m4.1a"><mrow id="S2.Thmtheorem1.p1.4.4.m4.1.1" xref="S2.Thmtheorem1.p1.4.4.m4.1.1.cmml"><mi id="S2.Thmtheorem1.p1.4.4.m4.1.1.2" xref="S2.Thmtheorem1.p1.4.4.m4.1.1.2.cmml">F</mi><mo id="S2.Thmtheorem1.p1.4.4.m4.1.1.1" xref="S2.Thmtheorem1.p1.4.4.m4.1.1.1.cmml">⁢</mo><mi id="S2.Thmtheorem1.p1.4.4.m4.1.1.3" xref="S2.Thmtheorem1.p1.4.4.m4.1.1.3.cmml">O</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem1.p1.4.4.m4.1b"><apply id="S2.Thmtheorem1.p1.4.4.m4.1.1.cmml" xref="S2.Thmtheorem1.p1.4.4.m4.1.1"><times id="S2.Thmtheorem1.p1.4.4.m4.1.1.1.cmml" xref="S2.Thmtheorem1.p1.4.4.m4.1.1.1"></times><ci id="S2.Thmtheorem1.p1.4.4.m4.1.1.2.cmml" xref="S2.Thmtheorem1.p1.4.4.m4.1.1.2">𝐹</ci><ci id="S2.Thmtheorem1.p1.4.4.m4.1.1.3.cmml" xref="S2.Thmtheorem1.p1.4.4.m4.1.1.3">𝑂</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem1.p1.4.4.m4.1c">FO</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem1.p1.4.4.m4.1d">italic_F italic_O</annotation></semantics></math>. Then <math alttext="\rho^{\max}(G)=R=\Delta=\Delta^{\min}(G)" class="ltx_Math" display="inline" id="S2.Thmtheorem1.p1.5.5.m5.2"><semantics id="S2.Thmtheorem1.p1.5.5.m5.2a"><mrow id="S2.Thmtheorem1.p1.5.5.m5.2.3" xref="S2.Thmtheorem1.p1.5.5.m5.2.3.cmml"><mrow id="S2.Thmtheorem1.p1.5.5.m5.2.3.2" xref="S2.Thmtheorem1.p1.5.5.m5.2.3.2.cmml"><msup id="S2.Thmtheorem1.p1.5.5.m5.2.3.2.2" xref="S2.Thmtheorem1.p1.5.5.m5.2.3.2.2.cmml"><mi id="S2.Thmtheorem1.p1.5.5.m5.2.3.2.2.2" xref="S2.Thmtheorem1.p1.5.5.m5.2.3.2.2.2.cmml">ρ</mi><mi id="S2.Thmtheorem1.p1.5.5.m5.2.3.2.2.3" xref="S2.Thmtheorem1.p1.5.5.m5.2.3.2.2.3.cmml">max</mi></msup><mo id="S2.Thmtheorem1.p1.5.5.m5.2.3.2.1" xref="S2.Thmtheorem1.p1.5.5.m5.2.3.2.1.cmml">⁢</mo><mrow id="S2.Thmtheorem1.p1.5.5.m5.2.3.2.3.2" xref="S2.Thmtheorem1.p1.5.5.m5.2.3.2.cmml"><mo id="S2.Thmtheorem1.p1.5.5.m5.2.3.2.3.2.1" stretchy="false" xref="S2.Thmtheorem1.p1.5.5.m5.2.3.2.cmml">(</mo><mi id="S2.Thmtheorem1.p1.5.5.m5.1.1" xref="S2.Thmtheorem1.p1.5.5.m5.1.1.cmml">G</mi><mo id="S2.Thmtheorem1.p1.5.5.m5.2.3.2.3.2.2" stretchy="false" xref="S2.Thmtheorem1.p1.5.5.m5.2.3.2.cmml">)</mo></mrow></mrow><mo id="S2.Thmtheorem1.p1.5.5.m5.2.3.3" xref="S2.Thmtheorem1.p1.5.5.m5.2.3.3.cmml">=</mo><mi id="S2.Thmtheorem1.p1.5.5.m5.2.3.4" xref="S2.Thmtheorem1.p1.5.5.m5.2.3.4.cmml">R</mi><mo id="S2.Thmtheorem1.p1.5.5.m5.2.3.5" xref="S2.Thmtheorem1.p1.5.5.m5.2.3.5.cmml">=</mo><mi id="S2.Thmtheorem1.p1.5.5.m5.2.3.6" mathvariant="normal" xref="S2.Thmtheorem1.p1.5.5.m5.2.3.6.cmml">Δ</mi><mo id="S2.Thmtheorem1.p1.5.5.m5.2.3.7" xref="S2.Thmtheorem1.p1.5.5.m5.2.3.7.cmml">=</mo><mrow id="S2.Thmtheorem1.p1.5.5.m5.2.3.8" xref="S2.Thmtheorem1.p1.5.5.m5.2.3.8.cmml"><msup id="S2.Thmtheorem1.p1.5.5.m5.2.3.8.2" xref="S2.Thmtheorem1.p1.5.5.m5.2.3.8.2.cmml"><mi id="S2.Thmtheorem1.p1.5.5.m5.2.3.8.2.2" mathvariant="normal" xref="S2.Thmtheorem1.p1.5.5.m5.2.3.8.2.2.cmml">Δ</mi><mi id="S2.Thmtheorem1.p1.5.5.m5.2.3.8.2.3" xref="S2.Thmtheorem1.p1.5.5.m5.2.3.8.2.3.cmml">min</mi></msup><mo id="S2.Thmtheorem1.p1.5.5.m5.2.3.8.1" xref="S2.Thmtheorem1.p1.5.5.m5.2.3.8.1.cmml">⁢</mo><mrow id="S2.Thmtheorem1.p1.5.5.m5.2.3.8.3.2" xref="S2.Thmtheorem1.p1.5.5.m5.2.3.8.cmml"><mo id="S2.Thmtheorem1.p1.5.5.m5.2.3.8.3.2.1" stretchy="false" xref="S2.Thmtheorem1.p1.5.5.m5.2.3.8.cmml">(</mo><mi id="S2.Thmtheorem1.p1.5.5.m5.2.2" xref="S2.Thmtheorem1.p1.5.5.m5.2.2.cmml">G</mi><mo id="S2.Thmtheorem1.p1.5.5.m5.2.3.8.3.2.2" stretchy="false" xref="S2.Thmtheorem1.p1.5.5.m5.2.3.8.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem1.p1.5.5.m5.2b"><apply id="S2.Thmtheorem1.p1.5.5.m5.2.3.cmml" xref="S2.Thmtheorem1.p1.5.5.m5.2.3"><and id="S2.Thmtheorem1.p1.5.5.m5.2.3a.cmml" xref="S2.Thmtheorem1.p1.5.5.m5.2.3"></and><apply id="S2.Thmtheorem1.p1.5.5.m5.2.3b.cmml" xref="S2.Thmtheorem1.p1.5.5.m5.2.3"><eq id="S2.Thmtheorem1.p1.5.5.m5.2.3.3.cmml" xref="S2.Thmtheorem1.p1.5.5.m5.2.3.3"></eq><apply id="S2.Thmtheorem1.p1.5.5.m5.2.3.2.cmml" xref="S2.Thmtheorem1.p1.5.5.m5.2.3.2"><times id="S2.Thmtheorem1.p1.5.5.m5.2.3.2.1.cmml" xref="S2.Thmtheorem1.p1.5.5.m5.2.3.2.1"></times><apply id="S2.Thmtheorem1.p1.5.5.m5.2.3.2.2.cmml" xref="S2.Thmtheorem1.p1.5.5.m5.2.3.2.2"><csymbol cd="ambiguous" id="S2.Thmtheorem1.p1.5.5.m5.2.3.2.2.1.cmml" xref="S2.Thmtheorem1.p1.5.5.m5.2.3.2.2">superscript</csymbol><ci id="S2.Thmtheorem1.p1.5.5.m5.2.3.2.2.2.cmml" xref="S2.Thmtheorem1.p1.5.5.m5.2.3.2.2.2">𝜌</ci><max id="S2.Thmtheorem1.p1.5.5.m5.2.3.2.2.3.cmml" xref="S2.Thmtheorem1.p1.5.5.m5.2.3.2.2.3"></max></apply><ci id="S2.Thmtheorem1.p1.5.5.m5.1.1.cmml" xref="S2.Thmtheorem1.p1.5.5.m5.1.1">𝐺</ci></apply><ci id="S2.Thmtheorem1.p1.5.5.m5.2.3.4.cmml" xref="S2.Thmtheorem1.p1.5.5.m5.2.3.4">𝑅</ci></apply><apply id="S2.Thmtheorem1.p1.5.5.m5.2.3c.cmml" xref="S2.Thmtheorem1.p1.5.5.m5.2.3"><eq id="S2.Thmtheorem1.p1.5.5.m5.2.3.5.cmml" xref="S2.Thmtheorem1.p1.5.5.m5.2.3.5"></eq><share href="https://arxiv.org/html/2411.12694v2#S2.Thmtheorem1.p1.5.5.m5.2.3.4.cmml" id="S2.Thmtheorem1.p1.5.5.m5.2.3d.cmml" xref="S2.Thmtheorem1.p1.5.5.m5.2.3"></share><ci id="S2.Thmtheorem1.p1.5.5.m5.2.3.6.cmml" xref="S2.Thmtheorem1.p1.5.5.m5.2.3.6">Δ</ci></apply><apply id="S2.Thmtheorem1.p1.5.5.m5.2.3e.cmml" xref="S2.Thmtheorem1.p1.5.5.m5.2.3"><eq id="S2.Thmtheorem1.p1.5.5.m5.2.3.7.cmml" xref="S2.Thmtheorem1.p1.5.5.m5.2.3.7"></eq><share href="https://arxiv.org/html/2411.12694v2#S2.Thmtheorem1.p1.5.5.m5.2.3.6.cmml" id="S2.Thmtheorem1.p1.5.5.m5.2.3f.cmml" xref="S2.Thmtheorem1.p1.5.5.m5.2.3"></share><apply id="S2.Thmtheorem1.p1.5.5.m5.2.3.8.cmml" xref="S2.Thmtheorem1.p1.5.5.m5.2.3.8"><times id="S2.Thmtheorem1.p1.5.5.m5.2.3.8.1.cmml" xref="S2.Thmtheorem1.p1.5.5.m5.2.3.8.1"></times><apply id="S2.Thmtheorem1.p1.5.5.m5.2.3.8.2.cmml" xref="S2.Thmtheorem1.p1.5.5.m5.2.3.8.2"><csymbol cd="ambiguous" id="S2.Thmtheorem1.p1.5.5.m5.2.3.8.2.1.cmml" xref="S2.Thmtheorem1.p1.5.5.m5.2.3.8.2">superscript</csymbol><ci id="S2.Thmtheorem1.p1.5.5.m5.2.3.8.2.2.cmml" xref="S2.Thmtheorem1.p1.5.5.m5.2.3.8.2.2">Δ</ci><min id="S2.Thmtheorem1.p1.5.5.m5.2.3.8.2.3.cmml" xref="S2.Thmtheorem1.p1.5.5.m5.2.3.8.2.3"></min></apply><ci id="S2.Thmtheorem1.p1.5.5.m5.2.2.cmml" xref="S2.Thmtheorem1.p1.5.5.m5.2.2">𝐺</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem1.p1.5.5.m5.2c">\rho^{\max}(G)=R=\Delta=\Delta^{\min}(G)</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem1.p1.5.5.m5.2d">italic_ρ start_POSTSUPERSCRIPT roman_max end_POSTSUPERSCRIPT ( italic_G ) = italic_R = roman_Δ = roman_Δ start_POSTSUPERSCRIPT roman_min end_POSTSUPERSCRIPT ( italic_G )</annotation></semantics></math>.</span></p> </div> </div> <div class="ltx_para ltx_noindent" id="S2.SS0.SSS0.P0.SPx1.p5"> <p class="ltx_p" id="S2.SS0.SSS0.P0.SPx1.p5.1">We show that this can be generalised to when <math alttext="G" class="ltx_Math" display="inline" id="S2.SS0.SSS0.P0.SPx1.p5.1.m1.1"><semantics id="S2.SS0.SSS0.P0.SPx1.p5.1.m1.1a"><mi id="S2.SS0.SSS0.P0.SPx1.p5.1.m1.1.1" xref="S2.SS0.SSS0.P0.SPx1.p5.1.m1.1.1.cmml">G</mi><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.P0.SPx1.p5.1.m1.1b"><ci id="S2.SS0.SSS0.P0.SPx1.p5.1.m1.1.1.cmml" xref="S2.SS0.SSS0.P0.SPx1.p5.1.m1.1.1">𝐺</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.P0.SPx1.p5.1.m1.1c">G</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.P0.SPx1.p5.1.m1.1d">italic_G</annotation></semantics></math> is a weighted graph:</p> </div> <div class="ltx_theorem ltx_theorem_lemma" 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">Lemma 2.2</span></span><span class="ltx_text ltx_font_bold" id="S2.Thmtheorem2.2.2">.</span> </h6> <div class="ltx_para" id="S2.Thmtheorem2.p1"> <p class="ltx_p" id="S2.Thmtheorem2.p1.5"><span class="ltx_text ltx_font_italic" id="S2.Thmtheorem2.p1.5.5">Let <math alttext="G" class="ltx_Math" display="inline" id="S2.Thmtheorem2.p1.1.1.m1.1"><semantics id="S2.Thmtheorem2.p1.1.1.m1.1a"><mi id="S2.Thmtheorem2.p1.1.1.m1.1.1" xref="S2.Thmtheorem2.p1.1.1.m1.1.1.cmml">G</mi><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem2.p1.1.1.m1.1b"><ci id="S2.Thmtheorem2.p1.1.1.m1.1.1.cmml" xref="S2.Thmtheorem2.p1.1.1.m1.1.1">𝐺</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem2.p1.1.1.m1.1c">G</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem2.p1.1.1.m1.1d">italic_G</annotation></semantics></math> be any weighted graph. Denote by <math alttext="R" class="ltx_Math" display="inline" id="S2.Thmtheorem2.p1.2.2.m2.1"><semantics id="S2.Thmtheorem2.p1.2.2.m2.1a"><mi id="S2.Thmtheorem2.p1.2.2.m2.1.1" xref="S2.Thmtheorem2.p1.2.2.m2.1.1.cmml">R</mi><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem2.p1.2.2.m2.1b"><ci id="S2.Thmtheorem2.p1.2.2.m2.1.1.cmml" xref="S2.Thmtheorem2.p1.2.2.m2.1.1">𝑅</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem2.p1.2.2.m2.1c">R</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem2.p1.2.2.m2.1d">italic_R</annotation></semantics></math> the optimal solution of DS and by <math alttext="D" class="ltx_Math" display="inline" id="S2.Thmtheorem2.p1.3.3.m3.1"><semantics id="S2.Thmtheorem2.p1.3.3.m3.1a"><mi id="S2.Thmtheorem2.p1.3.3.m3.1.1" xref="S2.Thmtheorem2.p1.3.3.m3.1.1.cmml">D</mi><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem2.p1.3.3.m3.1b"><ci id="S2.Thmtheorem2.p1.3.3.m3.1.1.cmml" xref="S2.Thmtheorem2.p1.3.3.m3.1.1">𝐷</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem2.p1.3.3.m3.1c">D</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem2.p1.3.3.m3.1d">italic_D</annotation></semantics></math> the optimal solution of <math alttext="FO" class="ltx_Math" display="inline" id="S2.Thmtheorem2.p1.4.4.m4.1"><semantics id="S2.Thmtheorem2.p1.4.4.m4.1a"><mrow id="S2.Thmtheorem2.p1.4.4.m4.1.1" xref="S2.Thmtheorem2.p1.4.4.m4.1.1.cmml"><mi id="S2.Thmtheorem2.p1.4.4.m4.1.1.2" xref="S2.Thmtheorem2.p1.4.4.m4.1.1.2.cmml">F</mi><mo id="S2.Thmtheorem2.p1.4.4.m4.1.1.1" xref="S2.Thmtheorem2.p1.4.4.m4.1.1.1.cmml">⁢</mo><mi id="S2.Thmtheorem2.p1.4.4.m4.1.1.3" xref="S2.Thmtheorem2.p1.4.4.m4.1.1.3.cmml">O</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem2.p1.4.4.m4.1b"><apply id="S2.Thmtheorem2.p1.4.4.m4.1.1.cmml" xref="S2.Thmtheorem2.p1.4.4.m4.1.1"><times id="S2.Thmtheorem2.p1.4.4.m4.1.1.1.cmml" xref="S2.Thmtheorem2.p1.4.4.m4.1.1.1"></times><ci id="S2.Thmtheorem2.p1.4.4.m4.1.1.2.cmml" xref="S2.Thmtheorem2.p1.4.4.m4.1.1.2">𝐹</ci><ci id="S2.Thmtheorem2.p1.4.4.m4.1.1.3.cmml" xref="S2.Thmtheorem2.p1.4.4.m4.1.1.3">𝑂</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem2.p1.4.4.m4.1c">FO</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem2.p1.4.4.m4.1d">italic_F italic_O</annotation></semantics></math>. Then <math alttext="\rho^{\max}(G)=R=D=\Delta^{\min}(G)" class="ltx_Math" display="inline" id="S2.Thmtheorem2.p1.5.5.m5.2"><semantics id="S2.Thmtheorem2.p1.5.5.m5.2a"><mrow id="S2.Thmtheorem2.p1.5.5.m5.2.3" xref="S2.Thmtheorem2.p1.5.5.m5.2.3.cmml"><mrow id="S2.Thmtheorem2.p1.5.5.m5.2.3.2" xref="S2.Thmtheorem2.p1.5.5.m5.2.3.2.cmml"><msup id="S2.Thmtheorem2.p1.5.5.m5.2.3.2.2" xref="S2.Thmtheorem2.p1.5.5.m5.2.3.2.2.cmml"><mi id="S2.Thmtheorem2.p1.5.5.m5.2.3.2.2.2" xref="S2.Thmtheorem2.p1.5.5.m5.2.3.2.2.2.cmml">ρ</mi><mi id="S2.Thmtheorem2.p1.5.5.m5.2.3.2.2.3" xref="S2.Thmtheorem2.p1.5.5.m5.2.3.2.2.3.cmml">max</mi></msup><mo id="S2.Thmtheorem2.p1.5.5.m5.2.3.2.1" xref="S2.Thmtheorem2.p1.5.5.m5.2.3.2.1.cmml">⁢</mo><mrow id="S2.Thmtheorem2.p1.5.5.m5.2.3.2.3.2" xref="S2.Thmtheorem2.p1.5.5.m5.2.3.2.cmml"><mo id="S2.Thmtheorem2.p1.5.5.m5.2.3.2.3.2.1" stretchy="false" xref="S2.Thmtheorem2.p1.5.5.m5.2.3.2.cmml">(</mo><mi id="S2.Thmtheorem2.p1.5.5.m5.1.1" xref="S2.Thmtheorem2.p1.5.5.m5.1.1.cmml">G</mi><mo id="S2.Thmtheorem2.p1.5.5.m5.2.3.2.3.2.2" stretchy="false" xref="S2.Thmtheorem2.p1.5.5.m5.2.3.2.cmml">)</mo></mrow></mrow><mo id="S2.Thmtheorem2.p1.5.5.m5.2.3.3" xref="S2.Thmtheorem2.p1.5.5.m5.2.3.3.cmml">=</mo><mi id="S2.Thmtheorem2.p1.5.5.m5.2.3.4" xref="S2.Thmtheorem2.p1.5.5.m5.2.3.4.cmml">R</mi><mo id="S2.Thmtheorem2.p1.5.5.m5.2.3.5" xref="S2.Thmtheorem2.p1.5.5.m5.2.3.5.cmml">=</mo><mi id="S2.Thmtheorem2.p1.5.5.m5.2.3.6" xref="S2.Thmtheorem2.p1.5.5.m5.2.3.6.cmml">D</mi><mo id="S2.Thmtheorem2.p1.5.5.m5.2.3.7" xref="S2.Thmtheorem2.p1.5.5.m5.2.3.7.cmml">=</mo><mrow id="S2.Thmtheorem2.p1.5.5.m5.2.3.8" xref="S2.Thmtheorem2.p1.5.5.m5.2.3.8.cmml"><msup id="S2.Thmtheorem2.p1.5.5.m5.2.3.8.2" xref="S2.Thmtheorem2.p1.5.5.m5.2.3.8.2.cmml"><mi id="S2.Thmtheorem2.p1.5.5.m5.2.3.8.2.2" mathvariant="normal" xref="S2.Thmtheorem2.p1.5.5.m5.2.3.8.2.2.cmml">Δ</mi><mi id="S2.Thmtheorem2.p1.5.5.m5.2.3.8.2.3" xref="S2.Thmtheorem2.p1.5.5.m5.2.3.8.2.3.cmml">min</mi></msup><mo id="S2.Thmtheorem2.p1.5.5.m5.2.3.8.1" xref="S2.Thmtheorem2.p1.5.5.m5.2.3.8.1.cmml">⁢</mo><mrow id="S2.Thmtheorem2.p1.5.5.m5.2.3.8.3.2" xref="S2.Thmtheorem2.p1.5.5.m5.2.3.8.cmml"><mo id="S2.Thmtheorem2.p1.5.5.m5.2.3.8.3.2.1" stretchy="false" xref="S2.Thmtheorem2.p1.5.5.m5.2.3.8.cmml">(</mo><mi id="S2.Thmtheorem2.p1.5.5.m5.2.2" xref="S2.Thmtheorem2.p1.5.5.m5.2.2.cmml">G</mi><mo id="S2.Thmtheorem2.p1.5.5.m5.2.3.8.3.2.2" stretchy="false" xref="S2.Thmtheorem2.p1.5.5.m5.2.3.8.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem2.p1.5.5.m5.2b"><apply id="S2.Thmtheorem2.p1.5.5.m5.2.3.cmml" xref="S2.Thmtheorem2.p1.5.5.m5.2.3"><and id="S2.Thmtheorem2.p1.5.5.m5.2.3a.cmml" xref="S2.Thmtheorem2.p1.5.5.m5.2.3"></and><apply id="S2.Thmtheorem2.p1.5.5.m5.2.3b.cmml" xref="S2.Thmtheorem2.p1.5.5.m5.2.3"><eq id="S2.Thmtheorem2.p1.5.5.m5.2.3.3.cmml" xref="S2.Thmtheorem2.p1.5.5.m5.2.3.3"></eq><apply id="S2.Thmtheorem2.p1.5.5.m5.2.3.2.cmml" xref="S2.Thmtheorem2.p1.5.5.m5.2.3.2"><times id="S2.Thmtheorem2.p1.5.5.m5.2.3.2.1.cmml" xref="S2.Thmtheorem2.p1.5.5.m5.2.3.2.1"></times><apply id="S2.Thmtheorem2.p1.5.5.m5.2.3.2.2.cmml" xref="S2.Thmtheorem2.p1.5.5.m5.2.3.2.2"><csymbol cd="ambiguous" id="S2.Thmtheorem2.p1.5.5.m5.2.3.2.2.1.cmml" xref="S2.Thmtheorem2.p1.5.5.m5.2.3.2.2">superscript</csymbol><ci id="S2.Thmtheorem2.p1.5.5.m5.2.3.2.2.2.cmml" xref="S2.Thmtheorem2.p1.5.5.m5.2.3.2.2.2">𝜌</ci><max id="S2.Thmtheorem2.p1.5.5.m5.2.3.2.2.3.cmml" xref="S2.Thmtheorem2.p1.5.5.m5.2.3.2.2.3"></max></apply><ci id="S2.Thmtheorem2.p1.5.5.m5.1.1.cmml" xref="S2.Thmtheorem2.p1.5.5.m5.1.1">𝐺</ci></apply><ci id="S2.Thmtheorem2.p1.5.5.m5.2.3.4.cmml" xref="S2.Thmtheorem2.p1.5.5.m5.2.3.4">𝑅</ci></apply><apply id="S2.Thmtheorem2.p1.5.5.m5.2.3c.cmml" xref="S2.Thmtheorem2.p1.5.5.m5.2.3"><eq id="S2.Thmtheorem2.p1.5.5.m5.2.3.5.cmml" xref="S2.Thmtheorem2.p1.5.5.m5.2.3.5"></eq><share href="https://arxiv.org/html/2411.12694v2#S2.Thmtheorem2.p1.5.5.m5.2.3.4.cmml" id="S2.Thmtheorem2.p1.5.5.m5.2.3d.cmml" xref="S2.Thmtheorem2.p1.5.5.m5.2.3"></share><ci id="S2.Thmtheorem2.p1.5.5.m5.2.3.6.cmml" xref="S2.Thmtheorem2.p1.5.5.m5.2.3.6">𝐷</ci></apply><apply id="S2.Thmtheorem2.p1.5.5.m5.2.3e.cmml" xref="S2.Thmtheorem2.p1.5.5.m5.2.3"><eq id="S2.Thmtheorem2.p1.5.5.m5.2.3.7.cmml" xref="S2.Thmtheorem2.p1.5.5.m5.2.3.7"></eq><share href="https://arxiv.org/html/2411.12694v2#S2.Thmtheorem2.p1.5.5.m5.2.3.6.cmml" id="S2.Thmtheorem2.p1.5.5.m5.2.3f.cmml" xref="S2.Thmtheorem2.p1.5.5.m5.2.3"></share><apply id="S2.Thmtheorem2.p1.5.5.m5.2.3.8.cmml" xref="S2.Thmtheorem2.p1.5.5.m5.2.3.8"><times id="S2.Thmtheorem2.p1.5.5.m5.2.3.8.1.cmml" xref="S2.Thmtheorem2.p1.5.5.m5.2.3.8.1"></times><apply id="S2.Thmtheorem2.p1.5.5.m5.2.3.8.2.cmml" xref="S2.Thmtheorem2.p1.5.5.m5.2.3.8.2"><csymbol cd="ambiguous" id="S2.Thmtheorem2.p1.5.5.m5.2.3.8.2.1.cmml" xref="S2.Thmtheorem2.p1.5.5.m5.2.3.8.2">superscript</csymbol><ci id="S2.Thmtheorem2.p1.5.5.m5.2.3.8.2.2.cmml" xref="S2.Thmtheorem2.p1.5.5.m5.2.3.8.2.2">Δ</ci><min id="S2.Thmtheorem2.p1.5.5.m5.2.3.8.2.3.cmml" xref="S2.Thmtheorem2.p1.5.5.m5.2.3.8.2.3"></min></apply><ci id="S2.Thmtheorem2.p1.5.5.m5.2.2.cmml" xref="S2.Thmtheorem2.p1.5.5.m5.2.2">𝐺</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem2.p1.5.5.m5.2c">\rho^{\max}(G)=R=D=\Delta^{\min}(G)</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem2.p1.5.5.m5.2d">italic_ρ start_POSTSUPERSCRIPT roman_max end_POSTSUPERSCRIPT ( italic_G ) = italic_R = italic_D = roman_Δ start_POSTSUPERSCRIPT roman_min end_POSTSUPERSCRIPT ( italic_G )</annotation></semantics></math>.</span></p> </div> </div> <div class="ltx_theorem ltx_theorem_proof" 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">Proof 2.3</span></span><span class="ltx_text ltx_font_bold" id="S2.Thmtheorem3.2.2">.</span> </h6> <div class="ltx_para" id="S2.Thmtheorem3.p1"> <p class="ltx_p" id="S2.Thmtheorem3.p1.10"><span class="ltx_text ltx_font_bold ltx_font_italic" id="S2.Thmtheorem3.p1.10.10">We show that <math alttext="R\geq\rho^{\max}(G)" class="ltx_Math" display="inline" id="S2.Thmtheorem3.p1.1.1.m1.1"><semantics id="S2.Thmtheorem3.p1.1.1.m1.1a"><mrow id="S2.Thmtheorem3.p1.1.1.m1.1.2" xref="S2.Thmtheorem3.p1.1.1.m1.1.2.cmml"><mi id="S2.Thmtheorem3.p1.1.1.m1.1.2.2" mathvariant="normal" xref="S2.Thmtheorem3.p1.1.1.m1.1.2.2.cmml">R</mi><mo id="S2.Thmtheorem3.p1.1.1.m1.1.2.1" xref="S2.Thmtheorem3.p1.1.1.m1.1.2.1.cmml">≥</mo><mrow id="S2.Thmtheorem3.p1.1.1.m1.1.2.3" xref="S2.Thmtheorem3.p1.1.1.m1.1.2.3.cmml"><msup id="S2.Thmtheorem3.p1.1.1.m1.1.2.3.2" xref="S2.Thmtheorem3.p1.1.1.m1.1.2.3.2.cmml"><mi id="S2.Thmtheorem3.p1.1.1.m1.1.2.3.2.2" mathvariant="normal" xref="S2.Thmtheorem3.p1.1.1.m1.1.2.3.2.2.cmml">ρ</mi><mi id="S2.Thmtheorem3.p1.1.1.m1.1.2.3.2.3" xref="S2.Thmtheorem3.p1.1.1.m1.1.2.3.2.3.cmml">max</mi></msup><mo id="S2.Thmtheorem3.p1.1.1.m1.1.2.3.1" xref="S2.Thmtheorem3.p1.1.1.m1.1.2.3.1.cmml">⁢</mo><mrow id="S2.Thmtheorem3.p1.1.1.m1.1.2.3.3.2" xref="S2.Thmtheorem3.p1.1.1.m1.1.2.3.cmml"><mo id="S2.Thmtheorem3.p1.1.1.m1.1.2.3.3.2.1" stretchy="false" xref="S2.Thmtheorem3.p1.1.1.m1.1.2.3.cmml">(</mo><mi id="S2.Thmtheorem3.p1.1.1.m1.1.1" mathvariant="normal" xref="S2.Thmtheorem3.p1.1.1.m1.1.1.cmml">G</mi><mo id="S2.Thmtheorem3.p1.1.1.m1.1.2.3.3.2.2" stretchy="false" xref="S2.Thmtheorem3.p1.1.1.m1.1.2.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem3.p1.1.1.m1.1b"><apply id="S2.Thmtheorem3.p1.1.1.m1.1.2.cmml" xref="S2.Thmtheorem3.p1.1.1.m1.1.2"><geq id="S2.Thmtheorem3.p1.1.1.m1.1.2.1.cmml" xref="S2.Thmtheorem3.p1.1.1.m1.1.2.1"></geq><ci id="S2.Thmtheorem3.p1.1.1.m1.1.2.2.cmml" xref="S2.Thmtheorem3.p1.1.1.m1.1.2.2">R</ci><apply id="S2.Thmtheorem3.p1.1.1.m1.1.2.3.cmml" xref="S2.Thmtheorem3.p1.1.1.m1.1.2.3"><times id="S2.Thmtheorem3.p1.1.1.m1.1.2.3.1.cmml" xref="S2.Thmtheorem3.p1.1.1.m1.1.2.3.1"></times><apply id="S2.Thmtheorem3.p1.1.1.m1.1.2.3.2.cmml" xref="S2.Thmtheorem3.p1.1.1.m1.1.2.3.2"><csymbol cd="ambiguous" id="S2.Thmtheorem3.p1.1.1.m1.1.2.3.2.1.cmml" xref="S2.Thmtheorem3.p1.1.1.m1.1.2.3.2">superscript</csymbol><ci id="S2.Thmtheorem3.p1.1.1.m1.1.2.3.2.2.cmml" xref="S2.Thmtheorem3.p1.1.1.m1.1.2.3.2.2">ρ</ci><max id="S2.Thmtheorem3.p1.1.1.m1.1.2.3.2.3.cmml" xref="S2.Thmtheorem3.p1.1.1.m1.1.2.3.2.3"></max></apply><ci id="S2.Thmtheorem3.p1.1.1.m1.1.1.cmml" xref="S2.Thmtheorem3.p1.1.1.m1.1.1">G</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem3.p1.1.1.m1.1c">R\geq\rho^{\max}(G)</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem3.p1.1.1.m1.1d">italic_R ≥ italic_ρ start_POSTSUPERSCRIPT roman_max end_POSTSUPERSCRIPT ( italic_G )</annotation></semantics></math>.<span class="ltx_text ltx_font_medium" id="S2.Thmtheorem3.p1.10.10.9"> First note that if <math alttext="H\subseteq G" class="ltx_Math" display="inline" id="S2.Thmtheorem3.p1.2.2.1.m1.1"><semantics id="S2.Thmtheorem3.p1.2.2.1.m1.1a"><mrow id="S2.Thmtheorem3.p1.2.2.1.m1.1.1" xref="S2.Thmtheorem3.p1.2.2.1.m1.1.1.cmml"><mi id="S2.Thmtheorem3.p1.2.2.1.m1.1.1.2" mathvariant="normal" xref="S2.Thmtheorem3.p1.2.2.1.m1.1.1.2.cmml">H</mi><mo id="S2.Thmtheorem3.p1.2.2.1.m1.1.1.1" xref="S2.Thmtheorem3.p1.2.2.1.m1.1.1.1.cmml">⊆</mo><mi id="S2.Thmtheorem3.p1.2.2.1.m1.1.1.3" mathvariant="normal" xref="S2.Thmtheorem3.p1.2.2.1.m1.1.1.3.cmml">G</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem3.p1.2.2.1.m1.1b"><apply id="S2.Thmtheorem3.p1.2.2.1.m1.1.1.cmml" xref="S2.Thmtheorem3.p1.2.2.1.m1.1.1"><subset id="S2.Thmtheorem3.p1.2.2.1.m1.1.1.1.cmml" xref="S2.Thmtheorem3.p1.2.2.1.m1.1.1.1"></subset><ci id="S2.Thmtheorem3.p1.2.2.1.m1.1.1.2.cmml" xref="S2.Thmtheorem3.p1.2.2.1.m1.1.1.2">H</ci><ci id="S2.Thmtheorem3.p1.2.2.1.m1.1.1.3.cmml" xref="S2.Thmtheorem3.p1.2.2.1.m1.1.1.3">G</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem3.p1.2.2.1.m1.1c">H\subseteq G</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem3.p1.2.2.1.m1.1d">italic_H ⊆ italic_G</annotation></semantics></math> is the densest subgraph of <math alttext="G" class="ltx_Math" display="inline" id="S2.Thmtheorem3.p1.3.3.2.m2.1"><semantics id="S2.Thmtheorem3.p1.3.3.2.m2.1a"><mi id="S2.Thmtheorem3.p1.3.3.2.m2.1.1" mathvariant="normal" xref="S2.Thmtheorem3.p1.3.3.2.m2.1.1.cmml">G</mi><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem3.p1.3.3.2.m2.1b"><ci id="S2.Thmtheorem3.p1.3.3.2.m2.1.1.cmml" xref="S2.Thmtheorem3.p1.3.3.2.m2.1.1">G</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem3.p1.3.3.2.m2.1c">G</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem3.p1.3.3.2.m2.1d">italic_G</annotation></semantics></math>, then for every edge <math alttext="e\in E(H)" class="ltx_Math" display="inline" id="S2.Thmtheorem3.p1.4.4.3.m3.1"><semantics id="S2.Thmtheorem3.p1.4.4.3.m3.1a"><mrow id="S2.Thmtheorem3.p1.4.4.3.m3.1.2" xref="S2.Thmtheorem3.p1.4.4.3.m3.1.2.cmml"><mi id="S2.Thmtheorem3.p1.4.4.3.m3.1.2.2" mathvariant="normal" xref="S2.Thmtheorem3.p1.4.4.3.m3.1.2.2.cmml">e</mi><mo id="S2.Thmtheorem3.p1.4.4.3.m3.1.2.1" xref="S2.Thmtheorem3.p1.4.4.3.m3.1.2.1.cmml">∈</mo><mrow id="S2.Thmtheorem3.p1.4.4.3.m3.1.2.3" xref="S2.Thmtheorem3.p1.4.4.3.m3.1.2.3.cmml"><mi id="S2.Thmtheorem3.p1.4.4.3.m3.1.2.3.2" mathvariant="normal" xref="S2.Thmtheorem3.p1.4.4.3.m3.1.2.3.2.cmml">E</mi><mo id="S2.Thmtheorem3.p1.4.4.3.m3.1.2.3.1" xref="S2.Thmtheorem3.p1.4.4.3.m3.1.2.3.1.cmml">⁢</mo><mrow id="S2.Thmtheorem3.p1.4.4.3.m3.1.2.3.3.2" xref="S2.Thmtheorem3.p1.4.4.3.m3.1.2.3.cmml"><mo id="S2.Thmtheorem3.p1.4.4.3.m3.1.2.3.3.2.1" stretchy="false" xref="S2.Thmtheorem3.p1.4.4.3.m3.1.2.3.cmml">(</mo><mi id="S2.Thmtheorem3.p1.4.4.3.m3.1.1" mathvariant="normal" xref="S2.Thmtheorem3.p1.4.4.3.m3.1.1.cmml">H</mi><mo id="S2.Thmtheorem3.p1.4.4.3.m3.1.2.3.3.2.2" stretchy="false" xref="S2.Thmtheorem3.p1.4.4.3.m3.1.2.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem3.p1.4.4.3.m3.1b"><apply id="S2.Thmtheorem3.p1.4.4.3.m3.1.2.cmml" xref="S2.Thmtheorem3.p1.4.4.3.m3.1.2"><in id="S2.Thmtheorem3.p1.4.4.3.m3.1.2.1.cmml" xref="S2.Thmtheorem3.p1.4.4.3.m3.1.2.1"></in><ci id="S2.Thmtheorem3.p1.4.4.3.m3.1.2.2.cmml" xref="S2.Thmtheorem3.p1.4.4.3.m3.1.2.2">e</ci><apply id="S2.Thmtheorem3.p1.4.4.3.m3.1.2.3.cmml" xref="S2.Thmtheorem3.p1.4.4.3.m3.1.2.3"><times id="S2.Thmtheorem3.p1.4.4.3.m3.1.2.3.1.cmml" xref="S2.Thmtheorem3.p1.4.4.3.m3.1.2.3.1"></times><ci id="S2.Thmtheorem3.p1.4.4.3.m3.1.2.3.2.cmml" xref="S2.Thmtheorem3.p1.4.4.3.m3.1.2.3.2">E</ci><ci id="S2.Thmtheorem3.p1.4.4.3.m3.1.1.cmml" xref="S2.Thmtheorem3.p1.4.4.3.m3.1.1">H</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem3.p1.4.4.3.m3.1c">e\in E(H)</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem3.p1.4.4.3.m3.1d">italic_e ∈ italic_E ( italic_H )</annotation></semantics></math>, <math alttext="\textsl{h}(e)=\textsl{g}(e)" class="ltx_Math" display="inline" id="S2.Thmtheorem3.p1.5.5.4.m4.2"><semantics id="S2.Thmtheorem3.p1.5.5.4.m4.2a"><mrow id="S2.Thmtheorem3.p1.5.5.4.m4.2.3" xref="S2.Thmtheorem3.p1.5.5.4.m4.2.3.cmml"><mrow id="S2.Thmtheorem3.p1.5.5.4.m4.2.3.2" xref="S2.Thmtheorem3.p1.5.5.4.m4.2.3.2.cmml"><mtext class="ltx_mathvariant_italic" id="S2.Thmtheorem3.p1.5.5.4.m4.2.3.2.2" xref="S2.Thmtheorem3.p1.5.5.4.m4.2.3.2.2a.cmml">h</mtext><mo id="S2.Thmtheorem3.p1.5.5.4.m4.2.3.2.1" xref="S2.Thmtheorem3.p1.5.5.4.m4.2.3.2.1.cmml">⁢</mo><mrow id="S2.Thmtheorem3.p1.5.5.4.m4.2.3.2.3.2" xref="S2.Thmtheorem3.p1.5.5.4.m4.2.3.2.cmml"><mo id="S2.Thmtheorem3.p1.5.5.4.m4.2.3.2.3.2.1" stretchy="false" xref="S2.Thmtheorem3.p1.5.5.4.m4.2.3.2.cmml">(</mo><mi id="S2.Thmtheorem3.p1.5.5.4.m4.1.1" mathvariant="normal" xref="S2.Thmtheorem3.p1.5.5.4.m4.1.1.cmml">e</mi><mo id="S2.Thmtheorem3.p1.5.5.4.m4.2.3.2.3.2.2" stretchy="false" xref="S2.Thmtheorem3.p1.5.5.4.m4.2.3.2.cmml">)</mo></mrow></mrow><mo id="S2.Thmtheorem3.p1.5.5.4.m4.2.3.1" xref="S2.Thmtheorem3.p1.5.5.4.m4.2.3.1.cmml">=</mo><mrow id="S2.Thmtheorem3.p1.5.5.4.m4.2.3.3" xref="S2.Thmtheorem3.p1.5.5.4.m4.2.3.3.cmml"><mtext class="ltx_mathvariant_italic" id="S2.Thmtheorem3.p1.5.5.4.m4.2.3.3.2" xref="S2.Thmtheorem3.p1.5.5.4.m4.2.3.3.2a.cmml">g</mtext><mo id="S2.Thmtheorem3.p1.5.5.4.m4.2.3.3.1" xref="S2.Thmtheorem3.p1.5.5.4.m4.2.3.3.1.cmml">⁢</mo><mrow id="S2.Thmtheorem3.p1.5.5.4.m4.2.3.3.3.2" xref="S2.Thmtheorem3.p1.5.5.4.m4.2.3.3.cmml"><mo id="S2.Thmtheorem3.p1.5.5.4.m4.2.3.3.3.2.1" stretchy="false" xref="S2.Thmtheorem3.p1.5.5.4.m4.2.3.3.cmml">(</mo><mi id="S2.Thmtheorem3.p1.5.5.4.m4.2.2" mathvariant="normal" xref="S2.Thmtheorem3.p1.5.5.4.m4.2.2.cmml">e</mi><mo id="S2.Thmtheorem3.p1.5.5.4.m4.2.3.3.3.2.2" stretchy="false" xref="S2.Thmtheorem3.p1.5.5.4.m4.2.3.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem3.p1.5.5.4.m4.2b"><apply id="S2.Thmtheorem3.p1.5.5.4.m4.2.3.cmml" xref="S2.Thmtheorem3.p1.5.5.4.m4.2.3"><eq id="S2.Thmtheorem3.p1.5.5.4.m4.2.3.1.cmml" xref="S2.Thmtheorem3.p1.5.5.4.m4.2.3.1"></eq><apply id="S2.Thmtheorem3.p1.5.5.4.m4.2.3.2.cmml" xref="S2.Thmtheorem3.p1.5.5.4.m4.2.3.2"><times id="S2.Thmtheorem3.p1.5.5.4.m4.2.3.2.1.cmml" xref="S2.Thmtheorem3.p1.5.5.4.m4.2.3.2.1"></times><ci id="S2.Thmtheorem3.p1.5.5.4.m4.2.3.2.2a.cmml" xref="S2.Thmtheorem3.p1.5.5.4.m4.2.3.2.2"><mtext class="ltx_mathvariant_italic" id="S2.Thmtheorem3.p1.5.5.4.m4.2.3.2.2.cmml" xref="S2.Thmtheorem3.p1.5.5.4.m4.2.3.2.2">h</mtext></ci><ci id="S2.Thmtheorem3.p1.5.5.4.m4.1.1.cmml" xref="S2.Thmtheorem3.p1.5.5.4.m4.1.1">e</ci></apply><apply id="S2.Thmtheorem3.p1.5.5.4.m4.2.3.3.cmml" xref="S2.Thmtheorem3.p1.5.5.4.m4.2.3.3"><times id="S2.Thmtheorem3.p1.5.5.4.m4.2.3.3.1.cmml" xref="S2.Thmtheorem3.p1.5.5.4.m4.2.3.3.1"></times><ci id="S2.Thmtheorem3.p1.5.5.4.m4.2.3.3.2a.cmml" xref="S2.Thmtheorem3.p1.5.5.4.m4.2.3.3.2"><mtext class="ltx_mathvariant_italic" id="S2.Thmtheorem3.p1.5.5.4.m4.2.3.3.2.cmml" xref="S2.Thmtheorem3.p1.5.5.4.m4.2.3.3.2">g</mtext></ci><ci id="S2.Thmtheorem3.p1.5.5.4.m4.2.2.cmml" xref="S2.Thmtheorem3.p1.5.5.4.m4.2.2">e</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem3.p1.5.5.4.m4.2c">\textsl{h}(e)=\textsl{g}(e)</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem3.p1.5.5.4.m4.2d">h ( italic_e ) = g ( italic_e )</annotation></semantics></math> (as decreasing weights only decreases the density of the subgraph). There are finitely many subgraphs of <math alttext="G" class="ltx_Math" display="inline" id="S2.Thmtheorem3.p1.6.6.5.m5.1"><semantics id="S2.Thmtheorem3.p1.6.6.5.m5.1a"><mi id="S2.Thmtheorem3.p1.6.6.5.m5.1.1" mathvariant="normal" xref="S2.Thmtheorem3.p1.6.6.5.m5.1.1.cmml">G</mi><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem3.p1.6.6.5.m5.1b"><ci id="S2.Thmtheorem3.p1.6.6.5.m5.1.1.cmml" xref="S2.Thmtheorem3.p1.6.6.5.m5.1.1">G</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem3.p1.6.6.5.m5.1c">G</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem3.p1.6.6.5.m5.1d">italic_G</annotation></semantics></math> with the same edge weights as <math alttext="G" class="ltx_Math" display="inline" id="S2.Thmtheorem3.p1.7.7.6.m6.1"><semantics id="S2.Thmtheorem3.p1.7.7.6.m6.1a"><mi id="S2.Thmtheorem3.p1.7.7.6.m6.1.1" mathvariant="normal" xref="S2.Thmtheorem3.p1.7.7.6.m6.1.1.cmml">G</mi><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem3.p1.7.7.6.m6.1b"><ci id="S2.Thmtheorem3.p1.7.7.6.m6.1.1.cmml" xref="S2.Thmtheorem3.p1.7.7.6.m6.1.1">G</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem3.p1.7.7.6.m6.1c">G</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem3.p1.7.7.6.m6.1d">italic_G</annotation></semantics></math>, and thus there exists a <math alttext="H\subseteq G" class="ltx_Math" display="inline" id="S2.Thmtheorem3.p1.8.8.7.m7.1"><semantics id="S2.Thmtheorem3.p1.8.8.7.m7.1a"><mrow id="S2.Thmtheorem3.p1.8.8.7.m7.1.1" xref="S2.Thmtheorem3.p1.8.8.7.m7.1.1.cmml"><mi id="S2.Thmtheorem3.p1.8.8.7.m7.1.1.2" mathvariant="normal" xref="S2.Thmtheorem3.p1.8.8.7.m7.1.1.2.cmml">H</mi><mo id="S2.Thmtheorem3.p1.8.8.7.m7.1.1.1" xref="S2.Thmtheorem3.p1.8.8.7.m7.1.1.1.cmml">⊆</mo><mi id="S2.Thmtheorem3.p1.8.8.7.m7.1.1.3" mathvariant="normal" xref="S2.Thmtheorem3.p1.8.8.7.m7.1.1.3.cmml">G</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem3.p1.8.8.7.m7.1b"><apply id="S2.Thmtheorem3.p1.8.8.7.m7.1.1.cmml" xref="S2.Thmtheorem3.p1.8.8.7.m7.1.1"><subset id="S2.Thmtheorem3.p1.8.8.7.m7.1.1.1.cmml" xref="S2.Thmtheorem3.p1.8.8.7.m7.1.1.1"></subset><ci id="S2.Thmtheorem3.p1.8.8.7.m7.1.1.2.cmml" xref="S2.Thmtheorem3.p1.8.8.7.m7.1.1.2">H</ci><ci id="S2.Thmtheorem3.p1.8.8.7.m7.1.1.3.cmml" xref="S2.Thmtheorem3.p1.8.8.7.m7.1.1.3">G</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem3.p1.8.8.7.m7.1c">H\subseteq G</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem3.p1.8.8.7.m7.1d">italic_H ⊆ italic_G</annotation></semantics></math> that is a densest subgraph of <math alttext="G" class="ltx_Math" display="inline" id="S2.Thmtheorem3.p1.9.9.8.m8.1"><semantics id="S2.Thmtheorem3.p1.9.9.8.m8.1a"><mi id="S2.Thmtheorem3.p1.9.9.8.m8.1.1" mathvariant="normal" xref="S2.Thmtheorem3.p1.9.9.8.m8.1.1.cmml">G</mi><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem3.p1.9.9.8.m8.1b"><ci id="S2.Thmtheorem3.p1.9.9.8.m8.1.1.cmml" xref="S2.Thmtheorem3.p1.9.9.8.m8.1.1">G</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem3.p1.9.9.8.m8.1c">G</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem3.p1.9.9.8.m8.1d">italic_G</annotation></semantics></math>. We use <math alttext="H" class="ltx_Math" display="inline" id="S2.Thmtheorem3.p1.10.10.9.m9.1"><semantics id="S2.Thmtheorem3.p1.10.10.9.m9.1a"><mi id="S2.Thmtheorem3.p1.10.10.9.m9.1.1" mathvariant="normal" xref="S2.Thmtheorem3.p1.10.10.9.m9.1.1.cmml">H</mi><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem3.p1.10.10.9.m9.1b"><ci id="S2.Thmtheorem3.p1.10.10.9.m9.1.1.cmml" xref="S2.Thmtheorem3.p1.10.10.9.m9.1.1">H</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem3.p1.10.10.9.m9.1c">H</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem3.p1.10.10.9.m9.1d">italic_H</annotation></semantics></math> to find a feasible solution to DS:</span></span></p> </div> <div class="ltx_para" id="S2.Thmtheorem3.p2"> <p class="ltx_p" id="S2.Thmtheorem3.p2.9"><span class="ltx_text ltx_font_italic" id="S2.Thmtheorem3.p2.9.9">For each <math alttext="v\in V(H)" class="ltx_Math" display="inline" id="S2.Thmtheorem3.p2.1.1.m1.1"><semantics id="S2.Thmtheorem3.p2.1.1.m1.1a"><mrow id="S2.Thmtheorem3.p2.1.1.m1.1.2" xref="S2.Thmtheorem3.p2.1.1.m1.1.2.cmml"><mi id="S2.Thmtheorem3.p2.1.1.m1.1.2.2" xref="S2.Thmtheorem3.p2.1.1.m1.1.2.2.cmml">v</mi><mo id="S2.Thmtheorem3.p2.1.1.m1.1.2.1" xref="S2.Thmtheorem3.p2.1.1.m1.1.2.1.cmml">∈</mo><mrow id="S2.Thmtheorem3.p2.1.1.m1.1.2.3" xref="S2.Thmtheorem3.p2.1.1.m1.1.2.3.cmml"><mi id="S2.Thmtheorem3.p2.1.1.m1.1.2.3.2" xref="S2.Thmtheorem3.p2.1.1.m1.1.2.3.2.cmml">V</mi><mo id="S2.Thmtheorem3.p2.1.1.m1.1.2.3.1" xref="S2.Thmtheorem3.p2.1.1.m1.1.2.3.1.cmml">⁢</mo><mrow id="S2.Thmtheorem3.p2.1.1.m1.1.2.3.3.2" xref="S2.Thmtheorem3.p2.1.1.m1.1.2.3.cmml"><mo id="S2.Thmtheorem3.p2.1.1.m1.1.2.3.3.2.1" stretchy="false" xref="S2.Thmtheorem3.p2.1.1.m1.1.2.3.cmml">(</mo><mi id="S2.Thmtheorem3.p2.1.1.m1.1.1" xref="S2.Thmtheorem3.p2.1.1.m1.1.1.cmml">H</mi><mo id="S2.Thmtheorem3.p2.1.1.m1.1.2.3.3.2.2" stretchy="false" xref="S2.Thmtheorem3.p2.1.1.m1.1.2.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem3.p2.1.1.m1.1b"><apply id="S2.Thmtheorem3.p2.1.1.m1.1.2.cmml" xref="S2.Thmtheorem3.p2.1.1.m1.1.2"><in id="S2.Thmtheorem3.p2.1.1.m1.1.2.1.cmml" xref="S2.Thmtheorem3.p2.1.1.m1.1.2.1"></in><ci id="S2.Thmtheorem3.p2.1.1.m1.1.2.2.cmml" xref="S2.Thmtheorem3.p2.1.1.m1.1.2.2">𝑣</ci><apply id="S2.Thmtheorem3.p2.1.1.m1.1.2.3.cmml" xref="S2.Thmtheorem3.p2.1.1.m1.1.2.3"><times id="S2.Thmtheorem3.p2.1.1.m1.1.2.3.1.cmml" xref="S2.Thmtheorem3.p2.1.1.m1.1.2.3.1"></times><ci id="S2.Thmtheorem3.p2.1.1.m1.1.2.3.2.cmml" xref="S2.Thmtheorem3.p2.1.1.m1.1.2.3.2">𝑉</ci><ci id="S2.Thmtheorem3.p2.1.1.m1.1.1.cmml" xref="S2.Thmtheorem3.p2.1.1.m1.1.1">𝐻</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem3.p2.1.1.m1.1c">v\in V(H)</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem3.p2.1.1.m1.1d">italic_v ∈ italic_V ( italic_H )</annotation></semantics></math>, we set <math alttext="x_{v}=\frac{1}{|V(H)|}" class="ltx_Math" display="inline" id="S2.Thmtheorem3.p2.2.2.m2.2"><semantics id="S2.Thmtheorem3.p2.2.2.m2.2a"><mrow id="S2.Thmtheorem3.p2.2.2.m2.2.3" xref="S2.Thmtheorem3.p2.2.2.m2.2.3.cmml"><msub id="S2.Thmtheorem3.p2.2.2.m2.2.3.2" xref="S2.Thmtheorem3.p2.2.2.m2.2.3.2.cmml"><mi id="S2.Thmtheorem3.p2.2.2.m2.2.3.2.2" xref="S2.Thmtheorem3.p2.2.2.m2.2.3.2.2.cmml">x</mi><mi id="S2.Thmtheorem3.p2.2.2.m2.2.3.2.3" xref="S2.Thmtheorem3.p2.2.2.m2.2.3.2.3.cmml">v</mi></msub><mo id="S2.Thmtheorem3.p2.2.2.m2.2.3.1" xref="S2.Thmtheorem3.p2.2.2.m2.2.3.1.cmml">=</mo><mfrac id="S2.Thmtheorem3.p2.2.2.m2.2.2" xref="S2.Thmtheorem3.p2.2.2.m2.2.2.cmml"><mn id="S2.Thmtheorem3.p2.2.2.m2.2.2.4" xref="S2.Thmtheorem3.p2.2.2.m2.2.2.4.cmml">1</mn><mrow id="S2.Thmtheorem3.p2.2.2.m2.2.2.2.2" xref="S2.Thmtheorem3.p2.2.2.m2.2.2.2.3.cmml"><mo id="S2.Thmtheorem3.p2.2.2.m2.2.2.2.2.2" stretchy="false" xref="S2.Thmtheorem3.p2.2.2.m2.2.2.2.3.1.cmml">|</mo><mrow id="S2.Thmtheorem3.p2.2.2.m2.2.2.2.2.1" xref="S2.Thmtheorem3.p2.2.2.m2.2.2.2.2.1.cmml"><mi id="S2.Thmtheorem3.p2.2.2.m2.2.2.2.2.1.2" xref="S2.Thmtheorem3.p2.2.2.m2.2.2.2.2.1.2.cmml">V</mi><mo id="S2.Thmtheorem3.p2.2.2.m2.2.2.2.2.1.1" xref="S2.Thmtheorem3.p2.2.2.m2.2.2.2.2.1.1.cmml">⁢</mo><mrow id="S2.Thmtheorem3.p2.2.2.m2.2.2.2.2.1.3.2" xref="S2.Thmtheorem3.p2.2.2.m2.2.2.2.2.1.cmml"><mo id="S2.Thmtheorem3.p2.2.2.m2.2.2.2.2.1.3.2.1" stretchy="false" xref="S2.Thmtheorem3.p2.2.2.m2.2.2.2.2.1.cmml">(</mo><mi id="S2.Thmtheorem3.p2.2.2.m2.1.1.1.1" xref="S2.Thmtheorem3.p2.2.2.m2.1.1.1.1.cmml">H</mi><mo id="S2.Thmtheorem3.p2.2.2.m2.2.2.2.2.1.3.2.2" stretchy="false" xref="S2.Thmtheorem3.p2.2.2.m2.2.2.2.2.1.cmml">)</mo></mrow></mrow><mo id="S2.Thmtheorem3.p2.2.2.m2.2.2.2.2.3" stretchy="false" xref="S2.Thmtheorem3.p2.2.2.m2.2.2.2.3.1.cmml">|</mo></mrow></mfrac></mrow><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem3.p2.2.2.m2.2b"><apply id="S2.Thmtheorem3.p2.2.2.m2.2.3.cmml" xref="S2.Thmtheorem3.p2.2.2.m2.2.3"><eq id="S2.Thmtheorem3.p2.2.2.m2.2.3.1.cmml" xref="S2.Thmtheorem3.p2.2.2.m2.2.3.1"></eq><apply id="S2.Thmtheorem3.p2.2.2.m2.2.3.2.cmml" xref="S2.Thmtheorem3.p2.2.2.m2.2.3.2"><csymbol cd="ambiguous" id="S2.Thmtheorem3.p2.2.2.m2.2.3.2.1.cmml" xref="S2.Thmtheorem3.p2.2.2.m2.2.3.2">subscript</csymbol><ci id="S2.Thmtheorem3.p2.2.2.m2.2.3.2.2.cmml" xref="S2.Thmtheorem3.p2.2.2.m2.2.3.2.2">𝑥</ci><ci id="S2.Thmtheorem3.p2.2.2.m2.2.3.2.3.cmml" xref="S2.Thmtheorem3.p2.2.2.m2.2.3.2.3">𝑣</ci></apply><apply id="S2.Thmtheorem3.p2.2.2.m2.2.2.cmml" xref="S2.Thmtheorem3.p2.2.2.m2.2.2"><divide id="S2.Thmtheorem3.p2.2.2.m2.2.2.3.cmml" xref="S2.Thmtheorem3.p2.2.2.m2.2.2"></divide><cn id="S2.Thmtheorem3.p2.2.2.m2.2.2.4.cmml" type="integer" xref="S2.Thmtheorem3.p2.2.2.m2.2.2.4">1</cn><apply id="S2.Thmtheorem3.p2.2.2.m2.2.2.2.3.cmml" xref="S2.Thmtheorem3.p2.2.2.m2.2.2.2.2"><abs id="S2.Thmtheorem3.p2.2.2.m2.2.2.2.3.1.cmml" xref="S2.Thmtheorem3.p2.2.2.m2.2.2.2.2.2"></abs><apply id="S2.Thmtheorem3.p2.2.2.m2.2.2.2.2.1.cmml" xref="S2.Thmtheorem3.p2.2.2.m2.2.2.2.2.1"><times id="S2.Thmtheorem3.p2.2.2.m2.2.2.2.2.1.1.cmml" xref="S2.Thmtheorem3.p2.2.2.m2.2.2.2.2.1.1"></times><ci id="S2.Thmtheorem3.p2.2.2.m2.2.2.2.2.1.2.cmml" xref="S2.Thmtheorem3.p2.2.2.m2.2.2.2.2.1.2">𝑉</ci><ci id="S2.Thmtheorem3.p2.2.2.m2.1.1.1.1.cmml" xref="S2.Thmtheorem3.p2.2.2.m2.1.1.1.1">𝐻</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem3.p2.2.2.m2.2c">x_{v}=\frac{1}{|V(H)|}</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem3.p2.2.2.m2.2d">italic_x start_POSTSUBSCRIPT italic_v end_POSTSUBSCRIPT = divide start_ARG 1 end_ARG start_ARG | italic_V ( italic_H ) | end_ARG</annotation></semantics></math>. For every other <math alttext="v\in V" class="ltx_Math" display="inline" id="S2.Thmtheorem3.p2.3.3.m3.1"><semantics id="S2.Thmtheorem3.p2.3.3.m3.1a"><mrow id="S2.Thmtheorem3.p2.3.3.m3.1.1" xref="S2.Thmtheorem3.p2.3.3.m3.1.1.cmml"><mi id="S2.Thmtheorem3.p2.3.3.m3.1.1.2" xref="S2.Thmtheorem3.p2.3.3.m3.1.1.2.cmml">v</mi><mo id="S2.Thmtheorem3.p2.3.3.m3.1.1.1" xref="S2.Thmtheorem3.p2.3.3.m3.1.1.1.cmml">∈</mo><mi id="S2.Thmtheorem3.p2.3.3.m3.1.1.3" xref="S2.Thmtheorem3.p2.3.3.m3.1.1.3.cmml">V</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem3.p2.3.3.m3.1b"><apply id="S2.Thmtheorem3.p2.3.3.m3.1.1.cmml" xref="S2.Thmtheorem3.p2.3.3.m3.1.1"><in id="S2.Thmtheorem3.p2.3.3.m3.1.1.1.cmml" xref="S2.Thmtheorem3.p2.3.3.m3.1.1.1"></in><ci id="S2.Thmtheorem3.p2.3.3.m3.1.1.2.cmml" xref="S2.Thmtheorem3.p2.3.3.m3.1.1.2">𝑣</ci><ci id="S2.Thmtheorem3.p2.3.3.m3.1.1.3.cmml" xref="S2.Thmtheorem3.p2.3.3.m3.1.1.3">𝑉</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem3.p2.3.3.m3.1c">v\in V</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem3.p2.3.3.m3.1d">italic_v ∈ italic_V</annotation></semantics></math>, we set <math alttext="x_{v}" class="ltx_Math" display="inline" id="S2.Thmtheorem3.p2.4.4.m4.1"><semantics id="S2.Thmtheorem3.p2.4.4.m4.1a"><msub id="S2.Thmtheorem3.p2.4.4.m4.1.1" xref="S2.Thmtheorem3.p2.4.4.m4.1.1.cmml"><mi id="S2.Thmtheorem3.p2.4.4.m4.1.1.2" xref="S2.Thmtheorem3.p2.4.4.m4.1.1.2.cmml">x</mi><mi id="S2.Thmtheorem3.p2.4.4.m4.1.1.3" xref="S2.Thmtheorem3.p2.4.4.m4.1.1.3.cmml">v</mi></msub><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem3.p2.4.4.m4.1b"><apply id="S2.Thmtheorem3.p2.4.4.m4.1.1.cmml" xref="S2.Thmtheorem3.p2.4.4.m4.1.1"><csymbol cd="ambiguous" id="S2.Thmtheorem3.p2.4.4.m4.1.1.1.cmml" xref="S2.Thmtheorem3.p2.4.4.m4.1.1">subscript</csymbol><ci id="S2.Thmtheorem3.p2.4.4.m4.1.1.2.cmml" xref="S2.Thmtheorem3.p2.4.4.m4.1.1.2">𝑥</ci><ci id="S2.Thmtheorem3.p2.4.4.m4.1.1.3.cmml" xref="S2.Thmtheorem3.p2.4.4.m4.1.1.3">𝑣</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem3.p2.4.4.m4.1c">x_{v}</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem3.p2.4.4.m4.1d">italic_x start_POSTSUBSCRIPT italic_v end_POSTSUBSCRIPT</annotation></semantics></math> to be <math alttext="0" class="ltx_Math" display="inline" id="S2.Thmtheorem3.p2.5.5.m5.1"><semantics id="S2.Thmtheorem3.p2.5.5.m5.1a"><mn id="S2.Thmtheorem3.p2.5.5.m5.1.1" xref="S2.Thmtheorem3.p2.5.5.m5.1.1.cmml">0</mn><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem3.p2.5.5.m5.1b"><cn id="S2.Thmtheorem3.p2.5.5.m5.1.1.cmml" type="integer" xref="S2.Thmtheorem3.p2.5.5.m5.1.1">0</cn></annotation-xml></semantics></math>. For every <math alttext="\overline{uv}\in E(H)" class="ltx_Math" display="inline" id="S2.Thmtheorem3.p2.6.6.m6.1"><semantics id="S2.Thmtheorem3.p2.6.6.m6.1a"><mrow id="S2.Thmtheorem3.p2.6.6.m6.1.2" xref="S2.Thmtheorem3.p2.6.6.m6.1.2.cmml"><mover accent="true" id="S2.Thmtheorem3.p2.6.6.m6.1.2.2" xref="S2.Thmtheorem3.p2.6.6.m6.1.2.2.cmml"><mrow id="S2.Thmtheorem3.p2.6.6.m6.1.2.2.2" xref="S2.Thmtheorem3.p2.6.6.m6.1.2.2.2.cmml"><mi id="S2.Thmtheorem3.p2.6.6.m6.1.2.2.2.2" xref="S2.Thmtheorem3.p2.6.6.m6.1.2.2.2.2.cmml">u</mi><mo id="S2.Thmtheorem3.p2.6.6.m6.1.2.2.2.1" xref="S2.Thmtheorem3.p2.6.6.m6.1.2.2.2.1.cmml">⁢</mo><mi id="S2.Thmtheorem3.p2.6.6.m6.1.2.2.2.3" xref="S2.Thmtheorem3.p2.6.6.m6.1.2.2.2.3.cmml">v</mi></mrow><mo id="S2.Thmtheorem3.p2.6.6.m6.1.2.2.1" xref="S2.Thmtheorem3.p2.6.6.m6.1.2.2.1.cmml">¯</mo></mover><mo id="S2.Thmtheorem3.p2.6.6.m6.1.2.1" xref="S2.Thmtheorem3.p2.6.6.m6.1.2.1.cmml">∈</mo><mrow id="S2.Thmtheorem3.p2.6.6.m6.1.2.3" xref="S2.Thmtheorem3.p2.6.6.m6.1.2.3.cmml"><mi id="S2.Thmtheorem3.p2.6.6.m6.1.2.3.2" xref="S2.Thmtheorem3.p2.6.6.m6.1.2.3.2.cmml">E</mi><mo id="S2.Thmtheorem3.p2.6.6.m6.1.2.3.1" xref="S2.Thmtheorem3.p2.6.6.m6.1.2.3.1.cmml">⁢</mo><mrow id="S2.Thmtheorem3.p2.6.6.m6.1.2.3.3.2" xref="S2.Thmtheorem3.p2.6.6.m6.1.2.3.cmml"><mo id="S2.Thmtheorem3.p2.6.6.m6.1.2.3.3.2.1" stretchy="false" xref="S2.Thmtheorem3.p2.6.6.m6.1.2.3.cmml">(</mo><mi id="S2.Thmtheorem3.p2.6.6.m6.1.1" xref="S2.Thmtheorem3.p2.6.6.m6.1.1.cmml">H</mi><mo id="S2.Thmtheorem3.p2.6.6.m6.1.2.3.3.2.2" stretchy="false" xref="S2.Thmtheorem3.p2.6.6.m6.1.2.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem3.p2.6.6.m6.1b"><apply id="S2.Thmtheorem3.p2.6.6.m6.1.2.cmml" xref="S2.Thmtheorem3.p2.6.6.m6.1.2"><in id="S2.Thmtheorem3.p2.6.6.m6.1.2.1.cmml" xref="S2.Thmtheorem3.p2.6.6.m6.1.2.1"></in><apply id="S2.Thmtheorem3.p2.6.6.m6.1.2.2.cmml" xref="S2.Thmtheorem3.p2.6.6.m6.1.2.2"><ci id="S2.Thmtheorem3.p2.6.6.m6.1.2.2.1.cmml" xref="S2.Thmtheorem3.p2.6.6.m6.1.2.2.1">¯</ci><apply id="S2.Thmtheorem3.p2.6.6.m6.1.2.2.2.cmml" xref="S2.Thmtheorem3.p2.6.6.m6.1.2.2.2"><times id="S2.Thmtheorem3.p2.6.6.m6.1.2.2.2.1.cmml" xref="S2.Thmtheorem3.p2.6.6.m6.1.2.2.2.1"></times><ci id="S2.Thmtheorem3.p2.6.6.m6.1.2.2.2.2.cmml" xref="S2.Thmtheorem3.p2.6.6.m6.1.2.2.2.2">𝑢</ci><ci id="S2.Thmtheorem3.p2.6.6.m6.1.2.2.2.3.cmml" xref="S2.Thmtheorem3.p2.6.6.m6.1.2.2.2.3">𝑣</ci></apply></apply><apply id="S2.Thmtheorem3.p2.6.6.m6.1.2.3.cmml" xref="S2.Thmtheorem3.p2.6.6.m6.1.2.3"><times id="S2.Thmtheorem3.p2.6.6.m6.1.2.3.1.cmml" xref="S2.Thmtheorem3.p2.6.6.m6.1.2.3.1"></times><ci id="S2.Thmtheorem3.p2.6.6.m6.1.2.3.2.cmml" xref="S2.Thmtheorem3.p2.6.6.m6.1.2.3.2">𝐸</ci><ci id="S2.Thmtheorem3.p2.6.6.m6.1.1.cmml" xref="S2.Thmtheorem3.p2.6.6.m6.1.1">𝐻</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem3.p2.6.6.m6.1c">\overline{uv}\in E(H)</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem3.p2.6.6.m6.1d">over¯ start_ARG italic_u italic_v end_ARG ∈ italic_E ( italic_H )</annotation></semantics></math>, we set <math alttext="y_{u,v}=\frac{1}{|V(H)|}" class="ltx_Math" display="inline" id="S2.Thmtheorem3.p2.7.7.m7.4"><semantics id="S2.Thmtheorem3.p2.7.7.m7.4a"><mrow id="S2.Thmtheorem3.p2.7.7.m7.4.5" xref="S2.Thmtheorem3.p2.7.7.m7.4.5.cmml"><msub id="S2.Thmtheorem3.p2.7.7.m7.4.5.2" xref="S2.Thmtheorem3.p2.7.7.m7.4.5.2.cmml"><mi id="S2.Thmtheorem3.p2.7.7.m7.4.5.2.2" xref="S2.Thmtheorem3.p2.7.7.m7.4.5.2.2.cmml">y</mi><mrow id="S2.Thmtheorem3.p2.7.7.m7.2.2.2.4" xref="S2.Thmtheorem3.p2.7.7.m7.2.2.2.3.cmml"><mi id="S2.Thmtheorem3.p2.7.7.m7.1.1.1.1" xref="S2.Thmtheorem3.p2.7.7.m7.1.1.1.1.cmml">u</mi><mo id="S2.Thmtheorem3.p2.7.7.m7.2.2.2.4.1" xref="S2.Thmtheorem3.p2.7.7.m7.2.2.2.3.cmml">,</mo><mi id="S2.Thmtheorem3.p2.7.7.m7.2.2.2.2" xref="S2.Thmtheorem3.p2.7.7.m7.2.2.2.2.cmml">v</mi></mrow></msub><mo id="S2.Thmtheorem3.p2.7.7.m7.4.5.1" xref="S2.Thmtheorem3.p2.7.7.m7.4.5.1.cmml">=</mo><mfrac id="S2.Thmtheorem3.p2.7.7.m7.4.4" xref="S2.Thmtheorem3.p2.7.7.m7.4.4.cmml"><mn id="S2.Thmtheorem3.p2.7.7.m7.4.4.4" xref="S2.Thmtheorem3.p2.7.7.m7.4.4.4.cmml">1</mn><mrow id="S2.Thmtheorem3.p2.7.7.m7.4.4.2.2" xref="S2.Thmtheorem3.p2.7.7.m7.4.4.2.3.cmml"><mo id="S2.Thmtheorem3.p2.7.7.m7.4.4.2.2.2" stretchy="false" xref="S2.Thmtheorem3.p2.7.7.m7.4.4.2.3.1.cmml">|</mo><mrow id="S2.Thmtheorem3.p2.7.7.m7.4.4.2.2.1" xref="S2.Thmtheorem3.p2.7.7.m7.4.4.2.2.1.cmml"><mi id="S2.Thmtheorem3.p2.7.7.m7.4.4.2.2.1.2" xref="S2.Thmtheorem3.p2.7.7.m7.4.4.2.2.1.2.cmml">V</mi><mo id="S2.Thmtheorem3.p2.7.7.m7.4.4.2.2.1.1" xref="S2.Thmtheorem3.p2.7.7.m7.4.4.2.2.1.1.cmml">⁢</mo><mrow id="S2.Thmtheorem3.p2.7.7.m7.4.4.2.2.1.3.2" xref="S2.Thmtheorem3.p2.7.7.m7.4.4.2.2.1.cmml"><mo id="S2.Thmtheorem3.p2.7.7.m7.4.4.2.2.1.3.2.1" stretchy="false" xref="S2.Thmtheorem3.p2.7.7.m7.4.4.2.2.1.cmml">(</mo><mi id="S2.Thmtheorem3.p2.7.7.m7.3.3.1.1" xref="S2.Thmtheorem3.p2.7.7.m7.3.3.1.1.cmml">H</mi><mo id="S2.Thmtheorem3.p2.7.7.m7.4.4.2.2.1.3.2.2" stretchy="false" xref="S2.Thmtheorem3.p2.7.7.m7.4.4.2.2.1.cmml">)</mo></mrow></mrow><mo id="S2.Thmtheorem3.p2.7.7.m7.4.4.2.2.3" stretchy="false" xref="S2.Thmtheorem3.p2.7.7.m7.4.4.2.3.1.cmml">|</mo></mrow></mfrac></mrow><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem3.p2.7.7.m7.4b"><apply id="S2.Thmtheorem3.p2.7.7.m7.4.5.cmml" xref="S2.Thmtheorem3.p2.7.7.m7.4.5"><eq id="S2.Thmtheorem3.p2.7.7.m7.4.5.1.cmml" xref="S2.Thmtheorem3.p2.7.7.m7.4.5.1"></eq><apply id="S2.Thmtheorem3.p2.7.7.m7.4.5.2.cmml" xref="S2.Thmtheorem3.p2.7.7.m7.4.5.2"><csymbol cd="ambiguous" id="S2.Thmtheorem3.p2.7.7.m7.4.5.2.1.cmml" xref="S2.Thmtheorem3.p2.7.7.m7.4.5.2">subscript</csymbol><ci id="S2.Thmtheorem3.p2.7.7.m7.4.5.2.2.cmml" xref="S2.Thmtheorem3.p2.7.7.m7.4.5.2.2">𝑦</ci><list id="S2.Thmtheorem3.p2.7.7.m7.2.2.2.3.cmml" xref="S2.Thmtheorem3.p2.7.7.m7.2.2.2.4"><ci id="S2.Thmtheorem3.p2.7.7.m7.1.1.1.1.cmml" xref="S2.Thmtheorem3.p2.7.7.m7.1.1.1.1">𝑢</ci><ci id="S2.Thmtheorem3.p2.7.7.m7.2.2.2.2.cmml" xref="S2.Thmtheorem3.p2.7.7.m7.2.2.2.2">𝑣</ci></list></apply><apply id="S2.Thmtheorem3.p2.7.7.m7.4.4.cmml" xref="S2.Thmtheorem3.p2.7.7.m7.4.4"><divide id="S2.Thmtheorem3.p2.7.7.m7.4.4.3.cmml" xref="S2.Thmtheorem3.p2.7.7.m7.4.4"></divide><cn id="S2.Thmtheorem3.p2.7.7.m7.4.4.4.cmml" type="integer" xref="S2.Thmtheorem3.p2.7.7.m7.4.4.4">1</cn><apply id="S2.Thmtheorem3.p2.7.7.m7.4.4.2.3.cmml" xref="S2.Thmtheorem3.p2.7.7.m7.4.4.2.2"><abs id="S2.Thmtheorem3.p2.7.7.m7.4.4.2.3.1.cmml" xref="S2.Thmtheorem3.p2.7.7.m7.4.4.2.2.2"></abs><apply id="S2.Thmtheorem3.p2.7.7.m7.4.4.2.2.1.cmml" xref="S2.Thmtheorem3.p2.7.7.m7.4.4.2.2.1"><times id="S2.Thmtheorem3.p2.7.7.m7.4.4.2.2.1.1.cmml" xref="S2.Thmtheorem3.p2.7.7.m7.4.4.2.2.1.1"></times><ci id="S2.Thmtheorem3.p2.7.7.m7.4.4.2.2.1.2.cmml" xref="S2.Thmtheorem3.p2.7.7.m7.4.4.2.2.1.2">𝑉</ci><ci id="S2.Thmtheorem3.p2.7.7.m7.3.3.1.1.cmml" xref="S2.Thmtheorem3.p2.7.7.m7.3.3.1.1">𝐻</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem3.p2.7.7.m7.4c">y_{u,v}=\frac{1}{|V(H)|}</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem3.p2.7.7.m7.4d">italic_y start_POSTSUBSCRIPT italic_u , italic_v end_POSTSUBSCRIPT = divide start_ARG 1 end_ARG start_ARG | italic_V ( italic_H ) | end_ARG</annotation></semantics></math>. For every other <math alttext="\overline{uv}\in E" class="ltx_Math" display="inline" id="S2.Thmtheorem3.p2.8.8.m8.1"><semantics id="S2.Thmtheorem3.p2.8.8.m8.1a"><mrow id="S2.Thmtheorem3.p2.8.8.m8.1.1" xref="S2.Thmtheorem3.p2.8.8.m8.1.1.cmml"><mover accent="true" id="S2.Thmtheorem3.p2.8.8.m8.1.1.2" xref="S2.Thmtheorem3.p2.8.8.m8.1.1.2.cmml"><mrow id="S2.Thmtheorem3.p2.8.8.m8.1.1.2.2" xref="S2.Thmtheorem3.p2.8.8.m8.1.1.2.2.cmml"><mi id="S2.Thmtheorem3.p2.8.8.m8.1.1.2.2.2" xref="S2.Thmtheorem3.p2.8.8.m8.1.1.2.2.2.cmml">u</mi><mo id="S2.Thmtheorem3.p2.8.8.m8.1.1.2.2.1" xref="S2.Thmtheorem3.p2.8.8.m8.1.1.2.2.1.cmml">⁢</mo><mi id="S2.Thmtheorem3.p2.8.8.m8.1.1.2.2.3" xref="S2.Thmtheorem3.p2.8.8.m8.1.1.2.2.3.cmml">v</mi></mrow><mo id="S2.Thmtheorem3.p2.8.8.m8.1.1.2.1" xref="S2.Thmtheorem3.p2.8.8.m8.1.1.2.1.cmml">¯</mo></mover><mo id="S2.Thmtheorem3.p2.8.8.m8.1.1.1" xref="S2.Thmtheorem3.p2.8.8.m8.1.1.1.cmml">∈</mo><mi id="S2.Thmtheorem3.p2.8.8.m8.1.1.3" xref="S2.Thmtheorem3.p2.8.8.m8.1.1.3.cmml">E</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem3.p2.8.8.m8.1b"><apply id="S2.Thmtheorem3.p2.8.8.m8.1.1.cmml" xref="S2.Thmtheorem3.p2.8.8.m8.1.1"><in id="S2.Thmtheorem3.p2.8.8.m8.1.1.1.cmml" xref="S2.Thmtheorem3.p2.8.8.m8.1.1.1"></in><apply id="S2.Thmtheorem3.p2.8.8.m8.1.1.2.cmml" xref="S2.Thmtheorem3.p2.8.8.m8.1.1.2"><ci id="S2.Thmtheorem3.p2.8.8.m8.1.1.2.1.cmml" xref="S2.Thmtheorem3.p2.8.8.m8.1.1.2.1">¯</ci><apply id="S2.Thmtheorem3.p2.8.8.m8.1.1.2.2.cmml" xref="S2.Thmtheorem3.p2.8.8.m8.1.1.2.2"><times id="S2.Thmtheorem3.p2.8.8.m8.1.1.2.2.1.cmml" xref="S2.Thmtheorem3.p2.8.8.m8.1.1.2.2.1"></times><ci id="S2.Thmtheorem3.p2.8.8.m8.1.1.2.2.2.cmml" xref="S2.Thmtheorem3.p2.8.8.m8.1.1.2.2.2">𝑢</ci><ci id="S2.Thmtheorem3.p2.8.8.m8.1.1.2.2.3.cmml" xref="S2.Thmtheorem3.p2.8.8.m8.1.1.2.2.3">𝑣</ci></apply></apply><ci id="S2.Thmtheorem3.p2.8.8.m8.1.1.3.cmml" xref="S2.Thmtheorem3.p2.8.8.m8.1.1.3">𝐸</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem3.p2.8.8.m8.1c">\overline{uv}\in E</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem3.p2.8.8.m8.1d">over¯ start_ARG italic_u italic_v end_ARG ∈ italic_E</annotation></semantics></math>, we set <math alttext="y_{u,v}=0" class="ltx_Math" display="inline" id="S2.Thmtheorem3.p2.9.9.m9.2"><semantics id="S2.Thmtheorem3.p2.9.9.m9.2a"><mrow id="S2.Thmtheorem3.p2.9.9.m9.2.3" xref="S2.Thmtheorem3.p2.9.9.m9.2.3.cmml"><msub id="S2.Thmtheorem3.p2.9.9.m9.2.3.2" xref="S2.Thmtheorem3.p2.9.9.m9.2.3.2.cmml"><mi id="S2.Thmtheorem3.p2.9.9.m9.2.3.2.2" xref="S2.Thmtheorem3.p2.9.9.m9.2.3.2.2.cmml">y</mi><mrow id="S2.Thmtheorem3.p2.9.9.m9.2.2.2.4" xref="S2.Thmtheorem3.p2.9.9.m9.2.2.2.3.cmml"><mi id="S2.Thmtheorem3.p2.9.9.m9.1.1.1.1" xref="S2.Thmtheorem3.p2.9.9.m9.1.1.1.1.cmml">u</mi><mo id="S2.Thmtheorem3.p2.9.9.m9.2.2.2.4.1" xref="S2.Thmtheorem3.p2.9.9.m9.2.2.2.3.cmml">,</mo><mi id="S2.Thmtheorem3.p2.9.9.m9.2.2.2.2" xref="S2.Thmtheorem3.p2.9.9.m9.2.2.2.2.cmml">v</mi></mrow></msub><mo id="S2.Thmtheorem3.p2.9.9.m9.2.3.1" xref="S2.Thmtheorem3.p2.9.9.m9.2.3.1.cmml">=</mo><mn id="S2.Thmtheorem3.p2.9.9.m9.2.3.3" xref="S2.Thmtheorem3.p2.9.9.m9.2.3.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem3.p2.9.9.m9.2b"><apply id="S2.Thmtheorem3.p2.9.9.m9.2.3.cmml" xref="S2.Thmtheorem3.p2.9.9.m9.2.3"><eq id="S2.Thmtheorem3.p2.9.9.m9.2.3.1.cmml" xref="S2.Thmtheorem3.p2.9.9.m9.2.3.1"></eq><apply id="S2.Thmtheorem3.p2.9.9.m9.2.3.2.cmml" xref="S2.Thmtheorem3.p2.9.9.m9.2.3.2"><csymbol cd="ambiguous" id="S2.Thmtheorem3.p2.9.9.m9.2.3.2.1.cmml" xref="S2.Thmtheorem3.p2.9.9.m9.2.3.2">subscript</csymbol><ci id="S2.Thmtheorem3.p2.9.9.m9.2.3.2.2.cmml" xref="S2.Thmtheorem3.p2.9.9.m9.2.3.2.2">𝑦</ci><list id="S2.Thmtheorem3.p2.9.9.m9.2.2.2.3.cmml" xref="S2.Thmtheorem3.p2.9.9.m9.2.2.2.4"><ci id="S2.Thmtheorem3.p2.9.9.m9.1.1.1.1.cmml" xref="S2.Thmtheorem3.p2.9.9.m9.1.1.1.1">𝑢</ci><ci id="S2.Thmtheorem3.p2.9.9.m9.2.2.2.2.cmml" xref="S2.Thmtheorem3.p2.9.9.m9.2.2.2.2">𝑣</ci></list></apply><cn id="S2.Thmtheorem3.p2.9.9.m9.2.3.3.cmml" type="integer" xref="S2.Thmtheorem3.p2.9.9.m9.2.3.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem3.p2.9.9.m9.2c">y_{u,v}=0</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem3.p2.9.9.m9.2d">italic_y start_POSTSUBSCRIPT italic_u , italic_v end_POSTSUBSCRIPT = 0</annotation></semantics></math>. This gives a feasible solution to DS. Moreover:</span></p> <table class="ltx_equation ltx_eqn_table" id="S2.Ex6"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_left_padleft"></td> <td class="ltx_eqn_cell ltx_align_left"><math alttext="\sum_{\overline{uv}\in E}\textsl{g}(\overline{uv})\cdot y_{u,v}=\sum_{% \overline{uv}\in E(H)}\textsl{g}(\overline{uv})\cdot\frac{1}{|V(H)|}=\rho(H)=% \rho^{\max}(G)." class="ltx_Math" display="block" id="S2.Ex6.m1.10"><semantics id="S2.Ex6.m1.10a"><mrow id="S2.Ex6.m1.10.10.1" xref="S2.Ex6.m1.10.10.1.1.cmml"><mrow id="S2.Ex6.m1.10.10.1.1" xref="S2.Ex6.m1.10.10.1.1.cmml"><mrow id="S2.Ex6.m1.10.10.1.1.2" xref="S2.Ex6.m1.10.10.1.1.2.cmml"><munder id="S2.Ex6.m1.10.10.1.1.2.1" xref="S2.Ex6.m1.10.10.1.1.2.1.cmml"><mo id="S2.Ex6.m1.10.10.1.1.2.1.2" movablelimits="false" xref="S2.Ex6.m1.10.10.1.1.2.1.2.cmml">∑</mo><mrow id="S2.Ex6.m1.10.10.1.1.2.1.3" xref="S2.Ex6.m1.10.10.1.1.2.1.3.cmml"><mover accent="true" id="S2.Ex6.m1.10.10.1.1.2.1.3.2" xref="S2.Ex6.m1.10.10.1.1.2.1.3.2.cmml"><mrow id="S2.Ex6.m1.10.10.1.1.2.1.3.2.2" xref="S2.Ex6.m1.10.10.1.1.2.1.3.2.2.cmml"><mi id="S2.Ex6.m1.10.10.1.1.2.1.3.2.2.2" xref="S2.Ex6.m1.10.10.1.1.2.1.3.2.2.2.cmml">u</mi><mo id="S2.Ex6.m1.10.10.1.1.2.1.3.2.2.1" xref="S2.Ex6.m1.10.10.1.1.2.1.3.2.2.1.cmml">⁢</mo><mi id="S2.Ex6.m1.10.10.1.1.2.1.3.2.2.3" xref="S2.Ex6.m1.10.10.1.1.2.1.3.2.2.3.cmml">v</mi></mrow><mo id="S2.Ex6.m1.10.10.1.1.2.1.3.2.1" xref="S2.Ex6.m1.10.10.1.1.2.1.3.2.1.cmml">¯</mo></mover><mo id="S2.Ex6.m1.10.10.1.1.2.1.3.1" xref="S2.Ex6.m1.10.10.1.1.2.1.3.1.cmml">∈</mo><mi id="S2.Ex6.m1.10.10.1.1.2.1.3.3" xref="S2.Ex6.m1.10.10.1.1.2.1.3.3.cmml">E</mi></mrow></munder><mrow id="S2.Ex6.m1.10.10.1.1.2.2" xref="S2.Ex6.m1.10.10.1.1.2.2.cmml"><mrow id="S2.Ex6.m1.10.10.1.1.2.2.2" xref="S2.Ex6.m1.10.10.1.1.2.2.2.cmml"><mtext class="ltx_mathvariant_italic" id="S2.Ex6.m1.10.10.1.1.2.2.2.2" xref="S2.Ex6.m1.10.10.1.1.2.2.2.2a.cmml">g</mtext><mo id="S2.Ex6.m1.10.10.1.1.2.2.2.1" xref="S2.Ex6.m1.10.10.1.1.2.2.2.1.cmml">⁢</mo><mrow id="S2.Ex6.m1.10.10.1.1.2.2.2.3.2" xref="S2.Ex6.m1.6.6.cmml"><mo id="S2.Ex6.m1.10.10.1.1.2.2.2.3.2.1" stretchy="false" xref="S2.Ex6.m1.6.6.cmml">(</mo><mover accent="true" id="S2.Ex6.m1.6.6" xref="S2.Ex6.m1.6.6.cmml"><mrow id="S2.Ex6.m1.6.6.2" xref="S2.Ex6.m1.6.6.2.cmml"><mi id="S2.Ex6.m1.6.6.2.2" xref="S2.Ex6.m1.6.6.2.2.cmml">u</mi><mo id="S2.Ex6.m1.6.6.2.1" xref="S2.Ex6.m1.6.6.2.1.cmml">⁢</mo><mi id="S2.Ex6.m1.6.6.2.3" xref="S2.Ex6.m1.6.6.2.3.cmml">v</mi></mrow><mo id="S2.Ex6.m1.6.6.1" xref="S2.Ex6.m1.6.6.1.cmml">¯</mo></mover><mo id="S2.Ex6.m1.10.10.1.1.2.2.2.3.2.2" rspace="0.055em" stretchy="false" xref="S2.Ex6.m1.6.6.cmml">)</mo></mrow></mrow><mo id="S2.Ex6.m1.10.10.1.1.2.2.1" rspace="0.222em" xref="S2.Ex6.m1.10.10.1.1.2.2.1.cmml">⋅</mo><msub id="S2.Ex6.m1.10.10.1.1.2.2.3" xref="S2.Ex6.m1.10.10.1.1.2.2.3.cmml"><mi id="S2.Ex6.m1.10.10.1.1.2.2.3.2" xref="S2.Ex6.m1.10.10.1.1.2.2.3.2.cmml">y</mi><mrow id="S2.Ex6.m1.2.2.2.4" xref="S2.Ex6.m1.2.2.2.3.cmml"><mi id="S2.Ex6.m1.1.1.1.1" xref="S2.Ex6.m1.1.1.1.1.cmml">u</mi><mo id="S2.Ex6.m1.2.2.2.4.1" xref="S2.Ex6.m1.2.2.2.3.cmml">,</mo><mi id="S2.Ex6.m1.2.2.2.2" xref="S2.Ex6.m1.2.2.2.2.cmml">v</mi></mrow></msub></mrow></mrow><mo id="S2.Ex6.m1.10.10.1.1.3" rspace="0.111em" xref="S2.Ex6.m1.10.10.1.1.3.cmml">=</mo><mrow id="S2.Ex6.m1.10.10.1.1.4" xref="S2.Ex6.m1.10.10.1.1.4.cmml"><munder id="S2.Ex6.m1.10.10.1.1.4.1" xref="S2.Ex6.m1.10.10.1.1.4.1.cmml"><mo id="S2.Ex6.m1.10.10.1.1.4.1.2" movablelimits="false" xref="S2.Ex6.m1.10.10.1.1.4.1.2.cmml">∑</mo><mrow id="S2.Ex6.m1.3.3.1" xref="S2.Ex6.m1.3.3.1.cmml"><mover accent="true" id="S2.Ex6.m1.3.3.1.3" xref="S2.Ex6.m1.3.3.1.3.cmml"><mrow id="S2.Ex6.m1.3.3.1.3.2" xref="S2.Ex6.m1.3.3.1.3.2.cmml"><mi id="S2.Ex6.m1.3.3.1.3.2.2" xref="S2.Ex6.m1.3.3.1.3.2.2.cmml">u</mi><mo id="S2.Ex6.m1.3.3.1.3.2.1" xref="S2.Ex6.m1.3.3.1.3.2.1.cmml">⁢</mo><mi id="S2.Ex6.m1.3.3.1.3.2.3" xref="S2.Ex6.m1.3.3.1.3.2.3.cmml">v</mi></mrow><mo id="S2.Ex6.m1.3.3.1.3.1" xref="S2.Ex6.m1.3.3.1.3.1.cmml">¯</mo></mover><mo id="S2.Ex6.m1.3.3.1.2" xref="S2.Ex6.m1.3.3.1.2.cmml">∈</mo><mrow id="S2.Ex6.m1.3.3.1.4" xref="S2.Ex6.m1.3.3.1.4.cmml"><mi id="S2.Ex6.m1.3.3.1.4.2" xref="S2.Ex6.m1.3.3.1.4.2.cmml">E</mi><mo id="S2.Ex6.m1.3.3.1.4.1" xref="S2.Ex6.m1.3.3.1.4.1.cmml">⁢</mo><mrow id="S2.Ex6.m1.3.3.1.4.3.2" xref="S2.Ex6.m1.3.3.1.4.cmml"><mo id="S2.Ex6.m1.3.3.1.4.3.2.1" stretchy="false" xref="S2.Ex6.m1.3.3.1.4.cmml">(</mo><mi id="S2.Ex6.m1.3.3.1.1" xref="S2.Ex6.m1.3.3.1.1.cmml">H</mi><mo id="S2.Ex6.m1.3.3.1.4.3.2.2" stretchy="false" xref="S2.Ex6.m1.3.3.1.4.cmml">)</mo></mrow></mrow></mrow></munder><mrow id="S2.Ex6.m1.10.10.1.1.4.2" xref="S2.Ex6.m1.10.10.1.1.4.2.cmml"><mrow id="S2.Ex6.m1.10.10.1.1.4.2.2" xref="S2.Ex6.m1.10.10.1.1.4.2.2.cmml"><mtext class="ltx_mathvariant_italic" id="S2.Ex6.m1.10.10.1.1.4.2.2.2" xref="S2.Ex6.m1.10.10.1.1.4.2.2.2a.cmml">g</mtext><mo id="S2.Ex6.m1.10.10.1.1.4.2.2.1" xref="S2.Ex6.m1.10.10.1.1.4.2.2.1.cmml">⁢</mo><mrow id="S2.Ex6.m1.10.10.1.1.4.2.2.3.2" xref="S2.Ex6.m1.7.7.cmml"><mo id="S2.Ex6.m1.10.10.1.1.4.2.2.3.2.1" stretchy="false" xref="S2.Ex6.m1.7.7.cmml">(</mo><mover accent="true" id="S2.Ex6.m1.7.7" xref="S2.Ex6.m1.7.7.cmml"><mrow id="S2.Ex6.m1.7.7.2" xref="S2.Ex6.m1.7.7.2.cmml"><mi id="S2.Ex6.m1.7.7.2.2" xref="S2.Ex6.m1.7.7.2.2.cmml">u</mi><mo id="S2.Ex6.m1.7.7.2.1" xref="S2.Ex6.m1.7.7.2.1.cmml">⁢</mo><mi id="S2.Ex6.m1.7.7.2.3" xref="S2.Ex6.m1.7.7.2.3.cmml">v</mi></mrow><mo id="S2.Ex6.m1.7.7.1" xref="S2.Ex6.m1.7.7.1.cmml">¯</mo></mover><mo id="S2.Ex6.m1.10.10.1.1.4.2.2.3.2.2" rspace="0.055em" stretchy="false" xref="S2.Ex6.m1.7.7.cmml">)</mo></mrow></mrow><mo id="S2.Ex6.m1.10.10.1.1.4.2.1" rspace="0.222em" xref="S2.Ex6.m1.10.10.1.1.4.2.1.cmml">⋅</mo><mfrac id="S2.Ex6.m1.5.5" xref="S2.Ex6.m1.5.5.cmml"><mn id="S2.Ex6.m1.5.5.4" xref="S2.Ex6.m1.5.5.4.cmml">1</mn><mrow id="S2.Ex6.m1.5.5.2.2" xref="S2.Ex6.m1.5.5.2.3.cmml"><mo 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id="S2.Ex6.m1.10.10.1.1.5.cmml" xref="S2.Ex6.m1.10.10.1.1.5"></eq><share href="https://arxiv.org/html/2411.12694v2#S2.Ex6.m1.10.10.1.1.4.cmml" id="S2.Ex6.m1.10.10.1.1d.cmml" xref="S2.Ex6.m1.10.10.1"></share><apply id="S2.Ex6.m1.10.10.1.1.6.cmml" xref="S2.Ex6.m1.10.10.1.1.6"><times id="S2.Ex6.m1.10.10.1.1.6.1.cmml" xref="S2.Ex6.m1.10.10.1.1.6.1"></times><ci id="S2.Ex6.m1.10.10.1.1.6.2.cmml" xref="S2.Ex6.m1.10.10.1.1.6.2">𝜌</ci><ci id="S2.Ex6.m1.8.8.cmml" xref="S2.Ex6.m1.8.8">𝐻</ci></apply></apply><apply id="S2.Ex6.m1.10.10.1.1e.cmml" xref="S2.Ex6.m1.10.10.1"><eq id="S2.Ex6.m1.10.10.1.1.7.cmml" xref="S2.Ex6.m1.10.10.1.1.7"></eq><share href="https://arxiv.org/html/2411.12694v2#S2.Ex6.m1.10.10.1.1.6.cmml" id="S2.Ex6.m1.10.10.1.1f.cmml" xref="S2.Ex6.m1.10.10.1"></share><apply id="S2.Ex6.m1.10.10.1.1.8.cmml" xref="S2.Ex6.m1.10.10.1.1.8"><times id="S2.Ex6.m1.10.10.1.1.8.1.cmml" xref="S2.Ex6.m1.10.10.1.1.8.1"></times><apply id="S2.Ex6.m1.10.10.1.1.8.2.cmml" xref="S2.Ex6.m1.10.10.1.1.8.2"><csymbol cd="ambiguous" id="S2.Ex6.m1.10.10.1.1.8.2.1.cmml" xref="S2.Ex6.m1.10.10.1.1.8.2">superscript</csymbol><ci id="S2.Ex6.m1.10.10.1.1.8.2.2.cmml" xref="S2.Ex6.m1.10.10.1.1.8.2.2">𝜌</ci><max id="S2.Ex6.m1.10.10.1.1.8.2.3.cmml" xref="S2.Ex6.m1.10.10.1.1.8.2.3"></max></apply><ci id="S2.Ex6.m1.9.9.cmml" xref="S2.Ex6.m1.9.9">𝐺</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Ex6.m1.10c">\sum_{\overline{uv}\in E}\textsl{g}(\overline{uv})\cdot y_{u,v}=\sum_{% \overline{uv}\in E(H)}\textsl{g}(\overline{uv})\cdot\frac{1}{|V(H)|}=\rho(H)=% \rho^{\max}(G).</annotation><annotation encoding="application/x-llamapun" id="S2.Ex6.m1.10d">∑ start_POSTSUBSCRIPT over¯ start_ARG italic_u italic_v end_ARG ∈ italic_E end_POSTSUBSCRIPT g ( over¯ start_ARG italic_u italic_v end_ARG ) ⋅ italic_y start_POSTSUBSCRIPT italic_u , italic_v end_POSTSUBSCRIPT = ∑ start_POSTSUBSCRIPT over¯ start_ARG italic_u italic_v end_ARG ∈ italic_E ( italic_H ) end_POSTSUBSCRIPT g ( over¯ start_ARG italic_u italic_v end_ARG ) ⋅ divide start_ARG 1 end_ARG start_ARG | italic_V ( italic_H ) | end_ARG = italic_ρ ( italic_H ) = italic_ρ start_POSTSUPERSCRIPT roman_max end_POSTSUPERSCRIPT ( italic_G ) .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_left_padright"></td> </tr></tbody> </table> </div> <div class="ltx_para ltx_noindent" id="S2.Thmtheorem3.p3"> <p class="ltx_p" id="S2.Thmtheorem3.p3.1"><span class="ltx_text ltx_font_italic" id="S2.Thmtheorem3.p3.1.1">It follows that </span><math alttext="R\geq\rho^{\max}(G)" class="ltx_Math" display="inline" id="S2.Thmtheorem3.p3.1.m1.1"><semantics id="S2.Thmtheorem3.p3.1.m1.1a"><mrow id="S2.Thmtheorem3.p3.1.m1.1.2" xref="S2.Thmtheorem3.p3.1.m1.1.2.cmml"><mi id="S2.Thmtheorem3.p3.1.m1.1.2.2" xref="S2.Thmtheorem3.p3.1.m1.1.2.2.cmml">R</mi><mo id="S2.Thmtheorem3.p3.1.m1.1.2.1" xref="S2.Thmtheorem3.p3.1.m1.1.2.1.cmml">≥</mo><mrow id="S2.Thmtheorem3.p3.1.m1.1.2.3" xref="S2.Thmtheorem3.p3.1.m1.1.2.3.cmml"><msup id="S2.Thmtheorem3.p3.1.m1.1.2.3.2" xref="S2.Thmtheorem3.p3.1.m1.1.2.3.2.cmml"><mi id="S2.Thmtheorem3.p3.1.m1.1.2.3.2.2" xref="S2.Thmtheorem3.p3.1.m1.1.2.3.2.2.cmml">ρ</mi><mi id="S2.Thmtheorem3.p3.1.m1.1.2.3.2.3" xref="S2.Thmtheorem3.p3.1.m1.1.2.3.2.3.cmml">max</mi></msup><mo id="S2.Thmtheorem3.p3.1.m1.1.2.3.1" xref="S2.Thmtheorem3.p3.1.m1.1.2.3.1.cmml">⁢</mo><mrow id="S2.Thmtheorem3.p3.1.m1.1.2.3.3.2" xref="S2.Thmtheorem3.p3.1.m1.1.2.3.cmml"><mo id="S2.Thmtheorem3.p3.1.m1.1.2.3.3.2.1" stretchy="false" xref="S2.Thmtheorem3.p3.1.m1.1.2.3.cmml">(</mo><mi id="S2.Thmtheorem3.p3.1.m1.1.1" xref="S2.Thmtheorem3.p3.1.m1.1.1.cmml">G</mi><mo id="S2.Thmtheorem3.p3.1.m1.1.2.3.3.2.2" stretchy="false" xref="S2.Thmtheorem3.p3.1.m1.1.2.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem3.p3.1.m1.1b"><apply id="S2.Thmtheorem3.p3.1.m1.1.2.cmml" xref="S2.Thmtheorem3.p3.1.m1.1.2"><geq id="S2.Thmtheorem3.p3.1.m1.1.2.1.cmml" xref="S2.Thmtheorem3.p3.1.m1.1.2.1"></geq><ci id="S2.Thmtheorem3.p3.1.m1.1.2.2.cmml" xref="S2.Thmtheorem3.p3.1.m1.1.2.2">𝑅</ci><apply id="S2.Thmtheorem3.p3.1.m1.1.2.3.cmml" xref="S2.Thmtheorem3.p3.1.m1.1.2.3"><times id="S2.Thmtheorem3.p3.1.m1.1.2.3.1.cmml" xref="S2.Thmtheorem3.p3.1.m1.1.2.3.1"></times><apply id="S2.Thmtheorem3.p3.1.m1.1.2.3.2.cmml" xref="S2.Thmtheorem3.p3.1.m1.1.2.3.2"><csymbol cd="ambiguous" id="S2.Thmtheorem3.p3.1.m1.1.2.3.2.1.cmml" xref="S2.Thmtheorem3.p3.1.m1.1.2.3.2">superscript</csymbol><ci id="S2.Thmtheorem3.p3.1.m1.1.2.3.2.2.cmml" xref="S2.Thmtheorem3.p3.1.m1.1.2.3.2.2">𝜌</ci><max id="S2.Thmtheorem3.p3.1.m1.1.2.3.2.3.cmml" xref="S2.Thmtheorem3.p3.1.m1.1.2.3.2.3"></max></apply><ci id="S2.Thmtheorem3.p3.1.m1.1.1.cmml" xref="S2.Thmtheorem3.p3.1.m1.1.1">𝐺</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem3.p3.1.m1.1c">R\geq\rho^{\max}(G)</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem3.p3.1.m1.1d">italic_R ≥ italic_ρ start_POSTSUPERSCRIPT roman_max end_POSTSUPERSCRIPT ( italic_G )</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S2.Thmtheorem3.p3.1.2">.</span></p> </div> <div class="ltx_para" id="S2.Thmtheorem3.p4"> <p class="ltx_p" id="S2.Thmtheorem3.p4.9"><span class="ltx_text ltx_font_bold ltx_font_italic" id="S2.Thmtheorem3.p4.9.9">We show that <math alttext="R\leq\rho^{\max}(G)" class="ltx_Math" display="inline" id="S2.Thmtheorem3.p4.1.1.m1.1"><semantics id="S2.Thmtheorem3.p4.1.1.m1.1a"><mrow id="S2.Thmtheorem3.p4.1.1.m1.1.2" xref="S2.Thmtheorem3.p4.1.1.m1.1.2.cmml"><mi id="S2.Thmtheorem3.p4.1.1.m1.1.2.2" mathvariant="normal" xref="S2.Thmtheorem3.p4.1.1.m1.1.2.2.cmml">R</mi><mo id="S2.Thmtheorem3.p4.1.1.m1.1.2.1" xref="S2.Thmtheorem3.p4.1.1.m1.1.2.1.cmml">≤</mo><mrow id="S2.Thmtheorem3.p4.1.1.m1.1.2.3" xref="S2.Thmtheorem3.p4.1.1.m1.1.2.3.cmml"><msup id="S2.Thmtheorem3.p4.1.1.m1.1.2.3.2" xref="S2.Thmtheorem3.p4.1.1.m1.1.2.3.2.cmml"><mi id="S2.Thmtheorem3.p4.1.1.m1.1.2.3.2.2" mathvariant="normal" xref="S2.Thmtheorem3.p4.1.1.m1.1.2.3.2.2.cmml">ρ</mi><mi id="S2.Thmtheorem3.p4.1.1.m1.1.2.3.2.3" xref="S2.Thmtheorem3.p4.1.1.m1.1.2.3.2.3.cmml">max</mi></msup><mo id="S2.Thmtheorem3.p4.1.1.m1.1.2.3.1" xref="S2.Thmtheorem3.p4.1.1.m1.1.2.3.1.cmml">⁢</mo><mrow id="S2.Thmtheorem3.p4.1.1.m1.1.2.3.3.2" xref="S2.Thmtheorem3.p4.1.1.m1.1.2.3.cmml"><mo id="S2.Thmtheorem3.p4.1.1.m1.1.2.3.3.2.1" stretchy="false" xref="S2.Thmtheorem3.p4.1.1.m1.1.2.3.cmml">(</mo><mi id="S2.Thmtheorem3.p4.1.1.m1.1.1" mathvariant="normal" xref="S2.Thmtheorem3.p4.1.1.m1.1.1.cmml">G</mi><mo id="S2.Thmtheorem3.p4.1.1.m1.1.2.3.3.2.2" stretchy="false" xref="S2.Thmtheorem3.p4.1.1.m1.1.2.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem3.p4.1.1.m1.1b"><apply id="S2.Thmtheorem3.p4.1.1.m1.1.2.cmml" xref="S2.Thmtheorem3.p4.1.1.m1.1.2"><leq id="S2.Thmtheorem3.p4.1.1.m1.1.2.1.cmml" xref="S2.Thmtheorem3.p4.1.1.m1.1.2.1"></leq><ci id="S2.Thmtheorem3.p4.1.1.m1.1.2.2.cmml" xref="S2.Thmtheorem3.p4.1.1.m1.1.2.2">R</ci><apply id="S2.Thmtheorem3.p4.1.1.m1.1.2.3.cmml" xref="S2.Thmtheorem3.p4.1.1.m1.1.2.3"><times id="S2.Thmtheorem3.p4.1.1.m1.1.2.3.1.cmml" xref="S2.Thmtheorem3.p4.1.1.m1.1.2.3.1"></times><apply id="S2.Thmtheorem3.p4.1.1.m1.1.2.3.2.cmml" xref="S2.Thmtheorem3.p4.1.1.m1.1.2.3.2"><csymbol cd="ambiguous" id="S2.Thmtheorem3.p4.1.1.m1.1.2.3.2.1.cmml" xref="S2.Thmtheorem3.p4.1.1.m1.1.2.3.2">superscript</csymbol><ci id="S2.Thmtheorem3.p4.1.1.m1.1.2.3.2.2.cmml" xref="S2.Thmtheorem3.p4.1.1.m1.1.2.3.2.2">ρ</ci><max id="S2.Thmtheorem3.p4.1.1.m1.1.2.3.2.3.cmml" xref="S2.Thmtheorem3.p4.1.1.m1.1.2.3.2.3"></max></apply><ci id="S2.Thmtheorem3.p4.1.1.m1.1.1.cmml" xref="S2.Thmtheorem3.p4.1.1.m1.1.1">G</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem3.p4.1.1.m1.1c">R\leq\rho^{\max}(G)</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem3.p4.1.1.m1.1d">italic_R ≤ italic_ρ start_POSTSUPERSCRIPT roman_max end_POSTSUPERSCRIPT ( italic_G )</annotation></semantics></math>.<span class="ltx_text ltx_font_medium" id="S2.Thmtheorem3.p4.9.9.8"> Consider any solution to DS that realises the value <math alttext="R" class="ltx_Math" display="inline" id="S2.Thmtheorem3.p4.2.2.1.m1.1"><semantics id="S2.Thmtheorem3.p4.2.2.1.m1.1a"><mi id="S2.Thmtheorem3.p4.2.2.1.m1.1.1" mathvariant="normal" xref="S2.Thmtheorem3.p4.2.2.1.m1.1.1.cmml">R</mi><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem3.p4.2.2.1.m1.1b"><ci id="S2.Thmtheorem3.p4.2.2.1.m1.1.1.cmml" xref="S2.Thmtheorem3.p4.2.2.1.m1.1.1">R</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem3.p4.2.2.1.m1.1c">R</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem3.p4.2.2.1.m1.1d">italic_R</annotation></semantics></math>. Denote for a real value <math alttext="r" class="ltx_Math" display="inline" id="S2.Thmtheorem3.p4.3.3.2.m2.1"><semantics id="S2.Thmtheorem3.p4.3.3.2.m2.1a"><mi id="S2.Thmtheorem3.p4.3.3.2.m2.1.1" mathvariant="normal" xref="S2.Thmtheorem3.p4.3.3.2.m2.1.1.cmml">r</mi><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem3.p4.3.3.2.m2.1b"><ci id="S2.Thmtheorem3.p4.3.3.2.m2.1.1.cmml" xref="S2.Thmtheorem3.p4.3.3.2.m2.1.1">r</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem3.p4.3.3.2.m2.1c">r</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem3.p4.3.3.2.m2.1d">italic_r</annotation></semantics></math> by <math alttext="V(r):=\{u\in V\mid x_{u}\geq r\}" class="ltx_Math" display="inline" id="S2.Thmtheorem3.p4.4.4.3.m3.3"><semantics id="S2.Thmtheorem3.p4.4.4.3.m3.3a"><mrow id="S2.Thmtheorem3.p4.4.4.3.m3.3.3" xref="S2.Thmtheorem3.p4.4.4.3.m3.3.3.cmml"><mrow id="S2.Thmtheorem3.p4.4.4.3.m3.3.3.4" 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xref="S2.Thmtheorem3.p4.4.4.3.m3.3.3.2.3.1.cmml">{</mo><mrow id="S2.Thmtheorem3.p4.4.4.3.m3.2.2.1.1.1" xref="S2.Thmtheorem3.p4.4.4.3.m3.2.2.1.1.1.cmml"><mi id="S2.Thmtheorem3.p4.4.4.3.m3.2.2.1.1.1.2" mathvariant="normal" xref="S2.Thmtheorem3.p4.4.4.3.m3.2.2.1.1.1.2.cmml">u</mi><mo id="S2.Thmtheorem3.p4.4.4.3.m3.2.2.1.1.1.1" xref="S2.Thmtheorem3.p4.4.4.3.m3.2.2.1.1.1.1.cmml">∈</mo><mi id="S2.Thmtheorem3.p4.4.4.3.m3.2.2.1.1.1.3" mathvariant="normal" xref="S2.Thmtheorem3.p4.4.4.3.m3.2.2.1.1.1.3.cmml">V</mi></mrow><mo fence="true" id="S2.Thmtheorem3.p4.4.4.3.m3.3.3.2.2.4" lspace="0em" rspace="0em" xref="S2.Thmtheorem3.p4.4.4.3.m3.3.3.2.3.1.cmml">∣</mo><mrow id="S2.Thmtheorem3.p4.4.4.3.m3.3.3.2.2.2" xref="S2.Thmtheorem3.p4.4.4.3.m3.3.3.2.2.2.cmml"><msub id="S2.Thmtheorem3.p4.4.4.3.m3.3.3.2.2.2.2" xref="S2.Thmtheorem3.p4.4.4.3.m3.3.3.2.2.2.2.cmml"><mi id="S2.Thmtheorem3.p4.4.4.3.m3.3.3.2.2.2.2.2" mathvariant="normal" xref="S2.Thmtheorem3.p4.4.4.3.m3.3.3.2.2.2.2.2.cmml">x</mi><mi id="S2.Thmtheorem3.p4.4.4.3.m3.3.3.2.2.2.2.3" mathvariant="normal" xref="S2.Thmtheorem3.p4.4.4.3.m3.3.3.2.2.2.2.3.cmml">u</mi></msub><mo id="S2.Thmtheorem3.p4.4.4.3.m3.3.3.2.2.2.1" xref="S2.Thmtheorem3.p4.4.4.3.m3.3.3.2.2.2.1.cmml">≥</mo><mi id="S2.Thmtheorem3.p4.4.4.3.m3.3.3.2.2.2.3" mathvariant="normal" xref="S2.Thmtheorem3.p4.4.4.3.m3.3.3.2.2.2.3.cmml">r</mi></mrow><mo id="S2.Thmtheorem3.p4.4.4.3.m3.3.3.2.2.5" stretchy="false" xref="S2.Thmtheorem3.p4.4.4.3.m3.3.3.2.3.1.cmml">}</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem3.p4.4.4.3.m3.3b"><apply id="S2.Thmtheorem3.p4.4.4.3.m3.3.3.cmml" xref="S2.Thmtheorem3.p4.4.4.3.m3.3.3"><csymbol cd="latexml" id="S2.Thmtheorem3.p4.4.4.3.m3.3.3.3.cmml" xref="S2.Thmtheorem3.p4.4.4.3.m3.3.3.3">assign</csymbol><apply id="S2.Thmtheorem3.p4.4.4.3.m3.3.3.4.cmml" xref="S2.Thmtheorem3.p4.4.4.3.m3.3.3.4"><times id="S2.Thmtheorem3.p4.4.4.3.m3.3.3.4.1.cmml" xref="S2.Thmtheorem3.p4.4.4.3.m3.3.3.4.1"></times><ci id="S2.Thmtheorem3.p4.4.4.3.m3.3.3.4.2.cmml" xref="S2.Thmtheorem3.p4.4.4.3.m3.3.3.4.2">V</ci><ci id="S2.Thmtheorem3.p4.4.4.3.m3.1.1.cmml" xref="S2.Thmtheorem3.p4.4.4.3.m3.1.1">r</ci></apply><apply id="S2.Thmtheorem3.p4.4.4.3.m3.3.3.2.3.cmml" xref="S2.Thmtheorem3.p4.4.4.3.m3.3.3.2.2"><csymbol cd="latexml" id="S2.Thmtheorem3.p4.4.4.3.m3.3.3.2.3.1.cmml" xref="S2.Thmtheorem3.p4.4.4.3.m3.3.3.2.2.3">conditional-set</csymbol><apply id="S2.Thmtheorem3.p4.4.4.3.m3.2.2.1.1.1.cmml" xref="S2.Thmtheorem3.p4.4.4.3.m3.2.2.1.1.1"><in id="S2.Thmtheorem3.p4.4.4.3.m3.2.2.1.1.1.1.cmml" xref="S2.Thmtheorem3.p4.4.4.3.m3.2.2.1.1.1.1"></in><ci id="S2.Thmtheorem3.p4.4.4.3.m3.2.2.1.1.1.2.cmml" xref="S2.Thmtheorem3.p4.4.4.3.m3.2.2.1.1.1.2">u</ci><ci id="S2.Thmtheorem3.p4.4.4.3.m3.2.2.1.1.1.3.cmml" xref="S2.Thmtheorem3.p4.4.4.3.m3.2.2.1.1.1.3">V</ci></apply><apply id="S2.Thmtheorem3.p4.4.4.3.m3.3.3.2.2.2.cmml" xref="S2.Thmtheorem3.p4.4.4.3.m3.3.3.2.2.2"><geq id="S2.Thmtheorem3.p4.4.4.3.m3.3.3.2.2.2.1.cmml" xref="S2.Thmtheorem3.p4.4.4.3.m3.3.3.2.2.2.1"></geq><apply id="S2.Thmtheorem3.p4.4.4.3.m3.3.3.2.2.2.2.cmml" xref="S2.Thmtheorem3.p4.4.4.3.m3.3.3.2.2.2.2"><csymbol cd="ambiguous" id="S2.Thmtheorem3.p4.4.4.3.m3.3.3.2.2.2.2.1.cmml" xref="S2.Thmtheorem3.p4.4.4.3.m3.3.3.2.2.2.2">subscript</csymbol><ci id="S2.Thmtheorem3.p4.4.4.3.m3.3.3.2.2.2.2.2.cmml" xref="S2.Thmtheorem3.p4.4.4.3.m3.3.3.2.2.2.2.2">x</ci><ci id="S2.Thmtheorem3.p4.4.4.3.m3.3.3.2.2.2.2.3.cmml" xref="S2.Thmtheorem3.p4.4.4.3.m3.3.3.2.2.2.2.3">u</ci></apply><ci id="S2.Thmtheorem3.p4.4.4.3.m3.3.3.2.2.2.3.cmml" xref="S2.Thmtheorem3.p4.4.4.3.m3.3.3.2.2.2.3">r</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem3.p4.4.4.3.m3.3c">V(r):=\{u\in V\mid x_{u}\geq r\}</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem3.p4.4.4.3.m3.3d">italic_V ( italic_r ) := { italic_u ∈ italic_V ∣ italic_x start_POSTSUBSCRIPT italic_u end_POSTSUBSCRIPT ≥ italic_r }</annotation></semantics></math>. Similarly, denote by <math alttext="E(r):=\{\overline{uv}\in E\mid y_{u,v}\geq r\}" class="ltx_Math" display="inline" id="S2.Thmtheorem3.p4.5.5.4.m4.5"><semantics id="S2.Thmtheorem3.p4.5.5.4.m4.5a"><mrow id="S2.Thmtheorem3.p4.5.5.4.m4.5.5" xref="S2.Thmtheorem3.p4.5.5.4.m4.5.5.cmml"><mrow id="S2.Thmtheorem3.p4.5.5.4.m4.5.5.4" xref="S2.Thmtheorem3.p4.5.5.4.m4.5.5.4.cmml"><mi id="S2.Thmtheorem3.p4.5.5.4.m4.5.5.4.2" mathvariant="normal" xref="S2.Thmtheorem3.p4.5.5.4.m4.5.5.4.2.cmml">E</mi><mo id="S2.Thmtheorem3.p4.5.5.4.m4.5.5.4.1" xref="S2.Thmtheorem3.p4.5.5.4.m4.5.5.4.1.cmml">⁢</mo><mrow id="S2.Thmtheorem3.p4.5.5.4.m4.5.5.4.3.2" xref="S2.Thmtheorem3.p4.5.5.4.m4.5.5.4.cmml"><mo id="S2.Thmtheorem3.p4.5.5.4.m4.5.5.4.3.2.1" stretchy="false" xref="S2.Thmtheorem3.p4.5.5.4.m4.5.5.4.cmml">(</mo><mi id="S2.Thmtheorem3.p4.5.5.4.m4.3.3" mathvariant="normal" xref="S2.Thmtheorem3.p4.5.5.4.m4.3.3.cmml">r</mi><mo id="S2.Thmtheorem3.p4.5.5.4.m4.5.5.4.3.2.2" rspace="0.278em" stretchy="false" xref="S2.Thmtheorem3.p4.5.5.4.m4.5.5.4.cmml">)</mo></mrow></mrow><mo id="S2.Thmtheorem3.p4.5.5.4.m4.5.5.3" rspace="0.278em" xref="S2.Thmtheorem3.p4.5.5.4.m4.5.5.3.cmml">:=</mo><mrow id="S2.Thmtheorem3.p4.5.5.4.m4.5.5.2.2" xref="S2.Thmtheorem3.p4.5.5.4.m4.5.5.2.3.cmml"><mo id="S2.Thmtheorem3.p4.5.5.4.m4.5.5.2.2.3" stretchy="false" xref="S2.Thmtheorem3.p4.5.5.4.m4.5.5.2.3.1.cmml">{</mo><mrow id="S2.Thmtheorem3.p4.5.5.4.m4.4.4.1.1.1" xref="S2.Thmtheorem3.p4.5.5.4.m4.4.4.1.1.1.cmml"><mover accent="true" id="S2.Thmtheorem3.p4.5.5.4.m4.4.4.1.1.1.2" xref="S2.Thmtheorem3.p4.5.5.4.m4.4.4.1.1.1.2.cmml"><mrow id="S2.Thmtheorem3.p4.5.5.4.m4.4.4.1.1.1.2.2" xref="S2.Thmtheorem3.p4.5.5.4.m4.4.4.1.1.1.2.2.cmml"><mi id="S2.Thmtheorem3.p4.5.5.4.m4.4.4.1.1.1.2.2.2" mathvariant="normal" xref="S2.Thmtheorem3.p4.5.5.4.m4.4.4.1.1.1.2.2.2.cmml">u</mi><mo id="S2.Thmtheorem3.p4.5.5.4.m4.4.4.1.1.1.2.2.1" xref="S2.Thmtheorem3.p4.5.5.4.m4.4.4.1.1.1.2.2.1.cmml">⁢</mo><mi id="S2.Thmtheorem3.p4.5.5.4.m4.4.4.1.1.1.2.2.3" mathvariant="normal" xref="S2.Thmtheorem3.p4.5.5.4.m4.4.4.1.1.1.2.2.3.cmml">v</mi></mrow><mo id="S2.Thmtheorem3.p4.5.5.4.m4.4.4.1.1.1.2.1" xref="S2.Thmtheorem3.p4.5.5.4.m4.4.4.1.1.1.2.1.cmml">¯</mo></mover><mo id="S2.Thmtheorem3.p4.5.5.4.m4.4.4.1.1.1.1" xref="S2.Thmtheorem3.p4.5.5.4.m4.4.4.1.1.1.1.cmml">∈</mo><mi id="S2.Thmtheorem3.p4.5.5.4.m4.4.4.1.1.1.3" mathvariant="normal" xref="S2.Thmtheorem3.p4.5.5.4.m4.4.4.1.1.1.3.cmml">E</mi></mrow><mo fence="true" id="S2.Thmtheorem3.p4.5.5.4.m4.5.5.2.2.4" lspace="0em" rspace="0em" xref="S2.Thmtheorem3.p4.5.5.4.m4.5.5.2.3.1.cmml">∣</mo><mrow id="S2.Thmtheorem3.p4.5.5.4.m4.5.5.2.2.2" xref="S2.Thmtheorem3.p4.5.5.4.m4.5.5.2.2.2.cmml"><msub id="S2.Thmtheorem3.p4.5.5.4.m4.5.5.2.2.2.2" xref="S2.Thmtheorem3.p4.5.5.4.m4.5.5.2.2.2.2.cmml"><mi id="S2.Thmtheorem3.p4.5.5.4.m4.5.5.2.2.2.2.2" mathvariant="normal" xref="S2.Thmtheorem3.p4.5.5.4.m4.5.5.2.2.2.2.2.cmml">y</mi><mrow id="S2.Thmtheorem3.p4.5.5.4.m4.2.2.2.4" xref="S2.Thmtheorem3.p4.5.5.4.m4.2.2.2.3.cmml"><mi id="S2.Thmtheorem3.p4.5.5.4.m4.1.1.1.1" mathvariant="normal" xref="S2.Thmtheorem3.p4.5.5.4.m4.1.1.1.1.cmml">u</mi><mo id="S2.Thmtheorem3.p4.5.5.4.m4.2.2.2.4.1" xref="S2.Thmtheorem3.p4.5.5.4.m4.2.2.2.3.cmml">,</mo><mi id="S2.Thmtheorem3.p4.5.5.4.m4.2.2.2.2" mathvariant="normal" xref="S2.Thmtheorem3.p4.5.5.4.m4.2.2.2.2.cmml">v</mi></mrow></msub><mo id="S2.Thmtheorem3.p4.5.5.4.m4.5.5.2.2.2.1" xref="S2.Thmtheorem3.p4.5.5.4.m4.5.5.2.2.2.1.cmml">≥</mo><mi id="S2.Thmtheorem3.p4.5.5.4.m4.5.5.2.2.2.3" mathvariant="normal" xref="S2.Thmtheorem3.p4.5.5.4.m4.5.5.2.2.2.3.cmml">r</mi></mrow><mo id="S2.Thmtheorem3.p4.5.5.4.m4.5.5.2.2.5" stretchy="false" xref="S2.Thmtheorem3.p4.5.5.4.m4.5.5.2.3.1.cmml">}</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem3.p4.5.5.4.m4.5b"><apply id="S2.Thmtheorem3.p4.5.5.4.m4.5.5.cmml" xref="S2.Thmtheorem3.p4.5.5.4.m4.5.5"><csymbol cd="latexml" id="S2.Thmtheorem3.p4.5.5.4.m4.5.5.3.cmml" xref="S2.Thmtheorem3.p4.5.5.4.m4.5.5.3">assign</csymbol><apply id="S2.Thmtheorem3.p4.5.5.4.m4.5.5.4.cmml" xref="S2.Thmtheorem3.p4.5.5.4.m4.5.5.4"><times id="S2.Thmtheorem3.p4.5.5.4.m4.5.5.4.1.cmml" xref="S2.Thmtheorem3.p4.5.5.4.m4.5.5.4.1"></times><ci id="S2.Thmtheorem3.p4.5.5.4.m4.5.5.4.2.cmml" xref="S2.Thmtheorem3.p4.5.5.4.m4.5.5.4.2">E</ci><ci id="S2.Thmtheorem3.p4.5.5.4.m4.3.3.cmml" xref="S2.Thmtheorem3.p4.5.5.4.m4.3.3">r</ci></apply><apply id="S2.Thmtheorem3.p4.5.5.4.m4.5.5.2.3.cmml" xref="S2.Thmtheorem3.p4.5.5.4.m4.5.5.2.2"><csymbol cd="latexml" id="S2.Thmtheorem3.p4.5.5.4.m4.5.5.2.3.1.cmml" xref="S2.Thmtheorem3.p4.5.5.4.m4.5.5.2.2.3">conditional-set</csymbol><apply id="S2.Thmtheorem3.p4.5.5.4.m4.4.4.1.1.1.cmml" xref="S2.Thmtheorem3.p4.5.5.4.m4.4.4.1.1.1"><in id="S2.Thmtheorem3.p4.5.5.4.m4.4.4.1.1.1.1.cmml" xref="S2.Thmtheorem3.p4.5.5.4.m4.4.4.1.1.1.1"></in><apply id="S2.Thmtheorem3.p4.5.5.4.m4.4.4.1.1.1.2.cmml" xref="S2.Thmtheorem3.p4.5.5.4.m4.4.4.1.1.1.2"><ci id="S2.Thmtheorem3.p4.5.5.4.m4.4.4.1.1.1.2.1.cmml" xref="S2.Thmtheorem3.p4.5.5.4.m4.4.4.1.1.1.2.1">¯</ci><apply id="S2.Thmtheorem3.p4.5.5.4.m4.4.4.1.1.1.2.2.cmml" xref="S2.Thmtheorem3.p4.5.5.4.m4.4.4.1.1.1.2.2"><times id="S2.Thmtheorem3.p4.5.5.4.m4.4.4.1.1.1.2.2.1.cmml" xref="S2.Thmtheorem3.p4.5.5.4.m4.4.4.1.1.1.2.2.1"></times><ci id="S2.Thmtheorem3.p4.5.5.4.m4.4.4.1.1.1.2.2.2.cmml" xref="S2.Thmtheorem3.p4.5.5.4.m4.4.4.1.1.1.2.2.2">u</ci><ci id="S2.Thmtheorem3.p4.5.5.4.m4.4.4.1.1.1.2.2.3.cmml" xref="S2.Thmtheorem3.p4.5.5.4.m4.4.4.1.1.1.2.2.3">v</ci></apply></apply><ci id="S2.Thmtheorem3.p4.5.5.4.m4.4.4.1.1.1.3.cmml" xref="S2.Thmtheorem3.p4.5.5.4.m4.4.4.1.1.1.3">E</ci></apply><apply id="S2.Thmtheorem3.p4.5.5.4.m4.5.5.2.2.2.cmml" xref="S2.Thmtheorem3.p4.5.5.4.m4.5.5.2.2.2"><geq id="S2.Thmtheorem3.p4.5.5.4.m4.5.5.2.2.2.1.cmml" xref="S2.Thmtheorem3.p4.5.5.4.m4.5.5.2.2.2.1"></geq><apply id="S2.Thmtheorem3.p4.5.5.4.m4.5.5.2.2.2.2.cmml" xref="S2.Thmtheorem3.p4.5.5.4.m4.5.5.2.2.2.2"><csymbol cd="ambiguous" id="S2.Thmtheorem3.p4.5.5.4.m4.5.5.2.2.2.2.1.cmml" xref="S2.Thmtheorem3.p4.5.5.4.m4.5.5.2.2.2.2">subscript</csymbol><ci id="S2.Thmtheorem3.p4.5.5.4.m4.5.5.2.2.2.2.2.cmml" xref="S2.Thmtheorem3.p4.5.5.4.m4.5.5.2.2.2.2.2">y</ci><list id="S2.Thmtheorem3.p4.5.5.4.m4.2.2.2.3.cmml" xref="S2.Thmtheorem3.p4.5.5.4.m4.2.2.2.4"><ci id="S2.Thmtheorem3.p4.5.5.4.m4.1.1.1.1.cmml" xref="S2.Thmtheorem3.p4.5.5.4.m4.1.1.1.1">u</ci><ci id="S2.Thmtheorem3.p4.5.5.4.m4.2.2.2.2.cmml" xref="S2.Thmtheorem3.p4.5.5.4.m4.2.2.2.2">v</ci></list></apply><ci id="S2.Thmtheorem3.p4.5.5.4.m4.5.5.2.2.2.3.cmml" xref="S2.Thmtheorem3.p4.5.5.4.m4.5.5.2.2.2.3">r</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem3.p4.5.5.4.m4.5c">E(r):=\{\overline{uv}\in E\mid y_{u,v}\geq r\}</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem3.p4.5.5.4.m4.5d">italic_E ( italic_r ) := { over¯ start_ARG italic_u italic_v end_ARG ∈ italic_E ∣ italic_y start_POSTSUBSCRIPT italic_u , italic_v end_POSTSUBSCRIPT ≥ italic_r }</annotation></semantics></math>. Denote by <math alttext="\theta[\cdot]" class="ltx_Math" display="inline" id="S2.Thmtheorem3.p4.6.6.5.m5.1"><semantics id="S2.Thmtheorem3.p4.6.6.5.m5.1a"><mrow id="S2.Thmtheorem3.p4.6.6.5.m5.1.2" xref="S2.Thmtheorem3.p4.6.6.5.m5.1.2.cmml"><mi id="S2.Thmtheorem3.p4.6.6.5.m5.1.2.2" mathvariant="normal" xref="S2.Thmtheorem3.p4.6.6.5.m5.1.2.2.cmml">θ</mi><mo id="S2.Thmtheorem3.p4.6.6.5.m5.1.2.1" xref="S2.Thmtheorem3.p4.6.6.5.m5.1.2.1.cmml">⁢</mo><mrow id="S2.Thmtheorem3.p4.6.6.5.m5.1.2.3.2" xref="S2.Thmtheorem3.p4.6.6.5.m5.1.2.3.1.cmml"><mo id="S2.Thmtheorem3.p4.6.6.5.m5.1.2.3.2.1" stretchy="false" xref="S2.Thmtheorem3.p4.6.6.5.m5.1.2.3.1.1.cmml">[</mo><mo id="S2.Thmtheorem3.p4.6.6.5.m5.1.1" lspace="0em" rspace="0em" xref="S2.Thmtheorem3.p4.6.6.5.m5.1.1.cmml">⋅</mo><mo id="S2.Thmtheorem3.p4.6.6.5.m5.1.2.3.2.2" stretchy="false" xref="S2.Thmtheorem3.p4.6.6.5.m5.1.2.3.1.1.cmml">]</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem3.p4.6.6.5.m5.1b"><apply id="S2.Thmtheorem3.p4.6.6.5.m5.1.2.cmml" xref="S2.Thmtheorem3.p4.6.6.5.m5.1.2"><times id="S2.Thmtheorem3.p4.6.6.5.m5.1.2.1.cmml" xref="S2.Thmtheorem3.p4.6.6.5.m5.1.2.1"></times><ci id="S2.Thmtheorem3.p4.6.6.5.m5.1.2.2.cmml" xref="S2.Thmtheorem3.p4.6.6.5.m5.1.2.2">θ</ci><apply id="S2.Thmtheorem3.p4.6.6.5.m5.1.2.3.1.cmml" xref="S2.Thmtheorem3.p4.6.6.5.m5.1.2.3.2"><csymbol cd="latexml" id="S2.Thmtheorem3.p4.6.6.5.m5.1.2.3.1.1.cmml" xref="S2.Thmtheorem3.p4.6.6.5.m5.1.2.3.2.1">delimited-[]</csymbol><ci id="S2.Thmtheorem3.p4.6.6.5.m5.1.1.cmml" xref="S2.Thmtheorem3.p4.6.6.5.m5.1.1">⋅</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem3.p4.6.6.5.m5.1c">\theta[\cdot]</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem3.p4.6.6.5.m5.1d">italic_θ [ ⋅ ]</annotation></semantics></math> the indicator function which gives <math alttext="1" class="ltx_Math" display="inline" id="S2.Thmtheorem3.p4.7.7.6.m6.1"><semantics id="S2.Thmtheorem3.p4.7.7.6.m6.1a"><mn id="S2.Thmtheorem3.p4.7.7.6.m6.1.1" xref="S2.Thmtheorem3.p4.7.7.6.m6.1.1.cmml">1</mn><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem3.p4.7.7.6.m6.1b"><cn id="S2.Thmtheorem3.p4.7.7.6.m6.1.1.cmml" type="integer" xref="S2.Thmtheorem3.p4.7.7.6.m6.1.1">1</cn></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem3.p4.7.7.6.m6.1c">1</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem3.p4.7.7.6.m6.1d">1</annotation></semantics></math> when the expression <math alttext="\cdot" class="ltx_Math" display="inline" id="S2.Thmtheorem3.p4.8.8.7.m7.1"><semantics id="S2.Thmtheorem3.p4.8.8.7.m7.1a"><mo id="S2.Thmtheorem3.p4.8.8.7.m7.1.1" xref="S2.Thmtheorem3.p4.8.8.7.m7.1.1.cmml">⋅</mo><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem3.p4.8.8.7.m7.1b"><ci id="S2.Thmtheorem3.p4.8.8.7.m7.1.1.cmml" xref="S2.Thmtheorem3.p4.8.8.7.m7.1.1">⋅</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem3.p4.8.8.7.m7.1c">\cdot</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem3.p4.8.8.7.m7.1d">⋅</annotation></semantics></math> is true and <math alttext="0" class="ltx_Math" display="inline" id="S2.Thmtheorem3.p4.9.9.8.m8.1"><semantics id="S2.Thmtheorem3.p4.9.9.8.m8.1a"><mn id="S2.Thmtheorem3.p4.9.9.8.m8.1.1" xref="S2.Thmtheorem3.p4.9.9.8.m8.1.1.cmml">0</mn><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem3.p4.9.9.8.m8.1b"><cn id="S2.Thmtheorem3.p4.9.9.8.m8.1.1.cmml" type="integer" xref="S2.Thmtheorem3.p4.9.9.8.m8.1.1">0</cn></annotation-xml></semantics></math> otherwise.</span></span></p> </div> <div class="ltx_para" id="S2.Thmtheorem3.p5"> <table class="ltx_equation ltx_eqn_table" id="S2.E1"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_left_padleft"></td> <td class="ltx_eqn_cell ltx_align_left"><math alttext="\int_{0}^{\infty}|V(r)|\,dr=\int_{0}^{\infty}\sum_{v}\theta[u_{v}\geq r]\,dr=% \sum_{u}\int_{0}^{x_{u}}1\,dr=\sum_{u}x_{u}\leq 1" class="ltx_Math" display="block" id="S2.E1.m1.3"><semantics id="S2.E1.m1.3a"><mrow id="S2.E1.m1.3.3" xref="S2.E1.m1.3.3.cmml"><mrow id="S2.E1.m1.2.2.1" xref="S2.E1.m1.2.2.1.cmml"><msubsup id="S2.E1.m1.2.2.1.2" xref="S2.E1.m1.2.2.1.2.cmml"><mo id="S2.E1.m1.2.2.1.2.2.2" xref="S2.E1.m1.2.2.1.2.2.2.cmml">∫</mo><mn id="S2.E1.m1.2.2.1.2.2.3" xref="S2.E1.m1.2.2.1.2.2.3.cmml">0</mn><mi id="S2.E1.m1.2.2.1.2.3" mathvariant="normal" xref="S2.E1.m1.2.2.1.2.3.cmml">∞</mi></msubsup><mrow id="S2.E1.m1.2.2.1.1" xref="S2.E1.m1.2.2.1.1.cmml"><mrow id="S2.E1.m1.2.2.1.1.1.1" xref="S2.E1.m1.2.2.1.1.1.2.cmml"><mo id="S2.E1.m1.2.2.1.1.1.1.2" lspace="0em" stretchy="false" xref="S2.E1.m1.2.2.1.1.1.2.1.cmml">|</mo><mrow id="S2.E1.m1.2.2.1.1.1.1.1" xref="S2.E1.m1.2.2.1.1.1.1.1.cmml"><mi id="S2.E1.m1.2.2.1.1.1.1.1.2" xref="S2.E1.m1.2.2.1.1.1.1.1.2.cmml">V</mi><mo id="S2.E1.m1.2.2.1.1.1.1.1.1" xref="S2.E1.m1.2.2.1.1.1.1.1.1.cmml">⁢</mo><mrow id="S2.E1.m1.2.2.1.1.1.1.1.3.2" xref="S2.E1.m1.2.2.1.1.1.1.1.cmml"><mo id="S2.E1.m1.2.2.1.1.1.1.1.3.2.1" stretchy="false" xref="S2.E1.m1.2.2.1.1.1.1.1.cmml">(</mo><mi id="S2.E1.m1.1.1" xref="S2.E1.m1.1.1.cmml">r</mi><mo id="S2.E1.m1.2.2.1.1.1.1.1.3.2.2" stretchy="false" xref="S2.E1.m1.2.2.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S2.E1.m1.2.2.1.1.1.1.3" stretchy="false" xref="S2.E1.m1.2.2.1.1.1.2.1.cmml">|</mo></mrow><mo id="S2.E1.m1.2.2.1.1.2" lspace="0.170em" xref="S2.E1.m1.2.2.1.1.2.cmml">⁢</mo><mrow id="S2.E1.m1.2.2.1.1.3" xref="S2.E1.m1.2.2.1.1.3.cmml"><mo id="S2.E1.m1.2.2.1.1.3.1" rspace="0em" xref="S2.E1.m1.2.2.1.1.3.1.cmml">𝑑</mo><mi id="S2.E1.m1.2.2.1.1.3.2" xref="S2.E1.m1.2.2.1.1.3.2.cmml">r</mi></mrow></mrow></mrow><mo id="S2.E1.m1.3.3.4" rspace="0.111em" xref="S2.E1.m1.3.3.4.cmml">=</mo><mrow id="S2.E1.m1.3.3.2" xref="S2.E1.m1.3.3.2.cmml"><msubsup id="S2.E1.m1.3.3.2.2" xref="S2.E1.m1.3.3.2.2.cmml"><mo id="S2.E1.m1.3.3.2.2.2.2" rspace="0em" xref="S2.E1.m1.3.3.2.2.2.2.cmml">∫</mo><mn id="S2.E1.m1.3.3.2.2.2.3" xref="S2.E1.m1.3.3.2.2.2.3.cmml">0</mn><mi id="S2.E1.m1.3.3.2.2.3" mathvariant="normal" xref="S2.E1.m1.3.3.2.2.3.cmml">∞</mi></msubsup><mrow id="S2.E1.m1.3.3.2.1" xref="S2.E1.m1.3.3.2.1.cmml"><munder id="S2.E1.m1.3.3.2.1.2" xref="S2.E1.m1.3.3.2.1.2.cmml"><mo id="S2.E1.m1.3.3.2.1.2.2" movablelimits="false" xref="S2.E1.m1.3.3.2.1.2.2.cmml">∑</mo><mi id="S2.E1.m1.3.3.2.1.2.3" xref="S2.E1.m1.3.3.2.1.2.3.cmml">v</mi></munder><mrow id="S2.E1.m1.3.3.2.1.1" xref="S2.E1.m1.3.3.2.1.1.cmml"><mi id="S2.E1.m1.3.3.2.1.1.3" xref="S2.E1.m1.3.3.2.1.1.3.cmml">θ</mi><mo id="S2.E1.m1.3.3.2.1.1.2" xref="S2.E1.m1.3.3.2.1.1.2.cmml">⁢</mo><mrow id="S2.E1.m1.3.3.2.1.1.1.1" xref="S2.E1.m1.3.3.2.1.1.1.2.cmml"><mo id="S2.E1.m1.3.3.2.1.1.1.1.2" stretchy="false" xref="S2.E1.m1.3.3.2.1.1.1.2.1.cmml">[</mo><mrow id="S2.E1.m1.3.3.2.1.1.1.1.1" xref="S2.E1.m1.3.3.2.1.1.1.1.1.cmml"><msub id="S2.E1.m1.3.3.2.1.1.1.1.1.2" xref="S2.E1.m1.3.3.2.1.1.1.1.1.2.cmml"><mi id="S2.E1.m1.3.3.2.1.1.1.1.1.2.2" xref="S2.E1.m1.3.3.2.1.1.1.1.1.2.2.cmml">u</mi><mi id="S2.E1.m1.3.3.2.1.1.1.1.1.2.3" xref="S2.E1.m1.3.3.2.1.1.1.1.1.2.3.cmml">v</mi></msub><mo id="S2.E1.m1.3.3.2.1.1.1.1.1.1" xref="S2.E1.m1.3.3.2.1.1.1.1.1.1.cmml">≥</mo><mi id="S2.E1.m1.3.3.2.1.1.1.1.1.3" xref="S2.E1.m1.3.3.2.1.1.1.1.1.3.cmml">r</mi></mrow><mo id="S2.E1.m1.3.3.2.1.1.1.1.3" stretchy="false" xref="S2.E1.m1.3.3.2.1.1.1.2.1.cmml">]</mo></mrow><mo id="S2.E1.m1.3.3.2.1.1.2a" lspace="0.170em" xref="S2.E1.m1.3.3.2.1.1.2.cmml">⁢</mo><mi id="S2.E1.m1.3.3.2.1.1.4" xref="S2.E1.m1.3.3.2.1.1.4.cmml">d</mi><mo id="S2.E1.m1.3.3.2.1.1.2b" xref="S2.E1.m1.3.3.2.1.1.2.cmml">⁢</mo><mi id="S2.E1.m1.3.3.2.1.1.5" xref="S2.E1.m1.3.3.2.1.1.5.cmml">r</mi></mrow></mrow></mrow><mo id="S2.E1.m1.3.3.5" rspace="0.111em" xref="S2.E1.m1.3.3.5.cmml">=</mo><mrow id="S2.E1.m1.3.3.6" xref="S2.E1.m1.3.3.6.cmml"><munder id="S2.E1.m1.3.3.6.1" xref="S2.E1.m1.3.3.6.1.cmml"><mo id="S2.E1.m1.3.3.6.1.2" movablelimits="false" rspace="0em" xref="S2.E1.m1.3.3.6.1.2.cmml">∑</mo><mi id="S2.E1.m1.3.3.6.1.3" xref="S2.E1.m1.3.3.6.1.3.cmml">u</mi></munder><mrow id="S2.E1.m1.3.3.6.2" xref="S2.E1.m1.3.3.6.2.cmml"><msubsup id="S2.E1.m1.3.3.6.2.1" xref="S2.E1.m1.3.3.6.2.1.cmml"><mo id="S2.E1.m1.3.3.6.2.1.2.2" xref="S2.E1.m1.3.3.6.2.1.2.2.cmml">∫</mo><mn id="S2.E1.m1.3.3.6.2.1.2.3" xref="S2.E1.m1.3.3.6.2.1.2.3.cmml">0</mn><msub id="S2.E1.m1.3.3.6.2.1.3" xref="S2.E1.m1.3.3.6.2.1.3.cmml"><mi id="S2.E1.m1.3.3.6.2.1.3.2" xref="S2.E1.m1.3.3.6.2.1.3.2.cmml">x</mi><mi id="S2.E1.m1.3.3.6.2.1.3.3" xref="S2.E1.m1.3.3.6.2.1.3.3.cmml">u</mi></msub></msubsup><mrow id="S2.E1.m1.3.3.6.2.2" xref="S2.E1.m1.3.3.6.2.2.cmml"><mn id="S2.E1.m1.3.3.6.2.2.2" xref="S2.E1.m1.3.3.6.2.2.2.cmml">1</mn><mo id="S2.E1.m1.3.3.6.2.2.1" lspace="0.170em" xref="S2.E1.m1.3.3.6.2.2.1.cmml">⁢</mo><mrow id="S2.E1.m1.3.3.6.2.2.3" xref="S2.E1.m1.3.3.6.2.2.3.cmml"><mo id="S2.E1.m1.3.3.6.2.2.3.1" rspace="0em" xref="S2.E1.m1.3.3.6.2.2.3.1.cmml">𝑑</mo><mi id="S2.E1.m1.3.3.6.2.2.3.2" xref="S2.E1.m1.3.3.6.2.2.3.2.cmml">r</mi></mrow></mrow></mrow></mrow><mo id="S2.E1.m1.3.3.7" rspace="0.111em" xref="S2.E1.m1.3.3.7.cmml">=</mo><mrow id="S2.E1.m1.3.3.8" xref="S2.E1.m1.3.3.8.cmml"><munder id="S2.E1.m1.3.3.8.1" xref="S2.E1.m1.3.3.8.1.cmml"><mo id="S2.E1.m1.3.3.8.1.2" movablelimits="false" xref="S2.E1.m1.3.3.8.1.2.cmml">∑</mo><mi id="S2.E1.m1.3.3.8.1.3" xref="S2.E1.m1.3.3.8.1.3.cmml">u</mi></munder><msub id="S2.E1.m1.3.3.8.2" xref="S2.E1.m1.3.3.8.2.cmml"><mi id="S2.E1.m1.3.3.8.2.2" xref="S2.E1.m1.3.3.8.2.2.cmml">x</mi><mi id="S2.E1.m1.3.3.8.2.3" xref="S2.E1.m1.3.3.8.2.3.cmml">u</mi></msub></mrow><mo id="S2.E1.m1.3.3.9" xref="S2.E1.m1.3.3.9.cmml">≤</mo><mn id="S2.E1.m1.3.3.10" xref="S2.E1.m1.3.3.10.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S2.E1.m1.3b"><apply id="S2.E1.m1.3.3.cmml" xref="S2.E1.m1.3.3"><and id="S2.E1.m1.3.3a.cmml" xref="S2.E1.m1.3.3"></and><apply 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encoding="application/x-tex" id="S2.E1.m1.3c">\int_{0}^{\infty}|V(r)|\,dr=\int_{0}^{\infty}\sum_{v}\theta[u_{v}\geq r]\,dr=% \sum_{u}\int_{0}^{x_{u}}1\,dr=\sum_{u}x_{u}\leq 1</annotation><annotation encoding="application/x-llamapun" id="S2.E1.m1.3d">∫ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ∞ end_POSTSUPERSCRIPT | italic_V ( italic_r ) | italic_d italic_r = ∫ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ∞ end_POSTSUPERSCRIPT ∑ start_POSTSUBSCRIPT italic_v end_POSTSUBSCRIPT italic_θ [ italic_u start_POSTSUBSCRIPT italic_v end_POSTSUBSCRIPT ≥ italic_r ] italic_d italic_r = ∑ start_POSTSUBSCRIPT italic_u end_POSTSUBSCRIPT ∫ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_x start_POSTSUBSCRIPT italic_u end_POSTSUBSCRIPT end_POSTSUPERSCRIPT 1 italic_d italic_r = ∑ start_POSTSUBSCRIPT italic_u end_POSTSUBSCRIPT italic_x start_POSTSUBSCRIPT italic_u end_POSTSUBSCRIPT ≤ 1</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_left_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(1)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S2.Thmtheorem3.p5.1"><span class="ltx_text ltx_font_italic" id="S2.Thmtheorem3.p5.1.1">Similarly, we can denote:</span></p> </div> <div class="ltx_para" id="S2.Thmtheorem3.p6"> <table class="ltx_equation ltx_eqn_table" id="S2.E2"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_left_padleft"></td> <td class="ltx_eqn_cell ltx_align_left"><math alttext="\int_{0}^{\infty}\sum_{\overline{uv}\in E(r)}\textsl{g}(\overline{uv})\,dr=% \int_{0}^{\infty}\sum_{\overline{uv}}\theta[y_{u,v}\geq r]\cdot\textsl{g}(% \overline{uv})\,dr=\sum_{\overline{uv}}\int_{0}^{y_{u,v}}\textsl{g}(\overline{% uv})\,dr=\sum_{\overline{uv}}\textsl{g}(\overline{uv})\cdot y_{u,v}=R" class="ltx_Math" display="block" id="S2.E2.m1.12"><semantics 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xref="S2.E2.m1.1.1.1.3.2.1.cmml">⁢</mo><mi id="S2.E2.m1.1.1.1.3.2.3" xref="S2.E2.m1.1.1.1.3.2.3.cmml">v</mi></mrow><mo id="S2.E2.m1.1.1.1.3.1" xref="S2.E2.m1.1.1.1.3.1.cmml">¯</mo></mover><mo id="S2.E2.m1.1.1.1.2" xref="S2.E2.m1.1.1.1.2.cmml">∈</mo><mrow id="S2.E2.m1.1.1.1.4" xref="S2.E2.m1.1.1.1.4.cmml"><mi id="S2.E2.m1.1.1.1.4.2" xref="S2.E2.m1.1.1.1.4.2.cmml">E</mi><mo id="S2.E2.m1.1.1.1.4.1" xref="S2.E2.m1.1.1.1.4.1.cmml">⁢</mo><mrow id="S2.E2.m1.1.1.1.4.3.2" xref="S2.E2.m1.1.1.1.4.cmml"><mo id="S2.E2.m1.1.1.1.4.3.2.1" stretchy="false" xref="S2.E2.m1.1.1.1.4.cmml">(</mo><mi id="S2.E2.m1.1.1.1.1" xref="S2.E2.m1.1.1.1.1.cmml">r</mi><mo id="S2.E2.m1.1.1.1.4.3.2.2" stretchy="false" xref="S2.E2.m1.1.1.1.4.cmml">)</mo></mrow></mrow></mrow></munder><mrow id="S2.E2.m1.12.12.3.2.2" xref="S2.E2.m1.12.12.3.2.2.cmml"><mtext class="ltx_mathvariant_italic" id="S2.E2.m1.12.12.3.2.2.2" xref="S2.E2.m1.12.12.3.2.2.2a.cmml">g</mtext><mo id="S2.E2.m1.12.12.3.2.2.1" xref="S2.E2.m1.12.12.3.2.2.1.cmml">⁢</mo><mrow id="S2.E2.m1.12.12.3.2.2.3.2" xref="S2.E2.m1.8.8.cmml"><mo id="S2.E2.m1.12.12.3.2.2.3.2.1" stretchy="false" xref="S2.E2.m1.8.8.cmml">(</mo><mover accent="true" id="S2.E2.m1.8.8" xref="S2.E2.m1.8.8.cmml"><mrow id="S2.E2.m1.8.8.2" xref="S2.E2.m1.8.8.2.cmml"><mi id="S2.E2.m1.8.8.2.2" xref="S2.E2.m1.8.8.2.2.cmml">u</mi><mo id="S2.E2.m1.8.8.2.1" xref="S2.E2.m1.8.8.2.1.cmml">⁢</mo><mi id="S2.E2.m1.8.8.2.3" xref="S2.E2.m1.8.8.2.3.cmml">v</mi></mrow><mo id="S2.E2.m1.8.8.1" xref="S2.E2.m1.8.8.1.cmml">¯</mo></mover><mo id="S2.E2.m1.12.12.3.2.2.3.2.2" stretchy="false" xref="S2.E2.m1.8.8.cmml">)</mo></mrow><mo id="S2.E2.m1.12.12.3.2.2.1a" lspace="0.170em" xref="S2.E2.m1.12.12.3.2.2.1.cmml">⁢</mo><mi id="S2.E2.m1.12.12.3.2.2.4" xref="S2.E2.m1.12.12.3.2.2.4.cmml">d</mi><mo id="S2.E2.m1.12.12.3.2.2.1b" xref="S2.E2.m1.12.12.3.2.2.1.cmml">⁢</mo><mi id="S2.E2.m1.12.12.3.2.2.5" xref="S2.E2.m1.12.12.3.2.2.5.cmml">r</mi></mrow></mrow></mrow><mo id="S2.E2.m1.12.12.4" rspace="0.111em" xref="S2.E2.m1.12.12.4.cmml">=</mo><mrow id="S2.E2.m1.12.12.1" xref="S2.E2.m1.12.12.1.cmml"><msubsup id="S2.E2.m1.12.12.1.2" xref="S2.E2.m1.12.12.1.2.cmml"><mo id="S2.E2.m1.12.12.1.2.2.2" rspace="0em" xref="S2.E2.m1.12.12.1.2.2.2.cmml">∫</mo><mn id="S2.E2.m1.12.12.1.2.2.3" xref="S2.E2.m1.12.12.1.2.2.3.cmml">0</mn><mi id="S2.E2.m1.12.12.1.2.3" mathvariant="normal" xref="S2.E2.m1.12.12.1.2.3.cmml">∞</mi></msubsup><mrow id="S2.E2.m1.12.12.1.1" xref="S2.E2.m1.12.12.1.1.cmml"><munder id="S2.E2.m1.12.12.1.1.2" xref="S2.E2.m1.12.12.1.1.2.cmml"><mo id="S2.E2.m1.12.12.1.1.2.2" movablelimits="false" xref="S2.E2.m1.12.12.1.1.2.2.cmml">∑</mo><mover accent="true" id="S2.E2.m1.12.12.1.1.2.3" xref="S2.E2.m1.12.12.1.1.2.3.cmml"><mrow id="S2.E2.m1.12.12.1.1.2.3.2" xref="S2.E2.m1.12.12.1.1.2.3.2.cmml"><mi id="S2.E2.m1.12.12.1.1.2.3.2.2" xref="S2.E2.m1.12.12.1.1.2.3.2.2.cmml">u</mi><mo id="S2.E2.m1.12.12.1.1.2.3.2.1" xref="S2.E2.m1.12.12.1.1.2.3.2.1.cmml">⁢</mo><mi id="S2.E2.m1.12.12.1.1.2.3.2.3" xref="S2.E2.m1.12.12.1.1.2.3.2.3.cmml">v</mi></mrow><mo id="S2.E2.m1.12.12.1.1.2.3.1" xref="S2.E2.m1.12.12.1.1.2.3.1.cmml">¯</mo></mover></munder><mrow id="S2.E2.m1.12.12.1.1.1" xref="S2.E2.m1.12.12.1.1.1.cmml"><mrow id="S2.E2.m1.12.12.1.1.1.1" xref="S2.E2.m1.12.12.1.1.1.1.cmml"><mrow id="S2.E2.m1.12.12.1.1.1.1.1" xref="S2.E2.m1.12.12.1.1.1.1.1.cmml"><mi id="S2.E2.m1.12.12.1.1.1.1.1.3" xref="S2.E2.m1.12.12.1.1.1.1.1.3.cmml">θ</mi><mo id="S2.E2.m1.12.12.1.1.1.1.1.2" xref="S2.E2.m1.12.12.1.1.1.1.1.2.cmml">⁢</mo><mrow id="S2.E2.m1.12.12.1.1.1.1.1.1.1" xref="S2.E2.m1.12.12.1.1.1.1.1.1.2.cmml"><mo id="S2.E2.m1.12.12.1.1.1.1.1.1.1.2" stretchy="false" xref="S2.E2.m1.12.12.1.1.1.1.1.1.2.1.cmml">[</mo><mrow id="S2.E2.m1.12.12.1.1.1.1.1.1.1.1" xref="S2.E2.m1.12.12.1.1.1.1.1.1.1.1.cmml"><msub id="S2.E2.m1.12.12.1.1.1.1.1.1.1.1.2" xref="S2.E2.m1.12.12.1.1.1.1.1.1.1.1.2.cmml"><mi id="S2.E2.m1.12.12.1.1.1.1.1.1.1.1.2.2" xref="S2.E2.m1.12.12.1.1.1.1.1.1.1.1.2.2.cmml">y</mi><mrow id="S2.E2.m1.3.3.2.4" xref="S2.E2.m1.3.3.2.3.cmml"><mi id="S2.E2.m1.2.2.1.1" xref="S2.E2.m1.2.2.1.1.cmml">u</mi><mo id="S2.E2.m1.3.3.2.4.1" xref="S2.E2.m1.3.3.2.3.cmml">,</mo><mi id="S2.E2.m1.3.3.2.2" xref="S2.E2.m1.3.3.2.2.cmml">v</mi></mrow></msub><mo id="S2.E2.m1.12.12.1.1.1.1.1.1.1.1.1" xref="S2.E2.m1.12.12.1.1.1.1.1.1.1.1.1.cmml">≥</mo><mi id="S2.E2.m1.12.12.1.1.1.1.1.1.1.1.3" xref="S2.E2.m1.12.12.1.1.1.1.1.1.1.1.3.cmml">r</mi></mrow><mo id="S2.E2.m1.12.12.1.1.1.1.1.1.1.3" rspace="0.055em" stretchy="false" xref="S2.E2.m1.12.12.1.1.1.1.1.1.2.1.cmml">]</mo></mrow></mrow><mo id="S2.E2.m1.12.12.1.1.1.1.2" rspace="0.222em" xref="S2.E2.m1.12.12.1.1.1.1.2.cmml">⋅</mo><mtext class="ltx_mathvariant_italic" id="S2.E2.m1.12.12.1.1.1.1.3" xref="S2.E2.m1.12.12.1.1.1.1.3a.cmml">g</mtext></mrow><mo id="S2.E2.m1.12.12.1.1.1.2" xref="S2.E2.m1.12.12.1.1.1.2.cmml">⁢</mo><mrow id="S2.E2.m1.12.12.1.1.1.3.2" xref="S2.E2.m1.9.9.cmml"><mo id="S2.E2.m1.12.12.1.1.1.3.2.1" stretchy="false" xref="S2.E2.m1.9.9.cmml">(</mo><mover accent="true" id="S2.E2.m1.9.9" xref="S2.E2.m1.9.9.cmml"><mrow id="S2.E2.m1.9.9.2" xref="S2.E2.m1.9.9.2.cmml"><mi id="S2.E2.m1.9.9.2.2" xref="S2.E2.m1.9.9.2.2.cmml">u</mi><mo id="S2.E2.m1.9.9.2.1" xref="S2.E2.m1.9.9.2.1.cmml">⁢</mo><mi id="S2.E2.m1.9.9.2.3" xref="S2.E2.m1.9.9.2.3.cmml">v</mi></mrow><mo id="S2.E2.m1.9.9.1" xref="S2.E2.m1.9.9.1.cmml">¯</mo></mover><mo id="S2.E2.m1.12.12.1.1.1.3.2.2" stretchy="false" xref="S2.E2.m1.9.9.cmml">)</mo></mrow><mo id="S2.E2.m1.12.12.1.1.1.2a" lspace="0.170em" xref="S2.E2.m1.12.12.1.1.1.2.cmml">⁢</mo><mi id="S2.E2.m1.12.12.1.1.1.4" xref="S2.E2.m1.12.12.1.1.1.4.cmml">d</mi><mo id="S2.E2.m1.12.12.1.1.1.2b" xref="S2.E2.m1.12.12.1.1.1.2.cmml">⁢</mo><mi id="S2.E2.m1.12.12.1.1.1.5" xref="S2.E2.m1.12.12.1.1.1.5.cmml">r</mi></mrow></mrow></mrow><mo id="S2.E2.m1.12.12.5" rspace="0.111em" xref="S2.E2.m1.12.12.5.cmml">=</mo><mrow id="S2.E2.m1.12.12.6" xref="S2.E2.m1.12.12.6.cmml"><munder id="S2.E2.m1.12.12.6.1" xref="S2.E2.m1.12.12.6.1.cmml"><mo id="S2.E2.m1.12.12.6.1.2" movablelimits="false" rspace="0em" xref="S2.E2.m1.12.12.6.1.2.cmml">∑</mo><mover accent="true" id="S2.E2.m1.12.12.6.1.3" xref="S2.E2.m1.12.12.6.1.3.cmml"><mrow id="S2.E2.m1.12.12.6.1.3.2" xref="S2.E2.m1.12.12.6.1.3.2.cmml"><mi id="S2.E2.m1.12.12.6.1.3.2.2" xref="S2.E2.m1.12.12.6.1.3.2.2.cmml">u</mi><mo id="S2.E2.m1.12.12.6.1.3.2.1" xref="S2.E2.m1.12.12.6.1.3.2.1.cmml">⁢</mo><mi id="S2.E2.m1.12.12.6.1.3.2.3" xref="S2.E2.m1.12.12.6.1.3.2.3.cmml">v</mi></mrow><mo id="S2.E2.m1.12.12.6.1.3.1" xref="S2.E2.m1.12.12.6.1.3.1.cmml">¯</mo></mover></munder><mrow id="S2.E2.m1.12.12.6.2" xref="S2.E2.m1.12.12.6.2.cmml"><msubsup id="S2.E2.m1.12.12.6.2.1" xref="S2.E2.m1.12.12.6.2.1.cmml"><mo id="S2.E2.m1.12.12.6.2.1.2.2" xref="S2.E2.m1.12.12.6.2.1.2.2.cmml">∫</mo><mn id="S2.E2.m1.12.12.6.2.1.2.3" xref="S2.E2.m1.12.12.6.2.1.2.3.cmml">0</mn><msub id="S2.E2.m1.5.5.2" xref="S2.E2.m1.5.5.2.cmml"><mi id="S2.E2.m1.5.5.2.4" xref="S2.E2.m1.5.5.2.4.cmml">y</mi><mrow id="S2.E2.m1.5.5.2.2.2.4" xref="S2.E2.m1.5.5.2.2.2.3.cmml"><mi id="S2.E2.m1.4.4.1.1.1.1" xref="S2.E2.m1.4.4.1.1.1.1.cmml">u</mi><mo id="S2.E2.m1.5.5.2.2.2.4.1" xref="S2.E2.m1.5.5.2.2.2.3.cmml">,</mo><mi id="S2.E2.m1.5.5.2.2.2.2" xref="S2.E2.m1.5.5.2.2.2.2.cmml">v</mi></mrow></msub></msubsup><mrow id="S2.E2.m1.12.12.6.2.2" xref="S2.E2.m1.12.12.6.2.2.cmml"><mtext class="ltx_mathvariant_italic" id="S2.E2.m1.12.12.6.2.2.2" xref="S2.E2.m1.12.12.6.2.2.2a.cmml">g</mtext><mo id="S2.E2.m1.12.12.6.2.2.1" xref="S2.E2.m1.12.12.6.2.2.1.cmml">⁢</mo><mrow id="S2.E2.m1.12.12.6.2.2.3.2" xref="S2.E2.m1.10.10.cmml"><mo id="S2.E2.m1.12.12.6.2.2.3.2.1" stretchy="false" xref="S2.E2.m1.10.10.cmml">(</mo><mover accent="true" id="S2.E2.m1.10.10" xref="S2.E2.m1.10.10.cmml"><mrow id="S2.E2.m1.10.10.2" xref="S2.E2.m1.10.10.2.cmml"><mi id="S2.E2.m1.10.10.2.2" xref="S2.E2.m1.10.10.2.2.cmml">u</mi><mo id="S2.E2.m1.10.10.2.1" xref="S2.E2.m1.10.10.2.1.cmml">⁢</mo><mi id="S2.E2.m1.10.10.2.3" xref="S2.E2.m1.10.10.2.3.cmml">v</mi></mrow><mo id="S2.E2.m1.10.10.1" xref="S2.E2.m1.10.10.1.cmml">¯</mo></mover><mo id="S2.E2.m1.12.12.6.2.2.3.2.2" stretchy="false" xref="S2.E2.m1.10.10.cmml">)</mo></mrow><mo id="S2.E2.m1.12.12.6.2.2.1a" lspace="0.170em" xref="S2.E2.m1.12.12.6.2.2.1.cmml">⁢</mo><mrow id="S2.E2.m1.12.12.6.2.2.4" xref="S2.E2.m1.12.12.6.2.2.4.cmml"><mo id="S2.E2.m1.12.12.6.2.2.4.1" rspace="0em" xref="S2.E2.m1.12.12.6.2.2.4.1.cmml">𝑑</mo><mi id="S2.E2.m1.12.12.6.2.2.4.2" xref="S2.E2.m1.12.12.6.2.2.4.2.cmml">r</mi></mrow></mrow></mrow></mrow><mo id="S2.E2.m1.12.12.7" rspace="0.111em" xref="S2.E2.m1.12.12.7.cmml">=</mo><mrow id="S2.E2.m1.12.12.8" xref="S2.E2.m1.12.12.8.cmml"><munder id="S2.E2.m1.12.12.8.1" xref="S2.E2.m1.12.12.8.1.cmml"><mo id="S2.E2.m1.12.12.8.1.2" movablelimits="false" xref="S2.E2.m1.12.12.8.1.2.cmml">∑</mo><mover accent="true" id="S2.E2.m1.12.12.8.1.3" 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uv})\,dr=\sum_{\overline{uv}}\textsl{g}(\overline{uv})\cdot y_{u,v}=R</annotation><annotation encoding="application/x-llamapun" id="S2.E2.m1.12d">∫ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ∞ end_POSTSUPERSCRIPT ∑ start_POSTSUBSCRIPT over¯ start_ARG italic_u italic_v end_ARG ∈ italic_E ( italic_r ) end_POSTSUBSCRIPT g ( over¯ start_ARG italic_u italic_v end_ARG ) italic_d italic_r = ∫ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ∞ end_POSTSUPERSCRIPT ∑ start_POSTSUBSCRIPT over¯ start_ARG italic_u italic_v end_ARG end_POSTSUBSCRIPT italic_θ [ italic_y start_POSTSUBSCRIPT italic_u , italic_v end_POSTSUBSCRIPT ≥ italic_r ] ⋅ g ( over¯ start_ARG italic_u italic_v end_ARG ) italic_d italic_r = ∑ start_POSTSUBSCRIPT over¯ start_ARG italic_u italic_v end_ARG end_POSTSUBSCRIPT ∫ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_y start_POSTSUBSCRIPT italic_u , italic_v end_POSTSUBSCRIPT end_POSTSUPERSCRIPT g ( over¯ start_ARG italic_u italic_v end_ARG ) italic_d italic_r = ∑ start_POSTSUBSCRIPT over¯ start_ARG italic_u italic_v end_ARG end_POSTSUBSCRIPT g ( over¯ start_ARG italic_u italic_v end_ARG ) ⋅ italic_y start_POSTSUBSCRIPT italic_u , italic_v end_POSTSUBSCRIPT = italic_R</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_left_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(2)</span></td> </tr></tbody> </table> </div> <div class="ltx_para" id="S2.Thmtheorem3.p7"> <p class="ltx_p" id="S2.Thmtheorem3.p7.3"><span class="ltx_text ltx_font_italic" id="S2.Thmtheorem3.p7.3.3">Now we claim that we can choose a value <math alttext="\hat{r}" class="ltx_Math" display="inline" id="S2.Thmtheorem3.p7.1.1.m1.1"><semantics id="S2.Thmtheorem3.p7.1.1.m1.1a"><mover accent="true" id="S2.Thmtheorem3.p7.1.1.m1.1.1" xref="S2.Thmtheorem3.p7.1.1.m1.1.1.cmml"><mi id="S2.Thmtheorem3.p7.1.1.m1.1.1.2" xref="S2.Thmtheorem3.p7.1.1.m1.1.1.2.cmml">r</mi><mo id="S2.Thmtheorem3.p7.1.1.m1.1.1.1" xref="S2.Thmtheorem3.p7.1.1.m1.1.1.1.cmml">^</mo></mover><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem3.p7.1.1.m1.1b"><apply id="S2.Thmtheorem3.p7.1.1.m1.1.1.cmml" xref="S2.Thmtheorem3.p7.1.1.m1.1.1"><ci id="S2.Thmtheorem3.p7.1.1.m1.1.1.1.cmml" xref="S2.Thmtheorem3.p7.1.1.m1.1.1.1">^</ci><ci id="S2.Thmtheorem3.p7.1.1.m1.1.1.2.cmml" xref="S2.Thmtheorem3.p7.1.1.m1.1.1.2">𝑟</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem3.p7.1.1.m1.1c">\hat{r}</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem3.p7.1.1.m1.1d">over^ start_ARG italic_r end_ARG</annotation></semantics></math> such that <math alttext="\frac{\sum_{\overline{uv}\in E(\hat{r})}\textsl{g}(\overline{uv})}{|V(\hat{r})% |}\geq R" class="ltx_Math" display="inline" id="S2.Thmtheorem3.p7.2.2.m2.4"><semantics id="S2.Thmtheorem3.p7.2.2.m2.4a"><mrow 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xref="S2.Thmtheorem3.p7.2.2.m2.4.4.4.2.2"></abs><apply id="S2.Thmtheorem3.p7.2.2.m2.4.4.4.2.1.cmml" xref="S2.Thmtheorem3.p7.2.2.m2.4.4.4.2.1"><times id="S2.Thmtheorem3.p7.2.2.m2.4.4.4.2.1.1.cmml" xref="S2.Thmtheorem3.p7.2.2.m2.4.4.4.2.1.1"></times><ci id="S2.Thmtheorem3.p7.2.2.m2.4.4.4.2.1.2.cmml" xref="S2.Thmtheorem3.p7.2.2.m2.4.4.4.2.1.2">𝑉</ci><apply id="S2.Thmtheorem3.p7.2.2.m2.3.3.3.1.cmml" xref="S2.Thmtheorem3.p7.2.2.m2.4.4.4.2.1.3.2"><ci id="S2.Thmtheorem3.p7.2.2.m2.3.3.3.1.1.cmml" xref="S2.Thmtheorem3.p7.2.2.m2.3.3.3.1.1">^</ci><ci id="S2.Thmtheorem3.p7.2.2.m2.3.3.3.1.2.cmml" xref="S2.Thmtheorem3.p7.2.2.m2.3.3.3.1.2">𝑟</ci></apply></apply></apply></apply><ci id="S2.Thmtheorem3.p7.2.2.m2.4.5.2.cmml" xref="S2.Thmtheorem3.p7.2.2.m2.4.5.2">𝑅</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem3.p7.2.2.m2.4c">\frac{\sum_{\overline{uv}\in E(\hat{r})}\textsl{g}(\overline{uv})}{|V(\hat{r})% |}\geq R</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem3.p7.2.2.m2.4d">divide start_ARG ∑ start_POSTSUBSCRIPT over¯ start_ARG italic_u italic_v end_ARG ∈ italic_E ( over^ start_ARG italic_r end_ARG ) end_POSTSUBSCRIPT g ( over¯ start_ARG italic_u italic_v end_ARG ) end_ARG start_ARG | italic_V ( over^ start_ARG italic_r end_ARG ) | end_ARG ≥ italic_R</annotation></semantics></math>. Observe that the existence of <math alttext="\hat{r}" class="ltx_Math" display="inline" id="S2.Thmtheorem3.p7.3.3.m3.1"><semantics id="S2.Thmtheorem3.p7.3.3.m3.1a"><mover accent="true" id="S2.Thmtheorem3.p7.3.3.m3.1.1" xref="S2.Thmtheorem3.p7.3.3.m3.1.1.cmml"><mi id="S2.Thmtheorem3.p7.3.3.m3.1.1.2" xref="S2.Thmtheorem3.p7.3.3.m3.1.1.2.cmml">r</mi><mo id="S2.Thmtheorem3.p7.3.3.m3.1.1.1" xref="S2.Thmtheorem3.p7.3.3.m3.1.1.1.cmml">^</mo></mover><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem3.p7.3.3.m3.1b"><apply id="S2.Thmtheorem3.p7.3.3.m3.1.1.cmml" xref="S2.Thmtheorem3.p7.3.3.m3.1.1"><ci id="S2.Thmtheorem3.p7.3.3.m3.1.1.1.cmml" xref="S2.Thmtheorem3.p7.3.3.m3.1.1.1">^</ci><ci id="S2.Thmtheorem3.p7.3.3.m3.1.1.2.cmml" xref="S2.Thmtheorem3.p7.3.3.m3.1.1.2">𝑟</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem3.p7.3.3.m3.1c">\hat{r}</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem3.p7.3.3.m3.1d">over^ start_ARG italic_r end_ARG</annotation></semantics></math> implies the lemma.</span></p> </div> <div class="ltx_para" id="S2.Thmtheorem3.p8"> <p class="ltx_p" id="S2.Thmtheorem3.p8.1"><span class="ltx_text ltx_font_italic" id="S2.Thmtheorem3.p8.1.1">Suppose for the sake of contradiction that such <math alttext="\hat{r}" class="ltx_Math" display="inline" id="S2.Thmtheorem3.p8.1.1.m1.1"><semantics id="S2.Thmtheorem3.p8.1.1.m1.1a"><mover accent="true" id="S2.Thmtheorem3.p8.1.1.m1.1.1" xref="S2.Thmtheorem3.p8.1.1.m1.1.1.cmml"><mi id="S2.Thmtheorem3.p8.1.1.m1.1.1.2" xref="S2.Thmtheorem3.p8.1.1.m1.1.1.2.cmml">r</mi><mo id="S2.Thmtheorem3.p8.1.1.m1.1.1.1" xref="S2.Thmtheorem3.p8.1.1.m1.1.1.1.cmml">^</mo></mover><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem3.p8.1.1.m1.1b"><apply id="S2.Thmtheorem3.p8.1.1.m1.1.1.cmml" xref="S2.Thmtheorem3.p8.1.1.m1.1.1"><ci id="S2.Thmtheorem3.p8.1.1.m1.1.1.1.cmml" xref="S2.Thmtheorem3.p8.1.1.m1.1.1.1">^</ci><ci id="S2.Thmtheorem3.p8.1.1.m1.1.1.2.cmml" xref="S2.Thmtheorem3.p8.1.1.m1.1.1.2">𝑟</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem3.p8.1.1.m1.1c">\hat{r}</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem3.p8.1.1.m1.1d">over^ start_ARG italic_r end_ARG</annotation></semantics></math> does not exist. Then per assumption:</span></p> <table class="ltx_equation ltx_eqn_table" id="S2.Ex7"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_left_padleft"></td> <td class="ltx_eqn_cell ltx_align_left"><math alttext="\forall r\in(0,1):\quad R\cdot|V(r)|\,\,-\sum_{\overline{uv}\in E(r)}\textsl{g% }(\overline{uv})>0\Rightarrow\int_{0}^{\infty}R\cdot|V(r)|-\sum_{\overline{uv}% \in E(r)}\textsl{g}(\overline{uv})\,dr>0" class="ltx_math_unparsed" display="block" id="S2.Ex7.m1.4"><semantics id="S2.Ex7.m1.4a"><mrow id="S2.Ex7.m1.4b"><mo id="S2.Ex7.m1.4.5" rspace="0.167em">∀</mo><mi id="S2.Ex7.m1.4.6">r</mi><mo id="S2.Ex7.m1.4.7">∈</mo><mrow id="S2.Ex7.m1.4.8"><mo id="S2.Ex7.m1.4.8.1" stretchy="false">(</mo><mn id="S2.Ex7.m1.3.3">0</mn><mo id="S2.Ex7.m1.4.8.2">,</mo><mn id="S2.Ex7.m1.4.4">1</mn><mo id="S2.Ex7.m1.4.8.3" rspace="0.278em" stretchy="false">)</mo></mrow><mo id="S2.Ex7.m1.4.9">:</mo><mspace id="S2.Ex7.m1.4.10" width="1.278em"></mspace><mi id="S2.Ex7.m1.4.11">R</mi><mo id="S2.Ex7.m1.4.12" lspace="0.222em" rspace="0em">⋅</mo><mo fence="false" id="S2.Ex7.m1.4.13" rspace="0.167em" stretchy="false">|</mo><mi id="S2.Ex7.m1.4.14">V</mi><mrow id="S2.Ex7.m1.4.15"><mo id="S2.Ex7.m1.4.15.1" stretchy="false">(</mo><mi id="S2.Ex7.m1.4.15.2">r</mi><mo id="S2.Ex7.m1.4.15.3" stretchy="false">)</mo></mrow><mo fence="false" id="S2.Ex7.m1.4.16" rspace="0.108em" stretchy="false">|</mo><mo id="S2.Ex7.m1.4.17" rspace="0.055em">−</mo><munder id="S2.Ex7.m1.4.18"><mo id="S2.Ex7.m1.4.18.2" movablelimits="false">∑</mo><mrow id="S2.Ex7.m1.1.1.1"><mover accent="true" id="S2.Ex7.m1.1.1.1.3"><mrow id="S2.Ex7.m1.1.1.1.3.2"><mi id="S2.Ex7.m1.1.1.1.3.2.2">u</mi><mo id="S2.Ex7.m1.1.1.1.3.2.1">⁢</mo><mi id="S2.Ex7.m1.1.1.1.3.2.3">v</mi></mrow><mo id="S2.Ex7.m1.1.1.1.3.1">¯</mo></mover><mo id="S2.Ex7.m1.1.1.1.2">∈</mo><mrow id="S2.Ex7.m1.1.1.1.4"><mi id="S2.Ex7.m1.1.1.1.4.2">E</mi><mo id="S2.Ex7.m1.1.1.1.4.1">⁢</mo><mrow id="S2.Ex7.m1.1.1.1.4.3.2"><mo id="S2.Ex7.m1.1.1.1.4.3.2.1" stretchy="false">(</mo><mi id="S2.Ex7.m1.1.1.1.1">r</mi><mo id="S2.Ex7.m1.1.1.1.4.3.2.2" stretchy="false">)</mo></mrow></mrow></mrow></munder><mtext class="ltx_mathvariant_italic" id="S2.Ex7.m1.4.19">g</mtext><mrow id="S2.Ex7.m1.4.20"><mo id="S2.Ex7.m1.4.20.1" stretchy="false">(</mo><mover accent="true" id="S2.Ex7.m1.4.20.2"><mrow id="S2.Ex7.m1.4.20.2.2"><mi id="S2.Ex7.m1.4.20.2.2.2">u</mi><mo id="S2.Ex7.m1.4.20.2.2.1">⁢</mo><mi id="S2.Ex7.m1.4.20.2.2.3">v</mi></mrow><mo id="S2.Ex7.m1.4.20.2.1">¯</mo></mover><mo id="S2.Ex7.m1.4.20.3" stretchy="false">)</mo></mrow><mo id="S2.Ex7.m1.4.21">></mo><mn id="S2.Ex7.m1.4.22">0</mn><mo id="S2.Ex7.m1.4.23" rspace="0.111em" stretchy="false">⇒</mo><msubsup id="S2.Ex7.m1.4.24"><mo id="S2.Ex7.m1.4.24.2.2">∫</mo><mn id="S2.Ex7.m1.4.24.2.3">0</mn><mi id="S2.Ex7.m1.4.24.3" mathvariant="normal">∞</mi></msubsup><mi id="S2.Ex7.m1.4.25">R</mi><mo id="S2.Ex7.m1.4.26" lspace="0.222em" rspace="0em">⋅</mo><mo fence="false" id="S2.Ex7.m1.4.27" rspace="0.167em" stretchy="false">|</mo><mi id="S2.Ex7.m1.4.28">V</mi><mrow id="S2.Ex7.m1.4.29"><mo id="S2.Ex7.m1.4.29.1" stretchy="false">(</mo><mi id="S2.Ex7.m1.4.29.2">r</mi><mo id="S2.Ex7.m1.4.29.3" stretchy="false">)</mo></mrow><mo fence="false" id="S2.Ex7.m1.4.30" stretchy="false">|</mo><mo id="S2.Ex7.m1.4.31" lspace="0em" rspace="0.055em">−</mo><munder id="S2.Ex7.m1.4.32"><mo id="S2.Ex7.m1.4.32.2" movablelimits="false">∑</mo><mrow id="S2.Ex7.m1.2.2.1"><mover accent="true" id="S2.Ex7.m1.2.2.1.3"><mrow id="S2.Ex7.m1.2.2.1.3.2"><mi id="S2.Ex7.m1.2.2.1.3.2.2">u</mi><mo id="S2.Ex7.m1.2.2.1.3.2.1">⁢</mo><mi id="S2.Ex7.m1.2.2.1.3.2.3">v</mi></mrow><mo id="S2.Ex7.m1.2.2.1.3.1">¯</mo></mover><mo id="S2.Ex7.m1.2.2.1.2">∈</mo><mrow id="S2.Ex7.m1.2.2.1.4"><mi id="S2.Ex7.m1.2.2.1.4.2">E</mi><mo id="S2.Ex7.m1.2.2.1.4.1">⁢</mo><mrow id="S2.Ex7.m1.2.2.1.4.3.2"><mo id="S2.Ex7.m1.2.2.1.4.3.2.1" stretchy="false">(</mo><mi id="S2.Ex7.m1.2.2.1.1">r</mi><mo id="S2.Ex7.m1.2.2.1.4.3.2.2" stretchy="false">)</mo></mrow></mrow></mrow></munder><mtext class="ltx_mathvariant_italic" id="S2.Ex7.m1.4.33">g</mtext><mrow id="S2.Ex7.m1.4.34"><mo id="S2.Ex7.m1.4.34.1" stretchy="false">(</mo><mover accent="true" id="S2.Ex7.m1.4.34.2"><mrow id="S2.Ex7.m1.4.34.2.2"><mi id="S2.Ex7.m1.4.34.2.2.2">u</mi><mo id="S2.Ex7.m1.4.34.2.2.1">⁢</mo><mi id="S2.Ex7.m1.4.34.2.2.3">v</mi></mrow><mo id="S2.Ex7.m1.4.34.2.1">¯</mo></mover><mo id="S2.Ex7.m1.4.34.3" rspace="0.170em" stretchy="false">)</mo></mrow><mi id="S2.Ex7.m1.4.35">d</mi><mi id="S2.Ex7.m1.4.36">r</mi><mo id="S2.Ex7.m1.4.37">></mo><mn id="S2.Ex7.m1.4.38">0</mn></mrow><annotation encoding="application/x-tex" id="S2.Ex7.m1.4c">\forall r\in(0,1):\quad R\cdot|V(r)|\,\,-\sum_{\overline{uv}\in E(r)}\textsl{g% }(\overline{uv})>0\Rightarrow\int_{0}^{\infty}R\cdot|V(r)|-\sum_{\overline{uv}% \in E(r)}\textsl{g}(\overline{uv})\,dr>0</annotation><annotation encoding="application/x-llamapun" id="S2.Ex7.m1.4d">∀ italic_r ∈ ( 0 , 1 ) : italic_R ⋅ | italic_V ( italic_r ) | - ∑ start_POSTSUBSCRIPT over¯ start_ARG italic_u italic_v end_ARG ∈ italic_E ( italic_r ) end_POSTSUBSCRIPT g ( over¯ start_ARG italic_u italic_v end_ARG ) > 0 ⇒ ∫ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ∞ end_POSTSUPERSCRIPT italic_R ⋅ | italic_V ( italic_r ) | - ∑ start_POSTSUBSCRIPT over¯ start_ARG italic_u italic_v end_ARG ∈ italic_E ( italic_r ) end_POSTSUBSCRIPT g ( over¯ start_ARG italic_u italic_v end_ARG ) italic_d italic_r > 0</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_left_padright"></td> </tr></tbody> </table> </div> <div class="ltx_para" id="S2.Thmtheorem3.p9"> <p class="ltx_p" id="S2.Thmtheorem3.p9.1"><span class="ltx_text ltx_font_italic" id="S2.Thmtheorem3.p9.1.1">However, this now gives a contradiction through Equations <a class="ltx_ref" href="https://arxiv.org/html/2411.12694v2#S2.E1" title="Equation 1 ‣ Proof 2.3. ‣ Global graph measures. ‣ 2 Preliminaries and related work ‣ Local Density and its Distributed Approximation"><span class="ltx_text ltx_ref_tag">1</span></a> and <a class="ltx_ref" href="https://arxiv.org/html/2411.12694v2#S2.E2" title="Equation 2 ‣ Proof 2.3. ‣ Global graph measures. ‣ 2 Preliminaries and related work ‣ Local Density and its Distributed Approximation"><span class="ltx_text ltx_ref_tag">2</span></a>:</span></p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="A1.EGx2"> <tbody id="S2.Ex8"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_left_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle 0<\int_{0}^{\infty}R\cdot|V(r)|\,\,-\sum_{\overline{uv}\in E(r)}% \textsl{g}(\overline{uv})\,dr=R\cdot\int_{0}^{\infty}|V(r)|dr-\int_{0}^{\infty% }\sum_{\overline{uv}\in E(r)}\textsl{g}(\overline{uv})\,dr\leq R-R=0" class="ltx_Math" display="inline" id="S2.Ex8.m1.8"><semantics id="S2.Ex8.m1.8a"><mrow id="S2.Ex8.m1.8.8" xref="S2.Ex8.m1.8.8.cmml"><mn id="S2.Ex8.m1.8.8.4" xref="S2.Ex8.m1.8.8.4.cmml">0</mn><mo id="S2.Ex8.m1.8.8.5" xref="S2.Ex8.m1.8.8.5.cmml"><</mo><mrow id="S2.Ex8.m1.7.7.1" xref="S2.Ex8.m1.7.7.1.cmml"><mrow id="S2.Ex8.m1.7.7.1.1" xref="S2.Ex8.m1.7.7.1.1.cmml"><mstyle displaystyle="true" id="S2.Ex8.m1.7.7.1.1.2" xref="S2.Ex8.m1.7.7.1.1.2.cmml"><msubsup id="S2.Ex8.m1.7.7.1.1.2a" xref="S2.Ex8.m1.7.7.1.1.2.cmml"><mo id="S2.Ex8.m1.7.7.1.1.2.2.2" xref="S2.Ex8.m1.7.7.1.1.2.2.2.cmml">∫</mo><mn id="S2.Ex8.m1.7.7.1.1.2.2.3" xref="S2.Ex8.m1.7.7.1.1.2.2.3.cmml">0</mn><mi id="S2.Ex8.m1.7.7.1.1.2.3" mathvariant="normal" xref="S2.Ex8.m1.7.7.1.1.2.3.cmml">∞</mi></msubsup></mstyle><mrow id="S2.Ex8.m1.7.7.1.1.1" xref="S2.Ex8.m1.7.7.1.1.1.cmml"><mi id="S2.Ex8.m1.7.7.1.1.1.3" xref="S2.Ex8.m1.7.7.1.1.1.3.cmml">R</mi><mo id="S2.Ex8.m1.7.7.1.1.1.2" lspace="0.222em" rspace="0.222em" xref="S2.Ex8.m1.7.7.1.1.1.2.cmml">⋅</mo><mrow id="S2.Ex8.m1.7.7.1.1.1.1.1" xref="S2.Ex8.m1.7.7.1.1.1.1.2.cmml"><mo id="S2.Ex8.m1.7.7.1.1.1.1.1.2" stretchy="false" xref="S2.Ex8.m1.7.7.1.1.1.1.2.1.cmml">|</mo><mrow id="S2.Ex8.m1.7.7.1.1.1.1.1.1" xref="S2.Ex8.m1.7.7.1.1.1.1.1.1.cmml"><mi id="S2.Ex8.m1.7.7.1.1.1.1.1.1.2" xref="S2.Ex8.m1.7.7.1.1.1.1.1.1.2.cmml">V</mi><mo id="S2.Ex8.m1.7.7.1.1.1.1.1.1.1" xref="S2.Ex8.m1.7.7.1.1.1.1.1.1.1.cmml">⁢</mo><mrow id="S2.Ex8.m1.7.7.1.1.1.1.1.1.3.2" xref="S2.Ex8.m1.7.7.1.1.1.1.1.1.cmml"><mo id="S2.Ex8.m1.7.7.1.1.1.1.1.1.3.2.1" stretchy="false" xref="S2.Ex8.m1.7.7.1.1.1.1.1.1.cmml">(</mo><mi id="S2.Ex8.m1.3.3" xref="S2.Ex8.m1.3.3.cmml">r</mi><mo id="S2.Ex8.m1.7.7.1.1.1.1.1.1.3.2.2" stretchy="false" xref="S2.Ex8.m1.7.7.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S2.Ex8.m1.7.7.1.1.1.1.1.3" rspace="0.330em" stretchy="false" xref="S2.Ex8.m1.7.7.1.1.1.1.2.1.cmml">|</mo></mrow></mrow></mrow><mo id="S2.Ex8.m1.7.7.1.2" xref="S2.Ex8.m1.7.7.1.2.cmml">−</mo><mrow id="S2.Ex8.m1.7.7.1.3" xref="S2.Ex8.m1.7.7.1.3.cmml"><mstyle displaystyle="true" id="S2.Ex8.m1.7.7.1.3.1" xref="S2.Ex8.m1.7.7.1.3.1.cmml"><munder id="S2.Ex8.m1.7.7.1.3.1a" xref="S2.Ex8.m1.7.7.1.3.1.cmml"><mo id="S2.Ex8.m1.7.7.1.3.1.2" movablelimits="false" xref="S2.Ex8.m1.7.7.1.3.1.2.cmml">∑</mo><mrow id="S2.Ex8.m1.1.1.1" xref="S2.Ex8.m1.1.1.1.cmml"><mover accent="true" id="S2.Ex8.m1.1.1.1.3" xref="S2.Ex8.m1.1.1.1.3.cmml"><mrow id="S2.Ex8.m1.1.1.1.3.2" xref="S2.Ex8.m1.1.1.1.3.2.cmml"><mi id="S2.Ex8.m1.1.1.1.3.2.2" xref="S2.Ex8.m1.1.1.1.3.2.2.cmml">u</mi><mo id="S2.Ex8.m1.1.1.1.3.2.1" xref="S2.Ex8.m1.1.1.1.3.2.1.cmml">⁢</mo><mi id="S2.Ex8.m1.1.1.1.3.2.3" xref="S2.Ex8.m1.1.1.1.3.2.3.cmml">v</mi></mrow><mo id="S2.Ex8.m1.1.1.1.3.1" xref="S2.Ex8.m1.1.1.1.3.1.cmml">¯</mo></mover><mo id="S2.Ex8.m1.1.1.1.2" xref="S2.Ex8.m1.1.1.1.2.cmml">∈</mo><mrow id="S2.Ex8.m1.1.1.1.4" xref="S2.Ex8.m1.1.1.1.4.cmml"><mi id="S2.Ex8.m1.1.1.1.4.2" xref="S2.Ex8.m1.1.1.1.4.2.cmml">E</mi><mo id="S2.Ex8.m1.1.1.1.4.1" xref="S2.Ex8.m1.1.1.1.4.1.cmml">⁢</mo><mrow id="S2.Ex8.m1.1.1.1.4.3.2" xref="S2.Ex8.m1.1.1.1.4.cmml"><mo id="S2.Ex8.m1.1.1.1.4.3.2.1" stretchy="false" xref="S2.Ex8.m1.1.1.1.4.cmml">(</mo><mi id="S2.Ex8.m1.1.1.1.1" xref="S2.Ex8.m1.1.1.1.1.cmml">r</mi><mo id="S2.Ex8.m1.1.1.1.4.3.2.2" stretchy="false" xref="S2.Ex8.m1.1.1.1.4.cmml">)</mo></mrow></mrow></mrow></munder></mstyle><mrow id="S2.Ex8.m1.7.7.1.3.2" xref="S2.Ex8.m1.7.7.1.3.2.cmml"><mtext class="ltx_mathvariant_italic" id="S2.Ex8.m1.7.7.1.3.2.2" xref="S2.Ex8.m1.7.7.1.3.2.2a.cmml">g</mtext><mo id="S2.Ex8.m1.7.7.1.3.2.1" xref="S2.Ex8.m1.7.7.1.3.2.1.cmml">⁢</mo><mrow id="S2.Ex8.m1.7.7.1.3.2.3.2" xref="S2.Ex8.m1.4.4.cmml"><mo id="S2.Ex8.m1.7.7.1.3.2.3.2.1" stretchy="false" xref="S2.Ex8.m1.4.4.cmml">(</mo><mover accent="true" id="S2.Ex8.m1.4.4" xref="S2.Ex8.m1.4.4.cmml"><mrow id="S2.Ex8.m1.4.4.2" xref="S2.Ex8.m1.4.4.2.cmml"><mi id="S2.Ex8.m1.4.4.2.2" xref="S2.Ex8.m1.4.4.2.2.cmml">u</mi><mo id="S2.Ex8.m1.4.4.2.1" xref="S2.Ex8.m1.4.4.2.1.cmml">⁢</mo><mi id="S2.Ex8.m1.4.4.2.3" xref="S2.Ex8.m1.4.4.2.3.cmml">v</mi></mrow><mo id="S2.Ex8.m1.4.4.1" xref="S2.Ex8.m1.4.4.1.cmml">¯</mo></mover><mo id="S2.Ex8.m1.7.7.1.3.2.3.2.2" stretchy="false" xref="S2.Ex8.m1.4.4.cmml">)</mo></mrow><mo id="S2.Ex8.m1.7.7.1.3.2.1a" lspace="0.170em" xref="S2.Ex8.m1.7.7.1.3.2.1.cmml">⁢</mo><mi id="S2.Ex8.m1.7.7.1.3.2.4" xref="S2.Ex8.m1.7.7.1.3.2.4.cmml">d</mi><mo id="S2.Ex8.m1.7.7.1.3.2.1b" xref="S2.Ex8.m1.7.7.1.3.2.1.cmml">⁢</mo><mi id="S2.Ex8.m1.7.7.1.3.2.5" xref="S2.Ex8.m1.7.7.1.3.2.5.cmml">r</mi></mrow></mrow></mrow><mo id="S2.Ex8.m1.8.8.6" xref="S2.Ex8.m1.8.8.6.cmml">=</mo><mrow id="S2.Ex8.m1.8.8.2" xref="S2.Ex8.m1.8.8.2.cmml"><mrow id="S2.Ex8.m1.8.8.2.1" xref="S2.Ex8.m1.8.8.2.1.cmml"><mi id="S2.Ex8.m1.8.8.2.1.3" xref="S2.Ex8.m1.8.8.2.1.3.cmml">R</mi><mo id="S2.Ex8.m1.8.8.2.1.2" lspace="0.222em" rspace="0.222em" xref="S2.Ex8.m1.8.8.2.1.2.cmml">⋅</mo><mrow id="S2.Ex8.m1.8.8.2.1.1" xref="S2.Ex8.m1.8.8.2.1.1.cmml"><mstyle displaystyle="true" id="S2.Ex8.m1.8.8.2.1.1.2" xref="S2.Ex8.m1.8.8.2.1.1.2.cmml"><msubsup id="S2.Ex8.m1.8.8.2.1.1.2a" xref="S2.Ex8.m1.8.8.2.1.1.2.cmml"><mo id="S2.Ex8.m1.8.8.2.1.1.2.2.2" xref="S2.Ex8.m1.8.8.2.1.1.2.2.2.cmml">∫</mo><mn id="S2.Ex8.m1.8.8.2.1.1.2.2.3" xref="S2.Ex8.m1.8.8.2.1.1.2.2.3.cmml">0</mn><mi id="S2.Ex8.m1.8.8.2.1.1.2.3" mathvariant="normal" xref="S2.Ex8.m1.8.8.2.1.1.2.3.cmml">∞</mi></msubsup></mstyle><mrow id="S2.Ex8.m1.8.8.2.1.1.1" xref="S2.Ex8.m1.8.8.2.1.1.1.cmml"><mrow id="S2.Ex8.m1.8.8.2.1.1.1.1.1" xref="S2.Ex8.m1.8.8.2.1.1.1.1.2.cmml"><mo id="S2.Ex8.m1.8.8.2.1.1.1.1.1.2" stretchy="false" xref="S2.Ex8.m1.8.8.2.1.1.1.1.2.1.cmml">|</mo><mrow id="S2.Ex8.m1.8.8.2.1.1.1.1.1.1" xref="S2.Ex8.m1.8.8.2.1.1.1.1.1.1.cmml"><mi id="S2.Ex8.m1.8.8.2.1.1.1.1.1.1.2" xref="S2.Ex8.m1.8.8.2.1.1.1.1.1.1.2.cmml">V</mi><mo id="S2.Ex8.m1.8.8.2.1.1.1.1.1.1.1" xref="S2.Ex8.m1.8.8.2.1.1.1.1.1.1.1.cmml">⁢</mo><mrow id="S2.Ex8.m1.8.8.2.1.1.1.1.1.1.3.2" xref="S2.Ex8.m1.8.8.2.1.1.1.1.1.1.cmml"><mo id="S2.Ex8.m1.8.8.2.1.1.1.1.1.1.3.2.1" stretchy="false" xref="S2.Ex8.m1.8.8.2.1.1.1.1.1.1.cmml">(</mo><mi 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xref="S2.Ex8.m1.8.8.10">0</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Ex8.m1.8c">\displaystyle 0<\int_{0}^{\infty}R\cdot|V(r)|\,\,-\sum_{\overline{uv}\in E(r)}% \textsl{g}(\overline{uv})\,dr=R\cdot\int_{0}^{\infty}|V(r)|dr-\int_{0}^{\infty% }\sum_{\overline{uv}\in E(r)}\textsl{g}(\overline{uv})\,dr\leq R-R=0</annotation><annotation encoding="application/x-llamapun" id="S2.Ex8.m1.8d">0 < ∫ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ∞ end_POSTSUPERSCRIPT italic_R ⋅ | italic_V ( italic_r ) | - ∑ start_POSTSUBSCRIPT over¯ start_ARG italic_u italic_v end_ARG ∈ italic_E ( italic_r ) end_POSTSUBSCRIPT g ( over¯ start_ARG italic_u italic_v end_ARG ) italic_d italic_r = italic_R ⋅ ∫ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ∞ end_POSTSUPERSCRIPT | italic_V ( italic_r ) | italic_d italic_r - ∫ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ∞ end_POSTSUPERSCRIPT ∑ start_POSTSUBSCRIPT over¯ start_ARG italic_u italic_v end_ARG ∈ italic_E ( italic_r ) end_POSTSUBSCRIPT g ( over¯ start_ARG italic_u italic_v end_ARG ) italic_d italic_r ≤ italic_R - italic_R = 0</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_left_padright"></td> </tr></tbody> </table> </div> <div class="ltx_para" id="S2.Thmtheorem3.p10"> <p class="ltx_p" id="S2.Thmtheorem3.p10.1"><span class="ltx_text ltx_font_italic" id="S2.Thmtheorem3.p10.1.1">The proof that <math alttext="D=\Delta^{\min}(G)" class="ltx_Math" display="inline" id="S2.Thmtheorem3.p10.1.1.m1.1"><semantics id="S2.Thmtheorem3.p10.1.1.m1.1a"><mrow id="S2.Thmtheorem3.p10.1.1.m1.1.2" xref="S2.Thmtheorem3.p10.1.1.m1.1.2.cmml"><mi id="S2.Thmtheorem3.p10.1.1.m1.1.2.2" xref="S2.Thmtheorem3.p10.1.1.m1.1.2.2.cmml">D</mi><mo id="S2.Thmtheorem3.p10.1.1.m1.1.2.1" xref="S2.Thmtheorem3.p10.1.1.m1.1.2.1.cmml">=</mo><mrow id="S2.Thmtheorem3.p10.1.1.m1.1.2.3" xref="S2.Thmtheorem3.p10.1.1.m1.1.2.3.cmml"><msup id="S2.Thmtheorem3.p10.1.1.m1.1.2.3.2" xref="S2.Thmtheorem3.p10.1.1.m1.1.2.3.2.cmml"><mi id="S2.Thmtheorem3.p10.1.1.m1.1.2.3.2.2" mathvariant="normal" xref="S2.Thmtheorem3.p10.1.1.m1.1.2.3.2.2.cmml">Δ</mi><mi id="S2.Thmtheorem3.p10.1.1.m1.1.2.3.2.3" xref="S2.Thmtheorem3.p10.1.1.m1.1.2.3.2.3.cmml">min</mi></msup><mo id="S2.Thmtheorem3.p10.1.1.m1.1.2.3.1" xref="S2.Thmtheorem3.p10.1.1.m1.1.2.3.1.cmml">⁢</mo><mrow id="S2.Thmtheorem3.p10.1.1.m1.1.2.3.3.2" xref="S2.Thmtheorem3.p10.1.1.m1.1.2.3.cmml"><mo id="S2.Thmtheorem3.p10.1.1.m1.1.2.3.3.2.1" stretchy="false" xref="S2.Thmtheorem3.p10.1.1.m1.1.2.3.cmml">(</mo><mi id="S2.Thmtheorem3.p10.1.1.m1.1.1" xref="S2.Thmtheorem3.p10.1.1.m1.1.1.cmml">G</mi><mo id="S2.Thmtheorem3.p10.1.1.m1.1.2.3.3.2.2" stretchy="false" xref="S2.Thmtheorem3.p10.1.1.m1.1.2.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem3.p10.1.1.m1.1b"><apply id="S2.Thmtheorem3.p10.1.1.m1.1.2.cmml" xref="S2.Thmtheorem3.p10.1.1.m1.1.2"><eq id="S2.Thmtheorem3.p10.1.1.m1.1.2.1.cmml" xref="S2.Thmtheorem3.p10.1.1.m1.1.2.1"></eq><ci id="S2.Thmtheorem3.p10.1.1.m1.1.2.2.cmml" xref="S2.Thmtheorem3.p10.1.1.m1.1.2.2">𝐷</ci><apply id="S2.Thmtheorem3.p10.1.1.m1.1.2.3.cmml" xref="S2.Thmtheorem3.p10.1.1.m1.1.2.3"><times id="S2.Thmtheorem3.p10.1.1.m1.1.2.3.1.cmml" xref="S2.Thmtheorem3.p10.1.1.m1.1.2.3.1"></times><apply id="S2.Thmtheorem3.p10.1.1.m1.1.2.3.2.cmml" xref="S2.Thmtheorem3.p10.1.1.m1.1.2.3.2"><csymbol cd="ambiguous" id="S2.Thmtheorem3.p10.1.1.m1.1.2.3.2.1.cmml" xref="S2.Thmtheorem3.p10.1.1.m1.1.2.3.2">superscript</csymbol><ci id="S2.Thmtheorem3.p10.1.1.m1.1.2.3.2.2.cmml" xref="S2.Thmtheorem3.p10.1.1.m1.1.2.3.2.2">Δ</ci><min id="S2.Thmtheorem3.p10.1.1.m1.1.2.3.2.3.cmml" xref="S2.Thmtheorem3.p10.1.1.m1.1.2.3.2.3"></min></apply><ci id="S2.Thmtheorem3.p10.1.1.m1.1.1.cmml" xref="S2.Thmtheorem3.p10.1.1.m1.1.1">𝐺</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem3.p10.1.1.m1.1c">D=\Delta^{\min}(G)</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem3.p10.1.1.m1.1d">italic_D = roman_Δ start_POSTSUPERSCRIPT roman_min end_POSTSUPERSCRIPT ( italic_G )</annotation></semantics></math> is more straightforward:</span></p> </div> <div class="ltx_para" id="S2.Thmtheorem3.p11"> <p class="ltx_p" id="S2.Thmtheorem3.p11.20"><span class="ltx_text ltx_font_italic" id="S2.Thmtheorem3.p11.20.20">Any orientation of <math alttext="G" class="ltx_Math" display="inline" id="S2.Thmtheorem3.p11.1.1.m1.1"><semantics id="S2.Thmtheorem3.p11.1.1.m1.1a"><mi id="S2.Thmtheorem3.p11.1.1.m1.1.1" xref="S2.Thmtheorem3.p11.1.1.m1.1.1.cmml">G</mi><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem3.p11.1.1.m1.1b"><ci id="S2.Thmtheorem3.p11.1.1.m1.1.1.cmml" xref="S2.Thmtheorem3.p11.1.1.m1.1.1">𝐺</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem3.p11.1.1.m1.1c">G</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem3.p11.1.1.m1.1d">italic_G</annotation></semantics></math> is per definition is a solution to FO and so <math alttext="\Delta^{\min}(G)\geq D" class="ltx_Math" display="inline" id="S2.Thmtheorem3.p11.2.2.m2.1"><semantics id="S2.Thmtheorem3.p11.2.2.m2.1a"><mrow id="S2.Thmtheorem3.p11.2.2.m2.1.2" xref="S2.Thmtheorem3.p11.2.2.m2.1.2.cmml"><mrow id="S2.Thmtheorem3.p11.2.2.m2.1.2.2" xref="S2.Thmtheorem3.p11.2.2.m2.1.2.2.cmml"><msup id="S2.Thmtheorem3.p11.2.2.m2.1.2.2.2" xref="S2.Thmtheorem3.p11.2.2.m2.1.2.2.2.cmml"><mi id="S2.Thmtheorem3.p11.2.2.m2.1.2.2.2.2" mathvariant="normal" xref="S2.Thmtheorem3.p11.2.2.m2.1.2.2.2.2.cmml">Δ</mi><mi id="S2.Thmtheorem3.p11.2.2.m2.1.2.2.2.3" xref="S2.Thmtheorem3.p11.2.2.m2.1.2.2.2.3.cmml">min</mi></msup><mo id="S2.Thmtheorem3.p11.2.2.m2.1.2.2.1" xref="S2.Thmtheorem3.p11.2.2.m2.1.2.2.1.cmml">⁢</mo><mrow id="S2.Thmtheorem3.p11.2.2.m2.1.2.2.3.2" xref="S2.Thmtheorem3.p11.2.2.m2.1.2.2.cmml"><mo id="S2.Thmtheorem3.p11.2.2.m2.1.2.2.3.2.1" stretchy="false" xref="S2.Thmtheorem3.p11.2.2.m2.1.2.2.cmml">(</mo><mi id="S2.Thmtheorem3.p11.2.2.m2.1.1" xref="S2.Thmtheorem3.p11.2.2.m2.1.1.cmml">G</mi><mo id="S2.Thmtheorem3.p11.2.2.m2.1.2.2.3.2.2" stretchy="false" xref="S2.Thmtheorem3.p11.2.2.m2.1.2.2.cmml">)</mo></mrow></mrow><mo id="S2.Thmtheorem3.p11.2.2.m2.1.2.1" xref="S2.Thmtheorem3.p11.2.2.m2.1.2.1.cmml">≥</mo><mi id="S2.Thmtheorem3.p11.2.2.m2.1.2.3" xref="S2.Thmtheorem3.p11.2.2.m2.1.2.3.cmml">D</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem3.p11.2.2.m2.1b"><apply id="S2.Thmtheorem3.p11.2.2.m2.1.2.cmml" xref="S2.Thmtheorem3.p11.2.2.m2.1.2"><geq id="S2.Thmtheorem3.p11.2.2.m2.1.2.1.cmml" xref="S2.Thmtheorem3.p11.2.2.m2.1.2.1"></geq><apply id="S2.Thmtheorem3.p11.2.2.m2.1.2.2.cmml" xref="S2.Thmtheorem3.p11.2.2.m2.1.2.2"><times id="S2.Thmtheorem3.p11.2.2.m2.1.2.2.1.cmml" xref="S2.Thmtheorem3.p11.2.2.m2.1.2.2.1"></times><apply id="S2.Thmtheorem3.p11.2.2.m2.1.2.2.2.cmml" xref="S2.Thmtheorem3.p11.2.2.m2.1.2.2.2"><csymbol cd="ambiguous" id="S2.Thmtheorem3.p11.2.2.m2.1.2.2.2.1.cmml" xref="S2.Thmtheorem3.p11.2.2.m2.1.2.2.2">superscript</csymbol><ci id="S2.Thmtheorem3.p11.2.2.m2.1.2.2.2.2.cmml" xref="S2.Thmtheorem3.p11.2.2.m2.1.2.2.2.2">Δ</ci><min id="S2.Thmtheorem3.p11.2.2.m2.1.2.2.2.3.cmml" xref="S2.Thmtheorem3.p11.2.2.m2.1.2.2.2.3"></min></apply><ci id="S2.Thmtheorem3.p11.2.2.m2.1.1.cmml" xref="S2.Thmtheorem3.p11.2.2.m2.1.1">𝐺</ci></apply><ci id="S2.Thmtheorem3.p11.2.2.m2.1.2.3.cmml" xref="S2.Thmtheorem3.p11.2.2.m2.1.2.3">𝐷</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem3.p11.2.2.m2.1c">\Delta^{\min}(G)\geq D</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem3.p11.2.2.m2.1d">roman_Δ start_POSTSUPERSCRIPT roman_min end_POSTSUPERSCRIPT ( italic_G ) ≥ italic_D</annotation></semantics></math>. What remains is to show that <math alttext="\Delta^{\min}(G)\leq D" class="ltx_Math" display="inline" id="S2.Thmtheorem3.p11.3.3.m3.1"><semantics id="S2.Thmtheorem3.p11.3.3.m3.1a"><mrow id="S2.Thmtheorem3.p11.3.3.m3.1.2" xref="S2.Thmtheorem3.p11.3.3.m3.1.2.cmml"><mrow id="S2.Thmtheorem3.p11.3.3.m3.1.2.2" xref="S2.Thmtheorem3.p11.3.3.m3.1.2.2.cmml"><msup id="S2.Thmtheorem3.p11.3.3.m3.1.2.2.2" xref="S2.Thmtheorem3.p11.3.3.m3.1.2.2.2.cmml"><mi id="S2.Thmtheorem3.p11.3.3.m3.1.2.2.2.2" mathvariant="normal" xref="S2.Thmtheorem3.p11.3.3.m3.1.2.2.2.2.cmml">Δ</mi><mi id="S2.Thmtheorem3.p11.3.3.m3.1.2.2.2.3" xref="S2.Thmtheorem3.p11.3.3.m3.1.2.2.2.3.cmml">min</mi></msup><mo id="S2.Thmtheorem3.p11.3.3.m3.1.2.2.1" xref="S2.Thmtheorem3.p11.3.3.m3.1.2.2.1.cmml">⁢</mo><mrow id="S2.Thmtheorem3.p11.3.3.m3.1.2.2.3.2" xref="S2.Thmtheorem3.p11.3.3.m3.1.2.2.cmml"><mo id="S2.Thmtheorem3.p11.3.3.m3.1.2.2.3.2.1" stretchy="false" xref="S2.Thmtheorem3.p11.3.3.m3.1.2.2.cmml">(</mo><mi id="S2.Thmtheorem3.p11.3.3.m3.1.1" xref="S2.Thmtheorem3.p11.3.3.m3.1.1.cmml">G</mi><mo id="S2.Thmtheorem3.p11.3.3.m3.1.2.2.3.2.2" stretchy="false" xref="S2.Thmtheorem3.p11.3.3.m3.1.2.2.cmml">)</mo></mrow></mrow><mo id="S2.Thmtheorem3.p11.3.3.m3.1.2.1" xref="S2.Thmtheorem3.p11.3.3.m3.1.2.1.cmml">≤</mo><mi id="S2.Thmtheorem3.p11.3.3.m3.1.2.3" xref="S2.Thmtheorem3.p11.3.3.m3.1.2.3.cmml">D</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem3.p11.3.3.m3.1b"><apply id="S2.Thmtheorem3.p11.3.3.m3.1.2.cmml" xref="S2.Thmtheorem3.p11.3.3.m3.1.2"><leq id="S2.Thmtheorem3.p11.3.3.m3.1.2.1.cmml" xref="S2.Thmtheorem3.p11.3.3.m3.1.2.1"></leq><apply id="S2.Thmtheorem3.p11.3.3.m3.1.2.2.cmml" xref="S2.Thmtheorem3.p11.3.3.m3.1.2.2"><times id="S2.Thmtheorem3.p11.3.3.m3.1.2.2.1.cmml" xref="S2.Thmtheorem3.p11.3.3.m3.1.2.2.1"></times><apply id="S2.Thmtheorem3.p11.3.3.m3.1.2.2.2.cmml" xref="S2.Thmtheorem3.p11.3.3.m3.1.2.2.2"><csymbol cd="ambiguous" id="S2.Thmtheorem3.p11.3.3.m3.1.2.2.2.1.cmml" xref="S2.Thmtheorem3.p11.3.3.m3.1.2.2.2">superscript</csymbol><ci id="S2.Thmtheorem3.p11.3.3.m3.1.2.2.2.2.cmml" xref="S2.Thmtheorem3.p11.3.3.m3.1.2.2.2.2">Δ</ci><min id="S2.Thmtheorem3.p11.3.3.m3.1.2.2.2.3.cmml" xref="S2.Thmtheorem3.p11.3.3.m3.1.2.2.2.3"></min></apply><ci id="S2.Thmtheorem3.p11.3.3.m3.1.1.cmml" xref="S2.Thmtheorem3.p11.3.3.m3.1.1">𝐺</ci></apply><ci id="S2.Thmtheorem3.p11.3.3.m3.1.2.3.cmml" xref="S2.Thmtheorem3.p11.3.3.m3.1.2.3">𝐷</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem3.p11.3.3.m3.1c">\Delta^{\min}(G)\leq D</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem3.p11.3.3.m3.1d">roman_Δ start_POSTSUPERSCRIPT roman_min end_POSTSUPERSCRIPT ( italic_G ) ≤ italic_D</annotation></semantics></math>. Suppose for the sake of contradiction that <math alttext="D<\Delta^{\min}(G)" class="ltx_Math" display="inline" id="S2.Thmtheorem3.p11.4.4.m4.1"><semantics id="S2.Thmtheorem3.p11.4.4.m4.1a"><mrow id="S2.Thmtheorem3.p11.4.4.m4.1.2" xref="S2.Thmtheorem3.p11.4.4.m4.1.2.cmml"><mi id="S2.Thmtheorem3.p11.4.4.m4.1.2.2" xref="S2.Thmtheorem3.p11.4.4.m4.1.2.2.cmml">D</mi><mo id="S2.Thmtheorem3.p11.4.4.m4.1.2.1" xref="S2.Thmtheorem3.p11.4.4.m4.1.2.1.cmml"><</mo><mrow id="S2.Thmtheorem3.p11.4.4.m4.1.2.3" xref="S2.Thmtheorem3.p11.4.4.m4.1.2.3.cmml"><msup id="S2.Thmtheorem3.p11.4.4.m4.1.2.3.2" xref="S2.Thmtheorem3.p11.4.4.m4.1.2.3.2.cmml"><mi id="S2.Thmtheorem3.p11.4.4.m4.1.2.3.2.2" mathvariant="normal" xref="S2.Thmtheorem3.p11.4.4.m4.1.2.3.2.2.cmml">Δ</mi><mi id="S2.Thmtheorem3.p11.4.4.m4.1.2.3.2.3" xref="S2.Thmtheorem3.p11.4.4.m4.1.2.3.2.3.cmml">min</mi></msup><mo id="S2.Thmtheorem3.p11.4.4.m4.1.2.3.1" xref="S2.Thmtheorem3.p11.4.4.m4.1.2.3.1.cmml">⁢</mo><mrow id="S2.Thmtheorem3.p11.4.4.m4.1.2.3.3.2" xref="S2.Thmtheorem3.p11.4.4.m4.1.2.3.cmml"><mo id="S2.Thmtheorem3.p11.4.4.m4.1.2.3.3.2.1" stretchy="false" xref="S2.Thmtheorem3.p11.4.4.m4.1.2.3.cmml">(</mo><mi id="S2.Thmtheorem3.p11.4.4.m4.1.1" xref="S2.Thmtheorem3.p11.4.4.m4.1.1.cmml">G</mi><mo id="S2.Thmtheorem3.p11.4.4.m4.1.2.3.3.2.2" stretchy="false" xref="S2.Thmtheorem3.p11.4.4.m4.1.2.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem3.p11.4.4.m4.1b"><apply id="S2.Thmtheorem3.p11.4.4.m4.1.2.cmml" xref="S2.Thmtheorem3.p11.4.4.m4.1.2"><lt id="S2.Thmtheorem3.p11.4.4.m4.1.2.1.cmml" xref="S2.Thmtheorem3.p11.4.4.m4.1.2.1"></lt><ci id="S2.Thmtheorem3.p11.4.4.m4.1.2.2.cmml" xref="S2.Thmtheorem3.p11.4.4.m4.1.2.2">𝐷</ci><apply id="S2.Thmtheorem3.p11.4.4.m4.1.2.3.cmml" xref="S2.Thmtheorem3.p11.4.4.m4.1.2.3"><times id="S2.Thmtheorem3.p11.4.4.m4.1.2.3.1.cmml" xref="S2.Thmtheorem3.p11.4.4.m4.1.2.3.1"></times><apply id="S2.Thmtheorem3.p11.4.4.m4.1.2.3.2.cmml" xref="S2.Thmtheorem3.p11.4.4.m4.1.2.3.2"><csymbol cd="ambiguous" id="S2.Thmtheorem3.p11.4.4.m4.1.2.3.2.1.cmml" xref="S2.Thmtheorem3.p11.4.4.m4.1.2.3.2">superscript</csymbol><ci id="S2.Thmtheorem3.p11.4.4.m4.1.2.3.2.2.cmml" xref="S2.Thmtheorem3.p11.4.4.m4.1.2.3.2.2">Δ</ci><min id="S2.Thmtheorem3.p11.4.4.m4.1.2.3.2.3.cmml" xref="S2.Thmtheorem3.p11.4.4.m4.1.2.3.2.3"></min></apply><ci id="S2.Thmtheorem3.p11.4.4.m4.1.1.cmml" xref="S2.Thmtheorem3.p11.4.4.m4.1.1">𝐺</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem3.p11.4.4.m4.1c">D<\Delta^{\min}(G)</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem3.p11.4.4.m4.1d">italic_D < roman_Δ start_POSTSUPERSCRIPT roman_min end_POSTSUPERSCRIPT ( italic_G )</annotation></semantics></math> and consider the solution to FO that realises the value <math alttext="D" class="ltx_Math" display="inline" id="S2.Thmtheorem3.p11.5.5.m5.1"><semantics id="S2.Thmtheorem3.p11.5.5.m5.1a"><mi id="S2.Thmtheorem3.p11.5.5.m5.1.1" xref="S2.Thmtheorem3.p11.5.5.m5.1.1.cmml">D</mi><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem3.p11.5.5.m5.1b"><ci id="S2.Thmtheorem3.p11.5.5.m5.1.1.cmml" xref="S2.Thmtheorem3.p11.5.5.m5.1.1">𝐷</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem3.p11.5.5.m5.1c">D</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem3.p11.5.5.m5.1d">italic_D</annotation></semantics></math>. Let this solution assign to every <math alttext="\overline{uv}\in E" class="ltx_Math" display="inline" id="S2.Thmtheorem3.p11.6.6.m6.1"><semantics id="S2.Thmtheorem3.p11.6.6.m6.1a"><mrow id="S2.Thmtheorem3.p11.6.6.m6.1.1" xref="S2.Thmtheorem3.p11.6.6.m6.1.1.cmml"><mover accent="true" id="S2.Thmtheorem3.p11.6.6.m6.1.1.2" xref="S2.Thmtheorem3.p11.6.6.m6.1.1.2.cmml"><mrow id="S2.Thmtheorem3.p11.6.6.m6.1.1.2.2" xref="S2.Thmtheorem3.p11.6.6.m6.1.1.2.2.cmml"><mi id="S2.Thmtheorem3.p11.6.6.m6.1.1.2.2.2" xref="S2.Thmtheorem3.p11.6.6.m6.1.1.2.2.2.cmml">u</mi><mo id="S2.Thmtheorem3.p11.6.6.m6.1.1.2.2.1" xref="S2.Thmtheorem3.p11.6.6.m6.1.1.2.2.1.cmml">⁢</mo><mi id="S2.Thmtheorem3.p11.6.6.m6.1.1.2.2.3" xref="S2.Thmtheorem3.p11.6.6.m6.1.1.2.2.3.cmml">v</mi></mrow><mo id="S2.Thmtheorem3.p11.6.6.m6.1.1.2.1" xref="S2.Thmtheorem3.p11.6.6.m6.1.1.2.1.cmml">¯</mo></mover><mo id="S2.Thmtheorem3.p11.6.6.m6.1.1.1" xref="S2.Thmtheorem3.p11.6.6.m6.1.1.1.cmml">∈</mo><mi id="S2.Thmtheorem3.p11.6.6.m6.1.1.3" xref="S2.Thmtheorem3.p11.6.6.m6.1.1.3.cmml">E</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem3.p11.6.6.m6.1b"><apply id="S2.Thmtheorem3.p11.6.6.m6.1.1.cmml" xref="S2.Thmtheorem3.p11.6.6.m6.1.1"><in id="S2.Thmtheorem3.p11.6.6.m6.1.1.1.cmml" xref="S2.Thmtheorem3.p11.6.6.m6.1.1.1"></in><apply id="S2.Thmtheorem3.p11.6.6.m6.1.1.2.cmml" xref="S2.Thmtheorem3.p11.6.6.m6.1.1.2"><ci id="S2.Thmtheorem3.p11.6.6.m6.1.1.2.1.cmml" xref="S2.Thmtheorem3.p11.6.6.m6.1.1.2.1">¯</ci><apply id="S2.Thmtheorem3.p11.6.6.m6.1.1.2.2.cmml" xref="S2.Thmtheorem3.p11.6.6.m6.1.1.2.2"><times id="S2.Thmtheorem3.p11.6.6.m6.1.1.2.2.1.cmml" xref="S2.Thmtheorem3.p11.6.6.m6.1.1.2.2.1"></times><ci id="S2.Thmtheorem3.p11.6.6.m6.1.1.2.2.2.cmml" xref="S2.Thmtheorem3.p11.6.6.m6.1.1.2.2.2">𝑢</ci><ci id="S2.Thmtheorem3.p11.6.6.m6.1.1.2.2.3.cmml" xref="S2.Thmtheorem3.p11.6.6.m6.1.1.2.2.3">𝑣</ci></apply></apply><ci id="S2.Thmtheorem3.p11.6.6.m6.1.1.3.cmml" xref="S2.Thmtheorem3.p11.6.6.m6.1.1.3">𝐸</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem3.p11.6.6.m6.1c">\overline{uv}\in E</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem3.p11.6.6.m6.1d">over¯ start_ARG italic_u italic_v end_ARG ∈ italic_E</annotation></semantics></math> weights <math alttext="\textsl{g}(u\!\to\!v)\geq 0" class="ltx_Math" display="inline" id="S2.Thmtheorem3.p11.7.7.m7.1"><semantics id="S2.Thmtheorem3.p11.7.7.m7.1a"><mrow id="S2.Thmtheorem3.p11.7.7.m7.1.1" xref="S2.Thmtheorem3.p11.7.7.m7.1.1.cmml"><mrow id="S2.Thmtheorem3.p11.7.7.m7.1.1.1" xref="S2.Thmtheorem3.p11.7.7.m7.1.1.1.cmml"><mtext class="ltx_mathvariant_italic" id="S2.Thmtheorem3.p11.7.7.m7.1.1.1.3" xref="S2.Thmtheorem3.p11.7.7.m7.1.1.1.3a.cmml">g</mtext><mo id="S2.Thmtheorem3.p11.7.7.m7.1.1.1.2" xref="S2.Thmtheorem3.p11.7.7.m7.1.1.1.2.cmml">⁢</mo><mrow id="S2.Thmtheorem3.p11.7.7.m7.1.1.1.1.1" xref="S2.Thmtheorem3.p11.7.7.m7.1.1.1.1.1.1.cmml"><mo id="S2.Thmtheorem3.p11.7.7.m7.1.1.1.1.1.2" stretchy="false" xref="S2.Thmtheorem3.p11.7.7.m7.1.1.1.1.1.1.cmml">(</mo><mrow id="S2.Thmtheorem3.p11.7.7.m7.1.1.1.1.1.1" xref="S2.Thmtheorem3.p11.7.7.m7.1.1.1.1.1.1.cmml"><mi id="S2.Thmtheorem3.p11.7.7.m7.1.1.1.1.1.1.2" xref="S2.Thmtheorem3.p11.7.7.m7.1.1.1.1.1.1.2.cmml">u</mi><mo id="S2.Thmtheorem3.p11.7.7.m7.1.1.1.1.1.1.1" lspace="0.108em" rspace="0.108em" stretchy="false" xref="S2.Thmtheorem3.p11.7.7.m7.1.1.1.1.1.1.1.cmml">→</mo><mi id="S2.Thmtheorem3.p11.7.7.m7.1.1.1.1.1.1.3" xref="S2.Thmtheorem3.p11.7.7.m7.1.1.1.1.1.1.3.cmml">v</mi></mrow><mo id="S2.Thmtheorem3.p11.7.7.m7.1.1.1.1.1.3" stretchy="false" xref="S2.Thmtheorem3.p11.7.7.m7.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S2.Thmtheorem3.p11.7.7.m7.1.1.2" xref="S2.Thmtheorem3.p11.7.7.m7.1.1.2.cmml">≥</mo><mn id="S2.Thmtheorem3.p11.7.7.m7.1.1.3" xref="S2.Thmtheorem3.p11.7.7.m7.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem3.p11.7.7.m7.1b"><apply id="S2.Thmtheorem3.p11.7.7.m7.1.1.cmml" xref="S2.Thmtheorem3.p11.7.7.m7.1.1"><geq id="S2.Thmtheorem3.p11.7.7.m7.1.1.2.cmml" xref="S2.Thmtheorem3.p11.7.7.m7.1.1.2"></geq><apply id="S2.Thmtheorem3.p11.7.7.m7.1.1.1.cmml" xref="S2.Thmtheorem3.p11.7.7.m7.1.1.1"><times id="S2.Thmtheorem3.p11.7.7.m7.1.1.1.2.cmml" xref="S2.Thmtheorem3.p11.7.7.m7.1.1.1.2"></times><ci id="S2.Thmtheorem3.p11.7.7.m7.1.1.1.3a.cmml" xref="S2.Thmtheorem3.p11.7.7.m7.1.1.1.3"><mtext class="ltx_mathvariant_italic" id="S2.Thmtheorem3.p11.7.7.m7.1.1.1.3.cmml" xref="S2.Thmtheorem3.p11.7.7.m7.1.1.1.3">g</mtext></ci><apply id="S2.Thmtheorem3.p11.7.7.m7.1.1.1.1.1.1.cmml" xref="S2.Thmtheorem3.p11.7.7.m7.1.1.1.1.1"><ci id="S2.Thmtheorem3.p11.7.7.m7.1.1.1.1.1.1.1.cmml" xref="S2.Thmtheorem3.p11.7.7.m7.1.1.1.1.1.1.1">→</ci><ci id="S2.Thmtheorem3.p11.7.7.m7.1.1.1.1.1.1.2.cmml" xref="S2.Thmtheorem3.p11.7.7.m7.1.1.1.1.1.1.2">𝑢</ci><ci id="S2.Thmtheorem3.p11.7.7.m7.1.1.1.1.1.1.3.cmml" xref="S2.Thmtheorem3.p11.7.7.m7.1.1.1.1.1.1.3">𝑣</ci></apply></apply><cn id="S2.Thmtheorem3.p11.7.7.m7.1.1.3.cmml" type="integer" xref="S2.Thmtheorem3.p11.7.7.m7.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem3.p11.7.7.m7.1c">\textsl{g}(u\!\to\!v)\geq 0</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem3.p11.7.7.m7.1d">g ( italic_u → italic_v ) ≥ 0</annotation></semantics></math> and <math alttext="\textsl{g}(v\!\to\!u)\geq 0" class="ltx_Math" display="inline" id="S2.Thmtheorem3.p11.8.8.m8.1"><semantics id="S2.Thmtheorem3.p11.8.8.m8.1a"><mrow id="S2.Thmtheorem3.p11.8.8.m8.1.1" xref="S2.Thmtheorem3.p11.8.8.m8.1.1.cmml"><mrow id="S2.Thmtheorem3.p11.8.8.m8.1.1.1" xref="S2.Thmtheorem3.p11.8.8.m8.1.1.1.cmml"><mtext class="ltx_mathvariant_italic" id="S2.Thmtheorem3.p11.8.8.m8.1.1.1.3" xref="S2.Thmtheorem3.p11.8.8.m8.1.1.1.3a.cmml">g</mtext><mo id="S2.Thmtheorem3.p11.8.8.m8.1.1.1.2" xref="S2.Thmtheorem3.p11.8.8.m8.1.1.1.2.cmml">⁢</mo><mrow id="S2.Thmtheorem3.p11.8.8.m8.1.1.1.1.1" xref="S2.Thmtheorem3.p11.8.8.m8.1.1.1.1.1.1.cmml"><mo id="S2.Thmtheorem3.p11.8.8.m8.1.1.1.1.1.2" stretchy="false" xref="S2.Thmtheorem3.p11.8.8.m8.1.1.1.1.1.1.cmml">(</mo><mrow id="S2.Thmtheorem3.p11.8.8.m8.1.1.1.1.1.1" xref="S2.Thmtheorem3.p11.8.8.m8.1.1.1.1.1.1.cmml"><mi id="S2.Thmtheorem3.p11.8.8.m8.1.1.1.1.1.1.2" xref="S2.Thmtheorem3.p11.8.8.m8.1.1.1.1.1.1.2.cmml">v</mi><mo id="S2.Thmtheorem3.p11.8.8.m8.1.1.1.1.1.1.1" lspace="0.108em" rspace="0.108em" stretchy="false" xref="S2.Thmtheorem3.p11.8.8.m8.1.1.1.1.1.1.1.cmml">→</mo><mi id="S2.Thmtheorem3.p11.8.8.m8.1.1.1.1.1.1.3" xref="S2.Thmtheorem3.p11.8.8.m8.1.1.1.1.1.1.3.cmml">u</mi></mrow><mo id="S2.Thmtheorem3.p11.8.8.m8.1.1.1.1.1.3" stretchy="false" xref="S2.Thmtheorem3.p11.8.8.m8.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S2.Thmtheorem3.p11.8.8.m8.1.1.2" xref="S2.Thmtheorem3.p11.8.8.m8.1.1.2.cmml">≥</mo><mn id="S2.Thmtheorem3.p11.8.8.m8.1.1.3" xref="S2.Thmtheorem3.p11.8.8.m8.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem3.p11.8.8.m8.1b"><apply id="S2.Thmtheorem3.p11.8.8.m8.1.1.cmml" xref="S2.Thmtheorem3.p11.8.8.m8.1.1"><geq id="S2.Thmtheorem3.p11.8.8.m8.1.1.2.cmml" xref="S2.Thmtheorem3.p11.8.8.m8.1.1.2"></geq><apply id="S2.Thmtheorem3.p11.8.8.m8.1.1.1.cmml" xref="S2.Thmtheorem3.p11.8.8.m8.1.1.1"><times id="S2.Thmtheorem3.p11.8.8.m8.1.1.1.2.cmml" xref="S2.Thmtheorem3.p11.8.8.m8.1.1.1.2"></times><ci id="S2.Thmtheorem3.p11.8.8.m8.1.1.1.3a.cmml" xref="S2.Thmtheorem3.p11.8.8.m8.1.1.1.3"><mtext class="ltx_mathvariant_italic" id="S2.Thmtheorem3.p11.8.8.m8.1.1.1.3.cmml" xref="S2.Thmtheorem3.p11.8.8.m8.1.1.1.3">g</mtext></ci><apply id="S2.Thmtheorem3.p11.8.8.m8.1.1.1.1.1.1.cmml" xref="S2.Thmtheorem3.p11.8.8.m8.1.1.1.1.1"><ci id="S2.Thmtheorem3.p11.8.8.m8.1.1.1.1.1.1.1.cmml" xref="S2.Thmtheorem3.p11.8.8.m8.1.1.1.1.1.1.1">→</ci><ci id="S2.Thmtheorem3.p11.8.8.m8.1.1.1.1.1.1.2.cmml" xref="S2.Thmtheorem3.p11.8.8.m8.1.1.1.1.1.1.2">𝑣</ci><ci id="S2.Thmtheorem3.p11.8.8.m8.1.1.1.1.1.1.3.cmml" xref="S2.Thmtheorem3.p11.8.8.m8.1.1.1.1.1.1.3">𝑢</ci></apply></apply><cn id="S2.Thmtheorem3.p11.8.8.m8.1.1.3.cmml" type="integer" xref="S2.Thmtheorem3.p11.8.8.m8.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem3.p11.8.8.m8.1c">\textsl{g}(v\!\to\!u)\geq 0</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem3.p11.8.8.m8.1d">g ( italic_v → italic_u ) ≥ 0</annotation></semantics></math>. Because <math alttext="D<\Delta^{\min}(G)" class="ltx_Math" display="inline" id="S2.Thmtheorem3.p11.9.9.m9.1"><semantics id="S2.Thmtheorem3.p11.9.9.m9.1a"><mrow id="S2.Thmtheorem3.p11.9.9.m9.1.2" xref="S2.Thmtheorem3.p11.9.9.m9.1.2.cmml"><mi id="S2.Thmtheorem3.p11.9.9.m9.1.2.2" xref="S2.Thmtheorem3.p11.9.9.m9.1.2.2.cmml">D</mi><mo id="S2.Thmtheorem3.p11.9.9.m9.1.2.1" xref="S2.Thmtheorem3.p11.9.9.m9.1.2.1.cmml"><</mo><mrow id="S2.Thmtheorem3.p11.9.9.m9.1.2.3" xref="S2.Thmtheorem3.p11.9.9.m9.1.2.3.cmml"><msup id="S2.Thmtheorem3.p11.9.9.m9.1.2.3.2" xref="S2.Thmtheorem3.p11.9.9.m9.1.2.3.2.cmml"><mi id="S2.Thmtheorem3.p11.9.9.m9.1.2.3.2.2" mathvariant="normal" xref="S2.Thmtheorem3.p11.9.9.m9.1.2.3.2.2.cmml">Δ</mi><mi id="S2.Thmtheorem3.p11.9.9.m9.1.2.3.2.3" xref="S2.Thmtheorem3.p11.9.9.m9.1.2.3.2.3.cmml">min</mi></msup><mo id="S2.Thmtheorem3.p11.9.9.m9.1.2.3.1" xref="S2.Thmtheorem3.p11.9.9.m9.1.2.3.1.cmml">⁢</mo><mrow id="S2.Thmtheorem3.p11.9.9.m9.1.2.3.3.2" xref="S2.Thmtheorem3.p11.9.9.m9.1.2.3.cmml"><mo id="S2.Thmtheorem3.p11.9.9.m9.1.2.3.3.2.1" stretchy="false" xref="S2.Thmtheorem3.p11.9.9.m9.1.2.3.cmml">(</mo><mi id="S2.Thmtheorem3.p11.9.9.m9.1.1" xref="S2.Thmtheorem3.p11.9.9.m9.1.1.cmml">G</mi><mo id="S2.Thmtheorem3.p11.9.9.m9.1.2.3.3.2.2" stretchy="false" xref="S2.Thmtheorem3.p11.9.9.m9.1.2.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem3.p11.9.9.m9.1b"><apply id="S2.Thmtheorem3.p11.9.9.m9.1.2.cmml" xref="S2.Thmtheorem3.p11.9.9.m9.1.2"><lt id="S2.Thmtheorem3.p11.9.9.m9.1.2.1.cmml" xref="S2.Thmtheorem3.p11.9.9.m9.1.2.1"></lt><ci id="S2.Thmtheorem3.p11.9.9.m9.1.2.2.cmml" xref="S2.Thmtheorem3.p11.9.9.m9.1.2.2">𝐷</ci><apply id="S2.Thmtheorem3.p11.9.9.m9.1.2.3.cmml" xref="S2.Thmtheorem3.p11.9.9.m9.1.2.3"><times id="S2.Thmtheorem3.p11.9.9.m9.1.2.3.1.cmml" xref="S2.Thmtheorem3.p11.9.9.m9.1.2.3.1"></times><apply id="S2.Thmtheorem3.p11.9.9.m9.1.2.3.2.cmml" xref="S2.Thmtheorem3.p11.9.9.m9.1.2.3.2"><csymbol cd="ambiguous" id="S2.Thmtheorem3.p11.9.9.m9.1.2.3.2.1.cmml" xref="S2.Thmtheorem3.p11.9.9.m9.1.2.3.2">superscript</csymbol><ci id="S2.Thmtheorem3.p11.9.9.m9.1.2.3.2.2.cmml" xref="S2.Thmtheorem3.p11.9.9.m9.1.2.3.2.2">Δ</ci><min id="S2.Thmtheorem3.p11.9.9.m9.1.2.3.2.3.cmml" xref="S2.Thmtheorem3.p11.9.9.m9.1.2.3.2.3"></min></apply><ci id="S2.Thmtheorem3.p11.9.9.m9.1.1.cmml" xref="S2.Thmtheorem3.p11.9.9.m9.1.1">𝐺</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem3.p11.9.9.m9.1c">D<\Delta^{\min}(G)</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem3.p11.9.9.m9.1d">italic_D < roman_Δ start_POSTSUPERSCRIPT roman_min end_POSTSUPERSCRIPT ( italic_G )</annotation></semantics></math>, it must be that the solution to <math alttext="FO" class="ltx_Math" display="inline" id="S2.Thmtheorem3.p11.10.10.m10.1"><semantics id="S2.Thmtheorem3.p11.10.10.m10.1a"><mrow id="S2.Thmtheorem3.p11.10.10.m10.1.1" xref="S2.Thmtheorem3.p11.10.10.m10.1.1.cmml"><mi id="S2.Thmtheorem3.p11.10.10.m10.1.1.2" xref="S2.Thmtheorem3.p11.10.10.m10.1.1.2.cmml">F</mi><mo id="S2.Thmtheorem3.p11.10.10.m10.1.1.1" xref="S2.Thmtheorem3.p11.10.10.m10.1.1.1.cmml">⁢</mo><mi id="S2.Thmtheorem3.p11.10.10.m10.1.1.3" xref="S2.Thmtheorem3.p11.10.10.m10.1.1.3.cmml">O</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem3.p11.10.10.m10.1b"><apply id="S2.Thmtheorem3.p11.10.10.m10.1.1.cmml" xref="S2.Thmtheorem3.p11.10.10.m10.1.1"><times id="S2.Thmtheorem3.p11.10.10.m10.1.1.1.cmml" xref="S2.Thmtheorem3.p11.10.10.m10.1.1.1"></times><ci id="S2.Thmtheorem3.p11.10.10.m10.1.1.2.cmml" xref="S2.Thmtheorem3.p11.10.10.m10.1.1.2">𝐹</ci><ci id="S2.Thmtheorem3.p11.10.10.m10.1.1.3.cmml" xref="S2.Thmtheorem3.p11.10.10.m10.1.1.3">𝑂</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem3.p11.10.10.m10.1c">FO</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem3.p11.10.10.m10.1d">italic_F italic_O</annotation></semantics></math> is not an orientation and so there must be edges <math alttext="\overline{uv}\in E" class="ltx_Math" display="inline" id="S2.Thmtheorem3.p11.11.11.m11.1"><semantics id="S2.Thmtheorem3.p11.11.11.m11.1a"><mrow id="S2.Thmtheorem3.p11.11.11.m11.1.1" xref="S2.Thmtheorem3.p11.11.11.m11.1.1.cmml"><mover accent="true" id="S2.Thmtheorem3.p11.11.11.m11.1.1.2" xref="S2.Thmtheorem3.p11.11.11.m11.1.1.2.cmml"><mrow id="S2.Thmtheorem3.p11.11.11.m11.1.1.2.2" xref="S2.Thmtheorem3.p11.11.11.m11.1.1.2.2.cmml"><mi id="S2.Thmtheorem3.p11.11.11.m11.1.1.2.2.2" xref="S2.Thmtheorem3.p11.11.11.m11.1.1.2.2.2.cmml">u</mi><mo id="S2.Thmtheorem3.p11.11.11.m11.1.1.2.2.1" xref="S2.Thmtheorem3.p11.11.11.m11.1.1.2.2.1.cmml">⁢</mo><mi id="S2.Thmtheorem3.p11.11.11.m11.1.1.2.2.3" xref="S2.Thmtheorem3.p11.11.11.m11.1.1.2.2.3.cmml">v</mi></mrow><mo id="S2.Thmtheorem3.p11.11.11.m11.1.1.2.1" xref="S2.Thmtheorem3.p11.11.11.m11.1.1.2.1.cmml">¯</mo></mover><mo id="S2.Thmtheorem3.p11.11.11.m11.1.1.1" xref="S2.Thmtheorem3.p11.11.11.m11.1.1.1.cmml">∈</mo><mi id="S2.Thmtheorem3.p11.11.11.m11.1.1.3" xref="S2.Thmtheorem3.p11.11.11.m11.1.1.3.cmml">E</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem3.p11.11.11.m11.1b"><apply id="S2.Thmtheorem3.p11.11.11.m11.1.1.cmml" xref="S2.Thmtheorem3.p11.11.11.m11.1.1"><in id="S2.Thmtheorem3.p11.11.11.m11.1.1.1.cmml" xref="S2.Thmtheorem3.p11.11.11.m11.1.1.1"></in><apply id="S2.Thmtheorem3.p11.11.11.m11.1.1.2.cmml" xref="S2.Thmtheorem3.p11.11.11.m11.1.1.2"><ci id="S2.Thmtheorem3.p11.11.11.m11.1.1.2.1.cmml" xref="S2.Thmtheorem3.p11.11.11.m11.1.1.2.1">¯</ci><apply id="S2.Thmtheorem3.p11.11.11.m11.1.1.2.2.cmml" xref="S2.Thmtheorem3.p11.11.11.m11.1.1.2.2"><times id="S2.Thmtheorem3.p11.11.11.m11.1.1.2.2.1.cmml" xref="S2.Thmtheorem3.p11.11.11.m11.1.1.2.2.1"></times><ci id="S2.Thmtheorem3.p11.11.11.m11.1.1.2.2.2.cmml" xref="S2.Thmtheorem3.p11.11.11.m11.1.1.2.2.2">𝑢</ci><ci id="S2.Thmtheorem3.p11.11.11.m11.1.1.2.2.3.cmml" xref="S2.Thmtheorem3.p11.11.11.m11.1.1.2.2.3">𝑣</ci></apply></apply><ci id="S2.Thmtheorem3.p11.11.11.m11.1.1.3.cmml" xref="S2.Thmtheorem3.p11.11.11.m11.1.1.3">𝐸</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem3.p11.11.11.m11.1c">\overline{uv}\in E</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem3.p11.11.11.m11.1d">over¯ start_ARG italic_u italic_v end_ARG ∈ italic_E</annotation></semantics></math> for which <math alttext="\textsl{g}(u\!\to\!v)+\textsl{g}(v\!\to\!u)>\textsl{g}(\overline{uv})" class="ltx_Math" display="inline" id="S2.Thmtheorem3.p11.12.12.m12.3"><semantics id="S2.Thmtheorem3.p11.12.12.m12.3a"><mrow id="S2.Thmtheorem3.p11.12.12.m12.3.3" xref="S2.Thmtheorem3.p11.12.12.m12.3.3.cmml"><mrow id="S2.Thmtheorem3.p11.12.12.m12.3.3.2" xref="S2.Thmtheorem3.p11.12.12.m12.3.3.2.cmml"><mrow id="S2.Thmtheorem3.p11.12.12.m12.2.2.1.1" xref="S2.Thmtheorem3.p11.12.12.m12.2.2.1.1.cmml"><mtext class="ltx_mathvariant_italic" id="S2.Thmtheorem3.p11.12.12.m12.2.2.1.1.3" xref="S2.Thmtheorem3.p11.12.12.m12.2.2.1.1.3a.cmml">g</mtext><mo id="S2.Thmtheorem3.p11.12.12.m12.2.2.1.1.2" xref="S2.Thmtheorem3.p11.12.12.m12.2.2.1.1.2.cmml">⁢</mo><mrow id="S2.Thmtheorem3.p11.12.12.m12.2.2.1.1.1.1" xref="S2.Thmtheorem3.p11.12.12.m12.2.2.1.1.1.1.1.cmml"><mo id="S2.Thmtheorem3.p11.12.12.m12.2.2.1.1.1.1.2" stretchy="false" xref="S2.Thmtheorem3.p11.12.12.m12.2.2.1.1.1.1.1.cmml">(</mo><mrow id="S2.Thmtheorem3.p11.12.12.m12.2.2.1.1.1.1.1" xref="S2.Thmtheorem3.p11.12.12.m12.2.2.1.1.1.1.1.cmml"><mi id="S2.Thmtheorem3.p11.12.12.m12.2.2.1.1.1.1.1.2" xref="S2.Thmtheorem3.p11.12.12.m12.2.2.1.1.1.1.1.2.cmml">u</mi><mo id="S2.Thmtheorem3.p11.12.12.m12.2.2.1.1.1.1.1.1" lspace="0.108em" rspace="0.108em" stretchy="false" xref="S2.Thmtheorem3.p11.12.12.m12.2.2.1.1.1.1.1.1.cmml">→</mo><mi id="S2.Thmtheorem3.p11.12.12.m12.2.2.1.1.1.1.1.3" xref="S2.Thmtheorem3.p11.12.12.m12.2.2.1.1.1.1.1.3.cmml">v</mi></mrow><mo id="S2.Thmtheorem3.p11.12.12.m12.2.2.1.1.1.1.3" stretchy="false" xref="S2.Thmtheorem3.p11.12.12.m12.2.2.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S2.Thmtheorem3.p11.12.12.m12.3.3.2.3" xref="S2.Thmtheorem3.p11.12.12.m12.3.3.2.3.cmml">+</mo><mrow id="S2.Thmtheorem3.p11.12.12.m12.3.3.2.2" xref="S2.Thmtheorem3.p11.12.12.m12.3.3.2.2.cmml"><mtext class="ltx_mathvariant_italic" id="S2.Thmtheorem3.p11.12.12.m12.3.3.2.2.3" xref="S2.Thmtheorem3.p11.12.12.m12.3.3.2.2.3a.cmml">g</mtext><mo id="S2.Thmtheorem3.p11.12.12.m12.3.3.2.2.2" xref="S2.Thmtheorem3.p11.12.12.m12.3.3.2.2.2.cmml">⁢</mo><mrow id="S2.Thmtheorem3.p11.12.12.m12.3.3.2.2.1.1" xref="S2.Thmtheorem3.p11.12.12.m12.3.3.2.2.1.1.1.cmml"><mo id="S2.Thmtheorem3.p11.12.12.m12.3.3.2.2.1.1.2" stretchy="false" xref="S2.Thmtheorem3.p11.12.12.m12.3.3.2.2.1.1.1.cmml">(</mo><mrow id="S2.Thmtheorem3.p11.12.12.m12.3.3.2.2.1.1.1" xref="S2.Thmtheorem3.p11.12.12.m12.3.3.2.2.1.1.1.cmml"><mi id="S2.Thmtheorem3.p11.12.12.m12.3.3.2.2.1.1.1.2" xref="S2.Thmtheorem3.p11.12.12.m12.3.3.2.2.1.1.1.2.cmml">v</mi><mo id="S2.Thmtheorem3.p11.12.12.m12.3.3.2.2.1.1.1.1" lspace="0.108em" rspace="0.108em" stretchy="false" xref="S2.Thmtheorem3.p11.12.12.m12.3.3.2.2.1.1.1.1.cmml">→</mo><mi id="S2.Thmtheorem3.p11.12.12.m12.3.3.2.2.1.1.1.3" xref="S2.Thmtheorem3.p11.12.12.m12.3.3.2.2.1.1.1.3.cmml">u</mi></mrow><mo id="S2.Thmtheorem3.p11.12.12.m12.3.3.2.2.1.1.3" stretchy="false" xref="S2.Thmtheorem3.p11.12.12.m12.3.3.2.2.1.1.1.cmml">)</mo></mrow></mrow></mrow><mo id="S2.Thmtheorem3.p11.12.12.m12.3.3.3" xref="S2.Thmtheorem3.p11.12.12.m12.3.3.3.cmml">></mo><mrow id="S2.Thmtheorem3.p11.12.12.m12.3.3.4" xref="S2.Thmtheorem3.p11.12.12.m12.3.3.4.cmml"><mtext class="ltx_mathvariant_italic" id="S2.Thmtheorem3.p11.12.12.m12.3.3.4.2" xref="S2.Thmtheorem3.p11.12.12.m12.3.3.4.2a.cmml">g</mtext><mo id="S2.Thmtheorem3.p11.12.12.m12.3.3.4.1" xref="S2.Thmtheorem3.p11.12.12.m12.3.3.4.1.cmml">⁢</mo><mrow id="S2.Thmtheorem3.p11.12.12.m12.3.3.4.3.2" xref="S2.Thmtheorem3.p11.12.12.m12.1.1.cmml"><mo id="S2.Thmtheorem3.p11.12.12.m12.3.3.4.3.2.1" stretchy="false" xref="S2.Thmtheorem3.p11.12.12.m12.1.1.cmml">(</mo><mover accent="true" id="S2.Thmtheorem3.p11.12.12.m12.1.1" xref="S2.Thmtheorem3.p11.12.12.m12.1.1.cmml"><mrow id="S2.Thmtheorem3.p11.12.12.m12.1.1.2" xref="S2.Thmtheorem3.p11.12.12.m12.1.1.2.cmml"><mi id="S2.Thmtheorem3.p11.12.12.m12.1.1.2.2" xref="S2.Thmtheorem3.p11.12.12.m12.1.1.2.2.cmml">u</mi><mo id="S2.Thmtheorem3.p11.12.12.m12.1.1.2.1" xref="S2.Thmtheorem3.p11.12.12.m12.1.1.2.1.cmml">⁢</mo><mi id="S2.Thmtheorem3.p11.12.12.m12.1.1.2.3" xref="S2.Thmtheorem3.p11.12.12.m12.1.1.2.3.cmml">v</mi></mrow><mo id="S2.Thmtheorem3.p11.12.12.m12.1.1.1" xref="S2.Thmtheorem3.p11.12.12.m12.1.1.1.cmml">¯</mo></mover><mo id="S2.Thmtheorem3.p11.12.12.m12.3.3.4.3.2.2" stretchy="false" xref="S2.Thmtheorem3.p11.12.12.m12.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem3.p11.12.12.m12.3b"><apply id="S2.Thmtheorem3.p11.12.12.m12.3.3.cmml" xref="S2.Thmtheorem3.p11.12.12.m12.3.3"><gt id="S2.Thmtheorem3.p11.12.12.m12.3.3.3.cmml" xref="S2.Thmtheorem3.p11.12.12.m12.3.3.3"></gt><apply id="S2.Thmtheorem3.p11.12.12.m12.3.3.2.cmml" xref="S2.Thmtheorem3.p11.12.12.m12.3.3.2"><plus id="S2.Thmtheorem3.p11.12.12.m12.3.3.2.3.cmml" xref="S2.Thmtheorem3.p11.12.12.m12.3.3.2.3"></plus><apply id="S2.Thmtheorem3.p11.12.12.m12.2.2.1.1.cmml" xref="S2.Thmtheorem3.p11.12.12.m12.2.2.1.1"><times id="S2.Thmtheorem3.p11.12.12.m12.2.2.1.1.2.cmml" xref="S2.Thmtheorem3.p11.12.12.m12.2.2.1.1.2"></times><ci id="S2.Thmtheorem3.p11.12.12.m12.2.2.1.1.3a.cmml" xref="S2.Thmtheorem3.p11.12.12.m12.2.2.1.1.3"><mtext class="ltx_mathvariant_italic" id="S2.Thmtheorem3.p11.12.12.m12.2.2.1.1.3.cmml" xref="S2.Thmtheorem3.p11.12.12.m12.2.2.1.1.3">g</mtext></ci><apply id="S2.Thmtheorem3.p11.12.12.m12.2.2.1.1.1.1.1.cmml" xref="S2.Thmtheorem3.p11.12.12.m12.2.2.1.1.1.1"><ci id="S2.Thmtheorem3.p11.12.12.m12.2.2.1.1.1.1.1.1.cmml" xref="S2.Thmtheorem3.p11.12.12.m12.2.2.1.1.1.1.1.1">→</ci><ci id="S2.Thmtheorem3.p11.12.12.m12.2.2.1.1.1.1.1.2.cmml" xref="S2.Thmtheorem3.p11.12.12.m12.2.2.1.1.1.1.1.2">𝑢</ci><ci id="S2.Thmtheorem3.p11.12.12.m12.2.2.1.1.1.1.1.3.cmml" xref="S2.Thmtheorem3.p11.12.12.m12.2.2.1.1.1.1.1.3">𝑣</ci></apply></apply><apply id="S2.Thmtheorem3.p11.12.12.m12.3.3.2.2.cmml" xref="S2.Thmtheorem3.p11.12.12.m12.3.3.2.2"><times id="S2.Thmtheorem3.p11.12.12.m12.3.3.2.2.2.cmml" xref="S2.Thmtheorem3.p11.12.12.m12.3.3.2.2.2"></times><ci id="S2.Thmtheorem3.p11.12.12.m12.3.3.2.2.3a.cmml" xref="S2.Thmtheorem3.p11.12.12.m12.3.3.2.2.3"><mtext class="ltx_mathvariant_italic" id="S2.Thmtheorem3.p11.12.12.m12.3.3.2.2.3.cmml" xref="S2.Thmtheorem3.p11.12.12.m12.3.3.2.2.3">g</mtext></ci><apply id="S2.Thmtheorem3.p11.12.12.m12.3.3.2.2.1.1.1.cmml" xref="S2.Thmtheorem3.p11.12.12.m12.3.3.2.2.1.1"><ci id="S2.Thmtheorem3.p11.12.12.m12.3.3.2.2.1.1.1.1.cmml" xref="S2.Thmtheorem3.p11.12.12.m12.3.3.2.2.1.1.1.1">→</ci><ci id="S2.Thmtheorem3.p11.12.12.m12.3.3.2.2.1.1.1.2.cmml" xref="S2.Thmtheorem3.p11.12.12.m12.3.3.2.2.1.1.1.2">𝑣</ci><ci id="S2.Thmtheorem3.p11.12.12.m12.3.3.2.2.1.1.1.3.cmml" xref="S2.Thmtheorem3.p11.12.12.m12.3.3.2.2.1.1.1.3">𝑢</ci></apply></apply></apply><apply id="S2.Thmtheorem3.p11.12.12.m12.3.3.4.cmml" xref="S2.Thmtheorem3.p11.12.12.m12.3.3.4"><times id="S2.Thmtheorem3.p11.12.12.m12.3.3.4.1.cmml" xref="S2.Thmtheorem3.p11.12.12.m12.3.3.4.1"></times><ci id="S2.Thmtheorem3.p11.12.12.m12.3.3.4.2a.cmml" xref="S2.Thmtheorem3.p11.12.12.m12.3.3.4.2"><mtext class="ltx_mathvariant_italic" id="S2.Thmtheorem3.p11.12.12.m12.3.3.4.2.cmml" xref="S2.Thmtheorem3.p11.12.12.m12.3.3.4.2">g</mtext></ci><apply id="S2.Thmtheorem3.p11.12.12.m12.1.1.cmml" xref="S2.Thmtheorem3.p11.12.12.m12.3.3.4.3.2"><ci id="S2.Thmtheorem3.p11.12.12.m12.1.1.1.cmml" xref="S2.Thmtheorem3.p11.12.12.m12.1.1.1">¯</ci><apply id="S2.Thmtheorem3.p11.12.12.m12.1.1.2.cmml" xref="S2.Thmtheorem3.p11.12.12.m12.1.1.2"><times id="S2.Thmtheorem3.p11.12.12.m12.1.1.2.1.cmml" xref="S2.Thmtheorem3.p11.12.12.m12.1.1.2.1"></times><ci id="S2.Thmtheorem3.p11.12.12.m12.1.1.2.2.cmml" xref="S2.Thmtheorem3.p11.12.12.m12.1.1.2.2">𝑢</ci><ci id="S2.Thmtheorem3.p11.12.12.m12.1.1.2.3.cmml" xref="S2.Thmtheorem3.p11.12.12.m12.1.1.2.3">𝑣</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem3.p11.12.12.m12.3c">\textsl{g}(u\!\to\!v)+\textsl{g}(v\!\to\!u)>\textsl{g}(\overline{uv})</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem3.p11.12.12.m12.3d">g ( italic_u → italic_v ) + g ( italic_v → italic_u ) > g ( over¯ start_ARG italic_u italic_v end_ARG )</annotation></semantics></math>. Let <math alttext="\textsl{g}(u\!\to\!v)\geq\textsl{g}(v\!\to\!u)" class="ltx_Math" display="inline" id="S2.Thmtheorem3.p11.13.13.m13.2"><semantics id="S2.Thmtheorem3.p11.13.13.m13.2a"><mrow id="S2.Thmtheorem3.p11.13.13.m13.2.2" xref="S2.Thmtheorem3.p11.13.13.m13.2.2.cmml"><mrow id="S2.Thmtheorem3.p11.13.13.m13.1.1.1" xref="S2.Thmtheorem3.p11.13.13.m13.1.1.1.cmml"><mtext class="ltx_mathvariant_italic" id="S2.Thmtheorem3.p11.13.13.m13.1.1.1.3" xref="S2.Thmtheorem3.p11.13.13.m13.1.1.1.3a.cmml">g</mtext><mo id="S2.Thmtheorem3.p11.13.13.m13.1.1.1.2" xref="S2.Thmtheorem3.p11.13.13.m13.1.1.1.2.cmml">⁢</mo><mrow id="S2.Thmtheorem3.p11.13.13.m13.1.1.1.1.1" xref="S2.Thmtheorem3.p11.13.13.m13.1.1.1.1.1.1.cmml"><mo id="S2.Thmtheorem3.p11.13.13.m13.1.1.1.1.1.2" stretchy="false" xref="S2.Thmtheorem3.p11.13.13.m13.1.1.1.1.1.1.cmml">(</mo><mrow id="S2.Thmtheorem3.p11.13.13.m13.1.1.1.1.1.1" xref="S2.Thmtheorem3.p11.13.13.m13.1.1.1.1.1.1.cmml"><mi id="S2.Thmtheorem3.p11.13.13.m13.1.1.1.1.1.1.2" xref="S2.Thmtheorem3.p11.13.13.m13.1.1.1.1.1.1.2.cmml">u</mi><mo id="S2.Thmtheorem3.p11.13.13.m13.1.1.1.1.1.1.1" lspace="0.108em" rspace="0.108em" stretchy="false" xref="S2.Thmtheorem3.p11.13.13.m13.1.1.1.1.1.1.1.cmml">→</mo><mi id="S2.Thmtheorem3.p11.13.13.m13.1.1.1.1.1.1.3" xref="S2.Thmtheorem3.p11.13.13.m13.1.1.1.1.1.1.3.cmml">v</mi></mrow><mo id="S2.Thmtheorem3.p11.13.13.m13.1.1.1.1.1.3" stretchy="false" xref="S2.Thmtheorem3.p11.13.13.m13.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S2.Thmtheorem3.p11.13.13.m13.2.2.3" xref="S2.Thmtheorem3.p11.13.13.m13.2.2.3.cmml">≥</mo><mrow id="S2.Thmtheorem3.p11.13.13.m13.2.2.2" xref="S2.Thmtheorem3.p11.13.13.m13.2.2.2.cmml"><mtext class="ltx_mathvariant_italic" id="S2.Thmtheorem3.p11.13.13.m13.2.2.2.3" xref="S2.Thmtheorem3.p11.13.13.m13.2.2.2.3a.cmml">g</mtext><mo id="S2.Thmtheorem3.p11.13.13.m13.2.2.2.2" xref="S2.Thmtheorem3.p11.13.13.m13.2.2.2.2.cmml">⁢</mo><mrow id="S2.Thmtheorem3.p11.13.13.m13.2.2.2.1.1" xref="S2.Thmtheorem3.p11.13.13.m13.2.2.2.1.1.1.cmml"><mo id="S2.Thmtheorem3.p11.13.13.m13.2.2.2.1.1.2" stretchy="false" xref="S2.Thmtheorem3.p11.13.13.m13.2.2.2.1.1.1.cmml">(</mo><mrow id="S2.Thmtheorem3.p11.13.13.m13.2.2.2.1.1.1" xref="S2.Thmtheorem3.p11.13.13.m13.2.2.2.1.1.1.cmml"><mi id="S2.Thmtheorem3.p11.13.13.m13.2.2.2.1.1.1.2" xref="S2.Thmtheorem3.p11.13.13.m13.2.2.2.1.1.1.2.cmml">v</mi><mo id="S2.Thmtheorem3.p11.13.13.m13.2.2.2.1.1.1.1" lspace="0.108em" rspace="0.108em" stretchy="false" xref="S2.Thmtheorem3.p11.13.13.m13.2.2.2.1.1.1.1.cmml">→</mo><mi id="S2.Thmtheorem3.p11.13.13.m13.2.2.2.1.1.1.3" xref="S2.Thmtheorem3.p11.13.13.m13.2.2.2.1.1.1.3.cmml">u</mi></mrow><mo id="S2.Thmtheorem3.p11.13.13.m13.2.2.2.1.1.3" stretchy="false" xref="S2.Thmtheorem3.p11.13.13.m13.2.2.2.1.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem3.p11.13.13.m13.2b"><apply id="S2.Thmtheorem3.p11.13.13.m13.2.2.cmml" xref="S2.Thmtheorem3.p11.13.13.m13.2.2"><geq id="S2.Thmtheorem3.p11.13.13.m13.2.2.3.cmml" xref="S2.Thmtheorem3.p11.13.13.m13.2.2.3"></geq><apply id="S2.Thmtheorem3.p11.13.13.m13.1.1.1.cmml" xref="S2.Thmtheorem3.p11.13.13.m13.1.1.1"><times id="S2.Thmtheorem3.p11.13.13.m13.1.1.1.2.cmml" xref="S2.Thmtheorem3.p11.13.13.m13.1.1.1.2"></times><ci id="S2.Thmtheorem3.p11.13.13.m13.1.1.1.3a.cmml" xref="S2.Thmtheorem3.p11.13.13.m13.1.1.1.3"><mtext class="ltx_mathvariant_italic" id="S2.Thmtheorem3.p11.13.13.m13.1.1.1.3.cmml" xref="S2.Thmtheorem3.p11.13.13.m13.1.1.1.3">g</mtext></ci><apply id="S2.Thmtheorem3.p11.13.13.m13.1.1.1.1.1.1.cmml" xref="S2.Thmtheorem3.p11.13.13.m13.1.1.1.1.1"><ci id="S2.Thmtheorem3.p11.13.13.m13.1.1.1.1.1.1.1.cmml" xref="S2.Thmtheorem3.p11.13.13.m13.1.1.1.1.1.1.1">→</ci><ci id="S2.Thmtheorem3.p11.13.13.m13.1.1.1.1.1.1.2.cmml" xref="S2.Thmtheorem3.p11.13.13.m13.1.1.1.1.1.1.2">𝑢</ci><ci id="S2.Thmtheorem3.p11.13.13.m13.1.1.1.1.1.1.3.cmml" xref="S2.Thmtheorem3.p11.13.13.m13.1.1.1.1.1.1.3">𝑣</ci></apply></apply><apply id="S2.Thmtheorem3.p11.13.13.m13.2.2.2.cmml" xref="S2.Thmtheorem3.p11.13.13.m13.2.2.2"><times id="S2.Thmtheorem3.p11.13.13.m13.2.2.2.2.cmml" xref="S2.Thmtheorem3.p11.13.13.m13.2.2.2.2"></times><ci id="S2.Thmtheorem3.p11.13.13.m13.2.2.2.3a.cmml" xref="S2.Thmtheorem3.p11.13.13.m13.2.2.2.3"><mtext class="ltx_mathvariant_italic" id="S2.Thmtheorem3.p11.13.13.m13.2.2.2.3.cmml" xref="S2.Thmtheorem3.p11.13.13.m13.2.2.2.3">g</mtext></ci><apply id="S2.Thmtheorem3.p11.13.13.m13.2.2.2.1.1.1.cmml" xref="S2.Thmtheorem3.p11.13.13.m13.2.2.2.1.1"><ci id="S2.Thmtheorem3.p11.13.13.m13.2.2.2.1.1.1.1.cmml" xref="S2.Thmtheorem3.p11.13.13.m13.2.2.2.1.1.1.1">→</ci><ci id="S2.Thmtheorem3.p11.13.13.m13.2.2.2.1.1.1.2.cmml" xref="S2.Thmtheorem3.p11.13.13.m13.2.2.2.1.1.1.2">𝑣</ci><ci id="S2.Thmtheorem3.p11.13.13.m13.2.2.2.1.1.1.3.cmml" xref="S2.Thmtheorem3.p11.13.13.m13.2.2.2.1.1.1.3">𝑢</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem3.p11.13.13.m13.2c">\textsl{g}(u\!\to\!v)\geq\textsl{g}(v\!\to\!u)</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem3.p11.13.13.m13.2d">g ( italic_u → italic_v ) ≥ g ( italic_v → italic_u )</annotation></semantics></math>. We may decrease <math alttext="\textsl{g}(u\!\to\!v)" class="ltx_Math" display="inline" id="S2.Thmtheorem3.p11.14.14.m14.1"><semantics id="S2.Thmtheorem3.p11.14.14.m14.1a"><mrow id="S2.Thmtheorem3.p11.14.14.m14.1.1" xref="S2.Thmtheorem3.p11.14.14.m14.1.1.cmml"><mtext class="ltx_mathvariant_italic" id="S2.Thmtheorem3.p11.14.14.m14.1.1.3" xref="S2.Thmtheorem3.p11.14.14.m14.1.1.3a.cmml">g</mtext><mo id="S2.Thmtheorem3.p11.14.14.m14.1.1.2" xref="S2.Thmtheorem3.p11.14.14.m14.1.1.2.cmml">⁢</mo><mrow id="S2.Thmtheorem3.p11.14.14.m14.1.1.1.1" xref="S2.Thmtheorem3.p11.14.14.m14.1.1.1.1.1.cmml"><mo id="S2.Thmtheorem3.p11.14.14.m14.1.1.1.1.2" stretchy="false" xref="S2.Thmtheorem3.p11.14.14.m14.1.1.1.1.1.cmml">(</mo><mrow id="S2.Thmtheorem3.p11.14.14.m14.1.1.1.1.1" xref="S2.Thmtheorem3.p11.14.14.m14.1.1.1.1.1.cmml"><mi id="S2.Thmtheorem3.p11.14.14.m14.1.1.1.1.1.2" xref="S2.Thmtheorem3.p11.14.14.m14.1.1.1.1.1.2.cmml">u</mi><mo id="S2.Thmtheorem3.p11.14.14.m14.1.1.1.1.1.1" lspace="0.108em" rspace="0.108em" stretchy="false" xref="S2.Thmtheorem3.p11.14.14.m14.1.1.1.1.1.1.cmml">→</mo><mi id="S2.Thmtheorem3.p11.14.14.m14.1.1.1.1.1.3" xref="S2.Thmtheorem3.p11.14.14.m14.1.1.1.1.1.3.cmml">v</mi></mrow><mo id="S2.Thmtheorem3.p11.14.14.m14.1.1.1.1.3" stretchy="false" xref="S2.Thmtheorem3.p11.14.14.m14.1.1.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem3.p11.14.14.m14.1b"><apply id="S2.Thmtheorem3.p11.14.14.m14.1.1.cmml" xref="S2.Thmtheorem3.p11.14.14.m14.1.1"><times id="S2.Thmtheorem3.p11.14.14.m14.1.1.2.cmml" xref="S2.Thmtheorem3.p11.14.14.m14.1.1.2"></times><ci id="S2.Thmtheorem3.p11.14.14.m14.1.1.3a.cmml" xref="S2.Thmtheorem3.p11.14.14.m14.1.1.3"><mtext class="ltx_mathvariant_italic" id="S2.Thmtheorem3.p11.14.14.m14.1.1.3.cmml" xref="S2.Thmtheorem3.p11.14.14.m14.1.1.3">g</mtext></ci><apply id="S2.Thmtheorem3.p11.14.14.m14.1.1.1.1.1.cmml" xref="S2.Thmtheorem3.p11.14.14.m14.1.1.1.1"><ci id="S2.Thmtheorem3.p11.14.14.m14.1.1.1.1.1.1.cmml" xref="S2.Thmtheorem3.p11.14.14.m14.1.1.1.1.1.1">→</ci><ci id="S2.Thmtheorem3.p11.14.14.m14.1.1.1.1.1.2.cmml" xref="S2.Thmtheorem3.p11.14.14.m14.1.1.1.1.1.2">𝑢</ci><ci id="S2.Thmtheorem3.p11.14.14.m14.1.1.1.1.1.3.cmml" xref="S2.Thmtheorem3.p11.14.14.m14.1.1.1.1.1.3">𝑣</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem3.p11.14.14.m14.1c">\textsl{g}(u\!\to\!v)</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem3.p11.14.14.m14.1d">g ( italic_u → italic_v )</annotation></semantics></math> until <math alttext="\textsl{g}(u\!\to\!v)+\textsl{g}(v\!\to\!u)=\textsl{g}(\overline{uv})" class="ltx_Math" display="inline" id="S2.Thmtheorem3.p11.15.15.m15.3"><semantics id="S2.Thmtheorem3.p11.15.15.m15.3a"><mrow id="S2.Thmtheorem3.p11.15.15.m15.3.3" xref="S2.Thmtheorem3.p11.15.15.m15.3.3.cmml"><mrow id="S2.Thmtheorem3.p11.15.15.m15.3.3.2" xref="S2.Thmtheorem3.p11.15.15.m15.3.3.2.cmml"><mrow id="S2.Thmtheorem3.p11.15.15.m15.2.2.1.1" xref="S2.Thmtheorem3.p11.15.15.m15.2.2.1.1.cmml"><mtext class="ltx_mathvariant_italic" id="S2.Thmtheorem3.p11.15.15.m15.2.2.1.1.3" xref="S2.Thmtheorem3.p11.15.15.m15.2.2.1.1.3a.cmml">g</mtext><mo id="S2.Thmtheorem3.p11.15.15.m15.2.2.1.1.2" xref="S2.Thmtheorem3.p11.15.15.m15.2.2.1.1.2.cmml">⁢</mo><mrow id="S2.Thmtheorem3.p11.15.15.m15.2.2.1.1.1.1" xref="S2.Thmtheorem3.p11.15.15.m15.2.2.1.1.1.1.1.cmml"><mo id="S2.Thmtheorem3.p11.15.15.m15.2.2.1.1.1.1.2" stretchy="false" xref="S2.Thmtheorem3.p11.15.15.m15.2.2.1.1.1.1.1.cmml">(</mo><mrow id="S2.Thmtheorem3.p11.15.15.m15.2.2.1.1.1.1.1" xref="S2.Thmtheorem3.p11.15.15.m15.2.2.1.1.1.1.1.cmml"><mi id="S2.Thmtheorem3.p11.15.15.m15.2.2.1.1.1.1.1.2" xref="S2.Thmtheorem3.p11.15.15.m15.2.2.1.1.1.1.1.2.cmml">u</mi><mo id="S2.Thmtheorem3.p11.15.15.m15.2.2.1.1.1.1.1.1" lspace="0.108em" rspace="0.108em" stretchy="false" xref="S2.Thmtheorem3.p11.15.15.m15.2.2.1.1.1.1.1.1.cmml">→</mo><mi id="S2.Thmtheorem3.p11.15.15.m15.2.2.1.1.1.1.1.3" xref="S2.Thmtheorem3.p11.15.15.m15.2.2.1.1.1.1.1.3.cmml">v</mi></mrow><mo id="S2.Thmtheorem3.p11.15.15.m15.2.2.1.1.1.1.3" stretchy="false" xref="S2.Thmtheorem3.p11.15.15.m15.2.2.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S2.Thmtheorem3.p11.15.15.m15.3.3.2.3" xref="S2.Thmtheorem3.p11.15.15.m15.3.3.2.3.cmml">+</mo><mrow id="S2.Thmtheorem3.p11.15.15.m15.3.3.2.2" xref="S2.Thmtheorem3.p11.15.15.m15.3.3.2.2.cmml"><mtext class="ltx_mathvariant_italic" id="S2.Thmtheorem3.p11.15.15.m15.3.3.2.2.3" xref="S2.Thmtheorem3.p11.15.15.m15.3.3.2.2.3a.cmml">g</mtext><mo id="S2.Thmtheorem3.p11.15.15.m15.3.3.2.2.2" xref="S2.Thmtheorem3.p11.15.15.m15.3.3.2.2.2.cmml">⁢</mo><mrow id="S2.Thmtheorem3.p11.15.15.m15.3.3.2.2.1.1" xref="S2.Thmtheorem3.p11.15.15.m15.3.3.2.2.1.1.1.cmml"><mo id="S2.Thmtheorem3.p11.15.15.m15.3.3.2.2.1.1.2" stretchy="false" xref="S2.Thmtheorem3.p11.15.15.m15.3.3.2.2.1.1.1.cmml">(</mo><mrow id="S2.Thmtheorem3.p11.15.15.m15.3.3.2.2.1.1.1" xref="S2.Thmtheorem3.p11.15.15.m15.3.3.2.2.1.1.1.cmml"><mi id="S2.Thmtheorem3.p11.15.15.m15.3.3.2.2.1.1.1.2" xref="S2.Thmtheorem3.p11.15.15.m15.3.3.2.2.1.1.1.2.cmml">v</mi><mo id="S2.Thmtheorem3.p11.15.15.m15.3.3.2.2.1.1.1.1" lspace="0.108em" rspace="0.108em" stretchy="false" xref="S2.Thmtheorem3.p11.15.15.m15.3.3.2.2.1.1.1.1.cmml">→</mo><mi id="S2.Thmtheorem3.p11.15.15.m15.3.3.2.2.1.1.1.3" xref="S2.Thmtheorem3.p11.15.15.m15.3.3.2.2.1.1.1.3.cmml">u</mi></mrow><mo id="S2.Thmtheorem3.p11.15.15.m15.3.3.2.2.1.1.3" stretchy="false" xref="S2.Thmtheorem3.p11.15.15.m15.3.3.2.2.1.1.1.cmml">)</mo></mrow></mrow></mrow><mo id="S2.Thmtheorem3.p11.15.15.m15.3.3.3" xref="S2.Thmtheorem3.p11.15.15.m15.3.3.3.cmml">=</mo><mrow id="S2.Thmtheorem3.p11.15.15.m15.3.3.4" xref="S2.Thmtheorem3.p11.15.15.m15.3.3.4.cmml"><mtext class="ltx_mathvariant_italic" id="S2.Thmtheorem3.p11.15.15.m15.3.3.4.2" xref="S2.Thmtheorem3.p11.15.15.m15.3.3.4.2a.cmml">g</mtext><mo id="S2.Thmtheorem3.p11.15.15.m15.3.3.4.1" xref="S2.Thmtheorem3.p11.15.15.m15.3.3.4.1.cmml">⁢</mo><mrow id="S2.Thmtheorem3.p11.15.15.m15.3.3.4.3.2" xref="S2.Thmtheorem3.p11.15.15.m15.1.1.cmml"><mo id="S2.Thmtheorem3.p11.15.15.m15.3.3.4.3.2.1" stretchy="false" xref="S2.Thmtheorem3.p11.15.15.m15.1.1.cmml">(</mo><mover accent="true" id="S2.Thmtheorem3.p11.15.15.m15.1.1" xref="S2.Thmtheorem3.p11.15.15.m15.1.1.cmml"><mrow id="S2.Thmtheorem3.p11.15.15.m15.1.1.2" xref="S2.Thmtheorem3.p11.15.15.m15.1.1.2.cmml"><mi id="S2.Thmtheorem3.p11.15.15.m15.1.1.2.2" xref="S2.Thmtheorem3.p11.15.15.m15.1.1.2.2.cmml">u</mi><mo id="S2.Thmtheorem3.p11.15.15.m15.1.1.2.1" xref="S2.Thmtheorem3.p11.15.15.m15.1.1.2.1.cmml">⁢</mo><mi id="S2.Thmtheorem3.p11.15.15.m15.1.1.2.3" xref="S2.Thmtheorem3.p11.15.15.m15.1.1.2.3.cmml">v</mi></mrow><mo id="S2.Thmtheorem3.p11.15.15.m15.1.1.1" xref="S2.Thmtheorem3.p11.15.15.m15.1.1.1.cmml">¯</mo></mover><mo id="S2.Thmtheorem3.p11.15.15.m15.3.3.4.3.2.2" stretchy="false" xref="S2.Thmtheorem3.p11.15.15.m15.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem3.p11.15.15.m15.3b"><apply id="S2.Thmtheorem3.p11.15.15.m15.3.3.cmml" xref="S2.Thmtheorem3.p11.15.15.m15.3.3"><eq id="S2.Thmtheorem3.p11.15.15.m15.3.3.3.cmml" xref="S2.Thmtheorem3.p11.15.15.m15.3.3.3"></eq><apply id="S2.Thmtheorem3.p11.15.15.m15.3.3.2.cmml" xref="S2.Thmtheorem3.p11.15.15.m15.3.3.2"><plus id="S2.Thmtheorem3.p11.15.15.m15.3.3.2.3.cmml" xref="S2.Thmtheorem3.p11.15.15.m15.3.3.2.3"></plus><apply id="S2.Thmtheorem3.p11.15.15.m15.2.2.1.1.cmml" xref="S2.Thmtheorem3.p11.15.15.m15.2.2.1.1"><times id="S2.Thmtheorem3.p11.15.15.m15.2.2.1.1.2.cmml" xref="S2.Thmtheorem3.p11.15.15.m15.2.2.1.1.2"></times><ci id="S2.Thmtheorem3.p11.15.15.m15.2.2.1.1.3a.cmml" xref="S2.Thmtheorem3.p11.15.15.m15.2.2.1.1.3"><mtext class="ltx_mathvariant_italic" id="S2.Thmtheorem3.p11.15.15.m15.2.2.1.1.3.cmml" xref="S2.Thmtheorem3.p11.15.15.m15.2.2.1.1.3">g</mtext></ci><apply id="S2.Thmtheorem3.p11.15.15.m15.2.2.1.1.1.1.1.cmml" xref="S2.Thmtheorem3.p11.15.15.m15.2.2.1.1.1.1"><ci id="S2.Thmtheorem3.p11.15.15.m15.2.2.1.1.1.1.1.1.cmml" xref="S2.Thmtheorem3.p11.15.15.m15.2.2.1.1.1.1.1.1">→</ci><ci id="S2.Thmtheorem3.p11.15.15.m15.2.2.1.1.1.1.1.2.cmml" xref="S2.Thmtheorem3.p11.15.15.m15.2.2.1.1.1.1.1.2">𝑢</ci><ci id="S2.Thmtheorem3.p11.15.15.m15.2.2.1.1.1.1.1.3.cmml" xref="S2.Thmtheorem3.p11.15.15.m15.2.2.1.1.1.1.1.3">𝑣</ci></apply></apply><apply id="S2.Thmtheorem3.p11.15.15.m15.3.3.2.2.cmml" xref="S2.Thmtheorem3.p11.15.15.m15.3.3.2.2"><times id="S2.Thmtheorem3.p11.15.15.m15.3.3.2.2.2.cmml" xref="S2.Thmtheorem3.p11.15.15.m15.3.3.2.2.2"></times><ci id="S2.Thmtheorem3.p11.15.15.m15.3.3.2.2.3a.cmml" xref="S2.Thmtheorem3.p11.15.15.m15.3.3.2.2.3"><mtext class="ltx_mathvariant_italic" id="S2.Thmtheorem3.p11.15.15.m15.3.3.2.2.3.cmml" xref="S2.Thmtheorem3.p11.15.15.m15.3.3.2.2.3">g</mtext></ci><apply id="S2.Thmtheorem3.p11.15.15.m15.3.3.2.2.1.1.1.cmml" xref="S2.Thmtheorem3.p11.15.15.m15.3.3.2.2.1.1"><ci id="S2.Thmtheorem3.p11.15.15.m15.3.3.2.2.1.1.1.1.cmml" xref="S2.Thmtheorem3.p11.15.15.m15.3.3.2.2.1.1.1.1">→</ci><ci id="S2.Thmtheorem3.p11.15.15.m15.3.3.2.2.1.1.1.2.cmml" xref="S2.Thmtheorem3.p11.15.15.m15.3.3.2.2.1.1.1.2">𝑣</ci><ci id="S2.Thmtheorem3.p11.15.15.m15.3.3.2.2.1.1.1.3.cmml" xref="S2.Thmtheorem3.p11.15.15.m15.3.3.2.2.1.1.1.3">𝑢</ci></apply></apply></apply><apply id="S2.Thmtheorem3.p11.15.15.m15.3.3.4.cmml" xref="S2.Thmtheorem3.p11.15.15.m15.3.3.4"><times id="S2.Thmtheorem3.p11.15.15.m15.3.3.4.1.cmml" xref="S2.Thmtheorem3.p11.15.15.m15.3.3.4.1"></times><ci id="S2.Thmtheorem3.p11.15.15.m15.3.3.4.2a.cmml" xref="S2.Thmtheorem3.p11.15.15.m15.3.3.4.2"><mtext class="ltx_mathvariant_italic" id="S2.Thmtheorem3.p11.15.15.m15.3.3.4.2.cmml" xref="S2.Thmtheorem3.p11.15.15.m15.3.3.4.2">g</mtext></ci><apply id="S2.Thmtheorem3.p11.15.15.m15.1.1.cmml" xref="S2.Thmtheorem3.p11.15.15.m15.3.3.4.3.2"><ci id="S2.Thmtheorem3.p11.15.15.m15.1.1.1.cmml" xref="S2.Thmtheorem3.p11.15.15.m15.1.1.1">¯</ci><apply id="S2.Thmtheorem3.p11.15.15.m15.1.1.2.cmml" xref="S2.Thmtheorem3.p11.15.15.m15.1.1.2"><times id="S2.Thmtheorem3.p11.15.15.m15.1.1.2.1.cmml" xref="S2.Thmtheorem3.p11.15.15.m15.1.1.2.1"></times><ci id="S2.Thmtheorem3.p11.15.15.m15.1.1.2.2.cmml" xref="S2.Thmtheorem3.p11.15.15.m15.1.1.2.2">𝑢</ci><ci id="S2.Thmtheorem3.p11.15.15.m15.1.1.2.3.cmml" xref="S2.Thmtheorem3.p11.15.15.m15.1.1.2.3">𝑣</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem3.p11.15.15.m15.3c">\textsl{g}(u\!\to\!v)+\textsl{g}(v\!\to\!u)=\textsl{g}(\overline{uv})</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem3.p11.15.15.m15.3d">g ( italic_u → italic_v ) + g ( italic_v → italic_u ) = g ( over¯ start_ARG italic_u italic_v end_ARG )</annotation></semantics></math>. By performing this action, we only reduce the sum <math alttext="\sum_{v\in V}\textsl{g}(u\!\to\!v)" class="ltx_Math" display="inline" id="S2.Thmtheorem3.p11.16.16.m16.1"><semantics id="S2.Thmtheorem3.p11.16.16.m16.1a"><mrow id="S2.Thmtheorem3.p11.16.16.m16.1.1" xref="S2.Thmtheorem3.p11.16.16.m16.1.1.cmml"><msub id="S2.Thmtheorem3.p11.16.16.m16.1.1.2" xref="S2.Thmtheorem3.p11.16.16.m16.1.1.2.cmml"><mo id="S2.Thmtheorem3.p11.16.16.m16.1.1.2.2" xref="S2.Thmtheorem3.p11.16.16.m16.1.1.2.2.cmml">∑</mo><mrow id="S2.Thmtheorem3.p11.16.16.m16.1.1.2.3" xref="S2.Thmtheorem3.p11.16.16.m16.1.1.2.3.cmml"><mi id="S2.Thmtheorem3.p11.16.16.m16.1.1.2.3.2" xref="S2.Thmtheorem3.p11.16.16.m16.1.1.2.3.2.cmml">v</mi><mo id="S2.Thmtheorem3.p11.16.16.m16.1.1.2.3.1" xref="S2.Thmtheorem3.p11.16.16.m16.1.1.2.3.1.cmml">∈</mo><mi id="S2.Thmtheorem3.p11.16.16.m16.1.1.2.3.3" xref="S2.Thmtheorem3.p11.16.16.m16.1.1.2.3.3.cmml">V</mi></mrow></msub><mrow id="S2.Thmtheorem3.p11.16.16.m16.1.1.1" xref="S2.Thmtheorem3.p11.16.16.m16.1.1.1.cmml"><mtext class="ltx_mathvariant_italic" id="S2.Thmtheorem3.p11.16.16.m16.1.1.1.3" xref="S2.Thmtheorem3.p11.16.16.m16.1.1.1.3a.cmml">g</mtext><mo id="S2.Thmtheorem3.p11.16.16.m16.1.1.1.2" xref="S2.Thmtheorem3.p11.16.16.m16.1.1.1.2.cmml">⁢</mo><mrow id="S2.Thmtheorem3.p11.16.16.m16.1.1.1.1.1" xref="S2.Thmtheorem3.p11.16.16.m16.1.1.1.1.1.1.cmml"><mo id="S2.Thmtheorem3.p11.16.16.m16.1.1.1.1.1.2" stretchy="false" xref="S2.Thmtheorem3.p11.16.16.m16.1.1.1.1.1.1.cmml">(</mo><mrow id="S2.Thmtheorem3.p11.16.16.m16.1.1.1.1.1.1" xref="S2.Thmtheorem3.p11.16.16.m16.1.1.1.1.1.1.cmml"><mi id="S2.Thmtheorem3.p11.16.16.m16.1.1.1.1.1.1.2" xref="S2.Thmtheorem3.p11.16.16.m16.1.1.1.1.1.1.2.cmml">u</mi><mo id="S2.Thmtheorem3.p11.16.16.m16.1.1.1.1.1.1.1" lspace="0.108em" rspace="0.108em" stretchy="false" xref="S2.Thmtheorem3.p11.16.16.m16.1.1.1.1.1.1.1.cmml">→</mo><mi id="S2.Thmtheorem3.p11.16.16.m16.1.1.1.1.1.1.3" xref="S2.Thmtheorem3.p11.16.16.m16.1.1.1.1.1.1.3.cmml">v</mi></mrow><mo id="S2.Thmtheorem3.p11.16.16.m16.1.1.1.1.1.3" stretchy="false" xref="S2.Thmtheorem3.p11.16.16.m16.1.1.1.1.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem3.p11.16.16.m16.1b"><apply id="S2.Thmtheorem3.p11.16.16.m16.1.1.cmml" xref="S2.Thmtheorem3.p11.16.16.m16.1.1"><apply id="S2.Thmtheorem3.p11.16.16.m16.1.1.2.cmml" xref="S2.Thmtheorem3.p11.16.16.m16.1.1.2"><csymbol cd="ambiguous" id="S2.Thmtheorem3.p11.16.16.m16.1.1.2.1.cmml" xref="S2.Thmtheorem3.p11.16.16.m16.1.1.2">subscript</csymbol><sum id="S2.Thmtheorem3.p11.16.16.m16.1.1.2.2.cmml" xref="S2.Thmtheorem3.p11.16.16.m16.1.1.2.2"></sum><apply id="S2.Thmtheorem3.p11.16.16.m16.1.1.2.3.cmml" xref="S2.Thmtheorem3.p11.16.16.m16.1.1.2.3"><in id="S2.Thmtheorem3.p11.16.16.m16.1.1.2.3.1.cmml" xref="S2.Thmtheorem3.p11.16.16.m16.1.1.2.3.1"></in><ci id="S2.Thmtheorem3.p11.16.16.m16.1.1.2.3.2.cmml" xref="S2.Thmtheorem3.p11.16.16.m16.1.1.2.3.2">𝑣</ci><ci id="S2.Thmtheorem3.p11.16.16.m16.1.1.2.3.3.cmml" xref="S2.Thmtheorem3.p11.16.16.m16.1.1.2.3.3">𝑉</ci></apply></apply><apply id="S2.Thmtheorem3.p11.16.16.m16.1.1.1.cmml" xref="S2.Thmtheorem3.p11.16.16.m16.1.1.1"><times id="S2.Thmtheorem3.p11.16.16.m16.1.1.1.2.cmml" xref="S2.Thmtheorem3.p11.16.16.m16.1.1.1.2"></times><ci id="S2.Thmtheorem3.p11.16.16.m16.1.1.1.3a.cmml" xref="S2.Thmtheorem3.p11.16.16.m16.1.1.1.3"><mtext class="ltx_mathvariant_italic" id="S2.Thmtheorem3.p11.16.16.m16.1.1.1.3.cmml" xref="S2.Thmtheorem3.p11.16.16.m16.1.1.1.3">g</mtext></ci><apply id="S2.Thmtheorem3.p11.16.16.m16.1.1.1.1.1.1.cmml" xref="S2.Thmtheorem3.p11.16.16.m16.1.1.1.1.1"><ci id="S2.Thmtheorem3.p11.16.16.m16.1.1.1.1.1.1.1.cmml" xref="S2.Thmtheorem3.p11.16.16.m16.1.1.1.1.1.1.1">→</ci><ci id="S2.Thmtheorem3.p11.16.16.m16.1.1.1.1.1.1.2.cmml" xref="S2.Thmtheorem3.p11.16.16.m16.1.1.1.1.1.1.2">𝑢</ci><ci id="S2.Thmtheorem3.p11.16.16.m16.1.1.1.1.1.1.3.cmml" xref="S2.Thmtheorem3.p11.16.16.m16.1.1.1.1.1.1.3">𝑣</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem3.p11.16.16.m16.1c">\sum_{v\in V}\textsl{g}(u\!\to\!v)</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem3.p11.16.16.m16.1d">∑ start_POSTSUBSCRIPT italic_v ∈ italic_V end_POSTSUBSCRIPT g ( italic_u → italic_v )</annotation></semantics></math> and thus we keep having a valid solution to FO. Doing this for every edge <math alttext="\overline{uv}\in E" class="ltx_Math" display="inline" id="S2.Thmtheorem3.p11.17.17.m17.1"><semantics id="S2.Thmtheorem3.p11.17.17.m17.1a"><mrow id="S2.Thmtheorem3.p11.17.17.m17.1.1" xref="S2.Thmtheorem3.p11.17.17.m17.1.1.cmml"><mover accent="true" id="S2.Thmtheorem3.p11.17.17.m17.1.1.2" xref="S2.Thmtheorem3.p11.17.17.m17.1.1.2.cmml"><mrow id="S2.Thmtheorem3.p11.17.17.m17.1.1.2.2" xref="S2.Thmtheorem3.p11.17.17.m17.1.1.2.2.cmml"><mi id="S2.Thmtheorem3.p11.17.17.m17.1.1.2.2.2" xref="S2.Thmtheorem3.p11.17.17.m17.1.1.2.2.2.cmml">u</mi><mo id="S2.Thmtheorem3.p11.17.17.m17.1.1.2.2.1" xref="S2.Thmtheorem3.p11.17.17.m17.1.1.2.2.1.cmml">⁢</mo><mi id="S2.Thmtheorem3.p11.17.17.m17.1.1.2.2.3" xref="S2.Thmtheorem3.p11.17.17.m17.1.1.2.2.3.cmml">v</mi></mrow><mo id="S2.Thmtheorem3.p11.17.17.m17.1.1.2.1" xref="S2.Thmtheorem3.p11.17.17.m17.1.1.2.1.cmml">¯</mo></mover><mo id="S2.Thmtheorem3.p11.17.17.m17.1.1.1" xref="S2.Thmtheorem3.p11.17.17.m17.1.1.1.cmml">∈</mo><mi id="S2.Thmtheorem3.p11.17.17.m17.1.1.3" xref="S2.Thmtheorem3.p11.17.17.m17.1.1.3.cmml">E</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem3.p11.17.17.m17.1b"><apply id="S2.Thmtheorem3.p11.17.17.m17.1.1.cmml" xref="S2.Thmtheorem3.p11.17.17.m17.1.1"><in id="S2.Thmtheorem3.p11.17.17.m17.1.1.1.cmml" xref="S2.Thmtheorem3.p11.17.17.m17.1.1.1"></in><apply id="S2.Thmtheorem3.p11.17.17.m17.1.1.2.cmml" xref="S2.Thmtheorem3.p11.17.17.m17.1.1.2"><ci id="S2.Thmtheorem3.p11.17.17.m17.1.1.2.1.cmml" xref="S2.Thmtheorem3.p11.17.17.m17.1.1.2.1">¯</ci><apply id="S2.Thmtheorem3.p11.17.17.m17.1.1.2.2.cmml" xref="S2.Thmtheorem3.p11.17.17.m17.1.1.2.2"><times id="S2.Thmtheorem3.p11.17.17.m17.1.1.2.2.1.cmml" xref="S2.Thmtheorem3.p11.17.17.m17.1.1.2.2.1"></times><ci id="S2.Thmtheorem3.p11.17.17.m17.1.1.2.2.2.cmml" xref="S2.Thmtheorem3.p11.17.17.m17.1.1.2.2.2">𝑢</ci><ci id="S2.Thmtheorem3.p11.17.17.m17.1.1.2.2.3.cmml" xref="S2.Thmtheorem3.p11.17.17.m17.1.1.2.2.3">𝑣</ci></apply></apply><ci id="S2.Thmtheorem3.p11.17.17.m17.1.1.3.cmml" xref="S2.Thmtheorem3.p11.17.17.m17.1.1.3">𝐸</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem3.p11.17.17.m17.1c">\overline{uv}\in E</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem3.p11.17.17.m17.1d">over¯ start_ARG italic_u italic_v end_ARG ∈ italic_E</annotation></semantics></math> creates an orientation where for all <math alttext="u" class="ltx_Math" display="inline" id="S2.Thmtheorem3.p11.18.18.m18.1"><semantics id="S2.Thmtheorem3.p11.18.18.m18.1a"><mi id="S2.Thmtheorem3.p11.18.18.m18.1.1" xref="S2.Thmtheorem3.p11.18.18.m18.1.1.cmml">u</mi><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem3.p11.18.18.m18.1b"><ci id="S2.Thmtheorem3.p11.18.18.m18.1.1.cmml" xref="S2.Thmtheorem3.p11.18.18.m18.1.1">𝑢</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem3.p11.18.18.m18.1c">u</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem3.p11.18.18.m18.1d">italic_u</annotation></semantics></math>, <math alttext="\textsl{g}(u)=\sum_{v\in V}\textsl{g}(u\!\to\!v)\leq R" class="ltx_Math" display="inline" id="S2.Thmtheorem3.p11.19.19.m19.2"><semantics id="S2.Thmtheorem3.p11.19.19.m19.2a"><mrow id="S2.Thmtheorem3.p11.19.19.m19.2.2" xref="S2.Thmtheorem3.p11.19.19.m19.2.2.cmml"><mrow id="S2.Thmtheorem3.p11.19.19.m19.2.2.3" xref="S2.Thmtheorem3.p11.19.19.m19.2.2.3.cmml"><mtext class="ltx_mathvariant_italic" id="S2.Thmtheorem3.p11.19.19.m19.2.2.3.2" xref="S2.Thmtheorem3.p11.19.19.m19.2.2.3.2a.cmml">g</mtext><mo id="S2.Thmtheorem3.p11.19.19.m19.2.2.3.1" xref="S2.Thmtheorem3.p11.19.19.m19.2.2.3.1.cmml">⁢</mo><mrow id="S2.Thmtheorem3.p11.19.19.m19.2.2.3.3.2" xref="S2.Thmtheorem3.p11.19.19.m19.2.2.3.cmml"><mo id="S2.Thmtheorem3.p11.19.19.m19.2.2.3.3.2.1" stretchy="false" xref="S2.Thmtheorem3.p11.19.19.m19.2.2.3.cmml">(</mo><mi id="S2.Thmtheorem3.p11.19.19.m19.1.1" xref="S2.Thmtheorem3.p11.19.19.m19.1.1.cmml">u</mi><mo id="S2.Thmtheorem3.p11.19.19.m19.2.2.3.3.2.2" stretchy="false" xref="S2.Thmtheorem3.p11.19.19.m19.2.2.3.cmml">)</mo></mrow></mrow><mo id="S2.Thmtheorem3.p11.19.19.m19.2.2.4" rspace="0.111em" xref="S2.Thmtheorem3.p11.19.19.m19.2.2.4.cmml">=</mo><mrow id="S2.Thmtheorem3.p11.19.19.m19.2.2.1" xref="S2.Thmtheorem3.p11.19.19.m19.2.2.1.cmml"><msub id="S2.Thmtheorem3.p11.19.19.m19.2.2.1.2" xref="S2.Thmtheorem3.p11.19.19.m19.2.2.1.2.cmml"><mo id="S2.Thmtheorem3.p11.19.19.m19.2.2.1.2.2" xref="S2.Thmtheorem3.p11.19.19.m19.2.2.1.2.2.cmml">∑</mo><mrow id="S2.Thmtheorem3.p11.19.19.m19.2.2.1.2.3" xref="S2.Thmtheorem3.p11.19.19.m19.2.2.1.2.3.cmml"><mi id="S2.Thmtheorem3.p11.19.19.m19.2.2.1.2.3.2" xref="S2.Thmtheorem3.p11.19.19.m19.2.2.1.2.3.2.cmml">v</mi><mo id="S2.Thmtheorem3.p11.19.19.m19.2.2.1.2.3.1" xref="S2.Thmtheorem3.p11.19.19.m19.2.2.1.2.3.1.cmml">∈</mo><mi id="S2.Thmtheorem3.p11.19.19.m19.2.2.1.2.3.3" xref="S2.Thmtheorem3.p11.19.19.m19.2.2.1.2.3.3.cmml">V</mi></mrow></msub><mrow id="S2.Thmtheorem3.p11.19.19.m19.2.2.1.1" xref="S2.Thmtheorem3.p11.19.19.m19.2.2.1.1.cmml"><mtext class="ltx_mathvariant_italic" id="S2.Thmtheorem3.p11.19.19.m19.2.2.1.1.3" xref="S2.Thmtheorem3.p11.19.19.m19.2.2.1.1.3a.cmml">g</mtext><mo id="S2.Thmtheorem3.p11.19.19.m19.2.2.1.1.2" xref="S2.Thmtheorem3.p11.19.19.m19.2.2.1.1.2.cmml">⁢</mo><mrow id="S2.Thmtheorem3.p11.19.19.m19.2.2.1.1.1.1" xref="S2.Thmtheorem3.p11.19.19.m19.2.2.1.1.1.1.1.cmml"><mo id="S2.Thmtheorem3.p11.19.19.m19.2.2.1.1.1.1.2" stretchy="false" xref="S2.Thmtheorem3.p11.19.19.m19.2.2.1.1.1.1.1.cmml">(</mo><mrow id="S2.Thmtheorem3.p11.19.19.m19.2.2.1.1.1.1.1" xref="S2.Thmtheorem3.p11.19.19.m19.2.2.1.1.1.1.1.cmml"><mi id="S2.Thmtheorem3.p11.19.19.m19.2.2.1.1.1.1.1.2" xref="S2.Thmtheorem3.p11.19.19.m19.2.2.1.1.1.1.1.2.cmml">u</mi><mo id="S2.Thmtheorem3.p11.19.19.m19.2.2.1.1.1.1.1.1" lspace="0.108em" rspace="0.108em" stretchy="false" xref="S2.Thmtheorem3.p11.19.19.m19.2.2.1.1.1.1.1.1.cmml">→</mo><mi id="S2.Thmtheorem3.p11.19.19.m19.2.2.1.1.1.1.1.3" xref="S2.Thmtheorem3.p11.19.19.m19.2.2.1.1.1.1.1.3.cmml">v</mi></mrow><mo id="S2.Thmtheorem3.p11.19.19.m19.2.2.1.1.1.1.3" stretchy="false" xref="S2.Thmtheorem3.p11.19.19.m19.2.2.1.1.1.1.1.cmml">)</mo></mrow></mrow></mrow><mo id="S2.Thmtheorem3.p11.19.19.m19.2.2.5" xref="S2.Thmtheorem3.p11.19.19.m19.2.2.5.cmml">≤</mo><mi id="S2.Thmtheorem3.p11.19.19.m19.2.2.6" xref="S2.Thmtheorem3.p11.19.19.m19.2.2.6.cmml">R</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem3.p11.19.19.m19.2b"><apply id="S2.Thmtheorem3.p11.19.19.m19.2.2.cmml" xref="S2.Thmtheorem3.p11.19.19.m19.2.2"><and id="S2.Thmtheorem3.p11.19.19.m19.2.2a.cmml" xref="S2.Thmtheorem3.p11.19.19.m19.2.2"></and><apply id="S2.Thmtheorem3.p11.19.19.m19.2.2b.cmml" xref="S2.Thmtheorem3.p11.19.19.m19.2.2"><eq id="S2.Thmtheorem3.p11.19.19.m19.2.2.4.cmml" xref="S2.Thmtheorem3.p11.19.19.m19.2.2.4"></eq><apply id="S2.Thmtheorem3.p11.19.19.m19.2.2.3.cmml" xref="S2.Thmtheorem3.p11.19.19.m19.2.2.3"><times id="S2.Thmtheorem3.p11.19.19.m19.2.2.3.1.cmml" xref="S2.Thmtheorem3.p11.19.19.m19.2.2.3.1"></times><ci id="S2.Thmtheorem3.p11.19.19.m19.2.2.3.2a.cmml" xref="S2.Thmtheorem3.p11.19.19.m19.2.2.3.2"><mtext class="ltx_mathvariant_italic" id="S2.Thmtheorem3.p11.19.19.m19.2.2.3.2.cmml" xref="S2.Thmtheorem3.p11.19.19.m19.2.2.3.2">g</mtext></ci><ci id="S2.Thmtheorem3.p11.19.19.m19.1.1.cmml" xref="S2.Thmtheorem3.p11.19.19.m19.1.1">𝑢</ci></apply><apply id="S2.Thmtheorem3.p11.19.19.m19.2.2.1.cmml" xref="S2.Thmtheorem3.p11.19.19.m19.2.2.1"><apply id="S2.Thmtheorem3.p11.19.19.m19.2.2.1.2.cmml" xref="S2.Thmtheorem3.p11.19.19.m19.2.2.1.2"><csymbol cd="ambiguous" id="S2.Thmtheorem3.p11.19.19.m19.2.2.1.2.1.cmml" xref="S2.Thmtheorem3.p11.19.19.m19.2.2.1.2">subscript</csymbol><sum id="S2.Thmtheorem3.p11.19.19.m19.2.2.1.2.2.cmml" xref="S2.Thmtheorem3.p11.19.19.m19.2.2.1.2.2"></sum><apply id="S2.Thmtheorem3.p11.19.19.m19.2.2.1.2.3.cmml" xref="S2.Thmtheorem3.p11.19.19.m19.2.2.1.2.3"><in id="S2.Thmtheorem3.p11.19.19.m19.2.2.1.2.3.1.cmml" xref="S2.Thmtheorem3.p11.19.19.m19.2.2.1.2.3.1"></in><ci id="S2.Thmtheorem3.p11.19.19.m19.2.2.1.2.3.2.cmml" xref="S2.Thmtheorem3.p11.19.19.m19.2.2.1.2.3.2">𝑣</ci><ci id="S2.Thmtheorem3.p11.19.19.m19.2.2.1.2.3.3.cmml" xref="S2.Thmtheorem3.p11.19.19.m19.2.2.1.2.3.3">𝑉</ci></apply></apply><apply id="S2.Thmtheorem3.p11.19.19.m19.2.2.1.1.cmml" xref="S2.Thmtheorem3.p11.19.19.m19.2.2.1.1"><times id="S2.Thmtheorem3.p11.19.19.m19.2.2.1.1.2.cmml" xref="S2.Thmtheorem3.p11.19.19.m19.2.2.1.1.2"></times><ci id="S2.Thmtheorem3.p11.19.19.m19.2.2.1.1.3a.cmml" xref="S2.Thmtheorem3.p11.19.19.m19.2.2.1.1.3"><mtext class="ltx_mathvariant_italic" id="S2.Thmtheorem3.p11.19.19.m19.2.2.1.1.3.cmml" xref="S2.Thmtheorem3.p11.19.19.m19.2.2.1.1.3">g</mtext></ci><apply id="S2.Thmtheorem3.p11.19.19.m19.2.2.1.1.1.1.1.cmml" xref="S2.Thmtheorem3.p11.19.19.m19.2.2.1.1.1.1"><ci id="S2.Thmtheorem3.p11.19.19.m19.2.2.1.1.1.1.1.1.cmml" xref="S2.Thmtheorem3.p11.19.19.m19.2.2.1.1.1.1.1.1">→</ci><ci id="S2.Thmtheorem3.p11.19.19.m19.2.2.1.1.1.1.1.2.cmml" xref="S2.Thmtheorem3.p11.19.19.m19.2.2.1.1.1.1.1.2">𝑢</ci><ci id="S2.Thmtheorem3.p11.19.19.m19.2.2.1.1.1.1.1.3.cmml" xref="S2.Thmtheorem3.p11.19.19.m19.2.2.1.1.1.1.1.3">𝑣</ci></apply></apply></apply></apply><apply id="S2.Thmtheorem3.p11.19.19.m19.2.2c.cmml" xref="S2.Thmtheorem3.p11.19.19.m19.2.2"><leq id="S2.Thmtheorem3.p11.19.19.m19.2.2.5.cmml" xref="S2.Thmtheorem3.p11.19.19.m19.2.2.5"></leq><share href="https://arxiv.org/html/2411.12694v2#S2.Thmtheorem3.p11.19.19.m19.2.2.1.cmml" id="S2.Thmtheorem3.p11.19.19.m19.2.2d.cmml" xref="S2.Thmtheorem3.p11.19.19.m19.2.2"></share><ci id="S2.Thmtheorem3.p11.19.19.m19.2.2.6.cmml" xref="S2.Thmtheorem3.p11.19.19.m19.2.2.6">𝑅</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem3.p11.19.19.m19.2c">\textsl{g}(u)=\sum_{v\in V}\textsl{g}(u\!\to\!v)\leq R</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem3.p11.19.19.m19.2d">g ( italic_u ) = ∑ start_POSTSUBSCRIPT italic_v ∈ italic_V end_POSTSUBSCRIPT g ( italic_u → italic_v ) ≤ italic_R</annotation></semantics></math>. Thus <math alttext="\Delta^{\min}(G)\leq R" class="ltx_Math" display="inline" id="S2.Thmtheorem3.p11.20.20.m20.1"><semantics id="S2.Thmtheorem3.p11.20.20.m20.1a"><mrow id="S2.Thmtheorem3.p11.20.20.m20.1.2" xref="S2.Thmtheorem3.p11.20.20.m20.1.2.cmml"><mrow id="S2.Thmtheorem3.p11.20.20.m20.1.2.2" xref="S2.Thmtheorem3.p11.20.20.m20.1.2.2.cmml"><msup id="S2.Thmtheorem3.p11.20.20.m20.1.2.2.2" xref="S2.Thmtheorem3.p11.20.20.m20.1.2.2.2.cmml"><mi id="S2.Thmtheorem3.p11.20.20.m20.1.2.2.2.2" mathvariant="normal" xref="S2.Thmtheorem3.p11.20.20.m20.1.2.2.2.2.cmml">Δ</mi><mi id="S2.Thmtheorem3.p11.20.20.m20.1.2.2.2.3" xref="S2.Thmtheorem3.p11.20.20.m20.1.2.2.2.3.cmml">min</mi></msup><mo id="S2.Thmtheorem3.p11.20.20.m20.1.2.2.1" xref="S2.Thmtheorem3.p11.20.20.m20.1.2.2.1.cmml">⁢</mo><mrow id="S2.Thmtheorem3.p11.20.20.m20.1.2.2.3.2" xref="S2.Thmtheorem3.p11.20.20.m20.1.2.2.cmml"><mo id="S2.Thmtheorem3.p11.20.20.m20.1.2.2.3.2.1" stretchy="false" xref="S2.Thmtheorem3.p11.20.20.m20.1.2.2.cmml">(</mo><mi id="S2.Thmtheorem3.p11.20.20.m20.1.1" xref="S2.Thmtheorem3.p11.20.20.m20.1.1.cmml">G</mi><mo id="S2.Thmtheorem3.p11.20.20.m20.1.2.2.3.2.2" stretchy="false" xref="S2.Thmtheorem3.p11.20.20.m20.1.2.2.cmml">)</mo></mrow></mrow><mo id="S2.Thmtheorem3.p11.20.20.m20.1.2.1" xref="S2.Thmtheorem3.p11.20.20.m20.1.2.1.cmml">≤</mo><mi id="S2.Thmtheorem3.p11.20.20.m20.1.2.3" xref="S2.Thmtheorem3.p11.20.20.m20.1.2.3.cmml">R</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem3.p11.20.20.m20.1b"><apply id="S2.Thmtheorem3.p11.20.20.m20.1.2.cmml" xref="S2.Thmtheorem3.p11.20.20.m20.1.2"><leq id="S2.Thmtheorem3.p11.20.20.m20.1.2.1.cmml" xref="S2.Thmtheorem3.p11.20.20.m20.1.2.1"></leq><apply id="S2.Thmtheorem3.p11.20.20.m20.1.2.2.cmml" xref="S2.Thmtheorem3.p11.20.20.m20.1.2.2"><times id="S2.Thmtheorem3.p11.20.20.m20.1.2.2.1.cmml" xref="S2.Thmtheorem3.p11.20.20.m20.1.2.2.1"></times><apply id="S2.Thmtheorem3.p11.20.20.m20.1.2.2.2.cmml" xref="S2.Thmtheorem3.p11.20.20.m20.1.2.2.2"><csymbol cd="ambiguous" id="S2.Thmtheorem3.p11.20.20.m20.1.2.2.2.1.cmml" xref="S2.Thmtheorem3.p11.20.20.m20.1.2.2.2">superscript</csymbol><ci id="S2.Thmtheorem3.p11.20.20.m20.1.2.2.2.2.cmml" xref="S2.Thmtheorem3.p11.20.20.m20.1.2.2.2.2">Δ</ci><min id="S2.Thmtheorem3.p11.20.20.m20.1.2.2.2.3.cmml" xref="S2.Thmtheorem3.p11.20.20.m20.1.2.2.2.3"></min></apply><ci id="S2.Thmtheorem3.p11.20.20.m20.1.1.cmml" xref="S2.Thmtheorem3.p11.20.20.m20.1.1">𝐺</ci></apply><ci id="S2.Thmtheorem3.p11.20.20.m20.1.2.3.cmml" xref="S2.Thmtheorem3.p11.20.20.m20.1.2.3">𝑅</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem3.p11.20.20.m20.1c">\Delta^{\min}(G)\leq R</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem3.p11.20.20.m20.1d">roman_Δ start_POSTSUPERSCRIPT roman_min end_POSTSUPERSCRIPT ( italic_G ) ≤ italic_R</annotation></semantics></math> — a contradiction.</span></p> </div> </div> </section> <section class="ltx_subsection" id="S2.SS1"> <h3 class="ltx_title ltx_title_subsection"> <span class="ltx_tag ltx_tag_subsection">2.1 </span>Densest subgraph in dynamic algorithms</h3> <div class="ltx_para" id="S2.SS1.p1"> <p class="ltx_p" id="S2.SS1.p1.1">In a classical, non-distributed model of computation we can immediately formalise both the value variant of the (approximate) densest subgraph:</p> </div> <div class="ltx_theorem ltx_theorem_problem" 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">Problem 2.4</span></span><span class="ltx_text ltx_font_bold" id="S2.Thmtheorem4.2.2">.</span> </h6> <div class="ltx_para" id="S2.Thmtheorem4.p1"> <p class="ltx_p" id="S2.Thmtheorem4.p1.3"><span class="ltx_text ltx_font_italic" id="S2.Thmtheorem4.p1.3.3">Given a graph <math alttext="G" class="ltx_Math" display="inline" id="S2.Thmtheorem4.p1.1.1.m1.1"><semantics id="S2.Thmtheorem4.p1.1.1.m1.1a"><mi id="S2.Thmtheorem4.p1.1.1.m1.1.1" xref="S2.Thmtheorem4.p1.1.1.m1.1.1.cmml">G</mi><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem4.p1.1.1.m1.1b"><ci id="S2.Thmtheorem4.p1.1.1.m1.1.1.cmml" xref="S2.Thmtheorem4.p1.1.1.m1.1.1">𝐺</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem4.p1.1.1.m1.1c">G</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem4.p1.1.1.m1.1d">italic_G</annotation></semantics></math> and an <math alttext="\varepsilon>0" class="ltx_Math" display="inline" id="S2.Thmtheorem4.p1.2.2.m2.1"><semantics id="S2.Thmtheorem4.p1.2.2.m2.1a"><mrow id="S2.Thmtheorem4.p1.2.2.m2.1.1" xref="S2.Thmtheorem4.p1.2.2.m2.1.1.cmml"><mi id="S2.Thmtheorem4.p1.2.2.m2.1.1.2" xref="S2.Thmtheorem4.p1.2.2.m2.1.1.2.cmml">ε</mi><mo id="S2.Thmtheorem4.p1.2.2.m2.1.1.1" xref="S2.Thmtheorem4.p1.2.2.m2.1.1.1.cmml">></mo><mn id="S2.Thmtheorem4.p1.2.2.m2.1.1.3" xref="S2.Thmtheorem4.p1.2.2.m2.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem4.p1.2.2.m2.1b"><apply id="S2.Thmtheorem4.p1.2.2.m2.1.1.cmml" xref="S2.Thmtheorem4.p1.2.2.m2.1.1"><gt id="S2.Thmtheorem4.p1.2.2.m2.1.1.1.cmml" xref="S2.Thmtheorem4.p1.2.2.m2.1.1.1"></gt><ci id="S2.Thmtheorem4.p1.2.2.m2.1.1.2.cmml" xref="S2.Thmtheorem4.p1.2.2.m2.1.1.2">𝜀</ci><cn id="S2.Thmtheorem4.p1.2.2.m2.1.1.3.cmml" type="integer" xref="S2.Thmtheorem4.p1.2.2.m2.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem4.p1.2.2.m2.1c">\varepsilon>0</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem4.p1.2.2.m2.1d">italic_ε > 0</annotation></semantics></math>, output <math alttext="\rho^{\prime}\in[(1+\varepsilon)^{-1}\rho^{\max}(G),(1+\varepsilon)\rho^{\max}% (G)]" class="ltx_Math" display="inline" id="S2.Thmtheorem4.p1.3.3.m3.4"><semantics id="S2.Thmtheorem4.p1.3.3.m3.4a"><mrow id="S2.Thmtheorem4.p1.3.3.m3.4.4" xref="S2.Thmtheorem4.p1.3.3.m3.4.4.cmml"><msup id="S2.Thmtheorem4.p1.3.3.m3.4.4.4" xref="S2.Thmtheorem4.p1.3.3.m3.4.4.4.cmml"><mi id="S2.Thmtheorem4.p1.3.3.m3.4.4.4.2" xref="S2.Thmtheorem4.p1.3.3.m3.4.4.4.2.cmml">ρ</mi><mo id="S2.Thmtheorem4.p1.3.3.m3.4.4.4.3" xref="S2.Thmtheorem4.p1.3.3.m3.4.4.4.3.cmml">′</mo></msup><mo id="S2.Thmtheorem4.p1.3.3.m3.4.4.3" xref="S2.Thmtheorem4.p1.3.3.m3.4.4.3.cmml">∈</mo><mrow id="S2.Thmtheorem4.p1.3.3.m3.4.4.2.2" xref="S2.Thmtheorem4.p1.3.3.m3.4.4.2.3.cmml"><mo id="S2.Thmtheorem4.p1.3.3.m3.4.4.2.2.3" stretchy="false" xref="S2.Thmtheorem4.p1.3.3.m3.4.4.2.3.cmml">[</mo><mrow id="S2.Thmtheorem4.p1.3.3.m3.3.3.1.1.1" xref="S2.Thmtheorem4.p1.3.3.m3.3.3.1.1.1.cmml"><msup id="S2.Thmtheorem4.p1.3.3.m3.3.3.1.1.1.1" xref="S2.Thmtheorem4.p1.3.3.m3.3.3.1.1.1.1.cmml"><mrow id="S2.Thmtheorem4.p1.3.3.m3.3.3.1.1.1.1.1.1" xref="S2.Thmtheorem4.p1.3.3.m3.3.3.1.1.1.1.1.1.1.cmml"><mo id="S2.Thmtheorem4.p1.3.3.m3.3.3.1.1.1.1.1.1.2" stretchy="false" xref="S2.Thmtheorem4.p1.3.3.m3.3.3.1.1.1.1.1.1.1.cmml">(</mo><mrow id="S2.Thmtheorem4.p1.3.3.m3.3.3.1.1.1.1.1.1.1" xref="S2.Thmtheorem4.p1.3.3.m3.3.3.1.1.1.1.1.1.1.cmml"><mn id="S2.Thmtheorem4.p1.3.3.m3.3.3.1.1.1.1.1.1.1.2" xref="S2.Thmtheorem4.p1.3.3.m3.3.3.1.1.1.1.1.1.1.2.cmml">1</mn><mo id="S2.Thmtheorem4.p1.3.3.m3.3.3.1.1.1.1.1.1.1.1" xref="S2.Thmtheorem4.p1.3.3.m3.3.3.1.1.1.1.1.1.1.1.cmml">+</mo><mi id="S2.Thmtheorem4.p1.3.3.m3.3.3.1.1.1.1.1.1.1.3" xref="S2.Thmtheorem4.p1.3.3.m3.3.3.1.1.1.1.1.1.1.3.cmml">ε</mi></mrow><mo id="S2.Thmtheorem4.p1.3.3.m3.3.3.1.1.1.1.1.1.3" stretchy="false" xref="S2.Thmtheorem4.p1.3.3.m3.3.3.1.1.1.1.1.1.1.cmml">)</mo></mrow><mrow id="S2.Thmtheorem4.p1.3.3.m3.3.3.1.1.1.1.3" xref="S2.Thmtheorem4.p1.3.3.m3.3.3.1.1.1.1.3.cmml"><mo id="S2.Thmtheorem4.p1.3.3.m3.3.3.1.1.1.1.3a" xref="S2.Thmtheorem4.p1.3.3.m3.3.3.1.1.1.1.3.cmml">−</mo><mn id="S2.Thmtheorem4.p1.3.3.m3.3.3.1.1.1.1.3.2" xref="S2.Thmtheorem4.p1.3.3.m3.3.3.1.1.1.1.3.2.cmml">1</mn></mrow></msup><mo id="S2.Thmtheorem4.p1.3.3.m3.3.3.1.1.1.2" xref="S2.Thmtheorem4.p1.3.3.m3.3.3.1.1.1.2.cmml">⁢</mo><msup id="S2.Thmtheorem4.p1.3.3.m3.3.3.1.1.1.3" xref="S2.Thmtheorem4.p1.3.3.m3.3.3.1.1.1.3.cmml"><mi id="S2.Thmtheorem4.p1.3.3.m3.3.3.1.1.1.3.2" xref="S2.Thmtheorem4.p1.3.3.m3.3.3.1.1.1.3.2.cmml">ρ</mi><mi id="S2.Thmtheorem4.p1.3.3.m3.3.3.1.1.1.3.3" xref="S2.Thmtheorem4.p1.3.3.m3.3.3.1.1.1.3.3.cmml">max</mi></msup><mo id="S2.Thmtheorem4.p1.3.3.m3.3.3.1.1.1.2a" xref="S2.Thmtheorem4.p1.3.3.m3.3.3.1.1.1.2.cmml">⁢</mo><mrow id="S2.Thmtheorem4.p1.3.3.m3.3.3.1.1.1.4.2" xref="S2.Thmtheorem4.p1.3.3.m3.3.3.1.1.1.cmml"><mo id="S2.Thmtheorem4.p1.3.3.m3.3.3.1.1.1.4.2.1" stretchy="false" xref="S2.Thmtheorem4.p1.3.3.m3.3.3.1.1.1.cmml">(</mo><mi id="S2.Thmtheorem4.p1.3.3.m3.1.1" xref="S2.Thmtheorem4.p1.3.3.m3.1.1.cmml">G</mi><mo id="S2.Thmtheorem4.p1.3.3.m3.3.3.1.1.1.4.2.2" stretchy="false" xref="S2.Thmtheorem4.p1.3.3.m3.3.3.1.1.1.cmml">)</mo></mrow></mrow><mo id="S2.Thmtheorem4.p1.3.3.m3.4.4.2.2.4" xref="S2.Thmtheorem4.p1.3.3.m3.4.4.2.3.cmml">,</mo><mrow id="S2.Thmtheorem4.p1.3.3.m3.4.4.2.2.2" xref="S2.Thmtheorem4.p1.3.3.m3.4.4.2.2.2.cmml"><mrow id="S2.Thmtheorem4.p1.3.3.m3.4.4.2.2.2.1.1" xref="S2.Thmtheorem4.p1.3.3.m3.4.4.2.2.2.1.1.1.cmml"><mo id="S2.Thmtheorem4.p1.3.3.m3.4.4.2.2.2.1.1.2" stretchy="false" xref="S2.Thmtheorem4.p1.3.3.m3.4.4.2.2.2.1.1.1.cmml">(</mo><mrow id="S2.Thmtheorem4.p1.3.3.m3.4.4.2.2.2.1.1.1" xref="S2.Thmtheorem4.p1.3.3.m3.4.4.2.2.2.1.1.1.cmml"><mn id="S2.Thmtheorem4.p1.3.3.m3.4.4.2.2.2.1.1.1.2" xref="S2.Thmtheorem4.p1.3.3.m3.4.4.2.2.2.1.1.1.2.cmml">1</mn><mo id="S2.Thmtheorem4.p1.3.3.m3.4.4.2.2.2.1.1.1.1" xref="S2.Thmtheorem4.p1.3.3.m3.4.4.2.2.2.1.1.1.1.cmml">+</mo><mi id="S2.Thmtheorem4.p1.3.3.m3.4.4.2.2.2.1.1.1.3" xref="S2.Thmtheorem4.p1.3.3.m3.4.4.2.2.2.1.1.1.3.cmml">ε</mi></mrow><mo id="S2.Thmtheorem4.p1.3.3.m3.4.4.2.2.2.1.1.3" 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xref="S2.Thmtheorem4.p1.3.3.m3.4.4.2.2.2.cmml">)</mo></mrow></mrow><mo id="S2.Thmtheorem4.p1.3.3.m3.4.4.2.2.5" stretchy="false" xref="S2.Thmtheorem4.p1.3.3.m3.4.4.2.3.cmml">]</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem4.p1.3.3.m3.4b"><apply id="S2.Thmtheorem4.p1.3.3.m3.4.4.cmml" xref="S2.Thmtheorem4.p1.3.3.m3.4.4"><in id="S2.Thmtheorem4.p1.3.3.m3.4.4.3.cmml" xref="S2.Thmtheorem4.p1.3.3.m3.4.4.3"></in><apply id="S2.Thmtheorem4.p1.3.3.m3.4.4.4.cmml" xref="S2.Thmtheorem4.p1.3.3.m3.4.4.4"><csymbol cd="ambiguous" id="S2.Thmtheorem4.p1.3.3.m3.4.4.4.1.cmml" xref="S2.Thmtheorem4.p1.3.3.m3.4.4.4">superscript</csymbol><ci id="S2.Thmtheorem4.p1.3.3.m3.4.4.4.2.cmml" xref="S2.Thmtheorem4.p1.3.3.m3.4.4.4.2">𝜌</ci><ci id="S2.Thmtheorem4.p1.3.3.m3.4.4.4.3.cmml" xref="S2.Thmtheorem4.p1.3.3.m3.4.4.4.3">′</ci></apply><interval closure="closed" id="S2.Thmtheorem4.p1.3.3.m3.4.4.2.3.cmml" xref="S2.Thmtheorem4.p1.3.3.m3.4.4.2.2"><apply id="S2.Thmtheorem4.p1.3.3.m3.3.3.1.1.1.cmml" xref="S2.Thmtheorem4.p1.3.3.m3.3.3.1.1.1"><times id="S2.Thmtheorem4.p1.3.3.m3.3.3.1.1.1.2.cmml" xref="S2.Thmtheorem4.p1.3.3.m3.3.3.1.1.1.2"></times><apply id="S2.Thmtheorem4.p1.3.3.m3.3.3.1.1.1.1.cmml" xref="S2.Thmtheorem4.p1.3.3.m3.3.3.1.1.1.1"><csymbol cd="ambiguous" id="S2.Thmtheorem4.p1.3.3.m3.3.3.1.1.1.1.2.cmml" xref="S2.Thmtheorem4.p1.3.3.m3.3.3.1.1.1.1">superscript</csymbol><apply id="S2.Thmtheorem4.p1.3.3.m3.3.3.1.1.1.1.1.1.1.cmml" xref="S2.Thmtheorem4.p1.3.3.m3.3.3.1.1.1.1.1.1"><plus id="S2.Thmtheorem4.p1.3.3.m3.3.3.1.1.1.1.1.1.1.1.cmml" xref="S2.Thmtheorem4.p1.3.3.m3.3.3.1.1.1.1.1.1.1.1"></plus><cn id="S2.Thmtheorem4.p1.3.3.m3.3.3.1.1.1.1.1.1.1.2.cmml" type="integer" xref="S2.Thmtheorem4.p1.3.3.m3.3.3.1.1.1.1.1.1.1.2">1</cn><ci id="S2.Thmtheorem4.p1.3.3.m3.3.3.1.1.1.1.1.1.1.3.cmml" xref="S2.Thmtheorem4.p1.3.3.m3.3.3.1.1.1.1.1.1.1.3">𝜀</ci></apply><apply id="S2.Thmtheorem4.p1.3.3.m3.3.3.1.1.1.1.3.cmml" xref="S2.Thmtheorem4.p1.3.3.m3.3.3.1.1.1.1.3"><minus id="S2.Thmtheorem4.p1.3.3.m3.3.3.1.1.1.1.3.1.cmml" xref="S2.Thmtheorem4.p1.3.3.m3.3.3.1.1.1.1.3"></minus><cn id="S2.Thmtheorem4.p1.3.3.m3.3.3.1.1.1.1.3.2.cmml" type="integer" xref="S2.Thmtheorem4.p1.3.3.m3.3.3.1.1.1.1.3.2">1</cn></apply></apply><apply id="S2.Thmtheorem4.p1.3.3.m3.3.3.1.1.1.3.cmml" xref="S2.Thmtheorem4.p1.3.3.m3.3.3.1.1.1.3"><csymbol cd="ambiguous" id="S2.Thmtheorem4.p1.3.3.m3.3.3.1.1.1.3.1.cmml" xref="S2.Thmtheorem4.p1.3.3.m3.3.3.1.1.1.3">superscript</csymbol><ci id="S2.Thmtheorem4.p1.3.3.m3.3.3.1.1.1.3.2.cmml" xref="S2.Thmtheorem4.p1.3.3.m3.3.3.1.1.1.3.2">𝜌</ci><max id="S2.Thmtheorem4.p1.3.3.m3.3.3.1.1.1.3.3.cmml" xref="S2.Thmtheorem4.p1.3.3.m3.3.3.1.1.1.3.3"></max></apply><ci id="S2.Thmtheorem4.p1.3.3.m3.1.1.cmml" xref="S2.Thmtheorem4.p1.3.3.m3.1.1">𝐺</ci></apply><apply id="S2.Thmtheorem4.p1.3.3.m3.4.4.2.2.2.cmml" xref="S2.Thmtheorem4.p1.3.3.m3.4.4.2.2.2"><times id="S2.Thmtheorem4.p1.3.3.m3.4.4.2.2.2.2.cmml" xref="S2.Thmtheorem4.p1.3.3.m3.4.4.2.2.2.2"></times><apply id="S2.Thmtheorem4.p1.3.3.m3.4.4.2.2.2.1.1.1.cmml" xref="S2.Thmtheorem4.p1.3.3.m3.4.4.2.2.2.1.1"><plus id="S2.Thmtheorem4.p1.3.3.m3.4.4.2.2.2.1.1.1.1.cmml" xref="S2.Thmtheorem4.p1.3.3.m3.4.4.2.2.2.1.1.1.1"></plus><cn id="S2.Thmtheorem4.p1.3.3.m3.4.4.2.2.2.1.1.1.2.cmml" type="integer" xref="S2.Thmtheorem4.p1.3.3.m3.4.4.2.2.2.1.1.1.2">1</cn><ci id="S2.Thmtheorem4.p1.3.3.m3.4.4.2.2.2.1.1.1.3.cmml" xref="S2.Thmtheorem4.p1.3.3.m3.4.4.2.2.2.1.1.1.3">𝜀</ci></apply><apply id="S2.Thmtheorem4.p1.3.3.m3.4.4.2.2.2.3.cmml" xref="S2.Thmtheorem4.p1.3.3.m3.4.4.2.2.2.3"><csymbol cd="ambiguous" id="S2.Thmtheorem4.p1.3.3.m3.4.4.2.2.2.3.1.cmml" xref="S2.Thmtheorem4.p1.3.3.m3.4.4.2.2.2.3">superscript</csymbol><ci id="S2.Thmtheorem4.p1.3.3.m3.4.4.2.2.2.3.2.cmml" xref="S2.Thmtheorem4.p1.3.3.m3.4.4.2.2.2.3.2">𝜌</ci><max id="S2.Thmtheorem4.p1.3.3.m3.4.4.2.2.2.3.3.cmml" xref="S2.Thmtheorem4.p1.3.3.m3.4.4.2.2.2.3.3"></max></apply><ci id="S2.Thmtheorem4.p1.3.3.m3.2.2.cmml" xref="S2.Thmtheorem4.p1.3.3.m3.2.2">𝐺</ci></apply></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem4.p1.3.3.m3.4c">\rho^{\prime}\in[(1+\varepsilon)^{-1}\rho^{\max}(G),(1+\varepsilon)\rho^{\max}% (G)]</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem4.p1.3.3.m3.4d">italic_ρ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ∈ [ ( 1 + italic_ε ) start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT italic_ρ start_POSTSUPERSCRIPT roman_max end_POSTSUPERSCRIPT ( italic_G ) , ( 1 + italic_ε ) italic_ρ start_POSTSUPERSCRIPT roman_max end_POSTSUPERSCRIPT ( italic_G ) ]</annotation></semantics></math>.</span></p> </div> </div> <div class="ltx_para" id="S2.SS1.p2"> <p class="ltx_p" id="S2.SS1.p2.4">Alternatively, in the Fractional Orientation (FO) problem the goal is to output a <math alttext="(1+\varepsilon)" class="ltx_Math" display="inline" id="S2.SS1.p2.1.m1.1"><semantics id="S2.SS1.p2.1.m1.1a"><mrow id="S2.SS1.p2.1.m1.1.1.1" xref="S2.SS1.p2.1.m1.1.1.1.1.cmml"><mo id="S2.SS1.p2.1.m1.1.1.1.2" stretchy="false" xref="S2.SS1.p2.1.m1.1.1.1.1.cmml">(</mo><mrow id="S2.SS1.p2.1.m1.1.1.1.1" xref="S2.SS1.p2.1.m1.1.1.1.1.cmml"><mn id="S2.SS1.p2.1.m1.1.1.1.1.2" xref="S2.SS1.p2.1.m1.1.1.1.1.2.cmml">1</mn><mo id="S2.SS1.p2.1.m1.1.1.1.1.1" xref="S2.SS1.p2.1.m1.1.1.1.1.1.cmml">+</mo><mi id="S2.SS1.p2.1.m1.1.1.1.1.3" xref="S2.SS1.p2.1.m1.1.1.1.1.3.cmml">ε</mi></mrow><mo id="S2.SS1.p2.1.m1.1.1.1.3" stretchy="false" xref="S2.SS1.p2.1.m1.1.1.1.1.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.p2.1.m1.1b"><apply id="S2.SS1.p2.1.m1.1.1.1.1.cmml" xref="S2.SS1.p2.1.m1.1.1.1"><plus id="S2.SS1.p2.1.m1.1.1.1.1.1.cmml" xref="S2.SS1.p2.1.m1.1.1.1.1.1"></plus><cn id="S2.SS1.p2.1.m1.1.1.1.1.2.cmml" type="integer" xref="S2.SS1.p2.1.m1.1.1.1.1.2">1</cn><ci id="S2.SS1.p2.1.m1.1.1.1.1.3.cmml" xref="S2.SS1.p2.1.m1.1.1.1.1.3">𝜀</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p2.1.m1.1c">(1+\varepsilon)</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p2.1.m1.1d">( 1 + italic_ε )</annotation></semantics></math>-approximation of <math alttext="\Delta^{\min}(G)" class="ltx_Math" display="inline" id="S2.SS1.p2.2.m2.1"><semantics id="S2.SS1.p2.2.m2.1a"><mrow id="S2.SS1.p2.2.m2.1.2" xref="S2.SS1.p2.2.m2.1.2.cmml"><msup id="S2.SS1.p2.2.m2.1.2.2" xref="S2.SS1.p2.2.m2.1.2.2.cmml"><mi id="S2.SS1.p2.2.m2.1.2.2.2" mathvariant="normal" xref="S2.SS1.p2.2.m2.1.2.2.2.cmml">Δ</mi><mi id="S2.SS1.p2.2.m2.1.2.2.3" xref="S2.SS1.p2.2.m2.1.2.2.3.cmml">min</mi></msup><mo id="S2.SS1.p2.2.m2.1.2.1" xref="S2.SS1.p2.2.m2.1.2.1.cmml">⁢</mo><mrow id="S2.SS1.p2.2.m2.1.2.3.2" xref="S2.SS1.p2.2.m2.1.2.cmml"><mo id="S2.SS1.p2.2.m2.1.2.3.2.1" stretchy="false" xref="S2.SS1.p2.2.m2.1.2.cmml">(</mo><mi id="S2.SS1.p2.2.m2.1.1" xref="S2.SS1.p2.2.m2.1.1.cmml">G</mi><mo id="S2.SS1.p2.2.m2.1.2.3.2.2" stretchy="false" xref="S2.SS1.p2.2.m2.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.p2.2.m2.1b"><apply id="S2.SS1.p2.2.m2.1.2.cmml" xref="S2.SS1.p2.2.m2.1.2"><times id="S2.SS1.p2.2.m2.1.2.1.cmml" xref="S2.SS1.p2.2.m2.1.2.1"></times><apply id="S2.SS1.p2.2.m2.1.2.2.cmml" xref="S2.SS1.p2.2.m2.1.2.2"><csymbol cd="ambiguous" id="S2.SS1.p2.2.m2.1.2.2.1.cmml" xref="S2.SS1.p2.2.m2.1.2.2">superscript</csymbol><ci id="S2.SS1.p2.2.m2.1.2.2.2.cmml" xref="S2.SS1.p2.2.m2.1.2.2.2">Δ</ci><min id="S2.SS1.p2.2.m2.1.2.2.3.cmml" xref="S2.SS1.p2.2.m2.1.2.2.3"></min></apply><ci id="S2.SS1.p2.2.m2.1.1.cmml" xref="S2.SS1.p2.2.m2.1.1">𝐺</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p2.2.m2.1c">\Delta^{\min}(G)</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p2.2.m2.1d">roman_Δ start_POSTSUPERSCRIPT roman_min end_POSTSUPERSCRIPT ( italic_G )</annotation></semantics></math>. It turns out that FO is a more accessible problem to study. The LP formulations allow for a straightforward way to compute <math alttext="\Delta^{\min}(G)" class="ltx_Math" display="inline" id="S2.SS1.p2.3.m3.1"><semantics id="S2.SS1.p2.3.m3.1a"><mrow id="S2.SS1.p2.3.m3.1.2" xref="S2.SS1.p2.3.m3.1.2.cmml"><msup id="S2.SS1.p2.3.m3.1.2.2" xref="S2.SS1.p2.3.m3.1.2.2.cmml"><mi id="S2.SS1.p2.3.m3.1.2.2.2" mathvariant="normal" xref="S2.SS1.p2.3.m3.1.2.2.2.cmml">Δ</mi><mi id="S2.SS1.p2.3.m3.1.2.2.3" xref="S2.SS1.p2.3.m3.1.2.2.3.cmml">min</mi></msup><mo id="S2.SS1.p2.3.m3.1.2.1" xref="S2.SS1.p2.3.m3.1.2.1.cmml">⁢</mo><mrow id="S2.SS1.p2.3.m3.1.2.3.2" xref="S2.SS1.p2.3.m3.1.2.cmml"><mo id="S2.SS1.p2.3.m3.1.2.3.2.1" stretchy="false" xref="S2.SS1.p2.3.m3.1.2.cmml">(</mo><mi id="S2.SS1.p2.3.m3.1.1" xref="S2.SS1.p2.3.m3.1.1.cmml">G</mi><mo id="S2.SS1.p2.3.m3.1.2.3.2.2" stretchy="false" xref="S2.SS1.p2.3.m3.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.p2.3.m3.1b"><apply id="S2.SS1.p2.3.m3.1.2.cmml" xref="S2.SS1.p2.3.m3.1.2"><times id="S2.SS1.p2.3.m3.1.2.1.cmml" xref="S2.SS1.p2.3.m3.1.2.1"></times><apply id="S2.SS1.p2.3.m3.1.2.2.cmml" xref="S2.SS1.p2.3.m3.1.2.2"><csymbol cd="ambiguous" id="S2.SS1.p2.3.m3.1.2.2.1.cmml" xref="S2.SS1.p2.3.m3.1.2.2">superscript</csymbol><ci id="S2.SS1.p2.3.m3.1.2.2.2.cmml" xref="S2.SS1.p2.3.m3.1.2.2.2">Δ</ci><min id="S2.SS1.p2.3.m3.1.2.2.3.cmml" xref="S2.SS1.p2.3.m3.1.2.2.3"></min></apply><ci id="S2.SS1.p2.3.m3.1.1.cmml" xref="S2.SS1.p2.3.m3.1.1">𝐺</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p2.3.m3.1c">\Delta^{\min}(G)</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p2.3.m3.1d">roman_Δ start_POSTSUPERSCRIPT roman_min end_POSTSUPERSCRIPT ( italic_G )</annotation></semantics></math> and/or <math alttext="\rho^{\max}(G)" class="ltx_Math" display="inline" id="S2.SS1.p2.4.m4.1"><semantics id="S2.SS1.p2.4.m4.1a"><mrow id="S2.SS1.p2.4.m4.1.2" xref="S2.SS1.p2.4.m4.1.2.cmml"><msup id="S2.SS1.p2.4.m4.1.2.2" xref="S2.SS1.p2.4.m4.1.2.2.cmml"><mi id="S2.SS1.p2.4.m4.1.2.2.2" xref="S2.SS1.p2.4.m4.1.2.2.2.cmml">ρ</mi><mi id="S2.SS1.p2.4.m4.1.2.2.3" xref="S2.SS1.p2.4.m4.1.2.2.3.cmml">max</mi></msup><mo id="S2.SS1.p2.4.m4.1.2.1" xref="S2.SS1.p2.4.m4.1.2.1.cmml">⁢</mo><mrow id="S2.SS1.p2.4.m4.1.2.3.2" xref="S2.SS1.p2.4.m4.1.2.cmml"><mo id="S2.SS1.p2.4.m4.1.2.3.2.1" stretchy="false" xref="S2.SS1.p2.4.m4.1.2.cmml">(</mo><mi id="S2.SS1.p2.4.m4.1.1" xref="S2.SS1.p2.4.m4.1.1.cmml">G</mi><mo id="S2.SS1.p2.4.m4.1.2.3.2.2" stretchy="false" xref="S2.SS1.p2.4.m4.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.p2.4.m4.1b"><apply id="S2.SS1.p2.4.m4.1.2.cmml" xref="S2.SS1.p2.4.m4.1.2"><times id="S2.SS1.p2.4.m4.1.2.1.cmml" xref="S2.SS1.p2.4.m4.1.2.1"></times><apply id="S2.SS1.p2.4.m4.1.2.2.cmml" xref="S2.SS1.p2.4.m4.1.2.2"><csymbol cd="ambiguous" id="S2.SS1.p2.4.m4.1.2.2.1.cmml" xref="S2.SS1.p2.4.m4.1.2.2">superscript</csymbol><ci id="S2.SS1.p2.4.m4.1.2.2.2.cmml" xref="S2.SS1.p2.4.m4.1.2.2.2">𝜌</ci><max id="S2.SS1.p2.4.m4.1.2.2.3.cmml" xref="S2.SS1.p2.4.m4.1.2.2.3"></max></apply><ci id="S2.SS1.p2.4.m4.1.1.cmml" xref="S2.SS1.p2.4.m4.1.1">𝐺</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p2.4.m4.1c">\rho^{\max}(G)</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p2.4.m4.1d">italic_ρ start_POSTSUPERSCRIPT roman_max end_POSTSUPERSCRIPT ( italic_G )</annotation></semantics></math>. However, solving the LP requires information about the entire graph, and this information is expensive to collect. Sawlani and Wang <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2411.12694v2#bib.bib20" title="">20</a>]</cite> get around this difficulty by instead solving an approximate version of (FO). They work with a concept we call <em class="ltx_emph ltx_font_italic" id="S2.SS1.p2.4.1">local fairness</em>.</p> </div> <div class="ltx_theorem ltx_theorem_definition" 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">Definition 2.5</span></span><span class="ltx_text ltx_font_bold" id="S2.Thmtheorem5.2.2">.</span> </h6> <div class="ltx_para" id="S2.Thmtheorem5.p1"> <p class="ltx_p" id="S2.Thmtheorem5.p1.5"><span class="ltx_text ltx_font_italic" id="S2.Thmtheorem5.p1.5.5">Let <math alttext="\overrightarrow{G}" class="ltx_Math" display="inline" id="S2.Thmtheorem5.p1.1.1.m1.1"><semantics id="S2.Thmtheorem5.p1.1.1.m1.1a"><mover accent="true" id="S2.Thmtheorem5.p1.1.1.m1.1.1" xref="S2.Thmtheorem5.p1.1.1.m1.1.1.cmml"><mi id="S2.Thmtheorem5.p1.1.1.m1.1.1.2" xref="S2.Thmtheorem5.p1.1.1.m1.1.1.2.cmml">G</mi><mo id="S2.Thmtheorem5.p1.1.1.m1.1.1.1" stretchy="false" xref="S2.Thmtheorem5.p1.1.1.m1.1.1.1.cmml">→</mo></mover><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem5.p1.1.1.m1.1b"><apply id="S2.Thmtheorem5.p1.1.1.m1.1.1.cmml" xref="S2.Thmtheorem5.p1.1.1.m1.1.1"><ci id="S2.Thmtheorem5.p1.1.1.m1.1.1.1.cmml" xref="S2.Thmtheorem5.p1.1.1.m1.1.1.1">→</ci><ci id="S2.Thmtheorem5.p1.1.1.m1.1.1.2.cmml" xref="S2.Thmtheorem5.p1.1.1.m1.1.1.2">𝐺</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem5.p1.1.1.m1.1c">\overrightarrow{G}</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem5.p1.1.1.m1.1d">over→ start_ARG italic_G end_ARG</annotation></semantics></math> be a fractional orientation of a graph <math alttext="G" class="ltx_Math" display="inline" id="S2.Thmtheorem5.p1.2.2.m2.1"><semantics id="S2.Thmtheorem5.p1.2.2.m2.1a"><mi id="S2.Thmtheorem5.p1.2.2.m2.1.1" xref="S2.Thmtheorem5.p1.2.2.m2.1.1.cmml">G</mi><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem5.p1.2.2.m2.1b"><ci id="S2.Thmtheorem5.p1.2.2.m2.1.1.cmml" xref="S2.Thmtheorem5.p1.2.2.m2.1.1">𝐺</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem5.p1.2.2.m2.1c">G</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem5.p1.2.2.m2.1d">italic_G</annotation></semantics></math>. We say that <math alttext="\overrightarrow{G}" class="ltx_Math" display="inline" id="S2.Thmtheorem5.p1.3.3.m3.1"><semantics id="S2.Thmtheorem5.p1.3.3.m3.1a"><mover accent="true" id="S2.Thmtheorem5.p1.3.3.m3.1.1" xref="S2.Thmtheorem5.p1.3.3.m3.1.1.cmml"><mi id="S2.Thmtheorem5.p1.3.3.m3.1.1.2" xref="S2.Thmtheorem5.p1.3.3.m3.1.1.2.cmml">G</mi><mo id="S2.Thmtheorem5.p1.3.3.m3.1.1.1" stretchy="false" xref="S2.Thmtheorem5.p1.3.3.m3.1.1.1.cmml">→</mo></mover><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem5.p1.3.3.m3.1b"><apply id="S2.Thmtheorem5.p1.3.3.m3.1.1.cmml" xref="S2.Thmtheorem5.p1.3.3.m3.1.1"><ci id="S2.Thmtheorem5.p1.3.3.m3.1.1.1.cmml" xref="S2.Thmtheorem5.p1.3.3.m3.1.1.1">→</ci><ci id="S2.Thmtheorem5.p1.3.3.m3.1.1.2.cmml" xref="S2.Thmtheorem5.p1.3.3.m3.1.1.2">𝐺</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem5.p1.3.3.m3.1c">\overrightarrow{G}</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem5.p1.3.3.m3.1d">over→ start_ARG italic_G end_ARG</annotation></semantics></math> is locally fair whenever <math alttext="\textsl{g}(u\!\to\!v)>0" class="ltx_Math" display="inline" id="S2.Thmtheorem5.p1.4.4.m4.1"><semantics id="S2.Thmtheorem5.p1.4.4.m4.1a"><mrow id="S2.Thmtheorem5.p1.4.4.m4.1.1" xref="S2.Thmtheorem5.p1.4.4.m4.1.1.cmml"><mrow id="S2.Thmtheorem5.p1.4.4.m4.1.1.1" xref="S2.Thmtheorem5.p1.4.4.m4.1.1.1.cmml"><mtext class="ltx_mathvariant_italic" id="S2.Thmtheorem5.p1.4.4.m4.1.1.1.3" xref="S2.Thmtheorem5.p1.4.4.m4.1.1.1.3a.cmml">g</mtext><mo id="S2.Thmtheorem5.p1.4.4.m4.1.1.1.2" xref="S2.Thmtheorem5.p1.4.4.m4.1.1.1.2.cmml">⁢</mo><mrow id="S2.Thmtheorem5.p1.4.4.m4.1.1.1.1.1" xref="S2.Thmtheorem5.p1.4.4.m4.1.1.1.1.1.1.cmml"><mo id="S2.Thmtheorem5.p1.4.4.m4.1.1.1.1.1.2" stretchy="false" xref="S2.Thmtheorem5.p1.4.4.m4.1.1.1.1.1.1.cmml">(</mo><mrow id="S2.Thmtheorem5.p1.4.4.m4.1.1.1.1.1.1" xref="S2.Thmtheorem5.p1.4.4.m4.1.1.1.1.1.1.cmml"><mi id="S2.Thmtheorem5.p1.4.4.m4.1.1.1.1.1.1.2" xref="S2.Thmtheorem5.p1.4.4.m4.1.1.1.1.1.1.2.cmml">u</mi><mo id="S2.Thmtheorem5.p1.4.4.m4.1.1.1.1.1.1.1" lspace="0.108em" rspace="0.108em" stretchy="false" xref="S2.Thmtheorem5.p1.4.4.m4.1.1.1.1.1.1.1.cmml">→</mo><mi id="S2.Thmtheorem5.p1.4.4.m4.1.1.1.1.1.1.3" xref="S2.Thmtheorem5.p1.4.4.m4.1.1.1.1.1.1.3.cmml">v</mi></mrow><mo id="S2.Thmtheorem5.p1.4.4.m4.1.1.1.1.1.3" stretchy="false" xref="S2.Thmtheorem5.p1.4.4.m4.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S2.Thmtheorem5.p1.4.4.m4.1.1.2" xref="S2.Thmtheorem5.p1.4.4.m4.1.1.2.cmml">></mo><mn id="S2.Thmtheorem5.p1.4.4.m4.1.1.3" xref="S2.Thmtheorem5.p1.4.4.m4.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem5.p1.4.4.m4.1b"><apply id="S2.Thmtheorem5.p1.4.4.m4.1.1.cmml" xref="S2.Thmtheorem5.p1.4.4.m4.1.1"><gt id="S2.Thmtheorem5.p1.4.4.m4.1.1.2.cmml" xref="S2.Thmtheorem5.p1.4.4.m4.1.1.2"></gt><apply id="S2.Thmtheorem5.p1.4.4.m4.1.1.1.cmml" xref="S2.Thmtheorem5.p1.4.4.m4.1.1.1"><times id="S2.Thmtheorem5.p1.4.4.m4.1.1.1.2.cmml" xref="S2.Thmtheorem5.p1.4.4.m4.1.1.1.2"></times><ci id="S2.Thmtheorem5.p1.4.4.m4.1.1.1.3a.cmml" xref="S2.Thmtheorem5.p1.4.4.m4.1.1.1.3"><mtext class="ltx_mathvariant_italic" id="S2.Thmtheorem5.p1.4.4.m4.1.1.1.3.cmml" xref="S2.Thmtheorem5.p1.4.4.m4.1.1.1.3">g</mtext></ci><apply id="S2.Thmtheorem5.p1.4.4.m4.1.1.1.1.1.1.cmml" xref="S2.Thmtheorem5.p1.4.4.m4.1.1.1.1.1"><ci id="S2.Thmtheorem5.p1.4.4.m4.1.1.1.1.1.1.1.cmml" xref="S2.Thmtheorem5.p1.4.4.m4.1.1.1.1.1.1.1">→</ci><ci id="S2.Thmtheorem5.p1.4.4.m4.1.1.1.1.1.1.2.cmml" xref="S2.Thmtheorem5.p1.4.4.m4.1.1.1.1.1.1.2">𝑢</ci><ci id="S2.Thmtheorem5.p1.4.4.m4.1.1.1.1.1.1.3.cmml" xref="S2.Thmtheorem5.p1.4.4.m4.1.1.1.1.1.1.3">𝑣</ci></apply></apply><cn id="S2.Thmtheorem5.p1.4.4.m4.1.1.3.cmml" type="integer" xref="S2.Thmtheorem5.p1.4.4.m4.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem5.p1.4.4.m4.1c">\textsl{g}(u\!\to\!v)>0</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem5.p1.4.4.m4.1d">g ( italic_u → italic_v ) > 0</annotation></semantics></math> implies <math alttext="\textsl{g}(u)\leq\textsl{g}(v)" class="ltx_Math" display="inline" id="S2.Thmtheorem5.p1.5.5.m5.2"><semantics id="S2.Thmtheorem5.p1.5.5.m5.2a"><mrow id="S2.Thmtheorem5.p1.5.5.m5.2.3" xref="S2.Thmtheorem5.p1.5.5.m5.2.3.cmml"><mrow id="S2.Thmtheorem5.p1.5.5.m5.2.3.2" xref="S2.Thmtheorem5.p1.5.5.m5.2.3.2.cmml"><mtext class="ltx_mathvariant_italic" id="S2.Thmtheorem5.p1.5.5.m5.2.3.2.2" xref="S2.Thmtheorem5.p1.5.5.m5.2.3.2.2a.cmml">g</mtext><mo id="S2.Thmtheorem5.p1.5.5.m5.2.3.2.1" xref="S2.Thmtheorem5.p1.5.5.m5.2.3.2.1.cmml">⁢</mo><mrow id="S2.Thmtheorem5.p1.5.5.m5.2.3.2.3.2" xref="S2.Thmtheorem5.p1.5.5.m5.2.3.2.cmml"><mo id="S2.Thmtheorem5.p1.5.5.m5.2.3.2.3.2.1" stretchy="false" xref="S2.Thmtheorem5.p1.5.5.m5.2.3.2.cmml">(</mo><mi id="S2.Thmtheorem5.p1.5.5.m5.1.1" xref="S2.Thmtheorem5.p1.5.5.m5.1.1.cmml">u</mi><mo id="S2.Thmtheorem5.p1.5.5.m5.2.3.2.3.2.2" stretchy="false" xref="S2.Thmtheorem5.p1.5.5.m5.2.3.2.cmml">)</mo></mrow></mrow><mo id="S2.Thmtheorem5.p1.5.5.m5.2.3.1" xref="S2.Thmtheorem5.p1.5.5.m5.2.3.1.cmml">≤</mo><mrow id="S2.Thmtheorem5.p1.5.5.m5.2.3.3" xref="S2.Thmtheorem5.p1.5.5.m5.2.3.3.cmml"><mtext class="ltx_mathvariant_italic" id="S2.Thmtheorem5.p1.5.5.m5.2.3.3.2" xref="S2.Thmtheorem5.p1.5.5.m5.2.3.3.2a.cmml">g</mtext><mo id="S2.Thmtheorem5.p1.5.5.m5.2.3.3.1" xref="S2.Thmtheorem5.p1.5.5.m5.2.3.3.1.cmml">⁢</mo><mrow id="S2.Thmtheorem5.p1.5.5.m5.2.3.3.3.2" xref="S2.Thmtheorem5.p1.5.5.m5.2.3.3.cmml"><mo id="S2.Thmtheorem5.p1.5.5.m5.2.3.3.3.2.1" stretchy="false" xref="S2.Thmtheorem5.p1.5.5.m5.2.3.3.cmml">(</mo><mi id="S2.Thmtheorem5.p1.5.5.m5.2.2" xref="S2.Thmtheorem5.p1.5.5.m5.2.2.cmml">v</mi><mo id="S2.Thmtheorem5.p1.5.5.m5.2.3.3.3.2.2" stretchy="false" xref="S2.Thmtheorem5.p1.5.5.m5.2.3.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem5.p1.5.5.m5.2b"><apply id="S2.Thmtheorem5.p1.5.5.m5.2.3.cmml" xref="S2.Thmtheorem5.p1.5.5.m5.2.3"><leq id="S2.Thmtheorem5.p1.5.5.m5.2.3.1.cmml" xref="S2.Thmtheorem5.p1.5.5.m5.2.3.1"></leq><apply id="S2.Thmtheorem5.p1.5.5.m5.2.3.2.cmml" xref="S2.Thmtheorem5.p1.5.5.m5.2.3.2"><times id="S2.Thmtheorem5.p1.5.5.m5.2.3.2.1.cmml" xref="S2.Thmtheorem5.p1.5.5.m5.2.3.2.1"></times><ci id="S2.Thmtheorem5.p1.5.5.m5.2.3.2.2a.cmml" xref="S2.Thmtheorem5.p1.5.5.m5.2.3.2.2"><mtext class="ltx_mathvariant_italic" id="S2.Thmtheorem5.p1.5.5.m5.2.3.2.2.cmml" xref="S2.Thmtheorem5.p1.5.5.m5.2.3.2.2">g</mtext></ci><ci id="S2.Thmtheorem5.p1.5.5.m5.1.1.cmml" xref="S2.Thmtheorem5.p1.5.5.m5.1.1">𝑢</ci></apply><apply id="S2.Thmtheorem5.p1.5.5.m5.2.3.3.cmml" xref="S2.Thmtheorem5.p1.5.5.m5.2.3.3"><times id="S2.Thmtheorem5.p1.5.5.m5.2.3.3.1.cmml" xref="S2.Thmtheorem5.p1.5.5.m5.2.3.3.1"></times><ci id="S2.Thmtheorem5.p1.5.5.m5.2.3.3.2a.cmml" xref="S2.Thmtheorem5.p1.5.5.m5.2.3.3.2"><mtext class="ltx_mathvariant_italic" id="S2.Thmtheorem5.p1.5.5.m5.2.3.3.2.cmml" xref="S2.Thmtheorem5.p1.5.5.m5.2.3.3.2">g</mtext></ci><ci id="S2.Thmtheorem5.p1.5.5.m5.2.2.cmml" xref="S2.Thmtheorem5.p1.5.5.m5.2.2">𝑣</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem5.p1.5.5.m5.2c">\textsl{g}(u)\leq\textsl{g}(v)</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem5.p1.5.5.m5.2d">g ( italic_u ) ≤ g ( italic_v )</annotation></semantics></math>.</span></p> </div> </div> <div class="ltx_para ltx_noindent" id="S2.SS1.p3"> <p class="ltx_p" id="S2.SS1.p3.1">Chekuri et al. <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2411.12694v2#bib.bib7" title="">7</a>]</cite> extend this definition to <math alttext="\eta" class="ltx_Math" display="inline" id="S2.SS1.p3.1.m1.1"><semantics id="S2.SS1.p3.1.m1.1a"><mi id="S2.SS1.p3.1.m1.1.1" xref="S2.SS1.p3.1.m1.1.1.cmml">η</mi><annotation-xml encoding="MathML-Content" id="S2.SS1.p3.1.m1.1b"><ci id="S2.SS1.p3.1.m1.1.1.cmml" xref="S2.SS1.p3.1.m1.1.1">𝜂</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p3.1.m1.1c">\eta</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p3.1.m1.1d">italic_η</annotation></semantics></math>-fairness:</p> </div> <div class="ltx_theorem ltx_theorem_definition" 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">Definition 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.p1.7"><span class="ltx_text ltx_font_italic" id="S2.Thmtheorem6.p1.7.7">Let <math alttext="\overrightarrow{G}" class="ltx_Math" display="inline" id="S2.Thmtheorem6.p1.1.1.m1.1"><semantics id="S2.Thmtheorem6.p1.1.1.m1.1a"><mover accent="true" id="S2.Thmtheorem6.p1.1.1.m1.1.1" xref="S2.Thmtheorem6.p1.1.1.m1.1.1.cmml"><mi id="S2.Thmtheorem6.p1.1.1.m1.1.1.2" xref="S2.Thmtheorem6.p1.1.1.m1.1.1.2.cmml">G</mi><mo id="S2.Thmtheorem6.p1.1.1.m1.1.1.1" stretchy="false" xref="S2.Thmtheorem6.p1.1.1.m1.1.1.1.cmml">→</mo></mover><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem6.p1.1.1.m1.1b"><apply id="S2.Thmtheorem6.p1.1.1.m1.1.1.cmml" xref="S2.Thmtheorem6.p1.1.1.m1.1.1"><ci id="S2.Thmtheorem6.p1.1.1.m1.1.1.1.cmml" xref="S2.Thmtheorem6.p1.1.1.m1.1.1.1">→</ci><ci id="S2.Thmtheorem6.p1.1.1.m1.1.1.2.cmml" xref="S2.Thmtheorem6.p1.1.1.m1.1.1.2">𝐺</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem6.p1.1.1.m1.1c">\overrightarrow{G}</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem6.p1.1.1.m1.1d">over→ start_ARG italic_G end_ARG</annotation></semantics></math> be a fractional orientation of a graph <math alttext="G" class="ltx_Math" display="inline" id="S2.Thmtheorem6.p1.2.2.m2.1"><semantics id="S2.Thmtheorem6.p1.2.2.m2.1a"><mi id="S2.Thmtheorem6.p1.2.2.m2.1.1" xref="S2.Thmtheorem6.p1.2.2.m2.1.1.cmml">G</mi><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem6.p1.2.2.m2.1b"><ci id="S2.Thmtheorem6.p1.2.2.m2.1.1.cmml" xref="S2.Thmtheorem6.p1.2.2.m2.1.1">𝐺</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem6.p1.2.2.m2.1c">G</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem6.p1.2.2.m2.1d">italic_G</annotation></semantics></math>. We say that <math alttext="\overrightarrow{G}" class="ltx_Math" display="inline" id="S2.Thmtheorem6.p1.3.3.m3.1"><semantics id="S2.Thmtheorem6.p1.3.3.m3.1a"><mover accent="true" id="S2.Thmtheorem6.p1.3.3.m3.1.1" xref="S2.Thmtheorem6.p1.3.3.m3.1.1.cmml"><mi id="S2.Thmtheorem6.p1.3.3.m3.1.1.2" xref="S2.Thmtheorem6.p1.3.3.m3.1.1.2.cmml">G</mi><mo id="S2.Thmtheorem6.p1.3.3.m3.1.1.1" stretchy="false" xref="S2.Thmtheorem6.p1.3.3.m3.1.1.1.cmml">→</mo></mover><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem6.p1.3.3.m3.1b"><apply id="S2.Thmtheorem6.p1.3.3.m3.1.1.cmml" xref="S2.Thmtheorem6.p1.3.3.m3.1.1"><ci id="S2.Thmtheorem6.p1.3.3.m3.1.1.1.cmml" xref="S2.Thmtheorem6.p1.3.3.m3.1.1.1">→</ci><ci id="S2.Thmtheorem6.p1.3.3.m3.1.1.2.cmml" xref="S2.Thmtheorem6.p1.3.3.m3.1.1.2">𝐺</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem6.p1.3.3.m3.1c">\overrightarrow{G}</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem6.p1.3.3.m3.1d">over→ start_ARG italic_G end_ARG</annotation></semantics></math> is <math alttext="\eta" class="ltx_Math" display="inline" id="S2.Thmtheorem6.p1.4.4.m4.1"><semantics id="S2.Thmtheorem6.p1.4.4.m4.1a"><mi id="S2.Thmtheorem6.p1.4.4.m4.1.1" xref="S2.Thmtheorem6.p1.4.4.m4.1.1.cmml">η</mi><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem6.p1.4.4.m4.1b"><ci id="S2.Thmtheorem6.p1.4.4.m4.1.1.cmml" xref="S2.Thmtheorem6.p1.4.4.m4.1.1">𝜂</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem6.p1.4.4.m4.1c">\eta</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem6.p1.4.4.m4.1d">italic_η</annotation></semantics></math>-fair (for <math alttext="\eta>0" class="ltx_Math" display="inline" id="S2.Thmtheorem6.p1.5.5.m5.1"><semantics id="S2.Thmtheorem6.p1.5.5.m5.1a"><mrow id="S2.Thmtheorem6.p1.5.5.m5.1.1" xref="S2.Thmtheorem6.p1.5.5.m5.1.1.cmml"><mi id="S2.Thmtheorem6.p1.5.5.m5.1.1.2" xref="S2.Thmtheorem6.p1.5.5.m5.1.1.2.cmml">η</mi><mo id="S2.Thmtheorem6.p1.5.5.m5.1.1.1" xref="S2.Thmtheorem6.p1.5.5.m5.1.1.1.cmml">></mo><mn id="S2.Thmtheorem6.p1.5.5.m5.1.1.3" xref="S2.Thmtheorem6.p1.5.5.m5.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem6.p1.5.5.m5.1b"><apply id="S2.Thmtheorem6.p1.5.5.m5.1.1.cmml" xref="S2.Thmtheorem6.p1.5.5.m5.1.1"><gt id="S2.Thmtheorem6.p1.5.5.m5.1.1.1.cmml" xref="S2.Thmtheorem6.p1.5.5.m5.1.1.1"></gt><ci id="S2.Thmtheorem6.p1.5.5.m5.1.1.2.cmml" xref="S2.Thmtheorem6.p1.5.5.m5.1.1.2">𝜂</ci><cn id="S2.Thmtheorem6.p1.5.5.m5.1.1.3.cmml" type="integer" xref="S2.Thmtheorem6.p1.5.5.m5.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem6.p1.5.5.m5.1c">\eta>0</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem6.p1.5.5.m5.1d">italic_η > 0</annotation></semantics></math>) whenever <math alttext="\textsl{g}(u\!\to\!v)>0" class="ltx_Math" display="inline" id="S2.Thmtheorem6.p1.6.6.m6.1"><semantics id="S2.Thmtheorem6.p1.6.6.m6.1a"><mrow id="S2.Thmtheorem6.p1.6.6.m6.1.1" xref="S2.Thmtheorem6.p1.6.6.m6.1.1.cmml"><mrow id="S2.Thmtheorem6.p1.6.6.m6.1.1.1" xref="S2.Thmtheorem6.p1.6.6.m6.1.1.1.cmml"><mtext class="ltx_mathvariant_italic" id="S2.Thmtheorem6.p1.6.6.m6.1.1.1.3" xref="S2.Thmtheorem6.p1.6.6.m6.1.1.1.3a.cmml">g</mtext><mo id="S2.Thmtheorem6.p1.6.6.m6.1.1.1.2" xref="S2.Thmtheorem6.p1.6.6.m6.1.1.1.2.cmml">⁢</mo><mrow id="S2.Thmtheorem6.p1.6.6.m6.1.1.1.1.1" xref="S2.Thmtheorem6.p1.6.6.m6.1.1.1.1.1.1.cmml"><mo id="S2.Thmtheorem6.p1.6.6.m6.1.1.1.1.1.2" stretchy="false" xref="S2.Thmtheorem6.p1.6.6.m6.1.1.1.1.1.1.cmml">(</mo><mrow id="S2.Thmtheorem6.p1.6.6.m6.1.1.1.1.1.1" xref="S2.Thmtheorem6.p1.6.6.m6.1.1.1.1.1.1.cmml"><mi id="S2.Thmtheorem6.p1.6.6.m6.1.1.1.1.1.1.2" xref="S2.Thmtheorem6.p1.6.6.m6.1.1.1.1.1.1.2.cmml">u</mi><mo id="S2.Thmtheorem6.p1.6.6.m6.1.1.1.1.1.1.1" lspace="0.108em" rspace="0.108em" stretchy="false" xref="S2.Thmtheorem6.p1.6.6.m6.1.1.1.1.1.1.1.cmml">→</mo><mi id="S2.Thmtheorem6.p1.6.6.m6.1.1.1.1.1.1.3" xref="S2.Thmtheorem6.p1.6.6.m6.1.1.1.1.1.1.3.cmml">v</mi></mrow><mo id="S2.Thmtheorem6.p1.6.6.m6.1.1.1.1.1.3" stretchy="false" xref="S2.Thmtheorem6.p1.6.6.m6.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S2.Thmtheorem6.p1.6.6.m6.1.1.2" xref="S2.Thmtheorem6.p1.6.6.m6.1.1.2.cmml">></mo><mn id="S2.Thmtheorem6.p1.6.6.m6.1.1.3" xref="S2.Thmtheorem6.p1.6.6.m6.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem6.p1.6.6.m6.1b"><apply id="S2.Thmtheorem6.p1.6.6.m6.1.1.cmml" xref="S2.Thmtheorem6.p1.6.6.m6.1.1"><gt id="S2.Thmtheorem6.p1.6.6.m6.1.1.2.cmml" xref="S2.Thmtheorem6.p1.6.6.m6.1.1.2"></gt><apply id="S2.Thmtheorem6.p1.6.6.m6.1.1.1.cmml" xref="S2.Thmtheorem6.p1.6.6.m6.1.1.1"><times id="S2.Thmtheorem6.p1.6.6.m6.1.1.1.2.cmml" xref="S2.Thmtheorem6.p1.6.6.m6.1.1.1.2"></times><ci id="S2.Thmtheorem6.p1.6.6.m6.1.1.1.3a.cmml" xref="S2.Thmtheorem6.p1.6.6.m6.1.1.1.3"><mtext class="ltx_mathvariant_italic" id="S2.Thmtheorem6.p1.6.6.m6.1.1.1.3.cmml" xref="S2.Thmtheorem6.p1.6.6.m6.1.1.1.3">g</mtext></ci><apply id="S2.Thmtheorem6.p1.6.6.m6.1.1.1.1.1.1.cmml" xref="S2.Thmtheorem6.p1.6.6.m6.1.1.1.1.1"><ci id="S2.Thmtheorem6.p1.6.6.m6.1.1.1.1.1.1.1.cmml" xref="S2.Thmtheorem6.p1.6.6.m6.1.1.1.1.1.1.1">→</ci><ci id="S2.Thmtheorem6.p1.6.6.m6.1.1.1.1.1.1.2.cmml" xref="S2.Thmtheorem6.p1.6.6.m6.1.1.1.1.1.1.2">𝑢</ci><ci id="S2.Thmtheorem6.p1.6.6.m6.1.1.1.1.1.1.3.cmml" xref="S2.Thmtheorem6.p1.6.6.m6.1.1.1.1.1.1.3">𝑣</ci></apply></apply><cn id="S2.Thmtheorem6.p1.6.6.m6.1.1.3.cmml" type="integer" xref="S2.Thmtheorem6.p1.6.6.m6.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem6.p1.6.6.m6.1c">\textsl{g}(u\!\to\!v)>0</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem6.p1.6.6.m6.1d">g ( italic_u → italic_v ) > 0</annotation></semantics></math> implies that <math alttext="\textsl{g}(u)\leq(1+\eta)\textsl{g}(v)" class="ltx_Math" display="inline" id="S2.Thmtheorem6.p1.7.7.m7.3"><semantics id="S2.Thmtheorem6.p1.7.7.m7.3a"><mrow id="S2.Thmtheorem6.p1.7.7.m7.3.3" xref="S2.Thmtheorem6.p1.7.7.m7.3.3.cmml"><mrow id="S2.Thmtheorem6.p1.7.7.m7.3.3.3" xref="S2.Thmtheorem6.p1.7.7.m7.3.3.3.cmml"><mtext class="ltx_mathvariant_italic" id="S2.Thmtheorem6.p1.7.7.m7.3.3.3.2" xref="S2.Thmtheorem6.p1.7.7.m7.3.3.3.2a.cmml">g</mtext><mo id="S2.Thmtheorem6.p1.7.7.m7.3.3.3.1" xref="S2.Thmtheorem6.p1.7.7.m7.3.3.3.1.cmml">⁢</mo><mrow id="S2.Thmtheorem6.p1.7.7.m7.3.3.3.3.2" xref="S2.Thmtheorem6.p1.7.7.m7.3.3.3.cmml"><mo id="S2.Thmtheorem6.p1.7.7.m7.3.3.3.3.2.1" stretchy="false" xref="S2.Thmtheorem6.p1.7.7.m7.3.3.3.cmml">(</mo><mi id="S2.Thmtheorem6.p1.7.7.m7.1.1" xref="S2.Thmtheorem6.p1.7.7.m7.1.1.cmml">u</mi><mo id="S2.Thmtheorem6.p1.7.7.m7.3.3.3.3.2.2" stretchy="false" xref="S2.Thmtheorem6.p1.7.7.m7.3.3.3.cmml">)</mo></mrow></mrow><mo id="S2.Thmtheorem6.p1.7.7.m7.3.3.2" xref="S2.Thmtheorem6.p1.7.7.m7.3.3.2.cmml">≤</mo><mrow id="S2.Thmtheorem6.p1.7.7.m7.3.3.1" xref="S2.Thmtheorem6.p1.7.7.m7.3.3.1.cmml"><mrow id="S2.Thmtheorem6.p1.7.7.m7.3.3.1.1.1" xref="S2.Thmtheorem6.p1.7.7.m7.3.3.1.1.1.1.cmml"><mo id="S2.Thmtheorem6.p1.7.7.m7.3.3.1.1.1.2" stretchy="false" xref="S2.Thmtheorem6.p1.7.7.m7.3.3.1.1.1.1.cmml">(</mo><mrow id="S2.Thmtheorem6.p1.7.7.m7.3.3.1.1.1.1" xref="S2.Thmtheorem6.p1.7.7.m7.3.3.1.1.1.1.cmml"><mn id="S2.Thmtheorem6.p1.7.7.m7.3.3.1.1.1.1.2" xref="S2.Thmtheorem6.p1.7.7.m7.3.3.1.1.1.1.2.cmml">1</mn><mo id="S2.Thmtheorem6.p1.7.7.m7.3.3.1.1.1.1.1" xref="S2.Thmtheorem6.p1.7.7.m7.3.3.1.1.1.1.1.cmml">+</mo><mi id="S2.Thmtheorem6.p1.7.7.m7.3.3.1.1.1.1.3" xref="S2.Thmtheorem6.p1.7.7.m7.3.3.1.1.1.1.3.cmml">η</mi></mrow><mo id="S2.Thmtheorem6.p1.7.7.m7.3.3.1.1.1.3" stretchy="false" xref="S2.Thmtheorem6.p1.7.7.m7.3.3.1.1.1.1.cmml">)</mo></mrow><mo id="S2.Thmtheorem6.p1.7.7.m7.3.3.1.2" xref="S2.Thmtheorem6.p1.7.7.m7.3.3.1.2.cmml">⁢</mo><mtext class="ltx_mathvariant_italic" id="S2.Thmtheorem6.p1.7.7.m7.3.3.1.3" xref="S2.Thmtheorem6.p1.7.7.m7.3.3.1.3a.cmml">g</mtext><mo id="S2.Thmtheorem6.p1.7.7.m7.3.3.1.2a" xref="S2.Thmtheorem6.p1.7.7.m7.3.3.1.2.cmml">⁢</mo><mrow id="S2.Thmtheorem6.p1.7.7.m7.3.3.1.4.2" xref="S2.Thmtheorem6.p1.7.7.m7.3.3.1.cmml"><mo id="S2.Thmtheorem6.p1.7.7.m7.3.3.1.4.2.1" stretchy="false" xref="S2.Thmtheorem6.p1.7.7.m7.3.3.1.cmml">(</mo><mi id="S2.Thmtheorem6.p1.7.7.m7.2.2" xref="S2.Thmtheorem6.p1.7.7.m7.2.2.cmml">v</mi><mo id="S2.Thmtheorem6.p1.7.7.m7.3.3.1.4.2.2" stretchy="false" xref="S2.Thmtheorem6.p1.7.7.m7.3.3.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem6.p1.7.7.m7.3b"><apply id="S2.Thmtheorem6.p1.7.7.m7.3.3.cmml" xref="S2.Thmtheorem6.p1.7.7.m7.3.3"><leq id="S2.Thmtheorem6.p1.7.7.m7.3.3.2.cmml" xref="S2.Thmtheorem6.p1.7.7.m7.3.3.2"></leq><apply id="S2.Thmtheorem6.p1.7.7.m7.3.3.3.cmml" xref="S2.Thmtheorem6.p1.7.7.m7.3.3.3"><times id="S2.Thmtheorem6.p1.7.7.m7.3.3.3.1.cmml" xref="S2.Thmtheorem6.p1.7.7.m7.3.3.3.1"></times><ci id="S2.Thmtheorem6.p1.7.7.m7.3.3.3.2a.cmml" xref="S2.Thmtheorem6.p1.7.7.m7.3.3.3.2"><mtext class="ltx_mathvariant_italic" id="S2.Thmtheorem6.p1.7.7.m7.3.3.3.2.cmml" xref="S2.Thmtheorem6.p1.7.7.m7.3.3.3.2">g</mtext></ci><ci id="S2.Thmtheorem6.p1.7.7.m7.1.1.cmml" xref="S2.Thmtheorem6.p1.7.7.m7.1.1">𝑢</ci></apply><apply id="S2.Thmtheorem6.p1.7.7.m7.3.3.1.cmml" xref="S2.Thmtheorem6.p1.7.7.m7.3.3.1"><times id="S2.Thmtheorem6.p1.7.7.m7.3.3.1.2.cmml" xref="S2.Thmtheorem6.p1.7.7.m7.3.3.1.2"></times><apply id="S2.Thmtheorem6.p1.7.7.m7.3.3.1.1.1.1.cmml" xref="S2.Thmtheorem6.p1.7.7.m7.3.3.1.1.1"><plus id="S2.Thmtheorem6.p1.7.7.m7.3.3.1.1.1.1.1.cmml" xref="S2.Thmtheorem6.p1.7.7.m7.3.3.1.1.1.1.1"></plus><cn id="S2.Thmtheorem6.p1.7.7.m7.3.3.1.1.1.1.2.cmml" type="integer" xref="S2.Thmtheorem6.p1.7.7.m7.3.3.1.1.1.1.2">1</cn><ci id="S2.Thmtheorem6.p1.7.7.m7.3.3.1.1.1.1.3.cmml" xref="S2.Thmtheorem6.p1.7.7.m7.3.3.1.1.1.1.3">𝜂</ci></apply><ci id="S2.Thmtheorem6.p1.7.7.m7.3.3.1.3a.cmml" xref="S2.Thmtheorem6.p1.7.7.m7.3.3.1.3"><mtext class="ltx_mathvariant_italic" id="S2.Thmtheorem6.p1.7.7.m7.3.3.1.3.cmml" xref="S2.Thmtheorem6.p1.7.7.m7.3.3.1.3">g</mtext></ci><ci id="S2.Thmtheorem6.p1.7.7.m7.2.2.cmml" xref="S2.Thmtheorem6.p1.7.7.m7.2.2">𝑣</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem6.p1.7.7.m7.3c">\textsl{g}(u)\leq(1+\eta)\textsl{g}(v)</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem6.p1.7.7.m7.3d">g ( italic_u ) ≤ ( 1 + italic_η ) g ( italic_v )</annotation></semantics></math>.</span></p> </div> </div> <section class="ltx_subparagraph" id="S2.SS1.SSS0.P0.SPx1"> <h5 class="ltx_title ltx_title_subparagraph">Related work in dynamic algorithms</h5> <div class="ltx_para" id="S2.SS1.SSS0.P0.SPx1.p1"> <p class="ltx_p" id="S2.SS1.SSS0.P0.SPx1.p1.12">Chekuri et al. <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2411.12694v2#bib.bib7" title="">7</a>]</cite> continue to focus on computing a <math alttext="(1+\varepsilon)" class="ltx_Math" display="inline" id="S2.SS1.SSS0.P0.SPx1.p1.1.m1.1"><semantics id="S2.SS1.SSS0.P0.SPx1.p1.1.m1.1a"><mrow id="S2.SS1.SSS0.P0.SPx1.p1.1.m1.1.1.1" xref="S2.SS1.SSS0.P0.SPx1.p1.1.m1.1.1.1.1.cmml"><mo id="S2.SS1.SSS0.P0.SPx1.p1.1.m1.1.1.1.2" stretchy="false" xref="S2.SS1.SSS0.P0.SPx1.p1.1.m1.1.1.1.1.cmml">(</mo><mrow id="S2.SS1.SSS0.P0.SPx1.p1.1.m1.1.1.1.1" xref="S2.SS1.SSS0.P0.SPx1.p1.1.m1.1.1.1.1.cmml"><mn id="S2.SS1.SSS0.P0.SPx1.p1.1.m1.1.1.1.1.2" xref="S2.SS1.SSS0.P0.SPx1.p1.1.m1.1.1.1.1.2.cmml">1</mn><mo id="S2.SS1.SSS0.P0.SPx1.p1.1.m1.1.1.1.1.1" xref="S2.SS1.SSS0.P0.SPx1.p1.1.m1.1.1.1.1.1.cmml">+</mo><mi id="S2.SS1.SSS0.P0.SPx1.p1.1.m1.1.1.1.1.3" xref="S2.SS1.SSS0.P0.SPx1.p1.1.m1.1.1.1.1.3.cmml">ε</mi></mrow><mo id="S2.SS1.SSS0.P0.SPx1.p1.1.m1.1.1.1.3" stretchy="false" xref="S2.SS1.SSS0.P0.SPx1.p1.1.m1.1.1.1.1.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.SSS0.P0.SPx1.p1.1.m1.1b"><apply id="S2.SS1.SSS0.P0.SPx1.p1.1.m1.1.1.1.1.cmml" xref="S2.SS1.SSS0.P0.SPx1.p1.1.m1.1.1.1"><plus id="S2.SS1.SSS0.P0.SPx1.p1.1.m1.1.1.1.1.1.cmml" xref="S2.SS1.SSS0.P0.SPx1.p1.1.m1.1.1.1.1.1"></plus><cn id="S2.SS1.SSS0.P0.SPx1.p1.1.m1.1.1.1.1.2.cmml" type="integer" xref="S2.SS1.SSS0.P0.SPx1.p1.1.m1.1.1.1.1.2">1</cn><ci id="S2.SS1.SSS0.P0.SPx1.p1.1.m1.1.1.1.1.3.cmml" xref="S2.SS1.SSS0.P0.SPx1.p1.1.m1.1.1.1.1.3">𝜀</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.SSS0.P0.SPx1.p1.1.m1.1c">(1+\varepsilon)</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.SSS0.P0.SPx1.p1.1.m1.1d">( 1 + italic_ε )</annotation></semantics></math>-approximation of the Densest Subgraph problem. They show that, if <math alttext="G" class="ltx_Math" display="inline" id="S2.SS1.SSS0.P0.SPx1.p1.2.m2.1"><semantics id="S2.SS1.SSS0.P0.SPx1.p1.2.m2.1a"><mi id="S2.SS1.SSS0.P0.SPx1.p1.2.m2.1.1" xref="S2.SS1.SSS0.P0.SPx1.p1.2.m2.1.1.cmml">G</mi><annotation-xml encoding="MathML-Content" id="S2.SS1.SSS0.P0.SPx1.p1.2.m2.1b"><ci id="S2.SS1.SSS0.P0.SPx1.p1.2.m2.1.1.cmml" xref="S2.SS1.SSS0.P0.SPx1.p1.2.m2.1.1">𝐺</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.SSS0.P0.SPx1.p1.2.m2.1c">G</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.SSS0.P0.SPx1.p1.2.m2.1d">italic_G</annotation></semantics></math> is a unit weight graph, there exists a <math alttext="(1+\varepsilon)" class="ltx_Math" display="inline" id="S2.SS1.SSS0.P0.SPx1.p1.3.m3.1"><semantics id="S2.SS1.SSS0.P0.SPx1.p1.3.m3.1a"><mrow id="S2.SS1.SSS0.P0.SPx1.p1.3.m3.1.1.1" xref="S2.SS1.SSS0.P0.SPx1.p1.3.m3.1.1.1.1.cmml"><mo id="S2.SS1.SSS0.P0.SPx1.p1.3.m3.1.1.1.2" stretchy="false" xref="S2.SS1.SSS0.P0.SPx1.p1.3.m3.1.1.1.1.cmml">(</mo><mrow id="S2.SS1.SSS0.P0.SPx1.p1.3.m3.1.1.1.1" xref="S2.SS1.SSS0.P0.SPx1.p1.3.m3.1.1.1.1.cmml"><mn id="S2.SS1.SSS0.P0.SPx1.p1.3.m3.1.1.1.1.2" xref="S2.SS1.SSS0.P0.SPx1.p1.3.m3.1.1.1.1.2.cmml">1</mn><mo id="S2.SS1.SSS0.P0.SPx1.p1.3.m3.1.1.1.1.1" xref="S2.SS1.SSS0.P0.SPx1.p1.3.m3.1.1.1.1.1.cmml">+</mo><mi id="S2.SS1.SSS0.P0.SPx1.p1.3.m3.1.1.1.1.3" xref="S2.SS1.SSS0.P0.SPx1.p1.3.m3.1.1.1.1.3.cmml">ε</mi></mrow><mo id="S2.SS1.SSS0.P0.SPx1.p1.3.m3.1.1.1.3" stretchy="false" xref="S2.SS1.SSS0.P0.SPx1.p1.3.m3.1.1.1.1.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.SSS0.P0.SPx1.p1.3.m3.1b"><apply id="S2.SS1.SSS0.P0.SPx1.p1.3.m3.1.1.1.1.cmml" xref="S2.SS1.SSS0.P0.SPx1.p1.3.m3.1.1.1"><plus id="S2.SS1.SSS0.P0.SPx1.p1.3.m3.1.1.1.1.1.cmml" xref="S2.SS1.SSS0.P0.SPx1.p1.3.m3.1.1.1.1.1"></plus><cn id="S2.SS1.SSS0.P0.SPx1.p1.3.m3.1.1.1.1.2.cmml" type="integer" xref="S2.SS1.SSS0.P0.SPx1.p1.3.m3.1.1.1.1.2">1</cn><ci id="S2.SS1.SSS0.P0.SPx1.p1.3.m3.1.1.1.1.3.cmml" xref="S2.SS1.SSS0.P0.SPx1.p1.3.m3.1.1.1.1.3">𝜀</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.SSS0.P0.SPx1.p1.3.m3.1c">(1+\varepsilon)</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.SSS0.P0.SPx1.p1.3.m3.1d">( 1 + italic_ε )</annotation></semantics></math>-approximate solution to FO that is <math alttext="\eta" class="ltx_Math" display="inline" id="S2.SS1.SSS0.P0.SPx1.p1.4.m4.1"><semantics id="S2.SS1.SSS0.P0.SPx1.p1.4.m4.1a"><mi id="S2.SS1.SSS0.P0.SPx1.p1.4.m4.1.1" xref="S2.SS1.SSS0.P0.SPx1.p1.4.m4.1.1.cmml">η</mi><annotation-xml encoding="MathML-Content" id="S2.SS1.SSS0.P0.SPx1.p1.4.m4.1b"><ci id="S2.SS1.SSS0.P0.SPx1.p1.4.m4.1.1.cmml" xref="S2.SS1.SSS0.P0.SPx1.p1.4.m4.1.1">𝜂</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.SSS0.P0.SPx1.p1.4.m4.1c">\eta</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.SSS0.P0.SPx1.p1.4.m4.1d">italic_η</annotation></semantics></math>-fair (for some smartly chosen <math alttext="\eta" class="ltx_Math" display="inline" id="S2.SS1.SSS0.P0.SPx1.p1.5.m5.1"><semantics id="S2.SS1.SSS0.P0.SPx1.p1.5.m5.1a"><mi id="S2.SS1.SSS0.P0.SPx1.p1.5.m5.1.1" xref="S2.SS1.SSS0.P0.SPx1.p1.5.m5.1.1.cmml">η</mi><annotation-xml encoding="MathML-Content" id="S2.SS1.SSS0.P0.SPx1.p1.5.m5.1b"><ci id="S2.SS1.SSS0.P0.SPx1.p1.5.m5.1.1.cmml" xref="S2.SS1.SSS0.P0.SPx1.p1.5.m5.1.1">𝜂</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.SSS0.P0.SPx1.p1.5.m5.1c">\eta</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.SSS0.P0.SPx1.p1.5.m5.1d">italic_η</annotation></semantics></math>). They subsequently prove that an <math alttext="\eta" class="ltx_Math" display="inline" id="S2.SS1.SSS0.P0.SPx1.p1.6.m6.1"><semantics id="S2.SS1.SSS0.P0.SPx1.p1.6.m6.1a"><mi id="S2.SS1.SSS0.P0.SPx1.p1.6.m6.1.1" xref="S2.SS1.SSS0.P0.SPx1.p1.6.m6.1.1.cmml">η</mi><annotation-xml encoding="MathML-Content" id="S2.SS1.SSS0.P0.SPx1.p1.6.m6.1b"><ci id="S2.SS1.SSS0.P0.SPx1.p1.6.m6.1.1.cmml" xref="S2.SS1.SSS0.P0.SPx1.p1.6.m6.1.1">𝜂</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.SSS0.P0.SPx1.p1.6.m6.1c">\eta</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.SSS0.P0.SPx1.p1.6.m6.1d">italic_η</annotation></semantics></math>-fair orientation allows you to find a <math alttext="(1+\varepsilon)" class="ltx_Math" display="inline" id="S2.SS1.SSS0.P0.SPx1.p1.7.m7.1"><semantics id="S2.SS1.SSS0.P0.SPx1.p1.7.m7.1a"><mrow id="S2.SS1.SSS0.P0.SPx1.p1.7.m7.1.1.1" xref="S2.SS1.SSS0.P0.SPx1.p1.7.m7.1.1.1.1.cmml"><mo id="S2.SS1.SSS0.P0.SPx1.p1.7.m7.1.1.1.2" stretchy="false" xref="S2.SS1.SSS0.P0.SPx1.p1.7.m7.1.1.1.1.cmml">(</mo><mrow id="S2.SS1.SSS0.P0.SPx1.p1.7.m7.1.1.1.1" xref="S2.SS1.SSS0.P0.SPx1.p1.7.m7.1.1.1.1.cmml"><mn id="S2.SS1.SSS0.P0.SPx1.p1.7.m7.1.1.1.1.2" xref="S2.SS1.SSS0.P0.SPx1.p1.7.m7.1.1.1.1.2.cmml">1</mn><mo id="S2.SS1.SSS0.P0.SPx1.p1.7.m7.1.1.1.1.1" xref="S2.SS1.SSS0.P0.SPx1.p1.7.m7.1.1.1.1.1.cmml">+</mo><mi id="S2.SS1.SSS0.P0.SPx1.p1.7.m7.1.1.1.1.3" xref="S2.SS1.SSS0.P0.SPx1.p1.7.m7.1.1.1.1.3.cmml">ε</mi></mrow><mo id="S2.SS1.SSS0.P0.SPx1.p1.7.m7.1.1.1.3" stretchy="false" xref="S2.SS1.SSS0.P0.SPx1.p1.7.m7.1.1.1.1.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.SSS0.P0.SPx1.p1.7.m7.1b"><apply id="S2.SS1.SSS0.P0.SPx1.p1.7.m7.1.1.1.1.cmml" xref="S2.SS1.SSS0.P0.SPx1.p1.7.m7.1.1.1"><plus id="S2.SS1.SSS0.P0.SPx1.p1.7.m7.1.1.1.1.1.cmml" xref="S2.SS1.SSS0.P0.SPx1.p1.7.m7.1.1.1.1.1"></plus><cn id="S2.SS1.SSS0.P0.SPx1.p1.7.m7.1.1.1.1.2.cmml" type="integer" xref="S2.SS1.SSS0.P0.SPx1.p1.7.m7.1.1.1.1.2">1</cn><ci id="S2.SS1.SSS0.P0.SPx1.p1.7.m7.1.1.1.1.3.cmml" xref="S2.SS1.SSS0.P0.SPx1.p1.7.m7.1.1.1.1.3">𝜀</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.SSS0.P0.SPx1.p1.7.m7.1c">(1+\varepsilon)</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.SSS0.P0.SPx1.p1.7.m7.1d">( 1 + italic_ε )</annotation></semantics></math>-approximate densest subgraph. This allows them to dynamically maintain the value of the densest subgraph of <math alttext="G" class="ltx_Math" display="inline" id="S2.SS1.SSS0.P0.SPx1.p1.8.m8.1"><semantics id="S2.SS1.SSS0.P0.SPx1.p1.8.m8.1a"><mi id="S2.SS1.SSS0.P0.SPx1.p1.8.m8.1.1" xref="S2.SS1.SSS0.P0.SPx1.p1.8.m8.1.1.cmml">G</mi><annotation-xml encoding="MathML-Content" id="S2.SS1.SSS0.P0.SPx1.p1.8.m8.1b"><ci id="S2.SS1.SSS0.P0.SPx1.p1.8.m8.1.1.cmml" xref="S2.SS1.SSS0.P0.SPx1.p1.8.m8.1.1">𝐺</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.SSS0.P0.SPx1.p1.8.m8.1c">G</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.SSS0.P0.SPx1.p1.8.m8.1d">italic_G</annotation></semantics></math> in <math alttext="O(\varepsilon^{-6}\log^{3}n\log\rho^{\max}(G))" class="ltx_Math" display="inline" id="S2.SS1.SSS0.P0.SPx1.p1.9.m9.2"><semantics id="S2.SS1.SSS0.P0.SPx1.p1.9.m9.2a"><mrow id="S2.SS1.SSS0.P0.SPx1.p1.9.m9.2.2" xref="S2.SS1.SSS0.P0.SPx1.p1.9.m9.2.2.cmml"><mi id="S2.SS1.SSS0.P0.SPx1.p1.9.m9.2.2.3" xref="S2.SS1.SSS0.P0.SPx1.p1.9.m9.2.2.3.cmml">O</mi><mo id="S2.SS1.SSS0.P0.SPx1.p1.9.m9.2.2.2" xref="S2.SS1.SSS0.P0.SPx1.p1.9.m9.2.2.2.cmml">⁢</mo><mrow id="S2.SS1.SSS0.P0.SPx1.p1.9.m9.2.2.1.1" xref="S2.SS1.SSS0.P0.SPx1.p1.9.m9.2.2.1.1.1.cmml"><mo id="S2.SS1.SSS0.P0.SPx1.p1.9.m9.2.2.1.1.2" stretchy="false" xref="S2.SS1.SSS0.P0.SPx1.p1.9.m9.2.2.1.1.1.cmml">(</mo><mrow id="S2.SS1.SSS0.P0.SPx1.p1.9.m9.2.2.1.1.1" xref="S2.SS1.SSS0.P0.SPx1.p1.9.m9.2.2.1.1.1.cmml"><msup id="S2.SS1.SSS0.P0.SPx1.p1.9.m9.2.2.1.1.1.2" xref="S2.SS1.SSS0.P0.SPx1.p1.9.m9.2.2.1.1.1.2.cmml"><mi id="S2.SS1.SSS0.P0.SPx1.p1.9.m9.2.2.1.1.1.2.2" xref="S2.SS1.SSS0.P0.SPx1.p1.9.m9.2.2.1.1.1.2.2.cmml">ε</mi><mrow id="S2.SS1.SSS0.P0.SPx1.p1.9.m9.2.2.1.1.1.2.3" xref="S2.SS1.SSS0.P0.SPx1.p1.9.m9.2.2.1.1.1.2.3.cmml"><mo id="S2.SS1.SSS0.P0.SPx1.p1.9.m9.2.2.1.1.1.2.3a" xref="S2.SS1.SSS0.P0.SPx1.p1.9.m9.2.2.1.1.1.2.3.cmml">−</mo><mn id="S2.SS1.SSS0.P0.SPx1.p1.9.m9.2.2.1.1.1.2.3.2" 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encoding="application/x-llamapun" id="S2.SS1.SSS0.P0.SPx1.p1.9.m9.2d">italic_O ( italic_ε start_POSTSUPERSCRIPT - 6 end_POSTSUPERSCRIPT roman_log start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT italic_n roman_log italic_ρ start_POSTSUPERSCRIPT roman_max end_POSTSUPERSCRIPT ( italic_G ) )</annotation></semantics></math> time per insertion or deletion of edges in <math alttext="G" class="ltx_Math" display="inline" id="S2.SS1.SSS0.P0.SPx1.p1.10.m10.1"><semantics id="S2.SS1.SSS0.P0.SPx1.p1.10.m10.1a"><mi id="S2.SS1.SSS0.P0.SPx1.p1.10.m10.1.1" xref="S2.SS1.SSS0.P0.SPx1.p1.10.m10.1.1.cmml">G</mi><annotation-xml encoding="MathML-Content" id="S2.SS1.SSS0.P0.SPx1.p1.10.m10.1b"><ci id="S2.SS1.SSS0.P0.SPx1.p1.10.m10.1.1.cmml" xref="S2.SS1.SSS0.P0.SPx1.p1.10.m10.1.1">𝐺</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.SSS0.P0.SPx1.p1.10.m10.1c">G</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.SSS0.P0.SPx1.p1.10.m10.1d">italic_G</annotation></semantics></math>. By leveraging the <math alttext="\eta" class="ltx_Math" display="inline" id="S2.SS1.SSS0.P0.SPx1.p1.11.m11.1"><semantics id="S2.SS1.SSS0.P0.SPx1.p1.11.m11.1a"><mi id="S2.SS1.SSS0.P0.SPx1.p1.11.m11.1.1" xref="S2.SS1.SSS0.P0.SPx1.p1.11.m11.1.1.cmml">η</mi><annotation-xml encoding="MathML-Content" id="S2.SS1.SSS0.P0.SPx1.p1.11.m11.1b"><ci id="S2.SS1.SSS0.P0.SPx1.p1.11.m11.1.1.cmml" xref="S2.SS1.SSS0.P0.SPx1.p1.11.m11.1.1">𝜂</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.SSS0.P0.SPx1.p1.11.m11.1c">\eta</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.SSS0.P0.SPx1.p1.11.m11.1d">italic_η</annotation></semantics></math>-fairness of the orientation, they can report a <math alttext="(1+\varepsilon)" class="ltx_Math" display="inline" id="S2.SS1.SSS0.P0.SPx1.p1.12.m12.1"><semantics id="S2.SS1.SSS0.P0.SPx1.p1.12.m12.1a"><mrow id="S2.SS1.SSS0.P0.SPx1.p1.12.m12.1.1.1" xref="S2.SS1.SSS0.P0.SPx1.p1.12.m12.1.1.1.1.cmml"><mo id="S2.SS1.SSS0.P0.SPx1.p1.12.m12.1.1.1.2" stretchy="false" xref="S2.SS1.SSS0.P0.SPx1.p1.12.m12.1.1.1.1.cmml">(</mo><mrow id="S2.SS1.SSS0.P0.SPx1.p1.12.m12.1.1.1.1" xref="S2.SS1.SSS0.P0.SPx1.p1.12.m12.1.1.1.1.cmml"><mn id="S2.SS1.SSS0.P0.SPx1.p1.12.m12.1.1.1.1.2" xref="S2.SS1.SSS0.P0.SPx1.p1.12.m12.1.1.1.1.2.cmml">1</mn><mo id="S2.SS1.SSS0.P0.SPx1.p1.12.m12.1.1.1.1.1" xref="S2.SS1.SSS0.P0.SPx1.p1.12.m12.1.1.1.1.1.cmml">+</mo><mi id="S2.SS1.SSS0.P0.SPx1.p1.12.m12.1.1.1.1.3" xref="S2.SS1.SSS0.P0.SPx1.p1.12.m12.1.1.1.1.3.cmml">ε</mi></mrow><mo id="S2.SS1.SSS0.P0.SPx1.p1.12.m12.1.1.1.3" stretchy="false" xref="S2.SS1.SSS0.P0.SPx1.p1.12.m12.1.1.1.1.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.SSS0.P0.SPx1.p1.12.m12.1b"><apply id="S2.SS1.SSS0.P0.SPx1.p1.12.m12.1.1.1.1.cmml" xref="S2.SS1.SSS0.P0.SPx1.p1.12.m12.1.1.1"><plus id="S2.SS1.SSS0.P0.SPx1.p1.12.m12.1.1.1.1.1.cmml" xref="S2.SS1.SSS0.P0.SPx1.p1.12.m12.1.1.1.1.1"></plus><cn id="S2.SS1.SSS0.P0.SPx1.p1.12.m12.1.1.1.1.2.cmml" type="integer" xref="S2.SS1.SSS0.P0.SPx1.p1.12.m12.1.1.1.1.2">1</cn><ci id="S2.SS1.SSS0.P0.SPx1.p1.12.m12.1.1.1.1.3.cmml" xref="S2.SS1.SSS0.P0.SPx1.p1.12.m12.1.1.1.1.3">𝜀</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.SSS0.P0.SPx1.p1.12.m12.1c">(1+\varepsilon)</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.SSS0.P0.SPx1.p1.12.m12.1d">( 1 + italic_ε )</annotation></semantics></math>-approximate densest subgraph in time proportional to the size of the subgraph.</p> </div> </section> </section> <section class="ltx_subsection" id="S2.SS2"> <h3 class="ltx_title ltx_title_subsection"> <span class="ltx_tag ltx_tag_subsection">2.2 </span>Approximate densest subgraph in LOCAL and CONGEST</h3> <div class="ltx_para" id="S2.SS2.p1"> <p class="ltx_p" id="S2.SS2.p1.1">We focus on the <em class="ltx_emph ltx_font_italic" id="S2.SS2.p1.1.1">value</em> variant of the problem, where each vertex outputs a value (as opposed to the <em class="ltx_emph ltx_font_italic" id="S2.SS2.p1.1.2">reporting</em> variant in Appendix <a class="ltx_ref" href="https://arxiv.org/html/2411.12694v2#A1" title="Appendix A Reporting a densest subgraph ‣ Local Density and its Distributed Approximation"><span class="ltx_text ltx_ref_tag">A</span></a>, where the goal is to report a densest subgraph).</p> </div> <div class="ltx_theorem ltx_theorem_problem" id="S2.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="S2.Thmtheorem7.1.1.1">Problem 2.7</span></span><span class="ltx_text ltx_font_bold" id="S2.Thmtheorem7.2.2">.</span> </h6> <div class="ltx_para" id="S2.Thmtheorem7.p1"> <p class="ltx_p" id="S2.Thmtheorem7.p1.4"><span class="ltx_text ltx_font_italic" id="S2.Thmtheorem7.p1.4.4">Given a graph <math alttext="G" class="ltx_Math" display="inline" id="S2.Thmtheorem7.p1.1.1.m1.1"><semantics id="S2.Thmtheorem7.p1.1.1.m1.1a"><mi id="S2.Thmtheorem7.p1.1.1.m1.1.1" xref="S2.Thmtheorem7.p1.1.1.m1.1.1.cmml">G</mi><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem7.p1.1.1.m1.1b"><ci id="S2.Thmtheorem7.p1.1.1.m1.1.1.cmml" xref="S2.Thmtheorem7.p1.1.1.m1.1.1">𝐺</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem7.p1.1.1.m1.1c">G</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem7.p1.1.1.m1.1d">italic_G</annotation></semantics></math> and an <math alttext="\varepsilon>0" class="ltx_Math" display="inline" id="S2.Thmtheorem7.p1.2.2.m2.1"><semantics id="S2.Thmtheorem7.p1.2.2.m2.1a"><mrow id="S2.Thmtheorem7.p1.2.2.m2.1.1" xref="S2.Thmtheorem7.p1.2.2.m2.1.1.cmml"><mi id="S2.Thmtheorem7.p1.2.2.m2.1.1.2" xref="S2.Thmtheorem7.p1.2.2.m2.1.1.2.cmml">ε</mi><mo id="S2.Thmtheorem7.p1.2.2.m2.1.1.1" xref="S2.Thmtheorem7.p1.2.2.m2.1.1.1.cmml">></mo><mn id="S2.Thmtheorem7.p1.2.2.m2.1.1.3" xref="S2.Thmtheorem7.p1.2.2.m2.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem7.p1.2.2.m2.1b"><apply id="S2.Thmtheorem7.p1.2.2.m2.1.1.cmml" xref="S2.Thmtheorem7.p1.2.2.m2.1.1"><gt id="S2.Thmtheorem7.p1.2.2.m2.1.1.1.cmml" xref="S2.Thmtheorem7.p1.2.2.m2.1.1.1"></gt><ci id="S2.Thmtheorem7.p1.2.2.m2.1.1.2.cmml" xref="S2.Thmtheorem7.p1.2.2.m2.1.1.2">𝜀</ci><cn id="S2.Thmtheorem7.p1.2.2.m2.1.1.3.cmml" type="integer" xref="S2.Thmtheorem7.p1.2.2.m2.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem7.p1.2.2.m2.1c">\varepsilon>0</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem7.p1.2.2.m2.1d">italic_ε > 0</annotation></semantics></math>, each vertex <math alttext="v" class="ltx_Math" display="inline" id="S2.Thmtheorem7.p1.3.3.m3.1"><semantics id="S2.Thmtheorem7.p1.3.3.m3.1a"><mi id="S2.Thmtheorem7.p1.3.3.m3.1.1" xref="S2.Thmtheorem7.p1.3.3.m3.1.1.cmml">v</mi><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem7.p1.3.3.m3.1b"><ci id="S2.Thmtheorem7.p1.3.3.m3.1.1.cmml" xref="S2.Thmtheorem7.p1.3.3.m3.1.1">𝑣</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem7.p1.3.3.m3.1c">v</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem7.p1.3.3.m3.1d">italic_v</annotation></semantics></math> outputs a value <math alttext="\rho_{v}" class="ltx_Math" display="inline" id="S2.Thmtheorem7.p1.4.4.m4.1"><semantics id="S2.Thmtheorem7.p1.4.4.m4.1a"><msub id="S2.Thmtheorem7.p1.4.4.m4.1.1" xref="S2.Thmtheorem7.p1.4.4.m4.1.1.cmml"><mi id="S2.Thmtheorem7.p1.4.4.m4.1.1.2" xref="S2.Thmtheorem7.p1.4.4.m4.1.1.2.cmml">ρ</mi><mi id="S2.Thmtheorem7.p1.4.4.m4.1.1.3" xref="S2.Thmtheorem7.p1.4.4.m4.1.1.3.cmml">v</mi></msub><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem7.p1.4.4.m4.1b"><apply id="S2.Thmtheorem7.p1.4.4.m4.1.1.cmml" xref="S2.Thmtheorem7.p1.4.4.m4.1.1"><csymbol cd="ambiguous" id="S2.Thmtheorem7.p1.4.4.m4.1.1.1.cmml" xref="S2.Thmtheorem7.p1.4.4.m4.1.1">subscript</csymbol><ci id="S2.Thmtheorem7.p1.4.4.m4.1.1.2.cmml" xref="S2.Thmtheorem7.p1.4.4.m4.1.1.2">𝜌</ci><ci id="S2.Thmtheorem7.p1.4.4.m4.1.1.3.cmml" xref="S2.Thmtheorem7.p1.4.4.m4.1.1.3">𝑣</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem7.p1.4.4.m4.1c">\rho_{v}</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem7.p1.4.4.m4.1d">italic_ρ start_POSTSUBSCRIPT italic_v end_POSTSUBSCRIPT</annotation></semantics></math> and either:</span></p> <ul class="ltx_itemize" id="S2.I1"> <li class="ltx_item" id="S2.I1.i1" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">•</span> <div class="ltx_para" id="S2.I1.i1.p1"> <p class="ltx_p" id="S2.I1.i1.p1.1"><span class="ltx_text ltx_font_bold ltx_font_italic" id="S2.I1.i1.p1.1.1">Problem 2.1:</span><span class="ltx_text ltx_font_italic" id="S2.I1.i1.p1.1.2"> we require that </span><math 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encoding="application/x-llamapun" id="S2.I1.i1.p1.1.m1.6d">∀ italic_v , italic_ρ start_POSTSUBSCRIPT italic_v end_POSTSUBSCRIPT ∈ [ ( 1 + italic_ε ) start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT italic_ρ start_POSTSUPERSCRIPT roman_max end_POSTSUPERSCRIPT ( italic_G ) , ( 1 + italic_ε ) italic_ρ start_POSTSUPERSCRIPT roman_max end_POSTSUPERSCRIPT ( italic_G ) ]</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S2.I1.i1.p1.1.3">, or</span></p> </div> </li> <li class="ltx_item" id="S2.I1.i2" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">•</span> <div class="ltx_para" id="S2.I1.i2.p1"> <p class="ltx_p" id="S2.I1.i2.p1.1"><span class="ltx_text ltx_font_bold ltx_font_italic" id="S2.I1.i2.p1.1.1">Problem 2.2</span><span class="ltx_text ltx_font_italic" id="S2.I1.i2.p1.1.2">: we require that </span><math alttext="\,\max_{v}\rho_{v}\in[(1+\varepsilon)^{-1}\rho^{\max}(G),(1+\varepsilon)\rho^{% \max}(G)]" class="ltx_Math" display="inline" 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italic_v end_POSTSUBSCRIPT ∈ [ ( 1 + italic_ε ) start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT italic_ρ start_POSTSUPERSCRIPT roman_max end_POSTSUPERSCRIPT ( italic_G ) , ( 1 + italic_ε ) italic_ρ start_POSTSUPERSCRIPT roman_max end_POSTSUPERSCRIPT ( italic_G ) ]</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S2.I1.i2.p1.1.3">.</span></p> </div> </li> </ul> </div> </div> <section class="ltx_subparagraph" id="S2.SS2.SSS0.P0.SPx1"> <h5 class="ltx_title ltx_title_subparagraph">Related work. </h5> <div class="ltx_para" id="S2.SS2.SSS0.P0.SPx1.p1"> <p class="ltx_p" id="S2.SS2.SSS0.P0.SPx1.p1.22">Problem <a class="ltx_ref" href="https://arxiv.org/html/2411.12694v2#S2.Thmtheorem7" title="Problem 2.7. ‣ 2.2 Approximate densest subgraph in LOCAL and CONGEST ‣ 2 Preliminaries and related work ‣ Local Density and its Distributed Approximation"><span class="ltx_text ltx_ref_tag">2.7</span></a>.1 has a trivial <math alttext="\Omega(D)" class="ltx_Math" display="inline" id="S2.SS2.SSS0.P0.SPx1.p1.1.m1.1"><semantics id="S2.SS2.SSS0.P0.SPx1.p1.1.m1.1a"><mrow id="S2.SS2.SSS0.P0.SPx1.p1.1.m1.1.2" xref="S2.SS2.SSS0.P0.SPx1.p1.1.m1.1.2.cmml"><mi id="S2.SS2.SSS0.P0.SPx1.p1.1.m1.1.2.2" mathvariant="normal" xref="S2.SS2.SSS0.P0.SPx1.p1.1.m1.1.2.2.cmml">Ω</mi><mo id="S2.SS2.SSS0.P0.SPx1.p1.1.m1.1.2.1" xref="S2.SS2.SSS0.P0.SPx1.p1.1.m1.1.2.1.cmml">⁢</mo><mrow id="S2.SS2.SSS0.P0.SPx1.p1.1.m1.1.2.3.2" xref="S2.SS2.SSS0.P0.SPx1.p1.1.m1.1.2.cmml"><mo id="S2.SS2.SSS0.P0.SPx1.p1.1.m1.1.2.3.2.1" stretchy="false" xref="S2.SS2.SSS0.P0.SPx1.p1.1.m1.1.2.cmml">(</mo><mi id="S2.SS2.SSS0.P0.SPx1.p1.1.m1.1.1" xref="S2.SS2.SSS0.P0.SPx1.p1.1.m1.1.1.cmml">D</mi><mo id="S2.SS2.SSS0.P0.SPx1.p1.1.m1.1.2.3.2.2" stretchy="false" xref="S2.SS2.SSS0.P0.SPx1.p1.1.m1.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.SSS0.P0.SPx1.p1.1.m1.1b"><apply id="S2.SS2.SSS0.P0.SPx1.p1.1.m1.1.2.cmml" xref="S2.SS2.SSS0.P0.SPx1.p1.1.m1.1.2"><times id="S2.SS2.SSS0.P0.SPx1.p1.1.m1.1.2.1.cmml" xref="S2.SS2.SSS0.P0.SPx1.p1.1.m1.1.2.1"></times><ci id="S2.SS2.SSS0.P0.SPx1.p1.1.m1.1.2.2.cmml" xref="S2.SS2.SSS0.P0.SPx1.p1.1.m1.1.2.2">Ω</ci><ci id="S2.SS2.SSS0.P0.SPx1.p1.1.m1.1.1.cmml" xref="S2.SS2.SSS0.P0.SPx1.p1.1.m1.1.1">𝐷</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.SSS0.P0.SPx1.p1.1.m1.1c">\Omega(D)</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.SSS0.P0.SPx1.p1.1.m1.1d">roman_Ω ( italic_D )</annotation></semantics></math> lower bound, obtained by constructing a lollipop graph (where <math alttext="D" class="ltx_Math" display="inline" id="S2.SS2.SSS0.P0.SPx1.p1.2.m2.1"><semantics id="S2.SS2.SSS0.P0.SPx1.p1.2.m2.1a"><mi id="S2.SS2.SSS0.P0.SPx1.p1.2.m2.1.1" xref="S2.SS2.SSS0.P0.SPx1.p1.2.m2.1.1.cmml">D</mi><annotation-xml encoding="MathML-Content" id="S2.SS2.SSS0.P0.SPx1.p1.2.m2.1b"><ci id="S2.SS2.SSS0.P0.SPx1.p1.2.m2.1.1.cmml" xref="S2.SS2.SSS0.P0.SPx1.p1.2.m2.1.1">𝐷</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.SSS0.P0.SPx1.p1.2.m2.1c">D</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.SSS0.P0.SPx1.p1.2.m2.1d">italic_D</annotation></semantics></math> denotes the diameter). In LOCAL, it is trivial to solve Problem <a class="ltx_ref" href="https://arxiv.org/html/2411.12694v2#S2.Thmtheorem7" title="Problem 2.7. ‣ 2.2 Approximate densest subgraph in LOCAL and CONGEST ‣ 2 Preliminaries and related work ‣ Local Density and its Distributed Approximation"><span class="ltx_text ltx_ref_tag">2.7</span></a>.1 in <math alttext="\Theta(D)" class="ltx_Math" display="inline" id="S2.SS2.SSS0.P0.SPx1.p1.3.m3.1"><semantics id="S2.SS2.SSS0.P0.SPx1.p1.3.m3.1a"><mrow id="S2.SS2.SSS0.P0.SPx1.p1.3.m3.1.2" xref="S2.SS2.SSS0.P0.SPx1.p1.3.m3.1.2.cmml"><mi id="S2.SS2.SSS0.P0.SPx1.p1.3.m3.1.2.2" mathvariant="normal" xref="S2.SS2.SSS0.P0.SPx1.p1.3.m3.1.2.2.cmml">Θ</mi><mo id="S2.SS2.SSS0.P0.SPx1.p1.3.m3.1.2.1" xref="S2.SS2.SSS0.P0.SPx1.p1.3.m3.1.2.1.cmml">⁢</mo><mrow id="S2.SS2.SSS0.P0.SPx1.p1.3.m3.1.2.3.2" xref="S2.SS2.SSS0.P0.SPx1.p1.3.m3.1.2.cmml"><mo id="S2.SS2.SSS0.P0.SPx1.p1.3.m3.1.2.3.2.1" stretchy="false" xref="S2.SS2.SSS0.P0.SPx1.p1.3.m3.1.2.cmml">(</mo><mi id="S2.SS2.SSS0.P0.SPx1.p1.3.m3.1.1" xref="S2.SS2.SSS0.P0.SPx1.p1.3.m3.1.1.cmml">D</mi><mo id="S2.SS2.SSS0.P0.SPx1.p1.3.m3.1.2.3.2.2" stretchy="false" xref="S2.SS2.SSS0.P0.SPx1.p1.3.m3.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.SSS0.P0.SPx1.p1.3.m3.1b"><apply id="S2.SS2.SSS0.P0.SPx1.p1.3.m3.1.2.cmml" xref="S2.SS2.SSS0.P0.SPx1.p1.3.m3.1.2"><times id="S2.SS2.SSS0.P0.SPx1.p1.3.m3.1.2.1.cmml" xref="S2.SS2.SSS0.P0.SPx1.p1.3.m3.1.2.1"></times><ci id="S2.SS2.SSS0.P0.SPx1.p1.3.m3.1.2.2.cmml" xref="S2.SS2.SSS0.P0.SPx1.p1.3.m3.1.2.2">Θ</ci><ci id="S2.SS2.SSS0.P0.SPx1.p1.3.m3.1.1.cmml" xref="S2.SS2.SSS0.P0.SPx1.p1.3.m3.1.1">𝐷</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.SSS0.P0.SPx1.p1.3.m3.1c">\Theta(D)</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.SSS0.P0.SPx1.p1.3.m3.1d">roman_Θ ( italic_D )</annotation></semantics></math> time. Problem <a class="ltx_ref" href="https://arxiv.org/html/2411.12694v2#S2.Thmtheorem7" title="Problem 2.7. ‣ 2.2 Approximate densest subgraph in LOCAL and CONGEST ‣ 2 Preliminaries and related work ‣ Local Density and its Distributed Approximation"><span class="ltx_text ltx_ref_tag">2.7</span></a>.2 was studied by Ghaffari and Su <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2411.12694v2#bib.bib14" title="">14</a>]</cite> who present a randomised <math alttext="(1+\varepsilon)" class="ltx_Math" display="inline" id="S2.SS2.SSS0.P0.SPx1.p1.4.m4.1"><semantics id="S2.SS2.SSS0.P0.SPx1.p1.4.m4.1a"><mrow id="S2.SS2.SSS0.P0.SPx1.p1.4.m4.1.1.1" xref="S2.SS2.SSS0.P0.SPx1.p1.4.m4.1.1.1.1.cmml"><mo id="S2.SS2.SSS0.P0.SPx1.p1.4.m4.1.1.1.2" stretchy="false" xref="S2.SS2.SSS0.P0.SPx1.p1.4.m4.1.1.1.1.cmml">(</mo><mrow id="S2.SS2.SSS0.P0.SPx1.p1.4.m4.1.1.1.1" xref="S2.SS2.SSS0.P0.SPx1.p1.4.m4.1.1.1.1.cmml"><mn id="S2.SS2.SSS0.P0.SPx1.p1.4.m4.1.1.1.1.2" 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id="S2.SS2.SSS0.P0.SPx1.p1.4.m4.1c">(1+\varepsilon)</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.SSS0.P0.SPx1.p1.4.m4.1d">( 1 + italic_ε )</annotation></semantics></math>-approximation in LOCAL that uses <math alttext="O(\varepsilon^{-3}\log^{4}n)" class="ltx_Math" display="inline" id="S2.SS2.SSS0.P0.SPx1.p1.5.m5.1"><semantics id="S2.SS2.SSS0.P0.SPx1.p1.5.m5.1a"><mrow id="S2.SS2.SSS0.P0.SPx1.p1.5.m5.1.1" xref="S2.SS2.SSS0.P0.SPx1.p1.5.m5.1.1.cmml"><mi id="S2.SS2.SSS0.P0.SPx1.p1.5.m5.1.1.3" xref="S2.SS2.SSS0.P0.SPx1.p1.5.m5.1.1.3.cmml">O</mi><mo id="S2.SS2.SSS0.P0.SPx1.p1.5.m5.1.1.2" xref="S2.SS2.SSS0.P0.SPx1.p1.5.m5.1.1.2.cmml">⁢</mo><mrow id="S2.SS2.SSS0.P0.SPx1.p1.5.m5.1.1.1.1" xref="S2.SS2.SSS0.P0.SPx1.p1.5.m5.1.1.1.1.1.cmml"><mo id="S2.SS2.SSS0.P0.SPx1.p1.5.m5.1.1.1.1.2" stretchy="false" xref="S2.SS2.SSS0.P0.SPx1.p1.5.m5.1.1.1.1.1.cmml">(</mo><mrow id="S2.SS2.SSS0.P0.SPx1.p1.5.m5.1.1.1.1.1" xref="S2.SS2.SSS0.P0.SPx1.p1.5.m5.1.1.1.1.1.cmml"><msup 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xref="S2.SS2.SSS0.P0.SPx1.p1.5.m5.1.1.1.1.1.3.1.2.cmml">log</mi><mn id="S2.SS2.SSS0.P0.SPx1.p1.5.m5.1.1.1.1.1.3.1.3" xref="S2.SS2.SSS0.P0.SPx1.p1.5.m5.1.1.1.1.1.3.1.3.cmml">4</mn></msup><mo id="S2.SS2.SSS0.P0.SPx1.p1.5.m5.1.1.1.1.1.3a" lspace="0.167em" xref="S2.SS2.SSS0.P0.SPx1.p1.5.m5.1.1.1.1.1.3.cmml">⁡</mo><mi id="S2.SS2.SSS0.P0.SPx1.p1.5.m5.1.1.1.1.1.3.2" xref="S2.SS2.SSS0.P0.SPx1.p1.5.m5.1.1.1.1.1.3.2.cmml">n</mi></mrow></mrow><mo id="S2.SS2.SSS0.P0.SPx1.p1.5.m5.1.1.1.1.3" stretchy="false" xref="S2.SS2.SSS0.P0.SPx1.p1.5.m5.1.1.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.SSS0.P0.SPx1.p1.5.m5.1b"><apply id="S2.SS2.SSS0.P0.SPx1.p1.5.m5.1.1.cmml" xref="S2.SS2.SSS0.P0.SPx1.p1.5.m5.1.1"><times id="S2.SS2.SSS0.P0.SPx1.p1.5.m5.1.1.2.cmml" xref="S2.SS2.SSS0.P0.SPx1.p1.5.m5.1.1.2"></times><ci id="S2.SS2.SSS0.P0.SPx1.p1.5.m5.1.1.3.cmml" xref="S2.SS2.SSS0.P0.SPx1.p1.5.m5.1.1.3">𝑂</ci><apply id="S2.SS2.SSS0.P0.SPx1.p1.5.m5.1.1.1.1.1.cmml" xref="S2.SS2.SSS0.P0.SPx1.p1.5.m5.1.1.1.1"><times id="S2.SS2.SSS0.P0.SPx1.p1.5.m5.1.1.1.1.1.1.cmml" xref="S2.SS2.SSS0.P0.SPx1.p1.5.m5.1.1.1.1.1.1"></times><apply id="S2.SS2.SSS0.P0.SPx1.p1.5.m5.1.1.1.1.1.2.cmml" xref="S2.SS2.SSS0.P0.SPx1.p1.5.m5.1.1.1.1.1.2"><csymbol cd="ambiguous" id="S2.SS2.SSS0.P0.SPx1.p1.5.m5.1.1.1.1.1.2.1.cmml" xref="S2.SS2.SSS0.P0.SPx1.p1.5.m5.1.1.1.1.1.2">superscript</csymbol><ci id="S2.SS2.SSS0.P0.SPx1.p1.5.m5.1.1.1.1.1.2.2.cmml" xref="S2.SS2.SSS0.P0.SPx1.p1.5.m5.1.1.1.1.1.2.2">𝜀</ci><apply id="S2.SS2.SSS0.P0.SPx1.p1.5.m5.1.1.1.1.1.2.3.cmml" xref="S2.SS2.SSS0.P0.SPx1.p1.5.m5.1.1.1.1.1.2.3"><minus id="S2.SS2.SSS0.P0.SPx1.p1.5.m5.1.1.1.1.1.2.3.1.cmml" xref="S2.SS2.SSS0.P0.SPx1.p1.5.m5.1.1.1.1.1.2.3"></minus><cn id="S2.SS2.SSS0.P0.SPx1.p1.5.m5.1.1.1.1.1.2.3.2.cmml" type="integer" xref="S2.SS2.SSS0.P0.SPx1.p1.5.m5.1.1.1.1.1.2.3.2">3</cn></apply></apply><apply id="S2.SS2.SSS0.P0.SPx1.p1.5.m5.1.1.1.1.1.3.cmml" xref="S2.SS2.SSS0.P0.SPx1.p1.5.m5.1.1.1.1.1.3"><apply id="S2.SS2.SSS0.P0.SPx1.p1.5.m5.1.1.1.1.1.3.1.cmml" xref="S2.SS2.SSS0.P0.SPx1.p1.5.m5.1.1.1.1.1.3.1"><csymbol cd="ambiguous" id="S2.SS2.SSS0.P0.SPx1.p1.5.m5.1.1.1.1.1.3.1.1.cmml" xref="S2.SS2.SSS0.P0.SPx1.p1.5.m5.1.1.1.1.1.3.1">superscript</csymbol><log id="S2.SS2.SSS0.P0.SPx1.p1.5.m5.1.1.1.1.1.3.1.2.cmml" xref="S2.SS2.SSS0.P0.SPx1.p1.5.m5.1.1.1.1.1.3.1.2"></log><cn id="S2.SS2.SSS0.P0.SPx1.p1.5.m5.1.1.1.1.1.3.1.3.cmml" type="integer" xref="S2.SS2.SSS0.P0.SPx1.p1.5.m5.1.1.1.1.1.3.1.3">4</cn></apply><ci id="S2.SS2.SSS0.P0.SPx1.p1.5.m5.1.1.1.1.1.3.2.cmml" xref="S2.SS2.SSS0.P0.SPx1.p1.5.m5.1.1.1.1.1.3.2">𝑛</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.SSS0.P0.SPx1.p1.5.m5.1c">O(\varepsilon^{-3}\log^{4}n)</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.SSS0.P0.SPx1.p1.5.m5.1d">italic_O ( italic_ε start_POSTSUPERSCRIPT - 3 end_POSTSUPERSCRIPT roman_log start_POSTSUPERSCRIPT 4 end_POSTSUPERSCRIPT italic_n )</annotation></semantics></math> rounds. Fischer et al. <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2411.12694v2#bib.bib12" title="">12</a>]</cite> present a deterministic <math alttext="(1+\varepsilon)" class="ltx_Math" display="inline" id="S2.SS2.SSS0.P0.SPx1.p1.6.m6.1"><semantics id="S2.SS2.SSS0.P0.SPx1.p1.6.m6.1a"><mrow id="S2.SS2.SSS0.P0.SPx1.p1.6.m6.1.1.1" xref="S2.SS2.SSS0.P0.SPx1.p1.6.m6.1.1.1.1.cmml"><mo id="S2.SS2.SSS0.P0.SPx1.p1.6.m6.1.1.1.2" stretchy="false" xref="S2.SS2.SSS0.P0.SPx1.p1.6.m6.1.1.1.1.cmml">(</mo><mrow id="S2.SS2.SSS0.P0.SPx1.p1.6.m6.1.1.1.1" xref="S2.SS2.SSS0.P0.SPx1.p1.6.m6.1.1.1.1.cmml"><mn id="S2.SS2.SSS0.P0.SPx1.p1.6.m6.1.1.1.1.2" xref="S2.SS2.SSS0.P0.SPx1.p1.6.m6.1.1.1.1.2.cmml">1</mn><mo id="S2.SS2.SSS0.P0.SPx1.p1.6.m6.1.1.1.1.1" xref="S2.SS2.SSS0.P0.SPx1.p1.6.m6.1.1.1.1.1.cmml">+</mo><mi id="S2.SS2.SSS0.P0.SPx1.p1.6.m6.1.1.1.1.3" xref="S2.SS2.SSS0.P0.SPx1.p1.6.m6.1.1.1.1.3.cmml">ε</mi></mrow><mo id="S2.SS2.SSS0.P0.SPx1.p1.6.m6.1.1.1.3" stretchy="false" xref="S2.SS2.SSS0.P0.SPx1.p1.6.m6.1.1.1.1.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.SSS0.P0.SPx1.p1.6.m6.1b"><apply id="S2.SS2.SSS0.P0.SPx1.p1.6.m6.1.1.1.1.cmml" xref="S2.SS2.SSS0.P0.SPx1.p1.6.m6.1.1.1"><plus id="S2.SS2.SSS0.P0.SPx1.p1.6.m6.1.1.1.1.1.cmml" xref="S2.SS2.SSS0.P0.SPx1.p1.6.m6.1.1.1.1.1"></plus><cn id="S2.SS2.SSS0.P0.SPx1.p1.6.m6.1.1.1.1.2.cmml" type="integer" xref="S2.SS2.SSS0.P0.SPx1.p1.6.m6.1.1.1.1.2">1</cn><ci id="S2.SS2.SSS0.P0.SPx1.p1.6.m6.1.1.1.1.3.cmml" xref="S2.SS2.SSS0.P0.SPx1.p1.6.m6.1.1.1.1.3">𝜀</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.SSS0.P0.SPx1.p1.6.m6.1c">(1+\varepsilon)</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.SSS0.P0.SPx1.p1.6.m6.1d">( 1 + italic_ε )</annotation></semantics></math>-approximation in LOCAL that uses <math alttext="2^{O(\log^{2}(\varepsilon^{-1}\log n))}" class="ltx_Math" display="inline" 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id="S2.SS2.SSS0.P0.SPx1.p1.7.m7.1.1.1.1.1.1.2.2.1.3.1.cmml" xref="S2.SS2.SSS0.P0.SPx1.p1.7.m7.1.1.1.1.1.1.2.2.1.3.1"></log><ci id="S2.SS2.SSS0.P0.SPx1.p1.7.m7.1.1.1.1.1.1.2.2.1.3.2.cmml" xref="S2.SS2.SSS0.P0.SPx1.p1.7.m7.1.1.1.1.1.1.2.2.1.3.2">𝑛</ci></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.SSS0.P0.SPx1.p1.7.m7.1c">2^{O(\log^{2}(\varepsilon^{-1}\log n))}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.SSS0.P0.SPx1.p1.7.m7.1d">2 start_POSTSUPERSCRIPT italic_O ( roman_log start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ( italic_ε start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT roman_log italic_n ) ) end_POSTSUPERSCRIPT</annotation></semantics></math> rounds. Ghaffari et al. <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2411.12694v2#bib.bib13" title="">13</a>]</cite> improve this to <math alttext="O(\varepsilon^{-9}\log^{15}n)" class="ltx_Math" display="inline" id="S2.SS2.SSS0.P0.SPx1.p1.8.m8.1"><semantics id="S2.SS2.SSS0.P0.SPx1.p1.8.m8.1a"><mrow id="S2.SS2.SSS0.P0.SPx1.p1.8.m8.1.1" xref="S2.SS2.SSS0.P0.SPx1.p1.8.m8.1.1.cmml"><mi id="S2.SS2.SSS0.P0.SPx1.p1.8.m8.1.1.3" xref="S2.SS2.SSS0.P0.SPx1.p1.8.m8.1.1.3.cmml">O</mi><mo id="S2.SS2.SSS0.P0.SPx1.p1.8.m8.1.1.2" xref="S2.SS2.SSS0.P0.SPx1.p1.8.m8.1.1.2.cmml">⁢</mo><mrow id="S2.SS2.SSS0.P0.SPx1.p1.8.m8.1.1.1.1" xref="S2.SS2.SSS0.P0.SPx1.p1.8.m8.1.1.1.1.1.cmml"><mo id="S2.SS2.SSS0.P0.SPx1.p1.8.m8.1.1.1.1.2" stretchy="false" xref="S2.SS2.SSS0.P0.SPx1.p1.8.m8.1.1.1.1.1.cmml">(</mo><mrow id="S2.SS2.SSS0.P0.SPx1.p1.8.m8.1.1.1.1.1" xref="S2.SS2.SSS0.P0.SPx1.p1.8.m8.1.1.1.1.1.cmml"><msup id="S2.SS2.SSS0.P0.SPx1.p1.8.m8.1.1.1.1.1.2" xref="S2.SS2.SSS0.P0.SPx1.p1.8.m8.1.1.1.1.1.2.cmml"><mi id="S2.SS2.SSS0.P0.SPx1.p1.8.m8.1.1.1.1.1.2.2" xref="S2.SS2.SSS0.P0.SPx1.p1.8.m8.1.1.1.1.1.2.2.cmml">ε</mi><mrow id="S2.SS2.SSS0.P0.SPx1.p1.8.m8.1.1.1.1.1.2.3" xref="S2.SS2.SSS0.P0.SPx1.p1.8.m8.1.1.1.1.1.2.3.cmml"><mo id="S2.SS2.SSS0.P0.SPx1.p1.8.m8.1.1.1.1.1.2.3a" xref="S2.SS2.SSS0.P0.SPx1.p1.8.m8.1.1.1.1.1.2.3.cmml">−</mo><mn id="S2.SS2.SSS0.P0.SPx1.p1.8.m8.1.1.1.1.1.2.3.2" xref="S2.SS2.SSS0.P0.SPx1.p1.8.m8.1.1.1.1.1.2.3.2.cmml">9</mn></mrow></msup><mo id="S2.SS2.SSS0.P0.SPx1.p1.8.m8.1.1.1.1.1.1" lspace="0.167em" xref="S2.SS2.SSS0.P0.SPx1.p1.8.m8.1.1.1.1.1.1.cmml">⁢</mo><mrow id="S2.SS2.SSS0.P0.SPx1.p1.8.m8.1.1.1.1.1.3" xref="S2.SS2.SSS0.P0.SPx1.p1.8.m8.1.1.1.1.1.3.cmml"><msup id="S2.SS2.SSS0.P0.SPx1.p1.8.m8.1.1.1.1.1.3.1" xref="S2.SS2.SSS0.P0.SPx1.p1.8.m8.1.1.1.1.1.3.1.cmml"><mi id="S2.SS2.SSS0.P0.SPx1.p1.8.m8.1.1.1.1.1.3.1.2" xref="S2.SS2.SSS0.P0.SPx1.p1.8.m8.1.1.1.1.1.3.1.2.cmml">log</mi><mn id="S2.SS2.SSS0.P0.SPx1.p1.8.m8.1.1.1.1.1.3.1.3" xref="S2.SS2.SSS0.P0.SPx1.p1.8.m8.1.1.1.1.1.3.1.3.cmml">15</mn></msup><mo id="S2.SS2.SSS0.P0.SPx1.p1.8.m8.1.1.1.1.1.3a" lspace="0.167em" xref="S2.SS2.SSS0.P0.SPx1.p1.8.m8.1.1.1.1.1.3.cmml">⁡</mo><mi id="S2.SS2.SSS0.P0.SPx1.p1.8.m8.1.1.1.1.1.3.2" xref="S2.SS2.SSS0.P0.SPx1.p1.8.m8.1.1.1.1.1.3.2.cmml">n</mi></mrow></mrow><mo id="S2.SS2.SSS0.P0.SPx1.p1.8.m8.1.1.1.1.3" stretchy="false" xref="S2.SS2.SSS0.P0.SPx1.p1.8.m8.1.1.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.SSS0.P0.SPx1.p1.8.m8.1b"><apply id="S2.SS2.SSS0.P0.SPx1.p1.8.m8.1.1.cmml" xref="S2.SS2.SSS0.P0.SPx1.p1.8.m8.1.1"><times id="S2.SS2.SSS0.P0.SPx1.p1.8.m8.1.1.2.cmml" xref="S2.SS2.SSS0.P0.SPx1.p1.8.m8.1.1.2"></times><ci id="S2.SS2.SSS0.P0.SPx1.p1.8.m8.1.1.3.cmml" xref="S2.SS2.SSS0.P0.SPx1.p1.8.m8.1.1.3">𝑂</ci><apply id="S2.SS2.SSS0.P0.SPx1.p1.8.m8.1.1.1.1.1.cmml" xref="S2.SS2.SSS0.P0.SPx1.p1.8.m8.1.1.1.1"><times id="S2.SS2.SSS0.P0.SPx1.p1.8.m8.1.1.1.1.1.1.cmml" xref="S2.SS2.SSS0.P0.SPx1.p1.8.m8.1.1.1.1.1.1"></times><apply id="S2.SS2.SSS0.P0.SPx1.p1.8.m8.1.1.1.1.1.2.cmml" xref="S2.SS2.SSS0.P0.SPx1.p1.8.m8.1.1.1.1.1.2"><csymbol cd="ambiguous" id="S2.SS2.SSS0.P0.SPx1.p1.8.m8.1.1.1.1.1.2.1.cmml" xref="S2.SS2.SSS0.P0.SPx1.p1.8.m8.1.1.1.1.1.2">superscript</csymbol><ci id="S2.SS2.SSS0.P0.SPx1.p1.8.m8.1.1.1.1.1.2.2.cmml" xref="S2.SS2.SSS0.P0.SPx1.p1.8.m8.1.1.1.1.1.2.2">𝜀</ci><apply id="S2.SS2.SSS0.P0.SPx1.p1.8.m8.1.1.1.1.1.2.3.cmml" xref="S2.SS2.SSS0.P0.SPx1.p1.8.m8.1.1.1.1.1.2.3"><minus id="S2.SS2.SSS0.P0.SPx1.p1.8.m8.1.1.1.1.1.2.3.1.cmml" xref="S2.SS2.SSS0.P0.SPx1.p1.8.m8.1.1.1.1.1.2.3"></minus><cn id="S2.SS2.SSS0.P0.SPx1.p1.8.m8.1.1.1.1.1.2.3.2.cmml" type="integer" xref="S2.SS2.SSS0.P0.SPx1.p1.8.m8.1.1.1.1.1.2.3.2">9</cn></apply></apply><apply id="S2.SS2.SSS0.P0.SPx1.p1.8.m8.1.1.1.1.1.3.cmml" xref="S2.SS2.SSS0.P0.SPx1.p1.8.m8.1.1.1.1.1.3"><apply id="S2.SS2.SSS0.P0.SPx1.p1.8.m8.1.1.1.1.1.3.1.cmml" xref="S2.SS2.SSS0.P0.SPx1.p1.8.m8.1.1.1.1.1.3.1"><csymbol cd="ambiguous" id="S2.SS2.SSS0.P0.SPx1.p1.8.m8.1.1.1.1.1.3.1.1.cmml" xref="S2.SS2.SSS0.P0.SPx1.p1.8.m8.1.1.1.1.1.3.1">superscript</csymbol><log id="S2.SS2.SSS0.P0.SPx1.p1.8.m8.1.1.1.1.1.3.1.2.cmml" xref="S2.SS2.SSS0.P0.SPx1.p1.8.m8.1.1.1.1.1.3.1.2"></log><cn id="S2.SS2.SSS0.P0.SPx1.p1.8.m8.1.1.1.1.1.3.1.3.cmml" type="integer" xref="S2.SS2.SSS0.P0.SPx1.p1.8.m8.1.1.1.1.1.3.1.3">15</cn></apply><ci id="S2.SS2.SSS0.P0.SPx1.p1.8.m8.1.1.1.1.1.3.2.cmml" xref="S2.SS2.SSS0.P0.SPx1.p1.8.m8.1.1.1.1.1.3.2">𝑛</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.SSS0.P0.SPx1.p1.8.m8.1c">O(\varepsilon^{-9}\log^{15}n)</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.SSS0.P0.SPx1.p1.8.m8.1d">italic_O ( italic_ε start_POSTSUPERSCRIPT - 9 end_POSTSUPERSCRIPT roman_log start_POSTSUPERSCRIPT 15 end_POSTSUPERSCRIPT italic_n )</annotation></semantics></math> rounds. The work by Harris <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2411.12694v2#bib.bib16" title="">16</a>]</cite> improves this to <math alttext="\tilde{O}(\varepsilon^{-6}\log^{6}n)" class="ltx_Math" display="inline" id="S2.SS2.SSS0.P0.SPx1.p1.9.m9.1"><semantics id="S2.SS2.SSS0.P0.SPx1.p1.9.m9.1a"><mrow id="S2.SS2.SSS0.P0.SPx1.p1.9.m9.1.1" xref="S2.SS2.SSS0.P0.SPx1.p1.9.m9.1.1.cmml"><mover accent="true" id="S2.SS2.SSS0.P0.SPx1.p1.9.m9.1.1.3" xref="S2.SS2.SSS0.P0.SPx1.p1.9.m9.1.1.3.cmml"><mi id="S2.SS2.SSS0.P0.SPx1.p1.9.m9.1.1.3.2" xref="S2.SS2.SSS0.P0.SPx1.p1.9.m9.1.1.3.2.cmml">O</mi><mo id="S2.SS2.SSS0.P0.SPx1.p1.9.m9.1.1.3.1" xref="S2.SS2.SSS0.P0.SPx1.p1.9.m9.1.1.3.1.cmml">~</mo></mover><mo id="S2.SS2.SSS0.P0.SPx1.p1.9.m9.1.1.2" xref="S2.SS2.SSS0.P0.SPx1.p1.9.m9.1.1.2.cmml">⁢</mo><mrow id="S2.SS2.SSS0.P0.SPx1.p1.9.m9.1.1.1.1" xref="S2.SS2.SSS0.P0.SPx1.p1.9.m9.1.1.1.1.1.cmml"><mo id="S2.SS2.SSS0.P0.SPx1.p1.9.m9.1.1.1.1.2" stretchy="false" 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encoding="application/x-tex" id="S2.SS2.SSS0.P0.SPx1.p1.9.m9.1c">\tilde{O}(\varepsilon^{-6}\log^{6}n)</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.SSS0.P0.SPx1.p1.9.m9.1d">over~ start_ARG italic_O end_ARG ( italic_ε start_POSTSUPERSCRIPT - 6 end_POSTSUPERSCRIPT roman_log start_POSTSUPERSCRIPT 6 end_POSTSUPERSCRIPT italic_n )</annotation></semantics></math> rounds. Su and Vu <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2411.12694v2#bib.bib21" title="">21</a>]</cite> present the state-of-the-art in this area. They prove that for any graph <math alttext="G" class="ltx_Math" display="inline" id="S2.SS2.SSS0.P0.SPx1.p1.10.m10.1"><semantics id="S2.SS2.SSS0.P0.SPx1.p1.10.m10.1a"><mi id="S2.SS2.SSS0.P0.SPx1.p1.10.m10.1.1" xref="S2.SS2.SSS0.P0.SPx1.p1.10.m10.1.1.cmml">G</mi><annotation-xml encoding="MathML-Content" id="S2.SS2.SSS0.P0.SPx1.p1.10.m10.1b"><ci id="S2.SS2.SSS0.P0.SPx1.p1.10.m10.1.1.cmml" xref="S2.SS2.SSS0.P0.SPx1.p1.10.m10.1.1">𝐺</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.SSS0.P0.SPx1.p1.10.m10.1c">G</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.SSS0.P0.SPx1.p1.10.m10.1d">italic_G</annotation></semantics></math>, there exists a vertex <math alttext="v" class="ltx_Math" display="inline" id="S2.SS2.SSS0.P0.SPx1.p1.11.m11.1"><semantics id="S2.SS2.SSS0.P0.SPx1.p1.11.m11.1a"><mi id="S2.SS2.SSS0.P0.SPx1.p1.11.m11.1.1" xref="S2.SS2.SSS0.P0.SPx1.p1.11.m11.1.1.cmml">v</mi><annotation-xml encoding="MathML-Content" id="S2.SS2.SSS0.P0.SPx1.p1.11.m11.1b"><ci id="S2.SS2.SSS0.P0.SPx1.p1.11.m11.1.1.cmml" xref="S2.SS2.SSS0.P0.SPx1.p1.11.m11.1.1">𝑣</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.SSS0.P0.SPx1.p1.11.m11.1c">v</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.SSS0.P0.SPx1.p1.11.m11.1d">italic_v</annotation></semantics></math> such that for the <math alttext="k" class="ltx_Math" display="inline" id="S2.SS2.SSS0.P0.SPx1.p1.12.m12.1"><semantics id="S2.SS2.SSS0.P0.SPx1.p1.12.m12.1a"><mi id="S2.SS2.SSS0.P0.SPx1.p1.12.m12.1.1" xref="S2.SS2.SSS0.P0.SPx1.p1.12.m12.1.1.cmml">k</mi><annotation-xml encoding="MathML-Content" id="S2.SS2.SSS0.P0.SPx1.p1.12.m12.1b"><ci id="S2.SS2.SSS0.P0.SPx1.p1.12.m12.1.1.cmml" xref="S2.SS2.SSS0.P0.SPx1.p1.12.m12.1.1">𝑘</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.SSS0.P0.SPx1.p1.12.m12.1c">k</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.SSS0.P0.SPx1.p1.12.m12.1d">italic_k</annotation></semantics></math>-hop neighbourhood <math alttext="H_{k}(v)" class="ltx_Math" display="inline" id="S2.SS2.SSS0.P0.SPx1.p1.13.m13.1"><semantics id="S2.SS2.SSS0.P0.SPx1.p1.13.m13.1a"><mrow id="S2.SS2.SSS0.P0.SPx1.p1.13.m13.1.2" xref="S2.SS2.SSS0.P0.SPx1.p1.13.m13.1.2.cmml"><msub id="S2.SS2.SSS0.P0.SPx1.p1.13.m13.1.2.2" xref="S2.SS2.SSS0.P0.SPx1.p1.13.m13.1.2.2.cmml"><mi id="S2.SS2.SSS0.P0.SPx1.p1.13.m13.1.2.2.2" xref="S2.SS2.SSS0.P0.SPx1.p1.13.m13.1.2.2.2.cmml">H</mi><mi id="S2.SS2.SSS0.P0.SPx1.p1.13.m13.1.2.2.3" xref="S2.SS2.SSS0.P0.SPx1.p1.13.m13.1.2.2.3.cmml">k</mi></msub><mo id="S2.SS2.SSS0.P0.SPx1.p1.13.m13.1.2.1" xref="S2.SS2.SSS0.P0.SPx1.p1.13.m13.1.2.1.cmml">⁢</mo><mrow id="S2.SS2.SSS0.P0.SPx1.p1.13.m13.1.2.3.2" xref="S2.SS2.SSS0.P0.SPx1.p1.13.m13.1.2.cmml"><mo id="S2.SS2.SSS0.P0.SPx1.p1.13.m13.1.2.3.2.1" stretchy="false" xref="S2.SS2.SSS0.P0.SPx1.p1.13.m13.1.2.cmml">(</mo><mi id="S2.SS2.SSS0.P0.SPx1.p1.13.m13.1.1" 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id="S2.SS2.SSS0.P0.SPx1.p1.14.m14.1.1.1.1.1.1.2.1.cmml" xref="S2.SS2.SSS0.P0.SPx1.p1.14.m14.1.1.1.1.1.1.2">superscript</csymbol><ci id="S2.SS2.SSS0.P0.SPx1.p1.14.m14.1.1.1.1.1.1.2.2.cmml" xref="S2.SS2.SSS0.P0.SPx1.p1.14.m14.1.1.1.1.1.1.2.2">𝜀</ci><apply id="S2.SS2.SSS0.P0.SPx1.p1.14.m14.1.1.1.1.1.1.2.3.cmml" xref="S2.SS2.SSS0.P0.SPx1.p1.14.m14.1.1.1.1.1.1.2.3"><minus id="S2.SS2.SSS0.P0.SPx1.p1.14.m14.1.1.1.1.1.1.2.3.1.cmml" xref="S2.SS2.SSS0.P0.SPx1.p1.14.m14.1.1.1.1.1.1.2.3"></minus><cn id="S2.SS2.SSS0.P0.SPx1.p1.14.m14.1.1.1.1.1.1.2.3.2.cmml" type="integer" xref="S2.SS2.SSS0.P0.SPx1.p1.14.m14.1.1.1.1.1.1.2.3.2">1</cn></apply></apply><apply id="S2.SS2.SSS0.P0.SPx1.p1.14.m14.1.1.1.1.1.1.3.cmml" xref="S2.SS2.SSS0.P0.SPx1.p1.14.m14.1.1.1.1.1.1.3"><log id="S2.SS2.SSS0.P0.SPx1.p1.14.m14.1.1.1.1.1.1.3.1.cmml" xref="S2.SS2.SSS0.P0.SPx1.p1.14.m14.1.1.1.1.1.1.3.1"></log><ci id="S2.SS2.SSS0.P0.SPx1.p1.14.m14.1.1.1.1.1.1.3.2.cmml" xref="S2.SS2.SSS0.P0.SPx1.p1.14.m14.1.1.1.1.1.1.3.2">𝑛</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.SSS0.P0.SPx1.p1.14.m14.1c">k\in O(\varepsilon^{-1}\log n)</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.SSS0.P0.SPx1.p1.14.m14.1d">italic_k ∈ italic_O ( italic_ε start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT roman_log italic_n )</annotation></semantics></math>) the density <math alttext="\rho^{\max}(H_{k})" class="ltx_Math" display="inline" id="S2.SS2.SSS0.P0.SPx1.p1.15.m15.1"><semantics id="S2.SS2.SSS0.P0.SPx1.p1.15.m15.1a"><mrow id="S2.SS2.SSS0.P0.SPx1.p1.15.m15.1.1" xref="S2.SS2.SSS0.P0.SPx1.p1.15.m15.1.1.cmml"><msup id="S2.SS2.SSS0.P0.SPx1.p1.15.m15.1.1.3" xref="S2.SS2.SSS0.P0.SPx1.p1.15.m15.1.1.3.cmml"><mi id="S2.SS2.SSS0.P0.SPx1.p1.15.m15.1.1.3.2" xref="S2.SS2.SSS0.P0.SPx1.p1.15.m15.1.1.3.2.cmml">ρ</mi><mi id="S2.SS2.SSS0.P0.SPx1.p1.15.m15.1.1.3.3" xref="S2.SS2.SSS0.P0.SPx1.p1.15.m15.1.1.3.3.cmml">max</mi></msup><mo id="S2.SS2.SSS0.P0.SPx1.p1.15.m15.1.1.2" xref="S2.SS2.SSS0.P0.SPx1.p1.15.m15.1.1.2.cmml">⁢</mo><mrow id="S2.SS2.SSS0.P0.SPx1.p1.15.m15.1.1.1.1" xref="S2.SS2.SSS0.P0.SPx1.p1.15.m15.1.1.1.1.1.cmml"><mo id="S2.SS2.SSS0.P0.SPx1.p1.15.m15.1.1.1.1.2" stretchy="false" xref="S2.SS2.SSS0.P0.SPx1.p1.15.m15.1.1.1.1.1.cmml">(</mo><msub id="S2.SS2.SSS0.P0.SPx1.p1.15.m15.1.1.1.1.1" xref="S2.SS2.SSS0.P0.SPx1.p1.15.m15.1.1.1.1.1.cmml"><mi id="S2.SS2.SSS0.P0.SPx1.p1.15.m15.1.1.1.1.1.2" xref="S2.SS2.SSS0.P0.SPx1.p1.15.m15.1.1.1.1.1.2.cmml">H</mi><mi id="S2.SS2.SSS0.P0.SPx1.p1.15.m15.1.1.1.1.1.3" xref="S2.SS2.SSS0.P0.SPx1.p1.15.m15.1.1.1.1.1.3.cmml">k</mi></msub><mo id="S2.SS2.SSS0.P0.SPx1.p1.15.m15.1.1.1.1.3" stretchy="false" xref="S2.SS2.SSS0.P0.SPx1.p1.15.m15.1.1.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.SSS0.P0.SPx1.p1.15.m15.1b"><apply id="S2.SS2.SSS0.P0.SPx1.p1.15.m15.1.1.cmml" xref="S2.SS2.SSS0.P0.SPx1.p1.15.m15.1.1"><times id="S2.SS2.SSS0.P0.SPx1.p1.15.m15.1.1.2.cmml" xref="S2.SS2.SSS0.P0.SPx1.p1.15.m15.1.1.2"></times><apply id="S2.SS2.SSS0.P0.SPx1.p1.15.m15.1.1.3.cmml" xref="S2.SS2.SSS0.P0.SPx1.p1.15.m15.1.1.3"><csymbol cd="ambiguous" id="S2.SS2.SSS0.P0.SPx1.p1.15.m15.1.1.3.1.cmml" xref="S2.SS2.SSS0.P0.SPx1.p1.15.m15.1.1.3">superscript</csymbol><ci id="S2.SS2.SSS0.P0.SPx1.p1.15.m15.1.1.3.2.cmml" xref="S2.SS2.SSS0.P0.SPx1.p1.15.m15.1.1.3.2">𝜌</ci><max id="S2.SS2.SSS0.P0.SPx1.p1.15.m15.1.1.3.3.cmml" xref="S2.SS2.SSS0.P0.SPx1.p1.15.m15.1.1.3.3"></max></apply><apply id="S2.SS2.SSS0.P0.SPx1.p1.15.m15.1.1.1.1.1.cmml" xref="S2.SS2.SSS0.P0.SPx1.p1.15.m15.1.1.1.1"><csymbol cd="ambiguous" id="S2.SS2.SSS0.P0.SPx1.p1.15.m15.1.1.1.1.1.1.cmml" xref="S2.SS2.SSS0.P0.SPx1.p1.15.m15.1.1.1.1">subscript</csymbol><ci id="S2.SS2.SSS0.P0.SPx1.p1.15.m15.1.1.1.1.1.2.cmml" xref="S2.SS2.SSS0.P0.SPx1.p1.15.m15.1.1.1.1.1.2">𝐻</ci><ci id="S2.SS2.SSS0.P0.SPx1.p1.15.m15.1.1.1.1.1.3.cmml" xref="S2.SS2.SSS0.P0.SPx1.p1.15.m15.1.1.1.1.1.3">𝑘</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.SSS0.P0.SPx1.p1.15.m15.1c">\rho^{\max}(H_{k})</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.SSS0.P0.SPx1.p1.15.m15.1d">italic_ρ start_POSTSUPERSCRIPT roman_max end_POSTSUPERSCRIPT ( italic_H start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT )</annotation></semantics></math> is a <math alttext="(1+\varepsilon)" class="ltx_Math" display="inline" id="S2.SS2.SSS0.P0.SPx1.p1.16.m16.1"><semantics id="S2.SS2.SSS0.P0.SPx1.p1.16.m16.1a"><mrow id="S2.SS2.SSS0.P0.SPx1.p1.16.m16.1.1.1" xref="S2.SS2.SSS0.P0.SPx1.p1.16.m16.1.1.1.1.cmml"><mo id="S2.SS2.SSS0.P0.SPx1.p1.16.m16.1.1.1.2" stretchy="false" xref="S2.SS2.SSS0.P0.SPx1.p1.16.m16.1.1.1.1.cmml">(</mo><mrow id="S2.SS2.SSS0.P0.SPx1.p1.16.m16.1.1.1.1" xref="S2.SS2.SSS0.P0.SPx1.p1.16.m16.1.1.1.1.cmml"><mn id="S2.SS2.SSS0.P0.SPx1.p1.16.m16.1.1.1.1.2" xref="S2.SS2.SSS0.P0.SPx1.p1.16.m16.1.1.1.1.2.cmml">1</mn><mo id="S2.SS2.SSS0.P0.SPx1.p1.16.m16.1.1.1.1.1" xref="S2.SS2.SSS0.P0.SPx1.p1.16.m16.1.1.1.1.1.cmml">+</mo><mi id="S2.SS2.SSS0.P0.SPx1.p1.16.m16.1.1.1.1.3" xref="S2.SS2.SSS0.P0.SPx1.p1.16.m16.1.1.1.1.3.cmml">ε</mi></mrow><mo id="S2.SS2.SSS0.P0.SPx1.p1.16.m16.1.1.1.3" stretchy="false" xref="S2.SS2.SSS0.P0.SPx1.p1.16.m16.1.1.1.1.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.SSS0.P0.SPx1.p1.16.m16.1b"><apply id="S2.SS2.SSS0.P0.SPx1.p1.16.m16.1.1.1.1.cmml" xref="S2.SS2.SSS0.P0.SPx1.p1.16.m16.1.1.1"><plus id="S2.SS2.SSS0.P0.SPx1.p1.16.m16.1.1.1.1.1.cmml" xref="S2.SS2.SSS0.P0.SPx1.p1.16.m16.1.1.1.1.1"></plus><cn id="S2.SS2.SSS0.P0.SPx1.p1.16.m16.1.1.1.1.2.cmml" type="integer" xref="S2.SS2.SSS0.P0.SPx1.p1.16.m16.1.1.1.1.2">1</cn><ci id="S2.SS2.SSS0.P0.SPx1.p1.16.m16.1.1.1.1.3.cmml" xref="S2.SS2.SSS0.P0.SPx1.p1.16.m16.1.1.1.1.3">𝜀</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.SSS0.P0.SPx1.p1.16.m16.1c">(1+\varepsilon)</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.SSS0.P0.SPx1.p1.16.m16.1d">( 1 + italic_ε )</annotation></semantics></math>-approximation of <math alttext="\rho^{\max}(G)" class="ltx_Math" display="inline" id="S2.SS2.SSS0.P0.SPx1.p1.17.m17.1"><semantics id="S2.SS2.SSS0.P0.SPx1.p1.17.m17.1a"><mrow id="S2.SS2.SSS0.P0.SPx1.p1.17.m17.1.2" xref="S2.SS2.SSS0.P0.SPx1.p1.17.m17.1.2.cmml"><msup id="S2.SS2.SSS0.P0.SPx1.p1.17.m17.1.2.2" xref="S2.SS2.SSS0.P0.SPx1.p1.17.m17.1.2.2.cmml"><mi id="S2.SS2.SSS0.P0.SPx1.p1.17.m17.1.2.2.2" xref="S2.SS2.SSS0.P0.SPx1.p1.17.m17.1.2.2.2.cmml">ρ</mi><mi id="S2.SS2.SSS0.P0.SPx1.p1.17.m17.1.2.2.3" xref="S2.SS2.SSS0.P0.SPx1.p1.17.m17.1.2.2.3.cmml">max</mi></msup><mo id="S2.SS2.SSS0.P0.SPx1.p1.17.m17.1.2.1" xref="S2.SS2.SSS0.P0.SPx1.p1.17.m17.1.2.1.cmml">⁢</mo><mrow id="S2.SS2.SSS0.P0.SPx1.p1.17.m17.1.2.3.2" xref="S2.SS2.SSS0.P0.SPx1.p1.17.m17.1.2.cmml"><mo id="S2.SS2.SSS0.P0.SPx1.p1.17.m17.1.2.3.2.1" stretchy="false" xref="S2.SS2.SSS0.P0.SPx1.p1.17.m17.1.2.cmml">(</mo><mi id="S2.SS2.SSS0.P0.SPx1.p1.17.m17.1.1" xref="S2.SS2.SSS0.P0.SPx1.p1.17.m17.1.1.cmml">G</mi><mo id="S2.SS2.SSS0.P0.SPx1.p1.17.m17.1.2.3.2.2" stretchy="false" xref="S2.SS2.SSS0.P0.SPx1.p1.17.m17.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.SSS0.P0.SPx1.p1.17.m17.1b"><apply id="S2.SS2.SSS0.P0.SPx1.p1.17.m17.1.2.cmml" xref="S2.SS2.SSS0.P0.SPx1.p1.17.m17.1.2"><times id="S2.SS2.SSS0.P0.SPx1.p1.17.m17.1.2.1.cmml" xref="S2.SS2.SSS0.P0.SPx1.p1.17.m17.1.2.1"></times><apply id="S2.SS2.SSS0.P0.SPx1.p1.17.m17.1.2.2.cmml" xref="S2.SS2.SSS0.P0.SPx1.p1.17.m17.1.2.2"><csymbol cd="ambiguous" id="S2.SS2.SSS0.P0.SPx1.p1.17.m17.1.2.2.1.cmml" xref="S2.SS2.SSS0.P0.SPx1.p1.17.m17.1.2.2">superscript</csymbol><ci id="S2.SS2.SSS0.P0.SPx1.p1.17.m17.1.2.2.2.cmml" xref="S2.SS2.SSS0.P0.SPx1.p1.17.m17.1.2.2.2">𝜌</ci><max id="S2.SS2.SSS0.P0.SPx1.p1.17.m17.1.2.2.3.cmml" xref="S2.SS2.SSS0.P0.SPx1.p1.17.m17.1.2.2.3"></max></apply><ci id="S2.SS2.SSS0.P0.SPx1.p1.17.m17.1.1.cmml" xref="S2.SS2.SSS0.P0.SPx1.p1.17.m17.1.1">𝐺</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.SSS0.P0.SPx1.p1.17.m17.1c">\rho^{\max}(G)</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.SSS0.P0.SPx1.p1.17.m17.1d">italic_ρ start_POSTSUPERSCRIPT roman_max end_POSTSUPERSCRIPT ( italic_G )</annotation></semantics></math>. This immediately leads to a trivial LOCAL algorithm: each vertex <math alttext="u" class="ltx_Math" display="inline" id="S2.SS2.SSS0.P0.SPx1.p1.18.m18.1"><semantics id="S2.SS2.SSS0.P0.SPx1.p1.18.m18.1a"><mi id="S2.SS2.SSS0.P0.SPx1.p1.18.m18.1.1" xref="S2.SS2.SSS0.P0.SPx1.p1.18.m18.1.1.cmml">u</mi><annotation-xml encoding="MathML-Content" id="S2.SS2.SSS0.P0.SPx1.p1.18.m18.1b"><ci id="S2.SS2.SSS0.P0.SPx1.p1.18.m18.1.1.cmml" xref="S2.SS2.SSS0.P0.SPx1.p1.18.m18.1.1">𝑢</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.SSS0.P0.SPx1.p1.18.m18.1c">u</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.SSS0.P0.SPx1.p1.18.m18.1d">italic_u</annotation></semantics></math> collects its <math alttext="k" class="ltx_Math" display="inline" id="S2.SS2.SSS0.P0.SPx1.p1.19.m19.1"><semantics id="S2.SS2.SSS0.P0.SPx1.p1.19.m19.1a"><mi id="S2.SS2.SSS0.P0.SPx1.p1.19.m19.1.1" xref="S2.SS2.SSS0.P0.SPx1.p1.19.m19.1.1.cmml">k</mi><annotation-xml encoding="MathML-Content" id="S2.SS2.SSS0.P0.SPx1.p1.19.m19.1b"><ci id="S2.SS2.SSS0.P0.SPx1.p1.19.m19.1.1.cmml" xref="S2.SS2.SSS0.P0.SPx1.p1.19.m19.1.1">𝑘</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.SSS0.P0.SPx1.p1.19.m19.1c">k</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.SSS0.P0.SPx1.p1.19.m19.1d">italic_k</annotation></semantics></math>-hop neighbourhood <math alttext="H_{k}(u)" class="ltx_Math" display="inline" id="S2.SS2.SSS0.P0.SPx1.p1.20.m20.1"><semantics id="S2.SS2.SSS0.P0.SPx1.p1.20.m20.1a"><mrow id="S2.SS2.SSS0.P0.SPx1.p1.20.m20.1.2" xref="S2.SS2.SSS0.P0.SPx1.p1.20.m20.1.2.cmml"><msub id="S2.SS2.SSS0.P0.SPx1.p1.20.m20.1.2.2" xref="S2.SS2.SSS0.P0.SPx1.p1.20.m20.1.2.2.cmml"><mi id="S2.SS2.SSS0.P0.SPx1.p1.20.m20.1.2.2.2" xref="S2.SS2.SSS0.P0.SPx1.p1.20.m20.1.2.2.2.cmml">H</mi><mi id="S2.SS2.SSS0.P0.SPx1.p1.20.m20.1.2.2.3" xref="S2.SS2.SSS0.P0.SPx1.p1.20.m20.1.2.2.3.cmml">k</mi></msub><mo id="S2.SS2.SSS0.P0.SPx1.p1.20.m20.1.2.1" 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Subgraph in its own node, and reports the value <math alttext="\rho^{\max}(H_{k}(u))" class="ltx_Math" display="inline" id="S2.SS2.SSS0.P0.SPx1.p1.22.m22.2"><semantics id="S2.SS2.SSS0.P0.SPx1.p1.22.m22.2a"><mrow id="S2.SS2.SSS0.P0.SPx1.p1.22.m22.2.2" xref="S2.SS2.SSS0.P0.SPx1.p1.22.m22.2.2.cmml"><msup id="S2.SS2.SSS0.P0.SPx1.p1.22.m22.2.2.3" xref="S2.SS2.SSS0.P0.SPx1.p1.22.m22.2.2.3.cmml"><mi id="S2.SS2.SSS0.P0.SPx1.p1.22.m22.2.2.3.2" xref="S2.SS2.SSS0.P0.SPx1.p1.22.m22.2.2.3.2.cmml">ρ</mi><mi id="S2.SS2.SSS0.P0.SPx1.p1.22.m22.2.2.3.3" xref="S2.SS2.SSS0.P0.SPx1.p1.22.m22.2.2.3.3.cmml">max</mi></msup><mo id="S2.SS2.SSS0.P0.SPx1.p1.22.m22.2.2.2" xref="S2.SS2.SSS0.P0.SPx1.p1.22.m22.2.2.2.cmml">⁢</mo><mrow id="S2.SS2.SSS0.P0.SPx1.p1.22.m22.2.2.1.1" xref="S2.SS2.SSS0.P0.SPx1.p1.22.m22.2.2.1.1.1.cmml"><mo id="S2.SS2.SSS0.P0.SPx1.p1.22.m22.2.2.1.1.2" stretchy="false" xref="S2.SS2.SSS0.P0.SPx1.p1.22.m22.2.2.1.1.1.cmml">(</mo><mrow id="S2.SS2.SSS0.P0.SPx1.p1.22.m22.2.2.1.1.1" 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start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ( italic_u ) )</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="S2.SS2.SSS0.P0.SPx1.p2"> <p class="ltx_p" id="S2.SS2.SSS0.P0.SPx1.p2.3">In CONGEST, the state-of-the-art deterministic algorithm for Problem <a class="ltx_ref" href="https://arxiv.org/html/2411.12694v2#S2.Thmtheorem7" title="Problem 2.7. ‣ 2.2 Approximate densest subgraph in LOCAL and CONGEST ‣ 2 Preliminaries and related work ‣ Local Density and its Distributed Approximation"><span class="ltx_text ltx_ref_tag">2.7</span></a>.1 and <a class="ltx_ref" href="https://arxiv.org/html/2411.12694v2#S2.Thmtheorem7" title="Problem 2.7. ‣ 2.2 Approximate densest subgraph in LOCAL and CONGEST ‣ 2 Preliminaries and related work ‣ Local Density and its Distributed Approximation"><span class="ltx_text ltx_ref_tag">2.7</span></a>.2 is by Das Sarma et al. <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2411.12694v2#bib.bib11" title="">11</a>]</cite> who present a <math alttext="(2+\varepsilon)" class="ltx_Math" display="inline" id="S2.SS2.SSS0.P0.SPx1.p2.1.m1.1"><semantics id="S2.SS2.SSS0.P0.SPx1.p2.1.m1.1a"><mrow id="S2.SS2.SSS0.P0.SPx1.p2.1.m1.1.1.1" xref="S2.SS2.SSS0.P0.SPx1.p2.1.m1.1.1.1.1.cmml"><mo id="S2.SS2.SSS0.P0.SPx1.p2.1.m1.1.1.1.2" stretchy="false" xref="S2.SS2.SSS0.P0.SPx1.p2.1.m1.1.1.1.1.cmml">(</mo><mrow id="S2.SS2.SSS0.P0.SPx1.p2.1.m1.1.1.1.1" xref="S2.SS2.SSS0.P0.SPx1.p2.1.m1.1.1.1.1.cmml"><mn id="S2.SS2.SSS0.P0.SPx1.p2.1.m1.1.1.1.1.2" xref="S2.SS2.SSS0.P0.SPx1.p2.1.m1.1.1.1.1.2.cmml">2</mn><mo id="S2.SS2.SSS0.P0.SPx1.p2.1.m1.1.1.1.1.1" xref="S2.SS2.SSS0.P0.SPx1.p2.1.m1.1.1.1.1.1.cmml">+</mo><mi id="S2.SS2.SSS0.P0.SPx1.p2.1.m1.1.1.1.1.3" xref="S2.SS2.SSS0.P0.SPx1.p2.1.m1.1.1.1.1.3.cmml">ε</mi></mrow><mo id="S2.SS2.SSS0.P0.SPx1.p2.1.m1.1.1.1.3" stretchy="false" xref="S2.SS2.SSS0.P0.SPx1.p2.1.m1.1.1.1.1.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.SSS0.P0.SPx1.p2.1.m1.1b"><apply id="S2.SS2.SSS0.P0.SPx1.p2.1.m1.1.1.1.1.cmml" xref="S2.SS2.SSS0.P0.SPx1.p2.1.m1.1.1.1"><plus id="S2.SS2.SSS0.P0.SPx1.p2.1.m1.1.1.1.1.1.cmml" xref="S2.SS2.SSS0.P0.SPx1.p2.1.m1.1.1.1.1.1"></plus><cn id="S2.SS2.SSS0.P0.SPx1.p2.1.m1.1.1.1.1.2.cmml" type="integer" xref="S2.SS2.SSS0.P0.SPx1.p2.1.m1.1.1.1.1.2">2</cn><ci id="S2.SS2.SSS0.P0.SPx1.p2.1.m1.1.1.1.1.3.cmml" xref="S2.SS2.SSS0.P0.SPx1.p2.1.m1.1.1.1.1.3">𝜀</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.SSS0.P0.SPx1.p2.1.m1.1c">(2+\varepsilon)</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.SSS0.P0.SPx1.p2.1.m1.1d">( 2 + italic_ε )</annotation></semantics></math>-approximation in <math alttext="O(D\cdot\varepsilon^{-1}\log n)" class="ltx_Math" display="inline" id="S2.SS2.SSS0.P0.SPx1.p2.2.m2.1"><semantics id="S2.SS2.SSS0.P0.SPx1.p2.2.m2.1a"><mrow id="S2.SS2.SSS0.P0.SPx1.p2.2.m2.1.1" xref="S2.SS2.SSS0.P0.SPx1.p2.2.m2.1.1.cmml"><mi id="S2.SS2.SSS0.P0.SPx1.p2.2.m2.1.1.3" xref="S2.SS2.SSS0.P0.SPx1.p2.2.m2.1.1.3.cmml">O</mi><mo id="S2.SS2.SSS0.P0.SPx1.p2.2.m2.1.1.2" xref="S2.SS2.SSS0.P0.SPx1.p2.2.m2.1.1.2.cmml">⁢</mo><mrow id="S2.SS2.SSS0.P0.SPx1.p2.2.m2.1.1.1.1" xref="S2.SS2.SSS0.P0.SPx1.p2.2.m2.1.1.1.1.1.cmml"><mo id="S2.SS2.SSS0.P0.SPx1.p2.2.m2.1.1.1.1.2" stretchy="false" xref="S2.SS2.SSS0.P0.SPx1.p2.2.m2.1.1.1.1.1.cmml">(</mo><mrow id="S2.SS2.SSS0.P0.SPx1.p2.2.m2.1.1.1.1.1" xref="S2.SS2.SSS0.P0.SPx1.p2.2.m2.1.1.1.1.1.cmml"><mrow id="S2.SS2.SSS0.P0.SPx1.p2.2.m2.1.1.1.1.1.2" xref="S2.SS2.SSS0.P0.SPx1.p2.2.m2.1.1.1.1.1.2.cmml"><mi id="S2.SS2.SSS0.P0.SPx1.p2.2.m2.1.1.1.1.1.2.2" xref="S2.SS2.SSS0.P0.SPx1.p2.2.m2.1.1.1.1.1.2.2.cmml">D</mi><mo id="S2.SS2.SSS0.P0.SPx1.p2.2.m2.1.1.1.1.1.2.1" lspace="0.222em" rspace="0.222em" xref="S2.SS2.SSS0.P0.SPx1.p2.2.m2.1.1.1.1.1.2.1.cmml">⋅</mo><msup id="S2.SS2.SSS0.P0.SPx1.p2.2.m2.1.1.1.1.1.2.3" xref="S2.SS2.SSS0.P0.SPx1.p2.2.m2.1.1.1.1.1.2.3.cmml"><mi 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id="S2.SS2.SSS0.P0.SPx1.p2.2.m2.1.1.1.1.1.2.2.cmml" xref="S2.SS2.SSS0.P0.SPx1.p2.2.m2.1.1.1.1.1.2.2">𝐷</ci><apply id="S2.SS2.SSS0.P0.SPx1.p2.2.m2.1.1.1.1.1.2.3.cmml" xref="S2.SS2.SSS0.P0.SPx1.p2.2.m2.1.1.1.1.1.2.3"><csymbol cd="ambiguous" id="S2.SS2.SSS0.P0.SPx1.p2.2.m2.1.1.1.1.1.2.3.1.cmml" xref="S2.SS2.SSS0.P0.SPx1.p2.2.m2.1.1.1.1.1.2.3">superscript</csymbol><ci id="S2.SS2.SSS0.P0.SPx1.p2.2.m2.1.1.1.1.1.2.3.2.cmml" xref="S2.SS2.SSS0.P0.SPx1.p2.2.m2.1.1.1.1.1.2.3.2">𝜀</ci><apply id="S2.SS2.SSS0.P0.SPx1.p2.2.m2.1.1.1.1.1.2.3.3.cmml" xref="S2.SS2.SSS0.P0.SPx1.p2.2.m2.1.1.1.1.1.2.3.3"><minus id="S2.SS2.SSS0.P0.SPx1.p2.2.m2.1.1.1.1.1.2.3.3.1.cmml" xref="S2.SS2.SSS0.P0.SPx1.p2.2.m2.1.1.1.1.1.2.3.3"></minus><cn id="S2.SS2.SSS0.P0.SPx1.p2.2.m2.1.1.1.1.1.2.3.3.2.cmml" type="integer" xref="S2.SS2.SSS0.P0.SPx1.p2.2.m2.1.1.1.1.1.2.3.3.2">1</cn></apply></apply></apply><apply id="S2.SS2.SSS0.P0.SPx1.p2.2.m2.1.1.1.1.1.3.cmml" xref="S2.SS2.SSS0.P0.SPx1.p2.2.m2.1.1.1.1.1.3"><log id="S2.SS2.SSS0.P0.SPx1.p2.2.m2.1.1.1.1.1.3.1.cmml" xref="S2.SS2.SSS0.P0.SPx1.p2.2.m2.1.1.1.1.1.3.1"></log><ci id="S2.SS2.SSS0.P0.SPx1.p2.2.m2.1.1.1.1.1.3.2.cmml" xref="S2.SS2.SSS0.P0.SPx1.p2.2.m2.1.1.1.1.1.3.2">𝑛</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.SSS0.P0.SPx1.p2.2.m2.1c">O(D\cdot\varepsilon^{-1}\log n)</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.SSS0.P0.SPx1.p2.2.m2.1d">italic_O ( italic_D ⋅ italic_ε start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT roman_log italic_n )</annotation></semantics></math> rounds. The best randomised work is by Su and Vu <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2411.12694v2#bib.bib21" title="">21</a>]</cite> who present a randomised algorithm for Problem <a class="ltx_ref" href="https://arxiv.org/html/2411.12694v2#S2.Thmtheorem7" title="Problem 2.7. ‣ 2.2 Approximate densest subgraph in LOCAL and CONGEST ‣ 2 Preliminaries and related work ‣ Local Density and its Distributed Approximation"><span class="ltx_text ltx_ref_tag">2.7</span></a>.2 that runs in <math alttext="O(\varepsilon^{-4}\log^{4}n)" class="ltx_Math" display="inline" id="S2.SS2.SSS0.P0.SPx1.p2.3.m3.1"><semantics id="S2.SS2.SSS0.P0.SPx1.p2.3.m3.1a"><mrow id="S2.SS2.SSS0.P0.SPx1.p2.3.m3.1.1" xref="S2.SS2.SSS0.P0.SPx1.p2.3.m3.1.1.cmml"><mi id="S2.SS2.SSS0.P0.SPx1.p2.3.m3.1.1.3" xref="S2.SS2.SSS0.P0.SPx1.p2.3.m3.1.1.3.cmml">O</mi><mo id="S2.SS2.SSS0.P0.SPx1.p2.3.m3.1.1.2" xref="S2.SS2.SSS0.P0.SPx1.p2.3.m3.1.1.2.cmml">⁢</mo><mrow id="S2.SS2.SSS0.P0.SPx1.p2.3.m3.1.1.1.1" xref="S2.SS2.SSS0.P0.SPx1.p2.3.m3.1.1.1.1.1.cmml"><mo id="S2.SS2.SSS0.P0.SPx1.p2.3.m3.1.1.1.1.2" stretchy="false" xref="S2.SS2.SSS0.P0.SPx1.p2.3.m3.1.1.1.1.1.cmml">(</mo><mrow id="S2.SS2.SSS0.P0.SPx1.p2.3.m3.1.1.1.1.1" xref="S2.SS2.SSS0.P0.SPx1.p2.3.m3.1.1.1.1.1.cmml"><msup id="S2.SS2.SSS0.P0.SPx1.p2.3.m3.1.1.1.1.1.2" xref="S2.SS2.SSS0.P0.SPx1.p2.3.m3.1.1.1.1.1.2.cmml"><mi id="S2.SS2.SSS0.P0.SPx1.p2.3.m3.1.1.1.1.1.2.2" xref="S2.SS2.SSS0.P0.SPx1.p2.3.m3.1.1.1.1.1.2.2.cmml">ε</mi><mrow id="S2.SS2.SSS0.P0.SPx1.p2.3.m3.1.1.1.1.1.2.3" xref="S2.SS2.SSS0.P0.SPx1.p2.3.m3.1.1.1.1.1.2.3.cmml"><mo id="S2.SS2.SSS0.P0.SPx1.p2.3.m3.1.1.1.1.1.2.3a" xref="S2.SS2.SSS0.P0.SPx1.p2.3.m3.1.1.1.1.1.2.3.cmml">−</mo><mn id="S2.SS2.SSS0.P0.SPx1.p2.3.m3.1.1.1.1.1.2.3.2" xref="S2.SS2.SSS0.P0.SPx1.p2.3.m3.1.1.1.1.1.2.3.2.cmml">4</mn></mrow></msup><mo id="S2.SS2.SSS0.P0.SPx1.p2.3.m3.1.1.1.1.1.1" lspace="0.167em" xref="S2.SS2.SSS0.P0.SPx1.p2.3.m3.1.1.1.1.1.1.cmml">⁢</mo><mrow id="S2.SS2.SSS0.P0.SPx1.p2.3.m3.1.1.1.1.1.3" xref="S2.SS2.SSS0.P0.SPx1.p2.3.m3.1.1.1.1.1.3.cmml"><msup id="S2.SS2.SSS0.P0.SPx1.p2.3.m3.1.1.1.1.1.3.1" xref="S2.SS2.SSS0.P0.SPx1.p2.3.m3.1.1.1.1.1.3.1.cmml"><mi id="S2.SS2.SSS0.P0.SPx1.p2.3.m3.1.1.1.1.1.3.1.2" xref="S2.SS2.SSS0.P0.SPx1.p2.3.m3.1.1.1.1.1.3.1.2.cmml">log</mi><mn id="S2.SS2.SSS0.P0.SPx1.p2.3.m3.1.1.1.1.1.3.1.3" xref="S2.SS2.SSS0.P0.SPx1.p2.3.m3.1.1.1.1.1.3.1.3.cmml">4</mn></msup><mo id="S2.SS2.SSS0.P0.SPx1.p2.3.m3.1.1.1.1.1.3a" lspace="0.167em" xref="S2.SS2.SSS0.P0.SPx1.p2.3.m3.1.1.1.1.1.3.cmml">⁡</mo><mi id="S2.SS2.SSS0.P0.SPx1.p2.3.m3.1.1.1.1.1.3.2" xref="S2.SS2.SSS0.P0.SPx1.p2.3.m3.1.1.1.1.1.3.2.cmml">n</mi></mrow></mrow><mo id="S2.SS2.SSS0.P0.SPx1.p2.3.m3.1.1.1.1.3" stretchy="false" xref="S2.SS2.SSS0.P0.SPx1.p2.3.m3.1.1.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.SSS0.P0.SPx1.p2.3.m3.1b"><apply id="S2.SS2.SSS0.P0.SPx1.p2.3.m3.1.1.cmml" xref="S2.SS2.SSS0.P0.SPx1.p2.3.m3.1.1"><times id="S2.SS2.SSS0.P0.SPx1.p2.3.m3.1.1.2.cmml" xref="S2.SS2.SSS0.P0.SPx1.p2.3.m3.1.1.2"></times><ci id="S2.SS2.SSS0.P0.SPx1.p2.3.m3.1.1.3.cmml" xref="S2.SS2.SSS0.P0.SPx1.p2.3.m3.1.1.3">𝑂</ci><apply id="S2.SS2.SSS0.P0.SPx1.p2.3.m3.1.1.1.1.1.cmml" xref="S2.SS2.SSS0.P0.SPx1.p2.3.m3.1.1.1.1"><times id="S2.SS2.SSS0.P0.SPx1.p2.3.m3.1.1.1.1.1.1.cmml" xref="S2.SS2.SSS0.P0.SPx1.p2.3.m3.1.1.1.1.1.1"></times><apply id="S2.SS2.SSS0.P0.SPx1.p2.3.m3.1.1.1.1.1.2.cmml" xref="S2.SS2.SSS0.P0.SPx1.p2.3.m3.1.1.1.1.1.2"><csymbol cd="ambiguous" id="S2.SS2.SSS0.P0.SPx1.p2.3.m3.1.1.1.1.1.2.1.cmml" xref="S2.SS2.SSS0.P0.SPx1.p2.3.m3.1.1.1.1.1.2">superscript</csymbol><ci id="S2.SS2.SSS0.P0.SPx1.p2.3.m3.1.1.1.1.1.2.2.cmml" xref="S2.SS2.SSS0.P0.SPx1.p2.3.m3.1.1.1.1.1.2.2">𝜀</ci><apply id="S2.SS2.SSS0.P0.SPx1.p2.3.m3.1.1.1.1.1.2.3.cmml" xref="S2.SS2.SSS0.P0.SPx1.p2.3.m3.1.1.1.1.1.2.3"><minus id="S2.SS2.SSS0.P0.SPx1.p2.3.m3.1.1.1.1.1.2.3.1.cmml" xref="S2.SS2.SSS0.P0.SPx1.p2.3.m3.1.1.1.1.1.2.3"></minus><cn id="S2.SS2.SSS0.P0.SPx1.p2.3.m3.1.1.1.1.1.2.3.2.cmml" type="integer" xref="S2.SS2.SSS0.P0.SPx1.p2.3.m3.1.1.1.1.1.2.3.2">4</cn></apply></apply><apply id="S2.SS2.SSS0.P0.SPx1.p2.3.m3.1.1.1.1.1.3.cmml" xref="S2.SS2.SSS0.P0.SPx1.p2.3.m3.1.1.1.1.1.3"><apply id="S2.SS2.SSS0.P0.SPx1.p2.3.m3.1.1.1.1.1.3.1.cmml" xref="S2.SS2.SSS0.P0.SPx1.p2.3.m3.1.1.1.1.1.3.1"><csymbol cd="ambiguous" id="S2.SS2.SSS0.P0.SPx1.p2.3.m3.1.1.1.1.1.3.1.1.cmml" xref="S2.SS2.SSS0.P0.SPx1.p2.3.m3.1.1.1.1.1.3.1">superscript</csymbol><log id="S2.SS2.SSS0.P0.SPx1.p2.3.m3.1.1.1.1.1.3.1.2.cmml" xref="S2.SS2.SSS0.P0.SPx1.p2.3.m3.1.1.1.1.1.3.1.2"></log><cn id="S2.SS2.SSS0.P0.SPx1.p2.3.m3.1.1.1.1.1.3.1.3.cmml" type="integer" xref="S2.SS2.SSS0.P0.SPx1.p2.3.m3.1.1.1.1.1.3.1.3">4</cn></apply><ci id="S2.SS2.SSS0.P0.SPx1.p2.3.m3.1.1.1.1.1.3.2.cmml" xref="S2.SS2.SSS0.P0.SPx1.p2.3.m3.1.1.1.1.1.3.2">𝑛</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.SSS0.P0.SPx1.p2.3.m3.1c">O(\varepsilon^{-4}\log^{4}n)</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.SSS0.P0.SPx1.p2.3.m3.1d">italic_O ( italic_ε start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT roman_log start_POSTSUPERSCRIPT 4 end_POSTSUPERSCRIPT italic_n )</annotation></semantics></math> rounds w.h.p. See also Table <a class="ltx_ref" href="https://arxiv.org/html/2411.12694v2#S1.T1" title="Table 1 ‣ D: Results in CONGEST ‣ 1.1 Local density and results. ‣ 1 Introduction ‣ Local Density and its Distributed Approximation"><span class="ltx_text ltx_ref_tag">1</span></a> for an overview.</p> </div> </section> </section> <section class="ltx_subsection" id="S2.SS3"> <h3 class="ltx_title ltx_title_subsection"> <span class="ltx_tag ltx_tag_subsection">2.3 </span>Local density</h3> <div class="ltx_para" id="S2.SS3.p1"> <p class="ltx_p" id="S2.SS3.p1.1">Danisch, Chan, and Sozio <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2411.12694v2#bib.bib10" title="">10</a>]</cite> introduce a more local measure which they call the <em class="ltx_emph ltx_font_italic" id="S2.SS3.p1.1.1">local density</em>. Its lengthy definition assigns to each vertex <math alttext="v" class="ltx_Math" display="inline" id="S2.SS3.p1.1.m1.1"><semantics id="S2.SS3.p1.1.m1.1a"><mi id="S2.SS3.p1.1.m1.1.1" xref="S2.SS3.p1.1.m1.1.1.cmml">v</mi><annotation-xml encoding="MathML-Content" id="S2.SS3.p1.1.m1.1b"><ci id="S2.SS3.p1.1.m1.1.1.cmml" xref="S2.SS3.p1.1.m1.1.1">𝑣</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.p1.1.m1.1c">v</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.p1.1.m1.1d">italic_v</annotation></semantics></math> a value. We note for the reader that we almost immediately define our local out-degree (Definition <a class="ltx_ref" href="https://arxiv.org/html/2411.12694v2#S3.Thmtheorem1" title="Definition 3.1. ‣ 3 Results and organisation ‣ Local Density and its Distributed Approximation"><span class="ltx_text ltx_ref_tag">3.1</span></a>), and only use local out-degree in proofs. Hence, the reader is not required to have a thorough understanding of the following:</p> </div> <div class="ltx_theorem ltx_theorem_definition" id="S2.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="S2.Thmtheorem8.1.1.1">Definition 2.8</span></span><span class="ltx_text ltx_font_bold" id="S2.Thmtheorem8.2.2"> </span>(Definition 2.2 in <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2411.12694v2#bib.bib10" title="">10</a>]</cite>)<span class="ltx_text ltx_font_bold" id="S2.Thmtheorem8.3.3">.</span> </h6> <div class="ltx_para" id="S2.Thmtheorem8.p1"> <p class="ltx_p" id="S2.Thmtheorem8.p1.10"><span class="ltx_text ltx_font_italic" id="S2.Thmtheorem8.p1.10.10">Let <math alttext="G=(V,E)" class="ltx_Math" display="inline" id="S2.Thmtheorem8.p1.1.1.m1.2"><semantics id="S2.Thmtheorem8.p1.1.1.m1.2a"><mrow id="S2.Thmtheorem8.p1.1.1.m1.2.3" xref="S2.Thmtheorem8.p1.1.1.m1.2.3.cmml"><mi id="S2.Thmtheorem8.p1.1.1.m1.2.3.2" xref="S2.Thmtheorem8.p1.1.1.m1.2.3.2.cmml">G</mi><mo id="S2.Thmtheorem8.p1.1.1.m1.2.3.1" xref="S2.Thmtheorem8.p1.1.1.m1.2.3.1.cmml">=</mo><mrow id="S2.Thmtheorem8.p1.1.1.m1.2.3.3.2" xref="S2.Thmtheorem8.p1.1.1.m1.2.3.3.1.cmml"><mo id="S2.Thmtheorem8.p1.1.1.m1.2.3.3.2.1" stretchy="false" xref="S2.Thmtheorem8.p1.1.1.m1.2.3.3.1.cmml">(</mo><mi id="S2.Thmtheorem8.p1.1.1.m1.1.1" xref="S2.Thmtheorem8.p1.1.1.m1.1.1.cmml">V</mi><mo id="S2.Thmtheorem8.p1.1.1.m1.2.3.3.2.2" xref="S2.Thmtheorem8.p1.1.1.m1.2.3.3.1.cmml">,</mo><mi id="S2.Thmtheorem8.p1.1.1.m1.2.2" xref="S2.Thmtheorem8.p1.1.1.m1.2.2.cmml">E</mi><mo id="S2.Thmtheorem8.p1.1.1.m1.2.3.3.2.3" stretchy="false" xref="S2.Thmtheorem8.p1.1.1.m1.2.3.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem8.p1.1.1.m1.2b"><apply id="S2.Thmtheorem8.p1.1.1.m1.2.3.cmml" xref="S2.Thmtheorem8.p1.1.1.m1.2.3"><eq id="S2.Thmtheorem8.p1.1.1.m1.2.3.1.cmml" xref="S2.Thmtheorem8.p1.1.1.m1.2.3.1"></eq><ci id="S2.Thmtheorem8.p1.1.1.m1.2.3.2.cmml" xref="S2.Thmtheorem8.p1.1.1.m1.2.3.2">𝐺</ci><interval closure="open" id="S2.Thmtheorem8.p1.1.1.m1.2.3.3.1.cmml" xref="S2.Thmtheorem8.p1.1.1.m1.2.3.3.2"><ci id="S2.Thmtheorem8.p1.1.1.m1.1.1.cmml" xref="S2.Thmtheorem8.p1.1.1.m1.1.1">𝑉</ci><ci id="S2.Thmtheorem8.p1.1.1.m1.2.2.cmml" xref="S2.Thmtheorem8.p1.1.1.m1.2.2">𝐸</ci></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem8.p1.1.1.m1.2c">G=(V,E)</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem8.p1.1.1.m1.2d">italic_G = ( italic_V , italic_E )</annotation></semantics></math> be a weighted graph where an edge <math alttext="e" class="ltx_Math" display="inline" id="S2.Thmtheorem8.p1.2.2.m2.1"><semantics id="S2.Thmtheorem8.p1.2.2.m2.1a"><mi id="S2.Thmtheorem8.p1.2.2.m2.1.1" xref="S2.Thmtheorem8.p1.2.2.m2.1.1.cmml">e</mi><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem8.p1.2.2.m2.1b"><ci id="S2.Thmtheorem8.p1.2.2.m2.1.1.cmml" xref="S2.Thmtheorem8.p1.2.2.m2.1.1">𝑒</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem8.p1.2.2.m2.1c">e</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem8.p1.2.2.m2.1d">italic_e</annotation></semantics></math> has weight <math alttext="\textsl{g}(e)" class="ltx_Math" display="inline" id="S2.Thmtheorem8.p1.3.3.m3.1"><semantics id="S2.Thmtheorem8.p1.3.3.m3.1a"><mrow id="S2.Thmtheorem8.p1.3.3.m3.1.2" xref="S2.Thmtheorem8.p1.3.3.m3.1.2.cmml"><mtext class="ltx_mathvariant_italic" id="S2.Thmtheorem8.p1.3.3.m3.1.2.2" xref="S2.Thmtheorem8.p1.3.3.m3.1.2.2a.cmml">g</mtext><mo id="S2.Thmtheorem8.p1.3.3.m3.1.2.1" xref="S2.Thmtheorem8.p1.3.3.m3.1.2.1.cmml">⁢</mo><mrow id="S2.Thmtheorem8.p1.3.3.m3.1.2.3.2" xref="S2.Thmtheorem8.p1.3.3.m3.1.2.cmml"><mo id="S2.Thmtheorem8.p1.3.3.m3.1.2.3.2.1" stretchy="false" xref="S2.Thmtheorem8.p1.3.3.m3.1.2.cmml">(</mo><mi id="S2.Thmtheorem8.p1.3.3.m3.1.1" xref="S2.Thmtheorem8.p1.3.3.m3.1.1.cmml">e</mi><mo id="S2.Thmtheorem8.p1.3.3.m3.1.2.3.2.2" stretchy="false" xref="S2.Thmtheorem8.p1.3.3.m3.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem8.p1.3.3.m3.1b"><apply id="S2.Thmtheorem8.p1.3.3.m3.1.2.cmml" xref="S2.Thmtheorem8.p1.3.3.m3.1.2"><times id="S2.Thmtheorem8.p1.3.3.m3.1.2.1.cmml" xref="S2.Thmtheorem8.p1.3.3.m3.1.2.1"></times><ci id="S2.Thmtheorem8.p1.3.3.m3.1.2.2a.cmml" xref="S2.Thmtheorem8.p1.3.3.m3.1.2.2"><mtext class="ltx_mathvariant_italic" id="S2.Thmtheorem8.p1.3.3.m3.1.2.2.cmml" xref="S2.Thmtheorem8.p1.3.3.m3.1.2.2">g</mtext></ci><ci id="S2.Thmtheorem8.p1.3.3.m3.1.1.cmml" xref="S2.Thmtheorem8.p1.3.3.m3.1.1">𝑒</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem8.p1.3.3.m3.1c">\textsl{g}(e)</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem8.p1.3.3.m3.1d">g ( italic_e )</annotation></semantics></math>. Let <math alttext="B\subseteq V" class="ltx_Math" display="inline" id="S2.Thmtheorem8.p1.4.4.m4.1"><semantics id="S2.Thmtheorem8.p1.4.4.m4.1a"><mrow id="S2.Thmtheorem8.p1.4.4.m4.1.1" xref="S2.Thmtheorem8.p1.4.4.m4.1.1.cmml"><mi id="S2.Thmtheorem8.p1.4.4.m4.1.1.2" xref="S2.Thmtheorem8.p1.4.4.m4.1.1.2.cmml">B</mi><mo id="S2.Thmtheorem8.p1.4.4.m4.1.1.1" xref="S2.Thmtheorem8.p1.4.4.m4.1.1.1.cmml">⊆</mo><mi id="S2.Thmtheorem8.p1.4.4.m4.1.1.3" xref="S2.Thmtheorem8.p1.4.4.m4.1.1.3.cmml">V</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem8.p1.4.4.m4.1b"><apply id="S2.Thmtheorem8.p1.4.4.m4.1.1.cmml" xref="S2.Thmtheorem8.p1.4.4.m4.1.1"><subset id="S2.Thmtheorem8.p1.4.4.m4.1.1.1.cmml" xref="S2.Thmtheorem8.p1.4.4.m4.1.1.1"></subset><ci id="S2.Thmtheorem8.p1.4.4.m4.1.1.2.cmml" xref="S2.Thmtheorem8.p1.4.4.m4.1.1.2">𝐵</ci><ci id="S2.Thmtheorem8.p1.4.4.m4.1.1.3.cmml" xref="S2.Thmtheorem8.p1.4.4.m4.1.1.3">𝑉</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem8.p1.4.4.m4.1c">B\subseteq V</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem8.p1.4.4.m4.1d">italic_B ⊆ italic_V</annotation></semantics></math>. For any <math alttext="X\subseteq V-B" class="ltx_Math" display="inline" id="S2.Thmtheorem8.p1.5.5.m5.1"><semantics id="S2.Thmtheorem8.p1.5.5.m5.1a"><mrow id="S2.Thmtheorem8.p1.5.5.m5.1.1" xref="S2.Thmtheorem8.p1.5.5.m5.1.1.cmml"><mi id="S2.Thmtheorem8.p1.5.5.m5.1.1.2" xref="S2.Thmtheorem8.p1.5.5.m5.1.1.2.cmml">X</mi><mo id="S2.Thmtheorem8.p1.5.5.m5.1.1.1" xref="S2.Thmtheorem8.p1.5.5.m5.1.1.1.cmml">⊆</mo><mrow id="S2.Thmtheorem8.p1.5.5.m5.1.1.3" xref="S2.Thmtheorem8.p1.5.5.m5.1.1.3.cmml"><mi id="S2.Thmtheorem8.p1.5.5.m5.1.1.3.2" xref="S2.Thmtheorem8.p1.5.5.m5.1.1.3.2.cmml">V</mi><mo id="S2.Thmtheorem8.p1.5.5.m5.1.1.3.1" xref="S2.Thmtheorem8.p1.5.5.m5.1.1.3.1.cmml">−</mo><mi id="S2.Thmtheorem8.p1.5.5.m5.1.1.3.3" xref="S2.Thmtheorem8.p1.5.5.m5.1.1.3.3.cmml">B</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem8.p1.5.5.m5.1b"><apply id="S2.Thmtheorem8.p1.5.5.m5.1.1.cmml" xref="S2.Thmtheorem8.p1.5.5.m5.1.1"><subset id="S2.Thmtheorem8.p1.5.5.m5.1.1.1.cmml" xref="S2.Thmtheorem8.p1.5.5.m5.1.1.1"></subset><ci id="S2.Thmtheorem8.p1.5.5.m5.1.1.2.cmml" xref="S2.Thmtheorem8.p1.5.5.m5.1.1.2">𝑋</ci><apply id="S2.Thmtheorem8.p1.5.5.m5.1.1.3.cmml" xref="S2.Thmtheorem8.p1.5.5.m5.1.1.3"><minus id="S2.Thmtheorem8.p1.5.5.m5.1.1.3.1.cmml" xref="S2.Thmtheorem8.p1.5.5.m5.1.1.3.1"></minus><ci id="S2.Thmtheorem8.p1.5.5.m5.1.1.3.2.cmml" xref="S2.Thmtheorem8.p1.5.5.m5.1.1.3.2">𝑉</ci><ci id="S2.Thmtheorem8.p1.5.5.m5.1.1.3.3.cmml" xref="S2.Thmtheorem8.p1.5.5.m5.1.1.3.3">𝐵</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem8.p1.5.5.m5.1c">X\subseteq V-B</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem8.p1.5.5.m5.1d">italic_X ⊆ italic_V - italic_B</annotation></semantics></math>, we define the <em class="ltx_emph ltx_font_upright" id="S2.Thmtheorem8.p1.10.10.1">quotient edges</em> <math alttext="\hat{E}_{B}(X)" class="ltx_Math" display="inline" id="S2.Thmtheorem8.p1.6.6.m6.1"><semantics id="S2.Thmtheorem8.p1.6.6.m6.1a"><mrow id="S2.Thmtheorem8.p1.6.6.m6.1.2" xref="S2.Thmtheorem8.p1.6.6.m6.1.2.cmml"><msub id="S2.Thmtheorem8.p1.6.6.m6.1.2.2" xref="S2.Thmtheorem8.p1.6.6.m6.1.2.2.cmml"><mover accent="true" id="S2.Thmtheorem8.p1.6.6.m6.1.2.2.2" xref="S2.Thmtheorem8.p1.6.6.m6.1.2.2.2.cmml"><mi id="S2.Thmtheorem8.p1.6.6.m6.1.2.2.2.2" xref="S2.Thmtheorem8.p1.6.6.m6.1.2.2.2.2.cmml">E</mi><mo id="S2.Thmtheorem8.p1.6.6.m6.1.2.2.2.1" xref="S2.Thmtheorem8.p1.6.6.m6.1.2.2.2.1.cmml">^</mo></mover><mi id="S2.Thmtheorem8.p1.6.6.m6.1.2.2.3" xref="S2.Thmtheorem8.p1.6.6.m6.1.2.2.3.cmml">B</mi></msub><mo id="S2.Thmtheorem8.p1.6.6.m6.1.2.1" xref="S2.Thmtheorem8.p1.6.6.m6.1.2.1.cmml">⁢</mo><mrow id="S2.Thmtheorem8.p1.6.6.m6.1.2.3.2" xref="S2.Thmtheorem8.p1.6.6.m6.1.2.cmml"><mo id="S2.Thmtheorem8.p1.6.6.m6.1.2.3.2.1" stretchy="false" xref="S2.Thmtheorem8.p1.6.6.m6.1.2.cmml">(</mo><mi id="S2.Thmtheorem8.p1.6.6.m6.1.1" xref="S2.Thmtheorem8.p1.6.6.m6.1.1.cmml">X</mi><mo id="S2.Thmtheorem8.p1.6.6.m6.1.2.3.2.2" stretchy="false" xref="S2.Thmtheorem8.p1.6.6.m6.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem8.p1.6.6.m6.1b"><apply id="S2.Thmtheorem8.p1.6.6.m6.1.2.cmml" xref="S2.Thmtheorem8.p1.6.6.m6.1.2"><times id="S2.Thmtheorem8.p1.6.6.m6.1.2.1.cmml" xref="S2.Thmtheorem8.p1.6.6.m6.1.2.1"></times><apply id="S2.Thmtheorem8.p1.6.6.m6.1.2.2.cmml" xref="S2.Thmtheorem8.p1.6.6.m6.1.2.2"><csymbol cd="ambiguous" id="S2.Thmtheorem8.p1.6.6.m6.1.2.2.1.cmml" xref="S2.Thmtheorem8.p1.6.6.m6.1.2.2">subscript</csymbol><apply id="S2.Thmtheorem8.p1.6.6.m6.1.2.2.2.cmml" xref="S2.Thmtheorem8.p1.6.6.m6.1.2.2.2"><ci id="S2.Thmtheorem8.p1.6.6.m6.1.2.2.2.1.cmml" xref="S2.Thmtheorem8.p1.6.6.m6.1.2.2.2.1">^</ci><ci id="S2.Thmtheorem8.p1.6.6.m6.1.2.2.2.2.cmml" xref="S2.Thmtheorem8.p1.6.6.m6.1.2.2.2.2">𝐸</ci></apply><ci id="S2.Thmtheorem8.p1.6.6.m6.1.2.2.3.cmml" xref="S2.Thmtheorem8.p1.6.6.m6.1.2.2.3">𝐵</ci></apply><ci id="S2.Thmtheorem8.p1.6.6.m6.1.1.cmml" xref="S2.Thmtheorem8.p1.6.6.m6.1.1">𝑋</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem8.p1.6.6.m6.1c">\hat{E}_{B}(X)</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem8.p1.6.6.m6.1d">over^ start_ARG italic_E end_ARG start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT ( italic_X )</annotation></semantics></math> as all edges in <math alttext="G" class="ltx_Math" display="inline" id="S2.Thmtheorem8.p1.7.7.m7.1"><semantics id="S2.Thmtheorem8.p1.7.7.m7.1a"><mi id="S2.Thmtheorem8.p1.7.7.m7.1.1" xref="S2.Thmtheorem8.p1.7.7.m7.1.1.cmml">G</mi><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem8.p1.7.7.m7.1b"><ci id="S2.Thmtheorem8.p1.7.7.m7.1.1.cmml" xref="S2.Thmtheorem8.p1.7.7.m7.1.1">𝐺</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem8.p1.7.7.m7.1c">G</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem8.p1.7.7.m7.1d">italic_G</annotation></semantics></math> with one endpoint in <math alttext="X" class="ltx_Math" display="inline" id="S2.Thmtheorem8.p1.8.8.m8.1"><semantics id="S2.Thmtheorem8.p1.8.8.m8.1a"><mi id="S2.Thmtheorem8.p1.8.8.m8.1.1" xref="S2.Thmtheorem8.p1.8.8.m8.1.1.cmml">X</mi><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem8.p1.8.8.m8.1b"><ci id="S2.Thmtheorem8.p1.8.8.m8.1.1.cmml" xref="S2.Thmtheorem8.p1.8.8.m8.1.1">𝑋</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem8.p1.8.8.m8.1c">X</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem8.p1.8.8.m8.1d">italic_X</annotation></semantics></math>, and the other endpoint in <math alttext="X" class="ltx_Math" display="inline" id="S2.Thmtheorem8.p1.9.9.m9.1"><semantics id="S2.Thmtheorem8.p1.9.9.m9.1a"><mi id="S2.Thmtheorem8.p1.9.9.m9.1.1" xref="S2.Thmtheorem8.p1.9.9.m9.1.1.cmml">X</mi><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem8.p1.9.9.m9.1b"><ci id="S2.Thmtheorem8.p1.9.9.m9.1.1.cmml" xref="S2.Thmtheorem8.p1.9.9.m9.1.1">𝑋</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem8.p1.9.9.m9.1c">X</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem8.p1.9.9.m9.1d">italic_X</annotation></semantics></math> or <math alttext="B" class="ltx_Math" display="inline" id="S2.Thmtheorem8.p1.10.10.m10.1"><semantics id="S2.Thmtheorem8.p1.10.10.m10.1a"><mi id="S2.Thmtheorem8.p1.10.10.m10.1.1" xref="S2.Thmtheorem8.p1.10.10.m10.1.1.cmml">B</mi><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem8.p1.10.10.m10.1b"><ci id="S2.Thmtheorem8.p1.10.10.m10.1.1.cmml" xref="S2.Thmtheorem8.p1.10.10.m10.1.1">𝐵</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem8.p1.10.10.m10.1c">B</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem8.p1.10.10.m10.1d">italic_B</annotation></semantics></math>. We define:</span></p> <ul class="ltx_itemize" id="S2.I2"> <li class="ltx_item" id="S2.I2.i1" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">•</span> <div class="ltx_para" id="S2.I2.i1.p1"> <p class="ltx_p" id="S2.I2.i1.p1.2"><span class="ltx_text ltx_font_italic" id="S2.I2.i1.p1.2.1">for </span><math alttext="X\subseteq V-B" class="ltx_Math" display="inline" id="S2.I2.i1.p1.1.m1.1"><semantics id="S2.I2.i1.p1.1.m1.1a"><mrow id="S2.I2.i1.p1.1.m1.1.1" xref="S2.I2.i1.p1.1.m1.1.1.cmml"><mi id="S2.I2.i1.p1.1.m1.1.1.2" xref="S2.I2.i1.p1.1.m1.1.1.2.cmml">X</mi><mo id="S2.I2.i1.p1.1.m1.1.1.1" xref="S2.I2.i1.p1.1.m1.1.1.1.cmml">⊆</mo><mrow id="S2.I2.i1.p1.1.m1.1.1.3" xref="S2.I2.i1.p1.1.m1.1.1.3.cmml"><mi id="S2.I2.i1.p1.1.m1.1.1.3.2" xref="S2.I2.i1.p1.1.m1.1.1.3.2.cmml">V</mi><mo id="S2.I2.i1.p1.1.m1.1.1.3.1" xref="S2.I2.i1.p1.1.m1.1.1.3.1.cmml">−</mo><mi id="S2.I2.i1.p1.1.m1.1.1.3.3" xref="S2.I2.i1.p1.1.m1.1.1.3.3.cmml">B</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.I2.i1.p1.1.m1.1b"><apply id="S2.I2.i1.p1.1.m1.1.1.cmml" xref="S2.I2.i1.p1.1.m1.1.1"><subset id="S2.I2.i1.p1.1.m1.1.1.1.cmml" xref="S2.I2.i1.p1.1.m1.1.1.1"></subset><ci id="S2.I2.i1.p1.1.m1.1.1.2.cmml" xref="S2.I2.i1.p1.1.m1.1.1.2">𝑋</ci><apply id="S2.I2.i1.p1.1.m1.1.1.3.cmml" xref="S2.I2.i1.p1.1.m1.1.1.3"><minus id="S2.I2.i1.p1.1.m1.1.1.3.1.cmml" xref="S2.I2.i1.p1.1.m1.1.1.3.1"></minus><ci id="S2.I2.i1.p1.1.m1.1.1.3.2.cmml" xref="S2.I2.i1.p1.1.m1.1.1.3.2">𝑉</ci><ci id="S2.I2.i1.p1.1.m1.1.1.3.3.cmml" xref="S2.I2.i1.p1.1.m1.1.1.3.3">𝐵</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.I2.i1.p1.1.m1.1c">X\subseteq V-B</annotation><annotation encoding="application/x-llamapun" id="S2.I2.i1.p1.1.m1.1d">italic_X ⊆ italic_V - italic_B</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S2.I2.i1.p1.2.2">, the </span><em class="ltx_emph" id="S2.I2.i1.p1.2.3">quotient subgraph density</em><span class="ltx_text ltx_font_italic" id="S2.I2.i1.p1.2.4"> </span><math alttext="\hat{\rho}_{B}(X):=\frac{1}{|X|}\sum\limits_{e\in\hat{E}_{B}(X)}\textsl{g}(e)" class="ltx_Math" display="inline" id="S2.I2.i1.p1.2.m2.4"><semantics id="S2.I2.i1.p1.2.m2.4a"><mrow id="S2.I2.i1.p1.2.m2.4.5" xref="S2.I2.i1.p1.2.m2.4.5.cmml"><mrow id="S2.I2.i1.p1.2.m2.4.5.2" xref="S2.I2.i1.p1.2.m2.4.5.2.cmml"><msub id="S2.I2.i1.p1.2.m2.4.5.2.2" xref="S2.I2.i1.p1.2.m2.4.5.2.2.cmml"><mover accent="true" id="S2.I2.i1.p1.2.m2.4.5.2.2.2" xref="S2.I2.i1.p1.2.m2.4.5.2.2.2.cmml"><mi id="S2.I2.i1.p1.2.m2.4.5.2.2.2.2" xref="S2.I2.i1.p1.2.m2.4.5.2.2.2.2.cmml">ρ</mi><mo id="S2.I2.i1.p1.2.m2.4.5.2.2.2.1" xref="S2.I2.i1.p1.2.m2.4.5.2.2.2.1.cmml">^</mo></mover><mi id="S2.I2.i1.p1.2.m2.4.5.2.2.3" xref="S2.I2.i1.p1.2.m2.4.5.2.2.3.cmml">B</mi></msub><mo id="S2.I2.i1.p1.2.m2.4.5.2.1" xref="S2.I2.i1.p1.2.m2.4.5.2.1.cmml">⁢</mo><mrow id="S2.I2.i1.p1.2.m2.4.5.2.3.2" xref="S2.I2.i1.p1.2.m2.4.5.2.cmml"><mo 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xref="S2.I2.i1.p1.2.m2.4.5.3.1.cmml">⁢</mo><mrow id="S2.I2.i1.p1.2.m2.4.5.3.2" xref="S2.I2.i1.p1.2.m2.4.5.3.2.cmml"><munder id="S2.I2.i1.p1.2.m2.4.5.3.2.1" xref="S2.I2.i1.p1.2.m2.4.5.3.2.1.cmml"><mo id="S2.I2.i1.p1.2.m2.4.5.3.2.1.2" movablelimits="false" xref="S2.I2.i1.p1.2.m2.4.5.3.2.1.2.cmml">∑</mo><mrow id="S2.I2.i1.p1.2.m2.2.2.1" xref="S2.I2.i1.p1.2.m2.2.2.1.cmml"><mi id="S2.I2.i1.p1.2.m2.2.2.1.3" xref="S2.I2.i1.p1.2.m2.2.2.1.3.cmml">e</mi><mo id="S2.I2.i1.p1.2.m2.2.2.1.2" xref="S2.I2.i1.p1.2.m2.2.2.1.2.cmml">∈</mo><mrow id="S2.I2.i1.p1.2.m2.2.2.1.4" xref="S2.I2.i1.p1.2.m2.2.2.1.4.cmml"><msub id="S2.I2.i1.p1.2.m2.2.2.1.4.2" xref="S2.I2.i1.p1.2.m2.2.2.1.4.2.cmml"><mover accent="true" id="S2.I2.i1.p1.2.m2.2.2.1.4.2.2" xref="S2.I2.i1.p1.2.m2.2.2.1.4.2.2.cmml"><mi id="S2.I2.i1.p1.2.m2.2.2.1.4.2.2.2" xref="S2.I2.i1.p1.2.m2.2.2.1.4.2.2.2.cmml">E</mi><mo id="S2.I2.i1.p1.2.m2.2.2.1.4.2.2.1" xref="S2.I2.i1.p1.2.m2.2.2.1.4.2.2.1.cmml">^</mo></mover><mi id="S2.I2.i1.p1.2.m2.2.2.1.4.2.3" xref="S2.I2.i1.p1.2.m2.2.2.1.4.2.3.cmml">B</mi></msub><mo id="S2.I2.i1.p1.2.m2.2.2.1.4.1" xref="S2.I2.i1.p1.2.m2.2.2.1.4.1.cmml">⁢</mo><mrow id="S2.I2.i1.p1.2.m2.2.2.1.4.3.2" xref="S2.I2.i1.p1.2.m2.2.2.1.4.cmml"><mo id="S2.I2.i1.p1.2.m2.2.2.1.4.3.2.1" stretchy="false" xref="S2.I2.i1.p1.2.m2.2.2.1.4.cmml">(</mo><mi id="S2.I2.i1.p1.2.m2.2.2.1.1" xref="S2.I2.i1.p1.2.m2.2.2.1.1.cmml">X</mi><mo id="S2.I2.i1.p1.2.m2.2.2.1.4.3.2.2" stretchy="false" xref="S2.I2.i1.p1.2.m2.2.2.1.4.cmml">)</mo></mrow></mrow></mrow></munder><mrow id="S2.I2.i1.p1.2.m2.4.5.3.2.2" xref="S2.I2.i1.p1.2.m2.4.5.3.2.2.cmml"><mtext class="ltx_mathvariant_italic" id="S2.I2.i1.p1.2.m2.4.5.3.2.2.2" xref="S2.I2.i1.p1.2.m2.4.5.3.2.2.2a.cmml">g</mtext><mo id="S2.I2.i1.p1.2.m2.4.5.3.2.2.1" xref="S2.I2.i1.p1.2.m2.4.5.3.2.2.1.cmml">⁢</mo><mrow id="S2.I2.i1.p1.2.m2.4.5.3.2.2.3.2" xref="S2.I2.i1.p1.2.m2.4.5.3.2.2.cmml"><mo id="S2.I2.i1.p1.2.m2.4.5.3.2.2.3.2.1" stretchy="false" xref="S2.I2.i1.p1.2.m2.4.5.3.2.2.cmml">(</mo><mi 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id="S2.I2.i1.p1.2.m2.2.2.1.4.2.2.2.cmml" xref="S2.I2.i1.p1.2.m2.2.2.1.4.2.2.2">𝐸</ci></apply><ci id="S2.I2.i1.p1.2.m2.2.2.1.4.2.3.cmml" xref="S2.I2.i1.p1.2.m2.2.2.1.4.2.3">𝐵</ci></apply><ci id="S2.I2.i1.p1.2.m2.2.2.1.1.cmml" xref="S2.I2.i1.p1.2.m2.2.2.1.1">𝑋</ci></apply></apply></apply><apply id="S2.I2.i1.p1.2.m2.4.5.3.2.2.cmml" xref="S2.I2.i1.p1.2.m2.4.5.3.2.2"><times id="S2.I2.i1.p1.2.m2.4.5.3.2.2.1.cmml" xref="S2.I2.i1.p1.2.m2.4.5.3.2.2.1"></times><ci id="S2.I2.i1.p1.2.m2.4.5.3.2.2.2a.cmml" xref="S2.I2.i1.p1.2.m2.4.5.3.2.2.2"><mtext class="ltx_mathvariant_italic" id="S2.I2.i1.p1.2.m2.4.5.3.2.2.2.cmml" xref="S2.I2.i1.p1.2.m2.4.5.3.2.2.2">g</mtext></ci><ci id="S2.I2.i1.p1.2.m2.4.4.cmml" xref="S2.I2.i1.p1.2.m2.4.4">𝑒</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.I2.i1.p1.2.m2.4c">\hat{\rho}_{B}(X):=\frac{1}{|X|}\sum\limits_{e\in\hat{E}_{B}(X)}\textsl{g}(e)</annotation><annotation encoding="application/x-llamapun" id="S2.I2.i1.p1.2.m2.4d">over^ start_ARG italic_ρ end_ARG start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT ( italic_X ) := divide start_ARG 1 end_ARG start_ARG | italic_X | end_ARG ∑ start_POSTSUBSCRIPT italic_e ∈ over^ start_ARG italic_E end_ARG start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT ( italic_X ) end_POSTSUBSCRIPT g ( italic_e )</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S2.I2.i1.p1.2.5">.</span></p> </div> </li> <li class="ltx_item" id="S2.I2.i2" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">•</span> <div class="ltx_para" id="S2.I2.i2.p1"> <p class="ltx_p" id="S2.I2.i2.p1.1"><span class="ltx_text ltx_font_italic" id="S2.I2.i2.p1.1.1">the </span><em class="ltx_emph" id="S2.I2.i2.p1.1.2">maximum quotient density</em><span class="ltx_text ltx_font_italic" id="S2.I2.i2.p1.1.3"> </span><math alttext="\hat{\rho}_{B}(G):=\max\limits_{X\subseteq V-B}\,\hat{\rho}_{B}(X)" class="ltx_Math" display="inline" 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xref="S2.I2.i2.p1.1.m1.2.3.3.2.1.3.3"><minus id="S2.I2.i2.p1.1.m1.2.3.3.2.1.3.3.1.cmml" xref="S2.I2.i2.p1.1.m1.2.3.3.2.1.3.3.1"></minus><ci id="S2.I2.i2.p1.1.m1.2.3.3.2.1.3.3.2.cmml" xref="S2.I2.i2.p1.1.m1.2.3.3.2.1.3.3.2">𝑉</ci><ci id="S2.I2.i2.p1.1.m1.2.3.3.2.1.3.3.3.cmml" xref="S2.I2.i2.p1.1.m1.2.3.3.2.1.3.3.3">𝐵</ci></apply></apply></apply><apply id="S2.I2.i2.p1.1.m1.2.3.3.2.2.cmml" xref="S2.I2.i2.p1.1.m1.2.3.3.2.2"><csymbol cd="ambiguous" id="S2.I2.i2.p1.1.m1.2.3.3.2.2.1.cmml" xref="S2.I2.i2.p1.1.m1.2.3.3.2.2">subscript</csymbol><apply id="S2.I2.i2.p1.1.m1.2.3.3.2.2.2.cmml" xref="S2.I2.i2.p1.1.m1.2.3.3.2.2.2"><ci id="S2.I2.i2.p1.1.m1.2.3.3.2.2.2.1.cmml" xref="S2.I2.i2.p1.1.m1.2.3.3.2.2.2.1">^</ci><ci id="S2.I2.i2.p1.1.m1.2.3.3.2.2.2.2.cmml" xref="S2.I2.i2.p1.1.m1.2.3.3.2.2.2.2">𝜌</ci></apply><ci id="S2.I2.i2.p1.1.m1.2.3.3.2.2.3.cmml" xref="S2.I2.i2.p1.1.m1.2.3.3.2.2.3">𝐵</ci></apply></apply><ci id="S2.I2.i2.p1.1.m1.2.2.cmml" xref="S2.I2.i2.p1.1.m1.2.2">𝑋</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.I2.i2.p1.1.m1.2c">\hat{\rho}_{B}(G):=\max\limits_{X\subseteq V-B}\,\hat{\rho}_{B}(X)</annotation><annotation encoding="application/x-llamapun" id="S2.I2.i2.p1.1.m1.2d">over^ start_ARG italic_ρ end_ARG start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT ( italic_G ) := roman_max start_POSTSUBSCRIPT italic_X ⊆ italic_V - italic_B end_POSTSUBSCRIPT over^ start_ARG italic_ρ end_ARG start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT ( italic_X )</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S2.I2.i2.p1.1.4">.</span></p> </div> </li> </ul> </div> </div> <div class="ltx_theorem ltx_theorem_definition" id="S2.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="S2.Thmtheorem9.1.1.1">Definition 2.9</span></span><span class="ltx_text ltx_font_bold" id="S2.Thmtheorem9.2.2"> </span>(Definition 2.3 in <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2411.12694v2#bib.bib10" title="">10</a>]</cite>)<span class="ltx_text ltx_font_bold" id="S2.Thmtheorem9.3.3">.</span> </h6> <div class="ltx_para" id="S2.Thmtheorem9.p1"> <p class="ltx_p" id="S2.Thmtheorem9.p1.4"><span class="ltx_text ltx_font_italic" id="S2.Thmtheorem9.p1.4.4">Given a weighted undirected graph <math alttext="G=(V,E)" class="ltx_Math" display="inline" id="S2.Thmtheorem9.p1.1.1.m1.2"><semantics id="S2.Thmtheorem9.p1.1.1.m1.2a"><mrow id="S2.Thmtheorem9.p1.1.1.m1.2.3" xref="S2.Thmtheorem9.p1.1.1.m1.2.3.cmml"><mi id="S2.Thmtheorem9.p1.1.1.m1.2.3.2" xref="S2.Thmtheorem9.p1.1.1.m1.2.3.2.cmml">G</mi><mo id="S2.Thmtheorem9.p1.1.1.m1.2.3.1" xref="S2.Thmtheorem9.p1.1.1.m1.2.3.1.cmml">=</mo><mrow id="S2.Thmtheorem9.p1.1.1.m1.2.3.3.2" xref="S2.Thmtheorem9.p1.1.1.m1.2.3.3.1.cmml"><mo id="S2.Thmtheorem9.p1.1.1.m1.2.3.3.2.1" stretchy="false" xref="S2.Thmtheorem9.p1.1.1.m1.2.3.3.1.cmml">(</mo><mi id="S2.Thmtheorem9.p1.1.1.m1.1.1" xref="S2.Thmtheorem9.p1.1.1.m1.1.1.cmml">V</mi><mo id="S2.Thmtheorem9.p1.1.1.m1.2.3.3.2.2" xref="S2.Thmtheorem9.p1.1.1.m1.2.3.3.1.cmml">,</mo><mi id="S2.Thmtheorem9.p1.1.1.m1.2.2" xref="S2.Thmtheorem9.p1.1.1.m1.2.2.cmml">E</mi><mo id="S2.Thmtheorem9.p1.1.1.m1.2.3.3.2.3" stretchy="false" xref="S2.Thmtheorem9.p1.1.1.m1.2.3.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem9.p1.1.1.m1.2b"><apply id="S2.Thmtheorem9.p1.1.1.m1.2.3.cmml" xref="S2.Thmtheorem9.p1.1.1.m1.2.3"><eq id="S2.Thmtheorem9.p1.1.1.m1.2.3.1.cmml" xref="S2.Thmtheorem9.p1.1.1.m1.2.3.1"></eq><ci id="S2.Thmtheorem9.p1.1.1.m1.2.3.2.cmml" xref="S2.Thmtheorem9.p1.1.1.m1.2.3.2">𝐺</ci><interval closure="open" id="S2.Thmtheorem9.p1.1.1.m1.2.3.3.1.cmml" xref="S2.Thmtheorem9.p1.1.1.m1.2.3.3.2"><ci id="S2.Thmtheorem9.p1.1.1.m1.1.1.cmml" xref="S2.Thmtheorem9.p1.1.1.m1.1.1">𝑉</ci><ci id="S2.Thmtheorem9.p1.1.1.m1.2.2.cmml" xref="S2.Thmtheorem9.p1.1.1.m1.2.2">𝐸</ci></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem9.p1.1.1.m1.2c">G=(V,E)</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem9.p1.1.1.m1.2d">italic_G = ( italic_V , italic_E )</annotation></semantics></math>, we define the diminishing-dense decomposition <math alttext="\mathcal{B}" class="ltx_Math" display="inline" id="S2.Thmtheorem9.p1.2.2.m2.1"><semantics id="S2.Thmtheorem9.p1.2.2.m2.1a"><mi class="ltx_font_mathcaligraphic" id="S2.Thmtheorem9.p1.2.2.m2.1.1" xref="S2.Thmtheorem9.p1.2.2.m2.1.1.cmml">ℬ</mi><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem9.p1.2.2.m2.1b"><ci id="S2.Thmtheorem9.p1.2.2.m2.1.1.cmml" xref="S2.Thmtheorem9.p1.2.2.m2.1.1">ℬ</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem9.p1.2.2.m2.1c">\mathcal{B}</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem9.p1.2.2.m2.1d">caligraphic_B</annotation></semantics></math> of <math alttext="G" class="ltx_Math" display="inline" id="S2.Thmtheorem9.p1.3.3.m3.1"><semantics id="S2.Thmtheorem9.p1.3.3.m3.1a"><mi id="S2.Thmtheorem9.p1.3.3.m3.1.1" xref="S2.Thmtheorem9.p1.3.3.m3.1.1.cmml">G</mi><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem9.p1.3.3.m3.1b"><ci id="S2.Thmtheorem9.p1.3.3.m3.1.1.cmml" xref="S2.Thmtheorem9.p1.3.3.m3.1.1">𝐺</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem9.p1.3.3.m3.1c">G</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem9.p1.3.3.m3.1d">italic_G</annotation></semantics></math> as the sequence <math alttext="B_{0}\subset B_{1}\ldots\subset B_{\ell}=V" class="ltx_Math" display="inline" id="S2.Thmtheorem9.p1.4.4.m4.1"><semantics id="S2.Thmtheorem9.p1.4.4.m4.1a"><mrow id="S2.Thmtheorem9.p1.4.4.m4.1.1" xref="S2.Thmtheorem9.p1.4.4.m4.1.1.cmml"><msub id="S2.Thmtheorem9.p1.4.4.m4.1.1.2" xref="S2.Thmtheorem9.p1.4.4.m4.1.1.2.cmml"><mi id="S2.Thmtheorem9.p1.4.4.m4.1.1.2.2" xref="S2.Thmtheorem9.p1.4.4.m4.1.1.2.2.cmml">B</mi><mn id="S2.Thmtheorem9.p1.4.4.m4.1.1.2.3" xref="S2.Thmtheorem9.p1.4.4.m4.1.1.2.3.cmml">0</mn></msub><mo id="S2.Thmtheorem9.p1.4.4.m4.1.1.3" xref="S2.Thmtheorem9.p1.4.4.m4.1.1.3.cmml">⊂</mo><mrow id="S2.Thmtheorem9.p1.4.4.m4.1.1.4" xref="S2.Thmtheorem9.p1.4.4.m4.1.1.4.cmml"><msub id="S2.Thmtheorem9.p1.4.4.m4.1.1.4.2" xref="S2.Thmtheorem9.p1.4.4.m4.1.1.4.2.cmml"><mi id="S2.Thmtheorem9.p1.4.4.m4.1.1.4.2.2" xref="S2.Thmtheorem9.p1.4.4.m4.1.1.4.2.2.cmml">B</mi><mn id="S2.Thmtheorem9.p1.4.4.m4.1.1.4.2.3" xref="S2.Thmtheorem9.p1.4.4.m4.1.1.4.2.3.cmml">1</mn></msub><mo id="S2.Thmtheorem9.p1.4.4.m4.1.1.4.1" xref="S2.Thmtheorem9.p1.4.4.m4.1.1.4.1.cmml">⁢</mo><mi id="S2.Thmtheorem9.p1.4.4.m4.1.1.4.3" mathvariant="normal" xref="S2.Thmtheorem9.p1.4.4.m4.1.1.4.3.cmml">…</mi></mrow><mo id="S2.Thmtheorem9.p1.4.4.m4.1.1.5" xref="S2.Thmtheorem9.p1.4.4.m4.1.1.5.cmml">⊂</mo><msub id="S2.Thmtheorem9.p1.4.4.m4.1.1.6" xref="S2.Thmtheorem9.p1.4.4.m4.1.1.6.cmml"><mi id="S2.Thmtheorem9.p1.4.4.m4.1.1.6.2" xref="S2.Thmtheorem9.p1.4.4.m4.1.1.6.2.cmml">B</mi><mi id="S2.Thmtheorem9.p1.4.4.m4.1.1.6.3" mathvariant="normal" xref="S2.Thmtheorem9.p1.4.4.m4.1.1.6.3.cmml">ℓ</mi></msub><mo id="S2.Thmtheorem9.p1.4.4.m4.1.1.7" xref="S2.Thmtheorem9.p1.4.4.m4.1.1.7.cmml">=</mo><mi id="S2.Thmtheorem9.p1.4.4.m4.1.1.8" xref="S2.Thmtheorem9.p1.4.4.m4.1.1.8.cmml">V</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem9.p1.4.4.m4.1b"><apply id="S2.Thmtheorem9.p1.4.4.m4.1.1.cmml" xref="S2.Thmtheorem9.p1.4.4.m4.1.1"><and id="S2.Thmtheorem9.p1.4.4.m4.1.1a.cmml" xref="S2.Thmtheorem9.p1.4.4.m4.1.1"></and><apply id="S2.Thmtheorem9.p1.4.4.m4.1.1b.cmml" xref="S2.Thmtheorem9.p1.4.4.m4.1.1"><subset id="S2.Thmtheorem9.p1.4.4.m4.1.1.3.cmml" xref="S2.Thmtheorem9.p1.4.4.m4.1.1.3"></subset><apply id="S2.Thmtheorem9.p1.4.4.m4.1.1.2.cmml" xref="S2.Thmtheorem9.p1.4.4.m4.1.1.2"><csymbol cd="ambiguous" id="S2.Thmtheorem9.p1.4.4.m4.1.1.2.1.cmml" xref="S2.Thmtheorem9.p1.4.4.m4.1.1.2">subscript</csymbol><ci id="S2.Thmtheorem9.p1.4.4.m4.1.1.2.2.cmml" xref="S2.Thmtheorem9.p1.4.4.m4.1.1.2.2">𝐵</ci><cn id="S2.Thmtheorem9.p1.4.4.m4.1.1.2.3.cmml" type="integer" xref="S2.Thmtheorem9.p1.4.4.m4.1.1.2.3">0</cn></apply><apply id="S2.Thmtheorem9.p1.4.4.m4.1.1.4.cmml" xref="S2.Thmtheorem9.p1.4.4.m4.1.1.4"><times id="S2.Thmtheorem9.p1.4.4.m4.1.1.4.1.cmml" xref="S2.Thmtheorem9.p1.4.4.m4.1.1.4.1"></times><apply id="S2.Thmtheorem9.p1.4.4.m4.1.1.4.2.cmml" xref="S2.Thmtheorem9.p1.4.4.m4.1.1.4.2"><csymbol cd="ambiguous" id="S2.Thmtheorem9.p1.4.4.m4.1.1.4.2.1.cmml" xref="S2.Thmtheorem9.p1.4.4.m4.1.1.4.2">subscript</csymbol><ci id="S2.Thmtheorem9.p1.4.4.m4.1.1.4.2.2.cmml" xref="S2.Thmtheorem9.p1.4.4.m4.1.1.4.2.2">𝐵</ci><cn id="S2.Thmtheorem9.p1.4.4.m4.1.1.4.2.3.cmml" type="integer" xref="S2.Thmtheorem9.p1.4.4.m4.1.1.4.2.3">1</cn></apply><ci id="S2.Thmtheorem9.p1.4.4.m4.1.1.4.3.cmml" xref="S2.Thmtheorem9.p1.4.4.m4.1.1.4.3">…</ci></apply></apply><apply id="S2.Thmtheorem9.p1.4.4.m4.1.1c.cmml" xref="S2.Thmtheorem9.p1.4.4.m4.1.1"><subset id="S2.Thmtheorem9.p1.4.4.m4.1.1.5.cmml" xref="S2.Thmtheorem9.p1.4.4.m4.1.1.5"></subset><share href="https://arxiv.org/html/2411.12694v2#S2.Thmtheorem9.p1.4.4.m4.1.1.4.cmml" id="S2.Thmtheorem9.p1.4.4.m4.1.1d.cmml" xref="S2.Thmtheorem9.p1.4.4.m4.1.1"></share><apply id="S2.Thmtheorem9.p1.4.4.m4.1.1.6.cmml" xref="S2.Thmtheorem9.p1.4.4.m4.1.1.6"><csymbol cd="ambiguous" id="S2.Thmtheorem9.p1.4.4.m4.1.1.6.1.cmml" xref="S2.Thmtheorem9.p1.4.4.m4.1.1.6">subscript</csymbol><ci id="S2.Thmtheorem9.p1.4.4.m4.1.1.6.2.cmml" xref="S2.Thmtheorem9.p1.4.4.m4.1.1.6.2">𝐵</ci><ci id="S2.Thmtheorem9.p1.4.4.m4.1.1.6.3.cmml" xref="S2.Thmtheorem9.p1.4.4.m4.1.1.6.3">ℓ</ci></apply></apply><apply id="S2.Thmtheorem9.p1.4.4.m4.1.1e.cmml" xref="S2.Thmtheorem9.p1.4.4.m4.1.1"><eq id="S2.Thmtheorem9.p1.4.4.m4.1.1.7.cmml" xref="S2.Thmtheorem9.p1.4.4.m4.1.1.7"></eq><share href="https://arxiv.org/html/2411.12694v2#S2.Thmtheorem9.p1.4.4.m4.1.1.6.cmml" id="S2.Thmtheorem9.p1.4.4.m4.1.1f.cmml" xref="S2.Thmtheorem9.p1.4.4.m4.1.1"></share><ci id="S2.Thmtheorem9.p1.4.4.m4.1.1.8.cmml" xref="S2.Thmtheorem9.p1.4.4.m4.1.1.8">𝑉</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem9.p1.4.4.m4.1c">B_{0}\subset B_{1}\ldots\subset B_{\ell}=V</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem9.p1.4.4.m4.1d">italic_B start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ⊂ italic_B start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT … ⊂ italic_B start_POSTSUBSCRIPT roman_ℓ end_POSTSUBSCRIPT = italic_V</annotation></semantics></math>:</span></p> </div> <div class="ltx_para" id="S2.Thmtheorem9.p2"> <p class="ltx_p" id="S2.Thmtheorem9.p2.4"><span class="ltx_text ltx_font_italic" id="S2.Thmtheorem9.p2.4.4">We define <math alttext="B_{0}=\emptyset" class="ltx_Math" display="inline" id="S2.Thmtheorem9.p2.1.1.m1.1"><semantics id="S2.Thmtheorem9.p2.1.1.m1.1a"><mrow id="S2.Thmtheorem9.p2.1.1.m1.1.1" xref="S2.Thmtheorem9.p2.1.1.m1.1.1.cmml"><msub id="S2.Thmtheorem9.p2.1.1.m1.1.1.2" xref="S2.Thmtheorem9.p2.1.1.m1.1.1.2.cmml"><mi id="S2.Thmtheorem9.p2.1.1.m1.1.1.2.2" xref="S2.Thmtheorem9.p2.1.1.m1.1.1.2.2.cmml">B</mi><mn id="S2.Thmtheorem9.p2.1.1.m1.1.1.2.3" xref="S2.Thmtheorem9.p2.1.1.m1.1.1.2.3.cmml">0</mn></msub><mo id="S2.Thmtheorem9.p2.1.1.m1.1.1.1" xref="S2.Thmtheorem9.p2.1.1.m1.1.1.1.cmml">=</mo><mi id="S2.Thmtheorem9.p2.1.1.m1.1.1.3" mathvariant="normal" xref="S2.Thmtheorem9.p2.1.1.m1.1.1.3.cmml">∅</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem9.p2.1.1.m1.1b"><apply id="S2.Thmtheorem9.p2.1.1.m1.1.1.cmml" xref="S2.Thmtheorem9.p2.1.1.m1.1.1"><eq id="S2.Thmtheorem9.p2.1.1.m1.1.1.1.cmml" xref="S2.Thmtheorem9.p2.1.1.m1.1.1.1"></eq><apply id="S2.Thmtheorem9.p2.1.1.m1.1.1.2.cmml" xref="S2.Thmtheorem9.p2.1.1.m1.1.1.2"><csymbol cd="ambiguous" id="S2.Thmtheorem9.p2.1.1.m1.1.1.2.1.cmml" xref="S2.Thmtheorem9.p2.1.1.m1.1.1.2">subscript</csymbol><ci id="S2.Thmtheorem9.p2.1.1.m1.1.1.2.2.cmml" xref="S2.Thmtheorem9.p2.1.1.m1.1.1.2.2">𝐵</ci><cn id="S2.Thmtheorem9.p2.1.1.m1.1.1.2.3.cmml" type="integer" xref="S2.Thmtheorem9.p2.1.1.m1.1.1.2.3">0</cn></apply><emptyset id="S2.Thmtheorem9.p2.1.1.m1.1.1.3.cmml" xref="S2.Thmtheorem9.p2.1.1.m1.1.1.3"></emptyset></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem9.p2.1.1.m1.1c">B_{0}=\emptyset</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem9.p2.1.1.m1.1d">italic_B start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT = ∅</annotation></semantics></math>. For <math alttext="i\geq 1" class="ltx_Math" display="inline" id="S2.Thmtheorem9.p2.2.2.m2.1"><semantics id="S2.Thmtheorem9.p2.2.2.m2.1a"><mrow id="S2.Thmtheorem9.p2.2.2.m2.1.1" xref="S2.Thmtheorem9.p2.2.2.m2.1.1.cmml"><mi id="S2.Thmtheorem9.p2.2.2.m2.1.1.2" xref="S2.Thmtheorem9.p2.2.2.m2.1.1.2.cmml">i</mi><mo id="S2.Thmtheorem9.p2.2.2.m2.1.1.1" xref="S2.Thmtheorem9.p2.2.2.m2.1.1.1.cmml">≥</mo><mn id="S2.Thmtheorem9.p2.2.2.m2.1.1.3" xref="S2.Thmtheorem9.p2.2.2.m2.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem9.p2.2.2.m2.1b"><apply id="S2.Thmtheorem9.p2.2.2.m2.1.1.cmml" xref="S2.Thmtheorem9.p2.2.2.m2.1.1"><geq id="S2.Thmtheorem9.p2.2.2.m2.1.1.1.cmml" xref="S2.Thmtheorem9.p2.2.2.m2.1.1.1"></geq><ci id="S2.Thmtheorem9.p2.2.2.m2.1.1.2.cmml" xref="S2.Thmtheorem9.p2.2.2.m2.1.1.2">𝑖</ci><cn id="S2.Thmtheorem9.p2.2.2.m2.1.1.3.cmml" type="integer" xref="S2.Thmtheorem9.p2.2.2.m2.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem9.p2.2.2.m2.1c">i\geq 1</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem9.p2.2.2.m2.1d">italic_i ≥ 1</annotation></semantics></math> if <math alttext="B_{i-1}=V" class="ltx_Math" display="inline" id="S2.Thmtheorem9.p2.3.3.m3.1"><semantics id="S2.Thmtheorem9.p2.3.3.m3.1a"><mrow id="S2.Thmtheorem9.p2.3.3.m3.1.1" xref="S2.Thmtheorem9.p2.3.3.m3.1.1.cmml"><msub id="S2.Thmtheorem9.p2.3.3.m3.1.1.2" xref="S2.Thmtheorem9.p2.3.3.m3.1.1.2.cmml"><mi id="S2.Thmtheorem9.p2.3.3.m3.1.1.2.2" xref="S2.Thmtheorem9.p2.3.3.m3.1.1.2.2.cmml">B</mi><mrow id="S2.Thmtheorem9.p2.3.3.m3.1.1.2.3" xref="S2.Thmtheorem9.p2.3.3.m3.1.1.2.3.cmml"><mi id="S2.Thmtheorem9.p2.3.3.m3.1.1.2.3.2" xref="S2.Thmtheorem9.p2.3.3.m3.1.1.2.3.2.cmml">i</mi><mo id="S2.Thmtheorem9.p2.3.3.m3.1.1.2.3.1" xref="S2.Thmtheorem9.p2.3.3.m3.1.1.2.3.1.cmml">−</mo><mn id="S2.Thmtheorem9.p2.3.3.m3.1.1.2.3.3" xref="S2.Thmtheorem9.p2.3.3.m3.1.1.2.3.3.cmml">1</mn></mrow></msub><mo id="S2.Thmtheorem9.p2.3.3.m3.1.1.1" xref="S2.Thmtheorem9.p2.3.3.m3.1.1.1.cmml">=</mo><mi id="S2.Thmtheorem9.p2.3.3.m3.1.1.3" xref="S2.Thmtheorem9.p2.3.3.m3.1.1.3.cmml">V</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem9.p2.3.3.m3.1b"><apply id="S2.Thmtheorem9.p2.3.3.m3.1.1.cmml" xref="S2.Thmtheorem9.p2.3.3.m3.1.1"><eq id="S2.Thmtheorem9.p2.3.3.m3.1.1.1.cmml" xref="S2.Thmtheorem9.p2.3.3.m3.1.1.1"></eq><apply id="S2.Thmtheorem9.p2.3.3.m3.1.1.2.cmml" xref="S2.Thmtheorem9.p2.3.3.m3.1.1.2"><csymbol cd="ambiguous" id="S2.Thmtheorem9.p2.3.3.m3.1.1.2.1.cmml" xref="S2.Thmtheorem9.p2.3.3.m3.1.1.2">subscript</csymbol><ci id="S2.Thmtheorem9.p2.3.3.m3.1.1.2.2.cmml" xref="S2.Thmtheorem9.p2.3.3.m3.1.1.2.2">𝐵</ci><apply id="S2.Thmtheorem9.p2.3.3.m3.1.1.2.3.cmml" xref="S2.Thmtheorem9.p2.3.3.m3.1.1.2.3"><minus id="S2.Thmtheorem9.p2.3.3.m3.1.1.2.3.1.cmml" xref="S2.Thmtheorem9.p2.3.3.m3.1.1.2.3.1"></minus><ci id="S2.Thmtheorem9.p2.3.3.m3.1.1.2.3.2.cmml" xref="S2.Thmtheorem9.p2.3.3.m3.1.1.2.3.2">𝑖</ci><cn id="S2.Thmtheorem9.p2.3.3.m3.1.1.2.3.3.cmml" type="integer" xref="S2.Thmtheorem9.p2.3.3.m3.1.1.2.3.3">1</cn></apply></apply><ci id="S2.Thmtheorem9.p2.3.3.m3.1.1.3.cmml" xref="S2.Thmtheorem9.p2.3.3.m3.1.1.3">𝑉</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem9.p2.3.3.m3.1c">B_{i-1}=V</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem9.p2.3.3.m3.1d">italic_B start_POSTSUBSCRIPT italic_i - 1 end_POSTSUBSCRIPT = italic_V</annotation></semantics></math> then <math alttext="\ell:=i" class="ltx_Math" display="inline" id="S2.Thmtheorem9.p2.4.4.m4.1"><semantics id="S2.Thmtheorem9.p2.4.4.m4.1a"><mrow id="S2.Thmtheorem9.p2.4.4.m4.1.1" xref="S2.Thmtheorem9.p2.4.4.m4.1.1.cmml"><mi id="S2.Thmtheorem9.p2.4.4.m4.1.1.2" mathvariant="normal" xref="S2.Thmtheorem9.p2.4.4.m4.1.1.2.cmml">ℓ</mi><mo id="S2.Thmtheorem9.p2.4.4.m4.1.1.1" lspace="0.278em" rspace="0.278em" xref="S2.Thmtheorem9.p2.4.4.m4.1.1.1.cmml">:=</mo><mi id="S2.Thmtheorem9.p2.4.4.m4.1.1.3" xref="S2.Thmtheorem9.p2.4.4.m4.1.1.3.cmml">i</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem9.p2.4.4.m4.1b"><apply id="S2.Thmtheorem9.p2.4.4.m4.1.1.cmml" xref="S2.Thmtheorem9.p2.4.4.m4.1.1"><csymbol cd="latexml" id="S2.Thmtheorem9.p2.4.4.m4.1.1.1.cmml" xref="S2.Thmtheorem9.p2.4.4.m4.1.1.1">assign</csymbol><ci id="S2.Thmtheorem9.p2.4.4.m4.1.1.2.cmml" xref="S2.Thmtheorem9.p2.4.4.m4.1.1.2">ℓ</ci><ci id="S2.Thmtheorem9.p2.4.4.m4.1.1.3.cmml" xref="S2.Thmtheorem9.p2.4.4.m4.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem9.p2.4.4.m4.1c">\ell:=i</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem9.p2.4.4.m4.1d">roman_ℓ := italic_i</annotation></semantics></math>. Otherwise:</span></p> <table class="ltx_equation ltx_eqn_table" id="S2.Ex9"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_left_padleft"></td> <td class="ltx_eqn_cell ltx_align_left"><math alttext="S_{i}:=arg\,\max_{X\subseteq V-B_{i-1}}\hat{\rho}_{B_{i-1}}(X),\textnormal{ % and }B_{i}:=B_{i-1}\cup S_{i}." class="ltx_Math" display="block" id="S2.Ex9.m1.2"><semantics id="S2.Ex9.m1.2a"><mrow id="S2.Ex9.m1.2.2.1"><mrow id="S2.Ex9.m1.2.2.1.1.2" xref="S2.Ex9.m1.2.2.1.1.3.cmml"><mrow id="S2.Ex9.m1.2.2.1.1.1.1" xref="S2.Ex9.m1.2.2.1.1.1.1.cmml"><msub id="S2.Ex9.m1.2.2.1.1.1.1.2" xref="S2.Ex9.m1.2.2.1.1.1.1.2.cmml"><mi id="S2.Ex9.m1.2.2.1.1.1.1.2.2" xref="S2.Ex9.m1.2.2.1.1.1.1.2.2.cmml">S</mi><mi id="S2.Ex9.m1.2.2.1.1.1.1.2.3" xref="S2.Ex9.m1.2.2.1.1.1.1.2.3.cmml">i</mi></msub><mo id="S2.Ex9.m1.2.2.1.1.1.1.1" lspace="0.278em" rspace="0.278em" xref="S2.Ex9.m1.2.2.1.1.1.1.1.cmml">:=</mo><mrow id="S2.Ex9.m1.2.2.1.1.1.1.3" xref="S2.Ex9.m1.2.2.1.1.1.1.3.cmml"><mi id="S2.Ex9.m1.2.2.1.1.1.1.3.2" xref="S2.Ex9.m1.2.2.1.1.1.1.3.2.cmml">a</mi><mo id="S2.Ex9.m1.2.2.1.1.1.1.3.1" xref="S2.Ex9.m1.2.2.1.1.1.1.3.1.cmml">⁢</mo><mi id="S2.Ex9.m1.2.2.1.1.1.1.3.3" xref="S2.Ex9.m1.2.2.1.1.1.1.3.3.cmml">r</mi><mo id="S2.Ex9.m1.2.2.1.1.1.1.3.1a" xref="S2.Ex9.m1.2.2.1.1.1.1.3.1.cmml">⁢</mo><mi id="S2.Ex9.m1.2.2.1.1.1.1.3.4" xref="S2.Ex9.m1.2.2.1.1.1.1.3.4.cmml">g</mi><mo id="S2.Ex9.m1.2.2.1.1.1.1.3.1b" lspace="0.337em" xref="S2.Ex9.m1.2.2.1.1.1.1.3.1.cmml">⁢</mo><mrow id="S2.Ex9.m1.2.2.1.1.1.1.3.5" xref="S2.Ex9.m1.2.2.1.1.1.1.3.5.cmml"><munder id="S2.Ex9.m1.2.2.1.1.1.1.3.5.1" xref="S2.Ex9.m1.2.2.1.1.1.1.3.5.1.cmml"><mi id="S2.Ex9.m1.2.2.1.1.1.1.3.5.1.2" xref="S2.Ex9.m1.2.2.1.1.1.1.3.5.1.2.cmml">max</mi><mrow id="S2.Ex9.m1.2.2.1.1.1.1.3.5.1.3" xref="S2.Ex9.m1.2.2.1.1.1.1.3.5.1.3.cmml"><mi id="S2.Ex9.m1.2.2.1.1.1.1.3.5.1.3.2" xref="S2.Ex9.m1.2.2.1.1.1.1.3.5.1.3.2.cmml">X</mi><mo id="S2.Ex9.m1.2.2.1.1.1.1.3.5.1.3.1" xref="S2.Ex9.m1.2.2.1.1.1.1.3.5.1.3.1.cmml">⊆</mo><mrow id="S2.Ex9.m1.2.2.1.1.1.1.3.5.1.3.3" xref="S2.Ex9.m1.2.2.1.1.1.1.3.5.1.3.3.cmml"><mi id="S2.Ex9.m1.2.2.1.1.1.1.3.5.1.3.3.2" xref="S2.Ex9.m1.2.2.1.1.1.1.3.5.1.3.3.2.cmml">V</mi><mo id="S2.Ex9.m1.2.2.1.1.1.1.3.5.1.3.3.1" xref="S2.Ex9.m1.2.2.1.1.1.1.3.5.1.3.3.1.cmml">−</mo><msub id="S2.Ex9.m1.2.2.1.1.1.1.3.5.1.3.3.3" xref="S2.Ex9.m1.2.2.1.1.1.1.3.5.1.3.3.3.cmml"><mi id="S2.Ex9.m1.2.2.1.1.1.1.3.5.1.3.3.3.2" xref="S2.Ex9.m1.2.2.1.1.1.1.3.5.1.3.3.3.2.cmml">B</mi><mrow id="S2.Ex9.m1.2.2.1.1.1.1.3.5.1.3.3.3.3" xref="S2.Ex9.m1.2.2.1.1.1.1.3.5.1.3.3.3.3.cmml"><mi id="S2.Ex9.m1.2.2.1.1.1.1.3.5.1.3.3.3.3.2" xref="S2.Ex9.m1.2.2.1.1.1.1.3.5.1.3.3.3.3.2.cmml">i</mi><mo id="S2.Ex9.m1.2.2.1.1.1.1.3.5.1.3.3.3.3.1" xref="S2.Ex9.m1.2.2.1.1.1.1.3.5.1.3.3.3.3.1.cmml">−</mo><mn id="S2.Ex9.m1.2.2.1.1.1.1.3.5.1.3.3.3.3.3" xref="S2.Ex9.m1.2.2.1.1.1.1.3.5.1.3.3.3.3.3.cmml">1</mn></mrow></msub></mrow></mrow></munder><mo id="S2.Ex9.m1.2.2.1.1.1.1.3.5a" 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start_POSTSUBSCRIPT italic_i - 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT over^ start_ARG italic_ρ end_ARG start_POSTSUBSCRIPT italic_B start_POSTSUBSCRIPT italic_i - 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_X ) , and italic_B start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT := italic_B start_POSTSUBSCRIPT italic_i - 1 end_POSTSUBSCRIPT ∪ italic_S start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_left_padright"></td> </tr></tbody> </table> </div> </div> <div class="ltx_theorem ltx_theorem_definition" id="S2.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="S2.Thmtheorem10.1.1.1">Definition 2.10</span></span><span class="ltx_text ltx_font_bold" id="S2.Thmtheorem10.2.2"> </span>(Definition 2.3 in <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2411.12694v2#bib.bib10" title="">10</a>]</cite>)<span class="ltx_text ltx_font_bold" id="S2.Thmtheorem10.3.3">.</span> </h6> <div class="ltx_para" id="S2.Thmtheorem10.p1"> <p class="ltx_p" id="S2.Thmtheorem10.p1.6"><span class="ltx_text ltx_font_italic" id="S2.Thmtheorem10.p1.6.6">Given a weighted undirected graph <math alttext="G=(V,E)" class="ltx_Math" display="inline" id="S2.Thmtheorem10.p1.1.1.m1.2"><semantics id="S2.Thmtheorem10.p1.1.1.m1.2a"><mrow id="S2.Thmtheorem10.p1.1.1.m1.2.3" xref="S2.Thmtheorem10.p1.1.1.m1.2.3.cmml"><mi id="S2.Thmtheorem10.p1.1.1.m1.2.3.2" xref="S2.Thmtheorem10.p1.1.1.m1.2.3.2.cmml">G</mi><mo id="S2.Thmtheorem10.p1.1.1.m1.2.3.1" xref="S2.Thmtheorem10.p1.1.1.m1.2.3.1.cmml">=</mo><mrow id="S2.Thmtheorem10.p1.1.1.m1.2.3.3.2" xref="S2.Thmtheorem10.p1.1.1.m1.2.3.3.1.cmml"><mo id="S2.Thmtheorem10.p1.1.1.m1.2.3.3.2.1" stretchy="false" xref="S2.Thmtheorem10.p1.1.1.m1.2.3.3.1.cmml">(</mo><mi id="S2.Thmtheorem10.p1.1.1.m1.1.1" xref="S2.Thmtheorem10.p1.1.1.m1.1.1.cmml">V</mi><mo id="S2.Thmtheorem10.p1.1.1.m1.2.3.3.2.2" xref="S2.Thmtheorem10.p1.1.1.m1.2.3.3.1.cmml">,</mo><mi id="S2.Thmtheorem10.p1.1.1.m1.2.2" xref="S2.Thmtheorem10.p1.1.1.m1.2.2.cmml">E</mi><mo id="S2.Thmtheorem10.p1.1.1.m1.2.3.3.2.3" stretchy="false" xref="S2.Thmtheorem10.p1.1.1.m1.2.3.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem10.p1.1.1.m1.2b"><apply id="S2.Thmtheorem10.p1.1.1.m1.2.3.cmml" xref="S2.Thmtheorem10.p1.1.1.m1.2.3"><eq id="S2.Thmtheorem10.p1.1.1.m1.2.3.1.cmml" xref="S2.Thmtheorem10.p1.1.1.m1.2.3.1"></eq><ci id="S2.Thmtheorem10.p1.1.1.m1.2.3.2.cmml" xref="S2.Thmtheorem10.p1.1.1.m1.2.3.2">𝐺</ci><interval closure="open" id="S2.Thmtheorem10.p1.1.1.m1.2.3.3.1.cmml" xref="S2.Thmtheorem10.p1.1.1.m1.2.3.3.2"><ci id="S2.Thmtheorem10.p1.1.1.m1.1.1.cmml" xref="S2.Thmtheorem10.p1.1.1.m1.1.1">𝑉</ci><ci id="S2.Thmtheorem10.p1.1.1.m1.2.2.cmml" xref="S2.Thmtheorem10.p1.1.1.m1.2.2">𝐸</ci></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem10.p1.1.1.m1.2c">G=(V,E)</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem10.p1.1.1.m1.2d">italic_G = ( italic_V , italic_E )</annotation></semantics></math> and a diminishing-dense decomposition <math alttext="\mathcal{B}" class="ltx_Math" display="inline" id="S2.Thmtheorem10.p1.2.2.m2.1"><semantics id="S2.Thmtheorem10.p1.2.2.m2.1a"><mi class="ltx_font_mathcaligraphic" id="S2.Thmtheorem10.p1.2.2.m2.1.1" xref="S2.Thmtheorem10.p1.2.2.m2.1.1.cmml">ℬ</mi><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem10.p1.2.2.m2.1b"><ci id="S2.Thmtheorem10.p1.2.2.m2.1.1.cmml" xref="S2.Thmtheorem10.p1.2.2.m2.1.1">ℬ</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem10.p1.2.2.m2.1c">\mathcal{B}</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem10.p1.2.2.m2.1d">caligraphic_B</annotation></semantics></math>, each vertex <math alttext="v\in V" class="ltx_Math" display="inline" id="S2.Thmtheorem10.p1.3.3.m3.1"><semantics id="S2.Thmtheorem10.p1.3.3.m3.1a"><mrow id="S2.Thmtheorem10.p1.3.3.m3.1.1" xref="S2.Thmtheorem10.p1.3.3.m3.1.1.cmml"><mi id="S2.Thmtheorem10.p1.3.3.m3.1.1.2" xref="S2.Thmtheorem10.p1.3.3.m3.1.1.2.cmml">v</mi><mo id="S2.Thmtheorem10.p1.3.3.m3.1.1.1" xref="S2.Thmtheorem10.p1.3.3.m3.1.1.1.cmml">∈</mo><mi id="S2.Thmtheorem10.p1.3.3.m3.1.1.3" xref="S2.Thmtheorem10.p1.3.3.m3.1.1.3.cmml">V</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem10.p1.3.3.m3.1b"><apply id="S2.Thmtheorem10.p1.3.3.m3.1.1.cmml" xref="S2.Thmtheorem10.p1.3.3.m3.1.1"><in id="S2.Thmtheorem10.p1.3.3.m3.1.1.1.cmml" xref="S2.Thmtheorem10.p1.3.3.m3.1.1.1"></in><ci id="S2.Thmtheorem10.p1.3.3.m3.1.1.2.cmml" xref="S2.Thmtheorem10.p1.3.3.m3.1.1.2">𝑣</ci><ci id="S2.Thmtheorem10.p1.3.3.m3.1.1.3.cmml" xref="S2.Thmtheorem10.p1.3.3.m3.1.1.3">𝑉</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem10.p1.3.3.m3.1c">v\in V</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem10.p1.3.3.m3.1d">italic_v ∈ italic_V</annotation></semantics></math> has one integer <math alttext="i" class="ltx_Math" display="inline" id="S2.Thmtheorem10.p1.4.4.m4.1"><semantics id="S2.Thmtheorem10.p1.4.4.m4.1a"><mi id="S2.Thmtheorem10.p1.4.4.m4.1.1" xref="S2.Thmtheorem10.p1.4.4.m4.1.1.cmml">i</mi><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem10.p1.4.4.m4.1b"><ci id="S2.Thmtheorem10.p1.4.4.m4.1.1.cmml" xref="S2.Thmtheorem10.p1.4.4.m4.1.1">𝑖</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem10.p1.4.4.m4.1c">i</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem10.p1.4.4.m4.1d">italic_i</annotation></semantics></math> where <math alttext="v\in S_{i}" class="ltx_Math" display="inline" id="S2.Thmtheorem10.p1.5.5.m5.1"><semantics id="S2.Thmtheorem10.p1.5.5.m5.1a"><mrow id="S2.Thmtheorem10.p1.5.5.m5.1.1" xref="S2.Thmtheorem10.p1.5.5.m5.1.1.cmml"><mi id="S2.Thmtheorem10.p1.5.5.m5.1.1.2" xref="S2.Thmtheorem10.p1.5.5.m5.1.1.2.cmml">v</mi><mo id="S2.Thmtheorem10.p1.5.5.m5.1.1.1" xref="S2.Thmtheorem10.p1.5.5.m5.1.1.1.cmml">∈</mo><msub id="S2.Thmtheorem10.p1.5.5.m5.1.1.3" xref="S2.Thmtheorem10.p1.5.5.m5.1.1.3.cmml"><mi id="S2.Thmtheorem10.p1.5.5.m5.1.1.3.2" xref="S2.Thmtheorem10.p1.5.5.m5.1.1.3.2.cmml">S</mi><mi id="S2.Thmtheorem10.p1.5.5.m5.1.1.3.3" xref="S2.Thmtheorem10.p1.5.5.m5.1.1.3.3.cmml">i</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem10.p1.5.5.m5.1b"><apply id="S2.Thmtheorem10.p1.5.5.m5.1.1.cmml" xref="S2.Thmtheorem10.p1.5.5.m5.1.1"><in id="S2.Thmtheorem10.p1.5.5.m5.1.1.1.cmml" xref="S2.Thmtheorem10.p1.5.5.m5.1.1.1"></in><ci id="S2.Thmtheorem10.p1.5.5.m5.1.1.2.cmml" xref="S2.Thmtheorem10.p1.5.5.m5.1.1.2">𝑣</ci><apply id="S2.Thmtheorem10.p1.5.5.m5.1.1.3.cmml" xref="S2.Thmtheorem10.p1.5.5.m5.1.1.3"><csymbol cd="ambiguous" id="S2.Thmtheorem10.p1.5.5.m5.1.1.3.1.cmml" xref="S2.Thmtheorem10.p1.5.5.m5.1.1.3">subscript</csymbol><ci id="S2.Thmtheorem10.p1.5.5.m5.1.1.3.2.cmml" xref="S2.Thmtheorem10.p1.5.5.m5.1.1.3.2">𝑆</ci><ci id="S2.Thmtheorem10.p1.5.5.m5.1.1.3.3.cmml" xref="S2.Thmtheorem10.p1.5.5.m5.1.1.3.3">𝑖</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem10.p1.5.5.m5.1c">v\in S_{i}</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem10.p1.5.5.m5.1d">italic_v ∈ italic_S start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math>. 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id="S2.Thmtheorem10.p1.6.6.m6.2.2.1.3.2.2.cmml" xref="S2.Thmtheorem10.p1.6.6.m6.2.2.1.3.2.2">𝜌</ci></apply><apply id="S2.Thmtheorem10.p1.6.6.m6.2.2.1.3.3.cmml" xref="S2.Thmtheorem10.p1.6.6.m6.2.2.1.3.3"><csymbol cd="ambiguous" id="S2.Thmtheorem10.p1.6.6.m6.2.2.1.3.3.1.cmml" xref="S2.Thmtheorem10.p1.6.6.m6.2.2.1.3.3">subscript</csymbol><ci id="S2.Thmtheorem10.p1.6.6.m6.2.2.1.3.3.2.cmml" xref="S2.Thmtheorem10.p1.6.6.m6.2.2.1.3.3.2">𝐵</ci><apply id="S2.Thmtheorem10.p1.6.6.m6.2.2.1.3.3.3.cmml" xref="S2.Thmtheorem10.p1.6.6.m6.2.2.1.3.3.3"><minus id="S2.Thmtheorem10.p1.6.6.m6.2.2.1.3.3.3.1.cmml" xref="S2.Thmtheorem10.p1.6.6.m6.2.2.1.3.3.3.1"></minus><ci id="S2.Thmtheorem10.p1.6.6.m6.2.2.1.3.3.3.2.cmml" xref="S2.Thmtheorem10.p1.6.6.m6.2.2.1.3.3.3.2">𝑖</ci><cn id="S2.Thmtheorem10.p1.6.6.m6.2.2.1.3.3.3.3.cmml" type="integer" xref="S2.Thmtheorem10.p1.6.6.m6.2.2.1.3.3.3.3">1</cn></apply></apply></apply><apply id="S2.Thmtheorem10.p1.6.6.m6.2.2.1.1.1.1.cmml" xref="S2.Thmtheorem10.p1.6.6.m6.2.2.1.1.1"><csymbol cd="ambiguous" id="S2.Thmtheorem10.p1.6.6.m6.2.2.1.1.1.1.1.cmml" xref="S2.Thmtheorem10.p1.6.6.m6.2.2.1.1.1">subscript</csymbol><ci id="S2.Thmtheorem10.p1.6.6.m6.2.2.1.1.1.1.2.cmml" xref="S2.Thmtheorem10.p1.6.6.m6.2.2.1.1.1.1.2">𝑆</ci><ci id="S2.Thmtheorem10.p1.6.6.m6.2.2.1.1.1.1.3.cmml" xref="S2.Thmtheorem10.p1.6.6.m6.2.2.1.1.1.1.3">𝑖</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem10.p1.6.6.m6.2c">\rho^{*}(v):=\hat{\rho}_{B_{i-1}}(S_{i})</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem10.p1.6.6.m6.2d">italic_ρ start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ( italic_v ) := over^ start_ARG italic_ρ end_ARG start_POSTSUBSCRIPT italic_B start_POSTSUBSCRIPT italic_i - 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_S start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT )</annotation></semantics></math>.</span></p> </div> </div> <section class="ltx_subparagraph" id="S2.SS3.SSS0.P0.SPx1"> <h5 class="ltx_title ltx_title_subparagraph">The benefit of local measures:</h5> <div class="ltx_para" id="S2.SS3.SSS0.P0.SPx1.p1"> <p class="ltx_p" id="S2.SS3.SSS0.P0.SPx1.p1.1">Problem <a class="ltx_ref" href="https://arxiv.org/html/2411.12694v2#S2.Thmtheorem7" title="Problem 2.7. ‣ 2.2 Approximate densest subgraph in LOCAL and CONGEST ‣ 2 Preliminaries and related work ‣ Local Density and its Distributed Approximation"><span class="ltx_text ltx_ref_tag">2.7</span></a>’s variants have drawbacks in a distributed model of computation. Problem <a class="ltx_ref" href="https://arxiv.org/html/2411.12694v2#S2.Thmtheorem7" title="Problem 2.7. ‣ 2.2 Approximate densest subgraph in LOCAL and CONGEST ‣ 2 Preliminaries and related work ‣ Local Density and its Distributed Approximation"><span class="ltx_text ltx_ref_tag">2.7</span></a>.1 has an <math alttext="\Omega(d)" class="ltx_Math" display="inline" id="S2.SS3.SSS0.P0.SPx1.p1.1.m1.1"><semantics id="S2.SS3.SSS0.P0.SPx1.p1.1.m1.1a"><mrow id="S2.SS3.SSS0.P0.SPx1.p1.1.m1.1.2" xref="S2.SS3.SSS0.P0.SPx1.p1.1.m1.1.2.cmml"><mi id="S2.SS3.SSS0.P0.SPx1.p1.1.m1.1.2.2" mathvariant="normal" xref="S2.SS3.SSS0.P0.SPx1.p1.1.m1.1.2.2.cmml">Ω</mi><mo id="S2.SS3.SSS0.P0.SPx1.p1.1.m1.1.2.1" xref="S2.SS3.SSS0.P0.SPx1.p1.1.m1.1.2.1.cmml">⁢</mo><mrow id="S2.SS3.SSS0.P0.SPx1.p1.1.m1.1.2.3.2" xref="S2.SS3.SSS0.P0.SPx1.p1.1.m1.1.2.cmml"><mo id="S2.SS3.SSS0.P0.SPx1.p1.1.m1.1.2.3.2.1" stretchy="false" xref="S2.SS3.SSS0.P0.SPx1.p1.1.m1.1.2.cmml">(</mo><mi id="S2.SS3.SSS0.P0.SPx1.p1.1.m1.1.1" xref="S2.SS3.SSS0.P0.SPx1.p1.1.m1.1.1.cmml">d</mi><mo id="S2.SS3.SSS0.P0.SPx1.p1.1.m1.1.2.3.2.2" stretchy="false" xref="S2.SS3.SSS0.P0.SPx1.p1.1.m1.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS3.SSS0.P0.SPx1.p1.1.m1.1b"><apply id="S2.SS3.SSS0.P0.SPx1.p1.1.m1.1.2.cmml" xref="S2.SS3.SSS0.P0.SPx1.p1.1.m1.1.2"><times id="S2.SS3.SSS0.P0.SPx1.p1.1.m1.1.2.1.cmml" xref="S2.SS3.SSS0.P0.SPx1.p1.1.m1.1.2.1"></times><ci id="S2.SS3.SSS0.P0.SPx1.p1.1.m1.1.2.2.cmml" xref="S2.SS3.SSS0.P0.SPx1.p1.1.m1.1.2.2">Ω</ci><ci id="S2.SS3.SSS0.P0.SPx1.p1.1.m1.1.1.cmml" xref="S2.SS3.SSS0.P0.SPx1.p1.1.m1.1.1">𝑑</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.SSS0.P0.SPx1.p1.1.m1.1c">\Omega(d)</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.SSS0.P0.SPx1.p1.1.m1.1d">roman_Ω ( italic_d )</annotation></semantics></math> lower bound (making it trivial in LOCAL). Problem <a class="ltx_ref" href="https://arxiv.org/html/2411.12694v2#S2.Thmtheorem7" title="Problem 2.7. ‣ 2.2 Approximate densest subgraph in LOCAL and CONGEST ‣ 2 Preliminaries and related work ‣ Local Density and its Distributed Approximation"><span class="ltx_text ltx_ref_tag">2.7</span></a>.2 allows some vertices to output nonsense. The definition of local density alleviates these issues, as we may define an algorithmic problem which we consider to be more natural:</p> </div> <div class="ltx_theorem ltx_theorem_problem" id="S2.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="S2.Thmtheorem11.1.1.1">Problem 2.11</span></span><span class="ltx_text ltx_font_bold" id="S2.Thmtheorem11.2.2">.</span> </h6> <div class="ltx_para" id="S2.Thmtheorem11.p1"> <p class="ltx_p" id="S2.Thmtheorem11.p1.3"><span class="ltx_text ltx_font_italic" id="S2.Thmtheorem11.p1.3.3">Given <math alttext="(G,\varepsilon)" class="ltx_Math" display="inline" id="S2.Thmtheorem11.p1.1.1.m1.2"><semantics id="S2.Thmtheorem11.p1.1.1.m1.2a"><mrow id="S2.Thmtheorem11.p1.1.1.m1.2.3.2" xref="S2.Thmtheorem11.p1.1.1.m1.2.3.1.cmml"><mo id="S2.Thmtheorem11.p1.1.1.m1.2.3.2.1" stretchy="false" xref="S2.Thmtheorem11.p1.1.1.m1.2.3.1.cmml">(</mo><mi id="S2.Thmtheorem11.p1.1.1.m1.1.1" xref="S2.Thmtheorem11.p1.1.1.m1.1.1.cmml">G</mi><mo id="S2.Thmtheorem11.p1.1.1.m1.2.3.2.2" xref="S2.Thmtheorem11.p1.1.1.m1.2.3.1.cmml">,</mo><mi id="S2.Thmtheorem11.p1.1.1.m1.2.2" xref="S2.Thmtheorem11.p1.1.1.m1.2.2.cmml">ε</mi><mo id="S2.Thmtheorem11.p1.1.1.m1.2.3.2.3" stretchy="false" xref="S2.Thmtheorem11.p1.1.1.m1.2.3.1.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem11.p1.1.1.m1.2b"><interval closure="open" id="S2.Thmtheorem11.p1.1.1.m1.2.3.1.cmml" xref="S2.Thmtheorem11.p1.1.1.m1.2.3.2"><ci id="S2.Thmtheorem11.p1.1.1.m1.1.1.cmml" xref="S2.Thmtheorem11.p1.1.1.m1.1.1">𝐺</ci><ci id="S2.Thmtheorem11.p1.1.1.m1.2.2.cmml" xref="S2.Thmtheorem11.p1.1.1.m1.2.2">𝜀</ci></interval></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem11.p1.1.1.m1.2c">(G,\varepsilon)</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem11.p1.1.1.m1.2d">( italic_G , italic_ε )</annotation></semantics></math>, each vertex <math alttext="v" class="ltx_Math" display="inline" id="S2.Thmtheorem11.p1.2.2.m2.1"><semantics id="S2.Thmtheorem11.p1.2.2.m2.1a"><mi id="S2.Thmtheorem11.p1.2.2.m2.1.1" xref="S2.Thmtheorem11.p1.2.2.m2.1.1.cmml">v</mi><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem11.p1.2.2.m2.1b"><ci id="S2.Thmtheorem11.p1.2.2.m2.1.1.cmml" xref="S2.Thmtheorem11.p1.2.2.m2.1.1">𝑣</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem11.p1.2.2.m2.1c">v</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem11.p1.2.2.m2.1d">italic_v</annotation></semantics></math> outputs <math alttext="\rho_{v}\in[(1+\varepsilon)^{-1}\rho^{*}(v),(1+\varepsilon)\rho^{*}(v)]" class="ltx_Math" display="inline" id="S2.Thmtheorem11.p1.3.3.m3.4"><semantics id="S2.Thmtheorem11.p1.3.3.m3.4a"><mrow id="S2.Thmtheorem11.p1.3.3.m3.4.4" xref="S2.Thmtheorem11.p1.3.3.m3.4.4.cmml"><msub id="S2.Thmtheorem11.p1.3.3.m3.4.4.4" 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id="S2.Thmtheorem11.p1.3.3.m3.4.4.2.2.2.1.1.1.cmml" xref="S2.Thmtheorem11.p1.3.3.m3.4.4.2.2.2.1.1"><plus id="S2.Thmtheorem11.p1.3.3.m3.4.4.2.2.2.1.1.1.1.cmml" xref="S2.Thmtheorem11.p1.3.3.m3.4.4.2.2.2.1.1.1.1"></plus><cn id="S2.Thmtheorem11.p1.3.3.m3.4.4.2.2.2.1.1.1.2.cmml" type="integer" xref="S2.Thmtheorem11.p1.3.3.m3.4.4.2.2.2.1.1.1.2">1</cn><ci id="S2.Thmtheorem11.p1.3.3.m3.4.4.2.2.2.1.1.1.3.cmml" xref="S2.Thmtheorem11.p1.3.3.m3.4.4.2.2.2.1.1.1.3">𝜀</ci></apply><apply id="S2.Thmtheorem11.p1.3.3.m3.4.4.2.2.2.3.cmml" xref="S2.Thmtheorem11.p1.3.3.m3.4.4.2.2.2.3"><csymbol cd="ambiguous" id="S2.Thmtheorem11.p1.3.3.m3.4.4.2.2.2.3.1.cmml" xref="S2.Thmtheorem11.p1.3.3.m3.4.4.2.2.2.3">superscript</csymbol><ci id="S2.Thmtheorem11.p1.3.3.m3.4.4.2.2.2.3.2.cmml" xref="S2.Thmtheorem11.p1.3.3.m3.4.4.2.2.2.3.2">𝜌</ci><times id="S2.Thmtheorem11.p1.3.3.m3.4.4.2.2.2.3.3.cmml" xref="S2.Thmtheorem11.p1.3.3.m3.4.4.2.2.2.3.3"></times></apply><ci id="S2.Thmtheorem11.p1.3.3.m3.2.2.cmml" xref="S2.Thmtheorem11.p1.3.3.m3.2.2">𝑣</ci></apply></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem11.p1.3.3.m3.4c">\rho_{v}\in[(1+\varepsilon)^{-1}\rho^{*}(v),(1+\varepsilon)\rho^{*}(v)]</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem11.p1.3.3.m3.4d">italic_ρ start_POSTSUBSCRIPT italic_v end_POSTSUBSCRIPT ∈ [ ( 1 + italic_ε ) start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT italic_ρ start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ( italic_v ) , ( 1 + italic_ε ) italic_ρ start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ( italic_v ) ]</annotation></semantics></math>.</span></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>Results and organisation</h2> <div class="ltx_para" id="S3.p1"> <p class="ltx_p" id="S3.p1.1">Now we are ready to formally state our contributions. Our primary contribution is that we show a dual definition to local density, which we call the local out-degree:</p> </div> <div class="ltx_theorem ltx_theorem_definition" 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">Definition 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.p1.1"><span class="ltx_text ltx_font_italic" id="S3.Thmtheorem1.p1.1.1">Given a graph <math alttext="G=(V,E)" class="ltx_Math" display="inline" id="S3.Thmtheorem1.p1.1.1.m1.2"><semantics id="S3.Thmtheorem1.p1.1.1.m1.2a"><mrow id="S3.Thmtheorem1.p1.1.1.m1.2.3" xref="S3.Thmtheorem1.p1.1.1.m1.2.3.cmml"><mi id="S3.Thmtheorem1.p1.1.1.m1.2.3.2" xref="S3.Thmtheorem1.p1.1.1.m1.2.3.2.cmml">G</mi><mo id="S3.Thmtheorem1.p1.1.1.m1.2.3.1" xref="S3.Thmtheorem1.p1.1.1.m1.2.3.1.cmml">=</mo><mrow id="S3.Thmtheorem1.p1.1.1.m1.2.3.3.2" xref="S3.Thmtheorem1.p1.1.1.m1.2.3.3.1.cmml"><mo id="S3.Thmtheorem1.p1.1.1.m1.2.3.3.2.1" stretchy="false" xref="S3.Thmtheorem1.p1.1.1.m1.2.3.3.1.cmml">(</mo><mi id="S3.Thmtheorem1.p1.1.1.m1.1.1" xref="S3.Thmtheorem1.p1.1.1.m1.1.1.cmml">V</mi><mo id="S3.Thmtheorem1.p1.1.1.m1.2.3.3.2.2" xref="S3.Thmtheorem1.p1.1.1.m1.2.3.3.1.cmml">,</mo><mi id="S3.Thmtheorem1.p1.1.1.m1.2.2" xref="S3.Thmtheorem1.p1.1.1.m1.2.2.cmml">E</mi><mo id="S3.Thmtheorem1.p1.1.1.m1.2.3.3.2.3" stretchy="false" xref="S3.Thmtheorem1.p1.1.1.m1.2.3.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem1.p1.1.1.m1.2b"><apply id="S3.Thmtheorem1.p1.1.1.m1.2.3.cmml" xref="S3.Thmtheorem1.p1.1.1.m1.2.3"><eq id="S3.Thmtheorem1.p1.1.1.m1.2.3.1.cmml" xref="S3.Thmtheorem1.p1.1.1.m1.2.3.1"></eq><ci id="S3.Thmtheorem1.p1.1.1.m1.2.3.2.cmml" xref="S3.Thmtheorem1.p1.1.1.m1.2.3.2">𝐺</ci><interval closure="open" id="S3.Thmtheorem1.p1.1.1.m1.2.3.3.1.cmml" xref="S3.Thmtheorem1.p1.1.1.m1.2.3.3.2"><ci id="S3.Thmtheorem1.p1.1.1.m1.1.1.cmml" xref="S3.Thmtheorem1.p1.1.1.m1.1.1">𝑉</ci><ci id="S3.Thmtheorem1.p1.1.1.m1.2.2.cmml" xref="S3.Thmtheorem1.p1.1.1.m1.2.2">𝐸</ci></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem1.p1.1.1.m1.2c">G=(V,E)</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem1.p1.1.1.m1.2d">italic_G = ( italic_V , italic_E )</annotation></semantics></math>, we define the <em class="ltx_emph ltx_font_upright" id="S3.Thmtheorem1.p1.1.1.1">local out-degree</em> as:</span></p> <table class="ltx_equation ltx_eqn_table" id="S3.Ex10"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_left_padleft"></td> <td class="ltx_eqn_cell ltx_align_left"><math alttext="\textsl{g}^{*}(u):=\textnormal{ the out-degree }\textsl{g}(u)\textnormal{ in % any locally fair fractional orientation of }G." class="ltx_Math" display="block" id="S3.Ex10.m1.3"><semantics id="S3.Ex10.m1.3a"><mrow id="S3.Ex10.m1.3.3.1" xref="S3.Ex10.m1.3.3.1.1.cmml"><mrow id="S3.Ex10.m1.3.3.1.1" xref="S3.Ex10.m1.3.3.1.1.cmml"><mrow id="S3.Ex10.m1.3.3.1.1.2" xref="S3.Ex10.m1.3.3.1.1.2.cmml"><msup id="S3.Ex10.m1.3.3.1.1.2.2" xref="S3.Ex10.m1.3.3.1.1.2.2.cmml"><mtext class="ltx_mathvariant_italic" id="S3.Ex10.m1.3.3.1.1.2.2.2" xref="S3.Ex10.m1.3.3.1.1.2.2.2a.cmml">g</mtext><mo id="S3.Ex10.m1.3.3.1.1.2.2.3" xref="S3.Ex10.m1.3.3.1.1.2.2.3.cmml">∗</mo></msup><mo id="S3.Ex10.m1.3.3.1.1.2.1" xref="S3.Ex10.m1.3.3.1.1.2.1.cmml">⁢</mo><mrow id="S3.Ex10.m1.3.3.1.1.2.3.2" xref="S3.Ex10.m1.3.3.1.1.2.cmml"><mo id="S3.Ex10.m1.3.3.1.1.2.3.2.1" stretchy="false" xref="S3.Ex10.m1.3.3.1.1.2.cmml">(</mo><mi id="S3.Ex10.m1.1.1" xref="S3.Ex10.m1.1.1.cmml">u</mi><mo id="S3.Ex10.m1.3.3.1.1.2.3.2.2" rspace="0.278em" stretchy="false" xref="S3.Ex10.m1.3.3.1.1.2.cmml">)</mo></mrow></mrow><mo id="S3.Ex10.m1.3.3.1.1.1" rspace="0.278em" xref="S3.Ex10.m1.3.3.1.1.1.cmml">:=</mo><mrow id="S3.Ex10.m1.3.3.1.1.3" xref="S3.Ex10.m1.3.3.1.1.3.cmml"><mrow id="S3.Ex10.m1.3.3.1.1.3.2" xref="S3.Ex10.m1.3.3.1.1.3.2c.cmml"><mtext id="S3.Ex10.m1.3.3.1.1.3.2a" xref="S3.Ex10.m1.3.3.1.1.3.2c.cmml"> the out-degree </mtext><mtext class="ltx_mathvariant_italic" id="S3.Ex10.m1.3.3.1.1.3.2b" xref="S3.Ex10.m1.3.3.1.1.3.2c.cmml">g</mtext></mrow><mo id="S3.Ex10.m1.3.3.1.1.3.1" xref="S3.Ex10.m1.3.3.1.1.3.1.cmml">⁢</mo><mrow id="S3.Ex10.m1.3.3.1.1.3.3.2" xref="S3.Ex10.m1.3.3.1.1.3.cmml"><mo id="S3.Ex10.m1.3.3.1.1.3.3.2.1" stretchy="false" xref="S3.Ex10.m1.3.3.1.1.3.cmml">(</mo><mi id="S3.Ex10.m1.2.2" xref="S3.Ex10.m1.2.2.cmml">u</mi><mo id="S3.Ex10.m1.3.3.1.1.3.3.2.2" stretchy="false" xref="S3.Ex10.m1.3.3.1.1.3.cmml">)</mo></mrow><mo id="S3.Ex10.m1.3.3.1.1.3.1a" xref="S3.Ex10.m1.3.3.1.1.3.1.cmml">⁢</mo><mtext id="S3.Ex10.m1.3.3.1.1.3.4" xref="S3.Ex10.m1.3.3.1.1.3.4a.cmml"> in any locally fair fractional orientation of </mtext><mo id="S3.Ex10.m1.3.3.1.1.3.1b" xref="S3.Ex10.m1.3.3.1.1.3.1.cmml">⁢</mo><mi id="S3.Ex10.m1.3.3.1.1.3.5" xref="S3.Ex10.m1.3.3.1.1.3.5.cmml">G</mi></mrow></mrow><mo id="S3.Ex10.m1.3.3.1.2" lspace="0em" xref="S3.Ex10.m1.3.3.1.1.cmml">.</mo></mrow><annotation-xml encoding="MathML-Content" id="S3.Ex10.m1.3b"><apply id="S3.Ex10.m1.3.3.1.1.cmml" xref="S3.Ex10.m1.3.3.1"><csymbol cd="latexml" id="S3.Ex10.m1.3.3.1.1.1.cmml" xref="S3.Ex10.m1.3.3.1.1.1">assign</csymbol><apply id="S3.Ex10.m1.3.3.1.1.2.cmml" xref="S3.Ex10.m1.3.3.1.1.2"><times id="S3.Ex10.m1.3.3.1.1.2.1.cmml" xref="S3.Ex10.m1.3.3.1.1.2.1"></times><apply id="S3.Ex10.m1.3.3.1.1.2.2.cmml" xref="S3.Ex10.m1.3.3.1.1.2.2"><csymbol cd="ambiguous" id="S3.Ex10.m1.3.3.1.1.2.2.1.cmml" xref="S3.Ex10.m1.3.3.1.1.2.2">superscript</csymbol><ci id="S3.Ex10.m1.3.3.1.1.2.2.2a.cmml" xref="S3.Ex10.m1.3.3.1.1.2.2.2"><mtext class="ltx_mathvariant_italic" id="S3.Ex10.m1.3.3.1.1.2.2.2.cmml" xref="S3.Ex10.m1.3.3.1.1.2.2.2">g</mtext></ci><times id="S3.Ex10.m1.3.3.1.1.2.2.3.cmml" xref="S3.Ex10.m1.3.3.1.1.2.2.3"></times></apply><ci id="S3.Ex10.m1.1.1.cmml" xref="S3.Ex10.m1.1.1">𝑢</ci></apply><apply id="S3.Ex10.m1.3.3.1.1.3.cmml" xref="S3.Ex10.m1.3.3.1.1.3"><times id="S3.Ex10.m1.3.3.1.1.3.1.cmml" xref="S3.Ex10.m1.3.3.1.1.3.1"></times><ci id="S3.Ex10.m1.3.3.1.1.3.2c.cmml" xref="S3.Ex10.m1.3.3.1.1.3.2"><mrow id="S3.Ex10.m1.3.3.1.1.3.2.cmml" xref="S3.Ex10.m1.3.3.1.1.3.2"><mtext id="S3.Ex10.m1.3.3.1.1.3.2a.cmml" xref="S3.Ex10.m1.3.3.1.1.3.2"> the out-degree </mtext><mtext class="ltx_mathvariant_italic" id="S3.Ex10.m1.3.3.1.1.3.2b.cmml" xref="S3.Ex10.m1.3.3.1.1.3.2">g</mtext></mrow></ci><ci id="S3.Ex10.m1.2.2.cmml" xref="S3.Ex10.m1.2.2">𝑢</ci><ci id="S3.Ex10.m1.3.3.1.1.3.4a.cmml" xref="S3.Ex10.m1.3.3.1.1.3.4"><mtext id="S3.Ex10.m1.3.3.1.1.3.4.cmml" xref="S3.Ex10.m1.3.3.1.1.3.4"> in any locally fair fractional orientation of </mtext></ci><ci id="S3.Ex10.m1.3.3.1.1.3.5.cmml" xref="S3.Ex10.m1.3.3.1.1.3.5">𝐺</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Ex10.m1.3c">\textsl{g}^{*}(u):=\textnormal{ the out-degree }\textsl{g}(u)\textnormal{ in % any locally fair fractional orientation of }G.</annotation><annotation encoding="application/x-llamapun" id="S3.Ex10.m1.3d">g start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ( italic_u ) := the out-degree slanted_g ( italic_u ) in any locally fair fractional orientation of italic_G .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_left_padright"></td> </tr></tbody> </table> </div> </div> <div class="ltx_para ltx_noindent" id="S3.p2"> <p class="ltx_p" id="S3.p2.3">It is not immediately clear that the local out-degree is well-defined. We prove (Theorem <a class="ltx_ref" href="https://arxiv.org/html/2411.12694v2#S3.Thmtheorem2" title="Theorem 3.2. ‣ 3.A Conceptual results for local density ‣ 3 Results and organisation ‣ Local Density and its Distributed Approximation"><span class="ltx_text ltx_ref_tag">3.2</span></a>) that each vertex in <math alttext="G" class="ltx_Math" display="inline" id="S3.p2.1.m1.1"><semantics id="S3.p2.1.m1.1a"><mi id="S3.p2.1.m1.1.1" xref="S3.p2.1.m1.1.1.cmml">G</mi><annotation-xml encoding="MathML-Content" id="S3.p2.1.m1.1b"><ci id="S3.p2.1.m1.1.1.cmml" xref="S3.p2.1.m1.1.1">𝐺</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.p2.1.m1.1c">G</annotation><annotation encoding="application/x-llamapun" id="S3.p2.1.m1.1d">italic_G</annotation></semantics></math> has the same out-degree across all locally fair orientations of <math alttext="G" 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">G</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">𝐺</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.p2.2.m2.1c">G</annotation><annotation encoding="application/x-llamapun" id="S3.p2.2.m2.1d">italic_G</annotation></semantics></math> (and thus, the set of all locally fair orientations of <math alttext="G" class="ltx_Math" display="inline" id="S3.p2.3.m3.1"><semantics id="S3.p2.3.m3.1a"><mi id="S3.p2.3.m3.1.1" xref="S3.p2.3.m3.1.1.cmml">G</mi><annotation-xml encoding="MathML-Content" id="S3.p2.3.m3.1b"><ci id="S3.p2.3.m3.1.1.cmml" xref="S3.p2.3.m3.1.1">𝐺</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.p2.3.m3.1c">G</annotation><annotation encoding="application/x-llamapun" id="S3.p2.3.m3.1d">italic_G</annotation></semantics></math> assigns to each vertex a real value). We believe that the local out-degree is conceptually simpler that the local density. Through this definition, we are able to show various algorithms to approximate the local density.</p> </div> <section class="ltx_subsubsection" id="S3.SS0.SSS1"> <h4 class="ltx_title ltx_title_subsubsection"> <span class="ltx_tag ltx_tag_subsubsection">3.A </span>Conceptual results for local density </h4> <div class="ltx_para" id="S3.SS0.SSS1.p1"> <p class="ltx_p" id="S3.SS0.SSS1.p1.1">We prove in Section <a class="ltx_ref" href="https://arxiv.org/html/2411.12694v2#S4" title="4 Conceptual results for local density ‣ Local Density and its Distributed Approximation"><span class="ltx_text ltx_ref_tag">4</span></a> that these local definitions generalise the global definition of subgraph density and out-degree, as they exhibit the same dual behaviour. We show several previously unknown properties of the local density, which we consider to be of independent interest:</p> </div> <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.p1.4"><span class="ltx_text ltx_font_italic" id="S3.Thmtheorem2.p1.4.4">For any weighted graph <math alttext="G" class="ltx_Math" display="inline" id="S3.Thmtheorem2.p1.1.1.m1.1"><semantics id="S3.Thmtheorem2.p1.1.1.m1.1a"><mi id="S3.Thmtheorem2.p1.1.1.m1.1.1" xref="S3.Thmtheorem2.p1.1.1.m1.1.1.cmml">G</mi><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem2.p1.1.1.m1.1b"><ci id="S3.Thmtheorem2.p1.1.1.m1.1.1.cmml" 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id="S3.Thmtheorem2.p1.3.3.m3.1d">g start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ( italic_v )</annotation></semantics></math> is well-defined and equals <math alttext="\rho^{*}(v)" class="ltx_Math" display="inline" id="S3.Thmtheorem2.p1.4.4.m4.1"><semantics id="S3.Thmtheorem2.p1.4.4.m4.1a"><mrow id="S3.Thmtheorem2.p1.4.4.m4.1.2" xref="S3.Thmtheorem2.p1.4.4.m4.1.2.cmml"><msup id="S3.Thmtheorem2.p1.4.4.m4.1.2.2" xref="S3.Thmtheorem2.p1.4.4.m4.1.2.2.cmml"><mi id="S3.Thmtheorem2.p1.4.4.m4.1.2.2.2" xref="S3.Thmtheorem2.p1.4.4.m4.1.2.2.2.cmml">ρ</mi><mo id="S3.Thmtheorem2.p1.4.4.m4.1.2.2.3" xref="S3.Thmtheorem2.p1.4.4.m4.1.2.2.3.cmml">∗</mo></msup><mo id="S3.Thmtheorem2.p1.4.4.m4.1.2.1" xref="S3.Thmtheorem2.p1.4.4.m4.1.2.1.cmml">⁢</mo><mrow id="S3.Thmtheorem2.p1.4.4.m4.1.2.3.2" xref="S3.Thmtheorem2.p1.4.4.m4.1.2.cmml"><mo id="S3.Thmtheorem2.p1.4.4.m4.1.2.3.2.1" stretchy="false" xref="S3.Thmtheorem2.p1.4.4.m4.1.2.cmml">(</mo><mi id="S3.Thmtheorem2.p1.4.4.m4.1.1" 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id="S3.Thmtheorem2.p1.4.4.m4.1c">\rho^{*}(v)</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem2.p1.4.4.m4.1d">italic_ρ start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ( italic_v )</annotation></semantics></math>.</span></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.p1.2"><span class="ltx_text ltx_font_italic" id="S3.Thmtheorem3.p1.2.2">Given a weighted graph <math alttext="G" class="ltx_Math" display="inline" id="S3.Thmtheorem3.p1.1.1.m1.1"><semantics id="S3.Thmtheorem3.p1.1.1.m1.1a"><mi id="S3.Thmtheorem3.p1.1.1.m1.1.1" xref="S3.Thmtheorem3.p1.1.1.m1.1.1.cmml">G</mi><annotation-xml 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id="S3.Thmtheorem3.p1.2.2.m2.3c">\rho^{\max}(G)=\Delta^{\min}(G)=\max_{v}\textsl{g}^{*}(v)</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem3.p1.2.2.m2.3d">italic_ρ start_POSTSUPERSCRIPT roman_max end_POSTSUPERSCRIPT ( italic_G ) = roman_Δ start_POSTSUPERSCRIPT roman_min end_POSTSUPERSCRIPT ( italic_G ) = roman_max start_POSTSUBSCRIPT italic_v end_POSTSUBSCRIPT g start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ( italic_v )</annotation></semantics></math>.</span></p> </div> </div> <div class="ltx_theorem ltx_theorem_corollary" 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">Corollary 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.p1.2"><span class="ltx_text ltx_font_italic" id="S3.Thmtheorem4.p1.2.2">For any graph <math alttext="G" class="ltx_Math" display="inline" id="S3.Thmtheorem4.p1.1.1.m1.1"><semantics id="S3.Thmtheorem4.p1.1.1.m1.1a"><mi id="S3.Thmtheorem4.p1.1.1.m1.1.1" xref="S3.Thmtheorem4.p1.1.1.m1.1.1.cmml">G</mi><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem4.p1.1.1.m1.1b"><ci id="S3.Thmtheorem4.p1.1.1.m1.1.1.cmml" xref="S3.Thmtheorem4.p1.1.1.m1.1.1">𝐺</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem4.p1.1.1.m1.1c">G</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem4.p1.1.1.m1.1d">italic_G</annotation></semantics></math>, there exists a fractional orientation <math alttext="\overrightarrow{G}" class="ltx_Math" display="inline" id="S3.Thmtheorem4.p1.2.2.m2.1"><semantics id="S3.Thmtheorem4.p1.2.2.m2.1a"><mover accent="true" id="S3.Thmtheorem4.p1.2.2.m2.1.1" xref="S3.Thmtheorem4.p1.2.2.m2.1.1.cmml"><mi id="S3.Thmtheorem4.p1.2.2.m2.1.1.2" xref="S3.Thmtheorem4.p1.2.2.m2.1.1.2.cmml">G</mi><mo id="S3.Thmtheorem4.p1.2.2.m2.1.1.1" stretchy="false" xref="S3.Thmtheorem4.p1.2.2.m2.1.1.1.cmml">→</mo></mover><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem4.p1.2.2.m2.1b"><apply id="S3.Thmtheorem4.p1.2.2.m2.1.1.cmml" xref="S3.Thmtheorem4.p1.2.2.m2.1.1"><ci id="S3.Thmtheorem4.p1.2.2.m2.1.1.1.cmml" xref="S3.Thmtheorem4.p1.2.2.m2.1.1.1">→</ci><ci id="S3.Thmtheorem4.p1.2.2.m2.1.1.2.cmml" xref="S3.Thmtheorem4.p1.2.2.m2.1.1.2">𝐺</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem4.p1.2.2.m2.1c">\overrightarrow{G}</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem4.p1.2.2.m2.1d">over→ start_ARG italic_G end_ARG</annotation></semantics></math> that is locally fair.</span></p> </div> </div> </section> <section class="ltx_subsubsection" id="S3.SS0.SSS2"> <h4 class="ltx_title ltx_title_subsubsection"> <span class="ltx_tag ltx_tag_subsubsection">3.B </span>Results for dynamic algorithms</h4> <div class="ltx_para" id="S3.SS0.SSS2.p1"> <p class="ltx_p" id="S3.SS0.SSS2.p1.1">We show in Section <a class="ltx_ref" href="https://arxiv.org/html/2411.12694v2#S5" title="5 Results for dynamic algorithms ‣ Local Density and its Distributed Approximation"><span class="ltx_text ltx_ref_tag">5</span></a> that <math alttext="\eta" class="ltx_Math" display="inline" id="S3.SS0.SSS2.p1.1.m1.1"><semantics id="S3.SS0.SSS2.p1.1.m1.1a"><mi id="S3.SS0.SSS2.p1.1.m1.1.1" xref="S3.SS0.SSS2.p1.1.m1.1.1.cmml">η</mi><annotation-xml encoding="MathML-Content" id="S3.SS0.SSS2.p1.1.m1.1b"><ci id="S3.SS0.SSS2.p1.1.m1.1.1.cmml" xref="S3.SS0.SSS2.p1.1.m1.1.1">𝜂</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS0.SSS2.p1.1.m1.1c">\eta</annotation><annotation encoding="application/x-llamapun" id="S3.SS0.SSS2.p1.1.m1.1d">italic_η</annotation></semantics></math>-fair orientations imply approximations for our local measures:</p> </div> <div class="ltx_theorem ltx_theorem_theorem" 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">Theorem 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.p1.6"><span class="ltx_text ltx_font_italic" id="S3.Thmtheorem5.p1.6.6">Let <math alttext="G" class="ltx_Math" display="inline" id="S3.Thmtheorem5.p1.1.1.m1.1"><semantics id="S3.Thmtheorem5.p1.1.1.m1.1a"><mi id="S3.Thmtheorem5.p1.1.1.m1.1.1" xref="S3.Thmtheorem5.p1.1.1.m1.1.1.cmml">G</mi><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem5.p1.1.1.m1.1b"><ci id="S3.Thmtheorem5.p1.1.1.m1.1.1.cmml" xref="S3.Thmtheorem5.p1.1.1.m1.1.1">𝐺</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem5.p1.1.1.m1.1c">G</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem5.p1.1.1.m1.1d">italic_G</annotation></semantics></math> be a weighted graph and <math alttext="\overrightarrow{G}" class="ltx_Math" display="inline" id="S3.Thmtheorem5.p1.2.2.m2.1"><semantics id="S3.Thmtheorem5.p1.2.2.m2.1a"><mover accent="true" id="S3.Thmtheorem5.p1.2.2.m2.1.1" xref="S3.Thmtheorem5.p1.2.2.m2.1.1.cmml"><mi id="S3.Thmtheorem5.p1.2.2.m2.1.1.2" xref="S3.Thmtheorem5.p1.2.2.m2.1.1.2.cmml">G</mi><mo id="S3.Thmtheorem5.p1.2.2.m2.1.1.1" stretchy="false" xref="S3.Thmtheorem5.p1.2.2.m2.1.1.1.cmml">→</mo></mover><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem5.p1.2.2.m2.1b"><apply id="S3.Thmtheorem5.p1.2.2.m2.1.1.cmml" xref="S3.Thmtheorem5.p1.2.2.m2.1.1"><ci id="S3.Thmtheorem5.p1.2.2.m2.1.1.1.cmml" xref="S3.Thmtheorem5.p1.2.2.m2.1.1.1">→</ci><ci id="S3.Thmtheorem5.p1.2.2.m2.1.1.2.cmml" xref="S3.Thmtheorem5.p1.2.2.m2.1.1.2">𝐺</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem5.p1.2.2.m2.1c">\overrightarrow{G}</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem5.p1.2.2.m2.1d">over→ start_ARG italic_G end_ARG</annotation></semantics></math> be an <math alttext="\eta" class="ltx_Math" display="inline" id="S3.Thmtheorem5.p1.3.3.m3.1"><semantics id="S3.Thmtheorem5.p1.3.3.m3.1a"><mi id="S3.Thmtheorem5.p1.3.3.m3.1.1" xref="S3.Thmtheorem5.p1.3.3.m3.1.1.cmml">η</mi><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem5.p1.3.3.m3.1b"><ci id="S3.Thmtheorem5.p1.3.3.m3.1.1.cmml" xref="S3.Thmtheorem5.p1.3.3.m3.1.1">𝜂</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem5.p1.3.3.m3.1c">\eta</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem5.p1.3.3.m3.1d">italic_η</annotation></semantics></math>-fair fractional orientation for <math alttext="\eta\leq\frac{\varepsilon^{2}}{128\cdot\log n}" class="ltx_Math" display="inline" id="S3.Thmtheorem5.p1.4.4.m4.1"><semantics id="S3.Thmtheorem5.p1.4.4.m4.1a"><mrow id="S3.Thmtheorem5.p1.4.4.m4.1.1" xref="S3.Thmtheorem5.p1.4.4.m4.1.1.cmml"><mi id="S3.Thmtheorem5.p1.4.4.m4.1.1.2" xref="S3.Thmtheorem5.p1.4.4.m4.1.1.2.cmml">η</mi><mo id="S3.Thmtheorem5.p1.4.4.m4.1.1.1" xref="S3.Thmtheorem5.p1.4.4.m4.1.1.1.cmml">≤</mo><mfrac id="S3.Thmtheorem5.p1.4.4.m4.1.1.3" xref="S3.Thmtheorem5.p1.4.4.m4.1.1.3.cmml"><msup id="S3.Thmtheorem5.p1.4.4.m4.1.1.3.2" xref="S3.Thmtheorem5.p1.4.4.m4.1.1.3.2.cmml"><mi id="S3.Thmtheorem5.p1.4.4.m4.1.1.3.2.2" xref="S3.Thmtheorem5.p1.4.4.m4.1.1.3.2.2.cmml">ε</mi><mn id="S3.Thmtheorem5.p1.4.4.m4.1.1.3.2.3" xref="S3.Thmtheorem5.p1.4.4.m4.1.1.3.2.3.cmml">2</mn></msup><mrow id="S3.Thmtheorem5.p1.4.4.m4.1.1.3.3" xref="S3.Thmtheorem5.p1.4.4.m4.1.1.3.3.cmml"><mn id="S3.Thmtheorem5.p1.4.4.m4.1.1.3.3.2" xref="S3.Thmtheorem5.p1.4.4.m4.1.1.3.3.2.cmml">128</mn><mo id="S3.Thmtheorem5.p1.4.4.m4.1.1.3.3.1" lspace="0.222em" rspace="0.222em" xref="S3.Thmtheorem5.p1.4.4.m4.1.1.3.3.1.cmml">⋅</mo><mrow id="S3.Thmtheorem5.p1.4.4.m4.1.1.3.3.3" xref="S3.Thmtheorem5.p1.4.4.m4.1.1.3.3.3.cmml"><mi id="S3.Thmtheorem5.p1.4.4.m4.1.1.3.3.3.1" xref="S3.Thmtheorem5.p1.4.4.m4.1.1.3.3.3.1.cmml">log</mi><mo id="S3.Thmtheorem5.p1.4.4.m4.1.1.3.3.3a" lspace="0.167em" xref="S3.Thmtheorem5.p1.4.4.m4.1.1.3.3.3.cmml">⁡</mo><mi id="S3.Thmtheorem5.p1.4.4.m4.1.1.3.3.3.2" xref="S3.Thmtheorem5.p1.4.4.m4.1.1.3.3.3.2.cmml">n</mi></mrow></mrow></mfrac></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem5.p1.4.4.m4.1b"><apply id="S3.Thmtheorem5.p1.4.4.m4.1.1.cmml" xref="S3.Thmtheorem5.p1.4.4.m4.1.1"><leq id="S3.Thmtheorem5.p1.4.4.m4.1.1.1.cmml" xref="S3.Thmtheorem5.p1.4.4.m4.1.1.1"></leq><ci id="S3.Thmtheorem5.p1.4.4.m4.1.1.2.cmml" xref="S3.Thmtheorem5.p1.4.4.m4.1.1.2">𝜂</ci><apply id="S3.Thmtheorem5.p1.4.4.m4.1.1.3.cmml" xref="S3.Thmtheorem5.p1.4.4.m4.1.1.3"><divide id="S3.Thmtheorem5.p1.4.4.m4.1.1.3.1.cmml" xref="S3.Thmtheorem5.p1.4.4.m4.1.1.3"></divide><apply id="S3.Thmtheorem5.p1.4.4.m4.1.1.3.2.cmml" xref="S3.Thmtheorem5.p1.4.4.m4.1.1.3.2"><csymbol cd="ambiguous" id="S3.Thmtheorem5.p1.4.4.m4.1.1.3.2.1.cmml" xref="S3.Thmtheorem5.p1.4.4.m4.1.1.3.2">superscript</csymbol><ci id="S3.Thmtheorem5.p1.4.4.m4.1.1.3.2.2.cmml" xref="S3.Thmtheorem5.p1.4.4.m4.1.1.3.2.2">𝜀</ci><cn id="S3.Thmtheorem5.p1.4.4.m4.1.1.3.2.3.cmml" type="integer" xref="S3.Thmtheorem5.p1.4.4.m4.1.1.3.2.3">2</cn></apply><apply id="S3.Thmtheorem5.p1.4.4.m4.1.1.3.3.cmml" xref="S3.Thmtheorem5.p1.4.4.m4.1.1.3.3"><ci id="S3.Thmtheorem5.p1.4.4.m4.1.1.3.3.1.cmml" xref="S3.Thmtheorem5.p1.4.4.m4.1.1.3.3.1">⋅</ci><cn id="S3.Thmtheorem5.p1.4.4.m4.1.1.3.3.2.cmml" type="integer" xref="S3.Thmtheorem5.p1.4.4.m4.1.1.3.3.2">128</cn><apply id="S3.Thmtheorem5.p1.4.4.m4.1.1.3.3.3.cmml" xref="S3.Thmtheorem5.p1.4.4.m4.1.1.3.3.3"><log id="S3.Thmtheorem5.p1.4.4.m4.1.1.3.3.3.1.cmml" xref="S3.Thmtheorem5.p1.4.4.m4.1.1.3.3.3.1"></log><ci id="S3.Thmtheorem5.p1.4.4.m4.1.1.3.3.3.2.cmml" xref="S3.Thmtheorem5.p1.4.4.m4.1.1.3.3.3.2">𝑛</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem5.p1.4.4.m4.1c">\eta\leq\frac{\varepsilon^{2}}{128\cdot\log n}</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem5.p1.4.4.m4.1d">italic_η ≤ divide start_ARG italic_ε start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG start_ARG 128 ⋅ roman_log italic_n end_ARG</annotation></semantics></math>. Then <math alttext="\forall v\in V" class="ltx_Math" display="inline" id="S3.Thmtheorem5.p1.5.5.m5.1"><semantics id="S3.Thmtheorem5.p1.5.5.m5.1a"><mrow id="S3.Thmtheorem5.p1.5.5.m5.1.1" xref="S3.Thmtheorem5.p1.5.5.m5.1.1.cmml"><mrow id="S3.Thmtheorem5.p1.5.5.m5.1.1.2" xref="S3.Thmtheorem5.p1.5.5.m5.1.1.2.cmml"><mo id="S3.Thmtheorem5.p1.5.5.m5.1.1.2.1" rspace="0.167em" xref="S3.Thmtheorem5.p1.5.5.m5.1.1.2.1.cmml">∀</mo><mi id="S3.Thmtheorem5.p1.5.5.m5.1.1.2.2" xref="S3.Thmtheorem5.p1.5.5.m5.1.1.2.2.cmml">v</mi></mrow><mo id="S3.Thmtheorem5.p1.5.5.m5.1.1.1" xref="S3.Thmtheorem5.p1.5.5.m5.1.1.1.cmml">∈</mo><mi id="S3.Thmtheorem5.p1.5.5.m5.1.1.3" xref="S3.Thmtheorem5.p1.5.5.m5.1.1.3.cmml">V</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem5.p1.5.5.m5.1b"><apply id="S3.Thmtheorem5.p1.5.5.m5.1.1.cmml" xref="S3.Thmtheorem5.p1.5.5.m5.1.1"><in id="S3.Thmtheorem5.p1.5.5.m5.1.1.1.cmml" xref="S3.Thmtheorem5.p1.5.5.m5.1.1.1"></in><apply id="S3.Thmtheorem5.p1.5.5.m5.1.1.2.cmml" xref="S3.Thmtheorem5.p1.5.5.m5.1.1.2"><csymbol cd="latexml" id="S3.Thmtheorem5.p1.5.5.m5.1.1.2.1.cmml" xref="S3.Thmtheorem5.p1.5.5.m5.1.1.2.1">for-all</csymbol><ci id="S3.Thmtheorem5.p1.5.5.m5.1.1.2.2.cmml" xref="S3.Thmtheorem5.p1.5.5.m5.1.1.2.2">𝑣</ci></apply><ci id="S3.Thmtheorem5.p1.5.5.m5.1.1.3.cmml" xref="S3.Thmtheorem5.p1.5.5.m5.1.1.3">𝑉</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem5.p1.5.5.m5.1c">\forall v\in V</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem5.p1.5.5.m5.1d">∀ italic_v ∈ italic_V</annotation></semantics></math>: <math alttext="(1+\varepsilon)^{-1}\rho^{*}(v)\leq\textsl{g}(v)\leq(1+\varepsilon)\rho^{*}(v)." class="ltx_Math" display="inline" id="S3.Thmtheorem5.p1.6.6.m6.4"><semantics id="S3.Thmtheorem5.p1.6.6.m6.4a"><mrow id="S3.Thmtheorem5.p1.6.6.m6.4.4.1" xref="S3.Thmtheorem5.p1.6.6.m6.4.4.1.1.cmml"><mrow id="S3.Thmtheorem5.p1.6.6.m6.4.4.1.1" xref="S3.Thmtheorem5.p1.6.6.m6.4.4.1.1.cmml"><mrow 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xref="S3.Thmtheorem5.p1.6.6.m6.4.4.1.1.2.1.1.1.1"></plus><cn id="S3.Thmtheorem5.p1.6.6.m6.4.4.1.1.2.1.1.1.2.cmml" type="integer" xref="S3.Thmtheorem5.p1.6.6.m6.4.4.1.1.2.1.1.1.2">1</cn><ci id="S3.Thmtheorem5.p1.6.6.m6.4.4.1.1.2.1.1.1.3.cmml" xref="S3.Thmtheorem5.p1.6.6.m6.4.4.1.1.2.1.1.1.3">𝜀</ci></apply><apply id="S3.Thmtheorem5.p1.6.6.m6.4.4.1.1.2.3.cmml" xref="S3.Thmtheorem5.p1.6.6.m6.4.4.1.1.2.3"><csymbol cd="ambiguous" id="S3.Thmtheorem5.p1.6.6.m6.4.4.1.1.2.3.1.cmml" xref="S3.Thmtheorem5.p1.6.6.m6.4.4.1.1.2.3">superscript</csymbol><ci id="S3.Thmtheorem5.p1.6.6.m6.4.4.1.1.2.3.2.cmml" xref="S3.Thmtheorem5.p1.6.6.m6.4.4.1.1.2.3.2">𝜌</ci><times id="S3.Thmtheorem5.p1.6.6.m6.4.4.1.1.2.3.3.cmml" xref="S3.Thmtheorem5.p1.6.6.m6.4.4.1.1.2.3.3"></times></apply><ci id="S3.Thmtheorem5.p1.6.6.m6.3.3.cmml" xref="S3.Thmtheorem5.p1.6.6.m6.3.3">𝑣</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem5.p1.6.6.m6.4c">(1+\varepsilon)^{-1}\rho^{*}(v)\leq\textsl{g}(v)\leq(1+\varepsilon)\rho^{*}(v).</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem5.p1.6.6.m6.4d">( 1 + italic_ε ) start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT italic_ρ start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ( italic_v ) ≤ g ( italic_v ) ≤ ( 1 + italic_ε ) italic_ρ start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ( italic_v ) .</annotation></semantics></math></span></p> </div> </div> <div class="ltx_para ltx_noindent" id="S3.SS0.SSS2.p2"> <p class="ltx_p" id="S3.SS0.SSS2.p2.1">This immediately implies the following Corollary by applying <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2411.12694v2#bib.bib7" title="">7</a>]</cite>:</p> </div> <div class="ltx_theorem ltx_theorem_corollary" 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">Corollary 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.p1.4"><span class="ltx_text ltx_font_italic" id="S3.Thmtheorem6.p1.4.4">There exists an algorithm <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2411.12694v2#bib.bib7" title="">7</a>]</cite> that can fractionally orient a dynamic unit-weight graph <math alttext="G" class="ltx_Math" display="inline" id="S3.Thmtheorem6.p1.1.1.m1.1"><semantics id="S3.Thmtheorem6.p1.1.1.m1.1a"><mi id="S3.Thmtheorem6.p1.1.1.m1.1.1" xref="S3.Thmtheorem6.p1.1.1.m1.1.1.cmml">G</mi><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem6.p1.1.1.m1.1b"><ci id="S3.Thmtheorem6.p1.1.1.m1.1.1.cmml" xref="S3.Thmtheorem6.p1.1.1.m1.1.1">𝐺</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem6.p1.1.1.m1.1c">G</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem6.p1.1.1.m1.1d">italic_G</annotation></semantics></math> with <math alttext="n" class="ltx_Math" display="inline" id="S3.Thmtheorem6.p1.2.2.m2.1"><semantics id="S3.Thmtheorem6.p1.2.2.m2.1a"><mi id="S3.Thmtheorem6.p1.2.2.m2.1.1" xref="S3.Thmtheorem6.p1.2.2.m2.1.1.cmml">n</mi><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem6.p1.2.2.m2.1b"><ci id="S3.Thmtheorem6.p1.2.2.m2.1.1.cmml" xref="S3.Thmtheorem6.p1.2.2.m2.1.1">𝑛</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem6.p1.2.2.m2.1c">n</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem6.p1.2.2.m2.1d">italic_n</annotation></semantics></math> vertices subject to edge insertions and deletions with deterministic worst-case <math alttext="O(\varepsilon^{-6}\log^{4}n))" class="ltx_math_unparsed" display="inline" id="S3.Thmtheorem6.p1.3.3.m3.1"><semantics id="S3.Thmtheorem6.p1.3.3.m3.1a"><mrow id="S3.Thmtheorem6.p1.3.3.m3.1b"><mi id="S3.Thmtheorem6.p1.3.3.m3.1.1">O</mi><mrow id="S3.Thmtheorem6.p1.3.3.m3.1.2"><mo id="S3.Thmtheorem6.p1.3.3.m3.1.2.1" stretchy="false">(</mo><msup id="S3.Thmtheorem6.p1.3.3.m3.1.2.2"><mi id="S3.Thmtheorem6.p1.3.3.m3.1.2.2.2">ε</mi><mrow id="S3.Thmtheorem6.p1.3.3.m3.1.2.2.3"><mo id="S3.Thmtheorem6.p1.3.3.m3.1.2.2.3a">−</mo><mn id="S3.Thmtheorem6.p1.3.3.m3.1.2.2.3.2">6</mn></mrow></msup><msup id="S3.Thmtheorem6.p1.3.3.m3.1.2.3"><mi id="S3.Thmtheorem6.p1.3.3.m3.1.2.3.2">log</mi><mn id="S3.Thmtheorem6.p1.3.3.m3.1.2.3.3">4</mn></msup><mi id="S3.Thmtheorem6.p1.3.3.m3.1.2.4">n</mi><mo id="S3.Thmtheorem6.p1.3.3.m3.1.2.5" stretchy="false">)</mo></mrow><mo id="S3.Thmtheorem6.p1.3.3.m3.1.3" stretchy="false">)</mo></mrow><annotation encoding="application/x-tex" id="S3.Thmtheorem6.p1.3.3.m3.1c">O(\varepsilon^{-6}\log^{4}n))</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem6.p1.3.3.m3.1d">italic_O ( italic_ε start_POSTSUPERSCRIPT - 6 end_POSTSUPERSCRIPT roman_log start_POSTSUPERSCRIPT 4 end_POSTSUPERSCRIPT italic_n ) )</annotation></semantics></math> update time such that for all <math alttext="v\in V" class="ltx_Math" display="inline" id="S3.Thmtheorem6.p1.4.4.m4.1"><semantics id="S3.Thmtheorem6.p1.4.4.m4.1a"><mrow id="S3.Thmtheorem6.p1.4.4.m4.1.1" xref="S3.Thmtheorem6.p1.4.4.m4.1.1.cmml"><mi id="S3.Thmtheorem6.p1.4.4.m4.1.1.2" xref="S3.Thmtheorem6.p1.4.4.m4.1.1.2.cmml">v</mi><mo id="S3.Thmtheorem6.p1.4.4.m4.1.1.1" xref="S3.Thmtheorem6.p1.4.4.m4.1.1.1.cmml">∈</mo><mi id="S3.Thmtheorem6.p1.4.4.m4.1.1.3" xref="S3.Thmtheorem6.p1.4.4.m4.1.1.3.cmml">V</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem6.p1.4.4.m4.1b"><apply id="S3.Thmtheorem6.p1.4.4.m4.1.1.cmml" xref="S3.Thmtheorem6.p1.4.4.m4.1.1"><in id="S3.Thmtheorem6.p1.4.4.m4.1.1.1.cmml" xref="S3.Thmtheorem6.p1.4.4.m4.1.1.1"></in><ci id="S3.Thmtheorem6.p1.4.4.m4.1.1.2.cmml" xref="S3.Thmtheorem6.p1.4.4.m4.1.1.2">𝑣</ci><ci id="S3.Thmtheorem6.p1.4.4.m4.1.1.3.cmml" xref="S3.Thmtheorem6.p1.4.4.m4.1.1.3">𝑉</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem6.p1.4.4.m4.1c">v\in V</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem6.p1.4.4.m4.1d">italic_v ∈ italic_V</annotation></semantics></math>:</span></p> <table class="ltx_equation ltx_eqn_table" id="S3.Ex11"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_left_padleft"></td> <td class="ltx_eqn_cell ltx_align_left"><math alttext="\textsl{g}(v)\in[(1+\varepsilon)^{-1}\rho^{*}(v),(1+\varepsilon)\rho^{*}(v)]." class="ltx_Math" display="block" id="S3.Ex11.m1.4"><semantics id="S3.Ex11.m1.4a"><mrow id="S3.Ex11.m1.4.4.1" xref="S3.Ex11.m1.4.4.1.1.cmml"><mrow id="S3.Ex11.m1.4.4.1.1" xref="S3.Ex11.m1.4.4.1.1.cmml"><mrow id="S3.Ex11.m1.4.4.1.1.4" xref="S3.Ex11.m1.4.4.1.1.4.cmml"><mtext class="ltx_mathvariant_italic" id="S3.Ex11.m1.4.4.1.1.4.2" xref="S3.Ex11.m1.4.4.1.1.4.2a.cmml">g</mtext><mo 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xref="S3.Ex11.m1.4.4.1.1.2.2.2.1.1.1.2">1</cn><ci id="S3.Ex11.m1.4.4.1.1.2.2.2.1.1.1.3.cmml" xref="S3.Ex11.m1.4.4.1.1.2.2.2.1.1.1.3">𝜀</ci></apply><apply id="S3.Ex11.m1.4.4.1.1.2.2.2.3.cmml" xref="S3.Ex11.m1.4.4.1.1.2.2.2.3"><csymbol cd="ambiguous" id="S3.Ex11.m1.4.4.1.1.2.2.2.3.1.cmml" xref="S3.Ex11.m1.4.4.1.1.2.2.2.3">superscript</csymbol><ci id="S3.Ex11.m1.4.4.1.1.2.2.2.3.2.cmml" xref="S3.Ex11.m1.4.4.1.1.2.2.2.3.2">𝜌</ci><times id="S3.Ex11.m1.4.4.1.1.2.2.2.3.3.cmml" xref="S3.Ex11.m1.4.4.1.1.2.2.2.3.3"></times></apply><ci id="S3.Ex11.m1.3.3.cmml" xref="S3.Ex11.m1.3.3">𝑣</ci></apply></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Ex11.m1.4c">\textsl{g}(v)\in[(1+\varepsilon)^{-1}\rho^{*}(v),(1+\varepsilon)\rho^{*}(v)].</annotation><annotation encoding="application/x-llamapun" id="S3.Ex11.m1.4d">g ( italic_v ) ∈ [ ( 1 + italic_ε ) start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT italic_ρ start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ( italic_v ) , ( 1 + italic_ε ) italic_ρ start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ( italic_v ) ] .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_left_padright"></td> </tr></tbody> </table> </div> </div> </section> <section class="ltx_subsubsection" id="S3.SS0.SSS3"> <h4 class="ltx_title ltx_title_subsubsection"> <span class="ltx_tag ltx_tag_subsubsection">3.C </span>Results in LOCAL</h4> <div class="ltx_para" id="S3.SS0.SSS3.p1"> <p class="ltx_p" id="S3.SS0.SSS3.p1.1">The local density as a measure is not entirely local. However, we prove in Section <a class="ltx_ref" href="https://arxiv.org/html/2411.12694v2#S6" title="6 Results in LOCAL ‣ Local Density and its Distributed Approximation"><span class="ltx_text ltx_ref_tag">6</span></a> that far-away subgraphs affect the local density of a vertex <math alttext="v" class="ltx_Math" display="inline" id="S3.SS0.SSS3.p1.1.m1.1"><semantics id="S3.SS0.SSS3.p1.1.m1.1a"><mi id="S3.SS0.SSS3.p1.1.m1.1.1" xref="S3.SS0.SSS3.p1.1.m1.1.1.cmml">v</mi><annotation-xml encoding="MathML-Content" id="S3.SS0.SSS3.p1.1.m1.1b"><ci id="S3.SS0.SSS3.p1.1.m1.1.1.cmml" xref="S3.SS0.SSS3.p1.1.m1.1.1">𝑣</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS0.SSS3.p1.1.m1.1c">v</annotation><annotation encoding="application/x-llamapun" id="S3.SS0.SSS3.p1.1.m1.1d">italic_v</annotation></semantics></math> only marginally:</p> </div> <div class="ltx_theorem ltx_theorem_theorem" 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">Theorem 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.p1.8"><span class="ltx_text ltx_font_italic" id="S3.Thmtheorem7.p1.8.8">Let <math alttext="G" class="ltx_Math" display="inline" id="S3.Thmtheorem7.p1.1.1.m1.1"><semantics id="S3.Thmtheorem7.p1.1.1.m1.1a"><mi id="S3.Thmtheorem7.p1.1.1.m1.1.1" xref="S3.Thmtheorem7.p1.1.1.m1.1.1.cmml">G</mi><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem7.p1.1.1.m1.1b"><ci id="S3.Thmtheorem7.p1.1.1.m1.1.1.cmml" xref="S3.Thmtheorem7.p1.1.1.m1.1.1">𝐺</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem7.p1.1.1.m1.1c">G</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem7.p1.1.1.m1.1d">italic_G</annotation></semantics></math> be a unit-weight graph. For any <math alttext="\varepsilon>0" class="ltx_Math" display="inline" id="S3.Thmtheorem7.p1.2.2.m2.1"><semantics id="S3.Thmtheorem7.p1.2.2.m2.1a"><mrow id="S3.Thmtheorem7.p1.2.2.m2.1.1" xref="S3.Thmtheorem7.p1.2.2.m2.1.1.cmml"><mi id="S3.Thmtheorem7.p1.2.2.m2.1.1.2" xref="S3.Thmtheorem7.p1.2.2.m2.1.1.2.cmml">ε</mi><mo id="S3.Thmtheorem7.p1.2.2.m2.1.1.1" xref="S3.Thmtheorem7.p1.2.2.m2.1.1.1.cmml">></mo><mn id="S3.Thmtheorem7.p1.2.2.m2.1.1.3" xref="S3.Thmtheorem7.p1.2.2.m2.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem7.p1.2.2.m2.1b"><apply id="S3.Thmtheorem7.p1.2.2.m2.1.1.cmml" xref="S3.Thmtheorem7.p1.2.2.m2.1.1"><gt id="S3.Thmtheorem7.p1.2.2.m2.1.1.1.cmml" xref="S3.Thmtheorem7.p1.2.2.m2.1.1.1"></gt><ci id="S3.Thmtheorem7.p1.2.2.m2.1.1.2.cmml" xref="S3.Thmtheorem7.p1.2.2.m2.1.1.2">𝜀</ci><cn id="S3.Thmtheorem7.p1.2.2.m2.1.1.3.cmml" type="integer" xref="S3.Thmtheorem7.p1.2.2.m2.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem7.p1.2.2.m2.1c">\varepsilon>0</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem7.p1.2.2.m2.1d">italic_ε > 0</annotation></semantics></math> and vertex <math alttext="v" class="ltx_Math" display="inline" id="S3.Thmtheorem7.p1.3.3.m3.1"><semantics id="S3.Thmtheorem7.p1.3.3.m3.1a"><mi id="S3.Thmtheorem7.p1.3.3.m3.1.1" xref="S3.Thmtheorem7.p1.3.3.m3.1.1.cmml">v</mi><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem7.p1.3.3.m3.1b"><ci id="S3.Thmtheorem7.p1.3.3.m3.1.1.cmml" xref="S3.Thmtheorem7.p1.3.3.m3.1.1">𝑣</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem7.p1.3.3.m3.1c">v</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem7.p1.3.3.m3.1d">italic_v</annotation></semantics></math>, denote by <math alttext="\rho^{*}(v)" class="ltx_Math" display="inline" id="S3.Thmtheorem7.p1.4.4.m4.1"><semantics id="S3.Thmtheorem7.p1.4.4.m4.1a"><mrow id="S3.Thmtheorem7.p1.4.4.m4.1.2" xref="S3.Thmtheorem7.p1.4.4.m4.1.2.cmml"><msup id="S3.Thmtheorem7.p1.4.4.m4.1.2.2" xref="S3.Thmtheorem7.p1.4.4.m4.1.2.2.cmml"><mi id="S3.Thmtheorem7.p1.4.4.m4.1.2.2.2" xref="S3.Thmtheorem7.p1.4.4.m4.1.2.2.2.cmml">ρ</mi><mo id="S3.Thmtheorem7.p1.4.4.m4.1.2.2.3" xref="S3.Thmtheorem7.p1.4.4.m4.1.2.2.3.cmml">∗</mo></msup><mo id="S3.Thmtheorem7.p1.4.4.m4.1.2.1" xref="S3.Thmtheorem7.p1.4.4.m4.1.2.1.cmml">⁢</mo><mrow id="S3.Thmtheorem7.p1.4.4.m4.1.2.3.2" xref="S3.Thmtheorem7.p1.4.4.m4.1.2.cmml"><mo id="S3.Thmtheorem7.p1.4.4.m4.1.2.3.2.1" stretchy="false" xref="S3.Thmtheorem7.p1.4.4.m4.1.2.cmml">(</mo><mi id="S3.Thmtheorem7.p1.4.4.m4.1.1" xref="S3.Thmtheorem7.p1.4.4.m4.1.1.cmml">v</mi><mo id="S3.Thmtheorem7.p1.4.4.m4.1.2.3.2.2" stretchy="false" xref="S3.Thmtheorem7.p1.4.4.m4.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem7.p1.4.4.m4.1b"><apply id="S3.Thmtheorem7.p1.4.4.m4.1.2.cmml" xref="S3.Thmtheorem7.p1.4.4.m4.1.2"><times id="S3.Thmtheorem7.p1.4.4.m4.1.2.1.cmml" xref="S3.Thmtheorem7.p1.4.4.m4.1.2.1"></times><apply id="S3.Thmtheorem7.p1.4.4.m4.1.2.2.cmml" xref="S3.Thmtheorem7.p1.4.4.m4.1.2.2"><csymbol cd="ambiguous" id="S3.Thmtheorem7.p1.4.4.m4.1.2.2.1.cmml" xref="S3.Thmtheorem7.p1.4.4.m4.1.2.2">superscript</csymbol><ci id="S3.Thmtheorem7.p1.4.4.m4.1.2.2.2.cmml" xref="S3.Thmtheorem7.p1.4.4.m4.1.2.2.2">𝜌</ci><times id="S3.Thmtheorem7.p1.4.4.m4.1.2.2.3.cmml" xref="S3.Thmtheorem7.p1.4.4.m4.1.2.2.3"></times></apply><ci id="S3.Thmtheorem7.p1.4.4.m4.1.1.cmml" xref="S3.Thmtheorem7.p1.4.4.m4.1.1">𝑣</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem7.p1.4.4.m4.1c">\rho^{*}(v)</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem7.p1.4.4.m4.1d">italic_ρ start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ( italic_v )</annotation></semantics></math> its local density and by <math alttext="\rho^{*}_{k}(v)" class="ltx_Math" display="inline" id="S3.Thmtheorem7.p1.5.5.m5.1"><semantics id="S3.Thmtheorem7.p1.5.5.m5.1a"><mrow id="S3.Thmtheorem7.p1.5.5.m5.1.2" xref="S3.Thmtheorem7.p1.5.5.m5.1.2.cmml"><msubsup id="S3.Thmtheorem7.p1.5.5.m5.1.2.2" xref="S3.Thmtheorem7.p1.5.5.m5.1.2.2.cmml"><mi id="S3.Thmtheorem7.p1.5.5.m5.1.2.2.2.2" xref="S3.Thmtheorem7.p1.5.5.m5.1.2.2.2.2.cmml">ρ</mi><mi id="S3.Thmtheorem7.p1.5.5.m5.1.2.2.3" xref="S3.Thmtheorem7.p1.5.5.m5.1.2.2.3.cmml">k</mi><mo id="S3.Thmtheorem7.p1.5.5.m5.1.2.2.2.3" xref="S3.Thmtheorem7.p1.5.5.m5.1.2.2.2.3.cmml">∗</mo></msubsup><mo id="S3.Thmtheorem7.p1.5.5.m5.1.2.1" xref="S3.Thmtheorem7.p1.5.5.m5.1.2.1.cmml">⁢</mo><mrow id="S3.Thmtheorem7.p1.5.5.m5.1.2.3.2" xref="S3.Thmtheorem7.p1.5.5.m5.1.2.cmml"><mo id="S3.Thmtheorem7.p1.5.5.m5.1.2.3.2.1" stretchy="false" xref="S3.Thmtheorem7.p1.5.5.m5.1.2.cmml">(</mo><mi id="S3.Thmtheorem7.p1.5.5.m5.1.1" xref="S3.Thmtheorem7.p1.5.5.m5.1.1.cmml">v</mi><mo id="S3.Thmtheorem7.p1.5.5.m5.1.2.3.2.2" stretchy="false" xref="S3.Thmtheorem7.p1.5.5.m5.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem7.p1.5.5.m5.1b"><apply id="S3.Thmtheorem7.p1.5.5.m5.1.2.cmml" xref="S3.Thmtheorem7.p1.5.5.m5.1.2"><times id="S3.Thmtheorem7.p1.5.5.m5.1.2.1.cmml" xref="S3.Thmtheorem7.p1.5.5.m5.1.2.1"></times><apply id="S3.Thmtheorem7.p1.5.5.m5.1.2.2.cmml" xref="S3.Thmtheorem7.p1.5.5.m5.1.2.2"><csymbol cd="ambiguous" id="S3.Thmtheorem7.p1.5.5.m5.1.2.2.1.cmml" xref="S3.Thmtheorem7.p1.5.5.m5.1.2.2">subscript</csymbol><apply id="S3.Thmtheorem7.p1.5.5.m5.1.2.2.2.cmml" xref="S3.Thmtheorem7.p1.5.5.m5.1.2.2"><csymbol cd="ambiguous" id="S3.Thmtheorem7.p1.5.5.m5.1.2.2.2.1.cmml" xref="S3.Thmtheorem7.p1.5.5.m5.1.2.2">superscript</csymbol><ci id="S3.Thmtheorem7.p1.5.5.m5.1.2.2.2.2.cmml" xref="S3.Thmtheorem7.p1.5.5.m5.1.2.2.2.2">𝜌</ci><times id="S3.Thmtheorem7.p1.5.5.m5.1.2.2.2.3.cmml" xref="S3.Thmtheorem7.p1.5.5.m5.1.2.2.2.3"></times></apply><ci id="S3.Thmtheorem7.p1.5.5.m5.1.2.2.3.cmml" xref="S3.Thmtheorem7.p1.5.5.m5.1.2.2.3">𝑘</ci></apply><ci id="S3.Thmtheorem7.p1.5.5.m5.1.1.cmml" xref="S3.Thmtheorem7.p1.5.5.m5.1.1">𝑣</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem7.p1.5.5.m5.1c">\rho^{*}_{k}(v)</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem7.p1.5.5.m5.1d">italic_ρ start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ( italic_v )</annotation></semantics></math> its local density in <math alttext="H_{k}(v)" class="ltx_Math" display="inline" id="S3.Thmtheorem7.p1.6.6.m6.1"><semantics id="S3.Thmtheorem7.p1.6.6.m6.1a"><mrow id="S3.Thmtheorem7.p1.6.6.m6.1.2" xref="S3.Thmtheorem7.p1.6.6.m6.1.2.cmml"><msub id="S3.Thmtheorem7.p1.6.6.m6.1.2.2" xref="S3.Thmtheorem7.p1.6.6.m6.1.2.2.cmml"><mi id="S3.Thmtheorem7.p1.6.6.m6.1.2.2.2" xref="S3.Thmtheorem7.p1.6.6.m6.1.2.2.2.cmml">H</mi><mi id="S3.Thmtheorem7.p1.6.6.m6.1.2.2.3" xref="S3.Thmtheorem7.p1.6.6.m6.1.2.2.3.cmml">k</mi></msub><mo id="S3.Thmtheorem7.p1.6.6.m6.1.2.1" xref="S3.Thmtheorem7.p1.6.6.m6.1.2.1.cmml">⁢</mo><mrow id="S3.Thmtheorem7.p1.6.6.m6.1.2.3.2" xref="S3.Thmtheorem7.p1.6.6.m6.1.2.cmml"><mo id="S3.Thmtheorem7.p1.6.6.m6.1.2.3.2.1" stretchy="false" xref="S3.Thmtheorem7.p1.6.6.m6.1.2.cmml">(</mo><mi id="S3.Thmtheorem7.p1.6.6.m6.1.1" xref="S3.Thmtheorem7.p1.6.6.m6.1.1.cmml">v</mi><mo id="S3.Thmtheorem7.p1.6.6.m6.1.2.3.2.2" stretchy="false" xref="S3.Thmtheorem7.p1.6.6.m6.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem7.p1.6.6.m6.1b"><apply id="S3.Thmtheorem7.p1.6.6.m6.1.2.cmml" xref="S3.Thmtheorem7.p1.6.6.m6.1.2"><times id="S3.Thmtheorem7.p1.6.6.m6.1.2.1.cmml" xref="S3.Thmtheorem7.p1.6.6.m6.1.2.1"></times><apply id="S3.Thmtheorem7.p1.6.6.m6.1.2.2.cmml" xref="S3.Thmtheorem7.p1.6.6.m6.1.2.2"><csymbol cd="ambiguous" id="S3.Thmtheorem7.p1.6.6.m6.1.2.2.1.cmml" xref="S3.Thmtheorem7.p1.6.6.m6.1.2.2">subscript</csymbol><ci id="S3.Thmtheorem7.p1.6.6.m6.1.2.2.2.cmml" xref="S3.Thmtheorem7.p1.6.6.m6.1.2.2.2">𝐻</ci><ci id="S3.Thmtheorem7.p1.6.6.m6.1.2.2.3.cmml" xref="S3.Thmtheorem7.p1.6.6.m6.1.2.2.3">𝑘</ci></apply><ci id="S3.Thmtheorem7.p1.6.6.m6.1.1.cmml" xref="S3.Thmtheorem7.p1.6.6.m6.1.1">𝑣</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem7.p1.6.6.m6.1c">H_{k}(v)</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem7.p1.6.6.m6.1d">italic_H start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ( italic_v )</annotation></semantics></math>. 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id="S3.Thmtheorem7.p1.7.7.m7.5.5.2.2.2.1.1.1.3.cmml" xref="S3.Thmtheorem7.p1.7.7.m7.5.5.2.2.2.1.1.1.3">𝜀</ci></apply><apply id="S3.Thmtheorem7.p1.7.7.m7.5.5.2.2.2.3.cmml" xref="S3.Thmtheorem7.p1.7.7.m7.5.5.2.2.2.3"><csymbol cd="ambiguous" id="S3.Thmtheorem7.p1.7.7.m7.5.5.2.2.2.3.1.cmml" xref="S3.Thmtheorem7.p1.7.7.m7.5.5.2.2.2.3">superscript</csymbol><ci id="S3.Thmtheorem7.p1.7.7.m7.5.5.2.2.2.3.2.cmml" xref="S3.Thmtheorem7.p1.7.7.m7.5.5.2.2.2.3.2">𝜌</ci><times id="S3.Thmtheorem7.p1.7.7.m7.5.5.2.2.2.3.3.cmml" xref="S3.Thmtheorem7.p1.7.7.m7.5.5.2.2.2.3.3"></times></apply><ci id="S3.Thmtheorem7.p1.7.7.m7.3.3.cmml" xref="S3.Thmtheorem7.p1.7.7.m7.3.3">𝑣</ci></apply></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem7.p1.7.7.m7.5c">\rho^{*}_{k}(v)\in[(1+\varepsilon)^{-1}\rho^{*}(v),(1+\varepsilon)\rho^{*}(v)]</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem7.p1.7.7.m7.5d">italic_ρ start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ( italic_v ) ∈ [ ( 1 + italic_ε ) start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT italic_ρ start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ( italic_v ) , ( 1 + italic_ε ) italic_ρ start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ( italic_v ) ]</annotation></semantics></math> for <math alttext="k\in\Theta(\varepsilon^{-2}\log^{2}n)" class="ltx_Math" display="inline" id="S3.Thmtheorem7.p1.8.8.m8.1"><semantics id="S3.Thmtheorem7.p1.8.8.m8.1a"><mrow id="S3.Thmtheorem7.p1.8.8.m8.1.1" xref="S3.Thmtheorem7.p1.8.8.m8.1.1.cmml"><mi id="S3.Thmtheorem7.p1.8.8.m8.1.1.3" xref="S3.Thmtheorem7.p1.8.8.m8.1.1.3.cmml">k</mi><mo id="S3.Thmtheorem7.p1.8.8.m8.1.1.2" xref="S3.Thmtheorem7.p1.8.8.m8.1.1.2.cmml">∈</mo><mrow id="S3.Thmtheorem7.p1.8.8.m8.1.1.1" xref="S3.Thmtheorem7.p1.8.8.m8.1.1.1.cmml"><mi id="S3.Thmtheorem7.p1.8.8.m8.1.1.1.3" mathvariant="normal" xref="S3.Thmtheorem7.p1.8.8.m8.1.1.1.3.cmml">Θ</mi><mo id="S3.Thmtheorem7.p1.8.8.m8.1.1.1.2" 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xref="S3.Thmtheorem7.p1.8.8.m8.1.1.1.1.1.1.1.cmml">⁢</mo><mrow id="S3.Thmtheorem7.p1.8.8.m8.1.1.1.1.1.1.3" xref="S3.Thmtheorem7.p1.8.8.m8.1.1.1.1.1.1.3.cmml"><msup id="S3.Thmtheorem7.p1.8.8.m8.1.1.1.1.1.1.3.1" xref="S3.Thmtheorem7.p1.8.8.m8.1.1.1.1.1.1.3.1.cmml"><mi id="S3.Thmtheorem7.p1.8.8.m8.1.1.1.1.1.1.3.1.2" xref="S3.Thmtheorem7.p1.8.8.m8.1.1.1.1.1.1.3.1.2.cmml">log</mi><mn id="S3.Thmtheorem7.p1.8.8.m8.1.1.1.1.1.1.3.1.3" xref="S3.Thmtheorem7.p1.8.8.m8.1.1.1.1.1.1.3.1.3.cmml">2</mn></msup><mo id="S3.Thmtheorem7.p1.8.8.m8.1.1.1.1.1.1.3a" lspace="0.167em" xref="S3.Thmtheorem7.p1.8.8.m8.1.1.1.1.1.1.3.cmml">⁡</mo><mi id="S3.Thmtheorem7.p1.8.8.m8.1.1.1.1.1.1.3.2" xref="S3.Thmtheorem7.p1.8.8.m8.1.1.1.1.1.1.3.2.cmml">n</mi></mrow></mrow><mo id="S3.Thmtheorem7.p1.8.8.m8.1.1.1.1.1.3" stretchy="false" xref="S3.Thmtheorem7.p1.8.8.m8.1.1.1.1.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem7.p1.8.8.m8.1b"><apply id="S3.Thmtheorem7.p1.8.8.m8.1.1.cmml" xref="S3.Thmtheorem7.p1.8.8.m8.1.1"><in id="S3.Thmtheorem7.p1.8.8.m8.1.1.2.cmml" xref="S3.Thmtheorem7.p1.8.8.m8.1.1.2"></in><ci id="S3.Thmtheorem7.p1.8.8.m8.1.1.3.cmml" xref="S3.Thmtheorem7.p1.8.8.m8.1.1.3">𝑘</ci><apply id="S3.Thmtheorem7.p1.8.8.m8.1.1.1.cmml" xref="S3.Thmtheorem7.p1.8.8.m8.1.1.1"><times id="S3.Thmtheorem7.p1.8.8.m8.1.1.1.2.cmml" xref="S3.Thmtheorem7.p1.8.8.m8.1.1.1.2"></times><ci id="S3.Thmtheorem7.p1.8.8.m8.1.1.1.3.cmml" xref="S3.Thmtheorem7.p1.8.8.m8.1.1.1.3">Θ</ci><apply id="S3.Thmtheorem7.p1.8.8.m8.1.1.1.1.1.1.cmml" xref="S3.Thmtheorem7.p1.8.8.m8.1.1.1.1.1"><times id="S3.Thmtheorem7.p1.8.8.m8.1.1.1.1.1.1.1.cmml" xref="S3.Thmtheorem7.p1.8.8.m8.1.1.1.1.1.1.1"></times><apply id="S3.Thmtheorem7.p1.8.8.m8.1.1.1.1.1.1.2.cmml" xref="S3.Thmtheorem7.p1.8.8.m8.1.1.1.1.1.1.2"><csymbol cd="ambiguous" id="S3.Thmtheorem7.p1.8.8.m8.1.1.1.1.1.1.2.1.cmml" xref="S3.Thmtheorem7.p1.8.8.m8.1.1.1.1.1.1.2">superscript</csymbol><ci id="S3.Thmtheorem7.p1.8.8.m8.1.1.1.1.1.1.2.2.cmml" xref="S3.Thmtheorem7.p1.8.8.m8.1.1.1.1.1.1.2.2">𝜀</ci><apply id="S3.Thmtheorem7.p1.8.8.m8.1.1.1.1.1.1.2.3.cmml" xref="S3.Thmtheorem7.p1.8.8.m8.1.1.1.1.1.1.2.3"><minus id="S3.Thmtheorem7.p1.8.8.m8.1.1.1.1.1.1.2.3.1.cmml" xref="S3.Thmtheorem7.p1.8.8.m8.1.1.1.1.1.1.2.3"></minus><cn id="S3.Thmtheorem7.p1.8.8.m8.1.1.1.1.1.1.2.3.2.cmml" type="integer" xref="S3.Thmtheorem7.p1.8.8.m8.1.1.1.1.1.1.2.3.2">2</cn></apply></apply><apply id="S3.Thmtheorem7.p1.8.8.m8.1.1.1.1.1.1.3.cmml" xref="S3.Thmtheorem7.p1.8.8.m8.1.1.1.1.1.1.3"><apply id="S3.Thmtheorem7.p1.8.8.m8.1.1.1.1.1.1.3.1.cmml" xref="S3.Thmtheorem7.p1.8.8.m8.1.1.1.1.1.1.3.1"><csymbol cd="ambiguous" id="S3.Thmtheorem7.p1.8.8.m8.1.1.1.1.1.1.3.1.1.cmml" xref="S3.Thmtheorem7.p1.8.8.m8.1.1.1.1.1.1.3.1">superscript</csymbol><log id="S3.Thmtheorem7.p1.8.8.m8.1.1.1.1.1.1.3.1.2.cmml" xref="S3.Thmtheorem7.p1.8.8.m8.1.1.1.1.1.1.3.1.2"></log><cn id="S3.Thmtheorem7.p1.8.8.m8.1.1.1.1.1.1.3.1.3.cmml" type="integer" xref="S3.Thmtheorem7.p1.8.8.m8.1.1.1.1.1.1.3.1.3">2</cn></apply><ci id="S3.Thmtheorem7.p1.8.8.m8.1.1.1.1.1.1.3.2.cmml" xref="S3.Thmtheorem7.p1.8.8.m8.1.1.1.1.1.1.3.2">𝑛</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem7.p1.8.8.m8.1c">k\in\Theta(\varepsilon^{-2}\log^{2}n)</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem7.p1.8.8.m8.1d">italic_k ∈ roman_Θ ( italic_ε start_POSTSUPERSCRIPT - 2 end_POSTSUPERSCRIPT roman_log start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_n )</annotation></semantics></math>.</span></p> </div> </div> <div class="ltx_para ltx_noindent" id="S3.SS0.SSS3.p2"> <p class="ltx_p" id="S3.SS0.SSS3.p2.5">This immediately implies a trivial algorithm for problem <a class="ltx_ref" href="https://arxiv.org/html/2411.12694v2#S2.Thmtheorem7" title="Problem 2.7. ‣ 2.2 Approximate densest subgraph in LOCAL and CONGEST ‣ 2 Preliminaries and related work ‣ Local Density and its Distributed Approximation"><span class="ltx_text ltx_ref_tag">2.7</span></a> in LOCAL (where each vertex <math alttext="v" class="ltx_Math" display="inline" id="S3.SS0.SSS3.p2.1.m1.1"><semantics id="S3.SS0.SSS3.p2.1.m1.1a"><mi id="S3.SS0.SSS3.p2.1.m1.1.1" xref="S3.SS0.SSS3.p2.1.m1.1.1.cmml">v</mi><annotation-xml encoding="MathML-Content" id="S3.SS0.SSS3.p2.1.m1.1b"><ci id="S3.SS0.SSS3.p2.1.m1.1.1.cmml" xref="S3.SS0.SSS3.p2.1.m1.1.1">𝑣</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS0.SSS3.p2.1.m1.1c">v</annotation><annotation encoding="application/x-llamapun" id="S3.SS0.SSS3.p2.1.m1.1d">italic_v</annotation></semantics></math> collects its <math alttext="k" class="ltx_Math" display="inline" id="S3.SS0.SSS3.p2.2.m2.1"><semantics id="S3.SS0.SSS3.p2.2.m2.1a"><mi id="S3.SS0.SSS3.p2.2.m2.1.1" xref="S3.SS0.SSS3.p2.2.m2.1.1.cmml">k</mi><annotation-xml encoding="MathML-Content" id="S3.SS0.SSS3.p2.2.m2.1b"><ci id="S3.SS0.SSS3.p2.2.m2.1.1.cmml" xref="S3.SS0.SSS3.p2.2.m2.1.1">𝑘</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS0.SSS3.p2.2.m2.1c">k</annotation><annotation encoding="application/x-llamapun" id="S3.SS0.SSS3.p2.2.m2.1d">italic_k</annotation></semantics></math>-hop neighbourhood <math alttext="H_{k}(v)" class="ltx_Math" display="inline" id="S3.SS0.SSS3.p2.3.m3.1"><semantics id="S3.SS0.SSS3.p2.3.m3.1a"><mrow id="S3.SS0.SSS3.p2.3.m3.1.2" xref="S3.SS0.SSS3.p2.3.m3.1.2.cmml"><msub id="S3.SS0.SSS3.p2.3.m3.1.2.2" xref="S3.SS0.SSS3.p2.3.m3.1.2.2.cmml"><mi id="S3.SS0.SSS3.p2.3.m3.1.2.2.2" xref="S3.SS0.SSS3.p2.3.m3.1.2.2.2.cmml">H</mi><mi id="S3.SS0.SSS3.p2.3.m3.1.2.2.3" xref="S3.SS0.SSS3.p2.3.m3.1.2.2.3.cmml">k</mi></msub><mo id="S3.SS0.SSS3.p2.3.m3.1.2.1" xref="S3.SS0.SSS3.p2.3.m3.1.2.1.cmml">⁢</mo><mrow id="S3.SS0.SSS3.p2.3.m3.1.2.3.2" xref="S3.SS0.SSS3.p2.3.m3.1.2.cmml"><mo id="S3.SS0.SSS3.p2.3.m3.1.2.3.2.1" stretchy="false" xref="S3.SS0.SSS3.p2.3.m3.1.2.cmml">(</mo><mi id="S3.SS0.SSS3.p2.3.m3.1.1" xref="S3.SS0.SSS3.p2.3.m3.1.1.cmml">v</mi><mo id="S3.SS0.SSS3.p2.3.m3.1.2.3.2.2" stretchy="false" xref="S3.SS0.SSS3.p2.3.m3.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS0.SSS3.p2.3.m3.1b"><apply id="S3.SS0.SSS3.p2.3.m3.1.2.cmml" xref="S3.SS0.SSS3.p2.3.m3.1.2"><times id="S3.SS0.SSS3.p2.3.m3.1.2.1.cmml" xref="S3.SS0.SSS3.p2.3.m3.1.2.1"></times><apply id="S3.SS0.SSS3.p2.3.m3.1.2.2.cmml" xref="S3.SS0.SSS3.p2.3.m3.1.2.2"><csymbol cd="ambiguous" id="S3.SS0.SSS3.p2.3.m3.1.2.2.1.cmml" xref="S3.SS0.SSS3.p2.3.m3.1.2.2">subscript</csymbol><ci id="S3.SS0.SSS3.p2.3.m3.1.2.2.2.cmml" xref="S3.SS0.SSS3.p2.3.m3.1.2.2.2">𝐻</ci><ci id="S3.SS0.SSS3.p2.3.m3.1.2.2.3.cmml" xref="S3.SS0.SSS3.p2.3.m3.1.2.2.3">𝑘</ci></apply><ci id="S3.SS0.SSS3.p2.3.m3.1.1.cmml" xref="S3.SS0.SSS3.p2.3.m3.1.1">𝑣</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS0.SSS3.p2.3.m3.1c">H_{k}(v)</annotation><annotation encoding="application/x-llamapun" id="S3.SS0.SSS3.p2.3.m3.1d">italic_H start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ( italic_v )</annotation></semantics></math> for <math alttext="k\in\Theta(\varepsilon^{-2}\log^{2}n)" class="ltx_Math" display="inline" id="S3.SS0.SSS3.p2.4.m4.1"><semantics id="S3.SS0.SSS3.p2.4.m4.1a"><mrow id="S3.SS0.SSS3.p2.4.m4.1.1" xref="S3.SS0.SSS3.p2.4.m4.1.1.cmml"><mi id="S3.SS0.SSS3.p2.4.m4.1.1.3" xref="S3.SS0.SSS3.p2.4.m4.1.1.3.cmml">k</mi><mo id="S3.SS0.SSS3.p2.4.m4.1.1.2" xref="S3.SS0.SSS3.p2.4.m4.1.1.2.cmml">∈</mo><mrow id="S3.SS0.SSS3.p2.4.m4.1.1.1" xref="S3.SS0.SSS3.p2.4.m4.1.1.1.cmml"><mi id="S3.SS0.SSS3.p2.4.m4.1.1.1.3" mathvariant="normal" xref="S3.SS0.SSS3.p2.4.m4.1.1.1.3.cmml">Θ</mi><mo id="S3.SS0.SSS3.p2.4.m4.1.1.1.2" xref="S3.SS0.SSS3.p2.4.m4.1.1.1.2.cmml">⁢</mo><mrow id="S3.SS0.SSS3.p2.4.m4.1.1.1.1.1" xref="S3.SS0.SSS3.p2.4.m4.1.1.1.1.1.1.cmml"><mo id="S3.SS0.SSS3.p2.4.m4.1.1.1.1.1.2" stretchy="false" xref="S3.SS0.SSS3.p2.4.m4.1.1.1.1.1.1.cmml">(</mo><mrow id="S3.SS0.SSS3.p2.4.m4.1.1.1.1.1.1" xref="S3.SS0.SSS3.p2.4.m4.1.1.1.1.1.1.cmml"><msup id="S3.SS0.SSS3.p2.4.m4.1.1.1.1.1.1.2" xref="S3.SS0.SSS3.p2.4.m4.1.1.1.1.1.1.2.cmml"><mi id="S3.SS0.SSS3.p2.4.m4.1.1.1.1.1.1.2.2" xref="S3.SS0.SSS3.p2.4.m4.1.1.1.1.1.1.2.2.cmml">ε</mi><mrow id="S3.SS0.SSS3.p2.4.m4.1.1.1.1.1.1.2.3" xref="S3.SS0.SSS3.p2.4.m4.1.1.1.1.1.1.2.3.cmml"><mo id="S3.SS0.SSS3.p2.4.m4.1.1.1.1.1.1.2.3a" xref="S3.SS0.SSS3.p2.4.m4.1.1.1.1.1.1.2.3.cmml">−</mo><mn id="S3.SS0.SSS3.p2.4.m4.1.1.1.1.1.1.2.3.2" xref="S3.SS0.SSS3.p2.4.m4.1.1.1.1.1.1.2.3.2.cmml">2</mn></mrow></msup><mo id="S3.SS0.SSS3.p2.4.m4.1.1.1.1.1.1.1" lspace="0.167em" xref="S3.SS0.SSS3.p2.4.m4.1.1.1.1.1.1.1.cmml">⁢</mo><mrow id="S3.SS0.SSS3.p2.4.m4.1.1.1.1.1.1.3" xref="S3.SS0.SSS3.p2.4.m4.1.1.1.1.1.1.3.cmml"><msup id="S3.SS0.SSS3.p2.4.m4.1.1.1.1.1.1.3.1" xref="S3.SS0.SSS3.p2.4.m4.1.1.1.1.1.1.3.1.cmml"><mi id="S3.SS0.SSS3.p2.4.m4.1.1.1.1.1.1.3.1.2" xref="S3.SS0.SSS3.p2.4.m4.1.1.1.1.1.1.3.1.2.cmml">log</mi><mn id="S3.SS0.SSS3.p2.4.m4.1.1.1.1.1.1.3.1.3" xref="S3.SS0.SSS3.p2.4.m4.1.1.1.1.1.1.3.1.3.cmml">2</mn></msup><mo id="S3.SS0.SSS3.p2.4.m4.1.1.1.1.1.1.3a" lspace="0.167em" xref="S3.SS0.SSS3.p2.4.m4.1.1.1.1.1.1.3.cmml">⁡</mo><mi id="S3.SS0.SSS3.p2.4.m4.1.1.1.1.1.1.3.2" xref="S3.SS0.SSS3.p2.4.m4.1.1.1.1.1.1.3.2.cmml">n</mi></mrow></mrow><mo id="S3.SS0.SSS3.p2.4.m4.1.1.1.1.1.3" stretchy="false" xref="S3.SS0.SSS3.p2.4.m4.1.1.1.1.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS0.SSS3.p2.4.m4.1b"><apply id="S3.SS0.SSS3.p2.4.m4.1.1.cmml" xref="S3.SS0.SSS3.p2.4.m4.1.1"><in id="S3.SS0.SSS3.p2.4.m4.1.1.2.cmml" xref="S3.SS0.SSS3.p2.4.m4.1.1.2"></in><ci id="S3.SS0.SSS3.p2.4.m4.1.1.3.cmml" xref="S3.SS0.SSS3.p2.4.m4.1.1.3">𝑘</ci><apply id="S3.SS0.SSS3.p2.4.m4.1.1.1.cmml" xref="S3.SS0.SSS3.p2.4.m4.1.1.1"><times id="S3.SS0.SSS3.p2.4.m4.1.1.1.2.cmml" xref="S3.SS0.SSS3.p2.4.m4.1.1.1.2"></times><ci id="S3.SS0.SSS3.p2.4.m4.1.1.1.3.cmml" xref="S3.SS0.SSS3.p2.4.m4.1.1.1.3">Θ</ci><apply id="S3.SS0.SSS3.p2.4.m4.1.1.1.1.1.1.cmml" xref="S3.SS0.SSS3.p2.4.m4.1.1.1.1.1"><times id="S3.SS0.SSS3.p2.4.m4.1.1.1.1.1.1.1.cmml" xref="S3.SS0.SSS3.p2.4.m4.1.1.1.1.1.1.1"></times><apply id="S3.SS0.SSS3.p2.4.m4.1.1.1.1.1.1.2.cmml" xref="S3.SS0.SSS3.p2.4.m4.1.1.1.1.1.1.2"><csymbol cd="ambiguous" id="S3.SS0.SSS3.p2.4.m4.1.1.1.1.1.1.2.1.cmml" xref="S3.SS0.SSS3.p2.4.m4.1.1.1.1.1.1.2">superscript</csymbol><ci id="S3.SS0.SSS3.p2.4.m4.1.1.1.1.1.1.2.2.cmml" xref="S3.SS0.SSS3.p2.4.m4.1.1.1.1.1.1.2.2">𝜀</ci><apply id="S3.SS0.SSS3.p2.4.m4.1.1.1.1.1.1.2.3.cmml" xref="S3.SS0.SSS3.p2.4.m4.1.1.1.1.1.1.2.3"><minus id="S3.SS0.SSS3.p2.4.m4.1.1.1.1.1.1.2.3.1.cmml" xref="S3.SS0.SSS3.p2.4.m4.1.1.1.1.1.1.2.3"></minus><cn id="S3.SS0.SSS3.p2.4.m4.1.1.1.1.1.1.2.3.2.cmml" type="integer" xref="S3.SS0.SSS3.p2.4.m4.1.1.1.1.1.1.2.3.2">2</cn></apply></apply><apply id="S3.SS0.SSS3.p2.4.m4.1.1.1.1.1.1.3.cmml" xref="S3.SS0.SSS3.p2.4.m4.1.1.1.1.1.1.3"><apply id="S3.SS0.SSS3.p2.4.m4.1.1.1.1.1.1.3.1.cmml" xref="S3.SS0.SSS3.p2.4.m4.1.1.1.1.1.1.3.1"><csymbol cd="ambiguous" id="S3.SS0.SSS3.p2.4.m4.1.1.1.1.1.1.3.1.1.cmml" xref="S3.SS0.SSS3.p2.4.m4.1.1.1.1.1.1.3.1">superscript</csymbol><log id="S3.SS0.SSS3.p2.4.m4.1.1.1.1.1.1.3.1.2.cmml" xref="S3.SS0.SSS3.p2.4.m4.1.1.1.1.1.1.3.1.2"></log><cn id="S3.SS0.SSS3.p2.4.m4.1.1.1.1.1.1.3.1.3.cmml" type="integer" xref="S3.SS0.SSS3.p2.4.m4.1.1.1.1.1.1.3.1.3">2</cn></apply><ci id="S3.SS0.SSS3.p2.4.m4.1.1.1.1.1.1.3.2.cmml" xref="S3.SS0.SSS3.p2.4.m4.1.1.1.1.1.1.3.2">𝑛</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS0.SSS3.p2.4.m4.1c">k\in\Theta(\varepsilon^{-2}\log^{2}n)</annotation><annotation encoding="application/x-llamapun" id="S3.SS0.SSS3.p2.4.m4.1d">italic_k ∈ roman_Θ ( italic_ε start_POSTSUPERSCRIPT - 2 end_POSTSUPERSCRIPT roman_log start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_n )</annotation></semantics></math> and then solves FO on <math alttext="H_{k}(v)" class="ltx_Math" display="inline" id="S3.SS0.SSS3.p2.5.m5.1"><semantics id="S3.SS0.SSS3.p2.5.m5.1a"><mrow id="S3.SS0.SSS3.p2.5.m5.1.2" xref="S3.SS0.SSS3.p2.5.m5.1.2.cmml"><msub id="S3.SS0.SSS3.p2.5.m5.1.2.2" xref="S3.SS0.SSS3.p2.5.m5.1.2.2.cmml"><mi id="S3.SS0.SSS3.p2.5.m5.1.2.2.2" xref="S3.SS0.SSS3.p2.5.m5.1.2.2.2.cmml">H</mi><mi id="S3.SS0.SSS3.p2.5.m5.1.2.2.3" xref="S3.SS0.SSS3.p2.5.m5.1.2.2.3.cmml">k</mi></msub><mo id="S3.SS0.SSS3.p2.5.m5.1.2.1" xref="S3.SS0.SSS3.p2.5.m5.1.2.1.cmml">⁢</mo><mrow id="S3.SS0.SSS3.p2.5.m5.1.2.3.2" xref="S3.SS0.SSS3.p2.5.m5.1.2.cmml"><mo id="S3.SS0.SSS3.p2.5.m5.1.2.3.2.1" stretchy="false" xref="S3.SS0.SSS3.p2.5.m5.1.2.cmml">(</mo><mi id="S3.SS0.SSS3.p2.5.m5.1.1" xref="S3.SS0.SSS3.p2.5.m5.1.1.cmml">v</mi><mo id="S3.SS0.SSS3.p2.5.m5.1.2.3.2.2" stretchy="false" xref="S3.SS0.SSS3.p2.5.m5.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS0.SSS3.p2.5.m5.1b"><apply id="S3.SS0.SSS3.p2.5.m5.1.2.cmml" xref="S3.SS0.SSS3.p2.5.m5.1.2"><times id="S3.SS0.SSS3.p2.5.m5.1.2.1.cmml" xref="S3.SS0.SSS3.p2.5.m5.1.2.1"></times><apply id="S3.SS0.SSS3.p2.5.m5.1.2.2.cmml" xref="S3.SS0.SSS3.p2.5.m5.1.2.2"><csymbol cd="ambiguous" id="S3.SS0.SSS3.p2.5.m5.1.2.2.1.cmml" xref="S3.SS0.SSS3.p2.5.m5.1.2.2">subscript</csymbol><ci id="S3.SS0.SSS3.p2.5.m5.1.2.2.2.cmml" xref="S3.SS0.SSS3.p2.5.m5.1.2.2.2">𝐻</ci><ci id="S3.SS0.SSS3.p2.5.m5.1.2.2.3.cmml" xref="S3.SS0.SSS3.p2.5.m5.1.2.2.3">𝑘</ci></apply><ci id="S3.SS0.SSS3.p2.5.m5.1.1.cmml" xref="S3.SS0.SSS3.p2.5.m5.1.1">𝑣</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS0.SSS3.p2.5.m5.1c">H_{k}(v)</annotation><annotation encoding="application/x-llamapun" id="S3.SS0.SSS3.p2.5.m5.1d">italic_H start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ( italic_v )</annotation></semantics></math>):</p> </div> <div class="ltx_theorem ltx_theorem_corollary" 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">Corollary 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.p1.5"><span class="ltx_text ltx_font_italic" id="S3.Thmtheorem8.p1.5.5">There exists an algorithm in LOCAL that given a unit graph <math alttext="G" class="ltx_Math" display="inline" id="S3.Thmtheorem8.p1.1.1.m1.1"><semantics id="S3.Thmtheorem8.p1.1.1.m1.1a"><mi id="S3.Thmtheorem8.p1.1.1.m1.1.1" xref="S3.Thmtheorem8.p1.1.1.m1.1.1.cmml">G</mi><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem8.p1.1.1.m1.1b"><ci id="S3.Thmtheorem8.p1.1.1.m1.1.1.cmml" xref="S3.Thmtheorem8.p1.1.1.m1.1.1">𝐺</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem8.p1.1.1.m1.1c">G</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem8.p1.1.1.m1.1d">italic_G</annotation></semantics></math> and <math alttext="\varepsilon>0" class="ltx_Math" display="inline" id="S3.Thmtheorem8.p1.2.2.m2.1"><semantics id="S3.Thmtheorem8.p1.2.2.m2.1a"><mrow id="S3.Thmtheorem8.p1.2.2.m2.1.1" xref="S3.Thmtheorem8.p1.2.2.m2.1.1.cmml"><mi id="S3.Thmtheorem8.p1.2.2.m2.1.1.2" xref="S3.Thmtheorem8.p1.2.2.m2.1.1.2.cmml">ε</mi><mo id="S3.Thmtheorem8.p1.2.2.m2.1.1.1" xref="S3.Thmtheorem8.p1.2.2.m2.1.1.1.cmml">></mo><mn id="S3.Thmtheorem8.p1.2.2.m2.1.1.3" xref="S3.Thmtheorem8.p1.2.2.m2.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem8.p1.2.2.m2.1b"><apply id="S3.Thmtheorem8.p1.2.2.m2.1.1.cmml" xref="S3.Thmtheorem8.p1.2.2.m2.1.1"><gt id="S3.Thmtheorem8.p1.2.2.m2.1.1.1.cmml" xref="S3.Thmtheorem8.p1.2.2.m2.1.1.1"></gt><ci id="S3.Thmtheorem8.p1.2.2.m2.1.1.2.cmml" xref="S3.Thmtheorem8.p1.2.2.m2.1.1.2">𝜀</ci><cn id="S3.Thmtheorem8.p1.2.2.m2.1.1.3.cmml" type="integer" xref="S3.Thmtheorem8.p1.2.2.m2.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem8.p1.2.2.m2.1c">\varepsilon>0</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem8.p1.2.2.m2.1d">italic_ε > 0</annotation></semantics></math> computes in <math alttext="O(\varepsilon^{-2}\log^{2}n)" class="ltx_Math" display="inline" id="S3.Thmtheorem8.p1.3.3.m3.1"><semantics id="S3.Thmtheorem8.p1.3.3.m3.1a"><mrow id="S3.Thmtheorem8.p1.3.3.m3.1.1" xref="S3.Thmtheorem8.p1.3.3.m3.1.1.cmml"><mi id="S3.Thmtheorem8.p1.3.3.m3.1.1.3" xref="S3.Thmtheorem8.p1.3.3.m3.1.1.3.cmml">O</mi><mo id="S3.Thmtheorem8.p1.3.3.m3.1.1.2" xref="S3.Thmtheorem8.p1.3.3.m3.1.1.2.cmml">⁢</mo><mrow id="S3.Thmtheorem8.p1.3.3.m3.1.1.1.1" xref="S3.Thmtheorem8.p1.3.3.m3.1.1.1.1.1.cmml"><mo id="S3.Thmtheorem8.p1.3.3.m3.1.1.1.1.2" stretchy="false" xref="S3.Thmtheorem8.p1.3.3.m3.1.1.1.1.1.cmml">(</mo><mrow id="S3.Thmtheorem8.p1.3.3.m3.1.1.1.1.1" xref="S3.Thmtheorem8.p1.3.3.m3.1.1.1.1.1.cmml"><msup id="S3.Thmtheorem8.p1.3.3.m3.1.1.1.1.1.2" xref="S3.Thmtheorem8.p1.3.3.m3.1.1.1.1.1.2.cmml"><mi id="S3.Thmtheorem8.p1.3.3.m3.1.1.1.1.1.2.2" xref="S3.Thmtheorem8.p1.3.3.m3.1.1.1.1.1.2.2.cmml">ε</mi><mrow id="S3.Thmtheorem8.p1.3.3.m3.1.1.1.1.1.2.3" xref="S3.Thmtheorem8.p1.3.3.m3.1.1.1.1.1.2.3.cmml"><mo id="S3.Thmtheorem8.p1.3.3.m3.1.1.1.1.1.2.3a" xref="S3.Thmtheorem8.p1.3.3.m3.1.1.1.1.1.2.3.cmml">−</mo><mn id="S3.Thmtheorem8.p1.3.3.m3.1.1.1.1.1.2.3.2" xref="S3.Thmtheorem8.p1.3.3.m3.1.1.1.1.1.2.3.2.cmml">2</mn></mrow></msup><mo id="S3.Thmtheorem8.p1.3.3.m3.1.1.1.1.1.1" lspace="0.167em" xref="S3.Thmtheorem8.p1.3.3.m3.1.1.1.1.1.1.cmml">⁢</mo><mrow id="S3.Thmtheorem8.p1.3.3.m3.1.1.1.1.1.3" xref="S3.Thmtheorem8.p1.3.3.m3.1.1.1.1.1.3.cmml"><msup id="S3.Thmtheorem8.p1.3.3.m3.1.1.1.1.1.3.1" xref="S3.Thmtheorem8.p1.3.3.m3.1.1.1.1.1.3.1.cmml"><mi id="S3.Thmtheorem8.p1.3.3.m3.1.1.1.1.1.3.1.2" xref="S3.Thmtheorem8.p1.3.3.m3.1.1.1.1.1.3.1.2.cmml">log</mi><mn id="S3.Thmtheorem8.p1.3.3.m3.1.1.1.1.1.3.1.3" xref="S3.Thmtheorem8.p1.3.3.m3.1.1.1.1.1.3.1.3.cmml">2</mn></msup><mo id="S3.Thmtheorem8.p1.3.3.m3.1.1.1.1.1.3a" lspace="0.167em" xref="S3.Thmtheorem8.p1.3.3.m3.1.1.1.1.1.3.cmml">⁡</mo><mi id="S3.Thmtheorem8.p1.3.3.m3.1.1.1.1.1.3.2" xref="S3.Thmtheorem8.p1.3.3.m3.1.1.1.1.1.3.2.cmml">n</mi></mrow></mrow><mo id="S3.Thmtheorem8.p1.3.3.m3.1.1.1.1.3" stretchy="false" xref="S3.Thmtheorem8.p1.3.3.m3.1.1.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem8.p1.3.3.m3.1b"><apply id="S3.Thmtheorem8.p1.3.3.m3.1.1.cmml" xref="S3.Thmtheorem8.p1.3.3.m3.1.1"><times id="S3.Thmtheorem8.p1.3.3.m3.1.1.2.cmml" xref="S3.Thmtheorem8.p1.3.3.m3.1.1.2"></times><ci id="S3.Thmtheorem8.p1.3.3.m3.1.1.3.cmml" xref="S3.Thmtheorem8.p1.3.3.m3.1.1.3">𝑂</ci><apply id="S3.Thmtheorem8.p1.3.3.m3.1.1.1.1.1.cmml" xref="S3.Thmtheorem8.p1.3.3.m3.1.1.1.1"><times id="S3.Thmtheorem8.p1.3.3.m3.1.1.1.1.1.1.cmml" xref="S3.Thmtheorem8.p1.3.3.m3.1.1.1.1.1.1"></times><apply id="S3.Thmtheorem8.p1.3.3.m3.1.1.1.1.1.2.cmml" xref="S3.Thmtheorem8.p1.3.3.m3.1.1.1.1.1.2"><csymbol cd="ambiguous" id="S3.Thmtheorem8.p1.3.3.m3.1.1.1.1.1.2.1.cmml" xref="S3.Thmtheorem8.p1.3.3.m3.1.1.1.1.1.2">superscript</csymbol><ci id="S3.Thmtheorem8.p1.3.3.m3.1.1.1.1.1.2.2.cmml" xref="S3.Thmtheorem8.p1.3.3.m3.1.1.1.1.1.2.2">𝜀</ci><apply id="S3.Thmtheorem8.p1.3.3.m3.1.1.1.1.1.2.3.cmml" xref="S3.Thmtheorem8.p1.3.3.m3.1.1.1.1.1.2.3"><minus id="S3.Thmtheorem8.p1.3.3.m3.1.1.1.1.1.2.3.1.cmml" xref="S3.Thmtheorem8.p1.3.3.m3.1.1.1.1.1.2.3"></minus><cn id="S3.Thmtheorem8.p1.3.3.m3.1.1.1.1.1.2.3.2.cmml" type="integer" xref="S3.Thmtheorem8.p1.3.3.m3.1.1.1.1.1.2.3.2">2</cn></apply></apply><apply id="S3.Thmtheorem8.p1.3.3.m3.1.1.1.1.1.3.cmml" xref="S3.Thmtheorem8.p1.3.3.m3.1.1.1.1.1.3"><apply id="S3.Thmtheorem8.p1.3.3.m3.1.1.1.1.1.3.1.cmml" xref="S3.Thmtheorem8.p1.3.3.m3.1.1.1.1.1.3.1"><csymbol cd="ambiguous" id="S3.Thmtheorem8.p1.3.3.m3.1.1.1.1.1.3.1.1.cmml" xref="S3.Thmtheorem8.p1.3.3.m3.1.1.1.1.1.3.1">superscript</csymbol><log id="S3.Thmtheorem8.p1.3.3.m3.1.1.1.1.1.3.1.2.cmml" xref="S3.Thmtheorem8.p1.3.3.m3.1.1.1.1.1.3.1.2"></log><cn id="S3.Thmtheorem8.p1.3.3.m3.1.1.1.1.1.3.1.3.cmml" type="integer" xref="S3.Thmtheorem8.p1.3.3.m3.1.1.1.1.1.3.1.3">2</cn></apply><ci id="S3.Thmtheorem8.p1.3.3.m3.1.1.1.1.1.3.2.cmml" xref="S3.Thmtheorem8.p1.3.3.m3.1.1.1.1.1.3.2">𝑛</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem8.p1.3.3.m3.1c">O(\varepsilon^{-2}\log^{2}n)</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem8.p1.3.3.m3.1d">italic_O ( italic_ε start_POSTSUPERSCRIPT - 2 end_POSTSUPERSCRIPT roman_log start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_n )</annotation></semantics></math> rounds for all <math alttext="v\in V" class="ltx_Math" display="inline" id="S3.Thmtheorem8.p1.4.4.m4.1"><semantics id="S3.Thmtheorem8.p1.4.4.m4.1a"><mrow id="S3.Thmtheorem8.p1.4.4.m4.1.1" xref="S3.Thmtheorem8.p1.4.4.m4.1.1.cmml"><mi id="S3.Thmtheorem8.p1.4.4.m4.1.1.2" xref="S3.Thmtheorem8.p1.4.4.m4.1.1.2.cmml">v</mi><mo id="S3.Thmtheorem8.p1.4.4.m4.1.1.1" xref="S3.Thmtheorem8.p1.4.4.m4.1.1.1.cmml">∈</mo><mi id="S3.Thmtheorem8.p1.4.4.m4.1.1.3" xref="S3.Thmtheorem8.p1.4.4.m4.1.1.3.cmml">V</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem8.p1.4.4.m4.1b"><apply id="S3.Thmtheorem8.p1.4.4.m4.1.1.cmml" xref="S3.Thmtheorem8.p1.4.4.m4.1.1"><in id="S3.Thmtheorem8.p1.4.4.m4.1.1.1.cmml" xref="S3.Thmtheorem8.p1.4.4.m4.1.1.1"></in><ci id="S3.Thmtheorem8.p1.4.4.m4.1.1.2.cmml" xref="S3.Thmtheorem8.p1.4.4.m4.1.1.2">𝑣</ci><ci id="S3.Thmtheorem8.p1.4.4.m4.1.1.3.cmml" xref="S3.Thmtheorem8.p1.4.4.m4.1.1.3">𝑉</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem8.p1.4.4.m4.1c">v\in V</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem8.p1.4.4.m4.1d">italic_v ∈ italic_V</annotation></semantics></math> a value <math alttext="\rho_{v}\in[(1+\varepsilon)^{-1}\rho^{*}(v),(1+\varepsilon)\rho^{*}(v)]" class="ltx_Math" display="inline" id="S3.Thmtheorem8.p1.5.5.m5.4"><semantics id="S3.Thmtheorem8.p1.5.5.m5.4a"><mrow id="S3.Thmtheorem8.p1.5.5.m5.4.4" xref="S3.Thmtheorem8.p1.5.5.m5.4.4.cmml"><msub id="S3.Thmtheorem8.p1.5.5.m5.4.4.4" xref="S3.Thmtheorem8.p1.5.5.m5.4.4.4.cmml"><mi id="S3.Thmtheorem8.p1.5.5.m5.4.4.4.2" xref="S3.Thmtheorem8.p1.5.5.m5.4.4.4.2.cmml">ρ</mi><mi id="S3.Thmtheorem8.p1.5.5.m5.4.4.4.3" xref="S3.Thmtheorem8.p1.5.5.m5.4.4.4.3.cmml">v</mi></msub><mo id="S3.Thmtheorem8.p1.5.5.m5.4.4.3" xref="S3.Thmtheorem8.p1.5.5.m5.4.4.3.cmml">∈</mo><mrow id="S3.Thmtheorem8.p1.5.5.m5.4.4.2.2" xref="S3.Thmtheorem8.p1.5.5.m5.4.4.2.3.cmml"><mo id="S3.Thmtheorem8.p1.5.5.m5.4.4.2.2.3" stretchy="false" xref="S3.Thmtheorem8.p1.5.5.m5.4.4.2.3.cmml">[</mo><mrow id="S3.Thmtheorem8.p1.5.5.m5.3.3.1.1.1" xref="S3.Thmtheorem8.p1.5.5.m5.3.3.1.1.1.cmml"><msup id="S3.Thmtheorem8.p1.5.5.m5.3.3.1.1.1.1" xref="S3.Thmtheorem8.p1.5.5.m5.3.3.1.1.1.1.cmml"><mrow id="S3.Thmtheorem8.p1.5.5.m5.3.3.1.1.1.1.1.1" xref="S3.Thmtheorem8.p1.5.5.m5.3.3.1.1.1.1.1.1.1.cmml"><mo id="S3.Thmtheorem8.p1.5.5.m5.3.3.1.1.1.1.1.1.2" stretchy="false" xref="S3.Thmtheorem8.p1.5.5.m5.3.3.1.1.1.1.1.1.1.cmml">(</mo><mrow id="S3.Thmtheorem8.p1.5.5.m5.3.3.1.1.1.1.1.1.1" xref="S3.Thmtheorem8.p1.5.5.m5.3.3.1.1.1.1.1.1.1.cmml"><mn id="S3.Thmtheorem8.p1.5.5.m5.3.3.1.1.1.1.1.1.1.2" xref="S3.Thmtheorem8.p1.5.5.m5.3.3.1.1.1.1.1.1.1.2.cmml">1</mn><mo id="S3.Thmtheorem8.p1.5.5.m5.3.3.1.1.1.1.1.1.1.1" xref="S3.Thmtheorem8.p1.5.5.m5.3.3.1.1.1.1.1.1.1.1.cmml">+</mo><mi id="S3.Thmtheorem8.p1.5.5.m5.3.3.1.1.1.1.1.1.1.3" xref="S3.Thmtheorem8.p1.5.5.m5.3.3.1.1.1.1.1.1.1.3.cmml">ε</mi></mrow><mo id="S3.Thmtheorem8.p1.5.5.m5.3.3.1.1.1.1.1.1.3" stretchy="false" xref="S3.Thmtheorem8.p1.5.5.m5.3.3.1.1.1.1.1.1.1.cmml">)</mo></mrow><mrow id="S3.Thmtheorem8.p1.5.5.m5.3.3.1.1.1.1.3" xref="S3.Thmtheorem8.p1.5.5.m5.3.3.1.1.1.1.3.cmml"><mo id="S3.Thmtheorem8.p1.5.5.m5.3.3.1.1.1.1.3a" xref="S3.Thmtheorem8.p1.5.5.m5.3.3.1.1.1.1.3.cmml">−</mo><mn id="S3.Thmtheorem8.p1.5.5.m5.3.3.1.1.1.1.3.2" xref="S3.Thmtheorem8.p1.5.5.m5.3.3.1.1.1.1.3.2.cmml">1</mn></mrow></msup><mo id="S3.Thmtheorem8.p1.5.5.m5.3.3.1.1.1.2" xref="S3.Thmtheorem8.p1.5.5.m5.3.3.1.1.1.2.cmml">⁢</mo><msup id="S3.Thmtheorem8.p1.5.5.m5.3.3.1.1.1.3" xref="S3.Thmtheorem8.p1.5.5.m5.3.3.1.1.1.3.cmml"><mi id="S3.Thmtheorem8.p1.5.5.m5.3.3.1.1.1.3.2" xref="S3.Thmtheorem8.p1.5.5.m5.3.3.1.1.1.3.2.cmml">ρ</mi><mo id="S3.Thmtheorem8.p1.5.5.m5.3.3.1.1.1.3.3" xref="S3.Thmtheorem8.p1.5.5.m5.3.3.1.1.1.3.3.cmml">∗</mo></msup><mo id="S3.Thmtheorem8.p1.5.5.m5.3.3.1.1.1.2a" xref="S3.Thmtheorem8.p1.5.5.m5.3.3.1.1.1.2.cmml">⁢</mo><mrow id="S3.Thmtheorem8.p1.5.5.m5.3.3.1.1.1.4.2" xref="S3.Thmtheorem8.p1.5.5.m5.3.3.1.1.1.cmml"><mo id="S3.Thmtheorem8.p1.5.5.m5.3.3.1.1.1.4.2.1" stretchy="false" xref="S3.Thmtheorem8.p1.5.5.m5.3.3.1.1.1.cmml">(</mo><mi id="S3.Thmtheorem8.p1.5.5.m5.1.1" xref="S3.Thmtheorem8.p1.5.5.m5.1.1.cmml">v</mi><mo id="S3.Thmtheorem8.p1.5.5.m5.3.3.1.1.1.4.2.2" stretchy="false" xref="S3.Thmtheorem8.p1.5.5.m5.3.3.1.1.1.cmml">)</mo></mrow></mrow><mo id="S3.Thmtheorem8.p1.5.5.m5.4.4.2.2.4" xref="S3.Thmtheorem8.p1.5.5.m5.4.4.2.3.cmml">,</mo><mrow id="S3.Thmtheorem8.p1.5.5.m5.4.4.2.2.2" xref="S3.Thmtheorem8.p1.5.5.m5.4.4.2.2.2.cmml"><mrow id="S3.Thmtheorem8.p1.5.5.m5.4.4.2.2.2.1.1" xref="S3.Thmtheorem8.p1.5.5.m5.4.4.2.2.2.1.1.1.cmml"><mo id="S3.Thmtheorem8.p1.5.5.m5.4.4.2.2.2.1.1.2" stretchy="false" xref="S3.Thmtheorem8.p1.5.5.m5.4.4.2.2.2.1.1.1.cmml">(</mo><mrow id="S3.Thmtheorem8.p1.5.5.m5.4.4.2.2.2.1.1.1" xref="S3.Thmtheorem8.p1.5.5.m5.4.4.2.2.2.1.1.1.cmml"><mn id="S3.Thmtheorem8.p1.5.5.m5.4.4.2.2.2.1.1.1.2" xref="S3.Thmtheorem8.p1.5.5.m5.4.4.2.2.2.1.1.1.2.cmml">1</mn><mo id="S3.Thmtheorem8.p1.5.5.m5.4.4.2.2.2.1.1.1.1" xref="S3.Thmtheorem8.p1.5.5.m5.4.4.2.2.2.1.1.1.1.cmml">+</mo><mi id="S3.Thmtheorem8.p1.5.5.m5.4.4.2.2.2.1.1.1.3" xref="S3.Thmtheorem8.p1.5.5.m5.4.4.2.2.2.1.1.1.3.cmml">ε</mi></mrow><mo id="S3.Thmtheorem8.p1.5.5.m5.4.4.2.2.2.1.1.3" stretchy="false" xref="S3.Thmtheorem8.p1.5.5.m5.4.4.2.2.2.1.1.1.cmml">)</mo></mrow><mo id="S3.Thmtheorem8.p1.5.5.m5.4.4.2.2.2.2" xref="S3.Thmtheorem8.p1.5.5.m5.4.4.2.2.2.2.cmml">⁢</mo><msup id="S3.Thmtheorem8.p1.5.5.m5.4.4.2.2.2.3" xref="S3.Thmtheorem8.p1.5.5.m5.4.4.2.2.2.3.cmml"><mi id="S3.Thmtheorem8.p1.5.5.m5.4.4.2.2.2.3.2" xref="S3.Thmtheorem8.p1.5.5.m5.4.4.2.2.2.3.2.cmml">ρ</mi><mo id="S3.Thmtheorem8.p1.5.5.m5.4.4.2.2.2.3.3" xref="S3.Thmtheorem8.p1.5.5.m5.4.4.2.2.2.3.3.cmml">∗</mo></msup><mo id="S3.Thmtheorem8.p1.5.5.m5.4.4.2.2.2.2a" xref="S3.Thmtheorem8.p1.5.5.m5.4.4.2.2.2.2.cmml">⁢</mo><mrow id="S3.Thmtheorem8.p1.5.5.m5.4.4.2.2.2.4.2" xref="S3.Thmtheorem8.p1.5.5.m5.4.4.2.2.2.cmml"><mo id="S3.Thmtheorem8.p1.5.5.m5.4.4.2.2.2.4.2.1" stretchy="false" xref="S3.Thmtheorem8.p1.5.5.m5.4.4.2.2.2.cmml">(</mo><mi id="S3.Thmtheorem8.p1.5.5.m5.2.2" xref="S3.Thmtheorem8.p1.5.5.m5.2.2.cmml">v</mi><mo id="S3.Thmtheorem8.p1.5.5.m5.4.4.2.2.2.4.2.2" stretchy="false" xref="S3.Thmtheorem8.p1.5.5.m5.4.4.2.2.2.cmml">)</mo></mrow></mrow><mo id="S3.Thmtheorem8.p1.5.5.m5.4.4.2.2.5" stretchy="false" xref="S3.Thmtheorem8.p1.5.5.m5.4.4.2.3.cmml">]</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem8.p1.5.5.m5.4b"><apply id="S3.Thmtheorem8.p1.5.5.m5.4.4.cmml" xref="S3.Thmtheorem8.p1.5.5.m5.4.4"><in id="S3.Thmtheorem8.p1.5.5.m5.4.4.3.cmml" xref="S3.Thmtheorem8.p1.5.5.m5.4.4.3"></in><apply id="S3.Thmtheorem8.p1.5.5.m5.4.4.4.cmml" xref="S3.Thmtheorem8.p1.5.5.m5.4.4.4"><csymbol cd="ambiguous" id="S3.Thmtheorem8.p1.5.5.m5.4.4.4.1.cmml" xref="S3.Thmtheorem8.p1.5.5.m5.4.4.4">subscript</csymbol><ci id="S3.Thmtheorem8.p1.5.5.m5.4.4.4.2.cmml" xref="S3.Thmtheorem8.p1.5.5.m5.4.4.4.2">𝜌</ci><ci id="S3.Thmtheorem8.p1.5.5.m5.4.4.4.3.cmml" xref="S3.Thmtheorem8.p1.5.5.m5.4.4.4.3">𝑣</ci></apply><interval closure="closed" 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xref="S3.Thmtheorem8.p1.5.5.m5.4.4.2.2.2.3.2">𝜌</ci><times id="S3.Thmtheorem8.p1.5.5.m5.4.4.2.2.2.3.3.cmml" xref="S3.Thmtheorem8.p1.5.5.m5.4.4.2.2.2.3.3"></times></apply><ci id="S3.Thmtheorem8.p1.5.5.m5.2.2.cmml" xref="S3.Thmtheorem8.p1.5.5.m5.2.2">𝑣</ci></apply></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem8.p1.5.5.m5.4c">\rho_{v}\in[(1+\varepsilon)^{-1}\rho^{*}(v),(1+\varepsilon)\rho^{*}(v)]</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem8.p1.5.5.m5.4d">italic_ρ start_POSTSUBSCRIPT italic_v end_POSTSUBSCRIPT ∈ [ ( 1 + italic_ε ) start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT italic_ρ start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ( italic_v ) , ( 1 + italic_ε ) italic_ρ start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ( italic_v ) ]</annotation></semantics></math>.</span></p> </div> </div> </section> <section class="ltx_subsubsection" id="S3.SS0.SSS4"> <h4 class="ltx_title ltx_title_subsubsection"> <span class="ltx_tag ltx_tag_subsubsection">3.D </span>Results in CONGEST</h4> <div class="ltx_para" id="S3.SS0.SSS4.p1"> <p class="ltx_p" id="S3.SS0.SSS4.p1.2">Finally in Section <a class="ltx_ref" href="https://arxiv.org/html/2411.12694v2#S7" title="7 Results in CONGEST ‣ Local Density and its Distributed Approximation"><span class="ltx_text ltx_ref_tag">7</span></a>, we solve Problem <a class="ltx_ref" href="https://arxiv.org/html/2411.12694v2#S2.Thmtheorem11" title="Problem 2.11. ‣ The benefit of local measures: ‣ 2.3 Local density ‣ 2 Preliminaries and related work ‣ Local Density and its Distributed Approximation"><span class="ltx_text ltx_ref_tag">2.11</span></a> in CONGEST by computing an <math alttext="\eta" class="ltx_Math" display="inline" id="S3.SS0.SSS4.p1.1.m1.1"><semantics id="S3.SS0.SSS4.p1.1.m1.1a"><mi id="S3.SS0.SSS4.p1.1.m1.1.1" xref="S3.SS0.SSS4.p1.1.m1.1.1.cmml">η</mi><annotation-xml encoding="MathML-Content" id="S3.SS0.SSS4.p1.1.m1.1b"><ci id="S3.SS0.SSS4.p1.1.m1.1.1.cmml" xref="S3.SS0.SSS4.p1.1.m1.1.1">𝜂</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS0.SSS4.p1.1.m1.1c">\eta</annotation><annotation encoding="application/x-llamapun" id="S3.SS0.SSS4.p1.1.m1.1d">italic_η</annotation></semantics></math>-fair orientation. We use as a subroutine algorithm to compute <em class="ltx_emph ltx_font_italic" id="S3.SS0.SSS4.p1.2.1">blocking flows</em> in an <math alttext="h" class="ltx_Math" display="inline" id="S3.SS0.SSS4.p1.2.m2.1"><semantics id="S3.SS0.SSS4.p1.2.m2.1a"><mi id="S3.SS0.SSS4.p1.2.m2.1.1" xref="S3.SS0.SSS4.p1.2.m2.1.1.cmml">h</mi><annotation-xml encoding="MathML-Content" id="S3.SS0.SSS4.p1.2.m2.1b"><ci id="S3.SS0.SSS4.p1.2.m2.1.1.cmml" xref="S3.SS0.SSS4.p1.2.m2.1.1">ℎ</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS0.SSS4.p1.2.m2.1c">h</annotation><annotation encoding="application/x-llamapun" id="S3.SS0.SSS4.p1.2.m2.1d">italic_h</annotation></semantics></math>-layered DAG <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2411.12694v2#bib.bib15" title="">15</a>]</cite>:</p> </div> <div class="ltx_theorem ltx_theorem_theorem" id="S3.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="S3.Thmtheorem9.1.1.1">Theorem 3.9</span></span><span class="ltx_text ltx_font_bold" id="S3.Thmtheorem9.2.2">.</span> </h6> <div class="ltx_para" id="S3.Thmtheorem9.p1"> <p class="ltx_p" id="S3.Thmtheorem9.p1.9"><span class="ltx_text ltx_font_italic" id="S3.Thmtheorem9.p1.9.9">Suppose one can compute a blocking flow in an <math alttext="n" class="ltx_Math" display="inline" id="S3.Thmtheorem9.p1.1.1.m1.1"><semantics id="S3.Thmtheorem9.p1.1.1.m1.1a"><mi id="S3.Thmtheorem9.p1.1.1.m1.1.1" xref="S3.Thmtheorem9.p1.1.1.m1.1.1.cmml">n</mi><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem9.p1.1.1.m1.1b"><ci id="S3.Thmtheorem9.p1.1.1.m1.1.1.cmml" xref="S3.Thmtheorem9.p1.1.1.m1.1.1">𝑛</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem9.p1.1.1.m1.1c">n</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem9.p1.1.1.m1.1d">italic_n</annotation></semantics></math>-node <math alttext="h" class="ltx_Math" display="inline" id="S3.Thmtheorem9.p1.2.2.m2.1"><semantics id="S3.Thmtheorem9.p1.2.2.m2.1a"><mi id="S3.Thmtheorem9.p1.2.2.m2.1.1" xref="S3.Thmtheorem9.p1.2.2.m2.1.1.cmml">h</mi><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem9.p1.2.2.m2.1b"><ci id="S3.Thmtheorem9.p1.2.2.m2.1.1.cmml" xref="S3.Thmtheorem9.p1.2.2.m2.1.1">ℎ</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem9.p1.2.2.m2.1c">h</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem9.p1.2.2.m2.1d">italic_h</annotation></semantics></math>-layered DAG in <math alttext="\textnormal{Blocking}(h,n)" class="ltx_Math" display="inline" id="S3.Thmtheorem9.p1.3.3.m3.2"><semantics id="S3.Thmtheorem9.p1.3.3.m3.2a"><mrow id="S3.Thmtheorem9.p1.3.3.m3.2.3" xref="S3.Thmtheorem9.p1.3.3.m3.2.3.cmml"><mtext id="S3.Thmtheorem9.p1.3.3.m3.2.3.2" xref="S3.Thmtheorem9.p1.3.3.m3.2.3.2a.cmml">Blocking</mtext><mo id="S3.Thmtheorem9.p1.3.3.m3.2.3.1" xref="S3.Thmtheorem9.p1.3.3.m3.2.3.1.cmml">⁢</mo><mrow id="S3.Thmtheorem9.p1.3.3.m3.2.3.3.2" xref="S3.Thmtheorem9.p1.3.3.m3.2.3.3.1.cmml"><mo id="S3.Thmtheorem9.p1.3.3.m3.2.3.3.2.1" stretchy="false" xref="S3.Thmtheorem9.p1.3.3.m3.2.3.3.1.cmml">(</mo><mi id="S3.Thmtheorem9.p1.3.3.m3.1.1" xref="S3.Thmtheorem9.p1.3.3.m3.1.1.cmml">h</mi><mo id="S3.Thmtheorem9.p1.3.3.m3.2.3.3.2.2" xref="S3.Thmtheorem9.p1.3.3.m3.2.3.3.1.cmml">,</mo><mi id="S3.Thmtheorem9.p1.3.3.m3.2.2" xref="S3.Thmtheorem9.p1.3.3.m3.2.2.cmml">n</mi><mo id="S3.Thmtheorem9.p1.3.3.m3.2.3.3.2.3" stretchy="false" xref="S3.Thmtheorem9.p1.3.3.m3.2.3.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem9.p1.3.3.m3.2b"><apply id="S3.Thmtheorem9.p1.3.3.m3.2.3.cmml" xref="S3.Thmtheorem9.p1.3.3.m3.2.3"><times id="S3.Thmtheorem9.p1.3.3.m3.2.3.1.cmml" xref="S3.Thmtheorem9.p1.3.3.m3.2.3.1"></times><ci id="S3.Thmtheorem9.p1.3.3.m3.2.3.2a.cmml" xref="S3.Thmtheorem9.p1.3.3.m3.2.3.2"><mtext id="S3.Thmtheorem9.p1.3.3.m3.2.3.2.cmml" xref="S3.Thmtheorem9.p1.3.3.m3.2.3.2">Blocking</mtext></ci><interval closure="open" id="S3.Thmtheorem9.p1.3.3.m3.2.3.3.1.cmml" xref="S3.Thmtheorem9.p1.3.3.m3.2.3.3.2"><ci id="S3.Thmtheorem9.p1.3.3.m3.1.1.cmml" xref="S3.Thmtheorem9.p1.3.3.m3.1.1">ℎ</ci><ci id="S3.Thmtheorem9.p1.3.3.m3.2.2.cmml" xref="S3.Thmtheorem9.p1.3.3.m3.2.2">𝑛</ci></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem9.p1.3.3.m3.2c">\textnormal{Blocking}(h,n)</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem9.p1.3.3.m3.2d">Blocking ( italic_h , italic_n )</annotation></semantics></math> rounds. There exists an algorithm in CONGEST that given a unit-weight graph <math alttext="G" class="ltx_Math" display="inline" id="S3.Thmtheorem9.p1.4.4.m4.1"><semantics id="S3.Thmtheorem9.p1.4.4.m4.1a"><mi id="S3.Thmtheorem9.p1.4.4.m4.1.1" xref="S3.Thmtheorem9.p1.4.4.m4.1.1.cmml">G</mi><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem9.p1.4.4.m4.1b"><ci id="S3.Thmtheorem9.p1.4.4.m4.1.1.cmml" xref="S3.Thmtheorem9.p1.4.4.m4.1.1">𝐺</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem9.p1.4.4.m4.1c">G</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem9.p1.4.4.m4.1d">italic_G</annotation></semantics></math> and <math alttext="\varepsilon>0" class="ltx_Math" display="inline" id="S3.Thmtheorem9.p1.5.5.m5.1"><semantics id="S3.Thmtheorem9.p1.5.5.m5.1a"><mrow id="S3.Thmtheorem9.p1.5.5.m5.1.1" xref="S3.Thmtheorem9.p1.5.5.m5.1.1.cmml"><mi id="S3.Thmtheorem9.p1.5.5.m5.1.1.2" xref="S3.Thmtheorem9.p1.5.5.m5.1.1.2.cmml">ε</mi><mo id="S3.Thmtheorem9.p1.5.5.m5.1.1.1" xref="S3.Thmtheorem9.p1.5.5.m5.1.1.1.cmml">></mo><mn id="S3.Thmtheorem9.p1.5.5.m5.1.1.3" xref="S3.Thmtheorem9.p1.5.5.m5.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem9.p1.5.5.m5.1b"><apply id="S3.Thmtheorem9.p1.5.5.m5.1.1.cmml" xref="S3.Thmtheorem9.p1.5.5.m5.1.1"><gt id="S3.Thmtheorem9.p1.5.5.m5.1.1.1.cmml" xref="S3.Thmtheorem9.p1.5.5.m5.1.1.1"></gt><ci id="S3.Thmtheorem9.p1.5.5.m5.1.1.2.cmml" xref="S3.Thmtheorem9.p1.5.5.m5.1.1.2">𝜀</ci><cn id="S3.Thmtheorem9.p1.5.5.m5.1.1.3.cmml" type="integer" xref="S3.Thmtheorem9.p1.5.5.m5.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem9.p1.5.5.m5.1c">\varepsilon>0</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem9.p1.5.5.m5.1d">italic_ε > 0</annotation></semantics></math> computes in <math alttext="O(\varepsilon^{-3}\log^{4}n\cdot(\varepsilon^{-2}\log^{2}n+\textnormal{% Blocking}(\varepsilon^{-2}\log^{2}n,n)))" class="ltx_Math" display="inline" id="S3.Thmtheorem9.p1.6.6.m6.2"><semantics id="S3.Thmtheorem9.p1.6.6.m6.2a"><mrow id="S3.Thmtheorem9.p1.6.6.m6.2.2" xref="S3.Thmtheorem9.p1.6.6.m6.2.2.cmml"><mi id="S3.Thmtheorem9.p1.6.6.m6.2.2.3" xref="S3.Thmtheorem9.p1.6.6.m6.2.2.3.cmml">O</mi><mo id="S3.Thmtheorem9.p1.6.6.m6.2.2.2" xref="S3.Thmtheorem9.p1.6.6.m6.2.2.2.cmml">⁢</mo><mrow id="S3.Thmtheorem9.p1.6.6.m6.2.2.1.1" xref="S3.Thmtheorem9.p1.6.6.m6.2.2.1.1.1.cmml"><mo id="S3.Thmtheorem9.p1.6.6.m6.2.2.1.1.2" stretchy="false" xref="S3.Thmtheorem9.p1.6.6.m6.2.2.1.1.1.cmml">(</mo><mrow id="S3.Thmtheorem9.p1.6.6.m6.2.2.1.1.1" xref="S3.Thmtheorem9.p1.6.6.m6.2.2.1.1.1.cmml"><mrow id="S3.Thmtheorem9.p1.6.6.m6.2.2.1.1.1.3" xref="S3.Thmtheorem9.p1.6.6.m6.2.2.1.1.1.3.cmml"><msup id="S3.Thmtheorem9.p1.6.6.m6.2.2.1.1.1.3.2" xref="S3.Thmtheorem9.p1.6.6.m6.2.2.1.1.1.3.2.cmml"><mi id="S3.Thmtheorem9.p1.6.6.m6.2.2.1.1.1.3.2.2" xref="S3.Thmtheorem9.p1.6.6.m6.2.2.1.1.1.3.2.2.cmml">ε</mi><mrow id="S3.Thmtheorem9.p1.6.6.m6.2.2.1.1.1.3.2.3" xref="S3.Thmtheorem9.p1.6.6.m6.2.2.1.1.1.3.2.3.cmml"><mo id="S3.Thmtheorem9.p1.6.6.m6.2.2.1.1.1.3.2.3a" xref="S3.Thmtheorem9.p1.6.6.m6.2.2.1.1.1.3.2.3.cmml">−</mo><mn id="S3.Thmtheorem9.p1.6.6.m6.2.2.1.1.1.3.2.3.2" xref="S3.Thmtheorem9.p1.6.6.m6.2.2.1.1.1.3.2.3.2.cmml">3</mn></mrow></msup><mo id="S3.Thmtheorem9.p1.6.6.m6.2.2.1.1.1.3.1" lspace="0.167em" xref="S3.Thmtheorem9.p1.6.6.m6.2.2.1.1.1.3.1.cmml">⁢</mo><mrow id="S3.Thmtheorem9.p1.6.6.m6.2.2.1.1.1.3.3" xref="S3.Thmtheorem9.p1.6.6.m6.2.2.1.1.1.3.3.cmml"><msup id="S3.Thmtheorem9.p1.6.6.m6.2.2.1.1.1.3.3.1" xref="S3.Thmtheorem9.p1.6.6.m6.2.2.1.1.1.3.3.1.cmml"><mi id="S3.Thmtheorem9.p1.6.6.m6.2.2.1.1.1.3.3.1.2" xref="S3.Thmtheorem9.p1.6.6.m6.2.2.1.1.1.3.3.1.2.cmml">log</mi><mn id="S3.Thmtheorem9.p1.6.6.m6.2.2.1.1.1.3.3.1.3" xref="S3.Thmtheorem9.p1.6.6.m6.2.2.1.1.1.3.3.1.3.cmml">4</mn></msup><mo id="S3.Thmtheorem9.p1.6.6.m6.2.2.1.1.1.3.3a" lspace="0.167em" xref="S3.Thmtheorem9.p1.6.6.m6.2.2.1.1.1.3.3.cmml">⁡</mo><mi id="S3.Thmtheorem9.p1.6.6.m6.2.2.1.1.1.3.3.2" xref="S3.Thmtheorem9.p1.6.6.m6.2.2.1.1.1.3.3.2.cmml">n</mi></mrow></mrow><mo id="S3.Thmtheorem9.p1.6.6.m6.2.2.1.1.1.2" lspace="0.222em" rspace="0.222em" xref="S3.Thmtheorem9.p1.6.6.m6.2.2.1.1.1.2.cmml">⋅</mo><mrow id="S3.Thmtheorem9.p1.6.6.m6.2.2.1.1.1.1.1" xref="S3.Thmtheorem9.p1.6.6.m6.2.2.1.1.1.1.1.1.cmml"><mo id="S3.Thmtheorem9.p1.6.6.m6.2.2.1.1.1.1.1.2" stretchy="false" xref="S3.Thmtheorem9.p1.6.6.m6.2.2.1.1.1.1.1.1.cmml">(</mo><mrow id="S3.Thmtheorem9.p1.6.6.m6.2.2.1.1.1.1.1.1" xref="S3.Thmtheorem9.p1.6.6.m6.2.2.1.1.1.1.1.1.cmml"><mrow id="S3.Thmtheorem9.p1.6.6.m6.2.2.1.1.1.1.1.1.3" xref="S3.Thmtheorem9.p1.6.6.m6.2.2.1.1.1.1.1.1.3.cmml"><msup id="S3.Thmtheorem9.p1.6.6.m6.2.2.1.1.1.1.1.1.3.2" xref="S3.Thmtheorem9.p1.6.6.m6.2.2.1.1.1.1.1.1.3.2.cmml"><mi id="S3.Thmtheorem9.p1.6.6.m6.2.2.1.1.1.1.1.1.3.2.2" xref="S3.Thmtheorem9.p1.6.6.m6.2.2.1.1.1.1.1.1.3.2.2.cmml">ε</mi><mrow id="S3.Thmtheorem9.p1.6.6.m6.2.2.1.1.1.1.1.1.3.2.3" xref="S3.Thmtheorem9.p1.6.6.m6.2.2.1.1.1.1.1.1.3.2.3.cmml"><mo id="S3.Thmtheorem9.p1.6.6.m6.2.2.1.1.1.1.1.1.3.2.3a" xref="S3.Thmtheorem9.p1.6.6.m6.2.2.1.1.1.1.1.1.3.2.3.cmml">−</mo><mn id="S3.Thmtheorem9.p1.6.6.m6.2.2.1.1.1.1.1.1.3.2.3.2" xref="S3.Thmtheorem9.p1.6.6.m6.2.2.1.1.1.1.1.1.3.2.3.2.cmml">2</mn></mrow></msup><mo id="S3.Thmtheorem9.p1.6.6.m6.2.2.1.1.1.1.1.1.3.1" lspace="0.167em" xref="S3.Thmtheorem9.p1.6.6.m6.2.2.1.1.1.1.1.1.3.1.cmml">⁢</mo><mrow id="S3.Thmtheorem9.p1.6.6.m6.2.2.1.1.1.1.1.1.3.3" xref="S3.Thmtheorem9.p1.6.6.m6.2.2.1.1.1.1.1.1.3.3.cmml"><msup id="S3.Thmtheorem9.p1.6.6.m6.2.2.1.1.1.1.1.1.3.3.1" xref="S3.Thmtheorem9.p1.6.6.m6.2.2.1.1.1.1.1.1.3.3.1.cmml"><mi id="S3.Thmtheorem9.p1.6.6.m6.2.2.1.1.1.1.1.1.3.3.1.2" xref="S3.Thmtheorem9.p1.6.6.m6.2.2.1.1.1.1.1.1.3.3.1.2.cmml">log</mi><mn id="S3.Thmtheorem9.p1.6.6.m6.2.2.1.1.1.1.1.1.3.3.1.3" xref="S3.Thmtheorem9.p1.6.6.m6.2.2.1.1.1.1.1.1.3.3.1.3.cmml">2</mn></msup><mo id="S3.Thmtheorem9.p1.6.6.m6.2.2.1.1.1.1.1.1.3.3a" lspace="0.167em" xref="S3.Thmtheorem9.p1.6.6.m6.2.2.1.1.1.1.1.1.3.3.cmml">⁡</mo><mi id="S3.Thmtheorem9.p1.6.6.m6.2.2.1.1.1.1.1.1.3.3.2" xref="S3.Thmtheorem9.p1.6.6.m6.2.2.1.1.1.1.1.1.3.3.2.cmml">n</mi></mrow></mrow><mo id="S3.Thmtheorem9.p1.6.6.m6.2.2.1.1.1.1.1.1.2" xref="S3.Thmtheorem9.p1.6.6.m6.2.2.1.1.1.1.1.1.2.cmml">+</mo><mrow id="S3.Thmtheorem9.p1.6.6.m6.2.2.1.1.1.1.1.1.1" xref="S3.Thmtheorem9.p1.6.6.m6.2.2.1.1.1.1.1.1.1.cmml"><mtext id="S3.Thmtheorem9.p1.6.6.m6.2.2.1.1.1.1.1.1.1.3" xref="S3.Thmtheorem9.p1.6.6.m6.2.2.1.1.1.1.1.1.1.3a.cmml">Blocking</mtext><mo id="S3.Thmtheorem9.p1.6.6.m6.2.2.1.1.1.1.1.1.1.2" xref="S3.Thmtheorem9.p1.6.6.m6.2.2.1.1.1.1.1.1.1.2.cmml">⁢</mo><mrow id="S3.Thmtheorem9.p1.6.6.m6.2.2.1.1.1.1.1.1.1.1.1" xref="S3.Thmtheorem9.p1.6.6.m6.2.2.1.1.1.1.1.1.1.1.2.cmml"><mo id="S3.Thmtheorem9.p1.6.6.m6.2.2.1.1.1.1.1.1.1.1.1.2" stretchy="false" xref="S3.Thmtheorem9.p1.6.6.m6.2.2.1.1.1.1.1.1.1.1.2.cmml">(</mo><mrow id="S3.Thmtheorem9.p1.6.6.m6.2.2.1.1.1.1.1.1.1.1.1.1" xref="S3.Thmtheorem9.p1.6.6.m6.2.2.1.1.1.1.1.1.1.1.1.1.cmml"><msup id="S3.Thmtheorem9.p1.6.6.m6.2.2.1.1.1.1.1.1.1.1.1.1.2" xref="S3.Thmtheorem9.p1.6.6.m6.2.2.1.1.1.1.1.1.1.1.1.1.2.cmml"><mi id="S3.Thmtheorem9.p1.6.6.m6.2.2.1.1.1.1.1.1.1.1.1.1.2.2" xref="S3.Thmtheorem9.p1.6.6.m6.2.2.1.1.1.1.1.1.1.1.1.1.2.2.cmml">ε</mi><mrow id="S3.Thmtheorem9.p1.6.6.m6.2.2.1.1.1.1.1.1.1.1.1.1.2.3" xref="S3.Thmtheorem9.p1.6.6.m6.2.2.1.1.1.1.1.1.1.1.1.1.2.3.cmml"><mo id="S3.Thmtheorem9.p1.6.6.m6.2.2.1.1.1.1.1.1.1.1.1.1.2.3a" xref="S3.Thmtheorem9.p1.6.6.m6.2.2.1.1.1.1.1.1.1.1.1.1.2.3.cmml">−</mo><mn id="S3.Thmtheorem9.p1.6.6.m6.2.2.1.1.1.1.1.1.1.1.1.1.2.3.2" xref="S3.Thmtheorem9.p1.6.6.m6.2.2.1.1.1.1.1.1.1.1.1.1.2.3.2.cmml">2</mn></mrow></msup><mo id="S3.Thmtheorem9.p1.6.6.m6.2.2.1.1.1.1.1.1.1.1.1.1.1" lspace="0.167em" 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xref="S3.Thmtheorem9.p1.6.6.m6.2.2.1.1"><ci id="S3.Thmtheorem9.p1.6.6.m6.2.2.1.1.1.2.cmml" xref="S3.Thmtheorem9.p1.6.6.m6.2.2.1.1.1.2">⋅</ci><apply id="S3.Thmtheorem9.p1.6.6.m6.2.2.1.1.1.3.cmml" xref="S3.Thmtheorem9.p1.6.6.m6.2.2.1.1.1.3"><times id="S3.Thmtheorem9.p1.6.6.m6.2.2.1.1.1.3.1.cmml" xref="S3.Thmtheorem9.p1.6.6.m6.2.2.1.1.1.3.1"></times><apply id="S3.Thmtheorem9.p1.6.6.m6.2.2.1.1.1.3.2.cmml" xref="S3.Thmtheorem9.p1.6.6.m6.2.2.1.1.1.3.2"><csymbol cd="ambiguous" id="S3.Thmtheorem9.p1.6.6.m6.2.2.1.1.1.3.2.1.cmml" xref="S3.Thmtheorem9.p1.6.6.m6.2.2.1.1.1.3.2">superscript</csymbol><ci id="S3.Thmtheorem9.p1.6.6.m6.2.2.1.1.1.3.2.2.cmml" xref="S3.Thmtheorem9.p1.6.6.m6.2.2.1.1.1.3.2.2">𝜀</ci><apply id="S3.Thmtheorem9.p1.6.6.m6.2.2.1.1.1.3.2.3.cmml" xref="S3.Thmtheorem9.p1.6.6.m6.2.2.1.1.1.3.2.3"><minus id="S3.Thmtheorem9.p1.6.6.m6.2.2.1.1.1.3.2.3.1.cmml" xref="S3.Thmtheorem9.p1.6.6.m6.2.2.1.1.1.3.2.3"></minus><cn id="S3.Thmtheorem9.p1.6.6.m6.2.2.1.1.1.3.2.3.2.cmml" type="integer" xref="S3.Thmtheorem9.p1.6.6.m6.2.2.1.1.1.3.2.3.2">3</cn></apply></apply><apply id="S3.Thmtheorem9.p1.6.6.m6.2.2.1.1.1.3.3.cmml" xref="S3.Thmtheorem9.p1.6.6.m6.2.2.1.1.1.3.3"><apply id="S3.Thmtheorem9.p1.6.6.m6.2.2.1.1.1.3.3.1.cmml" xref="S3.Thmtheorem9.p1.6.6.m6.2.2.1.1.1.3.3.1"><csymbol cd="ambiguous" id="S3.Thmtheorem9.p1.6.6.m6.2.2.1.1.1.3.3.1.1.cmml" xref="S3.Thmtheorem9.p1.6.6.m6.2.2.1.1.1.3.3.1">superscript</csymbol><log id="S3.Thmtheorem9.p1.6.6.m6.2.2.1.1.1.3.3.1.2.cmml" xref="S3.Thmtheorem9.p1.6.6.m6.2.2.1.1.1.3.3.1.2"></log><cn id="S3.Thmtheorem9.p1.6.6.m6.2.2.1.1.1.3.3.1.3.cmml" type="integer" xref="S3.Thmtheorem9.p1.6.6.m6.2.2.1.1.1.3.3.1.3">4</cn></apply><ci id="S3.Thmtheorem9.p1.6.6.m6.2.2.1.1.1.3.3.2.cmml" xref="S3.Thmtheorem9.p1.6.6.m6.2.2.1.1.1.3.3.2">𝑛</ci></apply></apply><apply id="S3.Thmtheorem9.p1.6.6.m6.2.2.1.1.1.1.1.1.cmml" xref="S3.Thmtheorem9.p1.6.6.m6.2.2.1.1.1.1.1"><plus id="S3.Thmtheorem9.p1.6.6.m6.2.2.1.1.1.1.1.1.2.cmml" xref="S3.Thmtheorem9.p1.6.6.m6.2.2.1.1.1.1.1.1.2"></plus><apply id="S3.Thmtheorem9.p1.6.6.m6.2.2.1.1.1.1.1.1.3.cmml" xref="S3.Thmtheorem9.p1.6.6.m6.2.2.1.1.1.1.1.1.3"><times id="S3.Thmtheorem9.p1.6.6.m6.2.2.1.1.1.1.1.1.3.1.cmml" xref="S3.Thmtheorem9.p1.6.6.m6.2.2.1.1.1.1.1.1.3.1"></times><apply id="S3.Thmtheorem9.p1.6.6.m6.2.2.1.1.1.1.1.1.3.2.cmml" xref="S3.Thmtheorem9.p1.6.6.m6.2.2.1.1.1.1.1.1.3.2"><csymbol cd="ambiguous" id="S3.Thmtheorem9.p1.6.6.m6.2.2.1.1.1.1.1.1.3.2.1.cmml" xref="S3.Thmtheorem9.p1.6.6.m6.2.2.1.1.1.1.1.1.3.2">superscript</csymbol><ci id="S3.Thmtheorem9.p1.6.6.m6.2.2.1.1.1.1.1.1.3.2.2.cmml" xref="S3.Thmtheorem9.p1.6.6.m6.2.2.1.1.1.1.1.1.3.2.2">𝜀</ci><apply id="S3.Thmtheorem9.p1.6.6.m6.2.2.1.1.1.1.1.1.3.2.3.cmml" xref="S3.Thmtheorem9.p1.6.6.m6.2.2.1.1.1.1.1.1.3.2.3"><minus id="S3.Thmtheorem9.p1.6.6.m6.2.2.1.1.1.1.1.1.3.2.3.1.cmml" xref="S3.Thmtheorem9.p1.6.6.m6.2.2.1.1.1.1.1.1.3.2.3"></minus><cn id="S3.Thmtheorem9.p1.6.6.m6.2.2.1.1.1.1.1.1.3.2.3.2.cmml" type="integer" xref="S3.Thmtheorem9.p1.6.6.m6.2.2.1.1.1.1.1.1.3.2.3.2">2</cn></apply></apply><apply id="S3.Thmtheorem9.p1.6.6.m6.2.2.1.1.1.1.1.1.3.3.cmml" xref="S3.Thmtheorem9.p1.6.6.m6.2.2.1.1.1.1.1.1.3.3"><apply id="S3.Thmtheorem9.p1.6.6.m6.2.2.1.1.1.1.1.1.3.3.1.cmml" xref="S3.Thmtheorem9.p1.6.6.m6.2.2.1.1.1.1.1.1.3.3.1"><csymbol cd="ambiguous" id="S3.Thmtheorem9.p1.6.6.m6.2.2.1.1.1.1.1.1.3.3.1.1.cmml" xref="S3.Thmtheorem9.p1.6.6.m6.2.2.1.1.1.1.1.1.3.3.1">superscript</csymbol><log id="S3.Thmtheorem9.p1.6.6.m6.2.2.1.1.1.1.1.1.3.3.1.2.cmml" xref="S3.Thmtheorem9.p1.6.6.m6.2.2.1.1.1.1.1.1.3.3.1.2"></log><cn id="S3.Thmtheorem9.p1.6.6.m6.2.2.1.1.1.1.1.1.3.3.1.3.cmml" type="integer" xref="S3.Thmtheorem9.p1.6.6.m6.2.2.1.1.1.1.1.1.3.3.1.3">2</cn></apply><ci id="S3.Thmtheorem9.p1.6.6.m6.2.2.1.1.1.1.1.1.3.3.2.cmml" xref="S3.Thmtheorem9.p1.6.6.m6.2.2.1.1.1.1.1.1.3.3.2">𝑛</ci></apply></apply><apply id="S3.Thmtheorem9.p1.6.6.m6.2.2.1.1.1.1.1.1.1.cmml" xref="S3.Thmtheorem9.p1.6.6.m6.2.2.1.1.1.1.1.1.1"><times id="S3.Thmtheorem9.p1.6.6.m6.2.2.1.1.1.1.1.1.1.2.cmml" xref="S3.Thmtheorem9.p1.6.6.m6.2.2.1.1.1.1.1.1.1.2"></times><ci id="S3.Thmtheorem9.p1.6.6.m6.2.2.1.1.1.1.1.1.1.3a.cmml" xref="S3.Thmtheorem9.p1.6.6.m6.2.2.1.1.1.1.1.1.1.3"><mtext id="S3.Thmtheorem9.p1.6.6.m6.2.2.1.1.1.1.1.1.1.3.cmml" xref="S3.Thmtheorem9.p1.6.6.m6.2.2.1.1.1.1.1.1.1.3">Blocking</mtext></ci><interval closure="open" id="S3.Thmtheorem9.p1.6.6.m6.2.2.1.1.1.1.1.1.1.1.2.cmml" xref="S3.Thmtheorem9.p1.6.6.m6.2.2.1.1.1.1.1.1.1.1.1"><apply id="S3.Thmtheorem9.p1.6.6.m6.2.2.1.1.1.1.1.1.1.1.1.1.cmml" xref="S3.Thmtheorem9.p1.6.6.m6.2.2.1.1.1.1.1.1.1.1.1.1"><times id="S3.Thmtheorem9.p1.6.6.m6.2.2.1.1.1.1.1.1.1.1.1.1.1.cmml" xref="S3.Thmtheorem9.p1.6.6.m6.2.2.1.1.1.1.1.1.1.1.1.1.1"></times><apply id="S3.Thmtheorem9.p1.6.6.m6.2.2.1.1.1.1.1.1.1.1.1.1.2.cmml" xref="S3.Thmtheorem9.p1.6.6.m6.2.2.1.1.1.1.1.1.1.1.1.1.2"><csymbol cd="ambiguous" id="S3.Thmtheorem9.p1.6.6.m6.2.2.1.1.1.1.1.1.1.1.1.1.2.1.cmml" xref="S3.Thmtheorem9.p1.6.6.m6.2.2.1.1.1.1.1.1.1.1.1.1.2">superscript</csymbol><ci id="S3.Thmtheorem9.p1.6.6.m6.2.2.1.1.1.1.1.1.1.1.1.1.2.2.cmml" xref="S3.Thmtheorem9.p1.6.6.m6.2.2.1.1.1.1.1.1.1.1.1.1.2.2">𝜀</ci><apply id="S3.Thmtheorem9.p1.6.6.m6.2.2.1.1.1.1.1.1.1.1.1.1.2.3.cmml" xref="S3.Thmtheorem9.p1.6.6.m6.2.2.1.1.1.1.1.1.1.1.1.1.2.3"><minus id="S3.Thmtheorem9.p1.6.6.m6.2.2.1.1.1.1.1.1.1.1.1.1.2.3.1.cmml" xref="S3.Thmtheorem9.p1.6.6.m6.2.2.1.1.1.1.1.1.1.1.1.1.2.3"></minus><cn id="S3.Thmtheorem9.p1.6.6.m6.2.2.1.1.1.1.1.1.1.1.1.1.2.3.2.cmml" type="integer" xref="S3.Thmtheorem9.p1.6.6.m6.2.2.1.1.1.1.1.1.1.1.1.1.2.3.2">2</cn></apply></apply><apply id="S3.Thmtheorem9.p1.6.6.m6.2.2.1.1.1.1.1.1.1.1.1.1.3.cmml" xref="S3.Thmtheorem9.p1.6.6.m6.2.2.1.1.1.1.1.1.1.1.1.1.3"><apply id="S3.Thmtheorem9.p1.6.6.m6.2.2.1.1.1.1.1.1.1.1.1.1.3.1.cmml" xref="S3.Thmtheorem9.p1.6.6.m6.2.2.1.1.1.1.1.1.1.1.1.1.3.1"><csymbol cd="ambiguous" id="S3.Thmtheorem9.p1.6.6.m6.2.2.1.1.1.1.1.1.1.1.1.1.3.1.1.cmml" xref="S3.Thmtheorem9.p1.6.6.m6.2.2.1.1.1.1.1.1.1.1.1.1.3.1">superscript</csymbol><log id="S3.Thmtheorem9.p1.6.6.m6.2.2.1.1.1.1.1.1.1.1.1.1.3.1.2.cmml" xref="S3.Thmtheorem9.p1.6.6.m6.2.2.1.1.1.1.1.1.1.1.1.1.3.1.2"></log><cn id="S3.Thmtheorem9.p1.6.6.m6.2.2.1.1.1.1.1.1.1.1.1.1.3.1.3.cmml" type="integer" xref="S3.Thmtheorem9.p1.6.6.m6.2.2.1.1.1.1.1.1.1.1.1.1.3.1.3">2</cn></apply><ci id="S3.Thmtheorem9.p1.6.6.m6.2.2.1.1.1.1.1.1.1.1.1.1.3.2.cmml" xref="S3.Thmtheorem9.p1.6.6.m6.2.2.1.1.1.1.1.1.1.1.1.1.3.2">𝑛</ci></apply></apply><ci id="S3.Thmtheorem9.p1.6.6.m6.1.1.cmml" xref="S3.Thmtheorem9.p1.6.6.m6.1.1">𝑛</ci></interval></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem9.p1.6.6.m6.2c">O(\varepsilon^{-3}\log^{4}n\cdot(\varepsilon^{-2}\log^{2}n+\textnormal{% Blocking}(\varepsilon^{-2}\log^{2}n,n)))</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem9.p1.6.6.m6.2d">italic_O ( italic_ε start_POSTSUPERSCRIPT - 3 end_POSTSUPERSCRIPT roman_log start_POSTSUPERSCRIPT 4 end_POSTSUPERSCRIPT italic_n ⋅ ( italic_ε start_POSTSUPERSCRIPT - 2 end_POSTSUPERSCRIPT roman_log start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_n + Blocking ( italic_ε start_POSTSUPERSCRIPT - 2 end_POSTSUPERSCRIPT roman_log start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_n , italic_n ) ) )</annotation></semantics></math> rounds an orientation <math alttext="\overrightarrow{G}" class="ltx_Math" display="inline" id="S3.Thmtheorem9.p1.7.7.m7.1"><semantics id="S3.Thmtheorem9.p1.7.7.m7.1a"><mover accent="true" id="S3.Thmtheorem9.p1.7.7.m7.1.1" xref="S3.Thmtheorem9.p1.7.7.m7.1.1.cmml"><mi id="S3.Thmtheorem9.p1.7.7.m7.1.1.2" xref="S3.Thmtheorem9.p1.7.7.m7.1.1.2.cmml">G</mi><mo id="S3.Thmtheorem9.p1.7.7.m7.1.1.1" stretchy="false" xref="S3.Thmtheorem9.p1.7.7.m7.1.1.1.cmml">→</mo></mover><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem9.p1.7.7.m7.1b"><apply id="S3.Thmtheorem9.p1.7.7.m7.1.1.cmml" xref="S3.Thmtheorem9.p1.7.7.m7.1.1"><ci id="S3.Thmtheorem9.p1.7.7.m7.1.1.1.cmml" xref="S3.Thmtheorem9.p1.7.7.m7.1.1.1">→</ci><ci id="S3.Thmtheorem9.p1.7.7.m7.1.1.2.cmml" xref="S3.Thmtheorem9.p1.7.7.m7.1.1.2">𝐺</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem9.p1.7.7.m7.1c">\overrightarrow{G}</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem9.p1.7.7.m7.1d">over→ start_ARG italic_G end_ARG</annotation></semantics></math> such that for all <math alttext="v\in V" class="ltx_Math" display="inline" id="S3.Thmtheorem9.p1.8.8.m8.1"><semantics id="S3.Thmtheorem9.p1.8.8.m8.1a"><mrow id="S3.Thmtheorem9.p1.8.8.m8.1.1" xref="S3.Thmtheorem9.p1.8.8.m8.1.1.cmml"><mi id="S3.Thmtheorem9.p1.8.8.m8.1.1.2" xref="S3.Thmtheorem9.p1.8.8.m8.1.1.2.cmml">v</mi><mo id="S3.Thmtheorem9.p1.8.8.m8.1.1.1" xref="S3.Thmtheorem9.p1.8.8.m8.1.1.1.cmml">∈</mo><mi id="S3.Thmtheorem9.p1.8.8.m8.1.1.3" xref="S3.Thmtheorem9.p1.8.8.m8.1.1.3.cmml">V</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem9.p1.8.8.m8.1b"><apply id="S3.Thmtheorem9.p1.8.8.m8.1.1.cmml" xref="S3.Thmtheorem9.p1.8.8.m8.1.1"><in id="S3.Thmtheorem9.p1.8.8.m8.1.1.1.cmml" xref="S3.Thmtheorem9.p1.8.8.m8.1.1.1"></in><ci id="S3.Thmtheorem9.p1.8.8.m8.1.1.2.cmml" xref="S3.Thmtheorem9.p1.8.8.m8.1.1.2">𝑣</ci><ci id="S3.Thmtheorem9.p1.8.8.m8.1.1.3.cmml" xref="S3.Thmtheorem9.p1.8.8.m8.1.1.3">𝑉</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem9.p1.8.8.m8.1c">v\in V</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem9.p1.8.8.m8.1d">italic_v ∈ italic_V</annotation></semantics></math>: <math alttext="\textsl{g}(v)\in[(1+\varepsilon)^{-1}\rho^{*}(v)),(1+\varepsilon)\rho^{*}(v)% \rho^{*}(v)]" class="ltx_math_unparsed" display="inline" id="S3.Thmtheorem9.p1.9.9.m9.2"><semantics id="S3.Thmtheorem9.p1.9.9.m9.2a"><mrow id="S3.Thmtheorem9.p1.9.9.m9.2b"><mtext class="ltx_mathvariant_italic" id="S3.Thmtheorem9.p1.9.9.m9.2.3">g</mtext><mrow id="S3.Thmtheorem9.p1.9.9.m9.2.4"><mo id="S3.Thmtheorem9.p1.9.9.m9.2.4.1" stretchy="false">(</mo><mi id="S3.Thmtheorem9.p1.9.9.m9.1.1">v</mi><mo id="S3.Thmtheorem9.p1.9.9.m9.2.4.2" stretchy="false">)</mo></mrow><mo id="S3.Thmtheorem9.p1.9.9.m9.2.5">∈</mo><mrow id="S3.Thmtheorem9.p1.9.9.m9.2.6"><mo id="S3.Thmtheorem9.p1.9.9.m9.2.6.1" stretchy="false">[</mo><msup id="S3.Thmtheorem9.p1.9.9.m9.2.6.2"><mrow id="S3.Thmtheorem9.p1.9.9.m9.2.6.2.2"><mo id="S3.Thmtheorem9.p1.9.9.m9.2.6.2.2.1" stretchy="false">(</mo><mn id="S3.Thmtheorem9.p1.9.9.m9.2.6.2.2.2">1</mn><mo id="S3.Thmtheorem9.p1.9.9.m9.2.6.2.2.3">+</mo><mi id="S3.Thmtheorem9.p1.9.9.m9.2.6.2.2.4">ε</mi><mo id="S3.Thmtheorem9.p1.9.9.m9.2.6.2.2.5" stretchy="false">)</mo></mrow><mrow id="S3.Thmtheorem9.p1.9.9.m9.2.6.2.3"><mo id="S3.Thmtheorem9.p1.9.9.m9.2.6.2.3a">−</mo><mn id="S3.Thmtheorem9.p1.9.9.m9.2.6.2.3.2">1</mn></mrow></msup><msup id="S3.Thmtheorem9.p1.9.9.m9.2.6.3"><mi id="S3.Thmtheorem9.p1.9.9.m9.2.6.3.2">ρ</mi><mo id="S3.Thmtheorem9.p1.9.9.m9.2.6.3.3">∗</mo></msup><mrow id="S3.Thmtheorem9.p1.9.9.m9.2.6.4"><mo id="S3.Thmtheorem9.p1.9.9.m9.2.6.4.1" stretchy="false">(</mo><mi id="S3.Thmtheorem9.p1.9.9.m9.2.2">v</mi><mo id="S3.Thmtheorem9.p1.9.9.m9.2.6.4.2" stretchy="false">)</mo></mrow><mo id="S3.Thmtheorem9.p1.9.9.m9.2.6.5" stretchy="false">)</mo></mrow><mo id="S3.Thmtheorem9.p1.9.9.m9.2.7">,</mo><mrow id="S3.Thmtheorem9.p1.9.9.m9.2.8"><mo id="S3.Thmtheorem9.p1.9.9.m9.2.8.1" stretchy="false">(</mo><mn id="S3.Thmtheorem9.p1.9.9.m9.2.8.2">1</mn><mo id="S3.Thmtheorem9.p1.9.9.m9.2.8.3">+</mo><mi id="S3.Thmtheorem9.p1.9.9.m9.2.8.4">ε</mi><mo id="S3.Thmtheorem9.p1.9.9.m9.2.8.5" stretchy="false">)</mo></mrow><msup id="S3.Thmtheorem9.p1.9.9.m9.2.9"><mi id="S3.Thmtheorem9.p1.9.9.m9.2.9.2">ρ</mi><mo id="S3.Thmtheorem9.p1.9.9.m9.2.9.3">∗</mo></msup><mrow id="S3.Thmtheorem9.p1.9.9.m9.2.10"><mo id="S3.Thmtheorem9.p1.9.9.m9.2.10.1" stretchy="false">(</mo><mi id="S3.Thmtheorem9.p1.9.9.m9.2.10.2">v</mi><mo id="S3.Thmtheorem9.p1.9.9.m9.2.10.3" stretchy="false">)</mo></mrow><msup id="S3.Thmtheorem9.p1.9.9.m9.2.11"><mi id="S3.Thmtheorem9.p1.9.9.m9.2.11.2">ρ</mi><mo id="S3.Thmtheorem9.p1.9.9.m9.2.11.3">∗</mo></msup><mrow id="S3.Thmtheorem9.p1.9.9.m9.2.12"><mo id="S3.Thmtheorem9.p1.9.9.m9.2.12.1" stretchy="false">(</mo><mi id="S3.Thmtheorem9.p1.9.9.m9.2.12.2">v</mi><mo id="S3.Thmtheorem9.p1.9.9.m9.2.12.3" stretchy="false">)</mo></mrow><mo id="S3.Thmtheorem9.p1.9.9.m9.2.13" stretchy="false">]</mo></mrow><annotation encoding="application/x-tex" id="S3.Thmtheorem9.p1.9.9.m9.2c">\textsl{g}(v)\in[(1+\varepsilon)^{-1}\rho^{*}(v)),(1+\varepsilon)\rho^{*}(v)% \rho^{*}(v)]</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem9.p1.9.9.m9.2d">g ( italic_v ) ∈ [ ( 1 + italic_ε ) start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT italic_ρ start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ( italic_v ) ) , ( 1 + italic_ε ) italic_ρ start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ( italic_v ) italic_ρ start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ( italic_v ) ]</annotation></semantics></math>.</span></p> </div> </div> <div class="ltx_para" id="S3.SS0.SSS4.p2"> <p class="ltx_p" id="S3.SS0.SSS4.p2.1">As a corollary, we obtain the first deterministic algorithm running in a sublinear number of rounds for <math alttext="(1+\varepsilon)" class="ltx_Math" display="inline" id="S3.SS0.SSS4.p2.1.m1.1"><semantics id="S3.SS0.SSS4.p2.1.m1.1a"><mrow id="S3.SS0.SSS4.p2.1.m1.1.1.1" xref="S3.SS0.SSS4.p2.1.m1.1.1.1.1.cmml"><mo id="S3.SS0.SSS4.p2.1.m1.1.1.1.2" stretchy="false" xref="S3.SS0.SSS4.p2.1.m1.1.1.1.1.cmml">(</mo><mrow id="S3.SS0.SSS4.p2.1.m1.1.1.1.1" xref="S3.SS0.SSS4.p2.1.m1.1.1.1.1.cmml"><mn id="S3.SS0.SSS4.p2.1.m1.1.1.1.1.2" xref="S3.SS0.SSS4.p2.1.m1.1.1.1.1.2.cmml">1</mn><mo id="S3.SS0.SSS4.p2.1.m1.1.1.1.1.1" xref="S3.SS0.SSS4.p2.1.m1.1.1.1.1.1.cmml">+</mo><mi id="S3.SS0.SSS4.p2.1.m1.1.1.1.1.3" xref="S3.SS0.SSS4.p2.1.m1.1.1.1.1.3.cmml">ε</mi></mrow><mo id="S3.SS0.SSS4.p2.1.m1.1.1.1.3" stretchy="false" xref="S3.SS0.SSS4.p2.1.m1.1.1.1.1.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S3.SS0.SSS4.p2.1.m1.1b"><apply id="S3.SS0.SSS4.p2.1.m1.1.1.1.1.cmml" xref="S3.SS0.SSS4.p2.1.m1.1.1.1"><plus id="S3.SS0.SSS4.p2.1.m1.1.1.1.1.1.cmml" xref="S3.SS0.SSS4.p2.1.m1.1.1.1.1.1"></plus><cn id="S3.SS0.SSS4.p2.1.m1.1.1.1.1.2.cmml" type="integer" xref="S3.SS0.SSS4.p2.1.m1.1.1.1.1.2">1</cn><ci id="S3.SS0.SSS4.p2.1.m1.1.1.1.1.3.cmml" xref="S3.SS0.SSS4.p2.1.m1.1.1.1.1.3">𝜀</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS0.SSS4.p2.1.m1.1c">(1+\varepsilon)</annotation><annotation encoding="application/x-llamapun" id="S3.SS0.SSS4.p2.1.m1.1d">( 1 + italic_ε )</annotation></semantics></math>-approximate dense subgraph detection in the CONGEST model (Table <a class="ltx_ref" href="https://arxiv.org/html/2411.12694v2#S1.T1" title="Table 1 ‣ D: Results in CONGEST ‣ 1.1 Local density and results. ‣ 1 Introduction ‣ Local Density and its Distributed Approximation"><span class="ltx_text ltx_ref_tag">1</span></a>).</p> </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>Conceptual results for local density</h2> <div class="ltx_para" id="S4.p1"> <p class="ltx_p" id="S4.p1.1">Our primary contribution is the definition of local out-degree as a dual to local density.</p> </div> <div class="ltx_theorem ltx_theorem_lemma" 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">Lemma 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.p1.6"><span class="ltx_text ltx_font_italic" id="S4.Thmtheorem1.p1.6.6">For any two locally fair orientations <math alttext="\overrightarrow{G}" class="ltx_Math" display="inline" id="S4.Thmtheorem1.p1.1.1.m1.1"><semantics id="S4.Thmtheorem1.p1.1.1.m1.1a"><mover accent="true" id="S4.Thmtheorem1.p1.1.1.m1.1.1" xref="S4.Thmtheorem1.p1.1.1.m1.1.1.cmml"><mi id="S4.Thmtheorem1.p1.1.1.m1.1.1.2" xref="S4.Thmtheorem1.p1.1.1.m1.1.1.2.cmml">G</mi><mo id="S4.Thmtheorem1.p1.1.1.m1.1.1.1" stretchy="false" xref="S4.Thmtheorem1.p1.1.1.m1.1.1.1.cmml">→</mo></mover><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem1.p1.1.1.m1.1b"><apply id="S4.Thmtheorem1.p1.1.1.m1.1.1.cmml" xref="S4.Thmtheorem1.p1.1.1.m1.1.1"><ci id="S4.Thmtheorem1.p1.1.1.m1.1.1.1.cmml" xref="S4.Thmtheorem1.p1.1.1.m1.1.1.1">→</ci><ci id="S4.Thmtheorem1.p1.1.1.m1.1.1.2.cmml" xref="S4.Thmtheorem1.p1.1.1.m1.1.1.2">𝐺</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem1.p1.1.1.m1.1c">\overrightarrow{G}</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem1.p1.1.1.m1.1d">over→ start_ARG italic_G end_ARG</annotation></semantics></math> or <math alttext="\overrightarrow{G}^{\prime}" class="ltx_Math" display="inline" id="S4.Thmtheorem1.p1.2.2.m2.1"><semantics id="S4.Thmtheorem1.p1.2.2.m2.1a"><msup id="S4.Thmtheorem1.p1.2.2.m2.1.1" xref="S4.Thmtheorem1.p1.2.2.m2.1.1.cmml"><mover accent="true" id="S4.Thmtheorem1.p1.2.2.m2.1.1.2" xref="S4.Thmtheorem1.p1.2.2.m2.1.1.2.cmml"><mi id="S4.Thmtheorem1.p1.2.2.m2.1.1.2.2" xref="S4.Thmtheorem1.p1.2.2.m2.1.1.2.2.cmml">G</mi><mo id="S4.Thmtheorem1.p1.2.2.m2.1.1.2.1" stretchy="false" xref="S4.Thmtheorem1.p1.2.2.m2.1.1.2.1.cmml">→</mo></mover><mo id="S4.Thmtheorem1.p1.2.2.m2.1.1.3" xref="S4.Thmtheorem1.p1.2.2.m2.1.1.3.cmml">′</mo></msup><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem1.p1.2.2.m2.1b"><apply id="S4.Thmtheorem1.p1.2.2.m2.1.1.cmml" xref="S4.Thmtheorem1.p1.2.2.m2.1.1"><csymbol cd="ambiguous" id="S4.Thmtheorem1.p1.2.2.m2.1.1.1.cmml" xref="S4.Thmtheorem1.p1.2.2.m2.1.1">superscript</csymbol><apply id="S4.Thmtheorem1.p1.2.2.m2.1.1.2.cmml" xref="S4.Thmtheorem1.p1.2.2.m2.1.1.2"><ci id="S4.Thmtheorem1.p1.2.2.m2.1.1.2.1.cmml" xref="S4.Thmtheorem1.p1.2.2.m2.1.1.2.1">→</ci><ci id="S4.Thmtheorem1.p1.2.2.m2.1.1.2.2.cmml" xref="S4.Thmtheorem1.p1.2.2.m2.1.1.2.2">𝐺</ci></apply><ci id="S4.Thmtheorem1.p1.2.2.m2.1.1.3.cmml" xref="S4.Thmtheorem1.p1.2.2.m2.1.1.3">′</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem1.p1.2.2.m2.1c">\overrightarrow{G}^{\prime}</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem1.p1.2.2.m2.1d">over→ start_ARG italic_G end_ARG start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math> where a vertex <math alttext="u" class="ltx_Math" display="inline" id="S4.Thmtheorem1.p1.3.3.m3.1"><semantics id="S4.Thmtheorem1.p1.3.3.m3.1a"><mi id="S4.Thmtheorem1.p1.3.3.m3.1.1" xref="S4.Thmtheorem1.p1.3.3.m3.1.1.cmml">u</mi><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem1.p1.3.3.m3.1b"><ci id="S4.Thmtheorem1.p1.3.3.m3.1.1.cmml" xref="S4.Thmtheorem1.p1.3.3.m3.1.1">𝑢</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem1.p1.3.3.m3.1c">u</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem1.p1.3.3.m3.1d">italic_u</annotation></semantics></math> has out-degree <math alttext="\textsl{g}(u)" class="ltx_Math" display="inline" id="S4.Thmtheorem1.p1.4.4.m4.1"><semantics id="S4.Thmtheorem1.p1.4.4.m4.1a"><mrow id="S4.Thmtheorem1.p1.4.4.m4.1.2" xref="S4.Thmtheorem1.p1.4.4.m4.1.2.cmml"><mtext class="ltx_mathvariant_italic" id="S4.Thmtheorem1.p1.4.4.m4.1.2.2" xref="S4.Thmtheorem1.p1.4.4.m4.1.2.2a.cmml">g</mtext><mo id="S4.Thmtheorem1.p1.4.4.m4.1.2.1" xref="S4.Thmtheorem1.p1.4.4.m4.1.2.1.cmml">⁢</mo><mrow id="S4.Thmtheorem1.p1.4.4.m4.1.2.3.2" xref="S4.Thmtheorem1.p1.4.4.m4.1.2.cmml"><mo id="S4.Thmtheorem1.p1.4.4.m4.1.2.3.2.1" stretchy="false" xref="S4.Thmtheorem1.p1.4.4.m4.1.2.cmml">(</mo><mi id="S4.Thmtheorem1.p1.4.4.m4.1.1" xref="S4.Thmtheorem1.p1.4.4.m4.1.1.cmml">u</mi><mo id="S4.Thmtheorem1.p1.4.4.m4.1.2.3.2.2" stretchy="false" xref="S4.Thmtheorem1.p1.4.4.m4.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem1.p1.4.4.m4.1b"><apply id="S4.Thmtheorem1.p1.4.4.m4.1.2.cmml" xref="S4.Thmtheorem1.p1.4.4.m4.1.2"><times id="S4.Thmtheorem1.p1.4.4.m4.1.2.1.cmml" xref="S4.Thmtheorem1.p1.4.4.m4.1.2.1"></times><ci id="S4.Thmtheorem1.p1.4.4.m4.1.2.2a.cmml" xref="S4.Thmtheorem1.p1.4.4.m4.1.2.2"><mtext class="ltx_mathvariant_italic" id="S4.Thmtheorem1.p1.4.4.m4.1.2.2.cmml" xref="S4.Thmtheorem1.p1.4.4.m4.1.2.2">g</mtext></ci><ci id="S4.Thmtheorem1.p1.4.4.m4.1.1.cmml" xref="S4.Thmtheorem1.p1.4.4.m4.1.1">𝑢</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem1.p1.4.4.m4.1c">\textsl{g}(u)</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem1.p1.4.4.m4.1d">g ( italic_u )</annotation></semantics></math> or <math alttext="\textsl{g}^{\prime}(u)" class="ltx_Math" display="inline" id="S4.Thmtheorem1.p1.5.5.m5.1"><semantics id="S4.Thmtheorem1.p1.5.5.m5.1a"><mrow id="S4.Thmtheorem1.p1.5.5.m5.1.2" xref="S4.Thmtheorem1.p1.5.5.m5.1.2.cmml"><msup id="S4.Thmtheorem1.p1.5.5.m5.1.2.2" xref="S4.Thmtheorem1.p1.5.5.m5.1.2.2.cmml"><mtext class="ltx_mathvariant_italic" id="S4.Thmtheorem1.p1.5.5.m5.1.2.2.2" xref="S4.Thmtheorem1.p1.5.5.m5.1.2.2.2a.cmml">g</mtext><mo id="S4.Thmtheorem1.p1.5.5.m5.1.2.2.3" xref="S4.Thmtheorem1.p1.5.5.m5.1.2.2.3.cmml">′</mo></msup><mo id="S4.Thmtheorem1.p1.5.5.m5.1.2.1" xref="S4.Thmtheorem1.p1.5.5.m5.1.2.1.cmml">⁢</mo><mrow id="S4.Thmtheorem1.p1.5.5.m5.1.2.3.2" xref="S4.Thmtheorem1.p1.5.5.m5.1.2.cmml"><mo id="S4.Thmtheorem1.p1.5.5.m5.1.2.3.2.1" stretchy="false" xref="S4.Thmtheorem1.p1.5.5.m5.1.2.cmml">(</mo><mi id="S4.Thmtheorem1.p1.5.5.m5.1.1" xref="S4.Thmtheorem1.p1.5.5.m5.1.1.cmml">u</mi><mo id="S4.Thmtheorem1.p1.5.5.m5.1.2.3.2.2" stretchy="false" xref="S4.Thmtheorem1.p1.5.5.m5.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem1.p1.5.5.m5.1b"><apply id="S4.Thmtheorem1.p1.5.5.m5.1.2.cmml" xref="S4.Thmtheorem1.p1.5.5.m5.1.2"><times id="S4.Thmtheorem1.p1.5.5.m5.1.2.1.cmml" xref="S4.Thmtheorem1.p1.5.5.m5.1.2.1"></times><apply id="S4.Thmtheorem1.p1.5.5.m5.1.2.2.cmml" xref="S4.Thmtheorem1.p1.5.5.m5.1.2.2"><csymbol cd="ambiguous" id="S4.Thmtheorem1.p1.5.5.m5.1.2.2.1.cmml" xref="S4.Thmtheorem1.p1.5.5.m5.1.2.2">superscript</csymbol><ci id="S4.Thmtheorem1.p1.5.5.m5.1.2.2.2a.cmml" xref="S4.Thmtheorem1.p1.5.5.m5.1.2.2.2"><mtext class="ltx_mathvariant_italic" id="S4.Thmtheorem1.p1.5.5.m5.1.2.2.2.cmml" xref="S4.Thmtheorem1.p1.5.5.m5.1.2.2.2">g</mtext></ci><ci id="S4.Thmtheorem1.p1.5.5.m5.1.2.2.3.cmml" xref="S4.Thmtheorem1.p1.5.5.m5.1.2.2.3">′</ci></apply><ci id="S4.Thmtheorem1.p1.5.5.m5.1.1.cmml" xref="S4.Thmtheorem1.p1.5.5.m5.1.1">𝑢</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem1.p1.5.5.m5.1c">\textsl{g}^{\prime}(u)</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem1.p1.5.5.m5.1d">g start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ( italic_u )</annotation></semantics></math> respectively, <math alttext="\textsl{g}(u)=\textsl{g}^{\prime}(u)" class="ltx_Math" display="inline" id="S4.Thmtheorem1.p1.6.6.m6.2"><semantics id="S4.Thmtheorem1.p1.6.6.m6.2a"><mrow id="S4.Thmtheorem1.p1.6.6.m6.2.3" xref="S4.Thmtheorem1.p1.6.6.m6.2.3.cmml"><mrow id="S4.Thmtheorem1.p1.6.6.m6.2.3.2" xref="S4.Thmtheorem1.p1.6.6.m6.2.3.2.cmml"><mtext class="ltx_mathvariant_italic" id="S4.Thmtheorem1.p1.6.6.m6.2.3.2.2" xref="S4.Thmtheorem1.p1.6.6.m6.2.3.2.2a.cmml">g</mtext><mo id="S4.Thmtheorem1.p1.6.6.m6.2.3.2.1" xref="S4.Thmtheorem1.p1.6.6.m6.2.3.2.1.cmml">⁢</mo><mrow id="S4.Thmtheorem1.p1.6.6.m6.2.3.2.3.2" xref="S4.Thmtheorem1.p1.6.6.m6.2.3.2.cmml"><mo id="S4.Thmtheorem1.p1.6.6.m6.2.3.2.3.2.1" stretchy="false" xref="S4.Thmtheorem1.p1.6.6.m6.2.3.2.cmml">(</mo><mi id="S4.Thmtheorem1.p1.6.6.m6.1.1" xref="S4.Thmtheorem1.p1.6.6.m6.1.1.cmml">u</mi><mo id="S4.Thmtheorem1.p1.6.6.m6.2.3.2.3.2.2" stretchy="false" xref="S4.Thmtheorem1.p1.6.6.m6.2.3.2.cmml">)</mo></mrow></mrow><mo id="S4.Thmtheorem1.p1.6.6.m6.2.3.1" xref="S4.Thmtheorem1.p1.6.6.m6.2.3.1.cmml">=</mo><mrow id="S4.Thmtheorem1.p1.6.6.m6.2.3.3" xref="S4.Thmtheorem1.p1.6.6.m6.2.3.3.cmml"><msup id="S4.Thmtheorem1.p1.6.6.m6.2.3.3.2" xref="S4.Thmtheorem1.p1.6.6.m6.2.3.3.2.cmml"><mtext class="ltx_mathvariant_italic" id="S4.Thmtheorem1.p1.6.6.m6.2.3.3.2.2" xref="S4.Thmtheorem1.p1.6.6.m6.2.3.3.2.2a.cmml">g</mtext><mo id="S4.Thmtheorem1.p1.6.6.m6.2.3.3.2.3" xref="S4.Thmtheorem1.p1.6.6.m6.2.3.3.2.3.cmml">′</mo></msup><mo id="S4.Thmtheorem1.p1.6.6.m6.2.3.3.1" xref="S4.Thmtheorem1.p1.6.6.m6.2.3.3.1.cmml">⁢</mo><mrow id="S4.Thmtheorem1.p1.6.6.m6.2.3.3.3.2" xref="S4.Thmtheorem1.p1.6.6.m6.2.3.3.cmml"><mo id="S4.Thmtheorem1.p1.6.6.m6.2.3.3.3.2.1" stretchy="false" xref="S4.Thmtheorem1.p1.6.6.m6.2.3.3.cmml">(</mo><mi id="S4.Thmtheorem1.p1.6.6.m6.2.2" xref="S4.Thmtheorem1.p1.6.6.m6.2.2.cmml">u</mi><mo id="S4.Thmtheorem1.p1.6.6.m6.2.3.3.3.2.2" stretchy="false" xref="S4.Thmtheorem1.p1.6.6.m6.2.3.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem1.p1.6.6.m6.2b"><apply id="S4.Thmtheorem1.p1.6.6.m6.2.3.cmml" xref="S4.Thmtheorem1.p1.6.6.m6.2.3"><eq id="S4.Thmtheorem1.p1.6.6.m6.2.3.1.cmml" xref="S4.Thmtheorem1.p1.6.6.m6.2.3.1"></eq><apply id="S4.Thmtheorem1.p1.6.6.m6.2.3.2.cmml" xref="S4.Thmtheorem1.p1.6.6.m6.2.3.2"><times id="S4.Thmtheorem1.p1.6.6.m6.2.3.2.1.cmml" xref="S4.Thmtheorem1.p1.6.6.m6.2.3.2.1"></times><ci id="S4.Thmtheorem1.p1.6.6.m6.2.3.2.2a.cmml" xref="S4.Thmtheorem1.p1.6.6.m6.2.3.2.2"><mtext class="ltx_mathvariant_italic" id="S4.Thmtheorem1.p1.6.6.m6.2.3.2.2.cmml" xref="S4.Thmtheorem1.p1.6.6.m6.2.3.2.2">g</mtext></ci><ci id="S4.Thmtheorem1.p1.6.6.m6.1.1.cmml" xref="S4.Thmtheorem1.p1.6.6.m6.1.1">𝑢</ci></apply><apply id="S4.Thmtheorem1.p1.6.6.m6.2.3.3.cmml" xref="S4.Thmtheorem1.p1.6.6.m6.2.3.3"><times id="S4.Thmtheorem1.p1.6.6.m6.2.3.3.1.cmml" xref="S4.Thmtheorem1.p1.6.6.m6.2.3.3.1"></times><apply id="S4.Thmtheorem1.p1.6.6.m6.2.3.3.2.cmml" xref="S4.Thmtheorem1.p1.6.6.m6.2.3.3.2"><csymbol cd="ambiguous" id="S4.Thmtheorem1.p1.6.6.m6.2.3.3.2.1.cmml" xref="S4.Thmtheorem1.p1.6.6.m6.2.3.3.2">superscript</csymbol><ci id="S4.Thmtheorem1.p1.6.6.m6.2.3.3.2.2a.cmml" xref="S4.Thmtheorem1.p1.6.6.m6.2.3.3.2.2"><mtext class="ltx_mathvariant_italic" id="S4.Thmtheorem1.p1.6.6.m6.2.3.3.2.2.cmml" xref="S4.Thmtheorem1.p1.6.6.m6.2.3.3.2.2">g</mtext></ci><ci id="S4.Thmtheorem1.p1.6.6.m6.2.3.3.2.3.cmml" xref="S4.Thmtheorem1.p1.6.6.m6.2.3.3.2.3">′</ci></apply><ci id="S4.Thmtheorem1.p1.6.6.m6.2.2.cmml" xref="S4.Thmtheorem1.p1.6.6.m6.2.2">𝑢</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem1.p1.6.6.m6.2c">\textsl{g}(u)=\textsl{g}^{\prime}(u)</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem1.p1.6.6.m6.2d">g ( italic_u ) = g start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ( italic_u )</annotation></semantics></math>.</span></p> </div> </div> <div class="ltx_theorem ltx_theorem_proof" 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">Proof 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.p1.8"><span class="ltx_text ltx_font_italic" id="S4.Thmtheorem2.p1.8.8">Suppose for the sake of contradiction that there exists two locally fair orientations <math alttext="(\overrightarrow{G},\overrightarrow{G}^{\prime})" class="ltx_Math" display="inline" id="S4.Thmtheorem2.p1.1.1.m1.2"><semantics id="S4.Thmtheorem2.p1.1.1.m1.2a"><mrow id="S4.Thmtheorem2.p1.1.1.m1.2.2.1" xref="S4.Thmtheorem2.p1.1.1.m1.2.2.2.cmml"><mo id="S4.Thmtheorem2.p1.1.1.m1.2.2.1.2" stretchy="false" xref="S4.Thmtheorem2.p1.1.1.m1.2.2.2.cmml">(</mo><mover accent="true" id="S4.Thmtheorem2.p1.1.1.m1.1.1" xref="S4.Thmtheorem2.p1.1.1.m1.1.1.cmml"><mi id="S4.Thmtheorem2.p1.1.1.m1.1.1.2" xref="S4.Thmtheorem2.p1.1.1.m1.1.1.2.cmml">G</mi><mo id="S4.Thmtheorem2.p1.1.1.m1.1.1.1" stretchy="false" xref="S4.Thmtheorem2.p1.1.1.m1.1.1.1.cmml">→</mo></mover><mo id="S4.Thmtheorem2.p1.1.1.m1.2.2.1.3" xref="S4.Thmtheorem2.p1.1.1.m1.2.2.2.cmml">,</mo><msup id="S4.Thmtheorem2.p1.1.1.m1.2.2.1.1" xref="S4.Thmtheorem2.p1.1.1.m1.2.2.1.1.cmml"><mover accent="true" id="S4.Thmtheorem2.p1.1.1.m1.2.2.1.1.2" xref="S4.Thmtheorem2.p1.1.1.m1.2.2.1.1.2.cmml"><mi id="S4.Thmtheorem2.p1.1.1.m1.2.2.1.1.2.2" xref="S4.Thmtheorem2.p1.1.1.m1.2.2.1.1.2.2.cmml">G</mi><mo id="S4.Thmtheorem2.p1.1.1.m1.2.2.1.1.2.1" stretchy="false" xref="S4.Thmtheorem2.p1.1.1.m1.2.2.1.1.2.1.cmml">→</mo></mover><mo id="S4.Thmtheorem2.p1.1.1.m1.2.2.1.1.3" xref="S4.Thmtheorem2.p1.1.1.m1.2.2.1.1.3.cmml">′</mo></msup><mo id="S4.Thmtheorem2.p1.1.1.m1.2.2.1.4" stretchy="false" xref="S4.Thmtheorem2.p1.1.1.m1.2.2.2.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem2.p1.1.1.m1.2b"><interval closure="open" id="S4.Thmtheorem2.p1.1.1.m1.2.2.2.cmml" xref="S4.Thmtheorem2.p1.1.1.m1.2.2.1"><apply id="S4.Thmtheorem2.p1.1.1.m1.1.1.cmml" xref="S4.Thmtheorem2.p1.1.1.m1.1.1"><ci id="S4.Thmtheorem2.p1.1.1.m1.1.1.1.cmml" xref="S4.Thmtheorem2.p1.1.1.m1.1.1.1">→</ci><ci id="S4.Thmtheorem2.p1.1.1.m1.1.1.2.cmml" xref="S4.Thmtheorem2.p1.1.1.m1.1.1.2">𝐺</ci></apply><apply id="S4.Thmtheorem2.p1.1.1.m1.2.2.1.1.cmml" xref="S4.Thmtheorem2.p1.1.1.m1.2.2.1.1"><csymbol cd="ambiguous" id="S4.Thmtheorem2.p1.1.1.m1.2.2.1.1.1.cmml" xref="S4.Thmtheorem2.p1.1.1.m1.2.2.1.1">superscript</csymbol><apply id="S4.Thmtheorem2.p1.1.1.m1.2.2.1.1.2.cmml" xref="S4.Thmtheorem2.p1.1.1.m1.2.2.1.1.2"><ci id="S4.Thmtheorem2.p1.1.1.m1.2.2.1.1.2.1.cmml" xref="S4.Thmtheorem2.p1.1.1.m1.2.2.1.1.2.1">→</ci><ci id="S4.Thmtheorem2.p1.1.1.m1.2.2.1.1.2.2.cmml" xref="S4.Thmtheorem2.p1.1.1.m1.2.2.1.1.2.2">𝐺</ci></apply><ci id="S4.Thmtheorem2.p1.1.1.m1.2.2.1.1.3.cmml" xref="S4.Thmtheorem2.p1.1.1.m1.2.2.1.1.3">′</ci></apply></interval></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem2.p1.1.1.m1.2c">(\overrightarrow{G},\overrightarrow{G}^{\prime})</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem2.p1.1.1.m1.2d">( over→ start_ARG italic_G end_ARG , over→ start_ARG italic_G end_ARG start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT )</annotation></semantics></math> and a vertex <math alttext="u\in V" class="ltx_Math" display="inline" id="S4.Thmtheorem2.p1.2.2.m2.1"><semantics id="S4.Thmtheorem2.p1.2.2.m2.1a"><mrow id="S4.Thmtheorem2.p1.2.2.m2.1.1" xref="S4.Thmtheorem2.p1.2.2.m2.1.1.cmml"><mi id="S4.Thmtheorem2.p1.2.2.m2.1.1.2" xref="S4.Thmtheorem2.p1.2.2.m2.1.1.2.cmml">u</mi><mo id="S4.Thmtheorem2.p1.2.2.m2.1.1.1" xref="S4.Thmtheorem2.p1.2.2.m2.1.1.1.cmml">∈</mo><mi id="S4.Thmtheorem2.p1.2.2.m2.1.1.3" xref="S4.Thmtheorem2.p1.2.2.m2.1.1.3.cmml">V</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem2.p1.2.2.m2.1b"><apply id="S4.Thmtheorem2.p1.2.2.m2.1.1.cmml" xref="S4.Thmtheorem2.p1.2.2.m2.1.1"><in id="S4.Thmtheorem2.p1.2.2.m2.1.1.1.cmml" xref="S4.Thmtheorem2.p1.2.2.m2.1.1.1"></in><ci id="S4.Thmtheorem2.p1.2.2.m2.1.1.2.cmml" xref="S4.Thmtheorem2.p1.2.2.m2.1.1.2">𝑢</ci><ci id="S4.Thmtheorem2.p1.2.2.m2.1.1.3.cmml" xref="S4.Thmtheorem2.p1.2.2.m2.1.1.3">𝑉</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem2.p1.2.2.m2.1c">u\in V</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem2.p1.2.2.m2.1d">italic_u ∈ italic_V</annotation></semantics></math> where <math alttext="\textsl{g}(u)>\textsl{g}^{\prime}(u)" class="ltx_Math" display="inline" id="S4.Thmtheorem2.p1.3.3.m3.2"><semantics id="S4.Thmtheorem2.p1.3.3.m3.2a"><mrow id="S4.Thmtheorem2.p1.3.3.m3.2.3" xref="S4.Thmtheorem2.p1.3.3.m3.2.3.cmml"><mrow id="S4.Thmtheorem2.p1.3.3.m3.2.3.2" xref="S4.Thmtheorem2.p1.3.3.m3.2.3.2.cmml"><mtext class="ltx_mathvariant_italic" id="S4.Thmtheorem2.p1.3.3.m3.2.3.2.2" xref="S4.Thmtheorem2.p1.3.3.m3.2.3.2.2a.cmml">g</mtext><mo id="S4.Thmtheorem2.p1.3.3.m3.2.3.2.1" xref="S4.Thmtheorem2.p1.3.3.m3.2.3.2.1.cmml">⁢</mo><mrow id="S4.Thmtheorem2.p1.3.3.m3.2.3.2.3.2" xref="S4.Thmtheorem2.p1.3.3.m3.2.3.2.cmml"><mo id="S4.Thmtheorem2.p1.3.3.m3.2.3.2.3.2.1" stretchy="false" xref="S4.Thmtheorem2.p1.3.3.m3.2.3.2.cmml">(</mo><mi id="S4.Thmtheorem2.p1.3.3.m3.1.1" xref="S4.Thmtheorem2.p1.3.3.m3.1.1.cmml">u</mi><mo id="S4.Thmtheorem2.p1.3.3.m3.2.3.2.3.2.2" stretchy="false" xref="S4.Thmtheorem2.p1.3.3.m3.2.3.2.cmml">)</mo></mrow></mrow><mo id="S4.Thmtheorem2.p1.3.3.m3.2.3.1" xref="S4.Thmtheorem2.p1.3.3.m3.2.3.1.cmml">></mo><mrow id="S4.Thmtheorem2.p1.3.3.m3.2.3.3" xref="S4.Thmtheorem2.p1.3.3.m3.2.3.3.cmml"><msup id="S4.Thmtheorem2.p1.3.3.m3.2.3.3.2" xref="S4.Thmtheorem2.p1.3.3.m3.2.3.3.2.cmml"><mtext class="ltx_mathvariant_italic" id="S4.Thmtheorem2.p1.3.3.m3.2.3.3.2.2" xref="S4.Thmtheorem2.p1.3.3.m3.2.3.3.2.2a.cmml">g</mtext><mo id="S4.Thmtheorem2.p1.3.3.m3.2.3.3.2.3" xref="S4.Thmtheorem2.p1.3.3.m3.2.3.3.2.3.cmml">′</mo></msup><mo id="S4.Thmtheorem2.p1.3.3.m3.2.3.3.1" xref="S4.Thmtheorem2.p1.3.3.m3.2.3.3.1.cmml">⁢</mo><mrow id="S4.Thmtheorem2.p1.3.3.m3.2.3.3.3.2" xref="S4.Thmtheorem2.p1.3.3.m3.2.3.3.cmml"><mo id="S4.Thmtheorem2.p1.3.3.m3.2.3.3.3.2.1" stretchy="false" xref="S4.Thmtheorem2.p1.3.3.m3.2.3.3.cmml">(</mo><mi id="S4.Thmtheorem2.p1.3.3.m3.2.2" xref="S4.Thmtheorem2.p1.3.3.m3.2.2.cmml">u</mi><mo id="S4.Thmtheorem2.p1.3.3.m3.2.3.3.3.2.2" stretchy="false" xref="S4.Thmtheorem2.p1.3.3.m3.2.3.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem2.p1.3.3.m3.2b"><apply id="S4.Thmtheorem2.p1.3.3.m3.2.3.cmml" xref="S4.Thmtheorem2.p1.3.3.m3.2.3"><gt id="S4.Thmtheorem2.p1.3.3.m3.2.3.1.cmml" xref="S4.Thmtheorem2.p1.3.3.m3.2.3.1"></gt><apply id="S4.Thmtheorem2.p1.3.3.m3.2.3.2.cmml" xref="S4.Thmtheorem2.p1.3.3.m3.2.3.2"><times id="S4.Thmtheorem2.p1.3.3.m3.2.3.2.1.cmml" xref="S4.Thmtheorem2.p1.3.3.m3.2.3.2.1"></times><ci id="S4.Thmtheorem2.p1.3.3.m3.2.3.2.2a.cmml" xref="S4.Thmtheorem2.p1.3.3.m3.2.3.2.2"><mtext class="ltx_mathvariant_italic" id="S4.Thmtheorem2.p1.3.3.m3.2.3.2.2.cmml" xref="S4.Thmtheorem2.p1.3.3.m3.2.3.2.2">g</mtext></ci><ci id="S4.Thmtheorem2.p1.3.3.m3.1.1.cmml" xref="S4.Thmtheorem2.p1.3.3.m3.1.1">𝑢</ci></apply><apply id="S4.Thmtheorem2.p1.3.3.m3.2.3.3.cmml" xref="S4.Thmtheorem2.p1.3.3.m3.2.3.3"><times id="S4.Thmtheorem2.p1.3.3.m3.2.3.3.1.cmml" xref="S4.Thmtheorem2.p1.3.3.m3.2.3.3.1"></times><apply id="S4.Thmtheorem2.p1.3.3.m3.2.3.3.2.cmml" xref="S4.Thmtheorem2.p1.3.3.m3.2.3.3.2"><csymbol cd="ambiguous" id="S4.Thmtheorem2.p1.3.3.m3.2.3.3.2.1.cmml" xref="S4.Thmtheorem2.p1.3.3.m3.2.3.3.2">superscript</csymbol><ci id="S4.Thmtheorem2.p1.3.3.m3.2.3.3.2.2a.cmml" xref="S4.Thmtheorem2.p1.3.3.m3.2.3.3.2.2"><mtext class="ltx_mathvariant_italic" id="S4.Thmtheorem2.p1.3.3.m3.2.3.3.2.2.cmml" xref="S4.Thmtheorem2.p1.3.3.m3.2.3.3.2.2">g</mtext></ci><ci id="S4.Thmtheorem2.p1.3.3.m3.2.3.3.2.3.cmml" xref="S4.Thmtheorem2.p1.3.3.m3.2.3.3.2.3">′</ci></apply><ci id="S4.Thmtheorem2.p1.3.3.m3.2.2.cmml" xref="S4.Thmtheorem2.p1.3.3.m3.2.2">𝑢</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem2.p1.3.3.m3.2c">\textsl{g}(u)>\textsl{g}^{\prime}(u)</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem2.p1.3.3.m3.2d">g ( italic_u ) > g start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ( italic_u )</annotation></semantics></math>. We define their <em class="ltx_emph ltx_font_upright" id="S4.Thmtheorem2.p1.8.8.1">symmetric difference</em> graph <math alttext="S" class="ltx_Math" display="inline" id="S4.Thmtheorem2.p1.4.4.m4.1"><semantics id="S4.Thmtheorem2.p1.4.4.m4.1a"><mi id="S4.Thmtheorem2.p1.4.4.m4.1.1" xref="S4.Thmtheorem2.p1.4.4.m4.1.1.cmml">S</mi><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem2.p1.4.4.m4.1b"><ci id="S4.Thmtheorem2.p1.4.4.m4.1.1.cmml" xref="S4.Thmtheorem2.p1.4.4.m4.1.1">𝑆</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem2.p1.4.4.m4.1c">S</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem2.p1.4.4.m4.1d">italic_S</annotation></semantics></math> as a digraph where the vertices are <math alttext="V" class="ltx_Math" display="inline" id="S4.Thmtheorem2.p1.5.5.m5.1"><semantics id="S4.Thmtheorem2.p1.5.5.m5.1a"><mi id="S4.Thmtheorem2.p1.5.5.m5.1.1" xref="S4.Thmtheorem2.p1.5.5.m5.1.1.cmml">V</mi><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem2.p1.5.5.m5.1b"><ci id="S4.Thmtheorem2.p1.5.5.m5.1.1.cmml" xref="S4.Thmtheorem2.p1.5.5.m5.1.1">𝑉</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem2.p1.5.5.m5.1c">V</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem2.p1.5.5.m5.1d">italic_V</annotation></semantics></math> and there exists an edge <math alttext="\overrightarrow{ab}" class="ltx_Math" display="inline" id="S4.Thmtheorem2.p1.6.6.m6.1"><semantics id="S4.Thmtheorem2.p1.6.6.m6.1a"><mover accent="true" id="S4.Thmtheorem2.p1.6.6.m6.1.1" xref="S4.Thmtheorem2.p1.6.6.m6.1.1.cmml"><mrow id="S4.Thmtheorem2.p1.6.6.m6.1.1.2" xref="S4.Thmtheorem2.p1.6.6.m6.1.1.2.cmml"><mi id="S4.Thmtheorem2.p1.6.6.m6.1.1.2.2" xref="S4.Thmtheorem2.p1.6.6.m6.1.1.2.2.cmml">a</mi><mo id="S4.Thmtheorem2.p1.6.6.m6.1.1.2.1" xref="S4.Thmtheorem2.p1.6.6.m6.1.1.2.1.cmml">⁢</mo><mi id="S4.Thmtheorem2.p1.6.6.m6.1.1.2.3" xref="S4.Thmtheorem2.p1.6.6.m6.1.1.2.3.cmml">b</mi></mrow><mo id="S4.Thmtheorem2.p1.6.6.m6.1.1.1" stretchy="false" xref="S4.Thmtheorem2.p1.6.6.m6.1.1.1.cmml">→</mo></mover><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem2.p1.6.6.m6.1b"><apply id="S4.Thmtheorem2.p1.6.6.m6.1.1.cmml" xref="S4.Thmtheorem2.p1.6.6.m6.1.1"><ci id="S4.Thmtheorem2.p1.6.6.m6.1.1.1.cmml" xref="S4.Thmtheorem2.p1.6.6.m6.1.1.1">→</ci><apply id="S4.Thmtheorem2.p1.6.6.m6.1.1.2.cmml" xref="S4.Thmtheorem2.p1.6.6.m6.1.1.2"><times id="S4.Thmtheorem2.p1.6.6.m6.1.1.2.1.cmml" xref="S4.Thmtheorem2.p1.6.6.m6.1.1.2.1"></times><ci id="S4.Thmtheorem2.p1.6.6.m6.1.1.2.2.cmml" xref="S4.Thmtheorem2.p1.6.6.m6.1.1.2.2">𝑎</ci><ci id="S4.Thmtheorem2.p1.6.6.m6.1.1.2.3.cmml" xref="S4.Thmtheorem2.p1.6.6.m6.1.1.2.3">𝑏</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem2.p1.6.6.m6.1c">\overrightarrow{ab}</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem2.p1.6.6.m6.1d">over→ start_ARG italic_a italic_b end_ARG</annotation></semantics></math> whenever <math alttext="\textsl{g}(a\!\to\!b)>\textsl{g}^{\prime}(a\!\to\!b)" class="ltx_Math" display="inline" id="S4.Thmtheorem2.p1.7.7.m7.2"><semantics id="S4.Thmtheorem2.p1.7.7.m7.2a"><mrow id="S4.Thmtheorem2.p1.7.7.m7.2.2" xref="S4.Thmtheorem2.p1.7.7.m7.2.2.cmml"><mrow id="S4.Thmtheorem2.p1.7.7.m7.1.1.1" xref="S4.Thmtheorem2.p1.7.7.m7.1.1.1.cmml"><mtext class="ltx_mathvariant_italic" id="S4.Thmtheorem2.p1.7.7.m7.1.1.1.3" xref="S4.Thmtheorem2.p1.7.7.m7.1.1.1.3a.cmml">g</mtext><mo id="S4.Thmtheorem2.p1.7.7.m7.1.1.1.2" xref="S4.Thmtheorem2.p1.7.7.m7.1.1.1.2.cmml">⁢</mo><mrow id="S4.Thmtheorem2.p1.7.7.m7.1.1.1.1.1" xref="S4.Thmtheorem2.p1.7.7.m7.1.1.1.1.1.1.cmml"><mo id="S4.Thmtheorem2.p1.7.7.m7.1.1.1.1.1.2" stretchy="false" xref="S4.Thmtheorem2.p1.7.7.m7.1.1.1.1.1.1.cmml">(</mo><mrow id="S4.Thmtheorem2.p1.7.7.m7.1.1.1.1.1.1" xref="S4.Thmtheorem2.p1.7.7.m7.1.1.1.1.1.1.cmml"><mi id="S4.Thmtheorem2.p1.7.7.m7.1.1.1.1.1.1.2" xref="S4.Thmtheorem2.p1.7.7.m7.1.1.1.1.1.1.2.cmml">a</mi><mo id="S4.Thmtheorem2.p1.7.7.m7.1.1.1.1.1.1.1" lspace="0.108em" rspace="0.108em" stretchy="false" xref="S4.Thmtheorem2.p1.7.7.m7.1.1.1.1.1.1.1.cmml">→</mo><mi id="S4.Thmtheorem2.p1.7.7.m7.1.1.1.1.1.1.3" xref="S4.Thmtheorem2.p1.7.7.m7.1.1.1.1.1.1.3.cmml">b</mi></mrow><mo id="S4.Thmtheorem2.p1.7.7.m7.1.1.1.1.1.3" stretchy="false" xref="S4.Thmtheorem2.p1.7.7.m7.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S4.Thmtheorem2.p1.7.7.m7.2.2.3" xref="S4.Thmtheorem2.p1.7.7.m7.2.2.3.cmml">></mo><mrow id="S4.Thmtheorem2.p1.7.7.m7.2.2.2" xref="S4.Thmtheorem2.p1.7.7.m7.2.2.2.cmml"><msup id="S4.Thmtheorem2.p1.7.7.m7.2.2.2.3" xref="S4.Thmtheorem2.p1.7.7.m7.2.2.2.3.cmml"><mtext class="ltx_mathvariant_italic" id="S4.Thmtheorem2.p1.7.7.m7.2.2.2.3.2" xref="S4.Thmtheorem2.p1.7.7.m7.2.2.2.3.2a.cmml">g</mtext><mo id="S4.Thmtheorem2.p1.7.7.m7.2.2.2.3.3" xref="S4.Thmtheorem2.p1.7.7.m7.2.2.2.3.3.cmml">′</mo></msup><mo id="S4.Thmtheorem2.p1.7.7.m7.2.2.2.2" xref="S4.Thmtheorem2.p1.7.7.m7.2.2.2.2.cmml">⁢</mo><mrow id="S4.Thmtheorem2.p1.7.7.m7.2.2.2.1.1" xref="S4.Thmtheorem2.p1.7.7.m7.2.2.2.1.1.1.cmml"><mo id="S4.Thmtheorem2.p1.7.7.m7.2.2.2.1.1.2" stretchy="false" xref="S4.Thmtheorem2.p1.7.7.m7.2.2.2.1.1.1.cmml">(</mo><mrow id="S4.Thmtheorem2.p1.7.7.m7.2.2.2.1.1.1" xref="S4.Thmtheorem2.p1.7.7.m7.2.2.2.1.1.1.cmml"><mi id="S4.Thmtheorem2.p1.7.7.m7.2.2.2.1.1.1.2" xref="S4.Thmtheorem2.p1.7.7.m7.2.2.2.1.1.1.2.cmml">a</mi><mo id="S4.Thmtheorem2.p1.7.7.m7.2.2.2.1.1.1.1" lspace="0.108em" rspace="0.108em" stretchy="false" xref="S4.Thmtheorem2.p1.7.7.m7.2.2.2.1.1.1.1.cmml">→</mo><mi id="S4.Thmtheorem2.p1.7.7.m7.2.2.2.1.1.1.3" xref="S4.Thmtheorem2.p1.7.7.m7.2.2.2.1.1.1.3.cmml">b</mi></mrow><mo id="S4.Thmtheorem2.p1.7.7.m7.2.2.2.1.1.3" stretchy="false" xref="S4.Thmtheorem2.p1.7.7.m7.2.2.2.1.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem2.p1.7.7.m7.2b"><apply id="S4.Thmtheorem2.p1.7.7.m7.2.2.cmml" xref="S4.Thmtheorem2.p1.7.7.m7.2.2"><gt id="S4.Thmtheorem2.p1.7.7.m7.2.2.3.cmml" xref="S4.Thmtheorem2.p1.7.7.m7.2.2.3"></gt><apply id="S4.Thmtheorem2.p1.7.7.m7.1.1.1.cmml" xref="S4.Thmtheorem2.p1.7.7.m7.1.1.1"><times id="S4.Thmtheorem2.p1.7.7.m7.1.1.1.2.cmml" xref="S4.Thmtheorem2.p1.7.7.m7.1.1.1.2"></times><ci id="S4.Thmtheorem2.p1.7.7.m7.1.1.1.3a.cmml" xref="S4.Thmtheorem2.p1.7.7.m7.1.1.1.3"><mtext class="ltx_mathvariant_italic" id="S4.Thmtheorem2.p1.7.7.m7.1.1.1.3.cmml" xref="S4.Thmtheorem2.p1.7.7.m7.1.1.1.3">g</mtext></ci><apply id="S4.Thmtheorem2.p1.7.7.m7.1.1.1.1.1.1.cmml" xref="S4.Thmtheorem2.p1.7.7.m7.1.1.1.1.1"><ci id="S4.Thmtheorem2.p1.7.7.m7.1.1.1.1.1.1.1.cmml" xref="S4.Thmtheorem2.p1.7.7.m7.1.1.1.1.1.1.1">→</ci><ci id="S4.Thmtheorem2.p1.7.7.m7.1.1.1.1.1.1.2.cmml" xref="S4.Thmtheorem2.p1.7.7.m7.1.1.1.1.1.1.2">𝑎</ci><ci id="S4.Thmtheorem2.p1.7.7.m7.1.1.1.1.1.1.3.cmml" xref="S4.Thmtheorem2.p1.7.7.m7.1.1.1.1.1.1.3">𝑏</ci></apply></apply><apply id="S4.Thmtheorem2.p1.7.7.m7.2.2.2.cmml" xref="S4.Thmtheorem2.p1.7.7.m7.2.2.2"><times id="S4.Thmtheorem2.p1.7.7.m7.2.2.2.2.cmml" xref="S4.Thmtheorem2.p1.7.7.m7.2.2.2.2"></times><apply id="S4.Thmtheorem2.p1.7.7.m7.2.2.2.3.cmml" xref="S4.Thmtheorem2.p1.7.7.m7.2.2.2.3"><csymbol cd="ambiguous" id="S4.Thmtheorem2.p1.7.7.m7.2.2.2.3.1.cmml" xref="S4.Thmtheorem2.p1.7.7.m7.2.2.2.3">superscript</csymbol><ci id="S4.Thmtheorem2.p1.7.7.m7.2.2.2.3.2a.cmml" xref="S4.Thmtheorem2.p1.7.7.m7.2.2.2.3.2"><mtext class="ltx_mathvariant_italic" id="S4.Thmtheorem2.p1.7.7.m7.2.2.2.3.2.cmml" xref="S4.Thmtheorem2.p1.7.7.m7.2.2.2.3.2">g</mtext></ci><ci id="S4.Thmtheorem2.p1.7.7.m7.2.2.2.3.3.cmml" xref="S4.Thmtheorem2.p1.7.7.m7.2.2.2.3.3">′</ci></apply><apply id="S4.Thmtheorem2.p1.7.7.m7.2.2.2.1.1.1.cmml" xref="S4.Thmtheorem2.p1.7.7.m7.2.2.2.1.1"><ci id="S4.Thmtheorem2.p1.7.7.m7.2.2.2.1.1.1.1.cmml" xref="S4.Thmtheorem2.p1.7.7.m7.2.2.2.1.1.1.1">→</ci><ci id="S4.Thmtheorem2.p1.7.7.m7.2.2.2.1.1.1.2.cmml" xref="S4.Thmtheorem2.p1.7.7.m7.2.2.2.1.1.1.2">𝑎</ci><ci id="S4.Thmtheorem2.p1.7.7.m7.2.2.2.1.1.1.3.cmml" xref="S4.Thmtheorem2.p1.7.7.m7.2.2.2.1.1.1.3">𝑏</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem2.p1.7.7.m7.2c">\textsl{g}(a\!\to\!b)>\textsl{g}^{\prime}(a\!\to\!b)</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem2.p1.7.7.m7.2d">g ( italic_a → italic_b ) > g start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ( italic_a → italic_b )</annotation></semantics></math>. We may assume that <math alttext="S" class="ltx_Math" display="inline" id="S4.Thmtheorem2.p1.8.8.m8.1"><semantics id="S4.Thmtheorem2.p1.8.8.m8.1a"><mi id="S4.Thmtheorem2.p1.8.8.m8.1.1" xref="S4.Thmtheorem2.p1.8.8.m8.1.1.cmml">S</mi><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem2.p1.8.8.m8.1b"><ci id="S4.Thmtheorem2.p1.8.8.m8.1.1.cmml" xref="S4.Thmtheorem2.p1.8.8.m8.1.1">𝑆</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem2.p1.8.8.m8.1c">S</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem2.p1.8.8.m8.1d">italic_S</annotation></semantics></math> contains no directed cycles:</span></p> </div> <div class="ltx_para" id="S4.Thmtheorem2.p2"> <p class="ltx_p" id="S4.Thmtheorem2.p2.9"><span class="ltx_text ltx_font_italic" id="S4.Thmtheorem2.p2.9.9">Indeed if <math alttext="S" class="ltx_Math" display="inline" id="S4.Thmtheorem2.p2.1.1.m1.1"><semantics id="S4.Thmtheorem2.p2.1.1.m1.1a"><mi id="S4.Thmtheorem2.p2.1.1.m1.1.1" xref="S4.Thmtheorem2.p2.1.1.m1.1.1.cmml">S</mi><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem2.p2.1.1.m1.1b"><ci id="S4.Thmtheorem2.p2.1.1.m1.1.1.cmml" xref="S4.Thmtheorem2.p2.1.1.m1.1.1">𝑆</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem2.p2.1.1.m1.1c">S</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem2.p2.1.1.m1.1d">italic_S</annotation></semantics></math> contains any directed cycle <math alttext="\pi" class="ltx_Math" display="inline" id="S4.Thmtheorem2.p2.2.2.m2.1"><semantics id="S4.Thmtheorem2.p2.2.2.m2.1a"><mi id="S4.Thmtheorem2.p2.2.2.m2.1.1" xref="S4.Thmtheorem2.p2.2.2.m2.1.1.cmml">π</mi><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem2.p2.2.2.m2.1b"><ci id="S4.Thmtheorem2.p2.2.2.m2.1.1.cmml" xref="S4.Thmtheorem2.p2.2.2.m2.1.1">𝜋</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem2.p2.2.2.m2.1c">\pi</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem2.p2.2.2.m2.1d">italic_π</annotation></semantics></math> we change <math alttext="\overrightarrow{G^{\prime}}" class="ltx_Math" display="inline" id="S4.Thmtheorem2.p2.3.3.m3.1"><semantics id="S4.Thmtheorem2.p2.3.3.m3.1a"><mover accent="true" id="S4.Thmtheorem2.p2.3.3.m3.1.1" xref="S4.Thmtheorem2.p2.3.3.m3.1.1.cmml"><msup id="S4.Thmtheorem2.p2.3.3.m3.1.1.2" xref="S4.Thmtheorem2.p2.3.3.m3.1.1.2.cmml"><mi id="S4.Thmtheorem2.p2.3.3.m3.1.1.2.2" xref="S4.Thmtheorem2.p2.3.3.m3.1.1.2.2.cmml">G</mi><mo id="S4.Thmtheorem2.p2.3.3.m3.1.1.2.3" xref="S4.Thmtheorem2.p2.3.3.m3.1.1.2.3.cmml">′</mo></msup><mo id="S4.Thmtheorem2.p2.3.3.m3.1.1.1" stretchy="false" xref="S4.Thmtheorem2.p2.3.3.m3.1.1.1.cmml">→</mo></mover><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem2.p2.3.3.m3.1b"><apply id="S4.Thmtheorem2.p2.3.3.m3.1.1.cmml" xref="S4.Thmtheorem2.p2.3.3.m3.1.1"><ci id="S4.Thmtheorem2.p2.3.3.m3.1.1.1.cmml" xref="S4.Thmtheorem2.p2.3.3.m3.1.1.1">→</ci><apply id="S4.Thmtheorem2.p2.3.3.m3.1.1.2.cmml" xref="S4.Thmtheorem2.p2.3.3.m3.1.1.2"><csymbol cd="ambiguous" id="S4.Thmtheorem2.p2.3.3.m3.1.1.2.1.cmml" xref="S4.Thmtheorem2.p2.3.3.m3.1.1.2">superscript</csymbol><ci id="S4.Thmtheorem2.p2.3.3.m3.1.1.2.2.cmml" xref="S4.Thmtheorem2.p2.3.3.m3.1.1.2.2">𝐺</ci><ci id="S4.Thmtheorem2.p2.3.3.m3.1.1.2.3.cmml" xref="S4.Thmtheorem2.p2.3.3.m3.1.1.2.3">′</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem2.p2.3.3.m3.1c">\overrightarrow{G^{\prime}}</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem2.p2.3.3.m3.1d">over→ start_ARG italic_G start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_ARG</annotation></semantics></math>, where for all <math alttext="\overrightarrow{ab}\in\pi" class="ltx_Math" display="inline" id="S4.Thmtheorem2.p2.4.4.m4.1"><semantics id="S4.Thmtheorem2.p2.4.4.m4.1a"><mrow id="S4.Thmtheorem2.p2.4.4.m4.1.1" xref="S4.Thmtheorem2.p2.4.4.m4.1.1.cmml"><mover accent="true" id="S4.Thmtheorem2.p2.4.4.m4.1.1.2" xref="S4.Thmtheorem2.p2.4.4.m4.1.1.2.cmml"><mrow id="S4.Thmtheorem2.p2.4.4.m4.1.1.2.2" xref="S4.Thmtheorem2.p2.4.4.m4.1.1.2.2.cmml"><mi id="S4.Thmtheorem2.p2.4.4.m4.1.1.2.2.2" xref="S4.Thmtheorem2.p2.4.4.m4.1.1.2.2.2.cmml">a</mi><mo id="S4.Thmtheorem2.p2.4.4.m4.1.1.2.2.1" xref="S4.Thmtheorem2.p2.4.4.m4.1.1.2.2.1.cmml">⁢</mo><mi id="S4.Thmtheorem2.p2.4.4.m4.1.1.2.2.3" xref="S4.Thmtheorem2.p2.4.4.m4.1.1.2.2.3.cmml">b</mi></mrow><mo id="S4.Thmtheorem2.p2.4.4.m4.1.1.2.1" stretchy="false" xref="S4.Thmtheorem2.p2.4.4.m4.1.1.2.1.cmml">→</mo></mover><mo id="S4.Thmtheorem2.p2.4.4.m4.1.1.1" xref="S4.Thmtheorem2.p2.4.4.m4.1.1.1.cmml">∈</mo><mi id="S4.Thmtheorem2.p2.4.4.m4.1.1.3" xref="S4.Thmtheorem2.p2.4.4.m4.1.1.3.cmml">π</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem2.p2.4.4.m4.1b"><apply id="S4.Thmtheorem2.p2.4.4.m4.1.1.cmml" xref="S4.Thmtheorem2.p2.4.4.m4.1.1"><in id="S4.Thmtheorem2.p2.4.4.m4.1.1.1.cmml" xref="S4.Thmtheorem2.p2.4.4.m4.1.1.1"></in><apply id="S4.Thmtheorem2.p2.4.4.m4.1.1.2.cmml" xref="S4.Thmtheorem2.p2.4.4.m4.1.1.2"><ci id="S4.Thmtheorem2.p2.4.4.m4.1.1.2.1.cmml" xref="S4.Thmtheorem2.p2.4.4.m4.1.1.2.1">→</ci><apply id="S4.Thmtheorem2.p2.4.4.m4.1.1.2.2.cmml" xref="S4.Thmtheorem2.p2.4.4.m4.1.1.2.2"><times id="S4.Thmtheorem2.p2.4.4.m4.1.1.2.2.1.cmml" xref="S4.Thmtheorem2.p2.4.4.m4.1.1.2.2.1"></times><ci id="S4.Thmtheorem2.p2.4.4.m4.1.1.2.2.2.cmml" xref="S4.Thmtheorem2.p2.4.4.m4.1.1.2.2.2">𝑎</ci><ci id="S4.Thmtheorem2.p2.4.4.m4.1.1.2.2.3.cmml" xref="S4.Thmtheorem2.p2.4.4.m4.1.1.2.2.3">𝑏</ci></apply></apply><ci id="S4.Thmtheorem2.p2.4.4.m4.1.1.3.cmml" xref="S4.Thmtheorem2.p2.4.4.m4.1.1.3">𝜋</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem2.p2.4.4.m4.1c">\overrightarrow{ab}\in\pi</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem2.p2.4.4.m4.1d">over→ start_ARG italic_a italic_b end_ARG ∈ italic_π</annotation></semantics></math> we slightly increase <math alttext="\textsl{g}^{\prime}(a\!\to\!b)" class="ltx_Math" display="inline" id="S4.Thmtheorem2.p2.5.5.m5.1"><semantics id="S4.Thmtheorem2.p2.5.5.m5.1a"><mrow id="S4.Thmtheorem2.p2.5.5.m5.1.1" xref="S4.Thmtheorem2.p2.5.5.m5.1.1.cmml"><msup id="S4.Thmtheorem2.p2.5.5.m5.1.1.3" xref="S4.Thmtheorem2.p2.5.5.m5.1.1.3.cmml"><mtext class="ltx_mathvariant_italic" id="S4.Thmtheorem2.p2.5.5.m5.1.1.3.2" xref="S4.Thmtheorem2.p2.5.5.m5.1.1.3.2a.cmml">g</mtext><mo id="S4.Thmtheorem2.p2.5.5.m5.1.1.3.3" xref="S4.Thmtheorem2.p2.5.5.m5.1.1.3.3.cmml">′</mo></msup><mo id="S4.Thmtheorem2.p2.5.5.m5.1.1.2" xref="S4.Thmtheorem2.p2.5.5.m5.1.1.2.cmml">⁢</mo><mrow id="S4.Thmtheorem2.p2.5.5.m5.1.1.1.1" xref="S4.Thmtheorem2.p2.5.5.m5.1.1.1.1.1.cmml"><mo id="S4.Thmtheorem2.p2.5.5.m5.1.1.1.1.2" stretchy="false" xref="S4.Thmtheorem2.p2.5.5.m5.1.1.1.1.1.cmml">(</mo><mrow id="S4.Thmtheorem2.p2.5.5.m5.1.1.1.1.1" xref="S4.Thmtheorem2.p2.5.5.m5.1.1.1.1.1.cmml"><mi id="S4.Thmtheorem2.p2.5.5.m5.1.1.1.1.1.2" xref="S4.Thmtheorem2.p2.5.5.m5.1.1.1.1.1.2.cmml">a</mi><mo id="S4.Thmtheorem2.p2.5.5.m5.1.1.1.1.1.1" lspace="0.108em" rspace="0.108em" stretchy="false" xref="S4.Thmtheorem2.p2.5.5.m5.1.1.1.1.1.1.cmml">→</mo><mi id="S4.Thmtheorem2.p2.5.5.m5.1.1.1.1.1.3" xref="S4.Thmtheorem2.p2.5.5.m5.1.1.1.1.1.3.cmml">b</mi></mrow><mo id="S4.Thmtheorem2.p2.5.5.m5.1.1.1.1.3" stretchy="false" xref="S4.Thmtheorem2.p2.5.5.m5.1.1.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem2.p2.5.5.m5.1b"><apply id="S4.Thmtheorem2.p2.5.5.m5.1.1.cmml" xref="S4.Thmtheorem2.p2.5.5.m5.1.1"><times id="S4.Thmtheorem2.p2.5.5.m5.1.1.2.cmml" xref="S4.Thmtheorem2.p2.5.5.m5.1.1.2"></times><apply id="S4.Thmtheorem2.p2.5.5.m5.1.1.3.cmml" xref="S4.Thmtheorem2.p2.5.5.m5.1.1.3"><csymbol cd="ambiguous" id="S4.Thmtheorem2.p2.5.5.m5.1.1.3.1.cmml" xref="S4.Thmtheorem2.p2.5.5.m5.1.1.3">superscript</csymbol><ci id="S4.Thmtheorem2.p2.5.5.m5.1.1.3.2a.cmml" xref="S4.Thmtheorem2.p2.5.5.m5.1.1.3.2"><mtext class="ltx_mathvariant_italic" id="S4.Thmtheorem2.p2.5.5.m5.1.1.3.2.cmml" xref="S4.Thmtheorem2.p2.5.5.m5.1.1.3.2">g</mtext></ci><ci id="S4.Thmtheorem2.p2.5.5.m5.1.1.3.3.cmml" xref="S4.Thmtheorem2.p2.5.5.m5.1.1.3.3">′</ci></apply><apply id="S4.Thmtheorem2.p2.5.5.m5.1.1.1.1.1.cmml" xref="S4.Thmtheorem2.p2.5.5.m5.1.1.1.1"><ci id="S4.Thmtheorem2.p2.5.5.m5.1.1.1.1.1.1.cmml" xref="S4.Thmtheorem2.p2.5.5.m5.1.1.1.1.1.1">→</ci><ci id="S4.Thmtheorem2.p2.5.5.m5.1.1.1.1.1.2.cmml" xref="S4.Thmtheorem2.p2.5.5.m5.1.1.1.1.1.2">𝑎</ci><ci id="S4.Thmtheorem2.p2.5.5.m5.1.1.1.1.1.3.cmml" xref="S4.Thmtheorem2.p2.5.5.m5.1.1.1.1.1.3">𝑏</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem2.p2.5.5.m5.1c">\textsl{g}^{\prime}(a\!\to\!b)</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem2.p2.5.5.m5.1d">g start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ( italic_a → italic_b )</annotation></semantics></math> until <math alttext="S" class="ltx_Math" display="inline" id="S4.Thmtheorem2.p2.6.6.m6.1"><semantics id="S4.Thmtheorem2.p2.6.6.m6.1a"><mi id="S4.Thmtheorem2.p2.6.6.m6.1.1" xref="S4.Thmtheorem2.p2.6.6.m6.1.1.cmml">S</mi><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem2.p2.6.6.m6.1b"><ci id="S4.Thmtheorem2.p2.6.6.m6.1.1.cmml" xref="S4.Thmtheorem2.p2.6.6.m6.1.1">𝑆</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem2.p2.6.6.m6.1c">S</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem2.p2.6.6.m6.1d">italic_S</annotation></semantics></math> loses an edge. This operation does not change the out-degree of any vertex in <math alttext="\overrightarrow{G}^{\prime}" class="ltx_Math" display="inline" id="S4.Thmtheorem2.p2.7.7.m7.1"><semantics id="S4.Thmtheorem2.p2.7.7.m7.1a"><msup id="S4.Thmtheorem2.p2.7.7.m7.1.1" xref="S4.Thmtheorem2.p2.7.7.m7.1.1.cmml"><mover accent="true" id="S4.Thmtheorem2.p2.7.7.m7.1.1.2" xref="S4.Thmtheorem2.p2.7.7.m7.1.1.2.cmml"><mi id="S4.Thmtheorem2.p2.7.7.m7.1.1.2.2" xref="S4.Thmtheorem2.p2.7.7.m7.1.1.2.2.cmml">G</mi><mo id="S4.Thmtheorem2.p2.7.7.m7.1.1.2.1" stretchy="false" xref="S4.Thmtheorem2.p2.7.7.m7.1.1.2.1.cmml">→</mo></mover><mo id="S4.Thmtheorem2.p2.7.7.m7.1.1.3" xref="S4.Thmtheorem2.p2.7.7.m7.1.1.3.cmml">′</mo></msup><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem2.p2.7.7.m7.1b"><apply id="S4.Thmtheorem2.p2.7.7.m7.1.1.cmml" xref="S4.Thmtheorem2.p2.7.7.m7.1.1"><csymbol cd="ambiguous" id="S4.Thmtheorem2.p2.7.7.m7.1.1.1.cmml" xref="S4.Thmtheorem2.p2.7.7.m7.1.1">superscript</csymbol><apply id="S4.Thmtheorem2.p2.7.7.m7.1.1.2.cmml" xref="S4.Thmtheorem2.p2.7.7.m7.1.1.2"><ci id="S4.Thmtheorem2.p2.7.7.m7.1.1.2.1.cmml" xref="S4.Thmtheorem2.p2.7.7.m7.1.1.2.1">→</ci><ci id="S4.Thmtheorem2.p2.7.7.m7.1.1.2.2.cmml" xref="S4.Thmtheorem2.p2.7.7.m7.1.1.2.2">𝐺</ci></apply><ci id="S4.Thmtheorem2.p2.7.7.m7.1.1.3.cmml" xref="S4.Thmtheorem2.p2.7.7.m7.1.1.3">′</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem2.p2.7.7.m7.1c">\overrightarrow{G}^{\prime}</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem2.p2.7.7.m7.1d">over→ start_ARG italic_G end_ARG start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math>. So, we still have two locally fair orientations <math alttext="(\overrightarrow{G},\overrightarrow{G}^{\prime})" class="ltx_Math" display="inline" id="S4.Thmtheorem2.p2.8.8.m8.2"><semantics id="S4.Thmtheorem2.p2.8.8.m8.2a"><mrow id="S4.Thmtheorem2.p2.8.8.m8.2.2.1" xref="S4.Thmtheorem2.p2.8.8.m8.2.2.2.cmml"><mo id="S4.Thmtheorem2.p2.8.8.m8.2.2.1.2" stretchy="false" xref="S4.Thmtheorem2.p2.8.8.m8.2.2.2.cmml">(</mo><mover accent="true" id="S4.Thmtheorem2.p2.8.8.m8.1.1" xref="S4.Thmtheorem2.p2.8.8.m8.1.1.cmml"><mi id="S4.Thmtheorem2.p2.8.8.m8.1.1.2" xref="S4.Thmtheorem2.p2.8.8.m8.1.1.2.cmml">G</mi><mo id="S4.Thmtheorem2.p2.8.8.m8.1.1.1" stretchy="false" xref="S4.Thmtheorem2.p2.8.8.m8.1.1.1.cmml">→</mo></mover><mo id="S4.Thmtheorem2.p2.8.8.m8.2.2.1.3" xref="S4.Thmtheorem2.p2.8.8.m8.2.2.2.cmml">,</mo><msup id="S4.Thmtheorem2.p2.8.8.m8.2.2.1.1" xref="S4.Thmtheorem2.p2.8.8.m8.2.2.1.1.cmml"><mover accent="true" id="S4.Thmtheorem2.p2.8.8.m8.2.2.1.1.2" xref="S4.Thmtheorem2.p2.8.8.m8.2.2.1.1.2.cmml"><mi id="S4.Thmtheorem2.p2.8.8.m8.2.2.1.1.2.2" xref="S4.Thmtheorem2.p2.8.8.m8.2.2.1.1.2.2.cmml">G</mi><mo id="S4.Thmtheorem2.p2.8.8.m8.2.2.1.1.2.1" stretchy="false" xref="S4.Thmtheorem2.p2.8.8.m8.2.2.1.1.2.1.cmml">→</mo></mover><mo id="S4.Thmtheorem2.p2.8.8.m8.2.2.1.1.3" xref="S4.Thmtheorem2.p2.8.8.m8.2.2.1.1.3.cmml">′</mo></msup><mo id="S4.Thmtheorem2.p2.8.8.m8.2.2.1.4" stretchy="false" xref="S4.Thmtheorem2.p2.8.8.m8.2.2.2.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem2.p2.8.8.m8.2b"><interval closure="open" id="S4.Thmtheorem2.p2.8.8.m8.2.2.2.cmml" xref="S4.Thmtheorem2.p2.8.8.m8.2.2.1"><apply id="S4.Thmtheorem2.p2.8.8.m8.1.1.cmml" xref="S4.Thmtheorem2.p2.8.8.m8.1.1"><ci id="S4.Thmtheorem2.p2.8.8.m8.1.1.1.cmml" xref="S4.Thmtheorem2.p2.8.8.m8.1.1.1">→</ci><ci id="S4.Thmtheorem2.p2.8.8.m8.1.1.2.cmml" xref="S4.Thmtheorem2.p2.8.8.m8.1.1.2">𝐺</ci></apply><apply id="S4.Thmtheorem2.p2.8.8.m8.2.2.1.1.cmml" xref="S4.Thmtheorem2.p2.8.8.m8.2.2.1.1"><csymbol cd="ambiguous" id="S4.Thmtheorem2.p2.8.8.m8.2.2.1.1.1.cmml" xref="S4.Thmtheorem2.p2.8.8.m8.2.2.1.1">superscript</csymbol><apply id="S4.Thmtheorem2.p2.8.8.m8.2.2.1.1.2.cmml" xref="S4.Thmtheorem2.p2.8.8.m8.2.2.1.1.2"><ci id="S4.Thmtheorem2.p2.8.8.m8.2.2.1.1.2.1.cmml" xref="S4.Thmtheorem2.p2.8.8.m8.2.2.1.1.2.1">→</ci><ci id="S4.Thmtheorem2.p2.8.8.m8.2.2.1.1.2.2.cmml" xref="S4.Thmtheorem2.p2.8.8.m8.2.2.1.1.2.2">𝐺</ci></apply><ci id="S4.Thmtheorem2.p2.8.8.m8.2.2.1.1.3.cmml" xref="S4.Thmtheorem2.p2.8.8.m8.2.2.1.1.3">′</ci></apply></interval></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem2.p2.8.8.m8.2c">(\overrightarrow{G},\overrightarrow{G}^{\prime})</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem2.p2.8.8.m8.2d">( over→ start_ARG italic_G end_ARG , over→ start_ARG italic_G end_ARG start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT )</annotation></semantics></math> with <math alttext="\textsl{g}(u)>\textsl{g}^{\prime}(u)" class="ltx_Math" display="inline" id="S4.Thmtheorem2.p2.9.9.m9.2"><semantics id="S4.Thmtheorem2.p2.9.9.m9.2a"><mrow id="S4.Thmtheorem2.p2.9.9.m9.2.3" xref="S4.Thmtheorem2.p2.9.9.m9.2.3.cmml"><mrow id="S4.Thmtheorem2.p2.9.9.m9.2.3.2" xref="S4.Thmtheorem2.p2.9.9.m9.2.3.2.cmml"><mtext class="ltx_mathvariant_italic" id="S4.Thmtheorem2.p2.9.9.m9.2.3.2.2" xref="S4.Thmtheorem2.p2.9.9.m9.2.3.2.2a.cmml">g</mtext><mo id="S4.Thmtheorem2.p2.9.9.m9.2.3.2.1" xref="S4.Thmtheorem2.p2.9.9.m9.2.3.2.1.cmml">⁢</mo><mrow id="S4.Thmtheorem2.p2.9.9.m9.2.3.2.3.2" xref="S4.Thmtheorem2.p2.9.9.m9.2.3.2.cmml"><mo id="S4.Thmtheorem2.p2.9.9.m9.2.3.2.3.2.1" stretchy="false" xref="S4.Thmtheorem2.p2.9.9.m9.2.3.2.cmml">(</mo><mi id="S4.Thmtheorem2.p2.9.9.m9.1.1" xref="S4.Thmtheorem2.p2.9.9.m9.1.1.cmml">u</mi><mo id="S4.Thmtheorem2.p2.9.9.m9.2.3.2.3.2.2" stretchy="false" xref="S4.Thmtheorem2.p2.9.9.m9.2.3.2.cmml">)</mo></mrow></mrow><mo id="S4.Thmtheorem2.p2.9.9.m9.2.3.1" xref="S4.Thmtheorem2.p2.9.9.m9.2.3.1.cmml">></mo><mrow id="S4.Thmtheorem2.p2.9.9.m9.2.3.3" xref="S4.Thmtheorem2.p2.9.9.m9.2.3.3.cmml"><msup id="S4.Thmtheorem2.p2.9.9.m9.2.3.3.2" xref="S4.Thmtheorem2.p2.9.9.m9.2.3.3.2.cmml"><mtext class="ltx_mathvariant_italic" id="S4.Thmtheorem2.p2.9.9.m9.2.3.3.2.2" xref="S4.Thmtheorem2.p2.9.9.m9.2.3.3.2.2a.cmml">g</mtext><mo id="S4.Thmtheorem2.p2.9.9.m9.2.3.3.2.3" xref="S4.Thmtheorem2.p2.9.9.m9.2.3.3.2.3.cmml">′</mo></msup><mo id="S4.Thmtheorem2.p2.9.9.m9.2.3.3.1" xref="S4.Thmtheorem2.p2.9.9.m9.2.3.3.1.cmml">⁢</mo><mrow id="S4.Thmtheorem2.p2.9.9.m9.2.3.3.3.2" xref="S4.Thmtheorem2.p2.9.9.m9.2.3.3.cmml"><mo id="S4.Thmtheorem2.p2.9.9.m9.2.3.3.3.2.1" stretchy="false" xref="S4.Thmtheorem2.p2.9.9.m9.2.3.3.cmml">(</mo><mi id="S4.Thmtheorem2.p2.9.9.m9.2.2" xref="S4.Thmtheorem2.p2.9.9.m9.2.2.cmml">u</mi><mo id="S4.Thmtheorem2.p2.9.9.m9.2.3.3.3.2.2" stretchy="false" xref="S4.Thmtheorem2.p2.9.9.m9.2.3.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem2.p2.9.9.m9.2b"><apply id="S4.Thmtheorem2.p2.9.9.m9.2.3.cmml" xref="S4.Thmtheorem2.p2.9.9.m9.2.3"><gt id="S4.Thmtheorem2.p2.9.9.m9.2.3.1.cmml" xref="S4.Thmtheorem2.p2.9.9.m9.2.3.1"></gt><apply id="S4.Thmtheorem2.p2.9.9.m9.2.3.2.cmml" xref="S4.Thmtheorem2.p2.9.9.m9.2.3.2"><times id="S4.Thmtheorem2.p2.9.9.m9.2.3.2.1.cmml" xref="S4.Thmtheorem2.p2.9.9.m9.2.3.2.1"></times><ci id="S4.Thmtheorem2.p2.9.9.m9.2.3.2.2a.cmml" xref="S4.Thmtheorem2.p2.9.9.m9.2.3.2.2"><mtext class="ltx_mathvariant_italic" id="S4.Thmtheorem2.p2.9.9.m9.2.3.2.2.cmml" xref="S4.Thmtheorem2.p2.9.9.m9.2.3.2.2">g</mtext></ci><ci id="S4.Thmtheorem2.p2.9.9.m9.1.1.cmml" xref="S4.Thmtheorem2.p2.9.9.m9.1.1">𝑢</ci></apply><apply id="S4.Thmtheorem2.p2.9.9.m9.2.3.3.cmml" xref="S4.Thmtheorem2.p2.9.9.m9.2.3.3"><times id="S4.Thmtheorem2.p2.9.9.m9.2.3.3.1.cmml" xref="S4.Thmtheorem2.p2.9.9.m9.2.3.3.1"></times><apply id="S4.Thmtheorem2.p2.9.9.m9.2.3.3.2.cmml" xref="S4.Thmtheorem2.p2.9.9.m9.2.3.3.2"><csymbol cd="ambiguous" id="S4.Thmtheorem2.p2.9.9.m9.2.3.3.2.1.cmml" xref="S4.Thmtheorem2.p2.9.9.m9.2.3.3.2">superscript</csymbol><ci id="S4.Thmtheorem2.p2.9.9.m9.2.3.3.2.2a.cmml" xref="S4.Thmtheorem2.p2.9.9.m9.2.3.3.2.2"><mtext class="ltx_mathvariant_italic" id="S4.Thmtheorem2.p2.9.9.m9.2.3.3.2.2.cmml" xref="S4.Thmtheorem2.p2.9.9.m9.2.3.3.2.2">g</mtext></ci><ci id="S4.Thmtheorem2.p2.9.9.m9.2.3.3.2.3.cmml" xref="S4.Thmtheorem2.p2.9.9.m9.2.3.3.2.3">′</ci></apply><ci id="S4.Thmtheorem2.p2.9.9.m9.2.2.cmml" xref="S4.Thmtheorem2.p2.9.9.m9.2.2">𝑢</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem2.p2.9.9.m9.2c">\textsl{g}(u)>\textsl{g}^{\prime}(u)</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem2.p2.9.9.m9.2d">g ( italic_u ) > g start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ( italic_u )</annotation></semantics></math>.</span></p> </div> <div class="ltx_para" id="S4.Thmtheorem2.p3"> <p class="ltx_p" id="S4.Thmtheorem2.p3.10"><span class="ltx_text ltx_font_italic" id="S4.Thmtheorem2.p3.10.10">Since <math alttext="\textsl{g}(u)>\textsl{g}^{\prime}(u)" class="ltx_Math" display="inline" id="S4.Thmtheorem2.p3.1.1.m1.2"><semantics id="S4.Thmtheorem2.p3.1.1.m1.2a"><mrow id="S4.Thmtheorem2.p3.1.1.m1.2.3" xref="S4.Thmtheorem2.p3.1.1.m1.2.3.cmml"><mrow id="S4.Thmtheorem2.p3.1.1.m1.2.3.2" xref="S4.Thmtheorem2.p3.1.1.m1.2.3.2.cmml"><mtext class="ltx_mathvariant_italic" id="S4.Thmtheorem2.p3.1.1.m1.2.3.2.2" xref="S4.Thmtheorem2.p3.1.1.m1.2.3.2.2a.cmml">g</mtext><mo id="S4.Thmtheorem2.p3.1.1.m1.2.3.2.1" xref="S4.Thmtheorem2.p3.1.1.m1.2.3.2.1.cmml">⁢</mo><mrow id="S4.Thmtheorem2.p3.1.1.m1.2.3.2.3.2" xref="S4.Thmtheorem2.p3.1.1.m1.2.3.2.cmml"><mo id="S4.Thmtheorem2.p3.1.1.m1.2.3.2.3.2.1" stretchy="false" xref="S4.Thmtheorem2.p3.1.1.m1.2.3.2.cmml">(</mo><mi id="S4.Thmtheorem2.p3.1.1.m1.1.1" xref="S4.Thmtheorem2.p3.1.1.m1.1.1.cmml">u</mi><mo id="S4.Thmtheorem2.p3.1.1.m1.2.3.2.3.2.2" stretchy="false" xref="S4.Thmtheorem2.p3.1.1.m1.2.3.2.cmml">)</mo></mrow></mrow><mo id="S4.Thmtheorem2.p3.1.1.m1.2.3.1" xref="S4.Thmtheorem2.p3.1.1.m1.2.3.1.cmml">></mo><mrow id="S4.Thmtheorem2.p3.1.1.m1.2.3.3" xref="S4.Thmtheorem2.p3.1.1.m1.2.3.3.cmml"><msup id="S4.Thmtheorem2.p3.1.1.m1.2.3.3.2" xref="S4.Thmtheorem2.p3.1.1.m1.2.3.3.2.cmml"><mtext class="ltx_mathvariant_italic" id="S4.Thmtheorem2.p3.1.1.m1.2.3.3.2.2" xref="S4.Thmtheorem2.p3.1.1.m1.2.3.3.2.2a.cmml">g</mtext><mo id="S4.Thmtheorem2.p3.1.1.m1.2.3.3.2.3" xref="S4.Thmtheorem2.p3.1.1.m1.2.3.3.2.3.cmml">′</mo></msup><mo id="S4.Thmtheorem2.p3.1.1.m1.2.3.3.1" xref="S4.Thmtheorem2.p3.1.1.m1.2.3.3.1.cmml">⁢</mo><mrow id="S4.Thmtheorem2.p3.1.1.m1.2.3.3.3.2" xref="S4.Thmtheorem2.p3.1.1.m1.2.3.3.cmml"><mo id="S4.Thmtheorem2.p3.1.1.m1.2.3.3.3.2.1" stretchy="false" xref="S4.Thmtheorem2.p3.1.1.m1.2.3.3.cmml">(</mo><mi id="S4.Thmtheorem2.p3.1.1.m1.2.2" xref="S4.Thmtheorem2.p3.1.1.m1.2.2.cmml">u</mi><mo id="S4.Thmtheorem2.p3.1.1.m1.2.3.3.3.2.2" stretchy="false" xref="S4.Thmtheorem2.p3.1.1.m1.2.3.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem2.p3.1.1.m1.2b"><apply id="S4.Thmtheorem2.p3.1.1.m1.2.3.cmml" xref="S4.Thmtheorem2.p3.1.1.m1.2.3"><gt id="S4.Thmtheorem2.p3.1.1.m1.2.3.1.cmml" xref="S4.Thmtheorem2.p3.1.1.m1.2.3.1"></gt><apply id="S4.Thmtheorem2.p3.1.1.m1.2.3.2.cmml" xref="S4.Thmtheorem2.p3.1.1.m1.2.3.2"><times id="S4.Thmtheorem2.p3.1.1.m1.2.3.2.1.cmml" xref="S4.Thmtheorem2.p3.1.1.m1.2.3.2.1"></times><ci id="S4.Thmtheorem2.p3.1.1.m1.2.3.2.2a.cmml" xref="S4.Thmtheorem2.p3.1.1.m1.2.3.2.2"><mtext class="ltx_mathvariant_italic" id="S4.Thmtheorem2.p3.1.1.m1.2.3.2.2.cmml" xref="S4.Thmtheorem2.p3.1.1.m1.2.3.2.2">g</mtext></ci><ci id="S4.Thmtheorem2.p3.1.1.m1.1.1.cmml" xref="S4.Thmtheorem2.p3.1.1.m1.1.1">𝑢</ci></apply><apply id="S4.Thmtheorem2.p3.1.1.m1.2.3.3.cmml" xref="S4.Thmtheorem2.p3.1.1.m1.2.3.3"><times id="S4.Thmtheorem2.p3.1.1.m1.2.3.3.1.cmml" xref="S4.Thmtheorem2.p3.1.1.m1.2.3.3.1"></times><apply id="S4.Thmtheorem2.p3.1.1.m1.2.3.3.2.cmml" xref="S4.Thmtheorem2.p3.1.1.m1.2.3.3.2"><csymbol cd="ambiguous" id="S4.Thmtheorem2.p3.1.1.m1.2.3.3.2.1.cmml" xref="S4.Thmtheorem2.p3.1.1.m1.2.3.3.2">superscript</csymbol><ci id="S4.Thmtheorem2.p3.1.1.m1.2.3.3.2.2a.cmml" xref="S4.Thmtheorem2.p3.1.1.m1.2.3.3.2.2"><mtext class="ltx_mathvariant_italic" id="S4.Thmtheorem2.p3.1.1.m1.2.3.3.2.2.cmml" xref="S4.Thmtheorem2.p3.1.1.m1.2.3.3.2.2">g</mtext></ci><ci id="S4.Thmtheorem2.p3.1.1.m1.2.3.3.2.3.cmml" xref="S4.Thmtheorem2.p3.1.1.m1.2.3.3.2.3">′</ci></apply><ci id="S4.Thmtheorem2.p3.1.1.m1.2.2.cmml" xref="S4.Thmtheorem2.p3.1.1.m1.2.2">𝑢</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem2.p3.1.1.m1.2c">\textsl{g}(u)>\textsl{g}^{\prime}(u)</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem2.p3.1.1.m1.2d">g ( italic_u ) > g start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ( italic_u )</annotation></semantics></math>, the vertex <math alttext="u" class="ltx_Math" display="inline" id="S4.Thmtheorem2.p3.2.2.m2.1"><semantics id="S4.Thmtheorem2.p3.2.2.m2.1a"><mi id="S4.Thmtheorem2.p3.2.2.m2.1.1" xref="S4.Thmtheorem2.p3.2.2.m2.1.1.cmml">u</mi><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem2.p3.2.2.m2.1b"><ci id="S4.Thmtheorem2.p3.2.2.m2.1.1.cmml" xref="S4.Thmtheorem2.p3.2.2.m2.1.1">𝑢</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem2.p3.2.2.m2.1c">u</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem2.p3.2.2.m2.1d">italic_u</annotation></semantics></math> must have at least one out-edge in <math alttext="S" class="ltx_Math" display="inline" id="S4.Thmtheorem2.p3.3.3.m3.1"><semantics id="S4.Thmtheorem2.p3.3.3.m3.1a"><mi id="S4.Thmtheorem2.p3.3.3.m3.1.1" xref="S4.Thmtheorem2.p3.3.3.m3.1.1.cmml">S</mi><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem2.p3.3.3.m3.1b"><ci id="S4.Thmtheorem2.p3.3.3.m3.1.1.cmml" xref="S4.Thmtheorem2.p3.3.3.m3.1.1">𝑆</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem2.p3.3.3.m3.1c">S</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem2.p3.3.3.m3.1d">italic_S</annotation></semantics></math> and since <math alttext="S" class="ltx_Math" display="inline" id="S4.Thmtheorem2.p3.4.4.m4.1"><semantics id="S4.Thmtheorem2.p3.4.4.m4.1a"><mi id="S4.Thmtheorem2.p3.4.4.m4.1.1" xref="S4.Thmtheorem2.p3.4.4.m4.1.1.cmml">S</mi><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem2.p3.4.4.m4.1b"><ci id="S4.Thmtheorem2.p3.4.4.m4.1.1.cmml" xref="S4.Thmtheorem2.p3.4.4.m4.1.1">𝑆</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem2.p3.4.4.m4.1c">S</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem2.p3.4.4.m4.1d">italic_S</annotation></semantics></math> has no cycles, it follows that <math alttext="u" class="ltx_Math" display="inline" id="S4.Thmtheorem2.p3.5.5.m5.1"><semantics id="S4.Thmtheorem2.p3.5.5.m5.1a"><mi id="S4.Thmtheorem2.p3.5.5.m5.1.1" xref="S4.Thmtheorem2.p3.5.5.m5.1.1.cmml">u</mi><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem2.p3.5.5.m5.1b"><ci id="S4.Thmtheorem2.p3.5.5.m5.1.1.cmml" xref="S4.Thmtheorem2.p3.5.5.m5.1.1">𝑢</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem2.p3.5.5.m5.1c">u</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem2.p3.5.5.m5.1d">italic_u</annotation></semantics></math> must have some directed path <math alttext="\pi_{v}" class="ltx_Math" display="inline" id="S4.Thmtheorem2.p3.6.6.m6.1"><semantics id="S4.Thmtheorem2.p3.6.6.m6.1a"><msub id="S4.Thmtheorem2.p3.6.6.m6.1.1" xref="S4.Thmtheorem2.p3.6.6.m6.1.1.cmml"><mi id="S4.Thmtheorem2.p3.6.6.m6.1.1.2" xref="S4.Thmtheorem2.p3.6.6.m6.1.1.2.cmml">π</mi><mi id="S4.Thmtheorem2.p3.6.6.m6.1.1.3" xref="S4.Thmtheorem2.p3.6.6.m6.1.1.3.cmml">v</mi></msub><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem2.p3.6.6.m6.1b"><apply id="S4.Thmtheorem2.p3.6.6.m6.1.1.cmml" xref="S4.Thmtheorem2.p3.6.6.m6.1.1"><csymbol cd="ambiguous" id="S4.Thmtheorem2.p3.6.6.m6.1.1.1.cmml" xref="S4.Thmtheorem2.p3.6.6.m6.1.1">subscript</csymbol><ci id="S4.Thmtheorem2.p3.6.6.m6.1.1.2.cmml" xref="S4.Thmtheorem2.p3.6.6.m6.1.1.2">𝜋</ci><ci id="S4.Thmtheorem2.p3.6.6.m6.1.1.3.cmml" xref="S4.Thmtheorem2.p3.6.6.m6.1.1.3">𝑣</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem2.p3.6.6.m6.1c">\pi_{v}</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem2.p3.6.6.m6.1d">italic_π start_POSTSUBSCRIPT italic_v end_POSTSUBSCRIPT</annotation></semantics></math> to a sink <math alttext="v" class="ltx_Math" display="inline" id="S4.Thmtheorem2.p3.7.7.m7.1"><semantics id="S4.Thmtheorem2.p3.7.7.m7.1a"><mi id="S4.Thmtheorem2.p3.7.7.m7.1.1" xref="S4.Thmtheorem2.p3.7.7.m7.1.1.cmml">v</mi><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem2.p3.7.7.m7.1b"><ci id="S4.Thmtheorem2.p3.7.7.m7.1.1.cmml" xref="S4.Thmtheorem2.p3.7.7.m7.1.1">𝑣</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem2.p3.7.7.m7.1c">v</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem2.p3.7.7.m7.1d">italic_v</annotation></semantics></math> in <math alttext="S" class="ltx_Math" display="inline" id="S4.Thmtheorem2.p3.8.8.m8.1"><semantics id="S4.Thmtheorem2.p3.8.8.m8.1a"><mi id="S4.Thmtheorem2.p3.8.8.m8.1.1" xref="S4.Thmtheorem2.p3.8.8.m8.1.1.cmml">S</mi><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem2.p3.8.8.m8.1b"><ci id="S4.Thmtheorem2.p3.8.8.m8.1.1.cmml" xref="S4.Thmtheorem2.p3.8.8.m8.1.1">𝑆</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem2.p3.8.8.m8.1c">S</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem2.p3.8.8.m8.1d">italic_S</annotation></semantics></math>. Since <math alttext="v" class="ltx_Math" display="inline" id="S4.Thmtheorem2.p3.9.9.m9.1"><semantics id="S4.Thmtheorem2.p3.9.9.m9.1a"><mi id="S4.Thmtheorem2.p3.9.9.m9.1.1" xref="S4.Thmtheorem2.p3.9.9.m9.1.1.cmml">v</mi><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem2.p3.9.9.m9.1b"><ci id="S4.Thmtheorem2.p3.9.9.m9.1.1.cmml" xref="S4.Thmtheorem2.p3.9.9.m9.1.1">𝑣</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem2.p3.9.9.m9.1c">v</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem2.p3.9.9.m9.1d">italic_v</annotation></semantics></math> is a sink in the symmetric difference graph it follows that <math alttext="\textsl{g}(v)<\textsl{g}^{\prime}(v)" class="ltx_Math" display="inline" id="S4.Thmtheorem2.p3.10.10.m10.2"><semantics id="S4.Thmtheorem2.p3.10.10.m10.2a"><mrow id="S4.Thmtheorem2.p3.10.10.m10.2.3" xref="S4.Thmtheorem2.p3.10.10.m10.2.3.cmml"><mrow id="S4.Thmtheorem2.p3.10.10.m10.2.3.2" xref="S4.Thmtheorem2.p3.10.10.m10.2.3.2.cmml"><mtext class="ltx_mathvariant_italic" id="S4.Thmtheorem2.p3.10.10.m10.2.3.2.2" xref="S4.Thmtheorem2.p3.10.10.m10.2.3.2.2a.cmml">g</mtext><mo id="S4.Thmtheorem2.p3.10.10.m10.2.3.2.1" xref="S4.Thmtheorem2.p3.10.10.m10.2.3.2.1.cmml">⁢</mo><mrow id="S4.Thmtheorem2.p3.10.10.m10.2.3.2.3.2" xref="S4.Thmtheorem2.p3.10.10.m10.2.3.2.cmml"><mo id="S4.Thmtheorem2.p3.10.10.m10.2.3.2.3.2.1" stretchy="false" xref="S4.Thmtheorem2.p3.10.10.m10.2.3.2.cmml">(</mo><mi id="S4.Thmtheorem2.p3.10.10.m10.1.1" xref="S4.Thmtheorem2.p3.10.10.m10.1.1.cmml">v</mi><mo id="S4.Thmtheorem2.p3.10.10.m10.2.3.2.3.2.2" stretchy="false" xref="S4.Thmtheorem2.p3.10.10.m10.2.3.2.cmml">)</mo></mrow></mrow><mo id="S4.Thmtheorem2.p3.10.10.m10.2.3.1" xref="S4.Thmtheorem2.p3.10.10.m10.2.3.1.cmml"><</mo><mrow id="S4.Thmtheorem2.p3.10.10.m10.2.3.3" xref="S4.Thmtheorem2.p3.10.10.m10.2.3.3.cmml"><msup id="S4.Thmtheorem2.p3.10.10.m10.2.3.3.2" xref="S4.Thmtheorem2.p3.10.10.m10.2.3.3.2.cmml"><mtext class="ltx_mathvariant_italic" id="S4.Thmtheorem2.p3.10.10.m10.2.3.3.2.2" xref="S4.Thmtheorem2.p3.10.10.m10.2.3.3.2.2a.cmml">g</mtext><mo id="S4.Thmtheorem2.p3.10.10.m10.2.3.3.2.3" xref="S4.Thmtheorem2.p3.10.10.m10.2.3.3.2.3.cmml">′</mo></msup><mo id="S4.Thmtheorem2.p3.10.10.m10.2.3.3.1" xref="S4.Thmtheorem2.p3.10.10.m10.2.3.3.1.cmml">⁢</mo><mrow id="S4.Thmtheorem2.p3.10.10.m10.2.3.3.3.2" xref="S4.Thmtheorem2.p3.10.10.m10.2.3.3.cmml"><mo id="S4.Thmtheorem2.p3.10.10.m10.2.3.3.3.2.1" stretchy="false" xref="S4.Thmtheorem2.p3.10.10.m10.2.3.3.cmml">(</mo><mi id="S4.Thmtheorem2.p3.10.10.m10.2.2" xref="S4.Thmtheorem2.p3.10.10.m10.2.2.cmml">v</mi><mo id="S4.Thmtheorem2.p3.10.10.m10.2.3.3.3.2.2" stretchy="false" xref="S4.Thmtheorem2.p3.10.10.m10.2.3.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem2.p3.10.10.m10.2b"><apply id="S4.Thmtheorem2.p3.10.10.m10.2.3.cmml" xref="S4.Thmtheorem2.p3.10.10.m10.2.3"><lt id="S4.Thmtheorem2.p3.10.10.m10.2.3.1.cmml" xref="S4.Thmtheorem2.p3.10.10.m10.2.3.1"></lt><apply id="S4.Thmtheorem2.p3.10.10.m10.2.3.2.cmml" xref="S4.Thmtheorem2.p3.10.10.m10.2.3.2"><times id="S4.Thmtheorem2.p3.10.10.m10.2.3.2.1.cmml" xref="S4.Thmtheorem2.p3.10.10.m10.2.3.2.1"></times><ci id="S4.Thmtheorem2.p3.10.10.m10.2.3.2.2a.cmml" xref="S4.Thmtheorem2.p3.10.10.m10.2.3.2.2"><mtext class="ltx_mathvariant_italic" id="S4.Thmtheorem2.p3.10.10.m10.2.3.2.2.cmml" xref="S4.Thmtheorem2.p3.10.10.m10.2.3.2.2">g</mtext></ci><ci id="S4.Thmtheorem2.p3.10.10.m10.1.1.cmml" xref="S4.Thmtheorem2.p3.10.10.m10.1.1">𝑣</ci></apply><apply id="S4.Thmtheorem2.p3.10.10.m10.2.3.3.cmml" xref="S4.Thmtheorem2.p3.10.10.m10.2.3.3"><times id="S4.Thmtheorem2.p3.10.10.m10.2.3.3.1.cmml" xref="S4.Thmtheorem2.p3.10.10.m10.2.3.3.1"></times><apply id="S4.Thmtheorem2.p3.10.10.m10.2.3.3.2.cmml" xref="S4.Thmtheorem2.p3.10.10.m10.2.3.3.2"><csymbol cd="ambiguous" id="S4.Thmtheorem2.p3.10.10.m10.2.3.3.2.1.cmml" xref="S4.Thmtheorem2.p3.10.10.m10.2.3.3.2">superscript</csymbol><ci id="S4.Thmtheorem2.p3.10.10.m10.2.3.3.2.2a.cmml" xref="S4.Thmtheorem2.p3.10.10.m10.2.3.3.2.2"><mtext class="ltx_mathvariant_italic" id="S4.Thmtheorem2.p3.10.10.m10.2.3.3.2.2.cmml" xref="S4.Thmtheorem2.p3.10.10.m10.2.3.3.2.2">g</mtext></ci><ci id="S4.Thmtheorem2.p3.10.10.m10.2.3.3.2.3.cmml" xref="S4.Thmtheorem2.p3.10.10.m10.2.3.3.2.3">′</ci></apply><ci id="S4.Thmtheorem2.p3.10.10.m10.2.2.cmml" xref="S4.Thmtheorem2.p3.10.10.m10.2.2">𝑣</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem2.p3.10.10.m10.2c">\textsl{g}(v)<\textsl{g}^{\prime}(v)</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem2.p3.10.10.m10.2d">g ( italic_v ) < g start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ( italic_v )</annotation></semantics></math>.</span></p> </div> <div class="ltx_para ltx_noindent" id="S4.Thmtheorem2.p4"> <p class="ltx_p" id="S4.Thmtheorem2.p4.1"><span class="ltx_text ltx_font_italic" id="S4.Thmtheorem2.p4.1.1">However we now observe the following property of the path </span><math alttext="\pi_{v}" class="ltx_Math" display="inline" id="S4.Thmtheorem2.p4.1.m1.1"><semantics id="S4.Thmtheorem2.p4.1.m1.1a"><msub id="S4.Thmtheorem2.p4.1.m1.1.1" xref="S4.Thmtheorem2.p4.1.m1.1.1.cmml"><mi id="S4.Thmtheorem2.p4.1.m1.1.1.2" xref="S4.Thmtheorem2.p4.1.m1.1.1.2.cmml">π</mi><mi id="S4.Thmtheorem2.p4.1.m1.1.1.3" xref="S4.Thmtheorem2.p4.1.m1.1.1.3.cmml">v</mi></msub><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem2.p4.1.m1.1b"><apply id="S4.Thmtheorem2.p4.1.m1.1.1.cmml" xref="S4.Thmtheorem2.p4.1.m1.1.1"><csymbol cd="ambiguous" id="S4.Thmtheorem2.p4.1.m1.1.1.1.cmml" xref="S4.Thmtheorem2.p4.1.m1.1.1">subscript</csymbol><ci id="S4.Thmtheorem2.p4.1.m1.1.1.2.cmml" xref="S4.Thmtheorem2.p4.1.m1.1.1.2">𝜋</ci><ci id="S4.Thmtheorem2.p4.1.m1.1.1.3.cmml" xref="S4.Thmtheorem2.p4.1.m1.1.1.3">𝑣</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem2.p4.1.m1.1c">\pi_{v}</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem2.p4.1.m1.1d">italic_π start_POSTSUBSCRIPT italic_v end_POSTSUBSCRIPT</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S4.Thmtheorem2.p4.1.2">:</span></p> <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.p1.8"><math alttext="\forall\overrightarrow{ab}\in\pi_{v}" 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"><mrow id="S4.I1.i1.p1.1.m1.1.1.2" xref="S4.I1.i1.p1.1.m1.1.1.2.cmml"><mo id="S4.I1.i1.p1.1.m1.1.1.2.1" rspace="0.167em" xref="S4.I1.i1.p1.1.m1.1.1.2.1.cmml">∀</mo><mover accent="true" id="S4.I1.i1.p1.1.m1.1.1.2.2" xref="S4.I1.i1.p1.1.m1.1.1.2.2.cmml"><mrow id="S4.I1.i1.p1.1.m1.1.1.2.2.2" xref="S4.I1.i1.p1.1.m1.1.1.2.2.2.cmml"><mi id="S4.I1.i1.p1.1.m1.1.1.2.2.2.2" xref="S4.I1.i1.p1.1.m1.1.1.2.2.2.2.cmml">a</mi><mo id="S4.I1.i1.p1.1.m1.1.1.2.2.2.1" xref="S4.I1.i1.p1.1.m1.1.1.2.2.2.1.cmml">⁢</mo><mi id="S4.I1.i1.p1.1.m1.1.1.2.2.2.3" xref="S4.I1.i1.p1.1.m1.1.1.2.2.2.3.cmml">b</mi></mrow><mo id="S4.I1.i1.p1.1.m1.1.1.2.2.1" stretchy="false" xref="S4.I1.i1.p1.1.m1.1.1.2.2.1.cmml">→</mo></mover></mrow><mo id="S4.I1.i1.p1.1.m1.1.1.1" xref="S4.I1.i1.p1.1.m1.1.1.1.cmml">∈</mo><msub id="S4.I1.i1.p1.1.m1.1.1.3" xref="S4.I1.i1.p1.1.m1.1.1.3.cmml"><mi id="S4.I1.i1.p1.1.m1.1.1.3.2" xref="S4.I1.i1.p1.1.m1.1.1.3.2.cmml">π</mi><mi id="S4.I1.i1.p1.1.m1.1.1.3.3" xref="S4.I1.i1.p1.1.m1.1.1.3.3.cmml">v</mi></msub></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"><in id="S4.I1.i1.p1.1.m1.1.1.1.cmml" xref="S4.I1.i1.p1.1.m1.1.1.1"></in><apply id="S4.I1.i1.p1.1.m1.1.1.2.cmml" xref="S4.I1.i1.p1.1.m1.1.1.2"><csymbol cd="latexml" id="S4.I1.i1.p1.1.m1.1.1.2.1.cmml" xref="S4.I1.i1.p1.1.m1.1.1.2.1">for-all</csymbol><apply id="S4.I1.i1.p1.1.m1.1.1.2.2.cmml" xref="S4.I1.i1.p1.1.m1.1.1.2.2"><ci id="S4.I1.i1.p1.1.m1.1.1.2.2.1.cmml" xref="S4.I1.i1.p1.1.m1.1.1.2.2.1">→</ci><apply id="S4.I1.i1.p1.1.m1.1.1.2.2.2.cmml" xref="S4.I1.i1.p1.1.m1.1.1.2.2.2"><times id="S4.I1.i1.p1.1.m1.1.1.2.2.2.1.cmml" xref="S4.I1.i1.p1.1.m1.1.1.2.2.2.1"></times><ci id="S4.I1.i1.p1.1.m1.1.1.2.2.2.2.cmml" xref="S4.I1.i1.p1.1.m1.1.1.2.2.2.2">𝑎</ci><ci id="S4.I1.i1.p1.1.m1.1.1.2.2.2.3.cmml" xref="S4.I1.i1.p1.1.m1.1.1.2.2.2.3">𝑏</ci></apply></apply></apply><apply id="S4.I1.i1.p1.1.m1.1.1.3.cmml" xref="S4.I1.i1.p1.1.m1.1.1.3"><csymbol cd="ambiguous" id="S4.I1.i1.p1.1.m1.1.1.3.1.cmml" xref="S4.I1.i1.p1.1.m1.1.1.3">subscript</csymbol><ci id="S4.I1.i1.p1.1.m1.1.1.3.2.cmml" xref="S4.I1.i1.p1.1.m1.1.1.3.2">𝜋</ci><ci id="S4.I1.i1.p1.1.m1.1.1.3.3.cmml" xref="S4.I1.i1.p1.1.m1.1.1.3.3">𝑣</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.I1.i1.p1.1.m1.1c">\forall\overrightarrow{ab}\in\pi_{v}</annotation><annotation encoding="application/x-llamapun" id="S4.I1.i1.p1.1.m1.1d">∀ over→ start_ARG italic_a italic_b end_ARG ∈ italic_π start_POSTSUBSCRIPT italic_v end_POSTSUBSCRIPT</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S4.I1.i1.p1.8.1">, </span><math alttext="\textsl{g}(a\!\to\!b)>\textsl{g}^{\prime}(a\!\to\!b)" class="ltx_Math" display="inline" id="S4.I1.i1.p1.2.m2.2"><semantics id="S4.I1.i1.p1.2.m2.2a"><mrow id="S4.I1.i1.p1.2.m2.2.2" xref="S4.I1.i1.p1.2.m2.2.2.cmml"><mrow id="S4.I1.i1.p1.2.m2.1.1.1" xref="S4.I1.i1.p1.2.m2.1.1.1.cmml"><mtext class="ltx_mathvariant_italic" id="S4.I1.i1.p1.2.m2.1.1.1.3" xref="S4.I1.i1.p1.2.m2.1.1.1.3a.cmml">g</mtext><mo id="S4.I1.i1.p1.2.m2.1.1.1.2" xref="S4.I1.i1.p1.2.m2.1.1.1.2.cmml">⁢</mo><mrow id="S4.I1.i1.p1.2.m2.1.1.1.1.1" xref="S4.I1.i1.p1.2.m2.1.1.1.1.1.1.cmml"><mo id="S4.I1.i1.p1.2.m2.1.1.1.1.1.2" stretchy="false" xref="S4.I1.i1.p1.2.m2.1.1.1.1.1.1.cmml">(</mo><mrow id="S4.I1.i1.p1.2.m2.1.1.1.1.1.1" xref="S4.I1.i1.p1.2.m2.1.1.1.1.1.1.cmml"><mi id="S4.I1.i1.p1.2.m2.1.1.1.1.1.1.2" xref="S4.I1.i1.p1.2.m2.1.1.1.1.1.1.2.cmml">a</mi><mo id="S4.I1.i1.p1.2.m2.1.1.1.1.1.1.1" lspace="0.108em" rspace="0.108em" stretchy="false" xref="S4.I1.i1.p1.2.m2.1.1.1.1.1.1.1.cmml">→</mo><mi id="S4.I1.i1.p1.2.m2.1.1.1.1.1.1.3" xref="S4.I1.i1.p1.2.m2.1.1.1.1.1.1.3.cmml">b</mi></mrow><mo id="S4.I1.i1.p1.2.m2.1.1.1.1.1.3" stretchy="false" xref="S4.I1.i1.p1.2.m2.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S4.I1.i1.p1.2.m2.2.2.3" xref="S4.I1.i1.p1.2.m2.2.2.3.cmml">></mo><mrow id="S4.I1.i1.p1.2.m2.2.2.2" xref="S4.I1.i1.p1.2.m2.2.2.2.cmml"><msup id="S4.I1.i1.p1.2.m2.2.2.2.3" xref="S4.I1.i1.p1.2.m2.2.2.2.3.cmml"><mtext class="ltx_mathvariant_italic" id="S4.I1.i1.p1.2.m2.2.2.2.3.2" xref="S4.I1.i1.p1.2.m2.2.2.2.3.2a.cmml">g</mtext><mo id="S4.I1.i1.p1.2.m2.2.2.2.3.3" xref="S4.I1.i1.p1.2.m2.2.2.2.3.3.cmml">′</mo></msup><mo id="S4.I1.i1.p1.2.m2.2.2.2.2" xref="S4.I1.i1.p1.2.m2.2.2.2.2.cmml">⁢</mo><mrow id="S4.I1.i1.p1.2.m2.2.2.2.1.1" xref="S4.I1.i1.p1.2.m2.2.2.2.1.1.1.cmml"><mo id="S4.I1.i1.p1.2.m2.2.2.2.1.1.2" stretchy="false" xref="S4.I1.i1.p1.2.m2.2.2.2.1.1.1.cmml">(</mo><mrow id="S4.I1.i1.p1.2.m2.2.2.2.1.1.1" xref="S4.I1.i1.p1.2.m2.2.2.2.1.1.1.cmml"><mi id="S4.I1.i1.p1.2.m2.2.2.2.1.1.1.2" xref="S4.I1.i1.p1.2.m2.2.2.2.1.1.1.2.cmml">a</mi><mo id="S4.I1.i1.p1.2.m2.2.2.2.1.1.1.1" lspace="0.108em" rspace="0.108em" stretchy="false" xref="S4.I1.i1.p1.2.m2.2.2.2.1.1.1.1.cmml">→</mo><mi id="S4.I1.i1.p1.2.m2.2.2.2.1.1.1.3" xref="S4.I1.i1.p1.2.m2.2.2.2.1.1.1.3.cmml">b</mi></mrow><mo id="S4.I1.i1.p1.2.m2.2.2.2.1.1.3" stretchy="false" xref="S4.I1.i1.p1.2.m2.2.2.2.1.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.I1.i1.p1.2.m2.2b"><apply id="S4.I1.i1.p1.2.m2.2.2.cmml" xref="S4.I1.i1.p1.2.m2.2.2"><gt id="S4.I1.i1.p1.2.m2.2.2.3.cmml" xref="S4.I1.i1.p1.2.m2.2.2.3"></gt><apply id="S4.I1.i1.p1.2.m2.1.1.1.cmml" xref="S4.I1.i1.p1.2.m2.1.1.1"><times id="S4.I1.i1.p1.2.m2.1.1.1.2.cmml" xref="S4.I1.i1.p1.2.m2.1.1.1.2"></times><ci id="S4.I1.i1.p1.2.m2.1.1.1.3a.cmml" xref="S4.I1.i1.p1.2.m2.1.1.1.3"><mtext class="ltx_mathvariant_italic" id="S4.I1.i1.p1.2.m2.1.1.1.3.cmml" xref="S4.I1.i1.p1.2.m2.1.1.1.3">g</mtext></ci><apply id="S4.I1.i1.p1.2.m2.1.1.1.1.1.1.cmml" xref="S4.I1.i1.p1.2.m2.1.1.1.1.1"><ci id="S4.I1.i1.p1.2.m2.1.1.1.1.1.1.1.cmml" xref="S4.I1.i1.p1.2.m2.1.1.1.1.1.1.1">→</ci><ci id="S4.I1.i1.p1.2.m2.1.1.1.1.1.1.2.cmml" xref="S4.I1.i1.p1.2.m2.1.1.1.1.1.1.2">𝑎</ci><ci id="S4.I1.i1.p1.2.m2.1.1.1.1.1.1.3.cmml" xref="S4.I1.i1.p1.2.m2.1.1.1.1.1.1.3">𝑏</ci></apply></apply><apply id="S4.I1.i1.p1.2.m2.2.2.2.cmml" xref="S4.I1.i1.p1.2.m2.2.2.2"><times id="S4.I1.i1.p1.2.m2.2.2.2.2.cmml" xref="S4.I1.i1.p1.2.m2.2.2.2.2"></times><apply id="S4.I1.i1.p1.2.m2.2.2.2.3.cmml" xref="S4.I1.i1.p1.2.m2.2.2.2.3"><csymbol cd="ambiguous" id="S4.I1.i1.p1.2.m2.2.2.2.3.1.cmml" xref="S4.I1.i1.p1.2.m2.2.2.2.3">superscript</csymbol><ci id="S4.I1.i1.p1.2.m2.2.2.2.3.2a.cmml" xref="S4.I1.i1.p1.2.m2.2.2.2.3.2"><mtext class="ltx_mathvariant_italic" id="S4.I1.i1.p1.2.m2.2.2.2.3.2.cmml" xref="S4.I1.i1.p1.2.m2.2.2.2.3.2">g</mtext></ci><ci id="S4.I1.i1.p1.2.m2.2.2.2.3.3.cmml" xref="S4.I1.i1.p1.2.m2.2.2.2.3.3">′</ci></apply><apply id="S4.I1.i1.p1.2.m2.2.2.2.1.1.1.cmml" xref="S4.I1.i1.p1.2.m2.2.2.2.1.1"><ci id="S4.I1.i1.p1.2.m2.2.2.2.1.1.1.1.cmml" xref="S4.I1.i1.p1.2.m2.2.2.2.1.1.1.1">→</ci><ci id="S4.I1.i1.p1.2.m2.2.2.2.1.1.1.2.cmml" xref="S4.I1.i1.p1.2.m2.2.2.2.1.1.1.2">𝑎</ci><ci id="S4.I1.i1.p1.2.m2.2.2.2.1.1.1.3.cmml" xref="S4.I1.i1.p1.2.m2.2.2.2.1.1.1.3">𝑏</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.I1.i1.p1.2.m2.2c">\textsl{g}(a\!\to\!b)>\textsl{g}^{\prime}(a\!\to\!b)</annotation><annotation encoding="application/x-llamapun" id="S4.I1.i1.p1.2.m2.2d">g ( italic_a → italic_b ) > g start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ( italic_a → italic_b )</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S4.I1.i1.p1.8.2">. Thus, </span><math alttext="\textsl{g}(a\!\to\!b)>0" class="ltx_Math" display="inline" id="S4.I1.i1.p1.3.m3.1"><semantics id="S4.I1.i1.p1.3.m3.1a"><mrow id="S4.I1.i1.p1.3.m3.1.1" xref="S4.I1.i1.p1.3.m3.1.1.cmml"><mrow id="S4.I1.i1.p1.3.m3.1.1.1" xref="S4.I1.i1.p1.3.m3.1.1.1.cmml"><mtext class="ltx_mathvariant_italic" id="S4.I1.i1.p1.3.m3.1.1.1.3" xref="S4.I1.i1.p1.3.m3.1.1.1.3a.cmml">g</mtext><mo id="S4.I1.i1.p1.3.m3.1.1.1.2" xref="S4.I1.i1.p1.3.m3.1.1.1.2.cmml">⁢</mo><mrow id="S4.I1.i1.p1.3.m3.1.1.1.1.1" xref="S4.I1.i1.p1.3.m3.1.1.1.1.1.1.cmml"><mo id="S4.I1.i1.p1.3.m3.1.1.1.1.1.2" stretchy="false" xref="S4.I1.i1.p1.3.m3.1.1.1.1.1.1.cmml">(</mo><mrow id="S4.I1.i1.p1.3.m3.1.1.1.1.1.1" xref="S4.I1.i1.p1.3.m3.1.1.1.1.1.1.cmml"><mi id="S4.I1.i1.p1.3.m3.1.1.1.1.1.1.2" xref="S4.I1.i1.p1.3.m3.1.1.1.1.1.1.2.cmml">a</mi><mo id="S4.I1.i1.p1.3.m3.1.1.1.1.1.1.1" lspace="0.108em" rspace="0.108em" stretchy="false" xref="S4.I1.i1.p1.3.m3.1.1.1.1.1.1.1.cmml">→</mo><mi id="S4.I1.i1.p1.3.m3.1.1.1.1.1.1.3" xref="S4.I1.i1.p1.3.m3.1.1.1.1.1.1.3.cmml">b</mi></mrow><mo id="S4.I1.i1.p1.3.m3.1.1.1.1.1.3" stretchy="false" xref="S4.I1.i1.p1.3.m3.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S4.I1.i1.p1.3.m3.1.1.2" xref="S4.I1.i1.p1.3.m3.1.1.2.cmml">></mo><mn id="S4.I1.i1.p1.3.m3.1.1.3" xref="S4.I1.i1.p1.3.m3.1.1.3.cmml">0</mn></mrow><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"><gt id="S4.I1.i1.p1.3.m3.1.1.2.cmml" xref="S4.I1.i1.p1.3.m3.1.1.2"></gt><apply id="S4.I1.i1.p1.3.m3.1.1.1.cmml" xref="S4.I1.i1.p1.3.m3.1.1.1"><times id="S4.I1.i1.p1.3.m3.1.1.1.2.cmml" xref="S4.I1.i1.p1.3.m3.1.1.1.2"></times><ci id="S4.I1.i1.p1.3.m3.1.1.1.3a.cmml" xref="S4.I1.i1.p1.3.m3.1.1.1.3"><mtext class="ltx_mathvariant_italic" id="S4.I1.i1.p1.3.m3.1.1.1.3.cmml" xref="S4.I1.i1.p1.3.m3.1.1.1.3">g</mtext></ci><apply id="S4.I1.i1.p1.3.m3.1.1.1.1.1.1.cmml" xref="S4.I1.i1.p1.3.m3.1.1.1.1.1"><ci id="S4.I1.i1.p1.3.m3.1.1.1.1.1.1.1.cmml" xref="S4.I1.i1.p1.3.m3.1.1.1.1.1.1.1">→</ci><ci id="S4.I1.i1.p1.3.m3.1.1.1.1.1.1.2.cmml" xref="S4.I1.i1.p1.3.m3.1.1.1.1.1.1.2">𝑎</ci><ci id="S4.I1.i1.p1.3.m3.1.1.1.1.1.1.3.cmml" xref="S4.I1.i1.p1.3.m3.1.1.1.1.1.1.3">𝑏</ci></apply></apply><cn id="S4.I1.i1.p1.3.m3.1.1.3.cmml" type="integer" xref="S4.I1.i1.p1.3.m3.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.I1.i1.p1.3.m3.1c">\textsl{g}(a\!\to\!b)>0</annotation><annotation encoding="application/x-llamapun" id="S4.I1.i1.p1.3.m3.1d">g ( italic_a → italic_b ) > 0</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S4.I1.i1.p1.8.3"> and so there exists a directed path from </span><math alttext="u" class="ltx_Math" display="inline" id="S4.I1.i1.p1.4.m4.1"><semantics id="S4.I1.i1.p1.4.m4.1a"><mi id="S4.I1.i1.p1.4.m4.1.1" xref="S4.I1.i1.p1.4.m4.1.1.cmml">u</mi><annotation-xml encoding="MathML-Content" id="S4.I1.i1.p1.4.m4.1b"><ci id="S4.I1.i1.p1.4.m4.1.1.cmml" xref="S4.I1.i1.p1.4.m4.1.1">𝑢</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.I1.i1.p1.4.m4.1c">u</annotation><annotation encoding="application/x-llamapun" id="S4.I1.i1.p1.4.m4.1d">italic_u</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S4.I1.i1.p1.8.4"> to </span><math alttext="v" class="ltx_Math" display="inline" id="S4.I1.i1.p1.5.m5.1"><semantics id="S4.I1.i1.p1.5.m5.1a"><mi id="S4.I1.i1.p1.5.m5.1.1" xref="S4.I1.i1.p1.5.m5.1.1.cmml">v</mi><annotation-xml encoding="MathML-Content" id="S4.I1.i1.p1.5.m5.1b"><ci id="S4.I1.i1.p1.5.m5.1.1.cmml" xref="S4.I1.i1.p1.5.m5.1.1">𝑣</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.I1.i1.p1.5.m5.1c">v</annotation><annotation encoding="application/x-llamapun" id="S4.I1.i1.p1.5.m5.1d">italic_v</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S4.I1.i1.p1.8.5"> in </span><math alttext="\overrightarrow{G}" class="ltx_Math" display="inline" id="S4.I1.i1.p1.6.m6.1"><semantics id="S4.I1.i1.p1.6.m6.1a"><mover accent="true" id="S4.I1.i1.p1.6.m6.1.1" xref="S4.I1.i1.p1.6.m6.1.1.cmml"><mi id="S4.I1.i1.p1.6.m6.1.1.2" xref="S4.I1.i1.p1.6.m6.1.1.2.cmml">G</mi><mo id="S4.I1.i1.p1.6.m6.1.1.1" stretchy="false" xref="S4.I1.i1.p1.6.m6.1.1.1.cmml">→</mo></mover><annotation-xml encoding="MathML-Content" id="S4.I1.i1.p1.6.m6.1b"><apply id="S4.I1.i1.p1.6.m6.1.1.cmml" xref="S4.I1.i1.p1.6.m6.1.1"><ci id="S4.I1.i1.p1.6.m6.1.1.1.cmml" xref="S4.I1.i1.p1.6.m6.1.1.1">→</ci><ci id="S4.I1.i1.p1.6.m6.1.1.2.cmml" xref="S4.I1.i1.p1.6.m6.1.1.2">𝐺</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.I1.i1.p1.6.m6.1c">\overrightarrow{G}</annotation><annotation encoding="application/x-llamapun" id="S4.I1.i1.p1.6.m6.1d">over→ start_ARG italic_G end_ARG</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S4.I1.i1.p1.8.6">. Since </span><math alttext="\overrightarrow{G}" class="ltx_Math" display="inline" id="S4.I1.i1.p1.7.m7.1"><semantics id="S4.I1.i1.p1.7.m7.1a"><mover accent="true" 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><mo id="S4.I1.i1.p1.7.m7.1.1.1" stretchy="false" xref="S4.I1.i1.p1.7.m7.1.1.1.cmml">→</mo></mover><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"><ci id="S4.I1.i1.p1.7.m7.1.1.1.cmml" xref="S4.I1.i1.p1.7.m7.1.1.1">→</ci><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></annotation-xml><annotation encoding="application/x-tex" id="S4.I1.i1.p1.7.m7.1c">\overrightarrow{G}</annotation><annotation encoding="application/x-llamapun" id="S4.I1.i1.p1.7.m7.1d">over→ start_ARG italic_G end_ARG</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S4.I1.i1.p1.8.7"> is locally fair this implies that </span><math alttext="\textsl{g}(u)\leq\textsl{g}(v)" class="ltx_Math" display="inline" id="S4.I1.i1.p1.8.m8.2"><semantics id="S4.I1.i1.p1.8.m8.2a"><mrow id="S4.I1.i1.p1.8.m8.2.3" xref="S4.I1.i1.p1.8.m8.2.3.cmml"><mrow id="S4.I1.i1.p1.8.m8.2.3.2" xref="S4.I1.i1.p1.8.m8.2.3.2.cmml"><mtext class="ltx_mathvariant_italic" id="S4.I1.i1.p1.8.m8.2.3.2.2" xref="S4.I1.i1.p1.8.m8.2.3.2.2a.cmml">g</mtext><mo id="S4.I1.i1.p1.8.m8.2.3.2.1" xref="S4.I1.i1.p1.8.m8.2.3.2.1.cmml">⁢</mo><mrow id="S4.I1.i1.p1.8.m8.2.3.2.3.2" xref="S4.I1.i1.p1.8.m8.2.3.2.cmml"><mo id="S4.I1.i1.p1.8.m8.2.3.2.3.2.1" stretchy="false" xref="S4.I1.i1.p1.8.m8.2.3.2.cmml">(</mo><mi id="S4.I1.i1.p1.8.m8.1.1" xref="S4.I1.i1.p1.8.m8.1.1.cmml">u</mi><mo id="S4.I1.i1.p1.8.m8.2.3.2.3.2.2" stretchy="false" xref="S4.I1.i1.p1.8.m8.2.3.2.cmml">)</mo></mrow></mrow><mo id="S4.I1.i1.p1.8.m8.2.3.1" xref="S4.I1.i1.p1.8.m8.2.3.1.cmml">≤</mo><mrow id="S4.I1.i1.p1.8.m8.2.3.3" xref="S4.I1.i1.p1.8.m8.2.3.3.cmml"><mtext class="ltx_mathvariant_italic" id="S4.I1.i1.p1.8.m8.2.3.3.2" xref="S4.I1.i1.p1.8.m8.2.3.3.2a.cmml">g</mtext><mo id="S4.I1.i1.p1.8.m8.2.3.3.1" xref="S4.I1.i1.p1.8.m8.2.3.3.1.cmml">⁢</mo><mrow id="S4.I1.i1.p1.8.m8.2.3.3.3.2" xref="S4.I1.i1.p1.8.m8.2.3.3.cmml"><mo id="S4.I1.i1.p1.8.m8.2.3.3.3.2.1" stretchy="false" xref="S4.I1.i1.p1.8.m8.2.3.3.cmml">(</mo><mi id="S4.I1.i1.p1.8.m8.2.2" xref="S4.I1.i1.p1.8.m8.2.2.cmml">v</mi><mo id="S4.I1.i1.p1.8.m8.2.3.3.3.2.2" stretchy="false" xref="S4.I1.i1.p1.8.m8.2.3.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.I1.i1.p1.8.m8.2b"><apply id="S4.I1.i1.p1.8.m8.2.3.cmml" xref="S4.I1.i1.p1.8.m8.2.3"><leq id="S4.I1.i1.p1.8.m8.2.3.1.cmml" xref="S4.I1.i1.p1.8.m8.2.3.1"></leq><apply id="S4.I1.i1.p1.8.m8.2.3.2.cmml" xref="S4.I1.i1.p1.8.m8.2.3.2"><times id="S4.I1.i1.p1.8.m8.2.3.2.1.cmml" xref="S4.I1.i1.p1.8.m8.2.3.2.1"></times><ci id="S4.I1.i1.p1.8.m8.2.3.2.2a.cmml" xref="S4.I1.i1.p1.8.m8.2.3.2.2"><mtext class="ltx_mathvariant_italic" id="S4.I1.i1.p1.8.m8.2.3.2.2.cmml" xref="S4.I1.i1.p1.8.m8.2.3.2.2">g</mtext></ci><ci id="S4.I1.i1.p1.8.m8.1.1.cmml" xref="S4.I1.i1.p1.8.m8.1.1">𝑢</ci></apply><apply id="S4.I1.i1.p1.8.m8.2.3.3.cmml" xref="S4.I1.i1.p1.8.m8.2.3.3"><times id="S4.I1.i1.p1.8.m8.2.3.3.1.cmml" xref="S4.I1.i1.p1.8.m8.2.3.3.1"></times><ci id="S4.I1.i1.p1.8.m8.2.3.3.2a.cmml" xref="S4.I1.i1.p1.8.m8.2.3.3.2"><mtext class="ltx_mathvariant_italic" id="S4.I1.i1.p1.8.m8.2.3.3.2.cmml" xref="S4.I1.i1.p1.8.m8.2.3.3.2">g</mtext></ci><ci id="S4.I1.i1.p1.8.m8.2.2.cmml" xref="S4.I1.i1.p1.8.m8.2.2">𝑣</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.I1.i1.p1.8.m8.2c">\textsl{g}(u)\leq\textsl{g}(v)</annotation><annotation encoding="application/x-llamapun" id="S4.I1.i1.p1.8.m8.2d">g ( italic_u ) ≤ g ( italic_v )</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S4.I1.i1.p1.8.8">.</span></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.p1.8"><math alttext="\forall\overrightarrow{ab}\in\pi_{v}" 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"><mrow id="S4.I1.i2.p1.1.m1.1.1.2" xref="S4.I1.i2.p1.1.m1.1.1.2.cmml"><mo id="S4.I1.i2.p1.1.m1.1.1.2.1" rspace="0.167em" xref="S4.I1.i2.p1.1.m1.1.1.2.1.cmml">∀</mo><mover accent="true" id="S4.I1.i2.p1.1.m1.1.1.2.2" xref="S4.I1.i2.p1.1.m1.1.1.2.2.cmml"><mrow id="S4.I1.i2.p1.1.m1.1.1.2.2.2" xref="S4.I1.i2.p1.1.m1.1.1.2.2.2.cmml"><mi id="S4.I1.i2.p1.1.m1.1.1.2.2.2.2" xref="S4.I1.i2.p1.1.m1.1.1.2.2.2.2.cmml">a</mi><mo id="S4.I1.i2.p1.1.m1.1.1.2.2.2.1" xref="S4.I1.i2.p1.1.m1.1.1.2.2.2.1.cmml">⁢</mo><mi id="S4.I1.i2.p1.1.m1.1.1.2.2.2.3" xref="S4.I1.i2.p1.1.m1.1.1.2.2.2.3.cmml">b</mi></mrow><mo id="S4.I1.i2.p1.1.m1.1.1.2.2.1" stretchy="false" xref="S4.I1.i2.p1.1.m1.1.1.2.2.1.cmml">→</mo></mover></mrow><mo id="S4.I1.i2.p1.1.m1.1.1.1" xref="S4.I1.i2.p1.1.m1.1.1.1.cmml">∈</mo><msub id="S4.I1.i2.p1.1.m1.1.1.3" xref="S4.I1.i2.p1.1.m1.1.1.3.cmml"><mi id="S4.I1.i2.p1.1.m1.1.1.3.2" xref="S4.I1.i2.p1.1.m1.1.1.3.2.cmml">π</mi><mi id="S4.I1.i2.p1.1.m1.1.1.3.3" xref="S4.I1.i2.p1.1.m1.1.1.3.3.cmml">v</mi></msub></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"><in id="S4.I1.i2.p1.1.m1.1.1.1.cmml" xref="S4.I1.i2.p1.1.m1.1.1.1"></in><apply id="S4.I1.i2.p1.1.m1.1.1.2.cmml" xref="S4.I1.i2.p1.1.m1.1.1.2"><csymbol cd="latexml" id="S4.I1.i2.p1.1.m1.1.1.2.1.cmml" xref="S4.I1.i2.p1.1.m1.1.1.2.1">for-all</csymbol><apply id="S4.I1.i2.p1.1.m1.1.1.2.2.cmml" xref="S4.I1.i2.p1.1.m1.1.1.2.2"><ci id="S4.I1.i2.p1.1.m1.1.1.2.2.1.cmml" xref="S4.I1.i2.p1.1.m1.1.1.2.2.1">→</ci><apply id="S4.I1.i2.p1.1.m1.1.1.2.2.2.cmml" xref="S4.I1.i2.p1.1.m1.1.1.2.2.2"><times id="S4.I1.i2.p1.1.m1.1.1.2.2.2.1.cmml" xref="S4.I1.i2.p1.1.m1.1.1.2.2.2.1"></times><ci id="S4.I1.i2.p1.1.m1.1.1.2.2.2.2.cmml" xref="S4.I1.i2.p1.1.m1.1.1.2.2.2.2">𝑎</ci><ci id="S4.I1.i2.p1.1.m1.1.1.2.2.2.3.cmml" xref="S4.I1.i2.p1.1.m1.1.1.2.2.2.3">𝑏</ci></apply></apply></apply><apply id="S4.I1.i2.p1.1.m1.1.1.3.cmml" xref="S4.I1.i2.p1.1.m1.1.1.3"><csymbol cd="ambiguous" id="S4.I1.i2.p1.1.m1.1.1.3.1.cmml" xref="S4.I1.i2.p1.1.m1.1.1.3">subscript</csymbol><ci id="S4.I1.i2.p1.1.m1.1.1.3.2.cmml" xref="S4.I1.i2.p1.1.m1.1.1.3.2">𝜋</ci><ci id="S4.I1.i2.p1.1.m1.1.1.3.3.cmml" xref="S4.I1.i2.p1.1.m1.1.1.3.3">𝑣</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.I1.i2.p1.1.m1.1c">\forall\overrightarrow{ab}\in\pi_{v}</annotation><annotation encoding="application/x-llamapun" id="S4.I1.i2.p1.1.m1.1d">∀ over→ start_ARG italic_a italic_b end_ARG ∈ italic_π start_POSTSUBSCRIPT italic_v end_POSTSUBSCRIPT</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S4.I1.i2.p1.8.1">, </span><math alttext="\textsl{g}(a\!\to\!b)>\textsl{g}^{\prime}(a\!\to\!b)" class="ltx_Math" display="inline" id="S4.I1.i2.p1.2.m2.2"><semantics id="S4.I1.i2.p1.2.m2.2a"><mrow id="S4.I1.i2.p1.2.m2.2.2" xref="S4.I1.i2.p1.2.m2.2.2.cmml"><mrow id="S4.I1.i2.p1.2.m2.1.1.1" xref="S4.I1.i2.p1.2.m2.1.1.1.cmml"><mtext class="ltx_mathvariant_italic" id="S4.I1.i2.p1.2.m2.1.1.1.3" xref="S4.I1.i2.p1.2.m2.1.1.1.3a.cmml">g</mtext><mo id="S4.I1.i2.p1.2.m2.1.1.1.2" xref="S4.I1.i2.p1.2.m2.1.1.1.2.cmml">⁢</mo><mrow id="S4.I1.i2.p1.2.m2.1.1.1.1.1" xref="S4.I1.i2.p1.2.m2.1.1.1.1.1.1.cmml"><mo id="S4.I1.i2.p1.2.m2.1.1.1.1.1.2" stretchy="false" xref="S4.I1.i2.p1.2.m2.1.1.1.1.1.1.cmml">(</mo><mrow id="S4.I1.i2.p1.2.m2.1.1.1.1.1.1" xref="S4.I1.i2.p1.2.m2.1.1.1.1.1.1.cmml"><mi id="S4.I1.i2.p1.2.m2.1.1.1.1.1.1.2" xref="S4.I1.i2.p1.2.m2.1.1.1.1.1.1.2.cmml">a</mi><mo id="S4.I1.i2.p1.2.m2.1.1.1.1.1.1.1" lspace="0.108em" rspace="0.108em" stretchy="false" xref="S4.I1.i2.p1.2.m2.1.1.1.1.1.1.1.cmml">→</mo><mi id="S4.I1.i2.p1.2.m2.1.1.1.1.1.1.3" xref="S4.I1.i2.p1.2.m2.1.1.1.1.1.1.3.cmml">b</mi></mrow><mo id="S4.I1.i2.p1.2.m2.1.1.1.1.1.3" stretchy="false" xref="S4.I1.i2.p1.2.m2.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S4.I1.i2.p1.2.m2.2.2.3" xref="S4.I1.i2.p1.2.m2.2.2.3.cmml">></mo><mrow id="S4.I1.i2.p1.2.m2.2.2.2" xref="S4.I1.i2.p1.2.m2.2.2.2.cmml"><msup id="S4.I1.i2.p1.2.m2.2.2.2.3" xref="S4.I1.i2.p1.2.m2.2.2.2.3.cmml"><mtext class="ltx_mathvariant_italic" id="S4.I1.i2.p1.2.m2.2.2.2.3.2" xref="S4.I1.i2.p1.2.m2.2.2.2.3.2a.cmml">g</mtext><mo id="S4.I1.i2.p1.2.m2.2.2.2.3.3" xref="S4.I1.i2.p1.2.m2.2.2.2.3.3.cmml">′</mo></msup><mo id="S4.I1.i2.p1.2.m2.2.2.2.2" xref="S4.I1.i2.p1.2.m2.2.2.2.2.cmml">⁢</mo><mrow id="S4.I1.i2.p1.2.m2.2.2.2.1.1" xref="S4.I1.i2.p1.2.m2.2.2.2.1.1.1.cmml"><mo id="S4.I1.i2.p1.2.m2.2.2.2.1.1.2" stretchy="false" xref="S4.I1.i2.p1.2.m2.2.2.2.1.1.1.cmml">(</mo><mrow id="S4.I1.i2.p1.2.m2.2.2.2.1.1.1" xref="S4.I1.i2.p1.2.m2.2.2.2.1.1.1.cmml"><mi id="S4.I1.i2.p1.2.m2.2.2.2.1.1.1.2" xref="S4.I1.i2.p1.2.m2.2.2.2.1.1.1.2.cmml">a</mi><mo id="S4.I1.i2.p1.2.m2.2.2.2.1.1.1.1" lspace="0.108em" rspace="0.108em" stretchy="false" xref="S4.I1.i2.p1.2.m2.2.2.2.1.1.1.1.cmml">→</mo><mi id="S4.I1.i2.p1.2.m2.2.2.2.1.1.1.3" xref="S4.I1.i2.p1.2.m2.2.2.2.1.1.1.3.cmml">b</mi></mrow><mo id="S4.I1.i2.p1.2.m2.2.2.2.1.1.3" stretchy="false" xref="S4.I1.i2.p1.2.m2.2.2.2.1.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.I1.i2.p1.2.m2.2b"><apply id="S4.I1.i2.p1.2.m2.2.2.cmml" xref="S4.I1.i2.p1.2.m2.2.2"><gt id="S4.I1.i2.p1.2.m2.2.2.3.cmml" xref="S4.I1.i2.p1.2.m2.2.2.3"></gt><apply id="S4.I1.i2.p1.2.m2.1.1.1.cmml" xref="S4.I1.i2.p1.2.m2.1.1.1"><times id="S4.I1.i2.p1.2.m2.1.1.1.2.cmml" xref="S4.I1.i2.p1.2.m2.1.1.1.2"></times><ci id="S4.I1.i2.p1.2.m2.1.1.1.3a.cmml" xref="S4.I1.i2.p1.2.m2.1.1.1.3"><mtext class="ltx_mathvariant_italic" id="S4.I1.i2.p1.2.m2.1.1.1.3.cmml" xref="S4.I1.i2.p1.2.m2.1.1.1.3">g</mtext></ci><apply id="S4.I1.i2.p1.2.m2.1.1.1.1.1.1.cmml" xref="S4.I1.i2.p1.2.m2.1.1.1.1.1"><ci id="S4.I1.i2.p1.2.m2.1.1.1.1.1.1.1.cmml" xref="S4.I1.i2.p1.2.m2.1.1.1.1.1.1.1">→</ci><ci id="S4.I1.i2.p1.2.m2.1.1.1.1.1.1.2.cmml" xref="S4.I1.i2.p1.2.m2.1.1.1.1.1.1.2">𝑎</ci><ci id="S4.I1.i2.p1.2.m2.1.1.1.1.1.1.3.cmml" xref="S4.I1.i2.p1.2.m2.1.1.1.1.1.1.3">𝑏</ci></apply></apply><apply id="S4.I1.i2.p1.2.m2.2.2.2.cmml" xref="S4.I1.i2.p1.2.m2.2.2.2"><times id="S4.I1.i2.p1.2.m2.2.2.2.2.cmml" xref="S4.I1.i2.p1.2.m2.2.2.2.2"></times><apply id="S4.I1.i2.p1.2.m2.2.2.2.3.cmml" xref="S4.I1.i2.p1.2.m2.2.2.2.3"><csymbol cd="ambiguous" id="S4.I1.i2.p1.2.m2.2.2.2.3.1.cmml" xref="S4.I1.i2.p1.2.m2.2.2.2.3">superscript</csymbol><ci id="S4.I1.i2.p1.2.m2.2.2.2.3.2a.cmml" xref="S4.I1.i2.p1.2.m2.2.2.2.3.2"><mtext class="ltx_mathvariant_italic" id="S4.I1.i2.p1.2.m2.2.2.2.3.2.cmml" xref="S4.I1.i2.p1.2.m2.2.2.2.3.2">g</mtext></ci><ci id="S4.I1.i2.p1.2.m2.2.2.2.3.3.cmml" xref="S4.I1.i2.p1.2.m2.2.2.2.3.3">′</ci></apply><apply id="S4.I1.i2.p1.2.m2.2.2.2.1.1.1.cmml" xref="S4.I1.i2.p1.2.m2.2.2.2.1.1"><ci id="S4.I1.i2.p1.2.m2.2.2.2.1.1.1.1.cmml" xref="S4.I1.i2.p1.2.m2.2.2.2.1.1.1.1">→</ci><ci id="S4.I1.i2.p1.2.m2.2.2.2.1.1.1.2.cmml" xref="S4.I1.i2.p1.2.m2.2.2.2.1.1.1.2">𝑎</ci><ci id="S4.I1.i2.p1.2.m2.2.2.2.1.1.1.3.cmml" xref="S4.I1.i2.p1.2.m2.2.2.2.1.1.1.3">𝑏</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.I1.i2.p1.2.m2.2c">\textsl{g}(a\!\to\!b)>\textsl{g}^{\prime}(a\!\to\!b)</annotation><annotation encoding="application/x-llamapun" id="S4.I1.i2.p1.2.m2.2d">g ( italic_a → italic_b ) > g start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ( italic_a → italic_b )</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S4.I1.i2.p1.8.2">. Thus, </span><math alttext="\textsl{g}^{\prime}(a\!\to\!b)<\textsl{g}(ab)" class="ltx_Math" display="inline" id="S4.I1.i2.p1.3.m3.2"><semantics id="S4.I1.i2.p1.3.m3.2a"><mrow id="S4.I1.i2.p1.3.m3.2.2" xref="S4.I1.i2.p1.3.m3.2.2.cmml"><mrow id="S4.I1.i2.p1.3.m3.1.1.1" xref="S4.I1.i2.p1.3.m3.1.1.1.cmml"><msup id="S4.I1.i2.p1.3.m3.1.1.1.3" xref="S4.I1.i2.p1.3.m3.1.1.1.3.cmml"><mtext class="ltx_mathvariant_italic" id="S4.I1.i2.p1.3.m3.1.1.1.3.2" xref="S4.I1.i2.p1.3.m3.1.1.1.3.2a.cmml">g</mtext><mo id="S4.I1.i2.p1.3.m3.1.1.1.3.3" xref="S4.I1.i2.p1.3.m3.1.1.1.3.3.cmml">′</mo></msup><mo id="S4.I1.i2.p1.3.m3.1.1.1.2" xref="S4.I1.i2.p1.3.m3.1.1.1.2.cmml">⁢</mo><mrow id="S4.I1.i2.p1.3.m3.1.1.1.1.1" xref="S4.I1.i2.p1.3.m3.1.1.1.1.1.1.cmml"><mo id="S4.I1.i2.p1.3.m3.1.1.1.1.1.2" stretchy="false" xref="S4.I1.i2.p1.3.m3.1.1.1.1.1.1.cmml">(</mo><mrow id="S4.I1.i2.p1.3.m3.1.1.1.1.1.1" xref="S4.I1.i2.p1.3.m3.1.1.1.1.1.1.cmml"><mi id="S4.I1.i2.p1.3.m3.1.1.1.1.1.1.2" xref="S4.I1.i2.p1.3.m3.1.1.1.1.1.1.2.cmml">a</mi><mo id="S4.I1.i2.p1.3.m3.1.1.1.1.1.1.1" lspace="0.108em" rspace="0.108em" stretchy="false" xref="S4.I1.i2.p1.3.m3.1.1.1.1.1.1.1.cmml">→</mo><mi id="S4.I1.i2.p1.3.m3.1.1.1.1.1.1.3" xref="S4.I1.i2.p1.3.m3.1.1.1.1.1.1.3.cmml">b</mi></mrow><mo id="S4.I1.i2.p1.3.m3.1.1.1.1.1.3" stretchy="false" xref="S4.I1.i2.p1.3.m3.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S4.I1.i2.p1.3.m3.2.2.3" xref="S4.I1.i2.p1.3.m3.2.2.3.cmml"><</mo><mrow id="S4.I1.i2.p1.3.m3.2.2.2" xref="S4.I1.i2.p1.3.m3.2.2.2.cmml"><mtext class="ltx_mathvariant_italic" id="S4.I1.i2.p1.3.m3.2.2.2.3" xref="S4.I1.i2.p1.3.m3.2.2.2.3a.cmml">g</mtext><mo id="S4.I1.i2.p1.3.m3.2.2.2.2" xref="S4.I1.i2.p1.3.m3.2.2.2.2.cmml">⁢</mo><mrow id="S4.I1.i2.p1.3.m3.2.2.2.1.1" xref="S4.I1.i2.p1.3.m3.2.2.2.1.1.1.cmml"><mo id="S4.I1.i2.p1.3.m3.2.2.2.1.1.2" stretchy="false" xref="S4.I1.i2.p1.3.m3.2.2.2.1.1.1.cmml">(</mo><mrow id="S4.I1.i2.p1.3.m3.2.2.2.1.1.1" xref="S4.I1.i2.p1.3.m3.2.2.2.1.1.1.cmml"><mi id="S4.I1.i2.p1.3.m3.2.2.2.1.1.1.2" xref="S4.I1.i2.p1.3.m3.2.2.2.1.1.1.2.cmml">a</mi><mo id="S4.I1.i2.p1.3.m3.2.2.2.1.1.1.1" xref="S4.I1.i2.p1.3.m3.2.2.2.1.1.1.1.cmml">⁢</mo><mi id="S4.I1.i2.p1.3.m3.2.2.2.1.1.1.3" xref="S4.I1.i2.p1.3.m3.2.2.2.1.1.1.3.cmml">b</mi></mrow><mo id="S4.I1.i2.p1.3.m3.2.2.2.1.1.3" stretchy="false" xref="S4.I1.i2.p1.3.m3.2.2.2.1.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.I1.i2.p1.3.m3.2b"><apply id="S4.I1.i2.p1.3.m3.2.2.cmml" xref="S4.I1.i2.p1.3.m3.2.2"><lt id="S4.I1.i2.p1.3.m3.2.2.3.cmml" xref="S4.I1.i2.p1.3.m3.2.2.3"></lt><apply id="S4.I1.i2.p1.3.m3.1.1.1.cmml" xref="S4.I1.i2.p1.3.m3.1.1.1"><times id="S4.I1.i2.p1.3.m3.1.1.1.2.cmml" xref="S4.I1.i2.p1.3.m3.1.1.1.2"></times><apply id="S4.I1.i2.p1.3.m3.1.1.1.3.cmml" xref="S4.I1.i2.p1.3.m3.1.1.1.3"><csymbol cd="ambiguous" id="S4.I1.i2.p1.3.m3.1.1.1.3.1.cmml" xref="S4.I1.i2.p1.3.m3.1.1.1.3">superscript</csymbol><ci id="S4.I1.i2.p1.3.m3.1.1.1.3.2a.cmml" xref="S4.I1.i2.p1.3.m3.1.1.1.3.2"><mtext class="ltx_mathvariant_italic" id="S4.I1.i2.p1.3.m3.1.1.1.3.2.cmml" xref="S4.I1.i2.p1.3.m3.1.1.1.3.2">g</mtext></ci><ci id="S4.I1.i2.p1.3.m3.1.1.1.3.3.cmml" xref="S4.I1.i2.p1.3.m3.1.1.1.3.3">′</ci></apply><apply id="S4.I1.i2.p1.3.m3.1.1.1.1.1.1.cmml" xref="S4.I1.i2.p1.3.m3.1.1.1.1.1"><ci id="S4.I1.i2.p1.3.m3.1.1.1.1.1.1.1.cmml" xref="S4.I1.i2.p1.3.m3.1.1.1.1.1.1.1">→</ci><ci id="S4.I1.i2.p1.3.m3.1.1.1.1.1.1.2.cmml" xref="S4.I1.i2.p1.3.m3.1.1.1.1.1.1.2">𝑎</ci><ci id="S4.I1.i2.p1.3.m3.1.1.1.1.1.1.3.cmml" xref="S4.I1.i2.p1.3.m3.1.1.1.1.1.1.3">𝑏</ci></apply></apply><apply id="S4.I1.i2.p1.3.m3.2.2.2.cmml" xref="S4.I1.i2.p1.3.m3.2.2.2"><times id="S4.I1.i2.p1.3.m3.2.2.2.2.cmml" xref="S4.I1.i2.p1.3.m3.2.2.2.2"></times><ci id="S4.I1.i2.p1.3.m3.2.2.2.3a.cmml" xref="S4.I1.i2.p1.3.m3.2.2.2.3"><mtext class="ltx_mathvariant_italic" id="S4.I1.i2.p1.3.m3.2.2.2.3.cmml" xref="S4.I1.i2.p1.3.m3.2.2.2.3">g</mtext></ci><apply id="S4.I1.i2.p1.3.m3.2.2.2.1.1.1.cmml" xref="S4.I1.i2.p1.3.m3.2.2.2.1.1"><times id="S4.I1.i2.p1.3.m3.2.2.2.1.1.1.1.cmml" xref="S4.I1.i2.p1.3.m3.2.2.2.1.1.1.1"></times><ci id="S4.I1.i2.p1.3.m3.2.2.2.1.1.1.2.cmml" xref="S4.I1.i2.p1.3.m3.2.2.2.1.1.1.2">𝑎</ci><ci id="S4.I1.i2.p1.3.m3.2.2.2.1.1.1.3.cmml" xref="S4.I1.i2.p1.3.m3.2.2.2.1.1.1.3">𝑏</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.I1.i2.p1.3.m3.2c">\textsl{g}^{\prime}(a\!\to\!b)<\textsl{g}(ab)</annotation><annotation encoding="application/x-llamapun" id="S4.I1.i2.p1.3.m3.2d">g start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ( italic_a → italic_b ) < g ( italic_a italic_b )</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S4.I1.i2.p1.8.3"> and so </span><math alttext="\textsl{g}^{\prime}(b\!\to\!a)>0" class="ltx_Math" display="inline" id="S4.I1.i2.p1.4.m4.1"><semantics id="S4.I1.i2.p1.4.m4.1a"><mrow id="S4.I1.i2.p1.4.m4.1.1" xref="S4.I1.i2.p1.4.m4.1.1.cmml"><mrow id="S4.I1.i2.p1.4.m4.1.1.1" xref="S4.I1.i2.p1.4.m4.1.1.1.cmml"><msup id="S4.I1.i2.p1.4.m4.1.1.1.3" xref="S4.I1.i2.p1.4.m4.1.1.1.3.cmml"><mtext class="ltx_mathvariant_italic" id="S4.I1.i2.p1.4.m4.1.1.1.3.2" xref="S4.I1.i2.p1.4.m4.1.1.1.3.2a.cmml">g</mtext><mo id="S4.I1.i2.p1.4.m4.1.1.1.3.3" xref="S4.I1.i2.p1.4.m4.1.1.1.3.3.cmml">′</mo></msup><mo id="S4.I1.i2.p1.4.m4.1.1.1.2" xref="S4.I1.i2.p1.4.m4.1.1.1.2.cmml">⁢</mo><mrow id="S4.I1.i2.p1.4.m4.1.1.1.1.1" xref="S4.I1.i2.p1.4.m4.1.1.1.1.1.1.cmml"><mo id="S4.I1.i2.p1.4.m4.1.1.1.1.1.2" stretchy="false" xref="S4.I1.i2.p1.4.m4.1.1.1.1.1.1.cmml">(</mo><mrow id="S4.I1.i2.p1.4.m4.1.1.1.1.1.1" xref="S4.I1.i2.p1.4.m4.1.1.1.1.1.1.cmml"><mi id="S4.I1.i2.p1.4.m4.1.1.1.1.1.1.2" xref="S4.I1.i2.p1.4.m4.1.1.1.1.1.1.2.cmml">b</mi><mo id="S4.I1.i2.p1.4.m4.1.1.1.1.1.1.1" lspace="0.108em" rspace="0.108em" stretchy="false" xref="S4.I1.i2.p1.4.m4.1.1.1.1.1.1.1.cmml">→</mo><mi id="S4.I1.i2.p1.4.m4.1.1.1.1.1.1.3" xref="S4.I1.i2.p1.4.m4.1.1.1.1.1.1.3.cmml">a</mi></mrow><mo id="S4.I1.i2.p1.4.m4.1.1.1.1.1.3" stretchy="false" xref="S4.I1.i2.p1.4.m4.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S4.I1.i2.p1.4.m4.1.1.2" xref="S4.I1.i2.p1.4.m4.1.1.2.cmml">></mo><mn id="S4.I1.i2.p1.4.m4.1.1.3" xref="S4.I1.i2.p1.4.m4.1.1.3.cmml">0</mn></mrow><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"><gt id="S4.I1.i2.p1.4.m4.1.1.2.cmml" xref="S4.I1.i2.p1.4.m4.1.1.2"></gt><apply id="S4.I1.i2.p1.4.m4.1.1.1.cmml" xref="S4.I1.i2.p1.4.m4.1.1.1"><times id="S4.I1.i2.p1.4.m4.1.1.1.2.cmml" xref="S4.I1.i2.p1.4.m4.1.1.1.2"></times><apply id="S4.I1.i2.p1.4.m4.1.1.1.3.cmml" xref="S4.I1.i2.p1.4.m4.1.1.1.3"><csymbol cd="ambiguous" id="S4.I1.i2.p1.4.m4.1.1.1.3.1.cmml" xref="S4.I1.i2.p1.4.m4.1.1.1.3">superscript</csymbol><ci id="S4.I1.i2.p1.4.m4.1.1.1.3.2a.cmml" xref="S4.I1.i2.p1.4.m4.1.1.1.3.2"><mtext class="ltx_mathvariant_italic" id="S4.I1.i2.p1.4.m4.1.1.1.3.2.cmml" xref="S4.I1.i2.p1.4.m4.1.1.1.3.2">g</mtext></ci><ci id="S4.I1.i2.p1.4.m4.1.1.1.3.3.cmml" xref="S4.I1.i2.p1.4.m4.1.1.1.3.3">′</ci></apply><apply id="S4.I1.i2.p1.4.m4.1.1.1.1.1.1.cmml" xref="S4.I1.i2.p1.4.m4.1.1.1.1.1"><ci id="S4.I1.i2.p1.4.m4.1.1.1.1.1.1.1.cmml" xref="S4.I1.i2.p1.4.m4.1.1.1.1.1.1.1">→</ci><ci id="S4.I1.i2.p1.4.m4.1.1.1.1.1.1.2.cmml" xref="S4.I1.i2.p1.4.m4.1.1.1.1.1.1.2">𝑏</ci><ci id="S4.I1.i2.p1.4.m4.1.1.1.1.1.1.3.cmml" xref="S4.I1.i2.p1.4.m4.1.1.1.1.1.1.3">𝑎</ci></apply></apply><cn id="S4.I1.i2.p1.4.m4.1.1.3.cmml" type="integer" xref="S4.I1.i2.p1.4.m4.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.I1.i2.p1.4.m4.1c">\textsl{g}^{\prime}(b\!\to\!a)>0</annotation><annotation encoding="application/x-llamapun" id="S4.I1.i2.p1.4.m4.1d">g start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ( italic_b → italic_a ) > 0</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S4.I1.i2.p1.8.4">. It follows that there exists a directed path from </span><math alttext="v" class="ltx_Math" display="inline" id="S4.I1.i2.p1.5.m5.1"><semantics id="S4.I1.i2.p1.5.m5.1a"><mi id="S4.I1.i2.p1.5.m5.1.1" xref="S4.I1.i2.p1.5.m5.1.1.cmml">v</mi><annotation-xml encoding="MathML-Content" id="S4.I1.i2.p1.5.m5.1b"><ci id="S4.I1.i2.p1.5.m5.1.1.cmml" xref="S4.I1.i2.p1.5.m5.1.1">𝑣</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.I1.i2.p1.5.m5.1c">v</annotation><annotation encoding="application/x-llamapun" id="S4.I1.i2.p1.5.m5.1d">italic_v</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S4.I1.i2.p1.8.5"> to </span><math alttext="u" 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">u</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">𝑢</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.I1.i2.p1.6.m6.1c">u</annotation><annotation encoding="application/x-llamapun" id="S4.I1.i2.p1.6.m6.1d">italic_u</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S4.I1.i2.p1.8.6"> in </span><math alttext="\overrightarrow{G}^{\prime}" class="ltx_Math" display="inline" id="S4.I1.i2.p1.7.m7.1"><semantics id="S4.I1.i2.p1.7.m7.1a"><msup id="S4.I1.i2.p1.7.m7.1.1" xref="S4.I1.i2.p1.7.m7.1.1.cmml"><mover accent="true" id="S4.I1.i2.p1.7.m7.1.1.2" xref="S4.I1.i2.p1.7.m7.1.1.2.cmml"><mi id="S4.I1.i2.p1.7.m7.1.1.2.2" xref="S4.I1.i2.p1.7.m7.1.1.2.2.cmml">G</mi><mo id="S4.I1.i2.p1.7.m7.1.1.2.1" stretchy="false" xref="S4.I1.i2.p1.7.m7.1.1.2.1.cmml">→</mo></mover><mo id="S4.I1.i2.p1.7.m7.1.1.3" xref="S4.I1.i2.p1.7.m7.1.1.3.cmml">′</mo></msup><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"><csymbol cd="ambiguous" id="S4.I1.i2.p1.7.m7.1.1.1.cmml" xref="S4.I1.i2.p1.7.m7.1.1">superscript</csymbol><apply id="S4.I1.i2.p1.7.m7.1.1.2.cmml" xref="S4.I1.i2.p1.7.m7.1.1.2"><ci id="S4.I1.i2.p1.7.m7.1.1.2.1.cmml" xref="S4.I1.i2.p1.7.m7.1.1.2.1">→</ci><ci id="S4.I1.i2.p1.7.m7.1.1.2.2.cmml" xref="S4.I1.i2.p1.7.m7.1.1.2.2">𝐺</ci></apply><ci id="S4.I1.i2.p1.7.m7.1.1.3.cmml" xref="S4.I1.i2.p1.7.m7.1.1.3">′</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.I1.i2.p1.7.m7.1c">\overrightarrow{G}^{\prime}</annotation><annotation encoding="application/x-llamapun" id="S4.I1.i2.p1.7.m7.1d">over→ start_ARG italic_G end_ARG start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S4.I1.i2.p1.8.7">. Local fairness implies </span><math alttext="\textsl{g}^{\prime}(v)\leq\textsl{g}^{\prime}(u)" class="ltx_Math" display="inline" id="S4.I1.i2.p1.8.m8.2"><semantics id="S4.I1.i2.p1.8.m8.2a"><mrow id="S4.I1.i2.p1.8.m8.2.3" xref="S4.I1.i2.p1.8.m8.2.3.cmml"><mrow id="S4.I1.i2.p1.8.m8.2.3.2" xref="S4.I1.i2.p1.8.m8.2.3.2.cmml"><msup id="S4.I1.i2.p1.8.m8.2.3.2.2" xref="S4.I1.i2.p1.8.m8.2.3.2.2.cmml"><mtext class="ltx_mathvariant_italic" id="S4.I1.i2.p1.8.m8.2.3.2.2.2" xref="S4.I1.i2.p1.8.m8.2.3.2.2.2a.cmml">g</mtext><mo id="S4.I1.i2.p1.8.m8.2.3.2.2.3" xref="S4.I1.i2.p1.8.m8.2.3.2.2.3.cmml">′</mo></msup><mo id="S4.I1.i2.p1.8.m8.2.3.2.1" xref="S4.I1.i2.p1.8.m8.2.3.2.1.cmml">⁢</mo><mrow id="S4.I1.i2.p1.8.m8.2.3.2.3.2" xref="S4.I1.i2.p1.8.m8.2.3.2.cmml"><mo id="S4.I1.i2.p1.8.m8.2.3.2.3.2.1" stretchy="false" xref="S4.I1.i2.p1.8.m8.2.3.2.cmml">(</mo><mi id="S4.I1.i2.p1.8.m8.1.1" xref="S4.I1.i2.p1.8.m8.1.1.cmml">v</mi><mo id="S4.I1.i2.p1.8.m8.2.3.2.3.2.2" stretchy="false" xref="S4.I1.i2.p1.8.m8.2.3.2.cmml">)</mo></mrow></mrow><mo id="S4.I1.i2.p1.8.m8.2.3.1" xref="S4.I1.i2.p1.8.m8.2.3.1.cmml">≤</mo><mrow id="S4.I1.i2.p1.8.m8.2.3.3" xref="S4.I1.i2.p1.8.m8.2.3.3.cmml"><msup id="S4.I1.i2.p1.8.m8.2.3.3.2" xref="S4.I1.i2.p1.8.m8.2.3.3.2.cmml"><mtext class="ltx_mathvariant_italic" id="S4.I1.i2.p1.8.m8.2.3.3.2.2" xref="S4.I1.i2.p1.8.m8.2.3.3.2.2a.cmml">g</mtext><mo id="S4.I1.i2.p1.8.m8.2.3.3.2.3" xref="S4.I1.i2.p1.8.m8.2.3.3.2.3.cmml">′</mo></msup><mo id="S4.I1.i2.p1.8.m8.2.3.3.1" xref="S4.I1.i2.p1.8.m8.2.3.3.1.cmml">⁢</mo><mrow id="S4.I1.i2.p1.8.m8.2.3.3.3.2" xref="S4.I1.i2.p1.8.m8.2.3.3.cmml"><mo id="S4.I1.i2.p1.8.m8.2.3.3.3.2.1" stretchy="false" xref="S4.I1.i2.p1.8.m8.2.3.3.cmml">(</mo><mi id="S4.I1.i2.p1.8.m8.2.2" xref="S4.I1.i2.p1.8.m8.2.2.cmml">u</mi><mo id="S4.I1.i2.p1.8.m8.2.3.3.3.2.2" stretchy="false" xref="S4.I1.i2.p1.8.m8.2.3.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.I1.i2.p1.8.m8.2b"><apply id="S4.I1.i2.p1.8.m8.2.3.cmml" xref="S4.I1.i2.p1.8.m8.2.3"><leq id="S4.I1.i2.p1.8.m8.2.3.1.cmml" xref="S4.I1.i2.p1.8.m8.2.3.1"></leq><apply id="S4.I1.i2.p1.8.m8.2.3.2.cmml" xref="S4.I1.i2.p1.8.m8.2.3.2"><times id="S4.I1.i2.p1.8.m8.2.3.2.1.cmml" xref="S4.I1.i2.p1.8.m8.2.3.2.1"></times><apply id="S4.I1.i2.p1.8.m8.2.3.2.2.cmml" xref="S4.I1.i2.p1.8.m8.2.3.2.2"><csymbol cd="ambiguous" id="S4.I1.i2.p1.8.m8.2.3.2.2.1.cmml" xref="S4.I1.i2.p1.8.m8.2.3.2.2">superscript</csymbol><ci id="S4.I1.i2.p1.8.m8.2.3.2.2.2a.cmml" xref="S4.I1.i2.p1.8.m8.2.3.2.2.2"><mtext class="ltx_mathvariant_italic" id="S4.I1.i2.p1.8.m8.2.3.2.2.2.cmml" xref="S4.I1.i2.p1.8.m8.2.3.2.2.2">g</mtext></ci><ci id="S4.I1.i2.p1.8.m8.2.3.2.2.3.cmml" xref="S4.I1.i2.p1.8.m8.2.3.2.2.3">′</ci></apply><ci id="S4.I1.i2.p1.8.m8.1.1.cmml" xref="S4.I1.i2.p1.8.m8.1.1">𝑣</ci></apply><apply id="S4.I1.i2.p1.8.m8.2.3.3.cmml" xref="S4.I1.i2.p1.8.m8.2.3.3"><times id="S4.I1.i2.p1.8.m8.2.3.3.1.cmml" xref="S4.I1.i2.p1.8.m8.2.3.3.1"></times><apply id="S4.I1.i2.p1.8.m8.2.3.3.2.cmml" xref="S4.I1.i2.p1.8.m8.2.3.3.2"><csymbol cd="ambiguous" id="S4.I1.i2.p1.8.m8.2.3.3.2.1.cmml" xref="S4.I1.i2.p1.8.m8.2.3.3.2">superscript</csymbol><ci id="S4.I1.i2.p1.8.m8.2.3.3.2.2a.cmml" xref="S4.I1.i2.p1.8.m8.2.3.3.2.2"><mtext class="ltx_mathvariant_italic" id="S4.I1.i2.p1.8.m8.2.3.3.2.2.cmml" xref="S4.I1.i2.p1.8.m8.2.3.3.2.2">g</mtext></ci><ci id="S4.I1.i2.p1.8.m8.2.3.3.2.3.cmml" xref="S4.I1.i2.p1.8.m8.2.3.3.2.3">′</ci></apply><ci id="S4.I1.i2.p1.8.m8.2.2.cmml" xref="S4.I1.i2.p1.8.m8.2.2">𝑢</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.I1.i2.p1.8.m8.2c">\textsl{g}^{\prime}(v)\leq\textsl{g}^{\prime}(u)</annotation><annotation encoding="application/x-llamapun" id="S4.I1.i2.p1.8.m8.2d">g start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ( italic_v ) ≤ g start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ( italic_u )</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S4.I1.i2.p1.8.8">.</span></p> </div> </li> </ul> <p class="ltx_p" id="S4.Thmtheorem2.p4.5"><span class="ltx_text ltx_font_italic" id="S4.Thmtheorem2.p4.5.1">The 4 equations: </span><math alttext="\textsl{g}(u)>\textsl{g}^{\prime}(u)" class="ltx_Math" display="inline" id="S4.Thmtheorem2.p4.2.m1.2"><semantics id="S4.Thmtheorem2.p4.2.m1.2a"><mrow id="S4.Thmtheorem2.p4.2.m1.2.3" xref="S4.Thmtheorem2.p4.2.m1.2.3.cmml"><mrow id="S4.Thmtheorem2.p4.2.m1.2.3.2" xref="S4.Thmtheorem2.p4.2.m1.2.3.2.cmml"><mtext class="ltx_mathvariant_italic" id="S4.Thmtheorem2.p4.2.m1.2.3.2.2" xref="S4.Thmtheorem2.p4.2.m1.2.3.2.2a.cmml">g</mtext><mo id="S4.Thmtheorem2.p4.2.m1.2.3.2.1" xref="S4.Thmtheorem2.p4.2.m1.2.3.2.1.cmml">⁢</mo><mrow id="S4.Thmtheorem2.p4.2.m1.2.3.2.3.2" xref="S4.Thmtheorem2.p4.2.m1.2.3.2.cmml"><mo id="S4.Thmtheorem2.p4.2.m1.2.3.2.3.2.1" stretchy="false" xref="S4.Thmtheorem2.p4.2.m1.2.3.2.cmml">(</mo><mi id="S4.Thmtheorem2.p4.2.m1.1.1" xref="S4.Thmtheorem2.p4.2.m1.1.1.cmml">u</mi><mo id="S4.Thmtheorem2.p4.2.m1.2.3.2.3.2.2" stretchy="false" xref="S4.Thmtheorem2.p4.2.m1.2.3.2.cmml">)</mo></mrow></mrow><mo id="S4.Thmtheorem2.p4.2.m1.2.3.1" xref="S4.Thmtheorem2.p4.2.m1.2.3.1.cmml">></mo><mrow id="S4.Thmtheorem2.p4.2.m1.2.3.3" xref="S4.Thmtheorem2.p4.2.m1.2.3.3.cmml"><msup id="S4.Thmtheorem2.p4.2.m1.2.3.3.2" xref="S4.Thmtheorem2.p4.2.m1.2.3.3.2.cmml"><mtext class="ltx_mathvariant_italic" id="S4.Thmtheorem2.p4.2.m1.2.3.3.2.2" xref="S4.Thmtheorem2.p4.2.m1.2.3.3.2.2a.cmml">g</mtext><mo id="S4.Thmtheorem2.p4.2.m1.2.3.3.2.3" xref="S4.Thmtheorem2.p4.2.m1.2.3.3.2.3.cmml">′</mo></msup><mo id="S4.Thmtheorem2.p4.2.m1.2.3.3.1" xref="S4.Thmtheorem2.p4.2.m1.2.3.3.1.cmml">⁢</mo><mrow id="S4.Thmtheorem2.p4.2.m1.2.3.3.3.2" xref="S4.Thmtheorem2.p4.2.m1.2.3.3.cmml"><mo id="S4.Thmtheorem2.p4.2.m1.2.3.3.3.2.1" stretchy="false" xref="S4.Thmtheorem2.p4.2.m1.2.3.3.cmml">(</mo><mi id="S4.Thmtheorem2.p4.2.m1.2.2" xref="S4.Thmtheorem2.p4.2.m1.2.2.cmml">u</mi><mo id="S4.Thmtheorem2.p4.2.m1.2.3.3.3.2.2" stretchy="false" xref="S4.Thmtheorem2.p4.2.m1.2.3.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem2.p4.2.m1.2b"><apply id="S4.Thmtheorem2.p4.2.m1.2.3.cmml" xref="S4.Thmtheorem2.p4.2.m1.2.3"><gt id="S4.Thmtheorem2.p4.2.m1.2.3.1.cmml" xref="S4.Thmtheorem2.p4.2.m1.2.3.1"></gt><apply id="S4.Thmtheorem2.p4.2.m1.2.3.2.cmml" xref="S4.Thmtheorem2.p4.2.m1.2.3.2"><times id="S4.Thmtheorem2.p4.2.m1.2.3.2.1.cmml" xref="S4.Thmtheorem2.p4.2.m1.2.3.2.1"></times><ci id="S4.Thmtheorem2.p4.2.m1.2.3.2.2a.cmml" xref="S4.Thmtheorem2.p4.2.m1.2.3.2.2"><mtext class="ltx_mathvariant_italic" id="S4.Thmtheorem2.p4.2.m1.2.3.2.2.cmml" xref="S4.Thmtheorem2.p4.2.m1.2.3.2.2">g</mtext></ci><ci id="S4.Thmtheorem2.p4.2.m1.1.1.cmml" xref="S4.Thmtheorem2.p4.2.m1.1.1">𝑢</ci></apply><apply id="S4.Thmtheorem2.p4.2.m1.2.3.3.cmml" xref="S4.Thmtheorem2.p4.2.m1.2.3.3"><times id="S4.Thmtheorem2.p4.2.m1.2.3.3.1.cmml" xref="S4.Thmtheorem2.p4.2.m1.2.3.3.1"></times><apply id="S4.Thmtheorem2.p4.2.m1.2.3.3.2.cmml" xref="S4.Thmtheorem2.p4.2.m1.2.3.3.2"><csymbol cd="ambiguous" id="S4.Thmtheorem2.p4.2.m1.2.3.3.2.1.cmml" xref="S4.Thmtheorem2.p4.2.m1.2.3.3.2">superscript</csymbol><ci id="S4.Thmtheorem2.p4.2.m1.2.3.3.2.2a.cmml" xref="S4.Thmtheorem2.p4.2.m1.2.3.3.2.2"><mtext class="ltx_mathvariant_italic" id="S4.Thmtheorem2.p4.2.m1.2.3.3.2.2.cmml" xref="S4.Thmtheorem2.p4.2.m1.2.3.3.2.2">g</mtext></ci><ci id="S4.Thmtheorem2.p4.2.m1.2.3.3.2.3.cmml" xref="S4.Thmtheorem2.p4.2.m1.2.3.3.2.3">′</ci></apply><ci id="S4.Thmtheorem2.p4.2.m1.2.2.cmml" xref="S4.Thmtheorem2.p4.2.m1.2.2">𝑢</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem2.p4.2.m1.2c">\textsl{g}(u)>\textsl{g}^{\prime}(u)</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem2.p4.2.m1.2d">g ( italic_u ) > g start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ( italic_u )</annotation></semantics></math><span class="ltx_text ltx_font_italic" 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xref="S4.Thmtheorem2.p4.5.m4.2.3.3.1.cmml">⁢</mo><mrow id="S4.Thmtheorem2.p4.5.m4.2.3.3.3.2" xref="S4.Thmtheorem2.p4.5.m4.2.3.3.cmml"><mo id="S4.Thmtheorem2.p4.5.m4.2.3.3.3.2.1" stretchy="false" xref="S4.Thmtheorem2.p4.5.m4.2.3.3.cmml">(</mo><mi id="S4.Thmtheorem2.p4.5.m4.2.2" xref="S4.Thmtheorem2.p4.5.m4.2.2.cmml">v</mi><mo id="S4.Thmtheorem2.p4.5.m4.2.3.3.3.2.2" stretchy="false" xref="S4.Thmtheorem2.p4.5.m4.2.3.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem2.p4.5.m4.2b"><apply id="S4.Thmtheorem2.p4.5.m4.2.3.cmml" xref="S4.Thmtheorem2.p4.5.m4.2.3"><geq id="S4.Thmtheorem2.p4.5.m4.2.3.1.cmml" xref="S4.Thmtheorem2.p4.5.m4.2.3.1"></geq><apply id="S4.Thmtheorem2.p4.5.m4.2.3.2.cmml" xref="S4.Thmtheorem2.p4.5.m4.2.3.2"><times id="S4.Thmtheorem2.p4.5.m4.2.3.2.1.cmml" xref="S4.Thmtheorem2.p4.5.m4.2.3.2.1"></times><apply id="S4.Thmtheorem2.p4.5.m4.2.3.2.2.cmml" xref="S4.Thmtheorem2.p4.5.m4.2.3.2.2"><csymbol cd="ambiguous" id="S4.Thmtheorem2.p4.5.m4.2.3.2.2.1.cmml" xref="S4.Thmtheorem2.p4.5.m4.2.3.2.2">superscript</csymbol><ci id="S4.Thmtheorem2.p4.5.m4.2.3.2.2.2a.cmml" xref="S4.Thmtheorem2.p4.5.m4.2.3.2.2.2"><mtext class="ltx_mathvariant_italic" id="S4.Thmtheorem2.p4.5.m4.2.3.2.2.2.cmml" xref="S4.Thmtheorem2.p4.5.m4.2.3.2.2.2">g</mtext></ci><ci id="S4.Thmtheorem2.p4.5.m4.2.3.2.2.3.cmml" xref="S4.Thmtheorem2.p4.5.m4.2.3.2.2.3">′</ci></apply><ci id="S4.Thmtheorem2.p4.5.m4.1.1.cmml" xref="S4.Thmtheorem2.p4.5.m4.1.1">𝑢</ci></apply><apply id="S4.Thmtheorem2.p4.5.m4.2.3.3.cmml" xref="S4.Thmtheorem2.p4.5.m4.2.3.3"><times id="S4.Thmtheorem2.p4.5.m4.2.3.3.1.cmml" xref="S4.Thmtheorem2.p4.5.m4.2.3.3.1"></times><apply id="S4.Thmtheorem2.p4.5.m4.2.3.3.2.cmml" xref="S4.Thmtheorem2.p4.5.m4.2.3.3.2"><csymbol cd="ambiguous" id="S4.Thmtheorem2.p4.5.m4.2.3.3.2.1.cmml" xref="S4.Thmtheorem2.p4.5.m4.2.3.3.2">superscript</csymbol><ci id="S4.Thmtheorem2.p4.5.m4.2.3.3.2.2a.cmml" xref="S4.Thmtheorem2.p4.5.m4.2.3.3.2.2"><mtext class="ltx_mathvariant_italic" id="S4.Thmtheorem2.p4.5.m4.2.3.3.2.2.cmml" xref="S4.Thmtheorem2.p4.5.m4.2.3.3.2.2">g</mtext></ci><ci id="S4.Thmtheorem2.p4.5.m4.2.3.3.2.3.cmml" xref="S4.Thmtheorem2.p4.5.m4.2.3.3.2.3">′</ci></apply><ci id="S4.Thmtheorem2.p4.5.m4.2.2.cmml" xref="S4.Thmtheorem2.p4.5.m4.2.2">𝑣</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem2.p4.5.m4.2c">\,\textsl{g}^{\prime}(u)\geq\textsl{g}^{\prime}(v)</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem2.p4.5.m4.2d">g start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ( italic_u ) ≥ g start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ( italic_v )</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S4.Thmtheorem2.p4.5.5"> give a contradiction.</span></p> </div> </div> <div class="ltx_para" id="S4.p2"> <p class="ltx_p" id="S4.p2.6">Lemma <a class="ltx_ref" href="https://arxiv.org/html/2411.12694v2#S4.Thmtheorem1" title="Lemma 4.1. ‣ 4 Conceptual results for local density ‣ Local Density and its Distributed Approximation"><span class="ltx_text ltx_ref_tag">4.1</span></a> would imply that the local out-degree is well-defined, <em class="ltx_emph ltx_font_italic" id="S4.p2.6.1">if</em> the set of locally fair orientations is non-empty. Bera, Bhattacharya, Choudhari and Ghosh already aim to prove this in Section 4.1 of <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2411.12694v2#bib.bib3" title="">3</a>]</cite> (right above Equation 9). They claim that a locally fair orientation always exist by the following argument: They consider an arbitrary orientation <math alttext="\overrightarrow{G}" class="ltx_Math" display="inline" id="S4.p2.1.m1.1"><semantics id="S4.p2.1.m1.1a"><mover accent="true" id="S4.p2.1.m1.1.1" xref="S4.p2.1.m1.1.1.cmml"><mi id="S4.p2.1.m1.1.1.2" xref="S4.p2.1.m1.1.1.2.cmml">G</mi><mo id="S4.p2.1.m1.1.1.1" stretchy="false" xref="S4.p2.1.m1.1.1.1.cmml">→</mo></mover><annotation-xml encoding="MathML-Content" id="S4.p2.1.m1.1b"><apply id="S4.p2.1.m1.1.1.cmml" xref="S4.p2.1.m1.1.1"><ci id="S4.p2.1.m1.1.1.1.cmml" xref="S4.p2.1.m1.1.1.1">→</ci><ci id="S4.p2.1.m1.1.1.2.cmml" xref="S4.p2.1.m1.1.1.2">𝐺</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.p2.1.m1.1c">\overrightarrow{G}</annotation><annotation encoding="application/x-llamapun" id="S4.p2.1.m1.1d">over→ start_ARG italic_G end_ARG</annotation></semantics></math> that is not locally fair. They claim that for any pair <math alttext="(u,v)" class="ltx_Math" display="inline" id="S4.p2.2.m2.2"><semantics id="S4.p2.2.m2.2a"><mrow id="S4.p2.2.m2.2.3.2" xref="S4.p2.2.m2.2.3.1.cmml"><mo id="S4.p2.2.m2.2.3.2.1" stretchy="false" xref="S4.p2.2.m2.2.3.1.cmml">(</mo><mi id="S4.p2.2.m2.1.1" xref="S4.p2.2.m2.1.1.cmml">u</mi><mo id="S4.p2.2.m2.2.3.2.2" xref="S4.p2.2.m2.2.3.1.cmml">,</mo><mi id="S4.p2.2.m2.2.2" xref="S4.p2.2.m2.2.2.cmml">v</mi><mo id="S4.p2.2.m2.2.3.2.3" stretchy="false" xref="S4.p2.2.m2.2.3.1.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S4.p2.2.m2.2b"><interval closure="open" id="S4.p2.2.m2.2.3.1.cmml" xref="S4.p2.2.m2.2.3.2"><ci id="S4.p2.2.m2.1.1.cmml" xref="S4.p2.2.m2.1.1">𝑢</ci><ci id="S4.p2.2.m2.2.2.cmml" xref="S4.p2.2.m2.2.2">𝑣</ci></interval></annotation-xml><annotation encoding="application/x-tex" id="S4.p2.2.m2.2c">(u,v)</annotation><annotation encoding="application/x-llamapun" id="S4.p2.2.m2.2d">( italic_u , italic_v )</annotation></semantics></math> where <math alttext="\textsl{g}(u)>\textsl{g}(v)" class="ltx_Math" display="inline" id="S4.p2.3.m3.2"><semantics id="S4.p2.3.m3.2a"><mrow id="S4.p2.3.m3.2.3" xref="S4.p2.3.m3.2.3.cmml"><mrow id="S4.p2.3.m3.2.3.2" xref="S4.p2.3.m3.2.3.2.cmml"><mtext class="ltx_mathvariant_italic" id="S4.p2.3.m3.2.3.2.2" xref="S4.p2.3.m3.2.3.2.2a.cmml">g</mtext><mo id="S4.p2.3.m3.2.3.2.1" xref="S4.p2.3.m3.2.3.2.1.cmml">⁢</mo><mrow id="S4.p2.3.m3.2.3.2.3.2" xref="S4.p2.3.m3.2.3.2.cmml"><mo id="S4.p2.3.m3.2.3.2.3.2.1" stretchy="false" xref="S4.p2.3.m3.2.3.2.cmml">(</mo><mi id="S4.p2.3.m3.1.1" xref="S4.p2.3.m3.1.1.cmml">u</mi><mo id="S4.p2.3.m3.2.3.2.3.2.2" stretchy="false" xref="S4.p2.3.m3.2.3.2.cmml">)</mo></mrow></mrow><mo id="S4.p2.3.m3.2.3.1" xref="S4.p2.3.m3.2.3.1.cmml">></mo><mrow id="S4.p2.3.m3.2.3.3" xref="S4.p2.3.m3.2.3.3.cmml"><mtext class="ltx_mathvariant_italic" id="S4.p2.3.m3.2.3.3.2" xref="S4.p2.3.m3.2.3.3.2a.cmml">g</mtext><mo id="S4.p2.3.m3.2.3.3.1" xref="S4.p2.3.m3.2.3.3.1.cmml">⁢</mo><mrow id="S4.p2.3.m3.2.3.3.3.2" xref="S4.p2.3.m3.2.3.3.cmml"><mo id="S4.p2.3.m3.2.3.3.3.2.1" stretchy="false" xref="S4.p2.3.m3.2.3.3.cmml">(</mo><mi id="S4.p2.3.m3.2.2" xref="S4.p2.3.m3.2.2.cmml">v</mi><mo id="S4.p2.3.m3.2.3.3.3.2.2" stretchy="false" xref="S4.p2.3.m3.2.3.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.p2.3.m3.2b"><apply id="S4.p2.3.m3.2.3.cmml" xref="S4.p2.3.m3.2.3"><gt id="S4.p2.3.m3.2.3.1.cmml" xref="S4.p2.3.m3.2.3.1"></gt><apply id="S4.p2.3.m3.2.3.2.cmml" xref="S4.p2.3.m3.2.3.2"><times id="S4.p2.3.m3.2.3.2.1.cmml" xref="S4.p2.3.m3.2.3.2.1"></times><ci id="S4.p2.3.m3.2.3.2.2a.cmml" xref="S4.p2.3.m3.2.3.2.2"><mtext class="ltx_mathvariant_italic" id="S4.p2.3.m3.2.3.2.2.cmml" xref="S4.p2.3.m3.2.3.2.2">g</mtext></ci><ci id="S4.p2.3.m3.1.1.cmml" xref="S4.p2.3.m3.1.1">𝑢</ci></apply><apply id="S4.p2.3.m3.2.3.3.cmml" xref="S4.p2.3.m3.2.3.3"><times id="S4.p2.3.m3.2.3.3.1.cmml" xref="S4.p2.3.m3.2.3.3.1"></times><ci id="S4.p2.3.m3.2.3.3.2a.cmml" xref="S4.p2.3.m3.2.3.3.2"><mtext class="ltx_mathvariant_italic" id="S4.p2.3.m3.2.3.3.2.cmml" xref="S4.p2.3.m3.2.3.3.2">g</mtext></ci><ci id="S4.p2.3.m3.2.2.cmml" xref="S4.p2.3.m3.2.2">𝑣</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.p2.3.m3.2c">\textsl{g}(u)>\textsl{g}(v)</annotation><annotation encoding="application/x-llamapun" id="S4.p2.3.m3.2d">g ( italic_u ) > g ( italic_v )</annotation></semantics></math> and <math alttext="\textsl{g}(u\!\to\!v)>0" class="ltx_Math" display="inline" id="S4.p2.4.m4.1"><semantics id="S4.p2.4.m4.1a"><mrow id="S4.p2.4.m4.1.1" xref="S4.p2.4.m4.1.1.cmml"><mrow id="S4.p2.4.m4.1.1.1" xref="S4.p2.4.m4.1.1.1.cmml"><mtext class="ltx_mathvariant_italic" id="S4.p2.4.m4.1.1.1.3" xref="S4.p2.4.m4.1.1.1.3a.cmml">g</mtext><mo id="S4.p2.4.m4.1.1.1.2" xref="S4.p2.4.m4.1.1.1.2.cmml">⁢</mo><mrow id="S4.p2.4.m4.1.1.1.1.1" xref="S4.p2.4.m4.1.1.1.1.1.1.cmml"><mo id="S4.p2.4.m4.1.1.1.1.1.2" stretchy="false" xref="S4.p2.4.m4.1.1.1.1.1.1.cmml">(</mo><mrow id="S4.p2.4.m4.1.1.1.1.1.1" xref="S4.p2.4.m4.1.1.1.1.1.1.cmml"><mi id="S4.p2.4.m4.1.1.1.1.1.1.2" xref="S4.p2.4.m4.1.1.1.1.1.1.2.cmml">u</mi><mo id="S4.p2.4.m4.1.1.1.1.1.1.1" lspace="0.108em" rspace="0.108em" stretchy="false" xref="S4.p2.4.m4.1.1.1.1.1.1.1.cmml">→</mo><mi id="S4.p2.4.m4.1.1.1.1.1.1.3" xref="S4.p2.4.m4.1.1.1.1.1.1.3.cmml">v</mi></mrow><mo id="S4.p2.4.m4.1.1.1.1.1.3" stretchy="false" xref="S4.p2.4.m4.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S4.p2.4.m4.1.1.2" xref="S4.p2.4.m4.1.1.2.cmml">></mo><mn id="S4.p2.4.m4.1.1.3" xref="S4.p2.4.m4.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.p2.4.m4.1b"><apply id="S4.p2.4.m4.1.1.cmml" xref="S4.p2.4.m4.1.1"><gt id="S4.p2.4.m4.1.1.2.cmml" xref="S4.p2.4.m4.1.1.2"></gt><apply id="S4.p2.4.m4.1.1.1.cmml" xref="S4.p2.4.m4.1.1.1"><times id="S4.p2.4.m4.1.1.1.2.cmml" xref="S4.p2.4.m4.1.1.1.2"></times><ci id="S4.p2.4.m4.1.1.1.3a.cmml" xref="S4.p2.4.m4.1.1.1.3"><mtext class="ltx_mathvariant_italic" id="S4.p2.4.m4.1.1.1.3.cmml" xref="S4.p2.4.m4.1.1.1.3">g</mtext></ci><apply id="S4.p2.4.m4.1.1.1.1.1.1.cmml" xref="S4.p2.4.m4.1.1.1.1.1"><ci id="S4.p2.4.m4.1.1.1.1.1.1.1.cmml" xref="S4.p2.4.m4.1.1.1.1.1.1.1">→</ci><ci id="S4.p2.4.m4.1.1.1.1.1.1.2.cmml" xref="S4.p2.4.m4.1.1.1.1.1.1.2">𝑢</ci><ci id="S4.p2.4.m4.1.1.1.1.1.1.3.cmml" xref="S4.p2.4.m4.1.1.1.1.1.1.3">𝑣</ci></apply></apply><cn id="S4.p2.4.m4.1.1.3.cmml" type="integer" xref="S4.p2.4.m4.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.p2.4.m4.1c">\textsl{g}(u\!\to\!v)>0</annotation><annotation encoding="application/x-llamapun" id="S4.p2.4.m4.1d">g ( italic_u → italic_v ) > 0</annotation></semantics></math> it is possible to transfer some out-degree from <math alttext="\textsl{g}(u)" class="ltx_Math" display="inline" id="S4.p2.5.m5.1"><semantics id="S4.p2.5.m5.1a"><mrow id="S4.p2.5.m5.1.2" xref="S4.p2.5.m5.1.2.cmml"><mtext class="ltx_mathvariant_italic" id="S4.p2.5.m5.1.2.2" xref="S4.p2.5.m5.1.2.2a.cmml">g</mtext><mo id="S4.p2.5.m5.1.2.1" xref="S4.p2.5.m5.1.2.1.cmml">⁢</mo><mrow id="S4.p2.5.m5.1.2.3.2" xref="S4.p2.5.m5.1.2.cmml"><mo id="S4.p2.5.m5.1.2.3.2.1" stretchy="false" xref="S4.p2.5.m5.1.2.cmml">(</mo><mi id="S4.p2.5.m5.1.1" xref="S4.p2.5.m5.1.1.cmml">u</mi><mo id="S4.p2.5.m5.1.2.3.2.2" stretchy="false" xref="S4.p2.5.m5.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.p2.5.m5.1b"><apply id="S4.p2.5.m5.1.2.cmml" xref="S4.p2.5.m5.1.2"><times id="S4.p2.5.m5.1.2.1.cmml" xref="S4.p2.5.m5.1.2.1"></times><ci id="S4.p2.5.m5.1.2.2a.cmml" xref="S4.p2.5.m5.1.2.2"><mtext class="ltx_mathvariant_italic" id="S4.p2.5.m5.1.2.2.cmml" xref="S4.p2.5.m5.1.2.2">g</mtext></ci><ci id="S4.p2.5.m5.1.1.cmml" xref="S4.p2.5.m5.1.1">𝑢</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.p2.5.m5.1c">\textsl{g}(u)</annotation><annotation encoding="application/x-llamapun" id="S4.p2.5.m5.1d">g ( italic_u )</annotation></semantics></math> to <math alttext="\textsl{g}(v)" class="ltx_Math" display="inline" id="S4.p2.6.m6.1"><semantics id="S4.p2.6.m6.1a"><mrow id="S4.p2.6.m6.1.2" xref="S4.p2.6.m6.1.2.cmml"><mtext class="ltx_mathvariant_italic" id="S4.p2.6.m6.1.2.2" xref="S4.p2.6.m6.1.2.2a.cmml">g</mtext><mo id="S4.p2.6.m6.1.2.1" xref="S4.p2.6.m6.1.2.1.cmml">⁢</mo><mrow id="S4.p2.6.m6.1.2.3.2" xref="S4.p2.6.m6.1.2.cmml"><mo id="S4.p2.6.m6.1.2.3.2.1" stretchy="false" xref="S4.p2.6.m6.1.2.cmml">(</mo><mi id="S4.p2.6.m6.1.1" xref="S4.p2.6.m6.1.1.cmml">v</mi><mo id="S4.p2.6.m6.1.2.3.2.2" stretchy="false" xref="S4.p2.6.m6.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.p2.6.m6.1b"><apply id="S4.p2.6.m6.1.2.cmml" xref="S4.p2.6.m6.1.2"><times id="S4.p2.6.m6.1.2.1.cmml" xref="S4.p2.6.m6.1.2.1"></times><ci id="S4.p2.6.m6.1.2.2a.cmml" xref="S4.p2.6.m6.1.2.2"><mtext class="ltx_mathvariant_italic" id="S4.p2.6.m6.1.2.2.cmml" xref="S4.p2.6.m6.1.2.2">g</mtext></ci><ci id="S4.p2.6.m6.1.1.cmml" xref="S4.p2.6.m6.1.1">𝑣</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.p2.6.m6.1c">\textsl{g}(v)</annotation><annotation encoding="application/x-llamapun" id="S4.p2.6.m6.1d">g ( italic_v )</annotation></semantics></math>. The existence of a locally fair orientation would follow, if it can be shown that this procedure converges to a locally fair orientation. Indeed, since the space of all orientations is a compact polytope, the limit of a converging sequence over this domain must lie within the space.</p> </div> <div class="ltx_para" id="S4.p3"> <p class="ltx_p" id="S4.p3.6">However, it is intuitive but not clear that this procedure indeed converges. Indeed, decreasing <math alttext="\textsl{g}(u)" class="ltx_Math" display="inline" id="S4.p3.1.m1.1"><semantics id="S4.p3.1.m1.1a"><mrow id="S4.p3.1.m1.1.2" xref="S4.p3.1.m1.1.2.cmml"><mtext class="ltx_mathvariant_italic" id="S4.p3.1.m1.1.2.2" xref="S4.p3.1.m1.1.2.2a.cmml">g</mtext><mo id="S4.p3.1.m1.1.2.1" xref="S4.p3.1.m1.1.2.1.cmml">⁢</mo><mrow id="S4.p3.1.m1.1.2.3.2" xref="S4.p3.1.m1.1.2.cmml"><mo id="S4.p3.1.m1.1.2.3.2.1" stretchy="false" xref="S4.p3.1.m1.1.2.cmml">(</mo><mi id="S4.p3.1.m1.1.1" xref="S4.p3.1.m1.1.1.cmml">u</mi><mo id="S4.p3.1.m1.1.2.3.2.2" stretchy="false" xref="S4.p3.1.m1.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.p3.1.m1.1b"><apply id="S4.p3.1.m1.1.2.cmml" xref="S4.p3.1.m1.1.2"><times id="S4.p3.1.m1.1.2.1.cmml" xref="S4.p3.1.m1.1.2.1"></times><ci id="S4.p3.1.m1.1.2.2a.cmml" xref="S4.p3.1.m1.1.2.2"><mtext class="ltx_mathvariant_italic" id="S4.p3.1.m1.1.2.2.cmml" xref="S4.p3.1.m1.1.2.2">g</mtext></ci><ci id="S4.p3.1.m1.1.1.cmml" xref="S4.p3.1.m1.1.1">𝑢</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.p3.1.m1.1c">\textsl{g}(u)</annotation><annotation encoding="application/x-llamapun" id="S4.p3.1.m1.1d">g ( italic_u )</annotation></semantics></math> and increasing <math alttext="\textsl{g}(v)" class="ltx_Math" display="inline" id="S4.p3.2.m2.1"><semantics id="S4.p3.2.m2.1a"><mrow id="S4.p3.2.m2.1.2" xref="S4.p3.2.m2.1.2.cmml"><mtext class="ltx_mathvariant_italic" id="S4.p3.2.m2.1.2.2" xref="S4.p3.2.m2.1.2.2a.cmml">g</mtext><mo id="S4.p3.2.m2.1.2.1" xref="S4.p3.2.m2.1.2.1.cmml">⁢</mo><mrow id="S4.p3.2.m2.1.2.3.2" xref="S4.p3.2.m2.1.2.cmml"><mo id="S4.p3.2.m2.1.2.3.2.1" stretchy="false" xref="S4.p3.2.m2.1.2.cmml">(</mo><mi id="S4.p3.2.m2.1.1" xref="S4.p3.2.m2.1.1.cmml">v</mi><mo id="S4.p3.2.m2.1.2.3.2.2" stretchy="false" xref="S4.p3.2.m2.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.p3.2.m2.1b"><apply id="S4.p3.2.m2.1.2.cmml" xref="S4.p3.2.m2.1.2"><times id="S4.p3.2.m2.1.2.1.cmml" xref="S4.p3.2.m2.1.2.1"></times><ci id="S4.p3.2.m2.1.2.2a.cmml" xref="S4.p3.2.m2.1.2.2"><mtext class="ltx_mathvariant_italic" id="S4.p3.2.m2.1.2.2.cmml" xref="S4.p3.2.m2.1.2.2">g</mtext></ci><ci id="S4.p3.2.m2.1.1.cmml" xref="S4.p3.2.m2.1.1">𝑣</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.p3.2.m2.1c">\textsl{g}(v)</annotation><annotation encoding="application/x-llamapun" id="S4.p3.2.m2.1d">g ( italic_v )</annotation></semantics></math> may cause some other edge <math alttext="(w,u)" class="ltx_Math" display="inline" id="S4.p3.3.m3.2"><semantics id="S4.p3.3.m3.2a"><mrow id="S4.p3.3.m3.2.3.2" xref="S4.p3.3.m3.2.3.1.cmml"><mo id="S4.p3.3.m3.2.3.2.1" stretchy="false" xref="S4.p3.3.m3.2.3.1.cmml">(</mo><mi id="S4.p3.3.m3.1.1" xref="S4.p3.3.m3.1.1.cmml">w</mi><mo id="S4.p3.3.m3.2.3.2.2" xref="S4.p3.3.m3.2.3.1.cmml">,</mo><mi id="S4.p3.3.m3.2.2" xref="S4.p3.3.m3.2.2.cmml">u</mi><mo id="S4.p3.3.m3.2.3.2.3" stretchy="false" xref="S4.p3.3.m3.2.3.1.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S4.p3.3.m3.2b"><interval closure="open" id="S4.p3.3.m3.2.3.1.cmml" xref="S4.p3.3.m3.2.3.2"><ci id="S4.p3.3.m3.1.1.cmml" xref="S4.p3.3.m3.1.1">𝑤</ci><ci id="S4.p3.3.m3.2.2.cmml" xref="S4.p3.3.m3.2.2">𝑢</ci></interval></annotation-xml><annotation encoding="application/x-tex" id="S4.p3.3.m3.2c">(w,u)</annotation><annotation encoding="application/x-llamapun" id="S4.p3.3.m3.2d">( italic_w , italic_u )</annotation></semantics></math> or <math alttext="(v,w)" class="ltx_Math" display="inline" id="S4.p3.4.m4.2"><semantics id="S4.p3.4.m4.2a"><mrow id="S4.p3.4.m4.2.3.2" xref="S4.p3.4.m4.2.3.1.cmml"><mo id="S4.p3.4.m4.2.3.2.1" stretchy="false" xref="S4.p3.4.m4.2.3.1.cmml">(</mo><mi id="S4.p3.4.m4.1.1" xref="S4.p3.4.m4.1.1.cmml">v</mi><mo id="S4.p3.4.m4.2.3.2.2" xref="S4.p3.4.m4.2.3.1.cmml">,</mo><mi id="S4.p3.4.m4.2.2" xref="S4.p3.4.m4.2.2.cmml">w</mi><mo id="S4.p3.4.m4.2.3.2.3" stretchy="false" xref="S4.p3.4.m4.2.3.1.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S4.p3.4.m4.2b"><interval closure="open" id="S4.p3.4.m4.2.3.1.cmml" xref="S4.p3.4.m4.2.3.2"><ci id="S4.p3.4.m4.1.1.cmml" xref="S4.p3.4.m4.1.1">𝑣</ci><ci id="S4.p3.4.m4.2.2.cmml" xref="S4.p3.4.m4.2.2">𝑤</ci></interval></annotation-xml><annotation encoding="application/x-tex" id="S4.p3.4.m4.2c">(v,w)</annotation><annotation encoding="application/x-llamapun" id="S4.p3.4.m4.2d">( italic_v , italic_w )</annotation></semantics></math> to start violating local fairness. One way to show convergence is to define a potential function and to show that such a transfer always decreases the potential. We define the potential function <math alttext="\sum_{v\in V}\textsl{g}^{2}(v)" class="ltx_Math" display="inline" id="S4.p3.5.m5.1"><semantics id="S4.p3.5.m5.1a"><mrow id="S4.p3.5.m5.1.2" xref="S4.p3.5.m5.1.2.cmml"><msub id="S4.p3.5.m5.1.2.1" xref="S4.p3.5.m5.1.2.1.cmml"><mo id="S4.p3.5.m5.1.2.1.2" xref="S4.p3.5.m5.1.2.1.2.cmml">∑</mo><mrow id="S4.p3.5.m5.1.2.1.3" xref="S4.p3.5.m5.1.2.1.3.cmml"><mi id="S4.p3.5.m5.1.2.1.3.2" xref="S4.p3.5.m5.1.2.1.3.2.cmml">v</mi><mo id="S4.p3.5.m5.1.2.1.3.1" xref="S4.p3.5.m5.1.2.1.3.1.cmml">∈</mo><mi id="S4.p3.5.m5.1.2.1.3.3" xref="S4.p3.5.m5.1.2.1.3.3.cmml">V</mi></mrow></msub><mrow id="S4.p3.5.m5.1.2.2" xref="S4.p3.5.m5.1.2.2.cmml"><msup id="S4.p3.5.m5.1.2.2.2" xref="S4.p3.5.m5.1.2.2.2.cmml"><mtext class="ltx_mathvariant_italic" id="S4.p3.5.m5.1.2.2.2.2" xref="S4.p3.5.m5.1.2.2.2.2a.cmml">g</mtext><mn id="S4.p3.5.m5.1.2.2.2.3" xref="S4.p3.5.m5.1.2.2.2.3.cmml">2</mn></msup><mo id="S4.p3.5.m5.1.2.2.1" xref="S4.p3.5.m5.1.2.2.1.cmml">⁢</mo><mrow id="S4.p3.5.m5.1.2.2.3.2" xref="S4.p3.5.m5.1.2.2.cmml"><mo id="S4.p3.5.m5.1.2.2.3.2.1" stretchy="false" xref="S4.p3.5.m5.1.2.2.cmml">(</mo><mi id="S4.p3.5.m5.1.1" xref="S4.p3.5.m5.1.1.cmml">v</mi><mo id="S4.p3.5.m5.1.2.2.3.2.2" stretchy="false" xref="S4.p3.5.m5.1.2.2.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.p3.5.m5.1b"><apply id="S4.p3.5.m5.1.2.cmml" xref="S4.p3.5.m5.1.2"><apply id="S4.p3.5.m5.1.2.1.cmml" xref="S4.p3.5.m5.1.2.1"><csymbol cd="ambiguous" id="S4.p3.5.m5.1.2.1.1.cmml" xref="S4.p3.5.m5.1.2.1">subscript</csymbol><sum id="S4.p3.5.m5.1.2.1.2.cmml" xref="S4.p3.5.m5.1.2.1.2"></sum><apply id="S4.p3.5.m5.1.2.1.3.cmml" xref="S4.p3.5.m5.1.2.1.3"><in id="S4.p3.5.m5.1.2.1.3.1.cmml" xref="S4.p3.5.m5.1.2.1.3.1"></in><ci id="S4.p3.5.m5.1.2.1.3.2.cmml" xref="S4.p3.5.m5.1.2.1.3.2">𝑣</ci><ci id="S4.p3.5.m5.1.2.1.3.3.cmml" xref="S4.p3.5.m5.1.2.1.3.3">𝑉</ci></apply></apply><apply id="S4.p3.5.m5.1.2.2.cmml" xref="S4.p3.5.m5.1.2.2"><times id="S4.p3.5.m5.1.2.2.1.cmml" xref="S4.p3.5.m5.1.2.2.1"></times><apply id="S4.p3.5.m5.1.2.2.2.cmml" xref="S4.p3.5.m5.1.2.2.2"><csymbol cd="ambiguous" id="S4.p3.5.m5.1.2.2.2.1.cmml" xref="S4.p3.5.m5.1.2.2.2">superscript</csymbol><ci id="S4.p3.5.m5.1.2.2.2.2a.cmml" xref="S4.p3.5.m5.1.2.2.2.2"><mtext class="ltx_mathvariant_italic" id="S4.p3.5.m5.1.2.2.2.2.cmml" xref="S4.p3.5.m5.1.2.2.2.2">g</mtext></ci><cn id="S4.p3.5.m5.1.2.2.2.3.cmml" type="integer" xref="S4.p3.5.m5.1.2.2.2.3">2</cn></apply><ci id="S4.p3.5.m5.1.1.cmml" xref="S4.p3.5.m5.1.1">𝑣</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.p3.5.m5.1c">\sum_{v\in V}\textsl{g}^{2}(v)</annotation><annotation encoding="application/x-llamapun" id="S4.p3.5.m5.1d">∑ start_POSTSUBSCRIPT italic_v ∈ italic_V end_POSTSUBSCRIPT g start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ( italic_v )</annotation></semantics></math>, thereby creating a quadratic program where the domain is the space of all orientations of <math alttext="G" class="ltx_Math" display="inline" id="S4.p3.6.m6.1"><semantics id="S4.p3.6.m6.1a"><mi id="S4.p3.6.m6.1.1" xref="S4.p3.6.m6.1.1.cmml">G</mi><annotation-xml encoding="MathML-Content" id="S4.p3.6.m6.1b"><ci id="S4.p3.6.m6.1.1.cmml" xref="S4.p3.6.m6.1.1">𝐺</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.p3.6.m6.1c">G</annotation><annotation encoding="application/x-llamapun" id="S4.p3.6.m6.1d">italic_G</annotation></semantics></math>. We prove that any optimal solution to this quadratic program must be a locally fair orientation. Any quadratic function over a compact domain has an optimum and so the existence of a locally fair orientation follows.</p> </div> <div class="ltx_para" id="S4.p4"> <p class="ltx_p" id="S4.p4.1">See <a class="ltx_ref" href="https://arxiv.org/html/2411.12694v2#S3.Thmtheorem2" title="Theorem 3.2. ‣ 3.A Conceptual results for local density ‣ 3 Results and organisation ‣ Local Density and its Distributed Approximation"><span class="ltx_text ltx_ref_tag">3.2</span></a></p> </div> <div class="ltx_theorem ltx_theorem_proof" 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">Proof 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.p1.2"><span class="ltx_text ltx_font_italic" id="S4.Thmtheorem3.p1.2.2">We consider the following quadratic program <math alttext="\textbf{FO}^{2}" class="ltx_Math" display="inline" id="S4.Thmtheorem3.p1.1.1.m1.1"><semantics id="S4.Thmtheorem3.p1.1.1.m1.1a"><msup id="S4.Thmtheorem3.p1.1.1.m1.1.1" xref="S4.Thmtheorem3.p1.1.1.m1.1.1.cmml"><mtext class="ltx_mathvariant_bold" id="S4.Thmtheorem3.p1.1.1.m1.1.1.2" xref="S4.Thmtheorem3.p1.1.1.m1.1.1.2a.cmml">FO</mtext><mn id="S4.Thmtheorem3.p1.1.1.m1.1.1.3" xref="S4.Thmtheorem3.p1.1.1.m1.1.1.3.cmml">2</mn></msup><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem3.p1.1.1.m1.1b"><apply id="S4.Thmtheorem3.p1.1.1.m1.1.1.cmml" xref="S4.Thmtheorem3.p1.1.1.m1.1.1"><csymbol cd="ambiguous" id="S4.Thmtheorem3.p1.1.1.m1.1.1.1.cmml" xref="S4.Thmtheorem3.p1.1.1.m1.1.1">superscript</csymbol><ci id="S4.Thmtheorem3.p1.1.1.m1.1.1.2a.cmml" xref="S4.Thmtheorem3.p1.1.1.m1.1.1.2"><mtext class="ltx_mathvariant_bold" id="S4.Thmtheorem3.p1.1.1.m1.1.1.2.cmml" xref="S4.Thmtheorem3.p1.1.1.m1.1.1.2">FO</mtext></ci><cn id="S4.Thmtheorem3.p1.1.1.m1.1.1.3.cmml" type="integer" xref="S4.Thmtheorem3.p1.1.1.m1.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem3.p1.1.1.m1.1c">\textbf{FO}^{2}</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem3.p1.1.1.m1.1d">FO start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT</annotation></semantics></math> from <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2411.12694v2#bib.bib10" title="">10</a>, Section 4]</cite> where we compute a fractional orientation of the graph <math alttext="G" class="ltx_Math" display="inline" id="S4.Thmtheorem3.p1.2.2.m2.1"><semantics id="S4.Thmtheorem3.p1.2.2.m2.1a"><mi id="S4.Thmtheorem3.p1.2.2.m2.1.1" xref="S4.Thmtheorem3.p1.2.2.m2.1.1.cmml">G</mi><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem3.p1.2.2.m2.1b"><ci id="S4.Thmtheorem3.p1.2.2.m2.1.1.cmml" xref="S4.Thmtheorem3.p1.2.2.m2.1.1">𝐺</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem3.p1.2.2.m2.1c">G</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem3.p1.2.2.m2.1d">italic_G</annotation></semantics></math> subject to a quadratic cost function:</span></p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="A1.EGx3"> <tbody id="S4.Ex12"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_left_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle\min\sum\textsl{g}(u)^{2}" class="ltx_Math" display="inline" id="S4.Ex12.m1.1"><semantics id="S4.Ex12.m1.1a"><mrow id="S4.Ex12.m1.1.2" xref="S4.Ex12.m1.1.2.cmml"><mi id="S4.Ex12.m1.1.2.2" xref="S4.Ex12.m1.1.2.2.cmml">min</mi><mo id="S4.Ex12.m1.1.2.1" lspace="0.167em" xref="S4.Ex12.m1.1.2.1.cmml">⁢</mo><mstyle displaystyle="true" id="S4.Ex12.m1.1.2.3" xref="S4.Ex12.m1.1.2.3.cmml"><mrow id="S4.Ex12.m1.1.2.3a" xref="S4.Ex12.m1.1.2.3.cmml"><mo id="S4.Ex12.m1.1.2.3.1" movablelimits="false" xref="S4.Ex12.m1.1.2.3.1.cmml">∑</mo><mrow id="S4.Ex12.m1.1.2.3.2" xref="S4.Ex12.m1.1.2.3.2.cmml"><mtext class="ltx_mathvariant_italic" id="S4.Ex12.m1.1.2.3.2.2" xref="S4.Ex12.m1.1.2.3.2.2a.cmml">g</mtext><mo id="S4.Ex12.m1.1.2.3.2.1" xref="S4.Ex12.m1.1.2.3.2.1.cmml">⁢</mo><msup id="S4.Ex12.m1.1.2.3.2.3" xref="S4.Ex12.m1.1.2.3.2.3.cmml"><mrow id="S4.Ex12.m1.1.2.3.2.3.2.2" xref="S4.Ex12.m1.1.2.3.2.3.cmml"><mo id="S4.Ex12.m1.1.2.3.2.3.2.2.1" stretchy="false" xref="S4.Ex12.m1.1.2.3.2.3.cmml">(</mo><mi id="S4.Ex12.m1.1.1" xref="S4.Ex12.m1.1.1.cmml">u</mi><mo id="S4.Ex12.m1.1.2.3.2.3.2.2.2" stretchy="false" xref="S4.Ex12.m1.1.2.3.2.3.cmml">)</mo></mrow><mn id="S4.Ex12.m1.1.2.3.2.3.3" xref="S4.Ex12.m1.1.2.3.2.3.3.cmml">2</mn></msup></mrow></mrow></mstyle></mrow><annotation-xml encoding="MathML-Content" id="S4.Ex12.m1.1b"><apply id="S4.Ex12.m1.1.2.cmml" xref="S4.Ex12.m1.1.2"><times id="S4.Ex12.m1.1.2.1.cmml" xref="S4.Ex12.m1.1.2.1"></times><min id="S4.Ex12.m1.1.2.2.cmml" xref="S4.Ex12.m1.1.2.2"></min><apply id="S4.Ex12.m1.1.2.3.cmml" xref="S4.Ex12.m1.1.2.3"><sum id="S4.Ex12.m1.1.2.3.1.cmml" xref="S4.Ex12.m1.1.2.3.1"></sum><apply id="S4.Ex12.m1.1.2.3.2.cmml" xref="S4.Ex12.m1.1.2.3.2"><times id="S4.Ex12.m1.1.2.3.2.1.cmml" xref="S4.Ex12.m1.1.2.3.2.1"></times><ci id="S4.Ex12.m1.1.2.3.2.2a.cmml" xref="S4.Ex12.m1.1.2.3.2.2"><mtext class="ltx_mathvariant_italic" id="S4.Ex12.m1.1.2.3.2.2.cmml" xref="S4.Ex12.m1.1.2.3.2.2">g</mtext></ci><apply id="S4.Ex12.m1.1.2.3.2.3.cmml" xref="S4.Ex12.m1.1.2.3.2.3"><csymbol cd="ambiguous" id="S4.Ex12.m1.1.2.3.2.3.1.cmml" xref="S4.Ex12.m1.1.2.3.2.3">superscript</csymbol><ci id="S4.Ex12.m1.1.1.cmml" xref="S4.Ex12.m1.1.1">𝑢</ci><cn id="S4.Ex12.m1.1.2.3.2.3.3.cmml" type="integer" xref="S4.Ex12.m1.1.2.3.2.3.3">2</cn></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Ex12.m1.1c">\displaystyle\min\sum\textsl{g}(u)^{2}</annotation><annotation encoding="application/x-llamapun" id="S4.Ex12.m1.1d">roman_min ∑ g ( italic_u ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT</annotation></semantics></math></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><span class="ltx_text ltx_markedasmath ltx_font_italic" id="S4.Ex12.3.2.2.1">s.t.</span></td> <td class="ltx_eqn_cell ltx_eqn_left_padright"></td> </tr></tbody> <tbody id="S4.Ex13"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_left_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle\textsl{g}(u\!\to\!v)+\textsl{g}(v\!\to\!u)\geq\textsl{g}(% \overline{uv})" class="ltx_Math" display="inline" id="S4.Ex13.m1.3"><semantics id="S4.Ex13.m1.3a"><mrow id="S4.Ex13.m1.3.3" xref="S4.Ex13.m1.3.3.cmml"><mrow id="S4.Ex13.m1.3.3.2" xref="S4.Ex13.m1.3.3.2.cmml"><mrow id="S4.Ex13.m1.2.2.1.1" xref="S4.Ex13.m1.2.2.1.1.cmml"><mtext class="ltx_mathvariant_italic" id="S4.Ex13.m1.2.2.1.1.3" xref="S4.Ex13.m1.2.2.1.1.3a.cmml">g</mtext><mo id="S4.Ex13.m1.2.2.1.1.2" xref="S4.Ex13.m1.2.2.1.1.2.cmml">⁢</mo><mrow id="S4.Ex13.m1.2.2.1.1.1.1" xref="S4.Ex13.m1.2.2.1.1.1.1.1.cmml"><mo id="S4.Ex13.m1.2.2.1.1.1.1.2" stretchy="false" xref="S4.Ex13.m1.2.2.1.1.1.1.1.cmml">(</mo><mrow id="S4.Ex13.m1.2.2.1.1.1.1.1" xref="S4.Ex13.m1.2.2.1.1.1.1.1.cmml"><mi id="S4.Ex13.m1.2.2.1.1.1.1.1.2" xref="S4.Ex13.m1.2.2.1.1.1.1.1.2.cmml">u</mi><mo id="S4.Ex13.m1.2.2.1.1.1.1.1.1" lspace="0.108em" rspace="0.108em" stretchy="false" xref="S4.Ex13.m1.2.2.1.1.1.1.1.1.cmml">→</mo><mi id="S4.Ex13.m1.2.2.1.1.1.1.1.3" xref="S4.Ex13.m1.2.2.1.1.1.1.1.3.cmml">v</mi></mrow><mo id="S4.Ex13.m1.2.2.1.1.1.1.3" stretchy="false" xref="S4.Ex13.m1.2.2.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S4.Ex13.m1.3.3.2.3" xref="S4.Ex13.m1.3.3.2.3.cmml">+</mo><mrow id="S4.Ex13.m1.3.3.2.2" xref="S4.Ex13.m1.3.3.2.2.cmml"><mtext class="ltx_mathvariant_italic" id="S4.Ex13.m1.3.3.2.2.3" xref="S4.Ex13.m1.3.3.2.2.3a.cmml">g</mtext><mo id="S4.Ex13.m1.3.3.2.2.2" xref="S4.Ex13.m1.3.3.2.2.2.cmml">⁢</mo><mrow id="S4.Ex13.m1.3.3.2.2.1.1" xref="S4.Ex13.m1.3.3.2.2.1.1.1.cmml"><mo id="S4.Ex13.m1.3.3.2.2.1.1.2" stretchy="false" xref="S4.Ex13.m1.3.3.2.2.1.1.1.cmml">(</mo><mrow id="S4.Ex13.m1.3.3.2.2.1.1.1" xref="S4.Ex13.m1.3.3.2.2.1.1.1.cmml"><mi id="S4.Ex13.m1.3.3.2.2.1.1.1.2" xref="S4.Ex13.m1.3.3.2.2.1.1.1.2.cmml">v</mi><mo id="S4.Ex13.m1.3.3.2.2.1.1.1.1" lspace="0.108em" rspace="0.108em" stretchy="false" xref="S4.Ex13.m1.3.3.2.2.1.1.1.1.cmml">→</mo><mi id="S4.Ex13.m1.3.3.2.2.1.1.1.3" xref="S4.Ex13.m1.3.3.2.2.1.1.1.3.cmml">u</mi></mrow><mo id="S4.Ex13.m1.3.3.2.2.1.1.3" stretchy="false" xref="S4.Ex13.m1.3.3.2.2.1.1.1.cmml">)</mo></mrow></mrow></mrow><mo id="S4.Ex13.m1.3.3.3" xref="S4.Ex13.m1.3.3.3.cmml">≥</mo><mrow id="S4.Ex13.m1.3.3.4" xref="S4.Ex13.m1.3.3.4.cmml"><mtext class="ltx_mathvariant_italic" id="S4.Ex13.m1.3.3.4.2" xref="S4.Ex13.m1.3.3.4.2a.cmml">g</mtext><mo id="S4.Ex13.m1.3.3.4.1" xref="S4.Ex13.m1.3.3.4.1.cmml">⁢</mo><mrow id="S4.Ex13.m1.3.3.4.3.2" xref="S4.Ex13.m1.1.1.cmml"><mo id="S4.Ex13.m1.3.3.4.3.2.1" stretchy="false" xref="S4.Ex13.m1.1.1.cmml">(</mo><mover accent="true" id="S4.Ex13.m1.1.1" xref="S4.Ex13.m1.1.1.cmml"><mrow id="S4.Ex13.m1.1.1.2" xref="S4.Ex13.m1.1.1.2.cmml"><mi id="S4.Ex13.m1.1.1.2.2" xref="S4.Ex13.m1.1.1.2.2.cmml">u</mi><mo id="S4.Ex13.m1.1.1.2.1" xref="S4.Ex13.m1.1.1.2.1.cmml">⁢</mo><mi id="S4.Ex13.m1.1.1.2.3" xref="S4.Ex13.m1.1.1.2.3.cmml">v</mi></mrow><mo id="S4.Ex13.m1.1.1.1" xref="S4.Ex13.m1.1.1.1.cmml">¯</mo></mover><mo id="S4.Ex13.m1.3.3.4.3.2.2" stretchy="false" xref="S4.Ex13.m1.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.Ex13.m1.3b"><apply id="S4.Ex13.m1.3.3.cmml" xref="S4.Ex13.m1.3.3"><geq id="S4.Ex13.m1.3.3.3.cmml" xref="S4.Ex13.m1.3.3.3"></geq><apply id="S4.Ex13.m1.3.3.2.cmml" xref="S4.Ex13.m1.3.3.2"><plus id="S4.Ex13.m1.3.3.2.3.cmml" xref="S4.Ex13.m1.3.3.2.3"></plus><apply id="S4.Ex13.m1.2.2.1.1.cmml" xref="S4.Ex13.m1.2.2.1.1"><times id="S4.Ex13.m1.2.2.1.1.2.cmml" xref="S4.Ex13.m1.2.2.1.1.2"></times><ci id="S4.Ex13.m1.2.2.1.1.3a.cmml" xref="S4.Ex13.m1.2.2.1.1.3"><mtext class="ltx_mathvariant_italic" id="S4.Ex13.m1.2.2.1.1.3.cmml" xref="S4.Ex13.m1.2.2.1.1.3">g</mtext></ci><apply id="S4.Ex13.m1.2.2.1.1.1.1.1.cmml" xref="S4.Ex13.m1.2.2.1.1.1.1"><ci id="S4.Ex13.m1.2.2.1.1.1.1.1.1.cmml" xref="S4.Ex13.m1.2.2.1.1.1.1.1.1">→</ci><ci id="S4.Ex13.m1.2.2.1.1.1.1.1.2.cmml" xref="S4.Ex13.m1.2.2.1.1.1.1.1.2">𝑢</ci><ci id="S4.Ex13.m1.2.2.1.1.1.1.1.3.cmml" xref="S4.Ex13.m1.2.2.1.1.1.1.1.3">𝑣</ci></apply></apply><apply id="S4.Ex13.m1.3.3.2.2.cmml" xref="S4.Ex13.m1.3.3.2.2"><times id="S4.Ex13.m1.3.3.2.2.2.cmml" xref="S4.Ex13.m1.3.3.2.2.2"></times><ci id="S4.Ex13.m1.3.3.2.2.3a.cmml" xref="S4.Ex13.m1.3.3.2.2.3"><mtext class="ltx_mathvariant_italic" id="S4.Ex13.m1.3.3.2.2.3.cmml" xref="S4.Ex13.m1.3.3.2.2.3">g</mtext></ci><apply id="S4.Ex13.m1.3.3.2.2.1.1.1.cmml" xref="S4.Ex13.m1.3.3.2.2.1.1"><ci id="S4.Ex13.m1.3.3.2.2.1.1.1.1.cmml" xref="S4.Ex13.m1.3.3.2.2.1.1.1.1">→</ci><ci id="S4.Ex13.m1.3.3.2.2.1.1.1.2.cmml" xref="S4.Ex13.m1.3.3.2.2.1.1.1.2">𝑣</ci><ci id="S4.Ex13.m1.3.3.2.2.1.1.1.3.cmml" xref="S4.Ex13.m1.3.3.2.2.1.1.1.3">𝑢</ci></apply></apply></apply><apply id="S4.Ex13.m1.3.3.4.cmml" xref="S4.Ex13.m1.3.3.4"><times id="S4.Ex13.m1.3.3.4.1.cmml" xref="S4.Ex13.m1.3.3.4.1"></times><ci id="S4.Ex13.m1.3.3.4.2a.cmml" xref="S4.Ex13.m1.3.3.4.2"><mtext class="ltx_mathvariant_italic" id="S4.Ex13.m1.3.3.4.2.cmml" xref="S4.Ex13.m1.3.3.4.2">g</mtext></ci><apply id="S4.Ex13.m1.1.1.cmml" xref="S4.Ex13.m1.3.3.4.3.2"><ci id="S4.Ex13.m1.1.1.1.cmml" xref="S4.Ex13.m1.1.1.1">¯</ci><apply id="S4.Ex13.m1.1.1.2.cmml" xref="S4.Ex13.m1.1.1.2"><times id="S4.Ex13.m1.1.1.2.1.cmml" xref="S4.Ex13.m1.1.1.2.1"></times><ci id="S4.Ex13.m1.1.1.2.2.cmml" xref="S4.Ex13.m1.1.1.2.2">𝑢</ci><ci id="S4.Ex13.m1.1.1.2.3.cmml" xref="S4.Ex13.m1.1.1.2.3">𝑣</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Ex13.m1.3c">\displaystyle\textsl{g}(u\!\to\!v)+\textsl{g}(v\!\to\!u)\geq\textsl{g}(% \overline{uv})</annotation><annotation encoding="application/x-llamapun" id="S4.Ex13.m1.3d">g ( italic_u → italic_v ) + g ( italic_v → italic_u ) ≥ g ( over¯ start_ARG italic_u italic_v end_ARG )</annotation></semantics></math></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle\forall\overline{uv}\in E" class="ltx_Math" display="inline" id="S4.Ex13.m2.1"><semantics id="S4.Ex13.m2.1a"><mrow id="S4.Ex13.m2.1.1" xref="S4.Ex13.m2.1.1.cmml"><mrow id="S4.Ex13.m2.1.1.2" xref="S4.Ex13.m2.1.1.2.cmml"><mo id="S4.Ex13.m2.1.1.2.1" rspace="0.167em" xref="S4.Ex13.m2.1.1.2.1.cmml">∀</mo><mover accent="true" id="S4.Ex13.m2.1.1.2.2" xref="S4.Ex13.m2.1.1.2.2.cmml"><mrow id="S4.Ex13.m2.1.1.2.2.2" xref="S4.Ex13.m2.1.1.2.2.2.cmml"><mi id="S4.Ex13.m2.1.1.2.2.2.2" xref="S4.Ex13.m2.1.1.2.2.2.2.cmml">u</mi><mo id="S4.Ex13.m2.1.1.2.2.2.1" xref="S4.Ex13.m2.1.1.2.2.2.1.cmml">⁢</mo><mi id="S4.Ex13.m2.1.1.2.2.2.3" xref="S4.Ex13.m2.1.1.2.2.2.3.cmml">v</mi></mrow><mo id="S4.Ex13.m2.1.1.2.2.1" xref="S4.Ex13.m2.1.1.2.2.1.cmml">¯</mo></mover></mrow><mo id="S4.Ex13.m2.1.1.1" xref="S4.Ex13.m2.1.1.1.cmml">∈</mo><mi id="S4.Ex13.m2.1.1.3" xref="S4.Ex13.m2.1.1.3.cmml">E</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.Ex13.m2.1b"><apply id="S4.Ex13.m2.1.1.cmml" xref="S4.Ex13.m2.1.1"><in id="S4.Ex13.m2.1.1.1.cmml" xref="S4.Ex13.m2.1.1.1"></in><apply id="S4.Ex13.m2.1.1.2.cmml" xref="S4.Ex13.m2.1.1.2"><csymbol cd="latexml" id="S4.Ex13.m2.1.1.2.1.cmml" xref="S4.Ex13.m2.1.1.2.1">for-all</csymbol><apply id="S4.Ex13.m2.1.1.2.2.cmml" xref="S4.Ex13.m2.1.1.2.2"><ci id="S4.Ex13.m2.1.1.2.2.1.cmml" xref="S4.Ex13.m2.1.1.2.2.1">¯</ci><apply id="S4.Ex13.m2.1.1.2.2.2.cmml" xref="S4.Ex13.m2.1.1.2.2.2"><times id="S4.Ex13.m2.1.1.2.2.2.1.cmml" xref="S4.Ex13.m2.1.1.2.2.2.1"></times><ci id="S4.Ex13.m2.1.1.2.2.2.2.cmml" xref="S4.Ex13.m2.1.1.2.2.2.2">𝑢</ci><ci id="S4.Ex13.m2.1.1.2.2.2.3.cmml" xref="S4.Ex13.m2.1.1.2.2.2.3">𝑣</ci></apply></apply></apply><ci id="S4.Ex13.m2.1.1.3.cmml" xref="S4.Ex13.m2.1.1.3">𝐸</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Ex13.m2.1c">\displaystyle\forall\overline{uv}\in E</annotation><annotation encoding="application/x-llamapun" id="S4.Ex13.m2.1d">∀ over¯ start_ARG italic_u italic_v end_ARG ∈ italic_E</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_left_padright"></td> </tr></tbody> <tbody id="S4.Ex14"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_left_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle\textsl{g}(u)\geq\sum_{v\in V}\textsl{g}(u\!\to\!v)" class="ltx_Math" display="inline" id="S4.Ex14.m1.2"><semantics id="S4.Ex14.m1.2a"><mrow id="S4.Ex14.m1.2.2" xref="S4.Ex14.m1.2.2.cmml"><mrow id="S4.Ex14.m1.2.2.3" xref="S4.Ex14.m1.2.2.3.cmml"><mtext class="ltx_mathvariant_italic" id="S4.Ex14.m1.2.2.3.2" xref="S4.Ex14.m1.2.2.3.2a.cmml">g</mtext><mo id="S4.Ex14.m1.2.2.3.1" xref="S4.Ex14.m1.2.2.3.1.cmml">⁢</mo><mrow id="S4.Ex14.m1.2.2.3.3.2" xref="S4.Ex14.m1.2.2.3.cmml"><mo id="S4.Ex14.m1.2.2.3.3.2.1" stretchy="false" xref="S4.Ex14.m1.2.2.3.cmml">(</mo><mi id="S4.Ex14.m1.1.1" xref="S4.Ex14.m1.1.1.cmml">u</mi><mo id="S4.Ex14.m1.2.2.3.3.2.2" stretchy="false" xref="S4.Ex14.m1.2.2.3.cmml">)</mo></mrow></mrow><mo id="S4.Ex14.m1.2.2.2" xref="S4.Ex14.m1.2.2.2.cmml">≥</mo><mrow id="S4.Ex14.m1.2.2.1" xref="S4.Ex14.m1.2.2.1.cmml"><mstyle displaystyle="true" id="S4.Ex14.m1.2.2.1.2" xref="S4.Ex14.m1.2.2.1.2.cmml"><munder id="S4.Ex14.m1.2.2.1.2a" xref="S4.Ex14.m1.2.2.1.2.cmml"><mo id="S4.Ex14.m1.2.2.1.2.2" movablelimits="false" xref="S4.Ex14.m1.2.2.1.2.2.cmml">∑</mo><mrow id="S4.Ex14.m1.2.2.1.2.3" xref="S4.Ex14.m1.2.2.1.2.3.cmml"><mi id="S4.Ex14.m1.2.2.1.2.3.2" xref="S4.Ex14.m1.2.2.1.2.3.2.cmml">v</mi><mo id="S4.Ex14.m1.2.2.1.2.3.1" xref="S4.Ex14.m1.2.2.1.2.3.1.cmml">∈</mo><mi id="S4.Ex14.m1.2.2.1.2.3.3" xref="S4.Ex14.m1.2.2.1.2.3.3.cmml">V</mi></mrow></munder></mstyle><mrow id="S4.Ex14.m1.2.2.1.1" xref="S4.Ex14.m1.2.2.1.1.cmml"><mtext class="ltx_mathvariant_italic" id="S4.Ex14.m1.2.2.1.1.3" xref="S4.Ex14.m1.2.2.1.1.3a.cmml">g</mtext><mo id="S4.Ex14.m1.2.2.1.1.2" xref="S4.Ex14.m1.2.2.1.1.2.cmml">⁢</mo><mrow id="S4.Ex14.m1.2.2.1.1.1.1" xref="S4.Ex14.m1.2.2.1.1.1.1.1.cmml"><mo id="S4.Ex14.m1.2.2.1.1.1.1.2" stretchy="false" xref="S4.Ex14.m1.2.2.1.1.1.1.1.cmml">(</mo><mrow id="S4.Ex14.m1.2.2.1.1.1.1.1" xref="S4.Ex14.m1.2.2.1.1.1.1.1.cmml"><mi id="S4.Ex14.m1.2.2.1.1.1.1.1.2" xref="S4.Ex14.m1.2.2.1.1.1.1.1.2.cmml">u</mi><mo id="S4.Ex14.m1.2.2.1.1.1.1.1.1" lspace="0.108em" rspace="0.108em" stretchy="false" xref="S4.Ex14.m1.2.2.1.1.1.1.1.1.cmml">→</mo><mi id="S4.Ex14.m1.2.2.1.1.1.1.1.3" xref="S4.Ex14.m1.2.2.1.1.1.1.1.3.cmml">v</mi></mrow><mo id="S4.Ex14.m1.2.2.1.1.1.1.3" stretchy="false" xref="S4.Ex14.m1.2.2.1.1.1.1.1.cmml">)</mo></mrow></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.Ex14.m1.2b"><apply id="S4.Ex14.m1.2.2.cmml" xref="S4.Ex14.m1.2.2"><geq id="S4.Ex14.m1.2.2.2.cmml" xref="S4.Ex14.m1.2.2.2"></geq><apply id="S4.Ex14.m1.2.2.3.cmml" xref="S4.Ex14.m1.2.2.3"><times id="S4.Ex14.m1.2.2.3.1.cmml" xref="S4.Ex14.m1.2.2.3.1"></times><ci id="S4.Ex14.m1.2.2.3.2a.cmml" xref="S4.Ex14.m1.2.2.3.2"><mtext class="ltx_mathvariant_italic" id="S4.Ex14.m1.2.2.3.2.cmml" xref="S4.Ex14.m1.2.2.3.2">g</mtext></ci><ci id="S4.Ex14.m1.1.1.cmml" xref="S4.Ex14.m1.1.1">𝑢</ci></apply><apply id="S4.Ex14.m1.2.2.1.cmml" xref="S4.Ex14.m1.2.2.1"><apply id="S4.Ex14.m1.2.2.1.2.cmml" xref="S4.Ex14.m1.2.2.1.2"><csymbol cd="ambiguous" id="S4.Ex14.m1.2.2.1.2.1.cmml" xref="S4.Ex14.m1.2.2.1.2">subscript</csymbol><sum id="S4.Ex14.m1.2.2.1.2.2.cmml" xref="S4.Ex14.m1.2.2.1.2.2"></sum><apply id="S4.Ex14.m1.2.2.1.2.3.cmml" xref="S4.Ex14.m1.2.2.1.2.3"><in id="S4.Ex14.m1.2.2.1.2.3.1.cmml" xref="S4.Ex14.m1.2.2.1.2.3.1"></in><ci id="S4.Ex14.m1.2.2.1.2.3.2.cmml" xref="S4.Ex14.m1.2.2.1.2.3.2">𝑣</ci><ci id="S4.Ex14.m1.2.2.1.2.3.3.cmml" xref="S4.Ex14.m1.2.2.1.2.3.3">𝑉</ci></apply></apply><apply id="S4.Ex14.m1.2.2.1.1.cmml" xref="S4.Ex14.m1.2.2.1.1"><times id="S4.Ex14.m1.2.2.1.1.2.cmml" xref="S4.Ex14.m1.2.2.1.1.2"></times><ci id="S4.Ex14.m1.2.2.1.1.3a.cmml" xref="S4.Ex14.m1.2.2.1.1.3"><mtext class="ltx_mathvariant_italic" id="S4.Ex14.m1.2.2.1.1.3.cmml" xref="S4.Ex14.m1.2.2.1.1.3">g</mtext></ci><apply id="S4.Ex14.m1.2.2.1.1.1.1.1.cmml" xref="S4.Ex14.m1.2.2.1.1.1.1"><ci id="S4.Ex14.m1.2.2.1.1.1.1.1.1.cmml" xref="S4.Ex14.m1.2.2.1.1.1.1.1.1">→</ci><ci id="S4.Ex14.m1.2.2.1.1.1.1.1.2.cmml" xref="S4.Ex14.m1.2.2.1.1.1.1.1.2">𝑢</ci><ci id="S4.Ex14.m1.2.2.1.1.1.1.1.3.cmml" xref="S4.Ex14.m1.2.2.1.1.1.1.1.3">𝑣</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Ex14.m1.2c">\displaystyle\textsl{g}(u)\geq\sum_{v\in V}\textsl{g}(u\!\to\!v)</annotation><annotation encoding="application/x-llamapun" id="S4.Ex14.m1.2d">g ( italic_u ) ≥ ∑ start_POSTSUBSCRIPT italic_v ∈ italic_V end_POSTSUBSCRIPT g ( italic_u → italic_v )</annotation></semantics></math></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle\forall u\in V" class="ltx_Math" display="inline" id="S4.Ex14.m2.1"><semantics id="S4.Ex14.m2.1a"><mrow id="S4.Ex14.m2.1.1" xref="S4.Ex14.m2.1.1.cmml"><mrow id="S4.Ex14.m2.1.1.2" xref="S4.Ex14.m2.1.1.2.cmml"><mo id="S4.Ex14.m2.1.1.2.1" rspace="0.167em" xref="S4.Ex14.m2.1.1.2.1.cmml">∀</mo><mi id="S4.Ex14.m2.1.1.2.2" xref="S4.Ex14.m2.1.1.2.2.cmml">u</mi></mrow><mo id="S4.Ex14.m2.1.1.1" xref="S4.Ex14.m2.1.1.1.cmml">∈</mo><mi id="S4.Ex14.m2.1.1.3" xref="S4.Ex14.m2.1.1.3.cmml">V</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.Ex14.m2.1b"><apply id="S4.Ex14.m2.1.1.cmml" xref="S4.Ex14.m2.1.1"><in id="S4.Ex14.m2.1.1.1.cmml" xref="S4.Ex14.m2.1.1.1"></in><apply id="S4.Ex14.m2.1.1.2.cmml" xref="S4.Ex14.m2.1.1.2"><csymbol cd="latexml" id="S4.Ex14.m2.1.1.2.1.cmml" xref="S4.Ex14.m2.1.1.2.1">for-all</csymbol><ci id="S4.Ex14.m2.1.1.2.2.cmml" xref="S4.Ex14.m2.1.1.2.2">𝑢</ci></apply><ci id="S4.Ex14.m2.1.1.3.cmml" xref="S4.Ex14.m2.1.1.3">𝑉</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Ex14.m2.1c">\displaystyle\forall u\in V</annotation><annotation encoding="application/x-llamapun" id="S4.Ex14.m2.1d">∀ italic_u ∈ italic_V</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_left_padright"></td> </tr></tbody> <tbody id="S4.Ex15"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_left_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle\textsl{g}(u\!\to\!v),\textsl{g}(v\!\to\!u)\geq 0" class="ltx_Math" display="inline" id="S4.Ex15.m1.2"><semantics id="S4.Ex15.m1.2a"><mrow id="S4.Ex15.m1.2.2" xref="S4.Ex15.m1.2.2.cmml"><mrow id="S4.Ex15.m1.2.2.2.2" xref="S4.Ex15.m1.2.2.2.3.cmml"><mrow id="S4.Ex15.m1.1.1.1.1.1" xref="S4.Ex15.m1.1.1.1.1.1.cmml"><mtext class="ltx_mathvariant_italic" id="S4.Ex15.m1.1.1.1.1.1.3" xref="S4.Ex15.m1.1.1.1.1.1.3a.cmml">g</mtext><mo id="S4.Ex15.m1.1.1.1.1.1.2" xref="S4.Ex15.m1.1.1.1.1.1.2.cmml">⁢</mo><mrow id="S4.Ex15.m1.1.1.1.1.1.1.1" xref="S4.Ex15.m1.1.1.1.1.1.1.1.1.cmml"><mo id="S4.Ex15.m1.1.1.1.1.1.1.1.2" stretchy="false" xref="S4.Ex15.m1.1.1.1.1.1.1.1.1.cmml">(</mo><mrow id="S4.Ex15.m1.1.1.1.1.1.1.1.1" xref="S4.Ex15.m1.1.1.1.1.1.1.1.1.cmml"><mi id="S4.Ex15.m1.1.1.1.1.1.1.1.1.2" xref="S4.Ex15.m1.1.1.1.1.1.1.1.1.2.cmml">u</mi><mo id="S4.Ex15.m1.1.1.1.1.1.1.1.1.1" lspace="0.108em" rspace="0.108em" stretchy="false" xref="S4.Ex15.m1.1.1.1.1.1.1.1.1.1.cmml">→</mo><mi id="S4.Ex15.m1.1.1.1.1.1.1.1.1.3" xref="S4.Ex15.m1.1.1.1.1.1.1.1.1.3.cmml">v</mi></mrow><mo id="S4.Ex15.m1.1.1.1.1.1.1.1.3" stretchy="false" xref="S4.Ex15.m1.1.1.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S4.Ex15.m1.2.2.2.2.3" xref="S4.Ex15.m1.2.2.2.3.cmml">,</mo><mrow id="S4.Ex15.m1.2.2.2.2.2" xref="S4.Ex15.m1.2.2.2.2.2.cmml"><mtext class="ltx_mathvariant_italic" id="S4.Ex15.m1.2.2.2.2.2.3" xref="S4.Ex15.m1.2.2.2.2.2.3a.cmml">g</mtext><mo id="S4.Ex15.m1.2.2.2.2.2.2" xref="S4.Ex15.m1.2.2.2.2.2.2.cmml">⁢</mo><mrow id="S4.Ex15.m1.2.2.2.2.2.1.1" xref="S4.Ex15.m1.2.2.2.2.2.1.1.1.cmml"><mo id="S4.Ex15.m1.2.2.2.2.2.1.1.2" stretchy="false" xref="S4.Ex15.m1.2.2.2.2.2.1.1.1.cmml">(</mo><mrow id="S4.Ex15.m1.2.2.2.2.2.1.1.1" xref="S4.Ex15.m1.2.2.2.2.2.1.1.1.cmml"><mi id="S4.Ex15.m1.2.2.2.2.2.1.1.1.2" xref="S4.Ex15.m1.2.2.2.2.2.1.1.1.2.cmml">v</mi><mo id="S4.Ex15.m1.2.2.2.2.2.1.1.1.1" lspace="0.108em" rspace="0.108em" stretchy="false" xref="S4.Ex15.m1.2.2.2.2.2.1.1.1.1.cmml">→</mo><mi id="S4.Ex15.m1.2.2.2.2.2.1.1.1.3" xref="S4.Ex15.m1.2.2.2.2.2.1.1.1.3.cmml">u</mi></mrow><mo id="S4.Ex15.m1.2.2.2.2.2.1.1.3" stretchy="false" xref="S4.Ex15.m1.2.2.2.2.2.1.1.1.cmml">)</mo></mrow></mrow></mrow><mo id="S4.Ex15.m1.2.2.3" xref="S4.Ex15.m1.2.2.3.cmml">≥</mo><mn id="S4.Ex15.m1.2.2.4" xref="S4.Ex15.m1.2.2.4.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.Ex15.m1.2b"><apply id="S4.Ex15.m1.2.2.cmml" xref="S4.Ex15.m1.2.2"><geq id="S4.Ex15.m1.2.2.3.cmml" xref="S4.Ex15.m1.2.2.3"></geq><list id="S4.Ex15.m1.2.2.2.3.cmml" xref="S4.Ex15.m1.2.2.2.2"><apply id="S4.Ex15.m1.1.1.1.1.1.cmml" xref="S4.Ex15.m1.1.1.1.1.1"><times id="S4.Ex15.m1.1.1.1.1.1.2.cmml" xref="S4.Ex15.m1.1.1.1.1.1.2"></times><ci id="S4.Ex15.m1.1.1.1.1.1.3a.cmml" xref="S4.Ex15.m1.1.1.1.1.1.3"><mtext class="ltx_mathvariant_italic" id="S4.Ex15.m1.1.1.1.1.1.3.cmml" xref="S4.Ex15.m1.1.1.1.1.1.3">g</mtext></ci><apply id="S4.Ex15.m1.1.1.1.1.1.1.1.1.cmml" xref="S4.Ex15.m1.1.1.1.1.1.1.1"><ci id="S4.Ex15.m1.1.1.1.1.1.1.1.1.1.cmml" xref="S4.Ex15.m1.1.1.1.1.1.1.1.1.1">→</ci><ci id="S4.Ex15.m1.1.1.1.1.1.1.1.1.2.cmml" xref="S4.Ex15.m1.1.1.1.1.1.1.1.1.2">𝑢</ci><ci id="S4.Ex15.m1.1.1.1.1.1.1.1.1.3.cmml" xref="S4.Ex15.m1.1.1.1.1.1.1.1.1.3">𝑣</ci></apply></apply><apply id="S4.Ex15.m1.2.2.2.2.2.cmml" xref="S4.Ex15.m1.2.2.2.2.2"><times id="S4.Ex15.m1.2.2.2.2.2.2.cmml" xref="S4.Ex15.m1.2.2.2.2.2.2"></times><ci id="S4.Ex15.m1.2.2.2.2.2.3a.cmml" xref="S4.Ex15.m1.2.2.2.2.2.3"><mtext class="ltx_mathvariant_italic" id="S4.Ex15.m1.2.2.2.2.2.3.cmml" xref="S4.Ex15.m1.2.2.2.2.2.3">g</mtext></ci><apply id="S4.Ex15.m1.2.2.2.2.2.1.1.1.cmml" xref="S4.Ex15.m1.2.2.2.2.2.1.1"><ci id="S4.Ex15.m1.2.2.2.2.2.1.1.1.1.cmml" xref="S4.Ex15.m1.2.2.2.2.2.1.1.1.1">→</ci><ci id="S4.Ex15.m1.2.2.2.2.2.1.1.1.2.cmml" xref="S4.Ex15.m1.2.2.2.2.2.1.1.1.2">𝑣</ci><ci id="S4.Ex15.m1.2.2.2.2.2.1.1.1.3.cmml" xref="S4.Ex15.m1.2.2.2.2.2.1.1.1.3">𝑢</ci></apply></apply></list><cn id="S4.Ex15.m1.2.2.4.cmml" type="integer" xref="S4.Ex15.m1.2.2.4">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Ex15.m1.2c">\displaystyle\textsl{g}(u\!\to\!v),\textsl{g}(v\!\to\!u)\geq 0</annotation><annotation encoding="application/x-llamapun" id="S4.Ex15.m1.2d">g ( italic_u → italic_v ) , g ( italic_v → italic_u ) ≥ 0</annotation></semantics></math></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle\forall u,v\in V" class="ltx_Math" display="inline" id="S4.Ex15.m2.2"><semantics id="S4.Ex15.m2.2a"><mrow id="S4.Ex15.m2.2.2" xref="S4.Ex15.m2.2.2.cmml"><mrow id="S4.Ex15.m2.2.2.1.1" xref="S4.Ex15.m2.2.2.1.2.cmml"><mrow id="S4.Ex15.m2.2.2.1.1.1" xref="S4.Ex15.m2.2.2.1.1.1.cmml"><mo id="S4.Ex15.m2.2.2.1.1.1.1" rspace="0.167em" xref="S4.Ex15.m2.2.2.1.1.1.1.cmml">∀</mo><mi id="S4.Ex15.m2.2.2.1.1.1.2" xref="S4.Ex15.m2.2.2.1.1.1.2.cmml">u</mi></mrow><mo id="S4.Ex15.m2.2.2.1.1.2" xref="S4.Ex15.m2.2.2.1.2.cmml">,</mo><mi id="S4.Ex15.m2.1.1" xref="S4.Ex15.m2.1.1.cmml">v</mi></mrow><mo id="S4.Ex15.m2.2.2.2" xref="S4.Ex15.m2.2.2.2.cmml">∈</mo><mi id="S4.Ex15.m2.2.2.3" xref="S4.Ex15.m2.2.2.3.cmml">V</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.Ex15.m2.2b"><apply id="S4.Ex15.m2.2.2.cmml" xref="S4.Ex15.m2.2.2"><in id="S4.Ex15.m2.2.2.2.cmml" xref="S4.Ex15.m2.2.2.2"></in><list id="S4.Ex15.m2.2.2.1.2.cmml" xref="S4.Ex15.m2.2.2.1.1"><apply id="S4.Ex15.m2.2.2.1.1.1.cmml" xref="S4.Ex15.m2.2.2.1.1.1"><csymbol cd="latexml" id="S4.Ex15.m2.2.2.1.1.1.1.cmml" xref="S4.Ex15.m2.2.2.1.1.1.1">for-all</csymbol><ci id="S4.Ex15.m2.2.2.1.1.1.2.cmml" xref="S4.Ex15.m2.2.2.1.1.1.2">𝑢</ci></apply><ci id="S4.Ex15.m2.1.1.cmml" xref="S4.Ex15.m2.1.1">𝑣</ci></list><ci id="S4.Ex15.m2.2.2.3.cmml" xref="S4.Ex15.m2.2.2.3">𝑉</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Ex15.m2.2c">\displaystyle\forall u,v\in V</annotation><annotation encoding="application/x-llamapun" id="S4.Ex15.m2.2d">∀ italic_u , italic_v ∈ italic_V</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_left_padright"></td> </tr></tbody> </table> </div> <div class="ltx_para" id="S4.Thmtheorem3.p2"> <p class="ltx_p" id="S4.Thmtheorem3.p2.3"><span class="ltx_text ltx_font_italic" id="S4.Thmtheorem3.p2.3.3">Consider any optimal solution to the quadratic program. It must be that <math alttext="\textsl{g}(u)=\sum_{v\in V}\textsl{g}(u\!\to\!v)" class="ltx_Math" display="inline" id="S4.Thmtheorem3.p2.1.1.m1.2"><semantics id="S4.Thmtheorem3.p2.1.1.m1.2a"><mrow id="S4.Thmtheorem3.p2.1.1.m1.2.2" xref="S4.Thmtheorem3.p2.1.1.m1.2.2.cmml"><mrow id="S4.Thmtheorem3.p2.1.1.m1.2.2.3" xref="S4.Thmtheorem3.p2.1.1.m1.2.2.3.cmml"><mtext class="ltx_mathvariant_italic" id="S4.Thmtheorem3.p2.1.1.m1.2.2.3.2" xref="S4.Thmtheorem3.p2.1.1.m1.2.2.3.2a.cmml">g</mtext><mo id="S4.Thmtheorem3.p2.1.1.m1.2.2.3.1" xref="S4.Thmtheorem3.p2.1.1.m1.2.2.3.1.cmml">⁢</mo><mrow id="S4.Thmtheorem3.p2.1.1.m1.2.2.3.3.2" xref="S4.Thmtheorem3.p2.1.1.m1.2.2.3.cmml"><mo id="S4.Thmtheorem3.p2.1.1.m1.2.2.3.3.2.1" stretchy="false" xref="S4.Thmtheorem3.p2.1.1.m1.2.2.3.cmml">(</mo><mi id="S4.Thmtheorem3.p2.1.1.m1.1.1" xref="S4.Thmtheorem3.p2.1.1.m1.1.1.cmml">u</mi><mo id="S4.Thmtheorem3.p2.1.1.m1.2.2.3.3.2.2" stretchy="false" xref="S4.Thmtheorem3.p2.1.1.m1.2.2.3.cmml">)</mo></mrow></mrow><mo id="S4.Thmtheorem3.p2.1.1.m1.2.2.2" rspace="0.111em" xref="S4.Thmtheorem3.p2.1.1.m1.2.2.2.cmml">=</mo><mrow id="S4.Thmtheorem3.p2.1.1.m1.2.2.1" xref="S4.Thmtheorem3.p2.1.1.m1.2.2.1.cmml"><msub id="S4.Thmtheorem3.p2.1.1.m1.2.2.1.2" xref="S4.Thmtheorem3.p2.1.1.m1.2.2.1.2.cmml"><mo id="S4.Thmtheorem3.p2.1.1.m1.2.2.1.2.2" xref="S4.Thmtheorem3.p2.1.1.m1.2.2.1.2.2.cmml">∑</mo><mrow id="S4.Thmtheorem3.p2.1.1.m1.2.2.1.2.3" xref="S4.Thmtheorem3.p2.1.1.m1.2.2.1.2.3.cmml"><mi id="S4.Thmtheorem3.p2.1.1.m1.2.2.1.2.3.2" xref="S4.Thmtheorem3.p2.1.1.m1.2.2.1.2.3.2.cmml">v</mi><mo id="S4.Thmtheorem3.p2.1.1.m1.2.2.1.2.3.1" xref="S4.Thmtheorem3.p2.1.1.m1.2.2.1.2.3.1.cmml">∈</mo><mi id="S4.Thmtheorem3.p2.1.1.m1.2.2.1.2.3.3" xref="S4.Thmtheorem3.p2.1.1.m1.2.2.1.2.3.3.cmml">V</mi></mrow></msub><mrow id="S4.Thmtheorem3.p2.1.1.m1.2.2.1.1" xref="S4.Thmtheorem3.p2.1.1.m1.2.2.1.1.cmml"><mtext class="ltx_mathvariant_italic" id="S4.Thmtheorem3.p2.1.1.m1.2.2.1.1.3" xref="S4.Thmtheorem3.p2.1.1.m1.2.2.1.1.3a.cmml">g</mtext><mo id="S4.Thmtheorem3.p2.1.1.m1.2.2.1.1.2" xref="S4.Thmtheorem3.p2.1.1.m1.2.2.1.1.2.cmml">⁢</mo><mrow id="S4.Thmtheorem3.p2.1.1.m1.2.2.1.1.1.1" xref="S4.Thmtheorem3.p2.1.1.m1.2.2.1.1.1.1.1.cmml"><mo id="S4.Thmtheorem3.p2.1.1.m1.2.2.1.1.1.1.2" stretchy="false" xref="S4.Thmtheorem3.p2.1.1.m1.2.2.1.1.1.1.1.cmml">(</mo><mrow id="S4.Thmtheorem3.p2.1.1.m1.2.2.1.1.1.1.1" xref="S4.Thmtheorem3.p2.1.1.m1.2.2.1.1.1.1.1.cmml"><mi id="S4.Thmtheorem3.p2.1.1.m1.2.2.1.1.1.1.1.2" xref="S4.Thmtheorem3.p2.1.1.m1.2.2.1.1.1.1.1.2.cmml">u</mi><mo id="S4.Thmtheorem3.p2.1.1.m1.2.2.1.1.1.1.1.1" lspace="0.108em" rspace="0.108em" stretchy="false" xref="S4.Thmtheorem3.p2.1.1.m1.2.2.1.1.1.1.1.1.cmml">→</mo><mi id="S4.Thmtheorem3.p2.1.1.m1.2.2.1.1.1.1.1.3" xref="S4.Thmtheorem3.p2.1.1.m1.2.2.1.1.1.1.1.3.cmml">v</mi></mrow><mo id="S4.Thmtheorem3.p2.1.1.m1.2.2.1.1.1.1.3" stretchy="false" xref="S4.Thmtheorem3.p2.1.1.m1.2.2.1.1.1.1.1.cmml">)</mo></mrow></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem3.p2.1.1.m1.2b"><apply id="S4.Thmtheorem3.p2.1.1.m1.2.2.cmml" xref="S4.Thmtheorem3.p2.1.1.m1.2.2"><eq id="S4.Thmtheorem3.p2.1.1.m1.2.2.2.cmml" xref="S4.Thmtheorem3.p2.1.1.m1.2.2.2"></eq><apply id="S4.Thmtheorem3.p2.1.1.m1.2.2.3.cmml" xref="S4.Thmtheorem3.p2.1.1.m1.2.2.3"><times id="S4.Thmtheorem3.p2.1.1.m1.2.2.3.1.cmml" xref="S4.Thmtheorem3.p2.1.1.m1.2.2.3.1"></times><ci id="S4.Thmtheorem3.p2.1.1.m1.2.2.3.2a.cmml" xref="S4.Thmtheorem3.p2.1.1.m1.2.2.3.2"><mtext class="ltx_mathvariant_italic" id="S4.Thmtheorem3.p2.1.1.m1.2.2.3.2.cmml" xref="S4.Thmtheorem3.p2.1.1.m1.2.2.3.2">g</mtext></ci><ci id="S4.Thmtheorem3.p2.1.1.m1.1.1.cmml" xref="S4.Thmtheorem3.p2.1.1.m1.1.1">𝑢</ci></apply><apply id="S4.Thmtheorem3.p2.1.1.m1.2.2.1.cmml" xref="S4.Thmtheorem3.p2.1.1.m1.2.2.1"><apply id="S4.Thmtheorem3.p2.1.1.m1.2.2.1.2.cmml" xref="S4.Thmtheorem3.p2.1.1.m1.2.2.1.2"><csymbol cd="ambiguous" id="S4.Thmtheorem3.p2.1.1.m1.2.2.1.2.1.cmml" xref="S4.Thmtheorem3.p2.1.1.m1.2.2.1.2">subscript</csymbol><sum id="S4.Thmtheorem3.p2.1.1.m1.2.2.1.2.2.cmml" xref="S4.Thmtheorem3.p2.1.1.m1.2.2.1.2.2"></sum><apply id="S4.Thmtheorem3.p2.1.1.m1.2.2.1.2.3.cmml" xref="S4.Thmtheorem3.p2.1.1.m1.2.2.1.2.3"><in id="S4.Thmtheorem3.p2.1.1.m1.2.2.1.2.3.1.cmml" xref="S4.Thmtheorem3.p2.1.1.m1.2.2.1.2.3.1"></in><ci id="S4.Thmtheorem3.p2.1.1.m1.2.2.1.2.3.2.cmml" xref="S4.Thmtheorem3.p2.1.1.m1.2.2.1.2.3.2">𝑣</ci><ci id="S4.Thmtheorem3.p2.1.1.m1.2.2.1.2.3.3.cmml" xref="S4.Thmtheorem3.p2.1.1.m1.2.2.1.2.3.3">𝑉</ci></apply></apply><apply id="S4.Thmtheorem3.p2.1.1.m1.2.2.1.1.cmml" xref="S4.Thmtheorem3.p2.1.1.m1.2.2.1.1"><times id="S4.Thmtheorem3.p2.1.1.m1.2.2.1.1.2.cmml" xref="S4.Thmtheorem3.p2.1.1.m1.2.2.1.1.2"></times><ci id="S4.Thmtheorem3.p2.1.1.m1.2.2.1.1.3a.cmml" xref="S4.Thmtheorem3.p2.1.1.m1.2.2.1.1.3"><mtext class="ltx_mathvariant_italic" id="S4.Thmtheorem3.p2.1.1.m1.2.2.1.1.3.cmml" xref="S4.Thmtheorem3.p2.1.1.m1.2.2.1.1.3">g</mtext></ci><apply id="S4.Thmtheorem3.p2.1.1.m1.2.2.1.1.1.1.1.cmml" xref="S4.Thmtheorem3.p2.1.1.m1.2.2.1.1.1.1"><ci id="S4.Thmtheorem3.p2.1.1.m1.2.2.1.1.1.1.1.1.cmml" xref="S4.Thmtheorem3.p2.1.1.m1.2.2.1.1.1.1.1.1">→</ci><ci id="S4.Thmtheorem3.p2.1.1.m1.2.2.1.1.1.1.1.2.cmml" xref="S4.Thmtheorem3.p2.1.1.m1.2.2.1.1.1.1.1.2">𝑢</ci><ci id="S4.Thmtheorem3.p2.1.1.m1.2.2.1.1.1.1.1.3.cmml" xref="S4.Thmtheorem3.p2.1.1.m1.2.2.1.1.1.1.1.3">𝑣</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem3.p2.1.1.m1.2c">\textsl{g}(u)=\sum_{v\in V}\textsl{g}(u\!\to\!v)</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem3.p2.1.1.m1.2d">g ( italic_u ) = ∑ start_POSTSUBSCRIPT italic_v ∈ italic_V end_POSTSUBSCRIPT g ( italic_u → italic_v )</annotation></semantics></math>. Danisch, Chan, and Sozio <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2411.12694v2#bib.bib10" title="">10</a>, Corollary 4.4]</cite> prove that for any vertex <math alttext="u" class="ltx_Math" display="inline" id="S4.Thmtheorem3.p2.2.2.m2.1"><semantics id="S4.Thmtheorem3.p2.2.2.m2.1a"><mi id="S4.Thmtheorem3.p2.2.2.m2.1.1" xref="S4.Thmtheorem3.p2.2.2.m2.1.1.cmml">u</mi><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem3.p2.2.2.m2.1b"><ci id="S4.Thmtheorem3.p2.2.2.m2.1.1.cmml" xref="S4.Thmtheorem3.p2.2.2.m2.1.1">𝑢</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem3.p2.2.2.m2.1c">u</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem3.p2.2.2.m2.1d">italic_u</annotation></semantics></math>, the local density <math alttext="\rho^{*}(u)=\textsl{g}(u)" class="ltx_Math" display="inline" id="S4.Thmtheorem3.p2.3.3.m3.2"><semantics id="S4.Thmtheorem3.p2.3.3.m3.2a"><mrow id="S4.Thmtheorem3.p2.3.3.m3.2.3" xref="S4.Thmtheorem3.p2.3.3.m3.2.3.cmml"><mrow id="S4.Thmtheorem3.p2.3.3.m3.2.3.2" xref="S4.Thmtheorem3.p2.3.3.m3.2.3.2.cmml"><msup id="S4.Thmtheorem3.p2.3.3.m3.2.3.2.2" xref="S4.Thmtheorem3.p2.3.3.m3.2.3.2.2.cmml"><mi id="S4.Thmtheorem3.p2.3.3.m3.2.3.2.2.2" xref="S4.Thmtheorem3.p2.3.3.m3.2.3.2.2.2.cmml">ρ</mi><mo id="S4.Thmtheorem3.p2.3.3.m3.2.3.2.2.3" xref="S4.Thmtheorem3.p2.3.3.m3.2.3.2.2.3.cmml">∗</mo></msup><mo id="S4.Thmtheorem3.p2.3.3.m3.2.3.2.1" xref="S4.Thmtheorem3.p2.3.3.m3.2.3.2.1.cmml">⁢</mo><mrow id="S4.Thmtheorem3.p2.3.3.m3.2.3.2.3.2" xref="S4.Thmtheorem3.p2.3.3.m3.2.3.2.cmml"><mo id="S4.Thmtheorem3.p2.3.3.m3.2.3.2.3.2.1" stretchy="false" xref="S4.Thmtheorem3.p2.3.3.m3.2.3.2.cmml">(</mo><mi id="S4.Thmtheorem3.p2.3.3.m3.1.1" xref="S4.Thmtheorem3.p2.3.3.m3.1.1.cmml">u</mi><mo id="S4.Thmtheorem3.p2.3.3.m3.2.3.2.3.2.2" stretchy="false" xref="S4.Thmtheorem3.p2.3.3.m3.2.3.2.cmml">)</mo></mrow></mrow><mo id="S4.Thmtheorem3.p2.3.3.m3.2.3.1" xref="S4.Thmtheorem3.p2.3.3.m3.2.3.1.cmml">=</mo><mrow id="S4.Thmtheorem3.p2.3.3.m3.2.3.3" xref="S4.Thmtheorem3.p2.3.3.m3.2.3.3.cmml"><mtext class="ltx_mathvariant_italic" id="S4.Thmtheorem3.p2.3.3.m3.2.3.3.2" xref="S4.Thmtheorem3.p2.3.3.m3.2.3.3.2a.cmml">g</mtext><mo id="S4.Thmtheorem3.p2.3.3.m3.2.3.3.1" xref="S4.Thmtheorem3.p2.3.3.m3.2.3.3.1.cmml">⁢</mo><mrow id="S4.Thmtheorem3.p2.3.3.m3.2.3.3.3.2" xref="S4.Thmtheorem3.p2.3.3.m3.2.3.3.cmml"><mo id="S4.Thmtheorem3.p2.3.3.m3.2.3.3.3.2.1" stretchy="false" xref="S4.Thmtheorem3.p2.3.3.m3.2.3.3.cmml">(</mo><mi id="S4.Thmtheorem3.p2.3.3.m3.2.2" xref="S4.Thmtheorem3.p2.3.3.m3.2.2.cmml">u</mi><mo id="S4.Thmtheorem3.p2.3.3.m3.2.3.3.3.2.2" stretchy="false" xref="S4.Thmtheorem3.p2.3.3.m3.2.3.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem3.p2.3.3.m3.2b"><apply id="S4.Thmtheorem3.p2.3.3.m3.2.3.cmml" xref="S4.Thmtheorem3.p2.3.3.m3.2.3"><eq id="S4.Thmtheorem3.p2.3.3.m3.2.3.1.cmml" xref="S4.Thmtheorem3.p2.3.3.m3.2.3.1"></eq><apply id="S4.Thmtheorem3.p2.3.3.m3.2.3.2.cmml" xref="S4.Thmtheorem3.p2.3.3.m3.2.3.2"><times id="S4.Thmtheorem3.p2.3.3.m3.2.3.2.1.cmml" xref="S4.Thmtheorem3.p2.3.3.m3.2.3.2.1"></times><apply id="S4.Thmtheorem3.p2.3.3.m3.2.3.2.2.cmml" xref="S4.Thmtheorem3.p2.3.3.m3.2.3.2.2"><csymbol cd="ambiguous" id="S4.Thmtheorem3.p2.3.3.m3.2.3.2.2.1.cmml" xref="S4.Thmtheorem3.p2.3.3.m3.2.3.2.2">superscript</csymbol><ci id="S4.Thmtheorem3.p2.3.3.m3.2.3.2.2.2.cmml" xref="S4.Thmtheorem3.p2.3.3.m3.2.3.2.2.2">𝜌</ci><times id="S4.Thmtheorem3.p2.3.3.m3.2.3.2.2.3.cmml" xref="S4.Thmtheorem3.p2.3.3.m3.2.3.2.2.3"></times></apply><ci id="S4.Thmtheorem3.p2.3.3.m3.1.1.cmml" xref="S4.Thmtheorem3.p2.3.3.m3.1.1">𝑢</ci></apply><apply id="S4.Thmtheorem3.p2.3.3.m3.2.3.3.cmml" xref="S4.Thmtheorem3.p2.3.3.m3.2.3.3"><times id="S4.Thmtheorem3.p2.3.3.m3.2.3.3.1.cmml" xref="S4.Thmtheorem3.p2.3.3.m3.2.3.3.1"></times><ci id="S4.Thmtheorem3.p2.3.3.m3.2.3.3.2a.cmml" xref="S4.Thmtheorem3.p2.3.3.m3.2.3.3.2"><mtext class="ltx_mathvariant_italic" id="S4.Thmtheorem3.p2.3.3.m3.2.3.3.2.cmml" xref="S4.Thmtheorem3.p2.3.3.m3.2.3.3.2">g</mtext></ci><ci id="S4.Thmtheorem3.p2.3.3.m3.2.2.cmml" xref="S4.Thmtheorem3.p2.3.3.m3.2.2">𝑢</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem3.p2.3.3.m3.2c">\rho^{*}(u)=\textsl{g}(u)</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem3.p2.3.3.m3.2d">italic_ρ start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ( italic_u ) = g ( italic_u )</annotation></semantics></math>.</span></p> </div> <div class="ltx_para" id="S4.Thmtheorem3.p3"> <p class="ltx_p" id="S4.Thmtheorem3.p3.6"><span class="ltx_text ltx_font_italic" id="S4.Thmtheorem3.p3.6.6">We first note that any solution to the quadratic program is an orientation. Indeed, suppose for the sake of contradiction that there exists an edge <math alttext="\overline{uv}\in E" class="ltx_Math" display="inline" id="S4.Thmtheorem3.p3.1.1.m1.1"><semantics id="S4.Thmtheorem3.p3.1.1.m1.1a"><mrow id="S4.Thmtheorem3.p3.1.1.m1.1.1" xref="S4.Thmtheorem3.p3.1.1.m1.1.1.cmml"><mover accent="true" id="S4.Thmtheorem3.p3.1.1.m1.1.1.2" xref="S4.Thmtheorem3.p3.1.1.m1.1.1.2.cmml"><mrow id="S4.Thmtheorem3.p3.1.1.m1.1.1.2.2" xref="S4.Thmtheorem3.p3.1.1.m1.1.1.2.2.cmml"><mi id="S4.Thmtheorem3.p3.1.1.m1.1.1.2.2.2" xref="S4.Thmtheorem3.p3.1.1.m1.1.1.2.2.2.cmml">u</mi><mo id="S4.Thmtheorem3.p3.1.1.m1.1.1.2.2.1" xref="S4.Thmtheorem3.p3.1.1.m1.1.1.2.2.1.cmml">⁢</mo><mi id="S4.Thmtheorem3.p3.1.1.m1.1.1.2.2.3" xref="S4.Thmtheorem3.p3.1.1.m1.1.1.2.2.3.cmml">v</mi></mrow><mo id="S4.Thmtheorem3.p3.1.1.m1.1.1.2.1" xref="S4.Thmtheorem3.p3.1.1.m1.1.1.2.1.cmml">¯</mo></mover><mo id="S4.Thmtheorem3.p3.1.1.m1.1.1.1" xref="S4.Thmtheorem3.p3.1.1.m1.1.1.1.cmml">∈</mo><mi id="S4.Thmtheorem3.p3.1.1.m1.1.1.3" xref="S4.Thmtheorem3.p3.1.1.m1.1.1.3.cmml">E</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem3.p3.1.1.m1.1b"><apply id="S4.Thmtheorem3.p3.1.1.m1.1.1.cmml" xref="S4.Thmtheorem3.p3.1.1.m1.1.1"><in id="S4.Thmtheorem3.p3.1.1.m1.1.1.1.cmml" xref="S4.Thmtheorem3.p3.1.1.m1.1.1.1"></in><apply id="S4.Thmtheorem3.p3.1.1.m1.1.1.2.cmml" xref="S4.Thmtheorem3.p3.1.1.m1.1.1.2"><ci id="S4.Thmtheorem3.p3.1.1.m1.1.1.2.1.cmml" xref="S4.Thmtheorem3.p3.1.1.m1.1.1.2.1">¯</ci><apply id="S4.Thmtheorem3.p3.1.1.m1.1.1.2.2.cmml" xref="S4.Thmtheorem3.p3.1.1.m1.1.1.2.2"><times id="S4.Thmtheorem3.p3.1.1.m1.1.1.2.2.1.cmml" xref="S4.Thmtheorem3.p3.1.1.m1.1.1.2.2.1"></times><ci id="S4.Thmtheorem3.p3.1.1.m1.1.1.2.2.2.cmml" xref="S4.Thmtheorem3.p3.1.1.m1.1.1.2.2.2">𝑢</ci><ci id="S4.Thmtheorem3.p3.1.1.m1.1.1.2.2.3.cmml" xref="S4.Thmtheorem3.p3.1.1.m1.1.1.2.2.3">𝑣</ci></apply></apply><ci id="S4.Thmtheorem3.p3.1.1.m1.1.1.3.cmml" xref="S4.Thmtheorem3.p3.1.1.m1.1.1.3">𝐸</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem3.p3.1.1.m1.1c">\overline{uv}\in E</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem3.p3.1.1.m1.1d">over¯ start_ARG italic_u italic_v end_ARG ∈ italic_E</annotation></semantics></math> where <math alttext="\textsl{g}(u\!\to\!v)+\textsl{g}(v\!\to\!u)>\textsl{g}(\overline{uv})" class="ltx_Math" display="inline" id="S4.Thmtheorem3.p3.2.2.m2.3"><semantics id="S4.Thmtheorem3.p3.2.2.m2.3a"><mrow id="S4.Thmtheorem3.p3.2.2.m2.3.3" xref="S4.Thmtheorem3.p3.2.2.m2.3.3.cmml"><mrow id="S4.Thmtheorem3.p3.2.2.m2.3.3.2" xref="S4.Thmtheorem3.p3.2.2.m2.3.3.2.cmml"><mrow id="S4.Thmtheorem3.p3.2.2.m2.2.2.1.1" xref="S4.Thmtheorem3.p3.2.2.m2.2.2.1.1.cmml"><mtext class="ltx_mathvariant_italic" id="S4.Thmtheorem3.p3.2.2.m2.2.2.1.1.3" xref="S4.Thmtheorem3.p3.2.2.m2.2.2.1.1.3a.cmml">g</mtext><mo id="S4.Thmtheorem3.p3.2.2.m2.2.2.1.1.2" xref="S4.Thmtheorem3.p3.2.2.m2.2.2.1.1.2.cmml">⁢</mo><mrow id="S4.Thmtheorem3.p3.2.2.m2.2.2.1.1.1.1" xref="S4.Thmtheorem3.p3.2.2.m2.2.2.1.1.1.1.1.cmml"><mo id="S4.Thmtheorem3.p3.2.2.m2.2.2.1.1.1.1.2" stretchy="false" xref="S4.Thmtheorem3.p3.2.2.m2.2.2.1.1.1.1.1.cmml">(</mo><mrow id="S4.Thmtheorem3.p3.2.2.m2.2.2.1.1.1.1.1" xref="S4.Thmtheorem3.p3.2.2.m2.2.2.1.1.1.1.1.cmml"><mi id="S4.Thmtheorem3.p3.2.2.m2.2.2.1.1.1.1.1.2" xref="S4.Thmtheorem3.p3.2.2.m2.2.2.1.1.1.1.1.2.cmml">u</mi><mo id="S4.Thmtheorem3.p3.2.2.m2.2.2.1.1.1.1.1.1" lspace="0.108em" rspace="0.108em" stretchy="false" xref="S4.Thmtheorem3.p3.2.2.m2.2.2.1.1.1.1.1.1.cmml">→</mo><mi id="S4.Thmtheorem3.p3.2.2.m2.2.2.1.1.1.1.1.3" xref="S4.Thmtheorem3.p3.2.2.m2.2.2.1.1.1.1.1.3.cmml">v</mi></mrow><mo id="S4.Thmtheorem3.p3.2.2.m2.2.2.1.1.1.1.3" stretchy="false" xref="S4.Thmtheorem3.p3.2.2.m2.2.2.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S4.Thmtheorem3.p3.2.2.m2.3.3.2.3" xref="S4.Thmtheorem3.p3.2.2.m2.3.3.2.3.cmml">+</mo><mrow id="S4.Thmtheorem3.p3.2.2.m2.3.3.2.2" xref="S4.Thmtheorem3.p3.2.2.m2.3.3.2.2.cmml"><mtext class="ltx_mathvariant_italic" id="S4.Thmtheorem3.p3.2.2.m2.3.3.2.2.3" xref="S4.Thmtheorem3.p3.2.2.m2.3.3.2.2.3a.cmml">g</mtext><mo id="S4.Thmtheorem3.p3.2.2.m2.3.3.2.2.2" xref="S4.Thmtheorem3.p3.2.2.m2.3.3.2.2.2.cmml">⁢</mo><mrow id="S4.Thmtheorem3.p3.2.2.m2.3.3.2.2.1.1" xref="S4.Thmtheorem3.p3.2.2.m2.3.3.2.2.1.1.1.cmml"><mo id="S4.Thmtheorem3.p3.2.2.m2.3.3.2.2.1.1.2" stretchy="false" xref="S4.Thmtheorem3.p3.2.2.m2.3.3.2.2.1.1.1.cmml">(</mo><mrow id="S4.Thmtheorem3.p3.2.2.m2.3.3.2.2.1.1.1" xref="S4.Thmtheorem3.p3.2.2.m2.3.3.2.2.1.1.1.cmml"><mi id="S4.Thmtheorem3.p3.2.2.m2.3.3.2.2.1.1.1.2" xref="S4.Thmtheorem3.p3.2.2.m2.3.3.2.2.1.1.1.2.cmml">v</mi><mo id="S4.Thmtheorem3.p3.2.2.m2.3.3.2.2.1.1.1.1" lspace="0.108em" rspace="0.108em" stretchy="false" xref="S4.Thmtheorem3.p3.2.2.m2.3.3.2.2.1.1.1.1.cmml">→</mo><mi id="S4.Thmtheorem3.p3.2.2.m2.3.3.2.2.1.1.1.3" xref="S4.Thmtheorem3.p3.2.2.m2.3.3.2.2.1.1.1.3.cmml">u</mi></mrow><mo id="S4.Thmtheorem3.p3.2.2.m2.3.3.2.2.1.1.3" stretchy="false" xref="S4.Thmtheorem3.p3.2.2.m2.3.3.2.2.1.1.1.cmml">)</mo></mrow></mrow></mrow><mo id="S4.Thmtheorem3.p3.2.2.m2.3.3.3" xref="S4.Thmtheorem3.p3.2.2.m2.3.3.3.cmml">></mo><mrow id="S4.Thmtheorem3.p3.2.2.m2.3.3.4" xref="S4.Thmtheorem3.p3.2.2.m2.3.3.4.cmml"><mtext class="ltx_mathvariant_italic" id="S4.Thmtheorem3.p3.2.2.m2.3.3.4.2" xref="S4.Thmtheorem3.p3.2.2.m2.3.3.4.2a.cmml">g</mtext><mo id="S4.Thmtheorem3.p3.2.2.m2.3.3.4.1" xref="S4.Thmtheorem3.p3.2.2.m2.3.3.4.1.cmml">⁢</mo><mrow id="S4.Thmtheorem3.p3.2.2.m2.3.3.4.3.2" xref="S4.Thmtheorem3.p3.2.2.m2.1.1.cmml"><mo id="S4.Thmtheorem3.p3.2.2.m2.3.3.4.3.2.1" stretchy="false" xref="S4.Thmtheorem3.p3.2.2.m2.1.1.cmml">(</mo><mover accent="true" id="S4.Thmtheorem3.p3.2.2.m2.1.1" xref="S4.Thmtheorem3.p3.2.2.m2.1.1.cmml"><mrow id="S4.Thmtheorem3.p3.2.2.m2.1.1.2" xref="S4.Thmtheorem3.p3.2.2.m2.1.1.2.cmml"><mi id="S4.Thmtheorem3.p3.2.2.m2.1.1.2.2" xref="S4.Thmtheorem3.p3.2.2.m2.1.1.2.2.cmml">u</mi><mo id="S4.Thmtheorem3.p3.2.2.m2.1.1.2.1" xref="S4.Thmtheorem3.p3.2.2.m2.1.1.2.1.cmml">⁢</mo><mi id="S4.Thmtheorem3.p3.2.2.m2.1.1.2.3" xref="S4.Thmtheorem3.p3.2.2.m2.1.1.2.3.cmml">v</mi></mrow><mo id="S4.Thmtheorem3.p3.2.2.m2.1.1.1" xref="S4.Thmtheorem3.p3.2.2.m2.1.1.1.cmml">¯</mo></mover><mo id="S4.Thmtheorem3.p3.2.2.m2.3.3.4.3.2.2" stretchy="false" xref="S4.Thmtheorem3.p3.2.2.m2.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem3.p3.2.2.m2.3b"><apply id="S4.Thmtheorem3.p3.2.2.m2.3.3.cmml" xref="S4.Thmtheorem3.p3.2.2.m2.3.3"><gt id="S4.Thmtheorem3.p3.2.2.m2.3.3.3.cmml" xref="S4.Thmtheorem3.p3.2.2.m2.3.3.3"></gt><apply id="S4.Thmtheorem3.p3.2.2.m2.3.3.2.cmml" xref="S4.Thmtheorem3.p3.2.2.m2.3.3.2"><plus id="S4.Thmtheorem3.p3.2.2.m2.3.3.2.3.cmml" xref="S4.Thmtheorem3.p3.2.2.m2.3.3.2.3"></plus><apply id="S4.Thmtheorem3.p3.2.2.m2.2.2.1.1.cmml" xref="S4.Thmtheorem3.p3.2.2.m2.2.2.1.1"><times id="S4.Thmtheorem3.p3.2.2.m2.2.2.1.1.2.cmml" xref="S4.Thmtheorem3.p3.2.2.m2.2.2.1.1.2"></times><ci id="S4.Thmtheorem3.p3.2.2.m2.2.2.1.1.3a.cmml" xref="S4.Thmtheorem3.p3.2.2.m2.2.2.1.1.3"><mtext class="ltx_mathvariant_italic" id="S4.Thmtheorem3.p3.2.2.m2.2.2.1.1.3.cmml" xref="S4.Thmtheorem3.p3.2.2.m2.2.2.1.1.3">g</mtext></ci><apply id="S4.Thmtheorem3.p3.2.2.m2.2.2.1.1.1.1.1.cmml" xref="S4.Thmtheorem3.p3.2.2.m2.2.2.1.1.1.1"><ci id="S4.Thmtheorem3.p3.2.2.m2.2.2.1.1.1.1.1.1.cmml" xref="S4.Thmtheorem3.p3.2.2.m2.2.2.1.1.1.1.1.1">→</ci><ci id="S4.Thmtheorem3.p3.2.2.m2.2.2.1.1.1.1.1.2.cmml" xref="S4.Thmtheorem3.p3.2.2.m2.2.2.1.1.1.1.1.2">𝑢</ci><ci id="S4.Thmtheorem3.p3.2.2.m2.2.2.1.1.1.1.1.3.cmml" xref="S4.Thmtheorem3.p3.2.2.m2.2.2.1.1.1.1.1.3">𝑣</ci></apply></apply><apply id="S4.Thmtheorem3.p3.2.2.m2.3.3.2.2.cmml" xref="S4.Thmtheorem3.p3.2.2.m2.3.3.2.2"><times id="S4.Thmtheorem3.p3.2.2.m2.3.3.2.2.2.cmml" xref="S4.Thmtheorem3.p3.2.2.m2.3.3.2.2.2"></times><ci id="S4.Thmtheorem3.p3.2.2.m2.3.3.2.2.3a.cmml" xref="S4.Thmtheorem3.p3.2.2.m2.3.3.2.2.3"><mtext class="ltx_mathvariant_italic" id="S4.Thmtheorem3.p3.2.2.m2.3.3.2.2.3.cmml" xref="S4.Thmtheorem3.p3.2.2.m2.3.3.2.2.3">g</mtext></ci><apply id="S4.Thmtheorem3.p3.2.2.m2.3.3.2.2.1.1.1.cmml" xref="S4.Thmtheorem3.p3.2.2.m2.3.3.2.2.1.1"><ci id="S4.Thmtheorem3.p3.2.2.m2.3.3.2.2.1.1.1.1.cmml" xref="S4.Thmtheorem3.p3.2.2.m2.3.3.2.2.1.1.1.1">→</ci><ci id="S4.Thmtheorem3.p3.2.2.m2.3.3.2.2.1.1.1.2.cmml" xref="S4.Thmtheorem3.p3.2.2.m2.3.3.2.2.1.1.1.2">𝑣</ci><ci id="S4.Thmtheorem3.p3.2.2.m2.3.3.2.2.1.1.1.3.cmml" xref="S4.Thmtheorem3.p3.2.2.m2.3.3.2.2.1.1.1.3">𝑢</ci></apply></apply></apply><apply id="S4.Thmtheorem3.p3.2.2.m2.3.3.4.cmml" xref="S4.Thmtheorem3.p3.2.2.m2.3.3.4"><times id="S4.Thmtheorem3.p3.2.2.m2.3.3.4.1.cmml" xref="S4.Thmtheorem3.p3.2.2.m2.3.3.4.1"></times><ci id="S4.Thmtheorem3.p3.2.2.m2.3.3.4.2a.cmml" xref="S4.Thmtheorem3.p3.2.2.m2.3.3.4.2"><mtext class="ltx_mathvariant_italic" id="S4.Thmtheorem3.p3.2.2.m2.3.3.4.2.cmml" xref="S4.Thmtheorem3.p3.2.2.m2.3.3.4.2">g</mtext></ci><apply id="S4.Thmtheorem3.p3.2.2.m2.1.1.cmml" xref="S4.Thmtheorem3.p3.2.2.m2.3.3.4.3.2"><ci id="S4.Thmtheorem3.p3.2.2.m2.1.1.1.cmml" xref="S4.Thmtheorem3.p3.2.2.m2.1.1.1">¯</ci><apply id="S4.Thmtheorem3.p3.2.2.m2.1.1.2.cmml" xref="S4.Thmtheorem3.p3.2.2.m2.1.1.2"><times id="S4.Thmtheorem3.p3.2.2.m2.1.1.2.1.cmml" xref="S4.Thmtheorem3.p3.2.2.m2.1.1.2.1"></times><ci id="S4.Thmtheorem3.p3.2.2.m2.1.1.2.2.cmml" xref="S4.Thmtheorem3.p3.2.2.m2.1.1.2.2">𝑢</ci><ci id="S4.Thmtheorem3.p3.2.2.m2.1.1.2.3.cmml" xref="S4.Thmtheorem3.p3.2.2.m2.1.1.2.3">𝑣</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem3.p3.2.2.m2.3c">\textsl{g}(u\!\to\!v)+\textsl{g}(v\!\to\!u)>\textsl{g}(\overline{uv})</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem3.p3.2.2.m2.3d">g ( italic_u → italic_v ) + g ( italic_v → italic_u ) > g ( over¯ start_ARG italic_u italic_v end_ARG )</annotation></semantics></math>. We may now decrease either <math alttext="\textsl{g}(u\!\to\!v)" class="ltx_Math" display="inline" id="S4.Thmtheorem3.p3.3.3.m3.1"><semantics id="S4.Thmtheorem3.p3.3.3.m3.1a"><mrow id="S4.Thmtheorem3.p3.3.3.m3.1.1" xref="S4.Thmtheorem3.p3.3.3.m3.1.1.cmml"><mtext class="ltx_mathvariant_italic" id="S4.Thmtheorem3.p3.3.3.m3.1.1.3" xref="S4.Thmtheorem3.p3.3.3.m3.1.1.3a.cmml">g</mtext><mo id="S4.Thmtheorem3.p3.3.3.m3.1.1.2" xref="S4.Thmtheorem3.p3.3.3.m3.1.1.2.cmml">⁢</mo><mrow id="S4.Thmtheorem3.p3.3.3.m3.1.1.1.1" xref="S4.Thmtheorem3.p3.3.3.m3.1.1.1.1.1.cmml"><mo id="S4.Thmtheorem3.p3.3.3.m3.1.1.1.1.2" stretchy="false" xref="S4.Thmtheorem3.p3.3.3.m3.1.1.1.1.1.cmml">(</mo><mrow id="S4.Thmtheorem3.p3.3.3.m3.1.1.1.1.1" xref="S4.Thmtheorem3.p3.3.3.m3.1.1.1.1.1.cmml"><mi id="S4.Thmtheorem3.p3.3.3.m3.1.1.1.1.1.2" xref="S4.Thmtheorem3.p3.3.3.m3.1.1.1.1.1.2.cmml">u</mi><mo id="S4.Thmtheorem3.p3.3.3.m3.1.1.1.1.1.1" lspace="0.108em" rspace="0.108em" stretchy="false" xref="S4.Thmtheorem3.p3.3.3.m3.1.1.1.1.1.1.cmml">→</mo><mi id="S4.Thmtheorem3.p3.3.3.m3.1.1.1.1.1.3" xref="S4.Thmtheorem3.p3.3.3.m3.1.1.1.1.1.3.cmml">v</mi></mrow><mo id="S4.Thmtheorem3.p3.3.3.m3.1.1.1.1.3" stretchy="false" xref="S4.Thmtheorem3.p3.3.3.m3.1.1.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem3.p3.3.3.m3.1b"><apply id="S4.Thmtheorem3.p3.3.3.m3.1.1.cmml" xref="S4.Thmtheorem3.p3.3.3.m3.1.1"><times id="S4.Thmtheorem3.p3.3.3.m3.1.1.2.cmml" xref="S4.Thmtheorem3.p3.3.3.m3.1.1.2"></times><ci id="S4.Thmtheorem3.p3.3.3.m3.1.1.3a.cmml" xref="S4.Thmtheorem3.p3.3.3.m3.1.1.3"><mtext class="ltx_mathvariant_italic" id="S4.Thmtheorem3.p3.3.3.m3.1.1.3.cmml" xref="S4.Thmtheorem3.p3.3.3.m3.1.1.3">g</mtext></ci><apply id="S4.Thmtheorem3.p3.3.3.m3.1.1.1.1.1.cmml" xref="S4.Thmtheorem3.p3.3.3.m3.1.1.1.1"><ci id="S4.Thmtheorem3.p3.3.3.m3.1.1.1.1.1.1.cmml" xref="S4.Thmtheorem3.p3.3.3.m3.1.1.1.1.1.1">→</ci><ci id="S4.Thmtheorem3.p3.3.3.m3.1.1.1.1.1.2.cmml" xref="S4.Thmtheorem3.p3.3.3.m3.1.1.1.1.1.2">𝑢</ci><ci id="S4.Thmtheorem3.p3.3.3.m3.1.1.1.1.1.3.cmml" xref="S4.Thmtheorem3.p3.3.3.m3.1.1.1.1.1.3">𝑣</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem3.p3.3.3.m3.1c">\textsl{g}(u\!\to\!v)</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem3.p3.3.3.m3.1d">g ( italic_u → italic_v )</annotation></semantics></math> or <math alttext="\textsl{g}(v\!\to\!u)" class="ltx_Math" display="inline" id="S4.Thmtheorem3.p3.4.4.m4.1"><semantics id="S4.Thmtheorem3.p3.4.4.m4.1a"><mrow id="S4.Thmtheorem3.p3.4.4.m4.1.1" xref="S4.Thmtheorem3.p3.4.4.m4.1.1.cmml"><mtext class="ltx_mathvariant_italic" id="S4.Thmtheorem3.p3.4.4.m4.1.1.3" xref="S4.Thmtheorem3.p3.4.4.m4.1.1.3a.cmml">g</mtext><mo id="S4.Thmtheorem3.p3.4.4.m4.1.1.2" xref="S4.Thmtheorem3.p3.4.4.m4.1.1.2.cmml">⁢</mo><mrow id="S4.Thmtheorem3.p3.4.4.m4.1.1.1.1" xref="S4.Thmtheorem3.p3.4.4.m4.1.1.1.1.1.cmml"><mo id="S4.Thmtheorem3.p3.4.4.m4.1.1.1.1.2" stretchy="false" xref="S4.Thmtheorem3.p3.4.4.m4.1.1.1.1.1.cmml">(</mo><mrow id="S4.Thmtheorem3.p3.4.4.m4.1.1.1.1.1" xref="S4.Thmtheorem3.p3.4.4.m4.1.1.1.1.1.cmml"><mi id="S4.Thmtheorem3.p3.4.4.m4.1.1.1.1.1.2" xref="S4.Thmtheorem3.p3.4.4.m4.1.1.1.1.1.2.cmml">v</mi><mo id="S4.Thmtheorem3.p3.4.4.m4.1.1.1.1.1.1" lspace="0.108em" rspace="0.108em" stretchy="false" xref="S4.Thmtheorem3.p3.4.4.m4.1.1.1.1.1.1.cmml">→</mo><mi id="S4.Thmtheorem3.p3.4.4.m4.1.1.1.1.1.3" xref="S4.Thmtheorem3.p3.4.4.m4.1.1.1.1.1.3.cmml">u</mi></mrow><mo id="S4.Thmtheorem3.p3.4.4.m4.1.1.1.1.3" stretchy="false" xref="S4.Thmtheorem3.p3.4.4.m4.1.1.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem3.p3.4.4.m4.1b"><apply id="S4.Thmtheorem3.p3.4.4.m4.1.1.cmml" xref="S4.Thmtheorem3.p3.4.4.m4.1.1"><times id="S4.Thmtheorem3.p3.4.4.m4.1.1.2.cmml" xref="S4.Thmtheorem3.p3.4.4.m4.1.1.2"></times><ci id="S4.Thmtheorem3.p3.4.4.m4.1.1.3a.cmml" xref="S4.Thmtheorem3.p3.4.4.m4.1.1.3"><mtext class="ltx_mathvariant_italic" id="S4.Thmtheorem3.p3.4.4.m4.1.1.3.cmml" xref="S4.Thmtheorem3.p3.4.4.m4.1.1.3">g</mtext></ci><apply id="S4.Thmtheorem3.p3.4.4.m4.1.1.1.1.1.cmml" xref="S4.Thmtheorem3.p3.4.4.m4.1.1.1.1"><ci id="S4.Thmtheorem3.p3.4.4.m4.1.1.1.1.1.1.cmml" xref="S4.Thmtheorem3.p3.4.4.m4.1.1.1.1.1.1">→</ci><ci id="S4.Thmtheorem3.p3.4.4.m4.1.1.1.1.1.2.cmml" xref="S4.Thmtheorem3.p3.4.4.m4.1.1.1.1.1.2">𝑣</ci><ci id="S4.Thmtheorem3.p3.4.4.m4.1.1.1.1.1.3.cmml" xref="S4.Thmtheorem3.p3.4.4.m4.1.1.1.1.1.3">𝑢</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem3.p3.4.4.m4.1c">\textsl{g}(v\!\to\!u)</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem3.p3.4.4.m4.1d">g ( italic_v → italic_u )</annotation></semantics></math> to obtain another viable solution to the program. Consider decreasing <math alttext="\textsl{g}(u\!\to\!v)" class="ltx_Math" display="inline" id="S4.Thmtheorem3.p3.5.5.m5.1"><semantics id="S4.Thmtheorem3.p3.5.5.m5.1a"><mrow id="S4.Thmtheorem3.p3.5.5.m5.1.1" xref="S4.Thmtheorem3.p3.5.5.m5.1.1.cmml"><mtext class="ltx_mathvariant_italic" id="S4.Thmtheorem3.p3.5.5.m5.1.1.3" xref="S4.Thmtheorem3.p3.5.5.m5.1.1.3a.cmml">g</mtext><mo id="S4.Thmtheorem3.p3.5.5.m5.1.1.2" xref="S4.Thmtheorem3.p3.5.5.m5.1.1.2.cmml">⁢</mo><mrow id="S4.Thmtheorem3.p3.5.5.m5.1.1.1.1" xref="S4.Thmtheorem3.p3.5.5.m5.1.1.1.1.1.cmml"><mo id="S4.Thmtheorem3.p3.5.5.m5.1.1.1.1.2" stretchy="false" xref="S4.Thmtheorem3.p3.5.5.m5.1.1.1.1.1.cmml">(</mo><mrow id="S4.Thmtheorem3.p3.5.5.m5.1.1.1.1.1" xref="S4.Thmtheorem3.p3.5.5.m5.1.1.1.1.1.cmml"><mi id="S4.Thmtheorem3.p3.5.5.m5.1.1.1.1.1.2" xref="S4.Thmtheorem3.p3.5.5.m5.1.1.1.1.1.2.cmml">u</mi><mo id="S4.Thmtheorem3.p3.5.5.m5.1.1.1.1.1.1" lspace="0.108em" rspace="0.108em" stretchy="false" xref="S4.Thmtheorem3.p3.5.5.m5.1.1.1.1.1.1.cmml">→</mo><mi id="S4.Thmtheorem3.p3.5.5.m5.1.1.1.1.1.3" xref="S4.Thmtheorem3.p3.5.5.m5.1.1.1.1.1.3.cmml">v</mi></mrow><mo id="S4.Thmtheorem3.p3.5.5.m5.1.1.1.1.3" stretchy="false" xref="S4.Thmtheorem3.p3.5.5.m5.1.1.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem3.p3.5.5.m5.1b"><apply id="S4.Thmtheorem3.p3.5.5.m5.1.1.cmml" xref="S4.Thmtheorem3.p3.5.5.m5.1.1"><times id="S4.Thmtheorem3.p3.5.5.m5.1.1.2.cmml" xref="S4.Thmtheorem3.p3.5.5.m5.1.1.2"></times><ci id="S4.Thmtheorem3.p3.5.5.m5.1.1.3a.cmml" xref="S4.Thmtheorem3.p3.5.5.m5.1.1.3"><mtext class="ltx_mathvariant_italic" id="S4.Thmtheorem3.p3.5.5.m5.1.1.3.cmml" xref="S4.Thmtheorem3.p3.5.5.m5.1.1.3">g</mtext></ci><apply id="S4.Thmtheorem3.p3.5.5.m5.1.1.1.1.1.cmml" xref="S4.Thmtheorem3.p3.5.5.m5.1.1.1.1"><ci id="S4.Thmtheorem3.p3.5.5.m5.1.1.1.1.1.1.cmml" xref="S4.Thmtheorem3.p3.5.5.m5.1.1.1.1.1.1">→</ci><ci id="S4.Thmtheorem3.p3.5.5.m5.1.1.1.1.1.2.cmml" xref="S4.Thmtheorem3.p3.5.5.m5.1.1.1.1.1.2">𝑢</ci><ci id="S4.Thmtheorem3.p3.5.5.m5.1.1.1.1.1.3.cmml" xref="S4.Thmtheorem3.p3.5.5.m5.1.1.1.1.1.3">𝑣</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem3.p3.5.5.m5.1c">\textsl{g}(u\!\to\!v)</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem3.p3.5.5.m5.1d">g ( italic_u → italic_v )</annotation></semantics></math>, then we may decrease <math alttext="\textsl{g}(u)" class="ltx_Math" display="inline" id="S4.Thmtheorem3.p3.6.6.m6.1"><semantics id="S4.Thmtheorem3.p3.6.6.m6.1a"><mrow id="S4.Thmtheorem3.p3.6.6.m6.1.2" xref="S4.Thmtheorem3.p3.6.6.m6.1.2.cmml"><mtext class="ltx_mathvariant_italic" id="S4.Thmtheorem3.p3.6.6.m6.1.2.2" xref="S4.Thmtheorem3.p3.6.6.m6.1.2.2a.cmml">g</mtext><mo id="S4.Thmtheorem3.p3.6.6.m6.1.2.1" xref="S4.Thmtheorem3.p3.6.6.m6.1.2.1.cmml">⁢</mo><mrow id="S4.Thmtheorem3.p3.6.6.m6.1.2.3.2" xref="S4.Thmtheorem3.p3.6.6.m6.1.2.cmml"><mo id="S4.Thmtheorem3.p3.6.6.m6.1.2.3.2.1" stretchy="false" xref="S4.Thmtheorem3.p3.6.6.m6.1.2.cmml">(</mo><mi id="S4.Thmtheorem3.p3.6.6.m6.1.1" xref="S4.Thmtheorem3.p3.6.6.m6.1.1.cmml">u</mi><mo id="S4.Thmtheorem3.p3.6.6.m6.1.2.3.2.2" stretchy="false" xref="S4.Thmtheorem3.p3.6.6.m6.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem3.p3.6.6.m6.1b"><apply id="S4.Thmtheorem3.p3.6.6.m6.1.2.cmml" xref="S4.Thmtheorem3.p3.6.6.m6.1.2"><times id="S4.Thmtheorem3.p3.6.6.m6.1.2.1.cmml" xref="S4.Thmtheorem3.p3.6.6.m6.1.2.1"></times><ci id="S4.Thmtheorem3.p3.6.6.m6.1.2.2a.cmml" xref="S4.Thmtheorem3.p3.6.6.m6.1.2.2"><mtext class="ltx_mathvariant_italic" id="S4.Thmtheorem3.p3.6.6.m6.1.2.2.cmml" xref="S4.Thmtheorem3.p3.6.6.m6.1.2.2">g</mtext></ci><ci id="S4.Thmtheorem3.p3.6.6.m6.1.1.cmml" xref="S4.Thmtheorem3.p3.6.6.m6.1.1">𝑢</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem3.p3.6.6.m6.1c">\textsl{g}(u)</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem3.p3.6.6.m6.1d">g ( italic_u )</annotation></semantics></math> and maintain a viable and better solution to the program – a contradiction.</span></p> </div> <div class="ltx_para" id="S4.Thmtheorem3.p4"> <p class="ltx_p" id="S4.Thmtheorem3.p4.8"><span class="ltx_text ltx_font_italic" id="S4.Thmtheorem3.p4.8.8">Secondly, we claim that the optimal solution to the quadratic program is a locally fair orientation. Suppose for the sake of contradiction that there exist <math alttext="u,v" class="ltx_Math" display="inline" id="S4.Thmtheorem3.p4.1.1.m1.2"><semantics id="S4.Thmtheorem3.p4.1.1.m1.2a"><mrow id="S4.Thmtheorem3.p4.1.1.m1.2.3.2" xref="S4.Thmtheorem3.p4.1.1.m1.2.3.1.cmml"><mi id="S4.Thmtheorem3.p4.1.1.m1.1.1" xref="S4.Thmtheorem3.p4.1.1.m1.1.1.cmml">u</mi><mo id="S4.Thmtheorem3.p4.1.1.m1.2.3.2.1" xref="S4.Thmtheorem3.p4.1.1.m1.2.3.1.cmml">,</mo><mi id="S4.Thmtheorem3.p4.1.1.m1.2.2" xref="S4.Thmtheorem3.p4.1.1.m1.2.2.cmml">v</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem3.p4.1.1.m1.2b"><list id="S4.Thmtheorem3.p4.1.1.m1.2.3.1.cmml" xref="S4.Thmtheorem3.p4.1.1.m1.2.3.2"><ci id="S4.Thmtheorem3.p4.1.1.m1.1.1.cmml" xref="S4.Thmtheorem3.p4.1.1.m1.1.1">𝑢</ci><ci id="S4.Thmtheorem3.p4.1.1.m1.2.2.cmml" xref="S4.Thmtheorem3.p4.1.1.m1.2.2">𝑣</ci></list></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem3.p4.1.1.m1.2c">u,v</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem3.p4.1.1.m1.2d">italic_u , italic_v</annotation></semantics></math> with <math alttext="\textsl{g}(u)=\textsl{g}(v)+\delta^{\prime}" class="ltx_Math" display="inline" id="S4.Thmtheorem3.p4.2.2.m2.2"><semantics id="S4.Thmtheorem3.p4.2.2.m2.2a"><mrow id="S4.Thmtheorem3.p4.2.2.m2.2.3" xref="S4.Thmtheorem3.p4.2.2.m2.2.3.cmml"><mrow id="S4.Thmtheorem3.p4.2.2.m2.2.3.2" xref="S4.Thmtheorem3.p4.2.2.m2.2.3.2.cmml"><mtext class="ltx_mathvariant_italic" id="S4.Thmtheorem3.p4.2.2.m2.2.3.2.2" xref="S4.Thmtheorem3.p4.2.2.m2.2.3.2.2a.cmml">g</mtext><mo id="S4.Thmtheorem3.p4.2.2.m2.2.3.2.1" xref="S4.Thmtheorem3.p4.2.2.m2.2.3.2.1.cmml">⁢</mo><mrow id="S4.Thmtheorem3.p4.2.2.m2.2.3.2.3.2" xref="S4.Thmtheorem3.p4.2.2.m2.2.3.2.cmml"><mo id="S4.Thmtheorem3.p4.2.2.m2.2.3.2.3.2.1" stretchy="false" xref="S4.Thmtheorem3.p4.2.2.m2.2.3.2.cmml">(</mo><mi id="S4.Thmtheorem3.p4.2.2.m2.1.1" xref="S4.Thmtheorem3.p4.2.2.m2.1.1.cmml">u</mi><mo id="S4.Thmtheorem3.p4.2.2.m2.2.3.2.3.2.2" stretchy="false" xref="S4.Thmtheorem3.p4.2.2.m2.2.3.2.cmml">)</mo></mrow></mrow><mo id="S4.Thmtheorem3.p4.2.2.m2.2.3.1" xref="S4.Thmtheorem3.p4.2.2.m2.2.3.1.cmml">=</mo><mrow id="S4.Thmtheorem3.p4.2.2.m2.2.3.3" xref="S4.Thmtheorem3.p4.2.2.m2.2.3.3.cmml"><mrow id="S4.Thmtheorem3.p4.2.2.m2.2.3.3.2" xref="S4.Thmtheorem3.p4.2.2.m2.2.3.3.2.cmml"><mtext class="ltx_mathvariant_italic" id="S4.Thmtheorem3.p4.2.2.m2.2.3.3.2.2" xref="S4.Thmtheorem3.p4.2.2.m2.2.3.3.2.2a.cmml">g</mtext><mo id="S4.Thmtheorem3.p4.2.2.m2.2.3.3.2.1" xref="S4.Thmtheorem3.p4.2.2.m2.2.3.3.2.1.cmml">⁢</mo><mrow id="S4.Thmtheorem3.p4.2.2.m2.2.3.3.2.3.2" xref="S4.Thmtheorem3.p4.2.2.m2.2.3.3.2.cmml"><mo id="S4.Thmtheorem3.p4.2.2.m2.2.3.3.2.3.2.1" stretchy="false" xref="S4.Thmtheorem3.p4.2.2.m2.2.3.3.2.cmml">(</mo><mi id="S4.Thmtheorem3.p4.2.2.m2.2.2" xref="S4.Thmtheorem3.p4.2.2.m2.2.2.cmml">v</mi><mo id="S4.Thmtheorem3.p4.2.2.m2.2.3.3.2.3.2.2" stretchy="false" xref="S4.Thmtheorem3.p4.2.2.m2.2.3.3.2.cmml">)</mo></mrow></mrow><mo id="S4.Thmtheorem3.p4.2.2.m2.2.3.3.1" xref="S4.Thmtheorem3.p4.2.2.m2.2.3.3.1.cmml">+</mo><msup id="S4.Thmtheorem3.p4.2.2.m2.2.3.3.3" xref="S4.Thmtheorem3.p4.2.2.m2.2.3.3.3.cmml"><mi id="S4.Thmtheorem3.p4.2.2.m2.2.3.3.3.2" xref="S4.Thmtheorem3.p4.2.2.m2.2.3.3.3.2.cmml">δ</mi><mo id="S4.Thmtheorem3.p4.2.2.m2.2.3.3.3.3" xref="S4.Thmtheorem3.p4.2.2.m2.2.3.3.3.3.cmml">′</mo></msup></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem3.p4.2.2.m2.2b"><apply id="S4.Thmtheorem3.p4.2.2.m2.2.3.cmml" xref="S4.Thmtheorem3.p4.2.2.m2.2.3"><eq id="S4.Thmtheorem3.p4.2.2.m2.2.3.1.cmml" xref="S4.Thmtheorem3.p4.2.2.m2.2.3.1"></eq><apply id="S4.Thmtheorem3.p4.2.2.m2.2.3.2.cmml" xref="S4.Thmtheorem3.p4.2.2.m2.2.3.2"><times id="S4.Thmtheorem3.p4.2.2.m2.2.3.2.1.cmml" xref="S4.Thmtheorem3.p4.2.2.m2.2.3.2.1"></times><ci id="S4.Thmtheorem3.p4.2.2.m2.2.3.2.2a.cmml" xref="S4.Thmtheorem3.p4.2.2.m2.2.3.2.2"><mtext class="ltx_mathvariant_italic" id="S4.Thmtheorem3.p4.2.2.m2.2.3.2.2.cmml" xref="S4.Thmtheorem3.p4.2.2.m2.2.3.2.2">g</mtext></ci><ci id="S4.Thmtheorem3.p4.2.2.m2.1.1.cmml" xref="S4.Thmtheorem3.p4.2.2.m2.1.1">𝑢</ci></apply><apply id="S4.Thmtheorem3.p4.2.2.m2.2.3.3.cmml" xref="S4.Thmtheorem3.p4.2.2.m2.2.3.3"><plus id="S4.Thmtheorem3.p4.2.2.m2.2.3.3.1.cmml" xref="S4.Thmtheorem3.p4.2.2.m2.2.3.3.1"></plus><apply id="S4.Thmtheorem3.p4.2.2.m2.2.3.3.2.cmml" xref="S4.Thmtheorem3.p4.2.2.m2.2.3.3.2"><times id="S4.Thmtheorem3.p4.2.2.m2.2.3.3.2.1.cmml" xref="S4.Thmtheorem3.p4.2.2.m2.2.3.3.2.1"></times><ci id="S4.Thmtheorem3.p4.2.2.m2.2.3.3.2.2a.cmml" xref="S4.Thmtheorem3.p4.2.2.m2.2.3.3.2.2"><mtext class="ltx_mathvariant_italic" id="S4.Thmtheorem3.p4.2.2.m2.2.3.3.2.2.cmml" xref="S4.Thmtheorem3.p4.2.2.m2.2.3.3.2.2">g</mtext></ci><ci id="S4.Thmtheorem3.p4.2.2.m2.2.2.cmml" xref="S4.Thmtheorem3.p4.2.2.m2.2.2">𝑣</ci></apply><apply id="S4.Thmtheorem3.p4.2.2.m2.2.3.3.3.cmml" xref="S4.Thmtheorem3.p4.2.2.m2.2.3.3.3"><csymbol cd="ambiguous" id="S4.Thmtheorem3.p4.2.2.m2.2.3.3.3.1.cmml" xref="S4.Thmtheorem3.p4.2.2.m2.2.3.3.3">superscript</csymbol><ci id="S4.Thmtheorem3.p4.2.2.m2.2.3.3.3.2.cmml" xref="S4.Thmtheorem3.p4.2.2.m2.2.3.3.3.2">𝛿</ci><ci id="S4.Thmtheorem3.p4.2.2.m2.2.3.3.3.3.cmml" xref="S4.Thmtheorem3.p4.2.2.m2.2.3.3.3.3">′</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem3.p4.2.2.m2.2c">\textsl{g}(u)=\textsl{g}(v)+\delta^{\prime}</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem3.p4.2.2.m2.2d">g ( italic_u ) = g ( italic_v ) + italic_δ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math> and <math alttext="\textsl{g}(u\!\to\!v)=\delta" class="ltx_Math" display="inline" id="S4.Thmtheorem3.p4.3.3.m3.1"><semantics id="S4.Thmtheorem3.p4.3.3.m3.1a"><mrow id="S4.Thmtheorem3.p4.3.3.m3.1.1" xref="S4.Thmtheorem3.p4.3.3.m3.1.1.cmml"><mrow id="S4.Thmtheorem3.p4.3.3.m3.1.1.1" xref="S4.Thmtheorem3.p4.3.3.m3.1.1.1.cmml"><mtext class="ltx_mathvariant_italic" id="S4.Thmtheorem3.p4.3.3.m3.1.1.1.3" xref="S4.Thmtheorem3.p4.3.3.m3.1.1.1.3a.cmml">g</mtext><mo id="S4.Thmtheorem3.p4.3.3.m3.1.1.1.2" xref="S4.Thmtheorem3.p4.3.3.m3.1.1.1.2.cmml">⁢</mo><mrow id="S4.Thmtheorem3.p4.3.3.m3.1.1.1.1.1" xref="S4.Thmtheorem3.p4.3.3.m3.1.1.1.1.1.1.cmml"><mo id="S4.Thmtheorem3.p4.3.3.m3.1.1.1.1.1.2" stretchy="false" xref="S4.Thmtheorem3.p4.3.3.m3.1.1.1.1.1.1.cmml">(</mo><mrow id="S4.Thmtheorem3.p4.3.3.m3.1.1.1.1.1.1" xref="S4.Thmtheorem3.p4.3.3.m3.1.1.1.1.1.1.cmml"><mi id="S4.Thmtheorem3.p4.3.3.m3.1.1.1.1.1.1.2" xref="S4.Thmtheorem3.p4.3.3.m3.1.1.1.1.1.1.2.cmml">u</mi><mo id="S4.Thmtheorem3.p4.3.3.m3.1.1.1.1.1.1.1" lspace="0.108em" rspace="0.108em" stretchy="false" xref="S4.Thmtheorem3.p4.3.3.m3.1.1.1.1.1.1.1.cmml">→</mo><mi id="S4.Thmtheorem3.p4.3.3.m3.1.1.1.1.1.1.3" xref="S4.Thmtheorem3.p4.3.3.m3.1.1.1.1.1.1.3.cmml">v</mi></mrow><mo id="S4.Thmtheorem3.p4.3.3.m3.1.1.1.1.1.3" stretchy="false" xref="S4.Thmtheorem3.p4.3.3.m3.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S4.Thmtheorem3.p4.3.3.m3.1.1.2" xref="S4.Thmtheorem3.p4.3.3.m3.1.1.2.cmml">=</mo><mi id="S4.Thmtheorem3.p4.3.3.m3.1.1.3" xref="S4.Thmtheorem3.p4.3.3.m3.1.1.3.cmml">δ</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem3.p4.3.3.m3.1b"><apply id="S4.Thmtheorem3.p4.3.3.m3.1.1.cmml" xref="S4.Thmtheorem3.p4.3.3.m3.1.1"><eq id="S4.Thmtheorem3.p4.3.3.m3.1.1.2.cmml" xref="S4.Thmtheorem3.p4.3.3.m3.1.1.2"></eq><apply id="S4.Thmtheorem3.p4.3.3.m3.1.1.1.cmml" xref="S4.Thmtheorem3.p4.3.3.m3.1.1.1"><times id="S4.Thmtheorem3.p4.3.3.m3.1.1.1.2.cmml" xref="S4.Thmtheorem3.p4.3.3.m3.1.1.1.2"></times><ci id="S4.Thmtheorem3.p4.3.3.m3.1.1.1.3a.cmml" xref="S4.Thmtheorem3.p4.3.3.m3.1.1.1.3"><mtext class="ltx_mathvariant_italic" id="S4.Thmtheorem3.p4.3.3.m3.1.1.1.3.cmml" xref="S4.Thmtheorem3.p4.3.3.m3.1.1.1.3">g</mtext></ci><apply id="S4.Thmtheorem3.p4.3.3.m3.1.1.1.1.1.1.cmml" xref="S4.Thmtheorem3.p4.3.3.m3.1.1.1.1.1"><ci id="S4.Thmtheorem3.p4.3.3.m3.1.1.1.1.1.1.1.cmml" xref="S4.Thmtheorem3.p4.3.3.m3.1.1.1.1.1.1.1">→</ci><ci id="S4.Thmtheorem3.p4.3.3.m3.1.1.1.1.1.1.2.cmml" xref="S4.Thmtheorem3.p4.3.3.m3.1.1.1.1.1.1.2">𝑢</ci><ci id="S4.Thmtheorem3.p4.3.3.m3.1.1.1.1.1.1.3.cmml" xref="S4.Thmtheorem3.p4.3.3.m3.1.1.1.1.1.1.3">𝑣</ci></apply></apply><ci id="S4.Thmtheorem3.p4.3.3.m3.1.1.3.cmml" xref="S4.Thmtheorem3.p4.3.3.m3.1.1.3">𝛿</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem3.p4.3.3.m3.1c">\textsl{g}(u\!\to\!v)=\delta</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem3.p4.3.3.m3.1d">g ( italic_u → italic_v ) = italic_δ</annotation></semantics></math> for <math alttext="\delta,\delta^{\prime}>0" class="ltx_Math" display="inline" id="S4.Thmtheorem3.p4.4.4.m4.2"><semantics id="S4.Thmtheorem3.p4.4.4.m4.2a"><mrow id="S4.Thmtheorem3.p4.4.4.m4.2.2" xref="S4.Thmtheorem3.p4.4.4.m4.2.2.cmml"><mrow id="S4.Thmtheorem3.p4.4.4.m4.2.2.1.1" xref="S4.Thmtheorem3.p4.4.4.m4.2.2.1.2.cmml"><mi id="S4.Thmtheorem3.p4.4.4.m4.1.1" xref="S4.Thmtheorem3.p4.4.4.m4.1.1.cmml">δ</mi><mo id="S4.Thmtheorem3.p4.4.4.m4.2.2.1.1.2" xref="S4.Thmtheorem3.p4.4.4.m4.2.2.1.2.cmml">,</mo><msup id="S4.Thmtheorem3.p4.4.4.m4.2.2.1.1.1" xref="S4.Thmtheorem3.p4.4.4.m4.2.2.1.1.1.cmml"><mi id="S4.Thmtheorem3.p4.4.4.m4.2.2.1.1.1.2" xref="S4.Thmtheorem3.p4.4.4.m4.2.2.1.1.1.2.cmml">δ</mi><mo id="S4.Thmtheorem3.p4.4.4.m4.2.2.1.1.1.3" xref="S4.Thmtheorem3.p4.4.4.m4.2.2.1.1.1.3.cmml">′</mo></msup></mrow><mo id="S4.Thmtheorem3.p4.4.4.m4.2.2.2" xref="S4.Thmtheorem3.p4.4.4.m4.2.2.2.cmml">></mo><mn id="S4.Thmtheorem3.p4.4.4.m4.2.2.3" xref="S4.Thmtheorem3.p4.4.4.m4.2.2.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem3.p4.4.4.m4.2b"><apply id="S4.Thmtheorem3.p4.4.4.m4.2.2.cmml" xref="S4.Thmtheorem3.p4.4.4.m4.2.2"><gt id="S4.Thmtheorem3.p4.4.4.m4.2.2.2.cmml" xref="S4.Thmtheorem3.p4.4.4.m4.2.2.2"></gt><list id="S4.Thmtheorem3.p4.4.4.m4.2.2.1.2.cmml" xref="S4.Thmtheorem3.p4.4.4.m4.2.2.1.1"><ci id="S4.Thmtheorem3.p4.4.4.m4.1.1.cmml" xref="S4.Thmtheorem3.p4.4.4.m4.1.1">𝛿</ci><apply id="S4.Thmtheorem3.p4.4.4.m4.2.2.1.1.1.cmml" xref="S4.Thmtheorem3.p4.4.4.m4.2.2.1.1.1"><csymbol cd="ambiguous" id="S4.Thmtheorem3.p4.4.4.m4.2.2.1.1.1.1.cmml" xref="S4.Thmtheorem3.p4.4.4.m4.2.2.1.1.1">superscript</csymbol><ci id="S4.Thmtheorem3.p4.4.4.m4.2.2.1.1.1.2.cmml" xref="S4.Thmtheorem3.p4.4.4.m4.2.2.1.1.1.2">𝛿</ci><ci id="S4.Thmtheorem3.p4.4.4.m4.2.2.1.1.1.3.cmml" xref="S4.Thmtheorem3.p4.4.4.m4.2.2.1.1.1.3">′</ci></apply></list><cn id="S4.Thmtheorem3.p4.4.4.m4.2.2.3.cmml" type="integer" xref="S4.Thmtheorem3.p4.4.4.m4.2.2.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem3.p4.4.4.m4.2c">\delta,\delta^{\prime}>0</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem3.p4.4.4.m4.2d">italic_δ , italic_δ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT > 0</annotation></semantics></math>. We can decrease <math alttext="\textsl{g}(u\!\to\!v)" class="ltx_Math" display="inline" id="S4.Thmtheorem3.p4.5.5.m5.1"><semantics id="S4.Thmtheorem3.p4.5.5.m5.1a"><mrow id="S4.Thmtheorem3.p4.5.5.m5.1.1" xref="S4.Thmtheorem3.p4.5.5.m5.1.1.cmml"><mtext class="ltx_mathvariant_italic" id="S4.Thmtheorem3.p4.5.5.m5.1.1.3" xref="S4.Thmtheorem3.p4.5.5.m5.1.1.3a.cmml">g</mtext><mo id="S4.Thmtheorem3.p4.5.5.m5.1.1.2" xref="S4.Thmtheorem3.p4.5.5.m5.1.1.2.cmml">⁢</mo><mrow id="S4.Thmtheorem3.p4.5.5.m5.1.1.1.1" xref="S4.Thmtheorem3.p4.5.5.m5.1.1.1.1.1.cmml"><mo id="S4.Thmtheorem3.p4.5.5.m5.1.1.1.1.2" stretchy="false" xref="S4.Thmtheorem3.p4.5.5.m5.1.1.1.1.1.cmml">(</mo><mrow id="S4.Thmtheorem3.p4.5.5.m5.1.1.1.1.1" xref="S4.Thmtheorem3.p4.5.5.m5.1.1.1.1.1.cmml"><mi id="S4.Thmtheorem3.p4.5.5.m5.1.1.1.1.1.2" xref="S4.Thmtheorem3.p4.5.5.m5.1.1.1.1.1.2.cmml">u</mi><mo id="S4.Thmtheorem3.p4.5.5.m5.1.1.1.1.1.1" lspace="0.108em" rspace="0.108em" stretchy="false" xref="S4.Thmtheorem3.p4.5.5.m5.1.1.1.1.1.1.cmml">→</mo><mi id="S4.Thmtheorem3.p4.5.5.m5.1.1.1.1.1.3" xref="S4.Thmtheorem3.p4.5.5.m5.1.1.1.1.1.3.cmml">v</mi></mrow><mo id="S4.Thmtheorem3.p4.5.5.m5.1.1.1.1.3" stretchy="false" xref="S4.Thmtheorem3.p4.5.5.m5.1.1.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem3.p4.5.5.m5.1b"><apply id="S4.Thmtheorem3.p4.5.5.m5.1.1.cmml" xref="S4.Thmtheorem3.p4.5.5.m5.1.1"><times id="S4.Thmtheorem3.p4.5.5.m5.1.1.2.cmml" xref="S4.Thmtheorem3.p4.5.5.m5.1.1.2"></times><ci id="S4.Thmtheorem3.p4.5.5.m5.1.1.3a.cmml" xref="S4.Thmtheorem3.p4.5.5.m5.1.1.3"><mtext class="ltx_mathvariant_italic" id="S4.Thmtheorem3.p4.5.5.m5.1.1.3.cmml" xref="S4.Thmtheorem3.p4.5.5.m5.1.1.3">g</mtext></ci><apply id="S4.Thmtheorem3.p4.5.5.m5.1.1.1.1.1.cmml" xref="S4.Thmtheorem3.p4.5.5.m5.1.1.1.1"><ci id="S4.Thmtheorem3.p4.5.5.m5.1.1.1.1.1.1.cmml" xref="S4.Thmtheorem3.p4.5.5.m5.1.1.1.1.1.1">→</ci><ci id="S4.Thmtheorem3.p4.5.5.m5.1.1.1.1.1.2.cmml" xref="S4.Thmtheorem3.p4.5.5.m5.1.1.1.1.1.2">𝑢</ci><ci id="S4.Thmtheorem3.p4.5.5.m5.1.1.1.1.1.3.cmml" xref="S4.Thmtheorem3.p4.5.5.m5.1.1.1.1.1.3">𝑣</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem3.p4.5.5.m5.1c">\textsl{g}(u\!\to\!v)</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem3.p4.5.5.m5.1d">g ( italic_u → italic_v )</annotation></semantics></math> to zero by increasing <math alttext="\textsl{g}(v\!\to\!u)" class="ltx_Math" display="inline" id="S4.Thmtheorem3.p4.6.6.m6.1"><semantics id="S4.Thmtheorem3.p4.6.6.m6.1a"><mrow id="S4.Thmtheorem3.p4.6.6.m6.1.1" xref="S4.Thmtheorem3.p4.6.6.m6.1.1.cmml"><mtext class="ltx_mathvariant_italic" id="S4.Thmtheorem3.p4.6.6.m6.1.1.3" xref="S4.Thmtheorem3.p4.6.6.m6.1.1.3a.cmml">g</mtext><mo id="S4.Thmtheorem3.p4.6.6.m6.1.1.2" xref="S4.Thmtheorem3.p4.6.6.m6.1.1.2.cmml">⁢</mo><mrow id="S4.Thmtheorem3.p4.6.6.m6.1.1.1.1" xref="S4.Thmtheorem3.p4.6.6.m6.1.1.1.1.1.cmml"><mo id="S4.Thmtheorem3.p4.6.6.m6.1.1.1.1.2" stretchy="false" xref="S4.Thmtheorem3.p4.6.6.m6.1.1.1.1.1.cmml">(</mo><mrow id="S4.Thmtheorem3.p4.6.6.m6.1.1.1.1.1" xref="S4.Thmtheorem3.p4.6.6.m6.1.1.1.1.1.cmml"><mi id="S4.Thmtheorem3.p4.6.6.m6.1.1.1.1.1.2" xref="S4.Thmtheorem3.p4.6.6.m6.1.1.1.1.1.2.cmml">v</mi><mo id="S4.Thmtheorem3.p4.6.6.m6.1.1.1.1.1.1" lspace="0.108em" rspace="0.108em" stretchy="false" xref="S4.Thmtheorem3.p4.6.6.m6.1.1.1.1.1.1.cmml">→</mo><mi id="S4.Thmtheorem3.p4.6.6.m6.1.1.1.1.1.3" xref="S4.Thmtheorem3.p4.6.6.m6.1.1.1.1.1.3.cmml">u</mi></mrow><mo id="S4.Thmtheorem3.p4.6.6.m6.1.1.1.1.3" stretchy="false" xref="S4.Thmtheorem3.p4.6.6.m6.1.1.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem3.p4.6.6.m6.1b"><apply id="S4.Thmtheorem3.p4.6.6.m6.1.1.cmml" xref="S4.Thmtheorem3.p4.6.6.m6.1.1"><times id="S4.Thmtheorem3.p4.6.6.m6.1.1.2.cmml" xref="S4.Thmtheorem3.p4.6.6.m6.1.1.2"></times><ci id="S4.Thmtheorem3.p4.6.6.m6.1.1.3a.cmml" xref="S4.Thmtheorem3.p4.6.6.m6.1.1.3"><mtext class="ltx_mathvariant_italic" id="S4.Thmtheorem3.p4.6.6.m6.1.1.3.cmml" xref="S4.Thmtheorem3.p4.6.6.m6.1.1.3">g</mtext></ci><apply id="S4.Thmtheorem3.p4.6.6.m6.1.1.1.1.1.cmml" xref="S4.Thmtheorem3.p4.6.6.m6.1.1.1.1"><ci id="S4.Thmtheorem3.p4.6.6.m6.1.1.1.1.1.1.cmml" xref="S4.Thmtheorem3.p4.6.6.m6.1.1.1.1.1.1">→</ci><ci id="S4.Thmtheorem3.p4.6.6.m6.1.1.1.1.1.2.cmml" xref="S4.Thmtheorem3.p4.6.6.m6.1.1.1.1.1.2">𝑣</ci><ci id="S4.Thmtheorem3.p4.6.6.m6.1.1.1.1.1.3.cmml" xref="S4.Thmtheorem3.p4.6.6.m6.1.1.1.1.1.3">𝑢</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem3.p4.6.6.m6.1c">\textsl{g}(v\!\to\!u)</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem3.p4.6.6.m6.1d">g ( italic_v → italic_u )</annotation></semantics></math> by <math alttext="\Delta=\min\{\delta,\delta^{\prime}\}" class="ltx_Math" display="inline" id="S4.Thmtheorem3.p4.7.7.m7.3"><semantics id="S4.Thmtheorem3.p4.7.7.m7.3a"><mrow id="S4.Thmtheorem3.p4.7.7.m7.3.3" xref="S4.Thmtheorem3.p4.7.7.m7.3.3.cmml"><mi id="S4.Thmtheorem3.p4.7.7.m7.3.3.3" mathvariant="normal" xref="S4.Thmtheorem3.p4.7.7.m7.3.3.3.cmml">Δ</mi><mo id="S4.Thmtheorem3.p4.7.7.m7.3.3.2" xref="S4.Thmtheorem3.p4.7.7.m7.3.3.2.cmml">=</mo><mrow id="S4.Thmtheorem3.p4.7.7.m7.3.3.1.1" xref="S4.Thmtheorem3.p4.7.7.m7.3.3.1.2.cmml"><mi id="S4.Thmtheorem3.p4.7.7.m7.1.1" xref="S4.Thmtheorem3.p4.7.7.m7.1.1.cmml">min</mi><mo id="S4.Thmtheorem3.p4.7.7.m7.3.3.1.1a" xref="S4.Thmtheorem3.p4.7.7.m7.3.3.1.2.cmml">⁡</mo><mrow id="S4.Thmtheorem3.p4.7.7.m7.3.3.1.1.1" xref="S4.Thmtheorem3.p4.7.7.m7.3.3.1.2.cmml"><mo id="S4.Thmtheorem3.p4.7.7.m7.3.3.1.1.1.2" stretchy="false" xref="S4.Thmtheorem3.p4.7.7.m7.3.3.1.2.cmml">{</mo><mi id="S4.Thmtheorem3.p4.7.7.m7.2.2" xref="S4.Thmtheorem3.p4.7.7.m7.2.2.cmml">δ</mi><mo id="S4.Thmtheorem3.p4.7.7.m7.3.3.1.1.1.3" xref="S4.Thmtheorem3.p4.7.7.m7.3.3.1.2.cmml">,</mo><msup id="S4.Thmtheorem3.p4.7.7.m7.3.3.1.1.1.1" xref="S4.Thmtheorem3.p4.7.7.m7.3.3.1.1.1.1.cmml"><mi id="S4.Thmtheorem3.p4.7.7.m7.3.3.1.1.1.1.2" xref="S4.Thmtheorem3.p4.7.7.m7.3.3.1.1.1.1.2.cmml">δ</mi><mo id="S4.Thmtheorem3.p4.7.7.m7.3.3.1.1.1.1.3" xref="S4.Thmtheorem3.p4.7.7.m7.3.3.1.1.1.1.3.cmml">′</mo></msup><mo id="S4.Thmtheorem3.p4.7.7.m7.3.3.1.1.1.4" stretchy="false" xref="S4.Thmtheorem3.p4.7.7.m7.3.3.1.2.cmml">}</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem3.p4.7.7.m7.3b"><apply id="S4.Thmtheorem3.p4.7.7.m7.3.3.cmml" xref="S4.Thmtheorem3.p4.7.7.m7.3.3"><eq id="S4.Thmtheorem3.p4.7.7.m7.3.3.2.cmml" xref="S4.Thmtheorem3.p4.7.7.m7.3.3.2"></eq><ci id="S4.Thmtheorem3.p4.7.7.m7.3.3.3.cmml" xref="S4.Thmtheorem3.p4.7.7.m7.3.3.3">Δ</ci><apply id="S4.Thmtheorem3.p4.7.7.m7.3.3.1.2.cmml" xref="S4.Thmtheorem3.p4.7.7.m7.3.3.1.1"><min id="S4.Thmtheorem3.p4.7.7.m7.1.1.cmml" xref="S4.Thmtheorem3.p4.7.7.m7.1.1"></min><ci id="S4.Thmtheorem3.p4.7.7.m7.2.2.cmml" xref="S4.Thmtheorem3.p4.7.7.m7.2.2">𝛿</ci><apply id="S4.Thmtheorem3.p4.7.7.m7.3.3.1.1.1.1.cmml" xref="S4.Thmtheorem3.p4.7.7.m7.3.3.1.1.1.1"><csymbol cd="ambiguous" id="S4.Thmtheorem3.p4.7.7.m7.3.3.1.1.1.1.1.cmml" xref="S4.Thmtheorem3.p4.7.7.m7.3.3.1.1.1.1">superscript</csymbol><ci id="S4.Thmtheorem3.p4.7.7.m7.3.3.1.1.1.1.2.cmml" xref="S4.Thmtheorem3.p4.7.7.m7.3.3.1.1.1.1.2">𝛿</ci><ci id="S4.Thmtheorem3.p4.7.7.m7.3.3.1.1.1.1.3.cmml" xref="S4.Thmtheorem3.p4.7.7.m7.3.3.1.1.1.1.3">′</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem3.p4.7.7.m7.3c">\Delta=\min\{\delta,\delta^{\prime}\}</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem3.p4.7.7.m7.3d">roman_Δ = roman_min { italic_δ , italic_δ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT }</annotation></semantics></math> and still maintain a solution to the program. This reduces the solution’s value by <math alttext="(\textsl{g}(u)-\Delta)^{2}-(\textsl{g}(v)+\Delta)^{2}" class="ltx_Math" display="inline" id="S4.Thmtheorem3.p4.8.8.m8.4"><semantics id="S4.Thmtheorem3.p4.8.8.m8.4a"><mrow id="S4.Thmtheorem3.p4.8.8.m8.4.4" xref="S4.Thmtheorem3.p4.8.8.m8.4.4.cmml"><msup id="S4.Thmtheorem3.p4.8.8.m8.3.3.1" xref="S4.Thmtheorem3.p4.8.8.m8.3.3.1.cmml"><mrow id="S4.Thmtheorem3.p4.8.8.m8.3.3.1.1.1" xref="S4.Thmtheorem3.p4.8.8.m8.3.3.1.1.1.1.cmml"><mo id="S4.Thmtheorem3.p4.8.8.m8.3.3.1.1.1.2" stretchy="false" xref="S4.Thmtheorem3.p4.8.8.m8.3.3.1.1.1.1.cmml">(</mo><mrow id="S4.Thmtheorem3.p4.8.8.m8.3.3.1.1.1.1" xref="S4.Thmtheorem3.p4.8.8.m8.3.3.1.1.1.1.cmml"><mrow id="S4.Thmtheorem3.p4.8.8.m8.3.3.1.1.1.1.2" xref="S4.Thmtheorem3.p4.8.8.m8.3.3.1.1.1.1.2.cmml"><mtext class="ltx_mathvariant_italic" id="S4.Thmtheorem3.p4.8.8.m8.3.3.1.1.1.1.2.2" xref="S4.Thmtheorem3.p4.8.8.m8.3.3.1.1.1.1.2.2a.cmml">g</mtext><mo id="S4.Thmtheorem3.p4.8.8.m8.3.3.1.1.1.1.2.1" xref="S4.Thmtheorem3.p4.8.8.m8.3.3.1.1.1.1.2.1.cmml">⁢</mo><mrow id="S4.Thmtheorem3.p4.8.8.m8.3.3.1.1.1.1.2.3.2" xref="S4.Thmtheorem3.p4.8.8.m8.3.3.1.1.1.1.2.cmml"><mo id="S4.Thmtheorem3.p4.8.8.m8.3.3.1.1.1.1.2.3.2.1" stretchy="false" xref="S4.Thmtheorem3.p4.8.8.m8.3.3.1.1.1.1.2.cmml">(</mo><mi id="S4.Thmtheorem3.p4.8.8.m8.1.1" xref="S4.Thmtheorem3.p4.8.8.m8.1.1.cmml">u</mi><mo id="S4.Thmtheorem3.p4.8.8.m8.3.3.1.1.1.1.2.3.2.2" stretchy="false" xref="S4.Thmtheorem3.p4.8.8.m8.3.3.1.1.1.1.2.cmml">)</mo></mrow></mrow><mo id="S4.Thmtheorem3.p4.8.8.m8.3.3.1.1.1.1.1" xref="S4.Thmtheorem3.p4.8.8.m8.3.3.1.1.1.1.1.cmml">−</mo><mi id="S4.Thmtheorem3.p4.8.8.m8.3.3.1.1.1.1.3" mathvariant="normal" xref="S4.Thmtheorem3.p4.8.8.m8.3.3.1.1.1.1.3.cmml">Δ</mi></mrow><mo id="S4.Thmtheorem3.p4.8.8.m8.3.3.1.1.1.3" stretchy="false" xref="S4.Thmtheorem3.p4.8.8.m8.3.3.1.1.1.1.cmml">)</mo></mrow><mn id="S4.Thmtheorem3.p4.8.8.m8.3.3.1.3" xref="S4.Thmtheorem3.p4.8.8.m8.3.3.1.3.cmml">2</mn></msup><mo id="S4.Thmtheorem3.p4.8.8.m8.4.4.3" xref="S4.Thmtheorem3.p4.8.8.m8.4.4.3.cmml">−</mo><msup id="S4.Thmtheorem3.p4.8.8.m8.4.4.2" xref="S4.Thmtheorem3.p4.8.8.m8.4.4.2.cmml"><mrow id="S4.Thmtheorem3.p4.8.8.m8.4.4.2.1.1" xref="S4.Thmtheorem3.p4.8.8.m8.4.4.2.1.1.1.cmml"><mo id="S4.Thmtheorem3.p4.8.8.m8.4.4.2.1.1.2" stretchy="false" xref="S4.Thmtheorem3.p4.8.8.m8.4.4.2.1.1.1.cmml">(</mo><mrow id="S4.Thmtheorem3.p4.8.8.m8.4.4.2.1.1.1" xref="S4.Thmtheorem3.p4.8.8.m8.4.4.2.1.1.1.cmml"><mrow id="S4.Thmtheorem3.p4.8.8.m8.4.4.2.1.1.1.2" xref="S4.Thmtheorem3.p4.8.8.m8.4.4.2.1.1.1.2.cmml"><mtext class="ltx_mathvariant_italic" id="S4.Thmtheorem3.p4.8.8.m8.4.4.2.1.1.1.2.2" xref="S4.Thmtheorem3.p4.8.8.m8.4.4.2.1.1.1.2.2a.cmml">g</mtext><mo id="S4.Thmtheorem3.p4.8.8.m8.4.4.2.1.1.1.2.1" xref="S4.Thmtheorem3.p4.8.8.m8.4.4.2.1.1.1.2.1.cmml">⁢</mo><mrow id="S4.Thmtheorem3.p4.8.8.m8.4.4.2.1.1.1.2.3.2" xref="S4.Thmtheorem3.p4.8.8.m8.4.4.2.1.1.1.2.cmml"><mo id="S4.Thmtheorem3.p4.8.8.m8.4.4.2.1.1.1.2.3.2.1" stretchy="false" xref="S4.Thmtheorem3.p4.8.8.m8.4.4.2.1.1.1.2.cmml">(</mo><mi id="S4.Thmtheorem3.p4.8.8.m8.2.2" xref="S4.Thmtheorem3.p4.8.8.m8.2.2.cmml">v</mi><mo id="S4.Thmtheorem3.p4.8.8.m8.4.4.2.1.1.1.2.3.2.2" stretchy="false" xref="S4.Thmtheorem3.p4.8.8.m8.4.4.2.1.1.1.2.cmml">)</mo></mrow></mrow><mo id="S4.Thmtheorem3.p4.8.8.m8.4.4.2.1.1.1.1" xref="S4.Thmtheorem3.p4.8.8.m8.4.4.2.1.1.1.1.cmml">+</mo><mi id="S4.Thmtheorem3.p4.8.8.m8.4.4.2.1.1.1.3" mathvariant="normal" xref="S4.Thmtheorem3.p4.8.8.m8.4.4.2.1.1.1.3.cmml">Δ</mi></mrow><mo id="S4.Thmtheorem3.p4.8.8.m8.4.4.2.1.1.3" stretchy="false" xref="S4.Thmtheorem3.p4.8.8.m8.4.4.2.1.1.1.cmml">)</mo></mrow><mn id="S4.Thmtheorem3.p4.8.8.m8.4.4.2.3" xref="S4.Thmtheorem3.p4.8.8.m8.4.4.2.3.cmml">2</mn></msup></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem3.p4.8.8.m8.4b"><apply id="S4.Thmtheorem3.p4.8.8.m8.4.4.cmml" xref="S4.Thmtheorem3.p4.8.8.m8.4.4"><minus id="S4.Thmtheorem3.p4.8.8.m8.4.4.3.cmml" xref="S4.Thmtheorem3.p4.8.8.m8.4.4.3"></minus><apply id="S4.Thmtheorem3.p4.8.8.m8.3.3.1.cmml" xref="S4.Thmtheorem3.p4.8.8.m8.3.3.1"><csymbol cd="ambiguous" id="S4.Thmtheorem3.p4.8.8.m8.3.3.1.2.cmml" xref="S4.Thmtheorem3.p4.8.8.m8.3.3.1">superscript</csymbol><apply id="S4.Thmtheorem3.p4.8.8.m8.3.3.1.1.1.1.cmml" xref="S4.Thmtheorem3.p4.8.8.m8.3.3.1.1.1"><minus id="S4.Thmtheorem3.p4.8.8.m8.3.3.1.1.1.1.1.cmml" xref="S4.Thmtheorem3.p4.8.8.m8.3.3.1.1.1.1.1"></minus><apply id="S4.Thmtheorem3.p4.8.8.m8.3.3.1.1.1.1.2.cmml" xref="S4.Thmtheorem3.p4.8.8.m8.3.3.1.1.1.1.2"><times id="S4.Thmtheorem3.p4.8.8.m8.3.3.1.1.1.1.2.1.cmml" xref="S4.Thmtheorem3.p4.8.8.m8.3.3.1.1.1.1.2.1"></times><ci id="S4.Thmtheorem3.p4.8.8.m8.3.3.1.1.1.1.2.2a.cmml" xref="S4.Thmtheorem3.p4.8.8.m8.3.3.1.1.1.1.2.2"><mtext class="ltx_mathvariant_italic" id="S4.Thmtheorem3.p4.8.8.m8.3.3.1.1.1.1.2.2.cmml" xref="S4.Thmtheorem3.p4.8.8.m8.3.3.1.1.1.1.2.2">g</mtext></ci><ci id="S4.Thmtheorem3.p4.8.8.m8.1.1.cmml" xref="S4.Thmtheorem3.p4.8.8.m8.1.1">𝑢</ci></apply><ci id="S4.Thmtheorem3.p4.8.8.m8.3.3.1.1.1.1.3.cmml" xref="S4.Thmtheorem3.p4.8.8.m8.3.3.1.1.1.1.3">Δ</ci></apply><cn id="S4.Thmtheorem3.p4.8.8.m8.3.3.1.3.cmml" type="integer" xref="S4.Thmtheorem3.p4.8.8.m8.3.3.1.3">2</cn></apply><apply id="S4.Thmtheorem3.p4.8.8.m8.4.4.2.cmml" xref="S4.Thmtheorem3.p4.8.8.m8.4.4.2"><csymbol cd="ambiguous" id="S4.Thmtheorem3.p4.8.8.m8.4.4.2.2.cmml" xref="S4.Thmtheorem3.p4.8.8.m8.4.4.2">superscript</csymbol><apply id="S4.Thmtheorem3.p4.8.8.m8.4.4.2.1.1.1.cmml" xref="S4.Thmtheorem3.p4.8.8.m8.4.4.2.1.1"><plus id="S4.Thmtheorem3.p4.8.8.m8.4.4.2.1.1.1.1.cmml" xref="S4.Thmtheorem3.p4.8.8.m8.4.4.2.1.1.1.1"></plus><apply id="S4.Thmtheorem3.p4.8.8.m8.4.4.2.1.1.1.2.cmml" xref="S4.Thmtheorem3.p4.8.8.m8.4.4.2.1.1.1.2"><times id="S4.Thmtheorem3.p4.8.8.m8.4.4.2.1.1.1.2.1.cmml" xref="S4.Thmtheorem3.p4.8.8.m8.4.4.2.1.1.1.2.1"></times><ci id="S4.Thmtheorem3.p4.8.8.m8.4.4.2.1.1.1.2.2a.cmml" xref="S4.Thmtheorem3.p4.8.8.m8.4.4.2.1.1.1.2.2"><mtext class="ltx_mathvariant_italic" id="S4.Thmtheorem3.p4.8.8.m8.4.4.2.1.1.1.2.2.cmml" xref="S4.Thmtheorem3.p4.8.8.m8.4.4.2.1.1.1.2.2">g</mtext></ci><ci id="S4.Thmtheorem3.p4.8.8.m8.2.2.cmml" xref="S4.Thmtheorem3.p4.8.8.m8.2.2">𝑣</ci></apply><ci id="S4.Thmtheorem3.p4.8.8.m8.4.4.2.1.1.1.3.cmml" xref="S4.Thmtheorem3.p4.8.8.m8.4.4.2.1.1.1.3">Δ</ci></apply><cn id="S4.Thmtheorem3.p4.8.8.m8.4.4.2.3.cmml" type="integer" xref="S4.Thmtheorem3.p4.8.8.m8.4.4.2.3">2</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem3.p4.8.8.m8.4c">(\textsl{g}(u)-\Delta)^{2}-(\textsl{g}(v)+\Delta)^{2}</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem3.p4.8.8.m8.4d">( g ( italic_u ) - roman_Δ ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT - ( g ( italic_v ) + roman_Δ ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT</annotation></semantics></math>. However, we now found a solution to the quadratic program with a lower value than the optimal solution – a contradiction.</span></p> </div> <div class="ltx_para" id="S4.Thmtheorem3.p5"> <p class="ltx_p" id="S4.Thmtheorem3.p5.4"><span class="ltx_text ltx_font_italic" id="S4.Thmtheorem3.p5.4.4">Thus, the solution to <math alttext="\textbf{FO}^{2}" class="ltx_Math" display="inline" id="S4.Thmtheorem3.p5.1.1.m1.1"><semantics id="S4.Thmtheorem3.p5.1.1.m1.1a"><msup id="S4.Thmtheorem3.p5.1.1.m1.1.1" xref="S4.Thmtheorem3.p5.1.1.m1.1.1.cmml"><mtext class="ltx_mathvariant_bold" id="S4.Thmtheorem3.p5.1.1.m1.1.1.2" xref="S4.Thmtheorem3.p5.1.1.m1.1.1.2a.cmml">FO</mtext><mn id="S4.Thmtheorem3.p5.1.1.m1.1.1.3" xref="S4.Thmtheorem3.p5.1.1.m1.1.1.3.cmml">2</mn></msup><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem3.p5.1.1.m1.1b"><apply id="S4.Thmtheorem3.p5.1.1.m1.1.1.cmml" xref="S4.Thmtheorem3.p5.1.1.m1.1.1"><csymbol cd="ambiguous" id="S4.Thmtheorem3.p5.1.1.m1.1.1.1.cmml" xref="S4.Thmtheorem3.p5.1.1.m1.1.1">superscript</csymbol><ci id="S4.Thmtheorem3.p5.1.1.m1.1.1.2a.cmml" xref="S4.Thmtheorem3.p5.1.1.m1.1.1.2"><mtext class="ltx_mathvariant_bold" id="S4.Thmtheorem3.p5.1.1.m1.1.1.2.cmml" xref="S4.Thmtheorem3.p5.1.1.m1.1.1.2">FO</mtext></ci><cn id="S4.Thmtheorem3.p5.1.1.m1.1.1.3.cmml" type="integer" xref="S4.Thmtheorem3.p5.1.1.m1.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem3.p5.1.1.m1.1c">\textbf{FO}^{2}</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem3.p5.1.1.m1.1d">FO start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT</annotation></semantics></math> gives a locally fair orientation <math alttext="\overrightarrow{G}" class="ltx_Math" display="inline" id="S4.Thmtheorem3.p5.2.2.m2.1"><semantics id="S4.Thmtheorem3.p5.2.2.m2.1a"><mover accent="true" id="S4.Thmtheorem3.p5.2.2.m2.1.1" xref="S4.Thmtheorem3.p5.2.2.m2.1.1.cmml"><mi id="S4.Thmtheorem3.p5.2.2.m2.1.1.2" xref="S4.Thmtheorem3.p5.2.2.m2.1.1.2.cmml">G</mi><mo id="S4.Thmtheorem3.p5.2.2.m2.1.1.1" stretchy="false" xref="S4.Thmtheorem3.p5.2.2.m2.1.1.1.cmml">→</mo></mover><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem3.p5.2.2.m2.1b"><apply id="S4.Thmtheorem3.p5.2.2.m2.1.1.cmml" xref="S4.Thmtheorem3.p5.2.2.m2.1.1"><ci id="S4.Thmtheorem3.p5.2.2.m2.1.1.1.cmml" xref="S4.Thmtheorem3.p5.2.2.m2.1.1.1">→</ci><ci id="S4.Thmtheorem3.p5.2.2.m2.1.1.2.cmml" xref="S4.Thmtheorem3.p5.2.2.m2.1.1.2">𝐺</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem3.p5.2.2.m2.1c">\overrightarrow{G}</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem3.p5.2.2.m2.1d">over→ start_ARG italic_G end_ARG</annotation></semantics></math> where each vertex <math alttext="u" class="ltx_Math" display="inline" id="S4.Thmtheorem3.p5.3.3.m3.1"><semantics id="S4.Thmtheorem3.p5.3.3.m3.1a"><mi id="S4.Thmtheorem3.p5.3.3.m3.1.1" xref="S4.Thmtheorem3.p5.3.3.m3.1.1.cmml">u</mi><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem3.p5.3.3.m3.1b"><ci id="S4.Thmtheorem3.p5.3.3.m3.1.1.cmml" xref="S4.Thmtheorem3.p5.3.3.m3.1.1">𝑢</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem3.p5.3.3.m3.1c">u</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem3.p5.3.3.m3.1d">italic_u</annotation></semantics></math> has out-degree <math alttext="\textsl{g}(u)" class="ltx_Math" display="inline" id="S4.Thmtheorem3.p5.4.4.m4.1"><semantics id="S4.Thmtheorem3.p5.4.4.m4.1a"><mrow id="S4.Thmtheorem3.p5.4.4.m4.1.2" xref="S4.Thmtheorem3.p5.4.4.m4.1.2.cmml"><mtext class="ltx_mathvariant_italic" id="S4.Thmtheorem3.p5.4.4.m4.1.2.2" xref="S4.Thmtheorem3.p5.4.4.m4.1.2.2a.cmml">g</mtext><mo id="S4.Thmtheorem3.p5.4.4.m4.1.2.1" xref="S4.Thmtheorem3.p5.4.4.m4.1.2.1.cmml">⁢</mo><mrow id="S4.Thmtheorem3.p5.4.4.m4.1.2.3.2" xref="S4.Thmtheorem3.p5.4.4.m4.1.2.cmml"><mo id="S4.Thmtheorem3.p5.4.4.m4.1.2.3.2.1" stretchy="false" xref="S4.Thmtheorem3.p5.4.4.m4.1.2.cmml">(</mo><mi id="S4.Thmtheorem3.p5.4.4.m4.1.1" xref="S4.Thmtheorem3.p5.4.4.m4.1.1.cmml">u</mi><mo id="S4.Thmtheorem3.p5.4.4.m4.1.2.3.2.2" stretchy="false" xref="S4.Thmtheorem3.p5.4.4.m4.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem3.p5.4.4.m4.1b"><apply id="S4.Thmtheorem3.p5.4.4.m4.1.2.cmml" xref="S4.Thmtheorem3.p5.4.4.m4.1.2"><times id="S4.Thmtheorem3.p5.4.4.m4.1.2.1.cmml" xref="S4.Thmtheorem3.p5.4.4.m4.1.2.1"></times><ci id="S4.Thmtheorem3.p5.4.4.m4.1.2.2a.cmml" xref="S4.Thmtheorem3.p5.4.4.m4.1.2.2"><mtext class="ltx_mathvariant_italic" id="S4.Thmtheorem3.p5.4.4.m4.1.2.2.cmml" xref="S4.Thmtheorem3.p5.4.4.m4.1.2.2">g</mtext></ci><ci id="S4.Thmtheorem3.p5.4.4.m4.1.1.cmml" xref="S4.Thmtheorem3.p5.4.4.m4.1.1">𝑢</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem3.p5.4.4.m4.1c">\textsl{g}(u)</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem3.p5.4.4.m4.1d">g ( italic_u )</annotation></semantics></math>.</span></p> </div> <div class="ltx_para" id="S4.Thmtheorem3.p6"> <p class="ltx_p" id="S4.Thmtheorem3.p6.2"><span class="ltx_text ltx_font_italic" id="S4.Thmtheorem3.p6.2.2">The local density <math alttext="\textsl{g}^{*}(u)=\textsl{g}(u)" class="ltx_Math" display="inline" id="S4.Thmtheorem3.p6.1.1.m1.2"><semantics id="S4.Thmtheorem3.p6.1.1.m1.2a"><mrow id="S4.Thmtheorem3.p6.1.1.m1.2.3" xref="S4.Thmtheorem3.p6.1.1.m1.2.3.cmml"><mrow id="S4.Thmtheorem3.p6.1.1.m1.2.3.2" xref="S4.Thmtheorem3.p6.1.1.m1.2.3.2.cmml"><msup id="S4.Thmtheorem3.p6.1.1.m1.2.3.2.2" xref="S4.Thmtheorem3.p6.1.1.m1.2.3.2.2.cmml"><mtext class="ltx_mathvariant_italic" id="S4.Thmtheorem3.p6.1.1.m1.2.3.2.2.2" xref="S4.Thmtheorem3.p6.1.1.m1.2.3.2.2.2a.cmml">g</mtext><mo id="S4.Thmtheorem3.p6.1.1.m1.2.3.2.2.3" xref="S4.Thmtheorem3.p6.1.1.m1.2.3.2.2.3.cmml">∗</mo></msup><mo id="S4.Thmtheorem3.p6.1.1.m1.2.3.2.1" xref="S4.Thmtheorem3.p6.1.1.m1.2.3.2.1.cmml">⁢</mo><mrow id="S4.Thmtheorem3.p6.1.1.m1.2.3.2.3.2" xref="S4.Thmtheorem3.p6.1.1.m1.2.3.2.cmml"><mo id="S4.Thmtheorem3.p6.1.1.m1.2.3.2.3.2.1" stretchy="false" xref="S4.Thmtheorem3.p6.1.1.m1.2.3.2.cmml">(</mo><mi id="S4.Thmtheorem3.p6.1.1.m1.1.1" xref="S4.Thmtheorem3.p6.1.1.m1.1.1.cmml">u</mi><mo id="S4.Thmtheorem3.p6.1.1.m1.2.3.2.3.2.2" stretchy="false" xref="S4.Thmtheorem3.p6.1.1.m1.2.3.2.cmml">)</mo></mrow></mrow><mo id="S4.Thmtheorem3.p6.1.1.m1.2.3.1" xref="S4.Thmtheorem3.p6.1.1.m1.2.3.1.cmml">=</mo><mrow id="S4.Thmtheorem3.p6.1.1.m1.2.3.3" xref="S4.Thmtheorem3.p6.1.1.m1.2.3.3.cmml"><mtext class="ltx_mathvariant_italic" id="S4.Thmtheorem3.p6.1.1.m1.2.3.3.2" xref="S4.Thmtheorem3.p6.1.1.m1.2.3.3.2a.cmml">g</mtext><mo id="S4.Thmtheorem3.p6.1.1.m1.2.3.3.1" xref="S4.Thmtheorem3.p6.1.1.m1.2.3.3.1.cmml">⁢</mo><mrow id="S4.Thmtheorem3.p6.1.1.m1.2.3.3.3.2" xref="S4.Thmtheorem3.p6.1.1.m1.2.3.3.cmml"><mo id="S4.Thmtheorem3.p6.1.1.m1.2.3.3.3.2.1" stretchy="false" xref="S4.Thmtheorem3.p6.1.1.m1.2.3.3.cmml">(</mo><mi id="S4.Thmtheorem3.p6.1.1.m1.2.2" xref="S4.Thmtheorem3.p6.1.1.m1.2.2.cmml">u</mi><mo id="S4.Thmtheorem3.p6.1.1.m1.2.3.3.3.2.2" stretchy="false" xref="S4.Thmtheorem3.p6.1.1.m1.2.3.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem3.p6.1.1.m1.2b"><apply id="S4.Thmtheorem3.p6.1.1.m1.2.3.cmml" xref="S4.Thmtheorem3.p6.1.1.m1.2.3"><eq id="S4.Thmtheorem3.p6.1.1.m1.2.3.1.cmml" xref="S4.Thmtheorem3.p6.1.1.m1.2.3.1"></eq><apply id="S4.Thmtheorem3.p6.1.1.m1.2.3.2.cmml" xref="S4.Thmtheorem3.p6.1.1.m1.2.3.2"><times id="S4.Thmtheorem3.p6.1.1.m1.2.3.2.1.cmml" xref="S4.Thmtheorem3.p6.1.1.m1.2.3.2.1"></times><apply id="S4.Thmtheorem3.p6.1.1.m1.2.3.2.2.cmml" xref="S4.Thmtheorem3.p6.1.1.m1.2.3.2.2"><csymbol cd="ambiguous" id="S4.Thmtheorem3.p6.1.1.m1.2.3.2.2.1.cmml" xref="S4.Thmtheorem3.p6.1.1.m1.2.3.2.2">superscript</csymbol><ci id="S4.Thmtheorem3.p6.1.1.m1.2.3.2.2.2a.cmml" xref="S4.Thmtheorem3.p6.1.1.m1.2.3.2.2.2"><mtext class="ltx_mathvariant_italic" id="S4.Thmtheorem3.p6.1.1.m1.2.3.2.2.2.cmml" xref="S4.Thmtheorem3.p6.1.1.m1.2.3.2.2.2">g</mtext></ci><times id="S4.Thmtheorem3.p6.1.1.m1.2.3.2.2.3.cmml" xref="S4.Thmtheorem3.p6.1.1.m1.2.3.2.2.3"></times></apply><ci id="S4.Thmtheorem3.p6.1.1.m1.1.1.cmml" xref="S4.Thmtheorem3.p6.1.1.m1.1.1">𝑢</ci></apply><apply id="S4.Thmtheorem3.p6.1.1.m1.2.3.3.cmml" xref="S4.Thmtheorem3.p6.1.1.m1.2.3.3"><times id="S4.Thmtheorem3.p6.1.1.m1.2.3.3.1.cmml" xref="S4.Thmtheorem3.p6.1.1.m1.2.3.3.1"></times><ci id="S4.Thmtheorem3.p6.1.1.m1.2.3.3.2a.cmml" xref="S4.Thmtheorem3.p6.1.1.m1.2.3.3.2"><mtext class="ltx_mathvariant_italic" id="S4.Thmtheorem3.p6.1.1.m1.2.3.3.2.cmml" xref="S4.Thmtheorem3.p6.1.1.m1.2.3.3.2">g</mtext></ci><ci id="S4.Thmtheorem3.p6.1.1.m1.2.2.cmml" xref="S4.Thmtheorem3.p6.1.1.m1.2.2">𝑢</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem3.p6.1.1.m1.2c">\textsl{g}^{*}(u)=\textsl{g}(u)</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem3.p6.1.1.m1.2d">g start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ( italic_u ) = g ( italic_u )</annotation></semantics></math> is by Lemma <a class="ltx_ref" href="https://arxiv.org/html/2411.12694v2#S4.Thmtheorem1" title="Lemma 4.1. ‣ 4 Conceptual results for local density ‣ Local Density and its Distributed Approximation"><span class="ltx_text ltx_ref_tag">4.1</span></a> well-defined. Danisch, Chan and Sozio <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2411.12694v2#bib.bib10" title="">10</a>, Corollary 4.4]</cite> show that <math alttext="\rho^{*}(u)=\textsl{g}(u)" class="ltx_Math" display="inline" id="S4.Thmtheorem3.p6.2.2.m2.2"><semantics id="S4.Thmtheorem3.p6.2.2.m2.2a"><mrow id="S4.Thmtheorem3.p6.2.2.m2.2.3" xref="S4.Thmtheorem3.p6.2.2.m2.2.3.cmml"><mrow id="S4.Thmtheorem3.p6.2.2.m2.2.3.2" xref="S4.Thmtheorem3.p6.2.2.m2.2.3.2.cmml"><msup id="S4.Thmtheorem3.p6.2.2.m2.2.3.2.2" xref="S4.Thmtheorem3.p6.2.2.m2.2.3.2.2.cmml"><mi id="S4.Thmtheorem3.p6.2.2.m2.2.3.2.2.2" xref="S4.Thmtheorem3.p6.2.2.m2.2.3.2.2.2.cmml">ρ</mi><mo id="S4.Thmtheorem3.p6.2.2.m2.2.3.2.2.3" xref="S4.Thmtheorem3.p6.2.2.m2.2.3.2.2.3.cmml">∗</mo></msup><mo id="S4.Thmtheorem3.p6.2.2.m2.2.3.2.1" xref="S4.Thmtheorem3.p6.2.2.m2.2.3.2.1.cmml">⁢</mo><mrow id="S4.Thmtheorem3.p6.2.2.m2.2.3.2.3.2" xref="S4.Thmtheorem3.p6.2.2.m2.2.3.2.cmml"><mo id="S4.Thmtheorem3.p6.2.2.m2.2.3.2.3.2.1" stretchy="false" xref="S4.Thmtheorem3.p6.2.2.m2.2.3.2.cmml">(</mo><mi id="S4.Thmtheorem3.p6.2.2.m2.1.1" xref="S4.Thmtheorem3.p6.2.2.m2.1.1.cmml">u</mi><mo id="S4.Thmtheorem3.p6.2.2.m2.2.3.2.3.2.2" stretchy="false" xref="S4.Thmtheorem3.p6.2.2.m2.2.3.2.cmml">)</mo></mrow></mrow><mo id="S4.Thmtheorem3.p6.2.2.m2.2.3.1" xref="S4.Thmtheorem3.p6.2.2.m2.2.3.1.cmml">=</mo><mrow id="S4.Thmtheorem3.p6.2.2.m2.2.3.3" xref="S4.Thmtheorem3.p6.2.2.m2.2.3.3.cmml"><mtext class="ltx_mathvariant_italic" id="S4.Thmtheorem3.p6.2.2.m2.2.3.3.2" xref="S4.Thmtheorem3.p6.2.2.m2.2.3.3.2a.cmml">g</mtext><mo id="S4.Thmtheorem3.p6.2.2.m2.2.3.3.1" xref="S4.Thmtheorem3.p6.2.2.m2.2.3.3.1.cmml">⁢</mo><mrow id="S4.Thmtheorem3.p6.2.2.m2.2.3.3.3.2" xref="S4.Thmtheorem3.p6.2.2.m2.2.3.3.cmml"><mo id="S4.Thmtheorem3.p6.2.2.m2.2.3.3.3.2.1" stretchy="false" xref="S4.Thmtheorem3.p6.2.2.m2.2.3.3.cmml">(</mo><mi id="S4.Thmtheorem3.p6.2.2.m2.2.2" xref="S4.Thmtheorem3.p6.2.2.m2.2.2.cmml">u</mi><mo id="S4.Thmtheorem3.p6.2.2.m2.2.3.3.3.2.2" stretchy="false" xref="S4.Thmtheorem3.p6.2.2.m2.2.3.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem3.p6.2.2.m2.2b"><apply id="S4.Thmtheorem3.p6.2.2.m2.2.3.cmml" xref="S4.Thmtheorem3.p6.2.2.m2.2.3"><eq id="S4.Thmtheorem3.p6.2.2.m2.2.3.1.cmml" xref="S4.Thmtheorem3.p6.2.2.m2.2.3.1"></eq><apply id="S4.Thmtheorem3.p6.2.2.m2.2.3.2.cmml" xref="S4.Thmtheorem3.p6.2.2.m2.2.3.2"><times id="S4.Thmtheorem3.p6.2.2.m2.2.3.2.1.cmml" xref="S4.Thmtheorem3.p6.2.2.m2.2.3.2.1"></times><apply id="S4.Thmtheorem3.p6.2.2.m2.2.3.2.2.cmml" xref="S4.Thmtheorem3.p6.2.2.m2.2.3.2.2"><csymbol cd="ambiguous" id="S4.Thmtheorem3.p6.2.2.m2.2.3.2.2.1.cmml" xref="S4.Thmtheorem3.p6.2.2.m2.2.3.2.2">superscript</csymbol><ci id="S4.Thmtheorem3.p6.2.2.m2.2.3.2.2.2.cmml" xref="S4.Thmtheorem3.p6.2.2.m2.2.3.2.2.2">𝜌</ci><times id="S4.Thmtheorem3.p6.2.2.m2.2.3.2.2.3.cmml" xref="S4.Thmtheorem3.p6.2.2.m2.2.3.2.2.3"></times></apply><ci id="S4.Thmtheorem3.p6.2.2.m2.1.1.cmml" xref="S4.Thmtheorem3.p6.2.2.m2.1.1">𝑢</ci></apply><apply id="S4.Thmtheorem3.p6.2.2.m2.2.3.3.cmml" xref="S4.Thmtheorem3.p6.2.2.m2.2.3.3"><times id="S4.Thmtheorem3.p6.2.2.m2.2.3.3.1.cmml" xref="S4.Thmtheorem3.p6.2.2.m2.2.3.3.1"></times><ci id="S4.Thmtheorem3.p6.2.2.m2.2.3.3.2a.cmml" xref="S4.Thmtheorem3.p6.2.2.m2.2.3.3.2"><mtext class="ltx_mathvariant_italic" id="S4.Thmtheorem3.p6.2.2.m2.2.3.3.2.cmml" xref="S4.Thmtheorem3.p6.2.2.m2.2.3.3.2">g</mtext></ci><ci id="S4.Thmtheorem3.p6.2.2.m2.2.2.cmml" xref="S4.Thmtheorem3.p6.2.2.m2.2.2">𝑢</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem3.p6.2.2.m2.2c">\rho^{*}(u)=\textsl{g}(u)</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem3.p6.2.2.m2.2d">italic_ρ start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ( italic_u ) = g ( italic_u )</annotation></semantics></math>, which proves the theorem.</span></p> </div> </div> <div class="ltx_para ltx_noindent" id="S4.p5"> <p class="ltx_p" id="S4.p5.1">Since the local density equals the local out-degree, we conclude from <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2411.12694v2#bib.bib10" title="">10</a>]</cite> that:</p> </div> <div class="ltx_para" id="S4.p6"> <p class="ltx_p" id="S4.p6.1">See <a class="ltx_ref" href="https://arxiv.org/html/2411.12694v2#S3.Thmtheorem3" title="Corollary 3.3. ‣ 3.A Conceptual results for local density ‣ 3 Results and organisation ‣ Local Density and its Distributed Approximation"><span class="ltx_text ltx_ref_tag">3.3</span></a></p> </div> <div class="ltx_para" id="S4.p7"> <p class="ltx_p" id="S4.p7.1">Since a quadratic program over a convex domain always has a solution, we may also note the following interesting fact:</p> </div> <div class="ltx_para" id="S4.p8"> <p class="ltx_p" id="S4.p8.1">See <a class="ltx_ref" href="https://arxiv.org/html/2411.12694v2#S3.Thmtheorem4" title="Corollary 3.4. ‣ 3.A Conceptual results for local density ‣ 3 Results and organisation ‣ Local Density and its Distributed Approximation"><span class="ltx_text ltx_ref_tag">3.4</span></a></p> </div> </section> <section class="ltx_section" id="S5"> <h2 class="ltx_title ltx_title_section"> <span class="ltx_tag ltx_tag_section">5 </span>Results for dynamic algorithms</h2> <div class="ltx_para" id="S5.p1"> <p class="ltx_p" id="S5.p1.14">We use our definition of local out-degree to show that there already exist dynamic algorithms that approximate the local density of each vertex. Recall that an orientation <math alttext="\overrightarrow{G}" class="ltx_Math" display="inline" id="S5.p1.1.m1.1"><semantics id="S5.p1.1.m1.1a"><mover accent="true" id="S5.p1.1.m1.1.1" xref="S5.p1.1.m1.1.1.cmml"><mi id="S5.p1.1.m1.1.1.2" xref="S5.p1.1.m1.1.1.2.cmml">G</mi><mo id="S5.p1.1.m1.1.1.1" stretchy="false" xref="S5.p1.1.m1.1.1.1.cmml">→</mo></mover><annotation-xml encoding="MathML-Content" id="S5.p1.1.m1.1b"><apply id="S5.p1.1.m1.1.1.cmml" xref="S5.p1.1.m1.1.1"><ci id="S5.p1.1.m1.1.1.1.cmml" xref="S5.p1.1.m1.1.1.1">→</ci><ci id="S5.p1.1.m1.1.1.2.cmml" xref="S5.p1.1.m1.1.1.2">𝐺</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.p1.1.m1.1c">\overrightarrow{G}</annotation><annotation encoding="application/x-llamapun" id="S5.p1.1.m1.1d">over→ start_ARG italic_G end_ARG</annotation></semantics></math> is <math alttext="\eta" class="ltx_Math" display="inline" id="S5.p1.2.m2.1"><semantics id="S5.p1.2.m2.1a"><mi id="S5.p1.2.m2.1.1" xref="S5.p1.2.m2.1.1.cmml">η</mi><annotation-xml encoding="MathML-Content" id="S5.p1.2.m2.1b"><ci id="S5.p1.2.m2.1.1.cmml" xref="S5.p1.2.m2.1.1">𝜂</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.p1.2.m2.1c">\eta</annotation><annotation encoding="application/x-llamapun" id="S5.p1.2.m2.1d">italic_η</annotation></semantics></math>-fair whenever for all <math alttext="\overline{uv}\in E(\overrightarrow{G})" class="ltx_Math" display="inline" id="S5.p1.3.m3.1"><semantics id="S5.p1.3.m3.1a"><mrow id="S5.p1.3.m3.1.2" xref="S5.p1.3.m3.1.2.cmml"><mover accent="true" id="S5.p1.3.m3.1.2.2" xref="S5.p1.3.m3.1.2.2.cmml"><mrow id="S5.p1.3.m3.1.2.2.2" xref="S5.p1.3.m3.1.2.2.2.cmml"><mi id="S5.p1.3.m3.1.2.2.2.2" xref="S5.p1.3.m3.1.2.2.2.2.cmml">u</mi><mo id="S5.p1.3.m3.1.2.2.2.1" xref="S5.p1.3.m3.1.2.2.2.1.cmml">⁢</mo><mi id="S5.p1.3.m3.1.2.2.2.3" xref="S5.p1.3.m3.1.2.2.2.3.cmml">v</mi></mrow><mo id="S5.p1.3.m3.1.2.2.1" xref="S5.p1.3.m3.1.2.2.1.cmml">¯</mo></mover><mo id="S5.p1.3.m3.1.2.1" xref="S5.p1.3.m3.1.2.1.cmml">∈</mo><mrow id="S5.p1.3.m3.1.2.3" xref="S5.p1.3.m3.1.2.3.cmml"><mi id="S5.p1.3.m3.1.2.3.2" xref="S5.p1.3.m3.1.2.3.2.cmml">E</mi><mo id="S5.p1.3.m3.1.2.3.1" xref="S5.p1.3.m3.1.2.3.1.cmml">⁢</mo><mrow id="S5.p1.3.m3.1.2.3.3.2" xref="S5.p1.3.m3.1.1.cmml"><mo id="S5.p1.3.m3.1.2.3.3.2.1" stretchy="false" xref="S5.p1.3.m3.1.1.cmml">(</mo><mover accent="true" id="S5.p1.3.m3.1.1" xref="S5.p1.3.m3.1.1.cmml"><mi id="S5.p1.3.m3.1.1.2" xref="S5.p1.3.m3.1.1.2.cmml">G</mi><mo id="S5.p1.3.m3.1.1.1" stretchy="false" xref="S5.p1.3.m3.1.1.1.cmml">→</mo></mover><mo id="S5.p1.3.m3.1.2.3.3.2.2" stretchy="false" xref="S5.p1.3.m3.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.p1.3.m3.1b"><apply id="S5.p1.3.m3.1.2.cmml" xref="S5.p1.3.m3.1.2"><in id="S5.p1.3.m3.1.2.1.cmml" xref="S5.p1.3.m3.1.2.1"></in><apply id="S5.p1.3.m3.1.2.2.cmml" xref="S5.p1.3.m3.1.2.2"><ci id="S5.p1.3.m3.1.2.2.1.cmml" xref="S5.p1.3.m3.1.2.2.1">¯</ci><apply id="S5.p1.3.m3.1.2.2.2.cmml" xref="S5.p1.3.m3.1.2.2.2"><times id="S5.p1.3.m3.1.2.2.2.1.cmml" xref="S5.p1.3.m3.1.2.2.2.1"></times><ci id="S5.p1.3.m3.1.2.2.2.2.cmml" xref="S5.p1.3.m3.1.2.2.2.2">𝑢</ci><ci id="S5.p1.3.m3.1.2.2.2.3.cmml" xref="S5.p1.3.m3.1.2.2.2.3">𝑣</ci></apply></apply><apply id="S5.p1.3.m3.1.2.3.cmml" xref="S5.p1.3.m3.1.2.3"><times id="S5.p1.3.m3.1.2.3.1.cmml" xref="S5.p1.3.m3.1.2.3.1"></times><ci id="S5.p1.3.m3.1.2.3.2.cmml" xref="S5.p1.3.m3.1.2.3.2">𝐸</ci><apply id="S5.p1.3.m3.1.1.cmml" xref="S5.p1.3.m3.1.2.3.3.2"><ci id="S5.p1.3.m3.1.1.1.cmml" xref="S5.p1.3.m3.1.1.1">→</ci><ci id="S5.p1.3.m3.1.1.2.cmml" xref="S5.p1.3.m3.1.1.2">𝐺</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.p1.3.m3.1c">\overline{uv}\in E(\overrightarrow{G})</annotation><annotation encoding="application/x-llamapun" id="S5.p1.3.m3.1d">over¯ start_ARG italic_u italic_v end_ARG ∈ italic_E ( over→ start_ARG italic_G end_ARG )</annotation></semantics></math>, <math alttext="\textsl{g}(u\!\to\!v)>0" class="ltx_Math" display="inline" id="S5.p1.4.m4.1"><semantics id="S5.p1.4.m4.1a"><mrow id="S5.p1.4.m4.1.1" xref="S5.p1.4.m4.1.1.cmml"><mrow id="S5.p1.4.m4.1.1.1" xref="S5.p1.4.m4.1.1.1.cmml"><mtext class="ltx_mathvariant_italic" id="S5.p1.4.m4.1.1.1.3" xref="S5.p1.4.m4.1.1.1.3a.cmml">g</mtext><mo id="S5.p1.4.m4.1.1.1.2" xref="S5.p1.4.m4.1.1.1.2.cmml">⁢</mo><mrow id="S5.p1.4.m4.1.1.1.1.1" xref="S5.p1.4.m4.1.1.1.1.1.1.cmml"><mo id="S5.p1.4.m4.1.1.1.1.1.2" stretchy="false" xref="S5.p1.4.m4.1.1.1.1.1.1.cmml">(</mo><mrow id="S5.p1.4.m4.1.1.1.1.1.1" xref="S5.p1.4.m4.1.1.1.1.1.1.cmml"><mi id="S5.p1.4.m4.1.1.1.1.1.1.2" xref="S5.p1.4.m4.1.1.1.1.1.1.2.cmml">u</mi><mo id="S5.p1.4.m4.1.1.1.1.1.1.1" lspace="0.108em" rspace="0.108em" stretchy="false" xref="S5.p1.4.m4.1.1.1.1.1.1.1.cmml">→</mo><mi id="S5.p1.4.m4.1.1.1.1.1.1.3" xref="S5.p1.4.m4.1.1.1.1.1.1.3.cmml">v</mi></mrow><mo id="S5.p1.4.m4.1.1.1.1.1.3" stretchy="false" xref="S5.p1.4.m4.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S5.p1.4.m4.1.1.2" xref="S5.p1.4.m4.1.1.2.cmml">></mo><mn id="S5.p1.4.m4.1.1.3" xref="S5.p1.4.m4.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S5.p1.4.m4.1b"><apply id="S5.p1.4.m4.1.1.cmml" xref="S5.p1.4.m4.1.1"><gt id="S5.p1.4.m4.1.1.2.cmml" xref="S5.p1.4.m4.1.1.2"></gt><apply id="S5.p1.4.m4.1.1.1.cmml" xref="S5.p1.4.m4.1.1.1"><times id="S5.p1.4.m4.1.1.1.2.cmml" xref="S5.p1.4.m4.1.1.1.2"></times><ci id="S5.p1.4.m4.1.1.1.3a.cmml" xref="S5.p1.4.m4.1.1.1.3"><mtext class="ltx_mathvariant_italic" id="S5.p1.4.m4.1.1.1.3.cmml" xref="S5.p1.4.m4.1.1.1.3">g</mtext></ci><apply id="S5.p1.4.m4.1.1.1.1.1.1.cmml" xref="S5.p1.4.m4.1.1.1.1.1"><ci id="S5.p1.4.m4.1.1.1.1.1.1.1.cmml" xref="S5.p1.4.m4.1.1.1.1.1.1.1">→</ci><ci id="S5.p1.4.m4.1.1.1.1.1.1.2.cmml" xref="S5.p1.4.m4.1.1.1.1.1.1.2">𝑢</ci><ci id="S5.p1.4.m4.1.1.1.1.1.1.3.cmml" xref="S5.p1.4.m4.1.1.1.1.1.1.3">𝑣</ci></apply></apply><cn id="S5.p1.4.m4.1.1.3.cmml" type="integer" xref="S5.p1.4.m4.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.p1.4.m4.1c">\textsl{g}(u\!\to\!v)>0</annotation><annotation encoding="application/x-llamapun" id="S5.p1.4.m4.1d">g ( italic_u → italic_v ) > 0</annotation></semantics></math> implies that <math alttext="\textsl{g}(u)\leq(1+\eta)\textsl{g}(v)" class="ltx_Math" display="inline" id="S5.p1.5.m5.3"><semantics id="S5.p1.5.m5.3a"><mrow id="S5.p1.5.m5.3.3" xref="S5.p1.5.m5.3.3.cmml"><mrow id="S5.p1.5.m5.3.3.3" xref="S5.p1.5.m5.3.3.3.cmml"><mtext class="ltx_mathvariant_italic" id="S5.p1.5.m5.3.3.3.2" xref="S5.p1.5.m5.3.3.3.2a.cmml">g</mtext><mo id="S5.p1.5.m5.3.3.3.1" xref="S5.p1.5.m5.3.3.3.1.cmml">⁢</mo><mrow id="S5.p1.5.m5.3.3.3.3.2" xref="S5.p1.5.m5.3.3.3.cmml"><mo id="S5.p1.5.m5.3.3.3.3.2.1" stretchy="false" xref="S5.p1.5.m5.3.3.3.cmml">(</mo><mi id="S5.p1.5.m5.1.1" xref="S5.p1.5.m5.1.1.cmml">u</mi><mo id="S5.p1.5.m5.3.3.3.3.2.2" stretchy="false" xref="S5.p1.5.m5.3.3.3.cmml">)</mo></mrow></mrow><mo id="S5.p1.5.m5.3.3.2" xref="S5.p1.5.m5.3.3.2.cmml">≤</mo><mrow id="S5.p1.5.m5.3.3.1" xref="S5.p1.5.m5.3.3.1.cmml"><mrow id="S5.p1.5.m5.3.3.1.1.1" xref="S5.p1.5.m5.3.3.1.1.1.1.cmml"><mo id="S5.p1.5.m5.3.3.1.1.1.2" stretchy="false" xref="S5.p1.5.m5.3.3.1.1.1.1.cmml">(</mo><mrow id="S5.p1.5.m5.3.3.1.1.1.1" xref="S5.p1.5.m5.3.3.1.1.1.1.cmml"><mn id="S5.p1.5.m5.3.3.1.1.1.1.2" xref="S5.p1.5.m5.3.3.1.1.1.1.2.cmml">1</mn><mo id="S5.p1.5.m5.3.3.1.1.1.1.1" xref="S5.p1.5.m5.3.3.1.1.1.1.1.cmml">+</mo><mi id="S5.p1.5.m5.3.3.1.1.1.1.3" xref="S5.p1.5.m5.3.3.1.1.1.1.3.cmml">η</mi></mrow><mo id="S5.p1.5.m5.3.3.1.1.1.3" stretchy="false" xref="S5.p1.5.m5.3.3.1.1.1.1.cmml">)</mo></mrow><mo id="S5.p1.5.m5.3.3.1.2" xref="S5.p1.5.m5.3.3.1.2.cmml">⁢</mo><mtext class="ltx_mathvariant_italic" id="S5.p1.5.m5.3.3.1.3" xref="S5.p1.5.m5.3.3.1.3a.cmml">g</mtext><mo id="S5.p1.5.m5.3.3.1.2a" xref="S5.p1.5.m5.3.3.1.2.cmml">⁢</mo><mrow id="S5.p1.5.m5.3.3.1.4.2" xref="S5.p1.5.m5.3.3.1.cmml"><mo id="S5.p1.5.m5.3.3.1.4.2.1" stretchy="false" xref="S5.p1.5.m5.3.3.1.cmml">(</mo><mi id="S5.p1.5.m5.2.2" xref="S5.p1.5.m5.2.2.cmml">v</mi><mo id="S5.p1.5.m5.3.3.1.4.2.2" stretchy="false" xref="S5.p1.5.m5.3.3.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.p1.5.m5.3b"><apply id="S5.p1.5.m5.3.3.cmml" xref="S5.p1.5.m5.3.3"><leq id="S5.p1.5.m5.3.3.2.cmml" xref="S5.p1.5.m5.3.3.2"></leq><apply id="S5.p1.5.m5.3.3.3.cmml" xref="S5.p1.5.m5.3.3.3"><times id="S5.p1.5.m5.3.3.3.1.cmml" xref="S5.p1.5.m5.3.3.3.1"></times><ci id="S5.p1.5.m5.3.3.3.2a.cmml" xref="S5.p1.5.m5.3.3.3.2"><mtext class="ltx_mathvariant_italic" id="S5.p1.5.m5.3.3.3.2.cmml" xref="S5.p1.5.m5.3.3.3.2">g</mtext></ci><ci id="S5.p1.5.m5.1.1.cmml" xref="S5.p1.5.m5.1.1">𝑢</ci></apply><apply id="S5.p1.5.m5.3.3.1.cmml" xref="S5.p1.5.m5.3.3.1"><times id="S5.p1.5.m5.3.3.1.2.cmml" xref="S5.p1.5.m5.3.3.1.2"></times><apply id="S5.p1.5.m5.3.3.1.1.1.1.cmml" xref="S5.p1.5.m5.3.3.1.1.1"><plus id="S5.p1.5.m5.3.3.1.1.1.1.1.cmml" xref="S5.p1.5.m5.3.3.1.1.1.1.1"></plus><cn id="S5.p1.5.m5.3.3.1.1.1.1.2.cmml" type="integer" xref="S5.p1.5.m5.3.3.1.1.1.1.2">1</cn><ci id="S5.p1.5.m5.3.3.1.1.1.1.3.cmml" xref="S5.p1.5.m5.3.3.1.1.1.1.3">𝜂</ci></apply><ci id="S5.p1.5.m5.3.3.1.3a.cmml" xref="S5.p1.5.m5.3.3.1.3"><mtext class="ltx_mathvariant_italic" id="S5.p1.5.m5.3.3.1.3.cmml" xref="S5.p1.5.m5.3.3.1.3">g</mtext></ci><ci id="S5.p1.5.m5.2.2.cmml" xref="S5.p1.5.m5.2.2">𝑣</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.p1.5.m5.3c">\textsl{g}(u)\leq(1+\eta)\textsl{g}(v)</annotation><annotation encoding="application/x-llamapun" id="S5.p1.5.m5.3d">g ( italic_u ) ≤ ( 1 + italic_η ) g ( italic_v )</annotation></semantics></math>. We show that if we choose <math alttext="\eta\leq\frac{\varepsilon^{2}}{128\cdot\log n}" class="ltx_Math" display="inline" id="S5.p1.6.m6.1"><semantics id="S5.p1.6.m6.1a"><mrow id="S5.p1.6.m6.1.1" xref="S5.p1.6.m6.1.1.cmml"><mi id="S5.p1.6.m6.1.1.2" xref="S5.p1.6.m6.1.1.2.cmml">η</mi><mo id="S5.p1.6.m6.1.1.1" xref="S5.p1.6.m6.1.1.1.cmml">≤</mo><mfrac id="S5.p1.6.m6.1.1.3" xref="S5.p1.6.m6.1.1.3.cmml"><msup id="S5.p1.6.m6.1.1.3.2" xref="S5.p1.6.m6.1.1.3.2.cmml"><mi id="S5.p1.6.m6.1.1.3.2.2" xref="S5.p1.6.m6.1.1.3.2.2.cmml">ε</mi><mn id="S5.p1.6.m6.1.1.3.2.3" xref="S5.p1.6.m6.1.1.3.2.3.cmml">2</mn></msup><mrow id="S5.p1.6.m6.1.1.3.3" xref="S5.p1.6.m6.1.1.3.3.cmml"><mn id="S5.p1.6.m6.1.1.3.3.2" xref="S5.p1.6.m6.1.1.3.3.2.cmml">128</mn><mo id="S5.p1.6.m6.1.1.3.3.1" lspace="0.222em" rspace="0.222em" xref="S5.p1.6.m6.1.1.3.3.1.cmml">⋅</mo><mrow id="S5.p1.6.m6.1.1.3.3.3" xref="S5.p1.6.m6.1.1.3.3.3.cmml"><mi id="S5.p1.6.m6.1.1.3.3.3.1" xref="S5.p1.6.m6.1.1.3.3.3.1.cmml">log</mi><mo id="S5.p1.6.m6.1.1.3.3.3a" lspace="0.167em" xref="S5.p1.6.m6.1.1.3.3.3.cmml">⁡</mo><mi id="S5.p1.6.m6.1.1.3.3.3.2" xref="S5.p1.6.m6.1.1.3.3.3.2.cmml">n</mi></mrow></mrow></mfrac></mrow><annotation-xml encoding="MathML-Content" id="S5.p1.6.m6.1b"><apply id="S5.p1.6.m6.1.1.cmml" xref="S5.p1.6.m6.1.1"><leq id="S5.p1.6.m6.1.1.1.cmml" xref="S5.p1.6.m6.1.1.1"></leq><ci id="S5.p1.6.m6.1.1.2.cmml" xref="S5.p1.6.m6.1.1.2">𝜂</ci><apply id="S5.p1.6.m6.1.1.3.cmml" xref="S5.p1.6.m6.1.1.3"><divide id="S5.p1.6.m6.1.1.3.1.cmml" xref="S5.p1.6.m6.1.1.3"></divide><apply id="S5.p1.6.m6.1.1.3.2.cmml" xref="S5.p1.6.m6.1.1.3.2"><csymbol cd="ambiguous" id="S5.p1.6.m6.1.1.3.2.1.cmml" xref="S5.p1.6.m6.1.1.3.2">superscript</csymbol><ci id="S5.p1.6.m6.1.1.3.2.2.cmml" xref="S5.p1.6.m6.1.1.3.2.2">𝜀</ci><cn id="S5.p1.6.m6.1.1.3.2.3.cmml" type="integer" xref="S5.p1.6.m6.1.1.3.2.3">2</cn></apply><apply id="S5.p1.6.m6.1.1.3.3.cmml" xref="S5.p1.6.m6.1.1.3.3"><ci id="S5.p1.6.m6.1.1.3.3.1.cmml" xref="S5.p1.6.m6.1.1.3.3.1">⋅</ci><cn id="S5.p1.6.m6.1.1.3.3.2.cmml" type="integer" xref="S5.p1.6.m6.1.1.3.3.2">128</cn><apply id="S5.p1.6.m6.1.1.3.3.3.cmml" xref="S5.p1.6.m6.1.1.3.3.3"><log id="S5.p1.6.m6.1.1.3.3.3.1.cmml" xref="S5.p1.6.m6.1.1.3.3.3.1"></log><ci id="S5.p1.6.m6.1.1.3.3.3.2.cmml" xref="S5.p1.6.m6.1.1.3.3.3.2">𝑛</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.p1.6.m6.1c">\eta\leq\frac{\varepsilon^{2}}{128\cdot\log n}</annotation><annotation encoding="application/x-llamapun" id="S5.p1.6.m6.1d">italic_η ≤ divide start_ARG italic_ε start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG start_ARG 128 ⋅ roman_log italic_n end_ARG</annotation></semantics></math>, then for any <math alttext="\eta" class="ltx_Math" display="inline" id="S5.p1.7.m7.1"><semantics id="S5.p1.7.m7.1a"><mi id="S5.p1.7.m7.1.1" xref="S5.p1.7.m7.1.1.cmml">η</mi><annotation-xml encoding="MathML-Content" id="S5.p1.7.m7.1b"><ci id="S5.p1.7.m7.1.1.cmml" xref="S5.p1.7.m7.1.1">𝜂</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.p1.7.m7.1c">\eta</annotation><annotation encoding="application/x-llamapun" id="S5.p1.7.m7.1d">italic_η</annotation></semantics></math>-fair orientation, for all <math alttext="v" class="ltx_Math" display="inline" id="S5.p1.8.m8.1"><semantics id="S5.p1.8.m8.1a"><mi id="S5.p1.8.m8.1.1" xref="S5.p1.8.m8.1.1.cmml">v</mi><annotation-xml encoding="MathML-Content" id="S5.p1.8.m8.1b"><ci id="S5.p1.8.m8.1.1.cmml" xref="S5.p1.8.m8.1.1">𝑣</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.p1.8.m8.1c">v</annotation><annotation encoding="application/x-llamapun" id="S5.p1.8.m8.1d">italic_v</annotation></semantics></math>, the out-degree <math alttext="\textsl{g}(v)" class="ltx_Math" display="inline" id="S5.p1.9.m9.1"><semantics id="S5.p1.9.m9.1a"><mrow id="S5.p1.9.m9.1.2" xref="S5.p1.9.m9.1.2.cmml"><mtext class="ltx_mathvariant_italic" id="S5.p1.9.m9.1.2.2" xref="S5.p1.9.m9.1.2.2a.cmml">g</mtext><mo id="S5.p1.9.m9.1.2.1" xref="S5.p1.9.m9.1.2.1.cmml">⁢</mo><mrow id="S5.p1.9.m9.1.2.3.2" xref="S5.p1.9.m9.1.2.cmml"><mo id="S5.p1.9.m9.1.2.3.2.1" stretchy="false" xref="S5.p1.9.m9.1.2.cmml">(</mo><mi id="S5.p1.9.m9.1.1" xref="S5.p1.9.m9.1.1.cmml">v</mi><mo id="S5.p1.9.m9.1.2.3.2.2" stretchy="false" xref="S5.p1.9.m9.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.p1.9.m9.1b"><apply id="S5.p1.9.m9.1.2.cmml" xref="S5.p1.9.m9.1.2"><times id="S5.p1.9.m9.1.2.1.cmml" xref="S5.p1.9.m9.1.2.1"></times><ci id="S5.p1.9.m9.1.2.2a.cmml" xref="S5.p1.9.m9.1.2.2"><mtext class="ltx_mathvariant_italic" id="S5.p1.9.m9.1.2.2.cmml" xref="S5.p1.9.m9.1.2.2">g</mtext></ci><ci id="S5.p1.9.m9.1.1.cmml" xref="S5.p1.9.m9.1.1">𝑣</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.p1.9.m9.1c">\textsl{g}(v)</annotation><annotation encoding="application/x-llamapun" id="S5.p1.9.m9.1d">g ( italic_v )</annotation></semantics></math> is a <math alttext="(1+\varepsilon)" class="ltx_Math" display="inline" id="S5.p1.10.m10.1"><semantics id="S5.p1.10.m10.1a"><mrow id="S5.p1.10.m10.1.1.1" xref="S5.p1.10.m10.1.1.1.1.cmml"><mo id="S5.p1.10.m10.1.1.1.2" stretchy="false" xref="S5.p1.10.m10.1.1.1.1.cmml">(</mo><mrow id="S5.p1.10.m10.1.1.1.1" xref="S5.p1.10.m10.1.1.1.1.cmml"><mn id="S5.p1.10.m10.1.1.1.1.2" xref="S5.p1.10.m10.1.1.1.1.2.cmml">1</mn><mo id="S5.p1.10.m10.1.1.1.1.1" xref="S5.p1.10.m10.1.1.1.1.1.cmml">+</mo><mi id="S5.p1.10.m10.1.1.1.1.3" xref="S5.p1.10.m10.1.1.1.1.3.cmml">ε</mi></mrow><mo id="S5.p1.10.m10.1.1.1.3" stretchy="false" xref="S5.p1.10.m10.1.1.1.1.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S5.p1.10.m10.1b"><apply id="S5.p1.10.m10.1.1.1.1.cmml" xref="S5.p1.10.m10.1.1.1"><plus id="S5.p1.10.m10.1.1.1.1.1.cmml" xref="S5.p1.10.m10.1.1.1.1.1"></plus><cn id="S5.p1.10.m10.1.1.1.1.2.cmml" type="integer" xref="S5.p1.10.m10.1.1.1.1.2">1</cn><ci id="S5.p1.10.m10.1.1.1.1.3.cmml" xref="S5.p1.10.m10.1.1.1.1.3">𝜀</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.p1.10.m10.1c">(1+\varepsilon)</annotation><annotation encoding="application/x-llamapun" id="S5.p1.10.m10.1d">( 1 + italic_ε )</annotation></semantics></math> approximation of <math alttext="\textsl{g}^{*}(v)=\rho^{*}(v)" class="ltx_Math" display="inline" id="S5.p1.11.m11.2"><semantics id="S5.p1.11.m11.2a"><mrow id="S5.p1.11.m11.2.3" xref="S5.p1.11.m11.2.3.cmml"><mrow id="S5.p1.11.m11.2.3.2" xref="S5.p1.11.m11.2.3.2.cmml"><msup id="S5.p1.11.m11.2.3.2.2" xref="S5.p1.11.m11.2.3.2.2.cmml"><mtext class="ltx_mathvariant_italic" id="S5.p1.11.m11.2.3.2.2.2" xref="S5.p1.11.m11.2.3.2.2.2a.cmml">g</mtext><mo id="S5.p1.11.m11.2.3.2.2.3" xref="S5.p1.11.m11.2.3.2.2.3.cmml">∗</mo></msup><mo id="S5.p1.11.m11.2.3.2.1" xref="S5.p1.11.m11.2.3.2.1.cmml">⁢</mo><mrow id="S5.p1.11.m11.2.3.2.3.2" xref="S5.p1.11.m11.2.3.2.cmml"><mo id="S5.p1.11.m11.2.3.2.3.2.1" stretchy="false" xref="S5.p1.11.m11.2.3.2.cmml">(</mo><mi id="S5.p1.11.m11.1.1" xref="S5.p1.11.m11.1.1.cmml">v</mi><mo id="S5.p1.11.m11.2.3.2.3.2.2" stretchy="false" xref="S5.p1.11.m11.2.3.2.cmml">)</mo></mrow></mrow><mo id="S5.p1.11.m11.2.3.1" xref="S5.p1.11.m11.2.3.1.cmml">=</mo><mrow id="S5.p1.11.m11.2.3.3" xref="S5.p1.11.m11.2.3.3.cmml"><msup id="S5.p1.11.m11.2.3.3.2" xref="S5.p1.11.m11.2.3.3.2.cmml"><mi id="S5.p1.11.m11.2.3.3.2.2" xref="S5.p1.11.m11.2.3.3.2.2.cmml">ρ</mi><mo id="S5.p1.11.m11.2.3.3.2.3" xref="S5.p1.11.m11.2.3.3.2.3.cmml">∗</mo></msup><mo id="S5.p1.11.m11.2.3.3.1" xref="S5.p1.11.m11.2.3.3.1.cmml">⁢</mo><mrow id="S5.p1.11.m11.2.3.3.3.2" xref="S5.p1.11.m11.2.3.3.cmml"><mo id="S5.p1.11.m11.2.3.3.3.2.1" stretchy="false" xref="S5.p1.11.m11.2.3.3.cmml">(</mo><mi id="S5.p1.11.m11.2.2" xref="S5.p1.11.m11.2.2.cmml">v</mi><mo id="S5.p1.11.m11.2.3.3.3.2.2" stretchy="false" xref="S5.p1.11.m11.2.3.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.p1.11.m11.2b"><apply id="S5.p1.11.m11.2.3.cmml" xref="S5.p1.11.m11.2.3"><eq id="S5.p1.11.m11.2.3.1.cmml" xref="S5.p1.11.m11.2.3.1"></eq><apply id="S5.p1.11.m11.2.3.2.cmml" xref="S5.p1.11.m11.2.3.2"><times id="S5.p1.11.m11.2.3.2.1.cmml" xref="S5.p1.11.m11.2.3.2.1"></times><apply id="S5.p1.11.m11.2.3.2.2.cmml" xref="S5.p1.11.m11.2.3.2.2"><csymbol cd="ambiguous" id="S5.p1.11.m11.2.3.2.2.1.cmml" xref="S5.p1.11.m11.2.3.2.2">superscript</csymbol><ci id="S5.p1.11.m11.2.3.2.2.2a.cmml" xref="S5.p1.11.m11.2.3.2.2.2"><mtext class="ltx_mathvariant_italic" id="S5.p1.11.m11.2.3.2.2.2.cmml" xref="S5.p1.11.m11.2.3.2.2.2">g</mtext></ci><times id="S5.p1.11.m11.2.3.2.2.3.cmml" xref="S5.p1.11.m11.2.3.2.2.3"></times></apply><ci id="S5.p1.11.m11.1.1.cmml" xref="S5.p1.11.m11.1.1">𝑣</ci></apply><apply id="S5.p1.11.m11.2.3.3.cmml" xref="S5.p1.11.m11.2.3.3"><times id="S5.p1.11.m11.2.3.3.1.cmml" xref="S5.p1.11.m11.2.3.3.1"></times><apply id="S5.p1.11.m11.2.3.3.2.cmml" xref="S5.p1.11.m11.2.3.3.2"><csymbol cd="ambiguous" id="S5.p1.11.m11.2.3.3.2.1.cmml" xref="S5.p1.11.m11.2.3.3.2">superscript</csymbol><ci id="S5.p1.11.m11.2.3.3.2.2.cmml" xref="S5.p1.11.m11.2.3.3.2.2">𝜌</ci><times id="S5.p1.11.m11.2.3.3.2.3.cmml" xref="S5.p1.11.m11.2.3.3.2.3"></times></apply><ci id="S5.p1.11.m11.2.2.cmml" xref="S5.p1.11.m11.2.2">𝑣</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.p1.11.m11.2c">\textsl{g}^{*}(v)=\rho^{*}(v)</annotation><annotation encoding="application/x-llamapun" id="S5.p1.11.m11.2d">g start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ( italic_v ) = italic_ρ start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ( italic_v )</annotation></semantics></math>. Moreover, we prove that the maximal local out-degree (i.e. <math alttext="\max_{u\in V}\textsl{g}(u)" class="ltx_Math" display="inline" id="S5.p1.12.m12.1"><semantics id="S5.p1.12.m12.1a"><mrow id="S5.p1.12.m12.1.2" xref="S5.p1.12.m12.1.2.cmml"><mrow id="S5.p1.12.m12.1.2.2" xref="S5.p1.12.m12.1.2.2.cmml"><msub id="S5.p1.12.m12.1.2.2.1" xref="S5.p1.12.m12.1.2.2.1.cmml"><mi id="S5.p1.12.m12.1.2.2.1.2" xref="S5.p1.12.m12.1.2.2.1.2.cmml">max</mi><mrow id="S5.p1.12.m12.1.2.2.1.3" xref="S5.p1.12.m12.1.2.2.1.3.cmml"><mi id="S5.p1.12.m12.1.2.2.1.3.2" xref="S5.p1.12.m12.1.2.2.1.3.2.cmml">u</mi><mo id="S5.p1.12.m12.1.2.2.1.3.1" xref="S5.p1.12.m12.1.2.2.1.3.1.cmml">∈</mo><mi id="S5.p1.12.m12.1.2.2.1.3.3" xref="S5.p1.12.m12.1.2.2.1.3.3.cmml">V</mi></mrow></msub><mo id="S5.p1.12.m12.1.2.2a" lspace="0.167em" xref="S5.p1.12.m12.1.2.2.cmml">⁡</mo><mtext class="ltx_mathvariant_italic" id="S5.p1.12.m12.1.2.2.2" xref="S5.p1.12.m12.1.2.2.2a.cmml">g</mtext></mrow><mo id="S5.p1.12.m12.1.2.1" xref="S5.p1.12.m12.1.2.1.cmml">⁢</mo><mrow id="S5.p1.12.m12.1.2.3.2" xref="S5.p1.12.m12.1.2.cmml"><mo id="S5.p1.12.m12.1.2.3.2.1" stretchy="false" xref="S5.p1.12.m12.1.2.cmml">(</mo><mi id="S5.p1.12.m12.1.1" xref="S5.p1.12.m12.1.1.cmml">u</mi><mo id="S5.p1.12.m12.1.2.3.2.2" stretchy="false" xref="S5.p1.12.m12.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.p1.12.m12.1b"><apply id="S5.p1.12.m12.1.2.cmml" xref="S5.p1.12.m12.1.2"><times id="S5.p1.12.m12.1.2.1.cmml" xref="S5.p1.12.m12.1.2.1"></times><apply id="S5.p1.12.m12.1.2.2.cmml" xref="S5.p1.12.m12.1.2.2"><apply id="S5.p1.12.m12.1.2.2.1.cmml" xref="S5.p1.12.m12.1.2.2.1"><csymbol cd="ambiguous" id="S5.p1.12.m12.1.2.2.1.1.cmml" xref="S5.p1.12.m12.1.2.2.1">subscript</csymbol><max id="S5.p1.12.m12.1.2.2.1.2.cmml" xref="S5.p1.12.m12.1.2.2.1.2"></max><apply id="S5.p1.12.m12.1.2.2.1.3.cmml" xref="S5.p1.12.m12.1.2.2.1.3"><in id="S5.p1.12.m12.1.2.2.1.3.1.cmml" xref="S5.p1.12.m12.1.2.2.1.3.1"></in><ci id="S5.p1.12.m12.1.2.2.1.3.2.cmml" xref="S5.p1.12.m12.1.2.2.1.3.2">𝑢</ci><ci id="S5.p1.12.m12.1.2.2.1.3.3.cmml" xref="S5.p1.12.m12.1.2.2.1.3.3">𝑉</ci></apply></apply><ci id="S5.p1.12.m12.1.2.2.2a.cmml" xref="S5.p1.12.m12.1.2.2.2"><mtext class="ltx_mathvariant_italic" id="S5.p1.12.m12.1.2.2.2.cmml" xref="S5.p1.12.m12.1.2.2.2">g</mtext></ci></apply><ci id="S5.p1.12.m12.1.1.cmml" xref="S5.p1.12.m12.1.1">𝑢</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.p1.12.m12.1c">\max_{u\in V}\textsl{g}(u)</annotation><annotation encoding="application/x-llamapun" id="S5.p1.12.m12.1d">roman_max start_POSTSUBSCRIPT italic_u ∈ italic_V end_POSTSUBSCRIPT g ( italic_u )</annotation></semantics></math>) is a <math alttext="(1+\varepsilon)" class="ltx_Math" display="inline" id="S5.p1.13.m13.1"><semantics id="S5.p1.13.m13.1a"><mrow id="S5.p1.13.m13.1.1.1" xref="S5.p1.13.m13.1.1.1.1.cmml"><mo id="S5.p1.13.m13.1.1.1.2" stretchy="false" xref="S5.p1.13.m13.1.1.1.1.cmml">(</mo><mrow id="S5.p1.13.m13.1.1.1.1" xref="S5.p1.13.m13.1.1.1.1.cmml"><mn id="S5.p1.13.m13.1.1.1.1.2" xref="S5.p1.13.m13.1.1.1.1.2.cmml">1</mn><mo id="S5.p1.13.m13.1.1.1.1.1" xref="S5.p1.13.m13.1.1.1.1.1.cmml">+</mo><mi id="S5.p1.13.m13.1.1.1.1.3" xref="S5.p1.13.m13.1.1.1.1.3.cmml">ε</mi></mrow><mo id="S5.p1.13.m13.1.1.1.3" stretchy="false" xref="S5.p1.13.m13.1.1.1.1.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S5.p1.13.m13.1b"><apply id="S5.p1.13.m13.1.1.1.1.cmml" xref="S5.p1.13.m13.1.1.1"><plus id="S5.p1.13.m13.1.1.1.1.1.cmml" xref="S5.p1.13.m13.1.1.1.1.1"></plus><cn id="S5.p1.13.m13.1.1.1.1.2.cmml" type="integer" xref="S5.p1.13.m13.1.1.1.1.2">1</cn><ci id="S5.p1.13.m13.1.1.1.1.3.cmml" xref="S5.p1.13.m13.1.1.1.1.3">𝜀</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.p1.13.m13.1c">(1+\varepsilon)</annotation><annotation encoding="application/x-llamapun" id="S5.p1.13.m13.1d">( 1 + italic_ε )</annotation></semantics></math> approximation of <math alttext="\Delta^{\min}(G)=\rho^{\max}(G)" class="ltx_Math" display="inline" id="S5.p1.14.m14.2"><semantics id="S5.p1.14.m14.2a"><mrow id="S5.p1.14.m14.2.3" xref="S5.p1.14.m14.2.3.cmml"><mrow id="S5.p1.14.m14.2.3.2" xref="S5.p1.14.m14.2.3.2.cmml"><msup id="S5.p1.14.m14.2.3.2.2" xref="S5.p1.14.m14.2.3.2.2.cmml"><mi id="S5.p1.14.m14.2.3.2.2.2" mathvariant="normal" xref="S5.p1.14.m14.2.3.2.2.2.cmml">Δ</mi><mi id="S5.p1.14.m14.2.3.2.2.3" xref="S5.p1.14.m14.2.3.2.2.3.cmml">min</mi></msup><mo id="S5.p1.14.m14.2.3.2.1" xref="S5.p1.14.m14.2.3.2.1.cmml">⁢</mo><mrow id="S5.p1.14.m14.2.3.2.3.2" xref="S5.p1.14.m14.2.3.2.cmml"><mo id="S5.p1.14.m14.2.3.2.3.2.1" stretchy="false" xref="S5.p1.14.m14.2.3.2.cmml">(</mo><mi id="S5.p1.14.m14.1.1" xref="S5.p1.14.m14.1.1.cmml">G</mi><mo id="S5.p1.14.m14.2.3.2.3.2.2" stretchy="false" xref="S5.p1.14.m14.2.3.2.cmml">)</mo></mrow></mrow><mo id="S5.p1.14.m14.2.3.1" xref="S5.p1.14.m14.2.3.1.cmml">=</mo><mrow id="S5.p1.14.m14.2.3.3" xref="S5.p1.14.m14.2.3.3.cmml"><msup id="S5.p1.14.m14.2.3.3.2" xref="S5.p1.14.m14.2.3.3.2.cmml"><mi id="S5.p1.14.m14.2.3.3.2.2" xref="S5.p1.14.m14.2.3.3.2.2.cmml">ρ</mi><mi id="S5.p1.14.m14.2.3.3.2.3" xref="S5.p1.14.m14.2.3.3.2.3.cmml">max</mi></msup><mo id="S5.p1.14.m14.2.3.3.1" xref="S5.p1.14.m14.2.3.3.1.cmml">⁢</mo><mrow id="S5.p1.14.m14.2.3.3.3.2" xref="S5.p1.14.m14.2.3.3.cmml"><mo id="S5.p1.14.m14.2.3.3.3.2.1" stretchy="false" xref="S5.p1.14.m14.2.3.3.cmml">(</mo><mi id="S5.p1.14.m14.2.2" xref="S5.p1.14.m14.2.2.cmml">G</mi><mo id="S5.p1.14.m14.2.3.3.3.2.2" stretchy="false" xref="S5.p1.14.m14.2.3.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.p1.14.m14.2b"><apply id="S5.p1.14.m14.2.3.cmml" xref="S5.p1.14.m14.2.3"><eq id="S5.p1.14.m14.2.3.1.cmml" xref="S5.p1.14.m14.2.3.1"></eq><apply id="S5.p1.14.m14.2.3.2.cmml" xref="S5.p1.14.m14.2.3.2"><times id="S5.p1.14.m14.2.3.2.1.cmml" xref="S5.p1.14.m14.2.3.2.1"></times><apply id="S5.p1.14.m14.2.3.2.2.cmml" xref="S5.p1.14.m14.2.3.2.2"><csymbol cd="ambiguous" id="S5.p1.14.m14.2.3.2.2.1.cmml" xref="S5.p1.14.m14.2.3.2.2">superscript</csymbol><ci id="S5.p1.14.m14.2.3.2.2.2.cmml" xref="S5.p1.14.m14.2.3.2.2.2">Δ</ci><min id="S5.p1.14.m14.2.3.2.2.3.cmml" xref="S5.p1.14.m14.2.3.2.2.3"></min></apply><ci id="S5.p1.14.m14.1.1.cmml" xref="S5.p1.14.m14.1.1">𝐺</ci></apply><apply id="S5.p1.14.m14.2.3.3.cmml" xref="S5.p1.14.m14.2.3.3"><times id="S5.p1.14.m14.2.3.3.1.cmml" xref="S5.p1.14.m14.2.3.3.1"></times><apply id="S5.p1.14.m14.2.3.3.2.cmml" xref="S5.p1.14.m14.2.3.3.2"><csymbol cd="ambiguous" id="S5.p1.14.m14.2.3.3.2.1.cmml" xref="S5.p1.14.m14.2.3.3.2">superscript</csymbol><ci id="S5.p1.14.m14.2.3.3.2.2.cmml" xref="S5.p1.14.m14.2.3.3.2.2">𝜌</ci><max id="S5.p1.14.m14.2.3.3.2.3.cmml" xref="S5.p1.14.m14.2.3.3.2.3"></max></apply><ci id="S5.p1.14.m14.2.2.cmml" xref="S5.p1.14.m14.2.2">𝐺</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.p1.14.m14.2c">\Delta^{\min}(G)=\rho^{\max}(G)</annotation><annotation encoding="application/x-llamapun" id="S5.p1.14.m14.2d">roman_Δ start_POSTSUPERSCRIPT roman_min end_POSTSUPERSCRIPT ( italic_G ) = italic_ρ start_POSTSUPERSCRIPT roman_max end_POSTSUPERSCRIPT ( italic_G )</annotation></semantics></math>; illustrating that approximating the local measures is a strictly more general problem.</p> </div> <div class="ltx_theorem ltx_theorem_lemma" 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">Lemma 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.p1.3"><span class="ltx_text ltx_font_italic" id="S5.Thmtheorem1.p1.3.3">Let <math alttext="\eta\leq\frac{\varepsilon^{2}}{128\cdot\log n}" class="ltx_Math" display="inline" id="S5.Thmtheorem1.p1.1.1.m1.1"><semantics id="S5.Thmtheorem1.p1.1.1.m1.1a"><mrow id="S5.Thmtheorem1.p1.1.1.m1.1.1" xref="S5.Thmtheorem1.p1.1.1.m1.1.1.cmml"><mi id="S5.Thmtheorem1.p1.1.1.m1.1.1.2" xref="S5.Thmtheorem1.p1.1.1.m1.1.1.2.cmml">η</mi><mo id="S5.Thmtheorem1.p1.1.1.m1.1.1.1" xref="S5.Thmtheorem1.p1.1.1.m1.1.1.1.cmml">≤</mo><mfrac id="S5.Thmtheorem1.p1.1.1.m1.1.1.3" xref="S5.Thmtheorem1.p1.1.1.m1.1.1.3.cmml"><msup id="S5.Thmtheorem1.p1.1.1.m1.1.1.3.2" xref="S5.Thmtheorem1.p1.1.1.m1.1.1.3.2.cmml"><mi id="S5.Thmtheorem1.p1.1.1.m1.1.1.3.2.2" xref="S5.Thmtheorem1.p1.1.1.m1.1.1.3.2.2.cmml">ε</mi><mn id="S5.Thmtheorem1.p1.1.1.m1.1.1.3.2.3" xref="S5.Thmtheorem1.p1.1.1.m1.1.1.3.2.3.cmml">2</mn></msup><mrow id="S5.Thmtheorem1.p1.1.1.m1.1.1.3.3" xref="S5.Thmtheorem1.p1.1.1.m1.1.1.3.3.cmml"><mn id="S5.Thmtheorem1.p1.1.1.m1.1.1.3.3.2" xref="S5.Thmtheorem1.p1.1.1.m1.1.1.3.3.2.cmml">128</mn><mo id="S5.Thmtheorem1.p1.1.1.m1.1.1.3.3.1" lspace="0.222em" rspace="0.222em" xref="S5.Thmtheorem1.p1.1.1.m1.1.1.3.3.1.cmml">⋅</mo><mrow id="S5.Thmtheorem1.p1.1.1.m1.1.1.3.3.3" xref="S5.Thmtheorem1.p1.1.1.m1.1.1.3.3.3.cmml"><mi id="S5.Thmtheorem1.p1.1.1.m1.1.1.3.3.3.1" xref="S5.Thmtheorem1.p1.1.1.m1.1.1.3.3.3.1.cmml">log</mi><mo id="S5.Thmtheorem1.p1.1.1.m1.1.1.3.3.3a" lspace="0.167em" xref="S5.Thmtheorem1.p1.1.1.m1.1.1.3.3.3.cmml">⁡</mo><mi id="S5.Thmtheorem1.p1.1.1.m1.1.1.3.3.3.2" xref="S5.Thmtheorem1.p1.1.1.m1.1.1.3.3.3.2.cmml">n</mi></mrow></mrow></mfrac></mrow><annotation-xml encoding="MathML-Content" id="S5.Thmtheorem1.p1.1.1.m1.1b"><apply id="S5.Thmtheorem1.p1.1.1.m1.1.1.cmml" xref="S5.Thmtheorem1.p1.1.1.m1.1.1"><leq id="S5.Thmtheorem1.p1.1.1.m1.1.1.1.cmml" xref="S5.Thmtheorem1.p1.1.1.m1.1.1.1"></leq><ci id="S5.Thmtheorem1.p1.1.1.m1.1.1.2.cmml" xref="S5.Thmtheorem1.p1.1.1.m1.1.1.2">𝜂</ci><apply id="S5.Thmtheorem1.p1.1.1.m1.1.1.3.cmml" xref="S5.Thmtheorem1.p1.1.1.m1.1.1.3"><divide id="S5.Thmtheorem1.p1.1.1.m1.1.1.3.1.cmml" xref="S5.Thmtheorem1.p1.1.1.m1.1.1.3"></divide><apply id="S5.Thmtheorem1.p1.1.1.m1.1.1.3.2.cmml" xref="S5.Thmtheorem1.p1.1.1.m1.1.1.3.2"><csymbol cd="ambiguous" id="S5.Thmtheorem1.p1.1.1.m1.1.1.3.2.1.cmml" xref="S5.Thmtheorem1.p1.1.1.m1.1.1.3.2">superscript</csymbol><ci id="S5.Thmtheorem1.p1.1.1.m1.1.1.3.2.2.cmml" xref="S5.Thmtheorem1.p1.1.1.m1.1.1.3.2.2">𝜀</ci><cn id="S5.Thmtheorem1.p1.1.1.m1.1.1.3.2.3.cmml" type="integer" xref="S5.Thmtheorem1.p1.1.1.m1.1.1.3.2.3">2</cn></apply><apply id="S5.Thmtheorem1.p1.1.1.m1.1.1.3.3.cmml" xref="S5.Thmtheorem1.p1.1.1.m1.1.1.3.3"><ci id="S5.Thmtheorem1.p1.1.1.m1.1.1.3.3.1.cmml" xref="S5.Thmtheorem1.p1.1.1.m1.1.1.3.3.1">⋅</ci><cn id="S5.Thmtheorem1.p1.1.1.m1.1.1.3.3.2.cmml" type="integer" xref="S5.Thmtheorem1.p1.1.1.m1.1.1.3.3.2">128</cn><apply id="S5.Thmtheorem1.p1.1.1.m1.1.1.3.3.3.cmml" xref="S5.Thmtheorem1.p1.1.1.m1.1.1.3.3.3"><log id="S5.Thmtheorem1.p1.1.1.m1.1.1.3.3.3.1.cmml" xref="S5.Thmtheorem1.p1.1.1.m1.1.1.3.3.3.1"></log><ci id="S5.Thmtheorem1.p1.1.1.m1.1.1.3.3.3.2.cmml" xref="S5.Thmtheorem1.p1.1.1.m1.1.1.3.3.3.2">𝑛</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmtheorem1.p1.1.1.m1.1c">\eta\leq\frac{\varepsilon^{2}}{128\cdot\log n}</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem1.p1.1.1.m1.1d">italic_η ≤ divide start_ARG italic_ε start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG start_ARG 128 ⋅ roman_log italic_n end_ARG</annotation></semantics></math> and <math alttext="k\leq\log_{(1+\frac{1}{16}\varepsilon)}n" class="ltx_Math" display="inline" id="S5.Thmtheorem1.p1.2.2.m2.1"><semantics id="S5.Thmtheorem1.p1.2.2.m2.1a"><mrow id="S5.Thmtheorem1.p1.2.2.m2.1.2" xref="S5.Thmtheorem1.p1.2.2.m2.1.2.cmml"><mi id="S5.Thmtheorem1.p1.2.2.m2.1.2.2" xref="S5.Thmtheorem1.p1.2.2.m2.1.2.2.cmml">k</mi><mo id="S5.Thmtheorem1.p1.2.2.m2.1.2.1" xref="S5.Thmtheorem1.p1.2.2.m2.1.2.1.cmml">≤</mo><mrow id="S5.Thmtheorem1.p1.2.2.m2.1.2.3" xref="S5.Thmtheorem1.p1.2.2.m2.1.2.3.cmml"><msub id="S5.Thmtheorem1.p1.2.2.m2.1.2.3.1" xref="S5.Thmtheorem1.p1.2.2.m2.1.2.3.1.cmml"><mi id="S5.Thmtheorem1.p1.2.2.m2.1.2.3.1.2" xref="S5.Thmtheorem1.p1.2.2.m2.1.2.3.1.2.cmml">log</mi><mrow id="S5.Thmtheorem1.p1.2.2.m2.1.1.1.1" xref="S5.Thmtheorem1.p1.2.2.m2.1.1.1.1.1.cmml"><mo id="S5.Thmtheorem1.p1.2.2.m2.1.1.1.1.2" stretchy="false" xref="S5.Thmtheorem1.p1.2.2.m2.1.1.1.1.1.cmml">(</mo><mrow id="S5.Thmtheorem1.p1.2.2.m2.1.1.1.1.1" xref="S5.Thmtheorem1.p1.2.2.m2.1.1.1.1.1.cmml"><mn id="S5.Thmtheorem1.p1.2.2.m2.1.1.1.1.1.2" xref="S5.Thmtheorem1.p1.2.2.m2.1.1.1.1.1.2.cmml">1</mn><mo id="S5.Thmtheorem1.p1.2.2.m2.1.1.1.1.1.1" xref="S5.Thmtheorem1.p1.2.2.m2.1.1.1.1.1.1.cmml">+</mo><mrow id="S5.Thmtheorem1.p1.2.2.m2.1.1.1.1.1.3" xref="S5.Thmtheorem1.p1.2.2.m2.1.1.1.1.1.3.cmml"><mfrac id="S5.Thmtheorem1.p1.2.2.m2.1.1.1.1.1.3.2" xref="S5.Thmtheorem1.p1.2.2.m2.1.1.1.1.1.3.2.cmml"><mn id="S5.Thmtheorem1.p1.2.2.m2.1.1.1.1.1.3.2.2" xref="S5.Thmtheorem1.p1.2.2.m2.1.1.1.1.1.3.2.2.cmml">1</mn><mn id="S5.Thmtheorem1.p1.2.2.m2.1.1.1.1.1.3.2.3" xref="S5.Thmtheorem1.p1.2.2.m2.1.1.1.1.1.3.2.3.cmml">16</mn></mfrac><mo id="S5.Thmtheorem1.p1.2.2.m2.1.1.1.1.1.3.1" xref="S5.Thmtheorem1.p1.2.2.m2.1.1.1.1.1.3.1.cmml">⁢</mo><mi id="S5.Thmtheorem1.p1.2.2.m2.1.1.1.1.1.3.3" xref="S5.Thmtheorem1.p1.2.2.m2.1.1.1.1.1.3.3.cmml">ε</mi></mrow></mrow><mo id="S5.Thmtheorem1.p1.2.2.m2.1.1.1.1.3" stretchy="false" xref="S5.Thmtheorem1.p1.2.2.m2.1.1.1.1.1.cmml">)</mo></mrow></msub><mo id="S5.Thmtheorem1.p1.2.2.m2.1.2.3a" lspace="0.167em" xref="S5.Thmtheorem1.p1.2.2.m2.1.2.3.cmml">⁡</mo><mi id="S5.Thmtheorem1.p1.2.2.m2.1.2.3.2" xref="S5.Thmtheorem1.p1.2.2.m2.1.2.3.2.cmml">n</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.Thmtheorem1.p1.2.2.m2.1b"><apply id="S5.Thmtheorem1.p1.2.2.m2.1.2.cmml" xref="S5.Thmtheorem1.p1.2.2.m2.1.2"><leq id="S5.Thmtheorem1.p1.2.2.m2.1.2.1.cmml" xref="S5.Thmtheorem1.p1.2.2.m2.1.2.1"></leq><ci id="S5.Thmtheorem1.p1.2.2.m2.1.2.2.cmml" xref="S5.Thmtheorem1.p1.2.2.m2.1.2.2">𝑘</ci><apply id="S5.Thmtheorem1.p1.2.2.m2.1.2.3.cmml" xref="S5.Thmtheorem1.p1.2.2.m2.1.2.3"><apply id="S5.Thmtheorem1.p1.2.2.m2.1.2.3.1.cmml" xref="S5.Thmtheorem1.p1.2.2.m2.1.2.3.1"><csymbol cd="ambiguous" id="S5.Thmtheorem1.p1.2.2.m2.1.2.3.1.1.cmml" xref="S5.Thmtheorem1.p1.2.2.m2.1.2.3.1">subscript</csymbol><log id="S5.Thmtheorem1.p1.2.2.m2.1.2.3.1.2.cmml" xref="S5.Thmtheorem1.p1.2.2.m2.1.2.3.1.2"></log><apply id="S5.Thmtheorem1.p1.2.2.m2.1.1.1.1.1.cmml" xref="S5.Thmtheorem1.p1.2.2.m2.1.1.1.1"><plus id="S5.Thmtheorem1.p1.2.2.m2.1.1.1.1.1.1.cmml" xref="S5.Thmtheorem1.p1.2.2.m2.1.1.1.1.1.1"></plus><cn id="S5.Thmtheorem1.p1.2.2.m2.1.1.1.1.1.2.cmml" type="integer" xref="S5.Thmtheorem1.p1.2.2.m2.1.1.1.1.1.2">1</cn><apply id="S5.Thmtheorem1.p1.2.2.m2.1.1.1.1.1.3.cmml" xref="S5.Thmtheorem1.p1.2.2.m2.1.1.1.1.1.3"><times id="S5.Thmtheorem1.p1.2.2.m2.1.1.1.1.1.3.1.cmml" xref="S5.Thmtheorem1.p1.2.2.m2.1.1.1.1.1.3.1"></times><apply id="S5.Thmtheorem1.p1.2.2.m2.1.1.1.1.1.3.2.cmml" xref="S5.Thmtheorem1.p1.2.2.m2.1.1.1.1.1.3.2"><divide id="S5.Thmtheorem1.p1.2.2.m2.1.1.1.1.1.3.2.1.cmml" xref="S5.Thmtheorem1.p1.2.2.m2.1.1.1.1.1.3.2"></divide><cn id="S5.Thmtheorem1.p1.2.2.m2.1.1.1.1.1.3.2.2.cmml" type="integer" xref="S5.Thmtheorem1.p1.2.2.m2.1.1.1.1.1.3.2.2">1</cn><cn id="S5.Thmtheorem1.p1.2.2.m2.1.1.1.1.1.3.2.3.cmml" type="integer" xref="S5.Thmtheorem1.p1.2.2.m2.1.1.1.1.1.3.2.3">16</cn></apply><ci id="S5.Thmtheorem1.p1.2.2.m2.1.1.1.1.1.3.3.cmml" xref="S5.Thmtheorem1.p1.2.2.m2.1.1.1.1.1.3.3">𝜀</ci></apply></apply></apply><ci id="S5.Thmtheorem1.p1.2.2.m2.1.2.3.2.cmml" xref="S5.Thmtheorem1.p1.2.2.m2.1.2.3.2">𝑛</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmtheorem1.p1.2.2.m2.1c">k\leq\log_{(1+\frac{1}{16}\varepsilon)}n</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem1.p1.2.2.m2.1d">italic_k ≤ roman_log start_POSTSUBSCRIPT ( 1 + divide start_ARG 1 end_ARG start_ARG 16 end_ARG italic_ε ) end_POSTSUBSCRIPT italic_n</annotation></semantics></math>. Then <math alttext="(1+\eta)^{-k}\geq(1+0.5\varepsilon)^{-1}" class="ltx_Math" display="inline" id="S5.Thmtheorem1.p1.3.3.m3.2"><semantics id="S5.Thmtheorem1.p1.3.3.m3.2a"><mrow id="S5.Thmtheorem1.p1.3.3.m3.2.2" xref="S5.Thmtheorem1.p1.3.3.m3.2.2.cmml"><msup id="S5.Thmtheorem1.p1.3.3.m3.1.1.1" xref="S5.Thmtheorem1.p1.3.3.m3.1.1.1.cmml"><mrow id="S5.Thmtheorem1.p1.3.3.m3.1.1.1.1.1" xref="S5.Thmtheorem1.p1.3.3.m3.1.1.1.1.1.1.cmml"><mo id="S5.Thmtheorem1.p1.3.3.m3.1.1.1.1.1.2" stretchy="false" xref="S5.Thmtheorem1.p1.3.3.m3.1.1.1.1.1.1.cmml">(</mo><mrow id="S5.Thmtheorem1.p1.3.3.m3.1.1.1.1.1.1" xref="S5.Thmtheorem1.p1.3.3.m3.1.1.1.1.1.1.cmml"><mn id="S5.Thmtheorem1.p1.3.3.m3.1.1.1.1.1.1.2" xref="S5.Thmtheorem1.p1.3.3.m3.1.1.1.1.1.1.2.cmml">1</mn><mo id="S5.Thmtheorem1.p1.3.3.m3.1.1.1.1.1.1.1" xref="S5.Thmtheorem1.p1.3.3.m3.1.1.1.1.1.1.1.cmml">+</mo><mi id="S5.Thmtheorem1.p1.3.3.m3.1.1.1.1.1.1.3" xref="S5.Thmtheorem1.p1.3.3.m3.1.1.1.1.1.1.3.cmml">η</mi></mrow><mo id="S5.Thmtheorem1.p1.3.3.m3.1.1.1.1.1.3" stretchy="false" xref="S5.Thmtheorem1.p1.3.3.m3.1.1.1.1.1.1.cmml">)</mo></mrow><mrow id="S5.Thmtheorem1.p1.3.3.m3.1.1.1.3" xref="S5.Thmtheorem1.p1.3.3.m3.1.1.1.3.cmml"><mo id="S5.Thmtheorem1.p1.3.3.m3.1.1.1.3a" xref="S5.Thmtheorem1.p1.3.3.m3.1.1.1.3.cmml">−</mo><mi id="S5.Thmtheorem1.p1.3.3.m3.1.1.1.3.2" xref="S5.Thmtheorem1.p1.3.3.m3.1.1.1.3.2.cmml">k</mi></mrow></msup><mo id="S5.Thmtheorem1.p1.3.3.m3.2.2.3" xref="S5.Thmtheorem1.p1.3.3.m3.2.2.3.cmml">≥</mo><msup id="S5.Thmtheorem1.p1.3.3.m3.2.2.2" xref="S5.Thmtheorem1.p1.3.3.m3.2.2.2.cmml"><mrow id="S5.Thmtheorem1.p1.3.3.m3.2.2.2.1.1" xref="S5.Thmtheorem1.p1.3.3.m3.2.2.2.1.1.1.cmml"><mo id="S5.Thmtheorem1.p1.3.3.m3.2.2.2.1.1.2" stretchy="false" xref="S5.Thmtheorem1.p1.3.3.m3.2.2.2.1.1.1.cmml">(</mo><mrow id="S5.Thmtheorem1.p1.3.3.m3.2.2.2.1.1.1" xref="S5.Thmtheorem1.p1.3.3.m3.2.2.2.1.1.1.cmml"><mn id="S5.Thmtheorem1.p1.3.3.m3.2.2.2.1.1.1.2" xref="S5.Thmtheorem1.p1.3.3.m3.2.2.2.1.1.1.2.cmml">1</mn><mo id="S5.Thmtheorem1.p1.3.3.m3.2.2.2.1.1.1.1" xref="S5.Thmtheorem1.p1.3.3.m3.2.2.2.1.1.1.1.cmml">+</mo><mrow id="S5.Thmtheorem1.p1.3.3.m3.2.2.2.1.1.1.3" xref="S5.Thmtheorem1.p1.3.3.m3.2.2.2.1.1.1.3.cmml"><mn id="S5.Thmtheorem1.p1.3.3.m3.2.2.2.1.1.1.3.2" xref="S5.Thmtheorem1.p1.3.3.m3.2.2.2.1.1.1.3.2.cmml">0.5</mn><mo id="S5.Thmtheorem1.p1.3.3.m3.2.2.2.1.1.1.3.1" xref="S5.Thmtheorem1.p1.3.3.m3.2.2.2.1.1.1.3.1.cmml">⁢</mo><mi id="S5.Thmtheorem1.p1.3.3.m3.2.2.2.1.1.1.3.3" xref="S5.Thmtheorem1.p1.3.3.m3.2.2.2.1.1.1.3.3.cmml">ε</mi></mrow></mrow><mo id="S5.Thmtheorem1.p1.3.3.m3.2.2.2.1.1.3" stretchy="false" xref="S5.Thmtheorem1.p1.3.3.m3.2.2.2.1.1.1.cmml">)</mo></mrow><mrow id="S5.Thmtheorem1.p1.3.3.m3.2.2.2.3" xref="S5.Thmtheorem1.p1.3.3.m3.2.2.2.3.cmml"><mo id="S5.Thmtheorem1.p1.3.3.m3.2.2.2.3a" xref="S5.Thmtheorem1.p1.3.3.m3.2.2.2.3.cmml">−</mo><mn id="S5.Thmtheorem1.p1.3.3.m3.2.2.2.3.2" xref="S5.Thmtheorem1.p1.3.3.m3.2.2.2.3.2.cmml">1</mn></mrow></msup></mrow><annotation-xml encoding="MathML-Content" id="S5.Thmtheorem1.p1.3.3.m3.2b"><apply id="S5.Thmtheorem1.p1.3.3.m3.2.2.cmml" xref="S5.Thmtheorem1.p1.3.3.m3.2.2"><geq id="S5.Thmtheorem1.p1.3.3.m3.2.2.3.cmml" xref="S5.Thmtheorem1.p1.3.3.m3.2.2.3"></geq><apply id="S5.Thmtheorem1.p1.3.3.m3.1.1.1.cmml" xref="S5.Thmtheorem1.p1.3.3.m3.1.1.1"><csymbol cd="ambiguous" id="S5.Thmtheorem1.p1.3.3.m3.1.1.1.2.cmml" xref="S5.Thmtheorem1.p1.3.3.m3.1.1.1">superscript</csymbol><apply id="S5.Thmtheorem1.p1.3.3.m3.1.1.1.1.1.1.cmml" xref="S5.Thmtheorem1.p1.3.3.m3.1.1.1.1.1"><plus id="S5.Thmtheorem1.p1.3.3.m3.1.1.1.1.1.1.1.cmml" xref="S5.Thmtheorem1.p1.3.3.m3.1.1.1.1.1.1.1"></plus><cn id="S5.Thmtheorem1.p1.3.3.m3.1.1.1.1.1.1.2.cmml" type="integer" xref="S5.Thmtheorem1.p1.3.3.m3.1.1.1.1.1.1.2">1</cn><ci id="S5.Thmtheorem1.p1.3.3.m3.1.1.1.1.1.1.3.cmml" xref="S5.Thmtheorem1.p1.3.3.m3.1.1.1.1.1.1.3">𝜂</ci></apply><apply id="S5.Thmtheorem1.p1.3.3.m3.1.1.1.3.cmml" xref="S5.Thmtheorem1.p1.3.3.m3.1.1.1.3"><minus id="S5.Thmtheorem1.p1.3.3.m3.1.1.1.3.1.cmml" xref="S5.Thmtheorem1.p1.3.3.m3.1.1.1.3"></minus><ci id="S5.Thmtheorem1.p1.3.3.m3.1.1.1.3.2.cmml" xref="S5.Thmtheorem1.p1.3.3.m3.1.1.1.3.2">𝑘</ci></apply></apply><apply id="S5.Thmtheorem1.p1.3.3.m3.2.2.2.cmml" xref="S5.Thmtheorem1.p1.3.3.m3.2.2.2"><csymbol cd="ambiguous" id="S5.Thmtheorem1.p1.3.3.m3.2.2.2.2.cmml" xref="S5.Thmtheorem1.p1.3.3.m3.2.2.2">superscript</csymbol><apply id="S5.Thmtheorem1.p1.3.3.m3.2.2.2.1.1.1.cmml" xref="S5.Thmtheorem1.p1.3.3.m3.2.2.2.1.1"><plus id="S5.Thmtheorem1.p1.3.3.m3.2.2.2.1.1.1.1.cmml" xref="S5.Thmtheorem1.p1.3.3.m3.2.2.2.1.1.1.1"></plus><cn id="S5.Thmtheorem1.p1.3.3.m3.2.2.2.1.1.1.2.cmml" type="integer" xref="S5.Thmtheorem1.p1.3.3.m3.2.2.2.1.1.1.2">1</cn><apply id="S5.Thmtheorem1.p1.3.3.m3.2.2.2.1.1.1.3.cmml" xref="S5.Thmtheorem1.p1.3.3.m3.2.2.2.1.1.1.3"><times id="S5.Thmtheorem1.p1.3.3.m3.2.2.2.1.1.1.3.1.cmml" xref="S5.Thmtheorem1.p1.3.3.m3.2.2.2.1.1.1.3.1"></times><cn id="S5.Thmtheorem1.p1.3.3.m3.2.2.2.1.1.1.3.2.cmml" type="float" xref="S5.Thmtheorem1.p1.3.3.m3.2.2.2.1.1.1.3.2">0.5</cn><ci id="S5.Thmtheorem1.p1.3.3.m3.2.2.2.1.1.1.3.3.cmml" xref="S5.Thmtheorem1.p1.3.3.m3.2.2.2.1.1.1.3.3">𝜀</ci></apply></apply><apply id="S5.Thmtheorem1.p1.3.3.m3.2.2.2.3.cmml" xref="S5.Thmtheorem1.p1.3.3.m3.2.2.2.3"><minus id="S5.Thmtheorem1.p1.3.3.m3.2.2.2.3.1.cmml" xref="S5.Thmtheorem1.p1.3.3.m3.2.2.2.3"></minus><cn id="S5.Thmtheorem1.p1.3.3.m3.2.2.2.3.2.cmml" type="integer" xref="S5.Thmtheorem1.p1.3.3.m3.2.2.2.3.2">1</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmtheorem1.p1.3.3.m3.2c">(1+\eta)^{-k}\geq(1+0.5\varepsilon)^{-1}</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem1.p1.3.3.m3.2d">( 1 + italic_η ) start_POSTSUPERSCRIPT - italic_k end_POSTSUPERSCRIPT ≥ ( 1 + 0.5 italic_ε ) start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT</annotation></semantics></math>.</span></p> </div> </div> <div class="ltx_theorem ltx_theorem_proof" 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">Proof 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.p1.2"><span class="ltx_text ltx_font_italic" id="S5.Thmtheorem2.p1.2.2">Using <math alttext="\log(1+x)\geq x/2" class="ltx_Math" display="inline" id="S5.Thmtheorem2.p1.1.1.m1.2"><semantics id="S5.Thmtheorem2.p1.1.1.m1.2a"><mrow id="S5.Thmtheorem2.p1.1.1.m1.2.2" xref="S5.Thmtheorem2.p1.1.1.m1.2.2.cmml"><mrow id="S5.Thmtheorem2.p1.1.1.m1.2.2.1.1" xref="S5.Thmtheorem2.p1.1.1.m1.2.2.1.2.cmml"><mi id="S5.Thmtheorem2.p1.1.1.m1.1.1" xref="S5.Thmtheorem2.p1.1.1.m1.1.1.cmml">log</mi><mo id="S5.Thmtheorem2.p1.1.1.m1.2.2.1.1a" xref="S5.Thmtheorem2.p1.1.1.m1.2.2.1.2.cmml">⁡</mo><mrow id="S5.Thmtheorem2.p1.1.1.m1.2.2.1.1.1" xref="S5.Thmtheorem2.p1.1.1.m1.2.2.1.2.cmml"><mo id="S5.Thmtheorem2.p1.1.1.m1.2.2.1.1.1.2" stretchy="false" xref="S5.Thmtheorem2.p1.1.1.m1.2.2.1.2.cmml">(</mo><mrow id="S5.Thmtheorem2.p1.1.1.m1.2.2.1.1.1.1" xref="S5.Thmtheorem2.p1.1.1.m1.2.2.1.1.1.1.cmml"><mn id="S5.Thmtheorem2.p1.1.1.m1.2.2.1.1.1.1.2" xref="S5.Thmtheorem2.p1.1.1.m1.2.2.1.1.1.1.2.cmml">1</mn><mo id="S5.Thmtheorem2.p1.1.1.m1.2.2.1.1.1.1.1" xref="S5.Thmtheorem2.p1.1.1.m1.2.2.1.1.1.1.1.cmml">+</mo><mi id="S5.Thmtheorem2.p1.1.1.m1.2.2.1.1.1.1.3" xref="S5.Thmtheorem2.p1.1.1.m1.2.2.1.1.1.1.3.cmml">x</mi></mrow><mo id="S5.Thmtheorem2.p1.1.1.m1.2.2.1.1.1.3" stretchy="false" xref="S5.Thmtheorem2.p1.1.1.m1.2.2.1.2.cmml">)</mo></mrow></mrow><mo id="S5.Thmtheorem2.p1.1.1.m1.2.2.2" xref="S5.Thmtheorem2.p1.1.1.m1.2.2.2.cmml">≥</mo><mrow id="S5.Thmtheorem2.p1.1.1.m1.2.2.3" xref="S5.Thmtheorem2.p1.1.1.m1.2.2.3.cmml"><mi id="S5.Thmtheorem2.p1.1.1.m1.2.2.3.2" xref="S5.Thmtheorem2.p1.1.1.m1.2.2.3.2.cmml">x</mi><mo id="S5.Thmtheorem2.p1.1.1.m1.2.2.3.1" xref="S5.Thmtheorem2.p1.1.1.m1.2.2.3.1.cmml">/</mo><mn id="S5.Thmtheorem2.p1.1.1.m1.2.2.3.3" xref="S5.Thmtheorem2.p1.1.1.m1.2.2.3.3.cmml">2</mn></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.Thmtheorem2.p1.1.1.m1.2b"><apply id="S5.Thmtheorem2.p1.1.1.m1.2.2.cmml" xref="S5.Thmtheorem2.p1.1.1.m1.2.2"><geq id="S5.Thmtheorem2.p1.1.1.m1.2.2.2.cmml" xref="S5.Thmtheorem2.p1.1.1.m1.2.2.2"></geq><apply id="S5.Thmtheorem2.p1.1.1.m1.2.2.1.2.cmml" xref="S5.Thmtheorem2.p1.1.1.m1.2.2.1.1"><log id="S5.Thmtheorem2.p1.1.1.m1.1.1.cmml" xref="S5.Thmtheorem2.p1.1.1.m1.1.1"></log><apply id="S5.Thmtheorem2.p1.1.1.m1.2.2.1.1.1.1.cmml" xref="S5.Thmtheorem2.p1.1.1.m1.2.2.1.1.1.1"><plus id="S5.Thmtheorem2.p1.1.1.m1.2.2.1.1.1.1.1.cmml" xref="S5.Thmtheorem2.p1.1.1.m1.2.2.1.1.1.1.1"></plus><cn id="S5.Thmtheorem2.p1.1.1.m1.2.2.1.1.1.1.2.cmml" type="integer" xref="S5.Thmtheorem2.p1.1.1.m1.2.2.1.1.1.1.2">1</cn><ci id="S5.Thmtheorem2.p1.1.1.m1.2.2.1.1.1.1.3.cmml" xref="S5.Thmtheorem2.p1.1.1.m1.2.2.1.1.1.1.3">𝑥</ci></apply></apply><apply id="S5.Thmtheorem2.p1.1.1.m1.2.2.3.cmml" xref="S5.Thmtheorem2.p1.1.1.m1.2.2.3"><divide id="S5.Thmtheorem2.p1.1.1.m1.2.2.3.1.cmml" xref="S5.Thmtheorem2.p1.1.1.m1.2.2.3.1"></divide><ci id="S5.Thmtheorem2.p1.1.1.m1.2.2.3.2.cmml" xref="S5.Thmtheorem2.p1.1.1.m1.2.2.3.2">𝑥</ci><cn id="S5.Thmtheorem2.p1.1.1.m1.2.2.3.3.cmml" type="integer" xref="S5.Thmtheorem2.p1.1.1.m1.2.2.3.3">2</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmtheorem2.p1.1.1.m1.2c">\log(1+x)\geq x/2</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem2.p1.1.1.m1.2d">roman_log ( 1 + italic_x ) ≥ italic_x / 2</annotation></semantics></math> whenever <math alttext="x<1" class="ltx_Math" display="inline" id="S5.Thmtheorem2.p1.2.2.m2.1"><semantics id="S5.Thmtheorem2.p1.2.2.m2.1a"><mrow id="S5.Thmtheorem2.p1.2.2.m2.1.1" xref="S5.Thmtheorem2.p1.2.2.m2.1.1.cmml"><mi id="S5.Thmtheorem2.p1.2.2.m2.1.1.2" xref="S5.Thmtheorem2.p1.2.2.m2.1.1.2.cmml">x</mi><mo id="S5.Thmtheorem2.p1.2.2.m2.1.1.1" xref="S5.Thmtheorem2.p1.2.2.m2.1.1.1.cmml"><</mo><mn id="S5.Thmtheorem2.p1.2.2.m2.1.1.3" xref="S5.Thmtheorem2.p1.2.2.m2.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S5.Thmtheorem2.p1.2.2.m2.1b"><apply id="S5.Thmtheorem2.p1.2.2.m2.1.1.cmml" xref="S5.Thmtheorem2.p1.2.2.m2.1.1"><lt id="S5.Thmtheorem2.p1.2.2.m2.1.1.1.cmml" xref="S5.Thmtheorem2.p1.2.2.m2.1.1.1"></lt><ci id="S5.Thmtheorem2.p1.2.2.m2.1.1.2.cmml" xref="S5.Thmtheorem2.p1.2.2.m2.1.1.2">𝑥</ci><cn id="S5.Thmtheorem2.p1.2.2.m2.1.1.3.cmml" type="integer" xref="S5.Thmtheorem2.p1.2.2.m2.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmtheorem2.p1.2.2.m2.1c">x<1</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem2.p1.2.2.m2.1d">italic_x < 1</annotation></semantics></math>, we obtain:</span></p> <table class="ltx_equation ltx_eqn_table" id="S5.Ex16"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_left_padleft"></td> <td class="ltx_eqn_cell ltx_align_left"><math alttext="-\log_{1+\frac{\varepsilon}{16}}(n)=-\frac{\log(n)}{\log 1+\frac{\varepsilon}{% 16}}\geq-\frac{\log(n)}{\frac{\varepsilon}{32}}\geq-\frac{\varepsilon}{4}\cdot% {}\frac{128\log(n)}{\varepsilon^{2}}" class="ltx_Math" display="block" id="S5.Ex16.m1.8"><semantics id="S5.Ex16.m1.8a"><mrow id="S5.Ex16.m1.8.8" xref="S5.Ex16.m1.8.8.cmml"><mrow id="S5.Ex16.m1.8.8.1" xref="S5.Ex16.m1.8.8.1.cmml"><mo id="S5.Ex16.m1.8.8.1a" rspace="0.167em" xref="S5.Ex16.m1.8.8.1.cmml">−</mo><mrow id="S5.Ex16.m1.8.8.1.1.1" xref="S5.Ex16.m1.8.8.1.1.2.cmml"><msub id="S5.Ex16.m1.8.8.1.1.1.1" xref="S5.Ex16.m1.8.8.1.1.1.1.cmml"><mi id="S5.Ex16.m1.8.8.1.1.1.1.2" xref="S5.Ex16.m1.8.8.1.1.1.1.2.cmml">log</mi><mrow id="S5.Ex16.m1.8.8.1.1.1.1.3" xref="S5.Ex16.m1.8.8.1.1.1.1.3.cmml"><mn id="S5.Ex16.m1.8.8.1.1.1.1.3.2" xref="S5.Ex16.m1.8.8.1.1.1.1.3.2.cmml">1</mn><mo id="S5.Ex16.m1.8.8.1.1.1.1.3.1" xref="S5.Ex16.m1.8.8.1.1.1.1.3.1.cmml">+</mo><mfrac id="S5.Ex16.m1.8.8.1.1.1.1.3.3" xref="S5.Ex16.m1.8.8.1.1.1.1.3.3.cmml"><mi id="S5.Ex16.m1.8.8.1.1.1.1.3.3.2" xref="S5.Ex16.m1.8.8.1.1.1.1.3.3.2.cmml">ε</mi><mn id="S5.Ex16.m1.8.8.1.1.1.1.3.3.3" xref="S5.Ex16.m1.8.8.1.1.1.1.3.3.3.cmml">16</mn></mfrac></mrow></msub><mo id="S5.Ex16.m1.8.8.1.1.1a" xref="S5.Ex16.m1.8.8.1.1.2.cmml">⁡</mo><mrow id="S5.Ex16.m1.8.8.1.1.1.2" xref="S5.Ex16.m1.8.8.1.1.2.cmml"><mo id="S5.Ex16.m1.8.8.1.1.1.2.1" stretchy="false" xref="S5.Ex16.m1.8.8.1.1.2.cmml">(</mo><mi id="S5.Ex16.m1.7.7" xref="S5.Ex16.m1.7.7.cmml">n</mi><mo id="S5.Ex16.m1.8.8.1.1.1.2.2" stretchy="false" xref="S5.Ex16.m1.8.8.1.1.2.cmml">)</mo></mrow></mrow></mrow><mo id="S5.Ex16.m1.8.8.3" xref="S5.Ex16.m1.8.8.3.cmml">=</mo><mrow id="S5.Ex16.m1.8.8.4" xref="S5.Ex16.m1.8.8.4.cmml"><mo id="S5.Ex16.m1.8.8.4a" xref="S5.Ex16.m1.8.8.4.cmml">−</mo><mfrac id="S5.Ex16.m1.2.2" xref="S5.Ex16.m1.2.2.cmml"><mrow id="S5.Ex16.m1.2.2.2.4" xref="S5.Ex16.m1.2.2.2.3.cmml"><mi id="S5.Ex16.m1.1.1.1.1" xref="S5.Ex16.m1.1.1.1.1.cmml">log</mi><mo id="S5.Ex16.m1.2.2.2.4a" xref="S5.Ex16.m1.2.2.2.3.cmml">⁡</mo><mrow id="S5.Ex16.m1.2.2.2.4.1" xref="S5.Ex16.m1.2.2.2.3.cmml"><mo id="S5.Ex16.m1.2.2.2.4.1.1" stretchy="false" xref="S5.Ex16.m1.2.2.2.3.cmml">(</mo><mi id="S5.Ex16.m1.2.2.2.2" xref="S5.Ex16.m1.2.2.2.2.cmml">n</mi><mo id="S5.Ex16.m1.2.2.2.4.1.2" stretchy="false" xref="S5.Ex16.m1.2.2.2.3.cmml">)</mo></mrow></mrow><mrow id="S5.Ex16.m1.2.2.4" xref="S5.Ex16.m1.2.2.4.cmml"><mrow id="S5.Ex16.m1.2.2.4.2" xref="S5.Ex16.m1.2.2.4.2.cmml"><mi id="S5.Ex16.m1.2.2.4.2.1" xref="S5.Ex16.m1.2.2.4.2.1.cmml">log</mi><mo id="S5.Ex16.m1.2.2.4.2a" lspace="0.167em" xref="S5.Ex16.m1.2.2.4.2.cmml">⁡</mo><mn id="S5.Ex16.m1.2.2.4.2.2" xref="S5.Ex16.m1.2.2.4.2.2.cmml">1</mn></mrow><mo id="S5.Ex16.m1.2.2.4.1" xref="S5.Ex16.m1.2.2.4.1.cmml">+</mo><mfrac id="S5.Ex16.m1.2.2.4.3" xref="S5.Ex16.m1.2.2.4.3.cmml"><mi id="S5.Ex16.m1.2.2.4.3.2" xref="S5.Ex16.m1.2.2.4.3.2.cmml">ε</mi><mn id="S5.Ex16.m1.2.2.4.3.3" xref="S5.Ex16.m1.2.2.4.3.3.cmml">16</mn></mfrac></mrow></mfrac></mrow><mo id="S5.Ex16.m1.8.8.5" xref="S5.Ex16.m1.8.8.5.cmml">≥</mo><mrow id="S5.Ex16.m1.8.8.6" xref="S5.Ex16.m1.8.8.6.cmml"><mo id="S5.Ex16.m1.8.8.6a" xref="S5.Ex16.m1.8.8.6.cmml">−</mo><mfrac id="S5.Ex16.m1.4.4" xref="S5.Ex16.m1.4.4.cmml"><mrow id="S5.Ex16.m1.4.4.2.4" xref="S5.Ex16.m1.4.4.2.3.cmml"><mi id="S5.Ex16.m1.3.3.1.1" xref="S5.Ex16.m1.3.3.1.1.cmml">log</mi><mo id="S5.Ex16.m1.4.4.2.4a" xref="S5.Ex16.m1.4.4.2.3.cmml">⁡</mo><mrow id="S5.Ex16.m1.4.4.2.4.1" xref="S5.Ex16.m1.4.4.2.3.cmml"><mo id="S5.Ex16.m1.4.4.2.4.1.1" stretchy="false" xref="S5.Ex16.m1.4.4.2.3.cmml">(</mo><mi id="S5.Ex16.m1.4.4.2.2" xref="S5.Ex16.m1.4.4.2.2.cmml">n</mi><mo id="S5.Ex16.m1.4.4.2.4.1.2" stretchy="false" 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xref="S5.Ex16.m1.6.6.4">superscript</csymbol><ci id="S5.Ex16.m1.6.6.4.2.cmml" xref="S5.Ex16.m1.6.6.4.2">𝜀</ci><cn id="S5.Ex16.m1.6.6.4.3.cmml" type="integer" xref="S5.Ex16.m1.6.6.4.3">2</cn></apply></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Ex16.m1.8c">-\log_{1+\frac{\varepsilon}{16}}(n)=-\frac{\log(n)}{\log 1+\frac{\varepsilon}{% 16}}\geq-\frac{\log(n)}{\frac{\varepsilon}{32}}\geq-\frac{\varepsilon}{4}\cdot% {}\frac{128\log(n)}{\varepsilon^{2}}</annotation><annotation encoding="application/x-llamapun" id="S5.Ex16.m1.8d">- roman_log start_POSTSUBSCRIPT 1 + divide start_ARG italic_ε end_ARG start_ARG 16 end_ARG end_POSTSUBSCRIPT ( italic_n ) = - divide start_ARG roman_log ( italic_n ) end_ARG start_ARG roman_log 1 + divide start_ARG italic_ε end_ARG start_ARG 16 end_ARG end_ARG ≥ - divide start_ARG roman_log ( italic_n ) end_ARG start_ARG divide start_ARG italic_ε end_ARG start_ARG 32 end_ARG end_ARG ≥ - divide start_ARG italic_ε end_ARG start_ARG 4 end_ARG ⋅ divide start_ARG 128 roman_log ( italic_n ) end_ARG start_ARG italic_ε start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_left_padright"></td> </tr></tbody> </table> </div> <div class="ltx_para" id="S5.Thmtheorem2.p2"> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="A1.EGx4"> <tbody id="S5.Ex17"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_left_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle(1+\eta)^{-k}\geq(1+\frac{\varepsilon^{2}}{128\cdot c\cdot\log n}% )^{-c\cdot\log_{1+\frac{\varepsilon}{16}}(n)}\geq(1+\frac{\varepsilon^{2}}{128% \cdot c\log n})^{-\frac{\varepsilon}{4}\cdot{}\frac{128\cdot c\cdot\log(n)}{% \varepsilon^{2}}}" class="ltx_Math" display="inline" id="S5.Ex17.m1.7"><semantics id="S5.Ex17.m1.7a"><mrow id="S5.Ex17.m1.7.7" xref="S5.Ex17.m1.7.7.cmml"><msup 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xref="S5.Ex17.m1.4.4.2.4.2.3">4</cn></apply><apply id="S5.Ex17.m1.4.4.2.2.cmml" xref="S5.Ex17.m1.4.4.2.2"><divide id="S5.Ex17.m1.4.4.2.2.3.cmml" xref="S5.Ex17.m1.4.4.2.2"></divide><apply id="S5.Ex17.m1.4.4.2.2.2.cmml" xref="S5.Ex17.m1.4.4.2.2.2"><ci id="S5.Ex17.m1.4.4.2.2.2.3.cmml" xref="S5.Ex17.m1.4.4.2.2.2.3">⋅</ci><cn id="S5.Ex17.m1.4.4.2.2.2.4.cmml" type="integer" xref="S5.Ex17.m1.4.4.2.2.2.4">128</cn><ci id="S5.Ex17.m1.4.4.2.2.2.5.cmml" xref="S5.Ex17.m1.4.4.2.2.2.5">𝑐</ci><apply id="S5.Ex17.m1.4.4.2.2.2.6.1.cmml" xref="S5.Ex17.m1.4.4.2.2.2.6.2"><log id="S5.Ex17.m1.3.3.1.1.1.1.cmml" xref="S5.Ex17.m1.3.3.1.1.1.1"></log><ci id="S5.Ex17.m1.4.4.2.2.2.2.cmml" xref="S5.Ex17.m1.4.4.2.2.2.2">𝑛</ci></apply></apply><apply id="S5.Ex17.m1.4.4.2.2.4.cmml" xref="S5.Ex17.m1.4.4.2.2.4"><csymbol cd="ambiguous" id="S5.Ex17.m1.4.4.2.2.4.1.cmml" xref="S5.Ex17.m1.4.4.2.2.4">superscript</csymbol><ci id="S5.Ex17.m1.4.4.2.2.4.2.cmml" xref="S5.Ex17.m1.4.4.2.2.4.2">𝜀</ci><cn id="S5.Ex17.m1.4.4.2.2.4.3.cmml" type="integer" xref="S5.Ex17.m1.4.4.2.2.4.3">2</cn></apply></apply></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Ex17.m1.7c">\displaystyle(1+\eta)^{-k}\geq(1+\frac{\varepsilon^{2}}{128\cdot c\cdot\log n}% )^{-c\cdot\log_{1+\frac{\varepsilon}{16}}(n)}\geq(1+\frac{\varepsilon^{2}}{128% \cdot c\log n})^{-\frac{\varepsilon}{4}\cdot{}\frac{128\cdot c\cdot\log(n)}{% \varepsilon^{2}}}</annotation><annotation encoding="application/x-llamapun" id="S5.Ex17.m1.7d">( 1 + italic_η ) start_POSTSUPERSCRIPT - italic_k end_POSTSUPERSCRIPT ≥ ( 1 + divide start_ARG italic_ε start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG start_ARG 128 ⋅ italic_c ⋅ roman_log italic_n end_ARG ) start_POSTSUPERSCRIPT - italic_c ⋅ roman_log start_POSTSUBSCRIPT 1 + divide start_ARG italic_ε end_ARG start_ARG 16 end_ARG end_POSTSUBSCRIPT ( italic_n ) end_POSTSUPERSCRIPT ≥ ( 1 + divide start_ARG italic_ε start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG start_ARG 128 ⋅ italic_c roman_log italic_n end_ARG ) start_POSTSUPERSCRIPT - divide start_ARG italic_ε end_ARG start_ARG 4 end_ARG ⋅ divide start_ARG 128 ⋅ italic_c ⋅ roman_log ( italic_n ) end_ARG start_ARG italic_ε start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG end_POSTSUPERSCRIPT</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_left_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_left_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle(1+\eta)^{-k}\geq EXP\left[-\frac{\varepsilon}{4}\right]\geq EXP% \left[-\log(1+\frac{\varepsilon}{2})\right]\geq(1+\frac{\varepsilon}{2})^{-1}" class="ltx_Math" display="inline" id="S5.Ex18.m1.5"><semantics id="S5.Ex18.m1.5a"><mrow id="S5.Ex18.m1.5.5" xref="S5.Ex18.m1.5.5.cmml"><msup id="S5.Ex18.m1.2.2.1" xref="S5.Ex18.m1.2.2.1.cmml"><mrow id="S5.Ex18.m1.2.2.1.1.1" xref="S5.Ex18.m1.2.2.1.1.1.1.cmml"><mo 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xref="S5.Ex18.m1.5.5.4.1.1.1.3"></divide><ci id="S5.Ex18.m1.5.5.4.1.1.1.3.2.cmml" xref="S5.Ex18.m1.5.5.4.1.1.1.3.2">𝜀</ci><cn id="S5.Ex18.m1.5.5.4.1.1.1.3.3.cmml" type="integer" xref="S5.Ex18.m1.5.5.4.1.1.1.3.3">2</cn></apply></apply><apply id="S5.Ex18.m1.5.5.4.3.cmml" xref="S5.Ex18.m1.5.5.4.3"><minus id="S5.Ex18.m1.5.5.4.3.1.cmml" xref="S5.Ex18.m1.5.5.4.3"></minus><cn id="S5.Ex18.m1.5.5.4.3.2.cmml" type="integer" xref="S5.Ex18.m1.5.5.4.3.2">1</cn></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Ex18.m1.5c">\displaystyle(1+\eta)^{-k}\geq EXP\left[-\frac{\varepsilon}{4}\right]\geq EXP% \left[-\log(1+\frac{\varepsilon}{2})\right]\geq(1+\frac{\varepsilon}{2})^{-1}</annotation><annotation encoding="application/x-llamapun" id="S5.Ex18.m1.5d">( 1 + italic_η ) start_POSTSUPERSCRIPT - italic_k end_POSTSUPERSCRIPT ≥ italic_E italic_X italic_P [ - divide start_ARG italic_ε end_ARG start_ARG 4 end_ARG ] ≥ italic_E italic_X italic_P [ - roman_log ( 1 + divide start_ARG italic_ε end_ARG start_ARG 2 end_ARG ) ] ≥ ( 1 + divide start_ARG italic_ε end_ARG start_ARG 2 end_ARG ) start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_left_padright"></td> </tr></tbody> </table> </div> </div> <div class="ltx_para" id="S5.p2"> <p class="ltx_p" id="S5.p2.1">See <a class="ltx_ref" href="https://arxiv.org/html/2411.12694v2#S3.Thmtheorem5" title="Theorem 3.5. ‣ 3.B Results for dynamic algorithms ‣ 3 Results and organisation ‣ Local Density and its Distributed Approximation"><span class="ltx_text ltx_ref_tag">3.5</span></a></p> </div> <div class="ltx_theorem ltx_theorem_proof" 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">Proof 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.p1.2"><span class="ltx_text ltx_font_bold ltx_font_italic" id="S5.Thmtheorem3.p1.2.2">First, we show that for all vertices <math alttext="v" class="ltx_Math" display="inline" id="S5.Thmtheorem3.p1.1.1.m1.1"><semantics id="S5.Thmtheorem3.p1.1.1.m1.1a"><mi id="S5.Thmtheorem3.p1.1.1.m1.1.1" mathvariant="normal" xref="S5.Thmtheorem3.p1.1.1.m1.1.1.cmml">v</mi><annotation-xml encoding="MathML-Content" id="S5.Thmtheorem3.p1.1.1.m1.1b"><ci id="S5.Thmtheorem3.p1.1.1.m1.1.1.cmml" xref="S5.Thmtheorem3.p1.1.1.m1.1.1">v</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmtheorem3.p1.1.1.m1.1c">v</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem3.p1.1.1.m1.1d">italic_v</annotation></semantics></math>, <math alttext="\textsl{g}(v)\leq(1+\varepsilon)\rho^{*}(v)" class="ltx_Math" display="inline" id="S5.Thmtheorem3.p1.2.2.m2.3"><semantics id="S5.Thmtheorem3.p1.2.2.m2.3a"><mrow id="S5.Thmtheorem3.p1.2.2.m2.3.3" xref="S5.Thmtheorem3.p1.2.2.m2.3.3.cmml"><mrow id="S5.Thmtheorem3.p1.2.2.m2.3.3.3" xref="S5.Thmtheorem3.p1.2.2.m2.3.3.3.cmml"><mtext class="ltx_mathvariant_italic" id="S5.Thmtheorem3.p1.2.2.m2.3.3.3.2" xref="S5.Thmtheorem3.p1.2.2.m2.3.3.3.2a.cmml">g</mtext><mo id="S5.Thmtheorem3.p1.2.2.m2.3.3.3.1" xref="S5.Thmtheorem3.p1.2.2.m2.3.3.3.1.cmml">⁢</mo><mrow id="S5.Thmtheorem3.p1.2.2.m2.3.3.3.3.2" xref="S5.Thmtheorem3.p1.2.2.m2.3.3.3.cmml"><mo id="S5.Thmtheorem3.p1.2.2.m2.3.3.3.3.2.1" stretchy="false" xref="S5.Thmtheorem3.p1.2.2.m2.3.3.3.cmml">(</mo><mi id="S5.Thmtheorem3.p1.2.2.m2.1.1" mathvariant="normal" xref="S5.Thmtheorem3.p1.2.2.m2.1.1.cmml">v</mi><mo id="S5.Thmtheorem3.p1.2.2.m2.3.3.3.3.2.2" stretchy="false" xref="S5.Thmtheorem3.p1.2.2.m2.3.3.3.cmml">)</mo></mrow></mrow><mo id="S5.Thmtheorem3.p1.2.2.m2.3.3.2" xref="S5.Thmtheorem3.p1.2.2.m2.3.3.2.cmml">≤</mo><mrow id="S5.Thmtheorem3.p1.2.2.m2.3.3.1" xref="S5.Thmtheorem3.p1.2.2.m2.3.3.1.cmml"><mrow id="S5.Thmtheorem3.p1.2.2.m2.3.3.1.1.1" xref="S5.Thmtheorem3.p1.2.2.m2.3.3.1.1.1.1.cmml"><mo id="S5.Thmtheorem3.p1.2.2.m2.3.3.1.1.1.2" stretchy="false" xref="S5.Thmtheorem3.p1.2.2.m2.3.3.1.1.1.1.cmml">(</mo><mrow id="S5.Thmtheorem3.p1.2.2.m2.3.3.1.1.1.1" xref="S5.Thmtheorem3.p1.2.2.m2.3.3.1.1.1.1.cmml"><mn id="S5.Thmtheorem3.p1.2.2.m2.3.3.1.1.1.1.2" xref="S5.Thmtheorem3.p1.2.2.m2.3.3.1.1.1.1.2.cmml">1</mn><mo id="S5.Thmtheorem3.p1.2.2.m2.3.3.1.1.1.1.1" xref="S5.Thmtheorem3.p1.2.2.m2.3.3.1.1.1.1.1.cmml">+</mo><mi id="S5.Thmtheorem3.p1.2.2.m2.3.3.1.1.1.1.3" mathvariant="normal" xref="S5.Thmtheorem3.p1.2.2.m2.3.3.1.1.1.1.3.cmml">ε</mi></mrow><mo id="S5.Thmtheorem3.p1.2.2.m2.3.3.1.1.1.3" stretchy="false" xref="S5.Thmtheorem3.p1.2.2.m2.3.3.1.1.1.1.cmml">)</mo></mrow><mo id="S5.Thmtheorem3.p1.2.2.m2.3.3.1.2" xref="S5.Thmtheorem3.p1.2.2.m2.3.3.1.2.cmml">⁢</mo><msup id="S5.Thmtheorem3.p1.2.2.m2.3.3.1.3" xref="S5.Thmtheorem3.p1.2.2.m2.3.3.1.3.cmml"><mi id="S5.Thmtheorem3.p1.2.2.m2.3.3.1.3.2" mathvariant="normal" xref="S5.Thmtheorem3.p1.2.2.m2.3.3.1.3.2.cmml">ρ</mi><mo id="S5.Thmtheorem3.p1.2.2.m2.3.3.1.3.3" xref="S5.Thmtheorem3.p1.2.2.m2.3.3.1.3.3.cmml">∗</mo></msup><mo id="S5.Thmtheorem3.p1.2.2.m2.3.3.1.2a" xref="S5.Thmtheorem3.p1.2.2.m2.3.3.1.2.cmml">⁢</mo><mrow id="S5.Thmtheorem3.p1.2.2.m2.3.3.1.4.2" xref="S5.Thmtheorem3.p1.2.2.m2.3.3.1.cmml"><mo id="S5.Thmtheorem3.p1.2.2.m2.3.3.1.4.2.1" stretchy="false" xref="S5.Thmtheorem3.p1.2.2.m2.3.3.1.cmml">(</mo><mi id="S5.Thmtheorem3.p1.2.2.m2.2.2" mathvariant="normal" xref="S5.Thmtheorem3.p1.2.2.m2.2.2.cmml">v</mi><mo id="S5.Thmtheorem3.p1.2.2.m2.3.3.1.4.2.2" stretchy="false" xref="S5.Thmtheorem3.p1.2.2.m2.3.3.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.Thmtheorem3.p1.2.2.m2.3b"><apply id="S5.Thmtheorem3.p1.2.2.m2.3.3.cmml" xref="S5.Thmtheorem3.p1.2.2.m2.3.3"><leq id="S5.Thmtheorem3.p1.2.2.m2.3.3.2.cmml" xref="S5.Thmtheorem3.p1.2.2.m2.3.3.2"></leq><apply id="S5.Thmtheorem3.p1.2.2.m2.3.3.3.cmml" xref="S5.Thmtheorem3.p1.2.2.m2.3.3.3"><times id="S5.Thmtheorem3.p1.2.2.m2.3.3.3.1.cmml" xref="S5.Thmtheorem3.p1.2.2.m2.3.3.3.1"></times><ci id="S5.Thmtheorem3.p1.2.2.m2.3.3.3.2a.cmml" xref="S5.Thmtheorem3.p1.2.2.m2.3.3.3.2"><mtext class="ltx_mathvariant_italic" id="S5.Thmtheorem3.p1.2.2.m2.3.3.3.2.cmml" xref="S5.Thmtheorem3.p1.2.2.m2.3.3.3.2">g</mtext></ci><ci id="S5.Thmtheorem3.p1.2.2.m2.1.1.cmml" xref="S5.Thmtheorem3.p1.2.2.m2.1.1">v</ci></apply><apply id="S5.Thmtheorem3.p1.2.2.m2.3.3.1.cmml" xref="S5.Thmtheorem3.p1.2.2.m2.3.3.1"><times id="S5.Thmtheorem3.p1.2.2.m2.3.3.1.2.cmml" xref="S5.Thmtheorem3.p1.2.2.m2.3.3.1.2"></times><apply id="S5.Thmtheorem3.p1.2.2.m2.3.3.1.1.1.1.cmml" xref="S5.Thmtheorem3.p1.2.2.m2.3.3.1.1.1"><plus id="S5.Thmtheorem3.p1.2.2.m2.3.3.1.1.1.1.1.cmml" xref="S5.Thmtheorem3.p1.2.2.m2.3.3.1.1.1.1.1"></plus><cn id="S5.Thmtheorem3.p1.2.2.m2.3.3.1.1.1.1.2.cmml" type="integer" xref="S5.Thmtheorem3.p1.2.2.m2.3.3.1.1.1.1.2">1</cn><ci id="S5.Thmtheorem3.p1.2.2.m2.3.3.1.1.1.1.3.cmml" xref="S5.Thmtheorem3.p1.2.2.m2.3.3.1.1.1.1.3">ε</ci></apply><apply id="S5.Thmtheorem3.p1.2.2.m2.3.3.1.3.cmml" xref="S5.Thmtheorem3.p1.2.2.m2.3.3.1.3"><csymbol cd="ambiguous" id="S5.Thmtheorem3.p1.2.2.m2.3.3.1.3.1.cmml" xref="S5.Thmtheorem3.p1.2.2.m2.3.3.1.3">superscript</csymbol><ci id="S5.Thmtheorem3.p1.2.2.m2.3.3.1.3.2.cmml" xref="S5.Thmtheorem3.p1.2.2.m2.3.3.1.3.2">ρ</ci><times id="S5.Thmtheorem3.p1.2.2.m2.3.3.1.3.3.cmml" xref="S5.Thmtheorem3.p1.2.2.m2.3.3.1.3.3"></times></apply><ci id="S5.Thmtheorem3.p1.2.2.m2.2.2.cmml" xref="S5.Thmtheorem3.p1.2.2.m2.2.2">v</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmtheorem3.p1.2.2.m2.3c">\textsl{g}(v)\leq(1+\varepsilon)\rho^{*}(v)</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem3.p1.2.2.m2.3d">g ( italic_v ) ≤ ( 1 + italic_ε ) italic_ρ start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ( italic_v )</annotation></semantics></math>.<span class="ltx_text ltx_font_medium" id="S5.Thmtheorem3.p1.2.2.1"></span></span></p> </div> <div class="ltx_para" id="S5.Thmtheorem3.p2"> <p class="ltx_p" id="S5.Thmtheorem3.p2.9"><span class="ltx_text ltx_font_italic" id="S5.Thmtheorem3.p2.9.9">Suppose for the sake of contradiction that there exists a vertex <math alttext="u" class="ltx_Math" display="inline" id="S5.Thmtheorem3.p2.1.1.m1.1"><semantics id="S5.Thmtheorem3.p2.1.1.m1.1a"><mi id="S5.Thmtheorem3.p2.1.1.m1.1.1" xref="S5.Thmtheorem3.p2.1.1.m1.1.1.cmml">u</mi><annotation-xml encoding="MathML-Content" id="S5.Thmtheorem3.p2.1.1.m1.1b"><ci id="S5.Thmtheorem3.p2.1.1.m1.1.1.cmml" xref="S5.Thmtheorem3.p2.1.1.m1.1.1">𝑢</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmtheorem3.p2.1.1.m1.1c">u</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem3.p2.1.1.m1.1d">italic_u</annotation></semantics></math> with <math alttext="\textsl{g}(u)>(1+\varepsilon)\rho^{*}(u)" class="ltx_Math" display="inline" id="S5.Thmtheorem3.p2.2.2.m2.3"><semantics id="S5.Thmtheorem3.p2.2.2.m2.3a"><mrow id="S5.Thmtheorem3.p2.2.2.m2.3.3" xref="S5.Thmtheorem3.p2.2.2.m2.3.3.cmml"><mrow id="S5.Thmtheorem3.p2.2.2.m2.3.3.3" xref="S5.Thmtheorem3.p2.2.2.m2.3.3.3.cmml"><mtext class="ltx_mathvariant_italic" id="S5.Thmtheorem3.p2.2.2.m2.3.3.3.2" xref="S5.Thmtheorem3.p2.2.2.m2.3.3.3.2a.cmml">g</mtext><mo id="S5.Thmtheorem3.p2.2.2.m2.3.3.3.1" xref="S5.Thmtheorem3.p2.2.2.m2.3.3.3.1.cmml">⁢</mo><mrow id="S5.Thmtheorem3.p2.2.2.m2.3.3.3.3.2" xref="S5.Thmtheorem3.p2.2.2.m2.3.3.3.cmml"><mo id="S5.Thmtheorem3.p2.2.2.m2.3.3.3.3.2.1" stretchy="false" xref="S5.Thmtheorem3.p2.2.2.m2.3.3.3.cmml">(</mo><mi id="S5.Thmtheorem3.p2.2.2.m2.1.1" xref="S5.Thmtheorem3.p2.2.2.m2.1.1.cmml">u</mi><mo id="S5.Thmtheorem3.p2.2.2.m2.3.3.3.3.2.2" stretchy="false" xref="S5.Thmtheorem3.p2.2.2.m2.3.3.3.cmml">)</mo></mrow></mrow><mo id="S5.Thmtheorem3.p2.2.2.m2.3.3.2" xref="S5.Thmtheorem3.p2.2.2.m2.3.3.2.cmml">></mo><mrow id="S5.Thmtheorem3.p2.2.2.m2.3.3.1" xref="S5.Thmtheorem3.p2.2.2.m2.3.3.1.cmml"><mrow id="S5.Thmtheorem3.p2.2.2.m2.3.3.1.1.1" xref="S5.Thmtheorem3.p2.2.2.m2.3.3.1.1.1.1.cmml"><mo id="S5.Thmtheorem3.p2.2.2.m2.3.3.1.1.1.2" stretchy="false" xref="S5.Thmtheorem3.p2.2.2.m2.3.3.1.1.1.1.cmml">(</mo><mrow id="S5.Thmtheorem3.p2.2.2.m2.3.3.1.1.1.1" xref="S5.Thmtheorem3.p2.2.2.m2.3.3.1.1.1.1.cmml"><mn id="S5.Thmtheorem3.p2.2.2.m2.3.3.1.1.1.1.2" xref="S5.Thmtheorem3.p2.2.2.m2.3.3.1.1.1.1.2.cmml">1</mn><mo id="S5.Thmtheorem3.p2.2.2.m2.3.3.1.1.1.1.1" xref="S5.Thmtheorem3.p2.2.2.m2.3.3.1.1.1.1.1.cmml">+</mo><mi id="S5.Thmtheorem3.p2.2.2.m2.3.3.1.1.1.1.3" xref="S5.Thmtheorem3.p2.2.2.m2.3.3.1.1.1.1.3.cmml">ε</mi></mrow><mo id="S5.Thmtheorem3.p2.2.2.m2.3.3.1.1.1.3" stretchy="false" xref="S5.Thmtheorem3.p2.2.2.m2.3.3.1.1.1.1.cmml">)</mo></mrow><mo id="S5.Thmtheorem3.p2.2.2.m2.3.3.1.2" xref="S5.Thmtheorem3.p2.2.2.m2.3.3.1.2.cmml">⁢</mo><msup id="S5.Thmtheorem3.p2.2.2.m2.3.3.1.3" xref="S5.Thmtheorem3.p2.2.2.m2.3.3.1.3.cmml"><mi id="S5.Thmtheorem3.p2.2.2.m2.3.3.1.3.2" xref="S5.Thmtheorem3.p2.2.2.m2.3.3.1.3.2.cmml">ρ</mi><mo id="S5.Thmtheorem3.p2.2.2.m2.3.3.1.3.3" xref="S5.Thmtheorem3.p2.2.2.m2.3.3.1.3.3.cmml">∗</mo></msup><mo id="S5.Thmtheorem3.p2.2.2.m2.3.3.1.2a" xref="S5.Thmtheorem3.p2.2.2.m2.3.3.1.2.cmml">⁢</mo><mrow id="S5.Thmtheorem3.p2.2.2.m2.3.3.1.4.2" xref="S5.Thmtheorem3.p2.2.2.m2.3.3.1.cmml"><mo id="S5.Thmtheorem3.p2.2.2.m2.3.3.1.4.2.1" stretchy="false" xref="S5.Thmtheorem3.p2.2.2.m2.3.3.1.cmml">(</mo><mi id="S5.Thmtheorem3.p2.2.2.m2.2.2" xref="S5.Thmtheorem3.p2.2.2.m2.2.2.cmml">u</mi><mo id="S5.Thmtheorem3.p2.2.2.m2.3.3.1.4.2.2" stretchy="false" xref="S5.Thmtheorem3.p2.2.2.m2.3.3.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.Thmtheorem3.p2.2.2.m2.3b"><apply id="S5.Thmtheorem3.p2.2.2.m2.3.3.cmml" xref="S5.Thmtheorem3.p2.2.2.m2.3.3"><gt id="S5.Thmtheorem3.p2.2.2.m2.3.3.2.cmml" xref="S5.Thmtheorem3.p2.2.2.m2.3.3.2"></gt><apply id="S5.Thmtheorem3.p2.2.2.m2.3.3.3.cmml" xref="S5.Thmtheorem3.p2.2.2.m2.3.3.3"><times id="S5.Thmtheorem3.p2.2.2.m2.3.3.3.1.cmml" xref="S5.Thmtheorem3.p2.2.2.m2.3.3.3.1"></times><ci id="S5.Thmtheorem3.p2.2.2.m2.3.3.3.2a.cmml" xref="S5.Thmtheorem3.p2.2.2.m2.3.3.3.2"><mtext class="ltx_mathvariant_italic" id="S5.Thmtheorem3.p2.2.2.m2.3.3.3.2.cmml" xref="S5.Thmtheorem3.p2.2.2.m2.3.3.3.2">g</mtext></ci><ci id="S5.Thmtheorem3.p2.2.2.m2.1.1.cmml" xref="S5.Thmtheorem3.p2.2.2.m2.1.1">𝑢</ci></apply><apply id="S5.Thmtheorem3.p2.2.2.m2.3.3.1.cmml" xref="S5.Thmtheorem3.p2.2.2.m2.3.3.1"><times id="S5.Thmtheorem3.p2.2.2.m2.3.3.1.2.cmml" xref="S5.Thmtheorem3.p2.2.2.m2.3.3.1.2"></times><apply id="S5.Thmtheorem3.p2.2.2.m2.3.3.1.1.1.1.cmml" xref="S5.Thmtheorem3.p2.2.2.m2.3.3.1.1.1"><plus id="S5.Thmtheorem3.p2.2.2.m2.3.3.1.1.1.1.1.cmml" xref="S5.Thmtheorem3.p2.2.2.m2.3.3.1.1.1.1.1"></plus><cn id="S5.Thmtheorem3.p2.2.2.m2.3.3.1.1.1.1.2.cmml" type="integer" xref="S5.Thmtheorem3.p2.2.2.m2.3.3.1.1.1.1.2">1</cn><ci id="S5.Thmtheorem3.p2.2.2.m2.3.3.1.1.1.1.3.cmml" xref="S5.Thmtheorem3.p2.2.2.m2.3.3.1.1.1.1.3">𝜀</ci></apply><apply id="S5.Thmtheorem3.p2.2.2.m2.3.3.1.3.cmml" xref="S5.Thmtheorem3.p2.2.2.m2.3.3.1.3"><csymbol cd="ambiguous" id="S5.Thmtheorem3.p2.2.2.m2.3.3.1.3.1.cmml" xref="S5.Thmtheorem3.p2.2.2.m2.3.3.1.3">superscript</csymbol><ci id="S5.Thmtheorem3.p2.2.2.m2.3.3.1.3.2.cmml" xref="S5.Thmtheorem3.p2.2.2.m2.3.3.1.3.2">𝜌</ci><times id="S5.Thmtheorem3.p2.2.2.m2.3.3.1.3.3.cmml" xref="S5.Thmtheorem3.p2.2.2.m2.3.3.1.3.3"></times></apply><ci id="S5.Thmtheorem3.p2.2.2.m2.2.2.cmml" xref="S5.Thmtheorem3.p2.2.2.m2.2.2">𝑢</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmtheorem3.p2.2.2.m2.3c">\textsl{g}(u)>(1+\varepsilon)\rho^{*}(u)</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem3.p2.2.2.m2.3d">g ( italic_u ) > ( 1 + italic_ε ) italic_ρ start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ( italic_u )</annotation></semantics></math>. We fix <math alttext="\rho^{*}(u)=\textsl{g}^{*}(u)=\gamma" class="ltx_Math" display="inline" id="S5.Thmtheorem3.p2.3.3.m3.2"><semantics id="S5.Thmtheorem3.p2.3.3.m3.2a"><mrow id="S5.Thmtheorem3.p2.3.3.m3.2.3" xref="S5.Thmtheorem3.p2.3.3.m3.2.3.cmml"><mrow id="S5.Thmtheorem3.p2.3.3.m3.2.3.2" xref="S5.Thmtheorem3.p2.3.3.m3.2.3.2.cmml"><msup id="S5.Thmtheorem3.p2.3.3.m3.2.3.2.2" xref="S5.Thmtheorem3.p2.3.3.m3.2.3.2.2.cmml"><mi id="S5.Thmtheorem3.p2.3.3.m3.2.3.2.2.2" xref="S5.Thmtheorem3.p2.3.3.m3.2.3.2.2.2.cmml">ρ</mi><mo id="S5.Thmtheorem3.p2.3.3.m3.2.3.2.2.3" xref="S5.Thmtheorem3.p2.3.3.m3.2.3.2.2.3.cmml">∗</mo></msup><mo id="S5.Thmtheorem3.p2.3.3.m3.2.3.2.1" xref="S5.Thmtheorem3.p2.3.3.m3.2.3.2.1.cmml">⁢</mo><mrow id="S5.Thmtheorem3.p2.3.3.m3.2.3.2.3.2" xref="S5.Thmtheorem3.p2.3.3.m3.2.3.2.cmml"><mo id="S5.Thmtheorem3.p2.3.3.m3.2.3.2.3.2.1" stretchy="false" xref="S5.Thmtheorem3.p2.3.3.m3.2.3.2.cmml">(</mo><mi id="S5.Thmtheorem3.p2.3.3.m3.1.1" xref="S5.Thmtheorem3.p2.3.3.m3.1.1.cmml">u</mi><mo id="S5.Thmtheorem3.p2.3.3.m3.2.3.2.3.2.2" stretchy="false" xref="S5.Thmtheorem3.p2.3.3.m3.2.3.2.cmml">)</mo></mrow></mrow><mo id="S5.Thmtheorem3.p2.3.3.m3.2.3.3" xref="S5.Thmtheorem3.p2.3.3.m3.2.3.3.cmml">=</mo><mrow id="S5.Thmtheorem3.p2.3.3.m3.2.3.4" xref="S5.Thmtheorem3.p2.3.3.m3.2.3.4.cmml"><msup id="S5.Thmtheorem3.p2.3.3.m3.2.3.4.2" xref="S5.Thmtheorem3.p2.3.3.m3.2.3.4.2.cmml"><mtext class="ltx_mathvariant_italic" id="S5.Thmtheorem3.p2.3.3.m3.2.3.4.2.2" xref="S5.Thmtheorem3.p2.3.3.m3.2.3.4.2.2a.cmml">g</mtext><mo id="S5.Thmtheorem3.p2.3.3.m3.2.3.4.2.3" xref="S5.Thmtheorem3.p2.3.3.m3.2.3.4.2.3.cmml">∗</mo></msup><mo id="S5.Thmtheorem3.p2.3.3.m3.2.3.4.1" xref="S5.Thmtheorem3.p2.3.3.m3.2.3.4.1.cmml">⁢</mo><mrow id="S5.Thmtheorem3.p2.3.3.m3.2.3.4.3.2" xref="S5.Thmtheorem3.p2.3.3.m3.2.3.4.cmml"><mo id="S5.Thmtheorem3.p2.3.3.m3.2.3.4.3.2.1" stretchy="false" xref="S5.Thmtheorem3.p2.3.3.m3.2.3.4.cmml">(</mo><mi id="S5.Thmtheorem3.p2.3.3.m3.2.2" xref="S5.Thmtheorem3.p2.3.3.m3.2.2.cmml">u</mi><mo id="S5.Thmtheorem3.p2.3.3.m3.2.3.4.3.2.2" stretchy="false" xref="S5.Thmtheorem3.p2.3.3.m3.2.3.4.cmml">)</mo></mrow></mrow><mo id="S5.Thmtheorem3.p2.3.3.m3.2.3.5" xref="S5.Thmtheorem3.p2.3.3.m3.2.3.5.cmml">=</mo><mi id="S5.Thmtheorem3.p2.3.3.m3.2.3.6" xref="S5.Thmtheorem3.p2.3.3.m3.2.3.6.cmml">γ</mi></mrow><annotation-xml encoding="MathML-Content" id="S5.Thmtheorem3.p2.3.3.m3.2b"><apply id="S5.Thmtheorem3.p2.3.3.m3.2.3.cmml" xref="S5.Thmtheorem3.p2.3.3.m3.2.3"><and id="S5.Thmtheorem3.p2.3.3.m3.2.3a.cmml" xref="S5.Thmtheorem3.p2.3.3.m3.2.3"></and><apply id="S5.Thmtheorem3.p2.3.3.m3.2.3b.cmml" xref="S5.Thmtheorem3.p2.3.3.m3.2.3"><eq id="S5.Thmtheorem3.p2.3.3.m3.2.3.3.cmml" xref="S5.Thmtheorem3.p2.3.3.m3.2.3.3"></eq><apply id="S5.Thmtheorem3.p2.3.3.m3.2.3.2.cmml" xref="S5.Thmtheorem3.p2.3.3.m3.2.3.2"><times id="S5.Thmtheorem3.p2.3.3.m3.2.3.2.1.cmml" xref="S5.Thmtheorem3.p2.3.3.m3.2.3.2.1"></times><apply id="S5.Thmtheorem3.p2.3.3.m3.2.3.2.2.cmml" xref="S5.Thmtheorem3.p2.3.3.m3.2.3.2.2"><csymbol cd="ambiguous" id="S5.Thmtheorem3.p2.3.3.m3.2.3.2.2.1.cmml" xref="S5.Thmtheorem3.p2.3.3.m3.2.3.2.2">superscript</csymbol><ci id="S5.Thmtheorem3.p2.3.3.m3.2.3.2.2.2.cmml" xref="S5.Thmtheorem3.p2.3.3.m3.2.3.2.2.2">𝜌</ci><times id="S5.Thmtheorem3.p2.3.3.m3.2.3.2.2.3.cmml" xref="S5.Thmtheorem3.p2.3.3.m3.2.3.2.2.3"></times></apply><ci id="S5.Thmtheorem3.p2.3.3.m3.1.1.cmml" xref="S5.Thmtheorem3.p2.3.3.m3.1.1">𝑢</ci></apply><apply id="S5.Thmtheorem3.p2.3.3.m3.2.3.4.cmml" xref="S5.Thmtheorem3.p2.3.3.m3.2.3.4"><times id="S5.Thmtheorem3.p2.3.3.m3.2.3.4.1.cmml" xref="S5.Thmtheorem3.p2.3.3.m3.2.3.4.1"></times><apply id="S5.Thmtheorem3.p2.3.3.m3.2.3.4.2.cmml" xref="S5.Thmtheorem3.p2.3.3.m3.2.3.4.2"><csymbol cd="ambiguous" id="S5.Thmtheorem3.p2.3.3.m3.2.3.4.2.1.cmml" xref="S5.Thmtheorem3.p2.3.3.m3.2.3.4.2">superscript</csymbol><ci id="S5.Thmtheorem3.p2.3.3.m3.2.3.4.2.2a.cmml" xref="S5.Thmtheorem3.p2.3.3.m3.2.3.4.2.2"><mtext class="ltx_mathvariant_italic" id="S5.Thmtheorem3.p2.3.3.m3.2.3.4.2.2.cmml" xref="S5.Thmtheorem3.p2.3.3.m3.2.3.4.2.2">g</mtext></ci><times id="S5.Thmtheorem3.p2.3.3.m3.2.3.4.2.3.cmml" xref="S5.Thmtheorem3.p2.3.3.m3.2.3.4.2.3"></times></apply><ci id="S5.Thmtheorem3.p2.3.3.m3.2.2.cmml" xref="S5.Thmtheorem3.p2.3.3.m3.2.2">𝑢</ci></apply></apply><apply id="S5.Thmtheorem3.p2.3.3.m3.2.3c.cmml" xref="S5.Thmtheorem3.p2.3.3.m3.2.3"><eq id="S5.Thmtheorem3.p2.3.3.m3.2.3.5.cmml" xref="S5.Thmtheorem3.p2.3.3.m3.2.3.5"></eq><share href="https://arxiv.org/html/2411.12694v2#S5.Thmtheorem3.p2.3.3.m3.2.3.4.cmml" id="S5.Thmtheorem3.p2.3.3.m3.2.3d.cmml" xref="S5.Thmtheorem3.p2.3.3.m3.2.3"></share><ci id="S5.Thmtheorem3.p2.3.3.m3.2.3.6.cmml" xref="S5.Thmtheorem3.p2.3.3.m3.2.3.6">𝛾</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmtheorem3.p2.3.3.m3.2c">\rho^{*}(u)=\textsl{g}^{*}(u)=\gamma</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem3.p2.3.3.m3.2d">italic_ρ start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ( italic_u ) = g start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ( italic_u ) = italic_γ</annotation></semantics></math> and work with <math alttext="\gamma" class="ltx_Math" display="inline" id="S5.Thmtheorem3.p2.4.4.m4.1"><semantics id="S5.Thmtheorem3.p2.4.4.m4.1a"><mi id="S5.Thmtheorem3.p2.4.4.m4.1.1" xref="S5.Thmtheorem3.p2.4.4.m4.1.1.cmml">γ</mi><annotation-xml encoding="MathML-Content" id="S5.Thmtheorem3.p2.4.4.m4.1b"><ci id="S5.Thmtheorem3.p2.4.4.m4.1.1.cmml" xref="S5.Thmtheorem3.p2.4.4.m4.1.1">𝛾</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmtheorem3.p2.4.4.m4.1c">\gamma</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem3.p2.4.4.m4.1d">italic_γ</annotation></semantics></math> throughout the remainder of this proof to show a contradiction. By Corollary <a class="ltx_ref" href="https://arxiv.org/html/2411.12694v2#S3.Thmtheorem4" title="Corollary 3.4. ‣ 3.A Conceptual results for local density ‣ 3 Results and organisation ‣ Local Density and its Distributed Approximation"><span class="ltx_text ltx_ref_tag">3.4</span></a>, there exists at least one locally fair fractional orientation <math alttext="\overrightarrow{G}_{x}" class="ltx_Math" display="inline" id="S5.Thmtheorem3.p2.5.5.m5.1"><semantics id="S5.Thmtheorem3.p2.5.5.m5.1a"><msub id="S5.Thmtheorem3.p2.5.5.m5.1.1" xref="S5.Thmtheorem3.p2.5.5.m5.1.1.cmml"><mover accent="true" id="S5.Thmtheorem3.p2.5.5.m5.1.1.2" xref="S5.Thmtheorem3.p2.5.5.m5.1.1.2.cmml"><mi id="S5.Thmtheorem3.p2.5.5.m5.1.1.2.2" xref="S5.Thmtheorem3.p2.5.5.m5.1.1.2.2.cmml">G</mi><mo id="S5.Thmtheorem3.p2.5.5.m5.1.1.2.1" stretchy="false" xref="S5.Thmtheorem3.p2.5.5.m5.1.1.2.1.cmml">→</mo></mover><mi id="S5.Thmtheorem3.p2.5.5.m5.1.1.3" xref="S5.Thmtheorem3.p2.5.5.m5.1.1.3.cmml">x</mi></msub><annotation-xml encoding="MathML-Content" id="S5.Thmtheorem3.p2.5.5.m5.1b"><apply id="S5.Thmtheorem3.p2.5.5.m5.1.1.cmml" xref="S5.Thmtheorem3.p2.5.5.m5.1.1"><csymbol cd="ambiguous" id="S5.Thmtheorem3.p2.5.5.m5.1.1.1.cmml" xref="S5.Thmtheorem3.p2.5.5.m5.1.1">subscript</csymbol><apply id="S5.Thmtheorem3.p2.5.5.m5.1.1.2.cmml" xref="S5.Thmtheorem3.p2.5.5.m5.1.1.2"><ci id="S5.Thmtheorem3.p2.5.5.m5.1.1.2.1.cmml" xref="S5.Thmtheorem3.p2.5.5.m5.1.1.2.1">→</ci><ci id="S5.Thmtheorem3.p2.5.5.m5.1.1.2.2.cmml" xref="S5.Thmtheorem3.p2.5.5.m5.1.1.2.2">𝐺</ci></apply><ci id="S5.Thmtheorem3.p2.5.5.m5.1.1.3.cmml" xref="S5.Thmtheorem3.p2.5.5.m5.1.1.3">𝑥</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmtheorem3.p2.5.5.m5.1c">\overrightarrow{G}_{x}</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem3.p2.5.5.m5.1d">over→ start_ARG italic_G end_ARG start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT</annotation></semantics></math>. By Corollary <a class="ltx_ref" href="https://arxiv.org/html/2411.12694v2#S3.Thmtheorem3" title="Corollary 3.3. ‣ 3.A Conceptual results for local density ‣ 3 Results and organisation ‣ Local Density and its Distributed Approximation"><span class="ltx_text ltx_ref_tag">3.3</span></a>, every vertex <math alttext="v" class="ltx_Math" display="inline" id="S5.Thmtheorem3.p2.6.6.m6.1"><semantics id="S5.Thmtheorem3.p2.6.6.m6.1a"><mi id="S5.Thmtheorem3.p2.6.6.m6.1.1" xref="S5.Thmtheorem3.p2.6.6.m6.1.1.cmml">v</mi><annotation-xml encoding="MathML-Content" id="S5.Thmtheorem3.p2.6.6.m6.1b"><ci id="S5.Thmtheorem3.p2.6.6.m6.1.1.cmml" xref="S5.Thmtheorem3.p2.6.6.m6.1.1">𝑣</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmtheorem3.p2.6.6.m6.1c">v</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem3.p2.6.6.m6.1d">italic_v</annotation></semantics></math> in this orientation has out-degree <math alttext="\textsl{g}_{x}(v)=\textsl{g}^{*}(v)=\rho^{*}(v)" class="ltx_Math" display="inline" id="S5.Thmtheorem3.p2.7.7.m7.3"><semantics id="S5.Thmtheorem3.p2.7.7.m7.3a"><mrow id="S5.Thmtheorem3.p2.7.7.m7.3.4" xref="S5.Thmtheorem3.p2.7.7.m7.3.4.cmml"><mrow id="S5.Thmtheorem3.p2.7.7.m7.3.4.2" xref="S5.Thmtheorem3.p2.7.7.m7.3.4.2.cmml"><msub id="S5.Thmtheorem3.p2.7.7.m7.3.4.2.2" xref="S5.Thmtheorem3.p2.7.7.m7.3.4.2.2.cmml"><mtext class="ltx_mathvariant_italic" id="S5.Thmtheorem3.p2.7.7.m7.3.4.2.2.2" xref="S5.Thmtheorem3.p2.7.7.m7.3.4.2.2.2a.cmml">g</mtext><mi id="S5.Thmtheorem3.p2.7.7.m7.3.4.2.2.3" xref="S5.Thmtheorem3.p2.7.7.m7.3.4.2.2.3.cmml">x</mi></msub><mo id="S5.Thmtheorem3.p2.7.7.m7.3.4.2.1" xref="S5.Thmtheorem3.p2.7.7.m7.3.4.2.1.cmml">⁢</mo><mrow id="S5.Thmtheorem3.p2.7.7.m7.3.4.2.3.2" xref="S5.Thmtheorem3.p2.7.7.m7.3.4.2.cmml"><mo id="S5.Thmtheorem3.p2.7.7.m7.3.4.2.3.2.1" stretchy="false" xref="S5.Thmtheorem3.p2.7.7.m7.3.4.2.cmml">(</mo><mi id="S5.Thmtheorem3.p2.7.7.m7.1.1" xref="S5.Thmtheorem3.p2.7.7.m7.1.1.cmml">v</mi><mo id="S5.Thmtheorem3.p2.7.7.m7.3.4.2.3.2.2" stretchy="false" xref="S5.Thmtheorem3.p2.7.7.m7.3.4.2.cmml">)</mo></mrow></mrow><mo id="S5.Thmtheorem3.p2.7.7.m7.3.4.3" xref="S5.Thmtheorem3.p2.7.7.m7.3.4.3.cmml">=</mo><mrow id="S5.Thmtheorem3.p2.7.7.m7.3.4.4" xref="S5.Thmtheorem3.p2.7.7.m7.3.4.4.cmml"><msup id="S5.Thmtheorem3.p2.7.7.m7.3.4.4.2" xref="S5.Thmtheorem3.p2.7.7.m7.3.4.4.2.cmml"><mtext class="ltx_mathvariant_italic" id="S5.Thmtheorem3.p2.7.7.m7.3.4.4.2.2" xref="S5.Thmtheorem3.p2.7.7.m7.3.4.4.2.2a.cmml">g</mtext><mo id="S5.Thmtheorem3.p2.7.7.m7.3.4.4.2.3" xref="S5.Thmtheorem3.p2.7.7.m7.3.4.4.2.3.cmml">∗</mo></msup><mo id="S5.Thmtheorem3.p2.7.7.m7.3.4.4.1" xref="S5.Thmtheorem3.p2.7.7.m7.3.4.4.1.cmml">⁢</mo><mrow id="S5.Thmtheorem3.p2.7.7.m7.3.4.4.3.2" xref="S5.Thmtheorem3.p2.7.7.m7.3.4.4.cmml"><mo id="S5.Thmtheorem3.p2.7.7.m7.3.4.4.3.2.1" stretchy="false" xref="S5.Thmtheorem3.p2.7.7.m7.3.4.4.cmml">(</mo><mi id="S5.Thmtheorem3.p2.7.7.m7.2.2" xref="S5.Thmtheorem3.p2.7.7.m7.2.2.cmml">v</mi><mo id="S5.Thmtheorem3.p2.7.7.m7.3.4.4.3.2.2" stretchy="false" xref="S5.Thmtheorem3.p2.7.7.m7.3.4.4.cmml">)</mo></mrow></mrow><mo id="S5.Thmtheorem3.p2.7.7.m7.3.4.5" xref="S5.Thmtheorem3.p2.7.7.m7.3.4.5.cmml">=</mo><mrow id="S5.Thmtheorem3.p2.7.7.m7.3.4.6" xref="S5.Thmtheorem3.p2.7.7.m7.3.4.6.cmml"><msup id="S5.Thmtheorem3.p2.7.7.m7.3.4.6.2" xref="S5.Thmtheorem3.p2.7.7.m7.3.4.6.2.cmml"><mi id="S5.Thmtheorem3.p2.7.7.m7.3.4.6.2.2" xref="S5.Thmtheorem3.p2.7.7.m7.3.4.6.2.2.cmml">ρ</mi><mo id="S5.Thmtheorem3.p2.7.7.m7.3.4.6.2.3" xref="S5.Thmtheorem3.p2.7.7.m7.3.4.6.2.3.cmml">∗</mo></msup><mo id="S5.Thmtheorem3.p2.7.7.m7.3.4.6.1" xref="S5.Thmtheorem3.p2.7.7.m7.3.4.6.1.cmml">⁢</mo><mrow id="S5.Thmtheorem3.p2.7.7.m7.3.4.6.3.2" xref="S5.Thmtheorem3.p2.7.7.m7.3.4.6.cmml"><mo id="S5.Thmtheorem3.p2.7.7.m7.3.4.6.3.2.1" stretchy="false" xref="S5.Thmtheorem3.p2.7.7.m7.3.4.6.cmml">(</mo><mi id="S5.Thmtheorem3.p2.7.7.m7.3.3" xref="S5.Thmtheorem3.p2.7.7.m7.3.3.cmml">v</mi><mo id="S5.Thmtheorem3.p2.7.7.m7.3.4.6.3.2.2" stretchy="false" xref="S5.Thmtheorem3.p2.7.7.m7.3.4.6.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.Thmtheorem3.p2.7.7.m7.3b"><apply id="S5.Thmtheorem3.p2.7.7.m7.3.4.cmml" xref="S5.Thmtheorem3.p2.7.7.m7.3.4"><and id="S5.Thmtheorem3.p2.7.7.m7.3.4a.cmml" xref="S5.Thmtheorem3.p2.7.7.m7.3.4"></and><apply id="S5.Thmtheorem3.p2.7.7.m7.3.4b.cmml" xref="S5.Thmtheorem3.p2.7.7.m7.3.4"><eq id="S5.Thmtheorem3.p2.7.7.m7.3.4.3.cmml" xref="S5.Thmtheorem3.p2.7.7.m7.3.4.3"></eq><apply id="S5.Thmtheorem3.p2.7.7.m7.3.4.2.cmml" xref="S5.Thmtheorem3.p2.7.7.m7.3.4.2"><times id="S5.Thmtheorem3.p2.7.7.m7.3.4.2.1.cmml" xref="S5.Thmtheorem3.p2.7.7.m7.3.4.2.1"></times><apply id="S5.Thmtheorem3.p2.7.7.m7.3.4.2.2.cmml" xref="S5.Thmtheorem3.p2.7.7.m7.3.4.2.2"><csymbol cd="ambiguous" id="S5.Thmtheorem3.p2.7.7.m7.3.4.2.2.1.cmml" xref="S5.Thmtheorem3.p2.7.7.m7.3.4.2.2">subscript</csymbol><ci id="S5.Thmtheorem3.p2.7.7.m7.3.4.2.2.2a.cmml" xref="S5.Thmtheorem3.p2.7.7.m7.3.4.2.2.2"><mtext class="ltx_mathvariant_italic" id="S5.Thmtheorem3.p2.7.7.m7.3.4.2.2.2.cmml" xref="S5.Thmtheorem3.p2.7.7.m7.3.4.2.2.2">g</mtext></ci><ci id="S5.Thmtheorem3.p2.7.7.m7.3.4.2.2.3.cmml" xref="S5.Thmtheorem3.p2.7.7.m7.3.4.2.2.3">𝑥</ci></apply><ci id="S5.Thmtheorem3.p2.7.7.m7.1.1.cmml" xref="S5.Thmtheorem3.p2.7.7.m7.1.1">𝑣</ci></apply><apply id="S5.Thmtheorem3.p2.7.7.m7.3.4.4.cmml" xref="S5.Thmtheorem3.p2.7.7.m7.3.4.4"><times id="S5.Thmtheorem3.p2.7.7.m7.3.4.4.1.cmml" xref="S5.Thmtheorem3.p2.7.7.m7.3.4.4.1"></times><apply id="S5.Thmtheorem3.p2.7.7.m7.3.4.4.2.cmml" xref="S5.Thmtheorem3.p2.7.7.m7.3.4.4.2"><csymbol cd="ambiguous" id="S5.Thmtheorem3.p2.7.7.m7.3.4.4.2.1.cmml" xref="S5.Thmtheorem3.p2.7.7.m7.3.4.4.2">superscript</csymbol><ci id="S5.Thmtheorem3.p2.7.7.m7.3.4.4.2.2a.cmml" xref="S5.Thmtheorem3.p2.7.7.m7.3.4.4.2.2"><mtext class="ltx_mathvariant_italic" id="S5.Thmtheorem3.p2.7.7.m7.3.4.4.2.2.cmml" xref="S5.Thmtheorem3.p2.7.7.m7.3.4.4.2.2">g</mtext></ci><times id="S5.Thmtheorem3.p2.7.7.m7.3.4.4.2.3.cmml" xref="S5.Thmtheorem3.p2.7.7.m7.3.4.4.2.3"></times></apply><ci id="S5.Thmtheorem3.p2.7.7.m7.2.2.cmml" xref="S5.Thmtheorem3.p2.7.7.m7.2.2">𝑣</ci></apply></apply><apply id="S5.Thmtheorem3.p2.7.7.m7.3.4c.cmml" xref="S5.Thmtheorem3.p2.7.7.m7.3.4"><eq id="S5.Thmtheorem3.p2.7.7.m7.3.4.5.cmml" xref="S5.Thmtheorem3.p2.7.7.m7.3.4.5"></eq><share href="https://arxiv.org/html/2411.12694v2#S5.Thmtheorem3.p2.7.7.m7.3.4.4.cmml" id="S5.Thmtheorem3.p2.7.7.m7.3.4d.cmml" xref="S5.Thmtheorem3.p2.7.7.m7.3.4"></share><apply id="S5.Thmtheorem3.p2.7.7.m7.3.4.6.cmml" xref="S5.Thmtheorem3.p2.7.7.m7.3.4.6"><times id="S5.Thmtheorem3.p2.7.7.m7.3.4.6.1.cmml" xref="S5.Thmtheorem3.p2.7.7.m7.3.4.6.1"></times><apply id="S5.Thmtheorem3.p2.7.7.m7.3.4.6.2.cmml" xref="S5.Thmtheorem3.p2.7.7.m7.3.4.6.2"><csymbol cd="ambiguous" id="S5.Thmtheorem3.p2.7.7.m7.3.4.6.2.1.cmml" xref="S5.Thmtheorem3.p2.7.7.m7.3.4.6.2">superscript</csymbol><ci id="S5.Thmtheorem3.p2.7.7.m7.3.4.6.2.2.cmml" xref="S5.Thmtheorem3.p2.7.7.m7.3.4.6.2.2">𝜌</ci><times id="S5.Thmtheorem3.p2.7.7.m7.3.4.6.2.3.cmml" xref="S5.Thmtheorem3.p2.7.7.m7.3.4.6.2.3"></times></apply><ci id="S5.Thmtheorem3.p2.7.7.m7.3.3.cmml" xref="S5.Thmtheorem3.p2.7.7.m7.3.3">𝑣</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmtheorem3.p2.7.7.m7.3c">\textsl{g}_{x}(v)=\textsl{g}^{*}(v)=\rho^{*}(v)</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem3.p2.7.7.m7.3d">g start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT ( italic_v ) = g start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ( italic_v ) = italic_ρ start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ( italic_v )</annotation></semantics></math>. And thus, the fractional orientation <math alttext="\overrightarrow{G}_{x}" class="ltx_Math" display="inline" id="S5.Thmtheorem3.p2.8.8.m8.1"><semantics id="S5.Thmtheorem3.p2.8.8.m8.1a"><msub id="S5.Thmtheorem3.p2.8.8.m8.1.1" xref="S5.Thmtheorem3.p2.8.8.m8.1.1.cmml"><mover accent="true" id="S5.Thmtheorem3.p2.8.8.m8.1.1.2" xref="S5.Thmtheorem3.p2.8.8.m8.1.1.2.cmml"><mi id="S5.Thmtheorem3.p2.8.8.m8.1.1.2.2" xref="S5.Thmtheorem3.p2.8.8.m8.1.1.2.2.cmml">G</mi><mo id="S5.Thmtheorem3.p2.8.8.m8.1.1.2.1" stretchy="false" xref="S5.Thmtheorem3.p2.8.8.m8.1.1.2.1.cmml">→</mo></mover><mi id="S5.Thmtheorem3.p2.8.8.m8.1.1.3" xref="S5.Thmtheorem3.p2.8.8.m8.1.1.3.cmml">x</mi></msub><annotation-xml encoding="MathML-Content" id="S5.Thmtheorem3.p2.8.8.m8.1b"><apply id="S5.Thmtheorem3.p2.8.8.m8.1.1.cmml" xref="S5.Thmtheorem3.p2.8.8.m8.1.1"><csymbol cd="ambiguous" id="S5.Thmtheorem3.p2.8.8.m8.1.1.1.cmml" xref="S5.Thmtheorem3.p2.8.8.m8.1.1">subscript</csymbol><apply id="S5.Thmtheorem3.p2.8.8.m8.1.1.2.cmml" xref="S5.Thmtheorem3.p2.8.8.m8.1.1.2"><ci id="S5.Thmtheorem3.p2.8.8.m8.1.1.2.1.cmml" xref="S5.Thmtheorem3.p2.8.8.m8.1.1.2.1">→</ci><ci id="S5.Thmtheorem3.p2.8.8.m8.1.1.2.2.cmml" xref="S5.Thmtheorem3.p2.8.8.m8.1.1.2.2">𝐺</ci></apply><ci id="S5.Thmtheorem3.p2.8.8.m8.1.1.3.cmml" xref="S5.Thmtheorem3.p2.8.8.m8.1.1.3">𝑥</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmtheorem3.p2.8.8.m8.1c">\overrightarrow{G}_{x}</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem3.p2.8.8.m8.1d">over→ start_ARG italic_G end_ARG start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT</annotation></semantics></math> is not equal to <math alttext="\overrightarrow{G}" class="ltx_Math" display="inline" id="S5.Thmtheorem3.p2.9.9.m9.1"><semantics id="S5.Thmtheorem3.p2.9.9.m9.1a"><mover accent="true" id="S5.Thmtheorem3.p2.9.9.m9.1.1" xref="S5.Thmtheorem3.p2.9.9.m9.1.1.cmml"><mi id="S5.Thmtheorem3.p2.9.9.m9.1.1.2" xref="S5.Thmtheorem3.p2.9.9.m9.1.1.2.cmml">G</mi><mo id="S5.Thmtheorem3.p2.9.9.m9.1.1.1" stretchy="false" xref="S5.Thmtheorem3.p2.9.9.m9.1.1.1.cmml">→</mo></mover><annotation-xml encoding="MathML-Content" id="S5.Thmtheorem3.p2.9.9.m9.1b"><apply id="S5.Thmtheorem3.p2.9.9.m9.1.1.cmml" xref="S5.Thmtheorem3.p2.9.9.m9.1.1"><ci id="S5.Thmtheorem3.p2.9.9.m9.1.1.1.cmml" xref="S5.Thmtheorem3.p2.9.9.m9.1.1.1">→</ci><ci id="S5.Thmtheorem3.p2.9.9.m9.1.1.2.cmml" xref="S5.Thmtheorem3.p2.9.9.m9.1.1.2">𝐺</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmtheorem3.p2.9.9.m9.1c">\overrightarrow{G}</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem3.p2.9.9.m9.1d">over→ start_ARG italic_G end_ARG</annotation></semantics></math>.</span></p> </div> <div class="ltx_para" id="S5.Thmtheorem3.p3"> <p class="ltx_p" id="S5.Thmtheorem3.p3.2"><span class="ltx_text ltx_font_italic" id="S5.Thmtheorem3.p3.2.2">Given <math alttext="\overrightarrow{G}" class="ltx_Math" display="inline" id="S5.Thmtheorem3.p3.1.1.m1.1"><semantics id="S5.Thmtheorem3.p3.1.1.m1.1a"><mover accent="true" id="S5.Thmtheorem3.p3.1.1.m1.1.1" xref="S5.Thmtheorem3.p3.1.1.m1.1.1.cmml"><mi id="S5.Thmtheorem3.p3.1.1.m1.1.1.2" xref="S5.Thmtheorem3.p3.1.1.m1.1.1.2.cmml">G</mi><mo id="S5.Thmtheorem3.p3.1.1.m1.1.1.1" stretchy="false" xref="S5.Thmtheorem3.p3.1.1.m1.1.1.1.cmml">→</mo></mover><annotation-xml encoding="MathML-Content" id="S5.Thmtheorem3.p3.1.1.m1.1b"><apply id="S5.Thmtheorem3.p3.1.1.m1.1.1.cmml" xref="S5.Thmtheorem3.p3.1.1.m1.1.1"><ci id="S5.Thmtheorem3.p3.1.1.m1.1.1.1.cmml" xref="S5.Thmtheorem3.p3.1.1.m1.1.1.1">→</ci><ci id="S5.Thmtheorem3.p3.1.1.m1.1.1.2.cmml" xref="S5.Thmtheorem3.p3.1.1.m1.1.1.2">𝐺</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmtheorem3.p3.1.1.m1.1c">\overrightarrow{G}</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem3.p3.1.1.m1.1d">over→ start_ARG italic_G end_ARG</annotation></semantics></math> and a locally fair fractional orientation <math alttext="\overrightarrow{G}_{x}" class="ltx_Math" display="inline" id="S5.Thmtheorem3.p3.2.2.m2.1"><semantics id="S5.Thmtheorem3.p3.2.2.m2.1a"><msub id="S5.Thmtheorem3.p3.2.2.m2.1.1" xref="S5.Thmtheorem3.p3.2.2.m2.1.1.cmml"><mover accent="true" id="S5.Thmtheorem3.p3.2.2.m2.1.1.2" xref="S5.Thmtheorem3.p3.2.2.m2.1.1.2.cmml"><mi id="S5.Thmtheorem3.p3.2.2.m2.1.1.2.2" xref="S5.Thmtheorem3.p3.2.2.m2.1.1.2.2.cmml">G</mi><mo id="S5.Thmtheorem3.p3.2.2.m2.1.1.2.1" stretchy="false" xref="S5.Thmtheorem3.p3.2.2.m2.1.1.2.1.cmml">→</mo></mover><mi id="S5.Thmtheorem3.p3.2.2.m2.1.1.3" xref="S5.Thmtheorem3.p3.2.2.m2.1.1.3.cmml">x</mi></msub><annotation-xml encoding="MathML-Content" id="S5.Thmtheorem3.p3.2.2.m2.1b"><apply id="S5.Thmtheorem3.p3.2.2.m2.1.1.cmml" xref="S5.Thmtheorem3.p3.2.2.m2.1.1"><csymbol cd="ambiguous" id="S5.Thmtheorem3.p3.2.2.m2.1.1.1.cmml" xref="S5.Thmtheorem3.p3.2.2.m2.1.1">subscript</csymbol><apply id="S5.Thmtheorem3.p3.2.2.m2.1.1.2.cmml" xref="S5.Thmtheorem3.p3.2.2.m2.1.1.2"><ci id="S5.Thmtheorem3.p3.2.2.m2.1.1.2.1.cmml" xref="S5.Thmtheorem3.p3.2.2.m2.1.1.2.1">→</ci><ci id="S5.Thmtheorem3.p3.2.2.m2.1.1.2.2.cmml" xref="S5.Thmtheorem3.p3.2.2.m2.1.1.2.2">𝐺</ci></apply><ci id="S5.Thmtheorem3.p3.2.2.m2.1.1.3.cmml" xref="S5.Thmtheorem3.p3.2.2.m2.1.1.3">𝑥</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmtheorem3.p3.2.2.m2.1c">\overrightarrow{G}_{x}</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem3.p3.2.2.m2.1d">over→ start_ARG italic_G end_ARG start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT</annotation></semantics></math>, we do three steps:</span></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.p1.10"><span class="ltx_text ltx_font_italic" id="S5.I1.i1.p1.10.1">We partition the vertices of </span><math alttext="G" class="ltx_Math" display="inline" id="S5.I1.i1.p1.1.m1.1"><semantics id="S5.I1.i1.p1.1.m1.1a"><mi id="S5.I1.i1.p1.1.m1.1.1" xref="S5.I1.i1.p1.1.m1.1.1.cmml">G</mi><annotation-xml encoding="MathML-Content" id="S5.I1.i1.p1.1.m1.1b"><ci id="S5.I1.i1.p1.1.m1.1.1.cmml" xref="S5.I1.i1.p1.1.m1.1.1">𝐺</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.I1.i1.p1.1.m1.1c">G</annotation><annotation encoding="application/x-llamapun" id="S5.I1.i1.p1.1.m1.1d">italic_G</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S5.I1.i1.p1.10.2"> to create two graphs </span><math alttext="G_{1}" class="ltx_Math" display="inline" id="S5.I1.i1.p1.2.m2.1"><semantics id="S5.I1.i1.p1.2.m2.1a"><msub id="S5.I1.i1.p1.2.m2.1.1" xref="S5.I1.i1.p1.2.m2.1.1.cmml"><mi id="S5.I1.i1.p1.2.m2.1.1.2" xref="S5.I1.i1.p1.2.m2.1.1.2.cmml">G</mi><mn id="S5.I1.i1.p1.2.m2.1.1.3" xref="S5.I1.i1.p1.2.m2.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="S5.I1.i1.p1.2.m2.1b"><apply id="S5.I1.i1.p1.2.m2.1.1.cmml" xref="S5.I1.i1.p1.2.m2.1.1"><csymbol cd="ambiguous" id="S5.I1.i1.p1.2.m2.1.1.1.cmml" xref="S5.I1.i1.p1.2.m2.1.1">subscript</csymbol><ci id="S5.I1.i1.p1.2.m2.1.1.2.cmml" xref="S5.I1.i1.p1.2.m2.1.1.2">𝐺</ci><cn id="S5.I1.i1.p1.2.m2.1.1.3.cmml" type="integer" xref="S5.I1.i1.p1.2.m2.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.I1.i1.p1.2.m2.1c">G_{1}</annotation><annotation encoding="application/x-llamapun" id="S5.I1.i1.p1.2.m2.1d">italic_G start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S5.I1.i1.p1.10.3"> and </span><math alttext="G_{2}" class="ltx_Math" display="inline" id="S5.I1.i1.p1.3.m3.1"><semantics id="S5.I1.i1.p1.3.m3.1a"><msub id="S5.I1.i1.p1.3.m3.1.1" xref="S5.I1.i1.p1.3.m3.1.1.cmml"><mi id="S5.I1.i1.p1.3.m3.1.1.2" xref="S5.I1.i1.p1.3.m3.1.1.2.cmml">G</mi><mn id="S5.I1.i1.p1.3.m3.1.1.3" xref="S5.I1.i1.p1.3.m3.1.1.3.cmml">2</mn></msub><annotation-xml encoding="MathML-Content" id="S5.I1.i1.p1.3.m3.1b"><apply id="S5.I1.i1.p1.3.m3.1.1.cmml" xref="S5.I1.i1.p1.3.m3.1.1"><csymbol cd="ambiguous" id="S5.I1.i1.p1.3.m3.1.1.1.cmml" xref="S5.I1.i1.p1.3.m3.1.1">subscript</csymbol><ci id="S5.I1.i1.p1.3.m3.1.1.2.cmml" xref="S5.I1.i1.p1.3.m3.1.1.2">𝐺</ci><cn id="S5.I1.i1.p1.3.m3.1.1.3.cmml" type="integer" xref="S5.I1.i1.p1.3.m3.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.I1.i1.p1.3.m3.1c">G_{2}</annotation><annotation encoding="application/x-llamapun" id="S5.I1.i1.p1.3.m3.1d">italic_G start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S5.I1.i1.p1.10.4">. The partition is based on the orientation </span><math alttext="\overrightarrow{G}_{x}" class="ltx_Math" display="inline" id="S5.I1.i1.p1.4.m4.1"><semantics id="S5.I1.i1.p1.4.m4.1a"><msub id="S5.I1.i1.p1.4.m4.1.1" xref="S5.I1.i1.p1.4.m4.1.1.cmml"><mover accent="true" id="S5.I1.i1.p1.4.m4.1.1.2" xref="S5.I1.i1.p1.4.m4.1.1.2.cmml"><mi id="S5.I1.i1.p1.4.m4.1.1.2.2" xref="S5.I1.i1.p1.4.m4.1.1.2.2.cmml">G</mi><mo id="S5.I1.i1.p1.4.m4.1.1.2.1" stretchy="false" xref="S5.I1.i1.p1.4.m4.1.1.2.1.cmml">→</mo></mover><mi id="S5.I1.i1.p1.4.m4.1.1.3" xref="S5.I1.i1.p1.4.m4.1.1.3.cmml">x</mi></msub><annotation-xml encoding="MathML-Content" id="S5.I1.i1.p1.4.m4.1b"><apply id="S5.I1.i1.p1.4.m4.1.1.cmml" xref="S5.I1.i1.p1.4.m4.1.1"><csymbol cd="ambiguous" id="S5.I1.i1.p1.4.m4.1.1.1.cmml" xref="S5.I1.i1.p1.4.m4.1.1">subscript</csymbol><apply id="S5.I1.i1.p1.4.m4.1.1.2.cmml" xref="S5.I1.i1.p1.4.m4.1.1.2"><ci id="S5.I1.i1.p1.4.m4.1.1.2.1.cmml" xref="S5.I1.i1.p1.4.m4.1.1.2.1">→</ci><ci id="S5.I1.i1.p1.4.m4.1.1.2.2.cmml" xref="S5.I1.i1.p1.4.m4.1.1.2.2">𝐺</ci></apply><ci id="S5.I1.i1.p1.4.m4.1.1.3.cmml" xref="S5.I1.i1.p1.4.m4.1.1.3">𝑥</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.I1.i1.p1.4.m4.1c">\overrightarrow{G}_{x}</annotation><annotation encoding="application/x-llamapun" id="S5.I1.i1.p1.4.m4.1d">over→ start_ARG italic_G end_ARG start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S5.I1.i1.p1.10.5"> as we set: </span><math alttext="G^{1}=G[v\in V\mid\textsl{g}_{x}(v)\leq\gamma]" class="ltx_Math" display="inline" id="S5.I1.i1.p1.5.m5.2"><semantics id="S5.I1.i1.p1.5.m5.2a"><mrow id="S5.I1.i1.p1.5.m5.2.2" xref="S5.I1.i1.p1.5.m5.2.2.cmml"><msup id="S5.I1.i1.p1.5.m5.2.2.3" xref="S5.I1.i1.p1.5.m5.2.2.3.cmml"><mi id="S5.I1.i1.p1.5.m5.2.2.3.2" xref="S5.I1.i1.p1.5.m5.2.2.3.2.cmml">G</mi><mn id="S5.I1.i1.p1.5.m5.2.2.3.3" xref="S5.I1.i1.p1.5.m5.2.2.3.3.cmml">1</mn></msup><mo id="S5.I1.i1.p1.5.m5.2.2.2" xref="S5.I1.i1.p1.5.m5.2.2.2.cmml">=</mo><mrow id="S5.I1.i1.p1.5.m5.2.2.1" xref="S5.I1.i1.p1.5.m5.2.2.1.cmml"><mi id="S5.I1.i1.p1.5.m5.2.2.1.3" xref="S5.I1.i1.p1.5.m5.2.2.1.3.cmml">G</mi><mo id="S5.I1.i1.p1.5.m5.2.2.1.2" xref="S5.I1.i1.p1.5.m5.2.2.1.2.cmml">⁢</mo><mrow id="S5.I1.i1.p1.5.m5.2.2.1.1.1" xref="S5.I1.i1.p1.5.m5.2.2.1.1.2.cmml"><mo id="S5.I1.i1.p1.5.m5.2.2.1.1.1.2" stretchy="false" xref="S5.I1.i1.p1.5.m5.2.2.1.1.2.1.cmml">[</mo><mrow id="S5.I1.i1.p1.5.m5.2.2.1.1.1.1" xref="S5.I1.i1.p1.5.m5.2.2.1.1.1.1.cmml"><mi id="S5.I1.i1.p1.5.m5.2.2.1.1.1.1.2" xref="S5.I1.i1.p1.5.m5.2.2.1.1.1.1.2.cmml">v</mi><mo id="S5.I1.i1.p1.5.m5.2.2.1.1.1.1.3" xref="S5.I1.i1.p1.5.m5.2.2.1.1.1.1.3.cmml">∈</mo><mrow id="S5.I1.i1.p1.5.m5.2.2.1.1.1.1.4" xref="S5.I1.i1.p1.5.m5.2.2.1.1.1.1.4.cmml"><mi id="S5.I1.i1.p1.5.m5.2.2.1.1.1.1.4.2" xref="S5.I1.i1.p1.5.m5.2.2.1.1.1.1.4.2.cmml">V</mi><mo id="S5.I1.i1.p1.5.m5.2.2.1.1.1.1.4.1" xref="S5.I1.i1.p1.5.m5.2.2.1.1.1.1.4.1.cmml">∣</mo><mrow id="S5.I1.i1.p1.5.m5.2.2.1.1.1.1.4.3" xref="S5.I1.i1.p1.5.m5.2.2.1.1.1.1.4.3.cmml"><msub id="S5.I1.i1.p1.5.m5.2.2.1.1.1.1.4.3.2" xref="S5.I1.i1.p1.5.m5.2.2.1.1.1.1.4.3.2.cmml"><mtext class="ltx_mathvariant_italic" id="S5.I1.i1.p1.5.m5.2.2.1.1.1.1.4.3.2.2" xref="S5.I1.i1.p1.5.m5.2.2.1.1.1.1.4.3.2.2a.cmml">g</mtext><mi id="S5.I1.i1.p1.5.m5.2.2.1.1.1.1.4.3.2.3" xref="S5.I1.i1.p1.5.m5.2.2.1.1.1.1.4.3.2.3.cmml">x</mi></msub><mo id="S5.I1.i1.p1.5.m5.2.2.1.1.1.1.4.3.1" xref="S5.I1.i1.p1.5.m5.2.2.1.1.1.1.4.3.1.cmml">⁢</mo><mrow id="S5.I1.i1.p1.5.m5.2.2.1.1.1.1.4.3.3.2" xref="S5.I1.i1.p1.5.m5.2.2.1.1.1.1.4.3.cmml"><mo id="S5.I1.i1.p1.5.m5.2.2.1.1.1.1.4.3.3.2.1" stretchy="false" xref="S5.I1.i1.p1.5.m5.2.2.1.1.1.1.4.3.cmml">(</mo><mi id="S5.I1.i1.p1.5.m5.1.1" xref="S5.I1.i1.p1.5.m5.1.1.cmml">v</mi><mo id="S5.I1.i1.p1.5.m5.2.2.1.1.1.1.4.3.3.2.2" stretchy="false" xref="S5.I1.i1.p1.5.m5.2.2.1.1.1.1.4.3.cmml">)</mo></mrow></mrow></mrow><mo id="S5.I1.i1.p1.5.m5.2.2.1.1.1.1.5" xref="S5.I1.i1.p1.5.m5.2.2.1.1.1.1.5.cmml">≤</mo><mi id="S5.I1.i1.p1.5.m5.2.2.1.1.1.1.6" xref="S5.I1.i1.p1.5.m5.2.2.1.1.1.1.6.cmml">γ</mi></mrow><mo id="S5.I1.i1.p1.5.m5.2.2.1.1.1.3" stretchy="false" xref="S5.I1.i1.p1.5.m5.2.2.1.1.2.1.cmml">]</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.I1.i1.p1.5.m5.2b"><apply id="S5.I1.i1.p1.5.m5.2.2.cmml" xref="S5.I1.i1.p1.5.m5.2.2"><eq id="S5.I1.i1.p1.5.m5.2.2.2.cmml" xref="S5.I1.i1.p1.5.m5.2.2.2"></eq><apply id="S5.I1.i1.p1.5.m5.2.2.3.cmml" xref="S5.I1.i1.p1.5.m5.2.2.3"><csymbol cd="ambiguous" id="S5.I1.i1.p1.5.m5.2.2.3.1.cmml" xref="S5.I1.i1.p1.5.m5.2.2.3">superscript</csymbol><ci id="S5.I1.i1.p1.5.m5.2.2.3.2.cmml" xref="S5.I1.i1.p1.5.m5.2.2.3.2">𝐺</ci><cn id="S5.I1.i1.p1.5.m5.2.2.3.3.cmml" type="integer" xref="S5.I1.i1.p1.5.m5.2.2.3.3">1</cn></apply><apply id="S5.I1.i1.p1.5.m5.2.2.1.cmml" xref="S5.I1.i1.p1.5.m5.2.2.1"><times id="S5.I1.i1.p1.5.m5.2.2.1.2.cmml" xref="S5.I1.i1.p1.5.m5.2.2.1.2"></times><ci id="S5.I1.i1.p1.5.m5.2.2.1.3.cmml" xref="S5.I1.i1.p1.5.m5.2.2.1.3">𝐺</ci><apply id="S5.I1.i1.p1.5.m5.2.2.1.1.2.cmml" xref="S5.I1.i1.p1.5.m5.2.2.1.1.1"><csymbol cd="latexml" id="S5.I1.i1.p1.5.m5.2.2.1.1.2.1.cmml" xref="S5.I1.i1.p1.5.m5.2.2.1.1.1.2">delimited-[]</csymbol><apply id="S5.I1.i1.p1.5.m5.2.2.1.1.1.1.cmml" xref="S5.I1.i1.p1.5.m5.2.2.1.1.1.1"><and id="S5.I1.i1.p1.5.m5.2.2.1.1.1.1a.cmml" xref="S5.I1.i1.p1.5.m5.2.2.1.1.1.1"></and><apply id="S5.I1.i1.p1.5.m5.2.2.1.1.1.1b.cmml" xref="S5.I1.i1.p1.5.m5.2.2.1.1.1.1"><in id="S5.I1.i1.p1.5.m5.2.2.1.1.1.1.3.cmml" xref="S5.I1.i1.p1.5.m5.2.2.1.1.1.1.3"></in><ci id="S5.I1.i1.p1.5.m5.2.2.1.1.1.1.2.cmml" xref="S5.I1.i1.p1.5.m5.2.2.1.1.1.1.2">𝑣</ci><apply id="S5.I1.i1.p1.5.m5.2.2.1.1.1.1.4.cmml" xref="S5.I1.i1.p1.5.m5.2.2.1.1.1.1.4"><csymbol cd="latexml" id="S5.I1.i1.p1.5.m5.2.2.1.1.1.1.4.1.cmml" xref="S5.I1.i1.p1.5.m5.2.2.1.1.1.1.4.1">conditional</csymbol><ci id="S5.I1.i1.p1.5.m5.2.2.1.1.1.1.4.2.cmml" xref="S5.I1.i1.p1.5.m5.2.2.1.1.1.1.4.2">𝑉</ci><apply id="S5.I1.i1.p1.5.m5.2.2.1.1.1.1.4.3.cmml" xref="S5.I1.i1.p1.5.m5.2.2.1.1.1.1.4.3"><times id="S5.I1.i1.p1.5.m5.2.2.1.1.1.1.4.3.1.cmml" xref="S5.I1.i1.p1.5.m5.2.2.1.1.1.1.4.3.1"></times><apply id="S5.I1.i1.p1.5.m5.2.2.1.1.1.1.4.3.2.cmml" xref="S5.I1.i1.p1.5.m5.2.2.1.1.1.1.4.3.2"><csymbol cd="ambiguous" id="S5.I1.i1.p1.5.m5.2.2.1.1.1.1.4.3.2.1.cmml" xref="S5.I1.i1.p1.5.m5.2.2.1.1.1.1.4.3.2">subscript</csymbol><ci id="S5.I1.i1.p1.5.m5.2.2.1.1.1.1.4.3.2.2a.cmml" xref="S5.I1.i1.p1.5.m5.2.2.1.1.1.1.4.3.2.2"><mtext class="ltx_mathvariant_italic" id="S5.I1.i1.p1.5.m5.2.2.1.1.1.1.4.3.2.2.cmml" xref="S5.I1.i1.p1.5.m5.2.2.1.1.1.1.4.3.2.2">g</mtext></ci><ci id="S5.I1.i1.p1.5.m5.2.2.1.1.1.1.4.3.2.3.cmml" xref="S5.I1.i1.p1.5.m5.2.2.1.1.1.1.4.3.2.3">𝑥</ci></apply><ci id="S5.I1.i1.p1.5.m5.1.1.cmml" xref="S5.I1.i1.p1.5.m5.1.1">𝑣</ci></apply></apply></apply><apply id="S5.I1.i1.p1.5.m5.2.2.1.1.1.1c.cmml" xref="S5.I1.i1.p1.5.m5.2.2.1.1.1.1"><leq id="S5.I1.i1.p1.5.m5.2.2.1.1.1.1.5.cmml" xref="S5.I1.i1.p1.5.m5.2.2.1.1.1.1.5"></leq><share href="https://arxiv.org/html/2411.12694v2#S5.I1.i1.p1.5.m5.2.2.1.1.1.1.4.cmml" id="S5.I1.i1.p1.5.m5.2.2.1.1.1.1d.cmml" xref="S5.I1.i1.p1.5.m5.2.2.1.1.1.1"></share><ci id="S5.I1.i1.p1.5.m5.2.2.1.1.1.1.6.cmml" xref="S5.I1.i1.p1.5.m5.2.2.1.1.1.1.6">𝛾</ci></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.I1.i1.p1.5.m5.2c">G^{1}=G[v\in V\mid\textsl{g}_{x}(v)\leq\gamma]</annotation><annotation encoding="application/x-llamapun" id="S5.I1.i1.p1.5.m5.2d">italic_G start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT = italic_G [ italic_v ∈ italic_V ∣ g start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT ( italic_v ) ≤ italic_γ ]</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S5.I1.i1.p1.10.6"> and by </span><math alttext="G^{2}=G[v\in V\mid\textsl{g}_{x}(v)>\gamma]" class="ltx_Math" display="inline" id="S5.I1.i1.p1.6.m6.2"><semantics id="S5.I1.i1.p1.6.m6.2a"><mrow id="S5.I1.i1.p1.6.m6.2.2" xref="S5.I1.i1.p1.6.m6.2.2.cmml"><msup id="S5.I1.i1.p1.6.m6.2.2.3" xref="S5.I1.i1.p1.6.m6.2.2.3.cmml"><mi id="S5.I1.i1.p1.6.m6.2.2.3.2" xref="S5.I1.i1.p1.6.m6.2.2.3.2.cmml">G</mi><mn id="S5.I1.i1.p1.6.m6.2.2.3.3" xref="S5.I1.i1.p1.6.m6.2.2.3.3.cmml">2</mn></msup><mo id="S5.I1.i1.p1.6.m6.2.2.2" xref="S5.I1.i1.p1.6.m6.2.2.2.cmml">=</mo><mrow id="S5.I1.i1.p1.6.m6.2.2.1" xref="S5.I1.i1.p1.6.m6.2.2.1.cmml"><mi id="S5.I1.i1.p1.6.m6.2.2.1.3" xref="S5.I1.i1.p1.6.m6.2.2.1.3.cmml">G</mi><mo id="S5.I1.i1.p1.6.m6.2.2.1.2" xref="S5.I1.i1.p1.6.m6.2.2.1.2.cmml">⁢</mo><mrow id="S5.I1.i1.p1.6.m6.2.2.1.1.1" xref="S5.I1.i1.p1.6.m6.2.2.1.1.2.cmml"><mo id="S5.I1.i1.p1.6.m6.2.2.1.1.1.2" stretchy="false" xref="S5.I1.i1.p1.6.m6.2.2.1.1.2.1.cmml">[</mo><mrow id="S5.I1.i1.p1.6.m6.2.2.1.1.1.1" xref="S5.I1.i1.p1.6.m6.2.2.1.1.1.1.cmml"><mi id="S5.I1.i1.p1.6.m6.2.2.1.1.1.1.3" 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id="S5.I1.i1.p1.6.m6.2.2.1.1.1.1.1.1.1.1.1.cmml" xref="S5.I1.i1.p1.6.m6.2.2.1.1.1.1.1.1.1.1.1"></times><apply id="S5.I1.i1.p1.6.m6.2.2.1.1.1.1.1.1.1.1.2.cmml" xref="S5.I1.i1.p1.6.m6.2.2.1.1.1.1.1.1.1.1.2"><csymbol cd="ambiguous" id="S5.I1.i1.p1.6.m6.2.2.1.1.1.1.1.1.1.1.2.1.cmml" xref="S5.I1.i1.p1.6.m6.2.2.1.1.1.1.1.1.1.1.2">subscript</csymbol><ci id="S5.I1.i1.p1.6.m6.2.2.1.1.1.1.1.1.1.1.2.2a.cmml" xref="S5.I1.i1.p1.6.m6.2.2.1.1.1.1.1.1.1.1.2.2"><mtext class="ltx_mathvariant_italic" id="S5.I1.i1.p1.6.m6.2.2.1.1.1.1.1.1.1.1.2.2.cmml" xref="S5.I1.i1.p1.6.m6.2.2.1.1.1.1.1.1.1.1.2.2">g</mtext></ci><ci id="S5.I1.i1.p1.6.m6.2.2.1.1.1.1.1.1.1.1.2.3.cmml" xref="S5.I1.i1.p1.6.m6.2.2.1.1.1.1.1.1.1.1.2.3">𝑥</ci></apply><ci id="S5.I1.i1.p1.6.m6.1.1.cmml" xref="S5.I1.i1.p1.6.m6.1.1">𝑣</ci></apply></apply><ci id="S5.I1.i1.p1.6.m6.2.2.1.1.1.1.1.4.cmml" xref="S5.I1.i1.p1.6.m6.2.2.1.1.1.1.1.4">𝛾</ci></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.I1.i1.p1.6.m6.2c">G^{2}=G[v\in V\mid\textsl{g}_{x}(v)>\gamma]</annotation><annotation encoding="application/x-llamapun" id="S5.I1.i1.p1.6.m6.2d">italic_G start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT = italic_G [ italic_v ∈ italic_V ∣ g start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT ( italic_v ) > italic_γ ]</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S5.I1.i1.p1.10.7">. For ease of notation, we write any edge with one endpoint </span><math alttext="a\in V(G^{1})" class="ltx_Math" display="inline" id="S5.I1.i1.p1.7.m7.1"><semantics id="S5.I1.i1.p1.7.m7.1a"><mrow id="S5.I1.i1.p1.7.m7.1.1" xref="S5.I1.i1.p1.7.m7.1.1.cmml"><mi id="S5.I1.i1.p1.7.m7.1.1.3" xref="S5.I1.i1.p1.7.m7.1.1.3.cmml">a</mi><mo id="S5.I1.i1.p1.7.m7.1.1.2" xref="S5.I1.i1.p1.7.m7.1.1.2.cmml">∈</mo><mrow id="S5.I1.i1.p1.7.m7.1.1.1" xref="S5.I1.i1.p1.7.m7.1.1.1.cmml"><mi id="S5.I1.i1.p1.7.m7.1.1.1.3" xref="S5.I1.i1.p1.7.m7.1.1.1.3.cmml">V</mi><mo id="S5.I1.i1.p1.7.m7.1.1.1.2" xref="S5.I1.i1.p1.7.m7.1.1.1.2.cmml">⁢</mo><mrow id="S5.I1.i1.p1.7.m7.1.1.1.1.1" xref="S5.I1.i1.p1.7.m7.1.1.1.1.1.1.cmml"><mo id="S5.I1.i1.p1.7.m7.1.1.1.1.1.2" stretchy="false" xref="S5.I1.i1.p1.7.m7.1.1.1.1.1.1.cmml">(</mo><msup id="S5.I1.i1.p1.7.m7.1.1.1.1.1.1" xref="S5.I1.i1.p1.7.m7.1.1.1.1.1.1.cmml"><mi id="S5.I1.i1.p1.7.m7.1.1.1.1.1.1.2" xref="S5.I1.i1.p1.7.m7.1.1.1.1.1.1.2.cmml">G</mi><mn id="S5.I1.i1.p1.7.m7.1.1.1.1.1.1.3" xref="S5.I1.i1.p1.7.m7.1.1.1.1.1.1.3.cmml">1</mn></msup><mo id="S5.I1.i1.p1.7.m7.1.1.1.1.1.3" stretchy="false" xref="S5.I1.i1.p1.7.m7.1.1.1.1.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.I1.i1.p1.7.m7.1b"><apply id="S5.I1.i1.p1.7.m7.1.1.cmml" xref="S5.I1.i1.p1.7.m7.1.1"><in id="S5.I1.i1.p1.7.m7.1.1.2.cmml" xref="S5.I1.i1.p1.7.m7.1.1.2"></in><ci id="S5.I1.i1.p1.7.m7.1.1.3.cmml" xref="S5.I1.i1.p1.7.m7.1.1.3">𝑎</ci><apply id="S5.I1.i1.p1.7.m7.1.1.1.cmml" xref="S5.I1.i1.p1.7.m7.1.1.1"><times id="S5.I1.i1.p1.7.m7.1.1.1.2.cmml" xref="S5.I1.i1.p1.7.m7.1.1.1.2"></times><ci id="S5.I1.i1.p1.7.m7.1.1.1.3.cmml" xref="S5.I1.i1.p1.7.m7.1.1.1.3">𝑉</ci><apply id="S5.I1.i1.p1.7.m7.1.1.1.1.1.1.cmml" xref="S5.I1.i1.p1.7.m7.1.1.1.1.1"><csymbol cd="ambiguous" id="S5.I1.i1.p1.7.m7.1.1.1.1.1.1.1.cmml" xref="S5.I1.i1.p1.7.m7.1.1.1.1.1">superscript</csymbol><ci id="S5.I1.i1.p1.7.m7.1.1.1.1.1.1.2.cmml" xref="S5.I1.i1.p1.7.m7.1.1.1.1.1.1.2">𝐺</ci><cn id="S5.I1.i1.p1.7.m7.1.1.1.1.1.1.3.cmml" type="integer" xref="S5.I1.i1.p1.7.m7.1.1.1.1.1.1.3">1</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.I1.i1.p1.7.m7.1c">a\in V(G^{1})</annotation><annotation encoding="application/x-llamapun" id="S5.I1.i1.p1.7.m7.1d">italic_a ∈ italic_V ( italic_G start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT )</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S5.I1.i1.p1.10.8"> and one endpoint </span><math alttext="b\in V(G^{2})" 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"><mi id="S5.I1.i1.p1.8.m8.1.1.3" xref="S5.I1.i1.p1.8.m8.1.1.3.cmml">b</mi><mo id="S5.I1.i1.p1.8.m8.1.1.2" xref="S5.I1.i1.p1.8.m8.1.1.2.cmml">∈</mo><mrow id="S5.I1.i1.p1.8.m8.1.1.1" xref="S5.I1.i1.p1.8.m8.1.1.1.cmml"><mi id="S5.I1.i1.p1.8.m8.1.1.1.3" xref="S5.I1.i1.p1.8.m8.1.1.1.3.cmml">V</mi><mo id="S5.I1.i1.p1.8.m8.1.1.1.2" xref="S5.I1.i1.p1.8.m8.1.1.1.2.cmml">⁢</mo><mrow id="S5.I1.i1.p1.8.m8.1.1.1.1.1" xref="S5.I1.i1.p1.8.m8.1.1.1.1.1.1.cmml"><mo id="S5.I1.i1.p1.8.m8.1.1.1.1.1.2" stretchy="false" xref="S5.I1.i1.p1.8.m8.1.1.1.1.1.1.cmml">(</mo><msup id="S5.I1.i1.p1.8.m8.1.1.1.1.1.1" xref="S5.I1.i1.p1.8.m8.1.1.1.1.1.1.cmml"><mi id="S5.I1.i1.p1.8.m8.1.1.1.1.1.1.2" xref="S5.I1.i1.p1.8.m8.1.1.1.1.1.1.2.cmml">G</mi><mn id="S5.I1.i1.p1.8.m8.1.1.1.1.1.1.3" xref="S5.I1.i1.p1.8.m8.1.1.1.1.1.1.3.cmml">2</mn></msup><mo id="S5.I1.i1.p1.8.m8.1.1.1.1.1.3" stretchy="false" xref="S5.I1.i1.p1.8.m8.1.1.1.1.1.1.cmml">)</mo></mrow></mrow></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"><in id="S5.I1.i1.p1.8.m8.1.1.2.cmml" xref="S5.I1.i1.p1.8.m8.1.1.2"></in><ci id="S5.I1.i1.p1.8.m8.1.1.3.cmml" xref="S5.I1.i1.p1.8.m8.1.1.3">𝑏</ci><apply id="S5.I1.i1.p1.8.m8.1.1.1.cmml" xref="S5.I1.i1.p1.8.m8.1.1.1"><times id="S5.I1.i1.p1.8.m8.1.1.1.2.cmml" xref="S5.I1.i1.p1.8.m8.1.1.1.2"></times><ci id="S5.I1.i1.p1.8.m8.1.1.1.3.cmml" xref="S5.I1.i1.p1.8.m8.1.1.1.3">𝑉</ci><apply id="S5.I1.i1.p1.8.m8.1.1.1.1.1.1.cmml" xref="S5.I1.i1.p1.8.m8.1.1.1.1.1"><csymbol cd="ambiguous" id="S5.I1.i1.p1.8.m8.1.1.1.1.1.1.1.cmml" xref="S5.I1.i1.p1.8.m8.1.1.1.1.1">superscript</csymbol><ci id="S5.I1.i1.p1.8.m8.1.1.1.1.1.1.2.cmml" xref="S5.I1.i1.p1.8.m8.1.1.1.1.1.1.2">𝐺</ci><cn id="S5.I1.i1.p1.8.m8.1.1.1.1.1.1.3.cmml" type="integer" xref="S5.I1.i1.p1.8.m8.1.1.1.1.1.1.3">2</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.I1.i1.p1.8.m8.1c">b\in V(G^{2})</annotation><annotation encoding="application/x-llamapun" id="S5.I1.i1.p1.8.m8.1d">italic_b ∈ italic_V ( italic_G start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT )</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S5.I1.i1.p1.10.9"> as </span><math alttext="(a,b)" class="ltx_Math" display="inline" id="S5.I1.i1.p1.9.m9.2"><semantics id="S5.I1.i1.p1.9.m9.2a"><mrow id="S5.I1.i1.p1.9.m9.2.3.2" xref="S5.I1.i1.p1.9.m9.2.3.1.cmml"><mo id="S5.I1.i1.p1.9.m9.2.3.2.1" stretchy="false" xref="S5.I1.i1.p1.9.m9.2.3.1.cmml">(</mo><mi id="S5.I1.i1.p1.9.m9.1.1" xref="S5.I1.i1.p1.9.m9.1.1.cmml">a</mi><mo id="S5.I1.i1.p1.9.m9.2.3.2.2" xref="S5.I1.i1.p1.9.m9.2.3.1.cmml">,</mo><mi id="S5.I1.i1.p1.9.m9.2.2" xref="S5.I1.i1.p1.9.m9.2.2.cmml">b</mi><mo id="S5.I1.i1.p1.9.m9.2.3.2.3" stretchy="false" xref="S5.I1.i1.p1.9.m9.2.3.1.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S5.I1.i1.p1.9.m9.2b"><interval closure="open" id="S5.I1.i1.p1.9.m9.2.3.1.cmml" xref="S5.I1.i1.p1.9.m9.2.3.2"><ci id="S5.I1.i1.p1.9.m9.1.1.cmml" xref="S5.I1.i1.p1.9.m9.1.1">𝑎</ci><ci id="S5.I1.i1.p1.9.m9.2.2.cmml" xref="S5.I1.i1.p1.9.m9.2.2">𝑏</ci></interval></annotation-xml><annotation encoding="application/x-tex" id="S5.I1.i1.p1.9.m9.2c">(a,b)</annotation><annotation encoding="application/x-llamapun" id="S5.I1.i1.p1.9.m9.2d">( italic_a , italic_b )</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S5.I1.i1.p1.10.10"> and never as </span><math alttext="(b,a)" class="ltx_Math" display="inline" id="S5.I1.i1.p1.10.m10.2"><semantics id="S5.I1.i1.p1.10.m10.2a"><mrow id="S5.I1.i1.p1.10.m10.2.3.2" xref="S5.I1.i1.p1.10.m10.2.3.1.cmml"><mo id="S5.I1.i1.p1.10.m10.2.3.2.1" stretchy="false" xref="S5.I1.i1.p1.10.m10.2.3.1.cmml">(</mo><mi id="S5.I1.i1.p1.10.m10.1.1" xref="S5.I1.i1.p1.10.m10.1.1.cmml">b</mi><mo id="S5.I1.i1.p1.10.m10.2.3.2.2" xref="S5.I1.i1.p1.10.m10.2.3.1.cmml">,</mo><mi id="S5.I1.i1.p1.10.m10.2.2" xref="S5.I1.i1.p1.10.m10.2.2.cmml">a</mi><mo id="S5.I1.i1.p1.10.m10.2.3.2.3" stretchy="false" xref="S5.I1.i1.p1.10.m10.2.3.1.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S5.I1.i1.p1.10.m10.2b"><interval closure="open" id="S5.I1.i1.p1.10.m10.2.3.1.cmml" xref="S5.I1.i1.p1.10.m10.2.3.2"><ci id="S5.I1.i1.p1.10.m10.1.1.cmml" xref="S5.I1.i1.p1.10.m10.1.1">𝑏</ci><ci id="S5.I1.i1.p1.10.m10.2.2.cmml" xref="S5.I1.i1.p1.10.m10.2.2">𝑎</ci></interval></annotation-xml><annotation encoding="application/x-tex" id="S5.I1.i1.p1.10.m10.2c">(b,a)</annotation><annotation encoding="application/x-llamapun" id="S5.I1.i1.p1.10.m10.2d">( italic_b , italic_a )</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S5.I1.i1.p1.10.11">.</span></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.p1.6"><span class="ltx_text ltx_font_italic" id="S5.I1.i2.p1.6.1">From </span><math alttext="G^{1}" class="ltx_Math" display="inline" id="S5.I1.i2.p1.1.m1.1"><semantics id="S5.I1.i2.p1.1.m1.1a"><msup id="S5.I1.i2.p1.1.m1.1.1" xref="S5.I1.i2.p1.1.m1.1.1.cmml"><mi id="S5.I1.i2.p1.1.m1.1.1.2" xref="S5.I1.i2.p1.1.m1.1.1.2.cmml">G</mi><mn id="S5.I1.i2.p1.1.m1.1.1.3" xref="S5.I1.i2.p1.1.m1.1.1.3.cmml">1</mn></msup><annotation-xml encoding="MathML-Content" id="S5.I1.i2.p1.1.m1.1b"><apply id="S5.I1.i2.p1.1.m1.1.1.cmml" xref="S5.I1.i2.p1.1.m1.1.1"><csymbol cd="ambiguous" id="S5.I1.i2.p1.1.m1.1.1.1.cmml" xref="S5.I1.i2.p1.1.m1.1.1">superscript</csymbol><ci id="S5.I1.i2.p1.1.m1.1.1.2.cmml" xref="S5.I1.i2.p1.1.m1.1.1.2">𝐺</ci><cn id="S5.I1.i2.p1.1.m1.1.1.3.cmml" type="integer" xref="S5.I1.i2.p1.1.m1.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.I1.i2.p1.1.m1.1c">G^{1}</annotation><annotation encoding="application/x-llamapun" id="S5.I1.i2.p1.1.m1.1d">italic_G start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S5.I1.i2.p1.6.2">, we create a family of nested subgraphs using </span><math alttext="\overrightarrow{G}" class="ltx_Math" display="inline" id="S5.I1.i2.p1.2.m2.1"><semantics id="S5.I1.i2.p1.2.m2.1a"><mover accent="true" id="S5.I1.i2.p1.2.m2.1.1" xref="S5.I1.i2.p1.2.m2.1.1.cmml"><mi id="S5.I1.i2.p1.2.m2.1.1.2" xref="S5.I1.i2.p1.2.m2.1.1.2.cmml">G</mi><mo id="S5.I1.i2.p1.2.m2.1.1.1" stretchy="false" xref="S5.I1.i2.p1.2.m2.1.1.1.cmml">→</mo></mover><annotation-xml encoding="MathML-Content" id="S5.I1.i2.p1.2.m2.1b"><apply id="S5.I1.i2.p1.2.m2.1.1.cmml" xref="S5.I1.i2.p1.2.m2.1.1"><ci id="S5.I1.i2.p1.2.m2.1.1.1.cmml" xref="S5.I1.i2.p1.2.m2.1.1.1">→</ci><ci id="S5.I1.i2.p1.2.m2.1.1.2.cmml" xref="S5.I1.i2.p1.2.m2.1.1.2">𝐺</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.I1.i2.p1.2.m2.1c">\overrightarrow{G}</annotation><annotation encoding="application/x-llamapun" id="S5.I1.i2.p1.2.m2.1d">over→ start_ARG italic_G end_ARG</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S5.I1.i2.p1.6.3">. We define graphs </span><math alttext="G^{1}_{i}:=G^{1}[v\in V(G^{1})\mid\textsl{g}(v)\geq\frac{\textsl{g}(u)}{(1+% \eta)^{i}}]" class="ltx_Math" display="inline" id="S5.I1.i2.p1.3.m3.4"><semantics id="S5.I1.i2.p1.3.m3.4a"><mrow id="S5.I1.i2.p1.3.m3.4.4" xref="S5.I1.i2.p1.3.m3.4.4.cmml"><msubsup id="S5.I1.i2.p1.3.m3.4.4.3" xref="S5.I1.i2.p1.3.m3.4.4.3.cmml"><mi id="S5.I1.i2.p1.3.m3.4.4.3.2.2" xref="S5.I1.i2.p1.3.m3.4.4.3.2.2.cmml">G</mi><mi id="S5.I1.i2.p1.3.m3.4.4.3.3" xref="S5.I1.i2.p1.3.m3.4.4.3.3.cmml">i</mi><mn id="S5.I1.i2.p1.3.m3.4.4.3.2.3" xref="S5.I1.i2.p1.3.m3.4.4.3.2.3.cmml">1</mn></msubsup><mo id="S5.I1.i2.p1.3.m3.4.4.2" lspace="0.278em" rspace="0.278em" xref="S5.I1.i2.p1.3.m3.4.4.2.cmml">:=</mo><mrow id="S5.I1.i2.p1.3.m3.4.4.1" xref="S5.I1.i2.p1.3.m3.4.4.1.cmml"><msup id="S5.I1.i2.p1.3.m3.4.4.1.3" xref="S5.I1.i2.p1.3.m3.4.4.1.3.cmml"><mi id="S5.I1.i2.p1.3.m3.4.4.1.3.2" xref="S5.I1.i2.p1.3.m3.4.4.1.3.2.cmml">G</mi><mn id="S5.I1.i2.p1.3.m3.4.4.1.3.3" xref="S5.I1.i2.p1.3.m3.4.4.1.3.3.cmml">1</mn></msup><mo id="S5.I1.i2.p1.3.m3.4.4.1.2" xref="S5.I1.i2.p1.3.m3.4.4.1.2.cmml">⁢</mo><mrow id="S5.I1.i2.p1.3.m3.4.4.1.1.1" xref="S5.I1.i2.p1.3.m3.4.4.1.1.2.cmml"><mo id="S5.I1.i2.p1.3.m3.4.4.1.1.1.2" stretchy="false" xref="S5.I1.i2.p1.3.m3.4.4.1.1.2.1.cmml">[</mo><mrow id="S5.I1.i2.p1.3.m3.4.4.1.1.1.1" xref="S5.I1.i2.p1.3.m3.4.4.1.1.1.1.cmml"><mi id="S5.I1.i2.p1.3.m3.4.4.1.1.1.1.3" xref="S5.I1.i2.p1.3.m3.4.4.1.1.1.1.3.cmml">v</mi><mo id="S5.I1.i2.p1.3.m3.4.4.1.1.1.1.4" xref="S5.I1.i2.p1.3.m3.4.4.1.1.1.1.4.cmml">∈</mo><mrow id="S5.I1.i2.p1.3.m3.4.4.1.1.1.1.1" xref="S5.I1.i2.p1.3.m3.4.4.1.1.1.1.1.cmml"><mrow id="S5.I1.i2.p1.3.m3.4.4.1.1.1.1.1.1" xref="S5.I1.i2.p1.3.m3.4.4.1.1.1.1.1.1.cmml"><mi id="S5.I1.i2.p1.3.m3.4.4.1.1.1.1.1.1.3" xref="S5.I1.i2.p1.3.m3.4.4.1.1.1.1.1.1.3.cmml">V</mi><mo id="S5.I1.i2.p1.3.m3.4.4.1.1.1.1.1.1.2" xref="S5.I1.i2.p1.3.m3.4.4.1.1.1.1.1.1.2.cmml">⁢</mo><mrow id="S5.I1.i2.p1.3.m3.4.4.1.1.1.1.1.1.1.1" xref="S5.I1.i2.p1.3.m3.4.4.1.1.1.1.1.1.1.1.1.cmml"><mo id="S5.I1.i2.p1.3.m3.4.4.1.1.1.1.1.1.1.1.2" stretchy="false" xref="S5.I1.i2.p1.3.m3.4.4.1.1.1.1.1.1.1.1.1.cmml">(</mo><msup id="S5.I1.i2.p1.3.m3.4.4.1.1.1.1.1.1.1.1.1" xref="S5.I1.i2.p1.3.m3.4.4.1.1.1.1.1.1.1.1.1.cmml"><mi id="S5.I1.i2.p1.3.m3.4.4.1.1.1.1.1.1.1.1.1.2" xref="S5.I1.i2.p1.3.m3.4.4.1.1.1.1.1.1.1.1.1.2.cmml">G</mi><mn id="S5.I1.i2.p1.3.m3.4.4.1.1.1.1.1.1.1.1.1.3" xref="S5.I1.i2.p1.3.m3.4.4.1.1.1.1.1.1.1.1.1.3.cmml">1</mn></msup><mo id="S5.I1.i2.p1.3.m3.4.4.1.1.1.1.1.1.1.1.3" stretchy="false" xref="S5.I1.i2.p1.3.m3.4.4.1.1.1.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S5.I1.i2.p1.3.m3.4.4.1.1.1.1.1.2" xref="S5.I1.i2.p1.3.m3.4.4.1.1.1.1.1.2.cmml">∣</mo><mrow id="S5.I1.i2.p1.3.m3.4.4.1.1.1.1.1.3" xref="S5.I1.i2.p1.3.m3.4.4.1.1.1.1.1.3.cmml"><mtext class="ltx_mathvariant_italic" id="S5.I1.i2.p1.3.m3.4.4.1.1.1.1.1.3.2" xref="S5.I1.i2.p1.3.m3.4.4.1.1.1.1.1.3.2a.cmml">g</mtext><mo id="S5.I1.i2.p1.3.m3.4.4.1.1.1.1.1.3.1" 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href="https://arxiv.org/html/2411.12694v2#S5.I1.i2.p1.3.m3.4.4.1.1.1.1.1.cmml" id="S5.I1.i2.p1.3.m3.4.4.1.1.1.1d.cmml" xref="S5.I1.i2.p1.3.m3.4.4.1.1.1.1"></share><apply id="S5.I1.i2.p1.3.m3.2.2.cmml" xref="S5.I1.i2.p1.3.m3.2.2"><divide id="S5.I1.i2.p1.3.m3.2.2.3.cmml" xref="S5.I1.i2.p1.3.m3.2.2"></divide><apply id="S5.I1.i2.p1.3.m3.1.1.1.cmml" xref="S5.I1.i2.p1.3.m3.1.1.1"><times id="S5.I1.i2.p1.3.m3.1.1.1.2.cmml" xref="S5.I1.i2.p1.3.m3.1.1.1.2"></times><ci id="S5.I1.i2.p1.3.m3.1.1.1.3a.cmml" xref="S5.I1.i2.p1.3.m3.1.1.1.3"><mtext class="ltx_mathvariant_italic" id="S5.I1.i2.p1.3.m3.1.1.1.3.cmml" mathsize="70%" xref="S5.I1.i2.p1.3.m3.1.1.1.3">g</mtext></ci><ci id="S5.I1.i2.p1.3.m3.1.1.1.1.cmml" xref="S5.I1.i2.p1.3.m3.1.1.1.1">𝑢</ci></apply><apply id="S5.I1.i2.p1.3.m3.2.2.2.cmml" xref="S5.I1.i2.p1.3.m3.2.2.2"><csymbol cd="ambiguous" id="S5.I1.i2.p1.3.m3.2.2.2.2.cmml" xref="S5.I1.i2.p1.3.m3.2.2.2">superscript</csymbol><apply id="S5.I1.i2.p1.3.m3.2.2.2.1.1.1.cmml" xref="S5.I1.i2.p1.3.m3.2.2.2.1.1"><plus id="S5.I1.i2.p1.3.m3.2.2.2.1.1.1.1.cmml" xref="S5.I1.i2.p1.3.m3.2.2.2.1.1.1.1"></plus><cn id="S5.I1.i2.p1.3.m3.2.2.2.1.1.1.2.cmml" type="integer" xref="S5.I1.i2.p1.3.m3.2.2.2.1.1.1.2">1</cn><ci id="S5.I1.i2.p1.3.m3.2.2.2.1.1.1.3.cmml" xref="S5.I1.i2.p1.3.m3.2.2.2.1.1.1.3">𝜂</ci></apply><ci id="S5.I1.i2.p1.3.m3.2.2.2.3.cmml" xref="S5.I1.i2.p1.3.m3.2.2.2.3">𝑖</ci></apply></apply></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.I1.i2.p1.3.m3.4c">G^{1}_{i}:=G^{1}[v\in V(G^{1})\mid\textsl{g}(v)\geq\frac{\textsl{g}(u)}{(1+% \eta)^{i}}]</annotation><annotation encoding="application/x-llamapun" id="S5.I1.i2.p1.3.m3.4d">italic_G start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT := italic_G start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT [ italic_v ∈ italic_V ( italic_G start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT ) ∣ g ( italic_v ) ≥ divide start_ARG g ( italic_u ) end_ARG start_ARG ( 1 + italic_η ) start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT end_ARG ]</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S5.I1.i2.p1.6.4">. We denote by </span><math alttext="k" class="ltx_Math" display="inline" id="S5.I1.i2.p1.4.m4.1"><semantics id="S5.I1.i2.p1.4.m4.1a"><mi id="S5.I1.i2.p1.4.m4.1.1" xref="S5.I1.i2.p1.4.m4.1.1.cmml">k</mi><annotation-xml encoding="MathML-Content" id="S5.I1.i2.p1.4.m4.1b"><ci id="S5.I1.i2.p1.4.m4.1.1.cmml" xref="S5.I1.i2.p1.4.m4.1.1">𝑘</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.I1.i2.p1.4.m4.1c">k</annotation><annotation encoding="application/x-llamapun" id="S5.I1.i2.p1.4.m4.1d">italic_k</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S5.I1.i2.p1.6.5"> the lowest integer such that </span><math alttext="|V(G^{1}_{k+1})|<(1+\frac{\varepsilon}{16})|V(G^{1}_{k})|" class="ltx_Math" display="inline" id="S5.I1.i2.p1.5.m5.3"><semantics id="S5.I1.i2.p1.5.m5.3a"><mrow id="S5.I1.i2.p1.5.m5.3.3" xref="S5.I1.i2.p1.5.m5.3.3.cmml"><mrow id="S5.I1.i2.p1.5.m5.1.1.1.1" xref="S5.I1.i2.p1.5.m5.1.1.1.2.cmml"><mo id="S5.I1.i2.p1.5.m5.1.1.1.1.2" stretchy="false" xref="S5.I1.i2.p1.5.m5.1.1.1.2.1.cmml">|</mo><mrow id="S5.I1.i2.p1.5.m5.1.1.1.1.1" xref="S5.I1.i2.p1.5.m5.1.1.1.1.1.cmml"><mi id="S5.I1.i2.p1.5.m5.1.1.1.1.1.3" xref="S5.I1.i2.p1.5.m5.1.1.1.1.1.3.cmml">V</mi><mo id="S5.I1.i2.p1.5.m5.1.1.1.1.1.2" xref="S5.I1.i2.p1.5.m5.1.1.1.1.1.2.cmml">⁢</mo><mrow id="S5.I1.i2.p1.5.m5.1.1.1.1.1.1.1" xref="S5.I1.i2.p1.5.m5.1.1.1.1.1.1.1.1.cmml"><mo id="S5.I1.i2.p1.5.m5.1.1.1.1.1.1.1.2" stretchy="false" xref="S5.I1.i2.p1.5.m5.1.1.1.1.1.1.1.1.cmml">(</mo><msubsup id="S5.I1.i2.p1.5.m5.1.1.1.1.1.1.1.1" xref="S5.I1.i2.p1.5.m5.1.1.1.1.1.1.1.1.cmml"><mi id="S5.I1.i2.p1.5.m5.1.1.1.1.1.1.1.1.2.2" xref="S5.I1.i2.p1.5.m5.1.1.1.1.1.1.1.1.2.2.cmml">G</mi><mrow id="S5.I1.i2.p1.5.m5.1.1.1.1.1.1.1.1.3" xref="S5.I1.i2.p1.5.m5.1.1.1.1.1.1.1.1.3.cmml"><mi id="S5.I1.i2.p1.5.m5.1.1.1.1.1.1.1.1.3.2" xref="S5.I1.i2.p1.5.m5.1.1.1.1.1.1.1.1.3.2.cmml">k</mi><mo id="S5.I1.i2.p1.5.m5.1.1.1.1.1.1.1.1.3.1" xref="S5.I1.i2.p1.5.m5.1.1.1.1.1.1.1.1.3.1.cmml">+</mo><mn id="S5.I1.i2.p1.5.m5.1.1.1.1.1.1.1.1.3.3" xref="S5.I1.i2.p1.5.m5.1.1.1.1.1.1.1.1.3.3.cmml">1</mn></mrow><mn id="S5.I1.i2.p1.5.m5.1.1.1.1.1.1.1.1.2.3" xref="S5.I1.i2.p1.5.m5.1.1.1.1.1.1.1.1.2.3.cmml">1</mn></msubsup><mo id="S5.I1.i2.p1.5.m5.1.1.1.1.1.1.1.3" stretchy="false" xref="S5.I1.i2.p1.5.m5.1.1.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S5.I1.i2.p1.5.m5.1.1.1.1.3" stretchy="false" xref="S5.I1.i2.p1.5.m5.1.1.1.2.1.cmml">|</mo></mrow><mo id="S5.I1.i2.p1.5.m5.3.3.4" xref="S5.I1.i2.p1.5.m5.3.3.4.cmml"><</mo><mrow id="S5.I1.i2.p1.5.m5.3.3.3" xref="S5.I1.i2.p1.5.m5.3.3.3.cmml"><mrow id="S5.I1.i2.p1.5.m5.2.2.2.1.1" xref="S5.I1.i2.p1.5.m5.2.2.2.1.1.1.cmml"><mo id="S5.I1.i2.p1.5.m5.2.2.2.1.1.2" stretchy="false" xref="S5.I1.i2.p1.5.m5.2.2.2.1.1.1.cmml">(</mo><mrow id="S5.I1.i2.p1.5.m5.2.2.2.1.1.1" xref="S5.I1.i2.p1.5.m5.2.2.2.1.1.1.cmml"><mn id="S5.I1.i2.p1.5.m5.2.2.2.1.1.1.2" xref="S5.I1.i2.p1.5.m5.2.2.2.1.1.1.2.cmml">1</mn><mo id="S5.I1.i2.p1.5.m5.2.2.2.1.1.1.1" xref="S5.I1.i2.p1.5.m5.2.2.2.1.1.1.1.cmml">+</mo><mfrac id="S5.I1.i2.p1.5.m5.2.2.2.1.1.1.3" xref="S5.I1.i2.p1.5.m5.2.2.2.1.1.1.3.cmml"><mi id="S5.I1.i2.p1.5.m5.2.2.2.1.1.1.3.2" xref="S5.I1.i2.p1.5.m5.2.2.2.1.1.1.3.2.cmml">ε</mi><mn id="S5.I1.i2.p1.5.m5.2.2.2.1.1.1.3.3" xref="S5.I1.i2.p1.5.m5.2.2.2.1.1.1.3.3.cmml">16</mn></mfrac></mrow><mo id="S5.I1.i2.p1.5.m5.2.2.2.1.1.3" stretchy="false" xref="S5.I1.i2.p1.5.m5.2.2.2.1.1.1.cmml">)</mo></mrow><mo id="S5.I1.i2.p1.5.m5.3.3.3.3" xref="S5.I1.i2.p1.5.m5.3.3.3.3.cmml">⁢</mo><mrow id="S5.I1.i2.p1.5.m5.3.3.3.2.1" xref="S5.I1.i2.p1.5.m5.3.3.3.2.2.cmml"><mo id="S5.I1.i2.p1.5.m5.3.3.3.2.1.2" stretchy="false" xref="S5.I1.i2.p1.5.m5.3.3.3.2.2.1.cmml">|</mo><mrow id="S5.I1.i2.p1.5.m5.3.3.3.2.1.1" xref="S5.I1.i2.p1.5.m5.3.3.3.2.1.1.cmml"><mi id="S5.I1.i2.p1.5.m5.3.3.3.2.1.1.3" xref="S5.I1.i2.p1.5.m5.3.3.3.2.1.1.3.cmml">V</mi><mo id="S5.I1.i2.p1.5.m5.3.3.3.2.1.1.2" xref="S5.I1.i2.p1.5.m5.3.3.3.2.1.1.2.cmml">⁢</mo><mrow id="S5.I1.i2.p1.5.m5.3.3.3.2.1.1.1.1" xref="S5.I1.i2.p1.5.m5.3.3.3.2.1.1.1.1.1.cmml"><mo id="S5.I1.i2.p1.5.m5.3.3.3.2.1.1.1.1.2" stretchy="false" xref="S5.I1.i2.p1.5.m5.3.3.3.2.1.1.1.1.1.cmml">(</mo><msubsup id="S5.I1.i2.p1.5.m5.3.3.3.2.1.1.1.1.1" xref="S5.I1.i2.p1.5.m5.3.3.3.2.1.1.1.1.1.cmml"><mi id="S5.I1.i2.p1.5.m5.3.3.3.2.1.1.1.1.1.2.2" xref="S5.I1.i2.p1.5.m5.3.3.3.2.1.1.1.1.1.2.2.cmml">G</mi><mi id="S5.I1.i2.p1.5.m5.3.3.3.2.1.1.1.1.1.3" xref="S5.I1.i2.p1.5.m5.3.3.3.2.1.1.1.1.1.3.cmml">k</mi><mn id="S5.I1.i2.p1.5.m5.3.3.3.2.1.1.1.1.1.2.3" xref="S5.I1.i2.p1.5.m5.3.3.3.2.1.1.1.1.1.2.3.cmml">1</mn></msubsup><mo id="S5.I1.i2.p1.5.m5.3.3.3.2.1.1.1.1.3" stretchy="false" xref="S5.I1.i2.p1.5.m5.3.3.3.2.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S5.I1.i2.p1.5.m5.3.3.3.2.1.3" stretchy="false" xref="S5.I1.i2.p1.5.m5.3.3.3.2.2.1.cmml">|</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.I1.i2.p1.5.m5.3b"><apply id="S5.I1.i2.p1.5.m5.3.3.cmml" xref="S5.I1.i2.p1.5.m5.3.3"><lt id="S5.I1.i2.p1.5.m5.3.3.4.cmml" xref="S5.I1.i2.p1.5.m5.3.3.4"></lt><apply id="S5.I1.i2.p1.5.m5.1.1.1.2.cmml" xref="S5.I1.i2.p1.5.m5.1.1.1.1"><abs id="S5.I1.i2.p1.5.m5.1.1.1.2.1.cmml" xref="S5.I1.i2.p1.5.m5.1.1.1.1.2"></abs><apply id="S5.I1.i2.p1.5.m5.1.1.1.1.1.cmml" xref="S5.I1.i2.p1.5.m5.1.1.1.1.1"><times id="S5.I1.i2.p1.5.m5.1.1.1.1.1.2.cmml" xref="S5.I1.i2.p1.5.m5.1.1.1.1.1.2"></times><ci id="S5.I1.i2.p1.5.m5.1.1.1.1.1.3.cmml" xref="S5.I1.i2.p1.5.m5.1.1.1.1.1.3">𝑉</ci><apply id="S5.I1.i2.p1.5.m5.1.1.1.1.1.1.1.1.cmml" xref="S5.I1.i2.p1.5.m5.1.1.1.1.1.1.1"><csymbol cd="ambiguous" id="S5.I1.i2.p1.5.m5.1.1.1.1.1.1.1.1.1.cmml" xref="S5.I1.i2.p1.5.m5.1.1.1.1.1.1.1">subscript</csymbol><apply id="S5.I1.i2.p1.5.m5.1.1.1.1.1.1.1.1.2.cmml" xref="S5.I1.i2.p1.5.m5.1.1.1.1.1.1.1"><csymbol cd="ambiguous" id="S5.I1.i2.p1.5.m5.1.1.1.1.1.1.1.1.2.1.cmml" xref="S5.I1.i2.p1.5.m5.1.1.1.1.1.1.1">superscript</csymbol><ci id="S5.I1.i2.p1.5.m5.1.1.1.1.1.1.1.1.2.2.cmml" xref="S5.I1.i2.p1.5.m5.1.1.1.1.1.1.1.1.2.2">𝐺</ci><cn id="S5.I1.i2.p1.5.m5.1.1.1.1.1.1.1.1.2.3.cmml" type="integer" xref="S5.I1.i2.p1.5.m5.1.1.1.1.1.1.1.1.2.3">1</cn></apply><apply id="S5.I1.i2.p1.5.m5.1.1.1.1.1.1.1.1.3.cmml" xref="S5.I1.i2.p1.5.m5.1.1.1.1.1.1.1.1.3"><plus id="S5.I1.i2.p1.5.m5.1.1.1.1.1.1.1.1.3.1.cmml" xref="S5.I1.i2.p1.5.m5.1.1.1.1.1.1.1.1.3.1"></plus><ci id="S5.I1.i2.p1.5.m5.1.1.1.1.1.1.1.1.3.2.cmml" xref="S5.I1.i2.p1.5.m5.1.1.1.1.1.1.1.1.3.2">𝑘</ci><cn id="S5.I1.i2.p1.5.m5.1.1.1.1.1.1.1.1.3.3.cmml" type="integer" xref="S5.I1.i2.p1.5.m5.1.1.1.1.1.1.1.1.3.3">1</cn></apply></apply></apply></apply><apply id="S5.I1.i2.p1.5.m5.3.3.3.cmml" xref="S5.I1.i2.p1.5.m5.3.3.3"><times id="S5.I1.i2.p1.5.m5.3.3.3.3.cmml" xref="S5.I1.i2.p1.5.m5.3.3.3.3"></times><apply id="S5.I1.i2.p1.5.m5.2.2.2.1.1.1.cmml" xref="S5.I1.i2.p1.5.m5.2.2.2.1.1"><plus id="S5.I1.i2.p1.5.m5.2.2.2.1.1.1.1.cmml" xref="S5.I1.i2.p1.5.m5.2.2.2.1.1.1.1"></plus><cn id="S5.I1.i2.p1.5.m5.2.2.2.1.1.1.2.cmml" type="integer" xref="S5.I1.i2.p1.5.m5.2.2.2.1.1.1.2">1</cn><apply id="S5.I1.i2.p1.5.m5.2.2.2.1.1.1.3.cmml" xref="S5.I1.i2.p1.5.m5.2.2.2.1.1.1.3"><divide id="S5.I1.i2.p1.5.m5.2.2.2.1.1.1.3.1.cmml" xref="S5.I1.i2.p1.5.m5.2.2.2.1.1.1.3"></divide><ci id="S5.I1.i2.p1.5.m5.2.2.2.1.1.1.3.2.cmml" xref="S5.I1.i2.p1.5.m5.2.2.2.1.1.1.3.2">𝜀</ci><cn id="S5.I1.i2.p1.5.m5.2.2.2.1.1.1.3.3.cmml" type="integer" xref="S5.I1.i2.p1.5.m5.2.2.2.1.1.1.3.3">16</cn></apply></apply><apply id="S5.I1.i2.p1.5.m5.3.3.3.2.2.cmml" xref="S5.I1.i2.p1.5.m5.3.3.3.2.1"><abs id="S5.I1.i2.p1.5.m5.3.3.3.2.2.1.cmml" xref="S5.I1.i2.p1.5.m5.3.3.3.2.1.2"></abs><apply id="S5.I1.i2.p1.5.m5.3.3.3.2.1.1.cmml" xref="S5.I1.i2.p1.5.m5.3.3.3.2.1.1"><times id="S5.I1.i2.p1.5.m5.3.3.3.2.1.1.2.cmml" xref="S5.I1.i2.p1.5.m5.3.3.3.2.1.1.2"></times><ci id="S5.I1.i2.p1.5.m5.3.3.3.2.1.1.3.cmml" xref="S5.I1.i2.p1.5.m5.3.3.3.2.1.1.3">𝑉</ci><apply id="S5.I1.i2.p1.5.m5.3.3.3.2.1.1.1.1.1.cmml" xref="S5.I1.i2.p1.5.m5.3.3.3.2.1.1.1.1"><csymbol cd="ambiguous" id="S5.I1.i2.p1.5.m5.3.3.3.2.1.1.1.1.1.1.cmml" xref="S5.I1.i2.p1.5.m5.3.3.3.2.1.1.1.1">subscript</csymbol><apply id="S5.I1.i2.p1.5.m5.3.3.3.2.1.1.1.1.1.2.cmml" xref="S5.I1.i2.p1.5.m5.3.3.3.2.1.1.1.1"><csymbol cd="ambiguous" id="S5.I1.i2.p1.5.m5.3.3.3.2.1.1.1.1.1.2.1.cmml" xref="S5.I1.i2.p1.5.m5.3.3.3.2.1.1.1.1">superscript</csymbol><ci id="S5.I1.i2.p1.5.m5.3.3.3.2.1.1.1.1.1.2.2.cmml" xref="S5.I1.i2.p1.5.m5.3.3.3.2.1.1.1.1.1.2.2">𝐺</ci><cn id="S5.I1.i2.p1.5.m5.3.3.3.2.1.1.1.1.1.2.3.cmml" type="integer" xref="S5.I1.i2.p1.5.m5.3.3.3.2.1.1.1.1.1.2.3">1</cn></apply><ci id="S5.I1.i2.p1.5.m5.3.3.3.2.1.1.1.1.1.3.cmml" xref="S5.I1.i2.p1.5.m5.3.3.3.2.1.1.1.1.1.3">𝑘</ci></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.I1.i2.p1.5.m5.3c">|V(G^{1}_{k+1})|<(1+\frac{\varepsilon}{16})|V(G^{1}_{k})|</annotation><annotation encoding="application/x-llamapun" id="S5.I1.i2.p1.5.m5.3d">| italic_V ( italic_G start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_k + 1 end_POSTSUBSCRIPT ) | < ( 1 + divide start_ARG italic_ε end_ARG start_ARG 16 end_ARG ) | italic_V ( italic_G start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ) |</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S5.I1.i2.p1.6.6">. We apply Lemma </span><a class="ltx_ref ltx_font_italic" href="https://arxiv.org/html/2411.12694v2#S5.Thmtheorem1" title="Lemma 5.1. ‣ 5 Results for dynamic algorithms ‣ Local Density and its Distributed Approximation"><span class="ltx_text ltx_ref_tag">5.1</span></a><span class="ltx_text ltx_font_italic" id="S5.I1.i2.p1.6.7"> to observe that </span><math alttext="(1+\eta)^{-k}\geq(1+\varepsilon/2)^{-1}" class="ltx_Math" display="inline" id="S5.I1.i2.p1.6.m6.2"><semantics id="S5.I1.i2.p1.6.m6.2a"><mrow id="S5.I1.i2.p1.6.m6.2.2" xref="S5.I1.i2.p1.6.m6.2.2.cmml"><msup id="S5.I1.i2.p1.6.m6.1.1.1" xref="S5.I1.i2.p1.6.m6.1.1.1.cmml"><mrow id="S5.I1.i2.p1.6.m6.1.1.1.1.1" xref="S5.I1.i2.p1.6.m6.1.1.1.1.1.1.cmml"><mo id="S5.I1.i2.p1.6.m6.1.1.1.1.1.2" stretchy="false" xref="S5.I1.i2.p1.6.m6.1.1.1.1.1.1.cmml">(</mo><mrow id="S5.I1.i2.p1.6.m6.1.1.1.1.1.1" xref="S5.I1.i2.p1.6.m6.1.1.1.1.1.1.cmml"><mn id="S5.I1.i2.p1.6.m6.1.1.1.1.1.1.2" xref="S5.I1.i2.p1.6.m6.1.1.1.1.1.1.2.cmml">1</mn><mo id="S5.I1.i2.p1.6.m6.1.1.1.1.1.1.1" xref="S5.I1.i2.p1.6.m6.1.1.1.1.1.1.1.cmml">+</mo><mi id="S5.I1.i2.p1.6.m6.1.1.1.1.1.1.3" xref="S5.I1.i2.p1.6.m6.1.1.1.1.1.1.3.cmml">η</mi></mrow><mo id="S5.I1.i2.p1.6.m6.1.1.1.1.1.3" stretchy="false" xref="S5.I1.i2.p1.6.m6.1.1.1.1.1.1.cmml">)</mo></mrow><mrow id="S5.I1.i2.p1.6.m6.1.1.1.3" xref="S5.I1.i2.p1.6.m6.1.1.1.3.cmml"><mo id="S5.I1.i2.p1.6.m6.1.1.1.3a" xref="S5.I1.i2.p1.6.m6.1.1.1.3.cmml">−</mo><mi id="S5.I1.i2.p1.6.m6.1.1.1.3.2" xref="S5.I1.i2.p1.6.m6.1.1.1.3.2.cmml">k</mi></mrow></msup><mo id="S5.I1.i2.p1.6.m6.2.2.3" xref="S5.I1.i2.p1.6.m6.2.2.3.cmml">≥</mo><msup id="S5.I1.i2.p1.6.m6.2.2.2" xref="S5.I1.i2.p1.6.m6.2.2.2.cmml"><mrow id="S5.I1.i2.p1.6.m6.2.2.2.1.1" xref="S5.I1.i2.p1.6.m6.2.2.2.1.1.1.cmml"><mo id="S5.I1.i2.p1.6.m6.2.2.2.1.1.2" stretchy="false" xref="S5.I1.i2.p1.6.m6.2.2.2.1.1.1.cmml">(</mo><mrow id="S5.I1.i2.p1.6.m6.2.2.2.1.1.1" xref="S5.I1.i2.p1.6.m6.2.2.2.1.1.1.cmml"><mn id="S5.I1.i2.p1.6.m6.2.2.2.1.1.1.2" xref="S5.I1.i2.p1.6.m6.2.2.2.1.1.1.2.cmml">1</mn><mo id="S5.I1.i2.p1.6.m6.2.2.2.1.1.1.1" xref="S5.I1.i2.p1.6.m6.2.2.2.1.1.1.1.cmml">+</mo><mrow id="S5.I1.i2.p1.6.m6.2.2.2.1.1.1.3" xref="S5.I1.i2.p1.6.m6.2.2.2.1.1.1.3.cmml"><mi id="S5.I1.i2.p1.6.m6.2.2.2.1.1.1.3.2" xref="S5.I1.i2.p1.6.m6.2.2.2.1.1.1.3.2.cmml">ε</mi><mo id="S5.I1.i2.p1.6.m6.2.2.2.1.1.1.3.1" xref="S5.I1.i2.p1.6.m6.2.2.2.1.1.1.3.1.cmml">/</mo><mn id="S5.I1.i2.p1.6.m6.2.2.2.1.1.1.3.3" xref="S5.I1.i2.p1.6.m6.2.2.2.1.1.1.3.3.cmml">2</mn></mrow></mrow><mo id="S5.I1.i2.p1.6.m6.2.2.2.1.1.3" stretchy="false" xref="S5.I1.i2.p1.6.m6.2.2.2.1.1.1.cmml">)</mo></mrow><mrow id="S5.I1.i2.p1.6.m6.2.2.2.3" xref="S5.I1.i2.p1.6.m6.2.2.2.3.cmml"><mo id="S5.I1.i2.p1.6.m6.2.2.2.3a" xref="S5.I1.i2.p1.6.m6.2.2.2.3.cmml">−</mo><mn id="S5.I1.i2.p1.6.m6.2.2.2.3.2" xref="S5.I1.i2.p1.6.m6.2.2.2.3.2.cmml">1</mn></mrow></msup></mrow><annotation-xml encoding="MathML-Content" id="S5.I1.i2.p1.6.m6.2b"><apply id="S5.I1.i2.p1.6.m6.2.2.cmml" xref="S5.I1.i2.p1.6.m6.2.2"><geq id="S5.I1.i2.p1.6.m6.2.2.3.cmml" xref="S5.I1.i2.p1.6.m6.2.2.3"></geq><apply id="S5.I1.i2.p1.6.m6.1.1.1.cmml" xref="S5.I1.i2.p1.6.m6.1.1.1"><csymbol cd="ambiguous" id="S5.I1.i2.p1.6.m6.1.1.1.2.cmml" xref="S5.I1.i2.p1.6.m6.1.1.1">superscript</csymbol><apply id="S5.I1.i2.p1.6.m6.1.1.1.1.1.1.cmml" xref="S5.I1.i2.p1.6.m6.1.1.1.1.1"><plus id="S5.I1.i2.p1.6.m6.1.1.1.1.1.1.1.cmml" xref="S5.I1.i2.p1.6.m6.1.1.1.1.1.1.1"></plus><cn id="S5.I1.i2.p1.6.m6.1.1.1.1.1.1.2.cmml" type="integer" xref="S5.I1.i2.p1.6.m6.1.1.1.1.1.1.2">1</cn><ci id="S5.I1.i2.p1.6.m6.1.1.1.1.1.1.3.cmml" xref="S5.I1.i2.p1.6.m6.1.1.1.1.1.1.3">𝜂</ci></apply><apply id="S5.I1.i2.p1.6.m6.1.1.1.3.cmml" xref="S5.I1.i2.p1.6.m6.1.1.1.3"><minus id="S5.I1.i2.p1.6.m6.1.1.1.3.1.cmml" xref="S5.I1.i2.p1.6.m6.1.1.1.3"></minus><ci id="S5.I1.i2.p1.6.m6.1.1.1.3.2.cmml" xref="S5.I1.i2.p1.6.m6.1.1.1.3.2">𝑘</ci></apply></apply><apply id="S5.I1.i2.p1.6.m6.2.2.2.cmml" xref="S5.I1.i2.p1.6.m6.2.2.2"><csymbol cd="ambiguous" id="S5.I1.i2.p1.6.m6.2.2.2.2.cmml" xref="S5.I1.i2.p1.6.m6.2.2.2">superscript</csymbol><apply id="S5.I1.i2.p1.6.m6.2.2.2.1.1.1.cmml" xref="S5.I1.i2.p1.6.m6.2.2.2.1.1"><plus id="S5.I1.i2.p1.6.m6.2.2.2.1.1.1.1.cmml" xref="S5.I1.i2.p1.6.m6.2.2.2.1.1.1.1"></plus><cn id="S5.I1.i2.p1.6.m6.2.2.2.1.1.1.2.cmml" type="integer" xref="S5.I1.i2.p1.6.m6.2.2.2.1.1.1.2">1</cn><apply id="S5.I1.i2.p1.6.m6.2.2.2.1.1.1.3.cmml" xref="S5.I1.i2.p1.6.m6.2.2.2.1.1.1.3"><divide id="S5.I1.i2.p1.6.m6.2.2.2.1.1.1.3.1.cmml" xref="S5.I1.i2.p1.6.m6.2.2.2.1.1.1.3.1"></divide><ci id="S5.I1.i2.p1.6.m6.2.2.2.1.1.1.3.2.cmml" xref="S5.I1.i2.p1.6.m6.2.2.2.1.1.1.3.2">𝜀</ci><cn id="S5.I1.i2.p1.6.m6.2.2.2.1.1.1.3.3.cmml" type="integer" xref="S5.I1.i2.p1.6.m6.2.2.2.1.1.1.3.3">2</cn></apply></apply><apply id="S5.I1.i2.p1.6.m6.2.2.2.3.cmml" xref="S5.I1.i2.p1.6.m6.2.2.2.3"><minus id="S5.I1.i2.p1.6.m6.2.2.2.3.1.cmml" xref="S5.I1.i2.p1.6.m6.2.2.2.3"></minus><cn id="S5.I1.i2.p1.6.m6.2.2.2.3.2.cmml" type="integer" xref="S5.I1.i2.p1.6.m6.2.2.2.3.2">1</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.I1.i2.p1.6.m6.2c">(1+\eta)^{-k}\geq(1+\varepsilon/2)^{-1}</annotation><annotation encoding="application/x-llamapun" id="S5.I1.i2.p1.6.m6.2d">( 1 + italic_η ) start_POSTSUPERSCRIPT - italic_k end_POSTSUPERSCRIPT ≥ ( 1 + italic_ε / 2 ) start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S5.I1.i2.p1.6.8">.</span></p> </div> </li> <li class="ltx_item" id="S5.I1.i3" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">•</span> <div class="ltx_para" id="S5.I1.i3.p1"> <p class="ltx_p" id="S5.I1.i3.p1.1"><span class="ltx_text ltx_font_italic" id="S5.I1.i3.p1.1.1">Finally, we use both orientations to create three claims that contradict one another.</span></p> </div> </li> </ul> </div> <div class="ltx_para ltx_noindent" id="S5.Thmtheorem3.p4"> <p class="ltx_p" id="S5.Thmtheorem3.p4.4"><em class="ltx_emph" id="S5.Thmtheorem3.p4.4.1">The first claim. </em><span class="ltx_text ltx_font_italic" id="S5.Thmtheorem3.p4.4.2"> We denote by </span><math alttext="E_{\uparrow}" class="ltx_Math" display="inline" id="S5.Thmtheorem3.p4.1.m1.1"><semantics id="S5.Thmtheorem3.p4.1.m1.1a"><msub id="S5.Thmtheorem3.p4.1.m1.1.1" xref="S5.Thmtheorem3.p4.1.m1.1.1.cmml"><mi id="S5.Thmtheorem3.p4.1.m1.1.1.2" xref="S5.Thmtheorem3.p4.1.m1.1.1.2.cmml">E</mi><mo id="S5.Thmtheorem3.p4.1.m1.1.1.3" stretchy="false" xref="S5.Thmtheorem3.p4.1.m1.1.1.3.cmml">↑</mo></msub><annotation-xml encoding="MathML-Content" id="S5.Thmtheorem3.p4.1.m1.1b"><apply id="S5.Thmtheorem3.p4.1.m1.1.1.cmml" xref="S5.Thmtheorem3.p4.1.m1.1.1"><csymbol cd="ambiguous" id="S5.Thmtheorem3.p4.1.m1.1.1.1.cmml" xref="S5.Thmtheorem3.p4.1.m1.1.1">subscript</csymbol><ci id="S5.Thmtheorem3.p4.1.m1.1.1.2.cmml" xref="S5.Thmtheorem3.p4.1.m1.1.1.2">𝐸</ci><ci id="S5.Thmtheorem3.p4.1.m1.1.1.3.cmml" xref="S5.Thmtheorem3.p4.1.m1.1.1.3">↑</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmtheorem3.p4.1.m1.1c">E_{\uparrow}</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem3.p4.1.m1.1d">italic_E start_POSTSUBSCRIPT ↑ end_POSTSUBSCRIPT</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S5.Thmtheorem3.p4.4.3"> the set of all edges </span><math alttext="e=(a,b)" class="ltx_Math" display="inline" id="S5.Thmtheorem3.p4.2.m2.2"><semantics id="S5.Thmtheorem3.p4.2.m2.2a"><mrow id="S5.Thmtheorem3.p4.2.m2.2.3" xref="S5.Thmtheorem3.p4.2.m2.2.3.cmml"><mi id="S5.Thmtheorem3.p4.2.m2.2.3.2" xref="S5.Thmtheorem3.p4.2.m2.2.3.2.cmml">e</mi><mo id="S5.Thmtheorem3.p4.2.m2.2.3.1" xref="S5.Thmtheorem3.p4.2.m2.2.3.1.cmml">=</mo><mrow id="S5.Thmtheorem3.p4.2.m2.2.3.3.2" xref="S5.Thmtheorem3.p4.2.m2.2.3.3.1.cmml"><mo id="S5.Thmtheorem3.p4.2.m2.2.3.3.2.1" stretchy="false" xref="S5.Thmtheorem3.p4.2.m2.2.3.3.1.cmml">(</mo><mi id="S5.Thmtheorem3.p4.2.m2.1.1" xref="S5.Thmtheorem3.p4.2.m2.1.1.cmml">a</mi><mo id="S5.Thmtheorem3.p4.2.m2.2.3.3.2.2" xref="S5.Thmtheorem3.p4.2.m2.2.3.3.1.cmml">,</mo><mi id="S5.Thmtheorem3.p4.2.m2.2.2" xref="S5.Thmtheorem3.p4.2.m2.2.2.cmml">b</mi><mo id="S5.Thmtheorem3.p4.2.m2.2.3.3.2.3" stretchy="false" xref="S5.Thmtheorem3.p4.2.m2.2.3.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.Thmtheorem3.p4.2.m2.2b"><apply id="S5.Thmtheorem3.p4.2.m2.2.3.cmml" xref="S5.Thmtheorem3.p4.2.m2.2.3"><eq id="S5.Thmtheorem3.p4.2.m2.2.3.1.cmml" xref="S5.Thmtheorem3.p4.2.m2.2.3.1"></eq><ci id="S5.Thmtheorem3.p4.2.m2.2.3.2.cmml" xref="S5.Thmtheorem3.p4.2.m2.2.3.2">𝑒</ci><interval closure="open" id="S5.Thmtheorem3.p4.2.m2.2.3.3.1.cmml" xref="S5.Thmtheorem3.p4.2.m2.2.3.3.2"><ci id="S5.Thmtheorem3.p4.2.m2.1.1.cmml" xref="S5.Thmtheorem3.p4.2.m2.1.1">𝑎</ci><ci id="S5.Thmtheorem3.p4.2.m2.2.2.cmml" xref="S5.Thmtheorem3.p4.2.m2.2.2">𝑏</ci></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmtheorem3.p4.2.m2.2c">e=(a,b)</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem3.p4.2.m2.2d">italic_e = ( italic_a , italic_b )</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S5.Thmtheorem3.p4.4.4"> with </span><math alttext="a\in V(G^{1}_{k+1})" class="ltx_Math" display="inline" id="S5.Thmtheorem3.p4.3.m3.1"><semantics id="S5.Thmtheorem3.p4.3.m3.1a"><mrow id="S5.Thmtheorem3.p4.3.m3.1.1" xref="S5.Thmtheorem3.p4.3.m3.1.1.cmml"><mi id="S5.Thmtheorem3.p4.3.m3.1.1.3" xref="S5.Thmtheorem3.p4.3.m3.1.1.3.cmml">a</mi><mo id="S5.Thmtheorem3.p4.3.m3.1.1.2" xref="S5.Thmtheorem3.p4.3.m3.1.1.2.cmml">∈</mo><mrow id="S5.Thmtheorem3.p4.3.m3.1.1.1" xref="S5.Thmtheorem3.p4.3.m3.1.1.1.cmml"><mi id="S5.Thmtheorem3.p4.3.m3.1.1.1.3" xref="S5.Thmtheorem3.p4.3.m3.1.1.1.3.cmml">V</mi><mo id="S5.Thmtheorem3.p4.3.m3.1.1.1.2" xref="S5.Thmtheorem3.p4.3.m3.1.1.1.2.cmml">⁢</mo><mrow id="S5.Thmtheorem3.p4.3.m3.1.1.1.1.1" xref="S5.Thmtheorem3.p4.3.m3.1.1.1.1.1.1.cmml"><mo id="S5.Thmtheorem3.p4.3.m3.1.1.1.1.1.2" stretchy="false" xref="S5.Thmtheorem3.p4.3.m3.1.1.1.1.1.1.cmml">(</mo><msubsup id="S5.Thmtheorem3.p4.3.m3.1.1.1.1.1.1" xref="S5.Thmtheorem3.p4.3.m3.1.1.1.1.1.1.cmml"><mi id="S5.Thmtheorem3.p4.3.m3.1.1.1.1.1.1.2.2" xref="S5.Thmtheorem3.p4.3.m3.1.1.1.1.1.1.2.2.cmml">G</mi><mrow id="S5.Thmtheorem3.p4.3.m3.1.1.1.1.1.1.3" xref="S5.Thmtheorem3.p4.3.m3.1.1.1.1.1.1.3.cmml"><mi id="S5.Thmtheorem3.p4.3.m3.1.1.1.1.1.1.3.2" xref="S5.Thmtheorem3.p4.3.m3.1.1.1.1.1.1.3.2.cmml">k</mi><mo id="S5.Thmtheorem3.p4.3.m3.1.1.1.1.1.1.3.1" xref="S5.Thmtheorem3.p4.3.m3.1.1.1.1.1.1.3.1.cmml">+</mo><mn id="S5.Thmtheorem3.p4.3.m3.1.1.1.1.1.1.3.3" xref="S5.Thmtheorem3.p4.3.m3.1.1.1.1.1.1.3.3.cmml">1</mn></mrow><mn id="S5.Thmtheorem3.p4.3.m3.1.1.1.1.1.1.2.3" xref="S5.Thmtheorem3.p4.3.m3.1.1.1.1.1.1.2.3.cmml">1</mn></msubsup><mo id="S5.Thmtheorem3.p4.3.m3.1.1.1.1.1.3" stretchy="false" xref="S5.Thmtheorem3.p4.3.m3.1.1.1.1.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.Thmtheorem3.p4.3.m3.1b"><apply id="S5.Thmtheorem3.p4.3.m3.1.1.cmml" xref="S5.Thmtheorem3.p4.3.m3.1.1"><in id="S5.Thmtheorem3.p4.3.m3.1.1.2.cmml" xref="S5.Thmtheorem3.p4.3.m3.1.1.2"></in><ci id="S5.Thmtheorem3.p4.3.m3.1.1.3.cmml" xref="S5.Thmtheorem3.p4.3.m3.1.1.3">𝑎</ci><apply id="S5.Thmtheorem3.p4.3.m3.1.1.1.cmml" xref="S5.Thmtheorem3.p4.3.m3.1.1.1"><times id="S5.Thmtheorem3.p4.3.m3.1.1.1.2.cmml" xref="S5.Thmtheorem3.p4.3.m3.1.1.1.2"></times><ci id="S5.Thmtheorem3.p4.3.m3.1.1.1.3.cmml" xref="S5.Thmtheorem3.p4.3.m3.1.1.1.3">𝑉</ci><apply id="S5.Thmtheorem3.p4.3.m3.1.1.1.1.1.1.cmml" xref="S5.Thmtheorem3.p4.3.m3.1.1.1.1.1"><csymbol cd="ambiguous" id="S5.Thmtheorem3.p4.3.m3.1.1.1.1.1.1.1.cmml" xref="S5.Thmtheorem3.p4.3.m3.1.1.1.1.1">subscript</csymbol><apply id="S5.Thmtheorem3.p4.3.m3.1.1.1.1.1.1.2.cmml" xref="S5.Thmtheorem3.p4.3.m3.1.1.1.1.1"><csymbol cd="ambiguous" id="S5.Thmtheorem3.p4.3.m3.1.1.1.1.1.1.2.1.cmml" xref="S5.Thmtheorem3.p4.3.m3.1.1.1.1.1">superscript</csymbol><ci id="S5.Thmtheorem3.p4.3.m3.1.1.1.1.1.1.2.2.cmml" xref="S5.Thmtheorem3.p4.3.m3.1.1.1.1.1.1.2.2">𝐺</ci><cn id="S5.Thmtheorem3.p4.3.m3.1.1.1.1.1.1.2.3.cmml" type="integer" xref="S5.Thmtheorem3.p4.3.m3.1.1.1.1.1.1.2.3">1</cn></apply><apply id="S5.Thmtheorem3.p4.3.m3.1.1.1.1.1.1.3.cmml" xref="S5.Thmtheorem3.p4.3.m3.1.1.1.1.1.1.3"><plus id="S5.Thmtheorem3.p4.3.m3.1.1.1.1.1.1.3.1.cmml" xref="S5.Thmtheorem3.p4.3.m3.1.1.1.1.1.1.3.1"></plus><ci id="S5.Thmtheorem3.p4.3.m3.1.1.1.1.1.1.3.2.cmml" xref="S5.Thmtheorem3.p4.3.m3.1.1.1.1.1.1.3.2">𝑘</ci><cn id="S5.Thmtheorem3.p4.3.m3.1.1.1.1.1.1.3.3.cmml" type="integer" xref="S5.Thmtheorem3.p4.3.m3.1.1.1.1.1.1.3.3">1</cn></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmtheorem3.p4.3.m3.1c">a\in V(G^{1}_{k+1})</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem3.p4.3.m3.1d">italic_a ∈ italic_V ( italic_G start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_k + 1 end_POSTSUBSCRIPT )</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S5.Thmtheorem3.p4.4.5"> and </span><math alttext="b\in V(G^{2})" class="ltx_Math" display="inline" id="S5.Thmtheorem3.p4.4.m4.1"><semantics id="S5.Thmtheorem3.p4.4.m4.1a"><mrow id="S5.Thmtheorem3.p4.4.m4.1.1" xref="S5.Thmtheorem3.p4.4.m4.1.1.cmml"><mi id="S5.Thmtheorem3.p4.4.m4.1.1.3" xref="S5.Thmtheorem3.p4.4.m4.1.1.3.cmml">b</mi><mo id="S5.Thmtheorem3.p4.4.m4.1.1.2" xref="S5.Thmtheorem3.p4.4.m4.1.1.2.cmml">∈</mo><mrow id="S5.Thmtheorem3.p4.4.m4.1.1.1" xref="S5.Thmtheorem3.p4.4.m4.1.1.1.cmml"><mi id="S5.Thmtheorem3.p4.4.m4.1.1.1.3" xref="S5.Thmtheorem3.p4.4.m4.1.1.1.3.cmml">V</mi><mo id="S5.Thmtheorem3.p4.4.m4.1.1.1.2" xref="S5.Thmtheorem3.p4.4.m4.1.1.1.2.cmml">⁢</mo><mrow id="S5.Thmtheorem3.p4.4.m4.1.1.1.1.1" xref="S5.Thmtheorem3.p4.4.m4.1.1.1.1.1.1.cmml"><mo id="S5.Thmtheorem3.p4.4.m4.1.1.1.1.1.2" stretchy="false" xref="S5.Thmtheorem3.p4.4.m4.1.1.1.1.1.1.cmml">(</mo><msup id="S5.Thmtheorem3.p4.4.m4.1.1.1.1.1.1" xref="S5.Thmtheorem3.p4.4.m4.1.1.1.1.1.1.cmml"><mi id="S5.Thmtheorem3.p4.4.m4.1.1.1.1.1.1.2" xref="S5.Thmtheorem3.p4.4.m4.1.1.1.1.1.1.2.cmml">G</mi><mn id="S5.Thmtheorem3.p4.4.m4.1.1.1.1.1.1.3" xref="S5.Thmtheorem3.p4.4.m4.1.1.1.1.1.1.3.cmml">2</mn></msup><mo id="S5.Thmtheorem3.p4.4.m4.1.1.1.1.1.3" stretchy="false" xref="S5.Thmtheorem3.p4.4.m4.1.1.1.1.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.Thmtheorem3.p4.4.m4.1b"><apply id="S5.Thmtheorem3.p4.4.m4.1.1.cmml" xref="S5.Thmtheorem3.p4.4.m4.1.1"><in id="S5.Thmtheorem3.p4.4.m4.1.1.2.cmml" xref="S5.Thmtheorem3.p4.4.m4.1.1.2"></in><ci id="S5.Thmtheorem3.p4.4.m4.1.1.3.cmml" xref="S5.Thmtheorem3.p4.4.m4.1.1.3">𝑏</ci><apply id="S5.Thmtheorem3.p4.4.m4.1.1.1.cmml" xref="S5.Thmtheorem3.p4.4.m4.1.1.1"><times id="S5.Thmtheorem3.p4.4.m4.1.1.1.2.cmml" xref="S5.Thmtheorem3.p4.4.m4.1.1.1.2"></times><ci id="S5.Thmtheorem3.p4.4.m4.1.1.1.3.cmml" xref="S5.Thmtheorem3.p4.4.m4.1.1.1.3">𝑉</ci><apply id="S5.Thmtheorem3.p4.4.m4.1.1.1.1.1.1.cmml" xref="S5.Thmtheorem3.p4.4.m4.1.1.1.1.1"><csymbol cd="ambiguous" id="S5.Thmtheorem3.p4.4.m4.1.1.1.1.1.1.1.cmml" xref="S5.Thmtheorem3.p4.4.m4.1.1.1.1.1">superscript</csymbol><ci id="S5.Thmtheorem3.p4.4.m4.1.1.1.1.1.1.2.cmml" xref="S5.Thmtheorem3.p4.4.m4.1.1.1.1.1.1.2">𝐺</ci><cn id="S5.Thmtheorem3.p4.4.m4.1.1.1.1.1.1.3.cmml" type="integer" xref="S5.Thmtheorem3.p4.4.m4.1.1.1.1.1.1.3">2</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmtheorem3.p4.4.m4.1c">b\in V(G^{2})</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem3.p4.4.m4.1d">italic_b ∈ italic_V ( italic_G start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT )</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S5.Thmtheorem3.p4.4.6"> and claim that:</span></p> <table class="ltx_equation ltx_eqn_table" id="S5.E3"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_left_padleft"></td> <td class="ltx_eqn_cell ltx_align_left"><math alttext="\sum_{e\in E(G^{1}_{k+1})}\textsl{g}(e)+\sum_{e\in E_{\uparrow}}\textsl{g}(e)% \leq\sum_{v\in V(G_{k+1}^{1})}\textsl{g}_{x}(v)." class="ltx_Math" display="block" id="S5.E3.m1.6"><semantics id="S5.E3.m1.6a"><mrow id="S5.E3.m1.6.6.1" xref="S5.E3.m1.6.6.1.1.cmml"><mrow id="S5.E3.m1.6.6.1.1" xref="S5.E3.m1.6.6.1.1.cmml"><mrow id="S5.E3.m1.6.6.1.1.2" xref="S5.E3.m1.6.6.1.1.2.cmml"><mrow id="S5.E3.m1.6.6.1.1.2.2" xref="S5.E3.m1.6.6.1.1.2.2.cmml"><munder id="S5.E3.m1.6.6.1.1.2.2.1" xref="S5.E3.m1.6.6.1.1.2.2.1.cmml"><mo id="S5.E3.m1.6.6.1.1.2.2.1.2" movablelimits="false" xref="S5.E3.m1.6.6.1.1.2.2.1.2.cmml">∑</mo><mrow id="S5.E3.m1.1.1.1" xref="S5.E3.m1.1.1.1.cmml"><mi id="S5.E3.m1.1.1.1.3" xref="S5.E3.m1.1.1.1.3.cmml">e</mi><mo id="S5.E3.m1.1.1.1.2" xref="S5.E3.m1.1.1.1.2.cmml">∈</mo><mrow id="S5.E3.m1.1.1.1.1" xref="S5.E3.m1.1.1.1.1.cmml"><mi id="S5.E3.m1.1.1.1.1.3" xref="S5.E3.m1.1.1.1.1.3.cmml">E</mi><mo id="S5.E3.m1.1.1.1.1.2" xref="S5.E3.m1.1.1.1.1.2.cmml">⁢</mo><mrow id="S5.E3.m1.1.1.1.1.1.1" xref="S5.E3.m1.1.1.1.1.1.1.1.cmml"><mo id="S5.E3.m1.1.1.1.1.1.1.2" stretchy="false" xref="S5.E3.m1.1.1.1.1.1.1.1.cmml">(</mo><msubsup id="S5.E3.m1.1.1.1.1.1.1.1" xref="S5.E3.m1.1.1.1.1.1.1.1.cmml"><mi id="S5.E3.m1.1.1.1.1.1.1.1.2.2" xref="S5.E3.m1.1.1.1.1.1.1.1.2.2.cmml">G</mi><mrow id="S5.E3.m1.1.1.1.1.1.1.1.3" xref="S5.E3.m1.1.1.1.1.1.1.1.3.cmml"><mi id="S5.E3.m1.1.1.1.1.1.1.1.3.2" xref="S5.E3.m1.1.1.1.1.1.1.1.3.2.cmml">k</mi><mo id="S5.E3.m1.1.1.1.1.1.1.1.3.1" xref="S5.E3.m1.1.1.1.1.1.1.1.3.1.cmml">+</mo><mn id="S5.E3.m1.1.1.1.1.1.1.1.3.3" xref="S5.E3.m1.1.1.1.1.1.1.1.3.3.cmml">1</mn></mrow><mn id="S5.E3.m1.1.1.1.1.1.1.1.2.3" xref="S5.E3.m1.1.1.1.1.1.1.1.2.3.cmml">1</mn></msubsup><mo id="S5.E3.m1.1.1.1.1.1.1.3" stretchy="false" xref="S5.E3.m1.1.1.1.1.1.1.1.cmml">)</mo></mrow></mrow></mrow></munder><mrow id="S5.E3.m1.6.6.1.1.2.2.2" xref="S5.E3.m1.6.6.1.1.2.2.2.cmml"><mtext class="ltx_mathvariant_italic" id="S5.E3.m1.6.6.1.1.2.2.2.2" xref="S5.E3.m1.6.6.1.1.2.2.2.2a.cmml">g</mtext><mo id="S5.E3.m1.6.6.1.1.2.2.2.1" xref="S5.E3.m1.6.6.1.1.2.2.2.1.cmml">⁢</mo><mrow id="S5.E3.m1.6.6.1.1.2.2.2.3.2" xref="S5.E3.m1.6.6.1.1.2.2.2.cmml"><mo id="S5.E3.m1.6.6.1.1.2.2.2.3.2.1" stretchy="false" xref="S5.E3.m1.6.6.1.1.2.2.2.cmml">(</mo><mi id="S5.E3.m1.3.3" xref="S5.E3.m1.3.3.cmml">e</mi><mo id="S5.E3.m1.6.6.1.1.2.2.2.3.2.2" stretchy="false" xref="S5.E3.m1.6.6.1.1.2.2.2.cmml">)</mo></mrow></mrow></mrow><mo id="S5.E3.m1.6.6.1.1.2.1" rspace="0.055em" xref="S5.E3.m1.6.6.1.1.2.1.cmml">+</mo><mrow id="S5.E3.m1.6.6.1.1.2.3" xref="S5.E3.m1.6.6.1.1.2.3.cmml"><munder id="S5.E3.m1.6.6.1.1.2.3.1" xref="S5.E3.m1.6.6.1.1.2.3.1.cmml"><mo id="S5.E3.m1.6.6.1.1.2.3.1.2" movablelimits="false" xref="S5.E3.m1.6.6.1.1.2.3.1.2.cmml">∑</mo><mrow id="S5.E3.m1.6.6.1.1.2.3.1.3" xref="S5.E3.m1.6.6.1.1.2.3.1.3.cmml"><mi id="S5.E3.m1.6.6.1.1.2.3.1.3.2" xref="S5.E3.m1.6.6.1.1.2.3.1.3.2.cmml">e</mi><mo id="S5.E3.m1.6.6.1.1.2.3.1.3.1" xref="S5.E3.m1.6.6.1.1.2.3.1.3.1.cmml">∈</mo><msub id="S5.E3.m1.6.6.1.1.2.3.1.3.3" xref="S5.E3.m1.6.6.1.1.2.3.1.3.3.cmml"><mi id="S5.E3.m1.6.6.1.1.2.3.1.3.3.2" xref="S5.E3.m1.6.6.1.1.2.3.1.3.3.2.cmml">E</mi><mo id="S5.E3.m1.6.6.1.1.2.3.1.3.3.3" stretchy="false" xref="S5.E3.m1.6.6.1.1.2.3.1.3.3.3.cmml">↑</mo></msub></mrow></munder><mrow id="S5.E3.m1.6.6.1.1.2.3.2" xref="S5.E3.m1.6.6.1.1.2.3.2.cmml"><mtext class="ltx_mathvariant_italic" id="S5.E3.m1.6.6.1.1.2.3.2.2" xref="S5.E3.m1.6.6.1.1.2.3.2.2a.cmml">g</mtext><mo id="S5.E3.m1.6.6.1.1.2.3.2.1" xref="S5.E3.m1.6.6.1.1.2.3.2.1.cmml">⁢</mo><mrow id="S5.E3.m1.6.6.1.1.2.3.2.3.2" xref="S5.E3.m1.6.6.1.1.2.3.2.cmml"><mo id="S5.E3.m1.6.6.1.1.2.3.2.3.2.1" stretchy="false" xref="S5.E3.m1.6.6.1.1.2.3.2.cmml">(</mo><mi id="S5.E3.m1.4.4" xref="S5.E3.m1.4.4.cmml">e</mi><mo id="S5.E3.m1.6.6.1.1.2.3.2.3.2.2" stretchy="false" xref="S5.E3.m1.6.6.1.1.2.3.2.cmml">)</mo></mrow></mrow></mrow></mrow><mo id="S5.E3.m1.6.6.1.1.1" rspace="0.111em" xref="S5.E3.m1.6.6.1.1.1.cmml">≤</mo><mrow id="S5.E3.m1.6.6.1.1.3" xref="S5.E3.m1.6.6.1.1.3.cmml"><munder id="S5.E3.m1.6.6.1.1.3.1" xref="S5.E3.m1.6.6.1.1.3.1.cmml"><mo id="S5.E3.m1.6.6.1.1.3.1.2" movablelimits="false" xref="S5.E3.m1.6.6.1.1.3.1.2.cmml">∑</mo><mrow id="S5.E3.m1.2.2.1" xref="S5.E3.m1.2.2.1.cmml"><mi id="S5.E3.m1.2.2.1.3" xref="S5.E3.m1.2.2.1.3.cmml">v</mi><mo id="S5.E3.m1.2.2.1.2" xref="S5.E3.m1.2.2.1.2.cmml">∈</mo><mrow id="S5.E3.m1.2.2.1.1" xref="S5.E3.m1.2.2.1.1.cmml"><mi id="S5.E3.m1.2.2.1.1.3" xref="S5.E3.m1.2.2.1.1.3.cmml">V</mi><mo id="S5.E3.m1.2.2.1.1.2" xref="S5.E3.m1.2.2.1.1.2.cmml">⁢</mo><mrow id="S5.E3.m1.2.2.1.1.1.1" xref="S5.E3.m1.2.2.1.1.1.1.1.cmml"><mo id="S5.E3.m1.2.2.1.1.1.1.2" stretchy="false" xref="S5.E3.m1.2.2.1.1.1.1.1.cmml">(</mo><msubsup id="S5.E3.m1.2.2.1.1.1.1.1" xref="S5.E3.m1.2.2.1.1.1.1.1.cmml"><mi id="S5.E3.m1.2.2.1.1.1.1.1.2.2" xref="S5.E3.m1.2.2.1.1.1.1.1.2.2.cmml">G</mi><mrow id="S5.E3.m1.2.2.1.1.1.1.1.2.3" xref="S5.E3.m1.2.2.1.1.1.1.1.2.3.cmml"><mi id="S5.E3.m1.2.2.1.1.1.1.1.2.3.2" xref="S5.E3.m1.2.2.1.1.1.1.1.2.3.2.cmml">k</mi><mo id="S5.E3.m1.2.2.1.1.1.1.1.2.3.1" xref="S5.E3.m1.2.2.1.1.1.1.1.2.3.1.cmml">+</mo><mn id="S5.E3.m1.2.2.1.1.1.1.1.2.3.3" xref="S5.E3.m1.2.2.1.1.1.1.1.2.3.3.cmml">1</mn></mrow><mn id="S5.E3.m1.2.2.1.1.1.1.1.3" xref="S5.E3.m1.2.2.1.1.1.1.1.3.cmml">1</mn></msubsup><mo id="S5.E3.m1.2.2.1.1.1.1.3" stretchy="false" xref="S5.E3.m1.2.2.1.1.1.1.1.cmml">)</mo></mrow></mrow></mrow></munder><mrow id="S5.E3.m1.6.6.1.1.3.2" xref="S5.E3.m1.6.6.1.1.3.2.cmml"><msub id="S5.E3.m1.6.6.1.1.3.2.2" xref="S5.E3.m1.6.6.1.1.3.2.2.cmml"><mtext class="ltx_mathvariant_italic" 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encoding="application/x-llamapun" id="S5.E3.m1.6d">∑ start_POSTSUBSCRIPT italic_e ∈ italic_E ( italic_G start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_k + 1 end_POSTSUBSCRIPT ) end_POSTSUBSCRIPT g ( italic_e ) + ∑ start_POSTSUBSCRIPT italic_e ∈ italic_E start_POSTSUBSCRIPT ↑ end_POSTSUBSCRIPT end_POSTSUBSCRIPT g ( italic_e ) ≤ ∑ start_POSTSUBSCRIPT italic_v ∈ italic_V ( italic_G start_POSTSUBSCRIPT italic_k + 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT ) end_POSTSUBSCRIPT g start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT ( italic_v ) .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_left_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(3)</span></td> </tr></tbody> </table> </div> <div class="ltx_para ltx_noindent" id="S5.Thmtheorem3.p5"> <p class="ltx_p" id="S5.Thmtheorem3.p5.9"><span class="ltx_text ltx_font_italic" id="S5.Thmtheorem3.p5.9.1">Indeed for </span><math alttext="\overline{ab}\in E(G^{1}_{k+1})" class="ltx_Math" display="inline" id="S5.Thmtheorem3.p5.1.m1.1"><semantics id="S5.Thmtheorem3.p5.1.m1.1a"><mrow id="S5.Thmtheorem3.p5.1.m1.1.1" xref="S5.Thmtheorem3.p5.1.m1.1.1.cmml"><mover accent="true" id="S5.Thmtheorem3.p5.1.m1.1.1.3" xref="S5.Thmtheorem3.p5.1.m1.1.1.3.cmml"><mrow id="S5.Thmtheorem3.p5.1.m1.1.1.3.2" xref="S5.Thmtheorem3.p5.1.m1.1.1.3.2.cmml"><mi id="S5.Thmtheorem3.p5.1.m1.1.1.3.2.2" xref="S5.Thmtheorem3.p5.1.m1.1.1.3.2.2.cmml">a</mi><mo id="S5.Thmtheorem3.p5.1.m1.1.1.3.2.1" xref="S5.Thmtheorem3.p5.1.m1.1.1.3.2.1.cmml">⁢</mo><mi id="S5.Thmtheorem3.p5.1.m1.1.1.3.2.3" xref="S5.Thmtheorem3.p5.1.m1.1.1.3.2.3.cmml">b</mi></mrow><mo id="S5.Thmtheorem3.p5.1.m1.1.1.3.1" xref="S5.Thmtheorem3.p5.1.m1.1.1.3.1.cmml">¯</mo></mover><mo id="S5.Thmtheorem3.p5.1.m1.1.1.2" xref="S5.Thmtheorem3.p5.1.m1.1.1.2.cmml">∈</mo><mrow id="S5.Thmtheorem3.p5.1.m1.1.1.1" xref="S5.Thmtheorem3.p5.1.m1.1.1.1.cmml"><mi id="S5.Thmtheorem3.p5.1.m1.1.1.1.3" xref="S5.Thmtheorem3.p5.1.m1.1.1.1.3.cmml">E</mi><mo id="S5.Thmtheorem3.p5.1.m1.1.1.1.2" xref="S5.Thmtheorem3.p5.1.m1.1.1.1.2.cmml">⁢</mo><mrow id="S5.Thmtheorem3.p5.1.m1.1.1.1.1.1" xref="S5.Thmtheorem3.p5.1.m1.1.1.1.1.1.1.cmml"><mo id="S5.Thmtheorem3.p5.1.m1.1.1.1.1.1.2" stretchy="false" xref="S5.Thmtheorem3.p5.1.m1.1.1.1.1.1.1.cmml">(</mo><msubsup id="S5.Thmtheorem3.p5.1.m1.1.1.1.1.1.1" xref="S5.Thmtheorem3.p5.1.m1.1.1.1.1.1.1.cmml"><mi id="S5.Thmtheorem3.p5.1.m1.1.1.1.1.1.1.2.2" xref="S5.Thmtheorem3.p5.1.m1.1.1.1.1.1.1.2.2.cmml">G</mi><mrow id="S5.Thmtheorem3.p5.1.m1.1.1.1.1.1.1.3" xref="S5.Thmtheorem3.p5.1.m1.1.1.1.1.1.1.3.cmml"><mi id="S5.Thmtheorem3.p5.1.m1.1.1.1.1.1.1.3.2" xref="S5.Thmtheorem3.p5.1.m1.1.1.1.1.1.1.3.2.cmml">k</mi><mo id="S5.Thmtheorem3.p5.1.m1.1.1.1.1.1.1.3.1" xref="S5.Thmtheorem3.p5.1.m1.1.1.1.1.1.1.3.1.cmml">+</mo><mn id="S5.Thmtheorem3.p5.1.m1.1.1.1.1.1.1.3.3" xref="S5.Thmtheorem3.p5.1.m1.1.1.1.1.1.1.3.3.cmml">1</mn></mrow><mn id="S5.Thmtheorem3.p5.1.m1.1.1.1.1.1.1.2.3" xref="S5.Thmtheorem3.p5.1.m1.1.1.1.1.1.1.2.3.cmml">1</mn></msubsup><mo id="S5.Thmtheorem3.p5.1.m1.1.1.1.1.1.3" stretchy="false" xref="S5.Thmtheorem3.p5.1.m1.1.1.1.1.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.Thmtheorem3.p5.1.m1.1b"><apply id="S5.Thmtheorem3.p5.1.m1.1.1.cmml" xref="S5.Thmtheorem3.p5.1.m1.1.1"><in id="S5.Thmtheorem3.p5.1.m1.1.1.2.cmml" xref="S5.Thmtheorem3.p5.1.m1.1.1.2"></in><apply id="S5.Thmtheorem3.p5.1.m1.1.1.3.cmml" xref="S5.Thmtheorem3.p5.1.m1.1.1.3"><ci id="S5.Thmtheorem3.p5.1.m1.1.1.3.1.cmml" xref="S5.Thmtheorem3.p5.1.m1.1.1.3.1">¯</ci><apply id="S5.Thmtheorem3.p5.1.m1.1.1.3.2.cmml" xref="S5.Thmtheorem3.p5.1.m1.1.1.3.2"><times id="S5.Thmtheorem3.p5.1.m1.1.1.3.2.1.cmml" xref="S5.Thmtheorem3.p5.1.m1.1.1.3.2.1"></times><ci id="S5.Thmtheorem3.p5.1.m1.1.1.3.2.2.cmml" xref="S5.Thmtheorem3.p5.1.m1.1.1.3.2.2">𝑎</ci><ci id="S5.Thmtheorem3.p5.1.m1.1.1.3.2.3.cmml" xref="S5.Thmtheorem3.p5.1.m1.1.1.3.2.3">𝑏</ci></apply></apply><apply id="S5.Thmtheorem3.p5.1.m1.1.1.1.cmml" xref="S5.Thmtheorem3.p5.1.m1.1.1.1"><times id="S5.Thmtheorem3.p5.1.m1.1.1.1.2.cmml" xref="S5.Thmtheorem3.p5.1.m1.1.1.1.2"></times><ci id="S5.Thmtheorem3.p5.1.m1.1.1.1.3.cmml" xref="S5.Thmtheorem3.p5.1.m1.1.1.1.3">𝐸</ci><apply id="S5.Thmtheorem3.p5.1.m1.1.1.1.1.1.1.cmml" xref="S5.Thmtheorem3.p5.1.m1.1.1.1.1.1"><csymbol cd="ambiguous" id="S5.Thmtheorem3.p5.1.m1.1.1.1.1.1.1.1.cmml" xref="S5.Thmtheorem3.p5.1.m1.1.1.1.1.1">subscript</csymbol><apply id="S5.Thmtheorem3.p5.1.m1.1.1.1.1.1.1.2.cmml" xref="S5.Thmtheorem3.p5.1.m1.1.1.1.1.1"><csymbol cd="ambiguous" id="S5.Thmtheorem3.p5.1.m1.1.1.1.1.1.1.2.1.cmml" xref="S5.Thmtheorem3.p5.1.m1.1.1.1.1.1">superscript</csymbol><ci id="S5.Thmtheorem3.p5.1.m1.1.1.1.1.1.1.2.2.cmml" xref="S5.Thmtheorem3.p5.1.m1.1.1.1.1.1.1.2.2">𝐺</ci><cn id="S5.Thmtheorem3.p5.1.m1.1.1.1.1.1.1.2.3.cmml" type="integer" xref="S5.Thmtheorem3.p5.1.m1.1.1.1.1.1.1.2.3">1</cn></apply><apply id="S5.Thmtheorem3.p5.1.m1.1.1.1.1.1.1.3.cmml" xref="S5.Thmtheorem3.p5.1.m1.1.1.1.1.1.1.3"><plus id="S5.Thmtheorem3.p5.1.m1.1.1.1.1.1.1.3.1.cmml" xref="S5.Thmtheorem3.p5.1.m1.1.1.1.1.1.1.3.1"></plus><ci id="S5.Thmtheorem3.p5.1.m1.1.1.1.1.1.1.3.2.cmml" xref="S5.Thmtheorem3.p5.1.m1.1.1.1.1.1.1.3.2">𝑘</ci><cn id="S5.Thmtheorem3.p5.1.m1.1.1.1.1.1.1.3.3.cmml" type="integer" xref="S5.Thmtheorem3.p5.1.m1.1.1.1.1.1.1.3.3">1</cn></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmtheorem3.p5.1.m1.1c">\overline{ab}\in E(G^{1}_{k+1})</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem3.p5.1.m1.1d">over¯ start_ARG italic_a italic_b end_ARG ∈ italic_E ( italic_G start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_k + 1 end_POSTSUBSCRIPT )</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S5.Thmtheorem3.p5.9.2">, both endpoints are in </span><math alttext="V(G_{k+1})" class="ltx_Math" display="inline" id="S5.Thmtheorem3.p5.2.m2.1"><semantics id="S5.Thmtheorem3.p5.2.m2.1a"><mrow id="S5.Thmtheorem3.p5.2.m2.1.1" xref="S5.Thmtheorem3.p5.2.m2.1.1.cmml"><mi id="S5.Thmtheorem3.p5.2.m2.1.1.3" xref="S5.Thmtheorem3.p5.2.m2.1.1.3.cmml">V</mi><mo id="S5.Thmtheorem3.p5.2.m2.1.1.2" xref="S5.Thmtheorem3.p5.2.m2.1.1.2.cmml">⁢</mo><mrow id="S5.Thmtheorem3.p5.2.m2.1.1.1.1" xref="S5.Thmtheorem3.p5.2.m2.1.1.1.1.1.cmml"><mo id="S5.Thmtheorem3.p5.2.m2.1.1.1.1.2" stretchy="false" xref="S5.Thmtheorem3.p5.2.m2.1.1.1.1.1.cmml">(</mo><msub id="S5.Thmtheorem3.p5.2.m2.1.1.1.1.1" xref="S5.Thmtheorem3.p5.2.m2.1.1.1.1.1.cmml"><mi id="S5.Thmtheorem3.p5.2.m2.1.1.1.1.1.2" xref="S5.Thmtheorem3.p5.2.m2.1.1.1.1.1.2.cmml">G</mi><mrow id="S5.Thmtheorem3.p5.2.m2.1.1.1.1.1.3" xref="S5.Thmtheorem3.p5.2.m2.1.1.1.1.1.3.cmml"><mi id="S5.Thmtheorem3.p5.2.m2.1.1.1.1.1.3.2" xref="S5.Thmtheorem3.p5.2.m2.1.1.1.1.1.3.2.cmml">k</mi><mo id="S5.Thmtheorem3.p5.2.m2.1.1.1.1.1.3.1" xref="S5.Thmtheorem3.p5.2.m2.1.1.1.1.1.3.1.cmml">+</mo><mn id="S5.Thmtheorem3.p5.2.m2.1.1.1.1.1.3.3" xref="S5.Thmtheorem3.p5.2.m2.1.1.1.1.1.3.3.cmml">1</mn></mrow></msub><mo id="S5.Thmtheorem3.p5.2.m2.1.1.1.1.3" stretchy="false" xref="S5.Thmtheorem3.p5.2.m2.1.1.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.Thmtheorem3.p5.2.m2.1b"><apply id="S5.Thmtheorem3.p5.2.m2.1.1.cmml" xref="S5.Thmtheorem3.p5.2.m2.1.1"><times id="S5.Thmtheorem3.p5.2.m2.1.1.2.cmml" xref="S5.Thmtheorem3.p5.2.m2.1.1.2"></times><ci id="S5.Thmtheorem3.p5.2.m2.1.1.3.cmml" xref="S5.Thmtheorem3.p5.2.m2.1.1.3">𝑉</ci><apply id="S5.Thmtheorem3.p5.2.m2.1.1.1.1.1.cmml" xref="S5.Thmtheorem3.p5.2.m2.1.1.1.1"><csymbol cd="ambiguous" id="S5.Thmtheorem3.p5.2.m2.1.1.1.1.1.1.cmml" xref="S5.Thmtheorem3.p5.2.m2.1.1.1.1">subscript</csymbol><ci id="S5.Thmtheorem3.p5.2.m2.1.1.1.1.1.2.cmml" xref="S5.Thmtheorem3.p5.2.m2.1.1.1.1.1.2">𝐺</ci><apply id="S5.Thmtheorem3.p5.2.m2.1.1.1.1.1.3.cmml" xref="S5.Thmtheorem3.p5.2.m2.1.1.1.1.1.3"><plus id="S5.Thmtheorem3.p5.2.m2.1.1.1.1.1.3.1.cmml" xref="S5.Thmtheorem3.p5.2.m2.1.1.1.1.1.3.1"></plus><ci id="S5.Thmtheorem3.p5.2.m2.1.1.1.1.1.3.2.cmml" xref="S5.Thmtheorem3.p5.2.m2.1.1.1.1.1.3.2">𝑘</ci><cn id="S5.Thmtheorem3.p5.2.m2.1.1.1.1.1.3.3.cmml" type="integer" xref="S5.Thmtheorem3.p5.2.m2.1.1.1.1.1.3.3">1</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmtheorem3.p5.2.m2.1c">V(G_{k+1})</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem3.p5.2.m2.1d">italic_V ( italic_G start_POSTSUBSCRIPT italic_k + 1 end_POSTSUBSCRIPT )</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S5.Thmtheorem3.p5.9.3">. Because </span><math alttext="\overrightarrow{G}_{x}" class="ltx_Math" display="inline" id="S5.Thmtheorem3.p5.3.m3.1"><semantics id="S5.Thmtheorem3.p5.3.m3.1a"><msub id="S5.Thmtheorem3.p5.3.m3.1.1" xref="S5.Thmtheorem3.p5.3.m3.1.1.cmml"><mover accent="true" id="S5.Thmtheorem3.p5.3.m3.1.1.2" xref="S5.Thmtheorem3.p5.3.m3.1.1.2.cmml"><mi id="S5.Thmtheorem3.p5.3.m3.1.1.2.2" xref="S5.Thmtheorem3.p5.3.m3.1.1.2.2.cmml">G</mi><mo id="S5.Thmtheorem3.p5.3.m3.1.1.2.1" stretchy="false" xref="S5.Thmtheorem3.p5.3.m3.1.1.2.1.cmml">→</mo></mover><mi id="S5.Thmtheorem3.p5.3.m3.1.1.3" xref="S5.Thmtheorem3.p5.3.m3.1.1.3.cmml">x</mi></msub><annotation-xml encoding="MathML-Content" id="S5.Thmtheorem3.p5.3.m3.1b"><apply id="S5.Thmtheorem3.p5.3.m3.1.1.cmml" xref="S5.Thmtheorem3.p5.3.m3.1.1"><csymbol cd="ambiguous" id="S5.Thmtheorem3.p5.3.m3.1.1.1.cmml" xref="S5.Thmtheorem3.p5.3.m3.1.1">subscript</csymbol><apply id="S5.Thmtheorem3.p5.3.m3.1.1.2.cmml" xref="S5.Thmtheorem3.p5.3.m3.1.1.2"><ci id="S5.Thmtheorem3.p5.3.m3.1.1.2.1.cmml" xref="S5.Thmtheorem3.p5.3.m3.1.1.2.1">→</ci><ci id="S5.Thmtheorem3.p5.3.m3.1.1.2.2.cmml" xref="S5.Thmtheorem3.p5.3.m3.1.1.2.2">𝐺</ci></apply><ci id="S5.Thmtheorem3.p5.3.m3.1.1.3.cmml" xref="S5.Thmtheorem3.p5.3.m3.1.1.3">𝑥</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmtheorem3.p5.3.m3.1c">\overrightarrow{G}_{x}</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem3.p5.3.m3.1d">over→ start_ARG italic_G end_ARG start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S5.Thmtheorem3.p5.9.4"> is locally fair, this implies that </span><math alttext="\textsl{g}_{x}(a\!\to\!b)+\textsl{g}_{x}(b\!\to\!a)=\textsl{g}(\overline{ab})" class="ltx_Math" display="inline" id="S5.Thmtheorem3.p5.4.m4.3"><semantics id="S5.Thmtheorem3.p5.4.m4.3a"><mrow id="S5.Thmtheorem3.p5.4.m4.3.3" xref="S5.Thmtheorem3.p5.4.m4.3.3.cmml"><mrow id="S5.Thmtheorem3.p5.4.m4.3.3.2" xref="S5.Thmtheorem3.p5.4.m4.3.3.2.cmml"><mrow id="S5.Thmtheorem3.p5.4.m4.2.2.1.1" xref="S5.Thmtheorem3.p5.4.m4.2.2.1.1.cmml"><msub id="S5.Thmtheorem3.p5.4.m4.2.2.1.1.3" xref="S5.Thmtheorem3.p5.4.m4.2.2.1.1.3.cmml"><mtext class="ltx_mathvariant_italic" id="S5.Thmtheorem3.p5.4.m4.2.2.1.1.3.2" xref="S5.Thmtheorem3.p5.4.m4.2.2.1.1.3.2a.cmml">g</mtext><mi id="S5.Thmtheorem3.p5.4.m4.2.2.1.1.3.3" xref="S5.Thmtheorem3.p5.4.m4.2.2.1.1.3.3.cmml">x</mi></msub><mo id="S5.Thmtheorem3.p5.4.m4.2.2.1.1.2" xref="S5.Thmtheorem3.p5.4.m4.2.2.1.1.2.cmml">⁢</mo><mrow id="S5.Thmtheorem3.p5.4.m4.2.2.1.1.1.1" xref="S5.Thmtheorem3.p5.4.m4.2.2.1.1.1.1.1.cmml"><mo id="S5.Thmtheorem3.p5.4.m4.2.2.1.1.1.1.2" stretchy="false" xref="S5.Thmtheorem3.p5.4.m4.2.2.1.1.1.1.1.cmml">(</mo><mrow id="S5.Thmtheorem3.p5.4.m4.2.2.1.1.1.1.1" xref="S5.Thmtheorem3.p5.4.m4.2.2.1.1.1.1.1.cmml"><mi id="S5.Thmtheorem3.p5.4.m4.2.2.1.1.1.1.1.2" xref="S5.Thmtheorem3.p5.4.m4.2.2.1.1.1.1.1.2.cmml">a</mi><mo id="S5.Thmtheorem3.p5.4.m4.2.2.1.1.1.1.1.1" lspace="0.108em" rspace="0.108em" stretchy="false" xref="S5.Thmtheorem3.p5.4.m4.2.2.1.1.1.1.1.1.cmml">→</mo><mi id="S5.Thmtheorem3.p5.4.m4.2.2.1.1.1.1.1.3" xref="S5.Thmtheorem3.p5.4.m4.2.2.1.1.1.1.1.3.cmml">b</mi></mrow><mo id="S5.Thmtheorem3.p5.4.m4.2.2.1.1.1.1.3" stretchy="false" xref="S5.Thmtheorem3.p5.4.m4.2.2.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S5.Thmtheorem3.p5.4.m4.3.3.2.3" xref="S5.Thmtheorem3.p5.4.m4.3.3.2.3.cmml">+</mo><mrow id="S5.Thmtheorem3.p5.4.m4.3.3.2.2" xref="S5.Thmtheorem3.p5.4.m4.3.3.2.2.cmml"><msub id="S5.Thmtheorem3.p5.4.m4.3.3.2.2.3" xref="S5.Thmtheorem3.p5.4.m4.3.3.2.2.3.cmml"><mtext class="ltx_mathvariant_italic" id="S5.Thmtheorem3.p5.4.m4.3.3.2.2.3.2" xref="S5.Thmtheorem3.p5.4.m4.3.3.2.2.3.2a.cmml">g</mtext><mi id="S5.Thmtheorem3.p5.4.m4.3.3.2.2.3.3" xref="S5.Thmtheorem3.p5.4.m4.3.3.2.2.3.3.cmml">x</mi></msub><mo id="S5.Thmtheorem3.p5.4.m4.3.3.2.2.2" xref="S5.Thmtheorem3.p5.4.m4.3.3.2.2.2.cmml">⁢</mo><mrow id="S5.Thmtheorem3.p5.4.m4.3.3.2.2.1.1" xref="S5.Thmtheorem3.p5.4.m4.3.3.2.2.1.1.1.cmml"><mo id="S5.Thmtheorem3.p5.4.m4.3.3.2.2.1.1.2" stretchy="false" xref="S5.Thmtheorem3.p5.4.m4.3.3.2.2.1.1.1.cmml">(</mo><mrow id="S5.Thmtheorem3.p5.4.m4.3.3.2.2.1.1.1" xref="S5.Thmtheorem3.p5.4.m4.3.3.2.2.1.1.1.cmml"><mi id="S5.Thmtheorem3.p5.4.m4.3.3.2.2.1.1.1.2" xref="S5.Thmtheorem3.p5.4.m4.3.3.2.2.1.1.1.2.cmml">b</mi><mo id="S5.Thmtheorem3.p5.4.m4.3.3.2.2.1.1.1.1" lspace="0.108em" rspace="0.108em" stretchy="false" xref="S5.Thmtheorem3.p5.4.m4.3.3.2.2.1.1.1.1.cmml">→</mo><mi id="S5.Thmtheorem3.p5.4.m4.3.3.2.2.1.1.1.3" xref="S5.Thmtheorem3.p5.4.m4.3.3.2.2.1.1.1.3.cmml">a</mi></mrow><mo id="S5.Thmtheorem3.p5.4.m4.3.3.2.2.1.1.3" stretchy="false" xref="S5.Thmtheorem3.p5.4.m4.3.3.2.2.1.1.1.cmml">)</mo></mrow></mrow></mrow><mo id="S5.Thmtheorem3.p5.4.m4.3.3.3" xref="S5.Thmtheorem3.p5.4.m4.3.3.3.cmml">=</mo><mrow id="S5.Thmtheorem3.p5.4.m4.3.3.4" xref="S5.Thmtheorem3.p5.4.m4.3.3.4.cmml"><mtext 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xref="S5.Thmtheorem3.p5.4.m4.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.Thmtheorem3.p5.4.m4.3b"><apply id="S5.Thmtheorem3.p5.4.m4.3.3.cmml" xref="S5.Thmtheorem3.p5.4.m4.3.3"><eq id="S5.Thmtheorem3.p5.4.m4.3.3.3.cmml" xref="S5.Thmtheorem3.p5.4.m4.3.3.3"></eq><apply id="S5.Thmtheorem3.p5.4.m4.3.3.2.cmml" xref="S5.Thmtheorem3.p5.4.m4.3.3.2"><plus id="S5.Thmtheorem3.p5.4.m4.3.3.2.3.cmml" xref="S5.Thmtheorem3.p5.4.m4.3.3.2.3"></plus><apply id="S5.Thmtheorem3.p5.4.m4.2.2.1.1.cmml" xref="S5.Thmtheorem3.p5.4.m4.2.2.1.1"><times id="S5.Thmtheorem3.p5.4.m4.2.2.1.1.2.cmml" xref="S5.Thmtheorem3.p5.4.m4.2.2.1.1.2"></times><apply id="S5.Thmtheorem3.p5.4.m4.2.2.1.1.3.cmml" xref="S5.Thmtheorem3.p5.4.m4.2.2.1.1.3"><csymbol cd="ambiguous" id="S5.Thmtheorem3.p5.4.m4.2.2.1.1.3.1.cmml" xref="S5.Thmtheorem3.p5.4.m4.2.2.1.1.3">subscript</csymbol><ci id="S5.Thmtheorem3.p5.4.m4.2.2.1.1.3.2a.cmml" xref="S5.Thmtheorem3.p5.4.m4.2.2.1.1.3.2"><mtext 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id="S5.Thmtheorem3.p5.4.m4.3.3.2.2.3.1.cmml" xref="S5.Thmtheorem3.p5.4.m4.3.3.2.2.3">subscript</csymbol><ci id="S5.Thmtheorem3.p5.4.m4.3.3.2.2.3.2a.cmml" xref="S5.Thmtheorem3.p5.4.m4.3.3.2.2.3.2"><mtext class="ltx_mathvariant_italic" id="S5.Thmtheorem3.p5.4.m4.3.3.2.2.3.2.cmml" xref="S5.Thmtheorem3.p5.4.m4.3.3.2.2.3.2">g</mtext></ci><ci id="S5.Thmtheorem3.p5.4.m4.3.3.2.2.3.3.cmml" xref="S5.Thmtheorem3.p5.4.m4.3.3.2.2.3.3">𝑥</ci></apply><apply id="S5.Thmtheorem3.p5.4.m4.3.3.2.2.1.1.1.cmml" xref="S5.Thmtheorem3.p5.4.m4.3.3.2.2.1.1"><ci id="S5.Thmtheorem3.p5.4.m4.3.3.2.2.1.1.1.1.cmml" xref="S5.Thmtheorem3.p5.4.m4.3.3.2.2.1.1.1.1">→</ci><ci id="S5.Thmtheorem3.p5.4.m4.3.3.2.2.1.1.1.2.cmml" xref="S5.Thmtheorem3.p5.4.m4.3.3.2.2.1.1.1.2">𝑏</ci><ci id="S5.Thmtheorem3.p5.4.m4.3.3.2.2.1.1.1.3.cmml" xref="S5.Thmtheorem3.p5.4.m4.3.3.2.2.1.1.1.3">𝑎</ci></apply></apply></apply><apply id="S5.Thmtheorem3.p5.4.m4.3.3.4.cmml" xref="S5.Thmtheorem3.p5.4.m4.3.3.4"><times id="S5.Thmtheorem3.p5.4.m4.3.3.4.1.cmml" xref="S5.Thmtheorem3.p5.4.m4.3.3.4.1"></times><ci id="S5.Thmtheorem3.p5.4.m4.3.3.4.2a.cmml" xref="S5.Thmtheorem3.p5.4.m4.3.3.4.2"><mtext class="ltx_mathvariant_italic" id="S5.Thmtheorem3.p5.4.m4.3.3.4.2.cmml" xref="S5.Thmtheorem3.p5.4.m4.3.3.4.2">g</mtext></ci><apply id="S5.Thmtheorem3.p5.4.m4.1.1.cmml" xref="S5.Thmtheorem3.p5.4.m4.3.3.4.3.2"><ci id="S5.Thmtheorem3.p5.4.m4.1.1.1.cmml" xref="S5.Thmtheorem3.p5.4.m4.1.1.1">¯</ci><apply id="S5.Thmtheorem3.p5.4.m4.1.1.2.cmml" xref="S5.Thmtheorem3.p5.4.m4.1.1.2"><times id="S5.Thmtheorem3.p5.4.m4.1.1.2.1.cmml" xref="S5.Thmtheorem3.p5.4.m4.1.1.2.1"></times><ci id="S5.Thmtheorem3.p5.4.m4.1.1.2.2.cmml" xref="S5.Thmtheorem3.p5.4.m4.1.1.2.2">𝑎</ci><ci id="S5.Thmtheorem3.p5.4.m4.1.1.2.3.cmml" xref="S5.Thmtheorem3.p5.4.m4.1.1.2.3">𝑏</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmtheorem3.p5.4.m4.3c">\textsl{g}_{x}(a\!\to\!b)+\textsl{g}_{x}(b\!\to\!a)=\textsl{g}(\overline{ab})</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem3.p5.4.m4.3d">g start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT ( italic_a → italic_b ) + g start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT ( italic_b → italic_a ) = g ( over¯ start_ARG italic_a italic_b end_ARG )</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S5.Thmtheorem3.p5.9.5">. For all </span><math alttext="\overline{ab}\in E_{\uparrow}" class="ltx_Math" display="inline" id="S5.Thmtheorem3.p5.5.m5.1"><semantics id="S5.Thmtheorem3.p5.5.m5.1a"><mrow id="S5.Thmtheorem3.p5.5.m5.1.1" xref="S5.Thmtheorem3.p5.5.m5.1.1.cmml"><mover accent="true" id="S5.Thmtheorem3.p5.5.m5.1.1.2" xref="S5.Thmtheorem3.p5.5.m5.1.1.2.cmml"><mrow id="S5.Thmtheorem3.p5.5.m5.1.1.2.2" xref="S5.Thmtheorem3.p5.5.m5.1.1.2.2.cmml"><mi id="S5.Thmtheorem3.p5.5.m5.1.1.2.2.2" xref="S5.Thmtheorem3.p5.5.m5.1.1.2.2.2.cmml">a</mi><mo id="S5.Thmtheorem3.p5.5.m5.1.1.2.2.1" xref="S5.Thmtheorem3.p5.5.m5.1.1.2.2.1.cmml">⁢</mo><mi id="S5.Thmtheorem3.p5.5.m5.1.1.2.2.3" xref="S5.Thmtheorem3.p5.5.m5.1.1.2.2.3.cmml">b</mi></mrow><mo id="S5.Thmtheorem3.p5.5.m5.1.1.2.1" xref="S5.Thmtheorem3.p5.5.m5.1.1.2.1.cmml">¯</mo></mover><mo id="S5.Thmtheorem3.p5.5.m5.1.1.1" xref="S5.Thmtheorem3.p5.5.m5.1.1.1.cmml">∈</mo><msub id="S5.Thmtheorem3.p5.5.m5.1.1.3" xref="S5.Thmtheorem3.p5.5.m5.1.1.3.cmml"><mi id="S5.Thmtheorem3.p5.5.m5.1.1.3.2" xref="S5.Thmtheorem3.p5.5.m5.1.1.3.2.cmml">E</mi><mo id="S5.Thmtheorem3.p5.5.m5.1.1.3.3" stretchy="false" xref="S5.Thmtheorem3.p5.5.m5.1.1.3.3.cmml">↑</mo></msub></mrow><annotation-xml encoding="MathML-Content" id="S5.Thmtheorem3.p5.5.m5.1b"><apply id="S5.Thmtheorem3.p5.5.m5.1.1.cmml" xref="S5.Thmtheorem3.p5.5.m5.1.1"><in id="S5.Thmtheorem3.p5.5.m5.1.1.1.cmml" xref="S5.Thmtheorem3.p5.5.m5.1.1.1"></in><apply id="S5.Thmtheorem3.p5.5.m5.1.1.2.cmml" xref="S5.Thmtheorem3.p5.5.m5.1.1.2"><ci id="S5.Thmtheorem3.p5.5.m5.1.1.2.1.cmml" xref="S5.Thmtheorem3.p5.5.m5.1.1.2.1">¯</ci><apply id="S5.Thmtheorem3.p5.5.m5.1.1.2.2.cmml" xref="S5.Thmtheorem3.p5.5.m5.1.1.2.2"><times id="S5.Thmtheorem3.p5.5.m5.1.1.2.2.1.cmml" xref="S5.Thmtheorem3.p5.5.m5.1.1.2.2.1"></times><ci id="S5.Thmtheorem3.p5.5.m5.1.1.2.2.2.cmml" xref="S5.Thmtheorem3.p5.5.m5.1.1.2.2.2">𝑎</ci><ci id="S5.Thmtheorem3.p5.5.m5.1.1.2.2.3.cmml" xref="S5.Thmtheorem3.p5.5.m5.1.1.2.2.3">𝑏</ci></apply></apply><apply id="S5.Thmtheorem3.p5.5.m5.1.1.3.cmml" xref="S5.Thmtheorem3.p5.5.m5.1.1.3"><csymbol cd="ambiguous" id="S5.Thmtheorem3.p5.5.m5.1.1.3.1.cmml" xref="S5.Thmtheorem3.p5.5.m5.1.1.3">subscript</csymbol><ci id="S5.Thmtheorem3.p5.5.m5.1.1.3.2.cmml" xref="S5.Thmtheorem3.p5.5.m5.1.1.3.2">𝐸</ci><ci id="S5.Thmtheorem3.p5.5.m5.1.1.3.3.cmml" xref="S5.Thmtheorem3.p5.5.m5.1.1.3.3">↑</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmtheorem3.p5.5.m5.1c">\overline{ab}\in E_{\uparrow}</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem3.p5.5.m5.1d">over¯ start_ARG italic_a italic_b end_ARG ∈ italic_E start_POSTSUBSCRIPT ↑ end_POSTSUBSCRIPT</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S5.Thmtheorem3.p5.9.6">, </span><math alttext="\textsl{g}_{x}(b)>\gamma\geq\textsl{g}_{x}(a)" class="ltx_Math" display="inline" id="S5.Thmtheorem3.p5.6.m6.2"><semantics id="S5.Thmtheorem3.p5.6.m6.2a"><mrow id="S5.Thmtheorem3.p5.6.m6.2.3" xref="S5.Thmtheorem3.p5.6.m6.2.3.cmml"><mrow id="S5.Thmtheorem3.p5.6.m6.2.3.2" xref="S5.Thmtheorem3.p5.6.m6.2.3.2.cmml"><msub id="S5.Thmtheorem3.p5.6.m6.2.3.2.2" xref="S5.Thmtheorem3.p5.6.m6.2.3.2.2.cmml"><mtext class="ltx_mathvariant_italic" id="S5.Thmtheorem3.p5.6.m6.2.3.2.2.2" xref="S5.Thmtheorem3.p5.6.m6.2.3.2.2.2a.cmml">g</mtext><mi id="S5.Thmtheorem3.p5.6.m6.2.3.2.2.3" xref="S5.Thmtheorem3.p5.6.m6.2.3.2.2.3.cmml">x</mi></msub><mo id="S5.Thmtheorem3.p5.6.m6.2.3.2.1" xref="S5.Thmtheorem3.p5.6.m6.2.3.2.1.cmml">⁢</mo><mrow id="S5.Thmtheorem3.p5.6.m6.2.3.2.3.2" xref="S5.Thmtheorem3.p5.6.m6.2.3.2.cmml"><mo id="S5.Thmtheorem3.p5.6.m6.2.3.2.3.2.1" stretchy="false" xref="S5.Thmtheorem3.p5.6.m6.2.3.2.cmml">(</mo><mi id="S5.Thmtheorem3.p5.6.m6.1.1" xref="S5.Thmtheorem3.p5.6.m6.1.1.cmml">b</mi><mo id="S5.Thmtheorem3.p5.6.m6.2.3.2.3.2.2" stretchy="false" xref="S5.Thmtheorem3.p5.6.m6.2.3.2.cmml">)</mo></mrow></mrow><mo id="S5.Thmtheorem3.p5.6.m6.2.3.3" xref="S5.Thmtheorem3.p5.6.m6.2.3.3.cmml">></mo><mi id="S5.Thmtheorem3.p5.6.m6.2.3.4" xref="S5.Thmtheorem3.p5.6.m6.2.3.4.cmml">γ</mi><mo id="S5.Thmtheorem3.p5.6.m6.2.3.5" xref="S5.Thmtheorem3.p5.6.m6.2.3.5.cmml">≥</mo><mrow id="S5.Thmtheorem3.p5.6.m6.2.3.6" xref="S5.Thmtheorem3.p5.6.m6.2.3.6.cmml"><msub id="S5.Thmtheorem3.p5.6.m6.2.3.6.2" xref="S5.Thmtheorem3.p5.6.m6.2.3.6.2.cmml"><mtext class="ltx_mathvariant_italic" id="S5.Thmtheorem3.p5.6.m6.2.3.6.2.2" xref="S5.Thmtheorem3.p5.6.m6.2.3.6.2.2a.cmml">g</mtext><mi id="S5.Thmtheorem3.p5.6.m6.2.3.6.2.3" xref="S5.Thmtheorem3.p5.6.m6.2.3.6.2.3.cmml">x</mi></msub><mo id="S5.Thmtheorem3.p5.6.m6.2.3.6.1" xref="S5.Thmtheorem3.p5.6.m6.2.3.6.1.cmml">⁢</mo><mrow id="S5.Thmtheorem3.p5.6.m6.2.3.6.3.2" xref="S5.Thmtheorem3.p5.6.m6.2.3.6.cmml"><mo id="S5.Thmtheorem3.p5.6.m6.2.3.6.3.2.1" stretchy="false" xref="S5.Thmtheorem3.p5.6.m6.2.3.6.cmml">(</mo><mi id="S5.Thmtheorem3.p5.6.m6.2.2" xref="S5.Thmtheorem3.p5.6.m6.2.2.cmml">a</mi><mo id="S5.Thmtheorem3.p5.6.m6.2.3.6.3.2.2" stretchy="false" xref="S5.Thmtheorem3.p5.6.m6.2.3.6.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.Thmtheorem3.p5.6.m6.2b"><apply id="S5.Thmtheorem3.p5.6.m6.2.3.cmml" xref="S5.Thmtheorem3.p5.6.m6.2.3"><and id="S5.Thmtheorem3.p5.6.m6.2.3a.cmml" xref="S5.Thmtheorem3.p5.6.m6.2.3"></and><apply id="S5.Thmtheorem3.p5.6.m6.2.3b.cmml" xref="S5.Thmtheorem3.p5.6.m6.2.3"><gt id="S5.Thmtheorem3.p5.6.m6.2.3.3.cmml" xref="S5.Thmtheorem3.p5.6.m6.2.3.3"></gt><apply id="S5.Thmtheorem3.p5.6.m6.2.3.2.cmml" xref="S5.Thmtheorem3.p5.6.m6.2.3.2"><times id="S5.Thmtheorem3.p5.6.m6.2.3.2.1.cmml" xref="S5.Thmtheorem3.p5.6.m6.2.3.2.1"></times><apply id="S5.Thmtheorem3.p5.6.m6.2.3.2.2.cmml" xref="S5.Thmtheorem3.p5.6.m6.2.3.2.2"><csymbol cd="ambiguous" id="S5.Thmtheorem3.p5.6.m6.2.3.2.2.1.cmml" xref="S5.Thmtheorem3.p5.6.m6.2.3.2.2">subscript</csymbol><ci id="S5.Thmtheorem3.p5.6.m6.2.3.2.2.2a.cmml" xref="S5.Thmtheorem3.p5.6.m6.2.3.2.2.2"><mtext class="ltx_mathvariant_italic" id="S5.Thmtheorem3.p5.6.m6.2.3.2.2.2.cmml" xref="S5.Thmtheorem3.p5.6.m6.2.3.2.2.2">g</mtext></ci><ci id="S5.Thmtheorem3.p5.6.m6.2.3.2.2.3.cmml" xref="S5.Thmtheorem3.p5.6.m6.2.3.2.2.3">𝑥</ci></apply><ci id="S5.Thmtheorem3.p5.6.m6.1.1.cmml" xref="S5.Thmtheorem3.p5.6.m6.1.1">𝑏</ci></apply><ci id="S5.Thmtheorem3.p5.6.m6.2.3.4.cmml" xref="S5.Thmtheorem3.p5.6.m6.2.3.4">𝛾</ci></apply><apply id="S5.Thmtheorem3.p5.6.m6.2.3c.cmml" xref="S5.Thmtheorem3.p5.6.m6.2.3"><geq id="S5.Thmtheorem3.p5.6.m6.2.3.5.cmml" xref="S5.Thmtheorem3.p5.6.m6.2.3.5"></geq><share href="https://arxiv.org/html/2411.12694v2#S5.Thmtheorem3.p5.6.m6.2.3.4.cmml" id="S5.Thmtheorem3.p5.6.m6.2.3d.cmml" xref="S5.Thmtheorem3.p5.6.m6.2.3"></share><apply id="S5.Thmtheorem3.p5.6.m6.2.3.6.cmml" xref="S5.Thmtheorem3.p5.6.m6.2.3.6"><times id="S5.Thmtheorem3.p5.6.m6.2.3.6.1.cmml" xref="S5.Thmtheorem3.p5.6.m6.2.3.6.1"></times><apply id="S5.Thmtheorem3.p5.6.m6.2.3.6.2.cmml" xref="S5.Thmtheorem3.p5.6.m6.2.3.6.2"><csymbol cd="ambiguous" id="S5.Thmtheorem3.p5.6.m6.2.3.6.2.1.cmml" xref="S5.Thmtheorem3.p5.6.m6.2.3.6.2">subscript</csymbol><ci id="S5.Thmtheorem3.p5.6.m6.2.3.6.2.2a.cmml" xref="S5.Thmtheorem3.p5.6.m6.2.3.6.2.2"><mtext class="ltx_mathvariant_italic" id="S5.Thmtheorem3.p5.6.m6.2.3.6.2.2.cmml" xref="S5.Thmtheorem3.p5.6.m6.2.3.6.2.2">g</mtext></ci><ci id="S5.Thmtheorem3.p5.6.m6.2.3.6.2.3.cmml" xref="S5.Thmtheorem3.p5.6.m6.2.3.6.2.3">𝑥</ci></apply><ci id="S5.Thmtheorem3.p5.6.m6.2.2.cmml" xref="S5.Thmtheorem3.p5.6.m6.2.2">𝑎</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmtheorem3.p5.6.m6.2c">\textsl{g}_{x}(b)>\gamma\geq\textsl{g}_{x}(a)</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem3.p5.6.m6.2d">g start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT ( italic_b ) > italic_γ ≥ g start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT ( italic_a )</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S5.Thmtheorem3.p5.9.7"> per definition of </span><math alttext="G^{1}" class="ltx_Math" display="inline" id="S5.Thmtheorem3.p5.7.m7.1"><semantics id="S5.Thmtheorem3.p5.7.m7.1a"><msup id="S5.Thmtheorem3.p5.7.m7.1.1" xref="S5.Thmtheorem3.p5.7.m7.1.1.cmml"><mi id="S5.Thmtheorem3.p5.7.m7.1.1.2" xref="S5.Thmtheorem3.p5.7.m7.1.1.2.cmml">G</mi><mn id="S5.Thmtheorem3.p5.7.m7.1.1.3" xref="S5.Thmtheorem3.p5.7.m7.1.1.3.cmml">1</mn></msup><annotation-xml encoding="MathML-Content" id="S5.Thmtheorem3.p5.7.m7.1b"><apply id="S5.Thmtheorem3.p5.7.m7.1.1.cmml" xref="S5.Thmtheorem3.p5.7.m7.1.1"><csymbol cd="ambiguous" id="S5.Thmtheorem3.p5.7.m7.1.1.1.cmml" xref="S5.Thmtheorem3.p5.7.m7.1.1">superscript</csymbol><ci id="S5.Thmtheorem3.p5.7.m7.1.1.2.cmml" xref="S5.Thmtheorem3.p5.7.m7.1.1.2">𝐺</ci><cn id="S5.Thmtheorem3.p5.7.m7.1.1.3.cmml" type="integer" xref="S5.Thmtheorem3.p5.7.m7.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmtheorem3.p5.7.m7.1c">G^{1}</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem3.p5.7.m7.1d">italic_G start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S5.Thmtheorem3.p5.9.8"> and </span><math alttext="G^{2}" class="ltx_Math" display="inline" id="S5.Thmtheorem3.p5.8.m8.1"><semantics id="S5.Thmtheorem3.p5.8.m8.1a"><msup id="S5.Thmtheorem3.p5.8.m8.1.1" xref="S5.Thmtheorem3.p5.8.m8.1.1.cmml"><mi id="S5.Thmtheorem3.p5.8.m8.1.1.2" xref="S5.Thmtheorem3.p5.8.m8.1.1.2.cmml">G</mi><mn id="S5.Thmtheorem3.p5.8.m8.1.1.3" xref="S5.Thmtheorem3.p5.8.m8.1.1.3.cmml">2</mn></msup><annotation-xml encoding="MathML-Content" id="S5.Thmtheorem3.p5.8.m8.1b"><apply id="S5.Thmtheorem3.p5.8.m8.1.1.cmml" xref="S5.Thmtheorem3.p5.8.m8.1.1"><csymbol cd="ambiguous" id="S5.Thmtheorem3.p5.8.m8.1.1.1.cmml" xref="S5.Thmtheorem3.p5.8.m8.1.1">superscript</csymbol><ci id="S5.Thmtheorem3.p5.8.m8.1.1.2.cmml" xref="S5.Thmtheorem3.p5.8.m8.1.1.2">𝐺</ci><cn id="S5.Thmtheorem3.p5.8.m8.1.1.3.cmml" type="integer" xref="S5.Thmtheorem3.p5.8.m8.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmtheorem3.p5.8.m8.1c">G^{2}</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem3.p5.8.m8.1d">italic_G start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S5.Thmtheorem3.p5.9.9">. By local fairness, </span><math alttext="\textsl{g}_{x}(a\!\to\!b)=\textsl{g}(\overline{ab})" class="ltx_Math" display="inline" id="S5.Thmtheorem3.p5.9.m9.2"><semantics id="S5.Thmtheorem3.p5.9.m9.2a"><mrow id="S5.Thmtheorem3.p5.9.m9.2.2" xref="S5.Thmtheorem3.p5.9.m9.2.2.cmml"><mrow id="S5.Thmtheorem3.p5.9.m9.2.2.1" xref="S5.Thmtheorem3.p5.9.m9.2.2.1.cmml"><msub id="S5.Thmtheorem3.p5.9.m9.2.2.1.3" xref="S5.Thmtheorem3.p5.9.m9.2.2.1.3.cmml"><mtext class="ltx_mathvariant_italic" id="S5.Thmtheorem3.p5.9.m9.2.2.1.3.2" xref="S5.Thmtheorem3.p5.9.m9.2.2.1.3.2a.cmml">g</mtext><mi id="S5.Thmtheorem3.p5.9.m9.2.2.1.3.3" xref="S5.Thmtheorem3.p5.9.m9.2.2.1.3.3.cmml">x</mi></msub><mo id="S5.Thmtheorem3.p5.9.m9.2.2.1.2" xref="S5.Thmtheorem3.p5.9.m9.2.2.1.2.cmml">⁢</mo><mrow id="S5.Thmtheorem3.p5.9.m9.2.2.1.1.1" xref="S5.Thmtheorem3.p5.9.m9.2.2.1.1.1.1.cmml"><mo id="S5.Thmtheorem3.p5.9.m9.2.2.1.1.1.2" stretchy="false" xref="S5.Thmtheorem3.p5.9.m9.2.2.1.1.1.1.cmml">(</mo><mrow id="S5.Thmtheorem3.p5.9.m9.2.2.1.1.1.1" xref="S5.Thmtheorem3.p5.9.m9.2.2.1.1.1.1.cmml"><mi id="S5.Thmtheorem3.p5.9.m9.2.2.1.1.1.1.2" xref="S5.Thmtheorem3.p5.9.m9.2.2.1.1.1.1.2.cmml">a</mi><mo id="S5.Thmtheorem3.p5.9.m9.2.2.1.1.1.1.1" lspace="0.108em" rspace="0.108em" stretchy="false" xref="S5.Thmtheorem3.p5.9.m9.2.2.1.1.1.1.1.cmml">→</mo><mi id="S5.Thmtheorem3.p5.9.m9.2.2.1.1.1.1.3" xref="S5.Thmtheorem3.p5.9.m9.2.2.1.1.1.1.3.cmml">b</mi></mrow><mo id="S5.Thmtheorem3.p5.9.m9.2.2.1.1.1.3" stretchy="false" xref="S5.Thmtheorem3.p5.9.m9.2.2.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S5.Thmtheorem3.p5.9.m9.2.2.2" xref="S5.Thmtheorem3.p5.9.m9.2.2.2.cmml">=</mo><mrow id="S5.Thmtheorem3.p5.9.m9.2.2.3" xref="S5.Thmtheorem3.p5.9.m9.2.2.3.cmml"><mtext class="ltx_mathvariant_italic" id="S5.Thmtheorem3.p5.9.m9.2.2.3.2" xref="S5.Thmtheorem3.p5.9.m9.2.2.3.2a.cmml">g</mtext><mo id="S5.Thmtheorem3.p5.9.m9.2.2.3.1" xref="S5.Thmtheorem3.p5.9.m9.2.2.3.1.cmml">⁢</mo><mrow id="S5.Thmtheorem3.p5.9.m9.2.2.3.3.2" xref="S5.Thmtheorem3.p5.9.m9.1.1.cmml"><mo id="S5.Thmtheorem3.p5.9.m9.2.2.3.3.2.1" stretchy="false" xref="S5.Thmtheorem3.p5.9.m9.1.1.cmml">(</mo><mover accent="true" id="S5.Thmtheorem3.p5.9.m9.1.1" xref="S5.Thmtheorem3.p5.9.m9.1.1.cmml"><mrow id="S5.Thmtheorem3.p5.9.m9.1.1.2" xref="S5.Thmtheorem3.p5.9.m9.1.1.2.cmml"><mi id="S5.Thmtheorem3.p5.9.m9.1.1.2.2" xref="S5.Thmtheorem3.p5.9.m9.1.1.2.2.cmml">a</mi><mo id="S5.Thmtheorem3.p5.9.m9.1.1.2.1" xref="S5.Thmtheorem3.p5.9.m9.1.1.2.1.cmml">⁢</mo><mi id="S5.Thmtheorem3.p5.9.m9.1.1.2.3" xref="S5.Thmtheorem3.p5.9.m9.1.1.2.3.cmml">b</mi></mrow><mo id="S5.Thmtheorem3.p5.9.m9.1.1.1" xref="S5.Thmtheorem3.p5.9.m9.1.1.1.cmml">¯</mo></mover><mo id="S5.Thmtheorem3.p5.9.m9.2.2.3.3.2.2" stretchy="false" xref="S5.Thmtheorem3.p5.9.m9.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.Thmtheorem3.p5.9.m9.2b"><apply id="S5.Thmtheorem3.p5.9.m9.2.2.cmml" xref="S5.Thmtheorem3.p5.9.m9.2.2"><eq id="S5.Thmtheorem3.p5.9.m9.2.2.2.cmml" xref="S5.Thmtheorem3.p5.9.m9.2.2.2"></eq><apply id="S5.Thmtheorem3.p5.9.m9.2.2.1.cmml" xref="S5.Thmtheorem3.p5.9.m9.2.2.1"><times id="S5.Thmtheorem3.p5.9.m9.2.2.1.2.cmml" xref="S5.Thmtheorem3.p5.9.m9.2.2.1.2"></times><apply id="S5.Thmtheorem3.p5.9.m9.2.2.1.3.cmml" xref="S5.Thmtheorem3.p5.9.m9.2.2.1.3"><csymbol cd="ambiguous" id="S5.Thmtheorem3.p5.9.m9.2.2.1.3.1.cmml" xref="S5.Thmtheorem3.p5.9.m9.2.2.1.3">subscript</csymbol><ci id="S5.Thmtheorem3.p5.9.m9.2.2.1.3.2a.cmml" xref="S5.Thmtheorem3.p5.9.m9.2.2.1.3.2"><mtext class="ltx_mathvariant_italic" id="S5.Thmtheorem3.p5.9.m9.2.2.1.3.2.cmml" xref="S5.Thmtheorem3.p5.9.m9.2.2.1.3.2">g</mtext></ci><ci id="S5.Thmtheorem3.p5.9.m9.2.2.1.3.3.cmml" xref="S5.Thmtheorem3.p5.9.m9.2.2.1.3.3">𝑥</ci></apply><apply id="S5.Thmtheorem3.p5.9.m9.2.2.1.1.1.1.cmml" xref="S5.Thmtheorem3.p5.9.m9.2.2.1.1.1"><ci id="S5.Thmtheorem3.p5.9.m9.2.2.1.1.1.1.1.cmml" xref="S5.Thmtheorem3.p5.9.m9.2.2.1.1.1.1.1">→</ci><ci id="S5.Thmtheorem3.p5.9.m9.2.2.1.1.1.1.2.cmml" xref="S5.Thmtheorem3.p5.9.m9.2.2.1.1.1.1.2">𝑎</ci><ci id="S5.Thmtheorem3.p5.9.m9.2.2.1.1.1.1.3.cmml" xref="S5.Thmtheorem3.p5.9.m9.2.2.1.1.1.1.3">𝑏</ci></apply></apply><apply id="S5.Thmtheorem3.p5.9.m9.2.2.3.cmml" xref="S5.Thmtheorem3.p5.9.m9.2.2.3"><times id="S5.Thmtheorem3.p5.9.m9.2.2.3.1.cmml" xref="S5.Thmtheorem3.p5.9.m9.2.2.3.1"></times><ci id="S5.Thmtheorem3.p5.9.m9.2.2.3.2a.cmml" xref="S5.Thmtheorem3.p5.9.m9.2.2.3.2"><mtext class="ltx_mathvariant_italic" id="S5.Thmtheorem3.p5.9.m9.2.2.3.2.cmml" xref="S5.Thmtheorem3.p5.9.m9.2.2.3.2">g</mtext></ci><apply id="S5.Thmtheorem3.p5.9.m9.1.1.cmml" xref="S5.Thmtheorem3.p5.9.m9.2.2.3.3.2"><ci id="S5.Thmtheorem3.p5.9.m9.1.1.1.cmml" xref="S5.Thmtheorem3.p5.9.m9.1.1.1">¯</ci><apply id="S5.Thmtheorem3.p5.9.m9.1.1.2.cmml" xref="S5.Thmtheorem3.p5.9.m9.1.1.2"><times id="S5.Thmtheorem3.p5.9.m9.1.1.2.1.cmml" xref="S5.Thmtheorem3.p5.9.m9.1.1.2.1"></times><ci id="S5.Thmtheorem3.p5.9.m9.1.1.2.2.cmml" xref="S5.Thmtheorem3.p5.9.m9.1.1.2.2">𝑎</ci><ci id="S5.Thmtheorem3.p5.9.m9.1.1.2.3.cmml" xref="S5.Thmtheorem3.p5.9.m9.1.1.2.3">𝑏</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmtheorem3.p5.9.m9.2c">\textsl{g}_{x}(a\!\to\!b)=\textsl{g}(\overline{ab})</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem3.p5.9.m9.2d">g start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT ( italic_a → italic_b ) = g ( over¯ start_ARG italic_a italic_b end_ARG )</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S5.Thmtheorem3.p5.9.10"> and the inequality follows.</span></p> </div> <div class="ltx_para ltx_noindent" id="S5.Thmtheorem3.p6"> <p class="ltx_p" id="S5.Thmtheorem3.p6.1"><em class="ltx_emph" id="S5.Thmtheorem3.p6.1.1">The second claim.</em><span class="ltx_text ltx_font_italic" id="S5.Thmtheorem3.p6.1.2"> We secondly claim that:</span></p> <table class="ltx_equation ltx_eqn_table" id="S5.E4"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_left_padleft"></td> <td class="ltx_eqn_cell ltx_align_left"><math alttext="\sum_{v\in V(G^{1}_{k})}\textsl{g}(v)>\sum_{v\in V(G_{k+1}^{1})}\textsl{g}_{x}% (v)." class="ltx_Math" display="block" id="S5.E4.m1.5"><semantics id="S5.E4.m1.5a"><mrow id="S5.E4.m1.5.5.1" xref="S5.E4.m1.5.5.1.1.cmml"><mrow id="S5.E4.m1.5.5.1.1" xref="S5.E4.m1.5.5.1.1.cmml"><mrow id="S5.E4.m1.5.5.1.1.2" xref="S5.E4.m1.5.5.1.1.2.cmml"><munder id="S5.E4.m1.5.5.1.1.2.1" xref="S5.E4.m1.5.5.1.1.2.1.cmml"><mo id="S5.E4.m1.5.5.1.1.2.1.2" movablelimits="false" xref="S5.E4.m1.5.5.1.1.2.1.2.cmml">∑</mo><mrow id="S5.E4.m1.1.1.1" xref="S5.E4.m1.1.1.1.cmml"><mi id="S5.E4.m1.1.1.1.3" xref="S5.E4.m1.1.1.1.3.cmml">v</mi><mo id="S5.E4.m1.1.1.1.2" xref="S5.E4.m1.1.1.1.2.cmml">∈</mo><mrow id="S5.E4.m1.1.1.1.1" xref="S5.E4.m1.1.1.1.1.cmml"><mi id="S5.E4.m1.1.1.1.1.3" xref="S5.E4.m1.1.1.1.1.3.cmml">V</mi><mo id="S5.E4.m1.1.1.1.1.2" xref="S5.E4.m1.1.1.1.1.2.cmml">⁢</mo><mrow id="S5.E4.m1.1.1.1.1.1.1" xref="S5.E4.m1.1.1.1.1.1.1.1.cmml"><mo id="S5.E4.m1.1.1.1.1.1.1.2" stretchy="false" xref="S5.E4.m1.1.1.1.1.1.1.1.cmml">(</mo><msubsup id="S5.E4.m1.1.1.1.1.1.1.1" xref="S5.E4.m1.1.1.1.1.1.1.1.cmml"><mi id="S5.E4.m1.1.1.1.1.1.1.1.2.2" xref="S5.E4.m1.1.1.1.1.1.1.1.2.2.cmml">G</mi><mi id="S5.E4.m1.1.1.1.1.1.1.1.3" xref="S5.E4.m1.1.1.1.1.1.1.1.3.cmml">k</mi><mn id="S5.E4.m1.1.1.1.1.1.1.1.2.3" xref="S5.E4.m1.1.1.1.1.1.1.1.2.3.cmml">1</mn></msubsup><mo id="S5.E4.m1.1.1.1.1.1.1.3" stretchy="false" xref="S5.E4.m1.1.1.1.1.1.1.1.cmml">)</mo></mrow></mrow></mrow></munder><mrow id="S5.E4.m1.5.5.1.1.2.2" xref="S5.E4.m1.5.5.1.1.2.2.cmml"><mtext class="ltx_mathvariant_italic" id="S5.E4.m1.5.5.1.1.2.2.2" xref="S5.E4.m1.5.5.1.1.2.2.2a.cmml">g</mtext><mo id="S5.E4.m1.5.5.1.1.2.2.1" xref="S5.E4.m1.5.5.1.1.2.2.1.cmml">⁢</mo><mrow 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V(G^{1}_{k})}\textsl{g}(v)>\sum_{v\in V(G_{k+1}^{1})}\textsl{g}_{x}% (v).</annotation><annotation encoding="application/x-llamapun" id="S5.E4.m1.5d">∑ start_POSTSUBSCRIPT italic_v ∈ italic_V ( italic_G start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ) end_POSTSUBSCRIPT g ( italic_v ) > ∑ start_POSTSUBSCRIPT italic_v ∈ italic_V ( italic_G start_POSTSUBSCRIPT italic_k + 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT ) end_POSTSUBSCRIPT g start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT ( italic_v ) .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_left_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(4)</span></td> </tr></tbody> </table> </div> <div class="ltx_para ltx_noindent" id="S5.Thmtheorem3.p7"> <p class="ltx_p" id="S5.Thmtheorem3.p7.1"><span class="ltx_text ltx_font_italic" id="S5.Thmtheorem3.p7.1.1">This is because we can lower bound </span><math alttext="\sum_{v\in V(G^{1}_{k})}\textsl{g}(v)" class="ltx_Math" display="inline" id="S5.Thmtheorem3.p7.1.m1.2"><semantics id="S5.Thmtheorem3.p7.1.m1.2a"><mrow id="S5.Thmtheorem3.p7.1.m1.2.3" xref="S5.Thmtheorem3.p7.1.m1.2.3.cmml"><msub id="S5.Thmtheorem3.p7.1.m1.2.3.1" xref="S5.Thmtheorem3.p7.1.m1.2.3.1.cmml"><mo id="S5.Thmtheorem3.p7.1.m1.2.3.1.2" xref="S5.Thmtheorem3.p7.1.m1.2.3.1.2.cmml">∑</mo><mrow id="S5.Thmtheorem3.p7.1.m1.1.1.1" xref="S5.Thmtheorem3.p7.1.m1.1.1.1.cmml"><mi id="S5.Thmtheorem3.p7.1.m1.1.1.1.3" xref="S5.Thmtheorem3.p7.1.m1.1.1.1.3.cmml">v</mi><mo id="S5.Thmtheorem3.p7.1.m1.1.1.1.2" xref="S5.Thmtheorem3.p7.1.m1.1.1.1.2.cmml">∈</mo><mrow id="S5.Thmtheorem3.p7.1.m1.1.1.1.1" xref="S5.Thmtheorem3.p7.1.m1.1.1.1.1.cmml"><mi id="S5.Thmtheorem3.p7.1.m1.1.1.1.1.3" xref="S5.Thmtheorem3.p7.1.m1.1.1.1.1.3.cmml">V</mi><mo id="S5.Thmtheorem3.p7.1.m1.1.1.1.1.2" xref="S5.Thmtheorem3.p7.1.m1.1.1.1.1.2.cmml">⁢</mo><mrow id="S5.Thmtheorem3.p7.1.m1.1.1.1.1.1.1" xref="S5.Thmtheorem3.p7.1.m1.1.1.1.1.1.1.1.cmml"><mo id="S5.Thmtheorem3.p7.1.m1.1.1.1.1.1.1.2" stretchy="false" xref="S5.Thmtheorem3.p7.1.m1.1.1.1.1.1.1.1.cmml">(</mo><msubsup id="S5.Thmtheorem3.p7.1.m1.1.1.1.1.1.1.1" xref="S5.Thmtheorem3.p7.1.m1.1.1.1.1.1.1.1.cmml"><mi id="S5.Thmtheorem3.p7.1.m1.1.1.1.1.1.1.1.2.2" xref="S5.Thmtheorem3.p7.1.m1.1.1.1.1.1.1.1.2.2.cmml">G</mi><mi id="S5.Thmtheorem3.p7.1.m1.1.1.1.1.1.1.1.3" xref="S5.Thmtheorem3.p7.1.m1.1.1.1.1.1.1.1.3.cmml">k</mi><mn id="S5.Thmtheorem3.p7.1.m1.1.1.1.1.1.1.1.2.3" xref="S5.Thmtheorem3.p7.1.m1.1.1.1.1.1.1.1.2.3.cmml">1</mn></msubsup><mo id="S5.Thmtheorem3.p7.1.m1.1.1.1.1.1.1.3" stretchy="false" xref="S5.Thmtheorem3.p7.1.m1.1.1.1.1.1.1.1.cmml">)</mo></mrow></mrow></mrow></msub><mrow id="S5.Thmtheorem3.p7.1.m1.2.3.2" xref="S5.Thmtheorem3.p7.1.m1.2.3.2.cmml"><mtext class="ltx_mathvariant_italic" id="S5.Thmtheorem3.p7.1.m1.2.3.2.2" xref="S5.Thmtheorem3.p7.1.m1.2.3.2.2a.cmml">g</mtext><mo id="S5.Thmtheorem3.p7.1.m1.2.3.2.1" xref="S5.Thmtheorem3.p7.1.m1.2.3.2.1.cmml">⁢</mo><mrow id="S5.Thmtheorem3.p7.1.m1.2.3.2.3.2" xref="S5.Thmtheorem3.p7.1.m1.2.3.2.cmml"><mo id="S5.Thmtheorem3.p7.1.m1.2.3.2.3.2.1" stretchy="false" xref="S5.Thmtheorem3.p7.1.m1.2.3.2.cmml">(</mo><mi id="S5.Thmtheorem3.p7.1.m1.2.2" xref="S5.Thmtheorem3.p7.1.m1.2.2.cmml">v</mi><mo id="S5.Thmtheorem3.p7.1.m1.2.3.2.3.2.2" stretchy="false" xref="S5.Thmtheorem3.p7.1.m1.2.3.2.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.Thmtheorem3.p7.1.m1.2b"><apply id="S5.Thmtheorem3.p7.1.m1.2.3.cmml" xref="S5.Thmtheorem3.p7.1.m1.2.3"><apply id="S5.Thmtheorem3.p7.1.m1.2.3.1.cmml" xref="S5.Thmtheorem3.p7.1.m1.2.3.1"><csymbol cd="ambiguous" id="S5.Thmtheorem3.p7.1.m1.2.3.1.1.cmml" xref="S5.Thmtheorem3.p7.1.m1.2.3.1">subscript</csymbol><sum id="S5.Thmtheorem3.p7.1.m1.2.3.1.2.cmml" xref="S5.Thmtheorem3.p7.1.m1.2.3.1.2"></sum><apply id="S5.Thmtheorem3.p7.1.m1.1.1.1.cmml" xref="S5.Thmtheorem3.p7.1.m1.1.1.1"><in id="S5.Thmtheorem3.p7.1.m1.1.1.1.2.cmml" xref="S5.Thmtheorem3.p7.1.m1.1.1.1.2"></in><ci id="S5.Thmtheorem3.p7.1.m1.1.1.1.3.cmml" xref="S5.Thmtheorem3.p7.1.m1.1.1.1.3">𝑣</ci><apply id="S5.Thmtheorem3.p7.1.m1.1.1.1.1.cmml" xref="S5.Thmtheorem3.p7.1.m1.1.1.1.1"><times id="S5.Thmtheorem3.p7.1.m1.1.1.1.1.2.cmml" xref="S5.Thmtheorem3.p7.1.m1.1.1.1.1.2"></times><ci id="S5.Thmtheorem3.p7.1.m1.1.1.1.1.3.cmml" xref="S5.Thmtheorem3.p7.1.m1.1.1.1.1.3">𝑉</ci><apply id="S5.Thmtheorem3.p7.1.m1.1.1.1.1.1.1.1.cmml" xref="S5.Thmtheorem3.p7.1.m1.1.1.1.1.1.1"><csymbol cd="ambiguous" id="S5.Thmtheorem3.p7.1.m1.1.1.1.1.1.1.1.1.cmml" xref="S5.Thmtheorem3.p7.1.m1.1.1.1.1.1.1">subscript</csymbol><apply id="S5.Thmtheorem3.p7.1.m1.1.1.1.1.1.1.1.2.cmml" xref="S5.Thmtheorem3.p7.1.m1.1.1.1.1.1.1"><csymbol cd="ambiguous" id="S5.Thmtheorem3.p7.1.m1.1.1.1.1.1.1.1.2.1.cmml" xref="S5.Thmtheorem3.p7.1.m1.1.1.1.1.1.1">superscript</csymbol><ci id="S5.Thmtheorem3.p7.1.m1.1.1.1.1.1.1.1.2.2.cmml" xref="S5.Thmtheorem3.p7.1.m1.1.1.1.1.1.1.1.2.2">𝐺</ci><cn id="S5.Thmtheorem3.p7.1.m1.1.1.1.1.1.1.1.2.3.cmml" type="integer" xref="S5.Thmtheorem3.p7.1.m1.1.1.1.1.1.1.1.2.3">1</cn></apply><ci id="S5.Thmtheorem3.p7.1.m1.1.1.1.1.1.1.1.3.cmml" xref="S5.Thmtheorem3.p7.1.m1.1.1.1.1.1.1.1.3">𝑘</ci></apply></apply></apply></apply><apply id="S5.Thmtheorem3.p7.1.m1.2.3.2.cmml" xref="S5.Thmtheorem3.p7.1.m1.2.3.2"><times id="S5.Thmtheorem3.p7.1.m1.2.3.2.1.cmml" xref="S5.Thmtheorem3.p7.1.m1.2.3.2.1"></times><ci id="S5.Thmtheorem3.p7.1.m1.2.3.2.2a.cmml" xref="S5.Thmtheorem3.p7.1.m1.2.3.2.2"><mtext class="ltx_mathvariant_italic" id="S5.Thmtheorem3.p7.1.m1.2.3.2.2.cmml" xref="S5.Thmtheorem3.p7.1.m1.2.3.2.2">g</mtext></ci><ci id="S5.Thmtheorem3.p7.1.m1.2.2.cmml" xref="S5.Thmtheorem3.p7.1.m1.2.2">𝑣</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmtheorem3.p7.1.m1.2c">\sum_{v\in V(G^{1}_{k})}\textsl{g}(v)</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem3.p7.1.m1.2d">∑ start_POSTSUBSCRIPT italic_v ∈ italic_V ( italic_G start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ) end_POSTSUBSCRIPT g ( italic_v )</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S5.Thmtheorem3.p7.1.2"> as follows:</span></p> </div> <div class="ltx_para" id="S5.Thmtheorem3.p8"> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="A1.EGx5"> <tbody id="S5.Ex19"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_left_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle\sum_{v\in V(G^{1}_{k})}\textsl{g}(v)" class="ltx_Math" display="inline" id="S5.Ex19.m1.2"><semantics id="S5.Ex19.m1.2a"><mrow id="S5.Ex19.m1.2.3" xref="S5.Ex19.m1.2.3.cmml"><mstyle displaystyle="true" id="S5.Ex19.m1.2.3.1" xref="S5.Ex19.m1.2.3.1.cmml"><munder id="S5.Ex19.m1.2.3.1a" xref="S5.Ex19.m1.2.3.1.cmml"><mo id="S5.Ex19.m1.2.3.1.2" movablelimits="false" xref="S5.Ex19.m1.2.3.1.2.cmml">∑</mo><mrow id="S5.Ex19.m1.1.1.1" xref="S5.Ex19.m1.1.1.1.cmml"><mi id="S5.Ex19.m1.1.1.1.3" xref="S5.Ex19.m1.1.1.1.3.cmml">v</mi><mo id="S5.Ex19.m1.1.1.1.2" xref="S5.Ex19.m1.1.1.1.2.cmml">∈</mo><mrow id="S5.Ex19.m1.1.1.1.1" xref="S5.Ex19.m1.1.1.1.1.cmml"><mi id="S5.Ex19.m1.1.1.1.1.3" xref="S5.Ex19.m1.1.1.1.1.3.cmml">V</mi><mo id="S5.Ex19.m1.1.1.1.1.2" xref="S5.Ex19.m1.1.1.1.1.2.cmml">⁢</mo><mrow id="S5.Ex19.m1.1.1.1.1.1.1" xref="S5.Ex19.m1.1.1.1.1.1.1.1.cmml"><mo id="S5.Ex19.m1.1.1.1.1.1.1.2" stretchy="false" xref="S5.Ex19.m1.1.1.1.1.1.1.1.cmml">(</mo><msubsup id="S5.Ex19.m1.1.1.1.1.1.1.1" xref="S5.Ex19.m1.1.1.1.1.1.1.1.cmml"><mi id="S5.Ex19.m1.1.1.1.1.1.1.1.2.2" xref="S5.Ex19.m1.1.1.1.1.1.1.1.2.2.cmml">G</mi><mi id="S5.Ex19.m1.1.1.1.1.1.1.1.3" xref="S5.Ex19.m1.1.1.1.1.1.1.1.3.cmml">k</mi><mn id="S5.Ex19.m1.1.1.1.1.1.1.1.2.3" xref="S5.Ex19.m1.1.1.1.1.1.1.1.2.3.cmml">1</mn></msubsup><mo id="S5.Ex19.m1.1.1.1.1.1.1.3" stretchy="false" xref="S5.Ex19.m1.1.1.1.1.1.1.1.cmml">)</mo></mrow></mrow></mrow></munder></mstyle><mrow id="S5.Ex19.m1.2.3.2" xref="S5.Ex19.m1.2.3.2.cmml"><mtext class="ltx_mathvariant_italic" id="S5.Ex19.m1.2.3.2.2" xref="S5.Ex19.m1.2.3.2.2a.cmml">g</mtext><mo id="S5.Ex19.m1.2.3.2.1" xref="S5.Ex19.m1.2.3.2.1.cmml">⁢</mo><mrow id="S5.Ex19.m1.2.3.2.3.2" xref="S5.Ex19.m1.2.3.2.cmml"><mo id="S5.Ex19.m1.2.3.2.3.2.1" stretchy="false" xref="S5.Ex19.m1.2.3.2.cmml">(</mo><mi id="S5.Ex19.m1.2.2" xref="S5.Ex19.m1.2.2.cmml">v</mi><mo id="S5.Ex19.m1.2.3.2.3.2.2" stretchy="false" xref="S5.Ex19.m1.2.3.2.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.Ex19.m1.2b"><apply id="S5.Ex19.m1.2.3.cmml" xref="S5.Ex19.m1.2.3"><apply id="S5.Ex19.m1.2.3.1.cmml" xref="S5.Ex19.m1.2.3.1"><csymbol cd="ambiguous" id="S5.Ex19.m1.2.3.1.1.cmml" 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type="integer" xref="S5.Ex19.m1.1.1.1.1.1.1.1.2.3">1</cn></apply><ci id="S5.Ex19.m1.1.1.1.1.1.1.1.3.cmml" xref="S5.Ex19.m1.1.1.1.1.1.1.1.3">𝑘</ci></apply></apply></apply></apply><apply id="S5.Ex19.m1.2.3.2.cmml" xref="S5.Ex19.m1.2.3.2"><times id="S5.Ex19.m1.2.3.2.1.cmml" xref="S5.Ex19.m1.2.3.2.1"></times><ci id="S5.Ex19.m1.2.3.2.2a.cmml" xref="S5.Ex19.m1.2.3.2.2"><mtext class="ltx_mathvariant_italic" id="S5.Ex19.m1.2.3.2.2.cmml" xref="S5.Ex19.m1.2.3.2.2">g</mtext></ci><ci id="S5.Ex19.m1.2.2.cmml" xref="S5.Ex19.m1.2.2">𝑣</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Ex19.m1.2c">\displaystyle\sum_{v\in V(G^{1}_{k})}\textsl{g}(v)</annotation><annotation encoding="application/x-llamapun" id="S5.Ex19.m1.2d">∑ start_POSTSUBSCRIPT italic_v ∈ italic_V ( italic_G start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ) end_POSTSUBSCRIPT g ( italic_v )</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle\geq(1+\eta)^{-k}\cdot\textsl{g}(u)\cdot|V(G^{1}_{k})|>(1+\eta)^{% -k}\cdot\textsl{g}(u)\cdot|V(G^{1}_{k+1})|\cdot(1+\frac{\varepsilon}{16})^{-1}" class="ltx_Math" display="inline" id="S5.Ex19.m2.7"><semantics id="S5.Ex19.m2.7a"><mrow id="S5.Ex19.m2.7.7" xref="S5.Ex19.m2.7.7.cmml"><mi id="S5.Ex19.m2.7.7.7" xref="S5.Ex19.m2.7.7.7.cmml"></mi><mo id="S5.Ex19.m2.7.7.8" xref="S5.Ex19.m2.7.7.8.cmml">≥</mo><mrow id="S5.Ex19.m2.4.4.2" xref="S5.Ex19.m2.4.4.2.cmml"><mrow id="S5.Ex19.m2.3.3.1.1" xref="S5.Ex19.m2.3.3.1.1.cmml"><mrow id="S5.Ex19.m2.3.3.1.1.1" xref="S5.Ex19.m2.3.3.1.1.1.cmml"><msup id="S5.Ex19.m2.3.3.1.1.1.1" xref="S5.Ex19.m2.3.3.1.1.1.1.cmml"><mrow id="S5.Ex19.m2.3.3.1.1.1.1.1.1" xref="S5.Ex19.m2.3.3.1.1.1.1.1.1.1.cmml"><mo id="S5.Ex19.m2.3.3.1.1.1.1.1.1.2" stretchy="false" xref="S5.Ex19.m2.3.3.1.1.1.1.1.1.1.cmml">(</mo><mrow id="S5.Ex19.m2.3.3.1.1.1.1.1.1.1" xref="S5.Ex19.m2.3.3.1.1.1.1.1.1.1.cmml"><mn id="S5.Ex19.m2.3.3.1.1.1.1.1.1.1.2" 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type="integer" xref="S5.Ex19.m2.7.7.5.3.1.1.1.3.3">16</cn></apply></apply><apply id="S5.Ex19.m2.7.7.5.3.3.cmml" xref="S5.Ex19.m2.7.7.5.3.3"><minus id="S5.Ex19.m2.7.7.5.3.3.1.cmml" xref="S5.Ex19.m2.7.7.5.3.3"></minus><cn id="S5.Ex19.m2.7.7.5.3.3.2.cmml" type="integer" xref="S5.Ex19.m2.7.7.5.3.3.2">1</cn></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Ex19.m2.7c">\displaystyle\geq(1+\eta)^{-k}\cdot\textsl{g}(u)\cdot|V(G^{1}_{k})|>(1+\eta)^{% -k}\cdot\textsl{g}(u)\cdot|V(G^{1}_{k+1})|\cdot(1+\frac{\varepsilon}{16})^{-1}</annotation><annotation encoding="application/x-llamapun" id="S5.Ex19.m2.7d">≥ ( 1 + italic_η ) start_POSTSUPERSCRIPT - italic_k end_POSTSUPERSCRIPT ⋅ g ( italic_u ) ⋅ | italic_V ( italic_G start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ) | > ( 1 + italic_η ) start_POSTSUPERSCRIPT - italic_k end_POSTSUPERSCRIPT ⋅ g ( italic_u ) ⋅ | italic_V ( italic_G start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_k + 1 end_POSTSUBSCRIPT ) | ⋅ ( 1 + divide start_ARG italic_ε end_ARG start_ARG 16 end_ARG ) start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_left_padright"></td> </tr></tbody> <tbody id="S5.Ex20"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_left_padleft"></td> <td class="ltx_td ltx_eqn_cell"></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle>(1+\frac{\varepsilon}{2})^{-1}\cdot(1+\varepsilon)\gamma\cdot|V(% G^{1}_{k+1})|\cdot(1+\frac{\varepsilon}{16})^{-1}\geq\gamma\cdot|V(G^{1}_{k+1})|" class="ltx_Math" display="inline" id="S5.Ex20.m1.5"><semantics id="S5.Ex20.m1.5a"><mrow id="S5.Ex20.m1.5.5" xref="S5.Ex20.m1.5.5.cmml"><mi id="S5.Ex20.m1.5.5.7" xref="S5.Ex20.m1.5.5.7.cmml"></mi><mo id="S5.Ex20.m1.5.5.8" xref="S5.Ex20.m1.5.5.8.cmml">></mo><mrow id="S5.Ex20.m1.4.4.4" 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xref="S5.Ex20.m1.5.5"></share><apply id="S5.Ex20.m1.5.5.5.cmml" xref="S5.Ex20.m1.5.5.5"><ci id="S5.Ex20.m1.5.5.5.2.cmml" xref="S5.Ex20.m1.5.5.5.2">⋅</ci><ci id="S5.Ex20.m1.5.5.5.3.cmml" xref="S5.Ex20.m1.5.5.5.3">𝛾</ci><apply id="S5.Ex20.m1.5.5.5.1.2.cmml" xref="S5.Ex20.m1.5.5.5.1.1"><abs id="S5.Ex20.m1.5.5.5.1.2.1.cmml" xref="S5.Ex20.m1.5.5.5.1.1.2"></abs><apply id="S5.Ex20.m1.5.5.5.1.1.1.cmml" xref="S5.Ex20.m1.5.5.5.1.1.1"><times id="S5.Ex20.m1.5.5.5.1.1.1.2.cmml" xref="S5.Ex20.m1.5.5.5.1.1.1.2"></times><ci id="S5.Ex20.m1.5.5.5.1.1.1.3.cmml" xref="S5.Ex20.m1.5.5.5.1.1.1.3">𝑉</ci><apply id="S5.Ex20.m1.5.5.5.1.1.1.1.1.1.cmml" xref="S5.Ex20.m1.5.5.5.1.1.1.1.1"><csymbol cd="ambiguous" id="S5.Ex20.m1.5.5.5.1.1.1.1.1.1.1.cmml" xref="S5.Ex20.m1.5.5.5.1.1.1.1.1">subscript</csymbol><apply id="S5.Ex20.m1.5.5.5.1.1.1.1.1.1.2.cmml" xref="S5.Ex20.m1.5.5.5.1.1.1.1.1"><csymbol cd="ambiguous" id="S5.Ex20.m1.5.5.5.1.1.1.1.1.1.2.1.cmml" xref="S5.Ex20.m1.5.5.5.1.1.1.1.1">superscript</csymbol><ci 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start_ARG 2 end_ARG ) start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT ⋅ ( 1 + italic_ε ) italic_γ ⋅ | italic_V ( italic_G start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_k + 1 end_POSTSUBSCRIPT ) | ⋅ ( 1 + divide start_ARG italic_ε end_ARG start_ARG 16 end_ARG ) start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT ≥ italic_γ ⋅ | italic_V ( italic_G start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_k + 1 end_POSTSUBSCRIPT ) |</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_left_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S5.Thmtheorem3.p8.3"><span class="ltx_text ltx_font_italic" id="S5.Thmtheorem3.p8.3.3">The claim follows by noting that per definition of <math alttext="G^{1}" class="ltx_Math" display="inline" id="S5.Thmtheorem3.p8.1.1.m1.1"><semantics id="S5.Thmtheorem3.p8.1.1.m1.1a"><msup id="S5.Thmtheorem3.p8.1.1.m1.1.1" xref="S5.Thmtheorem3.p8.1.1.m1.1.1.cmml"><mi id="S5.Thmtheorem3.p8.1.1.m1.1.1.2" xref="S5.Thmtheorem3.p8.1.1.m1.1.1.2.cmml">G</mi><mn id="S5.Thmtheorem3.p8.1.1.m1.1.1.3" xref="S5.Thmtheorem3.p8.1.1.m1.1.1.3.cmml">1</mn></msup><annotation-xml encoding="MathML-Content" id="S5.Thmtheorem3.p8.1.1.m1.1b"><apply id="S5.Thmtheorem3.p8.1.1.m1.1.1.cmml" xref="S5.Thmtheorem3.p8.1.1.m1.1.1"><csymbol cd="ambiguous" id="S5.Thmtheorem3.p8.1.1.m1.1.1.1.cmml" xref="S5.Thmtheorem3.p8.1.1.m1.1.1">superscript</csymbol><ci id="S5.Thmtheorem3.p8.1.1.m1.1.1.2.cmml" xref="S5.Thmtheorem3.p8.1.1.m1.1.1.2">𝐺</ci><cn id="S5.Thmtheorem3.p8.1.1.m1.1.1.3.cmml" type="integer" xref="S5.Thmtheorem3.p8.1.1.m1.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmtheorem3.p8.1.1.m1.1c">G^{1}</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem3.p8.1.1.m1.1d">italic_G start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT</annotation></semantics></math>, for all <math alttext="v\in V(G^{1}_{k+1})" class="ltx_Math" display="inline" id="S5.Thmtheorem3.p8.2.2.m2.1"><semantics id="S5.Thmtheorem3.p8.2.2.m2.1a"><mrow id="S5.Thmtheorem3.p8.2.2.m2.1.1" xref="S5.Thmtheorem3.p8.2.2.m2.1.1.cmml"><mi id="S5.Thmtheorem3.p8.2.2.m2.1.1.3" xref="S5.Thmtheorem3.p8.2.2.m2.1.1.3.cmml">v</mi><mo id="S5.Thmtheorem3.p8.2.2.m2.1.1.2" xref="S5.Thmtheorem3.p8.2.2.m2.1.1.2.cmml">∈</mo><mrow id="S5.Thmtheorem3.p8.2.2.m2.1.1.1" xref="S5.Thmtheorem3.p8.2.2.m2.1.1.1.cmml"><mi id="S5.Thmtheorem3.p8.2.2.m2.1.1.1.3" xref="S5.Thmtheorem3.p8.2.2.m2.1.1.1.3.cmml">V</mi><mo id="S5.Thmtheorem3.p8.2.2.m2.1.1.1.2" xref="S5.Thmtheorem3.p8.2.2.m2.1.1.1.2.cmml">⁢</mo><mrow id="S5.Thmtheorem3.p8.2.2.m2.1.1.1.1.1" xref="S5.Thmtheorem3.p8.2.2.m2.1.1.1.1.1.1.cmml"><mo id="S5.Thmtheorem3.p8.2.2.m2.1.1.1.1.1.2" stretchy="false" xref="S5.Thmtheorem3.p8.2.2.m2.1.1.1.1.1.1.cmml">(</mo><msubsup id="S5.Thmtheorem3.p8.2.2.m2.1.1.1.1.1.1" xref="S5.Thmtheorem3.p8.2.2.m2.1.1.1.1.1.1.cmml"><mi id="S5.Thmtheorem3.p8.2.2.m2.1.1.1.1.1.1.2.2" xref="S5.Thmtheorem3.p8.2.2.m2.1.1.1.1.1.1.2.2.cmml">G</mi><mrow id="S5.Thmtheorem3.p8.2.2.m2.1.1.1.1.1.1.3" xref="S5.Thmtheorem3.p8.2.2.m2.1.1.1.1.1.1.3.cmml"><mi id="S5.Thmtheorem3.p8.2.2.m2.1.1.1.1.1.1.3.2" xref="S5.Thmtheorem3.p8.2.2.m2.1.1.1.1.1.1.3.2.cmml">k</mi><mo id="S5.Thmtheorem3.p8.2.2.m2.1.1.1.1.1.1.3.1" xref="S5.Thmtheorem3.p8.2.2.m2.1.1.1.1.1.1.3.1.cmml">+</mo><mn id="S5.Thmtheorem3.p8.2.2.m2.1.1.1.1.1.1.3.3" xref="S5.Thmtheorem3.p8.2.2.m2.1.1.1.1.1.1.3.3.cmml">1</mn></mrow><mn id="S5.Thmtheorem3.p8.2.2.m2.1.1.1.1.1.1.2.3" xref="S5.Thmtheorem3.p8.2.2.m2.1.1.1.1.1.1.2.3.cmml">1</mn></msubsup><mo id="S5.Thmtheorem3.p8.2.2.m2.1.1.1.1.1.3" stretchy="false" xref="S5.Thmtheorem3.p8.2.2.m2.1.1.1.1.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.Thmtheorem3.p8.2.2.m2.1b"><apply id="S5.Thmtheorem3.p8.2.2.m2.1.1.cmml" xref="S5.Thmtheorem3.p8.2.2.m2.1.1"><in id="S5.Thmtheorem3.p8.2.2.m2.1.1.2.cmml" xref="S5.Thmtheorem3.p8.2.2.m2.1.1.2"></in><ci id="S5.Thmtheorem3.p8.2.2.m2.1.1.3.cmml" xref="S5.Thmtheorem3.p8.2.2.m2.1.1.3">𝑣</ci><apply id="S5.Thmtheorem3.p8.2.2.m2.1.1.1.cmml" xref="S5.Thmtheorem3.p8.2.2.m2.1.1.1"><times id="S5.Thmtheorem3.p8.2.2.m2.1.1.1.2.cmml" xref="S5.Thmtheorem3.p8.2.2.m2.1.1.1.2"></times><ci id="S5.Thmtheorem3.p8.2.2.m2.1.1.1.3.cmml" xref="S5.Thmtheorem3.p8.2.2.m2.1.1.1.3">𝑉</ci><apply id="S5.Thmtheorem3.p8.2.2.m2.1.1.1.1.1.1.cmml" xref="S5.Thmtheorem3.p8.2.2.m2.1.1.1.1.1"><csymbol cd="ambiguous" id="S5.Thmtheorem3.p8.2.2.m2.1.1.1.1.1.1.1.cmml" xref="S5.Thmtheorem3.p8.2.2.m2.1.1.1.1.1">subscript</csymbol><apply id="S5.Thmtheorem3.p8.2.2.m2.1.1.1.1.1.1.2.cmml" xref="S5.Thmtheorem3.p8.2.2.m2.1.1.1.1.1"><csymbol cd="ambiguous" id="S5.Thmtheorem3.p8.2.2.m2.1.1.1.1.1.1.2.1.cmml" xref="S5.Thmtheorem3.p8.2.2.m2.1.1.1.1.1">superscript</csymbol><ci id="S5.Thmtheorem3.p8.2.2.m2.1.1.1.1.1.1.2.2.cmml" xref="S5.Thmtheorem3.p8.2.2.m2.1.1.1.1.1.1.2.2">𝐺</ci><cn id="S5.Thmtheorem3.p8.2.2.m2.1.1.1.1.1.1.2.3.cmml" type="integer" xref="S5.Thmtheorem3.p8.2.2.m2.1.1.1.1.1.1.2.3">1</cn></apply><apply id="S5.Thmtheorem3.p8.2.2.m2.1.1.1.1.1.1.3.cmml" xref="S5.Thmtheorem3.p8.2.2.m2.1.1.1.1.1.1.3"><plus id="S5.Thmtheorem3.p8.2.2.m2.1.1.1.1.1.1.3.1.cmml" xref="S5.Thmtheorem3.p8.2.2.m2.1.1.1.1.1.1.3.1"></plus><ci id="S5.Thmtheorem3.p8.2.2.m2.1.1.1.1.1.1.3.2.cmml" xref="S5.Thmtheorem3.p8.2.2.m2.1.1.1.1.1.1.3.2">𝑘</ci><cn id="S5.Thmtheorem3.p8.2.2.m2.1.1.1.1.1.1.3.3.cmml" type="integer" xref="S5.Thmtheorem3.p8.2.2.m2.1.1.1.1.1.1.3.3">1</cn></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmtheorem3.p8.2.2.m2.1c">v\in V(G^{1}_{k+1})</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem3.p8.2.2.m2.1d">italic_v ∈ italic_V ( italic_G start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_k + 1 end_POSTSUBSCRIPT )</annotation></semantics></math>, <math alttext="\textsl{g}_{x}(v)\leq\gamma" class="ltx_Math" display="inline" id="S5.Thmtheorem3.p8.3.3.m3.1"><semantics id="S5.Thmtheorem3.p8.3.3.m3.1a"><mrow id="S5.Thmtheorem3.p8.3.3.m3.1.2" xref="S5.Thmtheorem3.p8.3.3.m3.1.2.cmml"><mrow id="S5.Thmtheorem3.p8.3.3.m3.1.2.2" xref="S5.Thmtheorem3.p8.3.3.m3.1.2.2.cmml"><msub id="S5.Thmtheorem3.p8.3.3.m3.1.2.2.2" xref="S5.Thmtheorem3.p8.3.3.m3.1.2.2.2.cmml"><mtext class="ltx_mathvariant_italic" id="S5.Thmtheorem3.p8.3.3.m3.1.2.2.2.2" xref="S5.Thmtheorem3.p8.3.3.m3.1.2.2.2.2a.cmml">g</mtext><mi id="S5.Thmtheorem3.p8.3.3.m3.1.2.2.2.3" xref="S5.Thmtheorem3.p8.3.3.m3.1.2.2.2.3.cmml">x</mi></msub><mo id="S5.Thmtheorem3.p8.3.3.m3.1.2.2.1" xref="S5.Thmtheorem3.p8.3.3.m3.1.2.2.1.cmml">⁢</mo><mrow id="S5.Thmtheorem3.p8.3.3.m3.1.2.2.3.2" xref="S5.Thmtheorem3.p8.3.3.m3.1.2.2.cmml"><mo id="S5.Thmtheorem3.p8.3.3.m3.1.2.2.3.2.1" stretchy="false" xref="S5.Thmtheorem3.p8.3.3.m3.1.2.2.cmml">(</mo><mi id="S5.Thmtheorem3.p8.3.3.m3.1.1" xref="S5.Thmtheorem3.p8.3.3.m3.1.1.cmml">v</mi><mo id="S5.Thmtheorem3.p8.3.3.m3.1.2.2.3.2.2" stretchy="false" xref="S5.Thmtheorem3.p8.3.3.m3.1.2.2.cmml">)</mo></mrow></mrow><mo id="S5.Thmtheorem3.p8.3.3.m3.1.2.1" xref="S5.Thmtheorem3.p8.3.3.m3.1.2.1.cmml">≤</mo><mi id="S5.Thmtheorem3.p8.3.3.m3.1.2.3" xref="S5.Thmtheorem3.p8.3.3.m3.1.2.3.cmml">γ</mi></mrow><annotation-xml encoding="MathML-Content" id="S5.Thmtheorem3.p8.3.3.m3.1b"><apply id="S5.Thmtheorem3.p8.3.3.m3.1.2.cmml" xref="S5.Thmtheorem3.p8.3.3.m3.1.2"><leq id="S5.Thmtheorem3.p8.3.3.m3.1.2.1.cmml" xref="S5.Thmtheorem3.p8.3.3.m3.1.2.1"></leq><apply id="S5.Thmtheorem3.p8.3.3.m3.1.2.2.cmml" xref="S5.Thmtheorem3.p8.3.3.m3.1.2.2"><times id="S5.Thmtheorem3.p8.3.3.m3.1.2.2.1.cmml" xref="S5.Thmtheorem3.p8.3.3.m3.1.2.2.1"></times><apply id="S5.Thmtheorem3.p8.3.3.m3.1.2.2.2.cmml" xref="S5.Thmtheorem3.p8.3.3.m3.1.2.2.2"><csymbol cd="ambiguous" id="S5.Thmtheorem3.p8.3.3.m3.1.2.2.2.1.cmml" xref="S5.Thmtheorem3.p8.3.3.m3.1.2.2.2">subscript</csymbol><ci id="S5.Thmtheorem3.p8.3.3.m3.1.2.2.2.2a.cmml" xref="S5.Thmtheorem3.p8.3.3.m3.1.2.2.2.2"><mtext class="ltx_mathvariant_italic" id="S5.Thmtheorem3.p8.3.3.m3.1.2.2.2.2.cmml" xref="S5.Thmtheorem3.p8.3.3.m3.1.2.2.2.2">g</mtext></ci><ci id="S5.Thmtheorem3.p8.3.3.m3.1.2.2.2.3.cmml" xref="S5.Thmtheorem3.p8.3.3.m3.1.2.2.2.3">𝑥</ci></apply><ci id="S5.Thmtheorem3.p8.3.3.m3.1.1.cmml" xref="S5.Thmtheorem3.p8.3.3.m3.1.1">𝑣</ci></apply><ci id="S5.Thmtheorem3.p8.3.3.m3.1.2.3.cmml" xref="S5.Thmtheorem3.p8.3.3.m3.1.2.3">𝛾</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmtheorem3.p8.3.3.m3.1c">\textsl{g}_{x}(v)\leq\gamma</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem3.p8.3.3.m3.1d">g start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT ( italic_v ) ≤ italic_γ</annotation></semantics></math>.</span></p> </div> <div class="ltx_para ltx_noindent" id="S5.Thmtheorem3.p9"> <p class="ltx_p" id="S5.Thmtheorem3.p9.1"><em class="ltx_emph" id="S5.Thmtheorem3.p9.1.1">The third claim.</em><span class="ltx_text ltx_font_italic" id="S5.Thmtheorem3.p9.1.2"> Lastly, we claim that:</span></p> </div> <div class="ltx_para" id="S5.Thmtheorem3.p10"> <table class="ltx_equation ltx_eqn_table" id="S5.E5"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_left_padleft"></td> <td class="ltx_eqn_cell ltx_align_left"><math alttext="\sum_{v\in V(G^{1}_{k})}\textsl{g}(v)\leq\sum_{e\in E(G^{1}_{k+1})}\textsl{g}(% e)+\sum_{e\in E_{\uparrow}}\textsl{g}(e)" class="ltx_Math" display="block" id="S5.E5.m1.5"><semantics id="S5.E5.m1.5a"><mrow id="S5.E5.m1.5.6" xref="S5.E5.m1.5.6.cmml"><mrow id="S5.E5.m1.5.6.2" xref="S5.E5.m1.5.6.2.cmml"><munder id="S5.E5.m1.5.6.2.1" xref="S5.E5.m1.5.6.2.1.cmml"><mo id="S5.E5.m1.5.6.2.1.2" movablelimits="false" xref="S5.E5.m1.5.6.2.1.2.cmml">∑</mo><mrow id="S5.E5.m1.1.1.1" xref="S5.E5.m1.1.1.1.cmml"><mi id="S5.E5.m1.1.1.1.3" xref="S5.E5.m1.1.1.1.3.cmml">v</mi><mo id="S5.E5.m1.1.1.1.2" xref="S5.E5.m1.1.1.1.2.cmml">∈</mo><mrow id="S5.E5.m1.1.1.1.1" xref="S5.E5.m1.1.1.1.1.cmml"><mi id="S5.E5.m1.1.1.1.1.3" xref="S5.E5.m1.1.1.1.1.3.cmml">V</mi><mo id="S5.E5.m1.1.1.1.1.2" xref="S5.E5.m1.1.1.1.1.2.cmml">⁢</mo><mrow id="S5.E5.m1.1.1.1.1.1.1" xref="S5.E5.m1.1.1.1.1.1.1.1.cmml"><mo id="S5.E5.m1.1.1.1.1.1.1.2" stretchy="false" xref="S5.E5.m1.1.1.1.1.1.1.1.cmml">(</mo><msubsup id="S5.E5.m1.1.1.1.1.1.1.1" xref="S5.E5.m1.1.1.1.1.1.1.1.cmml"><mi id="S5.E5.m1.1.1.1.1.1.1.1.2.2" xref="S5.E5.m1.1.1.1.1.1.1.1.2.2.cmml">G</mi><mi id="S5.E5.m1.1.1.1.1.1.1.1.3" xref="S5.E5.m1.1.1.1.1.1.1.1.3.cmml">k</mi><mn id="S5.E5.m1.1.1.1.1.1.1.1.2.3" xref="S5.E5.m1.1.1.1.1.1.1.1.2.3.cmml">1</mn></msubsup><mo id="S5.E5.m1.1.1.1.1.1.1.3" stretchy="false" xref="S5.E5.m1.1.1.1.1.1.1.1.cmml">)</mo></mrow></mrow></mrow></munder><mrow id="S5.E5.m1.5.6.2.2" xref="S5.E5.m1.5.6.2.2.cmml"><mtext class="ltx_mathvariant_italic" id="S5.E5.m1.5.6.2.2.2" xref="S5.E5.m1.5.6.2.2.2a.cmml">g</mtext><mo 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xref="S5.E5.m1.5.6.3.2.1">subscript</csymbol><sum id="S5.E5.m1.5.6.3.2.1.2.cmml" xref="S5.E5.m1.5.6.3.2.1.2"></sum><apply id="S5.E5.m1.2.2.1.cmml" xref="S5.E5.m1.2.2.1"><in id="S5.E5.m1.2.2.1.2.cmml" xref="S5.E5.m1.2.2.1.2"></in><ci id="S5.E5.m1.2.2.1.3.cmml" xref="S5.E5.m1.2.2.1.3">𝑒</ci><apply id="S5.E5.m1.2.2.1.1.cmml" xref="S5.E5.m1.2.2.1.1"><times id="S5.E5.m1.2.2.1.1.2.cmml" xref="S5.E5.m1.2.2.1.1.2"></times><ci id="S5.E5.m1.2.2.1.1.3.cmml" xref="S5.E5.m1.2.2.1.1.3">𝐸</ci><apply id="S5.E5.m1.2.2.1.1.1.1.1.cmml" xref="S5.E5.m1.2.2.1.1.1.1"><csymbol cd="ambiguous" id="S5.E5.m1.2.2.1.1.1.1.1.1.cmml" xref="S5.E5.m1.2.2.1.1.1.1">subscript</csymbol><apply id="S5.E5.m1.2.2.1.1.1.1.1.2.cmml" xref="S5.E5.m1.2.2.1.1.1.1"><csymbol cd="ambiguous" id="S5.E5.m1.2.2.1.1.1.1.1.2.1.cmml" xref="S5.E5.m1.2.2.1.1.1.1">superscript</csymbol><ci id="S5.E5.m1.2.2.1.1.1.1.1.2.2.cmml" xref="S5.E5.m1.2.2.1.1.1.1.1.2.2">𝐺</ci><cn id="S5.E5.m1.2.2.1.1.1.1.1.2.3.cmml" type="integer" xref="S5.E5.m1.2.2.1.1.1.1.1.2.3">1</cn></apply><apply id="S5.E5.m1.2.2.1.1.1.1.1.3.cmml" xref="S5.E5.m1.2.2.1.1.1.1.1.3"><plus id="S5.E5.m1.2.2.1.1.1.1.1.3.1.cmml" xref="S5.E5.m1.2.2.1.1.1.1.1.3.1"></plus><ci id="S5.E5.m1.2.2.1.1.1.1.1.3.2.cmml" xref="S5.E5.m1.2.2.1.1.1.1.1.3.2">𝑘</ci><cn id="S5.E5.m1.2.2.1.1.1.1.1.3.3.cmml" type="integer" xref="S5.E5.m1.2.2.1.1.1.1.1.3.3">1</cn></apply></apply></apply></apply></apply><apply id="S5.E5.m1.5.6.3.2.2.cmml" xref="S5.E5.m1.5.6.3.2.2"><times id="S5.E5.m1.5.6.3.2.2.1.cmml" xref="S5.E5.m1.5.6.3.2.2.1"></times><ci id="S5.E5.m1.5.6.3.2.2.2a.cmml" xref="S5.E5.m1.5.6.3.2.2.2"><mtext class="ltx_mathvariant_italic" id="S5.E5.m1.5.6.3.2.2.2.cmml" xref="S5.E5.m1.5.6.3.2.2.2">g</mtext></ci><ci id="S5.E5.m1.4.4.cmml" xref="S5.E5.m1.4.4">𝑒</ci></apply></apply><apply id="S5.E5.m1.5.6.3.3.cmml" xref="S5.E5.m1.5.6.3.3"><apply id="S5.E5.m1.5.6.3.3.1.cmml" xref="S5.E5.m1.5.6.3.3.1"><csymbol cd="ambiguous" id="S5.E5.m1.5.6.3.3.1.1.cmml" xref="S5.E5.m1.5.6.3.3.1">subscript</csymbol><sum id="S5.E5.m1.5.6.3.3.1.2.cmml" xref="S5.E5.m1.5.6.3.3.1.2"></sum><apply id="S5.E5.m1.5.6.3.3.1.3.cmml" xref="S5.E5.m1.5.6.3.3.1.3"><in id="S5.E5.m1.5.6.3.3.1.3.1.cmml" xref="S5.E5.m1.5.6.3.3.1.3.1"></in><ci id="S5.E5.m1.5.6.3.3.1.3.2.cmml" xref="S5.E5.m1.5.6.3.3.1.3.2">𝑒</ci><apply id="S5.E5.m1.5.6.3.3.1.3.3.cmml" xref="S5.E5.m1.5.6.3.3.1.3.3"><csymbol cd="ambiguous" id="S5.E5.m1.5.6.3.3.1.3.3.1.cmml" xref="S5.E5.m1.5.6.3.3.1.3.3">subscript</csymbol><ci id="S5.E5.m1.5.6.3.3.1.3.3.2.cmml" xref="S5.E5.m1.5.6.3.3.1.3.3.2">𝐸</ci><ci id="S5.E5.m1.5.6.3.3.1.3.3.3.cmml" xref="S5.E5.m1.5.6.3.3.1.3.3.3">↑</ci></apply></apply></apply><apply id="S5.E5.m1.5.6.3.3.2.cmml" xref="S5.E5.m1.5.6.3.3.2"><times id="S5.E5.m1.5.6.3.3.2.1.cmml" xref="S5.E5.m1.5.6.3.3.2.1"></times><ci id="S5.E5.m1.5.6.3.3.2.2a.cmml" xref="S5.E5.m1.5.6.3.3.2.2"><mtext class="ltx_mathvariant_italic" id="S5.E5.m1.5.6.3.3.2.2.cmml" xref="S5.E5.m1.5.6.3.3.2.2">g</mtext></ci><ci id="S5.E5.m1.5.5.cmml" xref="S5.E5.m1.5.5">𝑒</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.E5.m1.5c">\sum_{v\in V(G^{1}_{k})}\textsl{g}(v)\leq\sum_{e\in E(G^{1}_{k+1})}\textsl{g}(% e)+\sum_{e\in E_{\uparrow}}\textsl{g}(e)</annotation><annotation encoding="application/x-llamapun" id="S5.E5.m1.5d">∑ start_POSTSUBSCRIPT italic_v ∈ italic_V ( italic_G start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ) end_POSTSUBSCRIPT g ( italic_v ) ≤ ∑ start_POSTSUBSCRIPT italic_e ∈ italic_E ( italic_G start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_k + 1 end_POSTSUBSCRIPT ) end_POSTSUBSCRIPT g ( italic_e ) + ∑ start_POSTSUBSCRIPT italic_e ∈ italic_E start_POSTSUBSCRIPT ↑ end_POSTSUBSCRIPT end_POSTSUBSCRIPT g ( italic_e )</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_left_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(5)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S5.Thmtheorem3.p10.10"><span class="ltx_text ltx_font_italic" id="S5.Thmtheorem3.p10.10.10">Consider any <math alttext="v\in V(G^{1}_{k})" class="ltx_Math" display="inline" id="S5.Thmtheorem3.p10.1.1.m1.1"><semantics id="S5.Thmtheorem3.p10.1.1.m1.1a"><mrow id="S5.Thmtheorem3.p10.1.1.m1.1.1" xref="S5.Thmtheorem3.p10.1.1.m1.1.1.cmml"><mi id="S5.Thmtheorem3.p10.1.1.m1.1.1.3" xref="S5.Thmtheorem3.p10.1.1.m1.1.1.3.cmml">v</mi><mo id="S5.Thmtheorem3.p10.1.1.m1.1.1.2" xref="S5.Thmtheorem3.p10.1.1.m1.1.1.2.cmml">∈</mo><mrow id="S5.Thmtheorem3.p10.1.1.m1.1.1.1" xref="S5.Thmtheorem3.p10.1.1.m1.1.1.1.cmml"><mi id="S5.Thmtheorem3.p10.1.1.m1.1.1.1.3" xref="S5.Thmtheorem3.p10.1.1.m1.1.1.1.3.cmml">V</mi><mo id="S5.Thmtheorem3.p10.1.1.m1.1.1.1.2" xref="S5.Thmtheorem3.p10.1.1.m1.1.1.1.2.cmml">⁢</mo><mrow id="S5.Thmtheorem3.p10.1.1.m1.1.1.1.1.1" xref="S5.Thmtheorem3.p10.1.1.m1.1.1.1.1.1.1.cmml"><mo id="S5.Thmtheorem3.p10.1.1.m1.1.1.1.1.1.2" stretchy="false" xref="S5.Thmtheorem3.p10.1.1.m1.1.1.1.1.1.1.cmml">(</mo><msubsup id="S5.Thmtheorem3.p10.1.1.m1.1.1.1.1.1.1" xref="S5.Thmtheorem3.p10.1.1.m1.1.1.1.1.1.1.cmml"><mi id="S5.Thmtheorem3.p10.1.1.m1.1.1.1.1.1.1.2.2" xref="S5.Thmtheorem3.p10.1.1.m1.1.1.1.1.1.1.2.2.cmml">G</mi><mi id="S5.Thmtheorem3.p10.1.1.m1.1.1.1.1.1.1.3" xref="S5.Thmtheorem3.p10.1.1.m1.1.1.1.1.1.1.3.cmml">k</mi><mn id="S5.Thmtheorem3.p10.1.1.m1.1.1.1.1.1.1.2.3" xref="S5.Thmtheorem3.p10.1.1.m1.1.1.1.1.1.1.2.3.cmml">1</mn></msubsup><mo id="S5.Thmtheorem3.p10.1.1.m1.1.1.1.1.1.3" stretchy="false" xref="S5.Thmtheorem3.p10.1.1.m1.1.1.1.1.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.Thmtheorem3.p10.1.1.m1.1b"><apply id="S5.Thmtheorem3.p10.1.1.m1.1.1.cmml" xref="S5.Thmtheorem3.p10.1.1.m1.1.1"><in id="S5.Thmtheorem3.p10.1.1.m1.1.1.2.cmml" xref="S5.Thmtheorem3.p10.1.1.m1.1.1.2"></in><ci id="S5.Thmtheorem3.p10.1.1.m1.1.1.3.cmml" xref="S5.Thmtheorem3.p10.1.1.m1.1.1.3">𝑣</ci><apply id="S5.Thmtheorem3.p10.1.1.m1.1.1.1.cmml" xref="S5.Thmtheorem3.p10.1.1.m1.1.1.1"><times id="S5.Thmtheorem3.p10.1.1.m1.1.1.1.2.cmml" xref="S5.Thmtheorem3.p10.1.1.m1.1.1.1.2"></times><ci id="S5.Thmtheorem3.p10.1.1.m1.1.1.1.3.cmml" xref="S5.Thmtheorem3.p10.1.1.m1.1.1.1.3">𝑉</ci><apply id="S5.Thmtheorem3.p10.1.1.m1.1.1.1.1.1.1.cmml" xref="S5.Thmtheorem3.p10.1.1.m1.1.1.1.1.1"><csymbol cd="ambiguous" id="S5.Thmtheorem3.p10.1.1.m1.1.1.1.1.1.1.1.cmml" xref="S5.Thmtheorem3.p10.1.1.m1.1.1.1.1.1">subscript</csymbol><apply id="S5.Thmtheorem3.p10.1.1.m1.1.1.1.1.1.1.2.cmml" xref="S5.Thmtheorem3.p10.1.1.m1.1.1.1.1.1"><csymbol cd="ambiguous" id="S5.Thmtheorem3.p10.1.1.m1.1.1.1.1.1.1.2.1.cmml" xref="S5.Thmtheorem3.p10.1.1.m1.1.1.1.1.1">superscript</csymbol><ci id="S5.Thmtheorem3.p10.1.1.m1.1.1.1.1.1.1.2.2.cmml" xref="S5.Thmtheorem3.p10.1.1.m1.1.1.1.1.1.1.2.2">𝐺</ci><cn id="S5.Thmtheorem3.p10.1.1.m1.1.1.1.1.1.1.2.3.cmml" type="integer" xref="S5.Thmtheorem3.p10.1.1.m1.1.1.1.1.1.1.2.3">1</cn></apply><ci id="S5.Thmtheorem3.p10.1.1.m1.1.1.1.1.1.1.3.cmml" xref="S5.Thmtheorem3.p10.1.1.m1.1.1.1.1.1.1.3">𝑘</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmtheorem3.p10.1.1.m1.1c">v\in V(G^{1}_{k})</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem3.p10.1.1.m1.1d">italic_v ∈ italic_V ( italic_G start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT )</annotation></semantics></math> and any vertex <math alttext="a" class="ltx_Math" display="inline" id="S5.Thmtheorem3.p10.2.2.m2.1"><semantics id="S5.Thmtheorem3.p10.2.2.m2.1a"><mi id="S5.Thmtheorem3.p10.2.2.m2.1.1" xref="S5.Thmtheorem3.p10.2.2.m2.1.1.cmml">a</mi><annotation-xml encoding="MathML-Content" id="S5.Thmtheorem3.p10.2.2.m2.1b"><ci id="S5.Thmtheorem3.p10.2.2.m2.1.1.cmml" xref="S5.Thmtheorem3.p10.2.2.m2.1.1">𝑎</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmtheorem3.p10.2.2.m2.1c">a</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem3.p10.2.2.m2.1d">italic_a</annotation></semantics></math> with <math alttext="\textsl{g}(v\!\to\!a)>0" class="ltx_Math" display="inline" id="S5.Thmtheorem3.p10.3.3.m3.1"><semantics id="S5.Thmtheorem3.p10.3.3.m3.1a"><mrow id="S5.Thmtheorem3.p10.3.3.m3.1.1" xref="S5.Thmtheorem3.p10.3.3.m3.1.1.cmml"><mrow id="S5.Thmtheorem3.p10.3.3.m3.1.1.1" xref="S5.Thmtheorem3.p10.3.3.m3.1.1.1.cmml"><mtext class="ltx_mathvariant_italic" id="S5.Thmtheorem3.p10.3.3.m3.1.1.1.3" xref="S5.Thmtheorem3.p10.3.3.m3.1.1.1.3a.cmml">g</mtext><mo id="S5.Thmtheorem3.p10.3.3.m3.1.1.1.2" xref="S5.Thmtheorem3.p10.3.3.m3.1.1.1.2.cmml">⁢</mo><mrow id="S5.Thmtheorem3.p10.3.3.m3.1.1.1.1.1" xref="S5.Thmtheorem3.p10.3.3.m3.1.1.1.1.1.1.cmml"><mo id="S5.Thmtheorem3.p10.3.3.m3.1.1.1.1.1.2" stretchy="false" xref="S5.Thmtheorem3.p10.3.3.m3.1.1.1.1.1.1.cmml">(</mo><mrow id="S5.Thmtheorem3.p10.3.3.m3.1.1.1.1.1.1" xref="S5.Thmtheorem3.p10.3.3.m3.1.1.1.1.1.1.cmml"><mi id="S5.Thmtheorem3.p10.3.3.m3.1.1.1.1.1.1.2" xref="S5.Thmtheorem3.p10.3.3.m3.1.1.1.1.1.1.2.cmml">v</mi><mo id="S5.Thmtheorem3.p10.3.3.m3.1.1.1.1.1.1.1" lspace="0.108em" rspace="0.108em" stretchy="false" xref="S5.Thmtheorem3.p10.3.3.m3.1.1.1.1.1.1.1.cmml">→</mo><mi id="S5.Thmtheorem3.p10.3.3.m3.1.1.1.1.1.1.3" xref="S5.Thmtheorem3.p10.3.3.m3.1.1.1.1.1.1.3.cmml">a</mi></mrow><mo id="S5.Thmtheorem3.p10.3.3.m3.1.1.1.1.1.3" stretchy="false" xref="S5.Thmtheorem3.p10.3.3.m3.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S5.Thmtheorem3.p10.3.3.m3.1.1.2" xref="S5.Thmtheorem3.p10.3.3.m3.1.1.2.cmml">></mo><mn id="S5.Thmtheorem3.p10.3.3.m3.1.1.3" xref="S5.Thmtheorem3.p10.3.3.m3.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S5.Thmtheorem3.p10.3.3.m3.1b"><apply id="S5.Thmtheorem3.p10.3.3.m3.1.1.cmml" xref="S5.Thmtheorem3.p10.3.3.m3.1.1"><gt id="S5.Thmtheorem3.p10.3.3.m3.1.1.2.cmml" xref="S5.Thmtheorem3.p10.3.3.m3.1.1.2"></gt><apply id="S5.Thmtheorem3.p10.3.3.m3.1.1.1.cmml" xref="S5.Thmtheorem3.p10.3.3.m3.1.1.1"><times id="S5.Thmtheorem3.p10.3.3.m3.1.1.1.2.cmml" xref="S5.Thmtheorem3.p10.3.3.m3.1.1.1.2"></times><ci id="S5.Thmtheorem3.p10.3.3.m3.1.1.1.3a.cmml" xref="S5.Thmtheorem3.p10.3.3.m3.1.1.1.3"><mtext class="ltx_mathvariant_italic" id="S5.Thmtheorem3.p10.3.3.m3.1.1.1.3.cmml" xref="S5.Thmtheorem3.p10.3.3.m3.1.1.1.3">g</mtext></ci><apply id="S5.Thmtheorem3.p10.3.3.m3.1.1.1.1.1.1.cmml" xref="S5.Thmtheorem3.p10.3.3.m3.1.1.1.1.1"><ci id="S5.Thmtheorem3.p10.3.3.m3.1.1.1.1.1.1.1.cmml" xref="S5.Thmtheorem3.p10.3.3.m3.1.1.1.1.1.1.1">→</ci><ci id="S5.Thmtheorem3.p10.3.3.m3.1.1.1.1.1.1.2.cmml" xref="S5.Thmtheorem3.p10.3.3.m3.1.1.1.1.1.1.2">𝑣</ci><ci id="S5.Thmtheorem3.p10.3.3.m3.1.1.1.1.1.1.3.cmml" xref="S5.Thmtheorem3.p10.3.3.m3.1.1.1.1.1.1.3">𝑎</ci></apply></apply><cn id="S5.Thmtheorem3.p10.3.3.m3.1.1.3.cmml" type="integer" xref="S5.Thmtheorem3.p10.3.3.m3.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmtheorem3.p10.3.3.m3.1c">\textsl{g}(v\!\to\!a)>0</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem3.p10.3.3.m3.1d">g ( italic_v → italic_a ) > 0</annotation></semantics></math>. Recall that <math alttext="\overrightarrow{G}" class="ltx_Math" display="inline" id="S5.Thmtheorem3.p10.4.4.m4.1"><semantics id="S5.Thmtheorem3.p10.4.4.m4.1a"><mover accent="true" id="S5.Thmtheorem3.p10.4.4.m4.1.1" xref="S5.Thmtheorem3.p10.4.4.m4.1.1.cmml"><mi id="S5.Thmtheorem3.p10.4.4.m4.1.1.2" xref="S5.Thmtheorem3.p10.4.4.m4.1.1.2.cmml">G</mi><mo id="S5.Thmtheorem3.p10.4.4.m4.1.1.1" stretchy="false" xref="S5.Thmtheorem3.p10.4.4.m4.1.1.1.cmml">→</mo></mover><annotation-xml encoding="MathML-Content" id="S5.Thmtheorem3.p10.4.4.m4.1b"><apply id="S5.Thmtheorem3.p10.4.4.m4.1.1.cmml" xref="S5.Thmtheorem3.p10.4.4.m4.1.1"><ci id="S5.Thmtheorem3.p10.4.4.m4.1.1.1.cmml" xref="S5.Thmtheorem3.p10.4.4.m4.1.1.1">→</ci><ci id="S5.Thmtheorem3.p10.4.4.m4.1.1.2.cmml" xref="S5.Thmtheorem3.p10.4.4.m4.1.1.2">𝐺</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmtheorem3.p10.4.4.m4.1c">\overrightarrow{G}</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem3.p10.4.4.m4.1d">over→ start_ARG italic_G end_ARG</annotation></semantics></math> is an <math alttext="\eta" class="ltx_Math" display="inline" id="S5.Thmtheorem3.p10.5.5.m5.1"><semantics id="S5.Thmtheorem3.p10.5.5.m5.1a"><mi id="S5.Thmtheorem3.p10.5.5.m5.1.1" xref="S5.Thmtheorem3.p10.5.5.m5.1.1.cmml">η</mi><annotation-xml encoding="MathML-Content" id="S5.Thmtheorem3.p10.5.5.m5.1b"><ci id="S5.Thmtheorem3.p10.5.5.m5.1.1.cmml" xref="S5.Thmtheorem3.p10.5.5.m5.1.1">𝜂</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmtheorem3.p10.5.5.m5.1c">\eta</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem3.p10.5.5.m5.1d">italic_η</annotation></semantics></math>-fair orientation. Thus, if <math alttext="a\in G^{1}" class="ltx_Math" display="inline" id="S5.Thmtheorem3.p10.6.6.m6.1"><semantics id="S5.Thmtheorem3.p10.6.6.m6.1a"><mrow id="S5.Thmtheorem3.p10.6.6.m6.1.1" xref="S5.Thmtheorem3.p10.6.6.m6.1.1.cmml"><mi id="S5.Thmtheorem3.p10.6.6.m6.1.1.2" xref="S5.Thmtheorem3.p10.6.6.m6.1.1.2.cmml">a</mi><mo id="S5.Thmtheorem3.p10.6.6.m6.1.1.1" xref="S5.Thmtheorem3.p10.6.6.m6.1.1.1.cmml">∈</mo><msup id="S5.Thmtheorem3.p10.6.6.m6.1.1.3" xref="S5.Thmtheorem3.p10.6.6.m6.1.1.3.cmml"><mi id="S5.Thmtheorem3.p10.6.6.m6.1.1.3.2" xref="S5.Thmtheorem3.p10.6.6.m6.1.1.3.2.cmml">G</mi><mn id="S5.Thmtheorem3.p10.6.6.m6.1.1.3.3" xref="S5.Thmtheorem3.p10.6.6.m6.1.1.3.3.cmml">1</mn></msup></mrow><annotation-xml encoding="MathML-Content" id="S5.Thmtheorem3.p10.6.6.m6.1b"><apply id="S5.Thmtheorem3.p10.6.6.m6.1.1.cmml" xref="S5.Thmtheorem3.p10.6.6.m6.1.1"><in id="S5.Thmtheorem3.p10.6.6.m6.1.1.1.cmml" xref="S5.Thmtheorem3.p10.6.6.m6.1.1.1"></in><ci id="S5.Thmtheorem3.p10.6.6.m6.1.1.2.cmml" xref="S5.Thmtheorem3.p10.6.6.m6.1.1.2">𝑎</ci><apply id="S5.Thmtheorem3.p10.6.6.m6.1.1.3.cmml" xref="S5.Thmtheorem3.p10.6.6.m6.1.1.3"><csymbol cd="ambiguous" id="S5.Thmtheorem3.p10.6.6.m6.1.1.3.1.cmml" xref="S5.Thmtheorem3.p10.6.6.m6.1.1.3">superscript</csymbol><ci id="S5.Thmtheorem3.p10.6.6.m6.1.1.3.2.cmml" xref="S5.Thmtheorem3.p10.6.6.m6.1.1.3.2">𝐺</ci><cn id="S5.Thmtheorem3.p10.6.6.m6.1.1.3.3.cmml" type="integer" xref="S5.Thmtheorem3.p10.6.6.m6.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmtheorem3.p10.6.6.m6.1c">a\in G^{1}</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem3.p10.6.6.m6.1d">italic_a ∈ italic_G start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT</annotation></semantics></math>, then <math alttext="\overline{va}\in E(G^{1}_{k+1})" class="ltx_Math" display="inline" id="S5.Thmtheorem3.p10.7.7.m7.1"><semantics id="S5.Thmtheorem3.p10.7.7.m7.1a"><mrow id="S5.Thmtheorem3.p10.7.7.m7.1.1" xref="S5.Thmtheorem3.p10.7.7.m7.1.1.cmml"><mover accent="true" id="S5.Thmtheorem3.p10.7.7.m7.1.1.3" xref="S5.Thmtheorem3.p10.7.7.m7.1.1.3.cmml"><mrow id="S5.Thmtheorem3.p10.7.7.m7.1.1.3.2" xref="S5.Thmtheorem3.p10.7.7.m7.1.1.3.2.cmml"><mi id="S5.Thmtheorem3.p10.7.7.m7.1.1.3.2.2" xref="S5.Thmtheorem3.p10.7.7.m7.1.1.3.2.2.cmml">v</mi><mo id="S5.Thmtheorem3.p10.7.7.m7.1.1.3.2.1" xref="S5.Thmtheorem3.p10.7.7.m7.1.1.3.2.1.cmml">⁢</mo><mi id="S5.Thmtheorem3.p10.7.7.m7.1.1.3.2.3" xref="S5.Thmtheorem3.p10.7.7.m7.1.1.3.2.3.cmml">a</mi></mrow><mo id="S5.Thmtheorem3.p10.7.7.m7.1.1.3.1" xref="S5.Thmtheorem3.p10.7.7.m7.1.1.3.1.cmml">¯</mo></mover><mo id="S5.Thmtheorem3.p10.7.7.m7.1.1.2" xref="S5.Thmtheorem3.p10.7.7.m7.1.1.2.cmml">∈</mo><mrow id="S5.Thmtheorem3.p10.7.7.m7.1.1.1" xref="S5.Thmtheorem3.p10.7.7.m7.1.1.1.cmml"><mi id="S5.Thmtheorem3.p10.7.7.m7.1.1.1.3" xref="S5.Thmtheorem3.p10.7.7.m7.1.1.1.3.cmml">E</mi><mo id="S5.Thmtheorem3.p10.7.7.m7.1.1.1.2" xref="S5.Thmtheorem3.p10.7.7.m7.1.1.1.2.cmml">⁢</mo><mrow id="S5.Thmtheorem3.p10.7.7.m7.1.1.1.1.1" xref="S5.Thmtheorem3.p10.7.7.m7.1.1.1.1.1.1.cmml"><mo id="S5.Thmtheorem3.p10.7.7.m7.1.1.1.1.1.2" stretchy="false" xref="S5.Thmtheorem3.p10.7.7.m7.1.1.1.1.1.1.cmml">(</mo><msubsup id="S5.Thmtheorem3.p10.7.7.m7.1.1.1.1.1.1" xref="S5.Thmtheorem3.p10.7.7.m7.1.1.1.1.1.1.cmml"><mi id="S5.Thmtheorem3.p10.7.7.m7.1.1.1.1.1.1.2.2" xref="S5.Thmtheorem3.p10.7.7.m7.1.1.1.1.1.1.2.2.cmml">G</mi><mrow id="S5.Thmtheorem3.p10.7.7.m7.1.1.1.1.1.1.3" xref="S5.Thmtheorem3.p10.7.7.m7.1.1.1.1.1.1.3.cmml"><mi id="S5.Thmtheorem3.p10.7.7.m7.1.1.1.1.1.1.3.2" xref="S5.Thmtheorem3.p10.7.7.m7.1.1.1.1.1.1.3.2.cmml">k</mi><mo id="S5.Thmtheorem3.p10.7.7.m7.1.1.1.1.1.1.3.1" xref="S5.Thmtheorem3.p10.7.7.m7.1.1.1.1.1.1.3.1.cmml">+</mo><mn id="S5.Thmtheorem3.p10.7.7.m7.1.1.1.1.1.1.3.3" xref="S5.Thmtheorem3.p10.7.7.m7.1.1.1.1.1.1.3.3.cmml">1</mn></mrow><mn id="S5.Thmtheorem3.p10.7.7.m7.1.1.1.1.1.1.2.3" xref="S5.Thmtheorem3.p10.7.7.m7.1.1.1.1.1.1.2.3.cmml">1</mn></msubsup><mo id="S5.Thmtheorem3.p10.7.7.m7.1.1.1.1.1.3" stretchy="false" xref="S5.Thmtheorem3.p10.7.7.m7.1.1.1.1.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.Thmtheorem3.p10.7.7.m7.1b"><apply id="S5.Thmtheorem3.p10.7.7.m7.1.1.cmml" xref="S5.Thmtheorem3.p10.7.7.m7.1.1"><in id="S5.Thmtheorem3.p10.7.7.m7.1.1.2.cmml" xref="S5.Thmtheorem3.p10.7.7.m7.1.1.2"></in><apply id="S5.Thmtheorem3.p10.7.7.m7.1.1.3.cmml" xref="S5.Thmtheorem3.p10.7.7.m7.1.1.3"><ci id="S5.Thmtheorem3.p10.7.7.m7.1.1.3.1.cmml" xref="S5.Thmtheorem3.p10.7.7.m7.1.1.3.1">¯</ci><apply id="S5.Thmtheorem3.p10.7.7.m7.1.1.3.2.cmml" xref="S5.Thmtheorem3.p10.7.7.m7.1.1.3.2"><times id="S5.Thmtheorem3.p10.7.7.m7.1.1.3.2.1.cmml" xref="S5.Thmtheorem3.p10.7.7.m7.1.1.3.2.1"></times><ci id="S5.Thmtheorem3.p10.7.7.m7.1.1.3.2.2.cmml" xref="S5.Thmtheorem3.p10.7.7.m7.1.1.3.2.2">𝑣</ci><ci id="S5.Thmtheorem3.p10.7.7.m7.1.1.3.2.3.cmml" xref="S5.Thmtheorem3.p10.7.7.m7.1.1.3.2.3">𝑎</ci></apply></apply><apply id="S5.Thmtheorem3.p10.7.7.m7.1.1.1.cmml" xref="S5.Thmtheorem3.p10.7.7.m7.1.1.1"><times id="S5.Thmtheorem3.p10.7.7.m7.1.1.1.2.cmml" xref="S5.Thmtheorem3.p10.7.7.m7.1.1.1.2"></times><ci id="S5.Thmtheorem3.p10.7.7.m7.1.1.1.3.cmml" xref="S5.Thmtheorem3.p10.7.7.m7.1.1.1.3">𝐸</ci><apply id="S5.Thmtheorem3.p10.7.7.m7.1.1.1.1.1.1.cmml" xref="S5.Thmtheorem3.p10.7.7.m7.1.1.1.1.1"><csymbol cd="ambiguous" id="S5.Thmtheorem3.p10.7.7.m7.1.1.1.1.1.1.1.cmml" xref="S5.Thmtheorem3.p10.7.7.m7.1.1.1.1.1">subscript</csymbol><apply id="S5.Thmtheorem3.p10.7.7.m7.1.1.1.1.1.1.2.cmml" xref="S5.Thmtheorem3.p10.7.7.m7.1.1.1.1.1"><csymbol cd="ambiguous" id="S5.Thmtheorem3.p10.7.7.m7.1.1.1.1.1.1.2.1.cmml" xref="S5.Thmtheorem3.p10.7.7.m7.1.1.1.1.1">superscript</csymbol><ci id="S5.Thmtheorem3.p10.7.7.m7.1.1.1.1.1.1.2.2.cmml" xref="S5.Thmtheorem3.p10.7.7.m7.1.1.1.1.1.1.2.2">𝐺</ci><cn id="S5.Thmtheorem3.p10.7.7.m7.1.1.1.1.1.1.2.3.cmml" type="integer" xref="S5.Thmtheorem3.p10.7.7.m7.1.1.1.1.1.1.2.3">1</cn></apply><apply id="S5.Thmtheorem3.p10.7.7.m7.1.1.1.1.1.1.3.cmml" xref="S5.Thmtheorem3.p10.7.7.m7.1.1.1.1.1.1.3"><plus id="S5.Thmtheorem3.p10.7.7.m7.1.1.1.1.1.1.3.1.cmml" xref="S5.Thmtheorem3.p10.7.7.m7.1.1.1.1.1.1.3.1"></plus><ci id="S5.Thmtheorem3.p10.7.7.m7.1.1.1.1.1.1.3.2.cmml" xref="S5.Thmtheorem3.p10.7.7.m7.1.1.1.1.1.1.3.2">𝑘</ci><cn id="S5.Thmtheorem3.p10.7.7.m7.1.1.1.1.1.1.3.3.cmml" type="integer" xref="S5.Thmtheorem3.p10.7.7.m7.1.1.1.1.1.1.3.3">1</cn></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmtheorem3.p10.7.7.m7.1c">\overline{va}\in E(G^{1}_{k+1})</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem3.p10.7.7.m7.1d">over¯ start_ARG italic_v italic_a end_ARG ∈ italic_E ( italic_G start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_k + 1 end_POSTSUBSCRIPT )</annotation></semantics></math>. If <math alttext="a\in G^{2}" class="ltx_Math" display="inline" id="S5.Thmtheorem3.p10.8.8.m8.1"><semantics id="S5.Thmtheorem3.p10.8.8.m8.1a"><mrow id="S5.Thmtheorem3.p10.8.8.m8.1.1" xref="S5.Thmtheorem3.p10.8.8.m8.1.1.cmml"><mi id="S5.Thmtheorem3.p10.8.8.m8.1.1.2" xref="S5.Thmtheorem3.p10.8.8.m8.1.1.2.cmml">a</mi><mo id="S5.Thmtheorem3.p10.8.8.m8.1.1.1" xref="S5.Thmtheorem3.p10.8.8.m8.1.1.1.cmml">∈</mo><msup id="S5.Thmtheorem3.p10.8.8.m8.1.1.3" xref="S5.Thmtheorem3.p10.8.8.m8.1.1.3.cmml"><mi id="S5.Thmtheorem3.p10.8.8.m8.1.1.3.2" xref="S5.Thmtheorem3.p10.8.8.m8.1.1.3.2.cmml">G</mi><mn id="S5.Thmtheorem3.p10.8.8.m8.1.1.3.3" xref="S5.Thmtheorem3.p10.8.8.m8.1.1.3.3.cmml">2</mn></msup></mrow><annotation-xml encoding="MathML-Content" id="S5.Thmtheorem3.p10.8.8.m8.1b"><apply id="S5.Thmtheorem3.p10.8.8.m8.1.1.cmml" xref="S5.Thmtheorem3.p10.8.8.m8.1.1"><in id="S5.Thmtheorem3.p10.8.8.m8.1.1.1.cmml" xref="S5.Thmtheorem3.p10.8.8.m8.1.1.1"></in><ci id="S5.Thmtheorem3.p10.8.8.m8.1.1.2.cmml" xref="S5.Thmtheorem3.p10.8.8.m8.1.1.2">𝑎</ci><apply id="S5.Thmtheorem3.p10.8.8.m8.1.1.3.cmml" xref="S5.Thmtheorem3.p10.8.8.m8.1.1.3"><csymbol cd="ambiguous" id="S5.Thmtheorem3.p10.8.8.m8.1.1.3.1.cmml" xref="S5.Thmtheorem3.p10.8.8.m8.1.1.3">superscript</csymbol><ci id="S5.Thmtheorem3.p10.8.8.m8.1.1.3.2.cmml" xref="S5.Thmtheorem3.p10.8.8.m8.1.1.3.2">𝐺</ci><cn id="S5.Thmtheorem3.p10.8.8.m8.1.1.3.3.cmml" type="integer" xref="S5.Thmtheorem3.p10.8.8.m8.1.1.3.3">2</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmtheorem3.p10.8.8.m8.1c">a\in G^{2}</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem3.p10.8.8.m8.1d">italic_a ∈ italic_G start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT</annotation></semantics></math>, then per definition <math alttext="\overline{va}\in E_{\uparrow}" class="ltx_Math" display="inline" id="S5.Thmtheorem3.p10.9.9.m9.1"><semantics id="S5.Thmtheorem3.p10.9.9.m9.1a"><mrow id="S5.Thmtheorem3.p10.9.9.m9.1.1" xref="S5.Thmtheorem3.p10.9.9.m9.1.1.cmml"><mover accent="true" id="S5.Thmtheorem3.p10.9.9.m9.1.1.2" xref="S5.Thmtheorem3.p10.9.9.m9.1.1.2.cmml"><mrow id="S5.Thmtheorem3.p10.9.9.m9.1.1.2.2" xref="S5.Thmtheorem3.p10.9.9.m9.1.1.2.2.cmml"><mi id="S5.Thmtheorem3.p10.9.9.m9.1.1.2.2.2" xref="S5.Thmtheorem3.p10.9.9.m9.1.1.2.2.2.cmml">v</mi><mo id="S5.Thmtheorem3.p10.9.9.m9.1.1.2.2.1" xref="S5.Thmtheorem3.p10.9.9.m9.1.1.2.2.1.cmml">⁢</mo><mi id="S5.Thmtheorem3.p10.9.9.m9.1.1.2.2.3" xref="S5.Thmtheorem3.p10.9.9.m9.1.1.2.2.3.cmml">a</mi></mrow><mo id="S5.Thmtheorem3.p10.9.9.m9.1.1.2.1" xref="S5.Thmtheorem3.p10.9.9.m9.1.1.2.1.cmml">¯</mo></mover><mo id="S5.Thmtheorem3.p10.9.9.m9.1.1.1" xref="S5.Thmtheorem3.p10.9.9.m9.1.1.1.cmml">∈</mo><msub id="S5.Thmtheorem3.p10.9.9.m9.1.1.3" xref="S5.Thmtheorem3.p10.9.9.m9.1.1.3.cmml"><mi id="S5.Thmtheorem3.p10.9.9.m9.1.1.3.2" xref="S5.Thmtheorem3.p10.9.9.m9.1.1.3.2.cmml">E</mi><mo id="S5.Thmtheorem3.p10.9.9.m9.1.1.3.3" stretchy="false" xref="S5.Thmtheorem3.p10.9.9.m9.1.1.3.3.cmml">↑</mo></msub></mrow><annotation-xml encoding="MathML-Content" id="S5.Thmtheorem3.p10.9.9.m9.1b"><apply id="S5.Thmtheorem3.p10.9.9.m9.1.1.cmml" xref="S5.Thmtheorem3.p10.9.9.m9.1.1"><in id="S5.Thmtheorem3.p10.9.9.m9.1.1.1.cmml" xref="S5.Thmtheorem3.p10.9.9.m9.1.1.1"></in><apply id="S5.Thmtheorem3.p10.9.9.m9.1.1.2.cmml" xref="S5.Thmtheorem3.p10.9.9.m9.1.1.2"><ci id="S5.Thmtheorem3.p10.9.9.m9.1.1.2.1.cmml" xref="S5.Thmtheorem3.p10.9.9.m9.1.1.2.1">¯</ci><apply id="S5.Thmtheorem3.p10.9.9.m9.1.1.2.2.cmml" xref="S5.Thmtheorem3.p10.9.9.m9.1.1.2.2"><times id="S5.Thmtheorem3.p10.9.9.m9.1.1.2.2.1.cmml" xref="S5.Thmtheorem3.p10.9.9.m9.1.1.2.2.1"></times><ci id="S5.Thmtheorem3.p10.9.9.m9.1.1.2.2.2.cmml" xref="S5.Thmtheorem3.p10.9.9.m9.1.1.2.2.2">𝑣</ci><ci id="S5.Thmtheorem3.p10.9.9.m9.1.1.2.2.3.cmml" xref="S5.Thmtheorem3.p10.9.9.m9.1.1.2.2.3">𝑎</ci></apply></apply><apply id="S5.Thmtheorem3.p10.9.9.m9.1.1.3.cmml" xref="S5.Thmtheorem3.p10.9.9.m9.1.1.3"><csymbol cd="ambiguous" id="S5.Thmtheorem3.p10.9.9.m9.1.1.3.1.cmml" xref="S5.Thmtheorem3.p10.9.9.m9.1.1.3">subscript</csymbol><ci id="S5.Thmtheorem3.p10.9.9.m9.1.1.3.2.cmml" xref="S5.Thmtheorem3.p10.9.9.m9.1.1.3.2">𝐸</ci><ci id="S5.Thmtheorem3.p10.9.9.m9.1.1.3.3.cmml" xref="S5.Thmtheorem3.p10.9.9.m9.1.1.3.3">↑</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmtheorem3.p10.9.9.m9.1c">\overline{va}\in E_{\uparrow}</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem3.p10.9.9.m9.1d">over¯ start_ARG italic_v italic_a end_ARG ∈ italic_E start_POSTSUBSCRIPT ↑ end_POSTSUBSCRIPT</annotation></semantics></math>. Per definition of a fractional orientation <math alttext="\textsl{g}(v\!\to\!a)\leq\textsl{g}(\overline{va})" class="ltx_Math" display="inline" id="S5.Thmtheorem3.p10.10.10.m10.2"><semantics id="S5.Thmtheorem3.p10.10.10.m10.2a"><mrow id="S5.Thmtheorem3.p10.10.10.m10.2.2" xref="S5.Thmtheorem3.p10.10.10.m10.2.2.cmml"><mrow id="S5.Thmtheorem3.p10.10.10.m10.2.2.1" xref="S5.Thmtheorem3.p10.10.10.m10.2.2.1.cmml"><mtext class="ltx_mathvariant_italic" id="S5.Thmtheorem3.p10.10.10.m10.2.2.1.3" xref="S5.Thmtheorem3.p10.10.10.m10.2.2.1.3a.cmml">g</mtext><mo id="S5.Thmtheorem3.p10.10.10.m10.2.2.1.2" xref="S5.Thmtheorem3.p10.10.10.m10.2.2.1.2.cmml">⁢</mo><mrow id="S5.Thmtheorem3.p10.10.10.m10.2.2.1.1.1" xref="S5.Thmtheorem3.p10.10.10.m10.2.2.1.1.1.1.cmml"><mo id="S5.Thmtheorem3.p10.10.10.m10.2.2.1.1.1.2" stretchy="false" xref="S5.Thmtheorem3.p10.10.10.m10.2.2.1.1.1.1.cmml">(</mo><mrow id="S5.Thmtheorem3.p10.10.10.m10.2.2.1.1.1.1" xref="S5.Thmtheorem3.p10.10.10.m10.2.2.1.1.1.1.cmml"><mi id="S5.Thmtheorem3.p10.10.10.m10.2.2.1.1.1.1.2" xref="S5.Thmtheorem3.p10.10.10.m10.2.2.1.1.1.1.2.cmml">v</mi><mo id="S5.Thmtheorem3.p10.10.10.m10.2.2.1.1.1.1.1" lspace="0.108em" rspace="0.108em" stretchy="false" xref="S5.Thmtheorem3.p10.10.10.m10.2.2.1.1.1.1.1.cmml">→</mo><mi id="S5.Thmtheorem3.p10.10.10.m10.2.2.1.1.1.1.3" xref="S5.Thmtheorem3.p10.10.10.m10.2.2.1.1.1.1.3.cmml">a</mi></mrow><mo id="S5.Thmtheorem3.p10.10.10.m10.2.2.1.1.1.3" stretchy="false" xref="S5.Thmtheorem3.p10.10.10.m10.2.2.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S5.Thmtheorem3.p10.10.10.m10.2.2.2" xref="S5.Thmtheorem3.p10.10.10.m10.2.2.2.cmml">≤</mo><mrow id="S5.Thmtheorem3.p10.10.10.m10.2.2.3" xref="S5.Thmtheorem3.p10.10.10.m10.2.2.3.cmml"><mtext class="ltx_mathvariant_italic" id="S5.Thmtheorem3.p10.10.10.m10.2.2.3.2" xref="S5.Thmtheorem3.p10.10.10.m10.2.2.3.2a.cmml">g</mtext><mo id="S5.Thmtheorem3.p10.10.10.m10.2.2.3.1" xref="S5.Thmtheorem3.p10.10.10.m10.2.2.3.1.cmml">⁢</mo><mrow id="S5.Thmtheorem3.p10.10.10.m10.2.2.3.3.2" xref="S5.Thmtheorem3.p10.10.10.m10.1.1.cmml"><mo id="S5.Thmtheorem3.p10.10.10.m10.2.2.3.3.2.1" stretchy="false" xref="S5.Thmtheorem3.p10.10.10.m10.1.1.cmml">(</mo><mover accent="true" id="S5.Thmtheorem3.p10.10.10.m10.1.1" xref="S5.Thmtheorem3.p10.10.10.m10.1.1.cmml"><mrow id="S5.Thmtheorem3.p10.10.10.m10.1.1.2" xref="S5.Thmtheorem3.p10.10.10.m10.1.1.2.cmml"><mi id="S5.Thmtheorem3.p10.10.10.m10.1.1.2.2" xref="S5.Thmtheorem3.p10.10.10.m10.1.1.2.2.cmml">v</mi><mo id="S5.Thmtheorem3.p10.10.10.m10.1.1.2.1" xref="S5.Thmtheorem3.p10.10.10.m10.1.1.2.1.cmml">⁢</mo><mi id="S5.Thmtheorem3.p10.10.10.m10.1.1.2.3" xref="S5.Thmtheorem3.p10.10.10.m10.1.1.2.3.cmml">a</mi></mrow><mo id="S5.Thmtheorem3.p10.10.10.m10.1.1.1" xref="S5.Thmtheorem3.p10.10.10.m10.1.1.1.cmml">¯</mo></mover><mo id="S5.Thmtheorem3.p10.10.10.m10.2.2.3.3.2.2" stretchy="false" xref="S5.Thmtheorem3.p10.10.10.m10.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.Thmtheorem3.p10.10.10.m10.2b"><apply id="S5.Thmtheorem3.p10.10.10.m10.2.2.cmml" xref="S5.Thmtheorem3.p10.10.10.m10.2.2"><leq id="S5.Thmtheorem3.p10.10.10.m10.2.2.2.cmml" xref="S5.Thmtheorem3.p10.10.10.m10.2.2.2"></leq><apply id="S5.Thmtheorem3.p10.10.10.m10.2.2.1.cmml" xref="S5.Thmtheorem3.p10.10.10.m10.2.2.1"><times id="S5.Thmtheorem3.p10.10.10.m10.2.2.1.2.cmml" xref="S5.Thmtheorem3.p10.10.10.m10.2.2.1.2"></times><ci id="S5.Thmtheorem3.p10.10.10.m10.2.2.1.3a.cmml" xref="S5.Thmtheorem3.p10.10.10.m10.2.2.1.3"><mtext class="ltx_mathvariant_italic" id="S5.Thmtheorem3.p10.10.10.m10.2.2.1.3.cmml" xref="S5.Thmtheorem3.p10.10.10.m10.2.2.1.3">g</mtext></ci><apply id="S5.Thmtheorem3.p10.10.10.m10.2.2.1.1.1.1.cmml" xref="S5.Thmtheorem3.p10.10.10.m10.2.2.1.1.1"><ci id="S5.Thmtheorem3.p10.10.10.m10.2.2.1.1.1.1.1.cmml" xref="S5.Thmtheorem3.p10.10.10.m10.2.2.1.1.1.1.1">→</ci><ci id="S5.Thmtheorem3.p10.10.10.m10.2.2.1.1.1.1.2.cmml" xref="S5.Thmtheorem3.p10.10.10.m10.2.2.1.1.1.1.2">𝑣</ci><ci id="S5.Thmtheorem3.p10.10.10.m10.2.2.1.1.1.1.3.cmml" xref="S5.Thmtheorem3.p10.10.10.m10.2.2.1.1.1.1.3">𝑎</ci></apply></apply><apply id="S5.Thmtheorem3.p10.10.10.m10.2.2.3.cmml" xref="S5.Thmtheorem3.p10.10.10.m10.2.2.3"><times id="S5.Thmtheorem3.p10.10.10.m10.2.2.3.1.cmml" xref="S5.Thmtheorem3.p10.10.10.m10.2.2.3.1"></times><ci id="S5.Thmtheorem3.p10.10.10.m10.2.2.3.2a.cmml" xref="S5.Thmtheorem3.p10.10.10.m10.2.2.3.2"><mtext class="ltx_mathvariant_italic" id="S5.Thmtheorem3.p10.10.10.m10.2.2.3.2.cmml" xref="S5.Thmtheorem3.p10.10.10.m10.2.2.3.2">g</mtext></ci><apply id="S5.Thmtheorem3.p10.10.10.m10.1.1.cmml" xref="S5.Thmtheorem3.p10.10.10.m10.2.2.3.3.2"><ci id="S5.Thmtheorem3.p10.10.10.m10.1.1.1.cmml" xref="S5.Thmtheorem3.p10.10.10.m10.1.1.1">¯</ci><apply id="S5.Thmtheorem3.p10.10.10.m10.1.1.2.cmml" xref="S5.Thmtheorem3.p10.10.10.m10.1.1.2"><times id="S5.Thmtheorem3.p10.10.10.m10.1.1.2.1.cmml" xref="S5.Thmtheorem3.p10.10.10.m10.1.1.2.1"></times><ci id="S5.Thmtheorem3.p10.10.10.m10.1.1.2.2.cmml" xref="S5.Thmtheorem3.p10.10.10.m10.1.1.2.2">𝑣</ci><ci id="S5.Thmtheorem3.p10.10.10.m10.1.1.2.3.cmml" xref="S5.Thmtheorem3.p10.10.10.m10.1.1.2.3">𝑎</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmtheorem3.p10.10.10.m10.2c">\textsl{g}(v\!\to\!a)\leq\textsl{g}(\overline{va})</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem3.p10.10.10.m10.2d">g ( italic_v → italic_a ) ≤ g ( over¯ start_ARG italic_v italic_a end_ARG )</annotation></semantics></math> and so the claim follows.</span></p> </div> <div class="ltx_para ltx_noindent" id="S5.Thmtheorem3.p11"> <p class="ltx_p" id="S5.Thmtheorem3.p11.2"><em class="ltx_emph" id="S5.Thmtheorem3.p11.2.1">A contradiction.</em><span class="ltx_text ltx_font_italic" id="S5.Thmtheorem3.p11.2.2"> Equation </span><a class="ltx_ref ltx_font_italic" href="https://arxiv.org/html/2411.12694v2#S5.E3" title="Equation 3 ‣ Proof 5.3. ‣ 5 Results for dynamic algorithms ‣ Local Density and its Distributed Approximation"><span class="ltx_text ltx_ref_tag">3</span></a><span class="ltx_text ltx_font_italic" id="S5.Thmtheorem3.p11.2.3">, </span><a class="ltx_ref ltx_font_italic" href="https://arxiv.org/html/2411.12694v2#S5.E4" title="Equation 4 ‣ Proof 5.3. ‣ 5 Results for dynamic algorithms ‣ Local Density and its Distributed Approximation"><span class="ltx_text ltx_ref_tag">4</span></a><span class="ltx_text ltx_font_italic" id="S5.Thmtheorem3.p11.2.4"> and </span><a class="ltx_ref ltx_font_italic" href="https://arxiv.org/html/2411.12694v2#S5.E5" title="Equation 5 ‣ Proof 5.3. ‣ 5 Results for dynamic algorithms ‣ Local Density and its Distributed Approximation"><span class="ltx_text ltx_ref_tag">5</span></a><span class="ltx_text ltx_font_italic" id="S5.Thmtheorem3.p11.2.5"> contradict each other. Thus, we have proven that for all vertices </span><math alttext="v" class="ltx_Math" display="inline" id="S5.Thmtheorem3.p11.1.m1.1"><semantics id="S5.Thmtheorem3.p11.1.m1.1a"><mi id="S5.Thmtheorem3.p11.1.m1.1.1" xref="S5.Thmtheorem3.p11.1.m1.1.1.cmml">v</mi><annotation-xml encoding="MathML-Content" id="S5.Thmtheorem3.p11.1.m1.1b"><ci id="S5.Thmtheorem3.p11.1.m1.1.1.cmml" xref="S5.Thmtheorem3.p11.1.m1.1.1">𝑣</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmtheorem3.p11.1.m1.1c">v</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem3.p11.1.m1.1d">italic_v</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S5.Thmtheorem3.p11.2.6">, </span><math alttext="\textsl{g}(v)\leq(1+\varepsilon)\rho^{*}(v)" class="ltx_Math" display="inline" id="S5.Thmtheorem3.p11.2.m2.3"><semantics id="S5.Thmtheorem3.p11.2.m2.3a"><mrow id="S5.Thmtheorem3.p11.2.m2.3.3" xref="S5.Thmtheorem3.p11.2.m2.3.3.cmml"><mrow id="S5.Thmtheorem3.p11.2.m2.3.3.3" xref="S5.Thmtheorem3.p11.2.m2.3.3.3.cmml"><mtext class="ltx_mathvariant_italic" id="S5.Thmtheorem3.p11.2.m2.3.3.3.2" xref="S5.Thmtheorem3.p11.2.m2.3.3.3.2a.cmml">g</mtext><mo id="S5.Thmtheorem3.p11.2.m2.3.3.3.1" xref="S5.Thmtheorem3.p11.2.m2.3.3.3.1.cmml">⁢</mo><mrow id="S5.Thmtheorem3.p11.2.m2.3.3.3.3.2" xref="S5.Thmtheorem3.p11.2.m2.3.3.3.cmml"><mo id="S5.Thmtheorem3.p11.2.m2.3.3.3.3.2.1" stretchy="false" xref="S5.Thmtheorem3.p11.2.m2.3.3.3.cmml">(</mo><mi id="S5.Thmtheorem3.p11.2.m2.1.1" xref="S5.Thmtheorem3.p11.2.m2.1.1.cmml">v</mi><mo id="S5.Thmtheorem3.p11.2.m2.3.3.3.3.2.2" stretchy="false" xref="S5.Thmtheorem3.p11.2.m2.3.3.3.cmml">)</mo></mrow></mrow><mo id="S5.Thmtheorem3.p11.2.m2.3.3.2" xref="S5.Thmtheorem3.p11.2.m2.3.3.2.cmml">≤</mo><mrow id="S5.Thmtheorem3.p11.2.m2.3.3.1" xref="S5.Thmtheorem3.p11.2.m2.3.3.1.cmml"><mrow id="S5.Thmtheorem3.p11.2.m2.3.3.1.1.1" xref="S5.Thmtheorem3.p11.2.m2.3.3.1.1.1.1.cmml"><mo id="S5.Thmtheorem3.p11.2.m2.3.3.1.1.1.2" stretchy="false" xref="S5.Thmtheorem3.p11.2.m2.3.3.1.1.1.1.cmml">(</mo><mrow id="S5.Thmtheorem3.p11.2.m2.3.3.1.1.1.1" xref="S5.Thmtheorem3.p11.2.m2.3.3.1.1.1.1.cmml"><mn id="S5.Thmtheorem3.p11.2.m2.3.3.1.1.1.1.2" xref="S5.Thmtheorem3.p11.2.m2.3.3.1.1.1.1.2.cmml">1</mn><mo id="S5.Thmtheorem3.p11.2.m2.3.3.1.1.1.1.1" xref="S5.Thmtheorem3.p11.2.m2.3.3.1.1.1.1.1.cmml">+</mo><mi id="S5.Thmtheorem3.p11.2.m2.3.3.1.1.1.1.3" xref="S5.Thmtheorem3.p11.2.m2.3.3.1.1.1.1.3.cmml">ε</mi></mrow><mo id="S5.Thmtheorem3.p11.2.m2.3.3.1.1.1.3" stretchy="false" xref="S5.Thmtheorem3.p11.2.m2.3.3.1.1.1.1.cmml">)</mo></mrow><mo id="S5.Thmtheorem3.p11.2.m2.3.3.1.2" xref="S5.Thmtheorem3.p11.2.m2.3.3.1.2.cmml">⁢</mo><msup id="S5.Thmtheorem3.p11.2.m2.3.3.1.3" xref="S5.Thmtheorem3.p11.2.m2.3.3.1.3.cmml"><mi id="S5.Thmtheorem3.p11.2.m2.3.3.1.3.2" xref="S5.Thmtheorem3.p11.2.m2.3.3.1.3.2.cmml">ρ</mi><mo id="S5.Thmtheorem3.p11.2.m2.3.3.1.3.3" xref="S5.Thmtheorem3.p11.2.m2.3.3.1.3.3.cmml">∗</mo></msup><mo id="S5.Thmtheorem3.p11.2.m2.3.3.1.2a" xref="S5.Thmtheorem3.p11.2.m2.3.3.1.2.cmml">⁢</mo><mrow id="S5.Thmtheorem3.p11.2.m2.3.3.1.4.2" xref="S5.Thmtheorem3.p11.2.m2.3.3.1.cmml"><mo id="S5.Thmtheorem3.p11.2.m2.3.3.1.4.2.1" stretchy="false" xref="S5.Thmtheorem3.p11.2.m2.3.3.1.cmml">(</mo><mi id="S5.Thmtheorem3.p11.2.m2.2.2" xref="S5.Thmtheorem3.p11.2.m2.2.2.cmml">v</mi><mo id="S5.Thmtheorem3.p11.2.m2.3.3.1.4.2.2" stretchy="false" xref="S5.Thmtheorem3.p11.2.m2.3.3.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.Thmtheorem3.p11.2.m2.3b"><apply id="S5.Thmtheorem3.p11.2.m2.3.3.cmml" xref="S5.Thmtheorem3.p11.2.m2.3.3"><leq id="S5.Thmtheorem3.p11.2.m2.3.3.2.cmml" xref="S5.Thmtheorem3.p11.2.m2.3.3.2"></leq><apply id="S5.Thmtheorem3.p11.2.m2.3.3.3.cmml" xref="S5.Thmtheorem3.p11.2.m2.3.3.3"><times id="S5.Thmtheorem3.p11.2.m2.3.3.3.1.cmml" xref="S5.Thmtheorem3.p11.2.m2.3.3.3.1"></times><ci id="S5.Thmtheorem3.p11.2.m2.3.3.3.2a.cmml" xref="S5.Thmtheorem3.p11.2.m2.3.3.3.2"><mtext class="ltx_mathvariant_italic" id="S5.Thmtheorem3.p11.2.m2.3.3.3.2.cmml" xref="S5.Thmtheorem3.p11.2.m2.3.3.3.2">g</mtext></ci><ci id="S5.Thmtheorem3.p11.2.m2.1.1.cmml" xref="S5.Thmtheorem3.p11.2.m2.1.1">𝑣</ci></apply><apply id="S5.Thmtheorem3.p11.2.m2.3.3.1.cmml" xref="S5.Thmtheorem3.p11.2.m2.3.3.1"><times id="S5.Thmtheorem3.p11.2.m2.3.3.1.2.cmml" xref="S5.Thmtheorem3.p11.2.m2.3.3.1.2"></times><apply id="S5.Thmtheorem3.p11.2.m2.3.3.1.1.1.1.cmml" xref="S5.Thmtheorem3.p11.2.m2.3.3.1.1.1"><plus id="S5.Thmtheorem3.p11.2.m2.3.3.1.1.1.1.1.cmml" xref="S5.Thmtheorem3.p11.2.m2.3.3.1.1.1.1.1"></plus><cn id="S5.Thmtheorem3.p11.2.m2.3.3.1.1.1.1.2.cmml" type="integer" xref="S5.Thmtheorem3.p11.2.m2.3.3.1.1.1.1.2">1</cn><ci id="S5.Thmtheorem3.p11.2.m2.3.3.1.1.1.1.3.cmml" xref="S5.Thmtheorem3.p11.2.m2.3.3.1.1.1.1.3">𝜀</ci></apply><apply id="S5.Thmtheorem3.p11.2.m2.3.3.1.3.cmml" xref="S5.Thmtheorem3.p11.2.m2.3.3.1.3"><csymbol cd="ambiguous" id="S5.Thmtheorem3.p11.2.m2.3.3.1.3.1.cmml" xref="S5.Thmtheorem3.p11.2.m2.3.3.1.3">superscript</csymbol><ci id="S5.Thmtheorem3.p11.2.m2.3.3.1.3.2.cmml" xref="S5.Thmtheorem3.p11.2.m2.3.3.1.3.2">𝜌</ci><times id="S5.Thmtheorem3.p11.2.m2.3.3.1.3.3.cmml" xref="S5.Thmtheorem3.p11.2.m2.3.3.1.3.3"></times></apply><ci id="S5.Thmtheorem3.p11.2.m2.2.2.cmml" xref="S5.Thmtheorem3.p11.2.m2.2.2">𝑣</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmtheorem3.p11.2.m2.3c">\textsl{g}(v)\leq(1+\varepsilon)\rho^{*}(v)</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem3.p11.2.m2.3d">g ( italic_v ) ≤ ( 1 + italic_ε ) italic_ρ start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ( italic_v )</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S5.Thmtheorem3.p11.2.7">.</span></p> </div> <div class="ltx_para" id="S5.Thmtheorem3.p12"> <p class="ltx_p" id="S5.Thmtheorem3.p12.8"><span class="ltx_text ltx_font_bold ltx_font_italic" id="S5.Thmtheorem3.p12.8.8">Next, we show that for all vertices <math alttext="v" class="ltx_Math" display="inline" id="S5.Thmtheorem3.p12.1.1.m1.1"><semantics id="S5.Thmtheorem3.p12.1.1.m1.1a"><mi id="S5.Thmtheorem3.p12.1.1.m1.1.1" mathvariant="normal" xref="S5.Thmtheorem3.p12.1.1.m1.1.1.cmml">v</mi><annotation-xml encoding="MathML-Content" id="S5.Thmtheorem3.p12.1.1.m1.1b"><ci id="S5.Thmtheorem3.p12.1.1.m1.1.1.cmml" xref="S5.Thmtheorem3.p12.1.1.m1.1.1">v</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmtheorem3.p12.1.1.m1.1c">v</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem3.p12.1.1.m1.1d">italic_v</annotation></semantics></math>, <math alttext="\textsl{g}(v)\geq(1+\varepsilon)^{-1}\rho^{*}(v)" class="ltx_Math" display="inline" id="S5.Thmtheorem3.p12.2.2.m2.3"><semantics id="S5.Thmtheorem3.p12.2.2.m2.3a"><mrow id="S5.Thmtheorem3.p12.2.2.m2.3.3" xref="S5.Thmtheorem3.p12.2.2.m2.3.3.cmml"><mrow id="S5.Thmtheorem3.p12.2.2.m2.3.3.3" xref="S5.Thmtheorem3.p12.2.2.m2.3.3.3.cmml"><mtext class="ltx_mathvariant_italic" id="S5.Thmtheorem3.p12.2.2.m2.3.3.3.2" xref="S5.Thmtheorem3.p12.2.2.m2.3.3.3.2a.cmml">g</mtext><mo id="S5.Thmtheorem3.p12.2.2.m2.3.3.3.1" xref="S5.Thmtheorem3.p12.2.2.m2.3.3.3.1.cmml">⁢</mo><mrow id="S5.Thmtheorem3.p12.2.2.m2.3.3.3.3.2" xref="S5.Thmtheorem3.p12.2.2.m2.3.3.3.cmml"><mo id="S5.Thmtheorem3.p12.2.2.m2.3.3.3.3.2.1" stretchy="false" xref="S5.Thmtheorem3.p12.2.2.m2.3.3.3.cmml">(</mo><mi id="S5.Thmtheorem3.p12.2.2.m2.1.1" mathvariant="normal" xref="S5.Thmtheorem3.p12.2.2.m2.1.1.cmml">v</mi><mo id="S5.Thmtheorem3.p12.2.2.m2.3.3.3.3.2.2" stretchy="false" xref="S5.Thmtheorem3.p12.2.2.m2.3.3.3.cmml">)</mo></mrow></mrow><mo id="S5.Thmtheorem3.p12.2.2.m2.3.3.2" xref="S5.Thmtheorem3.p12.2.2.m2.3.3.2.cmml">≥</mo><mrow id="S5.Thmtheorem3.p12.2.2.m2.3.3.1" xref="S5.Thmtheorem3.p12.2.2.m2.3.3.1.cmml"><msup id="S5.Thmtheorem3.p12.2.2.m2.3.3.1.1" xref="S5.Thmtheorem3.p12.2.2.m2.3.3.1.1.cmml"><mrow id="S5.Thmtheorem3.p12.2.2.m2.3.3.1.1.1.1" xref="S5.Thmtheorem3.p12.2.2.m2.3.3.1.1.1.1.1.cmml"><mo id="S5.Thmtheorem3.p12.2.2.m2.3.3.1.1.1.1.2" stretchy="false" xref="S5.Thmtheorem3.p12.2.2.m2.3.3.1.1.1.1.1.cmml">(</mo><mrow id="S5.Thmtheorem3.p12.2.2.m2.3.3.1.1.1.1.1" xref="S5.Thmtheorem3.p12.2.2.m2.3.3.1.1.1.1.1.cmml"><mn id="S5.Thmtheorem3.p12.2.2.m2.3.3.1.1.1.1.1.2" xref="S5.Thmtheorem3.p12.2.2.m2.3.3.1.1.1.1.1.2.cmml">1</mn><mo id="S5.Thmtheorem3.p12.2.2.m2.3.3.1.1.1.1.1.1" xref="S5.Thmtheorem3.p12.2.2.m2.3.3.1.1.1.1.1.1.cmml">+</mo><mi id="S5.Thmtheorem3.p12.2.2.m2.3.3.1.1.1.1.1.3" mathvariant="normal" xref="S5.Thmtheorem3.p12.2.2.m2.3.3.1.1.1.1.1.3.cmml">ε</mi></mrow><mo id="S5.Thmtheorem3.p12.2.2.m2.3.3.1.1.1.1.3" stretchy="false" xref="S5.Thmtheorem3.p12.2.2.m2.3.3.1.1.1.1.1.cmml">)</mo></mrow><mrow id="S5.Thmtheorem3.p12.2.2.m2.3.3.1.1.3" xref="S5.Thmtheorem3.p12.2.2.m2.3.3.1.1.3.cmml"><mo id="S5.Thmtheorem3.p12.2.2.m2.3.3.1.1.3a" xref="S5.Thmtheorem3.p12.2.2.m2.3.3.1.1.3.cmml">−</mo><mn id="S5.Thmtheorem3.p12.2.2.m2.3.3.1.1.3.2" xref="S5.Thmtheorem3.p12.2.2.m2.3.3.1.1.3.2.cmml">1</mn></mrow></msup><mo id="S5.Thmtheorem3.p12.2.2.m2.3.3.1.2" xref="S5.Thmtheorem3.p12.2.2.m2.3.3.1.2.cmml">⁢</mo><msup id="S5.Thmtheorem3.p12.2.2.m2.3.3.1.3" xref="S5.Thmtheorem3.p12.2.2.m2.3.3.1.3.cmml"><mi id="S5.Thmtheorem3.p12.2.2.m2.3.3.1.3.2" mathvariant="normal" xref="S5.Thmtheorem3.p12.2.2.m2.3.3.1.3.2.cmml">ρ</mi><mo id="S5.Thmtheorem3.p12.2.2.m2.3.3.1.3.3" xref="S5.Thmtheorem3.p12.2.2.m2.3.3.1.3.3.cmml">∗</mo></msup><mo id="S5.Thmtheorem3.p12.2.2.m2.3.3.1.2a" xref="S5.Thmtheorem3.p12.2.2.m2.3.3.1.2.cmml">⁢</mo><mrow id="S5.Thmtheorem3.p12.2.2.m2.3.3.1.4.2" xref="S5.Thmtheorem3.p12.2.2.m2.3.3.1.cmml"><mo id="S5.Thmtheorem3.p12.2.2.m2.3.3.1.4.2.1" stretchy="false" xref="S5.Thmtheorem3.p12.2.2.m2.3.3.1.cmml">(</mo><mi id="S5.Thmtheorem3.p12.2.2.m2.2.2" mathvariant="normal" xref="S5.Thmtheorem3.p12.2.2.m2.2.2.cmml">v</mi><mo id="S5.Thmtheorem3.p12.2.2.m2.3.3.1.4.2.2" stretchy="false" xref="S5.Thmtheorem3.p12.2.2.m2.3.3.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.Thmtheorem3.p12.2.2.m2.3b"><apply id="S5.Thmtheorem3.p12.2.2.m2.3.3.cmml" xref="S5.Thmtheorem3.p12.2.2.m2.3.3"><geq id="S5.Thmtheorem3.p12.2.2.m2.3.3.2.cmml" xref="S5.Thmtheorem3.p12.2.2.m2.3.3.2"></geq><apply id="S5.Thmtheorem3.p12.2.2.m2.3.3.3.cmml" xref="S5.Thmtheorem3.p12.2.2.m2.3.3.3"><times id="S5.Thmtheorem3.p12.2.2.m2.3.3.3.1.cmml" xref="S5.Thmtheorem3.p12.2.2.m2.3.3.3.1"></times><ci id="S5.Thmtheorem3.p12.2.2.m2.3.3.3.2a.cmml" xref="S5.Thmtheorem3.p12.2.2.m2.3.3.3.2"><mtext class="ltx_mathvariant_italic" id="S5.Thmtheorem3.p12.2.2.m2.3.3.3.2.cmml" xref="S5.Thmtheorem3.p12.2.2.m2.3.3.3.2">g</mtext></ci><ci id="S5.Thmtheorem3.p12.2.2.m2.1.1.cmml" xref="S5.Thmtheorem3.p12.2.2.m2.1.1">v</ci></apply><apply id="S5.Thmtheorem3.p12.2.2.m2.3.3.1.cmml" xref="S5.Thmtheorem3.p12.2.2.m2.3.3.1"><times id="S5.Thmtheorem3.p12.2.2.m2.3.3.1.2.cmml" xref="S5.Thmtheorem3.p12.2.2.m2.3.3.1.2"></times><apply id="S5.Thmtheorem3.p12.2.2.m2.3.3.1.1.cmml" xref="S5.Thmtheorem3.p12.2.2.m2.3.3.1.1"><csymbol cd="ambiguous" id="S5.Thmtheorem3.p12.2.2.m2.3.3.1.1.2.cmml" xref="S5.Thmtheorem3.p12.2.2.m2.3.3.1.1">superscript</csymbol><apply id="S5.Thmtheorem3.p12.2.2.m2.3.3.1.1.1.1.1.cmml" xref="S5.Thmtheorem3.p12.2.2.m2.3.3.1.1.1.1"><plus id="S5.Thmtheorem3.p12.2.2.m2.3.3.1.1.1.1.1.1.cmml" xref="S5.Thmtheorem3.p12.2.2.m2.3.3.1.1.1.1.1.1"></plus><cn id="S5.Thmtheorem3.p12.2.2.m2.3.3.1.1.1.1.1.2.cmml" type="integer" xref="S5.Thmtheorem3.p12.2.2.m2.3.3.1.1.1.1.1.2">1</cn><ci id="S5.Thmtheorem3.p12.2.2.m2.3.3.1.1.1.1.1.3.cmml" xref="S5.Thmtheorem3.p12.2.2.m2.3.3.1.1.1.1.1.3">ε</ci></apply><apply id="S5.Thmtheorem3.p12.2.2.m2.3.3.1.1.3.cmml" xref="S5.Thmtheorem3.p12.2.2.m2.3.3.1.1.3"><minus id="S5.Thmtheorem3.p12.2.2.m2.3.3.1.1.3.1.cmml" xref="S5.Thmtheorem3.p12.2.2.m2.3.3.1.1.3"></minus><cn id="S5.Thmtheorem3.p12.2.2.m2.3.3.1.1.3.2.cmml" type="integer" xref="S5.Thmtheorem3.p12.2.2.m2.3.3.1.1.3.2">1</cn></apply></apply><apply id="S5.Thmtheorem3.p12.2.2.m2.3.3.1.3.cmml" xref="S5.Thmtheorem3.p12.2.2.m2.3.3.1.3"><csymbol cd="ambiguous" id="S5.Thmtheorem3.p12.2.2.m2.3.3.1.3.1.cmml" xref="S5.Thmtheorem3.p12.2.2.m2.3.3.1.3">superscript</csymbol><ci id="S5.Thmtheorem3.p12.2.2.m2.3.3.1.3.2.cmml" xref="S5.Thmtheorem3.p12.2.2.m2.3.3.1.3.2">ρ</ci><times id="S5.Thmtheorem3.p12.2.2.m2.3.3.1.3.3.cmml" xref="S5.Thmtheorem3.p12.2.2.m2.3.3.1.3.3"></times></apply><ci id="S5.Thmtheorem3.p12.2.2.m2.2.2.cmml" xref="S5.Thmtheorem3.p12.2.2.m2.2.2">v</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmtheorem3.p12.2.2.m2.3c">\textsl{g}(v)\geq(1+\varepsilon)^{-1}\rho^{*}(v)</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem3.p12.2.2.m2.3d">g ( italic_v ) ≥ ( 1 + italic_ε ) start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT italic_ρ start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ( italic_v )</annotation></semantics></math><span class="ltx_text ltx_font_medium" id="S5.Thmtheorem3.p12.8.8.6"> Assume for the sake of contradiction that there exists a vertex <math alttext="u" class="ltx_Math" display="inline" id="S5.Thmtheorem3.p12.3.3.1.m1.1"><semantics id="S5.Thmtheorem3.p12.3.3.1.m1.1a"><mi id="S5.Thmtheorem3.p12.3.3.1.m1.1.1" mathvariant="normal" xref="S5.Thmtheorem3.p12.3.3.1.m1.1.1.cmml">u</mi><annotation-xml encoding="MathML-Content" id="S5.Thmtheorem3.p12.3.3.1.m1.1b"><ci id="S5.Thmtheorem3.p12.3.3.1.m1.1.1.cmml" xref="S5.Thmtheorem3.p12.3.3.1.m1.1.1">u</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmtheorem3.p12.3.3.1.m1.1c">u</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem3.p12.3.3.1.m1.1d">italic_u</annotation></semantics></math> with <math alttext="\textsl{g}(u)<(1+\varepsilon)^{-1}\rho^{*}(u)" class="ltx_Math" display="inline" id="S5.Thmtheorem3.p12.4.4.2.m2.3"><semantics id="S5.Thmtheorem3.p12.4.4.2.m2.3a"><mrow id="S5.Thmtheorem3.p12.4.4.2.m2.3.3" xref="S5.Thmtheorem3.p12.4.4.2.m2.3.3.cmml"><mrow id="S5.Thmtheorem3.p12.4.4.2.m2.3.3.3" xref="S5.Thmtheorem3.p12.4.4.2.m2.3.3.3.cmml"><mtext class="ltx_mathvariant_italic" id="S5.Thmtheorem3.p12.4.4.2.m2.3.3.3.2" xref="S5.Thmtheorem3.p12.4.4.2.m2.3.3.3.2a.cmml">g</mtext><mo id="S5.Thmtheorem3.p12.4.4.2.m2.3.3.3.1" xref="S5.Thmtheorem3.p12.4.4.2.m2.3.3.3.1.cmml">⁢</mo><mrow id="S5.Thmtheorem3.p12.4.4.2.m2.3.3.3.3.2" xref="S5.Thmtheorem3.p12.4.4.2.m2.3.3.3.cmml"><mo id="S5.Thmtheorem3.p12.4.4.2.m2.3.3.3.3.2.1" stretchy="false" xref="S5.Thmtheorem3.p12.4.4.2.m2.3.3.3.cmml">(</mo><mi id="S5.Thmtheorem3.p12.4.4.2.m2.1.1" mathvariant="normal" xref="S5.Thmtheorem3.p12.4.4.2.m2.1.1.cmml">u</mi><mo id="S5.Thmtheorem3.p12.4.4.2.m2.3.3.3.3.2.2" stretchy="false" xref="S5.Thmtheorem3.p12.4.4.2.m2.3.3.3.cmml">)</mo></mrow></mrow><mo id="S5.Thmtheorem3.p12.4.4.2.m2.3.3.2" xref="S5.Thmtheorem3.p12.4.4.2.m2.3.3.2.cmml"><</mo><mrow id="S5.Thmtheorem3.p12.4.4.2.m2.3.3.1" xref="S5.Thmtheorem3.p12.4.4.2.m2.3.3.1.cmml"><msup id="S5.Thmtheorem3.p12.4.4.2.m2.3.3.1.1" xref="S5.Thmtheorem3.p12.4.4.2.m2.3.3.1.1.cmml"><mrow id="S5.Thmtheorem3.p12.4.4.2.m2.3.3.1.1.1.1" xref="S5.Thmtheorem3.p12.4.4.2.m2.3.3.1.1.1.1.1.cmml"><mo id="S5.Thmtheorem3.p12.4.4.2.m2.3.3.1.1.1.1.2" stretchy="false" xref="S5.Thmtheorem3.p12.4.4.2.m2.3.3.1.1.1.1.1.cmml">(</mo><mrow id="S5.Thmtheorem3.p12.4.4.2.m2.3.3.1.1.1.1.1" xref="S5.Thmtheorem3.p12.4.4.2.m2.3.3.1.1.1.1.1.cmml"><mn id="S5.Thmtheorem3.p12.4.4.2.m2.3.3.1.1.1.1.1.2" xref="S5.Thmtheorem3.p12.4.4.2.m2.3.3.1.1.1.1.1.2.cmml">1</mn><mo id="S5.Thmtheorem3.p12.4.4.2.m2.3.3.1.1.1.1.1.1" xref="S5.Thmtheorem3.p12.4.4.2.m2.3.3.1.1.1.1.1.1.cmml">+</mo><mi id="S5.Thmtheorem3.p12.4.4.2.m2.3.3.1.1.1.1.1.3" mathvariant="normal" xref="S5.Thmtheorem3.p12.4.4.2.m2.3.3.1.1.1.1.1.3.cmml">ε</mi></mrow><mo id="S5.Thmtheorem3.p12.4.4.2.m2.3.3.1.1.1.1.3" stretchy="false" xref="S5.Thmtheorem3.p12.4.4.2.m2.3.3.1.1.1.1.1.cmml">)</mo></mrow><mrow id="S5.Thmtheorem3.p12.4.4.2.m2.3.3.1.1.3" xref="S5.Thmtheorem3.p12.4.4.2.m2.3.3.1.1.3.cmml"><mo id="S5.Thmtheorem3.p12.4.4.2.m2.3.3.1.1.3a" xref="S5.Thmtheorem3.p12.4.4.2.m2.3.3.1.1.3.cmml">−</mo><mn id="S5.Thmtheorem3.p12.4.4.2.m2.3.3.1.1.3.2" xref="S5.Thmtheorem3.p12.4.4.2.m2.3.3.1.1.3.2.cmml">1</mn></mrow></msup><mo id="S5.Thmtheorem3.p12.4.4.2.m2.3.3.1.2" xref="S5.Thmtheorem3.p12.4.4.2.m2.3.3.1.2.cmml">⁢</mo><msup id="S5.Thmtheorem3.p12.4.4.2.m2.3.3.1.3" xref="S5.Thmtheorem3.p12.4.4.2.m2.3.3.1.3.cmml"><mi id="S5.Thmtheorem3.p12.4.4.2.m2.3.3.1.3.2" mathvariant="normal" xref="S5.Thmtheorem3.p12.4.4.2.m2.3.3.1.3.2.cmml">ρ</mi><mo id="S5.Thmtheorem3.p12.4.4.2.m2.3.3.1.3.3" xref="S5.Thmtheorem3.p12.4.4.2.m2.3.3.1.3.3.cmml">∗</mo></msup><mo id="S5.Thmtheorem3.p12.4.4.2.m2.3.3.1.2a" xref="S5.Thmtheorem3.p12.4.4.2.m2.3.3.1.2.cmml">⁢</mo><mrow id="S5.Thmtheorem3.p12.4.4.2.m2.3.3.1.4.2" xref="S5.Thmtheorem3.p12.4.4.2.m2.3.3.1.cmml"><mo id="S5.Thmtheorem3.p12.4.4.2.m2.3.3.1.4.2.1" stretchy="false" xref="S5.Thmtheorem3.p12.4.4.2.m2.3.3.1.cmml">(</mo><mi id="S5.Thmtheorem3.p12.4.4.2.m2.2.2" mathvariant="normal" xref="S5.Thmtheorem3.p12.4.4.2.m2.2.2.cmml">u</mi><mo id="S5.Thmtheorem3.p12.4.4.2.m2.3.3.1.4.2.2" stretchy="false" xref="S5.Thmtheorem3.p12.4.4.2.m2.3.3.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.Thmtheorem3.p12.4.4.2.m2.3b"><apply id="S5.Thmtheorem3.p12.4.4.2.m2.3.3.cmml" xref="S5.Thmtheorem3.p12.4.4.2.m2.3.3"><lt id="S5.Thmtheorem3.p12.4.4.2.m2.3.3.2.cmml" xref="S5.Thmtheorem3.p12.4.4.2.m2.3.3.2"></lt><apply id="S5.Thmtheorem3.p12.4.4.2.m2.3.3.3.cmml" xref="S5.Thmtheorem3.p12.4.4.2.m2.3.3.3"><times id="S5.Thmtheorem3.p12.4.4.2.m2.3.3.3.1.cmml" xref="S5.Thmtheorem3.p12.4.4.2.m2.3.3.3.1"></times><ci id="S5.Thmtheorem3.p12.4.4.2.m2.3.3.3.2a.cmml" xref="S5.Thmtheorem3.p12.4.4.2.m2.3.3.3.2"><mtext class="ltx_mathvariant_italic" id="S5.Thmtheorem3.p12.4.4.2.m2.3.3.3.2.cmml" xref="S5.Thmtheorem3.p12.4.4.2.m2.3.3.3.2">g</mtext></ci><ci id="S5.Thmtheorem3.p12.4.4.2.m2.1.1.cmml" xref="S5.Thmtheorem3.p12.4.4.2.m2.1.1">u</ci></apply><apply id="S5.Thmtheorem3.p12.4.4.2.m2.3.3.1.cmml" xref="S5.Thmtheorem3.p12.4.4.2.m2.3.3.1"><times id="S5.Thmtheorem3.p12.4.4.2.m2.3.3.1.2.cmml" xref="S5.Thmtheorem3.p12.4.4.2.m2.3.3.1.2"></times><apply id="S5.Thmtheorem3.p12.4.4.2.m2.3.3.1.1.cmml" xref="S5.Thmtheorem3.p12.4.4.2.m2.3.3.1.1"><csymbol cd="ambiguous" id="S5.Thmtheorem3.p12.4.4.2.m2.3.3.1.1.2.cmml" xref="S5.Thmtheorem3.p12.4.4.2.m2.3.3.1.1">superscript</csymbol><apply id="S5.Thmtheorem3.p12.4.4.2.m2.3.3.1.1.1.1.1.cmml" xref="S5.Thmtheorem3.p12.4.4.2.m2.3.3.1.1.1.1"><plus id="S5.Thmtheorem3.p12.4.4.2.m2.3.3.1.1.1.1.1.1.cmml" xref="S5.Thmtheorem3.p12.4.4.2.m2.3.3.1.1.1.1.1.1"></plus><cn id="S5.Thmtheorem3.p12.4.4.2.m2.3.3.1.1.1.1.1.2.cmml" type="integer" xref="S5.Thmtheorem3.p12.4.4.2.m2.3.3.1.1.1.1.1.2">1</cn><ci id="S5.Thmtheorem3.p12.4.4.2.m2.3.3.1.1.1.1.1.3.cmml" xref="S5.Thmtheorem3.p12.4.4.2.m2.3.3.1.1.1.1.1.3">ε</ci></apply><apply id="S5.Thmtheorem3.p12.4.4.2.m2.3.3.1.1.3.cmml" xref="S5.Thmtheorem3.p12.4.4.2.m2.3.3.1.1.3"><minus id="S5.Thmtheorem3.p12.4.4.2.m2.3.3.1.1.3.1.cmml" xref="S5.Thmtheorem3.p12.4.4.2.m2.3.3.1.1.3"></minus><cn id="S5.Thmtheorem3.p12.4.4.2.m2.3.3.1.1.3.2.cmml" type="integer" xref="S5.Thmtheorem3.p12.4.4.2.m2.3.3.1.1.3.2">1</cn></apply></apply><apply id="S5.Thmtheorem3.p12.4.4.2.m2.3.3.1.3.cmml" xref="S5.Thmtheorem3.p12.4.4.2.m2.3.3.1.3"><csymbol cd="ambiguous" id="S5.Thmtheorem3.p12.4.4.2.m2.3.3.1.3.1.cmml" xref="S5.Thmtheorem3.p12.4.4.2.m2.3.3.1.3">superscript</csymbol><ci id="S5.Thmtheorem3.p12.4.4.2.m2.3.3.1.3.2.cmml" xref="S5.Thmtheorem3.p12.4.4.2.m2.3.3.1.3.2">ρ</ci><times id="S5.Thmtheorem3.p12.4.4.2.m2.3.3.1.3.3.cmml" xref="S5.Thmtheorem3.p12.4.4.2.m2.3.3.1.3.3"></times></apply><ci id="S5.Thmtheorem3.p12.4.4.2.m2.2.2.cmml" xref="S5.Thmtheorem3.p12.4.4.2.m2.2.2">u</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmtheorem3.p12.4.4.2.m2.3c">\textsl{g}(u)<(1+\varepsilon)^{-1}\rho^{*}(u)</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem3.p12.4.4.2.m2.3d">g ( italic_u ) < ( 1 + italic_ε ) start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT italic_ρ start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ( italic_u )</annotation></semantics></math>. We use the local density (Definition <a class="ltx_ref" href="https://arxiv.org/html/2411.12694v2#S2.Thmtheorem9" title="Definition 2.9 (Definition 2.3 in [10]). ‣ 2.3 Local density ‣ 2 Preliminaries and related work ‣ Local Density and its Distributed Approximation"><span class="ltx_text ltx_ref_tag">2.9</span></a>+<a class="ltx_ref" href="https://arxiv.org/html/2411.12694v2#S2.Thmtheorem10" title="Definition 2.10 (Definition 2.3 in [10]). ‣ 2.3 Local density ‣ 2 Preliminaries and related work ‣ Local Density and its Distributed Approximation"><span class="ltx_text ltx_ref_tag">2.10</span></a>). I.e., there exists a unique integer <math alttext="i" class="ltx_Math" display="inline" id="S5.Thmtheorem3.p12.5.5.3.m3.1"><semantics id="S5.Thmtheorem3.p12.5.5.3.m3.1a"><mi id="S5.Thmtheorem3.p12.5.5.3.m3.1.1" mathvariant="normal" xref="S5.Thmtheorem3.p12.5.5.3.m3.1.1.cmml">i</mi><annotation-xml encoding="MathML-Content" id="S5.Thmtheorem3.p12.5.5.3.m3.1b"><ci id="S5.Thmtheorem3.p12.5.5.3.m3.1.1.cmml" xref="S5.Thmtheorem3.p12.5.5.3.m3.1.1">i</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmtheorem3.p12.5.5.3.m3.1c">i</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem3.p12.5.5.3.m3.1d">italic_i</annotation></semantics></math> and two sets of vertices <math alttext="(S_{i},B_{i-1})" class="ltx_Math" display="inline" id="S5.Thmtheorem3.p12.6.6.4.m4.2"><semantics id="S5.Thmtheorem3.p12.6.6.4.m4.2a"><mrow id="S5.Thmtheorem3.p12.6.6.4.m4.2.2.2" xref="S5.Thmtheorem3.p12.6.6.4.m4.2.2.3.cmml"><mo id="S5.Thmtheorem3.p12.6.6.4.m4.2.2.2.3" stretchy="false" xref="S5.Thmtheorem3.p12.6.6.4.m4.2.2.3.cmml">(</mo><msub id="S5.Thmtheorem3.p12.6.6.4.m4.1.1.1.1" xref="S5.Thmtheorem3.p12.6.6.4.m4.1.1.1.1.cmml"><mi id="S5.Thmtheorem3.p12.6.6.4.m4.1.1.1.1.2" mathvariant="normal" xref="S5.Thmtheorem3.p12.6.6.4.m4.1.1.1.1.2.cmml">S</mi><mi id="S5.Thmtheorem3.p12.6.6.4.m4.1.1.1.1.3" mathvariant="normal" xref="S5.Thmtheorem3.p12.6.6.4.m4.1.1.1.1.3.cmml">i</mi></msub><mo id="S5.Thmtheorem3.p12.6.6.4.m4.2.2.2.4" xref="S5.Thmtheorem3.p12.6.6.4.m4.2.2.3.cmml">,</mo><msub id="S5.Thmtheorem3.p12.6.6.4.m4.2.2.2.2" xref="S5.Thmtheorem3.p12.6.6.4.m4.2.2.2.2.cmml"><mi id="S5.Thmtheorem3.p12.6.6.4.m4.2.2.2.2.2" mathvariant="normal" xref="S5.Thmtheorem3.p12.6.6.4.m4.2.2.2.2.2.cmml">B</mi><mrow id="S5.Thmtheorem3.p12.6.6.4.m4.2.2.2.2.3" xref="S5.Thmtheorem3.p12.6.6.4.m4.2.2.2.2.3.cmml"><mi id="S5.Thmtheorem3.p12.6.6.4.m4.2.2.2.2.3.2" mathvariant="normal" xref="S5.Thmtheorem3.p12.6.6.4.m4.2.2.2.2.3.2.cmml">i</mi><mo id="S5.Thmtheorem3.p12.6.6.4.m4.2.2.2.2.3.1" xref="S5.Thmtheorem3.p12.6.6.4.m4.2.2.2.2.3.1.cmml">−</mo><mn id="S5.Thmtheorem3.p12.6.6.4.m4.2.2.2.2.3.3" xref="S5.Thmtheorem3.p12.6.6.4.m4.2.2.2.2.3.3.cmml">1</mn></mrow></msub><mo id="S5.Thmtheorem3.p12.6.6.4.m4.2.2.2.5" stretchy="false" xref="S5.Thmtheorem3.p12.6.6.4.m4.2.2.3.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S5.Thmtheorem3.p12.6.6.4.m4.2b"><interval closure="open" id="S5.Thmtheorem3.p12.6.6.4.m4.2.2.3.cmml" xref="S5.Thmtheorem3.p12.6.6.4.m4.2.2.2"><apply id="S5.Thmtheorem3.p12.6.6.4.m4.1.1.1.1.cmml" xref="S5.Thmtheorem3.p12.6.6.4.m4.1.1.1.1"><csymbol cd="ambiguous" id="S5.Thmtheorem3.p12.6.6.4.m4.1.1.1.1.1.cmml" xref="S5.Thmtheorem3.p12.6.6.4.m4.1.1.1.1">subscript</csymbol><ci id="S5.Thmtheorem3.p12.6.6.4.m4.1.1.1.1.2.cmml" xref="S5.Thmtheorem3.p12.6.6.4.m4.1.1.1.1.2">S</ci><ci id="S5.Thmtheorem3.p12.6.6.4.m4.1.1.1.1.3.cmml" xref="S5.Thmtheorem3.p12.6.6.4.m4.1.1.1.1.3">i</ci></apply><apply id="S5.Thmtheorem3.p12.6.6.4.m4.2.2.2.2.cmml" xref="S5.Thmtheorem3.p12.6.6.4.m4.2.2.2.2"><csymbol cd="ambiguous" id="S5.Thmtheorem3.p12.6.6.4.m4.2.2.2.2.1.cmml" xref="S5.Thmtheorem3.p12.6.6.4.m4.2.2.2.2">subscript</csymbol><ci id="S5.Thmtheorem3.p12.6.6.4.m4.2.2.2.2.2.cmml" xref="S5.Thmtheorem3.p12.6.6.4.m4.2.2.2.2.2">B</ci><apply id="S5.Thmtheorem3.p12.6.6.4.m4.2.2.2.2.3.cmml" xref="S5.Thmtheorem3.p12.6.6.4.m4.2.2.2.2.3"><minus id="S5.Thmtheorem3.p12.6.6.4.m4.2.2.2.2.3.1.cmml" xref="S5.Thmtheorem3.p12.6.6.4.m4.2.2.2.2.3.1"></minus><ci id="S5.Thmtheorem3.p12.6.6.4.m4.2.2.2.2.3.2.cmml" xref="S5.Thmtheorem3.p12.6.6.4.m4.2.2.2.2.3.2">i</ci><cn id="S5.Thmtheorem3.p12.6.6.4.m4.2.2.2.2.3.3.cmml" type="integer" xref="S5.Thmtheorem3.p12.6.6.4.m4.2.2.2.2.3.3">1</cn></apply></apply></interval></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmtheorem3.p12.6.6.4.m4.2c">(S_{i},B_{i-1})</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem3.p12.6.6.4.m4.2d">( italic_S start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT , italic_B start_POSTSUBSCRIPT italic_i - 1 end_POSTSUBSCRIPT )</annotation></semantics></math> with <math alttext="u\in S_{i}" class="ltx_Math" display="inline" id="S5.Thmtheorem3.p12.7.7.5.m5.1"><semantics id="S5.Thmtheorem3.p12.7.7.5.m5.1a"><mrow id="S5.Thmtheorem3.p12.7.7.5.m5.1.1" xref="S5.Thmtheorem3.p12.7.7.5.m5.1.1.cmml"><mi id="S5.Thmtheorem3.p12.7.7.5.m5.1.1.2" mathvariant="normal" xref="S5.Thmtheorem3.p12.7.7.5.m5.1.1.2.cmml">u</mi><mo id="S5.Thmtheorem3.p12.7.7.5.m5.1.1.1" xref="S5.Thmtheorem3.p12.7.7.5.m5.1.1.1.cmml">∈</mo><msub id="S5.Thmtheorem3.p12.7.7.5.m5.1.1.3" xref="S5.Thmtheorem3.p12.7.7.5.m5.1.1.3.cmml"><mi id="S5.Thmtheorem3.p12.7.7.5.m5.1.1.3.2" mathvariant="normal" xref="S5.Thmtheorem3.p12.7.7.5.m5.1.1.3.2.cmml">S</mi><mi id="S5.Thmtheorem3.p12.7.7.5.m5.1.1.3.3" mathvariant="normal" xref="S5.Thmtheorem3.p12.7.7.5.m5.1.1.3.3.cmml">i</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S5.Thmtheorem3.p12.7.7.5.m5.1b"><apply id="S5.Thmtheorem3.p12.7.7.5.m5.1.1.cmml" xref="S5.Thmtheorem3.p12.7.7.5.m5.1.1"><in id="S5.Thmtheorem3.p12.7.7.5.m5.1.1.1.cmml" xref="S5.Thmtheorem3.p12.7.7.5.m5.1.1.1"></in><ci id="S5.Thmtheorem3.p12.7.7.5.m5.1.1.2.cmml" xref="S5.Thmtheorem3.p12.7.7.5.m5.1.1.2">u</ci><apply id="S5.Thmtheorem3.p12.7.7.5.m5.1.1.3.cmml" xref="S5.Thmtheorem3.p12.7.7.5.m5.1.1.3"><csymbol cd="ambiguous" id="S5.Thmtheorem3.p12.7.7.5.m5.1.1.3.1.cmml" xref="S5.Thmtheorem3.p12.7.7.5.m5.1.1.3">subscript</csymbol><ci id="S5.Thmtheorem3.p12.7.7.5.m5.1.1.3.2.cmml" xref="S5.Thmtheorem3.p12.7.7.5.m5.1.1.3.2">S</ci><ci id="S5.Thmtheorem3.p12.7.7.5.m5.1.1.3.3.cmml" xref="S5.Thmtheorem3.p12.7.7.5.m5.1.1.3.3">i</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmtheorem3.p12.7.7.5.m5.1c">u\in S_{i}</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem3.p12.7.7.5.m5.1d">italic_u ∈ italic_S start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math>. If we denote <math alttext="B=B_{i-1}" class="ltx_Math" display="inline" id="S5.Thmtheorem3.p12.8.8.6.m6.1"><semantics id="S5.Thmtheorem3.p12.8.8.6.m6.1a"><mrow id="S5.Thmtheorem3.p12.8.8.6.m6.1.1" xref="S5.Thmtheorem3.p12.8.8.6.m6.1.1.cmml"><mi id="S5.Thmtheorem3.p12.8.8.6.m6.1.1.2" mathvariant="normal" xref="S5.Thmtheorem3.p12.8.8.6.m6.1.1.2.cmml">B</mi><mo id="S5.Thmtheorem3.p12.8.8.6.m6.1.1.1" xref="S5.Thmtheorem3.p12.8.8.6.m6.1.1.1.cmml">=</mo><msub id="S5.Thmtheorem3.p12.8.8.6.m6.1.1.3" xref="S5.Thmtheorem3.p12.8.8.6.m6.1.1.3.cmml"><mi id="S5.Thmtheorem3.p12.8.8.6.m6.1.1.3.2" mathvariant="normal" xref="S5.Thmtheorem3.p12.8.8.6.m6.1.1.3.2.cmml">B</mi><mrow id="S5.Thmtheorem3.p12.8.8.6.m6.1.1.3.3" xref="S5.Thmtheorem3.p12.8.8.6.m6.1.1.3.3.cmml"><mi id="S5.Thmtheorem3.p12.8.8.6.m6.1.1.3.3.2" mathvariant="normal" xref="S5.Thmtheorem3.p12.8.8.6.m6.1.1.3.3.2.cmml">i</mi><mo id="S5.Thmtheorem3.p12.8.8.6.m6.1.1.3.3.1" xref="S5.Thmtheorem3.p12.8.8.6.m6.1.1.3.3.1.cmml">−</mo><mn id="S5.Thmtheorem3.p12.8.8.6.m6.1.1.3.3.3" xref="S5.Thmtheorem3.p12.8.8.6.m6.1.1.3.3.3.cmml">1</mn></mrow></msub></mrow><annotation-xml encoding="MathML-Content" id="S5.Thmtheorem3.p12.8.8.6.m6.1b"><apply id="S5.Thmtheorem3.p12.8.8.6.m6.1.1.cmml" xref="S5.Thmtheorem3.p12.8.8.6.m6.1.1"><eq id="S5.Thmtheorem3.p12.8.8.6.m6.1.1.1.cmml" xref="S5.Thmtheorem3.p12.8.8.6.m6.1.1.1"></eq><ci id="S5.Thmtheorem3.p12.8.8.6.m6.1.1.2.cmml" xref="S5.Thmtheorem3.p12.8.8.6.m6.1.1.2">B</ci><apply id="S5.Thmtheorem3.p12.8.8.6.m6.1.1.3.cmml" xref="S5.Thmtheorem3.p12.8.8.6.m6.1.1.3"><csymbol cd="ambiguous" id="S5.Thmtheorem3.p12.8.8.6.m6.1.1.3.1.cmml" xref="S5.Thmtheorem3.p12.8.8.6.m6.1.1.3">subscript</csymbol><ci id="S5.Thmtheorem3.p12.8.8.6.m6.1.1.3.2.cmml" xref="S5.Thmtheorem3.p12.8.8.6.m6.1.1.3.2">B</ci><apply id="S5.Thmtheorem3.p12.8.8.6.m6.1.1.3.3.cmml" xref="S5.Thmtheorem3.p12.8.8.6.m6.1.1.3.3"><minus id="S5.Thmtheorem3.p12.8.8.6.m6.1.1.3.3.1.cmml" xref="S5.Thmtheorem3.p12.8.8.6.m6.1.1.3.3.1"></minus><ci id="S5.Thmtheorem3.p12.8.8.6.m6.1.1.3.3.2.cmml" xref="S5.Thmtheorem3.p12.8.8.6.m6.1.1.3.3.2">i</ci><cn id="S5.Thmtheorem3.p12.8.8.6.m6.1.1.3.3.3.cmml" type="integer" xref="S5.Thmtheorem3.p12.8.8.6.m6.1.1.3.3.3">1</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmtheorem3.p12.8.8.6.m6.1c">B=B_{i-1}</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem3.p12.8.8.6.m6.1d">italic_B = italic_B start_POSTSUBSCRIPT italic_i - 1 end_POSTSUBSCRIPT</annotation></semantics></math> then:</span></span></p> <table class="ltx_equation ltx_eqn_table" id="S5.Ex21"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_left_padleft"></td> <td class="ltx_eqn_cell ltx_align_left"><math alttext="S_{i}=arg\,\max\limits_{X\subset V-B}\frac{1}{|X|}\sum_{\hat{E}_{B}(X)}\textsl% {g}(e)\textnormal{ and }\rho^{*}(u)=\frac{1}{|S_{i}|}\sum_{\hat{E}_{B}(S_{i})}% \textsl{g}(e)." class="ltx_Math" display="block" id="S5.Ex21.m1.8"><semantics id="S5.Ex21.m1.8a"><mrow id="S5.Ex21.m1.8.8.1" xref="S5.Ex21.m1.8.8.1.1.cmml"><mrow id="S5.Ex21.m1.8.8.1.1" xref="S5.Ex21.m1.8.8.1.1.cmml"><msub id="S5.Ex21.m1.8.8.1.1.2" xref="S5.Ex21.m1.8.8.1.1.2.cmml"><mi id="S5.Ex21.m1.8.8.1.1.2.2" xref="S5.Ex21.m1.8.8.1.1.2.2.cmml">S</mi><mi id="S5.Ex21.m1.8.8.1.1.2.3" xref="S5.Ex21.m1.8.8.1.1.2.3.cmml">i</mi></msub><mo id="S5.Ex21.m1.8.8.1.1.3" xref="S5.Ex21.m1.8.8.1.1.3.cmml">=</mo><mrow id="S5.Ex21.m1.8.8.1.1.4" xref="S5.Ex21.m1.8.8.1.1.4.cmml"><mi id="S5.Ex21.m1.8.8.1.1.4.2" xref="S5.Ex21.m1.8.8.1.1.4.2.cmml">a</mi><mo id="S5.Ex21.m1.8.8.1.1.4.1" xref="S5.Ex21.m1.8.8.1.1.4.1.cmml">⁢</mo><mi id="S5.Ex21.m1.8.8.1.1.4.3" xref="S5.Ex21.m1.8.8.1.1.4.3.cmml">r</mi><mo id="S5.Ex21.m1.8.8.1.1.4.1a" xref="S5.Ex21.m1.8.8.1.1.4.1.cmml">⁢</mo><mi id="S5.Ex21.m1.8.8.1.1.4.4" xref="S5.Ex21.m1.8.8.1.1.4.4.cmml">g</mi><mo 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xref="S5.Ex21.m1.8.8.1.1.6.2.2.1"></times><ci id="S5.Ex21.m1.8.8.1.1.6.2.2.2a.cmml" xref="S5.Ex21.m1.8.8.1.1.6.2.2.2"><mtext class="ltx_mathvariant_italic" id="S5.Ex21.m1.8.8.1.1.6.2.2.2.cmml" xref="S5.Ex21.m1.8.8.1.1.6.2.2.2">g</mtext></ci><ci id="S5.Ex21.m1.7.7.cmml" xref="S5.Ex21.m1.7.7">𝑒</ci></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Ex21.m1.8c">S_{i}=arg\,\max\limits_{X\subset V-B}\frac{1}{|X|}\sum_{\hat{E}_{B}(X)}\textsl% {g}(e)\textnormal{ and }\rho^{*}(u)=\frac{1}{|S_{i}|}\sum_{\hat{E}_{B}(S_{i})}% \textsl{g}(e).</annotation><annotation encoding="application/x-llamapun" id="S5.Ex21.m1.8d">italic_S start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT = italic_a italic_r italic_g roman_max start_POSTSUBSCRIPT italic_X ⊂ italic_V - italic_B end_POSTSUBSCRIPT divide start_ARG 1 end_ARG start_ARG | italic_X | end_ARG ∑ start_POSTSUBSCRIPT over^ start_ARG italic_E end_ARG start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT ( italic_X ) end_POSTSUBSCRIPT g ( italic_e ) and italic_ρ start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ( italic_u ) = divide start_ARG 1 end_ARG start_ARG | italic_S start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT | end_ARG ∑ start_POSTSUBSCRIPT over^ start_ARG italic_E end_ARG start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT ( italic_S start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ) end_POSTSUBSCRIPT g ( italic_e ) .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_left_padright"></td> </tr></tbody> </table> </div> <div class="ltx_para ltx_noindent" id="S5.Thmtheorem3.p13"> <p class="ltx_p" id="S5.Thmtheorem3.p13.1"><span class="ltx_text ltx_font_italic" id="S5.Thmtheorem3.p13.1.1">Next, we use our </span><math alttext="\eta" class="ltx_Math" display="inline" id="S5.Thmtheorem3.p13.1.m1.1"><semantics id="S5.Thmtheorem3.p13.1.m1.1a"><mi id="S5.Thmtheorem3.p13.1.m1.1.1" xref="S5.Thmtheorem3.p13.1.m1.1.1.cmml">η</mi><annotation-xml encoding="MathML-Content" 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xref="S5.Ex22.m1.3.3.1.1.2.2.2.1.2">⋅</ci><apply id="S5.Ex22.m1.3.3.1.1.2.2.2.1.3.cmml" xref="S5.Ex22.m1.3.3.1.1.2.2.2.1.3"><times id="S5.Ex22.m1.3.3.1.1.2.2.2.1.3.1.cmml" xref="S5.Ex22.m1.3.3.1.1.2.2.2.1.3.1"></times><ci id="S5.Ex22.m1.3.3.1.1.2.2.2.1.3.2a.cmml" xref="S5.Ex22.m1.3.3.1.1.2.2.2.1.3.2"><mtext class="ltx_mathvariant_italic" id="S5.Ex22.m1.3.3.1.1.2.2.2.1.3.2.cmml" xref="S5.Ex22.m1.3.3.1.1.2.2.2.1.3.2">g</mtext></ci><ci id="S5.Ex22.m1.2.2.cmml" xref="S5.Ex22.m1.2.2">𝑢</ci></apply><apply id="S5.Ex22.m1.3.3.1.1.2.2.2.1.1.cmml" xref="S5.Ex22.m1.3.3.1.1.2.2.2.1.1"><csymbol cd="ambiguous" id="S5.Ex22.m1.3.3.1.1.2.2.2.1.1.2.cmml" xref="S5.Ex22.m1.3.3.1.1.2.2.2.1.1">superscript</csymbol><apply id="S5.Ex22.m1.3.3.1.1.2.2.2.1.1.1.1.1.cmml" xref="S5.Ex22.m1.3.3.1.1.2.2.2.1.1.1.1"><plus id="S5.Ex22.m1.3.3.1.1.2.2.2.1.1.1.1.1.1.cmml" xref="S5.Ex22.m1.3.3.1.1.2.2.2.1.1.1.1.1.1"></plus><cn id="S5.Ex22.m1.3.3.1.1.2.2.2.1.1.1.1.1.2.cmml" type="integer" xref="S5.Ex22.m1.3.3.1.1.2.2.2.1.1.1.1.1.2">1</cn><ci id="S5.Ex22.m1.3.3.1.1.2.2.2.1.1.1.1.1.3.cmml" xref="S5.Ex22.m1.3.3.1.1.2.2.2.1.1.1.1.1.3">𝜂</ci></apply><ci id="S5.Ex22.m1.3.3.1.1.2.2.2.1.1.3.cmml" xref="S5.Ex22.m1.3.3.1.1.2.2.2.1.1.3">𝑗</ci></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Ex22.m1.3c">H_{j}:=\{v\in S_{i}\mid\textsl{g}(v)\leq\textsl{g}(u)\cdot(1+\eta)^{j}\}.</annotation><annotation encoding="application/x-llamapun" id="S5.Ex22.m1.3d">italic_H start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT := { italic_v ∈ italic_S start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ∣ g ( italic_v ) ≤ g ( italic_u ) ⋅ ( 1 + italic_η ) start_POSTSUPERSCRIPT italic_j end_POSTSUPERSCRIPT } .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_left_padright"></td> </tr></tbody> </table> </div> <div class="ltx_para ltx_noindent" id="S5.Thmtheorem3.p14"> <p class="ltx_p" id="S5.Thmtheorem3.p14.3"><span class="ltx_text ltx_font_italic" id="S5.Thmtheorem3.p14.3.1">For the minimum </span><math alttext="k" class="ltx_Math" display="inline" id="S5.Thmtheorem3.p14.1.m1.1"><semantics id="S5.Thmtheorem3.p14.1.m1.1a"><mi id="S5.Thmtheorem3.p14.1.m1.1.1" xref="S5.Thmtheorem3.p14.1.m1.1.1.cmml">k</mi><annotation-xml encoding="MathML-Content" id="S5.Thmtheorem3.p14.1.m1.1b"><ci id="S5.Thmtheorem3.p14.1.m1.1.1.cmml" xref="S5.Thmtheorem3.p14.1.m1.1.1">𝑘</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmtheorem3.p14.1.m1.1c">k</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem3.p14.1.m1.1d">italic_k</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S5.Thmtheorem3.p14.3.2"> where </span><math alttext="|H_{k+1}|<(1+\frac{1}{16}\varepsilon)|H_{k}|" class="ltx_Math" display="inline" id="S5.Thmtheorem3.p14.2.m2.3"><semantics id="S5.Thmtheorem3.p14.2.m2.3a"><mrow id="S5.Thmtheorem3.p14.2.m2.3.3" xref="S5.Thmtheorem3.p14.2.m2.3.3.cmml"><mrow 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xref="S5.Thmtheorem3.p14.2.m2.2.2.2.1.1.1.3"><times id="S5.Thmtheorem3.p14.2.m2.2.2.2.1.1.1.3.1.cmml" xref="S5.Thmtheorem3.p14.2.m2.2.2.2.1.1.1.3.1"></times><apply id="S5.Thmtheorem3.p14.2.m2.2.2.2.1.1.1.3.2.cmml" xref="S5.Thmtheorem3.p14.2.m2.2.2.2.1.1.1.3.2"><divide id="S5.Thmtheorem3.p14.2.m2.2.2.2.1.1.1.3.2.1.cmml" xref="S5.Thmtheorem3.p14.2.m2.2.2.2.1.1.1.3.2"></divide><cn id="S5.Thmtheorem3.p14.2.m2.2.2.2.1.1.1.3.2.2.cmml" type="integer" xref="S5.Thmtheorem3.p14.2.m2.2.2.2.1.1.1.3.2.2">1</cn><cn id="S5.Thmtheorem3.p14.2.m2.2.2.2.1.1.1.3.2.3.cmml" type="integer" xref="S5.Thmtheorem3.p14.2.m2.2.2.2.1.1.1.3.2.3">16</cn></apply><ci id="S5.Thmtheorem3.p14.2.m2.2.2.2.1.1.1.3.3.cmml" xref="S5.Thmtheorem3.p14.2.m2.2.2.2.1.1.1.3.3">𝜀</ci></apply></apply><apply id="S5.Thmtheorem3.p14.2.m2.3.3.3.2.2.cmml" xref="S5.Thmtheorem3.p14.2.m2.3.3.3.2.1"><abs id="S5.Thmtheorem3.p14.2.m2.3.3.3.2.2.1.cmml" xref="S5.Thmtheorem3.p14.2.m2.3.3.3.2.1.2"></abs><apply id="S5.Thmtheorem3.p14.2.m2.3.3.3.2.1.1.cmml" xref="S5.Thmtheorem3.p14.2.m2.3.3.3.2.1.1"><csymbol cd="ambiguous" id="S5.Thmtheorem3.p14.2.m2.3.3.3.2.1.1.1.cmml" xref="S5.Thmtheorem3.p14.2.m2.3.3.3.2.1.1">subscript</csymbol><ci id="S5.Thmtheorem3.p14.2.m2.3.3.3.2.1.1.2.cmml" xref="S5.Thmtheorem3.p14.2.m2.3.3.3.2.1.1.2">𝐻</ci><ci id="S5.Thmtheorem3.p14.2.m2.3.3.3.2.1.1.3.cmml" xref="S5.Thmtheorem3.p14.2.m2.3.3.3.2.1.1.3">𝑘</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmtheorem3.p14.2.m2.3c">|H_{k+1}|<(1+\frac{1}{16}\varepsilon)|H_{k}|</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem3.p14.2.m2.3d">| italic_H start_POSTSUBSCRIPT italic_k + 1 end_POSTSUBSCRIPT | < ( 1 + divide start_ARG 1 end_ARG start_ARG 16 end_ARG italic_ε ) | italic_H start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT |</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S5.Thmtheorem3.p14.3.3">. </span><math alttext="(1+\eta)^{k}\leq(1+\frac{\varepsilon}{2})" class="ltx_Math" display="inline" id="S5.Thmtheorem3.p14.3.m3.2"><semantics id="S5.Thmtheorem3.p14.3.m3.2a"><mrow id="S5.Thmtheorem3.p14.3.m3.2.2" xref="S5.Thmtheorem3.p14.3.m3.2.2.cmml"><msup id="S5.Thmtheorem3.p14.3.m3.1.1.1" xref="S5.Thmtheorem3.p14.3.m3.1.1.1.cmml"><mrow id="S5.Thmtheorem3.p14.3.m3.1.1.1.1.1" xref="S5.Thmtheorem3.p14.3.m3.1.1.1.1.1.1.cmml"><mo id="S5.Thmtheorem3.p14.3.m3.1.1.1.1.1.2" stretchy="false" xref="S5.Thmtheorem3.p14.3.m3.1.1.1.1.1.1.cmml">(</mo><mrow id="S5.Thmtheorem3.p14.3.m3.1.1.1.1.1.1" xref="S5.Thmtheorem3.p14.3.m3.1.1.1.1.1.1.cmml"><mn id="S5.Thmtheorem3.p14.3.m3.1.1.1.1.1.1.2" xref="S5.Thmtheorem3.p14.3.m3.1.1.1.1.1.1.2.cmml">1</mn><mo id="S5.Thmtheorem3.p14.3.m3.1.1.1.1.1.1.1" xref="S5.Thmtheorem3.p14.3.m3.1.1.1.1.1.1.1.cmml">+</mo><mi id="S5.Thmtheorem3.p14.3.m3.1.1.1.1.1.1.3" xref="S5.Thmtheorem3.p14.3.m3.1.1.1.1.1.1.3.cmml">η</mi></mrow><mo id="S5.Thmtheorem3.p14.3.m3.1.1.1.1.1.3" stretchy="false" xref="S5.Thmtheorem3.p14.3.m3.1.1.1.1.1.1.cmml">)</mo></mrow><mi id="S5.Thmtheorem3.p14.3.m3.1.1.1.3" xref="S5.Thmtheorem3.p14.3.m3.1.1.1.3.cmml">k</mi></msup><mo id="S5.Thmtheorem3.p14.3.m3.2.2.3" xref="S5.Thmtheorem3.p14.3.m3.2.2.3.cmml">≤</mo><mrow id="S5.Thmtheorem3.p14.3.m3.2.2.2.1" xref="S5.Thmtheorem3.p14.3.m3.2.2.2.1.1.cmml"><mo id="S5.Thmtheorem3.p14.3.m3.2.2.2.1.2" stretchy="false" xref="S5.Thmtheorem3.p14.3.m3.2.2.2.1.1.cmml">(</mo><mrow id="S5.Thmtheorem3.p14.3.m3.2.2.2.1.1" xref="S5.Thmtheorem3.p14.3.m3.2.2.2.1.1.cmml"><mn id="S5.Thmtheorem3.p14.3.m3.2.2.2.1.1.2" xref="S5.Thmtheorem3.p14.3.m3.2.2.2.1.1.2.cmml">1</mn><mo id="S5.Thmtheorem3.p14.3.m3.2.2.2.1.1.1" xref="S5.Thmtheorem3.p14.3.m3.2.2.2.1.1.1.cmml">+</mo><mfrac id="S5.Thmtheorem3.p14.3.m3.2.2.2.1.1.3" xref="S5.Thmtheorem3.p14.3.m3.2.2.2.1.1.3.cmml"><mi id="S5.Thmtheorem3.p14.3.m3.2.2.2.1.1.3.2" xref="S5.Thmtheorem3.p14.3.m3.2.2.2.1.1.3.2.cmml">ε</mi><mn id="S5.Thmtheorem3.p14.3.m3.2.2.2.1.1.3.3" xref="S5.Thmtheorem3.p14.3.m3.2.2.2.1.1.3.3.cmml">2</mn></mfrac></mrow><mo id="S5.Thmtheorem3.p14.3.m3.2.2.2.1.3" stretchy="false" xref="S5.Thmtheorem3.p14.3.m3.2.2.2.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.Thmtheorem3.p14.3.m3.2b"><apply id="S5.Thmtheorem3.p14.3.m3.2.2.cmml" xref="S5.Thmtheorem3.p14.3.m3.2.2"><leq id="S5.Thmtheorem3.p14.3.m3.2.2.3.cmml" xref="S5.Thmtheorem3.p14.3.m3.2.2.3"></leq><apply id="S5.Thmtheorem3.p14.3.m3.1.1.1.cmml" xref="S5.Thmtheorem3.p14.3.m3.1.1.1"><csymbol cd="ambiguous" id="S5.Thmtheorem3.p14.3.m3.1.1.1.2.cmml" xref="S5.Thmtheorem3.p14.3.m3.1.1.1">superscript</csymbol><apply id="S5.Thmtheorem3.p14.3.m3.1.1.1.1.1.1.cmml" xref="S5.Thmtheorem3.p14.3.m3.1.1.1.1.1"><plus id="S5.Thmtheorem3.p14.3.m3.1.1.1.1.1.1.1.cmml" xref="S5.Thmtheorem3.p14.3.m3.1.1.1.1.1.1.1"></plus><cn id="S5.Thmtheorem3.p14.3.m3.1.1.1.1.1.1.2.cmml" type="integer" xref="S5.Thmtheorem3.p14.3.m3.1.1.1.1.1.1.2">1</cn><ci id="S5.Thmtheorem3.p14.3.m3.1.1.1.1.1.1.3.cmml" xref="S5.Thmtheorem3.p14.3.m3.1.1.1.1.1.1.3">𝜂</ci></apply><ci id="S5.Thmtheorem3.p14.3.m3.1.1.1.3.cmml" xref="S5.Thmtheorem3.p14.3.m3.1.1.1.3">𝑘</ci></apply><apply id="S5.Thmtheorem3.p14.3.m3.2.2.2.1.1.cmml" xref="S5.Thmtheorem3.p14.3.m3.2.2.2.1"><plus id="S5.Thmtheorem3.p14.3.m3.2.2.2.1.1.1.cmml" xref="S5.Thmtheorem3.p14.3.m3.2.2.2.1.1.1"></plus><cn id="S5.Thmtheorem3.p14.3.m3.2.2.2.1.1.2.cmml" type="integer" xref="S5.Thmtheorem3.p14.3.m3.2.2.2.1.1.2">1</cn><apply id="S5.Thmtheorem3.p14.3.m3.2.2.2.1.1.3.cmml" xref="S5.Thmtheorem3.p14.3.m3.2.2.2.1.1.3"><divide id="S5.Thmtheorem3.p14.3.m3.2.2.2.1.1.3.1.cmml" xref="S5.Thmtheorem3.p14.3.m3.2.2.2.1.1.3"></divide><ci id="S5.Thmtheorem3.p14.3.m3.2.2.2.1.1.3.2.cmml" xref="S5.Thmtheorem3.p14.3.m3.2.2.2.1.1.3.2">𝜀</ci><cn id="S5.Thmtheorem3.p14.3.m3.2.2.2.1.1.3.3.cmml" type="integer" xref="S5.Thmtheorem3.p14.3.m3.2.2.2.1.1.3.3">2</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmtheorem3.p14.3.m3.2c">(1+\eta)^{k}\leq(1+\frac{\varepsilon}{2})</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem3.p14.3.m3.2d">( 1 + italic_η ) start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT ≤ ( 1 + divide start_ARG italic_ε end_ARG start_ARG 2 end_ARG )</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S5.Thmtheorem3.p14.3.4"> (Lemma </span><a class="ltx_ref ltx_font_italic" href="https://arxiv.org/html/2411.12694v2#S5.Thmtheorem1" title="Lemma 5.1. ‣ 5 Results for dynamic algorithms ‣ Local Density and its Distributed Approximation"><span class="ltx_text ltx_ref_tag">5.1</span></a><span class="ltx_text ltx_font_italic" id="S5.Thmtheorem3.p14.3.5">).</span></p> </div> <div class="ltx_para" id="S5.Thmtheorem3.p15"> <p class="ltx_p" id="S5.Thmtheorem3.p15.9"><span class="ltx_text ltx_font_italic" id="S5.Thmtheorem3.p15.9.9">Denote by <math alttext="E_{\uparrow}" class="ltx_Math" display="inline" id="S5.Thmtheorem3.p15.1.1.m1.1"><semantics id="S5.Thmtheorem3.p15.1.1.m1.1a"><msub id="S5.Thmtheorem3.p15.1.1.m1.1.1" xref="S5.Thmtheorem3.p15.1.1.m1.1.1.cmml"><mi id="S5.Thmtheorem3.p15.1.1.m1.1.1.2" xref="S5.Thmtheorem3.p15.1.1.m1.1.1.2.cmml">E</mi><mo id="S5.Thmtheorem3.p15.1.1.m1.1.1.3" stretchy="false" xref="S5.Thmtheorem3.p15.1.1.m1.1.1.3.cmml">↑</mo></msub><annotation-xml encoding="MathML-Content" id="S5.Thmtheorem3.p15.1.1.m1.1b"><apply id="S5.Thmtheorem3.p15.1.1.m1.1.1.cmml" xref="S5.Thmtheorem3.p15.1.1.m1.1.1"><csymbol cd="ambiguous" id="S5.Thmtheorem3.p15.1.1.m1.1.1.1.cmml" xref="S5.Thmtheorem3.p15.1.1.m1.1.1">subscript</csymbol><ci id="S5.Thmtheorem3.p15.1.1.m1.1.1.2.cmml" xref="S5.Thmtheorem3.p15.1.1.m1.1.1.2">𝐸</ci><ci id="S5.Thmtheorem3.p15.1.1.m1.1.1.3.cmml" xref="S5.Thmtheorem3.p15.1.1.m1.1.1.3">↑</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmtheorem3.p15.1.1.m1.1c">E_{\uparrow}</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem3.p15.1.1.m1.1d">italic_E start_POSTSUBSCRIPT ↑ end_POSTSUBSCRIPT</annotation></semantics></math> all edges with one vertex in <math alttext="H_{k}" class="ltx_Math" display="inline" id="S5.Thmtheorem3.p15.2.2.m2.1"><semantics id="S5.Thmtheorem3.p15.2.2.m2.1a"><msub id="S5.Thmtheorem3.p15.2.2.m2.1.1" xref="S5.Thmtheorem3.p15.2.2.m2.1.1.cmml"><mi id="S5.Thmtheorem3.p15.2.2.m2.1.1.2" xref="S5.Thmtheorem3.p15.2.2.m2.1.1.2.cmml">H</mi><mi id="S5.Thmtheorem3.p15.2.2.m2.1.1.3" xref="S5.Thmtheorem3.p15.2.2.m2.1.1.3.cmml">k</mi></msub><annotation-xml encoding="MathML-Content" id="S5.Thmtheorem3.p15.2.2.m2.1b"><apply id="S5.Thmtheorem3.p15.2.2.m2.1.1.cmml" xref="S5.Thmtheorem3.p15.2.2.m2.1.1"><csymbol cd="ambiguous" id="S5.Thmtheorem3.p15.2.2.m2.1.1.1.cmml" xref="S5.Thmtheorem3.p15.2.2.m2.1.1">subscript</csymbol><ci id="S5.Thmtheorem3.p15.2.2.m2.1.1.2.cmml" xref="S5.Thmtheorem3.p15.2.2.m2.1.1.2">𝐻</ci><ci id="S5.Thmtheorem3.p15.2.2.m2.1.1.3.cmml" xref="S5.Thmtheorem3.p15.2.2.m2.1.1.3">𝑘</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmtheorem3.p15.2.2.m2.1c">H_{k}</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem3.p15.2.2.m2.1d">italic_H start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT</annotation></semantics></math> and the other in <math alttext="S_{i}-H_{k}" class="ltx_Math" display="inline" id="S5.Thmtheorem3.p15.3.3.m3.1"><semantics id="S5.Thmtheorem3.p15.3.3.m3.1a"><mrow id="S5.Thmtheorem3.p15.3.3.m3.1.1" xref="S5.Thmtheorem3.p15.3.3.m3.1.1.cmml"><msub id="S5.Thmtheorem3.p15.3.3.m3.1.1.2" xref="S5.Thmtheorem3.p15.3.3.m3.1.1.2.cmml"><mi id="S5.Thmtheorem3.p15.3.3.m3.1.1.2.2" xref="S5.Thmtheorem3.p15.3.3.m3.1.1.2.2.cmml">S</mi><mi id="S5.Thmtheorem3.p15.3.3.m3.1.1.2.3" xref="S5.Thmtheorem3.p15.3.3.m3.1.1.2.3.cmml">i</mi></msub><mo id="S5.Thmtheorem3.p15.3.3.m3.1.1.1" xref="S5.Thmtheorem3.p15.3.3.m3.1.1.1.cmml">−</mo><msub id="S5.Thmtheorem3.p15.3.3.m3.1.1.3" xref="S5.Thmtheorem3.p15.3.3.m3.1.1.3.cmml"><mi id="S5.Thmtheorem3.p15.3.3.m3.1.1.3.2" xref="S5.Thmtheorem3.p15.3.3.m3.1.1.3.2.cmml">H</mi><mi id="S5.Thmtheorem3.p15.3.3.m3.1.1.3.3" xref="S5.Thmtheorem3.p15.3.3.m3.1.1.3.3.cmml">k</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S5.Thmtheorem3.p15.3.3.m3.1b"><apply id="S5.Thmtheorem3.p15.3.3.m3.1.1.cmml" xref="S5.Thmtheorem3.p15.3.3.m3.1.1"><minus id="S5.Thmtheorem3.p15.3.3.m3.1.1.1.cmml" xref="S5.Thmtheorem3.p15.3.3.m3.1.1.1"></minus><apply id="S5.Thmtheorem3.p15.3.3.m3.1.1.2.cmml" xref="S5.Thmtheorem3.p15.3.3.m3.1.1.2"><csymbol cd="ambiguous" id="S5.Thmtheorem3.p15.3.3.m3.1.1.2.1.cmml" xref="S5.Thmtheorem3.p15.3.3.m3.1.1.2">subscript</csymbol><ci id="S5.Thmtheorem3.p15.3.3.m3.1.1.2.2.cmml" xref="S5.Thmtheorem3.p15.3.3.m3.1.1.2.2">𝑆</ci><ci id="S5.Thmtheorem3.p15.3.3.m3.1.1.2.3.cmml" xref="S5.Thmtheorem3.p15.3.3.m3.1.1.2.3">𝑖</ci></apply><apply id="S5.Thmtheorem3.p15.3.3.m3.1.1.3.cmml" xref="S5.Thmtheorem3.p15.3.3.m3.1.1.3"><csymbol cd="ambiguous" id="S5.Thmtheorem3.p15.3.3.m3.1.1.3.1.cmml" xref="S5.Thmtheorem3.p15.3.3.m3.1.1.3">subscript</csymbol><ci id="S5.Thmtheorem3.p15.3.3.m3.1.1.3.2.cmml" xref="S5.Thmtheorem3.p15.3.3.m3.1.1.3.2">𝐻</ci><ci id="S5.Thmtheorem3.p15.3.3.m3.1.1.3.3.cmml" xref="S5.Thmtheorem3.p15.3.3.m3.1.1.3.3">𝑘</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmtheorem3.p15.3.3.m3.1c">S_{i}-H_{k}</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem3.p15.3.3.m3.1d">italic_S start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT - italic_H start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT</annotation></semantics></math>. Recall that <math alttext="\hat{E}_{B}(X)" class="ltx_Math" display="inline" id="S5.Thmtheorem3.p15.4.4.m4.1"><semantics id="S5.Thmtheorem3.p15.4.4.m4.1a"><mrow id="S5.Thmtheorem3.p15.4.4.m4.1.2" xref="S5.Thmtheorem3.p15.4.4.m4.1.2.cmml"><msub id="S5.Thmtheorem3.p15.4.4.m4.1.2.2" xref="S5.Thmtheorem3.p15.4.4.m4.1.2.2.cmml"><mover accent="true" id="S5.Thmtheorem3.p15.4.4.m4.1.2.2.2" xref="S5.Thmtheorem3.p15.4.4.m4.1.2.2.2.cmml"><mi id="S5.Thmtheorem3.p15.4.4.m4.1.2.2.2.2" xref="S5.Thmtheorem3.p15.4.4.m4.1.2.2.2.2.cmml">E</mi><mo id="S5.Thmtheorem3.p15.4.4.m4.1.2.2.2.1" xref="S5.Thmtheorem3.p15.4.4.m4.1.2.2.2.1.cmml">^</mo></mover><mi id="S5.Thmtheorem3.p15.4.4.m4.1.2.2.3" xref="S5.Thmtheorem3.p15.4.4.m4.1.2.2.3.cmml">B</mi></msub><mo id="S5.Thmtheorem3.p15.4.4.m4.1.2.1" xref="S5.Thmtheorem3.p15.4.4.m4.1.2.1.cmml">⁢</mo><mrow id="S5.Thmtheorem3.p15.4.4.m4.1.2.3.2" xref="S5.Thmtheorem3.p15.4.4.m4.1.2.cmml"><mo id="S5.Thmtheorem3.p15.4.4.m4.1.2.3.2.1" stretchy="false" xref="S5.Thmtheorem3.p15.4.4.m4.1.2.cmml">(</mo><mi id="S5.Thmtheorem3.p15.4.4.m4.1.1" xref="S5.Thmtheorem3.p15.4.4.m4.1.1.cmml">X</mi><mo id="S5.Thmtheorem3.p15.4.4.m4.1.2.3.2.2" stretchy="false" xref="S5.Thmtheorem3.p15.4.4.m4.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.Thmtheorem3.p15.4.4.m4.1b"><apply id="S5.Thmtheorem3.p15.4.4.m4.1.2.cmml" xref="S5.Thmtheorem3.p15.4.4.m4.1.2"><times id="S5.Thmtheorem3.p15.4.4.m4.1.2.1.cmml" xref="S5.Thmtheorem3.p15.4.4.m4.1.2.1"></times><apply id="S5.Thmtheorem3.p15.4.4.m4.1.2.2.cmml" xref="S5.Thmtheorem3.p15.4.4.m4.1.2.2"><csymbol cd="ambiguous" id="S5.Thmtheorem3.p15.4.4.m4.1.2.2.1.cmml" xref="S5.Thmtheorem3.p15.4.4.m4.1.2.2">subscript</csymbol><apply id="S5.Thmtheorem3.p15.4.4.m4.1.2.2.2.cmml" xref="S5.Thmtheorem3.p15.4.4.m4.1.2.2.2"><ci id="S5.Thmtheorem3.p15.4.4.m4.1.2.2.2.1.cmml" xref="S5.Thmtheorem3.p15.4.4.m4.1.2.2.2.1">^</ci><ci id="S5.Thmtheorem3.p15.4.4.m4.1.2.2.2.2.cmml" xref="S5.Thmtheorem3.p15.4.4.m4.1.2.2.2.2">𝐸</ci></apply><ci id="S5.Thmtheorem3.p15.4.4.m4.1.2.2.3.cmml" xref="S5.Thmtheorem3.p15.4.4.m4.1.2.2.3">𝐵</ci></apply><ci id="S5.Thmtheorem3.p15.4.4.m4.1.1.cmml" xref="S5.Thmtheorem3.p15.4.4.m4.1.1">𝑋</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmtheorem3.p15.4.4.m4.1c">\hat{E}_{B}(X)</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem3.p15.4.4.m4.1d">over^ start_ARG italic_E end_ARG start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT ( italic_X )</annotation></semantics></math> denotes all edges that have one endpoint in <math alttext="X" class="ltx_Math" display="inline" id="S5.Thmtheorem3.p15.5.5.m5.1"><semantics id="S5.Thmtheorem3.p15.5.5.m5.1a"><mi id="S5.Thmtheorem3.p15.5.5.m5.1.1" xref="S5.Thmtheorem3.p15.5.5.m5.1.1.cmml">X</mi><annotation-xml encoding="MathML-Content" id="S5.Thmtheorem3.p15.5.5.m5.1b"><ci id="S5.Thmtheorem3.p15.5.5.m5.1.1.cmml" xref="S5.Thmtheorem3.p15.5.5.m5.1.1">𝑋</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmtheorem3.p15.5.5.m5.1c">X</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem3.p15.5.5.m5.1d">italic_X</annotation></semantics></math> and the other in <math alttext="X" class="ltx_Math" display="inline" id="S5.Thmtheorem3.p15.6.6.m6.1"><semantics id="S5.Thmtheorem3.p15.6.6.m6.1a"><mi id="S5.Thmtheorem3.p15.6.6.m6.1.1" xref="S5.Thmtheorem3.p15.6.6.m6.1.1.cmml">X</mi><annotation-xml encoding="MathML-Content" id="S5.Thmtheorem3.p15.6.6.m6.1b"><ci id="S5.Thmtheorem3.p15.6.6.m6.1.1.cmml" xref="S5.Thmtheorem3.p15.6.6.m6.1.1">𝑋</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmtheorem3.p15.6.6.m6.1c">X</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem3.p15.6.6.m6.1d">italic_X</annotation></semantics></math> or <math alttext="B" class="ltx_Math" display="inline" id="S5.Thmtheorem3.p15.7.7.m7.1"><semantics id="S5.Thmtheorem3.p15.7.7.m7.1a"><mi id="S5.Thmtheorem3.p15.7.7.m7.1.1" xref="S5.Thmtheorem3.p15.7.7.m7.1.1.cmml">B</mi><annotation-xml encoding="MathML-Content" id="S5.Thmtheorem3.p15.7.7.m7.1b"><ci id="S5.Thmtheorem3.p15.7.7.m7.1.1.cmml" xref="S5.Thmtheorem3.p15.7.7.m7.1.1">𝐵</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmtheorem3.p15.7.7.m7.1c">B</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem3.p15.7.7.m7.1d">italic_B</annotation></semantics></math>. Since <math alttext="H_{k}\cap B=\emptyset" class="ltx_Math" display="inline" id="S5.Thmtheorem3.p15.8.8.m8.1"><semantics id="S5.Thmtheorem3.p15.8.8.m8.1a"><mrow id="S5.Thmtheorem3.p15.8.8.m8.1.1" xref="S5.Thmtheorem3.p15.8.8.m8.1.1.cmml"><mrow id="S5.Thmtheorem3.p15.8.8.m8.1.1.2" xref="S5.Thmtheorem3.p15.8.8.m8.1.1.2.cmml"><msub id="S5.Thmtheorem3.p15.8.8.m8.1.1.2.2" xref="S5.Thmtheorem3.p15.8.8.m8.1.1.2.2.cmml"><mi id="S5.Thmtheorem3.p15.8.8.m8.1.1.2.2.2" xref="S5.Thmtheorem3.p15.8.8.m8.1.1.2.2.2.cmml">H</mi><mi id="S5.Thmtheorem3.p15.8.8.m8.1.1.2.2.3" xref="S5.Thmtheorem3.p15.8.8.m8.1.1.2.2.3.cmml">k</mi></msub><mo id="S5.Thmtheorem3.p15.8.8.m8.1.1.2.1" xref="S5.Thmtheorem3.p15.8.8.m8.1.1.2.1.cmml">∩</mo><mi id="S5.Thmtheorem3.p15.8.8.m8.1.1.2.3" xref="S5.Thmtheorem3.p15.8.8.m8.1.1.2.3.cmml">B</mi></mrow><mo id="S5.Thmtheorem3.p15.8.8.m8.1.1.1" xref="S5.Thmtheorem3.p15.8.8.m8.1.1.1.cmml">=</mo><mi id="S5.Thmtheorem3.p15.8.8.m8.1.1.3" mathvariant="normal" xref="S5.Thmtheorem3.p15.8.8.m8.1.1.3.cmml">∅</mi></mrow><annotation-xml encoding="MathML-Content" id="S5.Thmtheorem3.p15.8.8.m8.1b"><apply id="S5.Thmtheorem3.p15.8.8.m8.1.1.cmml" xref="S5.Thmtheorem3.p15.8.8.m8.1.1"><eq id="S5.Thmtheorem3.p15.8.8.m8.1.1.1.cmml" xref="S5.Thmtheorem3.p15.8.8.m8.1.1.1"></eq><apply id="S5.Thmtheorem3.p15.8.8.m8.1.1.2.cmml" xref="S5.Thmtheorem3.p15.8.8.m8.1.1.2"><intersect id="S5.Thmtheorem3.p15.8.8.m8.1.1.2.1.cmml" xref="S5.Thmtheorem3.p15.8.8.m8.1.1.2.1"></intersect><apply id="S5.Thmtheorem3.p15.8.8.m8.1.1.2.2.cmml" xref="S5.Thmtheorem3.p15.8.8.m8.1.1.2.2"><csymbol cd="ambiguous" id="S5.Thmtheorem3.p15.8.8.m8.1.1.2.2.1.cmml" xref="S5.Thmtheorem3.p15.8.8.m8.1.1.2.2">subscript</csymbol><ci id="S5.Thmtheorem3.p15.8.8.m8.1.1.2.2.2.cmml" xref="S5.Thmtheorem3.p15.8.8.m8.1.1.2.2.2">𝐻</ci><ci id="S5.Thmtheorem3.p15.8.8.m8.1.1.2.2.3.cmml" xref="S5.Thmtheorem3.p15.8.8.m8.1.1.2.2.3">𝑘</ci></apply><ci id="S5.Thmtheorem3.p15.8.8.m8.1.1.2.3.cmml" xref="S5.Thmtheorem3.p15.8.8.m8.1.1.2.3">𝐵</ci></apply><emptyset id="S5.Thmtheorem3.p15.8.8.m8.1.1.3.cmml" xref="S5.Thmtheorem3.p15.8.8.m8.1.1.3"></emptyset></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmtheorem3.p15.8.8.m8.1c">H_{k}\cap B=\emptyset</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem3.p15.8.8.m8.1d">italic_H start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ∩ italic_B = ∅</annotation></semantics></math>, <math alttext="\hat{E}_{B}(S_{i}-H_{k})=\hat{E}_{B}(S_{i})-(\hat{E}_{B}(H_{k})\cup E_{% \uparrow})" class="ltx_Math" display="inline" id="S5.Thmtheorem3.p15.9.9.m9.3"><semantics id="S5.Thmtheorem3.p15.9.9.m9.3a"><mrow id="S5.Thmtheorem3.p15.9.9.m9.3.3" xref="S5.Thmtheorem3.p15.9.9.m9.3.3.cmml"><mrow id="S5.Thmtheorem3.p15.9.9.m9.1.1.1" xref="S5.Thmtheorem3.p15.9.9.m9.1.1.1.cmml"><msub id="S5.Thmtheorem3.p15.9.9.m9.1.1.1.3" xref="S5.Thmtheorem3.p15.9.9.m9.1.1.1.3.cmml"><mover accent="true" id="S5.Thmtheorem3.p15.9.9.m9.1.1.1.3.2" xref="S5.Thmtheorem3.p15.9.9.m9.1.1.1.3.2.cmml"><mi id="S5.Thmtheorem3.p15.9.9.m9.1.1.1.3.2.2" xref="S5.Thmtheorem3.p15.9.9.m9.1.1.1.3.2.2.cmml">E</mi><mo id="S5.Thmtheorem3.p15.9.9.m9.1.1.1.3.2.1" xref="S5.Thmtheorem3.p15.9.9.m9.1.1.1.3.2.1.cmml">^</mo></mover><mi id="S5.Thmtheorem3.p15.9.9.m9.1.1.1.3.3" xref="S5.Thmtheorem3.p15.9.9.m9.1.1.1.3.3.cmml">B</mi></msub><mo id="S5.Thmtheorem3.p15.9.9.m9.1.1.1.2" xref="S5.Thmtheorem3.p15.9.9.m9.1.1.1.2.cmml">⁢</mo><mrow id="S5.Thmtheorem3.p15.9.9.m9.1.1.1.1.1" xref="S5.Thmtheorem3.p15.9.9.m9.1.1.1.1.1.1.cmml"><mo id="S5.Thmtheorem3.p15.9.9.m9.1.1.1.1.1.2" stretchy="false" xref="S5.Thmtheorem3.p15.9.9.m9.1.1.1.1.1.1.cmml">(</mo><mrow id="S5.Thmtheorem3.p15.9.9.m9.1.1.1.1.1.1" xref="S5.Thmtheorem3.p15.9.9.m9.1.1.1.1.1.1.cmml"><msub id="S5.Thmtheorem3.p15.9.9.m9.1.1.1.1.1.1.2" xref="S5.Thmtheorem3.p15.9.9.m9.1.1.1.1.1.1.2.cmml"><mi id="S5.Thmtheorem3.p15.9.9.m9.1.1.1.1.1.1.2.2" xref="S5.Thmtheorem3.p15.9.9.m9.1.1.1.1.1.1.2.2.cmml">S</mi><mi id="S5.Thmtheorem3.p15.9.9.m9.1.1.1.1.1.1.2.3" xref="S5.Thmtheorem3.p15.9.9.m9.1.1.1.1.1.1.2.3.cmml">i</mi></msub><mo id="S5.Thmtheorem3.p15.9.9.m9.1.1.1.1.1.1.1" xref="S5.Thmtheorem3.p15.9.9.m9.1.1.1.1.1.1.1.cmml">−</mo><msub id="S5.Thmtheorem3.p15.9.9.m9.1.1.1.1.1.1.3" xref="S5.Thmtheorem3.p15.9.9.m9.1.1.1.1.1.1.3.cmml"><mi id="S5.Thmtheorem3.p15.9.9.m9.1.1.1.1.1.1.3.2" xref="S5.Thmtheorem3.p15.9.9.m9.1.1.1.1.1.1.3.2.cmml">H</mi><mi id="S5.Thmtheorem3.p15.9.9.m9.1.1.1.1.1.1.3.3" xref="S5.Thmtheorem3.p15.9.9.m9.1.1.1.1.1.1.3.3.cmml">k</mi></msub></mrow><mo id="S5.Thmtheorem3.p15.9.9.m9.1.1.1.1.1.3" stretchy="false" xref="S5.Thmtheorem3.p15.9.9.m9.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S5.Thmtheorem3.p15.9.9.m9.3.3.4" xref="S5.Thmtheorem3.p15.9.9.m9.3.3.4.cmml">=</mo><mrow id="S5.Thmtheorem3.p15.9.9.m9.3.3.3" xref="S5.Thmtheorem3.p15.9.9.m9.3.3.3.cmml"><mrow id="S5.Thmtheorem3.p15.9.9.m9.2.2.2.1" xref="S5.Thmtheorem3.p15.9.9.m9.2.2.2.1.cmml"><msub id="S5.Thmtheorem3.p15.9.9.m9.2.2.2.1.3" xref="S5.Thmtheorem3.p15.9.9.m9.2.2.2.1.3.cmml"><mover accent="true" id="S5.Thmtheorem3.p15.9.9.m9.2.2.2.1.3.2" xref="S5.Thmtheorem3.p15.9.9.m9.2.2.2.1.3.2.cmml"><mi id="S5.Thmtheorem3.p15.9.9.m9.2.2.2.1.3.2.2" xref="S5.Thmtheorem3.p15.9.9.m9.2.2.2.1.3.2.2.cmml">E</mi><mo id="S5.Thmtheorem3.p15.9.9.m9.2.2.2.1.3.2.1" xref="S5.Thmtheorem3.p15.9.9.m9.2.2.2.1.3.2.1.cmml">^</mo></mover><mi id="S5.Thmtheorem3.p15.9.9.m9.2.2.2.1.3.3" xref="S5.Thmtheorem3.p15.9.9.m9.2.2.2.1.3.3.cmml">B</mi></msub><mo id="S5.Thmtheorem3.p15.9.9.m9.2.2.2.1.2" xref="S5.Thmtheorem3.p15.9.9.m9.2.2.2.1.2.cmml">⁢</mo><mrow id="S5.Thmtheorem3.p15.9.9.m9.2.2.2.1.1.1" xref="S5.Thmtheorem3.p15.9.9.m9.2.2.2.1.1.1.1.cmml"><mo id="S5.Thmtheorem3.p15.9.9.m9.2.2.2.1.1.1.2" stretchy="false" xref="S5.Thmtheorem3.p15.9.9.m9.2.2.2.1.1.1.1.cmml">(</mo><msub id="S5.Thmtheorem3.p15.9.9.m9.2.2.2.1.1.1.1" 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xref="S5.Thmtheorem3.p15.9.9.m9.3.3.3.2.1.1.3.2">𝐸</ci><ci id="S5.Thmtheorem3.p15.9.9.m9.3.3.3.2.1.1.3.3.cmml" xref="S5.Thmtheorem3.p15.9.9.m9.3.3.3.2.1.1.3.3">↑</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmtheorem3.p15.9.9.m9.3c">\hat{E}_{B}(S_{i}-H_{k})=\hat{E}_{B}(S_{i})-(\hat{E}_{B}(H_{k})\cup E_{% \uparrow})</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem3.p15.9.9.m9.3d">over^ start_ARG italic_E end_ARG start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT ( italic_S start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT - italic_H start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ) = over^ start_ARG italic_E end_ARG start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT ( italic_S start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ) - ( over^ start_ARG italic_E end_ARG start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT ( italic_H start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ) ∪ italic_E start_POSTSUBSCRIPT ↑ end_POSTSUBSCRIPT )</annotation></semantics></math> and so:</span></p> <table class="ltx_equation ltx_eqn_table" id="S5.E6"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_left_padleft"></td> <td class="ltx_eqn_cell ltx_align_left"><math alttext="\hat{\rho}_{B}(S_{i}-H_{k})\cdot(|S_{i}|-|H_{k}|)=\sum_{e\in\hat{E}_{B}(S_{i}-% H_{k})}\textsl{g}(e)=\left(\sum_{e\in\hat{E}_{B}(S_{i})}\textsl{g}(e)\right)-% \left(\sum_{e\in\hat{E}_{B}(H_{k})\cup E_{\uparrow}}\textsl{g}(e)\right)" class="ltx_Math" display="block" id="S5.E6.m1.10"><semantics id="S5.E6.m1.10a"><mrow id="S5.E6.m1.10.10" xref="S5.E6.m1.10.10.cmml"><mrow id="S5.E6.m1.8.8.2" xref="S5.E6.m1.8.8.2.cmml"><mrow id="S5.E6.m1.7.7.1.1" xref="S5.E6.m1.7.7.1.1.cmml"><msub id="S5.E6.m1.7.7.1.1.3" xref="S5.E6.m1.7.7.1.1.3.cmml"><mover accent="true" id="S5.E6.m1.7.7.1.1.3.2" xref="S5.E6.m1.7.7.1.1.3.2.cmml"><mi id="S5.E6.m1.7.7.1.1.3.2.2" xref="S5.E6.m1.7.7.1.1.3.2.2.cmml">ρ</mi><mo id="S5.E6.m1.7.7.1.1.3.2.1" xref="S5.E6.m1.7.7.1.1.3.2.1.cmml">^</mo></mover><mi id="S5.E6.m1.7.7.1.1.3.3" xref="S5.E6.m1.7.7.1.1.3.3.cmml">B</mi></msub><mo id="S5.E6.m1.7.7.1.1.2" xref="S5.E6.m1.7.7.1.1.2.cmml">⁢</mo><mrow id="S5.E6.m1.7.7.1.1.1.1" xref="S5.E6.m1.7.7.1.1.1.1.1.cmml"><mo id="S5.E6.m1.7.7.1.1.1.1.2" stretchy="false" xref="S5.E6.m1.7.7.1.1.1.1.1.cmml">(</mo><mrow id="S5.E6.m1.7.7.1.1.1.1.1" xref="S5.E6.m1.7.7.1.1.1.1.1.cmml"><msub id="S5.E6.m1.7.7.1.1.1.1.1.2" xref="S5.E6.m1.7.7.1.1.1.1.1.2.cmml"><mi id="S5.E6.m1.7.7.1.1.1.1.1.2.2" xref="S5.E6.m1.7.7.1.1.1.1.1.2.2.cmml">S</mi><mi id="S5.E6.m1.7.7.1.1.1.1.1.2.3" xref="S5.E6.m1.7.7.1.1.1.1.1.2.3.cmml">i</mi></msub><mo id="S5.E6.m1.7.7.1.1.1.1.1.1" xref="S5.E6.m1.7.7.1.1.1.1.1.1.cmml">−</mo><msub id="S5.E6.m1.7.7.1.1.1.1.1.3" xref="S5.E6.m1.7.7.1.1.1.1.1.3.cmml"><mi id="S5.E6.m1.7.7.1.1.1.1.1.3.2" xref="S5.E6.m1.7.7.1.1.1.1.1.3.2.cmml">H</mi><mi id="S5.E6.m1.7.7.1.1.1.1.1.3.3" xref="S5.E6.m1.7.7.1.1.1.1.1.3.3.cmml">k</mi></msub></mrow><mo id="S5.E6.m1.7.7.1.1.1.1.3" rspace="0.055em" stretchy="false" xref="S5.E6.m1.7.7.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S5.E6.m1.8.8.2.3" rspace="0.222em" xref="S5.E6.m1.8.8.2.3.cmml">⋅</mo><mrow id="S5.E6.m1.8.8.2.2.1" xref="S5.E6.m1.8.8.2.2.1.1.cmml"><mo id="S5.E6.m1.8.8.2.2.1.2" stretchy="false" xref="S5.E6.m1.8.8.2.2.1.1.cmml">(</mo><mrow id="S5.E6.m1.8.8.2.2.1.1" xref="S5.E6.m1.8.8.2.2.1.1.cmml"><mrow id="S5.E6.m1.8.8.2.2.1.1.1.1" xref="S5.E6.m1.8.8.2.2.1.1.1.2.cmml"><mo id="S5.E6.m1.8.8.2.2.1.1.1.1.2" stretchy="false" xref="S5.E6.m1.8.8.2.2.1.1.1.2.1.cmml">|</mo><msub id="S5.E6.m1.8.8.2.2.1.1.1.1.1" xref="S5.E6.m1.8.8.2.2.1.1.1.1.1.cmml"><mi id="S5.E6.m1.8.8.2.2.1.1.1.1.1.2" xref="S5.E6.m1.8.8.2.2.1.1.1.1.1.2.cmml">S</mi><mi id="S5.E6.m1.8.8.2.2.1.1.1.1.1.3" xref="S5.E6.m1.8.8.2.2.1.1.1.1.1.3.cmml">i</mi></msub><mo id="S5.E6.m1.8.8.2.2.1.1.1.1.3" stretchy="false" xref="S5.E6.m1.8.8.2.2.1.1.1.2.1.cmml">|</mo></mrow><mo id="S5.E6.m1.8.8.2.2.1.1.3" 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id="S5.E6.m1.3.3.1.1.3.3.cmml" xref="S5.E6.m1.3.3.1.1.3.3">↑</ci></apply></apply></apply></apply><apply id="S5.E6.m1.10.10.4.2.1.1.2.cmml" xref="S5.E6.m1.10.10.4.2.1.1.2"><times id="S5.E6.m1.10.10.4.2.1.1.2.1.cmml" xref="S5.E6.m1.10.10.4.2.1.1.2.1"></times><ci id="S5.E6.m1.10.10.4.2.1.1.2.2a.cmml" xref="S5.E6.m1.10.10.4.2.1.1.2.2"><mtext class="ltx_mathvariant_italic" id="S5.E6.m1.10.10.4.2.1.1.2.2.cmml" xref="S5.E6.m1.10.10.4.2.1.1.2.2">g</mtext></ci><ci id="S5.E6.m1.6.6.cmml" xref="S5.E6.m1.6.6">𝑒</ci></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.E6.m1.10c">\hat{\rho}_{B}(S_{i}-H_{k})\cdot(|S_{i}|-|H_{k}|)=\sum_{e\in\hat{E}_{B}(S_{i}-% H_{k})}\textsl{g}(e)=\left(\sum_{e\in\hat{E}_{B}(S_{i})}\textsl{g}(e)\right)-% \left(\sum_{e\in\hat{E}_{B}(H_{k})\cup E_{\uparrow}}\textsl{g}(e)\right)</annotation><annotation encoding="application/x-llamapun" id="S5.E6.m1.10d">over^ start_ARG italic_ρ end_ARG start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT ( italic_S start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT - italic_H start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ) ⋅ ( | italic_S start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT | - | italic_H start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT | ) = ∑ start_POSTSUBSCRIPT italic_e ∈ over^ start_ARG italic_E end_ARG start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT ( italic_S start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT - italic_H start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ) end_POSTSUBSCRIPT g ( italic_e ) = ( ∑ start_POSTSUBSCRIPT italic_e ∈ over^ start_ARG italic_E end_ARG start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT ( italic_S start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ) end_POSTSUBSCRIPT g ( italic_e ) ) - ( ∑ start_POSTSUBSCRIPT italic_e ∈ over^ start_ARG italic_E end_ARG start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT ( italic_H start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ) ∪ italic_E start_POSTSUBSCRIPT ↑ end_POSTSUBSCRIPT end_POSTSUBSCRIPT g ( italic_e ) )</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_left_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(6)</span></td> </tr></tbody> </table> </div> <div class="ltx_para" id="S5.Thmtheorem3.p16"> <p class="ltx_p" id="S5.Thmtheorem3.p16.1"><span class="ltx_text ltx_font_italic" id="S5.Thmtheorem3.p16.1.1">We claim that:</span></p> <table class="ltx_equation ltx_eqn_table" id="S5.E7"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_left_padleft"></td> <td class="ltx_eqn_cell ltx_align_left"><math alttext="\sum_{e\in\hat{E}_{B}(H_{k})\cup E_{\uparrow}}\textsl{g}(e)\leq\sum_{v\in V(H_% {k+1})}\textsl{g}(v)." class="ltx_Math" display="block" id="S5.E7.m1.5"><semantics id="S5.E7.m1.5a"><mrow id="S5.E7.m1.5.5.1" xref="S5.E7.m1.5.5.1.1.cmml"><mrow id="S5.E7.m1.5.5.1.1" xref="S5.E7.m1.5.5.1.1.cmml"><mrow 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encoding="application/x-llamapun" id="S5.E7.m1.5d">∑ start_POSTSUBSCRIPT italic_e ∈ over^ start_ARG italic_E end_ARG start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT ( italic_H start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ) ∪ italic_E start_POSTSUBSCRIPT ↑ end_POSTSUBSCRIPT end_POSTSUBSCRIPT g ( italic_e ) ≤ ∑ start_POSTSUBSCRIPT italic_v ∈ italic_V ( italic_H start_POSTSUBSCRIPT italic_k + 1 end_POSTSUBSCRIPT ) end_POSTSUBSCRIPT g ( italic_v ) .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_left_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(7)</span></td> </tr></tbody> </table> </div> <div class="ltx_para" id="S5.Thmtheorem3.p17"> <p class="ltx_p" id="S5.Thmtheorem3.p17.1"><span class="ltx_text ltx_font_italic" id="S5.Thmtheorem3.p17.1.1">Indeed:</span></p> <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.p1.5"><span class="ltx_text ltx_font_italic" id="S5.I2.i1.p1.5.1">if </span><math alttext="\overline{ab}\in E(H_{k})" class="ltx_Math" display="inline" id="S5.I2.i1.p1.1.m1.1"><semantics id="S5.I2.i1.p1.1.m1.1a"><mrow id="S5.I2.i1.p1.1.m1.1.1" xref="S5.I2.i1.p1.1.m1.1.1.cmml"><mover accent="true" id="S5.I2.i1.p1.1.m1.1.1.3" xref="S5.I2.i1.p1.1.m1.1.1.3.cmml"><mrow id="S5.I2.i1.p1.1.m1.1.1.3.2" xref="S5.I2.i1.p1.1.m1.1.1.3.2.cmml"><mi id="S5.I2.i1.p1.1.m1.1.1.3.2.2" xref="S5.I2.i1.p1.1.m1.1.1.3.2.2.cmml">a</mi><mo id="S5.I2.i1.p1.1.m1.1.1.3.2.1" xref="S5.I2.i1.p1.1.m1.1.1.3.2.1.cmml">⁢</mo><mi id="S5.I2.i1.p1.1.m1.1.1.3.2.3" xref="S5.I2.i1.p1.1.m1.1.1.3.2.3.cmml">b</mi></mrow><mo id="S5.I2.i1.p1.1.m1.1.1.3.1" xref="S5.I2.i1.p1.1.m1.1.1.3.1.cmml">¯</mo></mover><mo id="S5.I2.i1.p1.1.m1.1.1.2" xref="S5.I2.i1.p1.1.m1.1.1.2.cmml">∈</mo><mrow 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id="S5.I2.i1.p1.1.m1.1.1.3.cmml" xref="S5.I2.i1.p1.1.m1.1.1.3"><ci id="S5.I2.i1.p1.1.m1.1.1.3.1.cmml" xref="S5.I2.i1.p1.1.m1.1.1.3.1">¯</ci><apply id="S5.I2.i1.p1.1.m1.1.1.3.2.cmml" xref="S5.I2.i1.p1.1.m1.1.1.3.2"><times id="S5.I2.i1.p1.1.m1.1.1.3.2.1.cmml" xref="S5.I2.i1.p1.1.m1.1.1.3.2.1"></times><ci id="S5.I2.i1.p1.1.m1.1.1.3.2.2.cmml" xref="S5.I2.i1.p1.1.m1.1.1.3.2.2">𝑎</ci><ci id="S5.I2.i1.p1.1.m1.1.1.3.2.3.cmml" xref="S5.I2.i1.p1.1.m1.1.1.3.2.3">𝑏</ci></apply></apply><apply id="S5.I2.i1.p1.1.m1.1.1.1.cmml" xref="S5.I2.i1.p1.1.m1.1.1.1"><times id="S5.I2.i1.p1.1.m1.1.1.1.2.cmml" xref="S5.I2.i1.p1.1.m1.1.1.1.2"></times><ci id="S5.I2.i1.p1.1.m1.1.1.1.3.cmml" xref="S5.I2.i1.p1.1.m1.1.1.1.3">𝐸</ci><apply id="S5.I2.i1.p1.1.m1.1.1.1.1.1.1.cmml" xref="S5.I2.i1.p1.1.m1.1.1.1.1.1"><csymbol cd="ambiguous" id="S5.I2.i1.p1.1.m1.1.1.1.1.1.1.1.cmml" xref="S5.I2.i1.p1.1.m1.1.1.1.1.1">subscript</csymbol><ci id="S5.I2.i1.p1.1.m1.1.1.1.1.1.1.2.cmml" xref="S5.I2.i1.p1.1.m1.1.1.1.1.1.1.2">𝐻</ci><ci id="S5.I2.i1.p1.1.m1.1.1.1.1.1.1.3.cmml" xref="S5.I2.i1.p1.1.m1.1.1.1.1.1.1.3">𝑘</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.I2.i1.p1.1.m1.1c">\overline{ab}\in E(H_{k})</annotation><annotation encoding="application/x-llamapun" id="S5.I2.i1.p1.1.m1.1d">over¯ start_ARG italic_a italic_b end_ARG ∈ italic_E ( italic_H start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT )</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S5.I2.i1.p1.5.2"> then </span><math alttext="\textsl{g}(\overline{ab})" class="ltx_Math" display="inline" id="S5.I2.i1.p1.2.m2.1"><semantics id="S5.I2.i1.p1.2.m2.1a"><mrow id="S5.I2.i1.p1.2.m2.1.2" xref="S5.I2.i1.p1.2.m2.1.2.cmml"><mtext class="ltx_mathvariant_italic" id="S5.I2.i1.p1.2.m2.1.2.2" xref="S5.I2.i1.p1.2.m2.1.2.2a.cmml">g</mtext><mo id="S5.I2.i1.p1.2.m2.1.2.1" xref="S5.I2.i1.p1.2.m2.1.2.1.cmml">⁢</mo><mrow id="S5.I2.i1.p1.2.m2.1.2.3.2" xref="S5.I2.i1.p1.2.m2.1.1.cmml"><mo id="S5.I2.i1.p1.2.m2.1.2.3.2.1" stretchy="false" xref="S5.I2.i1.p1.2.m2.1.1.cmml">(</mo><mover accent="true" id="S5.I2.i1.p1.2.m2.1.1" xref="S5.I2.i1.p1.2.m2.1.1.cmml"><mrow id="S5.I2.i1.p1.2.m2.1.1.2" xref="S5.I2.i1.p1.2.m2.1.1.2.cmml"><mi id="S5.I2.i1.p1.2.m2.1.1.2.2" xref="S5.I2.i1.p1.2.m2.1.1.2.2.cmml">a</mi><mo id="S5.I2.i1.p1.2.m2.1.1.2.1" xref="S5.I2.i1.p1.2.m2.1.1.2.1.cmml">⁢</mo><mi id="S5.I2.i1.p1.2.m2.1.1.2.3" xref="S5.I2.i1.p1.2.m2.1.1.2.3.cmml">b</mi></mrow><mo id="S5.I2.i1.p1.2.m2.1.1.1" xref="S5.I2.i1.p1.2.m2.1.1.1.cmml">¯</mo></mover><mo id="S5.I2.i1.p1.2.m2.1.2.3.2.2" stretchy="false" xref="S5.I2.i1.p1.2.m2.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.I2.i1.p1.2.m2.1b"><apply id="S5.I2.i1.p1.2.m2.1.2.cmml" xref="S5.I2.i1.p1.2.m2.1.2"><times id="S5.I2.i1.p1.2.m2.1.2.1.cmml" xref="S5.I2.i1.p1.2.m2.1.2.1"></times><ci id="S5.I2.i1.p1.2.m2.1.2.2a.cmml" xref="S5.I2.i1.p1.2.m2.1.2.2"><mtext class="ltx_mathvariant_italic" id="S5.I2.i1.p1.2.m2.1.2.2.cmml" xref="S5.I2.i1.p1.2.m2.1.2.2">g</mtext></ci><apply id="S5.I2.i1.p1.2.m2.1.1.cmml" xref="S5.I2.i1.p1.2.m2.1.2.3.2"><ci id="S5.I2.i1.p1.2.m2.1.1.1.cmml" xref="S5.I2.i1.p1.2.m2.1.1.1">¯</ci><apply id="S5.I2.i1.p1.2.m2.1.1.2.cmml" xref="S5.I2.i1.p1.2.m2.1.1.2"><times id="S5.I2.i1.p1.2.m2.1.1.2.1.cmml" xref="S5.I2.i1.p1.2.m2.1.1.2.1"></times><ci id="S5.I2.i1.p1.2.m2.1.1.2.2.cmml" xref="S5.I2.i1.p1.2.m2.1.1.2.2">𝑎</ci><ci id="S5.I2.i1.p1.2.m2.1.1.2.3.cmml" xref="S5.I2.i1.p1.2.m2.1.1.2.3">𝑏</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.I2.i1.p1.2.m2.1c">\textsl{g}(\overline{ab})</annotation><annotation encoding="application/x-llamapun" id="S5.I2.i1.p1.2.m2.1d">g ( over¯ start_ARG italic_a italic_b end_ARG )</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S5.I2.i1.p1.5.3"> is distributed over </span><math alttext="\textsl{g}(a)" class="ltx_Math" display="inline" id="S5.I2.i1.p1.3.m3.1"><semantics id="S5.I2.i1.p1.3.m3.1a"><mrow id="S5.I2.i1.p1.3.m3.1.2" xref="S5.I2.i1.p1.3.m3.1.2.cmml"><mtext class="ltx_mathvariant_italic" id="S5.I2.i1.p1.3.m3.1.2.2" xref="S5.I2.i1.p1.3.m3.1.2.2a.cmml">g</mtext><mo id="S5.I2.i1.p1.3.m3.1.2.1" xref="S5.I2.i1.p1.3.m3.1.2.1.cmml">⁢</mo><mrow id="S5.I2.i1.p1.3.m3.1.2.3.2" xref="S5.I2.i1.p1.3.m3.1.2.cmml"><mo id="S5.I2.i1.p1.3.m3.1.2.3.2.1" stretchy="false" xref="S5.I2.i1.p1.3.m3.1.2.cmml">(</mo><mi id="S5.I2.i1.p1.3.m3.1.1" xref="S5.I2.i1.p1.3.m3.1.1.cmml">a</mi><mo id="S5.I2.i1.p1.3.m3.1.2.3.2.2" stretchy="false" xref="S5.I2.i1.p1.3.m3.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.I2.i1.p1.3.m3.1b"><apply id="S5.I2.i1.p1.3.m3.1.2.cmml" xref="S5.I2.i1.p1.3.m3.1.2"><times id="S5.I2.i1.p1.3.m3.1.2.1.cmml" xref="S5.I2.i1.p1.3.m3.1.2.1"></times><ci id="S5.I2.i1.p1.3.m3.1.2.2a.cmml" xref="S5.I2.i1.p1.3.m3.1.2.2"><mtext class="ltx_mathvariant_italic" id="S5.I2.i1.p1.3.m3.1.2.2.cmml" xref="S5.I2.i1.p1.3.m3.1.2.2">g</mtext></ci><ci id="S5.I2.i1.p1.3.m3.1.1.cmml" xref="S5.I2.i1.p1.3.m3.1.1">𝑎</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.I2.i1.p1.3.m3.1c">\textsl{g}(a)</annotation><annotation encoding="application/x-llamapun" id="S5.I2.i1.p1.3.m3.1d">g ( italic_a )</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S5.I2.i1.p1.5.4"> and </span><math alttext="\textsl{g}(b)" class="ltx_Math" display="inline" id="S5.I2.i1.p1.4.m4.1"><semantics id="S5.I2.i1.p1.4.m4.1a"><mrow id="S5.I2.i1.p1.4.m4.1.2" xref="S5.I2.i1.p1.4.m4.1.2.cmml"><mtext class="ltx_mathvariant_italic" id="S5.I2.i1.p1.4.m4.1.2.2" xref="S5.I2.i1.p1.4.m4.1.2.2a.cmml">g</mtext><mo id="S5.I2.i1.p1.4.m4.1.2.1" xref="S5.I2.i1.p1.4.m4.1.2.1.cmml">⁢</mo><mrow id="S5.I2.i1.p1.4.m4.1.2.3.2" xref="S5.I2.i1.p1.4.m4.1.2.cmml"><mo id="S5.I2.i1.p1.4.m4.1.2.3.2.1" stretchy="false" xref="S5.I2.i1.p1.4.m4.1.2.cmml">(</mo><mi id="S5.I2.i1.p1.4.m4.1.1" xref="S5.I2.i1.p1.4.m4.1.1.cmml">b</mi><mo id="S5.I2.i1.p1.4.m4.1.2.3.2.2" stretchy="false" xref="S5.I2.i1.p1.4.m4.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.I2.i1.p1.4.m4.1b"><apply id="S5.I2.i1.p1.4.m4.1.2.cmml" xref="S5.I2.i1.p1.4.m4.1.2"><times id="S5.I2.i1.p1.4.m4.1.2.1.cmml" xref="S5.I2.i1.p1.4.m4.1.2.1"></times><ci id="S5.I2.i1.p1.4.m4.1.2.2a.cmml" xref="S5.I2.i1.p1.4.m4.1.2.2"><mtext class="ltx_mathvariant_italic" id="S5.I2.i1.p1.4.m4.1.2.2.cmml" xref="S5.I2.i1.p1.4.m4.1.2.2">g</mtext></ci><ci id="S5.I2.i1.p1.4.m4.1.1.cmml" xref="S5.I2.i1.p1.4.m4.1.1">𝑏</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.I2.i1.p1.4.m4.1c">\textsl{g}(b)</annotation><annotation encoding="application/x-llamapun" id="S5.I2.i1.p1.4.m4.1d">g ( italic_b )</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S5.I2.i1.p1.5.5"> with </span><math alttext="a,b\in H_{k}\subset H_{k+1}" class="ltx_Math" display="inline" id="S5.I2.i1.p1.5.m5.2"><semantics id="S5.I2.i1.p1.5.m5.2a"><mrow id="S5.I2.i1.p1.5.m5.2.3" xref="S5.I2.i1.p1.5.m5.2.3.cmml"><mrow id="S5.I2.i1.p1.5.m5.2.3.2.2" xref="S5.I2.i1.p1.5.m5.2.3.2.1.cmml"><mi id="S5.I2.i1.p1.5.m5.1.1" xref="S5.I2.i1.p1.5.m5.1.1.cmml">a</mi><mo id="S5.I2.i1.p1.5.m5.2.3.2.2.1" xref="S5.I2.i1.p1.5.m5.2.3.2.1.cmml">,</mo><mi id="S5.I2.i1.p1.5.m5.2.2" xref="S5.I2.i1.p1.5.m5.2.2.cmml">b</mi></mrow><mo id="S5.I2.i1.p1.5.m5.2.3.3" xref="S5.I2.i1.p1.5.m5.2.3.3.cmml">∈</mo><msub id="S5.I2.i1.p1.5.m5.2.3.4" xref="S5.I2.i1.p1.5.m5.2.3.4.cmml"><mi id="S5.I2.i1.p1.5.m5.2.3.4.2" xref="S5.I2.i1.p1.5.m5.2.3.4.2.cmml">H</mi><mi id="S5.I2.i1.p1.5.m5.2.3.4.3" xref="S5.I2.i1.p1.5.m5.2.3.4.3.cmml">k</mi></msub><mo id="S5.I2.i1.p1.5.m5.2.3.5" xref="S5.I2.i1.p1.5.m5.2.3.5.cmml">⊂</mo><msub id="S5.I2.i1.p1.5.m5.2.3.6" xref="S5.I2.i1.p1.5.m5.2.3.6.cmml"><mi id="S5.I2.i1.p1.5.m5.2.3.6.2" xref="S5.I2.i1.p1.5.m5.2.3.6.2.cmml">H</mi><mrow id="S5.I2.i1.p1.5.m5.2.3.6.3" xref="S5.I2.i1.p1.5.m5.2.3.6.3.cmml"><mi id="S5.I2.i1.p1.5.m5.2.3.6.3.2" xref="S5.I2.i1.p1.5.m5.2.3.6.3.2.cmml">k</mi><mo id="S5.I2.i1.p1.5.m5.2.3.6.3.1" xref="S5.I2.i1.p1.5.m5.2.3.6.3.1.cmml">+</mo><mn id="S5.I2.i1.p1.5.m5.2.3.6.3.3" xref="S5.I2.i1.p1.5.m5.2.3.6.3.3.cmml">1</mn></mrow></msub></mrow><annotation-xml encoding="MathML-Content" id="S5.I2.i1.p1.5.m5.2b"><apply id="S5.I2.i1.p1.5.m5.2.3.cmml" xref="S5.I2.i1.p1.5.m5.2.3"><and id="S5.I2.i1.p1.5.m5.2.3a.cmml" xref="S5.I2.i1.p1.5.m5.2.3"></and><apply id="S5.I2.i1.p1.5.m5.2.3b.cmml" xref="S5.I2.i1.p1.5.m5.2.3"><in id="S5.I2.i1.p1.5.m5.2.3.3.cmml" xref="S5.I2.i1.p1.5.m5.2.3.3"></in><list id="S5.I2.i1.p1.5.m5.2.3.2.1.cmml" xref="S5.I2.i1.p1.5.m5.2.3.2.2"><ci id="S5.I2.i1.p1.5.m5.1.1.cmml" xref="S5.I2.i1.p1.5.m5.1.1">𝑎</ci><ci id="S5.I2.i1.p1.5.m5.2.2.cmml" xref="S5.I2.i1.p1.5.m5.2.2">𝑏</ci></list><apply id="S5.I2.i1.p1.5.m5.2.3.4.cmml" xref="S5.I2.i1.p1.5.m5.2.3.4"><csymbol cd="ambiguous" id="S5.I2.i1.p1.5.m5.2.3.4.1.cmml" xref="S5.I2.i1.p1.5.m5.2.3.4">subscript</csymbol><ci id="S5.I2.i1.p1.5.m5.2.3.4.2.cmml" xref="S5.I2.i1.p1.5.m5.2.3.4.2">𝐻</ci><ci id="S5.I2.i1.p1.5.m5.2.3.4.3.cmml" xref="S5.I2.i1.p1.5.m5.2.3.4.3">𝑘</ci></apply></apply><apply id="S5.I2.i1.p1.5.m5.2.3c.cmml" xref="S5.I2.i1.p1.5.m5.2.3"><subset id="S5.I2.i1.p1.5.m5.2.3.5.cmml" xref="S5.I2.i1.p1.5.m5.2.3.5"></subset><share href="https://arxiv.org/html/2411.12694v2#S5.I2.i1.p1.5.m5.2.3.4.cmml" id="S5.I2.i1.p1.5.m5.2.3d.cmml" xref="S5.I2.i1.p1.5.m5.2.3"></share><apply id="S5.I2.i1.p1.5.m5.2.3.6.cmml" xref="S5.I2.i1.p1.5.m5.2.3.6"><csymbol cd="ambiguous" id="S5.I2.i1.p1.5.m5.2.3.6.1.cmml" xref="S5.I2.i1.p1.5.m5.2.3.6">subscript</csymbol><ci id="S5.I2.i1.p1.5.m5.2.3.6.2.cmml" xref="S5.I2.i1.p1.5.m5.2.3.6.2">𝐻</ci><apply id="S5.I2.i1.p1.5.m5.2.3.6.3.cmml" xref="S5.I2.i1.p1.5.m5.2.3.6.3"><plus id="S5.I2.i1.p1.5.m5.2.3.6.3.1.cmml" xref="S5.I2.i1.p1.5.m5.2.3.6.3.1"></plus><ci id="S5.I2.i1.p1.5.m5.2.3.6.3.2.cmml" xref="S5.I2.i1.p1.5.m5.2.3.6.3.2">𝑘</ci><cn id="S5.I2.i1.p1.5.m5.2.3.6.3.3.cmml" type="integer" xref="S5.I2.i1.p1.5.m5.2.3.6.3.3">1</cn></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.I2.i1.p1.5.m5.2c">a,b\in H_{k}\subset H_{k+1}</annotation><annotation encoding="application/x-llamapun" id="S5.I2.i1.p1.5.m5.2d">italic_a , italic_b ∈ italic_H start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ⊂ italic_H start_POSTSUBSCRIPT italic_k + 1 end_POSTSUBSCRIPT</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S5.I2.i1.p1.5.6">.</span></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.p1.7"><span class="ltx_text ltx_font_italic" id="S5.I2.i2.p1.7.1">if </span><math alttext="\overline{ab}\in E_{\uparrow}" class="ltx_Math" display="inline" id="S5.I2.i2.p1.1.m1.1"><semantics id="S5.I2.i2.p1.1.m1.1a"><mrow id="S5.I2.i2.p1.1.m1.1.1" xref="S5.I2.i2.p1.1.m1.1.1.cmml"><mover accent="true" id="S5.I2.i2.p1.1.m1.1.1.2" xref="S5.I2.i2.p1.1.m1.1.1.2.cmml"><mrow id="S5.I2.i2.p1.1.m1.1.1.2.2" xref="S5.I2.i2.p1.1.m1.1.1.2.2.cmml"><mi id="S5.I2.i2.p1.1.m1.1.1.2.2.2" xref="S5.I2.i2.p1.1.m1.1.1.2.2.2.cmml">a</mi><mo id="S5.I2.i2.p1.1.m1.1.1.2.2.1" xref="S5.I2.i2.p1.1.m1.1.1.2.2.1.cmml">⁢</mo><mi id="S5.I2.i2.p1.1.m1.1.1.2.2.3" xref="S5.I2.i2.p1.1.m1.1.1.2.2.3.cmml">b</mi></mrow><mo id="S5.I2.i2.p1.1.m1.1.1.2.1" xref="S5.I2.i2.p1.1.m1.1.1.2.1.cmml">¯</mo></mover><mo id="S5.I2.i2.p1.1.m1.1.1.1" xref="S5.I2.i2.p1.1.m1.1.1.1.cmml">∈</mo><msub id="S5.I2.i2.p1.1.m1.1.1.3" xref="S5.I2.i2.p1.1.m1.1.1.3.cmml"><mi id="S5.I2.i2.p1.1.m1.1.1.3.2" xref="S5.I2.i2.p1.1.m1.1.1.3.2.cmml">E</mi><mo id="S5.I2.i2.p1.1.m1.1.1.3.3" stretchy="false" xref="S5.I2.i2.p1.1.m1.1.1.3.3.cmml">↑</mo></msub></mrow><annotation-xml encoding="MathML-Content" id="S5.I2.i2.p1.1.m1.1b"><apply id="S5.I2.i2.p1.1.m1.1.1.cmml" xref="S5.I2.i2.p1.1.m1.1.1"><in id="S5.I2.i2.p1.1.m1.1.1.1.cmml" xref="S5.I2.i2.p1.1.m1.1.1.1"></in><apply id="S5.I2.i2.p1.1.m1.1.1.2.cmml" xref="S5.I2.i2.p1.1.m1.1.1.2"><ci id="S5.I2.i2.p1.1.m1.1.1.2.1.cmml" xref="S5.I2.i2.p1.1.m1.1.1.2.1">¯</ci><apply id="S5.I2.i2.p1.1.m1.1.1.2.2.cmml" xref="S5.I2.i2.p1.1.m1.1.1.2.2"><times id="S5.I2.i2.p1.1.m1.1.1.2.2.1.cmml" xref="S5.I2.i2.p1.1.m1.1.1.2.2.1"></times><ci id="S5.I2.i2.p1.1.m1.1.1.2.2.2.cmml" xref="S5.I2.i2.p1.1.m1.1.1.2.2.2">𝑎</ci><ci id="S5.I2.i2.p1.1.m1.1.1.2.2.3.cmml" xref="S5.I2.i2.p1.1.m1.1.1.2.2.3">𝑏</ci></apply></apply><apply id="S5.I2.i2.p1.1.m1.1.1.3.cmml" xref="S5.I2.i2.p1.1.m1.1.1.3"><csymbol cd="ambiguous" id="S5.I2.i2.p1.1.m1.1.1.3.1.cmml" xref="S5.I2.i2.p1.1.m1.1.1.3">subscript</csymbol><ci id="S5.I2.i2.p1.1.m1.1.1.3.2.cmml" xref="S5.I2.i2.p1.1.m1.1.1.3.2">𝐸</ci><ci id="S5.I2.i2.p1.1.m1.1.1.3.3.cmml" xref="S5.I2.i2.p1.1.m1.1.1.3.3">↑</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.I2.i2.p1.1.m1.1c">\overline{ab}\in E_{\uparrow}</annotation><annotation encoding="application/x-llamapun" id="S5.I2.i2.p1.1.m1.1d">over¯ start_ARG italic_a italic_b end_ARG ∈ italic_E start_POSTSUBSCRIPT ↑ end_POSTSUBSCRIPT</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S5.I2.i2.p1.7.2"> with </span><math alttext="a\in H_{k}" class="ltx_Math" display="inline" id="S5.I2.i2.p1.2.m2.1"><semantics id="S5.I2.i2.p1.2.m2.1a"><mrow id="S5.I2.i2.p1.2.m2.1.1" xref="S5.I2.i2.p1.2.m2.1.1.cmml"><mi id="S5.I2.i2.p1.2.m2.1.1.2" xref="S5.I2.i2.p1.2.m2.1.1.2.cmml">a</mi><mo id="S5.I2.i2.p1.2.m2.1.1.1" xref="S5.I2.i2.p1.2.m2.1.1.1.cmml">∈</mo><msub id="S5.I2.i2.p1.2.m2.1.1.3" xref="S5.I2.i2.p1.2.m2.1.1.3.cmml"><mi id="S5.I2.i2.p1.2.m2.1.1.3.2" xref="S5.I2.i2.p1.2.m2.1.1.3.2.cmml">H</mi><mi id="S5.I2.i2.p1.2.m2.1.1.3.3" xref="S5.I2.i2.p1.2.m2.1.1.3.3.cmml">k</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S5.I2.i2.p1.2.m2.1b"><apply id="S5.I2.i2.p1.2.m2.1.1.cmml" xref="S5.I2.i2.p1.2.m2.1.1"><in id="S5.I2.i2.p1.2.m2.1.1.1.cmml" xref="S5.I2.i2.p1.2.m2.1.1.1"></in><ci id="S5.I2.i2.p1.2.m2.1.1.2.cmml" xref="S5.I2.i2.p1.2.m2.1.1.2">𝑎</ci><apply id="S5.I2.i2.p1.2.m2.1.1.3.cmml" xref="S5.I2.i2.p1.2.m2.1.1.3"><csymbol cd="ambiguous" id="S5.I2.i2.p1.2.m2.1.1.3.1.cmml" xref="S5.I2.i2.p1.2.m2.1.1.3">subscript</csymbol><ci id="S5.I2.i2.p1.2.m2.1.1.3.2.cmml" xref="S5.I2.i2.p1.2.m2.1.1.3.2">𝐻</ci><ci id="S5.I2.i2.p1.2.m2.1.1.3.3.cmml" xref="S5.I2.i2.p1.2.m2.1.1.3.3">𝑘</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.I2.i2.p1.2.m2.1c">a\in H_{k}</annotation><annotation encoding="application/x-llamapun" id="S5.I2.i2.p1.2.m2.1d">italic_a ∈ italic_H start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S5.I2.i2.p1.7.3"> and </span><math alttext="b\in S_{i}\cap H_{k+1}" class="ltx_Math" display="inline" id="S5.I2.i2.p1.3.m3.1"><semantics id="S5.I2.i2.p1.3.m3.1a"><mrow id="S5.I2.i2.p1.3.m3.1.1" xref="S5.I2.i2.p1.3.m3.1.1.cmml"><mi id="S5.I2.i2.p1.3.m3.1.1.2" xref="S5.I2.i2.p1.3.m3.1.1.2.cmml">b</mi><mo id="S5.I2.i2.p1.3.m3.1.1.1" xref="S5.I2.i2.p1.3.m3.1.1.1.cmml">∈</mo><mrow id="S5.I2.i2.p1.3.m3.1.1.3" xref="S5.I2.i2.p1.3.m3.1.1.3.cmml"><msub id="S5.I2.i2.p1.3.m3.1.1.3.2" xref="S5.I2.i2.p1.3.m3.1.1.3.2.cmml"><mi id="S5.I2.i2.p1.3.m3.1.1.3.2.2" xref="S5.I2.i2.p1.3.m3.1.1.3.2.2.cmml">S</mi><mi id="S5.I2.i2.p1.3.m3.1.1.3.2.3" xref="S5.I2.i2.p1.3.m3.1.1.3.2.3.cmml">i</mi></msub><mo id="S5.I2.i2.p1.3.m3.1.1.3.1" xref="S5.I2.i2.p1.3.m3.1.1.3.1.cmml">∩</mo><msub id="S5.I2.i2.p1.3.m3.1.1.3.3" xref="S5.I2.i2.p1.3.m3.1.1.3.3.cmml"><mi id="S5.I2.i2.p1.3.m3.1.1.3.3.2" xref="S5.I2.i2.p1.3.m3.1.1.3.3.2.cmml">H</mi><mrow id="S5.I2.i2.p1.3.m3.1.1.3.3.3" xref="S5.I2.i2.p1.3.m3.1.1.3.3.3.cmml"><mi id="S5.I2.i2.p1.3.m3.1.1.3.3.3.2" xref="S5.I2.i2.p1.3.m3.1.1.3.3.3.2.cmml">k</mi><mo id="S5.I2.i2.p1.3.m3.1.1.3.3.3.1" xref="S5.I2.i2.p1.3.m3.1.1.3.3.3.1.cmml">+</mo><mn id="S5.I2.i2.p1.3.m3.1.1.3.3.3.3" xref="S5.I2.i2.p1.3.m3.1.1.3.3.3.3.cmml">1</mn></mrow></msub></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.I2.i2.p1.3.m3.1b"><apply id="S5.I2.i2.p1.3.m3.1.1.cmml" xref="S5.I2.i2.p1.3.m3.1.1"><in id="S5.I2.i2.p1.3.m3.1.1.1.cmml" xref="S5.I2.i2.p1.3.m3.1.1.1"></in><ci id="S5.I2.i2.p1.3.m3.1.1.2.cmml" xref="S5.I2.i2.p1.3.m3.1.1.2">𝑏</ci><apply id="S5.I2.i2.p1.3.m3.1.1.3.cmml" xref="S5.I2.i2.p1.3.m3.1.1.3"><intersect id="S5.I2.i2.p1.3.m3.1.1.3.1.cmml" xref="S5.I2.i2.p1.3.m3.1.1.3.1"></intersect><apply id="S5.I2.i2.p1.3.m3.1.1.3.2.cmml" xref="S5.I2.i2.p1.3.m3.1.1.3.2"><csymbol cd="ambiguous" id="S5.I2.i2.p1.3.m3.1.1.3.2.1.cmml" xref="S5.I2.i2.p1.3.m3.1.1.3.2">subscript</csymbol><ci id="S5.I2.i2.p1.3.m3.1.1.3.2.2.cmml" xref="S5.I2.i2.p1.3.m3.1.1.3.2.2">𝑆</ci><ci id="S5.I2.i2.p1.3.m3.1.1.3.2.3.cmml" xref="S5.I2.i2.p1.3.m3.1.1.3.2.3">𝑖</ci></apply><apply id="S5.I2.i2.p1.3.m3.1.1.3.3.cmml" xref="S5.I2.i2.p1.3.m3.1.1.3.3"><csymbol cd="ambiguous" id="S5.I2.i2.p1.3.m3.1.1.3.3.1.cmml" xref="S5.I2.i2.p1.3.m3.1.1.3.3">subscript</csymbol><ci id="S5.I2.i2.p1.3.m3.1.1.3.3.2.cmml" xref="S5.I2.i2.p1.3.m3.1.1.3.3.2">𝐻</ci><apply id="S5.I2.i2.p1.3.m3.1.1.3.3.3.cmml" xref="S5.I2.i2.p1.3.m3.1.1.3.3.3"><plus id="S5.I2.i2.p1.3.m3.1.1.3.3.3.1.cmml" xref="S5.I2.i2.p1.3.m3.1.1.3.3.3.1"></plus><ci id="S5.I2.i2.p1.3.m3.1.1.3.3.3.2.cmml" xref="S5.I2.i2.p1.3.m3.1.1.3.3.3.2">𝑘</ci><cn id="S5.I2.i2.p1.3.m3.1.1.3.3.3.3.cmml" type="integer" xref="S5.I2.i2.p1.3.m3.1.1.3.3.3.3">1</cn></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.I2.i2.p1.3.m3.1c">b\in S_{i}\cap H_{k+1}</annotation><annotation encoding="application/x-llamapun" id="S5.I2.i2.p1.3.m3.1d">italic_b ∈ italic_S start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ∩ italic_H start_POSTSUBSCRIPT italic_k + 1 end_POSTSUBSCRIPT</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S5.I2.i2.p1.7.4">, then then </span><math alttext="\textsl{g}(\overline{ab})" class="ltx_Math" display="inline" id="S5.I2.i2.p1.4.m4.1"><semantics id="S5.I2.i2.p1.4.m4.1a"><mrow id="S5.I2.i2.p1.4.m4.1.2" xref="S5.I2.i2.p1.4.m4.1.2.cmml"><mtext class="ltx_mathvariant_italic" id="S5.I2.i2.p1.4.m4.1.2.2" xref="S5.I2.i2.p1.4.m4.1.2.2a.cmml">g</mtext><mo id="S5.I2.i2.p1.4.m4.1.2.1" xref="S5.I2.i2.p1.4.m4.1.2.1.cmml">⁢</mo><mrow id="S5.I2.i2.p1.4.m4.1.2.3.2" xref="S5.I2.i2.p1.4.m4.1.1.cmml"><mo id="S5.I2.i2.p1.4.m4.1.2.3.2.1" stretchy="false" xref="S5.I2.i2.p1.4.m4.1.1.cmml">(</mo><mover accent="true" id="S5.I2.i2.p1.4.m4.1.1" xref="S5.I2.i2.p1.4.m4.1.1.cmml"><mrow id="S5.I2.i2.p1.4.m4.1.1.2" xref="S5.I2.i2.p1.4.m4.1.1.2.cmml"><mi id="S5.I2.i2.p1.4.m4.1.1.2.2" xref="S5.I2.i2.p1.4.m4.1.1.2.2.cmml">a</mi><mo id="S5.I2.i2.p1.4.m4.1.1.2.1" xref="S5.I2.i2.p1.4.m4.1.1.2.1.cmml">⁢</mo><mi id="S5.I2.i2.p1.4.m4.1.1.2.3" xref="S5.I2.i2.p1.4.m4.1.1.2.3.cmml">b</mi></mrow><mo id="S5.I2.i2.p1.4.m4.1.1.1" xref="S5.I2.i2.p1.4.m4.1.1.1.cmml">¯</mo></mover><mo id="S5.I2.i2.p1.4.m4.1.2.3.2.2" stretchy="false" xref="S5.I2.i2.p1.4.m4.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.I2.i2.p1.4.m4.1b"><apply id="S5.I2.i2.p1.4.m4.1.2.cmml" xref="S5.I2.i2.p1.4.m4.1.2"><times id="S5.I2.i2.p1.4.m4.1.2.1.cmml" xref="S5.I2.i2.p1.4.m4.1.2.1"></times><ci id="S5.I2.i2.p1.4.m4.1.2.2a.cmml" xref="S5.I2.i2.p1.4.m4.1.2.2"><mtext class="ltx_mathvariant_italic" id="S5.I2.i2.p1.4.m4.1.2.2.cmml" xref="S5.I2.i2.p1.4.m4.1.2.2">g</mtext></ci><apply id="S5.I2.i2.p1.4.m4.1.1.cmml" xref="S5.I2.i2.p1.4.m4.1.2.3.2"><ci id="S5.I2.i2.p1.4.m4.1.1.1.cmml" xref="S5.I2.i2.p1.4.m4.1.1.1">¯</ci><apply id="S5.I2.i2.p1.4.m4.1.1.2.cmml" xref="S5.I2.i2.p1.4.m4.1.1.2"><times id="S5.I2.i2.p1.4.m4.1.1.2.1.cmml" xref="S5.I2.i2.p1.4.m4.1.1.2.1"></times><ci id="S5.I2.i2.p1.4.m4.1.1.2.2.cmml" xref="S5.I2.i2.p1.4.m4.1.1.2.2">𝑎</ci><ci id="S5.I2.i2.p1.4.m4.1.1.2.3.cmml" xref="S5.I2.i2.p1.4.m4.1.1.2.3">𝑏</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.I2.i2.p1.4.m4.1c">\textsl{g}(\overline{ab})</annotation><annotation encoding="application/x-llamapun" id="S5.I2.i2.p1.4.m4.1d">g ( over¯ start_ARG italic_a italic_b end_ARG )</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S5.I2.i2.p1.7.5"> is distributed over </span><math alttext="\textsl{g}(a)" class="ltx_Math" display="inline" id="S5.I2.i2.p1.5.m5.1"><semantics id="S5.I2.i2.p1.5.m5.1a"><mrow id="S5.I2.i2.p1.5.m5.1.2" xref="S5.I2.i2.p1.5.m5.1.2.cmml"><mtext class="ltx_mathvariant_italic" id="S5.I2.i2.p1.5.m5.1.2.2" xref="S5.I2.i2.p1.5.m5.1.2.2a.cmml">g</mtext><mo id="S5.I2.i2.p1.5.m5.1.2.1" xref="S5.I2.i2.p1.5.m5.1.2.1.cmml">⁢</mo><mrow id="S5.I2.i2.p1.5.m5.1.2.3.2" xref="S5.I2.i2.p1.5.m5.1.2.cmml"><mo id="S5.I2.i2.p1.5.m5.1.2.3.2.1" stretchy="false" xref="S5.I2.i2.p1.5.m5.1.2.cmml">(</mo><mi id="S5.I2.i2.p1.5.m5.1.1" xref="S5.I2.i2.p1.5.m5.1.1.cmml">a</mi><mo id="S5.I2.i2.p1.5.m5.1.2.3.2.2" stretchy="false" xref="S5.I2.i2.p1.5.m5.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.I2.i2.p1.5.m5.1b"><apply id="S5.I2.i2.p1.5.m5.1.2.cmml" xref="S5.I2.i2.p1.5.m5.1.2"><times id="S5.I2.i2.p1.5.m5.1.2.1.cmml" xref="S5.I2.i2.p1.5.m5.1.2.1"></times><ci id="S5.I2.i2.p1.5.m5.1.2.2a.cmml" xref="S5.I2.i2.p1.5.m5.1.2.2"><mtext class="ltx_mathvariant_italic" id="S5.I2.i2.p1.5.m5.1.2.2.cmml" xref="S5.I2.i2.p1.5.m5.1.2.2">g</mtext></ci><ci id="S5.I2.i2.p1.5.m5.1.1.cmml" xref="S5.I2.i2.p1.5.m5.1.1">𝑎</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.I2.i2.p1.5.m5.1c">\textsl{g}(a)</annotation><annotation encoding="application/x-llamapun" id="S5.I2.i2.p1.5.m5.1d">g ( italic_a )</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S5.I2.i2.p1.7.6"> and </span><math alttext="\textsl{g}(b)" class="ltx_Math" display="inline" id="S5.I2.i2.p1.6.m6.1"><semantics id="S5.I2.i2.p1.6.m6.1a"><mrow id="S5.I2.i2.p1.6.m6.1.2" xref="S5.I2.i2.p1.6.m6.1.2.cmml"><mtext class="ltx_mathvariant_italic" id="S5.I2.i2.p1.6.m6.1.2.2" xref="S5.I2.i2.p1.6.m6.1.2.2a.cmml">g</mtext><mo id="S5.I2.i2.p1.6.m6.1.2.1" xref="S5.I2.i2.p1.6.m6.1.2.1.cmml">⁢</mo><mrow id="S5.I2.i2.p1.6.m6.1.2.3.2" xref="S5.I2.i2.p1.6.m6.1.2.cmml"><mo id="S5.I2.i2.p1.6.m6.1.2.3.2.1" stretchy="false" xref="S5.I2.i2.p1.6.m6.1.2.cmml">(</mo><mi id="S5.I2.i2.p1.6.m6.1.1" xref="S5.I2.i2.p1.6.m6.1.1.cmml">b</mi><mo id="S5.I2.i2.p1.6.m6.1.2.3.2.2" stretchy="false" xref="S5.I2.i2.p1.6.m6.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.I2.i2.p1.6.m6.1b"><apply id="S5.I2.i2.p1.6.m6.1.2.cmml" xref="S5.I2.i2.p1.6.m6.1.2"><times id="S5.I2.i2.p1.6.m6.1.2.1.cmml" xref="S5.I2.i2.p1.6.m6.1.2.1"></times><ci id="S5.I2.i2.p1.6.m6.1.2.2a.cmml" xref="S5.I2.i2.p1.6.m6.1.2.2"><mtext class="ltx_mathvariant_italic" id="S5.I2.i2.p1.6.m6.1.2.2.cmml" xref="S5.I2.i2.p1.6.m6.1.2.2">g</mtext></ci><ci id="S5.I2.i2.p1.6.m6.1.1.cmml" xref="S5.I2.i2.p1.6.m6.1.1">𝑏</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.I2.i2.p1.6.m6.1c">\textsl{g}(b)</annotation><annotation encoding="application/x-llamapun" id="S5.I2.i2.p1.6.m6.1d">g ( italic_b )</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S5.I2.i2.p1.7.7"> with </span><math alttext="a,b\in H_{k+1}" class="ltx_Math" display="inline" id="S5.I2.i2.p1.7.m7.2"><semantics id="S5.I2.i2.p1.7.m7.2a"><mrow id="S5.I2.i2.p1.7.m7.2.3" xref="S5.I2.i2.p1.7.m7.2.3.cmml"><mrow id="S5.I2.i2.p1.7.m7.2.3.2.2" xref="S5.I2.i2.p1.7.m7.2.3.2.1.cmml"><mi id="S5.I2.i2.p1.7.m7.1.1" xref="S5.I2.i2.p1.7.m7.1.1.cmml">a</mi><mo id="S5.I2.i2.p1.7.m7.2.3.2.2.1" xref="S5.I2.i2.p1.7.m7.2.3.2.1.cmml">,</mo><mi id="S5.I2.i2.p1.7.m7.2.2" xref="S5.I2.i2.p1.7.m7.2.2.cmml">b</mi></mrow><mo id="S5.I2.i2.p1.7.m7.2.3.1" xref="S5.I2.i2.p1.7.m7.2.3.1.cmml">∈</mo><msub id="S5.I2.i2.p1.7.m7.2.3.3" xref="S5.I2.i2.p1.7.m7.2.3.3.cmml"><mi id="S5.I2.i2.p1.7.m7.2.3.3.2" xref="S5.I2.i2.p1.7.m7.2.3.3.2.cmml">H</mi><mrow id="S5.I2.i2.p1.7.m7.2.3.3.3" xref="S5.I2.i2.p1.7.m7.2.3.3.3.cmml"><mi id="S5.I2.i2.p1.7.m7.2.3.3.3.2" xref="S5.I2.i2.p1.7.m7.2.3.3.3.2.cmml">k</mi><mo id="S5.I2.i2.p1.7.m7.2.3.3.3.1" xref="S5.I2.i2.p1.7.m7.2.3.3.3.1.cmml">+</mo><mn id="S5.I2.i2.p1.7.m7.2.3.3.3.3" xref="S5.I2.i2.p1.7.m7.2.3.3.3.3.cmml">1</mn></mrow></msub></mrow><annotation-xml encoding="MathML-Content" id="S5.I2.i2.p1.7.m7.2b"><apply id="S5.I2.i2.p1.7.m7.2.3.cmml" xref="S5.I2.i2.p1.7.m7.2.3"><in id="S5.I2.i2.p1.7.m7.2.3.1.cmml" xref="S5.I2.i2.p1.7.m7.2.3.1"></in><list id="S5.I2.i2.p1.7.m7.2.3.2.1.cmml" xref="S5.I2.i2.p1.7.m7.2.3.2.2"><ci id="S5.I2.i2.p1.7.m7.1.1.cmml" xref="S5.I2.i2.p1.7.m7.1.1">𝑎</ci><ci id="S5.I2.i2.p1.7.m7.2.2.cmml" xref="S5.I2.i2.p1.7.m7.2.2">𝑏</ci></list><apply id="S5.I2.i2.p1.7.m7.2.3.3.cmml" xref="S5.I2.i2.p1.7.m7.2.3.3"><csymbol cd="ambiguous" id="S5.I2.i2.p1.7.m7.2.3.3.1.cmml" xref="S5.I2.i2.p1.7.m7.2.3.3">subscript</csymbol><ci id="S5.I2.i2.p1.7.m7.2.3.3.2.cmml" xref="S5.I2.i2.p1.7.m7.2.3.3.2">𝐻</ci><apply id="S5.I2.i2.p1.7.m7.2.3.3.3.cmml" xref="S5.I2.i2.p1.7.m7.2.3.3.3"><plus id="S5.I2.i2.p1.7.m7.2.3.3.3.1.cmml" xref="S5.I2.i2.p1.7.m7.2.3.3.3.1"></plus><ci id="S5.I2.i2.p1.7.m7.2.3.3.3.2.cmml" xref="S5.I2.i2.p1.7.m7.2.3.3.3.2">𝑘</ci><cn id="S5.I2.i2.p1.7.m7.2.3.3.3.3.cmml" type="integer" xref="S5.I2.i2.p1.7.m7.2.3.3.3.3">1</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.I2.i2.p1.7.m7.2c">a,b\in H_{k+1}</annotation><annotation encoding="application/x-llamapun" id="S5.I2.i2.p1.7.m7.2d">italic_a , italic_b ∈ italic_H start_POSTSUBSCRIPT italic_k + 1 end_POSTSUBSCRIPT</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S5.I2.i2.p1.7.8">.</span></p> </div> </li> <li class="ltx_item" id="S5.I2.i3" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">•</span> <div class="ltx_para" id="S5.I2.i3.p1"> <p class="ltx_p" id="S5.I2.i3.p1.6"><span class="ltx_text ltx_font_italic" id="S5.I2.i3.p1.6.1">if </span><math alttext="\overline{ab}\in E_{\uparrow}" class="ltx_Math" display="inline" id="S5.I2.i3.p1.1.m1.1"><semantics id="S5.I2.i3.p1.1.m1.1a"><mrow id="S5.I2.i3.p1.1.m1.1.1" xref="S5.I2.i3.p1.1.m1.1.1.cmml"><mover accent="true" id="S5.I2.i3.p1.1.m1.1.1.2" xref="S5.I2.i3.p1.1.m1.1.1.2.cmml"><mrow id="S5.I2.i3.p1.1.m1.1.1.2.2" xref="S5.I2.i3.p1.1.m1.1.1.2.2.cmml"><mi id="S5.I2.i3.p1.1.m1.1.1.2.2.2" xref="S5.I2.i3.p1.1.m1.1.1.2.2.2.cmml">a</mi><mo id="S5.I2.i3.p1.1.m1.1.1.2.2.1" xref="S5.I2.i3.p1.1.m1.1.1.2.2.1.cmml">⁢</mo><mi id="S5.I2.i3.p1.1.m1.1.1.2.2.3" xref="S5.I2.i3.p1.1.m1.1.1.2.2.3.cmml">b</mi></mrow><mo id="S5.I2.i3.p1.1.m1.1.1.2.1" xref="S5.I2.i3.p1.1.m1.1.1.2.1.cmml">¯</mo></mover><mo id="S5.I2.i3.p1.1.m1.1.1.1" xref="S5.I2.i3.p1.1.m1.1.1.1.cmml">∈</mo><msub id="S5.I2.i3.p1.1.m1.1.1.3" xref="S5.I2.i3.p1.1.m1.1.1.3.cmml"><mi id="S5.I2.i3.p1.1.m1.1.1.3.2" xref="S5.I2.i3.p1.1.m1.1.1.3.2.cmml">E</mi><mo id="S5.I2.i3.p1.1.m1.1.1.3.3" stretchy="false" xref="S5.I2.i3.p1.1.m1.1.1.3.3.cmml">↑</mo></msub></mrow><annotation-xml encoding="MathML-Content" id="S5.I2.i3.p1.1.m1.1b"><apply id="S5.I2.i3.p1.1.m1.1.1.cmml" xref="S5.I2.i3.p1.1.m1.1.1"><in id="S5.I2.i3.p1.1.m1.1.1.1.cmml" xref="S5.I2.i3.p1.1.m1.1.1.1"></in><apply id="S5.I2.i3.p1.1.m1.1.1.2.cmml" xref="S5.I2.i3.p1.1.m1.1.1.2"><ci id="S5.I2.i3.p1.1.m1.1.1.2.1.cmml" xref="S5.I2.i3.p1.1.m1.1.1.2.1">¯</ci><apply id="S5.I2.i3.p1.1.m1.1.1.2.2.cmml" xref="S5.I2.i3.p1.1.m1.1.1.2.2"><times id="S5.I2.i3.p1.1.m1.1.1.2.2.1.cmml" xref="S5.I2.i3.p1.1.m1.1.1.2.2.1"></times><ci id="S5.I2.i3.p1.1.m1.1.1.2.2.2.cmml" xref="S5.I2.i3.p1.1.m1.1.1.2.2.2">𝑎</ci><ci id="S5.I2.i3.p1.1.m1.1.1.2.2.3.cmml" xref="S5.I2.i3.p1.1.m1.1.1.2.2.3">𝑏</ci></apply></apply><apply id="S5.I2.i3.p1.1.m1.1.1.3.cmml" xref="S5.I2.i3.p1.1.m1.1.1.3"><csymbol cd="ambiguous" id="S5.I2.i3.p1.1.m1.1.1.3.1.cmml" xref="S5.I2.i3.p1.1.m1.1.1.3">subscript</csymbol><ci id="S5.I2.i3.p1.1.m1.1.1.3.2.cmml" xref="S5.I2.i3.p1.1.m1.1.1.3.2">𝐸</ci><ci id="S5.I2.i3.p1.1.m1.1.1.3.3.cmml" xref="S5.I2.i3.p1.1.m1.1.1.3.3">↑</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.I2.i3.p1.1.m1.1c">\overline{ab}\in E_{\uparrow}</annotation><annotation encoding="application/x-llamapun" id="S5.I2.i3.p1.1.m1.1d">over¯ start_ARG italic_a italic_b end_ARG ∈ italic_E start_POSTSUBSCRIPT ↑ end_POSTSUBSCRIPT</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S5.I2.i3.p1.6.2"> with </span><math alttext="a\in H_{k}" class="ltx_Math" display="inline" id="S5.I2.i3.p1.2.m2.1"><semantics id="S5.I2.i3.p1.2.m2.1a"><mrow id="S5.I2.i3.p1.2.m2.1.1" xref="S5.I2.i3.p1.2.m2.1.1.cmml"><mi id="S5.I2.i3.p1.2.m2.1.1.2" xref="S5.I2.i3.p1.2.m2.1.1.2.cmml">a</mi><mo id="S5.I2.i3.p1.2.m2.1.1.1" xref="S5.I2.i3.p1.2.m2.1.1.1.cmml">∈</mo><msub id="S5.I2.i3.p1.2.m2.1.1.3" xref="S5.I2.i3.p1.2.m2.1.1.3.cmml"><mi id="S5.I2.i3.p1.2.m2.1.1.3.2" xref="S5.I2.i3.p1.2.m2.1.1.3.2.cmml">H</mi><mi id="S5.I2.i3.p1.2.m2.1.1.3.3" xref="S5.I2.i3.p1.2.m2.1.1.3.3.cmml">k</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S5.I2.i3.p1.2.m2.1b"><apply id="S5.I2.i3.p1.2.m2.1.1.cmml" xref="S5.I2.i3.p1.2.m2.1.1"><in id="S5.I2.i3.p1.2.m2.1.1.1.cmml" xref="S5.I2.i3.p1.2.m2.1.1.1"></in><ci id="S5.I2.i3.p1.2.m2.1.1.2.cmml" xref="S5.I2.i3.p1.2.m2.1.1.2">𝑎</ci><apply id="S5.I2.i3.p1.2.m2.1.1.3.cmml" xref="S5.I2.i3.p1.2.m2.1.1.3"><csymbol cd="ambiguous" id="S5.I2.i3.p1.2.m2.1.1.3.1.cmml" xref="S5.I2.i3.p1.2.m2.1.1.3">subscript</csymbol><ci id="S5.I2.i3.p1.2.m2.1.1.3.2.cmml" xref="S5.I2.i3.p1.2.m2.1.1.3.2">𝐻</ci><ci id="S5.I2.i3.p1.2.m2.1.1.3.3.cmml" xref="S5.I2.i3.p1.2.m2.1.1.3.3">𝑘</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.I2.i3.p1.2.m2.1c">a\in H_{k}</annotation><annotation encoding="application/x-llamapun" id="S5.I2.i3.p1.2.m2.1d">italic_a ∈ italic_H start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S5.I2.i3.p1.6.3"> and </span><math alttext="b\in S_{i}-H_{k+1}" class="ltx_Math" display="inline" id="S5.I2.i3.p1.3.m3.1"><semantics id="S5.I2.i3.p1.3.m3.1a"><mrow id="S5.I2.i3.p1.3.m3.1.1" xref="S5.I2.i3.p1.3.m3.1.1.cmml"><mi id="S5.I2.i3.p1.3.m3.1.1.2" xref="S5.I2.i3.p1.3.m3.1.1.2.cmml">b</mi><mo id="S5.I2.i3.p1.3.m3.1.1.1" xref="S5.I2.i3.p1.3.m3.1.1.1.cmml">∈</mo><mrow id="S5.I2.i3.p1.3.m3.1.1.3" xref="S5.I2.i3.p1.3.m3.1.1.3.cmml"><msub id="S5.I2.i3.p1.3.m3.1.1.3.2" xref="S5.I2.i3.p1.3.m3.1.1.3.2.cmml"><mi id="S5.I2.i3.p1.3.m3.1.1.3.2.2" xref="S5.I2.i3.p1.3.m3.1.1.3.2.2.cmml">S</mi><mi id="S5.I2.i3.p1.3.m3.1.1.3.2.3" xref="S5.I2.i3.p1.3.m3.1.1.3.2.3.cmml">i</mi></msub><mo id="S5.I2.i3.p1.3.m3.1.1.3.1" xref="S5.I2.i3.p1.3.m3.1.1.3.1.cmml">−</mo><msub id="S5.I2.i3.p1.3.m3.1.1.3.3" xref="S5.I2.i3.p1.3.m3.1.1.3.3.cmml"><mi id="S5.I2.i3.p1.3.m3.1.1.3.3.2" xref="S5.I2.i3.p1.3.m3.1.1.3.3.2.cmml">H</mi><mrow id="S5.I2.i3.p1.3.m3.1.1.3.3.3" xref="S5.I2.i3.p1.3.m3.1.1.3.3.3.cmml"><mi id="S5.I2.i3.p1.3.m3.1.1.3.3.3.2" xref="S5.I2.i3.p1.3.m3.1.1.3.3.3.2.cmml">k</mi><mo id="S5.I2.i3.p1.3.m3.1.1.3.3.3.1" xref="S5.I2.i3.p1.3.m3.1.1.3.3.3.1.cmml">+</mo><mn id="S5.I2.i3.p1.3.m3.1.1.3.3.3.3" xref="S5.I2.i3.p1.3.m3.1.1.3.3.3.3.cmml">1</mn></mrow></msub></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.I2.i3.p1.3.m3.1b"><apply id="S5.I2.i3.p1.3.m3.1.1.cmml" xref="S5.I2.i3.p1.3.m3.1.1"><in id="S5.I2.i3.p1.3.m3.1.1.1.cmml" xref="S5.I2.i3.p1.3.m3.1.1.1"></in><ci id="S5.I2.i3.p1.3.m3.1.1.2.cmml" xref="S5.I2.i3.p1.3.m3.1.1.2">𝑏</ci><apply id="S5.I2.i3.p1.3.m3.1.1.3.cmml" xref="S5.I2.i3.p1.3.m3.1.1.3"><minus id="S5.I2.i3.p1.3.m3.1.1.3.1.cmml" xref="S5.I2.i3.p1.3.m3.1.1.3.1"></minus><apply id="S5.I2.i3.p1.3.m3.1.1.3.2.cmml" xref="S5.I2.i3.p1.3.m3.1.1.3.2"><csymbol cd="ambiguous" id="S5.I2.i3.p1.3.m3.1.1.3.2.1.cmml" xref="S5.I2.i3.p1.3.m3.1.1.3.2">subscript</csymbol><ci id="S5.I2.i3.p1.3.m3.1.1.3.2.2.cmml" xref="S5.I2.i3.p1.3.m3.1.1.3.2.2">𝑆</ci><ci id="S5.I2.i3.p1.3.m3.1.1.3.2.3.cmml" xref="S5.I2.i3.p1.3.m3.1.1.3.2.3">𝑖</ci></apply><apply id="S5.I2.i3.p1.3.m3.1.1.3.3.cmml" xref="S5.I2.i3.p1.3.m3.1.1.3.3"><csymbol cd="ambiguous" id="S5.I2.i3.p1.3.m3.1.1.3.3.1.cmml" xref="S5.I2.i3.p1.3.m3.1.1.3.3">subscript</csymbol><ci id="S5.I2.i3.p1.3.m3.1.1.3.3.2.cmml" xref="S5.I2.i3.p1.3.m3.1.1.3.3.2">𝐻</ci><apply id="S5.I2.i3.p1.3.m3.1.1.3.3.3.cmml" xref="S5.I2.i3.p1.3.m3.1.1.3.3.3"><plus id="S5.I2.i3.p1.3.m3.1.1.3.3.3.1.cmml" xref="S5.I2.i3.p1.3.m3.1.1.3.3.3.1"></plus><ci id="S5.I2.i3.p1.3.m3.1.1.3.3.3.2.cmml" xref="S5.I2.i3.p1.3.m3.1.1.3.3.3.2">𝑘</ci><cn id="S5.I2.i3.p1.3.m3.1.1.3.3.3.3.cmml" type="integer" xref="S5.I2.i3.p1.3.m3.1.1.3.3.3.3">1</cn></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.I2.i3.p1.3.m3.1c">b\in S_{i}-H_{k+1}</annotation><annotation encoding="application/x-llamapun" id="S5.I2.i3.p1.3.m3.1d">italic_b ∈ italic_S start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT - italic_H start_POSTSUBSCRIPT italic_k + 1 end_POSTSUBSCRIPT</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S5.I2.i3.p1.6.4">, then </span><math alttext="\textsl{g}(b)>(1+\eta)\textsl{g}(a)" class="ltx_Math" display="inline" id="S5.I2.i3.p1.4.m4.3"><semantics id="S5.I2.i3.p1.4.m4.3a"><mrow id="S5.I2.i3.p1.4.m4.3.3" xref="S5.I2.i3.p1.4.m4.3.3.cmml"><mrow id="S5.I2.i3.p1.4.m4.3.3.3" xref="S5.I2.i3.p1.4.m4.3.3.3.cmml"><mtext class="ltx_mathvariant_italic" id="S5.I2.i3.p1.4.m4.3.3.3.2" xref="S5.I2.i3.p1.4.m4.3.3.3.2a.cmml">g</mtext><mo id="S5.I2.i3.p1.4.m4.3.3.3.1" xref="S5.I2.i3.p1.4.m4.3.3.3.1.cmml">⁢</mo><mrow id="S5.I2.i3.p1.4.m4.3.3.3.3.2" xref="S5.I2.i3.p1.4.m4.3.3.3.cmml"><mo id="S5.I2.i3.p1.4.m4.3.3.3.3.2.1" stretchy="false" xref="S5.I2.i3.p1.4.m4.3.3.3.cmml">(</mo><mi id="S5.I2.i3.p1.4.m4.1.1" xref="S5.I2.i3.p1.4.m4.1.1.cmml">b</mi><mo id="S5.I2.i3.p1.4.m4.3.3.3.3.2.2" stretchy="false" xref="S5.I2.i3.p1.4.m4.3.3.3.cmml">)</mo></mrow></mrow><mo id="S5.I2.i3.p1.4.m4.3.3.2" xref="S5.I2.i3.p1.4.m4.3.3.2.cmml">></mo><mrow id="S5.I2.i3.p1.4.m4.3.3.1" xref="S5.I2.i3.p1.4.m4.3.3.1.cmml"><mrow id="S5.I2.i3.p1.4.m4.3.3.1.1.1" xref="S5.I2.i3.p1.4.m4.3.3.1.1.1.1.cmml"><mo id="S5.I2.i3.p1.4.m4.3.3.1.1.1.2" stretchy="false" xref="S5.I2.i3.p1.4.m4.3.3.1.1.1.1.cmml">(</mo><mrow id="S5.I2.i3.p1.4.m4.3.3.1.1.1.1" xref="S5.I2.i3.p1.4.m4.3.3.1.1.1.1.cmml"><mn id="S5.I2.i3.p1.4.m4.3.3.1.1.1.1.2" xref="S5.I2.i3.p1.4.m4.3.3.1.1.1.1.2.cmml">1</mn><mo id="S5.I2.i3.p1.4.m4.3.3.1.1.1.1.1" xref="S5.I2.i3.p1.4.m4.3.3.1.1.1.1.1.cmml">+</mo><mi id="S5.I2.i3.p1.4.m4.3.3.1.1.1.1.3" xref="S5.I2.i3.p1.4.m4.3.3.1.1.1.1.3.cmml">η</mi></mrow><mo id="S5.I2.i3.p1.4.m4.3.3.1.1.1.3" stretchy="false" xref="S5.I2.i3.p1.4.m4.3.3.1.1.1.1.cmml">)</mo></mrow><mo id="S5.I2.i3.p1.4.m4.3.3.1.2" xref="S5.I2.i3.p1.4.m4.3.3.1.2.cmml">⁢</mo><mtext class="ltx_mathvariant_italic" id="S5.I2.i3.p1.4.m4.3.3.1.3" xref="S5.I2.i3.p1.4.m4.3.3.1.3a.cmml">g</mtext><mo id="S5.I2.i3.p1.4.m4.3.3.1.2a" xref="S5.I2.i3.p1.4.m4.3.3.1.2.cmml">⁢</mo><mrow id="S5.I2.i3.p1.4.m4.3.3.1.4.2" xref="S5.I2.i3.p1.4.m4.3.3.1.cmml"><mo id="S5.I2.i3.p1.4.m4.3.3.1.4.2.1" stretchy="false" xref="S5.I2.i3.p1.4.m4.3.3.1.cmml">(</mo><mi id="S5.I2.i3.p1.4.m4.2.2" xref="S5.I2.i3.p1.4.m4.2.2.cmml">a</mi><mo id="S5.I2.i3.p1.4.m4.3.3.1.4.2.2" stretchy="false" xref="S5.I2.i3.p1.4.m4.3.3.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.I2.i3.p1.4.m4.3b"><apply id="S5.I2.i3.p1.4.m4.3.3.cmml" xref="S5.I2.i3.p1.4.m4.3.3"><gt id="S5.I2.i3.p1.4.m4.3.3.2.cmml" xref="S5.I2.i3.p1.4.m4.3.3.2"></gt><apply id="S5.I2.i3.p1.4.m4.3.3.3.cmml" xref="S5.I2.i3.p1.4.m4.3.3.3"><times id="S5.I2.i3.p1.4.m4.3.3.3.1.cmml" xref="S5.I2.i3.p1.4.m4.3.3.3.1"></times><ci id="S5.I2.i3.p1.4.m4.3.3.3.2a.cmml" xref="S5.I2.i3.p1.4.m4.3.3.3.2"><mtext class="ltx_mathvariant_italic" id="S5.I2.i3.p1.4.m4.3.3.3.2.cmml" xref="S5.I2.i3.p1.4.m4.3.3.3.2">g</mtext></ci><ci id="S5.I2.i3.p1.4.m4.1.1.cmml" xref="S5.I2.i3.p1.4.m4.1.1">𝑏</ci></apply><apply id="S5.I2.i3.p1.4.m4.3.3.1.cmml" xref="S5.I2.i3.p1.4.m4.3.3.1"><times id="S5.I2.i3.p1.4.m4.3.3.1.2.cmml" xref="S5.I2.i3.p1.4.m4.3.3.1.2"></times><apply id="S5.I2.i3.p1.4.m4.3.3.1.1.1.1.cmml" xref="S5.I2.i3.p1.4.m4.3.3.1.1.1"><plus id="S5.I2.i3.p1.4.m4.3.3.1.1.1.1.1.cmml" xref="S5.I2.i3.p1.4.m4.3.3.1.1.1.1.1"></plus><cn id="S5.I2.i3.p1.4.m4.3.3.1.1.1.1.2.cmml" type="integer" xref="S5.I2.i3.p1.4.m4.3.3.1.1.1.1.2">1</cn><ci id="S5.I2.i3.p1.4.m4.3.3.1.1.1.1.3.cmml" xref="S5.I2.i3.p1.4.m4.3.3.1.1.1.1.3">𝜂</ci></apply><ci id="S5.I2.i3.p1.4.m4.3.3.1.3a.cmml" xref="S5.I2.i3.p1.4.m4.3.3.1.3"><mtext class="ltx_mathvariant_italic" id="S5.I2.i3.p1.4.m4.3.3.1.3.cmml" xref="S5.I2.i3.p1.4.m4.3.3.1.3">g</mtext></ci><ci id="S5.I2.i3.p1.4.m4.2.2.cmml" xref="S5.I2.i3.p1.4.m4.2.2">𝑎</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.I2.i3.p1.4.m4.3c">\textsl{g}(b)>(1+\eta)\textsl{g}(a)</annotation><annotation encoding="application/x-llamapun" id="S5.I2.i3.p1.4.m4.3d">g ( italic_b ) > ( 1 + italic_η ) g ( italic_a )</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S5.I2.i3.p1.6.5"> and by definition of </span><math alttext="\eta" class="ltx_Math" display="inline" id="S5.I2.i3.p1.5.m5.1"><semantics id="S5.I2.i3.p1.5.m5.1a"><mi id="S5.I2.i3.p1.5.m5.1.1" xref="S5.I2.i3.p1.5.m5.1.1.cmml">η</mi><annotation-xml encoding="MathML-Content" id="S5.I2.i3.p1.5.m5.1b"><ci id="S5.I2.i3.p1.5.m5.1.1.cmml" xref="S5.I2.i3.p1.5.m5.1.1">𝜂</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.I2.i3.p1.5.m5.1c">\eta</annotation><annotation encoding="application/x-llamapun" id="S5.I2.i3.p1.5.m5.1d">italic_η</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S5.I2.i3.p1.6.6">-fairness, </span><math alttext="\textsl{g}(a\!\to\!b)=\textsl{g}(\overline{ab})" class="ltx_Math" display="inline" id="S5.I2.i3.p1.6.m6.2"><semantics id="S5.I2.i3.p1.6.m6.2a"><mrow id="S5.I2.i3.p1.6.m6.2.2" xref="S5.I2.i3.p1.6.m6.2.2.cmml"><mrow id="S5.I2.i3.p1.6.m6.2.2.1" xref="S5.I2.i3.p1.6.m6.2.2.1.cmml"><mtext class="ltx_mathvariant_italic" id="S5.I2.i3.p1.6.m6.2.2.1.3" xref="S5.I2.i3.p1.6.m6.2.2.1.3a.cmml">g</mtext><mo id="S5.I2.i3.p1.6.m6.2.2.1.2" xref="S5.I2.i3.p1.6.m6.2.2.1.2.cmml">⁢</mo><mrow id="S5.I2.i3.p1.6.m6.2.2.1.1.1" xref="S5.I2.i3.p1.6.m6.2.2.1.1.1.1.cmml"><mo id="S5.I2.i3.p1.6.m6.2.2.1.1.1.2" stretchy="false" xref="S5.I2.i3.p1.6.m6.2.2.1.1.1.1.cmml">(</mo><mrow id="S5.I2.i3.p1.6.m6.2.2.1.1.1.1" xref="S5.I2.i3.p1.6.m6.2.2.1.1.1.1.cmml"><mi id="S5.I2.i3.p1.6.m6.2.2.1.1.1.1.2" xref="S5.I2.i3.p1.6.m6.2.2.1.1.1.1.2.cmml">a</mi><mo id="S5.I2.i3.p1.6.m6.2.2.1.1.1.1.1" lspace="0.108em" rspace="0.108em" stretchy="false" xref="S5.I2.i3.p1.6.m6.2.2.1.1.1.1.1.cmml">→</mo><mi id="S5.I2.i3.p1.6.m6.2.2.1.1.1.1.3" xref="S5.I2.i3.p1.6.m6.2.2.1.1.1.1.3.cmml">b</mi></mrow><mo id="S5.I2.i3.p1.6.m6.2.2.1.1.1.3" stretchy="false" xref="S5.I2.i3.p1.6.m6.2.2.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S5.I2.i3.p1.6.m6.2.2.2" xref="S5.I2.i3.p1.6.m6.2.2.2.cmml">=</mo><mrow id="S5.I2.i3.p1.6.m6.2.2.3" xref="S5.I2.i3.p1.6.m6.2.2.3.cmml"><mtext class="ltx_mathvariant_italic" id="S5.I2.i3.p1.6.m6.2.2.3.2" xref="S5.I2.i3.p1.6.m6.2.2.3.2a.cmml">g</mtext><mo id="S5.I2.i3.p1.6.m6.2.2.3.1" xref="S5.I2.i3.p1.6.m6.2.2.3.1.cmml">⁢</mo><mrow id="S5.I2.i3.p1.6.m6.2.2.3.3.2" xref="S5.I2.i3.p1.6.m6.1.1.cmml"><mo id="S5.I2.i3.p1.6.m6.2.2.3.3.2.1" stretchy="false" xref="S5.I2.i3.p1.6.m6.1.1.cmml">(</mo><mover accent="true" id="S5.I2.i3.p1.6.m6.1.1" xref="S5.I2.i3.p1.6.m6.1.1.cmml"><mrow id="S5.I2.i3.p1.6.m6.1.1.2" xref="S5.I2.i3.p1.6.m6.1.1.2.cmml"><mi id="S5.I2.i3.p1.6.m6.1.1.2.2" xref="S5.I2.i3.p1.6.m6.1.1.2.2.cmml">a</mi><mo id="S5.I2.i3.p1.6.m6.1.1.2.1" xref="S5.I2.i3.p1.6.m6.1.1.2.1.cmml">⁢</mo><mi id="S5.I2.i3.p1.6.m6.1.1.2.3" xref="S5.I2.i3.p1.6.m6.1.1.2.3.cmml">b</mi></mrow><mo id="S5.I2.i3.p1.6.m6.1.1.1" xref="S5.I2.i3.p1.6.m6.1.1.1.cmml">¯</mo></mover><mo id="S5.I2.i3.p1.6.m6.2.2.3.3.2.2" stretchy="false" xref="S5.I2.i3.p1.6.m6.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.I2.i3.p1.6.m6.2b"><apply id="S5.I2.i3.p1.6.m6.2.2.cmml" xref="S5.I2.i3.p1.6.m6.2.2"><eq id="S5.I2.i3.p1.6.m6.2.2.2.cmml" xref="S5.I2.i3.p1.6.m6.2.2.2"></eq><apply id="S5.I2.i3.p1.6.m6.2.2.1.cmml" xref="S5.I2.i3.p1.6.m6.2.2.1"><times id="S5.I2.i3.p1.6.m6.2.2.1.2.cmml" xref="S5.I2.i3.p1.6.m6.2.2.1.2"></times><ci id="S5.I2.i3.p1.6.m6.2.2.1.3a.cmml" xref="S5.I2.i3.p1.6.m6.2.2.1.3"><mtext class="ltx_mathvariant_italic" id="S5.I2.i3.p1.6.m6.2.2.1.3.cmml" xref="S5.I2.i3.p1.6.m6.2.2.1.3">g</mtext></ci><apply id="S5.I2.i3.p1.6.m6.2.2.1.1.1.1.cmml" xref="S5.I2.i3.p1.6.m6.2.2.1.1.1"><ci id="S5.I2.i3.p1.6.m6.2.2.1.1.1.1.1.cmml" xref="S5.I2.i3.p1.6.m6.2.2.1.1.1.1.1">→</ci><ci id="S5.I2.i3.p1.6.m6.2.2.1.1.1.1.2.cmml" xref="S5.I2.i3.p1.6.m6.2.2.1.1.1.1.2">𝑎</ci><ci id="S5.I2.i3.p1.6.m6.2.2.1.1.1.1.3.cmml" xref="S5.I2.i3.p1.6.m6.2.2.1.1.1.1.3">𝑏</ci></apply></apply><apply id="S5.I2.i3.p1.6.m6.2.2.3.cmml" xref="S5.I2.i3.p1.6.m6.2.2.3"><times id="S5.I2.i3.p1.6.m6.2.2.3.1.cmml" xref="S5.I2.i3.p1.6.m6.2.2.3.1"></times><ci id="S5.I2.i3.p1.6.m6.2.2.3.2a.cmml" xref="S5.I2.i3.p1.6.m6.2.2.3.2"><mtext class="ltx_mathvariant_italic" id="S5.I2.i3.p1.6.m6.2.2.3.2.cmml" xref="S5.I2.i3.p1.6.m6.2.2.3.2">g</mtext></ci><apply id="S5.I2.i3.p1.6.m6.1.1.cmml" xref="S5.I2.i3.p1.6.m6.2.2.3.3.2"><ci id="S5.I2.i3.p1.6.m6.1.1.1.cmml" xref="S5.I2.i3.p1.6.m6.1.1.1">¯</ci><apply id="S5.I2.i3.p1.6.m6.1.1.2.cmml" xref="S5.I2.i3.p1.6.m6.1.1.2"><times id="S5.I2.i3.p1.6.m6.1.1.2.1.cmml" xref="S5.I2.i3.p1.6.m6.1.1.2.1"></times><ci id="S5.I2.i3.p1.6.m6.1.1.2.2.cmml" xref="S5.I2.i3.p1.6.m6.1.1.2.2">𝑎</ci><ci id="S5.I2.i3.p1.6.m6.1.1.2.3.cmml" xref="S5.I2.i3.p1.6.m6.1.1.2.3">𝑏</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.I2.i3.p1.6.m6.2c">\textsl{g}(a\!\to\!b)=\textsl{g}(\overline{ab})</annotation><annotation encoding="application/x-llamapun" id="S5.I2.i3.p1.6.m6.2d">g ( italic_a → italic_b ) = g ( over¯ start_ARG italic_a italic_b end_ARG )</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S5.I2.i3.p1.6.7">.</span></p> </div> </li> </ul> </div> <div class="ltx_para" id="S5.Thmtheorem3.p18"> <p class="ltx_p" id="S5.Thmtheorem3.p18.1"><span class="ltx_text ltx_font_italic" id="S5.Thmtheorem3.p18.1.1">Now:</span></p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="A1.EGx6"> <tbody id="S5.Ex23"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_left_padleft"></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle\sum_{v\in V(H_{k+1})}\textsl{g}(v)\leq\textsl{g}(u)\cdot(1+\eta)% ^{k+1}\cdot|V(H_{k+1})|\leq\textsl{g}(u)\cdot(1+\frac{\varepsilon}{2})\cdot|H_% {k}|\cdot(1+\frac{\varepsilon}{16})\Rightarrow" class="ltx_Math" display="inline" id="S5.Ex23.m1.9"><semantics id="S5.Ex23.m1.9a"><mrow id="S5.Ex23.m1.9.9" xref="S5.Ex23.m1.9.9.cmml"><mrow id="S5.Ex23.m1.9.9.7" xref="S5.Ex23.m1.9.9.7.cmml"><mstyle displaystyle="true" id="S5.Ex23.m1.9.9.7.1" xref="S5.Ex23.m1.9.9.7.1.cmml"><munder id="S5.Ex23.m1.9.9.7.1a" xref="S5.Ex23.m1.9.9.7.1.cmml"><mo id="S5.Ex23.m1.9.9.7.1.2" movablelimits="false" xref="S5.Ex23.m1.9.9.7.1.2.cmml">∑</mo><mrow id="S5.Ex23.m1.1.1.1" xref="S5.Ex23.m1.1.1.1.cmml"><mi id="S5.Ex23.m1.1.1.1.3" xref="S5.Ex23.m1.1.1.1.3.cmml">v</mi><mo id="S5.Ex23.m1.1.1.1.2" xref="S5.Ex23.m1.1.1.1.2.cmml">∈</mo><mrow id="S5.Ex23.m1.1.1.1.1" xref="S5.Ex23.m1.1.1.1.1.cmml"><mi id="S5.Ex23.m1.1.1.1.1.3" xref="S5.Ex23.m1.1.1.1.1.3.cmml">V</mi><mo id="S5.Ex23.m1.1.1.1.1.2" xref="S5.Ex23.m1.1.1.1.1.2.cmml">⁢</mo><mrow id="S5.Ex23.m1.1.1.1.1.1.1" xref="S5.Ex23.m1.1.1.1.1.1.1.1.cmml"><mo id="S5.Ex23.m1.1.1.1.1.1.1.2" stretchy="false" xref="S5.Ex23.m1.1.1.1.1.1.1.1.cmml">(</mo><msub id="S5.Ex23.m1.1.1.1.1.1.1.1" xref="S5.Ex23.m1.1.1.1.1.1.1.1.cmml"><mi id="S5.Ex23.m1.1.1.1.1.1.1.1.2" xref="S5.Ex23.m1.1.1.1.1.1.1.1.2.cmml">H</mi><mrow id="S5.Ex23.m1.1.1.1.1.1.1.1.3" xref="S5.Ex23.m1.1.1.1.1.1.1.1.3.cmml"><mi id="S5.Ex23.m1.1.1.1.1.1.1.1.3.2" xref="S5.Ex23.m1.1.1.1.1.1.1.1.3.2.cmml">k</mi><mo id="S5.Ex23.m1.1.1.1.1.1.1.1.3.1" xref="S5.Ex23.m1.1.1.1.1.1.1.1.3.1.cmml">+</mo><mn id="S5.Ex23.m1.1.1.1.1.1.1.1.3.3" xref="S5.Ex23.m1.1.1.1.1.1.1.1.3.3.cmml">1</mn></mrow></msub><mo id="S5.Ex23.m1.1.1.1.1.1.1.3" stretchy="false" xref="S5.Ex23.m1.1.1.1.1.1.1.1.cmml">)</mo></mrow></mrow></mrow></munder></mstyle><mrow id="S5.Ex23.m1.9.9.7.2" xref="S5.Ex23.m1.9.9.7.2.cmml"><mtext class="ltx_mathvariant_italic" id="S5.Ex23.m1.9.9.7.2.2" xref="S5.Ex23.m1.9.9.7.2.2a.cmml">g</mtext><mo id="S5.Ex23.m1.9.9.7.2.1" xref="S5.Ex23.m1.9.9.7.2.1.cmml">⁢</mo><mrow id="S5.Ex23.m1.9.9.7.2.3.2" xref="S5.Ex23.m1.9.9.7.2.cmml"><mo id="S5.Ex23.m1.9.9.7.2.3.2.1" stretchy="false" xref="S5.Ex23.m1.9.9.7.2.cmml">(</mo><mi id="S5.Ex23.m1.2.2" xref="S5.Ex23.m1.2.2.cmml">v</mi><mo id="S5.Ex23.m1.9.9.7.2.3.2.2" stretchy="false" xref="S5.Ex23.m1.9.9.7.2.cmml">)</mo></mrow></mrow></mrow><mo id="S5.Ex23.m1.9.9.8" xref="S5.Ex23.m1.9.9.8.cmml">≤</mo><mrow id="S5.Ex23.m1.6.6.2" xref="S5.Ex23.m1.6.6.2.cmml"><mrow id="S5.Ex23.m1.6.6.2.4" xref="S5.Ex23.m1.6.6.2.4.cmml"><mtext class="ltx_mathvariant_italic" id="S5.Ex23.m1.6.6.2.4.2" xref="S5.Ex23.m1.6.6.2.4.2a.cmml">g</mtext><mo id="S5.Ex23.m1.6.6.2.4.1" xref="S5.Ex23.m1.6.6.2.4.1.cmml">⁢</mo><mrow id="S5.Ex23.m1.6.6.2.4.3.2" xref="S5.Ex23.m1.6.6.2.4.cmml"><mo id="S5.Ex23.m1.6.6.2.4.3.2.1" stretchy="false" xref="S5.Ex23.m1.6.6.2.4.cmml">(</mo><mi id="S5.Ex23.m1.3.3" xref="S5.Ex23.m1.3.3.cmml">u</mi><mo id="S5.Ex23.m1.6.6.2.4.3.2.2" rspace="0.055em" stretchy="false" xref="S5.Ex23.m1.6.6.2.4.cmml">)</mo></mrow></mrow><mo id="S5.Ex23.m1.6.6.2.3" rspace="0.222em" xref="S5.Ex23.m1.6.6.2.3.cmml">⋅</mo><msup id="S5.Ex23.m1.5.5.1.1" xref="S5.Ex23.m1.5.5.1.1.cmml"><mrow id="S5.Ex23.m1.5.5.1.1.1.1" xref="S5.Ex23.m1.5.5.1.1.1.1.1.cmml"><mo id="S5.Ex23.m1.5.5.1.1.1.1.2" stretchy="false" xref="S5.Ex23.m1.5.5.1.1.1.1.1.cmml">(</mo><mrow id="S5.Ex23.m1.5.5.1.1.1.1.1" xref="S5.Ex23.m1.5.5.1.1.1.1.1.cmml"><mn id="S5.Ex23.m1.5.5.1.1.1.1.1.2" xref="S5.Ex23.m1.5.5.1.1.1.1.1.2.cmml">1</mn><mo id="S5.Ex23.m1.5.5.1.1.1.1.1.1" xref="S5.Ex23.m1.5.5.1.1.1.1.1.1.cmml">+</mo><mi id="S5.Ex23.m1.5.5.1.1.1.1.1.3" xref="S5.Ex23.m1.5.5.1.1.1.1.1.3.cmml">η</mi></mrow><mo id="S5.Ex23.m1.5.5.1.1.1.1.3" rspace="0.055em" stretchy="false" xref="S5.Ex23.m1.5.5.1.1.1.1.1.cmml">)</mo></mrow><mrow id="S5.Ex23.m1.5.5.1.1.3" xref="S5.Ex23.m1.5.5.1.1.3.cmml"><mi id="S5.Ex23.m1.5.5.1.1.3.2" xref="S5.Ex23.m1.5.5.1.1.3.2.cmml">k</mi><mo id="S5.Ex23.m1.5.5.1.1.3.1" xref="S5.Ex23.m1.5.5.1.1.3.1.cmml">+</mo><mn id="S5.Ex23.m1.5.5.1.1.3.3" xref="S5.Ex23.m1.5.5.1.1.3.3.cmml">1</mn></mrow></msup><mo id="S5.Ex23.m1.6.6.2.3a" rspace="0.222em" xref="S5.Ex23.m1.6.6.2.3.cmml">⋅</mo><mrow id="S5.Ex23.m1.6.6.2.2.1" xref="S5.Ex23.m1.6.6.2.2.2.cmml"><mo id="S5.Ex23.m1.6.6.2.2.1.2" stretchy="false" xref="S5.Ex23.m1.6.6.2.2.2.1.cmml">|</mo><mrow id="S5.Ex23.m1.6.6.2.2.1.1" xref="S5.Ex23.m1.6.6.2.2.1.1.cmml"><mi id="S5.Ex23.m1.6.6.2.2.1.1.3" xref="S5.Ex23.m1.6.6.2.2.1.1.3.cmml">V</mi><mo id="S5.Ex23.m1.6.6.2.2.1.1.2" xref="S5.Ex23.m1.6.6.2.2.1.1.2.cmml">⁢</mo><mrow id="S5.Ex23.m1.6.6.2.2.1.1.1.1" xref="S5.Ex23.m1.6.6.2.2.1.1.1.1.1.cmml"><mo id="S5.Ex23.m1.6.6.2.2.1.1.1.1.2" stretchy="false" xref="S5.Ex23.m1.6.6.2.2.1.1.1.1.1.cmml">(</mo><msub id="S5.Ex23.m1.6.6.2.2.1.1.1.1.1" xref="S5.Ex23.m1.6.6.2.2.1.1.1.1.1.cmml"><mi id="S5.Ex23.m1.6.6.2.2.1.1.1.1.1.2" xref="S5.Ex23.m1.6.6.2.2.1.1.1.1.1.2.cmml">H</mi><mrow id="S5.Ex23.m1.6.6.2.2.1.1.1.1.1.3" xref="S5.Ex23.m1.6.6.2.2.1.1.1.1.1.3.cmml"><mi id="S5.Ex23.m1.6.6.2.2.1.1.1.1.1.3.2" xref="S5.Ex23.m1.6.6.2.2.1.1.1.1.1.3.2.cmml">k</mi><mo id="S5.Ex23.m1.6.6.2.2.1.1.1.1.1.3.1" xref="S5.Ex23.m1.6.6.2.2.1.1.1.1.1.3.1.cmml">+</mo><mn id="S5.Ex23.m1.6.6.2.2.1.1.1.1.1.3.3" xref="S5.Ex23.m1.6.6.2.2.1.1.1.1.1.3.3.cmml">1</mn></mrow></msub><mo id="S5.Ex23.m1.6.6.2.2.1.1.1.1.3" stretchy="false" xref="S5.Ex23.m1.6.6.2.2.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S5.Ex23.m1.6.6.2.2.1.3" stretchy="false" xref="S5.Ex23.m1.6.6.2.2.2.1.cmml">|</mo></mrow></mrow><mo id="S5.Ex23.m1.9.9.9" xref="S5.Ex23.m1.9.9.9.cmml">≤</mo><mrow id="S5.Ex23.m1.9.9.5" xref="S5.Ex23.m1.9.9.5.cmml"><mrow id="S5.Ex23.m1.9.9.5.5" xref="S5.Ex23.m1.9.9.5.5.cmml"><mtext class="ltx_mathvariant_italic" id="S5.Ex23.m1.9.9.5.5.2" xref="S5.Ex23.m1.9.9.5.5.2a.cmml">g</mtext><mo id="S5.Ex23.m1.9.9.5.5.1" xref="S5.Ex23.m1.9.9.5.5.1.cmml">⁢</mo><mrow id="S5.Ex23.m1.9.9.5.5.3.2" xref="S5.Ex23.m1.9.9.5.5.cmml"><mo id="S5.Ex23.m1.9.9.5.5.3.2.1" stretchy="false" xref="S5.Ex23.m1.9.9.5.5.cmml">(</mo><mi id="S5.Ex23.m1.4.4" xref="S5.Ex23.m1.4.4.cmml">u</mi><mo id="S5.Ex23.m1.9.9.5.5.3.2.2" rspace="0.055em" stretchy="false" xref="S5.Ex23.m1.9.9.5.5.cmml">)</mo></mrow></mrow><mo id="S5.Ex23.m1.9.9.5.4" rspace="0.222em" xref="S5.Ex23.m1.9.9.5.4.cmml">⋅</mo><mrow id="S5.Ex23.m1.7.7.3.1.1" xref="S5.Ex23.m1.7.7.3.1.1.1.cmml"><mo id="S5.Ex23.m1.7.7.3.1.1.2" stretchy="false" xref="S5.Ex23.m1.7.7.3.1.1.1.cmml">(</mo><mrow id="S5.Ex23.m1.7.7.3.1.1.1" xref="S5.Ex23.m1.7.7.3.1.1.1.cmml"><mn id="S5.Ex23.m1.7.7.3.1.1.1.2" xref="S5.Ex23.m1.7.7.3.1.1.1.2.cmml">1</mn><mo id="S5.Ex23.m1.7.7.3.1.1.1.1" xref="S5.Ex23.m1.7.7.3.1.1.1.1.cmml">+</mo><mstyle displaystyle="true" 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xref="S5.Ex23.m1.8.8.4.2.1"><abs id="S5.Ex23.m1.8.8.4.2.2.1.cmml" xref="S5.Ex23.m1.8.8.4.2.1.2"></abs><apply id="S5.Ex23.m1.8.8.4.2.1.1.cmml" xref="S5.Ex23.m1.8.8.4.2.1.1"><csymbol cd="ambiguous" id="S5.Ex23.m1.8.8.4.2.1.1.1.cmml" xref="S5.Ex23.m1.8.8.4.2.1.1">subscript</csymbol><ci id="S5.Ex23.m1.8.8.4.2.1.1.2.cmml" xref="S5.Ex23.m1.8.8.4.2.1.1.2">𝐻</ci><ci id="S5.Ex23.m1.8.8.4.2.1.1.3.cmml" xref="S5.Ex23.m1.8.8.4.2.1.1.3">𝑘</ci></apply></apply><apply id="S5.Ex23.m1.9.9.5.3.1.1.cmml" xref="S5.Ex23.m1.9.9.5.3.1"><plus id="S5.Ex23.m1.9.9.5.3.1.1.1.cmml" xref="S5.Ex23.m1.9.9.5.3.1.1.1"></plus><cn id="S5.Ex23.m1.9.9.5.3.1.1.2.cmml" type="integer" xref="S5.Ex23.m1.9.9.5.3.1.1.2">1</cn><apply id="S5.Ex23.m1.9.9.5.3.1.1.3.cmml" xref="S5.Ex23.m1.9.9.5.3.1.1.3"><divide id="S5.Ex23.m1.9.9.5.3.1.1.3.1.cmml" xref="S5.Ex23.m1.9.9.5.3.1.1.3"></divide><ci id="S5.Ex23.m1.9.9.5.3.1.1.3.2.cmml" xref="S5.Ex23.m1.9.9.5.3.1.1.3.2">𝜀</ci><cn id="S5.Ex23.m1.9.9.5.3.1.1.3.3.cmml" type="integer" xref="S5.Ex23.m1.9.9.5.3.1.1.3.3">16</cn></apply></apply></apply></apply><apply id="S5.Ex23.m1.9.9e.cmml" xref="S5.Ex23.m1.9.9"><ci id="S5.Ex23.m1.9.9.10.cmml" xref="S5.Ex23.m1.9.9.10">⇒</ci><share href="https://arxiv.org/html/2411.12694v2#S5.Ex23.m1.9.9.5.cmml" id="S5.Ex23.m1.9.9f.cmml" xref="S5.Ex23.m1.9.9"></share><csymbol cd="latexml" id="S5.Ex23.m1.9.9.11.cmml" xref="S5.Ex23.m1.9.9.11">absent</csymbol></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Ex23.m1.9c">\displaystyle\sum_{v\in V(H_{k+1})}\textsl{g}(v)\leq\textsl{g}(u)\cdot(1+\eta)% ^{k+1}\cdot|V(H_{k+1})|\leq\textsl{g}(u)\cdot(1+\frac{\varepsilon}{2})\cdot|H_% {k}|\cdot(1+\frac{\varepsilon}{16})\Rightarrow</annotation><annotation encoding="application/x-llamapun" id="S5.Ex23.m1.9d">∑ start_POSTSUBSCRIPT italic_v ∈ italic_V ( italic_H start_POSTSUBSCRIPT italic_k + 1 end_POSTSUBSCRIPT ) end_POSTSUBSCRIPT g ( italic_v ) ≤ g ( italic_u ) ⋅ ( 1 + italic_η ) start_POSTSUPERSCRIPT italic_k + 1 end_POSTSUPERSCRIPT ⋅ | italic_V ( italic_H start_POSTSUBSCRIPT italic_k + 1 end_POSTSUBSCRIPT ) | ≤ g ( italic_u ) ⋅ ( 1 + divide start_ARG italic_ε end_ARG start_ARG 2 end_ARG ) ⋅ | italic_H start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT | ⋅ ( 1 + divide start_ARG italic_ε end_ARG start_ARG 16 end_ARG ) ⇒</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_left_padright"></td> </tr></tbody> <tbody id="S5.Ex24"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_left_padleft"></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle\sum_{v\in V(H_{k+1})}\textsl{g}(v)\leq(1+\varepsilon)\textsl{g}(% u)\cdot|H_{k}|<\rho^{*}(u)\cdot|H_{k}|\Rightarrow" class="ltx_Math" display="inline" id="S5.Ex24.m1.7"><semantics id="S5.Ex24.m1.7a"><mrow id="S5.Ex24.m1.7.7" xref="S5.Ex24.m1.7.7.cmml"><mrow id="S5.Ex24.m1.7.7.5" xref="S5.Ex24.m1.7.7.5.cmml"><mstyle displaystyle="true" id="S5.Ex24.m1.7.7.5.1" xref="S5.Ex24.m1.7.7.5.1.cmml"><munder id="S5.Ex24.m1.7.7.5.1a" xref="S5.Ex24.m1.7.7.5.1.cmml"><mo id="S5.Ex24.m1.7.7.5.1.2" movablelimits="false" xref="S5.Ex24.m1.7.7.5.1.2.cmml">∑</mo><mrow id="S5.Ex24.m1.1.1.1" xref="S5.Ex24.m1.1.1.1.cmml"><mi id="S5.Ex24.m1.1.1.1.3" xref="S5.Ex24.m1.1.1.1.3.cmml">v</mi><mo id="S5.Ex24.m1.1.1.1.2" xref="S5.Ex24.m1.1.1.1.2.cmml">∈</mo><mrow id="S5.Ex24.m1.1.1.1.1" xref="S5.Ex24.m1.1.1.1.1.cmml"><mi id="S5.Ex24.m1.1.1.1.1.3" xref="S5.Ex24.m1.1.1.1.1.3.cmml">V</mi><mo id="S5.Ex24.m1.1.1.1.1.2" xref="S5.Ex24.m1.1.1.1.1.2.cmml">⁢</mo><mrow id="S5.Ex24.m1.1.1.1.1.1.1" xref="S5.Ex24.m1.1.1.1.1.1.1.1.cmml"><mo id="S5.Ex24.m1.1.1.1.1.1.1.2" stretchy="false" xref="S5.Ex24.m1.1.1.1.1.1.1.1.cmml">(</mo><msub id="S5.Ex24.m1.1.1.1.1.1.1.1" xref="S5.Ex24.m1.1.1.1.1.1.1.1.cmml"><mi id="S5.Ex24.m1.1.1.1.1.1.1.1.2" xref="S5.Ex24.m1.1.1.1.1.1.1.1.2.cmml">H</mi><mrow id="S5.Ex24.m1.1.1.1.1.1.1.1.3" xref="S5.Ex24.m1.1.1.1.1.1.1.1.3.cmml"><mi id="S5.Ex24.m1.1.1.1.1.1.1.1.3.2" xref="S5.Ex24.m1.1.1.1.1.1.1.1.3.2.cmml">k</mi><mo id="S5.Ex24.m1.1.1.1.1.1.1.1.3.1" xref="S5.Ex24.m1.1.1.1.1.1.1.1.3.1.cmml">+</mo><mn id="S5.Ex24.m1.1.1.1.1.1.1.1.3.3" xref="S5.Ex24.m1.1.1.1.1.1.1.1.3.3.cmml">1</mn></mrow></msub><mo id="S5.Ex24.m1.1.1.1.1.1.1.3" stretchy="false" xref="S5.Ex24.m1.1.1.1.1.1.1.1.cmml">)</mo></mrow></mrow></mrow></munder></mstyle><mrow id="S5.Ex24.m1.7.7.5.2" xref="S5.Ex24.m1.7.7.5.2.cmml"><mtext class="ltx_mathvariant_italic" id="S5.Ex24.m1.7.7.5.2.2" xref="S5.Ex24.m1.7.7.5.2.2a.cmml">g</mtext><mo id="S5.Ex24.m1.7.7.5.2.1" xref="S5.Ex24.m1.7.7.5.2.1.cmml">⁢</mo><mrow id="S5.Ex24.m1.7.7.5.2.3.2" xref="S5.Ex24.m1.7.7.5.2.cmml"><mo id="S5.Ex24.m1.7.7.5.2.3.2.1" stretchy="false" xref="S5.Ex24.m1.7.7.5.2.cmml">(</mo><mi id="S5.Ex24.m1.2.2" xref="S5.Ex24.m1.2.2.cmml">v</mi><mo id="S5.Ex24.m1.7.7.5.2.3.2.2" stretchy="false" xref="S5.Ex24.m1.7.7.5.2.cmml">)</mo></mrow></mrow></mrow><mo id="S5.Ex24.m1.7.7.6" xref="S5.Ex24.m1.7.7.6.cmml">≤</mo><mrow id="S5.Ex24.m1.6.6.2" xref="S5.Ex24.m1.6.6.2.cmml"><mrow id="S5.Ex24.m1.5.5.1.1" xref="S5.Ex24.m1.5.5.1.1.cmml"><mrow id="S5.Ex24.m1.5.5.1.1.1.1" xref="S5.Ex24.m1.5.5.1.1.1.1.1.cmml"><mo id="S5.Ex24.m1.5.5.1.1.1.1.2" stretchy="false" xref="S5.Ex24.m1.5.5.1.1.1.1.1.cmml">(</mo><mrow id="S5.Ex24.m1.5.5.1.1.1.1.1" xref="S5.Ex24.m1.5.5.1.1.1.1.1.cmml"><mn id="S5.Ex24.m1.5.5.1.1.1.1.1.2" xref="S5.Ex24.m1.5.5.1.1.1.1.1.2.cmml">1</mn><mo id="S5.Ex24.m1.5.5.1.1.1.1.1.1" xref="S5.Ex24.m1.5.5.1.1.1.1.1.1.cmml">+</mo><mi id="S5.Ex24.m1.5.5.1.1.1.1.1.3" xref="S5.Ex24.m1.5.5.1.1.1.1.1.3.cmml">ε</mi></mrow><mo id="S5.Ex24.m1.5.5.1.1.1.1.3" stretchy="false" xref="S5.Ex24.m1.5.5.1.1.1.1.1.cmml">)</mo></mrow><mo id="S5.Ex24.m1.5.5.1.1.2" xref="S5.Ex24.m1.5.5.1.1.2.cmml">⁢</mo><mtext class="ltx_mathvariant_italic" id="S5.Ex24.m1.5.5.1.1.3" xref="S5.Ex24.m1.5.5.1.1.3a.cmml">g</mtext><mo id="S5.Ex24.m1.5.5.1.1.2a" xref="S5.Ex24.m1.5.5.1.1.2.cmml">⁢</mo><mrow id="S5.Ex24.m1.5.5.1.1.4.2" xref="S5.Ex24.m1.5.5.1.1.cmml"><mo id="S5.Ex24.m1.5.5.1.1.4.2.1" stretchy="false" xref="S5.Ex24.m1.5.5.1.1.cmml">(</mo><mi id="S5.Ex24.m1.3.3" xref="S5.Ex24.m1.3.3.cmml">u</mi><mo id="S5.Ex24.m1.5.5.1.1.4.2.2" rspace="0.055em" stretchy="false" xref="S5.Ex24.m1.5.5.1.1.cmml">)</mo></mrow></mrow><mo id="S5.Ex24.m1.6.6.2.3" rspace="0.222em" xref="S5.Ex24.m1.6.6.2.3.cmml">⋅</mo><mrow id="S5.Ex24.m1.6.6.2.2.1" xref="S5.Ex24.m1.6.6.2.2.2.cmml"><mo id="S5.Ex24.m1.6.6.2.2.1.2" stretchy="false" xref="S5.Ex24.m1.6.6.2.2.2.1.cmml">|</mo><msub id="S5.Ex24.m1.6.6.2.2.1.1" xref="S5.Ex24.m1.6.6.2.2.1.1.cmml"><mi id="S5.Ex24.m1.6.6.2.2.1.1.2" xref="S5.Ex24.m1.6.6.2.2.1.1.2.cmml">H</mi><mi id="S5.Ex24.m1.6.6.2.2.1.1.3" xref="S5.Ex24.m1.6.6.2.2.1.1.3.cmml">k</mi></msub><mo id="S5.Ex24.m1.6.6.2.2.1.3" stretchy="false" xref="S5.Ex24.m1.6.6.2.2.2.1.cmml">|</mo></mrow></mrow><mo id="S5.Ex24.m1.7.7.7" xref="S5.Ex24.m1.7.7.7.cmml"><</mo><mrow id="S5.Ex24.m1.7.7.3" xref="S5.Ex24.m1.7.7.3.cmml"><mrow id="S5.Ex24.m1.7.7.3.3" xref="S5.Ex24.m1.7.7.3.3.cmml"><msup id="S5.Ex24.m1.7.7.3.3.2" xref="S5.Ex24.m1.7.7.3.3.2.cmml"><mi id="S5.Ex24.m1.7.7.3.3.2.2" xref="S5.Ex24.m1.7.7.3.3.2.2.cmml">ρ</mi><mo id="S5.Ex24.m1.7.7.3.3.2.3" xref="S5.Ex24.m1.7.7.3.3.2.3.cmml">∗</mo></msup><mo id="S5.Ex24.m1.7.7.3.3.1" xref="S5.Ex24.m1.7.7.3.3.1.cmml">⁢</mo><mrow id="S5.Ex24.m1.7.7.3.3.3.2" xref="S5.Ex24.m1.7.7.3.3.cmml"><mo id="S5.Ex24.m1.7.7.3.3.3.2.1" stretchy="false" xref="S5.Ex24.m1.7.7.3.3.cmml">(</mo><mi id="S5.Ex24.m1.4.4" xref="S5.Ex24.m1.4.4.cmml">u</mi><mo id="S5.Ex24.m1.7.7.3.3.3.2.2" rspace="0.055em" stretchy="false" xref="S5.Ex24.m1.7.7.3.3.cmml">)</mo></mrow></mrow><mo id="S5.Ex24.m1.7.7.3.2" rspace="0.222em" xref="S5.Ex24.m1.7.7.3.2.cmml">⋅</mo><mrow id="S5.Ex24.m1.7.7.3.1.1" xref="S5.Ex24.m1.7.7.3.1.2.cmml"><mo id="S5.Ex24.m1.7.7.3.1.1.2" stretchy="false" xref="S5.Ex24.m1.7.7.3.1.2.1.cmml">|</mo><msub id="S5.Ex24.m1.7.7.3.1.1.1" xref="S5.Ex24.m1.7.7.3.1.1.1.cmml"><mi id="S5.Ex24.m1.7.7.3.1.1.1.2" xref="S5.Ex24.m1.7.7.3.1.1.1.2.cmml">H</mi><mi id="S5.Ex24.m1.7.7.3.1.1.1.3" xref="S5.Ex24.m1.7.7.3.1.1.1.3.cmml">k</mi></msub><mo id="S5.Ex24.m1.7.7.3.1.1.3" stretchy="false" xref="S5.Ex24.m1.7.7.3.1.2.1.cmml">|</mo></mrow></mrow><mo id="S5.Ex24.m1.7.7.8" stretchy="false" xref="S5.Ex24.m1.7.7.8.cmml">⇒</mo><mi id="S5.Ex24.m1.7.7.9" xref="S5.Ex24.m1.7.7.9.cmml"></mi></mrow><annotation-xml encoding="MathML-Content" id="S5.Ex24.m1.7b"><apply id="S5.Ex24.m1.7.7.cmml" xref="S5.Ex24.m1.7.7"><and id="S5.Ex24.m1.7.7a.cmml" xref="S5.Ex24.m1.7.7"></and><apply id="S5.Ex24.m1.7.7b.cmml" xref="S5.Ex24.m1.7.7"><leq id="S5.Ex24.m1.7.7.6.cmml" xref="S5.Ex24.m1.7.7.6"></leq><apply id="S5.Ex24.m1.7.7.5.cmml" xref="S5.Ex24.m1.7.7.5"><apply id="S5.Ex24.m1.7.7.5.1.cmml" xref="S5.Ex24.m1.7.7.5.1"><csymbol cd="ambiguous" id="S5.Ex24.m1.7.7.5.1.1.cmml" xref="S5.Ex24.m1.7.7.5.1">subscript</csymbol><sum id="S5.Ex24.m1.7.7.5.1.2.cmml" 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+ 1 end_POSTSUBSCRIPT ) end_POSTSUBSCRIPT g ( italic_v ) ≤ ( 1 + italic_ε ) g ( italic_u ) ⋅ | italic_H start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT | < italic_ρ start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ( italic_u ) ⋅ | italic_H start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT | ⇒</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_left_padright"></td> </tr></tbody> <tbody id="S5.Ex25"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_left_padleft"></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle\sum_{e\in\hat{E}_{B}(H_{k})\cup E_{\uparrow}}\textsl{g}(e)<\rho^% {*}(u)|H_{k}|\quad\quad\quad\quad\quad\quad\emph{by Equation\leavevmode% \nobreak\ \ref{eq:v-estimate}}" class="ltx_Math" display="inline" id="S5.Ex25.m1.5"><semantics id="S5.Ex25.m1.5a"><mrow id="S5.Ex25.m1.5.5" xref="S5.Ex25.m1.5.5.cmml"><mrow id="S5.Ex25.m1.5.5.3" xref="S5.Ex25.m1.5.5.3.cmml"><mstyle displaystyle="true" 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\ref{eq:v-estimate}}</annotation><annotation encoding="application/x-llamapun" id="S5.Ex25.m1.5d">∑ start_POSTSUBSCRIPT italic_e ∈ over^ start_ARG italic_E end_ARG start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT ( italic_H start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ) ∪ italic_E start_POSTSUBSCRIPT ↑ end_POSTSUBSCRIPT end_POSTSUBSCRIPT g ( italic_e ) < italic_ρ start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ( italic_u ) | italic_H start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT | by Equation</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_left_padright"></td> </tr></tbody> </table> </div> <div class="ltx_para ltx_noindent" id="S5.Thmtheorem3.p19"> <p class="ltx_p" id="S5.Thmtheorem3.p19.1"><span class="ltx_text ltx_font_italic" id="S5.Thmtheorem3.p19.1.1">We now apply the fact </span><math alttext="\rho^{*}(u)|S_{i}|=\sum\limits_{e\in\hat{E}_{B}(S_{i})}\textsl{g}(e)" class="ltx_Math" display="inline" id="S5.Thmtheorem3.p19.1.m1.4"><semantics 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id="S5.Thmtheorem3.p19.1.m1.1.1.1.1.1.1.1.2.cmml" xref="S5.Thmtheorem3.p19.1.m1.1.1.1.1.1.1.1.2">𝑆</ci><ci id="S5.Thmtheorem3.p19.1.m1.1.1.1.1.1.1.1.3.cmml" xref="S5.Thmtheorem3.p19.1.m1.1.1.1.1.1.1.1.3">𝑖</ci></apply></apply></apply></apply><apply id="S5.Thmtheorem3.p19.1.m1.4.4.3.2.cmml" xref="S5.Thmtheorem3.p19.1.m1.4.4.3.2"><times id="S5.Thmtheorem3.p19.1.m1.4.4.3.2.1.cmml" xref="S5.Thmtheorem3.p19.1.m1.4.4.3.2.1"></times><ci id="S5.Thmtheorem3.p19.1.m1.4.4.3.2.2a.cmml" xref="S5.Thmtheorem3.p19.1.m1.4.4.3.2.2"><mtext class="ltx_mathvariant_italic" id="S5.Thmtheorem3.p19.1.m1.4.4.3.2.2.cmml" xref="S5.Thmtheorem3.p19.1.m1.4.4.3.2.2">g</mtext></ci><ci id="S5.Thmtheorem3.p19.1.m1.3.3.cmml" xref="S5.Thmtheorem3.p19.1.m1.3.3">𝑒</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmtheorem3.p19.1.m1.4c">\rho^{*}(u)|S_{i}|=\sum\limits_{e\in\hat{E}_{B}(S_{i})}\textsl{g}(e)</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem3.p19.1.m1.4d">italic_ρ start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ( italic_u ) | italic_S start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT | = ∑ start_POSTSUBSCRIPT italic_e ∈ over^ start_ARG italic_E end_ARG start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT ( italic_S start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ) end_POSTSUBSCRIPT g ( italic_e )</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S5.Thmtheorem3.p19.1.2"> to note that:</span></p> <table class="ltx_equation ltx_eqn_table" id="S5.Ex26"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_left_padleft"></td> <td class="ltx_eqn_cell ltx_align_left"><math alttext="\sum_{e\in\hat{E}_{B}(S_{i}-H_{k})}\textsl{g}(e)=\sum_{e\in\hat{E}_{B}(S_{i})}% \textsl{g}(e)\quad-\quad\sum_{e\in\hat{E}_{B}(H_{k})\cup E_{\uparrow}}\textsl{% g}(e)>\rho^{*}(u)|S_{i}|-\rho^{*}(u)|H_{k}|" class="ltx_Math" display="block" id="S5.Ex26.m1.11"><semantics id="S5.Ex26.m1.11a"><mrow id="S5.Ex26.m1.11.11.2" xref="S5.Ex26.m1.11.11.3.cmml"><mrow id="S5.Ex26.m1.10.10.1.1" xref="S5.Ex26.m1.10.10.1.1.cmml"><mrow id="S5.Ex26.m1.10.10.1.1.3" xref="S5.Ex26.m1.10.10.1.1.3.cmml"><munder id="S5.Ex26.m1.10.10.1.1.3.1" xref="S5.Ex26.m1.10.10.1.1.3.1.cmml"><mo id="S5.Ex26.m1.10.10.1.1.3.1.2" movablelimits="false" xref="S5.Ex26.m1.10.10.1.1.3.1.2.cmml">∑</mo><mrow id="S5.Ex26.m1.1.1.1" xref="S5.Ex26.m1.1.1.1.cmml"><mi id="S5.Ex26.m1.1.1.1.3" xref="S5.Ex26.m1.1.1.1.3.cmml">e</mi><mo id="S5.Ex26.m1.1.1.1.2" xref="S5.Ex26.m1.1.1.1.2.cmml">∈</mo><mrow id="S5.Ex26.m1.1.1.1.1" xref="S5.Ex26.m1.1.1.1.1.cmml"><msub id="S5.Ex26.m1.1.1.1.1.3" xref="S5.Ex26.m1.1.1.1.1.3.cmml"><mover accent="true" id="S5.Ex26.m1.1.1.1.1.3.2" xref="S5.Ex26.m1.1.1.1.1.3.2.cmml"><mi id="S5.Ex26.m1.1.1.1.1.3.2.2" xref="S5.Ex26.m1.1.1.1.1.3.2.2.cmml">E</mi><mo id="S5.Ex26.m1.1.1.1.1.3.2.1" xref="S5.Ex26.m1.1.1.1.1.3.2.1.cmml">^</mo></mover><mi id="S5.Ex26.m1.1.1.1.1.3.3" xref="S5.Ex26.m1.1.1.1.1.3.3.cmml">B</mi></msub><mo id="S5.Ex26.m1.1.1.1.1.2" xref="S5.Ex26.m1.1.1.1.1.2.cmml">⁢</mo><mrow id="S5.Ex26.m1.1.1.1.1.1.1" xref="S5.Ex26.m1.1.1.1.1.1.1.1.cmml"><mo id="S5.Ex26.m1.1.1.1.1.1.1.2" stretchy="false" xref="S5.Ex26.m1.1.1.1.1.1.1.1.cmml">(</mo><mrow id="S5.Ex26.m1.1.1.1.1.1.1.1" xref="S5.Ex26.m1.1.1.1.1.1.1.1.cmml"><msub id="S5.Ex26.m1.1.1.1.1.1.1.1.2" xref="S5.Ex26.m1.1.1.1.1.1.1.1.2.cmml"><mi id="S5.Ex26.m1.1.1.1.1.1.1.1.2.2" xref="S5.Ex26.m1.1.1.1.1.1.1.1.2.2.cmml">S</mi><mi id="S5.Ex26.m1.1.1.1.1.1.1.1.2.3" xref="S5.Ex26.m1.1.1.1.1.1.1.1.2.3.cmml">i</mi></msub><mo id="S5.Ex26.m1.1.1.1.1.1.1.1.1" xref="S5.Ex26.m1.1.1.1.1.1.1.1.1.cmml">−</mo><msub id="S5.Ex26.m1.1.1.1.1.1.1.1.3" xref="S5.Ex26.m1.1.1.1.1.1.1.1.3.cmml"><mi id="S5.Ex26.m1.1.1.1.1.1.1.1.3.2" xref="S5.Ex26.m1.1.1.1.1.1.1.1.3.2.cmml">H</mi><mi id="S5.Ex26.m1.1.1.1.1.1.1.1.3.3" xref="S5.Ex26.m1.1.1.1.1.1.1.1.3.3.cmml">k</mi></msub></mrow><mo id="S5.Ex26.m1.1.1.1.1.1.1.3" stretchy="false" xref="S5.Ex26.m1.1.1.1.1.1.1.1.cmml">)</mo></mrow></mrow></mrow></munder><mrow id="S5.Ex26.m1.10.10.1.1.3.2" xref="S5.Ex26.m1.10.10.1.1.3.2.cmml"><mtext class="ltx_mathvariant_italic" id="S5.Ex26.m1.10.10.1.1.3.2.2" xref="S5.Ex26.m1.10.10.1.1.3.2.2a.cmml">g</mtext><mo id="S5.Ex26.m1.10.10.1.1.3.2.1" xref="S5.Ex26.m1.10.10.1.1.3.2.1.cmml">⁢</mo><mrow id="S5.Ex26.m1.10.10.1.1.3.2.3.2" xref="S5.Ex26.m1.10.10.1.1.3.2.cmml"><mo id="S5.Ex26.m1.10.10.1.1.3.2.3.2.1" stretchy="false" xref="S5.Ex26.m1.10.10.1.1.3.2.cmml">(</mo><mi id="S5.Ex26.m1.4.4" xref="S5.Ex26.m1.4.4.cmml">e</mi><mo id="S5.Ex26.m1.10.10.1.1.3.2.3.2.2" stretchy="false" xref="S5.Ex26.m1.10.10.1.1.3.2.cmml">)</mo></mrow></mrow></mrow><mo id="S5.Ex26.m1.10.10.1.1.2" rspace="0.111em" xref="S5.Ex26.m1.10.10.1.1.2.cmml">=</mo><mrow id="S5.Ex26.m1.10.10.1.1.1.1" xref="S5.Ex26.m1.10.10.1.1.1.2.cmml"><mrow id="S5.Ex26.m1.10.10.1.1.1.1.1" xref="S5.Ex26.m1.10.10.1.1.1.1.1.cmml"><munder id="S5.Ex26.m1.10.10.1.1.1.1.1.1" xref="S5.Ex26.m1.10.10.1.1.1.1.1.1.cmml"><mo id="S5.Ex26.m1.10.10.1.1.1.1.1.1.2" movablelimits="false" xref="S5.Ex26.m1.10.10.1.1.1.1.1.1.2.cmml">∑</mo><mrow id="S5.Ex26.m1.2.2.1" xref="S5.Ex26.m1.2.2.1.cmml"><mi id="S5.Ex26.m1.2.2.1.3" xref="S5.Ex26.m1.2.2.1.3.cmml">e</mi><mo id="S5.Ex26.m1.2.2.1.2" xref="S5.Ex26.m1.2.2.1.2.cmml">∈</mo><mrow id="S5.Ex26.m1.2.2.1.1" xref="S5.Ex26.m1.2.2.1.1.cmml"><msub id="S5.Ex26.m1.2.2.1.1.3" xref="S5.Ex26.m1.2.2.1.1.3.cmml"><mover accent="true" id="S5.Ex26.m1.2.2.1.1.3.2" xref="S5.Ex26.m1.2.2.1.1.3.2.cmml"><mi id="S5.Ex26.m1.2.2.1.1.3.2.2" xref="S5.Ex26.m1.2.2.1.1.3.2.2.cmml">E</mi><mo id="S5.Ex26.m1.2.2.1.1.3.2.1" xref="S5.Ex26.m1.2.2.1.1.3.2.1.cmml">^</mo></mover><mi id="S5.Ex26.m1.2.2.1.1.3.3" xref="S5.Ex26.m1.2.2.1.1.3.3.cmml">B</mi></msub><mo id="S5.Ex26.m1.2.2.1.1.2" xref="S5.Ex26.m1.2.2.1.1.2.cmml">⁢</mo><mrow id="S5.Ex26.m1.2.2.1.1.1.1" xref="S5.Ex26.m1.2.2.1.1.1.1.1.cmml"><mo id="S5.Ex26.m1.2.2.1.1.1.1.2" stretchy="false" xref="S5.Ex26.m1.2.2.1.1.1.1.1.cmml">(</mo><msub id="S5.Ex26.m1.2.2.1.1.1.1.1" xref="S5.Ex26.m1.2.2.1.1.1.1.1.cmml"><mi id="S5.Ex26.m1.2.2.1.1.1.1.1.2" xref="S5.Ex26.m1.2.2.1.1.1.1.1.2.cmml">S</mi><mi id="S5.Ex26.m1.2.2.1.1.1.1.1.3" xref="S5.Ex26.m1.2.2.1.1.1.1.1.3.cmml">i</mi></msub><mo id="S5.Ex26.m1.2.2.1.1.1.1.3" stretchy="false" xref="S5.Ex26.m1.2.2.1.1.1.1.1.cmml">)</mo></mrow></mrow></mrow></munder><mrow id="S5.Ex26.m1.10.10.1.1.1.1.1.2" xref="S5.Ex26.m1.10.10.1.1.1.1.1.2.cmml"><mtext class="ltx_mathvariant_italic" id="S5.Ex26.m1.10.10.1.1.1.1.1.2.2" xref="S5.Ex26.m1.10.10.1.1.1.1.1.2.2a.cmml">g</mtext><mo id="S5.Ex26.m1.10.10.1.1.1.1.1.2.1" xref="S5.Ex26.m1.10.10.1.1.1.1.1.2.1.cmml">⁢</mo><mrow id="S5.Ex26.m1.10.10.1.1.1.1.1.2.3.2" xref="S5.Ex26.m1.10.10.1.1.1.1.1.2.cmml"><mo id="S5.Ex26.m1.10.10.1.1.1.1.1.2.3.2.1" stretchy="false" xref="S5.Ex26.m1.10.10.1.1.1.1.1.2.cmml">(</mo><mi id="S5.Ex26.m1.5.5" xref="S5.Ex26.m1.5.5.cmml">e</mi><mo id="S5.Ex26.m1.10.10.1.1.1.1.1.2.3.2.2" stretchy="false" xref="S5.Ex26.m1.10.10.1.1.1.1.1.2.cmml">)</mo></mrow></mrow></mrow><mspace id="S5.Ex26.m1.10.10.1.1.1.1.2" width="1em" xref="S5.Ex26.m1.10.10.1.1.1.2.cmml"></mspace><mo id="S5.Ex26.m1.9.9" xref="S5.Ex26.m1.9.9.cmml">−</mo></mrow></mrow><mspace id="S5.Ex26.m1.11.11.2.3" width="1em" xref="S5.Ex26.m1.11.11.3a.cmml"></mspace><mrow id="S5.Ex26.m1.11.11.2.2" xref="S5.Ex26.m1.11.11.2.2.cmml"><mrow id="S5.Ex26.m1.11.11.2.2.4" xref="S5.Ex26.m1.11.11.2.2.4.cmml"><munder id="S5.Ex26.m1.11.11.2.2.4.1" xref="S5.Ex26.m1.11.11.2.2.4.1.cmml"><mo id="S5.Ex26.m1.11.11.2.2.4.1.2" movablelimits="false" xref="S5.Ex26.m1.11.11.2.2.4.1.2.cmml">∑</mo><mrow id="S5.Ex26.m1.3.3.1" xref="S5.Ex26.m1.3.3.1.cmml"><mi id="S5.Ex26.m1.3.3.1.3" xref="S5.Ex26.m1.3.3.1.3.cmml">e</mi><mo id="S5.Ex26.m1.3.3.1.2" xref="S5.Ex26.m1.3.3.1.2.cmml">∈</mo><mrow id="S5.Ex26.m1.3.3.1.1" xref="S5.Ex26.m1.3.3.1.1.cmml"><mrow id="S5.Ex26.m1.3.3.1.1.1" xref="S5.Ex26.m1.3.3.1.1.1.cmml"><msub id="S5.Ex26.m1.3.3.1.1.1.3" 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xref="S5.Ex26.m1.11.11.2.2.2.2.1.1.2"></abs><apply id="S5.Ex26.m1.11.11.2.2.2.2.1.1.1.cmml" xref="S5.Ex26.m1.11.11.2.2.2.2.1.1.1"><csymbol cd="ambiguous" id="S5.Ex26.m1.11.11.2.2.2.2.1.1.1.1.cmml" xref="S5.Ex26.m1.11.11.2.2.2.2.1.1.1">subscript</csymbol><ci id="S5.Ex26.m1.11.11.2.2.2.2.1.1.1.2.cmml" xref="S5.Ex26.m1.11.11.2.2.2.2.1.1.1.2">𝐻</ci><ci id="S5.Ex26.m1.11.11.2.2.2.2.1.1.1.3.cmml" xref="S5.Ex26.m1.11.11.2.2.2.2.1.1.1.3">𝑘</ci></apply></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Ex26.m1.11c">\sum_{e\in\hat{E}_{B}(S_{i}-H_{k})}\textsl{g}(e)=\sum_{e\in\hat{E}_{B}(S_{i})}% \textsl{g}(e)\quad-\quad\sum_{e\in\hat{E}_{B}(H_{k})\cup E_{\uparrow}}\textsl{% g}(e)>\rho^{*}(u)|S_{i}|-\rho^{*}(u)|H_{k}|</annotation><annotation encoding="application/x-llamapun" id="S5.Ex26.m1.11d">∑ start_POSTSUBSCRIPT italic_e ∈ over^ start_ARG italic_E end_ARG start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT ( italic_S start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT - italic_H start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ) end_POSTSUBSCRIPT g ( italic_e ) = ∑ start_POSTSUBSCRIPT italic_e ∈ over^ start_ARG italic_E end_ARG start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT ( italic_S start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ) end_POSTSUBSCRIPT g ( italic_e ) - ∑ start_POSTSUBSCRIPT italic_e ∈ over^ start_ARG italic_E end_ARG start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT ( italic_H start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ) ∪ italic_E start_POSTSUBSCRIPT ↑ end_POSTSUBSCRIPT end_POSTSUBSCRIPT g ( italic_e ) > italic_ρ start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ( italic_u ) | italic_S start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT | - italic_ρ start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ( italic_u ) | italic_H start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT |</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_left_padright"></td> </tr></tbody> </table> </div> <div class="ltx_para" id="S5.Thmtheorem3.p20"> <p class="ltx_p" id="S5.Thmtheorem3.p20.3"><span class="ltx_text ltx_font_italic" id="S5.Thmtheorem3.p20.3.3">However, this implies that the set <math alttext="S_{i}-H_{k}" class="ltx_Math" display="inline" id="S5.Thmtheorem3.p20.1.1.m1.1"><semantics id="S5.Thmtheorem3.p20.1.1.m1.1a"><mrow id="S5.Thmtheorem3.p20.1.1.m1.1.1" xref="S5.Thmtheorem3.p20.1.1.m1.1.1.cmml"><msub id="S5.Thmtheorem3.p20.1.1.m1.1.1.2" xref="S5.Thmtheorem3.p20.1.1.m1.1.1.2.cmml"><mi id="S5.Thmtheorem3.p20.1.1.m1.1.1.2.2" xref="S5.Thmtheorem3.p20.1.1.m1.1.1.2.2.cmml">S</mi><mi id="S5.Thmtheorem3.p20.1.1.m1.1.1.2.3" xref="S5.Thmtheorem3.p20.1.1.m1.1.1.2.3.cmml">i</mi></msub><mo id="S5.Thmtheorem3.p20.1.1.m1.1.1.1" xref="S5.Thmtheorem3.p20.1.1.m1.1.1.1.cmml">−</mo><msub id="S5.Thmtheorem3.p20.1.1.m1.1.1.3" xref="S5.Thmtheorem3.p20.1.1.m1.1.1.3.cmml"><mi id="S5.Thmtheorem3.p20.1.1.m1.1.1.3.2" xref="S5.Thmtheorem3.p20.1.1.m1.1.1.3.2.cmml">H</mi><mi id="S5.Thmtheorem3.p20.1.1.m1.1.1.3.3" xref="S5.Thmtheorem3.p20.1.1.m1.1.1.3.3.cmml">k</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S5.Thmtheorem3.p20.1.1.m1.1b"><apply id="S5.Thmtheorem3.p20.1.1.m1.1.1.cmml" xref="S5.Thmtheorem3.p20.1.1.m1.1.1"><minus id="S5.Thmtheorem3.p20.1.1.m1.1.1.1.cmml" xref="S5.Thmtheorem3.p20.1.1.m1.1.1.1"></minus><apply id="S5.Thmtheorem3.p20.1.1.m1.1.1.2.cmml" xref="S5.Thmtheorem3.p20.1.1.m1.1.1.2"><csymbol cd="ambiguous" id="S5.Thmtheorem3.p20.1.1.m1.1.1.2.1.cmml" xref="S5.Thmtheorem3.p20.1.1.m1.1.1.2">subscript</csymbol><ci id="S5.Thmtheorem3.p20.1.1.m1.1.1.2.2.cmml" xref="S5.Thmtheorem3.p20.1.1.m1.1.1.2.2">𝑆</ci><ci id="S5.Thmtheorem3.p20.1.1.m1.1.1.2.3.cmml" xref="S5.Thmtheorem3.p20.1.1.m1.1.1.2.3">𝑖</ci></apply><apply id="S5.Thmtheorem3.p20.1.1.m1.1.1.3.cmml" xref="S5.Thmtheorem3.p20.1.1.m1.1.1.3"><csymbol cd="ambiguous" id="S5.Thmtheorem3.p20.1.1.m1.1.1.3.1.cmml" xref="S5.Thmtheorem3.p20.1.1.m1.1.1.3">subscript</csymbol><ci id="S5.Thmtheorem3.p20.1.1.m1.1.1.3.2.cmml" xref="S5.Thmtheorem3.p20.1.1.m1.1.1.3.2">𝐻</ci><ci id="S5.Thmtheorem3.p20.1.1.m1.1.1.3.3.cmml" xref="S5.Thmtheorem3.p20.1.1.m1.1.1.3.3">𝑘</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmtheorem3.p20.1.1.m1.1c">S_{i}-H_{k}</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem3.p20.1.1.m1.1d">italic_S start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT - italic_H start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT</annotation></semantics></math> is not empty, since <math alttext="S_{i}=H_{k}" class="ltx_Math" display="inline" id="S5.Thmtheorem3.p20.2.2.m2.1"><semantics id="S5.Thmtheorem3.p20.2.2.m2.1a"><mrow id="S5.Thmtheorem3.p20.2.2.m2.1.1" xref="S5.Thmtheorem3.p20.2.2.m2.1.1.cmml"><msub id="S5.Thmtheorem3.p20.2.2.m2.1.1.2" xref="S5.Thmtheorem3.p20.2.2.m2.1.1.2.cmml"><mi id="S5.Thmtheorem3.p20.2.2.m2.1.1.2.2" xref="S5.Thmtheorem3.p20.2.2.m2.1.1.2.2.cmml">S</mi><mi id="S5.Thmtheorem3.p20.2.2.m2.1.1.2.3" xref="S5.Thmtheorem3.p20.2.2.m2.1.1.2.3.cmml">i</mi></msub><mo id="S5.Thmtheorem3.p20.2.2.m2.1.1.1" xref="S5.Thmtheorem3.p20.2.2.m2.1.1.1.cmml">=</mo><msub id="S5.Thmtheorem3.p20.2.2.m2.1.1.3" xref="S5.Thmtheorem3.p20.2.2.m2.1.1.3.cmml"><mi id="S5.Thmtheorem3.p20.2.2.m2.1.1.3.2" xref="S5.Thmtheorem3.p20.2.2.m2.1.1.3.2.cmml">H</mi><mi id="S5.Thmtheorem3.p20.2.2.m2.1.1.3.3" xref="S5.Thmtheorem3.p20.2.2.m2.1.1.3.3.cmml">k</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S5.Thmtheorem3.p20.2.2.m2.1b"><apply id="S5.Thmtheorem3.p20.2.2.m2.1.1.cmml" xref="S5.Thmtheorem3.p20.2.2.m2.1.1"><eq id="S5.Thmtheorem3.p20.2.2.m2.1.1.1.cmml" xref="S5.Thmtheorem3.p20.2.2.m2.1.1.1"></eq><apply id="S5.Thmtheorem3.p20.2.2.m2.1.1.2.cmml" xref="S5.Thmtheorem3.p20.2.2.m2.1.1.2"><csymbol cd="ambiguous" id="S5.Thmtheorem3.p20.2.2.m2.1.1.2.1.cmml" xref="S5.Thmtheorem3.p20.2.2.m2.1.1.2">subscript</csymbol><ci id="S5.Thmtheorem3.p20.2.2.m2.1.1.2.2.cmml" xref="S5.Thmtheorem3.p20.2.2.m2.1.1.2.2">𝑆</ci><ci id="S5.Thmtheorem3.p20.2.2.m2.1.1.2.3.cmml" xref="S5.Thmtheorem3.p20.2.2.m2.1.1.2.3">𝑖</ci></apply><apply id="S5.Thmtheorem3.p20.2.2.m2.1.1.3.cmml" xref="S5.Thmtheorem3.p20.2.2.m2.1.1.3"><csymbol cd="ambiguous" id="S5.Thmtheorem3.p20.2.2.m2.1.1.3.1.cmml" xref="S5.Thmtheorem3.p20.2.2.m2.1.1.3">subscript</csymbol><ci id="S5.Thmtheorem3.p20.2.2.m2.1.1.3.2.cmml" xref="S5.Thmtheorem3.p20.2.2.m2.1.1.3.2">𝐻</ci><ci id="S5.Thmtheorem3.p20.2.2.m2.1.1.3.3.cmml" xref="S5.Thmtheorem3.p20.2.2.m2.1.1.3.3">𝑘</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmtheorem3.p20.2.2.m2.1c">S_{i}=H_{k}</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem3.p20.2.2.m2.1d">italic_S start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT = italic_H start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT</annotation></semantics></math> would imply that <math alttext="0>0" class="ltx_Math" display="inline" id="S5.Thmtheorem3.p20.3.3.m3.1"><semantics id="S5.Thmtheorem3.p20.3.3.m3.1a"><mrow id="S5.Thmtheorem3.p20.3.3.m3.1.1" xref="S5.Thmtheorem3.p20.3.3.m3.1.1.cmml"><mn id="S5.Thmtheorem3.p20.3.3.m3.1.1.2" xref="S5.Thmtheorem3.p20.3.3.m3.1.1.2.cmml">0</mn><mo id="S5.Thmtheorem3.p20.3.3.m3.1.1.1" xref="S5.Thmtheorem3.p20.3.3.m3.1.1.1.cmml">></mo><mn id="S5.Thmtheorem3.p20.3.3.m3.1.1.3" xref="S5.Thmtheorem3.p20.3.3.m3.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S5.Thmtheorem3.p20.3.3.m3.1b"><apply id="S5.Thmtheorem3.p20.3.3.m3.1.1.cmml" xref="S5.Thmtheorem3.p20.3.3.m3.1.1"><gt id="S5.Thmtheorem3.p20.3.3.m3.1.1.1.cmml" xref="S5.Thmtheorem3.p20.3.3.m3.1.1.1"></gt><cn id="S5.Thmtheorem3.p20.3.3.m3.1.1.2.cmml" type="integer" xref="S5.Thmtheorem3.p20.3.3.m3.1.1.2">0</cn><cn id="S5.Thmtheorem3.p20.3.3.m3.1.1.3.cmml" type="integer" xref="S5.Thmtheorem3.p20.3.3.m3.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmtheorem3.p20.3.3.m3.1c">0>0</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem3.p20.3.3.m3.1d">0 > 0</annotation></semantics></math>. Therefore, we may apply Equation <a class="ltx_ref" href="https://arxiv.org/html/2411.12694v2#S5.E6" title="Equation 6 ‣ Proof 5.3. ‣ 5 Results for dynamic algorithms ‣ Local Density and its Distributed Approximation"><span class="ltx_text ltx_ref_tag">6</span></a> to claim that:</span></p> </div> <div class="ltx_para" id="S5.Thmtheorem3.p21"> <table class="ltx_equation ltx_eqn_table" id="S5.Ex27"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_left_padleft"></td> <td class="ltx_eqn_cell ltx_align_left"><math alttext="\hat{\rho}_{B}(S_{i}-H_{k})\cdot(|S_{i}|-|H_{k}|)>\rho^{*}(u)(|S_{i}|-|H_{k}|)% \Rightarrow\hat{\rho}_{B}(S_{i}-H_{k})>\rho^{*}(u)." class="ltx_Math" display="block" id="S5.Ex27.m1.3"><semantics id="S5.Ex27.m1.3a"><mrow id="S5.Ex27.m1.3.3.1" xref="S5.Ex27.m1.3.3.1.1.cmml"><mrow id="S5.Ex27.m1.3.3.1.1" xref="S5.Ex27.m1.3.3.1.1.cmml"><mrow id="S5.Ex27.m1.3.3.1.1.2" xref="S5.Ex27.m1.3.3.1.1.2.cmml"><mrow 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id="S5.Ex27.m1.3.3.1.1.3.4.2.1" stretchy="false" xref="S5.Ex27.m1.3.3.1.1.3.cmml">(</mo><mi id="S5.Ex27.m1.1.1" xref="S5.Ex27.m1.1.1.cmml">u</mi><mo id="S5.Ex27.m1.3.3.1.1.3.4.2.2" stretchy="false" xref="S5.Ex27.m1.3.3.1.1.3.cmml">)</mo></mrow><mo id="S5.Ex27.m1.3.3.1.1.3.2a" xref="S5.Ex27.m1.3.3.1.1.3.2.cmml">⁢</mo><mrow id="S5.Ex27.m1.3.3.1.1.3.1.1" xref="S5.Ex27.m1.3.3.1.1.3.1.1.1.cmml"><mo id="S5.Ex27.m1.3.3.1.1.3.1.1.2" stretchy="false" xref="S5.Ex27.m1.3.3.1.1.3.1.1.1.cmml">(</mo><mrow id="S5.Ex27.m1.3.3.1.1.3.1.1.1" xref="S5.Ex27.m1.3.3.1.1.3.1.1.1.cmml"><mrow id="S5.Ex27.m1.3.3.1.1.3.1.1.1.1.1" xref="S5.Ex27.m1.3.3.1.1.3.1.1.1.1.2.cmml"><mo id="S5.Ex27.m1.3.3.1.1.3.1.1.1.1.1.2" stretchy="false" xref="S5.Ex27.m1.3.3.1.1.3.1.1.1.1.2.1.cmml">|</mo><msub id="S5.Ex27.m1.3.3.1.1.3.1.1.1.1.1.1" xref="S5.Ex27.m1.3.3.1.1.3.1.1.1.1.1.1.cmml"><mi id="S5.Ex27.m1.3.3.1.1.3.1.1.1.1.1.1.2" xref="S5.Ex27.m1.3.3.1.1.3.1.1.1.1.1.1.2.cmml">S</mi><mi id="S5.Ex27.m1.3.3.1.1.3.1.1.1.1.1.1.3" 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encoding="application/x-tex" id="S5.Ex27.m1.3c">\hat{\rho}_{B}(S_{i}-H_{k})\cdot(|S_{i}|-|H_{k}|)>\rho^{*}(u)(|S_{i}|-|H_{k}|)% \Rightarrow\hat{\rho}_{B}(S_{i}-H_{k})>\rho^{*}(u).</annotation><annotation encoding="application/x-llamapun" id="S5.Ex27.m1.3d">over^ start_ARG italic_ρ end_ARG start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT ( italic_S start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT - italic_H start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ) ⋅ ( | italic_S start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT | - | italic_H start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT | ) > italic_ρ start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ( italic_u ) ( | italic_S start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT | - | italic_H start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT | ) ⇒ over^ start_ARG italic_ρ end_ARG start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT ( italic_S start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT - italic_H start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ) > italic_ρ start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ( italic_u ) .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_left_padright"></td> </tr></tbody> </table> </div> <div class="ltx_para ltx_noindent" id="S5.Thmtheorem3.p22"> <p class="ltx_p" id="S5.Thmtheorem3.p22.3"><span class="ltx_text ltx_font_italic" id="S5.Thmtheorem3.p22.3.1">However, we now have found a non-empty set </span><math alttext="X=S_{i}-H_{k}\subseteq V-B" class="ltx_Math" display="inline" id="S5.Thmtheorem3.p22.1.m1.1"><semantics id="S5.Thmtheorem3.p22.1.m1.1a"><mrow id="S5.Thmtheorem3.p22.1.m1.1.1" xref="S5.Thmtheorem3.p22.1.m1.1.1.cmml"><mi id="S5.Thmtheorem3.p22.1.m1.1.1.2" xref="S5.Thmtheorem3.p22.1.m1.1.1.2.cmml">X</mi><mo id="S5.Thmtheorem3.p22.1.m1.1.1.3" xref="S5.Thmtheorem3.p22.1.m1.1.1.3.cmml">=</mo><mrow id="S5.Thmtheorem3.p22.1.m1.1.1.4" xref="S5.Thmtheorem3.p22.1.m1.1.1.4.cmml"><msub id="S5.Thmtheorem3.p22.1.m1.1.1.4.2" xref="S5.Thmtheorem3.p22.1.m1.1.1.4.2.cmml"><mi id="S5.Thmtheorem3.p22.1.m1.1.1.4.2.2" xref="S5.Thmtheorem3.p22.1.m1.1.1.4.2.2.cmml">S</mi><mi id="S5.Thmtheorem3.p22.1.m1.1.1.4.2.3" xref="S5.Thmtheorem3.p22.1.m1.1.1.4.2.3.cmml">i</mi></msub><mo id="S5.Thmtheorem3.p22.1.m1.1.1.4.1" xref="S5.Thmtheorem3.p22.1.m1.1.1.4.1.cmml">−</mo><msub id="S5.Thmtheorem3.p22.1.m1.1.1.4.3" xref="S5.Thmtheorem3.p22.1.m1.1.1.4.3.cmml"><mi id="S5.Thmtheorem3.p22.1.m1.1.1.4.3.2" xref="S5.Thmtheorem3.p22.1.m1.1.1.4.3.2.cmml">H</mi><mi id="S5.Thmtheorem3.p22.1.m1.1.1.4.3.3" xref="S5.Thmtheorem3.p22.1.m1.1.1.4.3.3.cmml">k</mi></msub></mrow><mo id="S5.Thmtheorem3.p22.1.m1.1.1.5" xref="S5.Thmtheorem3.p22.1.m1.1.1.5.cmml">⊆</mo><mrow id="S5.Thmtheorem3.p22.1.m1.1.1.6" xref="S5.Thmtheorem3.p22.1.m1.1.1.6.cmml"><mi id="S5.Thmtheorem3.p22.1.m1.1.1.6.2" xref="S5.Thmtheorem3.p22.1.m1.1.1.6.2.cmml">V</mi><mo id="S5.Thmtheorem3.p22.1.m1.1.1.6.1" xref="S5.Thmtheorem3.p22.1.m1.1.1.6.1.cmml">−</mo><mi id="S5.Thmtheorem3.p22.1.m1.1.1.6.3" xref="S5.Thmtheorem3.p22.1.m1.1.1.6.3.cmml">B</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.Thmtheorem3.p22.1.m1.1b"><apply id="S5.Thmtheorem3.p22.1.m1.1.1.cmml" xref="S5.Thmtheorem3.p22.1.m1.1.1"><and id="S5.Thmtheorem3.p22.1.m1.1.1a.cmml" xref="S5.Thmtheorem3.p22.1.m1.1.1"></and><apply id="S5.Thmtheorem3.p22.1.m1.1.1b.cmml" xref="S5.Thmtheorem3.p22.1.m1.1.1"><eq id="S5.Thmtheorem3.p22.1.m1.1.1.3.cmml" xref="S5.Thmtheorem3.p22.1.m1.1.1.3"></eq><ci id="S5.Thmtheorem3.p22.1.m1.1.1.2.cmml" xref="S5.Thmtheorem3.p22.1.m1.1.1.2">𝑋</ci><apply id="S5.Thmtheorem3.p22.1.m1.1.1.4.cmml" xref="S5.Thmtheorem3.p22.1.m1.1.1.4"><minus id="S5.Thmtheorem3.p22.1.m1.1.1.4.1.cmml" xref="S5.Thmtheorem3.p22.1.m1.1.1.4.1"></minus><apply id="S5.Thmtheorem3.p22.1.m1.1.1.4.2.cmml" xref="S5.Thmtheorem3.p22.1.m1.1.1.4.2"><csymbol cd="ambiguous" id="S5.Thmtheorem3.p22.1.m1.1.1.4.2.1.cmml" xref="S5.Thmtheorem3.p22.1.m1.1.1.4.2">subscript</csymbol><ci id="S5.Thmtheorem3.p22.1.m1.1.1.4.2.2.cmml" xref="S5.Thmtheorem3.p22.1.m1.1.1.4.2.2">𝑆</ci><ci id="S5.Thmtheorem3.p22.1.m1.1.1.4.2.3.cmml" xref="S5.Thmtheorem3.p22.1.m1.1.1.4.2.3">𝑖</ci></apply><apply id="S5.Thmtheorem3.p22.1.m1.1.1.4.3.cmml" xref="S5.Thmtheorem3.p22.1.m1.1.1.4.3"><csymbol cd="ambiguous" id="S5.Thmtheorem3.p22.1.m1.1.1.4.3.1.cmml" xref="S5.Thmtheorem3.p22.1.m1.1.1.4.3">subscript</csymbol><ci id="S5.Thmtheorem3.p22.1.m1.1.1.4.3.2.cmml" xref="S5.Thmtheorem3.p22.1.m1.1.1.4.3.2">𝐻</ci><ci id="S5.Thmtheorem3.p22.1.m1.1.1.4.3.3.cmml" xref="S5.Thmtheorem3.p22.1.m1.1.1.4.3.3">𝑘</ci></apply></apply></apply><apply id="S5.Thmtheorem3.p22.1.m1.1.1c.cmml" xref="S5.Thmtheorem3.p22.1.m1.1.1"><subset id="S5.Thmtheorem3.p22.1.m1.1.1.5.cmml" xref="S5.Thmtheorem3.p22.1.m1.1.1.5"></subset><share href="https://arxiv.org/html/2411.12694v2#S5.Thmtheorem3.p22.1.m1.1.1.4.cmml" id="S5.Thmtheorem3.p22.1.m1.1.1d.cmml" xref="S5.Thmtheorem3.p22.1.m1.1.1"></share><apply id="S5.Thmtheorem3.p22.1.m1.1.1.6.cmml" xref="S5.Thmtheorem3.p22.1.m1.1.1.6"><minus id="S5.Thmtheorem3.p22.1.m1.1.1.6.1.cmml" xref="S5.Thmtheorem3.p22.1.m1.1.1.6.1"></minus><ci id="S5.Thmtheorem3.p22.1.m1.1.1.6.2.cmml" xref="S5.Thmtheorem3.p22.1.m1.1.1.6.2">𝑉</ci><ci id="S5.Thmtheorem3.p22.1.m1.1.1.6.3.cmml" xref="S5.Thmtheorem3.p22.1.m1.1.1.6.3">𝐵</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmtheorem3.p22.1.m1.1c">X=S_{i}-H_{k}\subseteq V-B</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem3.p22.1.m1.1d">italic_X = italic_S start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT - italic_H start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ⊆ italic_V - italic_B</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S5.Thmtheorem3.p22.3.2">, where </span><math alttext="\hat{\rho}_{B}(X)>\hat{\rho}_{B}(S_{i})" class="ltx_Math" display="inline" id="S5.Thmtheorem3.p22.2.m2.2"><semantics id="S5.Thmtheorem3.p22.2.m2.2a"><mrow 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xref="S5.Thmtheorem3.p22.2.m2.2.2.1.1.1.1.3">𝑖</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmtheorem3.p22.2.m2.2c">\hat{\rho}_{B}(X)>\hat{\rho}_{B}(S_{i})</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem3.p22.2.m2.2d">over^ start_ARG italic_ρ end_ARG start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT ( italic_X ) > over^ start_ARG italic_ρ end_ARG start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT ( italic_S start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT )</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S5.Thmtheorem3.p22.3.3"> which contradicts the definition of </span><math alttext="S_{i}" class="ltx_Math" display="inline" id="S5.Thmtheorem3.p22.3.m3.1"><semantics id="S5.Thmtheorem3.p22.3.m3.1a"><msub id="S5.Thmtheorem3.p22.3.m3.1.1" xref="S5.Thmtheorem3.p22.3.m3.1.1.cmml"><mi id="S5.Thmtheorem3.p22.3.m3.1.1.2" xref="S5.Thmtheorem3.p22.3.m3.1.1.2.cmml">S</mi><mi id="S5.Thmtheorem3.p22.3.m3.1.1.3" xref="S5.Thmtheorem3.p22.3.m3.1.1.3.cmml">i</mi></msub><annotation-xml encoding="MathML-Content" id="S5.Thmtheorem3.p22.3.m3.1b"><apply id="S5.Thmtheorem3.p22.3.m3.1.1.cmml" xref="S5.Thmtheorem3.p22.3.m3.1.1"><csymbol cd="ambiguous" id="S5.Thmtheorem3.p22.3.m3.1.1.1.cmml" xref="S5.Thmtheorem3.p22.3.m3.1.1">subscript</csymbol><ci id="S5.Thmtheorem3.p22.3.m3.1.1.2.cmml" xref="S5.Thmtheorem3.p22.3.m3.1.1.2">𝑆</ci><ci id="S5.Thmtheorem3.p22.3.m3.1.1.3.cmml" xref="S5.Thmtheorem3.p22.3.m3.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmtheorem3.p22.3.m3.1c">S_{i}</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem3.p22.3.m3.1d">italic_S start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S5.Thmtheorem3.p22.3.4">.</span></p> </div> </div> <div class="ltx_para" id="S5.p3"> <p class="ltx_p" id="S5.p3.3">If <math alttext="G" 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">G</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">𝐺</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.p3.1.m1.1c">G</annotation><annotation encoding="application/x-llamapun" id="S5.p3.1.m1.1d">italic_G</annotation></semantics></math> is a unit-weight graph, Chekuri et al. <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2411.12694v2#bib.bib7" title="">7</a>]</cite> present a dynamic algorithm to maintain an <math alttext="\eta" class="ltx_Math" display="inline" id="S5.p3.2.m2.1"><semantics id="S5.p3.2.m2.1a"><mi id="S5.p3.2.m2.1.1" xref="S5.p3.2.m2.1.1.cmml">η</mi><annotation-xml encoding="MathML-Content" id="S5.p3.2.m2.1b"><ci id="S5.p3.2.m2.1.1.cmml" xref="S5.p3.2.m2.1.1">𝜂</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.p3.2.m2.1c">\eta</annotation><annotation encoding="application/x-llamapun" id="S5.p3.2.m2.1d">italic_η</annotation></semantics></math>-fair orientation in a unit-weight graph with <math alttext="\eta\in O(\varepsilon^{-2}\log n)" class="ltx_Math" display="inline" id="S5.p3.3.m3.1"><semantics id="S5.p3.3.m3.1a"><mrow id="S5.p3.3.m3.1.1" xref="S5.p3.3.m3.1.1.cmml"><mi id="S5.p3.3.m3.1.1.3" xref="S5.p3.3.m3.1.1.3.cmml">η</mi><mo id="S5.p3.3.m3.1.1.2" xref="S5.p3.3.m3.1.1.2.cmml">∈</mo><mrow id="S5.p3.3.m3.1.1.1" xref="S5.p3.3.m3.1.1.1.cmml"><mi id="S5.p3.3.m3.1.1.1.3" xref="S5.p3.3.m3.1.1.1.3.cmml">O</mi><mo id="S5.p3.3.m3.1.1.1.2" xref="S5.p3.3.m3.1.1.1.2.cmml">⁢</mo><mrow id="S5.p3.3.m3.1.1.1.1.1" xref="S5.p3.3.m3.1.1.1.1.1.1.cmml"><mo id="S5.p3.3.m3.1.1.1.1.1.2" stretchy="false" xref="S5.p3.3.m3.1.1.1.1.1.1.cmml">(</mo><mrow id="S5.p3.3.m3.1.1.1.1.1.1" xref="S5.p3.3.m3.1.1.1.1.1.1.cmml"><msup id="S5.p3.3.m3.1.1.1.1.1.1.2" xref="S5.p3.3.m3.1.1.1.1.1.1.2.cmml"><mi id="S5.p3.3.m3.1.1.1.1.1.1.2.2" xref="S5.p3.3.m3.1.1.1.1.1.1.2.2.cmml">ε</mi><mrow id="S5.p3.3.m3.1.1.1.1.1.1.2.3" xref="S5.p3.3.m3.1.1.1.1.1.1.2.3.cmml"><mo id="S5.p3.3.m3.1.1.1.1.1.1.2.3a" xref="S5.p3.3.m3.1.1.1.1.1.1.2.3.cmml">−</mo><mn id="S5.p3.3.m3.1.1.1.1.1.1.2.3.2" xref="S5.p3.3.m3.1.1.1.1.1.1.2.3.2.cmml">2</mn></mrow></msup><mo id="S5.p3.3.m3.1.1.1.1.1.1.1" lspace="0.167em" xref="S5.p3.3.m3.1.1.1.1.1.1.1.cmml">⁢</mo><mrow id="S5.p3.3.m3.1.1.1.1.1.1.3" xref="S5.p3.3.m3.1.1.1.1.1.1.3.cmml"><mi id="S5.p3.3.m3.1.1.1.1.1.1.3.1" xref="S5.p3.3.m3.1.1.1.1.1.1.3.1.cmml">log</mi><mo id="S5.p3.3.m3.1.1.1.1.1.1.3a" lspace="0.167em" xref="S5.p3.3.m3.1.1.1.1.1.1.3.cmml">⁡</mo><mi id="S5.p3.3.m3.1.1.1.1.1.1.3.2" xref="S5.p3.3.m3.1.1.1.1.1.1.3.2.cmml">n</mi></mrow></mrow><mo id="S5.p3.3.m3.1.1.1.1.1.3" stretchy="false" xref="S5.p3.3.m3.1.1.1.1.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.p3.3.m3.1b"><apply id="S5.p3.3.m3.1.1.cmml" xref="S5.p3.3.m3.1.1"><in id="S5.p3.3.m3.1.1.2.cmml" xref="S5.p3.3.m3.1.1.2"></in><ci id="S5.p3.3.m3.1.1.3.cmml" xref="S5.p3.3.m3.1.1.3">𝜂</ci><apply id="S5.p3.3.m3.1.1.1.cmml" xref="S5.p3.3.m3.1.1.1"><times id="S5.p3.3.m3.1.1.1.2.cmml" xref="S5.p3.3.m3.1.1.1.2"></times><ci id="S5.p3.3.m3.1.1.1.3.cmml" xref="S5.p3.3.m3.1.1.1.3">𝑂</ci><apply id="S5.p3.3.m3.1.1.1.1.1.1.cmml" xref="S5.p3.3.m3.1.1.1.1.1"><times id="S5.p3.3.m3.1.1.1.1.1.1.1.cmml" xref="S5.p3.3.m3.1.1.1.1.1.1.1"></times><apply id="S5.p3.3.m3.1.1.1.1.1.1.2.cmml" xref="S5.p3.3.m3.1.1.1.1.1.1.2"><csymbol cd="ambiguous" id="S5.p3.3.m3.1.1.1.1.1.1.2.1.cmml" xref="S5.p3.3.m3.1.1.1.1.1.1.2">superscript</csymbol><ci id="S5.p3.3.m3.1.1.1.1.1.1.2.2.cmml" xref="S5.p3.3.m3.1.1.1.1.1.1.2.2">𝜀</ci><apply id="S5.p3.3.m3.1.1.1.1.1.1.2.3.cmml" xref="S5.p3.3.m3.1.1.1.1.1.1.2.3"><minus id="S5.p3.3.m3.1.1.1.1.1.1.2.3.1.cmml" xref="S5.p3.3.m3.1.1.1.1.1.1.2.3"></minus><cn id="S5.p3.3.m3.1.1.1.1.1.1.2.3.2.cmml" type="integer" xref="S5.p3.3.m3.1.1.1.1.1.1.2.3.2">2</cn></apply></apply><apply id="S5.p3.3.m3.1.1.1.1.1.1.3.cmml" xref="S5.p3.3.m3.1.1.1.1.1.1.3"><log id="S5.p3.3.m3.1.1.1.1.1.1.3.1.cmml" xref="S5.p3.3.m3.1.1.1.1.1.1.3.1"></log><ci id="S5.p3.3.m3.1.1.1.1.1.1.3.2.cmml" xref="S5.p3.3.m3.1.1.1.1.1.1.3.2">𝑛</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.p3.3.m3.1c">\eta\in O(\varepsilon^{-2}\log n)</annotation><annotation encoding="application/x-llamapun" id="S5.p3.3.m3.1d">italic_η ∈ italic_O ( italic_ε start_POSTSUPERSCRIPT - 2 end_POSTSUPERSCRIPT roman_log italic_n )</annotation></semantics></math>. Thus, by Theorem <a class="ltx_ref" href="https://arxiv.org/html/2411.12694v2#S3.Thmtheorem5" title="Theorem 3.5. ‣ 3.B Results for dynamic algorithms ‣ 3 Results and organisation ‣ Local Density and its Distributed Approximation"><span class="ltx_text ltx_ref_tag">3.5</span></a>, we may now conclude that they approximate the local density (and/or the local out-degree):</p> </div> <div class="ltx_para" id="S5.p4"> <p class="ltx_p" id="S5.p4.1">See <a class="ltx_ref" href="https://arxiv.org/html/2411.12694v2#S3.Thmtheorem6" title="Corollary 3.6. ‣ 3.B Results for dynamic algorithms ‣ 3 Results and organisation ‣ Local Density and its Distributed Approximation"><span class="ltx_text ltx_ref_tag">3.6</span></a></p> </div> </section> <section class="ltx_section" id="S6"> <h2 class="ltx_title ltx_title_section"> <span class="ltx_tag ltx_tag_section">6 </span>Results in LOCAL</h2> <div class="ltx_para ltx_noindent" id="S6.p1"> <p class="ltx_p" id="S6.p1.1">We prove that the local out-degree of each <math alttext="v\in V" class="ltx_Math" display="inline" id="S6.p1.1.m1.1"><semantics id="S6.p1.1.m1.1a"><mrow id="S6.p1.1.m1.1.1" xref="S6.p1.1.m1.1.1.cmml"><mi id="S6.p1.1.m1.1.1.2" xref="S6.p1.1.m1.1.1.2.cmml">v</mi><mo id="S6.p1.1.m1.1.1.1" xref="S6.p1.1.m1.1.1.1.cmml">∈</mo><mi id="S6.p1.1.m1.1.1.3" xref="S6.p1.1.m1.1.1.3.cmml">V</mi></mrow><annotation-xml encoding="MathML-Content" id="S6.p1.1.m1.1b"><apply id="S6.p1.1.m1.1.1.cmml" xref="S6.p1.1.m1.1.1"><in id="S6.p1.1.m1.1.1.1.cmml" xref="S6.p1.1.m1.1.1.1"></in><ci id="S6.p1.1.m1.1.1.2.cmml" xref="S6.p1.1.m1.1.1.2">𝑣</ci><ci id="S6.p1.1.m1.1.1.3.cmml" xref="S6.p1.1.m1.1.1.3">𝑉</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.p1.1.m1.1c">v\in V</annotation><annotation encoding="application/x-llamapun" id="S6.p1.1.m1.1d">italic_v ∈ italic_V</annotation></semantics></math> is (largely) determined by its local neighbourhood. As a result, we immediately get an algorithm to solve Problem <a class="ltx_ref" href="https://arxiv.org/html/2411.12694v2#S2.Thmtheorem11" title="Problem 2.11. ‣ The benefit of local measures: ‣ 2.3 Local density ‣ 2 Preliminaries and related work ‣ Local Density and its Distributed Approximation"><span class="ltx_text ltx_ref_tag">2.11</span></a> in LOCAL.</p> </div> <div class="ltx_para" id="S6.p2"> <p class="ltx_p" id="S6.p2.1">See <a class="ltx_ref" href="https://arxiv.org/html/2411.12694v2#S3.Thmtheorem7" title="Theorem 3.7. ‣ 3.C Results in LOCAL ‣ 3 Results and organisation ‣ Local Density and its Distributed Approximation"><span class="ltx_text ltx_ref_tag">3.7</span></a></p> </div> <div class="ltx_theorem ltx_theorem_proof" id="S6.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="S6.Thmtheorem1.1.1.1">Proof 6.1</span></span><span class="ltx_text ltx_font_bold" id="S6.Thmtheorem1.2.2">.</span> </h6> <div class="ltx_para" id="S6.Thmtheorem1.p1"> <p class="ltx_p" id="S6.Thmtheorem1.p1.3"><span class="ltx_text ltx_font_italic" id="S6.Thmtheorem1.p1.3.3">To prove the theorem, we design a simple deletion-only algorithm to maintain an <math alttext="\eta" class="ltx_Math" display="inline" id="S6.Thmtheorem1.p1.1.1.m1.1"><semantics id="S6.Thmtheorem1.p1.1.1.m1.1a"><mi id="S6.Thmtheorem1.p1.1.1.m1.1.1" xref="S6.Thmtheorem1.p1.1.1.m1.1.1.cmml">η</mi><annotation-xml encoding="MathML-Content" id="S6.Thmtheorem1.p1.1.1.m1.1b"><ci id="S6.Thmtheorem1.p1.1.1.m1.1.1.cmml" xref="S6.Thmtheorem1.p1.1.1.m1.1.1">𝜂</ci></annotation-xml><annotation encoding="application/x-tex" id="S6.Thmtheorem1.p1.1.1.m1.1c">\eta</annotation><annotation encoding="application/x-llamapun" id="S6.Thmtheorem1.p1.1.1.m1.1d">italic_η</annotation></semantics></math>-fair orientation. For <math alttext="\eta=\frac{\varepsilon^{2}}{128\log n}" class="ltx_Math" display="inline" id="S6.Thmtheorem1.p1.2.2.m2.1"><semantics id="S6.Thmtheorem1.p1.2.2.m2.1a"><mrow id="S6.Thmtheorem1.p1.2.2.m2.1.1" xref="S6.Thmtheorem1.p1.2.2.m2.1.1.cmml"><mi id="S6.Thmtheorem1.p1.2.2.m2.1.1.2" xref="S6.Thmtheorem1.p1.2.2.m2.1.1.2.cmml">η</mi><mo id="S6.Thmtheorem1.p1.2.2.m2.1.1.1" xref="S6.Thmtheorem1.p1.2.2.m2.1.1.1.cmml">=</mo><mfrac id="S6.Thmtheorem1.p1.2.2.m2.1.1.3" xref="S6.Thmtheorem1.p1.2.2.m2.1.1.3.cmml"><msup id="S6.Thmtheorem1.p1.2.2.m2.1.1.3.2" xref="S6.Thmtheorem1.p1.2.2.m2.1.1.3.2.cmml"><mi id="S6.Thmtheorem1.p1.2.2.m2.1.1.3.2.2" xref="S6.Thmtheorem1.p1.2.2.m2.1.1.3.2.2.cmml">ε</mi><mn id="S6.Thmtheorem1.p1.2.2.m2.1.1.3.2.3" xref="S6.Thmtheorem1.p1.2.2.m2.1.1.3.2.3.cmml">2</mn></msup><mrow id="S6.Thmtheorem1.p1.2.2.m2.1.1.3.3" xref="S6.Thmtheorem1.p1.2.2.m2.1.1.3.3.cmml"><mn id="S6.Thmtheorem1.p1.2.2.m2.1.1.3.3.2" xref="S6.Thmtheorem1.p1.2.2.m2.1.1.3.3.2.cmml">128</mn><mo id="S6.Thmtheorem1.p1.2.2.m2.1.1.3.3.1" lspace="0.167em" xref="S6.Thmtheorem1.p1.2.2.m2.1.1.3.3.1.cmml">⁢</mo><mrow id="S6.Thmtheorem1.p1.2.2.m2.1.1.3.3.3" xref="S6.Thmtheorem1.p1.2.2.m2.1.1.3.3.3.cmml"><mi id="S6.Thmtheorem1.p1.2.2.m2.1.1.3.3.3.1" xref="S6.Thmtheorem1.p1.2.2.m2.1.1.3.3.3.1.cmml">log</mi><mo id="S6.Thmtheorem1.p1.2.2.m2.1.1.3.3.3a" lspace="0.167em" xref="S6.Thmtheorem1.p1.2.2.m2.1.1.3.3.3.cmml">⁡</mo><mi id="S6.Thmtheorem1.p1.2.2.m2.1.1.3.3.3.2" xref="S6.Thmtheorem1.p1.2.2.m2.1.1.3.3.3.2.cmml">n</mi></mrow></mrow></mfrac></mrow><annotation-xml encoding="MathML-Content" id="S6.Thmtheorem1.p1.2.2.m2.1b"><apply id="S6.Thmtheorem1.p1.2.2.m2.1.1.cmml" xref="S6.Thmtheorem1.p1.2.2.m2.1.1"><eq id="S6.Thmtheorem1.p1.2.2.m2.1.1.1.cmml" xref="S6.Thmtheorem1.p1.2.2.m2.1.1.1"></eq><ci id="S6.Thmtheorem1.p1.2.2.m2.1.1.2.cmml" xref="S6.Thmtheorem1.p1.2.2.m2.1.1.2">𝜂</ci><apply id="S6.Thmtheorem1.p1.2.2.m2.1.1.3.cmml" xref="S6.Thmtheorem1.p1.2.2.m2.1.1.3"><divide id="S6.Thmtheorem1.p1.2.2.m2.1.1.3.1.cmml" xref="S6.Thmtheorem1.p1.2.2.m2.1.1.3"></divide><apply id="S6.Thmtheorem1.p1.2.2.m2.1.1.3.2.cmml" xref="S6.Thmtheorem1.p1.2.2.m2.1.1.3.2"><csymbol cd="ambiguous" id="S6.Thmtheorem1.p1.2.2.m2.1.1.3.2.1.cmml" xref="S6.Thmtheorem1.p1.2.2.m2.1.1.3.2">superscript</csymbol><ci id="S6.Thmtheorem1.p1.2.2.m2.1.1.3.2.2.cmml" xref="S6.Thmtheorem1.p1.2.2.m2.1.1.3.2.2">𝜀</ci><cn id="S6.Thmtheorem1.p1.2.2.m2.1.1.3.2.3.cmml" type="integer" xref="S6.Thmtheorem1.p1.2.2.m2.1.1.3.2.3">2</cn></apply><apply id="S6.Thmtheorem1.p1.2.2.m2.1.1.3.3.cmml" xref="S6.Thmtheorem1.p1.2.2.m2.1.1.3.3"><times id="S6.Thmtheorem1.p1.2.2.m2.1.1.3.3.1.cmml" xref="S6.Thmtheorem1.p1.2.2.m2.1.1.3.3.1"></times><cn id="S6.Thmtheorem1.p1.2.2.m2.1.1.3.3.2.cmml" type="integer" xref="S6.Thmtheorem1.p1.2.2.m2.1.1.3.3.2">128</cn><apply id="S6.Thmtheorem1.p1.2.2.m2.1.1.3.3.3.cmml" xref="S6.Thmtheorem1.p1.2.2.m2.1.1.3.3.3"><log id="S6.Thmtheorem1.p1.2.2.m2.1.1.3.3.3.1.cmml" xref="S6.Thmtheorem1.p1.2.2.m2.1.1.3.3.3.1"></log><ci id="S6.Thmtheorem1.p1.2.2.m2.1.1.3.3.3.2.cmml" xref="S6.Thmtheorem1.p1.2.2.m2.1.1.3.3.3.2">𝑛</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.Thmtheorem1.p1.2.2.m2.1c">\eta=\frac{\varepsilon^{2}}{128\log n}</annotation><annotation encoding="application/x-llamapun" id="S6.Thmtheorem1.p1.2.2.m2.1d">italic_η = divide start_ARG italic_ε start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG start_ARG 128 roman_log italic_n end_ARG</annotation></semantics></math>, this algorithm has a recursive depth of <math alttext="O(\varepsilon^{-2}\log^{2}n)" class="ltx_Math" display="inline" id="S6.Thmtheorem1.p1.3.3.m3.1"><semantics id="S6.Thmtheorem1.p1.3.3.m3.1a"><mrow id="S6.Thmtheorem1.p1.3.3.m3.1.1" xref="S6.Thmtheorem1.p1.3.3.m3.1.1.cmml"><mi id="S6.Thmtheorem1.p1.3.3.m3.1.1.3" xref="S6.Thmtheorem1.p1.3.3.m3.1.1.3.cmml">O</mi><mo id="S6.Thmtheorem1.p1.3.3.m3.1.1.2" xref="S6.Thmtheorem1.p1.3.3.m3.1.1.2.cmml">⁢</mo><mrow id="S6.Thmtheorem1.p1.3.3.m3.1.1.1.1" xref="S6.Thmtheorem1.p1.3.3.m3.1.1.1.1.1.cmml"><mo id="S6.Thmtheorem1.p1.3.3.m3.1.1.1.1.2" stretchy="false" xref="S6.Thmtheorem1.p1.3.3.m3.1.1.1.1.1.cmml">(</mo><mrow id="S6.Thmtheorem1.p1.3.3.m3.1.1.1.1.1" xref="S6.Thmtheorem1.p1.3.3.m3.1.1.1.1.1.cmml"><msup id="S6.Thmtheorem1.p1.3.3.m3.1.1.1.1.1.2" xref="S6.Thmtheorem1.p1.3.3.m3.1.1.1.1.1.2.cmml"><mi id="S6.Thmtheorem1.p1.3.3.m3.1.1.1.1.1.2.2" xref="S6.Thmtheorem1.p1.3.3.m3.1.1.1.1.1.2.2.cmml">ε</mi><mrow id="S6.Thmtheorem1.p1.3.3.m3.1.1.1.1.1.2.3" xref="S6.Thmtheorem1.p1.3.3.m3.1.1.1.1.1.2.3.cmml"><mo id="S6.Thmtheorem1.p1.3.3.m3.1.1.1.1.1.2.3a" xref="S6.Thmtheorem1.p1.3.3.m3.1.1.1.1.1.2.3.cmml">−</mo><mn id="S6.Thmtheorem1.p1.3.3.m3.1.1.1.1.1.2.3.2" xref="S6.Thmtheorem1.p1.3.3.m3.1.1.1.1.1.2.3.2.cmml">2</mn></mrow></msup><mo id="S6.Thmtheorem1.p1.3.3.m3.1.1.1.1.1.1" lspace="0.167em" xref="S6.Thmtheorem1.p1.3.3.m3.1.1.1.1.1.1.cmml">⁢</mo><mrow id="S6.Thmtheorem1.p1.3.3.m3.1.1.1.1.1.3" xref="S6.Thmtheorem1.p1.3.3.m3.1.1.1.1.1.3.cmml"><msup id="S6.Thmtheorem1.p1.3.3.m3.1.1.1.1.1.3.1" xref="S6.Thmtheorem1.p1.3.3.m3.1.1.1.1.1.3.1.cmml"><mi id="S6.Thmtheorem1.p1.3.3.m3.1.1.1.1.1.3.1.2" xref="S6.Thmtheorem1.p1.3.3.m3.1.1.1.1.1.3.1.2.cmml">log</mi><mn id="S6.Thmtheorem1.p1.3.3.m3.1.1.1.1.1.3.1.3" xref="S6.Thmtheorem1.p1.3.3.m3.1.1.1.1.1.3.1.3.cmml">2</mn></msup><mo id="S6.Thmtheorem1.p1.3.3.m3.1.1.1.1.1.3a" lspace="0.167em" xref="S6.Thmtheorem1.p1.3.3.m3.1.1.1.1.1.3.cmml">⁡</mo><mi id="S6.Thmtheorem1.p1.3.3.m3.1.1.1.1.1.3.2" xref="S6.Thmtheorem1.p1.3.3.m3.1.1.1.1.1.3.2.cmml">n</mi></mrow></mrow><mo id="S6.Thmtheorem1.p1.3.3.m3.1.1.1.1.3" stretchy="false" xref="S6.Thmtheorem1.p1.3.3.m3.1.1.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.Thmtheorem1.p1.3.3.m3.1b"><apply id="S6.Thmtheorem1.p1.3.3.m3.1.1.cmml" xref="S6.Thmtheorem1.p1.3.3.m3.1.1"><times id="S6.Thmtheorem1.p1.3.3.m3.1.1.2.cmml" xref="S6.Thmtheorem1.p1.3.3.m3.1.1.2"></times><ci id="S6.Thmtheorem1.p1.3.3.m3.1.1.3.cmml" xref="S6.Thmtheorem1.p1.3.3.m3.1.1.3">𝑂</ci><apply id="S6.Thmtheorem1.p1.3.3.m3.1.1.1.1.1.cmml" xref="S6.Thmtheorem1.p1.3.3.m3.1.1.1.1"><times id="S6.Thmtheorem1.p1.3.3.m3.1.1.1.1.1.1.cmml" xref="S6.Thmtheorem1.p1.3.3.m3.1.1.1.1.1.1"></times><apply id="S6.Thmtheorem1.p1.3.3.m3.1.1.1.1.1.2.cmml" xref="S6.Thmtheorem1.p1.3.3.m3.1.1.1.1.1.2"><csymbol cd="ambiguous" id="S6.Thmtheorem1.p1.3.3.m3.1.1.1.1.1.2.1.cmml" xref="S6.Thmtheorem1.p1.3.3.m3.1.1.1.1.1.2">superscript</csymbol><ci id="S6.Thmtheorem1.p1.3.3.m3.1.1.1.1.1.2.2.cmml" xref="S6.Thmtheorem1.p1.3.3.m3.1.1.1.1.1.2.2">𝜀</ci><apply id="S6.Thmtheorem1.p1.3.3.m3.1.1.1.1.1.2.3.cmml" xref="S6.Thmtheorem1.p1.3.3.m3.1.1.1.1.1.2.3"><minus id="S6.Thmtheorem1.p1.3.3.m3.1.1.1.1.1.2.3.1.cmml" xref="S6.Thmtheorem1.p1.3.3.m3.1.1.1.1.1.2.3"></minus><cn id="S6.Thmtheorem1.p1.3.3.m3.1.1.1.1.1.2.3.2.cmml" type="integer" xref="S6.Thmtheorem1.p1.3.3.m3.1.1.1.1.1.2.3.2">2</cn></apply></apply><apply id="S6.Thmtheorem1.p1.3.3.m3.1.1.1.1.1.3.cmml" xref="S6.Thmtheorem1.p1.3.3.m3.1.1.1.1.1.3"><apply id="S6.Thmtheorem1.p1.3.3.m3.1.1.1.1.1.3.1.cmml" xref="S6.Thmtheorem1.p1.3.3.m3.1.1.1.1.1.3.1"><csymbol cd="ambiguous" id="S6.Thmtheorem1.p1.3.3.m3.1.1.1.1.1.3.1.1.cmml" xref="S6.Thmtheorem1.p1.3.3.m3.1.1.1.1.1.3.1">superscript</csymbol><log id="S6.Thmtheorem1.p1.3.3.m3.1.1.1.1.1.3.1.2.cmml" xref="S6.Thmtheorem1.p1.3.3.m3.1.1.1.1.1.3.1.2"></log><cn id="S6.Thmtheorem1.p1.3.3.m3.1.1.1.1.1.3.1.3.cmml" type="integer" xref="S6.Thmtheorem1.p1.3.3.m3.1.1.1.1.1.3.1.3">2</cn></apply><ci id="S6.Thmtheorem1.p1.3.3.m3.1.1.1.1.1.3.2.cmml" xref="S6.Thmtheorem1.p1.3.3.m3.1.1.1.1.1.3.2">𝑛</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.Thmtheorem1.p1.3.3.m3.1c">O(\varepsilon^{-2}\log^{2}n)</annotation><annotation encoding="application/x-llamapun" id="S6.Thmtheorem1.p1.3.3.m3.1d">italic_O ( italic_ε start_POSTSUPERSCRIPT - 2 end_POSTSUPERSCRIPT roman_log start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_n )</annotation></semantics></math>.</span></p> </div> <div class="ltx_para" id="S6.Thmtheorem1.p2"> <p class="ltx_p" id="S6.Thmtheorem1.p2.13"><span class="ltx_text ltx_font_italic" id="S6.Thmtheorem1.p2.13.13">Specifically, we say that a directed edge <math alttext="\overline{uv}" class="ltx_Math" display="inline" id="S6.Thmtheorem1.p2.1.1.m1.1"><semantics id="S6.Thmtheorem1.p2.1.1.m1.1a"><mover accent="true" id="S6.Thmtheorem1.p2.1.1.m1.1.1" xref="S6.Thmtheorem1.p2.1.1.m1.1.1.cmml"><mrow id="S6.Thmtheorem1.p2.1.1.m1.1.1.2" xref="S6.Thmtheorem1.p2.1.1.m1.1.1.2.cmml"><mi id="S6.Thmtheorem1.p2.1.1.m1.1.1.2.2" xref="S6.Thmtheorem1.p2.1.1.m1.1.1.2.2.cmml">u</mi><mo id="S6.Thmtheorem1.p2.1.1.m1.1.1.2.1" xref="S6.Thmtheorem1.p2.1.1.m1.1.1.2.1.cmml">⁢</mo><mi id="S6.Thmtheorem1.p2.1.1.m1.1.1.2.3" xref="S6.Thmtheorem1.p2.1.1.m1.1.1.2.3.cmml">v</mi></mrow><mo id="S6.Thmtheorem1.p2.1.1.m1.1.1.1" xref="S6.Thmtheorem1.p2.1.1.m1.1.1.1.cmml">¯</mo></mover><annotation-xml encoding="MathML-Content" id="S6.Thmtheorem1.p2.1.1.m1.1b"><apply id="S6.Thmtheorem1.p2.1.1.m1.1.1.cmml" xref="S6.Thmtheorem1.p2.1.1.m1.1.1"><ci id="S6.Thmtheorem1.p2.1.1.m1.1.1.1.cmml" xref="S6.Thmtheorem1.p2.1.1.m1.1.1.1">¯</ci><apply id="S6.Thmtheorem1.p2.1.1.m1.1.1.2.cmml" xref="S6.Thmtheorem1.p2.1.1.m1.1.1.2"><times id="S6.Thmtheorem1.p2.1.1.m1.1.1.2.1.cmml" xref="S6.Thmtheorem1.p2.1.1.m1.1.1.2.1"></times><ci id="S6.Thmtheorem1.p2.1.1.m1.1.1.2.2.cmml" xref="S6.Thmtheorem1.p2.1.1.m1.1.1.2.2">𝑢</ci><ci id="S6.Thmtheorem1.p2.1.1.m1.1.1.2.3.cmml" xref="S6.Thmtheorem1.p2.1.1.m1.1.1.2.3">𝑣</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.Thmtheorem1.p2.1.1.m1.1c">\overline{uv}</annotation><annotation encoding="application/x-llamapun" id="S6.Thmtheorem1.p2.1.1.m1.1d">over¯ start_ARG italic_u italic_v end_ARG</annotation></semantics></math> is bad whenever <math alttext="\textsl{g}(u\!\to\!v)>0" class="ltx_Math" display="inline" id="S6.Thmtheorem1.p2.2.2.m2.1"><semantics id="S6.Thmtheorem1.p2.2.2.m2.1a"><mrow id="S6.Thmtheorem1.p2.2.2.m2.1.1" xref="S6.Thmtheorem1.p2.2.2.m2.1.1.cmml"><mrow id="S6.Thmtheorem1.p2.2.2.m2.1.1.1" xref="S6.Thmtheorem1.p2.2.2.m2.1.1.1.cmml"><mtext class="ltx_mathvariant_italic" id="S6.Thmtheorem1.p2.2.2.m2.1.1.1.3" xref="S6.Thmtheorem1.p2.2.2.m2.1.1.1.3a.cmml">g</mtext><mo id="S6.Thmtheorem1.p2.2.2.m2.1.1.1.2" xref="S6.Thmtheorem1.p2.2.2.m2.1.1.1.2.cmml">⁢</mo><mrow id="S6.Thmtheorem1.p2.2.2.m2.1.1.1.1.1" xref="S6.Thmtheorem1.p2.2.2.m2.1.1.1.1.1.1.cmml"><mo id="S6.Thmtheorem1.p2.2.2.m2.1.1.1.1.1.2" stretchy="false" xref="S6.Thmtheorem1.p2.2.2.m2.1.1.1.1.1.1.cmml">(</mo><mrow id="S6.Thmtheorem1.p2.2.2.m2.1.1.1.1.1.1" xref="S6.Thmtheorem1.p2.2.2.m2.1.1.1.1.1.1.cmml"><mi id="S6.Thmtheorem1.p2.2.2.m2.1.1.1.1.1.1.2" xref="S6.Thmtheorem1.p2.2.2.m2.1.1.1.1.1.1.2.cmml">u</mi><mo id="S6.Thmtheorem1.p2.2.2.m2.1.1.1.1.1.1.1" lspace="0.108em" rspace="0.108em" stretchy="false" xref="S6.Thmtheorem1.p2.2.2.m2.1.1.1.1.1.1.1.cmml">→</mo><mi id="S6.Thmtheorem1.p2.2.2.m2.1.1.1.1.1.1.3" xref="S6.Thmtheorem1.p2.2.2.m2.1.1.1.1.1.1.3.cmml">v</mi></mrow><mo id="S6.Thmtheorem1.p2.2.2.m2.1.1.1.1.1.3" stretchy="false" xref="S6.Thmtheorem1.p2.2.2.m2.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S6.Thmtheorem1.p2.2.2.m2.1.1.2" xref="S6.Thmtheorem1.p2.2.2.m2.1.1.2.cmml">></mo><mn id="S6.Thmtheorem1.p2.2.2.m2.1.1.3" xref="S6.Thmtheorem1.p2.2.2.m2.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S6.Thmtheorem1.p2.2.2.m2.1b"><apply id="S6.Thmtheorem1.p2.2.2.m2.1.1.cmml" xref="S6.Thmtheorem1.p2.2.2.m2.1.1"><gt id="S6.Thmtheorem1.p2.2.2.m2.1.1.2.cmml" xref="S6.Thmtheorem1.p2.2.2.m2.1.1.2"></gt><apply id="S6.Thmtheorem1.p2.2.2.m2.1.1.1.cmml" xref="S6.Thmtheorem1.p2.2.2.m2.1.1.1"><times id="S6.Thmtheorem1.p2.2.2.m2.1.1.1.2.cmml" xref="S6.Thmtheorem1.p2.2.2.m2.1.1.1.2"></times><ci id="S6.Thmtheorem1.p2.2.2.m2.1.1.1.3a.cmml" xref="S6.Thmtheorem1.p2.2.2.m2.1.1.1.3"><mtext class="ltx_mathvariant_italic" id="S6.Thmtheorem1.p2.2.2.m2.1.1.1.3.cmml" xref="S6.Thmtheorem1.p2.2.2.m2.1.1.1.3">g</mtext></ci><apply id="S6.Thmtheorem1.p2.2.2.m2.1.1.1.1.1.1.cmml" xref="S6.Thmtheorem1.p2.2.2.m2.1.1.1.1.1"><ci id="S6.Thmtheorem1.p2.2.2.m2.1.1.1.1.1.1.1.cmml" xref="S6.Thmtheorem1.p2.2.2.m2.1.1.1.1.1.1.1">→</ci><ci id="S6.Thmtheorem1.p2.2.2.m2.1.1.1.1.1.1.2.cmml" xref="S6.Thmtheorem1.p2.2.2.m2.1.1.1.1.1.1.2">𝑢</ci><ci id="S6.Thmtheorem1.p2.2.2.m2.1.1.1.1.1.1.3.cmml" xref="S6.Thmtheorem1.p2.2.2.m2.1.1.1.1.1.1.3">𝑣</ci></apply></apply><cn id="S6.Thmtheorem1.p2.2.2.m2.1.1.3.cmml" type="integer" xref="S6.Thmtheorem1.p2.2.2.m2.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.Thmtheorem1.p2.2.2.m2.1c">\textsl{g}(u\!\to\!v)>0</annotation><annotation encoding="application/x-llamapun" id="S6.Thmtheorem1.p2.2.2.m2.1d">g ( italic_u → italic_v ) > 0</annotation></semantics></math> and <math alttext="\textsl{g}(u)>(1+\eta)\textsl{g}(v)" class="ltx_Math" display="inline" id="S6.Thmtheorem1.p2.3.3.m3.3"><semantics id="S6.Thmtheorem1.p2.3.3.m3.3a"><mrow id="S6.Thmtheorem1.p2.3.3.m3.3.3" xref="S6.Thmtheorem1.p2.3.3.m3.3.3.cmml"><mrow id="S6.Thmtheorem1.p2.3.3.m3.3.3.3" xref="S6.Thmtheorem1.p2.3.3.m3.3.3.3.cmml"><mtext class="ltx_mathvariant_italic" id="S6.Thmtheorem1.p2.3.3.m3.3.3.3.2" xref="S6.Thmtheorem1.p2.3.3.m3.3.3.3.2a.cmml">g</mtext><mo id="S6.Thmtheorem1.p2.3.3.m3.3.3.3.1" xref="S6.Thmtheorem1.p2.3.3.m3.3.3.3.1.cmml">⁢</mo><mrow id="S6.Thmtheorem1.p2.3.3.m3.3.3.3.3.2" xref="S6.Thmtheorem1.p2.3.3.m3.3.3.3.cmml"><mo id="S6.Thmtheorem1.p2.3.3.m3.3.3.3.3.2.1" stretchy="false" xref="S6.Thmtheorem1.p2.3.3.m3.3.3.3.cmml">(</mo><mi id="S6.Thmtheorem1.p2.3.3.m3.1.1" xref="S6.Thmtheorem1.p2.3.3.m3.1.1.cmml">u</mi><mo id="S6.Thmtheorem1.p2.3.3.m3.3.3.3.3.2.2" stretchy="false" xref="S6.Thmtheorem1.p2.3.3.m3.3.3.3.cmml">)</mo></mrow></mrow><mo id="S6.Thmtheorem1.p2.3.3.m3.3.3.2" xref="S6.Thmtheorem1.p2.3.3.m3.3.3.2.cmml">></mo><mrow id="S6.Thmtheorem1.p2.3.3.m3.3.3.1" xref="S6.Thmtheorem1.p2.3.3.m3.3.3.1.cmml"><mrow id="S6.Thmtheorem1.p2.3.3.m3.3.3.1.1.1" xref="S6.Thmtheorem1.p2.3.3.m3.3.3.1.1.1.1.cmml"><mo id="S6.Thmtheorem1.p2.3.3.m3.3.3.1.1.1.2" stretchy="false" xref="S6.Thmtheorem1.p2.3.3.m3.3.3.1.1.1.1.cmml">(</mo><mrow id="S6.Thmtheorem1.p2.3.3.m3.3.3.1.1.1.1" xref="S6.Thmtheorem1.p2.3.3.m3.3.3.1.1.1.1.cmml"><mn id="S6.Thmtheorem1.p2.3.3.m3.3.3.1.1.1.1.2" xref="S6.Thmtheorem1.p2.3.3.m3.3.3.1.1.1.1.2.cmml">1</mn><mo id="S6.Thmtheorem1.p2.3.3.m3.3.3.1.1.1.1.1" xref="S6.Thmtheorem1.p2.3.3.m3.3.3.1.1.1.1.1.cmml">+</mo><mi id="S6.Thmtheorem1.p2.3.3.m3.3.3.1.1.1.1.3" xref="S6.Thmtheorem1.p2.3.3.m3.3.3.1.1.1.1.3.cmml">η</mi></mrow><mo id="S6.Thmtheorem1.p2.3.3.m3.3.3.1.1.1.3" stretchy="false" xref="S6.Thmtheorem1.p2.3.3.m3.3.3.1.1.1.1.cmml">)</mo></mrow><mo id="S6.Thmtheorem1.p2.3.3.m3.3.3.1.2" xref="S6.Thmtheorem1.p2.3.3.m3.3.3.1.2.cmml">⁢</mo><mtext class="ltx_mathvariant_italic" id="S6.Thmtheorem1.p2.3.3.m3.3.3.1.3" xref="S6.Thmtheorem1.p2.3.3.m3.3.3.1.3a.cmml">g</mtext><mo id="S6.Thmtheorem1.p2.3.3.m3.3.3.1.2a" xref="S6.Thmtheorem1.p2.3.3.m3.3.3.1.2.cmml">⁢</mo><mrow id="S6.Thmtheorem1.p2.3.3.m3.3.3.1.4.2" xref="S6.Thmtheorem1.p2.3.3.m3.3.3.1.cmml"><mo id="S6.Thmtheorem1.p2.3.3.m3.3.3.1.4.2.1" stretchy="false" xref="S6.Thmtheorem1.p2.3.3.m3.3.3.1.cmml">(</mo><mi id="S6.Thmtheorem1.p2.3.3.m3.2.2" xref="S6.Thmtheorem1.p2.3.3.m3.2.2.cmml">v</mi><mo id="S6.Thmtheorem1.p2.3.3.m3.3.3.1.4.2.2" stretchy="false" xref="S6.Thmtheorem1.p2.3.3.m3.3.3.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.Thmtheorem1.p2.3.3.m3.3b"><apply id="S6.Thmtheorem1.p2.3.3.m3.3.3.cmml" xref="S6.Thmtheorem1.p2.3.3.m3.3.3"><gt id="S6.Thmtheorem1.p2.3.3.m3.3.3.2.cmml" xref="S6.Thmtheorem1.p2.3.3.m3.3.3.2"></gt><apply id="S6.Thmtheorem1.p2.3.3.m3.3.3.3.cmml" xref="S6.Thmtheorem1.p2.3.3.m3.3.3.3"><times id="S6.Thmtheorem1.p2.3.3.m3.3.3.3.1.cmml" xref="S6.Thmtheorem1.p2.3.3.m3.3.3.3.1"></times><ci id="S6.Thmtheorem1.p2.3.3.m3.3.3.3.2a.cmml" xref="S6.Thmtheorem1.p2.3.3.m3.3.3.3.2"><mtext class="ltx_mathvariant_italic" id="S6.Thmtheorem1.p2.3.3.m3.3.3.3.2.cmml" xref="S6.Thmtheorem1.p2.3.3.m3.3.3.3.2">g</mtext></ci><ci id="S6.Thmtheorem1.p2.3.3.m3.1.1.cmml" xref="S6.Thmtheorem1.p2.3.3.m3.1.1">𝑢</ci></apply><apply id="S6.Thmtheorem1.p2.3.3.m3.3.3.1.cmml" xref="S6.Thmtheorem1.p2.3.3.m3.3.3.1"><times id="S6.Thmtheorem1.p2.3.3.m3.3.3.1.2.cmml" xref="S6.Thmtheorem1.p2.3.3.m3.3.3.1.2"></times><apply id="S6.Thmtheorem1.p2.3.3.m3.3.3.1.1.1.1.cmml" xref="S6.Thmtheorem1.p2.3.3.m3.3.3.1.1.1"><plus id="S6.Thmtheorem1.p2.3.3.m3.3.3.1.1.1.1.1.cmml" xref="S6.Thmtheorem1.p2.3.3.m3.3.3.1.1.1.1.1"></plus><cn id="S6.Thmtheorem1.p2.3.3.m3.3.3.1.1.1.1.2.cmml" type="integer" xref="S6.Thmtheorem1.p2.3.3.m3.3.3.1.1.1.1.2">1</cn><ci id="S6.Thmtheorem1.p2.3.3.m3.3.3.1.1.1.1.3.cmml" xref="S6.Thmtheorem1.p2.3.3.m3.3.3.1.1.1.1.3">𝜂</ci></apply><ci id="S6.Thmtheorem1.p2.3.3.m3.3.3.1.3a.cmml" xref="S6.Thmtheorem1.p2.3.3.m3.3.3.1.3"><mtext class="ltx_mathvariant_italic" id="S6.Thmtheorem1.p2.3.3.m3.3.3.1.3.cmml" xref="S6.Thmtheorem1.p2.3.3.m3.3.3.1.3">g</mtext></ci><ci id="S6.Thmtheorem1.p2.3.3.m3.2.2.cmml" xref="S6.Thmtheorem1.p2.3.3.m3.2.2">𝑣</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.Thmtheorem1.p2.3.3.m3.3c">\textsl{g}(u)>(1+\eta)\textsl{g}(v)</annotation><annotation encoding="application/x-llamapun" id="S6.Thmtheorem1.p2.3.3.m3.3d">g ( italic_u ) > ( 1 + italic_η ) g ( italic_v )</annotation></semantics></math>. In an <math alttext="\eta" class="ltx_Math" display="inline" id="S6.Thmtheorem1.p2.4.4.m4.1"><semantics id="S6.Thmtheorem1.p2.4.4.m4.1a"><mi id="S6.Thmtheorem1.p2.4.4.m4.1.1" xref="S6.Thmtheorem1.p2.4.4.m4.1.1.cmml">η</mi><annotation-xml encoding="MathML-Content" id="S6.Thmtheorem1.p2.4.4.m4.1b"><ci id="S6.Thmtheorem1.p2.4.4.m4.1.1.cmml" xref="S6.Thmtheorem1.p2.4.4.m4.1.1">𝜂</ci></annotation-xml><annotation encoding="application/x-tex" id="S6.Thmtheorem1.p2.4.4.m4.1c">\eta</annotation><annotation encoding="application/x-llamapun" id="S6.Thmtheorem1.p2.4.4.m4.1d">italic_η</annotation></semantics></math>-fair orientation no edge is bad. Whenever we delete an edge <math alttext="\overline{x_{1}x_{0}}" class="ltx_Math" display="inline" id="S6.Thmtheorem1.p2.5.5.m5.1"><semantics id="S6.Thmtheorem1.p2.5.5.m5.1a"><mover accent="true" id="S6.Thmtheorem1.p2.5.5.m5.1.1" xref="S6.Thmtheorem1.p2.5.5.m5.1.1.cmml"><mrow id="S6.Thmtheorem1.p2.5.5.m5.1.1.2" xref="S6.Thmtheorem1.p2.5.5.m5.1.1.2.cmml"><msub id="S6.Thmtheorem1.p2.5.5.m5.1.1.2.2" xref="S6.Thmtheorem1.p2.5.5.m5.1.1.2.2.cmml"><mi id="S6.Thmtheorem1.p2.5.5.m5.1.1.2.2.2" xref="S6.Thmtheorem1.p2.5.5.m5.1.1.2.2.2.cmml">x</mi><mn id="S6.Thmtheorem1.p2.5.5.m5.1.1.2.2.3" xref="S6.Thmtheorem1.p2.5.5.m5.1.1.2.2.3.cmml">1</mn></msub><mo id="S6.Thmtheorem1.p2.5.5.m5.1.1.2.1" xref="S6.Thmtheorem1.p2.5.5.m5.1.1.2.1.cmml">⁢</mo><msub id="S6.Thmtheorem1.p2.5.5.m5.1.1.2.3" xref="S6.Thmtheorem1.p2.5.5.m5.1.1.2.3.cmml"><mi id="S6.Thmtheorem1.p2.5.5.m5.1.1.2.3.2" xref="S6.Thmtheorem1.p2.5.5.m5.1.1.2.3.2.cmml">x</mi><mn id="S6.Thmtheorem1.p2.5.5.m5.1.1.2.3.3" xref="S6.Thmtheorem1.p2.5.5.m5.1.1.2.3.3.cmml">0</mn></msub></mrow><mo id="S6.Thmtheorem1.p2.5.5.m5.1.1.1" xref="S6.Thmtheorem1.p2.5.5.m5.1.1.1.cmml">¯</mo></mover><annotation-xml encoding="MathML-Content" id="S6.Thmtheorem1.p2.5.5.m5.1b"><apply id="S6.Thmtheorem1.p2.5.5.m5.1.1.cmml" xref="S6.Thmtheorem1.p2.5.5.m5.1.1"><ci id="S6.Thmtheorem1.p2.5.5.m5.1.1.1.cmml" xref="S6.Thmtheorem1.p2.5.5.m5.1.1.1">¯</ci><apply id="S6.Thmtheorem1.p2.5.5.m5.1.1.2.cmml" xref="S6.Thmtheorem1.p2.5.5.m5.1.1.2"><times id="S6.Thmtheorem1.p2.5.5.m5.1.1.2.1.cmml" xref="S6.Thmtheorem1.p2.5.5.m5.1.1.2.1"></times><apply id="S6.Thmtheorem1.p2.5.5.m5.1.1.2.2.cmml" xref="S6.Thmtheorem1.p2.5.5.m5.1.1.2.2"><csymbol cd="ambiguous" id="S6.Thmtheorem1.p2.5.5.m5.1.1.2.2.1.cmml" xref="S6.Thmtheorem1.p2.5.5.m5.1.1.2.2">subscript</csymbol><ci id="S6.Thmtheorem1.p2.5.5.m5.1.1.2.2.2.cmml" xref="S6.Thmtheorem1.p2.5.5.m5.1.1.2.2.2">𝑥</ci><cn id="S6.Thmtheorem1.p2.5.5.m5.1.1.2.2.3.cmml" type="integer" xref="S6.Thmtheorem1.p2.5.5.m5.1.1.2.2.3">1</cn></apply><apply id="S6.Thmtheorem1.p2.5.5.m5.1.1.2.3.cmml" xref="S6.Thmtheorem1.p2.5.5.m5.1.1.2.3"><csymbol cd="ambiguous" id="S6.Thmtheorem1.p2.5.5.m5.1.1.2.3.1.cmml" xref="S6.Thmtheorem1.p2.5.5.m5.1.1.2.3">subscript</csymbol><ci id="S6.Thmtheorem1.p2.5.5.m5.1.1.2.3.2.cmml" xref="S6.Thmtheorem1.p2.5.5.m5.1.1.2.3.2">𝑥</ci><cn id="S6.Thmtheorem1.p2.5.5.m5.1.1.2.3.3.cmml" type="integer" xref="S6.Thmtheorem1.p2.5.5.m5.1.1.2.3.3">0</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.Thmtheorem1.p2.5.5.m5.1c">\overline{x_{1}x_{0}}</annotation><annotation encoding="application/x-llamapun" id="S6.Thmtheorem1.p2.5.5.m5.1d">over¯ start_ARG italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_x start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_ARG</annotation></semantics></math>, the out-degree <math alttext="\textsl{g}(x_{1})" class="ltx_Math" display="inline" id="S6.Thmtheorem1.p2.6.6.m6.1"><semantics id="S6.Thmtheorem1.p2.6.6.m6.1a"><mrow id="S6.Thmtheorem1.p2.6.6.m6.1.1" xref="S6.Thmtheorem1.p2.6.6.m6.1.1.cmml"><mtext class="ltx_mathvariant_italic" id="S6.Thmtheorem1.p2.6.6.m6.1.1.3" xref="S6.Thmtheorem1.p2.6.6.m6.1.1.3a.cmml">g</mtext><mo id="S6.Thmtheorem1.p2.6.6.m6.1.1.2" xref="S6.Thmtheorem1.p2.6.6.m6.1.1.2.cmml">⁢</mo><mrow id="S6.Thmtheorem1.p2.6.6.m6.1.1.1.1" xref="S6.Thmtheorem1.p2.6.6.m6.1.1.1.1.1.cmml"><mo id="S6.Thmtheorem1.p2.6.6.m6.1.1.1.1.2" stretchy="false" xref="S6.Thmtheorem1.p2.6.6.m6.1.1.1.1.1.cmml">(</mo><msub id="S6.Thmtheorem1.p2.6.6.m6.1.1.1.1.1" xref="S6.Thmtheorem1.p2.6.6.m6.1.1.1.1.1.cmml"><mi id="S6.Thmtheorem1.p2.6.6.m6.1.1.1.1.1.2" xref="S6.Thmtheorem1.p2.6.6.m6.1.1.1.1.1.2.cmml">x</mi><mn id="S6.Thmtheorem1.p2.6.6.m6.1.1.1.1.1.3" xref="S6.Thmtheorem1.p2.6.6.m6.1.1.1.1.1.3.cmml">1</mn></msub><mo id="S6.Thmtheorem1.p2.6.6.m6.1.1.1.1.3" stretchy="false" xref="S6.Thmtheorem1.p2.6.6.m6.1.1.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.Thmtheorem1.p2.6.6.m6.1b"><apply id="S6.Thmtheorem1.p2.6.6.m6.1.1.cmml" xref="S6.Thmtheorem1.p2.6.6.m6.1.1"><times id="S6.Thmtheorem1.p2.6.6.m6.1.1.2.cmml" xref="S6.Thmtheorem1.p2.6.6.m6.1.1.2"></times><ci id="S6.Thmtheorem1.p2.6.6.m6.1.1.3a.cmml" xref="S6.Thmtheorem1.p2.6.6.m6.1.1.3"><mtext class="ltx_mathvariant_italic" id="S6.Thmtheorem1.p2.6.6.m6.1.1.3.cmml" xref="S6.Thmtheorem1.p2.6.6.m6.1.1.3">g</mtext></ci><apply id="S6.Thmtheorem1.p2.6.6.m6.1.1.1.1.1.cmml" xref="S6.Thmtheorem1.p2.6.6.m6.1.1.1.1"><csymbol cd="ambiguous" id="S6.Thmtheorem1.p2.6.6.m6.1.1.1.1.1.1.cmml" xref="S6.Thmtheorem1.p2.6.6.m6.1.1.1.1">subscript</csymbol><ci id="S6.Thmtheorem1.p2.6.6.m6.1.1.1.1.1.2.cmml" xref="S6.Thmtheorem1.p2.6.6.m6.1.1.1.1.1.2">𝑥</ci><cn id="S6.Thmtheorem1.p2.6.6.m6.1.1.1.1.1.3.cmml" type="integer" xref="S6.Thmtheorem1.p2.6.6.m6.1.1.1.1.1.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.Thmtheorem1.p2.6.6.m6.1c">\textsl{g}(x_{1})</annotation><annotation encoding="application/x-llamapun" id="S6.Thmtheorem1.p2.6.6.m6.1d">g ( italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT )</annotation></semantics></math> decreases by <math alttext="\textsl{g}(x_{1}\!\to\!x_{0})" class="ltx_Math" display="inline" id="S6.Thmtheorem1.p2.7.7.m7.1"><semantics id="S6.Thmtheorem1.p2.7.7.m7.1a"><mrow id="S6.Thmtheorem1.p2.7.7.m7.1.1" xref="S6.Thmtheorem1.p2.7.7.m7.1.1.cmml"><mtext class="ltx_mathvariant_italic" id="S6.Thmtheorem1.p2.7.7.m7.1.1.3" xref="S6.Thmtheorem1.p2.7.7.m7.1.1.3a.cmml">g</mtext><mo id="S6.Thmtheorem1.p2.7.7.m7.1.1.2" xref="S6.Thmtheorem1.p2.7.7.m7.1.1.2.cmml">⁢</mo><mrow id="S6.Thmtheorem1.p2.7.7.m7.1.1.1.1" xref="S6.Thmtheorem1.p2.7.7.m7.1.1.1.1.1.cmml"><mo id="S6.Thmtheorem1.p2.7.7.m7.1.1.1.1.2" stretchy="false" xref="S6.Thmtheorem1.p2.7.7.m7.1.1.1.1.1.cmml">(</mo><mrow id="S6.Thmtheorem1.p2.7.7.m7.1.1.1.1.1" xref="S6.Thmtheorem1.p2.7.7.m7.1.1.1.1.1.cmml"><msub id="S6.Thmtheorem1.p2.7.7.m7.1.1.1.1.1.2" xref="S6.Thmtheorem1.p2.7.7.m7.1.1.1.1.1.2.cmml"><mi id="S6.Thmtheorem1.p2.7.7.m7.1.1.1.1.1.2.2" xref="S6.Thmtheorem1.p2.7.7.m7.1.1.1.1.1.2.2.cmml">x</mi><mn id="S6.Thmtheorem1.p2.7.7.m7.1.1.1.1.1.2.3" xref="S6.Thmtheorem1.p2.7.7.m7.1.1.1.1.1.2.3.cmml">1</mn></msub><mo id="S6.Thmtheorem1.p2.7.7.m7.1.1.1.1.1.1" rspace="0.108em" stretchy="false" xref="S6.Thmtheorem1.p2.7.7.m7.1.1.1.1.1.1.cmml">→</mo><msub id="S6.Thmtheorem1.p2.7.7.m7.1.1.1.1.1.3" xref="S6.Thmtheorem1.p2.7.7.m7.1.1.1.1.1.3.cmml"><mi id="S6.Thmtheorem1.p2.7.7.m7.1.1.1.1.1.3.2" xref="S6.Thmtheorem1.p2.7.7.m7.1.1.1.1.1.3.2.cmml">x</mi><mn id="S6.Thmtheorem1.p2.7.7.m7.1.1.1.1.1.3.3" xref="S6.Thmtheorem1.p2.7.7.m7.1.1.1.1.1.3.3.cmml">0</mn></msub></mrow><mo id="S6.Thmtheorem1.p2.7.7.m7.1.1.1.1.3" stretchy="false" xref="S6.Thmtheorem1.p2.7.7.m7.1.1.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.Thmtheorem1.p2.7.7.m7.1b"><apply id="S6.Thmtheorem1.p2.7.7.m7.1.1.cmml" xref="S6.Thmtheorem1.p2.7.7.m7.1.1"><times id="S6.Thmtheorem1.p2.7.7.m7.1.1.2.cmml" xref="S6.Thmtheorem1.p2.7.7.m7.1.1.2"></times><ci id="S6.Thmtheorem1.p2.7.7.m7.1.1.3a.cmml" xref="S6.Thmtheorem1.p2.7.7.m7.1.1.3"><mtext class="ltx_mathvariant_italic" id="S6.Thmtheorem1.p2.7.7.m7.1.1.3.cmml" xref="S6.Thmtheorem1.p2.7.7.m7.1.1.3">g</mtext></ci><apply id="S6.Thmtheorem1.p2.7.7.m7.1.1.1.1.1.cmml" xref="S6.Thmtheorem1.p2.7.7.m7.1.1.1.1"><ci id="S6.Thmtheorem1.p2.7.7.m7.1.1.1.1.1.1.cmml" xref="S6.Thmtheorem1.p2.7.7.m7.1.1.1.1.1.1">→</ci><apply id="S6.Thmtheorem1.p2.7.7.m7.1.1.1.1.1.2.cmml" xref="S6.Thmtheorem1.p2.7.7.m7.1.1.1.1.1.2"><csymbol cd="ambiguous" id="S6.Thmtheorem1.p2.7.7.m7.1.1.1.1.1.2.1.cmml" xref="S6.Thmtheorem1.p2.7.7.m7.1.1.1.1.1.2">subscript</csymbol><ci id="S6.Thmtheorem1.p2.7.7.m7.1.1.1.1.1.2.2.cmml" xref="S6.Thmtheorem1.p2.7.7.m7.1.1.1.1.1.2.2">𝑥</ci><cn id="S6.Thmtheorem1.p2.7.7.m7.1.1.1.1.1.2.3.cmml" type="integer" xref="S6.Thmtheorem1.p2.7.7.m7.1.1.1.1.1.2.3">1</cn></apply><apply id="S6.Thmtheorem1.p2.7.7.m7.1.1.1.1.1.3.cmml" xref="S6.Thmtheorem1.p2.7.7.m7.1.1.1.1.1.3"><csymbol cd="ambiguous" id="S6.Thmtheorem1.p2.7.7.m7.1.1.1.1.1.3.1.cmml" xref="S6.Thmtheorem1.p2.7.7.m7.1.1.1.1.1.3">subscript</csymbol><ci id="S6.Thmtheorem1.p2.7.7.m7.1.1.1.1.1.3.2.cmml" xref="S6.Thmtheorem1.p2.7.7.m7.1.1.1.1.1.3.2">𝑥</ci><cn id="S6.Thmtheorem1.p2.7.7.m7.1.1.1.1.1.3.3.cmml" type="integer" xref="S6.Thmtheorem1.p2.7.7.m7.1.1.1.1.1.3.3">0</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.Thmtheorem1.p2.7.7.m7.1c">\textsl{g}(x_{1}\!\to\!x_{0})</annotation><annotation encoding="application/x-llamapun" id="S6.Thmtheorem1.p2.7.7.m7.1d">g ( italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT → italic_x start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT )</annotation></semantics></math>. For vertices <math alttext="x_{2}" class="ltx_Math" display="inline" id="S6.Thmtheorem1.p2.8.8.m8.1"><semantics id="S6.Thmtheorem1.p2.8.8.m8.1a"><msub id="S6.Thmtheorem1.p2.8.8.m8.1.1" xref="S6.Thmtheorem1.p2.8.8.m8.1.1.cmml"><mi id="S6.Thmtheorem1.p2.8.8.m8.1.1.2" xref="S6.Thmtheorem1.p2.8.8.m8.1.1.2.cmml">x</mi><mn id="S6.Thmtheorem1.p2.8.8.m8.1.1.3" xref="S6.Thmtheorem1.p2.8.8.m8.1.1.3.cmml">2</mn></msub><annotation-xml encoding="MathML-Content" id="S6.Thmtheorem1.p2.8.8.m8.1b"><apply id="S6.Thmtheorem1.p2.8.8.m8.1.1.cmml" xref="S6.Thmtheorem1.p2.8.8.m8.1.1"><csymbol cd="ambiguous" id="S6.Thmtheorem1.p2.8.8.m8.1.1.1.cmml" xref="S6.Thmtheorem1.p2.8.8.m8.1.1">subscript</csymbol><ci id="S6.Thmtheorem1.p2.8.8.m8.1.1.2.cmml" xref="S6.Thmtheorem1.p2.8.8.m8.1.1.2">𝑥</ci><cn id="S6.Thmtheorem1.p2.8.8.m8.1.1.3.cmml" type="integer" xref="S6.Thmtheorem1.p2.8.8.m8.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.Thmtheorem1.p2.8.8.m8.1c">x_{2}</annotation><annotation encoding="application/x-llamapun" id="S6.Thmtheorem1.p2.8.8.m8.1d">italic_x start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT</annotation></semantics></math> with <math alttext="\textsl{g}(x_{2}\!\to\!x_{1})>0" class="ltx_Math" display="inline" id="S6.Thmtheorem1.p2.9.9.m9.1"><semantics id="S6.Thmtheorem1.p2.9.9.m9.1a"><mrow id="S6.Thmtheorem1.p2.9.9.m9.1.1" xref="S6.Thmtheorem1.p2.9.9.m9.1.1.cmml"><mrow id="S6.Thmtheorem1.p2.9.9.m9.1.1.1" xref="S6.Thmtheorem1.p2.9.9.m9.1.1.1.cmml"><mtext class="ltx_mathvariant_italic" id="S6.Thmtheorem1.p2.9.9.m9.1.1.1.3" xref="S6.Thmtheorem1.p2.9.9.m9.1.1.1.3a.cmml">g</mtext><mo id="S6.Thmtheorem1.p2.9.9.m9.1.1.1.2" xref="S6.Thmtheorem1.p2.9.9.m9.1.1.1.2.cmml">⁢</mo><mrow id="S6.Thmtheorem1.p2.9.9.m9.1.1.1.1.1" xref="S6.Thmtheorem1.p2.9.9.m9.1.1.1.1.1.1.cmml"><mo id="S6.Thmtheorem1.p2.9.9.m9.1.1.1.1.1.2" stretchy="false" xref="S6.Thmtheorem1.p2.9.9.m9.1.1.1.1.1.1.cmml">(</mo><mrow id="S6.Thmtheorem1.p2.9.9.m9.1.1.1.1.1.1" xref="S6.Thmtheorem1.p2.9.9.m9.1.1.1.1.1.1.cmml"><msub id="S6.Thmtheorem1.p2.9.9.m9.1.1.1.1.1.1.2" xref="S6.Thmtheorem1.p2.9.9.m9.1.1.1.1.1.1.2.cmml"><mi id="S6.Thmtheorem1.p2.9.9.m9.1.1.1.1.1.1.2.2" xref="S6.Thmtheorem1.p2.9.9.m9.1.1.1.1.1.1.2.2.cmml">x</mi><mn id="S6.Thmtheorem1.p2.9.9.m9.1.1.1.1.1.1.2.3" xref="S6.Thmtheorem1.p2.9.9.m9.1.1.1.1.1.1.2.3.cmml">2</mn></msub><mo id="S6.Thmtheorem1.p2.9.9.m9.1.1.1.1.1.1.1" rspace="0.108em" stretchy="false" xref="S6.Thmtheorem1.p2.9.9.m9.1.1.1.1.1.1.1.cmml">→</mo><msub id="S6.Thmtheorem1.p2.9.9.m9.1.1.1.1.1.1.3" xref="S6.Thmtheorem1.p2.9.9.m9.1.1.1.1.1.1.3.cmml"><mi id="S6.Thmtheorem1.p2.9.9.m9.1.1.1.1.1.1.3.2" xref="S6.Thmtheorem1.p2.9.9.m9.1.1.1.1.1.1.3.2.cmml">x</mi><mn id="S6.Thmtheorem1.p2.9.9.m9.1.1.1.1.1.1.3.3" xref="S6.Thmtheorem1.p2.9.9.m9.1.1.1.1.1.1.3.3.cmml">1</mn></msub></mrow><mo id="S6.Thmtheorem1.p2.9.9.m9.1.1.1.1.1.3" stretchy="false" xref="S6.Thmtheorem1.p2.9.9.m9.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S6.Thmtheorem1.p2.9.9.m9.1.1.2" xref="S6.Thmtheorem1.p2.9.9.m9.1.1.2.cmml">></mo><mn id="S6.Thmtheorem1.p2.9.9.m9.1.1.3" xref="S6.Thmtheorem1.p2.9.9.m9.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S6.Thmtheorem1.p2.9.9.m9.1b"><apply id="S6.Thmtheorem1.p2.9.9.m9.1.1.cmml" xref="S6.Thmtheorem1.p2.9.9.m9.1.1"><gt id="S6.Thmtheorem1.p2.9.9.m9.1.1.2.cmml" xref="S6.Thmtheorem1.p2.9.9.m9.1.1.2"></gt><apply id="S6.Thmtheorem1.p2.9.9.m9.1.1.1.cmml" xref="S6.Thmtheorem1.p2.9.9.m9.1.1.1"><times id="S6.Thmtheorem1.p2.9.9.m9.1.1.1.2.cmml" xref="S6.Thmtheorem1.p2.9.9.m9.1.1.1.2"></times><ci id="S6.Thmtheorem1.p2.9.9.m9.1.1.1.3a.cmml" xref="S6.Thmtheorem1.p2.9.9.m9.1.1.1.3"><mtext class="ltx_mathvariant_italic" id="S6.Thmtheorem1.p2.9.9.m9.1.1.1.3.cmml" xref="S6.Thmtheorem1.p2.9.9.m9.1.1.1.3">g</mtext></ci><apply id="S6.Thmtheorem1.p2.9.9.m9.1.1.1.1.1.1.cmml" xref="S6.Thmtheorem1.p2.9.9.m9.1.1.1.1.1"><ci id="S6.Thmtheorem1.p2.9.9.m9.1.1.1.1.1.1.1.cmml" xref="S6.Thmtheorem1.p2.9.9.m9.1.1.1.1.1.1.1">→</ci><apply id="S6.Thmtheorem1.p2.9.9.m9.1.1.1.1.1.1.2.cmml" xref="S6.Thmtheorem1.p2.9.9.m9.1.1.1.1.1.1.2"><csymbol cd="ambiguous" id="S6.Thmtheorem1.p2.9.9.m9.1.1.1.1.1.1.2.1.cmml" xref="S6.Thmtheorem1.p2.9.9.m9.1.1.1.1.1.1.2">subscript</csymbol><ci id="S6.Thmtheorem1.p2.9.9.m9.1.1.1.1.1.1.2.2.cmml" xref="S6.Thmtheorem1.p2.9.9.m9.1.1.1.1.1.1.2.2">𝑥</ci><cn id="S6.Thmtheorem1.p2.9.9.m9.1.1.1.1.1.1.2.3.cmml" type="integer" xref="S6.Thmtheorem1.p2.9.9.m9.1.1.1.1.1.1.2.3">2</cn></apply><apply id="S6.Thmtheorem1.p2.9.9.m9.1.1.1.1.1.1.3.cmml" xref="S6.Thmtheorem1.p2.9.9.m9.1.1.1.1.1.1.3"><csymbol cd="ambiguous" id="S6.Thmtheorem1.p2.9.9.m9.1.1.1.1.1.1.3.1.cmml" xref="S6.Thmtheorem1.p2.9.9.m9.1.1.1.1.1.1.3">subscript</csymbol><ci id="S6.Thmtheorem1.p2.9.9.m9.1.1.1.1.1.1.3.2.cmml" xref="S6.Thmtheorem1.p2.9.9.m9.1.1.1.1.1.1.3.2">𝑥</ci><cn id="S6.Thmtheorem1.p2.9.9.m9.1.1.1.1.1.1.3.3.cmml" type="integer" xref="S6.Thmtheorem1.p2.9.9.m9.1.1.1.1.1.1.3.3">1</cn></apply></apply></apply><cn id="S6.Thmtheorem1.p2.9.9.m9.1.1.3.cmml" type="integer" xref="S6.Thmtheorem1.p2.9.9.m9.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.Thmtheorem1.p2.9.9.m9.1c">\textsl{g}(x_{2}\!\to\!x_{1})>0</annotation><annotation encoding="application/x-llamapun" id="S6.Thmtheorem1.p2.9.9.m9.1d">g ( italic_x start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT → italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) > 0</annotation></semantics></math>, it may now be that <math alttext="\textsl{g}(x_{2})>(1+\eta)\textsl{g}(x_{1})" class="ltx_Math" display="inline" id="S6.Thmtheorem1.p2.10.10.m10.3"><semantics id="S6.Thmtheorem1.p2.10.10.m10.3a"><mrow id="S6.Thmtheorem1.p2.10.10.m10.3.3" xref="S6.Thmtheorem1.p2.10.10.m10.3.3.cmml"><mrow id="S6.Thmtheorem1.p2.10.10.m10.1.1.1" xref="S6.Thmtheorem1.p2.10.10.m10.1.1.1.cmml"><mtext class="ltx_mathvariant_italic" id="S6.Thmtheorem1.p2.10.10.m10.1.1.1.3" xref="S6.Thmtheorem1.p2.10.10.m10.1.1.1.3a.cmml">g</mtext><mo id="S6.Thmtheorem1.p2.10.10.m10.1.1.1.2" xref="S6.Thmtheorem1.p2.10.10.m10.1.1.1.2.cmml">⁢</mo><mrow id="S6.Thmtheorem1.p2.10.10.m10.1.1.1.1.1" xref="S6.Thmtheorem1.p2.10.10.m10.1.1.1.1.1.1.cmml"><mo id="S6.Thmtheorem1.p2.10.10.m10.1.1.1.1.1.2" stretchy="false" xref="S6.Thmtheorem1.p2.10.10.m10.1.1.1.1.1.1.cmml">(</mo><msub id="S6.Thmtheorem1.p2.10.10.m10.1.1.1.1.1.1" xref="S6.Thmtheorem1.p2.10.10.m10.1.1.1.1.1.1.cmml"><mi id="S6.Thmtheorem1.p2.10.10.m10.1.1.1.1.1.1.2" xref="S6.Thmtheorem1.p2.10.10.m10.1.1.1.1.1.1.2.cmml">x</mi><mn id="S6.Thmtheorem1.p2.10.10.m10.1.1.1.1.1.1.3" xref="S6.Thmtheorem1.p2.10.10.m10.1.1.1.1.1.1.3.cmml">2</mn></msub><mo id="S6.Thmtheorem1.p2.10.10.m10.1.1.1.1.1.3" stretchy="false" xref="S6.Thmtheorem1.p2.10.10.m10.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S6.Thmtheorem1.p2.10.10.m10.3.3.4" xref="S6.Thmtheorem1.p2.10.10.m10.3.3.4.cmml">></mo><mrow id="S6.Thmtheorem1.p2.10.10.m10.3.3.3" xref="S6.Thmtheorem1.p2.10.10.m10.3.3.3.cmml"><mrow id="S6.Thmtheorem1.p2.10.10.m10.2.2.2.1.1" xref="S6.Thmtheorem1.p2.10.10.m10.2.2.2.1.1.1.cmml"><mo id="S6.Thmtheorem1.p2.10.10.m10.2.2.2.1.1.2" stretchy="false" xref="S6.Thmtheorem1.p2.10.10.m10.2.2.2.1.1.1.cmml">(</mo><mrow id="S6.Thmtheorem1.p2.10.10.m10.2.2.2.1.1.1" xref="S6.Thmtheorem1.p2.10.10.m10.2.2.2.1.1.1.cmml"><mn id="S6.Thmtheorem1.p2.10.10.m10.2.2.2.1.1.1.2" xref="S6.Thmtheorem1.p2.10.10.m10.2.2.2.1.1.1.2.cmml">1</mn><mo id="S6.Thmtheorem1.p2.10.10.m10.2.2.2.1.1.1.1" xref="S6.Thmtheorem1.p2.10.10.m10.2.2.2.1.1.1.1.cmml">+</mo><mi id="S6.Thmtheorem1.p2.10.10.m10.2.2.2.1.1.1.3" xref="S6.Thmtheorem1.p2.10.10.m10.2.2.2.1.1.1.3.cmml">η</mi></mrow><mo id="S6.Thmtheorem1.p2.10.10.m10.2.2.2.1.1.3" stretchy="false" xref="S6.Thmtheorem1.p2.10.10.m10.2.2.2.1.1.1.cmml">)</mo></mrow><mo id="S6.Thmtheorem1.p2.10.10.m10.3.3.3.3" xref="S6.Thmtheorem1.p2.10.10.m10.3.3.3.3.cmml">⁢</mo><mtext class="ltx_mathvariant_italic" id="S6.Thmtheorem1.p2.10.10.m10.3.3.3.4" xref="S6.Thmtheorem1.p2.10.10.m10.3.3.3.4a.cmml">g</mtext><mo id="S6.Thmtheorem1.p2.10.10.m10.3.3.3.3a" xref="S6.Thmtheorem1.p2.10.10.m10.3.3.3.3.cmml">⁢</mo><mrow id="S6.Thmtheorem1.p2.10.10.m10.3.3.3.2.1" xref="S6.Thmtheorem1.p2.10.10.m10.3.3.3.2.1.1.cmml"><mo id="S6.Thmtheorem1.p2.10.10.m10.3.3.3.2.1.2" stretchy="false" xref="S6.Thmtheorem1.p2.10.10.m10.3.3.3.2.1.1.cmml">(</mo><msub id="S6.Thmtheorem1.p2.10.10.m10.3.3.3.2.1.1" xref="S6.Thmtheorem1.p2.10.10.m10.3.3.3.2.1.1.cmml"><mi id="S6.Thmtheorem1.p2.10.10.m10.3.3.3.2.1.1.2" xref="S6.Thmtheorem1.p2.10.10.m10.3.3.3.2.1.1.2.cmml">x</mi><mn id="S6.Thmtheorem1.p2.10.10.m10.3.3.3.2.1.1.3" xref="S6.Thmtheorem1.p2.10.10.m10.3.3.3.2.1.1.3.cmml">1</mn></msub><mo id="S6.Thmtheorem1.p2.10.10.m10.3.3.3.2.1.3" stretchy="false" xref="S6.Thmtheorem1.p2.10.10.m10.3.3.3.2.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.Thmtheorem1.p2.10.10.m10.3b"><apply id="S6.Thmtheorem1.p2.10.10.m10.3.3.cmml" xref="S6.Thmtheorem1.p2.10.10.m10.3.3"><gt id="S6.Thmtheorem1.p2.10.10.m10.3.3.4.cmml" xref="S6.Thmtheorem1.p2.10.10.m10.3.3.4"></gt><apply id="S6.Thmtheorem1.p2.10.10.m10.1.1.1.cmml" xref="S6.Thmtheorem1.p2.10.10.m10.1.1.1"><times id="S6.Thmtheorem1.p2.10.10.m10.1.1.1.2.cmml" xref="S6.Thmtheorem1.p2.10.10.m10.1.1.1.2"></times><ci id="S6.Thmtheorem1.p2.10.10.m10.1.1.1.3a.cmml" xref="S6.Thmtheorem1.p2.10.10.m10.1.1.1.3"><mtext class="ltx_mathvariant_italic" id="S6.Thmtheorem1.p2.10.10.m10.1.1.1.3.cmml" xref="S6.Thmtheorem1.p2.10.10.m10.1.1.1.3">g</mtext></ci><apply id="S6.Thmtheorem1.p2.10.10.m10.1.1.1.1.1.1.cmml" xref="S6.Thmtheorem1.p2.10.10.m10.1.1.1.1.1"><csymbol cd="ambiguous" id="S6.Thmtheorem1.p2.10.10.m10.1.1.1.1.1.1.1.cmml" xref="S6.Thmtheorem1.p2.10.10.m10.1.1.1.1.1">subscript</csymbol><ci id="S6.Thmtheorem1.p2.10.10.m10.1.1.1.1.1.1.2.cmml" xref="S6.Thmtheorem1.p2.10.10.m10.1.1.1.1.1.1.2">𝑥</ci><cn id="S6.Thmtheorem1.p2.10.10.m10.1.1.1.1.1.1.3.cmml" type="integer" xref="S6.Thmtheorem1.p2.10.10.m10.1.1.1.1.1.1.3">2</cn></apply></apply><apply id="S6.Thmtheorem1.p2.10.10.m10.3.3.3.cmml" xref="S6.Thmtheorem1.p2.10.10.m10.3.3.3"><times id="S6.Thmtheorem1.p2.10.10.m10.3.3.3.3.cmml" xref="S6.Thmtheorem1.p2.10.10.m10.3.3.3.3"></times><apply id="S6.Thmtheorem1.p2.10.10.m10.2.2.2.1.1.1.cmml" xref="S6.Thmtheorem1.p2.10.10.m10.2.2.2.1.1"><plus id="S6.Thmtheorem1.p2.10.10.m10.2.2.2.1.1.1.1.cmml" xref="S6.Thmtheorem1.p2.10.10.m10.2.2.2.1.1.1.1"></plus><cn id="S6.Thmtheorem1.p2.10.10.m10.2.2.2.1.1.1.2.cmml" type="integer" xref="S6.Thmtheorem1.p2.10.10.m10.2.2.2.1.1.1.2">1</cn><ci id="S6.Thmtheorem1.p2.10.10.m10.2.2.2.1.1.1.3.cmml" xref="S6.Thmtheorem1.p2.10.10.m10.2.2.2.1.1.1.3">𝜂</ci></apply><ci id="S6.Thmtheorem1.p2.10.10.m10.3.3.3.4a.cmml" xref="S6.Thmtheorem1.p2.10.10.m10.3.3.3.4"><mtext class="ltx_mathvariant_italic" id="S6.Thmtheorem1.p2.10.10.m10.3.3.3.4.cmml" xref="S6.Thmtheorem1.p2.10.10.m10.3.3.3.4">g</mtext></ci><apply id="S6.Thmtheorem1.p2.10.10.m10.3.3.3.2.1.1.cmml" xref="S6.Thmtheorem1.p2.10.10.m10.3.3.3.2.1"><csymbol cd="ambiguous" id="S6.Thmtheorem1.p2.10.10.m10.3.3.3.2.1.1.1.cmml" xref="S6.Thmtheorem1.p2.10.10.m10.3.3.3.2.1">subscript</csymbol><ci id="S6.Thmtheorem1.p2.10.10.m10.3.3.3.2.1.1.2.cmml" xref="S6.Thmtheorem1.p2.10.10.m10.3.3.3.2.1.1.2">𝑥</ci><cn id="S6.Thmtheorem1.p2.10.10.m10.3.3.3.2.1.1.3.cmml" type="integer" xref="S6.Thmtheorem1.p2.10.10.m10.3.3.3.2.1.1.3">1</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.Thmtheorem1.p2.10.10.m10.3c">\textsl{g}(x_{2})>(1+\eta)\textsl{g}(x_{1})</annotation><annotation encoding="application/x-llamapun" id="S6.Thmtheorem1.p2.10.10.m10.3d">g ( italic_x start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) > ( 1 + italic_η ) g ( italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT )</annotation></semantics></math> (and thus the edge <math alttext="\overline{x_{2}x_{1}}" class="ltx_Math" display="inline" id="S6.Thmtheorem1.p2.11.11.m11.1"><semantics id="S6.Thmtheorem1.p2.11.11.m11.1a"><mover accent="true" id="S6.Thmtheorem1.p2.11.11.m11.1.1" xref="S6.Thmtheorem1.p2.11.11.m11.1.1.cmml"><mrow id="S6.Thmtheorem1.p2.11.11.m11.1.1.2" xref="S6.Thmtheorem1.p2.11.11.m11.1.1.2.cmml"><msub id="S6.Thmtheorem1.p2.11.11.m11.1.1.2.2" xref="S6.Thmtheorem1.p2.11.11.m11.1.1.2.2.cmml"><mi id="S6.Thmtheorem1.p2.11.11.m11.1.1.2.2.2" xref="S6.Thmtheorem1.p2.11.11.m11.1.1.2.2.2.cmml">x</mi><mn id="S6.Thmtheorem1.p2.11.11.m11.1.1.2.2.3" xref="S6.Thmtheorem1.p2.11.11.m11.1.1.2.2.3.cmml">2</mn></msub><mo id="S6.Thmtheorem1.p2.11.11.m11.1.1.2.1" xref="S6.Thmtheorem1.p2.11.11.m11.1.1.2.1.cmml">⁢</mo><msub id="S6.Thmtheorem1.p2.11.11.m11.1.1.2.3" xref="S6.Thmtheorem1.p2.11.11.m11.1.1.2.3.cmml"><mi id="S6.Thmtheorem1.p2.11.11.m11.1.1.2.3.2" xref="S6.Thmtheorem1.p2.11.11.m11.1.1.2.3.2.cmml">x</mi><mn id="S6.Thmtheorem1.p2.11.11.m11.1.1.2.3.3" xref="S6.Thmtheorem1.p2.11.11.m11.1.1.2.3.3.cmml">1</mn></msub></mrow><mo id="S6.Thmtheorem1.p2.11.11.m11.1.1.1" xref="S6.Thmtheorem1.p2.11.11.m11.1.1.1.cmml">¯</mo></mover><annotation-xml encoding="MathML-Content" id="S6.Thmtheorem1.p2.11.11.m11.1b"><apply id="S6.Thmtheorem1.p2.11.11.m11.1.1.cmml" xref="S6.Thmtheorem1.p2.11.11.m11.1.1"><ci id="S6.Thmtheorem1.p2.11.11.m11.1.1.1.cmml" xref="S6.Thmtheorem1.p2.11.11.m11.1.1.1">¯</ci><apply id="S6.Thmtheorem1.p2.11.11.m11.1.1.2.cmml" xref="S6.Thmtheorem1.p2.11.11.m11.1.1.2"><times id="S6.Thmtheorem1.p2.11.11.m11.1.1.2.1.cmml" xref="S6.Thmtheorem1.p2.11.11.m11.1.1.2.1"></times><apply id="S6.Thmtheorem1.p2.11.11.m11.1.1.2.2.cmml" xref="S6.Thmtheorem1.p2.11.11.m11.1.1.2.2"><csymbol cd="ambiguous" id="S6.Thmtheorem1.p2.11.11.m11.1.1.2.2.1.cmml" xref="S6.Thmtheorem1.p2.11.11.m11.1.1.2.2">subscript</csymbol><ci id="S6.Thmtheorem1.p2.11.11.m11.1.1.2.2.2.cmml" xref="S6.Thmtheorem1.p2.11.11.m11.1.1.2.2.2">𝑥</ci><cn id="S6.Thmtheorem1.p2.11.11.m11.1.1.2.2.3.cmml" type="integer" xref="S6.Thmtheorem1.p2.11.11.m11.1.1.2.2.3">2</cn></apply><apply id="S6.Thmtheorem1.p2.11.11.m11.1.1.2.3.cmml" xref="S6.Thmtheorem1.p2.11.11.m11.1.1.2.3"><csymbol cd="ambiguous" id="S6.Thmtheorem1.p2.11.11.m11.1.1.2.3.1.cmml" xref="S6.Thmtheorem1.p2.11.11.m11.1.1.2.3">subscript</csymbol><ci id="S6.Thmtheorem1.p2.11.11.m11.1.1.2.3.2.cmml" xref="S6.Thmtheorem1.p2.11.11.m11.1.1.2.3.2">𝑥</ci><cn id="S6.Thmtheorem1.p2.11.11.m11.1.1.2.3.3.cmml" type="integer" xref="S6.Thmtheorem1.p2.11.11.m11.1.1.2.3.3">1</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.Thmtheorem1.p2.11.11.m11.1c">\overline{x_{2}x_{1}}</annotation><annotation encoding="application/x-llamapun" id="S6.Thmtheorem1.p2.11.11.m11.1d">over¯ start_ARG italic_x start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_ARG</annotation></semantics></math> becomes bad). Note if there exists such a vertex <math alttext="x_{2}" class="ltx_Math" display="inline" id="S6.Thmtheorem1.p2.12.12.m12.1"><semantics id="S6.Thmtheorem1.p2.12.12.m12.1a"><msub id="S6.Thmtheorem1.p2.12.12.m12.1.1" xref="S6.Thmtheorem1.p2.12.12.m12.1.1.cmml"><mi id="S6.Thmtheorem1.p2.12.12.m12.1.1.2" xref="S6.Thmtheorem1.p2.12.12.m12.1.1.2.cmml">x</mi><mn id="S6.Thmtheorem1.p2.12.12.m12.1.1.3" xref="S6.Thmtheorem1.p2.12.12.m12.1.1.3.cmml">2</mn></msub><annotation-xml encoding="MathML-Content" id="S6.Thmtheorem1.p2.12.12.m12.1b"><apply id="S6.Thmtheorem1.p2.12.12.m12.1.1.cmml" xref="S6.Thmtheorem1.p2.12.12.m12.1.1"><csymbol cd="ambiguous" id="S6.Thmtheorem1.p2.12.12.m12.1.1.1.cmml" xref="S6.Thmtheorem1.p2.12.12.m12.1.1">subscript</csymbol><ci id="S6.Thmtheorem1.p2.12.12.m12.1.1.2.cmml" xref="S6.Thmtheorem1.p2.12.12.m12.1.1.2">𝑥</ci><cn id="S6.Thmtheorem1.p2.12.12.m12.1.1.3.cmml" type="integer" xref="S6.Thmtheorem1.p2.12.12.m12.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.Thmtheorem1.p2.12.12.m12.1c">x_{2}</annotation><annotation encoding="application/x-llamapun" id="S6.Thmtheorem1.p2.12.12.m12.1d">italic_x start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT</annotation></semantics></math>, then it must hold for the vertex <math alttext="x_{2}^{*}:=\arg\max_{x_{2}\in V}\{\textsl{g}(x_{2})\mid\textsl{g}(x_{2}\!\to\!% x_{1})>0\}" class="ltx_Math" display="inline" id="S6.Thmtheorem1.p2.13.13.m13.2"><semantics id="S6.Thmtheorem1.p2.13.13.m13.2a"><mrow id="S6.Thmtheorem1.p2.13.13.m13.2.2" xref="S6.Thmtheorem1.p2.13.13.m13.2.2.cmml"><msubsup id="S6.Thmtheorem1.p2.13.13.m13.2.2.4" xref="S6.Thmtheorem1.p2.13.13.m13.2.2.4.cmml"><mi id="S6.Thmtheorem1.p2.13.13.m13.2.2.4.2.2" xref="S6.Thmtheorem1.p2.13.13.m13.2.2.4.2.2.cmml">x</mi><mn id="S6.Thmtheorem1.p2.13.13.m13.2.2.4.2.3" xref="S6.Thmtheorem1.p2.13.13.m13.2.2.4.2.3.cmml">2</mn><mo id="S6.Thmtheorem1.p2.13.13.m13.2.2.4.3" xref="S6.Thmtheorem1.p2.13.13.m13.2.2.4.3.cmml">∗</mo></msubsup><mo id="S6.Thmtheorem1.p2.13.13.m13.2.2.3" lspace="0.278em" rspace="0.278em" xref="S6.Thmtheorem1.p2.13.13.m13.2.2.3.cmml">:=</mo><mrow id="S6.Thmtheorem1.p2.13.13.m13.2.2.2" xref="S6.Thmtheorem1.p2.13.13.m13.2.2.2.cmml"><mi id="S6.Thmtheorem1.p2.13.13.m13.2.2.2.3" xref="S6.Thmtheorem1.p2.13.13.m13.2.2.2.3.cmml">arg</mi><mo id="S6.Thmtheorem1.p2.13.13.m13.2.2.2a" lspace="0.167em" xref="S6.Thmtheorem1.p2.13.13.m13.2.2.2.cmml">⁡</mo><mrow id="S6.Thmtheorem1.p2.13.13.m13.2.2.2.2.2" xref="S6.Thmtheorem1.p2.13.13.m13.2.2.2.2.3.cmml"><msub id="S6.Thmtheorem1.p2.13.13.m13.1.1.1.1.1.1" xref="S6.Thmtheorem1.p2.13.13.m13.1.1.1.1.1.1.cmml"><mi id="S6.Thmtheorem1.p2.13.13.m13.1.1.1.1.1.1.2" xref="S6.Thmtheorem1.p2.13.13.m13.1.1.1.1.1.1.2.cmml">max</mi><mrow id="S6.Thmtheorem1.p2.13.13.m13.1.1.1.1.1.1.3" xref="S6.Thmtheorem1.p2.13.13.m13.1.1.1.1.1.1.3.cmml"><msub id="S6.Thmtheorem1.p2.13.13.m13.1.1.1.1.1.1.3.2" 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xref="S6.Thmtheorem1.p2.13.13.m13.2.2.2.2.2.2.1.2.1.1.1.1.1.3">subscript</csymbol><ci id="S6.Thmtheorem1.p2.13.13.m13.2.2.2.2.2.2.1.2.1.1.1.1.1.3.2.cmml" xref="S6.Thmtheorem1.p2.13.13.m13.2.2.2.2.2.2.1.2.1.1.1.1.1.3.2">𝑥</ci><cn id="S6.Thmtheorem1.p2.13.13.m13.2.2.2.2.2.2.1.2.1.1.1.1.1.3.3.cmml" type="integer" xref="S6.Thmtheorem1.p2.13.13.m13.2.2.2.2.2.2.1.2.1.1.1.1.1.3.3">1</cn></apply></apply></apply></apply><cn id="S6.Thmtheorem1.p2.13.13.m13.2.2.2.2.2.2.1.5.cmml" type="integer" xref="S6.Thmtheorem1.p2.13.13.m13.2.2.2.2.2.2.1.5">0</cn></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.Thmtheorem1.p2.13.13.m13.2c">x_{2}^{*}:=\arg\max_{x_{2}\in V}\{\textsl{g}(x_{2})\mid\textsl{g}(x_{2}\!\to\!% x_{1})>0\}</annotation><annotation encoding="application/x-llamapun" id="S6.Thmtheorem1.p2.13.13.m13.2d">italic_x start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT := roman_arg roman_max start_POSTSUBSCRIPT italic_x start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ∈ italic_V end_POSTSUBSCRIPT { g ( italic_x start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) ∣ g ( italic_x start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT → italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) > 0 }</annotation></semantics></math>. This leads to a recursive algorithm to decrease the out-degree of a vertex (Algorithm <a class="ltx_ref" href="https://arxiv.org/html/2411.12694v2#alg1" title="Algorithm 1 ‣ Proof 6.1. ‣ 6 Results in LOCAL ‣ Local Density and its Distributed Approximation"><span class="ltx_text ltx_ref_tag">1</span></a>).</span></p> </div> <figure class="ltx_float ltx_algorithm" id="alg1"> <div class="ltx_listing ltx_lst_numbers_left ltx_listing" id="alg1.8"> <div class="ltx_listingline" id="alg1.8.1"> <span class="ltx_text ltx_font_italic" id="alg1.8.1.1"> </span> <div class="ltx_listing ltx_listing" id="alg1.8.1.2"> <div class="ltx_listingline" id="alg1.l1"> <span class="ltx_text ltx_font_italic" id="alg1.l1.1"> </span><math alttext="w^{*}\leftarrow\arg\max_{w\in V}\{\textsl{g}(w)\mid\textsl{g}(w\!\to\!u)>0\}" class="ltx_Math" display="inline" id="alg1.l1.m1.3"><semantics id="alg1.l1.m1.3a"><mrow id="alg1.l1.m1.3.3" xref="alg1.l1.m1.3.3.cmml"><msup id="alg1.l1.m1.3.3.4" 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id="alg1.l1.m1.3.3.2.2.2.2.1.1.1.1.3.cmml" xref="alg1.l1.m1.3.3.2.2.2.2.1.1.1.1.3">g</mtext></ci><apply id="alg1.l1.m1.3.3.2.2.2.2.1.1.1.1.1.1.1.cmml" xref="alg1.l1.m1.3.3.2.2.2.2.1.1.1.1.1.1"><ci id="alg1.l1.m1.3.3.2.2.2.2.1.1.1.1.1.1.1.1.cmml" xref="alg1.l1.m1.3.3.2.2.2.2.1.1.1.1.1.1.1.1">→</ci><ci id="alg1.l1.m1.3.3.2.2.2.2.1.1.1.1.1.1.1.2.cmml" xref="alg1.l1.m1.3.3.2.2.2.2.1.1.1.1.1.1.1.2">𝑤</ci><ci id="alg1.l1.m1.3.3.2.2.2.2.1.1.1.1.1.1.1.3.cmml" xref="alg1.l1.m1.3.3.2.2.2.2.1.1.1.1.1.1.1.3">𝑢</ci></apply></apply></apply><cn id="alg1.l1.m1.3.3.2.2.2.2.1.5.cmml" type="integer" xref="alg1.l1.m1.3.3.2.2.2.2.1.5">0</cn></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="alg1.l1.m1.3c">w^{*}\leftarrow\arg\max_{w\in V}\{\textsl{g}(w)\mid\textsl{g}(w\!\to\!u)>0\}</annotation><annotation encoding="application/x-llamapun" id="alg1.l1.m1.3d">italic_w start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ← roman_arg roman_max start_POSTSUBSCRIPT italic_w ∈ italic_V end_POSTSUBSCRIPT { g ( italic_w ) ∣ g ( italic_w → italic_u ) > 0 }</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="alg1.l1.2"> </span> </div> <div class="ltx_listingline" id="alg1.l2"> <span class="ltx_text ltx_font_italic" id="alg1.l2.1"> </span><span class="ltx_text ltx_font_bold ltx_font_italic" id="alg1.l2.2">while</span><span class="ltx_text ltx_font_italic" id="alg1.l2.3"> </span><math alttext="\Delta>0" class="ltx_Math" display="inline" id="alg1.l2.m1.1"><semantics id="alg1.l2.m1.1a"><mrow id="alg1.l2.m1.1.1" xref="alg1.l2.m1.1.1.cmml"><mi id="alg1.l2.m1.1.1.2" mathvariant="normal" xref="alg1.l2.m1.1.1.2.cmml">Δ</mi><mo id="alg1.l2.m1.1.1.1" xref="alg1.l2.m1.1.1.1.cmml">></mo><mn id="alg1.l2.m1.1.1.3" xref="alg1.l2.m1.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="alg1.l2.m1.1b"><apply id="alg1.l2.m1.1.1.cmml" xref="alg1.l2.m1.1.1"><gt id="alg1.l2.m1.1.1.1.cmml" xref="alg1.l2.m1.1.1.1"></gt><ci id="alg1.l2.m1.1.1.2.cmml" xref="alg1.l2.m1.1.1.2">Δ</ci><cn id="alg1.l2.m1.1.1.3.cmml" type="integer" xref="alg1.l2.m1.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="alg1.l2.m1.1c">\Delta>0</annotation><annotation encoding="application/x-llamapun" id="alg1.l2.m1.1d">roman_Δ > 0</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="alg1.l2.4"> </span><span class="ltx_text ltx_font_bold ltx_font_italic" id="alg1.l2.5">and</span><span class="ltx_text ltx_font_italic" id="alg1.l2.6"> </span><math alttext="\textsl{g}(w^{*})>(1+\eta)(\textsl{g}(u)-\Delta)" class="ltx_Math" display="inline" id="alg1.l2.m2.4"><semantics id="alg1.l2.m2.4a"><mrow id="alg1.l2.m2.4.4" xref="alg1.l2.m2.4.4.cmml"><mrow id="alg1.l2.m2.2.2.1" xref="alg1.l2.m2.2.2.1.cmml"><mtext class="ltx_mathvariant_italic" id="alg1.l2.m2.2.2.1.3" xref="alg1.l2.m2.2.2.1.3a.cmml">g</mtext><mo id="alg1.l2.m2.2.2.1.2" xref="alg1.l2.m2.2.2.1.2.cmml">⁢</mo><mrow id="alg1.l2.m2.2.2.1.1.1" xref="alg1.l2.m2.2.2.1.1.1.1.cmml"><mo id="alg1.l2.m2.2.2.1.1.1.2" stretchy="false" xref="alg1.l2.m2.2.2.1.1.1.1.cmml">(</mo><msup id="alg1.l2.m2.2.2.1.1.1.1" xref="alg1.l2.m2.2.2.1.1.1.1.cmml"><mi id="alg1.l2.m2.2.2.1.1.1.1.2" xref="alg1.l2.m2.2.2.1.1.1.1.2.cmml">w</mi><mo id="alg1.l2.m2.2.2.1.1.1.1.3" xref="alg1.l2.m2.2.2.1.1.1.1.3.cmml">∗</mo></msup><mo id="alg1.l2.m2.2.2.1.1.1.3" stretchy="false" xref="alg1.l2.m2.2.2.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="alg1.l2.m2.4.4.4" xref="alg1.l2.m2.4.4.4.cmml">></mo><mrow id="alg1.l2.m2.4.4.3" xref="alg1.l2.m2.4.4.3.cmml"><mrow id="alg1.l2.m2.3.3.2.1.1" xref="alg1.l2.m2.3.3.2.1.1.1.cmml"><mo id="alg1.l2.m2.3.3.2.1.1.2" stretchy="false" xref="alg1.l2.m2.3.3.2.1.1.1.cmml">(</mo><mrow id="alg1.l2.m2.3.3.2.1.1.1" xref="alg1.l2.m2.3.3.2.1.1.1.cmml"><mn id="alg1.l2.m2.3.3.2.1.1.1.2" xref="alg1.l2.m2.3.3.2.1.1.1.2.cmml">1</mn><mo id="alg1.l2.m2.3.3.2.1.1.1.1" xref="alg1.l2.m2.3.3.2.1.1.1.1.cmml">+</mo><mi id="alg1.l2.m2.3.3.2.1.1.1.3" xref="alg1.l2.m2.3.3.2.1.1.1.3.cmml">η</mi></mrow><mo id="alg1.l2.m2.3.3.2.1.1.3" stretchy="false" xref="alg1.l2.m2.3.3.2.1.1.1.cmml">)</mo></mrow><mo id="alg1.l2.m2.4.4.3.3" xref="alg1.l2.m2.4.4.3.3.cmml">⁢</mo><mrow id="alg1.l2.m2.4.4.3.2.1" xref="alg1.l2.m2.4.4.3.2.1.1.cmml"><mo id="alg1.l2.m2.4.4.3.2.1.2" stretchy="false" xref="alg1.l2.m2.4.4.3.2.1.1.cmml">(</mo><mrow id="alg1.l2.m2.4.4.3.2.1.1" xref="alg1.l2.m2.4.4.3.2.1.1.cmml"><mrow id="alg1.l2.m2.4.4.3.2.1.1.2" xref="alg1.l2.m2.4.4.3.2.1.1.2.cmml"><mtext class="ltx_mathvariant_italic" id="alg1.l2.m2.4.4.3.2.1.1.2.2" xref="alg1.l2.m2.4.4.3.2.1.1.2.2a.cmml">g</mtext><mo id="alg1.l2.m2.4.4.3.2.1.1.2.1" xref="alg1.l2.m2.4.4.3.2.1.1.2.1.cmml">⁢</mo><mrow id="alg1.l2.m2.4.4.3.2.1.1.2.3.2" xref="alg1.l2.m2.4.4.3.2.1.1.2.cmml"><mo id="alg1.l2.m2.4.4.3.2.1.1.2.3.2.1" stretchy="false" xref="alg1.l2.m2.4.4.3.2.1.1.2.cmml">(</mo><mi id="alg1.l2.m2.1.1" xref="alg1.l2.m2.1.1.cmml">u</mi><mo id="alg1.l2.m2.4.4.3.2.1.1.2.3.2.2" stretchy="false" xref="alg1.l2.m2.4.4.3.2.1.1.2.cmml">)</mo></mrow></mrow><mo id="alg1.l2.m2.4.4.3.2.1.1.1" xref="alg1.l2.m2.4.4.3.2.1.1.1.cmml">−</mo><mi id="alg1.l2.m2.4.4.3.2.1.1.3" mathvariant="normal" xref="alg1.l2.m2.4.4.3.2.1.1.3.cmml">Δ</mi></mrow><mo id="alg1.l2.m2.4.4.3.2.1.3" stretchy="false" xref="alg1.l2.m2.4.4.3.2.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="alg1.l2.m2.4b"><apply id="alg1.l2.m2.4.4.cmml" xref="alg1.l2.m2.4.4"><gt id="alg1.l2.m2.4.4.4.cmml" xref="alg1.l2.m2.4.4.4"></gt><apply id="alg1.l2.m2.2.2.1.cmml" xref="alg1.l2.m2.2.2.1"><times id="alg1.l2.m2.2.2.1.2.cmml" xref="alg1.l2.m2.2.2.1.2"></times><ci id="alg1.l2.m2.2.2.1.3a.cmml" xref="alg1.l2.m2.2.2.1.3"><mtext class="ltx_mathvariant_italic" id="alg1.l2.m2.2.2.1.3.cmml" xref="alg1.l2.m2.2.2.1.3">g</mtext></ci><apply id="alg1.l2.m2.2.2.1.1.1.1.cmml" xref="alg1.l2.m2.2.2.1.1.1"><csymbol cd="ambiguous" id="alg1.l2.m2.2.2.1.1.1.1.1.cmml" xref="alg1.l2.m2.2.2.1.1.1">superscript</csymbol><ci id="alg1.l2.m2.2.2.1.1.1.1.2.cmml" xref="alg1.l2.m2.2.2.1.1.1.1.2">𝑤</ci><times id="alg1.l2.m2.2.2.1.1.1.1.3.cmml" xref="alg1.l2.m2.2.2.1.1.1.1.3"></times></apply></apply><apply id="alg1.l2.m2.4.4.3.cmml" xref="alg1.l2.m2.4.4.3"><times id="alg1.l2.m2.4.4.3.3.cmml" xref="alg1.l2.m2.4.4.3.3"></times><apply id="alg1.l2.m2.3.3.2.1.1.1.cmml" xref="alg1.l2.m2.3.3.2.1.1"><plus id="alg1.l2.m2.3.3.2.1.1.1.1.cmml" xref="alg1.l2.m2.3.3.2.1.1.1.1"></plus><cn id="alg1.l2.m2.3.3.2.1.1.1.2.cmml" type="integer" xref="alg1.l2.m2.3.3.2.1.1.1.2">1</cn><ci id="alg1.l2.m2.3.3.2.1.1.1.3.cmml" xref="alg1.l2.m2.3.3.2.1.1.1.3">𝜂</ci></apply><apply id="alg1.l2.m2.4.4.3.2.1.1.cmml" xref="alg1.l2.m2.4.4.3.2.1"><minus id="alg1.l2.m2.4.4.3.2.1.1.1.cmml" xref="alg1.l2.m2.4.4.3.2.1.1.1"></minus><apply id="alg1.l2.m2.4.4.3.2.1.1.2.cmml" xref="alg1.l2.m2.4.4.3.2.1.1.2"><times id="alg1.l2.m2.4.4.3.2.1.1.2.1.cmml" xref="alg1.l2.m2.4.4.3.2.1.1.2.1"></times><ci id="alg1.l2.m2.4.4.3.2.1.1.2.2a.cmml" xref="alg1.l2.m2.4.4.3.2.1.1.2.2"><mtext class="ltx_mathvariant_italic" id="alg1.l2.m2.4.4.3.2.1.1.2.2.cmml" xref="alg1.l2.m2.4.4.3.2.1.1.2.2">g</mtext></ci><ci id="alg1.l2.m2.1.1.cmml" xref="alg1.l2.m2.1.1">𝑢</ci></apply><ci id="alg1.l2.m2.4.4.3.2.1.1.3.cmml" xref="alg1.l2.m2.4.4.3.2.1.1.3">Δ</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="alg1.l2.m2.4c">\textsl{g}(w^{*})>(1+\eta)(\textsl{g}(u)-\Delta)</annotation><annotation encoding="application/x-llamapun" id="alg1.l2.m2.4d">g ( italic_w start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ) > ( 1 + italic_η ) ( g ( italic_u ) - roman_Δ )</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="alg1.l2.7"> </span><span class="ltx_text ltx_font_bold ltx_font_italic" id="alg1.l2.8">do</span><span class="ltx_text ltx_font_italic" id="alg1.l2.9"> </span> </div> <div class="ltx_listingline" id="alg1.l3"> <span class="ltx_text ltx_font_italic" id="alg1.l3.1"> </span><span class="ltx_text ltx_font_bold ltx_font_italic" id="alg1.l3.2">if</span><span class="ltx_text ltx_font_italic" id="alg1.l3.3"> </span><math alttext="\Delta>\textsl{g}(w^{*}\!\to\!u)" class="ltx_Math" display="inline" id="alg1.l3.m1.1"><semantics id="alg1.l3.m1.1a"><mrow id="alg1.l3.m1.1.1" xref="alg1.l3.m1.1.1.cmml"><mi id="alg1.l3.m1.1.1.3" mathvariant="normal" xref="alg1.l3.m1.1.1.3.cmml">Δ</mi><mo id="alg1.l3.m1.1.1.2" xref="alg1.l3.m1.1.1.2.cmml">></mo><mrow id="alg1.l3.m1.1.1.1" xref="alg1.l3.m1.1.1.1.cmml"><mtext class="ltx_mathvariant_italic" id="alg1.l3.m1.1.1.1.3" xref="alg1.l3.m1.1.1.1.3a.cmml">g</mtext><mo id="alg1.l3.m1.1.1.1.2" xref="alg1.l3.m1.1.1.1.2.cmml">⁢</mo><mrow id="alg1.l3.m1.1.1.1.1.1" xref="alg1.l3.m1.1.1.1.1.1.1.cmml"><mo id="alg1.l3.m1.1.1.1.1.1.2" stretchy="false" xref="alg1.l3.m1.1.1.1.1.1.1.cmml">(</mo><mrow id="alg1.l3.m1.1.1.1.1.1.1" xref="alg1.l3.m1.1.1.1.1.1.1.cmml"><msup id="alg1.l3.m1.1.1.1.1.1.1.2" xref="alg1.l3.m1.1.1.1.1.1.1.2.cmml"><mi id="alg1.l3.m1.1.1.1.1.1.1.2.2" xref="alg1.l3.m1.1.1.1.1.1.1.2.2.cmml">w</mi><mo id="alg1.l3.m1.1.1.1.1.1.1.2.3" xref="alg1.l3.m1.1.1.1.1.1.1.2.3.cmml">∗</mo></msup><mo id="alg1.l3.m1.1.1.1.1.1.1.1" rspace="0.108em" stretchy="false" xref="alg1.l3.m1.1.1.1.1.1.1.1.cmml">→</mo><mi id="alg1.l3.m1.1.1.1.1.1.1.3" xref="alg1.l3.m1.1.1.1.1.1.1.3.cmml">u</mi></mrow><mo id="alg1.l3.m1.1.1.1.1.1.3" stretchy="false" xref="alg1.l3.m1.1.1.1.1.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="alg1.l3.m1.1b"><apply id="alg1.l3.m1.1.1.cmml" xref="alg1.l3.m1.1.1"><gt id="alg1.l3.m1.1.1.2.cmml" xref="alg1.l3.m1.1.1.2"></gt><ci id="alg1.l3.m1.1.1.3.cmml" xref="alg1.l3.m1.1.1.3">Δ</ci><apply id="alg1.l3.m1.1.1.1.cmml" xref="alg1.l3.m1.1.1.1"><times id="alg1.l3.m1.1.1.1.2.cmml" xref="alg1.l3.m1.1.1.1.2"></times><ci id="alg1.l3.m1.1.1.1.3a.cmml" xref="alg1.l3.m1.1.1.1.3"><mtext class="ltx_mathvariant_italic" id="alg1.l3.m1.1.1.1.3.cmml" xref="alg1.l3.m1.1.1.1.3">g</mtext></ci><apply id="alg1.l3.m1.1.1.1.1.1.1.cmml" xref="alg1.l3.m1.1.1.1.1.1"><ci id="alg1.l3.m1.1.1.1.1.1.1.1.cmml" xref="alg1.l3.m1.1.1.1.1.1.1.1">→</ci><apply id="alg1.l3.m1.1.1.1.1.1.1.2.cmml" xref="alg1.l3.m1.1.1.1.1.1.1.2"><csymbol cd="ambiguous" id="alg1.l3.m1.1.1.1.1.1.1.2.1.cmml" xref="alg1.l3.m1.1.1.1.1.1.1.2">superscript</csymbol><ci id="alg1.l3.m1.1.1.1.1.1.1.2.2.cmml" xref="alg1.l3.m1.1.1.1.1.1.1.2.2">𝑤</ci><times id="alg1.l3.m1.1.1.1.1.1.1.2.3.cmml" xref="alg1.l3.m1.1.1.1.1.1.1.2.3"></times></apply><ci id="alg1.l3.m1.1.1.1.1.1.1.3.cmml" xref="alg1.l3.m1.1.1.1.1.1.1.3">𝑢</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="alg1.l3.m1.1c">\Delta>\textsl{g}(w^{*}\!\to\!u)</annotation><annotation encoding="application/x-llamapun" id="alg1.l3.m1.1d">roman_Δ > g ( italic_w start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT → italic_u )</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="alg1.l3.4"> </span><span class="ltx_text ltx_font_bold ltx_font_italic" id="alg1.l3.5">then</span><span class="ltx_text ltx_font_italic" id="alg1.l3.6"> </span> </div> <div class="ltx_listingline" id="alg1.l4"> <span class="ltx_text ltx_font_italic" id="alg1.l4.1"> </span><span class="ltx_text ltx_font_smallcaps" id="alg1.l4.2">Decrease</span><span class="ltx_text ltx_font_italic" id="alg1.l4.3">(</span><math alttext="\textsl{g}(w^{*}\!\to\!v)" class="ltx_Math" display="inline" id="alg1.l4.m1.1"><semantics id="alg1.l4.m1.1a"><mrow id="alg1.l4.m1.1.1" xref="alg1.l4.m1.1.1.cmml"><mtext class="ltx_mathvariant_italic" id="alg1.l4.m1.1.1.3" xref="alg1.l4.m1.1.1.3a.cmml">g</mtext><mo id="alg1.l4.m1.1.1.2" xref="alg1.l4.m1.1.1.2.cmml">⁢</mo><mrow id="alg1.l4.m1.1.1.1.1" xref="alg1.l4.m1.1.1.1.1.1.cmml"><mo id="alg1.l4.m1.1.1.1.1.2" stretchy="false" xref="alg1.l4.m1.1.1.1.1.1.cmml">(</mo><mrow id="alg1.l4.m1.1.1.1.1.1" xref="alg1.l4.m1.1.1.1.1.1.cmml"><msup id="alg1.l4.m1.1.1.1.1.1.2" xref="alg1.l4.m1.1.1.1.1.1.2.cmml"><mi id="alg1.l4.m1.1.1.1.1.1.2.2" xref="alg1.l4.m1.1.1.1.1.1.2.2.cmml">w</mi><mo id="alg1.l4.m1.1.1.1.1.1.2.3" xref="alg1.l4.m1.1.1.1.1.1.2.3.cmml">∗</mo></msup><mo id="alg1.l4.m1.1.1.1.1.1.1" rspace="0.108em" stretchy="false" xref="alg1.l4.m1.1.1.1.1.1.1.cmml">→</mo><mi id="alg1.l4.m1.1.1.1.1.1.3" xref="alg1.l4.m1.1.1.1.1.1.3.cmml">v</mi></mrow><mo id="alg1.l4.m1.1.1.1.1.3" stretchy="false" xref="alg1.l4.m1.1.1.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="alg1.l4.m1.1b"><apply id="alg1.l4.m1.1.1.cmml" xref="alg1.l4.m1.1.1"><times id="alg1.l4.m1.1.1.2.cmml" xref="alg1.l4.m1.1.1.2"></times><ci id="alg1.l4.m1.1.1.3a.cmml" xref="alg1.l4.m1.1.1.3"><mtext class="ltx_mathvariant_italic" id="alg1.l4.m1.1.1.3.cmml" xref="alg1.l4.m1.1.1.3">g</mtext></ci><apply id="alg1.l4.m1.1.1.1.1.1.cmml" xref="alg1.l4.m1.1.1.1.1"><ci id="alg1.l4.m1.1.1.1.1.1.1.cmml" xref="alg1.l4.m1.1.1.1.1.1.1">→</ci><apply id="alg1.l4.m1.1.1.1.1.1.2.cmml" xref="alg1.l4.m1.1.1.1.1.1.2"><csymbol cd="ambiguous" id="alg1.l4.m1.1.1.1.1.1.2.1.cmml" xref="alg1.l4.m1.1.1.1.1.1.2">superscript</csymbol><ci id="alg1.l4.m1.1.1.1.1.1.2.2.cmml" xref="alg1.l4.m1.1.1.1.1.1.2.2">𝑤</ci><times id="alg1.l4.m1.1.1.1.1.1.2.3.cmml" xref="alg1.l4.m1.1.1.1.1.1.2.3"></times></apply><ci id="alg1.l4.m1.1.1.1.1.1.3.cmml" xref="alg1.l4.m1.1.1.1.1.1.3">𝑣</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="alg1.l4.m1.1c">\textsl{g}(w^{*}\!\to\!v)</annotation><annotation encoding="application/x-llamapun" id="alg1.l4.m1.1d">g ( italic_w start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT → italic_v )</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="alg1.l4.4"> by </span><math alttext="\textsl{g}(w^{*}\!\to\!v)" class="ltx_Math" display="inline" id="alg1.l4.m2.1"><semantics id="alg1.l4.m2.1a"><mrow id="alg1.l4.m2.1.1" xref="alg1.l4.m2.1.1.cmml"><mtext class="ltx_mathvariant_italic" id="alg1.l4.m2.1.1.3" xref="alg1.l4.m2.1.1.3a.cmml">g</mtext><mo id="alg1.l4.m2.1.1.2" xref="alg1.l4.m2.1.1.2.cmml">⁢</mo><mrow id="alg1.l4.m2.1.1.1.1" xref="alg1.l4.m2.1.1.1.1.1.cmml"><mo id="alg1.l4.m2.1.1.1.1.2" stretchy="false" xref="alg1.l4.m2.1.1.1.1.1.cmml">(</mo><mrow id="alg1.l4.m2.1.1.1.1.1" xref="alg1.l4.m2.1.1.1.1.1.cmml"><msup id="alg1.l4.m2.1.1.1.1.1.2" xref="alg1.l4.m2.1.1.1.1.1.2.cmml"><mi id="alg1.l4.m2.1.1.1.1.1.2.2" xref="alg1.l4.m2.1.1.1.1.1.2.2.cmml">w</mi><mo id="alg1.l4.m2.1.1.1.1.1.2.3" xref="alg1.l4.m2.1.1.1.1.1.2.3.cmml">∗</mo></msup><mo id="alg1.l4.m2.1.1.1.1.1.1" rspace="0.108em" stretchy="false" xref="alg1.l4.m2.1.1.1.1.1.1.cmml">→</mo><mi id="alg1.l4.m2.1.1.1.1.1.3" xref="alg1.l4.m2.1.1.1.1.1.3.cmml">v</mi></mrow><mo id="alg1.l4.m2.1.1.1.1.3" stretchy="false" xref="alg1.l4.m2.1.1.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="alg1.l4.m2.1b"><apply id="alg1.l4.m2.1.1.cmml" xref="alg1.l4.m2.1.1"><times id="alg1.l4.m2.1.1.2.cmml" xref="alg1.l4.m2.1.1.2"></times><ci id="alg1.l4.m2.1.1.3a.cmml" xref="alg1.l4.m2.1.1.3"><mtext class="ltx_mathvariant_italic" id="alg1.l4.m2.1.1.3.cmml" xref="alg1.l4.m2.1.1.3">g</mtext></ci><apply id="alg1.l4.m2.1.1.1.1.1.cmml" xref="alg1.l4.m2.1.1.1.1"><ci id="alg1.l4.m2.1.1.1.1.1.1.cmml" xref="alg1.l4.m2.1.1.1.1.1.1">→</ci><apply id="alg1.l4.m2.1.1.1.1.1.2.cmml" xref="alg1.l4.m2.1.1.1.1.1.2"><csymbol cd="ambiguous" id="alg1.l4.m2.1.1.1.1.1.2.1.cmml" xref="alg1.l4.m2.1.1.1.1.1.2">superscript</csymbol><ci id="alg1.l4.m2.1.1.1.1.1.2.2.cmml" xref="alg1.l4.m2.1.1.1.1.1.2.2">𝑤</ci><times id="alg1.l4.m2.1.1.1.1.1.2.3.cmml" xref="alg1.l4.m2.1.1.1.1.1.2.3"></times></apply><ci id="alg1.l4.m2.1.1.1.1.1.3.cmml" xref="alg1.l4.m2.1.1.1.1.1.3">𝑣</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="alg1.l4.m2.1c">\textsl{g}(w^{*}\!\to\!v)</annotation><annotation encoding="application/x-llamapun" id="alg1.l4.m2.1d">g ( italic_w start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT → italic_v )</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="alg1.l4.5">) </span> </div> <div class="ltx_listingline" id="alg1.l5"> <span class="ltx_text ltx_font_italic" id="alg1.l5.1"> </span><math alttext="\Delta=\Delta-\textsl{g}(w^{*}\!\to\!u)" class="ltx_Math" display="inline" id="alg1.l5.m1.1"><semantics id="alg1.l5.m1.1a"><mrow id="alg1.l5.m1.1.1" xref="alg1.l5.m1.1.1.cmml"><mi id="alg1.l5.m1.1.1.3" mathvariant="normal" xref="alg1.l5.m1.1.1.3.cmml">Δ</mi><mo id="alg1.l5.m1.1.1.2" xref="alg1.l5.m1.1.1.2.cmml">=</mo><mrow id="alg1.l5.m1.1.1.1" xref="alg1.l5.m1.1.1.1.cmml"><mi id="alg1.l5.m1.1.1.1.3" mathvariant="normal" xref="alg1.l5.m1.1.1.1.3.cmml">Δ</mi><mo id="alg1.l5.m1.1.1.1.2" xref="alg1.l5.m1.1.1.1.2.cmml">−</mo><mrow id="alg1.l5.m1.1.1.1.1" xref="alg1.l5.m1.1.1.1.1.cmml"><mtext class="ltx_mathvariant_italic" id="alg1.l5.m1.1.1.1.1.3" xref="alg1.l5.m1.1.1.1.1.3a.cmml">g</mtext><mo id="alg1.l5.m1.1.1.1.1.2" xref="alg1.l5.m1.1.1.1.1.2.cmml">⁢</mo><mrow id="alg1.l5.m1.1.1.1.1.1.1" xref="alg1.l5.m1.1.1.1.1.1.1.1.cmml"><mo id="alg1.l5.m1.1.1.1.1.1.1.2" stretchy="false" xref="alg1.l5.m1.1.1.1.1.1.1.1.cmml">(</mo><mrow id="alg1.l5.m1.1.1.1.1.1.1.1" xref="alg1.l5.m1.1.1.1.1.1.1.1.cmml"><msup id="alg1.l5.m1.1.1.1.1.1.1.1.2" xref="alg1.l5.m1.1.1.1.1.1.1.1.2.cmml"><mi id="alg1.l5.m1.1.1.1.1.1.1.1.2.2" xref="alg1.l5.m1.1.1.1.1.1.1.1.2.2.cmml">w</mi><mo id="alg1.l5.m1.1.1.1.1.1.1.1.2.3" xref="alg1.l5.m1.1.1.1.1.1.1.1.2.3.cmml">∗</mo></msup><mo id="alg1.l5.m1.1.1.1.1.1.1.1.1" rspace="0.108em" stretchy="false" xref="alg1.l5.m1.1.1.1.1.1.1.1.1.cmml">→</mo><mi id="alg1.l5.m1.1.1.1.1.1.1.1.3" xref="alg1.l5.m1.1.1.1.1.1.1.1.3.cmml">u</mi></mrow><mo id="alg1.l5.m1.1.1.1.1.1.1.3" stretchy="false" xref="alg1.l5.m1.1.1.1.1.1.1.1.cmml">)</mo></mrow></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="alg1.l5.m1.1b"><apply id="alg1.l5.m1.1.1.cmml" xref="alg1.l5.m1.1.1"><eq id="alg1.l5.m1.1.1.2.cmml" xref="alg1.l5.m1.1.1.2"></eq><ci id="alg1.l5.m1.1.1.3.cmml" xref="alg1.l5.m1.1.1.3">Δ</ci><apply id="alg1.l5.m1.1.1.1.cmml" xref="alg1.l5.m1.1.1.1"><minus id="alg1.l5.m1.1.1.1.2.cmml" xref="alg1.l5.m1.1.1.1.2"></minus><ci id="alg1.l5.m1.1.1.1.3.cmml" xref="alg1.l5.m1.1.1.1.3">Δ</ci><apply id="alg1.l5.m1.1.1.1.1.cmml" xref="alg1.l5.m1.1.1.1.1"><times id="alg1.l5.m1.1.1.1.1.2.cmml" xref="alg1.l5.m1.1.1.1.1.2"></times><ci id="alg1.l5.m1.1.1.1.1.3a.cmml" xref="alg1.l5.m1.1.1.1.1.3"><mtext class="ltx_mathvariant_italic" id="alg1.l5.m1.1.1.1.1.3.cmml" xref="alg1.l5.m1.1.1.1.1.3">g</mtext></ci><apply id="alg1.l5.m1.1.1.1.1.1.1.1.cmml" xref="alg1.l5.m1.1.1.1.1.1.1"><ci id="alg1.l5.m1.1.1.1.1.1.1.1.1.cmml" xref="alg1.l5.m1.1.1.1.1.1.1.1.1">→</ci><apply id="alg1.l5.m1.1.1.1.1.1.1.1.2.cmml" xref="alg1.l5.m1.1.1.1.1.1.1.1.2"><csymbol cd="ambiguous" id="alg1.l5.m1.1.1.1.1.1.1.1.2.1.cmml" xref="alg1.l5.m1.1.1.1.1.1.1.1.2">superscript</csymbol><ci id="alg1.l5.m1.1.1.1.1.1.1.1.2.2.cmml" xref="alg1.l5.m1.1.1.1.1.1.1.1.2.2">𝑤</ci><times id="alg1.l5.m1.1.1.1.1.1.1.1.2.3.cmml" xref="alg1.l5.m1.1.1.1.1.1.1.1.2.3"></times></apply><ci id="alg1.l5.m1.1.1.1.1.1.1.1.3.cmml" xref="alg1.l5.m1.1.1.1.1.1.1.1.3">𝑢</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="alg1.l5.m1.1c">\Delta=\Delta-\textsl{g}(w^{*}\!\to\!u)</annotation><annotation encoding="application/x-llamapun" id="alg1.l5.m1.1d">roman_Δ = roman_Δ - g ( italic_w start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT → italic_u )</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="alg1.l5.2"> </span> </div> <div class="ltx_listingline" id="alg1.l6"> <span class="ltx_text ltx_font_italic" id="alg1.l6.1"> </span><span class="ltx_text ltx_font_bold ltx_font_italic" id="alg1.l6.2">else</span><span class="ltx_text ltx_font_italic" id="alg1.l6.3"> </span> </div> <div class="ltx_listingline" id="alg1.l7"> <span class="ltx_text ltx_font_italic" id="alg1.l7.1"> </span><span class="ltx_text ltx_font_smallcaps" id="alg1.l7.2">Decrease</span><span class="ltx_text ltx_font_italic" id="alg1.l7.3">(</span><math alttext="\textsl{g}(w^{*}\!\to\!v)" class="ltx_Math" display="inline" id="alg1.l7.m1.1"><semantics id="alg1.l7.m1.1a"><mrow id="alg1.l7.m1.1.1" xref="alg1.l7.m1.1.1.cmml"><mtext class="ltx_mathvariant_italic" id="alg1.l7.m1.1.1.3" xref="alg1.l7.m1.1.1.3a.cmml">g</mtext><mo id="alg1.l7.m1.1.1.2" xref="alg1.l7.m1.1.1.2.cmml">⁢</mo><mrow id="alg1.l7.m1.1.1.1.1" xref="alg1.l7.m1.1.1.1.1.1.cmml"><mo id="alg1.l7.m1.1.1.1.1.2" stretchy="false" xref="alg1.l7.m1.1.1.1.1.1.cmml">(</mo><mrow id="alg1.l7.m1.1.1.1.1.1" xref="alg1.l7.m1.1.1.1.1.1.cmml"><msup id="alg1.l7.m1.1.1.1.1.1.2" xref="alg1.l7.m1.1.1.1.1.1.2.cmml"><mi id="alg1.l7.m1.1.1.1.1.1.2.2" xref="alg1.l7.m1.1.1.1.1.1.2.2.cmml">w</mi><mo id="alg1.l7.m1.1.1.1.1.1.2.3" xref="alg1.l7.m1.1.1.1.1.1.2.3.cmml">∗</mo></msup><mo id="alg1.l7.m1.1.1.1.1.1.1" rspace="0.108em" stretchy="false" xref="alg1.l7.m1.1.1.1.1.1.1.cmml">→</mo><mi id="alg1.l7.m1.1.1.1.1.1.3" xref="alg1.l7.m1.1.1.1.1.1.3.cmml">v</mi></mrow><mo id="alg1.l7.m1.1.1.1.1.3" stretchy="false" xref="alg1.l7.m1.1.1.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="alg1.l7.m1.1b"><apply id="alg1.l7.m1.1.1.cmml" xref="alg1.l7.m1.1.1"><times id="alg1.l7.m1.1.1.2.cmml" xref="alg1.l7.m1.1.1.2"></times><ci id="alg1.l7.m1.1.1.3a.cmml" xref="alg1.l7.m1.1.1.3"><mtext class="ltx_mathvariant_italic" id="alg1.l7.m1.1.1.3.cmml" xref="alg1.l7.m1.1.1.3">g</mtext></ci><apply id="alg1.l7.m1.1.1.1.1.1.cmml" xref="alg1.l7.m1.1.1.1.1"><ci id="alg1.l7.m1.1.1.1.1.1.1.cmml" xref="alg1.l7.m1.1.1.1.1.1.1">→</ci><apply id="alg1.l7.m1.1.1.1.1.1.2.cmml" xref="alg1.l7.m1.1.1.1.1.1.2"><csymbol cd="ambiguous" id="alg1.l7.m1.1.1.1.1.1.2.1.cmml" xref="alg1.l7.m1.1.1.1.1.1.2">superscript</csymbol><ci id="alg1.l7.m1.1.1.1.1.1.2.2.cmml" xref="alg1.l7.m1.1.1.1.1.1.2.2">𝑤</ci><times id="alg1.l7.m1.1.1.1.1.1.2.3.cmml" xref="alg1.l7.m1.1.1.1.1.1.2.3"></times></apply><ci id="alg1.l7.m1.1.1.1.1.1.3.cmml" xref="alg1.l7.m1.1.1.1.1.1.3">𝑣</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="alg1.l7.m1.1c">\textsl{g}(w^{*}\!\to\!v)</annotation><annotation encoding="application/x-llamapun" id="alg1.l7.m1.1d">g ( italic_w start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT → italic_v )</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="alg1.l7.4"> by </span><math alttext="\Delta" class="ltx_Math" display="inline" id="alg1.l7.m2.1"><semantics id="alg1.l7.m2.1a"><mi id="alg1.l7.m2.1.1" mathvariant="normal" xref="alg1.l7.m2.1.1.cmml">Δ</mi><annotation-xml encoding="MathML-Content" id="alg1.l7.m2.1b"><ci id="alg1.l7.m2.1.1.cmml" xref="alg1.l7.m2.1.1">Δ</ci></annotation-xml><annotation encoding="application/x-tex" id="alg1.l7.m2.1c">\Delta</annotation><annotation encoding="application/x-llamapun" id="alg1.l7.m2.1d">roman_Δ</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="alg1.l7.5">) </span> </div> <div class="ltx_listingline" id="alg1.l8"> <span class="ltx_text ltx_font_italic" id="alg1.l8.1"> </span><math alttext="w^{*}\leftarrow\arg\max_{w\in V}\{\textsl{g}(w)\mid\textsl{g}(w\!\to\!u)>0\}" class="ltx_Math" display="inline" id="alg1.l8.m1.3"><semantics id="alg1.l8.m1.3a"><mrow id="alg1.l8.m1.3.3" xref="alg1.l8.m1.3.3.cmml"><msup id="alg1.l8.m1.3.3.4" xref="alg1.l8.m1.3.3.4.cmml"><mi id="alg1.l8.m1.3.3.4.2" xref="alg1.l8.m1.3.3.4.2.cmml">w</mi><mo id="alg1.l8.m1.3.3.4.3" xref="alg1.l8.m1.3.3.4.3.cmml">∗</mo></msup><mo id="alg1.l8.m1.3.3.3" stretchy="false" xref="alg1.l8.m1.3.3.3.cmml">←</mo><mrow id="alg1.l8.m1.3.3.2" xref="alg1.l8.m1.3.3.2.cmml"><mi id="alg1.l8.m1.3.3.2.3" xref="alg1.l8.m1.3.3.2.3.cmml">arg</mi><mo id="alg1.l8.m1.3.3.2a" lspace="0.167em" xref="alg1.l8.m1.3.3.2.cmml">⁡</mo><mrow id="alg1.l8.m1.3.3.2.2.2" 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xref="alg1.l8.m1.3.3.2.2.2.2.1.2.cmml">⁢</mo><mrow id="alg1.l8.m1.3.3.2.2.2.2.1.4.2" xref="alg1.l8.m1.3.3.2.2.2.2.1.cmml"><mo id="alg1.l8.m1.3.3.2.2.2.2.1.4.2.1" stretchy="false" xref="alg1.l8.m1.3.3.2.2.2.2.1.cmml">(</mo><mi id="alg1.l8.m1.1.1" xref="alg1.l8.m1.1.1.cmml">w</mi><mo id="alg1.l8.m1.3.3.2.2.2.2.1.4.2.2" stretchy="false" xref="alg1.l8.m1.3.3.2.2.2.2.1.cmml">)</mo></mrow><mo id="alg1.l8.m1.3.3.2.2.2.2.1.2a" lspace="0em" xref="alg1.l8.m1.3.3.2.2.2.2.1.2.cmml">⁢</mo><mrow id="alg1.l8.m1.3.3.2.2.2.2.1.1.1" xref="alg1.l8.m1.3.3.2.2.2.2.1.1.2.cmml"><mo fence="true" id="alg1.l8.m1.3.3.2.2.2.2.1.1.1.2" rspace="0em" xref="alg1.l8.m1.3.3.2.2.2.2.1.1.2.1.cmml">∣</mo><mrow id="alg1.l8.m1.3.3.2.2.2.2.1.1.1.1" xref="alg1.l8.m1.3.3.2.2.2.2.1.1.1.1.cmml"><mtext class="ltx_mathvariant_italic" id="alg1.l8.m1.3.3.2.2.2.2.1.1.1.1.3" xref="alg1.l8.m1.3.3.2.2.2.2.1.1.1.1.3a.cmml">g</mtext><mo id="alg1.l8.m1.3.3.2.2.2.2.1.1.1.1.2" xref="alg1.l8.m1.3.3.2.2.2.2.1.1.1.1.2.cmml">⁢</mo><mrow 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xref="alg1.l8.m1.3.3.2.2.2.2.1.2.cmml">⁢</mo><mn id="alg1.l8.m1.3.3.2.2.2.2.1.5" xref="alg1.l8.m1.3.3.2.2.2.2.1.5.cmml">0</mn></mrow><mo id="alg1.l8.m1.3.3.2.2.2.2.3" stretchy="false" xref="alg1.l8.m1.3.3.2.2.3.cmml">}</mo></mrow></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="alg1.l8.m1.3b"><apply id="alg1.l8.m1.3.3.cmml" xref="alg1.l8.m1.3.3"><ci id="alg1.l8.m1.3.3.3.cmml" xref="alg1.l8.m1.3.3.3">←</ci><apply id="alg1.l8.m1.3.3.4.cmml" xref="alg1.l8.m1.3.3.4"><csymbol cd="ambiguous" id="alg1.l8.m1.3.3.4.1.cmml" xref="alg1.l8.m1.3.3.4">superscript</csymbol><ci id="alg1.l8.m1.3.3.4.2.cmml" xref="alg1.l8.m1.3.3.4.2">𝑤</ci><times id="alg1.l8.m1.3.3.4.3.cmml" xref="alg1.l8.m1.3.3.4.3"></times></apply><apply id="alg1.l8.m1.3.3.2.cmml" xref="alg1.l8.m1.3.3.2"><arg id="alg1.l8.m1.3.3.2.3.cmml" xref="alg1.l8.m1.3.3.2.3"></arg><apply id="alg1.l8.m1.3.3.2.2.3.cmml" xref="alg1.l8.m1.3.3.2.2.2"><apply id="alg1.l8.m1.2.2.1.1.1.1.cmml" xref="alg1.l8.m1.2.2.1.1.1.1"><csymbol cd="ambiguous" id="alg1.l8.m1.2.2.1.1.1.1.1.cmml" xref="alg1.l8.m1.2.2.1.1.1.1">subscript</csymbol><max id="alg1.l8.m1.2.2.1.1.1.1.2.cmml" xref="alg1.l8.m1.2.2.1.1.1.1.2"></max><apply id="alg1.l8.m1.2.2.1.1.1.1.3.cmml" xref="alg1.l8.m1.2.2.1.1.1.1.3"><in id="alg1.l8.m1.2.2.1.1.1.1.3.1.cmml" xref="alg1.l8.m1.2.2.1.1.1.1.3.1"></in><ci id="alg1.l8.m1.2.2.1.1.1.1.3.2.cmml" xref="alg1.l8.m1.2.2.1.1.1.1.3.2">𝑤</ci><ci id="alg1.l8.m1.2.2.1.1.1.1.3.3.cmml" xref="alg1.l8.m1.2.2.1.1.1.1.3.3">𝑉</ci></apply></apply><apply id="alg1.l8.m1.3.3.2.2.2.2.1.cmml" xref="alg1.l8.m1.3.3.2.2.2.2.1"><times id="alg1.l8.m1.3.3.2.2.2.2.1.2.cmml" xref="alg1.l8.m1.3.3.2.2.2.2.1.2"></times><ci id="alg1.l8.m1.3.3.2.2.2.2.1.3a.cmml" xref="alg1.l8.m1.3.3.2.2.2.2.1.3"><mtext class="ltx_mathvariant_italic" id="alg1.l8.m1.3.3.2.2.2.2.1.3.cmml" xref="alg1.l8.m1.3.3.2.2.2.2.1.3">g</mtext></ci><ci id="alg1.l8.m1.1.1.cmml" xref="alg1.l8.m1.1.1">𝑤</ci><apply id="alg1.l8.m1.3.3.2.2.2.2.1.1.2.cmml" xref="alg1.l8.m1.3.3.2.2.2.2.1.1.1"><csymbol cd="latexml" id="alg1.l8.m1.3.3.2.2.2.2.1.1.2.1.cmml" xref="alg1.l8.m1.3.3.2.2.2.2.1.1.1.2">ket</csymbol><apply id="alg1.l8.m1.3.3.2.2.2.2.1.1.1.1.cmml" xref="alg1.l8.m1.3.3.2.2.2.2.1.1.1.1"><times id="alg1.l8.m1.3.3.2.2.2.2.1.1.1.1.2.cmml" xref="alg1.l8.m1.3.3.2.2.2.2.1.1.1.1.2"></times><ci id="alg1.l8.m1.3.3.2.2.2.2.1.1.1.1.3a.cmml" xref="alg1.l8.m1.3.3.2.2.2.2.1.1.1.1.3"><mtext class="ltx_mathvariant_italic" id="alg1.l8.m1.3.3.2.2.2.2.1.1.1.1.3.cmml" xref="alg1.l8.m1.3.3.2.2.2.2.1.1.1.1.3">g</mtext></ci><apply id="alg1.l8.m1.3.3.2.2.2.2.1.1.1.1.1.1.1.cmml" xref="alg1.l8.m1.3.3.2.2.2.2.1.1.1.1.1.1"><ci id="alg1.l8.m1.3.3.2.2.2.2.1.1.1.1.1.1.1.1.cmml" xref="alg1.l8.m1.3.3.2.2.2.2.1.1.1.1.1.1.1.1">→</ci><ci id="alg1.l8.m1.3.3.2.2.2.2.1.1.1.1.1.1.1.2.cmml" xref="alg1.l8.m1.3.3.2.2.2.2.1.1.1.1.1.1.1.2">𝑤</ci><ci id="alg1.l8.m1.3.3.2.2.2.2.1.1.1.1.1.1.1.3.cmml" xref="alg1.l8.m1.3.3.2.2.2.2.1.1.1.1.1.1.1.3">𝑢</ci></apply></apply></apply><cn id="alg1.l8.m1.3.3.2.2.2.2.1.5.cmml" type="integer" xref="alg1.l8.m1.3.3.2.2.2.2.1.5">0</cn></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="alg1.l8.m1.3c">w^{*}\leftarrow\arg\max_{w\in V}\{\textsl{g}(w)\mid\textsl{g}(w\!\to\!u)>0\}</annotation><annotation encoding="application/x-llamapun" id="alg1.l8.m1.3d">italic_w start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ← roman_arg roman_max start_POSTSUBSCRIPT italic_w ∈ italic_V end_POSTSUBSCRIPT { g ( italic_w ) ∣ g ( italic_w → italic_u ) > 0 }</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="alg1.l8.2"> </span> </div> </div> <span class="ltx_text ltx_font_italic" id="alg1.8.1.3"> </span> </div> </div> <figcaption class="ltx_caption ltx_font_italic"><span class="ltx_tag ltx_tag_float"><span class="ltx_text ltx_font_bold ltx_font_upright" id="alg1.15.1.1">Algorithm 1</span> </span><span class="ltx_text ltx_font_smallcaps" id="alg1.16.2">decrease</span>(<math alttext="\textsl{g}(u\!\to\!v)" class="ltx_Math" display="inline" id="alg1.4.m1.1"><semantics id="alg1.4.m1.1b"><mrow id="alg1.4.m1.1.1" xref="alg1.4.m1.1.1.cmml"><mtext class="ltx_mathvariant_italic" id="alg1.4.m1.1.1.3" xref="alg1.4.m1.1.1.3a.cmml">g</mtext><mo id="alg1.4.m1.1.1.2" xref="alg1.4.m1.1.1.2.cmml">⁢</mo><mrow id="alg1.4.m1.1.1.1.1" xref="alg1.4.m1.1.1.1.1.1.cmml"><mo id="alg1.4.m1.1.1.1.1.2" stretchy="false" xref="alg1.4.m1.1.1.1.1.1.cmml">(</mo><mrow id="alg1.4.m1.1.1.1.1.1" xref="alg1.4.m1.1.1.1.1.1.cmml"><mi id="alg1.4.m1.1.1.1.1.1.2" xref="alg1.4.m1.1.1.1.1.1.2.cmml">u</mi><mo id="alg1.4.m1.1.1.1.1.1.1" lspace="0.108em" rspace="0.108em" stretchy="false" xref="alg1.4.m1.1.1.1.1.1.1.cmml">→</mo><mi id="alg1.4.m1.1.1.1.1.1.3" xref="alg1.4.m1.1.1.1.1.1.3.cmml">v</mi></mrow><mo id="alg1.4.m1.1.1.1.1.3" stretchy="false" xref="alg1.4.m1.1.1.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="alg1.4.m1.1c"><apply id="alg1.4.m1.1.1.cmml" xref="alg1.4.m1.1.1"><times id="alg1.4.m1.1.1.2.cmml" xref="alg1.4.m1.1.1.2"></times><ci id="alg1.4.m1.1.1.3a.cmml" xref="alg1.4.m1.1.1.3"><mtext class="ltx_mathvariant_italic" id="alg1.4.m1.1.1.3.cmml" xref="alg1.4.m1.1.1.3">g</mtext></ci><apply id="alg1.4.m1.1.1.1.1.1.cmml" xref="alg1.4.m1.1.1.1.1"><ci id="alg1.4.m1.1.1.1.1.1.1.cmml" xref="alg1.4.m1.1.1.1.1.1.1">→</ci><ci id="alg1.4.m1.1.1.1.1.1.2.cmml" xref="alg1.4.m1.1.1.1.1.1.2">𝑢</ci><ci id="alg1.4.m1.1.1.1.1.1.3.cmml" xref="alg1.4.m1.1.1.1.1.1.3">𝑣</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="alg1.4.m1.1d">\textsl{g}(u\!\to\!v)</annotation><annotation encoding="application/x-llamapun" id="alg1.4.m1.1e">g ( italic_u → italic_v )</annotation></semantics></math> by <math alttext="\Delta" class="ltx_Math" display="inline" id="alg1.5.m2.1"><semantics id="alg1.5.m2.1b"><mi id="alg1.5.m2.1.1" mathvariant="normal" xref="alg1.5.m2.1.1.cmml">Δ</mi><annotation-xml encoding="MathML-Content" id="alg1.5.m2.1c"><ci id="alg1.5.m2.1.1.cmml" xref="alg1.5.m2.1.1">Δ</ci></annotation-xml><annotation encoding="application/x-tex" id="alg1.5.m2.1d">\Delta</annotation><annotation encoding="application/x-llamapun" id="alg1.5.m2.1e">roman_Δ</annotation></semantics></math> – assuming that <math alttext="\Delta\leq\textsl{g}(u\!\to\!v)" class="ltx_Math" display="inline" id="alg1.6.m3.1"><semantics id="alg1.6.m3.1b"><mrow id="alg1.6.m3.1.1" xref="alg1.6.m3.1.1.cmml"><mi id="alg1.6.m3.1.1.3" mathvariant="normal" xref="alg1.6.m3.1.1.3.cmml">Δ</mi><mo id="alg1.6.m3.1.1.2" xref="alg1.6.m3.1.1.2.cmml">≤</mo><mrow id="alg1.6.m3.1.1.1" xref="alg1.6.m3.1.1.1.cmml"><mtext class="ltx_mathvariant_italic" id="alg1.6.m3.1.1.1.3" xref="alg1.6.m3.1.1.1.3a.cmml">g</mtext><mo id="alg1.6.m3.1.1.1.2" xref="alg1.6.m3.1.1.1.2.cmml">⁢</mo><mrow id="alg1.6.m3.1.1.1.1.1" xref="alg1.6.m3.1.1.1.1.1.1.cmml"><mo id="alg1.6.m3.1.1.1.1.1.2" stretchy="false" xref="alg1.6.m3.1.1.1.1.1.1.cmml">(</mo><mrow id="alg1.6.m3.1.1.1.1.1.1" xref="alg1.6.m3.1.1.1.1.1.1.cmml"><mi id="alg1.6.m3.1.1.1.1.1.1.2" xref="alg1.6.m3.1.1.1.1.1.1.2.cmml">u</mi><mo id="alg1.6.m3.1.1.1.1.1.1.1" lspace="0.108em" rspace="0.108em" stretchy="false" xref="alg1.6.m3.1.1.1.1.1.1.1.cmml">→</mo><mi id="alg1.6.m3.1.1.1.1.1.1.3" xref="alg1.6.m3.1.1.1.1.1.1.3.cmml">v</mi></mrow><mo id="alg1.6.m3.1.1.1.1.1.3" stretchy="false" xref="alg1.6.m3.1.1.1.1.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="alg1.6.m3.1c"><apply id="alg1.6.m3.1.1.cmml" xref="alg1.6.m3.1.1"><leq id="alg1.6.m3.1.1.2.cmml" xref="alg1.6.m3.1.1.2"></leq><ci id="alg1.6.m3.1.1.3.cmml" xref="alg1.6.m3.1.1.3">Δ</ci><apply id="alg1.6.m3.1.1.1.cmml" xref="alg1.6.m3.1.1.1"><times id="alg1.6.m3.1.1.1.2.cmml" xref="alg1.6.m3.1.1.1.2"></times><ci id="alg1.6.m3.1.1.1.3a.cmml" xref="alg1.6.m3.1.1.1.3"><mtext class="ltx_mathvariant_italic" id="alg1.6.m3.1.1.1.3.cmml" xref="alg1.6.m3.1.1.1.3">g</mtext></ci><apply id="alg1.6.m3.1.1.1.1.1.1.cmml" xref="alg1.6.m3.1.1.1.1.1"><ci id="alg1.6.m3.1.1.1.1.1.1.1.cmml" xref="alg1.6.m3.1.1.1.1.1.1.1">→</ci><ci id="alg1.6.m3.1.1.1.1.1.1.2.cmml" xref="alg1.6.m3.1.1.1.1.1.1.2">𝑢</ci><ci id="alg1.6.m3.1.1.1.1.1.1.3.cmml" xref="alg1.6.m3.1.1.1.1.1.1.3">𝑣</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="alg1.6.m3.1d">\Delta\leq\textsl{g}(u\!\to\!v)</annotation><annotation encoding="application/x-llamapun" id="alg1.6.m3.1e">roman_Δ ≤ g ( italic_u → italic_v )</annotation></semantics></math>)</figcaption> </figure> <div class="ltx_para" id="S6.Thmtheorem1.p3"> <p class="ltx_p" id="S6.Thmtheorem1.p3.11"><span class="ltx_text ltx_font_italic" id="S6.Thmtheorem1.p3.11.11">We claim that this algorithm as a recursive depth of <math alttext="O(\log_{1+\eta}n)" class="ltx_Math" display="inline" id="S6.Thmtheorem1.p3.1.1.m1.1"><semantics id="S6.Thmtheorem1.p3.1.1.m1.1a"><mrow id="S6.Thmtheorem1.p3.1.1.m1.1.1" xref="S6.Thmtheorem1.p3.1.1.m1.1.1.cmml"><mi id="S6.Thmtheorem1.p3.1.1.m1.1.1.3" xref="S6.Thmtheorem1.p3.1.1.m1.1.1.3.cmml">O</mi><mo id="S6.Thmtheorem1.p3.1.1.m1.1.1.2" xref="S6.Thmtheorem1.p3.1.1.m1.1.1.2.cmml">⁢</mo><mrow id="S6.Thmtheorem1.p3.1.1.m1.1.1.1.1" xref="S6.Thmtheorem1.p3.1.1.m1.1.1.1.1.1.cmml"><mo id="S6.Thmtheorem1.p3.1.1.m1.1.1.1.1.2" stretchy="false" xref="S6.Thmtheorem1.p3.1.1.m1.1.1.1.1.1.cmml">(</mo><mrow id="S6.Thmtheorem1.p3.1.1.m1.1.1.1.1.1" xref="S6.Thmtheorem1.p3.1.1.m1.1.1.1.1.1.cmml"><msub id="S6.Thmtheorem1.p3.1.1.m1.1.1.1.1.1.1" xref="S6.Thmtheorem1.p3.1.1.m1.1.1.1.1.1.1.cmml"><mi id="S6.Thmtheorem1.p3.1.1.m1.1.1.1.1.1.1.2" xref="S6.Thmtheorem1.p3.1.1.m1.1.1.1.1.1.1.2.cmml">log</mi><mrow id="S6.Thmtheorem1.p3.1.1.m1.1.1.1.1.1.1.3" xref="S6.Thmtheorem1.p3.1.1.m1.1.1.1.1.1.1.3.cmml"><mn id="S6.Thmtheorem1.p3.1.1.m1.1.1.1.1.1.1.3.2" xref="S6.Thmtheorem1.p3.1.1.m1.1.1.1.1.1.1.3.2.cmml">1</mn><mo id="S6.Thmtheorem1.p3.1.1.m1.1.1.1.1.1.1.3.1" xref="S6.Thmtheorem1.p3.1.1.m1.1.1.1.1.1.1.3.1.cmml">+</mo><mi id="S6.Thmtheorem1.p3.1.1.m1.1.1.1.1.1.1.3.3" xref="S6.Thmtheorem1.p3.1.1.m1.1.1.1.1.1.1.3.3.cmml">η</mi></mrow></msub><mo id="S6.Thmtheorem1.p3.1.1.m1.1.1.1.1.1a" lspace="0.167em" xref="S6.Thmtheorem1.p3.1.1.m1.1.1.1.1.1.cmml">⁡</mo><mi id="S6.Thmtheorem1.p3.1.1.m1.1.1.1.1.1.2" xref="S6.Thmtheorem1.p3.1.1.m1.1.1.1.1.1.2.cmml">n</mi></mrow><mo id="S6.Thmtheorem1.p3.1.1.m1.1.1.1.1.3" stretchy="false" xref="S6.Thmtheorem1.p3.1.1.m1.1.1.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.Thmtheorem1.p3.1.1.m1.1b"><apply id="S6.Thmtheorem1.p3.1.1.m1.1.1.cmml" xref="S6.Thmtheorem1.p3.1.1.m1.1.1"><times id="S6.Thmtheorem1.p3.1.1.m1.1.1.2.cmml" xref="S6.Thmtheorem1.p3.1.1.m1.1.1.2"></times><ci id="S6.Thmtheorem1.p3.1.1.m1.1.1.3.cmml" xref="S6.Thmtheorem1.p3.1.1.m1.1.1.3">𝑂</ci><apply id="S6.Thmtheorem1.p3.1.1.m1.1.1.1.1.1.cmml" xref="S6.Thmtheorem1.p3.1.1.m1.1.1.1.1"><apply id="S6.Thmtheorem1.p3.1.1.m1.1.1.1.1.1.1.cmml" xref="S6.Thmtheorem1.p3.1.1.m1.1.1.1.1.1.1"><csymbol cd="ambiguous" id="S6.Thmtheorem1.p3.1.1.m1.1.1.1.1.1.1.1.cmml" xref="S6.Thmtheorem1.p3.1.1.m1.1.1.1.1.1.1">subscript</csymbol><log id="S6.Thmtheorem1.p3.1.1.m1.1.1.1.1.1.1.2.cmml" xref="S6.Thmtheorem1.p3.1.1.m1.1.1.1.1.1.1.2"></log><apply id="S6.Thmtheorem1.p3.1.1.m1.1.1.1.1.1.1.3.cmml" xref="S6.Thmtheorem1.p3.1.1.m1.1.1.1.1.1.1.3"><plus id="S6.Thmtheorem1.p3.1.1.m1.1.1.1.1.1.1.3.1.cmml" xref="S6.Thmtheorem1.p3.1.1.m1.1.1.1.1.1.1.3.1"></plus><cn id="S6.Thmtheorem1.p3.1.1.m1.1.1.1.1.1.1.3.2.cmml" type="integer" xref="S6.Thmtheorem1.p3.1.1.m1.1.1.1.1.1.1.3.2">1</cn><ci id="S6.Thmtheorem1.p3.1.1.m1.1.1.1.1.1.1.3.3.cmml" xref="S6.Thmtheorem1.p3.1.1.m1.1.1.1.1.1.1.3.3">𝜂</ci></apply></apply><ci id="S6.Thmtheorem1.p3.1.1.m1.1.1.1.1.1.2.cmml" xref="S6.Thmtheorem1.p3.1.1.m1.1.1.1.1.1.2">𝑛</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.Thmtheorem1.p3.1.1.m1.1c">O(\log_{1+\eta}n)</annotation><annotation encoding="application/x-llamapun" id="S6.Thmtheorem1.p3.1.1.m1.1d">italic_O ( roman_log start_POSTSUBSCRIPT 1 + italic_η end_POSTSUBSCRIPT italic_n )</annotation></semantics></math>. Indeed any sequence of recursive calls is a path in <math alttext="G" class="ltx_Math" display="inline" id="S6.Thmtheorem1.p3.2.2.m2.1"><semantics id="S6.Thmtheorem1.p3.2.2.m2.1a"><mi id="S6.Thmtheorem1.p3.2.2.m2.1.1" xref="S6.Thmtheorem1.p3.2.2.m2.1.1.cmml">G</mi><annotation-xml encoding="MathML-Content" id="S6.Thmtheorem1.p3.2.2.m2.1b"><ci id="S6.Thmtheorem1.p3.2.2.m2.1.1.cmml" xref="S6.Thmtheorem1.p3.2.2.m2.1.1">𝐺</ci></annotation-xml><annotation encoding="application/x-tex" id="S6.Thmtheorem1.p3.2.2.m2.1c">G</annotation><annotation encoding="application/x-llamapun" id="S6.Thmtheorem1.p3.2.2.m2.1d">italic_G</annotation></semantics></math>. Denote the path belonging to the longest sequence of recursive calls by <math alttext="x_{0},x_{1},\ldots x_{\ell}" class="ltx_Math" display="inline" id="S6.Thmtheorem1.p3.3.3.m3.3"><semantics id="S6.Thmtheorem1.p3.3.3.m3.3a"><mrow id="S6.Thmtheorem1.p3.3.3.m3.3.3.3" xref="S6.Thmtheorem1.p3.3.3.m3.3.3.4.cmml"><msub id="S6.Thmtheorem1.p3.3.3.m3.1.1.1.1" xref="S6.Thmtheorem1.p3.3.3.m3.1.1.1.1.cmml"><mi id="S6.Thmtheorem1.p3.3.3.m3.1.1.1.1.2" xref="S6.Thmtheorem1.p3.3.3.m3.1.1.1.1.2.cmml">x</mi><mn id="S6.Thmtheorem1.p3.3.3.m3.1.1.1.1.3" xref="S6.Thmtheorem1.p3.3.3.m3.1.1.1.1.3.cmml">0</mn></msub><mo id="S6.Thmtheorem1.p3.3.3.m3.3.3.3.4" xref="S6.Thmtheorem1.p3.3.3.m3.3.3.4.cmml">,</mo><msub id="S6.Thmtheorem1.p3.3.3.m3.2.2.2.2" xref="S6.Thmtheorem1.p3.3.3.m3.2.2.2.2.cmml"><mi id="S6.Thmtheorem1.p3.3.3.m3.2.2.2.2.2" xref="S6.Thmtheorem1.p3.3.3.m3.2.2.2.2.2.cmml">x</mi><mn id="S6.Thmtheorem1.p3.3.3.m3.2.2.2.2.3" xref="S6.Thmtheorem1.p3.3.3.m3.2.2.2.2.3.cmml">1</mn></msub><mo id="S6.Thmtheorem1.p3.3.3.m3.3.3.3.5" xref="S6.Thmtheorem1.p3.3.3.m3.3.3.4.cmml">,</mo><mrow id="S6.Thmtheorem1.p3.3.3.m3.3.3.3.3" xref="S6.Thmtheorem1.p3.3.3.m3.3.3.3.3.cmml"><mi id="S6.Thmtheorem1.p3.3.3.m3.3.3.3.3.2" mathvariant="normal" xref="S6.Thmtheorem1.p3.3.3.m3.3.3.3.3.2.cmml">…</mi><mo id="S6.Thmtheorem1.p3.3.3.m3.3.3.3.3.1" xref="S6.Thmtheorem1.p3.3.3.m3.3.3.3.3.1.cmml">⁢</mo><msub id="S6.Thmtheorem1.p3.3.3.m3.3.3.3.3.3" xref="S6.Thmtheorem1.p3.3.3.m3.3.3.3.3.3.cmml"><mi id="S6.Thmtheorem1.p3.3.3.m3.3.3.3.3.3.2" xref="S6.Thmtheorem1.p3.3.3.m3.3.3.3.3.3.2.cmml">x</mi><mi id="S6.Thmtheorem1.p3.3.3.m3.3.3.3.3.3.3" mathvariant="normal" xref="S6.Thmtheorem1.p3.3.3.m3.3.3.3.3.3.3.cmml">ℓ</mi></msub></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.Thmtheorem1.p3.3.3.m3.3b"><list id="S6.Thmtheorem1.p3.3.3.m3.3.3.4.cmml" xref="S6.Thmtheorem1.p3.3.3.m3.3.3.3"><apply id="S6.Thmtheorem1.p3.3.3.m3.1.1.1.1.cmml" xref="S6.Thmtheorem1.p3.3.3.m3.1.1.1.1"><csymbol cd="ambiguous" id="S6.Thmtheorem1.p3.3.3.m3.1.1.1.1.1.cmml" xref="S6.Thmtheorem1.p3.3.3.m3.1.1.1.1">subscript</csymbol><ci id="S6.Thmtheorem1.p3.3.3.m3.1.1.1.1.2.cmml" xref="S6.Thmtheorem1.p3.3.3.m3.1.1.1.1.2">𝑥</ci><cn id="S6.Thmtheorem1.p3.3.3.m3.1.1.1.1.3.cmml" type="integer" xref="S6.Thmtheorem1.p3.3.3.m3.1.1.1.1.3">0</cn></apply><apply id="S6.Thmtheorem1.p3.3.3.m3.2.2.2.2.cmml" xref="S6.Thmtheorem1.p3.3.3.m3.2.2.2.2"><csymbol cd="ambiguous" id="S6.Thmtheorem1.p3.3.3.m3.2.2.2.2.1.cmml" xref="S6.Thmtheorem1.p3.3.3.m3.2.2.2.2">subscript</csymbol><ci id="S6.Thmtheorem1.p3.3.3.m3.2.2.2.2.2.cmml" xref="S6.Thmtheorem1.p3.3.3.m3.2.2.2.2.2">𝑥</ci><cn id="S6.Thmtheorem1.p3.3.3.m3.2.2.2.2.3.cmml" type="integer" xref="S6.Thmtheorem1.p3.3.3.m3.2.2.2.2.3">1</cn></apply><apply id="S6.Thmtheorem1.p3.3.3.m3.3.3.3.3.cmml" xref="S6.Thmtheorem1.p3.3.3.m3.3.3.3.3"><times id="S6.Thmtheorem1.p3.3.3.m3.3.3.3.3.1.cmml" xref="S6.Thmtheorem1.p3.3.3.m3.3.3.3.3.1"></times><ci id="S6.Thmtheorem1.p3.3.3.m3.3.3.3.3.2.cmml" xref="S6.Thmtheorem1.p3.3.3.m3.3.3.3.3.2">…</ci><apply id="S6.Thmtheorem1.p3.3.3.m3.3.3.3.3.3.cmml" xref="S6.Thmtheorem1.p3.3.3.m3.3.3.3.3.3"><csymbol cd="ambiguous" id="S6.Thmtheorem1.p3.3.3.m3.3.3.3.3.3.1.cmml" xref="S6.Thmtheorem1.p3.3.3.m3.3.3.3.3.3">subscript</csymbol><ci id="S6.Thmtheorem1.p3.3.3.m3.3.3.3.3.3.2.cmml" xref="S6.Thmtheorem1.p3.3.3.m3.3.3.3.3.3.2">𝑥</ci><ci id="S6.Thmtheorem1.p3.3.3.m3.3.3.3.3.3.3.cmml" xref="S6.Thmtheorem1.p3.3.3.m3.3.3.3.3.3.3">ℓ</ci></apply></apply></list></annotation-xml><annotation encoding="application/x-tex" id="S6.Thmtheorem1.p3.3.3.m3.3c">x_{0},x_{1},\ldots x_{\ell}</annotation><annotation encoding="application/x-llamapun" id="S6.Thmtheorem1.p3.3.3.m3.3d">italic_x start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , … italic_x start_POSTSUBSCRIPT roman_ℓ end_POSTSUBSCRIPT</annotation></semantics></math>. Since <math alttext="\Delta" class="ltx_Math" display="inline" id="S6.Thmtheorem1.p3.4.4.m4.1"><semantics id="S6.Thmtheorem1.p3.4.4.m4.1a"><mi id="S6.Thmtheorem1.p3.4.4.m4.1.1" mathvariant="normal" xref="S6.Thmtheorem1.p3.4.4.m4.1.1.cmml">Δ</mi><annotation-xml encoding="MathML-Content" id="S6.Thmtheorem1.p3.4.4.m4.1b"><ci id="S6.Thmtheorem1.p3.4.4.m4.1.1.cmml" xref="S6.Thmtheorem1.p3.4.4.m4.1.1">Δ</ci></annotation-xml><annotation encoding="application/x-tex" id="S6.Thmtheorem1.p3.4.4.m4.1c">\Delta</annotation><annotation encoding="application/x-llamapun" id="S6.Thmtheorem1.p3.4.4.m4.1d">roman_Δ</annotation></semantics></math> is always at most 1, it must hold for all <math alttext="i" class="ltx_Math" display="inline" id="S6.Thmtheorem1.p3.5.5.m5.1"><semantics id="S6.Thmtheorem1.p3.5.5.m5.1a"><mi id="S6.Thmtheorem1.p3.5.5.m5.1.1" xref="S6.Thmtheorem1.p3.5.5.m5.1.1.cmml">i</mi><annotation-xml encoding="MathML-Content" id="S6.Thmtheorem1.p3.5.5.m5.1b"><ci id="S6.Thmtheorem1.p3.5.5.m5.1.1.cmml" xref="S6.Thmtheorem1.p3.5.5.m5.1.1">𝑖</ci></annotation-xml><annotation encoding="application/x-tex" id="S6.Thmtheorem1.p3.5.5.m5.1c">i</annotation><annotation encoding="application/x-llamapun" id="S6.Thmtheorem1.p3.5.5.m5.1d">italic_i</annotation></semantics></math> that: <math alttext="\textsl{g}(x_{i})>(1+\eta)(\textsl{g}(x_{i-1})-1))" class="ltx_math_unparsed" display="inline" id="S6.Thmtheorem1.p3.6.6.m6.1"><semantics id="S6.Thmtheorem1.p3.6.6.m6.1a"><mrow id="S6.Thmtheorem1.p3.6.6.m6.1b"><mtext class="ltx_mathvariant_italic" id="S6.Thmtheorem1.p3.6.6.m6.1.1">g</mtext><mrow id="S6.Thmtheorem1.p3.6.6.m6.1.2"><mo id="S6.Thmtheorem1.p3.6.6.m6.1.2.1" stretchy="false">(</mo><msub id="S6.Thmtheorem1.p3.6.6.m6.1.2.2"><mi id="S6.Thmtheorem1.p3.6.6.m6.1.2.2.2">x</mi><mi id="S6.Thmtheorem1.p3.6.6.m6.1.2.2.3">i</mi></msub><mo id="S6.Thmtheorem1.p3.6.6.m6.1.2.3" stretchy="false">)</mo></mrow><mo id="S6.Thmtheorem1.p3.6.6.m6.1.3">></mo><mrow id="S6.Thmtheorem1.p3.6.6.m6.1.4"><mo id="S6.Thmtheorem1.p3.6.6.m6.1.4.1" stretchy="false">(</mo><mn id="S6.Thmtheorem1.p3.6.6.m6.1.4.2">1</mn><mo id="S6.Thmtheorem1.p3.6.6.m6.1.4.3">+</mo><mi id="S6.Thmtheorem1.p3.6.6.m6.1.4.4">η</mi><mo id="S6.Thmtheorem1.p3.6.6.m6.1.4.5" stretchy="false">)</mo></mrow><mrow id="S6.Thmtheorem1.p3.6.6.m6.1.5"><mo id="S6.Thmtheorem1.p3.6.6.m6.1.5.1" stretchy="false">(</mo><mtext class="ltx_mathvariant_italic" id="S6.Thmtheorem1.p3.6.6.m6.1.5.2">g</mtext><mrow id="S6.Thmtheorem1.p3.6.6.m6.1.5.3"><mo id="S6.Thmtheorem1.p3.6.6.m6.1.5.3.1" stretchy="false">(</mo><msub id="S6.Thmtheorem1.p3.6.6.m6.1.5.3.2"><mi id="S6.Thmtheorem1.p3.6.6.m6.1.5.3.2.2">x</mi><mrow id="S6.Thmtheorem1.p3.6.6.m6.1.5.3.2.3"><mi id="S6.Thmtheorem1.p3.6.6.m6.1.5.3.2.3.2">i</mi><mo id="S6.Thmtheorem1.p3.6.6.m6.1.5.3.2.3.1">−</mo><mn id="S6.Thmtheorem1.p3.6.6.m6.1.5.3.2.3.3">1</mn></mrow></msub><mo id="S6.Thmtheorem1.p3.6.6.m6.1.5.3.3" stretchy="false">)</mo></mrow><mo id="S6.Thmtheorem1.p3.6.6.m6.1.5.4">−</mo><mn id="S6.Thmtheorem1.p3.6.6.m6.1.5.5">1</mn><mo id="S6.Thmtheorem1.p3.6.6.m6.1.5.6" stretchy="false">)</mo></mrow><mo id="S6.Thmtheorem1.p3.6.6.m6.1.6" stretchy="false">)</mo></mrow><annotation encoding="application/x-tex" id="S6.Thmtheorem1.p3.6.6.m6.1c">\textsl{g}(x_{i})>(1+\eta)(\textsl{g}(x_{i-1})-1))</annotation><annotation encoding="application/x-llamapun" id="S6.Thmtheorem1.p3.6.6.m6.1d">g ( italic_x start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ) > ( 1 + italic_η ) ( g ( italic_x start_POSTSUBSCRIPT italic_i - 1 end_POSTSUBSCRIPT ) - 1 ) )</annotation></semantics></math>. Since a graph may have at most <math alttext="n^{2}" class="ltx_Math" display="inline" id="S6.Thmtheorem1.p3.7.7.m7.1"><semantics id="S6.Thmtheorem1.p3.7.7.m7.1a"><msup id="S6.Thmtheorem1.p3.7.7.m7.1.1" xref="S6.Thmtheorem1.p3.7.7.m7.1.1.cmml"><mi id="S6.Thmtheorem1.p3.7.7.m7.1.1.2" xref="S6.Thmtheorem1.p3.7.7.m7.1.1.2.cmml">n</mi><mn id="S6.Thmtheorem1.p3.7.7.m7.1.1.3" xref="S6.Thmtheorem1.p3.7.7.m7.1.1.3.cmml">2</mn></msup><annotation-xml encoding="MathML-Content" id="S6.Thmtheorem1.p3.7.7.m7.1b"><apply id="S6.Thmtheorem1.p3.7.7.m7.1.1.cmml" xref="S6.Thmtheorem1.p3.7.7.m7.1.1"><csymbol cd="ambiguous" id="S6.Thmtheorem1.p3.7.7.m7.1.1.1.cmml" xref="S6.Thmtheorem1.p3.7.7.m7.1.1">superscript</csymbol><ci id="S6.Thmtheorem1.p3.7.7.m7.1.1.2.cmml" xref="S6.Thmtheorem1.p3.7.7.m7.1.1.2">𝑛</ci><cn id="S6.Thmtheorem1.p3.7.7.m7.1.1.3.cmml" type="integer" xref="S6.Thmtheorem1.p3.7.7.m7.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.Thmtheorem1.p3.7.7.m7.1c">n^{2}</annotation><annotation encoding="application/x-llamapun" id="S6.Thmtheorem1.p3.7.7.m7.1d">italic_n start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT</annotation></semantics></math> edges, <math alttext="\textsl{g}(x_{\ell})\leq n" class="ltx_Math" display="inline" id="S6.Thmtheorem1.p3.8.8.m8.1"><semantics id="S6.Thmtheorem1.p3.8.8.m8.1a"><mrow id="S6.Thmtheorem1.p3.8.8.m8.1.1" xref="S6.Thmtheorem1.p3.8.8.m8.1.1.cmml"><mrow id="S6.Thmtheorem1.p3.8.8.m8.1.1.1" xref="S6.Thmtheorem1.p3.8.8.m8.1.1.1.cmml"><mtext class="ltx_mathvariant_italic" id="S6.Thmtheorem1.p3.8.8.m8.1.1.1.3" xref="S6.Thmtheorem1.p3.8.8.m8.1.1.1.3a.cmml">g</mtext><mo id="S6.Thmtheorem1.p3.8.8.m8.1.1.1.2" xref="S6.Thmtheorem1.p3.8.8.m8.1.1.1.2.cmml">⁢</mo><mrow id="S6.Thmtheorem1.p3.8.8.m8.1.1.1.1.1" xref="S6.Thmtheorem1.p3.8.8.m8.1.1.1.1.1.1.cmml"><mo id="S6.Thmtheorem1.p3.8.8.m8.1.1.1.1.1.2" stretchy="false" xref="S6.Thmtheorem1.p3.8.8.m8.1.1.1.1.1.1.cmml">(</mo><msub id="S6.Thmtheorem1.p3.8.8.m8.1.1.1.1.1.1" xref="S6.Thmtheorem1.p3.8.8.m8.1.1.1.1.1.1.cmml"><mi id="S6.Thmtheorem1.p3.8.8.m8.1.1.1.1.1.1.2" xref="S6.Thmtheorem1.p3.8.8.m8.1.1.1.1.1.1.2.cmml">x</mi><mi id="S6.Thmtheorem1.p3.8.8.m8.1.1.1.1.1.1.3" mathvariant="normal" xref="S6.Thmtheorem1.p3.8.8.m8.1.1.1.1.1.1.3.cmml">ℓ</mi></msub><mo id="S6.Thmtheorem1.p3.8.8.m8.1.1.1.1.1.3" stretchy="false" xref="S6.Thmtheorem1.p3.8.8.m8.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S6.Thmtheorem1.p3.8.8.m8.1.1.2" xref="S6.Thmtheorem1.p3.8.8.m8.1.1.2.cmml">≤</mo><mi id="S6.Thmtheorem1.p3.8.8.m8.1.1.3" xref="S6.Thmtheorem1.p3.8.8.m8.1.1.3.cmml">n</mi></mrow><annotation-xml encoding="MathML-Content" id="S6.Thmtheorem1.p3.8.8.m8.1b"><apply id="S6.Thmtheorem1.p3.8.8.m8.1.1.cmml" xref="S6.Thmtheorem1.p3.8.8.m8.1.1"><leq id="S6.Thmtheorem1.p3.8.8.m8.1.1.2.cmml" xref="S6.Thmtheorem1.p3.8.8.m8.1.1.2"></leq><apply id="S6.Thmtheorem1.p3.8.8.m8.1.1.1.cmml" xref="S6.Thmtheorem1.p3.8.8.m8.1.1.1"><times id="S6.Thmtheorem1.p3.8.8.m8.1.1.1.2.cmml" xref="S6.Thmtheorem1.p3.8.8.m8.1.1.1.2"></times><ci id="S6.Thmtheorem1.p3.8.8.m8.1.1.1.3a.cmml" xref="S6.Thmtheorem1.p3.8.8.m8.1.1.1.3"><mtext class="ltx_mathvariant_italic" id="S6.Thmtheorem1.p3.8.8.m8.1.1.1.3.cmml" xref="S6.Thmtheorem1.p3.8.8.m8.1.1.1.3">g</mtext></ci><apply id="S6.Thmtheorem1.p3.8.8.m8.1.1.1.1.1.1.cmml" xref="S6.Thmtheorem1.p3.8.8.m8.1.1.1.1.1"><csymbol cd="ambiguous" id="S6.Thmtheorem1.p3.8.8.m8.1.1.1.1.1.1.1.cmml" xref="S6.Thmtheorem1.p3.8.8.m8.1.1.1.1.1">subscript</csymbol><ci id="S6.Thmtheorem1.p3.8.8.m8.1.1.1.1.1.1.2.cmml" xref="S6.Thmtheorem1.p3.8.8.m8.1.1.1.1.1.1.2">𝑥</ci><ci id="S6.Thmtheorem1.p3.8.8.m8.1.1.1.1.1.1.3.cmml" xref="S6.Thmtheorem1.p3.8.8.m8.1.1.1.1.1.1.3">ℓ</ci></apply></apply><ci id="S6.Thmtheorem1.p3.8.8.m8.1.1.3.cmml" xref="S6.Thmtheorem1.p3.8.8.m8.1.1.3">𝑛</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.Thmtheorem1.p3.8.8.m8.1c">\textsl{g}(x_{\ell})\leq n</annotation><annotation encoding="application/x-llamapun" id="S6.Thmtheorem1.p3.8.8.m8.1d">g ( italic_x start_POSTSUBSCRIPT roman_ℓ end_POSTSUBSCRIPT ) ≤ italic_n</annotation></semantics></math> and it follows that the recursive depth is at most <math alttext="\ell\in O(\log_{1+\eta}n)" class="ltx_Math" display="inline" id="S6.Thmtheorem1.p3.9.9.m9.1"><semantics id="S6.Thmtheorem1.p3.9.9.m9.1a"><mrow id="S6.Thmtheorem1.p3.9.9.m9.1.1" xref="S6.Thmtheorem1.p3.9.9.m9.1.1.cmml"><mi id="S6.Thmtheorem1.p3.9.9.m9.1.1.3" mathvariant="normal" xref="S6.Thmtheorem1.p3.9.9.m9.1.1.3.cmml">ℓ</mi><mo id="S6.Thmtheorem1.p3.9.9.m9.1.1.2" xref="S6.Thmtheorem1.p3.9.9.m9.1.1.2.cmml">∈</mo><mrow id="S6.Thmtheorem1.p3.9.9.m9.1.1.1" xref="S6.Thmtheorem1.p3.9.9.m9.1.1.1.cmml"><mi id="S6.Thmtheorem1.p3.9.9.m9.1.1.1.3" xref="S6.Thmtheorem1.p3.9.9.m9.1.1.1.3.cmml">O</mi><mo id="S6.Thmtheorem1.p3.9.9.m9.1.1.1.2" xref="S6.Thmtheorem1.p3.9.9.m9.1.1.1.2.cmml">⁢</mo><mrow id="S6.Thmtheorem1.p3.9.9.m9.1.1.1.1.1" xref="S6.Thmtheorem1.p3.9.9.m9.1.1.1.1.1.1.cmml"><mo id="S6.Thmtheorem1.p3.9.9.m9.1.1.1.1.1.2" stretchy="false" xref="S6.Thmtheorem1.p3.9.9.m9.1.1.1.1.1.1.cmml">(</mo><mrow id="S6.Thmtheorem1.p3.9.9.m9.1.1.1.1.1.1" xref="S6.Thmtheorem1.p3.9.9.m9.1.1.1.1.1.1.cmml"><msub id="S6.Thmtheorem1.p3.9.9.m9.1.1.1.1.1.1.1" xref="S6.Thmtheorem1.p3.9.9.m9.1.1.1.1.1.1.1.cmml"><mi id="S6.Thmtheorem1.p3.9.9.m9.1.1.1.1.1.1.1.2" xref="S6.Thmtheorem1.p3.9.9.m9.1.1.1.1.1.1.1.2.cmml">log</mi><mrow id="S6.Thmtheorem1.p3.9.9.m9.1.1.1.1.1.1.1.3" xref="S6.Thmtheorem1.p3.9.9.m9.1.1.1.1.1.1.1.3.cmml"><mn id="S6.Thmtheorem1.p3.9.9.m9.1.1.1.1.1.1.1.3.2" xref="S6.Thmtheorem1.p3.9.9.m9.1.1.1.1.1.1.1.3.2.cmml">1</mn><mo id="S6.Thmtheorem1.p3.9.9.m9.1.1.1.1.1.1.1.3.1" xref="S6.Thmtheorem1.p3.9.9.m9.1.1.1.1.1.1.1.3.1.cmml">+</mo><mi id="S6.Thmtheorem1.p3.9.9.m9.1.1.1.1.1.1.1.3.3" xref="S6.Thmtheorem1.p3.9.9.m9.1.1.1.1.1.1.1.3.3.cmml">η</mi></mrow></msub><mo id="S6.Thmtheorem1.p3.9.9.m9.1.1.1.1.1.1a" lspace="0.167em" xref="S6.Thmtheorem1.p3.9.9.m9.1.1.1.1.1.1.cmml">⁡</mo><mi id="S6.Thmtheorem1.p3.9.9.m9.1.1.1.1.1.1.2" xref="S6.Thmtheorem1.p3.9.9.m9.1.1.1.1.1.1.2.cmml">n</mi></mrow><mo id="S6.Thmtheorem1.p3.9.9.m9.1.1.1.1.1.3" stretchy="false" xref="S6.Thmtheorem1.p3.9.9.m9.1.1.1.1.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.Thmtheorem1.p3.9.9.m9.1b"><apply id="S6.Thmtheorem1.p3.9.9.m9.1.1.cmml" xref="S6.Thmtheorem1.p3.9.9.m9.1.1"><in id="S6.Thmtheorem1.p3.9.9.m9.1.1.2.cmml" xref="S6.Thmtheorem1.p3.9.9.m9.1.1.2"></in><ci id="S6.Thmtheorem1.p3.9.9.m9.1.1.3.cmml" xref="S6.Thmtheorem1.p3.9.9.m9.1.1.3">ℓ</ci><apply id="S6.Thmtheorem1.p3.9.9.m9.1.1.1.cmml" xref="S6.Thmtheorem1.p3.9.9.m9.1.1.1"><times id="S6.Thmtheorem1.p3.9.9.m9.1.1.1.2.cmml" xref="S6.Thmtheorem1.p3.9.9.m9.1.1.1.2"></times><ci id="S6.Thmtheorem1.p3.9.9.m9.1.1.1.3.cmml" xref="S6.Thmtheorem1.p3.9.9.m9.1.1.1.3">𝑂</ci><apply id="S6.Thmtheorem1.p3.9.9.m9.1.1.1.1.1.1.cmml" xref="S6.Thmtheorem1.p3.9.9.m9.1.1.1.1.1"><apply id="S6.Thmtheorem1.p3.9.9.m9.1.1.1.1.1.1.1.cmml" xref="S6.Thmtheorem1.p3.9.9.m9.1.1.1.1.1.1.1"><csymbol cd="ambiguous" id="S6.Thmtheorem1.p3.9.9.m9.1.1.1.1.1.1.1.1.cmml" xref="S6.Thmtheorem1.p3.9.9.m9.1.1.1.1.1.1.1">subscript</csymbol><log id="S6.Thmtheorem1.p3.9.9.m9.1.1.1.1.1.1.1.2.cmml" xref="S6.Thmtheorem1.p3.9.9.m9.1.1.1.1.1.1.1.2"></log><apply id="S6.Thmtheorem1.p3.9.9.m9.1.1.1.1.1.1.1.3.cmml" xref="S6.Thmtheorem1.p3.9.9.m9.1.1.1.1.1.1.1.3"><plus id="S6.Thmtheorem1.p3.9.9.m9.1.1.1.1.1.1.1.3.1.cmml" xref="S6.Thmtheorem1.p3.9.9.m9.1.1.1.1.1.1.1.3.1"></plus><cn id="S6.Thmtheorem1.p3.9.9.m9.1.1.1.1.1.1.1.3.2.cmml" type="integer" xref="S6.Thmtheorem1.p3.9.9.m9.1.1.1.1.1.1.1.3.2">1</cn><ci id="S6.Thmtheorem1.p3.9.9.m9.1.1.1.1.1.1.1.3.3.cmml" xref="S6.Thmtheorem1.p3.9.9.m9.1.1.1.1.1.1.1.3.3">𝜂</ci></apply></apply><ci id="S6.Thmtheorem1.p3.9.9.m9.1.1.1.1.1.1.2.cmml" xref="S6.Thmtheorem1.p3.9.9.m9.1.1.1.1.1.1.2">𝑛</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.Thmtheorem1.p3.9.9.m9.1c">\ell\in O(\log_{1+\eta}n)</annotation><annotation encoding="application/x-llamapun" id="S6.Thmtheorem1.p3.9.9.m9.1d">roman_ℓ ∈ italic_O ( roman_log start_POSTSUBSCRIPT 1 + italic_η end_POSTSUBSCRIPT italic_n )</annotation></semantics></math>. We now apply <math alttext="\log(1+x)\geq x/2" class="ltx_Math" display="inline" id="S6.Thmtheorem1.p3.10.10.m10.2"><semantics id="S6.Thmtheorem1.p3.10.10.m10.2a"><mrow id="S6.Thmtheorem1.p3.10.10.m10.2.2" xref="S6.Thmtheorem1.p3.10.10.m10.2.2.cmml"><mrow id="S6.Thmtheorem1.p3.10.10.m10.2.2.1.1" xref="S6.Thmtheorem1.p3.10.10.m10.2.2.1.2.cmml"><mi id="S6.Thmtheorem1.p3.10.10.m10.1.1" xref="S6.Thmtheorem1.p3.10.10.m10.1.1.cmml">log</mi><mo id="S6.Thmtheorem1.p3.10.10.m10.2.2.1.1a" xref="S6.Thmtheorem1.p3.10.10.m10.2.2.1.2.cmml">⁡</mo><mrow id="S6.Thmtheorem1.p3.10.10.m10.2.2.1.1.1" xref="S6.Thmtheorem1.p3.10.10.m10.2.2.1.2.cmml"><mo id="S6.Thmtheorem1.p3.10.10.m10.2.2.1.1.1.2" stretchy="false" xref="S6.Thmtheorem1.p3.10.10.m10.2.2.1.2.cmml">(</mo><mrow id="S6.Thmtheorem1.p3.10.10.m10.2.2.1.1.1.1" xref="S6.Thmtheorem1.p3.10.10.m10.2.2.1.1.1.1.cmml"><mn id="S6.Thmtheorem1.p3.10.10.m10.2.2.1.1.1.1.2" xref="S6.Thmtheorem1.p3.10.10.m10.2.2.1.1.1.1.2.cmml">1</mn><mo id="S6.Thmtheorem1.p3.10.10.m10.2.2.1.1.1.1.1" xref="S6.Thmtheorem1.p3.10.10.m10.2.2.1.1.1.1.1.cmml">+</mo><mi id="S6.Thmtheorem1.p3.10.10.m10.2.2.1.1.1.1.3" xref="S6.Thmtheorem1.p3.10.10.m10.2.2.1.1.1.1.3.cmml">x</mi></mrow><mo id="S6.Thmtheorem1.p3.10.10.m10.2.2.1.1.1.3" stretchy="false" xref="S6.Thmtheorem1.p3.10.10.m10.2.2.1.2.cmml">)</mo></mrow></mrow><mo id="S6.Thmtheorem1.p3.10.10.m10.2.2.2" xref="S6.Thmtheorem1.p3.10.10.m10.2.2.2.cmml">≥</mo><mrow id="S6.Thmtheorem1.p3.10.10.m10.2.2.3" xref="S6.Thmtheorem1.p3.10.10.m10.2.2.3.cmml"><mi id="S6.Thmtheorem1.p3.10.10.m10.2.2.3.2" xref="S6.Thmtheorem1.p3.10.10.m10.2.2.3.2.cmml">x</mi><mo id="S6.Thmtheorem1.p3.10.10.m10.2.2.3.1" xref="S6.Thmtheorem1.p3.10.10.m10.2.2.3.1.cmml">/</mo><mn id="S6.Thmtheorem1.p3.10.10.m10.2.2.3.3" xref="S6.Thmtheorem1.p3.10.10.m10.2.2.3.3.cmml">2</mn></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.Thmtheorem1.p3.10.10.m10.2b"><apply id="S6.Thmtheorem1.p3.10.10.m10.2.2.cmml" xref="S6.Thmtheorem1.p3.10.10.m10.2.2"><geq id="S6.Thmtheorem1.p3.10.10.m10.2.2.2.cmml" xref="S6.Thmtheorem1.p3.10.10.m10.2.2.2"></geq><apply id="S6.Thmtheorem1.p3.10.10.m10.2.2.1.2.cmml" xref="S6.Thmtheorem1.p3.10.10.m10.2.2.1.1"><log id="S6.Thmtheorem1.p3.10.10.m10.1.1.cmml" xref="S6.Thmtheorem1.p3.10.10.m10.1.1"></log><apply id="S6.Thmtheorem1.p3.10.10.m10.2.2.1.1.1.1.cmml" xref="S6.Thmtheorem1.p3.10.10.m10.2.2.1.1.1.1"><plus id="S6.Thmtheorem1.p3.10.10.m10.2.2.1.1.1.1.1.cmml" xref="S6.Thmtheorem1.p3.10.10.m10.2.2.1.1.1.1.1"></plus><cn id="S6.Thmtheorem1.p3.10.10.m10.2.2.1.1.1.1.2.cmml" type="integer" xref="S6.Thmtheorem1.p3.10.10.m10.2.2.1.1.1.1.2">1</cn><ci id="S6.Thmtheorem1.p3.10.10.m10.2.2.1.1.1.1.3.cmml" xref="S6.Thmtheorem1.p3.10.10.m10.2.2.1.1.1.1.3">𝑥</ci></apply></apply><apply id="S6.Thmtheorem1.p3.10.10.m10.2.2.3.cmml" xref="S6.Thmtheorem1.p3.10.10.m10.2.2.3"><divide id="S6.Thmtheorem1.p3.10.10.m10.2.2.3.1.cmml" xref="S6.Thmtheorem1.p3.10.10.m10.2.2.3.1"></divide><ci id="S6.Thmtheorem1.p3.10.10.m10.2.2.3.2.cmml" xref="S6.Thmtheorem1.p3.10.10.m10.2.2.3.2">𝑥</ci><cn id="S6.Thmtheorem1.p3.10.10.m10.2.2.3.3.cmml" type="integer" xref="S6.Thmtheorem1.p3.10.10.m10.2.2.3.3">2</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.Thmtheorem1.p3.10.10.m10.2c">\log(1+x)\geq x/2</annotation><annotation encoding="application/x-llamapun" id="S6.Thmtheorem1.p3.10.10.m10.2d">roman_log ( 1 + italic_x ) ≥ italic_x / 2</annotation></semantics></math> and note that: <math alttext="\ell\leq\frac{\log n}{\log(1+\eta)}\leq\frac{\log n}{\eta/2}\subseteq O(\frac{% \log n}{64\varepsilon^{2}/\log n})\subseteq O(\frac{\log^{2}n}{\varepsilon^{2}})" class="ltx_Math" display="inline" id="S6.Thmtheorem1.p3.11.11.m11.4"><semantics id="S6.Thmtheorem1.p3.11.11.m11.4a"><mrow id="S6.Thmtheorem1.p3.11.11.m11.4.5" xref="S6.Thmtheorem1.p3.11.11.m11.4.5.cmml"><mi id="S6.Thmtheorem1.p3.11.11.m11.4.5.2" mathvariant="normal" xref="S6.Thmtheorem1.p3.11.11.m11.4.5.2.cmml">ℓ</mi><mo id="S6.Thmtheorem1.p3.11.11.m11.4.5.3" xref="S6.Thmtheorem1.p3.11.11.m11.4.5.3.cmml">≤</mo><mfrac id="S6.Thmtheorem1.p3.11.11.m11.2.2" xref="S6.Thmtheorem1.p3.11.11.m11.2.2.cmml"><mrow id="S6.Thmtheorem1.p3.11.11.m11.2.2.4" xref="S6.Thmtheorem1.p3.11.11.m11.2.2.4.cmml"><mi id="S6.Thmtheorem1.p3.11.11.m11.2.2.4.1" xref="S6.Thmtheorem1.p3.11.11.m11.2.2.4.1.cmml">log</mi><mo id="S6.Thmtheorem1.p3.11.11.m11.2.2.4a" lspace="0.167em" xref="S6.Thmtheorem1.p3.11.11.m11.2.2.4.cmml">⁡</mo><mi id="S6.Thmtheorem1.p3.11.11.m11.2.2.4.2" xref="S6.Thmtheorem1.p3.11.11.m11.2.2.4.2.cmml">n</mi></mrow><mrow id="S6.Thmtheorem1.p3.11.11.m11.2.2.2.2" xref="S6.Thmtheorem1.p3.11.11.m11.2.2.2.3.cmml"><mi id="S6.Thmtheorem1.p3.11.11.m11.1.1.1.1" xref="S6.Thmtheorem1.p3.11.11.m11.1.1.1.1.cmml">log</mi><mo id="S6.Thmtheorem1.p3.11.11.m11.2.2.2.2a" xref="S6.Thmtheorem1.p3.11.11.m11.2.2.2.3.cmml">⁡</mo><mrow id="S6.Thmtheorem1.p3.11.11.m11.2.2.2.2.1" xref="S6.Thmtheorem1.p3.11.11.m11.2.2.2.3.cmml"><mo id="S6.Thmtheorem1.p3.11.11.m11.2.2.2.2.1.2" stretchy="false" xref="S6.Thmtheorem1.p3.11.11.m11.2.2.2.3.cmml">(</mo><mrow id="S6.Thmtheorem1.p3.11.11.m11.2.2.2.2.1.1" xref="S6.Thmtheorem1.p3.11.11.m11.2.2.2.2.1.1.cmml"><mn id="S6.Thmtheorem1.p3.11.11.m11.2.2.2.2.1.1.2" xref="S6.Thmtheorem1.p3.11.11.m11.2.2.2.2.1.1.2.cmml">1</mn><mo id="S6.Thmtheorem1.p3.11.11.m11.2.2.2.2.1.1.1" xref="S6.Thmtheorem1.p3.11.11.m11.2.2.2.2.1.1.1.cmml">+</mo><mi id="S6.Thmtheorem1.p3.11.11.m11.2.2.2.2.1.1.3" xref="S6.Thmtheorem1.p3.11.11.m11.2.2.2.2.1.1.3.cmml">η</mi></mrow><mo id="S6.Thmtheorem1.p3.11.11.m11.2.2.2.2.1.3" stretchy="false" xref="S6.Thmtheorem1.p3.11.11.m11.2.2.2.3.cmml">)</mo></mrow></mrow></mfrac><mo id="S6.Thmtheorem1.p3.11.11.m11.4.5.4" xref="S6.Thmtheorem1.p3.11.11.m11.4.5.4.cmml">≤</mo><mfrac id="S6.Thmtheorem1.p3.11.11.m11.4.5.5" xref="S6.Thmtheorem1.p3.11.11.m11.4.5.5.cmml"><mrow id="S6.Thmtheorem1.p3.11.11.m11.4.5.5.2" xref="S6.Thmtheorem1.p3.11.11.m11.4.5.5.2.cmml"><mi id="S6.Thmtheorem1.p3.11.11.m11.4.5.5.2.1" xref="S6.Thmtheorem1.p3.11.11.m11.4.5.5.2.1.cmml">log</mi><mo id="S6.Thmtheorem1.p3.11.11.m11.4.5.5.2a" lspace="0.167em" xref="S6.Thmtheorem1.p3.11.11.m11.4.5.5.2.cmml">⁡</mo><mi id="S6.Thmtheorem1.p3.11.11.m11.4.5.5.2.2" xref="S6.Thmtheorem1.p3.11.11.m11.4.5.5.2.2.cmml">n</mi></mrow><mrow id="S6.Thmtheorem1.p3.11.11.m11.4.5.5.3" xref="S6.Thmtheorem1.p3.11.11.m11.4.5.5.3.cmml"><mi id="S6.Thmtheorem1.p3.11.11.m11.4.5.5.3.2" xref="S6.Thmtheorem1.p3.11.11.m11.4.5.5.3.2.cmml">η</mi><mo id="S6.Thmtheorem1.p3.11.11.m11.4.5.5.3.1" xref="S6.Thmtheorem1.p3.11.11.m11.4.5.5.3.1.cmml">/</mo><mn id="S6.Thmtheorem1.p3.11.11.m11.4.5.5.3.3" xref="S6.Thmtheorem1.p3.11.11.m11.4.5.5.3.3.cmml">2</mn></mrow></mfrac><mo id="S6.Thmtheorem1.p3.11.11.m11.4.5.6" xref="S6.Thmtheorem1.p3.11.11.m11.4.5.6.cmml">⊆</mo><mrow id="S6.Thmtheorem1.p3.11.11.m11.4.5.7" xref="S6.Thmtheorem1.p3.11.11.m11.4.5.7.cmml"><mi 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id="S6.Thmtheorem1.p3.11.11.m11.3.3.3.2" xref="S6.Thmtheorem1.p3.11.11.m11.3.3.3.2.cmml"><mn id="S6.Thmtheorem1.p3.11.11.m11.3.3.3.2.2" xref="S6.Thmtheorem1.p3.11.11.m11.3.3.3.2.2.cmml">64</mn><mo id="S6.Thmtheorem1.p3.11.11.m11.3.3.3.2.1" xref="S6.Thmtheorem1.p3.11.11.m11.3.3.3.2.1.cmml">⁢</mo><msup id="S6.Thmtheorem1.p3.11.11.m11.3.3.3.2.3" xref="S6.Thmtheorem1.p3.11.11.m11.3.3.3.2.3.cmml"><mi id="S6.Thmtheorem1.p3.11.11.m11.3.3.3.2.3.2" xref="S6.Thmtheorem1.p3.11.11.m11.3.3.3.2.3.2.cmml">ε</mi><mn id="S6.Thmtheorem1.p3.11.11.m11.3.3.3.2.3.3" xref="S6.Thmtheorem1.p3.11.11.m11.3.3.3.2.3.3.cmml">2</mn></msup></mrow><mo id="S6.Thmtheorem1.p3.11.11.m11.3.3.3.1" xref="S6.Thmtheorem1.p3.11.11.m11.3.3.3.1.cmml">/</mo><mrow id="S6.Thmtheorem1.p3.11.11.m11.3.3.3.3" xref="S6.Thmtheorem1.p3.11.11.m11.3.3.3.3.cmml"><mi id="S6.Thmtheorem1.p3.11.11.m11.3.3.3.3.1" xref="S6.Thmtheorem1.p3.11.11.m11.3.3.3.3.1.cmml">log</mi><mo id="S6.Thmtheorem1.p3.11.11.m11.3.3.3.3a" lspace="0.167em" 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id="S6.Thmtheorem1.p3.11.11.m11.4.4.2" xref="S6.Thmtheorem1.p3.11.11.m11.4.4.2.cmml"><msup id="S6.Thmtheorem1.p3.11.11.m11.4.4.2.1" xref="S6.Thmtheorem1.p3.11.11.m11.4.4.2.1.cmml"><mi id="S6.Thmtheorem1.p3.11.11.m11.4.4.2.1.2" xref="S6.Thmtheorem1.p3.11.11.m11.4.4.2.1.2.cmml">log</mi><mn id="S6.Thmtheorem1.p3.11.11.m11.4.4.2.1.3" xref="S6.Thmtheorem1.p3.11.11.m11.4.4.2.1.3.cmml">2</mn></msup><mo id="S6.Thmtheorem1.p3.11.11.m11.4.4.2a" lspace="0.167em" xref="S6.Thmtheorem1.p3.11.11.m11.4.4.2.cmml">⁡</mo><mi id="S6.Thmtheorem1.p3.11.11.m11.4.4.2.2" xref="S6.Thmtheorem1.p3.11.11.m11.4.4.2.2.cmml">n</mi></mrow><msup id="S6.Thmtheorem1.p3.11.11.m11.4.4.3" xref="S6.Thmtheorem1.p3.11.11.m11.4.4.3.cmml"><mi id="S6.Thmtheorem1.p3.11.11.m11.4.4.3.2" xref="S6.Thmtheorem1.p3.11.11.m11.4.4.3.2.cmml">ε</mi><mn id="S6.Thmtheorem1.p3.11.11.m11.4.4.3.3" xref="S6.Thmtheorem1.p3.11.11.m11.4.4.3.3.cmml">2</mn></msup></mfrac><mo id="S6.Thmtheorem1.p3.11.11.m11.4.5.9.3.2.2" stretchy="false" xref="S6.Thmtheorem1.p3.11.11.m11.4.4.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.Thmtheorem1.p3.11.11.m11.4b"><apply id="S6.Thmtheorem1.p3.11.11.m11.4.5.cmml" xref="S6.Thmtheorem1.p3.11.11.m11.4.5"><and id="S6.Thmtheorem1.p3.11.11.m11.4.5a.cmml" xref="S6.Thmtheorem1.p3.11.11.m11.4.5"></and><apply id="S6.Thmtheorem1.p3.11.11.m11.4.5b.cmml" xref="S6.Thmtheorem1.p3.11.11.m11.4.5"><leq id="S6.Thmtheorem1.p3.11.11.m11.4.5.3.cmml" xref="S6.Thmtheorem1.p3.11.11.m11.4.5.3"></leq><ci id="S6.Thmtheorem1.p3.11.11.m11.4.5.2.cmml" xref="S6.Thmtheorem1.p3.11.11.m11.4.5.2">ℓ</ci><apply id="S6.Thmtheorem1.p3.11.11.m11.2.2.cmml" xref="S6.Thmtheorem1.p3.11.11.m11.2.2"><divide id="S6.Thmtheorem1.p3.11.11.m11.2.2.3.cmml" xref="S6.Thmtheorem1.p3.11.11.m11.2.2"></divide><apply id="S6.Thmtheorem1.p3.11.11.m11.2.2.4.cmml" xref="S6.Thmtheorem1.p3.11.11.m11.2.2.4"><log id="S6.Thmtheorem1.p3.11.11.m11.2.2.4.1.cmml" xref="S6.Thmtheorem1.p3.11.11.m11.2.2.4.1"></log><ci id="S6.Thmtheorem1.p3.11.11.m11.2.2.4.2.cmml" xref="S6.Thmtheorem1.p3.11.11.m11.2.2.4.2">𝑛</ci></apply><apply id="S6.Thmtheorem1.p3.11.11.m11.2.2.2.3.cmml" xref="S6.Thmtheorem1.p3.11.11.m11.2.2.2.2"><log id="S6.Thmtheorem1.p3.11.11.m11.1.1.1.1.cmml" xref="S6.Thmtheorem1.p3.11.11.m11.1.1.1.1"></log><apply id="S6.Thmtheorem1.p3.11.11.m11.2.2.2.2.1.1.cmml" xref="S6.Thmtheorem1.p3.11.11.m11.2.2.2.2.1.1"><plus id="S6.Thmtheorem1.p3.11.11.m11.2.2.2.2.1.1.1.cmml" xref="S6.Thmtheorem1.p3.11.11.m11.2.2.2.2.1.1.1"></plus><cn id="S6.Thmtheorem1.p3.11.11.m11.2.2.2.2.1.1.2.cmml" type="integer" xref="S6.Thmtheorem1.p3.11.11.m11.2.2.2.2.1.1.2">1</cn><ci id="S6.Thmtheorem1.p3.11.11.m11.2.2.2.2.1.1.3.cmml" xref="S6.Thmtheorem1.p3.11.11.m11.2.2.2.2.1.1.3">𝜂</ci></apply></apply></apply></apply><apply id="S6.Thmtheorem1.p3.11.11.m11.4.5c.cmml" xref="S6.Thmtheorem1.p3.11.11.m11.4.5"><leq id="S6.Thmtheorem1.p3.11.11.m11.4.5.4.cmml" xref="S6.Thmtheorem1.p3.11.11.m11.4.5.4"></leq><share href="https://arxiv.org/html/2411.12694v2#S6.Thmtheorem1.p3.11.11.m11.2.2.cmml" id="S6.Thmtheorem1.p3.11.11.m11.4.5d.cmml" xref="S6.Thmtheorem1.p3.11.11.m11.4.5"></share><apply id="S6.Thmtheorem1.p3.11.11.m11.4.5.5.cmml" xref="S6.Thmtheorem1.p3.11.11.m11.4.5.5"><divide id="S6.Thmtheorem1.p3.11.11.m11.4.5.5.1.cmml" xref="S6.Thmtheorem1.p3.11.11.m11.4.5.5"></divide><apply id="S6.Thmtheorem1.p3.11.11.m11.4.5.5.2.cmml" xref="S6.Thmtheorem1.p3.11.11.m11.4.5.5.2"><log id="S6.Thmtheorem1.p3.11.11.m11.4.5.5.2.1.cmml" xref="S6.Thmtheorem1.p3.11.11.m11.4.5.5.2.1"></log><ci id="S6.Thmtheorem1.p3.11.11.m11.4.5.5.2.2.cmml" xref="S6.Thmtheorem1.p3.11.11.m11.4.5.5.2.2">𝑛</ci></apply><apply id="S6.Thmtheorem1.p3.11.11.m11.4.5.5.3.cmml" xref="S6.Thmtheorem1.p3.11.11.m11.4.5.5.3"><divide id="S6.Thmtheorem1.p3.11.11.m11.4.5.5.3.1.cmml" xref="S6.Thmtheorem1.p3.11.11.m11.4.5.5.3.1"></divide><ci id="S6.Thmtheorem1.p3.11.11.m11.4.5.5.3.2.cmml" xref="S6.Thmtheorem1.p3.11.11.m11.4.5.5.3.2">𝜂</ci><cn id="S6.Thmtheorem1.p3.11.11.m11.4.5.5.3.3.cmml" type="integer" xref="S6.Thmtheorem1.p3.11.11.m11.4.5.5.3.3">2</cn></apply></apply></apply><apply id="S6.Thmtheorem1.p3.11.11.m11.4.5e.cmml" xref="S6.Thmtheorem1.p3.11.11.m11.4.5"><subset id="S6.Thmtheorem1.p3.11.11.m11.4.5.6.cmml" xref="S6.Thmtheorem1.p3.11.11.m11.4.5.6"></subset><share href="https://arxiv.org/html/2411.12694v2#S6.Thmtheorem1.p3.11.11.m11.4.5.5.cmml" id="S6.Thmtheorem1.p3.11.11.m11.4.5f.cmml" xref="S6.Thmtheorem1.p3.11.11.m11.4.5"></share><apply id="S6.Thmtheorem1.p3.11.11.m11.4.5.7.cmml" xref="S6.Thmtheorem1.p3.11.11.m11.4.5.7"><times id="S6.Thmtheorem1.p3.11.11.m11.4.5.7.1.cmml" xref="S6.Thmtheorem1.p3.11.11.m11.4.5.7.1"></times><ci id="S6.Thmtheorem1.p3.11.11.m11.4.5.7.2.cmml" xref="S6.Thmtheorem1.p3.11.11.m11.4.5.7.2">𝑂</ci><apply id="S6.Thmtheorem1.p3.11.11.m11.3.3.cmml" xref="S6.Thmtheorem1.p3.11.11.m11.4.5.7.3.2"><divide id="S6.Thmtheorem1.p3.11.11.m11.3.3.1.cmml" xref="S6.Thmtheorem1.p3.11.11.m11.4.5.7.3.2"></divide><apply id="S6.Thmtheorem1.p3.11.11.m11.3.3.2.cmml" xref="S6.Thmtheorem1.p3.11.11.m11.3.3.2"><log id="S6.Thmtheorem1.p3.11.11.m11.3.3.2.1.cmml" xref="S6.Thmtheorem1.p3.11.11.m11.3.3.2.1"></log><ci id="S6.Thmtheorem1.p3.11.11.m11.3.3.2.2.cmml" xref="S6.Thmtheorem1.p3.11.11.m11.3.3.2.2">𝑛</ci></apply><apply id="S6.Thmtheorem1.p3.11.11.m11.3.3.3.cmml" xref="S6.Thmtheorem1.p3.11.11.m11.3.3.3"><divide id="S6.Thmtheorem1.p3.11.11.m11.3.3.3.1.cmml" xref="S6.Thmtheorem1.p3.11.11.m11.3.3.3.1"></divide><apply id="S6.Thmtheorem1.p3.11.11.m11.3.3.3.2.cmml" xref="S6.Thmtheorem1.p3.11.11.m11.3.3.3.2"><times id="S6.Thmtheorem1.p3.11.11.m11.3.3.3.2.1.cmml" xref="S6.Thmtheorem1.p3.11.11.m11.3.3.3.2.1"></times><cn id="S6.Thmtheorem1.p3.11.11.m11.3.3.3.2.2.cmml" type="integer" xref="S6.Thmtheorem1.p3.11.11.m11.3.3.3.2.2">64</cn><apply id="S6.Thmtheorem1.p3.11.11.m11.3.3.3.2.3.cmml" xref="S6.Thmtheorem1.p3.11.11.m11.3.3.3.2.3"><csymbol cd="ambiguous" id="S6.Thmtheorem1.p3.11.11.m11.3.3.3.2.3.1.cmml" xref="S6.Thmtheorem1.p3.11.11.m11.3.3.3.2.3">superscript</csymbol><ci id="S6.Thmtheorem1.p3.11.11.m11.3.3.3.2.3.2.cmml" xref="S6.Thmtheorem1.p3.11.11.m11.3.3.3.2.3.2">𝜀</ci><cn id="S6.Thmtheorem1.p3.11.11.m11.3.3.3.2.3.3.cmml" type="integer" xref="S6.Thmtheorem1.p3.11.11.m11.3.3.3.2.3.3">2</cn></apply></apply><apply id="S6.Thmtheorem1.p3.11.11.m11.3.3.3.3.cmml" xref="S6.Thmtheorem1.p3.11.11.m11.3.3.3.3"><log id="S6.Thmtheorem1.p3.11.11.m11.3.3.3.3.1.cmml" xref="S6.Thmtheorem1.p3.11.11.m11.3.3.3.3.1"></log><ci id="S6.Thmtheorem1.p3.11.11.m11.3.3.3.3.2.cmml" xref="S6.Thmtheorem1.p3.11.11.m11.3.3.3.3.2">𝑛</ci></apply></apply></apply></apply></apply><apply id="S6.Thmtheorem1.p3.11.11.m11.4.5g.cmml" xref="S6.Thmtheorem1.p3.11.11.m11.4.5"><subset id="S6.Thmtheorem1.p3.11.11.m11.4.5.8.cmml" xref="S6.Thmtheorem1.p3.11.11.m11.4.5.8"></subset><share href="https://arxiv.org/html/2411.12694v2#S6.Thmtheorem1.p3.11.11.m11.4.5.7.cmml" id="S6.Thmtheorem1.p3.11.11.m11.4.5h.cmml" xref="S6.Thmtheorem1.p3.11.11.m11.4.5"></share><apply id="S6.Thmtheorem1.p3.11.11.m11.4.5.9.cmml" xref="S6.Thmtheorem1.p3.11.11.m11.4.5.9"><times id="S6.Thmtheorem1.p3.11.11.m11.4.5.9.1.cmml" xref="S6.Thmtheorem1.p3.11.11.m11.4.5.9.1"></times><ci id="S6.Thmtheorem1.p3.11.11.m11.4.5.9.2.cmml" xref="S6.Thmtheorem1.p3.11.11.m11.4.5.9.2">𝑂</ci><apply id="S6.Thmtheorem1.p3.11.11.m11.4.4.cmml" xref="S6.Thmtheorem1.p3.11.11.m11.4.5.9.3.2"><divide id="S6.Thmtheorem1.p3.11.11.m11.4.4.1.cmml" xref="S6.Thmtheorem1.p3.11.11.m11.4.5.9.3.2"></divide><apply id="S6.Thmtheorem1.p3.11.11.m11.4.4.2.cmml" xref="S6.Thmtheorem1.p3.11.11.m11.4.4.2"><apply id="S6.Thmtheorem1.p3.11.11.m11.4.4.2.1.cmml" xref="S6.Thmtheorem1.p3.11.11.m11.4.4.2.1"><csymbol cd="ambiguous" id="S6.Thmtheorem1.p3.11.11.m11.4.4.2.1.1.cmml" xref="S6.Thmtheorem1.p3.11.11.m11.4.4.2.1">superscript</csymbol><log id="S6.Thmtheorem1.p3.11.11.m11.4.4.2.1.2.cmml" xref="S6.Thmtheorem1.p3.11.11.m11.4.4.2.1.2"></log><cn id="S6.Thmtheorem1.p3.11.11.m11.4.4.2.1.3.cmml" type="integer" xref="S6.Thmtheorem1.p3.11.11.m11.4.4.2.1.3">2</cn></apply><ci id="S6.Thmtheorem1.p3.11.11.m11.4.4.2.2.cmml" xref="S6.Thmtheorem1.p3.11.11.m11.4.4.2.2">𝑛</ci></apply><apply id="S6.Thmtheorem1.p3.11.11.m11.4.4.3.cmml" xref="S6.Thmtheorem1.p3.11.11.m11.4.4.3"><csymbol cd="ambiguous" id="S6.Thmtheorem1.p3.11.11.m11.4.4.3.1.cmml" xref="S6.Thmtheorem1.p3.11.11.m11.4.4.3">superscript</csymbol><ci id="S6.Thmtheorem1.p3.11.11.m11.4.4.3.2.cmml" xref="S6.Thmtheorem1.p3.11.11.m11.4.4.3.2">𝜀</ci><cn id="S6.Thmtheorem1.p3.11.11.m11.4.4.3.3.cmml" type="integer" xref="S6.Thmtheorem1.p3.11.11.m11.4.4.3.3">2</cn></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.Thmtheorem1.p3.11.11.m11.4c">\ell\leq\frac{\log n}{\log(1+\eta)}\leq\frac{\log n}{\eta/2}\subseteq O(\frac{% \log n}{64\varepsilon^{2}/\log n})\subseteq O(\frac{\log^{2}n}{\varepsilon^{2}})</annotation><annotation encoding="application/x-llamapun" id="S6.Thmtheorem1.p3.11.11.m11.4d">roman_ℓ ≤ divide start_ARG roman_log italic_n end_ARG start_ARG roman_log ( 1 + italic_η ) end_ARG ≤ divide start_ARG roman_log italic_n end_ARG start_ARG italic_η / 2 end_ARG ⊆ italic_O ( divide start_ARG roman_log italic_n end_ARG start_ARG 64 italic_ε start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / roman_log italic_n end_ARG ) ⊆ italic_O ( divide start_ARG roman_log start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_n end_ARG start_ARG italic_ε start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG )</annotation></semantics></math>.</span></p> </div> <div class="ltx_para" id="S6.Thmtheorem1.p4"> <p class="ltx_p" id="S6.Thmtheorem1.p4.10"><span class="ltx_text ltx_font_italic" id="S6.Thmtheorem1.p4.10.10">Given this theoretical algorithm, we prove the lemma. Consider any fair orientation <math alttext="\overrightarrow{G}" class="ltx_Math" display="inline" id="S6.Thmtheorem1.p4.1.1.m1.1"><semantics id="S6.Thmtheorem1.p4.1.1.m1.1a"><mover accent="true" id="S6.Thmtheorem1.p4.1.1.m1.1.1" xref="S6.Thmtheorem1.p4.1.1.m1.1.1.cmml"><mi id="S6.Thmtheorem1.p4.1.1.m1.1.1.2" xref="S6.Thmtheorem1.p4.1.1.m1.1.1.2.cmml">G</mi><mo id="S6.Thmtheorem1.p4.1.1.m1.1.1.1" stretchy="false" xref="S6.Thmtheorem1.p4.1.1.m1.1.1.1.cmml">→</mo></mover><annotation-xml encoding="MathML-Content" id="S6.Thmtheorem1.p4.1.1.m1.1b"><apply id="S6.Thmtheorem1.p4.1.1.m1.1.1.cmml" xref="S6.Thmtheorem1.p4.1.1.m1.1.1"><ci id="S6.Thmtheorem1.p4.1.1.m1.1.1.1.cmml" xref="S6.Thmtheorem1.p4.1.1.m1.1.1.1">→</ci><ci id="S6.Thmtheorem1.p4.1.1.m1.1.1.2.cmml" xref="S6.Thmtheorem1.p4.1.1.m1.1.1.2">𝐺</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.Thmtheorem1.p4.1.1.m1.1c">\overrightarrow{G}</annotation><annotation encoding="application/x-llamapun" id="S6.Thmtheorem1.p4.1.1.m1.1d">over→ start_ARG italic_G end_ARG</annotation></semantics></math>. Then by Theorem <a class="ltx_ref" href="https://arxiv.org/html/2411.12694v2#S3.Thmtheorem2" title="Theorem 3.2. ‣ 3.A Conceptual results for local density ‣ 3 Results and organisation ‣ Local Density and its Distributed Approximation"><span class="ltx_text ltx_ref_tag">3.2</span></a> for any vertex <math alttext="v" class="ltx_Math" display="inline" id="S6.Thmtheorem1.p4.2.2.m2.1"><semantics id="S6.Thmtheorem1.p4.2.2.m2.1a"><mi id="S6.Thmtheorem1.p4.2.2.m2.1.1" xref="S6.Thmtheorem1.p4.2.2.m2.1.1.cmml">v</mi><annotation-xml encoding="MathML-Content" id="S6.Thmtheorem1.p4.2.2.m2.1b"><ci id="S6.Thmtheorem1.p4.2.2.m2.1.1.cmml" xref="S6.Thmtheorem1.p4.2.2.m2.1.1">𝑣</ci></annotation-xml><annotation encoding="application/x-tex" id="S6.Thmtheorem1.p4.2.2.m2.1c">v</annotation><annotation encoding="application/x-llamapun" id="S6.Thmtheorem1.p4.2.2.m2.1d">italic_v</annotation></semantics></math>, <math alttext="\textsl{g}(v)=\rho^{*}(v)" class="ltx_Math" display="inline" id="S6.Thmtheorem1.p4.3.3.m3.2"><semantics id="S6.Thmtheorem1.p4.3.3.m3.2a"><mrow id="S6.Thmtheorem1.p4.3.3.m3.2.3" xref="S6.Thmtheorem1.p4.3.3.m3.2.3.cmml"><mrow id="S6.Thmtheorem1.p4.3.3.m3.2.3.2" xref="S6.Thmtheorem1.p4.3.3.m3.2.3.2.cmml"><mtext class="ltx_mathvariant_italic" id="S6.Thmtheorem1.p4.3.3.m3.2.3.2.2" xref="S6.Thmtheorem1.p4.3.3.m3.2.3.2.2a.cmml">g</mtext><mo id="S6.Thmtheorem1.p4.3.3.m3.2.3.2.1" xref="S6.Thmtheorem1.p4.3.3.m3.2.3.2.1.cmml">⁢</mo><mrow id="S6.Thmtheorem1.p4.3.3.m3.2.3.2.3.2" xref="S6.Thmtheorem1.p4.3.3.m3.2.3.2.cmml"><mo id="S6.Thmtheorem1.p4.3.3.m3.2.3.2.3.2.1" stretchy="false" xref="S6.Thmtheorem1.p4.3.3.m3.2.3.2.cmml">(</mo><mi id="S6.Thmtheorem1.p4.3.3.m3.1.1" xref="S6.Thmtheorem1.p4.3.3.m3.1.1.cmml">v</mi><mo id="S6.Thmtheorem1.p4.3.3.m3.2.3.2.3.2.2" stretchy="false" xref="S6.Thmtheorem1.p4.3.3.m3.2.3.2.cmml">)</mo></mrow></mrow><mo id="S6.Thmtheorem1.p4.3.3.m3.2.3.1" xref="S6.Thmtheorem1.p4.3.3.m3.2.3.1.cmml">=</mo><mrow id="S6.Thmtheorem1.p4.3.3.m3.2.3.3" xref="S6.Thmtheorem1.p4.3.3.m3.2.3.3.cmml"><msup id="S6.Thmtheorem1.p4.3.3.m3.2.3.3.2" xref="S6.Thmtheorem1.p4.3.3.m3.2.3.3.2.cmml"><mi id="S6.Thmtheorem1.p4.3.3.m3.2.3.3.2.2" xref="S6.Thmtheorem1.p4.3.3.m3.2.3.3.2.2.cmml">ρ</mi><mo id="S6.Thmtheorem1.p4.3.3.m3.2.3.3.2.3" xref="S6.Thmtheorem1.p4.3.3.m3.2.3.3.2.3.cmml">∗</mo></msup><mo id="S6.Thmtheorem1.p4.3.3.m3.2.3.3.1" xref="S6.Thmtheorem1.p4.3.3.m3.2.3.3.1.cmml">⁢</mo><mrow id="S6.Thmtheorem1.p4.3.3.m3.2.3.3.3.2" xref="S6.Thmtheorem1.p4.3.3.m3.2.3.3.cmml"><mo id="S6.Thmtheorem1.p4.3.3.m3.2.3.3.3.2.1" stretchy="false" xref="S6.Thmtheorem1.p4.3.3.m3.2.3.3.cmml">(</mo><mi id="S6.Thmtheorem1.p4.3.3.m3.2.2" xref="S6.Thmtheorem1.p4.3.3.m3.2.2.cmml">v</mi><mo id="S6.Thmtheorem1.p4.3.3.m3.2.3.3.3.2.2" stretchy="false" xref="S6.Thmtheorem1.p4.3.3.m3.2.3.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.Thmtheorem1.p4.3.3.m3.2b"><apply id="S6.Thmtheorem1.p4.3.3.m3.2.3.cmml" xref="S6.Thmtheorem1.p4.3.3.m3.2.3"><eq id="S6.Thmtheorem1.p4.3.3.m3.2.3.1.cmml" xref="S6.Thmtheorem1.p4.3.3.m3.2.3.1"></eq><apply id="S6.Thmtheorem1.p4.3.3.m3.2.3.2.cmml" xref="S6.Thmtheorem1.p4.3.3.m3.2.3.2"><times id="S6.Thmtheorem1.p4.3.3.m3.2.3.2.1.cmml" xref="S6.Thmtheorem1.p4.3.3.m3.2.3.2.1"></times><ci id="S6.Thmtheorem1.p4.3.3.m3.2.3.2.2a.cmml" xref="S6.Thmtheorem1.p4.3.3.m3.2.3.2.2"><mtext class="ltx_mathvariant_italic" id="S6.Thmtheorem1.p4.3.3.m3.2.3.2.2.cmml" xref="S6.Thmtheorem1.p4.3.3.m3.2.3.2.2">g</mtext></ci><ci id="S6.Thmtheorem1.p4.3.3.m3.1.1.cmml" xref="S6.Thmtheorem1.p4.3.3.m3.1.1">𝑣</ci></apply><apply id="S6.Thmtheorem1.p4.3.3.m3.2.3.3.cmml" xref="S6.Thmtheorem1.p4.3.3.m3.2.3.3"><times id="S6.Thmtheorem1.p4.3.3.m3.2.3.3.1.cmml" xref="S6.Thmtheorem1.p4.3.3.m3.2.3.3.1"></times><apply id="S6.Thmtheorem1.p4.3.3.m3.2.3.3.2.cmml" xref="S6.Thmtheorem1.p4.3.3.m3.2.3.3.2"><csymbol cd="ambiguous" id="S6.Thmtheorem1.p4.3.3.m3.2.3.3.2.1.cmml" xref="S6.Thmtheorem1.p4.3.3.m3.2.3.3.2">superscript</csymbol><ci id="S6.Thmtheorem1.p4.3.3.m3.2.3.3.2.2.cmml" xref="S6.Thmtheorem1.p4.3.3.m3.2.3.3.2.2">𝜌</ci><times id="S6.Thmtheorem1.p4.3.3.m3.2.3.3.2.3.cmml" xref="S6.Thmtheorem1.p4.3.3.m3.2.3.3.2.3"></times></apply><ci id="S6.Thmtheorem1.p4.3.3.m3.2.2.cmml" xref="S6.Thmtheorem1.p4.3.3.m3.2.2">𝑣</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.Thmtheorem1.p4.3.3.m3.2c">\textsl{g}(v)=\rho^{*}(v)</annotation><annotation encoding="application/x-llamapun" id="S6.Thmtheorem1.p4.3.3.m3.2d">g ( italic_v ) = italic_ρ start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ( italic_v )</annotation></semantics></math>. Choose some <math alttext="k\in\Theta(\varepsilon^{-2}\log^{2}n)" class="ltx_Math" display="inline" id="S6.Thmtheorem1.p4.4.4.m4.1"><semantics id="S6.Thmtheorem1.p4.4.4.m4.1a"><mrow id="S6.Thmtheorem1.p4.4.4.m4.1.1" xref="S6.Thmtheorem1.p4.4.4.m4.1.1.cmml"><mi id="S6.Thmtheorem1.p4.4.4.m4.1.1.3" xref="S6.Thmtheorem1.p4.4.4.m4.1.1.3.cmml">k</mi><mo id="S6.Thmtheorem1.p4.4.4.m4.1.1.2" xref="S6.Thmtheorem1.p4.4.4.m4.1.1.2.cmml">∈</mo><mrow id="S6.Thmtheorem1.p4.4.4.m4.1.1.1" xref="S6.Thmtheorem1.p4.4.4.m4.1.1.1.cmml"><mi id="S6.Thmtheorem1.p4.4.4.m4.1.1.1.3" mathvariant="normal" xref="S6.Thmtheorem1.p4.4.4.m4.1.1.1.3.cmml">Θ</mi><mo id="S6.Thmtheorem1.p4.4.4.m4.1.1.1.2" xref="S6.Thmtheorem1.p4.4.4.m4.1.1.1.2.cmml">⁢</mo><mrow id="S6.Thmtheorem1.p4.4.4.m4.1.1.1.1.1" xref="S6.Thmtheorem1.p4.4.4.m4.1.1.1.1.1.1.cmml"><mo id="S6.Thmtheorem1.p4.4.4.m4.1.1.1.1.1.2" stretchy="false" xref="S6.Thmtheorem1.p4.4.4.m4.1.1.1.1.1.1.cmml">(</mo><mrow id="S6.Thmtheorem1.p4.4.4.m4.1.1.1.1.1.1" xref="S6.Thmtheorem1.p4.4.4.m4.1.1.1.1.1.1.cmml"><msup id="S6.Thmtheorem1.p4.4.4.m4.1.1.1.1.1.1.2" xref="S6.Thmtheorem1.p4.4.4.m4.1.1.1.1.1.1.2.cmml"><mi id="S6.Thmtheorem1.p4.4.4.m4.1.1.1.1.1.1.2.2" xref="S6.Thmtheorem1.p4.4.4.m4.1.1.1.1.1.1.2.2.cmml">ε</mi><mrow id="S6.Thmtheorem1.p4.4.4.m4.1.1.1.1.1.1.2.3" xref="S6.Thmtheorem1.p4.4.4.m4.1.1.1.1.1.1.2.3.cmml"><mo id="S6.Thmtheorem1.p4.4.4.m4.1.1.1.1.1.1.2.3a" xref="S6.Thmtheorem1.p4.4.4.m4.1.1.1.1.1.1.2.3.cmml">−</mo><mn id="S6.Thmtheorem1.p4.4.4.m4.1.1.1.1.1.1.2.3.2" xref="S6.Thmtheorem1.p4.4.4.m4.1.1.1.1.1.1.2.3.2.cmml">2</mn></mrow></msup><mo id="S6.Thmtheorem1.p4.4.4.m4.1.1.1.1.1.1.1" lspace="0.167em" xref="S6.Thmtheorem1.p4.4.4.m4.1.1.1.1.1.1.1.cmml">⁢</mo><mrow id="S6.Thmtheorem1.p4.4.4.m4.1.1.1.1.1.1.3" xref="S6.Thmtheorem1.p4.4.4.m4.1.1.1.1.1.1.3.cmml"><msup id="S6.Thmtheorem1.p4.4.4.m4.1.1.1.1.1.1.3.1" xref="S6.Thmtheorem1.p4.4.4.m4.1.1.1.1.1.1.3.1.cmml"><mi id="S6.Thmtheorem1.p4.4.4.m4.1.1.1.1.1.1.3.1.2" xref="S6.Thmtheorem1.p4.4.4.m4.1.1.1.1.1.1.3.1.2.cmml">log</mi><mn id="S6.Thmtheorem1.p4.4.4.m4.1.1.1.1.1.1.3.1.3" xref="S6.Thmtheorem1.p4.4.4.m4.1.1.1.1.1.1.3.1.3.cmml">2</mn></msup><mo id="S6.Thmtheorem1.p4.4.4.m4.1.1.1.1.1.1.3a" lspace="0.167em" xref="S6.Thmtheorem1.p4.4.4.m4.1.1.1.1.1.1.3.cmml">⁡</mo><mi id="S6.Thmtheorem1.p4.4.4.m4.1.1.1.1.1.1.3.2" xref="S6.Thmtheorem1.p4.4.4.m4.1.1.1.1.1.1.3.2.cmml">n</mi></mrow></mrow><mo id="S6.Thmtheorem1.p4.4.4.m4.1.1.1.1.1.3" stretchy="false" xref="S6.Thmtheorem1.p4.4.4.m4.1.1.1.1.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.Thmtheorem1.p4.4.4.m4.1b"><apply id="S6.Thmtheorem1.p4.4.4.m4.1.1.cmml" xref="S6.Thmtheorem1.p4.4.4.m4.1.1"><in id="S6.Thmtheorem1.p4.4.4.m4.1.1.2.cmml" xref="S6.Thmtheorem1.p4.4.4.m4.1.1.2"></in><ci id="S6.Thmtheorem1.p4.4.4.m4.1.1.3.cmml" xref="S6.Thmtheorem1.p4.4.4.m4.1.1.3">𝑘</ci><apply id="S6.Thmtheorem1.p4.4.4.m4.1.1.1.cmml" xref="S6.Thmtheorem1.p4.4.4.m4.1.1.1"><times id="S6.Thmtheorem1.p4.4.4.m4.1.1.1.2.cmml" xref="S6.Thmtheorem1.p4.4.4.m4.1.1.1.2"></times><ci id="S6.Thmtheorem1.p4.4.4.m4.1.1.1.3.cmml" xref="S6.Thmtheorem1.p4.4.4.m4.1.1.1.3">Θ</ci><apply id="S6.Thmtheorem1.p4.4.4.m4.1.1.1.1.1.1.cmml" xref="S6.Thmtheorem1.p4.4.4.m4.1.1.1.1.1"><times id="S6.Thmtheorem1.p4.4.4.m4.1.1.1.1.1.1.1.cmml" xref="S6.Thmtheorem1.p4.4.4.m4.1.1.1.1.1.1.1"></times><apply id="S6.Thmtheorem1.p4.4.4.m4.1.1.1.1.1.1.2.cmml" xref="S6.Thmtheorem1.p4.4.4.m4.1.1.1.1.1.1.2"><csymbol cd="ambiguous" id="S6.Thmtheorem1.p4.4.4.m4.1.1.1.1.1.1.2.1.cmml" xref="S6.Thmtheorem1.p4.4.4.m4.1.1.1.1.1.1.2">superscript</csymbol><ci id="S6.Thmtheorem1.p4.4.4.m4.1.1.1.1.1.1.2.2.cmml" xref="S6.Thmtheorem1.p4.4.4.m4.1.1.1.1.1.1.2.2">𝜀</ci><apply id="S6.Thmtheorem1.p4.4.4.m4.1.1.1.1.1.1.2.3.cmml" xref="S6.Thmtheorem1.p4.4.4.m4.1.1.1.1.1.1.2.3"><minus id="S6.Thmtheorem1.p4.4.4.m4.1.1.1.1.1.1.2.3.1.cmml" xref="S6.Thmtheorem1.p4.4.4.m4.1.1.1.1.1.1.2.3"></minus><cn id="S6.Thmtheorem1.p4.4.4.m4.1.1.1.1.1.1.2.3.2.cmml" type="integer" xref="S6.Thmtheorem1.p4.4.4.m4.1.1.1.1.1.1.2.3.2">2</cn></apply></apply><apply id="S6.Thmtheorem1.p4.4.4.m4.1.1.1.1.1.1.3.cmml" xref="S6.Thmtheorem1.p4.4.4.m4.1.1.1.1.1.1.3"><apply id="S6.Thmtheorem1.p4.4.4.m4.1.1.1.1.1.1.3.1.cmml" xref="S6.Thmtheorem1.p4.4.4.m4.1.1.1.1.1.1.3.1"><csymbol cd="ambiguous" id="S6.Thmtheorem1.p4.4.4.m4.1.1.1.1.1.1.3.1.1.cmml" xref="S6.Thmtheorem1.p4.4.4.m4.1.1.1.1.1.1.3.1">superscript</csymbol><log id="S6.Thmtheorem1.p4.4.4.m4.1.1.1.1.1.1.3.1.2.cmml" xref="S6.Thmtheorem1.p4.4.4.m4.1.1.1.1.1.1.3.1.2"></log><cn id="S6.Thmtheorem1.p4.4.4.m4.1.1.1.1.1.1.3.1.3.cmml" type="integer" xref="S6.Thmtheorem1.p4.4.4.m4.1.1.1.1.1.1.3.1.3">2</cn></apply><ci id="S6.Thmtheorem1.p4.4.4.m4.1.1.1.1.1.1.3.2.cmml" xref="S6.Thmtheorem1.p4.4.4.m4.1.1.1.1.1.1.3.2">𝑛</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.Thmtheorem1.p4.4.4.m4.1c">k\in\Theta(\varepsilon^{-2}\log^{2}n)</annotation><annotation encoding="application/x-llamapun" id="S6.Thmtheorem1.p4.4.4.m4.1d">italic_k ∈ roman_Θ ( italic_ε start_POSTSUPERSCRIPT - 2 end_POSTSUPERSCRIPT roman_log start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_n )</annotation></semantics></math> sufficiently large and let <math alttext="H_{k}(v)" class="ltx_Math" display="inline" id="S6.Thmtheorem1.p4.5.5.m5.1"><semantics id="S6.Thmtheorem1.p4.5.5.m5.1a"><mrow id="S6.Thmtheorem1.p4.5.5.m5.1.2" xref="S6.Thmtheorem1.p4.5.5.m5.1.2.cmml"><msub id="S6.Thmtheorem1.p4.5.5.m5.1.2.2" xref="S6.Thmtheorem1.p4.5.5.m5.1.2.2.cmml"><mi id="S6.Thmtheorem1.p4.5.5.m5.1.2.2.2" xref="S6.Thmtheorem1.p4.5.5.m5.1.2.2.2.cmml">H</mi><mi id="S6.Thmtheorem1.p4.5.5.m5.1.2.2.3" xref="S6.Thmtheorem1.p4.5.5.m5.1.2.2.3.cmml">k</mi></msub><mo id="S6.Thmtheorem1.p4.5.5.m5.1.2.1" xref="S6.Thmtheorem1.p4.5.5.m5.1.2.1.cmml">⁢</mo><mrow id="S6.Thmtheorem1.p4.5.5.m5.1.2.3.2" xref="S6.Thmtheorem1.p4.5.5.m5.1.2.cmml"><mo id="S6.Thmtheorem1.p4.5.5.m5.1.2.3.2.1" stretchy="false" xref="S6.Thmtheorem1.p4.5.5.m5.1.2.cmml">(</mo><mi id="S6.Thmtheorem1.p4.5.5.m5.1.1" xref="S6.Thmtheorem1.p4.5.5.m5.1.1.cmml">v</mi><mo id="S6.Thmtheorem1.p4.5.5.m5.1.2.3.2.2" stretchy="false" xref="S6.Thmtheorem1.p4.5.5.m5.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.Thmtheorem1.p4.5.5.m5.1b"><apply id="S6.Thmtheorem1.p4.5.5.m5.1.2.cmml" xref="S6.Thmtheorem1.p4.5.5.m5.1.2"><times id="S6.Thmtheorem1.p4.5.5.m5.1.2.1.cmml" xref="S6.Thmtheorem1.p4.5.5.m5.1.2.1"></times><apply id="S6.Thmtheorem1.p4.5.5.m5.1.2.2.cmml" xref="S6.Thmtheorem1.p4.5.5.m5.1.2.2"><csymbol cd="ambiguous" id="S6.Thmtheorem1.p4.5.5.m5.1.2.2.1.cmml" xref="S6.Thmtheorem1.p4.5.5.m5.1.2.2">subscript</csymbol><ci id="S6.Thmtheorem1.p4.5.5.m5.1.2.2.2.cmml" xref="S6.Thmtheorem1.p4.5.5.m5.1.2.2.2">𝐻</ci><ci id="S6.Thmtheorem1.p4.5.5.m5.1.2.2.3.cmml" xref="S6.Thmtheorem1.p4.5.5.m5.1.2.2.3">𝑘</ci></apply><ci id="S6.Thmtheorem1.p4.5.5.m5.1.1.cmml" xref="S6.Thmtheorem1.p4.5.5.m5.1.1">𝑣</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.Thmtheorem1.p4.5.5.m5.1c">H_{k}(v)</annotation><annotation encoding="application/x-llamapun" id="S6.Thmtheorem1.p4.5.5.m5.1d">italic_H start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ( italic_v )</annotation></semantics></math> be the <math alttext="k" class="ltx_Math" display="inline" id="S6.Thmtheorem1.p4.6.6.m6.1"><semantics id="S6.Thmtheorem1.p4.6.6.m6.1a"><mi id="S6.Thmtheorem1.p4.6.6.m6.1.1" xref="S6.Thmtheorem1.p4.6.6.m6.1.1.cmml">k</mi><annotation-xml encoding="MathML-Content" id="S6.Thmtheorem1.p4.6.6.m6.1b"><ci id="S6.Thmtheorem1.p4.6.6.m6.1.1.cmml" xref="S6.Thmtheorem1.p4.6.6.m6.1.1">𝑘</ci></annotation-xml><annotation encoding="application/x-tex" id="S6.Thmtheorem1.p4.6.6.m6.1c">k</annotation><annotation encoding="application/x-llamapun" id="S6.Thmtheorem1.p4.6.6.m6.1d">italic_k</annotation></semantics></math>-hop neighbourhood of <math alttext="v" class="ltx_Math" display="inline" id="S6.Thmtheorem1.p4.7.7.m7.1"><semantics id="S6.Thmtheorem1.p4.7.7.m7.1a"><mi id="S6.Thmtheorem1.p4.7.7.m7.1.1" xref="S6.Thmtheorem1.p4.7.7.m7.1.1.cmml">v</mi><annotation-xml encoding="MathML-Content" id="S6.Thmtheorem1.p4.7.7.m7.1b"><ci id="S6.Thmtheorem1.p4.7.7.m7.1.1.cmml" xref="S6.Thmtheorem1.p4.7.7.m7.1.1">𝑣</ci></annotation-xml><annotation encoding="application/x-tex" id="S6.Thmtheorem1.p4.7.7.m7.1c">v</annotation><annotation encoding="application/x-llamapun" id="S6.Thmtheorem1.p4.7.7.m7.1d">italic_v</annotation></semantics></math> and <math alttext="E_{k}" class="ltx_Math" display="inline" id="S6.Thmtheorem1.p4.8.8.m8.1"><semantics id="S6.Thmtheorem1.p4.8.8.m8.1a"><msub id="S6.Thmtheorem1.p4.8.8.m8.1.1" xref="S6.Thmtheorem1.p4.8.8.m8.1.1.cmml"><mi id="S6.Thmtheorem1.p4.8.8.m8.1.1.2" xref="S6.Thmtheorem1.p4.8.8.m8.1.1.2.cmml">E</mi><mi id="S6.Thmtheorem1.p4.8.8.m8.1.1.3" xref="S6.Thmtheorem1.p4.8.8.m8.1.1.3.cmml">k</mi></msub><annotation-xml encoding="MathML-Content" id="S6.Thmtheorem1.p4.8.8.m8.1b"><apply id="S6.Thmtheorem1.p4.8.8.m8.1.1.cmml" xref="S6.Thmtheorem1.p4.8.8.m8.1.1"><csymbol cd="ambiguous" id="S6.Thmtheorem1.p4.8.8.m8.1.1.1.cmml" xref="S6.Thmtheorem1.p4.8.8.m8.1.1">subscript</csymbol><ci id="S6.Thmtheorem1.p4.8.8.m8.1.1.2.cmml" xref="S6.Thmtheorem1.p4.8.8.m8.1.1.2">𝐸</ci><ci id="S6.Thmtheorem1.p4.8.8.m8.1.1.3.cmml" xref="S6.Thmtheorem1.p4.8.8.m8.1.1.3">𝑘</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.Thmtheorem1.p4.8.8.m8.1c">E_{k}</annotation><annotation encoding="application/x-llamapun" id="S6.Thmtheorem1.p4.8.8.m8.1d">italic_E start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT</annotation></semantics></math> be all the edges in <math alttext="H_{k+1}(v)" class="ltx_Math" display="inline" id="S6.Thmtheorem1.p4.9.9.m9.1"><semantics id="S6.Thmtheorem1.p4.9.9.m9.1a"><mrow id="S6.Thmtheorem1.p4.9.9.m9.1.2" xref="S6.Thmtheorem1.p4.9.9.m9.1.2.cmml"><msub id="S6.Thmtheorem1.p4.9.9.m9.1.2.2" xref="S6.Thmtheorem1.p4.9.9.m9.1.2.2.cmml"><mi id="S6.Thmtheorem1.p4.9.9.m9.1.2.2.2" xref="S6.Thmtheorem1.p4.9.9.m9.1.2.2.2.cmml">H</mi><mrow id="S6.Thmtheorem1.p4.9.9.m9.1.2.2.3" xref="S6.Thmtheorem1.p4.9.9.m9.1.2.2.3.cmml"><mi id="S6.Thmtheorem1.p4.9.9.m9.1.2.2.3.2" xref="S6.Thmtheorem1.p4.9.9.m9.1.2.2.3.2.cmml">k</mi><mo id="S6.Thmtheorem1.p4.9.9.m9.1.2.2.3.1" xref="S6.Thmtheorem1.p4.9.9.m9.1.2.2.3.1.cmml">+</mo><mn id="S6.Thmtheorem1.p4.9.9.m9.1.2.2.3.3" xref="S6.Thmtheorem1.p4.9.9.m9.1.2.2.3.3.cmml">1</mn></mrow></msub><mo id="S6.Thmtheorem1.p4.9.9.m9.1.2.1" xref="S6.Thmtheorem1.p4.9.9.m9.1.2.1.cmml">⁢</mo><mrow id="S6.Thmtheorem1.p4.9.9.m9.1.2.3.2" xref="S6.Thmtheorem1.p4.9.9.m9.1.2.cmml"><mo id="S6.Thmtheorem1.p4.9.9.m9.1.2.3.2.1" stretchy="false" xref="S6.Thmtheorem1.p4.9.9.m9.1.2.cmml">(</mo><mi id="S6.Thmtheorem1.p4.9.9.m9.1.1" xref="S6.Thmtheorem1.p4.9.9.m9.1.1.cmml">v</mi><mo id="S6.Thmtheorem1.p4.9.9.m9.1.2.3.2.2" stretchy="false" xref="S6.Thmtheorem1.p4.9.9.m9.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.Thmtheorem1.p4.9.9.m9.1b"><apply id="S6.Thmtheorem1.p4.9.9.m9.1.2.cmml" xref="S6.Thmtheorem1.p4.9.9.m9.1.2"><times id="S6.Thmtheorem1.p4.9.9.m9.1.2.1.cmml" xref="S6.Thmtheorem1.p4.9.9.m9.1.2.1"></times><apply id="S6.Thmtheorem1.p4.9.9.m9.1.2.2.cmml" xref="S6.Thmtheorem1.p4.9.9.m9.1.2.2"><csymbol cd="ambiguous" id="S6.Thmtheorem1.p4.9.9.m9.1.2.2.1.cmml" xref="S6.Thmtheorem1.p4.9.9.m9.1.2.2">subscript</csymbol><ci id="S6.Thmtheorem1.p4.9.9.m9.1.2.2.2.cmml" xref="S6.Thmtheorem1.p4.9.9.m9.1.2.2.2">𝐻</ci><apply id="S6.Thmtheorem1.p4.9.9.m9.1.2.2.3.cmml" xref="S6.Thmtheorem1.p4.9.9.m9.1.2.2.3"><plus id="S6.Thmtheorem1.p4.9.9.m9.1.2.2.3.1.cmml" xref="S6.Thmtheorem1.p4.9.9.m9.1.2.2.3.1"></plus><ci id="S6.Thmtheorem1.p4.9.9.m9.1.2.2.3.2.cmml" xref="S6.Thmtheorem1.p4.9.9.m9.1.2.2.3.2">𝑘</ci><cn id="S6.Thmtheorem1.p4.9.9.m9.1.2.2.3.3.cmml" type="integer" xref="S6.Thmtheorem1.p4.9.9.m9.1.2.2.3.3">1</cn></apply></apply><ci id="S6.Thmtheorem1.p4.9.9.m9.1.1.cmml" xref="S6.Thmtheorem1.p4.9.9.m9.1.1">𝑣</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.Thmtheorem1.p4.9.9.m9.1c">H_{k+1}(v)</annotation><annotation encoding="application/x-llamapun" id="S6.Thmtheorem1.p4.9.9.m9.1d">italic_H start_POSTSUBSCRIPT italic_k + 1 end_POSTSUBSCRIPT ( italic_v )</annotation></semantics></math> that are not in <math alttext="H_{k}" class="ltx_Math" display="inline" id="S6.Thmtheorem1.p4.10.10.m10.1"><semantics id="S6.Thmtheorem1.p4.10.10.m10.1a"><msub id="S6.Thmtheorem1.p4.10.10.m10.1.1" xref="S6.Thmtheorem1.p4.10.10.m10.1.1.cmml"><mi id="S6.Thmtheorem1.p4.10.10.m10.1.1.2" xref="S6.Thmtheorem1.p4.10.10.m10.1.1.2.cmml">H</mi><mi id="S6.Thmtheorem1.p4.10.10.m10.1.1.3" xref="S6.Thmtheorem1.p4.10.10.m10.1.1.3.cmml">k</mi></msub><annotation-xml encoding="MathML-Content" id="S6.Thmtheorem1.p4.10.10.m10.1b"><apply id="S6.Thmtheorem1.p4.10.10.m10.1.1.cmml" xref="S6.Thmtheorem1.p4.10.10.m10.1.1"><csymbol cd="ambiguous" id="S6.Thmtheorem1.p4.10.10.m10.1.1.1.cmml" xref="S6.Thmtheorem1.p4.10.10.m10.1.1">subscript</csymbol><ci id="S6.Thmtheorem1.p4.10.10.m10.1.1.2.cmml" xref="S6.Thmtheorem1.p4.10.10.m10.1.1.2">𝐻</ci><ci id="S6.Thmtheorem1.p4.10.10.m10.1.1.3.cmml" xref="S6.Thmtheorem1.p4.10.10.m10.1.1.3">𝑘</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.Thmtheorem1.p4.10.10.m10.1c">H_{k}</annotation><annotation encoding="application/x-llamapun" id="S6.Thmtheorem1.p4.10.10.m10.1d">italic_H start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT</annotation></semantics></math>.</span></p> </div> <div class="ltx_para" id="S6.Thmtheorem1.p5"> <p class="ltx_p" id="S6.Thmtheorem1.p5.10"><span class="ltx_text ltx_font_italic" id="S6.Thmtheorem1.p5.10.10">Choose <math alttext="\eta=\frac{\varepsilon^{2}}{128\log n}" class="ltx_Math" display="inline" id="S6.Thmtheorem1.p5.1.1.m1.1"><semantics id="S6.Thmtheorem1.p5.1.1.m1.1a"><mrow id="S6.Thmtheorem1.p5.1.1.m1.1.1" xref="S6.Thmtheorem1.p5.1.1.m1.1.1.cmml"><mi id="S6.Thmtheorem1.p5.1.1.m1.1.1.2" xref="S6.Thmtheorem1.p5.1.1.m1.1.1.2.cmml">η</mi><mo id="S6.Thmtheorem1.p5.1.1.m1.1.1.1" xref="S6.Thmtheorem1.p5.1.1.m1.1.1.1.cmml">=</mo><mfrac id="S6.Thmtheorem1.p5.1.1.m1.1.1.3" xref="S6.Thmtheorem1.p5.1.1.m1.1.1.3.cmml"><msup id="S6.Thmtheorem1.p5.1.1.m1.1.1.3.2" xref="S6.Thmtheorem1.p5.1.1.m1.1.1.3.2.cmml"><mi id="S6.Thmtheorem1.p5.1.1.m1.1.1.3.2.2" xref="S6.Thmtheorem1.p5.1.1.m1.1.1.3.2.2.cmml">ε</mi><mn id="S6.Thmtheorem1.p5.1.1.m1.1.1.3.2.3" xref="S6.Thmtheorem1.p5.1.1.m1.1.1.3.2.3.cmml">2</mn></msup><mrow id="S6.Thmtheorem1.p5.1.1.m1.1.1.3.3" xref="S6.Thmtheorem1.p5.1.1.m1.1.1.3.3.cmml"><mn id="S6.Thmtheorem1.p5.1.1.m1.1.1.3.3.2" xref="S6.Thmtheorem1.p5.1.1.m1.1.1.3.3.2.cmml">128</mn><mo id="S6.Thmtheorem1.p5.1.1.m1.1.1.3.3.1" lspace="0.167em" xref="S6.Thmtheorem1.p5.1.1.m1.1.1.3.3.1.cmml">⁢</mo><mrow id="S6.Thmtheorem1.p5.1.1.m1.1.1.3.3.3" xref="S6.Thmtheorem1.p5.1.1.m1.1.1.3.3.3.cmml"><mi id="S6.Thmtheorem1.p5.1.1.m1.1.1.3.3.3.1" xref="S6.Thmtheorem1.p5.1.1.m1.1.1.3.3.3.1.cmml">log</mi><mo id="S6.Thmtheorem1.p5.1.1.m1.1.1.3.3.3a" lspace="0.167em" xref="S6.Thmtheorem1.p5.1.1.m1.1.1.3.3.3.cmml">⁡</mo><mi id="S6.Thmtheorem1.p5.1.1.m1.1.1.3.3.3.2" xref="S6.Thmtheorem1.p5.1.1.m1.1.1.3.3.3.2.cmml">n</mi></mrow></mrow></mfrac></mrow><annotation-xml encoding="MathML-Content" id="S6.Thmtheorem1.p5.1.1.m1.1b"><apply id="S6.Thmtheorem1.p5.1.1.m1.1.1.cmml" xref="S6.Thmtheorem1.p5.1.1.m1.1.1"><eq id="S6.Thmtheorem1.p5.1.1.m1.1.1.1.cmml" xref="S6.Thmtheorem1.p5.1.1.m1.1.1.1"></eq><ci id="S6.Thmtheorem1.p5.1.1.m1.1.1.2.cmml" xref="S6.Thmtheorem1.p5.1.1.m1.1.1.2">𝜂</ci><apply id="S6.Thmtheorem1.p5.1.1.m1.1.1.3.cmml" xref="S6.Thmtheorem1.p5.1.1.m1.1.1.3"><divide id="S6.Thmtheorem1.p5.1.1.m1.1.1.3.1.cmml" xref="S6.Thmtheorem1.p5.1.1.m1.1.1.3"></divide><apply id="S6.Thmtheorem1.p5.1.1.m1.1.1.3.2.cmml" xref="S6.Thmtheorem1.p5.1.1.m1.1.1.3.2"><csymbol cd="ambiguous" id="S6.Thmtheorem1.p5.1.1.m1.1.1.3.2.1.cmml" xref="S6.Thmtheorem1.p5.1.1.m1.1.1.3.2">superscript</csymbol><ci id="S6.Thmtheorem1.p5.1.1.m1.1.1.3.2.2.cmml" xref="S6.Thmtheorem1.p5.1.1.m1.1.1.3.2.2">𝜀</ci><cn id="S6.Thmtheorem1.p5.1.1.m1.1.1.3.2.3.cmml" type="integer" xref="S6.Thmtheorem1.p5.1.1.m1.1.1.3.2.3">2</cn></apply><apply id="S6.Thmtheorem1.p5.1.1.m1.1.1.3.3.cmml" xref="S6.Thmtheorem1.p5.1.1.m1.1.1.3.3"><times id="S6.Thmtheorem1.p5.1.1.m1.1.1.3.3.1.cmml" xref="S6.Thmtheorem1.p5.1.1.m1.1.1.3.3.1"></times><cn id="S6.Thmtheorem1.p5.1.1.m1.1.1.3.3.2.cmml" type="integer" xref="S6.Thmtheorem1.p5.1.1.m1.1.1.3.3.2">128</cn><apply id="S6.Thmtheorem1.p5.1.1.m1.1.1.3.3.3.cmml" xref="S6.Thmtheorem1.p5.1.1.m1.1.1.3.3.3"><log id="S6.Thmtheorem1.p5.1.1.m1.1.1.3.3.3.1.cmml" xref="S6.Thmtheorem1.p5.1.1.m1.1.1.3.3.3.1"></log><ci id="S6.Thmtheorem1.p5.1.1.m1.1.1.3.3.3.2.cmml" xref="S6.Thmtheorem1.p5.1.1.m1.1.1.3.3.3.2">𝑛</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.Thmtheorem1.p5.1.1.m1.1c">\eta=\frac{\varepsilon^{2}}{128\log n}</annotation><annotation encoding="application/x-llamapun" id="S6.Thmtheorem1.p5.1.1.m1.1d">italic_η = divide start_ARG italic_ε start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG start_ARG 128 roman_log italic_n end_ARG</annotation></semantics></math>. The orientation <math alttext="\overrightarrow{G}" class="ltx_Math" display="inline" id="S6.Thmtheorem1.p5.2.2.m2.1"><semantics id="S6.Thmtheorem1.p5.2.2.m2.1a"><mover accent="true" id="S6.Thmtheorem1.p5.2.2.m2.1.1" xref="S6.Thmtheorem1.p5.2.2.m2.1.1.cmml"><mi id="S6.Thmtheorem1.p5.2.2.m2.1.1.2" xref="S6.Thmtheorem1.p5.2.2.m2.1.1.2.cmml">G</mi><mo id="S6.Thmtheorem1.p5.2.2.m2.1.1.1" stretchy="false" xref="S6.Thmtheorem1.p5.2.2.m2.1.1.1.cmml">→</mo></mover><annotation-xml encoding="MathML-Content" id="S6.Thmtheorem1.p5.2.2.m2.1b"><apply id="S6.Thmtheorem1.p5.2.2.m2.1.1.cmml" xref="S6.Thmtheorem1.p5.2.2.m2.1.1"><ci id="S6.Thmtheorem1.p5.2.2.m2.1.1.1.cmml" xref="S6.Thmtheorem1.p5.2.2.m2.1.1.1">→</ci><ci id="S6.Thmtheorem1.p5.2.2.m2.1.1.2.cmml" xref="S6.Thmtheorem1.p5.2.2.m2.1.1.2">𝐺</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.Thmtheorem1.p5.2.2.m2.1c">\overrightarrow{G}</annotation><annotation encoding="application/x-llamapun" id="S6.Thmtheorem1.p5.2.2.m2.1d">over→ start_ARG italic_G end_ARG</annotation></semantics></math> is per definition an <math alttext="\eta" class="ltx_Math" display="inline" id="S6.Thmtheorem1.p5.3.3.m3.1"><semantics id="S6.Thmtheorem1.p5.3.3.m3.1a"><mi id="S6.Thmtheorem1.p5.3.3.m3.1.1" xref="S6.Thmtheorem1.p5.3.3.m3.1.1.cmml">η</mi><annotation-xml encoding="MathML-Content" id="S6.Thmtheorem1.p5.3.3.m3.1b"><ci id="S6.Thmtheorem1.p5.3.3.m3.1.1.cmml" xref="S6.Thmtheorem1.p5.3.3.m3.1.1">𝜂</ci></annotation-xml><annotation encoding="application/x-tex" id="S6.Thmtheorem1.p5.3.3.m3.1c">\eta</annotation><annotation encoding="application/x-llamapun" id="S6.Thmtheorem1.p5.3.3.m3.1d">italic_η</annotation></semantics></math>-fair orientation. We run on <math alttext="\overrightarrow{G}" class="ltx_Math" display="inline" id="S6.Thmtheorem1.p5.4.4.m4.1"><semantics id="S6.Thmtheorem1.p5.4.4.m4.1a"><mover accent="true" id="S6.Thmtheorem1.p5.4.4.m4.1.1" xref="S6.Thmtheorem1.p5.4.4.m4.1.1.cmml"><mi id="S6.Thmtheorem1.p5.4.4.m4.1.1.2" xref="S6.Thmtheorem1.p5.4.4.m4.1.1.2.cmml">G</mi><mo id="S6.Thmtheorem1.p5.4.4.m4.1.1.1" stretchy="false" xref="S6.Thmtheorem1.p5.4.4.m4.1.1.1.cmml">→</mo></mover><annotation-xml encoding="MathML-Content" id="S6.Thmtheorem1.p5.4.4.m4.1b"><apply id="S6.Thmtheorem1.p5.4.4.m4.1.1.cmml" xref="S6.Thmtheorem1.p5.4.4.m4.1.1"><ci id="S6.Thmtheorem1.p5.4.4.m4.1.1.1.cmml" xref="S6.Thmtheorem1.p5.4.4.m4.1.1.1">→</ci><ci id="S6.Thmtheorem1.p5.4.4.m4.1.1.2.cmml" xref="S6.Thmtheorem1.p5.4.4.m4.1.1.2">𝐺</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.Thmtheorem1.p5.4.4.m4.1c">\overrightarrow{G}</annotation><annotation encoding="application/x-llamapun" id="S6.Thmtheorem1.p5.4.4.m4.1d">over→ start_ARG italic_G end_ARG</annotation></semantics></math> our deletion-only algorithm, deleting all edges in <math alttext="E_{k}" class="ltx_Math" display="inline" id="S6.Thmtheorem1.p5.5.5.m5.1"><semantics id="S6.Thmtheorem1.p5.5.5.m5.1a"><msub id="S6.Thmtheorem1.p5.5.5.m5.1.1" xref="S6.Thmtheorem1.p5.5.5.m5.1.1.cmml"><mi id="S6.Thmtheorem1.p5.5.5.m5.1.1.2" xref="S6.Thmtheorem1.p5.5.5.m5.1.1.2.cmml">E</mi><mi id="S6.Thmtheorem1.p5.5.5.m5.1.1.3" xref="S6.Thmtheorem1.p5.5.5.m5.1.1.3.cmml">k</mi></msub><annotation-xml encoding="MathML-Content" id="S6.Thmtheorem1.p5.5.5.m5.1b"><apply id="S6.Thmtheorem1.p5.5.5.m5.1.1.cmml" xref="S6.Thmtheorem1.p5.5.5.m5.1.1"><csymbol cd="ambiguous" id="S6.Thmtheorem1.p5.5.5.m5.1.1.1.cmml" xref="S6.Thmtheorem1.p5.5.5.m5.1.1">subscript</csymbol><ci id="S6.Thmtheorem1.p5.5.5.m5.1.1.2.cmml" xref="S6.Thmtheorem1.p5.5.5.m5.1.1.2">𝐸</ci><ci id="S6.Thmtheorem1.p5.5.5.m5.1.1.3.cmml" xref="S6.Thmtheorem1.p5.5.5.m5.1.1.3">𝑘</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.Thmtheorem1.p5.5.5.m5.1c">E_{k}</annotation><annotation encoding="application/x-llamapun" id="S6.Thmtheorem1.p5.5.5.m5.1d">italic_E start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT</annotation></semantics></math>. Since our algorithm has a recursive depth of <math alttext="\ell<k" class="ltx_Math" display="inline" id="S6.Thmtheorem1.p5.6.6.m6.1"><semantics id="S6.Thmtheorem1.p5.6.6.m6.1a"><mrow id="S6.Thmtheorem1.p5.6.6.m6.1.1" xref="S6.Thmtheorem1.p5.6.6.m6.1.1.cmml"><mi id="S6.Thmtheorem1.p5.6.6.m6.1.1.2" mathvariant="normal" xref="S6.Thmtheorem1.p5.6.6.m6.1.1.2.cmml">ℓ</mi><mo id="S6.Thmtheorem1.p5.6.6.m6.1.1.1" xref="S6.Thmtheorem1.p5.6.6.m6.1.1.1.cmml"><</mo><mi id="S6.Thmtheorem1.p5.6.6.m6.1.1.3" xref="S6.Thmtheorem1.p5.6.6.m6.1.1.3.cmml">k</mi></mrow><annotation-xml encoding="MathML-Content" id="S6.Thmtheorem1.p5.6.6.m6.1b"><apply id="S6.Thmtheorem1.p5.6.6.m6.1.1.cmml" xref="S6.Thmtheorem1.p5.6.6.m6.1.1"><lt id="S6.Thmtheorem1.p5.6.6.m6.1.1.1.cmml" xref="S6.Thmtheorem1.p5.6.6.m6.1.1.1"></lt><ci id="S6.Thmtheorem1.p5.6.6.m6.1.1.2.cmml" xref="S6.Thmtheorem1.p5.6.6.m6.1.1.2">ℓ</ci><ci id="S6.Thmtheorem1.p5.6.6.m6.1.1.3.cmml" xref="S6.Thmtheorem1.p5.6.6.m6.1.1.3">𝑘</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.Thmtheorem1.p5.6.6.m6.1c">\ell<k</annotation><annotation encoding="application/x-llamapun" id="S6.Thmtheorem1.p5.6.6.m6.1d">roman_ℓ < italic_k</annotation></semantics></math>, we end up with an <math alttext="\eta" class="ltx_Math" display="inline" id="S6.Thmtheorem1.p5.7.7.m7.1"><semantics id="S6.Thmtheorem1.p5.7.7.m7.1a"><mi id="S6.Thmtheorem1.p5.7.7.m7.1.1" xref="S6.Thmtheorem1.p5.7.7.m7.1.1.cmml">η</mi><annotation-xml encoding="MathML-Content" id="S6.Thmtheorem1.p5.7.7.m7.1b"><ci id="S6.Thmtheorem1.p5.7.7.m7.1.1.cmml" xref="S6.Thmtheorem1.p5.7.7.m7.1.1">𝜂</ci></annotation-xml><annotation encoding="application/x-tex" id="S6.Thmtheorem1.p5.7.7.m7.1c">\eta</annotation><annotation encoding="application/x-llamapun" id="S6.Thmtheorem1.p5.7.7.m7.1d">italic_η</annotation></semantics></math>-fair orientation of <math alttext="H_{k}(v)" class="ltx_Math" display="inline" id="S6.Thmtheorem1.p5.8.8.m8.1"><semantics id="S6.Thmtheorem1.p5.8.8.m8.1a"><mrow id="S6.Thmtheorem1.p5.8.8.m8.1.2" xref="S6.Thmtheorem1.p5.8.8.m8.1.2.cmml"><msub id="S6.Thmtheorem1.p5.8.8.m8.1.2.2" xref="S6.Thmtheorem1.p5.8.8.m8.1.2.2.cmml"><mi id="S6.Thmtheorem1.p5.8.8.m8.1.2.2.2" xref="S6.Thmtheorem1.p5.8.8.m8.1.2.2.2.cmml">H</mi><mi id="S6.Thmtheorem1.p5.8.8.m8.1.2.2.3" xref="S6.Thmtheorem1.p5.8.8.m8.1.2.2.3.cmml">k</mi></msub><mo id="S6.Thmtheorem1.p5.8.8.m8.1.2.1" xref="S6.Thmtheorem1.p5.8.8.m8.1.2.1.cmml">⁢</mo><mrow id="S6.Thmtheorem1.p5.8.8.m8.1.2.3.2" xref="S6.Thmtheorem1.p5.8.8.m8.1.2.cmml"><mo id="S6.Thmtheorem1.p5.8.8.m8.1.2.3.2.1" stretchy="false" xref="S6.Thmtheorem1.p5.8.8.m8.1.2.cmml">(</mo><mi id="S6.Thmtheorem1.p5.8.8.m8.1.1" xref="S6.Thmtheorem1.p5.8.8.m8.1.1.cmml">v</mi><mo id="S6.Thmtheorem1.p5.8.8.m8.1.2.3.2.2" stretchy="false" xref="S6.Thmtheorem1.p5.8.8.m8.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.Thmtheorem1.p5.8.8.m8.1b"><apply id="S6.Thmtheorem1.p5.8.8.m8.1.2.cmml" xref="S6.Thmtheorem1.p5.8.8.m8.1.2"><times id="S6.Thmtheorem1.p5.8.8.m8.1.2.1.cmml" xref="S6.Thmtheorem1.p5.8.8.m8.1.2.1"></times><apply id="S6.Thmtheorem1.p5.8.8.m8.1.2.2.cmml" xref="S6.Thmtheorem1.p5.8.8.m8.1.2.2"><csymbol cd="ambiguous" id="S6.Thmtheorem1.p5.8.8.m8.1.2.2.1.cmml" xref="S6.Thmtheorem1.p5.8.8.m8.1.2.2">subscript</csymbol><ci id="S6.Thmtheorem1.p5.8.8.m8.1.2.2.2.cmml" xref="S6.Thmtheorem1.p5.8.8.m8.1.2.2.2">𝐻</ci><ci id="S6.Thmtheorem1.p5.8.8.m8.1.2.2.3.cmml" xref="S6.Thmtheorem1.p5.8.8.m8.1.2.2.3">𝑘</ci></apply><ci id="S6.Thmtheorem1.p5.8.8.m8.1.1.cmml" xref="S6.Thmtheorem1.p5.8.8.m8.1.1">𝑣</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.Thmtheorem1.p5.8.8.m8.1c">H_{k}(v)</annotation><annotation encoding="application/x-llamapun" id="S6.Thmtheorem1.p5.8.8.m8.1d">italic_H start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ( italic_v )</annotation></semantics></math> where <math alttext="\textsl{g}(v)=\rho^{*}(v)" class="ltx_Math" display="inline" id="S6.Thmtheorem1.p5.9.9.m9.2"><semantics id="S6.Thmtheorem1.p5.9.9.m9.2a"><mrow id="S6.Thmtheorem1.p5.9.9.m9.2.3" xref="S6.Thmtheorem1.p5.9.9.m9.2.3.cmml"><mrow id="S6.Thmtheorem1.p5.9.9.m9.2.3.2" xref="S6.Thmtheorem1.p5.9.9.m9.2.3.2.cmml"><mtext class="ltx_mathvariant_italic" id="S6.Thmtheorem1.p5.9.9.m9.2.3.2.2" xref="S6.Thmtheorem1.p5.9.9.m9.2.3.2.2a.cmml">g</mtext><mo id="S6.Thmtheorem1.p5.9.9.m9.2.3.2.1" xref="S6.Thmtheorem1.p5.9.9.m9.2.3.2.1.cmml">⁢</mo><mrow id="S6.Thmtheorem1.p5.9.9.m9.2.3.2.3.2" xref="S6.Thmtheorem1.p5.9.9.m9.2.3.2.cmml"><mo id="S6.Thmtheorem1.p5.9.9.m9.2.3.2.3.2.1" stretchy="false" xref="S6.Thmtheorem1.p5.9.9.m9.2.3.2.cmml">(</mo><mi id="S6.Thmtheorem1.p5.9.9.m9.1.1" xref="S6.Thmtheorem1.p5.9.9.m9.1.1.cmml">v</mi><mo id="S6.Thmtheorem1.p5.9.9.m9.2.3.2.3.2.2" stretchy="false" xref="S6.Thmtheorem1.p5.9.9.m9.2.3.2.cmml">)</mo></mrow></mrow><mo id="S6.Thmtheorem1.p5.9.9.m9.2.3.1" xref="S6.Thmtheorem1.p5.9.9.m9.2.3.1.cmml">=</mo><mrow id="S6.Thmtheorem1.p5.9.9.m9.2.3.3" xref="S6.Thmtheorem1.p5.9.9.m9.2.3.3.cmml"><msup id="S6.Thmtheorem1.p5.9.9.m9.2.3.3.2" xref="S6.Thmtheorem1.p5.9.9.m9.2.3.3.2.cmml"><mi id="S6.Thmtheorem1.p5.9.9.m9.2.3.3.2.2" xref="S6.Thmtheorem1.p5.9.9.m9.2.3.3.2.2.cmml">ρ</mi><mo id="S6.Thmtheorem1.p5.9.9.m9.2.3.3.2.3" xref="S6.Thmtheorem1.p5.9.9.m9.2.3.3.2.3.cmml">∗</mo></msup><mo id="S6.Thmtheorem1.p5.9.9.m9.2.3.3.1" xref="S6.Thmtheorem1.p5.9.9.m9.2.3.3.1.cmml">⁢</mo><mrow id="S6.Thmtheorem1.p5.9.9.m9.2.3.3.3.2" xref="S6.Thmtheorem1.p5.9.9.m9.2.3.3.cmml"><mo id="S6.Thmtheorem1.p5.9.9.m9.2.3.3.3.2.1" stretchy="false" xref="S6.Thmtheorem1.p5.9.9.m9.2.3.3.cmml">(</mo><mi id="S6.Thmtheorem1.p5.9.9.m9.2.2" xref="S6.Thmtheorem1.p5.9.9.m9.2.2.cmml">v</mi><mo id="S6.Thmtheorem1.p5.9.9.m9.2.3.3.3.2.2" stretchy="false" xref="S6.Thmtheorem1.p5.9.9.m9.2.3.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.Thmtheorem1.p5.9.9.m9.2b"><apply id="S6.Thmtheorem1.p5.9.9.m9.2.3.cmml" xref="S6.Thmtheorem1.p5.9.9.m9.2.3"><eq id="S6.Thmtheorem1.p5.9.9.m9.2.3.1.cmml" xref="S6.Thmtheorem1.p5.9.9.m9.2.3.1"></eq><apply id="S6.Thmtheorem1.p5.9.9.m9.2.3.2.cmml" xref="S6.Thmtheorem1.p5.9.9.m9.2.3.2"><times id="S6.Thmtheorem1.p5.9.9.m9.2.3.2.1.cmml" xref="S6.Thmtheorem1.p5.9.9.m9.2.3.2.1"></times><ci id="S6.Thmtheorem1.p5.9.9.m9.2.3.2.2a.cmml" xref="S6.Thmtheorem1.p5.9.9.m9.2.3.2.2"><mtext class="ltx_mathvariant_italic" id="S6.Thmtheorem1.p5.9.9.m9.2.3.2.2.cmml" xref="S6.Thmtheorem1.p5.9.9.m9.2.3.2.2">g</mtext></ci><ci id="S6.Thmtheorem1.p5.9.9.m9.1.1.cmml" xref="S6.Thmtheorem1.p5.9.9.m9.1.1">𝑣</ci></apply><apply id="S6.Thmtheorem1.p5.9.9.m9.2.3.3.cmml" xref="S6.Thmtheorem1.p5.9.9.m9.2.3.3"><times id="S6.Thmtheorem1.p5.9.9.m9.2.3.3.1.cmml" xref="S6.Thmtheorem1.p5.9.9.m9.2.3.3.1"></times><apply id="S6.Thmtheorem1.p5.9.9.m9.2.3.3.2.cmml" xref="S6.Thmtheorem1.p5.9.9.m9.2.3.3.2"><csymbol cd="ambiguous" id="S6.Thmtheorem1.p5.9.9.m9.2.3.3.2.1.cmml" xref="S6.Thmtheorem1.p5.9.9.m9.2.3.3.2">superscript</csymbol><ci id="S6.Thmtheorem1.p5.9.9.m9.2.3.3.2.2.cmml" xref="S6.Thmtheorem1.p5.9.9.m9.2.3.3.2.2">𝜌</ci><times id="S6.Thmtheorem1.p5.9.9.m9.2.3.3.2.3.cmml" xref="S6.Thmtheorem1.p5.9.9.m9.2.3.3.2.3"></times></apply><ci id="S6.Thmtheorem1.p5.9.9.m9.2.2.cmml" xref="S6.Thmtheorem1.p5.9.9.m9.2.2">𝑣</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.Thmtheorem1.p5.9.9.m9.2c">\textsl{g}(v)=\rho^{*}(v)</annotation><annotation encoding="application/x-llamapun" id="S6.Thmtheorem1.p5.9.9.m9.2d">g ( italic_v ) = italic_ρ start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ( italic_v )</annotation></semantics></math>. We apply Theorem <a class="ltx_ref" href="https://arxiv.org/html/2411.12694v2#S3.Thmtheorem5" title="Theorem 3.5. ‣ 3.B Results for dynamic algorithms ‣ 3 Results and organisation ‣ Local Density and its Distributed Approximation"><span class="ltx_text ltx_ref_tag">3.5</span></a> to conclude that <math alttext="\rho^{*}(v)=\textsl{g}(v)\in[(1+\varepsilon)^{-1}\rho_{k}^{*}(v),(1+% \varepsilon)^{-1}\rho_{k}^{*}(v)]" class="ltx_Math" display="inline" id="S6.Thmtheorem1.p5.10.10.m10.6"><semantics id="S6.Thmtheorem1.p5.10.10.m10.6a"><mrow id="S6.Thmtheorem1.p5.10.10.m10.6.6" xref="S6.Thmtheorem1.p5.10.10.m10.6.6.cmml"><mrow id="S6.Thmtheorem1.p5.10.10.m10.6.6.4" xref="S6.Thmtheorem1.p5.10.10.m10.6.6.4.cmml"><msup id="S6.Thmtheorem1.p5.10.10.m10.6.6.4.2" xref="S6.Thmtheorem1.p5.10.10.m10.6.6.4.2.cmml"><mi id="S6.Thmtheorem1.p5.10.10.m10.6.6.4.2.2" xref="S6.Thmtheorem1.p5.10.10.m10.6.6.4.2.2.cmml">ρ</mi><mo id="S6.Thmtheorem1.p5.10.10.m10.6.6.4.2.3" xref="S6.Thmtheorem1.p5.10.10.m10.6.6.4.2.3.cmml">∗</mo></msup><mo id="S6.Thmtheorem1.p5.10.10.m10.6.6.4.1" xref="S6.Thmtheorem1.p5.10.10.m10.6.6.4.1.cmml">⁢</mo><mrow id="S6.Thmtheorem1.p5.10.10.m10.6.6.4.3.2" xref="S6.Thmtheorem1.p5.10.10.m10.6.6.4.cmml"><mo id="S6.Thmtheorem1.p5.10.10.m10.6.6.4.3.2.1" stretchy="false" xref="S6.Thmtheorem1.p5.10.10.m10.6.6.4.cmml">(</mo><mi id="S6.Thmtheorem1.p5.10.10.m10.1.1" xref="S6.Thmtheorem1.p5.10.10.m10.1.1.cmml">v</mi><mo id="S6.Thmtheorem1.p5.10.10.m10.6.6.4.3.2.2" stretchy="false" xref="S6.Thmtheorem1.p5.10.10.m10.6.6.4.cmml">)</mo></mrow></mrow><mo id="S6.Thmtheorem1.p5.10.10.m10.6.6.5" xref="S6.Thmtheorem1.p5.10.10.m10.6.6.5.cmml">=</mo><mrow id="S6.Thmtheorem1.p5.10.10.m10.6.6.6" xref="S6.Thmtheorem1.p5.10.10.m10.6.6.6.cmml"><mtext class="ltx_mathvariant_italic" id="S6.Thmtheorem1.p5.10.10.m10.6.6.6.2" xref="S6.Thmtheorem1.p5.10.10.m10.6.6.6.2a.cmml">g</mtext><mo id="S6.Thmtheorem1.p5.10.10.m10.6.6.6.1" xref="S6.Thmtheorem1.p5.10.10.m10.6.6.6.1.cmml">⁢</mo><mrow id="S6.Thmtheorem1.p5.10.10.m10.6.6.6.3.2" xref="S6.Thmtheorem1.p5.10.10.m10.6.6.6.cmml"><mo id="S6.Thmtheorem1.p5.10.10.m10.6.6.6.3.2.1" stretchy="false" xref="S6.Thmtheorem1.p5.10.10.m10.6.6.6.cmml">(</mo><mi id="S6.Thmtheorem1.p5.10.10.m10.2.2" xref="S6.Thmtheorem1.p5.10.10.m10.2.2.cmml">v</mi><mo id="S6.Thmtheorem1.p5.10.10.m10.6.6.6.3.2.2" stretchy="false" xref="S6.Thmtheorem1.p5.10.10.m10.6.6.6.cmml">)</mo></mrow></mrow><mo id="S6.Thmtheorem1.p5.10.10.m10.6.6.7" xref="S6.Thmtheorem1.p5.10.10.m10.6.6.7.cmml">∈</mo><mrow id="S6.Thmtheorem1.p5.10.10.m10.6.6.2.2" xref="S6.Thmtheorem1.p5.10.10.m10.6.6.2.3.cmml"><mo id="S6.Thmtheorem1.p5.10.10.m10.6.6.2.2.3" stretchy="false" xref="S6.Thmtheorem1.p5.10.10.m10.6.6.2.3.cmml">[</mo><mrow id="S6.Thmtheorem1.p5.10.10.m10.5.5.1.1.1" xref="S6.Thmtheorem1.p5.10.10.m10.5.5.1.1.1.cmml"><msup id="S6.Thmtheorem1.p5.10.10.m10.5.5.1.1.1.1" xref="S6.Thmtheorem1.p5.10.10.m10.5.5.1.1.1.1.cmml"><mrow id="S6.Thmtheorem1.p5.10.10.m10.5.5.1.1.1.1.1.1" xref="S6.Thmtheorem1.p5.10.10.m10.5.5.1.1.1.1.1.1.1.cmml"><mo id="S6.Thmtheorem1.p5.10.10.m10.5.5.1.1.1.1.1.1.2" stretchy="false" xref="S6.Thmtheorem1.p5.10.10.m10.5.5.1.1.1.1.1.1.1.cmml">(</mo><mrow id="S6.Thmtheorem1.p5.10.10.m10.5.5.1.1.1.1.1.1.1" xref="S6.Thmtheorem1.p5.10.10.m10.5.5.1.1.1.1.1.1.1.cmml"><mn id="S6.Thmtheorem1.p5.10.10.m10.5.5.1.1.1.1.1.1.1.2" xref="S6.Thmtheorem1.p5.10.10.m10.5.5.1.1.1.1.1.1.1.2.cmml">1</mn><mo id="S6.Thmtheorem1.p5.10.10.m10.5.5.1.1.1.1.1.1.1.1" xref="S6.Thmtheorem1.p5.10.10.m10.5.5.1.1.1.1.1.1.1.1.cmml">+</mo><mi id="S6.Thmtheorem1.p5.10.10.m10.5.5.1.1.1.1.1.1.1.3" xref="S6.Thmtheorem1.p5.10.10.m10.5.5.1.1.1.1.1.1.1.3.cmml">ε</mi></mrow><mo id="S6.Thmtheorem1.p5.10.10.m10.5.5.1.1.1.1.1.1.3" stretchy="false" xref="S6.Thmtheorem1.p5.10.10.m10.5.5.1.1.1.1.1.1.1.cmml">)</mo></mrow><mrow id="S6.Thmtheorem1.p5.10.10.m10.5.5.1.1.1.1.3" xref="S6.Thmtheorem1.p5.10.10.m10.5.5.1.1.1.1.3.cmml"><mo id="S6.Thmtheorem1.p5.10.10.m10.5.5.1.1.1.1.3a" xref="S6.Thmtheorem1.p5.10.10.m10.5.5.1.1.1.1.3.cmml">−</mo><mn id="S6.Thmtheorem1.p5.10.10.m10.5.5.1.1.1.1.3.2" xref="S6.Thmtheorem1.p5.10.10.m10.5.5.1.1.1.1.3.2.cmml">1</mn></mrow></msup><mo id="S6.Thmtheorem1.p5.10.10.m10.5.5.1.1.1.2" xref="S6.Thmtheorem1.p5.10.10.m10.5.5.1.1.1.2.cmml">⁢</mo><msubsup id="S6.Thmtheorem1.p5.10.10.m10.5.5.1.1.1.3" xref="S6.Thmtheorem1.p5.10.10.m10.5.5.1.1.1.3.cmml"><mi id="S6.Thmtheorem1.p5.10.10.m10.5.5.1.1.1.3.2.2" xref="S6.Thmtheorem1.p5.10.10.m10.5.5.1.1.1.3.2.2.cmml">ρ</mi><mi id="S6.Thmtheorem1.p5.10.10.m10.5.5.1.1.1.3.2.3" xref="S6.Thmtheorem1.p5.10.10.m10.5.5.1.1.1.3.2.3.cmml">k</mi><mo id="S6.Thmtheorem1.p5.10.10.m10.5.5.1.1.1.3.3" xref="S6.Thmtheorem1.p5.10.10.m10.5.5.1.1.1.3.3.cmml">∗</mo></msubsup><mo id="S6.Thmtheorem1.p5.10.10.m10.5.5.1.1.1.2a" xref="S6.Thmtheorem1.p5.10.10.m10.5.5.1.1.1.2.cmml">⁢</mo><mrow id="S6.Thmtheorem1.p5.10.10.m10.5.5.1.1.1.4.2" xref="S6.Thmtheorem1.p5.10.10.m10.5.5.1.1.1.cmml"><mo id="S6.Thmtheorem1.p5.10.10.m10.5.5.1.1.1.4.2.1" stretchy="false" xref="S6.Thmtheorem1.p5.10.10.m10.5.5.1.1.1.cmml">(</mo><mi id="S6.Thmtheorem1.p5.10.10.m10.3.3" xref="S6.Thmtheorem1.p5.10.10.m10.3.3.cmml">v</mi><mo id="S6.Thmtheorem1.p5.10.10.m10.5.5.1.1.1.4.2.2" stretchy="false" xref="S6.Thmtheorem1.p5.10.10.m10.5.5.1.1.1.cmml">)</mo></mrow></mrow><mo id="S6.Thmtheorem1.p5.10.10.m10.6.6.2.2.4" xref="S6.Thmtheorem1.p5.10.10.m10.6.6.2.3.cmml">,</mo><mrow id="S6.Thmtheorem1.p5.10.10.m10.6.6.2.2.2" xref="S6.Thmtheorem1.p5.10.10.m10.6.6.2.2.2.cmml"><msup id="S6.Thmtheorem1.p5.10.10.m10.6.6.2.2.2.1" xref="S6.Thmtheorem1.p5.10.10.m10.6.6.2.2.2.1.cmml"><mrow id="S6.Thmtheorem1.p5.10.10.m10.6.6.2.2.2.1.1.1" xref="S6.Thmtheorem1.p5.10.10.m10.6.6.2.2.2.1.1.1.1.cmml"><mo id="S6.Thmtheorem1.p5.10.10.m10.6.6.2.2.2.1.1.1.2" stretchy="false" 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xref="S6.Thmtheorem1.p5.10.10.m10.6.6.2.2.2.3.2.2">𝜌</ci><ci id="S6.Thmtheorem1.p5.10.10.m10.6.6.2.2.2.3.2.3.cmml" xref="S6.Thmtheorem1.p5.10.10.m10.6.6.2.2.2.3.2.3">𝑘</ci></apply><times id="S6.Thmtheorem1.p5.10.10.m10.6.6.2.2.2.3.3.cmml" xref="S6.Thmtheorem1.p5.10.10.m10.6.6.2.2.2.3.3"></times></apply><ci id="S6.Thmtheorem1.p5.10.10.m10.4.4.cmml" xref="S6.Thmtheorem1.p5.10.10.m10.4.4">𝑣</ci></apply></interval></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.Thmtheorem1.p5.10.10.m10.6c">\rho^{*}(v)=\textsl{g}(v)\in[(1+\varepsilon)^{-1}\rho_{k}^{*}(v),(1+% \varepsilon)^{-1}\rho_{k}^{*}(v)]</annotation><annotation encoding="application/x-llamapun" id="S6.Thmtheorem1.p5.10.10.m10.6d">italic_ρ start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ( italic_v ) = g ( italic_v ) ∈ [ ( 1 + italic_ε ) start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT italic_ρ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ( italic_v ) , ( 1 + italic_ε ) start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT italic_ρ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ( italic_v ) ]</annotation></semantics></math> which concludes the theorem.</span></p> </div> </div> <div class="ltx_para" id="S6.p3"> <p class="ltx_p" id="S6.p3.1">See <a class="ltx_ref" href="https://arxiv.org/html/2411.12694v2#S3.Thmtheorem8" title="Corollary 3.8. ‣ 3.C Results in LOCAL ‣ 3 Results and organisation ‣ Local Density and its Distributed Approximation"><span class="ltx_text ltx_ref_tag">3.8</span></a></p> </div> </section> <section class="ltx_section" id="S7"> <h2 class="ltx_title ltx_title_section"> <span class="ltx_tag ltx_tag_section">7 </span>Results in CONGEST</h2> <div class="ltx_para" id="S7.p1"> <p class="ltx_p" id="S7.p1.5">We now describe an algorithm in CONGEST that for any unit-weight graph <math alttext="G" class="ltx_Math" display="inline" id="S7.p1.1.m1.1"><semantics id="S7.p1.1.m1.1a"><mi id="S7.p1.1.m1.1.1" xref="S7.p1.1.m1.1.1.cmml">G</mi><annotation-xml encoding="MathML-Content" id="S7.p1.1.m1.1b"><ci id="S7.p1.1.m1.1.1.cmml" xref="S7.p1.1.m1.1.1">𝐺</ci></annotation-xml><annotation encoding="application/x-tex" id="S7.p1.1.m1.1c">G</annotation><annotation encoding="application/x-llamapun" id="S7.p1.1.m1.1d">italic_G</annotation></semantics></math>, creates an <math alttext="\eta" class="ltx_Math" display="inline" id="S7.p1.2.m2.1"><semantics id="S7.p1.2.m2.1a"><mi id="S7.p1.2.m2.1.1" xref="S7.p1.2.m2.1.1.cmml">η</mi><annotation-xml encoding="MathML-Content" id="S7.p1.2.m2.1b"><ci id="S7.p1.2.m2.1.1.cmml" xref="S7.p1.2.m2.1.1">𝜂</ci></annotation-xml><annotation encoding="application/x-tex" id="S7.p1.2.m2.1c">\eta</annotation><annotation encoding="application/x-llamapun" id="S7.p1.2.m2.1d">italic_η</annotation></semantics></math>-fair orientation (with <math alttext="\eta=\frac{\varepsilon^{2}}{128\cdot\log n}" class="ltx_Math" display="inline" id="S7.p1.3.m3.1"><semantics id="S7.p1.3.m3.1a"><mrow id="S7.p1.3.m3.1.1" xref="S7.p1.3.m3.1.1.cmml"><mi id="S7.p1.3.m3.1.1.2" xref="S7.p1.3.m3.1.1.2.cmml">η</mi><mo id="S7.p1.3.m3.1.1.1" xref="S7.p1.3.m3.1.1.1.cmml">=</mo><mfrac id="S7.p1.3.m3.1.1.3" xref="S7.p1.3.m3.1.1.3.cmml"><msup id="S7.p1.3.m3.1.1.3.2" xref="S7.p1.3.m3.1.1.3.2.cmml"><mi id="S7.p1.3.m3.1.1.3.2.2" xref="S7.p1.3.m3.1.1.3.2.2.cmml">ε</mi><mn id="S7.p1.3.m3.1.1.3.2.3" xref="S7.p1.3.m3.1.1.3.2.3.cmml">2</mn></msup><mrow id="S7.p1.3.m3.1.1.3.3" xref="S7.p1.3.m3.1.1.3.3.cmml"><mn id="S7.p1.3.m3.1.1.3.3.2" xref="S7.p1.3.m3.1.1.3.3.2.cmml">128</mn><mo id="S7.p1.3.m3.1.1.3.3.1" lspace="0.222em" rspace="0.222em" xref="S7.p1.3.m3.1.1.3.3.1.cmml">⋅</mo><mrow id="S7.p1.3.m3.1.1.3.3.3" xref="S7.p1.3.m3.1.1.3.3.3.cmml"><mi id="S7.p1.3.m3.1.1.3.3.3.1" xref="S7.p1.3.m3.1.1.3.3.3.1.cmml">log</mi><mo id="S7.p1.3.m3.1.1.3.3.3a" lspace="0.167em" xref="S7.p1.3.m3.1.1.3.3.3.cmml">⁡</mo><mi id="S7.p1.3.m3.1.1.3.3.3.2" xref="S7.p1.3.m3.1.1.3.3.3.2.cmml">n</mi></mrow></mrow></mfrac></mrow><annotation-xml encoding="MathML-Content" id="S7.p1.3.m3.1b"><apply id="S7.p1.3.m3.1.1.cmml" xref="S7.p1.3.m3.1.1"><eq id="S7.p1.3.m3.1.1.1.cmml" xref="S7.p1.3.m3.1.1.1"></eq><ci id="S7.p1.3.m3.1.1.2.cmml" xref="S7.p1.3.m3.1.1.2">𝜂</ci><apply id="S7.p1.3.m3.1.1.3.cmml" xref="S7.p1.3.m3.1.1.3"><divide id="S7.p1.3.m3.1.1.3.1.cmml" xref="S7.p1.3.m3.1.1.3"></divide><apply id="S7.p1.3.m3.1.1.3.2.cmml" xref="S7.p1.3.m3.1.1.3.2"><csymbol cd="ambiguous" id="S7.p1.3.m3.1.1.3.2.1.cmml" xref="S7.p1.3.m3.1.1.3.2">superscript</csymbol><ci id="S7.p1.3.m3.1.1.3.2.2.cmml" xref="S7.p1.3.m3.1.1.3.2.2">𝜀</ci><cn id="S7.p1.3.m3.1.1.3.2.3.cmml" type="integer" xref="S7.p1.3.m3.1.1.3.2.3">2</cn></apply><apply id="S7.p1.3.m3.1.1.3.3.cmml" xref="S7.p1.3.m3.1.1.3.3"><ci id="S7.p1.3.m3.1.1.3.3.1.cmml" xref="S7.p1.3.m3.1.1.3.3.1">⋅</ci><cn id="S7.p1.3.m3.1.1.3.3.2.cmml" type="integer" xref="S7.p1.3.m3.1.1.3.3.2">128</cn><apply id="S7.p1.3.m3.1.1.3.3.3.cmml" xref="S7.p1.3.m3.1.1.3.3.3"><log id="S7.p1.3.m3.1.1.3.3.3.1.cmml" xref="S7.p1.3.m3.1.1.3.3.3.1"></log><ci id="S7.p1.3.m3.1.1.3.3.3.2.cmml" xref="S7.p1.3.m3.1.1.3.3.3.2">𝑛</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.p1.3.m3.1c">\eta=\frac{\varepsilon^{2}}{128\cdot\log n}</annotation><annotation encoding="application/x-llamapun" id="S7.p1.3.m3.1d">italic_η = divide start_ARG italic_ε start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG start_ARG 128 ⋅ roman_log italic_n end_ARG</annotation></semantics></math>). Our algorithm uses as a subroutine a distributed algorithm to compute a <em class="ltx_emph ltx_font_italic" id="S7.p1.5.1">blocking flow</em> in an <math alttext="h" class="ltx_Math" display="inline" id="S7.p1.4.m4.1"><semantics id="S7.p1.4.m4.1a"><mi id="S7.p1.4.m4.1.1" xref="S7.p1.4.m4.1.1.cmml">h</mi><annotation-xml encoding="MathML-Content" id="S7.p1.4.m4.1b"><ci id="S7.p1.4.m4.1.1.cmml" xref="S7.p1.4.m4.1.1">ℎ</ci></annotation-xml><annotation encoding="application/x-tex" id="S7.p1.4.m4.1c">h</annotation><annotation encoding="application/x-llamapun" id="S7.p1.4.m4.1d">italic_h</annotation></semantics></math>-layered DAG in <math alttext="O(\textnormal{Blocking}(h,n))" class="ltx_Math" display="inline" id="S7.p1.5.m5.3"><semantics id="S7.p1.5.m5.3a"><mrow id="S7.p1.5.m5.3.3" xref="S7.p1.5.m5.3.3.cmml"><mi id="S7.p1.5.m5.3.3.3" xref="S7.p1.5.m5.3.3.3.cmml">O</mi><mo id="S7.p1.5.m5.3.3.2" xref="S7.p1.5.m5.3.3.2.cmml">⁢</mo><mrow id="S7.p1.5.m5.3.3.1.1" xref="S7.p1.5.m5.3.3.1.1.1.cmml"><mo id="S7.p1.5.m5.3.3.1.1.2" stretchy="false" xref="S7.p1.5.m5.3.3.1.1.1.cmml">(</mo><mrow id="S7.p1.5.m5.3.3.1.1.1" xref="S7.p1.5.m5.3.3.1.1.1.cmml"><mtext id="S7.p1.5.m5.3.3.1.1.1.2" xref="S7.p1.5.m5.3.3.1.1.1.2a.cmml">Blocking</mtext><mo id="S7.p1.5.m5.3.3.1.1.1.1" xref="S7.p1.5.m5.3.3.1.1.1.1.cmml">⁢</mo><mrow id="S7.p1.5.m5.3.3.1.1.1.3.2" xref="S7.p1.5.m5.3.3.1.1.1.3.1.cmml"><mo id="S7.p1.5.m5.3.3.1.1.1.3.2.1" stretchy="false" xref="S7.p1.5.m5.3.3.1.1.1.3.1.cmml">(</mo><mi id="S7.p1.5.m5.1.1" xref="S7.p1.5.m5.1.1.cmml">h</mi><mo id="S7.p1.5.m5.3.3.1.1.1.3.2.2" xref="S7.p1.5.m5.3.3.1.1.1.3.1.cmml">,</mo><mi id="S7.p1.5.m5.2.2" xref="S7.p1.5.m5.2.2.cmml">n</mi><mo id="S7.p1.5.m5.3.3.1.1.1.3.2.3" stretchy="false" xref="S7.p1.5.m5.3.3.1.1.1.3.1.cmml">)</mo></mrow></mrow><mo id="S7.p1.5.m5.3.3.1.1.3" stretchy="false" xref="S7.p1.5.m5.3.3.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S7.p1.5.m5.3b"><apply id="S7.p1.5.m5.3.3.cmml" xref="S7.p1.5.m5.3.3"><times id="S7.p1.5.m5.3.3.2.cmml" xref="S7.p1.5.m5.3.3.2"></times><ci id="S7.p1.5.m5.3.3.3.cmml" xref="S7.p1.5.m5.3.3.3">𝑂</ci><apply id="S7.p1.5.m5.3.3.1.1.1.cmml" xref="S7.p1.5.m5.3.3.1.1"><times id="S7.p1.5.m5.3.3.1.1.1.1.cmml" xref="S7.p1.5.m5.3.3.1.1.1.1"></times><ci id="S7.p1.5.m5.3.3.1.1.1.2a.cmml" xref="S7.p1.5.m5.3.3.1.1.1.2"><mtext id="S7.p1.5.m5.3.3.1.1.1.2.cmml" xref="S7.p1.5.m5.3.3.1.1.1.2">Blocking</mtext></ci><interval closure="open" id="S7.p1.5.m5.3.3.1.1.1.3.1.cmml" xref="S7.p1.5.m5.3.3.1.1.1.3.2"><ci id="S7.p1.5.m5.1.1.cmml" xref="S7.p1.5.m5.1.1">ℎ</ci><ci id="S7.p1.5.m5.2.2.cmml" xref="S7.p1.5.m5.2.2">𝑛</ci></interval></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.p1.5.m5.3c">O(\textnormal{Blocking}(h,n))</annotation><annotation encoding="application/x-llamapun" id="S7.p1.5.m5.3d">italic_O ( Blocking ( italic_h , italic_n ) )</annotation></semantics></math> rounds.</p> </div> <div class="ltx_theorem ltx_theorem_definition" id="S7.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="S7.Thmtheorem1.1.1.1">Definition 7.1</span></span><span class="ltx_text ltx_font_bold" id="S7.Thmtheorem1.2.2">.</span> </h6> <div class="ltx_para" id="S7.Thmtheorem1.p1"> <p class="ltx_p" id="S7.Thmtheorem1.p1.4"><span class="ltx_text ltx_font_italic" id="S7.Thmtheorem1.p1.4.4">An edge-capacitated DAG <math alttext="G" class="ltx_Math" display="inline" id="S7.Thmtheorem1.p1.1.1.m1.1"><semantics id="S7.Thmtheorem1.p1.1.1.m1.1a"><mi id="S7.Thmtheorem1.p1.1.1.m1.1.1" xref="S7.Thmtheorem1.p1.1.1.m1.1.1.cmml">G</mi><annotation-xml encoding="MathML-Content" id="S7.Thmtheorem1.p1.1.1.m1.1b"><ci id="S7.Thmtheorem1.p1.1.1.m1.1.1.cmml" xref="S7.Thmtheorem1.p1.1.1.m1.1.1">𝐺</ci></annotation-xml><annotation encoding="application/x-tex" id="S7.Thmtheorem1.p1.1.1.m1.1c">G</annotation><annotation encoding="application/x-llamapun" id="S7.Thmtheorem1.p1.1.1.m1.1d">italic_G</annotation></semantics></math> is <math alttext="h" class="ltx_Math" display="inline" id="S7.Thmtheorem1.p1.2.2.m2.1"><semantics id="S7.Thmtheorem1.p1.2.2.m2.1a"><mi id="S7.Thmtheorem1.p1.2.2.m2.1.1" xref="S7.Thmtheorem1.p1.2.2.m2.1.1.cmml">h</mi><annotation-xml encoding="MathML-Content" id="S7.Thmtheorem1.p1.2.2.m2.1b"><ci id="S7.Thmtheorem1.p1.2.2.m2.1.1.cmml" xref="S7.Thmtheorem1.p1.2.2.m2.1.1">ℎ</ci></annotation-xml><annotation encoding="application/x-tex" id="S7.Thmtheorem1.p1.2.2.m2.1c">h</annotation><annotation encoding="application/x-llamapun" id="S7.Thmtheorem1.p1.2.2.m2.1d">italic_h</annotation></semantics></math>-layered if the vertices can be embedded on a grid of height <math alttext="h" class="ltx_Math" display="inline" id="S7.Thmtheorem1.p1.3.3.m3.1"><semantics id="S7.Thmtheorem1.p1.3.3.m3.1a"><mi id="S7.Thmtheorem1.p1.3.3.m3.1.1" xref="S7.Thmtheorem1.p1.3.3.m3.1.1.cmml">h</mi><annotation-xml encoding="MathML-Content" id="S7.Thmtheorem1.p1.3.3.m3.1b"><ci id="S7.Thmtheorem1.p1.3.3.m3.1.1.cmml" xref="S7.Thmtheorem1.p1.3.3.m3.1.1">ℎ</ci></annotation-xml><annotation encoding="application/x-tex" id="S7.Thmtheorem1.p1.3.3.m3.1c">h</annotation><annotation encoding="application/x-llamapun" id="S7.Thmtheorem1.p1.3.3.m3.1d">italic_h</annotation></semantics></math>, such that every directed edge <math alttext="\overline{uv}" class="ltx_Math" display="inline" id="S7.Thmtheorem1.p1.4.4.m4.1"><semantics id="S7.Thmtheorem1.p1.4.4.m4.1a"><mover accent="true" id="S7.Thmtheorem1.p1.4.4.m4.1.1" xref="S7.Thmtheorem1.p1.4.4.m4.1.1.cmml"><mrow id="S7.Thmtheorem1.p1.4.4.m4.1.1.2" xref="S7.Thmtheorem1.p1.4.4.m4.1.1.2.cmml"><mi id="S7.Thmtheorem1.p1.4.4.m4.1.1.2.2" xref="S7.Thmtheorem1.p1.4.4.m4.1.1.2.2.cmml">u</mi><mo id="S7.Thmtheorem1.p1.4.4.m4.1.1.2.1" xref="S7.Thmtheorem1.p1.4.4.m4.1.1.2.1.cmml">⁢</mo><mi id="S7.Thmtheorem1.p1.4.4.m4.1.1.2.3" xref="S7.Thmtheorem1.p1.4.4.m4.1.1.2.3.cmml">v</mi></mrow><mo id="S7.Thmtheorem1.p1.4.4.m4.1.1.1" xref="S7.Thmtheorem1.p1.4.4.m4.1.1.1.cmml">¯</mo></mover><annotation-xml encoding="MathML-Content" id="S7.Thmtheorem1.p1.4.4.m4.1b"><apply id="S7.Thmtheorem1.p1.4.4.m4.1.1.cmml" xref="S7.Thmtheorem1.p1.4.4.m4.1.1"><ci id="S7.Thmtheorem1.p1.4.4.m4.1.1.1.cmml" xref="S7.Thmtheorem1.p1.4.4.m4.1.1.1">¯</ci><apply id="S7.Thmtheorem1.p1.4.4.m4.1.1.2.cmml" xref="S7.Thmtheorem1.p1.4.4.m4.1.1.2"><times id="S7.Thmtheorem1.p1.4.4.m4.1.1.2.1.cmml" xref="S7.Thmtheorem1.p1.4.4.m4.1.1.2.1"></times><ci id="S7.Thmtheorem1.p1.4.4.m4.1.1.2.2.cmml" xref="S7.Thmtheorem1.p1.4.4.m4.1.1.2.2">𝑢</ci><ci id="S7.Thmtheorem1.p1.4.4.m4.1.1.2.3.cmml" xref="S7.Thmtheorem1.p1.4.4.m4.1.1.2.3">𝑣</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.Thmtheorem1.p1.4.4.m4.1c">\overline{uv}</annotation><annotation encoding="application/x-llamapun" id="S7.Thmtheorem1.p1.4.4.m4.1d">over¯ start_ARG italic_u italic_v end_ARG</annotation></semantics></math> points downwards.</span></p> </div> </div> <div class="ltx_theorem ltx_theorem_definition" id="S7.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="S7.Thmtheorem2.1.1.1">Definition 7.2</span></span><span class="ltx_text ltx_font_bold" id="S7.Thmtheorem2.2.2">.</span> </h6> <div class="ltx_para" id="S7.Thmtheorem2.p1"> <p class="ltx_p" id="S7.Thmtheorem2.p1.7"><span class="ltx_text ltx_font_italic" id="S7.Thmtheorem2.p1.7.7">For an edge-capacitated DAG <math alttext="G" class="ltx_Math" display="inline" id="S7.Thmtheorem2.p1.1.1.m1.1"><semantics id="S7.Thmtheorem2.p1.1.1.m1.1a"><mi id="S7.Thmtheorem2.p1.1.1.m1.1.1" xref="S7.Thmtheorem2.p1.1.1.m1.1.1.cmml">G</mi><annotation-xml encoding="MathML-Content" id="S7.Thmtheorem2.p1.1.1.m1.1b"><ci id="S7.Thmtheorem2.p1.1.1.m1.1.1.cmml" xref="S7.Thmtheorem2.p1.1.1.m1.1.1">𝐺</ci></annotation-xml><annotation encoding="application/x-tex" id="S7.Thmtheorem2.p1.1.1.m1.1c">G</annotation><annotation encoding="application/x-llamapun" id="S7.Thmtheorem2.p1.1.1.m1.1d">italic_G</annotation></semantics></math> with sources <math alttext="S" class="ltx_Math" display="inline" id="S7.Thmtheorem2.p1.2.2.m2.1"><semantics id="S7.Thmtheorem2.p1.2.2.m2.1a"><mi id="S7.Thmtheorem2.p1.2.2.m2.1.1" xref="S7.Thmtheorem2.p1.2.2.m2.1.1.cmml">S</mi><annotation-xml encoding="MathML-Content" id="S7.Thmtheorem2.p1.2.2.m2.1b"><ci id="S7.Thmtheorem2.p1.2.2.m2.1.1.cmml" xref="S7.Thmtheorem2.p1.2.2.m2.1.1">𝑆</ci></annotation-xml><annotation encoding="application/x-tex" id="S7.Thmtheorem2.p1.2.2.m2.1c">S</annotation><annotation encoding="application/x-llamapun" id="S7.Thmtheorem2.p1.2.2.m2.1d">italic_S</annotation></semantics></math> and terminals <math alttext="T" class="ltx_Math" display="inline" id="S7.Thmtheorem2.p1.3.3.m3.1"><semantics id="S7.Thmtheorem2.p1.3.3.m3.1a"><mi id="S7.Thmtheorem2.p1.3.3.m3.1.1" xref="S7.Thmtheorem2.p1.3.3.m3.1.1.cmml">T</mi><annotation-xml encoding="MathML-Content" id="S7.Thmtheorem2.p1.3.3.m3.1b"><ci id="S7.Thmtheorem2.p1.3.3.m3.1.1.cmml" xref="S7.Thmtheorem2.p1.3.3.m3.1.1">𝑇</ci></annotation-xml><annotation encoding="application/x-tex" id="S7.Thmtheorem2.p1.3.3.m3.1c">T</annotation><annotation encoding="application/x-llamapun" id="S7.Thmtheorem2.p1.3.3.m3.1d">italic_T</annotation></semantics></math>, a flow <math alttext="f" class="ltx_Math" display="inline" id="S7.Thmtheorem2.p1.4.4.m4.1"><semantics id="S7.Thmtheorem2.p1.4.4.m4.1a"><mi id="S7.Thmtheorem2.p1.4.4.m4.1.1" xref="S7.Thmtheorem2.p1.4.4.m4.1.1.cmml">f</mi><annotation-xml encoding="MathML-Content" id="S7.Thmtheorem2.p1.4.4.m4.1b"><ci id="S7.Thmtheorem2.p1.4.4.m4.1.1.cmml" xref="S7.Thmtheorem2.p1.4.4.m4.1.1">𝑓</ci></annotation-xml><annotation encoding="application/x-tex" id="S7.Thmtheorem2.p1.4.4.m4.1c">f</annotation><annotation encoding="application/x-llamapun" id="S7.Thmtheorem2.p1.4.4.m4.1d">italic_f</annotation></semantics></math> from <math alttext="S" class="ltx_Math" display="inline" id="S7.Thmtheorem2.p1.5.5.m5.1"><semantics id="S7.Thmtheorem2.p1.5.5.m5.1a"><mi id="S7.Thmtheorem2.p1.5.5.m5.1.1" xref="S7.Thmtheorem2.p1.5.5.m5.1.1.cmml">S</mi><annotation-xml encoding="MathML-Content" id="S7.Thmtheorem2.p1.5.5.m5.1b"><ci id="S7.Thmtheorem2.p1.5.5.m5.1.1.cmml" xref="S7.Thmtheorem2.p1.5.5.m5.1.1">𝑆</ci></annotation-xml><annotation encoding="application/x-tex" id="S7.Thmtheorem2.p1.5.5.m5.1c">S</annotation><annotation encoding="application/x-llamapun" id="S7.Thmtheorem2.p1.5.5.m5.1d">italic_S</annotation></semantics></math> to <math alttext="T" class="ltx_Math" display="inline" id="S7.Thmtheorem2.p1.6.6.m6.1"><semantics id="S7.Thmtheorem2.p1.6.6.m6.1a"><mi id="S7.Thmtheorem2.p1.6.6.m6.1.1" xref="S7.Thmtheorem2.p1.6.6.m6.1.1.cmml">T</mi><annotation-xml encoding="MathML-Content" id="S7.Thmtheorem2.p1.6.6.m6.1b"><ci id="S7.Thmtheorem2.p1.6.6.m6.1.1.cmml" xref="S7.Thmtheorem2.p1.6.6.m6.1.1">𝑇</ci></annotation-xml><annotation encoding="application/x-tex" id="S7.Thmtheorem2.p1.6.6.m6.1c">T</annotation><annotation encoding="application/x-llamapun" id="S7.Thmtheorem2.p1.6.6.m6.1d">italic_T</annotation></semantics></math> is <em class="ltx_emph ltx_font_upright" id="S7.Thmtheorem2.p1.7.7.1">blocking</em> if every augmenting path of <math alttext="f" class="ltx_Math" display="inline" id="S7.Thmtheorem2.p1.7.7.m7.1"><semantics id="S7.Thmtheorem2.p1.7.7.m7.1a"><mi id="S7.Thmtheorem2.p1.7.7.m7.1.1" xref="S7.Thmtheorem2.p1.7.7.m7.1.1.cmml">f</mi><annotation-xml encoding="MathML-Content" id="S7.Thmtheorem2.p1.7.7.m7.1b"><ci id="S7.Thmtheorem2.p1.7.7.m7.1.1.cmml" xref="S7.Thmtheorem2.p1.7.7.m7.1.1">𝑓</ci></annotation-xml><annotation encoding="application/x-tex" id="S7.Thmtheorem2.p1.7.7.m7.1c">f</annotation><annotation encoding="application/x-llamapun" id="S7.Thmtheorem2.p1.7.7.m7.1d">italic_f</annotation></semantics></math> contains at least one saturated edge.</span></p> </div> </div> <div class="ltx_theorem ltx_theorem_lemma" id="S7.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="S7.Thmtheorem3.1.1.1">Lemma 7.3</span></span><span class="ltx_text ltx_font_bold" id="S7.Thmtheorem3.2.2"> </span>(Lemma 7.2 and 9.1 in <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2411.12694v2#bib.bib15" title="">15</a>]</cite>)<span class="ltx_text ltx_font_bold" id="S7.Thmtheorem3.3.3">.</span> </h6> <div class="ltx_para" id="S7.Thmtheorem3.p1"> <p class="ltx_p" id="S7.Thmtheorem3.p1.5"><span class="ltx_text ltx_font_italic" id="S7.Thmtheorem3.p1.5.5">There exists an algorithm which, given an <math alttext="n" class="ltx_Math" display="inline" id="S7.Thmtheorem3.p1.1.1.m1.1"><semantics id="S7.Thmtheorem3.p1.1.1.m1.1a"><mi id="S7.Thmtheorem3.p1.1.1.m1.1.1" xref="S7.Thmtheorem3.p1.1.1.m1.1.1.cmml">n</mi><annotation-xml encoding="MathML-Content" id="S7.Thmtheorem3.p1.1.1.m1.1b"><ci id="S7.Thmtheorem3.p1.1.1.m1.1.1.cmml" xref="S7.Thmtheorem3.p1.1.1.m1.1.1">𝑛</ci></annotation-xml><annotation encoding="application/x-tex" id="S7.Thmtheorem3.p1.1.1.m1.1c">n</annotation><annotation encoding="application/x-llamapun" id="S7.Thmtheorem3.p1.1.1.m1.1d">italic_n</annotation></semantics></math>-node <math alttext="h" class="ltx_Math" display="inline" id="S7.Thmtheorem3.p1.2.2.m2.1"><semantics id="S7.Thmtheorem3.p1.2.2.m2.1a"><mi id="S7.Thmtheorem3.p1.2.2.m2.1.1" xref="S7.Thmtheorem3.p1.2.2.m2.1.1.cmml">h</mi><annotation-xml encoding="MathML-Content" id="S7.Thmtheorem3.p1.2.2.m2.1b"><ci id="S7.Thmtheorem3.p1.2.2.m2.1.1.cmml" xref="S7.Thmtheorem3.p1.2.2.m2.1.1">ℎ</ci></annotation-xml><annotation encoding="application/x-tex" id="S7.Thmtheorem3.p1.2.2.m2.1c">h</annotation><annotation encoding="application/x-llamapun" id="S7.Thmtheorem3.p1.2.2.m2.1d">italic_h</annotation></semantics></math>-layer edge-capacitated DAG <math alttext="D" class="ltx_Math" display="inline" id="S7.Thmtheorem3.p1.3.3.m3.1"><semantics id="S7.Thmtheorem3.p1.3.3.m3.1a"><mi id="S7.Thmtheorem3.p1.3.3.m3.1.1" xref="S7.Thmtheorem3.p1.3.3.m3.1.1.cmml">D</mi><annotation-xml encoding="MathML-Content" id="S7.Thmtheorem3.p1.3.3.m3.1b"><ci id="S7.Thmtheorem3.p1.3.3.m3.1.1.cmml" xref="S7.Thmtheorem3.p1.3.3.m3.1.1">𝐷</ci></annotation-xml><annotation encoding="application/x-tex" id="S7.Thmtheorem3.p1.3.3.m3.1c">D</annotation><annotation encoding="application/x-llamapun" id="S7.Thmtheorem3.p1.3.3.m3.1d">italic_D</annotation></semantics></math> with sources <math alttext="S" class="ltx_Math" display="inline" id="S7.Thmtheorem3.p1.4.4.m4.1"><semantics id="S7.Thmtheorem3.p1.4.4.m4.1a"><mi id="S7.Thmtheorem3.p1.4.4.m4.1.1" xref="S7.Thmtheorem3.p1.4.4.m4.1.1.cmml">S</mi><annotation-xml encoding="MathML-Content" id="S7.Thmtheorem3.p1.4.4.m4.1b"><ci id="S7.Thmtheorem3.p1.4.4.m4.1.1.cmml" xref="S7.Thmtheorem3.p1.4.4.m4.1.1">𝑆</ci></annotation-xml><annotation encoding="application/x-tex" id="S7.Thmtheorem3.p1.4.4.m4.1c">S</annotation><annotation encoding="application/x-llamapun" id="S7.Thmtheorem3.p1.4.4.m4.1d">italic_S</annotation></semantics></math> and terminals <math alttext="T" class="ltx_Math" display="inline" id="S7.Thmtheorem3.p1.5.5.m5.1"><semantics id="S7.Thmtheorem3.p1.5.5.m5.1a"><mi id="S7.Thmtheorem3.p1.5.5.m5.1.1" xref="S7.Thmtheorem3.p1.5.5.m5.1.1.cmml">T</mi><annotation-xml encoding="MathML-Content" id="S7.Thmtheorem3.p1.5.5.m5.1b"><ci id="S7.Thmtheorem3.p1.5.5.m5.1.1.cmml" xref="S7.Thmtheorem3.p1.5.5.m5.1.1">𝑇</ci></annotation-xml><annotation encoding="application/x-tex" id="S7.Thmtheorem3.p1.5.5.m5.1c">T</annotation><annotation encoding="application/x-llamapun" id="S7.Thmtheorem3.p1.5.5.m5.1d">italic_T</annotation></semantics></math> computes a blocking ST-flow in CONGEST in:</span></p> <ul class="ltx_itemize" id="S7.I1"> <li class="ltx_item" id="S7.I1.i1" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">•</span> <div class="ltx_para" id="S7.I1.i1.p1"> <p class="ltx_p" id="S7.I1.i1.p1.1"><math alttext="\textnormal{Blocking}(h,n)=\tilde{O}(h^{4})" class="ltx_Math" display="inline" id="S7.I1.i1.p1.1.m1.3"><semantics id="S7.I1.i1.p1.1.m1.3a"><mrow id="S7.I1.i1.p1.1.m1.3.3" xref="S7.I1.i1.p1.1.m1.3.3.cmml"><mrow id="S7.I1.i1.p1.1.m1.3.3.3" xref="S7.I1.i1.p1.1.m1.3.3.3.cmml"><mtext id="S7.I1.i1.p1.1.m1.3.3.3.2" xref="S7.I1.i1.p1.1.m1.3.3.3.2a.cmml">Blocking</mtext><mo id="S7.I1.i1.p1.1.m1.3.3.3.1" xref="S7.I1.i1.p1.1.m1.3.3.3.1.cmml">⁢</mo><mrow id="S7.I1.i1.p1.1.m1.3.3.3.3.2" xref="S7.I1.i1.p1.1.m1.3.3.3.3.1.cmml"><mo id="S7.I1.i1.p1.1.m1.3.3.3.3.2.1" stretchy="false" xref="S7.I1.i1.p1.1.m1.3.3.3.3.1.cmml">(</mo><mi id="S7.I1.i1.p1.1.m1.1.1" xref="S7.I1.i1.p1.1.m1.1.1.cmml">h</mi><mo id="S7.I1.i1.p1.1.m1.3.3.3.3.2.2" xref="S7.I1.i1.p1.1.m1.3.3.3.3.1.cmml">,</mo><mi id="S7.I1.i1.p1.1.m1.2.2" xref="S7.I1.i1.p1.1.m1.2.2.cmml">n</mi><mo id="S7.I1.i1.p1.1.m1.3.3.3.3.2.3" stretchy="false" xref="S7.I1.i1.p1.1.m1.3.3.3.3.1.cmml">)</mo></mrow></mrow><mo id="S7.I1.i1.p1.1.m1.3.3.2" xref="S7.I1.i1.p1.1.m1.3.3.2.cmml">=</mo><mrow id="S7.I1.i1.p1.1.m1.3.3.1" xref="S7.I1.i1.p1.1.m1.3.3.1.cmml"><mover accent="true" id="S7.I1.i1.p1.1.m1.3.3.1.3" xref="S7.I1.i1.p1.1.m1.3.3.1.3.cmml"><mi id="S7.I1.i1.p1.1.m1.3.3.1.3.2" xref="S7.I1.i1.p1.1.m1.3.3.1.3.2.cmml">O</mi><mo id="S7.I1.i1.p1.1.m1.3.3.1.3.1" xref="S7.I1.i1.p1.1.m1.3.3.1.3.1.cmml">~</mo></mover><mo id="S7.I1.i1.p1.1.m1.3.3.1.2" xref="S7.I1.i1.p1.1.m1.3.3.1.2.cmml">⁢</mo><mrow id="S7.I1.i1.p1.1.m1.3.3.1.1.1" xref="S7.I1.i1.p1.1.m1.3.3.1.1.1.1.cmml"><mo id="S7.I1.i1.p1.1.m1.3.3.1.1.1.2" stretchy="false" xref="S7.I1.i1.p1.1.m1.3.3.1.1.1.1.cmml">(</mo><msup id="S7.I1.i1.p1.1.m1.3.3.1.1.1.1" xref="S7.I1.i1.p1.1.m1.3.3.1.1.1.1.cmml"><mi id="S7.I1.i1.p1.1.m1.3.3.1.1.1.1.2" xref="S7.I1.i1.p1.1.m1.3.3.1.1.1.1.2.cmml">h</mi><mn id="S7.I1.i1.p1.1.m1.3.3.1.1.1.1.3" xref="S7.I1.i1.p1.1.m1.3.3.1.1.1.1.3.cmml">4</mn></msup><mo id="S7.I1.i1.p1.1.m1.3.3.1.1.1.3" stretchy="false" xref="S7.I1.i1.p1.1.m1.3.3.1.1.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S7.I1.i1.p1.1.m1.3b"><apply id="S7.I1.i1.p1.1.m1.3.3.cmml" xref="S7.I1.i1.p1.1.m1.3.3"><eq id="S7.I1.i1.p1.1.m1.3.3.2.cmml" xref="S7.I1.i1.p1.1.m1.3.3.2"></eq><apply id="S7.I1.i1.p1.1.m1.3.3.3.cmml" xref="S7.I1.i1.p1.1.m1.3.3.3"><times id="S7.I1.i1.p1.1.m1.3.3.3.1.cmml" xref="S7.I1.i1.p1.1.m1.3.3.3.1"></times><ci id="S7.I1.i1.p1.1.m1.3.3.3.2a.cmml" xref="S7.I1.i1.p1.1.m1.3.3.3.2"><mtext id="S7.I1.i1.p1.1.m1.3.3.3.2.cmml" xref="S7.I1.i1.p1.1.m1.3.3.3.2">Blocking</mtext></ci><interval closure="open" id="S7.I1.i1.p1.1.m1.3.3.3.3.1.cmml" xref="S7.I1.i1.p1.1.m1.3.3.3.3.2"><ci id="S7.I1.i1.p1.1.m1.1.1.cmml" xref="S7.I1.i1.p1.1.m1.1.1">ℎ</ci><ci id="S7.I1.i1.p1.1.m1.2.2.cmml" xref="S7.I1.i1.p1.1.m1.2.2">𝑛</ci></interval></apply><apply id="S7.I1.i1.p1.1.m1.3.3.1.cmml" xref="S7.I1.i1.p1.1.m1.3.3.1"><times id="S7.I1.i1.p1.1.m1.3.3.1.2.cmml" xref="S7.I1.i1.p1.1.m1.3.3.1.2"></times><apply id="S7.I1.i1.p1.1.m1.3.3.1.3.cmml" xref="S7.I1.i1.p1.1.m1.3.3.1.3"><ci id="S7.I1.i1.p1.1.m1.3.3.1.3.1.cmml" xref="S7.I1.i1.p1.1.m1.3.3.1.3.1">~</ci><ci id="S7.I1.i1.p1.1.m1.3.3.1.3.2.cmml" xref="S7.I1.i1.p1.1.m1.3.3.1.3.2">𝑂</ci></apply><apply id="S7.I1.i1.p1.1.m1.3.3.1.1.1.1.cmml" xref="S7.I1.i1.p1.1.m1.3.3.1.1.1"><csymbol cd="ambiguous" id="S7.I1.i1.p1.1.m1.3.3.1.1.1.1.1.cmml" xref="S7.I1.i1.p1.1.m1.3.3.1.1.1">superscript</csymbol><ci id="S7.I1.i1.p1.1.m1.3.3.1.1.1.1.2.cmml" xref="S7.I1.i1.p1.1.m1.3.3.1.1.1.1.2">ℎ</ci><cn id="S7.I1.i1.p1.1.m1.3.3.1.1.1.1.3.cmml" type="integer" xref="S7.I1.i1.p1.1.m1.3.3.1.1.1.1.3">4</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.I1.i1.p1.1.m1.3c">\textnormal{Blocking}(h,n)=\tilde{O}(h^{4})</annotation><annotation encoding="application/x-llamapun" id="S7.I1.i1.p1.1.m1.3d">Blocking ( italic_h , italic_n ) = over~ start_ARG italic_O end_ARG ( italic_h start_POSTSUPERSCRIPT 4 end_POSTSUPERSCRIPT )</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S7.I1.i1.p1.1.1"> rounds with high probability,</span></p> </div> </li> <li class="ltx_item" id="S7.I1.i2" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">•</span> <div class="ltx_para" id="S7.I1.i2.p1"> <p class="ltx_p" id="S7.I1.i2.p1.2"><math alttext="\textnormal{Blocking}(h,n)=\tilde{O}(h^{6}\cdot 2^{c\sqrt{\log n}})" class="ltx_Math" display="inline" id="S7.I1.i2.p1.1.m1.3"><semantics id="S7.I1.i2.p1.1.m1.3a"><mrow id="S7.I1.i2.p1.1.m1.3.3" xref="S7.I1.i2.p1.1.m1.3.3.cmml"><mrow id="S7.I1.i2.p1.1.m1.3.3.3" xref="S7.I1.i2.p1.1.m1.3.3.3.cmml"><mtext id="S7.I1.i2.p1.1.m1.3.3.3.2" 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id="S7.I1.i2.p1.1.m1.3b"><apply id="S7.I1.i2.p1.1.m1.3.3.cmml" xref="S7.I1.i2.p1.1.m1.3.3"><eq id="S7.I1.i2.p1.1.m1.3.3.2.cmml" xref="S7.I1.i2.p1.1.m1.3.3.2"></eq><apply id="S7.I1.i2.p1.1.m1.3.3.3.cmml" xref="S7.I1.i2.p1.1.m1.3.3.3"><times id="S7.I1.i2.p1.1.m1.3.3.3.1.cmml" xref="S7.I1.i2.p1.1.m1.3.3.3.1"></times><ci id="S7.I1.i2.p1.1.m1.3.3.3.2a.cmml" xref="S7.I1.i2.p1.1.m1.3.3.3.2"><mtext id="S7.I1.i2.p1.1.m1.3.3.3.2.cmml" xref="S7.I1.i2.p1.1.m1.3.3.3.2">Blocking</mtext></ci><interval closure="open" id="S7.I1.i2.p1.1.m1.3.3.3.3.1.cmml" xref="S7.I1.i2.p1.1.m1.3.3.3.3.2"><ci id="S7.I1.i2.p1.1.m1.1.1.cmml" xref="S7.I1.i2.p1.1.m1.1.1">ℎ</ci><ci id="S7.I1.i2.p1.1.m1.2.2.cmml" xref="S7.I1.i2.p1.1.m1.2.2">𝑛</ci></interval></apply><apply id="S7.I1.i2.p1.1.m1.3.3.1.cmml" xref="S7.I1.i2.p1.1.m1.3.3.1"><times id="S7.I1.i2.p1.1.m1.3.3.1.2.cmml" xref="S7.I1.i2.p1.1.m1.3.3.1.2"></times><apply id="S7.I1.i2.p1.1.m1.3.3.1.3.cmml" xref="S7.I1.i2.p1.1.m1.3.3.1.3"><ci 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id="S7.I1.i2.p1.1.m1.3.3.1.1.1.1.3.2.cmml" type="integer" xref="S7.I1.i2.p1.1.m1.3.3.1.1.1.1.3.2">2</cn><apply id="S7.I1.i2.p1.1.m1.3.3.1.1.1.1.3.3.cmml" xref="S7.I1.i2.p1.1.m1.3.3.1.1.1.1.3.3"><times id="S7.I1.i2.p1.1.m1.3.3.1.1.1.1.3.3.1.cmml" xref="S7.I1.i2.p1.1.m1.3.3.1.1.1.1.3.3.1"></times><ci id="S7.I1.i2.p1.1.m1.3.3.1.1.1.1.3.3.2.cmml" xref="S7.I1.i2.p1.1.m1.3.3.1.1.1.1.3.3.2">𝑐</ci><apply id="S7.I1.i2.p1.1.m1.3.3.1.1.1.1.3.3.3.cmml" xref="S7.I1.i2.p1.1.m1.3.3.1.1.1.1.3.3.3"><root id="S7.I1.i2.p1.1.m1.3.3.1.1.1.1.3.3.3a.cmml" xref="S7.I1.i2.p1.1.m1.3.3.1.1.1.1.3.3.3"></root><apply id="S7.I1.i2.p1.1.m1.3.3.1.1.1.1.3.3.3.2.cmml" xref="S7.I1.i2.p1.1.m1.3.3.1.1.1.1.3.3.3.2"><log id="S7.I1.i2.p1.1.m1.3.3.1.1.1.1.3.3.3.2.1.cmml" xref="S7.I1.i2.p1.1.m1.3.3.1.1.1.1.3.3.3.2.1"></log><ci id="S7.I1.i2.p1.1.m1.3.3.1.1.1.1.3.3.3.2.2.cmml" xref="S7.I1.i2.p1.1.m1.3.3.1.1.1.1.3.3.3.2.2">𝑛</ci></apply></apply></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.I1.i2.p1.1.m1.3c">\textnormal{Blocking}(h,n)=\tilde{O}(h^{6}\cdot 2^{c\sqrt{\log n}})</annotation><annotation encoding="application/x-llamapun" id="S7.I1.i2.p1.1.m1.3d">Blocking ( italic_h , italic_n ) = over~ start_ARG italic_O end_ARG ( italic_h start_POSTSUPERSCRIPT 6 end_POSTSUPERSCRIPT ⋅ 2 start_POSTSUPERSCRIPT italic_c square-root start_ARG roman_log italic_n end_ARG end_POSTSUPERSCRIPT )</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S7.I1.i2.p1.2.1"> deterministic rounds for some constant </span><math alttext="c" class="ltx_Math" display="inline" id="S7.I1.i2.p1.2.m2.1"><semantics id="S7.I1.i2.p1.2.m2.1a"><mi id="S7.I1.i2.p1.2.m2.1.1" xref="S7.I1.i2.p1.2.m2.1.1.cmml">c</mi><annotation-xml encoding="MathML-Content" id="S7.I1.i2.p1.2.m2.1b"><ci id="S7.I1.i2.p1.2.m2.1.1.cmml" xref="S7.I1.i2.p1.2.m2.1.1">𝑐</ci></annotation-xml><annotation encoding="application/x-tex" id="S7.I1.i2.p1.2.m2.1c">c</annotation><annotation encoding="application/x-llamapun" id="S7.I1.i2.p1.2.m2.1d">italic_c</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S7.I1.i2.p1.2.2">.</span></p> </div> </li> </ul> </div> </div> <div class="ltx_para" id="S7.p2"> <p class="ltx_p" id="S7.p2.7">We compute an <math alttext="\eta" class="ltx_Math" display="inline" id="S7.p2.1.m1.1"><semantics id="S7.p2.1.m1.1a"><mi id="S7.p2.1.m1.1.1" xref="S7.p2.1.m1.1.1.cmml">η</mi><annotation-xml encoding="MathML-Content" id="S7.p2.1.m1.1b"><ci id="S7.p2.1.m1.1.1.cmml" xref="S7.p2.1.m1.1.1">𝜂</ci></annotation-xml><annotation encoding="application/x-tex" id="S7.p2.1.m1.1c">\eta</annotation><annotation encoding="application/x-llamapun" id="S7.p2.1.m1.1d">italic_η</annotation></semantics></math>-fair orientation by repeatedly constructing a DAG with <math alttext="h\in O(\varepsilon^{-2}\log^{2}n)" class="ltx_Math" display="inline" id="S7.p2.2.m2.1"><semantics id="S7.p2.2.m2.1a"><mrow id="S7.p2.2.m2.1.1" xref="S7.p2.2.m2.1.1.cmml"><mi id="S7.p2.2.m2.1.1.3" xref="S7.p2.2.m2.1.1.3.cmml">h</mi><mo id="S7.p2.2.m2.1.1.2" xref="S7.p2.2.m2.1.1.2.cmml">∈</mo><mrow id="S7.p2.2.m2.1.1.1" xref="S7.p2.2.m2.1.1.1.cmml"><mi id="S7.p2.2.m2.1.1.1.3" xref="S7.p2.2.m2.1.1.1.3.cmml">O</mi><mo id="S7.p2.2.m2.1.1.1.2" xref="S7.p2.2.m2.1.1.1.2.cmml">⁢</mo><mrow id="S7.p2.2.m2.1.1.1.1.1" xref="S7.p2.2.m2.1.1.1.1.1.1.cmml"><mo id="S7.p2.2.m2.1.1.1.1.1.2" stretchy="false" xref="S7.p2.2.m2.1.1.1.1.1.1.cmml">(</mo><mrow id="S7.p2.2.m2.1.1.1.1.1.1" xref="S7.p2.2.m2.1.1.1.1.1.1.cmml"><msup id="S7.p2.2.m2.1.1.1.1.1.1.2" xref="S7.p2.2.m2.1.1.1.1.1.1.2.cmml"><mi id="S7.p2.2.m2.1.1.1.1.1.1.2.2" xref="S7.p2.2.m2.1.1.1.1.1.1.2.2.cmml">ε</mi><mrow id="S7.p2.2.m2.1.1.1.1.1.1.2.3" xref="S7.p2.2.m2.1.1.1.1.1.1.2.3.cmml"><mo id="S7.p2.2.m2.1.1.1.1.1.1.2.3a" xref="S7.p2.2.m2.1.1.1.1.1.1.2.3.cmml">−</mo><mn id="S7.p2.2.m2.1.1.1.1.1.1.2.3.2" xref="S7.p2.2.m2.1.1.1.1.1.1.2.3.2.cmml">2</mn></mrow></msup><mo 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xref="S7.p2.2.m2.1.1.1.1.1.1.3.1"><csymbol cd="ambiguous" id="S7.p2.2.m2.1.1.1.1.1.1.3.1.1.cmml" xref="S7.p2.2.m2.1.1.1.1.1.1.3.1">superscript</csymbol><log id="S7.p2.2.m2.1.1.1.1.1.1.3.1.2.cmml" xref="S7.p2.2.m2.1.1.1.1.1.1.3.1.2"></log><cn id="S7.p2.2.m2.1.1.1.1.1.1.3.1.3.cmml" type="integer" xref="S7.p2.2.m2.1.1.1.1.1.1.3.1.3">2</cn></apply><ci id="S7.p2.2.m2.1.1.1.1.1.1.3.2.cmml" xref="S7.p2.2.m2.1.1.1.1.1.1.3.2">𝑛</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.p2.2.m2.1c">h\in O(\varepsilon^{-2}\log^{2}n)</annotation><annotation encoding="application/x-llamapun" id="S7.p2.2.m2.1d">italic_h ∈ italic_O ( italic_ε start_POSTSUPERSCRIPT - 2 end_POSTSUPERSCRIPT roman_log start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_n )</annotation></semantics></math> and computing blocking flows. Theorem <a class="ltx_ref" href="https://arxiv.org/html/2411.12694v2#S3.Thmtheorem5" title="Theorem 3.5. ‣ 3.B Results for dynamic algorithms ‣ 3 Results and organisation ‣ Local Density and its Distributed Approximation"><span class="ltx_text ltx_ref_tag">3.5</span></a> implies in an <math alttext="\eta" class="ltx_Math" display="inline" id="S7.p2.3.m3.1"><semantics id="S7.p2.3.m3.1a"><mi id="S7.p2.3.m3.1.1" xref="S7.p2.3.m3.1.1.cmml">η</mi><annotation-xml encoding="MathML-Content" id="S7.p2.3.m3.1b"><ci id="S7.p2.3.m3.1.1.cmml" xref="S7.p2.3.m3.1.1">𝜂</ci></annotation-xml><annotation encoding="application/x-tex" id="S7.p2.3.m3.1c">\eta</annotation><annotation encoding="application/x-llamapun" id="S7.p2.3.m3.1d">italic_η</annotation></semantics></math>-fair orientation (for our choice of <math alttext="\eta" class="ltx_Math" display="inline" id="S7.p2.4.m4.1"><semantics id="S7.p2.4.m4.1a"><mi id="S7.p2.4.m4.1.1" xref="S7.p2.4.m4.1.1.cmml">η</mi><annotation-xml encoding="MathML-Content" id="S7.p2.4.m4.1b"><ci id="S7.p2.4.m4.1.1.cmml" xref="S7.p2.4.m4.1.1">𝜂</ci></annotation-xml><annotation encoding="application/x-tex" id="S7.p2.4.m4.1c">\eta</annotation><annotation encoding="application/x-llamapun" id="S7.p2.4.m4.1d">italic_η</annotation></semantics></math>) the out-degree of each vertex <math alttext="v" class="ltx_Math" display="inline" id="S7.p2.5.m5.1"><semantics id="S7.p2.5.m5.1a"><mi id="S7.p2.5.m5.1.1" xref="S7.p2.5.m5.1.1.cmml">v</mi><annotation-xml encoding="MathML-Content" id="S7.p2.5.m5.1b"><ci id="S7.p2.5.m5.1.1.cmml" xref="S7.p2.5.m5.1.1">𝑣</ci></annotation-xml><annotation encoding="application/x-tex" id="S7.p2.5.m5.1c">v</annotation><annotation encoding="application/x-llamapun" id="S7.p2.5.m5.1d">italic_v</annotation></semantics></math> is a <math alttext="(1+\varepsilon)" class="ltx_Math" display="inline" id="S7.p2.6.m6.1"><semantics id="S7.p2.6.m6.1a"><mrow id="S7.p2.6.m6.1.1.1" xref="S7.p2.6.m6.1.1.1.1.cmml"><mo 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)</annotation></semantics></math>-approximation <math alttext="\rho^{*}(v)" class="ltx_Math" display="inline" id="S7.p2.7.m7.1"><semantics id="S7.p2.7.m7.1a"><mrow id="S7.p2.7.m7.1.2" xref="S7.p2.7.m7.1.2.cmml"><msup id="S7.p2.7.m7.1.2.2" xref="S7.p2.7.m7.1.2.2.cmml"><mi id="S7.p2.7.m7.1.2.2.2" xref="S7.p2.7.m7.1.2.2.2.cmml">ρ</mi><mo id="S7.p2.7.m7.1.2.2.3" xref="S7.p2.7.m7.1.2.2.3.cmml">∗</mo></msup><mo id="S7.p2.7.m7.1.2.1" xref="S7.p2.7.m7.1.2.1.cmml">⁢</mo><mrow id="S7.p2.7.m7.1.2.3.2" xref="S7.p2.7.m7.1.2.cmml"><mo id="S7.p2.7.m7.1.2.3.2.1" stretchy="false" xref="S7.p2.7.m7.1.2.cmml">(</mo><mi id="S7.p2.7.m7.1.1" xref="S7.p2.7.m7.1.1.cmml">v</mi><mo id="S7.p2.7.m7.1.2.3.2.2" stretchy="false" xref="S7.p2.7.m7.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S7.p2.7.m7.1b"><apply id="S7.p2.7.m7.1.2.cmml" xref="S7.p2.7.m7.1.2"><times id="S7.p2.7.m7.1.2.1.cmml" xref="S7.p2.7.m7.1.2.1"></times><apply id="S7.p2.7.m7.1.2.2.cmml" 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xref="S7.SS0.SSS0.P0.SPx1.p1.2.m2.2.2.1.1.1.1.1">→</ci><ci id="S7.SS0.SSS0.P0.SPx1.p1.2.m2.2.2.1.1.1.1.2.cmml" xref="S7.SS0.SSS0.P0.SPx1.p1.2.m2.2.2.1.1.1.1.2">𝑢</ci><ci id="S7.SS0.SSS0.P0.SPx1.p1.2.m2.2.2.1.1.1.1.3.cmml" xref="S7.SS0.SSS0.P0.SPx1.p1.2.m2.2.2.1.1.1.1.3">𝑣</ci></apply></apply><apply id="S7.SS0.SSS0.P0.SPx1.p1.2.m2.2.2.3.cmml" xref="S7.SS0.SSS0.P0.SPx1.p1.2.m2.2.2.3"><times id="S7.SS0.SSS0.P0.SPx1.p1.2.m2.2.2.3.1.cmml" xref="S7.SS0.SSS0.P0.SPx1.p1.2.m2.2.2.3.1"></times><apply id="S7.SS0.SSS0.P0.SPx1.p1.2.m2.2.2.3.2.cmml" xref="S7.SS0.SSS0.P0.SPx1.p1.2.m2.2.2.3.2"><divide id="S7.SS0.SSS0.P0.SPx1.p1.2.m2.2.2.3.2.1.cmml" xref="S7.SS0.SSS0.P0.SPx1.p1.2.m2.2.2.3.2"></divide><cn id="S7.SS0.SSS0.P0.SPx1.p1.2.m2.2.2.3.2.2.cmml" type="integer" xref="S7.SS0.SSS0.P0.SPx1.p1.2.m2.2.2.3.2.2">1</cn><cn id="S7.SS0.SSS0.P0.SPx1.p1.2.m2.2.2.3.2.3.cmml" type="integer" xref="S7.SS0.SSS0.P0.SPx1.p1.2.m2.2.2.3.2.3">2</cn></apply><ci id="S7.SS0.SSS0.P0.SPx1.p1.2.m2.2.2.3.3a.cmml" xref="S7.SS0.SSS0.P0.SPx1.p1.2.m2.2.2.3.3"><mtext class="ltx_mathvariant_italic" id="S7.SS0.SSS0.P0.SPx1.p1.2.m2.2.2.3.3.cmml" xref="S7.SS0.SSS0.P0.SPx1.p1.2.m2.2.2.3.3">g</mtext></ci><ci id="S7.SS0.SSS0.P0.SPx1.p1.2.m2.1.1.cmml" xref="S7.SS0.SSS0.P0.SPx1.p1.2.m2.1.1">𝑒</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.SS0.SSS0.P0.SPx1.p1.2.m2.2c">\textsl{g}(u\!\to\!v)=\frac{1}{2}\textsl{g}(e)</annotation><annotation encoding="application/x-llamapun" id="S7.SS0.SSS0.P0.SPx1.p1.2.m2.2d">g ( italic_u → italic_v ) = divide start_ARG 1 end_ARG start_ARG 2 end_ARG g ( italic_e )</annotation></semantics></math> and <math alttext="\textsl{g}(v\!\to\!u)=\frac{1}{2}\textsl{g}(e)" class="ltx_Math" display="inline" id="S7.SS0.SSS0.P0.SPx1.p1.3.m3.2"><semantics id="S7.SS0.SSS0.P0.SPx1.p1.3.m3.2a"><mrow id="S7.SS0.SSS0.P0.SPx1.p1.3.m3.2.2" xref="S7.SS0.SSS0.P0.SPx1.p1.3.m3.2.2.cmml"><mrow id="S7.SS0.SSS0.P0.SPx1.p1.3.m3.2.2.1" xref="S7.SS0.SSS0.P0.SPx1.p1.3.m3.2.2.1.cmml"><mtext class="ltx_mathvariant_italic" id="S7.SS0.SSS0.P0.SPx1.p1.3.m3.2.2.1.3" xref="S7.SS0.SSS0.P0.SPx1.p1.3.m3.2.2.1.3a.cmml">g</mtext><mo id="S7.SS0.SSS0.P0.SPx1.p1.3.m3.2.2.1.2" xref="S7.SS0.SSS0.P0.SPx1.p1.3.m3.2.2.1.2.cmml">⁢</mo><mrow id="S7.SS0.SSS0.P0.SPx1.p1.3.m3.2.2.1.1.1" xref="S7.SS0.SSS0.P0.SPx1.p1.3.m3.2.2.1.1.1.1.cmml"><mo id="S7.SS0.SSS0.P0.SPx1.p1.3.m3.2.2.1.1.1.2" stretchy="false" xref="S7.SS0.SSS0.P0.SPx1.p1.3.m3.2.2.1.1.1.1.cmml">(</mo><mrow id="S7.SS0.SSS0.P0.SPx1.p1.3.m3.2.2.1.1.1.1" xref="S7.SS0.SSS0.P0.SPx1.p1.3.m3.2.2.1.1.1.1.cmml"><mi id="S7.SS0.SSS0.P0.SPx1.p1.3.m3.2.2.1.1.1.1.2" xref="S7.SS0.SSS0.P0.SPx1.p1.3.m3.2.2.1.1.1.1.2.cmml">v</mi><mo id="S7.SS0.SSS0.P0.SPx1.p1.3.m3.2.2.1.1.1.1.1" lspace="0.108em" rspace="0.108em" stretchy="false" xref="S7.SS0.SSS0.P0.SPx1.p1.3.m3.2.2.1.1.1.1.1.cmml">→</mo><mi id="S7.SS0.SSS0.P0.SPx1.p1.3.m3.2.2.1.1.1.1.3" xref="S7.SS0.SSS0.P0.SPx1.p1.3.m3.2.2.1.1.1.1.3.cmml">u</mi></mrow><mo id="S7.SS0.SSS0.P0.SPx1.p1.3.m3.2.2.1.1.1.3" stretchy="false" xref="S7.SS0.SSS0.P0.SPx1.p1.3.m3.2.2.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S7.SS0.SSS0.P0.SPx1.p1.3.m3.2.2.2" xref="S7.SS0.SSS0.P0.SPx1.p1.3.m3.2.2.2.cmml">=</mo><mrow id="S7.SS0.SSS0.P0.SPx1.p1.3.m3.2.2.3" xref="S7.SS0.SSS0.P0.SPx1.p1.3.m3.2.2.3.cmml"><mfrac id="S7.SS0.SSS0.P0.SPx1.p1.3.m3.2.2.3.2" xref="S7.SS0.SSS0.P0.SPx1.p1.3.m3.2.2.3.2.cmml"><mn id="S7.SS0.SSS0.P0.SPx1.p1.3.m3.2.2.3.2.2" xref="S7.SS0.SSS0.P0.SPx1.p1.3.m3.2.2.3.2.2.cmml">1</mn><mn id="S7.SS0.SSS0.P0.SPx1.p1.3.m3.2.2.3.2.3" xref="S7.SS0.SSS0.P0.SPx1.p1.3.m3.2.2.3.2.3.cmml">2</mn></mfrac><mo id="S7.SS0.SSS0.P0.SPx1.p1.3.m3.2.2.3.1" xref="S7.SS0.SSS0.P0.SPx1.p1.3.m3.2.2.3.1.cmml">⁢</mo><mtext class="ltx_mathvariant_italic" id="S7.SS0.SSS0.P0.SPx1.p1.3.m3.2.2.3.3" xref="S7.SS0.SSS0.P0.SPx1.p1.3.m3.2.2.3.3a.cmml">g</mtext><mo id="S7.SS0.SSS0.P0.SPx1.p1.3.m3.2.2.3.1a" xref="S7.SS0.SSS0.P0.SPx1.p1.3.m3.2.2.3.1.cmml">⁢</mo><mrow id="S7.SS0.SSS0.P0.SPx1.p1.3.m3.2.2.3.4.2" xref="S7.SS0.SSS0.P0.SPx1.p1.3.m3.2.2.3.cmml"><mo id="S7.SS0.SSS0.P0.SPx1.p1.3.m3.2.2.3.4.2.1" stretchy="false" xref="S7.SS0.SSS0.P0.SPx1.p1.3.m3.2.2.3.cmml">(</mo><mi id="S7.SS0.SSS0.P0.SPx1.p1.3.m3.1.1" xref="S7.SS0.SSS0.P0.SPx1.p1.3.m3.1.1.cmml">e</mi><mo id="S7.SS0.SSS0.P0.SPx1.p1.3.m3.2.2.3.4.2.2" stretchy="false" xref="S7.SS0.SSS0.P0.SPx1.p1.3.m3.2.2.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S7.SS0.SSS0.P0.SPx1.p1.3.m3.2b"><apply id="S7.SS0.SSS0.P0.SPx1.p1.3.m3.2.2.cmml" xref="S7.SS0.SSS0.P0.SPx1.p1.3.m3.2.2"><eq id="S7.SS0.SSS0.P0.SPx1.p1.3.m3.2.2.2.cmml" xref="S7.SS0.SSS0.P0.SPx1.p1.3.m3.2.2.2"></eq><apply id="S7.SS0.SSS0.P0.SPx1.p1.3.m3.2.2.1.cmml" xref="S7.SS0.SSS0.P0.SPx1.p1.3.m3.2.2.1"><times id="S7.SS0.SSS0.P0.SPx1.p1.3.m3.2.2.1.2.cmml" xref="S7.SS0.SSS0.P0.SPx1.p1.3.m3.2.2.1.2"></times><ci id="S7.SS0.SSS0.P0.SPx1.p1.3.m3.2.2.1.3a.cmml" xref="S7.SS0.SSS0.P0.SPx1.p1.3.m3.2.2.1.3"><mtext class="ltx_mathvariant_italic" id="S7.SS0.SSS0.P0.SPx1.p1.3.m3.2.2.1.3.cmml" xref="S7.SS0.SSS0.P0.SPx1.p1.3.m3.2.2.1.3">g</mtext></ci><apply id="S7.SS0.SSS0.P0.SPx1.p1.3.m3.2.2.1.1.1.1.cmml" xref="S7.SS0.SSS0.P0.SPx1.p1.3.m3.2.2.1.1.1"><ci id="S7.SS0.SSS0.P0.SPx1.p1.3.m3.2.2.1.1.1.1.1.cmml" xref="S7.SS0.SSS0.P0.SPx1.p1.3.m3.2.2.1.1.1.1.1">→</ci><ci id="S7.SS0.SSS0.P0.SPx1.p1.3.m3.2.2.1.1.1.1.2.cmml" xref="S7.SS0.SSS0.P0.SPx1.p1.3.m3.2.2.1.1.1.1.2">𝑣</ci><ci id="S7.SS0.SSS0.P0.SPx1.p1.3.m3.2.2.1.1.1.1.3.cmml" xref="S7.SS0.SSS0.P0.SPx1.p1.3.m3.2.2.1.1.1.1.3">𝑢</ci></apply></apply><apply id="S7.SS0.SSS0.P0.SPx1.p1.3.m3.2.2.3.cmml" xref="S7.SS0.SSS0.P0.SPx1.p1.3.m3.2.2.3"><times id="S7.SS0.SSS0.P0.SPx1.p1.3.m3.2.2.3.1.cmml" xref="S7.SS0.SSS0.P0.SPx1.p1.3.m3.2.2.3.1"></times><apply id="S7.SS0.SSS0.P0.SPx1.p1.3.m3.2.2.3.2.cmml" xref="S7.SS0.SSS0.P0.SPx1.p1.3.m3.2.2.3.2"><divide id="S7.SS0.SSS0.P0.SPx1.p1.3.m3.2.2.3.2.1.cmml" xref="S7.SS0.SSS0.P0.SPx1.p1.3.m3.2.2.3.2"></divide><cn id="S7.SS0.SSS0.P0.SPx1.p1.3.m3.2.2.3.2.2.cmml" type="integer" xref="S7.SS0.SSS0.P0.SPx1.p1.3.m3.2.2.3.2.2">1</cn><cn id="S7.SS0.SSS0.P0.SPx1.p1.3.m3.2.2.3.2.3.cmml" type="integer" xref="S7.SS0.SSS0.P0.SPx1.p1.3.m3.2.2.3.2.3">2</cn></apply><ci id="S7.SS0.SSS0.P0.SPx1.p1.3.m3.2.2.3.3a.cmml" xref="S7.SS0.SSS0.P0.SPx1.p1.3.m3.2.2.3.3"><mtext class="ltx_mathvariant_italic" id="S7.SS0.SSS0.P0.SPx1.p1.3.m3.2.2.3.3.cmml" xref="S7.SS0.SSS0.P0.SPx1.p1.3.m3.2.2.3.3">g</mtext></ci><ci id="S7.SS0.SSS0.P0.SPx1.p1.3.m3.1.1.cmml" xref="S7.SS0.SSS0.P0.SPx1.p1.3.m3.1.1">𝑒</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.SS0.SSS0.P0.SPx1.p1.3.m3.2c">\textsl{g}(v\!\to\!u)=\frac{1}{2}\textsl{g}(e)</annotation><annotation encoding="application/x-llamapun" id="S7.SS0.SSS0.P0.SPx1.p1.3.m3.2d">g ( italic_v → italic_u ) = divide start_ARG 1 end_ARG start_ARG 2 end_ARG g ( italic_e )</annotation></semantics></math>. This gives each vertex <math alttext="u" class="ltx_Math" display="inline" id="S7.SS0.SSS0.P0.SPx1.p1.4.m4.1"><semantics id="S7.SS0.SSS0.P0.SPx1.p1.4.m4.1a"><mi id="S7.SS0.SSS0.P0.SPx1.p1.4.m4.1.1" xref="S7.SS0.SSS0.P0.SPx1.p1.4.m4.1.1.cmml">u</mi><annotation-xml encoding="MathML-Content" id="S7.SS0.SSS0.P0.SPx1.p1.4.m4.1b"><ci id="S7.SS0.SSS0.P0.SPx1.p1.4.m4.1.1.cmml" xref="S7.SS0.SSS0.P0.SPx1.p1.4.m4.1.1">𝑢</ci></annotation-xml><annotation encoding="application/x-tex" id="S7.SS0.SSS0.P0.SPx1.p1.4.m4.1c">u</annotation><annotation encoding="application/x-llamapun" id="S7.SS0.SSS0.P0.SPx1.p1.4.m4.1d">italic_u</annotation></semantics></math> <em class="ltx_emph ltx_font_italic" id="S7.SS0.SSS0.P0.SPx1.p1.5.1">some</em> out-degree <math alttext="\textsl{g}(u)" class="ltx_Math" display="inline" id="S7.SS0.SSS0.P0.SPx1.p1.5.m5.1"><semantics id="S7.SS0.SSS0.P0.SPx1.p1.5.m5.1a"><mrow id="S7.SS0.SSS0.P0.SPx1.p1.5.m5.1.2" xref="S7.SS0.SSS0.P0.SPx1.p1.5.m5.1.2.cmml"><mtext class="ltx_mathvariant_italic" id="S7.SS0.SSS0.P0.SPx1.p1.5.m5.1.2.2" xref="S7.SS0.SSS0.P0.SPx1.p1.5.m5.1.2.2a.cmml">g</mtext><mo id="S7.SS0.SSS0.P0.SPx1.p1.5.m5.1.2.1" xref="S7.SS0.SSS0.P0.SPx1.p1.5.m5.1.2.1.cmml">⁢</mo><mrow id="S7.SS0.SSS0.P0.SPx1.p1.5.m5.1.2.3.2" xref="S7.SS0.SSS0.P0.SPx1.p1.5.m5.1.2.cmml"><mo id="S7.SS0.SSS0.P0.SPx1.p1.5.m5.1.2.3.2.1" stretchy="false" xref="S7.SS0.SSS0.P0.SPx1.p1.5.m5.1.2.cmml">(</mo><mi id="S7.SS0.SSS0.P0.SPx1.p1.5.m5.1.1" xref="S7.SS0.SSS0.P0.SPx1.p1.5.m5.1.1.cmml">u</mi><mo id="S7.SS0.SSS0.P0.SPx1.p1.5.m5.1.2.3.2.2" stretchy="false" xref="S7.SS0.SSS0.P0.SPx1.p1.5.m5.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S7.SS0.SSS0.P0.SPx1.p1.5.m5.1b"><apply id="S7.SS0.SSS0.P0.SPx1.p1.5.m5.1.2.cmml" xref="S7.SS0.SSS0.P0.SPx1.p1.5.m5.1.2"><times id="S7.SS0.SSS0.P0.SPx1.p1.5.m5.1.2.1.cmml" xref="S7.SS0.SSS0.P0.SPx1.p1.5.m5.1.2.1"></times><ci id="S7.SS0.SSS0.P0.SPx1.p1.5.m5.1.2.2a.cmml" xref="S7.SS0.SSS0.P0.SPx1.p1.5.m5.1.2.2"><mtext class="ltx_mathvariant_italic" id="S7.SS0.SSS0.P0.SPx1.p1.5.m5.1.2.2.cmml" xref="S7.SS0.SSS0.P0.SPx1.p1.5.m5.1.2.2">g</mtext></ci><ci id="S7.SS0.SSS0.P0.SPx1.p1.5.m5.1.1.cmml" xref="S7.SS0.SSS0.P0.SPx1.p1.5.m5.1.1">𝑢</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.SS0.SSS0.P0.SPx1.p1.5.m5.1c">\textsl{g}(u)</annotation><annotation encoding="application/x-llamapun" id="S7.SS0.SSS0.P0.SPx1.p1.5.m5.1d">g ( italic_u )</annotation></semantics></math> which we partition:</p> </div> <div class="ltx_theorem ltx_theorem_definition" id="S7.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="S7.Thmtheorem4.1.1.1">Definition 7.4</span></span><span class="ltx_text ltx_font_bold" id="S7.Thmtheorem4.2.2">.</span> </h6> <div class="ltx_para" id="S7.Thmtheorem4.p1"> <p class="ltx_p" id="S7.Thmtheorem4.p1.3"><span class="ltx_text ltx_font_italic" id="S7.Thmtheorem4.p1.3.3">Let each vertex <math alttext="u" class="ltx_Math" display="inline" id="S7.Thmtheorem4.p1.1.1.m1.1"><semantics id="S7.Thmtheorem4.p1.1.1.m1.1a"><mi id="S7.Thmtheorem4.p1.1.1.m1.1.1" xref="S7.Thmtheorem4.p1.1.1.m1.1.1.cmml">u</mi><annotation-xml encoding="MathML-Content" id="S7.Thmtheorem4.p1.1.1.m1.1b"><ci id="S7.Thmtheorem4.p1.1.1.m1.1.1.cmml" xref="S7.Thmtheorem4.p1.1.1.m1.1.1">𝑢</ci></annotation-xml><annotation encoding="application/x-tex" id="S7.Thmtheorem4.p1.1.1.m1.1c">u</annotation><annotation encoding="application/x-llamapun" id="S7.Thmtheorem4.p1.1.1.m1.1d">italic_u</annotation></semantics></math> have out-degree <math alttext="\textsl{g}(u)" class="ltx_Math" display="inline" id="S7.Thmtheorem4.p1.2.2.m2.1"><semantics id="S7.Thmtheorem4.p1.2.2.m2.1a"><mrow id="S7.Thmtheorem4.p1.2.2.m2.1.2" xref="S7.Thmtheorem4.p1.2.2.m2.1.2.cmml"><mtext class="ltx_mathvariant_italic" id="S7.Thmtheorem4.p1.2.2.m2.1.2.2" xref="S7.Thmtheorem4.p1.2.2.m2.1.2.2a.cmml">g</mtext><mo id="S7.Thmtheorem4.p1.2.2.m2.1.2.1" xref="S7.Thmtheorem4.p1.2.2.m2.1.2.1.cmml">⁢</mo><mrow id="S7.Thmtheorem4.p1.2.2.m2.1.2.3.2" xref="S7.Thmtheorem4.p1.2.2.m2.1.2.cmml"><mo id="S7.Thmtheorem4.p1.2.2.m2.1.2.3.2.1" stretchy="false" xref="S7.Thmtheorem4.p1.2.2.m2.1.2.cmml">(</mo><mi id="S7.Thmtheorem4.p1.2.2.m2.1.1" xref="S7.Thmtheorem4.p1.2.2.m2.1.1.cmml">u</mi><mo id="S7.Thmtheorem4.p1.2.2.m2.1.2.3.2.2" stretchy="false" xref="S7.Thmtheorem4.p1.2.2.m2.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S7.Thmtheorem4.p1.2.2.m2.1b"><apply id="S7.Thmtheorem4.p1.2.2.m2.1.2.cmml" xref="S7.Thmtheorem4.p1.2.2.m2.1.2"><times id="S7.Thmtheorem4.p1.2.2.m2.1.2.1.cmml" xref="S7.Thmtheorem4.p1.2.2.m2.1.2.1"></times><ci id="S7.Thmtheorem4.p1.2.2.m2.1.2.2a.cmml" xref="S7.Thmtheorem4.p1.2.2.m2.1.2.2"><mtext class="ltx_mathvariant_italic" id="S7.Thmtheorem4.p1.2.2.m2.1.2.2.cmml" xref="S7.Thmtheorem4.p1.2.2.m2.1.2.2">g</mtext></ci><ci id="S7.Thmtheorem4.p1.2.2.m2.1.1.cmml" xref="S7.Thmtheorem4.p1.2.2.m2.1.1">𝑢</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.Thmtheorem4.p1.2.2.m2.1c">\textsl{g}(u)</annotation><annotation encoding="application/x-llamapun" id="S7.Thmtheorem4.p1.2.2.m2.1d">g ( italic_u )</annotation></semantics></math>. We define <em class="ltx_emph ltx_font_upright" id="S7.Thmtheorem4.p1.3.3.1">level</em> <math alttext="i" class="ltx_Math" display="inline" id="S7.Thmtheorem4.p1.3.3.m3.1"><semantics id="S7.Thmtheorem4.p1.3.3.m3.1a"><mi id="S7.Thmtheorem4.p1.3.3.m3.1.1" xref="S7.Thmtheorem4.p1.3.3.m3.1.1.cmml">i</mi><annotation-xml encoding="MathML-Content" id="S7.Thmtheorem4.p1.3.3.m3.1b"><ci id="S7.Thmtheorem4.p1.3.3.m3.1.1.cmml" xref="S7.Thmtheorem4.p1.3.3.m3.1.1">𝑖</ci></annotation-xml><annotation encoding="application/x-tex" id="S7.Thmtheorem4.p1.3.3.m3.1c">i</annotation><annotation encoding="application/x-llamapun" id="S7.Thmtheorem4.p1.3.3.m3.1d">italic_i</annotation></semantics></math> as:</span></p> <table class="ltx_equation ltx_eqn_table" id="S7.Ex28"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_left_padleft"></td> <td class="ltx_eqn_cell ltx_align_left"><math alttext="L_{i}:=\left\{u\in V\mid\textsl{g}(u)\in\left[(1+\frac{\eta}{2})^{i},(1+\frac{% \eta}{2})^{i+1}\right]\right\}." class="ltx_Math" display="block" id="S7.Ex28.m1.2"><semantics id="S7.Ex28.m1.2a"><mrow id="S7.Ex28.m1.2.2.1" xref="S7.Ex28.m1.2.2.1.1.cmml"><mrow id="S7.Ex28.m1.2.2.1.1" xref="S7.Ex28.m1.2.2.1.1.cmml"><msub id="S7.Ex28.m1.2.2.1.1.4" xref="S7.Ex28.m1.2.2.1.1.4.cmml"><mi id="S7.Ex28.m1.2.2.1.1.4.2" xref="S7.Ex28.m1.2.2.1.1.4.2.cmml">L</mi><mi id="S7.Ex28.m1.2.2.1.1.4.3" xref="S7.Ex28.m1.2.2.1.1.4.3.cmml">i</mi></msub><mo id="S7.Ex28.m1.2.2.1.1.3" lspace="0.278em" rspace="0.278em" xref="S7.Ex28.m1.2.2.1.1.3.cmml">:=</mo><mrow id="S7.Ex28.m1.2.2.1.1.2.2" xref="S7.Ex28.m1.2.2.1.1.2.3.cmml"><mo id="S7.Ex28.m1.2.2.1.1.2.2.3" xref="S7.Ex28.m1.2.2.1.1.2.3.1.cmml">{</mo><mrow id="S7.Ex28.m1.2.2.1.1.1.1.1" xref="S7.Ex28.m1.2.2.1.1.1.1.1.cmml"><mi id="S7.Ex28.m1.2.2.1.1.1.1.1.2" xref="S7.Ex28.m1.2.2.1.1.1.1.1.2.cmml">u</mi><mo id="S7.Ex28.m1.2.2.1.1.1.1.1.1" xref="S7.Ex28.m1.2.2.1.1.1.1.1.1.cmml">∈</mo><mi id="S7.Ex28.m1.2.2.1.1.1.1.1.3" xref="S7.Ex28.m1.2.2.1.1.1.1.1.3.cmml">V</mi></mrow><mo fence="true" id="S7.Ex28.m1.2.2.1.1.2.2.4" lspace="0em" rspace="0em" xref="S7.Ex28.m1.2.2.1.1.2.3.1.cmml">∣</mo><mrow id="S7.Ex28.m1.2.2.1.1.2.2.2" xref="S7.Ex28.m1.2.2.1.1.2.2.2.cmml"><mrow id="S7.Ex28.m1.2.2.1.1.2.2.2.4" xref="S7.Ex28.m1.2.2.1.1.2.2.2.4.cmml"><mtext class="ltx_mathvariant_italic" id="S7.Ex28.m1.2.2.1.1.2.2.2.4.2" xref="S7.Ex28.m1.2.2.1.1.2.2.2.4.2a.cmml">g</mtext><mo id="S7.Ex28.m1.2.2.1.1.2.2.2.4.1" xref="S7.Ex28.m1.2.2.1.1.2.2.2.4.1.cmml">⁢</mo><mrow id="S7.Ex28.m1.2.2.1.1.2.2.2.4.3.2" xref="S7.Ex28.m1.2.2.1.1.2.2.2.4.cmml"><mo id="S7.Ex28.m1.2.2.1.1.2.2.2.4.3.2.1" stretchy="false" xref="S7.Ex28.m1.2.2.1.1.2.2.2.4.cmml">(</mo><mi id="S7.Ex28.m1.1.1" xref="S7.Ex28.m1.1.1.cmml">u</mi><mo id="S7.Ex28.m1.2.2.1.1.2.2.2.4.3.2.2" stretchy="false" xref="S7.Ex28.m1.2.2.1.1.2.2.2.4.cmml">)</mo></mrow></mrow><mo id="S7.Ex28.m1.2.2.1.1.2.2.2.3" xref="S7.Ex28.m1.2.2.1.1.2.2.2.3.cmml">∈</mo><mrow id="S7.Ex28.m1.2.2.1.1.2.2.2.2.2" xref="S7.Ex28.m1.2.2.1.1.2.2.2.2.3.cmml"><mo id="S7.Ex28.m1.2.2.1.1.2.2.2.2.2.3" xref="S7.Ex28.m1.2.2.1.1.2.2.2.2.3.cmml">[</mo><msup id="S7.Ex28.m1.2.2.1.1.2.2.2.1.1.1" xref="S7.Ex28.m1.2.2.1.1.2.2.2.1.1.1.cmml"><mrow id="S7.Ex28.m1.2.2.1.1.2.2.2.1.1.1.1.1" xref="S7.Ex28.m1.2.2.1.1.2.2.2.1.1.1.1.1.1.cmml"><mo id="S7.Ex28.m1.2.2.1.1.2.2.2.1.1.1.1.1.2" stretchy="false" xref="S7.Ex28.m1.2.2.1.1.2.2.2.1.1.1.1.1.1.cmml">(</mo><mrow id="S7.Ex28.m1.2.2.1.1.2.2.2.1.1.1.1.1.1" xref="S7.Ex28.m1.2.2.1.1.2.2.2.1.1.1.1.1.1.cmml"><mn id="S7.Ex28.m1.2.2.1.1.2.2.2.1.1.1.1.1.1.2" xref="S7.Ex28.m1.2.2.1.1.2.2.2.1.1.1.1.1.1.2.cmml">1</mn><mo id="S7.Ex28.m1.2.2.1.1.2.2.2.1.1.1.1.1.1.1" xref="S7.Ex28.m1.2.2.1.1.2.2.2.1.1.1.1.1.1.1.cmml">+</mo><mfrac id="S7.Ex28.m1.2.2.1.1.2.2.2.1.1.1.1.1.1.3" xref="S7.Ex28.m1.2.2.1.1.2.2.2.1.1.1.1.1.1.3.cmml"><mi id="S7.Ex28.m1.2.2.1.1.2.2.2.1.1.1.1.1.1.3.2" xref="S7.Ex28.m1.2.2.1.1.2.2.2.1.1.1.1.1.1.3.2.cmml">η</mi><mn id="S7.Ex28.m1.2.2.1.1.2.2.2.1.1.1.1.1.1.3.3" xref="S7.Ex28.m1.2.2.1.1.2.2.2.1.1.1.1.1.1.3.3.cmml">2</mn></mfrac></mrow><mo id="S7.Ex28.m1.2.2.1.1.2.2.2.1.1.1.1.1.3" stretchy="false" xref="S7.Ex28.m1.2.2.1.1.2.2.2.1.1.1.1.1.1.cmml">)</mo></mrow><mi id="S7.Ex28.m1.2.2.1.1.2.2.2.1.1.1.3" xref="S7.Ex28.m1.2.2.1.1.2.2.2.1.1.1.3.cmml">i</mi></msup><mo id="S7.Ex28.m1.2.2.1.1.2.2.2.2.2.4" xref="S7.Ex28.m1.2.2.1.1.2.2.2.2.3.cmml">,</mo><msup id="S7.Ex28.m1.2.2.1.1.2.2.2.2.2.2" xref="S7.Ex28.m1.2.2.1.1.2.2.2.2.2.2.cmml"><mrow id="S7.Ex28.m1.2.2.1.1.2.2.2.2.2.2.1.1" xref="S7.Ex28.m1.2.2.1.1.2.2.2.2.2.2.1.1.1.cmml"><mo id="S7.Ex28.m1.2.2.1.1.2.2.2.2.2.2.1.1.2" stretchy="false" xref="S7.Ex28.m1.2.2.1.1.2.2.2.2.2.2.1.1.1.cmml">(</mo><mrow id="S7.Ex28.m1.2.2.1.1.2.2.2.2.2.2.1.1.1" xref="S7.Ex28.m1.2.2.1.1.2.2.2.2.2.2.1.1.1.cmml"><mn id="S7.Ex28.m1.2.2.1.1.2.2.2.2.2.2.1.1.1.2" xref="S7.Ex28.m1.2.2.1.1.2.2.2.2.2.2.1.1.1.2.cmml">1</mn><mo id="S7.Ex28.m1.2.2.1.1.2.2.2.2.2.2.1.1.1.1" xref="S7.Ex28.m1.2.2.1.1.2.2.2.2.2.2.1.1.1.1.cmml">+</mo><mfrac id="S7.Ex28.m1.2.2.1.1.2.2.2.2.2.2.1.1.1.3" xref="S7.Ex28.m1.2.2.1.1.2.2.2.2.2.2.1.1.1.3.cmml"><mi id="S7.Ex28.m1.2.2.1.1.2.2.2.2.2.2.1.1.1.3.2" xref="S7.Ex28.m1.2.2.1.1.2.2.2.2.2.2.1.1.1.3.2.cmml">η</mi><mn id="S7.Ex28.m1.2.2.1.1.2.2.2.2.2.2.1.1.1.3.3" xref="S7.Ex28.m1.2.2.1.1.2.2.2.2.2.2.1.1.1.3.3.cmml">2</mn></mfrac></mrow><mo id="S7.Ex28.m1.2.2.1.1.2.2.2.2.2.2.1.1.3" stretchy="false" xref="S7.Ex28.m1.2.2.1.1.2.2.2.2.2.2.1.1.1.cmml">)</mo></mrow><mrow id="S7.Ex28.m1.2.2.1.1.2.2.2.2.2.2.3" xref="S7.Ex28.m1.2.2.1.1.2.2.2.2.2.2.3.cmml"><mi id="S7.Ex28.m1.2.2.1.1.2.2.2.2.2.2.3.2" xref="S7.Ex28.m1.2.2.1.1.2.2.2.2.2.2.3.2.cmml">i</mi><mo id="S7.Ex28.m1.2.2.1.1.2.2.2.2.2.2.3.1" xref="S7.Ex28.m1.2.2.1.1.2.2.2.2.2.2.3.1.cmml">+</mo><mn id="S7.Ex28.m1.2.2.1.1.2.2.2.2.2.2.3.3" xref="S7.Ex28.m1.2.2.1.1.2.2.2.2.2.2.3.3.cmml">1</mn></mrow></msup><mo id="S7.Ex28.m1.2.2.1.1.2.2.2.2.2.5" xref="S7.Ex28.m1.2.2.1.1.2.2.2.2.3.cmml">]</mo></mrow></mrow><mo id="S7.Ex28.m1.2.2.1.1.2.2.5" xref="S7.Ex28.m1.2.2.1.1.2.3.1.cmml">}</mo></mrow></mrow><mo id="S7.Ex28.m1.2.2.1.2" lspace="0em" xref="S7.Ex28.m1.2.2.1.1.cmml">.</mo></mrow><annotation-xml encoding="MathML-Content" id="S7.Ex28.m1.2b"><apply id="S7.Ex28.m1.2.2.1.1.cmml" xref="S7.Ex28.m1.2.2.1"><csymbol cd="latexml" id="S7.Ex28.m1.2.2.1.1.3.cmml" xref="S7.Ex28.m1.2.2.1.1.3">assign</csymbol><apply id="S7.Ex28.m1.2.2.1.1.4.cmml" xref="S7.Ex28.m1.2.2.1.1.4"><csymbol cd="ambiguous" id="S7.Ex28.m1.2.2.1.1.4.1.cmml" xref="S7.Ex28.m1.2.2.1.1.4">subscript</csymbol><ci id="S7.Ex28.m1.2.2.1.1.4.2.cmml" xref="S7.Ex28.m1.2.2.1.1.4.2">𝐿</ci><ci id="S7.Ex28.m1.2.2.1.1.4.3.cmml" xref="S7.Ex28.m1.2.2.1.1.4.3">𝑖</ci></apply><apply id="S7.Ex28.m1.2.2.1.1.2.3.cmml" xref="S7.Ex28.m1.2.2.1.1.2.2"><csymbol cd="latexml" 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xref="S7.Ex28.m1.1.1">𝑢</ci></apply><interval closure="closed" id="S7.Ex28.m1.2.2.1.1.2.2.2.2.3.cmml" xref="S7.Ex28.m1.2.2.1.1.2.2.2.2.2"><apply id="S7.Ex28.m1.2.2.1.1.2.2.2.1.1.1.cmml" xref="S7.Ex28.m1.2.2.1.1.2.2.2.1.1.1"><csymbol cd="ambiguous" id="S7.Ex28.m1.2.2.1.1.2.2.2.1.1.1.2.cmml" xref="S7.Ex28.m1.2.2.1.1.2.2.2.1.1.1">superscript</csymbol><apply id="S7.Ex28.m1.2.2.1.1.2.2.2.1.1.1.1.1.1.cmml" xref="S7.Ex28.m1.2.2.1.1.2.2.2.1.1.1.1.1"><plus id="S7.Ex28.m1.2.2.1.1.2.2.2.1.1.1.1.1.1.1.cmml" xref="S7.Ex28.m1.2.2.1.1.2.2.2.1.1.1.1.1.1.1"></plus><cn id="S7.Ex28.m1.2.2.1.1.2.2.2.1.1.1.1.1.1.2.cmml" type="integer" xref="S7.Ex28.m1.2.2.1.1.2.2.2.1.1.1.1.1.1.2">1</cn><apply id="S7.Ex28.m1.2.2.1.1.2.2.2.1.1.1.1.1.1.3.cmml" xref="S7.Ex28.m1.2.2.1.1.2.2.2.1.1.1.1.1.1.3"><divide id="S7.Ex28.m1.2.2.1.1.2.2.2.1.1.1.1.1.1.3.1.cmml" xref="S7.Ex28.m1.2.2.1.1.2.2.2.1.1.1.1.1.1.3"></divide><ci id="S7.Ex28.m1.2.2.1.1.2.2.2.1.1.1.1.1.1.3.2.cmml" 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xref="S7.Ex28.m1.2.2.1.1.2.2.2.2.2.2.1.1.1.3"></divide><ci id="S7.Ex28.m1.2.2.1.1.2.2.2.2.2.2.1.1.1.3.2.cmml" xref="S7.Ex28.m1.2.2.1.1.2.2.2.2.2.2.1.1.1.3.2">𝜂</ci><cn id="S7.Ex28.m1.2.2.1.1.2.2.2.2.2.2.1.1.1.3.3.cmml" type="integer" xref="S7.Ex28.m1.2.2.1.1.2.2.2.2.2.2.1.1.1.3.3">2</cn></apply></apply><apply id="S7.Ex28.m1.2.2.1.1.2.2.2.2.2.2.3.cmml" xref="S7.Ex28.m1.2.2.1.1.2.2.2.2.2.2.3"><plus id="S7.Ex28.m1.2.2.1.1.2.2.2.2.2.2.3.1.cmml" xref="S7.Ex28.m1.2.2.1.1.2.2.2.2.2.2.3.1"></plus><ci id="S7.Ex28.m1.2.2.1.1.2.2.2.2.2.2.3.2.cmml" xref="S7.Ex28.m1.2.2.1.1.2.2.2.2.2.2.3.2">𝑖</ci><cn id="S7.Ex28.m1.2.2.1.1.2.2.2.2.2.2.3.3.cmml" type="integer" xref="S7.Ex28.m1.2.2.1.1.2.2.2.2.2.2.3.3">1</cn></apply></apply></interval></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.Ex28.m1.2c">L_{i}:=\left\{u\in V\mid\textsl{g}(u)\in\left[(1+\frac{\eta}{2})^{i},(1+\frac{% \eta}{2})^{i+1}\right]\right\}.</annotation><annotation encoding="application/x-llamapun" id="S7.Ex28.m1.2d">italic_L start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT := { italic_u ∈ italic_V ∣ g ( italic_u ) ∈ [ ( 1 + divide start_ARG italic_η end_ARG start_ARG 2 end_ARG ) start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT , ( 1 + divide start_ARG italic_η end_ARG start_ARG 2 end_ARG ) start_POSTSUPERSCRIPT italic_i + 1 end_POSTSUPERSCRIPT ] } .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_left_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S7.Thmtheorem4.p1.12"><span class="ltx_text ltx_font_italic" id="S7.Thmtheorem4.p1.12.9">A vertex <math alttext="u\in L_{i}" class="ltx_Math" display="inline" id="S7.Thmtheorem4.p1.4.1.m1.1"><semantics id="S7.Thmtheorem4.p1.4.1.m1.1a"><mrow id="S7.Thmtheorem4.p1.4.1.m1.1.1" xref="S7.Thmtheorem4.p1.4.1.m1.1.1.cmml"><mi id="S7.Thmtheorem4.p1.4.1.m1.1.1.2" xref="S7.Thmtheorem4.p1.4.1.m1.1.1.2.cmml">u</mi><mo id="S7.Thmtheorem4.p1.4.1.m1.1.1.1" xref="S7.Thmtheorem4.p1.4.1.m1.1.1.1.cmml">∈</mo><msub id="S7.Thmtheorem4.p1.4.1.m1.1.1.3" xref="S7.Thmtheorem4.p1.4.1.m1.1.1.3.cmml"><mi id="S7.Thmtheorem4.p1.4.1.m1.1.1.3.2" xref="S7.Thmtheorem4.p1.4.1.m1.1.1.3.2.cmml">L</mi><mi id="S7.Thmtheorem4.p1.4.1.m1.1.1.3.3" xref="S7.Thmtheorem4.p1.4.1.m1.1.1.3.3.cmml">i</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S7.Thmtheorem4.p1.4.1.m1.1b"><apply id="S7.Thmtheorem4.p1.4.1.m1.1.1.cmml" xref="S7.Thmtheorem4.p1.4.1.m1.1.1"><in id="S7.Thmtheorem4.p1.4.1.m1.1.1.1.cmml" xref="S7.Thmtheorem4.p1.4.1.m1.1.1.1"></in><ci id="S7.Thmtheorem4.p1.4.1.m1.1.1.2.cmml" xref="S7.Thmtheorem4.p1.4.1.m1.1.1.2">𝑢</ci><apply id="S7.Thmtheorem4.p1.4.1.m1.1.1.3.cmml" xref="S7.Thmtheorem4.p1.4.1.m1.1.1.3"><csymbol cd="ambiguous" id="S7.Thmtheorem4.p1.4.1.m1.1.1.3.1.cmml" xref="S7.Thmtheorem4.p1.4.1.m1.1.1.3">subscript</csymbol><ci id="S7.Thmtheorem4.p1.4.1.m1.1.1.3.2.cmml" xref="S7.Thmtheorem4.p1.4.1.m1.1.1.3.2">𝐿</ci><ci id="S7.Thmtheorem4.p1.4.1.m1.1.1.3.3.cmml" xref="S7.Thmtheorem4.p1.4.1.m1.1.1.3.3">𝑖</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.Thmtheorem4.p1.4.1.m1.1c">u\in L_{i}</annotation><annotation encoding="application/x-llamapun" id="S7.Thmtheorem4.p1.4.1.m1.1d">italic_u ∈ italic_L start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math> is <em class="ltx_emph ltx_font_upright" id="S7.Thmtheorem4.p1.12.9.1">at level</em> <math alttext="i" class="ltx_Math" display="inline" id="S7.Thmtheorem4.p1.5.2.m2.1"><semantics id="S7.Thmtheorem4.p1.5.2.m2.1a"><mi id="S7.Thmtheorem4.p1.5.2.m2.1.1" xref="S7.Thmtheorem4.p1.5.2.m2.1.1.cmml">i</mi><annotation-xml encoding="MathML-Content" id="S7.Thmtheorem4.p1.5.2.m2.1b"><ci id="S7.Thmtheorem4.p1.5.2.m2.1.1.cmml" xref="S7.Thmtheorem4.p1.5.2.m2.1.1">𝑖</ci></annotation-xml><annotation encoding="application/x-tex" id="S7.Thmtheorem4.p1.5.2.m2.1c">i</annotation><annotation encoding="application/x-llamapun" id="S7.Thmtheorem4.p1.5.2.m2.1d">italic_i</annotation></semantics></math> and <math alttext="\ell^{\prime}" class="ltx_Math" display="inline" id="S7.Thmtheorem4.p1.6.3.m3.1"><semantics id="S7.Thmtheorem4.p1.6.3.m3.1a"><msup id="S7.Thmtheorem4.p1.6.3.m3.1.1" xref="S7.Thmtheorem4.p1.6.3.m3.1.1.cmml"><mi id="S7.Thmtheorem4.p1.6.3.m3.1.1.2" mathvariant="normal" xref="S7.Thmtheorem4.p1.6.3.m3.1.1.2.cmml">ℓ</mi><mo id="S7.Thmtheorem4.p1.6.3.m3.1.1.3" xref="S7.Thmtheorem4.p1.6.3.m3.1.1.3.cmml">′</mo></msup><annotation-xml encoding="MathML-Content" id="S7.Thmtheorem4.p1.6.3.m3.1b"><apply id="S7.Thmtheorem4.p1.6.3.m3.1.1.cmml" xref="S7.Thmtheorem4.p1.6.3.m3.1.1"><csymbol cd="ambiguous" id="S7.Thmtheorem4.p1.6.3.m3.1.1.1.cmml" xref="S7.Thmtheorem4.p1.6.3.m3.1.1">superscript</csymbol><ci id="S7.Thmtheorem4.p1.6.3.m3.1.1.2.cmml" xref="S7.Thmtheorem4.p1.6.3.m3.1.1.2">ℓ</ci><ci id="S7.Thmtheorem4.p1.6.3.m3.1.1.3.cmml" xref="S7.Thmtheorem4.p1.6.3.m3.1.1.3">′</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.Thmtheorem4.p1.6.3.m3.1c">\ell^{\prime}</annotation><annotation encoding="application/x-llamapun" id="S7.Thmtheorem4.p1.6.3.m3.1d">roman_ℓ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math> denotes the highest level that is not empty. <br class="ltx_break"/>Whenever <math alttext="\textsl{g}(u)=(1+\frac{\eta}{2})^{i}" class="ltx_Math" display="inline" id="S7.Thmtheorem4.p1.7.4.m4.2"><semantics id="S7.Thmtheorem4.p1.7.4.m4.2a"><mrow id="S7.Thmtheorem4.p1.7.4.m4.2.2" xref="S7.Thmtheorem4.p1.7.4.m4.2.2.cmml"><mrow id="S7.Thmtheorem4.p1.7.4.m4.2.2.3" xref="S7.Thmtheorem4.p1.7.4.m4.2.2.3.cmml"><mtext class="ltx_mathvariant_italic" id="S7.Thmtheorem4.p1.7.4.m4.2.2.3.2" xref="S7.Thmtheorem4.p1.7.4.m4.2.2.3.2a.cmml">g</mtext><mo id="S7.Thmtheorem4.p1.7.4.m4.2.2.3.1" xref="S7.Thmtheorem4.p1.7.4.m4.2.2.3.1.cmml">⁢</mo><mrow id="S7.Thmtheorem4.p1.7.4.m4.2.2.3.3.2" xref="S7.Thmtheorem4.p1.7.4.m4.2.2.3.cmml"><mo id="S7.Thmtheorem4.p1.7.4.m4.2.2.3.3.2.1" stretchy="false" xref="S7.Thmtheorem4.p1.7.4.m4.2.2.3.cmml">(</mo><mi id="S7.Thmtheorem4.p1.7.4.m4.1.1" xref="S7.Thmtheorem4.p1.7.4.m4.1.1.cmml">u</mi><mo id="S7.Thmtheorem4.p1.7.4.m4.2.2.3.3.2.2" stretchy="false" xref="S7.Thmtheorem4.p1.7.4.m4.2.2.3.cmml">)</mo></mrow></mrow><mo id="S7.Thmtheorem4.p1.7.4.m4.2.2.2" xref="S7.Thmtheorem4.p1.7.4.m4.2.2.2.cmml">=</mo><msup id="S7.Thmtheorem4.p1.7.4.m4.2.2.1" xref="S7.Thmtheorem4.p1.7.4.m4.2.2.1.cmml"><mrow id="S7.Thmtheorem4.p1.7.4.m4.2.2.1.1.1" xref="S7.Thmtheorem4.p1.7.4.m4.2.2.1.1.1.1.cmml"><mo id="S7.Thmtheorem4.p1.7.4.m4.2.2.1.1.1.2" stretchy="false" xref="S7.Thmtheorem4.p1.7.4.m4.2.2.1.1.1.1.cmml">(</mo><mrow id="S7.Thmtheorem4.p1.7.4.m4.2.2.1.1.1.1" xref="S7.Thmtheorem4.p1.7.4.m4.2.2.1.1.1.1.cmml"><mn id="S7.Thmtheorem4.p1.7.4.m4.2.2.1.1.1.1.2" xref="S7.Thmtheorem4.p1.7.4.m4.2.2.1.1.1.1.2.cmml">1</mn><mo id="S7.Thmtheorem4.p1.7.4.m4.2.2.1.1.1.1.1" xref="S7.Thmtheorem4.p1.7.4.m4.2.2.1.1.1.1.1.cmml">+</mo><mfrac id="S7.Thmtheorem4.p1.7.4.m4.2.2.1.1.1.1.3" xref="S7.Thmtheorem4.p1.7.4.m4.2.2.1.1.1.1.3.cmml"><mi id="S7.Thmtheorem4.p1.7.4.m4.2.2.1.1.1.1.3.2" xref="S7.Thmtheorem4.p1.7.4.m4.2.2.1.1.1.1.3.2.cmml">η</mi><mn id="S7.Thmtheorem4.p1.7.4.m4.2.2.1.1.1.1.3.3" xref="S7.Thmtheorem4.p1.7.4.m4.2.2.1.1.1.1.3.3.cmml">2</mn></mfrac></mrow><mo id="S7.Thmtheorem4.p1.7.4.m4.2.2.1.1.1.3" stretchy="false" xref="S7.Thmtheorem4.p1.7.4.m4.2.2.1.1.1.1.cmml">)</mo></mrow><mi id="S7.Thmtheorem4.p1.7.4.m4.2.2.1.3" xref="S7.Thmtheorem4.p1.7.4.m4.2.2.1.3.cmml">i</mi></msup></mrow><annotation-xml encoding="MathML-Content" id="S7.Thmtheorem4.p1.7.4.m4.2b"><apply id="S7.Thmtheorem4.p1.7.4.m4.2.2.cmml" xref="S7.Thmtheorem4.p1.7.4.m4.2.2"><eq id="S7.Thmtheorem4.p1.7.4.m4.2.2.2.cmml" xref="S7.Thmtheorem4.p1.7.4.m4.2.2.2"></eq><apply id="S7.Thmtheorem4.p1.7.4.m4.2.2.3.cmml" xref="S7.Thmtheorem4.p1.7.4.m4.2.2.3"><times id="S7.Thmtheorem4.p1.7.4.m4.2.2.3.1.cmml" xref="S7.Thmtheorem4.p1.7.4.m4.2.2.3.1"></times><ci id="S7.Thmtheorem4.p1.7.4.m4.2.2.3.2a.cmml" xref="S7.Thmtheorem4.p1.7.4.m4.2.2.3.2"><mtext class="ltx_mathvariant_italic" id="S7.Thmtheorem4.p1.7.4.m4.2.2.3.2.cmml" xref="S7.Thmtheorem4.p1.7.4.m4.2.2.3.2">g</mtext></ci><ci id="S7.Thmtheorem4.p1.7.4.m4.1.1.cmml" xref="S7.Thmtheorem4.p1.7.4.m4.1.1">𝑢</ci></apply><apply id="S7.Thmtheorem4.p1.7.4.m4.2.2.1.cmml" xref="S7.Thmtheorem4.p1.7.4.m4.2.2.1"><csymbol cd="ambiguous" id="S7.Thmtheorem4.p1.7.4.m4.2.2.1.2.cmml" xref="S7.Thmtheorem4.p1.7.4.m4.2.2.1">superscript</csymbol><apply id="S7.Thmtheorem4.p1.7.4.m4.2.2.1.1.1.1.cmml" xref="S7.Thmtheorem4.p1.7.4.m4.2.2.1.1.1"><plus id="S7.Thmtheorem4.p1.7.4.m4.2.2.1.1.1.1.1.cmml" xref="S7.Thmtheorem4.p1.7.4.m4.2.2.1.1.1.1.1"></plus><cn id="S7.Thmtheorem4.p1.7.4.m4.2.2.1.1.1.1.2.cmml" type="integer" xref="S7.Thmtheorem4.p1.7.4.m4.2.2.1.1.1.1.2">1</cn><apply id="S7.Thmtheorem4.p1.7.4.m4.2.2.1.1.1.1.3.cmml" xref="S7.Thmtheorem4.p1.7.4.m4.2.2.1.1.1.1.3"><divide id="S7.Thmtheorem4.p1.7.4.m4.2.2.1.1.1.1.3.1.cmml" xref="S7.Thmtheorem4.p1.7.4.m4.2.2.1.1.1.1.3"></divide><ci id="S7.Thmtheorem4.p1.7.4.m4.2.2.1.1.1.1.3.2.cmml" xref="S7.Thmtheorem4.p1.7.4.m4.2.2.1.1.1.1.3.2">𝜂</ci><cn id="S7.Thmtheorem4.p1.7.4.m4.2.2.1.1.1.1.3.3.cmml" type="integer" xref="S7.Thmtheorem4.p1.7.4.m4.2.2.1.1.1.1.3.3">2</cn></apply></apply><ci id="S7.Thmtheorem4.p1.7.4.m4.2.2.1.3.cmml" xref="S7.Thmtheorem4.p1.7.4.m4.2.2.1.3">𝑖</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.Thmtheorem4.p1.7.4.m4.2c">\textsl{g}(u)=(1+\frac{\eta}{2})^{i}</annotation><annotation encoding="application/x-llamapun" id="S7.Thmtheorem4.p1.7.4.m4.2d">g ( italic_u ) = ( 1 + divide start_ARG italic_η end_ARG start_ARG 2 end_ARG ) start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT</annotation></semantics></math>, <math alttext="u" class="ltx_Math" display="inline" id="S7.Thmtheorem4.p1.8.5.m5.1"><semantics id="S7.Thmtheorem4.p1.8.5.m5.1a"><mi id="S7.Thmtheorem4.p1.8.5.m5.1.1" xref="S7.Thmtheorem4.p1.8.5.m5.1.1.cmml">u</mi><annotation-xml encoding="MathML-Content" id="S7.Thmtheorem4.p1.8.5.m5.1b"><ci id="S7.Thmtheorem4.p1.8.5.m5.1.1.cmml" xref="S7.Thmtheorem4.p1.8.5.m5.1.1">𝑢</ci></annotation-xml><annotation encoding="application/x-tex" id="S7.Thmtheorem4.p1.8.5.m5.1c">u</annotation><annotation encoding="application/x-llamapun" id="S7.Thmtheorem4.p1.8.5.m5.1d">italic_u</annotation></semantics></math> may decide whether <math alttext="u\in L_{i}" class="ltx_Math" display="inline" id="S7.Thmtheorem4.p1.9.6.m6.1"><semantics id="S7.Thmtheorem4.p1.9.6.m6.1a"><mrow id="S7.Thmtheorem4.p1.9.6.m6.1.1" xref="S7.Thmtheorem4.p1.9.6.m6.1.1.cmml"><mi id="S7.Thmtheorem4.p1.9.6.m6.1.1.2" xref="S7.Thmtheorem4.p1.9.6.m6.1.1.2.cmml">u</mi><mo id="S7.Thmtheorem4.p1.9.6.m6.1.1.1" xref="S7.Thmtheorem4.p1.9.6.m6.1.1.1.cmml">∈</mo><msub id="S7.Thmtheorem4.p1.9.6.m6.1.1.3" xref="S7.Thmtheorem4.p1.9.6.m6.1.1.3.cmml"><mi id="S7.Thmtheorem4.p1.9.6.m6.1.1.3.2" xref="S7.Thmtheorem4.p1.9.6.m6.1.1.3.2.cmml">L</mi><mi id="S7.Thmtheorem4.p1.9.6.m6.1.1.3.3" xref="S7.Thmtheorem4.p1.9.6.m6.1.1.3.3.cmml">i</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S7.Thmtheorem4.p1.9.6.m6.1b"><apply id="S7.Thmtheorem4.p1.9.6.m6.1.1.cmml" xref="S7.Thmtheorem4.p1.9.6.m6.1.1"><in id="S7.Thmtheorem4.p1.9.6.m6.1.1.1.cmml" xref="S7.Thmtheorem4.p1.9.6.m6.1.1.1"></in><ci id="S7.Thmtheorem4.p1.9.6.m6.1.1.2.cmml" xref="S7.Thmtheorem4.p1.9.6.m6.1.1.2">𝑢</ci><apply id="S7.Thmtheorem4.p1.9.6.m6.1.1.3.cmml" xref="S7.Thmtheorem4.p1.9.6.m6.1.1.3"><csymbol cd="ambiguous" id="S7.Thmtheorem4.p1.9.6.m6.1.1.3.1.cmml" xref="S7.Thmtheorem4.p1.9.6.m6.1.1.3">subscript</csymbol><ci id="S7.Thmtheorem4.p1.9.6.m6.1.1.3.2.cmml" xref="S7.Thmtheorem4.p1.9.6.m6.1.1.3.2">𝐿</ci><ci id="S7.Thmtheorem4.p1.9.6.m6.1.1.3.3.cmml" xref="S7.Thmtheorem4.p1.9.6.m6.1.1.3.3">𝑖</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.Thmtheorem4.p1.9.6.m6.1c">u\in L_{i}</annotation><annotation encoding="application/x-llamapun" id="S7.Thmtheorem4.p1.9.6.m6.1d">italic_u ∈ italic_L start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math> or <math alttext="u\in L_{i-1}" class="ltx_Math" display="inline" id="S7.Thmtheorem4.p1.10.7.m7.1"><semantics id="S7.Thmtheorem4.p1.10.7.m7.1a"><mrow id="S7.Thmtheorem4.p1.10.7.m7.1.1" xref="S7.Thmtheorem4.p1.10.7.m7.1.1.cmml"><mi id="S7.Thmtheorem4.p1.10.7.m7.1.1.2" xref="S7.Thmtheorem4.p1.10.7.m7.1.1.2.cmml">u</mi><mo id="S7.Thmtheorem4.p1.10.7.m7.1.1.1" xref="S7.Thmtheorem4.p1.10.7.m7.1.1.1.cmml">∈</mo><msub id="S7.Thmtheorem4.p1.10.7.m7.1.1.3" xref="S7.Thmtheorem4.p1.10.7.m7.1.1.3.cmml"><mi id="S7.Thmtheorem4.p1.10.7.m7.1.1.3.2" xref="S7.Thmtheorem4.p1.10.7.m7.1.1.3.2.cmml">L</mi><mrow id="S7.Thmtheorem4.p1.10.7.m7.1.1.3.3" xref="S7.Thmtheorem4.p1.10.7.m7.1.1.3.3.cmml"><mi id="S7.Thmtheorem4.p1.10.7.m7.1.1.3.3.2" xref="S7.Thmtheorem4.p1.10.7.m7.1.1.3.3.2.cmml">i</mi><mo id="S7.Thmtheorem4.p1.10.7.m7.1.1.3.3.1" xref="S7.Thmtheorem4.p1.10.7.m7.1.1.3.3.1.cmml">−</mo><mn id="S7.Thmtheorem4.p1.10.7.m7.1.1.3.3.3" xref="S7.Thmtheorem4.p1.10.7.m7.1.1.3.3.3.cmml">1</mn></mrow></msub></mrow><annotation-xml encoding="MathML-Content" id="S7.Thmtheorem4.p1.10.7.m7.1b"><apply id="S7.Thmtheorem4.p1.10.7.m7.1.1.cmml" xref="S7.Thmtheorem4.p1.10.7.m7.1.1"><in id="S7.Thmtheorem4.p1.10.7.m7.1.1.1.cmml" xref="S7.Thmtheorem4.p1.10.7.m7.1.1.1"></in><ci id="S7.Thmtheorem4.p1.10.7.m7.1.1.2.cmml" xref="S7.Thmtheorem4.p1.10.7.m7.1.1.2">𝑢</ci><apply id="S7.Thmtheorem4.p1.10.7.m7.1.1.3.cmml" xref="S7.Thmtheorem4.p1.10.7.m7.1.1.3"><csymbol cd="ambiguous" id="S7.Thmtheorem4.p1.10.7.m7.1.1.3.1.cmml" xref="S7.Thmtheorem4.p1.10.7.m7.1.1.3">subscript</csymbol><ci id="S7.Thmtheorem4.p1.10.7.m7.1.1.3.2.cmml" xref="S7.Thmtheorem4.p1.10.7.m7.1.1.3.2">𝐿</ci><apply id="S7.Thmtheorem4.p1.10.7.m7.1.1.3.3.cmml" xref="S7.Thmtheorem4.p1.10.7.m7.1.1.3.3"><minus id="S7.Thmtheorem4.p1.10.7.m7.1.1.3.3.1.cmml" xref="S7.Thmtheorem4.p1.10.7.m7.1.1.3.3.1"></minus><ci id="S7.Thmtheorem4.p1.10.7.m7.1.1.3.3.2.cmml" xref="S7.Thmtheorem4.p1.10.7.m7.1.1.3.3.2">𝑖</ci><cn id="S7.Thmtheorem4.p1.10.7.m7.1.1.3.3.3.cmml" type="integer" xref="S7.Thmtheorem4.p1.10.7.m7.1.1.3.3.3">1</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.Thmtheorem4.p1.10.7.m7.1c">u\in L_{i-1}</annotation><annotation encoding="application/x-llamapun" id="S7.Thmtheorem4.p1.10.7.m7.1d">italic_u ∈ italic_L start_POSTSUBSCRIPT italic_i - 1 end_POSTSUBSCRIPT</annotation></semantics></math>; whenever our algorithm increases <math alttext="\textsl{g}(u)" class="ltx_Math" display="inline" id="S7.Thmtheorem4.p1.11.8.m8.1"><semantics id="S7.Thmtheorem4.p1.11.8.m8.1a"><mrow id="S7.Thmtheorem4.p1.11.8.m8.1.2" xref="S7.Thmtheorem4.p1.11.8.m8.1.2.cmml"><mtext class="ltx_mathvariant_italic" id="S7.Thmtheorem4.p1.11.8.m8.1.2.2" xref="S7.Thmtheorem4.p1.11.8.m8.1.2.2a.cmml">g</mtext><mo id="S7.Thmtheorem4.p1.11.8.m8.1.2.1" xref="S7.Thmtheorem4.p1.11.8.m8.1.2.1.cmml">⁢</mo><mrow id="S7.Thmtheorem4.p1.11.8.m8.1.2.3.2" xref="S7.Thmtheorem4.p1.11.8.m8.1.2.cmml"><mo id="S7.Thmtheorem4.p1.11.8.m8.1.2.3.2.1" stretchy="false" xref="S7.Thmtheorem4.p1.11.8.m8.1.2.cmml">(</mo><mi id="S7.Thmtheorem4.p1.11.8.m8.1.1" xref="S7.Thmtheorem4.p1.11.8.m8.1.1.cmml">u</mi><mo id="S7.Thmtheorem4.p1.11.8.m8.1.2.3.2.2" stretchy="false" xref="S7.Thmtheorem4.p1.11.8.m8.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S7.Thmtheorem4.p1.11.8.m8.1b"><apply id="S7.Thmtheorem4.p1.11.8.m8.1.2.cmml" xref="S7.Thmtheorem4.p1.11.8.m8.1.2"><times id="S7.Thmtheorem4.p1.11.8.m8.1.2.1.cmml" xref="S7.Thmtheorem4.p1.11.8.m8.1.2.1"></times><ci id="S7.Thmtheorem4.p1.11.8.m8.1.2.2a.cmml" xref="S7.Thmtheorem4.p1.11.8.m8.1.2.2"><mtext class="ltx_mathvariant_italic" id="S7.Thmtheorem4.p1.11.8.m8.1.2.2.cmml" xref="S7.Thmtheorem4.p1.11.8.m8.1.2.2">g</mtext></ci><ci id="S7.Thmtheorem4.p1.11.8.m8.1.1.cmml" xref="S7.Thmtheorem4.p1.11.8.m8.1.1">𝑢</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.Thmtheorem4.p1.11.8.m8.1c">\textsl{g}(u)</annotation><annotation encoding="application/x-llamapun" id="S7.Thmtheorem4.p1.11.8.m8.1d">g ( italic_u )</annotation></semantics></math> the vertex <math alttext="u" class="ltx_Math" display="inline" id="S7.Thmtheorem4.p1.12.9.m9.1"><semantics id="S7.Thmtheorem4.p1.12.9.m9.1a"><mi id="S7.Thmtheorem4.p1.12.9.m9.1.1" xref="S7.Thmtheorem4.p1.12.9.m9.1.1.cmml">u</mi><annotation-xml encoding="MathML-Content" id="S7.Thmtheorem4.p1.12.9.m9.1b"><ci id="S7.Thmtheorem4.p1.12.9.m9.1.1.cmml" xref="S7.Thmtheorem4.p1.12.9.m9.1.1">𝑢</ci></annotation-xml><annotation encoding="application/x-tex" id="S7.Thmtheorem4.p1.12.9.m9.1c">u</annotation><annotation encoding="application/x-llamapun" id="S7.Thmtheorem4.p1.12.9.m9.1d">italic_u</annotation></semantics></math> defaults to the lowest possible level and vice versa.</span></p> </div> </div> <div class="ltx_theorem ltx_theorem_definition" id="S7.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="S7.Thmtheorem5.1.1.1">Definition 7.5</span></span><span class="ltx_text ltx_font_bold" id="S7.Thmtheorem5.2.2">.</span> </h6> <div class="ltx_para" id="S7.Thmtheorem5.p1"> <p class="ltx_p" id="S7.Thmtheorem5.p1.3"><span class="ltx_text ltx_font_italic" id="S7.Thmtheorem5.p1.3.3">Consider an edge <math alttext="\overline{uv}" class="ltx_Math" display="inline" id="S7.Thmtheorem5.p1.1.1.m1.1"><semantics id="S7.Thmtheorem5.p1.1.1.m1.1a"><mover accent="true" id="S7.Thmtheorem5.p1.1.1.m1.1.1" xref="S7.Thmtheorem5.p1.1.1.m1.1.1.cmml"><mrow id="S7.Thmtheorem5.p1.1.1.m1.1.1.2" xref="S7.Thmtheorem5.p1.1.1.m1.1.1.2.cmml"><mi id="S7.Thmtheorem5.p1.1.1.m1.1.1.2.2" xref="S7.Thmtheorem5.p1.1.1.m1.1.1.2.2.cmml">u</mi><mo id="S7.Thmtheorem5.p1.1.1.m1.1.1.2.1" xref="S7.Thmtheorem5.p1.1.1.m1.1.1.2.1.cmml">⁢</mo><mi id="S7.Thmtheorem5.p1.1.1.m1.1.1.2.3" xref="S7.Thmtheorem5.p1.1.1.m1.1.1.2.3.cmml">v</mi></mrow><mo id="S7.Thmtheorem5.p1.1.1.m1.1.1.1" xref="S7.Thmtheorem5.p1.1.1.m1.1.1.1.cmml">¯</mo></mover><annotation-xml encoding="MathML-Content" id="S7.Thmtheorem5.p1.1.1.m1.1b"><apply id="S7.Thmtheorem5.p1.1.1.m1.1.1.cmml" xref="S7.Thmtheorem5.p1.1.1.m1.1.1"><ci id="S7.Thmtheorem5.p1.1.1.m1.1.1.1.cmml" xref="S7.Thmtheorem5.p1.1.1.m1.1.1.1">¯</ci><apply id="S7.Thmtheorem5.p1.1.1.m1.1.1.2.cmml" xref="S7.Thmtheorem5.p1.1.1.m1.1.1.2"><times id="S7.Thmtheorem5.p1.1.1.m1.1.1.2.1.cmml" xref="S7.Thmtheorem5.p1.1.1.m1.1.1.2.1"></times><ci id="S7.Thmtheorem5.p1.1.1.m1.1.1.2.2.cmml" xref="S7.Thmtheorem5.p1.1.1.m1.1.1.2.2">𝑢</ci><ci id="S7.Thmtheorem5.p1.1.1.m1.1.1.2.3.cmml" xref="S7.Thmtheorem5.p1.1.1.m1.1.1.2.3">𝑣</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.Thmtheorem5.p1.1.1.m1.1c">\overline{uv}</annotation><annotation encoding="application/x-llamapun" id="S7.Thmtheorem5.p1.1.1.m1.1d">over¯ start_ARG italic_u italic_v end_ARG</annotation></semantics></math> with <math alttext="u\in L_{i}" class="ltx_Math" display="inline" id="S7.Thmtheorem5.p1.2.2.m2.1"><semantics id="S7.Thmtheorem5.p1.2.2.m2.1a"><mrow id="S7.Thmtheorem5.p1.2.2.m2.1.1" xref="S7.Thmtheorem5.p1.2.2.m2.1.1.cmml"><mi id="S7.Thmtheorem5.p1.2.2.m2.1.1.2" xref="S7.Thmtheorem5.p1.2.2.m2.1.1.2.cmml">u</mi><mo id="S7.Thmtheorem5.p1.2.2.m2.1.1.1" xref="S7.Thmtheorem5.p1.2.2.m2.1.1.1.cmml">∈</mo><msub id="S7.Thmtheorem5.p1.2.2.m2.1.1.3" xref="S7.Thmtheorem5.p1.2.2.m2.1.1.3.cmml"><mi id="S7.Thmtheorem5.p1.2.2.m2.1.1.3.2" xref="S7.Thmtheorem5.p1.2.2.m2.1.1.3.2.cmml">L</mi><mi id="S7.Thmtheorem5.p1.2.2.m2.1.1.3.3" xref="S7.Thmtheorem5.p1.2.2.m2.1.1.3.3.cmml">i</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S7.Thmtheorem5.p1.2.2.m2.1b"><apply id="S7.Thmtheorem5.p1.2.2.m2.1.1.cmml" xref="S7.Thmtheorem5.p1.2.2.m2.1.1"><in id="S7.Thmtheorem5.p1.2.2.m2.1.1.1.cmml" xref="S7.Thmtheorem5.p1.2.2.m2.1.1.1"></in><ci id="S7.Thmtheorem5.p1.2.2.m2.1.1.2.cmml" xref="S7.Thmtheorem5.p1.2.2.m2.1.1.2">𝑢</ci><apply id="S7.Thmtheorem5.p1.2.2.m2.1.1.3.cmml" xref="S7.Thmtheorem5.p1.2.2.m2.1.1.3"><csymbol cd="ambiguous" id="S7.Thmtheorem5.p1.2.2.m2.1.1.3.1.cmml" xref="S7.Thmtheorem5.p1.2.2.m2.1.1.3">subscript</csymbol><ci id="S7.Thmtheorem5.p1.2.2.m2.1.1.3.2.cmml" xref="S7.Thmtheorem5.p1.2.2.m2.1.1.3.2">𝐿</ci><ci id="S7.Thmtheorem5.p1.2.2.m2.1.1.3.3.cmml" xref="S7.Thmtheorem5.p1.2.2.m2.1.1.3.3">𝑖</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.Thmtheorem5.p1.2.2.m2.1c">u\in L_{i}</annotation><annotation encoding="application/x-llamapun" id="S7.Thmtheorem5.p1.2.2.m2.1d">italic_u ∈ italic_L start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="v\in L_{j}" class="ltx_Math" display="inline" id="S7.Thmtheorem5.p1.3.3.m3.1"><semantics id="S7.Thmtheorem5.p1.3.3.m3.1a"><mrow id="S7.Thmtheorem5.p1.3.3.m3.1.1" xref="S7.Thmtheorem5.p1.3.3.m3.1.1.cmml"><mi id="S7.Thmtheorem5.p1.3.3.m3.1.1.2" xref="S7.Thmtheorem5.p1.3.3.m3.1.1.2.cmml">v</mi><mo id="S7.Thmtheorem5.p1.3.3.m3.1.1.1" xref="S7.Thmtheorem5.p1.3.3.m3.1.1.1.cmml">∈</mo><msub id="S7.Thmtheorem5.p1.3.3.m3.1.1.3" xref="S7.Thmtheorem5.p1.3.3.m3.1.1.3.cmml"><mi id="S7.Thmtheorem5.p1.3.3.m3.1.1.3.2" xref="S7.Thmtheorem5.p1.3.3.m3.1.1.3.2.cmml">L</mi><mi id="S7.Thmtheorem5.p1.3.3.m3.1.1.3.3" xref="S7.Thmtheorem5.p1.3.3.m3.1.1.3.3.cmml">j</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S7.Thmtheorem5.p1.3.3.m3.1b"><apply id="S7.Thmtheorem5.p1.3.3.m3.1.1.cmml" xref="S7.Thmtheorem5.p1.3.3.m3.1.1"><in id="S7.Thmtheorem5.p1.3.3.m3.1.1.1.cmml" xref="S7.Thmtheorem5.p1.3.3.m3.1.1.1"></in><ci id="S7.Thmtheorem5.p1.3.3.m3.1.1.2.cmml" xref="S7.Thmtheorem5.p1.3.3.m3.1.1.2">𝑣</ci><apply id="S7.Thmtheorem5.p1.3.3.m3.1.1.3.cmml" xref="S7.Thmtheorem5.p1.3.3.m3.1.1.3"><csymbol cd="ambiguous" id="S7.Thmtheorem5.p1.3.3.m3.1.1.3.1.cmml" xref="S7.Thmtheorem5.p1.3.3.m3.1.1.3">subscript</csymbol><ci id="S7.Thmtheorem5.p1.3.3.m3.1.1.3.2.cmml" xref="S7.Thmtheorem5.p1.3.3.m3.1.1.3.2">𝐿</ci><ci id="S7.Thmtheorem5.p1.3.3.m3.1.1.3.3.cmml" xref="S7.Thmtheorem5.p1.3.3.m3.1.1.3.3">𝑗</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.Thmtheorem5.p1.3.3.m3.1c">v\in L_{j}</annotation><annotation encoding="application/x-llamapun" id="S7.Thmtheorem5.p1.3.3.m3.1d">italic_v ∈ italic_L start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT</annotation></semantics></math>. We say that:</span></p> <ul class="ltx_itemize" id="S7.I2"> <li class="ltx_item" id="S7.I2.i1" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">•</span> <div class="ltx_para" id="S7.I2.i1.p1"> <p class="ltx_p" id="S7.I2.i1.p1.5"><math alttext="(u,v)" class="ltx_Math" display="inline" id="S7.I2.i1.p1.1.m1.2"><semantics id="S7.I2.i1.p1.1.m1.2a"><mrow id="S7.I2.i1.p1.1.m1.2.3.2" xref="S7.I2.i1.p1.1.m1.2.3.1.cmml"><mo id="S7.I2.i1.p1.1.m1.2.3.2.1" stretchy="false" xref="S7.I2.i1.p1.1.m1.2.3.1.cmml">(</mo><mi id="S7.I2.i1.p1.1.m1.1.1" xref="S7.I2.i1.p1.1.m1.1.1.cmml">u</mi><mo id="S7.I2.i1.p1.1.m1.2.3.2.2" xref="S7.I2.i1.p1.1.m1.2.3.1.cmml">,</mo><mi id="S7.I2.i1.p1.1.m1.2.2" xref="S7.I2.i1.p1.1.m1.2.2.cmml">v</mi><mo id="S7.I2.i1.p1.1.m1.2.3.2.3" stretchy="false" xref="S7.I2.i1.p1.1.m1.2.3.1.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S7.I2.i1.p1.1.m1.2b"><interval closure="open" id="S7.I2.i1.p1.1.m1.2.3.1.cmml" xref="S7.I2.i1.p1.1.m1.2.3.2"><ci id="S7.I2.i1.p1.1.m1.1.1.cmml" xref="S7.I2.i1.p1.1.m1.1.1">𝑢</ci><ci id="S7.I2.i1.p1.1.m1.2.2.cmml" xref="S7.I2.i1.p1.1.m1.2.2">𝑣</ci></interval></annotation-xml><annotation encoding="application/x-tex" id="S7.I2.i1.p1.1.m1.2c">(u,v)</annotation><annotation encoding="application/x-llamapun" id="S7.I2.i1.p1.1.m1.2d">( italic_u , italic_v )</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S7.I2.i1.p1.5.1"> is an </span><em class="ltx_emph" id="S7.I2.i1.p1.5.2">out-edge</em><span class="ltx_text ltx_font_italic" id="S7.I2.i1.p1.5.3"> from </span><math alttext="u" class="ltx_Math" display="inline" id="S7.I2.i1.p1.2.m2.1"><semantics id="S7.I2.i1.p1.2.m2.1a"><mi id="S7.I2.i1.p1.2.m2.1.1" xref="S7.I2.i1.p1.2.m2.1.1.cmml">u</mi><annotation-xml encoding="MathML-Content" id="S7.I2.i1.p1.2.m2.1b"><ci id="S7.I2.i1.p1.2.m2.1.1.cmml" xref="S7.I2.i1.p1.2.m2.1.1">𝑢</ci></annotation-xml><annotation encoding="application/x-tex" id="S7.I2.i1.p1.2.m2.1c">u</annotation><annotation encoding="application/x-llamapun" id="S7.I2.i1.p1.2.m2.1d">italic_u</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S7.I2.i1.p1.5.4"> and an in-edge into </span><math alttext="v" class="ltx_Math" display="inline" id="S7.I2.i1.p1.3.m3.1"><semantics id="S7.I2.i1.p1.3.m3.1a"><mi id="S7.I2.i1.p1.3.m3.1.1" xref="S7.I2.i1.p1.3.m3.1.1.cmml">v</mi><annotation-xml encoding="MathML-Content" id="S7.I2.i1.p1.3.m3.1b"><ci id="S7.I2.i1.p1.3.m3.1.1.cmml" xref="S7.I2.i1.p1.3.m3.1.1">𝑣</ci></annotation-xml><annotation encoding="application/x-tex" id="S7.I2.i1.p1.3.m3.1c">v</annotation><annotation encoding="application/x-llamapun" id="S7.I2.i1.p1.3.m3.1d">italic_v</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S7.I2.i1.p1.5.5"> whenever </span><math alttext="i>j" class="ltx_Math" display="inline" id="S7.I2.i1.p1.4.m4.1"><semantics id="S7.I2.i1.p1.4.m4.1a"><mrow id="S7.I2.i1.p1.4.m4.1.1" xref="S7.I2.i1.p1.4.m4.1.1.cmml"><mi id="S7.I2.i1.p1.4.m4.1.1.2" xref="S7.I2.i1.p1.4.m4.1.1.2.cmml">i</mi><mo id="S7.I2.i1.p1.4.m4.1.1.1" xref="S7.I2.i1.p1.4.m4.1.1.1.cmml">></mo><mi id="S7.I2.i1.p1.4.m4.1.1.3" xref="S7.I2.i1.p1.4.m4.1.1.3.cmml">j</mi></mrow><annotation-xml encoding="MathML-Content" id="S7.I2.i1.p1.4.m4.1b"><apply id="S7.I2.i1.p1.4.m4.1.1.cmml" xref="S7.I2.i1.p1.4.m4.1.1"><gt id="S7.I2.i1.p1.4.m4.1.1.1.cmml" xref="S7.I2.i1.p1.4.m4.1.1.1"></gt><ci id="S7.I2.i1.p1.4.m4.1.1.2.cmml" xref="S7.I2.i1.p1.4.m4.1.1.2">𝑖</ci><ci id="S7.I2.i1.p1.4.m4.1.1.3.cmml" xref="S7.I2.i1.p1.4.m4.1.1.3">𝑗</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.I2.i1.p1.4.m4.1c">i>j</annotation><annotation encoding="application/x-llamapun" id="S7.I2.i1.p1.4.m4.1d">italic_i > italic_j</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S7.I2.i1.p1.5.6"> and </span><math alttext="\textsl{g}(u\!\to\!v)>0" class="ltx_Math" display="inline" id="S7.I2.i1.p1.5.m5.1"><semantics id="S7.I2.i1.p1.5.m5.1a"><mrow id="S7.I2.i1.p1.5.m5.1.1" xref="S7.I2.i1.p1.5.m5.1.1.cmml"><mrow id="S7.I2.i1.p1.5.m5.1.1.1" xref="S7.I2.i1.p1.5.m5.1.1.1.cmml"><mtext class="ltx_mathvariant_italic" id="S7.I2.i1.p1.5.m5.1.1.1.3" xref="S7.I2.i1.p1.5.m5.1.1.1.3a.cmml">g</mtext><mo id="S7.I2.i1.p1.5.m5.1.1.1.2" xref="S7.I2.i1.p1.5.m5.1.1.1.2.cmml">⁢</mo><mrow id="S7.I2.i1.p1.5.m5.1.1.1.1.1" xref="S7.I2.i1.p1.5.m5.1.1.1.1.1.1.cmml"><mo id="S7.I2.i1.p1.5.m5.1.1.1.1.1.2" stretchy="false" xref="S7.I2.i1.p1.5.m5.1.1.1.1.1.1.cmml">(</mo><mrow id="S7.I2.i1.p1.5.m5.1.1.1.1.1.1" xref="S7.I2.i1.p1.5.m5.1.1.1.1.1.1.cmml"><mi id="S7.I2.i1.p1.5.m5.1.1.1.1.1.1.2" xref="S7.I2.i1.p1.5.m5.1.1.1.1.1.1.2.cmml">u</mi><mo id="S7.I2.i1.p1.5.m5.1.1.1.1.1.1.1" lspace="0.108em" rspace="0.108em" stretchy="false" xref="S7.I2.i1.p1.5.m5.1.1.1.1.1.1.1.cmml">→</mo><mi id="S7.I2.i1.p1.5.m5.1.1.1.1.1.1.3" xref="S7.I2.i1.p1.5.m5.1.1.1.1.1.1.3.cmml">v</mi></mrow><mo id="S7.I2.i1.p1.5.m5.1.1.1.1.1.3" stretchy="false" xref="S7.I2.i1.p1.5.m5.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S7.I2.i1.p1.5.m5.1.1.2" xref="S7.I2.i1.p1.5.m5.1.1.2.cmml">></mo><mn id="S7.I2.i1.p1.5.m5.1.1.3" xref="S7.I2.i1.p1.5.m5.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S7.I2.i1.p1.5.m5.1b"><apply id="S7.I2.i1.p1.5.m5.1.1.cmml" xref="S7.I2.i1.p1.5.m5.1.1"><gt id="S7.I2.i1.p1.5.m5.1.1.2.cmml" xref="S7.I2.i1.p1.5.m5.1.1.2"></gt><apply id="S7.I2.i1.p1.5.m5.1.1.1.cmml" xref="S7.I2.i1.p1.5.m5.1.1.1"><times id="S7.I2.i1.p1.5.m5.1.1.1.2.cmml" xref="S7.I2.i1.p1.5.m5.1.1.1.2"></times><ci id="S7.I2.i1.p1.5.m5.1.1.1.3a.cmml" xref="S7.I2.i1.p1.5.m5.1.1.1.3"><mtext class="ltx_mathvariant_italic" id="S7.I2.i1.p1.5.m5.1.1.1.3.cmml" xref="S7.I2.i1.p1.5.m5.1.1.1.3">g</mtext></ci><apply id="S7.I2.i1.p1.5.m5.1.1.1.1.1.1.cmml" xref="S7.I2.i1.p1.5.m5.1.1.1.1.1"><ci id="S7.I2.i1.p1.5.m5.1.1.1.1.1.1.1.cmml" xref="S7.I2.i1.p1.5.m5.1.1.1.1.1.1.1">→</ci><ci id="S7.I2.i1.p1.5.m5.1.1.1.1.1.1.2.cmml" xref="S7.I2.i1.p1.5.m5.1.1.1.1.1.1.2">𝑢</ci><ci id="S7.I2.i1.p1.5.m5.1.1.1.1.1.1.3.cmml" xref="S7.I2.i1.p1.5.m5.1.1.1.1.1.1.3">𝑣</ci></apply></apply><cn id="S7.I2.i1.p1.5.m5.1.1.3.cmml" type="integer" xref="S7.I2.i1.p1.5.m5.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.I2.i1.p1.5.m5.1c">\textsl{g}(u\!\to\!v)>0</annotation><annotation encoding="application/x-llamapun" id="S7.I2.i1.p1.5.m5.1d">g ( italic_u → italic_v ) > 0</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S7.I2.i1.p1.5.7">, and</span></p> </div> </li> <li class="ltx_item" id="S7.I2.i2" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">•</span> <div class="ltx_para" id="S7.I2.i2.p1"> <p class="ltx_p" id="S7.I2.i2.p1.3"><math alttext="(u,v)" class="ltx_Math" display="inline" id="S7.I2.i2.p1.1.m1.2"><semantics id="S7.I2.i2.p1.1.m1.2a"><mrow id="S7.I2.i2.p1.1.m1.2.3.2" xref="S7.I2.i2.p1.1.m1.2.3.1.cmml"><mo id="S7.I2.i2.p1.1.m1.2.3.2.1" stretchy="false" xref="S7.I2.i2.p1.1.m1.2.3.1.cmml">(</mo><mi id="S7.I2.i2.p1.1.m1.1.1" xref="S7.I2.i2.p1.1.m1.1.1.cmml">u</mi><mo id="S7.I2.i2.p1.1.m1.2.3.2.2" xref="S7.I2.i2.p1.1.m1.2.3.1.cmml">,</mo><mi id="S7.I2.i2.p1.1.m1.2.2" xref="S7.I2.i2.p1.1.m1.2.2.cmml">v</mi><mo id="S7.I2.i2.p1.1.m1.2.3.2.3" stretchy="false" xref="S7.I2.i2.p1.1.m1.2.3.1.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S7.I2.i2.p1.1.m1.2b"><interval closure="open" id="S7.I2.i2.p1.1.m1.2.3.1.cmml" xref="S7.I2.i2.p1.1.m1.2.3.2"><ci id="S7.I2.i2.p1.1.m1.1.1.cmml" xref="S7.I2.i2.p1.1.m1.1.1">𝑢</ci><ci id="S7.I2.i2.p1.1.m1.2.2.cmml" xref="S7.I2.i2.p1.1.m1.2.2">𝑣</ci></interval></annotation-xml><annotation encoding="application/x-tex" id="S7.I2.i2.p1.1.m1.2c">(u,v)</annotation><annotation encoding="application/x-llamapun" id="S7.I2.i2.p1.1.m1.2d">( italic_u , italic_v )</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S7.I2.i2.p1.3.1"> is </span><em class="ltx_emph" id="S7.I2.i2.p1.3.2">violating</em><span class="ltx_text ltx_font_italic" id="S7.I2.i2.p1.3.3"> whenever </span><math alttext="i>j+1" class="ltx_Math" display="inline" id="S7.I2.i2.p1.2.m2.1"><semantics id="S7.I2.i2.p1.2.m2.1a"><mrow id="S7.I2.i2.p1.2.m2.1.1" xref="S7.I2.i2.p1.2.m2.1.1.cmml"><mi id="S7.I2.i2.p1.2.m2.1.1.2" xref="S7.I2.i2.p1.2.m2.1.1.2.cmml">i</mi><mo id="S7.I2.i2.p1.2.m2.1.1.1" xref="S7.I2.i2.p1.2.m2.1.1.1.cmml">></mo><mrow id="S7.I2.i2.p1.2.m2.1.1.3" xref="S7.I2.i2.p1.2.m2.1.1.3.cmml"><mi id="S7.I2.i2.p1.2.m2.1.1.3.2" xref="S7.I2.i2.p1.2.m2.1.1.3.2.cmml">j</mi><mo id="S7.I2.i2.p1.2.m2.1.1.3.1" xref="S7.I2.i2.p1.2.m2.1.1.3.1.cmml">+</mo><mn id="S7.I2.i2.p1.2.m2.1.1.3.3" xref="S7.I2.i2.p1.2.m2.1.1.3.3.cmml">1</mn></mrow></mrow><annotation-xml encoding="MathML-Content" id="S7.I2.i2.p1.2.m2.1b"><apply id="S7.I2.i2.p1.2.m2.1.1.cmml" xref="S7.I2.i2.p1.2.m2.1.1"><gt id="S7.I2.i2.p1.2.m2.1.1.1.cmml" xref="S7.I2.i2.p1.2.m2.1.1.1"></gt><ci id="S7.I2.i2.p1.2.m2.1.1.2.cmml" xref="S7.I2.i2.p1.2.m2.1.1.2">𝑖</ci><apply id="S7.I2.i2.p1.2.m2.1.1.3.cmml" xref="S7.I2.i2.p1.2.m2.1.1.3"><plus id="S7.I2.i2.p1.2.m2.1.1.3.1.cmml" xref="S7.I2.i2.p1.2.m2.1.1.3.1"></plus><ci id="S7.I2.i2.p1.2.m2.1.1.3.2.cmml" xref="S7.I2.i2.p1.2.m2.1.1.3.2">𝑗</ci><cn id="S7.I2.i2.p1.2.m2.1.1.3.3.cmml" type="integer" xref="S7.I2.i2.p1.2.m2.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.I2.i2.p1.2.m2.1c">i>j+1</annotation><annotation encoding="application/x-llamapun" id="S7.I2.i2.p1.2.m2.1d">italic_i > italic_j + 1</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S7.I2.i2.p1.3.4"> and </span><math alttext="\textsl{g}(u\!\to\!v)>0" class="ltx_Math" display="inline" id="S7.I2.i2.p1.3.m3.1"><semantics id="S7.I2.i2.p1.3.m3.1a"><mrow id="S7.I2.i2.p1.3.m3.1.1" xref="S7.I2.i2.p1.3.m3.1.1.cmml"><mrow id="S7.I2.i2.p1.3.m3.1.1.1" xref="S7.I2.i2.p1.3.m3.1.1.1.cmml"><mtext class="ltx_mathvariant_italic" id="S7.I2.i2.p1.3.m3.1.1.1.3" xref="S7.I2.i2.p1.3.m3.1.1.1.3a.cmml">g</mtext><mo id="S7.I2.i2.p1.3.m3.1.1.1.2" xref="S7.I2.i2.p1.3.m3.1.1.1.2.cmml">⁢</mo><mrow id="S7.I2.i2.p1.3.m3.1.1.1.1.1" xref="S7.I2.i2.p1.3.m3.1.1.1.1.1.1.cmml"><mo id="S7.I2.i2.p1.3.m3.1.1.1.1.1.2" stretchy="false" xref="S7.I2.i2.p1.3.m3.1.1.1.1.1.1.cmml">(</mo><mrow id="S7.I2.i2.p1.3.m3.1.1.1.1.1.1" xref="S7.I2.i2.p1.3.m3.1.1.1.1.1.1.cmml"><mi id="S7.I2.i2.p1.3.m3.1.1.1.1.1.1.2" xref="S7.I2.i2.p1.3.m3.1.1.1.1.1.1.2.cmml">u</mi><mo id="S7.I2.i2.p1.3.m3.1.1.1.1.1.1.1" lspace="0.108em" rspace="0.108em" stretchy="false" xref="S7.I2.i2.p1.3.m3.1.1.1.1.1.1.1.cmml">→</mo><mi id="S7.I2.i2.p1.3.m3.1.1.1.1.1.1.3" xref="S7.I2.i2.p1.3.m3.1.1.1.1.1.1.3.cmml">v</mi></mrow><mo id="S7.I2.i2.p1.3.m3.1.1.1.1.1.3" stretchy="false" xref="S7.I2.i2.p1.3.m3.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S7.I2.i2.p1.3.m3.1.1.2" xref="S7.I2.i2.p1.3.m3.1.1.2.cmml">></mo><mn id="S7.I2.i2.p1.3.m3.1.1.3" xref="S7.I2.i2.p1.3.m3.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S7.I2.i2.p1.3.m3.1b"><apply id="S7.I2.i2.p1.3.m3.1.1.cmml" xref="S7.I2.i2.p1.3.m3.1.1"><gt id="S7.I2.i2.p1.3.m3.1.1.2.cmml" xref="S7.I2.i2.p1.3.m3.1.1.2"></gt><apply id="S7.I2.i2.p1.3.m3.1.1.1.cmml" xref="S7.I2.i2.p1.3.m3.1.1.1"><times id="S7.I2.i2.p1.3.m3.1.1.1.2.cmml" xref="S7.I2.i2.p1.3.m3.1.1.1.2"></times><ci id="S7.I2.i2.p1.3.m3.1.1.1.3a.cmml" xref="S7.I2.i2.p1.3.m3.1.1.1.3"><mtext class="ltx_mathvariant_italic" id="S7.I2.i2.p1.3.m3.1.1.1.3.cmml" xref="S7.I2.i2.p1.3.m3.1.1.1.3">g</mtext></ci><apply id="S7.I2.i2.p1.3.m3.1.1.1.1.1.1.cmml" xref="S7.I2.i2.p1.3.m3.1.1.1.1.1"><ci id="S7.I2.i2.p1.3.m3.1.1.1.1.1.1.1.cmml" xref="S7.I2.i2.p1.3.m3.1.1.1.1.1.1.1">→</ci><ci id="S7.I2.i2.p1.3.m3.1.1.1.1.1.1.2.cmml" xref="S7.I2.i2.p1.3.m3.1.1.1.1.1.1.2">𝑢</ci><ci id="S7.I2.i2.p1.3.m3.1.1.1.1.1.1.3.cmml" xref="S7.I2.i2.p1.3.m3.1.1.1.1.1.1.3">𝑣</ci></apply></apply><cn id="S7.I2.i2.p1.3.m3.1.1.3.cmml" type="integer" xref="S7.I2.i2.p1.3.m3.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.I2.i2.p1.3.m3.1c">\textsl{g}(u\!\to\!v)>0</annotation><annotation encoding="application/x-llamapun" id="S7.I2.i2.p1.3.m3.1d">g ( italic_u → italic_v ) > 0</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S7.I2.i2.p1.3.5"></span></p> </div> </li> </ul> </div> </div> <div class="ltx_para ltx_noindent" id="S7.SS0.SSS0.P0.SPx1.p2"> <p class="ltx_p" id="S7.SS0.SSS0.P0.SPx1.p2.1">Note that the orientation is <math alttext="\eta" class="ltx_Math" display="inline" id="S7.SS0.SSS0.P0.SPx1.p2.1.m1.1"><semantics id="S7.SS0.SSS0.P0.SPx1.p2.1.m1.1a"><mi id="S7.SS0.SSS0.P0.SPx1.p2.1.m1.1.1" xref="S7.SS0.SSS0.P0.SPx1.p2.1.m1.1.1.cmml">η</mi><annotation-xml encoding="MathML-Content" id="S7.SS0.SSS0.P0.SPx1.p2.1.m1.1b"><ci id="S7.SS0.SSS0.P0.SPx1.p2.1.m1.1.1.cmml" xref="S7.SS0.SSS0.P0.SPx1.p2.1.m1.1.1">𝜂</ci></annotation-xml><annotation encoding="application/x-tex" id="S7.SS0.SSS0.P0.SPx1.p2.1.m1.1c">\eta</annotation><annotation encoding="application/x-llamapun" id="S7.SS0.SSS0.P0.SPx1.p2.1.m1.1d">italic_η</annotation></semantics></math>-fair whenever there exist no violating edges.</p> </div> <figure class="ltx_figure" id="S7.F1"><img alt="Refer to caption" class="ltx_graphics ltx_centering ltx_img_landscape" height="166" id="S7.F1.g1" src="x1.png" width="829"/> <figcaption class="ltx_caption ltx_centering"><span class="ltx_tag ltx_tag_figure"><span class="ltx_text" id="S7.F1.26.12.1" style="font-size:90%;">Figure 1</span>: </span><span class="ltx_text" id="S7.F1.22.11" style="font-size:90%;"> Given a graph <math alttext="G" class="ltx_Math" display="inline" id="S7.F1.12.1.m1.1"><semantics id="S7.F1.12.1.m1.1b"><mi id="S7.F1.12.1.m1.1.1" xref="S7.F1.12.1.m1.1.1.cmml">G</mi><annotation-xml encoding="MathML-Content" id="S7.F1.12.1.m1.1c"><ci id="S7.F1.12.1.m1.1.1.cmml" xref="S7.F1.12.1.m1.1.1">𝐺</ci></annotation-xml><annotation encoding="application/x-tex" id="S7.F1.12.1.m1.1d">G</annotation><annotation encoding="application/x-llamapun" id="S7.F1.12.1.m1.1e">italic_G</annotation></semantics></math>, we arbitrarily orient <math alttext="G" class="ltx_Math" display="inline" id="S7.F1.13.2.m2.1"><semantics id="S7.F1.13.2.m2.1b"><mi id="S7.F1.13.2.m2.1.1" xref="S7.F1.13.2.m2.1.1.cmml">G</mi><annotation-xml encoding="MathML-Content" id="S7.F1.13.2.m2.1c"><ci id="S7.F1.13.2.m2.1.1.cmml" xref="S7.F1.13.2.m2.1.1">𝐺</ci></annotation-xml><annotation encoding="application/x-tex" id="S7.F1.13.2.m2.1d">G</annotation><annotation encoding="application/x-llamapun" id="S7.F1.13.2.m2.1e">italic_G</annotation></semantics></math>. This allows us to partition the vertices of <math alttext="G" class="ltx_Math" display="inline" id="S7.F1.14.3.m3.1"><semantics id="S7.F1.14.3.m3.1b"><mi id="S7.F1.14.3.m3.1.1" xref="S7.F1.14.3.m3.1.1.cmml">G</mi><annotation-xml encoding="MathML-Content" id="S7.F1.14.3.m3.1c"><ci id="S7.F1.14.3.m3.1.1.cmml" xref="S7.F1.14.3.m3.1.1">𝐺</ci></annotation-xml><annotation encoding="application/x-tex" id="S7.F1.14.3.m3.1d">G</annotation><annotation encoding="application/x-llamapun" id="S7.F1.14.3.m3.1e">italic_G</annotation></semantics></math> into <em class="ltx_emph ltx_font_italic" id="S7.F1.22.11.1">levels</em> <math alttext="L_{1},\ldots L_{6}" class="ltx_Math" display="inline" id="S7.F1.15.4.m4.2"><semantics id="S7.F1.15.4.m4.2b"><mrow id="S7.F1.15.4.m4.2.2.2" xref="S7.F1.15.4.m4.2.2.3.cmml"><msub id="S7.F1.15.4.m4.1.1.1.1" xref="S7.F1.15.4.m4.1.1.1.1.cmml"><mi id="S7.F1.15.4.m4.1.1.1.1.2" xref="S7.F1.15.4.m4.1.1.1.1.2.cmml">L</mi><mn id="S7.F1.15.4.m4.1.1.1.1.3" xref="S7.F1.15.4.m4.1.1.1.1.3.cmml">1</mn></msub><mo id="S7.F1.15.4.m4.2.2.2.3" xref="S7.F1.15.4.m4.2.2.3.cmml">,</mo><mrow id="S7.F1.15.4.m4.2.2.2.2" xref="S7.F1.15.4.m4.2.2.2.2.cmml"><mi id="S7.F1.15.4.m4.2.2.2.2.2" mathvariant="normal" xref="S7.F1.15.4.m4.2.2.2.2.2.cmml">…</mi><mo id="S7.F1.15.4.m4.2.2.2.2.1" xref="S7.F1.15.4.m4.2.2.2.2.1.cmml">⁢</mo><msub id="S7.F1.15.4.m4.2.2.2.2.3" xref="S7.F1.15.4.m4.2.2.2.2.3.cmml"><mi id="S7.F1.15.4.m4.2.2.2.2.3.2" xref="S7.F1.15.4.m4.2.2.2.2.3.2.cmml">L</mi><mn id="S7.F1.15.4.m4.2.2.2.2.3.3" xref="S7.F1.15.4.m4.2.2.2.2.3.3.cmml">6</mn></msub></mrow></mrow><annotation-xml encoding="MathML-Content" id="S7.F1.15.4.m4.2c"><list id="S7.F1.15.4.m4.2.2.3.cmml" xref="S7.F1.15.4.m4.2.2.2"><apply id="S7.F1.15.4.m4.1.1.1.1.cmml" xref="S7.F1.15.4.m4.1.1.1.1"><csymbol cd="ambiguous" id="S7.F1.15.4.m4.1.1.1.1.1.cmml" xref="S7.F1.15.4.m4.1.1.1.1">subscript</csymbol><ci id="S7.F1.15.4.m4.1.1.1.1.2.cmml" xref="S7.F1.15.4.m4.1.1.1.1.2">𝐿</ci><cn id="S7.F1.15.4.m4.1.1.1.1.3.cmml" type="integer" xref="S7.F1.15.4.m4.1.1.1.1.3">1</cn></apply><apply id="S7.F1.15.4.m4.2.2.2.2.cmml" xref="S7.F1.15.4.m4.2.2.2.2"><times id="S7.F1.15.4.m4.2.2.2.2.1.cmml" xref="S7.F1.15.4.m4.2.2.2.2.1"></times><ci id="S7.F1.15.4.m4.2.2.2.2.2.cmml" xref="S7.F1.15.4.m4.2.2.2.2.2">…</ci><apply id="S7.F1.15.4.m4.2.2.2.2.3.cmml" xref="S7.F1.15.4.m4.2.2.2.2.3"><csymbol cd="ambiguous" id="S7.F1.15.4.m4.2.2.2.2.3.1.cmml" xref="S7.F1.15.4.m4.2.2.2.2.3">subscript</csymbol><ci id="S7.F1.15.4.m4.2.2.2.2.3.2.cmml" xref="S7.F1.15.4.m4.2.2.2.2.3.2">𝐿</ci><cn id="S7.F1.15.4.m4.2.2.2.2.3.3.cmml" type="integer" xref="S7.F1.15.4.m4.2.2.2.2.3.3">6</cn></apply></apply></list></annotation-xml><annotation encoding="application/x-tex" id="S7.F1.15.4.m4.2d">L_{1},\ldots L_{6}</annotation><annotation encoding="application/x-llamapun" id="S7.F1.15.4.m4.2e">italic_L start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , … italic_L start_POSTSUBSCRIPT 6 end_POSTSUBSCRIPT</annotation></semantics></math> based on their current out-degree. We say that an edge <math alttext="(u,v)" class="ltx_Math" display="inline" id="S7.F1.16.5.m5.2"><semantics id="S7.F1.16.5.m5.2b"><mrow id="S7.F1.16.5.m5.2.3.2" xref="S7.F1.16.5.m5.2.3.1.cmml"><mo id="S7.F1.16.5.m5.2.3.2.1" stretchy="false" xref="S7.F1.16.5.m5.2.3.1.cmml">(</mo><mi id="S7.F1.16.5.m5.1.1" xref="S7.F1.16.5.m5.1.1.cmml">u</mi><mo id="S7.F1.16.5.m5.2.3.2.2" xref="S7.F1.16.5.m5.2.3.1.cmml">,</mo><mi id="S7.F1.16.5.m5.2.2" xref="S7.F1.16.5.m5.2.2.cmml">v</mi><mo id="S7.F1.16.5.m5.2.3.2.3" stretchy="false" xref="S7.F1.16.5.m5.2.3.1.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S7.F1.16.5.m5.2c"><interval closure="open" id="S7.F1.16.5.m5.2.3.1.cmml" xref="S7.F1.16.5.m5.2.3.2"><ci id="S7.F1.16.5.m5.1.1.cmml" xref="S7.F1.16.5.m5.1.1">𝑢</ci><ci id="S7.F1.16.5.m5.2.2.cmml" xref="S7.F1.16.5.m5.2.2">𝑣</ci></interval></annotation-xml><annotation encoding="application/x-tex" id="S7.F1.16.5.m5.2d">(u,v)</annotation><annotation encoding="application/x-llamapun" id="S7.F1.16.5.m5.2e">( italic_u , italic_v )</annotation></semantics></math> is <em class="ltx_emph ltx_font_italic" id="S7.F1.22.11.2">violating</em> whenever <math alttext="\textsl{g}(u\!\to\!v)>0" class="ltx_Math" display="inline" id="S7.F1.17.6.m6.1"><semantics id="S7.F1.17.6.m6.1b"><mrow id="S7.F1.17.6.m6.1.1" xref="S7.F1.17.6.m6.1.1.cmml"><mrow id="S7.F1.17.6.m6.1.1.1" xref="S7.F1.17.6.m6.1.1.1.cmml"><mtext class="ltx_mathvariant_italic" id="S7.F1.17.6.m6.1.1.1.3" xref="S7.F1.17.6.m6.1.1.1.3a.cmml">g</mtext><mo id="S7.F1.17.6.m6.1.1.1.2" xref="S7.F1.17.6.m6.1.1.1.2.cmml">⁢</mo><mrow id="S7.F1.17.6.m6.1.1.1.1.1" xref="S7.F1.17.6.m6.1.1.1.1.1.1.cmml"><mo id="S7.F1.17.6.m6.1.1.1.1.1.2" stretchy="false" xref="S7.F1.17.6.m6.1.1.1.1.1.1.cmml">(</mo><mrow id="S7.F1.17.6.m6.1.1.1.1.1.1" xref="S7.F1.17.6.m6.1.1.1.1.1.1.cmml"><mi id="S7.F1.17.6.m6.1.1.1.1.1.1.2" xref="S7.F1.17.6.m6.1.1.1.1.1.1.2.cmml">u</mi><mo id="S7.F1.17.6.m6.1.1.1.1.1.1.1" lspace="0.108em" rspace="0.108em" stretchy="false" xref="S7.F1.17.6.m6.1.1.1.1.1.1.1.cmml">→</mo><mi id="S7.F1.17.6.m6.1.1.1.1.1.1.3" xref="S7.F1.17.6.m6.1.1.1.1.1.1.3.cmml">v</mi></mrow><mo id="S7.F1.17.6.m6.1.1.1.1.1.3" stretchy="false" xref="S7.F1.17.6.m6.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S7.F1.17.6.m6.1.1.2" xref="S7.F1.17.6.m6.1.1.2.cmml">></mo><mn id="S7.F1.17.6.m6.1.1.3" xref="S7.F1.17.6.m6.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S7.F1.17.6.m6.1c"><apply id="S7.F1.17.6.m6.1.1.cmml" xref="S7.F1.17.6.m6.1.1"><gt id="S7.F1.17.6.m6.1.1.2.cmml" xref="S7.F1.17.6.m6.1.1.2"></gt><apply id="S7.F1.17.6.m6.1.1.1.cmml" xref="S7.F1.17.6.m6.1.1.1"><times id="S7.F1.17.6.m6.1.1.1.2.cmml" xref="S7.F1.17.6.m6.1.1.1.2"></times><ci id="S7.F1.17.6.m6.1.1.1.3a.cmml" xref="S7.F1.17.6.m6.1.1.1.3"><mtext class="ltx_mathvariant_italic" id="S7.F1.17.6.m6.1.1.1.3.cmml" xref="S7.F1.17.6.m6.1.1.1.3">g</mtext></ci><apply id="S7.F1.17.6.m6.1.1.1.1.1.1.cmml" xref="S7.F1.17.6.m6.1.1.1.1.1"><ci id="S7.F1.17.6.m6.1.1.1.1.1.1.1.cmml" xref="S7.F1.17.6.m6.1.1.1.1.1.1.1">→</ci><ci id="S7.F1.17.6.m6.1.1.1.1.1.1.2.cmml" xref="S7.F1.17.6.m6.1.1.1.1.1.1.2">𝑢</ci><ci id="S7.F1.17.6.m6.1.1.1.1.1.1.3.cmml" xref="S7.F1.17.6.m6.1.1.1.1.1.1.3">𝑣</ci></apply></apply><cn id="S7.F1.17.6.m6.1.1.3.cmml" type="integer" xref="S7.F1.17.6.m6.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.F1.17.6.m6.1d">\textsl{g}(u\!\to\!v)>0</annotation><annotation encoding="application/x-llamapun" id="S7.F1.17.6.m6.1e">g ( italic_u → italic_v ) > 0</annotation></semantics></math>, <math alttext="u\in L_{i}" class="ltx_Math" display="inline" id="S7.F1.18.7.m7.1"><semantics id="S7.F1.18.7.m7.1b"><mrow id="S7.F1.18.7.m7.1.1" xref="S7.F1.18.7.m7.1.1.cmml"><mi id="S7.F1.18.7.m7.1.1.2" xref="S7.F1.18.7.m7.1.1.2.cmml">u</mi><mo id="S7.F1.18.7.m7.1.1.1" xref="S7.F1.18.7.m7.1.1.1.cmml">∈</mo><msub id="S7.F1.18.7.m7.1.1.3" xref="S7.F1.18.7.m7.1.1.3.cmml"><mi id="S7.F1.18.7.m7.1.1.3.2" xref="S7.F1.18.7.m7.1.1.3.2.cmml">L</mi><mi id="S7.F1.18.7.m7.1.1.3.3" xref="S7.F1.18.7.m7.1.1.3.3.cmml">i</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S7.F1.18.7.m7.1c"><apply id="S7.F1.18.7.m7.1.1.cmml" xref="S7.F1.18.7.m7.1.1"><in id="S7.F1.18.7.m7.1.1.1.cmml" xref="S7.F1.18.7.m7.1.1.1"></in><ci id="S7.F1.18.7.m7.1.1.2.cmml" xref="S7.F1.18.7.m7.1.1.2">𝑢</ci><apply id="S7.F1.18.7.m7.1.1.3.cmml" xref="S7.F1.18.7.m7.1.1.3"><csymbol cd="ambiguous" id="S7.F1.18.7.m7.1.1.3.1.cmml" xref="S7.F1.18.7.m7.1.1.3">subscript</csymbol><ci id="S7.F1.18.7.m7.1.1.3.2.cmml" xref="S7.F1.18.7.m7.1.1.3.2">𝐿</ci><ci id="S7.F1.18.7.m7.1.1.3.3.cmml" xref="S7.F1.18.7.m7.1.1.3.3">𝑖</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.F1.18.7.m7.1d">u\in L_{i}</annotation><annotation encoding="application/x-llamapun" id="S7.F1.18.7.m7.1e">italic_u ∈ italic_L start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math>, <math alttext="v\in L_{j}" class="ltx_Math" display="inline" id="S7.F1.19.8.m8.1"><semantics id="S7.F1.19.8.m8.1b"><mrow id="S7.F1.19.8.m8.1.1" xref="S7.F1.19.8.m8.1.1.cmml"><mi id="S7.F1.19.8.m8.1.1.2" xref="S7.F1.19.8.m8.1.1.2.cmml">v</mi><mo id="S7.F1.19.8.m8.1.1.1" xref="S7.F1.19.8.m8.1.1.1.cmml">∈</mo><msub id="S7.F1.19.8.m8.1.1.3" xref="S7.F1.19.8.m8.1.1.3.cmml"><mi id="S7.F1.19.8.m8.1.1.3.2" xref="S7.F1.19.8.m8.1.1.3.2.cmml">L</mi><mi id="S7.F1.19.8.m8.1.1.3.3" xref="S7.F1.19.8.m8.1.1.3.3.cmml">j</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S7.F1.19.8.m8.1c"><apply id="S7.F1.19.8.m8.1.1.cmml" xref="S7.F1.19.8.m8.1.1"><in id="S7.F1.19.8.m8.1.1.1.cmml" xref="S7.F1.19.8.m8.1.1.1"></in><ci id="S7.F1.19.8.m8.1.1.2.cmml" xref="S7.F1.19.8.m8.1.1.2">𝑣</ci><apply id="S7.F1.19.8.m8.1.1.3.cmml" xref="S7.F1.19.8.m8.1.1.3"><csymbol cd="ambiguous" id="S7.F1.19.8.m8.1.1.3.1.cmml" xref="S7.F1.19.8.m8.1.1.3">subscript</csymbol><ci id="S7.F1.19.8.m8.1.1.3.2.cmml" xref="S7.F1.19.8.m8.1.1.3.2">𝐿</ci><ci id="S7.F1.19.8.m8.1.1.3.3.cmml" xref="S7.F1.19.8.m8.1.1.3.3">𝑗</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.F1.19.8.m8.1d">v\in L_{j}</annotation><annotation encoding="application/x-llamapun" id="S7.F1.19.8.m8.1e">italic_v ∈ italic_L start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="i>j+1" class="ltx_Math" display="inline" id="S7.F1.20.9.m9.1"><semantics id="S7.F1.20.9.m9.1b"><mrow id="S7.F1.20.9.m9.1.1" xref="S7.F1.20.9.m9.1.1.cmml"><mi id="S7.F1.20.9.m9.1.1.2" xref="S7.F1.20.9.m9.1.1.2.cmml">i</mi><mo id="S7.F1.20.9.m9.1.1.1" xref="S7.F1.20.9.m9.1.1.1.cmml">></mo><mrow id="S7.F1.20.9.m9.1.1.3" xref="S7.F1.20.9.m9.1.1.3.cmml"><mi id="S7.F1.20.9.m9.1.1.3.2" xref="S7.F1.20.9.m9.1.1.3.2.cmml">j</mi><mo id="S7.F1.20.9.m9.1.1.3.1" xref="S7.F1.20.9.m9.1.1.3.1.cmml">+</mo><mn id="S7.F1.20.9.m9.1.1.3.3" xref="S7.F1.20.9.m9.1.1.3.3.cmml">1</mn></mrow></mrow><annotation-xml encoding="MathML-Content" id="S7.F1.20.9.m9.1c"><apply id="S7.F1.20.9.m9.1.1.cmml" xref="S7.F1.20.9.m9.1.1"><gt id="S7.F1.20.9.m9.1.1.1.cmml" xref="S7.F1.20.9.m9.1.1.1"></gt><ci id="S7.F1.20.9.m9.1.1.2.cmml" xref="S7.F1.20.9.m9.1.1.2">𝑖</ci><apply id="S7.F1.20.9.m9.1.1.3.cmml" xref="S7.F1.20.9.m9.1.1.3"><plus id="S7.F1.20.9.m9.1.1.3.1.cmml" xref="S7.F1.20.9.m9.1.1.3.1"></plus><ci id="S7.F1.20.9.m9.1.1.3.2.cmml" xref="S7.F1.20.9.m9.1.1.3.2">𝑗</ci><cn id="S7.F1.20.9.m9.1.1.3.3.cmml" type="integer" xref="S7.F1.20.9.m9.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.F1.20.9.m9.1d">i>j+1</annotation><annotation encoding="application/x-llamapun" id="S7.F1.20.9.m9.1e">italic_i > italic_j + 1</annotation></semantics></math>. We show violating edges in red. Our algorithm iterates over an integer <math alttext="h" class="ltx_Math" display="inline" id="S7.F1.21.10.m10.1"><semantics id="S7.F1.21.10.m10.1b"><mi id="S7.F1.21.10.m10.1.1" xref="S7.F1.21.10.m10.1.1.cmml">h</mi><annotation-xml encoding="MathML-Content" id="S7.F1.21.10.m10.1c"><ci id="S7.F1.21.10.m10.1.1.cmml" xref="S7.F1.21.10.m10.1.1">ℎ</ci></annotation-xml><annotation encoding="application/x-tex" id="S7.F1.21.10.m10.1d">h</annotation><annotation encoding="application/x-llamapun" id="S7.F1.21.10.m10.1e">italic_h</annotation></semantics></math> from high to low, and tries to flip all violating edges from level <math alttext="L_{h}" class="ltx_Math" display="inline" id="S7.F1.22.11.m11.1"><semantics id="S7.F1.22.11.m11.1b"><msub id="S7.F1.22.11.m11.1.1" xref="S7.F1.22.11.m11.1.1.cmml"><mi id="S7.F1.22.11.m11.1.1.2" xref="S7.F1.22.11.m11.1.1.2.cmml">L</mi><mi id="S7.F1.22.11.m11.1.1.3" xref="S7.F1.22.11.m11.1.1.3.cmml">h</mi></msub><annotation-xml encoding="MathML-Content" id="S7.F1.22.11.m11.1c"><apply id="S7.F1.22.11.m11.1.1.cmml" xref="S7.F1.22.11.m11.1.1"><csymbol cd="ambiguous" id="S7.F1.22.11.m11.1.1.1.cmml" xref="S7.F1.22.11.m11.1.1">subscript</csymbol><ci id="S7.F1.22.11.m11.1.1.2.cmml" xref="S7.F1.22.11.m11.1.1.2">𝐿</ci><ci id="S7.F1.22.11.m11.1.1.3.cmml" xref="S7.F1.22.11.m11.1.1.3">ℎ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.F1.22.11.m11.1d">L_{h}</annotation><annotation encoding="application/x-llamapun" id="S7.F1.22.11.m11.1e">italic_L start_POSTSUBSCRIPT italic_h end_POSTSUBSCRIPT</annotation></semantics></math>. </span></figcaption> </figure> <div class="ltx_theorem ltx_theorem_lemma" id="S7.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="S7.Thmtheorem6.1.1.1">Lemma 7.6</span></span><span class="ltx_text ltx_font_bold" id="S7.Thmtheorem6.2.2">.</span> </h6> <div class="ltx_para" id="S7.Thmtheorem6.p1"> <p class="ltx_p" id="S7.Thmtheorem6.p1.4"><span class="ltx_text ltx_font_italic" id="S7.Thmtheorem6.p1.4.4">Let <math alttext="\eta\leq\frac{\varepsilon^{2}}{128\cdot\log n}" class="ltx_Math" display="inline" id="S7.Thmtheorem6.p1.1.1.m1.1"><semantics id="S7.Thmtheorem6.p1.1.1.m1.1a"><mrow id="S7.Thmtheorem6.p1.1.1.m1.1.1" xref="S7.Thmtheorem6.p1.1.1.m1.1.1.cmml"><mi id="S7.Thmtheorem6.p1.1.1.m1.1.1.2" xref="S7.Thmtheorem6.p1.1.1.m1.1.1.2.cmml">η</mi><mo id="S7.Thmtheorem6.p1.1.1.m1.1.1.1" xref="S7.Thmtheorem6.p1.1.1.m1.1.1.1.cmml">≤</mo><mfrac id="S7.Thmtheorem6.p1.1.1.m1.1.1.3" xref="S7.Thmtheorem6.p1.1.1.m1.1.1.3.cmml"><msup id="S7.Thmtheorem6.p1.1.1.m1.1.1.3.2" xref="S7.Thmtheorem6.p1.1.1.m1.1.1.3.2.cmml"><mi id="S7.Thmtheorem6.p1.1.1.m1.1.1.3.2.2" xref="S7.Thmtheorem6.p1.1.1.m1.1.1.3.2.2.cmml">ε</mi><mn id="S7.Thmtheorem6.p1.1.1.m1.1.1.3.2.3" xref="S7.Thmtheorem6.p1.1.1.m1.1.1.3.2.3.cmml">2</mn></msup><mrow id="S7.Thmtheorem6.p1.1.1.m1.1.1.3.3" xref="S7.Thmtheorem6.p1.1.1.m1.1.1.3.3.cmml"><mn id="S7.Thmtheorem6.p1.1.1.m1.1.1.3.3.2" xref="S7.Thmtheorem6.p1.1.1.m1.1.1.3.3.2.cmml">128</mn><mo id="S7.Thmtheorem6.p1.1.1.m1.1.1.3.3.1" lspace="0.222em" rspace="0.222em" xref="S7.Thmtheorem6.p1.1.1.m1.1.1.3.3.1.cmml">⋅</mo><mrow id="S7.Thmtheorem6.p1.1.1.m1.1.1.3.3.3" xref="S7.Thmtheorem6.p1.1.1.m1.1.1.3.3.3.cmml"><mi id="S7.Thmtheorem6.p1.1.1.m1.1.1.3.3.3.1" xref="S7.Thmtheorem6.p1.1.1.m1.1.1.3.3.3.1.cmml">log</mi><mo id="S7.Thmtheorem6.p1.1.1.m1.1.1.3.3.3a" lspace="0.167em" xref="S7.Thmtheorem6.p1.1.1.m1.1.1.3.3.3.cmml">⁡</mo><mi id="S7.Thmtheorem6.p1.1.1.m1.1.1.3.3.3.2" xref="S7.Thmtheorem6.p1.1.1.m1.1.1.3.3.3.2.cmml">n</mi></mrow></mrow></mfrac></mrow><annotation-xml encoding="MathML-Content" id="S7.Thmtheorem6.p1.1.1.m1.1b"><apply id="S7.Thmtheorem6.p1.1.1.m1.1.1.cmml" xref="S7.Thmtheorem6.p1.1.1.m1.1.1"><leq id="S7.Thmtheorem6.p1.1.1.m1.1.1.1.cmml" xref="S7.Thmtheorem6.p1.1.1.m1.1.1.1"></leq><ci id="S7.Thmtheorem6.p1.1.1.m1.1.1.2.cmml" xref="S7.Thmtheorem6.p1.1.1.m1.1.1.2">𝜂</ci><apply id="S7.Thmtheorem6.p1.1.1.m1.1.1.3.cmml" xref="S7.Thmtheorem6.p1.1.1.m1.1.1.3"><divide id="S7.Thmtheorem6.p1.1.1.m1.1.1.3.1.cmml" xref="S7.Thmtheorem6.p1.1.1.m1.1.1.3"></divide><apply id="S7.Thmtheorem6.p1.1.1.m1.1.1.3.2.cmml" xref="S7.Thmtheorem6.p1.1.1.m1.1.1.3.2"><csymbol cd="ambiguous" id="S7.Thmtheorem6.p1.1.1.m1.1.1.3.2.1.cmml" xref="S7.Thmtheorem6.p1.1.1.m1.1.1.3.2">superscript</csymbol><ci id="S7.Thmtheorem6.p1.1.1.m1.1.1.3.2.2.cmml" xref="S7.Thmtheorem6.p1.1.1.m1.1.1.3.2.2">𝜀</ci><cn id="S7.Thmtheorem6.p1.1.1.m1.1.1.3.2.3.cmml" type="integer" xref="S7.Thmtheorem6.p1.1.1.m1.1.1.3.2.3">2</cn></apply><apply id="S7.Thmtheorem6.p1.1.1.m1.1.1.3.3.cmml" xref="S7.Thmtheorem6.p1.1.1.m1.1.1.3.3"><ci id="S7.Thmtheorem6.p1.1.1.m1.1.1.3.3.1.cmml" xref="S7.Thmtheorem6.p1.1.1.m1.1.1.3.3.1">⋅</ci><cn id="S7.Thmtheorem6.p1.1.1.m1.1.1.3.3.2.cmml" type="integer" xref="S7.Thmtheorem6.p1.1.1.m1.1.1.3.3.2">128</cn><apply id="S7.Thmtheorem6.p1.1.1.m1.1.1.3.3.3.cmml" xref="S7.Thmtheorem6.p1.1.1.m1.1.1.3.3.3"><log id="S7.Thmtheorem6.p1.1.1.m1.1.1.3.3.3.1.cmml" xref="S7.Thmtheorem6.p1.1.1.m1.1.1.3.3.3.1"></log><ci id="S7.Thmtheorem6.p1.1.1.m1.1.1.3.3.3.2.cmml" xref="S7.Thmtheorem6.p1.1.1.m1.1.1.3.3.3.2">𝑛</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.Thmtheorem6.p1.1.1.m1.1c">\eta\leq\frac{\varepsilon^{2}}{128\cdot\log n}</annotation><annotation encoding="application/x-llamapun" id="S7.Thmtheorem6.p1.1.1.m1.1d">italic_η ≤ divide start_ARG italic_ε start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG start_ARG 128 ⋅ roman_log italic_n end_ARG</annotation></semantics></math>. For our orientation, let <math alttext="\ell^{\prime}" class="ltx_Math" display="inline" id="S7.Thmtheorem6.p1.2.2.m2.1"><semantics id="S7.Thmtheorem6.p1.2.2.m2.1a"><msup id="S7.Thmtheorem6.p1.2.2.m2.1.1" xref="S7.Thmtheorem6.p1.2.2.m2.1.1.cmml"><mi id="S7.Thmtheorem6.p1.2.2.m2.1.1.2" mathvariant="normal" xref="S7.Thmtheorem6.p1.2.2.m2.1.1.2.cmml">ℓ</mi><mo id="S7.Thmtheorem6.p1.2.2.m2.1.1.3" xref="S7.Thmtheorem6.p1.2.2.m2.1.1.3.cmml">′</mo></msup><annotation-xml encoding="MathML-Content" id="S7.Thmtheorem6.p1.2.2.m2.1b"><apply id="S7.Thmtheorem6.p1.2.2.m2.1.1.cmml" xref="S7.Thmtheorem6.p1.2.2.m2.1.1"><csymbol cd="ambiguous" id="S7.Thmtheorem6.p1.2.2.m2.1.1.1.cmml" xref="S7.Thmtheorem6.p1.2.2.m2.1.1">superscript</csymbol><ci id="S7.Thmtheorem6.p1.2.2.m2.1.1.2.cmml" xref="S7.Thmtheorem6.p1.2.2.m2.1.1.2">ℓ</ci><ci id="S7.Thmtheorem6.p1.2.2.m2.1.1.3.cmml" xref="S7.Thmtheorem6.p1.2.2.m2.1.1.3">′</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.Thmtheorem6.p1.2.2.m2.1c">\ell^{\prime}</annotation><annotation encoding="application/x-llamapun" id="S7.Thmtheorem6.p1.2.2.m2.1d">roman_ℓ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math> be the highest level such that <math alttext="L_{\ell^{\prime}}" class="ltx_Math" display="inline" id="S7.Thmtheorem6.p1.3.3.m3.1"><semantics id="S7.Thmtheorem6.p1.3.3.m3.1a"><msub id="S7.Thmtheorem6.p1.3.3.m3.1.1" xref="S7.Thmtheorem6.p1.3.3.m3.1.1.cmml"><mi id="S7.Thmtheorem6.p1.3.3.m3.1.1.2" xref="S7.Thmtheorem6.p1.3.3.m3.1.1.2.cmml">L</mi><msup id="S7.Thmtheorem6.p1.3.3.m3.1.1.3" xref="S7.Thmtheorem6.p1.3.3.m3.1.1.3.cmml"><mi id="S7.Thmtheorem6.p1.3.3.m3.1.1.3.2" mathvariant="normal" xref="S7.Thmtheorem6.p1.3.3.m3.1.1.3.2.cmml">ℓ</mi><mo id="S7.Thmtheorem6.p1.3.3.m3.1.1.3.3" xref="S7.Thmtheorem6.p1.3.3.m3.1.1.3.3.cmml">′</mo></msup></msub><annotation-xml encoding="MathML-Content" id="S7.Thmtheorem6.p1.3.3.m3.1b"><apply id="S7.Thmtheorem6.p1.3.3.m3.1.1.cmml" xref="S7.Thmtheorem6.p1.3.3.m3.1.1"><csymbol cd="ambiguous" id="S7.Thmtheorem6.p1.3.3.m3.1.1.1.cmml" xref="S7.Thmtheorem6.p1.3.3.m3.1.1">subscript</csymbol><ci id="S7.Thmtheorem6.p1.3.3.m3.1.1.2.cmml" xref="S7.Thmtheorem6.p1.3.3.m3.1.1.2">𝐿</ci><apply id="S7.Thmtheorem6.p1.3.3.m3.1.1.3.cmml" xref="S7.Thmtheorem6.p1.3.3.m3.1.1.3"><csymbol cd="ambiguous" id="S7.Thmtheorem6.p1.3.3.m3.1.1.3.1.cmml" xref="S7.Thmtheorem6.p1.3.3.m3.1.1.3">superscript</csymbol><ci id="S7.Thmtheorem6.p1.3.3.m3.1.1.3.2.cmml" xref="S7.Thmtheorem6.p1.3.3.m3.1.1.3.2">ℓ</ci><ci id="S7.Thmtheorem6.p1.3.3.m3.1.1.3.3.cmml" xref="S7.Thmtheorem6.p1.3.3.m3.1.1.3.3">′</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.Thmtheorem6.p1.3.3.m3.1c">L_{\ell^{\prime}}</annotation><annotation encoding="application/x-llamapun" id="S7.Thmtheorem6.p1.3.3.m3.1d">italic_L start_POSTSUBSCRIPT roman_ℓ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT</annotation></semantics></math> is not empty then <math alttext="\ell^{\prime}\leq\varepsilon^{-2}\log^{2}n" class="ltx_Math" display="inline" id="S7.Thmtheorem6.p1.4.4.m4.1"><semantics id="S7.Thmtheorem6.p1.4.4.m4.1a"><mrow id="S7.Thmtheorem6.p1.4.4.m4.1.1" xref="S7.Thmtheorem6.p1.4.4.m4.1.1.cmml"><msup id="S7.Thmtheorem6.p1.4.4.m4.1.1.2" xref="S7.Thmtheorem6.p1.4.4.m4.1.1.2.cmml"><mi id="S7.Thmtheorem6.p1.4.4.m4.1.1.2.2" mathvariant="normal" xref="S7.Thmtheorem6.p1.4.4.m4.1.1.2.2.cmml">ℓ</mi><mo id="S7.Thmtheorem6.p1.4.4.m4.1.1.2.3" xref="S7.Thmtheorem6.p1.4.4.m4.1.1.2.3.cmml">′</mo></msup><mo id="S7.Thmtheorem6.p1.4.4.m4.1.1.1" xref="S7.Thmtheorem6.p1.4.4.m4.1.1.1.cmml">≤</mo><mrow id="S7.Thmtheorem6.p1.4.4.m4.1.1.3" xref="S7.Thmtheorem6.p1.4.4.m4.1.1.3.cmml"><msup id="S7.Thmtheorem6.p1.4.4.m4.1.1.3.2" xref="S7.Thmtheorem6.p1.4.4.m4.1.1.3.2.cmml"><mi id="S7.Thmtheorem6.p1.4.4.m4.1.1.3.2.2" xref="S7.Thmtheorem6.p1.4.4.m4.1.1.3.2.2.cmml">ε</mi><mrow id="S7.Thmtheorem6.p1.4.4.m4.1.1.3.2.3" xref="S7.Thmtheorem6.p1.4.4.m4.1.1.3.2.3.cmml"><mo id="S7.Thmtheorem6.p1.4.4.m4.1.1.3.2.3a" xref="S7.Thmtheorem6.p1.4.4.m4.1.1.3.2.3.cmml">−</mo><mn id="S7.Thmtheorem6.p1.4.4.m4.1.1.3.2.3.2" xref="S7.Thmtheorem6.p1.4.4.m4.1.1.3.2.3.2.cmml">2</mn></mrow></msup><mo id="S7.Thmtheorem6.p1.4.4.m4.1.1.3.1" lspace="0.167em" xref="S7.Thmtheorem6.p1.4.4.m4.1.1.3.1.cmml">⁢</mo><mrow id="S7.Thmtheorem6.p1.4.4.m4.1.1.3.3" xref="S7.Thmtheorem6.p1.4.4.m4.1.1.3.3.cmml"><msup id="S7.Thmtheorem6.p1.4.4.m4.1.1.3.3.1" xref="S7.Thmtheorem6.p1.4.4.m4.1.1.3.3.1.cmml"><mi id="S7.Thmtheorem6.p1.4.4.m4.1.1.3.3.1.2" xref="S7.Thmtheorem6.p1.4.4.m4.1.1.3.3.1.2.cmml">log</mi><mn id="S7.Thmtheorem6.p1.4.4.m4.1.1.3.3.1.3" xref="S7.Thmtheorem6.p1.4.4.m4.1.1.3.3.1.3.cmml">2</mn></msup><mo id="S7.Thmtheorem6.p1.4.4.m4.1.1.3.3a" lspace="0.167em" xref="S7.Thmtheorem6.p1.4.4.m4.1.1.3.3.cmml">⁡</mo><mi id="S7.Thmtheorem6.p1.4.4.m4.1.1.3.3.2" xref="S7.Thmtheorem6.p1.4.4.m4.1.1.3.3.2.cmml">n</mi></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S7.Thmtheorem6.p1.4.4.m4.1b"><apply id="S7.Thmtheorem6.p1.4.4.m4.1.1.cmml" xref="S7.Thmtheorem6.p1.4.4.m4.1.1"><leq id="S7.Thmtheorem6.p1.4.4.m4.1.1.1.cmml" xref="S7.Thmtheorem6.p1.4.4.m4.1.1.1"></leq><apply id="S7.Thmtheorem6.p1.4.4.m4.1.1.2.cmml" xref="S7.Thmtheorem6.p1.4.4.m4.1.1.2"><csymbol cd="ambiguous" id="S7.Thmtheorem6.p1.4.4.m4.1.1.2.1.cmml" xref="S7.Thmtheorem6.p1.4.4.m4.1.1.2">superscript</csymbol><ci id="S7.Thmtheorem6.p1.4.4.m4.1.1.2.2.cmml" xref="S7.Thmtheorem6.p1.4.4.m4.1.1.2.2">ℓ</ci><ci id="S7.Thmtheorem6.p1.4.4.m4.1.1.2.3.cmml" xref="S7.Thmtheorem6.p1.4.4.m4.1.1.2.3">′</ci></apply><apply id="S7.Thmtheorem6.p1.4.4.m4.1.1.3.cmml" xref="S7.Thmtheorem6.p1.4.4.m4.1.1.3"><times id="S7.Thmtheorem6.p1.4.4.m4.1.1.3.1.cmml" xref="S7.Thmtheorem6.p1.4.4.m4.1.1.3.1"></times><apply id="S7.Thmtheorem6.p1.4.4.m4.1.1.3.2.cmml" xref="S7.Thmtheorem6.p1.4.4.m4.1.1.3.2"><csymbol cd="ambiguous" id="S7.Thmtheorem6.p1.4.4.m4.1.1.3.2.1.cmml" xref="S7.Thmtheorem6.p1.4.4.m4.1.1.3.2">superscript</csymbol><ci id="S7.Thmtheorem6.p1.4.4.m4.1.1.3.2.2.cmml" xref="S7.Thmtheorem6.p1.4.4.m4.1.1.3.2.2">𝜀</ci><apply id="S7.Thmtheorem6.p1.4.4.m4.1.1.3.2.3.cmml" xref="S7.Thmtheorem6.p1.4.4.m4.1.1.3.2.3"><minus id="S7.Thmtheorem6.p1.4.4.m4.1.1.3.2.3.1.cmml" xref="S7.Thmtheorem6.p1.4.4.m4.1.1.3.2.3"></minus><cn id="S7.Thmtheorem6.p1.4.4.m4.1.1.3.2.3.2.cmml" type="integer" xref="S7.Thmtheorem6.p1.4.4.m4.1.1.3.2.3.2">2</cn></apply></apply><apply id="S7.Thmtheorem6.p1.4.4.m4.1.1.3.3.cmml" xref="S7.Thmtheorem6.p1.4.4.m4.1.1.3.3"><apply id="S7.Thmtheorem6.p1.4.4.m4.1.1.3.3.1.cmml" xref="S7.Thmtheorem6.p1.4.4.m4.1.1.3.3.1"><csymbol cd="ambiguous" id="S7.Thmtheorem6.p1.4.4.m4.1.1.3.3.1.1.cmml" xref="S7.Thmtheorem6.p1.4.4.m4.1.1.3.3.1">superscript</csymbol><log id="S7.Thmtheorem6.p1.4.4.m4.1.1.3.3.1.2.cmml" xref="S7.Thmtheorem6.p1.4.4.m4.1.1.3.3.1.2"></log><cn id="S7.Thmtheorem6.p1.4.4.m4.1.1.3.3.1.3.cmml" type="integer" xref="S7.Thmtheorem6.p1.4.4.m4.1.1.3.3.1.3">2</cn></apply><ci id="S7.Thmtheorem6.p1.4.4.m4.1.1.3.3.2.cmml" xref="S7.Thmtheorem6.p1.4.4.m4.1.1.3.3.2">𝑛</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.Thmtheorem6.p1.4.4.m4.1c">\ell^{\prime}\leq\varepsilon^{-2}\log^{2}n</annotation><annotation encoding="application/x-llamapun" id="S7.Thmtheorem6.p1.4.4.m4.1d">roman_ℓ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ≤ italic_ε start_POSTSUPERSCRIPT - 2 end_POSTSUPERSCRIPT roman_log start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_n</annotation></semantics></math>.</span></p> </div> </div> <div class="ltx_theorem ltx_theorem_proof" id="S7.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="S7.Thmtheorem7.1.1.1">Proof 7.7</span></span><span class="ltx_text ltx_font_bold" id="S7.Thmtheorem7.2.2">.</span> </h6> <div class="ltx_para" id="S7.Thmtheorem7.p1"> <p class="ltx_p" id="S7.Thmtheorem7.p1.4"><span class="ltx_text ltx_font_italic" id="S7.Thmtheorem7.p1.4.4">The maximal out-degree of a vertex is <math alttext="n" class="ltx_Math" display="inline" id="S7.Thmtheorem7.p1.1.1.m1.1"><semantics id="S7.Thmtheorem7.p1.1.1.m1.1a"><mi id="S7.Thmtheorem7.p1.1.1.m1.1.1" xref="S7.Thmtheorem7.p1.1.1.m1.1.1.cmml">n</mi><annotation-xml encoding="MathML-Content" id="S7.Thmtheorem7.p1.1.1.m1.1b"><ci id="S7.Thmtheorem7.p1.1.1.m1.1.1.cmml" xref="S7.Thmtheorem7.p1.1.1.m1.1.1">𝑛</ci></annotation-xml><annotation encoding="application/x-tex" id="S7.Thmtheorem7.p1.1.1.m1.1c">n</annotation><annotation encoding="application/x-llamapun" id="S7.Thmtheorem7.p1.1.1.m1.1d">italic_n</annotation></semantics></math>. Thus, <math alttext="\ell^{\prime}\leq\frac{\log n}{\log(1+\frac{\eta}{2})}" class="ltx_Math" display="inline" id="S7.Thmtheorem7.p1.2.2.m2.2"><semantics id="S7.Thmtheorem7.p1.2.2.m2.2a"><mrow id="S7.Thmtheorem7.p1.2.2.m2.2.3" xref="S7.Thmtheorem7.p1.2.2.m2.2.3.cmml"><msup id="S7.Thmtheorem7.p1.2.2.m2.2.3.2" xref="S7.Thmtheorem7.p1.2.2.m2.2.3.2.cmml"><mi id="S7.Thmtheorem7.p1.2.2.m2.2.3.2.2" mathvariant="normal" xref="S7.Thmtheorem7.p1.2.2.m2.2.3.2.2.cmml">ℓ</mi><mo id="S7.Thmtheorem7.p1.2.2.m2.2.3.2.3" xref="S7.Thmtheorem7.p1.2.2.m2.2.3.2.3.cmml">′</mo></msup><mo id="S7.Thmtheorem7.p1.2.2.m2.2.3.1" xref="S7.Thmtheorem7.p1.2.2.m2.2.3.1.cmml">≤</mo><mfrac id="S7.Thmtheorem7.p1.2.2.m2.2.2" xref="S7.Thmtheorem7.p1.2.2.m2.2.2.cmml"><mrow id="S7.Thmtheorem7.p1.2.2.m2.2.2.4" xref="S7.Thmtheorem7.p1.2.2.m2.2.2.4.cmml"><mi id="S7.Thmtheorem7.p1.2.2.m2.2.2.4.1" xref="S7.Thmtheorem7.p1.2.2.m2.2.2.4.1.cmml">log</mi><mo id="S7.Thmtheorem7.p1.2.2.m2.2.2.4a" lspace="0.167em" xref="S7.Thmtheorem7.p1.2.2.m2.2.2.4.cmml">⁡</mo><mi id="S7.Thmtheorem7.p1.2.2.m2.2.2.4.2" xref="S7.Thmtheorem7.p1.2.2.m2.2.2.4.2.cmml">n</mi></mrow><mrow id="S7.Thmtheorem7.p1.2.2.m2.2.2.2.2" xref="S7.Thmtheorem7.p1.2.2.m2.2.2.2.3.cmml"><mi id="S7.Thmtheorem7.p1.2.2.m2.1.1.1.1" xref="S7.Thmtheorem7.p1.2.2.m2.1.1.1.1.cmml">log</mi><mo id="S7.Thmtheorem7.p1.2.2.m2.2.2.2.2a" xref="S7.Thmtheorem7.p1.2.2.m2.2.2.2.3.cmml">⁡</mo><mrow id="S7.Thmtheorem7.p1.2.2.m2.2.2.2.2.1" xref="S7.Thmtheorem7.p1.2.2.m2.2.2.2.3.cmml"><mo id="S7.Thmtheorem7.p1.2.2.m2.2.2.2.2.1.2" stretchy="false" xref="S7.Thmtheorem7.p1.2.2.m2.2.2.2.3.cmml">(</mo><mrow id="S7.Thmtheorem7.p1.2.2.m2.2.2.2.2.1.1" xref="S7.Thmtheorem7.p1.2.2.m2.2.2.2.2.1.1.cmml"><mn id="S7.Thmtheorem7.p1.2.2.m2.2.2.2.2.1.1.2" xref="S7.Thmtheorem7.p1.2.2.m2.2.2.2.2.1.1.2.cmml">1</mn><mo id="S7.Thmtheorem7.p1.2.2.m2.2.2.2.2.1.1.1" xref="S7.Thmtheorem7.p1.2.2.m2.2.2.2.2.1.1.1.cmml">+</mo><mfrac id="S7.Thmtheorem7.p1.2.2.m2.2.2.2.2.1.1.3" xref="S7.Thmtheorem7.p1.2.2.m2.2.2.2.2.1.1.3.cmml"><mi id="S7.Thmtheorem7.p1.2.2.m2.2.2.2.2.1.1.3.2" xref="S7.Thmtheorem7.p1.2.2.m2.2.2.2.2.1.1.3.2.cmml">η</mi><mn id="S7.Thmtheorem7.p1.2.2.m2.2.2.2.2.1.1.3.3" xref="S7.Thmtheorem7.p1.2.2.m2.2.2.2.2.1.1.3.3.cmml">2</mn></mfrac></mrow><mo id="S7.Thmtheorem7.p1.2.2.m2.2.2.2.2.1.3" stretchy="false" xref="S7.Thmtheorem7.p1.2.2.m2.2.2.2.3.cmml">)</mo></mrow></mrow></mfrac></mrow><annotation-xml encoding="MathML-Content" id="S7.Thmtheorem7.p1.2.2.m2.2b"><apply id="S7.Thmtheorem7.p1.2.2.m2.2.3.cmml" xref="S7.Thmtheorem7.p1.2.2.m2.2.3"><leq id="S7.Thmtheorem7.p1.2.2.m2.2.3.1.cmml" xref="S7.Thmtheorem7.p1.2.2.m2.2.3.1"></leq><apply id="S7.Thmtheorem7.p1.2.2.m2.2.3.2.cmml" xref="S7.Thmtheorem7.p1.2.2.m2.2.3.2"><csymbol cd="ambiguous" id="S7.Thmtheorem7.p1.2.2.m2.2.3.2.1.cmml" xref="S7.Thmtheorem7.p1.2.2.m2.2.3.2">superscript</csymbol><ci id="S7.Thmtheorem7.p1.2.2.m2.2.3.2.2.cmml" xref="S7.Thmtheorem7.p1.2.2.m2.2.3.2.2">ℓ</ci><ci id="S7.Thmtheorem7.p1.2.2.m2.2.3.2.3.cmml" xref="S7.Thmtheorem7.p1.2.2.m2.2.3.2.3">′</ci></apply><apply id="S7.Thmtheorem7.p1.2.2.m2.2.2.cmml" xref="S7.Thmtheorem7.p1.2.2.m2.2.2"><divide id="S7.Thmtheorem7.p1.2.2.m2.2.2.3.cmml" xref="S7.Thmtheorem7.p1.2.2.m2.2.2"></divide><apply id="S7.Thmtheorem7.p1.2.2.m2.2.2.4.cmml" xref="S7.Thmtheorem7.p1.2.2.m2.2.2.4"><log id="S7.Thmtheorem7.p1.2.2.m2.2.2.4.1.cmml" xref="S7.Thmtheorem7.p1.2.2.m2.2.2.4.1"></log><ci id="S7.Thmtheorem7.p1.2.2.m2.2.2.4.2.cmml" xref="S7.Thmtheorem7.p1.2.2.m2.2.2.4.2">𝑛</ci></apply><apply id="S7.Thmtheorem7.p1.2.2.m2.2.2.2.3.cmml" xref="S7.Thmtheorem7.p1.2.2.m2.2.2.2.2"><log id="S7.Thmtheorem7.p1.2.2.m2.1.1.1.1.cmml" xref="S7.Thmtheorem7.p1.2.2.m2.1.1.1.1"></log><apply id="S7.Thmtheorem7.p1.2.2.m2.2.2.2.2.1.1.cmml" xref="S7.Thmtheorem7.p1.2.2.m2.2.2.2.2.1.1"><plus id="S7.Thmtheorem7.p1.2.2.m2.2.2.2.2.1.1.1.cmml" xref="S7.Thmtheorem7.p1.2.2.m2.2.2.2.2.1.1.1"></plus><cn id="S7.Thmtheorem7.p1.2.2.m2.2.2.2.2.1.1.2.cmml" type="integer" xref="S7.Thmtheorem7.p1.2.2.m2.2.2.2.2.1.1.2">1</cn><apply id="S7.Thmtheorem7.p1.2.2.m2.2.2.2.2.1.1.3.cmml" xref="S7.Thmtheorem7.p1.2.2.m2.2.2.2.2.1.1.3"><divide id="S7.Thmtheorem7.p1.2.2.m2.2.2.2.2.1.1.3.1.cmml" xref="S7.Thmtheorem7.p1.2.2.m2.2.2.2.2.1.1.3"></divide><ci id="S7.Thmtheorem7.p1.2.2.m2.2.2.2.2.1.1.3.2.cmml" xref="S7.Thmtheorem7.p1.2.2.m2.2.2.2.2.1.1.3.2">𝜂</ci><cn id="S7.Thmtheorem7.p1.2.2.m2.2.2.2.2.1.1.3.3.cmml" type="integer" xref="S7.Thmtheorem7.p1.2.2.m2.2.2.2.2.1.1.3.3">2</cn></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.Thmtheorem7.p1.2.2.m2.2c">\ell^{\prime}\leq\frac{\log n}{\log(1+\frac{\eta}{2})}</annotation><annotation encoding="application/x-llamapun" id="S7.Thmtheorem7.p1.2.2.m2.2d">roman_ℓ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ≤ divide start_ARG roman_log italic_n end_ARG start_ARG roman_log ( 1 + divide start_ARG italic_η end_ARG start_ARG 2 end_ARG ) end_ARG</annotation></semantics></math>. We now apply <math alttext="\log(1+x)\geq x/2" class="ltx_Math" display="inline" id="S7.Thmtheorem7.p1.3.3.m3.2"><semantics id="S7.Thmtheorem7.p1.3.3.m3.2a"><mrow id="S7.Thmtheorem7.p1.3.3.m3.2.2" xref="S7.Thmtheorem7.p1.3.3.m3.2.2.cmml"><mrow id="S7.Thmtheorem7.p1.3.3.m3.2.2.1.1" xref="S7.Thmtheorem7.p1.3.3.m3.2.2.1.2.cmml"><mi id="S7.Thmtheorem7.p1.3.3.m3.1.1" xref="S7.Thmtheorem7.p1.3.3.m3.1.1.cmml">log</mi><mo id="S7.Thmtheorem7.p1.3.3.m3.2.2.1.1a" xref="S7.Thmtheorem7.p1.3.3.m3.2.2.1.2.cmml">⁡</mo><mrow id="S7.Thmtheorem7.p1.3.3.m3.2.2.1.1.1" xref="S7.Thmtheorem7.p1.3.3.m3.2.2.1.2.cmml"><mo id="S7.Thmtheorem7.p1.3.3.m3.2.2.1.1.1.2" stretchy="false" xref="S7.Thmtheorem7.p1.3.3.m3.2.2.1.2.cmml">(</mo><mrow id="S7.Thmtheorem7.p1.3.3.m3.2.2.1.1.1.1" xref="S7.Thmtheorem7.p1.3.3.m3.2.2.1.1.1.1.cmml"><mn id="S7.Thmtheorem7.p1.3.3.m3.2.2.1.1.1.1.2" xref="S7.Thmtheorem7.p1.3.3.m3.2.2.1.1.1.1.2.cmml">1</mn><mo id="S7.Thmtheorem7.p1.3.3.m3.2.2.1.1.1.1.1" xref="S7.Thmtheorem7.p1.3.3.m3.2.2.1.1.1.1.1.cmml">+</mo><mi id="S7.Thmtheorem7.p1.3.3.m3.2.2.1.1.1.1.3" xref="S7.Thmtheorem7.p1.3.3.m3.2.2.1.1.1.1.3.cmml">x</mi></mrow><mo id="S7.Thmtheorem7.p1.3.3.m3.2.2.1.1.1.3" stretchy="false" xref="S7.Thmtheorem7.p1.3.3.m3.2.2.1.2.cmml">)</mo></mrow></mrow><mo id="S7.Thmtheorem7.p1.3.3.m3.2.2.2" xref="S7.Thmtheorem7.p1.3.3.m3.2.2.2.cmml">≥</mo><mrow id="S7.Thmtheorem7.p1.3.3.m3.2.2.3" xref="S7.Thmtheorem7.p1.3.3.m3.2.2.3.cmml"><mi id="S7.Thmtheorem7.p1.3.3.m3.2.2.3.2" xref="S7.Thmtheorem7.p1.3.3.m3.2.2.3.2.cmml">x</mi><mo id="S7.Thmtheorem7.p1.3.3.m3.2.2.3.1" xref="S7.Thmtheorem7.p1.3.3.m3.2.2.3.1.cmml">/</mo><mn id="S7.Thmtheorem7.p1.3.3.m3.2.2.3.3" xref="S7.Thmtheorem7.p1.3.3.m3.2.2.3.3.cmml">2</mn></mrow></mrow><annotation-xml encoding="MathML-Content" id="S7.Thmtheorem7.p1.3.3.m3.2b"><apply id="S7.Thmtheorem7.p1.3.3.m3.2.2.cmml" xref="S7.Thmtheorem7.p1.3.3.m3.2.2"><geq id="S7.Thmtheorem7.p1.3.3.m3.2.2.2.cmml" xref="S7.Thmtheorem7.p1.3.3.m3.2.2.2"></geq><apply id="S7.Thmtheorem7.p1.3.3.m3.2.2.1.2.cmml" xref="S7.Thmtheorem7.p1.3.3.m3.2.2.1.1"><log id="S7.Thmtheorem7.p1.3.3.m3.1.1.cmml" xref="S7.Thmtheorem7.p1.3.3.m3.1.1"></log><apply id="S7.Thmtheorem7.p1.3.3.m3.2.2.1.1.1.1.cmml" xref="S7.Thmtheorem7.p1.3.3.m3.2.2.1.1.1.1"><plus id="S7.Thmtheorem7.p1.3.3.m3.2.2.1.1.1.1.1.cmml" xref="S7.Thmtheorem7.p1.3.3.m3.2.2.1.1.1.1.1"></plus><cn id="S7.Thmtheorem7.p1.3.3.m3.2.2.1.1.1.1.2.cmml" type="integer" xref="S7.Thmtheorem7.p1.3.3.m3.2.2.1.1.1.1.2">1</cn><ci id="S7.Thmtheorem7.p1.3.3.m3.2.2.1.1.1.1.3.cmml" xref="S7.Thmtheorem7.p1.3.3.m3.2.2.1.1.1.1.3">𝑥</ci></apply></apply><apply id="S7.Thmtheorem7.p1.3.3.m3.2.2.3.cmml" xref="S7.Thmtheorem7.p1.3.3.m3.2.2.3"><divide id="S7.Thmtheorem7.p1.3.3.m3.2.2.3.1.cmml" xref="S7.Thmtheorem7.p1.3.3.m3.2.2.3.1"></divide><ci id="S7.Thmtheorem7.p1.3.3.m3.2.2.3.2.cmml" xref="S7.Thmtheorem7.p1.3.3.m3.2.2.3.2">𝑥</ci><cn id="S7.Thmtheorem7.p1.3.3.m3.2.2.3.3.cmml" type="integer" xref="S7.Thmtheorem7.p1.3.3.m3.2.2.3.3">2</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.Thmtheorem7.p1.3.3.m3.2c">\log(1+x)\geq x/2</annotation><annotation encoding="application/x-llamapun" id="S7.Thmtheorem7.p1.3.3.m3.2d">roman_log ( 1 + italic_x ) ≥ italic_x / 2</annotation></semantics></math> and note that: <math alttext="\ell^{\prime}\leq\frac{\log n}{\log(1+\frac{\eta}{2})}\leq\frac{\log n}{\eta/4% }=\frac{\log n}{32\varepsilon^{2}/\log n}\leq\frac{\log^{2}n}{\varepsilon^{2}}" class="ltx_Math" display="inline" id="S7.Thmtheorem7.p1.4.4.m4.2"><semantics id="S7.Thmtheorem7.p1.4.4.m4.2a"><mrow id="S7.Thmtheorem7.p1.4.4.m4.2.3" xref="S7.Thmtheorem7.p1.4.4.m4.2.3.cmml"><msup id="S7.Thmtheorem7.p1.4.4.m4.2.3.2" xref="S7.Thmtheorem7.p1.4.4.m4.2.3.2.cmml"><mi id="S7.Thmtheorem7.p1.4.4.m4.2.3.2.2" mathvariant="normal" xref="S7.Thmtheorem7.p1.4.4.m4.2.3.2.2.cmml">ℓ</mi><mo id="S7.Thmtheorem7.p1.4.4.m4.2.3.2.3" xref="S7.Thmtheorem7.p1.4.4.m4.2.3.2.3.cmml">′</mo></msup><mo id="S7.Thmtheorem7.p1.4.4.m4.2.3.3" xref="S7.Thmtheorem7.p1.4.4.m4.2.3.3.cmml">≤</mo><mfrac id="S7.Thmtheorem7.p1.4.4.m4.2.2" xref="S7.Thmtheorem7.p1.4.4.m4.2.2.cmml"><mrow id="S7.Thmtheorem7.p1.4.4.m4.2.2.4" xref="S7.Thmtheorem7.p1.4.4.m4.2.2.4.cmml"><mi id="S7.Thmtheorem7.p1.4.4.m4.2.2.4.1" xref="S7.Thmtheorem7.p1.4.4.m4.2.2.4.1.cmml">log</mi><mo id="S7.Thmtheorem7.p1.4.4.m4.2.2.4a" lspace="0.167em" xref="S7.Thmtheorem7.p1.4.4.m4.2.2.4.cmml">⁡</mo><mi id="S7.Thmtheorem7.p1.4.4.m4.2.2.4.2" xref="S7.Thmtheorem7.p1.4.4.m4.2.2.4.2.cmml">n</mi></mrow><mrow id="S7.Thmtheorem7.p1.4.4.m4.2.2.2.2" xref="S7.Thmtheorem7.p1.4.4.m4.2.2.2.3.cmml"><mi id="S7.Thmtheorem7.p1.4.4.m4.1.1.1.1" xref="S7.Thmtheorem7.p1.4.4.m4.1.1.1.1.cmml">log</mi><mo id="S7.Thmtheorem7.p1.4.4.m4.2.2.2.2a" xref="S7.Thmtheorem7.p1.4.4.m4.2.2.2.3.cmml">⁡</mo><mrow id="S7.Thmtheorem7.p1.4.4.m4.2.2.2.2.1" xref="S7.Thmtheorem7.p1.4.4.m4.2.2.2.3.cmml"><mo id="S7.Thmtheorem7.p1.4.4.m4.2.2.2.2.1.2" stretchy="false" xref="S7.Thmtheorem7.p1.4.4.m4.2.2.2.3.cmml">(</mo><mrow id="S7.Thmtheorem7.p1.4.4.m4.2.2.2.2.1.1" xref="S7.Thmtheorem7.p1.4.4.m4.2.2.2.2.1.1.cmml"><mn id="S7.Thmtheorem7.p1.4.4.m4.2.2.2.2.1.1.2" xref="S7.Thmtheorem7.p1.4.4.m4.2.2.2.2.1.1.2.cmml">1</mn><mo id="S7.Thmtheorem7.p1.4.4.m4.2.2.2.2.1.1.1" xref="S7.Thmtheorem7.p1.4.4.m4.2.2.2.2.1.1.1.cmml">+</mo><mfrac id="S7.Thmtheorem7.p1.4.4.m4.2.2.2.2.1.1.3" xref="S7.Thmtheorem7.p1.4.4.m4.2.2.2.2.1.1.3.cmml"><mi id="S7.Thmtheorem7.p1.4.4.m4.2.2.2.2.1.1.3.2" xref="S7.Thmtheorem7.p1.4.4.m4.2.2.2.2.1.1.3.2.cmml">η</mi><mn id="S7.Thmtheorem7.p1.4.4.m4.2.2.2.2.1.1.3.3" xref="S7.Thmtheorem7.p1.4.4.m4.2.2.2.2.1.1.3.3.cmml">2</mn></mfrac></mrow><mo id="S7.Thmtheorem7.p1.4.4.m4.2.2.2.2.1.3" stretchy="false" xref="S7.Thmtheorem7.p1.4.4.m4.2.2.2.3.cmml">)</mo></mrow></mrow></mfrac><mo id="S7.Thmtheorem7.p1.4.4.m4.2.3.4" xref="S7.Thmtheorem7.p1.4.4.m4.2.3.4.cmml">≤</mo><mfrac 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id="S7.Thmtheorem7.p1.4.4.m4.2.3.7.3.2.3.2" xref="S7.Thmtheorem7.p1.4.4.m4.2.3.7.3.2.3.2.cmml">ε</mi><mn id="S7.Thmtheorem7.p1.4.4.m4.2.3.7.3.2.3.3" xref="S7.Thmtheorem7.p1.4.4.m4.2.3.7.3.2.3.3.cmml">2</mn></msup></mrow><mo id="S7.Thmtheorem7.p1.4.4.m4.2.3.7.3.1" xref="S7.Thmtheorem7.p1.4.4.m4.2.3.7.3.1.cmml">/</mo><mrow id="S7.Thmtheorem7.p1.4.4.m4.2.3.7.3.3" xref="S7.Thmtheorem7.p1.4.4.m4.2.3.7.3.3.cmml"><mi id="S7.Thmtheorem7.p1.4.4.m4.2.3.7.3.3.1" xref="S7.Thmtheorem7.p1.4.4.m4.2.3.7.3.3.1.cmml">log</mi><mo id="S7.Thmtheorem7.p1.4.4.m4.2.3.7.3.3a" lspace="0.167em" xref="S7.Thmtheorem7.p1.4.4.m4.2.3.7.3.3.cmml">⁡</mo><mi id="S7.Thmtheorem7.p1.4.4.m4.2.3.7.3.3.2" xref="S7.Thmtheorem7.p1.4.4.m4.2.3.7.3.3.2.cmml">n</mi></mrow></mrow></mfrac><mo id="S7.Thmtheorem7.p1.4.4.m4.2.3.8" xref="S7.Thmtheorem7.p1.4.4.m4.2.3.8.cmml">≤</mo><mfrac id="S7.Thmtheorem7.p1.4.4.m4.2.3.9" xref="S7.Thmtheorem7.p1.4.4.m4.2.3.9.cmml"><mrow id="S7.Thmtheorem7.p1.4.4.m4.2.3.9.2" xref="S7.Thmtheorem7.p1.4.4.m4.2.3.9.2.cmml"><msup id="S7.Thmtheorem7.p1.4.4.m4.2.3.9.2.1" xref="S7.Thmtheorem7.p1.4.4.m4.2.3.9.2.1.cmml"><mi id="S7.Thmtheorem7.p1.4.4.m4.2.3.9.2.1.2" xref="S7.Thmtheorem7.p1.4.4.m4.2.3.9.2.1.2.cmml">log</mi><mn id="S7.Thmtheorem7.p1.4.4.m4.2.3.9.2.1.3" xref="S7.Thmtheorem7.p1.4.4.m4.2.3.9.2.1.3.cmml">2</mn></msup><mo id="S7.Thmtheorem7.p1.4.4.m4.2.3.9.2a" lspace="0.167em" xref="S7.Thmtheorem7.p1.4.4.m4.2.3.9.2.cmml">⁡</mo><mi id="S7.Thmtheorem7.p1.4.4.m4.2.3.9.2.2" xref="S7.Thmtheorem7.p1.4.4.m4.2.3.9.2.2.cmml">n</mi></mrow><msup id="S7.Thmtheorem7.p1.4.4.m4.2.3.9.3" xref="S7.Thmtheorem7.p1.4.4.m4.2.3.9.3.cmml"><mi id="S7.Thmtheorem7.p1.4.4.m4.2.3.9.3.2" xref="S7.Thmtheorem7.p1.4.4.m4.2.3.9.3.2.cmml">ε</mi><mn id="S7.Thmtheorem7.p1.4.4.m4.2.3.9.3.3" xref="S7.Thmtheorem7.p1.4.4.m4.2.3.9.3.3.cmml">2</mn></msup></mfrac></mrow><annotation-xml encoding="MathML-Content" id="S7.Thmtheorem7.p1.4.4.m4.2b"><apply id="S7.Thmtheorem7.p1.4.4.m4.2.3.cmml" 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xref="S7.Thmtheorem7.p1.4.4.m4.2.2.2.2.1.1.3.2">𝜂</ci><cn id="S7.Thmtheorem7.p1.4.4.m4.2.2.2.2.1.1.3.3.cmml" type="integer" xref="S7.Thmtheorem7.p1.4.4.m4.2.2.2.2.1.1.3.3">2</cn></apply></apply></apply></apply></apply><apply id="S7.Thmtheorem7.p1.4.4.m4.2.3c.cmml" xref="S7.Thmtheorem7.p1.4.4.m4.2.3"><leq id="S7.Thmtheorem7.p1.4.4.m4.2.3.4.cmml" xref="S7.Thmtheorem7.p1.4.4.m4.2.3.4"></leq><share href="https://arxiv.org/html/2411.12694v2#S7.Thmtheorem7.p1.4.4.m4.2.2.cmml" id="S7.Thmtheorem7.p1.4.4.m4.2.3d.cmml" xref="S7.Thmtheorem7.p1.4.4.m4.2.3"></share><apply id="S7.Thmtheorem7.p1.4.4.m4.2.3.5.cmml" xref="S7.Thmtheorem7.p1.4.4.m4.2.3.5"><divide id="S7.Thmtheorem7.p1.4.4.m4.2.3.5.1.cmml" xref="S7.Thmtheorem7.p1.4.4.m4.2.3.5"></divide><apply id="S7.Thmtheorem7.p1.4.4.m4.2.3.5.2.cmml" xref="S7.Thmtheorem7.p1.4.4.m4.2.3.5.2"><log id="S7.Thmtheorem7.p1.4.4.m4.2.3.5.2.1.cmml" xref="S7.Thmtheorem7.p1.4.4.m4.2.3.5.2.1"></log><ci id="S7.Thmtheorem7.p1.4.4.m4.2.3.5.2.2.cmml" xref="S7.Thmtheorem7.p1.4.4.m4.2.3.5.2.2">𝑛</ci></apply><apply id="S7.Thmtheorem7.p1.4.4.m4.2.3.5.3.cmml" xref="S7.Thmtheorem7.p1.4.4.m4.2.3.5.3"><divide id="S7.Thmtheorem7.p1.4.4.m4.2.3.5.3.1.cmml" xref="S7.Thmtheorem7.p1.4.4.m4.2.3.5.3.1"></divide><ci id="S7.Thmtheorem7.p1.4.4.m4.2.3.5.3.2.cmml" xref="S7.Thmtheorem7.p1.4.4.m4.2.3.5.3.2">𝜂</ci><cn id="S7.Thmtheorem7.p1.4.4.m4.2.3.5.3.3.cmml" type="integer" xref="S7.Thmtheorem7.p1.4.4.m4.2.3.5.3.3">4</cn></apply></apply></apply><apply id="S7.Thmtheorem7.p1.4.4.m4.2.3e.cmml" xref="S7.Thmtheorem7.p1.4.4.m4.2.3"><eq id="S7.Thmtheorem7.p1.4.4.m4.2.3.6.cmml" xref="S7.Thmtheorem7.p1.4.4.m4.2.3.6"></eq><share href="https://arxiv.org/html/2411.12694v2#S7.Thmtheorem7.p1.4.4.m4.2.3.5.cmml" id="S7.Thmtheorem7.p1.4.4.m4.2.3f.cmml" xref="S7.Thmtheorem7.p1.4.4.m4.2.3"></share><apply id="S7.Thmtheorem7.p1.4.4.m4.2.3.7.cmml" xref="S7.Thmtheorem7.p1.4.4.m4.2.3.7"><divide id="S7.Thmtheorem7.p1.4.4.m4.2.3.7.1.cmml" xref="S7.Thmtheorem7.p1.4.4.m4.2.3.7"></divide><apply id="S7.Thmtheorem7.p1.4.4.m4.2.3.7.2.cmml" xref="S7.Thmtheorem7.p1.4.4.m4.2.3.7.2"><log id="S7.Thmtheorem7.p1.4.4.m4.2.3.7.2.1.cmml" xref="S7.Thmtheorem7.p1.4.4.m4.2.3.7.2.1"></log><ci id="S7.Thmtheorem7.p1.4.4.m4.2.3.7.2.2.cmml" xref="S7.Thmtheorem7.p1.4.4.m4.2.3.7.2.2">𝑛</ci></apply><apply id="S7.Thmtheorem7.p1.4.4.m4.2.3.7.3.cmml" xref="S7.Thmtheorem7.p1.4.4.m4.2.3.7.3"><divide id="S7.Thmtheorem7.p1.4.4.m4.2.3.7.3.1.cmml" xref="S7.Thmtheorem7.p1.4.4.m4.2.3.7.3.1"></divide><apply id="S7.Thmtheorem7.p1.4.4.m4.2.3.7.3.2.cmml" xref="S7.Thmtheorem7.p1.4.4.m4.2.3.7.3.2"><times id="S7.Thmtheorem7.p1.4.4.m4.2.3.7.3.2.1.cmml" xref="S7.Thmtheorem7.p1.4.4.m4.2.3.7.3.2.1"></times><cn id="S7.Thmtheorem7.p1.4.4.m4.2.3.7.3.2.2.cmml" type="integer" xref="S7.Thmtheorem7.p1.4.4.m4.2.3.7.3.2.2">32</cn><apply id="S7.Thmtheorem7.p1.4.4.m4.2.3.7.3.2.3.cmml" xref="S7.Thmtheorem7.p1.4.4.m4.2.3.7.3.2.3"><csymbol cd="ambiguous" 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xref="S7.Thmtheorem7.p1.4.4.m4.2.3"></share><apply id="S7.Thmtheorem7.p1.4.4.m4.2.3.9.cmml" xref="S7.Thmtheorem7.p1.4.4.m4.2.3.9"><divide id="S7.Thmtheorem7.p1.4.4.m4.2.3.9.1.cmml" xref="S7.Thmtheorem7.p1.4.4.m4.2.3.9"></divide><apply id="S7.Thmtheorem7.p1.4.4.m4.2.3.9.2.cmml" xref="S7.Thmtheorem7.p1.4.4.m4.2.3.9.2"><apply id="S7.Thmtheorem7.p1.4.4.m4.2.3.9.2.1.cmml" xref="S7.Thmtheorem7.p1.4.4.m4.2.3.9.2.1"><csymbol cd="ambiguous" id="S7.Thmtheorem7.p1.4.4.m4.2.3.9.2.1.1.cmml" xref="S7.Thmtheorem7.p1.4.4.m4.2.3.9.2.1">superscript</csymbol><log id="S7.Thmtheorem7.p1.4.4.m4.2.3.9.2.1.2.cmml" xref="S7.Thmtheorem7.p1.4.4.m4.2.3.9.2.1.2"></log><cn id="S7.Thmtheorem7.p1.4.4.m4.2.3.9.2.1.3.cmml" type="integer" xref="S7.Thmtheorem7.p1.4.4.m4.2.3.9.2.1.3">2</cn></apply><ci id="S7.Thmtheorem7.p1.4.4.m4.2.3.9.2.2.cmml" xref="S7.Thmtheorem7.p1.4.4.m4.2.3.9.2.2">𝑛</ci></apply><apply id="S7.Thmtheorem7.p1.4.4.m4.2.3.9.3.cmml" xref="S7.Thmtheorem7.p1.4.4.m4.2.3.9.3"><csymbol cd="ambiguous" id="S7.Thmtheorem7.p1.4.4.m4.2.3.9.3.1.cmml" xref="S7.Thmtheorem7.p1.4.4.m4.2.3.9.3">superscript</csymbol><ci id="S7.Thmtheorem7.p1.4.4.m4.2.3.9.3.2.cmml" xref="S7.Thmtheorem7.p1.4.4.m4.2.3.9.3.2">𝜀</ci><cn id="S7.Thmtheorem7.p1.4.4.m4.2.3.9.3.3.cmml" type="integer" xref="S7.Thmtheorem7.p1.4.4.m4.2.3.9.3.3">2</cn></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.Thmtheorem7.p1.4.4.m4.2c">\ell^{\prime}\leq\frac{\log n}{\log(1+\frac{\eta}{2})}\leq\frac{\log n}{\eta/4% }=\frac{\log n}{32\varepsilon^{2}/\log n}\leq\frac{\log^{2}n}{\varepsilon^{2}}</annotation><annotation encoding="application/x-llamapun" id="S7.Thmtheorem7.p1.4.4.m4.2d">roman_ℓ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ≤ divide start_ARG roman_log italic_n end_ARG start_ARG roman_log ( 1 + divide start_ARG italic_η end_ARG start_ARG 2 end_ARG ) end_ARG ≤ divide start_ARG roman_log italic_n end_ARG start_ARG italic_η / 4 end_ARG = divide start_ARG roman_log italic_n end_ARG start_ARG 32 italic_ε start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / roman_log italic_n end_ARG ≤ divide start_ARG roman_log start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_n end_ARG start_ARG italic_ε start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG</annotation></semantics></math>.</span></p> </div> </div> </section> <section class="ltx_subsection" id="S7.SS1"> <h3 class="ltx_title ltx_title_subsection"> <span class="ltx_tag ltx_tag_subsection">7.1 </span>Algorithm overview</h3> <div class="ltx_para" id="S7.SS1.p1"> <p class="ltx_p" id="S7.SS1.p1.2">Denote <math alttext="\ell=\lceil\varepsilon^{-2}\log^{2}n\rceil" class="ltx_Math" display="inline" id="S7.SS1.p1.1.m1.1"><semantics id="S7.SS1.p1.1.m1.1a"><mrow id="S7.SS1.p1.1.m1.1.1" xref="S7.SS1.p1.1.m1.1.1.cmml"><mi id="S7.SS1.p1.1.m1.1.1.3" mathvariant="normal" xref="S7.SS1.p1.1.m1.1.1.3.cmml">ℓ</mi><mo id="S7.SS1.p1.1.m1.1.1.2" xref="S7.SS1.p1.1.m1.1.1.2.cmml">=</mo><mrow id="S7.SS1.p1.1.m1.1.1.1.1" xref="S7.SS1.p1.1.m1.1.1.1.2.cmml"><mo id="S7.SS1.p1.1.m1.1.1.1.1.2" stretchy="false" xref="S7.SS1.p1.1.m1.1.1.1.2.1.cmml">⌈</mo><mrow id="S7.SS1.p1.1.m1.1.1.1.1.1" xref="S7.SS1.p1.1.m1.1.1.1.1.1.cmml"><msup id="S7.SS1.p1.1.m1.1.1.1.1.1.2" xref="S7.SS1.p1.1.m1.1.1.1.1.1.2.cmml"><mi id="S7.SS1.p1.1.m1.1.1.1.1.1.2.2" xref="S7.SS1.p1.1.m1.1.1.1.1.1.2.2.cmml">ε</mi><mrow id="S7.SS1.p1.1.m1.1.1.1.1.1.2.3" xref="S7.SS1.p1.1.m1.1.1.1.1.1.2.3.cmml"><mo id="S7.SS1.p1.1.m1.1.1.1.1.1.2.3a" xref="S7.SS1.p1.1.m1.1.1.1.1.1.2.3.cmml">−</mo><mn id="S7.SS1.p1.1.m1.1.1.1.1.1.2.3.2" xref="S7.SS1.p1.1.m1.1.1.1.1.1.2.3.2.cmml">2</mn></mrow></msup><mo id="S7.SS1.p1.1.m1.1.1.1.1.1.1" lspace="0.167em" xref="S7.SS1.p1.1.m1.1.1.1.1.1.1.cmml">⁢</mo><mrow id="S7.SS1.p1.1.m1.1.1.1.1.1.3" xref="S7.SS1.p1.1.m1.1.1.1.1.1.3.cmml"><msup id="S7.SS1.p1.1.m1.1.1.1.1.1.3.1" xref="S7.SS1.p1.1.m1.1.1.1.1.1.3.1.cmml"><mi id="S7.SS1.p1.1.m1.1.1.1.1.1.3.1.2" xref="S7.SS1.p1.1.m1.1.1.1.1.1.3.1.2.cmml">log</mi><mn id="S7.SS1.p1.1.m1.1.1.1.1.1.3.1.3" xref="S7.SS1.p1.1.m1.1.1.1.1.1.3.1.3.cmml">2</mn></msup><mo id="S7.SS1.p1.1.m1.1.1.1.1.1.3a" lspace="0.167em" xref="S7.SS1.p1.1.m1.1.1.1.1.1.3.cmml">⁡</mo><mi id="S7.SS1.p1.1.m1.1.1.1.1.1.3.2" xref="S7.SS1.p1.1.m1.1.1.1.1.1.3.2.cmml">n</mi></mrow></mrow><mo id="S7.SS1.p1.1.m1.1.1.1.1.3" stretchy="false" xref="S7.SS1.p1.1.m1.1.1.1.2.1.cmml">⌉</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S7.SS1.p1.1.m1.1b"><apply id="S7.SS1.p1.1.m1.1.1.cmml" xref="S7.SS1.p1.1.m1.1.1"><eq id="S7.SS1.p1.1.m1.1.1.2.cmml" xref="S7.SS1.p1.1.m1.1.1.2"></eq><ci id="S7.SS1.p1.1.m1.1.1.3.cmml" xref="S7.SS1.p1.1.m1.1.1.3">ℓ</ci><apply id="S7.SS1.p1.1.m1.1.1.1.2.cmml" xref="S7.SS1.p1.1.m1.1.1.1.1"><ceiling id="S7.SS1.p1.1.m1.1.1.1.2.1.cmml" xref="S7.SS1.p1.1.m1.1.1.1.1.2"></ceiling><apply id="S7.SS1.p1.1.m1.1.1.1.1.1.cmml" xref="S7.SS1.p1.1.m1.1.1.1.1.1"><times id="S7.SS1.p1.1.m1.1.1.1.1.1.1.cmml" xref="S7.SS1.p1.1.m1.1.1.1.1.1.1"></times><apply id="S7.SS1.p1.1.m1.1.1.1.1.1.2.cmml" xref="S7.SS1.p1.1.m1.1.1.1.1.1.2"><csymbol cd="ambiguous" id="S7.SS1.p1.1.m1.1.1.1.1.1.2.1.cmml" xref="S7.SS1.p1.1.m1.1.1.1.1.1.2">superscript</csymbol><ci id="S7.SS1.p1.1.m1.1.1.1.1.1.2.2.cmml" xref="S7.SS1.p1.1.m1.1.1.1.1.1.2.2">𝜀</ci><apply id="S7.SS1.p1.1.m1.1.1.1.1.1.2.3.cmml" xref="S7.SS1.p1.1.m1.1.1.1.1.1.2.3"><minus id="S7.SS1.p1.1.m1.1.1.1.1.1.2.3.1.cmml" xref="S7.SS1.p1.1.m1.1.1.1.1.1.2.3"></minus><cn id="S7.SS1.p1.1.m1.1.1.1.1.1.2.3.2.cmml" type="integer" xref="S7.SS1.p1.1.m1.1.1.1.1.1.2.3.2">2</cn></apply></apply><apply id="S7.SS1.p1.1.m1.1.1.1.1.1.3.cmml" xref="S7.SS1.p1.1.m1.1.1.1.1.1.3"><apply id="S7.SS1.p1.1.m1.1.1.1.1.1.3.1.cmml" xref="S7.SS1.p1.1.m1.1.1.1.1.1.3.1"><csymbol cd="ambiguous" id="S7.SS1.p1.1.m1.1.1.1.1.1.3.1.1.cmml" xref="S7.SS1.p1.1.m1.1.1.1.1.1.3.1">superscript</csymbol><log id="S7.SS1.p1.1.m1.1.1.1.1.1.3.1.2.cmml" xref="S7.SS1.p1.1.m1.1.1.1.1.1.3.1.2"></log><cn id="S7.SS1.p1.1.m1.1.1.1.1.1.3.1.3.cmml" type="integer" xref="S7.SS1.p1.1.m1.1.1.1.1.1.3.1.3">2</cn></apply><ci id="S7.SS1.p1.1.m1.1.1.1.1.1.3.2.cmml" xref="S7.SS1.p1.1.m1.1.1.1.1.1.3.2">𝑛</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.SS1.p1.1.m1.1c">\ell=\lceil\varepsilon^{-2}\log^{2}n\rceil</annotation><annotation encoding="application/x-llamapun" id="S7.SS1.p1.1.m1.1d">roman_ℓ = ⌈ italic_ε start_POSTSUPERSCRIPT - 2 end_POSTSUPERSCRIPT roman_log start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_n ⌉</annotation></semantics></math>. Our algorithm runs on a ‘clock’ denoted by <math alttext="(h:m:s)" class="ltx_Math" display="inline" id="S7.SS1.p1.2.m2.1"><semantics id="S7.SS1.p1.2.m2.1a"><mrow id="S7.SS1.p1.2.m2.1.1.1" xref="S7.SS1.p1.2.m2.1.1.1.1.cmml"><mo id="S7.SS1.p1.2.m2.1.1.1.2" stretchy="false" xref="S7.SS1.p1.2.m2.1.1.1.1.cmml">(</mo><mrow id="S7.SS1.p1.2.m2.1.1.1.1" xref="S7.SS1.p1.2.m2.1.1.1.1.cmml"><mi id="S7.SS1.p1.2.m2.1.1.1.1.2" xref="S7.SS1.p1.2.m2.1.1.1.1.2.cmml">h</mi><mo id="S7.SS1.p1.2.m2.1.1.1.1.3" lspace="0.278em" rspace="0.278em" xref="S7.SS1.p1.2.m2.1.1.1.1.3.cmml">:</mo><mi id="S7.SS1.p1.2.m2.1.1.1.1.4" xref="S7.SS1.p1.2.m2.1.1.1.1.4.cmml">m</mi><mo id="S7.SS1.p1.2.m2.1.1.1.1.5" lspace="0.278em" rspace="0.278em" xref="S7.SS1.p1.2.m2.1.1.1.1.5.cmml">:</mo><mi id="S7.SS1.p1.2.m2.1.1.1.1.6" xref="S7.SS1.p1.2.m2.1.1.1.1.6.cmml">s</mi></mrow><mo id="S7.SS1.p1.2.m2.1.1.1.3" stretchy="false" xref="S7.SS1.p1.2.m2.1.1.1.1.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S7.SS1.p1.2.m2.1b"><apply id="S7.SS1.p1.2.m2.1.1.1.1.cmml" xref="S7.SS1.p1.2.m2.1.1.1"><and id="S7.SS1.p1.2.m2.1.1.1.1a.cmml" xref="S7.SS1.p1.2.m2.1.1.1"></and><apply id="S7.SS1.p1.2.m2.1.1.1.1b.cmml" xref="S7.SS1.p1.2.m2.1.1.1"><ci id="S7.SS1.p1.2.m2.1.1.1.1.3.cmml" xref="S7.SS1.p1.2.m2.1.1.1.1.3">:</ci><ci id="S7.SS1.p1.2.m2.1.1.1.1.2.cmml" xref="S7.SS1.p1.2.m2.1.1.1.1.2">ℎ</ci><ci id="S7.SS1.p1.2.m2.1.1.1.1.4.cmml" xref="S7.SS1.p1.2.m2.1.1.1.1.4">𝑚</ci></apply><apply id="S7.SS1.p1.2.m2.1.1.1.1c.cmml" xref="S7.SS1.p1.2.m2.1.1.1"><ci id="S7.SS1.p1.2.m2.1.1.1.1.5.cmml" xref="S7.SS1.p1.2.m2.1.1.1.1.5">:</ci><share href="https://arxiv.org/html/2411.12694v2#S7.SS1.p1.2.m2.1.1.1.1.4.cmml" id="S7.SS1.p1.2.m2.1.1.1.1d.cmml" xref="S7.SS1.p1.2.m2.1.1.1"></share><ci id="S7.SS1.p1.2.m2.1.1.1.1.6.cmml" xref="S7.SS1.p1.2.m2.1.1.1.1.6">𝑠</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.SS1.p1.2.m2.1c">(h:m:s)</annotation><annotation encoding="application/x-llamapun" id="S7.SS1.p1.2.m2.1d">( italic_h : italic_m : italic_s )</annotation></semantics></math> where:</p> <ul class="ltx_itemize" id="S7.I3"> <li class="ltx_item" id="S7.I3.i1" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">•</span> <div class="ltx_para" id="S7.I3.i1.p1"> <p class="ltx_p" id="S7.I3.i1.p1.2">Each <span class="ltx_text ltx_font_smallcaps" id="S7.I3.i1.p1.2.1">hour</span> <math alttext="h" class="ltx_Math" display="inline" id="S7.I3.i1.p1.1.m1.1"><semantics id="S7.I3.i1.p1.1.m1.1a"><mi id="S7.I3.i1.p1.1.m1.1.1" xref="S7.I3.i1.p1.1.m1.1.1.cmml">h</mi><annotation-xml encoding="MathML-Content" id="S7.I3.i1.p1.1.m1.1b"><ci id="S7.I3.i1.p1.1.m1.1.1.cmml" xref="S7.I3.i1.p1.1.m1.1.1">ℎ</ci></annotation-xml><annotation encoding="application/x-tex" id="S7.I3.i1.p1.1.m1.1c">h</annotation><annotation encoding="application/x-llamapun" id="S7.I3.i1.p1.1.m1.1d">italic_h</annotation></semantics></math> lasts <math alttext="2\lceil\eta^{-1}\rceil+2" class="ltx_Math" display="inline" id="S7.I3.i1.p1.2.m2.1"><semantics id="S7.I3.i1.p1.2.m2.1a"><mrow id="S7.I3.i1.p1.2.m2.1.1" xref="S7.I3.i1.p1.2.m2.1.1.cmml"><mrow id="S7.I3.i1.p1.2.m2.1.1.1" xref="S7.I3.i1.p1.2.m2.1.1.1.cmml"><mn id="S7.I3.i1.p1.2.m2.1.1.1.3" xref="S7.I3.i1.p1.2.m2.1.1.1.3.cmml">2</mn><mo id="S7.I3.i1.p1.2.m2.1.1.1.2" xref="S7.I3.i1.p1.2.m2.1.1.1.2.cmml">⁢</mo><mrow id="S7.I3.i1.p1.2.m2.1.1.1.1.1" xref="S7.I3.i1.p1.2.m2.1.1.1.1.2.cmml"><mo id="S7.I3.i1.p1.2.m2.1.1.1.1.1.2" stretchy="false" xref="S7.I3.i1.p1.2.m2.1.1.1.1.2.1.cmml">⌈</mo><msup id="S7.I3.i1.p1.2.m2.1.1.1.1.1.1" xref="S7.I3.i1.p1.2.m2.1.1.1.1.1.1.cmml"><mi id="S7.I3.i1.p1.2.m2.1.1.1.1.1.1.2" xref="S7.I3.i1.p1.2.m2.1.1.1.1.1.1.2.cmml">η</mi><mrow id="S7.I3.i1.p1.2.m2.1.1.1.1.1.1.3" xref="S7.I3.i1.p1.2.m2.1.1.1.1.1.1.3.cmml"><mo id="S7.I3.i1.p1.2.m2.1.1.1.1.1.1.3a" xref="S7.I3.i1.p1.2.m2.1.1.1.1.1.1.3.cmml">−</mo><mn id="S7.I3.i1.p1.2.m2.1.1.1.1.1.1.3.2" xref="S7.I3.i1.p1.2.m2.1.1.1.1.1.1.3.2.cmml">1</mn></mrow></msup><mo id="S7.I3.i1.p1.2.m2.1.1.1.1.1.3" stretchy="false" xref="S7.I3.i1.p1.2.m2.1.1.1.1.2.1.cmml">⌉</mo></mrow></mrow><mo id="S7.I3.i1.p1.2.m2.1.1.2" xref="S7.I3.i1.p1.2.m2.1.1.2.cmml">+</mo><mn id="S7.I3.i1.p1.2.m2.1.1.3" xref="S7.I3.i1.p1.2.m2.1.1.3.cmml">2</mn></mrow><annotation-xml encoding="MathML-Content" id="S7.I3.i1.p1.2.m2.1b"><apply id="S7.I3.i1.p1.2.m2.1.1.cmml" xref="S7.I3.i1.p1.2.m2.1.1"><plus id="S7.I3.i1.p1.2.m2.1.1.2.cmml" xref="S7.I3.i1.p1.2.m2.1.1.2"></plus><apply id="S7.I3.i1.p1.2.m2.1.1.1.cmml" xref="S7.I3.i1.p1.2.m2.1.1.1"><times id="S7.I3.i1.p1.2.m2.1.1.1.2.cmml" xref="S7.I3.i1.p1.2.m2.1.1.1.2"></times><cn id="S7.I3.i1.p1.2.m2.1.1.1.3.cmml" type="integer" xref="S7.I3.i1.p1.2.m2.1.1.1.3">2</cn><apply id="S7.I3.i1.p1.2.m2.1.1.1.1.2.cmml" xref="S7.I3.i1.p1.2.m2.1.1.1.1.1"><ceiling id="S7.I3.i1.p1.2.m2.1.1.1.1.2.1.cmml" xref="S7.I3.i1.p1.2.m2.1.1.1.1.1.2"></ceiling><apply id="S7.I3.i1.p1.2.m2.1.1.1.1.1.1.cmml" xref="S7.I3.i1.p1.2.m2.1.1.1.1.1.1"><csymbol cd="ambiguous" id="S7.I3.i1.p1.2.m2.1.1.1.1.1.1.1.cmml" xref="S7.I3.i1.p1.2.m2.1.1.1.1.1.1">superscript</csymbol><ci id="S7.I3.i1.p1.2.m2.1.1.1.1.1.1.2.cmml" xref="S7.I3.i1.p1.2.m2.1.1.1.1.1.1.2">𝜂</ci><apply id="S7.I3.i1.p1.2.m2.1.1.1.1.1.1.3.cmml" xref="S7.I3.i1.p1.2.m2.1.1.1.1.1.1.3"><minus id="S7.I3.i1.p1.2.m2.1.1.1.1.1.1.3.1.cmml" xref="S7.I3.i1.p1.2.m2.1.1.1.1.1.1.3"></minus><cn id="S7.I3.i1.p1.2.m2.1.1.1.1.1.1.3.2.cmml" type="integer" xref="S7.I3.i1.p1.2.m2.1.1.1.1.1.1.3.2">1</cn></apply></apply></apply></apply><cn id="S7.I3.i1.p1.2.m2.1.1.3.cmml" type="integer" xref="S7.I3.i1.p1.2.m2.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.I3.i1.p1.2.m2.1c">2\lceil\eta^{-1}\rceil+2</annotation><annotation encoding="application/x-llamapun" id="S7.I3.i1.p1.2.m2.1d">2 ⌈ italic_η start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT ⌉ + 2</annotation></semantics></math> <span class="ltx_text ltx_font_smallcaps" id="S7.I3.i1.p1.2.2">minutes</span>,</p> </div> </li> <li class="ltx_item" id="S7.I3.i2" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">•</span> <div class="ltx_para" id="S7.I3.i2.p1"> <p class="ltx_p" id="S7.I3.i2.p1.2">Each even <span class="ltx_text ltx_font_smallcaps" id="S7.I3.i2.p1.2.1">minute</span> <math alttext="m" class="ltx_Math" display="inline" id="S7.I3.i2.p1.1.m1.1"><semantics id="S7.I3.i2.p1.1.m1.1a"><mi id="S7.I3.i2.p1.1.m1.1.1" xref="S7.I3.i2.p1.1.m1.1.1.cmml">m</mi><annotation-xml encoding="MathML-Content" id="S7.I3.i2.p1.1.m1.1b"><ci id="S7.I3.i2.p1.1.m1.1.1.cmml" xref="S7.I3.i2.p1.1.m1.1.1">𝑚</ci></annotation-xml><annotation encoding="application/x-tex" id="S7.I3.i2.p1.1.m1.1c">m</annotation><annotation encoding="application/x-llamapun" id="S7.I3.i2.p1.1.m1.1d">italic_m</annotation></semantics></math> lasts <math alttext="4\lceil\log_{8/7}n\rceil+2" class="ltx_Math" display="inline" id="S7.I3.i2.p1.2.m2.1"><semantics id="S7.I3.i2.p1.2.m2.1a"><mrow id="S7.I3.i2.p1.2.m2.1.1" xref="S7.I3.i2.p1.2.m2.1.1.cmml"><mrow id="S7.I3.i2.p1.2.m2.1.1.1" xref="S7.I3.i2.p1.2.m2.1.1.1.cmml"><mn id="S7.I3.i2.p1.2.m2.1.1.1.3" xref="S7.I3.i2.p1.2.m2.1.1.1.3.cmml">4</mn><mo id="S7.I3.i2.p1.2.m2.1.1.1.2" xref="S7.I3.i2.p1.2.m2.1.1.1.2.cmml">⁢</mo><mrow id="S7.I3.i2.p1.2.m2.1.1.1.1.1" xref="S7.I3.i2.p1.2.m2.1.1.1.1.2.cmml"><mo id="S7.I3.i2.p1.2.m2.1.1.1.1.1.2" stretchy="false" xref="S7.I3.i2.p1.2.m2.1.1.1.1.2.1.cmml">⌈</mo><mrow id="S7.I3.i2.p1.2.m2.1.1.1.1.1.1" xref="S7.I3.i2.p1.2.m2.1.1.1.1.1.1.cmml"><msub id="S7.I3.i2.p1.2.m2.1.1.1.1.1.1.1" xref="S7.I3.i2.p1.2.m2.1.1.1.1.1.1.1.cmml"><mi id="S7.I3.i2.p1.2.m2.1.1.1.1.1.1.1.2" xref="S7.I3.i2.p1.2.m2.1.1.1.1.1.1.1.2.cmml">log</mi><mrow id="S7.I3.i2.p1.2.m2.1.1.1.1.1.1.1.3" xref="S7.I3.i2.p1.2.m2.1.1.1.1.1.1.1.3.cmml"><mn id="S7.I3.i2.p1.2.m2.1.1.1.1.1.1.1.3.2" xref="S7.I3.i2.p1.2.m2.1.1.1.1.1.1.1.3.2.cmml">8</mn><mo id="S7.I3.i2.p1.2.m2.1.1.1.1.1.1.1.3.1" xref="S7.I3.i2.p1.2.m2.1.1.1.1.1.1.1.3.1.cmml">/</mo><mn id="S7.I3.i2.p1.2.m2.1.1.1.1.1.1.1.3.3" xref="S7.I3.i2.p1.2.m2.1.1.1.1.1.1.1.3.3.cmml">7</mn></mrow></msub><mo id="S7.I3.i2.p1.2.m2.1.1.1.1.1.1a" lspace="0.167em" xref="S7.I3.i2.p1.2.m2.1.1.1.1.1.1.cmml">⁡</mo><mi id="S7.I3.i2.p1.2.m2.1.1.1.1.1.1.2" xref="S7.I3.i2.p1.2.m2.1.1.1.1.1.1.2.cmml">n</mi></mrow><mo id="S7.I3.i2.p1.2.m2.1.1.1.1.1.3" stretchy="false" xref="S7.I3.i2.p1.2.m2.1.1.1.1.2.1.cmml">⌉</mo></mrow></mrow><mo id="S7.I3.i2.p1.2.m2.1.1.2" xref="S7.I3.i2.p1.2.m2.1.1.2.cmml">+</mo><mn id="S7.I3.i2.p1.2.m2.1.1.3" xref="S7.I3.i2.p1.2.m2.1.1.3.cmml">2</mn></mrow><annotation-xml encoding="MathML-Content" id="S7.I3.i2.p1.2.m2.1b"><apply id="S7.I3.i2.p1.2.m2.1.1.cmml" xref="S7.I3.i2.p1.2.m2.1.1"><plus id="S7.I3.i2.p1.2.m2.1.1.2.cmml" xref="S7.I3.i2.p1.2.m2.1.1.2"></plus><apply id="S7.I3.i2.p1.2.m2.1.1.1.cmml" xref="S7.I3.i2.p1.2.m2.1.1.1"><times id="S7.I3.i2.p1.2.m2.1.1.1.2.cmml" xref="S7.I3.i2.p1.2.m2.1.1.1.2"></times><cn id="S7.I3.i2.p1.2.m2.1.1.1.3.cmml" type="integer" xref="S7.I3.i2.p1.2.m2.1.1.1.3">4</cn><apply id="S7.I3.i2.p1.2.m2.1.1.1.1.2.cmml" xref="S7.I3.i2.p1.2.m2.1.1.1.1.1"><ceiling id="S7.I3.i2.p1.2.m2.1.1.1.1.2.1.cmml" xref="S7.I3.i2.p1.2.m2.1.1.1.1.1.2"></ceiling><apply id="S7.I3.i2.p1.2.m2.1.1.1.1.1.1.cmml" xref="S7.I3.i2.p1.2.m2.1.1.1.1.1.1"><apply id="S7.I3.i2.p1.2.m2.1.1.1.1.1.1.1.cmml" xref="S7.I3.i2.p1.2.m2.1.1.1.1.1.1.1"><csymbol cd="ambiguous" id="S7.I3.i2.p1.2.m2.1.1.1.1.1.1.1.1.cmml" xref="S7.I3.i2.p1.2.m2.1.1.1.1.1.1.1">subscript</csymbol><log id="S7.I3.i2.p1.2.m2.1.1.1.1.1.1.1.2.cmml" xref="S7.I3.i2.p1.2.m2.1.1.1.1.1.1.1.2"></log><apply id="S7.I3.i2.p1.2.m2.1.1.1.1.1.1.1.3.cmml" xref="S7.I3.i2.p1.2.m2.1.1.1.1.1.1.1.3"><divide id="S7.I3.i2.p1.2.m2.1.1.1.1.1.1.1.3.1.cmml" xref="S7.I3.i2.p1.2.m2.1.1.1.1.1.1.1.3.1"></divide><cn id="S7.I3.i2.p1.2.m2.1.1.1.1.1.1.1.3.2.cmml" type="integer" xref="S7.I3.i2.p1.2.m2.1.1.1.1.1.1.1.3.2">8</cn><cn id="S7.I3.i2.p1.2.m2.1.1.1.1.1.1.1.3.3.cmml" type="integer" xref="S7.I3.i2.p1.2.m2.1.1.1.1.1.1.1.3.3">7</cn></apply></apply><ci id="S7.I3.i2.p1.2.m2.1.1.1.1.1.1.2.cmml" xref="S7.I3.i2.p1.2.m2.1.1.1.1.1.1.2">𝑛</ci></apply></apply></apply><cn id="S7.I3.i2.p1.2.m2.1.1.3.cmml" type="integer" xref="S7.I3.i2.p1.2.m2.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.I3.i2.p1.2.m2.1c">4\lceil\log_{8/7}n\rceil+2</annotation><annotation encoding="application/x-llamapun" id="S7.I3.i2.p1.2.m2.1d">4 ⌈ roman_log start_POSTSUBSCRIPT 8 / 7 end_POSTSUBSCRIPT italic_n ⌉ + 2</annotation></semantics></math> rounds,</p> </div> </li> <li class="ltx_item" id="S7.I3.i3" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">•</span> <div class="ltx_para" id="S7.I3.i3.p1"> <p class="ltx_p" id="S7.I3.i3.p1.3">Each odd <span class="ltx_text ltx_font_smallcaps" id="S7.I3.i3.p1.3.1">minute</span> <math alttext="m" class="ltx_Math" display="inline" id="S7.I3.i3.p1.1.m1.1"><semantics id="S7.I3.i3.p1.1.m1.1a"><mi id="S7.I3.i3.p1.1.m1.1.1" xref="S7.I3.i3.p1.1.m1.1.1.cmml">m</mi><annotation-xml encoding="MathML-Content" id="S7.I3.i3.p1.1.m1.1b"><ci id="S7.I3.i3.p1.1.m1.1.1.cmml" xref="S7.I3.i3.p1.1.m1.1.1">𝑚</ci></annotation-xml><annotation encoding="application/x-tex" id="S7.I3.i3.p1.1.m1.1c">m</annotation><annotation encoding="application/x-llamapun" id="S7.I3.i3.p1.1.m1.1d">italic_m</annotation></semantics></math> in <span class="ltx_text ltx_font_smallcaps" id="S7.I3.i3.p1.3.2">hour</span> <math alttext="h" class="ltx_Math" display="inline" id="S7.I3.i3.p1.2.m2.1"><semantics id="S7.I3.i3.p1.2.m2.1a"><mi id="S7.I3.i3.p1.2.m2.1.1" xref="S7.I3.i3.p1.2.m2.1.1.cmml">h</mi><annotation-xml encoding="MathML-Content" id="S7.I3.i3.p1.2.m2.1b"><ci id="S7.I3.i3.p1.2.m2.1.1.cmml" xref="S7.I3.i3.p1.2.m2.1.1">ℎ</ci></annotation-xml><annotation encoding="application/x-tex" id="S7.I3.i3.p1.2.m2.1c">h</annotation><annotation encoding="application/x-llamapun" id="S7.I3.i3.p1.2.m2.1d">italic_h</annotation></semantics></math> lasts <math alttext="\ell-h+1" class="ltx_Math" display="inline" id="S7.I3.i3.p1.3.m3.1"><semantics id="S7.I3.i3.p1.3.m3.1a"><mrow id="S7.I3.i3.p1.3.m3.1.1" xref="S7.I3.i3.p1.3.m3.1.1.cmml"><mrow id="S7.I3.i3.p1.3.m3.1.1.2" xref="S7.I3.i3.p1.3.m3.1.1.2.cmml"><mi id="S7.I3.i3.p1.3.m3.1.1.2.2" mathvariant="normal" xref="S7.I3.i3.p1.3.m3.1.1.2.2.cmml">ℓ</mi><mo id="S7.I3.i3.p1.3.m3.1.1.2.1" xref="S7.I3.i3.p1.3.m3.1.1.2.1.cmml">−</mo><mi id="S7.I3.i3.p1.3.m3.1.1.2.3" xref="S7.I3.i3.p1.3.m3.1.1.2.3.cmml">h</mi></mrow><mo id="S7.I3.i3.p1.3.m3.1.1.1" xref="S7.I3.i3.p1.3.m3.1.1.1.cmml">+</mo><mn id="S7.I3.i3.p1.3.m3.1.1.3" xref="S7.I3.i3.p1.3.m3.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S7.I3.i3.p1.3.m3.1b"><apply id="S7.I3.i3.p1.3.m3.1.1.cmml" xref="S7.I3.i3.p1.3.m3.1.1"><plus id="S7.I3.i3.p1.3.m3.1.1.1.cmml" xref="S7.I3.i3.p1.3.m3.1.1.1"></plus><apply id="S7.I3.i3.p1.3.m3.1.1.2.cmml" xref="S7.I3.i3.p1.3.m3.1.1.2"><minus id="S7.I3.i3.p1.3.m3.1.1.2.1.cmml" xref="S7.I3.i3.p1.3.m3.1.1.2.1"></minus><ci id="S7.I3.i3.p1.3.m3.1.1.2.2.cmml" xref="S7.I3.i3.p1.3.m3.1.1.2.2">ℓ</ci><ci id="S7.I3.i3.p1.3.m3.1.1.2.3.cmml" xref="S7.I3.i3.p1.3.m3.1.1.2.3">ℎ</ci></apply><cn id="S7.I3.i3.p1.3.m3.1.1.3.cmml" type="integer" xref="S7.I3.i3.p1.3.m3.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.I3.i3.p1.3.m3.1c">\ell-h+1</annotation><annotation encoding="application/x-llamapun" id="S7.I3.i3.p1.3.m3.1d">roman_ℓ - italic_h + 1</annotation></semantics></math> <span class="ltx_text ltx_font_smallcaps" id="S7.I3.i3.p1.3.3">seconds</span>,</p> </div> </li> <li class="ltx_item" id="S7.I3.i4" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">•</span> <div class="ltx_para" id="S7.I3.i4.p1"> <p class="ltx_p" id="S7.I3.i4.p1.2">Each <span class="ltx_text ltx_font_smallcaps" id="S7.I3.i4.p1.2.1">second</span> <math alttext="s" class="ltx_Math" display="inline" id="S7.I3.i4.p1.1.m1.1"><semantics id="S7.I3.i4.p1.1.m1.1a"><mi id="S7.I3.i4.p1.1.m1.1.1" xref="S7.I3.i4.p1.1.m1.1.1.cmml">s</mi><annotation-xml encoding="MathML-Content" id="S7.I3.i4.p1.1.m1.1b"><ci id="S7.I3.i4.p1.1.m1.1.1.cmml" xref="S7.I3.i4.p1.1.m1.1.1">𝑠</ci></annotation-xml><annotation encoding="application/x-tex" id="S7.I3.i4.p1.1.m1.1c">s</annotation><annotation encoding="application/x-llamapun" id="S7.I3.i4.p1.1.m1.1d">italic_s</annotation></semantics></math> lasts <math alttext="\ell+\textnormal{Blocking}(\ell,n)" class="ltx_Math" display="inline" id="S7.I3.i4.p1.2.m2.2"><semantics id="S7.I3.i4.p1.2.m2.2a"><mrow id="S7.I3.i4.p1.2.m2.2.3" xref="S7.I3.i4.p1.2.m2.2.3.cmml"><mi id="S7.I3.i4.p1.2.m2.2.3.2" mathvariant="normal" xref="S7.I3.i4.p1.2.m2.2.3.2.cmml">ℓ</mi><mo id="S7.I3.i4.p1.2.m2.2.3.1" xref="S7.I3.i4.p1.2.m2.2.3.1.cmml">+</mo><mrow id="S7.I3.i4.p1.2.m2.2.3.3" xref="S7.I3.i4.p1.2.m2.2.3.3.cmml"><mtext id="S7.I3.i4.p1.2.m2.2.3.3.2" xref="S7.I3.i4.p1.2.m2.2.3.3.2a.cmml">Blocking</mtext><mo id="S7.I3.i4.p1.2.m2.2.3.3.1" xref="S7.I3.i4.p1.2.m2.2.3.3.1.cmml">⁢</mo><mrow id="S7.I3.i4.p1.2.m2.2.3.3.3.2" xref="S7.I3.i4.p1.2.m2.2.3.3.3.1.cmml"><mo id="S7.I3.i4.p1.2.m2.2.3.3.3.2.1" stretchy="false" xref="S7.I3.i4.p1.2.m2.2.3.3.3.1.cmml">(</mo><mi id="S7.I3.i4.p1.2.m2.1.1" mathvariant="normal" xref="S7.I3.i4.p1.2.m2.1.1.cmml">ℓ</mi><mo id="S7.I3.i4.p1.2.m2.2.3.3.3.2.2" xref="S7.I3.i4.p1.2.m2.2.3.3.3.1.cmml">,</mo><mi id="S7.I3.i4.p1.2.m2.2.2" xref="S7.I3.i4.p1.2.m2.2.2.cmml">n</mi><mo id="S7.I3.i4.p1.2.m2.2.3.3.3.2.3" stretchy="false" xref="S7.I3.i4.p1.2.m2.2.3.3.3.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S7.I3.i4.p1.2.m2.2b"><apply id="S7.I3.i4.p1.2.m2.2.3.cmml" xref="S7.I3.i4.p1.2.m2.2.3"><plus id="S7.I3.i4.p1.2.m2.2.3.1.cmml" xref="S7.I3.i4.p1.2.m2.2.3.1"></plus><ci id="S7.I3.i4.p1.2.m2.2.3.2.cmml" xref="S7.I3.i4.p1.2.m2.2.3.2">ℓ</ci><apply id="S7.I3.i4.p1.2.m2.2.3.3.cmml" xref="S7.I3.i4.p1.2.m2.2.3.3"><times id="S7.I3.i4.p1.2.m2.2.3.3.1.cmml" xref="S7.I3.i4.p1.2.m2.2.3.3.1"></times><ci id="S7.I3.i4.p1.2.m2.2.3.3.2a.cmml" xref="S7.I3.i4.p1.2.m2.2.3.3.2"><mtext id="S7.I3.i4.p1.2.m2.2.3.3.2.cmml" xref="S7.I3.i4.p1.2.m2.2.3.3.2">Blocking</mtext></ci><interval closure="open" id="S7.I3.i4.p1.2.m2.2.3.3.3.1.cmml" xref="S7.I3.i4.p1.2.m2.2.3.3.3.2"><ci id="S7.I3.i4.p1.2.m2.1.1.cmml" xref="S7.I3.i4.p1.2.m2.1.1">ℓ</ci><ci id="S7.I3.i4.p1.2.m2.2.2.cmml" xref="S7.I3.i4.p1.2.m2.2.2">𝑛</ci></interval></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.I3.i4.p1.2.m2.2c">\ell+\textnormal{Blocking}(\ell,n)</annotation><annotation encoding="application/x-llamapun" id="S7.I3.i4.p1.2.m2.2d">roman_ℓ + Blocking ( roman_ℓ , italic_n )</annotation></semantics></math> rounds.</p> </div> </li> </ul> </div> <div class="ltx_para ltx_noindent" id="S7.SS1.p2"> <p class="ltx_p" id="S7.SS1.p2.4">Each vertex tracks the clock to know which actions of our algorithm it should execute (if any). Our clock is special, in the sense that hours tick downwards. Minutes and seconds tick upwards, starting from zero. Each vertex <math alttext="v" class="ltx_Math" display="inline" id="S7.SS1.p2.1.m1.1"><semantics id="S7.SS1.p2.1.m1.1a"><mi id="S7.SS1.p2.1.m1.1.1" xref="S7.SS1.p2.1.m1.1.1.cmml">v</mi><annotation-xml encoding="MathML-Content" id="S7.SS1.p2.1.m1.1b"><ci id="S7.SS1.p2.1.m1.1.1.cmml" xref="S7.SS1.p2.1.m1.1.1">𝑣</ci></annotation-xml><annotation encoding="application/x-tex" id="S7.SS1.p2.1.m1.1c">v</annotation><annotation encoding="application/x-llamapun" id="S7.SS1.p2.1.m1.1d">italic_v</annotation></semantics></math> in our graph keeps track of the current time, measured in the current <span class="ltx_text ltx_font_smallcaps" id="S7.SS1.p2.4.1">hour</span> <math alttext="h" class="ltx_Math" display="inline" id="S7.SS1.p2.2.m2.1"><semantics id="S7.SS1.p2.2.m2.1a"><mi id="S7.SS1.p2.2.m2.1.1" xref="S7.SS1.p2.2.m2.1.1.cmml">h</mi><annotation-xml encoding="MathML-Content" id="S7.SS1.p2.2.m2.1b"><ci id="S7.SS1.p2.2.m2.1.1.cmml" xref="S7.SS1.p2.2.m2.1.1">ℎ</ci></annotation-xml><annotation encoding="application/x-tex" id="S7.SS1.p2.2.m2.1c">h</annotation><annotation encoding="application/x-llamapun" id="S7.SS1.p2.2.m2.1d">italic_h</annotation></semantics></math>, <span class="ltx_text ltx_font_smallcaps" id="S7.SS1.p2.4.2">minute</span> <math alttext="m" class="ltx_Math" display="inline" id="S7.SS1.p2.3.m3.1"><semantics id="S7.SS1.p2.3.m3.1a"><mi id="S7.SS1.p2.3.m3.1.1" xref="S7.SS1.p2.3.m3.1.1.cmml">m</mi><annotation-xml encoding="MathML-Content" id="S7.SS1.p2.3.m3.1b"><ci id="S7.SS1.p2.3.m3.1.1.cmml" xref="S7.SS1.p2.3.m3.1.1">𝑚</ci></annotation-xml><annotation encoding="application/x-tex" id="S7.SS1.p2.3.m3.1c">m</annotation><annotation encoding="application/x-llamapun" id="S7.SS1.p2.3.m3.1d">italic_m</annotation></semantics></math> and <span class="ltx_text ltx_font_smallcaps" id="S7.SS1.p2.4.3">second</span> <math alttext="s" class="ltx_Math" display="inline" id="S7.SS1.p2.4.m4.1"><semantics id="S7.SS1.p2.4.m4.1a"><mi id="S7.SS1.p2.4.m4.1.1" xref="S7.SS1.p2.4.m4.1.1.cmml">s</mi><annotation-xml encoding="MathML-Content" id="S7.SS1.p2.4.m4.1b"><ci id="S7.SS1.p2.4.m4.1.1.cmml" xref="S7.SS1.p2.4.m4.1.1">𝑠</ci></annotation-xml><annotation encoding="application/x-tex" id="S7.SS1.p2.4.m4.1c">s</annotation><annotation encoding="application/x-llamapun" id="S7.SS1.p2.4.m4.1d">italic_s</annotation></semantics></math>. We maintain the following:</p> </div> <div class="ltx_theorem ltx_theorem_invariant" id="Thminvariant1"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="Thminvariant1.1.1.1">Invariant 1</span></span><span class="ltx_text ltx_font_bold" id="Thminvariant1.2.2">.</span> </h6> <div class="ltx_para" id="Thminvariant1.p1"> <p class="ltx_p" id="Thminvariant1.p1.6"><span class="ltx_text ltx_font_italic" id="Thminvariant1.p1.6.6">During <span class="ltx_text ltx_font_smallcaps" id="Thminvariant1.p1.6.6.1">hour</span> <math alttext="h" class="ltx_Math" display="inline" id="Thminvariant1.p1.1.1.m1.1"><semantics id="Thminvariant1.p1.1.1.m1.1a"><mi id="Thminvariant1.p1.1.1.m1.1.1" xref="Thminvariant1.p1.1.1.m1.1.1.cmml">h</mi><annotation-xml encoding="MathML-Content" id="Thminvariant1.p1.1.1.m1.1b"><ci id="Thminvariant1.p1.1.1.m1.1.1.cmml" xref="Thminvariant1.p1.1.1.m1.1.1">ℎ</ci></annotation-xml><annotation encoding="application/x-tex" id="Thminvariant1.p1.1.1.m1.1c">h</annotation><annotation encoding="application/x-llamapun" id="Thminvariant1.p1.1.1.m1.1d">italic_h</annotation></semantics></math>, there are no violating out-edges from level <math alttext="k" class="ltx_Math" display="inline" id="Thminvariant1.p1.2.2.m2.1"><semantics id="Thminvariant1.p1.2.2.m2.1a"><mi id="Thminvariant1.p1.2.2.m2.1.1" xref="Thminvariant1.p1.2.2.m2.1.1.cmml">k</mi><annotation-xml encoding="MathML-Content" id="Thminvariant1.p1.2.2.m2.1b"><ci id="Thminvariant1.p1.2.2.m2.1.1.cmml" xref="Thminvariant1.p1.2.2.m2.1.1">𝑘</ci></annotation-xml><annotation encoding="application/x-tex" id="Thminvariant1.p1.2.2.m2.1c">k</annotation><annotation encoding="application/x-llamapun" id="Thminvariant1.p1.2.2.m2.1d">italic_k</annotation></semantics></math> for all <math alttext="k>h+1" class="ltx_Math" display="inline" id="Thminvariant1.p1.3.3.m3.1"><semantics id="Thminvariant1.p1.3.3.m3.1a"><mrow id="Thminvariant1.p1.3.3.m3.1.1" xref="Thminvariant1.p1.3.3.m3.1.1.cmml"><mi id="Thminvariant1.p1.3.3.m3.1.1.2" xref="Thminvariant1.p1.3.3.m3.1.1.2.cmml">k</mi><mo id="Thminvariant1.p1.3.3.m3.1.1.1" xref="Thminvariant1.p1.3.3.m3.1.1.1.cmml">></mo><mrow id="Thminvariant1.p1.3.3.m3.1.1.3" xref="Thminvariant1.p1.3.3.m3.1.1.3.cmml"><mi id="Thminvariant1.p1.3.3.m3.1.1.3.2" xref="Thminvariant1.p1.3.3.m3.1.1.3.2.cmml">h</mi><mo id="Thminvariant1.p1.3.3.m3.1.1.3.1" xref="Thminvariant1.p1.3.3.m3.1.1.3.1.cmml">+</mo><mn id="Thminvariant1.p1.3.3.m3.1.1.3.3" xref="Thminvariant1.p1.3.3.m3.1.1.3.3.cmml">1</mn></mrow></mrow><annotation-xml encoding="MathML-Content" id="Thminvariant1.p1.3.3.m3.1b"><apply id="Thminvariant1.p1.3.3.m3.1.1.cmml" xref="Thminvariant1.p1.3.3.m3.1.1"><gt id="Thminvariant1.p1.3.3.m3.1.1.1.cmml" xref="Thminvariant1.p1.3.3.m3.1.1.1"></gt><ci id="Thminvariant1.p1.3.3.m3.1.1.2.cmml" xref="Thminvariant1.p1.3.3.m3.1.1.2">𝑘</ci><apply id="Thminvariant1.p1.3.3.m3.1.1.3.cmml" xref="Thminvariant1.p1.3.3.m3.1.1.3"><plus id="Thminvariant1.p1.3.3.m3.1.1.3.1.cmml" xref="Thminvariant1.p1.3.3.m3.1.1.3.1"></plus><ci id="Thminvariant1.p1.3.3.m3.1.1.3.2.cmml" xref="Thminvariant1.p1.3.3.m3.1.1.3.2">ℎ</ci><cn id="Thminvariant1.p1.3.3.m3.1.1.3.3.cmml" type="integer" xref="Thminvariant1.p1.3.3.m3.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thminvariant1.p1.3.3.m3.1c">k>h+1</annotation><annotation encoding="application/x-llamapun" id="Thminvariant1.p1.3.3.m3.1d">italic_k > italic_h + 1</annotation></semantics></math>. At the start of each even <span class="ltx_text ltx_font_smallcaps" id="Thminvariant1.p1.6.6.2">minute</span> in <span class="ltx_text ltx_font_smallcaps" id="Thminvariant1.p1.6.6.3">hour</span> <math alttext="h" class="ltx_Math" display="inline" id="Thminvariant1.p1.4.4.m4.1"><semantics id="Thminvariant1.p1.4.4.m4.1a"><mi id="Thminvariant1.p1.4.4.m4.1.1" xref="Thminvariant1.p1.4.4.m4.1.1.cmml">h</mi><annotation-xml encoding="MathML-Content" id="Thminvariant1.p1.4.4.m4.1b"><ci id="Thminvariant1.p1.4.4.m4.1.1.cmml" xref="Thminvariant1.p1.4.4.m4.1.1">ℎ</ci></annotation-xml><annotation encoding="application/x-tex" id="Thminvariant1.p1.4.4.m4.1c">h</annotation><annotation encoding="application/x-llamapun" id="Thminvariant1.p1.4.4.m4.1d">italic_h</annotation></semantics></math>, there are no violating out-edges from level <math alttext="k" class="ltx_Math" display="inline" id="Thminvariant1.p1.5.5.m5.1"><semantics id="Thminvariant1.p1.5.5.m5.1a"><mi id="Thminvariant1.p1.5.5.m5.1.1" xref="Thminvariant1.p1.5.5.m5.1.1.cmml">k</mi><annotation-xml encoding="MathML-Content" id="Thminvariant1.p1.5.5.m5.1b"><ci id="Thminvariant1.p1.5.5.m5.1.1.cmml" xref="Thminvariant1.p1.5.5.m5.1.1">𝑘</ci></annotation-xml><annotation encoding="application/x-tex" id="Thminvariant1.p1.5.5.m5.1c">k</annotation><annotation encoding="application/x-llamapun" id="Thminvariant1.p1.5.5.m5.1d">italic_k</annotation></semantics></math> for all <math alttext="k>h" class="ltx_Math" display="inline" id="Thminvariant1.p1.6.6.m6.1"><semantics id="Thminvariant1.p1.6.6.m6.1a"><mrow id="Thminvariant1.p1.6.6.m6.1.1" xref="Thminvariant1.p1.6.6.m6.1.1.cmml"><mi id="Thminvariant1.p1.6.6.m6.1.1.2" xref="Thminvariant1.p1.6.6.m6.1.1.2.cmml">k</mi><mo id="Thminvariant1.p1.6.6.m6.1.1.1" xref="Thminvariant1.p1.6.6.m6.1.1.1.cmml">></mo><mi id="Thminvariant1.p1.6.6.m6.1.1.3" xref="Thminvariant1.p1.6.6.m6.1.1.3.cmml">h</mi></mrow><annotation-xml encoding="MathML-Content" id="Thminvariant1.p1.6.6.m6.1b"><apply id="Thminvariant1.p1.6.6.m6.1.1.cmml" xref="Thminvariant1.p1.6.6.m6.1.1"><gt id="Thminvariant1.p1.6.6.m6.1.1.1.cmml" xref="Thminvariant1.p1.6.6.m6.1.1.1"></gt><ci id="Thminvariant1.p1.6.6.m6.1.1.2.cmml" xref="Thminvariant1.p1.6.6.m6.1.1.2">𝑘</ci><ci id="Thminvariant1.p1.6.6.m6.1.1.3.cmml" xref="Thminvariant1.p1.6.6.m6.1.1.3">ℎ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="Thminvariant1.p1.6.6.m6.1c">k>h</annotation><annotation encoding="application/x-llamapun" id="Thminvariant1.p1.6.6.m6.1d">italic_k > italic_h</annotation></semantics></math>.</span></p> </div> </div> <div class="ltx_para ltx_noindent" id="S7.SS1.p3"> <p class="ltx_p" id="S7.SS1.p3.4">This invariant implies that we compute an <math alttext="\eta" class="ltx_Math" display="inline" id="S7.SS1.p3.1.m1.1"><semantics id="S7.SS1.p3.1.m1.1a"><mi id="S7.SS1.p3.1.m1.1.1" xref="S7.SS1.p3.1.m1.1.1.cmml">η</mi><annotation-xml encoding="MathML-Content" id="S7.SS1.p3.1.m1.1b"><ci id="S7.SS1.p3.1.m1.1.1.cmml" xref="S7.SS1.p3.1.m1.1.1">𝜂</ci></annotation-xml><annotation encoding="application/x-tex" id="S7.SS1.p3.1.m1.1c">\eta</annotation><annotation encoding="application/x-llamapun" id="S7.SS1.p3.1.m1.1d">italic_η</annotation></semantics></math>-fair orientation when the clock reaches <math alttext="(0:0:0)" class="ltx_Math" display="inline" id="S7.SS1.p3.2.m2.1"><semantics id="S7.SS1.p3.2.m2.1a"><mrow id="S7.SS1.p3.2.m2.1.1.1" xref="S7.SS1.p3.2.m2.1.1.1.1.cmml"><mo id="S7.SS1.p3.2.m2.1.1.1.2" stretchy="false" xref="S7.SS1.p3.2.m2.1.1.1.1.cmml">(</mo><mrow id="S7.SS1.p3.2.m2.1.1.1.1" xref="S7.SS1.p3.2.m2.1.1.1.1.cmml"><mn id="S7.SS1.p3.2.m2.1.1.1.1.2" xref="S7.SS1.p3.2.m2.1.1.1.1.2.cmml">0</mn><mo id="S7.SS1.p3.2.m2.1.1.1.1.3" lspace="0.278em" rspace="0.278em" xref="S7.SS1.p3.2.m2.1.1.1.1.3.cmml">:</mo><mn id="S7.SS1.p3.2.m2.1.1.1.1.4" xref="S7.SS1.p3.2.m2.1.1.1.1.4.cmml">0</mn><mo id="S7.SS1.p3.2.m2.1.1.1.1.5" lspace="0.278em" rspace="0.278em" xref="S7.SS1.p3.2.m2.1.1.1.1.5.cmml">:</mo><mn id="S7.SS1.p3.2.m2.1.1.1.1.6" xref="S7.SS1.p3.2.m2.1.1.1.1.6.cmml">0</mn></mrow><mo id="S7.SS1.p3.2.m2.1.1.1.3" stretchy="false" xref="S7.SS1.p3.2.m2.1.1.1.1.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S7.SS1.p3.2.m2.1b"><apply id="S7.SS1.p3.2.m2.1.1.1.1.cmml" xref="S7.SS1.p3.2.m2.1.1.1"><and id="S7.SS1.p3.2.m2.1.1.1.1a.cmml" xref="S7.SS1.p3.2.m2.1.1.1"></and><apply id="S7.SS1.p3.2.m2.1.1.1.1b.cmml" xref="S7.SS1.p3.2.m2.1.1.1"><ci id="S7.SS1.p3.2.m2.1.1.1.1.3.cmml" xref="S7.SS1.p3.2.m2.1.1.1.1.3">:</ci><cn id="S7.SS1.p3.2.m2.1.1.1.1.2.cmml" type="integer" xref="S7.SS1.p3.2.m2.1.1.1.1.2">0</cn><cn id="S7.SS1.p3.2.m2.1.1.1.1.4.cmml" type="integer" xref="S7.SS1.p3.2.m2.1.1.1.1.4">0</cn></apply><apply id="S7.SS1.p3.2.m2.1.1.1.1c.cmml" xref="S7.SS1.p3.2.m2.1.1.1"><ci id="S7.SS1.p3.2.m2.1.1.1.1.5.cmml" xref="S7.SS1.p3.2.m2.1.1.1.1.5">:</ci><share href="https://arxiv.org/html/2411.12694v2#S7.SS1.p3.2.m2.1.1.1.1.4.cmml" id="S7.SS1.p3.2.m2.1.1.1.1d.cmml" xref="S7.SS1.p3.2.m2.1.1.1"></share><cn id="S7.SS1.p3.2.m2.1.1.1.1.6.cmml" type="integer" xref="S7.SS1.p3.2.m2.1.1.1.1.6">0</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.SS1.p3.2.m2.1c">(0:0:0)</annotation><annotation encoding="application/x-llamapun" id="S7.SS1.p3.2.m2.1d">( 0 : 0 : 0 )</annotation></semantics></math>. Going from <span class="ltx_text ltx_font_smallcaps" id="S7.SS1.p3.4.1">hour</span> <math alttext="h" class="ltx_Math" display="inline" id="S7.SS1.p3.3.m3.1"><semantics id="S7.SS1.p3.3.m3.1a"><mi id="S7.SS1.p3.3.m3.1.1" xref="S7.SS1.p3.3.m3.1.1.cmml">h</mi><annotation-xml encoding="MathML-Content" id="S7.SS1.p3.3.m3.1b"><ci id="S7.SS1.p3.3.m3.1.1.cmml" xref="S7.SS1.p3.3.m3.1.1">ℎ</ci></annotation-xml><annotation encoding="application/x-tex" id="S7.SS1.p3.3.m3.1c">h</annotation><annotation encoding="application/x-llamapun" id="S7.SS1.p3.3.m3.1d">italic_h</annotation></semantics></math> to <span class="ltx_text ltx_font_smallcaps" id="S7.SS1.p3.4.2">hour</span> <math alttext="h-1" class="ltx_Math" display="inline" id="S7.SS1.p3.4.m4.1"><semantics id="S7.SS1.p3.4.m4.1a"><mrow id="S7.SS1.p3.4.m4.1.1" xref="S7.SS1.p3.4.m4.1.1.cmml"><mi id="S7.SS1.p3.4.m4.1.1.2" xref="S7.SS1.p3.4.m4.1.1.2.cmml">h</mi><mo id="S7.SS1.p3.4.m4.1.1.1" xref="S7.SS1.p3.4.m4.1.1.1.cmml">−</mo><mn id="S7.SS1.p3.4.m4.1.1.3" xref="S7.SS1.p3.4.m4.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S7.SS1.p3.4.m4.1b"><apply id="S7.SS1.p3.4.m4.1.1.cmml" xref="S7.SS1.p3.4.m4.1.1"><minus id="S7.SS1.p3.4.m4.1.1.1.cmml" xref="S7.SS1.p3.4.m4.1.1.1"></minus><ci id="S7.SS1.p3.4.m4.1.1.2.cmml" xref="S7.SS1.p3.4.m4.1.1.2">ℎ</ci><cn id="S7.SS1.p3.4.m4.1.1.3.cmml" type="integer" xref="S7.SS1.p3.4.m4.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.SS1.p3.4.m4.1c">h-1</annotation><annotation encoding="application/x-llamapun" id="S7.SS1.p3.4.m4.1d">italic_h - 1</annotation></semantics></math> we maintain this invariant by flipping directed paths:</p> </div> <div class="ltx_theorem ltx_theorem_definition" id="S7.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="S7.Thmtheorem8.1.1.1">Definition 7.8</span></span><span class="ltx_text ltx_font_bold" id="S7.Thmtheorem8.2.2">.</span> </h6> <div class="ltx_para" id="S7.Thmtheorem8.p1"> <p class="ltx_p" id="S7.Thmtheorem8.p1.8"><span class="ltx_text ltx_font_italic" id="S7.Thmtheorem8.p1.8.8">For any edge <math alttext="\overline{uv}" class="ltx_Math" display="inline" id="S7.Thmtheorem8.p1.1.1.m1.1"><semantics id="S7.Thmtheorem8.p1.1.1.m1.1a"><mover accent="true" id="S7.Thmtheorem8.p1.1.1.m1.1.1" xref="S7.Thmtheorem8.p1.1.1.m1.1.1.cmml"><mrow id="S7.Thmtheorem8.p1.1.1.m1.1.1.2" xref="S7.Thmtheorem8.p1.1.1.m1.1.1.2.cmml"><mi id="S7.Thmtheorem8.p1.1.1.m1.1.1.2.2" xref="S7.Thmtheorem8.p1.1.1.m1.1.1.2.2.cmml">u</mi><mo id="S7.Thmtheorem8.p1.1.1.m1.1.1.2.1" xref="S7.Thmtheorem8.p1.1.1.m1.1.1.2.1.cmml">⁢</mo><mi id="S7.Thmtheorem8.p1.1.1.m1.1.1.2.3" xref="S7.Thmtheorem8.p1.1.1.m1.1.1.2.3.cmml">v</mi></mrow><mo id="S7.Thmtheorem8.p1.1.1.m1.1.1.1" xref="S7.Thmtheorem8.p1.1.1.m1.1.1.1.cmml">¯</mo></mover><annotation-xml encoding="MathML-Content" id="S7.Thmtheorem8.p1.1.1.m1.1b"><apply id="S7.Thmtheorem8.p1.1.1.m1.1.1.cmml" xref="S7.Thmtheorem8.p1.1.1.m1.1.1"><ci id="S7.Thmtheorem8.p1.1.1.m1.1.1.1.cmml" xref="S7.Thmtheorem8.p1.1.1.m1.1.1.1">¯</ci><apply id="S7.Thmtheorem8.p1.1.1.m1.1.1.2.cmml" xref="S7.Thmtheorem8.p1.1.1.m1.1.1.2"><times id="S7.Thmtheorem8.p1.1.1.m1.1.1.2.1.cmml" xref="S7.Thmtheorem8.p1.1.1.m1.1.1.2.1"></times><ci id="S7.Thmtheorem8.p1.1.1.m1.1.1.2.2.cmml" xref="S7.Thmtheorem8.p1.1.1.m1.1.1.2.2">𝑢</ci><ci id="S7.Thmtheorem8.p1.1.1.m1.1.1.2.3.cmml" xref="S7.Thmtheorem8.p1.1.1.m1.1.1.2.3">𝑣</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.Thmtheorem8.p1.1.1.m1.1c">\overline{uv}</annotation><annotation encoding="application/x-llamapun" id="S7.Thmtheorem8.p1.1.1.m1.1d">over¯ start_ARG italic_u italic_v end_ARG</annotation></semantics></math> with <math alttext="\textsl{g}(u\!\to\!v)>0" class="ltx_Math" display="inline" id="S7.Thmtheorem8.p1.2.2.m2.1"><semantics id="S7.Thmtheorem8.p1.2.2.m2.1a"><mrow id="S7.Thmtheorem8.p1.2.2.m2.1.1" xref="S7.Thmtheorem8.p1.2.2.m2.1.1.cmml"><mrow id="S7.Thmtheorem8.p1.2.2.m2.1.1.1" xref="S7.Thmtheorem8.p1.2.2.m2.1.1.1.cmml"><mtext class="ltx_mathvariant_italic" id="S7.Thmtheorem8.p1.2.2.m2.1.1.1.3" xref="S7.Thmtheorem8.p1.2.2.m2.1.1.1.3a.cmml">g</mtext><mo id="S7.Thmtheorem8.p1.2.2.m2.1.1.1.2" xref="S7.Thmtheorem8.p1.2.2.m2.1.1.1.2.cmml">⁢</mo><mrow id="S7.Thmtheorem8.p1.2.2.m2.1.1.1.1.1" xref="S7.Thmtheorem8.p1.2.2.m2.1.1.1.1.1.1.cmml"><mo id="S7.Thmtheorem8.p1.2.2.m2.1.1.1.1.1.2" stretchy="false" xref="S7.Thmtheorem8.p1.2.2.m2.1.1.1.1.1.1.cmml">(</mo><mrow id="S7.Thmtheorem8.p1.2.2.m2.1.1.1.1.1.1" xref="S7.Thmtheorem8.p1.2.2.m2.1.1.1.1.1.1.cmml"><mi id="S7.Thmtheorem8.p1.2.2.m2.1.1.1.1.1.1.2" xref="S7.Thmtheorem8.p1.2.2.m2.1.1.1.1.1.1.2.cmml">u</mi><mo id="S7.Thmtheorem8.p1.2.2.m2.1.1.1.1.1.1.1" lspace="0.108em" rspace="0.108em" stretchy="false" xref="S7.Thmtheorem8.p1.2.2.m2.1.1.1.1.1.1.1.cmml">→</mo><mi id="S7.Thmtheorem8.p1.2.2.m2.1.1.1.1.1.1.3" xref="S7.Thmtheorem8.p1.2.2.m2.1.1.1.1.1.1.3.cmml">v</mi></mrow><mo id="S7.Thmtheorem8.p1.2.2.m2.1.1.1.1.1.3" stretchy="false" xref="S7.Thmtheorem8.p1.2.2.m2.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S7.Thmtheorem8.p1.2.2.m2.1.1.2" xref="S7.Thmtheorem8.p1.2.2.m2.1.1.2.cmml">></mo><mn id="S7.Thmtheorem8.p1.2.2.m2.1.1.3" xref="S7.Thmtheorem8.p1.2.2.m2.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S7.Thmtheorem8.p1.2.2.m2.1b"><apply id="S7.Thmtheorem8.p1.2.2.m2.1.1.cmml" xref="S7.Thmtheorem8.p1.2.2.m2.1.1"><gt id="S7.Thmtheorem8.p1.2.2.m2.1.1.2.cmml" xref="S7.Thmtheorem8.p1.2.2.m2.1.1.2"></gt><apply id="S7.Thmtheorem8.p1.2.2.m2.1.1.1.cmml" xref="S7.Thmtheorem8.p1.2.2.m2.1.1.1"><times id="S7.Thmtheorem8.p1.2.2.m2.1.1.1.2.cmml" xref="S7.Thmtheorem8.p1.2.2.m2.1.1.1.2"></times><ci id="S7.Thmtheorem8.p1.2.2.m2.1.1.1.3a.cmml" xref="S7.Thmtheorem8.p1.2.2.m2.1.1.1.3"><mtext class="ltx_mathvariant_italic" id="S7.Thmtheorem8.p1.2.2.m2.1.1.1.3.cmml" xref="S7.Thmtheorem8.p1.2.2.m2.1.1.1.3">g</mtext></ci><apply id="S7.Thmtheorem8.p1.2.2.m2.1.1.1.1.1.1.cmml" xref="S7.Thmtheorem8.p1.2.2.m2.1.1.1.1.1"><ci id="S7.Thmtheorem8.p1.2.2.m2.1.1.1.1.1.1.1.cmml" xref="S7.Thmtheorem8.p1.2.2.m2.1.1.1.1.1.1.1">→</ci><ci id="S7.Thmtheorem8.p1.2.2.m2.1.1.1.1.1.1.2.cmml" xref="S7.Thmtheorem8.p1.2.2.m2.1.1.1.1.1.1.2">𝑢</ci><ci id="S7.Thmtheorem8.p1.2.2.m2.1.1.1.1.1.1.3.cmml" xref="S7.Thmtheorem8.p1.2.2.m2.1.1.1.1.1.1.3">𝑣</ci></apply></apply><cn id="S7.Thmtheorem8.p1.2.2.m2.1.1.3.cmml" type="integer" xref="S7.Thmtheorem8.p1.2.2.m2.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.Thmtheorem8.p1.2.2.m2.1c">\textsl{g}(u\!\to\!v)>0</annotation><annotation encoding="application/x-llamapun" id="S7.Thmtheorem8.p1.2.2.m2.1d">g ( italic_u → italic_v ) > 0</annotation></semantics></math>, we say that our algorithm is <em class="ltx_emph ltx_font_upright" id="S7.Thmtheorem8.p1.8.8.1">flipping</em> <math alttext="\overline{uv}" class="ltx_Math" display="inline" id="S7.Thmtheorem8.p1.3.3.m3.1"><semantics id="S7.Thmtheorem8.p1.3.3.m3.1a"><mover accent="true" id="S7.Thmtheorem8.p1.3.3.m3.1.1" xref="S7.Thmtheorem8.p1.3.3.m3.1.1.cmml"><mrow id="S7.Thmtheorem8.p1.3.3.m3.1.1.2" xref="S7.Thmtheorem8.p1.3.3.m3.1.1.2.cmml"><mi id="S7.Thmtheorem8.p1.3.3.m3.1.1.2.2" xref="S7.Thmtheorem8.p1.3.3.m3.1.1.2.2.cmml">u</mi><mo id="S7.Thmtheorem8.p1.3.3.m3.1.1.2.1" xref="S7.Thmtheorem8.p1.3.3.m3.1.1.2.1.cmml">⁢</mo><mi id="S7.Thmtheorem8.p1.3.3.m3.1.1.2.3" xref="S7.Thmtheorem8.p1.3.3.m3.1.1.2.3.cmml">v</mi></mrow><mo id="S7.Thmtheorem8.p1.3.3.m3.1.1.1" xref="S7.Thmtheorem8.p1.3.3.m3.1.1.1.cmml">¯</mo></mover><annotation-xml encoding="MathML-Content" id="S7.Thmtheorem8.p1.3.3.m3.1b"><apply id="S7.Thmtheorem8.p1.3.3.m3.1.1.cmml" xref="S7.Thmtheorem8.p1.3.3.m3.1.1"><ci id="S7.Thmtheorem8.p1.3.3.m3.1.1.1.cmml" xref="S7.Thmtheorem8.p1.3.3.m3.1.1.1">¯</ci><apply id="S7.Thmtheorem8.p1.3.3.m3.1.1.2.cmml" xref="S7.Thmtheorem8.p1.3.3.m3.1.1.2"><times id="S7.Thmtheorem8.p1.3.3.m3.1.1.2.1.cmml" xref="S7.Thmtheorem8.p1.3.3.m3.1.1.2.1"></times><ci id="S7.Thmtheorem8.p1.3.3.m3.1.1.2.2.cmml" xref="S7.Thmtheorem8.p1.3.3.m3.1.1.2.2">𝑢</ci><ci id="S7.Thmtheorem8.p1.3.3.m3.1.1.2.3.cmml" xref="S7.Thmtheorem8.p1.3.3.m3.1.1.2.3">𝑣</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.Thmtheorem8.p1.3.3.m3.1c">\overline{uv}</annotation><annotation encoding="application/x-llamapun" id="S7.Thmtheorem8.p1.3.3.m3.1d">over¯ start_ARG italic_u italic_v end_ARG</annotation></semantics></math> whenever it decreases <math alttext="\textsl{g}(u\!\to\!v)" class="ltx_Math" display="inline" id="S7.Thmtheorem8.p1.4.4.m4.1"><semantics id="S7.Thmtheorem8.p1.4.4.m4.1a"><mrow id="S7.Thmtheorem8.p1.4.4.m4.1.1" xref="S7.Thmtheorem8.p1.4.4.m4.1.1.cmml"><mtext class="ltx_mathvariant_italic" id="S7.Thmtheorem8.p1.4.4.m4.1.1.3" xref="S7.Thmtheorem8.p1.4.4.m4.1.1.3a.cmml">g</mtext><mo id="S7.Thmtheorem8.p1.4.4.m4.1.1.2" xref="S7.Thmtheorem8.p1.4.4.m4.1.1.2.cmml">⁢</mo><mrow id="S7.Thmtheorem8.p1.4.4.m4.1.1.1.1" xref="S7.Thmtheorem8.p1.4.4.m4.1.1.1.1.1.cmml"><mo id="S7.Thmtheorem8.p1.4.4.m4.1.1.1.1.2" stretchy="false" xref="S7.Thmtheorem8.p1.4.4.m4.1.1.1.1.1.cmml">(</mo><mrow id="S7.Thmtheorem8.p1.4.4.m4.1.1.1.1.1" xref="S7.Thmtheorem8.p1.4.4.m4.1.1.1.1.1.cmml"><mi id="S7.Thmtheorem8.p1.4.4.m4.1.1.1.1.1.2" xref="S7.Thmtheorem8.p1.4.4.m4.1.1.1.1.1.2.cmml">u</mi><mo id="S7.Thmtheorem8.p1.4.4.m4.1.1.1.1.1.1" lspace="0.108em" rspace="0.108em" stretchy="false" xref="S7.Thmtheorem8.p1.4.4.m4.1.1.1.1.1.1.cmml">→</mo><mi id="S7.Thmtheorem8.p1.4.4.m4.1.1.1.1.1.3" xref="S7.Thmtheorem8.p1.4.4.m4.1.1.1.1.1.3.cmml">v</mi></mrow><mo id="S7.Thmtheorem8.p1.4.4.m4.1.1.1.1.3" stretchy="false" xref="S7.Thmtheorem8.p1.4.4.m4.1.1.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S7.Thmtheorem8.p1.4.4.m4.1b"><apply id="S7.Thmtheorem8.p1.4.4.m4.1.1.cmml" xref="S7.Thmtheorem8.p1.4.4.m4.1.1"><times id="S7.Thmtheorem8.p1.4.4.m4.1.1.2.cmml" xref="S7.Thmtheorem8.p1.4.4.m4.1.1.2"></times><ci id="S7.Thmtheorem8.p1.4.4.m4.1.1.3a.cmml" xref="S7.Thmtheorem8.p1.4.4.m4.1.1.3"><mtext class="ltx_mathvariant_italic" id="S7.Thmtheorem8.p1.4.4.m4.1.1.3.cmml" xref="S7.Thmtheorem8.p1.4.4.m4.1.1.3">g</mtext></ci><apply id="S7.Thmtheorem8.p1.4.4.m4.1.1.1.1.1.cmml" xref="S7.Thmtheorem8.p1.4.4.m4.1.1.1.1"><ci id="S7.Thmtheorem8.p1.4.4.m4.1.1.1.1.1.1.cmml" xref="S7.Thmtheorem8.p1.4.4.m4.1.1.1.1.1.1">→</ci><ci id="S7.Thmtheorem8.p1.4.4.m4.1.1.1.1.1.2.cmml" xref="S7.Thmtheorem8.p1.4.4.m4.1.1.1.1.1.2">𝑢</ci><ci id="S7.Thmtheorem8.p1.4.4.m4.1.1.1.1.1.3.cmml" xref="S7.Thmtheorem8.p1.4.4.m4.1.1.1.1.1.3">𝑣</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.Thmtheorem8.p1.4.4.m4.1c">\textsl{g}(u\!\to\!v)</annotation><annotation encoding="application/x-llamapun" id="S7.Thmtheorem8.p1.4.4.m4.1d">g ( italic_u → italic_v )</annotation></semantics></math> (increasing <math alttext="\textsl{g}(v\!\to\!u)" class="ltx_Math" display="inline" id="S7.Thmtheorem8.p1.5.5.m5.1"><semantics id="S7.Thmtheorem8.p1.5.5.m5.1a"><mrow id="S7.Thmtheorem8.p1.5.5.m5.1.1" xref="S7.Thmtheorem8.p1.5.5.m5.1.1.cmml"><mtext class="ltx_mathvariant_italic" id="S7.Thmtheorem8.p1.5.5.m5.1.1.3" xref="S7.Thmtheorem8.p1.5.5.m5.1.1.3a.cmml">g</mtext><mo id="S7.Thmtheorem8.p1.5.5.m5.1.1.2" xref="S7.Thmtheorem8.p1.5.5.m5.1.1.2.cmml">⁢</mo><mrow id="S7.Thmtheorem8.p1.5.5.m5.1.1.1.1" xref="S7.Thmtheorem8.p1.5.5.m5.1.1.1.1.1.cmml"><mo id="S7.Thmtheorem8.p1.5.5.m5.1.1.1.1.2" stretchy="false" xref="S7.Thmtheorem8.p1.5.5.m5.1.1.1.1.1.cmml">(</mo><mrow id="S7.Thmtheorem8.p1.5.5.m5.1.1.1.1.1" xref="S7.Thmtheorem8.p1.5.5.m5.1.1.1.1.1.cmml"><mi id="S7.Thmtheorem8.p1.5.5.m5.1.1.1.1.1.2" xref="S7.Thmtheorem8.p1.5.5.m5.1.1.1.1.1.2.cmml">v</mi><mo id="S7.Thmtheorem8.p1.5.5.m5.1.1.1.1.1.1" lspace="0.108em" rspace="0.108em" stretchy="false" xref="S7.Thmtheorem8.p1.5.5.m5.1.1.1.1.1.1.cmml">→</mo><mi id="S7.Thmtheorem8.p1.5.5.m5.1.1.1.1.1.3" xref="S7.Thmtheorem8.p1.5.5.m5.1.1.1.1.1.3.cmml">u</mi></mrow><mo id="S7.Thmtheorem8.p1.5.5.m5.1.1.1.1.3" stretchy="false" xref="S7.Thmtheorem8.p1.5.5.m5.1.1.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S7.Thmtheorem8.p1.5.5.m5.1b"><apply id="S7.Thmtheorem8.p1.5.5.m5.1.1.cmml" xref="S7.Thmtheorem8.p1.5.5.m5.1.1"><times id="S7.Thmtheorem8.p1.5.5.m5.1.1.2.cmml" xref="S7.Thmtheorem8.p1.5.5.m5.1.1.2"></times><ci id="S7.Thmtheorem8.p1.5.5.m5.1.1.3a.cmml" xref="S7.Thmtheorem8.p1.5.5.m5.1.1.3"><mtext class="ltx_mathvariant_italic" id="S7.Thmtheorem8.p1.5.5.m5.1.1.3.cmml" xref="S7.Thmtheorem8.p1.5.5.m5.1.1.3">g</mtext></ci><apply id="S7.Thmtheorem8.p1.5.5.m5.1.1.1.1.1.cmml" xref="S7.Thmtheorem8.p1.5.5.m5.1.1.1.1"><ci id="S7.Thmtheorem8.p1.5.5.m5.1.1.1.1.1.1.cmml" xref="S7.Thmtheorem8.p1.5.5.m5.1.1.1.1.1.1">→</ci><ci id="S7.Thmtheorem8.p1.5.5.m5.1.1.1.1.1.2.cmml" xref="S7.Thmtheorem8.p1.5.5.m5.1.1.1.1.1.2">𝑣</ci><ci id="S7.Thmtheorem8.p1.5.5.m5.1.1.1.1.1.3.cmml" xref="S7.Thmtheorem8.p1.5.5.m5.1.1.1.1.1.3">𝑢</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.Thmtheorem8.p1.5.5.m5.1c">\textsl{g}(v\!\to\!u)</annotation><annotation encoding="application/x-llamapun" id="S7.Thmtheorem8.p1.5.5.m5.1d">g ( italic_v → italic_u )</annotation></semantics></math>). Moreover, we say that <math alttext="\overline{uv}" class="ltx_Math" display="inline" id="S7.Thmtheorem8.p1.6.6.m6.1"><semantics id="S7.Thmtheorem8.p1.6.6.m6.1a"><mover accent="true" id="S7.Thmtheorem8.p1.6.6.m6.1.1" xref="S7.Thmtheorem8.p1.6.6.m6.1.1.cmml"><mrow id="S7.Thmtheorem8.p1.6.6.m6.1.1.2" xref="S7.Thmtheorem8.p1.6.6.m6.1.1.2.cmml"><mi id="S7.Thmtheorem8.p1.6.6.m6.1.1.2.2" xref="S7.Thmtheorem8.p1.6.6.m6.1.1.2.2.cmml">u</mi><mo id="S7.Thmtheorem8.p1.6.6.m6.1.1.2.1" xref="S7.Thmtheorem8.p1.6.6.m6.1.1.2.1.cmml">⁢</mo><mi id="S7.Thmtheorem8.p1.6.6.m6.1.1.2.3" xref="S7.Thmtheorem8.p1.6.6.m6.1.1.2.3.cmml">v</mi></mrow><mo id="S7.Thmtheorem8.p1.6.6.m6.1.1.1" xref="S7.Thmtheorem8.p1.6.6.m6.1.1.1.cmml">¯</mo></mover><annotation-xml encoding="MathML-Content" id="S7.Thmtheorem8.p1.6.6.m6.1b"><apply id="S7.Thmtheorem8.p1.6.6.m6.1.1.cmml" xref="S7.Thmtheorem8.p1.6.6.m6.1.1"><ci id="S7.Thmtheorem8.p1.6.6.m6.1.1.1.cmml" xref="S7.Thmtheorem8.p1.6.6.m6.1.1.1">¯</ci><apply id="S7.Thmtheorem8.p1.6.6.m6.1.1.2.cmml" xref="S7.Thmtheorem8.p1.6.6.m6.1.1.2"><times id="S7.Thmtheorem8.p1.6.6.m6.1.1.2.1.cmml" xref="S7.Thmtheorem8.p1.6.6.m6.1.1.2.1"></times><ci id="S7.Thmtheorem8.p1.6.6.m6.1.1.2.2.cmml" xref="S7.Thmtheorem8.p1.6.6.m6.1.1.2.2">𝑢</ci><ci id="S7.Thmtheorem8.p1.6.6.m6.1.1.2.3.cmml" xref="S7.Thmtheorem8.p1.6.6.m6.1.1.2.3">𝑣</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.Thmtheorem8.p1.6.6.m6.1c">\overline{uv}</annotation><annotation encoding="application/x-llamapun" id="S7.Thmtheorem8.p1.6.6.m6.1d">over¯ start_ARG italic_u italic_v end_ARG</annotation></semantics></math> is flipped whenever our algorithm has decreased <math alttext="\textsl{g}(u\!\to\!v)" class="ltx_Math" display="inline" id="S7.Thmtheorem8.p1.7.7.m7.1"><semantics id="S7.Thmtheorem8.p1.7.7.m7.1a"><mrow id="S7.Thmtheorem8.p1.7.7.m7.1.1" xref="S7.Thmtheorem8.p1.7.7.m7.1.1.cmml"><mtext class="ltx_mathvariant_italic" id="S7.Thmtheorem8.p1.7.7.m7.1.1.3" xref="S7.Thmtheorem8.p1.7.7.m7.1.1.3a.cmml">g</mtext><mo id="S7.Thmtheorem8.p1.7.7.m7.1.1.2" xref="S7.Thmtheorem8.p1.7.7.m7.1.1.2.cmml">⁢</mo><mrow id="S7.Thmtheorem8.p1.7.7.m7.1.1.1.1" xref="S7.Thmtheorem8.p1.7.7.m7.1.1.1.1.1.cmml"><mo id="S7.Thmtheorem8.p1.7.7.m7.1.1.1.1.2" stretchy="false" xref="S7.Thmtheorem8.p1.7.7.m7.1.1.1.1.1.cmml">(</mo><mrow id="S7.Thmtheorem8.p1.7.7.m7.1.1.1.1.1" xref="S7.Thmtheorem8.p1.7.7.m7.1.1.1.1.1.cmml"><mi id="S7.Thmtheorem8.p1.7.7.m7.1.1.1.1.1.2" xref="S7.Thmtheorem8.p1.7.7.m7.1.1.1.1.1.2.cmml">u</mi><mo id="S7.Thmtheorem8.p1.7.7.m7.1.1.1.1.1.1" lspace="0.108em" rspace="0.108em" stretchy="false" xref="S7.Thmtheorem8.p1.7.7.m7.1.1.1.1.1.1.cmml">→</mo><mi id="S7.Thmtheorem8.p1.7.7.m7.1.1.1.1.1.3" xref="S7.Thmtheorem8.p1.7.7.m7.1.1.1.1.1.3.cmml">v</mi></mrow><mo id="S7.Thmtheorem8.p1.7.7.m7.1.1.1.1.3" stretchy="false" xref="S7.Thmtheorem8.p1.7.7.m7.1.1.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S7.Thmtheorem8.p1.7.7.m7.1b"><apply id="S7.Thmtheorem8.p1.7.7.m7.1.1.cmml" xref="S7.Thmtheorem8.p1.7.7.m7.1.1"><times id="S7.Thmtheorem8.p1.7.7.m7.1.1.2.cmml" xref="S7.Thmtheorem8.p1.7.7.m7.1.1.2"></times><ci id="S7.Thmtheorem8.p1.7.7.m7.1.1.3a.cmml" xref="S7.Thmtheorem8.p1.7.7.m7.1.1.3"><mtext class="ltx_mathvariant_italic" id="S7.Thmtheorem8.p1.7.7.m7.1.1.3.cmml" xref="S7.Thmtheorem8.p1.7.7.m7.1.1.3">g</mtext></ci><apply id="S7.Thmtheorem8.p1.7.7.m7.1.1.1.1.1.cmml" xref="S7.Thmtheorem8.p1.7.7.m7.1.1.1.1"><ci id="S7.Thmtheorem8.p1.7.7.m7.1.1.1.1.1.1.cmml" xref="S7.Thmtheorem8.p1.7.7.m7.1.1.1.1.1.1">→</ci><ci id="S7.Thmtheorem8.p1.7.7.m7.1.1.1.1.1.2.cmml" xref="S7.Thmtheorem8.p1.7.7.m7.1.1.1.1.1.2">𝑢</ci><ci id="S7.Thmtheorem8.p1.7.7.m7.1.1.1.1.1.3.cmml" xref="S7.Thmtheorem8.p1.7.7.m7.1.1.1.1.1.3">𝑣</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.Thmtheorem8.p1.7.7.m7.1c">\textsl{g}(u\!\to\!v)</annotation><annotation encoding="application/x-llamapun" id="S7.Thmtheorem8.p1.7.7.m7.1d">g ( italic_u → italic_v )</annotation></semantics></math> such that <math alttext="\textsl{g}(u\!\to\!v)=0" class="ltx_Math" display="inline" id="S7.Thmtheorem8.p1.8.8.m8.1"><semantics id="S7.Thmtheorem8.p1.8.8.m8.1a"><mrow id="S7.Thmtheorem8.p1.8.8.m8.1.1" xref="S7.Thmtheorem8.p1.8.8.m8.1.1.cmml"><mrow id="S7.Thmtheorem8.p1.8.8.m8.1.1.1" xref="S7.Thmtheorem8.p1.8.8.m8.1.1.1.cmml"><mtext class="ltx_mathvariant_italic" id="S7.Thmtheorem8.p1.8.8.m8.1.1.1.3" xref="S7.Thmtheorem8.p1.8.8.m8.1.1.1.3a.cmml">g</mtext><mo id="S7.Thmtheorem8.p1.8.8.m8.1.1.1.2" xref="S7.Thmtheorem8.p1.8.8.m8.1.1.1.2.cmml">⁢</mo><mrow id="S7.Thmtheorem8.p1.8.8.m8.1.1.1.1.1" xref="S7.Thmtheorem8.p1.8.8.m8.1.1.1.1.1.1.cmml"><mo id="S7.Thmtheorem8.p1.8.8.m8.1.1.1.1.1.2" stretchy="false" xref="S7.Thmtheorem8.p1.8.8.m8.1.1.1.1.1.1.cmml">(</mo><mrow id="S7.Thmtheorem8.p1.8.8.m8.1.1.1.1.1.1" xref="S7.Thmtheorem8.p1.8.8.m8.1.1.1.1.1.1.cmml"><mi id="S7.Thmtheorem8.p1.8.8.m8.1.1.1.1.1.1.2" xref="S7.Thmtheorem8.p1.8.8.m8.1.1.1.1.1.1.2.cmml">u</mi><mo id="S7.Thmtheorem8.p1.8.8.m8.1.1.1.1.1.1.1" lspace="0.108em" rspace="0.108em" stretchy="false" xref="S7.Thmtheorem8.p1.8.8.m8.1.1.1.1.1.1.1.cmml">→</mo><mi id="S7.Thmtheorem8.p1.8.8.m8.1.1.1.1.1.1.3" xref="S7.Thmtheorem8.p1.8.8.m8.1.1.1.1.1.1.3.cmml">v</mi></mrow><mo id="S7.Thmtheorem8.p1.8.8.m8.1.1.1.1.1.3" stretchy="false" xref="S7.Thmtheorem8.p1.8.8.m8.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S7.Thmtheorem8.p1.8.8.m8.1.1.2" xref="S7.Thmtheorem8.p1.8.8.m8.1.1.2.cmml">=</mo><mn id="S7.Thmtheorem8.p1.8.8.m8.1.1.3" xref="S7.Thmtheorem8.p1.8.8.m8.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S7.Thmtheorem8.p1.8.8.m8.1b"><apply id="S7.Thmtheorem8.p1.8.8.m8.1.1.cmml" xref="S7.Thmtheorem8.p1.8.8.m8.1.1"><eq id="S7.Thmtheorem8.p1.8.8.m8.1.1.2.cmml" xref="S7.Thmtheorem8.p1.8.8.m8.1.1.2"></eq><apply id="S7.Thmtheorem8.p1.8.8.m8.1.1.1.cmml" xref="S7.Thmtheorem8.p1.8.8.m8.1.1.1"><times id="S7.Thmtheorem8.p1.8.8.m8.1.1.1.2.cmml" xref="S7.Thmtheorem8.p1.8.8.m8.1.1.1.2"></times><ci id="S7.Thmtheorem8.p1.8.8.m8.1.1.1.3a.cmml" xref="S7.Thmtheorem8.p1.8.8.m8.1.1.1.3"><mtext class="ltx_mathvariant_italic" id="S7.Thmtheorem8.p1.8.8.m8.1.1.1.3.cmml" xref="S7.Thmtheorem8.p1.8.8.m8.1.1.1.3">g</mtext></ci><apply id="S7.Thmtheorem8.p1.8.8.m8.1.1.1.1.1.1.cmml" xref="S7.Thmtheorem8.p1.8.8.m8.1.1.1.1.1"><ci id="S7.Thmtheorem8.p1.8.8.m8.1.1.1.1.1.1.1.cmml" xref="S7.Thmtheorem8.p1.8.8.m8.1.1.1.1.1.1.1">→</ci><ci id="S7.Thmtheorem8.p1.8.8.m8.1.1.1.1.1.1.2.cmml" xref="S7.Thmtheorem8.p1.8.8.m8.1.1.1.1.1.1.2">𝑢</ci><ci id="S7.Thmtheorem8.p1.8.8.m8.1.1.1.1.1.1.3.cmml" xref="S7.Thmtheorem8.p1.8.8.m8.1.1.1.1.1.1.3">𝑣</ci></apply></apply><cn id="S7.Thmtheorem8.p1.8.8.m8.1.1.3.cmml" type="integer" xref="S7.Thmtheorem8.p1.8.8.m8.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.Thmtheorem8.p1.8.8.m8.1c">\textsl{g}(u\!\to\!v)=0</annotation><annotation encoding="application/x-llamapun" id="S7.Thmtheorem8.p1.8.8.m8.1d">g ( italic_u → italic_v ) = 0</annotation></semantics></math>.</span></p> </div> </div> <section class="ltx_subparagraph" id="S7.SS1.SSS0.P0.SPx1"> <h5 class="ltx_title ltx_title_subparagraph">Algorithm (see also Figure <a class="ltx_ref" href="https://arxiv.org/html/2411.12694v2#S7.F2" title="Figure 2 ‣ Algorithm (see also Figure 2 and Algorithms 2 and 3 and 4 and 5) ‣ 7.1 Algorithm overview ‣ 7 Results in CONGEST ‣ Local Density and its Distributed Approximation"><span class="ltx_text ltx_ref_tag">2</span></a> and Algorithms <a class="ltx_ref" href="https://arxiv.org/html/2411.12694v2#alg2" title="Algorithm 2 ‣ 7.2 Formal algorithm definition. ‣ 7 Results in CONGEST ‣ Local Density and its Distributed Approximation"><span class="ltx_text ltx_ref_tag">2</span></a> and <a class="ltx_ref" href="https://arxiv.org/html/2411.12694v2#alg3" title="Algorithm 3 ‣ 7.2 Formal algorithm definition. ‣ 7 Results in CONGEST ‣ Local Density and its Distributed Approximation"><span class="ltx_text ltx_ref_tag">3</span></a> and <a class="ltx_ref" href="https://arxiv.org/html/2411.12694v2#alg4" title="Algorithm 4 ‣ 7.2 Formal algorithm definition. ‣ 7 Results in CONGEST ‣ Local Density and its Distributed Approximation"><span class="ltx_text ltx_ref_tag">4</span></a> and <a class="ltx_ref" href="https://arxiv.org/html/2411.12694v2#alg5" title="Algorithm 5 ‣ 7.2 Formal algorithm definition. ‣ 7 Results in CONGEST ‣ Local Density and its Distributed Approximation"><span class="ltx_text ltx_ref_tag">5</span></a>)</h5> <div class="ltx_para" id="S7.SS1.SSS0.P0.SPx1.p1"> <p class="ltx_p" id="S7.SS1.SSS0.P0.SPx1.p1.1">Each time frame has a purpose:</p> <ul class="ltx_itemize" id="S7.I4"> <li class="ltx_item" id="S7.I4.i1" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">•</span> <div class="ltx_para" id="S7.I4.i1.p1"> <p class="ltx_p" id="S7.I4.i1.p1.4">Each <span class="ltx_text ltx_font_smallcaps" id="S7.I4.i1.p1.4.1">hour</span> <math alttext="h" class="ltx_Math" display="inline" id="S7.I4.i1.p1.1.m1.1"><semantics id="S7.I4.i1.p1.1.m1.1a"><mi id="S7.I4.i1.p1.1.m1.1.1" xref="S7.I4.i1.p1.1.m1.1.1.cmml">h</mi><annotation-xml encoding="MathML-Content" id="S7.I4.i1.p1.1.m1.1b"><ci id="S7.I4.i1.p1.1.m1.1.1.cmml" xref="S7.I4.i1.p1.1.m1.1.1">ℎ</ci></annotation-xml><annotation encoding="application/x-tex" id="S7.I4.i1.p1.1.m1.1c">h</annotation><annotation encoding="application/x-llamapun" id="S7.I4.i1.p1.1.m1.1d">italic_h</annotation></semantics></math>, the goal is to identify and ‘fix’ all violating out-edges from vertices in <math alttext="L_{h}" class="ltx_Math" display="inline" id="S7.I4.i1.p1.2.m2.1"><semantics id="S7.I4.i1.p1.2.m2.1a"><msub id="S7.I4.i1.p1.2.m2.1.1" xref="S7.I4.i1.p1.2.m2.1.1.cmml"><mi id="S7.I4.i1.p1.2.m2.1.1.2" xref="S7.I4.i1.p1.2.m2.1.1.2.cmml">L</mi><mi id="S7.I4.i1.p1.2.m2.1.1.3" xref="S7.I4.i1.p1.2.m2.1.1.3.cmml">h</mi></msub><annotation-xml encoding="MathML-Content" id="S7.I4.i1.p1.2.m2.1b"><apply id="S7.I4.i1.p1.2.m2.1.1.cmml" xref="S7.I4.i1.p1.2.m2.1.1"><csymbol cd="ambiguous" id="S7.I4.i1.p1.2.m2.1.1.1.cmml" xref="S7.I4.i1.p1.2.m2.1.1">subscript</csymbol><ci id="S7.I4.i1.p1.2.m2.1.1.2.cmml" xref="S7.I4.i1.p1.2.m2.1.1.2">𝐿</ci><ci id="S7.I4.i1.p1.2.m2.1.1.3.cmml" xref="S7.I4.i1.p1.2.m2.1.1.3">ℎ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.I4.i1.p1.2.m2.1c">L_{h}</annotation><annotation encoding="application/x-llamapun" id="S7.I4.i1.p1.2.m2.1d">italic_L start_POSTSUBSCRIPT italic_h end_POSTSUBSCRIPT</annotation></semantics></math>; without introducing violating out-edges from vertices in a level <math alttext="L_{k}" class="ltx_Math" display="inline" id="S7.I4.i1.p1.3.m3.1"><semantics id="S7.I4.i1.p1.3.m3.1a"><msub id="S7.I4.i1.p1.3.m3.1.1" xref="S7.I4.i1.p1.3.m3.1.1.cmml"><mi id="S7.I4.i1.p1.3.m3.1.1.2" xref="S7.I4.i1.p1.3.m3.1.1.2.cmml">L</mi><mi id="S7.I4.i1.p1.3.m3.1.1.3" xref="S7.I4.i1.p1.3.m3.1.1.3.cmml">k</mi></msub><annotation-xml encoding="MathML-Content" id="S7.I4.i1.p1.3.m3.1b"><apply id="S7.I4.i1.p1.3.m3.1.1.cmml" xref="S7.I4.i1.p1.3.m3.1.1"><csymbol cd="ambiguous" id="S7.I4.i1.p1.3.m3.1.1.1.cmml" xref="S7.I4.i1.p1.3.m3.1.1">subscript</csymbol><ci id="S7.I4.i1.p1.3.m3.1.1.2.cmml" xref="S7.I4.i1.p1.3.m3.1.1.2">𝐿</ci><ci id="S7.I4.i1.p1.3.m3.1.1.3.cmml" xref="S7.I4.i1.p1.3.m3.1.1.3">𝑘</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.I4.i1.p1.3.m3.1c">L_{k}</annotation><annotation encoding="application/x-llamapun" id="S7.I4.i1.p1.3.m3.1d">italic_L start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT</annotation></semantics></math> with <math alttext="k>h" class="ltx_Math" display="inline" id="S7.I4.i1.p1.4.m4.1"><semantics id="S7.I4.i1.p1.4.m4.1a"><mrow id="S7.I4.i1.p1.4.m4.1.1" xref="S7.I4.i1.p1.4.m4.1.1.cmml"><mi id="S7.I4.i1.p1.4.m4.1.1.2" xref="S7.I4.i1.p1.4.m4.1.1.2.cmml">k</mi><mo id="S7.I4.i1.p1.4.m4.1.1.1" xref="S7.I4.i1.p1.4.m4.1.1.1.cmml">></mo><mi id="S7.I4.i1.p1.4.m4.1.1.3" xref="S7.I4.i1.p1.4.m4.1.1.3.cmml">h</mi></mrow><annotation-xml encoding="MathML-Content" id="S7.I4.i1.p1.4.m4.1b"><apply id="S7.I4.i1.p1.4.m4.1.1.cmml" xref="S7.I4.i1.p1.4.m4.1.1"><gt id="S7.I4.i1.p1.4.m4.1.1.1.cmml" xref="S7.I4.i1.p1.4.m4.1.1.1"></gt><ci id="S7.I4.i1.p1.4.m4.1.1.2.cmml" xref="S7.I4.i1.p1.4.m4.1.1.2">𝑘</ci><ci id="S7.I4.i1.p1.4.m4.1.1.3.cmml" xref="S7.I4.i1.p1.4.m4.1.1.3">ℎ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.I4.i1.p1.4.m4.1c">k>h</annotation><annotation encoding="application/x-llamapun" id="S7.I4.i1.p1.4.m4.1d">italic_k > italic_h</annotation></semantics></math>. We do this iteratively, using two different steps:</p> <ul class="ltx_itemize" id="S7.I4.i1.I1"> <li class="ltx_item" id="S7.I4.i1.I1.i1" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item"><span class="ltx_text ltx_font_bold" id="S7.I4.i1.I1.i1.1.1.1">–</span></span> <div class="ltx_para" id="S7.I4.i1.I1.i1.p1"> <p class="ltx_p" id="S7.I4.i1.I1.i1.p1.3">Each even <span class="ltx_text ltx_font_smallcaps" id="S7.I4.i1.I1.i1.p1.3.1">minute</span>, we fix all violating out-edges from vertices in <math alttext="L_{h}" class="ltx_Math" display="inline" id="S7.I4.i1.I1.i1.p1.1.m1.1"><semantics id="S7.I4.i1.I1.i1.p1.1.m1.1a"><msub id="S7.I4.i1.I1.i1.p1.1.m1.1.1" xref="S7.I4.i1.I1.i1.p1.1.m1.1.1.cmml"><mi id="S7.I4.i1.I1.i1.p1.1.m1.1.1.2" xref="S7.I4.i1.I1.i1.p1.1.m1.1.1.2.cmml">L</mi><mi id="S7.I4.i1.I1.i1.p1.1.m1.1.1.3" xref="S7.I4.i1.I1.i1.p1.1.m1.1.1.3.cmml">h</mi></msub><annotation-xml encoding="MathML-Content" id="S7.I4.i1.I1.i1.p1.1.m1.1b"><apply id="S7.I4.i1.I1.i1.p1.1.m1.1.1.cmml" xref="S7.I4.i1.I1.i1.p1.1.m1.1.1"><csymbol cd="ambiguous" id="S7.I4.i1.I1.i1.p1.1.m1.1.1.1.cmml" xref="S7.I4.i1.I1.i1.p1.1.m1.1.1">subscript</csymbol><ci id="S7.I4.i1.I1.i1.p1.1.m1.1.1.2.cmml" xref="S7.I4.i1.I1.i1.p1.1.m1.1.1.2">𝐿</ci><ci id="S7.I4.i1.I1.i1.p1.1.m1.1.1.3.cmml" xref="S7.I4.i1.I1.i1.p1.1.m1.1.1.3">ℎ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.I4.i1.I1.i1.p1.1.m1.1c">L_{h}</annotation><annotation encoding="application/x-llamapun" id="S7.I4.i1.I1.i1.p1.1.m1.1d">italic_L start_POSTSUBSCRIPT italic_h end_POSTSUBSCRIPT</annotation></semantics></math>, possibly creating violating out-edges from vertices in <math alttext="L_{h+1}" class="ltx_Math" display="inline" id="S7.I4.i1.I1.i1.p1.2.m2.1"><semantics id="S7.I4.i1.I1.i1.p1.2.m2.1a"><msub id="S7.I4.i1.I1.i1.p1.2.m2.1.1" xref="S7.I4.i1.I1.i1.p1.2.m2.1.1.cmml"><mi id="S7.I4.i1.I1.i1.p1.2.m2.1.1.2" xref="S7.I4.i1.I1.i1.p1.2.m2.1.1.2.cmml">L</mi><mrow id="S7.I4.i1.I1.i1.p1.2.m2.1.1.3" xref="S7.I4.i1.I1.i1.p1.2.m2.1.1.3.cmml"><mi id="S7.I4.i1.I1.i1.p1.2.m2.1.1.3.2" xref="S7.I4.i1.I1.i1.p1.2.m2.1.1.3.2.cmml">h</mi><mo id="S7.I4.i1.I1.i1.p1.2.m2.1.1.3.1" xref="S7.I4.i1.I1.i1.p1.2.m2.1.1.3.1.cmml">+</mo><mn id="S7.I4.i1.I1.i1.p1.2.m2.1.1.3.3" xref="S7.I4.i1.I1.i1.p1.2.m2.1.1.3.3.cmml">1</mn></mrow></msub><annotation-xml encoding="MathML-Content" id="S7.I4.i1.I1.i1.p1.2.m2.1b"><apply id="S7.I4.i1.I1.i1.p1.2.m2.1.1.cmml" xref="S7.I4.i1.I1.i1.p1.2.m2.1.1"><csymbol cd="ambiguous" id="S7.I4.i1.I1.i1.p1.2.m2.1.1.1.cmml" xref="S7.I4.i1.I1.i1.p1.2.m2.1.1">subscript</csymbol><ci id="S7.I4.i1.I1.i1.p1.2.m2.1.1.2.cmml" xref="S7.I4.i1.I1.i1.p1.2.m2.1.1.2">𝐿</ci><apply id="S7.I4.i1.I1.i1.p1.2.m2.1.1.3.cmml" xref="S7.I4.i1.I1.i1.p1.2.m2.1.1.3"><plus id="S7.I4.i1.I1.i1.p1.2.m2.1.1.3.1.cmml" xref="S7.I4.i1.I1.i1.p1.2.m2.1.1.3.1"></plus><ci id="S7.I4.i1.I1.i1.p1.2.m2.1.1.3.2.cmml" xref="S7.I4.i1.I1.i1.p1.2.m2.1.1.3.2">ℎ</ci><cn id="S7.I4.i1.I1.i1.p1.2.m2.1.1.3.3.cmml" type="integer" xref="S7.I4.i1.I1.i1.p1.2.m2.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.I4.i1.I1.i1.p1.2.m2.1c">L_{h+1}</annotation><annotation encoding="application/x-llamapun" id="S7.I4.i1.I1.i1.p1.2.m2.1d">italic_L start_POSTSUBSCRIPT italic_h + 1 end_POSTSUBSCRIPT</annotation></semantics></math> to vetices in <math alttext="L_{h-1}" class="ltx_Math" display="inline" id="S7.I4.i1.I1.i1.p1.3.m3.1"><semantics id="S7.I4.i1.I1.i1.p1.3.m3.1a"><msub id="S7.I4.i1.I1.i1.p1.3.m3.1.1" xref="S7.I4.i1.I1.i1.p1.3.m3.1.1.cmml"><mi id="S7.I4.i1.I1.i1.p1.3.m3.1.1.2" xref="S7.I4.i1.I1.i1.p1.3.m3.1.1.2.cmml">L</mi><mrow id="S7.I4.i1.I1.i1.p1.3.m3.1.1.3" xref="S7.I4.i1.I1.i1.p1.3.m3.1.1.3.cmml"><mi id="S7.I4.i1.I1.i1.p1.3.m3.1.1.3.2" xref="S7.I4.i1.I1.i1.p1.3.m3.1.1.3.2.cmml">h</mi><mo id="S7.I4.i1.I1.i1.p1.3.m3.1.1.3.1" xref="S7.I4.i1.I1.i1.p1.3.m3.1.1.3.1.cmml">−</mo><mn id="S7.I4.i1.I1.i1.p1.3.m3.1.1.3.3" xref="S7.I4.i1.I1.i1.p1.3.m3.1.1.3.3.cmml">1</mn></mrow></msub><annotation-xml encoding="MathML-Content" id="S7.I4.i1.I1.i1.p1.3.m3.1b"><apply id="S7.I4.i1.I1.i1.p1.3.m3.1.1.cmml" xref="S7.I4.i1.I1.i1.p1.3.m3.1.1"><csymbol cd="ambiguous" id="S7.I4.i1.I1.i1.p1.3.m3.1.1.1.cmml" xref="S7.I4.i1.I1.i1.p1.3.m3.1.1">subscript</csymbol><ci id="S7.I4.i1.I1.i1.p1.3.m3.1.1.2.cmml" xref="S7.I4.i1.I1.i1.p1.3.m3.1.1.2">𝐿</ci><apply id="S7.I4.i1.I1.i1.p1.3.m3.1.1.3.cmml" xref="S7.I4.i1.I1.i1.p1.3.m3.1.1.3"><minus id="S7.I4.i1.I1.i1.p1.3.m3.1.1.3.1.cmml" xref="S7.I4.i1.I1.i1.p1.3.m3.1.1.3.1"></minus><ci id="S7.I4.i1.I1.i1.p1.3.m3.1.1.3.2.cmml" xref="S7.I4.i1.I1.i1.p1.3.m3.1.1.3.2">ℎ</ci><cn id="S7.I4.i1.I1.i1.p1.3.m3.1.1.3.3.cmml" type="integer" xref="S7.I4.i1.I1.i1.p1.3.m3.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.I4.i1.I1.i1.p1.3.m3.1c">L_{h-1}</annotation><annotation encoding="application/x-llamapun" id="S7.I4.i1.I1.i1.p1.3.m3.1d">italic_L start_POSTSUBSCRIPT italic_h - 1 end_POSTSUBSCRIPT</annotation></semantics></math>.</p> </div> </li> <li class="ltx_item" id="S7.I4.i1.I1.i2" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item"><span class="ltx_text ltx_font_bold" id="S7.I4.i1.I1.i2.1.1.1">–</span></span> <div class="ltx_para" id="S7.I4.i1.I1.i2.p1"> <p class="ltx_p" id="S7.I4.i1.I1.i2.p1.5">Each odd <span class="ltx_text ltx_font_smallcaps" id="S7.I4.i1.I1.i2.p1.5.1">minute</span>, we fix all violating out-edges from vertices in <math alttext="L_{h+1}" class="ltx_Math" display="inline" id="S7.I4.i1.I1.i2.p1.1.m1.1"><semantics id="S7.I4.i1.I1.i2.p1.1.m1.1a"><msub id="S7.I4.i1.I1.i2.p1.1.m1.1.1" xref="S7.I4.i1.I1.i2.p1.1.m1.1.1.cmml"><mi id="S7.I4.i1.I1.i2.p1.1.m1.1.1.2" xref="S7.I4.i1.I1.i2.p1.1.m1.1.1.2.cmml">L</mi><mrow id="S7.I4.i1.I1.i2.p1.1.m1.1.1.3" xref="S7.I4.i1.I1.i2.p1.1.m1.1.1.3.cmml"><mi id="S7.I4.i1.I1.i2.p1.1.m1.1.1.3.2" xref="S7.I4.i1.I1.i2.p1.1.m1.1.1.3.2.cmml">h</mi><mo id="S7.I4.i1.I1.i2.p1.1.m1.1.1.3.1" xref="S7.I4.i1.I1.i2.p1.1.m1.1.1.3.1.cmml">+</mo><mn id="S7.I4.i1.I1.i2.p1.1.m1.1.1.3.3" xref="S7.I4.i1.I1.i2.p1.1.m1.1.1.3.3.cmml">1</mn></mrow></msub><annotation-xml encoding="MathML-Content" id="S7.I4.i1.I1.i2.p1.1.m1.1b"><apply id="S7.I4.i1.I1.i2.p1.1.m1.1.1.cmml" xref="S7.I4.i1.I1.i2.p1.1.m1.1.1"><csymbol cd="ambiguous" id="S7.I4.i1.I1.i2.p1.1.m1.1.1.1.cmml" xref="S7.I4.i1.I1.i2.p1.1.m1.1.1">subscript</csymbol><ci id="S7.I4.i1.I1.i2.p1.1.m1.1.1.2.cmml" xref="S7.I4.i1.I1.i2.p1.1.m1.1.1.2">𝐿</ci><apply id="S7.I4.i1.I1.i2.p1.1.m1.1.1.3.cmml" xref="S7.I4.i1.I1.i2.p1.1.m1.1.1.3"><plus id="S7.I4.i1.I1.i2.p1.1.m1.1.1.3.1.cmml" xref="S7.I4.i1.I1.i2.p1.1.m1.1.1.3.1"></plus><ci id="S7.I4.i1.I1.i2.p1.1.m1.1.1.3.2.cmml" xref="S7.I4.i1.I1.i2.p1.1.m1.1.1.3.2">ℎ</ci><cn id="S7.I4.i1.I1.i2.p1.1.m1.1.1.3.3.cmml" type="integer" xref="S7.I4.i1.I1.i2.p1.1.m1.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.I4.i1.I1.i2.p1.1.m1.1c">L_{h+1}</annotation><annotation encoding="application/x-llamapun" id="S7.I4.i1.I1.i2.p1.1.m1.1d">italic_L start_POSTSUBSCRIPT italic_h + 1 end_POSTSUBSCRIPT</annotation></semantics></math> to vertices in <math alttext="L_{h-1}" class="ltx_Math" display="inline" id="S7.I4.i1.I1.i2.p1.2.m2.1"><semantics id="S7.I4.i1.I1.i2.p1.2.m2.1a"><msub id="S7.I4.i1.I1.i2.p1.2.m2.1.1" xref="S7.I4.i1.I1.i2.p1.2.m2.1.1.cmml"><mi id="S7.I4.i1.I1.i2.p1.2.m2.1.1.2" xref="S7.I4.i1.I1.i2.p1.2.m2.1.1.2.cmml">L</mi><mrow id="S7.I4.i1.I1.i2.p1.2.m2.1.1.3" xref="S7.I4.i1.I1.i2.p1.2.m2.1.1.3.cmml"><mi id="S7.I4.i1.I1.i2.p1.2.m2.1.1.3.2" xref="S7.I4.i1.I1.i2.p1.2.m2.1.1.3.2.cmml">h</mi><mo id="S7.I4.i1.I1.i2.p1.2.m2.1.1.3.1" xref="S7.I4.i1.I1.i2.p1.2.m2.1.1.3.1.cmml">−</mo><mn id="S7.I4.i1.I1.i2.p1.2.m2.1.1.3.3" xref="S7.I4.i1.I1.i2.p1.2.m2.1.1.3.3.cmml">1</mn></mrow></msub><annotation-xml encoding="MathML-Content" id="S7.I4.i1.I1.i2.p1.2.m2.1b"><apply id="S7.I4.i1.I1.i2.p1.2.m2.1.1.cmml" xref="S7.I4.i1.I1.i2.p1.2.m2.1.1"><csymbol cd="ambiguous" id="S7.I4.i1.I1.i2.p1.2.m2.1.1.1.cmml" xref="S7.I4.i1.I1.i2.p1.2.m2.1.1">subscript</csymbol><ci id="S7.I4.i1.I1.i2.p1.2.m2.1.1.2.cmml" xref="S7.I4.i1.I1.i2.p1.2.m2.1.1.2">𝐿</ci><apply id="S7.I4.i1.I1.i2.p1.2.m2.1.1.3.cmml" xref="S7.I4.i1.I1.i2.p1.2.m2.1.1.3"><minus id="S7.I4.i1.I1.i2.p1.2.m2.1.1.3.1.cmml" xref="S7.I4.i1.I1.i2.p1.2.m2.1.1.3.1"></minus><ci id="S7.I4.i1.I1.i2.p1.2.m2.1.1.3.2.cmml" xref="S7.I4.i1.I1.i2.p1.2.m2.1.1.3.2">ℎ</ci><cn id="S7.I4.i1.I1.i2.p1.2.m2.1.1.3.3.cmml" type="integer" xref="S7.I4.i1.I1.i2.p1.2.m2.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.I4.i1.I1.i2.p1.2.m2.1c">L_{h-1}</annotation><annotation encoding="application/x-llamapun" id="S7.I4.i1.I1.i2.p1.2.m2.1d">italic_L start_POSTSUBSCRIPT italic_h - 1 end_POSTSUBSCRIPT</annotation></semantics></math>. We do this in such a manner, that we create no new violating out-edges from vertices in <math alttext="L_{k}" class="ltx_Math" display="inline" id="S7.I4.i1.I1.i2.p1.3.m3.1"><semantics id="S7.I4.i1.I1.i2.p1.3.m3.1a"><msub id="S7.I4.i1.I1.i2.p1.3.m3.1.1" xref="S7.I4.i1.I1.i2.p1.3.m3.1.1.cmml"><mi id="S7.I4.i1.I1.i2.p1.3.m3.1.1.2" xref="S7.I4.i1.I1.i2.p1.3.m3.1.1.2.cmml">L</mi><mi id="S7.I4.i1.I1.i2.p1.3.m3.1.1.3" xref="S7.I4.i1.I1.i2.p1.3.m3.1.1.3.cmml">k</mi></msub><annotation-xml encoding="MathML-Content" id="S7.I4.i1.I1.i2.p1.3.m3.1b"><apply id="S7.I4.i1.I1.i2.p1.3.m3.1.1.cmml" xref="S7.I4.i1.I1.i2.p1.3.m3.1.1"><csymbol cd="ambiguous" id="S7.I4.i1.I1.i2.p1.3.m3.1.1.1.cmml" xref="S7.I4.i1.I1.i2.p1.3.m3.1.1">subscript</csymbol><ci id="S7.I4.i1.I1.i2.p1.3.m3.1.1.2.cmml" xref="S7.I4.i1.I1.i2.p1.3.m3.1.1.2">𝐿</ci><ci id="S7.I4.i1.I1.i2.p1.3.m3.1.1.3.cmml" xref="S7.I4.i1.I1.i2.p1.3.m3.1.1.3">𝑘</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.I4.i1.I1.i2.p1.3.m3.1c">L_{k}</annotation><annotation encoding="application/x-llamapun" id="S7.I4.i1.I1.i2.p1.3.m3.1d">italic_L start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT</annotation></semantics></math> for <math alttext="k>h" class="ltx_Math" display="inline" id="S7.I4.i1.I1.i2.p1.4.m4.1"><semantics id="S7.I4.i1.I1.i2.p1.4.m4.1a"><mrow id="S7.I4.i1.I1.i2.p1.4.m4.1.1" xref="S7.I4.i1.I1.i2.p1.4.m4.1.1.cmml"><mi id="S7.I4.i1.I1.i2.p1.4.m4.1.1.2" xref="S7.I4.i1.I1.i2.p1.4.m4.1.1.2.cmml">k</mi><mo id="S7.I4.i1.I1.i2.p1.4.m4.1.1.1" xref="S7.I4.i1.I1.i2.p1.4.m4.1.1.1.cmml">></mo><mi id="S7.I4.i1.I1.i2.p1.4.m4.1.1.3" xref="S7.I4.i1.I1.i2.p1.4.m4.1.1.3.cmml">h</mi></mrow><annotation-xml encoding="MathML-Content" id="S7.I4.i1.I1.i2.p1.4.m4.1b"><apply id="S7.I4.i1.I1.i2.p1.4.m4.1.1.cmml" xref="S7.I4.i1.I1.i2.p1.4.m4.1.1"><gt id="S7.I4.i1.I1.i2.p1.4.m4.1.1.1.cmml" xref="S7.I4.i1.I1.i2.p1.4.m4.1.1.1"></gt><ci id="S7.I4.i1.I1.i2.p1.4.m4.1.1.2.cmml" xref="S7.I4.i1.I1.i2.p1.4.m4.1.1.2">𝑘</ci><ci id="S7.I4.i1.I1.i2.p1.4.m4.1.1.3.cmml" xref="S7.I4.i1.I1.i2.p1.4.m4.1.1.3">ℎ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.I4.i1.I1.i2.p1.4.m4.1c">k>h</annotation><annotation encoding="application/x-llamapun" id="S7.I4.i1.I1.i2.p1.4.m4.1d">italic_k > italic_h</annotation></semantics></math>. However, we may create violating out-edges from level <math alttext="L_{h}" class="ltx_Math" display="inline" id="S7.I4.i1.I1.i2.p1.5.m5.1"><semantics id="S7.I4.i1.I1.i2.p1.5.m5.1a"><msub id="S7.I4.i1.I1.i2.p1.5.m5.1.1" xref="S7.I4.i1.I1.i2.p1.5.m5.1.1.cmml"><mi id="S7.I4.i1.I1.i2.p1.5.m5.1.1.2" xref="S7.I4.i1.I1.i2.p1.5.m5.1.1.2.cmml">L</mi><mi id="S7.I4.i1.I1.i2.p1.5.m5.1.1.3" xref="S7.I4.i1.I1.i2.p1.5.m5.1.1.3.cmml">h</mi></msub><annotation-xml encoding="MathML-Content" id="S7.I4.i1.I1.i2.p1.5.m5.1b"><apply id="S7.I4.i1.I1.i2.p1.5.m5.1.1.cmml" xref="S7.I4.i1.I1.i2.p1.5.m5.1.1"><csymbol cd="ambiguous" id="S7.I4.i1.I1.i2.p1.5.m5.1.1.1.cmml" xref="S7.I4.i1.I1.i2.p1.5.m5.1.1">subscript</csymbol><ci id="S7.I4.i1.I1.i2.p1.5.m5.1.1.2.cmml" xref="S7.I4.i1.I1.i2.p1.5.m5.1.1.2">𝐿</ci><ci id="S7.I4.i1.I1.i2.p1.5.m5.1.1.3.cmml" xref="S7.I4.i1.I1.i2.p1.5.m5.1.1.3">ℎ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.I4.i1.I1.i2.p1.5.m5.1c">L_{h}</annotation><annotation encoding="application/x-llamapun" id="S7.I4.i1.I1.i2.p1.5.m5.1d">italic_L start_POSTSUBSCRIPT italic_h end_POSTSUBSCRIPT</annotation></semantics></math>.</p> </div> </li> </ul> </div> </li> <li class="ltx_item" id="S7.I4.i2" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">•</span> <div class="ltx_para" id="S7.I4.i2.p1"> <p class="ltx_p" id="S7.I4.i2.p1.5">Each even <span class="ltx_text ltx_font_smallcaps" id="S7.I4.i2.p1.5.1">minute</span> <math alttext="m^{\prime}" class="ltx_Math" display="inline" id="S7.I4.i2.p1.1.m1.1"><semantics id="S7.I4.i2.p1.1.m1.1a"><msup id="S7.I4.i2.p1.1.m1.1.1" xref="S7.I4.i2.p1.1.m1.1.1.cmml"><mi id="S7.I4.i2.p1.1.m1.1.1.2" xref="S7.I4.i2.p1.1.m1.1.1.2.cmml">m</mi><mo id="S7.I4.i2.p1.1.m1.1.1.3" xref="S7.I4.i2.p1.1.m1.1.1.3.cmml">′</mo></msup><annotation-xml encoding="MathML-Content" id="S7.I4.i2.p1.1.m1.1b"><apply id="S7.I4.i2.p1.1.m1.1.1.cmml" xref="S7.I4.i2.p1.1.m1.1.1"><csymbol cd="ambiguous" id="S7.I4.i2.p1.1.m1.1.1.1.cmml" xref="S7.I4.i2.p1.1.m1.1.1">superscript</csymbol><ci id="S7.I4.i2.p1.1.m1.1.1.2.cmml" xref="S7.I4.i2.p1.1.m1.1.1.2">𝑚</ci><ci id="S7.I4.i2.p1.1.m1.1.1.3.cmml" xref="S7.I4.i2.p1.1.m1.1.1.3">′</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.I4.i2.p1.1.m1.1c">m^{\prime}</annotation><annotation encoding="application/x-llamapun" id="S7.I4.i2.p1.1.m1.1d">italic_m start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math>, we define the set <math alttext="V_{m^{\prime}}\subset L_{h}" class="ltx_Math" display="inline" id="S7.I4.i2.p1.2.m2.1"><semantics id="S7.I4.i2.p1.2.m2.1a"><mrow id="S7.I4.i2.p1.2.m2.1.1" xref="S7.I4.i2.p1.2.m2.1.1.cmml"><msub id="S7.I4.i2.p1.2.m2.1.1.2" xref="S7.I4.i2.p1.2.m2.1.1.2.cmml"><mi id="S7.I4.i2.p1.2.m2.1.1.2.2" xref="S7.I4.i2.p1.2.m2.1.1.2.2.cmml">V</mi><msup id="S7.I4.i2.p1.2.m2.1.1.2.3" xref="S7.I4.i2.p1.2.m2.1.1.2.3.cmml"><mi id="S7.I4.i2.p1.2.m2.1.1.2.3.2" xref="S7.I4.i2.p1.2.m2.1.1.2.3.2.cmml">m</mi><mo id="S7.I4.i2.p1.2.m2.1.1.2.3.3" xref="S7.I4.i2.p1.2.m2.1.1.2.3.3.cmml">′</mo></msup></msub><mo id="S7.I4.i2.p1.2.m2.1.1.1" xref="S7.I4.i2.p1.2.m2.1.1.1.cmml">⊂</mo><msub id="S7.I4.i2.p1.2.m2.1.1.3" xref="S7.I4.i2.p1.2.m2.1.1.3.cmml"><mi id="S7.I4.i2.p1.2.m2.1.1.3.2" xref="S7.I4.i2.p1.2.m2.1.1.3.2.cmml">L</mi><mi id="S7.I4.i2.p1.2.m2.1.1.3.3" xref="S7.I4.i2.p1.2.m2.1.1.3.3.cmml">h</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S7.I4.i2.p1.2.m2.1b"><apply id="S7.I4.i2.p1.2.m2.1.1.cmml" xref="S7.I4.i2.p1.2.m2.1.1"><subset id="S7.I4.i2.p1.2.m2.1.1.1.cmml" xref="S7.I4.i2.p1.2.m2.1.1.1"></subset><apply id="S7.I4.i2.p1.2.m2.1.1.2.cmml" xref="S7.I4.i2.p1.2.m2.1.1.2"><csymbol cd="ambiguous" id="S7.I4.i2.p1.2.m2.1.1.2.1.cmml" xref="S7.I4.i2.p1.2.m2.1.1.2">subscript</csymbol><ci id="S7.I4.i2.p1.2.m2.1.1.2.2.cmml" xref="S7.I4.i2.p1.2.m2.1.1.2.2">𝑉</ci><apply id="S7.I4.i2.p1.2.m2.1.1.2.3.cmml" xref="S7.I4.i2.p1.2.m2.1.1.2.3"><csymbol cd="ambiguous" id="S7.I4.i2.p1.2.m2.1.1.2.3.1.cmml" xref="S7.I4.i2.p1.2.m2.1.1.2.3">superscript</csymbol><ci id="S7.I4.i2.p1.2.m2.1.1.2.3.2.cmml" xref="S7.I4.i2.p1.2.m2.1.1.2.3.2">𝑚</ci><ci id="S7.I4.i2.p1.2.m2.1.1.2.3.3.cmml" xref="S7.I4.i2.p1.2.m2.1.1.2.3.3">′</ci></apply></apply><apply id="S7.I4.i2.p1.2.m2.1.1.3.cmml" xref="S7.I4.i2.p1.2.m2.1.1.3"><csymbol cd="ambiguous" id="S7.I4.i2.p1.2.m2.1.1.3.1.cmml" xref="S7.I4.i2.p1.2.m2.1.1.3">subscript</csymbol><ci id="S7.I4.i2.p1.2.m2.1.1.3.2.cmml" xref="S7.I4.i2.p1.2.m2.1.1.3.2">𝐿</ci><ci id="S7.I4.i2.p1.2.m2.1.1.3.3.cmml" xref="S7.I4.i2.p1.2.m2.1.1.3.3">ℎ</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.I4.i2.p1.2.m2.1c">V_{m^{\prime}}\subset L_{h}</annotation><annotation encoding="application/x-llamapun" id="S7.I4.i2.p1.2.m2.1d">italic_V start_POSTSUBSCRIPT italic_m start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT ⊂ italic_L start_POSTSUBSCRIPT italic_h end_POSTSUBSCRIPT</annotation></semantics></math> of vertices that have at least one violating out-edge. Over <math alttext="4\lceil\log_{8/7}n\rceil+2" class="ltx_Math" display="inline" id="S7.I4.i2.p1.3.m3.1"><semantics id="S7.I4.i2.p1.3.m3.1a"><mrow id="S7.I4.i2.p1.3.m3.1.1" xref="S7.I4.i2.p1.3.m3.1.1.cmml"><mrow id="S7.I4.i2.p1.3.m3.1.1.1" xref="S7.I4.i2.p1.3.m3.1.1.1.cmml"><mn id="S7.I4.i2.p1.3.m3.1.1.1.3" xref="S7.I4.i2.p1.3.m3.1.1.1.3.cmml">4</mn><mo id="S7.I4.i2.p1.3.m3.1.1.1.2" xref="S7.I4.i2.p1.3.m3.1.1.1.2.cmml">⁢</mo><mrow id="S7.I4.i2.p1.3.m3.1.1.1.1.1" xref="S7.I4.i2.p1.3.m3.1.1.1.1.2.cmml"><mo id="S7.I4.i2.p1.3.m3.1.1.1.1.1.2" stretchy="false" xref="S7.I4.i2.p1.3.m3.1.1.1.1.2.1.cmml">⌈</mo><mrow id="S7.I4.i2.p1.3.m3.1.1.1.1.1.1" xref="S7.I4.i2.p1.3.m3.1.1.1.1.1.1.cmml"><msub id="S7.I4.i2.p1.3.m3.1.1.1.1.1.1.1" xref="S7.I4.i2.p1.3.m3.1.1.1.1.1.1.1.cmml"><mi id="S7.I4.i2.p1.3.m3.1.1.1.1.1.1.1.2" xref="S7.I4.i2.p1.3.m3.1.1.1.1.1.1.1.2.cmml">log</mi><mrow id="S7.I4.i2.p1.3.m3.1.1.1.1.1.1.1.3" xref="S7.I4.i2.p1.3.m3.1.1.1.1.1.1.1.3.cmml"><mn id="S7.I4.i2.p1.3.m3.1.1.1.1.1.1.1.3.2" xref="S7.I4.i2.p1.3.m3.1.1.1.1.1.1.1.3.2.cmml">8</mn><mo id="S7.I4.i2.p1.3.m3.1.1.1.1.1.1.1.3.1" xref="S7.I4.i2.p1.3.m3.1.1.1.1.1.1.1.3.1.cmml">/</mo><mn id="S7.I4.i2.p1.3.m3.1.1.1.1.1.1.1.3.3" xref="S7.I4.i2.p1.3.m3.1.1.1.1.1.1.1.3.3.cmml">7</mn></mrow></msub><mo id="S7.I4.i2.p1.3.m3.1.1.1.1.1.1a" lspace="0.167em" xref="S7.I4.i2.p1.3.m3.1.1.1.1.1.1.cmml">⁡</mo><mi id="S7.I4.i2.p1.3.m3.1.1.1.1.1.1.2" xref="S7.I4.i2.p1.3.m3.1.1.1.1.1.1.2.cmml">n</mi></mrow><mo id="S7.I4.i2.p1.3.m3.1.1.1.1.1.3" stretchy="false" xref="S7.I4.i2.p1.3.m3.1.1.1.1.2.1.cmml">⌉</mo></mrow></mrow><mo id="S7.I4.i2.p1.3.m3.1.1.2" xref="S7.I4.i2.p1.3.m3.1.1.2.cmml">+</mo><mn id="S7.I4.i2.p1.3.m3.1.1.3" xref="S7.I4.i2.p1.3.m3.1.1.3.cmml">2</mn></mrow><annotation-xml encoding="MathML-Content" id="S7.I4.i2.p1.3.m3.1b"><apply id="S7.I4.i2.p1.3.m3.1.1.cmml" xref="S7.I4.i2.p1.3.m3.1.1"><plus id="S7.I4.i2.p1.3.m3.1.1.2.cmml" xref="S7.I4.i2.p1.3.m3.1.1.2"></plus><apply id="S7.I4.i2.p1.3.m3.1.1.1.cmml" xref="S7.I4.i2.p1.3.m3.1.1.1"><times id="S7.I4.i2.p1.3.m3.1.1.1.2.cmml" xref="S7.I4.i2.p1.3.m3.1.1.1.2"></times><cn id="S7.I4.i2.p1.3.m3.1.1.1.3.cmml" type="integer" xref="S7.I4.i2.p1.3.m3.1.1.1.3">4</cn><apply id="S7.I4.i2.p1.3.m3.1.1.1.1.2.cmml" xref="S7.I4.i2.p1.3.m3.1.1.1.1.1"><ceiling id="S7.I4.i2.p1.3.m3.1.1.1.1.2.1.cmml" xref="S7.I4.i2.p1.3.m3.1.1.1.1.1.2"></ceiling><apply id="S7.I4.i2.p1.3.m3.1.1.1.1.1.1.cmml" xref="S7.I4.i2.p1.3.m3.1.1.1.1.1.1"><apply id="S7.I4.i2.p1.3.m3.1.1.1.1.1.1.1.cmml" xref="S7.I4.i2.p1.3.m3.1.1.1.1.1.1.1"><csymbol cd="ambiguous" id="S7.I4.i2.p1.3.m3.1.1.1.1.1.1.1.1.cmml" xref="S7.I4.i2.p1.3.m3.1.1.1.1.1.1.1">subscript</csymbol><log id="S7.I4.i2.p1.3.m3.1.1.1.1.1.1.1.2.cmml" xref="S7.I4.i2.p1.3.m3.1.1.1.1.1.1.1.2"></log><apply id="S7.I4.i2.p1.3.m3.1.1.1.1.1.1.1.3.cmml" xref="S7.I4.i2.p1.3.m3.1.1.1.1.1.1.1.3"><divide id="S7.I4.i2.p1.3.m3.1.1.1.1.1.1.1.3.1.cmml" xref="S7.I4.i2.p1.3.m3.1.1.1.1.1.1.1.3.1"></divide><cn id="S7.I4.i2.p1.3.m3.1.1.1.1.1.1.1.3.2.cmml" type="integer" xref="S7.I4.i2.p1.3.m3.1.1.1.1.1.1.1.3.2">8</cn><cn id="S7.I4.i2.p1.3.m3.1.1.1.1.1.1.1.3.3.cmml" type="integer" xref="S7.I4.i2.p1.3.m3.1.1.1.1.1.1.1.3.3">7</cn></apply></apply><ci id="S7.I4.i2.p1.3.m3.1.1.1.1.1.1.2.cmml" xref="S7.I4.i2.p1.3.m3.1.1.1.1.1.1.2">𝑛</ci></apply></apply></apply><cn id="S7.I4.i2.p1.3.m3.1.1.3.cmml" type="integer" xref="S7.I4.i2.p1.3.m3.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.I4.i2.p1.3.m3.1c">4\lceil\log_{8/7}n\rceil+2</annotation><annotation encoding="application/x-llamapun" id="S7.I4.i2.p1.3.m3.1d">4 ⌈ roman_log start_POSTSUBSCRIPT 8 / 7 end_POSTSUBSCRIPT italic_n ⌉ + 2</annotation></semantics></math> rounds, each <math alttext="u\in V_{m^{\prime}}" class="ltx_Math" display="inline" id="S7.I4.i2.p1.4.m4.1"><semantics id="S7.I4.i2.p1.4.m4.1a"><mrow id="S7.I4.i2.p1.4.m4.1.1" xref="S7.I4.i2.p1.4.m4.1.1.cmml"><mi id="S7.I4.i2.p1.4.m4.1.1.2" xref="S7.I4.i2.p1.4.m4.1.1.2.cmml">u</mi><mo id="S7.I4.i2.p1.4.m4.1.1.1" xref="S7.I4.i2.p1.4.m4.1.1.1.cmml">∈</mo><msub id="S7.I4.i2.p1.4.m4.1.1.3" xref="S7.I4.i2.p1.4.m4.1.1.3.cmml"><mi id="S7.I4.i2.p1.4.m4.1.1.3.2" xref="S7.I4.i2.p1.4.m4.1.1.3.2.cmml">V</mi><msup id="S7.I4.i2.p1.4.m4.1.1.3.3" xref="S7.I4.i2.p1.4.m4.1.1.3.3.cmml"><mi id="S7.I4.i2.p1.4.m4.1.1.3.3.2" xref="S7.I4.i2.p1.4.m4.1.1.3.3.2.cmml">m</mi><mo id="S7.I4.i2.p1.4.m4.1.1.3.3.3" xref="S7.I4.i2.p1.4.m4.1.1.3.3.3.cmml">′</mo></msup></msub></mrow><annotation-xml encoding="MathML-Content" id="S7.I4.i2.p1.4.m4.1b"><apply id="S7.I4.i2.p1.4.m4.1.1.cmml" xref="S7.I4.i2.p1.4.m4.1.1"><in id="S7.I4.i2.p1.4.m4.1.1.1.cmml" xref="S7.I4.i2.p1.4.m4.1.1.1"></in><ci id="S7.I4.i2.p1.4.m4.1.1.2.cmml" xref="S7.I4.i2.p1.4.m4.1.1.2">𝑢</ci><apply id="S7.I4.i2.p1.4.m4.1.1.3.cmml" xref="S7.I4.i2.p1.4.m4.1.1.3"><csymbol cd="ambiguous" id="S7.I4.i2.p1.4.m4.1.1.3.1.cmml" xref="S7.I4.i2.p1.4.m4.1.1.3">subscript</csymbol><ci id="S7.I4.i2.p1.4.m4.1.1.3.2.cmml" xref="S7.I4.i2.p1.4.m4.1.1.3.2">𝑉</ci><apply id="S7.I4.i2.p1.4.m4.1.1.3.3.cmml" xref="S7.I4.i2.p1.4.m4.1.1.3.3"><csymbol cd="ambiguous" id="S7.I4.i2.p1.4.m4.1.1.3.3.1.cmml" xref="S7.I4.i2.p1.4.m4.1.1.3.3">superscript</csymbol><ci id="S7.I4.i2.p1.4.m4.1.1.3.3.2.cmml" xref="S7.I4.i2.p1.4.m4.1.1.3.3.2">𝑚</ci><ci id="S7.I4.i2.p1.4.m4.1.1.3.3.3.cmml" xref="S7.I4.i2.p1.4.m4.1.1.3.3.3">′</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.I4.i2.p1.4.m4.1c">u\in V_{m^{\prime}}</annotation><annotation encoding="application/x-llamapun" id="S7.I4.i2.p1.4.m4.1d">italic_u ∈ italic_V start_POSTSUBSCRIPT italic_m start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT</annotation></semantics></math> flips violating out-edges <math alttext="\overline{uv}" class="ltx_Math" display="inline" id="S7.I4.i2.p1.5.m5.1"><semantics id="S7.I4.i2.p1.5.m5.1a"><mover accent="true" id="S7.I4.i2.p1.5.m5.1.1" xref="S7.I4.i2.p1.5.m5.1.1.cmml"><mrow id="S7.I4.i2.p1.5.m5.1.1.2" xref="S7.I4.i2.p1.5.m5.1.1.2.cmml"><mi id="S7.I4.i2.p1.5.m5.1.1.2.2" xref="S7.I4.i2.p1.5.m5.1.1.2.2.cmml">u</mi><mo id="S7.I4.i2.p1.5.m5.1.1.2.1" xref="S7.I4.i2.p1.5.m5.1.1.2.1.cmml">⁢</mo><mi id="S7.I4.i2.p1.5.m5.1.1.2.3" xref="S7.I4.i2.p1.5.m5.1.1.2.3.cmml">v</mi></mrow><mo id="S7.I4.i2.p1.5.m5.1.1.1" xref="S7.I4.i2.p1.5.m5.1.1.1.cmml">¯</mo></mover><annotation-xml encoding="MathML-Content" id="S7.I4.i2.p1.5.m5.1b"><apply id="S7.I4.i2.p1.5.m5.1.1.cmml" xref="S7.I4.i2.p1.5.m5.1.1"><ci id="S7.I4.i2.p1.5.m5.1.1.1.cmml" xref="S7.I4.i2.p1.5.m5.1.1.1">¯</ci><apply id="S7.I4.i2.p1.5.m5.1.1.2.cmml" xref="S7.I4.i2.p1.5.m5.1.1.2"><times id="S7.I4.i2.p1.5.m5.1.1.2.1.cmml" xref="S7.I4.i2.p1.5.m5.1.1.2.1"></times><ci id="S7.I4.i2.p1.5.m5.1.1.2.2.cmml" xref="S7.I4.i2.p1.5.m5.1.1.2.2">𝑢</ci><ci id="S7.I4.i2.p1.5.m5.1.1.2.3.cmml" xref="S7.I4.i2.p1.5.m5.1.1.2.3">𝑣</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.I4.i2.p1.5.m5.1c">\overline{uv}</annotation><annotation encoding="application/x-llamapun" id="S7.I4.i2.p1.5.m5.1d">over¯ start_ARG italic_u italic_v end_ARG</annotation></semantics></math>. Moreover:</p> <ul class="ltx_itemize" id="S7.I4.i2.I1"> <li class="ltx_item" id="S7.I4.i2.I1.i1" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item"><span class="ltx_text ltx_font_bold" id="S7.I4.i2.I1.i1.1.1.1">–</span></span> <div class="ltx_para" id="S7.I4.i2.I1.i1.p1"> <p class="ltx_p" id="S7.I4.i2.I1.i1.p1.2">The out-degree <math alttext="\textsl{g}(u)" class="ltx_Math" display="inline" id="S7.I4.i2.I1.i1.p1.1.m1.1"><semantics id="S7.I4.i2.I1.i1.p1.1.m1.1a"><mrow id="S7.I4.i2.I1.i1.p1.1.m1.1.2" xref="S7.I4.i2.I1.i1.p1.1.m1.1.2.cmml"><mtext class="ltx_mathvariant_italic" id="S7.I4.i2.I1.i1.p1.1.m1.1.2.2" xref="S7.I4.i2.I1.i1.p1.1.m1.1.2.2a.cmml">g</mtext><mo id="S7.I4.i2.I1.i1.p1.1.m1.1.2.1" xref="S7.I4.i2.I1.i1.p1.1.m1.1.2.1.cmml">⁢</mo><mrow id="S7.I4.i2.I1.i1.p1.1.m1.1.2.3.2" xref="S7.I4.i2.I1.i1.p1.1.m1.1.2.cmml"><mo id="S7.I4.i2.I1.i1.p1.1.m1.1.2.3.2.1" stretchy="false" xref="S7.I4.i2.I1.i1.p1.1.m1.1.2.cmml">(</mo><mi id="S7.I4.i2.I1.i1.p1.1.m1.1.1" xref="S7.I4.i2.I1.i1.p1.1.m1.1.1.cmml">u</mi><mo id="S7.I4.i2.I1.i1.p1.1.m1.1.2.3.2.2" stretchy="false" xref="S7.I4.i2.I1.i1.p1.1.m1.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S7.I4.i2.I1.i1.p1.1.m1.1b"><apply id="S7.I4.i2.I1.i1.p1.1.m1.1.2.cmml" xref="S7.I4.i2.I1.i1.p1.1.m1.1.2"><times id="S7.I4.i2.I1.i1.p1.1.m1.1.2.1.cmml" xref="S7.I4.i2.I1.i1.p1.1.m1.1.2.1"></times><ci id="S7.I4.i2.I1.i1.p1.1.m1.1.2.2a.cmml" xref="S7.I4.i2.I1.i1.p1.1.m1.1.2.2"><mtext class="ltx_mathvariant_italic" id="S7.I4.i2.I1.i1.p1.1.m1.1.2.2.cmml" xref="S7.I4.i2.I1.i1.p1.1.m1.1.2.2">g</mtext></ci><ci id="S7.I4.i2.I1.i1.p1.1.m1.1.1.cmml" xref="S7.I4.i2.I1.i1.p1.1.m1.1.1">𝑢</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.I4.i2.I1.i1.p1.1.m1.1c">\textsl{g}(u)</annotation><annotation encoding="application/x-llamapun" id="S7.I4.i2.I1.i1.p1.1.m1.1d">g ( italic_u )</annotation></semantics></math> is decreased such that <math alttext="u" class="ltx_Math" display="inline" id="S7.I4.i2.I1.i1.p1.2.m2.1"><semantics id="S7.I4.i2.I1.i1.p1.2.m2.1a"><mi id="S7.I4.i2.I1.i1.p1.2.m2.1.1" xref="S7.I4.i2.I1.i1.p1.2.m2.1.1.cmml">u</mi><annotation-xml encoding="MathML-Content" id="S7.I4.i2.I1.i1.p1.2.m2.1b"><ci id="S7.I4.i2.I1.i1.p1.2.m2.1.1.cmml" xref="S7.I4.i2.I1.i1.p1.2.m2.1.1">𝑢</ci></annotation-xml><annotation encoding="application/x-tex" id="S7.I4.i2.I1.i1.p1.2.m2.1c">u</annotation><annotation encoding="application/x-llamapun" id="S7.I4.i2.I1.i1.p1.2.m2.1d">italic_u</annotation></semantics></math> drops at most one level, and</p> </div> </li> <li class="ltx_item" id="S7.I4.i2.I1.i2" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item"><span class="ltx_text ltx_font_bold" id="S7.I4.i2.I1.i2.1.1.1">–</span></span> <div class="ltx_para" id="S7.I4.i2.I1.i2.p1"> <p class="ltx_p" id="S7.I4.i2.I1.i2.p1.3">The out-degree <math alttext="\textsl{g}(v)" class="ltx_Math" display="inline" id="S7.I4.i2.I1.i2.p1.1.m1.1"><semantics id="S7.I4.i2.I1.i2.p1.1.m1.1a"><mrow id="S7.I4.i2.I1.i2.p1.1.m1.1.2" xref="S7.I4.i2.I1.i2.p1.1.m1.1.2.cmml"><mtext class="ltx_mathvariant_italic" id="S7.I4.i2.I1.i2.p1.1.m1.1.2.2" xref="S7.I4.i2.I1.i2.p1.1.m1.1.2.2a.cmml">g</mtext><mo id="S7.I4.i2.I1.i2.p1.1.m1.1.2.1" xref="S7.I4.i2.I1.i2.p1.1.m1.1.2.1.cmml">⁢</mo><mrow id="S7.I4.i2.I1.i2.p1.1.m1.1.2.3.2" xref="S7.I4.i2.I1.i2.p1.1.m1.1.2.cmml"><mo id="S7.I4.i2.I1.i2.p1.1.m1.1.2.3.2.1" stretchy="false" xref="S7.I4.i2.I1.i2.p1.1.m1.1.2.cmml">(</mo><mi id="S7.I4.i2.I1.i2.p1.1.m1.1.1" xref="S7.I4.i2.I1.i2.p1.1.m1.1.1.cmml">v</mi><mo id="S7.I4.i2.I1.i2.p1.1.m1.1.2.3.2.2" stretchy="false" xref="S7.I4.i2.I1.i2.p1.1.m1.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S7.I4.i2.I1.i2.p1.1.m1.1b"><apply id="S7.I4.i2.I1.i2.p1.1.m1.1.2.cmml" xref="S7.I4.i2.I1.i2.p1.1.m1.1.2"><times id="S7.I4.i2.I1.i2.p1.1.m1.1.2.1.cmml" xref="S7.I4.i2.I1.i2.p1.1.m1.1.2.1"></times><ci id="S7.I4.i2.I1.i2.p1.1.m1.1.2.2a.cmml" xref="S7.I4.i2.I1.i2.p1.1.m1.1.2.2"><mtext class="ltx_mathvariant_italic" id="S7.I4.i2.I1.i2.p1.1.m1.1.2.2.cmml" xref="S7.I4.i2.I1.i2.p1.1.m1.1.2.2">g</mtext></ci><ci id="S7.I4.i2.I1.i2.p1.1.m1.1.1.cmml" xref="S7.I4.i2.I1.i2.p1.1.m1.1.1">𝑣</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.I4.i2.I1.i2.p1.1.m1.1c">\textsl{g}(v)</annotation><annotation encoding="application/x-llamapun" id="S7.I4.i2.I1.i2.p1.1.m1.1d">g ( italic_v )</annotation></semantics></math> such that <math alttext="v" class="ltx_Math" display="inline" id="S7.I4.i2.I1.i2.p1.2.m2.1"><semantics id="S7.I4.i2.I1.i2.p1.2.m2.1a"><mi id="S7.I4.i2.I1.i2.p1.2.m2.1.1" xref="S7.I4.i2.I1.i2.p1.2.m2.1.1.cmml">v</mi><annotation-xml encoding="MathML-Content" id="S7.I4.i2.I1.i2.p1.2.m2.1b"><ci id="S7.I4.i2.I1.i2.p1.2.m2.1.1.cmml" xref="S7.I4.i2.I1.i2.p1.2.m2.1.1">𝑣</ci></annotation-xml><annotation encoding="application/x-tex" id="S7.I4.i2.I1.i2.p1.2.m2.1c">v</annotation><annotation encoding="application/x-llamapun" id="S7.I4.i2.I1.i2.p1.2.m2.1d">italic_v</annotation></semantics></math> increases its level up to at most <math alttext="h-1" class="ltx_Math" display="inline" id="S7.I4.i2.I1.i2.p1.3.m3.1"><semantics id="S7.I4.i2.I1.i2.p1.3.m3.1a"><mrow id="S7.I4.i2.I1.i2.p1.3.m3.1.1" xref="S7.I4.i2.I1.i2.p1.3.m3.1.1.cmml"><mi id="S7.I4.i2.I1.i2.p1.3.m3.1.1.2" xref="S7.I4.i2.I1.i2.p1.3.m3.1.1.2.cmml">h</mi><mo id="S7.I4.i2.I1.i2.p1.3.m3.1.1.1" xref="S7.I4.i2.I1.i2.p1.3.m3.1.1.1.cmml">−</mo><mn id="S7.I4.i2.I1.i2.p1.3.m3.1.1.3" xref="S7.I4.i2.I1.i2.p1.3.m3.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S7.I4.i2.I1.i2.p1.3.m3.1b"><apply id="S7.I4.i2.I1.i2.p1.3.m3.1.1.cmml" xref="S7.I4.i2.I1.i2.p1.3.m3.1.1"><minus id="S7.I4.i2.I1.i2.p1.3.m3.1.1.1.cmml" xref="S7.I4.i2.I1.i2.p1.3.m3.1.1.1"></minus><ci id="S7.I4.i2.I1.i2.p1.3.m3.1.1.2.cmml" xref="S7.I4.i2.I1.i2.p1.3.m3.1.1.2">ℎ</ci><cn id="S7.I4.i2.I1.i2.p1.3.m3.1.1.3.cmml" type="integer" xref="S7.I4.i2.I1.i2.p1.3.m3.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.I4.i2.I1.i2.p1.3.m3.1c">h-1</annotation><annotation encoding="application/x-llamapun" id="S7.I4.i2.I1.i2.p1.3.m3.1d">italic_h - 1</annotation></semantics></math>.</p> </div> </li> </ul> <p class="ltx_p" id="S7.I4.i2.p1.12">We prove that at the end of this <span class="ltx_text ltx_font_smallcaps" id="S7.I4.i2.p1.12.1">minute</span> <math alttext="m^{\prime}" class="ltx_Math" display="inline" id="S7.I4.i2.p1.6.m1.1"><semantics id="S7.I4.i2.p1.6.m1.1a"><msup id="S7.I4.i2.p1.6.m1.1.1" xref="S7.I4.i2.p1.6.m1.1.1.cmml"><mi id="S7.I4.i2.p1.6.m1.1.1.2" xref="S7.I4.i2.p1.6.m1.1.1.2.cmml">m</mi><mo id="S7.I4.i2.p1.6.m1.1.1.3" xref="S7.I4.i2.p1.6.m1.1.1.3.cmml">′</mo></msup><annotation-xml encoding="MathML-Content" id="S7.I4.i2.p1.6.m1.1b"><apply id="S7.I4.i2.p1.6.m1.1.1.cmml" xref="S7.I4.i2.p1.6.m1.1.1"><csymbol cd="ambiguous" id="S7.I4.i2.p1.6.m1.1.1.1.cmml" xref="S7.I4.i2.p1.6.m1.1.1">superscript</csymbol><ci id="S7.I4.i2.p1.6.m1.1.1.2.cmml" xref="S7.I4.i2.p1.6.m1.1.1.2">𝑚</ci><ci id="S7.I4.i2.p1.6.m1.1.1.3.cmml" xref="S7.I4.i2.p1.6.m1.1.1.3">′</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.I4.i2.p1.6.m1.1c">m^{\prime}</annotation><annotation encoding="application/x-llamapun" id="S7.I4.i2.p1.6.m1.1d">italic_m start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math>, there exist no more <math alttext="u\in V_{m^{\prime}}" class="ltx_Math" display="inline" id="S7.I4.i2.p1.7.m2.1"><semantics id="S7.I4.i2.p1.7.m2.1a"><mrow id="S7.I4.i2.p1.7.m2.1.1" xref="S7.I4.i2.p1.7.m2.1.1.cmml"><mi id="S7.I4.i2.p1.7.m2.1.1.2" xref="S7.I4.i2.p1.7.m2.1.1.2.cmml">u</mi><mo id="S7.I4.i2.p1.7.m2.1.1.1" xref="S7.I4.i2.p1.7.m2.1.1.1.cmml">∈</mo><msub id="S7.I4.i2.p1.7.m2.1.1.3" xref="S7.I4.i2.p1.7.m2.1.1.3.cmml"><mi id="S7.I4.i2.p1.7.m2.1.1.3.2" xref="S7.I4.i2.p1.7.m2.1.1.3.2.cmml">V</mi><msup id="S7.I4.i2.p1.7.m2.1.1.3.3" xref="S7.I4.i2.p1.7.m2.1.1.3.3.cmml"><mi id="S7.I4.i2.p1.7.m2.1.1.3.3.2" xref="S7.I4.i2.p1.7.m2.1.1.3.3.2.cmml">m</mi><mo id="S7.I4.i2.p1.7.m2.1.1.3.3.3" xref="S7.I4.i2.p1.7.m2.1.1.3.3.3.cmml">′</mo></msup></msub></mrow><annotation-xml encoding="MathML-Content" id="S7.I4.i2.p1.7.m2.1b"><apply id="S7.I4.i2.p1.7.m2.1.1.cmml" xref="S7.I4.i2.p1.7.m2.1.1"><in id="S7.I4.i2.p1.7.m2.1.1.1.cmml" xref="S7.I4.i2.p1.7.m2.1.1.1"></in><ci id="S7.I4.i2.p1.7.m2.1.1.2.cmml" xref="S7.I4.i2.p1.7.m2.1.1.2">𝑢</ci><apply id="S7.I4.i2.p1.7.m2.1.1.3.cmml" xref="S7.I4.i2.p1.7.m2.1.1.3"><csymbol cd="ambiguous" id="S7.I4.i2.p1.7.m2.1.1.3.1.cmml" xref="S7.I4.i2.p1.7.m2.1.1.3">subscript</csymbol><ci id="S7.I4.i2.p1.7.m2.1.1.3.2.cmml" xref="S7.I4.i2.p1.7.m2.1.1.3.2">𝑉</ci><apply id="S7.I4.i2.p1.7.m2.1.1.3.3.cmml" xref="S7.I4.i2.p1.7.m2.1.1.3.3"><csymbol cd="ambiguous" id="S7.I4.i2.p1.7.m2.1.1.3.3.1.cmml" xref="S7.I4.i2.p1.7.m2.1.1.3.3">superscript</csymbol><ci id="S7.I4.i2.p1.7.m2.1.1.3.3.2.cmml" xref="S7.I4.i2.p1.7.m2.1.1.3.3.2">𝑚</ci><ci id="S7.I4.i2.p1.7.m2.1.1.3.3.3.cmml" xref="S7.I4.i2.p1.7.m2.1.1.3.3.3">′</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.I4.i2.p1.7.m2.1c">u\in V_{m^{\prime}}</annotation><annotation encoding="application/x-llamapun" id="S7.I4.i2.p1.7.m2.1d">italic_u ∈ italic_V start_POSTSUBSCRIPT italic_m start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT</annotation></semantics></math> with a violating out-edge. Thus, for each vertex <math alttext="u\in V_{m^{\prime}}" class="ltx_Math" display="inline" id="S7.I4.i2.p1.8.m3.1"><semantics id="S7.I4.i2.p1.8.m3.1a"><mrow id="S7.I4.i2.p1.8.m3.1.1" xref="S7.I4.i2.p1.8.m3.1.1.cmml"><mi id="S7.I4.i2.p1.8.m3.1.1.2" xref="S7.I4.i2.p1.8.m3.1.1.2.cmml">u</mi><mo id="S7.I4.i2.p1.8.m3.1.1.1" xref="S7.I4.i2.p1.8.m3.1.1.1.cmml">∈</mo><msub id="S7.I4.i2.p1.8.m3.1.1.3" xref="S7.I4.i2.p1.8.m3.1.1.3.cmml"><mi id="S7.I4.i2.p1.8.m3.1.1.3.2" xref="S7.I4.i2.p1.8.m3.1.1.3.2.cmml">V</mi><msup id="S7.I4.i2.p1.8.m3.1.1.3.3" xref="S7.I4.i2.p1.8.m3.1.1.3.3.cmml"><mi id="S7.I4.i2.p1.8.m3.1.1.3.3.2" xref="S7.I4.i2.p1.8.m3.1.1.3.3.2.cmml">m</mi><mo id="S7.I4.i2.p1.8.m3.1.1.3.3.3" xref="S7.I4.i2.p1.8.m3.1.1.3.3.3.cmml">′</mo></msup></msub></mrow><annotation-xml encoding="MathML-Content" id="S7.I4.i2.p1.8.m3.1b"><apply id="S7.I4.i2.p1.8.m3.1.1.cmml" xref="S7.I4.i2.p1.8.m3.1.1"><in id="S7.I4.i2.p1.8.m3.1.1.1.cmml" xref="S7.I4.i2.p1.8.m3.1.1.1"></in><ci id="S7.I4.i2.p1.8.m3.1.1.2.cmml" xref="S7.I4.i2.p1.8.m3.1.1.2">𝑢</ci><apply id="S7.I4.i2.p1.8.m3.1.1.3.cmml" xref="S7.I4.i2.p1.8.m3.1.1.3"><csymbol cd="ambiguous" id="S7.I4.i2.p1.8.m3.1.1.3.1.cmml" xref="S7.I4.i2.p1.8.m3.1.1.3">subscript</csymbol><ci id="S7.I4.i2.p1.8.m3.1.1.3.2.cmml" xref="S7.I4.i2.p1.8.m3.1.1.3.2">𝑉</ci><apply id="S7.I4.i2.p1.8.m3.1.1.3.3.cmml" xref="S7.I4.i2.p1.8.m3.1.1.3.3"><csymbol cd="ambiguous" id="S7.I4.i2.p1.8.m3.1.1.3.3.1.cmml" xref="S7.I4.i2.p1.8.m3.1.1.3.3">superscript</csymbol><ci id="S7.I4.i2.p1.8.m3.1.1.3.3.2.cmml" xref="S7.I4.i2.p1.8.m3.1.1.3.3.2">𝑚</ci><ci id="S7.I4.i2.p1.8.m3.1.1.3.3.3.cmml" xref="S7.I4.i2.p1.8.m3.1.1.3.3.3">′</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.I4.i2.p1.8.m3.1c">u\in V_{m^{\prime}}</annotation><annotation encoding="application/x-llamapun" id="S7.I4.i2.p1.8.m3.1d">italic_u ∈ italic_V start_POSTSUBSCRIPT italic_m start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT</annotation></semantics></math>, either <math alttext="u" class="ltx_Math" display="inline" id="S7.I4.i2.p1.9.m4.1"><semantics id="S7.I4.i2.p1.9.m4.1a"><mi id="S7.I4.i2.p1.9.m4.1.1" xref="S7.I4.i2.p1.9.m4.1.1.cmml">u</mi><annotation-xml encoding="MathML-Content" id="S7.I4.i2.p1.9.m4.1b"><ci id="S7.I4.i2.p1.9.m4.1.1.cmml" xref="S7.I4.i2.p1.9.m4.1.1">𝑢</ci></annotation-xml><annotation encoding="application/x-tex" id="S7.I4.i2.p1.9.m4.1c">u</annotation><annotation encoding="application/x-llamapun" id="S7.I4.i2.p1.9.m4.1d">italic_u</annotation></semantics></math> decreased one level, or all <math alttext="\overline{uv}" class="ltx_Math" display="inline" id="S7.I4.i2.p1.10.m5.1"><semantics id="S7.I4.i2.p1.10.m5.1a"><mover accent="true" id="S7.I4.i2.p1.10.m5.1.1" xref="S7.I4.i2.p1.10.m5.1.1.cmml"><mrow id="S7.I4.i2.p1.10.m5.1.1.2" xref="S7.I4.i2.p1.10.m5.1.1.2.cmml"><mi id="S7.I4.i2.p1.10.m5.1.1.2.2" xref="S7.I4.i2.p1.10.m5.1.1.2.2.cmml">u</mi><mo id="S7.I4.i2.p1.10.m5.1.1.2.1" xref="S7.I4.i2.p1.10.m5.1.1.2.1.cmml">⁢</mo><mi id="S7.I4.i2.p1.10.m5.1.1.2.3" xref="S7.I4.i2.p1.10.m5.1.1.2.3.cmml">v</mi></mrow><mo id="S7.I4.i2.p1.10.m5.1.1.1" xref="S7.I4.i2.p1.10.m5.1.1.1.cmml">¯</mo></mover><annotation-xml encoding="MathML-Content" id="S7.I4.i2.p1.10.m5.1b"><apply id="S7.I4.i2.p1.10.m5.1.1.cmml" xref="S7.I4.i2.p1.10.m5.1.1"><ci id="S7.I4.i2.p1.10.m5.1.1.1.cmml" xref="S7.I4.i2.p1.10.m5.1.1.1">¯</ci><apply id="S7.I4.i2.p1.10.m5.1.1.2.cmml" xref="S7.I4.i2.p1.10.m5.1.1.2"><times id="S7.I4.i2.p1.10.m5.1.1.2.1.cmml" xref="S7.I4.i2.p1.10.m5.1.1.2.1"></times><ci id="S7.I4.i2.p1.10.m5.1.1.2.2.cmml" xref="S7.I4.i2.p1.10.m5.1.1.2.2">𝑢</ci><ci id="S7.I4.i2.p1.10.m5.1.1.2.3.cmml" xref="S7.I4.i2.p1.10.m5.1.1.2.3">𝑣</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.I4.i2.p1.10.m5.1c">\overline{uv}</annotation><annotation encoding="application/x-llamapun" id="S7.I4.i2.p1.10.m5.1d">over¯ start_ARG italic_u italic_v end_ARG</annotation></semantics></math> that were violating now have that <math alttext="\textsl{g}(u\!\to\!v)=0" class="ltx_Math" display="inline" id="S7.I4.i2.p1.11.m6.1"><semantics id="S7.I4.i2.p1.11.m6.1a"><mrow id="S7.I4.i2.p1.11.m6.1.1" xref="S7.I4.i2.p1.11.m6.1.1.cmml"><mrow id="S7.I4.i2.p1.11.m6.1.1.1" xref="S7.I4.i2.p1.11.m6.1.1.1.cmml"><mtext class="ltx_mathvariant_italic" id="S7.I4.i2.p1.11.m6.1.1.1.3" xref="S7.I4.i2.p1.11.m6.1.1.1.3a.cmml">g</mtext><mo id="S7.I4.i2.p1.11.m6.1.1.1.2" xref="S7.I4.i2.p1.11.m6.1.1.1.2.cmml">⁢</mo><mrow id="S7.I4.i2.p1.11.m6.1.1.1.1.1" xref="S7.I4.i2.p1.11.m6.1.1.1.1.1.1.cmml"><mo id="S7.I4.i2.p1.11.m6.1.1.1.1.1.2" stretchy="false" xref="S7.I4.i2.p1.11.m6.1.1.1.1.1.1.cmml">(</mo><mrow id="S7.I4.i2.p1.11.m6.1.1.1.1.1.1" xref="S7.I4.i2.p1.11.m6.1.1.1.1.1.1.cmml"><mi id="S7.I4.i2.p1.11.m6.1.1.1.1.1.1.2" xref="S7.I4.i2.p1.11.m6.1.1.1.1.1.1.2.cmml">u</mi><mo id="S7.I4.i2.p1.11.m6.1.1.1.1.1.1.1" lspace="0.108em" rspace="0.108em" stretchy="false" xref="S7.I4.i2.p1.11.m6.1.1.1.1.1.1.1.cmml">→</mo><mi id="S7.I4.i2.p1.11.m6.1.1.1.1.1.1.3" xref="S7.I4.i2.p1.11.m6.1.1.1.1.1.1.3.cmml">v</mi></mrow><mo id="S7.I4.i2.p1.11.m6.1.1.1.1.1.3" stretchy="false" xref="S7.I4.i2.p1.11.m6.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S7.I4.i2.p1.11.m6.1.1.2" xref="S7.I4.i2.p1.11.m6.1.1.2.cmml">=</mo><mn id="S7.I4.i2.p1.11.m6.1.1.3" xref="S7.I4.i2.p1.11.m6.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S7.I4.i2.p1.11.m6.1b"><apply id="S7.I4.i2.p1.11.m6.1.1.cmml" xref="S7.I4.i2.p1.11.m6.1.1"><eq id="S7.I4.i2.p1.11.m6.1.1.2.cmml" xref="S7.I4.i2.p1.11.m6.1.1.2"></eq><apply id="S7.I4.i2.p1.11.m6.1.1.1.cmml" xref="S7.I4.i2.p1.11.m6.1.1.1"><times id="S7.I4.i2.p1.11.m6.1.1.1.2.cmml" xref="S7.I4.i2.p1.11.m6.1.1.1.2"></times><ci id="S7.I4.i2.p1.11.m6.1.1.1.3a.cmml" xref="S7.I4.i2.p1.11.m6.1.1.1.3"><mtext class="ltx_mathvariant_italic" id="S7.I4.i2.p1.11.m6.1.1.1.3.cmml" xref="S7.I4.i2.p1.11.m6.1.1.1.3">g</mtext></ci><apply id="S7.I4.i2.p1.11.m6.1.1.1.1.1.1.cmml" xref="S7.I4.i2.p1.11.m6.1.1.1.1.1"><ci id="S7.I4.i2.p1.11.m6.1.1.1.1.1.1.1.cmml" xref="S7.I4.i2.p1.11.m6.1.1.1.1.1.1.1">→</ci><ci id="S7.I4.i2.p1.11.m6.1.1.1.1.1.1.2.cmml" xref="S7.I4.i2.p1.11.m6.1.1.1.1.1.1.2">𝑢</ci><ci id="S7.I4.i2.p1.11.m6.1.1.1.1.1.1.3.cmml" xref="S7.I4.i2.p1.11.m6.1.1.1.1.1.1.3">𝑣</ci></apply></apply><cn id="S7.I4.i2.p1.11.m6.1.1.3.cmml" type="integer" xref="S7.I4.i2.p1.11.m6.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.I4.i2.p1.11.m6.1c">\textsl{g}(u\!\to\!v)=0</annotation><annotation encoding="application/x-llamapun" id="S7.I4.i2.p1.11.m6.1d">g ( italic_u → italic_v ) = 0</annotation></semantics></math>, or <math alttext="v\in L_{h-1}" class="ltx_Math" display="inline" id="S7.I4.i2.p1.12.m7.1"><semantics id="S7.I4.i2.p1.12.m7.1a"><mrow id="S7.I4.i2.p1.12.m7.1.1" xref="S7.I4.i2.p1.12.m7.1.1.cmml"><mi id="S7.I4.i2.p1.12.m7.1.1.2" xref="S7.I4.i2.p1.12.m7.1.1.2.cmml">v</mi><mo id="S7.I4.i2.p1.12.m7.1.1.1" xref="S7.I4.i2.p1.12.m7.1.1.1.cmml">∈</mo><msub id="S7.I4.i2.p1.12.m7.1.1.3" xref="S7.I4.i2.p1.12.m7.1.1.3.cmml"><mi id="S7.I4.i2.p1.12.m7.1.1.3.2" xref="S7.I4.i2.p1.12.m7.1.1.3.2.cmml">L</mi><mrow id="S7.I4.i2.p1.12.m7.1.1.3.3" xref="S7.I4.i2.p1.12.m7.1.1.3.3.cmml"><mi id="S7.I4.i2.p1.12.m7.1.1.3.3.2" xref="S7.I4.i2.p1.12.m7.1.1.3.3.2.cmml">h</mi><mo id="S7.I4.i2.p1.12.m7.1.1.3.3.1" xref="S7.I4.i2.p1.12.m7.1.1.3.3.1.cmml">−</mo><mn id="S7.I4.i2.p1.12.m7.1.1.3.3.3" xref="S7.I4.i2.p1.12.m7.1.1.3.3.3.cmml">1</mn></mrow></msub></mrow><annotation-xml encoding="MathML-Content" id="S7.I4.i2.p1.12.m7.1b"><apply id="S7.I4.i2.p1.12.m7.1.1.cmml" xref="S7.I4.i2.p1.12.m7.1.1"><in id="S7.I4.i2.p1.12.m7.1.1.1.cmml" xref="S7.I4.i2.p1.12.m7.1.1.1"></in><ci id="S7.I4.i2.p1.12.m7.1.1.2.cmml" xref="S7.I4.i2.p1.12.m7.1.1.2">𝑣</ci><apply id="S7.I4.i2.p1.12.m7.1.1.3.cmml" xref="S7.I4.i2.p1.12.m7.1.1.3"><csymbol cd="ambiguous" id="S7.I4.i2.p1.12.m7.1.1.3.1.cmml" xref="S7.I4.i2.p1.12.m7.1.1.3">subscript</csymbol><ci id="S7.I4.i2.p1.12.m7.1.1.3.2.cmml" xref="S7.I4.i2.p1.12.m7.1.1.3.2">𝐿</ci><apply id="S7.I4.i2.p1.12.m7.1.1.3.3.cmml" xref="S7.I4.i2.p1.12.m7.1.1.3.3"><minus id="S7.I4.i2.p1.12.m7.1.1.3.3.1.cmml" xref="S7.I4.i2.p1.12.m7.1.1.3.3.1"></minus><ci id="S7.I4.i2.p1.12.m7.1.1.3.3.2.cmml" xref="S7.I4.i2.p1.12.m7.1.1.3.3.2">ℎ</ci><cn id="S7.I4.i2.p1.12.m7.1.1.3.3.3.cmml" type="integer" xref="S7.I4.i2.p1.12.m7.1.1.3.3.3">1</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.I4.i2.p1.12.m7.1c">v\in L_{h-1}</annotation><annotation encoding="application/x-llamapun" id="S7.I4.i2.p1.12.m7.1d">italic_v ∈ italic_L start_POSTSUBSCRIPT italic_h - 1 end_POSTSUBSCRIPT</annotation></semantics></math>.</p> </div> </li> <li class="ltx_item" id="S7.I4.i3" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">•</span> <div class="ltx_para" id="S7.I4.i3.p1"> <p class="ltx_p" id="S7.I4.i3.p1.21">In each odd <span class="ltx_text ltx_font_smallcaps" id="S7.I4.i3.p1.21.1">minute</span> <math alttext="m" class="ltx_Math" display="inline" id="S7.I4.i3.p1.1.m1.1"><semantics id="S7.I4.i3.p1.1.m1.1a"><mi id="S7.I4.i3.p1.1.m1.1.1" xref="S7.I4.i3.p1.1.m1.1.1.cmml">m</mi><annotation-xml encoding="MathML-Content" id="S7.I4.i3.p1.1.m1.1b"><ci id="S7.I4.i3.p1.1.m1.1.1.cmml" xref="S7.I4.i3.p1.1.m1.1.1">𝑚</ci></annotation-xml><annotation encoding="application/x-tex" id="S7.I4.i3.p1.1.m1.1c">m</annotation><annotation encoding="application/x-llamapun" id="S7.I4.i3.p1.1.m1.1d">italic_m</annotation></semantics></math>, we inspect the vertices <math alttext="T_{m}:=\{u\in V_{m-1}\textnormal{ that dropped a level}\}" class="ltx_Math" display="inline" id="S7.I4.i3.p1.2.m2.1"><semantics id="S7.I4.i3.p1.2.m2.1a"><mrow id="S7.I4.i3.p1.2.m2.1.1" xref="S7.I4.i3.p1.2.m2.1.1.cmml"><msub id="S7.I4.i3.p1.2.m2.1.1.3" xref="S7.I4.i3.p1.2.m2.1.1.3.cmml"><mi id="S7.I4.i3.p1.2.m2.1.1.3.2" xref="S7.I4.i3.p1.2.m2.1.1.3.2.cmml">T</mi><mi id="S7.I4.i3.p1.2.m2.1.1.3.3" xref="S7.I4.i3.p1.2.m2.1.1.3.3.cmml">m</mi></msub><mo id="S7.I4.i3.p1.2.m2.1.1.2" lspace="0.278em" rspace="0.278em" xref="S7.I4.i3.p1.2.m2.1.1.2.cmml">:=</mo><mrow id="S7.I4.i3.p1.2.m2.1.1.1.1" xref="S7.I4.i3.p1.2.m2.1.1.1.2.cmml"><mo id="S7.I4.i3.p1.2.m2.1.1.1.1.2" stretchy="false" xref="S7.I4.i3.p1.2.m2.1.1.1.2.cmml">{</mo><mrow id="S7.I4.i3.p1.2.m2.1.1.1.1.1" xref="S7.I4.i3.p1.2.m2.1.1.1.1.1.cmml"><mi id="S7.I4.i3.p1.2.m2.1.1.1.1.1.2" xref="S7.I4.i3.p1.2.m2.1.1.1.1.1.2.cmml">u</mi><mo id="S7.I4.i3.p1.2.m2.1.1.1.1.1.1" xref="S7.I4.i3.p1.2.m2.1.1.1.1.1.1.cmml">∈</mo><mrow id="S7.I4.i3.p1.2.m2.1.1.1.1.1.3" xref="S7.I4.i3.p1.2.m2.1.1.1.1.1.3.cmml"><msub id="S7.I4.i3.p1.2.m2.1.1.1.1.1.3.2" xref="S7.I4.i3.p1.2.m2.1.1.1.1.1.3.2.cmml"><mi id="S7.I4.i3.p1.2.m2.1.1.1.1.1.3.2.2" xref="S7.I4.i3.p1.2.m2.1.1.1.1.1.3.2.2.cmml">V</mi><mrow id="S7.I4.i3.p1.2.m2.1.1.1.1.1.3.2.3" xref="S7.I4.i3.p1.2.m2.1.1.1.1.1.3.2.3.cmml"><mi id="S7.I4.i3.p1.2.m2.1.1.1.1.1.3.2.3.2" xref="S7.I4.i3.p1.2.m2.1.1.1.1.1.3.2.3.2.cmml">m</mi><mo id="S7.I4.i3.p1.2.m2.1.1.1.1.1.3.2.3.1" xref="S7.I4.i3.p1.2.m2.1.1.1.1.1.3.2.3.1.cmml">−</mo><mn id="S7.I4.i3.p1.2.m2.1.1.1.1.1.3.2.3.3" xref="S7.I4.i3.p1.2.m2.1.1.1.1.1.3.2.3.3.cmml">1</mn></mrow></msub><mo id="S7.I4.i3.p1.2.m2.1.1.1.1.1.3.1" xref="S7.I4.i3.p1.2.m2.1.1.1.1.1.3.1.cmml">⁢</mo><mtext id="S7.I4.i3.p1.2.m2.1.1.1.1.1.3.3" xref="S7.I4.i3.p1.2.m2.1.1.1.1.1.3.3a.cmml"> that dropped a level</mtext></mrow></mrow><mo id="S7.I4.i3.p1.2.m2.1.1.1.1.3" stretchy="false" xref="S7.I4.i3.p1.2.m2.1.1.1.2.cmml">}</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S7.I4.i3.p1.2.m2.1b"><apply id="S7.I4.i3.p1.2.m2.1.1.cmml" xref="S7.I4.i3.p1.2.m2.1.1"><csymbol cd="latexml" id="S7.I4.i3.p1.2.m2.1.1.2.cmml" xref="S7.I4.i3.p1.2.m2.1.1.2">assign</csymbol><apply id="S7.I4.i3.p1.2.m2.1.1.3.cmml" xref="S7.I4.i3.p1.2.m2.1.1.3"><csymbol cd="ambiguous" id="S7.I4.i3.p1.2.m2.1.1.3.1.cmml" xref="S7.I4.i3.p1.2.m2.1.1.3">subscript</csymbol><ci id="S7.I4.i3.p1.2.m2.1.1.3.2.cmml" xref="S7.I4.i3.p1.2.m2.1.1.3.2">𝑇</ci><ci id="S7.I4.i3.p1.2.m2.1.1.3.3.cmml" xref="S7.I4.i3.p1.2.m2.1.1.3.3">𝑚</ci></apply><set id="S7.I4.i3.p1.2.m2.1.1.1.2.cmml" xref="S7.I4.i3.p1.2.m2.1.1.1.1"><apply id="S7.I4.i3.p1.2.m2.1.1.1.1.1.cmml" xref="S7.I4.i3.p1.2.m2.1.1.1.1.1"><in id="S7.I4.i3.p1.2.m2.1.1.1.1.1.1.cmml" xref="S7.I4.i3.p1.2.m2.1.1.1.1.1.1"></in><ci id="S7.I4.i3.p1.2.m2.1.1.1.1.1.2.cmml" xref="S7.I4.i3.p1.2.m2.1.1.1.1.1.2">𝑢</ci><apply id="S7.I4.i3.p1.2.m2.1.1.1.1.1.3.cmml" xref="S7.I4.i3.p1.2.m2.1.1.1.1.1.3"><times id="S7.I4.i3.p1.2.m2.1.1.1.1.1.3.1.cmml" xref="S7.I4.i3.p1.2.m2.1.1.1.1.1.3.1"></times><apply id="S7.I4.i3.p1.2.m2.1.1.1.1.1.3.2.cmml" xref="S7.I4.i3.p1.2.m2.1.1.1.1.1.3.2"><csymbol cd="ambiguous" id="S7.I4.i3.p1.2.m2.1.1.1.1.1.3.2.1.cmml" xref="S7.I4.i3.p1.2.m2.1.1.1.1.1.3.2">subscript</csymbol><ci id="S7.I4.i3.p1.2.m2.1.1.1.1.1.3.2.2.cmml" xref="S7.I4.i3.p1.2.m2.1.1.1.1.1.3.2.2">𝑉</ci><apply id="S7.I4.i3.p1.2.m2.1.1.1.1.1.3.2.3.cmml" xref="S7.I4.i3.p1.2.m2.1.1.1.1.1.3.2.3"><minus id="S7.I4.i3.p1.2.m2.1.1.1.1.1.3.2.3.1.cmml" xref="S7.I4.i3.p1.2.m2.1.1.1.1.1.3.2.3.1"></minus><ci id="S7.I4.i3.p1.2.m2.1.1.1.1.1.3.2.3.2.cmml" xref="S7.I4.i3.p1.2.m2.1.1.1.1.1.3.2.3.2">𝑚</ci><cn id="S7.I4.i3.p1.2.m2.1.1.1.1.1.3.2.3.3.cmml" type="integer" xref="S7.I4.i3.p1.2.m2.1.1.1.1.1.3.2.3.3">1</cn></apply></apply><ci id="S7.I4.i3.p1.2.m2.1.1.1.1.1.3.3a.cmml" xref="S7.I4.i3.p1.2.m2.1.1.1.1.1.3.3"><mtext id="S7.I4.i3.p1.2.m2.1.1.1.1.1.3.3.cmml" xref="S7.I4.i3.p1.2.m2.1.1.1.1.1.3.3"> that dropped a level</mtext></ci></apply></apply></set></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.I4.i3.p1.2.m2.1c">T_{m}:=\{u\in V_{m-1}\textnormal{ that dropped a level}\}</annotation><annotation encoding="application/x-llamapun" id="S7.I4.i3.p1.2.m2.1d">italic_T start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT := { italic_u ∈ italic_V start_POSTSUBSCRIPT italic_m - 1 end_POSTSUBSCRIPT that dropped a level }</annotation></semantics></math>. For each <math alttext="u\in T_{m}" class="ltx_Math" display="inline" id="S7.I4.i3.p1.3.m3.1"><semantics id="S7.I4.i3.p1.3.m3.1a"><mrow id="S7.I4.i3.p1.3.m3.1.1" xref="S7.I4.i3.p1.3.m3.1.1.cmml"><mi id="S7.I4.i3.p1.3.m3.1.1.2" xref="S7.I4.i3.p1.3.m3.1.1.2.cmml">u</mi><mo id="S7.I4.i3.p1.3.m3.1.1.1" xref="S7.I4.i3.p1.3.m3.1.1.1.cmml">∈</mo><msub id="S7.I4.i3.p1.3.m3.1.1.3" xref="S7.I4.i3.p1.3.m3.1.1.3.cmml"><mi id="S7.I4.i3.p1.3.m3.1.1.3.2" xref="S7.I4.i3.p1.3.m3.1.1.3.2.cmml">T</mi><mi id="S7.I4.i3.p1.3.m3.1.1.3.3" xref="S7.I4.i3.p1.3.m3.1.1.3.3.cmml">m</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S7.I4.i3.p1.3.m3.1b"><apply id="S7.I4.i3.p1.3.m3.1.1.cmml" xref="S7.I4.i3.p1.3.m3.1.1"><in id="S7.I4.i3.p1.3.m3.1.1.1.cmml" xref="S7.I4.i3.p1.3.m3.1.1.1"></in><ci id="S7.I4.i3.p1.3.m3.1.1.2.cmml" xref="S7.I4.i3.p1.3.m3.1.1.2">𝑢</ci><apply id="S7.I4.i3.p1.3.m3.1.1.3.cmml" xref="S7.I4.i3.p1.3.m3.1.1.3"><csymbol cd="ambiguous" id="S7.I4.i3.p1.3.m3.1.1.3.1.cmml" xref="S7.I4.i3.p1.3.m3.1.1.3">subscript</csymbol><ci id="S7.I4.i3.p1.3.m3.1.1.3.2.cmml" xref="S7.I4.i3.p1.3.m3.1.1.3.2">𝑇</ci><ci id="S7.I4.i3.p1.3.m3.1.1.3.3.cmml" xref="S7.I4.i3.p1.3.m3.1.1.3.3">𝑚</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.I4.i3.p1.3.m3.1c">u\in T_{m}</annotation><annotation encoding="application/x-llamapun" id="S7.I4.i3.p1.3.m3.1d">italic_u ∈ italic_T start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT</annotation></semantics></math>, for each edge <math alttext="\overline{wu}" class="ltx_Math" display="inline" id="S7.I4.i3.p1.4.m4.1"><semantics id="S7.I4.i3.p1.4.m4.1a"><mover accent="true" id="S7.I4.i3.p1.4.m4.1.1" xref="S7.I4.i3.p1.4.m4.1.1.cmml"><mrow id="S7.I4.i3.p1.4.m4.1.1.2" xref="S7.I4.i3.p1.4.m4.1.1.2.cmml"><mi id="S7.I4.i3.p1.4.m4.1.1.2.2" xref="S7.I4.i3.p1.4.m4.1.1.2.2.cmml">w</mi><mo id="S7.I4.i3.p1.4.m4.1.1.2.1" xref="S7.I4.i3.p1.4.m4.1.1.2.1.cmml">⁢</mo><mi id="S7.I4.i3.p1.4.m4.1.1.2.3" xref="S7.I4.i3.p1.4.m4.1.1.2.3.cmml">u</mi></mrow><mo id="S7.I4.i3.p1.4.m4.1.1.1" xref="S7.I4.i3.p1.4.m4.1.1.1.cmml">¯</mo></mover><annotation-xml encoding="MathML-Content" id="S7.I4.i3.p1.4.m4.1b"><apply id="S7.I4.i3.p1.4.m4.1.1.cmml" xref="S7.I4.i3.p1.4.m4.1.1"><ci id="S7.I4.i3.p1.4.m4.1.1.1.cmml" xref="S7.I4.i3.p1.4.m4.1.1.1">¯</ci><apply id="S7.I4.i3.p1.4.m4.1.1.2.cmml" xref="S7.I4.i3.p1.4.m4.1.1.2"><times id="S7.I4.i3.p1.4.m4.1.1.2.1.cmml" xref="S7.I4.i3.p1.4.m4.1.1.2.1"></times><ci id="S7.I4.i3.p1.4.m4.1.1.2.2.cmml" xref="S7.I4.i3.p1.4.m4.1.1.2.2">𝑤</ci><ci id="S7.I4.i3.p1.4.m4.1.1.2.3.cmml" xref="S7.I4.i3.p1.4.m4.1.1.2.3">𝑢</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.I4.i3.p1.4.m4.1c">\overline{wu}</annotation><annotation encoding="application/x-llamapun" id="S7.I4.i3.p1.4.m4.1d">over¯ start_ARG italic_w italic_u end_ARG</annotation></semantics></math> with <math alttext="w\in L_{h+1}" class="ltx_Math" display="inline" id="S7.I4.i3.p1.5.m5.1"><semantics id="S7.I4.i3.p1.5.m5.1a"><mrow id="S7.I4.i3.p1.5.m5.1.1" xref="S7.I4.i3.p1.5.m5.1.1.cmml"><mi id="S7.I4.i3.p1.5.m5.1.1.2" xref="S7.I4.i3.p1.5.m5.1.1.2.cmml">w</mi><mo id="S7.I4.i3.p1.5.m5.1.1.1" xref="S7.I4.i3.p1.5.m5.1.1.1.cmml">∈</mo><msub id="S7.I4.i3.p1.5.m5.1.1.3" xref="S7.I4.i3.p1.5.m5.1.1.3.cmml"><mi id="S7.I4.i3.p1.5.m5.1.1.3.2" xref="S7.I4.i3.p1.5.m5.1.1.3.2.cmml">L</mi><mrow id="S7.I4.i3.p1.5.m5.1.1.3.3" xref="S7.I4.i3.p1.5.m5.1.1.3.3.cmml"><mi id="S7.I4.i3.p1.5.m5.1.1.3.3.2" xref="S7.I4.i3.p1.5.m5.1.1.3.3.2.cmml">h</mi><mo id="S7.I4.i3.p1.5.m5.1.1.3.3.1" xref="S7.I4.i3.p1.5.m5.1.1.3.3.1.cmml">+</mo><mn id="S7.I4.i3.p1.5.m5.1.1.3.3.3" xref="S7.I4.i3.p1.5.m5.1.1.3.3.3.cmml">1</mn></mrow></msub></mrow><annotation-xml encoding="MathML-Content" id="S7.I4.i3.p1.5.m5.1b"><apply id="S7.I4.i3.p1.5.m5.1.1.cmml" xref="S7.I4.i3.p1.5.m5.1.1"><in id="S7.I4.i3.p1.5.m5.1.1.1.cmml" xref="S7.I4.i3.p1.5.m5.1.1.1"></in><ci id="S7.I4.i3.p1.5.m5.1.1.2.cmml" xref="S7.I4.i3.p1.5.m5.1.1.2">𝑤</ci><apply id="S7.I4.i3.p1.5.m5.1.1.3.cmml" xref="S7.I4.i3.p1.5.m5.1.1.3"><csymbol cd="ambiguous" id="S7.I4.i3.p1.5.m5.1.1.3.1.cmml" xref="S7.I4.i3.p1.5.m5.1.1.3">subscript</csymbol><ci id="S7.I4.i3.p1.5.m5.1.1.3.2.cmml" xref="S7.I4.i3.p1.5.m5.1.1.3.2">𝐿</ci><apply id="S7.I4.i3.p1.5.m5.1.1.3.3.cmml" xref="S7.I4.i3.p1.5.m5.1.1.3.3"><plus id="S7.I4.i3.p1.5.m5.1.1.3.3.1.cmml" xref="S7.I4.i3.p1.5.m5.1.1.3.3.1"></plus><ci id="S7.I4.i3.p1.5.m5.1.1.3.3.2.cmml" xref="S7.I4.i3.p1.5.m5.1.1.3.3.2">ℎ</ci><cn id="S7.I4.i3.p1.5.m5.1.1.3.3.3.cmml" type="integer" xref="S7.I4.i3.p1.5.m5.1.1.3.3.3">1</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.I4.i3.p1.5.m5.1c">w\in L_{h+1}</annotation><annotation encoding="application/x-llamapun" id="S7.I4.i3.p1.5.m5.1d">italic_w ∈ italic_L start_POSTSUBSCRIPT italic_h + 1 end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="\textsl{g}(w\!\to\!u)>0" class="ltx_Math" display="inline" id="S7.I4.i3.p1.6.m6.1"><semantics id="S7.I4.i3.p1.6.m6.1a"><mrow id="S7.I4.i3.p1.6.m6.1.1" xref="S7.I4.i3.p1.6.m6.1.1.cmml"><mrow id="S7.I4.i3.p1.6.m6.1.1.1" xref="S7.I4.i3.p1.6.m6.1.1.1.cmml"><mtext class="ltx_mathvariant_italic" id="S7.I4.i3.p1.6.m6.1.1.1.3" xref="S7.I4.i3.p1.6.m6.1.1.1.3a.cmml">g</mtext><mo id="S7.I4.i3.p1.6.m6.1.1.1.2" xref="S7.I4.i3.p1.6.m6.1.1.1.2.cmml">⁢</mo><mrow id="S7.I4.i3.p1.6.m6.1.1.1.1.1" xref="S7.I4.i3.p1.6.m6.1.1.1.1.1.1.cmml"><mo id="S7.I4.i3.p1.6.m6.1.1.1.1.1.2" stretchy="false" xref="S7.I4.i3.p1.6.m6.1.1.1.1.1.1.cmml">(</mo><mrow id="S7.I4.i3.p1.6.m6.1.1.1.1.1.1" xref="S7.I4.i3.p1.6.m6.1.1.1.1.1.1.cmml"><mi id="S7.I4.i3.p1.6.m6.1.1.1.1.1.1.2" xref="S7.I4.i3.p1.6.m6.1.1.1.1.1.1.2.cmml">w</mi><mo id="S7.I4.i3.p1.6.m6.1.1.1.1.1.1.1" lspace="0.108em" rspace="0.108em" stretchy="false" xref="S7.I4.i3.p1.6.m6.1.1.1.1.1.1.1.cmml">→</mo><mi id="S7.I4.i3.p1.6.m6.1.1.1.1.1.1.3" xref="S7.I4.i3.p1.6.m6.1.1.1.1.1.1.3.cmml">u</mi></mrow><mo id="S7.I4.i3.p1.6.m6.1.1.1.1.1.3" stretchy="false" xref="S7.I4.i3.p1.6.m6.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S7.I4.i3.p1.6.m6.1.1.2" xref="S7.I4.i3.p1.6.m6.1.1.2.cmml">></mo><mn id="S7.I4.i3.p1.6.m6.1.1.3" xref="S7.I4.i3.p1.6.m6.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S7.I4.i3.p1.6.m6.1b"><apply id="S7.I4.i3.p1.6.m6.1.1.cmml" xref="S7.I4.i3.p1.6.m6.1.1"><gt id="S7.I4.i3.p1.6.m6.1.1.2.cmml" xref="S7.I4.i3.p1.6.m6.1.1.2"></gt><apply id="S7.I4.i3.p1.6.m6.1.1.1.cmml" xref="S7.I4.i3.p1.6.m6.1.1.1"><times id="S7.I4.i3.p1.6.m6.1.1.1.2.cmml" xref="S7.I4.i3.p1.6.m6.1.1.1.2"></times><ci id="S7.I4.i3.p1.6.m6.1.1.1.3a.cmml" xref="S7.I4.i3.p1.6.m6.1.1.1.3"><mtext class="ltx_mathvariant_italic" id="S7.I4.i3.p1.6.m6.1.1.1.3.cmml" xref="S7.I4.i3.p1.6.m6.1.1.1.3">g</mtext></ci><apply id="S7.I4.i3.p1.6.m6.1.1.1.1.1.1.cmml" xref="S7.I4.i3.p1.6.m6.1.1.1.1.1"><ci id="S7.I4.i3.p1.6.m6.1.1.1.1.1.1.1.cmml" xref="S7.I4.i3.p1.6.m6.1.1.1.1.1.1.1">→</ci><ci id="S7.I4.i3.p1.6.m6.1.1.1.1.1.1.2.cmml" xref="S7.I4.i3.p1.6.m6.1.1.1.1.1.1.2">𝑤</ci><ci id="S7.I4.i3.p1.6.m6.1.1.1.1.1.1.3.cmml" xref="S7.I4.i3.p1.6.m6.1.1.1.1.1.1.3">𝑢</ci></apply></apply><cn id="S7.I4.i3.p1.6.m6.1.1.3.cmml" type="integer" xref="S7.I4.i3.p1.6.m6.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.I4.i3.p1.6.m6.1c">\textsl{g}(w\!\to\!u)>0</annotation><annotation encoding="application/x-llamapun" id="S7.I4.i3.p1.6.m6.1d">g ( italic_w → italic_u ) > 0</annotation></semantics></math> the edge <math alttext="(w,u)" class="ltx_Math" display="inline" id="S7.I4.i3.p1.7.m7.2"><semantics id="S7.I4.i3.p1.7.m7.2a"><mrow id="S7.I4.i3.p1.7.m7.2.3.2" xref="S7.I4.i3.p1.7.m7.2.3.1.cmml"><mo id="S7.I4.i3.p1.7.m7.2.3.2.1" stretchy="false" xref="S7.I4.i3.p1.7.m7.2.3.1.cmml">(</mo><mi id="S7.I4.i3.p1.7.m7.1.1" xref="S7.I4.i3.p1.7.m7.1.1.cmml">w</mi><mo id="S7.I4.i3.p1.7.m7.2.3.2.2" xref="S7.I4.i3.p1.7.m7.2.3.1.cmml">,</mo><mi id="S7.I4.i3.p1.7.m7.2.2" xref="S7.I4.i3.p1.7.m7.2.2.cmml">u</mi><mo id="S7.I4.i3.p1.7.m7.2.3.2.3" stretchy="false" xref="S7.I4.i3.p1.7.m7.2.3.1.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S7.I4.i3.p1.7.m7.2b"><interval closure="open" id="S7.I4.i3.p1.7.m7.2.3.1.cmml" xref="S7.I4.i3.p1.7.m7.2.3.2"><ci id="S7.I4.i3.p1.7.m7.1.1.cmml" xref="S7.I4.i3.p1.7.m7.1.1">𝑤</ci><ci id="S7.I4.i3.p1.7.m7.2.2.cmml" xref="S7.I4.i3.p1.7.m7.2.2">𝑢</ci></interval></annotation-xml><annotation encoding="application/x-tex" id="S7.I4.i3.p1.7.m7.2c">(w,u)</annotation><annotation encoding="application/x-llamapun" id="S7.I4.i3.p1.7.m7.2d">( italic_w , italic_u )</annotation></semantics></math> has become a violating in the previous <span class="ltx_text ltx_font_smallcaps" id="S7.I4.i3.p1.21.2">minute</span>. We want to fix these edges, whilst guaranteeing that we create no violating in-edges from vertices in level <math alttext="h+2" class="ltx_Math" display="inline" id="S7.I4.i3.p1.8.m8.1"><semantics id="S7.I4.i3.p1.8.m8.1a"><mrow id="S7.I4.i3.p1.8.m8.1.1" xref="S7.I4.i3.p1.8.m8.1.1.cmml"><mi id="S7.I4.i3.p1.8.m8.1.1.2" xref="S7.I4.i3.p1.8.m8.1.1.2.cmml">h</mi><mo id="S7.I4.i3.p1.8.m8.1.1.1" xref="S7.I4.i3.p1.8.m8.1.1.1.cmml">+</mo><mn id="S7.I4.i3.p1.8.m8.1.1.3" xref="S7.I4.i3.p1.8.m8.1.1.3.cmml">2</mn></mrow><annotation-xml encoding="MathML-Content" id="S7.I4.i3.p1.8.m8.1b"><apply id="S7.I4.i3.p1.8.m8.1.1.cmml" xref="S7.I4.i3.p1.8.m8.1.1"><plus id="S7.I4.i3.p1.8.m8.1.1.1.cmml" xref="S7.I4.i3.p1.8.m8.1.1.1"></plus><ci id="S7.I4.i3.p1.8.m8.1.1.2.cmml" xref="S7.I4.i3.p1.8.m8.1.1.2">ℎ</ci><cn id="S7.I4.i3.p1.8.m8.1.1.3.cmml" type="integer" xref="S7.I4.i3.p1.8.m8.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.I4.i3.p1.8.m8.1c">h+2</annotation><annotation encoding="application/x-llamapun" id="S7.I4.i3.p1.8.m8.1d">italic_h + 2</annotation></semantics></math> and above. <br class="ltx_break"/>We obtain this, by recording at the start of the <span class="ltx_text ltx_font_smallcaps" id="S7.I4.i3.p1.21.3">minute</span> for every vertex <math alttext="u" class="ltx_Math" display="inline" id="S7.I4.i3.p1.9.m9.1"><semantics id="S7.I4.i3.p1.9.m9.1a"><mi id="S7.I4.i3.p1.9.m9.1.1" xref="S7.I4.i3.p1.9.m9.1.1.cmml">u</mi><annotation-xml encoding="MathML-Content" id="S7.I4.i3.p1.9.m9.1b"><ci id="S7.I4.i3.p1.9.m9.1.1.cmml" xref="S7.I4.i3.p1.9.m9.1.1">𝑢</ci></annotation-xml><annotation encoding="application/x-tex" id="S7.I4.i3.p1.9.m9.1c">u</annotation><annotation encoding="application/x-llamapun" id="S7.I4.i3.p1.9.m9.1d">italic_u</annotation></semantics></math> its level <math alttext="l_{m}(u)" class="ltx_Math" display="inline" id="S7.I4.i3.p1.10.m10.1"><semantics id="S7.I4.i3.p1.10.m10.1a"><mrow id="S7.I4.i3.p1.10.m10.1.2" xref="S7.I4.i3.p1.10.m10.1.2.cmml"><msub id="S7.I4.i3.p1.10.m10.1.2.2" xref="S7.I4.i3.p1.10.m10.1.2.2.cmml"><mi id="S7.I4.i3.p1.10.m10.1.2.2.2" xref="S7.I4.i3.p1.10.m10.1.2.2.2.cmml">l</mi><mi id="S7.I4.i3.p1.10.m10.1.2.2.3" xref="S7.I4.i3.p1.10.m10.1.2.2.3.cmml">m</mi></msub><mo id="S7.I4.i3.p1.10.m10.1.2.1" xref="S7.I4.i3.p1.10.m10.1.2.1.cmml">⁢</mo><mrow id="S7.I4.i3.p1.10.m10.1.2.3.2" xref="S7.I4.i3.p1.10.m10.1.2.cmml"><mo id="S7.I4.i3.p1.10.m10.1.2.3.2.1" stretchy="false" xref="S7.I4.i3.p1.10.m10.1.2.cmml">(</mo><mi id="S7.I4.i3.p1.10.m10.1.1" xref="S7.I4.i3.p1.10.m10.1.1.cmml">u</mi><mo id="S7.I4.i3.p1.10.m10.1.2.3.2.2" stretchy="false" xref="S7.I4.i3.p1.10.m10.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S7.I4.i3.p1.10.m10.1b"><apply id="S7.I4.i3.p1.10.m10.1.2.cmml" xref="S7.I4.i3.p1.10.m10.1.2"><times id="S7.I4.i3.p1.10.m10.1.2.1.cmml" xref="S7.I4.i3.p1.10.m10.1.2.1"></times><apply id="S7.I4.i3.p1.10.m10.1.2.2.cmml" xref="S7.I4.i3.p1.10.m10.1.2.2"><csymbol cd="ambiguous" id="S7.I4.i3.p1.10.m10.1.2.2.1.cmml" xref="S7.I4.i3.p1.10.m10.1.2.2">subscript</csymbol><ci id="S7.I4.i3.p1.10.m10.1.2.2.2.cmml" xref="S7.I4.i3.p1.10.m10.1.2.2.2">𝑙</ci><ci id="S7.I4.i3.p1.10.m10.1.2.2.3.cmml" xref="S7.I4.i3.p1.10.m10.1.2.2.3">𝑚</ci></apply><ci id="S7.I4.i3.p1.10.m10.1.1.cmml" xref="S7.I4.i3.p1.10.m10.1.1">𝑢</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.I4.i3.p1.10.m10.1c">l_{m}(u)</annotation><annotation encoding="application/x-llamapun" id="S7.I4.i3.p1.10.m10.1d">italic_l start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT ( italic_u )</annotation></semantics></math>. We then invoke <math alttext="\ell-h+1" class="ltx_Math" display="inline" id="S7.I4.i3.p1.11.m11.1"><semantics id="S7.I4.i3.p1.11.m11.1a"><mrow id="S7.I4.i3.p1.11.m11.1.1" xref="S7.I4.i3.p1.11.m11.1.1.cmml"><mrow id="S7.I4.i3.p1.11.m11.1.1.2" xref="S7.I4.i3.p1.11.m11.1.1.2.cmml"><mi id="S7.I4.i3.p1.11.m11.1.1.2.2" mathvariant="normal" xref="S7.I4.i3.p1.11.m11.1.1.2.2.cmml">ℓ</mi><mo id="S7.I4.i3.p1.11.m11.1.1.2.1" xref="S7.I4.i3.p1.11.m11.1.1.2.1.cmml">−</mo><mi id="S7.I4.i3.p1.11.m11.1.1.2.3" xref="S7.I4.i3.p1.11.m11.1.1.2.3.cmml">h</mi></mrow><mo id="S7.I4.i3.p1.11.m11.1.1.1" xref="S7.I4.i3.p1.11.m11.1.1.1.cmml">+</mo><mn id="S7.I4.i3.p1.11.m11.1.1.3" xref="S7.I4.i3.p1.11.m11.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S7.I4.i3.p1.11.m11.1b"><apply id="S7.I4.i3.p1.11.m11.1.1.cmml" xref="S7.I4.i3.p1.11.m11.1.1"><plus id="S7.I4.i3.p1.11.m11.1.1.1.cmml" xref="S7.I4.i3.p1.11.m11.1.1.1"></plus><apply id="S7.I4.i3.p1.11.m11.1.1.2.cmml" xref="S7.I4.i3.p1.11.m11.1.1.2"><minus id="S7.I4.i3.p1.11.m11.1.1.2.1.cmml" xref="S7.I4.i3.p1.11.m11.1.1.2.1"></minus><ci id="S7.I4.i3.p1.11.m11.1.1.2.2.cmml" xref="S7.I4.i3.p1.11.m11.1.1.2.2">ℓ</ci><ci id="S7.I4.i3.p1.11.m11.1.1.2.3.cmml" xref="S7.I4.i3.p1.11.m11.1.1.2.3">ℎ</ci></apply><cn id="S7.I4.i3.p1.11.m11.1.1.3.cmml" type="integer" xref="S7.I4.i3.p1.11.m11.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.I4.i3.p1.11.m11.1c">\ell-h+1</annotation><annotation encoding="application/x-llamapun" id="S7.I4.i3.p1.11.m11.1d">roman_ℓ - italic_h + 1</annotation></semantics></math> <span class="ltx_text ltx_font_smallcaps" id="S7.I4.i3.p1.21.4">seconds</span>. Each <span class="ltx_text ltx_font_smallcaps" id="S7.I4.i3.p1.21.5">second</span> <math alttext="s" class="ltx_Math" display="inline" id="S7.I4.i3.p1.12.m12.1"><semantics id="S7.I4.i3.p1.12.m12.1a"><mi id="S7.I4.i3.p1.12.m12.1.1" xref="S7.I4.i3.p1.12.m12.1.1.cmml">s</mi><annotation-xml encoding="MathML-Content" id="S7.I4.i3.p1.12.m12.1b"><ci id="S7.I4.i3.p1.12.m12.1.1.cmml" xref="S7.I4.i3.p1.12.m12.1.1">𝑠</ci></annotation-xml><annotation encoding="application/x-tex" id="S7.I4.i3.p1.12.m12.1c">s</annotation><annotation encoding="application/x-llamapun" id="S7.I4.i3.p1.12.m12.1d">italic_s</annotation></semantics></math>, we create a DAG <math alttext="D_{s}" class="ltx_Math" display="inline" id="S7.I4.i3.p1.13.m13.1"><semantics id="S7.I4.i3.p1.13.m13.1a"><msub id="S7.I4.i3.p1.13.m13.1.1" xref="S7.I4.i3.p1.13.m13.1.1.cmml"><mi id="S7.I4.i3.p1.13.m13.1.1.2" xref="S7.I4.i3.p1.13.m13.1.1.2.cmml">D</mi><mi id="S7.I4.i3.p1.13.m13.1.1.3" xref="S7.I4.i3.p1.13.m13.1.1.3.cmml">s</mi></msub><annotation-xml encoding="MathML-Content" id="S7.I4.i3.p1.13.m13.1b"><apply id="S7.I4.i3.p1.13.m13.1.1.cmml" xref="S7.I4.i3.p1.13.m13.1.1"><csymbol cd="ambiguous" id="S7.I4.i3.p1.13.m13.1.1.1.cmml" xref="S7.I4.i3.p1.13.m13.1.1">subscript</csymbol><ci id="S7.I4.i3.p1.13.m13.1.1.2.cmml" xref="S7.I4.i3.p1.13.m13.1.1.2">𝐷</ci><ci id="S7.I4.i3.p1.13.m13.1.1.3.cmml" xref="S7.I4.i3.p1.13.m13.1.1.3">𝑠</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.I4.i3.p1.13.m13.1c">D_{s}</annotation><annotation encoding="application/x-llamapun" id="S7.I4.i3.p1.13.m13.1d">italic_D start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT</annotation></semantics></math> where the sinks are <math alttext="T_{m}" class="ltx_Math" display="inline" id="S7.I4.i3.p1.14.m14.1"><semantics id="S7.I4.i3.p1.14.m14.1a"><msub id="S7.I4.i3.p1.14.m14.1.1" xref="S7.I4.i3.p1.14.m14.1.1.cmml"><mi id="S7.I4.i3.p1.14.m14.1.1.2" xref="S7.I4.i3.p1.14.m14.1.1.2.cmml">T</mi><mi id="S7.I4.i3.p1.14.m14.1.1.3" xref="S7.I4.i3.p1.14.m14.1.1.3.cmml">m</mi></msub><annotation-xml encoding="MathML-Content" id="S7.I4.i3.p1.14.m14.1b"><apply id="S7.I4.i3.p1.14.m14.1.1.cmml" xref="S7.I4.i3.p1.14.m14.1.1"><csymbol cd="ambiguous" id="S7.I4.i3.p1.14.m14.1.1.1.cmml" xref="S7.I4.i3.p1.14.m14.1.1">subscript</csymbol><ci id="S7.I4.i3.p1.14.m14.1.1.2.cmml" xref="S7.I4.i3.p1.14.m14.1.1.2">𝑇</ci><ci id="S7.I4.i3.p1.14.m14.1.1.3.cmml" xref="S7.I4.i3.p1.14.m14.1.1.3">𝑚</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.I4.i3.p1.14.m14.1c">T_{m}</annotation><annotation encoding="application/x-llamapun" id="S7.I4.i3.p1.14.m14.1d">italic_T start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT</annotation></semantics></math>. We increase for all <math alttext="u\in T_{m}" class="ltx_Math" display="inline" id="S7.I4.i3.p1.15.m15.1"><semantics id="S7.I4.i3.p1.15.m15.1a"><mrow id="S7.I4.i3.p1.15.m15.1.1" xref="S7.I4.i3.p1.15.m15.1.1.cmml"><mi id="S7.I4.i3.p1.15.m15.1.1.2" xref="S7.I4.i3.p1.15.m15.1.1.2.cmml">u</mi><mo id="S7.I4.i3.p1.15.m15.1.1.1" xref="S7.I4.i3.p1.15.m15.1.1.1.cmml">∈</mo><msub id="S7.I4.i3.p1.15.m15.1.1.3" xref="S7.I4.i3.p1.15.m15.1.1.3.cmml"><mi id="S7.I4.i3.p1.15.m15.1.1.3.2" xref="S7.I4.i3.p1.15.m15.1.1.3.2.cmml">T</mi><mi id="S7.I4.i3.p1.15.m15.1.1.3.3" xref="S7.I4.i3.p1.15.m15.1.1.3.3.cmml">m</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S7.I4.i3.p1.15.m15.1b"><apply id="S7.I4.i3.p1.15.m15.1.1.cmml" xref="S7.I4.i3.p1.15.m15.1.1"><in id="S7.I4.i3.p1.15.m15.1.1.1.cmml" xref="S7.I4.i3.p1.15.m15.1.1.1"></in><ci id="S7.I4.i3.p1.15.m15.1.1.2.cmml" xref="S7.I4.i3.p1.15.m15.1.1.2">𝑢</ci><apply id="S7.I4.i3.p1.15.m15.1.1.3.cmml" xref="S7.I4.i3.p1.15.m15.1.1.3"><csymbol cd="ambiguous" id="S7.I4.i3.p1.15.m15.1.1.3.1.cmml" xref="S7.I4.i3.p1.15.m15.1.1.3">subscript</csymbol><ci id="S7.I4.i3.p1.15.m15.1.1.3.2.cmml" xref="S7.I4.i3.p1.15.m15.1.1.3.2">𝑇</ci><ci id="S7.I4.i3.p1.15.m15.1.1.3.3.cmml" xref="S7.I4.i3.p1.15.m15.1.1.3.3">𝑚</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.I4.i3.p1.15.m15.1c">u\in T_{m}</annotation><annotation encoding="application/x-llamapun" id="S7.I4.i3.p1.15.m15.1d">italic_u ∈ italic_T start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT</annotation></semantics></math> the out-degree <math alttext="\textsl{g}(u)" class="ltx_Math" display="inline" id="S7.I4.i3.p1.16.m16.1"><semantics id="S7.I4.i3.p1.16.m16.1a"><mrow id="S7.I4.i3.p1.16.m16.1.2" xref="S7.I4.i3.p1.16.m16.1.2.cmml"><mtext class="ltx_mathvariant_italic" id="S7.I4.i3.p1.16.m16.1.2.2" xref="S7.I4.i3.p1.16.m16.1.2.2a.cmml">g</mtext><mo id="S7.I4.i3.p1.16.m16.1.2.1" xref="S7.I4.i3.p1.16.m16.1.2.1.cmml">⁢</mo><mrow id="S7.I4.i3.p1.16.m16.1.2.3.2" xref="S7.I4.i3.p1.16.m16.1.2.cmml"><mo id="S7.I4.i3.p1.16.m16.1.2.3.2.1" stretchy="false" xref="S7.I4.i3.p1.16.m16.1.2.cmml">(</mo><mi id="S7.I4.i3.p1.16.m16.1.1" xref="S7.I4.i3.p1.16.m16.1.1.cmml">u</mi><mo id="S7.I4.i3.p1.16.m16.1.2.3.2.2" stretchy="false" xref="S7.I4.i3.p1.16.m16.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S7.I4.i3.p1.16.m16.1b"><apply id="S7.I4.i3.p1.16.m16.1.2.cmml" xref="S7.I4.i3.p1.16.m16.1.2"><times id="S7.I4.i3.p1.16.m16.1.2.1.cmml" xref="S7.I4.i3.p1.16.m16.1.2.1"></times><ci id="S7.I4.i3.p1.16.m16.1.2.2a.cmml" xref="S7.I4.i3.p1.16.m16.1.2.2"><mtext class="ltx_mathvariant_italic" id="S7.I4.i3.p1.16.m16.1.2.2.cmml" xref="S7.I4.i3.p1.16.m16.1.2.2">g</mtext></ci><ci id="S7.I4.i3.p1.16.m16.1.1.cmml" xref="S7.I4.i3.p1.16.m16.1.1">𝑢</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.I4.i3.p1.16.m16.1c">\textsl{g}(u)</annotation><annotation encoding="application/x-llamapun" id="S7.I4.i3.p1.16.m16.1d">g ( italic_u )</annotation></semantics></math> by flipping a directed path from a source in <math alttext="D_{s}" class="ltx_Math" display="inline" id="S7.I4.i3.p1.17.m17.1"><semantics id="S7.I4.i3.p1.17.m17.1a"><msub id="S7.I4.i3.p1.17.m17.1.1" xref="S7.I4.i3.p1.17.m17.1.1.cmml"><mi id="S7.I4.i3.p1.17.m17.1.1.2" xref="S7.I4.i3.p1.17.m17.1.1.2.cmml">D</mi><mi id="S7.I4.i3.p1.17.m17.1.1.3" xref="S7.I4.i3.p1.17.m17.1.1.3.cmml">s</mi></msub><annotation-xml encoding="MathML-Content" id="S7.I4.i3.p1.17.m17.1b"><apply id="S7.I4.i3.p1.17.m17.1.1.cmml" xref="S7.I4.i3.p1.17.m17.1.1"><csymbol cd="ambiguous" id="S7.I4.i3.p1.17.m17.1.1.1.cmml" xref="S7.I4.i3.p1.17.m17.1.1">subscript</csymbol><ci id="S7.I4.i3.p1.17.m17.1.1.2.cmml" xref="S7.I4.i3.p1.17.m17.1.1.2">𝐷</ci><ci id="S7.I4.i3.p1.17.m17.1.1.3.cmml" xref="S7.I4.i3.p1.17.m17.1.1.3">𝑠</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.I4.i3.p1.17.m17.1c">D_{s}</annotation><annotation encoding="application/x-llamapun" id="S7.I4.i3.p1.17.m17.1d">italic_D start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT</annotation></semantics></math> to <math alttext="T_{m}" class="ltx_Math" display="inline" id="S7.I4.i3.p1.18.m18.1"><semantics id="S7.I4.i3.p1.18.m18.1a"><msub id="S7.I4.i3.p1.18.m18.1.1" xref="S7.I4.i3.p1.18.m18.1.1.cmml"><mi id="S7.I4.i3.p1.18.m18.1.1.2" xref="S7.I4.i3.p1.18.m18.1.1.2.cmml">T</mi><mi id="S7.I4.i3.p1.18.m18.1.1.3" xref="S7.I4.i3.p1.18.m18.1.1.3.cmml">m</mi></msub><annotation-xml encoding="MathML-Content" id="S7.I4.i3.p1.18.m18.1b"><apply id="S7.I4.i3.p1.18.m18.1.1.cmml" xref="S7.I4.i3.p1.18.m18.1.1"><csymbol cd="ambiguous" id="S7.I4.i3.p1.18.m18.1.1.1.cmml" xref="S7.I4.i3.p1.18.m18.1.1">subscript</csymbol><ci id="S7.I4.i3.p1.18.m18.1.1.2.cmml" xref="S7.I4.i3.p1.18.m18.1.1.2">𝑇</ci><ci id="S7.I4.i3.p1.18.m18.1.1.3.cmml" xref="S7.I4.i3.p1.18.m18.1.1.3">𝑚</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.I4.i3.p1.18.m18.1c">T_{m}</annotation><annotation encoding="application/x-llamapun" id="S7.I4.i3.p1.18.m18.1d">italic_T start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT</annotation></semantics></math>. We construct our DAG in such a manner that this procedure does not create new violating in-edges, and that for all <span class="ltx_text ltx_font_smallcaps" id="S7.I4.i3.p1.21.6">seconds</span> <math alttext="s" class="ltx_Math" display="inline" id="S7.I4.i3.p1.19.m19.1"><semantics id="S7.I4.i3.p1.19.m19.1a"><mi id="S7.I4.i3.p1.19.m19.1.1" xref="S7.I4.i3.p1.19.m19.1.1.cmml">s</mi><annotation-xml encoding="MathML-Content" id="S7.I4.i3.p1.19.m19.1b"><ci id="S7.I4.i3.p1.19.m19.1.1.cmml" xref="S7.I4.i3.p1.19.m19.1.1">𝑠</ci></annotation-xml><annotation encoding="application/x-tex" id="S7.I4.i3.p1.19.m19.1c">s</annotation><annotation encoding="application/x-llamapun" id="S7.I4.i3.p1.19.m19.1d">italic_s</annotation></semantics></math>, <math alttext="D_{s+1}" class="ltx_Math" display="inline" id="S7.I4.i3.p1.20.m20.1"><semantics id="S7.I4.i3.p1.20.m20.1a"><msub id="S7.I4.i3.p1.20.m20.1.1" xref="S7.I4.i3.p1.20.m20.1.1.cmml"><mi id="S7.I4.i3.p1.20.m20.1.1.2" xref="S7.I4.i3.p1.20.m20.1.1.2.cmml">D</mi><mrow id="S7.I4.i3.p1.20.m20.1.1.3" xref="S7.I4.i3.p1.20.m20.1.1.3.cmml"><mi id="S7.I4.i3.p1.20.m20.1.1.3.2" xref="S7.I4.i3.p1.20.m20.1.1.3.2.cmml">s</mi><mo id="S7.I4.i3.p1.20.m20.1.1.3.1" xref="S7.I4.i3.p1.20.m20.1.1.3.1.cmml">+</mo><mn id="S7.I4.i3.p1.20.m20.1.1.3.3" xref="S7.I4.i3.p1.20.m20.1.1.3.3.cmml">1</mn></mrow></msub><annotation-xml encoding="MathML-Content" id="S7.I4.i3.p1.20.m20.1b"><apply id="S7.I4.i3.p1.20.m20.1.1.cmml" xref="S7.I4.i3.p1.20.m20.1.1"><csymbol cd="ambiguous" id="S7.I4.i3.p1.20.m20.1.1.1.cmml" xref="S7.I4.i3.p1.20.m20.1.1">subscript</csymbol><ci id="S7.I4.i3.p1.20.m20.1.1.2.cmml" xref="S7.I4.i3.p1.20.m20.1.1.2">𝐷</ci><apply id="S7.I4.i3.p1.20.m20.1.1.3.cmml" xref="S7.I4.i3.p1.20.m20.1.1.3"><plus id="S7.I4.i3.p1.20.m20.1.1.3.1.cmml" xref="S7.I4.i3.p1.20.m20.1.1.3.1"></plus><ci id="S7.I4.i3.p1.20.m20.1.1.3.2.cmml" xref="S7.I4.i3.p1.20.m20.1.1.3.2">𝑠</ci><cn id="S7.I4.i3.p1.20.m20.1.1.3.3.cmml" type="integer" xref="S7.I4.i3.p1.20.m20.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.I4.i3.p1.20.m20.1c">D_{s+1}</annotation><annotation encoding="application/x-llamapun" id="S7.I4.i3.p1.20.m20.1d">italic_D start_POSTSUBSCRIPT italic_s + 1 end_POSTSUBSCRIPT</annotation></semantics></math> is a subgraph of <math alttext="D_{s}" class="ltx_Math" display="inline" id="S7.I4.i3.p1.21.m21.1"><semantics id="S7.I4.i3.p1.21.m21.1a"><msub id="S7.I4.i3.p1.21.m21.1.1" xref="S7.I4.i3.p1.21.m21.1.1.cmml"><mi id="S7.I4.i3.p1.21.m21.1.1.2" xref="S7.I4.i3.p1.21.m21.1.1.2.cmml">D</mi><mi id="S7.I4.i3.p1.21.m21.1.1.3" xref="S7.I4.i3.p1.21.m21.1.1.3.cmml">s</mi></msub><annotation-xml encoding="MathML-Content" id="S7.I4.i3.p1.21.m21.1b"><apply id="S7.I4.i3.p1.21.m21.1.1.cmml" xref="S7.I4.i3.p1.21.m21.1.1"><csymbol cd="ambiguous" id="S7.I4.i3.p1.21.m21.1.1.1.cmml" xref="S7.I4.i3.p1.21.m21.1.1">subscript</csymbol><ci id="S7.I4.i3.p1.21.m21.1.1.2.cmml" xref="S7.I4.i3.p1.21.m21.1.1.2">𝐷</ci><ci id="S7.I4.i3.p1.21.m21.1.1.3.cmml" xref="S7.I4.i3.p1.21.m21.1.1.3">𝑠</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.I4.i3.p1.21.m21.1c">D_{s}</annotation><annotation encoding="application/x-llamapun" id="S7.I4.i3.p1.21.m21.1d">italic_D start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT</annotation></semantics></math>.</p> </div> </li> <li class="ltx_item" id="S7.I4.i4" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">•</span> <div class="ltx_para" id="S7.I4.i4.p1"> <p class="ltx_p" id="S7.I4.i4.p1.9">At the start of <span class="ltx_text ltx_font_smallcaps" id="S7.I4.i4.p1.9.1">second</span> <math alttext="s" class="ltx_Math" display="inline" id="S7.I4.i4.p1.1.m1.1"><semantics id="S7.I4.i4.p1.1.m1.1a"><mi id="S7.I4.i4.p1.1.m1.1.1" xref="S7.I4.i4.p1.1.m1.1.1.cmml">s</mi><annotation-xml encoding="MathML-Content" id="S7.I4.i4.p1.1.m1.1b"><ci id="S7.I4.i4.p1.1.m1.1.1.cmml" xref="S7.I4.i4.p1.1.m1.1.1">𝑠</ci></annotation-xml><annotation encoding="application/x-tex" id="S7.I4.i4.p1.1.m1.1c">s</annotation><annotation encoding="application/x-llamapun" id="S7.I4.i4.p1.1.m1.1d">italic_s</annotation></semantics></math> of each <span class="ltx_text ltx_font_smallcaps" id="S7.I4.i4.p1.9.2">minute</span> <math alttext="m" class="ltx_Math" display="inline" id="S7.I4.i4.p1.2.m2.1"><semantics id="S7.I4.i4.p1.2.m2.1a"><mi id="S7.I4.i4.p1.2.m2.1.1" xref="S7.I4.i4.p1.2.m2.1.1.cmml">m</mi><annotation-xml encoding="MathML-Content" id="S7.I4.i4.p1.2.m2.1b"><ci id="S7.I4.i4.p1.2.m2.1.1.cmml" xref="S7.I4.i4.p1.2.m2.1.1">𝑚</ci></annotation-xml><annotation encoding="application/x-tex" id="S7.I4.i4.p1.2.m2.1c">m</annotation><annotation encoding="application/x-llamapun" id="S7.I4.i4.p1.2.m2.1d">italic_m</annotation></semantics></math>, we fix for every edge <math alttext="\overline{uv}" class="ltx_Math" display="inline" id="S7.I4.i4.p1.3.m3.1"><semantics id="S7.I4.i4.p1.3.m3.1a"><mover accent="true" id="S7.I4.i4.p1.3.m3.1.1" xref="S7.I4.i4.p1.3.m3.1.1.cmml"><mrow id="S7.I4.i4.p1.3.m3.1.1.2" xref="S7.I4.i4.p1.3.m3.1.1.2.cmml"><mi id="S7.I4.i4.p1.3.m3.1.1.2.2" xref="S7.I4.i4.p1.3.m3.1.1.2.2.cmml">u</mi><mo id="S7.I4.i4.p1.3.m3.1.1.2.1" xref="S7.I4.i4.p1.3.m3.1.1.2.1.cmml">⁢</mo><mi id="S7.I4.i4.p1.3.m3.1.1.2.3" xref="S7.I4.i4.p1.3.m3.1.1.2.3.cmml">v</mi></mrow><mo id="S7.I4.i4.p1.3.m3.1.1.1" xref="S7.I4.i4.p1.3.m3.1.1.1.cmml">¯</mo></mover><annotation-xml encoding="MathML-Content" id="S7.I4.i4.p1.3.m3.1b"><apply id="S7.I4.i4.p1.3.m3.1.1.cmml" xref="S7.I4.i4.p1.3.m3.1.1"><ci id="S7.I4.i4.p1.3.m3.1.1.1.cmml" xref="S7.I4.i4.p1.3.m3.1.1.1">¯</ci><apply id="S7.I4.i4.p1.3.m3.1.1.2.cmml" xref="S7.I4.i4.p1.3.m3.1.1.2"><times id="S7.I4.i4.p1.3.m3.1.1.2.1.cmml" xref="S7.I4.i4.p1.3.m3.1.1.2.1"></times><ci id="S7.I4.i4.p1.3.m3.1.1.2.2.cmml" xref="S7.I4.i4.p1.3.m3.1.1.2.2">𝑢</ci><ci id="S7.I4.i4.p1.3.m3.1.1.2.3.cmml" xref="S7.I4.i4.p1.3.m3.1.1.2.3">𝑣</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.I4.i4.p1.3.m3.1c">\overline{uv}</annotation><annotation encoding="application/x-llamapun" id="S7.I4.i4.p1.3.m3.1d">over¯ start_ARG italic_u italic_v end_ARG</annotation></semantics></math> the values <math alttext="\textsl{g}_{s}(u\!\to\!v)" class="ltx_Math" display="inline" id="S7.I4.i4.p1.4.m4.1"><semantics id="S7.I4.i4.p1.4.m4.1a"><mrow id="S7.I4.i4.p1.4.m4.1.1" xref="S7.I4.i4.p1.4.m4.1.1.cmml"><msub id="S7.I4.i4.p1.4.m4.1.1.3" xref="S7.I4.i4.p1.4.m4.1.1.3.cmml"><mtext class="ltx_mathvariant_italic" id="S7.I4.i4.p1.4.m4.1.1.3.2" xref="S7.I4.i4.p1.4.m4.1.1.3.2a.cmml">g</mtext><mi id="S7.I4.i4.p1.4.m4.1.1.3.3" xref="S7.I4.i4.p1.4.m4.1.1.3.3.cmml">s</mi></msub><mo id="S7.I4.i4.p1.4.m4.1.1.2" xref="S7.I4.i4.p1.4.m4.1.1.2.cmml">⁢</mo><mrow id="S7.I4.i4.p1.4.m4.1.1.1.1" xref="S7.I4.i4.p1.4.m4.1.1.1.1.1.cmml"><mo id="S7.I4.i4.p1.4.m4.1.1.1.1.2" stretchy="false" xref="S7.I4.i4.p1.4.m4.1.1.1.1.1.cmml">(</mo><mrow id="S7.I4.i4.p1.4.m4.1.1.1.1.1" xref="S7.I4.i4.p1.4.m4.1.1.1.1.1.cmml"><mi id="S7.I4.i4.p1.4.m4.1.1.1.1.1.2" xref="S7.I4.i4.p1.4.m4.1.1.1.1.1.2.cmml">u</mi><mo id="S7.I4.i4.p1.4.m4.1.1.1.1.1.1" lspace="0.108em" rspace="0.108em" stretchy="false" xref="S7.I4.i4.p1.4.m4.1.1.1.1.1.1.cmml">→</mo><mi id="S7.I4.i4.p1.4.m4.1.1.1.1.1.3" xref="S7.I4.i4.p1.4.m4.1.1.1.1.1.3.cmml">v</mi></mrow><mo id="S7.I4.i4.p1.4.m4.1.1.1.1.3" stretchy="false" xref="S7.I4.i4.p1.4.m4.1.1.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S7.I4.i4.p1.4.m4.1b"><apply id="S7.I4.i4.p1.4.m4.1.1.cmml" xref="S7.I4.i4.p1.4.m4.1.1"><times id="S7.I4.i4.p1.4.m4.1.1.2.cmml" xref="S7.I4.i4.p1.4.m4.1.1.2"></times><apply id="S7.I4.i4.p1.4.m4.1.1.3.cmml" xref="S7.I4.i4.p1.4.m4.1.1.3"><csymbol cd="ambiguous" id="S7.I4.i4.p1.4.m4.1.1.3.1.cmml" xref="S7.I4.i4.p1.4.m4.1.1.3">subscript</csymbol><ci id="S7.I4.i4.p1.4.m4.1.1.3.2a.cmml" xref="S7.I4.i4.p1.4.m4.1.1.3.2"><mtext class="ltx_mathvariant_italic" id="S7.I4.i4.p1.4.m4.1.1.3.2.cmml" xref="S7.I4.i4.p1.4.m4.1.1.3.2">g</mtext></ci><ci id="S7.I4.i4.p1.4.m4.1.1.3.3.cmml" xref="S7.I4.i4.p1.4.m4.1.1.3.3">𝑠</ci></apply><apply id="S7.I4.i4.p1.4.m4.1.1.1.1.1.cmml" xref="S7.I4.i4.p1.4.m4.1.1.1.1"><ci id="S7.I4.i4.p1.4.m4.1.1.1.1.1.1.cmml" xref="S7.I4.i4.p1.4.m4.1.1.1.1.1.1">→</ci><ci id="S7.I4.i4.p1.4.m4.1.1.1.1.1.2.cmml" xref="S7.I4.i4.p1.4.m4.1.1.1.1.1.2">𝑢</ci><ci id="S7.I4.i4.p1.4.m4.1.1.1.1.1.3.cmml" xref="S7.I4.i4.p1.4.m4.1.1.1.1.1.3">𝑣</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.I4.i4.p1.4.m4.1c">\textsl{g}_{s}(u\!\to\!v)</annotation><annotation encoding="application/x-llamapun" id="S7.I4.i4.p1.4.m4.1d">g start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_u → italic_v )</annotation></semantics></math>, and the levels <math alttext="l_{s}(u)" class="ltx_Math" display="inline" id="S7.I4.i4.p1.5.m5.1"><semantics id="S7.I4.i4.p1.5.m5.1a"><mrow id="S7.I4.i4.p1.5.m5.1.2" xref="S7.I4.i4.p1.5.m5.1.2.cmml"><msub id="S7.I4.i4.p1.5.m5.1.2.2" xref="S7.I4.i4.p1.5.m5.1.2.2.cmml"><mi id="S7.I4.i4.p1.5.m5.1.2.2.2" xref="S7.I4.i4.p1.5.m5.1.2.2.2.cmml">l</mi><mi id="S7.I4.i4.p1.5.m5.1.2.2.3" xref="S7.I4.i4.p1.5.m5.1.2.2.3.cmml">s</mi></msub><mo id="S7.I4.i4.p1.5.m5.1.2.1" xref="S7.I4.i4.p1.5.m5.1.2.1.cmml">⁢</mo><mrow id="S7.I4.i4.p1.5.m5.1.2.3.2" xref="S7.I4.i4.p1.5.m5.1.2.cmml"><mo id="S7.I4.i4.p1.5.m5.1.2.3.2.1" stretchy="false" xref="S7.I4.i4.p1.5.m5.1.2.cmml">(</mo><mi id="S7.I4.i4.p1.5.m5.1.1" xref="S7.I4.i4.p1.5.m5.1.1.cmml">u</mi><mo id="S7.I4.i4.p1.5.m5.1.2.3.2.2" stretchy="false" xref="S7.I4.i4.p1.5.m5.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S7.I4.i4.p1.5.m5.1b"><apply id="S7.I4.i4.p1.5.m5.1.2.cmml" xref="S7.I4.i4.p1.5.m5.1.2"><times id="S7.I4.i4.p1.5.m5.1.2.1.cmml" xref="S7.I4.i4.p1.5.m5.1.2.1"></times><apply id="S7.I4.i4.p1.5.m5.1.2.2.cmml" xref="S7.I4.i4.p1.5.m5.1.2.2"><csymbol cd="ambiguous" id="S7.I4.i4.p1.5.m5.1.2.2.1.cmml" xref="S7.I4.i4.p1.5.m5.1.2.2">subscript</csymbol><ci id="S7.I4.i4.p1.5.m5.1.2.2.2.cmml" xref="S7.I4.i4.p1.5.m5.1.2.2.2">𝑙</ci><ci id="S7.I4.i4.p1.5.m5.1.2.2.3.cmml" xref="S7.I4.i4.p1.5.m5.1.2.2.3">𝑠</ci></apply><ci id="S7.I4.i4.p1.5.m5.1.1.cmml" xref="S7.I4.i4.p1.5.m5.1.1">𝑢</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.I4.i4.p1.5.m5.1c">l_{s}(u)</annotation><annotation encoding="application/x-llamapun" id="S7.I4.i4.p1.5.m5.1d">italic_l start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_u )</annotation></semantics></math> and <math alttext="l_{s}(v)" class="ltx_Math" display="inline" id="S7.I4.i4.p1.6.m6.1"><semantics id="S7.I4.i4.p1.6.m6.1a"><mrow id="S7.I4.i4.p1.6.m6.1.2" xref="S7.I4.i4.p1.6.m6.1.2.cmml"><msub id="S7.I4.i4.p1.6.m6.1.2.2" xref="S7.I4.i4.p1.6.m6.1.2.2.cmml"><mi id="S7.I4.i4.p1.6.m6.1.2.2.2" xref="S7.I4.i4.p1.6.m6.1.2.2.2.cmml">l</mi><mi id="S7.I4.i4.p1.6.m6.1.2.2.3" xref="S7.I4.i4.p1.6.m6.1.2.2.3.cmml">s</mi></msub><mo id="S7.I4.i4.p1.6.m6.1.2.1" xref="S7.I4.i4.p1.6.m6.1.2.1.cmml">⁢</mo><mrow id="S7.I4.i4.p1.6.m6.1.2.3.2" xref="S7.I4.i4.p1.6.m6.1.2.cmml"><mo id="S7.I4.i4.p1.6.m6.1.2.3.2.1" stretchy="false" xref="S7.I4.i4.p1.6.m6.1.2.cmml">(</mo><mi id="S7.I4.i4.p1.6.m6.1.1" xref="S7.I4.i4.p1.6.m6.1.1.cmml">v</mi><mo id="S7.I4.i4.p1.6.m6.1.2.3.2.2" stretchy="false" xref="S7.I4.i4.p1.6.m6.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S7.I4.i4.p1.6.m6.1b"><apply id="S7.I4.i4.p1.6.m6.1.2.cmml" xref="S7.I4.i4.p1.6.m6.1.2"><times id="S7.I4.i4.p1.6.m6.1.2.1.cmml" xref="S7.I4.i4.p1.6.m6.1.2.1"></times><apply id="S7.I4.i4.p1.6.m6.1.2.2.cmml" xref="S7.I4.i4.p1.6.m6.1.2.2"><csymbol cd="ambiguous" id="S7.I4.i4.p1.6.m6.1.2.2.1.cmml" xref="S7.I4.i4.p1.6.m6.1.2.2">subscript</csymbol><ci id="S7.I4.i4.p1.6.m6.1.2.2.2.cmml" xref="S7.I4.i4.p1.6.m6.1.2.2.2">𝑙</ci><ci id="S7.I4.i4.p1.6.m6.1.2.2.3.cmml" xref="S7.I4.i4.p1.6.m6.1.2.2.3">𝑠</ci></apply><ci id="S7.I4.i4.p1.6.m6.1.1.cmml" xref="S7.I4.i4.p1.6.m6.1.1">𝑣</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.I4.i4.p1.6.m6.1c">l_{s}(v)</annotation><annotation encoding="application/x-llamapun" id="S7.I4.i4.p1.6.m6.1d">italic_l start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_v )</annotation></semantics></math> of <math alttext="u" class="ltx_Math" display="inline" id="S7.I4.i4.p1.7.m7.1"><semantics id="S7.I4.i4.p1.7.m7.1a"><mi id="S7.I4.i4.p1.7.m7.1.1" xref="S7.I4.i4.p1.7.m7.1.1.cmml">u</mi><annotation-xml encoding="MathML-Content" id="S7.I4.i4.p1.7.m7.1b"><ci id="S7.I4.i4.p1.7.m7.1.1.cmml" xref="S7.I4.i4.p1.7.m7.1.1">𝑢</ci></annotation-xml><annotation encoding="application/x-tex" id="S7.I4.i4.p1.7.m7.1c">u</annotation><annotation encoding="application/x-llamapun" id="S7.I4.i4.p1.7.m7.1d">italic_u</annotation></semantics></math> and <math alttext="v" class="ltx_Math" display="inline" id="S7.I4.i4.p1.8.m8.1"><semantics id="S7.I4.i4.p1.8.m8.1a"><mi id="S7.I4.i4.p1.8.m8.1.1" xref="S7.I4.i4.p1.8.m8.1.1.cmml">v</mi><annotation-xml encoding="MathML-Content" id="S7.I4.i4.p1.8.m8.1b"><ci id="S7.I4.i4.p1.8.m8.1.1.cmml" xref="S7.I4.i4.p1.8.m8.1.1">𝑣</ci></annotation-xml><annotation encoding="application/x-tex" id="S7.I4.i4.p1.8.m8.1c">v</annotation><annotation encoding="application/x-llamapun" id="S7.I4.i4.p1.8.m8.1d">italic_v</annotation></semantics></math>. We then define a graph <math alttext="D_{s}=(V_{s},E_{s})" class="ltx_Math" display="inline" id="S7.I4.i4.p1.9.m9.2"><semantics id="S7.I4.i4.p1.9.m9.2a"><mrow id="S7.I4.i4.p1.9.m9.2.2" xref="S7.I4.i4.p1.9.m9.2.2.cmml"><msub id="S7.I4.i4.p1.9.m9.2.2.4" xref="S7.I4.i4.p1.9.m9.2.2.4.cmml"><mi id="S7.I4.i4.p1.9.m9.2.2.4.2" xref="S7.I4.i4.p1.9.m9.2.2.4.2.cmml">D</mi><mi id="S7.I4.i4.p1.9.m9.2.2.4.3" xref="S7.I4.i4.p1.9.m9.2.2.4.3.cmml">s</mi></msub><mo id="S7.I4.i4.p1.9.m9.2.2.3" xref="S7.I4.i4.p1.9.m9.2.2.3.cmml">=</mo><mrow id="S7.I4.i4.p1.9.m9.2.2.2.2" xref="S7.I4.i4.p1.9.m9.2.2.2.3.cmml"><mo id="S7.I4.i4.p1.9.m9.2.2.2.2.3" stretchy="false" xref="S7.I4.i4.p1.9.m9.2.2.2.3.cmml">(</mo><msub id="S7.I4.i4.p1.9.m9.1.1.1.1.1" xref="S7.I4.i4.p1.9.m9.1.1.1.1.1.cmml"><mi id="S7.I4.i4.p1.9.m9.1.1.1.1.1.2" xref="S7.I4.i4.p1.9.m9.1.1.1.1.1.2.cmml">V</mi><mi id="S7.I4.i4.p1.9.m9.1.1.1.1.1.3" xref="S7.I4.i4.p1.9.m9.1.1.1.1.1.3.cmml">s</mi></msub><mo id="S7.I4.i4.p1.9.m9.2.2.2.2.4" xref="S7.I4.i4.p1.9.m9.2.2.2.3.cmml">,</mo><msub id="S7.I4.i4.p1.9.m9.2.2.2.2.2" xref="S7.I4.i4.p1.9.m9.2.2.2.2.2.cmml"><mi id="S7.I4.i4.p1.9.m9.2.2.2.2.2.2" xref="S7.I4.i4.p1.9.m9.2.2.2.2.2.2.cmml">E</mi><mi id="S7.I4.i4.p1.9.m9.2.2.2.2.2.3" xref="S7.I4.i4.p1.9.m9.2.2.2.2.2.3.cmml">s</mi></msub><mo id="S7.I4.i4.p1.9.m9.2.2.2.2.5" stretchy="false" xref="S7.I4.i4.p1.9.m9.2.2.2.3.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S7.I4.i4.p1.9.m9.2b"><apply id="S7.I4.i4.p1.9.m9.2.2.cmml" xref="S7.I4.i4.p1.9.m9.2.2"><eq id="S7.I4.i4.p1.9.m9.2.2.3.cmml" xref="S7.I4.i4.p1.9.m9.2.2.3"></eq><apply id="S7.I4.i4.p1.9.m9.2.2.4.cmml" xref="S7.I4.i4.p1.9.m9.2.2.4"><csymbol cd="ambiguous" id="S7.I4.i4.p1.9.m9.2.2.4.1.cmml" xref="S7.I4.i4.p1.9.m9.2.2.4">subscript</csymbol><ci id="S7.I4.i4.p1.9.m9.2.2.4.2.cmml" xref="S7.I4.i4.p1.9.m9.2.2.4.2">𝐷</ci><ci id="S7.I4.i4.p1.9.m9.2.2.4.3.cmml" xref="S7.I4.i4.p1.9.m9.2.2.4.3">𝑠</ci></apply><interval closure="open" id="S7.I4.i4.p1.9.m9.2.2.2.3.cmml" xref="S7.I4.i4.p1.9.m9.2.2.2.2"><apply id="S7.I4.i4.p1.9.m9.1.1.1.1.1.cmml" xref="S7.I4.i4.p1.9.m9.1.1.1.1.1"><csymbol cd="ambiguous" id="S7.I4.i4.p1.9.m9.1.1.1.1.1.1.cmml" xref="S7.I4.i4.p1.9.m9.1.1.1.1.1">subscript</csymbol><ci id="S7.I4.i4.p1.9.m9.1.1.1.1.1.2.cmml" xref="S7.I4.i4.p1.9.m9.1.1.1.1.1.2">𝑉</ci><ci id="S7.I4.i4.p1.9.m9.1.1.1.1.1.3.cmml" xref="S7.I4.i4.p1.9.m9.1.1.1.1.1.3">𝑠</ci></apply><apply id="S7.I4.i4.p1.9.m9.2.2.2.2.2.cmml" xref="S7.I4.i4.p1.9.m9.2.2.2.2.2"><csymbol cd="ambiguous" id="S7.I4.i4.p1.9.m9.2.2.2.2.2.1.cmml" xref="S7.I4.i4.p1.9.m9.2.2.2.2.2">subscript</csymbol><ci id="S7.I4.i4.p1.9.m9.2.2.2.2.2.2.cmml" xref="S7.I4.i4.p1.9.m9.2.2.2.2.2.2">𝐸</ci><ci id="S7.I4.i4.p1.9.m9.2.2.2.2.2.3.cmml" xref="S7.I4.i4.p1.9.m9.2.2.2.2.2.3">𝑠</ci></apply></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.I4.i4.p1.9.m9.2c">D_{s}=(V_{s},E_{s})</annotation><annotation encoding="application/x-llamapun" id="S7.I4.i4.p1.9.m9.2d">italic_D start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT = ( italic_V start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT , italic_E start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT )</annotation></semantics></math> as follows:</p> <ul class="ltx_itemize" id="S7.I4.i4.I1"> <li class="ltx_item" id="S7.I4.i4.I1.i1" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item"><span class="ltx_text ltx_font_bold" id="S7.I4.i4.I1.i1.1.1.1">–</span></span> <div class="ltx_para" id="S7.I4.i4.I1.i1.p1"> <p class="ltx_p" id="S7.I4.i4.I1.i1.p1.6">The edges <math alttext="E_{s}" class="ltx_Math" display="inline" id="S7.I4.i4.I1.i1.p1.1.m1.1"><semantics id="S7.I4.i4.I1.i1.p1.1.m1.1a"><msub id="S7.I4.i4.I1.i1.p1.1.m1.1.1" xref="S7.I4.i4.I1.i1.p1.1.m1.1.1.cmml"><mi id="S7.I4.i4.I1.i1.p1.1.m1.1.1.2" xref="S7.I4.i4.I1.i1.p1.1.m1.1.1.2.cmml">E</mi><mi id="S7.I4.i4.I1.i1.p1.1.m1.1.1.3" xref="S7.I4.i4.I1.i1.p1.1.m1.1.1.3.cmml">s</mi></msub><annotation-xml encoding="MathML-Content" id="S7.I4.i4.I1.i1.p1.1.m1.1b"><apply id="S7.I4.i4.I1.i1.p1.1.m1.1.1.cmml" xref="S7.I4.i4.I1.i1.p1.1.m1.1.1"><csymbol cd="ambiguous" id="S7.I4.i4.I1.i1.p1.1.m1.1.1.1.cmml" xref="S7.I4.i4.I1.i1.p1.1.m1.1.1">subscript</csymbol><ci id="S7.I4.i4.I1.i1.p1.1.m1.1.1.2.cmml" xref="S7.I4.i4.I1.i1.p1.1.m1.1.1.2">𝐸</ci><ci id="S7.I4.i4.I1.i1.p1.1.m1.1.1.3.cmml" xref="S7.I4.i4.I1.i1.p1.1.m1.1.1.3">𝑠</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.I4.i4.I1.i1.p1.1.m1.1c">E_{s}</annotation><annotation encoding="application/x-llamapun" id="S7.I4.i4.I1.i1.p1.1.m1.1d">italic_E start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT</annotation></semantics></math> are all violating in-edges to vertices in <math alttext="T_{m}" class="ltx_Math" display="inline" id="S7.I4.i4.I1.i1.p1.2.m2.1"><semantics id="S7.I4.i4.I1.i1.p1.2.m2.1a"><msub id="S7.I4.i4.I1.i1.p1.2.m2.1.1" xref="S7.I4.i4.I1.i1.p1.2.m2.1.1.cmml"><mi id="S7.I4.i4.I1.i1.p1.2.m2.1.1.2" xref="S7.I4.i4.I1.i1.p1.2.m2.1.1.2.cmml">T</mi><mi id="S7.I4.i4.I1.i1.p1.2.m2.1.1.3" xref="S7.I4.i4.I1.i1.p1.2.m2.1.1.3.cmml">m</mi></msub><annotation-xml encoding="MathML-Content" id="S7.I4.i4.I1.i1.p1.2.m2.1b"><apply id="S7.I4.i4.I1.i1.p1.2.m2.1.1.cmml" xref="S7.I4.i4.I1.i1.p1.2.m2.1.1"><csymbol cd="ambiguous" id="S7.I4.i4.I1.i1.p1.2.m2.1.1.1.cmml" xref="S7.I4.i4.I1.i1.p1.2.m2.1.1">subscript</csymbol><ci id="S7.I4.i4.I1.i1.p1.2.m2.1.1.2.cmml" xref="S7.I4.i4.I1.i1.p1.2.m2.1.1.2">𝑇</ci><ci id="S7.I4.i4.I1.i1.p1.2.m2.1.1.3.cmml" xref="S7.I4.i4.I1.i1.p1.2.m2.1.1.3">𝑚</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.I4.i4.I1.i1.p1.2.m2.1c">T_{m}</annotation><annotation encoding="application/x-llamapun" id="S7.I4.i4.I1.i1.p1.2.m2.1d">italic_T start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT</annotation></semantics></math> plus all <math alttext="\overline{uv}" class="ltx_Math" display="inline" id="S7.I4.i4.I1.i1.p1.3.m3.1"><semantics id="S7.I4.i4.I1.i1.p1.3.m3.1a"><mover accent="true" id="S7.I4.i4.I1.i1.p1.3.m3.1.1" xref="S7.I4.i4.I1.i1.p1.3.m3.1.1.cmml"><mrow id="S7.I4.i4.I1.i1.p1.3.m3.1.1.2" xref="S7.I4.i4.I1.i1.p1.3.m3.1.1.2.cmml"><mi id="S7.I4.i4.I1.i1.p1.3.m3.1.1.2.2" xref="S7.I4.i4.I1.i1.p1.3.m3.1.1.2.2.cmml">u</mi><mo id="S7.I4.i4.I1.i1.p1.3.m3.1.1.2.1" xref="S7.I4.i4.I1.i1.p1.3.m3.1.1.2.1.cmml">⁢</mo><mi id="S7.I4.i4.I1.i1.p1.3.m3.1.1.2.3" xref="S7.I4.i4.I1.i1.p1.3.m3.1.1.2.3.cmml">v</mi></mrow><mo id="S7.I4.i4.I1.i1.p1.3.m3.1.1.1" xref="S7.I4.i4.I1.i1.p1.3.m3.1.1.1.cmml">¯</mo></mover><annotation-xml encoding="MathML-Content" id="S7.I4.i4.I1.i1.p1.3.m3.1b"><apply id="S7.I4.i4.I1.i1.p1.3.m3.1.1.cmml" xref="S7.I4.i4.I1.i1.p1.3.m3.1.1"><ci id="S7.I4.i4.I1.i1.p1.3.m3.1.1.1.cmml" xref="S7.I4.i4.I1.i1.p1.3.m3.1.1.1">¯</ci><apply id="S7.I4.i4.I1.i1.p1.3.m3.1.1.2.cmml" xref="S7.I4.i4.I1.i1.p1.3.m3.1.1.2"><times id="S7.I4.i4.I1.i1.p1.3.m3.1.1.2.1.cmml" xref="S7.I4.i4.I1.i1.p1.3.m3.1.1.2.1"></times><ci id="S7.I4.i4.I1.i1.p1.3.m3.1.1.2.2.cmml" xref="S7.I4.i4.I1.i1.p1.3.m3.1.1.2.2">𝑢</ci><ci id="S7.I4.i4.I1.i1.p1.3.m3.1.1.2.3.cmml" xref="S7.I4.i4.I1.i1.p1.3.m3.1.1.2.3">𝑣</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.I4.i4.I1.i1.p1.3.m3.1c">\overline{uv}</annotation><annotation encoding="application/x-llamapun" id="S7.I4.i4.I1.i1.p1.3.m3.1d">over¯ start_ARG italic_u italic_v end_ARG</annotation></semantics></math> with: <br class="ltx_break"/><math alttext="l_{s}(u)=l_{m}(u)>h+1" class="ltx_Math" display="inline" id="S7.I4.i4.I1.i1.p1.4.m4.2"><semantics id="S7.I4.i4.I1.i1.p1.4.m4.2a"><mrow id="S7.I4.i4.I1.i1.p1.4.m4.2.3" xref="S7.I4.i4.I1.i1.p1.4.m4.2.3.cmml"><mrow id="S7.I4.i4.I1.i1.p1.4.m4.2.3.2" xref="S7.I4.i4.I1.i1.p1.4.m4.2.3.2.cmml"><msub id="S7.I4.i4.I1.i1.p1.4.m4.2.3.2.2" xref="S7.I4.i4.I1.i1.p1.4.m4.2.3.2.2.cmml"><mi id="S7.I4.i4.I1.i1.p1.4.m4.2.3.2.2.2" xref="S7.I4.i4.I1.i1.p1.4.m4.2.3.2.2.2.cmml">l</mi><mi id="S7.I4.i4.I1.i1.p1.4.m4.2.3.2.2.3" xref="S7.I4.i4.I1.i1.p1.4.m4.2.3.2.2.3.cmml">s</mi></msub><mo id="S7.I4.i4.I1.i1.p1.4.m4.2.3.2.1" xref="S7.I4.i4.I1.i1.p1.4.m4.2.3.2.1.cmml">⁢</mo><mrow id="S7.I4.i4.I1.i1.p1.4.m4.2.3.2.3.2" xref="S7.I4.i4.I1.i1.p1.4.m4.2.3.2.cmml"><mo id="S7.I4.i4.I1.i1.p1.4.m4.2.3.2.3.2.1" stretchy="false" xref="S7.I4.i4.I1.i1.p1.4.m4.2.3.2.cmml">(</mo><mi id="S7.I4.i4.I1.i1.p1.4.m4.1.1" xref="S7.I4.i4.I1.i1.p1.4.m4.1.1.cmml">u</mi><mo id="S7.I4.i4.I1.i1.p1.4.m4.2.3.2.3.2.2" stretchy="false" xref="S7.I4.i4.I1.i1.p1.4.m4.2.3.2.cmml">)</mo></mrow></mrow><mo id="S7.I4.i4.I1.i1.p1.4.m4.2.3.3" xref="S7.I4.i4.I1.i1.p1.4.m4.2.3.3.cmml">=</mo><mrow id="S7.I4.i4.I1.i1.p1.4.m4.2.3.4" xref="S7.I4.i4.I1.i1.p1.4.m4.2.3.4.cmml"><msub id="S7.I4.i4.I1.i1.p1.4.m4.2.3.4.2" xref="S7.I4.i4.I1.i1.p1.4.m4.2.3.4.2.cmml"><mi id="S7.I4.i4.I1.i1.p1.4.m4.2.3.4.2.2" xref="S7.I4.i4.I1.i1.p1.4.m4.2.3.4.2.2.cmml">l</mi><mi id="S7.I4.i4.I1.i1.p1.4.m4.2.3.4.2.3" xref="S7.I4.i4.I1.i1.p1.4.m4.2.3.4.2.3.cmml">m</mi></msub><mo id="S7.I4.i4.I1.i1.p1.4.m4.2.3.4.1" xref="S7.I4.i4.I1.i1.p1.4.m4.2.3.4.1.cmml">⁢</mo><mrow id="S7.I4.i4.I1.i1.p1.4.m4.2.3.4.3.2" xref="S7.I4.i4.I1.i1.p1.4.m4.2.3.4.cmml"><mo id="S7.I4.i4.I1.i1.p1.4.m4.2.3.4.3.2.1" stretchy="false" xref="S7.I4.i4.I1.i1.p1.4.m4.2.3.4.cmml">(</mo><mi id="S7.I4.i4.I1.i1.p1.4.m4.2.2" xref="S7.I4.i4.I1.i1.p1.4.m4.2.2.cmml">u</mi><mo id="S7.I4.i4.I1.i1.p1.4.m4.2.3.4.3.2.2" stretchy="false" xref="S7.I4.i4.I1.i1.p1.4.m4.2.3.4.cmml">)</mo></mrow></mrow><mo id="S7.I4.i4.I1.i1.p1.4.m4.2.3.5" xref="S7.I4.i4.I1.i1.p1.4.m4.2.3.5.cmml">></mo><mrow id="S7.I4.i4.I1.i1.p1.4.m4.2.3.6" xref="S7.I4.i4.I1.i1.p1.4.m4.2.3.6.cmml"><mi id="S7.I4.i4.I1.i1.p1.4.m4.2.3.6.2" xref="S7.I4.i4.I1.i1.p1.4.m4.2.3.6.2.cmml">h</mi><mo id="S7.I4.i4.I1.i1.p1.4.m4.2.3.6.1" xref="S7.I4.i4.I1.i1.p1.4.m4.2.3.6.1.cmml">+</mo><mn id="S7.I4.i4.I1.i1.p1.4.m4.2.3.6.3" xref="S7.I4.i4.I1.i1.p1.4.m4.2.3.6.3.cmml">1</mn></mrow></mrow><annotation-xml encoding="MathML-Content" id="S7.I4.i4.I1.i1.p1.4.m4.2b"><apply id="S7.I4.i4.I1.i1.p1.4.m4.2.3.cmml" xref="S7.I4.i4.I1.i1.p1.4.m4.2.3"><and id="S7.I4.i4.I1.i1.p1.4.m4.2.3a.cmml" xref="S7.I4.i4.I1.i1.p1.4.m4.2.3"></and><apply id="S7.I4.i4.I1.i1.p1.4.m4.2.3b.cmml" xref="S7.I4.i4.I1.i1.p1.4.m4.2.3"><eq id="S7.I4.i4.I1.i1.p1.4.m4.2.3.3.cmml" xref="S7.I4.i4.I1.i1.p1.4.m4.2.3.3"></eq><apply id="S7.I4.i4.I1.i1.p1.4.m4.2.3.2.cmml" xref="S7.I4.i4.I1.i1.p1.4.m4.2.3.2"><times id="S7.I4.i4.I1.i1.p1.4.m4.2.3.2.1.cmml" xref="S7.I4.i4.I1.i1.p1.4.m4.2.3.2.1"></times><apply id="S7.I4.i4.I1.i1.p1.4.m4.2.3.2.2.cmml" xref="S7.I4.i4.I1.i1.p1.4.m4.2.3.2.2"><csymbol cd="ambiguous" id="S7.I4.i4.I1.i1.p1.4.m4.2.3.2.2.1.cmml" xref="S7.I4.i4.I1.i1.p1.4.m4.2.3.2.2">subscript</csymbol><ci id="S7.I4.i4.I1.i1.p1.4.m4.2.3.2.2.2.cmml" xref="S7.I4.i4.I1.i1.p1.4.m4.2.3.2.2.2">𝑙</ci><ci id="S7.I4.i4.I1.i1.p1.4.m4.2.3.2.2.3.cmml" xref="S7.I4.i4.I1.i1.p1.4.m4.2.3.2.2.3">𝑠</ci></apply><ci id="S7.I4.i4.I1.i1.p1.4.m4.1.1.cmml" xref="S7.I4.i4.I1.i1.p1.4.m4.1.1">𝑢</ci></apply><apply id="S7.I4.i4.I1.i1.p1.4.m4.2.3.4.cmml" xref="S7.I4.i4.I1.i1.p1.4.m4.2.3.4"><times id="S7.I4.i4.I1.i1.p1.4.m4.2.3.4.1.cmml" xref="S7.I4.i4.I1.i1.p1.4.m4.2.3.4.1"></times><apply id="S7.I4.i4.I1.i1.p1.4.m4.2.3.4.2.cmml" xref="S7.I4.i4.I1.i1.p1.4.m4.2.3.4.2"><csymbol cd="ambiguous" id="S7.I4.i4.I1.i1.p1.4.m4.2.3.4.2.1.cmml" xref="S7.I4.i4.I1.i1.p1.4.m4.2.3.4.2">subscript</csymbol><ci id="S7.I4.i4.I1.i1.p1.4.m4.2.3.4.2.2.cmml" xref="S7.I4.i4.I1.i1.p1.4.m4.2.3.4.2.2">𝑙</ci><ci id="S7.I4.i4.I1.i1.p1.4.m4.2.3.4.2.3.cmml" xref="S7.I4.i4.I1.i1.p1.4.m4.2.3.4.2.3">𝑚</ci></apply><ci id="S7.I4.i4.I1.i1.p1.4.m4.2.2.cmml" xref="S7.I4.i4.I1.i1.p1.4.m4.2.2">𝑢</ci></apply></apply><apply id="S7.I4.i4.I1.i1.p1.4.m4.2.3c.cmml" xref="S7.I4.i4.I1.i1.p1.4.m4.2.3"><gt id="S7.I4.i4.I1.i1.p1.4.m4.2.3.5.cmml" xref="S7.I4.i4.I1.i1.p1.4.m4.2.3.5"></gt><share href="https://arxiv.org/html/2411.12694v2#S7.I4.i4.I1.i1.p1.4.m4.2.3.4.cmml" id="S7.I4.i4.I1.i1.p1.4.m4.2.3d.cmml" xref="S7.I4.i4.I1.i1.p1.4.m4.2.3"></share><apply id="S7.I4.i4.I1.i1.p1.4.m4.2.3.6.cmml" xref="S7.I4.i4.I1.i1.p1.4.m4.2.3.6"><plus id="S7.I4.i4.I1.i1.p1.4.m4.2.3.6.1.cmml" xref="S7.I4.i4.I1.i1.p1.4.m4.2.3.6.1"></plus><ci id="S7.I4.i4.I1.i1.p1.4.m4.2.3.6.2.cmml" xref="S7.I4.i4.I1.i1.p1.4.m4.2.3.6.2">ℎ</ci><cn id="S7.I4.i4.I1.i1.p1.4.m4.2.3.6.3.cmml" type="integer" xref="S7.I4.i4.I1.i1.p1.4.m4.2.3.6.3">1</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.I4.i4.I1.i1.p1.4.m4.2c">l_{s}(u)=l_{m}(u)>h+1</annotation><annotation encoding="application/x-llamapun" id="S7.I4.i4.I1.i1.p1.4.m4.2d">italic_l start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_u ) = italic_l start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT ( italic_u ) > italic_h + 1</annotation></semantics></math>, <math alttext="l_{s}(u)>l_{s}(v)" class="ltx_Math" display="inline" id="S7.I4.i4.I1.i1.p1.5.m5.2"><semantics id="S7.I4.i4.I1.i1.p1.5.m5.2a"><mrow id="S7.I4.i4.I1.i1.p1.5.m5.2.3" xref="S7.I4.i4.I1.i1.p1.5.m5.2.3.cmml"><mrow id="S7.I4.i4.I1.i1.p1.5.m5.2.3.2" xref="S7.I4.i4.I1.i1.p1.5.m5.2.3.2.cmml"><msub id="S7.I4.i4.I1.i1.p1.5.m5.2.3.2.2" xref="S7.I4.i4.I1.i1.p1.5.m5.2.3.2.2.cmml"><mi id="S7.I4.i4.I1.i1.p1.5.m5.2.3.2.2.2" xref="S7.I4.i4.I1.i1.p1.5.m5.2.3.2.2.2.cmml">l</mi><mi id="S7.I4.i4.I1.i1.p1.5.m5.2.3.2.2.3" xref="S7.I4.i4.I1.i1.p1.5.m5.2.3.2.2.3.cmml">s</mi></msub><mo id="S7.I4.i4.I1.i1.p1.5.m5.2.3.2.1" xref="S7.I4.i4.I1.i1.p1.5.m5.2.3.2.1.cmml">⁢</mo><mrow id="S7.I4.i4.I1.i1.p1.5.m5.2.3.2.3.2" xref="S7.I4.i4.I1.i1.p1.5.m5.2.3.2.cmml"><mo id="S7.I4.i4.I1.i1.p1.5.m5.2.3.2.3.2.1" stretchy="false" xref="S7.I4.i4.I1.i1.p1.5.m5.2.3.2.cmml">(</mo><mi id="S7.I4.i4.I1.i1.p1.5.m5.1.1" xref="S7.I4.i4.I1.i1.p1.5.m5.1.1.cmml">u</mi><mo id="S7.I4.i4.I1.i1.p1.5.m5.2.3.2.3.2.2" stretchy="false" 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italic_v )</annotation></semantics></math>, and <math alttext="\textsl{g}_{s}(u\!\to\!v)>0" class="ltx_Math" display="inline" id="S7.I4.i4.I1.i1.p1.6.m6.1"><semantics id="S7.I4.i4.I1.i1.p1.6.m6.1a"><mrow id="S7.I4.i4.I1.i1.p1.6.m6.1.1" xref="S7.I4.i4.I1.i1.p1.6.m6.1.1.cmml"><mrow id="S7.I4.i4.I1.i1.p1.6.m6.1.1.1" xref="S7.I4.i4.I1.i1.p1.6.m6.1.1.1.cmml"><msub id="S7.I4.i4.I1.i1.p1.6.m6.1.1.1.3" xref="S7.I4.i4.I1.i1.p1.6.m6.1.1.1.3.cmml"><mtext class="ltx_mathvariant_italic" id="S7.I4.i4.I1.i1.p1.6.m6.1.1.1.3.2" xref="S7.I4.i4.I1.i1.p1.6.m6.1.1.1.3.2a.cmml">g</mtext><mi id="S7.I4.i4.I1.i1.p1.6.m6.1.1.1.3.3" xref="S7.I4.i4.I1.i1.p1.6.m6.1.1.1.3.3.cmml">s</mi></msub><mo id="S7.I4.i4.I1.i1.p1.6.m6.1.1.1.2" xref="S7.I4.i4.I1.i1.p1.6.m6.1.1.1.2.cmml">⁢</mo><mrow id="S7.I4.i4.I1.i1.p1.6.m6.1.1.1.1.1" xref="S7.I4.i4.I1.i1.p1.6.m6.1.1.1.1.1.1.cmml"><mo id="S7.I4.i4.I1.i1.p1.6.m6.1.1.1.1.1.2" stretchy="false" xref="S7.I4.i4.I1.i1.p1.6.m6.1.1.1.1.1.1.cmml">(</mo><mrow id="S7.I4.i4.I1.i1.p1.6.m6.1.1.1.1.1.1" xref="S7.I4.i4.I1.i1.p1.6.m6.1.1.1.1.1.1.cmml"><mi id="S7.I4.i4.I1.i1.p1.6.m6.1.1.1.1.1.1.2" xref="S7.I4.i4.I1.i1.p1.6.m6.1.1.1.1.1.1.2.cmml">u</mi><mo id="S7.I4.i4.I1.i1.p1.6.m6.1.1.1.1.1.1.1" lspace="0.108em" rspace="0.108em" stretchy="false" xref="S7.I4.i4.I1.i1.p1.6.m6.1.1.1.1.1.1.1.cmml">→</mo><mi id="S7.I4.i4.I1.i1.p1.6.m6.1.1.1.1.1.1.3" xref="S7.I4.i4.I1.i1.p1.6.m6.1.1.1.1.1.1.3.cmml">v</mi></mrow><mo id="S7.I4.i4.I1.i1.p1.6.m6.1.1.1.1.1.3" stretchy="false" xref="S7.I4.i4.I1.i1.p1.6.m6.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S7.I4.i4.I1.i1.p1.6.m6.1.1.2" xref="S7.I4.i4.I1.i1.p1.6.m6.1.1.2.cmml">></mo><mn id="S7.I4.i4.I1.i1.p1.6.m6.1.1.3" xref="S7.I4.i4.I1.i1.p1.6.m6.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S7.I4.i4.I1.i1.p1.6.m6.1b"><apply id="S7.I4.i4.I1.i1.p1.6.m6.1.1.cmml" xref="S7.I4.i4.I1.i1.p1.6.m6.1.1"><gt id="S7.I4.i4.I1.i1.p1.6.m6.1.1.2.cmml" xref="S7.I4.i4.I1.i1.p1.6.m6.1.1.2"></gt><apply id="S7.I4.i4.I1.i1.p1.6.m6.1.1.1.cmml" xref="S7.I4.i4.I1.i1.p1.6.m6.1.1.1"><times id="S7.I4.i4.I1.i1.p1.6.m6.1.1.1.2.cmml" xref="S7.I4.i4.I1.i1.p1.6.m6.1.1.1.2"></times><apply id="S7.I4.i4.I1.i1.p1.6.m6.1.1.1.3.cmml" xref="S7.I4.i4.I1.i1.p1.6.m6.1.1.1.3"><csymbol cd="ambiguous" id="S7.I4.i4.I1.i1.p1.6.m6.1.1.1.3.1.cmml" xref="S7.I4.i4.I1.i1.p1.6.m6.1.1.1.3">subscript</csymbol><ci id="S7.I4.i4.I1.i1.p1.6.m6.1.1.1.3.2a.cmml" xref="S7.I4.i4.I1.i1.p1.6.m6.1.1.1.3.2"><mtext class="ltx_mathvariant_italic" id="S7.I4.i4.I1.i1.p1.6.m6.1.1.1.3.2.cmml" xref="S7.I4.i4.I1.i1.p1.6.m6.1.1.1.3.2">g</mtext></ci><ci id="S7.I4.i4.I1.i1.p1.6.m6.1.1.1.3.3.cmml" xref="S7.I4.i4.I1.i1.p1.6.m6.1.1.1.3.3">𝑠</ci></apply><apply id="S7.I4.i4.I1.i1.p1.6.m6.1.1.1.1.1.1.cmml" xref="S7.I4.i4.I1.i1.p1.6.m6.1.1.1.1.1"><ci id="S7.I4.i4.I1.i1.p1.6.m6.1.1.1.1.1.1.1.cmml" xref="S7.I4.i4.I1.i1.p1.6.m6.1.1.1.1.1.1.1">→</ci><ci id="S7.I4.i4.I1.i1.p1.6.m6.1.1.1.1.1.1.2.cmml" xref="S7.I4.i4.I1.i1.p1.6.m6.1.1.1.1.1.1.2">𝑢</ci><ci id="S7.I4.i4.I1.i1.p1.6.m6.1.1.1.1.1.1.3.cmml" xref="S7.I4.i4.I1.i1.p1.6.m6.1.1.1.1.1.1.3">𝑣</ci></apply></apply><cn id="S7.I4.i4.I1.i1.p1.6.m6.1.1.3.cmml" type="integer" xref="S7.I4.i4.I1.i1.p1.6.m6.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.I4.i4.I1.i1.p1.6.m6.1c">\textsl{g}_{s}(u\!\to\!v)>0</annotation><annotation encoding="application/x-llamapun" id="S7.I4.i4.I1.i1.p1.6.m6.1d">g start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_u → italic_v ) > 0</annotation></semantics></math>.</p> </div> </li> <li class="ltx_item" id="S7.I4.i4.I1.i2" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item"><span class="ltx_text ltx_font_bold" id="S7.I4.i4.I1.i2.1.1.1">–</span></span> <div class="ltx_para" id="S7.I4.i4.I1.i2.p1"> <p class="ltx_p" id="S7.I4.i4.I1.i2.p1.5">The vertices <math alttext="V_{s}" class="ltx_Math" display="inline" id="S7.I4.i4.I1.i2.p1.1.m1.1"><semantics id="S7.I4.i4.I1.i2.p1.1.m1.1a"><msub id="S7.I4.i4.I1.i2.p1.1.m1.1.1" xref="S7.I4.i4.I1.i2.p1.1.m1.1.1.cmml"><mi id="S7.I4.i4.I1.i2.p1.1.m1.1.1.2" xref="S7.I4.i4.I1.i2.p1.1.m1.1.1.2.cmml">V</mi><mi id="S7.I4.i4.I1.i2.p1.1.m1.1.1.3" xref="S7.I4.i4.I1.i2.p1.1.m1.1.1.3.cmml">s</mi></msub><annotation-xml encoding="MathML-Content" id="S7.I4.i4.I1.i2.p1.1.m1.1b"><apply id="S7.I4.i4.I1.i2.p1.1.m1.1.1.cmml" xref="S7.I4.i4.I1.i2.p1.1.m1.1.1"><csymbol cd="ambiguous" id="S7.I4.i4.I1.i2.p1.1.m1.1.1.1.cmml" xref="S7.I4.i4.I1.i2.p1.1.m1.1.1">subscript</csymbol><ci id="S7.I4.i4.I1.i2.p1.1.m1.1.1.2.cmml" xref="S7.I4.i4.I1.i2.p1.1.m1.1.1.2">𝑉</ci><ci id="S7.I4.i4.I1.i2.p1.1.m1.1.1.3.cmml" xref="S7.I4.i4.I1.i2.p1.1.m1.1.1.3">𝑠</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.I4.i4.I1.i2.p1.1.m1.1c">V_{s}</annotation><annotation encoding="application/x-llamapun" id="S7.I4.i4.I1.i2.p1.1.m1.1d">italic_V start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT</annotation></semantics></math> include <math alttext="T_{m}" class="ltx_Math" display="inline" id="S7.I4.i4.I1.i2.p1.2.m2.1"><semantics id="S7.I4.i4.I1.i2.p1.2.m2.1a"><msub id="S7.I4.i4.I1.i2.p1.2.m2.1.1" xref="S7.I4.i4.I1.i2.p1.2.m2.1.1.cmml"><mi id="S7.I4.i4.I1.i2.p1.2.m2.1.1.2" xref="S7.I4.i4.I1.i2.p1.2.m2.1.1.2.cmml">T</mi><mi id="S7.I4.i4.I1.i2.p1.2.m2.1.1.3" xref="S7.I4.i4.I1.i2.p1.2.m2.1.1.3.cmml">m</mi></msub><annotation-xml encoding="MathML-Content" id="S7.I4.i4.I1.i2.p1.2.m2.1b"><apply id="S7.I4.i4.I1.i2.p1.2.m2.1.1.cmml" xref="S7.I4.i4.I1.i2.p1.2.m2.1.1"><csymbol cd="ambiguous" id="S7.I4.i4.I1.i2.p1.2.m2.1.1.1.cmml" xref="S7.I4.i4.I1.i2.p1.2.m2.1.1">subscript</csymbol><ci id="S7.I4.i4.I1.i2.p1.2.m2.1.1.2.cmml" xref="S7.I4.i4.I1.i2.p1.2.m2.1.1.2">𝑇</ci><ci id="S7.I4.i4.I1.i2.p1.2.m2.1.1.3.cmml" xref="S7.I4.i4.I1.i2.p1.2.m2.1.1.3">𝑚</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.I4.i4.I1.i2.p1.2.m2.1c">T_{m}</annotation><annotation encoding="application/x-llamapun" id="S7.I4.i4.I1.i2.p1.2.m2.1d">italic_T start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT</annotation></semantics></math> plus all vertices in <math alttext="G" class="ltx_Math" display="inline" id="S7.I4.i4.I1.i2.p1.3.m3.1"><semantics id="S7.I4.i4.I1.i2.p1.3.m3.1a"><mi id="S7.I4.i4.I1.i2.p1.3.m3.1.1" xref="S7.I4.i4.I1.i2.p1.3.m3.1.1.cmml">G</mi><annotation-xml encoding="MathML-Content" id="S7.I4.i4.I1.i2.p1.3.m3.1b"><ci id="S7.I4.i4.I1.i2.p1.3.m3.1.1.cmml" xref="S7.I4.i4.I1.i2.p1.3.m3.1.1">𝐺</ci></annotation-xml><annotation encoding="application/x-tex" id="S7.I4.i4.I1.i2.p1.3.m3.1c">G</annotation><annotation encoding="application/x-llamapun" id="S7.I4.i4.I1.i2.p1.3.m3.1d">italic_G</annotation></semantics></math> with a directed path to <math alttext="T_{m}" class="ltx_Math" display="inline" id="S7.I4.i4.I1.i2.p1.4.m4.1"><semantics id="S7.I4.i4.I1.i2.p1.4.m4.1a"><msub id="S7.I4.i4.I1.i2.p1.4.m4.1.1" xref="S7.I4.i4.I1.i2.p1.4.m4.1.1.cmml"><mi id="S7.I4.i4.I1.i2.p1.4.m4.1.1.2" xref="S7.I4.i4.I1.i2.p1.4.m4.1.1.2.cmml">T</mi><mi id="S7.I4.i4.I1.i2.p1.4.m4.1.1.3" xref="S7.I4.i4.I1.i2.p1.4.m4.1.1.3.cmml">m</mi></msub><annotation-xml encoding="MathML-Content" id="S7.I4.i4.I1.i2.p1.4.m4.1b"><apply id="S7.I4.i4.I1.i2.p1.4.m4.1.1.cmml" xref="S7.I4.i4.I1.i2.p1.4.m4.1.1"><csymbol cd="ambiguous" id="S7.I4.i4.I1.i2.p1.4.m4.1.1.1.cmml" xref="S7.I4.i4.I1.i2.p1.4.m4.1.1">subscript</csymbol><ci id="S7.I4.i4.I1.i2.p1.4.m4.1.1.2.cmml" xref="S7.I4.i4.I1.i2.p1.4.m4.1.1.2">𝑇</ci><ci id="S7.I4.i4.I1.i2.p1.4.m4.1.1.3.cmml" xref="S7.I4.i4.I1.i2.p1.4.m4.1.1.3">𝑚</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.I4.i4.I1.i2.p1.4.m4.1c">T_{m}</annotation><annotation encoding="application/x-llamapun" id="S7.I4.i4.I1.i2.p1.4.m4.1d">italic_T start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT</annotation></semantics></math> in <math alttext="E_{s}" class="ltx_Math" display="inline" id="S7.I4.i4.I1.i2.p1.5.m5.1"><semantics id="S7.I4.i4.I1.i2.p1.5.m5.1a"><msub id="S7.I4.i4.I1.i2.p1.5.m5.1.1" xref="S7.I4.i4.I1.i2.p1.5.m5.1.1.cmml"><mi id="S7.I4.i4.I1.i2.p1.5.m5.1.1.2" xref="S7.I4.i4.I1.i2.p1.5.m5.1.1.2.cmml">E</mi><mi id="S7.I4.i4.I1.i2.p1.5.m5.1.1.3" xref="S7.I4.i4.I1.i2.p1.5.m5.1.1.3.cmml">s</mi></msub><annotation-xml encoding="MathML-Content" id="S7.I4.i4.I1.i2.p1.5.m5.1b"><apply id="S7.I4.i4.I1.i2.p1.5.m5.1.1.cmml" xref="S7.I4.i4.I1.i2.p1.5.m5.1.1"><csymbol cd="ambiguous" id="S7.I4.i4.I1.i2.p1.5.m5.1.1.1.cmml" xref="S7.I4.i4.I1.i2.p1.5.m5.1.1">subscript</csymbol><ci id="S7.I4.i4.I1.i2.p1.5.m5.1.1.2.cmml" xref="S7.I4.i4.I1.i2.p1.5.m5.1.1.2">𝐸</ci><ci id="S7.I4.i4.I1.i2.p1.5.m5.1.1.3.cmml" xref="S7.I4.i4.I1.i2.p1.5.m5.1.1.3">𝑠</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.I4.i4.I1.i2.p1.5.m5.1c">E_{s}</annotation><annotation encoding="application/x-llamapun" id="S7.I4.i4.I1.i2.p1.5.m5.1d">italic_E start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT</annotation></semantics></math>.</p> </div> </li> <li class="ltx_item" id="S7.I4.i4.I1.i3" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item"><span class="ltx_text ltx_font_bold" id="S7.I4.i4.I1.i3.1.1.1">–</span></span> <div class="ltx_para" id="S7.I4.i4.I1.i3.p1"> <p class="ltx_p" id="S7.I4.i4.I1.i3.p1.3">The vertices <math alttext="S_{s}\subset V_{s}" class="ltx_Math" display="inline" id="S7.I4.i4.I1.i3.p1.1.m1.1"><semantics id="S7.I4.i4.I1.i3.p1.1.m1.1a"><mrow id="S7.I4.i4.I1.i3.p1.1.m1.1.1" xref="S7.I4.i4.I1.i3.p1.1.m1.1.1.cmml"><msub id="S7.I4.i4.I1.i3.p1.1.m1.1.1.2" xref="S7.I4.i4.I1.i3.p1.1.m1.1.1.2.cmml"><mi id="S7.I4.i4.I1.i3.p1.1.m1.1.1.2.2" xref="S7.I4.i4.I1.i3.p1.1.m1.1.1.2.2.cmml">S</mi><mi id="S7.I4.i4.I1.i3.p1.1.m1.1.1.2.3" xref="S7.I4.i4.I1.i3.p1.1.m1.1.1.2.3.cmml">s</mi></msub><mo id="S7.I4.i4.I1.i3.p1.1.m1.1.1.1" xref="S7.I4.i4.I1.i3.p1.1.m1.1.1.1.cmml">⊂</mo><msub id="S7.I4.i4.I1.i3.p1.1.m1.1.1.3" xref="S7.I4.i4.I1.i3.p1.1.m1.1.1.3.cmml"><mi id="S7.I4.i4.I1.i3.p1.1.m1.1.1.3.2" xref="S7.I4.i4.I1.i3.p1.1.m1.1.1.3.2.cmml">V</mi><mi id="S7.I4.i4.I1.i3.p1.1.m1.1.1.3.3" xref="S7.I4.i4.I1.i3.p1.1.m1.1.1.3.3.cmml">s</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S7.I4.i4.I1.i3.p1.1.m1.1b"><apply id="S7.I4.i4.I1.i3.p1.1.m1.1.1.cmml" xref="S7.I4.i4.I1.i3.p1.1.m1.1.1"><subset id="S7.I4.i4.I1.i3.p1.1.m1.1.1.1.cmml" xref="S7.I4.i4.I1.i3.p1.1.m1.1.1.1"></subset><apply id="S7.I4.i4.I1.i3.p1.1.m1.1.1.2.cmml" xref="S7.I4.i4.I1.i3.p1.1.m1.1.1.2"><csymbol cd="ambiguous" id="S7.I4.i4.I1.i3.p1.1.m1.1.1.2.1.cmml" xref="S7.I4.i4.I1.i3.p1.1.m1.1.1.2">subscript</csymbol><ci id="S7.I4.i4.I1.i3.p1.1.m1.1.1.2.2.cmml" xref="S7.I4.i4.I1.i3.p1.1.m1.1.1.2.2">𝑆</ci><ci id="S7.I4.i4.I1.i3.p1.1.m1.1.1.2.3.cmml" xref="S7.I4.i4.I1.i3.p1.1.m1.1.1.2.3">𝑠</ci></apply><apply id="S7.I4.i4.I1.i3.p1.1.m1.1.1.3.cmml" xref="S7.I4.i4.I1.i3.p1.1.m1.1.1.3"><csymbol cd="ambiguous" id="S7.I4.i4.I1.i3.p1.1.m1.1.1.3.1.cmml" xref="S7.I4.i4.I1.i3.p1.1.m1.1.1.3">subscript</csymbol><ci id="S7.I4.i4.I1.i3.p1.1.m1.1.1.3.2.cmml" xref="S7.I4.i4.I1.i3.p1.1.m1.1.1.3.2">𝑉</ci><ci id="S7.I4.i4.I1.i3.p1.1.m1.1.1.3.3.cmml" xref="S7.I4.i4.I1.i3.p1.1.m1.1.1.3.3">𝑠</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.I4.i4.I1.i3.p1.1.m1.1c">S_{s}\subset V_{s}</annotation><annotation encoding="application/x-llamapun" id="S7.I4.i4.I1.i3.p1.1.m1.1d">italic_S start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ⊂ italic_V start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT</annotation></semantics></math> are all sources in <math alttext="D_{s}" class="ltx_Math" display="inline" id="S7.I4.i4.I1.i3.p1.2.m2.1"><semantics id="S7.I4.i4.I1.i3.p1.2.m2.1a"><msub id="S7.I4.i4.I1.i3.p1.2.m2.1.1" xref="S7.I4.i4.I1.i3.p1.2.m2.1.1.cmml"><mi id="S7.I4.i4.I1.i3.p1.2.m2.1.1.2" xref="S7.I4.i4.I1.i3.p1.2.m2.1.1.2.cmml">D</mi><mi id="S7.I4.i4.I1.i3.p1.2.m2.1.1.3" xref="S7.I4.i4.I1.i3.p1.2.m2.1.1.3.cmml">s</mi></msub><annotation-xml encoding="MathML-Content" id="S7.I4.i4.I1.i3.p1.2.m2.1b"><apply id="S7.I4.i4.I1.i3.p1.2.m2.1.1.cmml" xref="S7.I4.i4.I1.i3.p1.2.m2.1.1"><csymbol cd="ambiguous" id="S7.I4.i4.I1.i3.p1.2.m2.1.1.1.cmml" xref="S7.I4.i4.I1.i3.p1.2.m2.1.1">subscript</csymbol><ci id="S7.I4.i4.I1.i3.p1.2.m2.1.1.2.cmml" xref="S7.I4.i4.I1.i3.p1.2.m2.1.1.2">𝐷</ci><ci id="S7.I4.i4.I1.i3.p1.2.m2.1.1.3.cmml" xref="S7.I4.i4.I1.i3.p1.2.m2.1.1.3">𝑠</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.I4.i4.I1.i3.p1.2.m2.1c">D_{s}</annotation><annotation encoding="application/x-llamapun" id="S7.I4.i4.I1.i3.p1.2.m2.1d">italic_D start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT</annotation></semantics></math> (these are not in <math alttext="T_{m}" class="ltx_Math" display="inline" id="S7.I4.i4.I1.i3.p1.3.m3.1"><semantics id="S7.I4.i4.I1.i3.p1.3.m3.1a"><msub id="S7.I4.i4.I1.i3.p1.3.m3.1.1" xref="S7.I4.i4.I1.i3.p1.3.m3.1.1.cmml"><mi id="S7.I4.i4.I1.i3.p1.3.m3.1.1.2" xref="S7.I4.i4.I1.i3.p1.3.m3.1.1.2.cmml">T</mi><mi id="S7.I4.i4.I1.i3.p1.3.m3.1.1.3" xref="S7.I4.i4.I1.i3.p1.3.m3.1.1.3.cmml">m</mi></msub><annotation-xml encoding="MathML-Content" id="S7.I4.i4.I1.i3.p1.3.m3.1b"><apply id="S7.I4.i4.I1.i3.p1.3.m3.1.1.cmml" xref="S7.I4.i4.I1.i3.p1.3.m3.1.1"><csymbol cd="ambiguous" id="S7.I4.i4.I1.i3.p1.3.m3.1.1.1.cmml" xref="S7.I4.i4.I1.i3.p1.3.m3.1.1">subscript</csymbol><ci id="S7.I4.i4.I1.i3.p1.3.m3.1.1.2.cmml" xref="S7.I4.i4.I1.i3.p1.3.m3.1.1.2">𝑇</ci><ci id="S7.I4.i4.I1.i3.p1.3.m3.1.1.3.cmml" xref="S7.I4.i4.I1.i3.p1.3.m3.1.1.3">𝑚</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.I4.i4.I1.i3.p1.3.m3.1c">T_{m}</annotation><annotation encoding="application/x-llamapun" id="S7.I4.i4.I1.i3.p1.3.m3.1d">italic_T start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT</annotation></semantics></math>).</p> </div> </li> </ul> <p class="ltx_p" id="S7.I4.i4.p1.19">For each <math alttext="v\in T_{m}" class="ltx_Math" display="inline" id="S7.I4.i4.p1.10.m1.1"><semantics id="S7.I4.i4.p1.10.m1.1a"><mrow id="S7.I4.i4.p1.10.m1.1.1" xref="S7.I4.i4.p1.10.m1.1.1.cmml"><mi id="S7.I4.i4.p1.10.m1.1.1.2" xref="S7.I4.i4.p1.10.m1.1.1.2.cmml">v</mi><mo id="S7.I4.i4.p1.10.m1.1.1.1" xref="S7.I4.i4.p1.10.m1.1.1.1.cmml">∈</mo><msub id="S7.I4.i4.p1.10.m1.1.1.3" xref="S7.I4.i4.p1.10.m1.1.1.3.cmml"><mi id="S7.I4.i4.p1.10.m1.1.1.3.2" xref="S7.I4.i4.p1.10.m1.1.1.3.2.cmml">T</mi><mi id="S7.I4.i4.p1.10.m1.1.1.3.3" xref="S7.I4.i4.p1.10.m1.1.1.3.3.cmml">m</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S7.I4.i4.p1.10.m1.1b"><apply id="S7.I4.i4.p1.10.m1.1.1.cmml" xref="S7.I4.i4.p1.10.m1.1.1"><in id="S7.I4.i4.p1.10.m1.1.1.1.cmml" xref="S7.I4.i4.p1.10.m1.1.1.1"></in><ci id="S7.I4.i4.p1.10.m1.1.1.2.cmml" xref="S7.I4.i4.p1.10.m1.1.1.2">𝑣</ci><apply id="S7.I4.i4.p1.10.m1.1.1.3.cmml" xref="S7.I4.i4.p1.10.m1.1.1.3"><csymbol cd="ambiguous" id="S7.I4.i4.p1.10.m1.1.1.3.1.cmml" xref="S7.I4.i4.p1.10.m1.1.1.3">subscript</csymbol><ci id="S7.I4.i4.p1.10.m1.1.1.3.2.cmml" xref="S7.I4.i4.p1.10.m1.1.1.3.2">𝑇</ci><ci id="S7.I4.i4.p1.10.m1.1.1.3.3.cmml" xref="S7.I4.i4.p1.10.m1.1.1.3.3">𝑚</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.I4.i4.p1.10.m1.1c">v\in T_{m}</annotation><annotation encoding="application/x-llamapun" id="S7.I4.i4.p1.10.m1.1d">italic_v ∈ italic_T start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT</annotation></semantics></math> we increase <math alttext="\textsl{g}(v)" class="ltx_Math" display="inline" id="S7.I4.i4.p1.11.m2.1"><semantics id="S7.I4.i4.p1.11.m2.1a"><mrow id="S7.I4.i4.p1.11.m2.1.2" xref="S7.I4.i4.p1.11.m2.1.2.cmml"><mtext class="ltx_mathvariant_italic" id="S7.I4.i4.p1.11.m2.1.2.2" xref="S7.I4.i4.p1.11.m2.1.2.2a.cmml">g</mtext><mo id="S7.I4.i4.p1.11.m2.1.2.1" xref="S7.I4.i4.p1.11.m2.1.2.1.cmml">⁢</mo><mrow id="S7.I4.i4.p1.11.m2.1.2.3.2" xref="S7.I4.i4.p1.11.m2.1.2.cmml"><mo id="S7.I4.i4.p1.11.m2.1.2.3.2.1" stretchy="false" xref="S7.I4.i4.p1.11.m2.1.2.cmml">(</mo><mi id="S7.I4.i4.p1.11.m2.1.1" xref="S7.I4.i4.p1.11.m2.1.1.cmml">v</mi><mo id="S7.I4.i4.p1.11.m2.1.2.3.2.2" stretchy="false" xref="S7.I4.i4.p1.11.m2.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S7.I4.i4.p1.11.m2.1b"><apply id="S7.I4.i4.p1.11.m2.1.2.cmml" xref="S7.I4.i4.p1.11.m2.1.2"><times id="S7.I4.i4.p1.11.m2.1.2.1.cmml" xref="S7.I4.i4.p1.11.m2.1.2.1"></times><ci id="S7.I4.i4.p1.11.m2.1.2.2a.cmml" xref="S7.I4.i4.p1.11.m2.1.2.2"><mtext class="ltx_mathvariant_italic" id="S7.I4.i4.p1.11.m2.1.2.2.cmml" xref="S7.I4.i4.p1.11.m2.1.2.2">g</mtext></ci><ci id="S7.I4.i4.p1.11.m2.1.1.cmml" xref="S7.I4.i4.p1.11.m2.1.1">𝑣</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.I4.i4.p1.11.m2.1c">\textsl{g}(v)</annotation><annotation encoding="application/x-llamapun" id="S7.I4.i4.p1.11.m2.1d">g ( italic_v )</annotation></semantics></math> by flipping a directed path from a vertex in <math alttext="S_{s}" class="ltx_Math" display="inline" id="S7.I4.i4.p1.12.m3.1"><semantics id="S7.I4.i4.p1.12.m3.1a"><msub id="S7.I4.i4.p1.12.m3.1.1" xref="S7.I4.i4.p1.12.m3.1.1.cmml"><mi id="S7.I4.i4.p1.12.m3.1.1.2" xref="S7.I4.i4.p1.12.m3.1.1.2.cmml">S</mi><mi id="S7.I4.i4.p1.12.m3.1.1.3" xref="S7.I4.i4.p1.12.m3.1.1.3.cmml">s</mi></msub><annotation-xml encoding="MathML-Content" id="S7.I4.i4.p1.12.m3.1b"><apply id="S7.I4.i4.p1.12.m3.1.1.cmml" xref="S7.I4.i4.p1.12.m3.1.1"><csymbol cd="ambiguous" id="S7.I4.i4.p1.12.m3.1.1.1.cmml" xref="S7.I4.i4.p1.12.m3.1.1">subscript</csymbol><ci id="S7.I4.i4.p1.12.m3.1.1.2.cmml" xref="S7.I4.i4.p1.12.m3.1.1.2">𝑆</ci><ci id="S7.I4.i4.p1.12.m3.1.1.3.cmml" xref="S7.I4.i4.p1.12.m3.1.1.3">𝑠</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.I4.i4.p1.12.m3.1c">S_{s}</annotation><annotation encoding="application/x-llamapun" id="S7.I4.i4.p1.12.m3.1d">italic_S start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT</annotation></semantics></math> to <math alttext="v" class="ltx_Math" display="inline" id="S7.I4.i4.p1.13.m4.1"><semantics id="S7.I4.i4.p1.13.m4.1a"><mi id="S7.I4.i4.p1.13.m4.1.1" xref="S7.I4.i4.p1.13.m4.1.1.cmml">v</mi><annotation-xml encoding="MathML-Content" id="S7.I4.i4.p1.13.m4.1b"><ci id="S7.I4.i4.p1.13.m4.1.1.cmml" xref="S7.I4.i4.p1.13.m4.1.1">𝑣</ci></annotation-xml><annotation encoding="application/x-tex" id="S7.I4.i4.p1.13.m4.1c">v</annotation><annotation encoding="application/x-llamapun" id="S7.I4.i4.p1.13.m4.1d">italic_v</annotation></semantics></math>. We continue this until either <math alttext="\textsl{g}(v)=(1+\frac{\eta}{2})^{h+1}" class="ltx_Math" display="inline" id="S7.I4.i4.p1.14.m5.2"><semantics id="S7.I4.i4.p1.14.m5.2a"><mrow id="S7.I4.i4.p1.14.m5.2.2" xref="S7.I4.i4.p1.14.m5.2.2.cmml"><mrow id="S7.I4.i4.p1.14.m5.2.2.3" xref="S7.I4.i4.p1.14.m5.2.2.3.cmml"><mtext class="ltx_mathvariant_italic" id="S7.I4.i4.p1.14.m5.2.2.3.2" xref="S7.I4.i4.p1.14.m5.2.2.3.2a.cmml">g</mtext><mo id="S7.I4.i4.p1.14.m5.2.2.3.1" xref="S7.I4.i4.p1.14.m5.2.2.3.1.cmml">⁢</mo><mrow id="S7.I4.i4.p1.14.m5.2.2.3.3.2" xref="S7.I4.i4.p1.14.m5.2.2.3.cmml"><mo id="S7.I4.i4.p1.14.m5.2.2.3.3.2.1" stretchy="false" xref="S7.I4.i4.p1.14.m5.2.2.3.cmml">(</mo><mi id="S7.I4.i4.p1.14.m5.1.1" xref="S7.I4.i4.p1.14.m5.1.1.cmml">v</mi><mo id="S7.I4.i4.p1.14.m5.2.2.3.3.2.2" stretchy="false" xref="S7.I4.i4.p1.14.m5.2.2.3.cmml">)</mo></mrow></mrow><mo id="S7.I4.i4.p1.14.m5.2.2.2" xref="S7.I4.i4.p1.14.m5.2.2.2.cmml">=</mo><msup id="S7.I4.i4.p1.14.m5.2.2.1" xref="S7.I4.i4.p1.14.m5.2.2.1.cmml"><mrow id="S7.I4.i4.p1.14.m5.2.2.1.1.1" xref="S7.I4.i4.p1.14.m5.2.2.1.1.1.1.cmml"><mo id="S7.I4.i4.p1.14.m5.2.2.1.1.1.2" stretchy="false" xref="S7.I4.i4.p1.14.m5.2.2.1.1.1.1.cmml">(</mo><mrow id="S7.I4.i4.p1.14.m5.2.2.1.1.1.1" xref="S7.I4.i4.p1.14.m5.2.2.1.1.1.1.cmml"><mn id="S7.I4.i4.p1.14.m5.2.2.1.1.1.1.2" xref="S7.I4.i4.p1.14.m5.2.2.1.1.1.1.2.cmml">1</mn><mo id="S7.I4.i4.p1.14.m5.2.2.1.1.1.1.1" xref="S7.I4.i4.p1.14.m5.2.2.1.1.1.1.1.cmml">+</mo><mfrac id="S7.I4.i4.p1.14.m5.2.2.1.1.1.1.3" xref="S7.I4.i4.p1.14.m5.2.2.1.1.1.1.3.cmml"><mi id="S7.I4.i4.p1.14.m5.2.2.1.1.1.1.3.2" xref="S7.I4.i4.p1.14.m5.2.2.1.1.1.1.3.2.cmml">η</mi><mn id="S7.I4.i4.p1.14.m5.2.2.1.1.1.1.3.3" xref="S7.I4.i4.p1.14.m5.2.2.1.1.1.1.3.3.cmml">2</mn></mfrac></mrow><mo id="S7.I4.i4.p1.14.m5.2.2.1.1.1.3" stretchy="false" xref="S7.I4.i4.p1.14.m5.2.2.1.1.1.1.cmml">)</mo></mrow><mrow id="S7.I4.i4.p1.14.m5.2.2.1.3" xref="S7.I4.i4.p1.14.m5.2.2.1.3.cmml"><mi id="S7.I4.i4.p1.14.m5.2.2.1.3.2" xref="S7.I4.i4.p1.14.m5.2.2.1.3.2.cmml">h</mi><mo id="S7.I4.i4.p1.14.m5.2.2.1.3.1" xref="S7.I4.i4.p1.14.m5.2.2.1.3.1.cmml">+</mo><mn id="S7.I4.i4.p1.14.m5.2.2.1.3.3" xref="S7.I4.i4.p1.14.m5.2.2.1.3.3.cmml">1</mn></mrow></msup></mrow><annotation-xml encoding="MathML-Content" id="S7.I4.i4.p1.14.m5.2b"><apply id="S7.I4.i4.p1.14.m5.2.2.cmml" xref="S7.I4.i4.p1.14.m5.2.2"><eq id="S7.I4.i4.p1.14.m5.2.2.2.cmml" xref="S7.I4.i4.p1.14.m5.2.2.2"></eq><apply id="S7.I4.i4.p1.14.m5.2.2.3.cmml" xref="S7.I4.i4.p1.14.m5.2.2.3"><times id="S7.I4.i4.p1.14.m5.2.2.3.1.cmml" xref="S7.I4.i4.p1.14.m5.2.2.3.1"></times><ci id="S7.I4.i4.p1.14.m5.2.2.3.2a.cmml" xref="S7.I4.i4.p1.14.m5.2.2.3.2"><mtext class="ltx_mathvariant_italic" id="S7.I4.i4.p1.14.m5.2.2.3.2.cmml" xref="S7.I4.i4.p1.14.m5.2.2.3.2">g</mtext></ci><ci id="S7.I4.i4.p1.14.m5.1.1.cmml" xref="S7.I4.i4.p1.14.m5.1.1">𝑣</ci></apply><apply id="S7.I4.i4.p1.14.m5.2.2.1.cmml" xref="S7.I4.i4.p1.14.m5.2.2.1"><csymbol cd="ambiguous" id="S7.I4.i4.p1.14.m5.2.2.1.2.cmml" xref="S7.I4.i4.p1.14.m5.2.2.1">superscript</csymbol><apply id="S7.I4.i4.p1.14.m5.2.2.1.1.1.1.cmml" xref="S7.I4.i4.p1.14.m5.2.2.1.1.1"><plus id="S7.I4.i4.p1.14.m5.2.2.1.1.1.1.1.cmml" xref="S7.I4.i4.p1.14.m5.2.2.1.1.1.1.1"></plus><cn id="S7.I4.i4.p1.14.m5.2.2.1.1.1.1.2.cmml" type="integer" xref="S7.I4.i4.p1.14.m5.2.2.1.1.1.1.2">1</cn><apply id="S7.I4.i4.p1.14.m5.2.2.1.1.1.1.3.cmml" xref="S7.I4.i4.p1.14.m5.2.2.1.1.1.1.3"><divide id="S7.I4.i4.p1.14.m5.2.2.1.1.1.1.3.1.cmml" xref="S7.I4.i4.p1.14.m5.2.2.1.1.1.1.3"></divide><ci id="S7.I4.i4.p1.14.m5.2.2.1.1.1.1.3.2.cmml" xref="S7.I4.i4.p1.14.m5.2.2.1.1.1.1.3.2">𝜂</ci><cn id="S7.I4.i4.p1.14.m5.2.2.1.1.1.1.3.3.cmml" type="integer" xref="S7.I4.i4.p1.14.m5.2.2.1.1.1.1.3.3">2</cn></apply></apply><apply id="S7.I4.i4.p1.14.m5.2.2.1.3.cmml" xref="S7.I4.i4.p1.14.m5.2.2.1.3"><plus id="S7.I4.i4.p1.14.m5.2.2.1.3.1.cmml" xref="S7.I4.i4.p1.14.m5.2.2.1.3.1"></plus><ci id="S7.I4.i4.p1.14.m5.2.2.1.3.2.cmml" xref="S7.I4.i4.p1.14.m5.2.2.1.3.2">ℎ</ci><cn id="S7.I4.i4.p1.14.m5.2.2.1.3.3.cmml" type="integer" xref="S7.I4.i4.p1.14.m5.2.2.1.3.3">1</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.I4.i4.p1.14.m5.2c">\textsl{g}(v)=(1+\frac{\eta}{2})^{h+1}</annotation><annotation encoding="application/x-llamapun" id="S7.I4.i4.p1.14.m5.2d">g ( italic_v ) = ( 1 + divide start_ARG italic_η end_ARG start_ARG 2 end_ARG ) start_POSTSUPERSCRIPT italic_h + 1 end_POSTSUPERSCRIPT</annotation></semantics></math>, or, there exist no more edges <math alttext="(u,v)\in E_{s}" class="ltx_Math" display="inline" id="S7.I4.i4.p1.15.m6.2"><semantics id="S7.I4.i4.p1.15.m6.2a"><mrow id="S7.I4.i4.p1.15.m6.2.3" xref="S7.I4.i4.p1.15.m6.2.3.cmml"><mrow id="S7.I4.i4.p1.15.m6.2.3.2.2" xref="S7.I4.i4.p1.15.m6.2.3.2.1.cmml"><mo id="S7.I4.i4.p1.15.m6.2.3.2.2.1" stretchy="false" xref="S7.I4.i4.p1.15.m6.2.3.2.1.cmml">(</mo><mi id="S7.I4.i4.p1.15.m6.1.1" xref="S7.I4.i4.p1.15.m6.1.1.cmml">u</mi><mo id="S7.I4.i4.p1.15.m6.2.3.2.2.2" xref="S7.I4.i4.p1.15.m6.2.3.2.1.cmml">,</mo><mi id="S7.I4.i4.p1.15.m6.2.2" xref="S7.I4.i4.p1.15.m6.2.2.cmml">v</mi><mo id="S7.I4.i4.p1.15.m6.2.3.2.2.3" stretchy="false" xref="S7.I4.i4.p1.15.m6.2.3.2.1.cmml">)</mo></mrow><mo id="S7.I4.i4.p1.15.m6.2.3.1" xref="S7.I4.i4.p1.15.m6.2.3.1.cmml">∈</mo><msub id="S7.I4.i4.p1.15.m6.2.3.3" xref="S7.I4.i4.p1.15.m6.2.3.3.cmml"><mi id="S7.I4.i4.p1.15.m6.2.3.3.2" xref="S7.I4.i4.p1.15.m6.2.3.3.2.cmml">E</mi><mi id="S7.I4.i4.p1.15.m6.2.3.3.3" xref="S7.I4.i4.p1.15.m6.2.3.3.3.cmml">s</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S7.I4.i4.p1.15.m6.2b"><apply id="S7.I4.i4.p1.15.m6.2.3.cmml" xref="S7.I4.i4.p1.15.m6.2.3"><in id="S7.I4.i4.p1.15.m6.2.3.1.cmml" xref="S7.I4.i4.p1.15.m6.2.3.1"></in><interval closure="open" id="S7.I4.i4.p1.15.m6.2.3.2.1.cmml" xref="S7.I4.i4.p1.15.m6.2.3.2.2"><ci id="S7.I4.i4.p1.15.m6.1.1.cmml" xref="S7.I4.i4.p1.15.m6.1.1">𝑢</ci><ci id="S7.I4.i4.p1.15.m6.2.2.cmml" xref="S7.I4.i4.p1.15.m6.2.2">𝑣</ci></interval><apply id="S7.I4.i4.p1.15.m6.2.3.3.cmml" xref="S7.I4.i4.p1.15.m6.2.3.3"><csymbol cd="ambiguous" id="S7.I4.i4.p1.15.m6.2.3.3.1.cmml" xref="S7.I4.i4.p1.15.m6.2.3.3">subscript</csymbol><ci id="S7.I4.i4.p1.15.m6.2.3.3.2.cmml" xref="S7.I4.i4.p1.15.m6.2.3.3.2">𝐸</ci><ci id="S7.I4.i4.p1.15.m6.2.3.3.3.cmml" xref="S7.I4.i4.p1.15.m6.2.3.3.3">𝑠</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.I4.i4.p1.15.m6.2c">(u,v)\in E_{s}</annotation><annotation encoding="application/x-llamapun" id="S7.I4.i4.p1.15.m6.2d">( italic_u , italic_v ) ∈ italic_E start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT</annotation></semantics></math> with <math alttext="\textsl{g}(u\!\to\!v)>0" class="ltx_Math" display="inline" id="S7.I4.i4.p1.16.m7.1"><semantics id="S7.I4.i4.p1.16.m7.1a"><mrow id="S7.I4.i4.p1.16.m7.1.1" xref="S7.I4.i4.p1.16.m7.1.1.cmml"><mrow id="S7.I4.i4.p1.16.m7.1.1.1" xref="S7.I4.i4.p1.16.m7.1.1.1.cmml"><mtext class="ltx_mathvariant_italic" id="S7.I4.i4.p1.16.m7.1.1.1.3" xref="S7.I4.i4.p1.16.m7.1.1.1.3a.cmml">g</mtext><mo id="S7.I4.i4.p1.16.m7.1.1.1.2" xref="S7.I4.i4.p1.16.m7.1.1.1.2.cmml">⁢</mo><mrow id="S7.I4.i4.p1.16.m7.1.1.1.1.1" xref="S7.I4.i4.p1.16.m7.1.1.1.1.1.1.cmml"><mo id="S7.I4.i4.p1.16.m7.1.1.1.1.1.2" stretchy="false" xref="S7.I4.i4.p1.16.m7.1.1.1.1.1.1.cmml">(</mo><mrow id="S7.I4.i4.p1.16.m7.1.1.1.1.1.1" xref="S7.I4.i4.p1.16.m7.1.1.1.1.1.1.cmml"><mi id="S7.I4.i4.p1.16.m7.1.1.1.1.1.1.2" xref="S7.I4.i4.p1.16.m7.1.1.1.1.1.1.2.cmml">u</mi><mo id="S7.I4.i4.p1.16.m7.1.1.1.1.1.1.1" lspace="0.108em" rspace="0.108em" stretchy="false" xref="S7.I4.i4.p1.16.m7.1.1.1.1.1.1.1.cmml">→</mo><mi id="S7.I4.i4.p1.16.m7.1.1.1.1.1.1.3" xref="S7.I4.i4.p1.16.m7.1.1.1.1.1.1.3.cmml">v</mi></mrow><mo id="S7.I4.i4.p1.16.m7.1.1.1.1.1.3" stretchy="false" xref="S7.I4.i4.p1.16.m7.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S7.I4.i4.p1.16.m7.1.1.2" xref="S7.I4.i4.p1.16.m7.1.1.2.cmml">></mo><mn id="S7.I4.i4.p1.16.m7.1.1.3" xref="S7.I4.i4.p1.16.m7.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S7.I4.i4.p1.16.m7.1b"><apply id="S7.I4.i4.p1.16.m7.1.1.cmml" xref="S7.I4.i4.p1.16.m7.1.1"><gt id="S7.I4.i4.p1.16.m7.1.1.2.cmml" xref="S7.I4.i4.p1.16.m7.1.1.2"></gt><apply id="S7.I4.i4.p1.16.m7.1.1.1.cmml" xref="S7.I4.i4.p1.16.m7.1.1.1"><times id="S7.I4.i4.p1.16.m7.1.1.1.2.cmml" xref="S7.I4.i4.p1.16.m7.1.1.1.2"></times><ci id="S7.I4.i4.p1.16.m7.1.1.1.3a.cmml" xref="S7.I4.i4.p1.16.m7.1.1.1.3"><mtext class="ltx_mathvariant_italic" id="S7.I4.i4.p1.16.m7.1.1.1.3.cmml" xref="S7.I4.i4.p1.16.m7.1.1.1.3">g</mtext></ci><apply id="S7.I4.i4.p1.16.m7.1.1.1.1.1.1.cmml" xref="S7.I4.i4.p1.16.m7.1.1.1.1.1"><ci id="S7.I4.i4.p1.16.m7.1.1.1.1.1.1.1.cmml" xref="S7.I4.i4.p1.16.m7.1.1.1.1.1.1.1">→</ci><ci id="S7.I4.i4.p1.16.m7.1.1.1.1.1.1.2.cmml" xref="S7.I4.i4.p1.16.m7.1.1.1.1.1.1.2">𝑢</ci><ci id="S7.I4.i4.p1.16.m7.1.1.1.1.1.1.3.cmml" xref="S7.I4.i4.p1.16.m7.1.1.1.1.1.1.3">𝑣</ci></apply></apply><cn id="S7.I4.i4.p1.16.m7.1.1.3.cmml" type="integer" xref="S7.I4.i4.p1.16.m7.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.I4.i4.p1.16.m7.1c">\textsl{g}(u\!\to\!v)>0</annotation><annotation encoding="application/x-llamapun" id="S7.I4.i4.p1.16.m7.1d">g ( italic_u → italic_v ) > 0</annotation></semantics></math>. In both cases, <math alttext="v" class="ltx_Math" display="inline" id="S7.I4.i4.p1.17.m8.1"><semantics id="S7.I4.i4.p1.17.m8.1a"><mi id="S7.I4.i4.p1.17.m8.1.1" xref="S7.I4.i4.p1.17.m8.1.1.cmml">v</mi><annotation-xml encoding="MathML-Content" id="S7.I4.i4.p1.17.m8.1b"><ci id="S7.I4.i4.p1.17.m8.1.1.cmml" xref="S7.I4.i4.p1.17.m8.1.1">𝑣</ci></annotation-xml><annotation encoding="application/x-tex" id="S7.I4.i4.p1.17.m8.1c">v</annotation><annotation encoding="application/x-llamapun" id="S7.I4.i4.p1.17.m8.1d">italic_v</annotation></semantics></math> has no more violating in-edges. To find these directed paths, we create a flow problem on a graph <math alttext="D_{s}^{*}" class="ltx_Math" display="inline" id="S7.I4.i4.p1.18.m9.1"><semantics id="S7.I4.i4.p1.18.m9.1a"><msubsup id="S7.I4.i4.p1.18.m9.1.1" xref="S7.I4.i4.p1.18.m9.1.1.cmml"><mi id="S7.I4.i4.p1.18.m9.1.1.2.2" xref="S7.I4.i4.p1.18.m9.1.1.2.2.cmml">D</mi><mi id="S7.I4.i4.p1.18.m9.1.1.2.3" xref="S7.I4.i4.p1.18.m9.1.1.2.3.cmml">s</mi><mo id="S7.I4.i4.p1.18.m9.1.1.3" xref="S7.I4.i4.p1.18.m9.1.1.3.cmml">∗</mo></msubsup><annotation-xml encoding="MathML-Content" id="S7.I4.i4.p1.18.m9.1b"><apply id="S7.I4.i4.p1.18.m9.1.1.cmml" xref="S7.I4.i4.p1.18.m9.1.1"><csymbol cd="ambiguous" id="S7.I4.i4.p1.18.m9.1.1.1.cmml" xref="S7.I4.i4.p1.18.m9.1.1">superscript</csymbol><apply id="S7.I4.i4.p1.18.m9.1.1.2.cmml" xref="S7.I4.i4.p1.18.m9.1.1"><csymbol cd="ambiguous" id="S7.I4.i4.p1.18.m9.1.1.2.1.cmml" xref="S7.I4.i4.p1.18.m9.1.1">subscript</csymbol><ci id="S7.I4.i4.p1.18.m9.1.1.2.2.cmml" xref="S7.I4.i4.p1.18.m9.1.1.2.2">𝐷</ci><ci id="S7.I4.i4.p1.18.m9.1.1.2.3.cmml" xref="S7.I4.i4.p1.18.m9.1.1.2.3">𝑠</ci></apply><times id="S7.I4.i4.p1.18.m9.1.1.3.cmml" xref="S7.I4.i4.p1.18.m9.1.1.3"></times></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.I4.i4.p1.18.m9.1c">D_{s}^{*}</annotation><annotation encoding="application/x-llamapun" id="S7.I4.i4.p1.18.m9.1d">italic_D start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT</annotation></semantics></math> where the maximal path has length <math alttext="\ell+2" class="ltx_Math" display="inline" id="S7.I4.i4.p1.19.m10.1"><semantics id="S7.I4.i4.p1.19.m10.1a"><mrow id="S7.I4.i4.p1.19.m10.1.1" xref="S7.I4.i4.p1.19.m10.1.1.cmml"><mi id="S7.I4.i4.p1.19.m10.1.1.2" mathvariant="normal" xref="S7.I4.i4.p1.19.m10.1.1.2.cmml">ℓ</mi><mo id="S7.I4.i4.p1.19.m10.1.1.1" xref="S7.I4.i4.p1.19.m10.1.1.1.cmml">+</mo><mn id="S7.I4.i4.p1.19.m10.1.1.3" xref="S7.I4.i4.p1.19.m10.1.1.3.cmml">2</mn></mrow><annotation-xml encoding="MathML-Content" id="S7.I4.i4.p1.19.m10.1b"><apply id="S7.I4.i4.p1.19.m10.1.1.cmml" xref="S7.I4.i4.p1.19.m10.1.1"><plus id="S7.I4.i4.p1.19.m10.1.1.1.cmml" xref="S7.I4.i4.p1.19.m10.1.1.1"></plus><ci id="S7.I4.i4.p1.19.m10.1.1.2.cmml" xref="S7.I4.i4.p1.19.m10.1.1.2">ℓ</ci><cn id="S7.I4.i4.p1.19.m10.1.1.3.cmml" type="integer" xref="S7.I4.i4.p1.19.m10.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.I4.i4.p1.19.m10.1c">\ell+2</annotation><annotation encoding="application/x-llamapun" id="S7.I4.i4.p1.19.m10.1d">roman_ℓ + 2</annotation></semantics></math>:</p> <ul class="ltx_itemize" id="S7.I4.i4.I2"> <li class="ltx_item" id="S7.I4.i4.I2.i1" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item"><span class="ltx_text ltx_font_bold" id="S7.I4.i4.I2.i1.1.1.1">–</span></span> <div class="ltx_para" id="S7.I4.i4.I2.i1.p1"> <p class="ltx_p" id="S7.I4.i4.I2.i1.p1.8">For each <math alttext="u\in S_{s}" class="ltx_Math" display="inline" id="S7.I4.i4.I2.i1.p1.1.m1.1"><semantics id="S7.I4.i4.I2.i1.p1.1.m1.1a"><mrow id="S7.I4.i4.I2.i1.p1.1.m1.1.1" xref="S7.I4.i4.I2.i1.p1.1.m1.1.1.cmml"><mi id="S7.I4.i4.I2.i1.p1.1.m1.1.1.2" xref="S7.I4.i4.I2.i1.p1.1.m1.1.1.2.cmml">u</mi><mo id="S7.I4.i4.I2.i1.p1.1.m1.1.1.1" xref="S7.I4.i4.I2.i1.p1.1.m1.1.1.1.cmml">∈</mo><msub id="S7.I4.i4.I2.i1.p1.1.m1.1.1.3" xref="S7.I4.i4.I2.i1.p1.1.m1.1.1.3.cmml"><mi id="S7.I4.i4.I2.i1.p1.1.m1.1.1.3.2" xref="S7.I4.i4.I2.i1.p1.1.m1.1.1.3.2.cmml">S</mi><mi id="S7.I4.i4.I2.i1.p1.1.m1.1.1.3.3" xref="S7.I4.i4.I2.i1.p1.1.m1.1.1.3.3.cmml">s</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S7.I4.i4.I2.i1.p1.1.m1.1b"><apply id="S7.I4.i4.I2.i1.p1.1.m1.1.1.cmml" xref="S7.I4.i4.I2.i1.p1.1.m1.1.1"><in id="S7.I4.i4.I2.i1.p1.1.m1.1.1.1.cmml" xref="S7.I4.i4.I2.i1.p1.1.m1.1.1.1"></in><ci id="S7.I4.i4.I2.i1.p1.1.m1.1.1.2.cmml" xref="S7.I4.i4.I2.i1.p1.1.m1.1.1.2">𝑢</ci><apply id="S7.I4.i4.I2.i1.p1.1.m1.1.1.3.cmml" xref="S7.I4.i4.I2.i1.p1.1.m1.1.1.3"><csymbol cd="ambiguous" id="S7.I4.i4.I2.i1.p1.1.m1.1.1.3.1.cmml" xref="S7.I4.i4.I2.i1.p1.1.m1.1.1.3">subscript</csymbol><ci id="S7.I4.i4.I2.i1.p1.1.m1.1.1.3.2.cmml" xref="S7.I4.i4.I2.i1.p1.1.m1.1.1.3.2">𝑆</ci><ci id="S7.I4.i4.I2.i1.p1.1.m1.1.1.3.3.cmml" xref="S7.I4.i4.I2.i1.p1.1.m1.1.1.3.3">𝑠</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.I4.i4.I2.i1.p1.1.m1.1c">u\in S_{s}</annotation><annotation encoding="application/x-llamapun" id="S7.I4.i4.I2.i1.p1.1.m1.1d">italic_u ∈ italic_S start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT</annotation></semantics></math>, we define <math alttext="\sigma(u)=\textsl{g}(u)-(1+\frac{\eta}{2})^{l_{m}(u)-1}" class="ltx_Math" display="inline" id="S7.I4.i4.I2.i1.p1.2.m2.4"><semantics id="S7.I4.i4.I2.i1.p1.2.m2.4a"><mrow id="S7.I4.i4.I2.i1.p1.2.m2.4.4" xref="S7.I4.i4.I2.i1.p1.2.m2.4.4.cmml"><mrow id="S7.I4.i4.I2.i1.p1.2.m2.4.4.3" xref="S7.I4.i4.I2.i1.p1.2.m2.4.4.3.cmml"><mi id="S7.I4.i4.I2.i1.p1.2.m2.4.4.3.2" xref="S7.I4.i4.I2.i1.p1.2.m2.4.4.3.2.cmml">σ</mi><mo id="S7.I4.i4.I2.i1.p1.2.m2.4.4.3.1" xref="S7.I4.i4.I2.i1.p1.2.m2.4.4.3.1.cmml">⁢</mo><mrow id="S7.I4.i4.I2.i1.p1.2.m2.4.4.3.3.2" xref="S7.I4.i4.I2.i1.p1.2.m2.4.4.3.cmml"><mo id="S7.I4.i4.I2.i1.p1.2.m2.4.4.3.3.2.1" stretchy="false" xref="S7.I4.i4.I2.i1.p1.2.m2.4.4.3.cmml">(</mo><mi id="S7.I4.i4.I2.i1.p1.2.m2.2.2" xref="S7.I4.i4.I2.i1.p1.2.m2.2.2.cmml">u</mi><mo id="S7.I4.i4.I2.i1.p1.2.m2.4.4.3.3.2.2" stretchy="false" xref="S7.I4.i4.I2.i1.p1.2.m2.4.4.3.cmml">)</mo></mrow></mrow><mo id="S7.I4.i4.I2.i1.p1.2.m2.4.4.2" xref="S7.I4.i4.I2.i1.p1.2.m2.4.4.2.cmml">=</mo><mrow id="S7.I4.i4.I2.i1.p1.2.m2.4.4.1" xref="S7.I4.i4.I2.i1.p1.2.m2.4.4.1.cmml"><mrow id="S7.I4.i4.I2.i1.p1.2.m2.4.4.1.3" xref="S7.I4.i4.I2.i1.p1.2.m2.4.4.1.3.cmml"><mtext class="ltx_mathvariant_italic" id="S7.I4.i4.I2.i1.p1.2.m2.4.4.1.3.2" xref="S7.I4.i4.I2.i1.p1.2.m2.4.4.1.3.2a.cmml">g</mtext><mo id="S7.I4.i4.I2.i1.p1.2.m2.4.4.1.3.1" xref="S7.I4.i4.I2.i1.p1.2.m2.4.4.1.3.1.cmml">⁢</mo><mrow id="S7.I4.i4.I2.i1.p1.2.m2.4.4.1.3.3.2" xref="S7.I4.i4.I2.i1.p1.2.m2.4.4.1.3.cmml"><mo id="S7.I4.i4.I2.i1.p1.2.m2.4.4.1.3.3.2.1" stretchy="false" xref="S7.I4.i4.I2.i1.p1.2.m2.4.4.1.3.cmml">(</mo><mi id="S7.I4.i4.I2.i1.p1.2.m2.3.3" xref="S7.I4.i4.I2.i1.p1.2.m2.3.3.cmml">u</mi><mo id="S7.I4.i4.I2.i1.p1.2.m2.4.4.1.3.3.2.2" stretchy="false" xref="S7.I4.i4.I2.i1.p1.2.m2.4.4.1.3.cmml">)</mo></mrow></mrow><mo id="S7.I4.i4.I2.i1.p1.2.m2.4.4.1.2" xref="S7.I4.i4.I2.i1.p1.2.m2.4.4.1.2.cmml">−</mo><msup id="S7.I4.i4.I2.i1.p1.2.m2.4.4.1.1" xref="S7.I4.i4.I2.i1.p1.2.m2.4.4.1.1.cmml"><mrow id="S7.I4.i4.I2.i1.p1.2.m2.4.4.1.1.1.1" xref="S7.I4.i4.I2.i1.p1.2.m2.4.4.1.1.1.1.1.cmml"><mo id="S7.I4.i4.I2.i1.p1.2.m2.4.4.1.1.1.1.2" stretchy="false" xref="S7.I4.i4.I2.i1.p1.2.m2.4.4.1.1.1.1.1.cmml">(</mo><mrow id="S7.I4.i4.I2.i1.p1.2.m2.4.4.1.1.1.1.1" xref="S7.I4.i4.I2.i1.p1.2.m2.4.4.1.1.1.1.1.cmml"><mn 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xref="S7.I4.i4.I2.i1.p1.2.m2.4.4.1.1.1.1.1.3.2">𝜂</ci><cn id="S7.I4.i4.I2.i1.p1.2.m2.4.4.1.1.1.1.1.3.3.cmml" type="integer" xref="S7.I4.i4.I2.i1.p1.2.m2.4.4.1.1.1.1.1.3.3">2</cn></apply></apply><apply id="S7.I4.i4.I2.i1.p1.2.m2.1.1.1.cmml" xref="S7.I4.i4.I2.i1.p1.2.m2.1.1.1"><minus id="S7.I4.i4.I2.i1.p1.2.m2.1.1.1.2.cmml" xref="S7.I4.i4.I2.i1.p1.2.m2.1.1.1.2"></minus><apply id="S7.I4.i4.I2.i1.p1.2.m2.1.1.1.3.cmml" xref="S7.I4.i4.I2.i1.p1.2.m2.1.1.1.3"><times id="S7.I4.i4.I2.i1.p1.2.m2.1.1.1.3.1.cmml" xref="S7.I4.i4.I2.i1.p1.2.m2.1.1.1.3.1"></times><apply id="S7.I4.i4.I2.i1.p1.2.m2.1.1.1.3.2.cmml" xref="S7.I4.i4.I2.i1.p1.2.m2.1.1.1.3.2"><csymbol cd="ambiguous" id="S7.I4.i4.I2.i1.p1.2.m2.1.1.1.3.2.1.cmml" xref="S7.I4.i4.I2.i1.p1.2.m2.1.1.1.3.2">subscript</csymbol><ci id="S7.I4.i4.I2.i1.p1.2.m2.1.1.1.3.2.2.cmml" xref="S7.I4.i4.I2.i1.p1.2.m2.1.1.1.3.2.2">𝑙</ci><ci id="S7.I4.i4.I2.i1.p1.2.m2.1.1.1.3.2.3.cmml" xref="S7.I4.i4.I2.i1.p1.2.m2.1.1.1.3.2.3">𝑚</ci></apply><ci id="S7.I4.i4.I2.i1.p1.2.m2.1.1.1.1.cmml" xref="S7.I4.i4.I2.i1.p1.2.m2.1.1.1.1">𝑢</ci></apply><cn id="S7.I4.i4.I2.i1.p1.2.m2.1.1.1.4.cmml" type="integer" xref="S7.I4.i4.I2.i1.p1.2.m2.1.1.1.4">1</cn></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.I4.i4.I2.i1.p1.2.m2.4c">\sigma(u)=\textsl{g}(u)-(1+\frac{\eta}{2})^{l_{m}(u)-1}</annotation><annotation encoding="application/x-llamapun" id="S7.I4.i4.I2.i1.p1.2.m2.4d">italic_σ ( italic_u ) = g ( italic_u ) - ( 1 + divide start_ARG italic_η end_ARG start_ARG 2 end_ARG ) start_POSTSUPERSCRIPT italic_l start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT ( italic_u ) - 1 end_POSTSUPERSCRIPT</annotation></semantics></math> (the maximal amount we can decrease <math alttext="\textsl{g}(u)" class="ltx_Math" display="inline" id="S7.I4.i4.I2.i1.p1.3.m3.1"><semantics id="S7.I4.i4.I2.i1.p1.3.m3.1a"><mrow id="S7.I4.i4.I2.i1.p1.3.m3.1.2" xref="S7.I4.i4.I2.i1.p1.3.m3.1.2.cmml"><mtext class="ltx_mathvariant_italic" id="S7.I4.i4.I2.i1.p1.3.m3.1.2.2" xref="S7.I4.i4.I2.i1.p1.3.m3.1.2.2a.cmml">g</mtext><mo id="S7.I4.i4.I2.i1.p1.3.m3.1.2.1" xref="S7.I4.i4.I2.i1.p1.3.m3.1.2.1.cmml">⁢</mo><mrow id="S7.I4.i4.I2.i1.p1.3.m3.1.2.3.2" xref="S7.I4.i4.I2.i1.p1.3.m3.1.2.cmml"><mo id="S7.I4.i4.I2.i1.p1.3.m3.1.2.3.2.1" stretchy="false" xref="S7.I4.i4.I2.i1.p1.3.m3.1.2.cmml">(</mo><mi id="S7.I4.i4.I2.i1.p1.3.m3.1.1" xref="S7.I4.i4.I2.i1.p1.3.m3.1.1.cmml">u</mi><mo id="S7.I4.i4.I2.i1.p1.3.m3.1.2.3.2.2" stretchy="false" xref="S7.I4.i4.I2.i1.p1.3.m3.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S7.I4.i4.I2.i1.p1.3.m3.1b"><apply id="S7.I4.i4.I2.i1.p1.3.m3.1.2.cmml" xref="S7.I4.i4.I2.i1.p1.3.m3.1.2"><times id="S7.I4.i4.I2.i1.p1.3.m3.1.2.1.cmml" xref="S7.I4.i4.I2.i1.p1.3.m3.1.2.1"></times><ci id="S7.I4.i4.I2.i1.p1.3.m3.1.2.2a.cmml" xref="S7.I4.i4.I2.i1.p1.3.m3.1.2.2"><mtext class="ltx_mathvariant_italic" id="S7.I4.i4.I2.i1.p1.3.m3.1.2.2.cmml" xref="S7.I4.i4.I2.i1.p1.3.m3.1.2.2">g</mtext></ci><ci id="S7.I4.i4.I2.i1.p1.3.m3.1.1.cmml" xref="S7.I4.i4.I2.i1.p1.3.m3.1.1">𝑢</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.I4.i4.I2.i1.p1.3.m3.1c">\textsl{g}(u)</annotation><annotation encoding="application/x-llamapun" id="S7.I4.i4.I2.i1.p1.3.m3.1d">g ( italic_u )</annotation></semantics></math> such that is does not arrive in level <math alttext="l_{m}(u)-2" class="ltx_Math" display="inline" id="S7.I4.i4.I2.i1.p1.4.m4.1"><semantics id="S7.I4.i4.I2.i1.p1.4.m4.1a"><mrow id="S7.I4.i4.I2.i1.p1.4.m4.1.2" xref="S7.I4.i4.I2.i1.p1.4.m4.1.2.cmml"><mrow id="S7.I4.i4.I2.i1.p1.4.m4.1.2.2" xref="S7.I4.i4.I2.i1.p1.4.m4.1.2.2.cmml"><msub id="S7.I4.i4.I2.i1.p1.4.m4.1.2.2.2" xref="S7.I4.i4.I2.i1.p1.4.m4.1.2.2.2.cmml"><mi id="S7.I4.i4.I2.i1.p1.4.m4.1.2.2.2.2" xref="S7.I4.i4.I2.i1.p1.4.m4.1.2.2.2.2.cmml">l</mi><mi id="S7.I4.i4.I2.i1.p1.4.m4.1.2.2.2.3" xref="S7.I4.i4.I2.i1.p1.4.m4.1.2.2.2.3.cmml">m</mi></msub><mo id="S7.I4.i4.I2.i1.p1.4.m4.1.2.2.1" xref="S7.I4.i4.I2.i1.p1.4.m4.1.2.2.1.cmml">⁢</mo><mrow id="S7.I4.i4.I2.i1.p1.4.m4.1.2.2.3.2" xref="S7.I4.i4.I2.i1.p1.4.m4.1.2.2.cmml"><mo id="S7.I4.i4.I2.i1.p1.4.m4.1.2.2.3.2.1" stretchy="false" xref="S7.I4.i4.I2.i1.p1.4.m4.1.2.2.cmml">(</mo><mi id="S7.I4.i4.I2.i1.p1.4.m4.1.1" xref="S7.I4.i4.I2.i1.p1.4.m4.1.1.cmml">u</mi><mo id="S7.I4.i4.I2.i1.p1.4.m4.1.2.2.3.2.2" stretchy="false" xref="S7.I4.i4.I2.i1.p1.4.m4.1.2.2.cmml">)</mo></mrow></mrow><mo id="S7.I4.i4.I2.i1.p1.4.m4.1.2.1" xref="S7.I4.i4.I2.i1.p1.4.m4.1.2.1.cmml">−</mo><mn id="S7.I4.i4.I2.i1.p1.4.m4.1.2.3" xref="S7.I4.i4.I2.i1.p1.4.m4.1.2.3.cmml">2</mn></mrow><annotation-xml encoding="MathML-Content" id="S7.I4.i4.I2.i1.p1.4.m4.1b"><apply id="S7.I4.i4.I2.i1.p1.4.m4.1.2.cmml" xref="S7.I4.i4.I2.i1.p1.4.m4.1.2"><minus id="S7.I4.i4.I2.i1.p1.4.m4.1.2.1.cmml" xref="S7.I4.i4.I2.i1.p1.4.m4.1.2.1"></minus><apply id="S7.I4.i4.I2.i1.p1.4.m4.1.2.2.cmml" xref="S7.I4.i4.I2.i1.p1.4.m4.1.2.2"><times id="S7.I4.i4.I2.i1.p1.4.m4.1.2.2.1.cmml" xref="S7.I4.i4.I2.i1.p1.4.m4.1.2.2.1"></times><apply id="S7.I4.i4.I2.i1.p1.4.m4.1.2.2.2.cmml" xref="S7.I4.i4.I2.i1.p1.4.m4.1.2.2.2"><csymbol cd="ambiguous" id="S7.I4.i4.I2.i1.p1.4.m4.1.2.2.2.1.cmml" xref="S7.I4.i4.I2.i1.p1.4.m4.1.2.2.2">subscript</csymbol><ci id="S7.I4.i4.I2.i1.p1.4.m4.1.2.2.2.2.cmml" xref="S7.I4.i4.I2.i1.p1.4.m4.1.2.2.2.2">𝑙</ci><ci id="S7.I4.i4.I2.i1.p1.4.m4.1.2.2.2.3.cmml" xref="S7.I4.i4.I2.i1.p1.4.m4.1.2.2.2.3">𝑚</ci></apply><ci id="S7.I4.i4.I2.i1.p1.4.m4.1.1.cmml" xref="S7.I4.i4.I2.i1.p1.4.m4.1.1">𝑢</ci></apply><cn id="S7.I4.i4.I2.i1.p1.4.m4.1.2.3.cmml" type="integer" xref="S7.I4.i4.I2.i1.p1.4.m4.1.2.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.I4.i4.I2.i1.p1.4.m4.1c">l_{m}(u)-2</annotation><annotation encoding="application/x-llamapun" id="S7.I4.i4.I2.i1.p1.4.m4.1d">italic_l start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT ( italic_u ) - 2</annotation></semantics></math>). We connect every <math alttext="u\in S_{s}" class="ltx_Math" display="inline" id="S7.I4.i4.I2.i1.p1.5.m5.1"><semantics id="S7.I4.i4.I2.i1.p1.5.m5.1a"><mrow id="S7.I4.i4.I2.i1.p1.5.m5.1.1" xref="S7.I4.i4.I2.i1.p1.5.m5.1.1.cmml"><mi id="S7.I4.i4.I2.i1.p1.5.m5.1.1.2" xref="S7.I4.i4.I2.i1.p1.5.m5.1.1.2.cmml">u</mi><mo id="S7.I4.i4.I2.i1.p1.5.m5.1.1.1" xref="S7.I4.i4.I2.i1.p1.5.m5.1.1.1.cmml">∈</mo><msub id="S7.I4.i4.I2.i1.p1.5.m5.1.1.3" xref="S7.I4.i4.I2.i1.p1.5.m5.1.1.3.cmml"><mi id="S7.I4.i4.I2.i1.p1.5.m5.1.1.3.2" xref="S7.I4.i4.I2.i1.p1.5.m5.1.1.3.2.cmml">S</mi><mi id="S7.I4.i4.I2.i1.p1.5.m5.1.1.3.3" xref="S7.I4.i4.I2.i1.p1.5.m5.1.1.3.3.cmml">s</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S7.I4.i4.I2.i1.p1.5.m5.1b"><apply id="S7.I4.i4.I2.i1.p1.5.m5.1.1.cmml" xref="S7.I4.i4.I2.i1.p1.5.m5.1.1"><in id="S7.I4.i4.I2.i1.p1.5.m5.1.1.1.cmml" xref="S7.I4.i4.I2.i1.p1.5.m5.1.1.1"></in><ci id="S7.I4.i4.I2.i1.p1.5.m5.1.1.2.cmml" xref="S7.I4.i4.I2.i1.p1.5.m5.1.1.2">𝑢</ci><apply id="S7.I4.i4.I2.i1.p1.5.m5.1.1.3.cmml" xref="S7.I4.i4.I2.i1.p1.5.m5.1.1.3"><csymbol cd="ambiguous" id="S7.I4.i4.I2.i1.p1.5.m5.1.1.3.1.cmml" xref="S7.I4.i4.I2.i1.p1.5.m5.1.1.3">subscript</csymbol><ci id="S7.I4.i4.I2.i1.p1.5.m5.1.1.3.2.cmml" xref="S7.I4.i4.I2.i1.p1.5.m5.1.1.3.2">𝑆</ci><ci id="S7.I4.i4.I2.i1.p1.5.m5.1.1.3.3.cmml" xref="S7.I4.i4.I2.i1.p1.5.m5.1.1.3.3">𝑠</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.I4.i4.I2.i1.p1.5.m5.1c">u\in S_{s}</annotation><annotation encoding="application/x-llamapun" id="S7.I4.i4.I2.i1.p1.5.m5.1d">italic_u ∈ italic_S start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT</annotation></semantics></math> to a unique source <math alttext="s_{u}" class="ltx_Math" display="inline" id="S7.I4.i4.I2.i1.p1.6.m6.1"><semantics id="S7.I4.i4.I2.i1.p1.6.m6.1a"><msub id="S7.I4.i4.I2.i1.p1.6.m6.1.1" xref="S7.I4.i4.I2.i1.p1.6.m6.1.1.cmml"><mi id="S7.I4.i4.I2.i1.p1.6.m6.1.1.2" xref="S7.I4.i4.I2.i1.p1.6.m6.1.1.2.cmml">s</mi><mi id="S7.I4.i4.I2.i1.p1.6.m6.1.1.3" xref="S7.I4.i4.I2.i1.p1.6.m6.1.1.3.cmml">u</mi></msub><annotation-xml encoding="MathML-Content" id="S7.I4.i4.I2.i1.p1.6.m6.1b"><apply id="S7.I4.i4.I2.i1.p1.6.m6.1.1.cmml" xref="S7.I4.i4.I2.i1.p1.6.m6.1.1"><csymbol cd="ambiguous" id="S7.I4.i4.I2.i1.p1.6.m6.1.1.1.cmml" xref="S7.I4.i4.I2.i1.p1.6.m6.1.1">subscript</csymbol><ci id="S7.I4.i4.I2.i1.p1.6.m6.1.1.2.cmml" xref="S7.I4.i4.I2.i1.p1.6.m6.1.1.2">𝑠</ci><ci id="S7.I4.i4.I2.i1.p1.6.m6.1.1.3.cmml" xref="S7.I4.i4.I2.i1.p1.6.m6.1.1.3">𝑢</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.I4.i4.I2.i1.p1.6.m6.1c">s_{u}</annotation><annotation encoding="application/x-llamapun" id="S7.I4.i4.I2.i1.p1.6.m6.1d">italic_s start_POSTSUBSCRIPT italic_u end_POSTSUBSCRIPT</annotation></semantics></math> where the edge <math alttext="\overline{s_{u}u}" class="ltx_Math" display="inline" id="S7.I4.i4.I2.i1.p1.7.m7.1"><semantics id="S7.I4.i4.I2.i1.p1.7.m7.1a"><mover accent="true" id="S7.I4.i4.I2.i1.p1.7.m7.1.1" xref="S7.I4.i4.I2.i1.p1.7.m7.1.1.cmml"><mrow id="S7.I4.i4.I2.i1.p1.7.m7.1.1.2" xref="S7.I4.i4.I2.i1.p1.7.m7.1.1.2.cmml"><msub id="S7.I4.i4.I2.i1.p1.7.m7.1.1.2.2" xref="S7.I4.i4.I2.i1.p1.7.m7.1.1.2.2.cmml"><mi id="S7.I4.i4.I2.i1.p1.7.m7.1.1.2.2.2" xref="S7.I4.i4.I2.i1.p1.7.m7.1.1.2.2.2.cmml">s</mi><mi id="S7.I4.i4.I2.i1.p1.7.m7.1.1.2.2.3" xref="S7.I4.i4.I2.i1.p1.7.m7.1.1.2.2.3.cmml">u</mi></msub><mo id="S7.I4.i4.I2.i1.p1.7.m7.1.1.2.1" xref="S7.I4.i4.I2.i1.p1.7.m7.1.1.2.1.cmml">⁢</mo><mi id="S7.I4.i4.I2.i1.p1.7.m7.1.1.2.3" xref="S7.I4.i4.I2.i1.p1.7.m7.1.1.2.3.cmml">u</mi></mrow><mo id="S7.I4.i4.I2.i1.p1.7.m7.1.1.1" xref="S7.I4.i4.I2.i1.p1.7.m7.1.1.1.cmml">¯</mo></mover><annotation-xml encoding="MathML-Content" id="S7.I4.i4.I2.i1.p1.7.m7.1b"><apply id="S7.I4.i4.I2.i1.p1.7.m7.1.1.cmml" xref="S7.I4.i4.I2.i1.p1.7.m7.1.1"><ci id="S7.I4.i4.I2.i1.p1.7.m7.1.1.1.cmml" xref="S7.I4.i4.I2.i1.p1.7.m7.1.1.1">¯</ci><apply id="S7.I4.i4.I2.i1.p1.7.m7.1.1.2.cmml" xref="S7.I4.i4.I2.i1.p1.7.m7.1.1.2"><times id="S7.I4.i4.I2.i1.p1.7.m7.1.1.2.1.cmml" xref="S7.I4.i4.I2.i1.p1.7.m7.1.1.2.1"></times><apply id="S7.I4.i4.I2.i1.p1.7.m7.1.1.2.2.cmml" xref="S7.I4.i4.I2.i1.p1.7.m7.1.1.2.2"><csymbol cd="ambiguous" id="S7.I4.i4.I2.i1.p1.7.m7.1.1.2.2.1.cmml" xref="S7.I4.i4.I2.i1.p1.7.m7.1.1.2.2">subscript</csymbol><ci id="S7.I4.i4.I2.i1.p1.7.m7.1.1.2.2.2.cmml" xref="S7.I4.i4.I2.i1.p1.7.m7.1.1.2.2.2">𝑠</ci><ci id="S7.I4.i4.I2.i1.p1.7.m7.1.1.2.2.3.cmml" xref="S7.I4.i4.I2.i1.p1.7.m7.1.1.2.2.3">𝑢</ci></apply><ci id="S7.I4.i4.I2.i1.p1.7.m7.1.1.2.3.cmml" xref="S7.I4.i4.I2.i1.p1.7.m7.1.1.2.3">𝑢</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.I4.i4.I2.i1.p1.7.m7.1c">\overline{s_{u}u}</annotation><annotation encoding="application/x-llamapun" id="S7.I4.i4.I2.i1.p1.7.m7.1d">over¯ start_ARG italic_s start_POSTSUBSCRIPT italic_u end_POSTSUBSCRIPT italic_u end_ARG</annotation></semantics></math> has capacity <math alttext="\sigma(u)" class="ltx_Math" display="inline" id="S7.I4.i4.I2.i1.p1.8.m8.1"><semantics id="S7.I4.i4.I2.i1.p1.8.m8.1a"><mrow id="S7.I4.i4.I2.i1.p1.8.m8.1.2" xref="S7.I4.i4.I2.i1.p1.8.m8.1.2.cmml"><mi id="S7.I4.i4.I2.i1.p1.8.m8.1.2.2" xref="S7.I4.i4.I2.i1.p1.8.m8.1.2.2.cmml">σ</mi><mo id="S7.I4.i4.I2.i1.p1.8.m8.1.2.1" xref="S7.I4.i4.I2.i1.p1.8.m8.1.2.1.cmml">⁢</mo><mrow id="S7.I4.i4.I2.i1.p1.8.m8.1.2.3.2" xref="S7.I4.i4.I2.i1.p1.8.m8.1.2.cmml"><mo id="S7.I4.i4.I2.i1.p1.8.m8.1.2.3.2.1" stretchy="false" xref="S7.I4.i4.I2.i1.p1.8.m8.1.2.cmml">(</mo><mi id="S7.I4.i4.I2.i1.p1.8.m8.1.1" xref="S7.I4.i4.I2.i1.p1.8.m8.1.1.cmml">u</mi><mo id="S7.I4.i4.I2.i1.p1.8.m8.1.2.3.2.2" stretchy="false" xref="S7.I4.i4.I2.i1.p1.8.m8.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S7.I4.i4.I2.i1.p1.8.m8.1b"><apply id="S7.I4.i4.I2.i1.p1.8.m8.1.2.cmml" xref="S7.I4.i4.I2.i1.p1.8.m8.1.2"><times id="S7.I4.i4.I2.i1.p1.8.m8.1.2.1.cmml" xref="S7.I4.i4.I2.i1.p1.8.m8.1.2.1"></times><ci id="S7.I4.i4.I2.i1.p1.8.m8.1.2.2.cmml" xref="S7.I4.i4.I2.i1.p1.8.m8.1.2.2">𝜎</ci><ci id="S7.I4.i4.I2.i1.p1.8.m8.1.1.cmml" xref="S7.I4.i4.I2.i1.p1.8.m8.1.1">𝑢</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.I4.i4.I2.i1.p1.8.m8.1c">\sigma(u)</annotation><annotation encoding="application/x-llamapun" id="S7.I4.i4.I2.i1.p1.8.m8.1d">italic_σ ( italic_u )</annotation></semantics></math>.</p> </div> </li> <li class="ltx_item" id="S7.I4.i4.I2.i2" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item"><span class="ltx_text ltx_font_bold" id="S7.I4.i4.I2.i2.1.1.1">–</span></span> <div class="ltx_para" id="S7.I4.i4.I2.i2.p1"> <p class="ltx_p" id="S7.I4.i4.I2.i2.p1.8">For each <math alttext="v\in T_{m}" class="ltx_Math" display="inline" id="S7.I4.i4.I2.i2.p1.1.m1.1"><semantics id="S7.I4.i4.I2.i2.p1.1.m1.1a"><mrow id="S7.I4.i4.I2.i2.p1.1.m1.1.1" xref="S7.I4.i4.I2.i2.p1.1.m1.1.1.cmml"><mi id="S7.I4.i4.I2.i2.p1.1.m1.1.1.2" xref="S7.I4.i4.I2.i2.p1.1.m1.1.1.2.cmml">v</mi><mo id="S7.I4.i4.I2.i2.p1.1.m1.1.1.1" xref="S7.I4.i4.I2.i2.p1.1.m1.1.1.1.cmml">∈</mo><msub id="S7.I4.i4.I2.i2.p1.1.m1.1.1.3" xref="S7.I4.i4.I2.i2.p1.1.m1.1.1.3.cmml"><mi id="S7.I4.i4.I2.i2.p1.1.m1.1.1.3.2" xref="S7.I4.i4.I2.i2.p1.1.m1.1.1.3.2.cmml">T</mi><mi id="S7.I4.i4.I2.i2.p1.1.m1.1.1.3.3" xref="S7.I4.i4.I2.i2.p1.1.m1.1.1.3.3.cmml">m</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S7.I4.i4.I2.i2.p1.1.m1.1b"><apply id="S7.I4.i4.I2.i2.p1.1.m1.1.1.cmml" xref="S7.I4.i4.I2.i2.p1.1.m1.1.1"><in id="S7.I4.i4.I2.i2.p1.1.m1.1.1.1.cmml" xref="S7.I4.i4.I2.i2.p1.1.m1.1.1.1"></in><ci id="S7.I4.i4.I2.i2.p1.1.m1.1.1.2.cmml" xref="S7.I4.i4.I2.i2.p1.1.m1.1.1.2">𝑣</ci><apply id="S7.I4.i4.I2.i2.p1.1.m1.1.1.3.cmml" xref="S7.I4.i4.I2.i2.p1.1.m1.1.1.3"><csymbol cd="ambiguous" id="S7.I4.i4.I2.i2.p1.1.m1.1.1.3.1.cmml" xref="S7.I4.i4.I2.i2.p1.1.m1.1.1.3">subscript</csymbol><ci id="S7.I4.i4.I2.i2.p1.1.m1.1.1.3.2.cmml" xref="S7.I4.i4.I2.i2.p1.1.m1.1.1.3.2">𝑇</ci><ci id="S7.I4.i4.I2.i2.p1.1.m1.1.1.3.3.cmml" xref="S7.I4.i4.I2.i2.p1.1.m1.1.1.3.3">𝑚</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.I4.i4.I2.i2.p1.1.m1.1c">v\in T_{m}</annotation><annotation encoding="application/x-llamapun" id="S7.I4.i4.I2.i2.p1.1.m1.1d">italic_v ∈ italic_T start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT</annotation></semantics></math>, we define <math alttext="\delta(v)=(1+\frac{\eta}{2})^{h+1}-\textsl{g}(v)" class="ltx_Math" display="inline" id="S7.I4.i4.I2.i2.p1.2.m2.3"><semantics id="S7.I4.i4.I2.i2.p1.2.m2.3a"><mrow id="S7.I4.i4.I2.i2.p1.2.m2.3.3" xref="S7.I4.i4.I2.i2.p1.2.m2.3.3.cmml"><mrow id="S7.I4.i4.I2.i2.p1.2.m2.3.3.3" xref="S7.I4.i4.I2.i2.p1.2.m2.3.3.3.cmml"><mi id="S7.I4.i4.I2.i2.p1.2.m2.3.3.3.2" xref="S7.I4.i4.I2.i2.p1.2.m2.3.3.3.2.cmml">δ</mi><mo id="S7.I4.i4.I2.i2.p1.2.m2.3.3.3.1" xref="S7.I4.i4.I2.i2.p1.2.m2.3.3.3.1.cmml">⁢</mo><mrow id="S7.I4.i4.I2.i2.p1.2.m2.3.3.3.3.2" xref="S7.I4.i4.I2.i2.p1.2.m2.3.3.3.cmml"><mo id="S7.I4.i4.I2.i2.p1.2.m2.3.3.3.3.2.1" stretchy="false" xref="S7.I4.i4.I2.i2.p1.2.m2.3.3.3.cmml">(</mo><mi id="S7.I4.i4.I2.i2.p1.2.m2.1.1" xref="S7.I4.i4.I2.i2.p1.2.m2.1.1.cmml">v</mi><mo id="S7.I4.i4.I2.i2.p1.2.m2.3.3.3.3.2.2" stretchy="false" xref="S7.I4.i4.I2.i2.p1.2.m2.3.3.3.cmml">)</mo></mrow></mrow><mo id="S7.I4.i4.I2.i2.p1.2.m2.3.3.2" xref="S7.I4.i4.I2.i2.p1.2.m2.3.3.2.cmml">=</mo><mrow id="S7.I4.i4.I2.i2.p1.2.m2.3.3.1" xref="S7.I4.i4.I2.i2.p1.2.m2.3.3.1.cmml"><msup id="S7.I4.i4.I2.i2.p1.2.m2.3.3.1.1" xref="S7.I4.i4.I2.i2.p1.2.m2.3.3.1.1.cmml"><mrow id="S7.I4.i4.I2.i2.p1.2.m2.3.3.1.1.1.1" xref="S7.I4.i4.I2.i2.p1.2.m2.3.3.1.1.1.1.1.cmml"><mo id="S7.I4.i4.I2.i2.p1.2.m2.3.3.1.1.1.1.2" stretchy="false" xref="S7.I4.i4.I2.i2.p1.2.m2.3.3.1.1.1.1.1.cmml">(</mo><mrow id="S7.I4.i4.I2.i2.p1.2.m2.3.3.1.1.1.1.1" xref="S7.I4.i4.I2.i2.p1.2.m2.3.3.1.1.1.1.1.cmml"><mn id="S7.I4.i4.I2.i2.p1.2.m2.3.3.1.1.1.1.1.2" xref="S7.I4.i4.I2.i2.p1.2.m2.3.3.1.1.1.1.1.2.cmml">1</mn><mo id="S7.I4.i4.I2.i2.p1.2.m2.3.3.1.1.1.1.1.1" xref="S7.I4.i4.I2.i2.p1.2.m2.3.3.1.1.1.1.1.1.cmml">+</mo><mfrac id="S7.I4.i4.I2.i2.p1.2.m2.3.3.1.1.1.1.1.3" xref="S7.I4.i4.I2.i2.p1.2.m2.3.3.1.1.1.1.1.3.cmml"><mi id="S7.I4.i4.I2.i2.p1.2.m2.3.3.1.1.1.1.1.3.2" xref="S7.I4.i4.I2.i2.p1.2.m2.3.3.1.1.1.1.1.3.2.cmml">η</mi><mn id="S7.I4.i4.I2.i2.p1.2.m2.3.3.1.1.1.1.1.3.3" xref="S7.I4.i4.I2.i2.p1.2.m2.3.3.1.1.1.1.1.3.3.cmml">2</mn></mfrac></mrow><mo id="S7.I4.i4.I2.i2.p1.2.m2.3.3.1.1.1.1.3" stretchy="false" xref="S7.I4.i4.I2.i2.p1.2.m2.3.3.1.1.1.1.1.cmml">)</mo></mrow><mrow id="S7.I4.i4.I2.i2.p1.2.m2.3.3.1.1.3" xref="S7.I4.i4.I2.i2.p1.2.m2.3.3.1.1.3.cmml"><mi id="S7.I4.i4.I2.i2.p1.2.m2.3.3.1.1.3.2" xref="S7.I4.i4.I2.i2.p1.2.m2.3.3.1.1.3.2.cmml">h</mi><mo id="S7.I4.i4.I2.i2.p1.2.m2.3.3.1.1.3.1" xref="S7.I4.i4.I2.i2.p1.2.m2.3.3.1.1.3.1.cmml">+</mo><mn id="S7.I4.i4.I2.i2.p1.2.m2.3.3.1.1.3.3" xref="S7.I4.i4.I2.i2.p1.2.m2.3.3.1.1.3.3.cmml">1</mn></mrow></msup><mo id="S7.I4.i4.I2.i2.p1.2.m2.3.3.1.2" xref="S7.I4.i4.I2.i2.p1.2.m2.3.3.1.2.cmml">−</mo><mrow id="S7.I4.i4.I2.i2.p1.2.m2.3.3.1.3" xref="S7.I4.i4.I2.i2.p1.2.m2.3.3.1.3.cmml"><mtext class="ltx_mathvariant_italic" id="S7.I4.i4.I2.i2.p1.2.m2.3.3.1.3.2" xref="S7.I4.i4.I2.i2.p1.2.m2.3.3.1.3.2a.cmml">g</mtext><mo id="S7.I4.i4.I2.i2.p1.2.m2.3.3.1.3.1" xref="S7.I4.i4.I2.i2.p1.2.m2.3.3.1.3.1.cmml">⁢</mo><mrow id="S7.I4.i4.I2.i2.p1.2.m2.3.3.1.3.3.2" xref="S7.I4.i4.I2.i2.p1.2.m2.3.3.1.3.cmml"><mo id="S7.I4.i4.I2.i2.p1.2.m2.3.3.1.3.3.2.1" stretchy="false" xref="S7.I4.i4.I2.i2.p1.2.m2.3.3.1.3.cmml">(</mo><mi id="S7.I4.i4.I2.i2.p1.2.m2.2.2" xref="S7.I4.i4.I2.i2.p1.2.m2.2.2.cmml">v</mi><mo id="S7.I4.i4.I2.i2.p1.2.m2.3.3.1.3.3.2.2" stretchy="false" xref="S7.I4.i4.I2.i2.p1.2.m2.3.3.1.3.cmml">)</mo></mrow></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S7.I4.i4.I2.i2.p1.2.m2.3b"><apply id="S7.I4.i4.I2.i2.p1.2.m2.3.3.cmml" xref="S7.I4.i4.I2.i2.p1.2.m2.3.3"><eq id="S7.I4.i4.I2.i2.p1.2.m2.3.3.2.cmml" xref="S7.I4.i4.I2.i2.p1.2.m2.3.3.2"></eq><apply id="S7.I4.i4.I2.i2.p1.2.m2.3.3.3.cmml" xref="S7.I4.i4.I2.i2.p1.2.m2.3.3.3"><times id="S7.I4.i4.I2.i2.p1.2.m2.3.3.3.1.cmml" xref="S7.I4.i4.I2.i2.p1.2.m2.3.3.3.1"></times><ci id="S7.I4.i4.I2.i2.p1.2.m2.3.3.3.2.cmml" xref="S7.I4.i4.I2.i2.p1.2.m2.3.3.3.2">𝛿</ci><ci id="S7.I4.i4.I2.i2.p1.2.m2.1.1.cmml" xref="S7.I4.i4.I2.i2.p1.2.m2.1.1">𝑣</ci></apply><apply id="S7.I4.i4.I2.i2.p1.2.m2.3.3.1.cmml" xref="S7.I4.i4.I2.i2.p1.2.m2.3.3.1"><minus id="S7.I4.i4.I2.i2.p1.2.m2.3.3.1.2.cmml" xref="S7.I4.i4.I2.i2.p1.2.m2.3.3.1.2"></minus><apply id="S7.I4.i4.I2.i2.p1.2.m2.3.3.1.1.cmml" xref="S7.I4.i4.I2.i2.p1.2.m2.3.3.1.1"><csymbol cd="ambiguous" id="S7.I4.i4.I2.i2.p1.2.m2.3.3.1.1.2.cmml" xref="S7.I4.i4.I2.i2.p1.2.m2.3.3.1.1">superscript</csymbol><apply id="S7.I4.i4.I2.i2.p1.2.m2.3.3.1.1.1.1.1.cmml" xref="S7.I4.i4.I2.i2.p1.2.m2.3.3.1.1.1.1"><plus id="S7.I4.i4.I2.i2.p1.2.m2.3.3.1.1.1.1.1.1.cmml" xref="S7.I4.i4.I2.i2.p1.2.m2.3.3.1.1.1.1.1.1"></plus><cn id="S7.I4.i4.I2.i2.p1.2.m2.3.3.1.1.1.1.1.2.cmml" type="integer" xref="S7.I4.i4.I2.i2.p1.2.m2.3.3.1.1.1.1.1.2">1</cn><apply id="S7.I4.i4.I2.i2.p1.2.m2.3.3.1.1.1.1.1.3.cmml" xref="S7.I4.i4.I2.i2.p1.2.m2.3.3.1.1.1.1.1.3"><divide id="S7.I4.i4.I2.i2.p1.2.m2.3.3.1.1.1.1.1.3.1.cmml" xref="S7.I4.i4.I2.i2.p1.2.m2.3.3.1.1.1.1.1.3"></divide><ci id="S7.I4.i4.I2.i2.p1.2.m2.3.3.1.1.1.1.1.3.2.cmml" xref="S7.I4.i4.I2.i2.p1.2.m2.3.3.1.1.1.1.1.3.2">𝜂</ci><cn id="S7.I4.i4.I2.i2.p1.2.m2.3.3.1.1.1.1.1.3.3.cmml" type="integer" xref="S7.I4.i4.I2.i2.p1.2.m2.3.3.1.1.1.1.1.3.3">2</cn></apply></apply><apply id="S7.I4.i4.I2.i2.p1.2.m2.3.3.1.1.3.cmml" xref="S7.I4.i4.I2.i2.p1.2.m2.3.3.1.1.3"><plus id="S7.I4.i4.I2.i2.p1.2.m2.3.3.1.1.3.1.cmml" xref="S7.I4.i4.I2.i2.p1.2.m2.3.3.1.1.3.1"></plus><ci id="S7.I4.i4.I2.i2.p1.2.m2.3.3.1.1.3.2.cmml" xref="S7.I4.i4.I2.i2.p1.2.m2.3.3.1.1.3.2">ℎ</ci><cn id="S7.I4.i4.I2.i2.p1.2.m2.3.3.1.1.3.3.cmml" type="integer" xref="S7.I4.i4.I2.i2.p1.2.m2.3.3.1.1.3.3">1</cn></apply></apply><apply id="S7.I4.i4.I2.i2.p1.2.m2.3.3.1.3.cmml" xref="S7.I4.i4.I2.i2.p1.2.m2.3.3.1.3"><times id="S7.I4.i4.I2.i2.p1.2.m2.3.3.1.3.1.cmml" xref="S7.I4.i4.I2.i2.p1.2.m2.3.3.1.3.1"></times><ci id="S7.I4.i4.I2.i2.p1.2.m2.3.3.1.3.2a.cmml" xref="S7.I4.i4.I2.i2.p1.2.m2.3.3.1.3.2"><mtext class="ltx_mathvariant_italic" id="S7.I4.i4.I2.i2.p1.2.m2.3.3.1.3.2.cmml" xref="S7.I4.i4.I2.i2.p1.2.m2.3.3.1.3.2">g</mtext></ci><ci id="S7.I4.i4.I2.i2.p1.2.m2.2.2.cmml" xref="S7.I4.i4.I2.i2.p1.2.m2.2.2">𝑣</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.I4.i4.I2.i2.p1.2.m2.3c">\delta(v)=(1+\frac{\eta}{2})^{h+1}-\textsl{g}(v)</annotation><annotation encoding="application/x-llamapun" id="S7.I4.i4.I2.i2.p1.2.m2.3d">italic_δ ( italic_v ) = ( 1 + divide start_ARG italic_η end_ARG start_ARG 2 end_ARG ) start_POSTSUPERSCRIPT italic_h + 1 end_POSTSUPERSCRIPT - g ( italic_v )</annotation></semantics></math> (the maximal amount we can increase <math alttext="\textsl{g}(v)" class="ltx_Math" display="inline" id="S7.I4.i4.I2.i2.p1.3.m3.1"><semantics id="S7.I4.i4.I2.i2.p1.3.m3.1a"><mrow id="S7.I4.i4.I2.i2.p1.3.m3.1.2" xref="S7.I4.i4.I2.i2.p1.3.m3.1.2.cmml"><mtext class="ltx_mathvariant_italic" id="S7.I4.i4.I2.i2.p1.3.m3.1.2.2" xref="S7.I4.i4.I2.i2.p1.3.m3.1.2.2a.cmml">g</mtext><mo id="S7.I4.i4.I2.i2.p1.3.m3.1.2.1" xref="S7.I4.i4.I2.i2.p1.3.m3.1.2.1.cmml">⁢</mo><mrow id="S7.I4.i4.I2.i2.p1.3.m3.1.2.3.2" xref="S7.I4.i4.I2.i2.p1.3.m3.1.2.cmml"><mo id="S7.I4.i4.I2.i2.p1.3.m3.1.2.3.2.1" stretchy="false" xref="S7.I4.i4.I2.i2.p1.3.m3.1.2.cmml">(</mo><mi id="S7.I4.i4.I2.i2.p1.3.m3.1.1" xref="S7.I4.i4.I2.i2.p1.3.m3.1.1.cmml">v</mi><mo id="S7.I4.i4.I2.i2.p1.3.m3.1.2.3.2.2" stretchy="false" xref="S7.I4.i4.I2.i2.p1.3.m3.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S7.I4.i4.I2.i2.p1.3.m3.1b"><apply id="S7.I4.i4.I2.i2.p1.3.m3.1.2.cmml" xref="S7.I4.i4.I2.i2.p1.3.m3.1.2"><times id="S7.I4.i4.I2.i2.p1.3.m3.1.2.1.cmml" xref="S7.I4.i4.I2.i2.p1.3.m3.1.2.1"></times><ci id="S7.I4.i4.I2.i2.p1.3.m3.1.2.2a.cmml" xref="S7.I4.i4.I2.i2.p1.3.m3.1.2.2"><mtext class="ltx_mathvariant_italic" id="S7.I4.i4.I2.i2.p1.3.m3.1.2.2.cmml" xref="S7.I4.i4.I2.i2.p1.3.m3.1.2.2">g</mtext></ci><ci id="S7.I4.i4.I2.i2.p1.3.m3.1.1.cmml" xref="S7.I4.i4.I2.i2.p1.3.m3.1.1">𝑣</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.I4.i4.I2.i2.p1.3.m3.1c">\textsl{g}(v)</annotation><annotation encoding="application/x-llamapun" id="S7.I4.i4.I2.i2.p1.3.m3.1d">g ( italic_v )</annotation></semantics></math> such that it does not arrive in level <math alttext="h+1" class="ltx_Math" display="inline" id="S7.I4.i4.I2.i2.p1.4.m4.1"><semantics id="S7.I4.i4.I2.i2.p1.4.m4.1a"><mrow id="S7.I4.i4.I2.i2.p1.4.m4.1.1" xref="S7.I4.i4.I2.i2.p1.4.m4.1.1.cmml"><mi id="S7.I4.i4.I2.i2.p1.4.m4.1.1.2" xref="S7.I4.i4.I2.i2.p1.4.m4.1.1.2.cmml">h</mi><mo id="S7.I4.i4.I2.i2.p1.4.m4.1.1.1" xref="S7.I4.i4.I2.i2.p1.4.m4.1.1.1.cmml">+</mo><mn id="S7.I4.i4.I2.i2.p1.4.m4.1.1.3" xref="S7.I4.i4.I2.i2.p1.4.m4.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S7.I4.i4.I2.i2.p1.4.m4.1b"><apply id="S7.I4.i4.I2.i2.p1.4.m4.1.1.cmml" xref="S7.I4.i4.I2.i2.p1.4.m4.1.1"><plus id="S7.I4.i4.I2.i2.p1.4.m4.1.1.1.cmml" xref="S7.I4.i4.I2.i2.p1.4.m4.1.1.1"></plus><ci id="S7.I4.i4.I2.i2.p1.4.m4.1.1.2.cmml" xref="S7.I4.i4.I2.i2.p1.4.m4.1.1.2">ℎ</ci><cn id="S7.I4.i4.I2.i2.p1.4.m4.1.1.3.cmml" type="integer" xref="S7.I4.i4.I2.i2.p1.4.m4.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.I4.i4.I2.i2.p1.4.m4.1c">h+1</annotation><annotation encoding="application/x-llamapun" id="S7.I4.i4.I2.i2.p1.4.m4.1d">italic_h + 1</annotation></semantics></math>). We connect every <math alttext="v\in T_{m}" class="ltx_Math" display="inline" id="S7.I4.i4.I2.i2.p1.5.m5.1"><semantics id="S7.I4.i4.I2.i2.p1.5.m5.1a"><mrow id="S7.I4.i4.I2.i2.p1.5.m5.1.1" xref="S7.I4.i4.I2.i2.p1.5.m5.1.1.cmml"><mi id="S7.I4.i4.I2.i2.p1.5.m5.1.1.2" xref="S7.I4.i4.I2.i2.p1.5.m5.1.1.2.cmml">v</mi><mo id="S7.I4.i4.I2.i2.p1.5.m5.1.1.1" xref="S7.I4.i4.I2.i2.p1.5.m5.1.1.1.cmml">∈</mo><msub id="S7.I4.i4.I2.i2.p1.5.m5.1.1.3" xref="S7.I4.i4.I2.i2.p1.5.m5.1.1.3.cmml"><mi id="S7.I4.i4.I2.i2.p1.5.m5.1.1.3.2" xref="S7.I4.i4.I2.i2.p1.5.m5.1.1.3.2.cmml">T</mi><mi id="S7.I4.i4.I2.i2.p1.5.m5.1.1.3.3" xref="S7.I4.i4.I2.i2.p1.5.m5.1.1.3.3.cmml">m</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S7.I4.i4.I2.i2.p1.5.m5.1b"><apply id="S7.I4.i4.I2.i2.p1.5.m5.1.1.cmml" xref="S7.I4.i4.I2.i2.p1.5.m5.1.1"><in id="S7.I4.i4.I2.i2.p1.5.m5.1.1.1.cmml" xref="S7.I4.i4.I2.i2.p1.5.m5.1.1.1"></in><ci id="S7.I4.i4.I2.i2.p1.5.m5.1.1.2.cmml" xref="S7.I4.i4.I2.i2.p1.5.m5.1.1.2">𝑣</ci><apply id="S7.I4.i4.I2.i2.p1.5.m5.1.1.3.cmml" xref="S7.I4.i4.I2.i2.p1.5.m5.1.1.3"><csymbol cd="ambiguous" id="S7.I4.i4.I2.i2.p1.5.m5.1.1.3.1.cmml" xref="S7.I4.i4.I2.i2.p1.5.m5.1.1.3">subscript</csymbol><ci id="S7.I4.i4.I2.i2.p1.5.m5.1.1.3.2.cmml" xref="S7.I4.i4.I2.i2.p1.5.m5.1.1.3.2">𝑇</ci><ci id="S7.I4.i4.I2.i2.p1.5.m5.1.1.3.3.cmml" xref="S7.I4.i4.I2.i2.p1.5.m5.1.1.3.3">𝑚</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.I4.i4.I2.i2.p1.5.m5.1c">v\in T_{m}</annotation><annotation encoding="application/x-llamapun" id="S7.I4.i4.I2.i2.p1.5.m5.1d">italic_v ∈ italic_T start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT</annotation></semantics></math> to a unique sink <math alttext="t_{v}" class="ltx_Math" display="inline" id="S7.I4.i4.I2.i2.p1.6.m6.1"><semantics id="S7.I4.i4.I2.i2.p1.6.m6.1a"><msub id="S7.I4.i4.I2.i2.p1.6.m6.1.1" xref="S7.I4.i4.I2.i2.p1.6.m6.1.1.cmml"><mi id="S7.I4.i4.I2.i2.p1.6.m6.1.1.2" xref="S7.I4.i4.I2.i2.p1.6.m6.1.1.2.cmml">t</mi><mi id="S7.I4.i4.I2.i2.p1.6.m6.1.1.3" xref="S7.I4.i4.I2.i2.p1.6.m6.1.1.3.cmml">v</mi></msub><annotation-xml encoding="MathML-Content" id="S7.I4.i4.I2.i2.p1.6.m6.1b"><apply id="S7.I4.i4.I2.i2.p1.6.m6.1.1.cmml" xref="S7.I4.i4.I2.i2.p1.6.m6.1.1"><csymbol cd="ambiguous" id="S7.I4.i4.I2.i2.p1.6.m6.1.1.1.cmml" xref="S7.I4.i4.I2.i2.p1.6.m6.1.1">subscript</csymbol><ci id="S7.I4.i4.I2.i2.p1.6.m6.1.1.2.cmml" xref="S7.I4.i4.I2.i2.p1.6.m6.1.1.2">𝑡</ci><ci id="S7.I4.i4.I2.i2.p1.6.m6.1.1.3.cmml" xref="S7.I4.i4.I2.i2.p1.6.m6.1.1.3">𝑣</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.I4.i4.I2.i2.p1.6.m6.1c">t_{v}</annotation><annotation encoding="application/x-llamapun" id="S7.I4.i4.I2.i2.p1.6.m6.1d">italic_t start_POSTSUBSCRIPT italic_v end_POSTSUBSCRIPT</annotation></semantics></math> where the edge <math alttext="\overline{vt_{v}}" class="ltx_Math" display="inline" id="S7.I4.i4.I2.i2.p1.7.m7.1"><semantics id="S7.I4.i4.I2.i2.p1.7.m7.1a"><mover accent="true" id="S7.I4.i4.I2.i2.p1.7.m7.1.1" xref="S7.I4.i4.I2.i2.p1.7.m7.1.1.cmml"><mrow id="S7.I4.i4.I2.i2.p1.7.m7.1.1.2" xref="S7.I4.i4.I2.i2.p1.7.m7.1.1.2.cmml"><mi id="S7.I4.i4.I2.i2.p1.7.m7.1.1.2.2" xref="S7.I4.i4.I2.i2.p1.7.m7.1.1.2.2.cmml">v</mi><mo id="S7.I4.i4.I2.i2.p1.7.m7.1.1.2.1" xref="S7.I4.i4.I2.i2.p1.7.m7.1.1.2.1.cmml">⁢</mo><msub id="S7.I4.i4.I2.i2.p1.7.m7.1.1.2.3" xref="S7.I4.i4.I2.i2.p1.7.m7.1.1.2.3.cmml"><mi id="S7.I4.i4.I2.i2.p1.7.m7.1.1.2.3.2" xref="S7.I4.i4.I2.i2.p1.7.m7.1.1.2.3.2.cmml">t</mi><mi id="S7.I4.i4.I2.i2.p1.7.m7.1.1.2.3.3" xref="S7.I4.i4.I2.i2.p1.7.m7.1.1.2.3.3.cmml">v</mi></msub></mrow><mo id="S7.I4.i4.I2.i2.p1.7.m7.1.1.1" xref="S7.I4.i4.I2.i2.p1.7.m7.1.1.1.cmml">¯</mo></mover><annotation-xml encoding="MathML-Content" id="S7.I4.i4.I2.i2.p1.7.m7.1b"><apply id="S7.I4.i4.I2.i2.p1.7.m7.1.1.cmml" xref="S7.I4.i4.I2.i2.p1.7.m7.1.1"><ci id="S7.I4.i4.I2.i2.p1.7.m7.1.1.1.cmml" xref="S7.I4.i4.I2.i2.p1.7.m7.1.1.1">¯</ci><apply id="S7.I4.i4.I2.i2.p1.7.m7.1.1.2.cmml" xref="S7.I4.i4.I2.i2.p1.7.m7.1.1.2"><times id="S7.I4.i4.I2.i2.p1.7.m7.1.1.2.1.cmml" xref="S7.I4.i4.I2.i2.p1.7.m7.1.1.2.1"></times><ci id="S7.I4.i4.I2.i2.p1.7.m7.1.1.2.2.cmml" xref="S7.I4.i4.I2.i2.p1.7.m7.1.1.2.2">𝑣</ci><apply id="S7.I4.i4.I2.i2.p1.7.m7.1.1.2.3.cmml" xref="S7.I4.i4.I2.i2.p1.7.m7.1.1.2.3"><csymbol cd="ambiguous" id="S7.I4.i4.I2.i2.p1.7.m7.1.1.2.3.1.cmml" xref="S7.I4.i4.I2.i2.p1.7.m7.1.1.2.3">subscript</csymbol><ci id="S7.I4.i4.I2.i2.p1.7.m7.1.1.2.3.2.cmml" xref="S7.I4.i4.I2.i2.p1.7.m7.1.1.2.3.2">𝑡</ci><ci id="S7.I4.i4.I2.i2.p1.7.m7.1.1.2.3.3.cmml" xref="S7.I4.i4.I2.i2.p1.7.m7.1.1.2.3.3">𝑣</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.I4.i4.I2.i2.p1.7.m7.1c">\overline{vt_{v}}</annotation><annotation encoding="application/x-llamapun" id="S7.I4.i4.I2.i2.p1.7.m7.1d">over¯ start_ARG italic_v italic_t start_POSTSUBSCRIPT italic_v end_POSTSUBSCRIPT end_ARG</annotation></semantics></math> has capacity <math alttext="\delta(v)" class="ltx_Math" display="inline" id="S7.I4.i4.I2.i2.p1.8.m8.1"><semantics id="S7.I4.i4.I2.i2.p1.8.m8.1a"><mrow id="S7.I4.i4.I2.i2.p1.8.m8.1.2" xref="S7.I4.i4.I2.i2.p1.8.m8.1.2.cmml"><mi id="S7.I4.i4.I2.i2.p1.8.m8.1.2.2" xref="S7.I4.i4.I2.i2.p1.8.m8.1.2.2.cmml">δ</mi><mo id="S7.I4.i4.I2.i2.p1.8.m8.1.2.1" xref="S7.I4.i4.I2.i2.p1.8.m8.1.2.1.cmml">⁢</mo><mrow id="S7.I4.i4.I2.i2.p1.8.m8.1.2.3.2" xref="S7.I4.i4.I2.i2.p1.8.m8.1.2.cmml"><mo id="S7.I4.i4.I2.i2.p1.8.m8.1.2.3.2.1" stretchy="false" xref="S7.I4.i4.I2.i2.p1.8.m8.1.2.cmml">(</mo><mi id="S7.I4.i4.I2.i2.p1.8.m8.1.1" xref="S7.I4.i4.I2.i2.p1.8.m8.1.1.cmml">v</mi><mo id="S7.I4.i4.I2.i2.p1.8.m8.1.2.3.2.2" stretchy="false" xref="S7.I4.i4.I2.i2.p1.8.m8.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S7.I4.i4.I2.i2.p1.8.m8.1b"><apply id="S7.I4.i4.I2.i2.p1.8.m8.1.2.cmml" xref="S7.I4.i4.I2.i2.p1.8.m8.1.2"><times id="S7.I4.i4.I2.i2.p1.8.m8.1.2.1.cmml" xref="S7.I4.i4.I2.i2.p1.8.m8.1.2.1"></times><ci id="S7.I4.i4.I2.i2.p1.8.m8.1.2.2.cmml" xref="S7.I4.i4.I2.i2.p1.8.m8.1.2.2">𝛿</ci><ci id="S7.I4.i4.I2.i2.p1.8.m8.1.1.cmml" xref="S7.I4.i4.I2.i2.p1.8.m8.1.1">𝑣</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.I4.i4.I2.i2.p1.8.m8.1c">\delta(v)</annotation><annotation encoding="application/x-llamapun" id="S7.I4.i4.I2.i2.p1.8.m8.1d">italic_δ ( italic_v )</annotation></semantics></math>.</p> </div> </li> <li class="ltx_item" id="S7.I4.i4.I2.i3" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item"><span class="ltx_text ltx_font_bold" id="S7.I4.i4.I2.i3.1.1.1">–</span></span> <div class="ltx_para" id="S7.I4.i4.I2.i3.p1"> <p class="ltx_p" id="S7.I4.i4.I2.i3.p1.2">Each other edge <math alttext="\overline{uv}\in E_{s}" class="ltx_Math" display="inline" id="S7.I4.i4.I2.i3.p1.1.m1.1"><semantics id="S7.I4.i4.I2.i3.p1.1.m1.1a"><mrow id="S7.I4.i4.I2.i3.p1.1.m1.1.1" xref="S7.I4.i4.I2.i3.p1.1.m1.1.1.cmml"><mover accent="true" id="S7.I4.i4.I2.i3.p1.1.m1.1.1.2" xref="S7.I4.i4.I2.i3.p1.1.m1.1.1.2.cmml"><mrow id="S7.I4.i4.I2.i3.p1.1.m1.1.1.2.2" xref="S7.I4.i4.I2.i3.p1.1.m1.1.1.2.2.cmml"><mi id="S7.I4.i4.I2.i3.p1.1.m1.1.1.2.2.2" xref="S7.I4.i4.I2.i3.p1.1.m1.1.1.2.2.2.cmml">u</mi><mo id="S7.I4.i4.I2.i3.p1.1.m1.1.1.2.2.1" xref="S7.I4.i4.I2.i3.p1.1.m1.1.1.2.2.1.cmml">⁢</mo><mi id="S7.I4.i4.I2.i3.p1.1.m1.1.1.2.2.3" xref="S7.I4.i4.I2.i3.p1.1.m1.1.1.2.2.3.cmml">v</mi></mrow><mo id="S7.I4.i4.I2.i3.p1.1.m1.1.1.2.1" xref="S7.I4.i4.I2.i3.p1.1.m1.1.1.2.1.cmml">¯</mo></mover><mo id="S7.I4.i4.I2.i3.p1.1.m1.1.1.1" xref="S7.I4.i4.I2.i3.p1.1.m1.1.1.1.cmml">∈</mo><msub id="S7.I4.i4.I2.i3.p1.1.m1.1.1.3" xref="S7.I4.i4.I2.i3.p1.1.m1.1.1.3.cmml"><mi id="S7.I4.i4.I2.i3.p1.1.m1.1.1.3.2" xref="S7.I4.i4.I2.i3.p1.1.m1.1.1.3.2.cmml">E</mi><mi id="S7.I4.i4.I2.i3.p1.1.m1.1.1.3.3" xref="S7.I4.i4.I2.i3.p1.1.m1.1.1.3.3.cmml">s</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S7.I4.i4.I2.i3.p1.1.m1.1b"><apply id="S7.I4.i4.I2.i3.p1.1.m1.1.1.cmml" xref="S7.I4.i4.I2.i3.p1.1.m1.1.1"><in id="S7.I4.i4.I2.i3.p1.1.m1.1.1.1.cmml" xref="S7.I4.i4.I2.i3.p1.1.m1.1.1.1"></in><apply id="S7.I4.i4.I2.i3.p1.1.m1.1.1.2.cmml" xref="S7.I4.i4.I2.i3.p1.1.m1.1.1.2"><ci id="S7.I4.i4.I2.i3.p1.1.m1.1.1.2.1.cmml" xref="S7.I4.i4.I2.i3.p1.1.m1.1.1.2.1">¯</ci><apply id="S7.I4.i4.I2.i3.p1.1.m1.1.1.2.2.cmml" xref="S7.I4.i4.I2.i3.p1.1.m1.1.1.2.2"><times id="S7.I4.i4.I2.i3.p1.1.m1.1.1.2.2.1.cmml" xref="S7.I4.i4.I2.i3.p1.1.m1.1.1.2.2.1"></times><ci id="S7.I4.i4.I2.i3.p1.1.m1.1.1.2.2.2.cmml" xref="S7.I4.i4.I2.i3.p1.1.m1.1.1.2.2.2">𝑢</ci><ci id="S7.I4.i4.I2.i3.p1.1.m1.1.1.2.2.3.cmml" xref="S7.I4.i4.I2.i3.p1.1.m1.1.1.2.2.3">𝑣</ci></apply></apply><apply id="S7.I4.i4.I2.i3.p1.1.m1.1.1.3.cmml" xref="S7.I4.i4.I2.i3.p1.1.m1.1.1.3"><csymbol cd="ambiguous" id="S7.I4.i4.I2.i3.p1.1.m1.1.1.3.1.cmml" xref="S7.I4.i4.I2.i3.p1.1.m1.1.1.3">subscript</csymbol><ci id="S7.I4.i4.I2.i3.p1.1.m1.1.1.3.2.cmml" xref="S7.I4.i4.I2.i3.p1.1.m1.1.1.3.2">𝐸</ci><ci id="S7.I4.i4.I2.i3.p1.1.m1.1.1.3.3.cmml" xref="S7.I4.i4.I2.i3.p1.1.m1.1.1.3.3">𝑠</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.I4.i4.I2.i3.p1.1.m1.1c">\overline{uv}\in E_{s}</annotation><annotation encoding="application/x-llamapun" id="S7.I4.i4.I2.i3.p1.1.m1.1d">over¯ start_ARG italic_u italic_v end_ARG ∈ italic_E start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT</annotation></semantics></math> has capacity <math alttext="\textsl{g}(u\!\to\!v)" class="ltx_Math" display="inline" id="S7.I4.i4.I2.i3.p1.2.m2.1"><semantics id="S7.I4.i4.I2.i3.p1.2.m2.1a"><mrow id="S7.I4.i4.I2.i3.p1.2.m2.1.1" xref="S7.I4.i4.I2.i3.p1.2.m2.1.1.cmml"><mtext class="ltx_mathvariant_italic" id="S7.I4.i4.I2.i3.p1.2.m2.1.1.3" xref="S7.I4.i4.I2.i3.p1.2.m2.1.1.3a.cmml">g</mtext><mo id="S7.I4.i4.I2.i3.p1.2.m2.1.1.2" xref="S7.I4.i4.I2.i3.p1.2.m2.1.1.2.cmml">⁢</mo><mrow id="S7.I4.i4.I2.i3.p1.2.m2.1.1.1.1" xref="S7.I4.i4.I2.i3.p1.2.m2.1.1.1.1.1.cmml"><mo id="S7.I4.i4.I2.i3.p1.2.m2.1.1.1.1.2" stretchy="false" xref="S7.I4.i4.I2.i3.p1.2.m2.1.1.1.1.1.cmml">(</mo><mrow id="S7.I4.i4.I2.i3.p1.2.m2.1.1.1.1.1" xref="S7.I4.i4.I2.i3.p1.2.m2.1.1.1.1.1.cmml"><mi id="S7.I4.i4.I2.i3.p1.2.m2.1.1.1.1.1.2" xref="S7.I4.i4.I2.i3.p1.2.m2.1.1.1.1.1.2.cmml">u</mi><mo id="S7.I4.i4.I2.i3.p1.2.m2.1.1.1.1.1.1" lspace="0.108em" rspace="0.108em" stretchy="false" xref="S7.I4.i4.I2.i3.p1.2.m2.1.1.1.1.1.1.cmml">→</mo><mi id="S7.I4.i4.I2.i3.p1.2.m2.1.1.1.1.1.3" xref="S7.I4.i4.I2.i3.p1.2.m2.1.1.1.1.1.3.cmml">v</mi></mrow><mo id="S7.I4.i4.I2.i3.p1.2.m2.1.1.1.1.3" stretchy="false" xref="S7.I4.i4.I2.i3.p1.2.m2.1.1.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S7.I4.i4.I2.i3.p1.2.m2.1b"><apply id="S7.I4.i4.I2.i3.p1.2.m2.1.1.cmml" xref="S7.I4.i4.I2.i3.p1.2.m2.1.1"><times id="S7.I4.i4.I2.i3.p1.2.m2.1.1.2.cmml" xref="S7.I4.i4.I2.i3.p1.2.m2.1.1.2"></times><ci id="S7.I4.i4.I2.i3.p1.2.m2.1.1.3a.cmml" xref="S7.I4.i4.I2.i3.p1.2.m2.1.1.3"><mtext class="ltx_mathvariant_italic" id="S7.I4.i4.I2.i3.p1.2.m2.1.1.3.cmml" xref="S7.I4.i4.I2.i3.p1.2.m2.1.1.3">g</mtext></ci><apply id="S7.I4.i4.I2.i3.p1.2.m2.1.1.1.1.1.cmml" xref="S7.I4.i4.I2.i3.p1.2.m2.1.1.1.1"><ci id="S7.I4.i4.I2.i3.p1.2.m2.1.1.1.1.1.1.cmml" xref="S7.I4.i4.I2.i3.p1.2.m2.1.1.1.1.1.1">→</ci><ci id="S7.I4.i4.I2.i3.p1.2.m2.1.1.1.1.1.2.cmml" xref="S7.I4.i4.I2.i3.p1.2.m2.1.1.1.1.1.2">𝑢</ci><ci id="S7.I4.i4.I2.i3.p1.2.m2.1.1.1.1.1.3.cmml" xref="S7.I4.i4.I2.i3.p1.2.m2.1.1.1.1.1.3">𝑣</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.I4.i4.I2.i3.p1.2.m2.1c">\textsl{g}(u\!\to\!v)</annotation><annotation encoding="application/x-llamapun" id="S7.I4.i4.I2.i3.p1.2.m2.1d">g ( italic_u → italic_v )</annotation></semantics></math>.</p> </div> </li> </ul> </div> </li> </ul> </div> <figure class="ltx_figure" id="S7.F2"><img alt="Refer to caption" class="ltx_graphics ltx_centering ltx_img_landscape" height="632" id="S7.F2.g1" src="x2.png" width="830"/> <figcaption class="ltx_caption ltx_centering"><span class="ltx_tag ltx_tag_figure"><span class="ltx_text" id="S7.F2.52.23.1" style="font-size:90%;">Figure 2</span>: </span><span class="ltx_text" id="S7.F2.44.22" style="font-size:90%;"> <math alttext="(4:0:0)" class="ltx_Math" display="inline" id="S7.F2.23.1.m1.1"><semantics id="S7.F2.23.1.m1.1b"><mrow id="S7.F2.23.1.m1.1.1.1" xref="S7.F2.23.1.m1.1.1.1.1.cmml"><mo id="S7.F2.23.1.m1.1.1.1.2" stretchy="false" xref="S7.F2.23.1.m1.1.1.1.1.cmml">(</mo><mrow id="S7.F2.23.1.m1.1.1.1.1" xref="S7.F2.23.1.m1.1.1.1.1.cmml"><mn id="S7.F2.23.1.m1.1.1.1.1.2" xref="S7.F2.23.1.m1.1.1.1.1.2.cmml">4</mn><mo id="S7.F2.23.1.m1.1.1.1.1.3" lspace="0.278em" rspace="0.278em" xref="S7.F2.23.1.m1.1.1.1.1.3.cmml">:</mo><mn id="S7.F2.23.1.m1.1.1.1.1.4" xref="S7.F2.23.1.m1.1.1.1.1.4.cmml">0</mn><mo id="S7.F2.23.1.m1.1.1.1.1.5" lspace="0.278em" rspace="0.278em" xref="S7.F2.23.1.m1.1.1.1.1.5.cmml">:</mo><mn id="S7.F2.23.1.m1.1.1.1.1.6" xref="S7.F2.23.1.m1.1.1.1.1.6.cmml">0</mn></mrow><mo id="S7.F2.23.1.m1.1.1.1.3" stretchy="false" xref="S7.F2.23.1.m1.1.1.1.1.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S7.F2.23.1.m1.1c"><apply id="S7.F2.23.1.m1.1.1.1.1.cmml" xref="S7.F2.23.1.m1.1.1.1"><and id="S7.F2.23.1.m1.1.1.1.1a.cmml" xref="S7.F2.23.1.m1.1.1.1"></and><apply id="S7.F2.23.1.m1.1.1.1.1b.cmml" xref="S7.F2.23.1.m1.1.1.1"><ci id="S7.F2.23.1.m1.1.1.1.1.3.cmml" xref="S7.F2.23.1.m1.1.1.1.1.3">:</ci><cn id="S7.F2.23.1.m1.1.1.1.1.2.cmml" type="integer" xref="S7.F2.23.1.m1.1.1.1.1.2">4</cn><cn id="S7.F2.23.1.m1.1.1.1.1.4.cmml" type="integer" xref="S7.F2.23.1.m1.1.1.1.1.4">0</cn></apply><apply id="S7.F2.23.1.m1.1.1.1.1c.cmml" xref="S7.F2.23.1.m1.1.1.1"><ci id="S7.F2.23.1.m1.1.1.1.1.5.cmml" xref="S7.F2.23.1.m1.1.1.1.1.5">:</ci><share href="https://arxiv.org/html/2411.12694v2#S7.F2.23.1.m1.1.1.1.1.4.cmml" id="S7.F2.23.1.m1.1.1.1.1d.cmml" xref="S7.F2.23.1.m1.1.1.1"></share><cn id="S7.F2.23.1.m1.1.1.1.1.6.cmml" type="integer" xref="S7.F2.23.1.m1.1.1.1.1.6">0</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.F2.23.1.m1.1d">(4:0:0)</annotation><annotation encoding="application/x-llamapun" id="S7.F2.23.1.m1.1e">( 4 : 0 : 0 )</annotation></semantics></math> - at the first <span class="ltx_text ltx_font_smallcaps" id="S7.F2.44.22.1">minute</span> start in <span class="ltx_text ltx_font_smallcaps" id="S7.F2.44.22.2">hour</span> <math alttext="4" class="ltx_Math" display="inline" id="S7.F2.24.2.m2.1"><semantics id="S7.F2.24.2.m2.1b"><mn id="S7.F2.24.2.m2.1.1" xref="S7.F2.24.2.m2.1.1.cmml">4</mn><annotation-xml encoding="MathML-Content" id="S7.F2.24.2.m2.1c"><cn id="S7.F2.24.2.m2.1.1.cmml" type="integer" xref="S7.F2.24.2.m2.1.1">4</cn></annotation-xml><annotation encoding="application/x-tex" id="S7.F2.24.2.m2.1d">4</annotation><annotation encoding="application/x-llamapun" id="S7.F2.24.2.m2.1e">4</annotation></semantics></math>, we consider all violating out-edges <math alttext="\overline{uv}" class="ltx_Math" display="inline" id="S7.F2.25.3.m3.1"><semantics id="S7.F2.25.3.m3.1b"><mover accent="true" id="S7.F2.25.3.m3.1.1" xref="S7.F2.25.3.m3.1.1.cmml"><mrow id="S7.F2.25.3.m3.1.1.2" xref="S7.F2.25.3.m3.1.1.2.cmml"><mi id="S7.F2.25.3.m3.1.1.2.2" xref="S7.F2.25.3.m3.1.1.2.2.cmml">u</mi><mo id="S7.F2.25.3.m3.1.1.2.1" xref="S7.F2.25.3.m3.1.1.2.1.cmml">⁢</mo><mi id="S7.F2.25.3.m3.1.1.2.3" xref="S7.F2.25.3.m3.1.1.2.3.cmml">v</mi></mrow><mo id="S7.F2.25.3.m3.1.1.1" xref="S7.F2.25.3.m3.1.1.1.cmml">¯</mo></mover><annotation-xml encoding="MathML-Content" id="S7.F2.25.3.m3.1c"><apply id="S7.F2.25.3.m3.1.1.cmml" xref="S7.F2.25.3.m3.1.1"><ci id="S7.F2.25.3.m3.1.1.1.cmml" xref="S7.F2.25.3.m3.1.1.1">¯</ci><apply id="S7.F2.25.3.m3.1.1.2.cmml" xref="S7.F2.25.3.m3.1.1.2"><times id="S7.F2.25.3.m3.1.1.2.1.cmml" xref="S7.F2.25.3.m3.1.1.2.1"></times><ci id="S7.F2.25.3.m3.1.1.2.2.cmml" xref="S7.F2.25.3.m3.1.1.2.2">𝑢</ci><ci id="S7.F2.25.3.m3.1.1.2.3.cmml" xref="S7.F2.25.3.m3.1.1.2.3">𝑣</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.F2.25.3.m3.1d">\overline{uv}</annotation><annotation encoding="application/x-llamapun" id="S7.F2.25.3.m3.1e">over¯ start_ARG italic_u italic_v end_ARG</annotation></semantics></math> from level <math alttext="4" class="ltx_Math" display="inline" id="S7.F2.26.4.m4.1"><semantics id="S7.F2.26.4.m4.1b"><mn id="S7.F2.26.4.m4.1.1" xref="S7.F2.26.4.m4.1.1.cmml">4</mn><annotation-xml encoding="MathML-Content" id="S7.F2.26.4.m4.1c"><cn id="S7.F2.26.4.m4.1.1.cmml" type="integer" xref="S7.F2.26.4.m4.1.1">4</cn></annotation-xml><annotation encoding="application/x-tex" id="S7.F2.26.4.m4.1d">4</annotation><annotation encoding="application/x-llamapun" id="S7.F2.26.4.m4.1e">4</annotation></semantics></math> (red). Per definition, these edges point to level <math alttext="2" class="ltx_Math" display="inline" id="S7.F2.27.5.m5.1"><semantics id="S7.F2.27.5.m5.1b"><mn id="S7.F2.27.5.m5.1.1" xref="S7.F2.27.5.m5.1.1.cmml">2</mn><annotation-xml encoding="MathML-Content" id="S7.F2.27.5.m5.1c"><cn id="S7.F2.27.5.m5.1.1.cmml" type="integer" xref="S7.F2.27.5.m5.1.1">2</cn></annotation-xml><annotation encoding="application/x-tex" id="S7.F2.27.5.m5.1d">2</annotation><annotation encoding="application/x-llamapun" id="S7.F2.27.5.m5.1e">2</annotation></semantics></math> or lower. <br class="ltx_break"/><math alttext="(4:0:0)" class="ltx_Math" display="inline" id="S7.F2.28.6.m6.1"><semantics id="S7.F2.28.6.m6.1b"><mrow id="S7.F2.28.6.m6.1.1.1" xref="S7.F2.28.6.m6.1.1.1.1.cmml"><mo id="S7.F2.28.6.m6.1.1.1.2" stretchy="false" xref="S7.F2.28.6.m6.1.1.1.1.cmml">(</mo><mrow id="S7.F2.28.6.m6.1.1.1.1" xref="S7.F2.28.6.m6.1.1.1.1.cmml"><mn id="S7.F2.28.6.m6.1.1.1.1.2" xref="S7.F2.28.6.m6.1.1.1.1.2.cmml">4</mn><mo id="S7.F2.28.6.m6.1.1.1.1.3" lspace="0.278em" rspace="0.278em" xref="S7.F2.28.6.m6.1.1.1.1.3.cmml">:</mo><mn id="S7.F2.28.6.m6.1.1.1.1.4" xref="S7.F2.28.6.m6.1.1.1.1.4.cmml">0</mn><mo id="S7.F2.28.6.m6.1.1.1.1.5" lspace="0.278em" rspace="0.278em" xref="S7.F2.28.6.m6.1.1.1.1.5.cmml">:</mo><mn id="S7.F2.28.6.m6.1.1.1.1.6" xref="S7.F2.28.6.m6.1.1.1.1.6.cmml">0</mn></mrow><mo id="S7.F2.28.6.m6.1.1.1.3" stretchy="false" xref="S7.F2.28.6.m6.1.1.1.1.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S7.F2.28.6.m6.1c"><apply id="S7.F2.28.6.m6.1.1.1.1.cmml" xref="S7.F2.28.6.m6.1.1.1"><and id="S7.F2.28.6.m6.1.1.1.1a.cmml" xref="S7.F2.28.6.m6.1.1.1"></and><apply id="S7.F2.28.6.m6.1.1.1.1b.cmml" xref="S7.F2.28.6.m6.1.1.1"><ci id="S7.F2.28.6.m6.1.1.1.1.3.cmml" xref="S7.F2.28.6.m6.1.1.1.1.3">:</ci><cn id="S7.F2.28.6.m6.1.1.1.1.2.cmml" type="integer" xref="S7.F2.28.6.m6.1.1.1.1.2">4</cn><cn id="S7.F2.28.6.m6.1.1.1.1.4.cmml" type="integer" xref="S7.F2.28.6.m6.1.1.1.1.4">0</cn></apply><apply id="S7.F2.28.6.m6.1.1.1.1c.cmml" xref="S7.F2.28.6.m6.1.1.1"><ci id="S7.F2.28.6.m6.1.1.1.1.5.cmml" xref="S7.F2.28.6.m6.1.1.1.1.5">:</ci><share href="https://arxiv.org/html/2411.12694v2#S7.F2.28.6.m6.1.1.1.1.4.cmml" id="S7.F2.28.6.m6.1.1.1.1d.cmml" xref="S7.F2.28.6.m6.1.1.1"></share><cn id="S7.F2.28.6.m6.1.1.1.1.6.cmml" type="integer" xref="S7.F2.28.6.m6.1.1.1.1.6">0</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.F2.28.6.m6.1d">(4:0:0)</annotation><annotation encoding="application/x-llamapun" id="S7.F2.28.6.m6.1e">( 4 : 0 : 0 )</annotation></semantics></math> - at the first <span class="ltx_text ltx_font_smallcaps" id="S7.F2.44.22.3">minute</span> end, either <math alttext="u" class="ltx_Math" display="inline" id="S7.F2.29.7.m7.1"><semantics id="S7.F2.29.7.m7.1b"><mi id="S7.F2.29.7.m7.1.1" xref="S7.F2.29.7.m7.1.1.cmml">u</mi><annotation-xml encoding="MathML-Content" id="S7.F2.29.7.m7.1c"><ci id="S7.F2.29.7.m7.1.1.cmml" xref="S7.F2.29.7.m7.1.1">𝑢</ci></annotation-xml><annotation encoding="application/x-tex" id="S7.F2.29.7.m7.1d">u</annotation><annotation encoding="application/x-llamapun" id="S7.F2.29.7.m7.1e">italic_u</annotation></semantics></math> has dropped a level (pink), <math alttext="v" class="ltx_Math" display="inline" id="S7.F2.30.8.m8.1"><semantics id="S7.F2.30.8.m8.1b"><mi id="S7.F2.30.8.m8.1.1" xref="S7.F2.30.8.m8.1.1.cmml">v</mi><annotation-xml encoding="MathML-Content" id="S7.F2.30.8.m8.1c"><ci id="S7.F2.30.8.m8.1.1.cmml" xref="S7.F2.30.8.m8.1.1">𝑣</ci></annotation-xml><annotation encoding="application/x-tex" id="S7.F2.30.8.m8.1d">v</annotation><annotation encoding="application/x-llamapun" id="S7.F2.30.8.m8.1e">italic_v</annotation></semantics></math> increased their level to <math alttext="h-1" class="ltx_Math" display="inline" id="S7.F2.31.9.m9.1"><semantics id="S7.F2.31.9.m9.1b"><mrow id="S7.F2.31.9.m9.1.1" xref="S7.F2.31.9.m9.1.1.cmml"><mi id="S7.F2.31.9.m9.1.1.2" xref="S7.F2.31.9.m9.1.1.2.cmml">h</mi><mo id="S7.F2.31.9.m9.1.1.1" xref="S7.F2.31.9.m9.1.1.1.cmml">−</mo><mn id="S7.F2.31.9.m9.1.1.3" xref="S7.F2.31.9.m9.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S7.F2.31.9.m9.1c"><apply id="S7.F2.31.9.m9.1.1.cmml" xref="S7.F2.31.9.m9.1.1"><minus id="S7.F2.31.9.m9.1.1.1.cmml" xref="S7.F2.31.9.m9.1.1.1"></minus><ci id="S7.F2.31.9.m9.1.1.2.cmml" xref="S7.F2.31.9.m9.1.1.2">ℎ</ci><cn id="S7.F2.31.9.m9.1.1.3.cmml" type="integer" xref="S7.F2.31.9.m9.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.F2.31.9.m9.1d">h-1</annotation><annotation encoding="application/x-llamapun" id="S7.F2.31.9.m9.1e">italic_h - 1</annotation></semantics></math> (green) or the edge <math alttext="\overline{uv}" class="ltx_Math" display="inline" id="S7.F2.32.10.m10.1"><semantics id="S7.F2.32.10.m10.1b"><mover accent="true" id="S7.F2.32.10.m10.1.1" xref="S7.F2.32.10.m10.1.1.cmml"><mrow id="S7.F2.32.10.m10.1.1.2" xref="S7.F2.32.10.m10.1.1.2.cmml"><mi id="S7.F2.32.10.m10.1.1.2.2" xref="S7.F2.32.10.m10.1.1.2.2.cmml">u</mi><mo id="S7.F2.32.10.m10.1.1.2.1" xref="S7.F2.32.10.m10.1.1.2.1.cmml">⁢</mo><mi id="S7.F2.32.10.m10.1.1.2.3" xref="S7.F2.32.10.m10.1.1.2.3.cmml">v</mi></mrow><mo id="S7.F2.32.10.m10.1.1.1" xref="S7.F2.32.10.m10.1.1.1.cmml">¯</mo></mover><annotation-xml encoding="MathML-Content" id="S7.F2.32.10.m10.1c"><apply id="S7.F2.32.10.m10.1.1.cmml" xref="S7.F2.32.10.m10.1.1"><ci id="S7.F2.32.10.m10.1.1.1.cmml" xref="S7.F2.32.10.m10.1.1.1">¯</ci><apply id="S7.F2.32.10.m10.1.1.2.cmml" xref="S7.F2.32.10.m10.1.1.2"><times id="S7.F2.32.10.m10.1.1.2.1.cmml" xref="S7.F2.32.10.m10.1.1.2.1"></times><ci id="S7.F2.32.10.m10.1.1.2.2.cmml" xref="S7.F2.32.10.m10.1.1.2.2">𝑢</ci><ci id="S7.F2.32.10.m10.1.1.2.3.cmml" xref="S7.F2.32.10.m10.1.1.2.3">𝑣</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.F2.32.10.m10.1d">\overline{uv}</annotation><annotation encoding="application/x-llamapun" id="S7.F2.32.10.m10.1e">over¯ start_ARG italic_u italic_v end_ARG</annotation></semantics></math> is flipped (blue). We consider violating in-edges to pink vertices (orange) <br class="ltx_break"/><math alttext="(4:1:0)" class="ltx_Math" display="inline" id="S7.F2.33.11.m11.1"><semantics id="S7.F2.33.11.m11.1b"><mrow id="S7.F2.33.11.m11.1.1.1" xref="S7.F2.33.11.m11.1.1.1.1.cmml"><mo id="S7.F2.33.11.m11.1.1.1.2" stretchy="false" xref="S7.F2.33.11.m11.1.1.1.1.cmml">(</mo><mrow id="S7.F2.33.11.m11.1.1.1.1" xref="S7.F2.33.11.m11.1.1.1.1.cmml"><mn id="S7.F2.33.11.m11.1.1.1.1.2" xref="S7.F2.33.11.m11.1.1.1.1.2.cmml">4</mn><mo id="S7.F2.33.11.m11.1.1.1.1.3" lspace="0.278em" rspace="0.278em" xref="S7.F2.33.11.m11.1.1.1.1.3.cmml">:</mo><mn id="S7.F2.33.11.m11.1.1.1.1.4" xref="S7.F2.33.11.m11.1.1.1.1.4.cmml">1</mn><mo id="S7.F2.33.11.m11.1.1.1.1.5" lspace="0.278em" rspace="0.278em" xref="S7.F2.33.11.m11.1.1.1.1.5.cmml">:</mo><mn id="S7.F2.33.11.m11.1.1.1.1.6" xref="S7.F2.33.11.m11.1.1.1.1.6.cmml">0</mn></mrow><mo id="S7.F2.33.11.m11.1.1.1.3" stretchy="false" xref="S7.F2.33.11.m11.1.1.1.1.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S7.F2.33.11.m11.1c"><apply id="S7.F2.33.11.m11.1.1.1.1.cmml" xref="S7.F2.33.11.m11.1.1.1"><and id="S7.F2.33.11.m11.1.1.1.1a.cmml" xref="S7.F2.33.11.m11.1.1.1"></and><apply id="S7.F2.33.11.m11.1.1.1.1b.cmml" xref="S7.F2.33.11.m11.1.1.1"><ci id="S7.F2.33.11.m11.1.1.1.1.3.cmml" xref="S7.F2.33.11.m11.1.1.1.1.3">:</ci><cn id="S7.F2.33.11.m11.1.1.1.1.2.cmml" type="integer" xref="S7.F2.33.11.m11.1.1.1.1.2">4</cn><cn id="S7.F2.33.11.m11.1.1.1.1.4.cmml" type="integer" xref="S7.F2.33.11.m11.1.1.1.1.4">1</cn></apply><apply id="S7.F2.33.11.m11.1.1.1.1c.cmml" xref="S7.F2.33.11.m11.1.1.1"><ci id="S7.F2.33.11.m11.1.1.1.1.5.cmml" xref="S7.F2.33.11.m11.1.1.1.1.5">:</ci><share href="https://arxiv.org/html/2411.12694v2#S7.F2.33.11.m11.1.1.1.1.4.cmml" id="S7.F2.33.11.m11.1.1.1.1d.cmml" xref="S7.F2.33.11.m11.1.1.1"></share><cn id="S7.F2.33.11.m11.1.1.1.1.6.cmml" type="integer" xref="S7.F2.33.11.m11.1.1.1.1.6">0</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.F2.33.11.m11.1d">(4:1:0)</annotation><annotation encoding="application/x-llamapun" id="S7.F2.33.11.m11.1e">( 4 : 1 : 0 )</annotation></semantics></math> - at the first <span class="ltx_text ltx_font_smallcaps" id="S7.F2.44.22.4">second</span> start, we construct a DAG <math alttext="D_{0}" class="ltx_Math" display="inline" id="S7.F2.34.12.m12.1"><semantics id="S7.F2.34.12.m12.1b"><msub id="S7.F2.34.12.m12.1.1" xref="S7.F2.34.12.m12.1.1.cmml"><mi id="S7.F2.34.12.m12.1.1.2" xref="S7.F2.34.12.m12.1.1.2.cmml">D</mi><mn id="S7.F2.34.12.m12.1.1.3" xref="S7.F2.34.12.m12.1.1.3.cmml">0</mn></msub><annotation-xml encoding="MathML-Content" id="S7.F2.34.12.m12.1c"><apply id="S7.F2.34.12.m12.1.1.cmml" xref="S7.F2.34.12.m12.1.1"><csymbol cd="ambiguous" id="S7.F2.34.12.m12.1.1.1.cmml" xref="S7.F2.34.12.m12.1.1">subscript</csymbol><ci id="S7.F2.34.12.m12.1.1.2.cmml" xref="S7.F2.34.12.m12.1.1.2">𝐷</ci><cn id="S7.F2.34.12.m12.1.1.3.cmml" type="integer" xref="S7.F2.34.12.m12.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.F2.34.12.m12.1d">D_{0}</annotation><annotation encoding="application/x-llamapun" id="S7.F2.34.12.m12.1e">italic_D start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT</annotation></semantics></math> where the edges are the orange edges plus black edges. The vertex set are all <math alttext="u" class="ltx_Math" display="inline" id="S7.F2.35.13.m13.1"><semantics id="S7.F2.35.13.m13.1b"><mi id="S7.F2.35.13.m13.1.1" xref="S7.F2.35.13.m13.1.1.cmml">u</mi><annotation-xml encoding="MathML-Content" id="S7.F2.35.13.m13.1c"><ci id="S7.F2.35.13.m13.1.1.cmml" xref="S7.F2.35.13.m13.1.1">𝑢</ci></annotation-xml><annotation encoding="application/x-tex" id="S7.F2.35.13.m13.1d">u</annotation><annotation encoding="application/x-llamapun" id="S7.F2.35.13.m13.1e">italic_u</annotation></semantics></math> with a directed path to a pink vertex (<math alttext="v\in T_{1}" class="ltx_Math" display="inline" id="S7.F2.36.14.m14.1"><semantics id="S7.F2.36.14.m14.1b"><mrow id="S7.F2.36.14.m14.1.1" xref="S7.F2.36.14.m14.1.1.cmml"><mi id="S7.F2.36.14.m14.1.1.2" xref="S7.F2.36.14.m14.1.1.2.cmml">v</mi><mo id="S7.F2.36.14.m14.1.1.1" xref="S7.F2.36.14.m14.1.1.1.cmml">∈</mo><msub id="S7.F2.36.14.m14.1.1.3" xref="S7.F2.36.14.m14.1.1.3.cmml"><mi id="S7.F2.36.14.m14.1.1.3.2" xref="S7.F2.36.14.m14.1.1.3.2.cmml">T</mi><mn id="S7.F2.36.14.m14.1.1.3.3" xref="S7.F2.36.14.m14.1.1.3.3.cmml">1</mn></msub></mrow><annotation-xml encoding="MathML-Content" id="S7.F2.36.14.m14.1c"><apply id="S7.F2.36.14.m14.1.1.cmml" xref="S7.F2.36.14.m14.1.1"><in id="S7.F2.36.14.m14.1.1.1.cmml" xref="S7.F2.36.14.m14.1.1.1"></in><ci id="S7.F2.36.14.m14.1.1.2.cmml" xref="S7.F2.36.14.m14.1.1.2">𝑣</ci><apply id="S7.F2.36.14.m14.1.1.3.cmml" xref="S7.F2.36.14.m14.1.1.3"><csymbol cd="ambiguous" id="S7.F2.36.14.m14.1.1.3.1.cmml" xref="S7.F2.36.14.m14.1.1.3">subscript</csymbol><ci id="S7.F2.36.14.m14.1.1.3.2.cmml" xref="S7.F2.36.14.m14.1.1.3.2">𝑇</ci><cn id="S7.F2.36.14.m14.1.1.3.3.cmml" type="integer" xref="S7.F2.36.14.m14.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.F2.36.14.m14.1d">v\in T_{1}</annotation><annotation encoding="application/x-llamapun" id="S7.F2.36.14.m14.1e">italic_v ∈ italic_T start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math>). The yellow vertices are <math alttext="S_{0}" class="ltx_Math" display="inline" id="S7.F2.37.15.m15.1"><semantics id="S7.F2.37.15.m15.1b"><msub id="S7.F2.37.15.m15.1.1" xref="S7.F2.37.15.m15.1.1.cmml"><mi id="S7.F2.37.15.m15.1.1.2" xref="S7.F2.37.15.m15.1.1.2.cmml">S</mi><mn id="S7.F2.37.15.m15.1.1.3" xref="S7.F2.37.15.m15.1.1.3.cmml">0</mn></msub><annotation-xml encoding="MathML-Content" id="S7.F2.37.15.m15.1c"><apply id="S7.F2.37.15.m15.1.1.cmml" xref="S7.F2.37.15.m15.1.1"><csymbol cd="ambiguous" id="S7.F2.37.15.m15.1.1.1.cmml" xref="S7.F2.37.15.m15.1.1">subscript</csymbol><ci id="S7.F2.37.15.m15.1.1.2.cmml" xref="S7.F2.37.15.m15.1.1.2">𝑆</ci><cn id="S7.F2.37.15.m15.1.1.3.cmml" type="integer" xref="S7.F2.37.15.m15.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.F2.37.15.m15.1d">S_{0}</annotation><annotation encoding="application/x-llamapun" id="S7.F2.37.15.m15.1e">italic_S start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT</annotation></semantics></math>. <br class="ltx_break"/><math alttext="(4:1:0)" class="ltx_Math" display="inline" id="S7.F2.38.16.m16.1"><semantics id="S7.F2.38.16.m16.1b"><mrow id="S7.F2.38.16.m16.1.1.1" xref="S7.F2.38.16.m16.1.1.1.1.cmml"><mo id="S7.F2.38.16.m16.1.1.1.2" stretchy="false" xref="S7.F2.38.16.m16.1.1.1.1.cmml">(</mo><mrow id="S7.F2.38.16.m16.1.1.1.1" xref="S7.F2.38.16.m16.1.1.1.1.cmml"><mn id="S7.F2.38.16.m16.1.1.1.1.2" xref="S7.F2.38.16.m16.1.1.1.1.2.cmml">4</mn><mo id="S7.F2.38.16.m16.1.1.1.1.3" lspace="0.278em" rspace="0.278em" xref="S7.F2.38.16.m16.1.1.1.1.3.cmml">:</mo><mn id="S7.F2.38.16.m16.1.1.1.1.4" xref="S7.F2.38.16.m16.1.1.1.1.4.cmml">1</mn><mo id="S7.F2.38.16.m16.1.1.1.1.5" lspace="0.278em" rspace="0.278em" xref="S7.F2.38.16.m16.1.1.1.1.5.cmml">:</mo><mn id="S7.F2.38.16.m16.1.1.1.1.6" xref="S7.F2.38.16.m16.1.1.1.1.6.cmml">0</mn></mrow><mo id="S7.F2.38.16.m16.1.1.1.3" stretchy="false" xref="S7.F2.38.16.m16.1.1.1.1.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S7.F2.38.16.m16.1c"><apply id="S7.F2.38.16.m16.1.1.1.1.cmml" xref="S7.F2.38.16.m16.1.1.1"><and id="S7.F2.38.16.m16.1.1.1.1a.cmml" xref="S7.F2.38.16.m16.1.1.1"></and><apply id="S7.F2.38.16.m16.1.1.1.1b.cmml" xref="S7.F2.38.16.m16.1.1.1"><ci id="S7.F2.38.16.m16.1.1.1.1.3.cmml" xref="S7.F2.38.16.m16.1.1.1.1.3">:</ci><cn id="S7.F2.38.16.m16.1.1.1.1.2.cmml" type="integer" xref="S7.F2.38.16.m16.1.1.1.1.2">4</cn><cn id="S7.F2.38.16.m16.1.1.1.1.4.cmml" type="integer" xref="S7.F2.38.16.m16.1.1.1.1.4">1</cn></apply><apply id="S7.F2.38.16.m16.1.1.1.1c.cmml" xref="S7.F2.38.16.m16.1.1.1"><ci id="S7.F2.38.16.m16.1.1.1.1.5.cmml" xref="S7.F2.38.16.m16.1.1.1.1.5">:</ci><share href="https://arxiv.org/html/2411.12694v2#S7.F2.38.16.m16.1.1.1.1.4.cmml" id="S7.F2.38.16.m16.1.1.1.1d.cmml" xref="S7.F2.38.16.m16.1.1.1"></share><cn id="S7.F2.38.16.m16.1.1.1.1.6.cmml" type="integer" xref="S7.F2.38.16.m16.1.1.1.1.6">0</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.F2.38.16.m16.1d">(4:1:0)</annotation><annotation encoding="application/x-llamapun" id="S7.F2.38.16.m16.1e">( 4 : 1 : 0 )</annotation></semantics></math> - at the first <span class="ltx_text ltx_font_smallcaps" id="S7.F2.44.22.5">second</span> end, edges in the DAG may have flipped (blue), vertices in <math alttext="S_{0}" class="ltx_Math" display="inline" id="S7.F2.39.17.m17.1"><semantics id="S7.F2.39.17.m17.1b"><msub id="S7.F2.39.17.m17.1.1" xref="S7.F2.39.17.m17.1.1.cmml"><mi id="S7.F2.39.17.m17.1.1.2" xref="S7.F2.39.17.m17.1.1.2.cmml">S</mi><mn id="S7.F2.39.17.m17.1.1.3" xref="S7.F2.39.17.m17.1.1.3.cmml">0</mn></msub><annotation-xml encoding="MathML-Content" id="S7.F2.39.17.m17.1c"><apply id="S7.F2.39.17.m17.1.1.cmml" xref="S7.F2.39.17.m17.1.1"><csymbol cd="ambiguous" id="S7.F2.39.17.m17.1.1.1.cmml" xref="S7.F2.39.17.m17.1.1">subscript</csymbol><ci id="S7.F2.39.17.m17.1.1.2.cmml" xref="S7.F2.39.17.m17.1.1.2">𝑆</ci><cn id="S7.F2.39.17.m17.1.1.3.cmml" type="integer" xref="S7.F2.39.17.m17.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.F2.39.17.m17.1d">S_{0}</annotation><annotation encoding="application/x-llamapun" id="S7.F2.39.17.m17.1e">italic_S start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT</annotation></semantics></math> may have dropped a level or vertices in <math alttext="T_{1}" class="ltx_Math" display="inline" id="S7.F2.40.18.m18.1"><semantics id="S7.F2.40.18.m18.1b"><msub id="S7.F2.40.18.m18.1.1" xref="S7.F2.40.18.m18.1.1.cmml"><mi id="S7.F2.40.18.m18.1.1.2" xref="S7.F2.40.18.m18.1.1.2.cmml">T</mi><mn id="S7.F2.40.18.m18.1.1.3" xref="S7.F2.40.18.m18.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="S7.F2.40.18.m18.1c"><apply id="S7.F2.40.18.m18.1.1.cmml" xref="S7.F2.40.18.m18.1.1"><csymbol cd="ambiguous" id="S7.F2.40.18.m18.1.1.1.cmml" xref="S7.F2.40.18.m18.1.1">subscript</csymbol><ci id="S7.F2.40.18.m18.1.1.2.cmml" xref="S7.F2.40.18.m18.1.1.2">𝑇</ci><cn id="S7.F2.40.18.m18.1.1.3.cmml" type="integer" xref="S7.F2.40.18.m18.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.F2.40.18.m18.1d">T_{1}</annotation><annotation encoding="application/x-llamapun" id="S7.F2.40.18.m18.1e">italic_T start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math> may have increased a level (making some edges no longer violating – purple). <br class="ltx_break"/><math alttext="(4:1:1)" class="ltx_Math" display="inline" id="S7.F2.41.19.m19.1"><semantics id="S7.F2.41.19.m19.1b"><mrow id="S7.F2.41.19.m19.1.1.1" xref="S7.F2.41.19.m19.1.1.1.1.cmml"><mo id="S7.F2.41.19.m19.1.1.1.2" stretchy="false" xref="S7.F2.41.19.m19.1.1.1.1.cmml">(</mo><mrow id="S7.F2.41.19.m19.1.1.1.1" xref="S7.F2.41.19.m19.1.1.1.1.cmml"><mn id="S7.F2.41.19.m19.1.1.1.1.2" xref="S7.F2.41.19.m19.1.1.1.1.2.cmml">4</mn><mo id="S7.F2.41.19.m19.1.1.1.1.3" lspace="0.278em" rspace="0.278em" xref="S7.F2.41.19.m19.1.1.1.1.3.cmml">:</mo><mn id="S7.F2.41.19.m19.1.1.1.1.4" xref="S7.F2.41.19.m19.1.1.1.1.4.cmml">1</mn><mo id="S7.F2.41.19.m19.1.1.1.1.5" lspace="0.278em" rspace="0.278em" xref="S7.F2.41.19.m19.1.1.1.1.5.cmml">:</mo><mn id="S7.F2.41.19.m19.1.1.1.1.6" xref="S7.F2.41.19.m19.1.1.1.1.6.cmml">1</mn></mrow><mo id="S7.F2.41.19.m19.1.1.1.3" stretchy="false" xref="S7.F2.41.19.m19.1.1.1.1.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S7.F2.41.19.m19.1c"><apply id="S7.F2.41.19.m19.1.1.1.1.cmml" xref="S7.F2.41.19.m19.1.1.1"><and id="S7.F2.41.19.m19.1.1.1.1a.cmml" xref="S7.F2.41.19.m19.1.1.1"></and><apply id="S7.F2.41.19.m19.1.1.1.1b.cmml" xref="S7.F2.41.19.m19.1.1.1"><ci id="S7.F2.41.19.m19.1.1.1.1.3.cmml" xref="S7.F2.41.19.m19.1.1.1.1.3">:</ci><cn id="S7.F2.41.19.m19.1.1.1.1.2.cmml" type="integer" xref="S7.F2.41.19.m19.1.1.1.1.2">4</cn><cn id="S7.F2.41.19.m19.1.1.1.1.4.cmml" type="integer" xref="S7.F2.41.19.m19.1.1.1.1.4">1</cn></apply><apply id="S7.F2.41.19.m19.1.1.1.1c.cmml" xref="S7.F2.41.19.m19.1.1.1"><ci id="S7.F2.41.19.m19.1.1.1.1.5.cmml" xref="S7.F2.41.19.m19.1.1.1.1.5">:</ci><share href="https://arxiv.org/html/2411.12694v2#S7.F2.41.19.m19.1.1.1.1.4.cmml" id="S7.F2.41.19.m19.1.1.1.1d.cmml" xref="S7.F2.41.19.m19.1.1.1"></share><cn id="S7.F2.41.19.m19.1.1.1.1.6.cmml" type="integer" xref="S7.F2.41.19.m19.1.1.1.1.6">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.F2.41.19.m19.1d">(4:1:1)</annotation><annotation encoding="application/x-llamapun" id="S7.F2.41.19.m19.1e">( 4 : 1 : 1 )</annotation></semantics></math> - at the second <span class="ltx_text ltx_font_smallcaps" id="S7.F2.44.22.6">second</span> start, we construct a DAG <math alttext="D_{1}" class="ltx_Math" display="inline" id="S7.F2.42.20.m20.1"><semantics id="S7.F2.42.20.m20.1b"><msub id="S7.F2.42.20.m20.1.1" xref="S7.F2.42.20.m20.1.1.cmml"><mi id="S7.F2.42.20.m20.1.1.2" xref="S7.F2.42.20.m20.1.1.2.cmml">D</mi><mn id="S7.F2.42.20.m20.1.1.3" xref="S7.F2.42.20.m20.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="S7.F2.42.20.m20.1c"><apply id="S7.F2.42.20.m20.1.1.cmml" xref="S7.F2.42.20.m20.1.1"><csymbol cd="ambiguous" id="S7.F2.42.20.m20.1.1.1.cmml" xref="S7.F2.42.20.m20.1.1">subscript</csymbol><ci id="S7.F2.42.20.m20.1.1.2.cmml" xref="S7.F2.42.20.m20.1.1.2">𝐷</ci><cn id="S7.F2.42.20.m20.1.1.3.cmml" type="integer" xref="S7.F2.42.20.m20.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.F2.42.20.m20.1d">D_{1}</annotation><annotation encoding="application/x-llamapun" id="S7.F2.42.20.m20.1e">italic_D start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math>. Note that <math alttext="D_{1}" class="ltx_Math" display="inline" id="S7.F2.43.21.m21.1"><semantics id="S7.F2.43.21.m21.1b"><msub id="S7.F2.43.21.m21.1.1" xref="S7.F2.43.21.m21.1.1.cmml"><mi id="S7.F2.43.21.m21.1.1.2" xref="S7.F2.43.21.m21.1.1.2.cmml">D</mi><mn id="S7.F2.43.21.m21.1.1.3" xref="S7.F2.43.21.m21.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="S7.F2.43.21.m21.1c"><apply id="S7.F2.43.21.m21.1.1.cmml" xref="S7.F2.43.21.m21.1.1"><csymbol cd="ambiguous" id="S7.F2.43.21.m21.1.1.1.cmml" xref="S7.F2.43.21.m21.1.1">subscript</csymbol><ci id="S7.F2.43.21.m21.1.1.2.cmml" xref="S7.F2.43.21.m21.1.1.2">𝐷</ci><cn id="S7.F2.43.21.m21.1.1.3.cmml" type="integer" xref="S7.F2.43.21.m21.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.F2.43.21.m21.1d">D_{1}</annotation><annotation encoding="application/x-llamapun" id="S7.F2.43.21.m21.1e">italic_D start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math> is a subgraph of <math alttext="D_{0}" class="ltx_Math" display="inline" id="S7.F2.44.22.m22.1"><semantics id="S7.F2.44.22.m22.1b"><msub id="S7.F2.44.22.m22.1.1" xref="S7.F2.44.22.m22.1.1.cmml"><mi id="S7.F2.44.22.m22.1.1.2" xref="S7.F2.44.22.m22.1.1.2.cmml">D</mi><mn id="S7.F2.44.22.m22.1.1.3" xref="S7.F2.44.22.m22.1.1.3.cmml">0</mn></msub><annotation-xml encoding="MathML-Content" id="S7.F2.44.22.m22.1c"><apply id="S7.F2.44.22.m22.1.1.cmml" xref="S7.F2.44.22.m22.1.1"><csymbol cd="ambiguous" id="S7.F2.44.22.m22.1.1.1.cmml" xref="S7.F2.44.22.m22.1.1">subscript</csymbol><ci id="S7.F2.44.22.m22.1.1.2.cmml" xref="S7.F2.44.22.m22.1.1.2">𝐷</ci><cn id="S7.F2.44.22.m22.1.1.3.cmml" type="integer" xref="S7.F2.44.22.m22.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.F2.44.22.m22.1d">D_{0}</annotation><annotation encoding="application/x-llamapun" id="S7.F2.44.22.m22.1e">italic_D start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT</annotation></semantics></math>. </span></figcaption> </figure> </section> </section> <section class="ltx_subsection" id="S7.SS2"> <h3 class="ltx_title ltx_title_subsection"> <span class="ltx_tag ltx_tag_subsection">7.2 </span>Formal algorithm definition.</h3> <div class="ltx_para" id="S7.SS2.p1"> <p class="ltx_p" id="S7.SS2.p1.1">We now formalise our algorithm top-down, starting with defining variables.</p> </div> <div class="ltx_theorem ltx_theorem_definition" id="S7.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="S7.Thmtheorem9.1.1.1">Definition 7.9</span></span><span class="ltx_text ltx_font_bold" id="S7.Thmtheorem9.2.2">.</span> </h6> <div class="ltx_para" id="S7.Thmtheorem9.p1"> <p class="ltx_p" id="S7.Thmtheorem9.p1.5"><span class="ltx_text ltx_font_italic" id="S7.Thmtheorem9.p1.5.5">At the start of <math alttext="(h,m^{\prime},0)" class="ltx_Math" display="inline" id="S7.Thmtheorem9.p1.1.1.m1.3"><semantics id="S7.Thmtheorem9.p1.1.1.m1.3a"><mrow id="S7.Thmtheorem9.p1.1.1.m1.3.3.1" xref="S7.Thmtheorem9.p1.1.1.m1.3.3.2.cmml"><mo id="S7.Thmtheorem9.p1.1.1.m1.3.3.1.2" stretchy="false" xref="S7.Thmtheorem9.p1.1.1.m1.3.3.2.cmml">(</mo><mi id="S7.Thmtheorem9.p1.1.1.m1.1.1" xref="S7.Thmtheorem9.p1.1.1.m1.1.1.cmml">h</mi><mo id="S7.Thmtheorem9.p1.1.1.m1.3.3.1.3" xref="S7.Thmtheorem9.p1.1.1.m1.3.3.2.cmml">,</mo><msup id="S7.Thmtheorem9.p1.1.1.m1.3.3.1.1" xref="S7.Thmtheorem9.p1.1.1.m1.3.3.1.1.cmml"><mi id="S7.Thmtheorem9.p1.1.1.m1.3.3.1.1.2" xref="S7.Thmtheorem9.p1.1.1.m1.3.3.1.1.2.cmml">m</mi><mo id="S7.Thmtheorem9.p1.1.1.m1.3.3.1.1.3" xref="S7.Thmtheorem9.p1.1.1.m1.3.3.1.1.3.cmml">′</mo></msup><mo id="S7.Thmtheorem9.p1.1.1.m1.3.3.1.4" xref="S7.Thmtheorem9.p1.1.1.m1.3.3.2.cmml">,</mo><mn id="S7.Thmtheorem9.p1.1.1.m1.2.2" xref="S7.Thmtheorem9.p1.1.1.m1.2.2.cmml">0</mn><mo id="S7.Thmtheorem9.p1.1.1.m1.3.3.1.5" stretchy="false" xref="S7.Thmtheorem9.p1.1.1.m1.3.3.2.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S7.Thmtheorem9.p1.1.1.m1.3b"><vector id="S7.Thmtheorem9.p1.1.1.m1.3.3.2.cmml" xref="S7.Thmtheorem9.p1.1.1.m1.3.3.1"><ci id="S7.Thmtheorem9.p1.1.1.m1.1.1.cmml" xref="S7.Thmtheorem9.p1.1.1.m1.1.1">ℎ</ci><apply id="S7.Thmtheorem9.p1.1.1.m1.3.3.1.1.cmml" xref="S7.Thmtheorem9.p1.1.1.m1.3.3.1.1"><csymbol cd="ambiguous" id="S7.Thmtheorem9.p1.1.1.m1.3.3.1.1.1.cmml" xref="S7.Thmtheorem9.p1.1.1.m1.3.3.1.1">superscript</csymbol><ci id="S7.Thmtheorem9.p1.1.1.m1.3.3.1.1.2.cmml" xref="S7.Thmtheorem9.p1.1.1.m1.3.3.1.1.2">𝑚</ci><ci id="S7.Thmtheorem9.p1.1.1.m1.3.3.1.1.3.cmml" xref="S7.Thmtheorem9.p1.1.1.m1.3.3.1.1.3">′</ci></apply><cn id="S7.Thmtheorem9.p1.1.1.m1.2.2.cmml" type="integer" xref="S7.Thmtheorem9.p1.1.1.m1.2.2">0</cn></vector></annotation-xml><annotation encoding="application/x-tex" id="S7.Thmtheorem9.p1.1.1.m1.3c">(h,m^{\prime},0)</annotation><annotation encoding="application/x-llamapun" id="S7.Thmtheorem9.p1.1.1.m1.3d">( italic_h , italic_m start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT , 0 )</annotation></semantics></math>, where <math alttext="m^{\prime}" class="ltx_Math" display="inline" id="S7.Thmtheorem9.p1.2.2.m2.1"><semantics id="S7.Thmtheorem9.p1.2.2.m2.1a"><msup id="S7.Thmtheorem9.p1.2.2.m2.1.1" xref="S7.Thmtheorem9.p1.2.2.m2.1.1.cmml"><mi id="S7.Thmtheorem9.p1.2.2.m2.1.1.2" xref="S7.Thmtheorem9.p1.2.2.m2.1.1.2.cmml">m</mi><mo id="S7.Thmtheorem9.p1.2.2.m2.1.1.3" xref="S7.Thmtheorem9.p1.2.2.m2.1.1.3.cmml">′</mo></msup><annotation-xml encoding="MathML-Content" id="S7.Thmtheorem9.p1.2.2.m2.1b"><apply id="S7.Thmtheorem9.p1.2.2.m2.1.1.cmml" xref="S7.Thmtheorem9.p1.2.2.m2.1.1"><csymbol cd="ambiguous" id="S7.Thmtheorem9.p1.2.2.m2.1.1.1.cmml" xref="S7.Thmtheorem9.p1.2.2.m2.1.1">superscript</csymbol><ci id="S7.Thmtheorem9.p1.2.2.m2.1.1.2.cmml" xref="S7.Thmtheorem9.p1.2.2.m2.1.1.2">𝑚</ci><ci id="S7.Thmtheorem9.p1.2.2.m2.1.1.3.cmml" xref="S7.Thmtheorem9.p1.2.2.m2.1.1.3">′</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.Thmtheorem9.p1.2.2.m2.1c">m^{\prime}</annotation><annotation encoding="application/x-llamapun" id="S7.Thmtheorem9.p1.2.2.m2.1d">italic_m start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math> is even, we fix the set <math alttext="V_{m^{\prime}}:=\{v\in L_{h}\mid v\textnormal{ has at least one violating out-% edge}\}." class="ltx_Math" display="inline" id="S7.Thmtheorem9.p1.3.3.m3.1"><semantics id="S7.Thmtheorem9.p1.3.3.m3.1a"><mrow id="S7.Thmtheorem9.p1.3.3.m3.1.1.1" xref="S7.Thmtheorem9.p1.3.3.m3.1.1.1.1.cmml"><mrow id="S7.Thmtheorem9.p1.3.3.m3.1.1.1.1" xref="S7.Thmtheorem9.p1.3.3.m3.1.1.1.1.cmml"><msub id="S7.Thmtheorem9.p1.3.3.m3.1.1.1.1.4" xref="S7.Thmtheorem9.p1.3.3.m3.1.1.1.1.4.cmml"><mi id="S7.Thmtheorem9.p1.3.3.m3.1.1.1.1.4.2" xref="S7.Thmtheorem9.p1.3.3.m3.1.1.1.1.4.2.cmml">V</mi><msup id="S7.Thmtheorem9.p1.3.3.m3.1.1.1.1.4.3" xref="S7.Thmtheorem9.p1.3.3.m3.1.1.1.1.4.3.cmml"><mi 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id="S7.Thmtheorem9.p1.3.3.m3.1c">V_{m^{\prime}}:=\{v\in L_{h}\mid v\textnormal{ has at least one violating out-% edge}\}.</annotation><annotation encoding="application/x-llamapun" id="S7.Thmtheorem9.p1.3.3.m3.1d">italic_V start_POSTSUBSCRIPT italic_m start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT := { italic_v ∈ italic_L start_POSTSUBSCRIPT italic_h end_POSTSUBSCRIPT ∣ italic_v has at least one violating out-edge } .</annotation></semantics></math> At the start of <math alttext="(h,m,0)" class="ltx_Math" display="inline" id="S7.Thmtheorem9.p1.4.4.m4.3"><semantics id="S7.Thmtheorem9.p1.4.4.m4.3a"><mrow id="S7.Thmtheorem9.p1.4.4.m4.3.4.2" xref="S7.Thmtheorem9.p1.4.4.m4.3.4.1.cmml"><mo id="S7.Thmtheorem9.p1.4.4.m4.3.4.2.1" stretchy="false" xref="S7.Thmtheorem9.p1.4.4.m4.3.4.1.cmml">(</mo><mi id="S7.Thmtheorem9.p1.4.4.m4.1.1" xref="S7.Thmtheorem9.p1.4.4.m4.1.1.cmml">h</mi><mo id="S7.Thmtheorem9.p1.4.4.m4.3.4.2.2" xref="S7.Thmtheorem9.p1.4.4.m4.3.4.1.cmml">,</mo><mi id="S7.Thmtheorem9.p1.4.4.m4.2.2" xref="S7.Thmtheorem9.p1.4.4.m4.2.2.cmml">m</mi><mo id="S7.Thmtheorem9.p1.4.4.m4.3.4.2.3" xref="S7.Thmtheorem9.p1.4.4.m4.3.4.1.cmml">,</mo><mn id="S7.Thmtheorem9.p1.4.4.m4.3.3" xref="S7.Thmtheorem9.p1.4.4.m4.3.3.cmml">0</mn><mo id="S7.Thmtheorem9.p1.4.4.m4.3.4.2.4" stretchy="false" xref="S7.Thmtheorem9.p1.4.4.m4.3.4.1.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S7.Thmtheorem9.p1.4.4.m4.3b"><vector id="S7.Thmtheorem9.p1.4.4.m4.3.4.1.cmml" xref="S7.Thmtheorem9.p1.4.4.m4.3.4.2"><ci id="S7.Thmtheorem9.p1.4.4.m4.1.1.cmml" xref="S7.Thmtheorem9.p1.4.4.m4.1.1">ℎ</ci><ci id="S7.Thmtheorem9.p1.4.4.m4.2.2.cmml" xref="S7.Thmtheorem9.p1.4.4.m4.2.2">𝑚</ci><cn id="S7.Thmtheorem9.p1.4.4.m4.3.3.cmml" type="integer" xref="S7.Thmtheorem9.p1.4.4.m4.3.3">0</cn></vector></annotation-xml><annotation encoding="application/x-tex" id="S7.Thmtheorem9.p1.4.4.m4.3c">(h,m,0)</annotation><annotation encoding="application/x-llamapun" id="S7.Thmtheorem9.p1.4.4.m4.3d">( italic_h , italic_m , 0 )</annotation></semantics></math> where <math alttext="m" class="ltx_Math" display="inline" id="S7.Thmtheorem9.p1.5.5.m5.1"><semantics id="S7.Thmtheorem9.p1.5.5.m5.1a"><mi id="S7.Thmtheorem9.p1.5.5.m5.1.1" xref="S7.Thmtheorem9.p1.5.5.m5.1.1.cmml">m</mi><annotation-xml encoding="MathML-Content" id="S7.Thmtheorem9.p1.5.5.m5.1b"><ci id="S7.Thmtheorem9.p1.5.5.m5.1.1.cmml" xref="S7.Thmtheorem9.p1.5.5.m5.1.1">𝑚</ci></annotation-xml><annotation encoding="application/x-tex" id="S7.Thmtheorem9.p1.5.5.m5.1c">m</annotation><annotation encoding="application/x-llamapun" id="S7.Thmtheorem9.p1.5.5.m5.1d">italic_m</annotation></semantics></math> is odd, we fix:</span></p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="A1.EGx7"> <tbody id="S7.Ex29"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_left_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle T_{m}:=\{v\in V_{m-1}\mid\quad" class="ltx_math_unparsed" display="inline" id="S7.Ex29.m1.1"><semantics id="S7.Ex29.m1.1a"><mrow id="S7.Ex29.m1.1b"><msub id="S7.Ex29.m1.1.1"><mi id="S7.Ex29.m1.1.1.2">T</mi><mi id="S7.Ex29.m1.1.1.3">m</mi></msub><mo id="S7.Ex29.m1.1.2" lspace="0.278em" rspace="0.278em">:=</mo><mrow id="S7.Ex29.m1.1.3"><mo id="S7.Ex29.m1.1.3.1" stretchy="false">{</mo><mi id="S7.Ex29.m1.1.3.2">v</mi><mo id="S7.Ex29.m1.1.3.3">∈</mo><msub id="S7.Ex29.m1.1.3.4"><mi id="S7.Ex29.m1.1.3.4.2">V</mi><mrow id="S7.Ex29.m1.1.3.4.3"><mi id="S7.Ex29.m1.1.3.4.3.2">m</mi><mo id="S7.Ex29.m1.1.3.4.3.1">−</mo><mn id="S7.Ex29.m1.1.3.4.3.3">1</mn></mrow></msub><mo id="S7.Ex29.m1.1.3.5" lspace="0em">∣</mo><mspace id="S7.Ex29.m1.1.3.6" width="1.167em"></mspace></mrow></mrow><annotation encoding="application/x-tex" id="S7.Ex29.m1.1c">\displaystyle T_{m}:=\{v\in V_{m-1}\mid\quad</annotation><annotation encoding="application/x-llamapun" id="S7.Ex29.m1.1d">italic_T start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT := { italic_v ∈ italic_V start_POSTSUBSCRIPT italic_m - 1 end_POSTSUBSCRIPT ∣</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle v\textnormal{ decreased by one level in the previous }\textsc{% minute}\textnormal{ and }" class="ltx_Math" display="inline" id="S7.Ex29.m2.1"><semantics id="S7.Ex29.m2.1a"><mrow id="S7.Ex29.m2.1.1" xref="S7.Ex29.m2.1.1.cmml"><mi id="S7.Ex29.m2.1.1.2" xref="S7.Ex29.m2.1.1.2.cmml">v</mi><mo id="S7.Ex29.m2.1.1.1" xref="S7.Ex29.m2.1.1.1.cmml">⁢</mo><mrow id="S7.Ex29.m2.1.1.3" xref="S7.Ex29.m2.1.1.3d.cmml"><mtext id="S7.Ex29.m2.1.1.3a" xref="S7.Ex29.m2.1.1.3d.cmml"> decreased by one level in the previous </mtext><mtext class="ltx_font_smallcaps" id="S7.Ex29.m2.1.1.3b" xref="S7.Ex29.m2.1.1.3d.cmml">minute</mtext><mtext id="S7.Ex29.m2.1.1.3c" xref="S7.Ex29.m2.1.1.3d.cmml"> and</mtext></mrow></mrow><annotation-xml encoding="MathML-Content" id="S7.Ex29.m2.1b"><apply id="S7.Ex29.m2.1.1.cmml" xref="S7.Ex29.m2.1.1"><times id="S7.Ex29.m2.1.1.1.cmml" xref="S7.Ex29.m2.1.1.1"></times><ci id="S7.Ex29.m2.1.1.2.cmml" xref="S7.Ex29.m2.1.1.2">𝑣</ci><ci id="S7.Ex29.m2.1.1.3d.cmml" xref="S7.Ex29.m2.1.1.3"><mrow id="S7.Ex29.m2.1.1.3.cmml" xref="S7.Ex29.m2.1.1.3"><mtext id="S7.Ex29.m2.1.1.3a.cmml" xref="S7.Ex29.m2.1.1.3"> decreased by one level in the previous </mtext><mtext class="ltx_font_smallcaps" id="S7.Ex29.m2.1.1.3b.cmml" xref="S7.Ex29.m2.1.1.3">minute</mtext><mtext id="S7.Ex29.m2.1.1.3c.cmml" xref="S7.Ex29.m2.1.1.3"> and</mtext></mrow></ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.Ex29.m2.1c">\displaystyle v\textnormal{ decreased by one level in the previous }\textsc{% minute}\textnormal{ and }</annotation><annotation encoding="application/x-llamapun" id="S7.Ex29.m2.1d">italic_v decreased by one level in the previous smallcaps_minute and</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_left_padright"></td> </tr></tbody> <tbody id="S7.Ex30"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_left_padleft"></td> <td class="ltx_td ltx_eqn_cell"></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle v\textnormal{ has at least one violating in-edge}\}." class="ltx_math_unparsed" display="inline" id="S7.Ex30.m1.1"><semantics id="S7.Ex30.m1.1a"><mrow id="S7.Ex30.m1.1b"><mi id="S7.Ex30.m1.1.1">v</mi><mtext id="S7.Ex30.m1.1.2"> has at least one violating in-edge</mtext><mo id="S7.Ex30.m1.1.3" stretchy="false">}</mo><mo id="S7.Ex30.m1.1.4" lspace="0em">.</mo></mrow><annotation encoding="application/x-tex" id="S7.Ex30.m1.1c">\displaystyle v\textnormal{ has at least one violating in-edge}\}.</annotation><annotation encoding="application/x-llamapun" id="S7.Ex30.m1.1d">italic_v has at least one violating in-edge } .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_left_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S7.Thmtheorem9.p1.8"><span class="ltx_text ltx_font_italic" id="S7.Thmtheorem9.p1.8.3">Finally, we denote for any vertex <math alttext="u" class="ltx_Math" display="inline" id="S7.Thmtheorem9.p1.6.1.m1.1"><semantics id="S7.Thmtheorem9.p1.6.1.m1.1a"><mi id="S7.Thmtheorem9.p1.6.1.m1.1.1" xref="S7.Thmtheorem9.p1.6.1.m1.1.1.cmml">u</mi><annotation-xml encoding="MathML-Content" id="S7.Thmtheorem9.p1.6.1.m1.1b"><ci id="S7.Thmtheorem9.p1.6.1.m1.1.1.cmml" xref="S7.Thmtheorem9.p1.6.1.m1.1.1">𝑢</ci></annotation-xml><annotation encoding="application/x-tex" id="S7.Thmtheorem9.p1.6.1.m1.1c">u</annotation><annotation encoding="application/x-llamapun" id="S7.Thmtheorem9.p1.6.1.m1.1d">italic_u</annotation></semantics></math> by <math alttext="l_{m}(u)" class="ltx_Math" display="inline" id="S7.Thmtheorem9.p1.7.2.m2.1"><semantics id="S7.Thmtheorem9.p1.7.2.m2.1a"><mrow id="S7.Thmtheorem9.p1.7.2.m2.1.2" xref="S7.Thmtheorem9.p1.7.2.m2.1.2.cmml"><msub id="S7.Thmtheorem9.p1.7.2.m2.1.2.2" xref="S7.Thmtheorem9.p1.7.2.m2.1.2.2.cmml"><mi id="S7.Thmtheorem9.p1.7.2.m2.1.2.2.2" xref="S7.Thmtheorem9.p1.7.2.m2.1.2.2.2.cmml">l</mi><mi id="S7.Thmtheorem9.p1.7.2.m2.1.2.2.3" xref="S7.Thmtheorem9.p1.7.2.m2.1.2.2.3.cmml">m</mi></msub><mo id="S7.Thmtheorem9.p1.7.2.m2.1.2.1" xref="S7.Thmtheorem9.p1.7.2.m2.1.2.1.cmml">⁢</mo><mrow id="S7.Thmtheorem9.p1.7.2.m2.1.2.3.2" xref="S7.Thmtheorem9.p1.7.2.m2.1.2.cmml"><mo id="S7.Thmtheorem9.p1.7.2.m2.1.2.3.2.1" stretchy="false" xref="S7.Thmtheorem9.p1.7.2.m2.1.2.cmml">(</mo><mi id="S7.Thmtheorem9.p1.7.2.m2.1.1" xref="S7.Thmtheorem9.p1.7.2.m2.1.1.cmml">u</mi><mo id="S7.Thmtheorem9.p1.7.2.m2.1.2.3.2.2" stretchy="false" xref="S7.Thmtheorem9.p1.7.2.m2.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S7.Thmtheorem9.p1.7.2.m2.1b"><apply id="S7.Thmtheorem9.p1.7.2.m2.1.2.cmml" xref="S7.Thmtheorem9.p1.7.2.m2.1.2"><times id="S7.Thmtheorem9.p1.7.2.m2.1.2.1.cmml" xref="S7.Thmtheorem9.p1.7.2.m2.1.2.1"></times><apply id="S7.Thmtheorem9.p1.7.2.m2.1.2.2.cmml" xref="S7.Thmtheorem9.p1.7.2.m2.1.2.2"><csymbol cd="ambiguous" id="S7.Thmtheorem9.p1.7.2.m2.1.2.2.1.cmml" xref="S7.Thmtheorem9.p1.7.2.m2.1.2.2">subscript</csymbol><ci id="S7.Thmtheorem9.p1.7.2.m2.1.2.2.2.cmml" xref="S7.Thmtheorem9.p1.7.2.m2.1.2.2.2">𝑙</ci><ci id="S7.Thmtheorem9.p1.7.2.m2.1.2.2.3.cmml" xref="S7.Thmtheorem9.p1.7.2.m2.1.2.2.3">𝑚</ci></apply><ci id="S7.Thmtheorem9.p1.7.2.m2.1.1.cmml" xref="S7.Thmtheorem9.p1.7.2.m2.1.1">𝑢</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.Thmtheorem9.p1.7.2.m2.1c">l_{m}(u)</annotation><annotation encoding="application/x-llamapun" id="S7.Thmtheorem9.p1.7.2.m2.1d">italic_l start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT ( italic_u )</annotation></semantics></math> its level <em class="ltx_emph ltx_font_upright" id="S7.Thmtheorem9.p1.8.3.1">at the start of the </em><span class="ltx_text ltx_font_smallcaps" id="S7.Thmtheorem9.p1.8.3.2">minute</span> <math alttext="m" class="ltx_Math" display="inline" id="S7.Thmtheorem9.p1.8.3.m3.1"><semantics id="S7.Thmtheorem9.p1.8.3.m3.1a"><mi id="S7.Thmtheorem9.p1.8.3.m3.1.1" xref="S7.Thmtheorem9.p1.8.3.m3.1.1.cmml">m</mi><annotation-xml encoding="MathML-Content" id="S7.Thmtheorem9.p1.8.3.m3.1b"><ci id="S7.Thmtheorem9.p1.8.3.m3.1.1.cmml" xref="S7.Thmtheorem9.p1.8.3.m3.1.1">𝑚</ci></annotation-xml><annotation encoding="application/x-tex" id="S7.Thmtheorem9.p1.8.3.m3.1c">m</annotation><annotation encoding="application/x-llamapun" id="S7.Thmtheorem9.p1.8.3.m3.1d">italic_m</annotation></semantics></math>.</span></p> </div> </div> <div class="ltx_theorem ltx_theorem_definition" id="S7.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="S7.Thmtheorem10.1.1.1">Definition 7.10</span></span><span class="ltx_text ltx_font_bold" id="S7.Thmtheorem10.2.2">.</span> </h6> <div class="ltx_para" id="S7.Thmtheorem10.p1"> <p class="ltx_p" id="S7.Thmtheorem10.p1.2"><span class="ltx_text ltx_font_italic" id="S7.Thmtheorem10.p1.2.2">At the start of <math alttext="(h:m:s)" class="ltx_Math" display="inline" id="S7.Thmtheorem10.p1.1.1.m1.1"><semantics id="S7.Thmtheorem10.p1.1.1.m1.1a"><mrow id="S7.Thmtheorem10.p1.1.1.m1.1.1.1" xref="S7.Thmtheorem10.p1.1.1.m1.1.1.1.1.cmml"><mo id="S7.Thmtheorem10.p1.1.1.m1.1.1.1.2" stretchy="false" xref="S7.Thmtheorem10.p1.1.1.m1.1.1.1.1.cmml">(</mo><mrow id="S7.Thmtheorem10.p1.1.1.m1.1.1.1.1" xref="S7.Thmtheorem10.p1.1.1.m1.1.1.1.1.cmml"><mi id="S7.Thmtheorem10.p1.1.1.m1.1.1.1.1.2" xref="S7.Thmtheorem10.p1.1.1.m1.1.1.1.1.2.cmml">h</mi><mo id="S7.Thmtheorem10.p1.1.1.m1.1.1.1.1.3" lspace="0.278em" rspace="0.278em" xref="S7.Thmtheorem10.p1.1.1.m1.1.1.1.1.3.cmml">:</mo><mi id="S7.Thmtheorem10.p1.1.1.m1.1.1.1.1.4" xref="S7.Thmtheorem10.p1.1.1.m1.1.1.1.1.4.cmml">m</mi><mo id="S7.Thmtheorem10.p1.1.1.m1.1.1.1.1.5" lspace="0.278em" rspace="0.278em" xref="S7.Thmtheorem10.p1.1.1.m1.1.1.1.1.5.cmml">:</mo><mi id="S7.Thmtheorem10.p1.1.1.m1.1.1.1.1.6" xref="S7.Thmtheorem10.p1.1.1.m1.1.1.1.1.6.cmml">s</mi></mrow><mo id="S7.Thmtheorem10.p1.1.1.m1.1.1.1.3" stretchy="false" xref="S7.Thmtheorem10.p1.1.1.m1.1.1.1.1.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S7.Thmtheorem10.p1.1.1.m1.1b"><apply id="S7.Thmtheorem10.p1.1.1.m1.1.1.1.1.cmml" xref="S7.Thmtheorem10.p1.1.1.m1.1.1.1"><and id="S7.Thmtheorem10.p1.1.1.m1.1.1.1.1a.cmml" xref="S7.Thmtheorem10.p1.1.1.m1.1.1.1"></and><apply id="S7.Thmtheorem10.p1.1.1.m1.1.1.1.1b.cmml" xref="S7.Thmtheorem10.p1.1.1.m1.1.1.1"><ci id="S7.Thmtheorem10.p1.1.1.m1.1.1.1.1.3.cmml" xref="S7.Thmtheorem10.p1.1.1.m1.1.1.1.1.3">:</ci><ci id="S7.Thmtheorem10.p1.1.1.m1.1.1.1.1.2.cmml" xref="S7.Thmtheorem10.p1.1.1.m1.1.1.1.1.2">ℎ</ci><ci id="S7.Thmtheorem10.p1.1.1.m1.1.1.1.1.4.cmml" xref="S7.Thmtheorem10.p1.1.1.m1.1.1.1.1.4">𝑚</ci></apply><apply id="S7.Thmtheorem10.p1.1.1.m1.1.1.1.1c.cmml" xref="S7.Thmtheorem10.p1.1.1.m1.1.1.1"><ci id="S7.Thmtheorem10.p1.1.1.m1.1.1.1.1.5.cmml" xref="S7.Thmtheorem10.p1.1.1.m1.1.1.1.1.5">:</ci><share href="https://arxiv.org/html/2411.12694v2#S7.Thmtheorem10.p1.1.1.m1.1.1.1.1.4.cmml" id="S7.Thmtheorem10.p1.1.1.m1.1.1.1.1d.cmml" xref="S7.Thmtheorem10.p1.1.1.m1.1.1.1"></share><ci id="S7.Thmtheorem10.p1.1.1.m1.1.1.1.1.6.cmml" xref="S7.Thmtheorem10.p1.1.1.m1.1.1.1.1.6">𝑠</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.Thmtheorem10.p1.1.1.m1.1c">(h:m:s)</annotation><annotation encoding="application/x-llamapun" id="S7.Thmtheorem10.p1.1.1.m1.1d">( italic_h : italic_m : italic_s )</annotation></semantics></math>, where <math alttext="m" class="ltx_Math" display="inline" id="S7.Thmtheorem10.p1.2.2.m2.1"><semantics id="S7.Thmtheorem10.p1.2.2.m2.1a"><mi id="S7.Thmtheorem10.p1.2.2.m2.1.1" xref="S7.Thmtheorem10.p1.2.2.m2.1.1.cmml">m</mi><annotation-xml encoding="MathML-Content" id="S7.Thmtheorem10.p1.2.2.m2.1b"><ci id="S7.Thmtheorem10.p1.2.2.m2.1.1.cmml" xref="S7.Thmtheorem10.p1.2.2.m2.1.1">𝑚</ci></annotation-xml><annotation encoding="application/x-tex" id="S7.Thmtheorem10.p1.2.2.m2.1c">m</annotation><annotation encoding="application/x-llamapun" id="S7.Thmtheorem10.p1.2.2.m2.1d">italic_m</annotation></semantics></math> is odd, we fix the following:</span></p> <ul class="ltx_itemize" id="S7.I5"> <li class="ltx_item" id="S7.I5.i1" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">•</span> <div class="ltx_para" id="S7.I5.i1.p1"> <p class="ltx_p" id="S7.I5.i1.p1.2"><span class="ltx_text ltx_font_italic" id="S7.I5.i1.p1.2.1">For any vertex </span><math alttext="u" class="ltx_Math" display="inline" id="S7.I5.i1.p1.1.m1.1"><semantics id="S7.I5.i1.p1.1.m1.1a"><mi id="S7.I5.i1.p1.1.m1.1.1" xref="S7.I5.i1.p1.1.m1.1.1.cmml">u</mi><annotation-xml encoding="MathML-Content" id="S7.I5.i1.p1.1.m1.1b"><ci id="S7.I5.i1.p1.1.m1.1.1.cmml" xref="S7.I5.i1.p1.1.m1.1.1">𝑢</ci></annotation-xml><annotation encoding="application/x-tex" id="S7.I5.i1.p1.1.m1.1c">u</annotation><annotation encoding="application/x-llamapun" id="S7.I5.i1.p1.1.m1.1d">italic_u</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S7.I5.i1.p1.2.2">, we denote by </span><math alttext="l_{s}(u)" class="ltx_Math" display="inline" id="S7.I5.i1.p1.2.m2.1"><semantics id="S7.I5.i1.p1.2.m2.1a"><mrow id="S7.I5.i1.p1.2.m2.1.2" xref="S7.I5.i1.p1.2.m2.1.2.cmml"><msub id="S7.I5.i1.p1.2.m2.1.2.2" xref="S7.I5.i1.p1.2.m2.1.2.2.cmml"><mi id="S7.I5.i1.p1.2.m2.1.2.2.2" xref="S7.I5.i1.p1.2.m2.1.2.2.2.cmml">l</mi><mi id="S7.I5.i1.p1.2.m2.1.2.2.3" xref="S7.I5.i1.p1.2.m2.1.2.2.3.cmml">s</mi></msub><mo id="S7.I5.i1.p1.2.m2.1.2.1" xref="S7.I5.i1.p1.2.m2.1.2.1.cmml">⁢</mo><mrow id="S7.I5.i1.p1.2.m2.1.2.3.2" xref="S7.I5.i1.p1.2.m2.1.2.cmml"><mo id="S7.I5.i1.p1.2.m2.1.2.3.2.1" stretchy="false" xref="S7.I5.i1.p1.2.m2.1.2.cmml">(</mo><mi id="S7.I5.i1.p1.2.m2.1.1" xref="S7.I5.i1.p1.2.m2.1.1.cmml">u</mi><mo id="S7.I5.i1.p1.2.m2.1.2.3.2.2" stretchy="false" xref="S7.I5.i1.p1.2.m2.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S7.I5.i1.p1.2.m2.1b"><apply id="S7.I5.i1.p1.2.m2.1.2.cmml" xref="S7.I5.i1.p1.2.m2.1.2"><times id="S7.I5.i1.p1.2.m2.1.2.1.cmml" xref="S7.I5.i1.p1.2.m2.1.2.1"></times><apply id="S7.I5.i1.p1.2.m2.1.2.2.cmml" xref="S7.I5.i1.p1.2.m2.1.2.2"><csymbol cd="ambiguous" id="S7.I5.i1.p1.2.m2.1.2.2.1.cmml" xref="S7.I5.i1.p1.2.m2.1.2.2">subscript</csymbol><ci id="S7.I5.i1.p1.2.m2.1.2.2.2.cmml" xref="S7.I5.i1.p1.2.m2.1.2.2.2">𝑙</ci><ci id="S7.I5.i1.p1.2.m2.1.2.2.3.cmml" xref="S7.I5.i1.p1.2.m2.1.2.2.3">𝑠</ci></apply><ci id="S7.I5.i1.p1.2.m2.1.1.cmml" xref="S7.I5.i1.p1.2.m2.1.1">𝑢</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.I5.i1.p1.2.m2.1c">l_{s}(u)</annotation><annotation encoding="application/x-llamapun" id="S7.I5.i1.p1.2.m2.1d">italic_l start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_u )</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S7.I5.i1.p1.2.3"> its level at the start of the </span><span class="ltx_text ltx_font_smallcaps" id="S7.I5.i1.p1.2.4">second</span><span class="ltx_text ltx_font_italic" id="S7.I5.i1.p1.2.5">.</span></p> </div> </li> <li class="ltx_item" id="S7.I5.i2" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">•</span> <div class="ltx_para" id="S7.I5.i2.p1"> <p class="ltx_p" id="S7.I5.i2.p1.4"><span class="ltx_text ltx_font_italic" id="S7.I5.i2.p1.4.1">For any edge </span><math alttext="\overline{uv}" class="ltx_Math" display="inline" id="S7.I5.i2.p1.1.m1.1"><semantics id="S7.I5.i2.p1.1.m1.1a"><mover accent="true" id="S7.I5.i2.p1.1.m1.1.1" xref="S7.I5.i2.p1.1.m1.1.1.cmml"><mrow id="S7.I5.i2.p1.1.m1.1.1.2" xref="S7.I5.i2.p1.1.m1.1.1.2.cmml"><mi id="S7.I5.i2.p1.1.m1.1.1.2.2" xref="S7.I5.i2.p1.1.m1.1.1.2.2.cmml">u</mi><mo id="S7.I5.i2.p1.1.m1.1.1.2.1" xref="S7.I5.i2.p1.1.m1.1.1.2.1.cmml">⁢</mo><mi id="S7.I5.i2.p1.1.m1.1.1.2.3" xref="S7.I5.i2.p1.1.m1.1.1.2.3.cmml">v</mi></mrow><mo id="S7.I5.i2.p1.1.m1.1.1.1" xref="S7.I5.i2.p1.1.m1.1.1.1.cmml">¯</mo></mover><annotation-xml encoding="MathML-Content" id="S7.I5.i2.p1.1.m1.1b"><apply id="S7.I5.i2.p1.1.m1.1.1.cmml" xref="S7.I5.i2.p1.1.m1.1.1"><ci id="S7.I5.i2.p1.1.m1.1.1.1.cmml" xref="S7.I5.i2.p1.1.m1.1.1.1">¯</ci><apply id="S7.I5.i2.p1.1.m1.1.1.2.cmml" xref="S7.I5.i2.p1.1.m1.1.1.2"><times id="S7.I5.i2.p1.1.m1.1.1.2.1.cmml" xref="S7.I5.i2.p1.1.m1.1.1.2.1"></times><ci id="S7.I5.i2.p1.1.m1.1.1.2.2.cmml" xref="S7.I5.i2.p1.1.m1.1.1.2.2">𝑢</ci><ci id="S7.I5.i2.p1.1.m1.1.1.2.3.cmml" xref="S7.I5.i2.p1.1.m1.1.1.2.3">𝑣</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.I5.i2.p1.1.m1.1c">\overline{uv}</annotation><annotation encoding="application/x-llamapun" id="S7.I5.i2.p1.1.m1.1d">over¯ start_ARG italic_u italic_v end_ARG</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S7.I5.i2.p1.4.2"> denote by </span><math alttext="\textsl{g}_{s}(u\!\to\!v)" class="ltx_Math" display="inline" id="S7.I5.i2.p1.2.m2.1"><semantics id="S7.I5.i2.p1.2.m2.1a"><mrow id="S7.I5.i2.p1.2.m2.1.1" xref="S7.I5.i2.p1.2.m2.1.1.cmml"><msub id="S7.I5.i2.p1.2.m2.1.1.3" xref="S7.I5.i2.p1.2.m2.1.1.3.cmml"><mtext class="ltx_mathvariant_italic" id="S7.I5.i2.p1.2.m2.1.1.3.2" xref="S7.I5.i2.p1.2.m2.1.1.3.2a.cmml">g</mtext><mi id="S7.I5.i2.p1.2.m2.1.1.3.3" xref="S7.I5.i2.p1.2.m2.1.1.3.3.cmml">s</mi></msub><mo id="S7.I5.i2.p1.2.m2.1.1.2" xref="S7.I5.i2.p1.2.m2.1.1.2.cmml">⁢</mo><mrow id="S7.I5.i2.p1.2.m2.1.1.1.1" xref="S7.I5.i2.p1.2.m2.1.1.1.1.1.cmml"><mo id="S7.I5.i2.p1.2.m2.1.1.1.1.2" stretchy="false" xref="S7.I5.i2.p1.2.m2.1.1.1.1.1.cmml">(</mo><mrow id="S7.I5.i2.p1.2.m2.1.1.1.1.1" xref="S7.I5.i2.p1.2.m2.1.1.1.1.1.cmml"><mi id="S7.I5.i2.p1.2.m2.1.1.1.1.1.2" xref="S7.I5.i2.p1.2.m2.1.1.1.1.1.2.cmml">u</mi><mo id="S7.I5.i2.p1.2.m2.1.1.1.1.1.1" lspace="0.108em" rspace="0.108em" stretchy="false" xref="S7.I5.i2.p1.2.m2.1.1.1.1.1.1.cmml">→</mo><mi id="S7.I5.i2.p1.2.m2.1.1.1.1.1.3" xref="S7.I5.i2.p1.2.m2.1.1.1.1.1.3.cmml">v</mi></mrow><mo id="S7.I5.i2.p1.2.m2.1.1.1.1.3" stretchy="false" xref="S7.I5.i2.p1.2.m2.1.1.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S7.I5.i2.p1.2.m2.1b"><apply id="S7.I5.i2.p1.2.m2.1.1.cmml" xref="S7.I5.i2.p1.2.m2.1.1"><times id="S7.I5.i2.p1.2.m2.1.1.2.cmml" xref="S7.I5.i2.p1.2.m2.1.1.2"></times><apply id="S7.I5.i2.p1.2.m2.1.1.3.cmml" xref="S7.I5.i2.p1.2.m2.1.1.3"><csymbol cd="ambiguous" id="S7.I5.i2.p1.2.m2.1.1.3.1.cmml" xref="S7.I5.i2.p1.2.m2.1.1.3">subscript</csymbol><ci id="S7.I5.i2.p1.2.m2.1.1.3.2a.cmml" xref="S7.I5.i2.p1.2.m2.1.1.3.2"><mtext class="ltx_mathvariant_italic" id="S7.I5.i2.p1.2.m2.1.1.3.2.cmml" xref="S7.I5.i2.p1.2.m2.1.1.3.2">g</mtext></ci><ci id="S7.I5.i2.p1.2.m2.1.1.3.3.cmml" xref="S7.I5.i2.p1.2.m2.1.1.3.3">𝑠</ci></apply><apply id="S7.I5.i2.p1.2.m2.1.1.1.1.1.cmml" xref="S7.I5.i2.p1.2.m2.1.1.1.1"><ci id="S7.I5.i2.p1.2.m2.1.1.1.1.1.1.cmml" xref="S7.I5.i2.p1.2.m2.1.1.1.1.1.1">→</ci><ci id="S7.I5.i2.p1.2.m2.1.1.1.1.1.2.cmml" xref="S7.I5.i2.p1.2.m2.1.1.1.1.1.2">𝑢</ci><ci id="S7.I5.i2.p1.2.m2.1.1.1.1.1.3.cmml" xref="S7.I5.i2.p1.2.m2.1.1.1.1.1.3">𝑣</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.I5.i2.p1.2.m2.1c">\textsl{g}_{s}(u\!\to\!v)</annotation><annotation encoding="application/x-llamapun" id="S7.I5.i2.p1.2.m2.1d">g start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_u → italic_v )</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S7.I5.i2.p1.4.3"> the out-degree from </span><math alttext="u" class="ltx_Math" display="inline" id="S7.I5.i2.p1.3.m3.1"><semantics id="S7.I5.i2.p1.3.m3.1a"><mi id="S7.I5.i2.p1.3.m3.1.1" xref="S7.I5.i2.p1.3.m3.1.1.cmml">u</mi><annotation-xml encoding="MathML-Content" id="S7.I5.i2.p1.3.m3.1b"><ci id="S7.I5.i2.p1.3.m3.1.1.cmml" xref="S7.I5.i2.p1.3.m3.1.1">𝑢</ci></annotation-xml><annotation encoding="application/x-tex" id="S7.I5.i2.p1.3.m3.1c">u</annotation><annotation encoding="application/x-llamapun" id="S7.I5.i2.p1.3.m3.1d">italic_u</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S7.I5.i2.p1.4.4"> to </span><math alttext="v" class="ltx_Math" display="inline" id="S7.I5.i2.p1.4.m4.1"><semantics id="S7.I5.i2.p1.4.m4.1a"><mi id="S7.I5.i2.p1.4.m4.1.1" xref="S7.I5.i2.p1.4.m4.1.1.cmml">v</mi><annotation-xml encoding="MathML-Content" id="S7.I5.i2.p1.4.m4.1b"><ci id="S7.I5.i2.p1.4.m4.1.1.cmml" xref="S7.I5.i2.p1.4.m4.1.1">𝑣</ci></annotation-xml><annotation encoding="application/x-tex" id="S7.I5.i2.p1.4.m4.1c">v</annotation><annotation encoding="application/x-llamapun" id="S7.I5.i2.p1.4.m4.1d">italic_v</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S7.I5.i2.p1.4.5"> at the start of the </span><span class="ltx_text ltx_font_smallcaps" id="S7.I5.i2.p1.4.6">second</span><span class="ltx_text ltx_font_italic" id="S7.I5.i2.p1.4.7">.</span></p> </div> </li> <li class="ltx_item" id="S7.I5.i3" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">•</span> <div class="ltx_para" id="S7.I5.i3.p1"> <p class="ltx_p" id="S7.I5.i3.p1.6"><span class="ltx_text ltx_font_italic" id="S7.I5.i3.p1.6.1">We define the edge set </span><math alttext="E_{s}" class="ltx_Math" display="inline" id="S7.I5.i3.p1.1.m1.1"><semantics id="S7.I5.i3.p1.1.m1.1a"><msub id="S7.I5.i3.p1.1.m1.1.1" xref="S7.I5.i3.p1.1.m1.1.1.cmml"><mi id="S7.I5.i3.p1.1.m1.1.1.2" xref="S7.I5.i3.p1.1.m1.1.1.2.cmml">E</mi><mi id="S7.I5.i3.p1.1.m1.1.1.3" xref="S7.I5.i3.p1.1.m1.1.1.3.cmml">s</mi></msub><annotation-xml encoding="MathML-Content" id="S7.I5.i3.p1.1.m1.1b"><apply id="S7.I5.i3.p1.1.m1.1.1.cmml" xref="S7.I5.i3.p1.1.m1.1.1"><csymbol cd="ambiguous" id="S7.I5.i3.p1.1.m1.1.1.1.cmml" xref="S7.I5.i3.p1.1.m1.1.1">subscript</csymbol><ci id="S7.I5.i3.p1.1.m1.1.1.2.cmml" xref="S7.I5.i3.p1.1.m1.1.1.2">𝐸</ci><ci id="S7.I5.i3.p1.1.m1.1.1.3.cmml" xref="S7.I5.i3.p1.1.m1.1.1.3">𝑠</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.I5.i3.p1.1.m1.1c">E_{s}</annotation><annotation encoding="application/x-llamapun" id="S7.I5.i3.p1.1.m1.1d">italic_E start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S7.I5.i3.p1.6.2"> as all violating in-edges to vertices in </span><math alttext="T_{m}" class="ltx_Math" display="inline" id="S7.I5.i3.p1.2.m2.1"><semantics id="S7.I5.i3.p1.2.m2.1a"><msub id="S7.I5.i3.p1.2.m2.1.1" xref="S7.I5.i3.p1.2.m2.1.1.cmml"><mi id="S7.I5.i3.p1.2.m2.1.1.2" xref="S7.I5.i3.p1.2.m2.1.1.2.cmml">T</mi><mi id="S7.I5.i3.p1.2.m2.1.1.3" xref="S7.I5.i3.p1.2.m2.1.1.3.cmml">m</mi></msub><annotation-xml encoding="MathML-Content" id="S7.I5.i3.p1.2.m2.1b"><apply id="S7.I5.i3.p1.2.m2.1.1.cmml" xref="S7.I5.i3.p1.2.m2.1.1"><csymbol cd="ambiguous" id="S7.I5.i3.p1.2.m2.1.1.1.cmml" xref="S7.I5.i3.p1.2.m2.1.1">subscript</csymbol><ci id="S7.I5.i3.p1.2.m2.1.1.2.cmml" xref="S7.I5.i3.p1.2.m2.1.1.2">𝑇</ci><ci id="S7.I5.i3.p1.2.m2.1.1.3.cmml" xref="S7.I5.i3.p1.2.m2.1.1.3">𝑚</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.I5.i3.p1.2.m2.1c">T_{m}</annotation><annotation encoding="application/x-llamapun" id="S7.I5.i3.p1.2.m2.1d">italic_T start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S7.I5.i3.p1.6.3"> plus all </span><math alttext="\overline{uv}" class="ltx_Math" display="inline" id="S7.I5.i3.p1.3.m3.1"><semantics id="S7.I5.i3.p1.3.m3.1a"><mover accent="true" id="S7.I5.i3.p1.3.m3.1.1" xref="S7.I5.i3.p1.3.m3.1.1.cmml"><mrow id="S7.I5.i3.p1.3.m3.1.1.2" xref="S7.I5.i3.p1.3.m3.1.1.2.cmml"><mi id="S7.I5.i3.p1.3.m3.1.1.2.2" xref="S7.I5.i3.p1.3.m3.1.1.2.2.cmml">u</mi><mo id="S7.I5.i3.p1.3.m3.1.1.2.1" xref="S7.I5.i3.p1.3.m3.1.1.2.1.cmml">⁢</mo><mi id="S7.I5.i3.p1.3.m3.1.1.2.3" xref="S7.I5.i3.p1.3.m3.1.1.2.3.cmml">v</mi></mrow><mo id="S7.I5.i3.p1.3.m3.1.1.1" xref="S7.I5.i3.p1.3.m3.1.1.1.cmml">¯</mo></mover><annotation-xml encoding="MathML-Content" id="S7.I5.i3.p1.3.m3.1b"><apply id="S7.I5.i3.p1.3.m3.1.1.cmml" xref="S7.I5.i3.p1.3.m3.1.1"><ci id="S7.I5.i3.p1.3.m3.1.1.1.cmml" xref="S7.I5.i3.p1.3.m3.1.1.1">¯</ci><apply id="S7.I5.i3.p1.3.m3.1.1.2.cmml" xref="S7.I5.i3.p1.3.m3.1.1.2"><times id="S7.I5.i3.p1.3.m3.1.1.2.1.cmml" xref="S7.I5.i3.p1.3.m3.1.1.2.1"></times><ci id="S7.I5.i3.p1.3.m3.1.1.2.2.cmml" xref="S7.I5.i3.p1.3.m3.1.1.2.2">𝑢</ci><ci id="S7.I5.i3.p1.3.m3.1.1.2.3.cmml" xref="S7.I5.i3.p1.3.m3.1.1.2.3">𝑣</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.I5.i3.p1.3.m3.1c">\overline{uv}</annotation><annotation encoding="application/x-llamapun" id="S7.I5.i3.p1.3.m3.1d">over¯ start_ARG italic_u italic_v end_ARG</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S7.I5.i3.p1.6.4"> with: </span> <br class="ltx_break"/><math alttext="l_{s}(u)=l_{m}(u)>h+1" class="ltx_Math" display="inline" id="S7.I5.i3.p1.4.m4.2"><semantics id="S7.I5.i3.p1.4.m4.2a"><mrow id="S7.I5.i3.p1.4.m4.2.3" xref="S7.I5.i3.p1.4.m4.2.3.cmml"><mrow id="S7.I5.i3.p1.4.m4.2.3.2" xref="S7.I5.i3.p1.4.m4.2.3.2.cmml"><msub id="S7.I5.i3.p1.4.m4.2.3.2.2" xref="S7.I5.i3.p1.4.m4.2.3.2.2.cmml"><mi id="S7.I5.i3.p1.4.m4.2.3.2.2.2" xref="S7.I5.i3.p1.4.m4.2.3.2.2.2.cmml">l</mi><mi id="S7.I5.i3.p1.4.m4.2.3.2.2.3" xref="S7.I5.i3.p1.4.m4.2.3.2.2.3.cmml">s</mi></msub><mo id="S7.I5.i3.p1.4.m4.2.3.2.1" xref="S7.I5.i3.p1.4.m4.2.3.2.1.cmml">⁢</mo><mrow id="S7.I5.i3.p1.4.m4.2.3.2.3.2" xref="S7.I5.i3.p1.4.m4.2.3.2.cmml"><mo id="S7.I5.i3.p1.4.m4.2.3.2.3.2.1" stretchy="false" xref="S7.I5.i3.p1.4.m4.2.3.2.cmml">(</mo><mi id="S7.I5.i3.p1.4.m4.1.1" xref="S7.I5.i3.p1.4.m4.1.1.cmml">u</mi><mo id="S7.I5.i3.p1.4.m4.2.3.2.3.2.2" stretchy="false" xref="S7.I5.i3.p1.4.m4.2.3.2.cmml">)</mo></mrow></mrow><mo id="S7.I5.i3.p1.4.m4.2.3.3" xref="S7.I5.i3.p1.4.m4.2.3.3.cmml">=</mo><mrow id="S7.I5.i3.p1.4.m4.2.3.4" xref="S7.I5.i3.p1.4.m4.2.3.4.cmml"><msub id="S7.I5.i3.p1.4.m4.2.3.4.2" xref="S7.I5.i3.p1.4.m4.2.3.4.2.cmml"><mi id="S7.I5.i3.p1.4.m4.2.3.4.2.2" xref="S7.I5.i3.p1.4.m4.2.3.4.2.2.cmml">l</mi><mi id="S7.I5.i3.p1.4.m4.2.3.4.2.3" xref="S7.I5.i3.p1.4.m4.2.3.4.2.3.cmml">m</mi></msub><mo id="S7.I5.i3.p1.4.m4.2.3.4.1" xref="S7.I5.i3.p1.4.m4.2.3.4.1.cmml">⁢</mo><mrow id="S7.I5.i3.p1.4.m4.2.3.4.3.2" xref="S7.I5.i3.p1.4.m4.2.3.4.cmml"><mo id="S7.I5.i3.p1.4.m4.2.3.4.3.2.1" stretchy="false" xref="S7.I5.i3.p1.4.m4.2.3.4.cmml">(</mo><mi id="S7.I5.i3.p1.4.m4.2.2" xref="S7.I5.i3.p1.4.m4.2.2.cmml">u</mi><mo id="S7.I5.i3.p1.4.m4.2.3.4.3.2.2" stretchy="false" xref="S7.I5.i3.p1.4.m4.2.3.4.cmml">)</mo></mrow></mrow><mo id="S7.I5.i3.p1.4.m4.2.3.5" xref="S7.I5.i3.p1.4.m4.2.3.5.cmml">></mo><mrow 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id="S7.I5.i3.p1.4.m4.2.3.2.2.2.cmml" xref="S7.I5.i3.p1.4.m4.2.3.2.2.2">𝑙</ci><ci id="S7.I5.i3.p1.4.m4.2.3.2.2.3.cmml" xref="S7.I5.i3.p1.4.m4.2.3.2.2.3">𝑠</ci></apply><ci id="S7.I5.i3.p1.4.m4.1.1.cmml" xref="S7.I5.i3.p1.4.m4.1.1">𝑢</ci></apply><apply id="S7.I5.i3.p1.4.m4.2.3.4.cmml" xref="S7.I5.i3.p1.4.m4.2.3.4"><times id="S7.I5.i3.p1.4.m4.2.3.4.1.cmml" xref="S7.I5.i3.p1.4.m4.2.3.4.1"></times><apply id="S7.I5.i3.p1.4.m4.2.3.4.2.cmml" xref="S7.I5.i3.p1.4.m4.2.3.4.2"><csymbol cd="ambiguous" id="S7.I5.i3.p1.4.m4.2.3.4.2.1.cmml" xref="S7.I5.i3.p1.4.m4.2.3.4.2">subscript</csymbol><ci id="S7.I5.i3.p1.4.m4.2.3.4.2.2.cmml" xref="S7.I5.i3.p1.4.m4.2.3.4.2.2">𝑙</ci><ci id="S7.I5.i3.p1.4.m4.2.3.4.2.3.cmml" xref="S7.I5.i3.p1.4.m4.2.3.4.2.3">𝑚</ci></apply><ci id="S7.I5.i3.p1.4.m4.2.2.cmml" xref="S7.I5.i3.p1.4.m4.2.2">𝑢</ci></apply></apply><apply id="S7.I5.i3.p1.4.m4.2.3c.cmml" xref="S7.I5.i3.p1.4.m4.2.3"><gt id="S7.I5.i3.p1.4.m4.2.3.5.cmml" xref="S7.I5.i3.p1.4.m4.2.3.5"></gt><share href="https://arxiv.org/html/2411.12694v2#S7.I5.i3.p1.4.m4.2.3.4.cmml" id="S7.I5.i3.p1.4.m4.2.3d.cmml" xref="S7.I5.i3.p1.4.m4.2.3"></share><apply id="S7.I5.i3.p1.4.m4.2.3.6.cmml" xref="S7.I5.i3.p1.4.m4.2.3.6"><plus id="S7.I5.i3.p1.4.m4.2.3.6.1.cmml" xref="S7.I5.i3.p1.4.m4.2.3.6.1"></plus><ci id="S7.I5.i3.p1.4.m4.2.3.6.2.cmml" xref="S7.I5.i3.p1.4.m4.2.3.6.2">ℎ</ci><cn id="S7.I5.i3.p1.4.m4.2.3.6.3.cmml" type="integer" xref="S7.I5.i3.p1.4.m4.2.3.6.3">1</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.I5.i3.p1.4.m4.2c">l_{s}(u)=l_{m}(u)>h+1</annotation><annotation encoding="application/x-llamapun" id="S7.I5.i3.p1.4.m4.2d">italic_l start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_u ) = italic_l start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT ( italic_u ) > italic_h + 1</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S7.I5.i3.p1.6.5">, </span><math alttext="l_{s}(u)>l_{s}(v)" class="ltx_Math" display="inline" id="S7.I5.i3.p1.5.m5.2"><semantics id="S7.I5.i3.p1.5.m5.2a"><mrow id="S7.I5.i3.p1.5.m5.2.3" xref="S7.I5.i3.p1.5.m5.2.3.cmml"><mrow id="S7.I5.i3.p1.5.m5.2.3.2" xref="S7.I5.i3.p1.5.m5.2.3.2.cmml"><msub id="S7.I5.i3.p1.5.m5.2.3.2.2" xref="S7.I5.i3.p1.5.m5.2.3.2.2.cmml"><mi id="S7.I5.i3.p1.5.m5.2.3.2.2.2" xref="S7.I5.i3.p1.5.m5.2.3.2.2.2.cmml">l</mi><mi id="S7.I5.i3.p1.5.m5.2.3.2.2.3" xref="S7.I5.i3.p1.5.m5.2.3.2.2.3.cmml">s</mi></msub><mo id="S7.I5.i3.p1.5.m5.2.3.2.1" xref="S7.I5.i3.p1.5.m5.2.3.2.1.cmml">⁢</mo><mrow id="S7.I5.i3.p1.5.m5.2.3.2.3.2" xref="S7.I5.i3.p1.5.m5.2.3.2.cmml"><mo id="S7.I5.i3.p1.5.m5.2.3.2.3.2.1" stretchy="false" xref="S7.I5.i3.p1.5.m5.2.3.2.cmml">(</mo><mi id="S7.I5.i3.p1.5.m5.1.1" xref="S7.I5.i3.p1.5.m5.1.1.cmml">u</mi><mo id="S7.I5.i3.p1.5.m5.2.3.2.3.2.2" stretchy="false" xref="S7.I5.i3.p1.5.m5.2.3.2.cmml">)</mo></mrow></mrow><mo id="S7.I5.i3.p1.5.m5.2.3.1" xref="S7.I5.i3.p1.5.m5.2.3.1.cmml">></mo><mrow id="S7.I5.i3.p1.5.m5.2.3.3" xref="S7.I5.i3.p1.5.m5.2.3.3.cmml"><msub id="S7.I5.i3.p1.5.m5.2.3.3.2" xref="S7.I5.i3.p1.5.m5.2.3.3.2.cmml"><mi id="S7.I5.i3.p1.5.m5.2.3.3.2.2" xref="S7.I5.i3.p1.5.m5.2.3.3.2.2.cmml">l</mi><mi id="S7.I5.i3.p1.5.m5.2.3.3.2.3" xref="S7.I5.i3.p1.5.m5.2.3.3.2.3.cmml">s</mi></msub><mo id="S7.I5.i3.p1.5.m5.2.3.3.1" xref="S7.I5.i3.p1.5.m5.2.3.3.1.cmml">⁢</mo><mrow id="S7.I5.i3.p1.5.m5.2.3.3.3.2" xref="S7.I5.i3.p1.5.m5.2.3.3.cmml"><mo id="S7.I5.i3.p1.5.m5.2.3.3.3.2.1" stretchy="false" xref="S7.I5.i3.p1.5.m5.2.3.3.cmml">(</mo><mi id="S7.I5.i3.p1.5.m5.2.2" xref="S7.I5.i3.p1.5.m5.2.2.cmml">v</mi><mo id="S7.I5.i3.p1.5.m5.2.3.3.3.2.2" stretchy="false" xref="S7.I5.i3.p1.5.m5.2.3.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S7.I5.i3.p1.5.m5.2b"><apply id="S7.I5.i3.p1.5.m5.2.3.cmml" xref="S7.I5.i3.p1.5.m5.2.3"><gt id="S7.I5.i3.p1.5.m5.2.3.1.cmml" xref="S7.I5.i3.p1.5.m5.2.3.1"></gt><apply id="S7.I5.i3.p1.5.m5.2.3.2.cmml" xref="S7.I5.i3.p1.5.m5.2.3.2"><times id="S7.I5.i3.p1.5.m5.2.3.2.1.cmml" xref="S7.I5.i3.p1.5.m5.2.3.2.1"></times><apply id="S7.I5.i3.p1.5.m5.2.3.2.2.cmml" xref="S7.I5.i3.p1.5.m5.2.3.2.2"><csymbol cd="ambiguous" id="S7.I5.i3.p1.5.m5.2.3.2.2.1.cmml" xref="S7.I5.i3.p1.5.m5.2.3.2.2">subscript</csymbol><ci id="S7.I5.i3.p1.5.m5.2.3.2.2.2.cmml" xref="S7.I5.i3.p1.5.m5.2.3.2.2.2">𝑙</ci><ci id="S7.I5.i3.p1.5.m5.2.3.2.2.3.cmml" xref="S7.I5.i3.p1.5.m5.2.3.2.2.3">𝑠</ci></apply><ci id="S7.I5.i3.p1.5.m5.1.1.cmml" xref="S7.I5.i3.p1.5.m5.1.1">𝑢</ci></apply><apply id="S7.I5.i3.p1.5.m5.2.3.3.cmml" xref="S7.I5.i3.p1.5.m5.2.3.3"><times id="S7.I5.i3.p1.5.m5.2.3.3.1.cmml" xref="S7.I5.i3.p1.5.m5.2.3.3.1"></times><apply id="S7.I5.i3.p1.5.m5.2.3.3.2.cmml" xref="S7.I5.i3.p1.5.m5.2.3.3.2"><csymbol cd="ambiguous" id="S7.I5.i3.p1.5.m5.2.3.3.2.1.cmml" xref="S7.I5.i3.p1.5.m5.2.3.3.2">subscript</csymbol><ci id="S7.I5.i3.p1.5.m5.2.3.3.2.2.cmml" xref="S7.I5.i3.p1.5.m5.2.3.3.2.2">𝑙</ci><ci id="S7.I5.i3.p1.5.m5.2.3.3.2.3.cmml" xref="S7.I5.i3.p1.5.m5.2.3.3.2.3">𝑠</ci></apply><ci id="S7.I5.i3.p1.5.m5.2.2.cmml" xref="S7.I5.i3.p1.5.m5.2.2">𝑣</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.I5.i3.p1.5.m5.2c">l_{s}(u)>l_{s}(v)</annotation><annotation encoding="application/x-llamapun" id="S7.I5.i3.p1.5.m5.2d">italic_l start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_u ) > italic_l start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_v )</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S7.I5.i3.p1.6.6">, and </span><math alttext="\textsl{g}_{s}(u\!\to\!v)>0" class="ltx_Math" display="inline" id="S7.I5.i3.p1.6.m6.1"><semantics id="S7.I5.i3.p1.6.m6.1a"><mrow id="S7.I5.i3.p1.6.m6.1.1" xref="S7.I5.i3.p1.6.m6.1.1.cmml"><mrow id="S7.I5.i3.p1.6.m6.1.1.1" xref="S7.I5.i3.p1.6.m6.1.1.1.cmml"><msub id="S7.I5.i3.p1.6.m6.1.1.1.3" xref="S7.I5.i3.p1.6.m6.1.1.1.3.cmml"><mtext class="ltx_mathvariant_italic" id="S7.I5.i3.p1.6.m6.1.1.1.3.2" xref="S7.I5.i3.p1.6.m6.1.1.1.3.2a.cmml">g</mtext><mi id="S7.I5.i3.p1.6.m6.1.1.1.3.3" xref="S7.I5.i3.p1.6.m6.1.1.1.3.3.cmml">s</mi></msub><mo id="S7.I5.i3.p1.6.m6.1.1.1.2" xref="S7.I5.i3.p1.6.m6.1.1.1.2.cmml">⁢</mo><mrow id="S7.I5.i3.p1.6.m6.1.1.1.1.1" xref="S7.I5.i3.p1.6.m6.1.1.1.1.1.1.cmml"><mo id="S7.I5.i3.p1.6.m6.1.1.1.1.1.2" stretchy="false" xref="S7.I5.i3.p1.6.m6.1.1.1.1.1.1.cmml">(</mo><mrow id="S7.I5.i3.p1.6.m6.1.1.1.1.1.1" xref="S7.I5.i3.p1.6.m6.1.1.1.1.1.1.cmml"><mi id="S7.I5.i3.p1.6.m6.1.1.1.1.1.1.2" xref="S7.I5.i3.p1.6.m6.1.1.1.1.1.1.2.cmml">u</mi><mo id="S7.I5.i3.p1.6.m6.1.1.1.1.1.1.1" lspace="0.108em" rspace="0.108em" stretchy="false" xref="S7.I5.i3.p1.6.m6.1.1.1.1.1.1.1.cmml">→</mo><mi id="S7.I5.i3.p1.6.m6.1.1.1.1.1.1.3" xref="S7.I5.i3.p1.6.m6.1.1.1.1.1.1.3.cmml">v</mi></mrow><mo id="S7.I5.i3.p1.6.m6.1.1.1.1.1.3" stretchy="false" xref="S7.I5.i3.p1.6.m6.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S7.I5.i3.p1.6.m6.1.1.2" xref="S7.I5.i3.p1.6.m6.1.1.2.cmml">></mo><mn id="S7.I5.i3.p1.6.m6.1.1.3" xref="S7.I5.i3.p1.6.m6.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S7.I5.i3.p1.6.m6.1b"><apply id="S7.I5.i3.p1.6.m6.1.1.cmml" xref="S7.I5.i3.p1.6.m6.1.1"><gt id="S7.I5.i3.p1.6.m6.1.1.2.cmml" xref="S7.I5.i3.p1.6.m6.1.1.2"></gt><apply id="S7.I5.i3.p1.6.m6.1.1.1.cmml" xref="S7.I5.i3.p1.6.m6.1.1.1"><times id="S7.I5.i3.p1.6.m6.1.1.1.2.cmml" xref="S7.I5.i3.p1.6.m6.1.1.1.2"></times><apply id="S7.I5.i3.p1.6.m6.1.1.1.3.cmml" xref="S7.I5.i3.p1.6.m6.1.1.1.3"><csymbol cd="ambiguous" id="S7.I5.i3.p1.6.m6.1.1.1.3.1.cmml" xref="S7.I5.i3.p1.6.m6.1.1.1.3">subscript</csymbol><ci id="S7.I5.i3.p1.6.m6.1.1.1.3.2a.cmml" xref="S7.I5.i3.p1.6.m6.1.1.1.3.2"><mtext class="ltx_mathvariant_italic" id="S7.I5.i3.p1.6.m6.1.1.1.3.2.cmml" xref="S7.I5.i3.p1.6.m6.1.1.1.3.2">g</mtext></ci><ci id="S7.I5.i3.p1.6.m6.1.1.1.3.3.cmml" xref="S7.I5.i3.p1.6.m6.1.1.1.3.3">𝑠</ci></apply><apply id="S7.I5.i3.p1.6.m6.1.1.1.1.1.1.cmml" xref="S7.I5.i3.p1.6.m6.1.1.1.1.1"><ci id="S7.I5.i3.p1.6.m6.1.1.1.1.1.1.1.cmml" xref="S7.I5.i3.p1.6.m6.1.1.1.1.1.1.1">→</ci><ci id="S7.I5.i3.p1.6.m6.1.1.1.1.1.1.2.cmml" xref="S7.I5.i3.p1.6.m6.1.1.1.1.1.1.2">𝑢</ci><ci id="S7.I5.i3.p1.6.m6.1.1.1.1.1.1.3.cmml" xref="S7.I5.i3.p1.6.m6.1.1.1.1.1.1.3">𝑣</ci></apply></apply><cn id="S7.I5.i3.p1.6.m6.1.1.3.cmml" type="integer" xref="S7.I5.i3.p1.6.m6.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.I5.i3.p1.6.m6.1c">\textsl{g}_{s}(u\!\to\!v)>0</annotation><annotation encoding="application/x-llamapun" id="S7.I5.i3.p1.6.m6.1d">g start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_u → italic_v ) > 0</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S7.I5.i3.p1.6.7">.</span></p> </div> </li> <li class="ltx_item" id="S7.I5.i4" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">•</span> <div class="ltx_para" id="S7.I5.i4.p1"> <p class="ltx_p" id="S7.I5.i4.p1.2"><math alttext="V_{s}" class="ltx_Math" display="inline" id="S7.I5.i4.p1.1.m1.1"><semantics id="S7.I5.i4.p1.1.m1.1a"><msub id="S7.I5.i4.p1.1.m1.1.1" xref="S7.I5.i4.p1.1.m1.1.1.cmml"><mi id="S7.I5.i4.p1.1.m1.1.1.2" xref="S7.I5.i4.p1.1.m1.1.1.2.cmml">V</mi><mi id="S7.I5.i4.p1.1.m1.1.1.3" xref="S7.I5.i4.p1.1.m1.1.1.3.cmml">s</mi></msub><annotation-xml encoding="MathML-Content" id="S7.I5.i4.p1.1.m1.1b"><apply id="S7.I5.i4.p1.1.m1.1.1.cmml" xref="S7.I5.i4.p1.1.m1.1.1"><csymbol cd="ambiguous" id="S7.I5.i4.p1.1.m1.1.1.1.cmml" xref="S7.I5.i4.p1.1.m1.1.1">subscript</csymbol><ci id="S7.I5.i4.p1.1.m1.1.1.2.cmml" xref="S7.I5.i4.p1.1.m1.1.1.2">𝑉</ci><ci id="S7.I5.i4.p1.1.m1.1.1.3.cmml" xref="S7.I5.i4.p1.1.m1.1.1.3">𝑠</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.I5.i4.p1.1.m1.1c">V_{s}</annotation><annotation encoding="application/x-llamapun" id="S7.I5.i4.p1.1.m1.1d">italic_V start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S7.I5.i4.p1.2.1"> are all vertices with a directed path to a vertex in </span><math alttext="T_{m}" class="ltx_Math" display="inline" id="S7.I5.i4.p1.2.m2.1"><semantics id="S7.I5.i4.p1.2.m2.1a"><msub id="S7.I5.i4.p1.2.m2.1.1" xref="S7.I5.i4.p1.2.m2.1.1.cmml"><mi id="S7.I5.i4.p1.2.m2.1.1.2" xref="S7.I5.i4.p1.2.m2.1.1.2.cmml">T</mi><mi id="S7.I5.i4.p1.2.m2.1.1.3" xref="S7.I5.i4.p1.2.m2.1.1.3.cmml">m</mi></msub><annotation-xml encoding="MathML-Content" id="S7.I5.i4.p1.2.m2.1b"><apply id="S7.I5.i4.p1.2.m2.1.1.cmml" xref="S7.I5.i4.p1.2.m2.1.1"><csymbol cd="ambiguous" id="S7.I5.i4.p1.2.m2.1.1.1.cmml" xref="S7.I5.i4.p1.2.m2.1.1">subscript</csymbol><ci id="S7.I5.i4.p1.2.m2.1.1.2.cmml" xref="S7.I5.i4.p1.2.m2.1.1.2">𝑇</ci><ci id="S7.I5.i4.p1.2.m2.1.1.3.cmml" xref="S7.I5.i4.p1.2.m2.1.1.3">𝑚</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.I5.i4.p1.2.m2.1c">T_{m}</annotation><annotation encoding="application/x-llamapun" id="S7.I5.i4.p1.2.m2.1d">italic_T start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S7.I5.i4.p1.2.2">.</span></p> </div> </li> <li class="ltx_item" id="S7.I5.i5" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">•</span> <div class="ltx_para" id="S7.I5.i5.p1"> <p class="ltx_p" id="S7.I5.i5.p1.3"><math alttext="D_{s}=(V_{s},E_{s})" class="ltx_Math" display="inline" id="S7.I5.i5.p1.1.m1.2"><semantics id="S7.I5.i5.p1.1.m1.2a"><mrow id="S7.I5.i5.p1.1.m1.2.2" xref="S7.I5.i5.p1.1.m1.2.2.cmml"><msub id="S7.I5.i5.p1.1.m1.2.2.4" xref="S7.I5.i5.p1.1.m1.2.2.4.cmml"><mi id="S7.I5.i5.p1.1.m1.2.2.4.2" xref="S7.I5.i5.p1.1.m1.2.2.4.2.cmml">D</mi><mi id="S7.I5.i5.p1.1.m1.2.2.4.3" xref="S7.I5.i5.p1.1.m1.2.2.4.3.cmml">s</mi></msub><mo id="S7.I5.i5.p1.1.m1.2.2.3" xref="S7.I5.i5.p1.1.m1.2.2.3.cmml">=</mo><mrow id="S7.I5.i5.p1.1.m1.2.2.2.2" xref="S7.I5.i5.p1.1.m1.2.2.2.3.cmml"><mo id="S7.I5.i5.p1.1.m1.2.2.2.2.3" stretchy="false" xref="S7.I5.i5.p1.1.m1.2.2.2.3.cmml">(</mo><msub id="S7.I5.i5.p1.1.m1.1.1.1.1.1" xref="S7.I5.i5.p1.1.m1.1.1.1.1.1.cmml"><mi id="S7.I5.i5.p1.1.m1.1.1.1.1.1.2" xref="S7.I5.i5.p1.1.m1.1.1.1.1.1.2.cmml">V</mi><mi id="S7.I5.i5.p1.1.m1.1.1.1.1.1.3" xref="S7.I5.i5.p1.1.m1.1.1.1.1.1.3.cmml">s</mi></msub><mo id="S7.I5.i5.p1.1.m1.2.2.2.2.4" xref="S7.I5.i5.p1.1.m1.2.2.2.3.cmml">,</mo><msub id="S7.I5.i5.p1.1.m1.2.2.2.2.2" xref="S7.I5.i5.p1.1.m1.2.2.2.2.2.cmml"><mi id="S7.I5.i5.p1.1.m1.2.2.2.2.2.2" xref="S7.I5.i5.p1.1.m1.2.2.2.2.2.2.cmml">E</mi><mi id="S7.I5.i5.p1.1.m1.2.2.2.2.2.3" xref="S7.I5.i5.p1.1.m1.2.2.2.2.2.3.cmml">s</mi></msub><mo id="S7.I5.i5.p1.1.m1.2.2.2.2.5" stretchy="false" xref="S7.I5.i5.p1.1.m1.2.2.2.3.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S7.I5.i5.p1.1.m1.2b"><apply id="S7.I5.i5.p1.1.m1.2.2.cmml" xref="S7.I5.i5.p1.1.m1.2.2"><eq id="S7.I5.i5.p1.1.m1.2.2.3.cmml" xref="S7.I5.i5.p1.1.m1.2.2.3"></eq><apply id="S7.I5.i5.p1.1.m1.2.2.4.cmml" xref="S7.I5.i5.p1.1.m1.2.2.4"><csymbol cd="ambiguous" id="S7.I5.i5.p1.1.m1.2.2.4.1.cmml" xref="S7.I5.i5.p1.1.m1.2.2.4">subscript</csymbol><ci id="S7.I5.i5.p1.1.m1.2.2.4.2.cmml" xref="S7.I5.i5.p1.1.m1.2.2.4.2">𝐷</ci><ci id="S7.I5.i5.p1.1.m1.2.2.4.3.cmml" xref="S7.I5.i5.p1.1.m1.2.2.4.3">𝑠</ci></apply><interval closure="open" id="S7.I5.i5.p1.1.m1.2.2.2.3.cmml" xref="S7.I5.i5.p1.1.m1.2.2.2.2"><apply id="S7.I5.i5.p1.1.m1.1.1.1.1.1.cmml" xref="S7.I5.i5.p1.1.m1.1.1.1.1.1"><csymbol cd="ambiguous" id="S7.I5.i5.p1.1.m1.1.1.1.1.1.1.cmml" xref="S7.I5.i5.p1.1.m1.1.1.1.1.1">subscript</csymbol><ci id="S7.I5.i5.p1.1.m1.1.1.1.1.1.2.cmml" xref="S7.I5.i5.p1.1.m1.1.1.1.1.1.2">𝑉</ci><ci id="S7.I5.i5.p1.1.m1.1.1.1.1.1.3.cmml" xref="S7.I5.i5.p1.1.m1.1.1.1.1.1.3">𝑠</ci></apply><apply id="S7.I5.i5.p1.1.m1.2.2.2.2.2.cmml" xref="S7.I5.i5.p1.1.m1.2.2.2.2.2"><csymbol cd="ambiguous" id="S7.I5.i5.p1.1.m1.2.2.2.2.2.1.cmml" xref="S7.I5.i5.p1.1.m1.2.2.2.2.2">subscript</csymbol><ci id="S7.I5.i5.p1.1.m1.2.2.2.2.2.2.cmml" xref="S7.I5.i5.p1.1.m1.2.2.2.2.2.2">𝐸</ci><ci id="S7.I5.i5.p1.1.m1.2.2.2.2.2.3.cmml" xref="S7.I5.i5.p1.1.m1.2.2.2.2.2.3">𝑠</ci></apply></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.I5.i5.p1.1.m1.2c">D_{s}=(V_{s},E_{s})</annotation><annotation encoding="application/x-llamapun" id="S7.I5.i5.p1.1.m1.2d">italic_D start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT = ( italic_V start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT , italic_E start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT )</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S7.I5.i5.p1.3.1"> is a DAG where </span><math alttext="S_{s}" class="ltx_Math" display="inline" id="S7.I5.i5.p1.2.m2.1"><semantics id="S7.I5.i5.p1.2.m2.1a"><msub id="S7.I5.i5.p1.2.m2.1.1" xref="S7.I5.i5.p1.2.m2.1.1.cmml"><mi id="S7.I5.i5.p1.2.m2.1.1.2" xref="S7.I5.i5.p1.2.m2.1.1.2.cmml">S</mi><mi id="S7.I5.i5.p1.2.m2.1.1.3" xref="S7.I5.i5.p1.2.m2.1.1.3.cmml">s</mi></msub><annotation-xml encoding="MathML-Content" id="S7.I5.i5.p1.2.m2.1b"><apply id="S7.I5.i5.p1.2.m2.1.1.cmml" xref="S7.I5.i5.p1.2.m2.1.1"><csymbol cd="ambiguous" id="S7.I5.i5.p1.2.m2.1.1.1.cmml" xref="S7.I5.i5.p1.2.m2.1.1">subscript</csymbol><ci id="S7.I5.i5.p1.2.m2.1.1.2.cmml" xref="S7.I5.i5.p1.2.m2.1.1.2">𝑆</ci><ci id="S7.I5.i5.p1.2.m2.1.1.3.cmml" xref="S7.I5.i5.p1.2.m2.1.1.3">𝑠</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.I5.i5.p1.2.m2.1c">S_{s}</annotation><annotation encoding="application/x-llamapun" id="S7.I5.i5.p1.2.m2.1d">italic_S start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S7.I5.i5.p1.3.2"> are all the sources (per definition </span><math alttext="S_{s}\cap T_{m}=\emptyset" class="ltx_Math" display="inline" id="S7.I5.i5.p1.3.m3.1"><semantics id="S7.I5.i5.p1.3.m3.1a"><mrow id="S7.I5.i5.p1.3.m3.1.1" xref="S7.I5.i5.p1.3.m3.1.1.cmml"><mrow id="S7.I5.i5.p1.3.m3.1.1.2" xref="S7.I5.i5.p1.3.m3.1.1.2.cmml"><msub id="S7.I5.i5.p1.3.m3.1.1.2.2" xref="S7.I5.i5.p1.3.m3.1.1.2.2.cmml"><mi id="S7.I5.i5.p1.3.m3.1.1.2.2.2" xref="S7.I5.i5.p1.3.m3.1.1.2.2.2.cmml">S</mi><mi id="S7.I5.i5.p1.3.m3.1.1.2.2.3" xref="S7.I5.i5.p1.3.m3.1.1.2.2.3.cmml">s</mi></msub><mo id="S7.I5.i5.p1.3.m3.1.1.2.1" xref="S7.I5.i5.p1.3.m3.1.1.2.1.cmml">∩</mo><msub id="S7.I5.i5.p1.3.m3.1.1.2.3" xref="S7.I5.i5.p1.3.m3.1.1.2.3.cmml"><mi id="S7.I5.i5.p1.3.m3.1.1.2.3.2" xref="S7.I5.i5.p1.3.m3.1.1.2.3.2.cmml">T</mi><mi id="S7.I5.i5.p1.3.m3.1.1.2.3.3" xref="S7.I5.i5.p1.3.m3.1.1.2.3.3.cmml">m</mi></msub></mrow><mo id="S7.I5.i5.p1.3.m3.1.1.1" xref="S7.I5.i5.p1.3.m3.1.1.1.cmml">=</mo><mi id="S7.I5.i5.p1.3.m3.1.1.3" mathvariant="normal" xref="S7.I5.i5.p1.3.m3.1.1.3.cmml">∅</mi></mrow><annotation-xml encoding="MathML-Content" id="S7.I5.i5.p1.3.m3.1b"><apply id="S7.I5.i5.p1.3.m3.1.1.cmml" xref="S7.I5.i5.p1.3.m3.1.1"><eq id="S7.I5.i5.p1.3.m3.1.1.1.cmml" xref="S7.I5.i5.p1.3.m3.1.1.1"></eq><apply id="S7.I5.i5.p1.3.m3.1.1.2.cmml" xref="S7.I5.i5.p1.3.m3.1.1.2"><intersect id="S7.I5.i5.p1.3.m3.1.1.2.1.cmml" xref="S7.I5.i5.p1.3.m3.1.1.2.1"></intersect><apply id="S7.I5.i5.p1.3.m3.1.1.2.2.cmml" xref="S7.I5.i5.p1.3.m3.1.1.2.2"><csymbol cd="ambiguous" id="S7.I5.i5.p1.3.m3.1.1.2.2.1.cmml" xref="S7.I5.i5.p1.3.m3.1.1.2.2">subscript</csymbol><ci id="S7.I5.i5.p1.3.m3.1.1.2.2.2.cmml" xref="S7.I5.i5.p1.3.m3.1.1.2.2.2">𝑆</ci><ci id="S7.I5.i5.p1.3.m3.1.1.2.2.3.cmml" xref="S7.I5.i5.p1.3.m3.1.1.2.2.3">𝑠</ci></apply><apply id="S7.I5.i5.p1.3.m3.1.1.2.3.cmml" xref="S7.I5.i5.p1.3.m3.1.1.2.3"><csymbol cd="ambiguous" id="S7.I5.i5.p1.3.m3.1.1.2.3.1.cmml" xref="S7.I5.i5.p1.3.m3.1.1.2.3">subscript</csymbol><ci id="S7.I5.i5.p1.3.m3.1.1.2.3.2.cmml" xref="S7.I5.i5.p1.3.m3.1.1.2.3.2">𝑇</ci><ci id="S7.I5.i5.p1.3.m3.1.1.2.3.3.cmml" xref="S7.I5.i5.p1.3.m3.1.1.2.3.3">𝑚</ci></apply></apply><emptyset id="S7.I5.i5.p1.3.m3.1.1.3.cmml" xref="S7.I5.i5.p1.3.m3.1.1.3"></emptyset></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.I5.i5.p1.3.m3.1c">S_{s}\cap T_{m}=\emptyset</annotation><annotation encoding="application/x-llamapun" id="S7.I5.i5.p1.3.m3.1d">italic_S start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ∩ italic_T start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT = ∅</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S7.I5.i5.p1.3.3">).</span></p> </div> </li> </ul> </div> </div> <div class="ltx_pagination ltx_role_newpage"></div> <div class="ltx_theorem ltx_theorem_definition" id="S7.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="S7.Thmtheorem11.1.1.1">Definition 7.11</span></span><span class="ltx_text ltx_font_bold" id="S7.Thmtheorem11.2.2">.</span> </h6> <div class="ltx_para" id="S7.Thmtheorem11.p1"> <p class="ltx_p" id="S7.Thmtheorem11.p1.10"><span class="ltx_text ltx_font_italic" id="S7.Thmtheorem11.p1.10.10">At the start of <math alttext="(h,m,s)" class="ltx_Math" display="inline" id="S7.Thmtheorem11.p1.1.1.m1.3"><semantics id="S7.Thmtheorem11.p1.1.1.m1.3a"><mrow id="S7.Thmtheorem11.p1.1.1.m1.3.4.2" xref="S7.Thmtheorem11.p1.1.1.m1.3.4.1.cmml"><mo id="S7.Thmtheorem11.p1.1.1.m1.3.4.2.1" stretchy="false" xref="S7.Thmtheorem11.p1.1.1.m1.3.4.1.cmml">(</mo><mi id="S7.Thmtheorem11.p1.1.1.m1.1.1" xref="S7.Thmtheorem11.p1.1.1.m1.1.1.cmml">h</mi><mo id="S7.Thmtheorem11.p1.1.1.m1.3.4.2.2" xref="S7.Thmtheorem11.p1.1.1.m1.3.4.1.cmml">,</mo><mi id="S7.Thmtheorem11.p1.1.1.m1.2.2" xref="S7.Thmtheorem11.p1.1.1.m1.2.2.cmml">m</mi><mo id="S7.Thmtheorem11.p1.1.1.m1.3.4.2.3" xref="S7.Thmtheorem11.p1.1.1.m1.3.4.1.cmml">,</mo><mi id="S7.Thmtheorem11.p1.1.1.m1.3.3" xref="S7.Thmtheorem11.p1.1.1.m1.3.3.cmml">s</mi><mo id="S7.Thmtheorem11.p1.1.1.m1.3.4.2.4" stretchy="false" xref="S7.Thmtheorem11.p1.1.1.m1.3.4.1.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S7.Thmtheorem11.p1.1.1.m1.3b"><vector id="S7.Thmtheorem11.p1.1.1.m1.3.4.1.cmml" xref="S7.Thmtheorem11.p1.1.1.m1.3.4.2"><ci id="S7.Thmtheorem11.p1.1.1.m1.1.1.cmml" xref="S7.Thmtheorem11.p1.1.1.m1.1.1">ℎ</ci><ci id="S7.Thmtheorem11.p1.1.1.m1.2.2.cmml" xref="S7.Thmtheorem11.p1.1.1.m1.2.2">𝑚</ci><ci id="S7.Thmtheorem11.p1.1.1.m1.3.3.cmml" xref="S7.Thmtheorem11.p1.1.1.m1.3.3">𝑠</ci></vector></annotation-xml><annotation encoding="application/x-tex" id="S7.Thmtheorem11.p1.1.1.m1.3c">(h,m,s)</annotation><annotation encoding="application/x-llamapun" id="S7.Thmtheorem11.p1.1.1.m1.3d">( italic_h , italic_m , italic_s )</annotation></semantics></math> where <math alttext="m" class="ltx_Math" display="inline" id="S7.Thmtheorem11.p1.2.2.m2.1"><semantics id="S7.Thmtheorem11.p1.2.2.m2.1a"><mi id="S7.Thmtheorem11.p1.2.2.m2.1.1" xref="S7.Thmtheorem11.p1.2.2.m2.1.1.cmml">m</mi><annotation-xml encoding="MathML-Content" id="S7.Thmtheorem11.p1.2.2.m2.1b"><ci id="S7.Thmtheorem11.p1.2.2.m2.1.1.cmml" xref="S7.Thmtheorem11.p1.2.2.m2.1.1">𝑚</ci></annotation-xml><annotation encoding="application/x-tex" id="S7.Thmtheorem11.p1.2.2.m2.1c">m</annotation><annotation encoding="application/x-llamapun" id="S7.Thmtheorem11.p1.2.2.m2.1d">italic_m</annotation></semantics></math> is odd, given <math alttext="T_{m}" class="ltx_Math" display="inline" id="S7.Thmtheorem11.p1.3.3.m3.1"><semantics id="S7.Thmtheorem11.p1.3.3.m3.1a"><msub id="S7.Thmtheorem11.p1.3.3.m3.1.1" xref="S7.Thmtheorem11.p1.3.3.m3.1.1.cmml"><mi id="S7.Thmtheorem11.p1.3.3.m3.1.1.2" xref="S7.Thmtheorem11.p1.3.3.m3.1.1.2.cmml">T</mi><mi id="S7.Thmtheorem11.p1.3.3.m3.1.1.3" xref="S7.Thmtheorem11.p1.3.3.m3.1.1.3.cmml">m</mi></msub><annotation-xml encoding="MathML-Content" id="S7.Thmtheorem11.p1.3.3.m3.1b"><apply id="S7.Thmtheorem11.p1.3.3.m3.1.1.cmml" xref="S7.Thmtheorem11.p1.3.3.m3.1.1"><csymbol cd="ambiguous" id="S7.Thmtheorem11.p1.3.3.m3.1.1.1.cmml" xref="S7.Thmtheorem11.p1.3.3.m3.1.1">subscript</csymbol><ci id="S7.Thmtheorem11.p1.3.3.m3.1.1.2.cmml" xref="S7.Thmtheorem11.p1.3.3.m3.1.1.2">𝑇</ci><ci id="S7.Thmtheorem11.p1.3.3.m3.1.1.3.cmml" xref="S7.Thmtheorem11.p1.3.3.m3.1.1.3">𝑚</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.Thmtheorem11.p1.3.3.m3.1c">T_{m}</annotation><annotation encoding="application/x-llamapun" id="S7.Thmtheorem11.p1.3.3.m3.1d">italic_T start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT</annotation></semantics></math>, <math alttext="S_{s}" class="ltx_Math" display="inline" id="S7.Thmtheorem11.p1.4.4.m4.1"><semantics id="S7.Thmtheorem11.p1.4.4.m4.1a"><msub id="S7.Thmtheorem11.p1.4.4.m4.1.1" xref="S7.Thmtheorem11.p1.4.4.m4.1.1.cmml"><mi id="S7.Thmtheorem11.p1.4.4.m4.1.1.2" xref="S7.Thmtheorem11.p1.4.4.m4.1.1.2.cmml">S</mi><mi id="S7.Thmtheorem11.p1.4.4.m4.1.1.3" xref="S7.Thmtheorem11.p1.4.4.m4.1.1.3.cmml">s</mi></msub><annotation-xml encoding="MathML-Content" id="S7.Thmtheorem11.p1.4.4.m4.1b"><apply id="S7.Thmtheorem11.p1.4.4.m4.1.1.cmml" xref="S7.Thmtheorem11.p1.4.4.m4.1.1"><csymbol cd="ambiguous" id="S7.Thmtheorem11.p1.4.4.m4.1.1.1.cmml" xref="S7.Thmtheorem11.p1.4.4.m4.1.1">subscript</csymbol><ci id="S7.Thmtheorem11.p1.4.4.m4.1.1.2.cmml" xref="S7.Thmtheorem11.p1.4.4.m4.1.1.2">𝑆</ci><ci id="S7.Thmtheorem11.p1.4.4.m4.1.1.3.cmml" xref="S7.Thmtheorem11.p1.4.4.m4.1.1.3">𝑠</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.Thmtheorem11.p1.4.4.m4.1c">S_{s}</annotation><annotation encoding="application/x-llamapun" id="S7.Thmtheorem11.p1.4.4.m4.1d">italic_S start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="D_{s}" class="ltx_Math" display="inline" id="S7.Thmtheorem11.p1.5.5.m5.1"><semantics id="S7.Thmtheorem11.p1.5.5.m5.1a"><msub id="S7.Thmtheorem11.p1.5.5.m5.1.1" xref="S7.Thmtheorem11.p1.5.5.m5.1.1.cmml"><mi id="S7.Thmtheorem11.p1.5.5.m5.1.1.2" xref="S7.Thmtheorem11.p1.5.5.m5.1.1.2.cmml">D</mi><mi id="S7.Thmtheorem11.p1.5.5.m5.1.1.3" xref="S7.Thmtheorem11.p1.5.5.m5.1.1.3.cmml">s</mi></msub><annotation-xml encoding="MathML-Content" id="S7.Thmtheorem11.p1.5.5.m5.1b"><apply id="S7.Thmtheorem11.p1.5.5.m5.1.1.cmml" xref="S7.Thmtheorem11.p1.5.5.m5.1.1"><csymbol cd="ambiguous" id="S7.Thmtheorem11.p1.5.5.m5.1.1.1.cmml" xref="S7.Thmtheorem11.p1.5.5.m5.1.1">subscript</csymbol><ci id="S7.Thmtheorem11.p1.5.5.m5.1.1.2.cmml" xref="S7.Thmtheorem11.p1.5.5.m5.1.1.2">𝐷</ci><ci id="S7.Thmtheorem11.p1.5.5.m5.1.1.3.cmml" xref="S7.Thmtheorem11.p1.5.5.m5.1.1.3">𝑠</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.Thmtheorem11.p1.5.5.m5.1c">D_{s}</annotation><annotation encoding="application/x-llamapun" id="S7.Thmtheorem11.p1.5.5.m5.1d">italic_D start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT</annotation></semantics></math>, we define the DAG <math alttext="D_{s}^{*}" class="ltx_Math" display="inline" id="S7.Thmtheorem11.p1.6.6.m6.1"><semantics id="S7.Thmtheorem11.p1.6.6.m6.1a"><msubsup id="S7.Thmtheorem11.p1.6.6.m6.1.1" xref="S7.Thmtheorem11.p1.6.6.m6.1.1.cmml"><mi id="S7.Thmtheorem11.p1.6.6.m6.1.1.2.2" xref="S7.Thmtheorem11.p1.6.6.m6.1.1.2.2.cmml">D</mi><mi id="S7.Thmtheorem11.p1.6.6.m6.1.1.2.3" xref="S7.Thmtheorem11.p1.6.6.m6.1.1.2.3.cmml">s</mi><mo id="S7.Thmtheorem11.p1.6.6.m6.1.1.3" xref="S7.Thmtheorem11.p1.6.6.m6.1.1.3.cmml">∗</mo></msubsup><annotation-xml encoding="MathML-Content" id="S7.Thmtheorem11.p1.6.6.m6.1b"><apply id="S7.Thmtheorem11.p1.6.6.m6.1.1.cmml" xref="S7.Thmtheorem11.p1.6.6.m6.1.1"><csymbol cd="ambiguous" id="S7.Thmtheorem11.p1.6.6.m6.1.1.1.cmml" xref="S7.Thmtheorem11.p1.6.6.m6.1.1">superscript</csymbol><apply id="S7.Thmtheorem11.p1.6.6.m6.1.1.2.cmml" xref="S7.Thmtheorem11.p1.6.6.m6.1.1"><csymbol cd="ambiguous" id="S7.Thmtheorem11.p1.6.6.m6.1.1.2.1.cmml" xref="S7.Thmtheorem11.p1.6.6.m6.1.1">subscript</csymbol><ci id="S7.Thmtheorem11.p1.6.6.m6.1.1.2.2.cmml" xref="S7.Thmtheorem11.p1.6.6.m6.1.1.2.2">𝐷</ci><ci id="S7.Thmtheorem11.p1.6.6.m6.1.1.2.3.cmml" xref="S7.Thmtheorem11.p1.6.6.m6.1.1.2.3">𝑠</ci></apply><times id="S7.Thmtheorem11.p1.6.6.m6.1.1.3.cmml" xref="S7.Thmtheorem11.p1.6.6.m6.1.1.3"></times></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.Thmtheorem11.p1.6.6.m6.1c">D_{s}^{*}</annotation><annotation encoding="application/x-llamapun" id="S7.Thmtheorem11.p1.6.6.m6.1d">italic_D start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT</annotation></semantics></math> by connecting all <math alttext="u\in S_{s}" class="ltx_Math" display="inline" id="S7.Thmtheorem11.p1.7.7.m7.1"><semantics id="S7.Thmtheorem11.p1.7.7.m7.1a"><mrow id="S7.Thmtheorem11.p1.7.7.m7.1.1" xref="S7.Thmtheorem11.p1.7.7.m7.1.1.cmml"><mi id="S7.Thmtheorem11.p1.7.7.m7.1.1.2" xref="S7.Thmtheorem11.p1.7.7.m7.1.1.2.cmml">u</mi><mo id="S7.Thmtheorem11.p1.7.7.m7.1.1.1" xref="S7.Thmtheorem11.p1.7.7.m7.1.1.1.cmml">∈</mo><msub id="S7.Thmtheorem11.p1.7.7.m7.1.1.3" xref="S7.Thmtheorem11.p1.7.7.m7.1.1.3.cmml"><mi id="S7.Thmtheorem11.p1.7.7.m7.1.1.3.2" xref="S7.Thmtheorem11.p1.7.7.m7.1.1.3.2.cmml">S</mi><mi id="S7.Thmtheorem11.p1.7.7.m7.1.1.3.3" xref="S7.Thmtheorem11.p1.7.7.m7.1.1.3.3.cmml">s</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S7.Thmtheorem11.p1.7.7.m7.1b"><apply id="S7.Thmtheorem11.p1.7.7.m7.1.1.cmml" xref="S7.Thmtheorem11.p1.7.7.m7.1.1"><in id="S7.Thmtheorem11.p1.7.7.m7.1.1.1.cmml" xref="S7.Thmtheorem11.p1.7.7.m7.1.1.1"></in><ci id="S7.Thmtheorem11.p1.7.7.m7.1.1.2.cmml" xref="S7.Thmtheorem11.p1.7.7.m7.1.1.2">𝑢</ci><apply id="S7.Thmtheorem11.p1.7.7.m7.1.1.3.cmml" xref="S7.Thmtheorem11.p1.7.7.m7.1.1.3"><csymbol cd="ambiguous" id="S7.Thmtheorem11.p1.7.7.m7.1.1.3.1.cmml" xref="S7.Thmtheorem11.p1.7.7.m7.1.1.3">subscript</csymbol><ci id="S7.Thmtheorem11.p1.7.7.m7.1.1.3.2.cmml" xref="S7.Thmtheorem11.p1.7.7.m7.1.1.3.2">𝑆</ci><ci id="S7.Thmtheorem11.p1.7.7.m7.1.1.3.3.cmml" xref="S7.Thmtheorem11.p1.7.7.m7.1.1.3.3">𝑠</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.Thmtheorem11.p1.7.7.m7.1c">u\in S_{s}</annotation><annotation encoding="application/x-llamapun" id="S7.Thmtheorem11.p1.7.7.m7.1d">italic_u ∈ italic_S start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT</annotation></semantics></math> to a unique sink <math alttext="s_{u}" class="ltx_Math" display="inline" id="S7.Thmtheorem11.p1.8.8.m8.1"><semantics id="S7.Thmtheorem11.p1.8.8.m8.1a"><msub id="S7.Thmtheorem11.p1.8.8.m8.1.1" xref="S7.Thmtheorem11.p1.8.8.m8.1.1.cmml"><mi id="S7.Thmtheorem11.p1.8.8.m8.1.1.2" xref="S7.Thmtheorem11.p1.8.8.m8.1.1.2.cmml">s</mi><mi id="S7.Thmtheorem11.p1.8.8.m8.1.1.3" xref="S7.Thmtheorem11.p1.8.8.m8.1.1.3.cmml">u</mi></msub><annotation-xml encoding="MathML-Content" id="S7.Thmtheorem11.p1.8.8.m8.1b"><apply id="S7.Thmtheorem11.p1.8.8.m8.1.1.cmml" xref="S7.Thmtheorem11.p1.8.8.m8.1.1"><csymbol cd="ambiguous" id="S7.Thmtheorem11.p1.8.8.m8.1.1.1.cmml" xref="S7.Thmtheorem11.p1.8.8.m8.1.1">subscript</csymbol><ci id="S7.Thmtheorem11.p1.8.8.m8.1.1.2.cmml" xref="S7.Thmtheorem11.p1.8.8.m8.1.1.2">𝑠</ci><ci id="S7.Thmtheorem11.p1.8.8.m8.1.1.3.cmml" xref="S7.Thmtheorem11.p1.8.8.m8.1.1.3">𝑢</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.Thmtheorem11.p1.8.8.m8.1c">s_{u}</annotation><annotation encoding="application/x-llamapun" id="S7.Thmtheorem11.p1.8.8.m8.1d">italic_s start_POSTSUBSCRIPT italic_u end_POSTSUBSCRIPT</annotation></semantics></math> and all <math alttext="v\in T_{m}" class="ltx_Math" display="inline" id="S7.Thmtheorem11.p1.9.9.m9.1"><semantics id="S7.Thmtheorem11.p1.9.9.m9.1a"><mrow id="S7.Thmtheorem11.p1.9.9.m9.1.1" xref="S7.Thmtheorem11.p1.9.9.m9.1.1.cmml"><mi id="S7.Thmtheorem11.p1.9.9.m9.1.1.2" xref="S7.Thmtheorem11.p1.9.9.m9.1.1.2.cmml">v</mi><mo id="S7.Thmtheorem11.p1.9.9.m9.1.1.1" xref="S7.Thmtheorem11.p1.9.9.m9.1.1.1.cmml">∈</mo><msub id="S7.Thmtheorem11.p1.9.9.m9.1.1.3" xref="S7.Thmtheorem11.p1.9.9.m9.1.1.3.cmml"><mi id="S7.Thmtheorem11.p1.9.9.m9.1.1.3.2" xref="S7.Thmtheorem11.p1.9.9.m9.1.1.3.2.cmml">T</mi><mi id="S7.Thmtheorem11.p1.9.9.m9.1.1.3.3" xref="S7.Thmtheorem11.p1.9.9.m9.1.1.3.3.cmml">m</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S7.Thmtheorem11.p1.9.9.m9.1b"><apply id="S7.Thmtheorem11.p1.9.9.m9.1.1.cmml" xref="S7.Thmtheorem11.p1.9.9.m9.1.1"><in id="S7.Thmtheorem11.p1.9.9.m9.1.1.1.cmml" xref="S7.Thmtheorem11.p1.9.9.m9.1.1.1"></in><ci id="S7.Thmtheorem11.p1.9.9.m9.1.1.2.cmml" xref="S7.Thmtheorem11.p1.9.9.m9.1.1.2">𝑣</ci><apply id="S7.Thmtheorem11.p1.9.9.m9.1.1.3.cmml" xref="S7.Thmtheorem11.p1.9.9.m9.1.1.3"><csymbol cd="ambiguous" id="S7.Thmtheorem11.p1.9.9.m9.1.1.3.1.cmml" xref="S7.Thmtheorem11.p1.9.9.m9.1.1.3">subscript</csymbol><ci id="S7.Thmtheorem11.p1.9.9.m9.1.1.3.2.cmml" xref="S7.Thmtheorem11.p1.9.9.m9.1.1.3.2">𝑇</ci><ci id="S7.Thmtheorem11.p1.9.9.m9.1.1.3.3.cmml" xref="S7.Thmtheorem11.p1.9.9.m9.1.1.3.3">𝑚</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.Thmtheorem11.p1.9.9.m9.1c">v\in T_{m}</annotation><annotation encoding="application/x-llamapun" id="S7.Thmtheorem11.p1.9.9.m9.1d">italic_v ∈ italic_T start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT</annotation></semantics></math> to a unique sink <math alttext="t_{v}" class="ltx_Math" display="inline" id="S7.Thmtheorem11.p1.10.10.m10.1"><semantics id="S7.Thmtheorem11.p1.10.10.m10.1a"><msub id="S7.Thmtheorem11.p1.10.10.m10.1.1" xref="S7.Thmtheorem11.p1.10.10.m10.1.1.cmml"><mi id="S7.Thmtheorem11.p1.10.10.m10.1.1.2" xref="S7.Thmtheorem11.p1.10.10.m10.1.1.2.cmml">t</mi><mi id="S7.Thmtheorem11.p1.10.10.m10.1.1.3" xref="S7.Thmtheorem11.p1.10.10.m10.1.1.3.cmml">v</mi></msub><annotation-xml encoding="MathML-Content" id="S7.Thmtheorem11.p1.10.10.m10.1b"><apply id="S7.Thmtheorem11.p1.10.10.m10.1.1.cmml" xref="S7.Thmtheorem11.p1.10.10.m10.1.1"><csymbol cd="ambiguous" id="S7.Thmtheorem11.p1.10.10.m10.1.1.1.cmml" xref="S7.Thmtheorem11.p1.10.10.m10.1.1">subscript</csymbol><ci id="S7.Thmtheorem11.p1.10.10.m10.1.1.2.cmml" xref="S7.Thmtheorem11.p1.10.10.m10.1.1.2">𝑡</ci><ci id="S7.Thmtheorem11.p1.10.10.m10.1.1.3.cmml" xref="S7.Thmtheorem11.p1.10.10.m10.1.1.3">𝑣</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.Thmtheorem11.p1.10.10.m10.1c">t_{v}</annotation><annotation encoding="application/x-llamapun" id="S7.Thmtheorem11.p1.10.10.m10.1d">italic_t start_POSTSUBSCRIPT italic_v end_POSTSUBSCRIPT</annotation></semantics></math>.</span></p> <ul class="ltx_itemize" id="S7.I6"> <li class="ltx_item" id="S7.I6.i1" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">•</span> <div class="ltx_para" id="S7.I6.i1.p1"> <p class="ltx_p" id="S7.I6.i1.p1.3"><span class="ltx_text ltx_font_italic" id="S7.I6.i1.p1.3.1">For </span><math alttext="u\in S_{s}" class="ltx_Math" display="inline" id="S7.I6.i1.p1.1.m1.1"><semantics id="S7.I6.i1.p1.1.m1.1a"><mrow id="S7.I6.i1.p1.1.m1.1.1" xref="S7.I6.i1.p1.1.m1.1.1.cmml"><mi id="S7.I6.i1.p1.1.m1.1.1.2" xref="S7.I6.i1.p1.1.m1.1.1.2.cmml">u</mi><mo id="S7.I6.i1.p1.1.m1.1.1.1" xref="S7.I6.i1.p1.1.m1.1.1.1.cmml">∈</mo><msub id="S7.I6.i1.p1.1.m1.1.1.3" xref="S7.I6.i1.p1.1.m1.1.1.3.cmml"><mi id="S7.I6.i1.p1.1.m1.1.1.3.2" xref="S7.I6.i1.p1.1.m1.1.1.3.2.cmml">S</mi><mi id="S7.I6.i1.p1.1.m1.1.1.3.3" xref="S7.I6.i1.p1.1.m1.1.1.3.3.cmml">s</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S7.I6.i1.p1.1.m1.1b"><apply id="S7.I6.i1.p1.1.m1.1.1.cmml" xref="S7.I6.i1.p1.1.m1.1.1"><in id="S7.I6.i1.p1.1.m1.1.1.1.cmml" xref="S7.I6.i1.p1.1.m1.1.1.1"></in><ci id="S7.I6.i1.p1.1.m1.1.1.2.cmml" xref="S7.I6.i1.p1.1.m1.1.1.2">𝑢</ci><apply id="S7.I6.i1.p1.1.m1.1.1.3.cmml" xref="S7.I6.i1.p1.1.m1.1.1.3"><csymbol cd="ambiguous" id="S7.I6.i1.p1.1.m1.1.1.3.1.cmml" xref="S7.I6.i1.p1.1.m1.1.1.3">subscript</csymbol><ci id="S7.I6.i1.p1.1.m1.1.1.3.2.cmml" xref="S7.I6.i1.p1.1.m1.1.1.3.2">𝑆</ci><ci id="S7.I6.i1.p1.1.m1.1.1.3.3.cmml" xref="S7.I6.i1.p1.1.m1.1.1.3.3">𝑠</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.I6.i1.p1.1.m1.1c">u\in S_{s}</annotation><annotation encoding="application/x-llamapun" id="S7.I6.i1.p1.1.m1.1d">italic_u ∈ italic_S start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S7.I6.i1.p1.3.2">, the edge </span><math alttext="\overline{s_{u}u}" class="ltx_Math" display="inline" id="S7.I6.i1.p1.2.m2.1"><semantics id="S7.I6.i1.p1.2.m2.1a"><mover accent="true" id="S7.I6.i1.p1.2.m2.1.1" xref="S7.I6.i1.p1.2.m2.1.1.cmml"><mrow id="S7.I6.i1.p1.2.m2.1.1.2" xref="S7.I6.i1.p1.2.m2.1.1.2.cmml"><msub id="S7.I6.i1.p1.2.m2.1.1.2.2" xref="S7.I6.i1.p1.2.m2.1.1.2.2.cmml"><mi id="S7.I6.i1.p1.2.m2.1.1.2.2.2" xref="S7.I6.i1.p1.2.m2.1.1.2.2.2.cmml">s</mi><mi id="S7.I6.i1.p1.2.m2.1.1.2.2.3" xref="S7.I6.i1.p1.2.m2.1.1.2.2.3.cmml">u</mi></msub><mo id="S7.I6.i1.p1.2.m2.1.1.2.1" xref="S7.I6.i1.p1.2.m2.1.1.2.1.cmml">⁢</mo><mi id="S7.I6.i1.p1.2.m2.1.1.2.3" xref="S7.I6.i1.p1.2.m2.1.1.2.3.cmml">u</mi></mrow><mo id="S7.I6.i1.p1.2.m2.1.1.1" xref="S7.I6.i1.p1.2.m2.1.1.1.cmml">¯</mo></mover><annotation-xml encoding="MathML-Content" id="S7.I6.i1.p1.2.m2.1b"><apply id="S7.I6.i1.p1.2.m2.1.1.cmml" xref="S7.I6.i1.p1.2.m2.1.1"><ci id="S7.I6.i1.p1.2.m2.1.1.1.cmml" xref="S7.I6.i1.p1.2.m2.1.1.1">¯</ci><apply id="S7.I6.i1.p1.2.m2.1.1.2.cmml" xref="S7.I6.i1.p1.2.m2.1.1.2"><times id="S7.I6.i1.p1.2.m2.1.1.2.1.cmml" xref="S7.I6.i1.p1.2.m2.1.1.2.1"></times><apply id="S7.I6.i1.p1.2.m2.1.1.2.2.cmml" xref="S7.I6.i1.p1.2.m2.1.1.2.2"><csymbol cd="ambiguous" id="S7.I6.i1.p1.2.m2.1.1.2.2.1.cmml" xref="S7.I6.i1.p1.2.m2.1.1.2.2">subscript</csymbol><ci id="S7.I6.i1.p1.2.m2.1.1.2.2.2.cmml" xref="S7.I6.i1.p1.2.m2.1.1.2.2.2">𝑠</ci><ci id="S7.I6.i1.p1.2.m2.1.1.2.2.3.cmml" xref="S7.I6.i1.p1.2.m2.1.1.2.2.3">𝑢</ci></apply><ci id="S7.I6.i1.p1.2.m2.1.1.2.3.cmml" xref="S7.I6.i1.p1.2.m2.1.1.2.3">𝑢</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.I6.i1.p1.2.m2.1c">\overline{s_{u}u}</annotation><annotation encoding="application/x-llamapun" id="S7.I6.i1.p1.2.m2.1d">over¯ start_ARG italic_s start_POSTSUBSCRIPT italic_u end_POSTSUBSCRIPT italic_u end_ARG</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S7.I6.i1.p1.3.3"> has capacity </span><math alttext="\sigma(u)=\textsl{g}_{s}(u)-(1+\frac{\eta}{2})^{l_{m}(u)-1}" class="ltx_Math" display="inline" id="S7.I6.i1.p1.3.m3.4"><semantics id="S7.I6.i1.p1.3.m3.4a"><mrow id="S7.I6.i1.p1.3.m3.4.4" xref="S7.I6.i1.p1.3.m3.4.4.cmml"><mrow id="S7.I6.i1.p1.3.m3.4.4.3" xref="S7.I6.i1.p1.3.m3.4.4.3.cmml"><mi id="S7.I6.i1.p1.3.m3.4.4.3.2" xref="S7.I6.i1.p1.3.m3.4.4.3.2.cmml">σ</mi><mo id="S7.I6.i1.p1.3.m3.4.4.3.1" xref="S7.I6.i1.p1.3.m3.4.4.3.1.cmml">⁢</mo><mrow id="S7.I6.i1.p1.3.m3.4.4.3.3.2" xref="S7.I6.i1.p1.3.m3.4.4.3.cmml"><mo id="S7.I6.i1.p1.3.m3.4.4.3.3.2.1" stretchy="false" xref="S7.I6.i1.p1.3.m3.4.4.3.cmml">(</mo><mi id="S7.I6.i1.p1.3.m3.2.2" xref="S7.I6.i1.p1.3.m3.2.2.cmml">u</mi><mo id="S7.I6.i1.p1.3.m3.4.4.3.3.2.2" stretchy="false" xref="S7.I6.i1.p1.3.m3.4.4.3.cmml">)</mo></mrow></mrow><mo id="S7.I6.i1.p1.3.m3.4.4.2" xref="S7.I6.i1.p1.3.m3.4.4.2.cmml">=</mo><mrow id="S7.I6.i1.p1.3.m3.4.4.1" xref="S7.I6.i1.p1.3.m3.4.4.1.cmml"><mrow id="S7.I6.i1.p1.3.m3.4.4.1.3" xref="S7.I6.i1.p1.3.m3.4.4.1.3.cmml"><msub id="S7.I6.i1.p1.3.m3.4.4.1.3.2" xref="S7.I6.i1.p1.3.m3.4.4.1.3.2.cmml"><mtext class="ltx_mathvariant_italic" id="S7.I6.i1.p1.3.m3.4.4.1.3.2.2" xref="S7.I6.i1.p1.3.m3.4.4.1.3.2.2a.cmml">g</mtext><mi id="S7.I6.i1.p1.3.m3.4.4.1.3.2.3" xref="S7.I6.i1.p1.3.m3.4.4.1.3.2.3.cmml">s</mi></msub><mo id="S7.I6.i1.p1.3.m3.4.4.1.3.1" xref="S7.I6.i1.p1.3.m3.4.4.1.3.1.cmml">⁢</mo><mrow id="S7.I6.i1.p1.3.m3.4.4.1.3.3.2" xref="S7.I6.i1.p1.3.m3.4.4.1.3.cmml"><mo id="S7.I6.i1.p1.3.m3.4.4.1.3.3.2.1" stretchy="false" xref="S7.I6.i1.p1.3.m3.4.4.1.3.cmml">(</mo><mi id="S7.I6.i1.p1.3.m3.3.3" xref="S7.I6.i1.p1.3.m3.3.3.cmml">u</mi><mo id="S7.I6.i1.p1.3.m3.4.4.1.3.3.2.2" stretchy="false" xref="S7.I6.i1.p1.3.m3.4.4.1.3.cmml">)</mo></mrow></mrow><mo id="S7.I6.i1.p1.3.m3.4.4.1.2" xref="S7.I6.i1.p1.3.m3.4.4.1.2.cmml">−</mo><msup id="S7.I6.i1.p1.3.m3.4.4.1.1" xref="S7.I6.i1.p1.3.m3.4.4.1.1.cmml"><mrow id="S7.I6.i1.p1.3.m3.4.4.1.1.1.1" xref="S7.I6.i1.p1.3.m3.4.4.1.1.1.1.1.cmml"><mo id="S7.I6.i1.p1.3.m3.4.4.1.1.1.1.2" stretchy="false" xref="S7.I6.i1.p1.3.m3.4.4.1.1.1.1.1.cmml">(</mo><mrow id="S7.I6.i1.p1.3.m3.4.4.1.1.1.1.1" xref="S7.I6.i1.p1.3.m3.4.4.1.1.1.1.1.cmml"><mn id="S7.I6.i1.p1.3.m3.4.4.1.1.1.1.1.2" xref="S7.I6.i1.p1.3.m3.4.4.1.1.1.1.1.2.cmml">1</mn><mo id="S7.I6.i1.p1.3.m3.4.4.1.1.1.1.1.1" xref="S7.I6.i1.p1.3.m3.4.4.1.1.1.1.1.1.cmml">+</mo><mfrac id="S7.I6.i1.p1.3.m3.4.4.1.1.1.1.1.3" xref="S7.I6.i1.p1.3.m3.4.4.1.1.1.1.1.3.cmml"><mi id="S7.I6.i1.p1.3.m3.4.4.1.1.1.1.1.3.2" xref="S7.I6.i1.p1.3.m3.4.4.1.1.1.1.1.3.2.cmml">η</mi><mn id="S7.I6.i1.p1.3.m3.4.4.1.1.1.1.1.3.3" xref="S7.I6.i1.p1.3.m3.4.4.1.1.1.1.1.3.3.cmml">2</mn></mfrac></mrow><mo id="S7.I6.i1.p1.3.m3.4.4.1.1.1.1.3" stretchy="false" xref="S7.I6.i1.p1.3.m3.4.4.1.1.1.1.1.cmml">)</mo></mrow><mrow id="S7.I6.i1.p1.3.m3.1.1.1" xref="S7.I6.i1.p1.3.m3.1.1.1.cmml"><mrow id="S7.I6.i1.p1.3.m3.1.1.1.3" xref="S7.I6.i1.p1.3.m3.1.1.1.3.cmml"><msub id="S7.I6.i1.p1.3.m3.1.1.1.3.2" xref="S7.I6.i1.p1.3.m3.1.1.1.3.2.cmml"><mi id="S7.I6.i1.p1.3.m3.1.1.1.3.2.2" xref="S7.I6.i1.p1.3.m3.1.1.1.3.2.2.cmml">l</mi><mi id="S7.I6.i1.p1.3.m3.1.1.1.3.2.3" xref="S7.I6.i1.p1.3.m3.1.1.1.3.2.3.cmml">m</mi></msub><mo id="S7.I6.i1.p1.3.m3.1.1.1.3.1" xref="S7.I6.i1.p1.3.m3.1.1.1.3.1.cmml">⁢</mo><mrow id="S7.I6.i1.p1.3.m3.1.1.1.3.3.2" xref="S7.I6.i1.p1.3.m3.1.1.1.3.cmml"><mo id="S7.I6.i1.p1.3.m3.1.1.1.3.3.2.1" stretchy="false" xref="S7.I6.i1.p1.3.m3.1.1.1.3.cmml">(</mo><mi id="S7.I6.i1.p1.3.m3.1.1.1.1" xref="S7.I6.i1.p1.3.m3.1.1.1.1.cmml">u</mi><mo id="S7.I6.i1.p1.3.m3.1.1.1.3.3.2.2" stretchy="false" xref="S7.I6.i1.p1.3.m3.1.1.1.3.cmml">)</mo></mrow></mrow><mo id="S7.I6.i1.p1.3.m3.1.1.1.2" xref="S7.I6.i1.p1.3.m3.1.1.1.2.cmml">−</mo><mn id="S7.I6.i1.p1.3.m3.1.1.1.4" xref="S7.I6.i1.p1.3.m3.1.1.1.4.cmml">1</mn></mrow></msup></mrow></mrow><annotation-xml encoding="MathML-Content" id="S7.I6.i1.p1.3.m3.4b"><apply id="S7.I6.i1.p1.3.m3.4.4.cmml" xref="S7.I6.i1.p1.3.m3.4.4"><eq id="S7.I6.i1.p1.3.m3.4.4.2.cmml" xref="S7.I6.i1.p1.3.m3.4.4.2"></eq><apply id="S7.I6.i1.p1.3.m3.4.4.3.cmml" xref="S7.I6.i1.p1.3.m3.4.4.3"><times id="S7.I6.i1.p1.3.m3.4.4.3.1.cmml" xref="S7.I6.i1.p1.3.m3.4.4.3.1"></times><ci id="S7.I6.i1.p1.3.m3.4.4.3.2.cmml" xref="S7.I6.i1.p1.3.m3.4.4.3.2">𝜎</ci><ci id="S7.I6.i1.p1.3.m3.2.2.cmml" xref="S7.I6.i1.p1.3.m3.2.2">𝑢</ci></apply><apply id="S7.I6.i1.p1.3.m3.4.4.1.cmml" xref="S7.I6.i1.p1.3.m3.4.4.1"><minus id="S7.I6.i1.p1.3.m3.4.4.1.2.cmml" xref="S7.I6.i1.p1.3.m3.4.4.1.2"></minus><apply id="S7.I6.i1.p1.3.m3.4.4.1.3.cmml" xref="S7.I6.i1.p1.3.m3.4.4.1.3"><times id="S7.I6.i1.p1.3.m3.4.4.1.3.1.cmml" xref="S7.I6.i1.p1.3.m3.4.4.1.3.1"></times><apply id="S7.I6.i1.p1.3.m3.4.4.1.3.2.cmml" xref="S7.I6.i1.p1.3.m3.4.4.1.3.2"><csymbol cd="ambiguous" id="S7.I6.i1.p1.3.m3.4.4.1.3.2.1.cmml" xref="S7.I6.i1.p1.3.m3.4.4.1.3.2">subscript</csymbol><ci id="S7.I6.i1.p1.3.m3.4.4.1.3.2.2a.cmml" xref="S7.I6.i1.p1.3.m3.4.4.1.3.2.2"><mtext class="ltx_mathvariant_italic" id="S7.I6.i1.p1.3.m3.4.4.1.3.2.2.cmml" xref="S7.I6.i1.p1.3.m3.4.4.1.3.2.2">g</mtext></ci><ci id="S7.I6.i1.p1.3.m3.4.4.1.3.2.3.cmml" xref="S7.I6.i1.p1.3.m3.4.4.1.3.2.3">𝑠</ci></apply><ci id="S7.I6.i1.p1.3.m3.3.3.cmml" xref="S7.I6.i1.p1.3.m3.3.3">𝑢</ci></apply><apply id="S7.I6.i1.p1.3.m3.4.4.1.1.cmml" xref="S7.I6.i1.p1.3.m3.4.4.1.1"><csymbol cd="ambiguous" id="S7.I6.i1.p1.3.m3.4.4.1.1.2.cmml" xref="S7.I6.i1.p1.3.m3.4.4.1.1">superscript</csymbol><apply id="S7.I6.i1.p1.3.m3.4.4.1.1.1.1.1.cmml" xref="S7.I6.i1.p1.3.m3.4.4.1.1.1.1"><plus id="S7.I6.i1.p1.3.m3.4.4.1.1.1.1.1.1.cmml" xref="S7.I6.i1.p1.3.m3.4.4.1.1.1.1.1.1"></plus><cn id="S7.I6.i1.p1.3.m3.4.4.1.1.1.1.1.2.cmml" type="integer" xref="S7.I6.i1.p1.3.m3.4.4.1.1.1.1.1.2">1</cn><apply id="S7.I6.i1.p1.3.m3.4.4.1.1.1.1.1.3.cmml" xref="S7.I6.i1.p1.3.m3.4.4.1.1.1.1.1.3"><divide id="S7.I6.i1.p1.3.m3.4.4.1.1.1.1.1.3.1.cmml" xref="S7.I6.i1.p1.3.m3.4.4.1.1.1.1.1.3"></divide><ci id="S7.I6.i1.p1.3.m3.4.4.1.1.1.1.1.3.2.cmml" xref="S7.I6.i1.p1.3.m3.4.4.1.1.1.1.1.3.2">𝜂</ci><cn id="S7.I6.i1.p1.3.m3.4.4.1.1.1.1.1.3.3.cmml" type="integer" xref="S7.I6.i1.p1.3.m3.4.4.1.1.1.1.1.3.3">2</cn></apply></apply><apply id="S7.I6.i1.p1.3.m3.1.1.1.cmml" xref="S7.I6.i1.p1.3.m3.1.1.1"><minus id="S7.I6.i1.p1.3.m3.1.1.1.2.cmml" xref="S7.I6.i1.p1.3.m3.1.1.1.2"></minus><apply id="S7.I6.i1.p1.3.m3.1.1.1.3.cmml" xref="S7.I6.i1.p1.3.m3.1.1.1.3"><times id="S7.I6.i1.p1.3.m3.1.1.1.3.1.cmml" xref="S7.I6.i1.p1.3.m3.1.1.1.3.1"></times><apply id="S7.I6.i1.p1.3.m3.1.1.1.3.2.cmml" xref="S7.I6.i1.p1.3.m3.1.1.1.3.2"><csymbol cd="ambiguous" id="S7.I6.i1.p1.3.m3.1.1.1.3.2.1.cmml" xref="S7.I6.i1.p1.3.m3.1.1.1.3.2">subscript</csymbol><ci id="S7.I6.i1.p1.3.m3.1.1.1.3.2.2.cmml" xref="S7.I6.i1.p1.3.m3.1.1.1.3.2.2">𝑙</ci><ci id="S7.I6.i1.p1.3.m3.1.1.1.3.2.3.cmml" xref="S7.I6.i1.p1.3.m3.1.1.1.3.2.3">𝑚</ci></apply><ci id="S7.I6.i1.p1.3.m3.1.1.1.1.cmml" xref="S7.I6.i1.p1.3.m3.1.1.1.1">𝑢</ci></apply><cn id="S7.I6.i1.p1.3.m3.1.1.1.4.cmml" type="integer" xref="S7.I6.i1.p1.3.m3.1.1.1.4">1</cn></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.I6.i1.p1.3.m3.4c">\sigma(u)=\textsl{g}_{s}(u)-(1+\frac{\eta}{2})^{l_{m}(u)-1}</annotation><annotation encoding="application/x-llamapun" id="S7.I6.i1.p1.3.m3.4d">italic_σ ( italic_u ) = g start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_u ) - ( 1 + divide start_ARG italic_η end_ARG start_ARG 2 end_ARG ) start_POSTSUPERSCRIPT italic_l start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT ( italic_u ) - 1 end_POSTSUPERSCRIPT</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S7.I6.i1.p1.3.4">.</span></p> </div> </li> <li class="ltx_item" id="S7.I6.i2" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">•</span> <div class="ltx_para" id="S7.I6.i2.p1"> <p class="ltx_p" id="S7.I6.i2.p1.3"><span class="ltx_text ltx_font_italic" id="S7.I6.i2.p1.3.1">For </span><math alttext="v\in T_{m}" class="ltx_Math" display="inline" id="S7.I6.i2.p1.1.m1.1"><semantics id="S7.I6.i2.p1.1.m1.1a"><mrow id="S7.I6.i2.p1.1.m1.1.1" xref="S7.I6.i2.p1.1.m1.1.1.cmml"><mi id="S7.I6.i2.p1.1.m1.1.1.2" xref="S7.I6.i2.p1.1.m1.1.1.2.cmml">v</mi><mo id="S7.I6.i2.p1.1.m1.1.1.1" xref="S7.I6.i2.p1.1.m1.1.1.1.cmml">∈</mo><msub id="S7.I6.i2.p1.1.m1.1.1.3" xref="S7.I6.i2.p1.1.m1.1.1.3.cmml"><mi id="S7.I6.i2.p1.1.m1.1.1.3.2" xref="S7.I6.i2.p1.1.m1.1.1.3.2.cmml">T</mi><mi id="S7.I6.i2.p1.1.m1.1.1.3.3" xref="S7.I6.i2.p1.1.m1.1.1.3.3.cmml">m</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S7.I6.i2.p1.1.m1.1b"><apply id="S7.I6.i2.p1.1.m1.1.1.cmml" xref="S7.I6.i2.p1.1.m1.1.1"><in id="S7.I6.i2.p1.1.m1.1.1.1.cmml" xref="S7.I6.i2.p1.1.m1.1.1.1"></in><ci id="S7.I6.i2.p1.1.m1.1.1.2.cmml" xref="S7.I6.i2.p1.1.m1.1.1.2">𝑣</ci><apply id="S7.I6.i2.p1.1.m1.1.1.3.cmml" xref="S7.I6.i2.p1.1.m1.1.1.3"><csymbol cd="ambiguous" id="S7.I6.i2.p1.1.m1.1.1.3.1.cmml" xref="S7.I6.i2.p1.1.m1.1.1.3">subscript</csymbol><ci id="S7.I6.i2.p1.1.m1.1.1.3.2.cmml" xref="S7.I6.i2.p1.1.m1.1.1.3.2">𝑇</ci><ci id="S7.I6.i2.p1.1.m1.1.1.3.3.cmml" xref="S7.I6.i2.p1.1.m1.1.1.3.3">𝑚</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.I6.i2.p1.1.m1.1c">v\in T_{m}</annotation><annotation encoding="application/x-llamapun" id="S7.I6.i2.p1.1.m1.1d">italic_v ∈ italic_T start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S7.I6.i2.p1.3.2">, the edge </span><math alttext="\overline{vt_{v}}" class="ltx_Math" display="inline" id="S7.I6.i2.p1.2.m2.1"><semantics id="S7.I6.i2.p1.2.m2.1a"><mover accent="true" id="S7.I6.i2.p1.2.m2.1.1" xref="S7.I6.i2.p1.2.m2.1.1.cmml"><mrow id="S7.I6.i2.p1.2.m2.1.1.2" xref="S7.I6.i2.p1.2.m2.1.1.2.cmml"><mi id="S7.I6.i2.p1.2.m2.1.1.2.2" xref="S7.I6.i2.p1.2.m2.1.1.2.2.cmml">v</mi><mo id="S7.I6.i2.p1.2.m2.1.1.2.1" xref="S7.I6.i2.p1.2.m2.1.1.2.1.cmml">⁢</mo><msub id="S7.I6.i2.p1.2.m2.1.1.2.3" xref="S7.I6.i2.p1.2.m2.1.1.2.3.cmml"><mi id="S7.I6.i2.p1.2.m2.1.1.2.3.2" xref="S7.I6.i2.p1.2.m2.1.1.2.3.2.cmml">t</mi><mi id="S7.I6.i2.p1.2.m2.1.1.2.3.3" xref="S7.I6.i2.p1.2.m2.1.1.2.3.3.cmml">v</mi></msub></mrow><mo id="S7.I6.i2.p1.2.m2.1.1.1" xref="S7.I6.i2.p1.2.m2.1.1.1.cmml">¯</mo></mover><annotation-xml encoding="MathML-Content" id="S7.I6.i2.p1.2.m2.1b"><apply id="S7.I6.i2.p1.2.m2.1.1.cmml" xref="S7.I6.i2.p1.2.m2.1.1"><ci id="S7.I6.i2.p1.2.m2.1.1.1.cmml" xref="S7.I6.i2.p1.2.m2.1.1.1">¯</ci><apply id="S7.I6.i2.p1.2.m2.1.1.2.cmml" xref="S7.I6.i2.p1.2.m2.1.1.2"><times id="S7.I6.i2.p1.2.m2.1.1.2.1.cmml" xref="S7.I6.i2.p1.2.m2.1.1.2.1"></times><ci id="S7.I6.i2.p1.2.m2.1.1.2.2.cmml" xref="S7.I6.i2.p1.2.m2.1.1.2.2">𝑣</ci><apply id="S7.I6.i2.p1.2.m2.1.1.2.3.cmml" xref="S7.I6.i2.p1.2.m2.1.1.2.3"><csymbol cd="ambiguous" id="S7.I6.i2.p1.2.m2.1.1.2.3.1.cmml" xref="S7.I6.i2.p1.2.m2.1.1.2.3">subscript</csymbol><ci id="S7.I6.i2.p1.2.m2.1.1.2.3.2.cmml" xref="S7.I6.i2.p1.2.m2.1.1.2.3.2">𝑡</ci><ci id="S7.I6.i2.p1.2.m2.1.1.2.3.3.cmml" xref="S7.I6.i2.p1.2.m2.1.1.2.3.3">𝑣</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.I6.i2.p1.2.m2.1c">\overline{vt_{v}}</annotation><annotation encoding="application/x-llamapun" id="S7.I6.i2.p1.2.m2.1d">over¯ start_ARG italic_v italic_t start_POSTSUBSCRIPT italic_v end_POSTSUBSCRIPT end_ARG</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S7.I6.i2.p1.3.3"> has capacity </span><math alttext="\delta(v)=(1+\frac{\eta}{2})^{h+1}-\textsl{g}_{s}(v)" class="ltx_Math" display="inline" id="S7.I6.i2.p1.3.m3.3"><semantics id="S7.I6.i2.p1.3.m3.3a"><mrow id="S7.I6.i2.p1.3.m3.3.3" xref="S7.I6.i2.p1.3.m3.3.3.cmml"><mrow id="S7.I6.i2.p1.3.m3.3.3.3" xref="S7.I6.i2.p1.3.m3.3.3.3.cmml"><mi id="S7.I6.i2.p1.3.m3.3.3.3.2" xref="S7.I6.i2.p1.3.m3.3.3.3.2.cmml">δ</mi><mo id="S7.I6.i2.p1.3.m3.3.3.3.1" xref="S7.I6.i2.p1.3.m3.3.3.3.1.cmml">⁢</mo><mrow id="S7.I6.i2.p1.3.m3.3.3.3.3.2" xref="S7.I6.i2.p1.3.m3.3.3.3.cmml"><mo id="S7.I6.i2.p1.3.m3.3.3.3.3.2.1" stretchy="false" xref="S7.I6.i2.p1.3.m3.3.3.3.cmml">(</mo><mi id="S7.I6.i2.p1.3.m3.1.1" xref="S7.I6.i2.p1.3.m3.1.1.cmml">v</mi><mo id="S7.I6.i2.p1.3.m3.3.3.3.3.2.2" stretchy="false" xref="S7.I6.i2.p1.3.m3.3.3.3.cmml">)</mo></mrow></mrow><mo id="S7.I6.i2.p1.3.m3.3.3.2" xref="S7.I6.i2.p1.3.m3.3.3.2.cmml">=</mo><mrow id="S7.I6.i2.p1.3.m3.3.3.1" xref="S7.I6.i2.p1.3.m3.3.3.1.cmml"><msup id="S7.I6.i2.p1.3.m3.3.3.1.1" xref="S7.I6.i2.p1.3.m3.3.3.1.1.cmml"><mrow id="S7.I6.i2.p1.3.m3.3.3.1.1.1.1" xref="S7.I6.i2.p1.3.m3.3.3.1.1.1.1.1.cmml"><mo id="S7.I6.i2.p1.3.m3.3.3.1.1.1.1.2" stretchy="false" xref="S7.I6.i2.p1.3.m3.3.3.1.1.1.1.1.cmml">(</mo><mrow id="S7.I6.i2.p1.3.m3.3.3.1.1.1.1.1" xref="S7.I6.i2.p1.3.m3.3.3.1.1.1.1.1.cmml"><mn id="S7.I6.i2.p1.3.m3.3.3.1.1.1.1.1.2" xref="S7.I6.i2.p1.3.m3.3.3.1.1.1.1.1.2.cmml">1</mn><mo id="S7.I6.i2.p1.3.m3.3.3.1.1.1.1.1.1" xref="S7.I6.i2.p1.3.m3.3.3.1.1.1.1.1.1.cmml">+</mo><mfrac id="S7.I6.i2.p1.3.m3.3.3.1.1.1.1.1.3" xref="S7.I6.i2.p1.3.m3.3.3.1.1.1.1.1.3.cmml"><mi id="S7.I6.i2.p1.3.m3.3.3.1.1.1.1.1.3.2" xref="S7.I6.i2.p1.3.m3.3.3.1.1.1.1.1.3.2.cmml">η</mi><mn id="S7.I6.i2.p1.3.m3.3.3.1.1.1.1.1.3.3" xref="S7.I6.i2.p1.3.m3.3.3.1.1.1.1.1.3.3.cmml">2</mn></mfrac></mrow><mo id="S7.I6.i2.p1.3.m3.3.3.1.1.1.1.3" stretchy="false" xref="S7.I6.i2.p1.3.m3.3.3.1.1.1.1.1.cmml">)</mo></mrow><mrow id="S7.I6.i2.p1.3.m3.3.3.1.1.3" xref="S7.I6.i2.p1.3.m3.3.3.1.1.3.cmml"><mi id="S7.I6.i2.p1.3.m3.3.3.1.1.3.2" xref="S7.I6.i2.p1.3.m3.3.3.1.1.3.2.cmml">h</mi><mo id="S7.I6.i2.p1.3.m3.3.3.1.1.3.1" xref="S7.I6.i2.p1.3.m3.3.3.1.1.3.1.cmml">+</mo><mn id="S7.I6.i2.p1.3.m3.3.3.1.1.3.3" xref="S7.I6.i2.p1.3.m3.3.3.1.1.3.3.cmml">1</mn></mrow></msup><mo id="S7.I6.i2.p1.3.m3.3.3.1.2" xref="S7.I6.i2.p1.3.m3.3.3.1.2.cmml">−</mo><mrow id="S7.I6.i2.p1.3.m3.3.3.1.3" xref="S7.I6.i2.p1.3.m3.3.3.1.3.cmml"><msub id="S7.I6.i2.p1.3.m3.3.3.1.3.2" xref="S7.I6.i2.p1.3.m3.3.3.1.3.2.cmml"><mtext class="ltx_mathvariant_italic" id="S7.I6.i2.p1.3.m3.3.3.1.3.2.2" xref="S7.I6.i2.p1.3.m3.3.3.1.3.2.2a.cmml">g</mtext><mi id="S7.I6.i2.p1.3.m3.3.3.1.3.2.3" xref="S7.I6.i2.p1.3.m3.3.3.1.3.2.3.cmml">s</mi></msub><mo id="S7.I6.i2.p1.3.m3.3.3.1.3.1" xref="S7.I6.i2.p1.3.m3.3.3.1.3.1.cmml">⁢</mo><mrow id="S7.I6.i2.p1.3.m3.3.3.1.3.3.2" xref="S7.I6.i2.p1.3.m3.3.3.1.3.cmml"><mo id="S7.I6.i2.p1.3.m3.3.3.1.3.3.2.1" stretchy="false" xref="S7.I6.i2.p1.3.m3.3.3.1.3.cmml">(</mo><mi id="S7.I6.i2.p1.3.m3.2.2" xref="S7.I6.i2.p1.3.m3.2.2.cmml">v</mi><mo id="S7.I6.i2.p1.3.m3.3.3.1.3.3.2.2" stretchy="false" xref="S7.I6.i2.p1.3.m3.3.3.1.3.cmml">)</mo></mrow></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S7.I6.i2.p1.3.m3.3b"><apply id="S7.I6.i2.p1.3.m3.3.3.cmml" xref="S7.I6.i2.p1.3.m3.3.3"><eq id="S7.I6.i2.p1.3.m3.3.3.2.cmml" xref="S7.I6.i2.p1.3.m3.3.3.2"></eq><apply id="S7.I6.i2.p1.3.m3.3.3.3.cmml" xref="S7.I6.i2.p1.3.m3.3.3.3"><times id="S7.I6.i2.p1.3.m3.3.3.3.1.cmml" xref="S7.I6.i2.p1.3.m3.3.3.3.1"></times><ci id="S7.I6.i2.p1.3.m3.3.3.3.2.cmml" xref="S7.I6.i2.p1.3.m3.3.3.3.2">𝛿</ci><ci id="S7.I6.i2.p1.3.m3.1.1.cmml" xref="S7.I6.i2.p1.3.m3.1.1">𝑣</ci></apply><apply id="S7.I6.i2.p1.3.m3.3.3.1.cmml" xref="S7.I6.i2.p1.3.m3.3.3.1"><minus id="S7.I6.i2.p1.3.m3.3.3.1.2.cmml" xref="S7.I6.i2.p1.3.m3.3.3.1.2"></minus><apply id="S7.I6.i2.p1.3.m3.3.3.1.1.cmml" xref="S7.I6.i2.p1.3.m3.3.3.1.1"><csymbol cd="ambiguous" 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id="S7.I6.i2.p1.3.m3.3.3.1.1.3.2.cmml" xref="S7.I6.i2.p1.3.m3.3.3.1.1.3.2">ℎ</ci><cn id="S7.I6.i2.p1.3.m3.3.3.1.1.3.3.cmml" type="integer" xref="S7.I6.i2.p1.3.m3.3.3.1.1.3.3">1</cn></apply></apply><apply id="S7.I6.i2.p1.3.m3.3.3.1.3.cmml" xref="S7.I6.i2.p1.3.m3.3.3.1.3"><times id="S7.I6.i2.p1.3.m3.3.3.1.3.1.cmml" xref="S7.I6.i2.p1.3.m3.3.3.1.3.1"></times><apply id="S7.I6.i2.p1.3.m3.3.3.1.3.2.cmml" xref="S7.I6.i2.p1.3.m3.3.3.1.3.2"><csymbol cd="ambiguous" id="S7.I6.i2.p1.3.m3.3.3.1.3.2.1.cmml" xref="S7.I6.i2.p1.3.m3.3.3.1.3.2">subscript</csymbol><ci id="S7.I6.i2.p1.3.m3.3.3.1.3.2.2a.cmml" xref="S7.I6.i2.p1.3.m3.3.3.1.3.2.2"><mtext class="ltx_mathvariant_italic" id="S7.I6.i2.p1.3.m3.3.3.1.3.2.2.cmml" xref="S7.I6.i2.p1.3.m3.3.3.1.3.2.2">g</mtext></ci><ci id="S7.I6.i2.p1.3.m3.3.3.1.3.2.3.cmml" xref="S7.I6.i2.p1.3.m3.3.3.1.3.2.3">𝑠</ci></apply><ci id="S7.I6.i2.p1.3.m3.2.2.cmml" xref="S7.I6.i2.p1.3.m3.2.2">𝑣</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.I6.i2.p1.3.m3.3c">\delta(v)=(1+\frac{\eta}{2})^{h+1}-\textsl{g}_{s}(v)</annotation><annotation encoding="application/x-llamapun" id="S7.I6.i2.p1.3.m3.3d">italic_δ ( italic_v ) = ( 1 + divide start_ARG italic_η end_ARG start_ARG 2 end_ARG ) start_POSTSUPERSCRIPT italic_h + 1 end_POSTSUPERSCRIPT - g start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_v )</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S7.I6.i2.p1.3.4">.</span></p> </div> </li> <li class="ltx_item" id="S7.I6.i3" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">•</span> <div class="ltx_para" id="S7.I6.i3.p1"> <p class="ltx_p" id="S7.I6.i3.p1.2"><span class="ltx_text ltx_font_italic" id="S7.I6.i3.p1.2.1">Each other edge </span><math alttext="\overline{uv}\in E_{s}" class="ltx_Math" display="inline" id="S7.I6.i3.p1.1.m1.1"><semantics id="S7.I6.i3.p1.1.m1.1a"><mrow id="S7.I6.i3.p1.1.m1.1.1" xref="S7.I6.i3.p1.1.m1.1.1.cmml"><mover accent="true" id="S7.I6.i3.p1.1.m1.1.1.2" xref="S7.I6.i3.p1.1.m1.1.1.2.cmml"><mrow id="S7.I6.i3.p1.1.m1.1.1.2.2" xref="S7.I6.i3.p1.1.m1.1.1.2.2.cmml"><mi id="S7.I6.i3.p1.1.m1.1.1.2.2.2" xref="S7.I6.i3.p1.1.m1.1.1.2.2.2.cmml">u</mi><mo id="S7.I6.i3.p1.1.m1.1.1.2.2.1" xref="S7.I6.i3.p1.1.m1.1.1.2.2.1.cmml">⁢</mo><mi id="S7.I6.i3.p1.1.m1.1.1.2.2.3" xref="S7.I6.i3.p1.1.m1.1.1.2.2.3.cmml">v</mi></mrow><mo id="S7.I6.i3.p1.1.m1.1.1.2.1" xref="S7.I6.i3.p1.1.m1.1.1.2.1.cmml">¯</mo></mover><mo id="S7.I6.i3.p1.1.m1.1.1.1" xref="S7.I6.i3.p1.1.m1.1.1.1.cmml">∈</mo><msub id="S7.I6.i3.p1.1.m1.1.1.3" xref="S7.I6.i3.p1.1.m1.1.1.3.cmml"><mi id="S7.I6.i3.p1.1.m1.1.1.3.2" xref="S7.I6.i3.p1.1.m1.1.1.3.2.cmml">E</mi><mi id="S7.I6.i3.p1.1.m1.1.1.3.3" xref="S7.I6.i3.p1.1.m1.1.1.3.3.cmml">s</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S7.I6.i3.p1.1.m1.1b"><apply id="S7.I6.i3.p1.1.m1.1.1.cmml" xref="S7.I6.i3.p1.1.m1.1.1"><in id="S7.I6.i3.p1.1.m1.1.1.1.cmml" xref="S7.I6.i3.p1.1.m1.1.1.1"></in><apply id="S7.I6.i3.p1.1.m1.1.1.2.cmml" xref="S7.I6.i3.p1.1.m1.1.1.2"><ci id="S7.I6.i3.p1.1.m1.1.1.2.1.cmml" xref="S7.I6.i3.p1.1.m1.1.1.2.1">¯</ci><apply id="S7.I6.i3.p1.1.m1.1.1.2.2.cmml" xref="S7.I6.i3.p1.1.m1.1.1.2.2"><times id="S7.I6.i3.p1.1.m1.1.1.2.2.1.cmml" xref="S7.I6.i3.p1.1.m1.1.1.2.2.1"></times><ci id="S7.I6.i3.p1.1.m1.1.1.2.2.2.cmml" xref="S7.I6.i3.p1.1.m1.1.1.2.2.2">𝑢</ci><ci id="S7.I6.i3.p1.1.m1.1.1.2.2.3.cmml" xref="S7.I6.i3.p1.1.m1.1.1.2.2.3">𝑣</ci></apply></apply><apply id="S7.I6.i3.p1.1.m1.1.1.3.cmml" xref="S7.I6.i3.p1.1.m1.1.1.3"><csymbol cd="ambiguous" id="S7.I6.i3.p1.1.m1.1.1.3.1.cmml" xref="S7.I6.i3.p1.1.m1.1.1.3">subscript</csymbol><ci id="S7.I6.i3.p1.1.m1.1.1.3.2.cmml" xref="S7.I6.i3.p1.1.m1.1.1.3.2">𝐸</ci><ci id="S7.I6.i3.p1.1.m1.1.1.3.3.cmml" xref="S7.I6.i3.p1.1.m1.1.1.3.3">𝑠</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.I6.i3.p1.1.m1.1c">\overline{uv}\in E_{s}</annotation><annotation encoding="application/x-llamapun" id="S7.I6.i3.p1.1.m1.1d">over¯ start_ARG italic_u italic_v end_ARG ∈ italic_E start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S7.I6.i3.p1.2.2"> has capacity </span><math alttext="\textsl{g}_{s}(u\!\to\!v)" class="ltx_Math" display="inline" id="S7.I6.i3.p1.2.m2.1"><semantics id="S7.I6.i3.p1.2.m2.1a"><mrow id="S7.I6.i3.p1.2.m2.1.1" xref="S7.I6.i3.p1.2.m2.1.1.cmml"><msub id="S7.I6.i3.p1.2.m2.1.1.3" xref="S7.I6.i3.p1.2.m2.1.1.3.cmml"><mtext class="ltx_mathvariant_italic" id="S7.I6.i3.p1.2.m2.1.1.3.2" xref="S7.I6.i3.p1.2.m2.1.1.3.2a.cmml">g</mtext><mi id="S7.I6.i3.p1.2.m2.1.1.3.3" xref="S7.I6.i3.p1.2.m2.1.1.3.3.cmml">s</mi></msub><mo id="S7.I6.i3.p1.2.m2.1.1.2" xref="S7.I6.i3.p1.2.m2.1.1.2.cmml">⁢</mo><mrow id="S7.I6.i3.p1.2.m2.1.1.1.1" xref="S7.I6.i3.p1.2.m2.1.1.1.1.1.cmml"><mo id="S7.I6.i3.p1.2.m2.1.1.1.1.2" stretchy="false" xref="S7.I6.i3.p1.2.m2.1.1.1.1.1.cmml">(</mo><mrow id="S7.I6.i3.p1.2.m2.1.1.1.1.1" xref="S7.I6.i3.p1.2.m2.1.1.1.1.1.cmml"><mi id="S7.I6.i3.p1.2.m2.1.1.1.1.1.2" xref="S7.I6.i3.p1.2.m2.1.1.1.1.1.2.cmml">u</mi><mo id="S7.I6.i3.p1.2.m2.1.1.1.1.1.1" lspace="0.108em" rspace="0.108em" stretchy="false" xref="S7.I6.i3.p1.2.m2.1.1.1.1.1.1.cmml">→</mo><mi id="S7.I6.i3.p1.2.m2.1.1.1.1.1.3" xref="S7.I6.i3.p1.2.m2.1.1.1.1.1.3.cmml">v</mi></mrow><mo id="S7.I6.i3.p1.2.m2.1.1.1.1.3" stretchy="false" xref="S7.I6.i3.p1.2.m2.1.1.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S7.I6.i3.p1.2.m2.1b"><apply id="S7.I6.i3.p1.2.m2.1.1.cmml" xref="S7.I6.i3.p1.2.m2.1.1"><times id="S7.I6.i3.p1.2.m2.1.1.2.cmml" xref="S7.I6.i3.p1.2.m2.1.1.2"></times><apply id="S7.I6.i3.p1.2.m2.1.1.3.cmml" xref="S7.I6.i3.p1.2.m2.1.1.3"><csymbol cd="ambiguous" id="S7.I6.i3.p1.2.m2.1.1.3.1.cmml" xref="S7.I6.i3.p1.2.m2.1.1.3">subscript</csymbol><ci id="S7.I6.i3.p1.2.m2.1.1.3.2a.cmml" xref="S7.I6.i3.p1.2.m2.1.1.3.2"><mtext class="ltx_mathvariant_italic" id="S7.I6.i3.p1.2.m2.1.1.3.2.cmml" xref="S7.I6.i3.p1.2.m2.1.1.3.2">g</mtext></ci><ci id="S7.I6.i3.p1.2.m2.1.1.3.3.cmml" xref="S7.I6.i3.p1.2.m2.1.1.3.3">𝑠</ci></apply><apply id="S7.I6.i3.p1.2.m2.1.1.1.1.1.cmml" xref="S7.I6.i3.p1.2.m2.1.1.1.1"><ci id="S7.I6.i3.p1.2.m2.1.1.1.1.1.1.cmml" xref="S7.I6.i3.p1.2.m2.1.1.1.1.1.1">→</ci><ci id="S7.I6.i3.p1.2.m2.1.1.1.1.1.2.cmml" xref="S7.I6.i3.p1.2.m2.1.1.1.1.1.2">𝑢</ci><ci id="S7.I6.i3.p1.2.m2.1.1.1.1.1.3.cmml" xref="S7.I6.i3.p1.2.m2.1.1.1.1.1.3">𝑣</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.I6.i3.p1.2.m2.1c">\textsl{g}_{s}(u\!\to\!v)</annotation><annotation encoding="application/x-llamapun" id="S7.I6.i3.p1.2.m2.1d">g start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_u → italic_v )</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S7.I6.i3.p1.2.3">.</span></p> </div> </li> </ul> </div> </div> <div class="ltx_para" id="S7.SS2.p2"> <span class="ltx_ERROR undefined" id="S7.SS2.p2.8">{observation}</span> <p class="ltx_p" id="S7.SS2.p2.7">At the start of <math alttext="(h:m:s)" class="ltx_Math" display="inline" id="S7.SS2.p2.1.m1.1"><semantics id="S7.SS2.p2.1.m1.1a"><mrow id="S7.SS2.p2.1.m1.1.1.1" xref="S7.SS2.p2.1.m1.1.1.1.1.cmml"><mo id="S7.SS2.p2.1.m1.1.1.1.2" stretchy="false" xref="S7.SS2.p2.1.m1.1.1.1.1.cmml">(</mo><mrow id="S7.SS2.p2.1.m1.1.1.1.1" xref="S7.SS2.p2.1.m1.1.1.1.1.cmml"><mi id="S7.SS2.p2.1.m1.1.1.1.1.2" xref="S7.SS2.p2.1.m1.1.1.1.1.2.cmml">h</mi><mo id="S7.SS2.p2.1.m1.1.1.1.1.3" lspace="0.278em" rspace="0.278em" xref="S7.SS2.p2.1.m1.1.1.1.1.3.cmml">:</mo><mi id="S7.SS2.p2.1.m1.1.1.1.1.4" xref="S7.SS2.p2.1.m1.1.1.1.1.4.cmml">m</mi><mo id="S7.SS2.p2.1.m1.1.1.1.1.5" lspace="0.278em" rspace="0.278em" xref="S7.SS2.p2.1.m1.1.1.1.1.5.cmml">:</mo><mi id="S7.SS2.p2.1.m1.1.1.1.1.6" xref="S7.SS2.p2.1.m1.1.1.1.1.6.cmml">s</mi></mrow><mo id="S7.SS2.p2.1.m1.1.1.1.3" stretchy="false" xref="S7.SS2.p2.1.m1.1.1.1.1.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S7.SS2.p2.1.m1.1b"><apply id="S7.SS2.p2.1.m1.1.1.1.1.cmml" xref="S7.SS2.p2.1.m1.1.1.1"><and id="S7.SS2.p2.1.m1.1.1.1.1a.cmml" xref="S7.SS2.p2.1.m1.1.1.1"></and><apply id="S7.SS2.p2.1.m1.1.1.1.1b.cmml" xref="S7.SS2.p2.1.m1.1.1.1"><ci id="S7.SS2.p2.1.m1.1.1.1.1.3.cmml" xref="S7.SS2.p2.1.m1.1.1.1.1.3">:</ci><ci id="S7.SS2.p2.1.m1.1.1.1.1.2.cmml" xref="S7.SS2.p2.1.m1.1.1.1.1.2">ℎ</ci><ci id="S7.SS2.p2.1.m1.1.1.1.1.4.cmml" xref="S7.SS2.p2.1.m1.1.1.1.1.4">𝑚</ci></apply><apply id="S7.SS2.p2.1.m1.1.1.1.1c.cmml" xref="S7.SS2.p2.1.m1.1.1.1"><ci id="S7.SS2.p2.1.m1.1.1.1.1.5.cmml" xref="S7.SS2.p2.1.m1.1.1.1.1.5">:</ci><share href="https://arxiv.org/html/2411.12694v2#S7.SS2.p2.1.m1.1.1.1.1.4.cmml" id="S7.SS2.p2.1.m1.1.1.1.1d.cmml" xref="S7.SS2.p2.1.m1.1.1.1"></share><ci id="S7.SS2.p2.1.m1.1.1.1.1.6.cmml" xref="S7.SS2.p2.1.m1.1.1.1.1.6">𝑠</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.SS2.p2.1.m1.1c">(h:m:s)</annotation><annotation encoding="application/x-llamapun" id="S7.SS2.p2.1.m1.1d">( italic_h : italic_m : italic_s )</annotation></semantics></math> with <math alttext="m" class="ltx_Math" display="inline" id="S7.SS2.p2.2.m2.1"><semantics id="S7.SS2.p2.2.m2.1a"><mi id="S7.SS2.p2.2.m2.1.1" xref="S7.SS2.p2.2.m2.1.1.cmml">m</mi><annotation-xml encoding="MathML-Content" id="S7.SS2.p2.2.m2.1b"><ci id="S7.SS2.p2.2.m2.1.1.cmml" xref="S7.SS2.p2.2.m2.1.1">𝑚</ci></annotation-xml><annotation encoding="application/x-tex" id="S7.SS2.p2.2.m2.1c">m</annotation><annotation encoding="application/x-llamapun" id="S7.SS2.p2.2.m2.1d">italic_m</annotation></semantics></math> odd, if every vertex <math alttext="u" class="ltx_Math" display="inline" id="S7.SS2.p2.3.m3.1"><semantics id="S7.SS2.p2.3.m3.1a"><mi id="S7.SS2.p2.3.m3.1.1" xref="S7.SS2.p2.3.m3.1.1.cmml">u</mi><annotation-xml encoding="MathML-Content" id="S7.SS2.p2.3.m3.1b"><ci id="S7.SS2.p2.3.m3.1.1.cmml" xref="S7.SS2.p2.3.m3.1.1">𝑢</ci></annotation-xml><annotation encoding="application/x-tex" id="S7.SS2.p2.3.m3.1c">u</annotation><annotation encoding="application/x-llamapun" id="S7.SS2.p2.3.m3.1d">italic_u</annotation></semantics></math> is given <math alttext="l_{m}(u)" class="ltx_Math" display="inline" id="S7.SS2.p2.4.m4.1"><semantics id="S7.SS2.p2.4.m4.1a"><mrow id="S7.SS2.p2.4.m4.1.2" xref="S7.SS2.p2.4.m4.1.2.cmml"><msub id="S7.SS2.p2.4.m4.1.2.2" xref="S7.SS2.p2.4.m4.1.2.2.cmml"><mi id="S7.SS2.p2.4.m4.1.2.2.2" xref="S7.SS2.p2.4.m4.1.2.2.2.cmml">l</mi><mi id="S7.SS2.p2.4.m4.1.2.2.3" xref="S7.SS2.p2.4.m4.1.2.2.3.cmml">m</mi></msub><mo id="S7.SS2.p2.4.m4.1.2.1" xref="S7.SS2.p2.4.m4.1.2.1.cmml">⁢</mo><mrow id="S7.SS2.p2.4.m4.1.2.3.2" xref="S7.SS2.p2.4.m4.1.2.cmml"><mo id="S7.SS2.p2.4.m4.1.2.3.2.1" stretchy="false" xref="S7.SS2.p2.4.m4.1.2.cmml">(</mo><mi id="S7.SS2.p2.4.m4.1.1" xref="S7.SS2.p2.4.m4.1.1.cmml">u</mi><mo id="S7.SS2.p2.4.m4.1.2.3.2.2" stretchy="false" xref="S7.SS2.p2.4.m4.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S7.SS2.p2.4.m4.1b"><apply id="S7.SS2.p2.4.m4.1.2.cmml" xref="S7.SS2.p2.4.m4.1.2"><times id="S7.SS2.p2.4.m4.1.2.1.cmml" xref="S7.SS2.p2.4.m4.1.2.1"></times><apply id="S7.SS2.p2.4.m4.1.2.2.cmml" xref="S7.SS2.p2.4.m4.1.2.2"><csymbol cd="ambiguous" id="S7.SS2.p2.4.m4.1.2.2.1.cmml" xref="S7.SS2.p2.4.m4.1.2.2">subscript</csymbol><ci id="S7.SS2.p2.4.m4.1.2.2.2.cmml" xref="S7.SS2.p2.4.m4.1.2.2.2">𝑙</ci><ci id="S7.SS2.p2.4.m4.1.2.2.3.cmml" xref="S7.SS2.p2.4.m4.1.2.2.3">𝑚</ci></apply><ci id="S7.SS2.p2.4.m4.1.1.cmml" xref="S7.SS2.p2.4.m4.1.1">𝑢</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.SS2.p2.4.m4.1c">l_{m}(u)</annotation><annotation encoding="application/x-llamapun" id="S7.SS2.p2.4.m4.1d">italic_l start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT ( italic_u )</annotation></semantics></math>, whether <math alttext="u\in T_{m}" class="ltx_Math" display="inline" id="S7.SS2.p2.5.m5.1"><semantics id="S7.SS2.p2.5.m5.1a"><mrow id="S7.SS2.p2.5.m5.1.1" xref="S7.SS2.p2.5.m5.1.1.cmml"><mi id="S7.SS2.p2.5.m5.1.1.2" xref="S7.SS2.p2.5.m5.1.1.2.cmml">u</mi><mo id="S7.SS2.p2.5.m5.1.1.1" xref="S7.SS2.p2.5.m5.1.1.1.cmml">∈</mo><msub id="S7.SS2.p2.5.m5.1.1.3" xref="S7.SS2.p2.5.m5.1.1.3.cmml"><mi id="S7.SS2.p2.5.m5.1.1.3.2" xref="S7.SS2.p2.5.m5.1.1.3.2.cmml">T</mi><mi id="S7.SS2.p2.5.m5.1.1.3.3" xref="S7.SS2.p2.5.m5.1.1.3.3.cmml">m</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S7.SS2.p2.5.m5.1b"><apply id="S7.SS2.p2.5.m5.1.1.cmml" xref="S7.SS2.p2.5.m5.1.1"><in id="S7.SS2.p2.5.m5.1.1.1.cmml" xref="S7.SS2.p2.5.m5.1.1.1"></in><ci id="S7.SS2.p2.5.m5.1.1.2.cmml" xref="S7.SS2.p2.5.m5.1.1.2">𝑢</ci><apply id="S7.SS2.p2.5.m5.1.1.3.cmml" xref="S7.SS2.p2.5.m5.1.1.3"><csymbol cd="ambiguous" id="S7.SS2.p2.5.m5.1.1.3.1.cmml" xref="S7.SS2.p2.5.m5.1.1.3">subscript</csymbol><ci id="S7.SS2.p2.5.m5.1.1.3.2.cmml" xref="S7.SS2.p2.5.m5.1.1.3.2">𝑇</ci><ci id="S7.SS2.p2.5.m5.1.1.3.3.cmml" xref="S7.SS2.p2.5.m5.1.1.3.3">𝑚</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.SS2.p2.5.m5.1c">u\in T_{m}</annotation><annotation encoding="application/x-llamapun" id="S7.SS2.p2.5.m5.1d">italic_u ∈ italic_T start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT</annotation></semantics></math>, and <math alttext="h" class="ltx_Math" display="inline" id="S7.SS2.p2.6.m6.1"><semantics id="S7.SS2.p2.6.m6.1a"><mi id="S7.SS2.p2.6.m6.1.1" xref="S7.SS2.p2.6.m6.1.1.cmml">h</mi><annotation-xml encoding="MathML-Content" id="S7.SS2.p2.6.m6.1b"><ci id="S7.SS2.p2.6.m6.1.1.cmml" xref="S7.SS2.p2.6.m6.1.1">ℎ</ci></annotation-xml><annotation encoding="application/x-tex" id="S7.SS2.p2.6.m6.1c">h</annotation><annotation encoding="application/x-llamapun" id="S7.SS2.p2.6.m6.1d">italic_h</annotation></semantics></math>, then we may compute all elements of Definitions <a class="ltx_ref" href="https://arxiv.org/html/2411.12694v2#S7.Thmtheorem10" title="Definition 7.10. ‣ 7.2 Formal algorithm definition. ‣ 7 Results in CONGEST ‣ Local Density and its Distributed Approximation"><span class="ltx_text ltx_ref_tag">7.10</span></a> and <a class="ltx_ref" href="https://arxiv.org/html/2411.12694v2#S7.Thmtheorem11" title="Definition 7.11. ‣ 7.2 Formal algorithm definition. ‣ 7 Results in CONGEST ‣ Local Density and its Distributed Approximation"><span class="ltx_text ltx_ref_tag">7.11</span></a> in <math alttext="\ell" class="ltx_Math" display="inline" id="S7.SS2.p2.7.m7.1"><semantics id="S7.SS2.p2.7.m7.1a"><mi id="S7.SS2.p2.7.m7.1.1" mathvariant="normal" xref="S7.SS2.p2.7.m7.1.1.cmml">ℓ</mi><annotation-xml encoding="MathML-Content" id="S7.SS2.p2.7.m7.1b"><ci id="S7.SS2.p2.7.m7.1.1.cmml" xref="S7.SS2.p2.7.m7.1.1">ℓ</ci></annotation-xml><annotation encoding="application/x-tex" id="S7.SS2.p2.7.m7.1c">\ell</annotation><annotation encoding="application/x-llamapun" id="S7.SS2.p2.7.m7.1d">roman_ℓ</annotation></semantics></math> rounds.</p> </div> <figure class="ltx_float ltx_algorithm" id="alg2"> <div class="ltx_listing ltx_lst_numbers_left ltx_listing" id="alg2.4"> <div class="ltx_listingline" id="alg2.4.1"> <div class="ltx_listing ltx_listing" id="alg2.4.1.1"> <div class="ltx_listingline" id="alg2.l1"> <span class="ltx_text ltx_font_bold" id="alg2.l1.1">for</span> <math alttext="m=0" class="ltx_Math" display="inline" id="alg2.l1.m1.1"><semantics id="alg2.l1.m1.1a"><mrow id="alg2.l1.m1.1.1" xref="alg2.l1.m1.1.1.cmml"><mi id="alg2.l1.m1.1.1.2" xref="alg2.l1.m1.1.1.2.cmml">m</mi><mo id="alg2.l1.m1.1.1.1" xref="alg2.l1.m1.1.1.1.cmml">=</mo><mn id="alg2.l1.m1.1.1.3" xref="alg2.l1.m1.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="alg2.l1.m1.1b"><apply id="alg2.l1.m1.1.1.cmml" xref="alg2.l1.m1.1.1"><eq id="alg2.l1.m1.1.1.1.cmml" xref="alg2.l1.m1.1.1.1"></eq><ci id="alg2.l1.m1.1.1.2.cmml" xref="alg2.l1.m1.1.1.2">𝑚</ci><cn id="alg2.l1.m1.1.1.3.cmml" type="integer" xref="alg2.l1.m1.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="alg2.l1.m1.1c">m=0</annotation><annotation encoding="application/x-llamapun" id="alg2.l1.m1.1d">italic_m = 0</annotation></semantics></math> to <math alttext="2\lceil\eta^{-1}\rceil+2" class="ltx_Math" display="inline" id="alg2.l1.m2.1"><semantics id="alg2.l1.m2.1a"><mrow id="alg2.l1.m2.1.1" xref="alg2.l1.m2.1.1.cmml"><mrow id="alg2.l1.m2.1.1.1" xref="alg2.l1.m2.1.1.1.cmml"><mn id="alg2.l1.m2.1.1.1.3" xref="alg2.l1.m2.1.1.1.3.cmml">2</mn><mo id="alg2.l1.m2.1.1.1.2" xref="alg2.l1.m2.1.1.1.2.cmml">⁢</mo><mrow id="alg2.l1.m2.1.1.1.1.1" xref="alg2.l1.m2.1.1.1.1.2.cmml"><mo id="alg2.l1.m2.1.1.1.1.1.2" stretchy="false" xref="alg2.l1.m2.1.1.1.1.2.1.cmml">⌈</mo><msup id="alg2.l1.m2.1.1.1.1.1.1" xref="alg2.l1.m2.1.1.1.1.1.1.cmml"><mi id="alg2.l1.m2.1.1.1.1.1.1.2" xref="alg2.l1.m2.1.1.1.1.1.1.2.cmml">η</mi><mrow id="alg2.l1.m2.1.1.1.1.1.1.3" xref="alg2.l1.m2.1.1.1.1.1.1.3.cmml"><mo id="alg2.l1.m2.1.1.1.1.1.1.3a" xref="alg2.l1.m2.1.1.1.1.1.1.3.cmml">−</mo><mn id="alg2.l1.m2.1.1.1.1.1.1.3.2" xref="alg2.l1.m2.1.1.1.1.1.1.3.2.cmml">1</mn></mrow></msup><mo id="alg2.l1.m2.1.1.1.1.1.3" stretchy="false" xref="alg2.l1.m2.1.1.1.1.2.1.cmml">⌉</mo></mrow></mrow><mo id="alg2.l1.m2.1.1.2" xref="alg2.l1.m2.1.1.2.cmml">+</mo><mn id="alg2.l1.m2.1.1.3" xref="alg2.l1.m2.1.1.3.cmml">2</mn></mrow><annotation-xml encoding="MathML-Content" id="alg2.l1.m2.1b"><apply id="alg2.l1.m2.1.1.cmml" xref="alg2.l1.m2.1.1"><plus id="alg2.l1.m2.1.1.2.cmml" xref="alg2.l1.m2.1.1.2"></plus><apply id="alg2.l1.m2.1.1.1.cmml" xref="alg2.l1.m2.1.1.1"><times id="alg2.l1.m2.1.1.1.2.cmml" xref="alg2.l1.m2.1.1.1.2"></times><cn id="alg2.l1.m2.1.1.1.3.cmml" type="integer" xref="alg2.l1.m2.1.1.1.3">2</cn><apply id="alg2.l1.m2.1.1.1.1.2.cmml" xref="alg2.l1.m2.1.1.1.1.1"><ceiling id="alg2.l1.m2.1.1.1.1.2.1.cmml" xref="alg2.l1.m2.1.1.1.1.1.2"></ceiling><apply id="alg2.l1.m2.1.1.1.1.1.1.cmml" xref="alg2.l1.m2.1.1.1.1.1.1"><csymbol cd="ambiguous" id="alg2.l1.m2.1.1.1.1.1.1.1.cmml" xref="alg2.l1.m2.1.1.1.1.1.1">superscript</csymbol><ci id="alg2.l1.m2.1.1.1.1.1.1.2.cmml" xref="alg2.l1.m2.1.1.1.1.1.1.2">𝜂</ci><apply id="alg2.l1.m2.1.1.1.1.1.1.3.cmml" xref="alg2.l1.m2.1.1.1.1.1.1.3"><minus id="alg2.l1.m2.1.1.1.1.1.1.3.1.cmml" xref="alg2.l1.m2.1.1.1.1.1.1.3"></minus><cn id="alg2.l1.m2.1.1.1.1.1.1.3.2.cmml" type="integer" xref="alg2.l1.m2.1.1.1.1.1.1.3.2">1</cn></apply></apply></apply></apply><cn id="alg2.l1.m2.1.1.3.cmml" type="integer" xref="alg2.l1.m2.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="alg2.l1.m2.1c">2\lceil\eta^{-1}\rceil+2</annotation><annotation encoding="application/x-llamapun" id="alg2.l1.m2.1d">2 ⌈ italic_η start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT ⌉ + 2</annotation></semantics></math> <span class="ltx_text ltx_font_bold" id="alg2.l1.2">do</span> </div> <div class="ltx_listingline" id="alg2.l2"> <span class="ltx_text ltx_font_bold" id="alg2.l2.1">if</span> <math alttext="m" class="ltx_Math" display="inline" id="alg2.l2.m1.1"><semantics id="alg2.l2.m1.1a"><mi id="alg2.l2.m1.1.1" xref="alg2.l2.m1.1.1.cmml">m</mi><annotation-xml encoding="MathML-Content" id="alg2.l2.m1.1b"><ci id="alg2.l2.m1.1.1.cmml" xref="alg2.l2.m1.1.1">𝑚</ci></annotation-xml><annotation encoding="application/x-tex" id="alg2.l2.m1.1c">m</annotation><annotation encoding="application/x-llamapun" id="alg2.l2.m1.1d">italic_m</annotation></semantics></math> is even <span class="ltx_text ltx_font_bold" id="alg2.l2.2">then</span> </div> <div class="ltx_listingline" id="alg2.l3"> <span class="ltx_text ltx_font_smallcaps" id="alg2.l3.1">evenminute</span>(<math alttext="h" class="ltx_Math" display="inline" id="alg2.l3.m1.1"><semantics id="alg2.l3.m1.1a"><mi id="alg2.l3.m1.1.1" xref="alg2.l3.m1.1.1.cmml">h</mi><annotation-xml encoding="MathML-Content" id="alg2.l3.m1.1b"><ci id="alg2.l3.m1.1.1.cmml" xref="alg2.l3.m1.1.1">ℎ</ci></annotation-xml><annotation encoding="application/x-tex" id="alg2.l3.m1.1c">h</annotation><annotation encoding="application/x-llamapun" id="alg2.l3.m1.1d">italic_h</annotation></semantics></math>, <math alttext="m" class="ltx_Math" display="inline" id="alg2.l3.m2.1"><semantics id="alg2.l3.m2.1a"><mi id="alg2.l3.m2.1.1" xref="alg2.l3.m2.1.1.cmml">m</mi><annotation-xml encoding="MathML-Content" id="alg2.l3.m2.1b"><ci id="alg2.l3.m2.1.1.cmml" xref="alg2.l3.m2.1.1">𝑚</ci></annotation-xml><annotation encoding="application/x-tex" id="alg2.l3.m2.1c">m</annotation><annotation encoding="application/x-llamapun" id="alg2.l3.m2.1d">italic_m</annotation></semantics></math>) </div> <div class="ltx_listingline" id="alg2.l4"> <span class="ltx_text ltx_font_bold" id="alg2.l4.1">else</span> </div> <div class="ltx_listingline" id="alg2.l5"> <span class="ltx_text ltx_font_smallcaps" id="alg2.l5.1">oddminute</span>(<math alttext="h" class="ltx_Math" display="inline" id="alg2.l5.m1.1"><semantics id="alg2.l5.m1.1a"><mi id="alg2.l5.m1.1.1" xref="alg2.l5.m1.1.1.cmml">h</mi><annotation-xml encoding="MathML-Content" id="alg2.l5.m1.1b"><ci id="alg2.l5.m1.1.1.cmml" xref="alg2.l5.m1.1.1">ℎ</ci></annotation-xml><annotation encoding="application/x-tex" id="alg2.l5.m1.1c">h</annotation><annotation encoding="application/x-llamapun" id="alg2.l5.m1.1d">italic_h</annotation></semantics></math>, <math alttext="m" class="ltx_Math" display="inline" id="alg2.l5.m2.1"><semantics id="alg2.l5.m2.1a"><mi id="alg2.l5.m2.1.1" xref="alg2.l5.m2.1.1.cmml">m</mi><annotation-xml encoding="MathML-Content" id="alg2.l5.m2.1b"><ci id="alg2.l5.m2.1.1.cmml" xref="alg2.l5.m2.1.1">𝑚</ci></annotation-xml><annotation encoding="application/x-tex" id="alg2.l5.m2.1c">m</annotation><annotation encoding="application/x-llamapun" id="alg2.l5.m2.1d">italic_m</annotation></semantics></math>) </div> <div class="ltx_listingline" id="alg2.l6"> <span class="ltx_text ltx_font_bold" id="alg2.l6.1">if</span> <math alttext="h>0" class="ltx_Math" display="inline" id="alg2.l6.m1.1"><semantics id="alg2.l6.m1.1a"><mrow id="alg2.l6.m1.1.1" xref="alg2.l6.m1.1.1.cmml"><mi id="alg2.l6.m1.1.1.2" xref="alg2.l6.m1.1.1.2.cmml">h</mi><mo id="alg2.l6.m1.1.1.1" xref="alg2.l6.m1.1.1.1.cmml">></mo><mn id="alg2.l6.m1.1.1.3" xref="alg2.l6.m1.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="alg2.l6.m1.1b"><apply id="alg2.l6.m1.1.1.cmml" xref="alg2.l6.m1.1.1"><gt id="alg2.l6.m1.1.1.1.cmml" xref="alg2.l6.m1.1.1.1"></gt><ci id="alg2.l6.m1.1.1.2.cmml" xref="alg2.l6.m1.1.1.2">ℎ</ci><cn id="alg2.l6.m1.1.1.3.cmml" type="integer" xref="alg2.l6.m1.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="alg2.l6.m1.1c">h>0</annotation><annotation encoding="application/x-llamapun" id="alg2.l6.m1.1d">italic_h > 0</annotation></semantics></math> <span class="ltx_text ltx_font_bold" id="alg2.l6.2">then</span> </div> <div class="ltx_listingline" id="alg2.l7"> <span class="ltx_text ltx_font_smallcaps" id="alg2.l7.1">hour<math alttext="(h-1)" class="ltx_Math" display="inline" id="alg2.l7.1.m1.1"><semantics id="alg2.l7.1.m1.1a"><mrow id="alg2.l7.1.m1.1.1.1" xref="alg2.l7.1.m1.1.1.1.1.cmml"><mo id="alg2.l7.1.m1.1.1.1.2" stretchy="false" xref="alg2.l7.1.m1.1.1.1.1.cmml">(</mo><mrow id="alg2.l7.1.m1.1.1.1.1" xref="alg2.l7.1.m1.1.1.1.1.cmml"><mi id="alg2.l7.1.m1.1.1.1.1.2" xref="alg2.l7.1.m1.1.1.1.1.2.cmml">h</mi><mo id="alg2.l7.1.m1.1.1.1.1.1" xref="alg2.l7.1.m1.1.1.1.1.1.cmml">−</mo><mn id="alg2.l7.1.m1.1.1.1.1.3" xref="alg2.l7.1.m1.1.1.1.1.3.cmml">1</mn></mrow><mo id="alg2.l7.1.m1.1.1.1.3" stretchy="false" xref="alg2.l7.1.m1.1.1.1.1.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="alg2.l7.1.m1.1b"><apply id="alg2.l7.1.m1.1.1.1.1.cmml" xref="alg2.l7.1.m1.1.1.1"><minus id="alg2.l7.1.m1.1.1.1.1.1.cmml" xref="alg2.l7.1.m1.1.1.1.1.1"></minus><ci id="alg2.l7.1.m1.1.1.1.1.2.cmml" xref="alg2.l7.1.m1.1.1.1.1.2">ℎ</ci><cn id="alg2.l7.1.m1.1.1.1.1.3.cmml" type="integer" xref="alg2.l7.1.m1.1.1.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="alg2.l7.1.m1.1c">(h-1)</annotation><annotation encoding="application/x-llamapun" id="alg2.l7.1.m1.1d">( italic_h - 1 )</annotation></semantics></math></span> </div> </div> </div> </div> <figcaption class="ltx_caption"><span class="ltx_tag ltx_tag_float"><span class="ltx_text ltx_font_bold" id="alg2.6.1.1">Algorithm 2</span> </span><span class="ltx_text ltx_font_smallcaps" id="alg2.7.2">hour</span>(<math alttext="h" class="ltx_Math" display="inline" id="alg2.2.m1.1"><semantics id="alg2.2.m1.1b"><mi id="alg2.2.m1.1.1" xref="alg2.2.m1.1.1.cmml">h</mi><annotation-xml encoding="MathML-Content" id="alg2.2.m1.1c"><ci id="alg2.2.m1.1.1.cmml" xref="alg2.2.m1.1.1">ℎ</ci></annotation-xml><annotation encoding="application/x-tex" id="alg2.2.m1.1d">h</annotation><annotation encoding="application/x-llamapun" id="alg2.2.m1.1e">italic_h</annotation></semantics></math>)</figcaption> </figure> <figure class="ltx_float ltx_algorithm" id="alg3"> <div class="ltx_listing ltx_lst_numbers_left ltx_listing" id="alg3.6"> <div class="ltx_listingline" id="alg3.6.1"> <div class="ltx_listing ltx_listing" id="alg3.6.1.1"> <div class="ltx_listingline" id="alg3.l1"> <math alttext="V_{m}:=\{v\in L_{h}\mid v\textnormal{ has at least one violating out-edge }\}" class="ltx_Math" display="inline" id="alg3.l1.m1.2"><semantics id="alg3.l1.m1.2a"><mrow id="alg3.l1.m1.2.2" xref="alg3.l1.m1.2.2.cmml"><msub id="alg3.l1.m1.2.2.4" xref="alg3.l1.m1.2.2.4.cmml"><mi id="alg3.l1.m1.2.2.4.2" xref="alg3.l1.m1.2.2.4.2.cmml">V</mi><mi id="alg3.l1.m1.2.2.4.3" xref="alg3.l1.m1.2.2.4.3.cmml">m</mi></msub><mo id="alg3.l1.m1.2.2.3" lspace="0.278em" rspace="0.278em" xref="alg3.l1.m1.2.2.3.cmml">:=</mo><mrow id="alg3.l1.m1.2.2.2.2" xref="alg3.l1.m1.2.2.2.3.cmml"><mo id="alg3.l1.m1.2.2.2.2.3" stretchy="false" xref="alg3.l1.m1.2.2.2.3.1.cmml">{</mo><mrow id="alg3.l1.m1.1.1.1.1.1" xref="alg3.l1.m1.1.1.1.1.1.cmml"><mi id="alg3.l1.m1.1.1.1.1.1.2" xref="alg3.l1.m1.1.1.1.1.1.2.cmml">v</mi><mo id="alg3.l1.m1.1.1.1.1.1.1" xref="alg3.l1.m1.1.1.1.1.1.1.cmml">∈</mo><msub id="alg3.l1.m1.1.1.1.1.1.3" xref="alg3.l1.m1.1.1.1.1.1.3.cmml"><mi id="alg3.l1.m1.1.1.1.1.1.3.2" xref="alg3.l1.m1.1.1.1.1.1.3.2.cmml">L</mi><mi id="alg3.l1.m1.1.1.1.1.1.3.3" xref="alg3.l1.m1.1.1.1.1.1.3.3.cmml">h</mi></msub></mrow><mo fence="true" id="alg3.l1.m1.2.2.2.2.4" lspace="0em" rspace="0em" xref="alg3.l1.m1.2.2.2.3.1.cmml">∣</mo><mrow id="alg3.l1.m1.2.2.2.2.2" xref="alg3.l1.m1.2.2.2.2.2.cmml"><mi id="alg3.l1.m1.2.2.2.2.2.2" xref="alg3.l1.m1.2.2.2.2.2.2.cmml">v</mi><mo id="alg3.l1.m1.2.2.2.2.2.1" xref="alg3.l1.m1.2.2.2.2.2.1.cmml">⁢</mo><mtext id="alg3.l1.m1.2.2.2.2.2.3" xref="alg3.l1.m1.2.2.2.2.2.3a.cmml"> has at least one violating out-edge </mtext></mrow><mo id="alg3.l1.m1.2.2.2.2.5" stretchy="false" xref="alg3.l1.m1.2.2.2.3.1.cmml">}</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="alg3.l1.m1.2b"><apply id="alg3.l1.m1.2.2.cmml" xref="alg3.l1.m1.2.2"><csymbol cd="latexml" id="alg3.l1.m1.2.2.3.cmml" xref="alg3.l1.m1.2.2.3">assign</csymbol><apply id="alg3.l1.m1.2.2.4.cmml" xref="alg3.l1.m1.2.2.4"><csymbol cd="ambiguous" id="alg3.l1.m1.2.2.4.1.cmml" xref="alg3.l1.m1.2.2.4">subscript</csymbol><ci id="alg3.l1.m1.2.2.4.2.cmml" xref="alg3.l1.m1.2.2.4.2">𝑉</ci><ci id="alg3.l1.m1.2.2.4.3.cmml" xref="alg3.l1.m1.2.2.4.3">𝑚</ci></apply><apply id="alg3.l1.m1.2.2.2.3.cmml" xref="alg3.l1.m1.2.2.2.2"><csymbol cd="latexml" id="alg3.l1.m1.2.2.2.3.1.cmml" xref="alg3.l1.m1.2.2.2.2.3">conditional-set</csymbol><apply id="alg3.l1.m1.1.1.1.1.1.cmml" xref="alg3.l1.m1.1.1.1.1.1"><in id="alg3.l1.m1.1.1.1.1.1.1.cmml" xref="alg3.l1.m1.1.1.1.1.1.1"></in><ci id="alg3.l1.m1.1.1.1.1.1.2.cmml" xref="alg3.l1.m1.1.1.1.1.1.2">𝑣</ci><apply id="alg3.l1.m1.1.1.1.1.1.3.cmml" xref="alg3.l1.m1.1.1.1.1.1.3"><csymbol cd="ambiguous" id="alg3.l1.m1.1.1.1.1.1.3.1.cmml" xref="alg3.l1.m1.1.1.1.1.1.3">subscript</csymbol><ci id="alg3.l1.m1.1.1.1.1.1.3.2.cmml" xref="alg3.l1.m1.1.1.1.1.1.3.2">𝐿</ci><ci id="alg3.l1.m1.1.1.1.1.1.3.3.cmml" xref="alg3.l1.m1.1.1.1.1.1.3.3">ℎ</ci></apply></apply><apply id="alg3.l1.m1.2.2.2.2.2.cmml" xref="alg3.l1.m1.2.2.2.2.2"><times id="alg3.l1.m1.2.2.2.2.2.1.cmml" xref="alg3.l1.m1.2.2.2.2.2.1"></times><ci id="alg3.l1.m1.2.2.2.2.2.2.cmml" xref="alg3.l1.m1.2.2.2.2.2.2">𝑣</ci><ci id="alg3.l1.m1.2.2.2.2.2.3a.cmml" xref="alg3.l1.m1.2.2.2.2.2.3"><mtext id="alg3.l1.m1.2.2.2.2.2.3.cmml" xref="alg3.l1.m1.2.2.2.2.2.3"> has at least one violating out-edge </mtext></ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="alg3.l1.m1.2c">V_{m}:=\{v\in L_{h}\mid v\textnormal{ has at least one violating out-edge }\}</annotation><annotation encoding="application/x-llamapun" id="alg3.l1.m1.2d">italic_V start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT := { italic_v ∈ italic_L start_POSTSUBSCRIPT italic_h end_POSTSUBSCRIPT ∣ italic_v has at least one violating out-edge }</annotation></semantics></math> </div> <div class="ltx_listingline" id="alg3.l2"> <span class="ltx_text ltx_font_bold" id="alg3.l2.1">for</span> <math alttext="t=0" class="ltx_Math" display="inline" id="alg3.l2.m1.1"><semantics id="alg3.l2.m1.1a"><mrow id="alg3.l2.m1.1.1" xref="alg3.l2.m1.1.1.cmml"><mi id="alg3.l2.m1.1.1.2" xref="alg3.l2.m1.1.1.2.cmml">t</mi><mo id="alg3.l2.m1.1.1.1" xref="alg3.l2.m1.1.1.1.cmml">=</mo><mn id="alg3.l2.m1.1.1.3" xref="alg3.l2.m1.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="alg3.l2.m1.1b"><apply id="alg3.l2.m1.1.1.cmml" xref="alg3.l2.m1.1.1"><eq id="alg3.l2.m1.1.1.1.cmml" xref="alg3.l2.m1.1.1.1"></eq><ci id="alg3.l2.m1.1.1.2.cmml" xref="alg3.l2.m1.1.1.2">𝑡</ci><cn id="alg3.l2.m1.1.1.3.cmml" type="integer" xref="alg3.l2.m1.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="alg3.l2.m1.1c">t=0</annotation><annotation encoding="application/x-llamapun" id="alg3.l2.m1.1d">italic_t = 0</annotation></semantics></math> to <math alttext="2\lceil\log_{8/7}n\rceil+1" class="ltx_Math" display="inline" id="alg3.l2.m2.1"><semantics id="alg3.l2.m2.1a"><mrow id="alg3.l2.m2.1.1" xref="alg3.l2.m2.1.1.cmml"><mrow id="alg3.l2.m2.1.1.1" xref="alg3.l2.m2.1.1.1.cmml"><mn id="alg3.l2.m2.1.1.1.3" xref="alg3.l2.m2.1.1.1.3.cmml">2</mn><mo id="alg3.l2.m2.1.1.1.2" xref="alg3.l2.m2.1.1.1.2.cmml">⁢</mo><mrow id="alg3.l2.m2.1.1.1.1.1" xref="alg3.l2.m2.1.1.1.1.2.cmml"><mo id="alg3.l2.m2.1.1.1.1.1.2" stretchy="false" xref="alg3.l2.m2.1.1.1.1.2.1.cmml">⌈</mo><mrow id="alg3.l2.m2.1.1.1.1.1.1" xref="alg3.l2.m2.1.1.1.1.1.1.cmml"><msub id="alg3.l2.m2.1.1.1.1.1.1.1" xref="alg3.l2.m2.1.1.1.1.1.1.1.cmml"><mi id="alg3.l2.m2.1.1.1.1.1.1.1.2" xref="alg3.l2.m2.1.1.1.1.1.1.1.2.cmml">log</mi><mrow id="alg3.l2.m2.1.1.1.1.1.1.1.3" xref="alg3.l2.m2.1.1.1.1.1.1.1.3.cmml"><mn id="alg3.l2.m2.1.1.1.1.1.1.1.3.2" xref="alg3.l2.m2.1.1.1.1.1.1.1.3.2.cmml">8</mn><mo id="alg3.l2.m2.1.1.1.1.1.1.1.3.1" xref="alg3.l2.m2.1.1.1.1.1.1.1.3.1.cmml">/</mo><mn id="alg3.l2.m2.1.1.1.1.1.1.1.3.3" xref="alg3.l2.m2.1.1.1.1.1.1.1.3.3.cmml">7</mn></mrow></msub><mo id="alg3.l2.m2.1.1.1.1.1.1a" lspace="0.167em" xref="alg3.l2.m2.1.1.1.1.1.1.cmml">⁡</mo><mi id="alg3.l2.m2.1.1.1.1.1.1.2" xref="alg3.l2.m2.1.1.1.1.1.1.2.cmml">n</mi></mrow><mo id="alg3.l2.m2.1.1.1.1.1.3" stretchy="false" xref="alg3.l2.m2.1.1.1.1.2.1.cmml">⌉</mo></mrow></mrow><mo id="alg3.l2.m2.1.1.2" xref="alg3.l2.m2.1.1.2.cmml">+</mo><mn id="alg3.l2.m2.1.1.3" xref="alg3.l2.m2.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="alg3.l2.m2.1b"><apply id="alg3.l2.m2.1.1.cmml" xref="alg3.l2.m2.1.1"><plus id="alg3.l2.m2.1.1.2.cmml" xref="alg3.l2.m2.1.1.2"></plus><apply id="alg3.l2.m2.1.1.1.cmml" xref="alg3.l2.m2.1.1.1"><times id="alg3.l2.m2.1.1.1.2.cmml" xref="alg3.l2.m2.1.1.1.2"></times><cn id="alg3.l2.m2.1.1.1.3.cmml" type="integer" xref="alg3.l2.m2.1.1.1.3">2</cn><apply id="alg3.l2.m2.1.1.1.1.2.cmml" xref="alg3.l2.m2.1.1.1.1.1"><ceiling id="alg3.l2.m2.1.1.1.1.2.1.cmml" xref="alg3.l2.m2.1.1.1.1.1.2"></ceiling><apply id="alg3.l2.m2.1.1.1.1.1.1.cmml" xref="alg3.l2.m2.1.1.1.1.1.1"><apply id="alg3.l2.m2.1.1.1.1.1.1.1.cmml" xref="alg3.l2.m2.1.1.1.1.1.1.1"><csymbol cd="ambiguous" id="alg3.l2.m2.1.1.1.1.1.1.1.1.cmml" xref="alg3.l2.m2.1.1.1.1.1.1.1">subscript</csymbol><log id="alg3.l2.m2.1.1.1.1.1.1.1.2.cmml" xref="alg3.l2.m2.1.1.1.1.1.1.1.2"></log><apply id="alg3.l2.m2.1.1.1.1.1.1.1.3.cmml" xref="alg3.l2.m2.1.1.1.1.1.1.1.3"><divide id="alg3.l2.m2.1.1.1.1.1.1.1.3.1.cmml" xref="alg3.l2.m2.1.1.1.1.1.1.1.3.1"></divide><cn id="alg3.l2.m2.1.1.1.1.1.1.1.3.2.cmml" type="integer" xref="alg3.l2.m2.1.1.1.1.1.1.1.3.2">8</cn><cn id="alg3.l2.m2.1.1.1.1.1.1.1.3.3.cmml" type="integer" xref="alg3.l2.m2.1.1.1.1.1.1.1.3.3">7</cn></apply></apply><ci id="alg3.l2.m2.1.1.1.1.1.1.2.cmml" xref="alg3.l2.m2.1.1.1.1.1.1.2">𝑛</ci></apply></apply></apply><cn id="alg3.l2.m2.1.1.3.cmml" type="integer" xref="alg3.l2.m2.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="alg3.l2.m2.1c">2\lceil\log_{8/7}n\rceil+1</annotation><annotation encoding="application/x-llamapun" id="alg3.l2.m2.1d">2 ⌈ roman_log start_POSTSUBSCRIPT 8 / 7 end_POSTSUBSCRIPT italic_n ⌉ + 1</annotation></semantics></math> <span class="ltx_text ltx_font_bold" id="alg3.l2.2">do</span> </div> <div class="ltx_listingline" id="alg3.l3"> Let <math alttext="A_{t}\subset V_{m}" class="ltx_Math" display="inline" id="alg3.l3.m1.1"><semantics id="alg3.l3.m1.1a"><mrow id="alg3.l3.m1.1.1" xref="alg3.l3.m1.1.1.cmml"><msub id="alg3.l3.m1.1.1.2" xref="alg3.l3.m1.1.1.2.cmml"><mi id="alg3.l3.m1.1.1.2.2" xref="alg3.l3.m1.1.1.2.2.cmml">A</mi><mi id="alg3.l3.m1.1.1.2.3" xref="alg3.l3.m1.1.1.2.3.cmml">t</mi></msub><mo id="alg3.l3.m1.1.1.1" xref="alg3.l3.m1.1.1.1.cmml">⊂</mo><msub id="alg3.l3.m1.1.1.3" xref="alg3.l3.m1.1.1.3.cmml"><mi id="alg3.l3.m1.1.1.3.2" xref="alg3.l3.m1.1.1.3.2.cmml">V</mi><mi id="alg3.l3.m1.1.1.3.3" xref="alg3.l3.m1.1.1.3.3.cmml">m</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="alg3.l3.m1.1b"><apply id="alg3.l3.m1.1.1.cmml" xref="alg3.l3.m1.1.1"><subset id="alg3.l3.m1.1.1.1.cmml" xref="alg3.l3.m1.1.1.1"></subset><apply id="alg3.l3.m1.1.1.2.cmml" xref="alg3.l3.m1.1.1.2"><csymbol cd="ambiguous" id="alg3.l3.m1.1.1.2.1.cmml" xref="alg3.l3.m1.1.1.2">subscript</csymbol><ci id="alg3.l3.m1.1.1.2.2.cmml" xref="alg3.l3.m1.1.1.2.2">𝐴</ci><ci id="alg3.l3.m1.1.1.2.3.cmml" xref="alg3.l3.m1.1.1.2.3">𝑡</ci></apply><apply id="alg3.l3.m1.1.1.3.cmml" xref="alg3.l3.m1.1.1.3"><csymbol cd="ambiguous" id="alg3.l3.m1.1.1.3.1.cmml" xref="alg3.l3.m1.1.1.3">subscript</csymbol><ci id="alg3.l3.m1.1.1.3.2.cmml" xref="alg3.l3.m1.1.1.3.2">𝑉</ci><ci id="alg3.l3.m1.1.1.3.3.cmml" xref="alg3.l3.m1.1.1.3.3">𝑚</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="alg3.l3.m1.1c">A_{t}\subset V_{m}</annotation><annotation encoding="application/x-llamapun" id="alg3.l3.m1.1d">italic_A start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT ⊂ italic_V start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT</annotation></semantics></math> be the set of vertices at level <math alttext="h" class="ltx_Math" display="inline" id="alg3.l3.m2.1"><semantics id="alg3.l3.m2.1a"><mi id="alg3.l3.m2.1.1" xref="alg3.l3.m2.1.1.cmml">h</mi><annotation-xml encoding="MathML-Content" id="alg3.l3.m2.1b"><ci id="alg3.l3.m2.1.1.cmml" xref="alg3.l3.m2.1.1">ℎ</ci></annotation-xml><annotation encoding="application/x-tex" id="alg3.l3.m2.1c">h</annotation><annotation encoding="application/x-llamapun" id="alg3.l3.m2.1d">italic_h</annotation></semantics></math> with at least one violating out-edge </div> <div class="ltx_listingline" id="alg3.l4"> Each <math alttext="a\in A_{t}" class="ltx_Math" display="inline" id="alg3.l4.m1.1"><semantics id="alg3.l4.m1.1a"><mrow id="alg3.l4.m1.1.1" xref="alg3.l4.m1.1.1.cmml"><mi id="alg3.l4.m1.1.1.2" xref="alg3.l4.m1.1.1.2.cmml">a</mi><mo id="alg3.l4.m1.1.1.1" xref="alg3.l4.m1.1.1.1.cmml">∈</mo><msub id="alg3.l4.m1.1.1.3" xref="alg3.l4.m1.1.1.3.cmml"><mi id="alg3.l4.m1.1.1.3.2" xref="alg3.l4.m1.1.1.3.2.cmml">A</mi><mi id="alg3.l4.m1.1.1.3.3" xref="alg3.l4.m1.1.1.3.3.cmml">t</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="alg3.l4.m1.1b"><apply id="alg3.l4.m1.1.1.cmml" xref="alg3.l4.m1.1.1"><in id="alg3.l4.m1.1.1.1.cmml" xref="alg3.l4.m1.1.1.1"></in><ci id="alg3.l4.m1.1.1.2.cmml" xref="alg3.l4.m1.1.1.2">𝑎</ci><apply id="alg3.l4.m1.1.1.3.cmml" xref="alg3.l4.m1.1.1.3"><csymbol cd="ambiguous" id="alg3.l4.m1.1.1.3.1.cmml" xref="alg3.l4.m1.1.1.3">subscript</csymbol><ci id="alg3.l4.m1.1.1.3.2.cmml" xref="alg3.l4.m1.1.1.3.2">𝐴</ci><ci id="alg3.l4.m1.1.1.3.3.cmml" xref="alg3.l4.m1.1.1.3.3">𝑡</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="alg3.l4.m1.1c">a\in A_{t}</annotation><annotation encoding="application/x-llamapun" id="alg3.l4.m1.1d">italic_a ∈ italic_A start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT</annotation></semantics></math> determines the set <math alttext="E_{t}(a)" class="ltx_Math" display="inline" id="alg3.l4.m2.1"><semantics id="alg3.l4.m2.1a"><mrow id="alg3.l4.m2.1.2" xref="alg3.l4.m2.1.2.cmml"><msub id="alg3.l4.m2.1.2.2" xref="alg3.l4.m2.1.2.2.cmml"><mi id="alg3.l4.m2.1.2.2.2" xref="alg3.l4.m2.1.2.2.2.cmml">E</mi><mi id="alg3.l4.m2.1.2.2.3" xref="alg3.l4.m2.1.2.2.3.cmml">t</mi></msub><mo id="alg3.l4.m2.1.2.1" xref="alg3.l4.m2.1.2.1.cmml">⁢</mo><mrow id="alg3.l4.m2.1.2.3.2" xref="alg3.l4.m2.1.2.cmml"><mo id="alg3.l4.m2.1.2.3.2.1" stretchy="false" xref="alg3.l4.m2.1.2.cmml">(</mo><mi id="alg3.l4.m2.1.1" xref="alg3.l4.m2.1.1.cmml">a</mi><mo id="alg3.l4.m2.1.2.3.2.2" stretchy="false" xref="alg3.l4.m2.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="alg3.l4.m2.1b"><apply id="alg3.l4.m2.1.2.cmml" xref="alg3.l4.m2.1.2"><times id="alg3.l4.m2.1.2.1.cmml" xref="alg3.l4.m2.1.2.1"></times><apply id="alg3.l4.m2.1.2.2.cmml" xref="alg3.l4.m2.1.2.2"><csymbol cd="ambiguous" id="alg3.l4.m2.1.2.2.1.cmml" xref="alg3.l4.m2.1.2.2">subscript</csymbol><ci id="alg3.l4.m2.1.2.2.2.cmml" xref="alg3.l4.m2.1.2.2.2">𝐸</ci><ci id="alg3.l4.m2.1.2.2.3.cmml" xref="alg3.l4.m2.1.2.2.3">𝑡</ci></apply><ci id="alg3.l4.m2.1.1.cmml" xref="alg3.l4.m2.1.1">𝑎</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="alg3.l4.m2.1c">E_{t}(a)</annotation><annotation encoding="application/x-llamapun" id="alg3.l4.m2.1d">italic_E start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT ( italic_a )</annotation></semantics></math> of violating out-edges. </div> <div class="ltx_listingline" id="alg3.l5"> Each <math alttext="a\in A_{t}" class="ltx_Math" display="inline" id="alg3.l5.m1.1"><semantics id="alg3.l5.m1.1a"><mrow id="alg3.l5.m1.1.1" xref="alg3.l5.m1.1.1.cmml"><mi id="alg3.l5.m1.1.1.2" xref="alg3.l5.m1.1.1.2.cmml">a</mi><mo id="alg3.l5.m1.1.1.1" xref="alg3.l5.m1.1.1.1.cmml">∈</mo><msub id="alg3.l5.m1.1.1.3" xref="alg3.l5.m1.1.1.3.cmml"><mi id="alg3.l5.m1.1.1.3.2" xref="alg3.l5.m1.1.1.3.2.cmml">A</mi><mi id="alg3.l5.m1.1.1.3.3" xref="alg3.l5.m1.1.1.3.3.cmml">t</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="alg3.l5.m1.1b"><apply id="alg3.l5.m1.1.1.cmml" xref="alg3.l5.m1.1.1"><in id="alg3.l5.m1.1.1.1.cmml" xref="alg3.l5.m1.1.1.1"></in><ci id="alg3.l5.m1.1.1.2.cmml" xref="alg3.l5.m1.1.1.2">𝑎</ci><apply id="alg3.l5.m1.1.1.3.cmml" xref="alg3.l5.m1.1.1.3"><csymbol cd="ambiguous" id="alg3.l5.m1.1.1.3.1.cmml" xref="alg3.l5.m1.1.1.3">subscript</csymbol><ci id="alg3.l5.m1.1.1.3.2.cmml" xref="alg3.l5.m1.1.1.3.2">𝐴</ci><ci id="alg3.l5.m1.1.1.3.3.cmml" xref="alg3.l5.m1.1.1.3.3">𝑡</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="alg3.l5.m1.1c">a\in A_{t}</annotation><annotation encoding="application/x-llamapun" id="alg3.l5.m1.1d">italic_a ∈ italic_A start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT</annotation></semantics></math> computes <math alttext="\delta_{t}(a)=\textsl{g}(a)-(1+\eta)^{h-1}" class="ltx_Math" display="inline" id="alg3.l5.m2.3"><semantics id="alg3.l5.m2.3a"><mrow id="alg3.l5.m2.3.3" xref="alg3.l5.m2.3.3.cmml"><mrow id="alg3.l5.m2.3.3.3" xref="alg3.l5.m2.3.3.3.cmml"><msub id="alg3.l5.m2.3.3.3.2" xref="alg3.l5.m2.3.3.3.2.cmml"><mi id="alg3.l5.m2.3.3.3.2.2" xref="alg3.l5.m2.3.3.3.2.2.cmml">δ</mi><mi id="alg3.l5.m2.3.3.3.2.3" xref="alg3.l5.m2.3.3.3.2.3.cmml">t</mi></msub><mo id="alg3.l5.m2.3.3.3.1" xref="alg3.l5.m2.3.3.3.1.cmml">⁢</mo><mrow id="alg3.l5.m2.3.3.3.3.2" xref="alg3.l5.m2.3.3.3.cmml"><mo id="alg3.l5.m2.3.3.3.3.2.1" stretchy="false" xref="alg3.l5.m2.3.3.3.cmml">(</mo><mi id="alg3.l5.m2.1.1" xref="alg3.l5.m2.1.1.cmml">a</mi><mo id="alg3.l5.m2.3.3.3.3.2.2" stretchy="false" xref="alg3.l5.m2.3.3.3.cmml">)</mo></mrow></mrow><mo id="alg3.l5.m2.3.3.2" xref="alg3.l5.m2.3.3.2.cmml">=</mo><mrow id="alg3.l5.m2.3.3.1" xref="alg3.l5.m2.3.3.1.cmml"><mrow id="alg3.l5.m2.3.3.1.3" xref="alg3.l5.m2.3.3.1.3.cmml"><mtext class="ltx_mathvariant_italic" id="alg3.l5.m2.3.3.1.3.2" xref="alg3.l5.m2.3.3.1.3.2a.cmml">g</mtext><mo id="alg3.l5.m2.3.3.1.3.1" xref="alg3.l5.m2.3.3.1.3.1.cmml">⁢</mo><mrow id="alg3.l5.m2.3.3.1.3.3.2" xref="alg3.l5.m2.3.3.1.3.cmml"><mo id="alg3.l5.m2.3.3.1.3.3.2.1" stretchy="false" xref="alg3.l5.m2.3.3.1.3.cmml">(</mo><mi id="alg3.l5.m2.2.2" xref="alg3.l5.m2.2.2.cmml">a</mi><mo id="alg3.l5.m2.3.3.1.3.3.2.2" stretchy="false" xref="alg3.l5.m2.3.3.1.3.cmml">)</mo></mrow></mrow><mo id="alg3.l5.m2.3.3.1.2" xref="alg3.l5.m2.3.3.1.2.cmml">−</mo><msup id="alg3.l5.m2.3.3.1.1" xref="alg3.l5.m2.3.3.1.1.cmml"><mrow id="alg3.l5.m2.3.3.1.1.1.1" xref="alg3.l5.m2.3.3.1.1.1.1.1.cmml"><mo id="alg3.l5.m2.3.3.1.1.1.1.2" stretchy="false" xref="alg3.l5.m2.3.3.1.1.1.1.1.cmml">(</mo><mrow id="alg3.l5.m2.3.3.1.1.1.1.1" xref="alg3.l5.m2.3.3.1.1.1.1.1.cmml"><mn id="alg3.l5.m2.3.3.1.1.1.1.1.2" xref="alg3.l5.m2.3.3.1.1.1.1.1.2.cmml">1</mn><mo id="alg3.l5.m2.3.3.1.1.1.1.1.1" xref="alg3.l5.m2.3.3.1.1.1.1.1.1.cmml">+</mo><mi id="alg3.l5.m2.3.3.1.1.1.1.1.3" xref="alg3.l5.m2.3.3.1.1.1.1.1.3.cmml">η</mi></mrow><mo id="alg3.l5.m2.3.3.1.1.1.1.3" stretchy="false" xref="alg3.l5.m2.3.3.1.1.1.1.1.cmml">)</mo></mrow><mrow id="alg3.l5.m2.3.3.1.1.3" xref="alg3.l5.m2.3.3.1.1.3.cmml"><mi id="alg3.l5.m2.3.3.1.1.3.2" xref="alg3.l5.m2.3.3.1.1.3.2.cmml">h</mi><mo id="alg3.l5.m2.3.3.1.1.3.1" xref="alg3.l5.m2.3.3.1.1.3.1.cmml">−</mo><mn id="alg3.l5.m2.3.3.1.1.3.3" xref="alg3.l5.m2.3.3.1.1.3.3.cmml">1</mn></mrow></msup></mrow></mrow><annotation-xml encoding="MathML-Content" id="alg3.l5.m2.3b"><apply id="alg3.l5.m2.3.3.cmml" xref="alg3.l5.m2.3.3"><eq id="alg3.l5.m2.3.3.2.cmml" xref="alg3.l5.m2.3.3.2"></eq><apply id="alg3.l5.m2.3.3.3.cmml" xref="alg3.l5.m2.3.3.3"><times id="alg3.l5.m2.3.3.3.1.cmml" xref="alg3.l5.m2.3.3.3.1"></times><apply id="alg3.l5.m2.3.3.3.2.cmml" xref="alg3.l5.m2.3.3.3.2"><csymbol cd="ambiguous" id="alg3.l5.m2.3.3.3.2.1.cmml" xref="alg3.l5.m2.3.3.3.2">subscript</csymbol><ci id="alg3.l5.m2.3.3.3.2.2.cmml" xref="alg3.l5.m2.3.3.3.2.2">𝛿</ci><ci id="alg3.l5.m2.3.3.3.2.3.cmml" xref="alg3.l5.m2.3.3.3.2.3">𝑡</ci></apply><ci id="alg3.l5.m2.1.1.cmml" xref="alg3.l5.m2.1.1">𝑎</ci></apply><apply id="alg3.l5.m2.3.3.1.cmml" xref="alg3.l5.m2.3.3.1"><minus id="alg3.l5.m2.3.3.1.2.cmml" xref="alg3.l5.m2.3.3.1.2"></minus><apply id="alg3.l5.m2.3.3.1.3.cmml" xref="alg3.l5.m2.3.3.1.3"><times id="alg3.l5.m2.3.3.1.3.1.cmml" xref="alg3.l5.m2.3.3.1.3.1"></times><ci id="alg3.l5.m2.3.3.1.3.2a.cmml" xref="alg3.l5.m2.3.3.1.3.2"><mtext class="ltx_mathvariant_italic" id="alg3.l5.m2.3.3.1.3.2.cmml" xref="alg3.l5.m2.3.3.1.3.2">g</mtext></ci><ci id="alg3.l5.m2.2.2.cmml" xref="alg3.l5.m2.2.2">𝑎</ci></apply><apply id="alg3.l5.m2.3.3.1.1.cmml" xref="alg3.l5.m2.3.3.1.1"><csymbol cd="ambiguous" id="alg3.l5.m2.3.3.1.1.2.cmml" xref="alg3.l5.m2.3.3.1.1">superscript</csymbol><apply id="alg3.l5.m2.3.3.1.1.1.1.1.cmml" xref="alg3.l5.m2.3.3.1.1.1.1"><plus id="alg3.l5.m2.3.3.1.1.1.1.1.1.cmml" xref="alg3.l5.m2.3.3.1.1.1.1.1.1"></plus><cn id="alg3.l5.m2.3.3.1.1.1.1.1.2.cmml" type="integer" xref="alg3.l5.m2.3.3.1.1.1.1.1.2">1</cn><ci id="alg3.l5.m2.3.3.1.1.1.1.1.3.cmml" xref="alg3.l5.m2.3.3.1.1.1.1.1.3">𝜂</ci></apply><apply id="alg3.l5.m2.3.3.1.1.3.cmml" xref="alg3.l5.m2.3.3.1.1.3"><minus id="alg3.l5.m2.3.3.1.1.3.1.cmml" xref="alg3.l5.m2.3.3.1.1.3.1"></minus><ci id="alg3.l5.m2.3.3.1.1.3.2.cmml" xref="alg3.l5.m2.3.3.1.1.3.2">ℎ</ci><cn id="alg3.l5.m2.3.3.1.1.3.3.cmml" type="integer" xref="alg3.l5.m2.3.3.1.1.3.3">1</cn></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="alg3.l5.m2.3c">\delta_{t}(a)=\textsl{g}(a)-(1+\eta)^{h-1}</annotation><annotation encoding="application/x-llamapun" id="alg3.l5.m2.3d">italic_δ start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT ( italic_a ) = g ( italic_a ) - ( 1 + italic_η ) start_POSTSUPERSCRIPT italic_h - 1 end_POSTSUPERSCRIPT</annotation></semantics></math> and reports <math alttext="\delta_{t}(a)/|E_{t}(a)|" class="ltx_Math" display="inline" id="alg3.l5.m3.3"><semantics id="alg3.l5.m3.3a"><mrow id="alg3.l5.m3.3.3" xref="alg3.l5.m3.3.3.cmml"><mrow id="alg3.l5.m3.3.3.3" xref="alg3.l5.m3.3.3.3.cmml"><msub id="alg3.l5.m3.3.3.3.2" xref="alg3.l5.m3.3.3.3.2.cmml"><mi id="alg3.l5.m3.3.3.3.2.2" xref="alg3.l5.m3.3.3.3.2.2.cmml">δ</mi><mi id="alg3.l5.m3.3.3.3.2.3" xref="alg3.l5.m3.3.3.3.2.3.cmml">t</mi></msub><mo id="alg3.l5.m3.3.3.3.1" xref="alg3.l5.m3.3.3.3.1.cmml">⁢</mo><mrow id="alg3.l5.m3.3.3.3.3.2" xref="alg3.l5.m3.3.3.3.cmml"><mo id="alg3.l5.m3.3.3.3.3.2.1" stretchy="false" xref="alg3.l5.m3.3.3.3.cmml">(</mo><mi id="alg3.l5.m3.1.1" xref="alg3.l5.m3.1.1.cmml">a</mi><mo id="alg3.l5.m3.3.3.3.3.2.2" stretchy="false" xref="alg3.l5.m3.3.3.3.cmml">)</mo></mrow></mrow><mo id="alg3.l5.m3.3.3.2" xref="alg3.l5.m3.3.3.2.cmml">/</mo><mrow id="alg3.l5.m3.3.3.1.1" xref="alg3.l5.m3.3.3.1.2.cmml"><mo id="alg3.l5.m3.3.3.1.1.2" stretchy="false" xref="alg3.l5.m3.3.3.1.2.1.cmml">|</mo><mrow id="alg3.l5.m3.3.3.1.1.1" xref="alg3.l5.m3.3.3.1.1.1.cmml"><msub id="alg3.l5.m3.3.3.1.1.1.2" xref="alg3.l5.m3.3.3.1.1.1.2.cmml"><mi id="alg3.l5.m3.3.3.1.1.1.2.2" xref="alg3.l5.m3.3.3.1.1.1.2.2.cmml">E</mi><mi id="alg3.l5.m3.3.3.1.1.1.2.3" xref="alg3.l5.m3.3.3.1.1.1.2.3.cmml">t</mi></msub><mo id="alg3.l5.m3.3.3.1.1.1.1" xref="alg3.l5.m3.3.3.1.1.1.1.cmml">⁢</mo><mrow id="alg3.l5.m3.3.3.1.1.1.3.2" xref="alg3.l5.m3.3.3.1.1.1.cmml"><mo id="alg3.l5.m3.3.3.1.1.1.3.2.1" stretchy="false" xref="alg3.l5.m3.3.3.1.1.1.cmml">(</mo><mi id="alg3.l5.m3.2.2" xref="alg3.l5.m3.2.2.cmml">a</mi><mo id="alg3.l5.m3.3.3.1.1.1.3.2.2" stretchy="false" xref="alg3.l5.m3.3.3.1.1.1.cmml">)</mo></mrow></mrow><mo id="alg3.l5.m3.3.3.1.1.3" stretchy="false" xref="alg3.l5.m3.3.3.1.2.1.cmml">|</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="alg3.l5.m3.3b"><apply id="alg3.l5.m3.3.3.cmml" xref="alg3.l5.m3.3.3"><divide id="alg3.l5.m3.3.3.2.cmml" xref="alg3.l5.m3.3.3.2"></divide><apply id="alg3.l5.m3.3.3.3.cmml" xref="alg3.l5.m3.3.3.3"><times id="alg3.l5.m3.3.3.3.1.cmml" xref="alg3.l5.m3.3.3.3.1"></times><apply id="alg3.l5.m3.3.3.3.2.cmml" xref="alg3.l5.m3.3.3.3.2"><csymbol cd="ambiguous" id="alg3.l5.m3.3.3.3.2.1.cmml" xref="alg3.l5.m3.3.3.3.2">subscript</csymbol><ci id="alg3.l5.m3.3.3.3.2.2.cmml" xref="alg3.l5.m3.3.3.3.2.2">𝛿</ci><ci id="alg3.l5.m3.3.3.3.2.3.cmml" xref="alg3.l5.m3.3.3.3.2.3">𝑡</ci></apply><ci id="alg3.l5.m3.1.1.cmml" xref="alg3.l5.m3.1.1">𝑎</ci></apply><apply id="alg3.l5.m3.3.3.1.2.cmml" xref="alg3.l5.m3.3.3.1.1"><abs id="alg3.l5.m3.3.3.1.2.1.cmml" xref="alg3.l5.m3.3.3.1.1.2"></abs><apply id="alg3.l5.m3.3.3.1.1.1.cmml" xref="alg3.l5.m3.3.3.1.1.1"><times id="alg3.l5.m3.3.3.1.1.1.1.cmml" xref="alg3.l5.m3.3.3.1.1.1.1"></times><apply id="alg3.l5.m3.3.3.1.1.1.2.cmml" xref="alg3.l5.m3.3.3.1.1.1.2"><csymbol cd="ambiguous" id="alg3.l5.m3.3.3.1.1.1.2.1.cmml" xref="alg3.l5.m3.3.3.1.1.1.2">subscript</csymbol><ci id="alg3.l5.m3.3.3.1.1.1.2.2.cmml" xref="alg3.l5.m3.3.3.1.1.1.2.2">𝐸</ci><ci id="alg3.l5.m3.3.3.1.1.1.2.3.cmml" xref="alg3.l5.m3.3.3.1.1.1.2.3">𝑡</ci></apply><ci id="alg3.l5.m3.2.2.cmml" xref="alg3.l5.m3.2.2">𝑎</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="alg3.l5.m3.3c">\delta_{t}(a)/|E_{t}(a)|</annotation><annotation encoding="application/x-llamapun" id="alg3.l5.m3.3d">italic_δ start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT ( italic_a ) / | italic_E start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT ( italic_a ) |</annotation></semantics></math> across <math alttext="E_{t}(a)" class="ltx_Math" display="inline" id="alg3.l5.m4.1"><semantics id="alg3.l5.m4.1a"><mrow id="alg3.l5.m4.1.2" xref="alg3.l5.m4.1.2.cmml"><msub id="alg3.l5.m4.1.2.2" xref="alg3.l5.m4.1.2.2.cmml"><mi id="alg3.l5.m4.1.2.2.2" xref="alg3.l5.m4.1.2.2.2.cmml">E</mi><mi id="alg3.l5.m4.1.2.2.3" xref="alg3.l5.m4.1.2.2.3.cmml">t</mi></msub><mo id="alg3.l5.m4.1.2.1" xref="alg3.l5.m4.1.2.1.cmml">⁢</mo><mrow id="alg3.l5.m4.1.2.3.2" xref="alg3.l5.m4.1.2.cmml"><mo id="alg3.l5.m4.1.2.3.2.1" stretchy="false" xref="alg3.l5.m4.1.2.cmml">(</mo><mi id="alg3.l5.m4.1.1" xref="alg3.l5.m4.1.1.cmml">a</mi><mo id="alg3.l5.m4.1.2.3.2.2" stretchy="false" xref="alg3.l5.m4.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="alg3.l5.m4.1b"><apply id="alg3.l5.m4.1.2.cmml" xref="alg3.l5.m4.1.2"><times id="alg3.l5.m4.1.2.1.cmml" xref="alg3.l5.m4.1.2.1"></times><apply id="alg3.l5.m4.1.2.2.cmml" xref="alg3.l5.m4.1.2.2"><csymbol cd="ambiguous" id="alg3.l5.m4.1.2.2.1.cmml" xref="alg3.l5.m4.1.2.2">subscript</csymbol><ci id="alg3.l5.m4.1.2.2.2.cmml" xref="alg3.l5.m4.1.2.2.2">𝐸</ci><ci id="alg3.l5.m4.1.2.2.3.cmml" xref="alg3.l5.m4.1.2.2.3">𝑡</ci></apply><ci id="alg3.l5.m4.1.1.cmml" xref="alg3.l5.m4.1.1">𝑎</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="alg3.l5.m4.1c">E_{t}(a)</annotation><annotation encoding="application/x-llamapun" id="alg3.l5.m4.1d">italic_E start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT ( italic_a )</annotation></semantics></math>. </div> <div class="ltx_listingline" id="alg3.6.1.1.1"> <span class="ltx_text ltx_font_typewriter" id="alg3.6.1.1.1.1">/* </span><span class="ltx_text ltx_font_typewriter" id="alg3.6.1.1.1.2">next round:<span class="ltx_text" id="alg3.6.1.1.1.2.1" style="float:right;"> */</span></span> </div> <div class="ltx_listingline" id="alg3.6.1.1.2"> </div> <div class="ltx_listingline" id="alg3.l6"> Let <math alttext="B_{t}" class="ltx_Math" display="inline" id="alg3.l6.m1.1"><semantics id="alg3.l6.m1.1a"><msub id="alg3.l6.m1.1.1" xref="alg3.l6.m1.1.1.cmml"><mi id="alg3.l6.m1.1.1.2" xref="alg3.l6.m1.1.1.2.cmml">B</mi><mi id="alg3.l6.m1.1.1.3" xref="alg3.l6.m1.1.1.3.cmml">t</mi></msub><annotation-xml encoding="MathML-Content" id="alg3.l6.m1.1b"><apply id="alg3.l6.m1.1.1.cmml" xref="alg3.l6.m1.1.1"><csymbol cd="ambiguous" id="alg3.l6.m1.1.1.1.cmml" xref="alg3.l6.m1.1.1">subscript</csymbol><ci id="alg3.l6.m1.1.1.2.cmml" xref="alg3.l6.m1.1.1.2">𝐵</ci><ci id="alg3.l6.m1.1.1.3.cmml" xref="alg3.l6.m1.1.1.3">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="alg3.l6.m1.1c">B_{t}</annotation><annotation encoding="application/x-llamapun" id="alg3.l6.m1.1d">italic_B start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT</annotation></semantics></math> be the set of vertices that receive at least one violating in-edge from <math alttext="A_{t}" class="ltx_Math" display="inline" id="alg3.l6.m2.1"><semantics id="alg3.l6.m2.1a"><msub id="alg3.l6.m2.1.1" xref="alg3.l6.m2.1.1.cmml"><mi id="alg3.l6.m2.1.1.2" xref="alg3.l6.m2.1.1.2.cmml">A</mi><mi id="alg3.l6.m2.1.1.3" xref="alg3.l6.m2.1.1.3.cmml">t</mi></msub><annotation-xml encoding="MathML-Content" id="alg3.l6.m2.1b"><apply id="alg3.l6.m2.1.1.cmml" xref="alg3.l6.m2.1.1"><csymbol cd="ambiguous" id="alg3.l6.m2.1.1.1.cmml" xref="alg3.l6.m2.1.1">subscript</csymbol><ci id="alg3.l6.m2.1.1.2.cmml" xref="alg3.l6.m2.1.1.2">𝐴</ci><ci id="alg3.l6.m2.1.1.3.cmml" xref="alg3.l6.m2.1.1.3">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="alg3.l6.m2.1c">A_{t}</annotation><annotation encoding="application/x-llamapun" id="alg3.l6.m2.1d">italic_A start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT</annotation></semantics></math>. </div> <div class="ltx_listingline" id="alg3.l7"> Each <math alttext="b\in B_{t}" class="ltx_Math" display="inline" id="alg3.l7.m1.1"><semantics id="alg3.l7.m1.1a"><mrow id="alg3.l7.m1.1.1" xref="alg3.l7.m1.1.1.cmml"><mi id="alg3.l7.m1.1.1.2" xref="alg3.l7.m1.1.1.2.cmml">b</mi><mo id="alg3.l7.m1.1.1.1" xref="alg3.l7.m1.1.1.1.cmml">∈</mo><msub id="alg3.l7.m1.1.1.3" xref="alg3.l7.m1.1.1.3.cmml"><mi id="alg3.l7.m1.1.1.3.2" xref="alg3.l7.m1.1.1.3.2.cmml">B</mi><mi id="alg3.l7.m1.1.1.3.3" xref="alg3.l7.m1.1.1.3.3.cmml">t</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="alg3.l7.m1.1b"><apply id="alg3.l7.m1.1.1.cmml" xref="alg3.l7.m1.1.1"><in id="alg3.l7.m1.1.1.1.cmml" xref="alg3.l7.m1.1.1.1"></in><ci id="alg3.l7.m1.1.1.2.cmml" xref="alg3.l7.m1.1.1.2">𝑏</ci><apply id="alg3.l7.m1.1.1.3.cmml" xref="alg3.l7.m1.1.1.3"><csymbol cd="ambiguous" id="alg3.l7.m1.1.1.3.1.cmml" xref="alg3.l7.m1.1.1.3">subscript</csymbol><ci id="alg3.l7.m1.1.1.3.2.cmml" xref="alg3.l7.m1.1.1.3.2">𝐵</ci><ci id="alg3.l7.m1.1.1.3.3.cmml" xref="alg3.l7.m1.1.1.3.3">𝑡</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="alg3.l7.m1.1c">b\in B_{t}</annotation><annotation encoding="application/x-llamapun" id="alg3.l7.m1.1d">italic_b ∈ italic_B start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT</annotation></semantics></math> determines the set <math alttext="I_{t}(b)" class="ltx_Math" display="inline" id="alg3.l7.m2.1"><semantics id="alg3.l7.m2.1a"><mrow id="alg3.l7.m2.1.2" xref="alg3.l7.m2.1.2.cmml"><msub id="alg3.l7.m2.1.2.2" xref="alg3.l7.m2.1.2.2.cmml"><mi id="alg3.l7.m2.1.2.2.2" xref="alg3.l7.m2.1.2.2.2.cmml">I</mi><mi id="alg3.l7.m2.1.2.2.3" xref="alg3.l7.m2.1.2.2.3.cmml">t</mi></msub><mo id="alg3.l7.m2.1.2.1" xref="alg3.l7.m2.1.2.1.cmml">⁢</mo><mrow id="alg3.l7.m2.1.2.3.2" xref="alg3.l7.m2.1.2.cmml"><mo id="alg3.l7.m2.1.2.3.2.1" stretchy="false" xref="alg3.l7.m2.1.2.cmml">(</mo><mi id="alg3.l7.m2.1.1" xref="alg3.l7.m2.1.1.cmml">b</mi><mo id="alg3.l7.m2.1.2.3.2.2" stretchy="false" xref="alg3.l7.m2.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="alg3.l7.m2.1b"><apply id="alg3.l7.m2.1.2.cmml" xref="alg3.l7.m2.1.2"><times id="alg3.l7.m2.1.2.1.cmml" xref="alg3.l7.m2.1.2.1"></times><apply id="alg3.l7.m2.1.2.2.cmml" xref="alg3.l7.m2.1.2.2"><csymbol cd="ambiguous" id="alg3.l7.m2.1.2.2.1.cmml" xref="alg3.l7.m2.1.2.2">subscript</csymbol><ci id="alg3.l7.m2.1.2.2.2.cmml" xref="alg3.l7.m2.1.2.2.2">𝐼</ci><ci id="alg3.l7.m2.1.2.2.3.cmml" xref="alg3.l7.m2.1.2.2.3">𝑡</ci></apply><ci id="alg3.l7.m2.1.1.cmml" xref="alg3.l7.m2.1.1">𝑏</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="alg3.l7.m2.1c">I_{t}(b)</annotation><annotation encoding="application/x-llamapun" id="alg3.l7.m2.1d">italic_I start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT ( italic_b )</annotation></semantics></math> of vertices that reported a value to <math alttext="v" class="ltx_Math" display="inline" id="alg3.l7.m3.1"><semantics id="alg3.l7.m3.1a"><mi id="alg3.l7.m3.1.1" xref="alg3.l7.m3.1.1.cmml">v</mi><annotation-xml encoding="MathML-Content" id="alg3.l7.m3.1b"><ci id="alg3.l7.m3.1.1.cmml" xref="alg3.l7.m3.1.1">𝑣</ci></annotation-xml><annotation encoding="application/x-tex" id="alg3.l7.m3.1c">v</annotation><annotation encoding="application/x-llamapun" id="alg3.l7.m3.1d">italic_v</annotation></semantics></math>. </div> <div class="ltx_listingline" id="alg3.l8"> Each <math alttext="b\in B_{t}" class="ltx_Math" display="inline" id="alg3.l8.m1.1"><semantics id="alg3.l8.m1.1a"><mrow id="alg3.l8.m1.1.1" xref="alg3.l8.m1.1.1.cmml"><mi id="alg3.l8.m1.1.1.2" xref="alg3.l8.m1.1.1.2.cmml">b</mi><mo id="alg3.l8.m1.1.1.1" xref="alg3.l8.m1.1.1.1.cmml">∈</mo><msub id="alg3.l8.m1.1.1.3" xref="alg3.l8.m1.1.1.3.cmml"><mi id="alg3.l8.m1.1.1.3.2" xref="alg3.l8.m1.1.1.3.2.cmml">B</mi><mi id="alg3.l8.m1.1.1.3.3" xref="alg3.l8.m1.1.1.3.3.cmml">t</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="alg3.l8.m1.1b"><apply id="alg3.l8.m1.1.1.cmml" xref="alg3.l8.m1.1.1"><in id="alg3.l8.m1.1.1.1.cmml" xref="alg3.l8.m1.1.1.1"></in><ci id="alg3.l8.m1.1.1.2.cmml" xref="alg3.l8.m1.1.1.2">𝑏</ci><apply id="alg3.l8.m1.1.1.3.cmml" xref="alg3.l8.m1.1.1.3"><csymbol cd="ambiguous" id="alg3.l8.m1.1.1.3.1.cmml" xref="alg3.l8.m1.1.1.3">subscript</csymbol><ci id="alg3.l8.m1.1.1.3.2.cmml" xref="alg3.l8.m1.1.1.3.2">𝐵</ci><ci id="alg3.l8.m1.1.1.3.3.cmml" xref="alg3.l8.m1.1.1.3.3">𝑡</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="alg3.l8.m1.1c">b\in B_{t}</annotation><annotation encoding="application/x-llamapun" id="alg3.l8.m1.1d">italic_b ∈ italic_B start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT</annotation></semantics></math> sorts the <math alttext="a\in I_{t}(b)" class="ltx_Math" display="inline" id="alg3.l8.m2.1"><semantics id="alg3.l8.m2.1a"><mrow id="alg3.l8.m2.1.2" xref="alg3.l8.m2.1.2.cmml"><mi id="alg3.l8.m2.1.2.2" xref="alg3.l8.m2.1.2.2.cmml">a</mi><mo id="alg3.l8.m2.1.2.1" xref="alg3.l8.m2.1.2.1.cmml">∈</mo><mrow id="alg3.l8.m2.1.2.3" xref="alg3.l8.m2.1.2.3.cmml"><msub id="alg3.l8.m2.1.2.3.2" xref="alg3.l8.m2.1.2.3.2.cmml"><mi id="alg3.l8.m2.1.2.3.2.2" xref="alg3.l8.m2.1.2.3.2.2.cmml">I</mi><mi id="alg3.l8.m2.1.2.3.2.3" xref="alg3.l8.m2.1.2.3.2.3.cmml">t</mi></msub><mo id="alg3.l8.m2.1.2.3.1" xref="alg3.l8.m2.1.2.3.1.cmml">⁢</mo><mrow id="alg3.l8.m2.1.2.3.3.2" xref="alg3.l8.m2.1.2.3.cmml"><mo id="alg3.l8.m2.1.2.3.3.2.1" stretchy="false" xref="alg3.l8.m2.1.2.3.cmml">(</mo><mi id="alg3.l8.m2.1.1" xref="alg3.l8.m2.1.1.cmml">b</mi><mo id="alg3.l8.m2.1.2.3.3.2.2" stretchy="false" xref="alg3.l8.m2.1.2.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="alg3.l8.m2.1b"><apply id="alg3.l8.m2.1.2.cmml" xref="alg3.l8.m2.1.2"><in id="alg3.l8.m2.1.2.1.cmml" xref="alg3.l8.m2.1.2.1"></in><ci id="alg3.l8.m2.1.2.2.cmml" xref="alg3.l8.m2.1.2.2">𝑎</ci><apply id="alg3.l8.m2.1.2.3.cmml" xref="alg3.l8.m2.1.2.3"><times id="alg3.l8.m2.1.2.3.1.cmml" xref="alg3.l8.m2.1.2.3.1"></times><apply id="alg3.l8.m2.1.2.3.2.cmml" xref="alg3.l8.m2.1.2.3.2"><csymbol cd="ambiguous" id="alg3.l8.m2.1.2.3.2.1.cmml" xref="alg3.l8.m2.1.2.3.2">subscript</csymbol><ci id="alg3.l8.m2.1.2.3.2.2.cmml" xref="alg3.l8.m2.1.2.3.2.2">𝐼</ci><ci id="alg3.l8.m2.1.2.3.2.3.cmml" xref="alg3.l8.m2.1.2.3.2.3">𝑡</ci></apply><ci id="alg3.l8.m2.1.1.cmml" xref="alg3.l8.m2.1.1">𝑏</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="alg3.l8.m2.1c">a\in I_{t}(b)</annotation><annotation encoding="application/x-llamapun" id="alg3.l8.m2.1d">italic_a ∈ italic_I start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT ( italic_b )</annotation></semantics></math> by <math alttext="\delta(a)" class="ltx_Math" display="inline" id="alg3.l8.m3.1"><semantics id="alg3.l8.m3.1a"><mrow id="alg3.l8.m3.1.2" xref="alg3.l8.m3.1.2.cmml"><mi id="alg3.l8.m3.1.2.2" xref="alg3.l8.m3.1.2.2.cmml">δ</mi><mo id="alg3.l8.m3.1.2.1" xref="alg3.l8.m3.1.2.1.cmml">⁢</mo><mrow id="alg3.l8.m3.1.2.3.2" xref="alg3.l8.m3.1.2.cmml"><mo id="alg3.l8.m3.1.2.3.2.1" stretchy="false" xref="alg3.l8.m3.1.2.cmml">(</mo><mi id="alg3.l8.m3.1.1" xref="alg3.l8.m3.1.1.cmml">a</mi><mo id="alg3.l8.m3.1.2.3.2.2" stretchy="false" xref="alg3.l8.m3.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="alg3.l8.m3.1b"><apply id="alg3.l8.m3.1.2.cmml" xref="alg3.l8.m3.1.2"><times id="alg3.l8.m3.1.2.1.cmml" xref="alg3.l8.m3.1.2.1"></times><ci id="alg3.l8.m3.1.2.2.cmml" xref="alg3.l8.m3.1.2.2">𝛿</ci><ci id="alg3.l8.m3.1.1.cmml" xref="alg3.l8.m3.1.1">𝑎</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="alg3.l8.m3.1c">\delta(a)</annotation><annotation encoding="application/x-llamapun" id="alg3.l8.m3.1d">italic_δ ( italic_a )</annotation></semantics></math>. </div> <div class="ltx_listingline" id="alg3.l9"> Each <math alttext="b\in B_{t}" class="ltx_Math" display="inline" id="alg3.l9.m1.1"><semantics id="alg3.l9.m1.1a"><mrow id="alg3.l9.m1.1.1" xref="alg3.l9.m1.1.1.cmml"><mi id="alg3.l9.m1.1.1.2" xref="alg3.l9.m1.1.1.2.cmml">b</mi><mo id="alg3.l9.m1.1.1.1" xref="alg3.l9.m1.1.1.1.cmml">∈</mo><msub id="alg3.l9.m1.1.1.3" xref="alg3.l9.m1.1.1.3.cmml"><mi id="alg3.l9.m1.1.1.3.2" xref="alg3.l9.m1.1.1.3.2.cmml">B</mi><mi id="alg3.l9.m1.1.1.3.3" xref="alg3.l9.m1.1.1.3.3.cmml">t</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="alg3.l9.m1.1b"><apply id="alg3.l9.m1.1.1.cmml" xref="alg3.l9.m1.1.1"><in id="alg3.l9.m1.1.1.1.cmml" xref="alg3.l9.m1.1.1.1"></in><ci id="alg3.l9.m1.1.1.2.cmml" xref="alg3.l9.m1.1.1.2">𝑏</ci><apply id="alg3.l9.m1.1.1.3.cmml" xref="alg3.l9.m1.1.1.3"><csymbol cd="ambiguous" id="alg3.l9.m1.1.1.3.1.cmml" xref="alg3.l9.m1.1.1.3">subscript</csymbol><ci id="alg3.l9.m1.1.1.3.2.cmml" xref="alg3.l9.m1.1.1.3.2">𝐵</ci><ci id="alg3.l9.m1.1.1.3.3.cmml" xref="alg3.l9.m1.1.1.3.3">𝑡</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="alg3.l9.m1.1c">b\in B_{t}</annotation><annotation encoding="application/x-llamapun" id="alg3.l9.m1.1d">italic_b ∈ italic_B start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT</annotation></semantics></math> greedily decreases <math alttext="\textsl{g}(a\!\to\!b)" class="ltx_Math" display="inline" id="alg3.l9.m2.1"><semantics id="alg3.l9.m2.1a"><mrow id="alg3.l9.m2.1.1" xref="alg3.l9.m2.1.1.cmml"><mtext class="ltx_mathvariant_italic" id="alg3.l9.m2.1.1.3" xref="alg3.l9.m2.1.1.3a.cmml">g</mtext><mo id="alg3.l9.m2.1.1.2" xref="alg3.l9.m2.1.1.2.cmml">⁢</mo><mrow id="alg3.l9.m2.1.1.1.1" xref="alg3.l9.m2.1.1.1.1.1.cmml"><mo id="alg3.l9.m2.1.1.1.1.2" stretchy="false" xref="alg3.l9.m2.1.1.1.1.1.cmml">(</mo><mrow id="alg3.l9.m2.1.1.1.1.1" xref="alg3.l9.m2.1.1.1.1.1.cmml"><mi id="alg3.l9.m2.1.1.1.1.1.2" xref="alg3.l9.m2.1.1.1.1.1.2.cmml">a</mi><mo id="alg3.l9.m2.1.1.1.1.1.1" lspace="0.108em" rspace="0.108em" stretchy="false" xref="alg3.l9.m2.1.1.1.1.1.1.cmml">→</mo><mi id="alg3.l9.m2.1.1.1.1.1.3" xref="alg3.l9.m2.1.1.1.1.1.3.cmml">b</mi></mrow><mo id="alg3.l9.m2.1.1.1.1.3" stretchy="false" xref="alg3.l9.m2.1.1.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="alg3.l9.m2.1b"><apply id="alg3.l9.m2.1.1.cmml" xref="alg3.l9.m2.1.1"><times id="alg3.l9.m2.1.1.2.cmml" xref="alg3.l9.m2.1.1.2"></times><ci id="alg3.l9.m2.1.1.3a.cmml" xref="alg3.l9.m2.1.1.3"><mtext class="ltx_mathvariant_italic" id="alg3.l9.m2.1.1.3.cmml" xref="alg3.l9.m2.1.1.3">g</mtext></ci><apply id="alg3.l9.m2.1.1.1.1.1.cmml" xref="alg3.l9.m2.1.1.1.1"><ci id="alg3.l9.m2.1.1.1.1.1.1.cmml" xref="alg3.l9.m2.1.1.1.1.1.1">→</ci><ci id="alg3.l9.m2.1.1.1.1.1.2.cmml" xref="alg3.l9.m2.1.1.1.1.1.2">𝑎</ci><ci id="alg3.l9.m2.1.1.1.1.1.3.cmml" xref="alg3.l9.m2.1.1.1.1.1.3">𝑏</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="alg3.l9.m2.1c">\textsl{g}(a\!\to\!b)</annotation><annotation encoding="application/x-llamapun" id="alg3.l9.m2.1d">g ( italic_a → italic_b )</annotation></semantics></math> by at most <math alttext="\delta_{t}(a)" class="ltx_Math" display="inline" id="alg3.l9.m3.1"><semantics id="alg3.l9.m3.1a"><mrow id="alg3.l9.m3.1.2" xref="alg3.l9.m3.1.2.cmml"><msub id="alg3.l9.m3.1.2.2" xref="alg3.l9.m3.1.2.2.cmml"><mi id="alg3.l9.m3.1.2.2.2" xref="alg3.l9.m3.1.2.2.2.cmml">δ</mi><mi id="alg3.l9.m3.1.2.2.3" xref="alg3.l9.m3.1.2.2.3.cmml">t</mi></msub><mo id="alg3.l9.m3.1.2.1" xref="alg3.l9.m3.1.2.1.cmml">⁢</mo><mrow id="alg3.l9.m3.1.2.3.2" xref="alg3.l9.m3.1.2.cmml"><mo id="alg3.l9.m3.1.2.3.2.1" stretchy="false" xref="alg3.l9.m3.1.2.cmml">(</mo><mi id="alg3.l9.m3.1.1" xref="alg3.l9.m3.1.1.cmml">a</mi><mo id="alg3.l9.m3.1.2.3.2.2" stretchy="false" xref="alg3.l9.m3.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="alg3.l9.m3.1b"><apply id="alg3.l9.m3.1.2.cmml" xref="alg3.l9.m3.1.2"><times id="alg3.l9.m3.1.2.1.cmml" xref="alg3.l9.m3.1.2.1"></times><apply id="alg3.l9.m3.1.2.2.cmml" xref="alg3.l9.m3.1.2.2"><csymbol cd="ambiguous" id="alg3.l9.m3.1.2.2.1.cmml" xref="alg3.l9.m3.1.2.2">subscript</csymbol><ci id="alg3.l9.m3.1.2.2.2.cmml" xref="alg3.l9.m3.1.2.2.2">𝛿</ci><ci id="alg3.l9.m3.1.2.2.3.cmml" xref="alg3.l9.m3.1.2.2.3">𝑡</ci></apply><ci id="alg3.l9.m3.1.1.cmml" xref="alg3.l9.m3.1.1">𝑎</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="alg3.l9.m3.1c">\delta_{t}(a)</annotation><annotation encoding="application/x-llamapun" id="alg3.l9.m3.1d">italic_δ start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT ( italic_a )</annotation></semantics></math> for <math alttext="a\in I_{t}(b)" class="ltx_Math" display="inline" id="alg3.l9.m4.1"><semantics id="alg3.l9.m4.1a"><mrow id="alg3.l9.m4.1.2" xref="alg3.l9.m4.1.2.cmml"><mi id="alg3.l9.m4.1.2.2" xref="alg3.l9.m4.1.2.2.cmml">a</mi><mo id="alg3.l9.m4.1.2.1" xref="alg3.l9.m4.1.2.1.cmml">∈</mo><mrow id="alg3.l9.m4.1.2.3" xref="alg3.l9.m4.1.2.3.cmml"><msub id="alg3.l9.m4.1.2.3.2" xref="alg3.l9.m4.1.2.3.2.cmml"><mi id="alg3.l9.m4.1.2.3.2.2" xref="alg3.l9.m4.1.2.3.2.2.cmml">I</mi><mi id="alg3.l9.m4.1.2.3.2.3" xref="alg3.l9.m4.1.2.3.2.3.cmml">t</mi></msub><mo id="alg3.l9.m4.1.2.3.1" xref="alg3.l9.m4.1.2.3.1.cmml">⁢</mo><mrow id="alg3.l9.m4.1.2.3.3.2" xref="alg3.l9.m4.1.2.3.cmml"><mo id="alg3.l9.m4.1.2.3.3.2.1" stretchy="false" xref="alg3.l9.m4.1.2.3.cmml">(</mo><mi id="alg3.l9.m4.1.1" xref="alg3.l9.m4.1.1.cmml">b</mi><mo id="alg3.l9.m4.1.2.3.3.2.2" stretchy="false" xref="alg3.l9.m4.1.2.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="alg3.l9.m4.1b"><apply id="alg3.l9.m4.1.2.cmml" xref="alg3.l9.m4.1.2"><in id="alg3.l9.m4.1.2.1.cmml" xref="alg3.l9.m4.1.2.1"></in><ci id="alg3.l9.m4.1.2.2.cmml" xref="alg3.l9.m4.1.2.2">𝑎</ci><apply id="alg3.l9.m4.1.2.3.cmml" xref="alg3.l9.m4.1.2.3"><times id="alg3.l9.m4.1.2.3.1.cmml" xref="alg3.l9.m4.1.2.3.1"></times><apply id="alg3.l9.m4.1.2.3.2.cmml" xref="alg3.l9.m4.1.2.3.2"><csymbol cd="ambiguous" id="alg3.l9.m4.1.2.3.2.1.cmml" xref="alg3.l9.m4.1.2.3.2">subscript</csymbol><ci id="alg3.l9.m4.1.2.3.2.2.cmml" xref="alg3.l9.m4.1.2.3.2.2">𝐼</ci><ci id="alg3.l9.m4.1.2.3.2.3.cmml" xref="alg3.l9.m4.1.2.3.2.3">𝑡</ci></apply><ci id="alg3.l9.m4.1.1.cmml" xref="alg3.l9.m4.1.1">𝑏</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="alg3.l9.m4.1c">a\in I_{t}(b)</annotation><annotation encoding="application/x-llamapun" id="alg3.l9.m4.1d">italic_a ∈ italic_I start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT ( italic_b )</annotation></semantics></math>; until <math alttext="\textsl{g}(b)=(1+\frac{\eta}{2})^{h}" class="ltx_Math" display="inline" id="alg3.l9.m5.2"><semantics id="alg3.l9.m5.2a"><mrow id="alg3.l9.m5.2.2" xref="alg3.l9.m5.2.2.cmml"><mrow id="alg3.l9.m5.2.2.3" xref="alg3.l9.m5.2.2.3.cmml"><mtext class="ltx_mathvariant_italic" id="alg3.l9.m5.2.2.3.2" xref="alg3.l9.m5.2.2.3.2a.cmml">g</mtext><mo id="alg3.l9.m5.2.2.3.1" xref="alg3.l9.m5.2.2.3.1.cmml">⁢</mo><mrow id="alg3.l9.m5.2.2.3.3.2" xref="alg3.l9.m5.2.2.3.cmml"><mo id="alg3.l9.m5.2.2.3.3.2.1" stretchy="false" xref="alg3.l9.m5.2.2.3.cmml">(</mo><mi id="alg3.l9.m5.1.1" xref="alg3.l9.m5.1.1.cmml">b</mi><mo id="alg3.l9.m5.2.2.3.3.2.2" stretchy="false" xref="alg3.l9.m5.2.2.3.cmml">)</mo></mrow></mrow><mo id="alg3.l9.m5.2.2.2" xref="alg3.l9.m5.2.2.2.cmml">=</mo><msup id="alg3.l9.m5.2.2.1" xref="alg3.l9.m5.2.2.1.cmml"><mrow id="alg3.l9.m5.2.2.1.1.1" xref="alg3.l9.m5.2.2.1.1.1.1.cmml"><mo id="alg3.l9.m5.2.2.1.1.1.2" stretchy="false" xref="alg3.l9.m5.2.2.1.1.1.1.cmml">(</mo><mrow id="alg3.l9.m5.2.2.1.1.1.1" xref="alg3.l9.m5.2.2.1.1.1.1.cmml"><mn id="alg3.l9.m5.2.2.1.1.1.1.2" xref="alg3.l9.m5.2.2.1.1.1.1.2.cmml">1</mn><mo id="alg3.l9.m5.2.2.1.1.1.1.1" xref="alg3.l9.m5.2.2.1.1.1.1.1.cmml">+</mo><mfrac id="alg3.l9.m5.2.2.1.1.1.1.3" xref="alg3.l9.m5.2.2.1.1.1.1.3.cmml"><mi id="alg3.l9.m5.2.2.1.1.1.1.3.2" xref="alg3.l9.m5.2.2.1.1.1.1.3.2.cmml">η</mi><mn id="alg3.l9.m5.2.2.1.1.1.1.3.3" xref="alg3.l9.m5.2.2.1.1.1.1.3.3.cmml">2</mn></mfrac></mrow><mo id="alg3.l9.m5.2.2.1.1.1.3" stretchy="false" xref="alg3.l9.m5.2.2.1.1.1.1.cmml">)</mo></mrow><mi id="alg3.l9.m5.2.2.1.3" xref="alg3.l9.m5.2.2.1.3.cmml">h</mi></msup></mrow><annotation-xml encoding="MathML-Content" id="alg3.l9.m5.2b"><apply id="alg3.l9.m5.2.2.cmml" xref="alg3.l9.m5.2.2"><eq id="alg3.l9.m5.2.2.2.cmml" xref="alg3.l9.m5.2.2.2"></eq><apply id="alg3.l9.m5.2.2.3.cmml" xref="alg3.l9.m5.2.2.3"><times id="alg3.l9.m5.2.2.3.1.cmml" xref="alg3.l9.m5.2.2.3.1"></times><ci id="alg3.l9.m5.2.2.3.2a.cmml" xref="alg3.l9.m5.2.2.3.2"><mtext class="ltx_mathvariant_italic" id="alg3.l9.m5.2.2.3.2.cmml" xref="alg3.l9.m5.2.2.3.2">g</mtext></ci><ci id="alg3.l9.m5.1.1.cmml" xref="alg3.l9.m5.1.1">𝑏</ci></apply><apply id="alg3.l9.m5.2.2.1.cmml" xref="alg3.l9.m5.2.2.1"><csymbol cd="ambiguous" id="alg3.l9.m5.2.2.1.2.cmml" xref="alg3.l9.m5.2.2.1">superscript</csymbol><apply id="alg3.l9.m5.2.2.1.1.1.1.cmml" xref="alg3.l9.m5.2.2.1.1.1"><plus id="alg3.l9.m5.2.2.1.1.1.1.1.cmml" xref="alg3.l9.m5.2.2.1.1.1.1.1"></plus><cn id="alg3.l9.m5.2.2.1.1.1.1.2.cmml" type="integer" xref="alg3.l9.m5.2.2.1.1.1.1.2">1</cn><apply id="alg3.l9.m5.2.2.1.1.1.1.3.cmml" xref="alg3.l9.m5.2.2.1.1.1.1.3"><divide id="alg3.l9.m5.2.2.1.1.1.1.3.1.cmml" xref="alg3.l9.m5.2.2.1.1.1.1.3"></divide><ci id="alg3.l9.m5.2.2.1.1.1.1.3.2.cmml" xref="alg3.l9.m5.2.2.1.1.1.1.3.2">𝜂</ci><cn id="alg3.l9.m5.2.2.1.1.1.1.3.3.cmml" type="integer" xref="alg3.l9.m5.2.2.1.1.1.1.3.3">2</cn></apply></apply><ci id="alg3.l9.m5.2.2.1.3.cmml" xref="alg3.l9.m5.2.2.1.3">ℎ</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="alg3.l9.m5.2c">\textsl{g}(b)=(1+\frac{\eta}{2})^{h}</annotation><annotation encoding="application/x-llamapun" id="alg3.l9.m5.2d">g ( italic_b ) = ( 1 + divide start_ARG italic_η end_ARG start_ARG 2 end_ARG ) start_POSTSUPERSCRIPT italic_h end_POSTSUPERSCRIPT</annotation></semantics></math>. </div> </div> </div> </div> <figcaption class="ltx_caption"><span class="ltx_tag ltx_tag_float"><span class="ltx_text ltx_font_bold" id="alg3.8.1.1">Algorithm 3</span> </span><span class="ltx_text ltx_font_smallcaps" id="alg3.9.2">evenminute</span>(int <math alttext="h" class="ltx_Math" display="inline" id="alg3.3.m1.1"><semantics id="alg3.3.m1.1b"><mi id="alg3.3.m1.1.1" xref="alg3.3.m1.1.1.cmml">h</mi><annotation-xml encoding="MathML-Content" id="alg3.3.m1.1c"><ci id="alg3.3.m1.1.1.cmml" xref="alg3.3.m1.1.1">ℎ</ci></annotation-xml><annotation encoding="application/x-tex" id="alg3.3.m1.1d">h</annotation><annotation encoding="application/x-llamapun" id="alg3.3.m1.1e">italic_h</annotation></semantics></math>, int <math alttext="m" class="ltx_Math" display="inline" id="alg3.4.m2.1"><semantics id="alg3.4.m2.1b"><mi id="alg3.4.m2.1.1" xref="alg3.4.m2.1.1.cmml">m</mi><annotation-xml encoding="MathML-Content" id="alg3.4.m2.1c"><ci id="alg3.4.m2.1.1.cmml" xref="alg3.4.m2.1.1">𝑚</ci></annotation-xml><annotation encoding="application/x-tex" id="alg3.4.m2.1d">m</annotation><annotation encoding="application/x-llamapun" id="alg3.4.m2.1e">italic_m</annotation></semantics></math>)</figcaption> </figure> <figure class="ltx_float ltx_algorithm" id="alg4"> <div class="ltx_listing ltx_lst_numbers_left ltx_listing" id="alg4.6"> <div class="ltx_listingline" id="alg4.6.1"> <div class="ltx_listing ltx_listing" id="alg4.6.1.1"> <div class="ltx_listingline" id="alg4.l1"> <math alttext="T_{m}:=\{v\in V_{m-1}\mid v\textnormal{ has at least one violating in-edge }\}" class="ltx_Math" display="inline" id="alg4.l1.m1.2"><semantics id="alg4.l1.m1.2a"><mrow id="alg4.l1.m1.2.2" xref="alg4.l1.m1.2.2.cmml"><msub id="alg4.l1.m1.2.2.4" xref="alg4.l1.m1.2.2.4.cmml"><mi id="alg4.l1.m1.2.2.4.2" xref="alg4.l1.m1.2.2.4.2.cmml">T</mi><mi id="alg4.l1.m1.2.2.4.3" xref="alg4.l1.m1.2.2.4.3.cmml">m</mi></msub><mo id="alg4.l1.m1.2.2.3" lspace="0.278em" rspace="0.278em" xref="alg4.l1.m1.2.2.3.cmml">:=</mo><mrow id="alg4.l1.m1.2.2.2.2" xref="alg4.l1.m1.2.2.2.3.cmml"><mo id="alg4.l1.m1.2.2.2.2.3" stretchy="false" xref="alg4.l1.m1.2.2.2.3.1.cmml">{</mo><mrow id="alg4.l1.m1.1.1.1.1.1" xref="alg4.l1.m1.1.1.1.1.1.cmml"><mi id="alg4.l1.m1.1.1.1.1.1.2" xref="alg4.l1.m1.1.1.1.1.1.2.cmml">v</mi><mo id="alg4.l1.m1.1.1.1.1.1.1" xref="alg4.l1.m1.1.1.1.1.1.1.cmml">∈</mo><msub id="alg4.l1.m1.1.1.1.1.1.3" xref="alg4.l1.m1.1.1.1.1.1.3.cmml"><mi id="alg4.l1.m1.1.1.1.1.1.3.2" xref="alg4.l1.m1.1.1.1.1.1.3.2.cmml">V</mi><mrow id="alg4.l1.m1.1.1.1.1.1.3.3" xref="alg4.l1.m1.1.1.1.1.1.3.3.cmml"><mi id="alg4.l1.m1.1.1.1.1.1.3.3.2" xref="alg4.l1.m1.1.1.1.1.1.3.3.2.cmml">m</mi><mo id="alg4.l1.m1.1.1.1.1.1.3.3.1" xref="alg4.l1.m1.1.1.1.1.1.3.3.1.cmml">−</mo><mn id="alg4.l1.m1.1.1.1.1.1.3.3.3" xref="alg4.l1.m1.1.1.1.1.1.3.3.3.cmml">1</mn></mrow></msub></mrow><mo fence="true" id="alg4.l1.m1.2.2.2.2.4" lspace="0em" rspace="0em" xref="alg4.l1.m1.2.2.2.3.1.cmml">∣</mo><mrow id="alg4.l1.m1.2.2.2.2.2" xref="alg4.l1.m1.2.2.2.2.2.cmml"><mi id="alg4.l1.m1.2.2.2.2.2.2" xref="alg4.l1.m1.2.2.2.2.2.2.cmml">v</mi><mo id="alg4.l1.m1.2.2.2.2.2.1" xref="alg4.l1.m1.2.2.2.2.2.1.cmml">⁢</mo><mtext id="alg4.l1.m1.2.2.2.2.2.3" xref="alg4.l1.m1.2.2.2.2.2.3a.cmml"> has at least one violating in-edge </mtext></mrow><mo id="alg4.l1.m1.2.2.2.2.5" stretchy="false" xref="alg4.l1.m1.2.2.2.3.1.cmml">}</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="alg4.l1.m1.2b"><apply id="alg4.l1.m1.2.2.cmml" xref="alg4.l1.m1.2.2"><csymbol cd="latexml" id="alg4.l1.m1.2.2.3.cmml" xref="alg4.l1.m1.2.2.3">assign</csymbol><apply id="alg4.l1.m1.2.2.4.cmml" xref="alg4.l1.m1.2.2.4"><csymbol cd="ambiguous" id="alg4.l1.m1.2.2.4.1.cmml" xref="alg4.l1.m1.2.2.4">subscript</csymbol><ci id="alg4.l1.m1.2.2.4.2.cmml" xref="alg4.l1.m1.2.2.4.2">𝑇</ci><ci id="alg4.l1.m1.2.2.4.3.cmml" xref="alg4.l1.m1.2.2.4.3">𝑚</ci></apply><apply id="alg4.l1.m1.2.2.2.3.cmml" xref="alg4.l1.m1.2.2.2.2"><csymbol cd="latexml" id="alg4.l1.m1.2.2.2.3.1.cmml" xref="alg4.l1.m1.2.2.2.2.3">conditional-set</csymbol><apply id="alg4.l1.m1.1.1.1.1.1.cmml" xref="alg4.l1.m1.1.1.1.1.1"><in id="alg4.l1.m1.1.1.1.1.1.1.cmml" xref="alg4.l1.m1.1.1.1.1.1.1"></in><ci id="alg4.l1.m1.1.1.1.1.1.2.cmml" xref="alg4.l1.m1.1.1.1.1.1.2">𝑣</ci><apply id="alg4.l1.m1.1.1.1.1.1.3.cmml" xref="alg4.l1.m1.1.1.1.1.1.3"><csymbol cd="ambiguous" id="alg4.l1.m1.1.1.1.1.1.3.1.cmml" xref="alg4.l1.m1.1.1.1.1.1.3">subscript</csymbol><ci id="alg4.l1.m1.1.1.1.1.1.3.2.cmml" xref="alg4.l1.m1.1.1.1.1.1.3.2">𝑉</ci><apply id="alg4.l1.m1.1.1.1.1.1.3.3.cmml" xref="alg4.l1.m1.1.1.1.1.1.3.3"><minus id="alg4.l1.m1.1.1.1.1.1.3.3.1.cmml" xref="alg4.l1.m1.1.1.1.1.1.3.3.1"></minus><ci id="alg4.l1.m1.1.1.1.1.1.3.3.2.cmml" xref="alg4.l1.m1.1.1.1.1.1.3.3.2">𝑚</ci><cn id="alg4.l1.m1.1.1.1.1.1.3.3.3.cmml" type="integer" xref="alg4.l1.m1.1.1.1.1.1.3.3.3">1</cn></apply></apply></apply><apply id="alg4.l1.m1.2.2.2.2.2.cmml" xref="alg4.l1.m1.2.2.2.2.2"><times id="alg4.l1.m1.2.2.2.2.2.1.cmml" xref="alg4.l1.m1.2.2.2.2.2.1"></times><ci id="alg4.l1.m1.2.2.2.2.2.2.cmml" xref="alg4.l1.m1.2.2.2.2.2.2">𝑣</ci><ci id="alg4.l1.m1.2.2.2.2.2.3a.cmml" xref="alg4.l1.m1.2.2.2.2.2.3"><mtext id="alg4.l1.m1.2.2.2.2.2.3.cmml" xref="alg4.l1.m1.2.2.2.2.2.3"> has at least one violating in-edge </mtext></ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="alg4.l1.m1.2c">T_{m}:=\{v\in V_{m-1}\mid v\textnormal{ has at least one violating in-edge }\}</annotation><annotation encoding="application/x-llamapun" id="alg4.l1.m1.2d">italic_T start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT := { italic_v ∈ italic_V start_POSTSUBSCRIPT italic_m - 1 end_POSTSUBSCRIPT ∣ italic_v has at least one violating in-edge }</annotation></semantics></math> </div> <div class="ltx_listingline" id="alg4.l2"> <span class="ltx_text ltx_font_bold" id="alg4.l2.1">for</span> <math alttext="s=\ell" class="ltx_Math" display="inline" id="alg4.l2.m1.1"><semantics id="alg4.l2.m1.1a"><mrow id="alg4.l2.m1.1.1" xref="alg4.l2.m1.1.1.cmml"><mi id="alg4.l2.m1.1.1.2" xref="alg4.l2.m1.1.1.2.cmml">s</mi><mo id="alg4.l2.m1.1.1.1" xref="alg4.l2.m1.1.1.1.cmml">=</mo><mi id="alg4.l2.m1.1.1.3" mathvariant="normal" xref="alg4.l2.m1.1.1.3.cmml">ℓ</mi></mrow><annotation-xml encoding="MathML-Content" id="alg4.l2.m1.1b"><apply id="alg4.l2.m1.1.1.cmml" xref="alg4.l2.m1.1.1"><eq id="alg4.l2.m1.1.1.1.cmml" xref="alg4.l2.m1.1.1.1"></eq><ci id="alg4.l2.m1.1.1.2.cmml" xref="alg4.l2.m1.1.1.2">𝑠</ci><ci id="alg4.l2.m1.1.1.3.cmml" xref="alg4.l2.m1.1.1.3">ℓ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="alg4.l2.m1.1c">s=\ell</annotation><annotation encoding="application/x-llamapun" id="alg4.l2.m1.1d">italic_s = roman_ℓ</annotation></semantics></math> down to <math alttext="h" class="ltx_Math" display="inline" id="alg4.l2.m2.1"><semantics id="alg4.l2.m2.1a"><mi id="alg4.l2.m2.1.1" xref="alg4.l2.m2.1.1.cmml">h</mi><annotation-xml encoding="MathML-Content" id="alg4.l2.m2.1b"><ci id="alg4.l2.m2.1.1.cmml" xref="alg4.l2.m2.1.1">ℎ</ci></annotation-xml><annotation encoding="application/x-tex" id="alg4.l2.m2.1c">h</annotation><annotation encoding="application/x-llamapun" id="alg4.l2.m2.1d">italic_h</annotation></semantics></math> <span class="ltx_text ltx_font_bold" id="alg4.l2.2">do</span> </div> <div class="ltx_listingline" id="alg4.l3"> <span class="ltx_text ltx_font_smallcaps" id="alg4.l3.1">second<math alttext="(h,m,s)" class="ltx_Math" display="inline" id="alg4.l3.1.m1.3"><semantics id="alg4.l3.1.m1.3a"><mrow id="alg4.l3.1.m1.3.4.2" xref="alg4.l3.1.m1.3.4.1.cmml"><mo id="alg4.l3.1.m1.3.4.2.1" stretchy="false" xref="alg4.l3.1.m1.3.4.1.cmml">(</mo><mi id="alg4.l3.1.m1.1.1" xref="alg4.l3.1.m1.1.1.cmml">h</mi><mo id="alg4.l3.1.m1.3.4.2.2" xref="alg4.l3.1.m1.3.4.1.cmml">,</mo><mi id="alg4.l3.1.m1.2.2" xref="alg4.l3.1.m1.2.2.cmml">m</mi><mo id="alg4.l3.1.m1.3.4.2.3" xref="alg4.l3.1.m1.3.4.1.cmml">,</mo><mi id="alg4.l3.1.m1.3.3" xref="alg4.l3.1.m1.3.3.cmml">s</mi><mo id="alg4.l3.1.m1.3.4.2.4" stretchy="false" xref="alg4.l3.1.m1.3.4.1.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="alg4.l3.1.m1.3b"><vector id="alg4.l3.1.m1.3.4.1.cmml" xref="alg4.l3.1.m1.3.4.2"><ci id="alg4.l3.1.m1.1.1.cmml" xref="alg4.l3.1.m1.1.1">ℎ</ci><ci id="alg4.l3.1.m1.2.2.cmml" xref="alg4.l3.1.m1.2.2">𝑚</ci><ci id="alg4.l3.1.m1.3.3.cmml" xref="alg4.l3.1.m1.3.3">𝑠</ci></vector></annotation-xml><annotation encoding="application/x-tex" id="alg4.l3.1.m1.3c">(h,m,s)</annotation><annotation encoding="application/x-llamapun" id="alg4.l3.1.m1.3d">( italic_h , italic_m , italic_s )</annotation></semantics></math></span> </div> </div> </div> </div> <figcaption class="ltx_caption"><span class="ltx_tag ltx_tag_float"><span class="ltx_text ltx_font_bold" id="alg4.8.1.1">Algorithm 4</span> </span><span class="ltx_text ltx_font_smallcaps" id="alg4.9.2">oddminute</span>(int <math alttext="h" class="ltx_Math" display="inline" id="alg4.3.m1.1"><semantics id="alg4.3.m1.1b"><mi id="alg4.3.m1.1.1" xref="alg4.3.m1.1.1.cmml">h</mi><annotation-xml encoding="MathML-Content" id="alg4.3.m1.1c"><ci id="alg4.3.m1.1.1.cmml" xref="alg4.3.m1.1.1">ℎ</ci></annotation-xml><annotation encoding="application/x-tex" id="alg4.3.m1.1d">h</annotation><annotation encoding="application/x-llamapun" id="alg4.3.m1.1e">italic_h</annotation></semantics></math>, int <math alttext="m" class="ltx_Math" display="inline" id="alg4.4.m2.1"><semantics id="alg4.4.m2.1b"><mi id="alg4.4.m2.1.1" xref="alg4.4.m2.1.1.cmml">m</mi><annotation-xml encoding="MathML-Content" id="alg4.4.m2.1c"><ci id="alg4.4.m2.1.1.cmml" xref="alg4.4.m2.1.1">𝑚</ci></annotation-xml><annotation encoding="application/x-tex" id="alg4.4.m2.1d">m</annotation><annotation encoding="application/x-llamapun" id="alg4.4.m2.1e">italic_m</annotation></semantics></math>)</figcaption> </figure> <figure class="ltx_float ltx_algorithm" id="alg5"> <div class="ltx_listing ltx_lst_numbers_left ltx_listing" id="alg5.8"> <div class="ltx_listingline" id="alg5.8.1"> <div class="ltx_listing ltx_listing" id="alg5.8.1.1"> <div class="ltx_listingline" id="alg5.l1"> Compute the graph <math alttext="D_{s}" class="ltx_Math" display="inline" id="alg5.l1.m1.1"><semantics id="alg5.l1.m1.1a"><msub id="alg5.l1.m1.1.1" xref="alg5.l1.m1.1.1.cmml"><mi id="alg5.l1.m1.1.1.2" xref="alg5.l1.m1.1.1.2.cmml">D</mi><mi id="alg5.l1.m1.1.1.3" xref="alg5.l1.m1.1.1.3.cmml">s</mi></msub><annotation-xml encoding="MathML-Content" id="alg5.l1.m1.1b"><apply id="alg5.l1.m1.1.1.cmml" xref="alg5.l1.m1.1.1"><csymbol cd="ambiguous" id="alg5.l1.m1.1.1.1.cmml" xref="alg5.l1.m1.1.1">subscript</csymbol><ci id="alg5.l1.m1.1.1.2.cmml" xref="alg5.l1.m1.1.1.2">𝐷</ci><ci id="alg5.l1.m1.1.1.3.cmml" xref="alg5.l1.m1.1.1.3">𝑠</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="alg5.l1.m1.1c">D_{s}</annotation><annotation encoding="application/x-llamapun" id="alg5.l1.m1.1d">italic_D start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT</annotation></semantics></math> in <math alttext="\ell" class="ltx_Math" display="inline" id="alg5.l1.m2.1"><semantics id="alg5.l1.m2.1a"><mi id="alg5.l1.m2.1.1" mathvariant="normal" xref="alg5.l1.m2.1.1.cmml">ℓ</mi><annotation-xml encoding="MathML-Content" id="alg5.l1.m2.1b"><ci id="alg5.l1.m2.1.1.cmml" xref="alg5.l1.m2.1.1">ℓ</ci></annotation-xml><annotation encoding="application/x-tex" id="alg5.l1.m2.1c">\ell</annotation><annotation encoding="application/x-llamapun" id="alg5.l1.m2.1d">roman_ℓ</annotation></semantics></math> rounds. </div> <div class="ltx_listingline" id="alg5.l2"> Compute the graph <math alttext="D_{s}^{*}" class="ltx_Math" display="inline" id="alg5.l2.m1.1"><semantics id="alg5.l2.m1.1a"><msubsup id="alg5.l2.m1.1.1" xref="alg5.l2.m1.1.1.cmml"><mi id="alg5.l2.m1.1.1.2.2" xref="alg5.l2.m1.1.1.2.2.cmml">D</mi><mi id="alg5.l2.m1.1.1.2.3" xref="alg5.l2.m1.1.1.2.3.cmml">s</mi><mo id="alg5.l2.m1.1.1.3" xref="alg5.l2.m1.1.1.3.cmml">∗</mo></msubsup><annotation-xml encoding="MathML-Content" id="alg5.l2.m1.1b"><apply id="alg5.l2.m1.1.1.cmml" xref="alg5.l2.m1.1.1"><csymbol cd="ambiguous" id="alg5.l2.m1.1.1.1.cmml" xref="alg5.l2.m1.1.1">superscript</csymbol><apply id="alg5.l2.m1.1.1.2.cmml" xref="alg5.l2.m1.1.1"><csymbol cd="ambiguous" id="alg5.l2.m1.1.1.2.1.cmml" xref="alg5.l2.m1.1.1">subscript</csymbol><ci id="alg5.l2.m1.1.1.2.2.cmml" xref="alg5.l2.m1.1.1.2.2">𝐷</ci><ci id="alg5.l2.m1.1.1.2.3.cmml" xref="alg5.l2.m1.1.1.2.3">𝑠</ci></apply><times id="alg5.l2.m1.1.1.3.cmml" xref="alg5.l2.m1.1.1.3"></times></apply></annotation-xml><annotation encoding="application/x-tex" id="alg5.l2.m1.1c">D_{s}^{*}</annotation><annotation encoding="application/x-llamapun" id="alg5.l2.m1.1d">italic_D start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT</annotation></semantics></math> by adding a shared source to <math alttext="S_{s}" class="ltx_Math" display="inline" id="alg5.l2.m2.1"><semantics id="alg5.l2.m2.1a"><msub id="alg5.l2.m2.1.1" xref="alg5.l2.m2.1.1.cmml"><mi id="alg5.l2.m2.1.1.2" xref="alg5.l2.m2.1.1.2.cmml">S</mi><mi id="alg5.l2.m2.1.1.3" xref="alg5.l2.m2.1.1.3.cmml">s</mi></msub><annotation-xml encoding="MathML-Content" id="alg5.l2.m2.1b"><apply id="alg5.l2.m2.1.1.cmml" xref="alg5.l2.m2.1.1"><csymbol cd="ambiguous" id="alg5.l2.m2.1.1.1.cmml" xref="alg5.l2.m2.1.1">subscript</csymbol><ci id="alg5.l2.m2.1.1.2.cmml" xref="alg5.l2.m2.1.1.2">𝑆</ci><ci id="alg5.l2.m2.1.1.3.cmml" xref="alg5.l2.m2.1.1.3">𝑠</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="alg5.l2.m2.1c">S_{s}</annotation><annotation encoding="application/x-llamapun" id="alg5.l2.m2.1d">italic_S start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT</annotation></semantics></math> and a shared sink to <math alttext="T_{m}" class="ltx_Math" display="inline" id="alg5.l2.m3.1"><semantics id="alg5.l2.m3.1a"><msub id="alg5.l2.m3.1.1" xref="alg5.l2.m3.1.1.cmml"><mi id="alg5.l2.m3.1.1.2" xref="alg5.l2.m3.1.1.2.cmml">T</mi><mi id="alg5.l2.m3.1.1.3" xref="alg5.l2.m3.1.1.3.cmml">m</mi></msub><annotation-xml encoding="MathML-Content" id="alg5.l2.m3.1b"><apply id="alg5.l2.m3.1.1.cmml" xref="alg5.l2.m3.1.1"><csymbol cd="ambiguous" id="alg5.l2.m3.1.1.1.cmml" xref="alg5.l2.m3.1.1">subscript</csymbol><ci id="alg5.l2.m3.1.1.2.cmml" xref="alg5.l2.m3.1.1.2">𝑇</ci><ci id="alg5.l2.m3.1.1.3.cmml" xref="alg5.l2.m3.1.1.3">𝑚</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="alg5.l2.m3.1c">T_{m}</annotation><annotation encoding="application/x-llamapun" id="alg5.l2.m3.1d">italic_T start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT</annotation></semantics></math>. </div> <div class="ltx_listingline" id="alg5.l3"> <math alttext="f=" class="ltx_Math" display="inline" id="alg5.l3.m1.1"><semantics id="alg5.l3.m1.1a"><mrow id="alg5.l3.m1.1.1" xref="alg5.l3.m1.1.1.cmml"><mi id="alg5.l3.m1.1.1.2" xref="alg5.l3.m1.1.1.2.cmml">f</mi><mo id="alg5.l3.m1.1.1.1" xref="alg5.l3.m1.1.1.1.cmml">=</mo><mi id="alg5.l3.m1.1.1.3" xref="alg5.l3.m1.1.1.3.cmml"></mi></mrow><annotation-xml encoding="MathML-Content" id="alg5.l3.m1.1b"><apply id="alg5.l3.m1.1.1.cmml" xref="alg5.l3.m1.1.1"><eq id="alg5.l3.m1.1.1.1.cmml" xref="alg5.l3.m1.1.1.1"></eq><ci id="alg5.l3.m1.1.1.2.cmml" xref="alg5.l3.m1.1.1.2">𝑓</ci><csymbol cd="latexml" id="alg5.l3.m1.1.1.3.cmml" xref="alg5.l3.m1.1.1.3">absent</csymbol></apply></annotation-xml><annotation encoding="application/x-tex" id="alg5.l3.m1.1c">f=</annotation><annotation encoding="application/x-llamapun" id="alg5.l3.m1.1d">italic_f =</annotation></semantics></math> <span class="ltx_text ltx_font_smallcaps" id="alg5.l3.1">ComputeBlockingFlow</span>(<math alttext="D_{s}^{*}" class="ltx_Math" display="inline" id="alg5.l3.m2.1"><semantics id="alg5.l3.m2.1a"><msubsup id="alg5.l3.m2.1.1" xref="alg5.l3.m2.1.1.cmml"><mi id="alg5.l3.m2.1.1.2.2" xref="alg5.l3.m2.1.1.2.2.cmml">D</mi><mi id="alg5.l3.m2.1.1.2.3" xref="alg5.l3.m2.1.1.2.3.cmml">s</mi><mo id="alg5.l3.m2.1.1.3" xref="alg5.l3.m2.1.1.3.cmml">∗</mo></msubsup><annotation-xml encoding="MathML-Content" id="alg5.l3.m2.1b"><apply id="alg5.l3.m2.1.1.cmml" xref="alg5.l3.m2.1.1"><csymbol cd="ambiguous" id="alg5.l3.m2.1.1.1.cmml" xref="alg5.l3.m2.1.1">superscript</csymbol><apply id="alg5.l3.m2.1.1.2.cmml" xref="alg5.l3.m2.1.1"><csymbol cd="ambiguous" id="alg5.l3.m2.1.1.2.1.cmml" xref="alg5.l3.m2.1.1">subscript</csymbol><ci id="alg5.l3.m2.1.1.2.2.cmml" xref="alg5.l3.m2.1.1.2.2">𝐷</ci><ci id="alg5.l3.m2.1.1.2.3.cmml" xref="alg5.l3.m2.1.1.2.3">𝑠</ci></apply><times id="alg5.l3.m2.1.1.3.cmml" xref="alg5.l3.m2.1.1.3"></times></apply></annotation-xml><annotation encoding="application/x-tex" id="alg5.l3.m2.1c">D_{s}^{*}</annotation><annotation encoding="application/x-llamapun" id="alg5.l3.m2.1d">italic_D start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT</annotation></semantics></math>) </div> <div class="ltx_listingline" id="alg5.l4"> Flip all edges in <math alttext="D_{s}" class="ltx_Math" display="inline" id="alg5.l4.m1.1"><semantics id="alg5.l4.m1.1a"><msub id="alg5.l4.m1.1.1" xref="alg5.l4.m1.1.1.cmml"><mi id="alg5.l4.m1.1.1.2" xref="alg5.l4.m1.1.1.2.cmml">D</mi><mi id="alg5.l4.m1.1.1.3" xref="alg5.l4.m1.1.1.3.cmml">s</mi></msub><annotation-xml encoding="MathML-Content" id="alg5.l4.m1.1b"><apply id="alg5.l4.m1.1.1.cmml" xref="alg5.l4.m1.1.1"><csymbol cd="ambiguous" id="alg5.l4.m1.1.1.1.cmml" xref="alg5.l4.m1.1.1">subscript</csymbol><ci id="alg5.l4.m1.1.1.2.cmml" xref="alg5.l4.m1.1.1.2">𝐷</ci><ci id="alg5.l4.m1.1.1.3.cmml" xref="alg5.l4.m1.1.1.3">𝑠</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="alg5.l4.m1.1c">D_{s}</annotation><annotation encoding="application/x-llamapun" id="alg5.l4.m1.1d">italic_D start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT</annotation></semantics></math> with the corresponding flow in <math alttext="f" class="ltx_Math" display="inline" id="alg5.l4.m2.1"><semantics id="alg5.l4.m2.1a"><mi id="alg5.l4.m2.1.1" xref="alg5.l4.m2.1.1.cmml">f</mi><annotation-xml encoding="MathML-Content" id="alg5.l4.m2.1b"><ci id="alg5.l4.m2.1.1.cmml" xref="alg5.l4.m2.1.1">𝑓</ci></annotation-xml><annotation encoding="application/x-tex" id="alg5.l4.m2.1c">f</annotation><annotation encoding="application/x-llamapun" id="alg5.l4.m2.1d">italic_f</annotation></semantics></math>. </div> </div> </div> </div> <figcaption class="ltx_caption"><span class="ltx_tag ltx_tag_float"><span class="ltx_text ltx_font_bold" id="alg5.10.1.1">Algorithm 5</span> </span><span class="ltx_text ltx_font_smallcaps" id="alg5.11.2">second</span>(int <math alttext="h" class="ltx_Math" display="inline" id="alg5.4.m1.1"><semantics id="alg5.4.m1.1b"><mi id="alg5.4.m1.1.1" xref="alg5.4.m1.1.1.cmml">h</mi><annotation-xml encoding="MathML-Content" id="alg5.4.m1.1c"><ci id="alg5.4.m1.1.1.cmml" xref="alg5.4.m1.1.1">ℎ</ci></annotation-xml><annotation encoding="application/x-tex" id="alg5.4.m1.1d">h</annotation><annotation encoding="application/x-llamapun" id="alg5.4.m1.1e">italic_h</annotation></semantics></math>, int <math alttext="m" class="ltx_Math" display="inline" id="alg5.5.m2.1"><semantics id="alg5.5.m2.1b"><mi id="alg5.5.m2.1.1" xref="alg5.5.m2.1.1.cmml">m</mi><annotation-xml encoding="MathML-Content" id="alg5.5.m2.1c"><ci id="alg5.5.m2.1.1.cmml" xref="alg5.5.m2.1.1">𝑚</ci></annotation-xml><annotation encoding="application/x-tex" id="alg5.5.m2.1d">m</annotation><annotation encoding="application/x-llamapun" id="alg5.5.m2.1e">italic_m</annotation></semantics></math>, int <math alttext="s" class="ltx_Math" display="inline" id="alg5.6.m3.1"><semantics id="alg5.6.m3.1b"><mi id="alg5.6.m3.1.1" xref="alg5.6.m3.1.1.cmml">s</mi><annotation-xml encoding="MathML-Content" id="alg5.6.m3.1c"><ci id="alg5.6.m3.1.1.cmml" xref="alg5.6.m3.1.1">𝑠</ci></annotation-xml><annotation encoding="application/x-tex" id="alg5.6.m3.1d">s</annotation><annotation encoding="application/x-llamapun" id="alg5.6.m3.1e">italic_s</annotation></semantics></math>)</figcaption> </figure> </section> <section class="ltx_subsection" id="S7.SS3"> <h3 class="ltx_title ltx_title_subsection"> <span class="ltx_tag ltx_tag_subsection">7.3 </span>Sketching our algorithm’s correctness.</h3> <div class="ltx_para" id="S7.SS3.p1"> <p class="ltx_p" id="S7.SS3.p1.2">Per definition, our algorithm runs in <math alttext="O(\eta^{-1}\log n\cdot\ell\cdot(\ell+\textnormal{Blocking}(\ell,n))=O(% \varepsilon^{-3}\log^{4}n\cdot(\varepsilon^{-2}\log^{2}n+\textnormal{Blocking}% (\varepsilon^{-2}\log^{2}n,n)))" class="ltx_math_unparsed" display="inline" id="S7.SS3.p1.1.m1.3"><semantics id="S7.SS3.p1.1.m1.3a"><mrow id="S7.SS3.p1.1.m1.3b"><mi id="S7.SS3.p1.1.m1.3.4">O</mi><mrow id="S7.SS3.p1.1.m1.3.5"><mo id="S7.SS3.p1.1.m1.3.5.1" stretchy="false">(</mo><msup id="S7.SS3.p1.1.m1.3.5.2"><mi id="S7.SS3.p1.1.m1.3.5.2.2">η</mi><mrow id="S7.SS3.p1.1.m1.3.5.2.3"><mo id="S7.SS3.p1.1.m1.3.5.2.3a">−</mo><mn id="S7.SS3.p1.1.m1.3.5.2.3.2">1</mn></mrow></msup><mi id="S7.SS3.p1.1.m1.3.5.3">log</mi><mi id="S7.SS3.p1.1.m1.3.5.4">n</mi><mo id="S7.SS3.p1.1.m1.3.5.5" lspace="0.222em" rspace="0.222em">⋅</mo><mi id="S7.SS3.p1.1.m1.3.5.6" mathvariant="normal">ℓ</mi><mo id="S7.SS3.p1.1.m1.3.5.7" lspace="0.222em" rspace="0.222em">⋅</mo><mrow id="S7.SS3.p1.1.m1.3.5.8"><mo id="S7.SS3.p1.1.m1.3.5.8.1" stretchy="false">(</mo><mi id="S7.SS3.p1.1.m1.3.5.8.2" mathvariant="normal">ℓ</mi><mo id="S7.SS3.p1.1.m1.3.5.8.3">+</mo><mtext id="S7.SS3.p1.1.m1.3.5.8.4">Blocking</mtext><mrow id="S7.SS3.p1.1.m1.3.5.8.5"><mo id="S7.SS3.p1.1.m1.3.5.8.5.1" stretchy="false">(</mo><mi id="S7.SS3.p1.1.m1.1.1" mathvariant="normal">ℓ</mi><mo id="S7.SS3.p1.1.m1.3.5.8.5.2">,</mo><mi id="S7.SS3.p1.1.m1.2.2">n</mi><mo id="S7.SS3.p1.1.m1.3.5.8.5.3" stretchy="false">)</mo></mrow><mo id="S7.SS3.p1.1.m1.3.5.8.6" stretchy="false">)</mo></mrow><mo id="S7.SS3.p1.1.m1.3.5.9">=</mo><mi id="S7.SS3.p1.1.m1.3.5.10">O</mi><mrow id="S7.SS3.p1.1.m1.3.5.11"><mo id="S7.SS3.p1.1.m1.3.5.11.1" stretchy="false">(</mo><msup id="S7.SS3.p1.1.m1.3.5.11.2"><mi id="S7.SS3.p1.1.m1.3.5.11.2.2">ε</mi><mrow id="S7.SS3.p1.1.m1.3.5.11.2.3"><mo id="S7.SS3.p1.1.m1.3.5.11.2.3a">−</mo><mn id="S7.SS3.p1.1.m1.3.5.11.2.3.2">3</mn></mrow></msup><msup id="S7.SS3.p1.1.m1.3.5.11.3"><mi id="S7.SS3.p1.1.m1.3.5.11.3.2">log</mi><mn id="S7.SS3.p1.1.m1.3.5.11.3.3">4</mn></msup><mi id="S7.SS3.p1.1.m1.3.5.11.4">n</mi><mo id="S7.SS3.p1.1.m1.3.5.11.5" lspace="0.222em" rspace="0.222em">⋅</mo><mrow id="S7.SS3.p1.1.m1.3.5.11.6"><mo id="S7.SS3.p1.1.m1.3.5.11.6.1" stretchy="false">(</mo><msup id="S7.SS3.p1.1.m1.3.5.11.6.2"><mi id="S7.SS3.p1.1.m1.3.5.11.6.2.2">ε</mi><mrow id="S7.SS3.p1.1.m1.3.5.11.6.2.3"><mo id="S7.SS3.p1.1.m1.3.5.11.6.2.3a">−</mo><mn id="S7.SS3.p1.1.m1.3.5.11.6.2.3.2">2</mn></mrow></msup><msup id="S7.SS3.p1.1.m1.3.5.11.6.3"><mi id="S7.SS3.p1.1.m1.3.5.11.6.3.2">log</mi><mn id="S7.SS3.p1.1.m1.3.5.11.6.3.3">2</mn></msup><mi id="S7.SS3.p1.1.m1.3.5.11.6.4">n</mi><mo id="S7.SS3.p1.1.m1.3.5.11.6.5">+</mo><mtext id="S7.SS3.p1.1.m1.3.5.11.6.6">Blocking</mtext><mrow id="S7.SS3.p1.1.m1.3.5.11.6.7"><mo id="S7.SS3.p1.1.m1.3.5.11.6.7.1" stretchy="false">(</mo><msup id="S7.SS3.p1.1.m1.3.5.11.6.7.2"><mi id="S7.SS3.p1.1.m1.3.5.11.6.7.2.2">ε</mi><mrow id="S7.SS3.p1.1.m1.3.5.11.6.7.2.3"><mo id="S7.SS3.p1.1.m1.3.5.11.6.7.2.3a">−</mo><mn id="S7.SS3.p1.1.m1.3.5.11.6.7.2.3.2">2</mn></mrow></msup><msup id="S7.SS3.p1.1.m1.3.5.11.6.7.3"><mi id="S7.SS3.p1.1.m1.3.5.11.6.7.3.2">log</mi><mn id="S7.SS3.p1.1.m1.3.5.11.6.7.3.3">2</mn></msup><mi id="S7.SS3.p1.1.m1.3.5.11.6.7.4">n</mi><mo id="S7.SS3.p1.1.m1.3.5.11.6.7.5">,</mo><mi id="S7.SS3.p1.1.m1.3.3">n</mi><mo id="S7.SS3.p1.1.m1.3.5.11.6.7.6" stretchy="false">)</mo></mrow><mo id="S7.SS3.p1.1.m1.3.5.11.6.8" stretchy="false">)</mo></mrow><mo id="S7.SS3.p1.1.m1.3.5.11.7" stretchy="false">)</mo></mrow></mrow></mrow><annotation encoding="application/x-tex" id="S7.SS3.p1.1.m1.3c">O(\eta^{-1}\log n\cdot\ell\cdot(\ell+\textnormal{Blocking}(\ell,n))=O(% \varepsilon^{-3}\log^{4}n\cdot(\varepsilon^{-2}\log^{2}n+\textnormal{Blocking}% (\varepsilon^{-2}\log^{2}n,n)))</annotation><annotation encoding="application/x-llamapun" id="S7.SS3.p1.1.m1.3d">italic_O ( italic_η start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT roman_log italic_n ⋅ roman_ℓ ⋅ ( roman_ℓ + Blocking ( roman_ℓ , italic_n ) ) = italic_O ( italic_ε start_POSTSUPERSCRIPT - 3 end_POSTSUPERSCRIPT roman_log start_POSTSUPERSCRIPT 4 end_POSTSUPERSCRIPT italic_n ⋅ ( italic_ε start_POSTSUPERSCRIPT - 2 end_POSTSUPERSCRIPT roman_log start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_n + Blocking ( italic_ε start_POSTSUPERSCRIPT - 2 end_POSTSUPERSCRIPT roman_log start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_n , italic_n ) ) )</annotation></semantics></math> rounds. What remains is to show that we maintain Invariant <a class="ltx_ref" href="https://arxiv.org/html/2411.12694v2#Thminvariant1" title="Invariant 1. ‣ 7.1 Algorithm overview ‣ 7 Results in CONGEST ‣ Local Density and its Distributed Approximation"><span class="ltx_text ltx_ref_tag">1</span></a>, which implies that we compute an <math alttext="\eta" class="ltx_Math" display="inline" id="S7.SS3.p1.2.m2.1"><semantics id="S7.SS3.p1.2.m2.1a"><mi id="S7.SS3.p1.2.m2.1.1" xref="S7.SS3.p1.2.m2.1.1.cmml">η</mi><annotation-xml encoding="MathML-Content" id="S7.SS3.p1.2.m2.1b"><ci id="S7.SS3.p1.2.m2.1.1.cmml" xref="S7.SS3.p1.2.m2.1.1">𝜂</ci></annotation-xml><annotation encoding="application/x-tex" id="S7.SS3.p1.2.m2.1c">\eta</annotation><annotation encoding="application/x-llamapun" id="S7.SS3.p1.2.m2.1d">italic_η</annotation></semantics></math>-fair orientation.</p> </div> <div class="ltx_para ltx_noindent" id="S7.SS3.p2"> <p class="ltx_p" id="S7.SS3.p2.1">We note that by the choice of our algorithm’s variables, we have the following property:</p> </div> <div class="ltx_para" id="S7.SS3.p3"> <span class="ltx_ERROR undefined" id="S7.SS3.p3.3">{observation}</span> <p class="ltx_p" id="S7.SS3.p3.2">For all times <math alttext="(h:m:s)" class="ltx_Math" display="inline" id="S7.SS3.p3.1.m1.1"><semantics id="S7.SS3.p3.1.m1.1a"><mrow id="S7.SS3.p3.1.m1.1.1.1" xref="S7.SS3.p3.1.m1.1.1.1.1.cmml"><mo id="S7.SS3.p3.1.m1.1.1.1.2" stretchy="false" xref="S7.SS3.p3.1.m1.1.1.1.1.cmml">(</mo><mrow id="S7.SS3.p3.1.m1.1.1.1.1" xref="S7.SS3.p3.1.m1.1.1.1.1.cmml"><mi id="S7.SS3.p3.1.m1.1.1.1.1.2" xref="S7.SS3.p3.1.m1.1.1.1.1.2.cmml">h</mi><mo id="S7.SS3.p3.1.m1.1.1.1.1.3" lspace="0.278em" rspace="0.278em" xref="S7.SS3.p3.1.m1.1.1.1.1.3.cmml">:</mo><mi id="S7.SS3.p3.1.m1.1.1.1.1.4" xref="S7.SS3.p3.1.m1.1.1.1.1.4.cmml">m</mi><mo id="S7.SS3.p3.1.m1.1.1.1.1.5" lspace="0.278em" rspace="0.278em" xref="S7.SS3.p3.1.m1.1.1.1.1.5.cmml">:</mo><mi id="S7.SS3.p3.1.m1.1.1.1.1.6" xref="S7.SS3.p3.1.m1.1.1.1.1.6.cmml">s</mi></mrow><mo id="S7.SS3.p3.1.m1.1.1.1.3" stretchy="false" xref="S7.SS3.p3.1.m1.1.1.1.1.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S7.SS3.p3.1.m1.1b"><apply id="S7.SS3.p3.1.m1.1.1.1.1.cmml" xref="S7.SS3.p3.1.m1.1.1.1"><and id="S7.SS3.p3.1.m1.1.1.1.1a.cmml" xref="S7.SS3.p3.1.m1.1.1.1"></and><apply id="S7.SS3.p3.1.m1.1.1.1.1b.cmml" xref="S7.SS3.p3.1.m1.1.1.1"><ci id="S7.SS3.p3.1.m1.1.1.1.1.3.cmml" xref="S7.SS3.p3.1.m1.1.1.1.1.3">:</ci><ci id="S7.SS3.p3.1.m1.1.1.1.1.2.cmml" xref="S7.SS3.p3.1.m1.1.1.1.1.2">ℎ</ci><ci id="S7.SS3.p3.1.m1.1.1.1.1.4.cmml" xref="S7.SS3.p3.1.m1.1.1.1.1.4">𝑚</ci></apply><apply id="S7.SS3.p3.1.m1.1.1.1.1c.cmml" xref="S7.SS3.p3.1.m1.1.1.1"><ci id="S7.SS3.p3.1.m1.1.1.1.1.5.cmml" xref="S7.SS3.p3.1.m1.1.1.1.1.5">:</ci><share href="https://arxiv.org/html/2411.12694v2#S7.SS3.p3.1.m1.1.1.1.1.4.cmml" id="S7.SS3.p3.1.m1.1.1.1.1d.cmml" xref="S7.SS3.p3.1.m1.1.1.1"></share><ci id="S7.SS3.p3.1.m1.1.1.1.1.6.cmml" xref="S7.SS3.p3.1.m1.1.1.1.1.6">𝑠</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.SS3.p3.1.m1.1c">(h:m:s)</annotation><annotation encoding="application/x-llamapun" id="S7.SS3.p3.1.m1.1d">( italic_h : italic_m : italic_s )</annotation></semantics></math> with <math alttext="m" class="ltx_Math" display="inline" id="S7.SS3.p3.2.m2.1"><semantics id="S7.SS3.p3.2.m2.1a"><mi id="S7.SS3.p3.2.m2.1.1" xref="S7.SS3.p3.2.m2.1.1.cmml">m</mi><annotation-xml encoding="MathML-Content" id="S7.SS3.p3.2.m2.1b"><ci id="S7.SS3.p3.2.m2.1.1.cmml" xref="S7.SS3.p3.2.m2.1.1">𝑚</ci></annotation-xml><annotation encoding="application/x-tex" id="S7.SS3.p3.2.m2.1c">m</annotation><annotation encoding="application/x-llamapun" id="S7.SS3.p3.2.m2.1d">italic_m</annotation></semantics></math> odd:</p> <ul class="ltx_itemize" id="S7.I7"> <li class="ltx_item" id="S7.I7.i1" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">•</span> <div class="ltx_para" id="S7.I7.i1.p1"> <p class="ltx_p" id="S7.I7.i1.p1.4">Vertices at level <math alttext="k>h" class="ltx_Math" display="inline" id="S7.I7.i1.p1.1.m1.1"><semantics id="S7.I7.i1.p1.1.m1.1a"><mrow id="S7.I7.i1.p1.1.m1.1.1" xref="S7.I7.i1.p1.1.m1.1.1.cmml"><mi id="S7.I7.i1.p1.1.m1.1.1.2" xref="S7.I7.i1.p1.1.m1.1.1.2.cmml">k</mi><mo id="S7.I7.i1.p1.1.m1.1.1.1" xref="S7.I7.i1.p1.1.m1.1.1.1.cmml">></mo><mi id="S7.I7.i1.p1.1.m1.1.1.3" xref="S7.I7.i1.p1.1.m1.1.1.3.cmml">h</mi></mrow><annotation-xml encoding="MathML-Content" id="S7.I7.i1.p1.1.m1.1b"><apply id="S7.I7.i1.p1.1.m1.1.1.cmml" xref="S7.I7.i1.p1.1.m1.1.1"><gt id="S7.I7.i1.p1.1.m1.1.1.1.cmml" xref="S7.I7.i1.p1.1.m1.1.1.1"></gt><ci id="S7.I7.i1.p1.1.m1.1.1.2.cmml" xref="S7.I7.i1.p1.1.m1.1.1.2">𝑘</ci><ci id="S7.I7.i1.p1.1.m1.1.1.3.cmml" xref="S7.I7.i1.p1.1.m1.1.1.3">ℎ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.I7.i1.p1.1.m1.1c">k>h</annotation><annotation encoding="application/x-llamapun" id="S7.I7.i1.p1.1.m1.1d">italic_k > italic_h</annotation></semantics></math> only decrease their level and by at most one (because afterwards, <math alttext="l_{s}(u)<l_{m}(u)" class="ltx_Math" display="inline" id="S7.I7.i1.p1.2.m2.2"><semantics id="S7.I7.i1.p1.2.m2.2a"><mrow id="S7.I7.i1.p1.2.m2.2.3" xref="S7.I7.i1.p1.2.m2.2.3.cmml"><mrow id="S7.I7.i1.p1.2.m2.2.3.2" xref="S7.I7.i1.p1.2.m2.2.3.2.cmml"><msub id="S7.I7.i1.p1.2.m2.2.3.2.2" xref="S7.I7.i1.p1.2.m2.2.3.2.2.cmml"><mi id="S7.I7.i1.p1.2.m2.2.3.2.2.2" xref="S7.I7.i1.p1.2.m2.2.3.2.2.2.cmml">l</mi><mi id="S7.I7.i1.p1.2.m2.2.3.2.2.3" xref="S7.I7.i1.p1.2.m2.2.3.2.2.3.cmml">s</mi></msub><mo id="S7.I7.i1.p1.2.m2.2.3.2.1" xref="S7.I7.i1.p1.2.m2.2.3.2.1.cmml">⁢</mo><mrow id="S7.I7.i1.p1.2.m2.2.3.2.3.2" xref="S7.I7.i1.p1.2.m2.2.3.2.cmml"><mo id="S7.I7.i1.p1.2.m2.2.3.2.3.2.1" stretchy="false" xref="S7.I7.i1.p1.2.m2.2.3.2.cmml">(</mo><mi id="S7.I7.i1.p1.2.m2.1.1" xref="S7.I7.i1.p1.2.m2.1.1.cmml">u</mi><mo id="S7.I7.i1.p1.2.m2.2.3.2.3.2.2" stretchy="false" xref="S7.I7.i1.p1.2.m2.2.3.2.cmml">)</mo></mrow></mrow><mo id="S7.I7.i1.p1.2.m2.2.3.1" xref="S7.I7.i1.p1.2.m2.2.3.1.cmml"><</mo><mrow id="S7.I7.i1.p1.2.m2.2.3.3" xref="S7.I7.i1.p1.2.m2.2.3.3.cmml"><msub id="S7.I7.i1.p1.2.m2.2.3.3.2" xref="S7.I7.i1.p1.2.m2.2.3.3.2.cmml"><mi id="S7.I7.i1.p1.2.m2.2.3.3.2.2" xref="S7.I7.i1.p1.2.m2.2.3.3.2.2.cmml">l</mi><mi id="S7.I7.i1.p1.2.m2.2.3.3.2.3" xref="S7.I7.i1.p1.2.m2.2.3.3.2.3.cmml">m</mi></msub><mo id="S7.I7.i1.p1.2.m2.2.3.3.1" xref="S7.I7.i1.p1.2.m2.2.3.3.1.cmml">⁢</mo><mrow id="S7.I7.i1.p1.2.m2.2.3.3.3.2" xref="S7.I7.i1.p1.2.m2.2.3.3.cmml"><mo id="S7.I7.i1.p1.2.m2.2.3.3.3.2.1" stretchy="false" xref="S7.I7.i1.p1.2.m2.2.3.3.cmml">(</mo><mi id="S7.I7.i1.p1.2.m2.2.2" xref="S7.I7.i1.p1.2.m2.2.2.cmml">u</mi><mo id="S7.I7.i1.p1.2.m2.2.3.3.3.2.2" stretchy="false" xref="S7.I7.i1.p1.2.m2.2.3.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S7.I7.i1.p1.2.m2.2b"><apply id="S7.I7.i1.p1.2.m2.2.3.cmml" xref="S7.I7.i1.p1.2.m2.2.3"><lt id="S7.I7.i1.p1.2.m2.2.3.1.cmml" xref="S7.I7.i1.p1.2.m2.2.3.1"></lt><apply id="S7.I7.i1.p1.2.m2.2.3.2.cmml" xref="S7.I7.i1.p1.2.m2.2.3.2"><times id="S7.I7.i1.p1.2.m2.2.3.2.1.cmml" xref="S7.I7.i1.p1.2.m2.2.3.2.1"></times><apply id="S7.I7.i1.p1.2.m2.2.3.2.2.cmml" xref="S7.I7.i1.p1.2.m2.2.3.2.2"><csymbol cd="ambiguous" id="S7.I7.i1.p1.2.m2.2.3.2.2.1.cmml" xref="S7.I7.i1.p1.2.m2.2.3.2.2">subscript</csymbol><ci id="S7.I7.i1.p1.2.m2.2.3.2.2.2.cmml" xref="S7.I7.i1.p1.2.m2.2.3.2.2.2">𝑙</ci><ci id="S7.I7.i1.p1.2.m2.2.3.2.2.3.cmml" xref="S7.I7.i1.p1.2.m2.2.3.2.2.3">𝑠</ci></apply><ci id="S7.I7.i1.p1.2.m2.1.1.cmml" xref="S7.I7.i1.p1.2.m2.1.1">𝑢</ci></apply><apply id="S7.I7.i1.p1.2.m2.2.3.3.cmml" xref="S7.I7.i1.p1.2.m2.2.3.3"><times id="S7.I7.i1.p1.2.m2.2.3.3.1.cmml" xref="S7.I7.i1.p1.2.m2.2.3.3.1"></times><apply id="S7.I7.i1.p1.2.m2.2.3.3.2.cmml" xref="S7.I7.i1.p1.2.m2.2.3.3.2"><csymbol cd="ambiguous" id="S7.I7.i1.p1.2.m2.2.3.3.2.1.cmml" xref="S7.I7.i1.p1.2.m2.2.3.3.2">subscript</csymbol><ci id="S7.I7.i1.p1.2.m2.2.3.3.2.2.cmml" xref="S7.I7.i1.p1.2.m2.2.3.3.2.2">𝑙</ci><ci id="S7.I7.i1.p1.2.m2.2.3.3.2.3.cmml" xref="S7.I7.i1.p1.2.m2.2.3.3.2.3">𝑚</ci></apply><ci id="S7.I7.i1.p1.2.m2.2.2.cmml" xref="S7.I7.i1.p1.2.m2.2.2">𝑢</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.I7.i1.p1.2.m2.2c">l_{s}(u)<l_{m}(u)</annotation><annotation encoding="application/x-llamapun" id="S7.I7.i1.p1.2.m2.2d">italic_l start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_u ) < italic_l start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT ( italic_u )</annotation></semantics></math> and thus <math alttext="u" class="ltx_Math" display="inline" id="S7.I7.i1.p1.3.m3.1"><semantics id="S7.I7.i1.p1.3.m3.1a"><mi id="S7.I7.i1.p1.3.m3.1.1" xref="S7.I7.i1.p1.3.m3.1.1.cmml">u</mi><annotation-xml encoding="MathML-Content" id="S7.I7.i1.p1.3.m3.1b"><ci id="S7.I7.i1.p1.3.m3.1.1.cmml" xref="S7.I7.i1.p1.3.m3.1.1">𝑢</ci></annotation-xml><annotation encoding="application/x-tex" id="S7.I7.i1.p1.3.m3.1c">u</annotation><annotation encoding="application/x-llamapun" id="S7.I7.i1.p1.3.m3.1d">italic_u</annotation></semantics></math> has no out-edges in <math alttext="E_{s}" class="ltx_Math" display="inline" id="S7.I7.i1.p1.4.m4.1"><semantics id="S7.I7.i1.p1.4.m4.1a"><msub id="S7.I7.i1.p1.4.m4.1.1" xref="S7.I7.i1.p1.4.m4.1.1.cmml"><mi id="S7.I7.i1.p1.4.m4.1.1.2" xref="S7.I7.i1.p1.4.m4.1.1.2.cmml">E</mi><mi id="S7.I7.i1.p1.4.m4.1.1.3" xref="S7.I7.i1.p1.4.m4.1.1.3.cmml">s</mi></msub><annotation-xml encoding="MathML-Content" id="S7.I7.i1.p1.4.m4.1b"><apply id="S7.I7.i1.p1.4.m4.1.1.cmml" xref="S7.I7.i1.p1.4.m4.1.1"><csymbol cd="ambiguous" id="S7.I7.i1.p1.4.m4.1.1.1.cmml" xref="S7.I7.i1.p1.4.m4.1.1">subscript</csymbol><ci id="S7.I7.i1.p1.4.m4.1.1.2.cmml" xref="S7.I7.i1.p1.4.m4.1.1.2">𝐸</ci><ci id="S7.I7.i1.p1.4.m4.1.1.3.cmml" xref="S7.I7.i1.p1.4.m4.1.1.3">𝑠</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.I7.i1.p1.4.m4.1c">E_{s}</annotation><annotation encoding="application/x-llamapun" id="S7.I7.i1.p1.4.m4.1d">italic_E start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT</annotation></semantics></math>),</p> </div> </li> <li class="ltx_item" id="S7.I7.i2" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">•</span> <div class="ltx_para" id="S7.I7.i2.p1"> <p class="ltx_p" id="S7.I7.i2.p1.2">vertices at level <math alttext="h-1" class="ltx_Math" display="inline" id="S7.I7.i2.p1.1.m1.1"><semantics id="S7.I7.i2.p1.1.m1.1a"><mrow id="S7.I7.i2.p1.1.m1.1.1" xref="S7.I7.i2.p1.1.m1.1.1.cmml"><mi id="S7.I7.i2.p1.1.m1.1.1.2" xref="S7.I7.i2.p1.1.m1.1.1.2.cmml">h</mi><mo id="S7.I7.i2.p1.1.m1.1.1.1" xref="S7.I7.i2.p1.1.m1.1.1.1.cmml">−</mo><mn id="S7.I7.i2.p1.1.m1.1.1.3" xref="S7.I7.i2.p1.1.m1.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S7.I7.i2.p1.1.m1.1b"><apply id="S7.I7.i2.p1.1.m1.1.1.cmml" xref="S7.I7.i2.p1.1.m1.1.1"><minus id="S7.I7.i2.p1.1.m1.1.1.1.cmml" xref="S7.I7.i2.p1.1.m1.1.1.1"></minus><ci id="S7.I7.i2.p1.1.m1.1.1.2.cmml" xref="S7.I7.i2.p1.1.m1.1.1.2">ℎ</ci><cn id="S7.I7.i2.p1.1.m1.1.1.3.cmml" type="integer" xref="S7.I7.i2.p1.1.m1.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.I7.i2.p1.1.m1.1c">h-1</annotation><annotation encoding="application/x-llamapun" id="S7.I7.i2.p1.1.m1.1d">italic_h - 1</annotation></semantics></math> only increase their level (by at most <math alttext="1" class="ltx_Math" display="inline" id="S7.I7.i2.p1.2.m2.1"><semantics id="S7.I7.i2.p1.2.m2.1a"><mn id="S7.I7.i2.p1.2.m2.1.1" xref="S7.I7.i2.p1.2.m2.1.1.cmml">1</mn><annotation-xml encoding="MathML-Content" id="S7.I7.i2.p1.2.m2.1b"><cn id="S7.I7.i2.p1.2.m2.1.1.cmml" type="integer" xref="S7.I7.i2.p1.2.m2.1.1">1</cn></annotation-xml><annotation encoding="application/x-tex" id="S7.I7.i2.p1.2.m2.1c">1</annotation><annotation encoding="application/x-llamapun" id="S7.I7.i2.p1.2.m2.1d">1</annotation></semantics></math>), and</p> </div> </li> <li class="ltx_item" id="S7.I7.i3" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">•</span> <div class="ltx_para" id="S7.I7.i3.p1"> <p class="ltx_p" id="S7.I7.i3.p1.2">vertices at level <math alttext="k^{\prime}<h-1" class="ltx_Math" display="inline" id="S7.I7.i3.p1.1.m1.1"><semantics id="S7.I7.i3.p1.1.m1.1a"><mrow id="S7.I7.i3.p1.1.m1.1.1" xref="S7.I7.i3.p1.1.m1.1.1.cmml"><msup id="S7.I7.i3.p1.1.m1.1.1.2" xref="S7.I7.i3.p1.1.m1.1.1.2.cmml"><mi id="S7.I7.i3.p1.1.m1.1.1.2.2" xref="S7.I7.i3.p1.1.m1.1.1.2.2.cmml">k</mi><mo id="S7.I7.i3.p1.1.m1.1.1.2.3" xref="S7.I7.i3.p1.1.m1.1.1.2.3.cmml">′</mo></msup><mo id="S7.I7.i3.p1.1.m1.1.1.1" xref="S7.I7.i3.p1.1.m1.1.1.1.cmml"><</mo><mrow id="S7.I7.i3.p1.1.m1.1.1.3" xref="S7.I7.i3.p1.1.m1.1.1.3.cmml"><mi id="S7.I7.i3.p1.1.m1.1.1.3.2" xref="S7.I7.i3.p1.1.m1.1.1.3.2.cmml">h</mi><mo id="S7.I7.i3.p1.1.m1.1.1.3.1" xref="S7.I7.i3.p1.1.m1.1.1.3.1.cmml">−</mo><mn id="S7.I7.i3.p1.1.m1.1.1.3.3" xref="S7.I7.i3.p1.1.m1.1.1.3.3.cmml">1</mn></mrow></mrow><annotation-xml encoding="MathML-Content" id="S7.I7.i3.p1.1.m1.1b"><apply id="S7.I7.i3.p1.1.m1.1.1.cmml" xref="S7.I7.i3.p1.1.m1.1.1"><lt id="S7.I7.i3.p1.1.m1.1.1.1.cmml" xref="S7.I7.i3.p1.1.m1.1.1.1"></lt><apply id="S7.I7.i3.p1.1.m1.1.1.2.cmml" xref="S7.I7.i3.p1.1.m1.1.1.2"><csymbol cd="ambiguous" id="S7.I7.i3.p1.1.m1.1.1.2.1.cmml" xref="S7.I7.i3.p1.1.m1.1.1.2">superscript</csymbol><ci id="S7.I7.i3.p1.1.m1.1.1.2.2.cmml" xref="S7.I7.i3.p1.1.m1.1.1.2.2">𝑘</ci><ci id="S7.I7.i3.p1.1.m1.1.1.2.3.cmml" xref="S7.I7.i3.p1.1.m1.1.1.2.3">′</ci></apply><apply id="S7.I7.i3.p1.1.m1.1.1.3.cmml" xref="S7.I7.i3.p1.1.m1.1.1.3"><minus id="S7.I7.i3.p1.1.m1.1.1.3.1.cmml" xref="S7.I7.i3.p1.1.m1.1.1.3.1"></minus><ci id="S7.I7.i3.p1.1.m1.1.1.3.2.cmml" xref="S7.I7.i3.p1.1.m1.1.1.3.2">ℎ</ci><cn id="S7.I7.i3.p1.1.m1.1.1.3.3.cmml" type="integer" xref="S7.I7.i3.p1.1.m1.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.I7.i3.p1.1.m1.1c">k^{\prime}<h-1</annotation><annotation encoding="application/x-llamapun" id="S7.I7.i3.p1.1.m1.1d">italic_k start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT < italic_h - 1</annotation></semantics></math> and level <math alttext="h" class="ltx_Math" display="inline" id="S7.I7.i3.p1.2.m2.1"><semantics id="S7.I7.i3.p1.2.m2.1a"><mi id="S7.I7.i3.p1.2.m2.1.1" xref="S7.I7.i3.p1.2.m2.1.1.cmml">h</mi><annotation-xml encoding="MathML-Content" id="S7.I7.i3.p1.2.m2.1b"><ci id="S7.I7.i3.p1.2.m2.1.1.cmml" xref="S7.I7.i3.p1.2.m2.1.1">ℎ</ci></annotation-xml><annotation encoding="application/x-tex" id="S7.I7.i3.p1.2.m2.1c">h</annotation><annotation encoding="application/x-llamapun" id="S7.I7.i3.p1.2.m2.1d">italic_h</annotation></semantics></math> do not change their level.</p> </div> </li> </ul> </div> <div class="ltx_para" id="S7.SS3.p4"> <p class="ltx_p" id="S7.SS3.p4.7">We use this observation to show that we maintain Invariant <a class="ltx_ref" href="https://arxiv.org/html/2411.12694v2#Thminvariant1" title="Invariant 1. ‣ 7.1 Algorithm overview ‣ 7 Results in CONGEST ‣ Local Density and its Distributed Approximation"><span class="ltx_text ltx_ref_tag">1</span></a> by induction. Trivially, Invariant <a class="ltx_ref" href="https://arxiv.org/html/2411.12694v2#Thminvariant1" title="Invariant 1. ‣ 7.1 Algorithm overview ‣ 7 Results in CONGEST ‣ Local Density and its Distributed Approximation"><span class="ltx_text ltx_ref_tag">1</span></a> holds at the start of <math alttext="(\ell:0:0)" class="ltx_Math" display="inline" id="S7.SS3.p4.1.m1.1"><semantics id="S7.SS3.p4.1.m1.1a"><mrow id="S7.SS3.p4.1.m1.1.1.1" xref="S7.SS3.p4.1.m1.1.1.1.1.cmml"><mo id="S7.SS3.p4.1.m1.1.1.1.2" stretchy="false" xref="S7.SS3.p4.1.m1.1.1.1.1.cmml">(</mo><mrow id="S7.SS3.p4.1.m1.1.1.1.1" xref="S7.SS3.p4.1.m1.1.1.1.1.cmml"><mi id="S7.SS3.p4.1.m1.1.1.1.1.2" mathvariant="normal" xref="S7.SS3.p4.1.m1.1.1.1.1.2.cmml">ℓ</mi><mo id="S7.SS3.p4.1.m1.1.1.1.1.3" lspace="0.278em" rspace="0.278em" xref="S7.SS3.p4.1.m1.1.1.1.1.3.cmml">:</mo><mn id="S7.SS3.p4.1.m1.1.1.1.1.4" xref="S7.SS3.p4.1.m1.1.1.1.1.4.cmml">0</mn><mo id="S7.SS3.p4.1.m1.1.1.1.1.5" lspace="0.278em" rspace="0.278em" xref="S7.SS3.p4.1.m1.1.1.1.1.5.cmml">:</mo><mn id="S7.SS3.p4.1.m1.1.1.1.1.6" xref="S7.SS3.p4.1.m1.1.1.1.1.6.cmml">0</mn></mrow><mo id="S7.SS3.p4.1.m1.1.1.1.3" stretchy="false" xref="S7.SS3.p4.1.m1.1.1.1.1.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S7.SS3.p4.1.m1.1b"><apply id="S7.SS3.p4.1.m1.1.1.1.1.cmml" xref="S7.SS3.p4.1.m1.1.1.1"><and id="S7.SS3.p4.1.m1.1.1.1.1a.cmml" xref="S7.SS3.p4.1.m1.1.1.1"></and><apply id="S7.SS3.p4.1.m1.1.1.1.1b.cmml" xref="S7.SS3.p4.1.m1.1.1.1"><ci id="S7.SS3.p4.1.m1.1.1.1.1.3.cmml" xref="S7.SS3.p4.1.m1.1.1.1.1.3">:</ci><ci id="S7.SS3.p4.1.m1.1.1.1.1.2.cmml" xref="S7.SS3.p4.1.m1.1.1.1.1.2">ℓ</ci><cn id="S7.SS3.p4.1.m1.1.1.1.1.4.cmml" type="integer" xref="S7.SS3.p4.1.m1.1.1.1.1.4">0</cn></apply><apply id="S7.SS3.p4.1.m1.1.1.1.1c.cmml" xref="S7.SS3.p4.1.m1.1.1.1"><ci id="S7.SS3.p4.1.m1.1.1.1.1.5.cmml" xref="S7.SS3.p4.1.m1.1.1.1.1.5">:</ci><share href="https://arxiv.org/html/2411.12694v2#S7.SS3.p4.1.m1.1.1.1.1.4.cmml" id="S7.SS3.p4.1.m1.1.1.1.1d.cmml" xref="S7.SS3.p4.1.m1.1.1.1"></share><cn id="S7.SS3.p4.1.m1.1.1.1.1.6.cmml" type="integer" xref="S7.SS3.p4.1.m1.1.1.1.1.6">0</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.SS3.p4.1.m1.1c">(\ell:0:0)</annotation><annotation encoding="application/x-llamapun" id="S7.SS3.p4.1.m1.1d">( roman_ℓ : 0 : 0 )</annotation></semantics></math>. We now assume that the invariant holds at <math alttext="(h:0:0)" class="ltx_Math" display="inline" id="S7.SS3.p4.2.m2.1"><semantics id="S7.SS3.p4.2.m2.1a"><mrow id="S7.SS3.p4.2.m2.1.1.1" xref="S7.SS3.p4.2.m2.1.1.1.1.cmml"><mo id="S7.SS3.p4.2.m2.1.1.1.2" stretchy="false" xref="S7.SS3.p4.2.m2.1.1.1.1.cmml">(</mo><mrow id="S7.SS3.p4.2.m2.1.1.1.1" xref="S7.SS3.p4.2.m2.1.1.1.1.cmml"><mi id="S7.SS3.p4.2.m2.1.1.1.1.2" xref="S7.SS3.p4.2.m2.1.1.1.1.2.cmml">h</mi><mo id="S7.SS3.p4.2.m2.1.1.1.1.3" lspace="0.278em" rspace="0.278em" xref="S7.SS3.p4.2.m2.1.1.1.1.3.cmml">:</mo><mn id="S7.SS3.p4.2.m2.1.1.1.1.4" xref="S7.SS3.p4.2.m2.1.1.1.1.4.cmml">0</mn><mo id="S7.SS3.p4.2.m2.1.1.1.1.5" lspace="0.278em" rspace="0.278em" xref="S7.SS3.p4.2.m2.1.1.1.1.5.cmml">:</mo><mn id="S7.SS3.p4.2.m2.1.1.1.1.6" xref="S7.SS3.p4.2.m2.1.1.1.1.6.cmml">0</mn></mrow><mo id="S7.SS3.p4.2.m2.1.1.1.3" stretchy="false" xref="S7.SS3.p4.2.m2.1.1.1.1.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S7.SS3.p4.2.m2.1b"><apply id="S7.SS3.p4.2.m2.1.1.1.1.cmml" xref="S7.SS3.p4.2.m2.1.1.1"><and id="S7.SS3.p4.2.m2.1.1.1.1a.cmml" xref="S7.SS3.p4.2.m2.1.1.1"></and><apply id="S7.SS3.p4.2.m2.1.1.1.1b.cmml" xref="S7.SS3.p4.2.m2.1.1.1"><ci id="S7.SS3.p4.2.m2.1.1.1.1.3.cmml" xref="S7.SS3.p4.2.m2.1.1.1.1.3">:</ci><ci id="S7.SS3.p4.2.m2.1.1.1.1.2.cmml" xref="S7.SS3.p4.2.m2.1.1.1.1.2">ℎ</ci><cn id="S7.SS3.p4.2.m2.1.1.1.1.4.cmml" type="integer" xref="S7.SS3.p4.2.m2.1.1.1.1.4">0</cn></apply><apply id="S7.SS3.p4.2.m2.1.1.1.1c.cmml" xref="S7.SS3.p4.2.m2.1.1.1"><ci id="S7.SS3.p4.2.m2.1.1.1.1.5.cmml" xref="S7.SS3.p4.2.m2.1.1.1.1.5">:</ci><share href="https://arxiv.org/html/2411.12694v2#S7.SS3.p4.2.m2.1.1.1.1.4.cmml" id="S7.SS3.p4.2.m2.1.1.1.1d.cmml" xref="S7.SS3.p4.2.m2.1.1.1"></share><cn id="S7.SS3.p4.2.m2.1.1.1.1.6.cmml" type="integer" xref="S7.SS3.p4.2.m2.1.1.1.1.6">0</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.SS3.p4.2.m2.1c">(h:0:0)</annotation><annotation encoding="application/x-llamapun" id="S7.SS3.p4.2.m2.1d">( italic_h : 0 : 0 )</annotation></semantics></math>, i.e. that there are no violating out-edges from level <math alttext="L_{k}" class="ltx_Math" display="inline" id="S7.SS3.p4.3.m3.1"><semantics id="S7.SS3.p4.3.m3.1a"><msub id="S7.SS3.p4.3.m3.1.1" xref="S7.SS3.p4.3.m3.1.1.cmml"><mi id="S7.SS3.p4.3.m3.1.1.2" xref="S7.SS3.p4.3.m3.1.1.2.cmml">L</mi><mi id="S7.SS3.p4.3.m3.1.1.3" xref="S7.SS3.p4.3.m3.1.1.3.cmml">k</mi></msub><annotation-xml encoding="MathML-Content" id="S7.SS3.p4.3.m3.1b"><apply id="S7.SS3.p4.3.m3.1.1.cmml" xref="S7.SS3.p4.3.m3.1.1"><csymbol cd="ambiguous" id="S7.SS3.p4.3.m3.1.1.1.cmml" xref="S7.SS3.p4.3.m3.1.1">subscript</csymbol><ci id="S7.SS3.p4.3.m3.1.1.2.cmml" xref="S7.SS3.p4.3.m3.1.1.2">𝐿</ci><ci id="S7.SS3.p4.3.m3.1.1.3.cmml" xref="S7.SS3.p4.3.m3.1.1.3">𝑘</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.SS3.p4.3.m3.1c">L_{k}</annotation><annotation encoding="application/x-llamapun" id="S7.SS3.p4.3.m3.1d">italic_L start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT</annotation></semantics></math> with <math alttext="k>h" class="ltx_Math" display="inline" id="S7.SS3.p4.4.m4.1"><semantics id="S7.SS3.p4.4.m4.1a"><mrow id="S7.SS3.p4.4.m4.1.1" xref="S7.SS3.p4.4.m4.1.1.cmml"><mi id="S7.SS3.p4.4.m4.1.1.2" xref="S7.SS3.p4.4.m4.1.1.2.cmml">k</mi><mo id="S7.SS3.p4.4.m4.1.1.1" xref="S7.SS3.p4.4.m4.1.1.1.cmml">></mo><mi id="S7.SS3.p4.4.m4.1.1.3" xref="S7.SS3.p4.4.m4.1.1.3.cmml">h</mi></mrow><annotation-xml encoding="MathML-Content" id="S7.SS3.p4.4.m4.1b"><apply id="S7.SS3.p4.4.m4.1.1.cmml" xref="S7.SS3.p4.4.m4.1.1"><gt id="S7.SS3.p4.4.m4.1.1.1.cmml" xref="S7.SS3.p4.4.m4.1.1.1"></gt><ci id="S7.SS3.p4.4.m4.1.1.2.cmml" xref="S7.SS3.p4.4.m4.1.1.2">𝑘</ci><ci id="S7.SS3.p4.4.m4.1.1.3.cmml" xref="S7.SS3.p4.4.m4.1.1.3">ℎ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.SS3.p4.4.m4.1c">k>h</annotation><annotation encoding="application/x-llamapun" id="S7.SS3.p4.4.m4.1d">italic_k > italic_h</annotation></semantics></math>. We prove that our algorithm ensures that at the start of <math alttext="(h-1:0:0)" class="ltx_Math" display="inline" id="S7.SS3.p4.5.m5.1"><semantics id="S7.SS3.p4.5.m5.1a"><mrow id="S7.SS3.p4.5.m5.1.1.1" xref="S7.SS3.p4.5.m5.1.1.1.1.cmml"><mo id="S7.SS3.p4.5.m5.1.1.1.2" stretchy="false" xref="S7.SS3.p4.5.m5.1.1.1.1.cmml">(</mo><mrow id="S7.SS3.p4.5.m5.1.1.1.1" xref="S7.SS3.p4.5.m5.1.1.1.1.cmml"><mrow id="S7.SS3.p4.5.m5.1.1.1.1.2" xref="S7.SS3.p4.5.m5.1.1.1.1.2.cmml"><mi id="S7.SS3.p4.5.m5.1.1.1.1.2.2" xref="S7.SS3.p4.5.m5.1.1.1.1.2.2.cmml">h</mi><mo id="S7.SS3.p4.5.m5.1.1.1.1.2.1" xref="S7.SS3.p4.5.m5.1.1.1.1.2.1.cmml">−</mo><mn id="S7.SS3.p4.5.m5.1.1.1.1.2.3" xref="S7.SS3.p4.5.m5.1.1.1.1.2.3.cmml">1</mn></mrow><mo id="S7.SS3.p4.5.m5.1.1.1.1.3" lspace="0.278em" rspace="0.278em" xref="S7.SS3.p4.5.m5.1.1.1.1.3.cmml">:</mo><mn id="S7.SS3.p4.5.m5.1.1.1.1.4" xref="S7.SS3.p4.5.m5.1.1.1.1.4.cmml">0</mn><mo id="S7.SS3.p4.5.m5.1.1.1.1.5" lspace="0.278em" rspace="0.278em" xref="S7.SS3.p4.5.m5.1.1.1.1.5.cmml">:</mo><mn id="S7.SS3.p4.5.m5.1.1.1.1.6" xref="S7.SS3.p4.5.m5.1.1.1.1.6.cmml">0</mn></mrow><mo id="S7.SS3.p4.5.m5.1.1.1.3" stretchy="false" xref="S7.SS3.p4.5.m5.1.1.1.1.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S7.SS3.p4.5.m5.1b"><apply id="S7.SS3.p4.5.m5.1.1.1.1.cmml" xref="S7.SS3.p4.5.m5.1.1.1"><and id="S7.SS3.p4.5.m5.1.1.1.1a.cmml" xref="S7.SS3.p4.5.m5.1.1.1"></and><apply id="S7.SS3.p4.5.m5.1.1.1.1b.cmml" xref="S7.SS3.p4.5.m5.1.1.1"><ci id="S7.SS3.p4.5.m5.1.1.1.1.3.cmml" xref="S7.SS3.p4.5.m5.1.1.1.1.3">:</ci><apply id="S7.SS3.p4.5.m5.1.1.1.1.2.cmml" xref="S7.SS3.p4.5.m5.1.1.1.1.2"><minus id="S7.SS3.p4.5.m5.1.1.1.1.2.1.cmml" xref="S7.SS3.p4.5.m5.1.1.1.1.2.1"></minus><ci id="S7.SS3.p4.5.m5.1.1.1.1.2.2.cmml" xref="S7.SS3.p4.5.m5.1.1.1.1.2.2">ℎ</ci><cn id="S7.SS3.p4.5.m5.1.1.1.1.2.3.cmml" type="integer" xref="S7.SS3.p4.5.m5.1.1.1.1.2.3">1</cn></apply><cn id="S7.SS3.p4.5.m5.1.1.1.1.4.cmml" type="integer" xref="S7.SS3.p4.5.m5.1.1.1.1.4">0</cn></apply><apply id="S7.SS3.p4.5.m5.1.1.1.1c.cmml" xref="S7.SS3.p4.5.m5.1.1.1"><ci id="S7.SS3.p4.5.m5.1.1.1.1.5.cmml" xref="S7.SS3.p4.5.m5.1.1.1.1.5">:</ci><share href="https://arxiv.org/html/2411.12694v2#S7.SS3.p4.5.m5.1.1.1.1.4.cmml" id="S7.SS3.p4.5.m5.1.1.1.1d.cmml" xref="S7.SS3.p4.5.m5.1.1.1"></share><cn id="S7.SS3.p4.5.m5.1.1.1.1.6.cmml" type="integer" xref="S7.SS3.p4.5.m5.1.1.1.1.6">0</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.SS3.p4.5.m5.1c">(h-1:0:0)</annotation><annotation encoding="application/x-llamapun" id="S7.SS3.p4.5.m5.1d">( italic_h - 1 : 0 : 0 )</annotation></semantics></math> there are no violating out-edges from level <math alttext="L_{k}" class="ltx_Math" display="inline" id="S7.SS3.p4.6.m6.1"><semantics id="S7.SS3.p4.6.m6.1a"><msub id="S7.SS3.p4.6.m6.1.1" xref="S7.SS3.p4.6.m6.1.1.cmml"><mi id="S7.SS3.p4.6.m6.1.1.2" xref="S7.SS3.p4.6.m6.1.1.2.cmml">L</mi><mi id="S7.SS3.p4.6.m6.1.1.3" xref="S7.SS3.p4.6.m6.1.1.3.cmml">k</mi></msub><annotation-xml encoding="MathML-Content" id="S7.SS3.p4.6.m6.1b"><apply id="S7.SS3.p4.6.m6.1.1.cmml" xref="S7.SS3.p4.6.m6.1.1"><csymbol cd="ambiguous" id="S7.SS3.p4.6.m6.1.1.1.cmml" xref="S7.SS3.p4.6.m6.1.1">subscript</csymbol><ci id="S7.SS3.p4.6.m6.1.1.2.cmml" xref="S7.SS3.p4.6.m6.1.1.2">𝐿</ci><ci id="S7.SS3.p4.6.m6.1.1.3.cmml" xref="S7.SS3.p4.6.m6.1.1.3">𝑘</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.SS3.p4.6.m6.1c">L_{k}</annotation><annotation encoding="application/x-llamapun" id="S7.SS3.p4.6.m6.1d">italic_L start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT</annotation></semantics></math> with <math alttext="k\geq h" class="ltx_Math" display="inline" id="S7.SS3.p4.7.m7.1"><semantics id="S7.SS3.p4.7.m7.1a"><mrow id="S7.SS3.p4.7.m7.1.1" xref="S7.SS3.p4.7.m7.1.1.cmml"><mi id="S7.SS3.p4.7.m7.1.1.2" xref="S7.SS3.p4.7.m7.1.1.2.cmml">k</mi><mo id="S7.SS3.p4.7.m7.1.1.1" xref="S7.SS3.p4.7.m7.1.1.1.cmml">≥</mo><mi id="S7.SS3.p4.7.m7.1.1.3" xref="S7.SS3.p4.7.m7.1.1.3.cmml">h</mi></mrow><annotation-xml encoding="MathML-Content" id="S7.SS3.p4.7.m7.1b"><apply id="S7.SS3.p4.7.m7.1.1.cmml" xref="S7.SS3.p4.7.m7.1.1"><geq id="S7.SS3.p4.7.m7.1.1.1.cmml" xref="S7.SS3.p4.7.m7.1.1.1"></geq><ci id="S7.SS3.p4.7.m7.1.1.2.cmml" xref="S7.SS3.p4.7.m7.1.1.2">𝑘</ci><ci id="S7.SS3.p4.7.m7.1.1.3.cmml" xref="S7.SS3.p4.7.m7.1.1.3">ℎ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.SS3.p4.7.m7.1c">k\geq h</annotation><annotation encoding="application/x-llamapun" id="S7.SS3.p4.7.m7.1d">italic_k ≥ italic_h</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="S7.SS3.p5"> <p class="ltx_p" id="S7.SS3.p5.3">Moreover, we maintain the invariant that throughout <span class="ltx_text ltx_font_smallcaps" id="S7.SS3.p5.3.1">hour</span> <math alttext="(h-1)" class="ltx_Math" display="inline" id="S7.SS3.p5.1.m1.1"><semantics id="S7.SS3.p5.1.m1.1a"><mrow id="S7.SS3.p5.1.m1.1.1.1" xref="S7.SS3.p5.1.m1.1.1.1.1.cmml"><mo id="S7.SS3.p5.1.m1.1.1.1.2" stretchy="false" xref="S7.SS3.p5.1.m1.1.1.1.1.cmml">(</mo><mrow id="S7.SS3.p5.1.m1.1.1.1.1" xref="S7.SS3.p5.1.m1.1.1.1.1.cmml"><mi id="S7.SS3.p5.1.m1.1.1.1.1.2" xref="S7.SS3.p5.1.m1.1.1.1.1.2.cmml">h</mi><mo id="S7.SS3.p5.1.m1.1.1.1.1.1" xref="S7.SS3.p5.1.m1.1.1.1.1.1.cmml">−</mo><mn id="S7.SS3.p5.1.m1.1.1.1.1.3" xref="S7.SS3.p5.1.m1.1.1.1.1.3.cmml">1</mn></mrow><mo id="S7.SS3.p5.1.m1.1.1.1.3" stretchy="false" xref="S7.SS3.p5.1.m1.1.1.1.1.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S7.SS3.p5.1.m1.1b"><apply id="S7.SS3.p5.1.m1.1.1.1.1.cmml" xref="S7.SS3.p5.1.m1.1.1.1"><minus id="S7.SS3.p5.1.m1.1.1.1.1.1.cmml" xref="S7.SS3.p5.1.m1.1.1.1.1.1"></minus><ci id="S7.SS3.p5.1.m1.1.1.1.1.2.cmml" xref="S7.SS3.p5.1.m1.1.1.1.1.2">ℎ</ci><cn id="S7.SS3.p5.1.m1.1.1.1.1.3.cmml" type="integer" xref="S7.SS3.p5.1.m1.1.1.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.SS3.p5.1.m1.1c">(h-1)</annotation><annotation encoding="application/x-llamapun" id="S7.SS3.p5.1.m1.1d">( italic_h - 1 )</annotation></semantics></math> there are no violating out-edges from level <math alttext="L_{k^{\prime}}" class="ltx_Math" display="inline" id="S7.SS3.p5.2.m2.1"><semantics id="S7.SS3.p5.2.m2.1a"><msub id="S7.SS3.p5.2.m2.1.1" xref="S7.SS3.p5.2.m2.1.1.cmml"><mi id="S7.SS3.p5.2.m2.1.1.2" xref="S7.SS3.p5.2.m2.1.1.2.cmml">L</mi><msup id="S7.SS3.p5.2.m2.1.1.3" xref="S7.SS3.p5.2.m2.1.1.3.cmml"><mi id="S7.SS3.p5.2.m2.1.1.3.2" xref="S7.SS3.p5.2.m2.1.1.3.2.cmml">k</mi><mo id="S7.SS3.p5.2.m2.1.1.3.3" xref="S7.SS3.p5.2.m2.1.1.3.3.cmml">′</mo></msup></msub><annotation-xml encoding="MathML-Content" id="S7.SS3.p5.2.m2.1b"><apply id="S7.SS3.p5.2.m2.1.1.cmml" xref="S7.SS3.p5.2.m2.1.1"><csymbol cd="ambiguous" id="S7.SS3.p5.2.m2.1.1.1.cmml" xref="S7.SS3.p5.2.m2.1.1">subscript</csymbol><ci id="S7.SS3.p5.2.m2.1.1.2.cmml" xref="S7.SS3.p5.2.m2.1.1.2">𝐿</ci><apply id="S7.SS3.p5.2.m2.1.1.3.cmml" xref="S7.SS3.p5.2.m2.1.1.3"><csymbol cd="ambiguous" id="S7.SS3.p5.2.m2.1.1.3.1.cmml" xref="S7.SS3.p5.2.m2.1.1.3">superscript</csymbol><ci id="S7.SS3.p5.2.m2.1.1.3.2.cmml" xref="S7.SS3.p5.2.m2.1.1.3.2">𝑘</ci><ci id="S7.SS3.p5.2.m2.1.1.3.3.cmml" xref="S7.SS3.p5.2.m2.1.1.3.3">′</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.SS3.p5.2.m2.1c">L_{k^{\prime}}</annotation><annotation encoding="application/x-llamapun" id="S7.SS3.p5.2.m2.1d">italic_L start_POSTSUBSCRIPT italic_k start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT</annotation></semantics></math> with <math alttext="k^{\prime}>h+1" class="ltx_Math" display="inline" id="S7.SS3.p5.3.m3.1"><semantics id="S7.SS3.p5.3.m3.1a"><mrow id="S7.SS3.p5.3.m3.1.1" xref="S7.SS3.p5.3.m3.1.1.cmml"><msup id="S7.SS3.p5.3.m3.1.1.2" xref="S7.SS3.p5.3.m3.1.1.2.cmml"><mi id="S7.SS3.p5.3.m3.1.1.2.2" xref="S7.SS3.p5.3.m3.1.1.2.2.cmml">k</mi><mo id="S7.SS3.p5.3.m3.1.1.2.3" xref="S7.SS3.p5.3.m3.1.1.2.3.cmml">′</mo></msup><mo id="S7.SS3.p5.3.m3.1.1.1" xref="S7.SS3.p5.3.m3.1.1.1.cmml">></mo><mrow id="S7.SS3.p5.3.m3.1.1.3" xref="S7.SS3.p5.3.m3.1.1.3.cmml"><mi id="S7.SS3.p5.3.m3.1.1.3.2" xref="S7.SS3.p5.3.m3.1.1.3.2.cmml">h</mi><mo id="S7.SS3.p5.3.m3.1.1.3.1" xref="S7.SS3.p5.3.m3.1.1.3.1.cmml">+</mo><mn id="S7.SS3.p5.3.m3.1.1.3.3" xref="S7.SS3.p5.3.m3.1.1.3.3.cmml">1</mn></mrow></mrow><annotation-xml encoding="MathML-Content" id="S7.SS3.p5.3.m3.1b"><apply id="S7.SS3.p5.3.m3.1.1.cmml" xref="S7.SS3.p5.3.m3.1.1"><gt id="S7.SS3.p5.3.m3.1.1.1.cmml" xref="S7.SS3.p5.3.m3.1.1.1"></gt><apply id="S7.SS3.p5.3.m3.1.1.2.cmml" xref="S7.SS3.p5.3.m3.1.1.2"><csymbol cd="ambiguous" id="S7.SS3.p5.3.m3.1.1.2.1.cmml" xref="S7.SS3.p5.3.m3.1.1.2">superscript</csymbol><ci id="S7.SS3.p5.3.m3.1.1.2.2.cmml" xref="S7.SS3.p5.3.m3.1.1.2.2">𝑘</ci><ci id="S7.SS3.p5.3.m3.1.1.2.3.cmml" xref="S7.SS3.p5.3.m3.1.1.2.3">′</ci></apply><apply id="S7.SS3.p5.3.m3.1.1.3.cmml" xref="S7.SS3.p5.3.m3.1.1.3"><plus id="S7.SS3.p5.3.m3.1.1.3.1.cmml" xref="S7.SS3.p5.3.m3.1.1.3.1"></plus><ci id="S7.SS3.p5.3.m3.1.1.3.2.cmml" xref="S7.SS3.p5.3.m3.1.1.3.2">ℎ</ci><cn id="S7.SS3.p5.3.m3.1.1.3.3.cmml" type="integer" xref="S7.SS3.p5.3.m3.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.SS3.p5.3.m3.1c">k^{\prime}>h+1</annotation><annotation encoding="application/x-llamapun" id="S7.SS3.p5.3.m3.1d">italic_k start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT > italic_h + 1</annotation></semantics></math>. We prove this in the following way:</p> </div> <div class="ltx_para" id="S7.SS3.p6"> <ul class="ltx_itemize" id="S7.I8"> <li class="ltx_item" id="S7.I8.i1" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">•</span> <div class="ltx_para" id="S7.I8.i1.p1"> <p class="ltx_p" id="S7.I8.i1.p1.9">During <math alttext="(h:m:s)" class="ltx_Math" display="inline" id="S7.I8.i1.p1.1.m1.1"><semantics id="S7.I8.i1.p1.1.m1.1a"><mrow id="S7.I8.i1.p1.1.m1.1.1.1" xref="S7.I8.i1.p1.1.m1.1.1.1.1.cmml"><mo id="S7.I8.i1.p1.1.m1.1.1.1.2" stretchy="false" xref="S7.I8.i1.p1.1.m1.1.1.1.1.cmml">(</mo><mrow id="S7.I8.i1.p1.1.m1.1.1.1.1" xref="S7.I8.i1.p1.1.m1.1.1.1.1.cmml"><mi id="S7.I8.i1.p1.1.m1.1.1.1.1.2" xref="S7.I8.i1.p1.1.m1.1.1.1.1.2.cmml">h</mi><mo id="S7.I8.i1.p1.1.m1.1.1.1.1.3" lspace="0.278em" rspace="0.278em" xref="S7.I8.i1.p1.1.m1.1.1.1.1.3.cmml">:</mo><mi id="S7.I8.i1.p1.1.m1.1.1.1.1.4" xref="S7.I8.i1.p1.1.m1.1.1.1.1.4.cmml">m</mi><mo id="S7.I8.i1.p1.1.m1.1.1.1.1.5" lspace="0.278em" rspace="0.278em" xref="S7.I8.i1.p1.1.m1.1.1.1.1.5.cmml">:</mo><mi id="S7.I8.i1.p1.1.m1.1.1.1.1.6" xref="S7.I8.i1.p1.1.m1.1.1.1.1.6.cmml">s</mi></mrow><mo id="S7.I8.i1.p1.1.m1.1.1.1.3" stretchy="false" xref="S7.I8.i1.p1.1.m1.1.1.1.1.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S7.I8.i1.p1.1.m1.1b"><apply id="S7.I8.i1.p1.1.m1.1.1.1.1.cmml" xref="S7.I8.i1.p1.1.m1.1.1.1"><and id="S7.I8.i1.p1.1.m1.1.1.1.1a.cmml" xref="S7.I8.i1.p1.1.m1.1.1.1"></and><apply id="S7.I8.i1.p1.1.m1.1.1.1.1b.cmml" xref="S7.I8.i1.p1.1.m1.1.1.1"><ci id="S7.I8.i1.p1.1.m1.1.1.1.1.3.cmml" xref="S7.I8.i1.p1.1.m1.1.1.1.1.3">:</ci><ci id="S7.I8.i1.p1.1.m1.1.1.1.1.2.cmml" xref="S7.I8.i1.p1.1.m1.1.1.1.1.2">ℎ</ci><ci id="S7.I8.i1.p1.1.m1.1.1.1.1.4.cmml" xref="S7.I8.i1.p1.1.m1.1.1.1.1.4">𝑚</ci></apply><apply id="S7.I8.i1.p1.1.m1.1.1.1.1c.cmml" xref="S7.I8.i1.p1.1.m1.1.1.1"><ci id="S7.I8.i1.p1.1.m1.1.1.1.1.5.cmml" xref="S7.I8.i1.p1.1.m1.1.1.1.1.5">:</ci><share href="https://arxiv.org/html/2411.12694v2#S7.I8.i1.p1.1.m1.1.1.1.1.4.cmml" id="S7.I8.i1.p1.1.m1.1.1.1.1d.cmml" xref="S7.I8.i1.p1.1.m1.1.1.1"></share><ci id="S7.I8.i1.p1.1.m1.1.1.1.1.6.cmml" xref="S7.I8.i1.p1.1.m1.1.1.1.1.6">𝑠</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.I8.i1.p1.1.m1.1c">(h:m:s)</annotation><annotation encoding="application/x-llamapun" id="S7.I8.i1.p1.1.m1.1d">( italic_h : italic_m : italic_s )</annotation></semantics></math> for <math alttext="m" class="ltx_Math" display="inline" id="S7.I8.i1.p1.2.m2.1"><semantics id="S7.I8.i1.p1.2.m2.1a"><mi id="S7.I8.i1.p1.2.m2.1.1" xref="S7.I8.i1.p1.2.m2.1.1.cmml">m</mi><annotation-xml encoding="MathML-Content" id="S7.I8.i1.p1.2.m2.1b"><ci id="S7.I8.i1.p1.2.m2.1.1.cmml" xref="S7.I8.i1.p1.2.m2.1.1">𝑚</ci></annotation-xml><annotation encoding="application/x-tex" id="S7.I8.i1.p1.2.m2.1c">m</annotation><annotation encoding="application/x-llamapun" id="S7.I8.i1.p1.2.m2.1d">italic_m</annotation></semantics></math> even, our algorithm eliminates all violating edges going from <math alttext="L_{h}" class="ltx_Math" display="inline" id="S7.I8.i1.p1.3.m3.1"><semantics id="S7.I8.i1.p1.3.m3.1a"><msub id="S7.I8.i1.p1.3.m3.1.1" xref="S7.I8.i1.p1.3.m3.1.1.cmml"><mi id="S7.I8.i1.p1.3.m3.1.1.2" xref="S7.I8.i1.p1.3.m3.1.1.2.cmml">L</mi><mi id="S7.I8.i1.p1.3.m3.1.1.3" xref="S7.I8.i1.p1.3.m3.1.1.3.cmml">h</mi></msub><annotation-xml encoding="MathML-Content" id="S7.I8.i1.p1.3.m3.1b"><apply id="S7.I8.i1.p1.3.m3.1.1.cmml" xref="S7.I8.i1.p1.3.m3.1.1"><csymbol cd="ambiguous" id="S7.I8.i1.p1.3.m3.1.1.1.cmml" xref="S7.I8.i1.p1.3.m3.1.1">subscript</csymbol><ci id="S7.I8.i1.p1.3.m3.1.1.2.cmml" xref="S7.I8.i1.p1.3.m3.1.1.2">𝐿</ci><ci id="S7.I8.i1.p1.3.m3.1.1.3.cmml" xref="S7.I8.i1.p1.3.m3.1.1.3">ℎ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.I8.i1.p1.3.m3.1c">L_{h}</annotation><annotation encoding="application/x-llamapun" id="S7.I8.i1.p1.3.m3.1d">italic_L start_POSTSUBSCRIPT italic_h end_POSTSUBSCRIPT</annotation></semantics></math>. We show that in each iteration, the number of violating edges from <math alttext="L_{h}" class="ltx_Math" display="inline" id="S7.I8.i1.p1.4.m4.1"><semantics id="S7.I8.i1.p1.4.m4.1a"><msub id="S7.I8.i1.p1.4.m4.1.1" xref="S7.I8.i1.p1.4.m4.1.1.cmml"><mi id="S7.I8.i1.p1.4.m4.1.1.2" xref="S7.I8.i1.p1.4.m4.1.1.2.cmml">L</mi><mi id="S7.I8.i1.p1.4.m4.1.1.3" xref="S7.I8.i1.p1.4.m4.1.1.3.cmml">h</mi></msub><annotation-xml encoding="MathML-Content" id="S7.I8.i1.p1.4.m4.1b"><apply id="S7.I8.i1.p1.4.m4.1.1.cmml" xref="S7.I8.i1.p1.4.m4.1.1"><csymbol cd="ambiguous" id="S7.I8.i1.p1.4.m4.1.1.1.cmml" xref="S7.I8.i1.p1.4.m4.1.1">subscript</csymbol><ci id="S7.I8.i1.p1.4.m4.1.1.2.cmml" xref="S7.I8.i1.p1.4.m4.1.1.2">𝐿</ci><ci id="S7.I8.i1.p1.4.m4.1.1.3.cmml" xref="S7.I8.i1.p1.4.m4.1.1.3">ℎ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.I8.i1.p1.4.m4.1c">L_{h}</annotation><annotation encoding="application/x-llamapun" id="S7.I8.i1.p1.4.m4.1d">italic_L start_POSTSUBSCRIPT italic_h end_POSTSUBSCRIPT</annotation></semantics></math> drops by a factor <math alttext="7/8" class="ltx_Math" display="inline" id="S7.I8.i1.p1.5.m5.1"><semantics id="S7.I8.i1.p1.5.m5.1a"><mrow id="S7.I8.i1.p1.5.m5.1.1" xref="S7.I8.i1.p1.5.m5.1.1.cmml"><mn id="S7.I8.i1.p1.5.m5.1.1.2" xref="S7.I8.i1.p1.5.m5.1.1.2.cmml">7</mn><mo id="S7.I8.i1.p1.5.m5.1.1.1" xref="S7.I8.i1.p1.5.m5.1.1.1.cmml">/</mo><mn id="S7.I8.i1.p1.5.m5.1.1.3" xref="S7.I8.i1.p1.5.m5.1.1.3.cmml">8</mn></mrow><annotation-xml encoding="MathML-Content" id="S7.I8.i1.p1.5.m5.1b"><apply id="S7.I8.i1.p1.5.m5.1.1.cmml" xref="S7.I8.i1.p1.5.m5.1.1"><divide id="S7.I8.i1.p1.5.m5.1.1.1.cmml" xref="S7.I8.i1.p1.5.m5.1.1.1"></divide><cn id="S7.I8.i1.p1.5.m5.1.1.2.cmml" type="integer" xref="S7.I8.i1.p1.5.m5.1.1.2">7</cn><cn id="S7.I8.i1.p1.5.m5.1.1.3.cmml" type="integer" xref="S7.I8.i1.p1.5.m5.1.1.3">8</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.I8.i1.p1.5.m5.1c">7/8</annotation><annotation encoding="application/x-llamapun" id="S7.I8.i1.p1.5.m5.1d">7 / 8</annotation></semantics></math> (Lemma <a class="ltx_ref" href="https://arxiv.org/html/2411.12694v2#S7.Thmtheorem20" title="Lemma 7.20. ‣ 7.4 Formally proving correctness ‣ 7 Results in CONGEST ‣ Local Density and its Distributed Approximation"><span class="ltx_text ltx_ref_tag">7.20</span></a>), since the graph has at most <math alttext="n^{2}" class="ltx_Math" display="inline" id="S7.I8.i1.p1.6.m6.1"><semantics id="S7.I8.i1.p1.6.m6.1a"><msup id="S7.I8.i1.p1.6.m6.1.1" xref="S7.I8.i1.p1.6.m6.1.1.cmml"><mi id="S7.I8.i1.p1.6.m6.1.1.2" xref="S7.I8.i1.p1.6.m6.1.1.2.cmml">n</mi><mn id="S7.I8.i1.p1.6.m6.1.1.3" xref="S7.I8.i1.p1.6.m6.1.1.3.cmml">2</mn></msup><annotation-xml encoding="MathML-Content" id="S7.I8.i1.p1.6.m6.1b"><apply id="S7.I8.i1.p1.6.m6.1.1.cmml" xref="S7.I8.i1.p1.6.m6.1.1"><csymbol cd="ambiguous" id="S7.I8.i1.p1.6.m6.1.1.1.cmml" xref="S7.I8.i1.p1.6.m6.1.1">superscript</csymbol><ci id="S7.I8.i1.p1.6.m6.1.1.2.cmml" xref="S7.I8.i1.p1.6.m6.1.1.2">𝑛</ci><cn id="S7.I8.i1.p1.6.m6.1.1.3.cmml" type="integer" xref="S7.I8.i1.p1.6.m6.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.I8.i1.p1.6.m6.1c">n^{2}</annotation><annotation encoding="application/x-llamapun" id="S7.I8.i1.p1.6.m6.1d">italic_n start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT</annotation></semantics></math> edges. This implies that after the <span class="ltx_text ltx_font_smallcaps" id="S7.I8.i1.p1.9.1">minute</span>, there are no more violating edges going from <math alttext="L_{h}" class="ltx_Math" display="inline" id="S7.I8.i1.p1.7.m7.1"><semantics id="S7.I8.i1.p1.7.m7.1a"><msub id="S7.I8.i1.p1.7.m7.1.1" xref="S7.I8.i1.p1.7.m7.1.1.cmml"><mi id="S7.I8.i1.p1.7.m7.1.1.2" xref="S7.I8.i1.p1.7.m7.1.1.2.cmml">L</mi><mi id="S7.I8.i1.p1.7.m7.1.1.3" xref="S7.I8.i1.p1.7.m7.1.1.3.cmml">h</mi></msub><annotation-xml encoding="MathML-Content" id="S7.I8.i1.p1.7.m7.1b"><apply id="S7.I8.i1.p1.7.m7.1.1.cmml" xref="S7.I8.i1.p1.7.m7.1.1"><csymbol cd="ambiguous" id="S7.I8.i1.p1.7.m7.1.1.1.cmml" xref="S7.I8.i1.p1.7.m7.1.1">subscript</csymbol><ci id="S7.I8.i1.p1.7.m7.1.1.2.cmml" xref="S7.I8.i1.p1.7.m7.1.1.2">𝐿</ci><ci id="S7.I8.i1.p1.7.m7.1.1.3.cmml" xref="S7.I8.i1.p1.7.m7.1.1.3">ℎ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.I8.i1.p1.7.m7.1c">L_{h}</annotation><annotation encoding="application/x-llamapun" id="S7.I8.i1.p1.7.m7.1d">italic_L start_POSTSUBSCRIPT italic_h end_POSTSUBSCRIPT</annotation></semantics></math> (Corollary <a class="ltx_ref" href="https://arxiv.org/html/2411.12694v2#S7.Thmtheorem22" title="Corollary 7.22. ‣ 7.4 Formally proving correctness ‣ 7 Results in CONGEST ‣ Local Density and its Distributed Approximation"><span class="ltx_text ltx_ref_tag">7.22</span></a>). However, there may now be violating out-edges from vertices in <math alttext="L_{h+1}" class="ltx_Math" display="inline" id="S7.I8.i1.p1.8.m8.1"><semantics id="S7.I8.i1.p1.8.m8.1a"><msub id="S7.I8.i1.p1.8.m8.1.1" xref="S7.I8.i1.p1.8.m8.1.1.cmml"><mi id="S7.I8.i1.p1.8.m8.1.1.2" xref="S7.I8.i1.p1.8.m8.1.1.2.cmml">L</mi><mrow id="S7.I8.i1.p1.8.m8.1.1.3" xref="S7.I8.i1.p1.8.m8.1.1.3.cmml"><mi id="S7.I8.i1.p1.8.m8.1.1.3.2" xref="S7.I8.i1.p1.8.m8.1.1.3.2.cmml">h</mi><mo id="S7.I8.i1.p1.8.m8.1.1.3.1" xref="S7.I8.i1.p1.8.m8.1.1.3.1.cmml">+</mo><mn id="S7.I8.i1.p1.8.m8.1.1.3.3" xref="S7.I8.i1.p1.8.m8.1.1.3.3.cmml">1</mn></mrow></msub><annotation-xml encoding="MathML-Content" id="S7.I8.i1.p1.8.m8.1b"><apply id="S7.I8.i1.p1.8.m8.1.1.cmml" xref="S7.I8.i1.p1.8.m8.1.1"><csymbol cd="ambiguous" id="S7.I8.i1.p1.8.m8.1.1.1.cmml" xref="S7.I8.i1.p1.8.m8.1.1">subscript</csymbol><ci id="S7.I8.i1.p1.8.m8.1.1.2.cmml" xref="S7.I8.i1.p1.8.m8.1.1.2">𝐿</ci><apply id="S7.I8.i1.p1.8.m8.1.1.3.cmml" xref="S7.I8.i1.p1.8.m8.1.1.3"><plus id="S7.I8.i1.p1.8.m8.1.1.3.1.cmml" xref="S7.I8.i1.p1.8.m8.1.1.3.1"></plus><ci id="S7.I8.i1.p1.8.m8.1.1.3.2.cmml" xref="S7.I8.i1.p1.8.m8.1.1.3.2">ℎ</ci><cn id="S7.I8.i1.p1.8.m8.1.1.3.3.cmml" type="integer" xref="S7.I8.i1.p1.8.m8.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.I8.i1.p1.8.m8.1c">L_{h+1}</annotation><annotation encoding="application/x-llamapun" id="S7.I8.i1.p1.8.m8.1d">italic_L start_POSTSUBSCRIPT italic_h + 1 end_POSTSUBSCRIPT</annotation></semantics></math> to vertices in <math alttext="T_{m+1}\subseteq L_{h-1}" class="ltx_Math" display="inline" id="S7.I8.i1.p1.9.m9.1"><semantics id="S7.I8.i1.p1.9.m9.1a"><mrow id="S7.I8.i1.p1.9.m9.1.1" xref="S7.I8.i1.p1.9.m9.1.1.cmml"><msub id="S7.I8.i1.p1.9.m9.1.1.2" xref="S7.I8.i1.p1.9.m9.1.1.2.cmml"><mi id="S7.I8.i1.p1.9.m9.1.1.2.2" xref="S7.I8.i1.p1.9.m9.1.1.2.2.cmml">T</mi><mrow id="S7.I8.i1.p1.9.m9.1.1.2.3" xref="S7.I8.i1.p1.9.m9.1.1.2.3.cmml"><mi id="S7.I8.i1.p1.9.m9.1.1.2.3.2" xref="S7.I8.i1.p1.9.m9.1.1.2.3.2.cmml">m</mi><mo id="S7.I8.i1.p1.9.m9.1.1.2.3.1" xref="S7.I8.i1.p1.9.m9.1.1.2.3.1.cmml">+</mo><mn id="S7.I8.i1.p1.9.m9.1.1.2.3.3" xref="S7.I8.i1.p1.9.m9.1.1.2.3.3.cmml">1</mn></mrow></msub><mo id="S7.I8.i1.p1.9.m9.1.1.1" xref="S7.I8.i1.p1.9.m9.1.1.1.cmml">⊆</mo><msub id="S7.I8.i1.p1.9.m9.1.1.3" xref="S7.I8.i1.p1.9.m9.1.1.3.cmml"><mi id="S7.I8.i1.p1.9.m9.1.1.3.2" xref="S7.I8.i1.p1.9.m9.1.1.3.2.cmml">L</mi><mrow id="S7.I8.i1.p1.9.m9.1.1.3.3" xref="S7.I8.i1.p1.9.m9.1.1.3.3.cmml"><mi id="S7.I8.i1.p1.9.m9.1.1.3.3.2" xref="S7.I8.i1.p1.9.m9.1.1.3.3.2.cmml">h</mi><mo id="S7.I8.i1.p1.9.m9.1.1.3.3.1" xref="S7.I8.i1.p1.9.m9.1.1.3.3.1.cmml">−</mo><mn id="S7.I8.i1.p1.9.m9.1.1.3.3.3" xref="S7.I8.i1.p1.9.m9.1.1.3.3.3.cmml">1</mn></mrow></msub></mrow><annotation-xml encoding="MathML-Content" id="S7.I8.i1.p1.9.m9.1b"><apply id="S7.I8.i1.p1.9.m9.1.1.cmml" xref="S7.I8.i1.p1.9.m9.1.1"><subset id="S7.I8.i1.p1.9.m9.1.1.1.cmml" xref="S7.I8.i1.p1.9.m9.1.1.1"></subset><apply id="S7.I8.i1.p1.9.m9.1.1.2.cmml" xref="S7.I8.i1.p1.9.m9.1.1.2"><csymbol cd="ambiguous" id="S7.I8.i1.p1.9.m9.1.1.2.1.cmml" xref="S7.I8.i1.p1.9.m9.1.1.2">subscript</csymbol><ci id="S7.I8.i1.p1.9.m9.1.1.2.2.cmml" xref="S7.I8.i1.p1.9.m9.1.1.2.2">𝑇</ci><apply id="S7.I8.i1.p1.9.m9.1.1.2.3.cmml" xref="S7.I8.i1.p1.9.m9.1.1.2.3"><plus id="S7.I8.i1.p1.9.m9.1.1.2.3.1.cmml" xref="S7.I8.i1.p1.9.m9.1.1.2.3.1"></plus><ci id="S7.I8.i1.p1.9.m9.1.1.2.3.2.cmml" xref="S7.I8.i1.p1.9.m9.1.1.2.3.2">𝑚</ci><cn id="S7.I8.i1.p1.9.m9.1.1.2.3.3.cmml" type="integer" xref="S7.I8.i1.p1.9.m9.1.1.2.3.3">1</cn></apply></apply><apply id="S7.I8.i1.p1.9.m9.1.1.3.cmml" xref="S7.I8.i1.p1.9.m9.1.1.3"><csymbol cd="ambiguous" id="S7.I8.i1.p1.9.m9.1.1.3.1.cmml" xref="S7.I8.i1.p1.9.m9.1.1.3">subscript</csymbol><ci id="S7.I8.i1.p1.9.m9.1.1.3.2.cmml" xref="S7.I8.i1.p1.9.m9.1.1.3.2">𝐿</ci><apply id="S7.I8.i1.p1.9.m9.1.1.3.3.cmml" xref="S7.I8.i1.p1.9.m9.1.1.3.3"><minus id="S7.I8.i1.p1.9.m9.1.1.3.3.1.cmml" xref="S7.I8.i1.p1.9.m9.1.1.3.3.1"></minus><ci id="S7.I8.i1.p1.9.m9.1.1.3.3.2.cmml" xref="S7.I8.i1.p1.9.m9.1.1.3.3.2">ℎ</ci><cn id="S7.I8.i1.p1.9.m9.1.1.3.3.3.cmml" type="integer" xref="S7.I8.i1.p1.9.m9.1.1.3.3.3">1</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.I8.i1.p1.9.m9.1c">T_{m+1}\subseteq L_{h-1}</annotation><annotation encoding="application/x-llamapun" id="S7.I8.i1.p1.9.m9.1d">italic_T start_POSTSUBSCRIPT italic_m + 1 end_POSTSUBSCRIPT ⊆ italic_L start_POSTSUBSCRIPT italic_h - 1 end_POSTSUBSCRIPT</annotation></semantics></math>.</p> </div> </li> <li class="ltx_item" id="S7.I8.i2" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">•</span> <div class="ltx_para" id="S7.I8.i2.p1"> <p class="ltx_p" id="S7.I8.i2.p1.12">During <math alttext="(h:m:s)" class="ltx_Math" display="inline" id="S7.I8.i2.p1.1.m1.1"><semantics id="S7.I8.i2.p1.1.m1.1a"><mrow id="S7.I8.i2.p1.1.m1.1.1.1" xref="S7.I8.i2.p1.1.m1.1.1.1.1.cmml"><mo id="S7.I8.i2.p1.1.m1.1.1.1.2" stretchy="false" xref="S7.I8.i2.p1.1.m1.1.1.1.1.cmml">(</mo><mrow id="S7.I8.i2.p1.1.m1.1.1.1.1" xref="S7.I8.i2.p1.1.m1.1.1.1.1.cmml"><mi id="S7.I8.i2.p1.1.m1.1.1.1.1.2" xref="S7.I8.i2.p1.1.m1.1.1.1.1.2.cmml">h</mi><mo id="S7.I8.i2.p1.1.m1.1.1.1.1.3" lspace="0.278em" rspace="0.278em" xref="S7.I8.i2.p1.1.m1.1.1.1.1.3.cmml">:</mo><mi id="S7.I8.i2.p1.1.m1.1.1.1.1.4" xref="S7.I8.i2.p1.1.m1.1.1.1.1.4.cmml">m</mi><mo id="S7.I8.i2.p1.1.m1.1.1.1.1.5" lspace="0.278em" rspace="0.278em" xref="S7.I8.i2.p1.1.m1.1.1.1.1.5.cmml">:</mo><mi id="S7.I8.i2.p1.1.m1.1.1.1.1.6" xref="S7.I8.i2.p1.1.m1.1.1.1.1.6.cmml">s</mi></mrow><mo id="S7.I8.i2.p1.1.m1.1.1.1.3" stretchy="false" xref="S7.I8.i2.p1.1.m1.1.1.1.1.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S7.I8.i2.p1.1.m1.1b"><apply id="S7.I8.i2.p1.1.m1.1.1.1.1.cmml" xref="S7.I8.i2.p1.1.m1.1.1.1"><and id="S7.I8.i2.p1.1.m1.1.1.1.1a.cmml" xref="S7.I8.i2.p1.1.m1.1.1.1"></and><apply id="S7.I8.i2.p1.1.m1.1.1.1.1b.cmml" xref="S7.I8.i2.p1.1.m1.1.1.1"><ci id="S7.I8.i2.p1.1.m1.1.1.1.1.3.cmml" xref="S7.I8.i2.p1.1.m1.1.1.1.1.3">:</ci><ci id="S7.I8.i2.p1.1.m1.1.1.1.1.2.cmml" xref="S7.I8.i2.p1.1.m1.1.1.1.1.2">ℎ</ci><ci id="S7.I8.i2.p1.1.m1.1.1.1.1.4.cmml" xref="S7.I8.i2.p1.1.m1.1.1.1.1.4">𝑚</ci></apply><apply id="S7.I8.i2.p1.1.m1.1.1.1.1c.cmml" xref="S7.I8.i2.p1.1.m1.1.1.1"><ci id="S7.I8.i2.p1.1.m1.1.1.1.1.5.cmml" xref="S7.I8.i2.p1.1.m1.1.1.1.1.5">:</ci><share href="https://arxiv.org/html/2411.12694v2#S7.I8.i2.p1.1.m1.1.1.1.1.4.cmml" id="S7.I8.i2.p1.1.m1.1.1.1.1d.cmml" xref="S7.I8.i2.p1.1.m1.1.1.1"></share><ci id="S7.I8.i2.p1.1.m1.1.1.1.1.6.cmml" xref="S7.I8.i2.p1.1.m1.1.1.1.1.6">𝑠</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.I8.i2.p1.1.m1.1c">(h:m:s)</annotation><annotation encoding="application/x-llamapun" id="S7.I8.i2.p1.1.m1.1d">( italic_h : italic_m : italic_s )</annotation></semantics></math> for <math alttext="m" class="ltx_Math" display="inline" id="S7.I8.i2.p1.2.m2.1"><semantics id="S7.I8.i2.p1.2.m2.1a"><mi id="S7.I8.i2.p1.2.m2.1.1" xref="S7.I8.i2.p1.2.m2.1.1.cmml">m</mi><annotation-xml encoding="MathML-Content" id="S7.I8.i2.p1.2.m2.1b"><ci id="S7.I8.i2.p1.2.m2.1.1.cmml" xref="S7.I8.i2.p1.2.m2.1.1">𝑚</ci></annotation-xml><annotation encoding="application/x-tex" id="S7.I8.i2.p1.2.m2.1c">m</annotation><annotation encoding="application/x-llamapun" id="S7.I8.i2.p1.2.m2.1d">italic_m</annotation></semantics></math> odd, our algorithm eliminates all violating edges going from <math alttext="L_{h+1}" class="ltx_Math" display="inline" id="S7.I8.i2.p1.3.m3.1"><semantics id="S7.I8.i2.p1.3.m3.1a"><msub id="S7.I8.i2.p1.3.m3.1.1" xref="S7.I8.i2.p1.3.m3.1.1.cmml"><mi id="S7.I8.i2.p1.3.m3.1.1.2" xref="S7.I8.i2.p1.3.m3.1.1.2.cmml">L</mi><mrow id="S7.I8.i2.p1.3.m3.1.1.3" xref="S7.I8.i2.p1.3.m3.1.1.3.cmml"><mi id="S7.I8.i2.p1.3.m3.1.1.3.2" xref="S7.I8.i2.p1.3.m3.1.1.3.2.cmml">h</mi><mo id="S7.I8.i2.p1.3.m3.1.1.3.1" xref="S7.I8.i2.p1.3.m3.1.1.3.1.cmml">+</mo><mn id="S7.I8.i2.p1.3.m3.1.1.3.3" xref="S7.I8.i2.p1.3.m3.1.1.3.3.cmml">1</mn></mrow></msub><annotation-xml encoding="MathML-Content" id="S7.I8.i2.p1.3.m3.1b"><apply id="S7.I8.i2.p1.3.m3.1.1.cmml" xref="S7.I8.i2.p1.3.m3.1.1"><csymbol cd="ambiguous" id="S7.I8.i2.p1.3.m3.1.1.1.cmml" xref="S7.I8.i2.p1.3.m3.1.1">subscript</csymbol><ci id="S7.I8.i2.p1.3.m3.1.1.2.cmml" xref="S7.I8.i2.p1.3.m3.1.1.2">𝐿</ci><apply id="S7.I8.i2.p1.3.m3.1.1.3.cmml" xref="S7.I8.i2.p1.3.m3.1.1.3"><plus id="S7.I8.i2.p1.3.m3.1.1.3.1.cmml" xref="S7.I8.i2.p1.3.m3.1.1.3.1"></plus><ci id="S7.I8.i2.p1.3.m3.1.1.3.2.cmml" xref="S7.I8.i2.p1.3.m3.1.1.3.2">ℎ</ci><cn id="S7.I8.i2.p1.3.m3.1.1.3.3.cmml" type="integer" xref="S7.I8.i2.p1.3.m3.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.I8.i2.p1.3.m3.1c">L_{h+1}</annotation><annotation encoding="application/x-llamapun" id="S7.I8.i2.p1.3.m3.1d">italic_L start_POSTSUBSCRIPT italic_h + 1 end_POSTSUBSCRIPT</annotation></semantics></math> to <math alttext="T_{m}" class="ltx_Math" display="inline" id="S7.I8.i2.p1.4.m4.1"><semantics id="S7.I8.i2.p1.4.m4.1a"><msub id="S7.I8.i2.p1.4.m4.1.1" xref="S7.I8.i2.p1.4.m4.1.1.cmml"><mi id="S7.I8.i2.p1.4.m4.1.1.2" xref="S7.I8.i2.p1.4.m4.1.1.2.cmml">T</mi><mi id="S7.I8.i2.p1.4.m4.1.1.3" xref="S7.I8.i2.p1.4.m4.1.1.3.cmml">m</mi></msub><annotation-xml encoding="MathML-Content" id="S7.I8.i2.p1.4.m4.1b"><apply id="S7.I8.i2.p1.4.m4.1.1.cmml" xref="S7.I8.i2.p1.4.m4.1.1"><csymbol cd="ambiguous" id="S7.I8.i2.p1.4.m4.1.1.1.cmml" xref="S7.I8.i2.p1.4.m4.1.1">subscript</csymbol><ci id="S7.I8.i2.p1.4.m4.1.1.2.cmml" xref="S7.I8.i2.p1.4.m4.1.1.2">𝑇</ci><ci id="S7.I8.i2.p1.4.m4.1.1.3.cmml" xref="S7.I8.i2.p1.4.m4.1.1.3">𝑚</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.I8.i2.p1.4.m4.1c">T_{m}</annotation><annotation encoding="application/x-llamapun" id="S7.I8.i2.p1.4.m4.1d">italic_T start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT</annotation></semantics></math>. We show that for all <math alttext="s" class="ltx_Math" display="inline" id="S7.I8.i2.p1.5.m5.1"><semantics id="S7.I8.i2.p1.5.m5.1a"><mi id="S7.I8.i2.p1.5.m5.1.1" xref="S7.I8.i2.p1.5.m5.1.1.cmml">s</mi><annotation-xml encoding="MathML-Content" id="S7.I8.i2.p1.5.m5.1b"><ci id="S7.I8.i2.p1.5.m5.1.1.cmml" xref="S7.I8.i2.p1.5.m5.1.1">𝑠</ci></annotation-xml><annotation encoding="application/x-tex" id="S7.I8.i2.p1.5.m5.1c">s</annotation><annotation encoding="application/x-llamapun" id="S7.I8.i2.p1.5.m5.1d">italic_s</annotation></semantics></math>, the DAG <math alttext="D_{s}" class="ltx_Math" display="inline" id="S7.I8.i2.p1.6.m6.1"><semantics id="S7.I8.i2.p1.6.m6.1a"><msub id="S7.I8.i2.p1.6.m6.1.1" xref="S7.I8.i2.p1.6.m6.1.1.cmml"><mi id="S7.I8.i2.p1.6.m6.1.1.2" xref="S7.I8.i2.p1.6.m6.1.1.2.cmml">D</mi><mi id="S7.I8.i2.p1.6.m6.1.1.3" xref="S7.I8.i2.p1.6.m6.1.1.3.cmml">s</mi></msub><annotation-xml encoding="MathML-Content" id="S7.I8.i2.p1.6.m6.1b"><apply id="S7.I8.i2.p1.6.m6.1.1.cmml" xref="S7.I8.i2.p1.6.m6.1.1"><csymbol cd="ambiguous" id="S7.I8.i2.p1.6.m6.1.1.1.cmml" xref="S7.I8.i2.p1.6.m6.1.1">subscript</csymbol><ci id="S7.I8.i2.p1.6.m6.1.1.2.cmml" xref="S7.I8.i2.p1.6.m6.1.1.2">𝐷</ci><ci id="S7.I8.i2.p1.6.m6.1.1.3.cmml" xref="S7.I8.i2.p1.6.m6.1.1.3">𝑠</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.I8.i2.p1.6.m6.1c">D_{s}</annotation><annotation encoding="application/x-llamapun" id="S7.I8.i2.p1.6.m6.1d">italic_D start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT</annotation></semantics></math> is a subgraph of <math alttext="D_{s-1}" class="ltx_Math" display="inline" id="S7.I8.i2.p1.7.m7.1"><semantics id="S7.I8.i2.p1.7.m7.1a"><msub id="S7.I8.i2.p1.7.m7.1.1" xref="S7.I8.i2.p1.7.m7.1.1.cmml"><mi id="S7.I8.i2.p1.7.m7.1.1.2" xref="S7.I8.i2.p1.7.m7.1.1.2.cmml">D</mi><mrow id="S7.I8.i2.p1.7.m7.1.1.3" xref="S7.I8.i2.p1.7.m7.1.1.3.cmml"><mi id="S7.I8.i2.p1.7.m7.1.1.3.2" xref="S7.I8.i2.p1.7.m7.1.1.3.2.cmml">s</mi><mo id="S7.I8.i2.p1.7.m7.1.1.3.1" xref="S7.I8.i2.p1.7.m7.1.1.3.1.cmml">−</mo><mn id="S7.I8.i2.p1.7.m7.1.1.3.3" xref="S7.I8.i2.p1.7.m7.1.1.3.3.cmml">1</mn></mrow></msub><annotation-xml encoding="MathML-Content" id="S7.I8.i2.p1.7.m7.1b"><apply id="S7.I8.i2.p1.7.m7.1.1.cmml" xref="S7.I8.i2.p1.7.m7.1.1"><csymbol cd="ambiguous" id="S7.I8.i2.p1.7.m7.1.1.1.cmml" xref="S7.I8.i2.p1.7.m7.1.1">subscript</csymbol><ci id="S7.I8.i2.p1.7.m7.1.1.2.cmml" xref="S7.I8.i2.p1.7.m7.1.1.2">𝐷</ci><apply id="S7.I8.i2.p1.7.m7.1.1.3.cmml" xref="S7.I8.i2.p1.7.m7.1.1.3"><minus id="S7.I8.i2.p1.7.m7.1.1.3.1.cmml" xref="S7.I8.i2.p1.7.m7.1.1.3.1"></minus><ci id="S7.I8.i2.p1.7.m7.1.1.3.2.cmml" xref="S7.I8.i2.p1.7.m7.1.1.3.2">𝑠</ci><cn id="S7.I8.i2.p1.7.m7.1.1.3.3.cmml" type="integer" xref="S7.I8.i2.p1.7.m7.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.I8.i2.p1.7.m7.1c">D_{s-1}</annotation><annotation encoding="application/x-llamapun" id="S7.I8.i2.p1.7.m7.1d">italic_D start_POSTSUBSCRIPT italic_s - 1 end_POSTSUBSCRIPT</annotation></semantics></math> where the height is one fewer (Lemma <a class="ltx_ref" href="https://arxiv.org/html/2411.12694v2#S7.Thmtheorem16" title="Lemma 7.16. ‣ 7.4 Formally proving correctness ‣ 7 Results in CONGEST ‣ Local Density and its Distributed Approximation"><span class="ltx_text ltx_ref_tag">7.16</span></a>). Since the height of <math alttext="D_{s}" class="ltx_Math" display="inline" id="S7.I8.i2.p1.8.m8.1"><semantics id="S7.I8.i2.p1.8.m8.1a"><msub id="S7.I8.i2.p1.8.m8.1.1" xref="S7.I8.i2.p1.8.m8.1.1.cmml"><mi id="S7.I8.i2.p1.8.m8.1.1.2" xref="S7.I8.i2.p1.8.m8.1.1.2.cmml">D</mi><mi id="S7.I8.i2.p1.8.m8.1.1.3" xref="S7.I8.i2.p1.8.m8.1.1.3.cmml">s</mi></msub><annotation-xml encoding="MathML-Content" id="S7.I8.i2.p1.8.m8.1b"><apply id="S7.I8.i2.p1.8.m8.1.1.cmml" xref="S7.I8.i2.p1.8.m8.1.1"><csymbol cd="ambiguous" id="S7.I8.i2.p1.8.m8.1.1.1.cmml" xref="S7.I8.i2.p1.8.m8.1.1">subscript</csymbol><ci id="S7.I8.i2.p1.8.m8.1.1.2.cmml" xref="S7.I8.i2.p1.8.m8.1.1.2">𝐷</ci><ci id="S7.I8.i2.p1.8.m8.1.1.3.cmml" xref="S7.I8.i2.p1.8.m8.1.1.3">𝑠</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.I8.i2.p1.8.m8.1c">D_{s}</annotation><annotation encoding="application/x-llamapun" id="S7.I8.i2.p1.8.m8.1d">italic_D start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT</annotation></semantics></math> is at most <math alttext="\ell-h+1" class="ltx_Math" display="inline" id="S7.I8.i2.p1.9.m9.1"><semantics id="S7.I8.i2.p1.9.m9.1a"><mrow id="S7.I8.i2.p1.9.m9.1.1" xref="S7.I8.i2.p1.9.m9.1.1.cmml"><mrow id="S7.I8.i2.p1.9.m9.1.1.2" xref="S7.I8.i2.p1.9.m9.1.1.2.cmml"><mi id="S7.I8.i2.p1.9.m9.1.1.2.2" mathvariant="normal" xref="S7.I8.i2.p1.9.m9.1.1.2.2.cmml">ℓ</mi><mo id="S7.I8.i2.p1.9.m9.1.1.2.1" xref="S7.I8.i2.p1.9.m9.1.1.2.1.cmml">−</mo><mi id="S7.I8.i2.p1.9.m9.1.1.2.3" xref="S7.I8.i2.p1.9.m9.1.1.2.3.cmml">h</mi></mrow><mo id="S7.I8.i2.p1.9.m9.1.1.1" xref="S7.I8.i2.p1.9.m9.1.1.1.cmml">+</mo><mn id="S7.I8.i2.p1.9.m9.1.1.3" xref="S7.I8.i2.p1.9.m9.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S7.I8.i2.p1.9.m9.1b"><apply id="S7.I8.i2.p1.9.m9.1.1.cmml" xref="S7.I8.i2.p1.9.m9.1.1"><plus id="S7.I8.i2.p1.9.m9.1.1.1.cmml" xref="S7.I8.i2.p1.9.m9.1.1.1"></plus><apply id="S7.I8.i2.p1.9.m9.1.1.2.cmml" xref="S7.I8.i2.p1.9.m9.1.1.2"><minus id="S7.I8.i2.p1.9.m9.1.1.2.1.cmml" xref="S7.I8.i2.p1.9.m9.1.1.2.1"></minus><ci id="S7.I8.i2.p1.9.m9.1.1.2.2.cmml" xref="S7.I8.i2.p1.9.m9.1.1.2.2">ℓ</ci><ci id="S7.I8.i2.p1.9.m9.1.1.2.3.cmml" xref="S7.I8.i2.p1.9.m9.1.1.2.3">ℎ</ci></apply><cn id="S7.I8.i2.p1.9.m9.1.1.3.cmml" type="integer" xref="S7.I8.i2.p1.9.m9.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.I8.i2.p1.9.m9.1c">\ell-h+1</annotation><annotation encoding="application/x-llamapun" id="S7.I8.i2.p1.9.m9.1d">roman_ℓ - italic_h + 1</annotation></semantics></math>, this implies that after the <span class="ltx_text ltx_font_smallcaps" id="S7.I8.i2.p1.12.1">minute</span> <math alttext="m" class="ltx_Math" display="inline" id="S7.I8.i2.p1.10.m10.1"><semantics id="S7.I8.i2.p1.10.m10.1a"><mi id="S7.I8.i2.p1.10.m10.1.1" xref="S7.I8.i2.p1.10.m10.1.1.cmml">m</mi><annotation-xml encoding="MathML-Content" id="S7.I8.i2.p1.10.m10.1b"><ci id="S7.I8.i2.p1.10.m10.1.1.cmml" xref="S7.I8.i2.p1.10.m10.1.1">𝑚</ci></annotation-xml><annotation encoding="application/x-tex" id="S7.I8.i2.p1.10.m10.1c">m</annotation><annotation encoding="application/x-llamapun" id="S7.I8.i2.p1.10.m10.1d">italic_m</annotation></semantics></math>, the graph <math alttext="D_{s}" class="ltx_Math" display="inline" id="S7.I8.i2.p1.11.m11.1"><semantics id="S7.I8.i2.p1.11.m11.1a"><msub id="S7.I8.i2.p1.11.m11.1.1" xref="S7.I8.i2.p1.11.m11.1.1.cmml"><mi id="S7.I8.i2.p1.11.m11.1.1.2" xref="S7.I8.i2.p1.11.m11.1.1.2.cmml">D</mi><mi id="S7.I8.i2.p1.11.m11.1.1.3" xref="S7.I8.i2.p1.11.m11.1.1.3.cmml">s</mi></msub><annotation-xml encoding="MathML-Content" id="S7.I8.i2.p1.11.m11.1b"><apply id="S7.I8.i2.p1.11.m11.1.1.cmml" xref="S7.I8.i2.p1.11.m11.1.1"><csymbol cd="ambiguous" id="S7.I8.i2.p1.11.m11.1.1.1.cmml" xref="S7.I8.i2.p1.11.m11.1.1">subscript</csymbol><ci id="S7.I8.i2.p1.11.m11.1.1.2.cmml" xref="S7.I8.i2.p1.11.m11.1.1.2">𝐷</ci><ci id="S7.I8.i2.p1.11.m11.1.1.3.cmml" xref="S7.I8.i2.p1.11.m11.1.1.3">𝑠</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.I8.i2.p1.11.m11.1c">D_{s}</annotation><annotation encoding="application/x-llamapun" id="S7.I8.i2.p1.11.m11.1d">italic_D start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT</annotation></semantics></math> is empty and there are no more violating in-edges to <math alttext="T_{m}" class="ltx_Math" display="inline" id="S7.I8.i2.p1.12.m12.1"><semantics id="S7.I8.i2.p1.12.m12.1a"><msub id="S7.I8.i2.p1.12.m12.1.1" xref="S7.I8.i2.p1.12.m12.1.1.cmml"><mi id="S7.I8.i2.p1.12.m12.1.1.2" xref="S7.I8.i2.p1.12.m12.1.1.2.cmml">T</mi><mi id="S7.I8.i2.p1.12.m12.1.1.3" xref="S7.I8.i2.p1.12.m12.1.1.3.cmml">m</mi></msub><annotation-xml encoding="MathML-Content" id="S7.I8.i2.p1.12.m12.1b"><apply id="S7.I8.i2.p1.12.m12.1.1.cmml" xref="S7.I8.i2.p1.12.m12.1.1"><csymbol cd="ambiguous" id="S7.I8.i2.p1.12.m12.1.1.1.cmml" xref="S7.I8.i2.p1.12.m12.1.1">subscript</csymbol><ci id="S7.I8.i2.p1.12.m12.1.1.2.cmml" xref="S7.I8.i2.p1.12.m12.1.1.2">𝑇</ci><ci id="S7.I8.i2.p1.12.m12.1.1.3.cmml" xref="S7.I8.i2.p1.12.m12.1.1.3">𝑚</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.I8.i2.p1.12.m12.1c">T_{m}</annotation><annotation encoding="application/x-llamapun" id="S7.I8.i2.p1.12.m12.1d">italic_T start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT</annotation></semantics></math> (Corollary <a class="ltx_ref" href="https://arxiv.org/html/2411.12694v2#S7.Thmtheorem18" title="Corollary 7.18. ‣ 7.4 Formally proving correctness ‣ 7 Results in CONGEST ‣ Local Density and its Distributed Approximation"><span class="ltx_text ltx_ref_tag">7.18</span></a>).</p> </div> </li> <li class="ltx_item" id="S7.I8.i3" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">•</span> <div class="ltx_para" id="S7.I8.i3.p1"> <p class="ltx_p" id="S7.I8.i3.p1.6">Each <span class="ltx_text ltx_font_smallcaps" id="S7.I8.i3.p1.6.1">minute</span>, our algorithm alternates between having violating edges from <math alttext="L_{h}" class="ltx_Math" display="inline" id="S7.I8.i3.p1.1.m1.1"><semantics id="S7.I8.i3.p1.1.m1.1a"><msub id="S7.I8.i3.p1.1.m1.1.1" xref="S7.I8.i3.p1.1.m1.1.1.cmml"><mi id="S7.I8.i3.p1.1.m1.1.1.2" xref="S7.I8.i3.p1.1.m1.1.1.2.cmml">L</mi><mi id="S7.I8.i3.p1.1.m1.1.1.3" xref="S7.I8.i3.p1.1.m1.1.1.3.cmml">h</mi></msub><annotation-xml encoding="MathML-Content" id="S7.I8.i3.p1.1.m1.1b"><apply id="S7.I8.i3.p1.1.m1.1.1.cmml" xref="S7.I8.i3.p1.1.m1.1.1"><csymbol cd="ambiguous" id="S7.I8.i3.p1.1.m1.1.1.1.cmml" xref="S7.I8.i3.p1.1.m1.1.1">subscript</csymbol><ci id="S7.I8.i3.p1.1.m1.1.1.2.cmml" xref="S7.I8.i3.p1.1.m1.1.1.2">𝐿</ci><ci id="S7.I8.i3.p1.1.m1.1.1.3.cmml" xref="S7.I8.i3.p1.1.m1.1.1.3">ℎ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.I8.i3.p1.1.m1.1c">L_{h}</annotation><annotation encoding="application/x-llamapun" id="S7.I8.i3.p1.1.m1.1d">italic_L start_POSTSUBSCRIPT italic_h end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="L_{h+1}" class="ltx_Math" display="inline" id="S7.I8.i3.p1.2.m2.1"><semantics id="S7.I8.i3.p1.2.m2.1a"><msub id="S7.I8.i3.p1.2.m2.1.1" xref="S7.I8.i3.p1.2.m2.1.1.cmml"><mi id="S7.I8.i3.p1.2.m2.1.1.2" xref="S7.I8.i3.p1.2.m2.1.1.2.cmml">L</mi><mrow id="S7.I8.i3.p1.2.m2.1.1.3" xref="S7.I8.i3.p1.2.m2.1.1.3.cmml"><mi id="S7.I8.i3.p1.2.m2.1.1.3.2" xref="S7.I8.i3.p1.2.m2.1.1.3.2.cmml">h</mi><mo id="S7.I8.i3.p1.2.m2.1.1.3.1" xref="S7.I8.i3.p1.2.m2.1.1.3.1.cmml">+</mo><mn id="S7.I8.i3.p1.2.m2.1.1.3.3" xref="S7.I8.i3.p1.2.m2.1.1.3.3.cmml">1</mn></mrow></msub><annotation-xml encoding="MathML-Content" id="S7.I8.i3.p1.2.m2.1b"><apply id="S7.I8.i3.p1.2.m2.1.1.cmml" xref="S7.I8.i3.p1.2.m2.1.1"><csymbol cd="ambiguous" id="S7.I8.i3.p1.2.m2.1.1.1.cmml" xref="S7.I8.i3.p1.2.m2.1.1">subscript</csymbol><ci id="S7.I8.i3.p1.2.m2.1.1.2.cmml" xref="S7.I8.i3.p1.2.m2.1.1.2">𝐿</ci><apply id="S7.I8.i3.p1.2.m2.1.1.3.cmml" xref="S7.I8.i3.p1.2.m2.1.1.3"><plus id="S7.I8.i3.p1.2.m2.1.1.3.1.cmml" xref="S7.I8.i3.p1.2.m2.1.1.3.1"></plus><ci id="S7.I8.i3.p1.2.m2.1.1.3.2.cmml" xref="S7.I8.i3.p1.2.m2.1.1.3.2">ℎ</ci><cn id="S7.I8.i3.p1.2.m2.1.1.3.3.cmml" type="integer" xref="S7.I8.i3.p1.2.m2.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.I8.i3.p1.2.m2.1c">L_{h+1}</annotation><annotation encoding="application/x-llamapun" id="S7.I8.i3.p1.2.m2.1d">italic_L start_POSTSUBSCRIPT italic_h + 1 end_POSTSUBSCRIPT</annotation></semantics></math>. We show that our algorithm can alternate at most <math alttext="\eta" class="ltx_Math" display="inline" id="S7.I8.i3.p1.3.m3.1"><semantics id="S7.I8.i3.p1.3.m3.1a"><mi id="S7.I8.i3.p1.3.m3.1.1" xref="S7.I8.i3.p1.3.m3.1.1.cmml">η</mi><annotation-xml encoding="MathML-Content" id="S7.I8.i3.p1.3.m3.1b"><ci id="S7.I8.i3.p1.3.m3.1.1.cmml" xref="S7.I8.i3.p1.3.m3.1.1">𝜂</ci></annotation-xml><annotation encoding="application/x-tex" id="S7.I8.i3.p1.3.m3.1c">\eta</annotation><annotation encoding="application/x-llamapun" id="S7.I8.i3.p1.3.m3.1d">italic_η</annotation></semantics></math> times before there are no violating edges from both <math alttext="L_{h}" class="ltx_Math" display="inline" id="S7.I8.i3.p1.4.m4.1"><semantics id="S7.I8.i3.p1.4.m4.1a"><msub id="S7.I8.i3.p1.4.m4.1.1" xref="S7.I8.i3.p1.4.m4.1.1.cmml"><mi id="S7.I8.i3.p1.4.m4.1.1.2" xref="S7.I8.i3.p1.4.m4.1.1.2.cmml">L</mi><mi id="S7.I8.i3.p1.4.m4.1.1.3" xref="S7.I8.i3.p1.4.m4.1.1.3.cmml">h</mi></msub><annotation-xml encoding="MathML-Content" id="S7.I8.i3.p1.4.m4.1b"><apply id="S7.I8.i3.p1.4.m4.1.1.cmml" xref="S7.I8.i3.p1.4.m4.1.1"><csymbol cd="ambiguous" id="S7.I8.i3.p1.4.m4.1.1.1.cmml" xref="S7.I8.i3.p1.4.m4.1.1">subscript</csymbol><ci id="S7.I8.i3.p1.4.m4.1.1.2.cmml" xref="S7.I8.i3.p1.4.m4.1.1.2">𝐿</ci><ci id="S7.I8.i3.p1.4.m4.1.1.3.cmml" xref="S7.I8.i3.p1.4.m4.1.1.3">ℎ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.I8.i3.p1.4.m4.1c">L_{h}</annotation><annotation encoding="application/x-llamapun" id="S7.I8.i3.p1.4.m4.1d">italic_L start_POSTSUBSCRIPT italic_h end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="L_{h+1}" class="ltx_Math" display="inline" id="S7.I8.i3.p1.5.m5.1"><semantics id="S7.I8.i3.p1.5.m5.1a"><msub id="S7.I8.i3.p1.5.m5.1.1" xref="S7.I8.i3.p1.5.m5.1.1.cmml"><mi id="S7.I8.i3.p1.5.m5.1.1.2" xref="S7.I8.i3.p1.5.m5.1.1.2.cmml">L</mi><mrow id="S7.I8.i3.p1.5.m5.1.1.3" xref="S7.I8.i3.p1.5.m5.1.1.3.cmml"><mi id="S7.I8.i3.p1.5.m5.1.1.3.2" xref="S7.I8.i3.p1.5.m5.1.1.3.2.cmml">h</mi><mo id="S7.I8.i3.p1.5.m5.1.1.3.1" xref="S7.I8.i3.p1.5.m5.1.1.3.1.cmml">+</mo><mn id="S7.I8.i3.p1.5.m5.1.1.3.3" xref="S7.I8.i3.p1.5.m5.1.1.3.3.cmml">1</mn></mrow></msub><annotation-xml encoding="MathML-Content" id="S7.I8.i3.p1.5.m5.1b"><apply id="S7.I8.i3.p1.5.m5.1.1.cmml" xref="S7.I8.i3.p1.5.m5.1.1"><csymbol cd="ambiguous" id="S7.I8.i3.p1.5.m5.1.1.1.cmml" xref="S7.I8.i3.p1.5.m5.1.1">subscript</csymbol><ci id="S7.I8.i3.p1.5.m5.1.1.2.cmml" xref="S7.I8.i3.p1.5.m5.1.1.2">𝐿</ci><apply id="S7.I8.i3.p1.5.m5.1.1.3.cmml" xref="S7.I8.i3.p1.5.m5.1.1.3"><plus id="S7.I8.i3.p1.5.m5.1.1.3.1.cmml" xref="S7.I8.i3.p1.5.m5.1.1.3.1"></plus><ci id="S7.I8.i3.p1.5.m5.1.1.3.2.cmml" xref="S7.I8.i3.p1.5.m5.1.1.3.2">ℎ</ci><cn id="S7.I8.i3.p1.5.m5.1.1.3.3.cmml" type="integer" xref="S7.I8.i3.p1.5.m5.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.I8.i3.p1.5.m5.1c">L_{h+1}</annotation><annotation encoding="application/x-llamapun" id="S7.I8.i3.p1.5.m5.1d">italic_L start_POSTSUBSCRIPT italic_h + 1 end_POSTSUBSCRIPT</annotation></semantics></math>, which implies Invariant <a class="ltx_ref" href="https://arxiv.org/html/2411.12694v2#Thminvariant1" title="Invariant 1. ‣ 7.1 Algorithm overview ‣ 7 Results in CONGEST ‣ Local Density and its Distributed Approximation"><span class="ltx_text ltx_ref_tag">1</span></a> at the start of <math alttext="(h-1:0:0)" class="ltx_Math" display="inline" id="S7.I8.i3.p1.6.m6.1"><semantics id="S7.I8.i3.p1.6.m6.1a"><mrow id="S7.I8.i3.p1.6.m6.1.1.1" xref="S7.I8.i3.p1.6.m6.1.1.1.1.cmml"><mo id="S7.I8.i3.p1.6.m6.1.1.1.2" stretchy="false" xref="S7.I8.i3.p1.6.m6.1.1.1.1.cmml">(</mo><mrow id="S7.I8.i3.p1.6.m6.1.1.1.1" xref="S7.I8.i3.p1.6.m6.1.1.1.1.cmml"><mrow id="S7.I8.i3.p1.6.m6.1.1.1.1.2" xref="S7.I8.i3.p1.6.m6.1.1.1.1.2.cmml"><mi id="S7.I8.i3.p1.6.m6.1.1.1.1.2.2" xref="S7.I8.i3.p1.6.m6.1.1.1.1.2.2.cmml">h</mi><mo id="S7.I8.i3.p1.6.m6.1.1.1.1.2.1" xref="S7.I8.i3.p1.6.m6.1.1.1.1.2.1.cmml">−</mo><mn id="S7.I8.i3.p1.6.m6.1.1.1.1.2.3" xref="S7.I8.i3.p1.6.m6.1.1.1.1.2.3.cmml">1</mn></mrow><mo id="S7.I8.i3.p1.6.m6.1.1.1.1.3" lspace="0.278em" rspace="0.278em" xref="S7.I8.i3.p1.6.m6.1.1.1.1.3.cmml">:</mo><mn id="S7.I8.i3.p1.6.m6.1.1.1.1.4" xref="S7.I8.i3.p1.6.m6.1.1.1.1.4.cmml">0</mn><mo id="S7.I8.i3.p1.6.m6.1.1.1.1.5" lspace="0.278em" rspace="0.278em" xref="S7.I8.i3.p1.6.m6.1.1.1.1.5.cmml">:</mo><mn id="S7.I8.i3.p1.6.m6.1.1.1.1.6" xref="S7.I8.i3.p1.6.m6.1.1.1.1.6.cmml">0</mn></mrow><mo id="S7.I8.i3.p1.6.m6.1.1.1.3" stretchy="false" xref="S7.I8.i3.p1.6.m6.1.1.1.1.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S7.I8.i3.p1.6.m6.1b"><apply id="S7.I8.i3.p1.6.m6.1.1.1.1.cmml" xref="S7.I8.i3.p1.6.m6.1.1.1"><and id="S7.I8.i3.p1.6.m6.1.1.1.1a.cmml" xref="S7.I8.i3.p1.6.m6.1.1.1"></and><apply id="S7.I8.i3.p1.6.m6.1.1.1.1b.cmml" xref="S7.I8.i3.p1.6.m6.1.1.1"><ci id="S7.I8.i3.p1.6.m6.1.1.1.1.3.cmml" xref="S7.I8.i3.p1.6.m6.1.1.1.1.3">:</ci><apply id="S7.I8.i3.p1.6.m6.1.1.1.1.2.cmml" xref="S7.I8.i3.p1.6.m6.1.1.1.1.2"><minus id="S7.I8.i3.p1.6.m6.1.1.1.1.2.1.cmml" xref="S7.I8.i3.p1.6.m6.1.1.1.1.2.1"></minus><ci id="S7.I8.i3.p1.6.m6.1.1.1.1.2.2.cmml" xref="S7.I8.i3.p1.6.m6.1.1.1.1.2.2">ℎ</ci><cn id="S7.I8.i3.p1.6.m6.1.1.1.1.2.3.cmml" type="integer" xref="S7.I8.i3.p1.6.m6.1.1.1.1.2.3">1</cn></apply><cn id="S7.I8.i3.p1.6.m6.1.1.1.1.4.cmml" type="integer" xref="S7.I8.i3.p1.6.m6.1.1.1.1.4">0</cn></apply><apply id="S7.I8.i3.p1.6.m6.1.1.1.1c.cmml" xref="S7.I8.i3.p1.6.m6.1.1.1"><ci id="S7.I8.i3.p1.6.m6.1.1.1.1.5.cmml" xref="S7.I8.i3.p1.6.m6.1.1.1.1.5">:</ci><share href="https://arxiv.org/html/2411.12694v2#S7.I8.i3.p1.6.m6.1.1.1.1.4.cmml" id="S7.I8.i3.p1.6.m6.1.1.1.1d.cmml" xref="S7.I8.i3.p1.6.m6.1.1.1"></share><cn id="S7.I8.i3.p1.6.m6.1.1.1.1.6.cmml" type="integer" xref="S7.I8.i3.p1.6.m6.1.1.1.1.6">0</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.I8.i3.p1.6.m6.1c">(h-1:0:0)</annotation><annotation encoding="application/x-llamapun" id="S7.I8.i3.p1.6.m6.1d">( italic_h - 1 : 0 : 0 )</annotation></semantics></math>. (Theorem <a class="ltx_ref" href="https://arxiv.org/html/2411.12694v2#S7.Thmtheorem26" title="Theorem 7.26. ‣ Case 2: ‣ 7.4 Formally proving correctness ‣ 7 Results in CONGEST ‣ Local Density and its Distributed Approximation"><span class="ltx_text ltx_ref_tag">7.26</span></a>).</p> </div> </li> </ul> <p class="ltx_p" id="S7.SS3.p6.2">Invariant <a class="ltx_ref" href="https://arxiv.org/html/2411.12694v2#Thminvariant1" title="Invariant 1. ‣ 7.1 Algorithm overview ‣ 7 Results in CONGEST ‣ Local Density and its Distributed Approximation"><span class="ltx_text ltx_ref_tag">1</span></a> implies that we compute an <math alttext="\eta" class="ltx_Math" display="inline" id="S7.SS3.p6.1.m1.1"><semantics id="S7.SS3.p6.1.m1.1a"><mi id="S7.SS3.p6.1.m1.1.1" xref="S7.SS3.p6.1.m1.1.1.cmml">η</mi><annotation-xml encoding="MathML-Content" id="S7.SS3.p6.1.m1.1b"><ci id="S7.SS3.p6.1.m1.1.1.cmml" xref="S7.SS3.p6.1.m1.1.1">𝜂</ci></annotation-xml><annotation encoding="application/x-tex" id="S7.SS3.p6.1.m1.1c">\eta</annotation><annotation encoding="application/x-llamapun" id="S7.SS3.p6.1.m1.1d">italic_η</annotation></semantics></math>-fair orientation at time <math alttext="(0:0:0)" class="ltx_Math" display="inline" id="S7.SS3.p6.2.m2.1"><semantics id="S7.SS3.p6.2.m2.1a"><mrow id="S7.SS3.p6.2.m2.1.1.1" xref="S7.SS3.p6.2.m2.1.1.1.1.cmml"><mo id="S7.SS3.p6.2.m2.1.1.1.2" stretchy="false" xref="S7.SS3.p6.2.m2.1.1.1.1.cmml">(</mo><mrow id="S7.SS3.p6.2.m2.1.1.1.1" xref="S7.SS3.p6.2.m2.1.1.1.1.cmml"><mn id="S7.SS3.p6.2.m2.1.1.1.1.2" xref="S7.SS3.p6.2.m2.1.1.1.1.2.cmml">0</mn><mo id="S7.SS3.p6.2.m2.1.1.1.1.3" lspace="0.278em" rspace="0.278em" xref="S7.SS3.p6.2.m2.1.1.1.1.3.cmml">:</mo><mn id="S7.SS3.p6.2.m2.1.1.1.1.4" xref="S7.SS3.p6.2.m2.1.1.1.1.4.cmml">0</mn><mo id="S7.SS3.p6.2.m2.1.1.1.1.5" lspace="0.278em" rspace="0.278em" xref="S7.SS3.p6.2.m2.1.1.1.1.5.cmml">:</mo><mn id="S7.SS3.p6.2.m2.1.1.1.1.6" xref="S7.SS3.p6.2.m2.1.1.1.1.6.cmml">0</mn></mrow><mo id="S7.SS3.p6.2.m2.1.1.1.3" stretchy="false" xref="S7.SS3.p6.2.m2.1.1.1.1.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S7.SS3.p6.2.m2.1b"><apply id="S7.SS3.p6.2.m2.1.1.1.1.cmml" xref="S7.SS3.p6.2.m2.1.1.1"><and id="S7.SS3.p6.2.m2.1.1.1.1a.cmml" xref="S7.SS3.p6.2.m2.1.1.1"></and><apply id="S7.SS3.p6.2.m2.1.1.1.1b.cmml" xref="S7.SS3.p6.2.m2.1.1.1"><ci id="S7.SS3.p6.2.m2.1.1.1.1.3.cmml" xref="S7.SS3.p6.2.m2.1.1.1.1.3">:</ci><cn id="S7.SS3.p6.2.m2.1.1.1.1.2.cmml" type="integer" xref="S7.SS3.p6.2.m2.1.1.1.1.2">0</cn><cn id="S7.SS3.p6.2.m2.1.1.1.1.4.cmml" type="integer" xref="S7.SS3.p6.2.m2.1.1.1.1.4">0</cn></apply><apply id="S7.SS3.p6.2.m2.1.1.1.1c.cmml" xref="S7.SS3.p6.2.m2.1.1.1"><ci id="S7.SS3.p6.2.m2.1.1.1.1.5.cmml" xref="S7.SS3.p6.2.m2.1.1.1.1.5">:</ci><share href="https://arxiv.org/html/2411.12694v2#S7.SS3.p6.2.m2.1.1.1.1.4.cmml" id="S7.SS3.p6.2.m2.1.1.1.1d.cmml" xref="S7.SS3.p6.2.m2.1.1.1"></share><cn id="S7.SS3.p6.2.m2.1.1.1.1.6.cmml" type="integer" xref="S7.SS3.p6.2.m2.1.1.1.1.6">0</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.SS3.p6.2.m2.1c">(0:0:0)</annotation><annotation encoding="application/x-llamapun" id="S7.SS3.p6.2.m2.1d">( 0 : 0 : 0 )</annotation></semantics></math>, thus: See <a class="ltx_ref" href="https://arxiv.org/html/2411.12694v2#S3.Thmtheorem9" title="Theorem 3.9. ‣ 3.D Results in CONGEST ‣ 3 Results and organisation ‣ Local Density and its Distributed Approximation"><span class="ltx_text ltx_ref_tag">3.9</span></a></p> </div> <div class="ltx_para ltx_noindent" id="S7.SS3.p7"> <p class="ltx_p" id="S7.SS3.p7.1">We plug in the runtime of Blocking<math alttext="(h,n)" class="ltx_Math" display="inline" id="S7.SS3.p7.1.m1.2"><semantics id="S7.SS3.p7.1.m1.2a"><mrow id="S7.SS3.p7.1.m1.2.3.2" xref="S7.SS3.p7.1.m1.2.3.1.cmml"><mo id="S7.SS3.p7.1.m1.2.3.2.1" stretchy="false" xref="S7.SS3.p7.1.m1.2.3.1.cmml">(</mo><mi id="S7.SS3.p7.1.m1.1.1" xref="S7.SS3.p7.1.m1.1.1.cmml">h</mi><mo id="S7.SS3.p7.1.m1.2.3.2.2" xref="S7.SS3.p7.1.m1.2.3.1.cmml">,</mo><mi id="S7.SS3.p7.1.m1.2.2" xref="S7.SS3.p7.1.m1.2.2.cmml">n</mi><mo id="S7.SS3.p7.1.m1.2.3.2.3" stretchy="false" xref="S7.SS3.p7.1.m1.2.3.1.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S7.SS3.p7.1.m1.2b"><interval closure="open" id="S7.SS3.p7.1.m1.2.3.1.cmml" xref="S7.SS3.p7.1.m1.2.3.2"><ci id="S7.SS3.p7.1.m1.1.1.cmml" xref="S7.SS3.p7.1.m1.1.1">ℎ</ci><ci id="S7.SS3.p7.1.m1.2.2.cmml" xref="S7.SS3.p7.1.m1.2.2">𝑛</ci></interval></annotation-xml><annotation encoding="application/x-tex" id="S7.SS3.p7.1.m1.2c">(h,n)</annotation><annotation encoding="application/x-llamapun" id="S7.SS3.p7.1.m1.2d">( italic_h , italic_n )</annotation></semantics></math> of Lemma <a class="ltx_ref" href="https://arxiv.org/html/2411.12694v2#S7.Thmtheorem3" title="Lemma 7.3 (Lemma 7.2 and 9.1 in [15]). ‣ 7 Results in CONGEST ‣ Local Density and its Distributed Approximation"><span class="ltx_text ltx_ref_tag">7.3</span></a> by Haeupler, Hershkowitz, and Saranurak <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2411.12694v2#bib.bib15" title="">15</a>]</cite> to obtain the following runtime:</p> </div> <div class="ltx_theorem ltx_theorem_corollary" id="S7.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="S7.Thmtheorem12.1.1.1">Corollary 7.12</span></span><span class="ltx_text ltx_font_bold" id="S7.Thmtheorem12.2.2">.</span> </h6> <div class="ltx_para" id="S7.Thmtheorem12.p1"> <p class="ltx_p" id="S7.Thmtheorem12.p1.5"><span class="ltx_text ltx_font_italic" id="S7.Thmtheorem12.p1.5.5">There is an algorithm in CONGEST that given a unit-weight graph <math alttext="G" class="ltx_Math" display="inline" id="S7.Thmtheorem12.p1.1.1.m1.1"><semantics id="S7.Thmtheorem12.p1.1.1.m1.1a"><mi id="S7.Thmtheorem12.p1.1.1.m1.1.1" xref="S7.Thmtheorem12.p1.1.1.m1.1.1.cmml">G</mi><annotation-xml encoding="MathML-Content" id="S7.Thmtheorem12.p1.1.1.m1.1b"><ci id="S7.Thmtheorem12.p1.1.1.m1.1.1.cmml" xref="S7.Thmtheorem12.p1.1.1.m1.1.1">𝐺</ci></annotation-xml><annotation encoding="application/x-tex" id="S7.Thmtheorem12.p1.1.1.m1.1c">G</annotation><annotation encoding="application/x-llamapun" id="S7.Thmtheorem12.p1.1.1.m1.1d">italic_G</annotation></semantics></math> and an <math alttext="\varepsilon>0" class="ltx_Math" display="inline" id="S7.Thmtheorem12.p1.2.2.m2.1"><semantics id="S7.Thmtheorem12.p1.2.2.m2.1a"><mrow id="S7.Thmtheorem12.p1.2.2.m2.1.1" xref="S7.Thmtheorem12.p1.2.2.m2.1.1.cmml"><mi id="S7.Thmtheorem12.p1.2.2.m2.1.1.2" xref="S7.Thmtheorem12.p1.2.2.m2.1.1.2.cmml">ε</mi><mo id="S7.Thmtheorem12.p1.2.2.m2.1.1.1" xref="S7.Thmtheorem12.p1.2.2.m2.1.1.1.cmml">></mo><mn id="S7.Thmtheorem12.p1.2.2.m2.1.1.3" xref="S7.Thmtheorem12.p1.2.2.m2.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S7.Thmtheorem12.p1.2.2.m2.1b"><apply id="S7.Thmtheorem12.p1.2.2.m2.1.1.cmml" xref="S7.Thmtheorem12.p1.2.2.m2.1.1"><gt id="S7.Thmtheorem12.p1.2.2.m2.1.1.1.cmml" xref="S7.Thmtheorem12.p1.2.2.m2.1.1.1"></gt><ci id="S7.Thmtheorem12.p1.2.2.m2.1.1.2.cmml" xref="S7.Thmtheorem12.p1.2.2.m2.1.1.2">𝜀</ci><cn id="S7.Thmtheorem12.p1.2.2.m2.1.1.3.cmml" type="integer" xref="S7.Thmtheorem12.p1.2.2.m2.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.Thmtheorem12.p1.2.2.m2.1c">\varepsilon>0</annotation><annotation encoding="application/x-llamapun" id="S7.Thmtheorem12.p1.2.2.m2.1d">italic_ε > 0</annotation></semantics></math> computes an orientation <math alttext="\overrightarrow{G}" class="ltx_Math" display="inline" id="S7.Thmtheorem12.p1.3.3.m3.1"><semantics id="S7.Thmtheorem12.p1.3.3.m3.1a"><mover accent="true" id="S7.Thmtheorem12.p1.3.3.m3.1.1" xref="S7.Thmtheorem12.p1.3.3.m3.1.1.cmml"><mi id="S7.Thmtheorem12.p1.3.3.m3.1.1.2" xref="S7.Thmtheorem12.p1.3.3.m3.1.1.2.cmml">G</mi><mo id="S7.Thmtheorem12.p1.3.3.m3.1.1.1" stretchy="false" xref="S7.Thmtheorem12.p1.3.3.m3.1.1.1.cmml">→</mo></mover><annotation-xml encoding="MathML-Content" id="S7.Thmtheorem12.p1.3.3.m3.1b"><apply id="S7.Thmtheorem12.p1.3.3.m3.1.1.cmml" xref="S7.Thmtheorem12.p1.3.3.m3.1.1"><ci id="S7.Thmtheorem12.p1.3.3.m3.1.1.1.cmml" xref="S7.Thmtheorem12.p1.3.3.m3.1.1.1">→</ci><ci id="S7.Thmtheorem12.p1.3.3.m3.1.1.2.cmml" xref="S7.Thmtheorem12.p1.3.3.m3.1.1.2">𝐺</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.Thmtheorem12.p1.3.3.m3.1c">\overrightarrow{G}</annotation><annotation encoding="application/x-llamapun" id="S7.Thmtheorem12.p1.3.3.m3.1d">over→ start_ARG italic_G end_ARG</annotation></semantics></math> such that <math alttext="\forall v\in V" class="ltx_Math" display="inline" id="S7.Thmtheorem12.p1.4.4.m4.1"><semantics id="S7.Thmtheorem12.p1.4.4.m4.1a"><mrow id="S7.Thmtheorem12.p1.4.4.m4.1.1" xref="S7.Thmtheorem12.p1.4.4.m4.1.1.cmml"><mrow id="S7.Thmtheorem12.p1.4.4.m4.1.1.2" xref="S7.Thmtheorem12.p1.4.4.m4.1.1.2.cmml"><mo id="S7.Thmtheorem12.p1.4.4.m4.1.1.2.1" rspace="0.167em" xref="S7.Thmtheorem12.p1.4.4.m4.1.1.2.1.cmml">∀</mo><mi id="S7.Thmtheorem12.p1.4.4.m4.1.1.2.2" xref="S7.Thmtheorem12.p1.4.4.m4.1.1.2.2.cmml">v</mi></mrow><mo id="S7.Thmtheorem12.p1.4.4.m4.1.1.1" xref="S7.Thmtheorem12.p1.4.4.m4.1.1.1.cmml">∈</mo><mi id="S7.Thmtheorem12.p1.4.4.m4.1.1.3" xref="S7.Thmtheorem12.p1.4.4.m4.1.1.3.cmml">V</mi></mrow><annotation-xml encoding="MathML-Content" id="S7.Thmtheorem12.p1.4.4.m4.1b"><apply id="S7.Thmtheorem12.p1.4.4.m4.1.1.cmml" xref="S7.Thmtheorem12.p1.4.4.m4.1.1"><in id="S7.Thmtheorem12.p1.4.4.m4.1.1.1.cmml" xref="S7.Thmtheorem12.p1.4.4.m4.1.1.1"></in><apply id="S7.Thmtheorem12.p1.4.4.m4.1.1.2.cmml" xref="S7.Thmtheorem12.p1.4.4.m4.1.1.2"><csymbol cd="latexml" id="S7.Thmtheorem12.p1.4.4.m4.1.1.2.1.cmml" xref="S7.Thmtheorem12.p1.4.4.m4.1.1.2.1">for-all</csymbol><ci id="S7.Thmtheorem12.p1.4.4.m4.1.1.2.2.cmml" 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id="S7.Thmtheorem12.p1.5.5.m5.4.4.1.1.1.1.3a" xref="S7.Thmtheorem12.p1.5.5.m5.4.4.1.1.1.1.3.cmml">−</mo><mn id="S7.Thmtheorem12.p1.5.5.m5.4.4.1.1.1.1.3.2" xref="S7.Thmtheorem12.p1.5.5.m5.4.4.1.1.1.1.3.2.cmml">1</mn></mrow></msup><mo id="S7.Thmtheorem12.p1.5.5.m5.4.4.1.1.1.2" xref="S7.Thmtheorem12.p1.5.5.m5.4.4.1.1.1.2.cmml">⁢</mo><msup id="S7.Thmtheorem12.p1.5.5.m5.4.4.1.1.1.3" xref="S7.Thmtheorem12.p1.5.5.m5.4.4.1.1.1.3.cmml"><mi id="S7.Thmtheorem12.p1.5.5.m5.4.4.1.1.1.3.2" xref="S7.Thmtheorem12.p1.5.5.m5.4.4.1.1.1.3.2.cmml">ρ</mi><mo id="S7.Thmtheorem12.p1.5.5.m5.4.4.1.1.1.3.3" xref="S7.Thmtheorem12.p1.5.5.m5.4.4.1.1.1.3.3.cmml">∗</mo></msup><mo id="S7.Thmtheorem12.p1.5.5.m5.4.4.1.1.1.2a" xref="S7.Thmtheorem12.p1.5.5.m5.4.4.1.1.1.2.cmml">⁢</mo><mrow id="S7.Thmtheorem12.p1.5.5.m5.4.4.1.1.1.4.2" xref="S7.Thmtheorem12.p1.5.5.m5.4.4.1.1.1.cmml"><mo id="S7.Thmtheorem12.p1.5.5.m5.4.4.1.1.1.4.2.1" stretchy="false" xref="S7.Thmtheorem12.p1.5.5.m5.4.4.1.1.1.cmml">(</mo><mi 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cd="ambiguous" id="S7.Thmtheorem12.p1.5.5.m5.5.5.2.2.2.3.1.cmml" xref="S7.Thmtheorem12.p1.5.5.m5.5.5.2.2.2.3">superscript</csymbol><ci id="S7.Thmtheorem12.p1.5.5.m5.5.5.2.2.2.3.2.cmml" xref="S7.Thmtheorem12.p1.5.5.m5.5.5.2.2.2.3.2">𝜌</ci><times id="S7.Thmtheorem12.p1.5.5.m5.5.5.2.2.2.3.3.cmml" xref="S7.Thmtheorem12.p1.5.5.m5.5.5.2.2.2.3.3"></times></apply><ci id="S7.Thmtheorem12.p1.5.5.m5.3.3.cmml" xref="S7.Thmtheorem12.p1.5.5.m5.3.3">𝑣</ci></apply></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.Thmtheorem12.p1.5.5.m5.5c">\textsl{g}(v)\in[(1+\varepsilon)^{-1}\rho^{*}(v),(1+\varepsilon)\rho^{*}(v)]</annotation><annotation encoding="application/x-llamapun" id="S7.Thmtheorem12.p1.5.5.m5.5d">g ( italic_v ) ∈ [ ( 1 + italic_ε ) start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT italic_ρ start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ( italic_v ) , ( 1 + italic_ε ) italic_ρ start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ( italic_v ) ]</annotation></semantics></math> in:</span></p> <ul class="ltx_itemize" id="S7.I9"> <li class="ltx_item" id="S7.I9.i1" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">•</span> <div class="ltx_para" id="S7.I9.i1.p1"> <p class="ltx_p" id="S7.I9.i1.p1.1"><math alttext="\tilde{O}(\varepsilon^{-11}\log^{12}n)" class="ltx_Math" display="inline" id="S7.I9.i1.p1.1.m1.1"><semantics id="S7.I9.i1.p1.1.m1.1a"><mrow id="S7.I9.i1.p1.1.m1.1.1" xref="S7.I9.i1.p1.1.m1.1.1.cmml"><mover accent="true" id="S7.I9.i1.p1.1.m1.1.1.3" xref="S7.I9.i1.p1.1.m1.1.1.3.cmml"><mi id="S7.I9.i1.p1.1.m1.1.1.3.2" xref="S7.I9.i1.p1.1.m1.1.1.3.2.cmml">O</mi><mo id="S7.I9.i1.p1.1.m1.1.1.3.1" xref="S7.I9.i1.p1.1.m1.1.1.3.1.cmml">~</mo></mover><mo id="S7.I9.i1.p1.1.m1.1.1.2" xref="S7.I9.i1.p1.1.m1.1.1.2.cmml">⁢</mo><mrow id="S7.I9.i1.p1.1.m1.1.1.1.1" xref="S7.I9.i1.p1.1.m1.1.1.1.1.1.cmml"><mo id="S7.I9.i1.p1.1.m1.1.1.1.1.2" stretchy="false" xref="S7.I9.i1.p1.1.m1.1.1.1.1.1.cmml">(</mo><mrow id="S7.I9.i1.p1.1.m1.1.1.1.1.1" xref="S7.I9.i1.p1.1.m1.1.1.1.1.1.cmml"><msup id="S7.I9.i1.p1.1.m1.1.1.1.1.1.2" xref="S7.I9.i1.p1.1.m1.1.1.1.1.1.2.cmml"><mi id="S7.I9.i1.p1.1.m1.1.1.1.1.1.2.2" xref="S7.I9.i1.p1.1.m1.1.1.1.1.1.2.2.cmml">ε</mi><mrow id="S7.I9.i1.p1.1.m1.1.1.1.1.1.2.3" xref="S7.I9.i1.p1.1.m1.1.1.1.1.1.2.3.cmml"><mo id="S7.I9.i1.p1.1.m1.1.1.1.1.1.2.3a" xref="S7.I9.i1.p1.1.m1.1.1.1.1.1.2.3.cmml">−</mo><mn id="S7.I9.i1.p1.1.m1.1.1.1.1.1.2.3.2" xref="S7.I9.i1.p1.1.m1.1.1.1.1.1.2.3.2.cmml">11</mn></mrow></msup><mo id="S7.I9.i1.p1.1.m1.1.1.1.1.1.1" lspace="0.167em" xref="S7.I9.i1.p1.1.m1.1.1.1.1.1.1.cmml">⁢</mo><mrow id="S7.I9.i1.p1.1.m1.1.1.1.1.1.3" xref="S7.I9.i1.p1.1.m1.1.1.1.1.1.3.cmml"><msup id="S7.I9.i1.p1.1.m1.1.1.1.1.1.3.1" xref="S7.I9.i1.p1.1.m1.1.1.1.1.1.3.1.cmml"><mi id="S7.I9.i1.p1.1.m1.1.1.1.1.1.3.1.2" xref="S7.I9.i1.p1.1.m1.1.1.1.1.1.3.1.2.cmml">log</mi><mn id="S7.I9.i1.p1.1.m1.1.1.1.1.1.3.1.3" xref="S7.I9.i1.p1.1.m1.1.1.1.1.1.3.1.3.cmml">12</mn></msup><mo id="S7.I9.i1.p1.1.m1.1.1.1.1.1.3a" 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id="S7.I9.i1.p1.1.m1.1.1.1.1.1.2.1.cmml" xref="S7.I9.i1.p1.1.m1.1.1.1.1.1.2">superscript</csymbol><ci id="S7.I9.i1.p1.1.m1.1.1.1.1.1.2.2.cmml" xref="S7.I9.i1.p1.1.m1.1.1.1.1.1.2.2">𝜀</ci><apply id="S7.I9.i1.p1.1.m1.1.1.1.1.1.2.3.cmml" xref="S7.I9.i1.p1.1.m1.1.1.1.1.1.2.3"><minus id="S7.I9.i1.p1.1.m1.1.1.1.1.1.2.3.1.cmml" xref="S7.I9.i1.p1.1.m1.1.1.1.1.1.2.3"></minus><cn id="S7.I9.i1.p1.1.m1.1.1.1.1.1.2.3.2.cmml" type="integer" xref="S7.I9.i1.p1.1.m1.1.1.1.1.1.2.3.2">11</cn></apply></apply><apply id="S7.I9.i1.p1.1.m1.1.1.1.1.1.3.cmml" xref="S7.I9.i1.p1.1.m1.1.1.1.1.1.3"><apply id="S7.I9.i1.p1.1.m1.1.1.1.1.1.3.1.cmml" xref="S7.I9.i1.p1.1.m1.1.1.1.1.1.3.1"><csymbol cd="ambiguous" id="S7.I9.i1.p1.1.m1.1.1.1.1.1.3.1.1.cmml" xref="S7.I9.i1.p1.1.m1.1.1.1.1.1.3.1">superscript</csymbol><log id="S7.I9.i1.p1.1.m1.1.1.1.1.1.3.1.2.cmml" xref="S7.I9.i1.p1.1.m1.1.1.1.1.1.3.1.2"></log><cn id="S7.I9.i1.p1.1.m1.1.1.1.1.1.3.1.3.cmml" type="integer" xref="S7.I9.i1.p1.1.m1.1.1.1.1.1.3.1.3">12</cn></apply><ci id="S7.I9.i1.p1.1.m1.1.1.1.1.1.3.2.cmml" xref="S7.I9.i1.p1.1.m1.1.1.1.1.1.3.2">𝑛</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.I9.i1.p1.1.m1.1c">\tilde{O}(\varepsilon^{-11}\log^{12}n)</annotation><annotation encoding="application/x-llamapun" id="S7.I9.i1.p1.1.m1.1d">over~ start_ARG italic_O end_ARG ( italic_ε start_POSTSUPERSCRIPT - 11 end_POSTSUPERSCRIPT roman_log start_POSTSUPERSCRIPT 12 end_POSTSUPERSCRIPT italic_n )</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S7.I9.i1.p1.1.1"> rounds with high probability, or</span></p> </div> </li> <li class="ltx_item" id="S7.I9.i2" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">•</span> <div class="ltx_para" id="S7.I9.i2.p1"> <p class="ltx_p" id="S7.I9.i2.p1.1"><math alttext="\tilde{O}(\varepsilon^{-15}\log^{16}n\cdot 2^{O(\sqrt{\log n})})" class="ltx_Math" display="inline" 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xref="S7.I9.i2.p1.1.m1.1.1.1.4.2"><root id="S7.I9.i2.p1.1.m1.1.1.1.1a.cmml" xref="S7.I9.i2.p1.1.m1.1.1.1.4.2"></root><apply id="S7.I9.i2.p1.1.m1.1.1.1.1.2.cmml" xref="S7.I9.i2.p1.1.m1.1.1.1.1.2"><log id="S7.I9.i2.p1.1.m1.1.1.1.1.2.1.cmml" xref="S7.I9.i2.p1.1.m1.1.1.1.1.2.1"></log><ci id="S7.I9.i2.p1.1.m1.1.1.1.1.2.2.cmml" xref="S7.I9.i2.p1.1.m1.1.1.1.1.2.2">𝑛</ci></apply></apply></apply></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.I9.i2.p1.1.m1.2c">\tilde{O}(\varepsilon^{-15}\log^{16}n\cdot 2^{O(\sqrt{\log n})})</annotation><annotation encoding="application/x-llamapun" id="S7.I9.i2.p1.1.m1.2d">over~ start_ARG italic_O end_ARG ( italic_ε start_POSTSUPERSCRIPT - 15 end_POSTSUPERSCRIPT roman_log start_POSTSUPERSCRIPT 16 end_POSTSUPERSCRIPT italic_n ⋅ 2 start_POSTSUPERSCRIPT italic_O ( square-root start_ARG roman_log italic_n end_ARG ) end_POSTSUPERSCRIPT )</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S7.I9.i2.p1.1.1"> deterministic rounds.</span></p> </div> </li> </ul> </div> </div> <div class="ltx_para" id="S7.SS3.p8"> <p class="ltx_p" id="S7.SS3.p8.1">Finally, we apply Corollary <a class="ltx_ref" href="https://arxiv.org/html/2411.12694v2#S3.Thmtheorem3" title="Corollary 3.3. ‣ 3.A Conceptual results for local density ‣ 3 Results and organisation ‣ Local Density and its Distributed Approximation"><span class="ltx_text ltx_ref_tag">3.3</span></a> which states that <math alttext="\rho^{\max}(G)=\Delta^{\min}(G)=\max_{v}\textsl{g}^{*}(v)" class="ltx_Math" display="inline" id="S7.SS3.p8.1.m1.3"><semantics id="S7.SS3.p8.1.m1.3a"><mrow id="S7.SS3.p8.1.m1.3.4" xref="S7.SS3.p8.1.m1.3.4.cmml"><mrow id="S7.SS3.p8.1.m1.3.4.2" xref="S7.SS3.p8.1.m1.3.4.2.cmml"><msup id="S7.SS3.p8.1.m1.3.4.2.2" xref="S7.SS3.p8.1.m1.3.4.2.2.cmml"><mi id="S7.SS3.p8.1.m1.3.4.2.2.2" xref="S7.SS3.p8.1.m1.3.4.2.2.2.cmml">ρ</mi><mi id="S7.SS3.p8.1.m1.3.4.2.2.3" 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id="S7.SS3.p8.1.m1.3.4.6.2.2.3" xref="S7.SS3.p8.1.m1.3.4.6.2.2.3.cmml">∗</mo></msup></mrow><mo id="S7.SS3.p8.1.m1.3.4.6.1" xref="S7.SS3.p8.1.m1.3.4.6.1.cmml">⁢</mo><mrow id="S7.SS3.p8.1.m1.3.4.6.3.2" xref="S7.SS3.p8.1.m1.3.4.6.cmml"><mo id="S7.SS3.p8.1.m1.3.4.6.3.2.1" stretchy="false" xref="S7.SS3.p8.1.m1.3.4.6.cmml">(</mo><mi id="S7.SS3.p8.1.m1.3.3" xref="S7.SS3.p8.1.m1.3.3.cmml">v</mi><mo id="S7.SS3.p8.1.m1.3.4.6.3.2.2" stretchy="false" xref="S7.SS3.p8.1.m1.3.4.6.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S7.SS3.p8.1.m1.3b"><apply id="S7.SS3.p8.1.m1.3.4.cmml" xref="S7.SS3.p8.1.m1.3.4"><and id="S7.SS3.p8.1.m1.3.4a.cmml" xref="S7.SS3.p8.1.m1.3.4"></and><apply id="S7.SS3.p8.1.m1.3.4b.cmml" xref="S7.SS3.p8.1.m1.3.4"><eq id="S7.SS3.p8.1.m1.3.4.3.cmml" xref="S7.SS3.p8.1.m1.3.4.3"></eq><apply id="S7.SS3.p8.1.m1.3.4.2.cmml" xref="S7.SS3.p8.1.m1.3.4.2"><times id="S7.SS3.p8.1.m1.3.4.2.1.cmml" xref="S7.SS3.p8.1.m1.3.4.2.1"></times><apply id="S7.SS3.p8.1.m1.3.4.2.2.cmml" xref="S7.SS3.p8.1.m1.3.4.2.2"><csymbol cd="ambiguous" id="S7.SS3.p8.1.m1.3.4.2.2.1.cmml" xref="S7.SS3.p8.1.m1.3.4.2.2">superscript</csymbol><ci id="S7.SS3.p8.1.m1.3.4.2.2.2.cmml" xref="S7.SS3.p8.1.m1.3.4.2.2.2">𝜌</ci><max id="S7.SS3.p8.1.m1.3.4.2.2.3.cmml" xref="S7.SS3.p8.1.m1.3.4.2.2.3"></max></apply><ci id="S7.SS3.p8.1.m1.1.1.cmml" xref="S7.SS3.p8.1.m1.1.1">𝐺</ci></apply><apply id="S7.SS3.p8.1.m1.3.4.4.cmml" xref="S7.SS3.p8.1.m1.3.4.4"><times id="S7.SS3.p8.1.m1.3.4.4.1.cmml" xref="S7.SS3.p8.1.m1.3.4.4.1"></times><apply id="S7.SS3.p8.1.m1.3.4.4.2.cmml" xref="S7.SS3.p8.1.m1.3.4.4.2"><csymbol cd="ambiguous" id="S7.SS3.p8.1.m1.3.4.4.2.1.cmml" xref="S7.SS3.p8.1.m1.3.4.4.2">superscript</csymbol><ci id="S7.SS3.p8.1.m1.3.4.4.2.2.cmml" xref="S7.SS3.p8.1.m1.3.4.4.2.2">Δ</ci><min id="S7.SS3.p8.1.m1.3.4.4.2.3.cmml" xref="S7.SS3.p8.1.m1.3.4.4.2.3"></min></apply><ci id="S7.SS3.p8.1.m1.2.2.cmml" xref="S7.SS3.p8.1.m1.2.2">𝐺</ci></apply></apply><apply id="S7.SS3.p8.1.m1.3.4c.cmml" xref="S7.SS3.p8.1.m1.3.4"><eq id="S7.SS3.p8.1.m1.3.4.5.cmml" xref="S7.SS3.p8.1.m1.3.4.5"></eq><share href="https://arxiv.org/html/2411.12694v2#S7.SS3.p8.1.m1.3.4.4.cmml" id="S7.SS3.p8.1.m1.3.4d.cmml" xref="S7.SS3.p8.1.m1.3.4"></share><apply id="S7.SS3.p8.1.m1.3.4.6.cmml" xref="S7.SS3.p8.1.m1.3.4.6"><times id="S7.SS3.p8.1.m1.3.4.6.1.cmml" xref="S7.SS3.p8.1.m1.3.4.6.1"></times><apply id="S7.SS3.p8.1.m1.3.4.6.2.cmml" xref="S7.SS3.p8.1.m1.3.4.6.2"><apply id="S7.SS3.p8.1.m1.3.4.6.2.1.cmml" xref="S7.SS3.p8.1.m1.3.4.6.2.1"><csymbol cd="ambiguous" id="S7.SS3.p8.1.m1.3.4.6.2.1.1.cmml" xref="S7.SS3.p8.1.m1.3.4.6.2.1">subscript</csymbol><max id="S7.SS3.p8.1.m1.3.4.6.2.1.2.cmml" xref="S7.SS3.p8.1.m1.3.4.6.2.1.2"></max><ci id="S7.SS3.p8.1.m1.3.4.6.2.1.3.cmml" xref="S7.SS3.p8.1.m1.3.4.6.2.1.3">𝑣</ci></apply><apply id="S7.SS3.p8.1.m1.3.4.6.2.2.cmml" xref="S7.SS3.p8.1.m1.3.4.6.2.2"><csymbol cd="ambiguous" id="S7.SS3.p8.1.m1.3.4.6.2.2.1.cmml" xref="S7.SS3.p8.1.m1.3.4.6.2.2">superscript</csymbol><ci id="S7.SS3.p8.1.m1.3.4.6.2.2.2a.cmml" xref="S7.SS3.p8.1.m1.3.4.6.2.2.2"><mtext class="ltx_mathvariant_italic" id="S7.SS3.p8.1.m1.3.4.6.2.2.2.cmml" xref="S7.SS3.p8.1.m1.3.4.6.2.2.2">g</mtext></ci><times id="S7.SS3.p8.1.m1.3.4.6.2.2.3.cmml" xref="S7.SS3.p8.1.m1.3.4.6.2.2.3"></times></apply></apply><ci id="S7.SS3.p8.1.m1.3.3.cmml" xref="S7.SS3.p8.1.m1.3.3">𝑣</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.SS3.p8.1.m1.3c">\rho^{\max}(G)=\Delta^{\min}(G)=\max_{v}\textsl{g}^{*}(v)</annotation><annotation encoding="application/x-llamapun" id="S7.SS3.p8.1.m1.3d">italic_ρ start_POSTSUPERSCRIPT roman_max end_POSTSUPERSCRIPT ( italic_G ) = roman_Δ start_POSTSUPERSCRIPT roman_min end_POSTSUPERSCRIPT ( italic_G ) = roman_max start_POSTSUBSCRIPT italic_v end_POSTSUBSCRIPT g start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ( italic_v )</annotation></semantics></math> to conclude:</p> </div> <div class="ltx_theorem ltx_theorem_corollary" id="S7.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="S7.Thmtheorem13.1.1.1">Corollary 7.13</span></span><span class="ltx_text ltx_font_bold" id="S7.Thmtheorem13.2.2">.</span> </h6> <div class="ltx_para" id="S7.Thmtheorem13.p1"> <p class="ltx_p" id="S7.Thmtheorem13.p1.3"><span class="ltx_text ltx_font_italic" id="S7.Thmtheorem13.p1.3.3">There is a deterministic algorithm in CONGEST that given a unit-weight graph <math alttext="G" class="ltx_Math" display="inline" id="S7.Thmtheorem13.p1.1.1.m1.1"><semantics id="S7.Thmtheorem13.p1.1.1.m1.1a"><mi id="S7.Thmtheorem13.p1.1.1.m1.1.1" xref="S7.Thmtheorem13.p1.1.1.m1.1.1.cmml">G</mi><annotation-xml encoding="MathML-Content" id="S7.Thmtheorem13.p1.1.1.m1.1b"><ci id="S7.Thmtheorem13.p1.1.1.m1.1.1.cmml" xref="S7.Thmtheorem13.p1.1.1.m1.1.1">𝐺</ci></annotation-xml><annotation encoding="application/x-tex" id="S7.Thmtheorem13.p1.1.1.m1.1c">G</annotation><annotation encoding="application/x-llamapun" id="S7.Thmtheorem13.p1.1.1.m1.1d">italic_G</annotation></semantics></math> and an <math alttext="\varepsilon>0" class="ltx_Math" display="inline" id="S7.Thmtheorem13.p1.2.2.m2.1"><semantics id="S7.Thmtheorem13.p1.2.2.m2.1a"><mrow id="S7.Thmtheorem13.p1.2.2.m2.1.1" xref="S7.Thmtheorem13.p1.2.2.m2.1.1.cmml"><mi id="S7.Thmtheorem13.p1.2.2.m2.1.1.2" xref="S7.Thmtheorem13.p1.2.2.m2.1.1.2.cmml">ε</mi><mo id="S7.Thmtheorem13.p1.2.2.m2.1.1.1" xref="S7.Thmtheorem13.p1.2.2.m2.1.1.1.cmml">></mo><mn id="S7.Thmtheorem13.p1.2.2.m2.1.1.3" xref="S7.Thmtheorem13.p1.2.2.m2.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S7.Thmtheorem13.p1.2.2.m2.1b"><apply id="S7.Thmtheorem13.p1.2.2.m2.1.1.cmml" xref="S7.Thmtheorem13.p1.2.2.m2.1.1"><gt id="S7.Thmtheorem13.p1.2.2.m2.1.1.1.cmml" xref="S7.Thmtheorem13.p1.2.2.m2.1.1.1"></gt><ci id="S7.Thmtheorem13.p1.2.2.m2.1.1.2.cmml" xref="S7.Thmtheorem13.p1.2.2.m2.1.1.2">𝜀</ci><cn id="S7.Thmtheorem13.p1.2.2.m2.1.1.3.cmml" type="integer" xref="S7.Thmtheorem13.p1.2.2.m2.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.Thmtheorem13.p1.2.2.m2.1c">\varepsilon>0</annotation><annotation encoding="application/x-llamapun" id="S7.Thmtheorem13.p1.2.2.m2.1d">italic_ε > 0</annotation></semantics></math>, that computes an orientation <math alttext="\overrightarrow{G}" class="ltx_Math" display="inline" id="S7.Thmtheorem13.p1.3.3.m3.1"><semantics id="S7.Thmtheorem13.p1.3.3.m3.1a"><mover accent="true" id="S7.Thmtheorem13.p1.3.3.m3.1.1" xref="S7.Thmtheorem13.p1.3.3.m3.1.1.cmml"><mi id="S7.Thmtheorem13.p1.3.3.m3.1.1.2" xref="S7.Thmtheorem13.p1.3.3.m3.1.1.2.cmml">G</mi><mo id="S7.Thmtheorem13.p1.3.3.m3.1.1.1" stretchy="false" xref="S7.Thmtheorem13.p1.3.3.m3.1.1.1.cmml">→</mo></mover><annotation-xml encoding="MathML-Content" id="S7.Thmtheorem13.p1.3.3.m3.1b"><apply id="S7.Thmtheorem13.p1.3.3.m3.1.1.cmml" xref="S7.Thmtheorem13.p1.3.3.m3.1.1"><ci id="S7.Thmtheorem13.p1.3.3.m3.1.1.1.cmml" xref="S7.Thmtheorem13.p1.3.3.m3.1.1.1">→</ci><ci id="S7.Thmtheorem13.p1.3.3.m3.1.1.2.cmml" xref="S7.Thmtheorem13.p1.3.3.m3.1.1.2">𝐺</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.Thmtheorem13.p1.3.3.m3.1c">\overrightarrow{G}</annotation><annotation encoding="application/x-llamapun" id="S7.Thmtheorem13.p1.3.3.m3.1d">over→ start_ARG italic_G end_ARG</annotation></semantics></math> such that:</span></p> <table class="ltx_equation ltx_eqn_table" id="S7.Ex31"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_left_padleft"></td> <td class="ltx_eqn_cell ltx_align_left"><math alttext="\max_{v\in V}\textsl{g}(v)\in[(1+\varepsilon)^{-1}\rho^{\max}(G),(1+% \varepsilon)\rho^{\max}(G)]," class="ltx_Math" display="block" id="S7.Ex31.m1.4"><semantics id="S7.Ex31.m1.4a"><mrow id="S7.Ex31.m1.4.4.1" 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</table> <p class="ltx_p" id="S7.Thmtheorem13.p1.4"><span class="ltx_text ltx_font_italic" id="S7.Thmtheorem13.p1.4.1">in a number of rounds that is sublinear in <math alttext="n" class="ltx_Math" display="inline" id="S7.Thmtheorem13.p1.4.1.m1.1"><semantics id="S7.Thmtheorem13.p1.4.1.m1.1a"><mi id="S7.Thmtheorem13.p1.4.1.m1.1.1" xref="S7.Thmtheorem13.p1.4.1.m1.1.1.cmml">n</mi><annotation-xml encoding="MathML-Content" id="S7.Thmtheorem13.p1.4.1.m1.1b"><ci id="S7.Thmtheorem13.p1.4.1.m1.1.1.cmml" xref="S7.Thmtheorem13.p1.4.1.m1.1.1">𝑛</ci></annotation-xml><annotation encoding="application/x-tex" id="S7.Thmtheorem13.p1.4.1.m1.1c">n</annotation><annotation encoding="application/x-llamapun" id="S7.Thmtheorem13.p1.4.1.m1.1d">italic_n</annotation></semantics></math>.</span></p> </div> </div> </section> <section class="ltx_subsection" id="S7.SS4"> <h3 class="ltx_title ltx_title_subsection"> <span class="ltx_tag ltx_tag_subsection">7.4 </span>Formally proving correctness</h3> <div class="ltx_para" id="S7.SS4.p1"> <p class="ltx_p" id="S7.SS4.p1.1">We formalise Algorithms <a class="ltx_ref" href="https://arxiv.org/html/2411.12694v2#alg2" title="Algorithm 2 ‣ 7.2 Formal algorithm definition. ‣ 7 Results in CONGEST ‣ Local Density and its Distributed Approximation"><span class="ltx_text ltx_ref_tag">2</span></a> and <a class="ltx_ref" href="https://arxiv.org/html/2411.12694v2#alg3" title="Algorithm 3 ‣ 7.2 Formal algorithm definition. ‣ 7 Results in CONGEST ‣ Local Density and its Distributed Approximation"><span class="ltx_text ltx_ref_tag">3</span></a> and <a class="ltx_ref" href="https://arxiv.org/html/2411.12694v2#alg4" title="Algorithm 4 ‣ 7.2 Formal algorithm definition. ‣ 7 Results in CONGEST ‣ Local Density and its Distributed Approximation"><span class="ltx_text ltx_ref_tag">4</span></a> and <a class="ltx_ref" href="https://arxiv.org/html/2411.12694v2#alg5" title="Algorithm 5 ‣ 7.2 Formal algorithm definition. ‣ 7 Results in CONGEST ‣ Local Density and its Distributed Approximation"><span class="ltx_text ltx_ref_tag">5</span></a> and prove that they maintain Invariant <a class="ltx_ref" href="https://arxiv.org/html/2411.12694v2#Thminvariant1" title="Invariant 1. ‣ 7.1 Algorithm overview ‣ 7 Results in CONGEST ‣ Local Density and its Distributed Approximation"><span class="ltx_text ltx_ref_tag">1</span></a>.</p> </div> <div class="ltx_para" id="S7.SS4.p2"> <p class="ltx_p" id="S7.SS4.p2.1">First, we prove in three steps that after each odd <span class="ltx_text ltx_font_smallcaps" id="S7.SS4.p2.1.1">minute</span> (i.e., at the start of each even <span class="ltx_text ltx_font_smallcaps" id="S7.SS4.p2.1.2">minute</span>) there are only violating out-edges from <math alttext="L_{h}" class="ltx_Math" display="inline" id="S7.SS4.p2.1.m1.1"><semantics id="S7.SS4.p2.1.m1.1a"><msub id="S7.SS4.p2.1.m1.1.1" xref="S7.SS4.p2.1.m1.1.1.cmml"><mi id="S7.SS4.p2.1.m1.1.1.2" xref="S7.SS4.p2.1.m1.1.1.2.cmml">L</mi><mi id="S7.SS4.p2.1.m1.1.1.3" xref="S7.SS4.p2.1.m1.1.1.3.cmml">h</mi></msub><annotation-xml encoding="MathML-Content" id="S7.SS4.p2.1.m1.1b"><apply id="S7.SS4.p2.1.m1.1.1.cmml" xref="S7.SS4.p2.1.m1.1.1"><csymbol cd="ambiguous" id="S7.SS4.p2.1.m1.1.1.1.cmml" xref="S7.SS4.p2.1.m1.1.1">subscript</csymbol><ci id="S7.SS4.p2.1.m1.1.1.2.cmml" xref="S7.SS4.p2.1.m1.1.1.2">𝐿</ci><ci id="S7.SS4.p2.1.m1.1.1.3.cmml" xref="S7.SS4.p2.1.m1.1.1.3">ℎ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.SS4.p2.1.m1.1c">L_{h}</annotation><annotation encoding="application/x-llamapun" id="S7.SS4.p2.1.m1.1d">italic_L start_POSTSUBSCRIPT italic_h end_POSTSUBSCRIPT</annotation></semantics></math> and below (Corollary <a class="ltx_ref" href="https://arxiv.org/html/2411.12694v2#S7.Thmtheorem18" title="Corollary 7.18. ‣ 7.4 Formally proving correctness ‣ 7 Results in CONGEST ‣ Local Density and its Distributed Approximation"><span class="ltx_text ltx_ref_tag">7.18</span></a>).</p> </div> <div class="ltx_theorem ltx_theorem_lemma" id="S7.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="S7.Thmtheorem14.1.1.1">Lemma 7.14</span></span><span class="ltx_text ltx_font_bold" id="S7.Thmtheorem14.2.2">.</span> </h6> <div class="ltx_para" id="S7.Thmtheorem14.p1"> <p class="ltx_p" id="S7.Thmtheorem14.p1.3"><span class="ltx_text ltx_font_italic" id="S7.Thmtheorem14.p1.3.3">For all times <math alttext="(h:m:s)" class="ltx_Math" display="inline" id="S7.Thmtheorem14.p1.1.1.m1.1"><semantics id="S7.Thmtheorem14.p1.1.1.m1.1a"><mrow id="S7.Thmtheorem14.p1.1.1.m1.1.1.1" xref="S7.Thmtheorem14.p1.1.1.m1.1.1.1.1.cmml"><mo id="S7.Thmtheorem14.p1.1.1.m1.1.1.1.2" stretchy="false" xref="S7.Thmtheorem14.p1.1.1.m1.1.1.1.1.cmml">(</mo><mrow id="S7.Thmtheorem14.p1.1.1.m1.1.1.1.1" xref="S7.Thmtheorem14.p1.1.1.m1.1.1.1.1.cmml"><mi id="S7.Thmtheorem14.p1.1.1.m1.1.1.1.1.2" xref="S7.Thmtheorem14.p1.1.1.m1.1.1.1.1.2.cmml">h</mi><mo id="S7.Thmtheorem14.p1.1.1.m1.1.1.1.1.3" lspace="0.278em" rspace="0.278em" xref="S7.Thmtheorem14.p1.1.1.m1.1.1.1.1.3.cmml">:</mo><mi id="S7.Thmtheorem14.p1.1.1.m1.1.1.1.1.4" xref="S7.Thmtheorem14.p1.1.1.m1.1.1.1.1.4.cmml">m</mi><mo id="S7.Thmtheorem14.p1.1.1.m1.1.1.1.1.5" lspace="0.278em" rspace="0.278em" xref="S7.Thmtheorem14.p1.1.1.m1.1.1.1.1.5.cmml">:</mo><mi id="S7.Thmtheorem14.p1.1.1.m1.1.1.1.1.6" xref="S7.Thmtheorem14.p1.1.1.m1.1.1.1.1.6.cmml">s</mi></mrow><mo id="S7.Thmtheorem14.p1.1.1.m1.1.1.1.3" stretchy="false" xref="S7.Thmtheorem14.p1.1.1.m1.1.1.1.1.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S7.Thmtheorem14.p1.1.1.m1.1b"><apply id="S7.Thmtheorem14.p1.1.1.m1.1.1.1.1.cmml" xref="S7.Thmtheorem14.p1.1.1.m1.1.1.1"><and id="S7.Thmtheorem14.p1.1.1.m1.1.1.1.1a.cmml" xref="S7.Thmtheorem14.p1.1.1.m1.1.1.1"></and><apply id="S7.Thmtheorem14.p1.1.1.m1.1.1.1.1b.cmml" xref="S7.Thmtheorem14.p1.1.1.m1.1.1.1"><ci id="S7.Thmtheorem14.p1.1.1.m1.1.1.1.1.3.cmml" xref="S7.Thmtheorem14.p1.1.1.m1.1.1.1.1.3">:</ci><ci id="S7.Thmtheorem14.p1.1.1.m1.1.1.1.1.2.cmml" xref="S7.Thmtheorem14.p1.1.1.m1.1.1.1.1.2">ℎ</ci><ci id="S7.Thmtheorem14.p1.1.1.m1.1.1.1.1.4.cmml" xref="S7.Thmtheorem14.p1.1.1.m1.1.1.1.1.4">𝑚</ci></apply><apply id="S7.Thmtheorem14.p1.1.1.m1.1.1.1.1c.cmml" xref="S7.Thmtheorem14.p1.1.1.m1.1.1.1"><ci id="S7.Thmtheorem14.p1.1.1.m1.1.1.1.1.5.cmml" xref="S7.Thmtheorem14.p1.1.1.m1.1.1.1.1.5">:</ci><share href="https://arxiv.org/html/2411.12694v2#S7.Thmtheorem14.p1.1.1.m1.1.1.1.1.4.cmml" id="S7.Thmtheorem14.p1.1.1.m1.1.1.1.1d.cmml" xref="S7.Thmtheorem14.p1.1.1.m1.1.1.1"></share><ci id="S7.Thmtheorem14.p1.1.1.m1.1.1.1.1.6.cmml" xref="S7.Thmtheorem14.p1.1.1.m1.1.1.1.1.6">𝑠</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.Thmtheorem14.p1.1.1.m1.1c">(h:m:s)</annotation><annotation encoding="application/x-llamapun" id="S7.Thmtheorem14.p1.1.1.m1.1d">( italic_h : italic_m : italic_s )</annotation></semantics></math> with <math alttext="m" class="ltx_Math" display="inline" id="S7.Thmtheorem14.p1.2.2.m2.1"><semantics id="S7.Thmtheorem14.p1.2.2.m2.1a"><mi id="S7.Thmtheorem14.p1.2.2.m2.1.1" xref="S7.Thmtheorem14.p1.2.2.m2.1.1.cmml">m</mi><annotation-xml encoding="MathML-Content" id="S7.Thmtheorem14.p1.2.2.m2.1b"><ci id="S7.Thmtheorem14.p1.2.2.m2.1.1.cmml" xref="S7.Thmtheorem14.p1.2.2.m2.1.1">𝑚</ci></annotation-xml><annotation encoding="application/x-tex" id="S7.Thmtheorem14.p1.2.2.m2.1c">m</annotation><annotation encoding="application/x-llamapun" id="S7.Thmtheorem14.p1.2.2.m2.1d">italic_m</annotation></semantics></math> odd, there are no violating edges from above level <math alttext="h+1" class="ltx_Math" display="inline" id="S7.Thmtheorem14.p1.3.3.m3.1"><semantics id="S7.Thmtheorem14.p1.3.3.m3.1a"><mrow id="S7.Thmtheorem14.p1.3.3.m3.1.1" xref="S7.Thmtheorem14.p1.3.3.m3.1.1.cmml"><mi id="S7.Thmtheorem14.p1.3.3.m3.1.1.2" xref="S7.Thmtheorem14.p1.3.3.m3.1.1.2.cmml">h</mi><mo id="S7.Thmtheorem14.p1.3.3.m3.1.1.1" xref="S7.Thmtheorem14.p1.3.3.m3.1.1.1.cmml">+</mo><mn id="S7.Thmtheorem14.p1.3.3.m3.1.1.3" xref="S7.Thmtheorem14.p1.3.3.m3.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S7.Thmtheorem14.p1.3.3.m3.1b"><apply id="S7.Thmtheorem14.p1.3.3.m3.1.1.cmml" xref="S7.Thmtheorem14.p1.3.3.m3.1.1"><plus id="S7.Thmtheorem14.p1.3.3.m3.1.1.1.cmml" xref="S7.Thmtheorem14.p1.3.3.m3.1.1.1"></plus><ci id="S7.Thmtheorem14.p1.3.3.m3.1.1.2.cmml" xref="S7.Thmtheorem14.p1.3.3.m3.1.1.2">ℎ</ci><cn id="S7.Thmtheorem14.p1.3.3.m3.1.1.3.cmml" type="integer" xref="S7.Thmtheorem14.p1.3.3.m3.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.Thmtheorem14.p1.3.3.m3.1c">h+1</annotation><annotation encoding="application/x-llamapun" id="S7.Thmtheorem14.p1.3.3.m3.1d">italic_h + 1</annotation></semantics></math>.</span></p> </div> </div> <div class="ltx_theorem ltx_theorem_proof" id="S7.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="S7.Thmtheorem15.1.1.1">Proof 7.15</span></span><span class="ltx_text ltx_font_bold" id="S7.Thmtheorem15.2.2">.</span> </h6> <div class="ltx_para" id="S7.Thmtheorem15.p1"> <p class="ltx_p" id="S7.Thmtheorem15.p1.21"><span class="ltx_text ltx_font_italic" id="S7.Thmtheorem15.p1.21.21">At the start of <math alttext="(h:m:0)" class="ltx_Math" display="inline" id="S7.Thmtheorem15.p1.1.1.m1.1"><semantics id="S7.Thmtheorem15.p1.1.1.m1.1a"><mrow id="S7.Thmtheorem15.p1.1.1.m1.1.1.1" xref="S7.Thmtheorem15.p1.1.1.m1.1.1.1.1.cmml"><mo id="S7.Thmtheorem15.p1.1.1.m1.1.1.1.2" stretchy="false" xref="S7.Thmtheorem15.p1.1.1.m1.1.1.1.1.cmml">(</mo><mrow id="S7.Thmtheorem15.p1.1.1.m1.1.1.1.1" xref="S7.Thmtheorem15.p1.1.1.m1.1.1.1.1.cmml"><mi id="S7.Thmtheorem15.p1.1.1.m1.1.1.1.1.2" xref="S7.Thmtheorem15.p1.1.1.m1.1.1.1.1.2.cmml">h</mi><mo id="S7.Thmtheorem15.p1.1.1.m1.1.1.1.1.3" lspace="0.278em" rspace="0.278em" xref="S7.Thmtheorem15.p1.1.1.m1.1.1.1.1.3.cmml">:</mo><mi id="S7.Thmtheorem15.p1.1.1.m1.1.1.1.1.4" xref="S7.Thmtheorem15.p1.1.1.m1.1.1.1.1.4.cmml">m</mi><mo id="S7.Thmtheorem15.p1.1.1.m1.1.1.1.1.5" lspace="0.278em" rspace="0.278em" xref="S7.Thmtheorem15.p1.1.1.m1.1.1.1.1.5.cmml">:</mo><mn id="S7.Thmtheorem15.p1.1.1.m1.1.1.1.1.6" xref="S7.Thmtheorem15.p1.1.1.m1.1.1.1.1.6.cmml">0</mn></mrow><mo id="S7.Thmtheorem15.p1.1.1.m1.1.1.1.3" stretchy="false" xref="S7.Thmtheorem15.p1.1.1.m1.1.1.1.1.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S7.Thmtheorem15.p1.1.1.m1.1b"><apply id="S7.Thmtheorem15.p1.1.1.m1.1.1.1.1.cmml" xref="S7.Thmtheorem15.p1.1.1.m1.1.1.1"><and id="S7.Thmtheorem15.p1.1.1.m1.1.1.1.1a.cmml" xref="S7.Thmtheorem15.p1.1.1.m1.1.1.1"></and><apply id="S7.Thmtheorem15.p1.1.1.m1.1.1.1.1b.cmml" xref="S7.Thmtheorem15.p1.1.1.m1.1.1.1"><ci id="S7.Thmtheorem15.p1.1.1.m1.1.1.1.1.3.cmml" xref="S7.Thmtheorem15.p1.1.1.m1.1.1.1.1.3">:</ci><ci id="S7.Thmtheorem15.p1.1.1.m1.1.1.1.1.2.cmml" xref="S7.Thmtheorem15.p1.1.1.m1.1.1.1.1.2">ℎ</ci><ci id="S7.Thmtheorem15.p1.1.1.m1.1.1.1.1.4.cmml" xref="S7.Thmtheorem15.p1.1.1.m1.1.1.1.1.4">𝑚</ci></apply><apply id="S7.Thmtheorem15.p1.1.1.m1.1.1.1.1c.cmml" xref="S7.Thmtheorem15.p1.1.1.m1.1.1.1"><ci id="S7.Thmtheorem15.p1.1.1.m1.1.1.1.1.5.cmml" xref="S7.Thmtheorem15.p1.1.1.m1.1.1.1.1.5">:</ci><share href="https://arxiv.org/html/2411.12694v2#S7.Thmtheorem15.p1.1.1.m1.1.1.1.1.4.cmml" id="S7.Thmtheorem15.p1.1.1.m1.1.1.1.1d.cmml" xref="S7.Thmtheorem15.p1.1.1.m1.1.1.1"></share><cn id="S7.Thmtheorem15.p1.1.1.m1.1.1.1.1.6.cmml" type="integer" xref="S7.Thmtheorem15.p1.1.1.m1.1.1.1.1.6">0</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.Thmtheorem15.p1.1.1.m1.1c">(h:m:0)</annotation><annotation encoding="application/x-llamapun" id="S7.Thmtheorem15.p1.1.1.m1.1d">( italic_h : italic_m : 0 )</annotation></semantics></math>, there can only be violating in-edges from vertices in <math alttext="L_{h+1}" class="ltx_Math" display="inline" id="S7.Thmtheorem15.p1.2.2.m2.1"><semantics id="S7.Thmtheorem15.p1.2.2.m2.1a"><msub id="S7.Thmtheorem15.p1.2.2.m2.1.1" xref="S7.Thmtheorem15.p1.2.2.m2.1.1.cmml"><mi id="S7.Thmtheorem15.p1.2.2.m2.1.1.2" xref="S7.Thmtheorem15.p1.2.2.m2.1.1.2.cmml">L</mi><mrow id="S7.Thmtheorem15.p1.2.2.m2.1.1.3" xref="S7.Thmtheorem15.p1.2.2.m2.1.1.3.cmml"><mi id="S7.Thmtheorem15.p1.2.2.m2.1.1.3.2" xref="S7.Thmtheorem15.p1.2.2.m2.1.1.3.2.cmml">h</mi><mo id="S7.Thmtheorem15.p1.2.2.m2.1.1.3.1" xref="S7.Thmtheorem15.p1.2.2.m2.1.1.3.1.cmml">+</mo><mn id="S7.Thmtheorem15.p1.2.2.m2.1.1.3.3" xref="S7.Thmtheorem15.p1.2.2.m2.1.1.3.3.cmml">1</mn></mrow></msub><annotation-xml encoding="MathML-Content" id="S7.Thmtheorem15.p1.2.2.m2.1b"><apply id="S7.Thmtheorem15.p1.2.2.m2.1.1.cmml" xref="S7.Thmtheorem15.p1.2.2.m2.1.1"><csymbol cd="ambiguous" id="S7.Thmtheorem15.p1.2.2.m2.1.1.1.cmml" xref="S7.Thmtheorem15.p1.2.2.m2.1.1">subscript</csymbol><ci id="S7.Thmtheorem15.p1.2.2.m2.1.1.2.cmml" xref="S7.Thmtheorem15.p1.2.2.m2.1.1.2">𝐿</ci><apply id="S7.Thmtheorem15.p1.2.2.m2.1.1.3.cmml" xref="S7.Thmtheorem15.p1.2.2.m2.1.1.3"><plus id="S7.Thmtheorem15.p1.2.2.m2.1.1.3.1.cmml" xref="S7.Thmtheorem15.p1.2.2.m2.1.1.3.1"></plus><ci id="S7.Thmtheorem15.p1.2.2.m2.1.1.3.2.cmml" xref="S7.Thmtheorem15.p1.2.2.m2.1.1.3.2">ℎ</ci><cn id="S7.Thmtheorem15.p1.2.2.m2.1.1.3.3.cmml" type="integer" xref="S7.Thmtheorem15.p1.2.2.m2.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.Thmtheorem15.p1.2.2.m2.1c">L_{h+1}</annotation><annotation encoding="application/x-llamapun" id="S7.Thmtheorem15.p1.2.2.m2.1d">italic_L start_POSTSUBSCRIPT italic_h + 1 end_POSTSUBSCRIPT</annotation></semantics></math> to vertices in <math alttext="T_{m}" class="ltx_Math" display="inline" id="S7.Thmtheorem15.p1.3.3.m3.1"><semantics id="S7.Thmtheorem15.p1.3.3.m3.1a"><msub id="S7.Thmtheorem15.p1.3.3.m3.1.1" xref="S7.Thmtheorem15.p1.3.3.m3.1.1.cmml"><mi id="S7.Thmtheorem15.p1.3.3.m3.1.1.2" xref="S7.Thmtheorem15.p1.3.3.m3.1.1.2.cmml">T</mi><mi id="S7.Thmtheorem15.p1.3.3.m3.1.1.3" xref="S7.Thmtheorem15.p1.3.3.m3.1.1.3.cmml">m</mi></msub><annotation-xml encoding="MathML-Content" id="S7.Thmtheorem15.p1.3.3.m3.1b"><apply id="S7.Thmtheorem15.p1.3.3.m3.1.1.cmml" xref="S7.Thmtheorem15.p1.3.3.m3.1.1"><csymbol cd="ambiguous" id="S7.Thmtheorem15.p1.3.3.m3.1.1.1.cmml" xref="S7.Thmtheorem15.p1.3.3.m3.1.1">subscript</csymbol><ci id="S7.Thmtheorem15.p1.3.3.m3.1.1.2.cmml" xref="S7.Thmtheorem15.p1.3.3.m3.1.1.2">𝑇</ci><ci id="S7.Thmtheorem15.p1.3.3.m3.1.1.3.cmml" xref="S7.Thmtheorem15.p1.3.3.m3.1.1.3">𝑚</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.Thmtheorem15.p1.3.3.m3.1c">T_{m}</annotation><annotation encoding="application/x-llamapun" id="S7.Thmtheorem15.p1.3.3.m3.1d">italic_T start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT</annotation></semantics></math>. Consider for the sake of contradiction the first second <math alttext="s>0" class="ltx_Math" display="inline" id="S7.Thmtheorem15.p1.4.4.m4.1"><semantics id="S7.Thmtheorem15.p1.4.4.m4.1a"><mrow id="S7.Thmtheorem15.p1.4.4.m4.1.1" xref="S7.Thmtheorem15.p1.4.4.m4.1.1.cmml"><mi id="S7.Thmtheorem15.p1.4.4.m4.1.1.2" xref="S7.Thmtheorem15.p1.4.4.m4.1.1.2.cmml">s</mi><mo id="S7.Thmtheorem15.p1.4.4.m4.1.1.1" xref="S7.Thmtheorem15.p1.4.4.m4.1.1.1.cmml">></mo><mn id="S7.Thmtheorem15.p1.4.4.m4.1.1.3" xref="S7.Thmtheorem15.p1.4.4.m4.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S7.Thmtheorem15.p1.4.4.m4.1b"><apply id="S7.Thmtheorem15.p1.4.4.m4.1.1.cmml" xref="S7.Thmtheorem15.p1.4.4.m4.1.1"><gt id="S7.Thmtheorem15.p1.4.4.m4.1.1.1.cmml" xref="S7.Thmtheorem15.p1.4.4.m4.1.1.1"></gt><ci id="S7.Thmtheorem15.p1.4.4.m4.1.1.2.cmml" xref="S7.Thmtheorem15.p1.4.4.m4.1.1.2">𝑠</ci><cn id="S7.Thmtheorem15.p1.4.4.m4.1.1.3.cmml" type="integer" xref="S7.Thmtheorem15.p1.4.4.m4.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.Thmtheorem15.p1.4.4.m4.1c">s>0</annotation><annotation encoding="application/x-llamapun" id="S7.Thmtheorem15.p1.4.4.m4.1d">italic_s > 0</annotation></semantics></math>, where after <math alttext="(h:m:s)" class="ltx_Math" display="inline" id="S7.Thmtheorem15.p1.5.5.m5.1"><semantics id="S7.Thmtheorem15.p1.5.5.m5.1a"><mrow id="S7.Thmtheorem15.p1.5.5.m5.1.1.1" xref="S7.Thmtheorem15.p1.5.5.m5.1.1.1.1.cmml"><mo id="S7.Thmtheorem15.p1.5.5.m5.1.1.1.2" stretchy="false" xref="S7.Thmtheorem15.p1.5.5.m5.1.1.1.1.cmml">(</mo><mrow id="S7.Thmtheorem15.p1.5.5.m5.1.1.1.1" xref="S7.Thmtheorem15.p1.5.5.m5.1.1.1.1.cmml"><mi id="S7.Thmtheorem15.p1.5.5.m5.1.1.1.1.2" xref="S7.Thmtheorem15.p1.5.5.m5.1.1.1.1.2.cmml">h</mi><mo id="S7.Thmtheorem15.p1.5.5.m5.1.1.1.1.3" lspace="0.278em" rspace="0.278em" xref="S7.Thmtheorem15.p1.5.5.m5.1.1.1.1.3.cmml">:</mo><mi id="S7.Thmtheorem15.p1.5.5.m5.1.1.1.1.4" xref="S7.Thmtheorem15.p1.5.5.m5.1.1.1.1.4.cmml">m</mi><mo id="S7.Thmtheorem15.p1.5.5.m5.1.1.1.1.5" lspace="0.278em" rspace="0.278em" xref="S7.Thmtheorem15.p1.5.5.m5.1.1.1.1.5.cmml">:</mo><mi id="S7.Thmtheorem15.p1.5.5.m5.1.1.1.1.6" xref="S7.Thmtheorem15.p1.5.5.m5.1.1.1.1.6.cmml">s</mi></mrow><mo id="S7.Thmtheorem15.p1.5.5.m5.1.1.1.3" stretchy="false" xref="S7.Thmtheorem15.p1.5.5.m5.1.1.1.1.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S7.Thmtheorem15.p1.5.5.m5.1b"><apply id="S7.Thmtheorem15.p1.5.5.m5.1.1.1.1.cmml" xref="S7.Thmtheorem15.p1.5.5.m5.1.1.1"><and id="S7.Thmtheorem15.p1.5.5.m5.1.1.1.1a.cmml" xref="S7.Thmtheorem15.p1.5.5.m5.1.1.1"></and><apply id="S7.Thmtheorem15.p1.5.5.m5.1.1.1.1b.cmml" xref="S7.Thmtheorem15.p1.5.5.m5.1.1.1"><ci id="S7.Thmtheorem15.p1.5.5.m5.1.1.1.1.3.cmml" xref="S7.Thmtheorem15.p1.5.5.m5.1.1.1.1.3">:</ci><ci id="S7.Thmtheorem15.p1.5.5.m5.1.1.1.1.2.cmml" xref="S7.Thmtheorem15.p1.5.5.m5.1.1.1.1.2">ℎ</ci><ci id="S7.Thmtheorem15.p1.5.5.m5.1.1.1.1.4.cmml" xref="S7.Thmtheorem15.p1.5.5.m5.1.1.1.1.4">𝑚</ci></apply><apply id="S7.Thmtheorem15.p1.5.5.m5.1.1.1.1c.cmml" xref="S7.Thmtheorem15.p1.5.5.m5.1.1.1"><ci id="S7.Thmtheorem15.p1.5.5.m5.1.1.1.1.5.cmml" xref="S7.Thmtheorem15.p1.5.5.m5.1.1.1.1.5">:</ci><share href="https://arxiv.org/html/2411.12694v2#S7.Thmtheorem15.p1.5.5.m5.1.1.1.1.4.cmml" id="S7.Thmtheorem15.p1.5.5.m5.1.1.1.1d.cmml" xref="S7.Thmtheorem15.p1.5.5.m5.1.1.1"></share><ci id="S7.Thmtheorem15.p1.5.5.m5.1.1.1.1.6.cmml" xref="S7.Thmtheorem15.p1.5.5.m5.1.1.1.1.6">𝑠</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.Thmtheorem15.p1.5.5.m5.1c">(h:m:s)</annotation><annotation encoding="application/x-llamapun" id="S7.Thmtheorem15.p1.5.5.m5.1d">( italic_h : italic_m : italic_s )</annotation></semantics></math> there exists an outgoing edge <math alttext="(u,v)" class="ltx_Math" display="inline" id="S7.Thmtheorem15.p1.6.6.m6.2"><semantics id="S7.Thmtheorem15.p1.6.6.m6.2a"><mrow id="S7.Thmtheorem15.p1.6.6.m6.2.3.2" xref="S7.Thmtheorem15.p1.6.6.m6.2.3.1.cmml"><mo id="S7.Thmtheorem15.p1.6.6.m6.2.3.2.1" stretchy="false" xref="S7.Thmtheorem15.p1.6.6.m6.2.3.1.cmml">(</mo><mi id="S7.Thmtheorem15.p1.6.6.m6.1.1" xref="S7.Thmtheorem15.p1.6.6.m6.1.1.cmml">u</mi><mo id="S7.Thmtheorem15.p1.6.6.m6.2.3.2.2" xref="S7.Thmtheorem15.p1.6.6.m6.2.3.1.cmml">,</mo><mi id="S7.Thmtheorem15.p1.6.6.m6.2.2" xref="S7.Thmtheorem15.p1.6.6.m6.2.2.cmml">v</mi><mo id="S7.Thmtheorem15.p1.6.6.m6.2.3.2.3" stretchy="false" xref="S7.Thmtheorem15.p1.6.6.m6.2.3.1.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S7.Thmtheorem15.p1.6.6.m6.2b"><interval closure="open" id="S7.Thmtheorem15.p1.6.6.m6.2.3.1.cmml" xref="S7.Thmtheorem15.p1.6.6.m6.2.3.2"><ci id="S7.Thmtheorem15.p1.6.6.m6.1.1.cmml" xref="S7.Thmtheorem15.p1.6.6.m6.1.1">𝑢</ci><ci id="S7.Thmtheorem15.p1.6.6.m6.2.2.cmml" xref="S7.Thmtheorem15.p1.6.6.m6.2.2">𝑣</ci></interval></annotation-xml><annotation encoding="application/x-tex" id="S7.Thmtheorem15.p1.6.6.m6.2c">(u,v)</annotation><annotation encoding="application/x-llamapun" id="S7.Thmtheorem15.p1.6.6.m6.2d">( italic_u , italic_v )</annotation></semantics></math> with <math alttext="u\in L_{k}" class="ltx_Math" display="inline" id="S7.Thmtheorem15.p1.7.7.m7.1"><semantics id="S7.Thmtheorem15.p1.7.7.m7.1a"><mrow id="S7.Thmtheorem15.p1.7.7.m7.1.1" xref="S7.Thmtheorem15.p1.7.7.m7.1.1.cmml"><mi id="S7.Thmtheorem15.p1.7.7.m7.1.1.2" xref="S7.Thmtheorem15.p1.7.7.m7.1.1.2.cmml">u</mi><mo id="S7.Thmtheorem15.p1.7.7.m7.1.1.1" xref="S7.Thmtheorem15.p1.7.7.m7.1.1.1.cmml">∈</mo><msub id="S7.Thmtheorem15.p1.7.7.m7.1.1.3" xref="S7.Thmtheorem15.p1.7.7.m7.1.1.3.cmml"><mi id="S7.Thmtheorem15.p1.7.7.m7.1.1.3.2" xref="S7.Thmtheorem15.p1.7.7.m7.1.1.3.2.cmml">L</mi><mi id="S7.Thmtheorem15.p1.7.7.m7.1.1.3.3" xref="S7.Thmtheorem15.p1.7.7.m7.1.1.3.3.cmml">k</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S7.Thmtheorem15.p1.7.7.m7.1b"><apply id="S7.Thmtheorem15.p1.7.7.m7.1.1.cmml" xref="S7.Thmtheorem15.p1.7.7.m7.1.1"><in id="S7.Thmtheorem15.p1.7.7.m7.1.1.1.cmml" xref="S7.Thmtheorem15.p1.7.7.m7.1.1.1"></in><ci id="S7.Thmtheorem15.p1.7.7.m7.1.1.2.cmml" xref="S7.Thmtheorem15.p1.7.7.m7.1.1.2">𝑢</ci><apply id="S7.Thmtheorem15.p1.7.7.m7.1.1.3.cmml" xref="S7.Thmtheorem15.p1.7.7.m7.1.1.3"><csymbol cd="ambiguous" id="S7.Thmtheorem15.p1.7.7.m7.1.1.3.1.cmml" xref="S7.Thmtheorem15.p1.7.7.m7.1.1.3">subscript</csymbol><ci id="S7.Thmtheorem15.p1.7.7.m7.1.1.3.2.cmml" xref="S7.Thmtheorem15.p1.7.7.m7.1.1.3.2">𝐿</ci><ci id="S7.Thmtheorem15.p1.7.7.m7.1.1.3.3.cmml" xref="S7.Thmtheorem15.p1.7.7.m7.1.1.3.3">𝑘</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.Thmtheorem15.p1.7.7.m7.1c">u\in L_{k}</annotation><annotation encoding="application/x-llamapun" id="S7.Thmtheorem15.p1.7.7.m7.1d">italic_u ∈ italic_L start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="k>h+2" class="ltx_Math" display="inline" id="S7.Thmtheorem15.p1.8.8.m8.1"><semantics id="S7.Thmtheorem15.p1.8.8.m8.1a"><mrow id="S7.Thmtheorem15.p1.8.8.m8.1.1" xref="S7.Thmtheorem15.p1.8.8.m8.1.1.cmml"><mi id="S7.Thmtheorem15.p1.8.8.m8.1.1.2" xref="S7.Thmtheorem15.p1.8.8.m8.1.1.2.cmml">k</mi><mo id="S7.Thmtheorem15.p1.8.8.m8.1.1.1" xref="S7.Thmtheorem15.p1.8.8.m8.1.1.1.cmml">></mo><mrow id="S7.Thmtheorem15.p1.8.8.m8.1.1.3" xref="S7.Thmtheorem15.p1.8.8.m8.1.1.3.cmml"><mi id="S7.Thmtheorem15.p1.8.8.m8.1.1.3.2" xref="S7.Thmtheorem15.p1.8.8.m8.1.1.3.2.cmml">h</mi><mo id="S7.Thmtheorem15.p1.8.8.m8.1.1.3.1" xref="S7.Thmtheorem15.p1.8.8.m8.1.1.3.1.cmml">+</mo><mn id="S7.Thmtheorem15.p1.8.8.m8.1.1.3.3" xref="S7.Thmtheorem15.p1.8.8.m8.1.1.3.3.cmml">2</mn></mrow></mrow><annotation-xml encoding="MathML-Content" id="S7.Thmtheorem15.p1.8.8.m8.1b"><apply id="S7.Thmtheorem15.p1.8.8.m8.1.1.cmml" xref="S7.Thmtheorem15.p1.8.8.m8.1.1"><gt id="S7.Thmtheorem15.p1.8.8.m8.1.1.1.cmml" xref="S7.Thmtheorem15.p1.8.8.m8.1.1.1"></gt><ci id="S7.Thmtheorem15.p1.8.8.m8.1.1.2.cmml" xref="S7.Thmtheorem15.p1.8.8.m8.1.1.2">𝑘</ci><apply id="S7.Thmtheorem15.p1.8.8.m8.1.1.3.cmml" xref="S7.Thmtheorem15.p1.8.8.m8.1.1.3"><plus id="S7.Thmtheorem15.p1.8.8.m8.1.1.3.1.cmml" xref="S7.Thmtheorem15.p1.8.8.m8.1.1.3.1"></plus><ci id="S7.Thmtheorem15.p1.8.8.m8.1.1.3.2.cmml" xref="S7.Thmtheorem15.p1.8.8.m8.1.1.3.2">ℎ</ci><cn id="S7.Thmtheorem15.p1.8.8.m8.1.1.3.3.cmml" type="integer" xref="S7.Thmtheorem15.p1.8.8.m8.1.1.3.3">2</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.Thmtheorem15.p1.8.8.m8.1c">k>h+2</annotation><annotation encoding="application/x-llamapun" id="S7.Thmtheorem15.p1.8.8.m8.1d">italic_k > italic_h + 2</annotation></semantics></math>. Each vertex <math alttext="v" class="ltx_Math" display="inline" id="S7.Thmtheorem15.p1.9.9.m9.1"><semantics id="S7.Thmtheorem15.p1.9.9.m9.1a"><mi id="S7.Thmtheorem15.p1.9.9.m9.1.1" xref="S7.Thmtheorem15.p1.9.9.m9.1.1.cmml">v</mi><annotation-xml encoding="MathML-Content" id="S7.Thmtheorem15.p1.9.9.m9.1b"><ci id="S7.Thmtheorem15.p1.9.9.m9.1.1.cmml" xref="S7.Thmtheorem15.p1.9.9.m9.1.1">𝑣</ci></annotation-xml><annotation encoding="application/x-tex" id="S7.Thmtheorem15.p1.9.9.m9.1c">v</annotation><annotation encoding="application/x-llamapun" id="S7.Thmtheorem15.p1.9.9.m9.1d">italic_v</annotation></semantics></math> decreases its out-degree by at most one level, and thus <math alttext="l_{s}(u)=l_{s}(v)+1" class="ltx_Math" display="inline" id="S7.Thmtheorem15.p1.10.10.m10.2"><semantics id="S7.Thmtheorem15.p1.10.10.m10.2a"><mrow id="S7.Thmtheorem15.p1.10.10.m10.2.3" xref="S7.Thmtheorem15.p1.10.10.m10.2.3.cmml"><mrow id="S7.Thmtheorem15.p1.10.10.m10.2.3.2" xref="S7.Thmtheorem15.p1.10.10.m10.2.3.2.cmml"><msub id="S7.Thmtheorem15.p1.10.10.m10.2.3.2.2" xref="S7.Thmtheorem15.p1.10.10.m10.2.3.2.2.cmml"><mi id="S7.Thmtheorem15.p1.10.10.m10.2.3.2.2.2" xref="S7.Thmtheorem15.p1.10.10.m10.2.3.2.2.2.cmml">l</mi><mi id="S7.Thmtheorem15.p1.10.10.m10.2.3.2.2.3" xref="S7.Thmtheorem15.p1.10.10.m10.2.3.2.2.3.cmml">s</mi></msub><mo id="S7.Thmtheorem15.p1.10.10.m10.2.3.2.1" xref="S7.Thmtheorem15.p1.10.10.m10.2.3.2.1.cmml">⁢</mo><mrow id="S7.Thmtheorem15.p1.10.10.m10.2.3.2.3.2" xref="S7.Thmtheorem15.p1.10.10.m10.2.3.2.cmml"><mo id="S7.Thmtheorem15.p1.10.10.m10.2.3.2.3.2.1" stretchy="false" xref="S7.Thmtheorem15.p1.10.10.m10.2.3.2.cmml">(</mo><mi id="S7.Thmtheorem15.p1.10.10.m10.1.1" xref="S7.Thmtheorem15.p1.10.10.m10.1.1.cmml">u</mi><mo id="S7.Thmtheorem15.p1.10.10.m10.2.3.2.3.2.2" stretchy="false" xref="S7.Thmtheorem15.p1.10.10.m10.2.3.2.cmml">)</mo></mrow></mrow><mo id="S7.Thmtheorem15.p1.10.10.m10.2.3.1" xref="S7.Thmtheorem15.p1.10.10.m10.2.3.1.cmml">=</mo><mrow id="S7.Thmtheorem15.p1.10.10.m10.2.3.3" xref="S7.Thmtheorem15.p1.10.10.m10.2.3.3.cmml"><mrow id="S7.Thmtheorem15.p1.10.10.m10.2.3.3.2" xref="S7.Thmtheorem15.p1.10.10.m10.2.3.3.2.cmml"><msub id="S7.Thmtheorem15.p1.10.10.m10.2.3.3.2.2" xref="S7.Thmtheorem15.p1.10.10.m10.2.3.3.2.2.cmml"><mi id="S7.Thmtheorem15.p1.10.10.m10.2.3.3.2.2.2" xref="S7.Thmtheorem15.p1.10.10.m10.2.3.3.2.2.2.cmml">l</mi><mi id="S7.Thmtheorem15.p1.10.10.m10.2.3.3.2.2.3" xref="S7.Thmtheorem15.p1.10.10.m10.2.3.3.2.2.3.cmml">s</mi></msub><mo id="S7.Thmtheorem15.p1.10.10.m10.2.3.3.2.1" xref="S7.Thmtheorem15.p1.10.10.m10.2.3.3.2.1.cmml">⁢</mo><mrow id="S7.Thmtheorem15.p1.10.10.m10.2.3.3.2.3.2" xref="S7.Thmtheorem15.p1.10.10.m10.2.3.3.2.cmml"><mo id="S7.Thmtheorem15.p1.10.10.m10.2.3.3.2.3.2.1" stretchy="false" xref="S7.Thmtheorem15.p1.10.10.m10.2.3.3.2.cmml">(</mo><mi id="S7.Thmtheorem15.p1.10.10.m10.2.2" xref="S7.Thmtheorem15.p1.10.10.m10.2.2.cmml">v</mi><mo id="S7.Thmtheorem15.p1.10.10.m10.2.3.3.2.3.2.2" stretchy="false" xref="S7.Thmtheorem15.p1.10.10.m10.2.3.3.2.cmml">)</mo></mrow></mrow><mo id="S7.Thmtheorem15.p1.10.10.m10.2.3.3.1" xref="S7.Thmtheorem15.p1.10.10.m10.2.3.3.1.cmml">+</mo><mn id="S7.Thmtheorem15.p1.10.10.m10.2.3.3.3" xref="S7.Thmtheorem15.p1.10.10.m10.2.3.3.3.cmml">1</mn></mrow></mrow><annotation-xml encoding="MathML-Content" id="S7.Thmtheorem15.p1.10.10.m10.2b"><apply id="S7.Thmtheorem15.p1.10.10.m10.2.3.cmml" xref="S7.Thmtheorem15.p1.10.10.m10.2.3"><eq id="S7.Thmtheorem15.p1.10.10.m10.2.3.1.cmml" xref="S7.Thmtheorem15.p1.10.10.m10.2.3.1"></eq><apply id="S7.Thmtheorem15.p1.10.10.m10.2.3.2.cmml" xref="S7.Thmtheorem15.p1.10.10.m10.2.3.2"><times id="S7.Thmtheorem15.p1.10.10.m10.2.3.2.1.cmml" xref="S7.Thmtheorem15.p1.10.10.m10.2.3.2.1"></times><apply id="S7.Thmtheorem15.p1.10.10.m10.2.3.2.2.cmml" xref="S7.Thmtheorem15.p1.10.10.m10.2.3.2.2"><csymbol cd="ambiguous" id="S7.Thmtheorem15.p1.10.10.m10.2.3.2.2.1.cmml" xref="S7.Thmtheorem15.p1.10.10.m10.2.3.2.2">subscript</csymbol><ci id="S7.Thmtheorem15.p1.10.10.m10.2.3.2.2.2.cmml" xref="S7.Thmtheorem15.p1.10.10.m10.2.3.2.2.2">𝑙</ci><ci id="S7.Thmtheorem15.p1.10.10.m10.2.3.2.2.3.cmml" xref="S7.Thmtheorem15.p1.10.10.m10.2.3.2.2.3">𝑠</ci></apply><ci id="S7.Thmtheorem15.p1.10.10.m10.1.1.cmml" xref="S7.Thmtheorem15.p1.10.10.m10.1.1">𝑢</ci></apply><apply id="S7.Thmtheorem15.p1.10.10.m10.2.3.3.cmml" xref="S7.Thmtheorem15.p1.10.10.m10.2.3.3"><plus id="S7.Thmtheorem15.p1.10.10.m10.2.3.3.1.cmml" xref="S7.Thmtheorem15.p1.10.10.m10.2.3.3.1"></plus><apply id="S7.Thmtheorem15.p1.10.10.m10.2.3.3.2.cmml" xref="S7.Thmtheorem15.p1.10.10.m10.2.3.3.2"><times id="S7.Thmtheorem15.p1.10.10.m10.2.3.3.2.1.cmml" xref="S7.Thmtheorem15.p1.10.10.m10.2.3.3.2.1"></times><apply id="S7.Thmtheorem15.p1.10.10.m10.2.3.3.2.2.cmml" xref="S7.Thmtheorem15.p1.10.10.m10.2.3.3.2.2"><csymbol cd="ambiguous" id="S7.Thmtheorem15.p1.10.10.m10.2.3.3.2.2.1.cmml" xref="S7.Thmtheorem15.p1.10.10.m10.2.3.3.2.2">subscript</csymbol><ci id="S7.Thmtheorem15.p1.10.10.m10.2.3.3.2.2.2.cmml" xref="S7.Thmtheorem15.p1.10.10.m10.2.3.3.2.2.2">𝑙</ci><ci id="S7.Thmtheorem15.p1.10.10.m10.2.3.3.2.2.3.cmml" xref="S7.Thmtheorem15.p1.10.10.m10.2.3.3.2.2.3">𝑠</ci></apply><ci id="S7.Thmtheorem15.p1.10.10.m10.2.2.cmml" xref="S7.Thmtheorem15.p1.10.10.m10.2.2">𝑣</ci></apply><cn id="S7.Thmtheorem15.p1.10.10.m10.2.3.3.3.cmml" type="integer" xref="S7.Thmtheorem15.p1.10.10.m10.2.3.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.Thmtheorem15.p1.10.10.m10.2c">l_{s}(u)=l_{s}(v)+1</annotation><annotation encoding="application/x-llamapun" id="S7.Thmtheorem15.p1.10.10.m10.2d">italic_l start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_u ) = italic_l start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_v ) + 1</annotation></semantics></math>, <math alttext="l_{s}(v)=l_{m}(v)" class="ltx_Math" display="inline" id="S7.Thmtheorem15.p1.11.11.m11.2"><semantics id="S7.Thmtheorem15.p1.11.11.m11.2a"><mrow id="S7.Thmtheorem15.p1.11.11.m11.2.3" xref="S7.Thmtheorem15.p1.11.11.m11.2.3.cmml"><mrow id="S7.Thmtheorem15.p1.11.11.m11.2.3.2" xref="S7.Thmtheorem15.p1.11.11.m11.2.3.2.cmml"><msub id="S7.Thmtheorem15.p1.11.11.m11.2.3.2.2" xref="S7.Thmtheorem15.p1.11.11.m11.2.3.2.2.cmml"><mi id="S7.Thmtheorem15.p1.11.11.m11.2.3.2.2.2" xref="S7.Thmtheorem15.p1.11.11.m11.2.3.2.2.2.cmml">l</mi><mi id="S7.Thmtheorem15.p1.11.11.m11.2.3.2.2.3" xref="S7.Thmtheorem15.p1.11.11.m11.2.3.2.2.3.cmml">s</mi></msub><mo id="S7.Thmtheorem15.p1.11.11.m11.2.3.2.1" xref="S7.Thmtheorem15.p1.11.11.m11.2.3.2.1.cmml">⁢</mo><mrow id="S7.Thmtheorem15.p1.11.11.m11.2.3.2.3.2" xref="S7.Thmtheorem15.p1.11.11.m11.2.3.2.cmml"><mo id="S7.Thmtheorem15.p1.11.11.m11.2.3.2.3.2.1" stretchy="false" xref="S7.Thmtheorem15.p1.11.11.m11.2.3.2.cmml">(</mo><mi id="S7.Thmtheorem15.p1.11.11.m11.1.1" xref="S7.Thmtheorem15.p1.11.11.m11.1.1.cmml">v</mi><mo id="S7.Thmtheorem15.p1.11.11.m11.2.3.2.3.2.2" stretchy="false" xref="S7.Thmtheorem15.p1.11.11.m11.2.3.2.cmml">)</mo></mrow></mrow><mo id="S7.Thmtheorem15.p1.11.11.m11.2.3.1" xref="S7.Thmtheorem15.p1.11.11.m11.2.3.1.cmml">=</mo><mrow id="S7.Thmtheorem15.p1.11.11.m11.2.3.3" xref="S7.Thmtheorem15.p1.11.11.m11.2.3.3.cmml"><msub id="S7.Thmtheorem15.p1.11.11.m11.2.3.3.2" xref="S7.Thmtheorem15.p1.11.11.m11.2.3.3.2.cmml"><mi id="S7.Thmtheorem15.p1.11.11.m11.2.3.3.2.2" xref="S7.Thmtheorem15.p1.11.11.m11.2.3.3.2.2.cmml">l</mi><mi id="S7.Thmtheorem15.p1.11.11.m11.2.3.3.2.3" xref="S7.Thmtheorem15.p1.11.11.m11.2.3.3.2.3.cmml">m</mi></msub><mo id="S7.Thmtheorem15.p1.11.11.m11.2.3.3.1" xref="S7.Thmtheorem15.p1.11.11.m11.2.3.3.1.cmml">⁢</mo><mrow id="S7.Thmtheorem15.p1.11.11.m11.2.3.3.3.2" xref="S7.Thmtheorem15.p1.11.11.m11.2.3.3.cmml"><mo id="S7.Thmtheorem15.p1.11.11.m11.2.3.3.3.2.1" stretchy="false" xref="S7.Thmtheorem15.p1.11.11.m11.2.3.3.cmml">(</mo><mi id="S7.Thmtheorem15.p1.11.11.m11.2.2" xref="S7.Thmtheorem15.p1.11.11.m11.2.2.cmml">v</mi><mo id="S7.Thmtheorem15.p1.11.11.m11.2.3.3.3.2.2" stretchy="false" xref="S7.Thmtheorem15.p1.11.11.m11.2.3.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S7.Thmtheorem15.p1.11.11.m11.2b"><apply id="S7.Thmtheorem15.p1.11.11.m11.2.3.cmml" xref="S7.Thmtheorem15.p1.11.11.m11.2.3"><eq id="S7.Thmtheorem15.p1.11.11.m11.2.3.1.cmml" xref="S7.Thmtheorem15.p1.11.11.m11.2.3.1"></eq><apply id="S7.Thmtheorem15.p1.11.11.m11.2.3.2.cmml" xref="S7.Thmtheorem15.p1.11.11.m11.2.3.2"><times id="S7.Thmtheorem15.p1.11.11.m11.2.3.2.1.cmml" xref="S7.Thmtheorem15.p1.11.11.m11.2.3.2.1"></times><apply id="S7.Thmtheorem15.p1.11.11.m11.2.3.2.2.cmml" xref="S7.Thmtheorem15.p1.11.11.m11.2.3.2.2"><csymbol cd="ambiguous" id="S7.Thmtheorem15.p1.11.11.m11.2.3.2.2.1.cmml" xref="S7.Thmtheorem15.p1.11.11.m11.2.3.2.2">subscript</csymbol><ci id="S7.Thmtheorem15.p1.11.11.m11.2.3.2.2.2.cmml" xref="S7.Thmtheorem15.p1.11.11.m11.2.3.2.2.2">𝑙</ci><ci id="S7.Thmtheorem15.p1.11.11.m11.2.3.2.2.3.cmml" xref="S7.Thmtheorem15.p1.11.11.m11.2.3.2.2.3">𝑠</ci></apply><ci id="S7.Thmtheorem15.p1.11.11.m11.1.1.cmml" xref="S7.Thmtheorem15.p1.11.11.m11.1.1">𝑣</ci></apply><apply id="S7.Thmtheorem15.p1.11.11.m11.2.3.3.cmml" xref="S7.Thmtheorem15.p1.11.11.m11.2.3.3"><times id="S7.Thmtheorem15.p1.11.11.m11.2.3.3.1.cmml" xref="S7.Thmtheorem15.p1.11.11.m11.2.3.3.1"></times><apply id="S7.Thmtheorem15.p1.11.11.m11.2.3.3.2.cmml" xref="S7.Thmtheorem15.p1.11.11.m11.2.3.3.2"><csymbol cd="ambiguous" id="S7.Thmtheorem15.p1.11.11.m11.2.3.3.2.1.cmml" xref="S7.Thmtheorem15.p1.11.11.m11.2.3.3.2">subscript</csymbol><ci id="S7.Thmtheorem15.p1.11.11.m11.2.3.3.2.2.cmml" xref="S7.Thmtheorem15.p1.11.11.m11.2.3.3.2.2">𝑙</ci><ci id="S7.Thmtheorem15.p1.11.11.m11.2.3.3.2.3.cmml" xref="S7.Thmtheorem15.p1.11.11.m11.2.3.3.2.3">𝑚</ci></apply><ci id="S7.Thmtheorem15.p1.11.11.m11.2.2.cmml" xref="S7.Thmtheorem15.p1.11.11.m11.2.2">𝑣</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.Thmtheorem15.p1.11.11.m11.2c">l_{s}(v)=l_{m}(v)</annotation><annotation encoding="application/x-llamapun" id="S7.Thmtheorem15.p1.11.11.m11.2d">italic_l start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_v ) = italic_l start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT ( italic_v )</annotation></semantics></math> and <math alttext="v" class="ltx_Math" display="inline" id="S7.Thmtheorem15.p1.12.12.m12.1"><semantics id="S7.Thmtheorem15.p1.12.12.m12.1a"><mi id="S7.Thmtheorem15.p1.12.12.m12.1.1" xref="S7.Thmtheorem15.p1.12.12.m12.1.1.cmml">v</mi><annotation-xml encoding="MathML-Content" id="S7.Thmtheorem15.p1.12.12.m12.1b"><ci id="S7.Thmtheorem15.p1.12.12.m12.1.1.cmml" xref="S7.Thmtheorem15.p1.12.12.m12.1.1">𝑣</ci></annotation-xml><annotation encoding="application/x-tex" id="S7.Thmtheorem15.p1.12.12.m12.1c">v</annotation><annotation encoding="application/x-llamapun" id="S7.Thmtheorem15.p1.12.12.m12.1d">italic_v</annotation></semantics></math> decreased its out-degree by one during the <span class="ltx_text ltx_font_smallcaps" id="S7.Thmtheorem15.p1.21.21.1">second</span> <math alttext="s" class="ltx_Math" display="inline" id="S7.Thmtheorem15.p1.13.13.m13.1"><semantics id="S7.Thmtheorem15.p1.13.13.m13.1a"><mi id="S7.Thmtheorem15.p1.13.13.m13.1.1" xref="S7.Thmtheorem15.p1.13.13.m13.1.1.cmml">s</mi><annotation-xml encoding="MathML-Content" id="S7.Thmtheorem15.p1.13.13.m13.1b"><ci id="S7.Thmtheorem15.p1.13.13.m13.1.1.cmml" xref="S7.Thmtheorem15.p1.13.13.m13.1.1">𝑠</ci></annotation-xml><annotation encoding="application/x-tex" id="S7.Thmtheorem15.p1.13.13.m13.1c">s</annotation><annotation encoding="application/x-llamapun" id="S7.Thmtheorem15.p1.13.13.m13.1d">italic_s</annotation></semantics></math>. Since the edge <math alttext="\overline{uv}" class="ltx_Math" display="inline" id="S7.Thmtheorem15.p1.14.14.m14.1"><semantics id="S7.Thmtheorem15.p1.14.14.m14.1a"><mover accent="true" id="S7.Thmtheorem15.p1.14.14.m14.1.1" xref="S7.Thmtheorem15.p1.14.14.m14.1.1.cmml"><mrow id="S7.Thmtheorem15.p1.14.14.m14.1.1.2" xref="S7.Thmtheorem15.p1.14.14.m14.1.1.2.cmml"><mi id="S7.Thmtheorem15.p1.14.14.m14.1.1.2.2" xref="S7.Thmtheorem15.p1.14.14.m14.1.1.2.2.cmml">u</mi><mo id="S7.Thmtheorem15.p1.14.14.m14.1.1.2.1" xref="S7.Thmtheorem15.p1.14.14.m14.1.1.2.1.cmml">⁢</mo><mi id="S7.Thmtheorem15.p1.14.14.m14.1.1.2.3" xref="S7.Thmtheorem15.p1.14.14.m14.1.1.2.3.cmml">v</mi></mrow><mo id="S7.Thmtheorem15.p1.14.14.m14.1.1.1" xref="S7.Thmtheorem15.p1.14.14.m14.1.1.1.cmml">¯</mo></mover><annotation-xml encoding="MathML-Content" id="S7.Thmtheorem15.p1.14.14.m14.1b"><apply id="S7.Thmtheorem15.p1.14.14.m14.1.1.cmml" xref="S7.Thmtheorem15.p1.14.14.m14.1.1"><ci id="S7.Thmtheorem15.p1.14.14.m14.1.1.1.cmml" xref="S7.Thmtheorem15.p1.14.14.m14.1.1.1">¯</ci><apply id="S7.Thmtheorem15.p1.14.14.m14.1.1.2.cmml" xref="S7.Thmtheorem15.p1.14.14.m14.1.1.2"><times id="S7.Thmtheorem15.p1.14.14.m14.1.1.2.1.cmml" xref="S7.Thmtheorem15.p1.14.14.m14.1.1.2.1"></times><ci id="S7.Thmtheorem15.p1.14.14.m14.1.1.2.2.cmml" xref="S7.Thmtheorem15.p1.14.14.m14.1.1.2.2">𝑢</ci><ci id="S7.Thmtheorem15.p1.14.14.m14.1.1.2.3.cmml" xref="S7.Thmtheorem15.p1.14.14.m14.1.1.2.3">𝑣</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.Thmtheorem15.p1.14.14.m14.1c">\overline{uv}</annotation><annotation encoding="application/x-llamapun" id="S7.Thmtheorem15.p1.14.14.m14.1d">over¯ start_ARG italic_u italic_v end_ARG</annotation></semantics></math> is violating at the end of <math alttext="(h:m:s)" class="ltx_Math" display="inline" id="S7.Thmtheorem15.p1.15.15.m15.1"><semantics id="S7.Thmtheorem15.p1.15.15.m15.1a"><mrow id="S7.Thmtheorem15.p1.15.15.m15.1.1.1" xref="S7.Thmtheorem15.p1.15.15.m15.1.1.1.1.cmml"><mo id="S7.Thmtheorem15.p1.15.15.m15.1.1.1.2" stretchy="false" xref="S7.Thmtheorem15.p1.15.15.m15.1.1.1.1.cmml">(</mo><mrow id="S7.Thmtheorem15.p1.15.15.m15.1.1.1.1" xref="S7.Thmtheorem15.p1.15.15.m15.1.1.1.1.cmml"><mi id="S7.Thmtheorem15.p1.15.15.m15.1.1.1.1.2" xref="S7.Thmtheorem15.p1.15.15.m15.1.1.1.1.2.cmml">h</mi><mo id="S7.Thmtheorem15.p1.15.15.m15.1.1.1.1.3" lspace="0.278em" rspace="0.278em" xref="S7.Thmtheorem15.p1.15.15.m15.1.1.1.1.3.cmml">:</mo><mi id="S7.Thmtheorem15.p1.15.15.m15.1.1.1.1.4" xref="S7.Thmtheorem15.p1.15.15.m15.1.1.1.1.4.cmml">m</mi><mo id="S7.Thmtheorem15.p1.15.15.m15.1.1.1.1.5" lspace="0.278em" rspace="0.278em" xref="S7.Thmtheorem15.p1.15.15.m15.1.1.1.1.5.cmml">:</mo><mi id="S7.Thmtheorem15.p1.15.15.m15.1.1.1.1.6" xref="S7.Thmtheorem15.p1.15.15.m15.1.1.1.1.6.cmml">s</mi></mrow><mo id="S7.Thmtheorem15.p1.15.15.m15.1.1.1.3" stretchy="false" xref="S7.Thmtheorem15.p1.15.15.m15.1.1.1.1.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S7.Thmtheorem15.p1.15.15.m15.1b"><apply id="S7.Thmtheorem15.p1.15.15.m15.1.1.1.1.cmml" xref="S7.Thmtheorem15.p1.15.15.m15.1.1.1"><and id="S7.Thmtheorem15.p1.15.15.m15.1.1.1.1a.cmml" xref="S7.Thmtheorem15.p1.15.15.m15.1.1.1"></and><apply id="S7.Thmtheorem15.p1.15.15.m15.1.1.1.1b.cmml" xref="S7.Thmtheorem15.p1.15.15.m15.1.1.1"><ci id="S7.Thmtheorem15.p1.15.15.m15.1.1.1.1.3.cmml" xref="S7.Thmtheorem15.p1.15.15.m15.1.1.1.1.3">:</ci><ci id="S7.Thmtheorem15.p1.15.15.m15.1.1.1.1.2.cmml" xref="S7.Thmtheorem15.p1.15.15.m15.1.1.1.1.2">ℎ</ci><ci id="S7.Thmtheorem15.p1.15.15.m15.1.1.1.1.4.cmml" xref="S7.Thmtheorem15.p1.15.15.m15.1.1.1.1.4">𝑚</ci></apply><apply id="S7.Thmtheorem15.p1.15.15.m15.1.1.1.1c.cmml" xref="S7.Thmtheorem15.p1.15.15.m15.1.1.1"><ci id="S7.Thmtheorem15.p1.15.15.m15.1.1.1.1.5.cmml" xref="S7.Thmtheorem15.p1.15.15.m15.1.1.1.1.5">:</ci><share href="https://arxiv.org/html/2411.12694v2#S7.Thmtheorem15.p1.15.15.m15.1.1.1.1.4.cmml" id="S7.Thmtheorem15.p1.15.15.m15.1.1.1.1d.cmml" xref="S7.Thmtheorem15.p1.15.15.m15.1.1.1"></share><ci id="S7.Thmtheorem15.p1.15.15.m15.1.1.1.1.6.cmml" xref="S7.Thmtheorem15.p1.15.15.m15.1.1.1.1.6">𝑠</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.Thmtheorem15.p1.15.15.m15.1c">(h:m:s)</annotation><annotation encoding="application/x-llamapun" id="S7.Thmtheorem15.p1.15.15.m15.1d">( italic_h : italic_m : italic_s )</annotation></semantics></math>, it must be that <math alttext="\textsl{g}_{s}(u\!\to\!v)>0" class="ltx_Math" display="inline" id="S7.Thmtheorem15.p1.16.16.m16.1"><semantics id="S7.Thmtheorem15.p1.16.16.m16.1a"><mrow id="S7.Thmtheorem15.p1.16.16.m16.1.1" xref="S7.Thmtheorem15.p1.16.16.m16.1.1.cmml"><mrow id="S7.Thmtheorem15.p1.16.16.m16.1.1.1" xref="S7.Thmtheorem15.p1.16.16.m16.1.1.1.cmml"><msub id="S7.Thmtheorem15.p1.16.16.m16.1.1.1.3" xref="S7.Thmtheorem15.p1.16.16.m16.1.1.1.3.cmml"><mtext class="ltx_mathvariant_italic" id="S7.Thmtheorem15.p1.16.16.m16.1.1.1.3.2" xref="S7.Thmtheorem15.p1.16.16.m16.1.1.1.3.2a.cmml">g</mtext><mi id="S7.Thmtheorem15.p1.16.16.m16.1.1.1.3.3" xref="S7.Thmtheorem15.p1.16.16.m16.1.1.1.3.3.cmml">s</mi></msub><mo id="S7.Thmtheorem15.p1.16.16.m16.1.1.1.2" xref="S7.Thmtheorem15.p1.16.16.m16.1.1.1.2.cmml">⁢</mo><mrow id="S7.Thmtheorem15.p1.16.16.m16.1.1.1.1.1" xref="S7.Thmtheorem15.p1.16.16.m16.1.1.1.1.1.1.cmml"><mo id="S7.Thmtheorem15.p1.16.16.m16.1.1.1.1.1.2" stretchy="false" xref="S7.Thmtheorem15.p1.16.16.m16.1.1.1.1.1.1.cmml">(</mo><mrow id="S7.Thmtheorem15.p1.16.16.m16.1.1.1.1.1.1" xref="S7.Thmtheorem15.p1.16.16.m16.1.1.1.1.1.1.cmml"><mi id="S7.Thmtheorem15.p1.16.16.m16.1.1.1.1.1.1.2" xref="S7.Thmtheorem15.p1.16.16.m16.1.1.1.1.1.1.2.cmml">u</mi><mo id="S7.Thmtheorem15.p1.16.16.m16.1.1.1.1.1.1.1" lspace="0.108em" rspace="0.108em" stretchy="false" xref="S7.Thmtheorem15.p1.16.16.m16.1.1.1.1.1.1.1.cmml">→</mo><mi id="S7.Thmtheorem15.p1.16.16.m16.1.1.1.1.1.1.3" xref="S7.Thmtheorem15.p1.16.16.m16.1.1.1.1.1.1.3.cmml">v</mi></mrow><mo id="S7.Thmtheorem15.p1.16.16.m16.1.1.1.1.1.3" stretchy="false" xref="S7.Thmtheorem15.p1.16.16.m16.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S7.Thmtheorem15.p1.16.16.m16.1.1.2" xref="S7.Thmtheorem15.p1.16.16.m16.1.1.2.cmml">></mo><mn id="S7.Thmtheorem15.p1.16.16.m16.1.1.3" xref="S7.Thmtheorem15.p1.16.16.m16.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S7.Thmtheorem15.p1.16.16.m16.1b"><apply id="S7.Thmtheorem15.p1.16.16.m16.1.1.cmml" xref="S7.Thmtheorem15.p1.16.16.m16.1.1"><gt id="S7.Thmtheorem15.p1.16.16.m16.1.1.2.cmml" xref="S7.Thmtheorem15.p1.16.16.m16.1.1.2"></gt><apply id="S7.Thmtheorem15.p1.16.16.m16.1.1.1.cmml" xref="S7.Thmtheorem15.p1.16.16.m16.1.1.1"><times id="S7.Thmtheorem15.p1.16.16.m16.1.1.1.2.cmml" xref="S7.Thmtheorem15.p1.16.16.m16.1.1.1.2"></times><apply id="S7.Thmtheorem15.p1.16.16.m16.1.1.1.3.cmml" xref="S7.Thmtheorem15.p1.16.16.m16.1.1.1.3"><csymbol cd="ambiguous" id="S7.Thmtheorem15.p1.16.16.m16.1.1.1.3.1.cmml" xref="S7.Thmtheorem15.p1.16.16.m16.1.1.1.3">subscript</csymbol><ci id="S7.Thmtheorem15.p1.16.16.m16.1.1.1.3.2a.cmml" xref="S7.Thmtheorem15.p1.16.16.m16.1.1.1.3.2"><mtext class="ltx_mathvariant_italic" id="S7.Thmtheorem15.p1.16.16.m16.1.1.1.3.2.cmml" xref="S7.Thmtheorem15.p1.16.16.m16.1.1.1.3.2">g</mtext></ci><ci id="S7.Thmtheorem15.p1.16.16.m16.1.1.1.3.3.cmml" xref="S7.Thmtheorem15.p1.16.16.m16.1.1.1.3.3">𝑠</ci></apply><apply id="S7.Thmtheorem15.p1.16.16.m16.1.1.1.1.1.1.cmml" xref="S7.Thmtheorem15.p1.16.16.m16.1.1.1.1.1"><ci id="S7.Thmtheorem15.p1.16.16.m16.1.1.1.1.1.1.1.cmml" xref="S7.Thmtheorem15.p1.16.16.m16.1.1.1.1.1.1.1">→</ci><ci id="S7.Thmtheorem15.p1.16.16.m16.1.1.1.1.1.1.2.cmml" xref="S7.Thmtheorem15.p1.16.16.m16.1.1.1.1.1.1.2">𝑢</ci><ci id="S7.Thmtheorem15.p1.16.16.m16.1.1.1.1.1.1.3.cmml" xref="S7.Thmtheorem15.p1.16.16.m16.1.1.1.1.1.1.3">𝑣</ci></apply></apply><cn id="S7.Thmtheorem15.p1.16.16.m16.1.1.3.cmml" type="integer" xref="S7.Thmtheorem15.p1.16.16.m16.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.Thmtheorem15.p1.16.16.m16.1c">\textsl{g}_{s}(u\!\to\!v)>0</annotation><annotation encoding="application/x-llamapun" id="S7.Thmtheorem15.p1.16.16.m16.1d">g start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_u → italic_v ) > 0</annotation></semantics></math>. For each edge <math alttext="\overline{ab}" class="ltx_Math" display="inline" id="S7.Thmtheorem15.p1.17.17.m17.1"><semantics id="S7.Thmtheorem15.p1.17.17.m17.1a"><mover accent="true" id="S7.Thmtheorem15.p1.17.17.m17.1.1" xref="S7.Thmtheorem15.p1.17.17.m17.1.1.cmml"><mrow id="S7.Thmtheorem15.p1.17.17.m17.1.1.2" xref="S7.Thmtheorem15.p1.17.17.m17.1.1.2.cmml"><mi id="S7.Thmtheorem15.p1.17.17.m17.1.1.2.2" xref="S7.Thmtheorem15.p1.17.17.m17.1.1.2.2.cmml">a</mi><mo id="S7.Thmtheorem15.p1.17.17.m17.1.1.2.1" xref="S7.Thmtheorem15.p1.17.17.m17.1.1.2.1.cmml">⁢</mo><mi id="S7.Thmtheorem15.p1.17.17.m17.1.1.2.3" xref="S7.Thmtheorem15.p1.17.17.m17.1.1.2.3.cmml">b</mi></mrow><mo id="S7.Thmtheorem15.p1.17.17.m17.1.1.1" xref="S7.Thmtheorem15.p1.17.17.m17.1.1.1.cmml">¯</mo></mover><annotation-xml encoding="MathML-Content" id="S7.Thmtheorem15.p1.17.17.m17.1b"><apply id="S7.Thmtheorem15.p1.17.17.m17.1.1.cmml" xref="S7.Thmtheorem15.p1.17.17.m17.1.1"><ci id="S7.Thmtheorem15.p1.17.17.m17.1.1.1.cmml" xref="S7.Thmtheorem15.p1.17.17.m17.1.1.1">¯</ci><apply id="S7.Thmtheorem15.p1.17.17.m17.1.1.2.cmml" xref="S7.Thmtheorem15.p1.17.17.m17.1.1.2"><times id="S7.Thmtheorem15.p1.17.17.m17.1.1.2.1.cmml" xref="S7.Thmtheorem15.p1.17.17.m17.1.1.2.1"></times><ci id="S7.Thmtheorem15.p1.17.17.m17.1.1.2.2.cmml" xref="S7.Thmtheorem15.p1.17.17.m17.1.1.2.2">𝑎</ci><ci id="S7.Thmtheorem15.p1.17.17.m17.1.1.2.3.cmml" xref="S7.Thmtheorem15.p1.17.17.m17.1.1.2.3">𝑏</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.Thmtheorem15.p1.17.17.m17.1c">\overline{ab}</annotation><annotation encoding="application/x-llamapun" id="S7.Thmtheorem15.p1.17.17.m17.1d">over¯ start_ARG italic_a italic_b end_ARG</annotation></semantics></math> with <math alttext="l_{m}(a)>l_{m}(b)" class="ltx_Math" display="inline" id="S7.Thmtheorem15.p1.18.18.m18.2"><semantics id="S7.Thmtheorem15.p1.18.18.m18.2a"><mrow id="S7.Thmtheorem15.p1.18.18.m18.2.3" xref="S7.Thmtheorem15.p1.18.18.m18.2.3.cmml"><mrow id="S7.Thmtheorem15.p1.18.18.m18.2.3.2" xref="S7.Thmtheorem15.p1.18.18.m18.2.3.2.cmml"><msub id="S7.Thmtheorem15.p1.18.18.m18.2.3.2.2" xref="S7.Thmtheorem15.p1.18.18.m18.2.3.2.2.cmml"><mi id="S7.Thmtheorem15.p1.18.18.m18.2.3.2.2.2" xref="S7.Thmtheorem15.p1.18.18.m18.2.3.2.2.2.cmml">l</mi><mi id="S7.Thmtheorem15.p1.18.18.m18.2.3.2.2.3" xref="S7.Thmtheorem15.p1.18.18.m18.2.3.2.2.3.cmml">m</mi></msub><mo id="S7.Thmtheorem15.p1.18.18.m18.2.3.2.1" xref="S7.Thmtheorem15.p1.18.18.m18.2.3.2.1.cmml">⁢</mo><mrow id="S7.Thmtheorem15.p1.18.18.m18.2.3.2.3.2" xref="S7.Thmtheorem15.p1.18.18.m18.2.3.2.cmml"><mo id="S7.Thmtheorem15.p1.18.18.m18.2.3.2.3.2.1" stretchy="false" xref="S7.Thmtheorem15.p1.18.18.m18.2.3.2.cmml">(</mo><mi id="S7.Thmtheorem15.p1.18.18.m18.1.1" xref="S7.Thmtheorem15.p1.18.18.m18.1.1.cmml">a</mi><mo id="S7.Thmtheorem15.p1.18.18.m18.2.3.2.3.2.2" stretchy="false" xref="S7.Thmtheorem15.p1.18.18.m18.2.3.2.cmml">)</mo></mrow></mrow><mo id="S7.Thmtheorem15.p1.18.18.m18.2.3.1" xref="S7.Thmtheorem15.p1.18.18.m18.2.3.1.cmml">></mo><mrow id="S7.Thmtheorem15.p1.18.18.m18.2.3.3" xref="S7.Thmtheorem15.p1.18.18.m18.2.3.3.cmml"><msub id="S7.Thmtheorem15.p1.18.18.m18.2.3.3.2" xref="S7.Thmtheorem15.p1.18.18.m18.2.3.3.2.cmml"><mi id="S7.Thmtheorem15.p1.18.18.m18.2.3.3.2.2" xref="S7.Thmtheorem15.p1.18.18.m18.2.3.3.2.2.cmml">l</mi><mi id="S7.Thmtheorem15.p1.18.18.m18.2.3.3.2.3" xref="S7.Thmtheorem15.p1.18.18.m18.2.3.3.2.3.cmml">m</mi></msub><mo id="S7.Thmtheorem15.p1.18.18.m18.2.3.3.1" xref="S7.Thmtheorem15.p1.18.18.m18.2.3.3.1.cmml">⁢</mo><mrow id="S7.Thmtheorem15.p1.18.18.m18.2.3.3.3.2" xref="S7.Thmtheorem15.p1.18.18.m18.2.3.3.cmml"><mo id="S7.Thmtheorem15.p1.18.18.m18.2.3.3.3.2.1" stretchy="false" xref="S7.Thmtheorem15.p1.18.18.m18.2.3.3.cmml">(</mo><mi id="S7.Thmtheorem15.p1.18.18.m18.2.2" xref="S7.Thmtheorem15.p1.18.18.m18.2.2.cmml">b</mi><mo id="S7.Thmtheorem15.p1.18.18.m18.2.3.3.3.2.2" stretchy="false" xref="S7.Thmtheorem15.p1.18.18.m18.2.3.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S7.Thmtheorem15.p1.18.18.m18.2b"><apply id="S7.Thmtheorem15.p1.18.18.m18.2.3.cmml" xref="S7.Thmtheorem15.p1.18.18.m18.2.3"><gt id="S7.Thmtheorem15.p1.18.18.m18.2.3.1.cmml" xref="S7.Thmtheorem15.p1.18.18.m18.2.3.1"></gt><apply id="S7.Thmtheorem15.p1.18.18.m18.2.3.2.cmml" xref="S7.Thmtheorem15.p1.18.18.m18.2.3.2"><times id="S7.Thmtheorem15.p1.18.18.m18.2.3.2.1.cmml" xref="S7.Thmtheorem15.p1.18.18.m18.2.3.2.1"></times><apply id="S7.Thmtheorem15.p1.18.18.m18.2.3.2.2.cmml" xref="S7.Thmtheorem15.p1.18.18.m18.2.3.2.2"><csymbol cd="ambiguous" id="S7.Thmtheorem15.p1.18.18.m18.2.3.2.2.1.cmml" xref="S7.Thmtheorem15.p1.18.18.m18.2.3.2.2">subscript</csymbol><ci id="S7.Thmtheorem15.p1.18.18.m18.2.3.2.2.2.cmml" xref="S7.Thmtheorem15.p1.18.18.m18.2.3.2.2.2">𝑙</ci><ci id="S7.Thmtheorem15.p1.18.18.m18.2.3.2.2.3.cmml" xref="S7.Thmtheorem15.p1.18.18.m18.2.3.2.2.3">𝑚</ci></apply><ci id="S7.Thmtheorem15.p1.18.18.m18.1.1.cmml" xref="S7.Thmtheorem15.p1.18.18.m18.1.1">𝑎</ci></apply><apply id="S7.Thmtheorem15.p1.18.18.m18.2.3.3.cmml" xref="S7.Thmtheorem15.p1.18.18.m18.2.3.3"><times id="S7.Thmtheorem15.p1.18.18.m18.2.3.3.1.cmml" xref="S7.Thmtheorem15.p1.18.18.m18.2.3.3.1"></times><apply id="S7.Thmtheorem15.p1.18.18.m18.2.3.3.2.cmml" xref="S7.Thmtheorem15.p1.18.18.m18.2.3.3.2"><csymbol cd="ambiguous" id="S7.Thmtheorem15.p1.18.18.m18.2.3.3.2.1.cmml" xref="S7.Thmtheorem15.p1.18.18.m18.2.3.3.2">subscript</csymbol><ci id="S7.Thmtheorem15.p1.18.18.m18.2.3.3.2.2.cmml" xref="S7.Thmtheorem15.p1.18.18.m18.2.3.3.2.2">𝑙</ci><ci id="S7.Thmtheorem15.p1.18.18.m18.2.3.3.2.3.cmml" xref="S7.Thmtheorem15.p1.18.18.m18.2.3.3.2.3">𝑚</ci></apply><ci id="S7.Thmtheorem15.p1.18.18.m18.2.2.cmml" xref="S7.Thmtheorem15.p1.18.18.m18.2.2">𝑏</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.Thmtheorem15.p1.18.18.m18.2c">l_{m}(a)>l_{m}(b)</annotation><annotation encoding="application/x-llamapun" id="S7.Thmtheorem15.p1.18.18.m18.2d">italic_l start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT ( italic_a ) > italic_l start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT ( italic_b )</annotation></semantics></math> the out-degree <math alttext="\textsl{g}(a\!\to\!b)" class="ltx_Math" display="inline" id="S7.Thmtheorem15.p1.19.19.m19.1"><semantics id="S7.Thmtheorem15.p1.19.19.m19.1a"><mrow id="S7.Thmtheorem15.p1.19.19.m19.1.1" xref="S7.Thmtheorem15.p1.19.19.m19.1.1.cmml"><mtext class="ltx_mathvariant_italic" id="S7.Thmtheorem15.p1.19.19.m19.1.1.3" xref="S7.Thmtheorem15.p1.19.19.m19.1.1.3a.cmml">g</mtext><mo id="S7.Thmtheorem15.p1.19.19.m19.1.1.2" xref="S7.Thmtheorem15.p1.19.19.m19.1.1.2.cmml">⁢</mo><mrow id="S7.Thmtheorem15.p1.19.19.m19.1.1.1.1" xref="S7.Thmtheorem15.p1.19.19.m19.1.1.1.1.1.cmml"><mo id="S7.Thmtheorem15.p1.19.19.m19.1.1.1.1.2" stretchy="false" xref="S7.Thmtheorem15.p1.19.19.m19.1.1.1.1.1.cmml">(</mo><mrow id="S7.Thmtheorem15.p1.19.19.m19.1.1.1.1.1" xref="S7.Thmtheorem15.p1.19.19.m19.1.1.1.1.1.cmml"><mi id="S7.Thmtheorem15.p1.19.19.m19.1.1.1.1.1.2" xref="S7.Thmtheorem15.p1.19.19.m19.1.1.1.1.1.2.cmml">a</mi><mo id="S7.Thmtheorem15.p1.19.19.m19.1.1.1.1.1.1" lspace="0.108em" rspace="0.108em" stretchy="false" xref="S7.Thmtheorem15.p1.19.19.m19.1.1.1.1.1.1.cmml">→</mo><mi id="S7.Thmtheorem15.p1.19.19.m19.1.1.1.1.1.3" xref="S7.Thmtheorem15.p1.19.19.m19.1.1.1.1.1.3.cmml">b</mi></mrow><mo id="S7.Thmtheorem15.p1.19.19.m19.1.1.1.1.3" stretchy="false" xref="S7.Thmtheorem15.p1.19.19.m19.1.1.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S7.Thmtheorem15.p1.19.19.m19.1b"><apply id="S7.Thmtheorem15.p1.19.19.m19.1.1.cmml" xref="S7.Thmtheorem15.p1.19.19.m19.1.1"><times id="S7.Thmtheorem15.p1.19.19.m19.1.1.2.cmml" xref="S7.Thmtheorem15.p1.19.19.m19.1.1.2"></times><ci id="S7.Thmtheorem15.p1.19.19.m19.1.1.3a.cmml" xref="S7.Thmtheorem15.p1.19.19.m19.1.1.3"><mtext class="ltx_mathvariant_italic" id="S7.Thmtheorem15.p1.19.19.m19.1.1.3.cmml" xref="S7.Thmtheorem15.p1.19.19.m19.1.1.3">g</mtext></ci><apply id="S7.Thmtheorem15.p1.19.19.m19.1.1.1.1.1.cmml" xref="S7.Thmtheorem15.p1.19.19.m19.1.1.1.1"><ci id="S7.Thmtheorem15.p1.19.19.m19.1.1.1.1.1.1.cmml" xref="S7.Thmtheorem15.p1.19.19.m19.1.1.1.1.1.1">→</ci><ci id="S7.Thmtheorem15.p1.19.19.m19.1.1.1.1.1.2.cmml" xref="S7.Thmtheorem15.p1.19.19.m19.1.1.1.1.1.2">𝑎</ci><ci id="S7.Thmtheorem15.p1.19.19.m19.1.1.1.1.1.3.cmml" xref="S7.Thmtheorem15.p1.19.19.m19.1.1.1.1.1.3">𝑏</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.Thmtheorem15.p1.19.19.m19.1c">\textsl{g}(a\!\to\!b)</annotation><annotation encoding="application/x-llamapun" id="S7.Thmtheorem15.p1.19.19.m19.1d">g ( italic_a → italic_b )</annotation></semantics></math> only decreases and thus <math alttext="\textsl{g}_{0}(u\!\to\!v)>0" class="ltx_Math" display="inline" id="S7.Thmtheorem15.p1.20.20.m20.1"><semantics id="S7.Thmtheorem15.p1.20.20.m20.1a"><mrow id="S7.Thmtheorem15.p1.20.20.m20.1.1" xref="S7.Thmtheorem15.p1.20.20.m20.1.1.cmml"><mrow id="S7.Thmtheorem15.p1.20.20.m20.1.1.1" xref="S7.Thmtheorem15.p1.20.20.m20.1.1.1.cmml"><msub id="S7.Thmtheorem15.p1.20.20.m20.1.1.1.3" xref="S7.Thmtheorem15.p1.20.20.m20.1.1.1.3.cmml"><mtext class="ltx_mathvariant_italic" id="S7.Thmtheorem15.p1.20.20.m20.1.1.1.3.2" xref="S7.Thmtheorem15.p1.20.20.m20.1.1.1.3.2a.cmml">g</mtext><mn id="S7.Thmtheorem15.p1.20.20.m20.1.1.1.3.3" xref="S7.Thmtheorem15.p1.20.20.m20.1.1.1.3.3.cmml">0</mn></msub><mo id="S7.Thmtheorem15.p1.20.20.m20.1.1.1.2" xref="S7.Thmtheorem15.p1.20.20.m20.1.1.1.2.cmml">⁢</mo><mrow id="S7.Thmtheorem15.p1.20.20.m20.1.1.1.1.1" xref="S7.Thmtheorem15.p1.20.20.m20.1.1.1.1.1.1.cmml"><mo id="S7.Thmtheorem15.p1.20.20.m20.1.1.1.1.1.2" stretchy="false" xref="S7.Thmtheorem15.p1.20.20.m20.1.1.1.1.1.1.cmml">(</mo><mrow id="S7.Thmtheorem15.p1.20.20.m20.1.1.1.1.1.1" xref="S7.Thmtheorem15.p1.20.20.m20.1.1.1.1.1.1.cmml"><mi id="S7.Thmtheorem15.p1.20.20.m20.1.1.1.1.1.1.2" xref="S7.Thmtheorem15.p1.20.20.m20.1.1.1.1.1.1.2.cmml">u</mi><mo id="S7.Thmtheorem15.p1.20.20.m20.1.1.1.1.1.1.1" lspace="0.108em" rspace="0.108em" stretchy="false" xref="S7.Thmtheorem15.p1.20.20.m20.1.1.1.1.1.1.1.cmml">→</mo><mi id="S7.Thmtheorem15.p1.20.20.m20.1.1.1.1.1.1.3" xref="S7.Thmtheorem15.p1.20.20.m20.1.1.1.1.1.1.3.cmml">v</mi></mrow><mo id="S7.Thmtheorem15.p1.20.20.m20.1.1.1.1.1.3" stretchy="false" xref="S7.Thmtheorem15.p1.20.20.m20.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S7.Thmtheorem15.p1.20.20.m20.1.1.2" xref="S7.Thmtheorem15.p1.20.20.m20.1.1.2.cmml">></mo><mn id="S7.Thmtheorem15.p1.20.20.m20.1.1.3" xref="S7.Thmtheorem15.p1.20.20.m20.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S7.Thmtheorem15.p1.20.20.m20.1b"><apply id="S7.Thmtheorem15.p1.20.20.m20.1.1.cmml" xref="S7.Thmtheorem15.p1.20.20.m20.1.1"><gt id="S7.Thmtheorem15.p1.20.20.m20.1.1.2.cmml" xref="S7.Thmtheorem15.p1.20.20.m20.1.1.2"></gt><apply id="S7.Thmtheorem15.p1.20.20.m20.1.1.1.cmml" xref="S7.Thmtheorem15.p1.20.20.m20.1.1.1"><times id="S7.Thmtheorem15.p1.20.20.m20.1.1.1.2.cmml" xref="S7.Thmtheorem15.p1.20.20.m20.1.1.1.2"></times><apply id="S7.Thmtheorem15.p1.20.20.m20.1.1.1.3.cmml" xref="S7.Thmtheorem15.p1.20.20.m20.1.1.1.3"><csymbol cd="ambiguous" id="S7.Thmtheorem15.p1.20.20.m20.1.1.1.3.1.cmml" xref="S7.Thmtheorem15.p1.20.20.m20.1.1.1.3">subscript</csymbol><ci id="S7.Thmtheorem15.p1.20.20.m20.1.1.1.3.2a.cmml" xref="S7.Thmtheorem15.p1.20.20.m20.1.1.1.3.2"><mtext class="ltx_mathvariant_italic" id="S7.Thmtheorem15.p1.20.20.m20.1.1.1.3.2.cmml" xref="S7.Thmtheorem15.p1.20.20.m20.1.1.1.3.2">g</mtext></ci><cn id="S7.Thmtheorem15.p1.20.20.m20.1.1.1.3.3.cmml" type="integer" xref="S7.Thmtheorem15.p1.20.20.m20.1.1.1.3.3">0</cn></apply><apply id="S7.Thmtheorem15.p1.20.20.m20.1.1.1.1.1.1.cmml" xref="S7.Thmtheorem15.p1.20.20.m20.1.1.1.1.1"><ci id="S7.Thmtheorem15.p1.20.20.m20.1.1.1.1.1.1.1.cmml" xref="S7.Thmtheorem15.p1.20.20.m20.1.1.1.1.1.1.1">→</ci><ci id="S7.Thmtheorem15.p1.20.20.m20.1.1.1.1.1.1.2.cmml" xref="S7.Thmtheorem15.p1.20.20.m20.1.1.1.1.1.1.2">𝑢</ci><ci id="S7.Thmtheorem15.p1.20.20.m20.1.1.1.1.1.1.3.cmml" xref="S7.Thmtheorem15.p1.20.20.m20.1.1.1.1.1.1.3">𝑣</ci></apply></apply><cn id="S7.Thmtheorem15.p1.20.20.m20.1.1.3.cmml" type="integer" xref="S7.Thmtheorem15.p1.20.20.m20.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.Thmtheorem15.p1.20.20.m20.1c">\textsl{g}_{0}(u\!\to\!v)>0</annotation><annotation encoding="application/x-llamapun" id="S7.Thmtheorem15.p1.20.20.m20.1d">g start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ( italic_u → italic_v ) > 0</annotation></semantics></math>. Moreover, by Observation <a class="ltx_ref" href="https://arxiv.org/html/2411.12694v2#S7.SS3" title="7.3 Sketching our algorithm’s correctness. ‣ 7 Results in CONGEST ‣ Local Density and its Distributed Approximation"><span class="ltx_text ltx_ref_tag">7.3</span></a>, <math alttext="l_{s}(v)=l_{m}(v)\geq h+1" class="ltx_Math" display="inline" id="S7.Thmtheorem15.p1.21.21.m21.2"><semantics id="S7.Thmtheorem15.p1.21.21.m21.2a"><mrow id="S7.Thmtheorem15.p1.21.21.m21.2.3" xref="S7.Thmtheorem15.p1.21.21.m21.2.3.cmml"><mrow id="S7.Thmtheorem15.p1.21.21.m21.2.3.2" xref="S7.Thmtheorem15.p1.21.21.m21.2.3.2.cmml"><msub id="S7.Thmtheorem15.p1.21.21.m21.2.3.2.2" xref="S7.Thmtheorem15.p1.21.21.m21.2.3.2.2.cmml"><mi id="S7.Thmtheorem15.p1.21.21.m21.2.3.2.2.2" xref="S7.Thmtheorem15.p1.21.21.m21.2.3.2.2.2.cmml">l</mi><mi id="S7.Thmtheorem15.p1.21.21.m21.2.3.2.2.3" xref="S7.Thmtheorem15.p1.21.21.m21.2.3.2.2.3.cmml">s</mi></msub><mo id="S7.Thmtheorem15.p1.21.21.m21.2.3.2.1" xref="S7.Thmtheorem15.p1.21.21.m21.2.3.2.1.cmml">⁢</mo><mrow id="S7.Thmtheorem15.p1.21.21.m21.2.3.2.3.2" xref="S7.Thmtheorem15.p1.21.21.m21.2.3.2.cmml"><mo id="S7.Thmtheorem15.p1.21.21.m21.2.3.2.3.2.1" stretchy="false" xref="S7.Thmtheorem15.p1.21.21.m21.2.3.2.cmml">(</mo><mi id="S7.Thmtheorem15.p1.21.21.m21.1.1" xref="S7.Thmtheorem15.p1.21.21.m21.1.1.cmml">v</mi><mo id="S7.Thmtheorem15.p1.21.21.m21.2.3.2.3.2.2" stretchy="false" xref="S7.Thmtheorem15.p1.21.21.m21.2.3.2.cmml">)</mo></mrow></mrow><mo id="S7.Thmtheorem15.p1.21.21.m21.2.3.3" xref="S7.Thmtheorem15.p1.21.21.m21.2.3.3.cmml">=</mo><mrow id="S7.Thmtheorem15.p1.21.21.m21.2.3.4" xref="S7.Thmtheorem15.p1.21.21.m21.2.3.4.cmml"><msub id="S7.Thmtheorem15.p1.21.21.m21.2.3.4.2" xref="S7.Thmtheorem15.p1.21.21.m21.2.3.4.2.cmml"><mi id="S7.Thmtheorem15.p1.21.21.m21.2.3.4.2.2" xref="S7.Thmtheorem15.p1.21.21.m21.2.3.4.2.2.cmml">l</mi><mi id="S7.Thmtheorem15.p1.21.21.m21.2.3.4.2.3" xref="S7.Thmtheorem15.p1.21.21.m21.2.3.4.2.3.cmml">m</mi></msub><mo id="S7.Thmtheorem15.p1.21.21.m21.2.3.4.1" xref="S7.Thmtheorem15.p1.21.21.m21.2.3.4.1.cmml">⁢</mo><mrow id="S7.Thmtheorem15.p1.21.21.m21.2.3.4.3.2" xref="S7.Thmtheorem15.p1.21.21.m21.2.3.4.cmml"><mo id="S7.Thmtheorem15.p1.21.21.m21.2.3.4.3.2.1" stretchy="false" xref="S7.Thmtheorem15.p1.21.21.m21.2.3.4.cmml">(</mo><mi id="S7.Thmtheorem15.p1.21.21.m21.2.2" xref="S7.Thmtheorem15.p1.21.21.m21.2.2.cmml">v</mi><mo id="S7.Thmtheorem15.p1.21.21.m21.2.3.4.3.2.2" stretchy="false" xref="S7.Thmtheorem15.p1.21.21.m21.2.3.4.cmml">)</mo></mrow></mrow><mo id="S7.Thmtheorem15.p1.21.21.m21.2.3.5" xref="S7.Thmtheorem15.p1.21.21.m21.2.3.5.cmml">≥</mo><mrow id="S7.Thmtheorem15.p1.21.21.m21.2.3.6" xref="S7.Thmtheorem15.p1.21.21.m21.2.3.6.cmml"><mi id="S7.Thmtheorem15.p1.21.21.m21.2.3.6.2" xref="S7.Thmtheorem15.p1.21.21.m21.2.3.6.2.cmml">h</mi><mo id="S7.Thmtheorem15.p1.21.21.m21.2.3.6.1" xref="S7.Thmtheorem15.p1.21.21.m21.2.3.6.1.cmml">+</mo><mn id="S7.Thmtheorem15.p1.21.21.m21.2.3.6.3" xref="S7.Thmtheorem15.p1.21.21.m21.2.3.6.3.cmml">1</mn></mrow></mrow><annotation-xml encoding="MathML-Content" id="S7.Thmtheorem15.p1.21.21.m21.2b"><apply id="S7.Thmtheorem15.p1.21.21.m21.2.3.cmml" xref="S7.Thmtheorem15.p1.21.21.m21.2.3"><and id="S7.Thmtheorem15.p1.21.21.m21.2.3a.cmml" xref="S7.Thmtheorem15.p1.21.21.m21.2.3"></and><apply id="S7.Thmtheorem15.p1.21.21.m21.2.3b.cmml" xref="S7.Thmtheorem15.p1.21.21.m21.2.3"><eq id="S7.Thmtheorem15.p1.21.21.m21.2.3.3.cmml" xref="S7.Thmtheorem15.p1.21.21.m21.2.3.3"></eq><apply id="S7.Thmtheorem15.p1.21.21.m21.2.3.2.cmml" xref="S7.Thmtheorem15.p1.21.21.m21.2.3.2"><times id="S7.Thmtheorem15.p1.21.21.m21.2.3.2.1.cmml" xref="S7.Thmtheorem15.p1.21.21.m21.2.3.2.1"></times><apply id="S7.Thmtheorem15.p1.21.21.m21.2.3.2.2.cmml" xref="S7.Thmtheorem15.p1.21.21.m21.2.3.2.2"><csymbol cd="ambiguous" id="S7.Thmtheorem15.p1.21.21.m21.2.3.2.2.1.cmml" xref="S7.Thmtheorem15.p1.21.21.m21.2.3.2.2">subscript</csymbol><ci id="S7.Thmtheorem15.p1.21.21.m21.2.3.2.2.2.cmml" xref="S7.Thmtheorem15.p1.21.21.m21.2.3.2.2.2">𝑙</ci><ci id="S7.Thmtheorem15.p1.21.21.m21.2.3.2.2.3.cmml" xref="S7.Thmtheorem15.p1.21.21.m21.2.3.2.2.3">𝑠</ci></apply><ci id="S7.Thmtheorem15.p1.21.21.m21.1.1.cmml" xref="S7.Thmtheorem15.p1.21.21.m21.1.1">𝑣</ci></apply><apply id="S7.Thmtheorem15.p1.21.21.m21.2.3.4.cmml" xref="S7.Thmtheorem15.p1.21.21.m21.2.3.4"><times id="S7.Thmtheorem15.p1.21.21.m21.2.3.4.1.cmml" xref="S7.Thmtheorem15.p1.21.21.m21.2.3.4.1"></times><apply id="S7.Thmtheorem15.p1.21.21.m21.2.3.4.2.cmml" xref="S7.Thmtheorem15.p1.21.21.m21.2.3.4.2"><csymbol cd="ambiguous" id="S7.Thmtheorem15.p1.21.21.m21.2.3.4.2.1.cmml" xref="S7.Thmtheorem15.p1.21.21.m21.2.3.4.2">subscript</csymbol><ci id="S7.Thmtheorem15.p1.21.21.m21.2.3.4.2.2.cmml" xref="S7.Thmtheorem15.p1.21.21.m21.2.3.4.2.2">𝑙</ci><ci id="S7.Thmtheorem15.p1.21.21.m21.2.3.4.2.3.cmml" xref="S7.Thmtheorem15.p1.21.21.m21.2.3.4.2.3">𝑚</ci></apply><ci id="S7.Thmtheorem15.p1.21.21.m21.2.2.cmml" xref="S7.Thmtheorem15.p1.21.21.m21.2.2">𝑣</ci></apply></apply><apply id="S7.Thmtheorem15.p1.21.21.m21.2.3c.cmml" xref="S7.Thmtheorem15.p1.21.21.m21.2.3"><geq id="S7.Thmtheorem15.p1.21.21.m21.2.3.5.cmml" xref="S7.Thmtheorem15.p1.21.21.m21.2.3.5"></geq><share href="https://arxiv.org/html/2411.12694v2#S7.Thmtheorem15.p1.21.21.m21.2.3.4.cmml" id="S7.Thmtheorem15.p1.21.21.m21.2.3d.cmml" xref="S7.Thmtheorem15.p1.21.21.m21.2.3"></share><apply id="S7.Thmtheorem15.p1.21.21.m21.2.3.6.cmml" xref="S7.Thmtheorem15.p1.21.21.m21.2.3.6"><plus id="S7.Thmtheorem15.p1.21.21.m21.2.3.6.1.cmml" xref="S7.Thmtheorem15.p1.21.21.m21.2.3.6.1"></plus><ci id="S7.Thmtheorem15.p1.21.21.m21.2.3.6.2.cmml" xref="S7.Thmtheorem15.p1.21.21.m21.2.3.6.2">ℎ</ci><cn id="S7.Thmtheorem15.p1.21.21.m21.2.3.6.3.cmml" type="integer" xref="S7.Thmtheorem15.p1.21.21.m21.2.3.6.3">1</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.Thmtheorem15.p1.21.21.m21.2c">l_{s}(v)=l_{m}(v)\geq h+1</annotation><annotation encoding="application/x-llamapun" id="S7.Thmtheorem15.p1.21.21.m21.2d">italic_l start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_v ) = italic_l start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT ( italic_v ) ≥ italic_h + 1</annotation></semantics></math>.</span></p> </div> <div class="ltx_para" id="S7.Thmtheorem15.p2"> <p class="ltx_p" id="S7.Thmtheorem15.p2.8"><span class="ltx_text ltx_font_italic" id="S7.Thmtheorem15.p2.8.8">At the start of <math alttext="(h:m:s)" class="ltx_Math" display="inline" id="S7.Thmtheorem15.p2.1.1.m1.1"><semantics id="S7.Thmtheorem15.p2.1.1.m1.1a"><mrow id="S7.Thmtheorem15.p2.1.1.m1.1.1.1" xref="S7.Thmtheorem15.p2.1.1.m1.1.1.1.1.cmml"><mo id="S7.Thmtheorem15.p2.1.1.m1.1.1.1.2" stretchy="false" xref="S7.Thmtheorem15.p2.1.1.m1.1.1.1.1.cmml">(</mo><mrow id="S7.Thmtheorem15.p2.1.1.m1.1.1.1.1" xref="S7.Thmtheorem15.p2.1.1.m1.1.1.1.1.cmml"><mi id="S7.Thmtheorem15.p2.1.1.m1.1.1.1.1.2" xref="S7.Thmtheorem15.p2.1.1.m1.1.1.1.1.2.cmml">h</mi><mo id="S7.Thmtheorem15.p2.1.1.m1.1.1.1.1.3" lspace="0.278em" rspace="0.278em" xref="S7.Thmtheorem15.p2.1.1.m1.1.1.1.1.3.cmml">:</mo><mi id="S7.Thmtheorem15.p2.1.1.m1.1.1.1.1.4" xref="S7.Thmtheorem15.p2.1.1.m1.1.1.1.1.4.cmml">m</mi><mo id="S7.Thmtheorem15.p2.1.1.m1.1.1.1.1.5" lspace="0.278em" rspace="0.278em" xref="S7.Thmtheorem15.p2.1.1.m1.1.1.1.1.5.cmml">:</mo><mi id="S7.Thmtheorem15.p2.1.1.m1.1.1.1.1.6" xref="S7.Thmtheorem15.p2.1.1.m1.1.1.1.1.6.cmml">s</mi></mrow><mo id="S7.Thmtheorem15.p2.1.1.m1.1.1.1.3" stretchy="false" xref="S7.Thmtheorem15.p2.1.1.m1.1.1.1.1.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S7.Thmtheorem15.p2.1.1.m1.1b"><apply id="S7.Thmtheorem15.p2.1.1.m1.1.1.1.1.cmml" xref="S7.Thmtheorem15.p2.1.1.m1.1.1.1"><and id="S7.Thmtheorem15.p2.1.1.m1.1.1.1.1a.cmml" xref="S7.Thmtheorem15.p2.1.1.m1.1.1.1"></and><apply id="S7.Thmtheorem15.p2.1.1.m1.1.1.1.1b.cmml" xref="S7.Thmtheorem15.p2.1.1.m1.1.1.1"><ci id="S7.Thmtheorem15.p2.1.1.m1.1.1.1.1.3.cmml" xref="S7.Thmtheorem15.p2.1.1.m1.1.1.1.1.3">:</ci><ci id="S7.Thmtheorem15.p2.1.1.m1.1.1.1.1.2.cmml" xref="S7.Thmtheorem15.p2.1.1.m1.1.1.1.1.2">ℎ</ci><ci id="S7.Thmtheorem15.p2.1.1.m1.1.1.1.1.4.cmml" xref="S7.Thmtheorem15.p2.1.1.m1.1.1.1.1.4">𝑚</ci></apply><apply id="S7.Thmtheorem15.p2.1.1.m1.1.1.1.1c.cmml" xref="S7.Thmtheorem15.p2.1.1.m1.1.1.1"><ci id="S7.Thmtheorem15.p2.1.1.m1.1.1.1.1.5.cmml" xref="S7.Thmtheorem15.p2.1.1.m1.1.1.1.1.5">:</ci><share href="https://arxiv.org/html/2411.12694v2#S7.Thmtheorem15.p2.1.1.m1.1.1.1.1.4.cmml" id="S7.Thmtheorem15.p2.1.1.m1.1.1.1.1d.cmml" xref="S7.Thmtheorem15.p2.1.1.m1.1.1.1"></share><ci id="S7.Thmtheorem15.p2.1.1.m1.1.1.1.1.6.cmml" xref="S7.Thmtheorem15.p2.1.1.m1.1.1.1.1.6">𝑠</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.Thmtheorem15.p2.1.1.m1.1c">(h:m:s)</annotation><annotation encoding="application/x-llamapun" id="S7.Thmtheorem15.p2.1.1.m1.1d">( italic_h : italic_m : italic_s )</annotation></semantics></math>, the vertex <math alttext="v" class="ltx_Math" display="inline" id="S7.Thmtheorem15.p2.2.2.m2.1"><semantics id="S7.Thmtheorem15.p2.2.2.m2.1a"><mi id="S7.Thmtheorem15.p2.2.2.m2.1.1" xref="S7.Thmtheorem15.p2.2.2.m2.1.1.cmml">v</mi><annotation-xml encoding="MathML-Content" id="S7.Thmtheorem15.p2.2.2.m2.1b"><ci id="S7.Thmtheorem15.p2.2.2.m2.1.1.cmml" xref="S7.Thmtheorem15.p2.2.2.m2.1.1">𝑣</ci></annotation-xml><annotation encoding="application/x-tex" id="S7.Thmtheorem15.p2.2.2.m2.1c">v</annotation><annotation encoding="application/x-llamapun" id="S7.Thmtheorem15.p2.2.2.m2.1d">italic_v</annotation></semantics></math> must a source in <math alttext="D_{s}" class="ltx_Math" display="inline" id="S7.Thmtheorem15.p2.3.3.m3.1"><semantics id="S7.Thmtheorem15.p2.3.3.m3.1a"><msub id="S7.Thmtheorem15.p2.3.3.m3.1.1" xref="S7.Thmtheorem15.p2.3.3.m3.1.1.cmml"><mi id="S7.Thmtheorem15.p2.3.3.m3.1.1.2" xref="S7.Thmtheorem15.p2.3.3.m3.1.1.2.cmml">D</mi><mi id="S7.Thmtheorem15.p2.3.3.m3.1.1.3" xref="S7.Thmtheorem15.p2.3.3.m3.1.1.3.cmml">s</mi></msub><annotation-xml encoding="MathML-Content" id="S7.Thmtheorem15.p2.3.3.m3.1b"><apply id="S7.Thmtheorem15.p2.3.3.m3.1.1.cmml" xref="S7.Thmtheorem15.p2.3.3.m3.1.1"><csymbol cd="ambiguous" id="S7.Thmtheorem15.p2.3.3.m3.1.1.1.cmml" xref="S7.Thmtheorem15.p2.3.3.m3.1.1">subscript</csymbol><ci id="S7.Thmtheorem15.p2.3.3.m3.1.1.2.cmml" xref="S7.Thmtheorem15.p2.3.3.m3.1.1.2">𝐷</ci><ci id="S7.Thmtheorem15.p2.3.3.m3.1.1.3.cmml" xref="S7.Thmtheorem15.p2.3.3.m3.1.1.3">𝑠</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.Thmtheorem15.p2.3.3.m3.1c">D_{s}</annotation><annotation encoding="application/x-llamapun" id="S7.Thmtheorem15.p2.3.3.m3.1d">italic_D start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT</annotation></semantics></math> (since only sources decrease their out-degree). However, by the definition of <math alttext="E_{s}" class="ltx_Math" display="inline" id="S7.Thmtheorem15.p2.4.4.m4.1"><semantics id="S7.Thmtheorem15.p2.4.4.m4.1a"><msub id="S7.Thmtheorem15.p2.4.4.m4.1.1" xref="S7.Thmtheorem15.p2.4.4.m4.1.1.cmml"><mi id="S7.Thmtheorem15.p2.4.4.m4.1.1.2" xref="S7.Thmtheorem15.p2.4.4.m4.1.1.2.cmml">E</mi><mi id="S7.Thmtheorem15.p2.4.4.m4.1.1.3" xref="S7.Thmtheorem15.p2.4.4.m4.1.1.3.cmml">s</mi></msub><annotation-xml encoding="MathML-Content" id="S7.Thmtheorem15.p2.4.4.m4.1b"><apply id="S7.Thmtheorem15.p2.4.4.m4.1.1.cmml" xref="S7.Thmtheorem15.p2.4.4.m4.1.1"><csymbol cd="ambiguous" id="S7.Thmtheorem15.p2.4.4.m4.1.1.1.cmml" xref="S7.Thmtheorem15.p2.4.4.m4.1.1">subscript</csymbol><ci id="S7.Thmtheorem15.p2.4.4.m4.1.1.2.cmml" xref="S7.Thmtheorem15.p2.4.4.m4.1.1.2">𝐸</ci><ci id="S7.Thmtheorem15.p2.4.4.m4.1.1.3.cmml" xref="S7.Thmtheorem15.p2.4.4.m4.1.1.3">𝑠</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.Thmtheorem15.p2.4.4.m4.1c">E_{s}</annotation><annotation encoding="application/x-llamapun" id="S7.Thmtheorem15.p2.4.4.m4.1d">italic_E start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT</annotation></semantics></math>, this implies that <math alttext="l_{s}(u)\neq l_{m}(u)" class="ltx_Math" display="inline" id="S7.Thmtheorem15.p2.5.5.m5.2"><semantics id="S7.Thmtheorem15.p2.5.5.m5.2a"><mrow id="S7.Thmtheorem15.p2.5.5.m5.2.3" xref="S7.Thmtheorem15.p2.5.5.m5.2.3.cmml"><mrow id="S7.Thmtheorem15.p2.5.5.m5.2.3.2" xref="S7.Thmtheorem15.p2.5.5.m5.2.3.2.cmml"><msub id="S7.Thmtheorem15.p2.5.5.m5.2.3.2.2" xref="S7.Thmtheorem15.p2.5.5.m5.2.3.2.2.cmml"><mi id="S7.Thmtheorem15.p2.5.5.m5.2.3.2.2.2" xref="S7.Thmtheorem15.p2.5.5.m5.2.3.2.2.2.cmml">l</mi><mi id="S7.Thmtheorem15.p2.5.5.m5.2.3.2.2.3" xref="S7.Thmtheorem15.p2.5.5.m5.2.3.2.2.3.cmml">s</mi></msub><mo id="S7.Thmtheorem15.p2.5.5.m5.2.3.2.1" xref="S7.Thmtheorem15.p2.5.5.m5.2.3.2.1.cmml">⁢</mo><mrow id="S7.Thmtheorem15.p2.5.5.m5.2.3.2.3.2" xref="S7.Thmtheorem15.p2.5.5.m5.2.3.2.cmml"><mo id="S7.Thmtheorem15.p2.5.5.m5.2.3.2.3.2.1" stretchy="false" xref="S7.Thmtheorem15.p2.5.5.m5.2.3.2.cmml">(</mo><mi id="S7.Thmtheorem15.p2.5.5.m5.1.1" xref="S7.Thmtheorem15.p2.5.5.m5.1.1.cmml">u</mi><mo id="S7.Thmtheorem15.p2.5.5.m5.2.3.2.3.2.2" stretchy="false" xref="S7.Thmtheorem15.p2.5.5.m5.2.3.2.cmml">)</mo></mrow></mrow><mo id="S7.Thmtheorem15.p2.5.5.m5.2.3.1" xref="S7.Thmtheorem15.p2.5.5.m5.2.3.1.cmml">≠</mo><mrow id="S7.Thmtheorem15.p2.5.5.m5.2.3.3" xref="S7.Thmtheorem15.p2.5.5.m5.2.3.3.cmml"><msub id="S7.Thmtheorem15.p2.5.5.m5.2.3.3.2" xref="S7.Thmtheorem15.p2.5.5.m5.2.3.3.2.cmml"><mi id="S7.Thmtheorem15.p2.5.5.m5.2.3.3.2.2" xref="S7.Thmtheorem15.p2.5.5.m5.2.3.3.2.2.cmml">l</mi><mi id="S7.Thmtheorem15.p2.5.5.m5.2.3.3.2.3" xref="S7.Thmtheorem15.p2.5.5.m5.2.3.3.2.3.cmml">m</mi></msub><mo id="S7.Thmtheorem15.p2.5.5.m5.2.3.3.1" xref="S7.Thmtheorem15.p2.5.5.m5.2.3.3.1.cmml">⁢</mo><mrow id="S7.Thmtheorem15.p2.5.5.m5.2.3.3.3.2" xref="S7.Thmtheorem15.p2.5.5.m5.2.3.3.cmml"><mo id="S7.Thmtheorem15.p2.5.5.m5.2.3.3.3.2.1" stretchy="false" xref="S7.Thmtheorem15.p2.5.5.m5.2.3.3.cmml">(</mo><mi id="S7.Thmtheorem15.p2.5.5.m5.2.2" xref="S7.Thmtheorem15.p2.5.5.m5.2.2.cmml">u</mi><mo id="S7.Thmtheorem15.p2.5.5.m5.2.3.3.3.2.2" stretchy="false" xref="S7.Thmtheorem15.p2.5.5.m5.2.3.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S7.Thmtheorem15.p2.5.5.m5.2b"><apply id="S7.Thmtheorem15.p2.5.5.m5.2.3.cmml" xref="S7.Thmtheorem15.p2.5.5.m5.2.3"><neq id="S7.Thmtheorem15.p2.5.5.m5.2.3.1.cmml" xref="S7.Thmtheorem15.p2.5.5.m5.2.3.1"></neq><apply id="S7.Thmtheorem15.p2.5.5.m5.2.3.2.cmml" xref="S7.Thmtheorem15.p2.5.5.m5.2.3.2"><times id="S7.Thmtheorem15.p2.5.5.m5.2.3.2.1.cmml" xref="S7.Thmtheorem15.p2.5.5.m5.2.3.2.1"></times><apply id="S7.Thmtheorem15.p2.5.5.m5.2.3.2.2.cmml" xref="S7.Thmtheorem15.p2.5.5.m5.2.3.2.2"><csymbol cd="ambiguous" id="S7.Thmtheorem15.p2.5.5.m5.2.3.2.2.1.cmml" xref="S7.Thmtheorem15.p2.5.5.m5.2.3.2.2">subscript</csymbol><ci id="S7.Thmtheorem15.p2.5.5.m5.2.3.2.2.2.cmml" xref="S7.Thmtheorem15.p2.5.5.m5.2.3.2.2.2">𝑙</ci><ci id="S7.Thmtheorem15.p2.5.5.m5.2.3.2.2.3.cmml" xref="S7.Thmtheorem15.p2.5.5.m5.2.3.2.2.3">𝑠</ci></apply><ci id="S7.Thmtheorem15.p2.5.5.m5.1.1.cmml" xref="S7.Thmtheorem15.p2.5.5.m5.1.1">𝑢</ci></apply><apply id="S7.Thmtheorem15.p2.5.5.m5.2.3.3.cmml" xref="S7.Thmtheorem15.p2.5.5.m5.2.3.3"><times id="S7.Thmtheorem15.p2.5.5.m5.2.3.3.1.cmml" xref="S7.Thmtheorem15.p2.5.5.m5.2.3.3.1"></times><apply id="S7.Thmtheorem15.p2.5.5.m5.2.3.3.2.cmml" xref="S7.Thmtheorem15.p2.5.5.m5.2.3.3.2"><csymbol cd="ambiguous" id="S7.Thmtheorem15.p2.5.5.m5.2.3.3.2.1.cmml" xref="S7.Thmtheorem15.p2.5.5.m5.2.3.3.2">subscript</csymbol><ci id="S7.Thmtheorem15.p2.5.5.m5.2.3.3.2.2.cmml" xref="S7.Thmtheorem15.p2.5.5.m5.2.3.3.2.2">𝑙</ci><ci id="S7.Thmtheorem15.p2.5.5.m5.2.3.3.2.3.cmml" xref="S7.Thmtheorem15.p2.5.5.m5.2.3.3.2.3">𝑚</ci></apply><ci id="S7.Thmtheorem15.p2.5.5.m5.2.2.cmml" xref="S7.Thmtheorem15.p2.5.5.m5.2.2">𝑢</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.Thmtheorem15.p2.5.5.m5.2c">l_{s}(u)\neq l_{m}(u)</annotation><annotation encoding="application/x-llamapun" id="S7.Thmtheorem15.p2.5.5.m5.2d">italic_l start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_u ) ≠ italic_l start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT ( italic_u )</annotation></semantics></math> (else, <math alttext="E_{s}" class="ltx_Math" display="inline" id="S7.Thmtheorem15.p2.6.6.m6.1"><semantics id="S7.Thmtheorem15.p2.6.6.m6.1a"><msub id="S7.Thmtheorem15.p2.6.6.m6.1.1" xref="S7.Thmtheorem15.p2.6.6.m6.1.1.cmml"><mi id="S7.Thmtheorem15.p2.6.6.m6.1.1.2" xref="S7.Thmtheorem15.p2.6.6.m6.1.1.2.cmml">E</mi><mi id="S7.Thmtheorem15.p2.6.6.m6.1.1.3" xref="S7.Thmtheorem15.p2.6.6.m6.1.1.3.cmml">s</mi></msub><annotation-xml encoding="MathML-Content" id="S7.Thmtheorem15.p2.6.6.m6.1b"><apply id="S7.Thmtheorem15.p2.6.6.m6.1.1.cmml" xref="S7.Thmtheorem15.p2.6.6.m6.1.1"><csymbol cd="ambiguous" id="S7.Thmtheorem15.p2.6.6.m6.1.1.1.cmml" xref="S7.Thmtheorem15.p2.6.6.m6.1.1">subscript</csymbol><ci id="S7.Thmtheorem15.p2.6.6.m6.1.1.2.cmml" xref="S7.Thmtheorem15.p2.6.6.m6.1.1.2">𝐸</ci><ci id="S7.Thmtheorem15.p2.6.6.m6.1.1.3.cmml" xref="S7.Thmtheorem15.p2.6.6.m6.1.1.3">𝑠</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.Thmtheorem15.p2.6.6.m6.1c">E_{s}</annotation><annotation encoding="application/x-llamapun" id="S7.Thmtheorem15.p2.6.6.m6.1d">italic_E start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT</annotation></semantics></math> must include the edge <math alttext="\overline{uv}" class="ltx_Math" display="inline" id="S7.Thmtheorem15.p2.7.7.m7.1"><semantics id="S7.Thmtheorem15.p2.7.7.m7.1a"><mover accent="true" id="S7.Thmtheorem15.p2.7.7.m7.1.1" xref="S7.Thmtheorem15.p2.7.7.m7.1.1.cmml"><mrow id="S7.Thmtheorem15.p2.7.7.m7.1.1.2" xref="S7.Thmtheorem15.p2.7.7.m7.1.1.2.cmml"><mi id="S7.Thmtheorem15.p2.7.7.m7.1.1.2.2" xref="S7.Thmtheorem15.p2.7.7.m7.1.1.2.2.cmml">u</mi><mo id="S7.Thmtheorem15.p2.7.7.m7.1.1.2.1" xref="S7.Thmtheorem15.p2.7.7.m7.1.1.2.1.cmml">⁢</mo><mi id="S7.Thmtheorem15.p2.7.7.m7.1.1.2.3" xref="S7.Thmtheorem15.p2.7.7.m7.1.1.2.3.cmml">v</mi></mrow><mo id="S7.Thmtheorem15.p2.7.7.m7.1.1.1" xref="S7.Thmtheorem15.p2.7.7.m7.1.1.1.cmml">¯</mo></mover><annotation-xml encoding="MathML-Content" id="S7.Thmtheorem15.p2.7.7.m7.1b"><apply id="S7.Thmtheorem15.p2.7.7.m7.1.1.cmml" xref="S7.Thmtheorem15.p2.7.7.m7.1.1"><ci id="S7.Thmtheorem15.p2.7.7.m7.1.1.1.cmml" xref="S7.Thmtheorem15.p2.7.7.m7.1.1.1">¯</ci><apply id="S7.Thmtheorem15.p2.7.7.m7.1.1.2.cmml" xref="S7.Thmtheorem15.p2.7.7.m7.1.1.2"><times id="S7.Thmtheorem15.p2.7.7.m7.1.1.2.1.cmml" xref="S7.Thmtheorem15.p2.7.7.m7.1.1.2.1"></times><ci id="S7.Thmtheorem15.p2.7.7.m7.1.1.2.2.cmml" xref="S7.Thmtheorem15.p2.7.7.m7.1.1.2.2">𝑢</ci><ci id="S7.Thmtheorem15.p2.7.7.m7.1.1.2.3.cmml" xref="S7.Thmtheorem15.p2.7.7.m7.1.1.2.3">𝑣</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.Thmtheorem15.p2.7.7.m7.1c">\overline{uv}</annotation><annotation encoding="application/x-llamapun" id="S7.Thmtheorem15.p2.7.7.m7.1d">over¯ start_ARG italic_u italic_v end_ARG</annotation></semantics></math>). Thus, <math alttext="l_{m}(u)=l_{s}(u)+1" class="ltx_Math" display="inline" id="S7.Thmtheorem15.p2.8.8.m8.2"><semantics id="S7.Thmtheorem15.p2.8.8.m8.2a"><mrow id="S7.Thmtheorem15.p2.8.8.m8.2.3" xref="S7.Thmtheorem15.p2.8.8.m8.2.3.cmml"><mrow id="S7.Thmtheorem15.p2.8.8.m8.2.3.2" xref="S7.Thmtheorem15.p2.8.8.m8.2.3.2.cmml"><msub id="S7.Thmtheorem15.p2.8.8.m8.2.3.2.2" xref="S7.Thmtheorem15.p2.8.8.m8.2.3.2.2.cmml"><mi id="S7.Thmtheorem15.p2.8.8.m8.2.3.2.2.2" xref="S7.Thmtheorem15.p2.8.8.m8.2.3.2.2.2.cmml">l</mi><mi id="S7.Thmtheorem15.p2.8.8.m8.2.3.2.2.3" xref="S7.Thmtheorem15.p2.8.8.m8.2.3.2.2.3.cmml">m</mi></msub><mo id="S7.Thmtheorem15.p2.8.8.m8.2.3.2.1" xref="S7.Thmtheorem15.p2.8.8.m8.2.3.2.1.cmml">⁢</mo><mrow id="S7.Thmtheorem15.p2.8.8.m8.2.3.2.3.2" xref="S7.Thmtheorem15.p2.8.8.m8.2.3.2.cmml"><mo id="S7.Thmtheorem15.p2.8.8.m8.2.3.2.3.2.1" stretchy="false" xref="S7.Thmtheorem15.p2.8.8.m8.2.3.2.cmml">(</mo><mi id="S7.Thmtheorem15.p2.8.8.m8.1.1" xref="S7.Thmtheorem15.p2.8.8.m8.1.1.cmml">u</mi><mo id="S7.Thmtheorem15.p2.8.8.m8.2.3.2.3.2.2" stretchy="false" xref="S7.Thmtheorem15.p2.8.8.m8.2.3.2.cmml">)</mo></mrow></mrow><mo id="S7.Thmtheorem15.p2.8.8.m8.2.3.1" xref="S7.Thmtheorem15.p2.8.8.m8.2.3.1.cmml">=</mo><mrow id="S7.Thmtheorem15.p2.8.8.m8.2.3.3" xref="S7.Thmtheorem15.p2.8.8.m8.2.3.3.cmml"><mrow id="S7.Thmtheorem15.p2.8.8.m8.2.3.3.2" xref="S7.Thmtheorem15.p2.8.8.m8.2.3.3.2.cmml"><msub id="S7.Thmtheorem15.p2.8.8.m8.2.3.3.2.2" xref="S7.Thmtheorem15.p2.8.8.m8.2.3.3.2.2.cmml"><mi id="S7.Thmtheorem15.p2.8.8.m8.2.3.3.2.2.2" xref="S7.Thmtheorem15.p2.8.8.m8.2.3.3.2.2.2.cmml">l</mi><mi id="S7.Thmtheorem15.p2.8.8.m8.2.3.3.2.2.3" xref="S7.Thmtheorem15.p2.8.8.m8.2.3.3.2.2.3.cmml">s</mi></msub><mo id="S7.Thmtheorem15.p2.8.8.m8.2.3.3.2.1" xref="S7.Thmtheorem15.p2.8.8.m8.2.3.3.2.1.cmml">⁢</mo><mrow id="S7.Thmtheorem15.p2.8.8.m8.2.3.3.2.3.2" xref="S7.Thmtheorem15.p2.8.8.m8.2.3.3.2.cmml"><mo id="S7.Thmtheorem15.p2.8.8.m8.2.3.3.2.3.2.1" stretchy="false" xref="S7.Thmtheorem15.p2.8.8.m8.2.3.3.2.cmml">(</mo><mi id="S7.Thmtheorem15.p2.8.8.m8.2.2" xref="S7.Thmtheorem15.p2.8.8.m8.2.2.cmml">u</mi><mo id="S7.Thmtheorem15.p2.8.8.m8.2.3.3.2.3.2.2" stretchy="false" xref="S7.Thmtheorem15.p2.8.8.m8.2.3.3.2.cmml">)</mo></mrow></mrow><mo id="S7.Thmtheorem15.p2.8.8.m8.2.3.3.1" xref="S7.Thmtheorem15.p2.8.8.m8.2.3.3.1.cmml">+</mo><mn id="S7.Thmtheorem15.p2.8.8.m8.2.3.3.3" xref="S7.Thmtheorem15.p2.8.8.m8.2.3.3.3.cmml">1</mn></mrow></mrow><annotation-xml encoding="MathML-Content" id="S7.Thmtheorem15.p2.8.8.m8.2b"><apply id="S7.Thmtheorem15.p2.8.8.m8.2.3.cmml" xref="S7.Thmtheorem15.p2.8.8.m8.2.3"><eq id="S7.Thmtheorem15.p2.8.8.m8.2.3.1.cmml" xref="S7.Thmtheorem15.p2.8.8.m8.2.3.1"></eq><apply id="S7.Thmtheorem15.p2.8.8.m8.2.3.2.cmml" xref="S7.Thmtheorem15.p2.8.8.m8.2.3.2"><times id="S7.Thmtheorem15.p2.8.8.m8.2.3.2.1.cmml" xref="S7.Thmtheorem15.p2.8.8.m8.2.3.2.1"></times><apply id="S7.Thmtheorem15.p2.8.8.m8.2.3.2.2.cmml" xref="S7.Thmtheorem15.p2.8.8.m8.2.3.2.2"><csymbol cd="ambiguous" id="S7.Thmtheorem15.p2.8.8.m8.2.3.2.2.1.cmml" xref="S7.Thmtheorem15.p2.8.8.m8.2.3.2.2">subscript</csymbol><ci id="S7.Thmtheorem15.p2.8.8.m8.2.3.2.2.2.cmml" xref="S7.Thmtheorem15.p2.8.8.m8.2.3.2.2.2">𝑙</ci><ci id="S7.Thmtheorem15.p2.8.8.m8.2.3.2.2.3.cmml" xref="S7.Thmtheorem15.p2.8.8.m8.2.3.2.2.3">𝑚</ci></apply><ci id="S7.Thmtheorem15.p2.8.8.m8.1.1.cmml" xref="S7.Thmtheorem15.p2.8.8.m8.1.1">𝑢</ci></apply><apply id="S7.Thmtheorem15.p2.8.8.m8.2.3.3.cmml" xref="S7.Thmtheorem15.p2.8.8.m8.2.3.3"><plus id="S7.Thmtheorem15.p2.8.8.m8.2.3.3.1.cmml" xref="S7.Thmtheorem15.p2.8.8.m8.2.3.3.1"></plus><apply id="S7.Thmtheorem15.p2.8.8.m8.2.3.3.2.cmml" xref="S7.Thmtheorem15.p2.8.8.m8.2.3.3.2"><times id="S7.Thmtheorem15.p2.8.8.m8.2.3.3.2.1.cmml" xref="S7.Thmtheorem15.p2.8.8.m8.2.3.3.2.1"></times><apply id="S7.Thmtheorem15.p2.8.8.m8.2.3.3.2.2.cmml" xref="S7.Thmtheorem15.p2.8.8.m8.2.3.3.2.2"><csymbol cd="ambiguous" id="S7.Thmtheorem15.p2.8.8.m8.2.3.3.2.2.1.cmml" xref="S7.Thmtheorem15.p2.8.8.m8.2.3.3.2.2">subscript</csymbol><ci id="S7.Thmtheorem15.p2.8.8.m8.2.3.3.2.2.2.cmml" xref="S7.Thmtheorem15.p2.8.8.m8.2.3.3.2.2.2">𝑙</ci><ci id="S7.Thmtheorem15.p2.8.8.m8.2.3.3.2.2.3.cmml" xref="S7.Thmtheorem15.p2.8.8.m8.2.3.3.2.2.3">𝑠</ci></apply><ci id="S7.Thmtheorem15.p2.8.8.m8.2.2.cmml" xref="S7.Thmtheorem15.p2.8.8.m8.2.2">𝑢</ci></apply><cn id="S7.Thmtheorem15.p2.8.8.m8.2.3.3.3.cmml" type="integer" xref="S7.Thmtheorem15.p2.8.8.m8.2.3.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.Thmtheorem15.p2.8.8.m8.2c">l_{m}(u)=l_{s}(u)+1</annotation><annotation encoding="application/x-llamapun" id="S7.Thmtheorem15.p2.8.8.m8.2d">italic_l start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT ( italic_u ) = italic_l start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_u ) + 1</annotation></semantics></math>.</span></p> </div> <div class="ltx_para" id="S7.Thmtheorem15.p3"> <p class="ltx_p" id="S7.Thmtheorem15.p3.6"><span class="ltx_text ltx_font_italic" id="S7.Thmtheorem15.p3.6.6">However, we now get that <math alttext="l_{m}(u)=l_{s}(u)+1=l_{s}(v)+2=l_{m}(v)+2" class="ltx_Math" display="inline" id="S7.Thmtheorem15.p3.1.1.m1.4"><semantics id="S7.Thmtheorem15.p3.1.1.m1.4a"><mrow id="S7.Thmtheorem15.p3.1.1.m1.4.5" xref="S7.Thmtheorem15.p3.1.1.m1.4.5.cmml"><mrow id="S7.Thmtheorem15.p3.1.1.m1.4.5.2" xref="S7.Thmtheorem15.p3.1.1.m1.4.5.2.cmml"><msub id="S7.Thmtheorem15.p3.1.1.m1.4.5.2.2" xref="S7.Thmtheorem15.p3.1.1.m1.4.5.2.2.cmml"><mi id="S7.Thmtheorem15.p3.1.1.m1.4.5.2.2.2" xref="S7.Thmtheorem15.p3.1.1.m1.4.5.2.2.2.cmml">l</mi><mi id="S7.Thmtheorem15.p3.1.1.m1.4.5.2.2.3" xref="S7.Thmtheorem15.p3.1.1.m1.4.5.2.2.3.cmml">m</mi></msub><mo id="S7.Thmtheorem15.p3.1.1.m1.4.5.2.1" xref="S7.Thmtheorem15.p3.1.1.m1.4.5.2.1.cmml">⁢</mo><mrow id="S7.Thmtheorem15.p3.1.1.m1.4.5.2.3.2" xref="S7.Thmtheorem15.p3.1.1.m1.4.5.2.cmml"><mo id="S7.Thmtheorem15.p3.1.1.m1.4.5.2.3.2.1" stretchy="false" xref="S7.Thmtheorem15.p3.1.1.m1.4.5.2.cmml">(</mo><mi id="S7.Thmtheorem15.p3.1.1.m1.1.1" xref="S7.Thmtheorem15.p3.1.1.m1.1.1.cmml">u</mi><mo id="S7.Thmtheorem15.p3.1.1.m1.4.5.2.3.2.2" stretchy="false" xref="S7.Thmtheorem15.p3.1.1.m1.4.5.2.cmml">)</mo></mrow></mrow><mo id="S7.Thmtheorem15.p3.1.1.m1.4.5.3" xref="S7.Thmtheorem15.p3.1.1.m1.4.5.3.cmml">=</mo><mrow id="S7.Thmtheorem15.p3.1.1.m1.4.5.4" xref="S7.Thmtheorem15.p3.1.1.m1.4.5.4.cmml"><mrow id="S7.Thmtheorem15.p3.1.1.m1.4.5.4.2" xref="S7.Thmtheorem15.p3.1.1.m1.4.5.4.2.cmml"><msub id="S7.Thmtheorem15.p3.1.1.m1.4.5.4.2.2" xref="S7.Thmtheorem15.p3.1.1.m1.4.5.4.2.2.cmml"><mi id="S7.Thmtheorem15.p3.1.1.m1.4.5.4.2.2.2" xref="S7.Thmtheorem15.p3.1.1.m1.4.5.4.2.2.2.cmml">l</mi><mi id="S7.Thmtheorem15.p3.1.1.m1.4.5.4.2.2.3" xref="S7.Thmtheorem15.p3.1.1.m1.4.5.4.2.2.3.cmml">s</mi></msub><mo id="S7.Thmtheorem15.p3.1.1.m1.4.5.4.2.1" xref="S7.Thmtheorem15.p3.1.1.m1.4.5.4.2.1.cmml">⁢</mo><mrow id="S7.Thmtheorem15.p3.1.1.m1.4.5.4.2.3.2" xref="S7.Thmtheorem15.p3.1.1.m1.4.5.4.2.cmml"><mo id="S7.Thmtheorem15.p3.1.1.m1.4.5.4.2.3.2.1" stretchy="false" xref="S7.Thmtheorem15.p3.1.1.m1.4.5.4.2.cmml">(</mo><mi id="S7.Thmtheorem15.p3.1.1.m1.2.2" xref="S7.Thmtheorem15.p3.1.1.m1.2.2.cmml">u</mi><mo id="S7.Thmtheorem15.p3.1.1.m1.4.5.4.2.3.2.2" stretchy="false" xref="S7.Thmtheorem15.p3.1.1.m1.4.5.4.2.cmml">)</mo></mrow></mrow><mo id="S7.Thmtheorem15.p3.1.1.m1.4.5.4.1" xref="S7.Thmtheorem15.p3.1.1.m1.4.5.4.1.cmml">+</mo><mn id="S7.Thmtheorem15.p3.1.1.m1.4.5.4.3" xref="S7.Thmtheorem15.p3.1.1.m1.4.5.4.3.cmml">1</mn></mrow><mo id="S7.Thmtheorem15.p3.1.1.m1.4.5.5" xref="S7.Thmtheorem15.p3.1.1.m1.4.5.5.cmml">=</mo><mrow id="S7.Thmtheorem15.p3.1.1.m1.4.5.6" xref="S7.Thmtheorem15.p3.1.1.m1.4.5.6.cmml"><mrow id="S7.Thmtheorem15.p3.1.1.m1.4.5.6.2" xref="S7.Thmtheorem15.p3.1.1.m1.4.5.6.2.cmml"><msub id="S7.Thmtheorem15.p3.1.1.m1.4.5.6.2.2" xref="S7.Thmtheorem15.p3.1.1.m1.4.5.6.2.2.cmml"><mi id="S7.Thmtheorem15.p3.1.1.m1.4.5.6.2.2.2" xref="S7.Thmtheorem15.p3.1.1.m1.4.5.6.2.2.2.cmml">l</mi><mi id="S7.Thmtheorem15.p3.1.1.m1.4.5.6.2.2.3" xref="S7.Thmtheorem15.p3.1.1.m1.4.5.6.2.2.3.cmml">s</mi></msub><mo id="S7.Thmtheorem15.p3.1.1.m1.4.5.6.2.1" xref="S7.Thmtheorem15.p3.1.1.m1.4.5.6.2.1.cmml">⁢</mo><mrow id="S7.Thmtheorem15.p3.1.1.m1.4.5.6.2.3.2" xref="S7.Thmtheorem15.p3.1.1.m1.4.5.6.2.cmml"><mo id="S7.Thmtheorem15.p3.1.1.m1.4.5.6.2.3.2.1" stretchy="false" xref="S7.Thmtheorem15.p3.1.1.m1.4.5.6.2.cmml">(</mo><mi id="S7.Thmtheorem15.p3.1.1.m1.3.3" xref="S7.Thmtheorem15.p3.1.1.m1.3.3.cmml">v</mi><mo id="S7.Thmtheorem15.p3.1.1.m1.4.5.6.2.3.2.2" stretchy="false" xref="S7.Thmtheorem15.p3.1.1.m1.4.5.6.2.cmml">)</mo></mrow></mrow><mo id="S7.Thmtheorem15.p3.1.1.m1.4.5.6.1" xref="S7.Thmtheorem15.p3.1.1.m1.4.5.6.1.cmml">+</mo><mn id="S7.Thmtheorem15.p3.1.1.m1.4.5.6.3" xref="S7.Thmtheorem15.p3.1.1.m1.4.5.6.3.cmml">2</mn></mrow><mo id="S7.Thmtheorem15.p3.1.1.m1.4.5.7" xref="S7.Thmtheorem15.p3.1.1.m1.4.5.7.cmml">=</mo><mrow id="S7.Thmtheorem15.p3.1.1.m1.4.5.8" xref="S7.Thmtheorem15.p3.1.1.m1.4.5.8.cmml"><mrow id="S7.Thmtheorem15.p3.1.1.m1.4.5.8.2" xref="S7.Thmtheorem15.p3.1.1.m1.4.5.8.2.cmml"><msub id="S7.Thmtheorem15.p3.1.1.m1.4.5.8.2.2" xref="S7.Thmtheorem15.p3.1.1.m1.4.5.8.2.2.cmml"><mi id="S7.Thmtheorem15.p3.1.1.m1.4.5.8.2.2.2" xref="S7.Thmtheorem15.p3.1.1.m1.4.5.8.2.2.2.cmml">l</mi><mi id="S7.Thmtheorem15.p3.1.1.m1.4.5.8.2.2.3" xref="S7.Thmtheorem15.p3.1.1.m1.4.5.8.2.2.3.cmml">m</mi></msub><mo id="S7.Thmtheorem15.p3.1.1.m1.4.5.8.2.1" xref="S7.Thmtheorem15.p3.1.1.m1.4.5.8.2.1.cmml">⁢</mo><mrow id="S7.Thmtheorem15.p3.1.1.m1.4.5.8.2.3.2" xref="S7.Thmtheorem15.p3.1.1.m1.4.5.8.2.cmml"><mo id="S7.Thmtheorem15.p3.1.1.m1.4.5.8.2.3.2.1" stretchy="false" xref="S7.Thmtheorem15.p3.1.1.m1.4.5.8.2.cmml">(</mo><mi id="S7.Thmtheorem15.p3.1.1.m1.4.4" xref="S7.Thmtheorem15.p3.1.1.m1.4.4.cmml">v</mi><mo id="S7.Thmtheorem15.p3.1.1.m1.4.5.8.2.3.2.2" stretchy="false" xref="S7.Thmtheorem15.p3.1.1.m1.4.5.8.2.cmml">)</mo></mrow></mrow><mo id="S7.Thmtheorem15.p3.1.1.m1.4.5.8.1" xref="S7.Thmtheorem15.p3.1.1.m1.4.5.8.1.cmml">+</mo><mn id="S7.Thmtheorem15.p3.1.1.m1.4.5.8.3" xref="S7.Thmtheorem15.p3.1.1.m1.4.5.8.3.cmml">2</mn></mrow></mrow><annotation-xml encoding="MathML-Content" id="S7.Thmtheorem15.p3.1.1.m1.4b"><apply id="S7.Thmtheorem15.p3.1.1.m1.4.5.cmml" xref="S7.Thmtheorem15.p3.1.1.m1.4.5"><and id="S7.Thmtheorem15.p3.1.1.m1.4.5a.cmml" xref="S7.Thmtheorem15.p3.1.1.m1.4.5"></and><apply id="S7.Thmtheorem15.p3.1.1.m1.4.5b.cmml" xref="S7.Thmtheorem15.p3.1.1.m1.4.5"><eq id="S7.Thmtheorem15.p3.1.1.m1.4.5.3.cmml" xref="S7.Thmtheorem15.p3.1.1.m1.4.5.3"></eq><apply id="S7.Thmtheorem15.p3.1.1.m1.4.5.2.cmml" xref="S7.Thmtheorem15.p3.1.1.m1.4.5.2"><times id="S7.Thmtheorem15.p3.1.1.m1.4.5.2.1.cmml" xref="S7.Thmtheorem15.p3.1.1.m1.4.5.2.1"></times><apply id="S7.Thmtheorem15.p3.1.1.m1.4.5.2.2.cmml" xref="S7.Thmtheorem15.p3.1.1.m1.4.5.2.2"><csymbol cd="ambiguous" id="S7.Thmtheorem15.p3.1.1.m1.4.5.2.2.1.cmml" xref="S7.Thmtheorem15.p3.1.1.m1.4.5.2.2">subscript</csymbol><ci id="S7.Thmtheorem15.p3.1.1.m1.4.5.2.2.2.cmml" xref="S7.Thmtheorem15.p3.1.1.m1.4.5.2.2.2">𝑙</ci><ci id="S7.Thmtheorem15.p3.1.1.m1.4.5.2.2.3.cmml" xref="S7.Thmtheorem15.p3.1.1.m1.4.5.2.2.3">𝑚</ci></apply><ci id="S7.Thmtheorem15.p3.1.1.m1.1.1.cmml" xref="S7.Thmtheorem15.p3.1.1.m1.1.1">𝑢</ci></apply><apply id="S7.Thmtheorem15.p3.1.1.m1.4.5.4.cmml" xref="S7.Thmtheorem15.p3.1.1.m1.4.5.4"><plus id="S7.Thmtheorem15.p3.1.1.m1.4.5.4.1.cmml" xref="S7.Thmtheorem15.p3.1.1.m1.4.5.4.1"></plus><apply id="S7.Thmtheorem15.p3.1.1.m1.4.5.4.2.cmml" xref="S7.Thmtheorem15.p3.1.1.m1.4.5.4.2"><times id="S7.Thmtheorem15.p3.1.1.m1.4.5.4.2.1.cmml" xref="S7.Thmtheorem15.p3.1.1.m1.4.5.4.2.1"></times><apply id="S7.Thmtheorem15.p3.1.1.m1.4.5.4.2.2.cmml" xref="S7.Thmtheorem15.p3.1.1.m1.4.5.4.2.2"><csymbol cd="ambiguous" id="S7.Thmtheorem15.p3.1.1.m1.4.5.4.2.2.1.cmml" xref="S7.Thmtheorem15.p3.1.1.m1.4.5.4.2.2">subscript</csymbol><ci id="S7.Thmtheorem15.p3.1.1.m1.4.5.4.2.2.2.cmml" xref="S7.Thmtheorem15.p3.1.1.m1.4.5.4.2.2.2">𝑙</ci><ci id="S7.Thmtheorem15.p3.1.1.m1.4.5.4.2.2.3.cmml" xref="S7.Thmtheorem15.p3.1.1.m1.4.5.4.2.2.3">𝑠</ci></apply><ci id="S7.Thmtheorem15.p3.1.1.m1.2.2.cmml" xref="S7.Thmtheorem15.p3.1.1.m1.2.2">𝑢</ci></apply><cn id="S7.Thmtheorem15.p3.1.1.m1.4.5.4.3.cmml" type="integer" xref="S7.Thmtheorem15.p3.1.1.m1.4.5.4.3">1</cn></apply></apply><apply id="S7.Thmtheorem15.p3.1.1.m1.4.5c.cmml" xref="S7.Thmtheorem15.p3.1.1.m1.4.5"><eq id="S7.Thmtheorem15.p3.1.1.m1.4.5.5.cmml" xref="S7.Thmtheorem15.p3.1.1.m1.4.5.5"></eq><share href="https://arxiv.org/html/2411.12694v2#S7.Thmtheorem15.p3.1.1.m1.4.5.4.cmml" id="S7.Thmtheorem15.p3.1.1.m1.4.5d.cmml" xref="S7.Thmtheorem15.p3.1.1.m1.4.5"></share><apply id="S7.Thmtheorem15.p3.1.1.m1.4.5.6.cmml" xref="S7.Thmtheorem15.p3.1.1.m1.4.5.6"><plus id="S7.Thmtheorem15.p3.1.1.m1.4.5.6.1.cmml" xref="S7.Thmtheorem15.p3.1.1.m1.4.5.6.1"></plus><apply id="S7.Thmtheorem15.p3.1.1.m1.4.5.6.2.cmml" xref="S7.Thmtheorem15.p3.1.1.m1.4.5.6.2"><times id="S7.Thmtheorem15.p3.1.1.m1.4.5.6.2.1.cmml" xref="S7.Thmtheorem15.p3.1.1.m1.4.5.6.2.1"></times><apply id="S7.Thmtheorem15.p3.1.1.m1.4.5.6.2.2.cmml" xref="S7.Thmtheorem15.p3.1.1.m1.4.5.6.2.2"><csymbol cd="ambiguous" id="S7.Thmtheorem15.p3.1.1.m1.4.5.6.2.2.1.cmml" xref="S7.Thmtheorem15.p3.1.1.m1.4.5.6.2.2">subscript</csymbol><ci id="S7.Thmtheorem15.p3.1.1.m1.4.5.6.2.2.2.cmml" xref="S7.Thmtheorem15.p3.1.1.m1.4.5.6.2.2.2">𝑙</ci><ci id="S7.Thmtheorem15.p3.1.1.m1.4.5.6.2.2.3.cmml" xref="S7.Thmtheorem15.p3.1.1.m1.4.5.6.2.2.3">𝑠</ci></apply><ci id="S7.Thmtheorem15.p3.1.1.m1.3.3.cmml" xref="S7.Thmtheorem15.p3.1.1.m1.3.3">𝑣</ci></apply><cn id="S7.Thmtheorem15.p3.1.1.m1.4.5.6.3.cmml" type="integer" xref="S7.Thmtheorem15.p3.1.1.m1.4.5.6.3">2</cn></apply></apply><apply id="S7.Thmtheorem15.p3.1.1.m1.4.5e.cmml" xref="S7.Thmtheorem15.p3.1.1.m1.4.5"><eq id="S7.Thmtheorem15.p3.1.1.m1.4.5.7.cmml" xref="S7.Thmtheorem15.p3.1.1.m1.4.5.7"></eq><share href="https://arxiv.org/html/2411.12694v2#S7.Thmtheorem15.p3.1.1.m1.4.5.6.cmml" id="S7.Thmtheorem15.p3.1.1.m1.4.5f.cmml" xref="S7.Thmtheorem15.p3.1.1.m1.4.5"></share><apply id="S7.Thmtheorem15.p3.1.1.m1.4.5.8.cmml" xref="S7.Thmtheorem15.p3.1.1.m1.4.5.8"><plus id="S7.Thmtheorem15.p3.1.1.m1.4.5.8.1.cmml" xref="S7.Thmtheorem15.p3.1.1.m1.4.5.8.1"></plus><apply id="S7.Thmtheorem15.p3.1.1.m1.4.5.8.2.cmml" xref="S7.Thmtheorem15.p3.1.1.m1.4.5.8.2"><times id="S7.Thmtheorem15.p3.1.1.m1.4.5.8.2.1.cmml" xref="S7.Thmtheorem15.p3.1.1.m1.4.5.8.2.1"></times><apply id="S7.Thmtheorem15.p3.1.1.m1.4.5.8.2.2.cmml" xref="S7.Thmtheorem15.p3.1.1.m1.4.5.8.2.2"><csymbol cd="ambiguous" id="S7.Thmtheorem15.p3.1.1.m1.4.5.8.2.2.1.cmml" xref="S7.Thmtheorem15.p3.1.1.m1.4.5.8.2.2">subscript</csymbol><ci id="S7.Thmtheorem15.p3.1.1.m1.4.5.8.2.2.2.cmml" xref="S7.Thmtheorem15.p3.1.1.m1.4.5.8.2.2.2">𝑙</ci><ci id="S7.Thmtheorem15.p3.1.1.m1.4.5.8.2.2.3.cmml" xref="S7.Thmtheorem15.p3.1.1.m1.4.5.8.2.2.3">𝑚</ci></apply><ci id="S7.Thmtheorem15.p3.1.1.m1.4.4.cmml" xref="S7.Thmtheorem15.p3.1.1.m1.4.4">𝑣</ci></apply><cn id="S7.Thmtheorem15.p3.1.1.m1.4.5.8.3.cmml" type="integer" xref="S7.Thmtheorem15.p3.1.1.m1.4.5.8.3">2</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.Thmtheorem15.p3.1.1.m1.4c">l_{m}(u)=l_{s}(u)+1=l_{s}(v)+2=l_{m}(v)+2</annotation><annotation encoding="application/x-llamapun" id="S7.Thmtheorem15.p3.1.1.m1.4d">italic_l start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT ( italic_u ) = italic_l start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_u ) + 1 = italic_l start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_v ) + 2 = italic_l start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT ( italic_v ) + 2</annotation></semantics></math> and <math alttext="\textsl{g}_{0}(u\!\to\!v)>0" class="ltx_Math" display="inline" id="S7.Thmtheorem15.p3.2.2.m2.1"><semantics id="S7.Thmtheorem15.p3.2.2.m2.1a"><mrow id="S7.Thmtheorem15.p3.2.2.m2.1.1" xref="S7.Thmtheorem15.p3.2.2.m2.1.1.cmml"><mrow id="S7.Thmtheorem15.p3.2.2.m2.1.1.1" xref="S7.Thmtheorem15.p3.2.2.m2.1.1.1.cmml"><msub id="S7.Thmtheorem15.p3.2.2.m2.1.1.1.3" xref="S7.Thmtheorem15.p3.2.2.m2.1.1.1.3.cmml"><mtext class="ltx_mathvariant_italic" id="S7.Thmtheorem15.p3.2.2.m2.1.1.1.3.2" xref="S7.Thmtheorem15.p3.2.2.m2.1.1.1.3.2a.cmml">g</mtext><mn id="S7.Thmtheorem15.p3.2.2.m2.1.1.1.3.3" xref="S7.Thmtheorem15.p3.2.2.m2.1.1.1.3.3.cmml">0</mn></msub><mo id="S7.Thmtheorem15.p3.2.2.m2.1.1.1.2" xref="S7.Thmtheorem15.p3.2.2.m2.1.1.1.2.cmml">⁢</mo><mrow id="S7.Thmtheorem15.p3.2.2.m2.1.1.1.1.1" xref="S7.Thmtheorem15.p3.2.2.m2.1.1.1.1.1.1.cmml"><mo id="S7.Thmtheorem15.p3.2.2.m2.1.1.1.1.1.2" stretchy="false" xref="S7.Thmtheorem15.p3.2.2.m2.1.1.1.1.1.1.cmml">(</mo><mrow id="S7.Thmtheorem15.p3.2.2.m2.1.1.1.1.1.1" xref="S7.Thmtheorem15.p3.2.2.m2.1.1.1.1.1.1.cmml"><mi id="S7.Thmtheorem15.p3.2.2.m2.1.1.1.1.1.1.2" xref="S7.Thmtheorem15.p3.2.2.m2.1.1.1.1.1.1.2.cmml">u</mi><mo id="S7.Thmtheorem15.p3.2.2.m2.1.1.1.1.1.1.1" lspace="0.108em" rspace="0.108em" stretchy="false" xref="S7.Thmtheorem15.p3.2.2.m2.1.1.1.1.1.1.1.cmml">→</mo><mi id="S7.Thmtheorem15.p3.2.2.m2.1.1.1.1.1.1.3" xref="S7.Thmtheorem15.p3.2.2.m2.1.1.1.1.1.1.3.cmml">v</mi></mrow><mo id="S7.Thmtheorem15.p3.2.2.m2.1.1.1.1.1.3" stretchy="false" xref="S7.Thmtheorem15.p3.2.2.m2.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S7.Thmtheorem15.p3.2.2.m2.1.1.2" xref="S7.Thmtheorem15.p3.2.2.m2.1.1.2.cmml">></mo><mn id="S7.Thmtheorem15.p3.2.2.m2.1.1.3" xref="S7.Thmtheorem15.p3.2.2.m2.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S7.Thmtheorem15.p3.2.2.m2.1b"><apply id="S7.Thmtheorem15.p3.2.2.m2.1.1.cmml" xref="S7.Thmtheorem15.p3.2.2.m2.1.1"><gt id="S7.Thmtheorem15.p3.2.2.m2.1.1.2.cmml" xref="S7.Thmtheorem15.p3.2.2.m2.1.1.2"></gt><apply id="S7.Thmtheorem15.p3.2.2.m2.1.1.1.cmml" xref="S7.Thmtheorem15.p3.2.2.m2.1.1.1"><times id="S7.Thmtheorem15.p3.2.2.m2.1.1.1.2.cmml" xref="S7.Thmtheorem15.p3.2.2.m2.1.1.1.2"></times><apply id="S7.Thmtheorem15.p3.2.2.m2.1.1.1.3.cmml" xref="S7.Thmtheorem15.p3.2.2.m2.1.1.1.3"><csymbol cd="ambiguous" id="S7.Thmtheorem15.p3.2.2.m2.1.1.1.3.1.cmml" xref="S7.Thmtheorem15.p3.2.2.m2.1.1.1.3">subscript</csymbol><ci id="S7.Thmtheorem15.p3.2.2.m2.1.1.1.3.2a.cmml" xref="S7.Thmtheorem15.p3.2.2.m2.1.1.1.3.2"><mtext class="ltx_mathvariant_italic" id="S7.Thmtheorem15.p3.2.2.m2.1.1.1.3.2.cmml" xref="S7.Thmtheorem15.p3.2.2.m2.1.1.1.3.2">g</mtext></ci><cn id="S7.Thmtheorem15.p3.2.2.m2.1.1.1.3.3.cmml" type="integer" xref="S7.Thmtheorem15.p3.2.2.m2.1.1.1.3.3">0</cn></apply><apply id="S7.Thmtheorem15.p3.2.2.m2.1.1.1.1.1.1.cmml" xref="S7.Thmtheorem15.p3.2.2.m2.1.1.1.1.1"><ci id="S7.Thmtheorem15.p3.2.2.m2.1.1.1.1.1.1.1.cmml" xref="S7.Thmtheorem15.p3.2.2.m2.1.1.1.1.1.1.1">→</ci><ci id="S7.Thmtheorem15.p3.2.2.m2.1.1.1.1.1.1.2.cmml" xref="S7.Thmtheorem15.p3.2.2.m2.1.1.1.1.1.1.2">𝑢</ci><ci id="S7.Thmtheorem15.p3.2.2.m2.1.1.1.1.1.1.3.cmml" xref="S7.Thmtheorem15.p3.2.2.m2.1.1.1.1.1.1.3">𝑣</ci></apply></apply><cn id="S7.Thmtheorem15.p3.2.2.m2.1.1.3.cmml" type="integer" xref="S7.Thmtheorem15.p3.2.2.m2.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.Thmtheorem15.p3.2.2.m2.1c">\textsl{g}_{0}(u\!\to\!v)>0</annotation><annotation encoding="application/x-llamapun" id="S7.Thmtheorem15.p3.2.2.m2.1d">g start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ( italic_u → italic_v ) > 0</annotation></semantics></math>. This contradicts that <math alttext="s" class="ltx_Math" display="inline" id="S7.Thmtheorem15.p3.3.3.m3.1"><semantics id="S7.Thmtheorem15.p3.3.3.m3.1a"><mi id="S7.Thmtheorem15.p3.3.3.m3.1.1" xref="S7.Thmtheorem15.p3.3.3.m3.1.1.cmml">s</mi><annotation-xml encoding="MathML-Content" id="S7.Thmtheorem15.p3.3.3.m3.1b"><ci id="S7.Thmtheorem15.p3.3.3.m3.1.1.cmml" xref="S7.Thmtheorem15.p3.3.3.m3.1.1">𝑠</ci></annotation-xml><annotation encoding="application/x-tex" id="S7.Thmtheorem15.p3.3.3.m3.1c">s</annotation><annotation encoding="application/x-llamapun" id="S7.Thmtheorem15.p3.3.3.m3.1d">italic_s</annotation></semantics></math> is the first second where there is a violating edge from level <math alttext="h+2" class="ltx_Math" display="inline" id="S7.Thmtheorem15.p3.4.4.m4.1"><semantics id="S7.Thmtheorem15.p3.4.4.m4.1a"><mrow id="S7.Thmtheorem15.p3.4.4.m4.1.1" xref="S7.Thmtheorem15.p3.4.4.m4.1.1.cmml"><mi id="S7.Thmtheorem15.p3.4.4.m4.1.1.2" xref="S7.Thmtheorem15.p3.4.4.m4.1.1.2.cmml">h</mi><mo id="S7.Thmtheorem15.p3.4.4.m4.1.1.1" xref="S7.Thmtheorem15.p3.4.4.m4.1.1.1.cmml">+</mo><mn id="S7.Thmtheorem15.p3.4.4.m4.1.1.3" xref="S7.Thmtheorem15.p3.4.4.m4.1.1.3.cmml">2</mn></mrow><annotation-xml encoding="MathML-Content" id="S7.Thmtheorem15.p3.4.4.m4.1b"><apply id="S7.Thmtheorem15.p3.4.4.m4.1.1.cmml" xref="S7.Thmtheorem15.p3.4.4.m4.1.1"><plus id="S7.Thmtheorem15.p3.4.4.m4.1.1.1.cmml" xref="S7.Thmtheorem15.p3.4.4.m4.1.1.1"></plus><ci id="S7.Thmtheorem15.p3.4.4.m4.1.1.2.cmml" xref="S7.Thmtheorem15.p3.4.4.m4.1.1.2">ℎ</ci><cn id="S7.Thmtheorem15.p3.4.4.m4.1.1.3.cmml" type="integer" xref="S7.Thmtheorem15.p3.4.4.m4.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.Thmtheorem15.p3.4.4.m4.1c">h+2</annotation><annotation encoding="application/x-llamapun" id="S7.Thmtheorem15.p3.4.4.m4.1d">italic_h + 2</annotation></semantics></math> or above, since at <math alttext="(h:m:0)" class="ltx_Math" display="inline" id="S7.Thmtheorem15.p3.5.5.m5.1"><semantics id="S7.Thmtheorem15.p3.5.5.m5.1a"><mrow id="S7.Thmtheorem15.p3.5.5.m5.1.1.1" xref="S7.Thmtheorem15.p3.5.5.m5.1.1.1.1.cmml"><mo id="S7.Thmtheorem15.p3.5.5.m5.1.1.1.2" stretchy="false" xref="S7.Thmtheorem15.p3.5.5.m5.1.1.1.1.cmml">(</mo><mrow id="S7.Thmtheorem15.p3.5.5.m5.1.1.1.1" xref="S7.Thmtheorem15.p3.5.5.m5.1.1.1.1.cmml"><mi id="S7.Thmtheorem15.p3.5.5.m5.1.1.1.1.2" xref="S7.Thmtheorem15.p3.5.5.m5.1.1.1.1.2.cmml">h</mi><mo id="S7.Thmtheorem15.p3.5.5.m5.1.1.1.1.3" lspace="0.278em" rspace="0.278em" xref="S7.Thmtheorem15.p3.5.5.m5.1.1.1.1.3.cmml">:</mo><mi id="S7.Thmtheorem15.p3.5.5.m5.1.1.1.1.4" xref="S7.Thmtheorem15.p3.5.5.m5.1.1.1.1.4.cmml">m</mi><mo id="S7.Thmtheorem15.p3.5.5.m5.1.1.1.1.5" lspace="0.278em" rspace="0.278em" xref="S7.Thmtheorem15.p3.5.5.m5.1.1.1.1.5.cmml">:</mo><mn id="S7.Thmtheorem15.p3.5.5.m5.1.1.1.1.6" xref="S7.Thmtheorem15.p3.5.5.m5.1.1.1.1.6.cmml">0</mn></mrow><mo id="S7.Thmtheorem15.p3.5.5.m5.1.1.1.3" stretchy="false" xref="S7.Thmtheorem15.p3.5.5.m5.1.1.1.1.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S7.Thmtheorem15.p3.5.5.m5.1b"><apply id="S7.Thmtheorem15.p3.5.5.m5.1.1.1.1.cmml" xref="S7.Thmtheorem15.p3.5.5.m5.1.1.1"><and id="S7.Thmtheorem15.p3.5.5.m5.1.1.1.1a.cmml" xref="S7.Thmtheorem15.p3.5.5.m5.1.1.1"></and><apply id="S7.Thmtheorem15.p3.5.5.m5.1.1.1.1b.cmml" xref="S7.Thmtheorem15.p3.5.5.m5.1.1.1"><ci id="S7.Thmtheorem15.p3.5.5.m5.1.1.1.1.3.cmml" xref="S7.Thmtheorem15.p3.5.5.m5.1.1.1.1.3">:</ci><ci id="S7.Thmtheorem15.p3.5.5.m5.1.1.1.1.2.cmml" xref="S7.Thmtheorem15.p3.5.5.m5.1.1.1.1.2">ℎ</ci><ci id="S7.Thmtheorem15.p3.5.5.m5.1.1.1.1.4.cmml" xref="S7.Thmtheorem15.p3.5.5.m5.1.1.1.1.4">𝑚</ci></apply><apply id="S7.Thmtheorem15.p3.5.5.m5.1.1.1.1c.cmml" xref="S7.Thmtheorem15.p3.5.5.m5.1.1.1"><ci id="S7.Thmtheorem15.p3.5.5.m5.1.1.1.1.5.cmml" xref="S7.Thmtheorem15.p3.5.5.m5.1.1.1.1.5">:</ci><share href="https://arxiv.org/html/2411.12694v2#S7.Thmtheorem15.p3.5.5.m5.1.1.1.1.4.cmml" id="S7.Thmtheorem15.p3.5.5.m5.1.1.1.1d.cmml" xref="S7.Thmtheorem15.p3.5.5.m5.1.1.1"></share><cn id="S7.Thmtheorem15.p3.5.5.m5.1.1.1.1.6.cmml" type="integer" xref="S7.Thmtheorem15.p3.5.5.m5.1.1.1.1.6">0</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.Thmtheorem15.p3.5.5.m5.1c">(h:m:0)</annotation><annotation encoding="application/x-llamapun" id="S7.Thmtheorem15.p3.5.5.m5.1d">( italic_h : italic_m : 0 )</annotation></semantics></math> the edge <math alttext="u\!\to\!v" class="ltx_Math" display="inline" id="S7.Thmtheorem15.p3.6.6.m6.1"><semantics id="S7.Thmtheorem15.p3.6.6.m6.1a"><mrow id="S7.Thmtheorem15.p3.6.6.m6.1.1" xref="S7.Thmtheorem15.p3.6.6.m6.1.1.cmml"><mi id="S7.Thmtheorem15.p3.6.6.m6.1.1.2" xref="S7.Thmtheorem15.p3.6.6.m6.1.1.2.cmml">u</mi><mo id="S7.Thmtheorem15.p3.6.6.m6.1.1.1" lspace="0.108em" rspace="0.108em" stretchy="false" xref="S7.Thmtheorem15.p3.6.6.m6.1.1.1.cmml">→</mo><mi id="S7.Thmtheorem15.p3.6.6.m6.1.1.3" xref="S7.Thmtheorem15.p3.6.6.m6.1.1.3.cmml">v</mi></mrow><annotation-xml encoding="MathML-Content" id="S7.Thmtheorem15.p3.6.6.m6.1b"><apply id="S7.Thmtheorem15.p3.6.6.m6.1.1.cmml" xref="S7.Thmtheorem15.p3.6.6.m6.1.1"><ci id="S7.Thmtheorem15.p3.6.6.m6.1.1.1.cmml" xref="S7.Thmtheorem15.p3.6.6.m6.1.1.1">→</ci><ci id="S7.Thmtheorem15.p3.6.6.m6.1.1.2.cmml" xref="S7.Thmtheorem15.p3.6.6.m6.1.1.2">𝑢</ci><ci id="S7.Thmtheorem15.p3.6.6.m6.1.1.3.cmml" xref="S7.Thmtheorem15.p3.6.6.m6.1.1.3">𝑣</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.Thmtheorem15.p3.6.6.m6.1c">u\!\to\!v</annotation><annotation encoding="application/x-llamapun" id="S7.Thmtheorem15.p3.6.6.m6.1d">italic_u → italic_v</annotation></semantics></math> is violating.</span></p> </div> </div> <div class="ltx_theorem ltx_theorem_lemma" id="S7.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="S7.Thmtheorem16.1.1.1">Lemma 7.16</span></span><span class="ltx_text ltx_font_bold" id="S7.Thmtheorem16.2.2">.</span> </h6> <div class="ltx_para" id="S7.Thmtheorem16.p1"> <p class="ltx_p" id="S7.Thmtheorem16.p1.5"><span class="ltx_text ltx_font_italic" id="S7.Thmtheorem16.p1.5.5">For each time <math alttext="(h:m:s)" class="ltx_Math" display="inline" id="S7.Thmtheorem16.p1.1.1.m1.1"><semantics id="S7.Thmtheorem16.p1.1.1.m1.1a"><mrow id="S7.Thmtheorem16.p1.1.1.m1.1.1.1" xref="S7.Thmtheorem16.p1.1.1.m1.1.1.1.1.cmml"><mo id="S7.Thmtheorem16.p1.1.1.m1.1.1.1.2" stretchy="false" xref="S7.Thmtheorem16.p1.1.1.m1.1.1.1.1.cmml">(</mo><mrow id="S7.Thmtheorem16.p1.1.1.m1.1.1.1.1" xref="S7.Thmtheorem16.p1.1.1.m1.1.1.1.1.cmml"><mi id="S7.Thmtheorem16.p1.1.1.m1.1.1.1.1.2" xref="S7.Thmtheorem16.p1.1.1.m1.1.1.1.1.2.cmml">h</mi><mo id="S7.Thmtheorem16.p1.1.1.m1.1.1.1.1.3" lspace="0.278em" rspace="0.278em" xref="S7.Thmtheorem16.p1.1.1.m1.1.1.1.1.3.cmml">:</mo><mi id="S7.Thmtheorem16.p1.1.1.m1.1.1.1.1.4" xref="S7.Thmtheorem16.p1.1.1.m1.1.1.1.1.4.cmml">m</mi><mo id="S7.Thmtheorem16.p1.1.1.m1.1.1.1.1.5" lspace="0.278em" rspace="0.278em" xref="S7.Thmtheorem16.p1.1.1.m1.1.1.1.1.5.cmml">:</mo><mi id="S7.Thmtheorem16.p1.1.1.m1.1.1.1.1.6" xref="S7.Thmtheorem16.p1.1.1.m1.1.1.1.1.6.cmml">s</mi></mrow><mo id="S7.Thmtheorem16.p1.1.1.m1.1.1.1.3" stretchy="false" xref="S7.Thmtheorem16.p1.1.1.m1.1.1.1.1.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S7.Thmtheorem16.p1.1.1.m1.1b"><apply id="S7.Thmtheorem16.p1.1.1.m1.1.1.1.1.cmml" xref="S7.Thmtheorem16.p1.1.1.m1.1.1.1"><and id="S7.Thmtheorem16.p1.1.1.m1.1.1.1.1a.cmml" xref="S7.Thmtheorem16.p1.1.1.m1.1.1.1"></and><apply id="S7.Thmtheorem16.p1.1.1.m1.1.1.1.1b.cmml" xref="S7.Thmtheorem16.p1.1.1.m1.1.1.1"><ci id="S7.Thmtheorem16.p1.1.1.m1.1.1.1.1.3.cmml" xref="S7.Thmtheorem16.p1.1.1.m1.1.1.1.1.3">:</ci><ci id="S7.Thmtheorem16.p1.1.1.m1.1.1.1.1.2.cmml" xref="S7.Thmtheorem16.p1.1.1.m1.1.1.1.1.2">ℎ</ci><ci id="S7.Thmtheorem16.p1.1.1.m1.1.1.1.1.4.cmml" xref="S7.Thmtheorem16.p1.1.1.m1.1.1.1.1.4">𝑚</ci></apply><apply id="S7.Thmtheorem16.p1.1.1.m1.1.1.1.1c.cmml" xref="S7.Thmtheorem16.p1.1.1.m1.1.1.1"><ci id="S7.Thmtheorem16.p1.1.1.m1.1.1.1.1.5.cmml" xref="S7.Thmtheorem16.p1.1.1.m1.1.1.1.1.5">:</ci><share href="https://arxiv.org/html/2411.12694v2#S7.Thmtheorem16.p1.1.1.m1.1.1.1.1.4.cmml" id="S7.Thmtheorem16.p1.1.1.m1.1.1.1.1d.cmml" xref="S7.Thmtheorem16.p1.1.1.m1.1.1.1"></share><ci id="S7.Thmtheorem16.p1.1.1.m1.1.1.1.1.6.cmml" xref="S7.Thmtheorem16.p1.1.1.m1.1.1.1.1.6">𝑠</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.Thmtheorem16.p1.1.1.m1.1c">(h:m:s)</annotation><annotation encoding="application/x-llamapun" id="S7.Thmtheorem16.p1.1.1.m1.1d">( italic_h : italic_m : italic_s )</annotation></semantics></math> with <math alttext="m" class="ltx_Math" display="inline" id="S7.Thmtheorem16.p1.2.2.m2.1"><semantics id="S7.Thmtheorem16.p1.2.2.m2.1a"><mi id="S7.Thmtheorem16.p1.2.2.m2.1.1" xref="S7.Thmtheorem16.p1.2.2.m2.1.1.cmml">m</mi><annotation-xml encoding="MathML-Content" id="S7.Thmtheorem16.p1.2.2.m2.1b"><ci id="S7.Thmtheorem16.p1.2.2.m2.1.1.cmml" xref="S7.Thmtheorem16.p1.2.2.m2.1.1">𝑚</ci></annotation-xml><annotation encoding="application/x-tex" id="S7.Thmtheorem16.p1.2.2.m2.1c">m</annotation><annotation encoding="application/x-llamapun" id="S7.Thmtheorem16.p1.2.2.m2.1d">italic_m</annotation></semantics></math> odd, <math alttext="D_{s}\subset D_{s+1}" class="ltx_Math" display="inline" id="S7.Thmtheorem16.p1.3.3.m3.1"><semantics id="S7.Thmtheorem16.p1.3.3.m3.1a"><mrow id="S7.Thmtheorem16.p1.3.3.m3.1.1" xref="S7.Thmtheorem16.p1.3.3.m3.1.1.cmml"><msub id="S7.Thmtheorem16.p1.3.3.m3.1.1.2" xref="S7.Thmtheorem16.p1.3.3.m3.1.1.2.cmml"><mi id="S7.Thmtheorem16.p1.3.3.m3.1.1.2.2" xref="S7.Thmtheorem16.p1.3.3.m3.1.1.2.2.cmml">D</mi><mi id="S7.Thmtheorem16.p1.3.3.m3.1.1.2.3" xref="S7.Thmtheorem16.p1.3.3.m3.1.1.2.3.cmml">s</mi></msub><mo id="S7.Thmtheorem16.p1.3.3.m3.1.1.1" xref="S7.Thmtheorem16.p1.3.3.m3.1.1.1.cmml">⊂</mo><msub id="S7.Thmtheorem16.p1.3.3.m3.1.1.3" xref="S7.Thmtheorem16.p1.3.3.m3.1.1.3.cmml"><mi id="S7.Thmtheorem16.p1.3.3.m3.1.1.3.2" xref="S7.Thmtheorem16.p1.3.3.m3.1.1.3.2.cmml">D</mi><mrow id="S7.Thmtheorem16.p1.3.3.m3.1.1.3.3" xref="S7.Thmtheorem16.p1.3.3.m3.1.1.3.3.cmml"><mi id="S7.Thmtheorem16.p1.3.3.m3.1.1.3.3.2" xref="S7.Thmtheorem16.p1.3.3.m3.1.1.3.3.2.cmml">s</mi><mo id="S7.Thmtheorem16.p1.3.3.m3.1.1.3.3.1" xref="S7.Thmtheorem16.p1.3.3.m3.1.1.3.3.1.cmml">+</mo><mn id="S7.Thmtheorem16.p1.3.3.m3.1.1.3.3.3" xref="S7.Thmtheorem16.p1.3.3.m3.1.1.3.3.3.cmml">1</mn></mrow></msub></mrow><annotation-xml encoding="MathML-Content" id="S7.Thmtheorem16.p1.3.3.m3.1b"><apply id="S7.Thmtheorem16.p1.3.3.m3.1.1.cmml" xref="S7.Thmtheorem16.p1.3.3.m3.1.1"><subset id="S7.Thmtheorem16.p1.3.3.m3.1.1.1.cmml" xref="S7.Thmtheorem16.p1.3.3.m3.1.1.1"></subset><apply id="S7.Thmtheorem16.p1.3.3.m3.1.1.2.cmml" xref="S7.Thmtheorem16.p1.3.3.m3.1.1.2"><csymbol cd="ambiguous" id="S7.Thmtheorem16.p1.3.3.m3.1.1.2.1.cmml" xref="S7.Thmtheorem16.p1.3.3.m3.1.1.2">subscript</csymbol><ci id="S7.Thmtheorem16.p1.3.3.m3.1.1.2.2.cmml" xref="S7.Thmtheorem16.p1.3.3.m3.1.1.2.2">𝐷</ci><ci id="S7.Thmtheorem16.p1.3.3.m3.1.1.2.3.cmml" xref="S7.Thmtheorem16.p1.3.3.m3.1.1.2.3">𝑠</ci></apply><apply id="S7.Thmtheorem16.p1.3.3.m3.1.1.3.cmml" xref="S7.Thmtheorem16.p1.3.3.m3.1.1.3"><csymbol cd="ambiguous" id="S7.Thmtheorem16.p1.3.3.m3.1.1.3.1.cmml" xref="S7.Thmtheorem16.p1.3.3.m3.1.1.3">subscript</csymbol><ci id="S7.Thmtheorem16.p1.3.3.m3.1.1.3.2.cmml" xref="S7.Thmtheorem16.p1.3.3.m3.1.1.3.2">𝐷</ci><apply id="S7.Thmtheorem16.p1.3.3.m3.1.1.3.3.cmml" xref="S7.Thmtheorem16.p1.3.3.m3.1.1.3.3"><plus id="S7.Thmtheorem16.p1.3.3.m3.1.1.3.3.1.cmml" xref="S7.Thmtheorem16.p1.3.3.m3.1.1.3.3.1"></plus><ci id="S7.Thmtheorem16.p1.3.3.m3.1.1.3.3.2.cmml" xref="S7.Thmtheorem16.p1.3.3.m3.1.1.3.3.2">𝑠</ci><cn id="S7.Thmtheorem16.p1.3.3.m3.1.1.3.3.3.cmml" type="integer" xref="S7.Thmtheorem16.p1.3.3.m3.1.1.3.3.3">1</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.Thmtheorem16.p1.3.3.m3.1c">D_{s}\subset D_{s+1}</annotation><annotation encoding="application/x-llamapun" id="S7.Thmtheorem16.p1.3.3.m3.1d">italic_D start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ⊂ italic_D start_POSTSUBSCRIPT italic_s + 1 end_POSTSUBSCRIPT</annotation></semantics></math> and the height of <math alttext="D_{s+1}" class="ltx_Math" display="inline" id="S7.Thmtheorem16.p1.4.4.m4.1"><semantics id="S7.Thmtheorem16.p1.4.4.m4.1a"><msub id="S7.Thmtheorem16.p1.4.4.m4.1.1" xref="S7.Thmtheorem16.p1.4.4.m4.1.1.cmml"><mi id="S7.Thmtheorem16.p1.4.4.m4.1.1.2" xref="S7.Thmtheorem16.p1.4.4.m4.1.1.2.cmml">D</mi><mrow id="S7.Thmtheorem16.p1.4.4.m4.1.1.3" xref="S7.Thmtheorem16.p1.4.4.m4.1.1.3.cmml"><mi id="S7.Thmtheorem16.p1.4.4.m4.1.1.3.2" xref="S7.Thmtheorem16.p1.4.4.m4.1.1.3.2.cmml">s</mi><mo id="S7.Thmtheorem16.p1.4.4.m4.1.1.3.1" xref="S7.Thmtheorem16.p1.4.4.m4.1.1.3.1.cmml">+</mo><mn id="S7.Thmtheorem16.p1.4.4.m4.1.1.3.3" xref="S7.Thmtheorem16.p1.4.4.m4.1.1.3.3.cmml">1</mn></mrow></msub><annotation-xml encoding="MathML-Content" id="S7.Thmtheorem16.p1.4.4.m4.1b"><apply id="S7.Thmtheorem16.p1.4.4.m4.1.1.cmml" xref="S7.Thmtheorem16.p1.4.4.m4.1.1"><csymbol cd="ambiguous" id="S7.Thmtheorem16.p1.4.4.m4.1.1.1.cmml" xref="S7.Thmtheorem16.p1.4.4.m4.1.1">subscript</csymbol><ci id="S7.Thmtheorem16.p1.4.4.m4.1.1.2.cmml" xref="S7.Thmtheorem16.p1.4.4.m4.1.1.2">𝐷</ci><apply id="S7.Thmtheorem16.p1.4.4.m4.1.1.3.cmml" xref="S7.Thmtheorem16.p1.4.4.m4.1.1.3"><plus id="S7.Thmtheorem16.p1.4.4.m4.1.1.3.1.cmml" xref="S7.Thmtheorem16.p1.4.4.m4.1.1.3.1"></plus><ci id="S7.Thmtheorem16.p1.4.4.m4.1.1.3.2.cmml" xref="S7.Thmtheorem16.p1.4.4.m4.1.1.3.2">𝑠</ci><cn id="S7.Thmtheorem16.p1.4.4.m4.1.1.3.3.cmml" type="integer" xref="S7.Thmtheorem16.p1.4.4.m4.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.Thmtheorem16.p1.4.4.m4.1c">D_{s+1}</annotation><annotation encoding="application/x-llamapun" id="S7.Thmtheorem16.p1.4.4.m4.1d">italic_D start_POSTSUBSCRIPT italic_s + 1 end_POSTSUBSCRIPT</annotation></semantics></math> is at least one fewer than the height of <math alttext="D_{s}" class="ltx_Math" display="inline" id="S7.Thmtheorem16.p1.5.5.m5.1"><semantics id="S7.Thmtheorem16.p1.5.5.m5.1a"><msub id="S7.Thmtheorem16.p1.5.5.m5.1.1" xref="S7.Thmtheorem16.p1.5.5.m5.1.1.cmml"><mi id="S7.Thmtheorem16.p1.5.5.m5.1.1.2" xref="S7.Thmtheorem16.p1.5.5.m5.1.1.2.cmml">D</mi><mi id="S7.Thmtheorem16.p1.5.5.m5.1.1.3" xref="S7.Thmtheorem16.p1.5.5.m5.1.1.3.cmml">s</mi></msub><annotation-xml encoding="MathML-Content" id="S7.Thmtheorem16.p1.5.5.m5.1b"><apply id="S7.Thmtheorem16.p1.5.5.m5.1.1.cmml" xref="S7.Thmtheorem16.p1.5.5.m5.1.1"><csymbol cd="ambiguous" id="S7.Thmtheorem16.p1.5.5.m5.1.1.1.cmml" xref="S7.Thmtheorem16.p1.5.5.m5.1.1">subscript</csymbol><ci id="S7.Thmtheorem16.p1.5.5.m5.1.1.2.cmml" xref="S7.Thmtheorem16.p1.5.5.m5.1.1.2">𝐷</ci><ci id="S7.Thmtheorem16.p1.5.5.m5.1.1.3.cmml" xref="S7.Thmtheorem16.p1.5.5.m5.1.1.3">𝑠</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.Thmtheorem16.p1.5.5.m5.1c">D_{s}</annotation><annotation encoding="application/x-llamapun" id="S7.Thmtheorem16.p1.5.5.m5.1d">italic_D start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT</annotation></semantics></math>.</span></p> </div> </div> <div class="ltx_theorem ltx_theorem_proof" id="S7.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="S7.Thmtheorem17.1.1.1">Proof 7.17</span></span><span class="ltx_text ltx_font_bold" id="S7.Thmtheorem17.2.2">.</span> </h6> <div class="ltx_para" id="S7.Thmtheorem17.p1"> <p class="ltx_p" id="S7.Thmtheorem17.p1.13"><span class="ltx_text ltx_font_italic" id="S7.Thmtheorem17.p1.13.13">For every edge <math alttext="\overline{uv}\in E_{s}" class="ltx_Math" display="inline" id="S7.Thmtheorem17.p1.1.1.m1.1"><semantics id="S7.Thmtheorem17.p1.1.1.m1.1a"><mrow id="S7.Thmtheorem17.p1.1.1.m1.1.1" xref="S7.Thmtheorem17.p1.1.1.m1.1.1.cmml"><mover accent="true" id="S7.Thmtheorem17.p1.1.1.m1.1.1.2" xref="S7.Thmtheorem17.p1.1.1.m1.1.1.2.cmml"><mrow id="S7.Thmtheorem17.p1.1.1.m1.1.1.2.2" xref="S7.Thmtheorem17.p1.1.1.m1.1.1.2.2.cmml"><mi id="S7.Thmtheorem17.p1.1.1.m1.1.1.2.2.2" xref="S7.Thmtheorem17.p1.1.1.m1.1.1.2.2.2.cmml">u</mi><mo id="S7.Thmtheorem17.p1.1.1.m1.1.1.2.2.1" xref="S7.Thmtheorem17.p1.1.1.m1.1.1.2.2.1.cmml">⁢</mo><mi id="S7.Thmtheorem17.p1.1.1.m1.1.1.2.2.3" xref="S7.Thmtheorem17.p1.1.1.m1.1.1.2.2.3.cmml">v</mi></mrow><mo id="S7.Thmtheorem17.p1.1.1.m1.1.1.2.1" xref="S7.Thmtheorem17.p1.1.1.m1.1.1.2.1.cmml">¯</mo></mover><mo id="S7.Thmtheorem17.p1.1.1.m1.1.1.1" xref="S7.Thmtheorem17.p1.1.1.m1.1.1.1.cmml">∈</mo><msub id="S7.Thmtheorem17.p1.1.1.m1.1.1.3" xref="S7.Thmtheorem17.p1.1.1.m1.1.1.3.cmml"><mi id="S7.Thmtheorem17.p1.1.1.m1.1.1.3.2" xref="S7.Thmtheorem17.p1.1.1.m1.1.1.3.2.cmml">E</mi><mi id="S7.Thmtheorem17.p1.1.1.m1.1.1.3.3" xref="S7.Thmtheorem17.p1.1.1.m1.1.1.3.3.cmml">s</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S7.Thmtheorem17.p1.1.1.m1.1b"><apply id="S7.Thmtheorem17.p1.1.1.m1.1.1.cmml" xref="S7.Thmtheorem17.p1.1.1.m1.1.1"><in id="S7.Thmtheorem17.p1.1.1.m1.1.1.1.cmml" xref="S7.Thmtheorem17.p1.1.1.m1.1.1.1"></in><apply id="S7.Thmtheorem17.p1.1.1.m1.1.1.2.cmml" xref="S7.Thmtheorem17.p1.1.1.m1.1.1.2"><ci id="S7.Thmtheorem17.p1.1.1.m1.1.1.2.1.cmml" xref="S7.Thmtheorem17.p1.1.1.m1.1.1.2.1">¯</ci><apply id="S7.Thmtheorem17.p1.1.1.m1.1.1.2.2.cmml" xref="S7.Thmtheorem17.p1.1.1.m1.1.1.2.2"><times id="S7.Thmtheorem17.p1.1.1.m1.1.1.2.2.1.cmml" xref="S7.Thmtheorem17.p1.1.1.m1.1.1.2.2.1"></times><ci id="S7.Thmtheorem17.p1.1.1.m1.1.1.2.2.2.cmml" xref="S7.Thmtheorem17.p1.1.1.m1.1.1.2.2.2">𝑢</ci><ci id="S7.Thmtheorem17.p1.1.1.m1.1.1.2.2.3.cmml" xref="S7.Thmtheorem17.p1.1.1.m1.1.1.2.2.3">𝑣</ci></apply></apply><apply id="S7.Thmtheorem17.p1.1.1.m1.1.1.3.cmml" xref="S7.Thmtheorem17.p1.1.1.m1.1.1.3"><csymbol cd="ambiguous" id="S7.Thmtheorem17.p1.1.1.m1.1.1.3.1.cmml" xref="S7.Thmtheorem17.p1.1.1.m1.1.1.3">subscript</csymbol><ci id="S7.Thmtheorem17.p1.1.1.m1.1.1.3.2.cmml" xref="S7.Thmtheorem17.p1.1.1.m1.1.1.3.2">𝐸</ci><ci id="S7.Thmtheorem17.p1.1.1.m1.1.1.3.3.cmml" xref="S7.Thmtheorem17.p1.1.1.m1.1.1.3.3">𝑠</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.Thmtheorem17.p1.1.1.m1.1c">\overline{uv}\in E_{s}</annotation><annotation encoding="application/x-llamapun" id="S7.Thmtheorem17.p1.1.1.m1.1d">over¯ start_ARG italic_u italic_v end_ARG ∈ italic_E start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT</annotation></semantics></math>, by Observation <a class="ltx_ref" href="https://arxiv.org/html/2411.12694v2#S7.SS3" title="7.3 Sketching our algorithm’s correctness. ‣ 7 Results in CONGEST ‣ Local Density and its Distributed Approximation"><span class="ltx_text ltx_ref_tag">7.3</span></a>, our algorithm only decreases <math alttext="\textsl{g}(u\!\to\!v)" class="ltx_Math" display="inline" id="S7.Thmtheorem17.p1.2.2.m2.1"><semantics id="S7.Thmtheorem17.p1.2.2.m2.1a"><mrow id="S7.Thmtheorem17.p1.2.2.m2.1.1" xref="S7.Thmtheorem17.p1.2.2.m2.1.1.cmml"><mtext class="ltx_mathvariant_italic" id="S7.Thmtheorem17.p1.2.2.m2.1.1.3" xref="S7.Thmtheorem17.p1.2.2.m2.1.1.3a.cmml">g</mtext><mo id="S7.Thmtheorem17.p1.2.2.m2.1.1.2" xref="S7.Thmtheorem17.p1.2.2.m2.1.1.2.cmml">⁢</mo><mrow id="S7.Thmtheorem17.p1.2.2.m2.1.1.1.1" xref="S7.Thmtheorem17.p1.2.2.m2.1.1.1.1.1.cmml"><mo id="S7.Thmtheorem17.p1.2.2.m2.1.1.1.1.2" stretchy="false" xref="S7.Thmtheorem17.p1.2.2.m2.1.1.1.1.1.cmml">(</mo><mrow id="S7.Thmtheorem17.p1.2.2.m2.1.1.1.1.1" xref="S7.Thmtheorem17.p1.2.2.m2.1.1.1.1.1.cmml"><mi id="S7.Thmtheorem17.p1.2.2.m2.1.1.1.1.1.2" xref="S7.Thmtheorem17.p1.2.2.m2.1.1.1.1.1.2.cmml">u</mi><mo id="S7.Thmtheorem17.p1.2.2.m2.1.1.1.1.1.1" lspace="0.108em" rspace="0.108em" stretchy="false" xref="S7.Thmtheorem17.p1.2.2.m2.1.1.1.1.1.1.cmml">→</mo><mi id="S7.Thmtheorem17.p1.2.2.m2.1.1.1.1.1.3" xref="S7.Thmtheorem17.p1.2.2.m2.1.1.1.1.1.3.cmml">v</mi></mrow><mo id="S7.Thmtheorem17.p1.2.2.m2.1.1.1.1.3" stretchy="false" xref="S7.Thmtheorem17.p1.2.2.m2.1.1.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S7.Thmtheorem17.p1.2.2.m2.1b"><apply id="S7.Thmtheorem17.p1.2.2.m2.1.1.cmml" xref="S7.Thmtheorem17.p1.2.2.m2.1.1"><times id="S7.Thmtheorem17.p1.2.2.m2.1.1.2.cmml" xref="S7.Thmtheorem17.p1.2.2.m2.1.1.2"></times><ci id="S7.Thmtheorem17.p1.2.2.m2.1.1.3a.cmml" xref="S7.Thmtheorem17.p1.2.2.m2.1.1.3"><mtext class="ltx_mathvariant_italic" id="S7.Thmtheorem17.p1.2.2.m2.1.1.3.cmml" xref="S7.Thmtheorem17.p1.2.2.m2.1.1.3">g</mtext></ci><apply id="S7.Thmtheorem17.p1.2.2.m2.1.1.1.1.1.cmml" xref="S7.Thmtheorem17.p1.2.2.m2.1.1.1.1"><ci id="S7.Thmtheorem17.p1.2.2.m2.1.1.1.1.1.1.cmml" xref="S7.Thmtheorem17.p1.2.2.m2.1.1.1.1.1.1">→</ci><ci id="S7.Thmtheorem17.p1.2.2.m2.1.1.1.1.1.2.cmml" xref="S7.Thmtheorem17.p1.2.2.m2.1.1.1.1.1.2">𝑢</ci><ci id="S7.Thmtheorem17.p1.2.2.m2.1.1.1.1.1.3.cmml" xref="S7.Thmtheorem17.p1.2.2.m2.1.1.1.1.1.3">𝑣</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.Thmtheorem17.p1.2.2.m2.1c">\textsl{g}(u\!\to\!v)</annotation><annotation encoding="application/x-llamapun" id="S7.Thmtheorem17.p1.2.2.m2.1d">g ( italic_u → italic_v )</annotation></semantics></math>, and only decreases the level of <math alttext="u" class="ltx_Math" display="inline" id="S7.Thmtheorem17.p1.3.3.m3.1"><semantics id="S7.Thmtheorem17.p1.3.3.m3.1a"><mi id="S7.Thmtheorem17.p1.3.3.m3.1.1" xref="S7.Thmtheorem17.p1.3.3.m3.1.1.cmml">u</mi><annotation-xml encoding="MathML-Content" id="S7.Thmtheorem17.p1.3.3.m3.1b"><ci id="S7.Thmtheorem17.p1.3.3.m3.1.1.cmml" xref="S7.Thmtheorem17.p1.3.3.m3.1.1">𝑢</ci></annotation-xml><annotation encoding="application/x-tex" id="S7.Thmtheorem17.p1.3.3.m3.1c">u</annotation><annotation encoding="application/x-llamapun" id="S7.Thmtheorem17.p1.3.3.m3.1d">italic_u</annotation></semantics></math>. It immediately follows from the definition of <math alttext="E_{s}" class="ltx_Math" display="inline" id="S7.Thmtheorem17.p1.4.4.m4.1"><semantics id="S7.Thmtheorem17.p1.4.4.m4.1a"><msub id="S7.Thmtheorem17.p1.4.4.m4.1.1" xref="S7.Thmtheorem17.p1.4.4.m4.1.1.cmml"><mi id="S7.Thmtheorem17.p1.4.4.m4.1.1.2" xref="S7.Thmtheorem17.p1.4.4.m4.1.1.2.cmml">E</mi><mi id="S7.Thmtheorem17.p1.4.4.m4.1.1.3" xref="S7.Thmtheorem17.p1.4.4.m4.1.1.3.cmml">s</mi></msub><annotation-xml encoding="MathML-Content" id="S7.Thmtheorem17.p1.4.4.m4.1b"><apply id="S7.Thmtheorem17.p1.4.4.m4.1.1.cmml" xref="S7.Thmtheorem17.p1.4.4.m4.1.1"><csymbol cd="ambiguous" id="S7.Thmtheorem17.p1.4.4.m4.1.1.1.cmml" xref="S7.Thmtheorem17.p1.4.4.m4.1.1">subscript</csymbol><ci id="S7.Thmtheorem17.p1.4.4.m4.1.1.2.cmml" xref="S7.Thmtheorem17.p1.4.4.m4.1.1.2">𝐸</ci><ci id="S7.Thmtheorem17.p1.4.4.m4.1.1.3.cmml" xref="S7.Thmtheorem17.p1.4.4.m4.1.1.3">𝑠</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.Thmtheorem17.p1.4.4.m4.1c">E_{s}</annotation><annotation encoding="application/x-llamapun" id="S7.Thmtheorem17.p1.4.4.m4.1d">italic_E start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT</annotation></semantics></math> that if <math alttext="\overline{uv}" class="ltx_Math" display="inline" id="S7.Thmtheorem17.p1.5.5.m5.1"><semantics id="S7.Thmtheorem17.p1.5.5.m5.1a"><mover accent="true" id="S7.Thmtheorem17.p1.5.5.m5.1.1" xref="S7.Thmtheorem17.p1.5.5.m5.1.1.cmml"><mrow id="S7.Thmtheorem17.p1.5.5.m5.1.1.2" xref="S7.Thmtheorem17.p1.5.5.m5.1.1.2.cmml"><mi id="S7.Thmtheorem17.p1.5.5.m5.1.1.2.2" xref="S7.Thmtheorem17.p1.5.5.m5.1.1.2.2.cmml">u</mi><mo id="S7.Thmtheorem17.p1.5.5.m5.1.1.2.1" xref="S7.Thmtheorem17.p1.5.5.m5.1.1.2.1.cmml">⁢</mo><mi id="S7.Thmtheorem17.p1.5.5.m5.1.1.2.3" xref="S7.Thmtheorem17.p1.5.5.m5.1.1.2.3.cmml">v</mi></mrow><mo id="S7.Thmtheorem17.p1.5.5.m5.1.1.1" xref="S7.Thmtheorem17.p1.5.5.m5.1.1.1.cmml">¯</mo></mover><annotation-xml encoding="MathML-Content" id="S7.Thmtheorem17.p1.5.5.m5.1b"><apply id="S7.Thmtheorem17.p1.5.5.m5.1.1.cmml" xref="S7.Thmtheorem17.p1.5.5.m5.1.1"><ci id="S7.Thmtheorem17.p1.5.5.m5.1.1.1.cmml" xref="S7.Thmtheorem17.p1.5.5.m5.1.1.1">¯</ci><apply id="S7.Thmtheorem17.p1.5.5.m5.1.1.2.cmml" xref="S7.Thmtheorem17.p1.5.5.m5.1.1.2"><times id="S7.Thmtheorem17.p1.5.5.m5.1.1.2.1.cmml" xref="S7.Thmtheorem17.p1.5.5.m5.1.1.2.1"></times><ci id="S7.Thmtheorem17.p1.5.5.m5.1.1.2.2.cmml" xref="S7.Thmtheorem17.p1.5.5.m5.1.1.2.2">𝑢</ci><ci id="S7.Thmtheorem17.p1.5.5.m5.1.1.2.3.cmml" xref="S7.Thmtheorem17.p1.5.5.m5.1.1.2.3">𝑣</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.Thmtheorem17.p1.5.5.m5.1c">\overline{uv}</annotation><annotation encoding="application/x-llamapun" id="S7.Thmtheorem17.p1.5.5.m5.1d">over¯ start_ARG italic_u italic_v end_ARG</annotation></semantics></math> is in <math alttext="E_{s}" class="ltx_Math" display="inline" id="S7.Thmtheorem17.p1.6.6.m6.1"><semantics id="S7.Thmtheorem17.p1.6.6.m6.1a"><msub id="S7.Thmtheorem17.p1.6.6.m6.1.1" xref="S7.Thmtheorem17.p1.6.6.m6.1.1.cmml"><mi id="S7.Thmtheorem17.p1.6.6.m6.1.1.2" xref="S7.Thmtheorem17.p1.6.6.m6.1.1.2.cmml">E</mi><mi id="S7.Thmtheorem17.p1.6.6.m6.1.1.3" xref="S7.Thmtheorem17.p1.6.6.m6.1.1.3.cmml">s</mi></msub><annotation-xml encoding="MathML-Content" id="S7.Thmtheorem17.p1.6.6.m6.1b"><apply id="S7.Thmtheorem17.p1.6.6.m6.1.1.cmml" xref="S7.Thmtheorem17.p1.6.6.m6.1.1"><csymbol cd="ambiguous" id="S7.Thmtheorem17.p1.6.6.m6.1.1.1.cmml" xref="S7.Thmtheorem17.p1.6.6.m6.1.1">subscript</csymbol><ci id="S7.Thmtheorem17.p1.6.6.m6.1.1.2.cmml" xref="S7.Thmtheorem17.p1.6.6.m6.1.1.2">𝐸</ci><ci id="S7.Thmtheorem17.p1.6.6.m6.1.1.3.cmml" xref="S7.Thmtheorem17.p1.6.6.m6.1.1.3">𝑠</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.Thmtheorem17.p1.6.6.m6.1c">E_{s}</annotation><annotation encoding="application/x-llamapun" id="S7.Thmtheorem17.p1.6.6.m6.1d">italic_E start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT</annotation></semantics></math> but not in <math alttext="E_{s+1}" class="ltx_Math" display="inline" id="S7.Thmtheorem17.p1.7.7.m7.1"><semantics id="S7.Thmtheorem17.p1.7.7.m7.1a"><msub id="S7.Thmtheorem17.p1.7.7.m7.1.1" xref="S7.Thmtheorem17.p1.7.7.m7.1.1.cmml"><mi id="S7.Thmtheorem17.p1.7.7.m7.1.1.2" xref="S7.Thmtheorem17.p1.7.7.m7.1.1.2.cmml">E</mi><mrow id="S7.Thmtheorem17.p1.7.7.m7.1.1.3" xref="S7.Thmtheorem17.p1.7.7.m7.1.1.3.cmml"><mi id="S7.Thmtheorem17.p1.7.7.m7.1.1.3.2" xref="S7.Thmtheorem17.p1.7.7.m7.1.1.3.2.cmml">s</mi><mo id="S7.Thmtheorem17.p1.7.7.m7.1.1.3.1" xref="S7.Thmtheorem17.p1.7.7.m7.1.1.3.1.cmml">+</mo><mn id="S7.Thmtheorem17.p1.7.7.m7.1.1.3.3" xref="S7.Thmtheorem17.p1.7.7.m7.1.1.3.3.cmml">1</mn></mrow></msub><annotation-xml encoding="MathML-Content" id="S7.Thmtheorem17.p1.7.7.m7.1b"><apply id="S7.Thmtheorem17.p1.7.7.m7.1.1.cmml" xref="S7.Thmtheorem17.p1.7.7.m7.1.1"><csymbol cd="ambiguous" id="S7.Thmtheorem17.p1.7.7.m7.1.1.1.cmml" xref="S7.Thmtheorem17.p1.7.7.m7.1.1">subscript</csymbol><ci id="S7.Thmtheorem17.p1.7.7.m7.1.1.2.cmml" xref="S7.Thmtheorem17.p1.7.7.m7.1.1.2">𝐸</ci><apply id="S7.Thmtheorem17.p1.7.7.m7.1.1.3.cmml" xref="S7.Thmtheorem17.p1.7.7.m7.1.1.3"><plus id="S7.Thmtheorem17.p1.7.7.m7.1.1.3.1.cmml" xref="S7.Thmtheorem17.p1.7.7.m7.1.1.3.1"></plus><ci id="S7.Thmtheorem17.p1.7.7.m7.1.1.3.2.cmml" xref="S7.Thmtheorem17.p1.7.7.m7.1.1.3.2">𝑠</ci><cn id="S7.Thmtheorem17.p1.7.7.m7.1.1.3.3.cmml" type="integer" xref="S7.Thmtheorem17.p1.7.7.m7.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.Thmtheorem17.p1.7.7.m7.1c">E_{s+1}</annotation><annotation encoding="application/x-llamapun" id="S7.Thmtheorem17.p1.7.7.m7.1d">italic_E start_POSTSUBSCRIPT italic_s + 1 end_POSTSUBSCRIPT</annotation></semantics></math>, then <math alttext="\overline{uv}\in E_{l}" class="ltx_Math" display="inline" id="S7.Thmtheorem17.p1.8.8.m8.1"><semantics id="S7.Thmtheorem17.p1.8.8.m8.1a"><mrow id="S7.Thmtheorem17.p1.8.8.m8.1.1" xref="S7.Thmtheorem17.p1.8.8.m8.1.1.cmml"><mover accent="true" id="S7.Thmtheorem17.p1.8.8.m8.1.1.2" xref="S7.Thmtheorem17.p1.8.8.m8.1.1.2.cmml"><mrow id="S7.Thmtheorem17.p1.8.8.m8.1.1.2.2" xref="S7.Thmtheorem17.p1.8.8.m8.1.1.2.2.cmml"><mi id="S7.Thmtheorem17.p1.8.8.m8.1.1.2.2.2" xref="S7.Thmtheorem17.p1.8.8.m8.1.1.2.2.2.cmml">u</mi><mo id="S7.Thmtheorem17.p1.8.8.m8.1.1.2.2.1" xref="S7.Thmtheorem17.p1.8.8.m8.1.1.2.2.1.cmml">⁢</mo><mi id="S7.Thmtheorem17.p1.8.8.m8.1.1.2.2.3" xref="S7.Thmtheorem17.p1.8.8.m8.1.1.2.2.3.cmml">v</mi></mrow><mo id="S7.Thmtheorem17.p1.8.8.m8.1.1.2.1" xref="S7.Thmtheorem17.p1.8.8.m8.1.1.2.1.cmml">¯</mo></mover><mo id="S7.Thmtheorem17.p1.8.8.m8.1.1.1" xref="S7.Thmtheorem17.p1.8.8.m8.1.1.1.cmml">∈</mo><msub id="S7.Thmtheorem17.p1.8.8.m8.1.1.3" xref="S7.Thmtheorem17.p1.8.8.m8.1.1.3.cmml"><mi id="S7.Thmtheorem17.p1.8.8.m8.1.1.3.2" xref="S7.Thmtheorem17.p1.8.8.m8.1.1.3.2.cmml">E</mi><mi id="S7.Thmtheorem17.p1.8.8.m8.1.1.3.3" xref="S7.Thmtheorem17.p1.8.8.m8.1.1.3.3.cmml">l</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S7.Thmtheorem17.p1.8.8.m8.1b"><apply id="S7.Thmtheorem17.p1.8.8.m8.1.1.cmml" xref="S7.Thmtheorem17.p1.8.8.m8.1.1"><in id="S7.Thmtheorem17.p1.8.8.m8.1.1.1.cmml" xref="S7.Thmtheorem17.p1.8.8.m8.1.1.1"></in><apply id="S7.Thmtheorem17.p1.8.8.m8.1.1.2.cmml" xref="S7.Thmtheorem17.p1.8.8.m8.1.1.2"><ci id="S7.Thmtheorem17.p1.8.8.m8.1.1.2.1.cmml" xref="S7.Thmtheorem17.p1.8.8.m8.1.1.2.1">¯</ci><apply id="S7.Thmtheorem17.p1.8.8.m8.1.1.2.2.cmml" xref="S7.Thmtheorem17.p1.8.8.m8.1.1.2.2"><times id="S7.Thmtheorem17.p1.8.8.m8.1.1.2.2.1.cmml" xref="S7.Thmtheorem17.p1.8.8.m8.1.1.2.2.1"></times><ci id="S7.Thmtheorem17.p1.8.8.m8.1.1.2.2.2.cmml" xref="S7.Thmtheorem17.p1.8.8.m8.1.1.2.2.2">𝑢</ci><ci id="S7.Thmtheorem17.p1.8.8.m8.1.1.2.2.3.cmml" xref="S7.Thmtheorem17.p1.8.8.m8.1.1.2.2.3">𝑣</ci></apply></apply><apply id="S7.Thmtheorem17.p1.8.8.m8.1.1.3.cmml" xref="S7.Thmtheorem17.p1.8.8.m8.1.1.3"><csymbol cd="ambiguous" id="S7.Thmtheorem17.p1.8.8.m8.1.1.3.1.cmml" xref="S7.Thmtheorem17.p1.8.8.m8.1.1.3">subscript</csymbol><ci id="S7.Thmtheorem17.p1.8.8.m8.1.1.3.2.cmml" xref="S7.Thmtheorem17.p1.8.8.m8.1.1.3.2">𝐸</ci><ci id="S7.Thmtheorem17.p1.8.8.m8.1.1.3.3.cmml" xref="S7.Thmtheorem17.p1.8.8.m8.1.1.3.3">𝑙</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.Thmtheorem17.p1.8.8.m8.1c">\overline{uv}\in E_{l}</annotation><annotation encoding="application/x-llamapun" id="S7.Thmtheorem17.p1.8.8.m8.1d">over¯ start_ARG italic_u italic_v end_ARG ∈ italic_E start_POSTSUBSCRIPT italic_l end_POSTSUBSCRIPT</annotation></semantics></math> for all <math alttext="l<s" class="ltx_Math" display="inline" id="S7.Thmtheorem17.p1.9.9.m9.1"><semantics id="S7.Thmtheorem17.p1.9.9.m9.1a"><mrow id="S7.Thmtheorem17.p1.9.9.m9.1.1" xref="S7.Thmtheorem17.p1.9.9.m9.1.1.cmml"><mi id="S7.Thmtheorem17.p1.9.9.m9.1.1.2" xref="S7.Thmtheorem17.p1.9.9.m9.1.1.2.cmml">l</mi><mo id="S7.Thmtheorem17.p1.9.9.m9.1.1.1" xref="S7.Thmtheorem17.p1.9.9.m9.1.1.1.cmml"><</mo><mi id="S7.Thmtheorem17.p1.9.9.m9.1.1.3" xref="S7.Thmtheorem17.p1.9.9.m9.1.1.3.cmml">s</mi></mrow><annotation-xml encoding="MathML-Content" id="S7.Thmtheorem17.p1.9.9.m9.1b"><apply id="S7.Thmtheorem17.p1.9.9.m9.1.1.cmml" xref="S7.Thmtheorem17.p1.9.9.m9.1.1"><lt id="S7.Thmtheorem17.p1.9.9.m9.1.1.1.cmml" xref="S7.Thmtheorem17.p1.9.9.m9.1.1.1"></lt><ci id="S7.Thmtheorem17.p1.9.9.m9.1.1.2.cmml" xref="S7.Thmtheorem17.p1.9.9.m9.1.1.2">𝑙</ci><ci id="S7.Thmtheorem17.p1.9.9.m9.1.1.3.cmml" xref="S7.Thmtheorem17.p1.9.9.m9.1.1.3">𝑠</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.Thmtheorem17.p1.9.9.m9.1c">l<s</annotation><annotation encoding="application/x-llamapun" id="S7.Thmtheorem17.p1.9.9.m9.1d">italic_l < italic_s</annotation></semantics></math> and <math alttext="\overline{uv}\not\in E_{k}" class="ltx_Math" display="inline" id="S7.Thmtheorem17.p1.10.10.m10.1"><semantics id="S7.Thmtheorem17.p1.10.10.m10.1a"><mrow id="S7.Thmtheorem17.p1.10.10.m10.1.1" xref="S7.Thmtheorem17.p1.10.10.m10.1.1.cmml"><mover accent="true" id="S7.Thmtheorem17.p1.10.10.m10.1.1.2" xref="S7.Thmtheorem17.p1.10.10.m10.1.1.2.cmml"><mrow id="S7.Thmtheorem17.p1.10.10.m10.1.1.2.2" xref="S7.Thmtheorem17.p1.10.10.m10.1.1.2.2.cmml"><mi id="S7.Thmtheorem17.p1.10.10.m10.1.1.2.2.2" xref="S7.Thmtheorem17.p1.10.10.m10.1.1.2.2.2.cmml">u</mi><mo id="S7.Thmtheorem17.p1.10.10.m10.1.1.2.2.1" xref="S7.Thmtheorem17.p1.10.10.m10.1.1.2.2.1.cmml">⁢</mo><mi id="S7.Thmtheorem17.p1.10.10.m10.1.1.2.2.3" xref="S7.Thmtheorem17.p1.10.10.m10.1.1.2.2.3.cmml">v</mi></mrow><mo id="S7.Thmtheorem17.p1.10.10.m10.1.1.2.1" xref="S7.Thmtheorem17.p1.10.10.m10.1.1.2.1.cmml">¯</mo></mover><mo id="S7.Thmtheorem17.p1.10.10.m10.1.1.1" xref="S7.Thmtheorem17.p1.10.10.m10.1.1.1.cmml">∉</mo><msub id="S7.Thmtheorem17.p1.10.10.m10.1.1.3" xref="S7.Thmtheorem17.p1.10.10.m10.1.1.3.cmml"><mi id="S7.Thmtheorem17.p1.10.10.m10.1.1.3.2" xref="S7.Thmtheorem17.p1.10.10.m10.1.1.3.2.cmml">E</mi><mi id="S7.Thmtheorem17.p1.10.10.m10.1.1.3.3" xref="S7.Thmtheorem17.p1.10.10.m10.1.1.3.3.cmml">k</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S7.Thmtheorem17.p1.10.10.m10.1b"><apply id="S7.Thmtheorem17.p1.10.10.m10.1.1.cmml" xref="S7.Thmtheorem17.p1.10.10.m10.1.1"><notin id="S7.Thmtheorem17.p1.10.10.m10.1.1.1.cmml" xref="S7.Thmtheorem17.p1.10.10.m10.1.1.1"></notin><apply id="S7.Thmtheorem17.p1.10.10.m10.1.1.2.cmml" xref="S7.Thmtheorem17.p1.10.10.m10.1.1.2"><ci id="S7.Thmtheorem17.p1.10.10.m10.1.1.2.1.cmml" xref="S7.Thmtheorem17.p1.10.10.m10.1.1.2.1">¯</ci><apply id="S7.Thmtheorem17.p1.10.10.m10.1.1.2.2.cmml" xref="S7.Thmtheorem17.p1.10.10.m10.1.1.2.2"><times id="S7.Thmtheorem17.p1.10.10.m10.1.1.2.2.1.cmml" xref="S7.Thmtheorem17.p1.10.10.m10.1.1.2.2.1"></times><ci id="S7.Thmtheorem17.p1.10.10.m10.1.1.2.2.2.cmml" xref="S7.Thmtheorem17.p1.10.10.m10.1.1.2.2.2">𝑢</ci><ci id="S7.Thmtheorem17.p1.10.10.m10.1.1.2.2.3.cmml" xref="S7.Thmtheorem17.p1.10.10.m10.1.1.2.2.3">𝑣</ci></apply></apply><apply id="S7.Thmtheorem17.p1.10.10.m10.1.1.3.cmml" xref="S7.Thmtheorem17.p1.10.10.m10.1.1.3"><csymbol cd="ambiguous" id="S7.Thmtheorem17.p1.10.10.m10.1.1.3.1.cmml" xref="S7.Thmtheorem17.p1.10.10.m10.1.1.3">subscript</csymbol><ci id="S7.Thmtheorem17.p1.10.10.m10.1.1.3.2.cmml" xref="S7.Thmtheorem17.p1.10.10.m10.1.1.3.2">𝐸</ci><ci id="S7.Thmtheorem17.p1.10.10.m10.1.1.3.3.cmml" xref="S7.Thmtheorem17.p1.10.10.m10.1.1.3.3">𝑘</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.Thmtheorem17.p1.10.10.m10.1c">\overline{uv}\not\in E_{k}</annotation><annotation encoding="application/x-llamapun" id="S7.Thmtheorem17.p1.10.10.m10.1d">over¯ start_ARG italic_u italic_v end_ARG ∉ italic_E start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT</annotation></semantics></math> for all <math alttext="k>s" class="ltx_Math" display="inline" id="S7.Thmtheorem17.p1.11.11.m11.1"><semantics id="S7.Thmtheorem17.p1.11.11.m11.1a"><mrow id="S7.Thmtheorem17.p1.11.11.m11.1.1" xref="S7.Thmtheorem17.p1.11.11.m11.1.1.cmml"><mi id="S7.Thmtheorem17.p1.11.11.m11.1.1.2" xref="S7.Thmtheorem17.p1.11.11.m11.1.1.2.cmml">k</mi><mo id="S7.Thmtheorem17.p1.11.11.m11.1.1.1" xref="S7.Thmtheorem17.p1.11.11.m11.1.1.1.cmml">></mo><mi id="S7.Thmtheorem17.p1.11.11.m11.1.1.3" xref="S7.Thmtheorem17.p1.11.11.m11.1.1.3.cmml">s</mi></mrow><annotation-xml encoding="MathML-Content" id="S7.Thmtheorem17.p1.11.11.m11.1b"><apply id="S7.Thmtheorem17.p1.11.11.m11.1.1.cmml" xref="S7.Thmtheorem17.p1.11.11.m11.1.1"><gt id="S7.Thmtheorem17.p1.11.11.m11.1.1.1.cmml" xref="S7.Thmtheorem17.p1.11.11.m11.1.1.1"></gt><ci id="S7.Thmtheorem17.p1.11.11.m11.1.1.2.cmml" xref="S7.Thmtheorem17.p1.11.11.m11.1.1.2">𝑘</ci><ci id="S7.Thmtheorem17.p1.11.11.m11.1.1.3.cmml" xref="S7.Thmtheorem17.p1.11.11.m11.1.1.3">𝑠</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.Thmtheorem17.p1.11.11.m11.1c">k>s</annotation><annotation encoding="application/x-llamapun" id="S7.Thmtheorem17.p1.11.11.m11.1d">italic_k > italic_s</annotation></semantics></math>. Thus, <math alttext="D_{s}" class="ltx_Math" display="inline" id="S7.Thmtheorem17.p1.12.12.m12.1"><semantics id="S7.Thmtheorem17.p1.12.12.m12.1a"><msub id="S7.Thmtheorem17.p1.12.12.m12.1.1" xref="S7.Thmtheorem17.p1.12.12.m12.1.1.cmml"><mi id="S7.Thmtheorem17.p1.12.12.m12.1.1.2" xref="S7.Thmtheorem17.p1.12.12.m12.1.1.2.cmml">D</mi><mi id="S7.Thmtheorem17.p1.12.12.m12.1.1.3" xref="S7.Thmtheorem17.p1.12.12.m12.1.1.3.cmml">s</mi></msub><annotation-xml encoding="MathML-Content" id="S7.Thmtheorem17.p1.12.12.m12.1b"><apply id="S7.Thmtheorem17.p1.12.12.m12.1.1.cmml" xref="S7.Thmtheorem17.p1.12.12.m12.1.1"><csymbol cd="ambiguous" id="S7.Thmtheorem17.p1.12.12.m12.1.1.1.cmml" xref="S7.Thmtheorem17.p1.12.12.m12.1.1">subscript</csymbol><ci id="S7.Thmtheorem17.p1.12.12.m12.1.1.2.cmml" xref="S7.Thmtheorem17.p1.12.12.m12.1.1.2">𝐷</ci><ci id="S7.Thmtheorem17.p1.12.12.m12.1.1.3.cmml" xref="S7.Thmtheorem17.p1.12.12.m12.1.1.3">𝑠</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.Thmtheorem17.p1.12.12.m12.1c">D_{s}</annotation><annotation encoding="application/x-llamapun" id="S7.Thmtheorem17.p1.12.12.m12.1d">italic_D start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT</annotation></semantics></math> is a subgraph of <math alttext="D_{s+1}" class="ltx_Math" display="inline" id="S7.Thmtheorem17.p1.13.13.m13.1"><semantics id="S7.Thmtheorem17.p1.13.13.m13.1a"><msub id="S7.Thmtheorem17.p1.13.13.m13.1.1" xref="S7.Thmtheorem17.p1.13.13.m13.1.1.cmml"><mi id="S7.Thmtheorem17.p1.13.13.m13.1.1.2" xref="S7.Thmtheorem17.p1.13.13.m13.1.1.2.cmml">D</mi><mrow id="S7.Thmtheorem17.p1.13.13.m13.1.1.3" xref="S7.Thmtheorem17.p1.13.13.m13.1.1.3.cmml"><mi id="S7.Thmtheorem17.p1.13.13.m13.1.1.3.2" xref="S7.Thmtheorem17.p1.13.13.m13.1.1.3.2.cmml">s</mi><mo id="S7.Thmtheorem17.p1.13.13.m13.1.1.3.1" xref="S7.Thmtheorem17.p1.13.13.m13.1.1.3.1.cmml">+</mo><mn id="S7.Thmtheorem17.p1.13.13.m13.1.1.3.3" xref="S7.Thmtheorem17.p1.13.13.m13.1.1.3.3.cmml">1</mn></mrow></msub><annotation-xml encoding="MathML-Content" id="S7.Thmtheorem17.p1.13.13.m13.1b"><apply id="S7.Thmtheorem17.p1.13.13.m13.1.1.cmml" xref="S7.Thmtheorem17.p1.13.13.m13.1.1"><csymbol cd="ambiguous" id="S7.Thmtheorem17.p1.13.13.m13.1.1.1.cmml" xref="S7.Thmtheorem17.p1.13.13.m13.1.1">subscript</csymbol><ci id="S7.Thmtheorem17.p1.13.13.m13.1.1.2.cmml" xref="S7.Thmtheorem17.p1.13.13.m13.1.1.2">𝐷</ci><apply id="S7.Thmtheorem17.p1.13.13.m13.1.1.3.cmml" xref="S7.Thmtheorem17.p1.13.13.m13.1.1.3"><plus id="S7.Thmtheorem17.p1.13.13.m13.1.1.3.1.cmml" xref="S7.Thmtheorem17.p1.13.13.m13.1.1.3.1"></plus><ci id="S7.Thmtheorem17.p1.13.13.m13.1.1.3.2.cmml" xref="S7.Thmtheorem17.p1.13.13.m13.1.1.3.2">𝑠</ci><cn id="S7.Thmtheorem17.p1.13.13.m13.1.1.3.3.cmml" type="integer" xref="S7.Thmtheorem17.p1.13.13.m13.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.Thmtheorem17.p1.13.13.m13.1c">D_{s+1}</annotation><annotation encoding="application/x-llamapun" id="S7.Thmtheorem17.p1.13.13.m13.1d">italic_D start_POSTSUBSCRIPT italic_s + 1 end_POSTSUBSCRIPT</annotation></semantics></math>.</span></p> </div> <div class="ltx_para" id="S7.Thmtheorem17.p2"> <p class="ltx_p" id="S7.Thmtheorem17.p2.16"><span class="ltx_text ltx_font_italic" id="S7.Thmtheorem17.p2.16.16">In <math alttext="(h:s:m)" class="ltx_Math" display="inline" id="S7.Thmtheorem17.p2.1.1.m1.1"><semantics id="S7.Thmtheorem17.p2.1.1.m1.1a"><mrow id="S7.Thmtheorem17.p2.1.1.m1.1.1.1" xref="S7.Thmtheorem17.p2.1.1.m1.1.1.1.1.cmml"><mo id="S7.Thmtheorem17.p2.1.1.m1.1.1.1.2" stretchy="false" xref="S7.Thmtheorem17.p2.1.1.m1.1.1.1.1.cmml">(</mo><mrow id="S7.Thmtheorem17.p2.1.1.m1.1.1.1.1" xref="S7.Thmtheorem17.p2.1.1.m1.1.1.1.1.cmml"><mi id="S7.Thmtheorem17.p2.1.1.m1.1.1.1.1.2" xref="S7.Thmtheorem17.p2.1.1.m1.1.1.1.1.2.cmml">h</mi><mo id="S7.Thmtheorem17.p2.1.1.m1.1.1.1.1.3" lspace="0.278em" rspace="0.278em" xref="S7.Thmtheorem17.p2.1.1.m1.1.1.1.1.3.cmml">:</mo><mi id="S7.Thmtheorem17.p2.1.1.m1.1.1.1.1.4" xref="S7.Thmtheorem17.p2.1.1.m1.1.1.1.1.4.cmml">s</mi><mo id="S7.Thmtheorem17.p2.1.1.m1.1.1.1.1.5" lspace="0.278em" rspace="0.278em" xref="S7.Thmtheorem17.p2.1.1.m1.1.1.1.1.5.cmml">:</mo><mi id="S7.Thmtheorem17.p2.1.1.m1.1.1.1.1.6" xref="S7.Thmtheorem17.p2.1.1.m1.1.1.1.1.6.cmml">m</mi></mrow><mo id="S7.Thmtheorem17.p2.1.1.m1.1.1.1.3" stretchy="false" xref="S7.Thmtheorem17.p2.1.1.m1.1.1.1.1.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S7.Thmtheorem17.p2.1.1.m1.1b"><apply id="S7.Thmtheorem17.p2.1.1.m1.1.1.1.1.cmml" xref="S7.Thmtheorem17.p2.1.1.m1.1.1.1"><and id="S7.Thmtheorem17.p2.1.1.m1.1.1.1.1a.cmml" xref="S7.Thmtheorem17.p2.1.1.m1.1.1.1"></and><apply id="S7.Thmtheorem17.p2.1.1.m1.1.1.1.1b.cmml" xref="S7.Thmtheorem17.p2.1.1.m1.1.1.1"><ci id="S7.Thmtheorem17.p2.1.1.m1.1.1.1.1.3.cmml" xref="S7.Thmtheorem17.p2.1.1.m1.1.1.1.1.3">:</ci><ci id="S7.Thmtheorem17.p2.1.1.m1.1.1.1.1.2.cmml" xref="S7.Thmtheorem17.p2.1.1.m1.1.1.1.1.2">ℎ</ci><ci id="S7.Thmtheorem17.p2.1.1.m1.1.1.1.1.4.cmml" xref="S7.Thmtheorem17.p2.1.1.m1.1.1.1.1.4">𝑠</ci></apply><apply id="S7.Thmtheorem17.p2.1.1.m1.1.1.1.1c.cmml" xref="S7.Thmtheorem17.p2.1.1.m1.1.1.1"><ci id="S7.Thmtheorem17.p2.1.1.m1.1.1.1.1.5.cmml" xref="S7.Thmtheorem17.p2.1.1.m1.1.1.1.1.5">:</ci><share href="https://arxiv.org/html/2411.12694v2#S7.Thmtheorem17.p2.1.1.m1.1.1.1.1.4.cmml" id="S7.Thmtheorem17.p2.1.1.m1.1.1.1.1d.cmml" xref="S7.Thmtheorem17.p2.1.1.m1.1.1.1"></share><ci id="S7.Thmtheorem17.p2.1.1.m1.1.1.1.1.6.cmml" xref="S7.Thmtheorem17.p2.1.1.m1.1.1.1.1.6">𝑚</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.Thmtheorem17.p2.1.1.m1.1c">(h:s:m)</annotation><annotation encoding="application/x-llamapun" id="S7.Thmtheorem17.p2.1.1.m1.1d">( italic_h : italic_s : italic_m )</annotation></semantics></math> we compute a blocking flow <math alttext="f" class="ltx_Math" display="inline" id="S7.Thmtheorem17.p2.2.2.m2.1"><semantics id="S7.Thmtheorem17.p2.2.2.m2.1a"><mi id="S7.Thmtheorem17.p2.2.2.m2.1.1" xref="S7.Thmtheorem17.p2.2.2.m2.1.1.cmml">f</mi><annotation-xml encoding="MathML-Content" id="S7.Thmtheorem17.p2.2.2.m2.1b"><ci id="S7.Thmtheorem17.p2.2.2.m2.1.1.cmml" xref="S7.Thmtheorem17.p2.2.2.m2.1.1">𝑓</ci></annotation-xml><annotation encoding="application/x-tex" id="S7.Thmtheorem17.p2.2.2.m2.1c">f</annotation><annotation encoding="application/x-llamapun" id="S7.Thmtheorem17.p2.2.2.m2.1d">italic_f</annotation></semantics></math> from <math alttext="S" class="ltx_Math" display="inline" id="S7.Thmtheorem17.p2.3.3.m3.1"><semantics id="S7.Thmtheorem17.p2.3.3.m3.1a"><mi id="S7.Thmtheorem17.p2.3.3.m3.1.1" xref="S7.Thmtheorem17.p2.3.3.m3.1.1.cmml">S</mi><annotation-xml encoding="MathML-Content" id="S7.Thmtheorem17.p2.3.3.m3.1b"><ci id="S7.Thmtheorem17.p2.3.3.m3.1.1.cmml" xref="S7.Thmtheorem17.p2.3.3.m3.1.1">𝑆</ci></annotation-xml><annotation encoding="application/x-tex" id="S7.Thmtheorem17.p2.3.3.m3.1c">S</annotation><annotation encoding="application/x-llamapun" id="S7.Thmtheorem17.p2.3.3.m3.1d">italic_S</annotation></semantics></math> to <math alttext="T" class="ltx_Math" display="inline" id="S7.Thmtheorem17.p2.4.4.m4.1"><semantics id="S7.Thmtheorem17.p2.4.4.m4.1a"><mi id="S7.Thmtheorem17.p2.4.4.m4.1.1" xref="S7.Thmtheorem17.p2.4.4.m4.1.1.cmml">T</mi><annotation-xml encoding="MathML-Content" id="S7.Thmtheorem17.p2.4.4.m4.1b"><ci id="S7.Thmtheorem17.p2.4.4.m4.1.1.cmml" xref="S7.Thmtheorem17.p2.4.4.m4.1.1">𝑇</ci></annotation-xml><annotation encoding="application/x-tex" id="S7.Thmtheorem17.p2.4.4.m4.1c">T</annotation><annotation encoding="application/x-llamapun" id="S7.Thmtheorem17.p2.4.4.m4.1d">italic_T</annotation></semantics></math> and flip edges according to the flow. We claim that this blocking flow implies that for all <math alttext="u\in S^{s}" class="ltx_Math" display="inline" id="S7.Thmtheorem17.p2.5.5.m5.1"><semantics id="S7.Thmtheorem17.p2.5.5.m5.1a"><mrow id="S7.Thmtheorem17.p2.5.5.m5.1.1" xref="S7.Thmtheorem17.p2.5.5.m5.1.1.cmml"><mi id="S7.Thmtheorem17.p2.5.5.m5.1.1.2" xref="S7.Thmtheorem17.p2.5.5.m5.1.1.2.cmml">u</mi><mo id="S7.Thmtheorem17.p2.5.5.m5.1.1.1" xref="S7.Thmtheorem17.p2.5.5.m5.1.1.1.cmml">∈</mo><msup id="S7.Thmtheorem17.p2.5.5.m5.1.1.3" xref="S7.Thmtheorem17.p2.5.5.m5.1.1.3.cmml"><mi id="S7.Thmtheorem17.p2.5.5.m5.1.1.3.2" xref="S7.Thmtheorem17.p2.5.5.m5.1.1.3.2.cmml">S</mi><mi id="S7.Thmtheorem17.p2.5.5.m5.1.1.3.3" xref="S7.Thmtheorem17.p2.5.5.m5.1.1.3.3.cmml">s</mi></msup></mrow><annotation-xml encoding="MathML-Content" id="S7.Thmtheorem17.p2.5.5.m5.1b"><apply id="S7.Thmtheorem17.p2.5.5.m5.1.1.cmml" xref="S7.Thmtheorem17.p2.5.5.m5.1.1"><in id="S7.Thmtheorem17.p2.5.5.m5.1.1.1.cmml" xref="S7.Thmtheorem17.p2.5.5.m5.1.1.1"></in><ci id="S7.Thmtheorem17.p2.5.5.m5.1.1.2.cmml" xref="S7.Thmtheorem17.p2.5.5.m5.1.1.2">𝑢</ci><apply id="S7.Thmtheorem17.p2.5.5.m5.1.1.3.cmml" xref="S7.Thmtheorem17.p2.5.5.m5.1.1.3"><csymbol cd="ambiguous" id="S7.Thmtheorem17.p2.5.5.m5.1.1.3.1.cmml" xref="S7.Thmtheorem17.p2.5.5.m5.1.1.3">superscript</csymbol><ci id="S7.Thmtheorem17.p2.5.5.m5.1.1.3.2.cmml" xref="S7.Thmtheorem17.p2.5.5.m5.1.1.3.2">𝑆</ci><ci id="S7.Thmtheorem17.p2.5.5.m5.1.1.3.3.cmml" xref="S7.Thmtheorem17.p2.5.5.m5.1.1.3.3">𝑠</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.Thmtheorem17.p2.5.5.m5.1c">u\in S^{s}</annotation><annotation encoding="application/x-llamapun" id="S7.Thmtheorem17.p2.5.5.m5.1d">italic_u ∈ italic_S start_POSTSUPERSCRIPT italic_s end_POSTSUPERSCRIPT</annotation></semantics></math>, there exists no path from <math alttext="u" class="ltx_Math" display="inline" id="S7.Thmtheorem17.p2.6.6.m6.1"><semantics id="S7.Thmtheorem17.p2.6.6.m6.1a"><mi id="S7.Thmtheorem17.p2.6.6.m6.1.1" xref="S7.Thmtheorem17.p2.6.6.m6.1.1.cmml">u</mi><annotation-xml encoding="MathML-Content" id="S7.Thmtheorem17.p2.6.6.m6.1b"><ci id="S7.Thmtheorem17.p2.6.6.m6.1.1.cmml" xref="S7.Thmtheorem17.p2.6.6.m6.1.1">𝑢</ci></annotation-xml><annotation encoding="application/x-tex" id="S7.Thmtheorem17.p2.6.6.m6.1c">u</annotation><annotation encoding="application/x-llamapun" id="S7.Thmtheorem17.p2.6.6.m6.1d">italic_u</annotation></semantics></math> to <math alttext="T_{m}" class="ltx_Math" display="inline" id="S7.Thmtheorem17.p2.7.7.m7.1"><semantics id="S7.Thmtheorem17.p2.7.7.m7.1a"><msub id="S7.Thmtheorem17.p2.7.7.m7.1.1" xref="S7.Thmtheorem17.p2.7.7.m7.1.1.cmml"><mi id="S7.Thmtheorem17.p2.7.7.m7.1.1.2" xref="S7.Thmtheorem17.p2.7.7.m7.1.1.2.cmml">T</mi><mi id="S7.Thmtheorem17.p2.7.7.m7.1.1.3" xref="S7.Thmtheorem17.p2.7.7.m7.1.1.3.cmml">m</mi></msub><annotation-xml encoding="MathML-Content" id="S7.Thmtheorem17.p2.7.7.m7.1b"><apply id="S7.Thmtheorem17.p2.7.7.m7.1.1.cmml" xref="S7.Thmtheorem17.p2.7.7.m7.1.1"><csymbol cd="ambiguous" id="S7.Thmtheorem17.p2.7.7.m7.1.1.1.cmml" xref="S7.Thmtheorem17.p2.7.7.m7.1.1">subscript</csymbol><ci id="S7.Thmtheorem17.p2.7.7.m7.1.1.2.cmml" xref="S7.Thmtheorem17.p2.7.7.m7.1.1.2">𝑇</ci><ci id="S7.Thmtheorem17.p2.7.7.m7.1.1.3.cmml" xref="S7.Thmtheorem17.p2.7.7.m7.1.1.3">𝑚</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.Thmtheorem17.p2.7.7.m7.1c">T_{m}</annotation><annotation encoding="application/x-llamapun" id="S7.Thmtheorem17.p2.7.7.m7.1d">italic_T start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT</annotation></semantics></math> in <math alttext="E_{s+1}" class="ltx_Math" display="inline" id="S7.Thmtheorem17.p2.8.8.m8.1"><semantics id="S7.Thmtheorem17.p2.8.8.m8.1a"><msub id="S7.Thmtheorem17.p2.8.8.m8.1.1" xref="S7.Thmtheorem17.p2.8.8.m8.1.1.cmml"><mi id="S7.Thmtheorem17.p2.8.8.m8.1.1.2" xref="S7.Thmtheorem17.p2.8.8.m8.1.1.2.cmml">E</mi><mrow id="S7.Thmtheorem17.p2.8.8.m8.1.1.3" xref="S7.Thmtheorem17.p2.8.8.m8.1.1.3.cmml"><mi id="S7.Thmtheorem17.p2.8.8.m8.1.1.3.2" xref="S7.Thmtheorem17.p2.8.8.m8.1.1.3.2.cmml">s</mi><mo id="S7.Thmtheorem17.p2.8.8.m8.1.1.3.1" xref="S7.Thmtheorem17.p2.8.8.m8.1.1.3.1.cmml">+</mo><mn id="S7.Thmtheorem17.p2.8.8.m8.1.1.3.3" xref="S7.Thmtheorem17.p2.8.8.m8.1.1.3.3.cmml">1</mn></mrow></msub><annotation-xml encoding="MathML-Content" id="S7.Thmtheorem17.p2.8.8.m8.1b"><apply id="S7.Thmtheorem17.p2.8.8.m8.1.1.cmml" xref="S7.Thmtheorem17.p2.8.8.m8.1.1"><csymbol cd="ambiguous" id="S7.Thmtheorem17.p2.8.8.m8.1.1.1.cmml" xref="S7.Thmtheorem17.p2.8.8.m8.1.1">subscript</csymbol><ci id="S7.Thmtheorem17.p2.8.8.m8.1.1.2.cmml" xref="S7.Thmtheorem17.p2.8.8.m8.1.1.2">𝐸</ci><apply id="S7.Thmtheorem17.p2.8.8.m8.1.1.3.cmml" xref="S7.Thmtheorem17.p2.8.8.m8.1.1.3"><plus id="S7.Thmtheorem17.p2.8.8.m8.1.1.3.1.cmml" xref="S7.Thmtheorem17.p2.8.8.m8.1.1.3.1"></plus><ci id="S7.Thmtheorem17.p2.8.8.m8.1.1.3.2.cmml" xref="S7.Thmtheorem17.p2.8.8.m8.1.1.3.2">𝑠</ci><cn id="S7.Thmtheorem17.p2.8.8.m8.1.1.3.3.cmml" type="integer" xref="S7.Thmtheorem17.p2.8.8.m8.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.Thmtheorem17.p2.8.8.m8.1c">E_{s+1}</annotation><annotation encoding="application/x-llamapun" id="S7.Thmtheorem17.p2.8.8.m8.1d">italic_E start_POSTSUBSCRIPT italic_s + 1 end_POSTSUBSCRIPT</annotation></semantics></math> (and thus <math alttext="u\not\in V_{k}" class="ltx_Math" display="inline" id="S7.Thmtheorem17.p2.9.9.m9.1"><semantics id="S7.Thmtheorem17.p2.9.9.m9.1a"><mrow id="S7.Thmtheorem17.p2.9.9.m9.1.1" xref="S7.Thmtheorem17.p2.9.9.m9.1.1.cmml"><mi id="S7.Thmtheorem17.p2.9.9.m9.1.1.2" xref="S7.Thmtheorem17.p2.9.9.m9.1.1.2.cmml">u</mi><mo id="S7.Thmtheorem17.p2.9.9.m9.1.1.1" xref="S7.Thmtheorem17.p2.9.9.m9.1.1.1.cmml">∉</mo><msub id="S7.Thmtheorem17.p2.9.9.m9.1.1.3" xref="S7.Thmtheorem17.p2.9.9.m9.1.1.3.cmml"><mi id="S7.Thmtheorem17.p2.9.9.m9.1.1.3.2" xref="S7.Thmtheorem17.p2.9.9.m9.1.1.3.2.cmml">V</mi><mi id="S7.Thmtheorem17.p2.9.9.m9.1.1.3.3" xref="S7.Thmtheorem17.p2.9.9.m9.1.1.3.3.cmml">k</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S7.Thmtheorem17.p2.9.9.m9.1b"><apply id="S7.Thmtheorem17.p2.9.9.m9.1.1.cmml" xref="S7.Thmtheorem17.p2.9.9.m9.1.1"><notin id="S7.Thmtheorem17.p2.9.9.m9.1.1.1.cmml" xref="S7.Thmtheorem17.p2.9.9.m9.1.1.1"></notin><ci id="S7.Thmtheorem17.p2.9.9.m9.1.1.2.cmml" xref="S7.Thmtheorem17.p2.9.9.m9.1.1.2">𝑢</ci><apply id="S7.Thmtheorem17.p2.9.9.m9.1.1.3.cmml" xref="S7.Thmtheorem17.p2.9.9.m9.1.1.3"><csymbol cd="ambiguous" id="S7.Thmtheorem17.p2.9.9.m9.1.1.3.1.cmml" xref="S7.Thmtheorem17.p2.9.9.m9.1.1.3">subscript</csymbol><ci id="S7.Thmtheorem17.p2.9.9.m9.1.1.3.2.cmml" xref="S7.Thmtheorem17.p2.9.9.m9.1.1.3.2">𝑉</ci><ci id="S7.Thmtheorem17.p2.9.9.m9.1.1.3.3.cmml" xref="S7.Thmtheorem17.p2.9.9.m9.1.1.3.3">𝑘</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.Thmtheorem17.p2.9.9.m9.1c">u\not\in V_{k}</annotation><annotation encoding="application/x-llamapun" id="S7.Thmtheorem17.p2.9.9.m9.1d">italic_u ∉ italic_V start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT</annotation></semantics></math> for <math alttext="k>s" class="ltx_Math" display="inline" id="S7.Thmtheorem17.p2.10.10.m10.1"><semantics id="S7.Thmtheorem17.p2.10.10.m10.1a"><mrow id="S7.Thmtheorem17.p2.10.10.m10.1.1" xref="S7.Thmtheorem17.p2.10.10.m10.1.1.cmml"><mi id="S7.Thmtheorem17.p2.10.10.m10.1.1.2" xref="S7.Thmtheorem17.p2.10.10.m10.1.1.2.cmml">k</mi><mo id="S7.Thmtheorem17.p2.10.10.m10.1.1.1" xref="S7.Thmtheorem17.p2.10.10.m10.1.1.1.cmml">></mo><mi id="S7.Thmtheorem17.p2.10.10.m10.1.1.3" xref="S7.Thmtheorem17.p2.10.10.m10.1.1.3.cmml">s</mi></mrow><annotation-xml encoding="MathML-Content" id="S7.Thmtheorem17.p2.10.10.m10.1b"><apply id="S7.Thmtheorem17.p2.10.10.m10.1.1.cmml" xref="S7.Thmtheorem17.p2.10.10.m10.1.1"><gt id="S7.Thmtheorem17.p2.10.10.m10.1.1.1.cmml" xref="S7.Thmtheorem17.p2.10.10.m10.1.1.1"></gt><ci id="S7.Thmtheorem17.p2.10.10.m10.1.1.2.cmml" xref="S7.Thmtheorem17.p2.10.10.m10.1.1.2">𝑘</ci><ci id="S7.Thmtheorem17.p2.10.10.m10.1.1.3.cmml" xref="S7.Thmtheorem17.p2.10.10.m10.1.1.3">𝑠</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.Thmtheorem17.p2.10.10.m10.1c">k>s</annotation><annotation encoding="application/x-llamapun" id="S7.Thmtheorem17.p2.10.10.m10.1d">italic_k > italic_s</annotation></semantics></math>). Indeed, for any <math alttext="u\in S^{s}" class="ltx_Math" display="inline" id="S7.Thmtheorem17.p2.11.11.m11.1"><semantics id="S7.Thmtheorem17.p2.11.11.m11.1a"><mrow id="S7.Thmtheorem17.p2.11.11.m11.1.1" xref="S7.Thmtheorem17.p2.11.11.m11.1.1.cmml"><mi id="S7.Thmtheorem17.p2.11.11.m11.1.1.2" xref="S7.Thmtheorem17.p2.11.11.m11.1.1.2.cmml">u</mi><mo id="S7.Thmtheorem17.p2.11.11.m11.1.1.1" xref="S7.Thmtheorem17.p2.11.11.m11.1.1.1.cmml">∈</mo><msup id="S7.Thmtheorem17.p2.11.11.m11.1.1.3" xref="S7.Thmtheorem17.p2.11.11.m11.1.1.3.cmml"><mi id="S7.Thmtheorem17.p2.11.11.m11.1.1.3.2" xref="S7.Thmtheorem17.p2.11.11.m11.1.1.3.2.cmml">S</mi><mi id="S7.Thmtheorem17.p2.11.11.m11.1.1.3.3" xref="S7.Thmtheorem17.p2.11.11.m11.1.1.3.3.cmml">s</mi></msup></mrow><annotation-xml encoding="MathML-Content" id="S7.Thmtheorem17.p2.11.11.m11.1b"><apply id="S7.Thmtheorem17.p2.11.11.m11.1.1.cmml" xref="S7.Thmtheorem17.p2.11.11.m11.1.1"><in id="S7.Thmtheorem17.p2.11.11.m11.1.1.1.cmml" xref="S7.Thmtheorem17.p2.11.11.m11.1.1.1"></in><ci id="S7.Thmtheorem17.p2.11.11.m11.1.1.2.cmml" xref="S7.Thmtheorem17.p2.11.11.m11.1.1.2">𝑢</ci><apply id="S7.Thmtheorem17.p2.11.11.m11.1.1.3.cmml" xref="S7.Thmtheorem17.p2.11.11.m11.1.1.3"><csymbol cd="ambiguous" id="S7.Thmtheorem17.p2.11.11.m11.1.1.3.1.cmml" xref="S7.Thmtheorem17.p2.11.11.m11.1.1.3">superscript</csymbol><ci id="S7.Thmtheorem17.p2.11.11.m11.1.1.3.2.cmml" xref="S7.Thmtheorem17.p2.11.11.m11.1.1.3.2">𝑆</ci><ci id="S7.Thmtheorem17.p2.11.11.m11.1.1.3.3.cmml" xref="S7.Thmtheorem17.p2.11.11.m11.1.1.3.3">𝑠</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.Thmtheorem17.p2.11.11.m11.1c">u\in S^{s}</annotation><annotation encoding="application/x-llamapun" id="S7.Thmtheorem17.p2.11.11.m11.1d">italic_u ∈ italic_S start_POSTSUPERSCRIPT italic_s end_POSTSUPERSCRIPT</annotation></semantics></math> and any <math alttext="v\in T_{m}" class="ltx_Math" display="inline" id="S7.Thmtheorem17.p2.12.12.m12.1"><semantics id="S7.Thmtheorem17.p2.12.12.m12.1a"><mrow id="S7.Thmtheorem17.p2.12.12.m12.1.1" xref="S7.Thmtheorem17.p2.12.12.m12.1.1.cmml"><mi id="S7.Thmtheorem17.p2.12.12.m12.1.1.2" xref="S7.Thmtheorem17.p2.12.12.m12.1.1.2.cmml">v</mi><mo id="S7.Thmtheorem17.p2.12.12.m12.1.1.1" xref="S7.Thmtheorem17.p2.12.12.m12.1.1.1.cmml">∈</mo><msub id="S7.Thmtheorem17.p2.12.12.m12.1.1.3" xref="S7.Thmtheorem17.p2.12.12.m12.1.1.3.cmml"><mi id="S7.Thmtheorem17.p2.12.12.m12.1.1.3.2" xref="S7.Thmtheorem17.p2.12.12.m12.1.1.3.2.cmml">T</mi><mi id="S7.Thmtheorem17.p2.12.12.m12.1.1.3.3" xref="S7.Thmtheorem17.p2.12.12.m12.1.1.3.3.cmml">m</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S7.Thmtheorem17.p2.12.12.m12.1b"><apply id="S7.Thmtheorem17.p2.12.12.m12.1.1.cmml" xref="S7.Thmtheorem17.p2.12.12.m12.1.1"><in id="S7.Thmtheorem17.p2.12.12.m12.1.1.1.cmml" xref="S7.Thmtheorem17.p2.12.12.m12.1.1.1"></in><ci id="S7.Thmtheorem17.p2.12.12.m12.1.1.2.cmml" xref="S7.Thmtheorem17.p2.12.12.m12.1.1.2">𝑣</ci><apply id="S7.Thmtheorem17.p2.12.12.m12.1.1.3.cmml" xref="S7.Thmtheorem17.p2.12.12.m12.1.1.3"><csymbol cd="ambiguous" id="S7.Thmtheorem17.p2.12.12.m12.1.1.3.1.cmml" xref="S7.Thmtheorem17.p2.12.12.m12.1.1.3">subscript</csymbol><ci id="S7.Thmtheorem17.p2.12.12.m12.1.1.3.2.cmml" xref="S7.Thmtheorem17.p2.12.12.m12.1.1.3.2">𝑇</ci><ci id="S7.Thmtheorem17.p2.12.12.m12.1.1.3.3.cmml" xref="S7.Thmtheorem17.p2.12.12.m12.1.1.3.3">𝑚</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.Thmtheorem17.p2.12.12.m12.1c">v\in T_{m}</annotation><annotation encoding="application/x-llamapun" id="S7.Thmtheorem17.p2.12.12.m12.1d">italic_v ∈ italic_T start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT</annotation></semantics></math>, consider an arbitrary directed path <math alttext="\pi" class="ltx_Math" display="inline" id="S7.Thmtheorem17.p2.13.13.m13.1"><semantics id="S7.Thmtheorem17.p2.13.13.m13.1a"><mi id="S7.Thmtheorem17.p2.13.13.m13.1.1" xref="S7.Thmtheorem17.p2.13.13.m13.1.1.cmml">π</mi><annotation-xml encoding="MathML-Content" id="S7.Thmtheorem17.p2.13.13.m13.1b"><ci id="S7.Thmtheorem17.p2.13.13.m13.1.1.cmml" xref="S7.Thmtheorem17.p2.13.13.m13.1.1">𝜋</ci></annotation-xml><annotation encoding="application/x-tex" id="S7.Thmtheorem17.p2.13.13.m13.1c">\pi</annotation><annotation encoding="application/x-llamapun" id="S7.Thmtheorem17.p2.13.13.m13.1d">italic_π</annotation></semantics></math> from <math alttext="u" class="ltx_Math" display="inline" id="S7.Thmtheorem17.p2.14.14.m14.1"><semantics id="S7.Thmtheorem17.p2.14.14.m14.1a"><mi id="S7.Thmtheorem17.p2.14.14.m14.1.1" xref="S7.Thmtheorem17.p2.14.14.m14.1.1.cmml">u</mi><annotation-xml encoding="MathML-Content" id="S7.Thmtheorem17.p2.14.14.m14.1b"><ci id="S7.Thmtheorem17.p2.14.14.m14.1.1.cmml" xref="S7.Thmtheorem17.p2.14.14.m14.1.1">𝑢</ci></annotation-xml><annotation encoding="application/x-tex" id="S7.Thmtheorem17.p2.14.14.m14.1c">u</annotation><annotation encoding="application/x-llamapun" id="S7.Thmtheorem17.p2.14.14.m14.1d">italic_u</annotation></semantics></math> to <math alttext="v" class="ltx_Math" display="inline" id="S7.Thmtheorem17.p2.15.15.m15.1"><semantics id="S7.Thmtheorem17.p2.15.15.m15.1a"><mi id="S7.Thmtheorem17.p2.15.15.m15.1.1" xref="S7.Thmtheorem17.p2.15.15.m15.1.1.cmml">v</mi><annotation-xml encoding="MathML-Content" id="S7.Thmtheorem17.p2.15.15.m15.1b"><ci id="S7.Thmtheorem17.p2.15.15.m15.1.1.cmml" xref="S7.Thmtheorem17.p2.15.15.m15.1.1">𝑣</ci></annotation-xml><annotation encoding="application/x-tex" id="S7.Thmtheorem17.p2.15.15.m15.1c">v</annotation><annotation encoding="application/x-llamapun" id="S7.Thmtheorem17.p2.15.15.m15.1d">italic_v</annotation></semantics></math> in <math alttext="D_{s}" class="ltx_Math" display="inline" id="S7.Thmtheorem17.p2.16.16.m16.1"><semantics id="S7.Thmtheorem17.p2.16.16.m16.1a"><msub id="S7.Thmtheorem17.p2.16.16.m16.1.1" xref="S7.Thmtheorem17.p2.16.16.m16.1.1.cmml"><mi id="S7.Thmtheorem17.p2.16.16.m16.1.1.2" xref="S7.Thmtheorem17.p2.16.16.m16.1.1.2.cmml">D</mi><mi id="S7.Thmtheorem17.p2.16.16.m16.1.1.3" xref="S7.Thmtheorem17.p2.16.16.m16.1.1.3.cmml">s</mi></msub><annotation-xml encoding="MathML-Content" id="S7.Thmtheorem17.p2.16.16.m16.1b"><apply id="S7.Thmtheorem17.p2.16.16.m16.1.1.cmml" xref="S7.Thmtheorem17.p2.16.16.m16.1.1"><csymbol cd="ambiguous" id="S7.Thmtheorem17.p2.16.16.m16.1.1.1.cmml" xref="S7.Thmtheorem17.p2.16.16.m16.1.1">subscript</csymbol><ci id="S7.Thmtheorem17.p2.16.16.m16.1.1.2.cmml" xref="S7.Thmtheorem17.p2.16.16.m16.1.1.2">𝐷</ci><ci id="S7.Thmtheorem17.p2.16.16.m16.1.1.3.cmml" xref="S7.Thmtheorem17.p2.16.16.m16.1.1.3">𝑠</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.Thmtheorem17.p2.16.16.m16.1c">D_{s}</annotation><annotation encoding="application/x-llamapun" id="S7.Thmtheorem17.p2.16.16.m16.1d">italic_D start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT</annotation></semantics></math>. One of three (not mutually exclusive) conditions must hold. Either:</span></p> <ul class="ltx_itemize" id="S7.I10"> <li class="ltx_item" id="S7.I10.i1" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">•</span> <div class="ltx_para" id="S7.I10.i1.p1"> <p class="ltx_p" id="S7.I10.i1.p1.4"><span class="ltx_text ltx_font_italic" id="S7.I10.i1.p1.4.1">There exists an </span><math alttext="\overline{ab}\in\pi" class="ltx_Math" display="inline" id="S7.I10.i1.p1.1.m1.1"><semantics id="S7.I10.i1.p1.1.m1.1a"><mrow id="S7.I10.i1.p1.1.m1.1.1" xref="S7.I10.i1.p1.1.m1.1.1.cmml"><mover accent="true" id="S7.I10.i1.p1.1.m1.1.1.2" xref="S7.I10.i1.p1.1.m1.1.1.2.cmml"><mrow id="S7.I10.i1.p1.1.m1.1.1.2.2" xref="S7.I10.i1.p1.1.m1.1.1.2.2.cmml"><mi id="S7.I10.i1.p1.1.m1.1.1.2.2.2" xref="S7.I10.i1.p1.1.m1.1.1.2.2.2.cmml">a</mi><mo id="S7.I10.i1.p1.1.m1.1.1.2.2.1" xref="S7.I10.i1.p1.1.m1.1.1.2.2.1.cmml">⁢</mo><mi id="S7.I10.i1.p1.1.m1.1.1.2.2.3" xref="S7.I10.i1.p1.1.m1.1.1.2.2.3.cmml">b</mi></mrow><mo id="S7.I10.i1.p1.1.m1.1.1.2.1" xref="S7.I10.i1.p1.1.m1.1.1.2.1.cmml">¯</mo></mover><mo id="S7.I10.i1.p1.1.m1.1.1.1" xref="S7.I10.i1.p1.1.m1.1.1.1.cmml">∈</mo><mi id="S7.I10.i1.p1.1.m1.1.1.3" xref="S7.I10.i1.p1.1.m1.1.1.3.cmml">π</mi></mrow><annotation-xml encoding="MathML-Content" id="S7.I10.i1.p1.1.m1.1b"><apply id="S7.I10.i1.p1.1.m1.1.1.cmml" xref="S7.I10.i1.p1.1.m1.1.1"><in id="S7.I10.i1.p1.1.m1.1.1.1.cmml" xref="S7.I10.i1.p1.1.m1.1.1.1"></in><apply id="S7.I10.i1.p1.1.m1.1.1.2.cmml" xref="S7.I10.i1.p1.1.m1.1.1.2"><ci id="S7.I10.i1.p1.1.m1.1.1.2.1.cmml" xref="S7.I10.i1.p1.1.m1.1.1.2.1">¯</ci><apply id="S7.I10.i1.p1.1.m1.1.1.2.2.cmml" xref="S7.I10.i1.p1.1.m1.1.1.2.2"><times id="S7.I10.i1.p1.1.m1.1.1.2.2.1.cmml" xref="S7.I10.i1.p1.1.m1.1.1.2.2.1"></times><ci id="S7.I10.i1.p1.1.m1.1.1.2.2.2.cmml" xref="S7.I10.i1.p1.1.m1.1.1.2.2.2">𝑎</ci><ci id="S7.I10.i1.p1.1.m1.1.1.2.2.3.cmml" xref="S7.I10.i1.p1.1.m1.1.1.2.2.3">𝑏</ci></apply></apply><ci id="S7.I10.i1.p1.1.m1.1.1.3.cmml" xref="S7.I10.i1.p1.1.m1.1.1.3">𝜋</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.I10.i1.p1.1.m1.1c">\overline{ab}\in\pi</annotation><annotation encoding="application/x-llamapun" id="S7.I10.i1.p1.1.m1.1d">over¯ start_ARG italic_a italic_b end_ARG ∈ italic_π</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S7.I10.i1.p1.4.2"> that is saturated in </span><math alttext="f" class="ltx_Math" display="inline" id="S7.I10.i1.p1.2.m2.1"><semantics id="S7.I10.i1.p1.2.m2.1a"><mi id="S7.I10.i1.p1.2.m2.1.1" xref="S7.I10.i1.p1.2.m2.1.1.cmml">f</mi><annotation-xml encoding="MathML-Content" id="S7.I10.i1.p1.2.m2.1b"><ci id="S7.I10.i1.p1.2.m2.1.1.cmml" xref="S7.I10.i1.p1.2.m2.1.1">𝑓</ci></annotation-xml><annotation encoding="application/x-tex" id="S7.I10.i1.p1.2.m2.1c">f</annotation><annotation encoding="application/x-llamapun" id="S7.I10.i1.p1.2.m2.1d">italic_f</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S7.I10.i1.p1.4.3">. Then </span><math alttext="\textsl{g}_{s+1}(a\!\to\!b)=0" class="ltx_Math" display="inline" id="S7.I10.i1.p1.3.m3.1"><semantics id="S7.I10.i1.p1.3.m3.1a"><mrow id="S7.I10.i1.p1.3.m3.1.1" xref="S7.I10.i1.p1.3.m3.1.1.cmml"><mrow id="S7.I10.i1.p1.3.m3.1.1.1" xref="S7.I10.i1.p1.3.m3.1.1.1.cmml"><msub id="S7.I10.i1.p1.3.m3.1.1.1.3" xref="S7.I10.i1.p1.3.m3.1.1.1.3.cmml"><mtext class="ltx_mathvariant_italic" id="S7.I10.i1.p1.3.m3.1.1.1.3.2" xref="S7.I10.i1.p1.3.m3.1.1.1.3.2a.cmml">g</mtext><mrow id="S7.I10.i1.p1.3.m3.1.1.1.3.3" xref="S7.I10.i1.p1.3.m3.1.1.1.3.3.cmml"><mi id="S7.I10.i1.p1.3.m3.1.1.1.3.3.2" xref="S7.I10.i1.p1.3.m3.1.1.1.3.3.2.cmml">s</mi><mo id="S7.I10.i1.p1.3.m3.1.1.1.3.3.1" xref="S7.I10.i1.p1.3.m3.1.1.1.3.3.1.cmml">+</mo><mn id="S7.I10.i1.p1.3.m3.1.1.1.3.3.3" xref="S7.I10.i1.p1.3.m3.1.1.1.3.3.3.cmml">1</mn></mrow></msub><mo id="S7.I10.i1.p1.3.m3.1.1.1.2" xref="S7.I10.i1.p1.3.m3.1.1.1.2.cmml">⁢</mo><mrow id="S7.I10.i1.p1.3.m3.1.1.1.1.1" xref="S7.I10.i1.p1.3.m3.1.1.1.1.1.1.cmml"><mo id="S7.I10.i1.p1.3.m3.1.1.1.1.1.2" stretchy="false" xref="S7.I10.i1.p1.3.m3.1.1.1.1.1.1.cmml">(</mo><mrow id="S7.I10.i1.p1.3.m3.1.1.1.1.1.1" xref="S7.I10.i1.p1.3.m3.1.1.1.1.1.1.cmml"><mi id="S7.I10.i1.p1.3.m3.1.1.1.1.1.1.2" xref="S7.I10.i1.p1.3.m3.1.1.1.1.1.1.2.cmml">a</mi><mo id="S7.I10.i1.p1.3.m3.1.1.1.1.1.1.1" lspace="0.108em" rspace="0.108em" stretchy="false" xref="S7.I10.i1.p1.3.m3.1.1.1.1.1.1.1.cmml">→</mo><mi id="S7.I10.i1.p1.3.m3.1.1.1.1.1.1.3" xref="S7.I10.i1.p1.3.m3.1.1.1.1.1.1.3.cmml">b</mi></mrow><mo id="S7.I10.i1.p1.3.m3.1.1.1.1.1.3" stretchy="false" xref="S7.I10.i1.p1.3.m3.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S7.I10.i1.p1.3.m3.1.1.2" xref="S7.I10.i1.p1.3.m3.1.1.2.cmml">=</mo><mn id="S7.I10.i1.p1.3.m3.1.1.3" xref="S7.I10.i1.p1.3.m3.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S7.I10.i1.p1.3.m3.1b"><apply id="S7.I10.i1.p1.3.m3.1.1.cmml" xref="S7.I10.i1.p1.3.m3.1.1"><eq id="S7.I10.i1.p1.3.m3.1.1.2.cmml" xref="S7.I10.i1.p1.3.m3.1.1.2"></eq><apply id="S7.I10.i1.p1.3.m3.1.1.1.cmml" xref="S7.I10.i1.p1.3.m3.1.1.1"><times id="S7.I10.i1.p1.3.m3.1.1.1.2.cmml" xref="S7.I10.i1.p1.3.m3.1.1.1.2"></times><apply id="S7.I10.i1.p1.3.m3.1.1.1.3.cmml" xref="S7.I10.i1.p1.3.m3.1.1.1.3"><csymbol cd="ambiguous" id="S7.I10.i1.p1.3.m3.1.1.1.3.1.cmml" xref="S7.I10.i1.p1.3.m3.1.1.1.3">subscript</csymbol><ci id="S7.I10.i1.p1.3.m3.1.1.1.3.2a.cmml" xref="S7.I10.i1.p1.3.m3.1.1.1.3.2"><mtext class="ltx_mathvariant_italic" id="S7.I10.i1.p1.3.m3.1.1.1.3.2.cmml" xref="S7.I10.i1.p1.3.m3.1.1.1.3.2">g</mtext></ci><apply id="S7.I10.i1.p1.3.m3.1.1.1.3.3.cmml" xref="S7.I10.i1.p1.3.m3.1.1.1.3.3"><plus id="S7.I10.i1.p1.3.m3.1.1.1.3.3.1.cmml" xref="S7.I10.i1.p1.3.m3.1.1.1.3.3.1"></plus><ci id="S7.I10.i1.p1.3.m3.1.1.1.3.3.2.cmml" xref="S7.I10.i1.p1.3.m3.1.1.1.3.3.2">𝑠</ci><cn id="S7.I10.i1.p1.3.m3.1.1.1.3.3.3.cmml" type="integer" xref="S7.I10.i1.p1.3.m3.1.1.1.3.3.3">1</cn></apply></apply><apply id="S7.I10.i1.p1.3.m3.1.1.1.1.1.1.cmml" xref="S7.I10.i1.p1.3.m3.1.1.1.1.1"><ci id="S7.I10.i1.p1.3.m3.1.1.1.1.1.1.1.cmml" xref="S7.I10.i1.p1.3.m3.1.1.1.1.1.1.1">→</ci><ci id="S7.I10.i1.p1.3.m3.1.1.1.1.1.1.2.cmml" xref="S7.I10.i1.p1.3.m3.1.1.1.1.1.1.2">𝑎</ci><ci id="S7.I10.i1.p1.3.m3.1.1.1.1.1.1.3.cmml" xref="S7.I10.i1.p1.3.m3.1.1.1.1.1.1.3">𝑏</ci></apply></apply><cn id="S7.I10.i1.p1.3.m3.1.1.3.cmml" type="integer" xref="S7.I10.i1.p1.3.m3.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.I10.i1.p1.3.m3.1c">\textsl{g}_{s+1}(a\!\to\!b)=0</annotation><annotation encoding="application/x-llamapun" id="S7.I10.i1.p1.3.m3.1d">g start_POSTSUBSCRIPT italic_s + 1 end_POSTSUBSCRIPT ( italic_a → italic_b ) = 0</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S7.I10.i1.p1.4.4"> and </span><math alttext="\pi\not\subseteq E_{s+1}" class="ltx_Math" display="inline" id="S7.I10.i1.p1.4.m4.1"><semantics id="S7.I10.i1.p1.4.m4.1a"><mrow id="S7.I10.i1.p1.4.m4.1.1" xref="S7.I10.i1.p1.4.m4.1.1.cmml"><mi id="S7.I10.i1.p1.4.m4.1.1.2" xref="S7.I10.i1.p1.4.m4.1.1.2.cmml">π</mi><mo id="S7.I10.i1.p1.4.m4.1.1.1" xref="S7.I10.i1.p1.4.m4.1.1.1.cmml">⊈</mo><msub id="S7.I10.i1.p1.4.m4.1.1.3" xref="S7.I10.i1.p1.4.m4.1.1.3.cmml"><mi id="S7.I10.i1.p1.4.m4.1.1.3.2" xref="S7.I10.i1.p1.4.m4.1.1.3.2.cmml">E</mi><mrow id="S7.I10.i1.p1.4.m4.1.1.3.3" xref="S7.I10.i1.p1.4.m4.1.1.3.3.cmml"><mi id="S7.I10.i1.p1.4.m4.1.1.3.3.2" xref="S7.I10.i1.p1.4.m4.1.1.3.3.2.cmml">s</mi><mo id="S7.I10.i1.p1.4.m4.1.1.3.3.1" xref="S7.I10.i1.p1.4.m4.1.1.3.3.1.cmml">+</mo><mn id="S7.I10.i1.p1.4.m4.1.1.3.3.3" xref="S7.I10.i1.p1.4.m4.1.1.3.3.3.cmml">1</mn></mrow></msub></mrow><annotation-xml encoding="MathML-Content" id="S7.I10.i1.p1.4.m4.1b"><apply id="S7.I10.i1.p1.4.m4.1.1.cmml" xref="S7.I10.i1.p1.4.m4.1.1"><csymbol cd="latexml" id="S7.I10.i1.p1.4.m4.1.1.1.cmml" xref="S7.I10.i1.p1.4.m4.1.1.1">not-subset-of-or-equals</csymbol><ci id="S7.I10.i1.p1.4.m4.1.1.2.cmml" xref="S7.I10.i1.p1.4.m4.1.1.2">𝜋</ci><apply id="S7.I10.i1.p1.4.m4.1.1.3.cmml" xref="S7.I10.i1.p1.4.m4.1.1.3"><csymbol cd="ambiguous" id="S7.I10.i1.p1.4.m4.1.1.3.1.cmml" xref="S7.I10.i1.p1.4.m4.1.1.3">subscript</csymbol><ci id="S7.I10.i1.p1.4.m4.1.1.3.2.cmml" xref="S7.I10.i1.p1.4.m4.1.1.3.2">𝐸</ci><apply id="S7.I10.i1.p1.4.m4.1.1.3.3.cmml" xref="S7.I10.i1.p1.4.m4.1.1.3.3"><plus id="S7.I10.i1.p1.4.m4.1.1.3.3.1.cmml" xref="S7.I10.i1.p1.4.m4.1.1.3.3.1"></plus><ci id="S7.I10.i1.p1.4.m4.1.1.3.3.2.cmml" xref="S7.I10.i1.p1.4.m4.1.1.3.3.2">𝑠</ci><cn id="S7.I10.i1.p1.4.m4.1.1.3.3.3.cmml" type="integer" xref="S7.I10.i1.p1.4.m4.1.1.3.3.3">1</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.I10.i1.p1.4.m4.1c">\pi\not\subseteq E_{s+1}</annotation><annotation encoding="application/x-llamapun" id="S7.I10.i1.p1.4.m4.1d">italic_π ⊈ italic_E start_POSTSUBSCRIPT italic_s + 1 end_POSTSUBSCRIPT</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S7.I10.i1.p1.4.5">.</span></p> </div> </li> <li class="ltx_item" id="S7.I10.i2" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">•</span> <div class="ltx_para" id="S7.I10.i2.p1"> <p class="ltx_p" id="S7.I10.i2.p1.7"><span class="ltx_text ltx_font_italic" id="S7.I10.i2.p1.7.1">The edge </span><math alttext="\overline{vt_{v}}" class="ltx_Math" display="inline" id="S7.I10.i2.p1.1.m1.1"><semantics id="S7.I10.i2.p1.1.m1.1a"><mover accent="true" id="S7.I10.i2.p1.1.m1.1.1" xref="S7.I10.i2.p1.1.m1.1.1.cmml"><mrow id="S7.I10.i2.p1.1.m1.1.1.2" xref="S7.I10.i2.p1.1.m1.1.1.2.cmml"><mi id="S7.I10.i2.p1.1.m1.1.1.2.2" xref="S7.I10.i2.p1.1.m1.1.1.2.2.cmml">v</mi><mo id="S7.I10.i2.p1.1.m1.1.1.2.1" xref="S7.I10.i2.p1.1.m1.1.1.2.1.cmml">⁢</mo><msub id="S7.I10.i2.p1.1.m1.1.1.2.3" xref="S7.I10.i2.p1.1.m1.1.1.2.3.cmml"><mi id="S7.I10.i2.p1.1.m1.1.1.2.3.2" xref="S7.I10.i2.p1.1.m1.1.1.2.3.2.cmml">t</mi><mi id="S7.I10.i2.p1.1.m1.1.1.2.3.3" xref="S7.I10.i2.p1.1.m1.1.1.2.3.3.cmml">v</mi></msub></mrow><mo id="S7.I10.i2.p1.1.m1.1.1.1" xref="S7.I10.i2.p1.1.m1.1.1.1.cmml">¯</mo></mover><annotation-xml encoding="MathML-Content" id="S7.I10.i2.p1.1.m1.1b"><apply id="S7.I10.i2.p1.1.m1.1.1.cmml" xref="S7.I10.i2.p1.1.m1.1.1"><ci id="S7.I10.i2.p1.1.m1.1.1.1.cmml" xref="S7.I10.i2.p1.1.m1.1.1.1">¯</ci><apply id="S7.I10.i2.p1.1.m1.1.1.2.cmml" xref="S7.I10.i2.p1.1.m1.1.1.2"><times id="S7.I10.i2.p1.1.m1.1.1.2.1.cmml" xref="S7.I10.i2.p1.1.m1.1.1.2.1"></times><ci id="S7.I10.i2.p1.1.m1.1.1.2.2.cmml" xref="S7.I10.i2.p1.1.m1.1.1.2.2">𝑣</ci><apply id="S7.I10.i2.p1.1.m1.1.1.2.3.cmml" xref="S7.I10.i2.p1.1.m1.1.1.2.3"><csymbol cd="ambiguous" id="S7.I10.i2.p1.1.m1.1.1.2.3.1.cmml" xref="S7.I10.i2.p1.1.m1.1.1.2.3">subscript</csymbol><ci id="S7.I10.i2.p1.1.m1.1.1.2.3.2.cmml" xref="S7.I10.i2.p1.1.m1.1.1.2.3.2">𝑡</ci><ci id="S7.I10.i2.p1.1.m1.1.1.2.3.3.cmml" xref="S7.I10.i2.p1.1.m1.1.1.2.3.3">𝑣</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.I10.i2.p1.1.m1.1c">\overline{vt_{v}}</annotation><annotation encoding="application/x-llamapun" id="S7.I10.i2.p1.1.m1.1d">over¯ start_ARG italic_v italic_t start_POSTSUBSCRIPT italic_v end_POSTSUBSCRIPT end_ARG</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S7.I10.i2.p1.7.2"> is saturated in </span><math alttext="f" class="ltx_Math" display="inline" id="S7.I10.i2.p1.2.m2.1"><semantics id="S7.I10.i2.p1.2.m2.1a"><mi id="S7.I10.i2.p1.2.m2.1.1" xref="S7.I10.i2.p1.2.m2.1.1.cmml">f</mi><annotation-xml encoding="MathML-Content" id="S7.I10.i2.p1.2.m2.1b"><ci id="S7.I10.i2.p1.2.m2.1.1.cmml" xref="S7.I10.i2.p1.2.m2.1.1">𝑓</ci></annotation-xml><annotation encoding="application/x-tex" id="S7.I10.i2.p1.2.m2.1c">f</annotation><annotation encoding="application/x-llamapun" id="S7.I10.i2.p1.2.m2.1d">italic_f</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S7.I10.i2.p1.7.3">. Then </span><math alttext="\textsl{g}_{s+1}(v)=(1+\frac{\eta}{2})^{h+1}" class="ltx_Math" display="inline" id="S7.I10.i2.p1.3.m3.2"><semantics id="S7.I10.i2.p1.3.m3.2a"><mrow id="S7.I10.i2.p1.3.m3.2.2" xref="S7.I10.i2.p1.3.m3.2.2.cmml"><mrow id="S7.I10.i2.p1.3.m3.2.2.3" xref="S7.I10.i2.p1.3.m3.2.2.3.cmml"><msub id="S7.I10.i2.p1.3.m3.2.2.3.2" xref="S7.I10.i2.p1.3.m3.2.2.3.2.cmml"><mtext class="ltx_mathvariant_italic" id="S7.I10.i2.p1.3.m3.2.2.3.2.2" xref="S7.I10.i2.p1.3.m3.2.2.3.2.2a.cmml">g</mtext><mrow id="S7.I10.i2.p1.3.m3.2.2.3.2.3" xref="S7.I10.i2.p1.3.m3.2.2.3.2.3.cmml"><mi id="S7.I10.i2.p1.3.m3.2.2.3.2.3.2" xref="S7.I10.i2.p1.3.m3.2.2.3.2.3.2.cmml">s</mi><mo id="S7.I10.i2.p1.3.m3.2.2.3.2.3.1" xref="S7.I10.i2.p1.3.m3.2.2.3.2.3.1.cmml">+</mo><mn id="S7.I10.i2.p1.3.m3.2.2.3.2.3.3" xref="S7.I10.i2.p1.3.m3.2.2.3.2.3.3.cmml">1</mn></mrow></msub><mo id="S7.I10.i2.p1.3.m3.2.2.3.1" xref="S7.I10.i2.p1.3.m3.2.2.3.1.cmml">⁢</mo><mrow id="S7.I10.i2.p1.3.m3.2.2.3.3.2" xref="S7.I10.i2.p1.3.m3.2.2.3.cmml"><mo id="S7.I10.i2.p1.3.m3.2.2.3.3.2.1" stretchy="false" xref="S7.I10.i2.p1.3.m3.2.2.3.cmml">(</mo><mi id="S7.I10.i2.p1.3.m3.1.1" xref="S7.I10.i2.p1.3.m3.1.1.cmml">v</mi><mo id="S7.I10.i2.p1.3.m3.2.2.3.3.2.2" stretchy="false" xref="S7.I10.i2.p1.3.m3.2.2.3.cmml">)</mo></mrow></mrow><mo id="S7.I10.i2.p1.3.m3.2.2.2" xref="S7.I10.i2.p1.3.m3.2.2.2.cmml">=</mo><msup id="S7.I10.i2.p1.3.m3.2.2.1" xref="S7.I10.i2.p1.3.m3.2.2.1.cmml"><mrow id="S7.I10.i2.p1.3.m3.2.2.1.1.1" xref="S7.I10.i2.p1.3.m3.2.2.1.1.1.1.cmml"><mo id="S7.I10.i2.p1.3.m3.2.2.1.1.1.2" stretchy="false" xref="S7.I10.i2.p1.3.m3.2.2.1.1.1.1.cmml">(</mo><mrow id="S7.I10.i2.p1.3.m3.2.2.1.1.1.1" xref="S7.I10.i2.p1.3.m3.2.2.1.1.1.1.cmml"><mn id="S7.I10.i2.p1.3.m3.2.2.1.1.1.1.2" xref="S7.I10.i2.p1.3.m3.2.2.1.1.1.1.2.cmml">1</mn><mo id="S7.I10.i2.p1.3.m3.2.2.1.1.1.1.1" xref="S7.I10.i2.p1.3.m3.2.2.1.1.1.1.1.cmml">+</mo><mfrac id="S7.I10.i2.p1.3.m3.2.2.1.1.1.1.3" xref="S7.I10.i2.p1.3.m3.2.2.1.1.1.1.3.cmml"><mi id="S7.I10.i2.p1.3.m3.2.2.1.1.1.1.3.2" xref="S7.I10.i2.p1.3.m3.2.2.1.1.1.1.3.2.cmml">η</mi><mn id="S7.I10.i2.p1.3.m3.2.2.1.1.1.1.3.3" xref="S7.I10.i2.p1.3.m3.2.2.1.1.1.1.3.3.cmml">2</mn></mfrac></mrow><mo id="S7.I10.i2.p1.3.m3.2.2.1.1.1.3" stretchy="false" xref="S7.I10.i2.p1.3.m3.2.2.1.1.1.1.cmml">)</mo></mrow><mrow id="S7.I10.i2.p1.3.m3.2.2.1.3" xref="S7.I10.i2.p1.3.m3.2.2.1.3.cmml"><mi id="S7.I10.i2.p1.3.m3.2.2.1.3.2" xref="S7.I10.i2.p1.3.m3.2.2.1.3.2.cmml">h</mi><mo id="S7.I10.i2.p1.3.m3.2.2.1.3.1" xref="S7.I10.i2.p1.3.m3.2.2.1.3.1.cmml">+</mo><mn id="S7.I10.i2.p1.3.m3.2.2.1.3.3" xref="S7.I10.i2.p1.3.m3.2.2.1.3.3.cmml">1</mn></mrow></msup></mrow><annotation-xml encoding="MathML-Content" id="S7.I10.i2.p1.3.m3.2b"><apply id="S7.I10.i2.p1.3.m3.2.2.cmml" xref="S7.I10.i2.p1.3.m3.2.2"><eq id="S7.I10.i2.p1.3.m3.2.2.2.cmml" xref="S7.I10.i2.p1.3.m3.2.2.2"></eq><apply id="S7.I10.i2.p1.3.m3.2.2.3.cmml" xref="S7.I10.i2.p1.3.m3.2.2.3"><times id="S7.I10.i2.p1.3.m3.2.2.3.1.cmml" xref="S7.I10.i2.p1.3.m3.2.2.3.1"></times><apply id="S7.I10.i2.p1.3.m3.2.2.3.2.cmml" xref="S7.I10.i2.p1.3.m3.2.2.3.2"><csymbol cd="ambiguous" id="S7.I10.i2.p1.3.m3.2.2.3.2.1.cmml" xref="S7.I10.i2.p1.3.m3.2.2.3.2">subscript</csymbol><ci id="S7.I10.i2.p1.3.m3.2.2.3.2.2a.cmml" xref="S7.I10.i2.p1.3.m3.2.2.3.2.2"><mtext class="ltx_mathvariant_italic" id="S7.I10.i2.p1.3.m3.2.2.3.2.2.cmml" xref="S7.I10.i2.p1.3.m3.2.2.3.2.2">g</mtext></ci><apply id="S7.I10.i2.p1.3.m3.2.2.3.2.3.cmml" xref="S7.I10.i2.p1.3.m3.2.2.3.2.3"><plus id="S7.I10.i2.p1.3.m3.2.2.3.2.3.1.cmml" xref="S7.I10.i2.p1.3.m3.2.2.3.2.3.1"></plus><ci id="S7.I10.i2.p1.3.m3.2.2.3.2.3.2.cmml" xref="S7.I10.i2.p1.3.m3.2.2.3.2.3.2">𝑠</ci><cn id="S7.I10.i2.p1.3.m3.2.2.3.2.3.3.cmml" type="integer" xref="S7.I10.i2.p1.3.m3.2.2.3.2.3.3">1</cn></apply></apply><ci id="S7.I10.i2.p1.3.m3.1.1.cmml" xref="S7.I10.i2.p1.3.m3.1.1">𝑣</ci></apply><apply id="S7.I10.i2.p1.3.m3.2.2.1.cmml" xref="S7.I10.i2.p1.3.m3.2.2.1"><csymbol cd="ambiguous" id="S7.I10.i2.p1.3.m3.2.2.1.2.cmml" xref="S7.I10.i2.p1.3.m3.2.2.1">superscript</csymbol><apply id="S7.I10.i2.p1.3.m3.2.2.1.1.1.1.cmml" xref="S7.I10.i2.p1.3.m3.2.2.1.1.1"><plus id="S7.I10.i2.p1.3.m3.2.2.1.1.1.1.1.cmml" xref="S7.I10.i2.p1.3.m3.2.2.1.1.1.1.1"></plus><cn id="S7.I10.i2.p1.3.m3.2.2.1.1.1.1.2.cmml" type="integer" xref="S7.I10.i2.p1.3.m3.2.2.1.1.1.1.2">1</cn><apply id="S7.I10.i2.p1.3.m3.2.2.1.1.1.1.3.cmml" xref="S7.I10.i2.p1.3.m3.2.2.1.1.1.1.3"><divide id="S7.I10.i2.p1.3.m3.2.2.1.1.1.1.3.1.cmml" xref="S7.I10.i2.p1.3.m3.2.2.1.1.1.1.3"></divide><ci id="S7.I10.i2.p1.3.m3.2.2.1.1.1.1.3.2.cmml" xref="S7.I10.i2.p1.3.m3.2.2.1.1.1.1.3.2">𝜂</ci><cn id="S7.I10.i2.p1.3.m3.2.2.1.1.1.1.3.3.cmml" type="integer" xref="S7.I10.i2.p1.3.m3.2.2.1.1.1.1.3.3">2</cn></apply></apply><apply id="S7.I10.i2.p1.3.m3.2.2.1.3.cmml" xref="S7.I10.i2.p1.3.m3.2.2.1.3"><plus id="S7.I10.i2.p1.3.m3.2.2.1.3.1.cmml" xref="S7.I10.i2.p1.3.m3.2.2.1.3.1"></plus><ci id="S7.I10.i2.p1.3.m3.2.2.1.3.2.cmml" xref="S7.I10.i2.p1.3.m3.2.2.1.3.2">ℎ</ci><cn id="S7.I10.i2.p1.3.m3.2.2.1.3.3.cmml" type="integer" xref="S7.I10.i2.p1.3.m3.2.2.1.3.3">1</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.I10.i2.p1.3.m3.2c">\textsl{g}_{s+1}(v)=(1+\frac{\eta}{2})^{h+1}</annotation><annotation encoding="application/x-llamapun" id="S7.I10.i2.p1.3.m3.2d">g start_POSTSUBSCRIPT italic_s + 1 end_POSTSUBSCRIPT ( italic_v ) = ( 1 + divide start_ARG italic_η end_ARG start_ARG 2 end_ARG ) start_POSTSUPERSCRIPT italic_h + 1 end_POSTSUPERSCRIPT</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S7.I10.i2.p1.7.4">, and </span><math alttext="v" class="ltx_Math" display="inline" id="S7.I10.i2.p1.4.m4.1"><semantics id="S7.I10.i2.p1.4.m4.1a"><mi id="S7.I10.i2.p1.4.m4.1.1" xref="S7.I10.i2.p1.4.m4.1.1.cmml">v</mi><annotation-xml encoding="MathML-Content" id="S7.I10.i2.p1.4.m4.1b"><ci id="S7.I10.i2.p1.4.m4.1.1.cmml" xref="S7.I10.i2.p1.4.m4.1.1">𝑣</ci></annotation-xml><annotation encoding="application/x-tex" id="S7.I10.i2.p1.4.m4.1c">v</annotation><annotation encoding="application/x-llamapun" id="S7.I10.i2.p1.4.m4.1d">italic_v</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S7.I10.i2.p1.7.5"> is in level </span><math alttext="h" class="ltx_Math" display="inline" id="S7.I10.i2.p1.5.m5.1"><semantics id="S7.I10.i2.p1.5.m5.1a"><mi id="S7.I10.i2.p1.5.m5.1.1" xref="S7.I10.i2.p1.5.m5.1.1.cmml">h</mi><annotation-xml encoding="MathML-Content" id="S7.I10.i2.p1.5.m5.1b"><ci id="S7.I10.i2.p1.5.m5.1.1.cmml" xref="S7.I10.i2.p1.5.m5.1.1">ℎ</ci></annotation-xml><annotation encoding="application/x-tex" id="S7.I10.i2.p1.5.m5.1c">h</annotation><annotation encoding="application/x-llamapun" id="S7.I10.i2.p1.5.m5.1d">italic_h</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S7.I10.i2.p1.7.6">. This implies that </span><math alttext="v" class="ltx_Math" display="inline" id="S7.I10.i2.p1.6.m6.1"><semantics id="S7.I10.i2.p1.6.m6.1a"><mi id="S7.I10.i2.p1.6.m6.1.1" xref="S7.I10.i2.p1.6.m6.1.1.cmml">v</mi><annotation-xml encoding="MathML-Content" id="S7.I10.i2.p1.6.m6.1b"><ci id="S7.I10.i2.p1.6.m6.1.1.cmml" xref="S7.I10.i2.p1.6.m6.1.1">𝑣</ci></annotation-xml><annotation encoding="application/x-tex" id="S7.I10.i2.p1.6.m6.1c">v</annotation><annotation encoding="application/x-llamapun" id="S7.I10.i2.p1.6.m6.1d">italic_v</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S7.I10.i2.p1.7.7"> has no incoming violating edges (and thus no incoming edges) in </span><math alttext="E_{s+1}" class="ltx_Math" display="inline" id="S7.I10.i2.p1.7.m7.1"><semantics id="S7.I10.i2.p1.7.m7.1a"><msub id="S7.I10.i2.p1.7.m7.1.1" xref="S7.I10.i2.p1.7.m7.1.1.cmml"><mi id="S7.I10.i2.p1.7.m7.1.1.2" xref="S7.I10.i2.p1.7.m7.1.1.2.cmml">E</mi><mrow id="S7.I10.i2.p1.7.m7.1.1.3" xref="S7.I10.i2.p1.7.m7.1.1.3.cmml"><mi id="S7.I10.i2.p1.7.m7.1.1.3.2" xref="S7.I10.i2.p1.7.m7.1.1.3.2.cmml">s</mi><mo id="S7.I10.i2.p1.7.m7.1.1.3.1" xref="S7.I10.i2.p1.7.m7.1.1.3.1.cmml">+</mo><mn id="S7.I10.i2.p1.7.m7.1.1.3.3" xref="S7.I10.i2.p1.7.m7.1.1.3.3.cmml">1</mn></mrow></msub><annotation-xml encoding="MathML-Content" id="S7.I10.i2.p1.7.m7.1b"><apply id="S7.I10.i2.p1.7.m7.1.1.cmml" xref="S7.I10.i2.p1.7.m7.1.1"><csymbol cd="ambiguous" id="S7.I10.i2.p1.7.m7.1.1.1.cmml" xref="S7.I10.i2.p1.7.m7.1.1">subscript</csymbol><ci id="S7.I10.i2.p1.7.m7.1.1.2.cmml" xref="S7.I10.i2.p1.7.m7.1.1.2">𝐸</ci><apply id="S7.I10.i2.p1.7.m7.1.1.3.cmml" xref="S7.I10.i2.p1.7.m7.1.1.3"><plus id="S7.I10.i2.p1.7.m7.1.1.3.1.cmml" xref="S7.I10.i2.p1.7.m7.1.1.3.1"></plus><ci id="S7.I10.i2.p1.7.m7.1.1.3.2.cmml" xref="S7.I10.i2.p1.7.m7.1.1.3.2">𝑠</ci><cn id="S7.I10.i2.p1.7.m7.1.1.3.3.cmml" type="integer" xref="S7.I10.i2.p1.7.m7.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.I10.i2.p1.7.m7.1c">E_{s+1}</annotation><annotation encoding="application/x-llamapun" id="S7.I10.i2.p1.7.m7.1d">italic_E start_POSTSUBSCRIPT italic_s + 1 end_POSTSUBSCRIPT</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S7.I10.i2.p1.7.8">.</span></p> </div> </li> <li class="ltx_item" id="S7.I10.i3" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">•</span> <div class="ltx_para" id="S7.I10.i3.p1"> <p class="ltx_p" id="S7.I10.i3.p1.7"><span class="ltx_text ltx_font_italic" id="S7.I10.i3.p1.7.1">The edge </span><math alttext="\overline{s_{u}u}" class="ltx_Math" display="inline" id="S7.I10.i3.p1.1.m1.1"><semantics id="S7.I10.i3.p1.1.m1.1a"><mover accent="true" id="S7.I10.i3.p1.1.m1.1.1" xref="S7.I10.i3.p1.1.m1.1.1.cmml"><mrow id="S7.I10.i3.p1.1.m1.1.1.2" xref="S7.I10.i3.p1.1.m1.1.1.2.cmml"><msub id="S7.I10.i3.p1.1.m1.1.1.2.2" xref="S7.I10.i3.p1.1.m1.1.1.2.2.cmml"><mi id="S7.I10.i3.p1.1.m1.1.1.2.2.2" xref="S7.I10.i3.p1.1.m1.1.1.2.2.2.cmml">s</mi><mi id="S7.I10.i3.p1.1.m1.1.1.2.2.3" xref="S7.I10.i3.p1.1.m1.1.1.2.2.3.cmml">u</mi></msub><mo id="S7.I10.i3.p1.1.m1.1.1.2.1" xref="S7.I10.i3.p1.1.m1.1.1.2.1.cmml">⁢</mo><mi id="S7.I10.i3.p1.1.m1.1.1.2.3" xref="S7.I10.i3.p1.1.m1.1.1.2.3.cmml">u</mi></mrow><mo id="S7.I10.i3.p1.1.m1.1.1.1" xref="S7.I10.i3.p1.1.m1.1.1.1.cmml">¯</mo></mover><annotation-xml encoding="MathML-Content" id="S7.I10.i3.p1.1.m1.1b"><apply id="S7.I10.i3.p1.1.m1.1.1.cmml" xref="S7.I10.i3.p1.1.m1.1.1"><ci id="S7.I10.i3.p1.1.m1.1.1.1.cmml" xref="S7.I10.i3.p1.1.m1.1.1.1">¯</ci><apply id="S7.I10.i3.p1.1.m1.1.1.2.cmml" xref="S7.I10.i3.p1.1.m1.1.1.2"><times id="S7.I10.i3.p1.1.m1.1.1.2.1.cmml" xref="S7.I10.i3.p1.1.m1.1.1.2.1"></times><apply id="S7.I10.i3.p1.1.m1.1.1.2.2.cmml" xref="S7.I10.i3.p1.1.m1.1.1.2.2"><csymbol cd="ambiguous" id="S7.I10.i3.p1.1.m1.1.1.2.2.1.cmml" xref="S7.I10.i3.p1.1.m1.1.1.2.2">subscript</csymbol><ci id="S7.I10.i3.p1.1.m1.1.1.2.2.2.cmml" xref="S7.I10.i3.p1.1.m1.1.1.2.2.2">𝑠</ci><ci id="S7.I10.i3.p1.1.m1.1.1.2.2.3.cmml" xref="S7.I10.i3.p1.1.m1.1.1.2.2.3">𝑢</ci></apply><ci id="S7.I10.i3.p1.1.m1.1.1.2.3.cmml" xref="S7.I10.i3.p1.1.m1.1.1.2.3">𝑢</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.I10.i3.p1.1.m1.1c">\overline{s_{u}u}</annotation><annotation encoding="application/x-llamapun" id="S7.I10.i3.p1.1.m1.1d">over¯ start_ARG italic_s start_POSTSUBSCRIPT italic_u end_POSTSUBSCRIPT italic_u end_ARG</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S7.I10.i3.p1.7.2"> is saturated in </span><math alttext="f" class="ltx_Math" display="inline" id="S7.I10.i3.p1.2.m2.1"><semantics id="S7.I10.i3.p1.2.m2.1a"><mi id="S7.I10.i3.p1.2.m2.1.1" xref="S7.I10.i3.p1.2.m2.1.1.cmml">f</mi><annotation-xml encoding="MathML-Content" id="S7.I10.i3.p1.2.m2.1b"><ci id="S7.I10.i3.p1.2.m2.1.1.cmml" xref="S7.I10.i3.p1.2.m2.1.1">𝑓</ci></annotation-xml><annotation encoding="application/x-tex" id="S7.I10.i3.p1.2.m2.1c">f</annotation><annotation encoding="application/x-llamapun" id="S7.I10.i3.p1.2.m2.1d">italic_f</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S7.I10.i3.p1.7.3">. Then </span><math alttext="\textsl{g}_{s+1}(u)=(1+\frac{\eta}{2})^{l_{m}(u)-1}" class="ltx_Math" display="inline" id="S7.I10.i3.p1.3.m3.3"><semantics id="S7.I10.i3.p1.3.m3.3a"><mrow id="S7.I10.i3.p1.3.m3.3.3" xref="S7.I10.i3.p1.3.m3.3.3.cmml"><mrow id="S7.I10.i3.p1.3.m3.3.3.3" xref="S7.I10.i3.p1.3.m3.3.3.3.cmml"><msub id="S7.I10.i3.p1.3.m3.3.3.3.2" xref="S7.I10.i3.p1.3.m3.3.3.3.2.cmml"><mtext class="ltx_mathvariant_italic" id="S7.I10.i3.p1.3.m3.3.3.3.2.2" xref="S7.I10.i3.p1.3.m3.3.3.3.2.2a.cmml">g</mtext><mrow id="S7.I10.i3.p1.3.m3.3.3.3.2.3" xref="S7.I10.i3.p1.3.m3.3.3.3.2.3.cmml"><mi id="S7.I10.i3.p1.3.m3.3.3.3.2.3.2" xref="S7.I10.i3.p1.3.m3.3.3.3.2.3.2.cmml">s</mi><mo id="S7.I10.i3.p1.3.m3.3.3.3.2.3.1" xref="S7.I10.i3.p1.3.m3.3.3.3.2.3.1.cmml">+</mo><mn id="S7.I10.i3.p1.3.m3.3.3.3.2.3.3" xref="S7.I10.i3.p1.3.m3.3.3.3.2.3.3.cmml">1</mn></mrow></msub><mo id="S7.I10.i3.p1.3.m3.3.3.3.1" xref="S7.I10.i3.p1.3.m3.3.3.3.1.cmml">⁢</mo><mrow id="S7.I10.i3.p1.3.m3.3.3.3.3.2" xref="S7.I10.i3.p1.3.m3.3.3.3.cmml"><mo id="S7.I10.i3.p1.3.m3.3.3.3.3.2.1" stretchy="false" xref="S7.I10.i3.p1.3.m3.3.3.3.cmml">(</mo><mi id="S7.I10.i3.p1.3.m3.2.2" xref="S7.I10.i3.p1.3.m3.2.2.cmml">u</mi><mo id="S7.I10.i3.p1.3.m3.3.3.3.3.2.2" stretchy="false" xref="S7.I10.i3.p1.3.m3.3.3.3.cmml">)</mo></mrow></mrow><mo id="S7.I10.i3.p1.3.m3.3.3.2" xref="S7.I10.i3.p1.3.m3.3.3.2.cmml">=</mo><msup id="S7.I10.i3.p1.3.m3.3.3.1" xref="S7.I10.i3.p1.3.m3.3.3.1.cmml"><mrow id="S7.I10.i3.p1.3.m3.3.3.1.1.1" xref="S7.I10.i3.p1.3.m3.3.3.1.1.1.1.cmml"><mo id="S7.I10.i3.p1.3.m3.3.3.1.1.1.2" stretchy="false" xref="S7.I10.i3.p1.3.m3.3.3.1.1.1.1.cmml">(</mo><mrow id="S7.I10.i3.p1.3.m3.3.3.1.1.1.1" xref="S7.I10.i3.p1.3.m3.3.3.1.1.1.1.cmml"><mn id="S7.I10.i3.p1.3.m3.3.3.1.1.1.1.2" xref="S7.I10.i3.p1.3.m3.3.3.1.1.1.1.2.cmml">1</mn><mo id="S7.I10.i3.p1.3.m3.3.3.1.1.1.1.1" xref="S7.I10.i3.p1.3.m3.3.3.1.1.1.1.1.cmml">+</mo><mfrac id="S7.I10.i3.p1.3.m3.3.3.1.1.1.1.3" xref="S7.I10.i3.p1.3.m3.3.3.1.1.1.1.3.cmml"><mi id="S7.I10.i3.p1.3.m3.3.3.1.1.1.1.3.2" xref="S7.I10.i3.p1.3.m3.3.3.1.1.1.1.3.2.cmml">η</mi><mn id="S7.I10.i3.p1.3.m3.3.3.1.1.1.1.3.3" xref="S7.I10.i3.p1.3.m3.3.3.1.1.1.1.3.3.cmml">2</mn></mfrac></mrow><mo id="S7.I10.i3.p1.3.m3.3.3.1.1.1.3" stretchy="false" xref="S7.I10.i3.p1.3.m3.3.3.1.1.1.1.cmml">)</mo></mrow><mrow id="S7.I10.i3.p1.3.m3.1.1.1" xref="S7.I10.i3.p1.3.m3.1.1.1.cmml"><mrow id="S7.I10.i3.p1.3.m3.1.1.1.3" xref="S7.I10.i3.p1.3.m3.1.1.1.3.cmml"><msub id="S7.I10.i3.p1.3.m3.1.1.1.3.2" xref="S7.I10.i3.p1.3.m3.1.1.1.3.2.cmml"><mi id="S7.I10.i3.p1.3.m3.1.1.1.3.2.2" xref="S7.I10.i3.p1.3.m3.1.1.1.3.2.2.cmml">l</mi><mi id="S7.I10.i3.p1.3.m3.1.1.1.3.2.3" xref="S7.I10.i3.p1.3.m3.1.1.1.3.2.3.cmml">m</mi></msub><mo id="S7.I10.i3.p1.3.m3.1.1.1.3.1" xref="S7.I10.i3.p1.3.m3.1.1.1.3.1.cmml">⁢</mo><mrow id="S7.I10.i3.p1.3.m3.1.1.1.3.3.2" xref="S7.I10.i3.p1.3.m3.1.1.1.3.cmml"><mo id="S7.I10.i3.p1.3.m3.1.1.1.3.3.2.1" stretchy="false" xref="S7.I10.i3.p1.3.m3.1.1.1.3.cmml">(</mo><mi id="S7.I10.i3.p1.3.m3.1.1.1.1" xref="S7.I10.i3.p1.3.m3.1.1.1.1.cmml">u</mi><mo id="S7.I10.i3.p1.3.m3.1.1.1.3.3.2.2" stretchy="false" xref="S7.I10.i3.p1.3.m3.1.1.1.3.cmml">)</mo></mrow></mrow><mo id="S7.I10.i3.p1.3.m3.1.1.1.2" xref="S7.I10.i3.p1.3.m3.1.1.1.2.cmml">−</mo><mn id="S7.I10.i3.p1.3.m3.1.1.1.4" xref="S7.I10.i3.p1.3.m3.1.1.1.4.cmml">1</mn></mrow></msup></mrow><annotation-xml encoding="MathML-Content" id="S7.I10.i3.p1.3.m3.3b"><apply id="S7.I10.i3.p1.3.m3.3.3.cmml" xref="S7.I10.i3.p1.3.m3.3.3"><eq id="S7.I10.i3.p1.3.m3.3.3.2.cmml" xref="S7.I10.i3.p1.3.m3.3.3.2"></eq><apply id="S7.I10.i3.p1.3.m3.3.3.3.cmml" xref="S7.I10.i3.p1.3.m3.3.3.3"><times id="S7.I10.i3.p1.3.m3.3.3.3.1.cmml" xref="S7.I10.i3.p1.3.m3.3.3.3.1"></times><apply id="S7.I10.i3.p1.3.m3.3.3.3.2.cmml" xref="S7.I10.i3.p1.3.m3.3.3.3.2"><csymbol cd="ambiguous" id="S7.I10.i3.p1.3.m3.3.3.3.2.1.cmml" xref="S7.I10.i3.p1.3.m3.3.3.3.2">subscript</csymbol><ci id="S7.I10.i3.p1.3.m3.3.3.3.2.2a.cmml" xref="S7.I10.i3.p1.3.m3.3.3.3.2.2"><mtext class="ltx_mathvariant_italic" id="S7.I10.i3.p1.3.m3.3.3.3.2.2.cmml" xref="S7.I10.i3.p1.3.m3.3.3.3.2.2">g</mtext></ci><apply id="S7.I10.i3.p1.3.m3.3.3.3.2.3.cmml" xref="S7.I10.i3.p1.3.m3.3.3.3.2.3"><plus id="S7.I10.i3.p1.3.m3.3.3.3.2.3.1.cmml" xref="S7.I10.i3.p1.3.m3.3.3.3.2.3.1"></plus><ci id="S7.I10.i3.p1.3.m3.3.3.3.2.3.2.cmml" xref="S7.I10.i3.p1.3.m3.3.3.3.2.3.2">𝑠</ci><cn id="S7.I10.i3.p1.3.m3.3.3.3.2.3.3.cmml" type="integer" xref="S7.I10.i3.p1.3.m3.3.3.3.2.3.3">1</cn></apply></apply><ci id="S7.I10.i3.p1.3.m3.2.2.cmml" xref="S7.I10.i3.p1.3.m3.2.2">𝑢</ci></apply><apply id="S7.I10.i3.p1.3.m3.3.3.1.cmml" xref="S7.I10.i3.p1.3.m3.3.3.1"><csymbol cd="ambiguous" id="S7.I10.i3.p1.3.m3.3.3.1.2.cmml" xref="S7.I10.i3.p1.3.m3.3.3.1">superscript</csymbol><apply id="S7.I10.i3.p1.3.m3.3.3.1.1.1.1.cmml" xref="S7.I10.i3.p1.3.m3.3.3.1.1.1"><plus id="S7.I10.i3.p1.3.m3.3.3.1.1.1.1.1.cmml" xref="S7.I10.i3.p1.3.m3.3.3.1.1.1.1.1"></plus><cn id="S7.I10.i3.p1.3.m3.3.3.1.1.1.1.2.cmml" type="integer" xref="S7.I10.i3.p1.3.m3.3.3.1.1.1.1.2">1</cn><apply id="S7.I10.i3.p1.3.m3.3.3.1.1.1.1.3.cmml" xref="S7.I10.i3.p1.3.m3.3.3.1.1.1.1.3"><divide id="S7.I10.i3.p1.3.m3.3.3.1.1.1.1.3.1.cmml" xref="S7.I10.i3.p1.3.m3.3.3.1.1.1.1.3"></divide><ci id="S7.I10.i3.p1.3.m3.3.3.1.1.1.1.3.2.cmml" xref="S7.I10.i3.p1.3.m3.3.3.1.1.1.1.3.2">𝜂</ci><cn id="S7.I10.i3.p1.3.m3.3.3.1.1.1.1.3.3.cmml" type="integer" xref="S7.I10.i3.p1.3.m3.3.3.1.1.1.1.3.3">2</cn></apply></apply><apply id="S7.I10.i3.p1.3.m3.1.1.1.cmml" xref="S7.I10.i3.p1.3.m3.1.1.1"><minus id="S7.I10.i3.p1.3.m3.1.1.1.2.cmml" xref="S7.I10.i3.p1.3.m3.1.1.1.2"></minus><apply id="S7.I10.i3.p1.3.m3.1.1.1.3.cmml" xref="S7.I10.i3.p1.3.m3.1.1.1.3"><times id="S7.I10.i3.p1.3.m3.1.1.1.3.1.cmml" xref="S7.I10.i3.p1.3.m3.1.1.1.3.1"></times><apply id="S7.I10.i3.p1.3.m3.1.1.1.3.2.cmml" xref="S7.I10.i3.p1.3.m3.1.1.1.3.2"><csymbol cd="ambiguous" id="S7.I10.i3.p1.3.m3.1.1.1.3.2.1.cmml" xref="S7.I10.i3.p1.3.m3.1.1.1.3.2">subscript</csymbol><ci id="S7.I10.i3.p1.3.m3.1.1.1.3.2.2.cmml" xref="S7.I10.i3.p1.3.m3.1.1.1.3.2.2">𝑙</ci><ci id="S7.I10.i3.p1.3.m3.1.1.1.3.2.3.cmml" xref="S7.I10.i3.p1.3.m3.1.1.1.3.2.3">𝑚</ci></apply><ci id="S7.I10.i3.p1.3.m3.1.1.1.1.cmml" xref="S7.I10.i3.p1.3.m3.1.1.1.1">𝑢</ci></apply><cn id="S7.I10.i3.p1.3.m3.1.1.1.4.cmml" type="integer" xref="S7.I10.i3.p1.3.m3.1.1.1.4">1</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.I10.i3.p1.3.m3.3c">\textsl{g}_{s+1}(u)=(1+\frac{\eta}{2})^{l_{m}(u)-1}</annotation><annotation encoding="application/x-llamapun" id="S7.I10.i3.p1.3.m3.3d">g start_POSTSUBSCRIPT italic_s + 1 end_POSTSUBSCRIPT ( italic_u ) = ( 1 + divide start_ARG italic_η end_ARG start_ARG 2 end_ARG ) start_POSTSUPERSCRIPT italic_l start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT ( italic_u ) - 1 end_POSTSUPERSCRIPT</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S7.I10.i3.p1.7.4"> and </span><math alttext="u" class="ltx_Math" display="inline" id="S7.I10.i3.p1.4.m4.1"><semantics id="S7.I10.i3.p1.4.m4.1a"><mi id="S7.I10.i3.p1.4.m4.1.1" xref="S7.I10.i3.p1.4.m4.1.1.cmml">u</mi><annotation-xml encoding="MathML-Content" id="S7.I10.i3.p1.4.m4.1b"><ci id="S7.I10.i3.p1.4.m4.1.1.cmml" xref="S7.I10.i3.p1.4.m4.1.1">𝑢</ci></annotation-xml><annotation encoding="application/x-tex" id="S7.I10.i3.p1.4.m4.1c">u</annotation><annotation encoding="application/x-llamapun" id="S7.I10.i3.p1.4.m4.1d">italic_u</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S7.I10.i3.p1.7.5"> is in level </span><math alttext="l_{m}(u)-1" class="ltx_Math" display="inline" id="S7.I10.i3.p1.5.m5.1"><semantics id="S7.I10.i3.p1.5.m5.1a"><mrow id="S7.I10.i3.p1.5.m5.1.2" xref="S7.I10.i3.p1.5.m5.1.2.cmml"><mrow id="S7.I10.i3.p1.5.m5.1.2.2" xref="S7.I10.i3.p1.5.m5.1.2.2.cmml"><msub id="S7.I10.i3.p1.5.m5.1.2.2.2" xref="S7.I10.i3.p1.5.m5.1.2.2.2.cmml"><mi id="S7.I10.i3.p1.5.m5.1.2.2.2.2" xref="S7.I10.i3.p1.5.m5.1.2.2.2.2.cmml">l</mi><mi id="S7.I10.i3.p1.5.m5.1.2.2.2.3" xref="S7.I10.i3.p1.5.m5.1.2.2.2.3.cmml">m</mi></msub><mo id="S7.I10.i3.p1.5.m5.1.2.2.1" xref="S7.I10.i3.p1.5.m5.1.2.2.1.cmml">⁢</mo><mrow id="S7.I10.i3.p1.5.m5.1.2.2.3.2" xref="S7.I10.i3.p1.5.m5.1.2.2.cmml"><mo id="S7.I10.i3.p1.5.m5.1.2.2.3.2.1" stretchy="false" xref="S7.I10.i3.p1.5.m5.1.2.2.cmml">(</mo><mi id="S7.I10.i3.p1.5.m5.1.1" xref="S7.I10.i3.p1.5.m5.1.1.cmml">u</mi><mo id="S7.I10.i3.p1.5.m5.1.2.2.3.2.2" stretchy="false" xref="S7.I10.i3.p1.5.m5.1.2.2.cmml">)</mo></mrow></mrow><mo id="S7.I10.i3.p1.5.m5.1.2.1" xref="S7.I10.i3.p1.5.m5.1.2.1.cmml">−</mo><mn id="S7.I10.i3.p1.5.m5.1.2.3" xref="S7.I10.i3.p1.5.m5.1.2.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S7.I10.i3.p1.5.m5.1b"><apply id="S7.I10.i3.p1.5.m5.1.2.cmml" xref="S7.I10.i3.p1.5.m5.1.2"><minus id="S7.I10.i3.p1.5.m5.1.2.1.cmml" xref="S7.I10.i3.p1.5.m5.1.2.1"></minus><apply id="S7.I10.i3.p1.5.m5.1.2.2.cmml" xref="S7.I10.i3.p1.5.m5.1.2.2"><times id="S7.I10.i3.p1.5.m5.1.2.2.1.cmml" xref="S7.I10.i3.p1.5.m5.1.2.2.1"></times><apply id="S7.I10.i3.p1.5.m5.1.2.2.2.cmml" xref="S7.I10.i3.p1.5.m5.1.2.2.2"><csymbol cd="ambiguous" id="S7.I10.i3.p1.5.m5.1.2.2.2.1.cmml" xref="S7.I10.i3.p1.5.m5.1.2.2.2">subscript</csymbol><ci id="S7.I10.i3.p1.5.m5.1.2.2.2.2.cmml" xref="S7.I10.i3.p1.5.m5.1.2.2.2.2">𝑙</ci><ci id="S7.I10.i3.p1.5.m5.1.2.2.2.3.cmml" xref="S7.I10.i3.p1.5.m5.1.2.2.2.3">𝑚</ci></apply><ci id="S7.I10.i3.p1.5.m5.1.1.cmml" xref="S7.I10.i3.p1.5.m5.1.1">𝑢</ci></apply><cn id="S7.I10.i3.p1.5.m5.1.2.3.cmml" type="integer" xref="S7.I10.i3.p1.5.m5.1.2.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.I10.i3.p1.5.m5.1c">l_{m}(u)-1</annotation><annotation encoding="application/x-llamapun" id="S7.I10.i3.p1.5.m5.1d">italic_l start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT ( italic_u ) - 1</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S7.I10.i3.p1.7.6">. This implies that </span><math alttext="u" class="ltx_Math" display="inline" id="S7.I10.i3.p1.6.m6.1"><semantics id="S7.I10.i3.p1.6.m6.1a"><mi id="S7.I10.i3.p1.6.m6.1.1" xref="S7.I10.i3.p1.6.m6.1.1.cmml">u</mi><annotation-xml encoding="MathML-Content" id="S7.I10.i3.p1.6.m6.1b"><ci id="S7.I10.i3.p1.6.m6.1.1.cmml" xref="S7.I10.i3.p1.6.m6.1.1">𝑢</ci></annotation-xml><annotation encoding="application/x-tex" id="S7.I10.i3.p1.6.m6.1c">u</annotation><annotation encoding="application/x-llamapun" id="S7.I10.i3.p1.6.m6.1d">italic_u</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S7.I10.i3.p1.7.7"> has no outgoing edges in </span><math alttext="E_{s+1}" class="ltx_Math" display="inline" id="S7.I10.i3.p1.7.m7.1"><semantics id="S7.I10.i3.p1.7.m7.1a"><msub id="S7.I10.i3.p1.7.m7.1.1" xref="S7.I10.i3.p1.7.m7.1.1.cmml"><mi id="S7.I10.i3.p1.7.m7.1.1.2" xref="S7.I10.i3.p1.7.m7.1.1.2.cmml">E</mi><mrow id="S7.I10.i3.p1.7.m7.1.1.3" xref="S7.I10.i3.p1.7.m7.1.1.3.cmml"><mi id="S7.I10.i3.p1.7.m7.1.1.3.2" xref="S7.I10.i3.p1.7.m7.1.1.3.2.cmml">s</mi><mo id="S7.I10.i3.p1.7.m7.1.1.3.1" xref="S7.I10.i3.p1.7.m7.1.1.3.1.cmml">+</mo><mn id="S7.I10.i3.p1.7.m7.1.1.3.3" xref="S7.I10.i3.p1.7.m7.1.1.3.3.cmml">1</mn></mrow></msub><annotation-xml encoding="MathML-Content" id="S7.I10.i3.p1.7.m7.1b"><apply id="S7.I10.i3.p1.7.m7.1.1.cmml" xref="S7.I10.i3.p1.7.m7.1.1"><csymbol cd="ambiguous" id="S7.I10.i3.p1.7.m7.1.1.1.cmml" xref="S7.I10.i3.p1.7.m7.1.1">subscript</csymbol><ci id="S7.I10.i3.p1.7.m7.1.1.2.cmml" xref="S7.I10.i3.p1.7.m7.1.1.2">𝐸</ci><apply id="S7.I10.i3.p1.7.m7.1.1.3.cmml" xref="S7.I10.i3.p1.7.m7.1.1.3"><plus id="S7.I10.i3.p1.7.m7.1.1.3.1.cmml" xref="S7.I10.i3.p1.7.m7.1.1.3.1"></plus><ci id="S7.I10.i3.p1.7.m7.1.1.3.2.cmml" xref="S7.I10.i3.p1.7.m7.1.1.3.2">𝑠</ci><cn id="S7.I10.i3.p1.7.m7.1.1.3.3.cmml" type="integer" xref="S7.I10.i3.p1.7.m7.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.I10.i3.p1.7.m7.1c">E_{s+1}</annotation><annotation encoding="application/x-llamapun" id="S7.I10.i3.p1.7.m7.1d">italic_E start_POSTSUBSCRIPT italic_s + 1 end_POSTSUBSCRIPT</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S7.I10.i3.p1.7.8">.</span></p> </div> </li> </ul> <p class="ltx_p" id="S7.Thmtheorem17.p2.22"><span class="ltx_text ltx_font_italic" id="S7.Thmtheorem17.p2.22.6">Since <math alttext="D_{s+1}" class="ltx_Math" display="inline" id="S7.Thmtheorem17.p2.17.1.m1.1"><semantics id="S7.Thmtheorem17.p2.17.1.m1.1a"><msub id="S7.Thmtheorem17.p2.17.1.m1.1.1" xref="S7.Thmtheorem17.p2.17.1.m1.1.1.cmml"><mi id="S7.Thmtheorem17.p2.17.1.m1.1.1.2" xref="S7.Thmtheorem17.p2.17.1.m1.1.1.2.cmml">D</mi><mrow id="S7.Thmtheorem17.p2.17.1.m1.1.1.3" xref="S7.Thmtheorem17.p2.17.1.m1.1.1.3.cmml"><mi id="S7.Thmtheorem17.p2.17.1.m1.1.1.3.2" xref="S7.Thmtheorem17.p2.17.1.m1.1.1.3.2.cmml">s</mi><mo id="S7.Thmtheorem17.p2.17.1.m1.1.1.3.1" xref="S7.Thmtheorem17.p2.17.1.m1.1.1.3.1.cmml">+</mo><mn id="S7.Thmtheorem17.p2.17.1.m1.1.1.3.3" xref="S7.Thmtheorem17.p2.17.1.m1.1.1.3.3.cmml">1</mn></mrow></msub><annotation-xml encoding="MathML-Content" id="S7.Thmtheorem17.p2.17.1.m1.1b"><apply id="S7.Thmtheorem17.p2.17.1.m1.1.1.cmml" xref="S7.Thmtheorem17.p2.17.1.m1.1.1"><csymbol cd="ambiguous" id="S7.Thmtheorem17.p2.17.1.m1.1.1.1.cmml" xref="S7.Thmtheorem17.p2.17.1.m1.1.1">subscript</csymbol><ci id="S7.Thmtheorem17.p2.17.1.m1.1.1.2.cmml" xref="S7.Thmtheorem17.p2.17.1.m1.1.1.2">𝐷</ci><apply id="S7.Thmtheorem17.p2.17.1.m1.1.1.3.cmml" xref="S7.Thmtheorem17.p2.17.1.m1.1.1.3"><plus id="S7.Thmtheorem17.p2.17.1.m1.1.1.3.1.cmml" xref="S7.Thmtheorem17.p2.17.1.m1.1.1.3.1"></plus><ci id="S7.Thmtheorem17.p2.17.1.m1.1.1.3.2.cmml" xref="S7.Thmtheorem17.p2.17.1.m1.1.1.3.2">𝑠</ci><cn id="S7.Thmtheorem17.p2.17.1.m1.1.1.3.3.cmml" type="integer" xref="S7.Thmtheorem17.p2.17.1.m1.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.Thmtheorem17.p2.17.1.m1.1c">D_{s+1}</annotation><annotation encoding="application/x-llamapun" id="S7.Thmtheorem17.p2.17.1.m1.1d">italic_D start_POSTSUBSCRIPT italic_s + 1 end_POSTSUBSCRIPT</annotation></semantics></math> is a subgraph of <math alttext="D_{s}" class="ltx_Math" display="inline" id="S7.Thmtheorem17.p2.18.2.m2.1"><semantics id="S7.Thmtheorem17.p2.18.2.m2.1a"><msub id="S7.Thmtheorem17.p2.18.2.m2.1.1" xref="S7.Thmtheorem17.p2.18.2.m2.1.1.cmml"><mi id="S7.Thmtheorem17.p2.18.2.m2.1.1.2" xref="S7.Thmtheorem17.p2.18.2.m2.1.1.2.cmml">D</mi><mi id="S7.Thmtheorem17.p2.18.2.m2.1.1.3" xref="S7.Thmtheorem17.p2.18.2.m2.1.1.3.cmml">s</mi></msub><annotation-xml encoding="MathML-Content" id="S7.Thmtheorem17.p2.18.2.m2.1b"><apply id="S7.Thmtheorem17.p2.18.2.m2.1.1.cmml" xref="S7.Thmtheorem17.p2.18.2.m2.1.1"><csymbol cd="ambiguous" id="S7.Thmtheorem17.p2.18.2.m2.1.1.1.cmml" xref="S7.Thmtheorem17.p2.18.2.m2.1.1">subscript</csymbol><ci id="S7.Thmtheorem17.p2.18.2.m2.1.1.2.cmml" xref="S7.Thmtheorem17.p2.18.2.m2.1.1.2">𝐷</ci><ci id="S7.Thmtheorem17.p2.18.2.m2.1.1.3.cmml" xref="S7.Thmtheorem17.p2.18.2.m2.1.1.3">𝑠</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.Thmtheorem17.p2.18.2.m2.1c">D_{s}</annotation><annotation encoding="application/x-llamapun" id="S7.Thmtheorem17.p2.18.2.m2.1d">italic_D start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT</annotation></semantics></math>, and no vertices in <math alttext="S_{s}" class="ltx_Math" display="inline" id="S7.Thmtheorem17.p2.19.3.m3.1"><semantics id="S7.Thmtheorem17.p2.19.3.m3.1a"><msub id="S7.Thmtheorem17.p2.19.3.m3.1.1" xref="S7.Thmtheorem17.p2.19.3.m3.1.1.cmml"><mi id="S7.Thmtheorem17.p2.19.3.m3.1.1.2" xref="S7.Thmtheorem17.p2.19.3.m3.1.1.2.cmml">S</mi><mi id="S7.Thmtheorem17.p2.19.3.m3.1.1.3" xref="S7.Thmtheorem17.p2.19.3.m3.1.1.3.cmml">s</mi></msub><annotation-xml encoding="MathML-Content" id="S7.Thmtheorem17.p2.19.3.m3.1b"><apply id="S7.Thmtheorem17.p2.19.3.m3.1.1.cmml" xref="S7.Thmtheorem17.p2.19.3.m3.1.1"><csymbol cd="ambiguous" id="S7.Thmtheorem17.p2.19.3.m3.1.1.1.cmml" xref="S7.Thmtheorem17.p2.19.3.m3.1.1">subscript</csymbol><ci id="S7.Thmtheorem17.p2.19.3.m3.1.1.2.cmml" xref="S7.Thmtheorem17.p2.19.3.m3.1.1.2">𝑆</ci><ci id="S7.Thmtheorem17.p2.19.3.m3.1.1.3.cmml" xref="S7.Thmtheorem17.p2.19.3.m3.1.1.3">𝑠</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.Thmtheorem17.p2.19.3.m3.1c">S_{s}</annotation><annotation encoding="application/x-llamapun" id="S7.Thmtheorem17.p2.19.3.m3.1d">italic_S start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT</annotation></semantics></math> are in <math alttext="D_{s+1}" class="ltx_Math" display="inline" id="S7.Thmtheorem17.p2.20.4.m4.1"><semantics id="S7.Thmtheorem17.p2.20.4.m4.1a"><msub id="S7.Thmtheorem17.p2.20.4.m4.1.1" xref="S7.Thmtheorem17.p2.20.4.m4.1.1.cmml"><mi id="S7.Thmtheorem17.p2.20.4.m4.1.1.2" xref="S7.Thmtheorem17.p2.20.4.m4.1.1.2.cmml">D</mi><mrow id="S7.Thmtheorem17.p2.20.4.m4.1.1.3" xref="S7.Thmtheorem17.p2.20.4.m4.1.1.3.cmml"><mi id="S7.Thmtheorem17.p2.20.4.m4.1.1.3.2" xref="S7.Thmtheorem17.p2.20.4.m4.1.1.3.2.cmml">s</mi><mo id="S7.Thmtheorem17.p2.20.4.m4.1.1.3.1" xref="S7.Thmtheorem17.p2.20.4.m4.1.1.3.1.cmml">+</mo><mn id="S7.Thmtheorem17.p2.20.4.m4.1.1.3.3" xref="S7.Thmtheorem17.p2.20.4.m4.1.1.3.3.cmml">1</mn></mrow></msub><annotation-xml encoding="MathML-Content" id="S7.Thmtheorem17.p2.20.4.m4.1b"><apply id="S7.Thmtheorem17.p2.20.4.m4.1.1.cmml" xref="S7.Thmtheorem17.p2.20.4.m4.1.1"><csymbol cd="ambiguous" id="S7.Thmtheorem17.p2.20.4.m4.1.1.1.cmml" xref="S7.Thmtheorem17.p2.20.4.m4.1.1">subscript</csymbol><ci id="S7.Thmtheorem17.p2.20.4.m4.1.1.2.cmml" xref="S7.Thmtheorem17.p2.20.4.m4.1.1.2">𝐷</ci><apply id="S7.Thmtheorem17.p2.20.4.m4.1.1.3.cmml" xref="S7.Thmtheorem17.p2.20.4.m4.1.1.3"><plus id="S7.Thmtheorem17.p2.20.4.m4.1.1.3.1.cmml" xref="S7.Thmtheorem17.p2.20.4.m4.1.1.3.1"></plus><ci id="S7.Thmtheorem17.p2.20.4.m4.1.1.3.2.cmml" xref="S7.Thmtheorem17.p2.20.4.m4.1.1.3.2">𝑠</ci><cn id="S7.Thmtheorem17.p2.20.4.m4.1.1.3.3.cmml" type="integer" xref="S7.Thmtheorem17.p2.20.4.m4.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.Thmtheorem17.p2.20.4.m4.1c">D_{s+1}</annotation><annotation encoding="application/x-llamapun" id="S7.Thmtheorem17.p2.20.4.m4.1d">italic_D start_POSTSUBSCRIPT italic_s + 1 end_POSTSUBSCRIPT</annotation></semantics></math>, the height of <math alttext="D_{s+1}" class="ltx_Math" display="inline" id="S7.Thmtheorem17.p2.21.5.m5.1"><semantics id="S7.Thmtheorem17.p2.21.5.m5.1a"><msub id="S7.Thmtheorem17.p2.21.5.m5.1.1" xref="S7.Thmtheorem17.p2.21.5.m5.1.1.cmml"><mi id="S7.Thmtheorem17.p2.21.5.m5.1.1.2" xref="S7.Thmtheorem17.p2.21.5.m5.1.1.2.cmml">D</mi><mrow id="S7.Thmtheorem17.p2.21.5.m5.1.1.3" xref="S7.Thmtheorem17.p2.21.5.m5.1.1.3.cmml"><mi id="S7.Thmtheorem17.p2.21.5.m5.1.1.3.2" xref="S7.Thmtheorem17.p2.21.5.m5.1.1.3.2.cmml">s</mi><mo id="S7.Thmtheorem17.p2.21.5.m5.1.1.3.1" xref="S7.Thmtheorem17.p2.21.5.m5.1.1.3.1.cmml">+</mo><mn id="S7.Thmtheorem17.p2.21.5.m5.1.1.3.3" xref="S7.Thmtheorem17.p2.21.5.m5.1.1.3.3.cmml">1</mn></mrow></msub><annotation-xml encoding="MathML-Content" id="S7.Thmtheorem17.p2.21.5.m5.1b"><apply id="S7.Thmtheorem17.p2.21.5.m5.1.1.cmml" xref="S7.Thmtheorem17.p2.21.5.m5.1.1"><csymbol cd="ambiguous" id="S7.Thmtheorem17.p2.21.5.m5.1.1.1.cmml" xref="S7.Thmtheorem17.p2.21.5.m5.1.1">subscript</csymbol><ci id="S7.Thmtheorem17.p2.21.5.m5.1.1.2.cmml" xref="S7.Thmtheorem17.p2.21.5.m5.1.1.2">𝐷</ci><apply id="S7.Thmtheorem17.p2.21.5.m5.1.1.3.cmml" xref="S7.Thmtheorem17.p2.21.5.m5.1.1.3"><plus id="S7.Thmtheorem17.p2.21.5.m5.1.1.3.1.cmml" xref="S7.Thmtheorem17.p2.21.5.m5.1.1.3.1"></plus><ci id="S7.Thmtheorem17.p2.21.5.m5.1.1.3.2.cmml" xref="S7.Thmtheorem17.p2.21.5.m5.1.1.3.2">𝑠</ci><cn id="S7.Thmtheorem17.p2.21.5.m5.1.1.3.3.cmml" type="integer" xref="S7.Thmtheorem17.p2.21.5.m5.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.Thmtheorem17.p2.21.5.m5.1c">D_{s+1}</annotation><annotation encoding="application/x-llamapun" id="S7.Thmtheorem17.p2.21.5.m5.1d">italic_D start_POSTSUBSCRIPT italic_s + 1 end_POSTSUBSCRIPT</annotation></semantics></math> is at least one fewer than the height in <math alttext="D_{s}" class="ltx_Math" display="inline" id="S7.Thmtheorem17.p2.22.6.m6.1"><semantics id="S7.Thmtheorem17.p2.22.6.m6.1a"><msub id="S7.Thmtheorem17.p2.22.6.m6.1.1" xref="S7.Thmtheorem17.p2.22.6.m6.1.1.cmml"><mi id="S7.Thmtheorem17.p2.22.6.m6.1.1.2" xref="S7.Thmtheorem17.p2.22.6.m6.1.1.2.cmml">D</mi><mi id="S7.Thmtheorem17.p2.22.6.m6.1.1.3" xref="S7.Thmtheorem17.p2.22.6.m6.1.1.3.cmml">s</mi></msub><annotation-xml encoding="MathML-Content" id="S7.Thmtheorem17.p2.22.6.m6.1b"><apply id="S7.Thmtheorem17.p2.22.6.m6.1.1.cmml" xref="S7.Thmtheorem17.p2.22.6.m6.1.1"><csymbol cd="ambiguous" id="S7.Thmtheorem17.p2.22.6.m6.1.1.1.cmml" xref="S7.Thmtheorem17.p2.22.6.m6.1.1">subscript</csymbol><ci id="S7.Thmtheorem17.p2.22.6.m6.1.1.2.cmml" xref="S7.Thmtheorem17.p2.22.6.m6.1.1.2">𝐷</ci><ci id="S7.Thmtheorem17.p2.22.6.m6.1.1.3.cmml" xref="S7.Thmtheorem17.p2.22.6.m6.1.1.3">𝑠</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.Thmtheorem17.p2.22.6.m6.1c">D_{s}</annotation><annotation encoding="application/x-llamapun" id="S7.Thmtheorem17.p2.22.6.m6.1d">italic_D start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT</annotation></semantics></math>.</span></p> </div> </div> <div class="ltx_theorem ltx_theorem_corollary" id="S7.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="S7.Thmtheorem18.1.1.1">Corollary 7.18</span></span><span class="ltx_text ltx_font_bold" id="S7.Thmtheorem18.2.2">.</span> </h6> <div class="ltx_para" id="S7.Thmtheorem18.p1"> <p class="ltx_p" id="S7.Thmtheorem18.p1.3"><span class="ltx_text ltx_font_italic" id="S7.Thmtheorem18.p1.3.3">At the start of <math alttext="(h:m:0)" class="ltx_Math" display="inline" id="S7.Thmtheorem18.p1.1.1.m1.1"><semantics id="S7.Thmtheorem18.p1.1.1.m1.1a"><mrow id="S7.Thmtheorem18.p1.1.1.m1.1.1.1" xref="S7.Thmtheorem18.p1.1.1.m1.1.1.1.1.cmml"><mo id="S7.Thmtheorem18.p1.1.1.m1.1.1.1.2" stretchy="false" xref="S7.Thmtheorem18.p1.1.1.m1.1.1.1.1.cmml">(</mo><mrow id="S7.Thmtheorem18.p1.1.1.m1.1.1.1.1" xref="S7.Thmtheorem18.p1.1.1.m1.1.1.1.1.cmml"><mi id="S7.Thmtheorem18.p1.1.1.m1.1.1.1.1.2" xref="S7.Thmtheorem18.p1.1.1.m1.1.1.1.1.2.cmml">h</mi><mo id="S7.Thmtheorem18.p1.1.1.m1.1.1.1.1.3" lspace="0.278em" rspace="0.278em" xref="S7.Thmtheorem18.p1.1.1.m1.1.1.1.1.3.cmml">:</mo><mi id="S7.Thmtheorem18.p1.1.1.m1.1.1.1.1.4" xref="S7.Thmtheorem18.p1.1.1.m1.1.1.1.1.4.cmml">m</mi><mo id="S7.Thmtheorem18.p1.1.1.m1.1.1.1.1.5" lspace="0.278em" rspace="0.278em" xref="S7.Thmtheorem18.p1.1.1.m1.1.1.1.1.5.cmml">:</mo><mn id="S7.Thmtheorem18.p1.1.1.m1.1.1.1.1.6" xref="S7.Thmtheorem18.p1.1.1.m1.1.1.1.1.6.cmml">0</mn></mrow><mo id="S7.Thmtheorem18.p1.1.1.m1.1.1.1.3" stretchy="false" xref="S7.Thmtheorem18.p1.1.1.m1.1.1.1.1.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S7.Thmtheorem18.p1.1.1.m1.1b"><apply id="S7.Thmtheorem18.p1.1.1.m1.1.1.1.1.cmml" xref="S7.Thmtheorem18.p1.1.1.m1.1.1.1"><and id="S7.Thmtheorem18.p1.1.1.m1.1.1.1.1a.cmml" xref="S7.Thmtheorem18.p1.1.1.m1.1.1.1"></and><apply id="S7.Thmtheorem18.p1.1.1.m1.1.1.1.1b.cmml" xref="S7.Thmtheorem18.p1.1.1.m1.1.1.1"><ci id="S7.Thmtheorem18.p1.1.1.m1.1.1.1.1.3.cmml" xref="S7.Thmtheorem18.p1.1.1.m1.1.1.1.1.3">:</ci><ci id="S7.Thmtheorem18.p1.1.1.m1.1.1.1.1.2.cmml" xref="S7.Thmtheorem18.p1.1.1.m1.1.1.1.1.2">ℎ</ci><ci id="S7.Thmtheorem18.p1.1.1.m1.1.1.1.1.4.cmml" xref="S7.Thmtheorem18.p1.1.1.m1.1.1.1.1.4">𝑚</ci></apply><apply id="S7.Thmtheorem18.p1.1.1.m1.1.1.1.1c.cmml" xref="S7.Thmtheorem18.p1.1.1.m1.1.1.1"><ci id="S7.Thmtheorem18.p1.1.1.m1.1.1.1.1.5.cmml" xref="S7.Thmtheorem18.p1.1.1.m1.1.1.1.1.5">:</ci><share href="https://arxiv.org/html/2411.12694v2#S7.Thmtheorem18.p1.1.1.m1.1.1.1.1.4.cmml" id="S7.Thmtheorem18.p1.1.1.m1.1.1.1.1d.cmml" xref="S7.Thmtheorem18.p1.1.1.m1.1.1.1"></share><cn id="S7.Thmtheorem18.p1.1.1.m1.1.1.1.1.6.cmml" type="integer" xref="S7.Thmtheorem18.p1.1.1.m1.1.1.1.1.6">0</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.Thmtheorem18.p1.1.1.m1.1c">(h:m:0)</annotation><annotation encoding="application/x-llamapun" id="S7.Thmtheorem18.p1.1.1.m1.1d">( italic_h : italic_m : 0 )</annotation></semantics></math> with <math alttext="m" class="ltx_Math" display="inline" id="S7.Thmtheorem18.p1.2.2.m2.1"><semantics id="S7.Thmtheorem18.p1.2.2.m2.1a"><mi id="S7.Thmtheorem18.p1.2.2.m2.1.1" xref="S7.Thmtheorem18.p1.2.2.m2.1.1.cmml">m</mi><annotation-xml encoding="MathML-Content" id="S7.Thmtheorem18.p1.2.2.m2.1b"><ci id="S7.Thmtheorem18.p1.2.2.m2.1.1.cmml" xref="S7.Thmtheorem18.p1.2.2.m2.1.1">𝑚</ci></annotation-xml><annotation encoding="application/x-tex" id="S7.Thmtheorem18.p1.2.2.m2.1c">m</annotation><annotation encoding="application/x-llamapun" id="S7.Thmtheorem18.p1.2.2.m2.1d">italic_m</annotation></semantics></math> even, there are no violating edges from levels <math alttext="k>h" class="ltx_Math" display="inline" id="S7.Thmtheorem18.p1.3.3.m3.1"><semantics id="S7.Thmtheorem18.p1.3.3.m3.1a"><mrow id="S7.Thmtheorem18.p1.3.3.m3.1.1" xref="S7.Thmtheorem18.p1.3.3.m3.1.1.cmml"><mi id="S7.Thmtheorem18.p1.3.3.m3.1.1.2" xref="S7.Thmtheorem18.p1.3.3.m3.1.1.2.cmml">k</mi><mo id="S7.Thmtheorem18.p1.3.3.m3.1.1.1" xref="S7.Thmtheorem18.p1.3.3.m3.1.1.1.cmml">></mo><mi id="S7.Thmtheorem18.p1.3.3.m3.1.1.3" xref="S7.Thmtheorem18.p1.3.3.m3.1.1.3.cmml">h</mi></mrow><annotation-xml encoding="MathML-Content" id="S7.Thmtheorem18.p1.3.3.m3.1b"><apply id="S7.Thmtheorem18.p1.3.3.m3.1.1.cmml" xref="S7.Thmtheorem18.p1.3.3.m3.1.1"><gt id="S7.Thmtheorem18.p1.3.3.m3.1.1.1.cmml" xref="S7.Thmtheorem18.p1.3.3.m3.1.1.1"></gt><ci id="S7.Thmtheorem18.p1.3.3.m3.1.1.2.cmml" xref="S7.Thmtheorem18.p1.3.3.m3.1.1.2">𝑘</ci><ci id="S7.Thmtheorem18.p1.3.3.m3.1.1.3.cmml" xref="S7.Thmtheorem18.p1.3.3.m3.1.1.3">ℎ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.Thmtheorem18.p1.3.3.m3.1c">k>h</annotation><annotation encoding="application/x-llamapun" id="S7.Thmtheorem18.p1.3.3.m3.1d">italic_k > italic_h</annotation></semantics></math>.</span></p> </div> </div> <div class="ltx_theorem ltx_theorem_proof" id="S7.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="S7.Thmtheorem19.1.1.1">Proof 7.19</span></span><span class="ltx_text ltx_font_bold" id="S7.Thmtheorem19.2.2">.</span> </h6> <div class="ltx_para" id="S7.Thmtheorem19.p1"> <p class="ltx_p" id="S7.Thmtheorem19.p1.5"><span class="ltx_text ltx_font_italic" id="S7.Thmtheorem19.p1.5.5">Consider the previous <span class="ltx_text ltx_font_smallcaps" id="S7.Thmtheorem19.p1.5.5.1">minute</span>. At time <math alttext="(h:m-1:0)" class="ltx_Math" display="inline" id="S7.Thmtheorem19.p1.1.1.m1.1"><semantics id="S7.Thmtheorem19.p1.1.1.m1.1a"><mrow id="S7.Thmtheorem19.p1.1.1.m1.1.1.1" xref="S7.Thmtheorem19.p1.1.1.m1.1.1.1.1.cmml"><mo id="S7.Thmtheorem19.p1.1.1.m1.1.1.1.2" stretchy="false" xref="S7.Thmtheorem19.p1.1.1.m1.1.1.1.1.cmml">(</mo><mrow id="S7.Thmtheorem19.p1.1.1.m1.1.1.1.1" xref="S7.Thmtheorem19.p1.1.1.m1.1.1.1.1.cmml"><mi id="S7.Thmtheorem19.p1.1.1.m1.1.1.1.1.2" xref="S7.Thmtheorem19.p1.1.1.m1.1.1.1.1.2.cmml">h</mi><mo id="S7.Thmtheorem19.p1.1.1.m1.1.1.1.1.3" lspace="0.278em" rspace="0.278em" xref="S7.Thmtheorem19.p1.1.1.m1.1.1.1.1.3.cmml">:</mo><mrow id="S7.Thmtheorem19.p1.1.1.m1.1.1.1.1.4" xref="S7.Thmtheorem19.p1.1.1.m1.1.1.1.1.4.cmml"><mi id="S7.Thmtheorem19.p1.1.1.m1.1.1.1.1.4.2" xref="S7.Thmtheorem19.p1.1.1.m1.1.1.1.1.4.2.cmml">m</mi><mo id="S7.Thmtheorem19.p1.1.1.m1.1.1.1.1.4.1" xref="S7.Thmtheorem19.p1.1.1.m1.1.1.1.1.4.1.cmml">−</mo><mn id="S7.Thmtheorem19.p1.1.1.m1.1.1.1.1.4.3" xref="S7.Thmtheorem19.p1.1.1.m1.1.1.1.1.4.3.cmml">1</mn></mrow><mo id="S7.Thmtheorem19.p1.1.1.m1.1.1.1.1.5" lspace="0.278em" rspace="0.278em" xref="S7.Thmtheorem19.p1.1.1.m1.1.1.1.1.5.cmml">:</mo><mn id="S7.Thmtheorem19.p1.1.1.m1.1.1.1.1.6" xref="S7.Thmtheorem19.p1.1.1.m1.1.1.1.1.6.cmml">0</mn></mrow><mo id="S7.Thmtheorem19.p1.1.1.m1.1.1.1.3" stretchy="false" xref="S7.Thmtheorem19.p1.1.1.m1.1.1.1.1.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S7.Thmtheorem19.p1.1.1.m1.1b"><apply id="S7.Thmtheorem19.p1.1.1.m1.1.1.1.1.cmml" xref="S7.Thmtheorem19.p1.1.1.m1.1.1.1"><and id="S7.Thmtheorem19.p1.1.1.m1.1.1.1.1a.cmml" xref="S7.Thmtheorem19.p1.1.1.m1.1.1.1"></and><apply id="S7.Thmtheorem19.p1.1.1.m1.1.1.1.1b.cmml" xref="S7.Thmtheorem19.p1.1.1.m1.1.1.1"><ci id="S7.Thmtheorem19.p1.1.1.m1.1.1.1.1.3.cmml" xref="S7.Thmtheorem19.p1.1.1.m1.1.1.1.1.3">:</ci><ci id="S7.Thmtheorem19.p1.1.1.m1.1.1.1.1.2.cmml" xref="S7.Thmtheorem19.p1.1.1.m1.1.1.1.1.2">ℎ</ci><apply id="S7.Thmtheorem19.p1.1.1.m1.1.1.1.1.4.cmml" xref="S7.Thmtheorem19.p1.1.1.m1.1.1.1.1.4"><minus id="S7.Thmtheorem19.p1.1.1.m1.1.1.1.1.4.1.cmml" xref="S7.Thmtheorem19.p1.1.1.m1.1.1.1.1.4.1"></minus><ci id="S7.Thmtheorem19.p1.1.1.m1.1.1.1.1.4.2.cmml" xref="S7.Thmtheorem19.p1.1.1.m1.1.1.1.1.4.2">𝑚</ci><cn id="S7.Thmtheorem19.p1.1.1.m1.1.1.1.1.4.3.cmml" type="integer" xref="S7.Thmtheorem19.p1.1.1.m1.1.1.1.1.4.3">1</cn></apply></apply><apply id="S7.Thmtheorem19.p1.1.1.m1.1.1.1.1c.cmml" xref="S7.Thmtheorem19.p1.1.1.m1.1.1.1"><ci id="S7.Thmtheorem19.p1.1.1.m1.1.1.1.1.5.cmml" xref="S7.Thmtheorem19.p1.1.1.m1.1.1.1.1.5">:</ci><share href="https://arxiv.org/html/2411.12694v2#S7.Thmtheorem19.p1.1.1.m1.1.1.1.1.4.cmml" id="S7.Thmtheorem19.p1.1.1.m1.1.1.1.1d.cmml" xref="S7.Thmtheorem19.p1.1.1.m1.1.1.1"></share><cn id="S7.Thmtheorem19.p1.1.1.m1.1.1.1.1.6.cmml" type="integer" xref="S7.Thmtheorem19.p1.1.1.m1.1.1.1.1.6">0</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.Thmtheorem19.p1.1.1.m1.1c">(h:m-1:0)</annotation><annotation encoding="application/x-llamapun" id="S7.Thmtheorem19.p1.1.1.m1.1d">( italic_h : italic_m - 1 : 0 )</annotation></semantics></math> the height of <math alttext="D_{0}" class="ltx_Math" display="inline" id="S7.Thmtheorem19.p1.2.2.m2.1"><semantics id="S7.Thmtheorem19.p1.2.2.m2.1a"><msub id="S7.Thmtheorem19.p1.2.2.m2.1.1" xref="S7.Thmtheorem19.p1.2.2.m2.1.1.cmml"><mi id="S7.Thmtheorem19.p1.2.2.m2.1.1.2" xref="S7.Thmtheorem19.p1.2.2.m2.1.1.2.cmml">D</mi><mn id="S7.Thmtheorem19.p1.2.2.m2.1.1.3" xref="S7.Thmtheorem19.p1.2.2.m2.1.1.3.cmml">0</mn></msub><annotation-xml encoding="MathML-Content" id="S7.Thmtheorem19.p1.2.2.m2.1b"><apply id="S7.Thmtheorem19.p1.2.2.m2.1.1.cmml" xref="S7.Thmtheorem19.p1.2.2.m2.1.1"><csymbol cd="ambiguous" id="S7.Thmtheorem19.p1.2.2.m2.1.1.1.cmml" xref="S7.Thmtheorem19.p1.2.2.m2.1.1">subscript</csymbol><ci id="S7.Thmtheorem19.p1.2.2.m2.1.1.2.cmml" xref="S7.Thmtheorem19.p1.2.2.m2.1.1.2">𝐷</ci><cn id="S7.Thmtheorem19.p1.2.2.m2.1.1.3.cmml" type="integer" xref="S7.Thmtheorem19.p1.2.2.m2.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.Thmtheorem19.p1.2.2.m2.1c">D_{0}</annotation><annotation encoding="application/x-llamapun" id="S7.Thmtheorem19.p1.2.2.m2.1d">italic_D start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT</annotation></semantics></math> is at most <math alttext="\ell-h+1" class="ltx_Math" display="inline" id="S7.Thmtheorem19.p1.3.3.m3.1"><semantics id="S7.Thmtheorem19.p1.3.3.m3.1a"><mrow id="S7.Thmtheorem19.p1.3.3.m3.1.1" xref="S7.Thmtheorem19.p1.3.3.m3.1.1.cmml"><mrow id="S7.Thmtheorem19.p1.3.3.m3.1.1.2" xref="S7.Thmtheorem19.p1.3.3.m3.1.1.2.cmml"><mi id="S7.Thmtheorem19.p1.3.3.m3.1.1.2.2" mathvariant="normal" xref="S7.Thmtheorem19.p1.3.3.m3.1.1.2.2.cmml">ℓ</mi><mo id="S7.Thmtheorem19.p1.3.3.m3.1.1.2.1" xref="S7.Thmtheorem19.p1.3.3.m3.1.1.2.1.cmml">−</mo><mi id="S7.Thmtheorem19.p1.3.3.m3.1.1.2.3" xref="S7.Thmtheorem19.p1.3.3.m3.1.1.2.3.cmml">h</mi></mrow><mo id="S7.Thmtheorem19.p1.3.3.m3.1.1.1" xref="S7.Thmtheorem19.p1.3.3.m3.1.1.1.cmml">+</mo><mn id="S7.Thmtheorem19.p1.3.3.m3.1.1.3" xref="S7.Thmtheorem19.p1.3.3.m3.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S7.Thmtheorem19.p1.3.3.m3.1b"><apply id="S7.Thmtheorem19.p1.3.3.m3.1.1.cmml" xref="S7.Thmtheorem19.p1.3.3.m3.1.1"><plus id="S7.Thmtheorem19.p1.3.3.m3.1.1.1.cmml" xref="S7.Thmtheorem19.p1.3.3.m3.1.1.1"></plus><apply id="S7.Thmtheorem19.p1.3.3.m3.1.1.2.cmml" xref="S7.Thmtheorem19.p1.3.3.m3.1.1.2"><minus id="S7.Thmtheorem19.p1.3.3.m3.1.1.2.1.cmml" xref="S7.Thmtheorem19.p1.3.3.m3.1.1.2.1"></minus><ci id="S7.Thmtheorem19.p1.3.3.m3.1.1.2.2.cmml" xref="S7.Thmtheorem19.p1.3.3.m3.1.1.2.2">ℓ</ci><ci id="S7.Thmtheorem19.p1.3.3.m3.1.1.2.3.cmml" xref="S7.Thmtheorem19.p1.3.3.m3.1.1.2.3">ℎ</ci></apply><cn id="S7.Thmtheorem19.p1.3.3.m3.1.1.3.cmml" type="integer" xref="S7.Thmtheorem19.p1.3.3.m3.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.Thmtheorem19.p1.3.3.m3.1c">\ell-h+1</annotation><annotation encoding="application/x-llamapun" id="S7.Thmtheorem19.p1.3.3.m3.1d">roman_ℓ - italic_h + 1</annotation></semantics></math>. We may now apply Lemma <a class="ltx_ref" href="https://arxiv.org/html/2411.12694v2#S7.Thmtheorem16" title="Lemma 7.16. ‣ 7.4 Formally proving correctness ‣ 7 Results in CONGEST ‣ Local Density and its Distributed Approximation"><span class="ltx_text ltx_ref_tag">7.16</span></a> to conclude that that at second <math alttext="s=\ell-h+1" class="ltx_Math" display="inline" id="S7.Thmtheorem19.p1.4.4.m4.1"><semantics id="S7.Thmtheorem19.p1.4.4.m4.1a"><mrow id="S7.Thmtheorem19.p1.4.4.m4.1.1" xref="S7.Thmtheorem19.p1.4.4.m4.1.1.cmml"><mi id="S7.Thmtheorem19.p1.4.4.m4.1.1.2" xref="S7.Thmtheorem19.p1.4.4.m4.1.1.2.cmml">s</mi><mo id="S7.Thmtheorem19.p1.4.4.m4.1.1.1" xref="S7.Thmtheorem19.p1.4.4.m4.1.1.1.cmml">=</mo><mrow id="S7.Thmtheorem19.p1.4.4.m4.1.1.3" xref="S7.Thmtheorem19.p1.4.4.m4.1.1.3.cmml"><mrow id="S7.Thmtheorem19.p1.4.4.m4.1.1.3.2" xref="S7.Thmtheorem19.p1.4.4.m4.1.1.3.2.cmml"><mi id="S7.Thmtheorem19.p1.4.4.m4.1.1.3.2.2" mathvariant="normal" xref="S7.Thmtheorem19.p1.4.4.m4.1.1.3.2.2.cmml">ℓ</mi><mo id="S7.Thmtheorem19.p1.4.4.m4.1.1.3.2.1" xref="S7.Thmtheorem19.p1.4.4.m4.1.1.3.2.1.cmml">−</mo><mi id="S7.Thmtheorem19.p1.4.4.m4.1.1.3.2.3" xref="S7.Thmtheorem19.p1.4.4.m4.1.1.3.2.3.cmml">h</mi></mrow><mo id="S7.Thmtheorem19.p1.4.4.m4.1.1.3.1" xref="S7.Thmtheorem19.p1.4.4.m4.1.1.3.1.cmml">+</mo><mn id="S7.Thmtheorem19.p1.4.4.m4.1.1.3.3" xref="S7.Thmtheorem19.p1.4.4.m4.1.1.3.3.cmml">1</mn></mrow></mrow><annotation-xml encoding="MathML-Content" id="S7.Thmtheorem19.p1.4.4.m4.1b"><apply id="S7.Thmtheorem19.p1.4.4.m4.1.1.cmml" xref="S7.Thmtheorem19.p1.4.4.m4.1.1"><eq id="S7.Thmtheorem19.p1.4.4.m4.1.1.1.cmml" xref="S7.Thmtheorem19.p1.4.4.m4.1.1.1"></eq><ci id="S7.Thmtheorem19.p1.4.4.m4.1.1.2.cmml" xref="S7.Thmtheorem19.p1.4.4.m4.1.1.2">𝑠</ci><apply id="S7.Thmtheorem19.p1.4.4.m4.1.1.3.cmml" xref="S7.Thmtheorem19.p1.4.4.m4.1.1.3"><plus id="S7.Thmtheorem19.p1.4.4.m4.1.1.3.1.cmml" xref="S7.Thmtheorem19.p1.4.4.m4.1.1.3.1"></plus><apply id="S7.Thmtheorem19.p1.4.4.m4.1.1.3.2.cmml" xref="S7.Thmtheorem19.p1.4.4.m4.1.1.3.2"><minus id="S7.Thmtheorem19.p1.4.4.m4.1.1.3.2.1.cmml" xref="S7.Thmtheorem19.p1.4.4.m4.1.1.3.2.1"></minus><ci id="S7.Thmtheorem19.p1.4.4.m4.1.1.3.2.2.cmml" xref="S7.Thmtheorem19.p1.4.4.m4.1.1.3.2.2">ℓ</ci><ci id="S7.Thmtheorem19.p1.4.4.m4.1.1.3.2.3.cmml" xref="S7.Thmtheorem19.p1.4.4.m4.1.1.3.2.3">ℎ</ci></apply><cn id="S7.Thmtheorem19.p1.4.4.m4.1.1.3.3.cmml" type="integer" xref="S7.Thmtheorem19.p1.4.4.m4.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.Thmtheorem19.p1.4.4.m4.1c">s=\ell-h+1</annotation><annotation encoding="application/x-llamapun" id="S7.Thmtheorem19.p1.4.4.m4.1d">italic_s = roman_ℓ - italic_h + 1</annotation></semantics></math>, <math alttext="V_{s}" class="ltx_Math" display="inline" id="S7.Thmtheorem19.p1.5.5.m5.1"><semantics id="S7.Thmtheorem19.p1.5.5.m5.1a"><msub id="S7.Thmtheorem19.p1.5.5.m5.1.1" xref="S7.Thmtheorem19.p1.5.5.m5.1.1.cmml"><mi id="S7.Thmtheorem19.p1.5.5.m5.1.1.2" xref="S7.Thmtheorem19.p1.5.5.m5.1.1.2.cmml">V</mi><mi id="S7.Thmtheorem19.p1.5.5.m5.1.1.3" xref="S7.Thmtheorem19.p1.5.5.m5.1.1.3.cmml">s</mi></msub><annotation-xml encoding="MathML-Content" id="S7.Thmtheorem19.p1.5.5.m5.1b"><apply id="S7.Thmtheorem19.p1.5.5.m5.1.1.cmml" xref="S7.Thmtheorem19.p1.5.5.m5.1.1"><csymbol cd="ambiguous" id="S7.Thmtheorem19.p1.5.5.m5.1.1.1.cmml" xref="S7.Thmtheorem19.p1.5.5.m5.1.1">subscript</csymbol><ci id="S7.Thmtheorem19.p1.5.5.m5.1.1.2.cmml" xref="S7.Thmtheorem19.p1.5.5.m5.1.1.2">𝑉</ci><ci id="S7.Thmtheorem19.p1.5.5.m5.1.1.3.cmml" xref="S7.Thmtheorem19.p1.5.5.m5.1.1.3">𝑠</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.Thmtheorem19.p1.5.5.m5.1c">V_{s}</annotation><annotation encoding="application/x-llamapun" id="S7.Thmtheorem19.p1.5.5.m5.1d">italic_V start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT</annotation></semantics></math> is empty.</span></p> </div> <div class="ltx_para" id="S7.Thmtheorem19.p2"> <p class="ltx_p" id="S7.Thmtheorem19.p2.11"><span class="ltx_text ltx_font_italic" id="S7.Thmtheorem19.p2.11.11">By Lemma <a class="ltx_ref" href="https://arxiv.org/html/2411.12694v2#S7.Thmtheorem14" title="Lemma 7.14. ‣ 7.4 Formally proving correctness ‣ 7 Results in CONGEST ‣ Local Density and its Distributed Approximation"><span class="ltx_text ltx_ref_tag">7.14</span></a>, the only violating out-edges from levels <math alttext="k>h" class="ltx_Math" display="inline" id="S7.Thmtheorem19.p2.1.1.m1.1"><semantics id="S7.Thmtheorem19.p2.1.1.m1.1a"><mrow id="S7.Thmtheorem19.p2.1.1.m1.1.1" xref="S7.Thmtheorem19.p2.1.1.m1.1.1.cmml"><mi id="S7.Thmtheorem19.p2.1.1.m1.1.1.2" xref="S7.Thmtheorem19.p2.1.1.m1.1.1.2.cmml">k</mi><mo id="S7.Thmtheorem19.p2.1.1.m1.1.1.1" xref="S7.Thmtheorem19.p2.1.1.m1.1.1.1.cmml">></mo><mi id="S7.Thmtheorem19.p2.1.1.m1.1.1.3" xref="S7.Thmtheorem19.p2.1.1.m1.1.1.3.cmml">h</mi></mrow><annotation-xml encoding="MathML-Content" id="S7.Thmtheorem19.p2.1.1.m1.1b"><apply id="S7.Thmtheorem19.p2.1.1.m1.1.1.cmml" xref="S7.Thmtheorem19.p2.1.1.m1.1.1"><gt id="S7.Thmtheorem19.p2.1.1.m1.1.1.1.cmml" xref="S7.Thmtheorem19.p2.1.1.m1.1.1.1"></gt><ci id="S7.Thmtheorem19.p2.1.1.m1.1.1.2.cmml" xref="S7.Thmtheorem19.p2.1.1.m1.1.1.2">𝑘</ci><ci id="S7.Thmtheorem19.p2.1.1.m1.1.1.3.cmml" xref="S7.Thmtheorem19.p2.1.1.m1.1.1.3">ℎ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.Thmtheorem19.p2.1.1.m1.1c">k>h</annotation><annotation encoding="application/x-llamapun" id="S7.Thmtheorem19.p2.1.1.m1.1d">italic_k > italic_h</annotation></semantics></math> come from level <math alttext="h+1" class="ltx_Math" display="inline" id="S7.Thmtheorem19.p2.2.2.m2.1"><semantics id="S7.Thmtheorem19.p2.2.2.m2.1a"><mrow id="S7.Thmtheorem19.p2.2.2.m2.1.1" xref="S7.Thmtheorem19.p2.2.2.m2.1.1.cmml"><mi id="S7.Thmtheorem19.p2.2.2.m2.1.1.2" xref="S7.Thmtheorem19.p2.2.2.m2.1.1.2.cmml">h</mi><mo id="S7.Thmtheorem19.p2.2.2.m2.1.1.1" xref="S7.Thmtheorem19.p2.2.2.m2.1.1.1.cmml">+</mo><mn id="S7.Thmtheorem19.p2.2.2.m2.1.1.3" xref="S7.Thmtheorem19.p2.2.2.m2.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S7.Thmtheorem19.p2.2.2.m2.1b"><apply id="S7.Thmtheorem19.p2.2.2.m2.1.1.cmml" xref="S7.Thmtheorem19.p2.2.2.m2.1.1"><plus id="S7.Thmtheorem19.p2.2.2.m2.1.1.1.cmml" xref="S7.Thmtheorem19.p2.2.2.m2.1.1.1"></plus><ci id="S7.Thmtheorem19.p2.2.2.m2.1.1.2.cmml" xref="S7.Thmtheorem19.p2.2.2.m2.1.1.2">ℎ</ci><cn id="S7.Thmtheorem19.p2.2.2.m2.1.1.3.cmml" type="integer" xref="S7.Thmtheorem19.p2.2.2.m2.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.Thmtheorem19.p2.2.2.m2.1c">h+1</annotation><annotation encoding="application/x-llamapun" id="S7.Thmtheorem19.p2.2.2.m2.1d">italic_h + 1</annotation></semantics></math>. By Observation <a class="ltx_ref" href="https://arxiv.org/html/2411.12694v2#S7.SS3" title="7.3 Sketching our algorithm’s correctness. ‣ 7 Results in CONGEST ‣ Local Density and its Distributed Approximation"><span class="ltx_text ltx_ref_tag">7.3</span></a>, for all <math alttext="u\in L_{h+1}" class="ltx_Math" display="inline" id="S7.Thmtheorem19.p2.3.3.m3.1"><semantics id="S7.Thmtheorem19.p2.3.3.m3.1a"><mrow id="S7.Thmtheorem19.p2.3.3.m3.1.1" xref="S7.Thmtheorem19.p2.3.3.m3.1.1.cmml"><mi id="S7.Thmtheorem19.p2.3.3.m3.1.1.2" xref="S7.Thmtheorem19.p2.3.3.m3.1.1.2.cmml">u</mi><mo id="S7.Thmtheorem19.p2.3.3.m3.1.1.1" xref="S7.Thmtheorem19.p2.3.3.m3.1.1.1.cmml">∈</mo><msub id="S7.Thmtheorem19.p2.3.3.m3.1.1.3" xref="S7.Thmtheorem19.p2.3.3.m3.1.1.3.cmml"><mi id="S7.Thmtheorem19.p2.3.3.m3.1.1.3.2" xref="S7.Thmtheorem19.p2.3.3.m3.1.1.3.2.cmml">L</mi><mrow id="S7.Thmtheorem19.p2.3.3.m3.1.1.3.3" xref="S7.Thmtheorem19.p2.3.3.m3.1.1.3.3.cmml"><mi id="S7.Thmtheorem19.p2.3.3.m3.1.1.3.3.2" xref="S7.Thmtheorem19.p2.3.3.m3.1.1.3.3.2.cmml">h</mi><mo id="S7.Thmtheorem19.p2.3.3.m3.1.1.3.3.1" xref="S7.Thmtheorem19.p2.3.3.m3.1.1.3.3.1.cmml">+</mo><mn id="S7.Thmtheorem19.p2.3.3.m3.1.1.3.3.3" xref="S7.Thmtheorem19.p2.3.3.m3.1.1.3.3.3.cmml">1</mn></mrow></msub></mrow><annotation-xml encoding="MathML-Content" id="S7.Thmtheorem19.p2.3.3.m3.1b"><apply id="S7.Thmtheorem19.p2.3.3.m3.1.1.cmml" xref="S7.Thmtheorem19.p2.3.3.m3.1.1"><in id="S7.Thmtheorem19.p2.3.3.m3.1.1.1.cmml" xref="S7.Thmtheorem19.p2.3.3.m3.1.1.1"></in><ci id="S7.Thmtheorem19.p2.3.3.m3.1.1.2.cmml" xref="S7.Thmtheorem19.p2.3.3.m3.1.1.2">𝑢</ci><apply id="S7.Thmtheorem19.p2.3.3.m3.1.1.3.cmml" xref="S7.Thmtheorem19.p2.3.3.m3.1.1.3"><csymbol cd="ambiguous" id="S7.Thmtheorem19.p2.3.3.m3.1.1.3.1.cmml" xref="S7.Thmtheorem19.p2.3.3.m3.1.1.3">subscript</csymbol><ci id="S7.Thmtheorem19.p2.3.3.m3.1.1.3.2.cmml" xref="S7.Thmtheorem19.p2.3.3.m3.1.1.3.2">𝐿</ci><apply id="S7.Thmtheorem19.p2.3.3.m3.1.1.3.3.cmml" xref="S7.Thmtheorem19.p2.3.3.m3.1.1.3.3"><plus id="S7.Thmtheorem19.p2.3.3.m3.1.1.3.3.1.cmml" xref="S7.Thmtheorem19.p2.3.3.m3.1.1.3.3.1"></plus><ci id="S7.Thmtheorem19.p2.3.3.m3.1.1.3.3.2.cmml" xref="S7.Thmtheorem19.p2.3.3.m3.1.1.3.3.2">ℎ</ci><cn id="S7.Thmtheorem19.p2.3.3.m3.1.1.3.3.3.cmml" type="integer" xref="S7.Thmtheorem19.p2.3.3.m3.1.1.3.3.3">1</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.Thmtheorem19.p2.3.3.m3.1c">u\in L_{h+1}</annotation><annotation encoding="application/x-llamapun" id="S7.Thmtheorem19.p2.3.3.m3.1d">italic_u ∈ italic_L start_POSTSUBSCRIPT italic_h + 1 end_POSTSUBSCRIPT</annotation></semantics></math> with a violating edge <math alttext="\overline{uv}" class="ltx_Math" display="inline" id="S7.Thmtheorem19.p2.4.4.m4.1"><semantics id="S7.Thmtheorem19.p2.4.4.m4.1a"><mover accent="true" id="S7.Thmtheorem19.p2.4.4.m4.1.1" xref="S7.Thmtheorem19.p2.4.4.m4.1.1.cmml"><mrow id="S7.Thmtheorem19.p2.4.4.m4.1.1.2" xref="S7.Thmtheorem19.p2.4.4.m4.1.1.2.cmml"><mi id="S7.Thmtheorem19.p2.4.4.m4.1.1.2.2" xref="S7.Thmtheorem19.p2.4.4.m4.1.1.2.2.cmml">u</mi><mo id="S7.Thmtheorem19.p2.4.4.m4.1.1.2.1" xref="S7.Thmtheorem19.p2.4.4.m4.1.1.2.1.cmml">⁢</mo><mi id="S7.Thmtheorem19.p2.4.4.m4.1.1.2.3" xref="S7.Thmtheorem19.p2.4.4.m4.1.1.2.3.cmml">v</mi></mrow><mo id="S7.Thmtheorem19.p2.4.4.m4.1.1.1" xref="S7.Thmtheorem19.p2.4.4.m4.1.1.1.cmml">¯</mo></mover><annotation-xml encoding="MathML-Content" id="S7.Thmtheorem19.p2.4.4.m4.1b"><apply id="S7.Thmtheorem19.p2.4.4.m4.1.1.cmml" xref="S7.Thmtheorem19.p2.4.4.m4.1.1"><ci id="S7.Thmtheorem19.p2.4.4.m4.1.1.1.cmml" xref="S7.Thmtheorem19.p2.4.4.m4.1.1.1">¯</ci><apply id="S7.Thmtheorem19.p2.4.4.m4.1.1.2.cmml" xref="S7.Thmtheorem19.p2.4.4.m4.1.1.2"><times id="S7.Thmtheorem19.p2.4.4.m4.1.1.2.1.cmml" xref="S7.Thmtheorem19.p2.4.4.m4.1.1.2.1"></times><ci id="S7.Thmtheorem19.p2.4.4.m4.1.1.2.2.cmml" xref="S7.Thmtheorem19.p2.4.4.m4.1.1.2.2">𝑢</ci><ci id="S7.Thmtheorem19.p2.4.4.m4.1.1.2.3.cmml" xref="S7.Thmtheorem19.p2.4.4.m4.1.1.2.3">𝑣</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.Thmtheorem19.p2.4.4.m4.1c">\overline{uv}</annotation><annotation encoding="application/x-llamapun" id="S7.Thmtheorem19.p2.4.4.m4.1d">over¯ start_ARG italic_u italic_v end_ARG</annotation></semantics></math>: <math alttext="l_{m}(u)=h+1" class="ltx_Math" display="inline" id="S7.Thmtheorem19.p2.5.5.m5.1"><semantics id="S7.Thmtheorem19.p2.5.5.m5.1a"><mrow id="S7.Thmtheorem19.p2.5.5.m5.1.2" xref="S7.Thmtheorem19.p2.5.5.m5.1.2.cmml"><mrow id="S7.Thmtheorem19.p2.5.5.m5.1.2.2" xref="S7.Thmtheorem19.p2.5.5.m5.1.2.2.cmml"><msub id="S7.Thmtheorem19.p2.5.5.m5.1.2.2.2" xref="S7.Thmtheorem19.p2.5.5.m5.1.2.2.2.cmml"><mi id="S7.Thmtheorem19.p2.5.5.m5.1.2.2.2.2" xref="S7.Thmtheorem19.p2.5.5.m5.1.2.2.2.2.cmml">l</mi><mi id="S7.Thmtheorem19.p2.5.5.m5.1.2.2.2.3" xref="S7.Thmtheorem19.p2.5.5.m5.1.2.2.2.3.cmml">m</mi></msub><mo id="S7.Thmtheorem19.p2.5.5.m5.1.2.2.1" xref="S7.Thmtheorem19.p2.5.5.m5.1.2.2.1.cmml">⁢</mo><mrow id="S7.Thmtheorem19.p2.5.5.m5.1.2.2.3.2" xref="S7.Thmtheorem19.p2.5.5.m5.1.2.2.cmml"><mo id="S7.Thmtheorem19.p2.5.5.m5.1.2.2.3.2.1" stretchy="false" xref="S7.Thmtheorem19.p2.5.5.m5.1.2.2.cmml">(</mo><mi id="S7.Thmtheorem19.p2.5.5.m5.1.1" xref="S7.Thmtheorem19.p2.5.5.m5.1.1.cmml">u</mi><mo id="S7.Thmtheorem19.p2.5.5.m5.1.2.2.3.2.2" stretchy="false" xref="S7.Thmtheorem19.p2.5.5.m5.1.2.2.cmml">)</mo></mrow></mrow><mo id="S7.Thmtheorem19.p2.5.5.m5.1.2.1" xref="S7.Thmtheorem19.p2.5.5.m5.1.2.1.cmml">=</mo><mrow id="S7.Thmtheorem19.p2.5.5.m5.1.2.3" xref="S7.Thmtheorem19.p2.5.5.m5.1.2.3.cmml"><mi id="S7.Thmtheorem19.p2.5.5.m5.1.2.3.2" xref="S7.Thmtheorem19.p2.5.5.m5.1.2.3.2.cmml">h</mi><mo id="S7.Thmtheorem19.p2.5.5.m5.1.2.3.1" xref="S7.Thmtheorem19.p2.5.5.m5.1.2.3.1.cmml">+</mo><mn id="S7.Thmtheorem19.p2.5.5.m5.1.2.3.3" xref="S7.Thmtheorem19.p2.5.5.m5.1.2.3.3.cmml">1</mn></mrow></mrow><annotation-xml encoding="MathML-Content" id="S7.Thmtheorem19.p2.5.5.m5.1b"><apply id="S7.Thmtheorem19.p2.5.5.m5.1.2.cmml" xref="S7.Thmtheorem19.p2.5.5.m5.1.2"><eq id="S7.Thmtheorem19.p2.5.5.m5.1.2.1.cmml" xref="S7.Thmtheorem19.p2.5.5.m5.1.2.1"></eq><apply id="S7.Thmtheorem19.p2.5.5.m5.1.2.2.cmml" xref="S7.Thmtheorem19.p2.5.5.m5.1.2.2"><times id="S7.Thmtheorem19.p2.5.5.m5.1.2.2.1.cmml" xref="S7.Thmtheorem19.p2.5.5.m5.1.2.2.1"></times><apply id="S7.Thmtheorem19.p2.5.5.m5.1.2.2.2.cmml" xref="S7.Thmtheorem19.p2.5.5.m5.1.2.2.2"><csymbol cd="ambiguous" id="S7.Thmtheorem19.p2.5.5.m5.1.2.2.2.1.cmml" xref="S7.Thmtheorem19.p2.5.5.m5.1.2.2.2">subscript</csymbol><ci id="S7.Thmtheorem19.p2.5.5.m5.1.2.2.2.2.cmml" xref="S7.Thmtheorem19.p2.5.5.m5.1.2.2.2.2">𝑙</ci><ci id="S7.Thmtheorem19.p2.5.5.m5.1.2.2.2.3.cmml" xref="S7.Thmtheorem19.p2.5.5.m5.1.2.2.2.3">𝑚</ci></apply><ci id="S7.Thmtheorem19.p2.5.5.m5.1.1.cmml" xref="S7.Thmtheorem19.p2.5.5.m5.1.1">𝑢</ci></apply><apply id="S7.Thmtheorem19.p2.5.5.m5.1.2.3.cmml" xref="S7.Thmtheorem19.p2.5.5.m5.1.2.3"><plus id="S7.Thmtheorem19.p2.5.5.m5.1.2.3.1.cmml" xref="S7.Thmtheorem19.p2.5.5.m5.1.2.3.1"></plus><ci id="S7.Thmtheorem19.p2.5.5.m5.1.2.3.2.cmml" xref="S7.Thmtheorem19.p2.5.5.m5.1.2.3.2">ℎ</ci><cn id="S7.Thmtheorem19.p2.5.5.m5.1.2.3.3.cmml" type="integer" xref="S7.Thmtheorem19.p2.5.5.m5.1.2.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.Thmtheorem19.p2.5.5.m5.1c">l_{m}(u)=h+1</annotation><annotation encoding="application/x-llamapun" id="S7.Thmtheorem19.p2.5.5.m5.1d">italic_l start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT ( italic_u ) = italic_h + 1</annotation></semantics></math> and <math alttext="v\in T_{m}" class="ltx_Math" display="inline" id="S7.Thmtheorem19.p2.6.6.m6.1"><semantics id="S7.Thmtheorem19.p2.6.6.m6.1a"><mrow id="S7.Thmtheorem19.p2.6.6.m6.1.1" xref="S7.Thmtheorem19.p2.6.6.m6.1.1.cmml"><mi id="S7.Thmtheorem19.p2.6.6.m6.1.1.2" xref="S7.Thmtheorem19.p2.6.6.m6.1.1.2.cmml">v</mi><mo id="S7.Thmtheorem19.p2.6.6.m6.1.1.1" xref="S7.Thmtheorem19.p2.6.6.m6.1.1.1.cmml">∈</mo><msub id="S7.Thmtheorem19.p2.6.6.m6.1.1.3" xref="S7.Thmtheorem19.p2.6.6.m6.1.1.3.cmml"><mi id="S7.Thmtheorem19.p2.6.6.m6.1.1.3.2" xref="S7.Thmtheorem19.p2.6.6.m6.1.1.3.2.cmml">T</mi><mi id="S7.Thmtheorem19.p2.6.6.m6.1.1.3.3" xref="S7.Thmtheorem19.p2.6.6.m6.1.1.3.3.cmml">m</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S7.Thmtheorem19.p2.6.6.m6.1b"><apply id="S7.Thmtheorem19.p2.6.6.m6.1.1.cmml" xref="S7.Thmtheorem19.p2.6.6.m6.1.1"><in id="S7.Thmtheorem19.p2.6.6.m6.1.1.1.cmml" xref="S7.Thmtheorem19.p2.6.6.m6.1.1.1"></in><ci id="S7.Thmtheorem19.p2.6.6.m6.1.1.2.cmml" xref="S7.Thmtheorem19.p2.6.6.m6.1.1.2">𝑣</ci><apply id="S7.Thmtheorem19.p2.6.6.m6.1.1.3.cmml" xref="S7.Thmtheorem19.p2.6.6.m6.1.1.3"><csymbol cd="ambiguous" id="S7.Thmtheorem19.p2.6.6.m6.1.1.3.1.cmml" xref="S7.Thmtheorem19.p2.6.6.m6.1.1.3">subscript</csymbol><ci id="S7.Thmtheorem19.p2.6.6.m6.1.1.3.2.cmml" xref="S7.Thmtheorem19.p2.6.6.m6.1.1.3.2">𝑇</ci><ci id="S7.Thmtheorem19.p2.6.6.m6.1.1.3.3.cmml" xref="S7.Thmtheorem19.p2.6.6.m6.1.1.3.3">𝑚</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.Thmtheorem19.p2.6.6.m6.1c">v\in T_{m}</annotation><annotation encoding="application/x-llamapun" id="S7.Thmtheorem19.p2.6.6.m6.1d">italic_v ∈ italic_T start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT</annotation></semantics></math> (since all vertices at a level below <math alttext="h+1" class="ltx_Math" display="inline" id="S7.Thmtheorem19.p2.7.7.m7.1"><semantics id="S7.Thmtheorem19.p2.7.7.m7.1a"><mrow id="S7.Thmtheorem19.p2.7.7.m7.1.1" xref="S7.Thmtheorem19.p2.7.7.m7.1.1.cmml"><mi id="S7.Thmtheorem19.p2.7.7.m7.1.1.2" xref="S7.Thmtheorem19.p2.7.7.m7.1.1.2.cmml">h</mi><mo id="S7.Thmtheorem19.p2.7.7.m7.1.1.1" xref="S7.Thmtheorem19.p2.7.7.m7.1.1.1.cmml">+</mo><mn id="S7.Thmtheorem19.p2.7.7.m7.1.1.3" xref="S7.Thmtheorem19.p2.7.7.m7.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S7.Thmtheorem19.p2.7.7.m7.1b"><apply id="S7.Thmtheorem19.p2.7.7.m7.1.1.cmml" xref="S7.Thmtheorem19.p2.7.7.m7.1.1"><plus id="S7.Thmtheorem19.p2.7.7.m7.1.1.1.cmml" xref="S7.Thmtheorem19.p2.7.7.m7.1.1.1"></plus><ci id="S7.Thmtheorem19.p2.7.7.m7.1.1.2.cmml" xref="S7.Thmtheorem19.p2.7.7.m7.1.1.2">ℎ</ci><cn id="S7.Thmtheorem19.p2.7.7.m7.1.1.3.cmml" type="integer" xref="S7.Thmtheorem19.p2.7.7.m7.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.Thmtheorem19.p2.7.7.m7.1c">h+1</annotation><annotation encoding="application/x-llamapun" id="S7.Thmtheorem19.p2.7.7.m7.1d">italic_h + 1</annotation></semantics></math> only increase their out-degree). Thus any violating edge <math alttext="(u,v)\in E_{s}" class="ltx_Math" display="inline" id="S7.Thmtheorem19.p2.8.8.m8.2"><semantics id="S7.Thmtheorem19.p2.8.8.m8.2a"><mrow id="S7.Thmtheorem19.p2.8.8.m8.2.3" xref="S7.Thmtheorem19.p2.8.8.m8.2.3.cmml"><mrow id="S7.Thmtheorem19.p2.8.8.m8.2.3.2.2" xref="S7.Thmtheorem19.p2.8.8.m8.2.3.2.1.cmml"><mo id="S7.Thmtheorem19.p2.8.8.m8.2.3.2.2.1" stretchy="false" xref="S7.Thmtheorem19.p2.8.8.m8.2.3.2.1.cmml">(</mo><mi id="S7.Thmtheorem19.p2.8.8.m8.1.1" xref="S7.Thmtheorem19.p2.8.8.m8.1.1.cmml">u</mi><mo id="S7.Thmtheorem19.p2.8.8.m8.2.3.2.2.2" xref="S7.Thmtheorem19.p2.8.8.m8.2.3.2.1.cmml">,</mo><mi id="S7.Thmtheorem19.p2.8.8.m8.2.2" xref="S7.Thmtheorem19.p2.8.8.m8.2.2.cmml">v</mi><mo id="S7.Thmtheorem19.p2.8.8.m8.2.3.2.2.3" stretchy="false" xref="S7.Thmtheorem19.p2.8.8.m8.2.3.2.1.cmml">)</mo></mrow><mo id="S7.Thmtheorem19.p2.8.8.m8.2.3.1" xref="S7.Thmtheorem19.p2.8.8.m8.2.3.1.cmml">∈</mo><msub id="S7.Thmtheorem19.p2.8.8.m8.2.3.3" xref="S7.Thmtheorem19.p2.8.8.m8.2.3.3.cmml"><mi id="S7.Thmtheorem19.p2.8.8.m8.2.3.3.2" xref="S7.Thmtheorem19.p2.8.8.m8.2.3.3.2.cmml">E</mi><mi id="S7.Thmtheorem19.p2.8.8.m8.2.3.3.3" xref="S7.Thmtheorem19.p2.8.8.m8.2.3.3.3.cmml">s</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S7.Thmtheorem19.p2.8.8.m8.2b"><apply id="S7.Thmtheorem19.p2.8.8.m8.2.3.cmml" xref="S7.Thmtheorem19.p2.8.8.m8.2.3"><in id="S7.Thmtheorem19.p2.8.8.m8.2.3.1.cmml" xref="S7.Thmtheorem19.p2.8.8.m8.2.3.1"></in><interval closure="open" id="S7.Thmtheorem19.p2.8.8.m8.2.3.2.1.cmml" xref="S7.Thmtheorem19.p2.8.8.m8.2.3.2.2"><ci id="S7.Thmtheorem19.p2.8.8.m8.1.1.cmml" xref="S7.Thmtheorem19.p2.8.8.m8.1.1">𝑢</ci><ci id="S7.Thmtheorem19.p2.8.8.m8.2.2.cmml" xref="S7.Thmtheorem19.p2.8.8.m8.2.2">𝑣</ci></interval><apply id="S7.Thmtheorem19.p2.8.8.m8.2.3.3.cmml" xref="S7.Thmtheorem19.p2.8.8.m8.2.3.3"><csymbol cd="ambiguous" id="S7.Thmtheorem19.p2.8.8.m8.2.3.3.1.cmml" xref="S7.Thmtheorem19.p2.8.8.m8.2.3.3">subscript</csymbol><ci id="S7.Thmtheorem19.p2.8.8.m8.2.3.3.2.cmml" xref="S7.Thmtheorem19.p2.8.8.m8.2.3.3.2">𝐸</ci><ci id="S7.Thmtheorem19.p2.8.8.m8.2.3.3.3.cmml" xref="S7.Thmtheorem19.p2.8.8.m8.2.3.3.3">𝑠</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.Thmtheorem19.p2.8.8.m8.2c">(u,v)\in E_{s}</annotation><annotation encoding="application/x-llamapun" id="S7.Thmtheorem19.p2.8.8.m8.2d">( italic_u , italic_v ) ∈ italic_E start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT</annotation></semantics></math> with <math alttext="u\in V_{s}" class="ltx_Math" display="inline" id="S7.Thmtheorem19.p2.9.9.m9.1"><semantics id="S7.Thmtheorem19.p2.9.9.m9.1a"><mrow id="S7.Thmtheorem19.p2.9.9.m9.1.1" xref="S7.Thmtheorem19.p2.9.9.m9.1.1.cmml"><mi id="S7.Thmtheorem19.p2.9.9.m9.1.1.2" xref="S7.Thmtheorem19.p2.9.9.m9.1.1.2.cmml">u</mi><mo id="S7.Thmtheorem19.p2.9.9.m9.1.1.1" xref="S7.Thmtheorem19.p2.9.9.m9.1.1.1.cmml">∈</mo><msub id="S7.Thmtheorem19.p2.9.9.m9.1.1.3" xref="S7.Thmtheorem19.p2.9.9.m9.1.1.3.cmml"><mi id="S7.Thmtheorem19.p2.9.9.m9.1.1.3.2" xref="S7.Thmtheorem19.p2.9.9.m9.1.1.3.2.cmml">V</mi><mi id="S7.Thmtheorem19.p2.9.9.m9.1.1.3.3" xref="S7.Thmtheorem19.p2.9.9.m9.1.1.3.3.cmml">s</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S7.Thmtheorem19.p2.9.9.m9.1b"><apply id="S7.Thmtheorem19.p2.9.9.m9.1.1.cmml" xref="S7.Thmtheorem19.p2.9.9.m9.1.1"><in id="S7.Thmtheorem19.p2.9.9.m9.1.1.1.cmml" xref="S7.Thmtheorem19.p2.9.9.m9.1.1.1"></in><ci id="S7.Thmtheorem19.p2.9.9.m9.1.1.2.cmml" xref="S7.Thmtheorem19.p2.9.9.m9.1.1.2">𝑢</ci><apply id="S7.Thmtheorem19.p2.9.9.m9.1.1.3.cmml" xref="S7.Thmtheorem19.p2.9.9.m9.1.1.3"><csymbol cd="ambiguous" id="S7.Thmtheorem19.p2.9.9.m9.1.1.3.1.cmml" xref="S7.Thmtheorem19.p2.9.9.m9.1.1.3">subscript</csymbol><ci id="S7.Thmtheorem19.p2.9.9.m9.1.1.3.2.cmml" xref="S7.Thmtheorem19.p2.9.9.m9.1.1.3.2">𝑉</ci><ci id="S7.Thmtheorem19.p2.9.9.m9.1.1.3.3.cmml" xref="S7.Thmtheorem19.p2.9.9.m9.1.1.3.3">𝑠</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.Thmtheorem19.p2.9.9.m9.1c">u\in V_{s}</annotation><annotation encoding="application/x-llamapun" id="S7.Thmtheorem19.p2.9.9.m9.1d">italic_u ∈ italic_V start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT</annotation></semantics></math>. Since for <math alttext="s=\ell-h+1" class="ltx_Math" display="inline" id="S7.Thmtheorem19.p2.10.10.m10.1"><semantics id="S7.Thmtheorem19.p2.10.10.m10.1a"><mrow id="S7.Thmtheorem19.p2.10.10.m10.1.1" xref="S7.Thmtheorem19.p2.10.10.m10.1.1.cmml"><mi id="S7.Thmtheorem19.p2.10.10.m10.1.1.2" xref="S7.Thmtheorem19.p2.10.10.m10.1.1.2.cmml">s</mi><mo id="S7.Thmtheorem19.p2.10.10.m10.1.1.1" xref="S7.Thmtheorem19.p2.10.10.m10.1.1.1.cmml">=</mo><mrow id="S7.Thmtheorem19.p2.10.10.m10.1.1.3" xref="S7.Thmtheorem19.p2.10.10.m10.1.1.3.cmml"><mrow id="S7.Thmtheorem19.p2.10.10.m10.1.1.3.2" xref="S7.Thmtheorem19.p2.10.10.m10.1.1.3.2.cmml"><mi id="S7.Thmtheorem19.p2.10.10.m10.1.1.3.2.2" mathvariant="normal" xref="S7.Thmtheorem19.p2.10.10.m10.1.1.3.2.2.cmml">ℓ</mi><mo id="S7.Thmtheorem19.p2.10.10.m10.1.1.3.2.1" xref="S7.Thmtheorem19.p2.10.10.m10.1.1.3.2.1.cmml">−</mo><mi id="S7.Thmtheorem19.p2.10.10.m10.1.1.3.2.3" xref="S7.Thmtheorem19.p2.10.10.m10.1.1.3.2.3.cmml">h</mi></mrow><mo id="S7.Thmtheorem19.p2.10.10.m10.1.1.3.1" xref="S7.Thmtheorem19.p2.10.10.m10.1.1.3.1.cmml">+</mo><mn id="S7.Thmtheorem19.p2.10.10.m10.1.1.3.3" xref="S7.Thmtheorem19.p2.10.10.m10.1.1.3.3.cmml">1</mn></mrow></mrow><annotation-xml encoding="MathML-Content" id="S7.Thmtheorem19.p2.10.10.m10.1b"><apply id="S7.Thmtheorem19.p2.10.10.m10.1.1.cmml" xref="S7.Thmtheorem19.p2.10.10.m10.1.1"><eq id="S7.Thmtheorem19.p2.10.10.m10.1.1.1.cmml" xref="S7.Thmtheorem19.p2.10.10.m10.1.1.1"></eq><ci id="S7.Thmtheorem19.p2.10.10.m10.1.1.2.cmml" xref="S7.Thmtheorem19.p2.10.10.m10.1.1.2">𝑠</ci><apply id="S7.Thmtheorem19.p2.10.10.m10.1.1.3.cmml" xref="S7.Thmtheorem19.p2.10.10.m10.1.1.3"><plus id="S7.Thmtheorem19.p2.10.10.m10.1.1.3.1.cmml" xref="S7.Thmtheorem19.p2.10.10.m10.1.1.3.1"></plus><apply id="S7.Thmtheorem19.p2.10.10.m10.1.1.3.2.cmml" xref="S7.Thmtheorem19.p2.10.10.m10.1.1.3.2"><minus id="S7.Thmtheorem19.p2.10.10.m10.1.1.3.2.1.cmml" xref="S7.Thmtheorem19.p2.10.10.m10.1.1.3.2.1"></minus><ci id="S7.Thmtheorem19.p2.10.10.m10.1.1.3.2.2.cmml" xref="S7.Thmtheorem19.p2.10.10.m10.1.1.3.2.2">ℓ</ci><ci id="S7.Thmtheorem19.p2.10.10.m10.1.1.3.2.3.cmml" xref="S7.Thmtheorem19.p2.10.10.m10.1.1.3.2.3">ℎ</ci></apply><cn id="S7.Thmtheorem19.p2.10.10.m10.1.1.3.3.cmml" type="integer" xref="S7.Thmtheorem19.p2.10.10.m10.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.Thmtheorem19.p2.10.10.m10.1c">s=\ell-h+1</annotation><annotation encoding="application/x-llamapun" id="S7.Thmtheorem19.p2.10.10.m10.1d">italic_s = roman_ℓ - italic_h + 1</annotation></semantics></math> the set <math alttext="V_{s}" class="ltx_Math" display="inline" id="S7.Thmtheorem19.p2.11.11.m11.1"><semantics id="S7.Thmtheorem19.p2.11.11.m11.1a"><msub id="S7.Thmtheorem19.p2.11.11.m11.1.1" xref="S7.Thmtheorem19.p2.11.11.m11.1.1.cmml"><mi id="S7.Thmtheorem19.p2.11.11.m11.1.1.2" xref="S7.Thmtheorem19.p2.11.11.m11.1.1.2.cmml">V</mi><mi id="S7.Thmtheorem19.p2.11.11.m11.1.1.3" xref="S7.Thmtheorem19.p2.11.11.m11.1.1.3.cmml">s</mi></msub><annotation-xml encoding="MathML-Content" id="S7.Thmtheorem19.p2.11.11.m11.1b"><apply id="S7.Thmtheorem19.p2.11.11.m11.1.1.cmml" xref="S7.Thmtheorem19.p2.11.11.m11.1.1"><csymbol cd="ambiguous" id="S7.Thmtheorem19.p2.11.11.m11.1.1.1.cmml" xref="S7.Thmtheorem19.p2.11.11.m11.1.1">subscript</csymbol><ci id="S7.Thmtheorem19.p2.11.11.m11.1.1.2.cmml" xref="S7.Thmtheorem19.p2.11.11.m11.1.1.2">𝑉</ci><ci id="S7.Thmtheorem19.p2.11.11.m11.1.1.3.cmml" xref="S7.Thmtheorem19.p2.11.11.m11.1.1.3">𝑠</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.Thmtheorem19.p2.11.11.m11.1c">V_{s}</annotation><annotation encoding="application/x-llamapun" id="S7.Thmtheorem19.p2.11.11.m11.1d">italic_V start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT</annotation></semantics></math> is empty, the lemma follows.</span></p> </div> </div> <div class="ltx_para" id="S7.SS4.p3"> <p class="ltx_p" id="S7.SS4.p3.3">Next, we show that after each even <span class="ltx_text ltx_font_smallcaps" id="S7.SS4.p3.3.1">minute</span> (i.e., at the start of each odd <span class="ltx_text ltx_font_smallcaps" id="S7.SS4.p3.3.2">minute</span>), there are only violating out-edges from <math alttext="L_{h-1}" class="ltx_Math" display="inline" id="S7.SS4.p3.1.m1.1"><semantics id="S7.SS4.p3.1.m1.1a"><msub id="S7.SS4.p3.1.m1.1.1" xref="S7.SS4.p3.1.m1.1.1.cmml"><mi id="S7.SS4.p3.1.m1.1.1.2" xref="S7.SS4.p3.1.m1.1.1.2.cmml">L</mi><mrow id="S7.SS4.p3.1.m1.1.1.3" xref="S7.SS4.p3.1.m1.1.1.3.cmml"><mi id="S7.SS4.p3.1.m1.1.1.3.2" xref="S7.SS4.p3.1.m1.1.1.3.2.cmml">h</mi><mo id="S7.SS4.p3.1.m1.1.1.3.1" xref="S7.SS4.p3.1.m1.1.1.3.1.cmml">−</mo><mn id="S7.SS4.p3.1.m1.1.1.3.3" xref="S7.SS4.p3.1.m1.1.1.3.3.cmml">1</mn></mrow></msub><annotation-xml encoding="MathML-Content" id="S7.SS4.p3.1.m1.1b"><apply id="S7.SS4.p3.1.m1.1.1.cmml" xref="S7.SS4.p3.1.m1.1.1"><csymbol cd="ambiguous" id="S7.SS4.p3.1.m1.1.1.1.cmml" xref="S7.SS4.p3.1.m1.1.1">subscript</csymbol><ci id="S7.SS4.p3.1.m1.1.1.2.cmml" xref="S7.SS4.p3.1.m1.1.1.2">𝐿</ci><apply id="S7.SS4.p3.1.m1.1.1.3.cmml" xref="S7.SS4.p3.1.m1.1.1.3"><minus id="S7.SS4.p3.1.m1.1.1.3.1.cmml" xref="S7.SS4.p3.1.m1.1.1.3.1"></minus><ci id="S7.SS4.p3.1.m1.1.1.3.2.cmml" xref="S7.SS4.p3.1.m1.1.1.3.2">ℎ</ci><cn id="S7.SS4.p3.1.m1.1.1.3.3.cmml" type="integer" xref="S7.SS4.p3.1.m1.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.SS4.p3.1.m1.1c">L_{h-1}</annotation><annotation encoding="application/x-llamapun" id="S7.SS4.p3.1.m1.1d">italic_L start_POSTSUBSCRIPT italic_h - 1 end_POSTSUBSCRIPT</annotation></semantics></math> and below, and from vertices in <math alttext="L_{h+1}" class="ltx_Math" display="inline" id="S7.SS4.p3.2.m2.1"><semantics id="S7.SS4.p3.2.m2.1a"><msub id="S7.SS4.p3.2.m2.1.1" xref="S7.SS4.p3.2.m2.1.1.cmml"><mi id="S7.SS4.p3.2.m2.1.1.2" xref="S7.SS4.p3.2.m2.1.1.2.cmml">L</mi><mrow id="S7.SS4.p3.2.m2.1.1.3" xref="S7.SS4.p3.2.m2.1.1.3.cmml"><mi id="S7.SS4.p3.2.m2.1.1.3.2" xref="S7.SS4.p3.2.m2.1.1.3.2.cmml">h</mi><mo id="S7.SS4.p3.2.m2.1.1.3.1" xref="S7.SS4.p3.2.m2.1.1.3.1.cmml">+</mo><mn id="S7.SS4.p3.2.m2.1.1.3.3" xref="S7.SS4.p3.2.m2.1.1.3.3.cmml">1</mn></mrow></msub><annotation-xml encoding="MathML-Content" id="S7.SS4.p3.2.m2.1b"><apply id="S7.SS4.p3.2.m2.1.1.cmml" xref="S7.SS4.p3.2.m2.1.1"><csymbol cd="ambiguous" id="S7.SS4.p3.2.m2.1.1.1.cmml" xref="S7.SS4.p3.2.m2.1.1">subscript</csymbol><ci id="S7.SS4.p3.2.m2.1.1.2.cmml" xref="S7.SS4.p3.2.m2.1.1.2">𝐿</ci><apply id="S7.SS4.p3.2.m2.1.1.3.cmml" xref="S7.SS4.p3.2.m2.1.1.3"><plus id="S7.SS4.p3.2.m2.1.1.3.1.cmml" xref="S7.SS4.p3.2.m2.1.1.3.1"></plus><ci id="S7.SS4.p3.2.m2.1.1.3.2.cmml" xref="S7.SS4.p3.2.m2.1.1.3.2">ℎ</ci><cn id="S7.SS4.p3.2.m2.1.1.3.3.cmml" type="integer" xref="S7.SS4.p3.2.m2.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.SS4.p3.2.m2.1c">L_{h+1}</annotation><annotation encoding="application/x-llamapun" id="S7.SS4.p3.2.m2.1d">italic_L start_POSTSUBSCRIPT italic_h + 1 end_POSTSUBSCRIPT</annotation></semantics></math> to vertices in <math alttext="L_{h_{1}}" class="ltx_Math" display="inline" id="S7.SS4.p3.3.m3.1"><semantics id="S7.SS4.p3.3.m3.1a"><msub id="S7.SS4.p3.3.m3.1.1" xref="S7.SS4.p3.3.m3.1.1.cmml"><mi id="S7.SS4.p3.3.m3.1.1.2" xref="S7.SS4.p3.3.m3.1.1.2.cmml">L</mi><msub id="S7.SS4.p3.3.m3.1.1.3" xref="S7.SS4.p3.3.m3.1.1.3.cmml"><mi id="S7.SS4.p3.3.m3.1.1.3.2" xref="S7.SS4.p3.3.m3.1.1.3.2.cmml">h</mi><mn id="S7.SS4.p3.3.m3.1.1.3.3" xref="S7.SS4.p3.3.m3.1.1.3.3.cmml">1</mn></msub></msub><annotation-xml encoding="MathML-Content" id="S7.SS4.p3.3.m3.1b"><apply id="S7.SS4.p3.3.m3.1.1.cmml" xref="S7.SS4.p3.3.m3.1.1"><csymbol cd="ambiguous" id="S7.SS4.p3.3.m3.1.1.1.cmml" xref="S7.SS4.p3.3.m3.1.1">subscript</csymbol><ci id="S7.SS4.p3.3.m3.1.1.2.cmml" xref="S7.SS4.p3.3.m3.1.1.2">𝐿</ci><apply id="S7.SS4.p3.3.m3.1.1.3.cmml" xref="S7.SS4.p3.3.m3.1.1.3"><csymbol cd="ambiguous" id="S7.SS4.p3.3.m3.1.1.3.1.cmml" xref="S7.SS4.p3.3.m3.1.1.3">subscript</csymbol><ci id="S7.SS4.p3.3.m3.1.1.3.2.cmml" xref="S7.SS4.p3.3.m3.1.1.3.2">ℎ</ci><cn id="S7.SS4.p3.3.m3.1.1.3.3.cmml" type="integer" xref="S7.SS4.p3.3.m3.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.SS4.p3.3.m3.1c">L_{h_{1}}</annotation><annotation encoding="application/x-llamapun" id="S7.SS4.p3.3.m3.1d">italic_L start_POSTSUBSCRIPT italic_h start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT</annotation></semantics></math> (Corollary <a class="ltx_ref" href="https://arxiv.org/html/2411.12694v2#S7.Thmtheorem22" title="Corollary 7.22. ‣ 7.4 Formally proving correctness ‣ 7 Results in CONGEST ‣ Local Density and its Distributed Approximation"><span class="ltx_text ltx_ref_tag">7.22</span></a>).</p> </div> <div class="ltx_theorem ltx_theorem_lemma" id="S7.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="S7.Thmtheorem20.1.1.1">Lemma 7.20</span></span><span class="ltx_text ltx_font_bold" id="S7.Thmtheorem20.2.2">.</span> </h6> <div class="ltx_para" id="S7.Thmtheorem20.p1"> <p class="ltx_p" id="S7.Thmtheorem20.p1.5"><span class="ltx_text ltx_font_italic" id="S7.Thmtheorem20.p1.5.5">At the start of <math alttext="(h:m:t)" class="ltx_Math" display="inline" id="S7.Thmtheorem20.p1.1.1.m1.1"><semantics id="S7.Thmtheorem20.p1.1.1.m1.1a"><mrow id="S7.Thmtheorem20.p1.1.1.m1.1.1.1" xref="S7.Thmtheorem20.p1.1.1.m1.1.1.1.1.cmml"><mo id="S7.Thmtheorem20.p1.1.1.m1.1.1.1.2" stretchy="false" xref="S7.Thmtheorem20.p1.1.1.m1.1.1.1.1.cmml">(</mo><mrow id="S7.Thmtheorem20.p1.1.1.m1.1.1.1.1" xref="S7.Thmtheorem20.p1.1.1.m1.1.1.1.1.cmml"><mi id="S7.Thmtheorem20.p1.1.1.m1.1.1.1.1.2" xref="S7.Thmtheorem20.p1.1.1.m1.1.1.1.1.2.cmml">h</mi><mo id="S7.Thmtheorem20.p1.1.1.m1.1.1.1.1.3" lspace="0.278em" rspace="0.278em" xref="S7.Thmtheorem20.p1.1.1.m1.1.1.1.1.3.cmml">:</mo><mi id="S7.Thmtheorem20.p1.1.1.m1.1.1.1.1.4" xref="S7.Thmtheorem20.p1.1.1.m1.1.1.1.1.4.cmml">m</mi><mo id="S7.Thmtheorem20.p1.1.1.m1.1.1.1.1.5" lspace="0.278em" rspace="0.278em" xref="S7.Thmtheorem20.p1.1.1.m1.1.1.1.1.5.cmml">:</mo><mi id="S7.Thmtheorem20.p1.1.1.m1.1.1.1.1.6" xref="S7.Thmtheorem20.p1.1.1.m1.1.1.1.1.6.cmml">t</mi></mrow><mo id="S7.Thmtheorem20.p1.1.1.m1.1.1.1.3" stretchy="false" xref="S7.Thmtheorem20.p1.1.1.m1.1.1.1.1.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S7.Thmtheorem20.p1.1.1.m1.1b"><apply id="S7.Thmtheorem20.p1.1.1.m1.1.1.1.1.cmml" xref="S7.Thmtheorem20.p1.1.1.m1.1.1.1"><and id="S7.Thmtheorem20.p1.1.1.m1.1.1.1.1a.cmml" xref="S7.Thmtheorem20.p1.1.1.m1.1.1.1"></and><apply id="S7.Thmtheorem20.p1.1.1.m1.1.1.1.1b.cmml" xref="S7.Thmtheorem20.p1.1.1.m1.1.1.1"><ci id="S7.Thmtheorem20.p1.1.1.m1.1.1.1.1.3.cmml" xref="S7.Thmtheorem20.p1.1.1.m1.1.1.1.1.3">:</ci><ci id="S7.Thmtheorem20.p1.1.1.m1.1.1.1.1.2.cmml" xref="S7.Thmtheorem20.p1.1.1.m1.1.1.1.1.2">ℎ</ci><ci id="S7.Thmtheorem20.p1.1.1.m1.1.1.1.1.4.cmml" xref="S7.Thmtheorem20.p1.1.1.m1.1.1.1.1.4">𝑚</ci></apply><apply id="S7.Thmtheorem20.p1.1.1.m1.1.1.1.1c.cmml" xref="S7.Thmtheorem20.p1.1.1.m1.1.1.1"><ci id="S7.Thmtheorem20.p1.1.1.m1.1.1.1.1.5.cmml" xref="S7.Thmtheorem20.p1.1.1.m1.1.1.1.1.5">:</ci><share href="https://arxiv.org/html/2411.12694v2#S7.Thmtheorem20.p1.1.1.m1.1.1.1.1.4.cmml" id="S7.Thmtheorem20.p1.1.1.m1.1.1.1.1d.cmml" xref="S7.Thmtheorem20.p1.1.1.m1.1.1.1"></share><ci id="S7.Thmtheorem20.p1.1.1.m1.1.1.1.1.6.cmml" xref="S7.Thmtheorem20.p1.1.1.m1.1.1.1.1.6">𝑡</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.Thmtheorem20.p1.1.1.m1.1c">(h:m:t)</annotation><annotation encoding="application/x-llamapun" id="S7.Thmtheorem20.p1.1.1.m1.1d">( italic_h : italic_m : italic_t )</annotation></semantics></math>, with <math alttext="m" class="ltx_Math" display="inline" id="S7.Thmtheorem20.p1.2.2.m2.1"><semantics id="S7.Thmtheorem20.p1.2.2.m2.1a"><mi id="S7.Thmtheorem20.p1.2.2.m2.1.1" xref="S7.Thmtheorem20.p1.2.2.m2.1.1.cmml">m</mi><annotation-xml encoding="MathML-Content" id="S7.Thmtheorem20.p1.2.2.m2.1b"><ci id="S7.Thmtheorem20.p1.2.2.m2.1.1.cmml" xref="S7.Thmtheorem20.p1.2.2.m2.1.1">𝑚</ci></annotation-xml><annotation encoding="application/x-tex" id="S7.Thmtheorem20.p1.2.2.m2.1c">m</annotation><annotation encoding="application/x-llamapun" id="S7.Thmtheorem20.p1.2.2.m2.1d">italic_m</annotation></semantics></math> even, denote by <math alttext="E_{t}" class="ltx_Math" display="inline" id="S7.Thmtheorem20.p1.3.3.m3.1"><semantics id="S7.Thmtheorem20.p1.3.3.m3.1a"><msub id="S7.Thmtheorem20.p1.3.3.m3.1.1" xref="S7.Thmtheorem20.p1.3.3.m3.1.1.cmml"><mi id="S7.Thmtheorem20.p1.3.3.m3.1.1.2" xref="S7.Thmtheorem20.p1.3.3.m3.1.1.2.cmml">E</mi><mi id="S7.Thmtheorem20.p1.3.3.m3.1.1.3" xref="S7.Thmtheorem20.p1.3.3.m3.1.1.3.cmml">t</mi></msub><annotation-xml encoding="MathML-Content" id="S7.Thmtheorem20.p1.3.3.m3.1b"><apply id="S7.Thmtheorem20.p1.3.3.m3.1.1.cmml" xref="S7.Thmtheorem20.p1.3.3.m3.1.1"><csymbol cd="ambiguous" id="S7.Thmtheorem20.p1.3.3.m3.1.1.1.cmml" xref="S7.Thmtheorem20.p1.3.3.m3.1.1">subscript</csymbol><ci id="S7.Thmtheorem20.p1.3.3.m3.1.1.2.cmml" xref="S7.Thmtheorem20.p1.3.3.m3.1.1.2">𝐸</ci><ci id="S7.Thmtheorem20.p1.3.3.m3.1.1.3.cmml" xref="S7.Thmtheorem20.p1.3.3.m3.1.1.3">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.Thmtheorem20.p1.3.3.m3.1c">E_{t}</annotation><annotation encoding="application/x-llamapun" id="S7.Thmtheorem20.p1.3.3.m3.1d">italic_E start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT</annotation></semantics></math> the number of violating edges going from <math alttext="L_{h}" class="ltx_Math" display="inline" id="S7.Thmtheorem20.p1.4.4.m4.1"><semantics id="S7.Thmtheorem20.p1.4.4.m4.1a"><msub id="S7.Thmtheorem20.p1.4.4.m4.1.1" xref="S7.Thmtheorem20.p1.4.4.m4.1.1.cmml"><mi id="S7.Thmtheorem20.p1.4.4.m4.1.1.2" xref="S7.Thmtheorem20.p1.4.4.m4.1.1.2.cmml">L</mi><mi id="S7.Thmtheorem20.p1.4.4.m4.1.1.3" xref="S7.Thmtheorem20.p1.4.4.m4.1.1.3.cmml">h</mi></msub><annotation-xml encoding="MathML-Content" id="S7.Thmtheorem20.p1.4.4.m4.1b"><apply id="S7.Thmtheorem20.p1.4.4.m4.1.1.cmml" xref="S7.Thmtheorem20.p1.4.4.m4.1.1"><csymbol cd="ambiguous" id="S7.Thmtheorem20.p1.4.4.m4.1.1.1.cmml" xref="S7.Thmtheorem20.p1.4.4.m4.1.1">subscript</csymbol><ci id="S7.Thmtheorem20.p1.4.4.m4.1.1.2.cmml" xref="S7.Thmtheorem20.p1.4.4.m4.1.1.2">𝐿</ci><ci id="S7.Thmtheorem20.p1.4.4.m4.1.1.3.cmml" xref="S7.Thmtheorem20.p1.4.4.m4.1.1.3">ℎ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.Thmtheorem20.p1.4.4.m4.1c">L_{h}</annotation><annotation encoding="application/x-llamapun" id="S7.Thmtheorem20.p1.4.4.m4.1d">italic_L start_POSTSUBSCRIPT italic_h end_POSTSUBSCRIPT</annotation></semantics></math>. Then <math alttext="|E_{t}|\leq\frac{7}{8}|E_{t-1}|" class="ltx_Math" display="inline" id="S7.Thmtheorem20.p1.5.5.m5.2"><semantics id="S7.Thmtheorem20.p1.5.5.m5.2a"><mrow id="S7.Thmtheorem20.p1.5.5.m5.2.2" xref="S7.Thmtheorem20.p1.5.5.m5.2.2.cmml"><mrow id="S7.Thmtheorem20.p1.5.5.m5.1.1.1.1" xref="S7.Thmtheorem20.p1.5.5.m5.1.1.1.2.cmml"><mo id="S7.Thmtheorem20.p1.5.5.m5.1.1.1.1.2" stretchy="false" xref="S7.Thmtheorem20.p1.5.5.m5.1.1.1.2.1.cmml">|</mo><msub id="S7.Thmtheorem20.p1.5.5.m5.1.1.1.1.1" xref="S7.Thmtheorem20.p1.5.5.m5.1.1.1.1.1.cmml"><mi id="S7.Thmtheorem20.p1.5.5.m5.1.1.1.1.1.2" xref="S7.Thmtheorem20.p1.5.5.m5.1.1.1.1.1.2.cmml">E</mi><mi id="S7.Thmtheorem20.p1.5.5.m5.1.1.1.1.1.3" xref="S7.Thmtheorem20.p1.5.5.m5.1.1.1.1.1.3.cmml">t</mi></msub><mo id="S7.Thmtheorem20.p1.5.5.m5.1.1.1.1.3" stretchy="false" xref="S7.Thmtheorem20.p1.5.5.m5.1.1.1.2.1.cmml">|</mo></mrow><mo id="S7.Thmtheorem20.p1.5.5.m5.2.2.3" xref="S7.Thmtheorem20.p1.5.5.m5.2.2.3.cmml">≤</mo><mrow id="S7.Thmtheorem20.p1.5.5.m5.2.2.2" xref="S7.Thmtheorem20.p1.5.5.m5.2.2.2.cmml"><mfrac id="S7.Thmtheorem20.p1.5.5.m5.2.2.2.3" xref="S7.Thmtheorem20.p1.5.5.m5.2.2.2.3.cmml"><mn id="S7.Thmtheorem20.p1.5.5.m5.2.2.2.3.2" xref="S7.Thmtheorem20.p1.5.5.m5.2.2.2.3.2.cmml">7</mn><mn id="S7.Thmtheorem20.p1.5.5.m5.2.2.2.3.3" xref="S7.Thmtheorem20.p1.5.5.m5.2.2.2.3.3.cmml">8</mn></mfrac><mo id="S7.Thmtheorem20.p1.5.5.m5.2.2.2.2" xref="S7.Thmtheorem20.p1.5.5.m5.2.2.2.2.cmml">⁢</mo><mrow id="S7.Thmtheorem20.p1.5.5.m5.2.2.2.1.1" xref="S7.Thmtheorem20.p1.5.5.m5.2.2.2.1.2.cmml"><mo id="S7.Thmtheorem20.p1.5.5.m5.2.2.2.1.1.2" stretchy="false" xref="S7.Thmtheorem20.p1.5.5.m5.2.2.2.1.2.1.cmml">|</mo><msub id="S7.Thmtheorem20.p1.5.5.m5.2.2.2.1.1.1" xref="S7.Thmtheorem20.p1.5.5.m5.2.2.2.1.1.1.cmml"><mi id="S7.Thmtheorem20.p1.5.5.m5.2.2.2.1.1.1.2" xref="S7.Thmtheorem20.p1.5.5.m5.2.2.2.1.1.1.2.cmml">E</mi><mrow id="S7.Thmtheorem20.p1.5.5.m5.2.2.2.1.1.1.3" xref="S7.Thmtheorem20.p1.5.5.m5.2.2.2.1.1.1.3.cmml"><mi id="S7.Thmtheorem20.p1.5.5.m5.2.2.2.1.1.1.3.2" xref="S7.Thmtheorem20.p1.5.5.m5.2.2.2.1.1.1.3.2.cmml">t</mi><mo id="S7.Thmtheorem20.p1.5.5.m5.2.2.2.1.1.1.3.1" xref="S7.Thmtheorem20.p1.5.5.m5.2.2.2.1.1.1.3.1.cmml">−</mo><mn id="S7.Thmtheorem20.p1.5.5.m5.2.2.2.1.1.1.3.3" xref="S7.Thmtheorem20.p1.5.5.m5.2.2.2.1.1.1.3.3.cmml">1</mn></mrow></msub><mo id="S7.Thmtheorem20.p1.5.5.m5.2.2.2.1.1.3" stretchy="false" xref="S7.Thmtheorem20.p1.5.5.m5.2.2.2.1.2.1.cmml">|</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S7.Thmtheorem20.p1.5.5.m5.2b"><apply id="S7.Thmtheorem20.p1.5.5.m5.2.2.cmml" xref="S7.Thmtheorem20.p1.5.5.m5.2.2"><leq id="S7.Thmtheorem20.p1.5.5.m5.2.2.3.cmml" xref="S7.Thmtheorem20.p1.5.5.m5.2.2.3"></leq><apply id="S7.Thmtheorem20.p1.5.5.m5.1.1.1.2.cmml" xref="S7.Thmtheorem20.p1.5.5.m5.1.1.1.1"><abs id="S7.Thmtheorem20.p1.5.5.m5.1.1.1.2.1.cmml" xref="S7.Thmtheorem20.p1.5.5.m5.1.1.1.1.2"></abs><apply id="S7.Thmtheorem20.p1.5.5.m5.1.1.1.1.1.cmml" xref="S7.Thmtheorem20.p1.5.5.m5.1.1.1.1.1"><csymbol cd="ambiguous" id="S7.Thmtheorem20.p1.5.5.m5.1.1.1.1.1.1.cmml" xref="S7.Thmtheorem20.p1.5.5.m5.1.1.1.1.1">subscript</csymbol><ci id="S7.Thmtheorem20.p1.5.5.m5.1.1.1.1.1.2.cmml" xref="S7.Thmtheorem20.p1.5.5.m5.1.1.1.1.1.2">𝐸</ci><ci id="S7.Thmtheorem20.p1.5.5.m5.1.1.1.1.1.3.cmml" xref="S7.Thmtheorem20.p1.5.5.m5.1.1.1.1.1.3">𝑡</ci></apply></apply><apply id="S7.Thmtheorem20.p1.5.5.m5.2.2.2.cmml" xref="S7.Thmtheorem20.p1.5.5.m5.2.2.2"><times id="S7.Thmtheorem20.p1.5.5.m5.2.2.2.2.cmml" xref="S7.Thmtheorem20.p1.5.5.m5.2.2.2.2"></times><apply id="S7.Thmtheorem20.p1.5.5.m5.2.2.2.3.cmml" xref="S7.Thmtheorem20.p1.5.5.m5.2.2.2.3"><divide id="S7.Thmtheorem20.p1.5.5.m5.2.2.2.3.1.cmml" xref="S7.Thmtheorem20.p1.5.5.m5.2.2.2.3"></divide><cn id="S7.Thmtheorem20.p1.5.5.m5.2.2.2.3.2.cmml" type="integer" xref="S7.Thmtheorem20.p1.5.5.m5.2.2.2.3.2">7</cn><cn id="S7.Thmtheorem20.p1.5.5.m5.2.2.2.3.3.cmml" type="integer" xref="S7.Thmtheorem20.p1.5.5.m5.2.2.2.3.3">8</cn></apply><apply id="S7.Thmtheorem20.p1.5.5.m5.2.2.2.1.2.cmml" xref="S7.Thmtheorem20.p1.5.5.m5.2.2.2.1.1"><abs id="S7.Thmtheorem20.p1.5.5.m5.2.2.2.1.2.1.cmml" xref="S7.Thmtheorem20.p1.5.5.m5.2.2.2.1.1.2"></abs><apply id="S7.Thmtheorem20.p1.5.5.m5.2.2.2.1.1.1.cmml" xref="S7.Thmtheorem20.p1.5.5.m5.2.2.2.1.1.1"><csymbol cd="ambiguous" id="S7.Thmtheorem20.p1.5.5.m5.2.2.2.1.1.1.1.cmml" xref="S7.Thmtheorem20.p1.5.5.m5.2.2.2.1.1.1">subscript</csymbol><ci id="S7.Thmtheorem20.p1.5.5.m5.2.2.2.1.1.1.2.cmml" xref="S7.Thmtheorem20.p1.5.5.m5.2.2.2.1.1.1.2">𝐸</ci><apply id="S7.Thmtheorem20.p1.5.5.m5.2.2.2.1.1.1.3.cmml" xref="S7.Thmtheorem20.p1.5.5.m5.2.2.2.1.1.1.3"><minus id="S7.Thmtheorem20.p1.5.5.m5.2.2.2.1.1.1.3.1.cmml" xref="S7.Thmtheorem20.p1.5.5.m5.2.2.2.1.1.1.3.1"></minus><ci id="S7.Thmtheorem20.p1.5.5.m5.2.2.2.1.1.1.3.2.cmml" xref="S7.Thmtheorem20.p1.5.5.m5.2.2.2.1.1.1.3.2">𝑡</ci><cn id="S7.Thmtheorem20.p1.5.5.m5.2.2.2.1.1.1.3.3.cmml" type="integer" xref="S7.Thmtheorem20.p1.5.5.m5.2.2.2.1.1.1.3.3">1</cn></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.Thmtheorem20.p1.5.5.m5.2c">|E_{t}|\leq\frac{7}{8}|E_{t-1}|</annotation><annotation encoding="application/x-llamapun" id="S7.Thmtheorem20.p1.5.5.m5.2d">| italic_E start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT | ≤ divide start_ARG 7 end_ARG start_ARG 8 end_ARG | italic_E start_POSTSUBSCRIPT italic_t - 1 end_POSTSUBSCRIPT |</annotation></semantics></math>.</span></p> </div> </div> <div class="ltx_theorem ltx_theorem_proof" id="S7.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="S7.Thmtheorem21.1.1.1">Proof 7.21</span></span><span class="ltx_text ltx_font_bold" id="S7.Thmtheorem21.2.2">.</span> </h6> <div class="ltx_para" id="S7.Thmtheorem21.p1"> <p class="ltx_p" id="S7.Thmtheorem21.p1.10"><span class="ltx_text ltx_font_italic" id="S7.Thmtheorem21.p1.10.10">Let <math alttext="E_{t}\subseteq A_{t}\times B_{t}" class="ltx_Math" display="inline" id="S7.Thmtheorem21.p1.1.1.m1.1"><semantics id="S7.Thmtheorem21.p1.1.1.m1.1a"><mrow id="S7.Thmtheorem21.p1.1.1.m1.1.1" xref="S7.Thmtheorem21.p1.1.1.m1.1.1.cmml"><msub id="S7.Thmtheorem21.p1.1.1.m1.1.1.2" xref="S7.Thmtheorem21.p1.1.1.m1.1.1.2.cmml"><mi id="S7.Thmtheorem21.p1.1.1.m1.1.1.2.2" xref="S7.Thmtheorem21.p1.1.1.m1.1.1.2.2.cmml">E</mi><mi id="S7.Thmtheorem21.p1.1.1.m1.1.1.2.3" xref="S7.Thmtheorem21.p1.1.1.m1.1.1.2.3.cmml">t</mi></msub><mo id="S7.Thmtheorem21.p1.1.1.m1.1.1.1" xref="S7.Thmtheorem21.p1.1.1.m1.1.1.1.cmml">⊆</mo><mrow id="S7.Thmtheorem21.p1.1.1.m1.1.1.3" xref="S7.Thmtheorem21.p1.1.1.m1.1.1.3.cmml"><msub id="S7.Thmtheorem21.p1.1.1.m1.1.1.3.2" xref="S7.Thmtheorem21.p1.1.1.m1.1.1.3.2.cmml"><mi id="S7.Thmtheorem21.p1.1.1.m1.1.1.3.2.2" xref="S7.Thmtheorem21.p1.1.1.m1.1.1.3.2.2.cmml">A</mi><mi id="S7.Thmtheorem21.p1.1.1.m1.1.1.3.2.3" xref="S7.Thmtheorem21.p1.1.1.m1.1.1.3.2.3.cmml">t</mi></msub><mo id="S7.Thmtheorem21.p1.1.1.m1.1.1.3.1" lspace="0.222em" rspace="0.222em" xref="S7.Thmtheorem21.p1.1.1.m1.1.1.3.1.cmml">×</mo><msub id="S7.Thmtheorem21.p1.1.1.m1.1.1.3.3" xref="S7.Thmtheorem21.p1.1.1.m1.1.1.3.3.cmml"><mi id="S7.Thmtheorem21.p1.1.1.m1.1.1.3.3.2" xref="S7.Thmtheorem21.p1.1.1.m1.1.1.3.3.2.cmml">B</mi><mi id="S7.Thmtheorem21.p1.1.1.m1.1.1.3.3.3" xref="S7.Thmtheorem21.p1.1.1.m1.1.1.3.3.3.cmml">t</mi></msub></mrow></mrow><annotation-xml encoding="MathML-Content" id="S7.Thmtheorem21.p1.1.1.m1.1b"><apply id="S7.Thmtheorem21.p1.1.1.m1.1.1.cmml" xref="S7.Thmtheorem21.p1.1.1.m1.1.1"><subset id="S7.Thmtheorem21.p1.1.1.m1.1.1.1.cmml" xref="S7.Thmtheorem21.p1.1.1.m1.1.1.1"></subset><apply id="S7.Thmtheorem21.p1.1.1.m1.1.1.2.cmml" xref="S7.Thmtheorem21.p1.1.1.m1.1.1.2"><csymbol cd="ambiguous" id="S7.Thmtheorem21.p1.1.1.m1.1.1.2.1.cmml" xref="S7.Thmtheorem21.p1.1.1.m1.1.1.2">subscript</csymbol><ci id="S7.Thmtheorem21.p1.1.1.m1.1.1.2.2.cmml" xref="S7.Thmtheorem21.p1.1.1.m1.1.1.2.2">𝐸</ci><ci id="S7.Thmtheorem21.p1.1.1.m1.1.1.2.3.cmml" xref="S7.Thmtheorem21.p1.1.1.m1.1.1.2.3">𝑡</ci></apply><apply id="S7.Thmtheorem21.p1.1.1.m1.1.1.3.cmml" xref="S7.Thmtheorem21.p1.1.1.m1.1.1.3"><times id="S7.Thmtheorem21.p1.1.1.m1.1.1.3.1.cmml" xref="S7.Thmtheorem21.p1.1.1.m1.1.1.3.1"></times><apply id="S7.Thmtheorem21.p1.1.1.m1.1.1.3.2.cmml" xref="S7.Thmtheorem21.p1.1.1.m1.1.1.3.2"><csymbol cd="ambiguous" id="S7.Thmtheorem21.p1.1.1.m1.1.1.3.2.1.cmml" xref="S7.Thmtheorem21.p1.1.1.m1.1.1.3.2">subscript</csymbol><ci id="S7.Thmtheorem21.p1.1.1.m1.1.1.3.2.2.cmml" xref="S7.Thmtheorem21.p1.1.1.m1.1.1.3.2.2">𝐴</ci><ci id="S7.Thmtheorem21.p1.1.1.m1.1.1.3.2.3.cmml" xref="S7.Thmtheorem21.p1.1.1.m1.1.1.3.2.3">𝑡</ci></apply><apply id="S7.Thmtheorem21.p1.1.1.m1.1.1.3.3.cmml" xref="S7.Thmtheorem21.p1.1.1.m1.1.1.3.3"><csymbol cd="ambiguous" id="S7.Thmtheorem21.p1.1.1.m1.1.1.3.3.1.cmml" xref="S7.Thmtheorem21.p1.1.1.m1.1.1.3.3">subscript</csymbol><ci id="S7.Thmtheorem21.p1.1.1.m1.1.1.3.3.2.cmml" xref="S7.Thmtheorem21.p1.1.1.m1.1.1.3.3.2">𝐵</ci><ci id="S7.Thmtheorem21.p1.1.1.m1.1.1.3.3.3.cmml" xref="S7.Thmtheorem21.p1.1.1.m1.1.1.3.3.3">𝑡</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.Thmtheorem21.p1.1.1.m1.1c">E_{t}\subseteq A_{t}\times B_{t}</annotation><annotation encoding="application/x-llamapun" id="S7.Thmtheorem21.p1.1.1.m1.1d">italic_E start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT ⊆ italic_A start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT × italic_B start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT</annotation></semantics></math>. 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xref="S7.Thmtheorem21.p1.2.2.m2.1.1.1.1.4.3">ℎ</ci></apply><apply id="S7.Thmtheorem21.p1.2.2.m2.1.1.1.1.2.cmml" xref="S7.Thmtheorem21.p1.2.2.m2.1.1.1.1.2"><minus id="S7.Thmtheorem21.p1.2.2.m2.1.1.1.1.2.3.cmml" xref="S7.Thmtheorem21.p1.2.2.m2.1.1.1.1.2.3"></minus><apply id="S7.Thmtheorem21.p1.2.2.m2.1.1.1.1.1.1.cmml" xref="S7.Thmtheorem21.p1.2.2.m2.1.1.1.1.1.1"><csymbol cd="ambiguous" id="S7.Thmtheorem21.p1.2.2.m2.1.1.1.1.1.1.2.cmml" xref="S7.Thmtheorem21.p1.2.2.m2.1.1.1.1.1.1">superscript</csymbol><apply id="S7.Thmtheorem21.p1.2.2.m2.1.1.1.1.1.1.1.1.1.cmml" xref="S7.Thmtheorem21.p1.2.2.m2.1.1.1.1.1.1.1.1"><plus id="S7.Thmtheorem21.p1.2.2.m2.1.1.1.1.1.1.1.1.1.1.cmml" xref="S7.Thmtheorem21.p1.2.2.m2.1.1.1.1.1.1.1.1.1.1"></plus><cn id="S7.Thmtheorem21.p1.2.2.m2.1.1.1.1.1.1.1.1.1.2.cmml" type="integer" xref="S7.Thmtheorem21.p1.2.2.m2.1.1.1.1.1.1.1.1.1.2">1</cn><apply id="S7.Thmtheorem21.p1.2.2.m2.1.1.1.1.1.1.1.1.1.3.cmml" xref="S7.Thmtheorem21.p1.2.2.m2.1.1.1.1.1.1.1.1.1.3"><divide 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id="S7.Thmtheorem21.p1.2.2.m2.1.1.1.1.2.2.1.1.1.2.cmml" type="integer" xref="S7.Thmtheorem21.p1.2.2.m2.1.1.1.1.2.2.1.1.1.2">1</cn><apply id="S7.Thmtheorem21.p1.2.2.m2.1.1.1.1.2.2.1.1.1.3.cmml" xref="S7.Thmtheorem21.p1.2.2.m2.1.1.1.1.2.2.1.1.1.3"><divide id="S7.Thmtheorem21.p1.2.2.m2.1.1.1.1.2.2.1.1.1.3.1.cmml" xref="S7.Thmtheorem21.p1.2.2.m2.1.1.1.1.2.2.1.1.1.3"></divide><ci id="S7.Thmtheorem21.p1.2.2.m2.1.1.1.1.2.2.1.1.1.3.2.cmml" xref="S7.Thmtheorem21.p1.2.2.m2.1.1.1.1.2.2.1.1.1.3.2">𝜂</ci><cn id="S7.Thmtheorem21.p1.2.2.m2.1.1.1.1.2.2.1.1.1.3.3.cmml" type="integer" xref="S7.Thmtheorem21.p1.2.2.m2.1.1.1.1.2.2.1.1.1.3.3">2</cn></apply></apply><apply id="S7.Thmtheorem21.p1.2.2.m2.1.1.1.1.2.2.3.cmml" xref="S7.Thmtheorem21.p1.2.2.m2.1.1.1.1.2.2.3"><minus id="S7.Thmtheorem21.p1.2.2.m2.1.1.1.1.2.2.3.1.cmml" xref="S7.Thmtheorem21.p1.2.2.m2.1.1.1.1.2.2.3.1"></minus><ci id="S7.Thmtheorem21.p1.2.2.m2.1.1.1.1.2.2.3.2.cmml" xref="S7.Thmtheorem21.p1.2.2.m2.1.1.1.1.2.2.3.2">ℎ</ci><cn id="S7.Thmtheorem21.p1.2.2.m2.1.1.1.1.2.2.3.3.cmml" type="integer" xref="S7.Thmtheorem21.p1.2.2.m2.1.1.1.1.2.2.3.3">1</cn></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.Thmtheorem21.p1.2.2.m2.1c">\Delta_{h}=(1+\frac{\eta}{2})^{h}-(1+\frac{\eta}{2})^{h-1}.</annotation><annotation encoding="application/x-llamapun" id="S7.Thmtheorem21.p1.2.2.m2.1d">roman_Δ start_POSTSUBSCRIPT italic_h end_POSTSUBSCRIPT = ( 1 + divide start_ARG italic_η end_ARG start_ARG 2 end_ARG ) start_POSTSUPERSCRIPT italic_h end_POSTSUPERSCRIPT - ( 1 + divide start_ARG italic_η end_ARG start_ARG 2 end_ARG ) start_POSTSUPERSCRIPT italic_h - 1 end_POSTSUPERSCRIPT .</annotation></semantics></math> We say that a vertex <math alttext="a\in A_{t}" class="ltx_Math" display="inline" id="S7.Thmtheorem21.p1.3.3.m3.1"><semantics id="S7.Thmtheorem21.p1.3.3.m3.1a"><mrow id="S7.Thmtheorem21.p1.3.3.m3.1.1" xref="S7.Thmtheorem21.p1.3.3.m3.1.1.cmml"><mi id="S7.Thmtheorem21.p1.3.3.m3.1.1.2" xref="S7.Thmtheorem21.p1.3.3.m3.1.1.2.cmml">a</mi><mo id="S7.Thmtheorem21.p1.3.3.m3.1.1.1" xref="S7.Thmtheorem21.p1.3.3.m3.1.1.1.cmml">∈</mo><msub id="S7.Thmtheorem21.p1.3.3.m3.1.1.3" xref="S7.Thmtheorem21.p1.3.3.m3.1.1.3.cmml"><mi id="S7.Thmtheorem21.p1.3.3.m3.1.1.3.2" xref="S7.Thmtheorem21.p1.3.3.m3.1.1.3.2.cmml">A</mi><mi id="S7.Thmtheorem21.p1.3.3.m3.1.1.3.3" xref="S7.Thmtheorem21.p1.3.3.m3.1.1.3.3.cmml">t</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S7.Thmtheorem21.p1.3.3.m3.1b"><apply id="S7.Thmtheorem21.p1.3.3.m3.1.1.cmml" xref="S7.Thmtheorem21.p1.3.3.m3.1.1"><in id="S7.Thmtheorem21.p1.3.3.m3.1.1.1.cmml" xref="S7.Thmtheorem21.p1.3.3.m3.1.1.1"></in><ci id="S7.Thmtheorem21.p1.3.3.m3.1.1.2.cmml" xref="S7.Thmtheorem21.p1.3.3.m3.1.1.2">𝑎</ci><apply id="S7.Thmtheorem21.p1.3.3.m3.1.1.3.cmml" xref="S7.Thmtheorem21.p1.3.3.m3.1.1.3"><csymbol cd="ambiguous" id="S7.Thmtheorem21.p1.3.3.m3.1.1.3.1.cmml" xref="S7.Thmtheorem21.p1.3.3.m3.1.1.3">subscript</csymbol><ci id="S7.Thmtheorem21.p1.3.3.m3.1.1.3.2.cmml" xref="S7.Thmtheorem21.p1.3.3.m3.1.1.3.2">𝐴</ci><ci id="S7.Thmtheorem21.p1.3.3.m3.1.1.3.3.cmml" xref="S7.Thmtheorem21.p1.3.3.m3.1.1.3.3">𝑡</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.Thmtheorem21.p1.3.3.m3.1c">a\in A_{t}</annotation><annotation encoding="application/x-llamapun" id="S7.Thmtheorem21.p1.3.3.m3.1d">italic_a ∈ italic_A start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT</annotation></semantics></math> is <em class="ltx_emph ltx_font_upright" id="S7.Thmtheorem21.p1.10.10.1">satisfied</em> if <math alttext="\textsl{g}(a)\leq(1+\frac{\eta}{2})^{h-1}+\frac{1}{2}\Delta_{h}" class="ltx_Math" display="inline" id="S7.Thmtheorem21.p1.4.4.m4.2"><semantics id="S7.Thmtheorem21.p1.4.4.m4.2a"><mrow id="S7.Thmtheorem21.p1.4.4.m4.2.2" xref="S7.Thmtheorem21.p1.4.4.m4.2.2.cmml"><mrow id="S7.Thmtheorem21.p1.4.4.m4.2.2.3" xref="S7.Thmtheorem21.p1.4.4.m4.2.2.3.cmml"><mtext class="ltx_mathvariant_italic" id="S7.Thmtheorem21.p1.4.4.m4.2.2.3.2" xref="S7.Thmtheorem21.p1.4.4.m4.2.2.3.2a.cmml">g</mtext><mo id="S7.Thmtheorem21.p1.4.4.m4.2.2.3.1" xref="S7.Thmtheorem21.p1.4.4.m4.2.2.3.1.cmml">⁢</mo><mrow id="S7.Thmtheorem21.p1.4.4.m4.2.2.3.3.2" xref="S7.Thmtheorem21.p1.4.4.m4.2.2.3.cmml"><mo id="S7.Thmtheorem21.p1.4.4.m4.2.2.3.3.2.1" stretchy="false" xref="S7.Thmtheorem21.p1.4.4.m4.2.2.3.cmml">(</mo><mi id="S7.Thmtheorem21.p1.4.4.m4.1.1" xref="S7.Thmtheorem21.p1.4.4.m4.1.1.cmml">a</mi><mo id="S7.Thmtheorem21.p1.4.4.m4.2.2.3.3.2.2" stretchy="false" xref="S7.Thmtheorem21.p1.4.4.m4.2.2.3.cmml">)</mo></mrow></mrow><mo id="S7.Thmtheorem21.p1.4.4.m4.2.2.2" xref="S7.Thmtheorem21.p1.4.4.m4.2.2.2.cmml">≤</mo><mrow id="S7.Thmtheorem21.p1.4.4.m4.2.2.1" xref="S7.Thmtheorem21.p1.4.4.m4.2.2.1.cmml"><msup id="S7.Thmtheorem21.p1.4.4.m4.2.2.1.1" xref="S7.Thmtheorem21.p1.4.4.m4.2.2.1.1.cmml"><mrow id="S7.Thmtheorem21.p1.4.4.m4.2.2.1.1.1.1" xref="S7.Thmtheorem21.p1.4.4.m4.2.2.1.1.1.1.1.cmml"><mo id="S7.Thmtheorem21.p1.4.4.m4.2.2.1.1.1.1.2" stretchy="false" xref="S7.Thmtheorem21.p1.4.4.m4.2.2.1.1.1.1.1.cmml">(</mo><mrow id="S7.Thmtheorem21.p1.4.4.m4.2.2.1.1.1.1.1" xref="S7.Thmtheorem21.p1.4.4.m4.2.2.1.1.1.1.1.cmml"><mn id="S7.Thmtheorem21.p1.4.4.m4.2.2.1.1.1.1.1.2" xref="S7.Thmtheorem21.p1.4.4.m4.2.2.1.1.1.1.1.2.cmml">1</mn><mo id="S7.Thmtheorem21.p1.4.4.m4.2.2.1.1.1.1.1.1" xref="S7.Thmtheorem21.p1.4.4.m4.2.2.1.1.1.1.1.1.cmml">+</mo><mfrac id="S7.Thmtheorem21.p1.4.4.m4.2.2.1.1.1.1.1.3" xref="S7.Thmtheorem21.p1.4.4.m4.2.2.1.1.1.1.1.3.cmml"><mi id="S7.Thmtheorem21.p1.4.4.m4.2.2.1.1.1.1.1.3.2" xref="S7.Thmtheorem21.p1.4.4.m4.2.2.1.1.1.1.1.3.2.cmml">η</mi><mn id="S7.Thmtheorem21.p1.4.4.m4.2.2.1.1.1.1.1.3.3" xref="S7.Thmtheorem21.p1.4.4.m4.2.2.1.1.1.1.1.3.3.cmml">2</mn></mfrac></mrow><mo id="S7.Thmtheorem21.p1.4.4.m4.2.2.1.1.1.1.3" stretchy="false" xref="S7.Thmtheorem21.p1.4.4.m4.2.2.1.1.1.1.1.cmml">)</mo></mrow><mrow id="S7.Thmtheorem21.p1.4.4.m4.2.2.1.1.3" xref="S7.Thmtheorem21.p1.4.4.m4.2.2.1.1.3.cmml"><mi id="S7.Thmtheorem21.p1.4.4.m4.2.2.1.1.3.2" xref="S7.Thmtheorem21.p1.4.4.m4.2.2.1.1.3.2.cmml">h</mi><mo id="S7.Thmtheorem21.p1.4.4.m4.2.2.1.1.3.1" xref="S7.Thmtheorem21.p1.4.4.m4.2.2.1.1.3.1.cmml">−</mo><mn id="S7.Thmtheorem21.p1.4.4.m4.2.2.1.1.3.3" xref="S7.Thmtheorem21.p1.4.4.m4.2.2.1.1.3.3.cmml">1</mn></mrow></msup><mo id="S7.Thmtheorem21.p1.4.4.m4.2.2.1.2" xref="S7.Thmtheorem21.p1.4.4.m4.2.2.1.2.cmml">+</mo><mrow id="S7.Thmtheorem21.p1.4.4.m4.2.2.1.3" xref="S7.Thmtheorem21.p1.4.4.m4.2.2.1.3.cmml"><mfrac id="S7.Thmtheorem21.p1.4.4.m4.2.2.1.3.2" xref="S7.Thmtheorem21.p1.4.4.m4.2.2.1.3.2.cmml"><mn id="S7.Thmtheorem21.p1.4.4.m4.2.2.1.3.2.2" xref="S7.Thmtheorem21.p1.4.4.m4.2.2.1.3.2.2.cmml">1</mn><mn id="S7.Thmtheorem21.p1.4.4.m4.2.2.1.3.2.3" xref="S7.Thmtheorem21.p1.4.4.m4.2.2.1.3.2.3.cmml">2</mn></mfrac><mo id="S7.Thmtheorem21.p1.4.4.m4.2.2.1.3.1" xref="S7.Thmtheorem21.p1.4.4.m4.2.2.1.3.1.cmml">⁢</mo><msub id="S7.Thmtheorem21.p1.4.4.m4.2.2.1.3.3" xref="S7.Thmtheorem21.p1.4.4.m4.2.2.1.3.3.cmml"><mi id="S7.Thmtheorem21.p1.4.4.m4.2.2.1.3.3.2" mathvariant="normal" xref="S7.Thmtheorem21.p1.4.4.m4.2.2.1.3.3.2.cmml">Δ</mi><mi id="S7.Thmtheorem21.p1.4.4.m4.2.2.1.3.3.3" xref="S7.Thmtheorem21.p1.4.4.m4.2.2.1.3.3.3.cmml">h</mi></msub></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S7.Thmtheorem21.p1.4.4.m4.2b"><apply id="S7.Thmtheorem21.p1.4.4.m4.2.2.cmml" xref="S7.Thmtheorem21.p1.4.4.m4.2.2"><leq id="S7.Thmtheorem21.p1.4.4.m4.2.2.2.cmml" xref="S7.Thmtheorem21.p1.4.4.m4.2.2.2"></leq><apply id="S7.Thmtheorem21.p1.4.4.m4.2.2.3.cmml" xref="S7.Thmtheorem21.p1.4.4.m4.2.2.3"><times id="S7.Thmtheorem21.p1.4.4.m4.2.2.3.1.cmml" xref="S7.Thmtheorem21.p1.4.4.m4.2.2.3.1"></times><ci id="S7.Thmtheorem21.p1.4.4.m4.2.2.3.2a.cmml" xref="S7.Thmtheorem21.p1.4.4.m4.2.2.3.2"><mtext class="ltx_mathvariant_italic" id="S7.Thmtheorem21.p1.4.4.m4.2.2.3.2.cmml" xref="S7.Thmtheorem21.p1.4.4.m4.2.2.3.2">g</mtext></ci><ci id="S7.Thmtheorem21.p1.4.4.m4.1.1.cmml" xref="S7.Thmtheorem21.p1.4.4.m4.1.1">𝑎</ci></apply><apply id="S7.Thmtheorem21.p1.4.4.m4.2.2.1.cmml" xref="S7.Thmtheorem21.p1.4.4.m4.2.2.1"><plus id="S7.Thmtheorem21.p1.4.4.m4.2.2.1.2.cmml" xref="S7.Thmtheorem21.p1.4.4.m4.2.2.1.2"></plus><apply id="S7.Thmtheorem21.p1.4.4.m4.2.2.1.1.cmml" xref="S7.Thmtheorem21.p1.4.4.m4.2.2.1.1"><csymbol cd="ambiguous" id="S7.Thmtheorem21.p1.4.4.m4.2.2.1.1.2.cmml" xref="S7.Thmtheorem21.p1.4.4.m4.2.2.1.1">superscript</csymbol><apply id="S7.Thmtheorem21.p1.4.4.m4.2.2.1.1.1.1.1.cmml" xref="S7.Thmtheorem21.p1.4.4.m4.2.2.1.1.1.1"><plus id="S7.Thmtheorem21.p1.4.4.m4.2.2.1.1.1.1.1.1.cmml" xref="S7.Thmtheorem21.p1.4.4.m4.2.2.1.1.1.1.1.1"></plus><cn id="S7.Thmtheorem21.p1.4.4.m4.2.2.1.1.1.1.1.2.cmml" type="integer" xref="S7.Thmtheorem21.p1.4.4.m4.2.2.1.1.1.1.1.2">1</cn><apply id="S7.Thmtheorem21.p1.4.4.m4.2.2.1.1.1.1.1.3.cmml" xref="S7.Thmtheorem21.p1.4.4.m4.2.2.1.1.1.1.1.3"><divide id="S7.Thmtheorem21.p1.4.4.m4.2.2.1.1.1.1.1.3.1.cmml" xref="S7.Thmtheorem21.p1.4.4.m4.2.2.1.1.1.1.1.3"></divide><ci id="S7.Thmtheorem21.p1.4.4.m4.2.2.1.1.1.1.1.3.2.cmml" xref="S7.Thmtheorem21.p1.4.4.m4.2.2.1.1.1.1.1.3.2">𝜂</ci><cn id="S7.Thmtheorem21.p1.4.4.m4.2.2.1.1.1.1.1.3.3.cmml" type="integer" xref="S7.Thmtheorem21.p1.4.4.m4.2.2.1.1.1.1.1.3.3">2</cn></apply></apply><apply id="S7.Thmtheorem21.p1.4.4.m4.2.2.1.1.3.cmml" xref="S7.Thmtheorem21.p1.4.4.m4.2.2.1.1.3"><minus id="S7.Thmtheorem21.p1.4.4.m4.2.2.1.1.3.1.cmml" xref="S7.Thmtheorem21.p1.4.4.m4.2.2.1.1.3.1"></minus><ci id="S7.Thmtheorem21.p1.4.4.m4.2.2.1.1.3.2.cmml" xref="S7.Thmtheorem21.p1.4.4.m4.2.2.1.1.3.2">ℎ</ci><cn id="S7.Thmtheorem21.p1.4.4.m4.2.2.1.1.3.3.cmml" type="integer" xref="S7.Thmtheorem21.p1.4.4.m4.2.2.1.1.3.3">1</cn></apply></apply><apply id="S7.Thmtheorem21.p1.4.4.m4.2.2.1.3.cmml" xref="S7.Thmtheorem21.p1.4.4.m4.2.2.1.3"><times id="S7.Thmtheorem21.p1.4.4.m4.2.2.1.3.1.cmml" xref="S7.Thmtheorem21.p1.4.4.m4.2.2.1.3.1"></times><apply id="S7.Thmtheorem21.p1.4.4.m4.2.2.1.3.2.cmml" xref="S7.Thmtheorem21.p1.4.4.m4.2.2.1.3.2"><divide id="S7.Thmtheorem21.p1.4.4.m4.2.2.1.3.2.1.cmml" xref="S7.Thmtheorem21.p1.4.4.m4.2.2.1.3.2"></divide><cn id="S7.Thmtheorem21.p1.4.4.m4.2.2.1.3.2.2.cmml" type="integer" xref="S7.Thmtheorem21.p1.4.4.m4.2.2.1.3.2.2">1</cn><cn id="S7.Thmtheorem21.p1.4.4.m4.2.2.1.3.2.3.cmml" type="integer" xref="S7.Thmtheorem21.p1.4.4.m4.2.2.1.3.2.3">2</cn></apply><apply id="S7.Thmtheorem21.p1.4.4.m4.2.2.1.3.3.cmml" xref="S7.Thmtheorem21.p1.4.4.m4.2.2.1.3.3"><csymbol cd="ambiguous" id="S7.Thmtheorem21.p1.4.4.m4.2.2.1.3.3.1.cmml" xref="S7.Thmtheorem21.p1.4.4.m4.2.2.1.3.3">subscript</csymbol><ci id="S7.Thmtheorem21.p1.4.4.m4.2.2.1.3.3.2.cmml" xref="S7.Thmtheorem21.p1.4.4.m4.2.2.1.3.3.2">Δ</ci><ci id="S7.Thmtheorem21.p1.4.4.m4.2.2.1.3.3.3.cmml" xref="S7.Thmtheorem21.p1.4.4.m4.2.2.1.3.3.3">ℎ</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.Thmtheorem21.p1.4.4.m4.2c">\textsl{g}(a)\leq(1+\frac{\eta}{2})^{h-1}+\frac{1}{2}\Delta_{h}</annotation><annotation encoding="application/x-llamapun" id="S7.Thmtheorem21.p1.4.4.m4.2d">g ( italic_a ) ≤ ( 1 + divide start_ARG italic_η end_ARG start_ARG 2 end_ARG ) start_POSTSUPERSCRIPT italic_h - 1 end_POSTSUPERSCRIPT + divide start_ARG 1 end_ARG start_ARG 2 end_ARG roman_Δ start_POSTSUBSCRIPT italic_h end_POSTSUBSCRIPT</annotation></semantics></math>. A vertex <math alttext="b\in B_{t}" class="ltx_Math" display="inline" id="S7.Thmtheorem21.p1.5.5.m5.1"><semantics id="S7.Thmtheorem21.p1.5.5.m5.1a"><mrow id="S7.Thmtheorem21.p1.5.5.m5.1.1" xref="S7.Thmtheorem21.p1.5.5.m5.1.1.cmml"><mi id="S7.Thmtheorem21.p1.5.5.m5.1.1.2" xref="S7.Thmtheorem21.p1.5.5.m5.1.1.2.cmml">b</mi><mo id="S7.Thmtheorem21.p1.5.5.m5.1.1.1" xref="S7.Thmtheorem21.p1.5.5.m5.1.1.1.cmml">∈</mo><msub id="S7.Thmtheorem21.p1.5.5.m5.1.1.3" xref="S7.Thmtheorem21.p1.5.5.m5.1.1.3.cmml"><mi id="S7.Thmtheorem21.p1.5.5.m5.1.1.3.2" xref="S7.Thmtheorem21.p1.5.5.m5.1.1.3.2.cmml">B</mi><mi id="S7.Thmtheorem21.p1.5.5.m5.1.1.3.3" xref="S7.Thmtheorem21.p1.5.5.m5.1.1.3.3.cmml">t</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S7.Thmtheorem21.p1.5.5.m5.1b"><apply id="S7.Thmtheorem21.p1.5.5.m5.1.1.cmml" xref="S7.Thmtheorem21.p1.5.5.m5.1.1"><in id="S7.Thmtheorem21.p1.5.5.m5.1.1.1.cmml" xref="S7.Thmtheorem21.p1.5.5.m5.1.1.1"></in><ci id="S7.Thmtheorem21.p1.5.5.m5.1.1.2.cmml" xref="S7.Thmtheorem21.p1.5.5.m5.1.1.2">𝑏</ci><apply id="S7.Thmtheorem21.p1.5.5.m5.1.1.3.cmml" xref="S7.Thmtheorem21.p1.5.5.m5.1.1.3"><csymbol cd="ambiguous" id="S7.Thmtheorem21.p1.5.5.m5.1.1.3.1.cmml" xref="S7.Thmtheorem21.p1.5.5.m5.1.1.3">subscript</csymbol><ci id="S7.Thmtheorem21.p1.5.5.m5.1.1.3.2.cmml" xref="S7.Thmtheorem21.p1.5.5.m5.1.1.3.2">𝐵</ci><ci id="S7.Thmtheorem21.p1.5.5.m5.1.1.3.3.cmml" xref="S7.Thmtheorem21.p1.5.5.m5.1.1.3.3">𝑡</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.Thmtheorem21.p1.5.5.m5.1c">b\in B_{t}</annotation><annotation encoding="application/x-llamapun" id="S7.Thmtheorem21.p1.5.5.m5.1d">italic_b ∈ italic_B start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT</annotation></semantics></math> is <em class="ltx_emph ltx_font_upright" id="S7.Thmtheorem21.p1.10.10.2">satisfied</em> whenever <math alttext="\textsl{g}(b)=(1+\frac{\eta}{2})^{h}" class="ltx_Math" display="inline" id="S7.Thmtheorem21.p1.6.6.m6.2"><semantics id="S7.Thmtheorem21.p1.6.6.m6.2a"><mrow id="S7.Thmtheorem21.p1.6.6.m6.2.2" xref="S7.Thmtheorem21.p1.6.6.m6.2.2.cmml"><mrow id="S7.Thmtheorem21.p1.6.6.m6.2.2.3" xref="S7.Thmtheorem21.p1.6.6.m6.2.2.3.cmml"><mtext class="ltx_mathvariant_italic" id="S7.Thmtheorem21.p1.6.6.m6.2.2.3.2" xref="S7.Thmtheorem21.p1.6.6.m6.2.2.3.2a.cmml">g</mtext><mo id="S7.Thmtheorem21.p1.6.6.m6.2.2.3.1" xref="S7.Thmtheorem21.p1.6.6.m6.2.2.3.1.cmml">⁢</mo><mrow id="S7.Thmtheorem21.p1.6.6.m6.2.2.3.3.2" xref="S7.Thmtheorem21.p1.6.6.m6.2.2.3.cmml"><mo id="S7.Thmtheorem21.p1.6.6.m6.2.2.3.3.2.1" stretchy="false" xref="S7.Thmtheorem21.p1.6.6.m6.2.2.3.cmml">(</mo><mi id="S7.Thmtheorem21.p1.6.6.m6.1.1" xref="S7.Thmtheorem21.p1.6.6.m6.1.1.cmml">b</mi><mo id="S7.Thmtheorem21.p1.6.6.m6.2.2.3.3.2.2" stretchy="false" xref="S7.Thmtheorem21.p1.6.6.m6.2.2.3.cmml">)</mo></mrow></mrow><mo id="S7.Thmtheorem21.p1.6.6.m6.2.2.2" xref="S7.Thmtheorem21.p1.6.6.m6.2.2.2.cmml">=</mo><msup id="S7.Thmtheorem21.p1.6.6.m6.2.2.1" xref="S7.Thmtheorem21.p1.6.6.m6.2.2.1.cmml"><mrow id="S7.Thmtheorem21.p1.6.6.m6.2.2.1.1.1" xref="S7.Thmtheorem21.p1.6.6.m6.2.2.1.1.1.1.cmml"><mo id="S7.Thmtheorem21.p1.6.6.m6.2.2.1.1.1.2" stretchy="false" xref="S7.Thmtheorem21.p1.6.6.m6.2.2.1.1.1.1.cmml">(</mo><mrow id="S7.Thmtheorem21.p1.6.6.m6.2.2.1.1.1.1" xref="S7.Thmtheorem21.p1.6.6.m6.2.2.1.1.1.1.cmml"><mn id="S7.Thmtheorem21.p1.6.6.m6.2.2.1.1.1.1.2" xref="S7.Thmtheorem21.p1.6.6.m6.2.2.1.1.1.1.2.cmml">1</mn><mo id="S7.Thmtheorem21.p1.6.6.m6.2.2.1.1.1.1.1" xref="S7.Thmtheorem21.p1.6.6.m6.2.2.1.1.1.1.1.cmml">+</mo><mfrac id="S7.Thmtheorem21.p1.6.6.m6.2.2.1.1.1.1.3" xref="S7.Thmtheorem21.p1.6.6.m6.2.2.1.1.1.1.3.cmml"><mi id="S7.Thmtheorem21.p1.6.6.m6.2.2.1.1.1.1.3.2" xref="S7.Thmtheorem21.p1.6.6.m6.2.2.1.1.1.1.3.2.cmml">η</mi><mn id="S7.Thmtheorem21.p1.6.6.m6.2.2.1.1.1.1.3.3" xref="S7.Thmtheorem21.p1.6.6.m6.2.2.1.1.1.1.3.3.cmml">2</mn></mfrac></mrow><mo id="S7.Thmtheorem21.p1.6.6.m6.2.2.1.1.1.3" stretchy="false" xref="S7.Thmtheorem21.p1.6.6.m6.2.2.1.1.1.1.cmml">)</mo></mrow><mi id="S7.Thmtheorem21.p1.6.6.m6.2.2.1.3" xref="S7.Thmtheorem21.p1.6.6.m6.2.2.1.3.cmml">h</mi></msup></mrow><annotation-xml encoding="MathML-Content" id="S7.Thmtheorem21.p1.6.6.m6.2b"><apply id="S7.Thmtheorem21.p1.6.6.m6.2.2.cmml" xref="S7.Thmtheorem21.p1.6.6.m6.2.2"><eq id="S7.Thmtheorem21.p1.6.6.m6.2.2.2.cmml" xref="S7.Thmtheorem21.p1.6.6.m6.2.2.2"></eq><apply id="S7.Thmtheorem21.p1.6.6.m6.2.2.3.cmml" xref="S7.Thmtheorem21.p1.6.6.m6.2.2.3"><times id="S7.Thmtheorem21.p1.6.6.m6.2.2.3.1.cmml" xref="S7.Thmtheorem21.p1.6.6.m6.2.2.3.1"></times><ci id="S7.Thmtheorem21.p1.6.6.m6.2.2.3.2a.cmml" xref="S7.Thmtheorem21.p1.6.6.m6.2.2.3.2"><mtext class="ltx_mathvariant_italic" id="S7.Thmtheorem21.p1.6.6.m6.2.2.3.2.cmml" xref="S7.Thmtheorem21.p1.6.6.m6.2.2.3.2">g</mtext></ci><ci id="S7.Thmtheorem21.p1.6.6.m6.1.1.cmml" xref="S7.Thmtheorem21.p1.6.6.m6.1.1">𝑏</ci></apply><apply id="S7.Thmtheorem21.p1.6.6.m6.2.2.1.cmml" xref="S7.Thmtheorem21.p1.6.6.m6.2.2.1"><csymbol cd="ambiguous" id="S7.Thmtheorem21.p1.6.6.m6.2.2.1.2.cmml" xref="S7.Thmtheorem21.p1.6.6.m6.2.2.1">superscript</csymbol><apply id="S7.Thmtheorem21.p1.6.6.m6.2.2.1.1.1.1.cmml" xref="S7.Thmtheorem21.p1.6.6.m6.2.2.1.1.1"><plus id="S7.Thmtheorem21.p1.6.6.m6.2.2.1.1.1.1.1.cmml" xref="S7.Thmtheorem21.p1.6.6.m6.2.2.1.1.1.1.1"></plus><cn id="S7.Thmtheorem21.p1.6.6.m6.2.2.1.1.1.1.2.cmml" type="integer" xref="S7.Thmtheorem21.p1.6.6.m6.2.2.1.1.1.1.2">1</cn><apply id="S7.Thmtheorem21.p1.6.6.m6.2.2.1.1.1.1.3.cmml" xref="S7.Thmtheorem21.p1.6.6.m6.2.2.1.1.1.1.3"><divide id="S7.Thmtheorem21.p1.6.6.m6.2.2.1.1.1.1.3.1.cmml" xref="S7.Thmtheorem21.p1.6.6.m6.2.2.1.1.1.1.3"></divide><ci id="S7.Thmtheorem21.p1.6.6.m6.2.2.1.1.1.1.3.2.cmml" xref="S7.Thmtheorem21.p1.6.6.m6.2.2.1.1.1.1.3.2">𝜂</ci><cn id="S7.Thmtheorem21.p1.6.6.m6.2.2.1.1.1.1.3.3.cmml" type="integer" xref="S7.Thmtheorem21.p1.6.6.m6.2.2.1.1.1.1.3.3">2</cn></apply></apply><ci id="S7.Thmtheorem21.p1.6.6.m6.2.2.1.3.cmml" xref="S7.Thmtheorem21.p1.6.6.m6.2.2.1.3">ℎ</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.Thmtheorem21.p1.6.6.m6.2c">\textsl{g}(b)=(1+\frac{\eta}{2})^{h}</annotation><annotation encoding="application/x-llamapun" id="S7.Thmtheorem21.p1.6.6.m6.2d">g ( italic_b ) = ( 1 + divide start_ARG italic_η end_ARG start_ARG 2 end_ARG ) start_POSTSUPERSCRIPT italic_h end_POSTSUPERSCRIPT</annotation></semantics></math>. We say that an edge <math alttext="\overline{uv}\in E_{t}" class="ltx_Math" display="inline" id="S7.Thmtheorem21.p1.7.7.m7.1"><semantics id="S7.Thmtheorem21.p1.7.7.m7.1a"><mrow id="S7.Thmtheorem21.p1.7.7.m7.1.1" xref="S7.Thmtheorem21.p1.7.7.m7.1.1.cmml"><mover accent="true" id="S7.Thmtheorem21.p1.7.7.m7.1.1.2" xref="S7.Thmtheorem21.p1.7.7.m7.1.1.2.cmml"><mrow id="S7.Thmtheorem21.p1.7.7.m7.1.1.2.2" xref="S7.Thmtheorem21.p1.7.7.m7.1.1.2.2.cmml"><mi id="S7.Thmtheorem21.p1.7.7.m7.1.1.2.2.2" xref="S7.Thmtheorem21.p1.7.7.m7.1.1.2.2.2.cmml">u</mi><mo id="S7.Thmtheorem21.p1.7.7.m7.1.1.2.2.1" xref="S7.Thmtheorem21.p1.7.7.m7.1.1.2.2.1.cmml">⁢</mo><mi id="S7.Thmtheorem21.p1.7.7.m7.1.1.2.2.3" xref="S7.Thmtheorem21.p1.7.7.m7.1.1.2.2.3.cmml">v</mi></mrow><mo id="S7.Thmtheorem21.p1.7.7.m7.1.1.2.1" xref="S7.Thmtheorem21.p1.7.7.m7.1.1.2.1.cmml">¯</mo></mover><mo id="S7.Thmtheorem21.p1.7.7.m7.1.1.1" xref="S7.Thmtheorem21.p1.7.7.m7.1.1.1.cmml">∈</mo><msub id="S7.Thmtheorem21.p1.7.7.m7.1.1.3" xref="S7.Thmtheorem21.p1.7.7.m7.1.1.3.cmml"><mi id="S7.Thmtheorem21.p1.7.7.m7.1.1.3.2" xref="S7.Thmtheorem21.p1.7.7.m7.1.1.3.2.cmml">E</mi><mi id="S7.Thmtheorem21.p1.7.7.m7.1.1.3.3" xref="S7.Thmtheorem21.p1.7.7.m7.1.1.3.3.cmml">t</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S7.Thmtheorem21.p1.7.7.m7.1b"><apply id="S7.Thmtheorem21.p1.7.7.m7.1.1.cmml" xref="S7.Thmtheorem21.p1.7.7.m7.1.1"><in id="S7.Thmtheorem21.p1.7.7.m7.1.1.1.cmml" xref="S7.Thmtheorem21.p1.7.7.m7.1.1.1"></in><apply id="S7.Thmtheorem21.p1.7.7.m7.1.1.2.cmml" xref="S7.Thmtheorem21.p1.7.7.m7.1.1.2"><ci id="S7.Thmtheorem21.p1.7.7.m7.1.1.2.1.cmml" xref="S7.Thmtheorem21.p1.7.7.m7.1.1.2.1">¯</ci><apply id="S7.Thmtheorem21.p1.7.7.m7.1.1.2.2.cmml" xref="S7.Thmtheorem21.p1.7.7.m7.1.1.2.2"><times id="S7.Thmtheorem21.p1.7.7.m7.1.1.2.2.1.cmml" xref="S7.Thmtheorem21.p1.7.7.m7.1.1.2.2.1"></times><ci id="S7.Thmtheorem21.p1.7.7.m7.1.1.2.2.2.cmml" xref="S7.Thmtheorem21.p1.7.7.m7.1.1.2.2.2">𝑢</ci><ci id="S7.Thmtheorem21.p1.7.7.m7.1.1.2.2.3.cmml" xref="S7.Thmtheorem21.p1.7.7.m7.1.1.2.2.3">𝑣</ci></apply></apply><apply id="S7.Thmtheorem21.p1.7.7.m7.1.1.3.cmml" xref="S7.Thmtheorem21.p1.7.7.m7.1.1.3"><csymbol cd="ambiguous" id="S7.Thmtheorem21.p1.7.7.m7.1.1.3.1.cmml" xref="S7.Thmtheorem21.p1.7.7.m7.1.1.3">subscript</csymbol><ci id="S7.Thmtheorem21.p1.7.7.m7.1.1.3.2.cmml" xref="S7.Thmtheorem21.p1.7.7.m7.1.1.3.2">𝐸</ci><ci id="S7.Thmtheorem21.p1.7.7.m7.1.1.3.3.cmml" xref="S7.Thmtheorem21.p1.7.7.m7.1.1.3.3">𝑡</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.Thmtheorem21.p1.7.7.m7.1c">\overline{uv}\in E_{t}</annotation><annotation encoding="application/x-llamapun" id="S7.Thmtheorem21.p1.7.7.m7.1d">over¯ start_ARG italic_u italic_v end_ARG ∈ italic_E start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT</annotation></semantics></math> is <em class="ltx_emph ltx_font_upright" id="S7.Thmtheorem21.p1.10.10.3">satisfied</em> whenever <math alttext="\textsl{g}(u\!\to\!v)=0" class="ltx_Math" display="inline" id="S7.Thmtheorem21.p1.8.8.m8.1"><semantics id="S7.Thmtheorem21.p1.8.8.m8.1a"><mrow id="S7.Thmtheorem21.p1.8.8.m8.1.1" xref="S7.Thmtheorem21.p1.8.8.m8.1.1.cmml"><mrow id="S7.Thmtheorem21.p1.8.8.m8.1.1.1" xref="S7.Thmtheorem21.p1.8.8.m8.1.1.1.cmml"><mtext class="ltx_mathvariant_italic" id="S7.Thmtheorem21.p1.8.8.m8.1.1.1.3" xref="S7.Thmtheorem21.p1.8.8.m8.1.1.1.3a.cmml">g</mtext><mo id="S7.Thmtheorem21.p1.8.8.m8.1.1.1.2" xref="S7.Thmtheorem21.p1.8.8.m8.1.1.1.2.cmml">⁢</mo><mrow id="S7.Thmtheorem21.p1.8.8.m8.1.1.1.1.1" xref="S7.Thmtheorem21.p1.8.8.m8.1.1.1.1.1.1.cmml"><mo id="S7.Thmtheorem21.p1.8.8.m8.1.1.1.1.1.2" stretchy="false" xref="S7.Thmtheorem21.p1.8.8.m8.1.1.1.1.1.1.cmml">(</mo><mrow id="S7.Thmtheorem21.p1.8.8.m8.1.1.1.1.1.1" xref="S7.Thmtheorem21.p1.8.8.m8.1.1.1.1.1.1.cmml"><mi id="S7.Thmtheorem21.p1.8.8.m8.1.1.1.1.1.1.2" xref="S7.Thmtheorem21.p1.8.8.m8.1.1.1.1.1.1.2.cmml">u</mi><mo id="S7.Thmtheorem21.p1.8.8.m8.1.1.1.1.1.1.1" lspace="0.108em" rspace="0.108em" stretchy="false" xref="S7.Thmtheorem21.p1.8.8.m8.1.1.1.1.1.1.1.cmml">→</mo><mi id="S7.Thmtheorem21.p1.8.8.m8.1.1.1.1.1.1.3" xref="S7.Thmtheorem21.p1.8.8.m8.1.1.1.1.1.1.3.cmml">v</mi></mrow><mo id="S7.Thmtheorem21.p1.8.8.m8.1.1.1.1.1.3" stretchy="false" xref="S7.Thmtheorem21.p1.8.8.m8.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S7.Thmtheorem21.p1.8.8.m8.1.1.2" xref="S7.Thmtheorem21.p1.8.8.m8.1.1.2.cmml">=</mo><mn id="S7.Thmtheorem21.p1.8.8.m8.1.1.3" xref="S7.Thmtheorem21.p1.8.8.m8.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S7.Thmtheorem21.p1.8.8.m8.1b"><apply id="S7.Thmtheorem21.p1.8.8.m8.1.1.cmml" xref="S7.Thmtheorem21.p1.8.8.m8.1.1"><eq id="S7.Thmtheorem21.p1.8.8.m8.1.1.2.cmml" xref="S7.Thmtheorem21.p1.8.8.m8.1.1.2"></eq><apply id="S7.Thmtheorem21.p1.8.8.m8.1.1.1.cmml" xref="S7.Thmtheorem21.p1.8.8.m8.1.1.1"><times id="S7.Thmtheorem21.p1.8.8.m8.1.1.1.2.cmml" xref="S7.Thmtheorem21.p1.8.8.m8.1.1.1.2"></times><ci id="S7.Thmtheorem21.p1.8.8.m8.1.1.1.3a.cmml" xref="S7.Thmtheorem21.p1.8.8.m8.1.1.1.3"><mtext class="ltx_mathvariant_italic" id="S7.Thmtheorem21.p1.8.8.m8.1.1.1.3.cmml" xref="S7.Thmtheorem21.p1.8.8.m8.1.1.1.3">g</mtext></ci><apply id="S7.Thmtheorem21.p1.8.8.m8.1.1.1.1.1.1.cmml" xref="S7.Thmtheorem21.p1.8.8.m8.1.1.1.1.1"><ci id="S7.Thmtheorem21.p1.8.8.m8.1.1.1.1.1.1.1.cmml" xref="S7.Thmtheorem21.p1.8.8.m8.1.1.1.1.1.1.1">→</ci><ci id="S7.Thmtheorem21.p1.8.8.m8.1.1.1.1.1.1.2.cmml" xref="S7.Thmtheorem21.p1.8.8.m8.1.1.1.1.1.1.2">𝑢</ci><ci id="S7.Thmtheorem21.p1.8.8.m8.1.1.1.1.1.1.3.cmml" xref="S7.Thmtheorem21.p1.8.8.m8.1.1.1.1.1.1.3">𝑣</ci></apply></apply><cn id="S7.Thmtheorem21.p1.8.8.m8.1.1.3.cmml" type="integer" xref="S7.Thmtheorem21.p1.8.8.m8.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.Thmtheorem21.p1.8.8.m8.1c">\textsl{g}(u\!\to\!v)=0</annotation><annotation encoding="application/x-llamapun" id="S7.Thmtheorem21.p1.8.8.m8.1d">g ( italic_u → italic_v ) = 0</annotation></semantics></math> or whenever at least <math alttext="u" class="ltx_Math" display="inline" id="S7.Thmtheorem21.p1.9.9.m9.1"><semantics id="S7.Thmtheorem21.p1.9.9.m9.1a"><mi id="S7.Thmtheorem21.p1.9.9.m9.1.1" xref="S7.Thmtheorem21.p1.9.9.m9.1.1.cmml">u</mi><annotation-xml encoding="MathML-Content" id="S7.Thmtheorem21.p1.9.9.m9.1b"><ci id="S7.Thmtheorem21.p1.9.9.m9.1.1.cmml" xref="S7.Thmtheorem21.p1.9.9.m9.1.1">𝑢</ci></annotation-xml><annotation encoding="application/x-tex" id="S7.Thmtheorem21.p1.9.9.m9.1c">u</annotation><annotation encoding="application/x-llamapun" id="S7.Thmtheorem21.p1.9.9.m9.1d">italic_u</annotation></semantics></math> or <math alttext="v" class="ltx_Math" display="inline" id="S7.Thmtheorem21.p1.10.10.m10.1"><semantics id="S7.Thmtheorem21.p1.10.10.m10.1a"><mi id="S7.Thmtheorem21.p1.10.10.m10.1.1" xref="S7.Thmtheorem21.p1.10.10.m10.1.1.cmml">v</mi><annotation-xml encoding="MathML-Content" id="S7.Thmtheorem21.p1.10.10.m10.1b"><ci id="S7.Thmtheorem21.p1.10.10.m10.1.1.cmml" xref="S7.Thmtheorem21.p1.10.10.m10.1.1">𝑣</ci></annotation-xml><annotation encoding="application/x-tex" id="S7.Thmtheorem21.p1.10.10.m10.1c">v</annotation><annotation encoding="application/x-llamapun" id="S7.Thmtheorem21.p1.10.10.m10.1d">italic_v</annotation></semantics></math> is satisfied. Observe that a satisfied edge is not violating.</span></p> </div> <div class="ltx_para ltx_noindent" id="S7.Thmtheorem21.p2"> <p class="ltx_p" id="S7.Thmtheorem21.p2.2"><span class="ltx_text ltx_font_italic" id="S7.Thmtheorem21.p2.2.1">Fix any edge </span><math alttext="\overline{ab}\in E_{t}" class="ltx_Math" display="inline" id="S7.Thmtheorem21.p2.1.m1.1"><semantics id="S7.Thmtheorem21.p2.1.m1.1a"><mrow id="S7.Thmtheorem21.p2.1.m1.1.1" xref="S7.Thmtheorem21.p2.1.m1.1.1.cmml"><mover accent="true" id="S7.Thmtheorem21.p2.1.m1.1.1.2" xref="S7.Thmtheorem21.p2.1.m1.1.1.2.cmml"><mrow id="S7.Thmtheorem21.p2.1.m1.1.1.2.2" xref="S7.Thmtheorem21.p2.1.m1.1.1.2.2.cmml"><mi id="S7.Thmtheorem21.p2.1.m1.1.1.2.2.2" xref="S7.Thmtheorem21.p2.1.m1.1.1.2.2.2.cmml">a</mi><mo id="S7.Thmtheorem21.p2.1.m1.1.1.2.2.1" xref="S7.Thmtheorem21.p2.1.m1.1.1.2.2.1.cmml">⁢</mo><mi id="S7.Thmtheorem21.p2.1.m1.1.1.2.2.3" xref="S7.Thmtheorem21.p2.1.m1.1.1.2.2.3.cmml">b</mi></mrow><mo id="S7.Thmtheorem21.p2.1.m1.1.1.2.1" xref="S7.Thmtheorem21.p2.1.m1.1.1.2.1.cmml">¯</mo></mover><mo id="S7.Thmtheorem21.p2.1.m1.1.1.1" xref="S7.Thmtheorem21.p2.1.m1.1.1.1.cmml">∈</mo><msub id="S7.Thmtheorem21.p2.1.m1.1.1.3" xref="S7.Thmtheorem21.p2.1.m1.1.1.3.cmml"><mi id="S7.Thmtheorem21.p2.1.m1.1.1.3.2" xref="S7.Thmtheorem21.p2.1.m1.1.1.3.2.cmml">E</mi><mi id="S7.Thmtheorem21.p2.1.m1.1.1.3.3" xref="S7.Thmtheorem21.p2.1.m1.1.1.3.3.cmml">t</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S7.Thmtheorem21.p2.1.m1.1b"><apply id="S7.Thmtheorem21.p2.1.m1.1.1.cmml" xref="S7.Thmtheorem21.p2.1.m1.1.1"><in id="S7.Thmtheorem21.p2.1.m1.1.1.1.cmml" xref="S7.Thmtheorem21.p2.1.m1.1.1.1"></in><apply id="S7.Thmtheorem21.p2.1.m1.1.1.2.cmml" xref="S7.Thmtheorem21.p2.1.m1.1.1.2"><ci id="S7.Thmtheorem21.p2.1.m1.1.1.2.1.cmml" xref="S7.Thmtheorem21.p2.1.m1.1.1.2.1">¯</ci><apply id="S7.Thmtheorem21.p2.1.m1.1.1.2.2.cmml" xref="S7.Thmtheorem21.p2.1.m1.1.1.2.2"><times id="S7.Thmtheorem21.p2.1.m1.1.1.2.2.1.cmml" xref="S7.Thmtheorem21.p2.1.m1.1.1.2.2.1"></times><ci id="S7.Thmtheorem21.p2.1.m1.1.1.2.2.2.cmml" xref="S7.Thmtheorem21.p2.1.m1.1.1.2.2.2">𝑎</ci><ci id="S7.Thmtheorem21.p2.1.m1.1.1.2.2.3.cmml" xref="S7.Thmtheorem21.p2.1.m1.1.1.2.2.3">𝑏</ci></apply></apply><apply id="S7.Thmtheorem21.p2.1.m1.1.1.3.cmml" xref="S7.Thmtheorem21.p2.1.m1.1.1.3"><csymbol cd="ambiguous" id="S7.Thmtheorem21.p2.1.m1.1.1.3.1.cmml" xref="S7.Thmtheorem21.p2.1.m1.1.1.3">subscript</csymbol><ci id="S7.Thmtheorem21.p2.1.m1.1.1.3.2.cmml" xref="S7.Thmtheorem21.p2.1.m1.1.1.3.2">𝐸</ci><ci id="S7.Thmtheorem21.p2.1.m1.1.1.3.3.cmml" xref="S7.Thmtheorem21.p2.1.m1.1.1.3.3">𝑡</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.Thmtheorem21.p2.1.m1.1c">\overline{ab}\in E_{t}</annotation><annotation encoding="application/x-llamapun" id="S7.Thmtheorem21.p2.1.m1.1d">over¯ start_ARG italic_a italic_b end_ARG ∈ italic_E start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S7.Thmtheorem21.p2.2.2">. At the end of iteration </span><math alttext="t" class="ltx_Math" display="inline" id="S7.Thmtheorem21.p2.2.m2.1"><semantics id="S7.Thmtheorem21.p2.2.m2.1a"><mi id="S7.Thmtheorem21.p2.2.m2.1.1" xref="S7.Thmtheorem21.p2.2.m2.1.1.cmml">t</mi><annotation-xml encoding="MathML-Content" id="S7.Thmtheorem21.p2.2.m2.1b"><ci id="S7.Thmtheorem21.p2.2.m2.1.1.cmml" xref="S7.Thmtheorem21.p2.2.m2.1.1">𝑡</ci></annotation-xml><annotation encoding="application/x-tex" id="S7.Thmtheorem21.p2.2.m2.1c">t</annotation><annotation encoding="application/x-llamapun" id="S7.Thmtheorem21.p2.2.m2.1d">italic_t</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S7.Thmtheorem21.p2.2.3">:</span></p> <ol class="ltx_enumerate" id="S7.I11"> <li class="ltx_item" id="S7.I11.i1" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">1.</span> <div class="ltx_para" id="S7.I11.i1.p1"> <p class="ltx_p" id="S7.I11.i1.p1.1"><span class="ltx_text ltx_font_italic" id="S7.I11.i1.p1.1.1">If </span><math alttext="\textsl{g}(a\!\to\!b)=0" class="ltx_Math" display="inline" id="S7.I11.i1.p1.1.m1.1"><semantics id="S7.I11.i1.p1.1.m1.1a"><mrow id="S7.I11.i1.p1.1.m1.1.1" xref="S7.I11.i1.p1.1.m1.1.1.cmml"><mrow id="S7.I11.i1.p1.1.m1.1.1.1" xref="S7.I11.i1.p1.1.m1.1.1.1.cmml"><mtext class="ltx_mathvariant_italic" id="S7.I11.i1.p1.1.m1.1.1.1.3" xref="S7.I11.i1.p1.1.m1.1.1.1.3a.cmml">g</mtext><mo id="S7.I11.i1.p1.1.m1.1.1.1.2" xref="S7.I11.i1.p1.1.m1.1.1.1.2.cmml">⁢</mo><mrow id="S7.I11.i1.p1.1.m1.1.1.1.1.1" xref="S7.I11.i1.p1.1.m1.1.1.1.1.1.1.cmml"><mo id="S7.I11.i1.p1.1.m1.1.1.1.1.1.2" stretchy="false" xref="S7.I11.i1.p1.1.m1.1.1.1.1.1.1.cmml">(</mo><mrow id="S7.I11.i1.p1.1.m1.1.1.1.1.1.1" xref="S7.I11.i1.p1.1.m1.1.1.1.1.1.1.cmml"><mi id="S7.I11.i1.p1.1.m1.1.1.1.1.1.1.2" xref="S7.I11.i1.p1.1.m1.1.1.1.1.1.1.2.cmml">a</mi><mo id="S7.I11.i1.p1.1.m1.1.1.1.1.1.1.1" lspace="0.108em" rspace="0.108em" stretchy="false" xref="S7.I11.i1.p1.1.m1.1.1.1.1.1.1.1.cmml">→</mo><mi id="S7.I11.i1.p1.1.m1.1.1.1.1.1.1.3" xref="S7.I11.i1.p1.1.m1.1.1.1.1.1.1.3.cmml">b</mi></mrow><mo id="S7.I11.i1.p1.1.m1.1.1.1.1.1.3" stretchy="false" xref="S7.I11.i1.p1.1.m1.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S7.I11.i1.p1.1.m1.1.1.2" xref="S7.I11.i1.p1.1.m1.1.1.2.cmml">=</mo><mn id="S7.I11.i1.p1.1.m1.1.1.3" xref="S7.I11.i1.p1.1.m1.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S7.I11.i1.p1.1.m1.1b"><apply id="S7.I11.i1.p1.1.m1.1.1.cmml" xref="S7.I11.i1.p1.1.m1.1.1"><eq id="S7.I11.i1.p1.1.m1.1.1.2.cmml" xref="S7.I11.i1.p1.1.m1.1.1.2"></eq><apply id="S7.I11.i1.p1.1.m1.1.1.1.cmml" xref="S7.I11.i1.p1.1.m1.1.1.1"><times id="S7.I11.i1.p1.1.m1.1.1.1.2.cmml" xref="S7.I11.i1.p1.1.m1.1.1.1.2"></times><ci id="S7.I11.i1.p1.1.m1.1.1.1.3a.cmml" xref="S7.I11.i1.p1.1.m1.1.1.1.3"><mtext class="ltx_mathvariant_italic" id="S7.I11.i1.p1.1.m1.1.1.1.3.cmml" xref="S7.I11.i1.p1.1.m1.1.1.1.3">g</mtext></ci><apply id="S7.I11.i1.p1.1.m1.1.1.1.1.1.1.cmml" xref="S7.I11.i1.p1.1.m1.1.1.1.1.1"><ci id="S7.I11.i1.p1.1.m1.1.1.1.1.1.1.1.cmml" xref="S7.I11.i1.p1.1.m1.1.1.1.1.1.1.1">→</ci><ci id="S7.I11.i1.p1.1.m1.1.1.1.1.1.1.2.cmml" xref="S7.I11.i1.p1.1.m1.1.1.1.1.1.1.2">𝑎</ci><ci id="S7.I11.i1.p1.1.m1.1.1.1.1.1.1.3.cmml" xref="S7.I11.i1.p1.1.m1.1.1.1.1.1.1.3">𝑏</ci></apply></apply><cn id="S7.I11.i1.p1.1.m1.1.1.3.cmml" type="integer" xref="S7.I11.i1.p1.1.m1.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.I11.i1.p1.1.m1.1c">\textsl{g}(a\!\to\!b)=0</annotation><annotation encoding="application/x-llamapun" id="S7.I11.i1.p1.1.m1.1d">g ( italic_a → italic_b ) = 0</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S7.I11.i1.p1.1.2"> then the edge is satisfied.</span></p> </div> </li> <li class="ltx_item" id="S7.I11.i2" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">2.</span> <div class="ltx_para" id="S7.I11.i2.p1"> <p class="ltx_p" id="S7.I11.i2.p1.5"><span class="ltx_text ltx_font_italic" id="S7.I11.i2.p1.5.1">If </span><math alttext="\textsl{g}(a\!\to\!b)>0" class="ltx_Math" display="inline" id="S7.I11.i2.p1.1.m1.1"><semantics id="S7.I11.i2.p1.1.m1.1a"><mrow id="S7.I11.i2.p1.1.m1.1.1" xref="S7.I11.i2.p1.1.m1.1.1.cmml"><mrow id="S7.I11.i2.p1.1.m1.1.1.1" xref="S7.I11.i2.p1.1.m1.1.1.1.cmml"><mtext class="ltx_mathvariant_italic" id="S7.I11.i2.p1.1.m1.1.1.1.3" xref="S7.I11.i2.p1.1.m1.1.1.1.3a.cmml">g</mtext><mo id="S7.I11.i2.p1.1.m1.1.1.1.2" xref="S7.I11.i2.p1.1.m1.1.1.1.2.cmml">⁢</mo><mrow id="S7.I11.i2.p1.1.m1.1.1.1.1.1" xref="S7.I11.i2.p1.1.m1.1.1.1.1.1.1.cmml"><mo id="S7.I11.i2.p1.1.m1.1.1.1.1.1.2" stretchy="false" xref="S7.I11.i2.p1.1.m1.1.1.1.1.1.1.cmml">(</mo><mrow id="S7.I11.i2.p1.1.m1.1.1.1.1.1.1" xref="S7.I11.i2.p1.1.m1.1.1.1.1.1.1.cmml"><mi id="S7.I11.i2.p1.1.m1.1.1.1.1.1.1.2" xref="S7.I11.i2.p1.1.m1.1.1.1.1.1.1.2.cmml">a</mi><mo id="S7.I11.i2.p1.1.m1.1.1.1.1.1.1.1" lspace="0.108em" rspace="0.108em" stretchy="false" xref="S7.I11.i2.p1.1.m1.1.1.1.1.1.1.1.cmml">→</mo><mi id="S7.I11.i2.p1.1.m1.1.1.1.1.1.1.3" xref="S7.I11.i2.p1.1.m1.1.1.1.1.1.1.3.cmml">b</mi></mrow><mo id="S7.I11.i2.p1.1.m1.1.1.1.1.1.3" stretchy="false" xref="S7.I11.i2.p1.1.m1.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S7.I11.i2.p1.1.m1.1.1.2" xref="S7.I11.i2.p1.1.m1.1.1.2.cmml">></mo><mn id="S7.I11.i2.p1.1.m1.1.1.3" xref="S7.I11.i2.p1.1.m1.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S7.I11.i2.p1.1.m1.1b"><apply id="S7.I11.i2.p1.1.m1.1.1.cmml" xref="S7.I11.i2.p1.1.m1.1.1"><gt id="S7.I11.i2.p1.1.m1.1.1.2.cmml" xref="S7.I11.i2.p1.1.m1.1.1.2"></gt><apply id="S7.I11.i2.p1.1.m1.1.1.1.cmml" xref="S7.I11.i2.p1.1.m1.1.1.1"><times id="S7.I11.i2.p1.1.m1.1.1.1.2.cmml" xref="S7.I11.i2.p1.1.m1.1.1.1.2"></times><ci id="S7.I11.i2.p1.1.m1.1.1.1.3a.cmml" xref="S7.I11.i2.p1.1.m1.1.1.1.3"><mtext class="ltx_mathvariant_italic" id="S7.I11.i2.p1.1.m1.1.1.1.3.cmml" xref="S7.I11.i2.p1.1.m1.1.1.1.3">g</mtext></ci><apply id="S7.I11.i2.p1.1.m1.1.1.1.1.1.1.cmml" xref="S7.I11.i2.p1.1.m1.1.1.1.1.1"><ci id="S7.I11.i2.p1.1.m1.1.1.1.1.1.1.1.cmml" xref="S7.I11.i2.p1.1.m1.1.1.1.1.1.1.1">→</ci><ci id="S7.I11.i2.p1.1.m1.1.1.1.1.1.1.2.cmml" xref="S7.I11.i2.p1.1.m1.1.1.1.1.1.1.2">𝑎</ci><ci id="S7.I11.i2.p1.1.m1.1.1.1.1.1.1.3.cmml" xref="S7.I11.i2.p1.1.m1.1.1.1.1.1.1.3">𝑏</ci></apply></apply><cn id="S7.I11.i2.p1.1.m1.1.1.3.cmml" type="integer" xref="S7.I11.i2.p1.1.m1.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.I11.i2.p1.1.m1.1c">\textsl{g}(a\!\to\!b)>0</annotation><annotation encoding="application/x-llamapun" id="S7.I11.i2.p1.1.m1.1d">g ( italic_a → italic_b ) > 0</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S7.I11.i2.p1.5.2"> and </span><math alttext="\textsl{g}(a\!\to\!b)" class="ltx_Math" display="inline" id="S7.I11.i2.p1.2.m2.1"><semantics id="S7.I11.i2.p1.2.m2.1a"><mrow id="S7.I11.i2.p1.2.m2.1.1" xref="S7.I11.i2.p1.2.m2.1.1.cmml"><mtext class="ltx_mathvariant_italic" id="S7.I11.i2.p1.2.m2.1.1.3" xref="S7.I11.i2.p1.2.m2.1.1.3a.cmml">g</mtext><mo id="S7.I11.i2.p1.2.m2.1.1.2" xref="S7.I11.i2.p1.2.m2.1.1.2.cmml">⁢</mo><mrow id="S7.I11.i2.p1.2.m2.1.1.1.1" xref="S7.I11.i2.p1.2.m2.1.1.1.1.1.cmml"><mo id="S7.I11.i2.p1.2.m2.1.1.1.1.2" stretchy="false" xref="S7.I11.i2.p1.2.m2.1.1.1.1.1.cmml">(</mo><mrow id="S7.I11.i2.p1.2.m2.1.1.1.1.1" xref="S7.I11.i2.p1.2.m2.1.1.1.1.1.cmml"><mi id="S7.I11.i2.p1.2.m2.1.1.1.1.1.2" xref="S7.I11.i2.p1.2.m2.1.1.1.1.1.2.cmml">a</mi><mo id="S7.I11.i2.p1.2.m2.1.1.1.1.1.1" lspace="0.108em" rspace="0.108em" stretchy="false" xref="S7.I11.i2.p1.2.m2.1.1.1.1.1.1.cmml">→</mo><mi id="S7.I11.i2.p1.2.m2.1.1.1.1.1.3" xref="S7.I11.i2.p1.2.m2.1.1.1.1.1.3.cmml">b</mi></mrow><mo id="S7.I11.i2.p1.2.m2.1.1.1.1.3" stretchy="false" xref="S7.I11.i2.p1.2.m2.1.1.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S7.I11.i2.p1.2.m2.1b"><apply id="S7.I11.i2.p1.2.m2.1.1.cmml" xref="S7.I11.i2.p1.2.m2.1.1"><times id="S7.I11.i2.p1.2.m2.1.1.2.cmml" xref="S7.I11.i2.p1.2.m2.1.1.2"></times><ci id="S7.I11.i2.p1.2.m2.1.1.3a.cmml" xref="S7.I11.i2.p1.2.m2.1.1.3"><mtext class="ltx_mathvariant_italic" id="S7.I11.i2.p1.2.m2.1.1.3.cmml" xref="S7.I11.i2.p1.2.m2.1.1.3">g</mtext></ci><apply id="S7.I11.i2.p1.2.m2.1.1.1.1.1.cmml" xref="S7.I11.i2.p1.2.m2.1.1.1.1"><ci id="S7.I11.i2.p1.2.m2.1.1.1.1.1.1.cmml" xref="S7.I11.i2.p1.2.m2.1.1.1.1.1.1">→</ci><ci id="S7.I11.i2.p1.2.m2.1.1.1.1.1.2.cmml" xref="S7.I11.i2.p1.2.m2.1.1.1.1.1.2">𝑎</ci><ci id="S7.I11.i2.p1.2.m2.1.1.1.1.1.3.cmml" xref="S7.I11.i2.p1.2.m2.1.1.1.1.1.3">𝑏</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.I11.i2.p1.2.m2.1c">\textsl{g}(a\!\to\!b)</annotation><annotation encoding="application/x-llamapun" id="S7.I11.i2.p1.2.m2.1d">g ( italic_a → italic_b )</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S7.I11.i2.p1.5.3"> is not decreased by </span><math alttext="\frac{\delta_{t}(a)}{|E_{t}(a)|}" class="ltx_Math" display="inline" id="S7.I11.i2.p1.3.m3.3"><semantics id="S7.I11.i2.p1.3.m3.3a"><mfrac id="S7.I11.i2.p1.3.m3.3.3" xref="S7.I11.i2.p1.3.m3.3.3.cmml"><mrow id="S7.I11.i2.p1.3.m3.1.1.1" xref="S7.I11.i2.p1.3.m3.1.1.1.cmml"><msub id="S7.I11.i2.p1.3.m3.1.1.1.3" xref="S7.I11.i2.p1.3.m3.1.1.1.3.cmml"><mi id="S7.I11.i2.p1.3.m3.1.1.1.3.2" xref="S7.I11.i2.p1.3.m3.1.1.1.3.2.cmml">δ</mi><mi id="S7.I11.i2.p1.3.m3.1.1.1.3.3" xref="S7.I11.i2.p1.3.m3.1.1.1.3.3.cmml">t</mi></msub><mo id="S7.I11.i2.p1.3.m3.1.1.1.2" xref="S7.I11.i2.p1.3.m3.1.1.1.2.cmml">⁢</mo><mrow id="S7.I11.i2.p1.3.m3.1.1.1.4.2" xref="S7.I11.i2.p1.3.m3.1.1.1.cmml"><mo id="S7.I11.i2.p1.3.m3.1.1.1.4.2.1" stretchy="false" xref="S7.I11.i2.p1.3.m3.1.1.1.cmml">(</mo><mi id="S7.I11.i2.p1.3.m3.1.1.1.1" xref="S7.I11.i2.p1.3.m3.1.1.1.1.cmml">a</mi><mo id="S7.I11.i2.p1.3.m3.1.1.1.4.2.2" stretchy="false" xref="S7.I11.i2.p1.3.m3.1.1.1.cmml">)</mo></mrow></mrow><mrow id="S7.I11.i2.p1.3.m3.3.3.3.2" xref="S7.I11.i2.p1.3.m3.3.3.3.3.cmml"><mo id="S7.I11.i2.p1.3.m3.3.3.3.2.2" stretchy="false" xref="S7.I11.i2.p1.3.m3.3.3.3.3.1.cmml">|</mo><mrow id="S7.I11.i2.p1.3.m3.3.3.3.2.1" xref="S7.I11.i2.p1.3.m3.3.3.3.2.1.cmml"><msub id="S7.I11.i2.p1.3.m3.3.3.3.2.1.2" xref="S7.I11.i2.p1.3.m3.3.3.3.2.1.2.cmml"><mi id="S7.I11.i2.p1.3.m3.3.3.3.2.1.2.2" xref="S7.I11.i2.p1.3.m3.3.3.3.2.1.2.2.cmml">E</mi><mi id="S7.I11.i2.p1.3.m3.3.3.3.2.1.2.3" xref="S7.I11.i2.p1.3.m3.3.3.3.2.1.2.3.cmml">t</mi></msub><mo id="S7.I11.i2.p1.3.m3.3.3.3.2.1.1" xref="S7.I11.i2.p1.3.m3.3.3.3.2.1.1.cmml">⁢</mo><mrow id="S7.I11.i2.p1.3.m3.3.3.3.2.1.3.2" xref="S7.I11.i2.p1.3.m3.3.3.3.2.1.cmml"><mo id="S7.I11.i2.p1.3.m3.3.3.3.2.1.3.2.1" stretchy="false" xref="S7.I11.i2.p1.3.m3.3.3.3.2.1.cmml">(</mo><mi id="S7.I11.i2.p1.3.m3.2.2.2.1" xref="S7.I11.i2.p1.3.m3.2.2.2.1.cmml">a</mi><mo id="S7.I11.i2.p1.3.m3.3.3.3.2.1.3.2.2" stretchy="false" xref="S7.I11.i2.p1.3.m3.3.3.3.2.1.cmml">)</mo></mrow></mrow><mo id="S7.I11.i2.p1.3.m3.3.3.3.2.3" stretchy="false" xref="S7.I11.i2.p1.3.m3.3.3.3.3.1.cmml">|</mo></mrow></mfrac><annotation-xml encoding="MathML-Content" id="S7.I11.i2.p1.3.m3.3b"><apply id="S7.I11.i2.p1.3.m3.3.3.cmml" xref="S7.I11.i2.p1.3.m3.3.3"><divide id="S7.I11.i2.p1.3.m3.3.3.4.cmml" xref="S7.I11.i2.p1.3.m3.3.3"></divide><apply id="S7.I11.i2.p1.3.m3.1.1.1.cmml" xref="S7.I11.i2.p1.3.m3.1.1.1"><times id="S7.I11.i2.p1.3.m3.1.1.1.2.cmml" xref="S7.I11.i2.p1.3.m3.1.1.1.2"></times><apply id="S7.I11.i2.p1.3.m3.1.1.1.3.cmml" xref="S7.I11.i2.p1.3.m3.1.1.1.3"><csymbol cd="ambiguous" id="S7.I11.i2.p1.3.m3.1.1.1.3.1.cmml" xref="S7.I11.i2.p1.3.m3.1.1.1.3">subscript</csymbol><ci id="S7.I11.i2.p1.3.m3.1.1.1.3.2.cmml" xref="S7.I11.i2.p1.3.m3.1.1.1.3.2">𝛿</ci><ci id="S7.I11.i2.p1.3.m3.1.1.1.3.3.cmml" xref="S7.I11.i2.p1.3.m3.1.1.1.3.3">𝑡</ci></apply><ci id="S7.I11.i2.p1.3.m3.1.1.1.1.cmml" xref="S7.I11.i2.p1.3.m3.1.1.1.1">𝑎</ci></apply><apply id="S7.I11.i2.p1.3.m3.3.3.3.3.cmml" xref="S7.I11.i2.p1.3.m3.3.3.3.2"><abs id="S7.I11.i2.p1.3.m3.3.3.3.3.1.cmml" xref="S7.I11.i2.p1.3.m3.3.3.3.2.2"></abs><apply id="S7.I11.i2.p1.3.m3.3.3.3.2.1.cmml" xref="S7.I11.i2.p1.3.m3.3.3.3.2.1"><times id="S7.I11.i2.p1.3.m3.3.3.3.2.1.1.cmml" xref="S7.I11.i2.p1.3.m3.3.3.3.2.1.1"></times><apply id="S7.I11.i2.p1.3.m3.3.3.3.2.1.2.cmml" xref="S7.I11.i2.p1.3.m3.3.3.3.2.1.2"><csymbol cd="ambiguous" id="S7.I11.i2.p1.3.m3.3.3.3.2.1.2.1.cmml" xref="S7.I11.i2.p1.3.m3.3.3.3.2.1.2">subscript</csymbol><ci id="S7.I11.i2.p1.3.m3.3.3.3.2.1.2.2.cmml" xref="S7.I11.i2.p1.3.m3.3.3.3.2.1.2.2">𝐸</ci><ci id="S7.I11.i2.p1.3.m3.3.3.3.2.1.2.3.cmml" xref="S7.I11.i2.p1.3.m3.3.3.3.2.1.2.3">𝑡</ci></apply><ci id="S7.I11.i2.p1.3.m3.2.2.2.1.cmml" xref="S7.I11.i2.p1.3.m3.2.2.2.1">𝑎</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.I11.i2.p1.3.m3.3c">\frac{\delta_{t}(a)}{|E_{t}(a)|}</annotation><annotation encoding="application/x-llamapun" id="S7.I11.i2.p1.3.m3.3d">divide start_ARG italic_δ start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT ( italic_a ) end_ARG start_ARG | italic_E start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT ( italic_a ) | end_ARG</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S7.I11.i2.p1.5.4">. This only occurs in our code if </span><math alttext="b" class="ltx_Math" display="inline" id="S7.I11.i2.p1.4.m4.1"><semantics id="S7.I11.i2.p1.4.m4.1a"><mi id="S7.I11.i2.p1.4.m4.1.1" xref="S7.I11.i2.p1.4.m4.1.1.cmml">b</mi><annotation-xml encoding="MathML-Content" id="S7.I11.i2.p1.4.m4.1b"><ci id="S7.I11.i2.p1.4.m4.1.1.cmml" xref="S7.I11.i2.p1.4.m4.1.1">𝑏</ci></annotation-xml><annotation encoding="application/x-tex" id="S7.I11.i2.p1.4.m4.1c">b</annotation><annotation encoding="application/x-llamapun" id="S7.I11.i2.p1.4.m4.1d">italic_b</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S7.I11.i2.p1.5.5"> is satisfied and thus </span><math alttext="\overline{ab}" class="ltx_Math" display="inline" id="S7.I11.i2.p1.5.m5.1"><semantics id="S7.I11.i2.p1.5.m5.1a"><mover accent="true" id="S7.I11.i2.p1.5.m5.1.1" xref="S7.I11.i2.p1.5.m5.1.1.cmml"><mrow id="S7.I11.i2.p1.5.m5.1.1.2" xref="S7.I11.i2.p1.5.m5.1.1.2.cmml"><mi id="S7.I11.i2.p1.5.m5.1.1.2.2" xref="S7.I11.i2.p1.5.m5.1.1.2.2.cmml">a</mi><mo id="S7.I11.i2.p1.5.m5.1.1.2.1" xref="S7.I11.i2.p1.5.m5.1.1.2.1.cmml">⁢</mo><mi id="S7.I11.i2.p1.5.m5.1.1.2.3" xref="S7.I11.i2.p1.5.m5.1.1.2.3.cmml">b</mi></mrow><mo id="S7.I11.i2.p1.5.m5.1.1.1" xref="S7.I11.i2.p1.5.m5.1.1.1.cmml">¯</mo></mover><annotation-xml encoding="MathML-Content" id="S7.I11.i2.p1.5.m5.1b"><apply id="S7.I11.i2.p1.5.m5.1.1.cmml" xref="S7.I11.i2.p1.5.m5.1.1"><ci id="S7.I11.i2.p1.5.m5.1.1.1.cmml" xref="S7.I11.i2.p1.5.m5.1.1.1">¯</ci><apply id="S7.I11.i2.p1.5.m5.1.1.2.cmml" xref="S7.I11.i2.p1.5.m5.1.1.2"><times id="S7.I11.i2.p1.5.m5.1.1.2.1.cmml" xref="S7.I11.i2.p1.5.m5.1.1.2.1"></times><ci id="S7.I11.i2.p1.5.m5.1.1.2.2.cmml" xref="S7.I11.i2.p1.5.m5.1.1.2.2">𝑎</ci><ci id="S7.I11.i2.p1.5.m5.1.1.2.3.cmml" xref="S7.I11.i2.p1.5.m5.1.1.2.3">𝑏</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.I11.i2.p1.5.m5.1c">\overline{ab}</annotation><annotation encoding="application/x-llamapun" id="S7.I11.i2.p1.5.m5.1d">over¯ start_ARG italic_a italic_b end_ARG</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S7.I11.i2.p1.5.6"> is satisfied.</span></p> </div> </li> <li class="ltx_item" id="S7.I11.i3" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">3.</span> <div class="ltx_para" id="S7.I11.i3.p1"> <p class="ltx_p" id="S7.I11.i3.p1.3"><span class="ltx_text ltx_font_italic" id="S7.I11.i3.p1.3.1">If otherwise </span><math alttext="\textsl{g}(a\!\to\!b)>0" class="ltx_Math" display="inline" id="S7.I11.i3.p1.1.m1.1"><semantics id="S7.I11.i3.p1.1.m1.1a"><mrow id="S7.I11.i3.p1.1.m1.1.1" xref="S7.I11.i3.p1.1.m1.1.1.cmml"><mrow id="S7.I11.i3.p1.1.m1.1.1.1" xref="S7.I11.i3.p1.1.m1.1.1.1.cmml"><mtext class="ltx_mathvariant_italic" id="S7.I11.i3.p1.1.m1.1.1.1.3" xref="S7.I11.i3.p1.1.m1.1.1.1.3a.cmml">g</mtext><mo id="S7.I11.i3.p1.1.m1.1.1.1.2" xref="S7.I11.i3.p1.1.m1.1.1.1.2.cmml">⁢</mo><mrow id="S7.I11.i3.p1.1.m1.1.1.1.1.1" xref="S7.I11.i3.p1.1.m1.1.1.1.1.1.1.cmml"><mo id="S7.I11.i3.p1.1.m1.1.1.1.1.1.2" stretchy="false" xref="S7.I11.i3.p1.1.m1.1.1.1.1.1.1.cmml">(</mo><mrow id="S7.I11.i3.p1.1.m1.1.1.1.1.1.1" xref="S7.I11.i3.p1.1.m1.1.1.1.1.1.1.cmml"><mi id="S7.I11.i3.p1.1.m1.1.1.1.1.1.1.2" xref="S7.I11.i3.p1.1.m1.1.1.1.1.1.1.2.cmml">a</mi><mo id="S7.I11.i3.p1.1.m1.1.1.1.1.1.1.1" lspace="0.108em" rspace="0.108em" stretchy="false" xref="S7.I11.i3.p1.1.m1.1.1.1.1.1.1.1.cmml">→</mo><mi id="S7.I11.i3.p1.1.m1.1.1.1.1.1.1.3" xref="S7.I11.i3.p1.1.m1.1.1.1.1.1.1.3.cmml">b</mi></mrow><mo id="S7.I11.i3.p1.1.m1.1.1.1.1.1.3" stretchy="false" xref="S7.I11.i3.p1.1.m1.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S7.I11.i3.p1.1.m1.1.1.2" xref="S7.I11.i3.p1.1.m1.1.1.2.cmml">></mo><mn id="S7.I11.i3.p1.1.m1.1.1.3" xref="S7.I11.i3.p1.1.m1.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S7.I11.i3.p1.1.m1.1b"><apply id="S7.I11.i3.p1.1.m1.1.1.cmml" xref="S7.I11.i3.p1.1.m1.1.1"><gt id="S7.I11.i3.p1.1.m1.1.1.2.cmml" xref="S7.I11.i3.p1.1.m1.1.1.2"></gt><apply id="S7.I11.i3.p1.1.m1.1.1.1.cmml" xref="S7.I11.i3.p1.1.m1.1.1.1"><times id="S7.I11.i3.p1.1.m1.1.1.1.2.cmml" xref="S7.I11.i3.p1.1.m1.1.1.1.2"></times><ci id="S7.I11.i3.p1.1.m1.1.1.1.3a.cmml" xref="S7.I11.i3.p1.1.m1.1.1.1.3"><mtext class="ltx_mathvariant_italic" id="S7.I11.i3.p1.1.m1.1.1.1.3.cmml" xref="S7.I11.i3.p1.1.m1.1.1.1.3">g</mtext></ci><apply id="S7.I11.i3.p1.1.m1.1.1.1.1.1.1.cmml" xref="S7.I11.i3.p1.1.m1.1.1.1.1.1"><ci id="S7.I11.i3.p1.1.m1.1.1.1.1.1.1.1.cmml" xref="S7.I11.i3.p1.1.m1.1.1.1.1.1.1.1">→</ci><ci id="S7.I11.i3.p1.1.m1.1.1.1.1.1.1.2.cmml" xref="S7.I11.i3.p1.1.m1.1.1.1.1.1.1.2">𝑎</ci><ci id="S7.I11.i3.p1.1.m1.1.1.1.1.1.1.3.cmml" xref="S7.I11.i3.p1.1.m1.1.1.1.1.1.1.3">𝑏</ci></apply></apply><cn id="S7.I11.i3.p1.1.m1.1.1.3.cmml" type="integer" xref="S7.I11.i3.p1.1.m1.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.I11.i3.p1.1.m1.1c">\textsl{g}(a\!\to\!b)>0</annotation><annotation encoding="application/x-llamapun" id="S7.I11.i3.p1.1.m1.1d">g ( italic_a → italic_b ) > 0</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S7.I11.i3.p1.3.2"> and </span><math alttext="\textsl{g}(a\!\to\!b)" class="ltx_Math" display="inline" id="S7.I11.i3.p1.2.m2.1"><semantics id="S7.I11.i3.p1.2.m2.1a"><mrow id="S7.I11.i3.p1.2.m2.1.1" xref="S7.I11.i3.p1.2.m2.1.1.cmml"><mtext class="ltx_mathvariant_italic" id="S7.I11.i3.p1.2.m2.1.1.3" xref="S7.I11.i3.p1.2.m2.1.1.3a.cmml">g</mtext><mo id="S7.I11.i3.p1.2.m2.1.1.2" xref="S7.I11.i3.p1.2.m2.1.1.2.cmml">⁢</mo><mrow id="S7.I11.i3.p1.2.m2.1.1.1.1" xref="S7.I11.i3.p1.2.m2.1.1.1.1.1.cmml"><mo id="S7.I11.i3.p1.2.m2.1.1.1.1.2" stretchy="false" xref="S7.I11.i3.p1.2.m2.1.1.1.1.1.cmml">(</mo><mrow id="S7.I11.i3.p1.2.m2.1.1.1.1.1" xref="S7.I11.i3.p1.2.m2.1.1.1.1.1.cmml"><mi id="S7.I11.i3.p1.2.m2.1.1.1.1.1.2" xref="S7.I11.i3.p1.2.m2.1.1.1.1.1.2.cmml">a</mi><mo id="S7.I11.i3.p1.2.m2.1.1.1.1.1.1" lspace="0.108em" rspace="0.108em" stretchy="false" xref="S7.I11.i3.p1.2.m2.1.1.1.1.1.1.cmml">→</mo><mi id="S7.I11.i3.p1.2.m2.1.1.1.1.1.3" xref="S7.I11.i3.p1.2.m2.1.1.1.1.1.3.cmml">b</mi></mrow><mo id="S7.I11.i3.p1.2.m2.1.1.1.1.3" stretchy="false" xref="S7.I11.i3.p1.2.m2.1.1.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S7.I11.i3.p1.2.m2.1b"><apply id="S7.I11.i3.p1.2.m2.1.1.cmml" xref="S7.I11.i3.p1.2.m2.1.1"><times id="S7.I11.i3.p1.2.m2.1.1.2.cmml" xref="S7.I11.i3.p1.2.m2.1.1.2"></times><ci id="S7.I11.i3.p1.2.m2.1.1.3a.cmml" xref="S7.I11.i3.p1.2.m2.1.1.3"><mtext class="ltx_mathvariant_italic" id="S7.I11.i3.p1.2.m2.1.1.3.cmml" xref="S7.I11.i3.p1.2.m2.1.1.3">g</mtext></ci><apply id="S7.I11.i3.p1.2.m2.1.1.1.1.1.cmml" xref="S7.I11.i3.p1.2.m2.1.1.1.1"><ci id="S7.I11.i3.p1.2.m2.1.1.1.1.1.1.cmml" xref="S7.I11.i3.p1.2.m2.1.1.1.1.1.1">→</ci><ci id="S7.I11.i3.p1.2.m2.1.1.1.1.1.2.cmml" xref="S7.I11.i3.p1.2.m2.1.1.1.1.1.2">𝑎</ci><ci id="S7.I11.i3.p1.2.m2.1.1.1.1.1.3.cmml" xref="S7.I11.i3.p1.2.m2.1.1.1.1.1.3">𝑏</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.I11.i3.p1.2.m2.1c">\textsl{g}(a\!\to\!b)</annotation><annotation encoding="application/x-llamapun" id="S7.I11.i3.p1.2.m2.1d">g ( italic_a → italic_b )</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S7.I11.i3.p1.3.3"> is decreased by </span><math alttext="\frac{\delta_{t}(a)}{|E_{t}(a)|}" class="ltx_Math" display="inline" id="S7.I11.i3.p1.3.m3.3"><semantics id="S7.I11.i3.p1.3.m3.3a"><mfrac id="S7.I11.i3.p1.3.m3.3.3" xref="S7.I11.i3.p1.3.m3.3.3.cmml"><mrow id="S7.I11.i3.p1.3.m3.1.1.1" xref="S7.I11.i3.p1.3.m3.1.1.1.cmml"><msub id="S7.I11.i3.p1.3.m3.1.1.1.3" xref="S7.I11.i3.p1.3.m3.1.1.1.3.cmml"><mi id="S7.I11.i3.p1.3.m3.1.1.1.3.2" xref="S7.I11.i3.p1.3.m3.1.1.1.3.2.cmml">δ</mi><mi id="S7.I11.i3.p1.3.m3.1.1.1.3.3" xref="S7.I11.i3.p1.3.m3.1.1.1.3.3.cmml">t</mi></msub><mo id="S7.I11.i3.p1.3.m3.1.1.1.2" xref="S7.I11.i3.p1.3.m3.1.1.1.2.cmml">⁢</mo><mrow id="S7.I11.i3.p1.3.m3.1.1.1.4.2" xref="S7.I11.i3.p1.3.m3.1.1.1.cmml"><mo id="S7.I11.i3.p1.3.m3.1.1.1.4.2.1" stretchy="false" xref="S7.I11.i3.p1.3.m3.1.1.1.cmml">(</mo><mi id="S7.I11.i3.p1.3.m3.1.1.1.1" xref="S7.I11.i3.p1.3.m3.1.1.1.1.cmml">a</mi><mo id="S7.I11.i3.p1.3.m3.1.1.1.4.2.2" stretchy="false" xref="S7.I11.i3.p1.3.m3.1.1.1.cmml">)</mo></mrow></mrow><mrow id="S7.I11.i3.p1.3.m3.3.3.3.2" xref="S7.I11.i3.p1.3.m3.3.3.3.3.cmml"><mo id="S7.I11.i3.p1.3.m3.3.3.3.2.2" stretchy="false" xref="S7.I11.i3.p1.3.m3.3.3.3.3.1.cmml">|</mo><mrow id="S7.I11.i3.p1.3.m3.3.3.3.2.1" xref="S7.I11.i3.p1.3.m3.3.3.3.2.1.cmml"><msub id="S7.I11.i3.p1.3.m3.3.3.3.2.1.2" xref="S7.I11.i3.p1.3.m3.3.3.3.2.1.2.cmml"><mi id="S7.I11.i3.p1.3.m3.3.3.3.2.1.2.2" xref="S7.I11.i3.p1.3.m3.3.3.3.2.1.2.2.cmml">E</mi><mi id="S7.I11.i3.p1.3.m3.3.3.3.2.1.2.3" xref="S7.I11.i3.p1.3.m3.3.3.3.2.1.2.3.cmml">t</mi></msub><mo id="S7.I11.i3.p1.3.m3.3.3.3.2.1.1" xref="S7.I11.i3.p1.3.m3.3.3.3.2.1.1.cmml">⁢</mo><mrow id="S7.I11.i3.p1.3.m3.3.3.3.2.1.3.2" xref="S7.I11.i3.p1.3.m3.3.3.3.2.1.cmml"><mo id="S7.I11.i3.p1.3.m3.3.3.3.2.1.3.2.1" stretchy="false" xref="S7.I11.i3.p1.3.m3.3.3.3.2.1.cmml">(</mo><mi id="S7.I11.i3.p1.3.m3.2.2.2.1" xref="S7.I11.i3.p1.3.m3.2.2.2.1.cmml">a</mi><mo id="S7.I11.i3.p1.3.m3.3.3.3.2.1.3.2.2" stretchy="false" xref="S7.I11.i3.p1.3.m3.3.3.3.2.1.cmml">)</mo></mrow></mrow><mo id="S7.I11.i3.p1.3.m3.3.3.3.2.3" stretchy="false" xref="S7.I11.i3.p1.3.m3.3.3.3.3.1.cmml">|</mo></mrow></mfrac><annotation-xml encoding="MathML-Content" id="S7.I11.i3.p1.3.m3.3b"><apply id="S7.I11.i3.p1.3.m3.3.3.cmml" xref="S7.I11.i3.p1.3.m3.3.3"><divide id="S7.I11.i3.p1.3.m3.3.3.4.cmml" xref="S7.I11.i3.p1.3.m3.3.3"></divide><apply id="S7.I11.i3.p1.3.m3.1.1.1.cmml" xref="S7.I11.i3.p1.3.m3.1.1.1"><times id="S7.I11.i3.p1.3.m3.1.1.1.2.cmml" xref="S7.I11.i3.p1.3.m3.1.1.1.2"></times><apply id="S7.I11.i3.p1.3.m3.1.1.1.3.cmml" xref="S7.I11.i3.p1.3.m3.1.1.1.3"><csymbol cd="ambiguous" id="S7.I11.i3.p1.3.m3.1.1.1.3.1.cmml" xref="S7.I11.i3.p1.3.m3.1.1.1.3">subscript</csymbol><ci id="S7.I11.i3.p1.3.m3.1.1.1.3.2.cmml" xref="S7.I11.i3.p1.3.m3.1.1.1.3.2">𝛿</ci><ci id="S7.I11.i3.p1.3.m3.1.1.1.3.3.cmml" xref="S7.I11.i3.p1.3.m3.1.1.1.3.3">𝑡</ci></apply><ci id="S7.I11.i3.p1.3.m3.1.1.1.1.cmml" xref="S7.I11.i3.p1.3.m3.1.1.1.1">𝑎</ci></apply><apply id="S7.I11.i3.p1.3.m3.3.3.3.3.cmml" xref="S7.I11.i3.p1.3.m3.3.3.3.2"><abs id="S7.I11.i3.p1.3.m3.3.3.3.3.1.cmml" xref="S7.I11.i3.p1.3.m3.3.3.3.2.2"></abs><apply id="S7.I11.i3.p1.3.m3.3.3.3.2.1.cmml" xref="S7.I11.i3.p1.3.m3.3.3.3.2.1"><times id="S7.I11.i3.p1.3.m3.3.3.3.2.1.1.cmml" xref="S7.I11.i3.p1.3.m3.3.3.3.2.1.1"></times><apply id="S7.I11.i3.p1.3.m3.3.3.3.2.1.2.cmml" xref="S7.I11.i3.p1.3.m3.3.3.3.2.1.2"><csymbol cd="ambiguous" id="S7.I11.i3.p1.3.m3.3.3.3.2.1.2.1.cmml" xref="S7.I11.i3.p1.3.m3.3.3.3.2.1.2">subscript</csymbol><ci id="S7.I11.i3.p1.3.m3.3.3.3.2.1.2.2.cmml" xref="S7.I11.i3.p1.3.m3.3.3.3.2.1.2.2">𝐸</ci><ci id="S7.I11.i3.p1.3.m3.3.3.3.2.1.2.3.cmml" xref="S7.I11.i3.p1.3.m3.3.3.3.2.1.2.3">𝑡</ci></apply><ci id="S7.I11.i3.p1.3.m3.2.2.2.1.cmml" xref="S7.I11.i3.p1.3.m3.2.2.2.1">𝑎</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.I11.i3.p1.3.m3.3c">\frac{\delta_{t}(a)}{|E_{t}(a)|}</annotation><annotation encoding="application/x-llamapun" id="S7.I11.i3.p1.3.m3.3d">divide start_ARG italic_δ start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT ( italic_a ) end_ARG start_ARG | italic_E start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT ( italic_a ) | end_ARG</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S7.I11.i3.p1.3.4">, we make a case distinction:</span></p> <ul class="ltx_itemize" id="S7.I11.i3.I1"> <li class="ltx_item" id="S7.I11.i3.I0.i1" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">•</span> <div class="ltx_para" id="S7.I11.i3.I0.i1.p1"> <p class="ltx_p" id="S7.I11.i3.I0.i1.p1.3"><span class="ltx_text ltx_font_italic" id="S7.I11.i3.I0.i1.p1.3.1">If </span><math alttext="a" class="ltx_Math" display="inline" id="S7.I11.i3.I0.i1.p1.1.m1.1"><semantics id="S7.I11.i3.I0.i1.p1.1.m1.1a"><mi id="S7.I11.i3.I0.i1.p1.1.m1.1.1" xref="S7.I11.i3.I0.i1.p1.1.m1.1.1.cmml">a</mi><annotation-xml encoding="MathML-Content" id="S7.I11.i3.I0.i1.p1.1.m1.1b"><ci id="S7.I11.i3.I0.i1.p1.1.m1.1.1.cmml" xref="S7.I11.i3.I0.i1.p1.1.m1.1.1">𝑎</ci></annotation-xml><annotation encoding="application/x-tex" id="S7.I11.i3.I0.i1.p1.1.m1.1c">a</annotation><annotation encoding="application/x-llamapun" id="S7.I11.i3.I0.i1.p1.1.m1.1d">italic_a</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S7.I11.i3.I0.i1.p1.3.2"> is satisfied at the end of iteration </span><math alttext="t" class="ltx_Math" display="inline" id="S7.I11.i3.I0.i1.p1.2.m2.1"><semantics id="S7.I11.i3.I0.i1.p1.2.m2.1a"><mi id="S7.I11.i3.I0.i1.p1.2.m2.1.1" xref="S7.I11.i3.I0.i1.p1.2.m2.1.1.cmml">t</mi><annotation-xml encoding="MathML-Content" id="S7.I11.i3.I0.i1.p1.2.m2.1b"><ci id="S7.I11.i3.I0.i1.p1.2.m2.1.1.cmml" xref="S7.I11.i3.I0.i1.p1.2.m2.1.1">𝑡</ci></annotation-xml><annotation encoding="application/x-tex" id="S7.I11.i3.I0.i1.p1.2.m2.1c">t</annotation><annotation encoding="application/x-llamapun" id="S7.I11.i3.I0.i1.p1.2.m2.1d">italic_t</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S7.I11.i3.I0.i1.p1.3.3">, then </span><math alttext="\overline{ab}" class="ltx_Math" display="inline" id="S7.I11.i3.I0.i1.p1.3.m3.1"><semantics id="S7.I11.i3.I0.i1.p1.3.m3.1a"><mover accent="true" id="S7.I11.i3.I0.i1.p1.3.m3.1.1" xref="S7.I11.i3.I0.i1.p1.3.m3.1.1.cmml"><mrow id="S7.I11.i3.I0.i1.p1.3.m3.1.1.2" xref="S7.I11.i3.I0.i1.p1.3.m3.1.1.2.cmml"><mi id="S7.I11.i3.I0.i1.p1.3.m3.1.1.2.2" xref="S7.I11.i3.I0.i1.p1.3.m3.1.1.2.2.cmml">a</mi><mo id="S7.I11.i3.I0.i1.p1.3.m3.1.1.2.1" xref="S7.I11.i3.I0.i1.p1.3.m3.1.1.2.1.cmml">⁢</mo><mi id="S7.I11.i3.I0.i1.p1.3.m3.1.1.2.3" xref="S7.I11.i3.I0.i1.p1.3.m3.1.1.2.3.cmml">b</mi></mrow><mo id="S7.I11.i3.I0.i1.p1.3.m3.1.1.1" xref="S7.I11.i3.I0.i1.p1.3.m3.1.1.1.cmml">¯</mo></mover><annotation-xml encoding="MathML-Content" id="S7.I11.i3.I0.i1.p1.3.m3.1b"><apply id="S7.I11.i3.I0.i1.p1.3.m3.1.1.cmml" xref="S7.I11.i3.I0.i1.p1.3.m3.1.1"><ci id="S7.I11.i3.I0.i1.p1.3.m3.1.1.1.cmml" xref="S7.I11.i3.I0.i1.p1.3.m3.1.1.1">¯</ci><apply id="S7.I11.i3.I0.i1.p1.3.m3.1.1.2.cmml" xref="S7.I11.i3.I0.i1.p1.3.m3.1.1.2"><times id="S7.I11.i3.I0.i1.p1.3.m3.1.1.2.1.cmml" xref="S7.I11.i3.I0.i1.p1.3.m3.1.1.2.1"></times><ci id="S7.I11.i3.I0.i1.p1.3.m3.1.1.2.2.cmml" xref="S7.I11.i3.I0.i1.p1.3.m3.1.1.2.2">𝑎</ci><ci id="S7.I11.i3.I0.i1.p1.3.m3.1.1.2.3.cmml" xref="S7.I11.i3.I0.i1.p1.3.m3.1.1.2.3">𝑏</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.I11.i3.I0.i1.p1.3.m3.1c">\overline{ab}</annotation><annotation encoding="application/x-llamapun" id="S7.I11.i3.I0.i1.p1.3.m3.1d">over¯ start_ARG italic_a italic_b end_ARG</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S7.I11.i3.I0.i1.p1.3.4"> is satisfied.</span></p> </div> </li> <li class="ltx_item" id="S7.I11.i3.I0.i2" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">•</span> <div class="ltx_para" id="S7.I11.i3.I0.i2.p1"> <p class="ltx_p" id="S7.I11.i3.I0.i2.p1.18"><span class="ltx_text ltx_font_italic" id="S7.I11.i3.I0.i2.p1.18.1">Otherwise, let </span><math alttext="\textsl{g}(a)" class="ltx_Math" display="inline" id="S7.I11.i3.I0.i2.p1.1.m1.1"><semantics id="S7.I11.i3.I0.i2.p1.1.m1.1a"><mrow id="S7.I11.i3.I0.i2.p1.1.m1.1.2" xref="S7.I11.i3.I0.i2.p1.1.m1.1.2.cmml"><mtext class="ltx_mathvariant_italic" id="S7.I11.i3.I0.i2.p1.1.m1.1.2.2" xref="S7.I11.i3.I0.i2.p1.1.m1.1.2.2a.cmml">g</mtext><mo id="S7.I11.i3.I0.i2.p1.1.m1.1.2.1" xref="S7.I11.i3.I0.i2.p1.1.m1.1.2.1.cmml">⁢</mo><mrow id="S7.I11.i3.I0.i2.p1.1.m1.1.2.3.2" xref="S7.I11.i3.I0.i2.p1.1.m1.1.2.cmml"><mo id="S7.I11.i3.I0.i2.p1.1.m1.1.2.3.2.1" stretchy="false" xref="S7.I11.i3.I0.i2.p1.1.m1.1.2.cmml">(</mo><mi id="S7.I11.i3.I0.i2.p1.1.m1.1.1" xref="S7.I11.i3.I0.i2.p1.1.m1.1.1.cmml">a</mi><mo id="S7.I11.i3.I0.i2.p1.1.m1.1.2.3.2.2" stretchy="false" xref="S7.I11.i3.I0.i2.p1.1.m1.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S7.I11.i3.I0.i2.p1.1.m1.1b"><apply id="S7.I11.i3.I0.i2.p1.1.m1.1.2.cmml" xref="S7.I11.i3.I0.i2.p1.1.m1.1.2"><times id="S7.I11.i3.I0.i2.p1.1.m1.1.2.1.cmml" xref="S7.I11.i3.I0.i2.p1.1.m1.1.2.1"></times><ci id="S7.I11.i3.I0.i2.p1.1.m1.1.2.2a.cmml" xref="S7.I11.i3.I0.i2.p1.1.m1.1.2.2"><mtext class="ltx_mathvariant_italic" id="S7.I11.i3.I0.i2.p1.1.m1.1.2.2.cmml" xref="S7.I11.i3.I0.i2.p1.1.m1.1.2.2">g</mtext></ci><ci id="S7.I11.i3.I0.i2.p1.1.m1.1.1.cmml" xref="S7.I11.i3.I0.i2.p1.1.m1.1.1">𝑎</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.I11.i3.I0.i2.p1.1.m1.1c">\textsl{g}(a)</annotation><annotation encoding="application/x-llamapun" id="S7.I11.i3.I0.i2.p1.1.m1.1d">g ( italic_a )</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S7.I11.i3.I0.i2.p1.18.2"> denote the out-degree of </span><math alttext="a" class="ltx_Math" display="inline" id="S7.I11.i3.I0.i2.p1.2.m2.1"><semantics id="S7.I11.i3.I0.i2.p1.2.m2.1a"><mi id="S7.I11.i3.I0.i2.p1.2.m2.1.1" xref="S7.I11.i3.I0.i2.p1.2.m2.1.1.cmml">a</mi><annotation-xml encoding="MathML-Content" id="S7.I11.i3.I0.i2.p1.2.m2.1b"><ci id="S7.I11.i3.I0.i2.p1.2.m2.1.1.cmml" xref="S7.I11.i3.I0.i2.p1.2.m2.1.1">𝑎</ci></annotation-xml><annotation encoding="application/x-tex" id="S7.I11.i3.I0.i2.p1.2.m2.1c">a</annotation><annotation encoding="application/x-llamapun" id="S7.I11.i3.I0.i2.p1.2.m2.1d">italic_a</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S7.I11.i3.I0.i2.p1.18.3"> at the beginning of iteration </span><math alttext="t" class="ltx_Math" display="inline" id="S7.I11.i3.I0.i2.p1.3.m3.1"><semantics id="S7.I11.i3.I0.i2.p1.3.m3.1a"><mi id="S7.I11.i3.I0.i2.p1.3.m3.1.1" xref="S7.I11.i3.I0.i2.p1.3.m3.1.1.cmml">t</mi><annotation-xml encoding="MathML-Content" id="S7.I11.i3.I0.i2.p1.3.m3.1b"><ci id="S7.I11.i3.I0.i2.p1.3.m3.1.1.cmml" xref="S7.I11.i3.I0.i2.p1.3.m3.1.1">𝑡</ci></annotation-xml><annotation encoding="application/x-tex" id="S7.I11.i3.I0.i2.p1.3.m3.1c">t</annotation><annotation encoding="application/x-llamapun" id="S7.I11.i3.I0.i2.p1.3.m3.1d">italic_t</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S7.I11.i3.I0.i2.p1.18.4"> and </span><math alttext="\textsl{g}^{\prime}(a)" class="ltx_Math" display="inline" id="S7.I11.i3.I0.i2.p1.4.m4.1"><semantics id="S7.I11.i3.I0.i2.p1.4.m4.1a"><mrow id="S7.I11.i3.I0.i2.p1.4.m4.1.2" xref="S7.I11.i3.I0.i2.p1.4.m4.1.2.cmml"><msup id="S7.I11.i3.I0.i2.p1.4.m4.1.2.2" xref="S7.I11.i3.I0.i2.p1.4.m4.1.2.2.cmml"><mtext class="ltx_mathvariant_italic" id="S7.I11.i3.I0.i2.p1.4.m4.1.2.2.2" xref="S7.I11.i3.I0.i2.p1.4.m4.1.2.2.2a.cmml">g</mtext><mo id="S7.I11.i3.I0.i2.p1.4.m4.1.2.2.3" xref="S7.I11.i3.I0.i2.p1.4.m4.1.2.2.3.cmml">′</mo></msup><mo id="S7.I11.i3.I0.i2.p1.4.m4.1.2.1" xref="S7.I11.i3.I0.i2.p1.4.m4.1.2.1.cmml">⁢</mo><mrow id="S7.I11.i3.I0.i2.p1.4.m4.1.2.3.2" xref="S7.I11.i3.I0.i2.p1.4.m4.1.2.cmml"><mo id="S7.I11.i3.I0.i2.p1.4.m4.1.2.3.2.1" stretchy="false" xref="S7.I11.i3.I0.i2.p1.4.m4.1.2.cmml">(</mo><mi id="S7.I11.i3.I0.i2.p1.4.m4.1.1" xref="S7.I11.i3.I0.i2.p1.4.m4.1.1.cmml">a</mi><mo id="S7.I11.i3.I0.i2.p1.4.m4.1.2.3.2.2" stretchy="false" xref="S7.I11.i3.I0.i2.p1.4.m4.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S7.I11.i3.I0.i2.p1.4.m4.1b"><apply id="S7.I11.i3.I0.i2.p1.4.m4.1.2.cmml" xref="S7.I11.i3.I0.i2.p1.4.m4.1.2"><times id="S7.I11.i3.I0.i2.p1.4.m4.1.2.1.cmml" xref="S7.I11.i3.I0.i2.p1.4.m4.1.2.1"></times><apply id="S7.I11.i3.I0.i2.p1.4.m4.1.2.2.cmml" xref="S7.I11.i3.I0.i2.p1.4.m4.1.2.2"><csymbol cd="ambiguous" id="S7.I11.i3.I0.i2.p1.4.m4.1.2.2.1.cmml" xref="S7.I11.i3.I0.i2.p1.4.m4.1.2.2">superscript</csymbol><ci id="S7.I11.i3.I0.i2.p1.4.m4.1.2.2.2a.cmml" xref="S7.I11.i3.I0.i2.p1.4.m4.1.2.2.2"><mtext class="ltx_mathvariant_italic" id="S7.I11.i3.I0.i2.p1.4.m4.1.2.2.2.cmml" xref="S7.I11.i3.I0.i2.p1.4.m4.1.2.2.2">g</mtext></ci><ci id="S7.I11.i3.I0.i2.p1.4.m4.1.2.2.3.cmml" xref="S7.I11.i3.I0.i2.p1.4.m4.1.2.2.3">′</ci></apply><ci id="S7.I11.i3.I0.i2.p1.4.m4.1.1.cmml" xref="S7.I11.i3.I0.i2.p1.4.m4.1.1">𝑎</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.I11.i3.I0.i2.p1.4.m4.1c">\textsl{g}^{\prime}(a)</annotation><annotation encoding="application/x-llamapun" id="S7.I11.i3.I0.i2.p1.4.m4.1d">g start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ( italic_a )</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S7.I11.i3.I0.i2.p1.18.5"> denote the out-degree of </span><math alttext="a" class="ltx_Math" display="inline" id="S7.I11.i3.I0.i2.p1.5.m5.1"><semantics id="S7.I11.i3.I0.i2.p1.5.m5.1a"><mi id="S7.I11.i3.I0.i2.p1.5.m5.1.1" xref="S7.I11.i3.I0.i2.p1.5.m5.1.1.cmml">a</mi><annotation-xml encoding="MathML-Content" id="S7.I11.i3.I0.i2.p1.5.m5.1b"><ci id="S7.I11.i3.I0.i2.p1.5.m5.1.1.cmml" xref="S7.I11.i3.I0.i2.p1.5.m5.1.1">𝑎</ci></annotation-xml><annotation encoding="application/x-tex" id="S7.I11.i3.I0.i2.p1.5.m5.1c">a</annotation><annotation encoding="application/x-llamapun" id="S7.I11.i3.I0.i2.p1.5.m5.1d">italic_a</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S7.I11.i3.I0.i2.p1.18.6"> at the end of iteration </span><math alttext="t" class="ltx_Math" display="inline" id="S7.I11.i3.I0.i2.p1.6.m6.1"><semantics id="S7.I11.i3.I0.i2.p1.6.m6.1a"><mi id="S7.I11.i3.I0.i2.p1.6.m6.1.1" xref="S7.I11.i3.I0.i2.p1.6.m6.1.1.cmml">t</mi><annotation-xml encoding="MathML-Content" id="S7.I11.i3.I0.i2.p1.6.m6.1b"><ci id="S7.I11.i3.I0.i2.p1.6.m6.1.1.cmml" xref="S7.I11.i3.I0.i2.p1.6.m6.1.1">𝑡</ci></annotation-xml><annotation encoding="application/x-tex" id="S7.I11.i3.I0.i2.p1.6.m6.1c">t</annotation><annotation encoding="application/x-llamapun" id="S7.I11.i3.I0.i2.p1.6.m6.1d">italic_t</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S7.I11.i3.I0.i2.p1.18.7">. Per definition of </span><math alttext="a\in L_{h}" class="ltx_Math" display="inline" id="S7.I11.i3.I0.i2.p1.7.m7.1"><semantics id="S7.I11.i3.I0.i2.p1.7.m7.1a"><mrow id="S7.I11.i3.I0.i2.p1.7.m7.1.1" xref="S7.I11.i3.I0.i2.p1.7.m7.1.1.cmml"><mi id="S7.I11.i3.I0.i2.p1.7.m7.1.1.2" xref="S7.I11.i3.I0.i2.p1.7.m7.1.1.2.cmml">a</mi><mo id="S7.I11.i3.I0.i2.p1.7.m7.1.1.1" xref="S7.I11.i3.I0.i2.p1.7.m7.1.1.1.cmml">∈</mo><msub id="S7.I11.i3.I0.i2.p1.7.m7.1.1.3" xref="S7.I11.i3.I0.i2.p1.7.m7.1.1.3.cmml"><mi id="S7.I11.i3.I0.i2.p1.7.m7.1.1.3.2" xref="S7.I11.i3.I0.i2.p1.7.m7.1.1.3.2.cmml">L</mi><mi id="S7.I11.i3.I0.i2.p1.7.m7.1.1.3.3" xref="S7.I11.i3.I0.i2.p1.7.m7.1.1.3.3.cmml">h</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S7.I11.i3.I0.i2.p1.7.m7.1b"><apply id="S7.I11.i3.I0.i2.p1.7.m7.1.1.cmml" xref="S7.I11.i3.I0.i2.p1.7.m7.1.1"><in id="S7.I11.i3.I0.i2.p1.7.m7.1.1.1.cmml" xref="S7.I11.i3.I0.i2.p1.7.m7.1.1.1"></in><ci id="S7.I11.i3.I0.i2.p1.7.m7.1.1.2.cmml" xref="S7.I11.i3.I0.i2.p1.7.m7.1.1.2">𝑎</ci><apply id="S7.I11.i3.I0.i2.p1.7.m7.1.1.3.cmml" xref="S7.I11.i3.I0.i2.p1.7.m7.1.1.3"><csymbol cd="ambiguous" id="S7.I11.i3.I0.i2.p1.7.m7.1.1.3.1.cmml" xref="S7.I11.i3.I0.i2.p1.7.m7.1.1.3">subscript</csymbol><ci id="S7.I11.i3.I0.i2.p1.7.m7.1.1.3.2.cmml" xref="S7.I11.i3.I0.i2.p1.7.m7.1.1.3.2">𝐿</ci><ci id="S7.I11.i3.I0.i2.p1.7.m7.1.1.3.3.cmml" xref="S7.I11.i3.I0.i2.p1.7.m7.1.1.3.3">ℎ</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.I11.i3.I0.i2.p1.7.m7.1c">a\in L_{h}</annotation><annotation encoding="application/x-llamapun" id="S7.I11.i3.I0.i2.p1.7.m7.1d">italic_a ∈ italic_L start_POSTSUBSCRIPT italic_h end_POSTSUBSCRIPT</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S7.I11.i3.I0.i2.p1.18.8">, </span><math alttext="(1+\frac{\eta}{2})^{h}\leq\textsl{g}(a)\leq(1+\frac{\eta}{2})^{h+1}" class="ltx_Math" display="inline" id="S7.I11.i3.I0.i2.p1.8.m8.3"><semantics id="S7.I11.i3.I0.i2.p1.8.m8.3a"><mrow id="S7.I11.i3.I0.i2.p1.8.m8.3.3" xref="S7.I11.i3.I0.i2.p1.8.m8.3.3.cmml"><msup id="S7.I11.i3.I0.i2.p1.8.m8.2.2.1" xref="S7.I11.i3.I0.i2.p1.8.m8.2.2.1.cmml"><mrow id="S7.I11.i3.I0.i2.p1.8.m8.2.2.1.1.1" xref="S7.I11.i3.I0.i2.p1.8.m8.2.2.1.1.1.1.cmml"><mo id="S7.I11.i3.I0.i2.p1.8.m8.2.2.1.1.1.2" stretchy="false" xref="S7.I11.i3.I0.i2.p1.8.m8.2.2.1.1.1.1.cmml">(</mo><mrow id="S7.I11.i3.I0.i2.p1.8.m8.2.2.1.1.1.1" xref="S7.I11.i3.I0.i2.p1.8.m8.2.2.1.1.1.1.cmml"><mn id="S7.I11.i3.I0.i2.p1.8.m8.2.2.1.1.1.1.2" xref="S7.I11.i3.I0.i2.p1.8.m8.2.2.1.1.1.1.2.cmml">1</mn><mo id="S7.I11.i3.I0.i2.p1.8.m8.2.2.1.1.1.1.1" xref="S7.I11.i3.I0.i2.p1.8.m8.2.2.1.1.1.1.1.cmml">+</mo><mfrac id="S7.I11.i3.I0.i2.p1.8.m8.2.2.1.1.1.1.3" xref="S7.I11.i3.I0.i2.p1.8.m8.2.2.1.1.1.1.3.cmml"><mi id="S7.I11.i3.I0.i2.p1.8.m8.2.2.1.1.1.1.3.2" xref="S7.I11.i3.I0.i2.p1.8.m8.2.2.1.1.1.1.3.2.cmml">η</mi><mn id="S7.I11.i3.I0.i2.p1.8.m8.2.2.1.1.1.1.3.3" xref="S7.I11.i3.I0.i2.p1.8.m8.2.2.1.1.1.1.3.3.cmml">2</mn></mfrac></mrow><mo id="S7.I11.i3.I0.i2.p1.8.m8.2.2.1.1.1.3" stretchy="false" xref="S7.I11.i3.I0.i2.p1.8.m8.2.2.1.1.1.1.cmml">)</mo></mrow><mi id="S7.I11.i3.I0.i2.p1.8.m8.2.2.1.3" xref="S7.I11.i3.I0.i2.p1.8.m8.2.2.1.3.cmml">h</mi></msup><mo id="S7.I11.i3.I0.i2.p1.8.m8.3.3.4" xref="S7.I11.i3.I0.i2.p1.8.m8.3.3.4.cmml">≤</mo><mrow id="S7.I11.i3.I0.i2.p1.8.m8.3.3.5" xref="S7.I11.i3.I0.i2.p1.8.m8.3.3.5.cmml"><mtext class="ltx_mathvariant_italic" id="S7.I11.i3.I0.i2.p1.8.m8.3.3.5.2" xref="S7.I11.i3.I0.i2.p1.8.m8.3.3.5.2a.cmml">g</mtext><mo id="S7.I11.i3.I0.i2.p1.8.m8.3.3.5.1" xref="S7.I11.i3.I0.i2.p1.8.m8.3.3.5.1.cmml">⁢</mo><mrow id="S7.I11.i3.I0.i2.p1.8.m8.3.3.5.3.2" xref="S7.I11.i3.I0.i2.p1.8.m8.3.3.5.cmml"><mo id="S7.I11.i3.I0.i2.p1.8.m8.3.3.5.3.2.1" stretchy="false" xref="S7.I11.i3.I0.i2.p1.8.m8.3.3.5.cmml">(</mo><mi id="S7.I11.i3.I0.i2.p1.8.m8.1.1" xref="S7.I11.i3.I0.i2.p1.8.m8.1.1.cmml">a</mi><mo id="S7.I11.i3.I0.i2.p1.8.m8.3.3.5.3.2.2" stretchy="false" xref="S7.I11.i3.I0.i2.p1.8.m8.3.3.5.cmml">)</mo></mrow></mrow><mo id="S7.I11.i3.I0.i2.p1.8.m8.3.3.6" xref="S7.I11.i3.I0.i2.p1.8.m8.3.3.6.cmml">≤</mo><msup id="S7.I11.i3.I0.i2.p1.8.m8.3.3.2" xref="S7.I11.i3.I0.i2.p1.8.m8.3.3.2.cmml"><mrow id="S7.I11.i3.I0.i2.p1.8.m8.3.3.2.1.1" xref="S7.I11.i3.I0.i2.p1.8.m8.3.3.2.1.1.1.cmml"><mo id="S7.I11.i3.I0.i2.p1.8.m8.3.3.2.1.1.2" stretchy="false" xref="S7.I11.i3.I0.i2.p1.8.m8.3.3.2.1.1.1.cmml">(</mo><mrow id="S7.I11.i3.I0.i2.p1.8.m8.3.3.2.1.1.1" xref="S7.I11.i3.I0.i2.p1.8.m8.3.3.2.1.1.1.cmml"><mn id="S7.I11.i3.I0.i2.p1.8.m8.3.3.2.1.1.1.2" xref="S7.I11.i3.I0.i2.p1.8.m8.3.3.2.1.1.1.2.cmml">1</mn><mo id="S7.I11.i3.I0.i2.p1.8.m8.3.3.2.1.1.1.1" xref="S7.I11.i3.I0.i2.p1.8.m8.3.3.2.1.1.1.1.cmml">+</mo><mfrac id="S7.I11.i3.I0.i2.p1.8.m8.3.3.2.1.1.1.3" xref="S7.I11.i3.I0.i2.p1.8.m8.3.3.2.1.1.1.3.cmml"><mi id="S7.I11.i3.I0.i2.p1.8.m8.3.3.2.1.1.1.3.2" 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id="S7.I11.i3.I0.i2.p1.8.m8.2.2.1.1.1.1.3.3.cmml" type="integer" xref="S7.I11.i3.I0.i2.p1.8.m8.2.2.1.1.1.1.3.3">2</cn></apply></apply><ci id="S7.I11.i3.I0.i2.p1.8.m8.2.2.1.3.cmml" xref="S7.I11.i3.I0.i2.p1.8.m8.2.2.1.3">ℎ</ci></apply><apply id="S7.I11.i3.I0.i2.p1.8.m8.3.3.5.cmml" xref="S7.I11.i3.I0.i2.p1.8.m8.3.3.5"><times id="S7.I11.i3.I0.i2.p1.8.m8.3.3.5.1.cmml" xref="S7.I11.i3.I0.i2.p1.8.m8.3.3.5.1"></times><ci id="S7.I11.i3.I0.i2.p1.8.m8.3.3.5.2a.cmml" xref="S7.I11.i3.I0.i2.p1.8.m8.3.3.5.2"><mtext class="ltx_mathvariant_italic" id="S7.I11.i3.I0.i2.p1.8.m8.3.3.5.2.cmml" xref="S7.I11.i3.I0.i2.p1.8.m8.3.3.5.2">g</mtext></ci><ci id="S7.I11.i3.I0.i2.p1.8.m8.1.1.cmml" xref="S7.I11.i3.I0.i2.p1.8.m8.1.1">𝑎</ci></apply></apply><apply id="S7.I11.i3.I0.i2.p1.8.m8.3.3c.cmml" xref="S7.I11.i3.I0.i2.p1.8.m8.3.3"><leq id="S7.I11.i3.I0.i2.p1.8.m8.3.3.6.cmml" xref="S7.I11.i3.I0.i2.p1.8.m8.3.3.6"></leq><share href="https://arxiv.org/html/2411.12694v2#S7.I11.i3.I0.i2.p1.8.m8.3.3.5.cmml" id="S7.I11.i3.I0.i2.p1.8.m8.3.3d.cmml" xref="S7.I11.i3.I0.i2.p1.8.m8.3.3"></share><apply id="S7.I11.i3.I0.i2.p1.8.m8.3.3.2.cmml" xref="S7.I11.i3.I0.i2.p1.8.m8.3.3.2"><csymbol cd="ambiguous" id="S7.I11.i3.I0.i2.p1.8.m8.3.3.2.2.cmml" xref="S7.I11.i3.I0.i2.p1.8.m8.3.3.2">superscript</csymbol><apply id="S7.I11.i3.I0.i2.p1.8.m8.3.3.2.1.1.1.cmml" xref="S7.I11.i3.I0.i2.p1.8.m8.3.3.2.1.1"><plus id="S7.I11.i3.I0.i2.p1.8.m8.3.3.2.1.1.1.1.cmml" xref="S7.I11.i3.I0.i2.p1.8.m8.3.3.2.1.1.1.1"></plus><cn id="S7.I11.i3.I0.i2.p1.8.m8.3.3.2.1.1.1.2.cmml" type="integer" xref="S7.I11.i3.I0.i2.p1.8.m8.3.3.2.1.1.1.2">1</cn><apply id="S7.I11.i3.I0.i2.p1.8.m8.3.3.2.1.1.1.3.cmml" xref="S7.I11.i3.I0.i2.p1.8.m8.3.3.2.1.1.1.3"><divide id="S7.I11.i3.I0.i2.p1.8.m8.3.3.2.1.1.1.3.1.cmml" xref="S7.I11.i3.I0.i2.p1.8.m8.3.3.2.1.1.1.3"></divide><ci id="S7.I11.i3.I0.i2.p1.8.m8.3.3.2.1.1.1.3.2.cmml" xref="S7.I11.i3.I0.i2.p1.8.m8.3.3.2.1.1.1.3.2">𝜂</ci><cn id="S7.I11.i3.I0.i2.p1.8.m8.3.3.2.1.1.1.3.3.cmml" type="integer" xref="S7.I11.i3.I0.i2.p1.8.m8.3.3.2.1.1.1.3.3">2</cn></apply></apply><apply id="S7.I11.i3.I0.i2.p1.8.m8.3.3.2.3.cmml" xref="S7.I11.i3.I0.i2.p1.8.m8.3.3.2.3"><plus id="S7.I11.i3.I0.i2.p1.8.m8.3.3.2.3.1.cmml" xref="S7.I11.i3.I0.i2.p1.8.m8.3.3.2.3.1"></plus><ci id="S7.I11.i3.I0.i2.p1.8.m8.3.3.2.3.2.cmml" xref="S7.I11.i3.I0.i2.p1.8.m8.3.3.2.3.2">ℎ</ci><cn id="S7.I11.i3.I0.i2.p1.8.m8.3.3.2.3.3.cmml" type="integer" xref="S7.I11.i3.I0.i2.p1.8.m8.3.3.2.3.3">1</cn></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.I11.i3.I0.i2.p1.8.m8.3c">(1+\frac{\eta}{2})^{h}\leq\textsl{g}(a)\leq(1+\frac{\eta}{2})^{h+1}</annotation><annotation encoding="application/x-llamapun" id="S7.I11.i3.I0.i2.p1.8.m8.3d">( 1 + divide start_ARG italic_η end_ARG start_ARG 2 end_ARG ) start_POSTSUPERSCRIPT italic_h end_POSTSUPERSCRIPT ≤ g ( italic_a ) ≤ ( 1 + divide start_ARG italic_η end_ARG start_ARG 2 end_ARG ) start_POSTSUPERSCRIPT italic_h + 1 end_POSTSUPERSCRIPT</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S7.I11.i3.I0.i2.p1.18.9">. Thus, </span><math alttext="\delta_{t}(a)" class="ltx_Math" display="inline" id="S7.I11.i3.I0.i2.p1.9.m9.1"><semantics id="S7.I11.i3.I0.i2.p1.9.m9.1a"><mrow id="S7.I11.i3.I0.i2.p1.9.m9.1.2" xref="S7.I11.i3.I0.i2.p1.9.m9.1.2.cmml"><msub id="S7.I11.i3.I0.i2.p1.9.m9.1.2.2" xref="S7.I11.i3.I0.i2.p1.9.m9.1.2.2.cmml"><mi id="S7.I11.i3.I0.i2.p1.9.m9.1.2.2.2" xref="S7.I11.i3.I0.i2.p1.9.m9.1.2.2.2.cmml">δ</mi><mi id="S7.I11.i3.I0.i2.p1.9.m9.1.2.2.3" xref="S7.I11.i3.I0.i2.p1.9.m9.1.2.2.3.cmml">t</mi></msub><mo id="S7.I11.i3.I0.i2.p1.9.m9.1.2.1" xref="S7.I11.i3.I0.i2.p1.9.m9.1.2.1.cmml">⁢</mo><mrow id="S7.I11.i3.I0.i2.p1.9.m9.1.2.3.2" xref="S7.I11.i3.I0.i2.p1.9.m9.1.2.cmml"><mo id="S7.I11.i3.I0.i2.p1.9.m9.1.2.3.2.1" stretchy="false" xref="S7.I11.i3.I0.i2.p1.9.m9.1.2.cmml">(</mo><mi id="S7.I11.i3.I0.i2.p1.9.m9.1.1" xref="S7.I11.i3.I0.i2.p1.9.m9.1.1.cmml">a</mi><mo id="S7.I11.i3.I0.i2.p1.9.m9.1.2.3.2.2" stretchy="false" xref="S7.I11.i3.I0.i2.p1.9.m9.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S7.I11.i3.I0.i2.p1.9.m9.1b"><apply id="S7.I11.i3.I0.i2.p1.9.m9.1.2.cmml" xref="S7.I11.i3.I0.i2.p1.9.m9.1.2"><times id="S7.I11.i3.I0.i2.p1.9.m9.1.2.1.cmml" xref="S7.I11.i3.I0.i2.p1.9.m9.1.2.1"></times><apply id="S7.I11.i3.I0.i2.p1.9.m9.1.2.2.cmml" xref="S7.I11.i3.I0.i2.p1.9.m9.1.2.2"><csymbol cd="ambiguous" id="S7.I11.i3.I0.i2.p1.9.m9.1.2.2.1.cmml" xref="S7.I11.i3.I0.i2.p1.9.m9.1.2.2">subscript</csymbol><ci id="S7.I11.i3.I0.i2.p1.9.m9.1.2.2.2.cmml" xref="S7.I11.i3.I0.i2.p1.9.m9.1.2.2.2">𝛿</ci><ci id="S7.I11.i3.I0.i2.p1.9.m9.1.2.2.3.cmml" xref="S7.I11.i3.I0.i2.p1.9.m9.1.2.2.3">𝑡</ci></apply><ci id="S7.I11.i3.I0.i2.p1.9.m9.1.1.cmml" xref="S7.I11.i3.I0.i2.p1.9.m9.1.1">𝑎</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.I11.i3.I0.i2.p1.9.m9.1c">\delta_{t}(a)</annotation><annotation encoding="application/x-llamapun" id="S7.I11.i3.I0.i2.p1.9.m9.1d">italic_δ start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT ( italic_a )</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S7.I11.i3.I0.i2.p1.18.10"> is at least </span><math alttext="\Delta_{h}" class="ltx_Math" display="inline" id="S7.I11.i3.I0.i2.p1.10.m10.1"><semantics id="S7.I11.i3.I0.i2.p1.10.m10.1a"><msub id="S7.I11.i3.I0.i2.p1.10.m10.1.1" xref="S7.I11.i3.I0.i2.p1.10.m10.1.1.cmml"><mi id="S7.I11.i3.I0.i2.p1.10.m10.1.1.2" mathvariant="normal" xref="S7.I11.i3.I0.i2.p1.10.m10.1.1.2.cmml">Δ</mi><mi id="S7.I11.i3.I0.i2.p1.10.m10.1.1.3" xref="S7.I11.i3.I0.i2.p1.10.m10.1.1.3.cmml">h</mi></msub><annotation-xml encoding="MathML-Content" id="S7.I11.i3.I0.i2.p1.10.m10.1b"><apply id="S7.I11.i3.I0.i2.p1.10.m10.1.1.cmml" xref="S7.I11.i3.I0.i2.p1.10.m10.1.1"><csymbol cd="ambiguous" id="S7.I11.i3.I0.i2.p1.10.m10.1.1.1.cmml" xref="S7.I11.i3.I0.i2.p1.10.m10.1.1">subscript</csymbol><ci id="S7.I11.i3.I0.i2.p1.10.m10.1.1.2.cmml" xref="S7.I11.i3.I0.i2.p1.10.m10.1.1.2">Δ</ci><ci id="S7.I11.i3.I0.i2.p1.10.m10.1.1.3.cmml" xref="S7.I11.i3.I0.i2.p1.10.m10.1.1.3">ℎ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.I11.i3.I0.i2.p1.10.m10.1c">\Delta_{h}</annotation><annotation encoding="application/x-llamapun" id="S7.I11.i3.I0.i2.p1.10.m10.1d">roman_Δ start_POSTSUBSCRIPT italic_h end_POSTSUBSCRIPT</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S7.I11.i3.I0.i2.p1.18.11"> and fewer than </span><math alttext="4\Delta_{h}" class="ltx_Math" display="inline" id="S7.I11.i3.I0.i2.p1.11.m11.1"><semantics id="S7.I11.i3.I0.i2.p1.11.m11.1a"><mrow id="S7.I11.i3.I0.i2.p1.11.m11.1.1" xref="S7.I11.i3.I0.i2.p1.11.m11.1.1.cmml"><mn id="S7.I11.i3.I0.i2.p1.11.m11.1.1.2" xref="S7.I11.i3.I0.i2.p1.11.m11.1.1.2.cmml">4</mn><mo id="S7.I11.i3.I0.i2.p1.11.m11.1.1.1" xref="S7.I11.i3.I0.i2.p1.11.m11.1.1.1.cmml">⁢</mo><msub id="S7.I11.i3.I0.i2.p1.11.m11.1.1.3" xref="S7.I11.i3.I0.i2.p1.11.m11.1.1.3.cmml"><mi id="S7.I11.i3.I0.i2.p1.11.m11.1.1.3.2" mathvariant="normal" 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id="S7.I11.i3.I0.i2.p1.11.m11.1c">4\Delta_{h}</annotation><annotation encoding="application/x-llamapun" id="S7.I11.i3.I0.i2.p1.11.m11.1d">4 roman_Δ start_POSTSUBSCRIPT italic_h end_POSTSUBSCRIPT</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S7.I11.i3.I0.i2.p1.18.12">. If follows that if </span><math alttext="a" class="ltx_Math" display="inline" id="S7.I11.i3.I0.i2.p1.12.m12.1"><semantics id="S7.I11.i3.I0.i2.p1.12.m12.1a"><mi id="S7.I11.i3.I0.i2.p1.12.m12.1.1" xref="S7.I11.i3.I0.i2.p1.12.m12.1.1.cmml">a</mi><annotation-xml encoding="MathML-Content" id="S7.I11.i3.I0.i2.p1.12.m12.1b"><ci id="S7.I11.i3.I0.i2.p1.12.m12.1.1.cmml" xref="S7.I11.i3.I0.i2.p1.12.m12.1.1">𝑎</ci></annotation-xml><annotation encoding="application/x-tex" id="S7.I11.i3.I0.i2.p1.12.m12.1c">a</annotation><annotation encoding="application/x-llamapun" id="S7.I11.i3.I0.i2.p1.12.m12.1d">italic_a</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S7.I11.i3.I0.i2.p1.18.13"> was not satisfied at the end of iteration </span><math alttext="t" class="ltx_Math" display="inline" id="S7.I11.i3.I0.i2.p1.13.m13.1"><semantics id="S7.I11.i3.I0.i2.p1.13.m13.1a"><mi id="S7.I11.i3.I0.i2.p1.13.m13.1.1" xref="S7.I11.i3.I0.i2.p1.13.m13.1.1.cmml">t</mi><annotation-xml encoding="MathML-Content" id="S7.I11.i3.I0.i2.p1.13.m13.1b"><ci id="S7.I11.i3.I0.i2.p1.13.m13.1.1.cmml" xref="S7.I11.i3.I0.i2.p1.13.m13.1.1">𝑡</ci></annotation-xml><annotation encoding="application/x-tex" id="S7.I11.i3.I0.i2.p1.13.m13.1c">t</annotation><annotation encoding="application/x-llamapun" id="S7.I11.i3.I0.i2.p1.13.m13.1d">italic_t</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S7.I11.i3.I0.i2.p1.18.14">, at least </span><math alttext="\frac{1}{8}" class="ltx_Math" display="inline" id="S7.I11.i3.I0.i2.p1.14.m14.1"><semantics id="S7.I11.i3.I0.i2.p1.14.m14.1a"><mfrac id="S7.I11.i3.I0.i2.p1.14.m14.1.1" xref="S7.I11.i3.I0.i2.p1.14.m14.1.1.cmml"><mn id="S7.I11.i3.I0.i2.p1.14.m14.1.1.2" xref="S7.I11.i3.I0.i2.p1.14.m14.1.1.2.cmml">1</mn><mn id="S7.I11.i3.I0.i2.p1.14.m14.1.1.3" xref="S7.I11.i3.I0.i2.p1.14.m14.1.1.3.cmml">8</mn></mfrac><annotation-xml encoding="MathML-Content" id="S7.I11.i3.I0.i2.p1.14.m14.1b"><apply id="S7.I11.i3.I0.i2.p1.14.m14.1.1.cmml" xref="S7.I11.i3.I0.i2.p1.14.m14.1.1"><divide id="S7.I11.i3.I0.i2.p1.14.m14.1.1.1.cmml" xref="S7.I11.i3.I0.i2.p1.14.m14.1.1"></divide><cn id="S7.I11.i3.I0.i2.p1.14.m14.1.1.2.cmml" type="integer" xref="S7.I11.i3.I0.i2.p1.14.m14.1.1.2">1</cn><cn id="S7.I11.i3.I0.i2.p1.14.m14.1.1.3.cmml" type="integer" xref="S7.I11.i3.I0.i2.p1.14.m14.1.1.3">8</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.I11.i3.I0.i2.p1.14.m14.1c">\frac{1}{8}</annotation><annotation encoding="application/x-llamapun" id="S7.I11.i3.I0.i2.p1.14.m14.1d">divide start_ARG 1 end_ARG start_ARG 8 end_ARG</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S7.I11.i3.I0.i2.p1.18.15">’th of the edges </span><math alttext="\overline{ac}\in E_{t}(a)" class="ltx_Math" display="inline" id="S7.I11.i3.I0.i2.p1.15.m15.1"><semantics id="S7.I11.i3.I0.i2.p1.15.m15.1a"><mrow id="S7.I11.i3.I0.i2.p1.15.m15.1.2" xref="S7.I11.i3.I0.i2.p1.15.m15.1.2.cmml"><mover accent="true" 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xref="S7.I11.i3.I0.i2.p1.15.m15.1.2.2.1">¯</ci><apply id="S7.I11.i3.I0.i2.p1.15.m15.1.2.2.2.cmml" xref="S7.I11.i3.I0.i2.p1.15.m15.1.2.2.2"><times id="S7.I11.i3.I0.i2.p1.15.m15.1.2.2.2.1.cmml" xref="S7.I11.i3.I0.i2.p1.15.m15.1.2.2.2.1"></times><ci id="S7.I11.i3.I0.i2.p1.15.m15.1.2.2.2.2.cmml" xref="S7.I11.i3.I0.i2.p1.15.m15.1.2.2.2.2">𝑎</ci><ci id="S7.I11.i3.I0.i2.p1.15.m15.1.2.2.2.3.cmml" xref="S7.I11.i3.I0.i2.p1.15.m15.1.2.2.2.3">𝑐</ci></apply></apply><apply id="S7.I11.i3.I0.i2.p1.15.m15.1.2.3.cmml" xref="S7.I11.i3.I0.i2.p1.15.m15.1.2.3"><times id="S7.I11.i3.I0.i2.p1.15.m15.1.2.3.1.cmml" xref="S7.I11.i3.I0.i2.p1.15.m15.1.2.3.1"></times><apply id="S7.I11.i3.I0.i2.p1.15.m15.1.2.3.2.cmml" xref="S7.I11.i3.I0.i2.p1.15.m15.1.2.3.2"><csymbol cd="ambiguous" id="S7.I11.i3.I0.i2.p1.15.m15.1.2.3.2.1.cmml" xref="S7.I11.i3.I0.i2.p1.15.m15.1.2.3.2">subscript</csymbol><ci id="S7.I11.i3.I0.i2.p1.15.m15.1.2.3.2.2.cmml" xref="S7.I11.i3.I0.i2.p1.15.m15.1.2.3.2.2">𝐸</ci><ci id="S7.I11.i3.I0.i2.p1.15.m15.1.2.3.2.3.cmml" xref="S7.I11.i3.I0.i2.p1.15.m15.1.2.3.2.3">𝑡</ci></apply><ci id="S7.I11.i3.I0.i2.p1.15.m15.1.1.cmml" xref="S7.I11.i3.I0.i2.p1.15.m15.1.1">𝑎</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.I11.i3.I0.i2.p1.15.m15.1c">\overline{ac}\in E_{t}(a)</annotation><annotation encoding="application/x-llamapun" id="S7.I11.i3.I0.i2.p1.15.m15.1d">over¯ start_ARG italic_a italic_c end_ARG ∈ italic_E start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT ( italic_a )</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S7.I11.i3.I0.i2.p1.18.16"> has not been decreased by </span><math alttext="\frac{\delta_{t}(a)}{|E_{t}(a)|}" class="ltx_Math" display="inline" id="S7.I11.i3.I0.i2.p1.16.m16.3"><semantics id="S7.I11.i3.I0.i2.p1.16.m16.3a"><mfrac id="S7.I11.i3.I0.i2.p1.16.m16.3.3" xref="S7.I11.i3.I0.i2.p1.16.m16.3.3.cmml"><mrow id="S7.I11.i3.I0.i2.p1.16.m16.1.1.1" xref="S7.I11.i3.I0.i2.p1.16.m16.1.1.1.cmml"><msub 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xref="S7.I11.i3.I0.i2.p1.16.m16.3.3.3.2.1.2.3">𝑡</ci></apply><ci id="S7.I11.i3.I0.i2.p1.16.m16.2.2.2.1.cmml" xref="S7.I11.i3.I0.i2.p1.16.m16.2.2.2.1">𝑎</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.I11.i3.I0.i2.p1.16.m16.3c">\frac{\delta_{t}(a)}{|E_{t}(a)|}</annotation><annotation encoding="application/x-llamapun" id="S7.I11.i3.I0.i2.p1.16.m16.3d">divide start_ARG italic_δ start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT ( italic_a ) end_ARG start_ARG | italic_E start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT ( italic_a ) | end_ARG</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S7.I11.i3.I0.i2.p1.18.17">. To all these edges </span><math alttext="\overline{ac}" class="ltx_Math" display="inline" id="S7.I11.i3.I0.i2.p1.17.m17.1"><semantics id="S7.I11.i3.I0.i2.p1.17.m17.1a"><mover accent="true" id="S7.I11.i3.I0.i2.p1.17.m17.1.1" xref="S7.I11.i3.I0.i2.p1.17.m17.1.1.cmml"><mrow id="S7.I11.i3.I0.i2.p1.17.m17.1.1.2" xref="S7.I11.i3.I0.i2.p1.17.m17.1.1.2.cmml"><mi id="S7.I11.i3.I0.i2.p1.17.m17.1.1.2.2" xref="S7.I11.i3.I0.i2.p1.17.m17.1.1.2.2.cmml">a</mi><mo id="S7.I11.i3.I0.i2.p1.17.m17.1.1.2.1" xref="S7.I11.i3.I0.i2.p1.17.m17.1.1.2.1.cmml">⁢</mo><mi id="S7.I11.i3.I0.i2.p1.17.m17.1.1.2.3" xref="S7.I11.i3.I0.i2.p1.17.m17.1.1.2.3.cmml">c</mi></mrow><mo id="S7.I11.i3.I0.i2.p1.17.m17.1.1.1" xref="S7.I11.i3.I0.i2.p1.17.m17.1.1.1.cmml">¯</mo></mover><annotation-xml encoding="MathML-Content" id="S7.I11.i3.I0.i2.p1.17.m17.1b"><apply id="S7.I11.i3.I0.i2.p1.17.m17.1.1.cmml" xref="S7.I11.i3.I0.i2.p1.17.m17.1.1"><ci id="S7.I11.i3.I0.i2.p1.17.m17.1.1.1.cmml" xref="S7.I11.i3.I0.i2.p1.17.m17.1.1.1">¯</ci><apply id="S7.I11.i3.I0.i2.p1.17.m17.1.1.2.cmml" xref="S7.I11.i3.I0.i2.p1.17.m17.1.1.2"><times id="S7.I11.i3.I0.i2.p1.17.m17.1.1.2.1.cmml" xref="S7.I11.i3.I0.i2.p1.17.m17.1.1.2.1"></times><ci id="S7.I11.i3.I0.i2.p1.17.m17.1.1.2.2.cmml" xref="S7.I11.i3.I0.i2.p1.17.m17.1.1.2.2">𝑎</ci><ci id="S7.I11.i3.I0.i2.p1.17.m17.1.1.2.3.cmml" xref="S7.I11.i3.I0.i2.p1.17.m17.1.1.2.3">𝑐</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.I11.i3.I0.i2.p1.17.m17.1c">\overline{ac}</annotation><annotation encoding="application/x-llamapun" id="S7.I11.i3.I0.i2.p1.17.m17.1d">over¯ start_ARG italic_a italic_c end_ARG</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S7.I11.i3.I0.i2.p1.18.18">, case </span><math alttext="2" class="ltx_Math" display="inline" id="S7.I11.i3.I0.i2.p1.18.m18.1"><semantics id="S7.I11.i3.I0.i2.p1.18.m18.1a"><mn id="S7.I11.i3.I0.i2.p1.18.m18.1.1" xref="S7.I11.i3.I0.i2.p1.18.m18.1.1.cmml">2</mn><annotation-xml encoding="MathML-Content" id="S7.I11.i3.I0.i2.p1.18.m18.1b"><cn id="S7.I11.i3.I0.i2.p1.18.m18.1.1.cmml" type="integer" xref="S7.I11.i3.I0.i2.p1.18.m18.1.1">2</cn></annotation-xml><annotation encoding="application/x-tex" id="S7.I11.i3.I0.i2.p1.18.m18.1c">2</annotation><annotation encoding="application/x-llamapun" id="S7.I11.i3.I0.i2.p1.18.m18.1d">2</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S7.I11.i3.I0.i2.p1.18.19"> applies.</span></p> </div> </li> </ul> </div> </li> </ol> </div> <div class="ltx_para" id="S7.Thmtheorem21.p3"> <p class="ltx_p" id="S7.Thmtheorem21.p3.1"><span class="ltx_text ltx_font_italic" id="S7.Thmtheorem21.p3.1.1">Since satisfied edges are no longer violating, it follows that <math alttext="|E_{t+1}|\leq\frac{7}{8}|E_{t}|" class="ltx_Math" display="inline" id="S7.Thmtheorem21.p3.1.1.m1.2"><semantics id="S7.Thmtheorem21.p3.1.1.m1.2a"><mrow id="S7.Thmtheorem21.p3.1.1.m1.2.2" 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type="integer" xref="S7.Thmtheorem21.p3.1.1.m1.2.2.2.3.2">7</cn><cn id="S7.Thmtheorem21.p3.1.1.m1.2.2.2.3.3.cmml" type="integer" xref="S7.Thmtheorem21.p3.1.1.m1.2.2.2.3.3">8</cn></apply><apply id="S7.Thmtheorem21.p3.1.1.m1.2.2.2.1.2.cmml" xref="S7.Thmtheorem21.p3.1.1.m1.2.2.2.1.1"><abs id="S7.Thmtheorem21.p3.1.1.m1.2.2.2.1.2.1.cmml" xref="S7.Thmtheorem21.p3.1.1.m1.2.2.2.1.1.2"></abs><apply id="S7.Thmtheorem21.p3.1.1.m1.2.2.2.1.1.1.cmml" xref="S7.Thmtheorem21.p3.1.1.m1.2.2.2.1.1.1"><csymbol cd="ambiguous" id="S7.Thmtheorem21.p3.1.1.m1.2.2.2.1.1.1.1.cmml" xref="S7.Thmtheorem21.p3.1.1.m1.2.2.2.1.1.1">subscript</csymbol><ci id="S7.Thmtheorem21.p3.1.1.m1.2.2.2.1.1.1.2.cmml" xref="S7.Thmtheorem21.p3.1.1.m1.2.2.2.1.1.1.2">𝐸</ci><ci id="S7.Thmtheorem21.p3.1.1.m1.2.2.2.1.1.1.3.cmml" xref="S7.Thmtheorem21.p3.1.1.m1.2.2.2.1.1.1.3">𝑡</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.Thmtheorem21.p3.1.1.m1.2c">|E_{t+1}|\leq\frac{7}{8}|E_{t}|</annotation><annotation encoding="application/x-llamapun" id="S7.Thmtheorem21.p3.1.1.m1.2d">| italic_E start_POSTSUBSCRIPT italic_t + 1 end_POSTSUBSCRIPT | ≤ divide start_ARG 7 end_ARG start_ARG 8 end_ARG | italic_E start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT |</annotation></semantics></math>.</span></p> </div> </div> <div class="ltx_theorem ltx_theorem_corollary" id="S7.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="S7.Thmtheorem22.1.1.1">Corollary 7.22</span></span><span class="ltx_text ltx_font_bold" id="S7.Thmtheorem22.2.2">.</span> </h6> <div class="ltx_para" id="S7.Thmtheorem22.p1"> <p class="ltx_p" id="S7.Thmtheorem22.p1.4"><span class="ltx_text ltx_font_italic" id="S7.Thmtheorem22.p1.4.4">At the start of <math alttext="(h:m:0)" class="ltx_Math" display="inline" id="S7.Thmtheorem22.p1.1.1.m1.1"><semantics id="S7.Thmtheorem22.p1.1.1.m1.1a"><mrow id="S7.Thmtheorem22.p1.1.1.m1.1.1.1" xref="S7.Thmtheorem22.p1.1.1.m1.1.1.1.1.cmml"><mo id="S7.Thmtheorem22.p1.1.1.m1.1.1.1.2" stretchy="false" xref="S7.Thmtheorem22.p1.1.1.m1.1.1.1.1.cmml">(</mo><mrow id="S7.Thmtheorem22.p1.1.1.m1.1.1.1.1" xref="S7.Thmtheorem22.p1.1.1.m1.1.1.1.1.cmml"><mi id="S7.Thmtheorem22.p1.1.1.m1.1.1.1.1.2" xref="S7.Thmtheorem22.p1.1.1.m1.1.1.1.1.2.cmml">h</mi><mo id="S7.Thmtheorem22.p1.1.1.m1.1.1.1.1.3" lspace="0.278em" rspace="0.278em" xref="S7.Thmtheorem22.p1.1.1.m1.1.1.1.1.3.cmml">:</mo><mi id="S7.Thmtheorem22.p1.1.1.m1.1.1.1.1.4" xref="S7.Thmtheorem22.p1.1.1.m1.1.1.1.1.4.cmml">m</mi><mo id="S7.Thmtheorem22.p1.1.1.m1.1.1.1.1.5" lspace="0.278em" rspace="0.278em" xref="S7.Thmtheorem22.p1.1.1.m1.1.1.1.1.5.cmml">:</mo><mn id="S7.Thmtheorem22.p1.1.1.m1.1.1.1.1.6" xref="S7.Thmtheorem22.p1.1.1.m1.1.1.1.1.6.cmml">0</mn></mrow><mo id="S7.Thmtheorem22.p1.1.1.m1.1.1.1.3" stretchy="false" xref="S7.Thmtheorem22.p1.1.1.m1.1.1.1.1.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S7.Thmtheorem22.p1.1.1.m1.1b"><apply id="S7.Thmtheorem22.p1.1.1.m1.1.1.1.1.cmml" xref="S7.Thmtheorem22.p1.1.1.m1.1.1.1"><and id="S7.Thmtheorem22.p1.1.1.m1.1.1.1.1a.cmml" xref="S7.Thmtheorem22.p1.1.1.m1.1.1.1"></and><apply id="S7.Thmtheorem22.p1.1.1.m1.1.1.1.1b.cmml" xref="S7.Thmtheorem22.p1.1.1.m1.1.1.1"><ci id="S7.Thmtheorem22.p1.1.1.m1.1.1.1.1.3.cmml" xref="S7.Thmtheorem22.p1.1.1.m1.1.1.1.1.3">:</ci><ci id="S7.Thmtheorem22.p1.1.1.m1.1.1.1.1.2.cmml" xref="S7.Thmtheorem22.p1.1.1.m1.1.1.1.1.2">ℎ</ci><ci id="S7.Thmtheorem22.p1.1.1.m1.1.1.1.1.4.cmml" xref="S7.Thmtheorem22.p1.1.1.m1.1.1.1.1.4">𝑚</ci></apply><apply id="S7.Thmtheorem22.p1.1.1.m1.1.1.1.1c.cmml" xref="S7.Thmtheorem22.p1.1.1.m1.1.1.1"><ci id="S7.Thmtheorem22.p1.1.1.m1.1.1.1.1.5.cmml" xref="S7.Thmtheorem22.p1.1.1.m1.1.1.1.1.5">:</ci><share href="https://arxiv.org/html/2411.12694v2#S7.Thmtheorem22.p1.1.1.m1.1.1.1.1.4.cmml" id="S7.Thmtheorem22.p1.1.1.m1.1.1.1.1d.cmml" xref="S7.Thmtheorem22.p1.1.1.m1.1.1.1"></share><cn id="S7.Thmtheorem22.p1.1.1.m1.1.1.1.1.6.cmml" type="integer" xref="S7.Thmtheorem22.p1.1.1.m1.1.1.1.1.6">0</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.Thmtheorem22.p1.1.1.m1.1c">(h:m:0)</annotation><annotation encoding="application/x-llamapun" id="S7.Thmtheorem22.p1.1.1.m1.1d">( italic_h : italic_m : 0 )</annotation></semantics></math> with <math alttext="m" class="ltx_Math" display="inline" id="S7.Thmtheorem22.p1.2.2.m2.1"><semantics id="S7.Thmtheorem22.p1.2.2.m2.1a"><mi id="S7.Thmtheorem22.p1.2.2.m2.1.1" xref="S7.Thmtheorem22.p1.2.2.m2.1.1.cmml">m</mi><annotation-xml encoding="MathML-Content" id="S7.Thmtheorem22.p1.2.2.m2.1b"><ci id="S7.Thmtheorem22.p1.2.2.m2.1.1.cmml" xref="S7.Thmtheorem22.p1.2.2.m2.1.1">𝑚</ci></annotation-xml><annotation encoding="application/x-tex" id="S7.Thmtheorem22.p1.2.2.m2.1c">m</annotation><annotation encoding="application/x-llamapun" id="S7.Thmtheorem22.p1.2.2.m2.1d">italic_m</annotation></semantics></math> odd, there are no violating edges from levels <math alttext="k>h+1" class="ltx_Math" display="inline" id="S7.Thmtheorem22.p1.3.3.m3.1"><semantics id="S7.Thmtheorem22.p1.3.3.m3.1a"><mrow id="S7.Thmtheorem22.p1.3.3.m3.1.1" xref="S7.Thmtheorem22.p1.3.3.m3.1.1.cmml"><mi id="S7.Thmtheorem22.p1.3.3.m3.1.1.2" xref="S7.Thmtheorem22.p1.3.3.m3.1.1.2.cmml">k</mi><mo id="S7.Thmtheorem22.p1.3.3.m3.1.1.1" xref="S7.Thmtheorem22.p1.3.3.m3.1.1.1.cmml">></mo><mrow id="S7.Thmtheorem22.p1.3.3.m3.1.1.3" xref="S7.Thmtheorem22.p1.3.3.m3.1.1.3.cmml"><mi id="S7.Thmtheorem22.p1.3.3.m3.1.1.3.2" xref="S7.Thmtheorem22.p1.3.3.m3.1.1.3.2.cmml">h</mi><mo id="S7.Thmtheorem22.p1.3.3.m3.1.1.3.1" xref="S7.Thmtheorem22.p1.3.3.m3.1.1.3.1.cmml">+</mo><mn id="S7.Thmtheorem22.p1.3.3.m3.1.1.3.3" xref="S7.Thmtheorem22.p1.3.3.m3.1.1.3.3.cmml">1</mn></mrow></mrow><annotation-xml encoding="MathML-Content" id="S7.Thmtheorem22.p1.3.3.m3.1b"><apply id="S7.Thmtheorem22.p1.3.3.m3.1.1.cmml" xref="S7.Thmtheorem22.p1.3.3.m3.1.1"><gt id="S7.Thmtheorem22.p1.3.3.m3.1.1.1.cmml" xref="S7.Thmtheorem22.p1.3.3.m3.1.1.1"></gt><ci id="S7.Thmtheorem22.p1.3.3.m3.1.1.2.cmml" xref="S7.Thmtheorem22.p1.3.3.m3.1.1.2">𝑘</ci><apply id="S7.Thmtheorem22.p1.3.3.m3.1.1.3.cmml" xref="S7.Thmtheorem22.p1.3.3.m3.1.1.3"><plus id="S7.Thmtheorem22.p1.3.3.m3.1.1.3.1.cmml" xref="S7.Thmtheorem22.p1.3.3.m3.1.1.3.1"></plus><ci id="S7.Thmtheorem22.p1.3.3.m3.1.1.3.2.cmml" xref="S7.Thmtheorem22.p1.3.3.m3.1.1.3.2">ℎ</ci><cn id="S7.Thmtheorem22.p1.3.3.m3.1.1.3.3.cmml" type="integer" xref="S7.Thmtheorem22.p1.3.3.m3.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.Thmtheorem22.p1.3.3.m3.1c">k>h+1</annotation><annotation encoding="application/x-llamapun" id="S7.Thmtheorem22.p1.3.3.m3.1d">italic_k > italic_h + 1</annotation></semantics></math>, or from level <math alttext="h" class="ltx_Math" display="inline" id="S7.Thmtheorem22.p1.4.4.m4.1"><semantics id="S7.Thmtheorem22.p1.4.4.m4.1a"><mi id="S7.Thmtheorem22.p1.4.4.m4.1.1" xref="S7.Thmtheorem22.p1.4.4.m4.1.1.cmml">h</mi><annotation-xml encoding="MathML-Content" id="S7.Thmtheorem22.p1.4.4.m4.1b"><ci id="S7.Thmtheorem22.p1.4.4.m4.1.1.cmml" xref="S7.Thmtheorem22.p1.4.4.m4.1.1">ℎ</ci></annotation-xml><annotation encoding="application/x-tex" id="S7.Thmtheorem22.p1.4.4.m4.1c">h</annotation><annotation encoding="application/x-llamapun" id="S7.Thmtheorem22.p1.4.4.m4.1d">italic_h</annotation></semantics></math>.</span></p> </div> </div> <div class="ltx_theorem ltx_theorem_proof" id="S7.Thmtheorem23"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="S7.Thmtheorem23.1.1.1">Proof 7.23</span></span><span class="ltx_text ltx_font_bold" id="S7.Thmtheorem23.2.2">.</span> </h6> <div class="ltx_para" id="S7.Thmtheorem23.p1"> <p class="ltx_p" id="S7.Thmtheorem23.p1.8"><span class="ltx_text ltx_font_italic" id="S7.Thmtheorem23.p1.8.8">The <em class="ltx_emph ltx_font_upright" id="S7.Thmtheorem23.p1.8.8.1">minute</em> is a for-loop of <math alttext="2\lceil\log_{8/7}n\rceil+1" class="ltx_Math" display="inline" id="S7.Thmtheorem23.p1.1.1.m1.1"><semantics id="S7.Thmtheorem23.p1.1.1.m1.1a"><mrow id="S7.Thmtheorem23.p1.1.1.m1.1.1" xref="S7.Thmtheorem23.p1.1.1.m1.1.1.cmml"><mrow id="S7.Thmtheorem23.p1.1.1.m1.1.1.1" xref="S7.Thmtheorem23.p1.1.1.m1.1.1.1.cmml"><mn id="S7.Thmtheorem23.p1.1.1.m1.1.1.1.3" xref="S7.Thmtheorem23.p1.1.1.m1.1.1.1.3.cmml">2</mn><mo id="S7.Thmtheorem23.p1.1.1.m1.1.1.1.2" xref="S7.Thmtheorem23.p1.1.1.m1.1.1.1.2.cmml">⁢</mo><mrow id="S7.Thmtheorem23.p1.1.1.m1.1.1.1.1.1" xref="S7.Thmtheorem23.p1.1.1.m1.1.1.1.1.2.cmml"><mo id="S7.Thmtheorem23.p1.1.1.m1.1.1.1.1.1.2" stretchy="false" xref="S7.Thmtheorem23.p1.1.1.m1.1.1.1.1.2.1.cmml">⌈</mo><mrow id="S7.Thmtheorem23.p1.1.1.m1.1.1.1.1.1.1" xref="S7.Thmtheorem23.p1.1.1.m1.1.1.1.1.1.1.cmml"><msub id="S7.Thmtheorem23.p1.1.1.m1.1.1.1.1.1.1.1" xref="S7.Thmtheorem23.p1.1.1.m1.1.1.1.1.1.1.1.cmml"><mi id="S7.Thmtheorem23.p1.1.1.m1.1.1.1.1.1.1.1.2" xref="S7.Thmtheorem23.p1.1.1.m1.1.1.1.1.1.1.1.2.cmml">log</mi><mrow id="S7.Thmtheorem23.p1.1.1.m1.1.1.1.1.1.1.1.3" xref="S7.Thmtheorem23.p1.1.1.m1.1.1.1.1.1.1.1.3.cmml"><mn id="S7.Thmtheorem23.p1.1.1.m1.1.1.1.1.1.1.1.3.2" xref="S7.Thmtheorem23.p1.1.1.m1.1.1.1.1.1.1.1.3.2.cmml">8</mn><mo id="S7.Thmtheorem23.p1.1.1.m1.1.1.1.1.1.1.1.3.1" xref="S7.Thmtheorem23.p1.1.1.m1.1.1.1.1.1.1.1.3.1.cmml">/</mo><mn id="S7.Thmtheorem23.p1.1.1.m1.1.1.1.1.1.1.1.3.3" xref="S7.Thmtheorem23.p1.1.1.m1.1.1.1.1.1.1.1.3.3.cmml">7</mn></mrow></msub><mo id="S7.Thmtheorem23.p1.1.1.m1.1.1.1.1.1.1a" lspace="0.167em" xref="S7.Thmtheorem23.p1.1.1.m1.1.1.1.1.1.1.cmml">⁡</mo><mi id="S7.Thmtheorem23.p1.1.1.m1.1.1.1.1.1.1.2" xref="S7.Thmtheorem23.p1.1.1.m1.1.1.1.1.1.1.2.cmml">n</mi></mrow><mo id="S7.Thmtheorem23.p1.1.1.m1.1.1.1.1.1.3" stretchy="false" xref="S7.Thmtheorem23.p1.1.1.m1.1.1.1.1.2.1.cmml">⌉</mo></mrow></mrow><mo id="S7.Thmtheorem23.p1.1.1.m1.1.1.2" xref="S7.Thmtheorem23.p1.1.1.m1.1.1.2.cmml">+</mo><mn id="S7.Thmtheorem23.p1.1.1.m1.1.1.3" xref="S7.Thmtheorem23.p1.1.1.m1.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S7.Thmtheorem23.p1.1.1.m1.1b"><apply id="S7.Thmtheorem23.p1.1.1.m1.1.1.cmml" xref="S7.Thmtheorem23.p1.1.1.m1.1.1"><plus id="S7.Thmtheorem23.p1.1.1.m1.1.1.2.cmml" xref="S7.Thmtheorem23.p1.1.1.m1.1.1.2"></plus><apply id="S7.Thmtheorem23.p1.1.1.m1.1.1.1.cmml" xref="S7.Thmtheorem23.p1.1.1.m1.1.1.1"><times id="S7.Thmtheorem23.p1.1.1.m1.1.1.1.2.cmml" xref="S7.Thmtheorem23.p1.1.1.m1.1.1.1.2"></times><cn id="S7.Thmtheorem23.p1.1.1.m1.1.1.1.3.cmml" type="integer" xref="S7.Thmtheorem23.p1.1.1.m1.1.1.1.3">2</cn><apply id="S7.Thmtheorem23.p1.1.1.m1.1.1.1.1.2.cmml" xref="S7.Thmtheorem23.p1.1.1.m1.1.1.1.1.1"><ceiling id="S7.Thmtheorem23.p1.1.1.m1.1.1.1.1.2.1.cmml" xref="S7.Thmtheorem23.p1.1.1.m1.1.1.1.1.1.2"></ceiling><apply id="S7.Thmtheorem23.p1.1.1.m1.1.1.1.1.1.1.cmml" xref="S7.Thmtheorem23.p1.1.1.m1.1.1.1.1.1.1"><apply id="S7.Thmtheorem23.p1.1.1.m1.1.1.1.1.1.1.1.cmml" xref="S7.Thmtheorem23.p1.1.1.m1.1.1.1.1.1.1.1"><csymbol cd="ambiguous" id="S7.Thmtheorem23.p1.1.1.m1.1.1.1.1.1.1.1.1.cmml" xref="S7.Thmtheorem23.p1.1.1.m1.1.1.1.1.1.1.1">subscript</csymbol><log id="S7.Thmtheorem23.p1.1.1.m1.1.1.1.1.1.1.1.2.cmml" xref="S7.Thmtheorem23.p1.1.1.m1.1.1.1.1.1.1.1.2"></log><apply id="S7.Thmtheorem23.p1.1.1.m1.1.1.1.1.1.1.1.3.cmml" xref="S7.Thmtheorem23.p1.1.1.m1.1.1.1.1.1.1.1.3"><divide id="S7.Thmtheorem23.p1.1.1.m1.1.1.1.1.1.1.1.3.1.cmml" xref="S7.Thmtheorem23.p1.1.1.m1.1.1.1.1.1.1.1.3.1"></divide><cn id="S7.Thmtheorem23.p1.1.1.m1.1.1.1.1.1.1.1.3.2.cmml" type="integer" xref="S7.Thmtheorem23.p1.1.1.m1.1.1.1.1.1.1.1.3.2">8</cn><cn id="S7.Thmtheorem23.p1.1.1.m1.1.1.1.1.1.1.1.3.3.cmml" type="integer" xref="S7.Thmtheorem23.p1.1.1.m1.1.1.1.1.1.1.1.3.3">7</cn></apply></apply><ci id="S7.Thmtheorem23.p1.1.1.m1.1.1.1.1.1.1.2.cmml" xref="S7.Thmtheorem23.p1.1.1.m1.1.1.1.1.1.1.2">𝑛</ci></apply></apply></apply><cn id="S7.Thmtheorem23.p1.1.1.m1.1.1.3.cmml" type="integer" xref="S7.Thmtheorem23.p1.1.1.m1.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.Thmtheorem23.p1.1.1.m1.1c">2\lceil\log_{8/7}n\rceil+1</annotation><annotation encoding="application/x-llamapun" id="S7.Thmtheorem23.p1.1.1.m1.1d">2 ⌈ roman_log start_POSTSUBSCRIPT 8 / 7 end_POSTSUBSCRIPT italic_n ⌉ + 1</annotation></semantics></math> iterations, each taking two rounds. Since a graph may have at most <math alttext="n^{2}" class="ltx_Math" display="inline" id="S7.Thmtheorem23.p1.2.2.m2.1"><semantics id="S7.Thmtheorem23.p1.2.2.m2.1a"><msup id="S7.Thmtheorem23.p1.2.2.m2.1.1" xref="S7.Thmtheorem23.p1.2.2.m2.1.1.cmml"><mi id="S7.Thmtheorem23.p1.2.2.m2.1.1.2" xref="S7.Thmtheorem23.p1.2.2.m2.1.1.2.cmml">n</mi><mn id="S7.Thmtheorem23.p1.2.2.m2.1.1.3" xref="S7.Thmtheorem23.p1.2.2.m2.1.1.3.cmml">2</mn></msup><annotation-xml encoding="MathML-Content" id="S7.Thmtheorem23.p1.2.2.m2.1b"><apply id="S7.Thmtheorem23.p1.2.2.m2.1.1.cmml" xref="S7.Thmtheorem23.p1.2.2.m2.1.1"><csymbol cd="ambiguous" id="S7.Thmtheorem23.p1.2.2.m2.1.1.1.cmml" xref="S7.Thmtheorem23.p1.2.2.m2.1.1">superscript</csymbol><ci id="S7.Thmtheorem23.p1.2.2.m2.1.1.2.cmml" xref="S7.Thmtheorem23.p1.2.2.m2.1.1.2">𝑛</ci><cn id="S7.Thmtheorem23.p1.2.2.m2.1.1.3.cmml" type="integer" xref="S7.Thmtheorem23.p1.2.2.m2.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.Thmtheorem23.p1.2.2.m2.1c">n^{2}</annotation><annotation encoding="application/x-llamapun" id="S7.Thmtheorem23.p1.2.2.m2.1d">italic_n start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT</annotation></semantics></math> edges, <math alttext="|E_{0}|\leq n^{2}" class="ltx_Math" display="inline" id="S7.Thmtheorem23.p1.3.3.m3.1"><semantics id="S7.Thmtheorem23.p1.3.3.m3.1a"><mrow id="S7.Thmtheorem23.p1.3.3.m3.1.1" xref="S7.Thmtheorem23.p1.3.3.m3.1.1.cmml"><mrow id="S7.Thmtheorem23.p1.3.3.m3.1.1.1.1" xref="S7.Thmtheorem23.p1.3.3.m3.1.1.1.2.cmml"><mo id="S7.Thmtheorem23.p1.3.3.m3.1.1.1.1.2" stretchy="false" xref="S7.Thmtheorem23.p1.3.3.m3.1.1.1.2.1.cmml">|</mo><msub id="S7.Thmtheorem23.p1.3.3.m3.1.1.1.1.1" xref="S7.Thmtheorem23.p1.3.3.m3.1.1.1.1.1.cmml"><mi id="S7.Thmtheorem23.p1.3.3.m3.1.1.1.1.1.2" xref="S7.Thmtheorem23.p1.3.3.m3.1.1.1.1.1.2.cmml">E</mi><mn id="S7.Thmtheorem23.p1.3.3.m3.1.1.1.1.1.3" xref="S7.Thmtheorem23.p1.3.3.m3.1.1.1.1.1.3.cmml">0</mn></msub><mo id="S7.Thmtheorem23.p1.3.3.m3.1.1.1.1.3" stretchy="false" xref="S7.Thmtheorem23.p1.3.3.m3.1.1.1.2.1.cmml">|</mo></mrow><mo id="S7.Thmtheorem23.p1.3.3.m3.1.1.2" xref="S7.Thmtheorem23.p1.3.3.m3.1.1.2.cmml">≤</mo><msup id="S7.Thmtheorem23.p1.3.3.m3.1.1.3" xref="S7.Thmtheorem23.p1.3.3.m3.1.1.3.cmml"><mi id="S7.Thmtheorem23.p1.3.3.m3.1.1.3.2" xref="S7.Thmtheorem23.p1.3.3.m3.1.1.3.2.cmml">n</mi><mn id="S7.Thmtheorem23.p1.3.3.m3.1.1.3.3" xref="S7.Thmtheorem23.p1.3.3.m3.1.1.3.3.cmml">2</mn></msup></mrow><annotation-xml encoding="MathML-Content" id="S7.Thmtheorem23.p1.3.3.m3.1b"><apply id="S7.Thmtheorem23.p1.3.3.m3.1.1.cmml" xref="S7.Thmtheorem23.p1.3.3.m3.1.1"><leq id="S7.Thmtheorem23.p1.3.3.m3.1.1.2.cmml" xref="S7.Thmtheorem23.p1.3.3.m3.1.1.2"></leq><apply id="S7.Thmtheorem23.p1.3.3.m3.1.1.1.2.cmml" xref="S7.Thmtheorem23.p1.3.3.m3.1.1.1.1"><abs id="S7.Thmtheorem23.p1.3.3.m3.1.1.1.2.1.cmml" xref="S7.Thmtheorem23.p1.3.3.m3.1.1.1.1.2"></abs><apply id="S7.Thmtheorem23.p1.3.3.m3.1.1.1.1.1.cmml" xref="S7.Thmtheorem23.p1.3.3.m3.1.1.1.1.1"><csymbol cd="ambiguous" id="S7.Thmtheorem23.p1.3.3.m3.1.1.1.1.1.1.cmml" xref="S7.Thmtheorem23.p1.3.3.m3.1.1.1.1.1">subscript</csymbol><ci id="S7.Thmtheorem23.p1.3.3.m3.1.1.1.1.1.2.cmml" xref="S7.Thmtheorem23.p1.3.3.m3.1.1.1.1.1.2">𝐸</ci><cn id="S7.Thmtheorem23.p1.3.3.m3.1.1.1.1.1.3.cmml" type="integer" xref="S7.Thmtheorem23.p1.3.3.m3.1.1.1.1.1.3">0</cn></apply></apply><apply id="S7.Thmtheorem23.p1.3.3.m3.1.1.3.cmml" xref="S7.Thmtheorem23.p1.3.3.m3.1.1.3"><csymbol cd="ambiguous" id="S7.Thmtheorem23.p1.3.3.m3.1.1.3.1.cmml" xref="S7.Thmtheorem23.p1.3.3.m3.1.1.3">superscript</csymbol><ci id="S7.Thmtheorem23.p1.3.3.m3.1.1.3.2.cmml" xref="S7.Thmtheorem23.p1.3.3.m3.1.1.3.2">𝑛</ci><cn id="S7.Thmtheorem23.p1.3.3.m3.1.1.3.3.cmml" type="integer" xref="S7.Thmtheorem23.p1.3.3.m3.1.1.3.3">2</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.Thmtheorem23.p1.3.3.m3.1c">|E_{0}|\leq n^{2}</annotation><annotation encoding="application/x-llamapun" id="S7.Thmtheorem23.p1.3.3.m3.1d">| italic_E start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT | ≤ italic_n start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT</annotation></semantics></math>. We now apply Lemma <a class="ltx_ref" href="https://arxiv.org/html/2411.12694v2#S7.Thmtheorem20" title="Lemma 7.20. ‣ 7.4 Formally proving correctness ‣ 7 Results in CONGEST ‣ Local Density and its Distributed Approximation"><span class="ltx_text ltx_ref_tag">7.20</span></a> to obtain that after the <span class="ltx_text ltx_font_smallcaps" id="S7.Thmtheorem23.p1.8.8.2">minute</span> there are no vertices in <math alttext="L_{h}" class="ltx_Math" display="inline" id="S7.Thmtheorem23.p1.4.4.m4.1"><semantics id="S7.Thmtheorem23.p1.4.4.m4.1a"><msub id="S7.Thmtheorem23.p1.4.4.m4.1.1" xref="S7.Thmtheorem23.p1.4.4.m4.1.1.cmml"><mi id="S7.Thmtheorem23.p1.4.4.m4.1.1.2" xref="S7.Thmtheorem23.p1.4.4.m4.1.1.2.cmml">L</mi><mi id="S7.Thmtheorem23.p1.4.4.m4.1.1.3" xref="S7.Thmtheorem23.p1.4.4.m4.1.1.3.cmml">h</mi></msub><annotation-xml encoding="MathML-Content" id="S7.Thmtheorem23.p1.4.4.m4.1b"><apply id="S7.Thmtheorem23.p1.4.4.m4.1.1.cmml" xref="S7.Thmtheorem23.p1.4.4.m4.1.1"><csymbol cd="ambiguous" id="S7.Thmtheorem23.p1.4.4.m4.1.1.1.cmml" xref="S7.Thmtheorem23.p1.4.4.m4.1.1">subscript</csymbol><ci id="S7.Thmtheorem23.p1.4.4.m4.1.1.2.cmml" xref="S7.Thmtheorem23.p1.4.4.m4.1.1.2">𝐿</ci><ci id="S7.Thmtheorem23.p1.4.4.m4.1.1.3.cmml" xref="S7.Thmtheorem23.p1.4.4.m4.1.1.3">ℎ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.Thmtheorem23.p1.4.4.m4.1c">L_{h}</annotation><annotation encoding="application/x-llamapun" id="S7.Thmtheorem23.p1.4.4.m4.1d">italic_L start_POSTSUBSCRIPT italic_h end_POSTSUBSCRIPT</annotation></semantics></math> with a violating out-edge. This may create vertices in <math alttext="L_{h+1}" class="ltx_Math" display="inline" id="S7.Thmtheorem23.p1.5.5.m5.1"><semantics id="S7.Thmtheorem23.p1.5.5.m5.1a"><msub id="S7.Thmtheorem23.p1.5.5.m5.1.1" xref="S7.Thmtheorem23.p1.5.5.m5.1.1.cmml"><mi id="S7.Thmtheorem23.p1.5.5.m5.1.1.2" xref="S7.Thmtheorem23.p1.5.5.m5.1.1.2.cmml">L</mi><mrow id="S7.Thmtheorem23.p1.5.5.m5.1.1.3" xref="S7.Thmtheorem23.p1.5.5.m5.1.1.3.cmml"><mi id="S7.Thmtheorem23.p1.5.5.m5.1.1.3.2" xref="S7.Thmtheorem23.p1.5.5.m5.1.1.3.2.cmml">h</mi><mo id="S7.Thmtheorem23.p1.5.5.m5.1.1.3.1" xref="S7.Thmtheorem23.p1.5.5.m5.1.1.3.1.cmml">+</mo><mn id="S7.Thmtheorem23.p1.5.5.m5.1.1.3.3" xref="S7.Thmtheorem23.p1.5.5.m5.1.1.3.3.cmml">1</mn></mrow></msub><annotation-xml encoding="MathML-Content" id="S7.Thmtheorem23.p1.5.5.m5.1b"><apply id="S7.Thmtheorem23.p1.5.5.m5.1.1.cmml" xref="S7.Thmtheorem23.p1.5.5.m5.1.1"><csymbol cd="ambiguous" id="S7.Thmtheorem23.p1.5.5.m5.1.1.1.cmml" xref="S7.Thmtheorem23.p1.5.5.m5.1.1">subscript</csymbol><ci id="S7.Thmtheorem23.p1.5.5.m5.1.1.2.cmml" xref="S7.Thmtheorem23.p1.5.5.m5.1.1.2">𝐿</ci><apply id="S7.Thmtheorem23.p1.5.5.m5.1.1.3.cmml" xref="S7.Thmtheorem23.p1.5.5.m5.1.1.3"><plus id="S7.Thmtheorem23.p1.5.5.m5.1.1.3.1.cmml" xref="S7.Thmtheorem23.p1.5.5.m5.1.1.3.1"></plus><ci id="S7.Thmtheorem23.p1.5.5.m5.1.1.3.2.cmml" xref="S7.Thmtheorem23.p1.5.5.m5.1.1.3.2">ℎ</ci><cn id="S7.Thmtheorem23.p1.5.5.m5.1.1.3.3.cmml" type="integer" xref="S7.Thmtheorem23.p1.5.5.m5.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.Thmtheorem23.p1.5.5.m5.1c">L_{h+1}</annotation><annotation encoding="application/x-llamapun" id="S7.Thmtheorem23.p1.5.5.m5.1d">italic_L start_POSTSUBSCRIPT italic_h + 1 end_POSTSUBSCRIPT</annotation></semantics></math> with a violating out-edge, but since vertices in <math alttext="L_{h}" class="ltx_Math" display="inline" id="S7.Thmtheorem23.p1.6.6.m6.1"><semantics id="S7.Thmtheorem23.p1.6.6.m6.1a"><msub id="S7.Thmtheorem23.p1.6.6.m6.1.1" xref="S7.Thmtheorem23.p1.6.6.m6.1.1.cmml"><mi id="S7.Thmtheorem23.p1.6.6.m6.1.1.2" xref="S7.Thmtheorem23.p1.6.6.m6.1.1.2.cmml">L</mi><mi id="S7.Thmtheorem23.p1.6.6.m6.1.1.3" xref="S7.Thmtheorem23.p1.6.6.m6.1.1.3.cmml">h</mi></msub><annotation-xml encoding="MathML-Content" id="S7.Thmtheorem23.p1.6.6.m6.1b"><apply id="S7.Thmtheorem23.p1.6.6.m6.1.1.cmml" xref="S7.Thmtheorem23.p1.6.6.m6.1.1"><csymbol cd="ambiguous" id="S7.Thmtheorem23.p1.6.6.m6.1.1.1.cmml" xref="S7.Thmtheorem23.p1.6.6.m6.1.1">subscript</csymbol><ci id="S7.Thmtheorem23.p1.6.6.m6.1.1.2.cmml" xref="S7.Thmtheorem23.p1.6.6.m6.1.1.2">𝐿</ci><ci id="S7.Thmtheorem23.p1.6.6.m6.1.1.3.cmml" xref="S7.Thmtheorem23.p1.6.6.m6.1.1.3">ℎ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.Thmtheorem23.p1.6.6.m6.1c">L_{h}</annotation><annotation encoding="application/x-llamapun" id="S7.Thmtheorem23.p1.6.6.m6.1d">italic_L start_POSTSUBSCRIPT italic_h end_POSTSUBSCRIPT</annotation></semantics></math> decreased their level by at most one, no vertices in <math alttext="L_{k}" class="ltx_Math" display="inline" id="S7.Thmtheorem23.p1.7.7.m7.1"><semantics id="S7.Thmtheorem23.p1.7.7.m7.1a"><msub id="S7.Thmtheorem23.p1.7.7.m7.1.1" xref="S7.Thmtheorem23.p1.7.7.m7.1.1.cmml"><mi id="S7.Thmtheorem23.p1.7.7.m7.1.1.2" xref="S7.Thmtheorem23.p1.7.7.m7.1.1.2.cmml">L</mi><mi id="S7.Thmtheorem23.p1.7.7.m7.1.1.3" xref="S7.Thmtheorem23.p1.7.7.m7.1.1.3.cmml">k</mi></msub><annotation-xml encoding="MathML-Content" id="S7.Thmtheorem23.p1.7.7.m7.1b"><apply id="S7.Thmtheorem23.p1.7.7.m7.1.1.cmml" xref="S7.Thmtheorem23.p1.7.7.m7.1.1"><csymbol cd="ambiguous" id="S7.Thmtheorem23.p1.7.7.m7.1.1.1.cmml" xref="S7.Thmtheorem23.p1.7.7.m7.1.1">subscript</csymbol><ci id="S7.Thmtheorem23.p1.7.7.m7.1.1.2.cmml" xref="S7.Thmtheorem23.p1.7.7.m7.1.1.2">𝐿</ci><ci id="S7.Thmtheorem23.p1.7.7.m7.1.1.3.cmml" xref="S7.Thmtheorem23.p1.7.7.m7.1.1.3">𝑘</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.Thmtheorem23.p1.7.7.m7.1c">L_{k}</annotation><annotation encoding="application/x-llamapun" id="S7.Thmtheorem23.p1.7.7.m7.1d">italic_L start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT</annotation></semantics></math> with <math alttext="k>h+1" class="ltx_Math" display="inline" id="S7.Thmtheorem23.p1.8.8.m8.1"><semantics id="S7.Thmtheorem23.p1.8.8.m8.1a"><mrow id="S7.Thmtheorem23.p1.8.8.m8.1.1" xref="S7.Thmtheorem23.p1.8.8.m8.1.1.cmml"><mi id="S7.Thmtheorem23.p1.8.8.m8.1.1.2" xref="S7.Thmtheorem23.p1.8.8.m8.1.1.2.cmml">k</mi><mo id="S7.Thmtheorem23.p1.8.8.m8.1.1.1" xref="S7.Thmtheorem23.p1.8.8.m8.1.1.1.cmml">></mo><mrow id="S7.Thmtheorem23.p1.8.8.m8.1.1.3" xref="S7.Thmtheorem23.p1.8.8.m8.1.1.3.cmml"><mi id="S7.Thmtheorem23.p1.8.8.m8.1.1.3.2" xref="S7.Thmtheorem23.p1.8.8.m8.1.1.3.2.cmml">h</mi><mo id="S7.Thmtheorem23.p1.8.8.m8.1.1.3.1" xref="S7.Thmtheorem23.p1.8.8.m8.1.1.3.1.cmml">+</mo><mn id="S7.Thmtheorem23.p1.8.8.m8.1.1.3.3" xref="S7.Thmtheorem23.p1.8.8.m8.1.1.3.3.cmml">1</mn></mrow></mrow><annotation-xml encoding="MathML-Content" id="S7.Thmtheorem23.p1.8.8.m8.1b"><apply id="S7.Thmtheorem23.p1.8.8.m8.1.1.cmml" xref="S7.Thmtheorem23.p1.8.8.m8.1.1"><gt id="S7.Thmtheorem23.p1.8.8.m8.1.1.1.cmml" xref="S7.Thmtheorem23.p1.8.8.m8.1.1.1"></gt><ci id="S7.Thmtheorem23.p1.8.8.m8.1.1.2.cmml" xref="S7.Thmtheorem23.p1.8.8.m8.1.1.2">𝑘</ci><apply id="S7.Thmtheorem23.p1.8.8.m8.1.1.3.cmml" xref="S7.Thmtheorem23.p1.8.8.m8.1.1.3"><plus id="S7.Thmtheorem23.p1.8.8.m8.1.1.3.1.cmml" xref="S7.Thmtheorem23.p1.8.8.m8.1.1.3.1"></plus><ci id="S7.Thmtheorem23.p1.8.8.m8.1.1.3.2.cmml" xref="S7.Thmtheorem23.p1.8.8.m8.1.1.3.2">ℎ</ci><cn id="S7.Thmtheorem23.p1.8.8.m8.1.1.3.3.cmml" type="integer" xref="S7.Thmtheorem23.p1.8.8.m8.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.Thmtheorem23.p1.8.8.m8.1c">k>h+1</annotation><annotation encoding="application/x-llamapun" id="S7.Thmtheorem23.p1.8.8.m8.1d">italic_k > italic_h + 1</annotation></semantics></math> may have a violating out-edge.</span></p> </div> </div> <div class="ltx_para" id="S7.SS4.p4"> <p class="ltx_p" id="S7.SS4.p4.1">Finally, we show one more helper lemma to imply our main theorem:</p> </div> <div class="ltx_theorem ltx_theorem_lemma" id="S7.Thmtheorem24"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="S7.Thmtheorem24.1.1.1">Lemma 7.24</span></span><span class="ltx_text ltx_font_bold" id="S7.Thmtheorem24.2.2">.</span> </h6> <div class="ltx_para" id="S7.Thmtheorem24.p1"> <p class="ltx_p" id="S7.Thmtheorem24.p1.7"><span class="ltx_text ltx_font_italic" id="S7.Thmtheorem24.p1.7.7">If at the start of <math alttext="(h:x:0)" class="ltx_Math" display="inline" id="S7.Thmtheorem24.p1.1.1.m1.1"><semantics id="S7.Thmtheorem24.p1.1.1.m1.1a"><mrow id="S7.Thmtheorem24.p1.1.1.m1.1.1.1" xref="S7.Thmtheorem24.p1.1.1.m1.1.1.1.1.cmml"><mo id="S7.Thmtheorem24.p1.1.1.m1.1.1.1.2" stretchy="false" xref="S7.Thmtheorem24.p1.1.1.m1.1.1.1.1.cmml">(</mo><mrow id="S7.Thmtheorem24.p1.1.1.m1.1.1.1.1" xref="S7.Thmtheorem24.p1.1.1.m1.1.1.1.1.cmml"><mi id="S7.Thmtheorem24.p1.1.1.m1.1.1.1.1.2" xref="S7.Thmtheorem24.p1.1.1.m1.1.1.1.1.2.cmml">h</mi><mo id="S7.Thmtheorem24.p1.1.1.m1.1.1.1.1.3" lspace="0.278em" rspace="0.278em" xref="S7.Thmtheorem24.p1.1.1.m1.1.1.1.1.3.cmml">:</mo><mi id="S7.Thmtheorem24.p1.1.1.m1.1.1.1.1.4" xref="S7.Thmtheorem24.p1.1.1.m1.1.1.1.1.4.cmml">x</mi><mo id="S7.Thmtheorem24.p1.1.1.m1.1.1.1.1.5" lspace="0.278em" rspace="0.278em" xref="S7.Thmtheorem24.p1.1.1.m1.1.1.1.1.5.cmml">:</mo><mn id="S7.Thmtheorem24.p1.1.1.m1.1.1.1.1.6" xref="S7.Thmtheorem24.p1.1.1.m1.1.1.1.1.6.cmml">0</mn></mrow><mo id="S7.Thmtheorem24.p1.1.1.m1.1.1.1.3" stretchy="false" xref="S7.Thmtheorem24.p1.1.1.m1.1.1.1.1.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S7.Thmtheorem24.p1.1.1.m1.1b"><apply id="S7.Thmtheorem24.p1.1.1.m1.1.1.1.1.cmml" xref="S7.Thmtheorem24.p1.1.1.m1.1.1.1"><and id="S7.Thmtheorem24.p1.1.1.m1.1.1.1.1a.cmml" xref="S7.Thmtheorem24.p1.1.1.m1.1.1.1"></and><apply id="S7.Thmtheorem24.p1.1.1.m1.1.1.1.1b.cmml" xref="S7.Thmtheorem24.p1.1.1.m1.1.1.1"><ci id="S7.Thmtheorem24.p1.1.1.m1.1.1.1.1.3.cmml" xref="S7.Thmtheorem24.p1.1.1.m1.1.1.1.1.3">:</ci><ci id="S7.Thmtheorem24.p1.1.1.m1.1.1.1.1.2.cmml" xref="S7.Thmtheorem24.p1.1.1.m1.1.1.1.1.2">ℎ</ci><ci id="S7.Thmtheorem24.p1.1.1.m1.1.1.1.1.4.cmml" xref="S7.Thmtheorem24.p1.1.1.m1.1.1.1.1.4">𝑥</ci></apply><apply id="S7.Thmtheorem24.p1.1.1.m1.1.1.1.1c.cmml" xref="S7.Thmtheorem24.p1.1.1.m1.1.1.1"><ci id="S7.Thmtheorem24.p1.1.1.m1.1.1.1.1.5.cmml" xref="S7.Thmtheorem24.p1.1.1.m1.1.1.1.1.5">:</ci><share href="https://arxiv.org/html/2411.12694v2#S7.Thmtheorem24.p1.1.1.m1.1.1.1.1.4.cmml" id="S7.Thmtheorem24.p1.1.1.m1.1.1.1.1d.cmml" xref="S7.Thmtheorem24.p1.1.1.m1.1.1.1"></share><cn id="S7.Thmtheorem24.p1.1.1.m1.1.1.1.1.6.cmml" type="integer" xref="S7.Thmtheorem24.p1.1.1.m1.1.1.1.1.6">0</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.Thmtheorem24.p1.1.1.m1.1c">(h:x:0)</annotation><annotation encoding="application/x-llamapun" id="S7.Thmtheorem24.p1.1.1.m1.1d">( italic_h : italic_x : 0 )</annotation></semantics></math> for even <math alttext="x" class="ltx_Math" display="inline" id="S7.Thmtheorem24.p1.2.2.m2.1"><semantics id="S7.Thmtheorem24.p1.2.2.m2.1a"><mi id="S7.Thmtheorem24.p1.2.2.m2.1.1" xref="S7.Thmtheorem24.p1.2.2.m2.1.1.cmml">x</mi><annotation-xml encoding="MathML-Content" id="S7.Thmtheorem24.p1.2.2.m2.1b"><ci id="S7.Thmtheorem24.p1.2.2.m2.1.1.cmml" xref="S7.Thmtheorem24.p1.2.2.m2.1.1">𝑥</ci></annotation-xml><annotation encoding="application/x-tex" id="S7.Thmtheorem24.p1.2.2.m2.1c">x</annotation><annotation encoding="application/x-llamapun" id="S7.Thmtheorem24.p1.2.2.m2.1d">italic_x</annotation></semantics></math>, a vertex <math alttext="v" class="ltx_Math" display="inline" id="S7.Thmtheorem24.p1.3.3.m3.1"><semantics id="S7.Thmtheorem24.p1.3.3.m3.1a"><mi id="S7.Thmtheorem24.p1.3.3.m3.1.1" xref="S7.Thmtheorem24.p1.3.3.m3.1.1.cmml">v</mi><annotation-xml encoding="MathML-Content" id="S7.Thmtheorem24.p1.3.3.m3.1b"><ci id="S7.Thmtheorem24.p1.3.3.m3.1.1.cmml" xref="S7.Thmtheorem24.p1.3.3.m3.1.1">𝑣</ci></annotation-xml><annotation encoding="application/x-tex" id="S7.Thmtheorem24.p1.3.3.m3.1c">v</annotation><annotation encoding="application/x-llamapun" id="S7.Thmtheorem24.p1.3.3.m3.1d">italic_v</annotation></semantics></math> has <math alttext="l_{x}(v)=h" class="ltx_Math" display="inline" id="S7.Thmtheorem24.p1.4.4.m4.1"><semantics id="S7.Thmtheorem24.p1.4.4.m4.1a"><mrow id="S7.Thmtheorem24.p1.4.4.m4.1.2" xref="S7.Thmtheorem24.p1.4.4.m4.1.2.cmml"><mrow id="S7.Thmtheorem24.p1.4.4.m4.1.2.2" xref="S7.Thmtheorem24.p1.4.4.m4.1.2.2.cmml"><msub id="S7.Thmtheorem24.p1.4.4.m4.1.2.2.2" xref="S7.Thmtheorem24.p1.4.4.m4.1.2.2.2.cmml"><mi id="S7.Thmtheorem24.p1.4.4.m4.1.2.2.2.2" xref="S7.Thmtheorem24.p1.4.4.m4.1.2.2.2.2.cmml">l</mi><mi id="S7.Thmtheorem24.p1.4.4.m4.1.2.2.2.3" xref="S7.Thmtheorem24.p1.4.4.m4.1.2.2.2.3.cmml">x</mi></msub><mo id="S7.Thmtheorem24.p1.4.4.m4.1.2.2.1" xref="S7.Thmtheorem24.p1.4.4.m4.1.2.2.1.cmml">⁢</mo><mrow id="S7.Thmtheorem24.p1.4.4.m4.1.2.2.3.2" xref="S7.Thmtheorem24.p1.4.4.m4.1.2.2.cmml"><mo id="S7.Thmtheorem24.p1.4.4.m4.1.2.2.3.2.1" stretchy="false" xref="S7.Thmtheorem24.p1.4.4.m4.1.2.2.cmml">(</mo><mi id="S7.Thmtheorem24.p1.4.4.m4.1.1" xref="S7.Thmtheorem24.p1.4.4.m4.1.1.cmml">v</mi><mo id="S7.Thmtheorem24.p1.4.4.m4.1.2.2.3.2.2" stretchy="false" xref="S7.Thmtheorem24.p1.4.4.m4.1.2.2.cmml">)</mo></mrow></mrow><mo id="S7.Thmtheorem24.p1.4.4.m4.1.2.1" xref="S7.Thmtheorem24.p1.4.4.m4.1.2.1.cmml">=</mo><mi id="S7.Thmtheorem24.p1.4.4.m4.1.2.3" xref="S7.Thmtheorem24.p1.4.4.m4.1.2.3.cmml">h</mi></mrow><annotation-xml encoding="MathML-Content" id="S7.Thmtheorem24.p1.4.4.m4.1b"><apply id="S7.Thmtheorem24.p1.4.4.m4.1.2.cmml" xref="S7.Thmtheorem24.p1.4.4.m4.1.2"><eq id="S7.Thmtheorem24.p1.4.4.m4.1.2.1.cmml" xref="S7.Thmtheorem24.p1.4.4.m4.1.2.1"></eq><apply id="S7.Thmtheorem24.p1.4.4.m4.1.2.2.cmml" xref="S7.Thmtheorem24.p1.4.4.m4.1.2.2"><times id="S7.Thmtheorem24.p1.4.4.m4.1.2.2.1.cmml" xref="S7.Thmtheorem24.p1.4.4.m4.1.2.2.1"></times><apply id="S7.Thmtheorem24.p1.4.4.m4.1.2.2.2.cmml" xref="S7.Thmtheorem24.p1.4.4.m4.1.2.2.2"><csymbol cd="ambiguous" id="S7.Thmtheorem24.p1.4.4.m4.1.2.2.2.1.cmml" xref="S7.Thmtheorem24.p1.4.4.m4.1.2.2.2">subscript</csymbol><ci id="S7.Thmtheorem24.p1.4.4.m4.1.2.2.2.2.cmml" xref="S7.Thmtheorem24.p1.4.4.m4.1.2.2.2.2">𝑙</ci><ci id="S7.Thmtheorem24.p1.4.4.m4.1.2.2.2.3.cmml" xref="S7.Thmtheorem24.p1.4.4.m4.1.2.2.2.3">𝑥</ci></apply><ci id="S7.Thmtheorem24.p1.4.4.m4.1.1.cmml" xref="S7.Thmtheorem24.p1.4.4.m4.1.1">𝑣</ci></apply><ci id="S7.Thmtheorem24.p1.4.4.m4.1.2.3.cmml" xref="S7.Thmtheorem24.p1.4.4.m4.1.2.3">ℎ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.Thmtheorem24.p1.4.4.m4.1c">l_{x}(v)=h</annotation><annotation encoding="application/x-llamapun" id="S7.Thmtheorem24.p1.4.4.m4.1d">italic_l start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT ( italic_v ) = italic_h</annotation></semantics></math> and <math alttext="v" class="ltx_Math" display="inline" id="S7.Thmtheorem24.p1.5.5.m5.1"><semantics id="S7.Thmtheorem24.p1.5.5.m5.1a"><mi id="S7.Thmtheorem24.p1.5.5.m5.1.1" xref="S7.Thmtheorem24.p1.5.5.m5.1.1.cmml">v</mi><annotation-xml encoding="MathML-Content" id="S7.Thmtheorem24.p1.5.5.m5.1b"><ci id="S7.Thmtheorem24.p1.5.5.m5.1.1.cmml" xref="S7.Thmtheorem24.p1.5.5.m5.1.1">𝑣</ci></annotation-xml><annotation encoding="application/x-tex" id="S7.Thmtheorem24.p1.5.5.m5.1c">v</annotation><annotation encoding="application/x-llamapun" id="S7.Thmtheorem24.p1.5.5.m5.1d">italic_v</annotation></semantics></math> has an outgoing violating edge, then for all even <math alttext="m<x" class="ltx_Math" display="inline" id="S7.Thmtheorem24.p1.6.6.m6.1"><semantics id="S7.Thmtheorem24.p1.6.6.m6.1a"><mrow id="S7.Thmtheorem24.p1.6.6.m6.1.1" xref="S7.Thmtheorem24.p1.6.6.m6.1.1.cmml"><mi id="S7.Thmtheorem24.p1.6.6.m6.1.1.2" xref="S7.Thmtheorem24.p1.6.6.m6.1.1.2.cmml">m</mi><mo id="S7.Thmtheorem24.p1.6.6.m6.1.1.1" xref="S7.Thmtheorem24.p1.6.6.m6.1.1.1.cmml"><</mo><mi id="S7.Thmtheorem24.p1.6.6.m6.1.1.3" xref="S7.Thmtheorem24.p1.6.6.m6.1.1.3.cmml">x</mi></mrow><annotation-xml encoding="MathML-Content" id="S7.Thmtheorem24.p1.6.6.m6.1b"><apply id="S7.Thmtheorem24.p1.6.6.m6.1.1.cmml" xref="S7.Thmtheorem24.p1.6.6.m6.1.1"><lt id="S7.Thmtheorem24.p1.6.6.m6.1.1.1.cmml" xref="S7.Thmtheorem24.p1.6.6.m6.1.1.1"></lt><ci id="S7.Thmtheorem24.p1.6.6.m6.1.1.2.cmml" xref="S7.Thmtheorem24.p1.6.6.m6.1.1.2">𝑚</ci><ci id="S7.Thmtheorem24.p1.6.6.m6.1.1.3.cmml" xref="S7.Thmtheorem24.p1.6.6.m6.1.1.3">𝑥</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.Thmtheorem24.p1.6.6.m6.1c">m<x</annotation><annotation encoding="application/x-llamapun" id="S7.Thmtheorem24.p1.6.6.m6.1d">italic_m < italic_x</annotation></semantics></math>, <math alttext="v\in T_{m}" class="ltx_Math" display="inline" id="S7.Thmtheorem24.p1.7.7.m7.1"><semantics id="S7.Thmtheorem24.p1.7.7.m7.1a"><mrow id="S7.Thmtheorem24.p1.7.7.m7.1.1" xref="S7.Thmtheorem24.p1.7.7.m7.1.1.cmml"><mi id="S7.Thmtheorem24.p1.7.7.m7.1.1.2" xref="S7.Thmtheorem24.p1.7.7.m7.1.1.2.cmml">v</mi><mo id="S7.Thmtheorem24.p1.7.7.m7.1.1.1" xref="S7.Thmtheorem24.p1.7.7.m7.1.1.1.cmml">∈</mo><msub id="S7.Thmtheorem24.p1.7.7.m7.1.1.3" xref="S7.Thmtheorem24.p1.7.7.m7.1.1.3.cmml"><mi id="S7.Thmtheorem24.p1.7.7.m7.1.1.3.2" xref="S7.Thmtheorem24.p1.7.7.m7.1.1.3.2.cmml">T</mi><mi id="S7.Thmtheorem24.p1.7.7.m7.1.1.3.3" xref="S7.Thmtheorem24.p1.7.7.m7.1.1.3.3.cmml">m</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S7.Thmtheorem24.p1.7.7.m7.1b"><apply id="S7.Thmtheorem24.p1.7.7.m7.1.1.cmml" xref="S7.Thmtheorem24.p1.7.7.m7.1.1"><in id="S7.Thmtheorem24.p1.7.7.m7.1.1.1.cmml" xref="S7.Thmtheorem24.p1.7.7.m7.1.1.1"></in><ci id="S7.Thmtheorem24.p1.7.7.m7.1.1.2.cmml" xref="S7.Thmtheorem24.p1.7.7.m7.1.1.2">𝑣</ci><apply id="S7.Thmtheorem24.p1.7.7.m7.1.1.3.cmml" xref="S7.Thmtheorem24.p1.7.7.m7.1.1.3"><csymbol cd="ambiguous" id="S7.Thmtheorem24.p1.7.7.m7.1.1.3.1.cmml" xref="S7.Thmtheorem24.p1.7.7.m7.1.1.3">subscript</csymbol><ci id="S7.Thmtheorem24.p1.7.7.m7.1.1.3.2.cmml" xref="S7.Thmtheorem24.p1.7.7.m7.1.1.3.2">𝑇</ci><ci id="S7.Thmtheorem24.p1.7.7.m7.1.1.3.3.cmml" xref="S7.Thmtheorem24.p1.7.7.m7.1.1.3.3">𝑚</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.Thmtheorem24.p1.7.7.m7.1c">v\in T_{m}</annotation><annotation encoding="application/x-llamapun" id="S7.Thmtheorem24.p1.7.7.m7.1d">italic_v ∈ italic_T start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT</annotation></semantics></math>.</span></p> </div> </div> <div class="ltx_theorem ltx_theorem_proof" id="S7.Thmtheorem25"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="S7.Thmtheorem25.1.1.1">Proof 7.25</span></span><span class="ltx_text ltx_font_bold" id="S7.Thmtheorem25.2.2">.</span> </h6> <div class="ltx_para" id="S7.Thmtheorem25.p1"> <p class="ltx_p" id="S7.Thmtheorem25.p1.3"><span class="ltx_text ltx_font_italic" id="S7.Thmtheorem25.p1.3.3">Consider any <math alttext="m<x" class="ltx_Math" display="inline" id="S7.Thmtheorem25.p1.1.1.m1.1"><semantics id="S7.Thmtheorem25.p1.1.1.m1.1a"><mrow id="S7.Thmtheorem25.p1.1.1.m1.1.1" xref="S7.Thmtheorem25.p1.1.1.m1.1.1.cmml"><mi id="S7.Thmtheorem25.p1.1.1.m1.1.1.2" xref="S7.Thmtheorem25.p1.1.1.m1.1.1.2.cmml">m</mi><mo id="S7.Thmtheorem25.p1.1.1.m1.1.1.1" xref="S7.Thmtheorem25.p1.1.1.m1.1.1.1.cmml"><</mo><mi id="S7.Thmtheorem25.p1.1.1.m1.1.1.3" xref="S7.Thmtheorem25.p1.1.1.m1.1.1.3.cmml">x</mi></mrow><annotation-xml encoding="MathML-Content" id="S7.Thmtheorem25.p1.1.1.m1.1b"><apply id="S7.Thmtheorem25.p1.1.1.m1.1.1.cmml" xref="S7.Thmtheorem25.p1.1.1.m1.1.1"><lt id="S7.Thmtheorem25.p1.1.1.m1.1.1.1.cmml" xref="S7.Thmtheorem25.p1.1.1.m1.1.1.1"></lt><ci id="S7.Thmtheorem25.p1.1.1.m1.1.1.2.cmml" xref="S7.Thmtheorem25.p1.1.1.m1.1.1.2">𝑚</ci><ci id="S7.Thmtheorem25.p1.1.1.m1.1.1.3.cmml" xref="S7.Thmtheorem25.p1.1.1.m1.1.1.3">𝑥</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.Thmtheorem25.p1.1.1.m1.1c">m<x</annotation><annotation encoding="application/x-llamapun" id="S7.Thmtheorem25.p1.1.1.m1.1d">italic_m < italic_x</annotation></semantics></math> that is odd. If at the start of <math alttext="(h:m:0)" class="ltx_Math" display="inline" id="S7.Thmtheorem25.p1.2.2.m2.1"><semantics id="S7.Thmtheorem25.p1.2.2.m2.1a"><mrow id="S7.Thmtheorem25.p1.2.2.m2.1.1.1" xref="S7.Thmtheorem25.p1.2.2.m2.1.1.1.1.cmml"><mo id="S7.Thmtheorem25.p1.2.2.m2.1.1.1.2" stretchy="false" xref="S7.Thmtheorem25.p1.2.2.m2.1.1.1.1.cmml">(</mo><mrow id="S7.Thmtheorem25.p1.2.2.m2.1.1.1.1" xref="S7.Thmtheorem25.p1.2.2.m2.1.1.1.1.cmml"><mi id="S7.Thmtheorem25.p1.2.2.m2.1.1.1.1.2" xref="S7.Thmtheorem25.p1.2.2.m2.1.1.1.1.2.cmml">h</mi><mo id="S7.Thmtheorem25.p1.2.2.m2.1.1.1.1.3" lspace="0.278em" rspace="0.278em" xref="S7.Thmtheorem25.p1.2.2.m2.1.1.1.1.3.cmml">:</mo><mi id="S7.Thmtheorem25.p1.2.2.m2.1.1.1.1.4" xref="S7.Thmtheorem25.p1.2.2.m2.1.1.1.1.4.cmml">m</mi><mo id="S7.Thmtheorem25.p1.2.2.m2.1.1.1.1.5" lspace="0.278em" rspace="0.278em" xref="S7.Thmtheorem25.p1.2.2.m2.1.1.1.1.5.cmml">:</mo><mn id="S7.Thmtheorem25.p1.2.2.m2.1.1.1.1.6" xref="S7.Thmtheorem25.p1.2.2.m2.1.1.1.1.6.cmml">0</mn></mrow><mo id="S7.Thmtheorem25.p1.2.2.m2.1.1.1.3" stretchy="false" xref="S7.Thmtheorem25.p1.2.2.m2.1.1.1.1.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S7.Thmtheorem25.p1.2.2.m2.1b"><apply id="S7.Thmtheorem25.p1.2.2.m2.1.1.1.1.cmml" xref="S7.Thmtheorem25.p1.2.2.m2.1.1.1"><and id="S7.Thmtheorem25.p1.2.2.m2.1.1.1.1a.cmml" xref="S7.Thmtheorem25.p1.2.2.m2.1.1.1"></and><apply id="S7.Thmtheorem25.p1.2.2.m2.1.1.1.1b.cmml" xref="S7.Thmtheorem25.p1.2.2.m2.1.1.1"><ci id="S7.Thmtheorem25.p1.2.2.m2.1.1.1.1.3.cmml" xref="S7.Thmtheorem25.p1.2.2.m2.1.1.1.1.3">:</ci><ci id="S7.Thmtheorem25.p1.2.2.m2.1.1.1.1.2.cmml" xref="S7.Thmtheorem25.p1.2.2.m2.1.1.1.1.2">ℎ</ci><ci id="S7.Thmtheorem25.p1.2.2.m2.1.1.1.1.4.cmml" xref="S7.Thmtheorem25.p1.2.2.m2.1.1.1.1.4">𝑚</ci></apply><apply id="S7.Thmtheorem25.p1.2.2.m2.1.1.1.1c.cmml" xref="S7.Thmtheorem25.p1.2.2.m2.1.1.1"><ci id="S7.Thmtheorem25.p1.2.2.m2.1.1.1.1.5.cmml" xref="S7.Thmtheorem25.p1.2.2.m2.1.1.1.1.5">:</ci><share href="https://arxiv.org/html/2411.12694v2#S7.Thmtheorem25.p1.2.2.m2.1.1.1.1.4.cmml" id="S7.Thmtheorem25.p1.2.2.m2.1.1.1.1d.cmml" xref="S7.Thmtheorem25.p1.2.2.m2.1.1.1"></share><cn id="S7.Thmtheorem25.p1.2.2.m2.1.1.1.1.6.cmml" type="integer" xref="S7.Thmtheorem25.p1.2.2.m2.1.1.1.1.6">0</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.Thmtheorem25.p1.2.2.m2.1c">(h:m:0)</annotation><annotation encoding="application/x-llamapun" id="S7.Thmtheorem25.p1.2.2.m2.1d">( italic_h : italic_m : 0 )</annotation></semantics></math>, a vertex <math alttext="v\not\in T_{m}" class="ltx_Math" display="inline" id="S7.Thmtheorem25.p1.3.3.m3.1"><semantics id="S7.Thmtheorem25.p1.3.3.m3.1a"><mrow id="S7.Thmtheorem25.p1.3.3.m3.1.1" xref="S7.Thmtheorem25.p1.3.3.m3.1.1.cmml"><mi id="S7.Thmtheorem25.p1.3.3.m3.1.1.2" xref="S7.Thmtheorem25.p1.3.3.m3.1.1.2.cmml">v</mi><mo id="S7.Thmtheorem25.p1.3.3.m3.1.1.1" xref="S7.Thmtheorem25.p1.3.3.m3.1.1.1.cmml">∉</mo><msub id="S7.Thmtheorem25.p1.3.3.m3.1.1.3" xref="S7.Thmtheorem25.p1.3.3.m3.1.1.3.cmml"><mi id="S7.Thmtheorem25.p1.3.3.m3.1.1.3.2" xref="S7.Thmtheorem25.p1.3.3.m3.1.1.3.2.cmml">T</mi><mi id="S7.Thmtheorem25.p1.3.3.m3.1.1.3.3" xref="S7.Thmtheorem25.p1.3.3.m3.1.1.3.3.cmml">m</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S7.Thmtheorem25.p1.3.3.m3.1b"><apply id="S7.Thmtheorem25.p1.3.3.m3.1.1.cmml" xref="S7.Thmtheorem25.p1.3.3.m3.1.1"><notin id="S7.Thmtheorem25.p1.3.3.m3.1.1.1.cmml" xref="S7.Thmtheorem25.p1.3.3.m3.1.1.1"></notin><ci id="S7.Thmtheorem25.p1.3.3.m3.1.1.2.cmml" xref="S7.Thmtheorem25.p1.3.3.m3.1.1.2">𝑣</ci><apply id="S7.Thmtheorem25.p1.3.3.m3.1.1.3.cmml" xref="S7.Thmtheorem25.p1.3.3.m3.1.1.3"><csymbol cd="ambiguous" id="S7.Thmtheorem25.p1.3.3.m3.1.1.3.1.cmml" xref="S7.Thmtheorem25.p1.3.3.m3.1.1.3">subscript</csymbol><ci id="S7.Thmtheorem25.p1.3.3.m3.1.1.3.2.cmml" xref="S7.Thmtheorem25.p1.3.3.m3.1.1.3.2">𝑇</ci><ci id="S7.Thmtheorem25.p1.3.3.m3.1.1.3.3.cmml" xref="S7.Thmtheorem25.p1.3.3.m3.1.1.3.3">𝑚</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.Thmtheorem25.p1.3.3.m3.1c">v\not\in T_{m}</annotation><annotation encoding="application/x-llamapun" id="S7.Thmtheorem25.p1.3.3.m3.1d">italic_v ∉ italic_T start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT</annotation></semantics></math> then this can because of two reasons. Either:</span></p> <ol class="ltx_enumerate" id="S7.I12"> <li class="ltx_item" id="S7.I12.i1" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">1.</span> <div class="ltx_para" id="S7.I12.i1.p1"> <p class="ltx_p" id="S7.I12.i1.p1.2"><math alttext="v" class="ltx_Math" display="inline" id="S7.I12.i1.p1.1.m1.1"><semantics id="S7.I12.i1.p1.1.m1.1a"><mi id="S7.I12.i1.p1.1.m1.1.1" xref="S7.I12.i1.p1.1.m1.1.1.cmml">v</mi><annotation-xml encoding="MathML-Content" id="S7.I12.i1.p1.1.m1.1b"><ci id="S7.I12.i1.p1.1.m1.1.1.cmml" xref="S7.I12.i1.p1.1.m1.1.1">𝑣</ci></annotation-xml><annotation encoding="application/x-tex" id="S7.I12.i1.p1.1.m1.1c">v</annotation><annotation encoding="application/x-llamapun" id="S7.I12.i1.p1.1.m1.1d">italic_v</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S7.I12.i1.p1.2.1"> did not decrease their level in </span><span class="ltx_text ltx_font_smallcaps" id="S7.I12.i1.p1.2.2">minute</span><span class="ltx_text ltx_font_italic" id="S7.I12.i1.p1.2.3"> </span><math alttext="m-1" class="ltx_Math" display="inline" id="S7.I12.i1.p1.2.m2.1"><semantics id="S7.I12.i1.p1.2.m2.1a"><mrow id="S7.I12.i1.p1.2.m2.1.1" xref="S7.I12.i1.p1.2.m2.1.1.cmml"><mi id="S7.I12.i1.p1.2.m2.1.1.2" xref="S7.I12.i1.p1.2.m2.1.1.2.cmml">m</mi><mo id="S7.I12.i1.p1.2.m2.1.1.1" xref="S7.I12.i1.p1.2.m2.1.1.1.cmml">−</mo><mn id="S7.I12.i1.p1.2.m2.1.1.3" xref="S7.I12.i1.p1.2.m2.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S7.I12.i1.p1.2.m2.1b"><apply id="S7.I12.i1.p1.2.m2.1.1.cmml" xref="S7.I12.i1.p1.2.m2.1.1"><minus id="S7.I12.i1.p1.2.m2.1.1.1.cmml" xref="S7.I12.i1.p1.2.m2.1.1.1"></minus><ci id="S7.I12.i1.p1.2.m2.1.1.2.cmml" xref="S7.I12.i1.p1.2.m2.1.1.2">𝑚</ci><cn id="S7.I12.i1.p1.2.m2.1.1.3.cmml" type="integer" xref="S7.I12.i1.p1.2.m2.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.I12.i1.p1.2.m2.1c">m-1</annotation><annotation encoding="application/x-llamapun" id="S7.I12.i1.p1.2.m2.1d">italic_m - 1</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S7.I12.i1.p1.2.4">, or</span></p> </div> </li> <li class="ltx_item" id="S7.I12.i2" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">2.</span> <div class="ltx_para" id="S7.I12.i2.p1"> <p class="ltx_p" id="S7.I12.i2.p1.2"><math alttext="v" class="ltx_Math" display="inline" id="S7.I12.i2.p1.1.m1.1"><semantics id="S7.I12.i2.p1.1.m1.1a"><mi id="S7.I12.i2.p1.1.m1.1.1" xref="S7.I12.i2.p1.1.m1.1.1.cmml">v</mi><annotation-xml encoding="MathML-Content" id="S7.I12.i2.p1.1.m1.1b"><ci id="S7.I12.i2.p1.1.m1.1.1.cmml" xref="S7.I12.i2.p1.1.m1.1.1">𝑣</ci></annotation-xml><annotation encoding="application/x-tex" id="S7.I12.i2.p1.1.m1.1c">v</annotation><annotation encoding="application/x-llamapun" id="S7.I12.i2.p1.1.m1.1d">italic_v</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S7.I12.i2.p1.2.1"> did decrease their level in </span><span class="ltx_text ltx_font_smallcaps" id="S7.I12.i2.p1.2.2">minute</span><span class="ltx_text ltx_font_italic" id="S7.I12.i2.p1.2.3"> </span><math alttext="m-1" class="ltx_Math" display="inline" id="S7.I12.i2.p1.2.m2.1"><semantics id="S7.I12.i2.p1.2.m2.1a"><mrow id="S7.I12.i2.p1.2.m2.1.1" xref="S7.I12.i2.p1.2.m2.1.1.cmml"><mi id="S7.I12.i2.p1.2.m2.1.1.2" xref="S7.I12.i2.p1.2.m2.1.1.2.cmml">m</mi><mo id="S7.I12.i2.p1.2.m2.1.1.1" xref="S7.I12.i2.p1.2.m2.1.1.1.cmml">−</mo><mn id="S7.I12.i2.p1.2.m2.1.1.3" xref="S7.I12.i2.p1.2.m2.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S7.I12.i2.p1.2.m2.1b"><apply id="S7.I12.i2.p1.2.m2.1.1.cmml" xref="S7.I12.i2.p1.2.m2.1.1"><minus id="S7.I12.i2.p1.2.m2.1.1.1.cmml" xref="S7.I12.i2.p1.2.m2.1.1.1"></minus><ci id="S7.I12.i2.p1.2.m2.1.1.2.cmml" xref="S7.I12.i2.p1.2.m2.1.1.2">𝑚</ci><cn id="S7.I12.i2.p1.2.m2.1.1.3.cmml" type="integer" xref="S7.I12.i2.p1.2.m2.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.I12.i2.p1.2.m2.1c">m-1</annotation><annotation encoding="application/x-llamapun" id="S7.I12.i2.p1.2.m2.1d">italic_m - 1</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S7.I12.i2.p1.2.4"> but then had no incoming violating edges.</span></p> </div> </li> </ol> </div> <div class="ltx_para" id="S7.Thmtheorem25.p2"> <p class="ltx_p" id="S7.Thmtheorem25.p2.3"><span class="ltx_text ltx_font_italic" id="S7.Thmtheorem25.p2.3.3">We show that Case 1 implies that <math alttext="v" class="ltx_Math" display="inline" id="S7.Thmtheorem25.p2.1.1.m1.1"><semantics id="S7.Thmtheorem25.p2.1.1.m1.1a"><mi id="S7.Thmtheorem25.p2.1.1.m1.1.1" xref="S7.Thmtheorem25.p2.1.1.m1.1.1.cmml">v</mi><annotation-xml encoding="MathML-Content" id="S7.Thmtheorem25.p2.1.1.m1.1b"><ci id="S7.Thmtheorem25.p2.1.1.m1.1.1.cmml" xref="S7.Thmtheorem25.p2.1.1.m1.1.1">𝑣</ci></annotation-xml><annotation encoding="application/x-tex" id="S7.Thmtheorem25.p2.1.1.m1.1c">v</annotation><annotation encoding="application/x-llamapun" id="S7.Thmtheorem25.p2.1.1.m1.1d">italic_v</annotation></semantics></math> has no violating out-edges at <math alttext="(h:x:0)" class="ltx_Math" display="inline" id="S7.Thmtheorem25.p2.2.2.m2.1"><semantics id="S7.Thmtheorem25.p2.2.2.m2.1a"><mrow id="S7.Thmtheorem25.p2.2.2.m2.1.1.1" xref="S7.Thmtheorem25.p2.2.2.m2.1.1.1.1.cmml"><mo id="S7.Thmtheorem25.p2.2.2.m2.1.1.1.2" stretchy="false" xref="S7.Thmtheorem25.p2.2.2.m2.1.1.1.1.cmml">(</mo><mrow id="S7.Thmtheorem25.p2.2.2.m2.1.1.1.1" xref="S7.Thmtheorem25.p2.2.2.m2.1.1.1.1.cmml"><mi id="S7.Thmtheorem25.p2.2.2.m2.1.1.1.1.2" xref="S7.Thmtheorem25.p2.2.2.m2.1.1.1.1.2.cmml">h</mi><mo id="S7.Thmtheorem25.p2.2.2.m2.1.1.1.1.3" lspace="0.278em" rspace="0.278em" xref="S7.Thmtheorem25.p2.2.2.m2.1.1.1.1.3.cmml">:</mo><mi id="S7.Thmtheorem25.p2.2.2.m2.1.1.1.1.4" xref="S7.Thmtheorem25.p2.2.2.m2.1.1.1.1.4.cmml">x</mi><mo id="S7.Thmtheorem25.p2.2.2.m2.1.1.1.1.5" lspace="0.278em" rspace="0.278em" xref="S7.Thmtheorem25.p2.2.2.m2.1.1.1.1.5.cmml">:</mo><mn id="S7.Thmtheorem25.p2.2.2.m2.1.1.1.1.6" xref="S7.Thmtheorem25.p2.2.2.m2.1.1.1.1.6.cmml">0</mn></mrow><mo id="S7.Thmtheorem25.p2.2.2.m2.1.1.1.3" stretchy="false" xref="S7.Thmtheorem25.p2.2.2.m2.1.1.1.1.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S7.Thmtheorem25.p2.2.2.m2.1b"><apply id="S7.Thmtheorem25.p2.2.2.m2.1.1.1.1.cmml" xref="S7.Thmtheorem25.p2.2.2.m2.1.1.1"><and id="S7.Thmtheorem25.p2.2.2.m2.1.1.1.1a.cmml" xref="S7.Thmtheorem25.p2.2.2.m2.1.1.1"></and><apply id="S7.Thmtheorem25.p2.2.2.m2.1.1.1.1b.cmml" xref="S7.Thmtheorem25.p2.2.2.m2.1.1.1"><ci id="S7.Thmtheorem25.p2.2.2.m2.1.1.1.1.3.cmml" xref="S7.Thmtheorem25.p2.2.2.m2.1.1.1.1.3">:</ci><ci id="S7.Thmtheorem25.p2.2.2.m2.1.1.1.1.2.cmml" xref="S7.Thmtheorem25.p2.2.2.m2.1.1.1.1.2">ℎ</ci><ci id="S7.Thmtheorem25.p2.2.2.m2.1.1.1.1.4.cmml" xref="S7.Thmtheorem25.p2.2.2.m2.1.1.1.1.4">𝑥</ci></apply><apply id="S7.Thmtheorem25.p2.2.2.m2.1.1.1.1c.cmml" xref="S7.Thmtheorem25.p2.2.2.m2.1.1.1"><ci id="S7.Thmtheorem25.p2.2.2.m2.1.1.1.1.5.cmml" xref="S7.Thmtheorem25.p2.2.2.m2.1.1.1.1.5">:</ci><share href="https://arxiv.org/html/2411.12694v2#S7.Thmtheorem25.p2.2.2.m2.1.1.1.1.4.cmml" id="S7.Thmtheorem25.p2.2.2.m2.1.1.1.1d.cmml" xref="S7.Thmtheorem25.p2.2.2.m2.1.1.1"></share><cn id="S7.Thmtheorem25.p2.2.2.m2.1.1.1.1.6.cmml" type="integer" xref="S7.Thmtheorem25.p2.2.2.m2.1.1.1.1.6">0</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.Thmtheorem25.p2.2.2.m2.1c">(h:x:0)</annotation><annotation encoding="application/x-llamapun" id="S7.Thmtheorem25.p2.2.2.m2.1d">( italic_h : italic_x : 0 )</annotation></semantics></math>. Case 2 implies that <math alttext="l_{x}(v)\neq h" class="ltx_Math" display="inline" id="S7.Thmtheorem25.p2.3.3.m3.1"><semantics id="S7.Thmtheorem25.p2.3.3.m3.1a"><mrow id="S7.Thmtheorem25.p2.3.3.m3.1.2" xref="S7.Thmtheorem25.p2.3.3.m3.1.2.cmml"><mrow id="S7.Thmtheorem25.p2.3.3.m3.1.2.2" xref="S7.Thmtheorem25.p2.3.3.m3.1.2.2.cmml"><msub id="S7.Thmtheorem25.p2.3.3.m3.1.2.2.2" xref="S7.Thmtheorem25.p2.3.3.m3.1.2.2.2.cmml"><mi id="S7.Thmtheorem25.p2.3.3.m3.1.2.2.2.2" xref="S7.Thmtheorem25.p2.3.3.m3.1.2.2.2.2.cmml">l</mi><mi id="S7.Thmtheorem25.p2.3.3.m3.1.2.2.2.3" xref="S7.Thmtheorem25.p2.3.3.m3.1.2.2.2.3.cmml">x</mi></msub><mo id="S7.Thmtheorem25.p2.3.3.m3.1.2.2.1" xref="S7.Thmtheorem25.p2.3.3.m3.1.2.2.1.cmml">⁢</mo><mrow id="S7.Thmtheorem25.p2.3.3.m3.1.2.2.3.2" xref="S7.Thmtheorem25.p2.3.3.m3.1.2.2.cmml"><mo id="S7.Thmtheorem25.p2.3.3.m3.1.2.2.3.2.1" stretchy="false" xref="S7.Thmtheorem25.p2.3.3.m3.1.2.2.cmml">(</mo><mi id="S7.Thmtheorem25.p2.3.3.m3.1.1" xref="S7.Thmtheorem25.p2.3.3.m3.1.1.cmml">v</mi><mo id="S7.Thmtheorem25.p2.3.3.m3.1.2.2.3.2.2" stretchy="false" xref="S7.Thmtheorem25.p2.3.3.m3.1.2.2.cmml">)</mo></mrow></mrow><mo id="S7.Thmtheorem25.p2.3.3.m3.1.2.1" xref="S7.Thmtheorem25.p2.3.3.m3.1.2.1.cmml">≠</mo><mi id="S7.Thmtheorem25.p2.3.3.m3.1.2.3" xref="S7.Thmtheorem25.p2.3.3.m3.1.2.3.cmml">h</mi></mrow><annotation-xml encoding="MathML-Content" id="S7.Thmtheorem25.p2.3.3.m3.1b"><apply id="S7.Thmtheorem25.p2.3.3.m3.1.2.cmml" xref="S7.Thmtheorem25.p2.3.3.m3.1.2"><neq id="S7.Thmtheorem25.p2.3.3.m3.1.2.1.cmml" xref="S7.Thmtheorem25.p2.3.3.m3.1.2.1"></neq><apply id="S7.Thmtheorem25.p2.3.3.m3.1.2.2.cmml" xref="S7.Thmtheorem25.p2.3.3.m3.1.2.2"><times id="S7.Thmtheorem25.p2.3.3.m3.1.2.2.1.cmml" xref="S7.Thmtheorem25.p2.3.3.m3.1.2.2.1"></times><apply id="S7.Thmtheorem25.p2.3.3.m3.1.2.2.2.cmml" xref="S7.Thmtheorem25.p2.3.3.m3.1.2.2.2"><csymbol cd="ambiguous" id="S7.Thmtheorem25.p2.3.3.m3.1.2.2.2.1.cmml" xref="S7.Thmtheorem25.p2.3.3.m3.1.2.2.2">subscript</csymbol><ci id="S7.Thmtheorem25.p2.3.3.m3.1.2.2.2.2.cmml" xref="S7.Thmtheorem25.p2.3.3.m3.1.2.2.2.2">𝑙</ci><ci id="S7.Thmtheorem25.p2.3.3.m3.1.2.2.2.3.cmml" xref="S7.Thmtheorem25.p2.3.3.m3.1.2.2.2.3">𝑥</ci></apply><ci id="S7.Thmtheorem25.p2.3.3.m3.1.1.cmml" xref="S7.Thmtheorem25.p2.3.3.m3.1.1">𝑣</ci></apply><ci id="S7.Thmtheorem25.p2.3.3.m3.1.2.3.cmml" xref="S7.Thmtheorem25.p2.3.3.m3.1.2.3">ℎ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.Thmtheorem25.p2.3.3.m3.1c">l_{x}(v)\neq h</annotation><annotation encoding="application/x-llamapun" id="S7.Thmtheorem25.p2.3.3.m3.1d">italic_l start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT ( italic_v ) ≠ italic_h</annotation></semantics></math> and this together implies the lemma.</span></p> </div> </div> <section class="ltx_subparagraph" id="S7.SS4.SSS0.P0.SPx1"> <h5 class="ltx_title ltx_font_italic ltx_title_subparagraph">Case 1:</h5> <div class="ltx_para" id="S7.SS4.SSS0.P0.SPx1.p1"> <p class="ltx_p" id="S7.SS4.SSS0.P0.SPx1.p1.13"><span class="ltx_text ltx_font_italic" id="S7.SS4.SSS0.P0.SPx1.p1.13.1">Suppose that </span><math alttext="l_{m-1}(v)=h" class="ltx_Math" display="inline" id="S7.SS4.SSS0.P0.SPx1.p1.1.m1.1"><semantics id="S7.SS4.SSS0.P0.SPx1.p1.1.m1.1a"><mrow id="S7.SS4.SSS0.P0.SPx1.p1.1.m1.1.2" xref="S7.SS4.SSS0.P0.SPx1.p1.1.m1.1.2.cmml"><mrow id="S7.SS4.SSS0.P0.SPx1.p1.1.m1.1.2.2" xref="S7.SS4.SSS0.P0.SPx1.p1.1.m1.1.2.2.cmml"><msub id="S7.SS4.SSS0.P0.SPx1.p1.1.m1.1.2.2.2" xref="S7.SS4.SSS0.P0.SPx1.p1.1.m1.1.2.2.2.cmml"><mi id="S7.SS4.SSS0.P0.SPx1.p1.1.m1.1.2.2.2.2" xref="S7.SS4.SSS0.P0.SPx1.p1.1.m1.1.2.2.2.2.cmml">l</mi><mrow id="S7.SS4.SSS0.P0.SPx1.p1.1.m1.1.2.2.2.3" xref="S7.SS4.SSS0.P0.SPx1.p1.1.m1.1.2.2.2.3.cmml"><mi id="S7.SS4.SSS0.P0.SPx1.p1.1.m1.1.2.2.2.3.2" xref="S7.SS4.SSS0.P0.SPx1.p1.1.m1.1.2.2.2.3.2.cmml">m</mi><mo id="S7.SS4.SSS0.P0.SPx1.p1.1.m1.1.2.2.2.3.1" xref="S7.SS4.SSS0.P0.SPx1.p1.1.m1.1.2.2.2.3.1.cmml">−</mo><mn id="S7.SS4.SSS0.P0.SPx1.p1.1.m1.1.2.2.2.3.3" xref="S7.SS4.SSS0.P0.SPx1.p1.1.m1.1.2.2.2.3.3.cmml">1</mn></mrow></msub><mo id="S7.SS4.SSS0.P0.SPx1.p1.1.m1.1.2.2.1" xref="S7.SS4.SSS0.P0.SPx1.p1.1.m1.1.2.2.1.cmml">⁢</mo><mrow id="S7.SS4.SSS0.P0.SPx1.p1.1.m1.1.2.2.3.2" xref="S7.SS4.SSS0.P0.SPx1.p1.1.m1.1.2.2.cmml"><mo id="S7.SS4.SSS0.P0.SPx1.p1.1.m1.1.2.2.3.2.1" stretchy="false" xref="S7.SS4.SSS0.P0.SPx1.p1.1.m1.1.2.2.cmml">(</mo><mi id="S7.SS4.SSS0.P0.SPx1.p1.1.m1.1.1" xref="S7.SS4.SSS0.P0.SPx1.p1.1.m1.1.1.cmml">v</mi><mo id="S7.SS4.SSS0.P0.SPx1.p1.1.m1.1.2.2.3.2.2" stretchy="false" xref="S7.SS4.SSS0.P0.SPx1.p1.1.m1.1.2.2.cmml">)</mo></mrow></mrow><mo id="S7.SS4.SSS0.P0.SPx1.p1.1.m1.1.2.1" xref="S7.SS4.SSS0.P0.SPx1.p1.1.m1.1.2.1.cmml">=</mo><mi id="S7.SS4.SSS0.P0.SPx1.p1.1.m1.1.2.3" xref="S7.SS4.SSS0.P0.SPx1.p1.1.m1.1.2.3.cmml">h</mi></mrow><annotation-xml encoding="MathML-Content" id="S7.SS4.SSS0.P0.SPx1.p1.1.m1.1b"><apply id="S7.SS4.SSS0.P0.SPx1.p1.1.m1.1.2.cmml" xref="S7.SS4.SSS0.P0.SPx1.p1.1.m1.1.2"><eq id="S7.SS4.SSS0.P0.SPx1.p1.1.m1.1.2.1.cmml" xref="S7.SS4.SSS0.P0.SPx1.p1.1.m1.1.2.1"></eq><apply id="S7.SS4.SSS0.P0.SPx1.p1.1.m1.1.2.2.cmml" xref="S7.SS4.SSS0.P0.SPx1.p1.1.m1.1.2.2"><times id="S7.SS4.SSS0.P0.SPx1.p1.1.m1.1.2.2.1.cmml" xref="S7.SS4.SSS0.P0.SPx1.p1.1.m1.1.2.2.1"></times><apply id="S7.SS4.SSS0.P0.SPx1.p1.1.m1.1.2.2.2.cmml" xref="S7.SS4.SSS0.P0.SPx1.p1.1.m1.1.2.2.2"><csymbol cd="ambiguous" id="S7.SS4.SSS0.P0.SPx1.p1.1.m1.1.2.2.2.1.cmml" xref="S7.SS4.SSS0.P0.SPx1.p1.1.m1.1.2.2.2">subscript</csymbol><ci id="S7.SS4.SSS0.P0.SPx1.p1.1.m1.1.2.2.2.2.cmml" xref="S7.SS4.SSS0.P0.SPx1.p1.1.m1.1.2.2.2.2">𝑙</ci><apply id="S7.SS4.SSS0.P0.SPx1.p1.1.m1.1.2.2.2.3.cmml" xref="S7.SS4.SSS0.P0.SPx1.p1.1.m1.1.2.2.2.3"><minus id="S7.SS4.SSS0.P0.SPx1.p1.1.m1.1.2.2.2.3.1.cmml" xref="S7.SS4.SSS0.P0.SPx1.p1.1.m1.1.2.2.2.3.1"></minus><ci id="S7.SS4.SSS0.P0.SPx1.p1.1.m1.1.2.2.2.3.2.cmml" xref="S7.SS4.SSS0.P0.SPx1.p1.1.m1.1.2.2.2.3.2">𝑚</ci><cn id="S7.SS4.SSS0.P0.SPx1.p1.1.m1.1.2.2.2.3.3.cmml" type="integer" xref="S7.SS4.SSS0.P0.SPx1.p1.1.m1.1.2.2.2.3.3">1</cn></apply></apply><ci id="S7.SS4.SSS0.P0.SPx1.p1.1.m1.1.1.cmml" xref="S7.SS4.SSS0.P0.SPx1.p1.1.m1.1.1">𝑣</ci></apply><ci id="S7.SS4.SSS0.P0.SPx1.p1.1.m1.1.2.3.cmml" xref="S7.SS4.SSS0.P0.SPx1.p1.1.m1.1.2.3">ℎ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.SS4.SSS0.P0.SPx1.p1.1.m1.1c">l_{m-1}(v)=h</annotation><annotation encoding="application/x-llamapun" id="S7.SS4.SSS0.P0.SPx1.p1.1.m1.1d">italic_l start_POSTSUBSCRIPT italic_m - 1 end_POSTSUBSCRIPT ( italic_v ) = italic_h</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S7.SS4.SSS0.P0.SPx1.p1.13.2"> and that at the start of </span><math alttext="(h:m-1:0)" class="ltx_Math" display="inline" id="S7.SS4.SSS0.P0.SPx1.p1.2.m2.1"><semantics id="S7.SS4.SSS0.P0.SPx1.p1.2.m2.1a"><mrow id="S7.SS4.SSS0.P0.SPx1.p1.2.m2.1.1.1" xref="S7.SS4.SSS0.P0.SPx1.p1.2.m2.1.1.1.1.cmml"><mo id="S7.SS4.SSS0.P0.SPx1.p1.2.m2.1.1.1.2" stretchy="false" xref="S7.SS4.SSS0.P0.SPx1.p1.2.m2.1.1.1.1.cmml">(</mo><mrow id="S7.SS4.SSS0.P0.SPx1.p1.2.m2.1.1.1.1" xref="S7.SS4.SSS0.P0.SPx1.p1.2.m2.1.1.1.1.cmml"><mi id="S7.SS4.SSS0.P0.SPx1.p1.2.m2.1.1.1.1.2" xref="S7.SS4.SSS0.P0.SPx1.p1.2.m2.1.1.1.1.2.cmml">h</mi><mo id="S7.SS4.SSS0.P0.SPx1.p1.2.m2.1.1.1.1.3" lspace="0.278em" rspace="0.278em" xref="S7.SS4.SSS0.P0.SPx1.p1.2.m2.1.1.1.1.3.cmml">:</mo><mrow id="S7.SS4.SSS0.P0.SPx1.p1.2.m2.1.1.1.1.4" xref="S7.SS4.SSS0.P0.SPx1.p1.2.m2.1.1.1.1.4.cmml"><mi id="S7.SS4.SSS0.P0.SPx1.p1.2.m2.1.1.1.1.4.2" xref="S7.SS4.SSS0.P0.SPx1.p1.2.m2.1.1.1.1.4.2.cmml">m</mi><mo id="S7.SS4.SSS0.P0.SPx1.p1.2.m2.1.1.1.1.4.1" xref="S7.SS4.SSS0.P0.SPx1.p1.2.m2.1.1.1.1.4.1.cmml">−</mo><mn id="S7.SS4.SSS0.P0.SPx1.p1.2.m2.1.1.1.1.4.3" xref="S7.SS4.SSS0.P0.SPx1.p1.2.m2.1.1.1.1.4.3.cmml">1</mn></mrow><mo id="S7.SS4.SSS0.P0.SPx1.p1.2.m2.1.1.1.1.5" lspace="0.278em" rspace="0.278em" xref="S7.SS4.SSS0.P0.SPx1.p1.2.m2.1.1.1.1.5.cmml">:</mo><mn id="S7.SS4.SSS0.P0.SPx1.p1.2.m2.1.1.1.1.6" xref="S7.SS4.SSS0.P0.SPx1.p1.2.m2.1.1.1.1.6.cmml">0</mn></mrow><mo id="S7.SS4.SSS0.P0.SPx1.p1.2.m2.1.1.1.3" stretchy="false" xref="S7.SS4.SSS0.P0.SPx1.p1.2.m2.1.1.1.1.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S7.SS4.SSS0.P0.SPx1.p1.2.m2.1b"><apply id="S7.SS4.SSS0.P0.SPx1.p1.2.m2.1.1.1.1.cmml" xref="S7.SS4.SSS0.P0.SPx1.p1.2.m2.1.1.1"><and id="S7.SS4.SSS0.P0.SPx1.p1.2.m2.1.1.1.1a.cmml" xref="S7.SS4.SSS0.P0.SPx1.p1.2.m2.1.1.1"></and><apply id="S7.SS4.SSS0.P0.SPx1.p1.2.m2.1.1.1.1b.cmml" xref="S7.SS4.SSS0.P0.SPx1.p1.2.m2.1.1.1"><ci id="S7.SS4.SSS0.P0.SPx1.p1.2.m2.1.1.1.1.3.cmml" xref="S7.SS4.SSS0.P0.SPx1.p1.2.m2.1.1.1.1.3">:</ci><ci id="S7.SS4.SSS0.P0.SPx1.p1.2.m2.1.1.1.1.2.cmml" xref="S7.SS4.SSS0.P0.SPx1.p1.2.m2.1.1.1.1.2">ℎ</ci><apply id="S7.SS4.SSS0.P0.SPx1.p1.2.m2.1.1.1.1.4.cmml" xref="S7.SS4.SSS0.P0.SPx1.p1.2.m2.1.1.1.1.4"><minus id="S7.SS4.SSS0.P0.SPx1.p1.2.m2.1.1.1.1.4.1.cmml" xref="S7.SS4.SSS0.P0.SPx1.p1.2.m2.1.1.1.1.4.1"></minus><ci id="S7.SS4.SSS0.P0.SPx1.p1.2.m2.1.1.1.1.4.2.cmml" xref="S7.SS4.SSS0.P0.SPx1.p1.2.m2.1.1.1.1.4.2">𝑚</ci><cn id="S7.SS4.SSS0.P0.SPx1.p1.2.m2.1.1.1.1.4.3.cmml" type="integer" xref="S7.SS4.SSS0.P0.SPx1.p1.2.m2.1.1.1.1.4.3">1</cn></apply></apply><apply id="S7.SS4.SSS0.P0.SPx1.p1.2.m2.1.1.1.1c.cmml" xref="S7.SS4.SSS0.P0.SPx1.p1.2.m2.1.1.1"><ci id="S7.SS4.SSS0.P0.SPx1.p1.2.m2.1.1.1.1.5.cmml" xref="S7.SS4.SSS0.P0.SPx1.p1.2.m2.1.1.1.1.5">:</ci><share href="https://arxiv.org/html/2411.12694v2#S7.SS4.SSS0.P0.SPx1.p1.2.m2.1.1.1.1.4.cmml" id="S7.SS4.SSS0.P0.SPx1.p1.2.m2.1.1.1.1d.cmml" xref="S7.SS4.SSS0.P0.SPx1.p1.2.m2.1.1.1"></share><cn id="S7.SS4.SSS0.P0.SPx1.p1.2.m2.1.1.1.1.6.cmml" type="integer" xref="S7.SS4.SSS0.P0.SPx1.p1.2.m2.1.1.1.1.6">0</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.SS4.SSS0.P0.SPx1.p1.2.m2.1c">(h:m-1:0)</annotation><annotation encoding="application/x-llamapun" id="S7.SS4.SSS0.P0.SPx1.p1.2.m2.1d">( italic_h : italic_m - 1 : 0 )</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S7.SS4.SSS0.P0.SPx1.p1.13.3"> the vertex </span><math alttext="v" class="ltx_Math" display="inline" id="S7.SS4.SSS0.P0.SPx1.p1.3.m3.1"><semantics id="S7.SS4.SSS0.P0.SPx1.p1.3.m3.1a"><mi id="S7.SS4.SSS0.P0.SPx1.p1.3.m3.1.1" xref="S7.SS4.SSS0.P0.SPx1.p1.3.m3.1.1.cmml">v</mi><annotation-xml encoding="MathML-Content" id="S7.SS4.SSS0.P0.SPx1.p1.3.m3.1b"><ci id="S7.SS4.SSS0.P0.SPx1.p1.3.m3.1.1.cmml" xref="S7.SS4.SSS0.P0.SPx1.p1.3.m3.1.1">𝑣</ci></annotation-xml><annotation encoding="application/x-tex" id="S7.SS4.SSS0.P0.SPx1.p1.3.m3.1c">v</annotation><annotation encoding="application/x-llamapun" id="S7.SS4.SSS0.P0.SPx1.p1.3.m3.1d">italic_v</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S7.SS4.SSS0.P0.SPx1.p1.13.4"> has no violating out-edges. Then </span><math alttext="v" class="ltx_Math" display="inline" id="S7.SS4.SSS0.P0.SPx1.p1.4.m4.1"><semantics id="S7.SS4.SSS0.P0.SPx1.p1.4.m4.1a"><mi id="S7.SS4.SSS0.P0.SPx1.p1.4.m4.1.1" xref="S7.SS4.SSS0.P0.SPx1.p1.4.m4.1.1.cmml">v</mi><annotation-xml encoding="MathML-Content" id="S7.SS4.SSS0.P0.SPx1.p1.4.m4.1b"><ci id="S7.SS4.SSS0.P0.SPx1.p1.4.m4.1.1.cmml" xref="S7.SS4.SSS0.P0.SPx1.p1.4.m4.1.1">𝑣</ci></annotation-xml><annotation encoding="application/x-tex" id="S7.SS4.SSS0.P0.SPx1.p1.4.m4.1c">v</annotation><annotation encoding="application/x-llamapun" id="S7.SS4.SSS0.P0.SPx1.p1.4.m4.1d">italic_v</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S7.SS4.SSS0.P0.SPx1.p1.13.5"> does not change its level during </span><span class="ltx_text ltx_font_smallcaps" id="S7.SS4.SSS0.P0.SPx1.p1.13.6">minute</span><span class="ltx_text ltx_font_italic" id="S7.SS4.SSS0.P0.SPx1.p1.13.7"> </span><math alttext="m-1" class="ltx_Math" display="inline" id="S7.SS4.SSS0.P0.SPx1.p1.5.m5.1"><semantics id="S7.SS4.SSS0.P0.SPx1.p1.5.m5.1a"><mrow id="S7.SS4.SSS0.P0.SPx1.p1.5.m5.1.1" xref="S7.SS4.SSS0.P0.SPx1.p1.5.m5.1.1.cmml"><mi id="S7.SS4.SSS0.P0.SPx1.p1.5.m5.1.1.2" xref="S7.SS4.SSS0.P0.SPx1.p1.5.m5.1.1.2.cmml">m</mi><mo id="S7.SS4.SSS0.P0.SPx1.p1.5.m5.1.1.1" xref="S7.SS4.SSS0.P0.SPx1.p1.5.m5.1.1.1.cmml">−</mo><mn id="S7.SS4.SSS0.P0.SPx1.p1.5.m5.1.1.3" xref="S7.SS4.SSS0.P0.SPx1.p1.5.m5.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S7.SS4.SSS0.P0.SPx1.p1.5.m5.1b"><apply id="S7.SS4.SSS0.P0.SPx1.p1.5.m5.1.1.cmml" xref="S7.SS4.SSS0.P0.SPx1.p1.5.m5.1.1"><minus id="S7.SS4.SSS0.P0.SPx1.p1.5.m5.1.1.1.cmml" xref="S7.SS4.SSS0.P0.SPx1.p1.5.m5.1.1.1"></minus><ci id="S7.SS4.SSS0.P0.SPx1.p1.5.m5.1.1.2.cmml" xref="S7.SS4.SSS0.P0.SPx1.p1.5.m5.1.1.2">𝑚</ci><cn id="S7.SS4.SSS0.P0.SPx1.p1.5.m5.1.1.3.cmml" type="integer" xref="S7.SS4.SSS0.P0.SPx1.p1.5.m5.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.SS4.SSS0.P0.SPx1.p1.5.m5.1c">m-1</annotation><annotation encoding="application/x-llamapun" id="S7.SS4.SSS0.P0.SPx1.p1.5.m5.1d">italic_m - 1</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S7.SS4.SSS0.P0.SPx1.p1.13.8"> and </span><math alttext="l_{m}(v)=h" class="ltx_Math" display="inline" id="S7.SS4.SSS0.P0.SPx1.p1.6.m6.1"><semantics id="S7.SS4.SSS0.P0.SPx1.p1.6.m6.1a"><mrow id="S7.SS4.SSS0.P0.SPx1.p1.6.m6.1.2" xref="S7.SS4.SSS0.P0.SPx1.p1.6.m6.1.2.cmml"><mrow id="S7.SS4.SSS0.P0.SPx1.p1.6.m6.1.2.2" xref="S7.SS4.SSS0.P0.SPx1.p1.6.m6.1.2.2.cmml"><msub id="S7.SS4.SSS0.P0.SPx1.p1.6.m6.1.2.2.2" xref="S7.SS4.SSS0.P0.SPx1.p1.6.m6.1.2.2.2.cmml"><mi id="S7.SS4.SSS0.P0.SPx1.p1.6.m6.1.2.2.2.2" xref="S7.SS4.SSS0.P0.SPx1.p1.6.m6.1.2.2.2.2.cmml">l</mi><mi id="S7.SS4.SSS0.P0.SPx1.p1.6.m6.1.2.2.2.3" xref="S7.SS4.SSS0.P0.SPx1.p1.6.m6.1.2.2.2.3.cmml">m</mi></msub><mo id="S7.SS4.SSS0.P0.SPx1.p1.6.m6.1.2.2.1" xref="S7.SS4.SSS0.P0.SPx1.p1.6.m6.1.2.2.1.cmml">⁢</mo><mrow id="S7.SS4.SSS0.P0.SPx1.p1.6.m6.1.2.2.3.2" xref="S7.SS4.SSS0.P0.SPx1.p1.6.m6.1.2.2.cmml"><mo id="S7.SS4.SSS0.P0.SPx1.p1.6.m6.1.2.2.3.2.1" stretchy="false" xref="S7.SS4.SSS0.P0.SPx1.p1.6.m6.1.2.2.cmml">(</mo><mi id="S7.SS4.SSS0.P0.SPx1.p1.6.m6.1.1" xref="S7.SS4.SSS0.P0.SPx1.p1.6.m6.1.1.cmml">v</mi><mo id="S7.SS4.SSS0.P0.SPx1.p1.6.m6.1.2.2.3.2.2" stretchy="false" xref="S7.SS4.SSS0.P0.SPx1.p1.6.m6.1.2.2.cmml">)</mo></mrow></mrow><mo id="S7.SS4.SSS0.P0.SPx1.p1.6.m6.1.2.1" xref="S7.SS4.SSS0.P0.SPx1.p1.6.m6.1.2.1.cmml">=</mo><mi id="S7.SS4.SSS0.P0.SPx1.p1.6.m6.1.2.3" xref="S7.SS4.SSS0.P0.SPx1.p1.6.m6.1.2.3.cmml">h</mi></mrow><annotation-xml encoding="MathML-Content" id="S7.SS4.SSS0.P0.SPx1.p1.6.m6.1b"><apply id="S7.SS4.SSS0.P0.SPx1.p1.6.m6.1.2.cmml" xref="S7.SS4.SSS0.P0.SPx1.p1.6.m6.1.2"><eq id="S7.SS4.SSS0.P0.SPx1.p1.6.m6.1.2.1.cmml" xref="S7.SS4.SSS0.P0.SPx1.p1.6.m6.1.2.1"></eq><apply id="S7.SS4.SSS0.P0.SPx1.p1.6.m6.1.2.2.cmml" xref="S7.SS4.SSS0.P0.SPx1.p1.6.m6.1.2.2"><times id="S7.SS4.SSS0.P0.SPx1.p1.6.m6.1.2.2.1.cmml" xref="S7.SS4.SSS0.P0.SPx1.p1.6.m6.1.2.2.1"></times><apply id="S7.SS4.SSS0.P0.SPx1.p1.6.m6.1.2.2.2.cmml" xref="S7.SS4.SSS0.P0.SPx1.p1.6.m6.1.2.2.2"><csymbol cd="ambiguous" id="S7.SS4.SSS0.P0.SPx1.p1.6.m6.1.2.2.2.1.cmml" xref="S7.SS4.SSS0.P0.SPx1.p1.6.m6.1.2.2.2">subscript</csymbol><ci id="S7.SS4.SSS0.P0.SPx1.p1.6.m6.1.2.2.2.2.cmml" xref="S7.SS4.SSS0.P0.SPx1.p1.6.m6.1.2.2.2.2">𝑙</ci><ci id="S7.SS4.SSS0.P0.SPx1.p1.6.m6.1.2.2.2.3.cmml" xref="S7.SS4.SSS0.P0.SPx1.p1.6.m6.1.2.2.2.3">𝑚</ci></apply><ci id="S7.SS4.SSS0.P0.SPx1.p1.6.m6.1.1.cmml" xref="S7.SS4.SSS0.P0.SPx1.p1.6.m6.1.1">𝑣</ci></apply><ci id="S7.SS4.SSS0.P0.SPx1.p1.6.m6.1.2.3.cmml" xref="S7.SS4.SSS0.P0.SPx1.p1.6.m6.1.2.3">ℎ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.SS4.SSS0.P0.SPx1.p1.6.m6.1c">l_{m}(v)=h</annotation><annotation encoding="application/x-llamapun" id="S7.SS4.SSS0.P0.SPx1.p1.6.m6.1d">italic_l start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT ( italic_v ) = italic_h</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S7.SS4.SSS0.P0.SPx1.p1.13.9">. During odd </span><span class="ltx_text ltx_font_smallcaps" id="S7.SS4.SSS0.P0.SPx1.p1.13.10">minutes</span><span class="ltx_text ltx_font_italic" id="S7.SS4.SSS0.P0.SPx1.p1.13.11">, vertices at level </span><math alttext="h" class="ltx_Math" display="inline" id="S7.SS4.SSS0.P0.SPx1.p1.7.m7.1"><semantics id="S7.SS4.SSS0.P0.SPx1.p1.7.m7.1a"><mi id="S7.SS4.SSS0.P0.SPx1.p1.7.m7.1.1" xref="S7.SS4.SSS0.P0.SPx1.p1.7.m7.1.1.cmml">h</mi><annotation-xml encoding="MathML-Content" id="S7.SS4.SSS0.P0.SPx1.p1.7.m7.1b"><ci id="S7.SS4.SSS0.P0.SPx1.p1.7.m7.1.1.cmml" xref="S7.SS4.SSS0.P0.SPx1.p1.7.m7.1.1">ℎ</ci></annotation-xml><annotation encoding="application/x-tex" id="S7.SS4.SSS0.P0.SPx1.p1.7.m7.1c">h</annotation><annotation encoding="application/x-llamapun" id="S7.SS4.SSS0.P0.SPx1.p1.7.m7.1d">italic_h</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S7.SS4.SSS0.P0.SPx1.p1.13.12"> and levels </span><math alttext="k<h-1" class="ltx_Math" display="inline" id="S7.SS4.SSS0.P0.SPx1.p1.8.m8.1"><semantics id="S7.SS4.SSS0.P0.SPx1.p1.8.m8.1a"><mrow id="S7.SS4.SSS0.P0.SPx1.p1.8.m8.1.1" xref="S7.SS4.SSS0.P0.SPx1.p1.8.m8.1.1.cmml"><mi id="S7.SS4.SSS0.P0.SPx1.p1.8.m8.1.1.2" xref="S7.SS4.SSS0.P0.SPx1.p1.8.m8.1.1.2.cmml">k</mi><mo id="S7.SS4.SSS0.P0.SPx1.p1.8.m8.1.1.1" xref="S7.SS4.SSS0.P0.SPx1.p1.8.m8.1.1.1.cmml"><</mo><mrow id="S7.SS4.SSS0.P0.SPx1.p1.8.m8.1.1.3" xref="S7.SS4.SSS0.P0.SPx1.p1.8.m8.1.1.3.cmml"><mi id="S7.SS4.SSS0.P0.SPx1.p1.8.m8.1.1.3.2" xref="S7.SS4.SSS0.P0.SPx1.p1.8.m8.1.1.3.2.cmml">h</mi><mo id="S7.SS4.SSS0.P0.SPx1.p1.8.m8.1.1.3.1" xref="S7.SS4.SSS0.P0.SPx1.p1.8.m8.1.1.3.1.cmml">−</mo><mn id="S7.SS4.SSS0.P0.SPx1.p1.8.m8.1.1.3.3" xref="S7.SS4.SSS0.P0.SPx1.p1.8.m8.1.1.3.3.cmml">1</mn></mrow></mrow><annotation-xml encoding="MathML-Content" id="S7.SS4.SSS0.P0.SPx1.p1.8.m8.1b"><apply id="S7.SS4.SSS0.P0.SPx1.p1.8.m8.1.1.cmml" xref="S7.SS4.SSS0.P0.SPx1.p1.8.m8.1.1"><lt id="S7.SS4.SSS0.P0.SPx1.p1.8.m8.1.1.1.cmml" xref="S7.SS4.SSS0.P0.SPx1.p1.8.m8.1.1.1"></lt><ci id="S7.SS4.SSS0.P0.SPx1.p1.8.m8.1.1.2.cmml" xref="S7.SS4.SSS0.P0.SPx1.p1.8.m8.1.1.2">𝑘</ci><apply id="S7.SS4.SSS0.P0.SPx1.p1.8.m8.1.1.3.cmml" xref="S7.SS4.SSS0.P0.SPx1.p1.8.m8.1.1.3"><minus id="S7.SS4.SSS0.P0.SPx1.p1.8.m8.1.1.3.1.cmml" xref="S7.SS4.SSS0.P0.SPx1.p1.8.m8.1.1.3.1"></minus><ci id="S7.SS4.SSS0.P0.SPx1.p1.8.m8.1.1.3.2.cmml" xref="S7.SS4.SSS0.P0.SPx1.p1.8.m8.1.1.3.2">ℎ</ci><cn id="S7.SS4.SSS0.P0.SPx1.p1.8.m8.1.1.3.3.cmml" type="integer" xref="S7.SS4.SSS0.P0.SPx1.p1.8.m8.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.SS4.SSS0.P0.SPx1.p1.8.m8.1c">k<h-1</annotation><annotation encoding="application/x-llamapun" id="S7.SS4.SSS0.P0.SPx1.p1.8.m8.1d">italic_k < italic_h - 1</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S7.SS4.SSS0.P0.SPx1.p1.13.13"> do not change their level. Thus, at the start of </span><math alttext="(h:m+1:0)" class="ltx_Math" display="inline" id="S7.SS4.SSS0.P0.SPx1.p1.9.m9.1"><semantics id="S7.SS4.SSS0.P0.SPx1.p1.9.m9.1a"><mrow id="S7.SS4.SSS0.P0.SPx1.p1.9.m9.1.1.1" xref="S7.SS4.SSS0.P0.SPx1.p1.9.m9.1.1.1.1.cmml"><mo id="S7.SS4.SSS0.P0.SPx1.p1.9.m9.1.1.1.2" stretchy="false" xref="S7.SS4.SSS0.P0.SPx1.p1.9.m9.1.1.1.1.cmml">(</mo><mrow id="S7.SS4.SSS0.P0.SPx1.p1.9.m9.1.1.1.1" xref="S7.SS4.SSS0.P0.SPx1.p1.9.m9.1.1.1.1.cmml"><mi id="S7.SS4.SSS0.P0.SPx1.p1.9.m9.1.1.1.1.2" xref="S7.SS4.SSS0.P0.SPx1.p1.9.m9.1.1.1.1.2.cmml">h</mi><mo id="S7.SS4.SSS0.P0.SPx1.p1.9.m9.1.1.1.1.3" lspace="0.278em" rspace="0.278em" xref="S7.SS4.SSS0.P0.SPx1.p1.9.m9.1.1.1.1.3.cmml">:</mo><mrow id="S7.SS4.SSS0.P0.SPx1.p1.9.m9.1.1.1.1.4" xref="S7.SS4.SSS0.P0.SPx1.p1.9.m9.1.1.1.1.4.cmml"><mi id="S7.SS4.SSS0.P0.SPx1.p1.9.m9.1.1.1.1.4.2" xref="S7.SS4.SSS0.P0.SPx1.p1.9.m9.1.1.1.1.4.2.cmml">m</mi><mo id="S7.SS4.SSS0.P0.SPx1.p1.9.m9.1.1.1.1.4.1" xref="S7.SS4.SSS0.P0.SPx1.p1.9.m9.1.1.1.1.4.1.cmml">+</mo><mn id="S7.SS4.SSS0.P0.SPx1.p1.9.m9.1.1.1.1.4.3" xref="S7.SS4.SSS0.P0.SPx1.p1.9.m9.1.1.1.1.4.3.cmml">1</mn></mrow><mo id="S7.SS4.SSS0.P0.SPx1.p1.9.m9.1.1.1.1.5" lspace="0.278em" rspace="0.278em" xref="S7.SS4.SSS0.P0.SPx1.p1.9.m9.1.1.1.1.5.cmml">:</mo><mn id="S7.SS4.SSS0.P0.SPx1.p1.9.m9.1.1.1.1.6" xref="S7.SS4.SSS0.P0.SPx1.p1.9.m9.1.1.1.1.6.cmml">0</mn></mrow><mo id="S7.SS4.SSS0.P0.SPx1.p1.9.m9.1.1.1.3" stretchy="false" xref="S7.SS4.SSS0.P0.SPx1.p1.9.m9.1.1.1.1.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S7.SS4.SSS0.P0.SPx1.p1.9.m9.1b"><apply id="S7.SS4.SSS0.P0.SPx1.p1.9.m9.1.1.1.1.cmml" xref="S7.SS4.SSS0.P0.SPx1.p1.9.m9.1.1.1"><and id="S7.SS4.SSS0.P0.SPx1.p1.9.m9.1.1.1.1a.cmml" xref="S7.SS4.SSS0.P0.SPx1.p1.9.m9.1.1.1"></and><apply id="S7.SS4.SSS0.P0.SPx1.p1.9.m9.1.1.1.1b.cmml" xref="S7.SS4.SSS0.P0.SPx1.p1.9.m9.1.1.1"><ci id="S7.SS4.SSS0.P0.SPx1.p1.9.m9.1.1.1.1.3.cmml" xref="S7.SS4.SSS0.P0.SPx1.p1.9.m9.1.1.1.1.3">:</ci><ci id="S7.SS4.SSS0.P0.SPx1.p1.9.m9.1.1.1.1.2.cmml" xref="S7.SS4.SSS0.P0.SPx1.p1.9.m9.1.1.1.1.2">ℎ</ci><apply id="S7.SS4.SSS0.P0.SPx1.p1.9.m9.1.1.1.1.4.cmml" xref="S7.SS4.SSS0.P0.SPx1.p1.9.m9.1.1.1.1.4"><plus id="S7.SS4.SSS0.P0.SPx1.p1.9.m9.1.1.1.1.4.1.cmml" xref="S7.SS4.SSS0.P0.SPx1.p1.9.m9.1.1.1.1.4.1"></plus><ci id="S7.SS4.SSS0.P0.SPx1.p1.9.m9.1.1.1.1.4.2.cmml" xref="S7.SS4.SSS0.P0.SPx1.p1.9.m9.1.1.1.1.4.2">𝑚</ci><cn id="S7.SS4.SSS0.P0.SPx1.p1.9.m9.1.1.1.1.4.3.cmml" type="integer" xref="S7.SS4.SSS0.P0.SPx1.p1.9.m9.1.1.1.1.4.3">1</cn></apply></apply><apply id="S7.SS4.SSS0.P0.SPx1.p1.9.m9.1.1.1.1c.cmml" xref="S7.SS4.SSS0.P0.SPx1.p1.9.m9.1.1.1"><ci id="S7.SS4.SSS0.P0.SPx1.p1.9.m9.1.1.1.1.5.cmml" xref="S7.SS4.SSS0.P0.SPx1.p1.9.m9.1.1.1.1.5">:</ci><share href="https://arxiv.org/html/2411.12694v2#S7.SS4.SSS0.P0.SPx1.p1.9.m9.1.1.1.1.4.cmml" id="S7.SS4.SSS0.P0.SPx1.p1.9.m9.1.1.1.1d.cmml" xref="S7.SS4.SSS0.P0.SPx1.p1.9.m9.1.1.1"></share><cn id="S7.SS4.SSS0.P0.SPx1.p1.9.m9.1.1.1.1.6.cmml" type="integer" xref="S7.SS4.SSS0.P0.SPx1.p1.9.m9.1.1.1.1.6">0</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.SS4.SSS0.P0.SPx1.p1.9.m9.1c">(h:m+1:0)</annotation><annotation encoding="application/x-llamapun" id="S7.SS4.SSS0.P0.SPx1.p1.9.m9.1d">( italic_h : italic_m + 1 : 0 )</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S7.SS4.SSS0.P0.SPx1.p1.13.14">, </span><math alttext="l_{m+1}(v)=h" class="ltx_Math" display="inline" id="S7.SS4.SSS0.P0.SPx1.p1.10.m10.1"><semantics id="S7.SS4.SSS0.P0.SPx1.p1.10.m10.1a"><mrow id="S7.SS4.SSS0.P0.SPx1.p1.10.m10.1.2" xref="S7.SS4.SSS0.P0.SPx1.p1.10.m10.1.2.cmml"><mrow id="S7.SS4.SSS0.P0.SPx1.p1.10.m10.1.2.2" xref="S7.SS4.SSS0.P0.SPx1.p1.10.m10.1.2.2.cmml"><msub id="S7.SS4.SSS0.P0.SPx1.p1.10.m10.1.2.2.2" xref="S7.SS4.SSS0.P0.SPx1.p1.10.m10.1.2.2.2.cmml"><mi id="S7.SS4.SSS0.P0.SPx1.p1.10.m10.1.2.2.2.2" xref="S7.SS4.SSS0.P0.SPx1.p1.10.m10.1.2.2.2.2.cmml">l</mi><mrow id="S7.SS4.SSS0.P0.SPx1.p1.10.m10.1.2.2.2.3" xref="S7.SS4.SSS0.P0.SPx1.p1.10.m10.1.2.2.2.3.cmml"><mi id="S7.SS4.SSS0.P0.SPx1.p1.10.m10.1.2.2.2.3.2" xref="S7.SS4.SSS0.P0.SPx1.p1.10.m10.1.2.2.2.3.2.cmml">m</mi><mo id="S7.SS4.SSS0.P0.SPx1.p1.10.m10.1.2.2.2.3.1" xref="S7.SS4.SSS0.P0.SPx1.p1.10.m10.1.2.2.2.3.1.cmml">+</mo><mn id="S7.SS4.SSS0.P0.SPx1.p1.10.m10.1.2.2.2.3.3" xref="S7.SS4.SSS0.P0.SPx1.p1.10.m10.1.2.2.2.3.3.cmml">1</mn></mrow></msub><mo id="S7.SS4.SSS0.P0.SPx1.p1.10.m10.1.2.2.1" xref="S7.SS4.SSS0.P0.SPx1.p1.10.m10.1.2.2.1.cmml">⁢</mo><mrow id="S7.SS4.SSS0.P0.SPx1.p1.10.m10.1.2.2.3.2" xref="S7.SS4.SSS0.P0.SPx1.p1.10.m10.1.2.2.cmml"><mo id="S7.SS4.SSS0.P0.SPx1.p1.10.m10.1.2.2.3.2.1" stretchy="false" xref="S7.SS4.SSS0.P0.SPx1.p1.10.m10.1.2.2.cmml">(</mo><mi id="S7.SS4.SSS0.P0.SPx1.p1.10.m10.1.1" xref="S7.SS4.SSS0.P0.SPx1.p1.10.m10.1.1.cmml">v</mi><mo id="S7.SS4.SSS0.P0.SPx1.p1.10.m10.1.2.2.3.2.2" stretchy="false" xref="S7.SS4.SSS0.P0.SPx1.p1.10.m10.1.2.2.cmml">)</mo></mrow></mrow><mo id="S7.SS4.SSS0.P0.SPx1.p1.10.m10.1.2.1" xref="S7.SS4.SSS0.P0.SPx1.p1.10.m10.1.2.1.cmml">=</mo><mi id="S7.SS4.SSS0.P0.SPx1.p1.10.m10.1.2.3" xref="S7.SS4.SSS0.P0.SPx1.p1.10.m10.1.2.3.cmml">h</mi></mrow><annotation-xml encoding="MathML-Content" id="S7.SS4.SSS0.P0.SPx1.p1.10.m10.1b"><apply id="S7.SS4.SSS0.P0.SPx1.p1.10.m10.1.2.cmml" xref="S7.SS4.SSS0.P0.SPx1.p1.10.m10.1.2"><eq id="S7.SS4.SSS0.P0.SPx1.p1.10.m10.1.2.1.cmml" xref="S7.SS4.SSS0.P0.SPx1.p1.10.m10.1.2.1"></eq><apply id="S7.SS4.SSS0.P0.SPx1.p1.10.m10.1.2.2.cmml" xref="S7.SS4.SSS0.P0.SPx1.p1.10.m10.1.2.2"><times id="S7.SS4.SSS0.P0.SPx1.p1.10.m10.1.2.2.1.cmml" xref="S7.SS4.SSS0.P0.SPx1.p1.10.m10.1.2.2.1"></times><apply id="S7.SS4.SSS0.P0.SPx1.p1.10.m10.1.2.2.2.cmml" xref="S7.SS4.SSS0.P0.SPx1.p1.10.m10.1.2.2.2"><csymbol cd="ambiguous" id="S7.SS4.SSS0.P0.SPx1.p1.10.m10.1.2.2.2.1.cmml" xref="S7.SS4.SSS0.P0.SPx1.p1.10.m10.1.2.2.2">subscript</csymbol><ci id="S7.SS4.SSS0.P0.SPx1.p1.10.m10.1.2.2.2.2.cmml" xref="S7.SS4.SSS0.P0.SPx1.p1.10.m10.1.2.2.2.2">𝑙</ci><apply id="S7.SS4.SSS0.P0.SPx1.p1.10.m10.1.2.2.2.3.cmml" xref="S7.SS4.SSS0.P0.SPx1.p1.10.m10.1.2.2.2.3"><plus id="S7.SS4.SSS0.P0.SPx1.p1.10.m10.1.2.2.2.3.1.cmml" xref="S7.SS4.SSS0.P0.SPx1.p1.10.m10.1.2.2.2.3.1"></plus><ci id="S7.SS4.SSS0.P0.SPx1.p1.10.m10.1.2.2.2.3.2.cmml" xref="S7.SS4.SSS0.P0.SPx1.p1.10.m10.1.2.2.2.3.2">𝑚</ci><cn id="S7.SS4.SSS0.P0.SPx1.p1.10.m10.1.2.2.2.3.3.cmml" type="integer" xref="S7.SS4.SSS0.P0.SPx1.p1.10.m10.1.2.2.2.3.3">1</cn></apply></apply><ci id="S7.SS4.SSS0.P0.SPx1.p1.10.m10.1.1.cmml" xref="S7.SS4.SSS0.P0.SPx1.p1.10.m10.1.1">𝑣</ci></apply><ci id="S7.SS4.SSS0.P0.SPx1.p1.10.m10.1.2.3.cmml" xref="S7.SS4.SSS0.P0.SPx1.p1.10.m10.1.2.3">ℎ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.SS4.SSS0.P0.SPx1.p1.10.m10.1c">l_{m+1}(v)=h</annotation><annotation encoding="application/x-llamapun" id="S7.SS4.SSS0.P0.SPx1.p1.10.m10.1d">italic_l start_POSTSUBSCRIPT italic_m + 1 end_POSTSUBSCRIPT ( italic_v ) = italic_h</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S7.SS4.SSS0.P0.SPx1.p1.13.15"> and </span><math alttext="v" class="ltx_Math" display="inline" id="S7.SS4.SSS0.P0.SPx1.p1.11.m11.1"><semantics id="S7.SS4.SSS0.P0.SPx1.p1.11.m11.1a"><mi id="S7.SS4.SSS0.P0.SPx1.p1.11.m11.1.1" xref="S7.SS4.SSS0.P0.SPx1.p1.11.m11.1.1.cmml">v</mi><annotation-xml encoding="MathML-Content" id="S7.SS4.SSS0.P0.SPx1.p1.11.m11.1b"><ci id="S7.SS4.SSS0.P0.SPx1.p1.11.m11.1.1.cmml" xref="S7.SS4.SSS0.P0.SPx1.p1.11.m11.1.1">𝑣</ci></annotation-xml><annotation encoding="application/x-tex" id="S7.SS4.SSS0.P0.SPx1.p1.11.m11.1c">v</annotation><annotation encoding="application/x-llamapun" id="S7.SS4.SSS0.P0.SPx1.p1.11.m11.1d">italic_v</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S7.SS4.SSS0.P0.SPx1.p1.13.16"> has not gained any violating out-edges. We recursively apply this argument to conclude that </span><math alttext="l_{x}(v)=h" class="ltx_Math" display="inline" id="S7.SS4.SSS0.P0.SPx1.p1.12.m12.1"><semantics id="S7.SS4.SSS0.P0.SPx1.p1.12.m12.1a"><mrow id="S7.SS4.SSS0.P0.SPx1.p1.12.m12.1.2" xref="S7.SS4.SSS0.P0.SPx1.p1.12.m12.1.2.cmml"><mrow id="S7.SS4.SSS0.P0.SPx1.p1.12.m12.1.2.2" xref="S7.SS4.SSS0.P0.SPx1.p1.12.m12.1.2.2.cmml"><msub id="S7.SS4.SSS0.P0.SPx1.p1.12.m12.1.2.2.2" xref="S7.SS4.SSS0.P0.SPx1.p1.12.m12.1.2.2.2.cmml"><mi id="S7.SS4.SSS0.P0.SPx1.p1.12.m12.1.2.2.2.2" xref="S7.SS4.SSS0.P0.SPx1.p1.12.m12.1.2.2.2.2.cmml">l</mi><mi id="S7.SS4.SSS0.P0.SPx1.p1.12.m12.1.2.2.2.3" xref="S7.SS4.SSS0.P0.SPx1.p1.12.m12.1.2.2.2.3.cmml">x</mi></msub><mo id="S7.SS4.SSS0.P0.SPx1.p1.12.m12.1.2.2.1" xref="S7.SS4.SSS0.P0.SPx1.p1.12.m12.1.2.2.1.cmml">⁢</mo><mrow id="S7.SS4.SSS0.P0.SPx1.p1.12.m12.1.2.2.3.2" xref="S7.SS4.SSS0.P0.SPx1.p1.12.m12.1.2.2.cmml"><mo id="S7.SS4.SSS0.P0.SPx1.p1.12.m12.1.2.2.3.2.1" stretchy="false" xref="S7.SS4.SSS0.P0.SPx1.p1.12.m12.1.2.2.cmml">(</mo><mi id="S7.SS4.SSS0.P0.SPx1.p1.12.m12.1.1" xref="S7.SS4.SSS0.P0.SPx1.p1.12.m12.1.1.cmml">v</mi><mo id="S7.SS4.SSS0.P0.SPx1.p1.12.m12.1.2.2.3.2.2" stretchy="false" xref="S7.SS4.SSS0.P0.SPx1.p1.12.m12.1.2.2.cmml">)</mo></mrow></mrow><mo id="S7.SS4.SSS0.P0.SPx1.p1.12.m12.1.2.1" xref="S7.SS4.SSS0.P0.SPx1.p1.12.m12.1.2.1.cmml">=</mo><mi id="S7.SS4.SSS0.P0.SPx1.p1.12.m12.1.2.3" xref="S7.SS4.SSS0.P0.SPx1.p1.12.m12.1.2.3.cmml">h</mi></mrow><annotation-xml encoding="MathML-Content" id="S7.SS4.SSS0.P0.SPx1.p1.12.m12.1b"><apply id="S7.SS4.SSS0.P0.SPx1.p1.12.m12.1.2.cmml" xref="S7.SS4.SSS0.P0.SPx1.p1.12.m12.1.2"><eq id="S7.SS4.SSS0.P0.SPx1.p1.12.m12.1.2.1.cmml" xref="S7.SS4.SSS0.P0.SPx1.p1.12.m12.1.2.1"></eq><apply id="S7.SS4.SSS0.P0.SPx1.p1.12.m12.1.2.2.cmml" xref="S7.SS4.SSS0.P0.SPx1.p1.12.m12.1.2.2"><times id="S7.SS4.SSS0.P0.SPx1.p1.12.m12.1.2.2.1.cmml" xref="S7.SS4.SSS0.P0.SPx1.p1.12.m12.1.2.2.1"></times><apply id="S7.SS4.SSS0.P0.SPx1.p1.12.m12.1.2.2.2.cmml" xref="S7.SS4.SSS0.P0.SPx1.p1.12.m12.1.2.2.2"><csymbol cd="ambiguous" id="S7.SS4.SSS0.P0.SPx1.p1.12.m12.1.2.2.2.1.cmml" xref="S7.SS4.SSS0.P0.SPx1.p1.12.m12.1.2.2.2">subscript</csymbol><ci id="S7.SS4.SSS0.P0.SPx1.p1.12.m12.1.2.2.2.2.cmml" xref="S7.SS4.SSS0.P0.SPx1.p1.12.m12.1.2.2.2.2">𝑙</ci><ci id="S7.SS4.SSS0.P0.SPx1.p1.12.m12.1.2.2.2.3.cmml" xref="S7.SS4.SSS0.P0.SPx1.p1.12.m12.1.2.2.2.3">𝑥</ci></apply><ci id="S7.SS4.SSS0.P0.SPx1.p1.12.m12.1.1.cmml" xref="S7.SS4.SSS0.P0.SPx1.p1.12.m12.1.1">𝑣</ci></apply><ci id="S7.SS4.SSS0.P0.SPx1.p1.12.m12.1.2.3.cmml" xref="S7.SS4.SSS0.P0.SPx1.p1.12.m12.1.2.3">ℎ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.SS4.SSS0.P0.SPx1.p1.12.m12.1c">l_{x}(v)=h</annotation><annotation encoding="application/x-llamapun" id="S7.SS4.SSS0.P0.SPx1.p1.12.m12.1d">italic_l start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT ( italic_v ) = italic_h</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S7.SS4.SSS0.P0.SPx1.p1.13.17"> and </span><math alttext="v" class="ltx_Math" display="inline" id="S7.SS4.SSS0.P0.SPx1.p1.13.m13.1"><semantics id="S7.SS4.SSS0.P0.SPx1.p1.13.m13.1a"><mi id="S7.SS4.SSS0.P0.SPx1.p1.13.m13.1.1" xref="S7.SS4.SSS0.P0.SPx1.p1.13.m13.1.1.cmml">v</mi><annotation-xml encoding="MathML-Content" id="S7.SS4.SSS0.P0.SPx1.p1.13.m13.1b"><ci id="S7.SS4.SSS0.P0.SPx1.p1.13.m13.1.1.cmml" xref="S7.SS4.SSS0.P0.SPx1.p1.13.m13.1.1">𝑣</ci></annotation-xml><annotation encoding="application/x-tex" id="S7.SS4.SSS0.P0.SPx1.p1.13.m13.1c">v</annotation><annotation encoding="application/x-llamapun" id="S7.SS4.SSS0.P0.SPx1.p1.13.m13.1d">italic_v</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S7.SS4.SSS0.P0.SPx1.p1.13.18"> has no violating out-edges.</span></p> </div> </section> <section class="ltx_subparagraph" id="S7.SS4.SSS0.P0.SPx2"> <h5 class="ltx_title ltx_font_italic ltx_title_subparagraph">Case 2:</h5> <div class="ltx_para" id="S7.SS4.SSS0.P0.SPx2.p1"> <p class="ltx_p" id="S7.SS4.SSS0.P0.SPx2.p1.17"><span class="ltx_text ltx_font_italic" id="S7.SS4.SSS0.P0.SPx2.p1.17.1">Suppose that </span><math alttext="l_{m}(v)=h-1" class="ltx_Math" display="inline" id="S7.SS4.SSS0.P0.SPx2.p1.1.m1.1"><semantics id="S7.SS4.SSS0.P0.SPx2.p1.1.m1.1a"><mrow id="S7.SS4.SSS0.P0.SPx2.p1.1.m1.1.2" xref="S7.SS4.SSS0.P0.SPx2.p1.1.m1.1.2.cmml"><mrow id="S7.SS4.SSS0.P0.SPx2.p1.1.m1.1.2.2" xref="S7.SS4.SSS0.P0.SPx2.p1.1.m1.1.2.2.cmml"><msub id="S7.SS4.SSS0.P0.SPx2.p1.1.m1.1.2.2.2" xref="S7.SS4.SSS0.P0.SPx2.p1.1.m1.1.2.2.2.cmml"><mi id="S7.SS4.SSS0.P0.SPx2.p1.1.m1.1.2.2.2.2" xref="S7.SS4.SSS0.P0.SPx2.p1.1.m1.1.2.2.2.2.cmml">l</mi><mi id="S7.SS4.SSS0.P0.SPx2.p1.1.m1.1.2.2.2.3" xref="S7.SS4.SSS0.P0.SPx2.p1.1.m1.1.2.2.2.3.cmml">m</mi></msub><mo id="S7.SS4.SSS0.P0.SPx2.p1.1.m1.1.2.2.1" xref="S7.SS4.SSS0.P0.SPx2.p1.1.m1.1.2.2.1.cmml">⁢</mo><mrow id="S7.SS4.SSS0.P0.SPx2.p1.1.m1.1.2.2.3.2" xref="S7.SS4.SSS0.P0.SPx2.p1.1.m1.1.2.2.cmml"><mo id="S7.SS4.SSS0.P0.SPx2.p1.1.m1.1.2.2.3.2.1" stretchy="false" xref="S7.SS4.SSS0.P0.SPx2.p1.1.m1.1.2.2.cmml">(</mo><mi id="S7.SS4.SSS0.P0.SPx2.p1.1.m1.1.1" xref="S7.SS4.SSS0.P0.SPx2.p1.1.m1.1.1.cmml">v</mi><mo id="S7.SS4.SSS0.P0.SPx2.p1.1.m1.1.2.2.3.2.2" stretchy="false" xref="S7.SS4.SSS0.P0.SPx2.p1.1.m1.1.2.2.cmml">)</mo></mrow></mrow><mo id="S7.SS4.SSS0.P0.SPx2.p1.1.m1.1.2.1" xref="S7.SS4.SSS0.P0.SPx2.p1.1.m1.1.2.1.cmml">=</mo><mrow id="S7.SS4.SSS0.P0.SPx2.p1.1.m1.1.2.3" xref="S7.SS4.SSS0.P0.SPx2.p1.1.m1.1.2.3.cmml"><mi id="S7.SS4.SSS0.P0.SPx2.p1.1.m1.1.2.3.2" xref="S7.SS4.SSS0.P0.SPx2.p1.1.m1.1.2.3.2.cmml">h</mi><mo id="S7.SS4.SSS0.P0.SPx2.p1.1.m1.1.2.3.1" xref="S7.SS4.SSS0.P0.SPx2.p1.1.m1.1.2.3.1.cmml">−</mo><mn id="S7.SS4.SSS0.P0.SPx2.p1.1.m1.1.2.3.3" xref="S7.SS4.SSS0.P0.SPx2.p1.1.m1.1.2.3.3.cmml">1</mn></mrow></mrow><annotation-xml encoding="MathML-Content" id="S7.SS4.SSS0.P0.SPx2.p1.1.m1.1b"><apply id="S7.SS4.SSS0.P0.SPx2.p1.1.m1.1.2.cmml" xref="S7.SS4.SSS0.P0.SPx2.p1.1.m1.1.2"><eq id="S7.SS4.SSS0.P0.SPx2.p1.1.m1.1.2.1.cmml" xref="S7.SS4.SSS0.P0.SPx2.p1.1.m1.1.2.1"></eq><apply id="S7.SS4.SSS0.P0.SPx2.p1.1.m1.1.2.2.cmml" xref="S7.SS4.SSS0.P0.SPx2.p1.1.m1.1.2.2"><times id="S7.SS4.SSS0.P0.SPx2.p1.1.m1.1.2.2.1.cmml" xref="S7.SS4.SSS0.P0.SPx2.p1.1.m1.1.2.2.1"></times><apply id="S7.SS4.SSS0.P0.SPx2.p1.1.m1.1.2.2.2.cmml" xref="S7.SS4.SSS0.P0.SPx2.p1.1.m1.1.2.2.2"><csymbol cd="ambiguous" id="S7.SS4.SSS0.P0.SPx2.p1.1.m1.1.2.2.2.1.cmml" xref="S7.SS4.SSS0.P0.SPx2.p1.1.m1.1.2.2.2">subscript</csymbol><ci id="S7.SS4.SSS0.P0.SPx2.p1.1.m1.1.2.2.2.2.cmml" xref="S7.SS4.SSS0.P0.SPx2.p1.1.m1.1.2.2.2.2">𝑙</ci><ci id="S7.SS4.SSS0.P0.SPx2.p1.1.m1.1.2.2.2.3.cmml" xref="S7.SS4.SSS0.P0.SPx2.p1.1.m1.1.2.2.2.3">𝑚</ci></apply><ci id="S7.SS4.SSS0.P0.SPx2.p1.1.m1.1.1.cmml" xref="S7.SS4.SSS0.P0.SPx2.p1.1.m1.1.1">𝑣</ci></apply><apply id="S7.SS4.SSS0.P0.SPx2.p1.1.m1.1.2.3.cmml" xref="S7.SS4.SSS0.P0.SPx2.p1.1.m1.1.2.3"><minus id="S7.SS4.SSS0.P0.SPx2.p1.1.m1.1.2.3.1.cmml" xref="S7.SS4.SSS0.P0.SPx2.p1.1.m1.1.2.3.1"></minus><ci id="S7.SS4.SSS0.P0.SPx2.p1.1.m1.1.2.3.2.cmml" xref="S7.SS4.SSS0.P0.SPx2.p1.1.m1.1.2.3.2">ℎ</ci><cn id="S7.SS4.SSS0.P0.SPx2.p1.1.m1.1.2.3.3.cmml" type="integer" xref="S7.SS4.SSS0.P0.SPx2.p1.1.m1.1.2.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.SS4.SSS0.P0.SPx2.p1.1.m1.1c">l_{m}(v)=h-1</annotation><annotation encoding="application/x-llamapun" id="S7.SS4.SSS0.P0.SPx2.p1.1.m1.1d">italic_l start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT ( italic_v ) = italic_h - 1</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S7.SS4.SSS0.P0.SPx2.p1.17.2"> and </span><math alttext="v" class="ltx_Math" display="inline" id="S7.SS4.SSS0.P0.SPx2.p1.2.m2.1"><semantics id="S7.SS4.SSS0.P0.SPx2.p1.2.m2.1a"><mi id="S7.SS4.SSS0.P0.SPx2.p1.2.m2.1.1" xref="S7.SS4.SSS0.P0.SPx2.p1.2.m2.1.1.cmml">v</mi><annotation-xml encoding="MathML-Content" id="S7.SS4.SSS0.P0.SPx2.p1.2.m2.1b"><ci id="S7.SS4.SSS0.P0.SPx2.p1.2.m2.1.1.cmml" xref="S7.SS4.SSS0.P0.SPx2.p1.2.m2.1.1">𝑣</ci></annotation-xml><annotation encoding="application/x-tex" id="S7.SS4.SSS0.P0.SPx2.p1.2.m2.1c">v</annotation><annotation encoding="application/x-llamapun" id="S7.SS4.SSS0.P0.SPx2.p1.2.m2.1d">italic_v</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S7.SS4.SSS0.P0.SPx2.p1.17.3"> has no violating in-edge at the start of </span><math alttext="(h:m:0)" class="ltx_Math" display="inline" id="S7.SS4.SSS0.P0.SPx2.p1.3.m3.1"><semantics id="S7.SS4.SSS0.P0.SPx2.p1.3.m3.1a"><mrow id="S7.SS4.SSS0.P0.SPx2.p1.3.m3.1.1.1" xref="S7.SS4.SSS0.P0.SPx2.p1.3.m3.1.1.1.1.cmml"><mo id="S7.SS4.SSS0.P0.SPx2.p1.3.m3.1.1.1.2" stretchy="false" xref="S7.SS4.SSS0.P0.SPx2.p1.3.m3.1.1.1.1.cmml">(</mo><mrow id="S7.SS4.SSS0.P0.SPx2.p1.3.m3.1.1.1.1" xref="S7.SS4.SSS0.P0.SPx2.p1.3.m3.1.1.1.1.cmml"><mi id="S7.SS4.SSS0.P0.SPx2.p1.3.m3.1.1.1.1.2" xref="S7.SS4.SSS0.P0.SPx2.p1.3.m3.1.1.1.1.2.cmml">h</mi><mo id="S7.SS4.SSS0.P0.SPx2.p1.3.m3.1.1.1.1.3" lspace="0.278em" rspace="0.278em" xref="S7.SS4.SSS0.P0.SPx2.p1.3.m3.1.1.1.1.3.cmml">:</mo><mi id="S7.SS4.SSS0.P0.SPx2.p1.3.m3.1.1.1.1.4" xref="S7.SS4.SSS0.P0.SPx2.p1.3.m3.1.1.1.1.4.cmml">m</mi><mo id="S7.SS4.SSS0.P0.SPx2.p1.3.m3.1.1.1.1.5" lspace="0.278em" rspace="0.278em" xref="S7.SS4.SSS0.P0.SPx2.p1.3.m3.1.1.1.1.5.cmml">:</mo><mn id="S7.SS4.SSS0.P0.SPx2.p1.3.m3.1.1.1.1.6" xref="S7.SS4.SSS0.P0.SPx2.p1.3.m3.1.1.1.1.6.cmml">0</mn></mrow><mo id="S7.SS4.SSS0.P0.SPx2.p1.3.m3.1.1.1.3" stretchy="false" xref="S7.SS4.SSS0.P0.SPx2.p1.3.m3.1.1.1.1.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S7.SS4.SSS0.P0.SPx2.p1.3.m3.1b"><apply id="S7.SS4.SSS0.P0.SPx2.p1.3.m3.1.1.1.1.cmml" xref="S7.SS4.SSS0.P0.SPx2.p1.3.m3.1.1.1"><and id="S7.SS4.SSS0.P0.SPx2.p1.3.m3.1.1.1.1a.cmml" xref="S7.SS4.SSS0.P0.SPx2.p1.3.m3.1.1.1"></and><apply id="S7.SS4.SSS0.P0.SPx2.p1.3.m3.1.1.1.1b.cmml" xref="S7.SS4.SSS0.P0.SPx2.p1.3.m3.1.1.1"><ci id="S7.SS4.SSS0.P0.SPx2.p1.3.m3.1.1.1.1.3.cmml" xref="S7.SS4.SSS0.P0.SPx2.p1.3.m3.1.1.1.1.3">:</ci><ci id="S7.SS4.SSS0.P0.SPx2.p1.3.m3.1.1.1.1.2.cmml" xref="S7.SS4.SSS0.P0.SPx2.p1.3.m3.1.1.1.1.2">ℎ</ci><ci id="S7.SS4.SSS0.P0.SPx2.p1.3.m3.1.1.1.1.4.cmml" xref="S7.SS4.SSS0.P0.SPx2.p1.3.m3.1.1.1.1.4">𝑚</ci></apply><apply id="S7.SS4.SSS0.P0.SPx2.p1.3.m3.1.1.1.1c.cmml" xref="S7.SS4.SSS0.P0.SPx2.p1.3.m3.1.1.1"><ci id="S7.SS4.SSS0.P0.SPx2.p1.3.m3.1.1.1.1.5.cmml" xref="S7.SS4.SSS0.P0.SPx2.p1.3.m3.1.1.1.1.5">:</ci><share href="https://arxiv.org/html/2411.12694v2#S7.SS4.SSS0.P0.SPx2.p1.3.m3.1.1.1.1.4.cmml" id="S7.SS4.SSS0.P0.SPx2.p1.3.m3.1.1.1.1d.cmml" xref="S7.SS4.SSS0.P0.SPx2.p1.3.m3.1.1.1"></share><cn id="S7.SS4.SSS0.P0.SPx2.p1.3.m3.1.1.1.1.6.cmml" type="integer" xref="S7.SS4.SSS0.P0.SPx2.p1.3.m3.1.1.1.1.6">0</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.SS4.SSS0.P0.SPx2.p1.3.m3.1c">(h:m:0)</annotation><annotation encoding="application/x-llamapun" id="S7.SS4.SSS0.P0.SPx2.p1.3.m3.1d">( italic_h : italic_m : 0 )</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S7.SS4.SSS0.P0.SPx2.p1.17.4">. Then by our algorithm, </span><math alttext="v" class="ltx_Math" display="inline" id="S7.SS4.SSS0.P0.SPx2.p1.4.m4.1"><semantics id="S7.SS4.SSS0.P0.SPx2.p1.4.m4.1a"><mi id="S7.SS4.SSS0.P0.SPx2.p1.4.m4.1.1" xref="S7.SS4.SSS0.P0.SPx2.p1.4.m4.1.1.cmml">v</mi><annotation-xml encoding="MathML-Content" id="S7.SS4.SSS0.P0.SPx2.p1.4.m4.1b"><ci id="S7.SS4.SSS0.P0.SPx2.p1.4.m4.1.1.cmml" xref="S7.SS4.SSS0.P0.SPx2.p1.4.m4.1.1">𝑣</ci></annotation-xml><annotation encoding="application/x-tex" id="S7.SS4.SSS0.P0.SPx2.p1.4.m4.1c">v</annotation><annotation encoding="application/x-llamapun" id="S7.SS4.SSS0.P0.SPx2.p1.4.m4.1d">italic_v</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S7.SS4.SSS0.P0.SPx2.p1.17.5"> does not change its level during </span><span class="ltx_text ltx_markedasmath ltx_font_smallcaps" id="S7.SS4.SSS0.P0.SPx2.p1.17.6">minute</span><span class="ltx_text ltx_font_italic" id="S7.SS4.SSS0.P0.SPx2.p1.17.7"> </span><math alttext="m" class="ltx_Math" display="inline" id="S7.SS4.SSS0.P0.SPx2.p1.6.m6.1"><semantics id="S7.SS4.SSS0.P0.SPx2.p1.6.m6.1a"><mi id="S7.SS4.SSS0.P0.SPx2.p1.6.m6.1.1" xref="S7.SS4.SSS0.P0.SPx2.p1.6.m6.1.1.cmml">m</mi><annotation-xml encoding="MathML-Content" id="S7.SS4.SSS0.P0.SPx2.p1.6.m6.1b"><ci id="S7.SS4.SSS0.P0.SPx2.p1.6.m6.1.1.cmml" xref="S7.SS4.SSS0.P0.SPx2.p1.6.m6.1.1">𝑚</ci></annotation-xml><annotation encoding="application/x-tex" id="S7.SS4.SSS0.P0.SPx2.p1.6.m6.1c">m</annotation><annotation encoding="application/x-llamapun" id="S7.SS4.SSS0.P0.SPx2.p1.6.m6.1d">italic_m</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S7.SS4.SSS0.P0.SPx2.p1.17.8">. But this means that in </span><span class="ltx_text ltx_font_smallcaps" id="S7.SS4.SSS0.P0.SPx2.p1.17.9">minute</span><span class="ltx_text ltx_font_italic" id="S7.SS4.SSS0.P0.SPx2.p1.17.10"> </span><math alttext="m+1" class="ltx_Math" display="inline" id="S7.SS4.SSS0.P0.SPx2.p1.7.m7.1"><semantics id="S7.SS4.SSS0.P0.SPx2.p1.7.m7.1a"><mrow id="S7.SS4.SSS0.P0.SPx2.p1.7.m7.1.1" xref="S7.SS4.SSS0.P0.SPx2.p1.7.m7.1.1.cmml"><mi id="S7.SS4.SSS0.P0.SPx2.p1.7.m7.1.1.2" xref="S7.SS4.SSS0.P0.SPx2.p1.7.m7.1.1.2.cmml">m</mi><mo id="S7.SS4.SSS0.P0.SPx2.p1.7.m7.1.1.1" xref="S7.SS4.SSS0.P0.SPx2.p1.7.m7.1.1.1.cmml">+</mo><mn id="S7.SS4.SSS0.P0.SPx2.p1.7.m7.1.1.3" xref="S7.SS4.SSS0.P0.SPx2.p1.7.m7.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S7.SS4.SSS0.P0.SPx2.p1.7.m7.1b"><apply id="S7.SS4.SSS0.P0.SPx2.p1.7.m7.1.1.cmml" xref="S7.SS4.SSS0.P0.SPx2.p1.7.m7.1.1"><plus id="S7.SS4.SSS0.P0.SPx2.p1.7.m7.1.1.1.cmml" xref="S7.SS4.SSS0.P0.SPx2.p1.7.m7.1.1.1"></plus><ci id="S7.SS4.SSS0.P0.SPx2.p1.7.m7.1.1.2.cmml" xref="S7.SS4.SSS0.P0.SPx2.p1.7.m7.1.1.2">𝑚</ci><cn id="S7.SS4.SSS0.P0.SPx2.p1.7.m7.1.1.3.cmml" type="integer" xref="S7.SS4.SSS0.P0.SPx2.p1.7.m7.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.SS4.SSS0.P0.SPx2.p1.7.m7.1c">m+1</annotation><annotation encoding="application/x-llamapun" id="S7.SS4.SSS0.P0.SPx2.p1.7.m7.1d">italic_m + 1</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S7.SS4.SSS0.P0.SPx2.p1.17.11"> the vertex is in level </span><math alttext="L_{h-1}" class="ltx_Math" display="inline" id="S7.SS4.SSS0.P0.SPx2.p1.8.m8.1"><semantics id="S7.SS4.SSS0.P0.SPx2.p1.8.m8.1a"><msub id="S7.SS4.SSS0.P0.SPx2.p1.8.m8.1.1" xref="S7.SS4.SSS0.P0.SPx2.p1.8.m8.1.1.cmml"><mi id="S7.SS4.SSS0.P0.SPx2.p1.8.m8.1.1.2" xref="S7.SS4.SSS0.P0.SPx2.p1.8.m8.1.1.2.cmml">L</mi><mrow id="S7.SS4.SSS0.P0.SPx2.p1.8.m8.1.1.3" xref="S7.SS4.SSS0.P0.SPx2.p1.8.m8.1.1.3.cmml"><mi id="S7.SS4.SSS0.P0.SPx2.p1.8.m8.1.1.3.2" xref="S7.SS4.SSS0.P0.SPx2.p1.8.m8.1.1.3.2.cmml">h</mi><mo id="S7.SS4.SSS0.P0.SPx2.p1.8.m8.1.1.3.1" xref="S7.SS4.SSS0.P0.SPx2.p1.8.m8.1.1.3.1.cmml">−</mo><mn id="S7.SS4.SSS0.P0.SPx2.p1.8.m8.1.1.3.3" xref="S7.SS4.SSS0.P0.SPx2.p1.8.m8.1.1.3.3.cmml">1</mn></mrow></msub><annotation-xml encoding="MathML-Content" id="S7.SS4.SSS0.P0.SPx2.p1.8.m8.1b"><apply id="S7.SS4.SSS0.P0.SPx2.p1.8.m8.1.1.cmml" xref="S7.SS4.SSS0.P0.SPx2.p1.8.m8.1.1"><csymbol cd="ambiguous" id="S7.SS4.SSS0.P0.SPx2.p1.8.m8.1.1.1.cmml" xref="S7.SS4.SSS0.P0.SPx2.p1.8.m8.1.1">subscript</csymbol><ci id="S7.SS4.SSS0.P0.SPx2.p1.8.m8.1.1.2.cmml" xref="S7.SS4.SSS0.P0.SPx2.p1.8.m8.1.1.2">𝐿</ci><apply id="S7.SS4.SSS0.P0.SPx2.p1.8.m8.1.1.3.cmml" xref="S7.SS4.SSS0.P0.SPx2.p1.8.m8.1.1.3"><minus id="S7.SS4.SSS0.P0.SPx2.p1.8.m8.1.1.3.1.cmml" xref="S7.SS4.SSS0.P0.SPx2.p1.8.m8.1.1.3.1"></minus><ci id="S7.SS4.SSS0.P0.SPx2.p1.8.m8.1.1.3.2.cmml" xref="S7.SS4.SSS0.P0.SPx2.p1.8.m8.1.1.3.2">ℎ</ci><cn id="S7.SS4.SSS0.P0.SPx2.p1.8.m8.1.1.3.3.cmml" type="integer" xref="S7.SS4.SSS0.P0.SPx2.p1.8.m8.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.SS4.SSS0.P0.SPx2.p1.8.m8.1c">L_{h-1}</annotation><annotation encoding="application/x-llamapun" id="S7.SS4.SSS0.P0.SPx2.p1.8.m8.1d">italic_L start_POSTSUBSCRIPT italic_h - 1 end_POSTSUBSCRIPT</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S7.SS4.SSS0.P0.SPx2.p1.17.12">. During even </span><span class="ltx_text ltx_font_smallcaps" id="S7.SS4.SSS0.P0.SPx2.p1.17.13">minutes</span><span class="ltx_text ltx_font_italic" id="S7.SS4.SSS0.P0.SPx2.p1.17.14">, vertices at level </span><math alttext="h-1" class="ltx_Math" display="inline" id="S7.SS4.SSS0.P0.SPx2.p1.9.m9.1"><semantics id="S7.SS4.SSS0.P0.SPx2.p1.9.m9.1a"><mrow id="S7.SS4.SSS0.P0.SPx2.p1.9.m9.1.1" xref="S7.SS4.SSS0.P0.SPx2.p1.9.m9.1.1.cmml"><mi id="S7.SS4.SSS0.P0.SPx2.p1.9.m9.1.1.2" xref="S7.SS4.SSS0.P0.SPx2.p1.9.m9.1.1.2.cmml">h</mi><mo id="S7.SS4.SSS0.P0.SPx2.p1.9.m9.1.1.1" xref="S7.SS4.SSS0.P0.SPx2.p1.9.m9.1.1.1.cmml">−</mo><mn id="S7.SS4.SSS0.P0.SPx2.p1.9.m9.1.1.3" xref="S7.SS4.SSS0.P0.SPx2.p1.9.m9.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S7.SS4.SSS0.P0.SPx2.p1.9.m9.1b"><apply id="S7.SS4.SSS0.P0.SPx2.p1.9.m9.1.1.cmml" xref="S7.SS4.SSS0.P0.SPx2.p1.9.m9.1.1"><minus id="S7.SS4.SSS0.P0.SPx2.p1.9.m9.1.1.1.cmml" xref="S7.SS4.SSS0.P0.SPx2.p1.9.m9.1.1.1"></minus><ci id="S7.SS4.SSS0.P0.SPx2.p1.9.m9.1.1.2.cmml" xref="S7.SS4.SSS0.P0.SPx2.p1.9.m9.1.1.2">ℎ</ci><cn id="S7.SS4.SSS0.P0.SPx2.p1.9.m9.1.1.3.cmml" type="integer" xref="S7.SS4.SSS0.P0.SPx2.p1.9.m9.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.SS4.SSS0.P0.SPx2.p1.9.m9.1c">h-1</annotation><annotation encoding="application/x-llamapun" id="S7.SS4.SSS0.P0.SPx2.p1.9.m9.1d">italic_h - 1</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S7.SS4.SSS0.P0.SPx2.p1.17.15"> do not change their out-degree and so </span><math alttext="l_{m+2}(v)=h-1" class="ltx_Math" display="inline" id="S7.SS4.SSS0.P0.SPx2.p1.10.m10.1"><semantics id="S7.SS4.SSS0.P0.SPx2.p1.10.m10.1a"><mrow id="S7.SS4.SSS0.P0.SPx2.p1.10.m10.1.2" xref="S7.SS4.SSS0.P0.SPx2.p1.10.m10.1.2.cmml"><mrow id="S7.SS4.SSS0.P0.SPx2.p1.10.m10.1.2.2" xref="S7.SS4.SSS0.P0.SPx2.p1.10.m10.1.2.2.cmml"><msub id="S7.SS4.SSS0.P0.SPx2.p1.10.m10.1.2.2.2" xref="S7.SS4.SSS0.P0.SPx2.p1.10.m10.1.2.2.2.cmml"><mi id="S7.SS4.SSS0.P0.SPx2.p1.10.m10.1.2.2.2.2" xref="S7.SS4.SSS0.P0.SPx2.p1.10.m10.1.2.2.2.2.cmml">l</mi><mrow id="S7.SS4.SSS0.P0.SPx2.p1.10.m10.1.2.2.2.3" xref="S7.SS4.SSS0.P0.SPx2.p1.10.m10.1.2.2.2.3.cmml"><mi id="S7.SS4.SSS0.P0.SPx2.p1.10.m10.1.2.2.2.3.2" xref="S7.SS4.SSS0.P0.SPx2.p1.10.m10.1.2.2.2.3.2.cmml">m</mi><mo id="S7.SS4.SSS0.P0.SPx2.p1.10.m10.1.2.2.2.3.1" xref="S7.SS4.SSS0.P0.SPx2.p1.10.m10.1.2.2.2.3.1.cmml">+</mo><mn id="S7.SS4.SSS0.P0.SPx2.p1.10.m10.1.2.2.2.3.3" xref="S7.SS4.SSS0.P0.SPx2.p1.10.m10.1.2.2.2.3.3.cmml">2</mn></mrow></msub><mo id="S7.SS4.SSS0.P0.SPx2.p1.10.m10.1.2.2.1" xref="S7.SS4.SSS0.P0.SPx2.p1.10.m10.1.2.2.1.cmml">⁢</mo><mrow id="S7.SS4.SSS0.P0.SPx2.p1.10.m10.1.2.2.3.2" xref="S7.SS4.SSS0.P0.SPx2.p1.10.m10.1.2.2.cmml"><mo id="S7.SS4.SSS0.P0.SPx2.p1.10.m10.1.2.2.3.2.1" stretchy="false" xref="S7.SS4.SSS0.P0.SPx2.p1.10.m10.1.2.2.cmml">(</mo><mi id="S7.SS4.SSS0.P0.SPx2.p1.10.m10.1.1" xref="S7.SS4.SSS0.P0.SPx2.p1.10.m10.1.1.cmml">v</mi><mo id="S7.SS4.SSS0.P0.SPx2.p1.10.m10.1.2.2.3.2.2" stretchy="false" xref="S7.SS4.SSS0.P0.SPx2.p1.10.m10.1.2.2.cmml">)</mo></mrow></mrow><mo id="S7.SS4.SSS0.P0.SPx2.p1.10.m10.1.2.1" xref="S7.SS4.SSS0.P0.SPx2.p1.10.m10.1.2.1.cmml">=</mo><mrow id="S7.SS4.SSS0.P0.SPx2.p1.10.m10.1.2.3" xref="S7.SS4.SSS0.P0.SPx2.p1.10.m10.1.2.3.cmml"><mi id="S7.SS4.SSS0.P0.SPx2.p1.10.m10.1.2.3.2" xref="S7.SS4.SSS0.P0.SPx2.p1.10.m10.1.2.3.2.cmml">h</mi><mo id="S7.SS4.SSS0.P0.SPx2.p1.10.m10.1.2.3.1" xref="S7.SS4.SSS0.P0.SPx2.p1.10.m10.1.2.3.1.cmml">−</mo><mn id="S7.SS4.SSS0.P0.SPx2.p1.10.m10.1.2.3.3" xref="S7.SS4.SSS0.P0.SPx2.p1.10.m10.1.2.3.3.cmml">1</mn></mrow></mrow><annotation-xml encoding="MathML-Content" id="S7.SS4.SSS0.P0.SPx2.p1.10.m10.1b"><apply id="S7.SS4.SSS0.P0.SPx2.p1.10.m10.1.2.cmml" xref="S7.SS4.SSS0.P0.SPx2.p1.10.m10.1.2"><eq id="S7.SS4.SSS0.P0.SPx2.p1.10.m10.1.2.1.cmml" xref="S7.SS4.SSS0.P0.SPx2.p1.10.m10.1.2.1"></eq><apply id="S7.SS4.SSS0.P0.SPx2.p1.10.m10.1.2.2.cmml" xref="S7.SS4.SSS0.P0.SPx2.p1.10.m10.1.2.2"><times id="S7.SS4.SSS0.P0.SPx2.p1.10.m10.1.2.2.1.cmml" xref="S7.SS4.SSS0.P0.SPx2.p1.10.m10.1.2.2.1"></times><apply id="S7.SS4.SSS0.P0.SPx2.p1.10.m10.1.2.2.2.cmml" xref="S7.SS4.SSS0.P0.SPx2.p1.10.m10.1.2.2.2"><csymbol cd="ambiguous" id="S7.SS4.SSS0.P0.SPx2.p1.10.m10.1.2.2.2.1.cmml" xref="S7.SS4.SSS0.P0.SPx2.p1.10.m10.1.2.2.2">subscript</csymbol><ci id="S7.SS4.SSS0.P0.SPx2.p1.10.m10.1.2.2.2.2.cmml" xref="S7.SS4.SSS0.P0.SPx2.p1.10.m10.1.2.2.2.2">𝑙</ci><apply id="S7.SS4.SSS0.P0.SPx2.p1.10.m10.1.2.2.2.3.cmml" xref="S7.SS4.SSS0.P0.SPx2.p1.10.m10.1.2.2.2.3"><plus id="S7.SS4.SSS0.P0.SPx2.p1.10.m10.1.2.2.2.3.1.cmml" xref="S7.SS4.SSS0.P0.SPx2.p1.10.m10.1.2.2.2.3.1"></plus><ci id="S7.SS4.SSS0.P0.SPx2.p1.10.m10.1.2.2.2.3.2.cmml" xref="S7.SS4.SSS0.P0.SPx2.p1.10.m10.1.2.2.2.3.2">𝑚</ci><cn id="S7.SS4.SSS0.P0.SPx2.p1.10.m10.1.2.2.2.3.3.cmml" type="integer" xref="S7.SS4.SSS0.P0.SPx2.p1.10.m10.1.2.2.2.3.3">2</cn></apply></apply><ci id="S7.SS4.SSS0.P0.SPx2.p1.10.m10.1.1.cmml" xref="S7.SS4.SSS0.P0.SPx2.p1.10.m10.1.1">𝑣</ci></apply><apply id="S7.SS4.SSS0.P0.SPx2.p1.10.m10.1.2.3.cmml" xref="S7.SS4.SSS0.P0.SPx2.p1.10.m10.1.2.3"><minus id="S7.SS4.SSS0.P0.SPx2.p1.10.m10.1.2.3.1.cmml" xref="S7.SS4.SSS0.P0.SPx2.p1.10.m10.1.2.3.1"></minus><ci id="S7.SS4.SSS0.P0.SPx2.p1.10.m10.1.2.3.2.cmml" xref="S7.SS4.SSS0.P0.SPx2.p1.10.m10.1.2.3.2">ℎ</ci><cn id="S7.SS4.SSS0.P0.SPx2.p1.10.m10.1.2.3.3.cmml" type="integer" xref="S7.SS4.SSS0.P0.SPx2.p1.10.m10.1.2.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.SS4.SSS0.P0.SPx2.p1.10.m10.1c">l_{m+2}(v)=h-1</annotation><annotation encoding="application/x-llamapun" id="S7.SS4.SSS0.P0.SPx2.p1.10.m10.1d">italic_l start_POSTSUBSCRIPT italic_m + 2 end_POSTSUBSCRIPT ( italic_v ) = italic_h - 1</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S7.SS4.SSS0.P0.SPx2.p1.17.16">. Moreover, vertices in </span><math alttext="L_{k}" class="ltx_Math" display="inline" id="S7.SS4.SSS0.P0.SPx2.p1.11.m11.1"><semantics id="S7.SS4.SSS0.P0.SPx2.p1.11.m11.1a"><msub id="S7.SS4.SSS0.P0.SPx2.p1.11.m11.1.1" xref="S7.SS4.SSS0.P0.SPx2.p1.11.m11.1.1.cmml"><mi id="S7.SS4.SSS0.P0.SPx2.p1.11.m11.1.1.2" xref="S7.SS4.SSS0.P0.SPx2.p1.11.m11.1.1.2.cmml">L</mi><mi id="S7.SS4.SSS0.P0.SPx2.p1.11.m11.1.1.3" xref="S7.SS4.SSS0.P0.SPx2.p1.11.m11.1.1.3.cmml">k</mi></msub><annotation-xml encoding="MathML-Content" id="S7.SS4.SSS0.P0.SPx2.p1.11.m11.1b"><apply id="S7.SS4.SSS0.P0.SPx2.p1.11.m11.1.1.cmml" xref="S7.SS4.SSS0.P0.SPx2.p1.11.m11.1.1"><csymbol cd="ambiguous" id="S7.SS4.SSS0.P0.SPx2.p1.11.m11.1.1.1.cmml" xref="S7.SS4.SSS0.P0.SPx2.p1.11.m11.1.1">subscript</csymbol><ci id="S7.SS4.SSS0.P0.SPx2.p1.11.m11.1.1.2.cmml" xref="S7.SS4.SSS0.P0.SPx2.p1.11.m11.1.1.2">𝐿</ci><ci id="S7.SS4.SSS0.P0.SPx2.p1.11.m11.1.1.3.cmml" xref="S7.SS4.SSS0.P0.SPx2.p1.11.m11.1.1.3">𝑘</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.SS4.SSS0.P0.SPx2.p1.11.m11.1c">L_{k}</annotation><annotation encoding="application/x-llamapun" id="S7.SS4.SSS0.P0.SPx2.p1.11.m11.1d">italic_L start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S7.SS4.SSS0.P0.SPx2.p1.17.17"> with </span><math alttext="k\geq h" class="ltx_Math" display="inline" id="S7.SS4.SSS0.P0.SPx2.p1.12.m12.1"><semantics id="S7.SS4.SSS0.P0.SPx2.p1.12.m12.1a"><mrow id="S7.SS4.SSS0.P0.SPx2.p1.12.m12.1.1" xref="S7.SS4.SSS0.P0.SPx2.p1.12.m12.1.1.cmml"><mi id="S7.SS4.SSS0.P0.SPx2.p1.12.m12.1.1.2" xref="S7.SS4.SSS0.P0.SPx2.p1.12.m12.1.1.2.cmml">k</mi><mo id="S7.SS4.SSS0.P0.SPx2.p1.12.m12.1.1.1" xref="S7.SS4.SSS0.P0.SPx2.p1.12.m12.1.1.1.cmml">≥</mo><mi id="S7.SS4.SSS0.P0.SPx2.p1.12.m12.1.1.3" xref="S7.SS4.SSS0.P0.SPx2.p1.12.m12.1.1.3.cmml">h</mi></mrow><annotation-xml encoding="MathML-Content" id="S7.SS4.SSS0.P0.SPx2.p1.12.m12.1b"><apply id="S7.SS4.SSS0.P0.SPx2.p1.12.m12.1.1.cmml" xref="S7.SS4.SSS0.P0.SPx2.p1.12.m12.1.1"><geq id="S7.SS4.SSS0.P0.SPx2.p1.12.m12.1.1.1.cmml" xref="S7.SS4.SSS0.P0.SPx2.p1.12.m12.1.1.1"></geq><ci id="S7.SS4.SSS0.P0.SPx2.p1.12.m12.1.1.2.cmml" xref="S7.SS4.SSS0.P0.SPx2.p1.12.m12.1.1.2">𝑘</ci><ci id="S7.SS4.SSS0.P0.SPx2.p1.12.m12.1.1.3.cmml" xref="S7.SS4.SSS0.P0.SPx2.p1.12.m12.1.1.3">ℎ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.SS4.SSS0.P0.SPx2.p1.12.m12.1c">k\geq h</annotation><annotation encoding="application/x-llamapun" id="S7.SS4.SSS0.P0.SPx2.p1.12.m12.1d">italic_k ≥ italic_h</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S7.SS4.SSS0.P0.SPx2.p1.17.18"> may only decrease their out-degree and thus cannot suddenly gain a violating in-edge to </span><math alttext="v" class="ltx_Math" display="inline" id="S7.SS4.SSS0.P0.SPx2.p1.13.m13.1"><semantics id="S7.SS4.SSS0.P0.SPx2.p1.13.m13.1a"><mi id="S7.SS4.SSS0.P0.SPx2.p1.13.m13.1.1" xref="S7.SS4.SSS0.P0.SPx2.p1.13.m13.1.1.cmml">v</mi><annotation-xml encoding="MathML-Content" id="S7.SS4.SSS0.P0.SPx2.p1.13.m13.1b"><ci id="S7.SS4.SSS0.P0.SPx2.p1.13.m13.1.1.cmml" xref="S7.SS4.SSS0.P0.SPx2.p1.13.m13.1.1">𝑣</ci></annotation-xml><annotation encoding="application/x-tex" id="S7.SS4.SSS0.P0.SPx2.p1.13.m13.1c">v</annotation><annotation encoding="application/x-llamapun" id="S7.SS4.SSS0.P0.SPx2.p1.13.m13.1d">italic_v</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S7.SS4.SSS0.P0.SPx2.p1.17.19">. Thus, </span><math alttext="l_{m+1}(v)=h-1" class="ltx_Math" display="inline" id="S7.SS4.SSS0.P0.SPx2.p1.14.m14.1"><semantics id="S7.SS4.SSS0.P0.SPx2.p1.14.m14.1a"><mrow id="S7.SS4.SSS0.P0.SPx2.p1.14.m14.1.2" xref="S7.SS4.SSS0.P0.SPx2.p1.14.m14.1.2.cmml"><mrow id="S7.SS4.SSS0.P0.SPx2.p1.14.m14.1.2.2" xref="S7.SS4.SSS0.P0.SPx2.p1.14.m14.1.2.2.cmml"><msub id="S7.SS4.SSS0.P0.SPx2.p1.14.m14.1.2.2.2" xref="S7.SS4.SSS0.P0.SPx2.p1.14.m14.1.2.2.2.cmml"><mi id="S7.SS4.SSS0.P0.SPx2.p1.14.m14.1.2.2.2.2" xref="S7.SS4.SSS0.P0.SPx2.p1.14.m14.1.2.2.2.2.cmml">l</mi><mrow id="S7.SS4.SSS0.P0.SPx2.p1.14.m14.1.2.2.2.3" xref="S7.SS4.SSS0.P0.SPx2.p1.14.m14.1.2.2.2.3.cmml"><mi id="S7.SS4.SSS0.P0.SPx2.p1.14.m14.1.2.2.2.3.2" xref="S7.SS4.SSS0.P0.SPx2.p1.14.m14.1.2.2.2.3.2.cmml">m</mi><mo id="S7.SS4.SSS0.P0.SPx2.p1.14.m14.1.2.2.2.3.1" xref="S7.SS4.SSS0.P0.SPx2.p1.14.m14.1.2.2.2.3.1.cmml">+</mo><mn id="S7.SS4.SSS0.P0.SPx2.p1.14.m14.1.2.2.2.3.3" xref="S7.SS4.SSS0.P0.SPx2.p1.14.m14.1.2.2.2.3.3.cmml">1</mn></mrow></msub><mo id="S7.SS4.SSS0.P0.SPx2.p1.14.m14.1.2.2.1" xref="S7.SS4.SSS0.P0.SPx2.p1.14.m14.1.2.2.1.cmml">⁢</mo><mrow id="S7.SS4.SSS0.P0.SPx2.p1.14.m14.1.2.2.3.2" xref="S7.SS4.SSS0.P0.SPx2.p1.14.m14.1.2.2.cmml"><mo id="S7.SS4.SSS0.P0.SPx2.p1.14.m14.1.2.2.3.2.1" stretchy="false" xref="S7.SS4.SSS0.P0.SPx2.p1.14.m14.1.2.2.cmml">(</mo><mi id="S7.SS4.SSS0.P0.SPx2.p1.14.m14.1.1" xref="S7.SS4.SSS0.P0.SPx2.p1.14.m14.1.1.cmml">v</mi><mo id="S7.SS4.SSS0.P0.SPx2.p1.14.m14.1.2.2.3.2.2" stretchy="false" xref="S7.SS4.SSS0.P0.SPx2.p1.14.m14.1.2.2.cmml">)</mo></mrow></mrow><mo id="S7.SS4.SSS0.P0.SPx2.p1.14.m14.1.2.1" xref="S7.SS4.SSS0.P0.SPx2.p1.14.m14.1.2.1.cmml">=</mo><mrow id="S7.SS4.SSS0.P0.SPx2.p1.14.m14.1.2.3" xref="S7.SS4.SSS0.P0.SPx2.p1.14.m14.1.2.3.cmml"><mi id="S7.SS4.SSS0.P0.SPx2.p1.14.m14.1.2.3.2" xref="S7.SS4.SSS0.P0.SPx2.p1.14.m14.1.2.3.2.cmml">h</mi><mo id="S7.SS4.SSS0.P0.SPx2.p1.14.m14.1.2.3.1" xref="S7.SS4.SSS0.P0.SPx2.p1.14.m14.1.2.3.1.cmml">−</mo><mn id="S7.SS4.SSS0.P0.SPx2.p1.14.m14.1.2.3.3" xref="S7.SS4.SSS0.P0.SPx2.p1.14.m14.1.2.3.3.cmml">1</mn></mrow></mrow><annotation-xml encoding="MathML-Content" id="S7.SS4.SSS0.P0.SPx2.p1.14.m14.1b"><apply id="S7.SS4.SSS0.P0.SPx2.p1.14.m14.1.2.cmml" xref="S7.SS4.SSS0.P0.SPx2.p1.14.m14.1.2"><eq id="S7.SS4.SSS0.P0.SPx2.p1.14.m14.1.2.1.cmml" xref="S7.SS4.SSS0.P0.SPx2.p1.14.m14.1.2.1"></eq><apply id="S7.SS4.SSS0.P0.SPx2.p1.14.m14.1.2.2.cmml" xref="S7.SS4.SSS0.P0.SPx2.p1.14.m14.1.2.2"><times id="S7.SS4.SSS0.P0.SPx2.p1.14.m14.1.2.2.1.cmml" xref="S7.SS4.SSS0.P0.SPx2.p1.14.m14.1.2.2.1"></times><apply id="S7.SS4.SSS0.P0.SPx2.p1.14.m14.1.2.2.2.cmml" xref="S7.SS4.SSS0.P0.SPx2.p1.14.m14.1.2.2.2"><csymbol cd="ambiguous" id="S7.SS4.SSS0.P0.SPx2.p1.14.m14.1.2.2.2.1.cmml" xref="S7.SS4.SSS0.P0.SPx2.p1.14.m14.1.2.2.2">subscript</csymbol><ci id="S7.SS4.SSS0.P0.SPx2.p1.14.m14.1.2.2.2.2.cmml" xref="S7.SS4.SSS0.P0.SPx2.p1.14.m14.1.2.2.2.2">𝑙</ci><apply id="S7.SS4.SSS0.P0.SPx2.p1.14.m14.1.2.2.2.3.cmml" xref="S7.SS4.SSS0.P0.SPx2.p1.14.m14.1.2.2.2.3"><plus id="S7.SS4.SSS0.P0.SPx2.p1.14.m14.1.2.2.2.3.1.cmml" xref="S7.SS4.SSS0.P0.SPx2.p1.14.m14.1.2.2.2.3.1"></plus><ci id="S7.SS4.SSS0.P0.SPx2.p1.14.m14.1.2.2.2.3.2.cmml" xref="S7.SS4.SSS0.P0.SPx2.p1.14.m14.1.2.2.2.3.2">𝑚</ci><cn id="S7.SS4.SSS0.P0.SPx2.p1.14.m14.1.2.2.2.3.3.cmml" type="integer" xref="S7.SS4.SSS0.P0.SPx2.p1.14.m14.1.2.2.2.3.3">1</cn></apply></apply><ci id="S7.SS4.SSS0.P0.SPx2.p1.14.m14.1.1.cmml" xref="S7.SS4.SSS0.P0.SPx2.p1.14.m14.1.1">𝑣</ci></apply><apply id="S7.SS4.SSS0.P0.SPx2.p1.14.m14.1.2.3.cmml" xref="S7.SS4.SSS0.P0.SPx2.p1.14.m14.1.2.3"><minus id="S7.SS4.SSS0.P0.SPx2.p1.14.m14.1.2.3.1.cmml" xref="S7.SS4.SSS0.P0.SPx2.p1.14.m14.1.2.3.1"></minus><ci id="S7.SS4.SSS0.P0.SPx2.p1.14.m14.1.2.3.2.cmml" xref="S7.SS4.SSS0.P0.SPx2.p1.14.m14.1.2.3.2">ℎ</ci><cn id="S7.SS4.SSS0.P0.SPx2.p1.14.m14.1.2.3.3.cmml" type="integer" xref="S7.SS4.SSS0.P0.SPx2.p1.14.m14.1.2.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.SS4.SSS0.P0.SPx2.p1.14.m14.1c">l_{m+1}(v)=h-1</annotation><annotation encoding="application/x-llamapun" id="S7.SS4.SSS0.P0.SPx2.p1.14.m14.1d">italic_l start_POSTSUBSCRIPT italic_m + 1 end_POSTSUBSCRIPT ( italic_v ) = italic_h - 1</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S7.SS4.SSS0.P0.SPx2.p1.17.20"> and </span><math alttext="v" class="ltx_Math" display="inline" id="S7.SS4.SSS0.P0.SPx2.p1.15.m15.1"><semantics id="S7.SS4.SSS0.P0.SPx2.p1.15.m15.1a"><mi id="S7.SS4.SSS0.P0.SPx2.p1.15.m15.1.1" xref="S7.SS4.SSS0.P0.SPx2.p1.15.m15.1.1.cmml">v</mi><annotation-xml encoding="MathML-Content" id="S7.SS4.SSS0.P0.SPx2.p1.15.m15.1b"><ci id="S7.SS4.SSS0.P0.SPx2.p1.15.m15.1.1.cmml" xref="S7.SS4.SSS0.P0.SPx2.p1.15.m15.1.1">𝑣</ci></annotation-xml><annotation encoding="application/x-tex" id="S7.SS4.SSS0.P0.SPx2.p1.15.m15.1c">v</annotation><annotation encoding="application/x-llamapun" id="S7.SS4.SSS0.P0.SPx2.p1.15.m15.1d">italic_v</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S7.SS4.SSS0.P0.SPx2.p1.17.21"> has no violating in-edge at the start of </span><math alttext="(h:m+1:0)" class="ltx_Math" display="inline" id="S7.SS4.SSS0.P0.SPx2.p1.16.m16.1"><semantics id="S7.SS4.SSS0.P0.SPx2.p1.16.m16.1a"><mrow id="S7.SS4.SSS0.P0.SPx2.p1.16.m16.1.1.1" xref="S7.SS4.SSS0.P0.SPx2.p1.16.m16.1.1.1.1.cmml"><mo id="S7.SS4.SSS0.P0.SPx2.p1.16.m16.1.1.1.2" stretchy="false" xref="S7.SS4.SSS0.P0.SPx2.p1.16.m16.1.1.1.1.cmml">(</mo><mrow id="S7.SS4.SSS0.P0.SPx2.p1.16.m16.1.1.1.1" xref="S7.SS4.SSS0.P0.SPx2.p1.16.m16.1.1.1.1.cmml"><mi id="S7.SS4.SSS0.P0.SPx2.p1.16.m16.1.1.1.1.2" xref="S7.SS4.SSS0.P0.SPx2.p1.16.m16.1.1.1.1.2.cmml">h</mi><mo id="S7.SS4.SSS0.P0.SPx2.p1.16.m16.1.1.1.1.3" lspace="0.278em" rspace="0.278em" xref="S7.SS4.SSS0.P0.SPx2.p1.16.m16.1.1.1.1.3.cmml">:</mo><mrow id="S7.SS4.SSS0.P0.SPx2.p1.16.m16.1.1.1.1.4" xref="S7.SS4.SSS0.P0.SPx2.p1.16.m16.1.1.1.1.4.cmml"><mi id="S7.SS4.SSS0.P0.SPx2.p1.16.m16.1.1.1.1.4.2" xref="S7.SS4.SSS0.P0.SPx2.p1.16.m16.1.1.1.1.4.2.cmml">m</mi><mo id="S7.SS4.SSS0.P0.SPx2.p1.16.m16.1.1.1.1.4.1" xref="S7.SS4.SSS0.P0.SPx2.p1.16.m16.1.1.1.1.4.1.cmml">+</mo><mn id="S7.SS4.SSS0.P0.SPx2.p1.16.m16.1.1.1.1.4.3" xref="S7.SS4.SSS0.P0.SPx2.p1.16.m16.1.1.1.1.4.3.cmml">1</mn></mrow><mo id="S7.SS4.SSS0.P0.SPx2.p1.16.m16.1.1.1.1.5" lspace="0.278em" rspace="0.278em" xref="S7.SS4.SSS0.P0.SPx2.p1.16.m16.1.1.1.1.5.cmml">:</mo><mn id="S7.SS4.SSS0.P0.SPx2.p1.16.m16.1.1.1.1.6" xref="S7.SS4.SSS0.P0.SPx2.p1.16.m16.1.1.1.1.6.cmml">0</mn></mrow><mo id="S7.SS4.SSS0.P0.SPx2.p1.16.m16.1.1.1.3" stretchy="false" xref="S7.SS4.SSS0.P0.SPx2.p1.16.m16.1.1.1.1.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S7.SS4.SSS0.P0.SPx2.p1.16.m16.1b"><apply id="S7.SS4.SSS0.P0.SPx2.p1.16.m16.1.1.1.1.cmml" xref="S7.SS4.SSS0.P0.SPx2.p1.16.m16.1.1.1"><and id="S7.SS4.SSS0.P0.SPx2.p1.16.m16.1.1.1.1a.cmml" xref="S7.SS4.SSS0.P0.SPx2.p1.16.m16.1.1.1"></and><apply id="S7.SS4.SSS0.P0.SPx2.p1.16.m16.1.1.1.1b.cmml" xref="S7.SS4.SSS0.P0.SPx2.p1.16.m16.1.1.1"><ci id="S7.SS4.SSS0.P0.SPx2.p1.16.m16.1.1.1.1.3.cmml" xref="S7.SS4.SSS0.P0.SPx2.p1.16.m16.1.1.1.1.3">:</ci><ci id="S7.SS4.SSS0.P0.SPx2.p1.16.m16.1.1.1.1.2.cmml" xref="S7.SS4.SSS0.P0.SPx2.p1.16.m16.1.1.1.1.2">ℎ</ci><apply id="S7.SS4.SSS0.P0.SPx2.p1.16.m16.1.1.1.1.4.cmml" xref="S7.SS4.SSS0.P0.SPx2.p1.16.m16.1.1.1.1.4"><plus id="S7.SS4.SSS0.P0.SPx2.p1.16.m16.1.1.1.1.4.1.cmml" xref="S7.SS4.SSS0.P0.SPx2.p1.16.m16.1.1.1.1.4.1"></plus><ci id="S7.SS4.SSS0.P0.SPx2.p1.16.m16.1.1.1.1.4.2.cmml" xref="S7.SS4.SSS0.P0.SPx2.p1.16.m16.1.1.1.1.4.2">𝑚</ci><cn id="S7.SS4.SSS0.P0.SPx2.p1.16.m16.1.1.1.1.4.3.cmml" type="integer" xref="S7.SS4.SSS0.P0.SPx2.p1.16.m16.1.1.1.1.4.3">1</cn></apply></apply><apply id="S7.SS4.SSS0.P0.SPx2.p1.16.m16.1.1.1.1c.cmml" xref="S7.SS4.SSS0.P0.SPx2.p1.16.m16.1.1.1"><ci id="S7.SS4.SSS0.P0.SPx2.p1.16.m16.1.1.1.1.5.cmml" xref="S7.SS4.SSS0.P0.SPx2.p1.16.m16.1.1.1.1.5">:</ci><share href="https://arxiv.org/html/2411.12694v2#S7.SS4.SSS0.P0.SPx2.p1.16.m16.1.1.1.1.4.cmml" id="S7.SS4.SSS0.P0.SPx2.p1.16.m16.1.1.1.1d.cmml" xref="S7.SS4.SSS0.P0.SPx2.p1.16.m16.1.1.1"></share><cn id="S7.SS4.SSS0.P0.SPx2.p1.16.m16.1.1.1.1.6.cmml" type="integer" xref="S7.SS4.SSS0.P0.SPx2.p1.16.m16.1.1.1.1.6">0</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.SS4.SSS0.P0.SPx2.p1.16.m16.1c">(h:m+1:0)</annotation><annotation encoding="application/x-llamapun" id="S7.SS4.SSS0.P0.SPx2.p1.16.m16.1d">( italic_h : italic_m + 1 : 0 )</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S7.SS4.SSS0.P0.SPx2.p1.17.22">. We may recursively apply this argument to conclude that </span><math alttext="l_{x}(v)=h-1" class="ltx_Math" display="inline" id="S7.SS4.SSS0.P0.SPx2.p1.17.m17.1"><semantics id="S7.SS4.SSS0.P0.SPx2.p1.17.m17.1a"><mrow id="S7.SS4.SSS0.P0.SPx2.p1.17.m17.1.2" xref="S7.SS4.SSS0.P0.SPx2.p1.17.m17.1.2.cmml"><mrow id="S7.SS4.SSS0.P0.SPx2.p1.17.m17.1.2.2" xref="S7.SS4.SSS0.P0.SPx2.p1.17.m17.1.2.2.cmml"><msub id="S7.SS4.SSS0.P0.SPx2.p1.17.m17.1.2.2.2" xref="S7.SS4.SSS0.P0.SPx2.p1.17.m17.1.2.2.2.cmml"><mi id="S7.SS4.SSS0.P0.SPx2.p1.17.m17.1.2.2.2.2" xref="S7.SS4.SSS0.P0.SPx2.p1.17.m17.1.2.2.2.2.cmml">l</mi><mi id="S7.SS4.SSS0.P0.SPx2.p1.17.m17.1.2.2.2.3" xref="S7.SS4.SSS0.P0.SPx2.p1.17.m17.1.2.2.2.3.cmml">x</mi></msub><mo id="S7.SS4.SSS0.P0.SPx2.p1.17.m17.1.2.2.1" xref="S7.SS4.SSS0.P0.SPx2.p1.17.m17.1.2.2.1.cmml">⁢</mo><mrow id="S7.SS4.SSS0.P0.SPx2.p1.17.m17.1.2.2.3.2" xref="S7.SS4.SSS0.P0.SPx2.p1.17.m17.1.2.2.cmml"><mo id="S7.SS4.SSS0.P0.SPx2.p1.17.m17.1.2.2.3.2.1" stretchy="false" xref="S7.SS4.SSS0.P0.SPx2.p1.17.m17.1.2.2.cmml">(</mo><mi id="S7.SS4.SSS0.P0.SPx2.p1.17.m17.1.1" xref="S7.SS4.SSS0.P0.SPx2.p1.17.m17.1.1.cmml">v</mi><mo id="S7.SS4.SSS0.P0.SPx2.p1.17.m17.1.2.2.3.2.2" stretchy="false" xref="S7.SS4.SSS0.P0.SPx2.p1.17.m17.1.2.2.cmml">)</mo></mrow></mrow><mo id="S7.SS4.SSS0.P0.SPx2.p1.17.m17.1.2.1" xref="S7.SS4.SSS0.P0.SPx2.p1.17.m17.1.2.1.cmml">=</mo><mrow id="S7.SS4.SSS0.P0.SPx2.p1.17.m17.1.2.3" xref="S7.SS4.SSS0.P0.SPx2.p1.17.m17.1.2.3.cmml"><mi id="S7.SS4.SSS0.P0.SPx2.p1.17.m17.1.2.3.2" xref="S7.SS4.SSS0.P0.SPx2.p1.17.m17.1.2.3.2.cmml">h</mi><mo id="S7.SS4.SSS0.P0.SPx2.p1.17.m17.1.2.3.1" xref="S7.SS4.SSS0.P0.SPx2.p1.17.m17.1.2.3.1.cmml">−</mo><mn id="S7.SS4.SSS0.P0.SPx2.p1.17.m17.1.2.3.3" xref="S7.SS4.SSS0.P0.SPx2.p1.17.m17.1.2.3.3.cmml">1</mn></mrow></mrow><annotation-xml encoding="MathML-Content" id="S7.SS4.SSS0.P0.SPx2.p1.17.m17.1b"><apply id="S7.SS4.SSS0.P0.SPx2.p1.17.m17.1.2.cmml" xref="S7.SS4.SSS0.P0.SPx2.p1.17.m17.1.2"><eq id="S7.SS4.SSS0.P0.SPx2.p1.17.m17.1.2.1.cmml" xref="S7.SS4.SSS0.P0.SPx2.p1.17.m17.1.2.1"></eq><apply id="S7.SS4.SSS0.P0.SPx2.p1.17.m17.1.2.2.cmml" xref="S7.SS4.SSS0.P0.SPx2.p1.17.m17.1.2.2"><times id="S7.SS4.SSS0.P0.SPx2.p1.17.m17.1.2.2.1.cmml" xref="S7.SS4.SSS0.P0.SPx2.p1.17.m17.1.2.2.1"></times><apply id="S7.SS4.SSS0.P0.SPx2.p1.17.m17.1.2.2.2.cmml" xref="S7.SS4.SSS0.P0.SPx2.p1.17.m17.1.2.2.2"><csymbol cd="ambiguous" id="S7.SS4.SSS0.P0.SPx2.p1.17.m17.1.2.2.2.1.cmml" xref="S7.SS4.SSS0.P0.SPx2.p1.17.m17.1.2.2.2">subscript</csymbol><ci id="S7.SS4.SSS0.P0.SPx2.p1.17.m17.1.2.2.2.2.cmml" xref="S7.SS4.SSS0.P0.SPx2.p1.17.m17.1.2.2.2.2">𝑙</ci><ci id="S7.SS4.SSS0.P0.SPx2.p1.17.m17.1.2.2.2.3.cmml" xref="S7.SS4.SSS0.P0.SPx2.p1.17.m17.1.2.2.2.3">𝑥</ci></apply><ci id="S7.SS4.SSS0.P0.SPx2.p1.17.m17.1.1.cmml" xref="S7.SS4.SSS0.P0.SPx2.p1.17.m17.1.1">𝑣</ci></apply><apply id="S7.SS4.SSS0.P0.SPx2.p1.17.m17.1.2.3.cmml" xref="S7.SS4.SSS0.P0.SPx2.p1.17.m17.1.2.3"><minus id="S7.SS4.SSS0.P0.SPx2.p1.17.m17.1.2.3.1.cmml" xref="S7.SS4.SSS0.P0.SPx2.p1.17.m17.1.2.3.1"></minus><ci id="S7.SS4.SSS0.P0.SPx2.p1.17.m17.1.2.3.2.cmml" xref="S7.SS4.SSS0.P0.SPx2.p1.17.m17.1.2.3.2">ℎ</ci><cn id="S7.SS4.SSS0.P0.SPx2.p1.17.m17.1.2.3.3.cmml" type="integer" xref="S7.SS4.SSS0.P0.SPx2.p1.17.m17.1.2.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.SS4.SSS0.P0.SPx2.p1.17.m17.1c">l_{x}(v)=h-1</annotation><annotation encoding="application/x-llamapun" id="S7.SS4.SSS0.P0.SPx2.p1.17.m17.1d">italic_l start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT ( italic_v ) = italic_h - 1</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S7.SS4.SSS0.P0.SPx2.p1.17.23">. </span><span class="ltx_text" id="S7.SS4.SSS0.P0.SPx2.p1.17.24"></span></p> </div> <div class="ltx_theorem ltx_theorem_theorem" id="S7.Thmtheorem26"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="S7.Thmtheorem26.1.1.1">Theorem 7.26</span></span><span class="ltx_text ltx_font_bold" id="S7.Thmtheorem26.2.2">.</span> </h6> <div class="ltx_para" id="S7.Thmtheorem26.p1"> <p class="ltx_p" id="S7.Thmtheorem26.p1.3"><span class="ltx_text ltx_font_italic" id="S7.Thmtheorem26.p1.3.3">Assume that Invariant <a class="ltx_ref" href="https://arxiv.org/html/2411.12694v2#Thminvariant1" title="Invariant 1. ‣ 7.1 Algorithm overview ‣ 7 Results in CONGEST ‣ Local Density and its Distributed Approximation"><span class="ltx_text ltx_ref_tag">1</span></a> holds at the start of <math alttext="(h:0:0)" class="ltx_Math" display="inline" id="S7.Thmtheorem26.p1.1.1.m1.1"><semantics id="S7.Thmtheorem26.p1.1.1.m1.1a"><mrow id="S7.Thmtheorem26.p1.1.1.m1.1.1.1" xref="S7.Thmtheorem26.p1.1.1.m1.1.1.1.1.cmml"><mo id="S7.Thmtheorem26.p1.1.1.m1.1.1.1.2" stretchy="false" xref="S7.Thmtheorem26.p1.1.1.m1.1.1.1.1.cmml">(</mo><mrow id="S7.Thmtheorem26.p1.1.1.m1.1.1.1.1" xref="S7.Thmtheorem26.p1.1.1.m1.1.1.1.1.cmml"><mi id="S7.Thmtheorem26.p1.1.1.m1.1.1.1.1.2" xref="S7.Thmtheorem26.p1.1.1.m1.1.1.1.1.2.cmml">h</mi><mo id="S7.Thmtheorem26.p1.1.1.m1.1.1.1.1.3" lspace="0.278em" rspace="0.278em" xref="S7.Thmtheorem26.p1.1.1.m1.1.1.1.1.3.cmml">:</mo><mn id="S7.Thmtheorem26.p1.1.1.m1.1.1.1.1.4" xref="S7.Thmtheorem26.p1.1.1.m1.1.1.1.1.4.cmml">0</mn><mo id="S7.Thmtheorem26.p1.1.1.m1.1.1.1.1.5" lspace="0.278em" rspace="0.278em" xref="S7.Thmtheorem26.p1.1.1.m1.1.1.1.1.5.cmml">:</mo><mn id="S7.Thmtheorem26.p1.1.1.m1.1.1.1.1.6" xref="S7.Thmtheorem26.p1.1.1.m1.1.1.1.1.6.cmml">0</mn></mrow><mo id="S7.Thmtheorem26.p1.1.1.m1.1.1.1.3" stretchy="false" xref="S7.Thmtheorem26.p1.1.1.m1.1.1.1.1.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S7.Thmtheorem26.p1.1.1.m1.1b"><apply id="S7.Thmtheorem26.p1.1.1.m1.1.1.1.1.cmml" xref="S7.Thmtheorem26.p1.1.1.m1.1.1.1"><and id="S7.Thmtheorem26.p1.1.1.m1.1.1.1.1a.cmml" xref="S7.Thmtheorem26.p1.1.1.m1.1.1.1"></and><apply id="S7.Thmtheorem26.p1.1.1.m1.1.1.1.1b.cmml" xref="S7.Thmtheorem26.p1.1.1.m1.1.1.1"><ci id="S7.Thmtheorem26.p1.1.1.m1.1.1.1.1.3.cmml" xref="S7.Thmtheorem26.p1.1.1.m1.1.1.1.1.3">:</ci><ci id="S7.Thmtheorem26.p1.1.1.m1.1.1.1.1.2.cmml" xref="S7.Thmtheorem26.p1.1.1.m1.1.1.1.1.2">ℎ</ci><cn id="S7.Thmtheorem26.p1.1.1.m1.1.1.1.1.4.cmml" type="integer" xref="S7.Thmtheorem26.p1.1.1.m1.1.1.1.1.4">0</cn></apply><apply id="S7.Thmtheorem26.p1.1.1.m1.1.1.1.1c.cmml" xref="S7.Thmtheorem26.p1.1.1.m1.1.1.1"><ci id="S7.Thmtheorem26.p1.1.1.m1.1.1.1.1.5.cmml" xref="S7.Thmtheorem26.p1.1.1.m1.1.1.1.1.5">:</ci><share href="https://arxiv.org/html/2411.12694v2#S7.Thmtheorem26.p1.1.1.m1.1.1.1.1.4.cmml" id="S7.Thmtheorem26.p1.1.1.m1.1.1.1.1d.cmml" xref="S7.Thmtheorem26.p1.1.1.m1.1.1.1"></share><cn id="S7.Thmtheorem26.p1.1.1.m1.1.1.1.1.6.cmml" type="integer" xref="S7.Thmtheorem26.p1.1.1.m1.1.1.1.1.6">0</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.Thmtheorem26.p1.1.1.m1.1c">(h:0:0)</annotation><annotation encoding="application/x-llamapun" id="S7.Thmtheorem26.p1.1.1.m1.1d">( italic_h : 0 : 0 )</annotation></semantics></math>, then our algorithm maintains Invariant <a class="ltx_ref" href="https://arxiv.org/html/2411.12694v2#Thminvariant1" title="Invariant 1. ‣ 7.1 Algorithm overview ‣ 7 Results in CONGEST ‣ Local Density and its Distributed Approximation"><span class="ltx_text ltx_ref_tag">1</span></a> throughout <span class="ltx_text ltx_font_smallcaps" id="S7.Thmtheorem26.p1.3.3.1">hour</span> <math alttext="h" class="ltx_Math" display="inline" id="S7.Thmtheorem26.p1.2.2.m2.1"><semantics id="S7.Thmtheorem26.p1.2.2.m2.1a"><mi id="S7.Thmtheorem26.p1.2.2.m2.1.1" xref="S7.Thmtheorem26.p1.2.2.m2.1.1.cmml">h</mi><annotation-xml encoding="MathML-Content" id="S7.Thmtheorem26.p1.2.2.m2.1b"><ci id="S7.Thmtheorem26.p1.2.2.m2.1.1.cmml" xref="S7.Thmtheorem26.p1.2.2.m2.1.1">ℎ</ci></annotation-xml><annotation encoding="application/x-tex" id="S7.Thmtheorem26.p1.2.2.m2.1c">h</annotation><annotation encoding="application/x-llamapun" id="S7.Thmtheorem26.p1.2.2.m2.1d">italic_h</annotation></semantics></math> until and including the start of <math alttext="(h-1:0:0)" class="ltx_Math" display="inline" id="S7.Thmtheorem26.p1.3.3.m3.1"><semantics id="S7.Thmtheorem26.p1.3.3.m3.1a"><mrow id="S7.Thmtheorem26.p1.3.3.m3.1.1.1" xref="S7.Thmtheorem26.p1.3.3.m3.1.1.1.1.cmml"><mo id="S7.Thmtheorem26.p1.3.3.m3.1.1.1.2" stretchy="false" xref="S7.Thmtheorem26.p1.3.3.m3.1.1.1.1.cmml">(</mo><mrow id="S7.Thmtheorem26.p1.3.3.m3.1.1.1.1" xref="S7.Thmtheorem26.p1.3.3.m3.1.1.1.1.cmml"><mrow id="S7.Thmtheorem26.p1.3.3.m3.1.1.1.1.2" xref="S7.Thmtheorem26.p1.3.3.m3.1.1.1.1.2.cmml"><mi id="S7.Thmtheorem26.p1.3.3.m3.1.1.1.1.2.2" xref="S7.Thmtheorem26.p1.3.3.m3.1.1.1.1.2.2.cmml">h</mi><mo id="S7.Thmtheorem26.p1.3.3.m3.1.1.1.1.2.1" xref="S7.Thmtheorem26.p1.3.3.m3.1.1.1.1.2.1.cmml">−</mo><mn id="S7.Thmtheorem26.p1.3.3.m3.1.1.1.1.2.3" xref="S7.Thmtheorem26.p1.3.3.m3.1.1.1.1.2.3.cmml">1</mn></mrow><mo id="S7.Thmtheorem26.p1.3.3.m3.1.1.1.1.3" lspace="0.278em" rspace="0.278em" xref="S7.Thmtheorem26.p1.3.3.m3.1.1.1.1.3.cmml">:</mo><mn id="S7.Thmtheorem26.p1.3.3.m3.1.1.1.1.4" xref="S7.Thmtheorem26.p1.3.3.m3.1.1.1.1.4.cmml">0</mn><mo id="S7.Thmtheorem26.p1.3.3.m3.1.1.1.1.5" lspace="0.278em" rspace="0.278em" xref="S7.Thmtheorem26.p1.3.3.m3.1.1.1.1.5.cmml">:</mo><mn id="S7.Thmtheorem26.p1.3.3.m3.1.1.1.1.6" xref="S7.Thmtheorem26.p1.3.3.m3.1.1.1.1.6.cmml">0</mn></mrow><mo id="S7.Thmtheorem26.p1.3.3.m3.1.1.1.3" stretchy="false" xref="S7.Thmtheorem26.p1.3.3.m3.1.1.1.1.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S7.Thmtheorem26.p1.3.3.m3.1b"><apply id="S7.Thmtheorem26.p1.3.3.m3.1.1.1.1.cmml" xref="S7.Thmtheorem26.p1.3.3.m3.1.1.1"><and id="S7.Thmtheorem26.p1.3.3.m3.1.1.1.1a.cmml" xref="S7.Thmtheorem26.p1.3.3.m3.1.1.1"></and><apply id="S7.Thmtheorem26.p1.3.3.m3.1.1.1.1b.cmml" xref="S7.Thmtheorem26.p1.3.3.m3.1.1.1"><ci id="S7.Thmtheorem26.p1.3.3.m3.1.1.1.1.3.cmml" xref="S7.Thmtheorem26.p1.3.3.m3.1.1.1.1.3">:</ci><apply id="S7.Thmtheorem26.p1.3.3.m3.1.1.1.1.2.cmml" xref="S7.Thmtheorem26.p1.3.3.m3.1.1.1.1.2"><minus id="S7.Thmtheorem26.p1.3.3.m3.1.1.1.1.2.1.cmml" xref="S7.Thmtheorem26.p1.3.3.m3.1.1.1.1.2.1"></minus><ci id="S7.Thmtheorem26.p1.3.3.m3.1.1.1.1.2.2.cmml" xref="S7.Thmtheorem26.p1.3.3.m3.1.1.1.1.2.2">ℎ</ci><cn id="S7.Thmtheorem26.p1.3.3.m3.1.1.1.1.2.3.cmml" type="integer" xref="S7.Thmtheorem26.p1.3.3.m3.1.1.1.1.2.3">1</cn></apply><cn id="S7.Thmtheorem26.p1.3.3.m3.1.1.1.1.4.cmml" type="integer" xref="S7.Thmtheorem26.p1.3.3.m3.1.1.1.1.4">0</cn></apply><apply id="S7.Thmtheorem26.p1.3.3.m3.1.1.1.1c.cmml" xref="S7.Thmtheorem26.p1.3.3.m3.1.1.1"><ci id="S7.Thmtheorem26.p1.3.3.m3.1.1.1.1.5.cmml" xref="S7.Thmtheorem26.p1.3.3.m3.1.1.1.1.5">:</ci><share href="https://arxiv.org/html/2411.12694v2#S7.Thmtheorem26.p1.3.3.m3.1.1.1.1.4.cmml" id="S7.Thmtheorem26.p1.3.3.m3.1.1.1.1d.cmml" xref="S7.Thmtheorem26.p1.3.3.m3.1.1.1"></share><cn id="S7.Thmtheorem26.p1.3.3.m3.1.1.1.1.6.cmml" type="integer" xref="S7.Thmtheorem26.p1.3.3.m3.1.1.1.1.6">0</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.Thmtheorem26.p1.3.3.m3.1c">(h-1:0:0)</annotation><annotation encoding="application/x-llamapun" id="S7.Thmtheorem26.p1.3.3.m3.1d">( italic_h - 1 : 0 : 0 )</annotation></semantics></math>.</span></p> </div> </div> <div class="ltx_theorem ltx_theorem_proof" id="S7.Thmtheorem27"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="S7.Thmtheorem27.1.1.1">Proof 7.27</span></span><span class="ltx_text ltx_font_bold" id="S7.Thmtheorem27.2.2">.</span> </h6> <div class="ltx_para" id="S7.Thmtheorem27.p1"> <p class="ltx_p" id="S7.Thmtheorem27.p1.2"><span class="ltx_text ltx_font_italic" id="S7.Thmtheorem27.p1.2.2">Suppose for the sake of contradiction that at the start of <math alttext="(h:2\lceil\eta^{-1}\rceil+2:0)" class="ltx_Math" display="inline" id="S7.Thmtheorem27.p1.1.1.m1.1"><semantics id="S7.Thmtheorem27.p1.1.1.m1.1a"><mrow id="S7.Thmtheorem27.p1.1.1.m1.1.1.1" xref="S7.Thmtheorem27.p1.1.1.m1.1.1.1.1.cmml"><mo id="S7.Thmtheorem27.p1.1.1.m1.1.1.1.2" stretchy="false" xref="S7.Thmtheorem27.p1.1.1.m1.1.1.1.1.cmml">(</mo><mrow id="S7.Thmtheorem27.p1.1.1.m1.1.1.1.1" xref="S7.Thmtheorem27.p1.1.1.m1.1.1.1.1.cmml"><mi id="S7.Thmtheorem27.p1.1.1.m1.1.1.1.1.3" xref="S7.Thmtheorem27.p1.1.1.m1.1.1.1.1.3.cmml">h</mi><mo id="S7.Thmtheorem27.p1.1.1.m1.1.1.1.1.4" lspace="0.278em" rspace="0.278em" xref="S7.Thmtheorem27.p1.1.1.m1.1.1.1.1.4.cmml">:</mo><mrow id="S7.Thmtheorem27.p1.1.1.m1.1.1.1.1.1" xref="S7.Thmtheorem27.p1.1.1.m1.1.1.1.1.1.cmml"><mrow id="S7.Thmtheorem27.p1.1.1.m1.1.1.1.1.1.1" xref="S7.Thmtheorem27.p1.1.1.m1.1.1.1.1.1.1.cmml"><mn id="S7.Thmtheorem27.p1.1.1.m1.1.1.1.1.1.1.3" xref="S7.Thmtheorem27.p1.1.1.m1.1.1.1.1.1.1.3.cmml">2</mn><mo id="S7.Thmtheorem27.p1.1.1.m1.1.1.1.1.1.1.2" xref="S7.Thmtheorem27.p1.1.1.m1.1.1.1.1.1.1.2.cmml">⁢</mo><mrow id="S7.Thmtheorem27.p1.1.1.m1.1.1.1.1.1.1.1.1" xref="S7.Thmtheorem27.p1.1.1.m1.1.1.1.1.1.1.1.2.cmml"><mo id="S7.Thmtheorem27.p1.1.1.m1.1.1.1.1.1.1.1.1.2" stretchy="false" xref="S7.Thmtheorem27.p1.1.1.m1.1.1.1.1.1.1.1.2.1.cmml">⌈</mo><msup id="S7.Thmtheorem27.p1.1.1.m1.1.1.1.1.1.1.1.1.1" xref="S7.Thmtheorem27.p1.1.1.m1.1.1.1.1.1.1.1.1.1.cmml"><mi id="S7.Thmtheorem27.p1.1.1.m1.1.1.1.1.1.1.1.1.1.2" xref="S7.Thmtheorem27.p1.1.1.m1.1.1.1.1.1.1.1.1.1.2.cmml">η</mi><mrow id="S7.Thmtheorem27.p1.1.1.m1.1.1.1.1.1.1.1.1.1.3" xref="S7.Thmtheorem27.p1.1.1.m1.1.1.1.1.1.1.1.1.1.3.cmml"><mo id="S7.Thmtheorem27.p1.1.1.m1.1.1.1.1.1.1.1.1.1.3a" xref="S7.Thmtheorem27.p1.1.1.m1.1.1.1.1.1.1.1.1.1.3.cmml">−</mo><mn id="S7.Thmtheorem27.p1.1.1.m1.1.1.1.1.1.1.1.1.1.3.2" xref="S7.Thmtheorem27.p1.1.1.m1.1.1.1.1.1.1.1.1.1.3.2.cmml">1</mn></mrow></msup><mo id="S7.Thmtheorem27.p1.1.1.m1.1.1.1.1.1.1.1.1.3" stretchy="false" xref="S7.Thmtheorem27.p1.1.1.m1.1.1.1.1.1.1.1.2.1.cmml">⌉</mo></mrow></mrow><mo id="S7.Thmtheorem27.p1.1.1.m1.1.1.1.1.1.2" xref="S7.Thmtheorem27.p1.1.1.m1.1.1.1.1.1.2.cmml">+</mo><mn id="S7.Thmtheorem27.p1.1.1.m1.1.1.1.1.1.3" xref="S7.Thmtheorem27.p1.1.1.m1.1.1.1.1.1.3.cmml">2</mn></mrow><mo id="S7.Thmtheorem27.p1.1.1.m1.1.1.1.1.5" lspace="0.278em" rspace="0.278em" xref="S7.Thmtheorem27.p1.1.1.m1.1.1.1.1.5.cmml">:</mo><mn id="S7.Thmtheorem27.p1.1.1.m1.1.1.1.1.6" xref="S7.Thmtheorem27.p1.1.1.m1.1.1.1.1.6.cmml">0</mn></mrow><mo id="S7.Thmtheorem27.p1.1.1.m1.1.1.1.3" stretchy="false" xref="S7.Thmtheorem27.p1.1.1.m1.1.1.1.1.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S7.Thmtheorem27.p1.1.1.m1.1b"><apply id="S7.Thmtheorem27.p1.1.1.m1.1.1.1.1.cmml" xref="S7.Thmtheorem27.p1.1.1.m1.1.1.1"><and id="S7.Thmtheorem27.p1.1.1.m1.1.1.1.1a.cmml" xref="S7.Thmtheorem27.p1.1.1.m1.1.1.1"></and><apply id="S7.Thmtheorem27.p1.1.1.m1.1.1.1.1b.cmml" xref="S7.Thmtheorem27.p1.1.1.m1.1.1.1"><ci id="S7.Thmtheorem27.p1.1.1.m1.1.1.1.1.4.cmml" xref="S7.Thmtheorem27.p1.1.1.m1.1.1.1.1.4">:</ci><ci id="S7.Thmtheorem27.p1.1.1.m1.1.1.1.1.3.cmml" xref="S7.Thmtheorem27.p1.1.1.m1.1.1.1.1.3">ℎ</ci><apply id="S7.Thmtheorem27.p1.1.1.m1.1.1.1.1.1.cmml" xref="S7.Thmtheorem27.p1.1.1.m1.1.1.1.1.1"><plus id="S7.Thmtheorem27.p1.1.1.m1.1.1.1.1.1.2.cmml" xref="S7.Thmtheorem27.p1.1.1.m1.1.1.1.1.1.2"></plus><apply id="S7.Thmtheorem27.p1.1.1.m1.1.1.1.1.1.1.cmml" xref="S7.Thmtheorem27.p1.1.1.m1.1.1.1.1.1.1"><times id="S7.Thmtheorem27.p1.1.1.m1.1.1.1.1.1.1.2.cmml" xref="S7.Thmtheorem27.p1.1.1.m1.1.1.1.1.1.1.2"></times><cn id="S7.Thmtheorem27.p1.1.1.m1.1.1.1.1.1.1.3.cmml" type="integer" xref="S7.Thmtheorem27.p1.1.1.m1.1.1.1.1.1.1.3">2</cn><apply id="S7.Thmtheorem27.p1.1.1.m1.1.1.1.1.1.1.1.2.cmml" xref="S7.Thmtheorem27.p1.1.1.m1.1.1.1.1.1.1.1.1"><ceiling id="S7.Thmtheorem27.p1.1.1.m1.1.1.1.1.1.1.1.2.1.cmml" xref="S7.Thmtheorem27.p1.1.1.m1.1.1.1.1.1.1.1.1.2"></ceiling><apply id="S7.Thmtheorem27.p1.1.1.m1.1.1.1.1.1.1.1.1.1.cmml" xref="S7.Thmtheorem27.p1.1.1.m1.1.1.1.1.1.1.1.1.1"><csymbol cd="ambiguous" id="S7.Thmtheorem27.p1.1.1.m1.1.1.1.1.1.1.1.1.1.1.cmml" xref="S7.Thmtheorem27.p1.1.1.m1.1.1.1.1.1.1.1.1.1">superscript</csymbol><ci id="S7.Thmtheorem27.p1.1.1.m1.1.1.1.1.1.1.1.1.1.2.cmml" xref="S7.Thmtheorem27.p1.1.1.m1.1.1.1.1.1.1.1.1.1.2">𝜂</ci><apply id="S7.Thmtheorem27.p1.1.1.m1.1.1.1.1.1.1.1.1.1.3.cmml" xref="S7.Thmtheorem27.p1.1.1.m1.1.1.1.1.1.1.1.1.1.3"><minus id="S7.Thmtheorem27.p1.1.1.m1.1.1.1.1.1.1.1.1.1.3.1.cmml" xref="S7.Thmtheorem27.p1.1.1.m1.1.1.1.1.1.1.1.1.1.3"></minus><cn id="S7.Thmtheorem27.p1.1.1.m1.1.1.1.1.1.1.1.1.1.3.2.cmml" type="integer" xref="S7.Thmtheorem27.p1.1.1.m1.1.1.1.1.1.1.1.1.1.3.2">1</cn></apply></apply></apply></apply><cn id="S7.Thmtheorem27.p1.1.1.m1.1.1.1.1.1.3.cmml" type="integer" xref="S7.Thmtheorem27.p1.1.1.m1.1.1.1.1.1.3">2</cn></apply></apply><apply id="S7.Thmtheorem27.p1.1.1.m1.1.1.1.1c.cmml" xref="S7.Thmtheorem27.p1.1.1.m1.1.1.1"><ci id="S7.Thmtheorem27.p1.1.1.m1.1.1.1.1.5.cmml" xref="S7.Thmtheorem27.p1.1.1.m1.1.1.1.1.5">:</ci><share href="https://arxiv.org/html/2411.12694v2#S7.Thmtheorem27.p1.1.1.m1.1.1.1.1.1.cmml" id="S7.Thmtheorem27.p1.1.1.m1.1.1.1.1d.cmml" xref="S7.Thmtheorem27.p1.1.1.m1.1.1.1"></share><cn id="S7.Thmtheorem27.p1.1.1.m1.1.1.1.1.6.cmml" type="integer" xref="S7.Thmtheorem27.p1.1.1.m1.1.1.1.1.6">0</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.Thmtheorem27.p1.1.1.m1.1c">(h:2\lceil\eta^{-1}\rceil+2:0)</annotation><annotation encoding="application/x-llamapun" id="S7.Thmtheorem27.p1.1.1.m1.1d">( italic_h : 2 ⌈ italic_η start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT ⌉ + 2 : 0 )</annotation></semantics></math> there exists a vertex <math alttext="v\in L_{h}" class="ltx_Math" display="inline" id="S7.Thmtheorem27.p1.2.2.m2.1"><semantics id="S7.Thmtheorem27.p1.2.2.m2.1a"><mrow id="S7.Thmtheorem27.p1.2.2.m2.1.1" xref="S7.Thmtheorem27.p1.2.2.m2.1.1.cmml"><mi id="S7.Thmtheorem27.p1.2.2.m2.1.1.2" xref="S7.Thmtheorem27.p1.2.2.m2.1.1.2.cmml">v</mi><mo id="S7.Thmtheorem27.p1.2.2.m2.1.1.1" xref="S7.Thmtheorem27.p1.2.2.m2.1.1.1.cmml">∈</mo><msub id="S7.Thmtheorem27.p1.2.2.m2.1.1.3" xref="S7.Thmtheorem27.p1.2.2.m2.1.1.3.cmml"><mi id="S7.Thmtheorem27.p1.2.2.m2.1.1.3.2" xref="S7.Thmtheorem27.p1.2.2.m2.1.1.3.2.cmml">L</mi><mi id="S7.Thmtheorem27.p1.2.2.m2.1.1.3.3" xref="S7.Thmtheorem27.p1.2.2.m2.1.1.3.3.cmml">h</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S7.Thmtheorem27.p1.2.2.m2.1b"><apply id="S7.Thmtheorem27.p1.2.2.m2.1.1.cmml" xref="S7.Thmtheorem27.p1.2.2.m2.1.1"><in id="S7.Thmtheorem27.p1.2.2.m2.1.1.1.cmml" xref="S7.Thmtheorem27.p1.2.2.m2.1.1.1"></in><ci id="S7.Thmtheorem27.p1.2.2.m2.1.1.2.cmml" xref="S7.Thmtheorem27.p1.2.2.m2.1.1.2">𝑣</ci><apply id="S7.Thmtheorem27.p1.2.2.m2.1.1.3.cmml" xref="S7.Thmtheorem27.p1.2.2.m2.1.1.3"><csymbol cd="ambiguous" id="S7.Thmtheorem27.p1.2.2.m2.1.1.3.1.cmml" xref="S7.Thmtheorem27.p1.2.2.m2.1.1.3">subscript</csymbol><ci id="S7.Thmtheorem27.p1.2.2.m2.1.1.3.2.cmml" xref="S7.Thmtheorem27.p1.2.2.m2.1.1.3.2">𝐿</ci><ci id="S7.Thmtheorem27.p1.2.2.m2.1.1.3.3.cmml" xref="S7.Thmtheorem27.p1.2.2.m2.1.1.3.3">ℎ</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.Thmtheorem27.p1.2.2.m2.1c">v\in L_{h}</annotation><annotation encoding="application/x-llamapun" id="S7.Thmtheorem27.p1.2.2.m2.1d">italic_v ∈ italic_L start_POSTSUBSCRIPT italic_h end_POSTSUBSCRIPT</annotation></semantics></math> with at least one violating out-edge. We may apply Lemma <a class="ltx_ref" href="https://arxiv.org/html/2411.12694v2#S7.Thmtheorem24" title="Lemma 7.24. ‣ 7.4 Formally proving correctness ‣ 7 Results in CONGEST ‣ Local Density and its Distributed Approximation"><span class="ltx_text ltx_ref_tag">7.24</span></a> so that:</span></p> </div> <div class="ltx_para" id="S7.Thmtheorem27.p2"> <ul class="ltx_itemize" id="S7.I13"> <li class="ltx_item" id="S7.I13.i1" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">•</span> <div class="ltx_para" id="S7.I13.i1.p1"> <p class="ltx_p" id="S7.I13.i1.p1.4"><span class="ltx_text ltx_font_italic" id="S7.I13.i1.p1.4.1">for even </span><math alttext="m" class="ltx_Math" display="inline" id="S7.I13.i1.p1.1.m1.1"><semantics id="S7.I13.i1.p1.1.m1.1a"><mi id="S7.I13.i1.p1.1.m1.1.1" xref="S7.I13.i1.p1.1.m1.1.1.cmml">m</mi><annotation-xml encoding="MathML-Content" id="S7.I13.i1.p1.1.m1.1b"><ci id="S7.I13.i1.p1.1.m1.1.1.cmml" xref="S7.I13.i1.p1.1.m1.1.1">𝑚</ci></annotation-xml><annotation encoding="application/x-tex" id="S7.I13.i1.p1.1.m1.1c">m</annotation><annotation encoding="application/x-llamapun" id="S7.I13.i1.p1.1.m1.1d">italic_m</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S7.I13.i1.p1.4.2">, at the start of </span><math alttext="(h:m:0)" class="ltx_Math" display="inline" id="S7.I13.i1.p1.2.m2.1"><semantics id="S7.I13.i1.p1.2.m2.1a"><mrow id="S7.I13.i1.p1.2.m2.1.1.1" xref="S7.I13.i1.p1.2.m2.1.1.1.1.cmml"><mo id="S7.I13.i1.p1.2.m2.1.1.1.2" stretchy="false" xref="S7.I13.i1.p1.2.m2.1.1.1.1.cmml">(</mo><mrow id="S7.I13.i1.p1.2.m2.1.1.1.1" xref="S7.I13.i1.p1.2.m2.1.1.1.1.cmml"><mi id="S7.I13.i1.p1.2.m2.1.1.1.1.2" xref="S7.I13.i1.p1.2.m2.1.1.1.1.2.cmml">h</mi><mo id="S7.I13.i1.p1.2.m2.1.1.1.1.3" lspace="0.278em" rspace="0.278em" xref="S7.I13.i1.p1.2.m2.1.1.1.1.3.cmml">:</mo><mi id="S7.I13.i1.p1.2.m2.1.1.1.1.4" xref="S7.I13.i1.p1.2.m2.1.1.1.1.4.cmml">m</mi><mo id="S7.I13.i1.p1.2.m2.1.1.1.1.5" lspace="0.278em" rspace="0.278em" xref="S7.I13.i1.p1.2.m2.1.1.1.1.5.cmml">:</mo><mn id="S7.I13.i1.p1.2.m2.1.1.1.1.6" xref="S7.I13.i1.p1.2.m2.1.1.1.1.6.cmml">0</mn></mrow><mo id="S7.I13.i1.p1.2.m2.1.1.1.3" stretchy="false" xref="S7.I13.i1.p1.2.m2.1.1.1.1.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S7.I13.i1.p1.2.m2.1b"><apply id="S7.I13.i1.p1.2.m2.1.1.1.1.cmml" xref="S7.I13.i1.p1.2.m2.1.1.1"><and id="S7.I13.i1.p1.2.m2.1.1.1.1a.cmml" xref="S7.I13.i1.p1.2.m2.1.1.1"></and><apply id="S7.I13.i1.p1.2.m2.1.1.1.1b.cmml" xref="S7.I13.i1.p1.2.m2.1.1.1"><ci id="S7.I13.i1.p1.2.m2.1.1.1.1.3.cmml" xref="S7.I13.i1.p1.2.m2.1.1.1.1.3">:</ci><ci id="S7.I13.i1.p1.2.m2.1.1.1.1.2.cmml" xref="S7.I13.i1.p1.2.m2.1.1.1.1.2">ℎ</ci><ci id="S7.I13.i1.p1.2.m2.1.1.1.1.4.cmml" xref="S7.I13.i1.p1.2.m2.1.1.1.1.4">𝑚</ci></apply><apply id="S7.I13.i1.p1.2.m2.1.1.1.1c.cmml" xref="S7.I13.i1.p1.2.m2.1.1.1"><ci id="S7.I13.i1.p1.2.m2.1.1.1.1.5.cmml" xref="S7.I13.i1.p1.2.m2.1.1.1.1.5">:</ci><share href="https://arxiv.org/html/2411.12694v2#S7.I13.i1.p1.2.m2.1.1.1.1.4.cmml" id="S7.I13.i1.p1.2.m2.1.1.1.1d.cmml" xref="S7.I13.i1.p1.2.m2.1.1.1"></share><cn id="S7.I13.i1.p1.2.m2.1.1.1.1.6.cmml" type="integer" xref="S7.I13.i1.p1.2.m2.1.1.1.1.6">0</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.I13.i1.p1.2.m2.1c">(h:m:0)</annotation><annotation encoding="application/x-llamapun" id="S7.I13.i1.p1.2.m2.1d">( italic_h : italic_m : 0 )</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S7.I13.i1.p1.4.3">, </span><math alttext="l_{m}(v)=h" class="ltx_Math" display="inline" id="S7.I13.i1.p1.3.m3.1"><semantics id="S7.I13.i1.p1.3.m3.1a"><mrow id="S7.I13.i1.p1.3.m3.1.2" xref="S7.I13.i1.p1.3.m3.1.2.cmml"><mrow id="S7.I13.i1.p1.3.m3.1.2.2" xref="S7.I13.i1.p1.3.m3.1.2.2.cmml"><msub id="S7.I13.i1.p1.3.m3.1.2.2.2" xref="S7.I13.i1.p1.3.m3.1.2.2.2.cmml"><mi id="S7.I13.i1.p1.3.m3.1.2.2.2.2" xref="S7.I13.i1.p1.3.m3.1.2.2.2.2.cmml">l</mi><mi id="S7.I13.i1.p1.3.m3.1.2.2.2.3" xref="S7.I13.i1.p1.3.m3.1.2.2.2.3.cmml">m</mi></msub><mo id="S7.I13.i1.p1.3.m3.1.2.2.1" xref="S7.I13.i1.p1.3.m3.1.2.2.1.cmml">⁢</mo><mrow id="S7.I13.i1.p1.3.m3.1.2.2.3.2" xref="S7.I13.i1.p1.3.m3.1.2.2.cmml"><mo id="S7.I13.i1.p1.3.m3.1.2.2.3.2.1" stretchy="false" xref="S7.I13.i1.p1.3.m3.1.2.2.cmml">(</mo><mi id="S7.I13.i1.p1.3.m3.1.1" xref="S7.I13.i1.p1.3.m3.1.1.cmml">v</mi><mo id="S7.I13.i1.p1.3.m3.1.2.2.3.2.2" stretchy="false" xref="S7.I13.i1.p1.3.m3.1.2.2.cmml">)</mo></mrow></mrow><mo id="S7.I13.i1.p1.3.m3.1.2.1" xref="S7.I13.i1.p1.3.m3.1.2.1.cmml">=</mo><mi id="S7.I13.i1.p1.3.m3.1.2.3" xref="S7.I13.i1.p1.3.m3.1.2.3.cmml">h</mi></mrow><annotation-xml encoding="MathML-Content" id="S7.I13.i1.p1.3.m3.1b"><apply id="S7.I13.i1.p1.3.m3.1.2.cmml" xref="S7.I13.i1.p1.3.m3.1.2"><eq id="S7.I13.i1.p1.3.m3.1.2.1.cmml" xref="S7.I13.i1.p1.3.m3.1.2.1"></eq><apply id="S7.I13.i1.p1.3.m3.1.2.2.cmml" xref="S7.I13.i1.p1.3.m3.1.2.2"><times id="S7.I13.i1.p1.3.m3.1.2.2.1.cmml" xref="S7.I13.i1.p1.3.m3.1.2.2.1"></times><apply id="S7.I13.i1.p1.3.m3.1.2.2.2.cmml" xref="S7.I13.i1.p1.3.m3.1.2.2.2"><csymbol cd="ambiguous" id="S7.I13.i1.p1.3.m3.1.2.2.2.1.cmml" xref="S7.I13.i1.p1.3.m3.1.2.2.2">subscript</csymbol><ci id="S7.I13.i1.p1.3.m3.1.2.2.2.2.cmml" xref="S7.I13.i1.p1.3.m3.1.2.2.2.2">𝑙</ci><ci id="S7.I13.i1.p1.3.m3.1.2.2.2.3.cmml" xref="S7.I13.i1.p1.3.m3.1.2.2.2.3">𝑚</ci></apply><ci id="S7.I13.i1.p1.3.m3.1.1.cmml" xref="S7.I13.i1.p1.3.m3.1.1">𝑣</ci></apply><ci id="S7.I13.i1.p1.3.m3.1.2.3.cmml" xref="S7.I13.i1.p1.3.m3.1.2.3">ℎ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.I13.i1.p1.3.m3.1c">l_{m}(v)=h</annotation><annotation encoding="application/x-llamapun" id="S7.I13.i1.p1.3.m3.1d">italic_l start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT ( italic_v ) = italic_h</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S7.I13.i1.p1.4.4"> and </span><math alttext="v" class="ltx_Math" display="inline" id="S7.I13.i1.p1.4.m4.1"><semantics id="S7.I13.i1.p1.4.m4.1a"><mi id="S7.I13.i1.p1.4.m4.1.1" xref="S7.I13.i1.p1.4.m4.1.1.cmml">v</mi><annotation-xml encoding="MathML-Content" id="S7.I13.i1.p1.4.m4.1b"><ci id="S7.I13.i1.p1.4.m4.1.1.cmml" xref="S7.I13.i1.p1.4.m4.1.1">𝑣</ci></annotation-xml><annotation encoding="application/x-tex" id="S7.I13.i1.p1.4.m4.1c">v</annotation><annotation encoding="application/x-llamapun" id="S7.I13.i1.p1.4.m4.1d">italic_v</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S7.I13.i1.p1.4.5"> had a violating out-edge</span></p> </div> </li> <li class="ltx_item" id="S7.I13.i2" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">•</span> <div class="ltx_para" id="S7.I13.i2.p1"> <p class="ltx_p" id="S7.I13.i2.p1.4"><span class="ltx_text ltx_font_italic" id="S7.I13.i2.p1.4.1">for odd </span><math alttext="m" class="ltx_Math" display="inline" id="S7.I13.i2.p1.1.m1.1"><semantics id="S7.I13.i2.p1.1.m1.1a"><mi id="S7.I13.i2.p1.1.m1.1.1" xref="S7.I13.i2.p1.1.m1.1.1.cmml">m</mi><annotation-xml encoding="MathML-Content" id="S7.I13.i2.p1.1.m1.1b"><ci id="S7.I13.i2.p1.1.m1.1.1.cmml" xref="S7.I13.i2.p1.1.m1.1.1">𝑚</ci></annotation-xml><annotation encoding="application/x-tex" id="S7.I13.i2.p1.1.m1.1c">m</annotation><annotation encoding="application/x-llamapun" id="S7.I13.i2.p1.1.m1.1d">italic_m</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S7.I13.i2.p1.4.2">, at the start of </span><math alttext="(h:m:0)" class="ltx_Math" display="inline" id="S7.I13.i2.p1.2.m2.1"><semantics id="S7.I13.i2.p1.2.m2.1a"><mrow id="S7.I13.i2.p1.2.m2.1.1.1" xref="S7.I13.i2.p1.2.m2.1.1.1.1.cmml"><mo id="S7.I13.i2.p1.2.m2.1.1.1.2" stretchy="false" xref="S7.I13.i2.p1.2.m2.1.1.1.1.cmml">(</mo><mrow id="S7.I13.i2.p1.2.m2.1.1.1.1" xref="S7.I13.i2.p1.2.m2.1.1.1.1.cmml"><mi id="S7.I13.i2.p1.2.m2.1.1.1.1.2" xref="S7.I13.i2.p1.2.m2.1.1.1.1.2.cmml">h</mi><mo id="S7.I13.i2.p1.2.m2.1.1.1.1.3" lspace="0.278em" rspace="0.278em" xref="S7.I13.i2.p1.2.m2.1.1.1.1.3.cmml">:</mo><mi id="S7.I13.i2.p1.2.m2.1.1.1.1.4" xref="S7.I13.i2.p1.2.m2.1.1.1.1.4.cmml">m</mi><mo id="S7.I13.i2.p1.2.m2.1.1.1.1.5" lspace="0.278em" rspace="0.278em" xref="S7.I13.i2.p1.2.m2.1.1.1.1.5.cmml">:</mo><mn id="S7.I13.i2.p1.2.m2.1.1.1.1.6" xref="S7.I13.i2.p1.2.m2.1.1.1.1.6.cmml">0</mn></mrow><mo id="S7.I13.i2.p1.2.m2.1.1.1.3" stretchy="false" xref="S7.I13.i2.p1.2.m2.1.1.1.1.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S7.I13.i2.p1.2.m2.1b"><apply id="S7.I13.i2.p1.2.m2.1.1.1.1.cmml" xref="S7.I13.i2.p1.2.m2.1.1.1"><and id="S7.I13.i2.p1.2.m2.1.1.1.1a.cmml" xref="S7.I13.i2.p1.2.m2.1.1.1"></and><apply id="S7.I13.i2.p1.2.m2.1.1.1.1b.cmml" xref="S7.I13.i2.p1.2.m2.1.1.1"><ci id="S7.I13.i2.p1.2.m2.1.1.1.1.3.cmml" xref="S7.I13.i2.p1.2.m2.1.1.1.1.3">:</ci><ci id="S7.I13.i2.p1.2.m2.1.1.1.1.2.cmml" xref="S7.I13.i2.p1.2.m2.1.1.1.1.2">ℎ</ci><ci id="S7.I13.i2.p1.2.m2.1.1.1.1.4.cmml" xref="S7.I13.i2.p1.2.m2.1.1.1.1.4">𝑚</ci></apply><apply id="S7.I13.i2.p1.2.m2.1.1.1.1c.cmml" xref="S7.I13.i2.p1.2.m2.1.1.1"><ci id="S7.I13.i2.p1.2.m2.1.1.1.1.5.cmml" xref="S7.I13.i2.p1.2.m2.1.1.1.1.5">:</ci><share href="https://arxiv.org/html/2411.12694v2#S7.I13.i2.p1.2.m2.1.1.1.1.4.cmml" id="S7.I13.i2.p1.2.m2.1.1.1.1d.cmml" xref="S7.I13.i2.p1.2.m2.1.1.1"></share><cn id="S7.I13.i2.p1.2.m2.1.1.1.1.6.cmml" type="integer" xref="S7.I13.i2.p1.2.m2.1.1.1.1.6">0</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.I13.i2.p1.2.m2.1c">(h:m:0)</annotation><annotation encoding="application/x-llamapun" id="S7.I13.i2.p1.2.m2.1d">( italic_h : italic_m : 0 )</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S7.I13.i2.p1.4.3">, </span><math alttext="l_{m}(v)=h-1" class="ltx_Math" display="inline" id="S7.I13.i2.p1.3.m3.1"><semantics id="S7.I13.i2.p1.3.m3.1a"><mrow id="S7.I13.i2.p1.3.m3.1.2" xref="S7.I13.i2.p1.3.m3.1.2.cmml"><mrow id="S7.I13.i2.p1.3.m3.1.2.2" xref="S7.I13.i2.p1.3.m3.1.2.2.cmml"><msub id="S7.I13.i2.p1.3.m3.1.2.2.2" xref="S7.I13.i2.p1.3.m3.1.2.2.2.cmml"><mi id="S7.I13.i2.p1.3.m3.1.2.2.2.2" xref="S7.I13.i2.p1.3.m3.1.2.2.2.2.cmml">l</mi><mi id="S7.I13.i2.p1.3.m3.1.2.2.2.3" xref="S7.I13.i2.p1.3.m3.1.2.2.2.3.cmml">m</mi></msub><mo id="S7.I13.i2.p1.3.m3.1.2.2.1" xref="S7.I13.i2.p1.3.m3.1.2.2.1.cmml">⁢</mo><mrow id="S7.I13.i2.p1.3.m3.1.2.2.3.2" xref="S7.I13.i2.p1.3.m3.1.2.2.cmml"><mo id="S7.I13.i2.p1.3.m3.1.2.2.3.2.1" stretchy="false" xref="S7.I13.i2.p1.3.m3.1.2.2.cmml">(</mo><mi id="S7.I13.i2.p1.3.m3.1.1" xref="S7.I13.i2.p1.3.m3.1.1.cmml">v</mi><mo id="S7.I13.i2.p1.3.m3.1.2.2.3.2.2" stretchy="false" xref="S7.I13.i2.p1.3.m3.1.2.2.cmml">)</mo></mrow></mrow><mo id="S7.I13.i2.p1.3.m3.1.2.1" xref="S7.I13.i2.p1.3.m3.1.2.1.cmml">=</mo><mrow id="S7.I13.i2.p1.3.m3.1.2.3" xref="S7.I13.i2.p1.3.m3.1.2.3.cmml"><mi id="S7.I13.i2.p1.3.m3.1.2.3.2" xref="S7.I13.i2.p1.3.m3.1.2.3.2.cmml">h</mi><mo id="S7.I13.i2.p1.3.m3.1.2.3.1" xref="S7.I13.i2.p1.3.m3.1.2.3.1.cmml">−</mo><mn id="S7.I13.i2.p1.3.m3.1.2.3.3" xref="S7.I13.i2.p1.3.m3.1.2.3.3.cmml">1</mn></mrow></mrow><annotation-xml encoding="MathML-Content" id="S7.I13.i2.p1.3.m3.1b"><apply id="S7.I13.i2.p1.3.m3.1.2.cmml" xref="S7.I13.i2.p1.3.m3.1.2"><eq id="S7.I13.i2.p1.3.m3.1.2.1.cmml" xref="S7.I13.i2.p1.3.m3.1.2.1"></eq><apply id="S7.I13.i2.p1.3.m3.1.2.2.cmml" xref="S7.I13.i2.p1.3.m3.1.2.2"><times id="S7.I13.i2.p1.3.m3.1.2.2.1.cmml" xref="S7.I13.i2.p1.3.m3.1.2.2.1"></times><apply id="S7.I13.i2.p1.3.m3.1.2.2.2.cmml" xref="S7.I13.i2.p1.3.m3.1.2.2.2"><csymbol cd="ambiguous" id="S7.I13.i2.p1.3.m3.1.2.2.2.1.cmml" xref="S7.I13.i2.p1.3.m3.1.2.2.2">subscript</csymbol><ci id="S7.I13.i2.p1.3.m3.1.2.2.2.2.cmml" xref="S7.I13.i2.p1.3.m3.1.2.2.2.2">𝑙</ci><ci id="S7.I13.i2.p1.3.m3.1.2.2.2.3.cmml" xref="S7.I13.i2.p1.3.m3.1.2.2.2.3">𝑚</ci></apply><ci id="S7.I13.i2.p1.3.m3.1.1.cmml" xref="S7.I13.i2.p1.3.m3.1.1">𝑣</ci></apply><apply id="S7.I13.i2.p1.3.m3.1.2.3.cmml" xref="S7.I13.i2.p1.3.m3.1.2.3"><minus id="S7.I13.i2.p1.3.m3.1.2.3.1.cmml" xref="S7.I13.i2.p1.3.m3.1.2.3.1"></minus><ci id="S7.I13.i2.p1.3.m3.1.2.3.2.cmml" xref="S7.I13.i2.p1.3.m3.1.2.3.2">ℎ</ci><cn id="S7.I13.i2.p1.3.m3.1.2.3.3.cmml" type="integer" xref="S7.I13.i2.p1.3.m3.1.2.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.I13.i2.p1.3.m3.1c">l_{m}(v)=h-1</annotation><annotation encoding="application/x-llamapun" id="S7.I13.i2.p1.3.m3.1d">italic_l start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT ( italic_v ) = italic_h - 1</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S7.I13.i2.p1.4.4"> and </span><math alttext="v" class="ltx_Math" display="inline" id="S7.I13.i2.p1.4.m4.1"><semantics id="S7.I13.i2.p1.4.m4.1a"><mi id="S7.I13.i2.p1.4.m4.1.1" xref="S7.I13.i2.p1.4.m4.1.1.cmml">v</mi><annotation-xml encoding="MathML-Content" id="S7.I13.i2.p1.4.m4.1b"><ci id="S7.I13.i2.p1.4.m4.1.1.cmml" xref="S7.I13.i2.p1.4.m4.1.1">𝑣</ci></annotation-xml><annotation encoding="application/x-tex" id="S7.I13.i2.p1.4.m4.1c">v</annotation><annotation encoding="application/x-llamapun" id="S7.I13.i2.p1.4.m4.1d">italic_v</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S7.I13.i2.p1.4.5"> had a violating in-edge.</span></p> </div> </li> </ul> </div> <div class="ltx_para" id="S7.Thmtheorem27.p3"> <p class="ltx_p" id="S7.Thmtheorem27.p3.6"><span class="ltx_text ltx_font_italic" id="S7.Thmtheorem27.p3.6.6">Our algorithms guarantee that at the start of <math alttext="(h:m:0)" class="ltx_Math" display="inline" id="S7.Thmtheorem27.p3.1.1.m1.1"><semantics id="S7.Thmtheorem27.p3.1.1.m1.1a"><mrow id="S7.Thmtheorem27.p3.1.1.m1.1.1.1" xref="S7.Thmtheorem27.p3.1.1.m1.1.1.1.1.cmml"><mo id="S7.Thmtheorem27.p3.1.1.m1.1.1.1.2" stretchy="false" xref="S7.Thmtheorem27.p3.1.1.m1.1.1.1.1.cmml">(</mo><mrow id="S7.Thmtheorem27.p3.1.1.m1.1.1.1.1" xref="S7.Thmtheorem27.p3.1.1.m1.1.1.1.1.cmml"><mi id="S7.Thmtheorem27.p3.1.1.m1.1.1.1.1.2" xref="S7.Thmtheorem27.p3.1.1.m1.1.1.1.1.2.cmml">h</mi><mo id="S7.Thmtheorem27.p3.1.1.m1.1.1.1.1.3" lspace="0.278em" rspace="0.278em" xref="S7.Thmtheorem27.p3.1.1.m1.1.1.1.1.3.cmml">:</mo><mi id="S7.Thmtheorem27.p3.1.1.m1.1.1.1.1.4" xref="S7.Thmtheorem27.p3.1.1.m1.1.1.1.1.4.cmml">m</mi><mo id="S7.Thmtheorem27.p3.1.1.m1.1.1.1.1.5" lspace="0.278em" rspace="0.278em" xref="S7.Thmtheorem27.p3.1.1.m1.1.1.1.1.5.cmml">:</mo><mn id="S7.Thmtheorem27.p3.1.1.m1.1.1.1.1.6" xref="S7.Thmtheorem27.p3.1.1.m1.1.1.1.1.6.cmml">0</mn></mrow><mo id="S7.Thmtheorem27.p3.1.1.m1.1.1.1.3" stretchy="false" xref="S7.Thmtheorem27.p3.1.1.m1.1.1.1.1.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S7.Thmtheorem27.p3.1.1.m1.1b"><apply id="S7.Thmtheorem27.p3.1.1.m1.1.1.1.1.cmml" xref="S7.Thmtheorem27.p3.1.1.m1.1.1.1"><and id="S7.Thmtheorem27.p3.1.1.m1.1.1.1.1a.cmml" xref="S7.Thmtheorem27.p3.1.1.m1.1.1.1"></and><apply id="S7.Thmtheorem27.p3.1.1.m1.1.1.1.1b.cmml" xref="S7.Thmtheorem27.p3.1.1.m1.1.1.1"><ci id="S7.Thmtheorem27.p3.1.1.m1.1.1.1.1.3.cmml" xref="S7.Thmtheorem27.p3.1.1.m1.1.1.1.1.3">:</ci><ci id="S7.Thmtheorem27.p3.1.1.m1.1.1.1.1.2.cmml" xref="S7.Thmtheorem27.p3.1.1.m1.1.1.1.1.2">ℎ</ci><ci id="S7.Thmtheorem27.p3.1.1.m1.1.1.1.1.4.cmml" xref="S7.Thmtheorem27.p3.1.1.m1.1.1.1.1.4">𝑚</ci></apply><apply id="S7.Thmtheorem27.p3.1.1.m1.1.1.1.1c.cmml" xref="S7.Thmtheorem27.p3.1.1.m1.1.1.1"><ci id="S7.Thmtheorem27.p3.1.1.m1.1.1.1.1.5.cmml" xref="S7.Thmtheorem27.p3.1.1.m1.1.1.1.1.5">:</ci><share href="https://arxiv.org/html/2411.12694v2#S7.Thmtheorem27.p3.1.1.m1.1.1.1.1.4.cmml" id="S7.Thmtheorem27.p3.1.1.m1.1.1.1.1d.cmml" xref="S7.Thmtheorem27.p3.1.1.m1.1.1.1"></share><cn id="S7.Thmtheorem27.p3.1.1.m1.1.1.1.1.6.cmml" type="integer" xref="S7.Thmtheorem27.p3.1.1.m1.1.1.1.1.6">0</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.Thmtheorem27.p3.1.1.m1.1c">(h:m:0)</annotation><annotation encoding="application/x-llamapun" id="S7.Thmtheorem27.p3.1.1.m1.1d">( italic_h : italic_m : 0 )</annotation></semantics></math> for <math alttext="m" class="ltx_Math" display="inline" id="S7.Thmtheorem27.p3.2.2.m2.1"><semantics id="S7.Thmtheorem27.p3.2.2.m2.1a"><mi id="S7.Thmtheorem27.p3.2.2.m2.1.1" xref="S7.Thmtheorem27.p3.2.2.m2.1.1.cmml">m</mi><annotation-xml encoding="MathML-Content" id="S7.Thmtheorem27.p3.2.2.m2.1b"><ci id="S7.Thmtheorem27.p3.2.2.m2.1.1.cmml" xref="S7.Thmtheorem27.p3.2.2.m2.1.1">𝑚</ci></annotation-xml><annotation encoding="application/x-tex" id="S7.Thmtheorem27.p3.2.2.m2.1c">m</annotation><annotation encoding="application/x-llamapun" id="S7.Thmtheorem27.p3.2.2.m2.1d">italic_m</annotation></semantics></math> even, <math alttext="\textsl{g}(v)=(1+\frac{\eta}{2})^{h+1}" class="ltx_Math" display="inline" id="S7.Thmtheorem27.p3.3.3.m3.2"><semantics id="S7.Thmtheorem27.p3.3.3.m3.2a"><mrow id="S7.Thmtheorem27.p3.3.3.m3.2.2" xref="S7.Thmtheorem27.p3.3.3.m3.2.2.cmml"><mrow id="S7.Thmtheorem27.p3.3.3.m3.2.2.3" xref="S7.Thmtheorem27.p3.3.3.m3.2.2.3.cmml"><mtext class="ltx_mathvariant_italic" id="S7.Thmtheorem27.p3.3.3.m3.2.2.3.2" xref="S7.Thmtheorem27.p3.3.3.m3.2.2.3.2a.cmml">g</mtext><mo id="S7.Thmtheorem27.p3.3.3.m3.2.2.3.1" xref="S7.Thmtheorem27.p3.3.3.m3.2.2.3.1.cmml">⁢</mo><mrow id="S7.Thmtheorem27.p3.3.3.m3.2.2.3.3.2" xref="S7.Thmtheorem27.p3.3.3.m3.2.2.3.cmml"><mo id="S7.Thmtheorem27.p3.3.3.m3.2.2.3.3.2.1" stretchy="false" xref="S7.Thmtheorem27.p3.3.3.m3.2.2.3.cmml">(</mo><mi id="S7.Thmtheorem27.p3.3.3.m3.1.1" xref="S7.Thmtheorem27.p3.3.3.m3.1.1.cmml">v</mi><mo id="S7.Thmtheorem27.p3.3.3.m3.2.2.3.3.2.2" stretchy="false" xref="S7.Thmtheorem27.p3.3.3.m3.2.2.3.cmml">)</mo></mrow></mrow><mo id="S7.Thmtheorem27.p3.3.3.m3.2.2.2" xref="S7.Thmtheorem27.p3.3.3.m3.2.2.2.cmml">=</mo><msup id="S7.Thmtheorem27.p3.3.3.m3.2.2.1" xref="S7.Thmtheorem27.p3.3.3.m3.2.2.1.cmml"><mrow id="S7.Thmtheorem27.p3.3.3.m3.2.2.1.1.1" xref="S7.Thmtheorem27.p3.3.3.m3.2.2.1.1.1.1.cmml"><mo id="S7.Thmtheorem27.p3.3.3.m3.2.2.1.1.1.2" stretchy="false" xref="S7.Thmtheorem27.p3.3.3.m3.2.2.1.1.1.1.cmml">(</mo><mrow id="S7.Thmtheorem27.p3.3.3.m3.2.2.1.1.1.1" xref="S7.Thmtheorem27.p3.3.3.m3.2.2.1.1.1.1.cmml"><mn id="S7.Thmtheorem27.p3.3.3.m3.2.2.1.1.1.1.2" xref="S7.Thmtheorem27.p3.3.3.m3.2.2.1.1.1.1.2.cmml">1</mn><mo id="S7.Thmtheorem27.p3.3.3.m3.2.2.1.1.1.1.1" xref="S7.Thmtheorem27.p3.3.3.m3.2.2.1.1.1.1.1.cmml">+</mo><mfrac id="S7.Thmtheorem27.p3.3.3.m3.2.2.1.1.1.1.3" xref="S7.Thmtheorem27.p3.3.3.m3.2.2.1.1.1.1.3.cmml"><mi id="S7.Thmtheorem27.p3.3.3.m3.2.2.1.1.1.1.3.2" xref="S7.Thmtheorem27.p3.3.3.m3.2.2.1.1.1.1.3.2.cmml">η</mi><mn id="S7.Thmtheorem27.p3.3.3.m3.2.2.1.1.1.1.3.3" xref="S7.Thmtheorem27.p3.3.3.m3.2.2.1.1.1.1.3.3.cmml">2</mn></mfrac></mrow><mo id="S7.Thmtheorem27.p3.3.3.m3.2.2.1.1.1.3" stretchy="false" xref="S7.Thmtheorem27.p3.3.3.m3.2.2.1.1.1.1.cmml">)</mo></mrow><mrow id="S7.Thmtheorem27.p3.3.3.m3.2.2.1.3" xref="S7.Thmtheorem27.p3.3.3.m3.2.2.1.3.cmml"><mi id="S7.Thmtheorem27.p3.3.3.m3.2.2.1.3.2" xref="S7.Thmtheorem27.p3.3.3.m3.2.2.1.3.2.cmml">h</mi><mo id="S7.Thmtheorem27.p3.3.3.m3.2.2.1.3.1" xref="S7.Thmtheorem27.p3.3.3.m3.2.2.1.3.1.cmml">+</mo><mn id="S7.Thmtheorem27.p3.3.3.m3.2.2.1.3.3" xref="S7.Thmtheorem27.p3.3.3.m3.2.2.1.3.3.cmml">1</mn></mrow></msup></mrow><annotation-xml encoding="MathML-Content" id="S7.Thmtheorem27.p3.3.3.m3.2b"><apply id="S7.Thmtheorem27.p3.3.3.m3.2.2.cmml" xref="S7.Thmtheorem27.p3.3.3.m3.2.2"><eq id="S7.Thmtheorem27.p3.3.3.m3.2.2.2.cmml" xref="S7.Thmtheorem27.p3.3.3.m3.2.2.2"></eq><apply id="S7.Thmtheorem27.p3.3.3.m3.2.2.3.cmml" xref="S7.Thmtheorem27.p3.3.3.m3.2.2.3"><times id="S7.Thmtheorem27.p3.3.3.m3.2.2.3.1.cmml" xref="S7.Thmtheorem27.p3.3.3.m3.2.2.3.1"></times><ci id="S7.Thmtheorem27.p3.3.3.m3.2.2.3.2a.cmml" xref="S7.Thmtheorem27.p3.3.3.m3.2.2.3.2"><mtext class="ltx_mathvariant_italic" id="S7.Thmtheorem27.p3.3.3.m3.2.2.3.2.cmml" xref="S7.Thmtheorem27.p3.3.3.m3.2.2.3.2">g</mtext></ci><ci id="S7.Thmtheorem27.p3.3.3.m3.1.1.cmml" xref="S7.Thmtheorem27.p3.3.3.m3.1.1">𝑣</ci></apply><apply id="S7.Thmtheorem27.p3.3.3.m3.2.2.1.cmml" xref="S7.Thmtheorem27.p3.3.3.m3.2.2.1"><csymbol cd="ambiguous" id="S7.Thmtheorem27.p3.3.3.m3.2.2.1.2.cmml" xref="S7.Thmtheorem27.p3.3.3.m3.2.2.1">superscript</csymbol><apply id="S7.Thmtheorem27.p3.3.3.m3.2.2.1.1.1.1.cmml" xref="S7.Thmtheorem27.p3.3.3.m3.2.2.1.1.1"><plus id="S7.Thmtheorem27.p3.3.3.m3.2.2.1.1.1.1.1.cmml" xref="S7.Thmtheorem27.p3.3.3.m3.2.2.1.1.1.1.1"></plus><cn id="S7.Thmtheorem27.p3.3.3.m3.2.2.1.1.1.1.2.cmml" type="integer" xref="S7.Thmtheorem27.p3.3.3.m3.2.2.1.1.1.1.2">1</cn><apply id="S7.Thmtheorem27.p3.3.3.m3.2.2.1.1.1.1.3.cmml" xref="S7.Thmtheorem27.p3.3.3.m3.2.2.1.1.1.1.3"><divide id="S7.Thmtheorem27.p3.3.3.m3.2.2.1.1.1.1.3.1.cmml" xref="S7.Thmtheorem27.p3.3.3.m3.2.2.1.1.1.1.3"></divide><ci id="S7.Thmtheorem27.p3.3.3.m3.2.2.1.1.1.1.3.2.cmml" xref="S7.Thmtheorem27.p3.3.3.m3.2.2.1.1.1.1.3.2">𝜂</ci><cn id="S7.Thmtheorem27.p3.3.3.m3.2.2.1.1.1.1.3.3.cmml" type="integer" xref="S7.Thmtheorem27.p3.3.3.m3.2.2.1.1.1.1.3.3">2</cn></apply></apply><apply id="S7.Thmtheorem27.p3.3.3.m3.2.2.1.3.cmml" xref="S7.Thmtheorem27.p3.3.3.m3.2.2.1.3"><plus id="S7.Thmtheorem27.p3.3.3.m3.2.2.1.3.1.cmml" xref="S7.Thmtheorem27.p3.3.3.m3.2.2.1.3.1"></plus><ci id="S7.Thmtheorem27.p3.3.3.m3.2.2.1.3.2.cmml" xref="S7.Thmtheorem27.p3.3.3.m3.2.2.1.3.2">ℎ</ci><cn id="S7.Thmtheorem27.p3.3.3.m3.2.2.1.3.3.cmml" type="integer" xref="S7.Thmtheorem27.p3.3.3.m3.2.2.1.3.3">1</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.Thmtheorem27.p3.3.3.m3.2c">\textsl{g}(v)=(1+\frac{\eta}{2})^{h+1}</annotation><annotation encoding="application/x-llamapun" id="S7.Thmtheorem27.p3.3.3.m3.2d">g ( italic_v ) = ( 1 + divide start_ARG italic_η end_ARG start_ARG 2 end_ARG ) start_POSTSUPERSCRIPT italic_h + 1 end_POSTSUPERSCRIPT</annotation></semantics></math> and at the end of <math alttext="(h:m:0)" class="ltx_Math" display="inline" id="S7.Thmtheorem27.p3.4.4.m4.1"><semantics id="S7.Thmtheorem27.p3.4.4.m4.1a"><mrow id="S7.Thmtheorem27.p3.4.4.m4.1.1.1" xref="S7.Thmtheorem27.p3.4.4.m4.1.1.1.1.cmml"><mo id="S7.Thmtheorem27.p3.4.4.m4.1.1.1.2" stretchy="false" xref="S7.Thmtheorem27.p3.4.4.m4.1.1.1.1.cmml">(</mo><mrow id="S7.Thmtheorem27.p3.4.4.m4.1.1.1.1" xref="S7.Thmtheorem27.p3.4.4.m4.1.1.1.1.cmml"><mi id="S7.Thmtheorem27.p3.4.4.m4.1.1.1.1.2" xref="S7.Thmtheorem27.p3.4.4.m4.1.1.1.1.2.cmml">h</mi><mo id="S7.Thmtheorem27.p3.4.4.m4.1.1.1.1.3" lspace="0.278em" rspace="0.278em" xref="S7.Thmtheorem27.p3.4.4.m4.1.1.1.1.3.cmml">:</mo><mi id="S7.Thmtheorem27.p3.4.4.m4.1.1.1.1.4" xref="S7.Thmtheorem27.p3.4.4.m4.1.1.1.1.4.cmml">m</mi><mo id="S7.Thmtheorem27.p3.4.4.m4.1.1.1.1.5" lspace="0.278em" rspace="0.278em" xref="S7.Thmtheorem27.p3.4.4.m4.1.1.1.1.5.cmml">:</mo><mn id="S7.Thmtheorem27.p3.4.4.m4.1.1.1.1.6" xref="S7.Thmtheorem27.p3.4.4.m4.1.1.1.1.6.cmml">0</mn></mrow><mo id="S7.Thmtheorem27.p3.4.4.m4.1.1.1.3" stretchy="false" xref="S7.Thmtheorem27.p3.4.4.m4.1.1.1.1.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S7.Thmtheorem27.p3.4.4.m4.1b"><apply id="S7.Thmtheorem27.p3.4.4.m4.1.1.1.1.cmml" xref="S7.Thmtheorem27.p3.4.4.m4.1.1.1"><and id="S7.Thmtheorem27.p3.4.4.m4.1.1.1.1a.cmml" xref="S7.Thmtheorem27.p3.4.4.m4.1.1.1"></and><apply id="S7.Thmtheorem27.p3.4.4.m4.1.1.1.1b.cmml" xref="S7.Thmtheorem27.p3.4.4.m4.1.1.1"><ci id="S7.Thmtheorem27.p3.4.4.m4.1.1.1.1.3.cmml" xref="S7.Thmtheorem27.p3.4.4.m4.1.1.1.1.3">:</ci><ci id="S7.Thmtheorem27.p3.4.4.m4.1.1.1.1.2.cmml" xref="S7.Thmtheorem27.p3.4.4.m4.1.1.1.1.2">ℎ</ci><ci id="S7.Thmtheorem27.p3.4.4.m4.1.1.1.1.4.cmml" xref="S7.Thmtheorem27.p3.4.4.m4.1.1.1.1.4">𝑚</ci></apply><apply id="S7.Thmtheorem27.p3.4.4.m4.1.1.1.1c.cmml" xref="S7.Thmtheorem27.p3.4.4.m4.1.1.1"><ci id="S7.Thmtheorem27.p3.4.4.m4.1.1.1.1.5.cmml" xref="S7.Thmtheorem27.p3.4.4.m4.1.1.1.1.5">:</ci><share href="https://arxiv.org/html/2411.12694v2#S7.Thmtheorem27.p3.4.4.m4.1.1.1.1.4.cmml" id="S7.Thmtheorem27.p3.4.4.m4.1.1.1.1d.cmml" xref="S7.Thmtheorem27.p3.4.4.m4.1.1.1"></share><cn id="S7.Thmtheorem27.p3.4.4.m4.1.1.1.1.6.cmml" type="integer" xref="S7.Thmtheorem27.p3.4.4.m4.1.1.1.1.6">0</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.Thmtheorem27.p3.4.4.m4.1c">(h:m:0)</annotation><annotation encoding="application/x-llamapun" id="S7.Thmtheorem27.p3.4.4.m4.1d">( italic_h : italic_m : 0 )</annotation></semantics></math> for <math alttext="m" class="ltx_Math" display="inline" id="S7.Thmtheorem27.p3.5.5.m5.1"><semantics id="S7.Thmtheorem27.p3.5.5.m5.1a"><mi id="S7.Thmtheorem27.p3.5.5.m5.1.1" xref="S7.Thmtheorem27.p3.5.5.m5.1.1.cmml">m</mi><annotation-xml encoding="MathML-Content" id="S7.Thmtheorem27.p3.5.5.m5.1b"><ci id="S7.Thmtheorem27.p3.5.5.m5.1.1.cmml" xref="S7.Thmtheorem27.p3.5.5.m5.1.1">𝑚</ci></annotation-xml><annotation encoding="application/x-tex" id="S7.Thmtheorem27.p3.5.5.m5.1c">m</annotation><annotation encoding="application/x-llamapun" id="S7.Thmtheorem27.p3.5.5.m5.1d">italic_m</annotation></semantics></math> odd, <math alttext="\textsl{g}(v)=\textsl{g}(v)=(1+\frac{\eta}{2})^{h-1}" class="ltx_Math" display="inline" id="S7.Thmtheorem27.p3.6.6.m6.3"><semantics id="S7.Thmtheorem27.p3.6.6.m6.3a"><mrow id="S7.Thmtheorem27.p3.6.6.m6.3.3" xref="S7.Thmtheorem27.p3.6.6.m6.3.3.cmml"><mrow id="S7.Thmtheorem27.p3.6.6.m6.3.3.3" xref="S7.Thmtheorem27.p3.6.6.m6.3.3.3.cmml"><mtext class="ltx_mathvariant_italic" id="S7.Thmtheorem27.p3.6.6.m6.3.3.3.2" xref="S7.Thmtheorem27.p3.6.6.m6.3.3.3.2a.cmml">g</mtext><mo id="S7.Thmtheorem27.p3.6.6.m6.3.3.3.1" xref="S7.Thmtheorem27.p3.6.6.m6.3.3.3.1.cmml">⁢</mo><mrow id="S7.Thmtheorem27.p3.6.6.m6.3.3.3.3.2" xref="S7.Thmtheorem27.p3.6.6.m6.3.3.3.cmml"><mo id="S7.Thmtheorem27.p3.6.6.m6.3.3.3.3.2.1" stretchy="false" xref="S7.Thmtheorem27.p3.6.6.m6.3.3.3.cmml">(</mo><mi id="S7.Thmtheorem27.p3.6.6.m6.1.1" xref="S7.Thmtheorem27.p3.6.6.m6.1.1.cmml">v</mi><mo id="S7.Thmtheorem27.p3.6.6.m6.3.3.3.3.2.2" stretchy="false" xref="S7.Thmtheorem27.p3.6.6.m6.3.3.3.cmml">)</mo></mrow></mrow><mo id="S7.Thmtheorem27.p3.6.6.m6.3.3.4" xref="S7.Thmtheorem27.p3.6.6.m6.3.3.4.cmml">=</mo><mrow id="S7.Thmtheorem27.p3.6.6.m6.3.3.5" xref="S7.Thmtheorem27.p3.6.6.m6.3.3.5.cmml"><mtext class="ltx_mathvariant_italic" id="S7.Thmtheorem27.p3.6.6.m6.3.3.5.2" xref="S7.Thmtheorem27.p3.6.6.m6.3.3.5.2a.cmml">g</mtext><mo id="S7.Thmtheorem27.p3.6.6.m6.3.3.5.1" xref="S7.Thmtheorem27.p3.6.6.m6.3.3.5.1.cmml">⁢</mo><mrow id="S7.Thmtheorem27.p3.6.6.m6.3.3.5.3.2" xref="S7.Thmtheorem27.p3.6.6.m6.3.3.5.cmml"><mo id="S7.Thmtheorem27.p3.6.6.m6.3.3.5.3.2.1" stretchy="false" xref="S7.Thmtheorem27.p3.6.6.m6.3.3.5.cmml">(</mo><mi id="S7.Thmtheorem27.p3.6.6.m6.2.2" xref="S7.Thmtheorem27.p3.6.6.m6.2.2.cmml">v</mi><mo id="S7.Thmtheorem27.p3.6.6.m6.3.3.5.3.2.2" stretchy="false" xref="S7.Thmtheorem27.p3.6.6.m6.3.3.5.cmml">)</mo></mrow></mrow><mo id="S7.Thmtheorem27.p3.6.6.m6.3.3.6" xref="S7.Thmtheorem27.p3.6.6.m6.3.3.6.cmml">=</mo><msup id="S7.Thmtheorem27.p3.6.6.m6.3.3.1" xref="S7.Thmtheorem27.p3.6.6.m6.3.3.1.cmml"><mrow id="S7.Thmtheorem27.p3.6.6.m6.3.3.1.1.1" xref="S7.Thmtheorem27.p3.6.6.m6.3.3.1.1.1.1.cmml"><mo id="S7.Thmtheorem27.p3.6.6.m6.3.3.1.1.1.2" stretchy="false" xref="S7.Thmtheorem27.p3.6.6.m6.3.3.1.1.1.1.cmml">(</mo><mrow id="S7.Thmtheorem27.p3.6.6.m6.3.3.1.1.1.1" xref="S7.Thmtheorem27.p3.6.6.m6.3.3.1.1.1.1.cmml"><mn id="S7.Thmtheorem27.p3.6.6.m6.3.3.1.1.1.1.2" xref="S7.Thmtheorem27.p3.6.6.m6.3.3.1.1.1.1.2.cmml">1</mn><mo id="S7.Thmtheorem27.p3.6.6.m6.3.3.1.1.1.1.1" xref="S7.Thmtheorem27.p3.6.6.m6.3.3.1.1.1.1.1.cmml">+</mo><mfrac id="S7.Thmtheorem27.p3.6.6.m6.3.3.1.1.1.1.3" xref="S7.Thmtheorem27.p3.6.6.m6.3.3.1.1.1.1.3.cmml"><mi id="S7.Thmtheorem27.p3.6.6.m6.3.3.1.1.1.1.3.2" xref="S7.Thmtheorem27.p3.6.6.m6.3.3.1.1.1.1.3.2.cmml">η</mi><mn id="S7.Thmtheorem27.p3.6.6.m6.3.3.1.1.1.1.3.3" xref="S7.Thmtheorem27.p3.6.6.m6.3.3.1.1.1.1.3.3.cmml">2</mn></mfrac></mrow><mo id="S7.Thmtheorem27.p3.6.6.m6.3.3.1.1.1.3" stretchy="false" xref="S7.Thmtheorem27.p3.6.6.m6.3.3.1.1.1.1.cmml">)</mo></mrow><mrow id="S7.Thmtheorem27.p3.6.6.m6.3.3.1.3" xref="S7.Thmtheorem27.p3.6.6.m6.3.3.1.3.cmml"><mi id="S7.Thmtheorem27.p3.6.6.m6.3.3.1.3.2" xref="S7.Thmtheorem27.p3.6.6.m6.3.3.1.3.2.cmml">h</mi><mo id="S7.Thmtheorem27.p3.6.6.m6.3.3.1.3.1" xref="S7.Thmtheorem27.p3.6.6.m6.3.3.1.3.1.cmml">−</mo><mn id="S7.Thmtheorem27.p3.6.6.m6.3.3.1.3.3" xref="S7.Thmtheorem27.p3.6.6.m6.3.3.1.3.3.cmml">1</mn></mrow></msup></mrow><annotation-xml encoding="MathML-Content" id="S7.Thmtheorem27.p3.6.6.m6.3b"><apply id="S7.Thmtheorem27.p3.6.6.m6.3.3.cmml" xref="S7.Thmtheorem27.p3.6.6.m6.3.3"><and id="S7.Thmtheorem27.p3.6.6.m6.3.3a.cmml" xref="S7.Thmtheorem27.p3.6.6.m6.3.3"></and><apply id="S7.Thmtheorem27.p3.6.6.m6.3.3b.cmml" xref="S7.Thmtheorem27.p3.6.6.m6.3.3"><eq id="S7.Thmtheorem27.p3.6.6.m6.3.3.4.cmml" xref="S7.Thmtheorem27.p3.6.6.m6.3.3.4"></eq><apply id="S7.Thmtheorem27.p3.6.6.m6.3.3.3.cmml" xref="S7.Thmtheorem27.p3.6.6.m6.3.3.3"><times id="S7.Thmtheorem27.p3.6.6.m6.3.3.3.1.cmml" xref="S7.Thmtheorem27.p3.6.6.m6.3.3.3.1"></times><ci id="S7.Thmtheorem27.p3.6.6.m6.3.3.3.2a.cmml" xref="S7.Thmtheorem27.p3.6.6.m6.3.3.3.2"><mtext class="ltx_mathvariant_italic" id="S7.Thmtheorem27.p3.6.6.m6.3.3.3.2.cmml" xref="S7.Thmtheorem27.p3.6.6.m6.3.3.3.2">g</mtext></ci><ci id="S7.Thmtheorem27.p3.6.6.m6.1.1.cmml" xref="S7.Thmtheorem27.p3.6.6.m6.1.1">𝑣</ci></apply><apply id="S7.Thmtheorem27.p3.6.6.m6.3.3.5.cmml" xref="S7.Thmtheorem27.p3.6.6.m6.3.3.5"><times id="S7.Thmtheorem27.p3.6.6.m6.3.3.5.1.cmml" xref="S7.Thmtheorem27.p3.6.6.m6.3.3.5.1"></times><ci id="S7.Thmtheorem27.p3.6.6.m6.3.3.5.2a.cmml" xref="S7.Thmtheorem27.p3.6.6.m6.3.3.5.2"><mtext class="ltx_mathvariant_italic" id="S7.Thmtheorem27.p3.6.6.m6.3.3.5.2.cmml" xref="S7.Thmtheorem27.p3.6.6.m6.3.3.5.2">g</mtext></ci><ci id="S7.Thmtheorem27.p3.6.6.m6.2.2.cmml" xref="S7.Thmtheorem27.p3.6.6.m6.2.2">𝑣</ci></apply></apply><apply id="S7.Thmtheorem27.p3.6.6.m6.3.3c.cmml" xref="S7.Thmtheorem27.p3.6.6.m6.3.3"><eq id="S7.Thmtheorem27.p3.6.6.m6.3.3.6.cmml" xref="S7.Thmtheorem27.p3.6.6.m6.3.3.6"></eq><share href="https://arxiv.org/html/2411.12694v2#S7.Thmtheorem27.p3.6.6.m6.3.3.5.cmml" id="S7.Thmtheorem27.p3.6.6.m6.3.3d.cmml" xref="S7.Thmtheorem27.p3.6.6.m6.3.3"></share><apply id="S7.Thmtheorem27.p3.6.6.m6.3.3.1.cmml" xref="S7.Thmtheorem27.p3.6.6.m6.3.3.1"><csymbol cd="ambiguous" id="S7.Thmtheorem27.p3.6.6.m6.3.3.1.2.cmml" xref="S7.Thmtheorem27.p3.6.6.m6.3.3.1">superscript</csymbol><apply id="S7.Thmtheorem27.p3.6.6.m6.3.3.1.1.1.1.cmml" xref="S7.Thmtheorem27.p3.6.6.m6.3.3.1.1.1"><plus id="S7.Thmtheorem27.p3.6.6.m6.3.3.1.1.1.1.1.cmml" xref="S7.Thmtheorem27.p3.6.6.m6.3.3.1.1.1.1.1"></plus><cn id="S7.Thmtheorem27.p3.6.6.m6.3.3.1.1.1.1.2.cmml" type="integer" xref="S7.Thmtheorem27.p3.6.6.m6.3.3.1.1.1.1.2">1</cn><apply id="S7.Thmtheorem27.p3.6.6.m6.3.3.1.1.1.1.3.cmml" xref="S7.Thmtheorem27.p3.6.6.m6.3.3.1.1.1.1.3"><divide id="S7.Thmtheorem27.p3.6.6.m6.3.3.1.1.1.1.3.1.cmml" xref="S7.Thmtheorem27.p3.6.6.m6.3.3.1.1.1.1.3"></divide><ci id="S7.Thmtheorem27.p3.6.6.m6.3.3.1.1.1.1.3.2.cmml" xref="S7.Thmtheorem27.p3.6.6.m6.3.3.1.1.1.1.3.2">𝜂</ci><cn id="S7.Thmtheorem27.p3.6.6.m6.3.3.1.1.1.1.3.3.cmml" type="integer" xref="S7.Thmtheorem27.p3.6.6.m6.3.3.1.1.1.1.3.3">2</cn></apply></apply><apply id="S7.Thmtheorem27.p3.6.6.m6.3.3.1.3.cmml" xref="S7.Thmtheorem27.p3.6.6.m6.3.3.1.3"><minus id="S7.Thmtheorem27.p3.6.6.m6.3.3.1.3.1.cmml" xref="S7.Thmtheorem27.p3.6.6.m6.3.3.1.3.1"></minus><ci id="S7.Thmtheorem27.p3.6.6.m6.3.3.1.3.2.cmml" xref="S7.Thmtheorem27.p3.6.6.m6.3.3.1.3.2">ℎ</ci><cn id="S7.Thmtheorem27.p3.6.6.m6.3.3.1.3.3.cmml" type="integer" xref="S7.Thmtheorem27.p3.6.6.m6.3.3.1.3.3">1</cn></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.Thmtheorem27.p3.6.6.m6.3c">\textsl{g}(v)=\textsl{g}(v)=(1+\frac{\eta}{2})^{h-1}</annotation><annotation encoding="application/x-llamapun" id="S7.Thmtheorem27.p3.6.6.m6.3d">g ( italic_v ) = g ( italic_v ) = ( 1 + divide start_ARG italic_η end_ARG start_ARG 2 end_ARG ) start_POSTSUPERSCRIPT italic_h - 1 end_POSTSUPERSCRIPT</annotation></semantics></math>.</span></p> </div> <div class="ltx_para" id="S7.Thmtheorem27.p4"> <p class="ltx_p" id="S7.Thmtheorem27.p4.20"><span class="ltx_text ltx_font_italic" id="S7.Thmtheorem27.p4.20.20">Denote for all even <math alttext="m" class="ltx_Math" display="inline" id="S7.Thmtheorem27.p4.1.1.m1.1"><semantics id="S7.Thmtheorem27.p4.1.1.m1.1a"><mi id="S7.Thmtheorem27.p4.1.1.m1.1.1" xref="S7.Thmtheorem27.p4.1.1.m1.1.1.cmml">m</mi><annotation-xml encoding="MathML-Content" id="S7.Thmtheorem27.p4.1.1.m1.1b"><ci id="S7.Thmtheorem27.p4.1.1.m1.1.1.cmml" xref="S7.Thmtheorem27.p4.1.1.m1.1.1">𝑚</ci></annotation-xml><annotation encoding="application/x-tex" id="S7.Thmtheorem27.p4.1.1.m1.1c">m</annotation><annotation encoding="application/x-llamapun" id="S7.Thmtheorem27.p4.1.1.m1.1d">italic_m</annotation></semantics></math> by <math alttext="E_{m}(v)" class="ltx_Math" display="inline" id="S7.Thmtheorem27.p4.2.2.m2.1"><semantics id="S7.Thmtheorem27.p4.2.2.m2.1a"><mrow id="S7.Thmtheorem27.p4.2.2.m2.1.2" xref="S7.Thmtheorem27.p4.2.2.m2.1.2.cmml"><msub id="S7.Thmtheorem27.p4.2.2.m2.1.2.2" xref="S7.Thmtheorem27.p4.2.2.m2.1.2.2.cmml"><mi id="S7.Thmtheorem27.p4.2.2.m2.1.2.2.2" xref="S7.Thmtheorem27.p4.2.2.m2.1.2.2.2.cmml">E</mi><mi id="S7.Thmtheorem27.p4.2.2.m2.1.2.2.3" xref="S7.Thmtheorem27.p4.2.2.m2.1.2.2.3.cmml">m</mi></msub><mo id="S7.Thmtheorem27.p4.2.2.m2.1.2.1" xref="S7.Thmtheorem27.p4.2.2.m2.1.2.1.cmml">⁢</mo><mrow id="S7.Thmtheorem27.p4.2.2.m2.1.2.3.2" xref="S7.Thmtheorem27.p4.2.2.m2.1.2.cmml"><mo id="S7.Thmtheorem27.p4.2.2.m2.1.2.3.2.1" stretchy="false" xref="S7.Thmtheorem27.p4.2.2.m2.1.2.cmml">(</mo><mi id="S7.Thmtheorem27.p4.2.2.m2.1.1" xref="S7.Thmtheorem27.p4.2.2.m2.1.1.cmml">v</mi><mo id="S7.Thmtheorem27.p4.2.2.m2.1.2.3.2.2" stretchy="false" xref="S7.Thmtheorem27.p4.2.2.m2.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S7.Thmtheorem27.p4.2.2.m2.1b"><apply id="S7.Thmtheorem27.p4.2.2.m2.1.2.cmml" xref="S7.Thmtheorem27.p4.2.2.m2.1.2"><times id="S7.Thmtheorem27.p4.2.2.m2.1.2.1.cmml" xref="S7.Thmtheorem27.p4.2.2.m2.1.2.1"></times><apply id="S7.Thmtheorem27.p4.2.2.m2.1.2.2.cmml" xref="S7.Thmtheorem27.p4.2.2.m2.1.2.2"><csymbol cd="ambiguous" id="S7.Thmtheorem27.p4.2.2.m2.1.2.2.1.cmml" xref="S7.Thmtheorem27.p4.2.2.m2.1.2.2">subscript</csymbol><ci id="S7.Thmtheorem27.p4.2.2.m2.1.2.2.2.cmml" xref="S7.Thmtheorem27.p4.2.2.m2.1.2.2.2">𝐸</ci><ci id="S7.Thmtheorem27.p4.2.2.m2.1.2.2.3.cmml" xref="S7.Thmtheorem27.p4.2.2.m2.1.2.2.3">𝑚</ci></apply><ci id="S7.Thmtheorem27.p4.2.2.m2.1.1.cmml" xref="S7.Thmtheorem27.p4.2.2.m2.1.1">𝑣</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.Thmtheorem27.p4.2.2.m2.1c">E_{m}(v)</annotation><annotation encoding="application/x-llamapun" id="S7.Thmtheorem27.p4.2.2.m2.1d">italic_E start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT ( italic_v )</annotation></semantics></math> the violating out-edges from <math alttext="v" class="ltx_Math" display="inline" id="S7.Thmtheorem27.p4.3.3.m3.1"><semantics id="S7.Thmtheorem27.p4.3.3.m3.1a"><mi id="S7.Thmtheorem27.p4.3.3.m3.1.1" xref="S7.Thmtheorem27.p4.3.3.m3.1.1.cmml">v</mi><annotation-xml encoding="MathML-Content" id="S7.Thmtheorem27.p4.3.3.m3.1b"><ci id="S7.Thmtheorem27.p4.3.3.m3.1.1.cmml" xref="S7.Thmtheorem27.p4.3.3.m3.1.1">𝑣</ci></annotation-xml><annotation encoding="application/x-tex" id="S7.Thmtheorem27.p4.3.3.m3.1c">v</annotation><annotation encoding="application/x-llamapun" id="S7.Thmtheorem27.p4.3.3.m3.1d">italic_v</annotation></semantics></math> and by <math alttext="\Delta_{m}=\sum_{\overline{vw}}\textsl{g}(v\!\to\!w)" class="ltx_Math" display="inline" id="S7.Thmtheorem27.p4.4.4.m4.1"><semantics id="S7.Thmtheorem27.p4.4.4.m4.1a"><mrow id="S7.Thmtheorem27.p4.4.4.m4.1.1" xref="S7.Thmtheorem27.p4.4.4.m4.1.1.cmml"><msub id="S7.Thmtheorem27.p4.4.4.m4.1.1.3" xref="S7.Thmtheorem27.p4.4.4.m4.1.1.3.cmml"><mi id="S7.Thmtheorem27.p4.4.4.m4.1.1.3.2" mathvariant="normal" xref="S7.Thmtheorem27.p4.4.4.m4.1.1.3.2.cmml">Δ</mi><mi id="S7.Thmtheorem27.p4.4.4.m4.1.1.3.3" xref="S7.Thmtheorem27.p4.4.4.m4.1.1.3.3.cmml">m</mi></msub><mo id="S7.Thmtheorem27.p4.4.4.m4.1.1.2" rspace="0.111em" xref="S7.Thmtheorem27.p4.4.4.m4.1.1.2.cmml">=</mo><mrow id="S7.Thmtheorem27.p4.4.4.m4.1.1.1" xref="S7.Thmtheorem27.p4.4.4.m4.1.1.1.cmml"><msub id="S7.Thmtheorem27.p4.4.4.m4.1.1.1.2" xref="S7.Thmtheorem27.p4.4.4.m4.1.1.1.2.cmml"><mo id="S7.Thmtheorem27.p4.4.4.m4.1.1.1.2.2" xref="S7.Thmtheorem27.p4.4.4.m4.1.1.1.2.2.cmml">∑</mo><mover accent="true" id="S7.Thmtheorem27.p4.4.4.m4.1.1.1.2.3" xref="S7.Thmtheorem27.p4.4.4.m4.1.1.1.2.3.cmml"><mrow id="S7.Thmtheorem27.p4.4.4.m4.1.1.1.2.3.2" xref="S7.Thmtheorem27.p4.4.4.m4.1.1.1.2.3.2.cmml"><mi id="S7.Thmtheorem27.p4.4.4.m4.1.1.1.2.3.2.2" xref="S7.Thmtheorem27.p4.4.4.m4.1.1.1.2.3.2.2.cmml">v</mi><mo id="S7.Thmtheorem27.p4.4.4.m4.1.1.1.2.3.2.1" xref="S7.Thmtheorem27.p4.4.4.m4.1.1.1.2.3.2.1.cmml">⁢</mo><mi id="S7.Thmtheorem27.p4.4.4.m4.1.1.1.2.3.2.3" xref="S7.Thmtheorem27.p4.4.4.m4.1.1.1.2.3.2.3.cmml">w</mi></mrow><mo id="S7.Thmtheorem27.p4.4.4.m4.1.1.1.2.3.1" xref="S7.Thmtheorem27.p4.4.4.m4.1.1.1.2.3.1.cmml">¯</mo></mover></msub><mrow id="S7.Thmtheorem27.p4.4.4.m4.1.1.1.1" xref="S7.Thmtheorem27.p4.4.4.m4.1.1.1.1.cmml"><mtext class="ltx_mathvariant_italic" id="S7.Thmtheorem27.p4.4.4.m4.1.1.1.1.3" xref="S7.Thmtheorem27.p4.4.4.m4.1.1.1.1.3a.cmml">g</mtext><mo id="S7.Thmtheorem27.p4.4.4.m4.1.1.1.1.2" xref="S7.Thmtheorem27.p4.4.4.m4.1.1.1.1.2.cmml">⁢</mo><mrow id="S7.Thmtheorem27.p4.4.4.m4.1.1.1.1.1.1" xref="S7.Thmtheorem27.p4.4.4.m4.1.1.1.1.1.1.1.cmml"><mo id="S7.Thmtheorem27.p4.4.4.m4.1.1.1.1.1.1.2" stretchy="false" xref="S7.Thmtheorem27.p4.4.4.m4.1.1.1.1.1.1.1.cmml">(</mo><mrow id="S7.Thmtheorem27.p4.4.4.m4.1.1.1.1.1.1.1" xref="S7.Thmtheorem27.p4.4.4.m4.1.1.1.1.1.1.1.cmml"><mi id="S7.Thmtheorem27.p4.4.4.m4.1.1.1.1.1.1.1.2" xref="S7.Thmtheorem27.p4.4.4.m4.1.1.1.1.1.1.1.2.cmml">v</mi><mo id="S7.Thmtheorem27.p4.4.4.m4.1.1.1.1.1.1.1.1" lspace="0.108em" rspace="0.108em" stretchy="false" xref="S7.Thmtheorem27.p4.4.4.m4.1.1.1.1.1.1.1.1.cmml">→</mo><mi id="S7.Thmtheorem27.p4.4.4.m4.1.1.1.1.1.1.1.3" xref="S7.Thmtheorem27.p4.4.4.m4.1.1.1.1.1.1.1.3.cmml">w</mi></mrow><mo id="S7.Thmtheorem27.p4.4.4.m4.1.1.1.1.1.1.3" stretchy="false" xref="S7.Thmtheorem27.p4.4.4.m4.1.1.1.1.1.1.1.cmml">)</mo></mrow></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S7.Thmtheorem27.p4.4.4.m4.1b"><apply id="S7.Thmtheorem27.p4.4.4.m4.1.1.cmml" xref="S7.Thmtheorem27.p4.4.4.m4.1.1"><eq id="S7.Thmtheorem27.p4.4.4.m4.1.1.2.cmml" xref="S7.Thmtheorem27.p4.4.4.m4.1.1.2"></eq><apply id="S7.Thmtheorem27.p4.4.4.m4.1.1.3.cmml" xref="S7.Thmtheorem27.p4.4.4.m4.1.1.3"><csymbol cd="ambiguous" id="S7.Thmtheorem27.p4.4.4.m4.1.1.3.1.cmml" xref="S7.Thmtheorem27.p4.4.4.m4.1.1.3">subscript</csymbol><ci id="S7.Thmtheorem27.p4.4.4.m4.1.1.3.2.cmml" xref="S7.Thmtheorem27.p4.4.4.m4.1.1.3.2">Δ</ci><ci id="S7.Thmtheorem27.p4.4.4.m4.1.1.3.3.cmml" xref="S7.Thmtheorem27.p4.4.4.m4.1.1.3.3">𝑚</ci></apply><apply id="S7.Thmtheorem27.p4.4.4.m4.1.1.1.cmml" xref="S7.Thmtheorem27.p4.4.4.m4.1.1.1"><apply id="S7.Thmtheorem27.p4.4.4.m4.1.1.1.2.cmml" xref="S7.Thmtheorem27.p4.4.4.m4.1.1.1.2"><csymbol cd="ambiguous" id="S7.Thmtheorem27.p4.4.4.m4.1.1.1.2.1.cmml" xref="S7.Thmtheorem27.p4.4.4.m4.1.1.1.2">subscript</csymbol><sum id="S7.Thmtheorem27.p4.4.4.m4.1.1.1.2.2.cmml" xref="S7.Thmtheorem27.p4.4.4.m4.1.1.1.2.2"></sum><apply id="S7.Thmtheorem27.p4.4.4.m4.1.1.1.2.3.cmml" xref="S7.Thmtheorem27.p4.4.4.m4.1.1.1.2.3"><ci id="S7.Thmtheorem27.p4.4.4.m4.1.1.1.2.3.1.cmml" xref="S7.Thmtheorem27.p4.4.4.m4.1.1.1.2.3.1">¯</ci><apply id="S7.Thmtheorem27.p4.4.4.m4.1.1.1.2.3.2.cmml" xref="S7.Thmtheorem27.p4.4.4.m4.1.1.1.2.3.2"><times id="S7.Thmtheorem27.p4.4.4.m4.1.1.1.2.3.2.1.cmml" xref="S7.Thmtheorem27.p4.4.4.m4.1.1.1.2.3.2.1"></times><ci id="S7.Thmtheorem27.p4.4.4.m4.1.1.1.2.3.2.2.cmml" xref="S7.Thmtheorem27.p4.4.4.m4.1.1.1.2.3.2.2">𝑣</ci><ci id="S7.Thmtheorem27.p4.4.4.m4.1.1.1.2.3.2.3.cmml" xref="S7.Thmtheorem27.p4.4.4.m4.1.1.1.2.3.2.3">𝑤</ci></apply></apply></apply><apply id="S7.Thmtheorem27.p4.4.4.m4.1.1.1.1.cmml" xref="S7.Thmtheorem27.p4.4.4.m4.1.1.1.1"><times id="S7.Thmtheorem27.p4.4.4.m4.1.1.1.1.2.cmml" xref="S7.Thmtheorem27.p4.4.4.m4.1.1.1.1.2"></times><ci id="S7.Thmtheorem27.p4.4.4.m4.1.1.1.1.3a.cmml" xref="S7.Thmtheorem27.p4.4.4.m4.1.1.1.1.3"><mtext class="ltx_mathvariant_italic" id="S7.Thmtheorem27.p4.4.4.m4.1.1.1.1.3.cmml" xref="S7.Thmtheorem27.p4.4.4.m4.1.1.1.1.3">g</mtext></ci><apply id="S7.Thmtheorem27.p4.4.4.m4.1.1.1.1.1.1.1.cmml" xref="S7.Thmtheorem27.p4.4.4.m4.1.1.1.1.1.1"><ci id="S7.Thmtheorem27.p4.4.4.m4.1.1.1.1.1.1.1.1.cmml" xref="S7.Thmtheorem27.p4.4.4.m4.1.1.1.1.1.1.1.1">→</ci><ci id="S7.Thmtheorem27.p4.4.4.m4.1.1.1.1.1.1.1.2.cmml" xref="S7.Thmtheorem27.p4.4.4.m4.1.1.1.1.1.1.1.2">𝑣</ci><ci id="S7.Thmtheorem27.p4.4.4.m4.1.1.1.1.1.1.1.3.cmml" xref="S7.Thmtheorem27.p4.4.4.m4.1.1.1.1.1.1.1.3">𝑤</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.Thmtheorem27.p4.4.4.m4.1c">\Delta_{m}=\sum_{\overline{vw}}\textsl{g}(v\!\to\!w)</annotation><annotation encoding="application/x-llamapun" id="S7.Thmtheorem27.p4.4.4.m4.1d">roman_Δ start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT = ∑ start_POSTSUBSCRIPT over¯ start_ARG italic_v italic_w end_ARG end_POSTSUBSCRIPT g ( italic_v → italic_w )</annotation></semantics></math>. Since <math alttext="v" class="ltx_Math" display="inline" id="S7.Thmtheorem27.p4.5.5.m5.1"><semantics id="S7.Thmtheorem27.p4.5.5.m5.1a"><mi id="S7.Thmtheorem27.p4.5.5.m5.1.1" xref="S7.Thmtheorem27.p4.5.5.m5.1.1.cmml">v</mi><annotation-xml encoding="MathML-Content" id="S7.Thmtheorem27.p4.5.5.m5.1b"><ci id="S7.Thmtheorem27.p4.5.5.m5.1.1.cmml" xref="S7.Thmtheorem27.p4.5.5.m5.1.1">𝑣</ci></annotation-xml><annotation encoding="application/x-tex" id="S7.Thmtheorem27.p4.5.5.m5.1c">v</annotation><annotation encoding="application/x-llamapun" id="S7.Thmtheorem27.p4.5.5.m5.1d">italic_v</annotation></semantics></math> is in level <math alttext="h" class="ltx_Math" display="inline" id="S7.Thmtheorem27.p4.6.6.m6.1"><semantics id="S7.Thmtheorem27.p4.6.6.m6.1a"><mi id="S7.Thmtheorem27.p4.6.6.m6.1.1" xref="S7.Thmtheorem27.p4.6.6.m6.1.1.cmml">h</mi><annotation-xml encoding="MathML-Content" id="S7.Thmtheorem27.p4.6.6.m6.1b"><ci id="S7.Thmtheorem27.p4.6.6.m6.1.1.cmml" xref="S7.Thmtheorem27.p4.6.6.m6.1.1">ℎ</ci></annotation-xml><annotation encoding="application/x-tex" id="S7.Thmtheorem27.p4.6.6.m6.1c">h</annotation><annotation encoding="application/x-llamapun" id="S7.Thmtheorem27.p4.6.6.m6.1d">italic_h</annotation></semantics></math> it must be that <math alttext="\Delta_{0}\leq(1+\frac{\eta}{2})^{h+1}" class="ltx_Math" display="inline" id="S7.Thmtheorem27.p4.7.7.m7.1"><semantics id="S7.Thmtheorem27.p4.7.7.m7.1a"><mrow id="S7.Thmtheorem27.p4.7.7.m7.1.1" xref="S7.Thmtheorem27.p4.7.7.m7.1.1.cmml"><msub id="S7.Thmtheorem27.p4.7.7.m7.1.1.3" xref="S7.Thmtheorem27.p4.7.7.m7.1.1.3.cmml"><mi id="S7.Thmtheorem27.p4.7.7.m7.1.1.3.2" mathvariant="normal" xref="S7.Thmtheorem27.p4.7.7.m7.1.1.3.2.cmml">Δ</mi><mn id="S7.Thmtheorem27.p4.7.7.m7.1.1.3.3" xref="S7.Thmtheorem27.p4.7.7.m7.1.1.3.3.cmml">0</mn></msub><mo id="S7.Thmtheorem27.p4.7.7.m7.1.1.2" xref="S7.Thmtheorem27.p4.7.7.m7.1.1.2.cmml">≤</mo><msup id="S7.Thmtheorem27.p4.7.7.m7.1.1.1" xref="S7.Thmtheorem27.p4.7.7.m7.1.1.1.cmml"><mrow id="S7.Thmtheorem27.p4.7.7.m7.1.1.1.1.1" xref="S7.Thmtheorem27.p4.7.7.m7.1.1.1.1.1.1.cmml"><mo id="S7.Thmtheorem27.p4.7.7.m7.1.1.1.1.1.2" stretchy="false" xref="S7.Thmtheorem27.p4.7.7.m7.1.1.1.1.1.1.cmml">(</mo><mrow id="S7.Thmtheorem27.p4.7.7.m7.1.1.1.1.1.1" xref="S7.Thmtheorem27.p4.7.7.m7.1.1.1.1.1.1.cmml"><mn id="S7.Thmtheorem27.p4.7.7.m7.1.1.1.1.1.1.2" xref="S7.Thmtheorem27.p4.7.7.m7.1.1.1.1.1.1.2.cmml">1</mn><mo id="S7.Thmtheorem27.p4.7.7.m7.1.1.1.1.1.1.1" xref="S7.Thmtheorem27.p4.7.7.m7.1.1.1.1.1.1.1.cmml">+</mo><mfrac id="S7.Thmtheorem27.p4.7.7.m7.1.1.1.1.1.1.3" xref="S7.Thmtheorem27.p4.7.7.m7.1.1.1.1.1.1.3.cmml"><mi id="S7.Thmtheorem27.p4.7.7.m7.1.1.1.1.1.1.3.2" xref="S7.Thmtheorem27.p4.7.7.m7.1.1.1.1.1.1.3.2.cmml">η</mi><mn id="S7.Thmtheorem27.p4.7.7.m7.1.1.1.1.1.1.3.3" xref="S7.Thmtheorem27.p4.7.7.m7.1.1.1.1.1.1.3.3.cmml">2</mn></mfrac></mrow><mo id="S7.Thmtheorem27.p4.7.7.m7.1.1.1.1.1.3" stretchy="false" xref="S7.Thmtheorem27.p4.7.7.m7.1.1.1.1.1.1.cmml">)</mo></mrow><mrow id="S7.Thmtheorem27.p4.7.7.m7.1.1.1.3" xref="S7.Thmtheorem27.p4.7.7.m7.1.1.1.3.cmml"><mi id="S7.Thmtheorem27.p4.7.7.m7.1.1.1.3.2" xref="S7.Thmtheorem27.p4.7.7.m7.1.1.1.3.2.cmml">h</mi><mo id="S7.Thmtheorem27.p4.7.7.m7.1.1.1.3.1" xref="S7.Thmtheorem27.p4.7.7.m7.1.1.1.3.1.cmml">+</mo><mn id="S7.Thmtheorem27.p4.7.7.m7.1.1.1.3.3" xref="S7.Thmtheorem27.p4.7.7.m7.1.1.1.3.3.cmml">1</mn></mrow></msup></mrow><annotation-xml encoding="MathML-Content" id="S7.Thmtheorem27.p4.7.7.m7.1b"><apply id="S7.Thmtheorem27.p4.7.7.m7.1.1.cmml" xref="S7.Thmtheorem27.p4.7.7.m7.1.1"><leq id="S7.Thmtheorem27.p4.7.7.m7.1.1.2.cmml" xref="S7.Thmtheorem27.p4.7.7.m7.1.1.2"></leq><apply id="S7.Thmtheorem27.p4.7.7.m7.1.1.3.cmml" xref="S7.Thmtheorem27.p4.7.7.m7.1.1.3"><csymbol cd="ambiguous" id="S7.Thmtheorem27.p4.7.7.m7.1.1.3.1.cmml" xref="S7.Thmtheorem27.p4.7.7.m7.1.1.3">subscript</csymbol><ci id="S7.Thmtheorem27.p4.7.7.m7.1.1.3.2.cmml" xref="S7.Thmtheorem27.p4.7.7.m7.1.1.3.2">Δ</ci><cn id="S7.Thmtheorem27.p4.7.7.m7.1.1.3.3.cmml" type="integer" xref="S7.Thmtheorem27.p4.7.7.m7.1.1.3.3">0</cn></apply><apply id="S7.Thmtheorem27.p4.7.7.m7.1.1.1.cmml" xref="S7.Thmtheorem27.p4.7.7.m7.1.1.1"><csymbol cd="ambiguous" id="S7.Thmtheorem27.p4.7.7.m7.1.1.1.2.cmml" xref="S7.Thmtheorem27.p4.7.7.m7.1.1.1">superscript</csymbol><apply id="S7.Thmtheorem27.p4.7.7.m7.1.1.1.1.1.1.cmml" xref="S7.Thmtheorem27.p4.7.7.m7.1.1.1.1.1"><plus id="S7.Thmtheorem27.p4.7.7.m7.1.1.1.1.1.1.1.cmml" xref="S7.Thmtheorem27.p4.7.7.m7.1.1.1.1.1.1.1"></plus><cn id="S7.Thmtheorem27.p4.7.7.m7.1.1.1.1.1.1.2.cmml" type="integer" xref="S7.Thmtheorem27.p4.7.7.m7.1.1.1.1.1.1.2">1</cn><apply id="S7.Thmtheorem27.p4.7.7.m7.1.1.1.1.1.1.3.cmml" xref="S7.Thmtheorem27.p4.7.7.m7.1.1.1.1.1.1.3"><divide id="S7.Thmtheorem27.p4.7.7.m7.1.1.1.1.1.1.3.1.cmml" xref="S7.Thmtheorem27.p4.7.7.m7.1.1.1.1.1.1.3"></divide><ci id="S7.Thmtheorem27.p4.7.7.m7.1.1.1.1.1.1.3.2.cmml" xref="S7.Thmtheorem27.p4.7.7.m7.1.1.1.1.1.1.3.2">𝜂</ci><cn id="S7.Thmtheorem27.p4.7.7.m7.1.1.1.1.1.1.3.3.cmml" type="integer" xref="S7.Thmtheorem27.p4.7.7.m7.1.1.1.1.1.1.3.3">2</cn></apply></apply><apply id="S7.Thmtheorem27.p4.7.7.m7.1.1.1.3.cmml" xref="S7.Thmtheorem27.p4.7.7.m7.1.1.1.3"><plus id="S7.Thmtheorem27.p4.7.7.m7.1.1.1.3.1.cmml" xref="S7.Thmtheorem27.p4.7.7.m7.1.1.1.3.1"></plus><ci id="S7.Thmtheorem27.p4.7.7.m7.1.1.1.3.2.cmml" xref="S7.Thmtheorem27.p4.7.7.m7.1.1.1.3.2">ℎ</ci><cn id="S7.Thmtheorem27.p4.7.7.m7.1.1.1.3.3.cmml" type="integer" xref="S7.Thmtheorem27.p4.7.7.m7.1.1.1.3.3">1</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.Thmtheorem27.p4.7.7.m7.1c">\Delta_{0}\leq(1+\frac{\eta}{2})^{h+1}</annotation><annotation encoding="application/x-llamapun" id="S7.Thmtheorem27.p4.7.7.m7.1d">roman_Δ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ≤ ( 1 + divide start_ARG italic_η end_ARG start_ARG 2 end_ARG ) start_POSTSUPERSCRIPT italic_h + 1 end_POSTSUPERSCRIPT</annotation></semantics></math>. When going from minute <math alttext="m" class="ltx_Math" display="inline" id="S7.Thmtheorem27.p4.8.8.m8.1"><semantics id="S7.Thmtheorem27.p4.8.8.m8.1a"><mi id="S7.Thmtheorem27.p4.8.8.m8.1.1" xref="S7.Thmtheorem27.p4.8.8.m8.1.1.cmml">m</mi><annotation-xml encoding="MathML-Content" id="S7.Thmtheorem27.p4.8.8.m8.1b"><ci id="S7.Thmtheorem27.p4.8.8.m8.1.1.cmml" xref="S7.Thmtheorem27.p4.8.8.m8.1.1">𝑚</ci></annotation-xml><annotation encoding="application/x-tex" id="S7.Thmtheorem27.p4.8.8.m8.1c">m</annotation><annotation encoding="application/x-llamapun" id="S7.Thmtheorem27.p4.8.8.m8.1d">italic_m</annotation></semantics></math> to <math alttext="m+1" class="ltx_Math" display="inline" id="S7.Thmtheorem27.p4.9.9.m9.1"><semantics id="S7.Thmtheorem27.p4.9.9.m9.1a"><mrow id="S7.Thmtheorem27.p4.9.9.m9.1.1" xref="S7.Thmtheorem27.p4.9.9.m9.1.1.cmml"><mi id="S7.Thmtheorem27.p4.9.9.m9.1.1.2" xref="S7.Thmtheorem27.p4.9.9.m9.1.1.2.cmml">m</mi><mo id="S7.Thmtheorem27.p4.9.9.m9.1.1.1" xref="S7.Thmtheorem27.p4.9.9.m9.1.1.1.cmml">+</mo><mn id="S7.Thmtheorem27.p4.9.9.m9.1.1.3" xref="S7.Thmtheorem27.p4.9.9.m9.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S7.Thmtheorem27.p4.9.9.m9.1b"><apply id="S7.Thmtheorem27.p4.9.9.m9.1.1.cmml" xref="S7.Thmtheorem27.p4.9.9.m9.1.1"><plus id="S7.Thmtheorem27.p4.9.9.m9.1.1.1.cmml" xref="S7.Thmtheorem27.p4.9.9.m9.1.1.1"></plus><ci id="S7.Thmtheorem27.p4.9.9.m9.1.1.2.cmml" xref="S7.Thmtheorem27.p4.9.9.m9.1.1.2">𝑚</ci><cn id="S7.Thmtheorem27.p4.9.9.m9.1.1.3.cmml" type="integer" xref="S7.Thmtheorem27.p4.9.9.m9.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.Thmtheorem27.p4.9.9.m9.1c">m+1</annotation><annotation encoding="application/x-llamapun" id="S7.Thmtheorem27.p4.9.9.m9.1d">italic_m + 1</annotation></semantics></math>, the out-degree of <math alttext="v" class="ltx_Math" display="inline" id="S7.Thmtheorem27.p4.10.10.m10.1"><semantics id="S7.Thmtheorem27.p4.10.10.m10.1a"><mi id="S7.Thmtheorem27.p4.10.10.m10.1.1" xref="S7.Thmtheorem27.p4.10.10.m10.1.1.cmml">v</mi><annotation-xml encoding="MathML-Content" id="S7.Thmtheorem27.p4.10.10.m10.1b"><ci id="S7.Thmtheorem27.p4.10.10.m10.1.1.cmml" xref="S7.Thmtheorem27.p4.10.10.m10.1.1">𝑣</ci></annotation-xml><annotation encoding="application/x-tex" id="S7.Thmtheorem27.p4.10.10.m10.1c">v</annotation><annotation encoding="application/x-llamapun" id="S7.Thmtheorem27.p4.10.10.m10.1d">italic_v</annotation></semantics></math> decreases by only decreasing <math alttext="\textsl{g}(v\!\to\!w)" class="ltx_Math" display="inline" id="S7.Thmtheorem27.p4.11.11.m11.1"><semantics id="S7.Thmtheorem27.p4.11.11.m11.1a"><mrow id="S7.Thmtheorem27.p4.11.11.m11.1.1" xref="S7.Thmtheorem27.p4.11.11.m11.1.1.cmml"><mtext class="ltx_mathvariant_italic" id="S7.Thmtheorem27.p4.11.11.m11.1.1.3" xref="S7.Thmtheorem27.p4.11.11.m11.1.1.3a.cmml">g</mtext><mo id="S7.Thmtheorem27.p4.11.11.m11.1.1.2" xref="S7.Thmtheorem27.p4.11.11.m11.1.1.2.cmml">⁢</mo><mrow id="S7.Thmtheorem27.p4.11.11.m11.1.1.1.1" xref="S7.Thmtheorem27.p4.11.11.m11.1.1.1.1.1.cmml"><mo id="S7.Thmtheorem27.p4.11.11.m11.1.1.1.1.2" stretchy="false" xref="S7.Thmtheorem27.p4.11.11.m11.1.1.1.1.1.cmml">(</mo><mrow id="S7.Thmtheorem27.p4.11.11.m11.1.1.1.1.1" xref="S7.Thmtheorem27.p4.11.11.m11.1.1.1.1.1.cmml"><mi id="S7.Thmtheorem27.p4.11.11.m11.1.1.1.1.1.2" xref="S7.Thmtheorem27.p4.11.11.m11.1.1.1.1.1.2.cmml">v</mi><mo id="S7.Thmtheorem27.p4.11.11.m11.1.1.1.1.1.1" lspace="0.108em" rspace="0.108em" stretchy="false" xref="S7.Thmtheorem27.p4.11.11.m11.1.1.1.1.1.1.cmml">→</mo><mi id="S7.Thmtheorem27.p4.11.11.m11.1.1.1.1.1.3" xref="S7.Thmtheorem27.p4.11.11.m11.1.1.1.1.1.3.cmml">w</mi></mrow><mo id="S7.Thmtheorem27.p4.11.11.m11.1.1.1.1.3" stretchy="false" xref="S7.Thmtheorem27.p4.11.11.m11.1.1.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S7.Thmtheorem27.p4.11.11.m11.1b"><apply id="S7.Thmtheorem27.p4.11.11.m11.1.1.cmml" xref="S7.Thmtheorem27.p4.11.11.m11.1.1"><times id="S7.Thmtheorem27.p4.11.11.m11.1.1.2.cmml" xref="S7.Thmtheorem27.p4.11.11.m11.1.1.2"></times><ci id="S7.Thmtheorem27.p4.11.11.m11.1.1.3a.cmml" xref="S7.Thmtheorem27.p4.11.11.m11.1.1.3"><mtext class="ltx_mathvariant_italic" id="S7.Thmtheorem27.p4.11.11.m11.1.1.3.cmml" xref="S7.Thmtheorem27.p4.11.11.m11.1.1.3">g</mtext></ci><apply id="S7.Thmtheorem27.p4.11.11.m11.1.1.1.1.1.cmml" xref="S7.Thmtheorem27.p4.11.11.m11.1.1.1.1"><ci id="S7.Thmtheorem27.p4.11.11.m11.1.1.1.1.1.1.cmml" xref="S7.Thmtheorem27.p4.11.11.m11.1.1.1.1.1.1">→</ci><ci id="S7.Thmtheorem27.p4.11.11.m11.1.1.1.1.1.2.cmml" xref="S7.Thmtheorem27.p4.11.11.m11.1.1.1.1.1.2">𝑣</ci><ci id="S7.Thmtheorem27.p4.11.11.m11.1.1.1.1.1.3.cmml" xref="S7.Thmtheorem27.p4.11.11.m11.1.1.1.1.1.3">𝑤</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.Thmtheorem27.p4.11.11.m11.1c">\textsl{g}(v\!\to\!w)</annotation><annotation encoding="application/x-llamapun" id="S7.Thmtheorem27.p4.11.11.m11.1d">g ( italic_v → italic_w )</annotation></semantics></math> for <math alttext="\overline{vw}\in E_{m}(v)" class="ltx_Math" display="inline" id="S7.Thmtheorem27.p4.12.12.m12.1"><semantics id="S7.Thmtheorem27.p4.12.12.m12.1a"><mrow id="S7.Thmtheorem27.p4.12.12.m12.1.2" xref="S7.Thmtheorem27.p4.12.12.m12.1.2.cmml"><mover accent="true" id="S7.Thmtheorem27.p4.12.12.m12.1.2.2" xref="S7.Thmtheorem27.p4.12.12.m12.1.2.2.cmml"><mrow id="S7.Thmtheorem27.p4.12.12.m12.1.2.2.2" xref="S7.Thmtheorem27.p4.12.12.m12.1.2.2.2.cmml"><mi id="S7.Thmtheorem27.p4.12.12.m12.1.2.2.2.2" xref="S7.Thmtheorem27.p4.12.12.m12.1.2.2.2.2.cmml">v</mi><mo id="S7.Thmtheorem27.p4.12.12.m12.1.2.2.2.1" xref="S7.Thmtheorem27.p4.12.12.m12.1.2.2.2.1.cmml">⁢</mo><mi id="S7.Thmtheorem27.p4.12.12.m12.1.2.2.2.3" xref="S7.Thmtheorem27.p4.12.12.m12.1.2.2.2.3.cmml">w</mi></mrow><mo id="S7.Thmtheorem27.p4.12.12.m12.1.2.2.1" xref="S7.Thmtheorem27.p4.12.12.m12.1.2.2.1.cmml">¯</mo></mover><mo id="S7.Thmtheorem27.p4.12.12.m12.1.2.1" xref="S7.Thmtheorem27.p4.12.12.m12.1.2.1.cmml">∈</mo><mrow id="S7.Thmtheorem27.p4.12.12.m12.1.2.3" xref="S7.Thmtheorem27.p4.12.12.m12.1.2.3.cmml"><msub id="S7.Thmtheorem27.p4.12.12.m12.1.2.3.2" xref="S7.Thmtheorem27.p4.12.12.m12.1.2.3.2.cmml"><mi id="S7.Thmtheorem27.p4.12.12.m12.1.2.3.2.2" xref="S7.Thmtheorem27.p4.12.12.m12.1.2.3.2.2.cmml">E</mi><mi id="S7.Thmtheorem27.p4.12.12.m12.1.2.3.2.3" xref="S7.Thmtheorem27.p4.12.12.m12.1.2.3.2.3.cmml">m</mi></msub><mo id="S7.Thmtheorem27.p4.12.12.m12.1.2.3.1" xref="S7.Thmtheorem27.p4.12.12.m12.1.2.3.1.cmml">⁢</mo><mrow id="S7.Thmtheorem27.p4.12.12.m12.1.2.3.3.2" xref="S7.Thmtheorem27.p4.12.12.m12.1.2.3.cmml"><mo id="S7.Thmtheorem27.p4.12.12.m12.1.2.3.3.2.1" stretchy="false" xref="S7.Thmtheorem27.p4.12.12.m12.1.2.3.cmml">(</mo><mi id="S7.Thmtheorem27.p4.12.12.m12.1.1" xref="S7.Thmtheorem27.p4.12.12.m12.1.1.cmml">v</mi><mo id="S7.Thmtheorem27.p4.12.12.m12.1.2.3.3.2.2" stretchy="false" xref="S7.Thmtheorem27.p4.12.12.m12.1.2.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S7.Thmtheorem27.p4.12.12.m12.1b"><apply id="S7.Thmtheorem27.p4.12.12.m12.1.2.cmml" xref="S7.Thmtheorem27.p4.12.12.m12.1.2"><in id="S7.Thmtheorem27.p4.12.12.m12.1.2.1.cmml" xref="S7.Thmtheorem27.p4.12.12.m12.1.2.1"></in><apply id="S7.Thmtheorem27.p4.12.12.m12.1.2.2.cmml" xref="S7.Thmtheorem27.p4.12.12.m12.1.2.2"><ci id="S7.Thmtheorem27.p4.12.12.m12.1.2.2.1.cmml" xref="S7.Thmtheorem27.p4.12.12.m12.1.2.2.1">¯</ci><apply id="S7.Thmtheorem27.p4.12.12.m12.1.2.2.2.cmml" xref="S7.Thmtheorem27.p4.12.12.m12.1.2.2.2"><times id="S7.Thmtheorem27.p4.12.12.m12.1.2.2.2.1.cmml" xref="S7.Thmtheorem27.p4.12.12.m12.1.2.2.2.1"></times><ci id="S7.Thmtheorem27.p4.12.12.m12.1.2.2.2.2.cmml" xref="S7.Thmtheorem27.p4.12.12.m12.1.2.2.2.2">𝑣</ci><ci id="S7.Thmtheorem27.p4.12.12.m12.1.2.2.2.3.cmml" xref="S7.Thmtheorem27.p4.12.12.m12.1.2.2.2.3">𝑤</ci></apply></apply><apply id="S7.Thmtheorem27.p4.12.12.m12.1.2.3.cmml" xref="S7.Thmtheorem27.p4.12.12.m12.1.2.3"><times id="S7.Thmtheorem27.p4.12.12.m12.1.2.3.1.cmml" xref="S7.Thmtheorem27.p4.12.12.m12.1.2.3.1"></times><apply id="S7.Thmtheorem27.p4.12.12.m12.1.2.3.2.cmml" xref="S7.Thmtheorem27.p4.12.12.m12.1.2.3.2"><csymbol cd="ambiguous" id="S7.Thmtheorem27.p4.12.12.m12.1.2.3.2.1.cmml" xref="S7.Thmtheorem27.p4.12.12.m12.1.2.3.2">subscript</csymbol><ci id="S7.Thmtheorem27.p4.12.12.m12.1.2.3.2.2.cmml" xref="S7.Thmtheorem27.p4.12.12.m12.1.2.3.2.2">𝐸</ci><ci id="S7.Thmtheorem27.p4.12.12.m12.1.2.3.2.3.cmml" xref="S7.Thmtheorem27.p4.12.12.m12.1.2.3.2.3">𝑚</ci></apply><ci id="S7.Thmtheorem27.p4.12.12.m12.1.1.cmml" xref="S7.Thmtheorem27.p4.12.12.m12.1.1">𝑣</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.Thmtheorem27.p4.12.12.m12.1c">\overline{vw}\in E_{m}(v)</annotation><annotation encoding="application/x-llamapun" id="S7.Thmtheorem27.p4.12.12.m12.1d">over¯ start_ARG italic_v italic_w end_ARG ∈ italic_E start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT ( italic_v )</annotation></semantics></math>. When going from minute <math alttext="m+1" class="ltx_Math" display="inline" id="S7.Thmtheorem27.p4.13.13.m13.1"><semantics id="S7.Thmtheorem27.p4.13.13.m13.1a"><mrow id="S7.Thmtheorem27.p4.13.13.m13.1.1" xref="S7.Thmtheorem27.p4.13.13.m13.1.1.cmml"><mi id="S7.Thmtheorem27.p4.13.13.m13.1.1.2" xref="S7.Thmtheorem27.p4.13.13.m13.1.1.2.cmml">m</mi><mo id="S7.Thmtheorem27.p4.13.13.m13.1.1.1" xref="S7.Thmtheorem27.p4.13.13.m13.1.1.1.cmml">+</mo><mn id="S7.Thmtheorem27.p4.13.13.m13.1.1.3" xref="S7.Thmtheorem27.p4.13.13.m13.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S7.Thmtheorem27.p4.13.13.m13.1b"><apply id="S7.Thmtheorem27.p4.13.13.m13.1.1.cmml" xref="S7.Thmtheorem27.p4.13.13.m13.1.1"><plus id="S7.Thmtheorem27.p4.13.13.m13.1.1.1.cmml" xref="S7.Thmtheorem27.p4.13.13.m13.1.1.1"></plus><ci id="S7.Thmtheorem27.p4.13.13.m13.1.1.2.cmml" xref="S7.Thmtheorem27.p4.13.13.m13.1.1.2">𝑚</ci><cn id="S7.Thmtheorem27.p4.13.13.m13.1.1.3.cmml" type="integer" xref="S7.Thmtheorem27.p4.13.13.m13.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.Thmtheorem27.p4.13.13.m13.1c">m+1</annotation><annotation encoding="application/x-llamapun" id="S7.Thmtheorem27.p4.13.13.m13.1d">italic_m + 1</annotation></semantics></math> to <math alttext="m" class="ltx_Math" display="inline" id="S7.Thmtheorem27.p4.14.14.m14.1"><semantics id="S7.Thmtheorem27.p4.14.14.m14.1a"><mi id="S7.Thmtheorem27.p4.14.14.m14.1.1" xref="S7.Thmtheorem27.p4.14.14.m14.1.1.cmml">m</mi><annotation-xml encoding="MathML-Content" id="S7.Thmtheorem27.p4.14.14.m14.1b"><ci id="S7.Thmtheorem27.p4.14.14.m14.1.1.cmml" xref="S7.Thmtheorem27.p4.14.14.m14.1.1">𝑚</ci></annotation-xml><annotation encoding="application/x-tex" id="S7.Thmtheorem27.p4.14.14.m14.1c">m</annotation><annotation encoding="application/x-llamapun" id="S7.Thmtheorem27.p4.14.14.m14.1d">italic_m</annotation></semantics></math>, the out-degree of <math alttext="v" class="ltx_Math" display="inline" id="S7.Thmtheorem27.p4.15.15.m15.1"><semantics id="S7.Thmtheorem27.p4.15.15.m15.1a"><mi id="S7.Thmtheorem27.p4.15.15.m15.1.1" xref="S7.Thmtheorem27.p4.15.15.m15.1.1.cmml">v</mi><annotation-xml encoding="MathML-Content" id="S7.Thmtheorem27.p4.15.15.m15.1b"><ci id="S7.Thmtheorem27.p4.15.15.m15.1.1.cmml" xref="S7.Thmtheorem27.p4.15.15.m15.1.1">𝑣</ci></annotation-xml><annotation encoding="application/x-tex" id="S7.Thmtheorem27.p4.15.15.m15.1c">v</annotation><annotation encoding="application/x-llamapun" id="S7.Thmtheorem27.p4.15.15.m15.1d">italic_v</annotation></semantics></math> increases by only increasing <math alttext="\textsl{g}(v\!\to\!u)" class="ltx_Math" display="inline" id="S7.Thmtheorem27.p4.16.16.m16.1"><semantics id="S7.Thmtheorem27.p4.16.16.m16.1a"><mrow id="S7.Thmtheorem27.p4.16.16.m16.1.1" xref="S7.Thmtheorem27.p4.16.16.m16.1.1.cmml"><mtext class="ltx_mathvariant_italic" id="S7.Thmtheorem27.p4.16.16.m16.1.1.3" xref="S7.Thmtheorem27.p4.16.16.m16.1.1.3a.cmml">g</mtext><mo id="S7.Thmtheorem27.p4.16.16.m16.1.1.2" xref="S7.Thmtheorem27.p4.16.16.m16.1.1.2.cmml">⁢</mo><mrow id="S7.Thmtheorem27.p4.16.16.m16.1.1.1.1" xref="S7.Thmtheorem27.p4.16.16.m16.1.1.1.1.1.cmml"><mo id="S7.Thmtheorem27.p4.16.16.m16.1.1.1.1.2" stretchy="false" xref="S7.Thmtheorem27.p4.16.16.m16.1.1.1.1.1.cmml">(</mo><mrow id="S7.Thmtheorem27.p4.16.16.m16.1.1.1.1.1" xref="S7.Thmtheorem27.p4.16.16.m16.1.1.1.1.1.cmml"><mi id="S7.Thmtheorem27.p4.16.16.m16.1.1.1.1.1.2" xref="S7.Thmtheorem27.p4.16.16.m16.1.1.1.1.1.2.cmml">v</mi><mo id="S7.Thmtheorem27.p4.16.16.m16.1.1.1.1.1.1" lspace="0.108em" rspace="0.108em" stretchy="false" xref="S7.Thmtheorem27.p4.16.16.m16.1.1.1.1.1.1.cmml">→</mo><mi id="S7.Thmtheorem27.p4.16.16.m16.1.1.1.1.1.3" xref="S7.Thmtheorem27.p4.16.16.m16.1.1.1.1.1.3.cmml">u</mi></mrow><mo id="S7.Thmtheorem27.p4.16.16.m16.1.1.1.1.3" stretchy="false" xref="S7.Thmtheorem27.p4.16.16.m16.1.1.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S7.Thmtheorem27.p4.16.16.m16.1b"><apply id="S7.Thmtheorem27.p4.16.16.m16.1.1.cmml" xref="S7.Thmtheorem27.p4.16.16.m16.1.1"><times id="S7.Thmtheorem27.p4.16.16.m16.1.1.2.cmml" xref="S7.Thmtheorem27.p4.16.16.m16.1.1.2"></times><ci id="S7.Thmtheorem27.p4.16.16.m16.1.1.3a.cmml" xref="S7.Thmtheorem27.p4.16.16.m16.1.1.3"><mtext class="ltx_mathvariant_italic" id="S7.Thmtheorem27.p4.16.16.m16.1.1.3.cmml" xref="S7.Thmtheorem27.p4.16.16.m16.1.1.3">g</mtext></ci><apply id="S7.Thmtheorem27.p4.16.16.m16.1.1.1.1.1.cmml" xref="S7.Thmtheorem27.p4.16.16.m16.1.1.1.1"><ci id="S7.Thmtheorem27.p4.16.16.m16.1.1.1.1.1.1.cmml" xref="S7.Thmtheorem27.p4.16.16.m16.1.1.1.1.1.1">→</ci><ci id="S7.Thmtheorem27.p4.16.16.m16.1.1.1.1.1.2.cmml" xref="S7.Thmtheorem27.p4.16.16.m16.1.1.1.1.1.2">𝑣</ci><ci id="S7.Thmtheorem27.p4.16.16.m16.1.1.1.1.1.3.cmml" xref="S7.Thmtheorem27.p4.16.16.m16.1.1.1.1.1.3">𝑢</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.Thmtheorem27.p4.16.16.m16.1c">\textsl{g}(v\!\to\!u)</annotation><annotation encoding="application/x-llamapun" id="S7.Thmtheorem27.p4.16.16.m16.1d">g ( italic_v → italic_u )</annotation></semantics></math> for <math alttext="\overline{vw}\not\in E_{m}(v)" class="ltx_Math" display="inline" id="S7.Thmtheorem27.p4.17.17.m17.1"><semantics id="S7.Thmtheorem27.p4.17.17.m17.1a"><mrow id="S7.Thmtheorem27.p4.17.17.m17.1.2" xref="S7.Thmtheorem27.p4.17.17.m17.1.2.cmml"><mover accent="true" id="S7.Thmtheorem27.p4.17.17.m17.1.2.2" xref="S7.Thmtheorem27.p4.17.17.m17.1.2.2.cmml"><mrow id="S7.Thmtheorem27.p4.17.17.m17.1.2.2.2" xref="S7.Thmtheorem27.p4.17.17.m17.1.2.2.2.cmml"><mi id="S7.Thmtheorem27.p4.17.17.m17.1.2.2.2.2" xref="S7.Thmtheorem27.p4.17.17.m17.1.2.2.2.2.cmml">v</mi><mo id="S7.Thmtheorem27.p4.17.17.m17.1.2.2.2.1" xref="S7.Thmtheorem27.p4.17.17.m17.1.2.2.2.1.cmml">⁢</mo><mi id="S7.Thmtheorem27.p4.17.17.m17.1.2.2.2.3" xref="S7.Thmtheorem27.p4.17.17.m17.1.2.2.2.3.cmml">w</mi></mrow><mo id="S7.Thmtheorem27.p4.17.17.m17.1.2.2.1" xref="S7.Thmtheorem27.p4.17.17.m17.1.2.2.1.cmml">¯</mo></mover><mo id="S7.Thmtheorem27.p4.17.17.m17.1.2.1" xref="S7.Thmtheorem27.p4.17.17.m17.1.2.1.cmml">∉</mo><mrow id="S7.Thmtheorem27.p4.17.17.m17.1.2.3" xref="S7.Thmtheorem27.p4.17.17.m17.1.2.3.cmml"><msub id="S7.Thmtheorem27.p4.17.17.m17.1.2.3.2" xref="S7.Thmtheorem27.p4.17.17.m17.1.2.3.2.cmml"><mi id="S7.Thmtheorem27.p4.17.17.m17.1.2.3.2.2" xref="S7.Thmtheorem27.p4.17.17.m17.1.2.3.2.2.cmml">E</mi><mi id="S7.Thmtheorem27.p4.17.17.m17.1.2.3.2.3" xref="S7.Thmtheorem27.p4.17.17.m17.1.2.3.2.3.cmml">m</mi></msub><mo id="S7.Thmtheorem27.p4.17.17.m17.1.2.3.1" xref="S7.Thmtheorem27.p4.17.17.m17.1.2.3.1.cmml">⁢</mo><mrow id="S7.Thmtheorem27.p4.17.17.m17.1.2.3.3.2" xref="S7.Thmtheorem27.p4.17.17.m17.1.2.3.cmml"><mo id="S7.Thmtheorem27.p4.17.17.m17.1.2.3.3.2.1" stretchy="false" xref="S7.Thmtheorem27.p4.17.17.m17.1.2.3.cmml">(</mo><mi id="S7.Thmtheorem27.p4.17.17.m17.1.1" xref="S7.Thmtheorem27.p4.17.17.m17.1.1.cmml">v</mi><mo id="S7.Thmtheorem27.p4.17.17.m17.1.2.3.3.2.2" stretchy="false" xref="S7.Thmtheorem27.p4.17.17.m17.1.2.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S7.Thmtheorem27.p4.17.17.m17.1b"><apply id="S7.Thmtheorem27.p4.17.17.m17.1.2.cmml" xref="S7.Thmtheorem27.p4.17.17.m17.1.2"><notin id="S7.Thmtheorem27.p4.17.17.m17.1.2.1.cmml" xref="S7.Thmtheorem27.p4.17.17.m17.1.2.1"></notin><apply id="S7.Thmtheorem27.p4.17.17.m17.1.2.2.cmml" xref="S7.Thmtheorem27.p4.17.17.m17.1.2.2"><ci id="S7.Thmtheorem27.p4.17.17.m17.1.2.2.1.cmml" xref="S7.Thmtheorem27.p4.17.17.m17.1.2.2.1">¯</ci><apply id="S7.Thmtheorem27.p4.17.17.m17.1.2.2.2.cmml" xref="S7.Thmtheorem27.p4.17.17.m17.1.2.2.2"><times id="S7.Thmtheorem27.p4.17.17.m17.1.2.2.2.1.cmml" xref="S7.Thmtheorem27.p4.17.17.m17.1.2.2.2.1"></times><ci id="S7.Thmtheorem27.p4.17.17.m17.1.2.2.2.2.cmml" xref="S7.Thmtheorem27.p4.17.17.m17.1.2.2.2.2">𝑣</ci><ci id="S7.Thmtheorem27.p4.17.17.m17.1.2.2.2.3.cmml" xref="S7.Thmtheorem27.p4.17.17.m17.1.2.2.2.3">𝑤</ci></apply></apply><apply id="S7.Thmtheorem27.p4.17.17.m17.1.2.3.cmml" xref="S7.Thmtheorem27.p4.17.17.m17.1.2.3"><times id="S7.Thmtheorem27.p4.17.17.m17.1.2.3.1.cmml" xref="S7.Thmtheorem27.p4.17.17.m17.1.2.3.1"></times><apply id="S7.Thmtheorem27.p4.17.17.m17.1.2.3.2.cmml" xref="S7.Thmtheorem27.p4.17.17.m17.1.2.3.2"><csymbol cd="ambiguous" id="S7.Thmtheorem27.p4.17.17.m17.1.2.3.2.1.cmml" xref="S7.Thmtheorem27.p4.17.17.m17.1.2.3.2">subscript</csymbol><ci id="S7.Thmtheorem27.p4.17.17.m17.1.2.3.2.2.cmml" xref="S7.Thmtheorem27.p4.17.17.m17.1.2.3.2.2">𝐸</ci><ci id="S7.Thmtheorem27.p4.17.17.m17.1.2.3.2.3.cmml" xref="S7.Thmtheorem27.p4.17.17.m17.1.2.3.2.3">𝑚</ci></apply><ci id="S7.Thmtheorem27.p4.17.17.m17.1.1.cmml" xref="S7.Thmtheorem27.p4.17.17.m17.1.1">𝑣</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.Thmtheorem27.p4.17.17.m17.1c">\overline{vw}\not\in E_{m}(v)</annotation><annotation encoding="application/x-llamapun" id="S7.Thmtheorem27.p4.17.17.m17.1d">over¯ start_ARG italic_v italic_w end_ARG ∉ italic_E start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT ( italic_v )</annotation></semantics></math>. Thus <math alttext="\Delta_{m}" class="ltx_Math" display="inline" id="S7.Thmtheorem27.p4.18.18.m18.1"><semantics id="S7.Thmtheorem27.p4.18.18.m18.1a"><msub id="S7.Thmtheorem27.p4.18.18.m18.1.1" xref="S7.Thmtheorem27.p4.18.18.m18.1.1.cmml"><mi id="S7.Thmtheorem27.p4.18.18.m18.1.1.2" mathvariant="normal" xref="S7.Thmtheorem27.p4.18.18.m18.1.1.2.cmml">Δ</mi><mi id="S7.Thmtheorem27.p4.18.18.m18.1.1.3" xref="S7.Thmtheorem27.p4.18.18.m18.1.1.3.cmml">m</mi></msub><annotation-xml encoding="MathML-Content" id="S7.Thmtheorem27.p4.18.18.m18.1b"><apply id="S7.Thmtheorem27.p4.18.18.m18.1.1.cmml" xref="S7.Thmtheorem27.p4.18.18.m18.1.1"><csymbol cd="ambiguous" id="S7.Thmtheorem27.p4.18.18.m18.1.1.1.cmml" xref="S7.Thmtheorem27.p4.18.18.m18.1.1">subscript</csymbol><ci id="S7.Thmtheorem27.p4.18.18.m18.1.1.2.cmml" xref="S7.Thmtheorem27.p4.18.18.m18.1.1.2">Δ</ci><ci id="S7.Thmtheorem27.p4.18.18.m18.1.1.3.cmml" xref="S7.Thmtheorem27.p4.18.18.m18.1.1.3">𝑚</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.Thmtheorem27.p4.18.18.m18.1c">\Delta_{m}</annotation><annotation encoding="application/x-llamapun" id="S7.Thmtheorem27.p4.18.18.m18.1d">roman_Δ start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT</annotation></semantics></math> decreases by <math alttext="(1+\frac{\eta}{2})^{h+1}-(1+\frac{\eta}{2})^{h-1}" class="ltx_Math" display="inline" id="S7.Thmtheorem27.p4.19.19.m19.2"><semantics id="S7.Thmtheorem27.p4.19.19.m19.2a"><mrow id="S7.Thmtheorem27.p4.19.19.m19.2.2" xref="S7.Thmtheorem27.p4.19.19.m19.2.2.cmml"><msup id="S7.Thmtheorem27.p4.19.19.m19.1.1.1" xref="S7.Thmtheorem27.p4.19.19.m19.1.1.1.cmml"><mrow id="S7.Thmtheorem27.p4.19.19.m19.1.1.1.1.1" xref="S7.Thmtheorem27.p4.19.19.m19.1.1.1.1.1.1.cmml"><mo id="S7.Thmtheorem27.p4.19.19.m19.1.1.1.1.1.2" stretchy="false" xref="S7.Thmtheorem27.p4.19.19.m19.1.1.1.1.1.1.cmml">(</mo><mrow id="S7.Thmtheorem27.p4.19.19.m19.1.1.1.1.1.1" xref="S7.Thmtheorem27.p4.19.19.m19.1.1.1.1.1.1.cmml"><mn id="S7.Thmtheorem27.p4.19.19.m19.1.1.1.1.1.1.2" xref="S7.Thmtheorem27.p4.19.19.m19.1.1.1.1.1.1.2.cmml">1</mn><mo id="S7.Thmtheorem27.p4.19.19.m19.1.1.1.1.1.1.1" xref="S7.Thmtheorem27.p4.19.19.m19.1.1.1.1.1.1.1.cmml">+</mo><mfrac id="S7.Thmtheorem27.p4.19.19.m19.1.1.1.1.1.1.3" xref="S7.Thmtheorem27.p4.19.19.m19.1.1.1.1.1.1.3.cmml"><mi id="S7.Thmtheorem27.p4.19.19.m19.1.1.1.1.1.1.3.2" xref="S7.Thmtheorem27.p4.19.19.m19.1.1.1.1.1.1.3.2.cmml">η</mi><mn id="S7.Thmtheorem27.p4.19.19.m19.1.1.1.1.1.1.3.3" xref="S7.Thmtheorem27.p4.19.19.m19.1.1.1.1.1.1.3.3.cmml">2</mn></mfrac></mrow><mo id="S7.Thmtheorem27.p4.19.19.m19.1.1.1.1.1.3" stretchy="false" xref="S7.Thmtheorem27.p4.19.19.m19.1.1.1.1.1.1.cmml">)</mo></mrow><mrow id="S7.Thmtheorem27.p4.19.19.m19.1.1.1.3" xref="S7.Thmtheorem27.p4.19.19.m19.1.1.1.3.cmml"><mi id="S7.Thmtheorem27.p4.19.19.m19.1.1.1.3.2" xref="S7.Thmtheorem27.p4.19.19.m19.1.1.1.3.2.cmml">h</mi><mo id="S7.Thmtheorem27.p4.19.19.m19.1.1.1.3.1" xref="S7.Thmtheorem27.p4.19.19.m19.1.1.1.3.1.cmml">+</mo><mn id="S7.Thmtheorem27.p4.19.19.m19.1.1.1.3.3" xref="S7.Thmtheorem27.p4.19.19.m19.1.1.1.3.3.cmml">1</mn></mrow></msup><mo id="S7.Thmtheorem27.p4.19.19.m19.2.2.3" xref="S7.Thmtheorem27.p4.19.19.m19.2.2.3.cmml">−</mo><msup id="S7.Thmtheorem27.p4.19.19.m19.2.2.2" xref="S7.Thmtheorem27.p4.19.19.m19.2.2.2.cmml"><mrow id="S7.Thmtheorem27.p4.19.19.m19.2.2.2.1.1" xref="S7.Thmtheorem27.p4.19.19.m19.2.2.2.1.1.1.cmml"><mo id="S7.Thmtheorem27.p4.19.19.m19.2.2.2.1.1.2" stretchy="false" xref="S7.Thmtheorem27.p4.19.19.m19.2.2.2.1.1.1.cmml">(</mo><mrow id="S7.Thmtheorem27.p4.19.19.m19.2.2.2.1.1.1" xref="S7.Thmtheorem27.p4.19.19.m19.2.2.2.1.1.1.cmml"><mn id="S7.Thmtheorem27.p4.19.19.m19.2.2.2.1.1.1.2" xref="S7.Thmtheorem27.p4.19.19.m19.2.2.2.1.1.1.2.cmml">1</mn><mo id="S7.Thmtheorem27.p4.19.19.m19.2.2.2.1.1.1.1" xref="S7.Thmtheorem27.p4.19.19.m19.2.2.2.1.1.1.1.cmml">+</mo><mfrac id="S7.Thmtheorem27.p4.19.19.m19.2.2.2.1.1.1.3" xref="S7.Thmtheorem27.p4.19.19.m19.2.2.2.1.1.1.3.cmml"><mi id="S7.Thmtheorem27.p4.19.19.m19.2.2.2.1.1.1.3.2" xref="S7.Thmtheorem27.p4.19.19.m19.2.2.2.1.1.1.3.2.cmml">η</mi><mn id="S7.Thmtheorem27.p4.19.19.m19.2.2.2.1.1.1.3.3" xref="S7.Thmtheorem27.p4.19.19.m19.2.2.2.1.1.1.3.3.cmml">2</mn></mfrac></mrow><mo id="S7.Thmtheorem27.p4.19.19.m19.2.2.2.1.1.3" stretchy="false" xref="S7.Thmtheorem27.p4.19.19.m19.2.2.2.1.1.1.cmml">)</mo></mrow><mrow id="S7.Thmtheorem27.p4.19.19.m19.2.2.2.3" xref="S7.Thmtheorem27.p4.19.19.m19.2.2.2.3.cmml"><mi id="S7.Thmtheorem27.p4.19.19.m19.2.2.2.3.2" xref="S7.Thmtheorem27.p4.19.19.m19.2.2.2.3.2.cmml">h</mi><mo id="S7.Thmtheorem27.p4.19.19.m19.2.2.2.3.1" xref="S7.Thmtheorem27.p4.19.19.m19.2.2.2.3.1.cmml">−</mo><mn id="S7.Thmtheorem27.p4.19.19.m19.2.2.2.3.3" xref="S7.Thmtheorem27.p4.19.19.m19.2.2.2.3.3.cmml">1</mn></mrow></msup></mrow><annotation-xml encoding="MathML-Content" id="S7.Thmtheorem27.p4.19.19.m19.2b"><apply id="S7.Thmtheorem27.p4.19.19.m19.2.2.cmml" xref="S7.Thmtheorem27.p4.19.19.m19.2.2"><minus id="S7.Thmtheorem27.p4.19.19.m19.2.2.3.cmml" xref="S7.Thmtheorem27.p4.19.19.m19.2.2.3"></minus><apply id="S7.Thmtheorem27.p4.19.19.m19.1.1.1.cmml" xref="S7.Thmtheorem27.p4.19.19.m19.1.1.1"><csymbol cd="ambiguous" id="S7.Thmtheorem27.p4.19.19.m19.1.1.1.2.cmml" xref="S7.Thmtheorem27.p4.19.19.m19.1.1.1">superscript</csymbol><apply id="S7.Thmtheorem27.p4.19.19.m19.1.1.1.1.1.1.cmml" xref="S7.Thmtheorem27.p4.19.19.m19.1.1.1.1.1"><plus id="S7.Thmtheorem27.p4.19.19.m19.1.1.1.1.1.1.1.cmml" xref="S7.Thmtheorem27.p4.19.19.m19.1.1.1.1.1.1.1"></plus><cn id="S7.Thmtheorem27.p4.19.19.m19.1.1.1.1.1.1.2.cmml" type="integer" xref="S7.Thmtheorem27.p4.19.19.m19.1.1.1.1.1.1.2">1</cn><apply id="S7.Thmtheorem27.p4.19.19.m19.1.1.1.1.1.1.3.cmml" xref="S7.Thmtheorem27.p4.19.19.m19.1.1.1.1.1.1.3"><divide id="S7.Thmtheorem27.p4.19.19.m19.1.1.1.1.1.1.3.1.cmml" xref="S7.Thmtheorem27.p4.19.19.m19.1.1.1.1.1.1.3"></divide><ci id="S7.Thmtheorem27.p4.19.19.m19.1.1.1.1.1.1.3.2.cmml" xref="S7.Thmtheorem27.p4.19.19.m19.1.1.1.1.1.1.3.2">𝜂</ci><cn id="S7.Thmtheorem27.p4.19.19.m19.1.1.1.1.1.1.3.3.cmml" type="integer" xref="S7.Thmtheorem27.p4.19.19.m19.1.1.1.1.1.1.3.3">2</cn></apply></apply><apply id="S7.Thmtheorem27.p4.19.19.m19.1.1.1.3.cmml" xref="S7.Thmtheorem27.p4.19.19.m19.1.1.1.3"><plus id="S7.Thmtheorem27.p4.19.19.m19.1.1.1.3.1.cmml" xref="S7.Thmtheorem27.p4.19.19.m19.1.1.1.3.1"></plus><ci id="S7.Thmtheorem27.p4.19.19.m19.1.1.1.3.2.cmml" xref="S7.Thmtheorem27.p4.19.19.m19.1.1.1.3.2">ℎ</ci><cn id="S7.Thmtheorem27.p4.19.19.m19.1.1.1.3.3.cmml" type="integer" xref="S7.Thmtheorem27.p4.19.19.m19.1.1.1.3.3">1</cn></apply></apply><apply id="S7.Thmtheorem27.p4.19.19.m19.2.2.2.cmml" xref="S7.Thmtheorem27.p4.19.19.m19.2.2.2"><csymbol cd="ambiguous" id="S7.Thmtheorem27.p4.19.19.m19.2.2.2.2.cmml" xref="S7.Thmtheorem27.p4.19.19.m19.2.2.2">superscript</csymbol><apply id="S7.Thmtheorem27.p4.19.19.m19.2.2.2.1.1.1.cmml" xref="S7.Thmtheorem27.p4.19.19.m19.2.2.2.1.1"><plus id="S7.Thmtheorem27.p4.19.19.m19.2.2.2.1.1.1.1.cmml" xref="S7.Thmtheorem27.p4.19.19.m19.2.2.2.1.1.1.1"></plus><cn id="S7.Thmtheorem27.p4.19.19.m19.2.2.2.1.1.1.2.cmml" type="integer" xref="S7.Thmtheorem27.p4.19.19.m19.2.2.2.1.1.1.2">1</cn><apply id="S7.Thmtheorem27.p4.19.19.m19.2.2.2.1.1.1.3.cmml" xref="S7.Thmtheorem27.p4.19.19.m19.2.2.2.1.1.1.3"><divide id="S7.Thmtheorem27.p4.19.19.m19.2.2.2.1.1.1.3.1.cmml" xref="S7.Thmtheorem27.p4.19.19.m19.2.2.2.1.1.1.3"></divide><ci id="S7.Thmtheorem27.p4.19.19.m19.2.2.2.1.1.1.3.2.cmml" xref="S7.Thmtheorem27.p4.19.19.m19.2.2.2.1.1.1.3.2">𝜂</ci><cn id="S7.Thmtheorem27.p4.19.19.m19.2.2.2.1.1.1.3.3.cmml" type="integer" xref="S7.Thmtheorem27.p4.19.19.m19.2.2.2.1.1.1.3.3">2</cn></apply></apply><apply id="S7.Thmtheorem27.p4.19.19.m19.2.2.2.3.cmml" xref="S7.Thmtheorem27.p4.19.19.m19.2.2.2.3"><minus id="S7.Thmtheorem27.p4.19.19.m19.2.2.2.3.1.cmml" xref="S7.Thmtheorem27.p4.19.19.m19.2.2.2.3.1"></minus><ci id="S7.Thmtheorem27.p4.19.19.m19.2.2.2.3.2.cmml" xref="S7.Thmtheorem27.p4.19.19.m19.2.2.2.3.2">ℎ</ci><cn id="S7.Thmtheorem27.p4.19.19.m19.2.2.2.3.3.cmml" type="integer" xref="S7.Thmtheorem27.p4.19.19.m19.2.2.2.3.3">1</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.Thmtheorem27.p4.19.19.m19.2c">(1+\frac{\eta}{2})^{h+1}-(1+\frac{\eta}{2})^{h-1}</annotation><annotation encoding="application/x-llamapun" id="S7.Thmtheorem27.p4.19.19.m19.2d">( 1 + divide start_ARG italic_η end_ARG start_ARG 2 end_ARG ) start_POSTSUPERSCRIPT italic_h + 1 end_POSTSUPERSCRIPT - ( 1 + divide start_ARG italic_η end_ARG start_ARG 2 end_ARG ) start_POSTSUPERSCRIPT italic_h - 1 end_POSTSUPERSCRIPT</annotation></semantics></math> during <span class="ltx_text ltx_font_smallcaps" id="S7.Thmtheorem27.p4.20.20.1">minute</span> <math alttext="m" class="ltx_Math" display="inline" id="S7.Thmtheorem27.p4.20.20.m20.1"><semantics id="S7.Thmtheorem27.p4.20.20.m20.1a"><mi id="S7.Thmtheorem27.p4.20.20.m20.1.1" xref="S7.Thmtheorem27.p4.20.20.m20.1.1.cmml">m</mi><annotation-xml encoding="MathML-Content" id="S7.Thmtheorem27.p4.20.20.m20.1b"><ci id="S7.Thmtheorem27.p4.20.20.m20.1.1.cmml" xref="S7.Thmtheorem27.p4.20.20.m20.1.1">𝑚</ci></annotation-xml><annotation encoding="application/x-tex" id="S7.Thmtheorem27.p4.20.20.m20.1c">m</annotation><annotation encoding="application/x-llamapun" id="S7.Thmtheorem27.p4.20.20.m20.1d">italic_m</annotation></semantics></math>. And, after at most</span></p> <table class="ltx_equation ltx_eqn_table" id="S7.Ex32"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_left_padleft"></td> <td class="ltx_eqn_cell ltx_align_left"><math alttext="\frac{\Delta_{0}}{(1+\frac{\eta}{2})^{h+1}-(1+\frac{\eta}{2})^{h-1}}\leq\frac{% (1+\frac{\eta}{2})^{h+1}}{(1+\frac{\eta}{2})^{h+1}-(1+\frac{\eta}{2})^{h-1}}% \leq\eta^{-1}\quad\textnormal{ odd }\textsc{ minutes}" class="ltx_Math" display="block" id="S7.Ex32.m1.7"><semantics id="S7.Ex32.m1.7a"><mrow id="S7.Ex32.m1.7.7.1" xref="S7.Ex32.m1.7.7.2.cmml"><mrow id="S7.Ex32.m1.7.7.1.1" xref="S7.Ex32.m1.7.7.1.1.cmml"><mfrac id="S7.Ex32.m1.2.2" xref="S7.Ex32.m1.2.2.cmml"><msub id="S7.Ex32.m1.2.2.4" xref="S7.Ex32.m1.2.2.4.cmml"><mi id="S7.Ex32.m1.2.2.4.2" mathvariant="normal" xref="S7.Ex32.m1.2.2.4.2.cmml">Δ</mi><mn id="S7.Ex32.m1.2.2.4.3" xref="S7.Ex32.m1.2.2.4.3.cmml">0</mn></msub><mrow id="S7.Ex32.m1.2.2.2" xref="S7.Ex32.m1.2.2.2.cmml"><msup id="S7.Ex32.m1.1.1.1.1" xref="S7.Ex32.m1.1.1.1.1.cmml"><mrow id="S7.Ex32.m1.1.1.1.1.1.1" xref="S7.Ex32.m1.1.1.1.1.1.1.1.cmml"><mo id="S7.Ex32.m1.1.1.1.1.1.1.2" stretchy="false" xref="S7.Ex32.m1.1.1.1.1.1.1.1.cmml">(</mo><mrow id="S7.Ex32.m1.1.1.1.1.1.1.1" xref="S7.Ex32.m1.1.1.1.1.1.1.1.cmml"><mn id="S7.Ex32.m1.1.1.1.1.1.1.1.2" xref="S7.Ex32.m1.1.1.1.1.1.1.1.2.cmml">1</mn><mo id="S7.Ex32.m1.1.1.1.1.1.1.1.1" xref="S7.Ex32.m1.1.1.1.1.1.1.1.1.cmml">+</mo><mfrac id="S7.Ex32.m1.1.1.1.1.1.1.1.3" xref="S7.Ex32.m1.1.1.1.1.1.1.1.3.cmml"><mi id="S7.Ex32.m1.1.1.1.1.1.1.1.3.2" xref="S7.Ex32.m1.1.1.1.1.1.1.1.3.2.cmml">η</mi><mn id="S7.Ex32.m1.1.1.1.1.1.1.1.3.3" xref="S7.Ex32.m1.1.1.1.1.1.1.1.3.3.cmml">2</mn></mfrac></mrow><mo id="S7.Ex32.m1.1.1.1.1.1.1.3" stretchy="false" xref="S7.Ex32.m1.1.1.1.1.1.1.1.cmml">)</mo></mrow><mrow id="S7.Ex32.m1.1.1.1.1.3" xref="S7.Ex32.m1.1.1.1.1.3.cmml"><mi id="S7.Ex32.m1.1.1.1.1.3.2" xref="S7.Ex32.m1.1.1.1.1.3.2.cmml">h</mi><mo id="S7.Ex32.m1.1.1.1.1.3.1" xref="S7.Ex32.m1.1.1.1.1.3.1.cmml">+</mo><mn id="S7.Ex32.m1.1.1.1.1.3.3" xref="S7.Ex32.m1.1.1.1.1.3.3.cmml">1</mn></mrow></msup><mo id="S7.Ex32.m1.2.2.2.3" xref="S7.Ex32.m1.2.2.2.3.cmml">−</mo><msup id="S7.Ex32.m1.2.2.2.2" xref="S7.Ex32.m1.2.2.2.2.cmml"><mrow id="S7.Ex32.m1.2.2.2.2.1.1" xref="S7.Ex32.m1.2.2.2.2.1.1.1.cmml"><mo id="S7.Ex32.m1.2.2.2.2.1.1.2" stretchy="false" xref="S7.Ex32.m1.2.2.2.2.1.1.1.cmml">(</mo><mrow id="S7.Ex32.m1.2.2.2.2.1.1.1" xref="S7.Ex32.m1.2.2.2.2.1.1.1.cmml"><mn id="S7.Ex32.m1.2.2.2.2.1.1.1.2" xref="S7.Ex32.m1.2.2.2.2.1.1.1.2.cmml">1</mn><mo id="S7.Ex32.m1.2.2.2.2.1.1.1.1" xref="S7.Ex32.m1.2.2.2.2.1.1.1.1.cmml">+</mo><mfrac id="S7.Ex32.m1.2.2.2.2.1.1.1.3" xref="S7.Ex32.m1.2.2.2.2.1.1.1.3.cmml"><mi id="S7.Ex32.m1.2.2.2.2.1.1.1.3.2" xref="S7.Ex32.m1.2.2.2.2.1.1.1.3.2.cmml">η</mi><mn id="S7.Ex32.m1.2.2.2.2.1.1.1.3.3" xref="S7.Ex32.m1.2.2.2.2.1.1.1.3.3.cmml">2</mn></mfrac></mrow><mo id="S7.Ex32.m1.2.2.2.2.1.1.3" stretchy="false" xref="S7.Ex32.m1.2.2.2.2.1.1.1.cmml">)</mo></mrow><mrow id="S7.Ex32.m1.2.2.2.2.3" xref="S7.Ex32.m1.2.2.2.2.3.cmml"><mi id="S7.Ex32.m1.2.2.2.2.3.2" xref="S7.Ex32.m1.2.2.2.2.3.2.cmml">h</mi><mo id="S7.Ex32.m1.2.2.2.2.3.1" xref="S7.Ex32.m1.2.2.2.2.3.1.cmml">−</mo><mn id="S7.Ex32.m1.2.2.2.2.3.3" xref="S7.Ex32.m1.2.2.2.2.3.3.cmml">1</mn></mrow></msup></mrow></mfrac><mo id="S7.Ex32.m1.7.7.1.1.2" xref="S7.Ex32.m1.7.7.1.1.2.cmml">≤</mo><mfrac id="S7.Ex32.m1.5.5" xref="S7.Ex32.m1.5.5.cmml"><msup id="S7.Ex32.m1.3.3.1" xref="S7.Ex32.m1.3.3.1.cmml"><mrow id="S7.Ex32.m1.3.3.1.1.1" xref="S7.Ex32.m1.3.3.1.1.1.1.cmml"><mo id="S7.Ex32.m1.3.3.1.1.1.2" stretchy="false" xref="S7.Ex32.m1.3.3.1.1.1.1.cmml">(</mo><mrow id="S7.Ex32.m1.3.3.1.1.1.1" xref="S7.Ex32.m1.3.3.1.1.1.1.cmml"><mn id="S7.Ex32.m1.3.3.1.1.1.1.2" xref="S7.Ex32.m1.3.3.1.1.1.1.2.cmml">1</mn><mo id="S7.Ex32.m1.3.3.1.1.1.1.1" xref="S7.Ex32.m1.3.3.1.1.1.1.1.cmml">+</mo><mfrac id="S7.Ex32.m1.3.3.1.1.1.1.3" xref="S7.Ex32.m1.3.3.1.1.1.1.3.cmml"><mi id="S7.Ex32.m1.3.3.1.1.1.1.3.2" xref="S7.Ex32.m1.3.3.1.1.1.1.3.2.cmml">η</mi><mn id="S7.Ex32.m1.3.3.1.1.1.1.3.3" xref="S7.Ex32.m1.3.3.1.1.1.1.3.3.cmml">2</mn></mfrac></mrow><mo id="S7.Ex32.m1.3.3.1.1.1.3" stretchy="false" xref="S7.Ex32.m1.3.3.1.1.1.1.cmml">)</mo></mrow><mrow id="S7.Ex32.m1.3.3.1.3" xref="S7.Ex32.m1.3.3.1.3.cmml"><mi id="S7.Ex32.m1.3.3.1.3.2" xref="S7.Ex32.m1.3.3.1.3.2.cmml">h</mi><mo id="S7.Ex32.m1.3.3.1.3.1" xref="S7.Ex32.m1.3.3.1.3.1.cmml">+</mo><mn id="S7.Ex32.m1.3.3.1.3.3" xref="S7.Ex32.m1.3.3.1.3.3.cmml">1</mn></mrow></msup><mrow id="S7.Ex32.m1.5.5.3" xref="S7.Ex32.m1.5.5.3.cmml"><msup id="S7.Ex32.m1.4.4.2.1" xref="S7.Ex32.m1.4.4.2.1.cmml"><mrow id="S7.Ex32.m1.4.4.2.1.1.1" xref="S7.Ex32.m1.4.4.2.1.1.1.1.cmml"><mo id="S7.Ex32.m1.4.4.2.1.1.1.2" stretchy="false" xref="S7.Ex32.m1.4.4.2.1.1.1.1.cmml">(</mo><mrow id="S7.Ex32.m1.4.4.2.1.1.1.1" xref="S7.Ex32.m1.4.4.2.1.1.1.1.cmml"><mn id="S7.Ex32.m1.4.4.2.1.1.1.1.2" xref="S7.Ex32.m1.4.4.2.1.1.1.1.2.cmml">1</mn><mo 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xref="S7.Ex32.m1.7.7.1.1.4"><csymbol cd="ambiguous" id="S7.Ex32.m1.7.7.1.1.4.1.cmml" xref="S7.Ex32.m1.7.7.1.1.4">superscript</csymbol><ci id="S7.Ex32.m1.7.7.1.1.4.2.cmml" xref="S7.Ex32.m1.7.7.1.1.4.2">𝜂</ci><apply id="S7.Ex32.m1.7.7.1.1.4.3.cmml" xref="S7.Ex32.m1.7.7.1.1.4.3"><minus id="S7.Ex32.m1.7.7.1.1.4.3.1.cmml" xref="S7.Ex32.m1.7.7.1.1.4.3"></minus><cn id="S7.Ex32.m1.7.7.1.1.4.3.2.cmml" type="integer" xref="S7.Ex32.m1.7.7.1.1.4.3.2">1</cn></apply></apply></apply></apply><ci id="S7.Ex32.m1.6.6c.cmml" xref="S7.Ex32.m1.6.6"><mrow id="S7.Ex32.m1.6.6.cmml" xref="S7.Ex32.m1.6.6"><mtext id="S7.Ex32.m1.6.6a.cmml" xref="S7.Ex32.m1.6.6"> odd </mtext><mtext class="ltx_font_smallcaps" id="S7.Ex32.m1.6.6b.cmml" xref="S7.Ex32.m1.6.6"> minutes</mtext></mrow></ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.Ex32.m1.7c">\frac{\Delta_{0}}{(1+\frac{\eta}{2})^{h+1}-(1+\frac{\eta}{2})^{h-1}}\leq\frac{% (1+\frac{\eta}{2})^{h+1}}{(1+\frac{\eta}{2})^{h+1}-(1+\frac{\eta}{2})^{h-1}}% \leq\eta^{-1}\quad\textnormal{ odd }\textsc{ minutes}</annotation><annotation encoding="application/x-llamapun" id="S7.Ex32.m1.7d">divide start_ARG roman_Δ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_ARG start_ARG ( 1 + divide start_ARG italic_η end_ARG start_ARG 2 end_ARG ) start_POSTSUPERSCRIPT italic_h + 1 end_POSTSUPERSCRIPT - ( 1 + divide start_ARG italic_η end_ARG start_ARG 2 end_ARG ) start_POSTSUPERSCRIPT italic_h - 1 end_POSTSUPERSCRIPT end_ARG ≤ divide start_ARG ( 1 + divide start_ARG italic_η end_ARG start_ARG 2 end_ARG ) start_POSTSUPERSCRIPT italic_h + 1 end_POSTSUPERSCRIPT end_ARG start_ARG ( 1 + divide start_ARG italic_η end_ARG start_ARG 2 end_ARG ) start_POSTSUPERSCRIPT italic_h + 1 end_POSTSUPERSCRIPT - ( 1 + divide start_ARG italic_η end_ARG start_ARG 2 end_ARG ) start_POSTSUPERSCRIPT italic_h - 1 end_POSTSUPERSCRIPT end_ARG ≤ italic_η start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT odd smallcaps_minutes</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_left_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S7.Thmtheorem27.p4.22"><span class="ltx_text ltx_font_italic" id="S7.Thmtheorem27.p4.22.2">the value <math alttext="\Delta_{m}=0" class="ltx_Math" display="inline" id="S7.Thmtheorem27.p4.21.1.m1.1"><semantics id="S7.Thmtheorem27.p4.21.1.m1.1a"><mrow id="S7.Thmtheorem27.p4.21.1.m1.1.1" xref="S7.Thmtheorem27.p4.21.1.m1.1.1.cmml"><msub id="S7.Thmtheorem27.p4.21.1.m1.1.1.2" xref="S7.Thmtheorem27.p4.21.1.m1.1.1.2.cmml"><mi id="S7.Thmtheorem27.p4.21.1.m1.1.1.2.2" mathvariant="normal" xref="S7.Thmtheorem27.p4.21.1.m1.1.1.2.2.cmml">Δ</mi><mi id="S7.Thmtheorem27.p4.21.1.m1.1.1.2.3" xref="S7.Thmtheorem27.p4.21.1.m1.1.1.2.3.cmml">m</mi></msub><mo id="S7.Thmtheorem27.p4.21.1.m1.1.1.1" xref="S7.Thmtheorem27.p4.21.1.m1.1.1.1.cmml">=</mo><mn id="S7.Thmtheorem27.p4.21.1.m1.1.1.3" xref="S7.Thmtheorem27.p4.21.1.m1.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S7.Thmtheorem27.p4.21.1.m1.1b"><apply id="S7.Thmtheorem27.p4.21.1.m1.1.1.cmml" xref="S7.Thmtheorem27.p4.21.1.m1.1.1"><eq id="S7.Thmtheorem27.p4.21.1.m1.1.1.1.cmml" xref="S7.Thmtheorem27.p4.21.1.m1.1.1.1"></eq><apply id="S7.Thmtheorem27.p4.21.1.m1.1.1.2.cmml" xref="S7.Thmtheorem27.p4.21.1.m1.1.1.2"><csymbol cd="ambiguous" id="S7.Thmtheorem27.p4.21.1.m1.1.1.2.1.cmml" xref="S7.Thmtheorem27.p4.21.1.m1.1.1.2">subscript</csymbol><ci id="S7.Thmtheorem27.p4.21.1.m1.1.1.2.2.cmml" xref="S7.Thmtheorem27.p4.21.1.m1.1.1.2.2">Δ</ci><ci id="S7.Thmtheorem27.p4.21.1.m1.1.1.2.3.cmml" xref="S7.Thmtheorem27.p4.21.1.m1.1.1.2.3">𝑚</ci></apply><cn id="S7.Thmtheorem27.p4.21.1.m1.1.1.3.cmml" type="integer" xref="S7.Thmtheorem27.p4.21.1.m1.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.Thmtheorem27.p4.21.1.m1.1c">\Delta_{m}=0</annotation><annotation encoding="application/x-llamapun" id="S7.Thmtheorem27.p4.21.1.m1.1d">roman_Δ start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT = 0</annotation></semantics></math> which contradicts the assumption that <math alttext="v" class="ltx_Math" display="inline" id="S7.Thmtheorem27.p4.22.2.m2.1"><semantics id="S7.Thmtheorem27.p4.22.2.m2.1a"><mi id="S7.Thmtheorem27.p4.22.2.m2.1.1" xref="S7.Thmtheorem27.p4.22.2.m2.1.1.cmml">v</mi><annotation-xml encoding="MathML-Content" id="S7.Thmtheorem27.p4.22.2.m2.1b"><ci id="S7.Thmtheorem27.p4.22.2.m2.1.1.cmml" xref="S7.Thmtheorem27.p4.22.2.m2.1.1">𝑣</ci></annotation-xml><annotation encoding="application/x-tex" id="S7.Thmtheorem27.p4.22.2.m2.1c">v</annotation><annotation encoding="application/x-llamapun" id="S7.Thmtheorem27.p4.22.2.m2.1d">italic_v</annotation></semantics></math> has an outgoing violating edge.</span></p> </div> </div> <div class="ltx_para ltx_noindent" id="S7.SS4.SSS0.P0.SPx2.p2"> <p class="ltx_p" id="S7.SS4.SSS0.P0.SPx2.p2.2">Invariant <a class="ltx_ref" href="https://arxiv.org/html/2411.12694v2#Thminvariant1" title="Invariant 1. ‣ 7.1 Algorithm overview ‣ 7 Results in CONGEST ‣ Local Density and its Distributed Approximation"><span class="ltx_text ltx_ref_tag">1</span></a> implies that we compute an <math alttext="\eta" class="ltx_Math" display="inline" id="S7.SS4.SSS0.P0.SPx2.p2.1.m1.1"><semantics id="S7.SS4.SSS0.P0.SPx2.p2.1.m1.1a"><mi id="S7.SS4.SSS0.P0.SPx2.p2.1.m1.1.1" xref="S7.SS4.SSS0.P0.SPx2.p2.1.m1.1.1.cmml">η</mi><annotation-xml encoding="MathML-Content" id="S7.SS4.SSS0.P0.SPx2.p2.1.m1.1b"><ci id="S7.SS4.SSS0.P0.SPx2.p2.1.m1.1.1.cmml" xref="S7.SS4.SSS0.P0.SPx2.p2.1.m1.1.1">𝜂</ci></annotation-xml><annotation encoding="application/x-tex" id="S7.SS4.SSS0.P0.SPx2.p2.1.m1.1c">\eta</annotation><annotation encoding="application/x-llamapun" id="S7.SS4.SSS0.P0.SPx2.p2.1.m1.1d">italic_η</annotation></semantics></math>-fair orientation when the clock reaches <math alttext="(0:0:0)" class="ltx_Math" display="inline" id="S7.SS4.SSS0.P0.SPx2.p2.2.m2.1"><semantics id="S7.SS4.SSS0.P0.SPx2.p2.2.m2.1a"><mrow id="S7.SS4.SSS0.P0.SPx2.p2.2.m2.1.1.1" xref="S7.SS4.SSS0.P0.SPx2.p2.2.m2.1.1.1.1.cmml"><mo id="S7.SS4.SSS0.P0.SPx2.p2.2.m2.1.1.1.2" stretchy="false" xref="S7.SS4.SSS0.P0.SPx2.p2.2.m2.1.1.1.1.cmml">(</mo><mrow id="S7.SS4.SSS0.P0.SPx2.p2.2.m2.1.1.1.1" xref="S7.SS4.SSS0.P0.SPx2.p2.2.m2.1.1.1.1.cmml"><mn id="S7.SS4.SSS0.P0.SPx2.p2.2.m2.1.1.1.1.2" xref="S7.SS4.SSS0.P0.SPx2.p2.2.m2.1.1.1.1.2.cmml">0</mn><mo id="S7.SS4.SSS0.P0.SPx2.p2.2.m2.1.1.1.1.3" lspace="0.278em" rspace="0.278em" xref="S7.SS4.SSS0.P0.SPx2.p2.2.m2.1.1.1.1.3.cmml">:</mo><mn id="S7.SS4.SSS0.P0.SPx2.p2.2.m2.1.1.1.1.4" xref="S7.SS4.SSS0.P0.SPx2.p2.2.m2.1.1.1.1.4.cmml">0</mn><mo id="S7.SS4.SSS0.P0.SPx2.p2.2.m2.1.1.1.1.5" lspace="0.278em" rspace="0.278em" xref="S7.SS4.SSS0.P0.SPx2.p2.2.m2.1.1.1.1.5.cmml">:</mo><mn id="S7.SS4.SSS0.P0.SPx2.p2.2.m2.1.1.1.1.6" xref="S7.SS4.SSS0.P0.SPx2.p2.2.m2.1.1.1.1.6.cmml">0</mn></mrow><mo id="S7.SS4.SSS0.P0.SPx2.p2.2.m2.1.1.1.3" stretchy="false" xref="S7.SS4.SSS0.P0.SPx2.p2.2.m2.1.1.1.1.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S7.SS4.SSS0.P0.SPx2.p2.2.m2.1b"><apply id="S7.SS4.SSS0.P0.SPx2.p2.2.m2.1.1.1.1.cmml" xref="S7.SS4.SSS0.P0.SPx2.p2.2.m2.1.1.1"><and id="S7.SS4.SSS0.P0.SPx2.p2.2.m2.1.1.1.1a.cmml" xref="S7.SS4.SSS0.P0.SPx2.p2.2.m2.1.1.1"></and><apply id="S7.SS4.SSS0.P0.SPx2.p2.2.m2.1.1.1.1b.cmml" xref="S7.SS4.SSS0.P0.SPx2.p2.2.m2.1.1.1"><ci id="S7.SS4.SSS0.P0.SPx2.p2.2.m2.1.1.1.1.3.cmml" xref="S7.SS4.SSS0.P0.SPx2.p2.2.m2.1.1.1.1.3">:</ci><cn id="S7.SS4.SSS0.P0.SPx2.p2.2.m2.1.1.1.1.2.cmml" type="integer" xref="S7.SS4.SSS0.P0.SPx2.p2.2.m2.1.1.1.1.2">0</cn><cn id="S7.SS4.SSS0.P0.SPx2.p2.2.m2.1.1.1.1.4.cmml" type="integer" xref="S7.SS4.SSS0.P0.SPx2.p2.2.m2.1.1.1.1.4">0</cn></apply><apply id="S7.SS4.SSS0.P0.SPx2.p2.2.m2.1.1.1.1c.cmml" xref="S7.SS4.SSS0.P0.SPx2.p2.2.m2.1.1.1"><ci id="S7.SS4.SSS0.P0.SPx2.p2.2.m2.1.1.1.1.5.cmml" xref="S7.SS4.SSS0.P0.SPx2.p2.2.m2.1.1.1.1.5">:</ci><share href="https://arxiv.org/html/2411.12694v2#S7.SS4.SSS0.P0.SPx2.p2.2.m2.1.1.1.1.4.cmml" id="S7.SS4.SSS0.P0.SPx2.p2.2.m2.1.1.1.1d.cmml" xref="S7.SS4.SSS0.P0.SPx2.p2.2.m2.1.1.1"></share><cn id="S7.SS4.SSS0.P0.SPx2.p2.2.m2.1.1.1.1.6.cmml" type="integer" xref="S7.SS4.SSS0.P0.SPx2.p2.2.m2.1.1.1.1.6">0</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.SS4.SSS0.P0.SPx2.p2.2.m2.1c">(0:0:0)</annotation><annotation encoding="application/x-llamapun" id="S7.SS4.SSS0.P0.SPx2.p2.2.m2.1d">( 0 : 0 : 0 )</annotation></semantics></math>, thus: See <a class="ltx_ref" href="https://arxiv.org/html/2411.12694v2#S3.Thmtheorem9" title="Theorem 3.9. ‣ 3.D Results in CONGEST ‣ 3 Results and organisation ‣ Local Density and its Distributed Approximation"><span class="ltx_text ltx_ref_tag">3.9</span></a></p> </div> <div class="ltx_para ltx_noindent" id="S7.SS4.SSS0.P0.SPx2.p3"> <p class="ltx_p" id="S7.SS4.SSS0.P0.SPx2.p3.1">We plug in the runtime of Blocking<math alttext="(h,n)" class="ltx_Math" display="inline" id="S7.SS4.SSS0.P0.SPx2.p3.1.m1.2"><semantics id="S7.SS4.SSS0.P0.SPx2.p3.1.m1.2a"><mrow id="S7.SS4.SSS0.P0.SPx2.p3.1.m1.2.3.2" xref="S7.SS4.SSS0.P0.SPx2.p3.1.m1.2.3.1.cmml"><mo id="S7.SS4.SSS0.P0.SPx2.p3.1.m1.2.3.2.1" stretchy="false" xref="S7.SS4.SSS0.P0.SPx2.p3.1.m1.2.3.1.cmml">(</mo><mi id="S7.SS4.SSS0.P0.SPx2.p3.1.m1.1.1" xref="S7.SS4.SSS0.P0.SPx2.p3.1.m1.1.1.cmml">h</mi><mo id="S7.SS4.SSS0.P0.SPx2.p3.1.m1.2.3.2.2" xref="S7.SS4.SSS0.P0.SPx2.p3.1.m1.2.3.1.cmml">,</mo><mi id="S7.SS4.SSS0.P0.SPx2.p3.1.m1.2.2" xref="S7.SS4.SSS0.P0.SPx2.p3.1.m1.2.2.cmml">n</mi><mo id="S7.SS4.SSS0.P0.SPx2.p3.1.m1.2.3.2.3" stretchy="false" xref="S7.SS4.SSS0.P0.SPx2.p3.1.m1.2.3.1.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S7.SS4.SSS0.P0.SPx2.p3.1.m1.2b"><interval closure="open" id="S7.SS4.SSS0.P0.SPx2.p3.1.m1.2.3.1.cmml" xref="S7.SS4.SSS0.P0.SPx2.p3.1.m1.2.3.2"><ci id="S7.SS4.SSS0.P0.SPx2.p3.1.m1.1.1.cmml" xref="S7.SS4.SSS0.P0.SPx2.p3.1.m1.1.1">ℎ</ci><ci id="S7.SS4.SSS0.P0.SPx2.p3.1.m1.2.2.cmml" xref="S7.SS4.SSS0.P0.SPx2.p3.1.m1.2.2">𝑛</ci></interval></annotation-xml><annotation encoding="application/x-tex" id="S7.SS4.SSS0.P0.SPx2.p3.1.m1.2c">(h,n)</annotation><annotation encoding="application/x-llamapun" id="S7.SS4.SSS0.P0.SPx2.p3.1.m1.2d">( italic_h , italic_n )</annotation></semantics></math> of Lemma <a class="ltx_ref" href="https://arxiv.org/html/2411.12694v2#S7.Thmtheorem3" title="Lemma 7.3 (Lemma 7.2 and 9.1 in [15]). ‣ 7 Results in CONGEST ‣ Local Density and its Distributed Approximation"><span class="ltx_text ltx_ref_tag">7.3</span></a> by Haeupler, Hershkowitz, and Saranurak <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2411.12694v2#bib.bib15" title="">15</a>]</cite>:</p> </div> <div class="ltx_para" id="S7.SS4.SSS0.P0.SPx2.p4"> <p class="ltx_p" id="S7.SS4.SSS0.P0.SPx2.p4.1"><span class="ltx_text" id="S7.SS4.SSS0.P0.SPx2.p4.1.1">See <a class="ltx_ref" href="https://arxiv.org/html/2411.12694v2#S7.Thmtheorem12" title="Corollary 7.12. ‣ 7.3 Sketching our algorithm’s correctness. ‣ 7 Results in CONGEST ‣ Local Density and its Distributed Approximation"><span class="ltx_text ltx_ref_tag">7.12</span></a></span></p> </div> <div class="ltx_pagination ltx_role_newpage"></div> </section> </section> </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.bib1"> <span class="ltx_tag ltx_tag_bibitem">[1]</span> <span class="ltx_bibblock"> Bahman Bahmani, Ravi Kumar, and Sergei Vassilvitskii. </span> <span class="ltx_bibblock">Densest subgraph in streaming and mapreduce. </span> <span class="ltx_bibblock"><span class="ltx_text ltx_font_italic" id="bib.bib1.1.1">Proc. 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International World Wide Web Conferences Steering Committee. </span> </li> </ul> </section> <div class="ltx_pagination ltx_role_newpage"></div> <section class="ltx_appendix" id="A1"> <h2 class="ltx_title ltx_title_appendix"> <span class="ltx_tag ltx_tag_appendix">Appendix A </span>Reporting a densest subgraph</h2> <div class="ltx_para" id="A1.p1"> <p class="ltx_p" id="A1.p1.1">Problems <a class="ltx_ref" href="https://arxiv.org/html/2411.12694v2#S2.Thmtheorem11" title="Problem 2.11. ‣ The benefit of local measures: ‣ 2.3 Local density ‣ 2 Preliminaries and related work ‣ Local Density and its Distributed Approximation"><span class="ltx_text ltx_ref_tag">2.11</span></a> and <a class="ltx_ref" href="https://arxiv.org/html/2411.12694v2#S2.Thmtheorem11" title="Problem 2.11. ‣ The benefit of local measures: ‣ 2.3 Local density ‣ 2 Preliminaries and related work ‣ Local Density and its Distributed Approximation"><span class="ltx_text ltx_ref_tag">2.11</span></a> are what we call the <em class="ltx_emph ltx_font_italic" id="A1.p1.1.1">value variant</em> of the subgraph density problem, where the goal is to output a value. The value variant has a natural alternative, where the goal is to actually report a densest subgraph.</p> </div> <section class="ltx_subsection" id="A1.SS1"> <h3 class="ltx_title ltx_title_subsection"> <span class="ltx_tag ltx_tag_subsection">A.1 </span>Related work: distributed densest subgraph reporting</h3> <div class="ltx_para" id="A1.SS1.p1"> <p class="ltx_p" id="A1.SS1.p1.1">The reporting variant of the subgraph density problem, has two subvariants in the distributed model of computation. Here, the second subvariant is by Harris <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2411.12694v2#bib.bib16" title="">16</a>]</cite>.</p> </div> <div class="ltx_theorem ltx_theorem_problem" id="A1.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="A1.Thmtheorem1.1.1.1">Problem A.1</span></span><span class="ltx_text ltx_font_bold" id="A1.Thmtheorem1.2.2">.</span> </h6> <div class="ltx_para" id="A1.Thmtheorem1.p1"> <p class="ltx_p" id="A1.Thmtheorem1.p1.4"><span class="ltx_text ltx_font_italic" id="A1.Thmtheorem1.p1.4.4">Given a graph <math alttext="G" class="ltx_Math" display="inline" id="A1.Thmtheorem1.p1.1.1.m1.1"><semantics id="A1.Thmtheorem1.p1.1.1.m1.1a"><mi id="A1.Thmtheorem1.p1.1.1.m1.1.1" xref="A1.Thmtheorem1.p1.1.1.m1.1.1.cmml">G</mi><annotation-xml encoding="MathML-Content" id="A1.Thmtheorem1.p1.1.1.m1.1b"><ci id="A1.Thmtheorem1.p1.1.1.m1.1.1.cmml" xref="A1.Thmtheorem1.p1.1.1.m1.1.1">𝐺</ci></annotation-xml><annotation encoding="application/x-tex" id="A1.Thmtheorem1.p1.1.1.m1.1c">G</annotation><annotation encoding="application/x-llamapun" id="A1.Thmtheorem1.p1.1.1.m1.1d">italic_G</annotation></semantics></math>, after a computation, each vertex outputs a bit <math alttext="b_{v}" class="ltx_Math" display="inline" id="A1.Thmtheorem1.p1.2.2.m2.1"><semantics id="A1.Thmtheorem1.p1.2.2.m2.1a"><msub id="A1.Thmtheorem1.p1.2.2.m2.1.1" xref="A1.Thmtheorem1.p1.2.2.m2.1.1.cmml"><mi id="A1.Thmtheorem1.p1.2.2.m2.1.1.2" xref="A1.Thmtheorem1.p1.2.2.m2.1.1.2.cmml">b</mi><mi id="A1.Thmtheorem1.p1.2.2.m2.1.1.3" xref="A1.Thmtheorem1.p1.2.2.m2.1.1.3.cmml">v</mi></msub><annotation-xml encoding="MathML-Content" id="A1.Thmtheorem1.p1.2.2.m2.1b"><apply id="A1.Thmtheorem1.p1.2.2.m2.1.1.cmml" xref="A1.Thmtheorem1.p1.2.2.m2.1.1"><csymbol cd="ambiguous" id="A1.Thmtheorem1.p1.2.2.m2.1.1.1.cmml" xref="A1.Thmtheorem1.p1.2.2.m2.1.1">subscript</csymbol><ci id="A1.Thmtheorem1.p1.2.2.m2.1.1.2.cmml" xref="A1.Thmtheorem1.p1.2.2.m2.1.1.2">𝑏</ci><ci id="A1.Thmtheorem1.p1.2.2.m2.1.1.3.cmml" xref="A1.Thmtheorem1.p1.2.2.m2.1.1.3">𝑣</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.Thmtheorem1.p1.2.2.m2.1c">b_{v}</annotation><annotation encoding="application/x-llamapun" id="A1.Thmtheorem1.p1.2.2.m2.1d">italic_b start_POSTSUBSCRIPT italic_v end_POSTSUBSCRIPT</annotation></semantics></math>. Denote by <math alttext="H" class="ltx_Math" display="inline" id="A1.Thmtheorem1.p1.3.3.m3.1"><semantics id="A1.Thmtheorem1.p1.3.3.m3.1a"><mi id="A1.Thmtheorem1.p1.3.3.m3.1.1" xref="A1.Thmtheorem1.p1.3.3.m3.1.1.cmml">H</mi><annotation-xml encoding="MathML-Content" id="A1.Thmtheorem1.p1.3.3.m3.1b"><ci id="A1.Thmtheorem1.p1.3.3.m3.1.1.cmml" xref="A1.Thmtheorem1.p1.3.3.m3.1.1">𝐻</ci></annotation-xml><annotation encoding="application/x-tex" id="A1.Thmtheorem1.p1.3.3.m3.1c">H</annotation><annotation encoding="application/x-llamapun" id="A1.Thmtheorem1.p1.3.3.m3.1d">italic_H</annotation></semantics></math> the subgraph induced by all vertices with a <math alttext="1" class="ltx_Math" display="inline" id="A1.Thmtheorem1.p1.4.4.m4.1"><semantics id="A1.Thmtheorem1.p1.4.4.m4.1a"><mn id="A1.Thmtheorem1.p1.4.4.m4.1.1" xref="A1.Thmtheorem1.p1.4.4.m4.1.1.cmml">1</mn><annotation-xml encoding="MathML-Content" id="A1.Thmtheorem1.p1.4.4.m4.1b"><cn id="A1.Thmtheorem1.p1.4.4.m4.1.1.cmml" type="integer" xref="A1.Thmtheorem1.p1.4.4.m4.1.1">1</cn></annotation-xml><annotation encoding="application/x-tex" id="A1.Thmtheorem1.p1.4.4.m4.1c">1</annotation><annotation encoding="application/x-llamapun" id="A1.Thmtheorem1.p1.4.4.m4.1d">1</annotation></semantics></math>-bit. We consider two variants where either:</span></p> <ul class="ltx_itemize" id="A1.I1"> <li class="ltx_item" id="A1.I1.i1" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">•</span> <div class="ltx_para" id="A1.I1.i1.p1"> <p class="ltx_p" id="A1.I1.i1.p1.1"><span class="ltx_text ltx_font_bold ltx_font_italic" id="A1.I1.i1.p1.1.1">Problem 4.1</span><span class="ltx_text ltx_font_italic" id="A1.I1.i1.p1.1.2">: We require that </span><math alttext="\rho(H)\leq\rho^{\max}(G)\leq(1+\varepsilon)\rho(H)" class="ltx_Math" display="inline" id="A1.I1.i1.p1.1.m1.4"><semantics id="A1.I1.i1.p1.1.m1.4a"><mrow id="A1.I1.i1.p1.1.m1.4.4" xref="A1.I1.i1.p1.1.m1.4.4.cmml"><mrow id="A1.I1.i1.p1.1.m1.4.4.3" xref="A1.I1.i1.p1.1.m1.4.4.3.cmml"><mi id="A1.I1.i1.p1.1.m1.4.4.3.2" xref="A1.I1.i1.p1.1.m1.4.4.3.2.cmml">ρ</mi><mo id="A1.I1.i1.p1.1.m1.4.4.3.1" xref="A1.I1.i1.p1.1.m1.4.4.3.1.cmml">⁢</mo><mrow id="A1.I1.i1.p1.1.m1.4.4.3.3.2" xref="A1.I1.i1.p1.1.m1.4.4.3.cmml"><mo 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id="A1.I1.i1.p1.1.m1.4.4.5.3.2.2" stretchy="false" xref="A1.I1.i1.p1.1.m1.4.4.5.cmml">)</mo></mrow></mrow><mo id="A1.I1.i1.p1.1.m1.4.4.6" xref="A1.I1.i1.p1.1.m1.4.4.6.cmml">≤</mo><mrow id="A1.I1.i1.p1.1.m1.4.4.1" xref="A1.I1.i1.p1.1.m1.4.4.1.cmml"><mrow id="A1.I1.i1.p1.1.m1.4.4.1.1.1" xref="A1.I1.i1.p1.1.m1.4.4.1.1.1.1.cmml"><mo id="A1.I1.i1.p1.1.m1.4.4.1.1.1.2" stretchy="false" xref="A1.I1.i1.p1.1.m1.4.4.1.1.1.1.cmml">(</mo><mrow id="A1.I1.i1.p1.1.m1.4.4.1.1.1.1" xref="A1.I1.i1.p1.1.m1.4.4.1.1.1.1.cmml"><mn id="A1.I1.i1.p1.1.m1.4.4.1.1.1.1.2" xref="A1.I1.i1.p1.1.m1.4.4.1.1.1.1.2.cmml">1</mn><mo id="A1.I1.i1.p1.1.m1.4.4.1.1.1.1.1" xref="A1.I1.i1.p1.1.m1.4.4.1.1.1.1.1.cmml">+</mo><mi id="A1.I1.i1.p1.1.m1.4.4.1.1.1.1.3" xref="A1.I1.i1.p1.1.m1.4.4.1.1.1.1.3.cmml">ε</mi></mrow><mo id="A1.I1.i1.p1.1.m1.4.4.1.1.1.3" stretchy="false" xref="A1.I1.i1.p1.1.m1.4.4.1.1.1.1.cmml">)</mo></mrow><mo id="A1.I1.i1.p1.1.m1.4.4.1.2" xref="A1.I1.i1.p1.1.m1.4.4.1.2.cmml">⁢</mo><mi 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href="https://arxiv.org/html/2411.12694v2#A1.I1.i1.p1.1.m1.4.4.5.cmml" id="A1.I1.i1.p1.1.m1.4.4d.cmml" xref="A1.I1.i1.p1.1.m1.4.4"></share><apply id="A1.I1.i1.p1.1.m1.4.4.1.cmml" xref="A1.I1.i1.p1.1.m1.4.4.1"><times id="A1.I1.i1.p1.1.m1.4.4.1.2.cmml" xref="A1.I1.i1.p1.1.m1.4.4.1.2"></times><apply id="A1.I1.i1.p1.1.m1.4.4.1.1.1.1.cmml" xref="A1.I1.i1.p1.1.m1.4.4.1.1.1"><plus id="A1.I1.i1.p1.1.m1.4.4.1.1.1.1.1.cmml" xref="A1.I1.i1.p1.1.m1.4.4.1.1.1.1.1"></plus><cn id="A1.I1.i1.p1.1.m1.4.4.1.1.1.1.2.cmml" type="integer" xref="A1.I1.i1.p1.1.m1.4.4.1.1.1.1.2">1</cn><ci id="A1.I1.i1.p1.1.m1.4.4.1.1.1.1.3.cmml" xref="A1.I1.i1.p1.1.m1.4.4.1.1.1.1.3">𝜀</ci></apply><ci id="A1.I1.i1.p1.1.m1.4.4.1.3.cmml" xref="A1.I1.i1.p1.1.m1.4.4.1.3">𝜌</ci><ci id="A1.I1.i1.p1.1.m1.3.3.cmml" xref="A1.I1.i1.p1.1.m1.3.3">𝐻</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.I1.i1.p1.1.m1.4c">\rho(H)\leq\rho^{\max}(G)\leq(1+\varepsilon)\rho(H)</annotation><annotation encoding="application/x-llamapun" id="A1.I1.i1.p1.1.m1.4d">italic_ρ ( italic_H ) ≤ italic_ρ start_POSTSUPERSCRIPT roman_max end_POSTSUPERSCRIPT ( italic_G ) ≤ ( 1 + italic_ε ) italic_ρ ( italic_H )</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="A1.I1.i1.p1.1.3">, or</span></p> </div> </li> <li class="ltx_item" id="A1.I1.i2" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">•</span> <div class="ltx_para" id="A1.I1.i2.p1"> <p class="ltx_p" id="A1.I1.i2.p1.5"><span class="ltx_text ltx_font_bold ltx_font_italic" id="A1.I1.i2.p1.5.1">Problem 4.2</span><span class="ltx_text ltx_font_italic" id="A1.I1.i2.p1.5.2">: Given an oracle that before outputting the bit, gives each vertex a value </span><math alttext="\tilde{D}\leq\rho^{\max}(G)" class="ltx_Math" display="inline" id="A1.I1.i2.p1.1.m1.1"><semantics id="A1.I1.i2.p1.1.m1.1a"><mrow id="A1.I1.i2.p1.1.m1.1.2" xref="A1.I1.i2.p1.1.m1.1.2.cmml"><mover accent="true" id="A1.I1.i2.p1.1.m1.1.2.2" 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xref="A1.I1.i2.p1.1.m1.1.2.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="A1.I1.i2.p1.1.m1.1b"><apply id="A1.I1.i2.p1.1.m1.1.2.cmml" xref="A1.I1.i2.p1.1.m1.1.2"><leq id="A1.I1.i2.p1.1.m1.1.2.1.cmml" xref="A1.I1.i2.p1.1.m1.1.2.1"></leq><apply id="A1.I1.i2.p1.1.m1.1.2.2.cmml" xref="A1.I1.i2.p1.1.m1.1.2.2"><ci id="A1.I1.i2.p1.1.m1.1.2.2.1.cmml" xref="A1.I1.i2.p1.1.m1.1.2.2.1">~</ci><ci id="A1.I1.i2.p1.1.m1.1.2.2.2.cmml" xref="A1.I1.i2.p1.1.m1.1.2.2.2">𝐷</ci></apply><apply id="A1.I1.i2.p1.1.m1.1.2.3.cmml" xref="A1.I1.i2.p1.1.m1.1.2.3"><times id="A1.I1.i2.p1.1.m1.1.2.3.1.cmml" xref="A1.I1.i2.p1.1.m1.1.2.3.1"></times><apply id="A1.I1.i2.p1.1.m1.1.2.3.2.cmml" xref="A1.I1.i2.p1.1.m1.1.2.3.2"><csymbol cd="ambiguous" id="A1.I1.i2.p1.1.m1.1.2.3.2.1.cmml" xref="A1.I1.i2.p1.1.m1.1.2.3.2">superscript</csymbol><ci id="A1.I1.i2.p1.1.m1.1.2.3.2.2.cmml" xref="A1.I1.i2.p1.1.m1.1.2.3.2.2">𝜌</ci><max id="A1.I1.i2.p1.1.m1.1.2.3.2.3.cmml" xref="A1.I1.i2.p1.1.m1.1.2.3.2.3"></max></apply><ci id="A1.I1.i2.p1.1.m1.1.1.cmml" xref="A1.I1.i2.p1.1.m1.1.1">𝐺</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.I1.i2.p1.1.m1.1c">\tilde{D}\leq\rho^{\max}(G)</annotation><annotation encoding="application/x-llamapun" id="A1.I1.i2.p1.1.m1.1d">over~ start_ARG italic_D end_ARG ≤ italic_ρ start_POSTSUPERSCRIPT roman_max end_POSTSUPERSCRIPT ( italic_G )</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="A1.I1.i2.p1.5.3">. Each vertex outputs a bit </span><math alttext="b_{v}" class="ltx_Math" display="inline" id="A1.I1.i2.p1.2.m2.1"><semantics id="A1.I1.i2.p1.2.m2.1a"><msub id="A1.I1.i2.p1.2.m2.1.1" xref="A1.I1.i2.p1.2.m2.1.1.cmml"><mi id="A1.I1.i2.p1.2.m2.1.1.2" xref="A1.I1.i2.p1.2.m2.1.1.2.cmml">b</mi><mi id="A1.I1.i2.p1.2.m2.1.1.3" xref="A1.I1.i2.p1.2.m2.1.1.3.cmml">v</mi></msub><annotation-xml encoding="MathML-Content" id="A1.I1.i2.p1.2.m2.1b"><apply id="A1.I1.i2.p1.2.m2.1.1.cmml" xref="A1.I1.i2.p1.2.m2.1.1"><csymbol cd="ambiguous" id="A1.I1.i2.p1.2.m2.1.1.1.cmml" xref="A1.I1.i2.p1.2.m2.1.1">subscript</csymbol><ci id="A1.I1.i2.p1.2.m2.1.1.2.cmml" xref="A1.I1.i2.p1.2.m2.1.1.2">𝑏</ci><ci id="A1.I1.i2.p1.2.m2.1.1.3.cmml" xref="A1.I1.i2.p1.2.m2.1.1.3">𝑣</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.I1.i2.p1.2.m2.1c">b_{v}</annotation><annotation encoding="application/x-llamapun" id="A1.I1.i2.p1.2.m2.1d">italic_b start_POSTSUBSCRIPT italic_v end_POSTSUBSCRIPT</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="A1.I1.i2.p1.5.4"> and </span><math alttext="\rho(H)\geq(1-\varepsilon)\tilde{D}" class="ltx_Math" display="inline" id="A1.I1.i2.p1.3.m3.2"><semantics id="A1.I1.i2.p1.3.m3.2a"><mrow id="A1.I1.i2.p1.3.m3.2.2" xref="A1.I1.i2.p1.3.m3.2.2.cmml"><mrow id="A1.I1.i2.p1.3.m3.2.2.3" xref="A1.I1.i2.p1.3.m3.2.2.3.cmml"><mi id="A1.I1.i2.p1.3.m3.2.2.3.2" xref="A1.I1.i2.p1.3.m3.2.2.3.2.cmml">ρ</mi><mo id="A1.I1.i2.p1.3.m3.2.2.3.1" xref="A1.I1.i2.p1.3.m3.2.2.3.1.cmml">⁢</mo><mrow id="A1.I1.i2.p1.3.m3.2.2.3.3.2" xref="A1.I1.i2.p1.3.m3.2.2.3.cmml"><mo id="A1.I1.i2.p1.3.m3.2.2.3.3.2.1" stretchy="false" xref="A1.I1.i2.p1.3.m3.2.2.3.cmml">(</mo><mi id="A1.I1.i2.p1.3.m3.1.1" xref="A1.I1.i2.p1.3.m3.1.1.cmml">H</mi><mo id="A1.I1.i2.p1.3.m3.2.2.3.3.2.2" stretchy="false" xref="A1.I1.i2.p1.3.m3.2.2.3.cmml">)</mo></mrow></mrow><mo id="A1.I1.i2.p1.3.m3.2.2.2" xref="A1.I1.i2.p1.3.m3.2.2.2.cmml">≥</mo><mrow id="A1.I1.i2.p1.3.m3.2.2.1" xref="A1.I1.i2.p1.3.m3.2.2.1.cmml"><mrow id="A1.I1.i2.p1.3.m3.2.2.1.1.1" xref="A1.I1.i2.p1.3.m3.2.2.1.1.1.1.cmml"><mo id="A1.I1.i2.p1.3.m3.2.2.1.1.1.2" stretchy="false" xref="A1.I1.i2.p1.3.m3.2.2.1.1.1.1.cmml">(</mo><mrow id="A1.I1.i2.p1.3.m3.2.2.1.1.1.1" xref="A1.I1.i2.p1.3.m3.2.2.1.1.1.1.cmml"><mn id="A1.I1.i2.p1.3.m3.2.2.1.1.1.1.2" xref="A1.I1.i2.p1.3.m3.2.2.1.1.1.1.2.cmml">1</mn><mo id="A1.I1.i2.p1.3.m3.2.2.1.1.1.1.1" xref="A1.I1.i2.p1.3.m3.2.2.1.1.1.1.1.cmml">−</mo><mi id="A1.I1.i2.p1.3.m3.2.2.1.1.1.1.3" xref="A1.I1.i2.p1.3.m3.2.2.1.1.1.1.3.cmml">ε</mi></mrow><mo id="A1.I1.i2.p1.3.m3.2.2.1.1.1.3" stretchy="false" xref="A1.I1.i2.p1.3.m3.2.2.1.1.1.1.cmml">)</mo></mrow><mo id="A1.I1.i2.p1.3.m3.2.2.1.2" xref="A1.I1.i2.p1.3.m3.2.2.1.2.cmml">⁢</mo><mover accent="true" id="A1.I1.i2.p1.3.m3.2.2.1.3" xref="A1.I1.i2.p1.3.m3.2.2.1.3.cmml"><mi id="A1.I1.i2.p1.3.m3.2.2.1.3.2" xref="A1.I1.i2.p1.3.m3.2.2.1.3.2.cmml">D</mi><mo id="A1.I1.i2.p1.3.m3.2.2.1.3.1" xref="A1.I1.i2.p1.3.m3.2.2.1.3.1.cmml">~</mo></mover></mrow></mrow><annotation-xml encoding="MathML-Content" id="A1.I1.i2.p1.3.m3.2b"><apply id="A1.I1.i2.p1.3.m3.2.2.cmml" xref="A1.I1.i2.p1.3.m3.2.2"><geq id="A1.I1.i2.p1.3.m3.2.2.2.cmml" xref="A1.I1.i2.p1.3.m3.2.2.2"></geq><apply id="A1.I1.i2.p1.3.m3.2.2.3.cmml" xref="A1.I1.i2.p1.3.m3.2.2.3"><times id="A1.I1.i2.p1.3.m3.2.2.3.1.cmml" xref="A1.I1.i2.p1.3.m3.2.2.3.1"></times><ci id="A1.I1.i2.p1.3.m3.2.2.3.2.cmml" xref="A1.I1.i2.p1.3.m3.2.2.3.2">𝜌</ci><ci id="A1.I1.i2.p1.3.m3.1.1.cmml" xref="A1.I1.i2.p1.3.m3.1.1">𝐻</ci></apply><apply id="A1.I1.i2.p1.3.m3.2.2.1.cmml" xref="A1.I1.i2.p1.3.m3.2.2.1"><times id="A1.I1.i2.p1.3.m3.2.2.1.2.cmml" xref="A1.I1.i2.p1.3.m3.2.2.1.2"></times><apply id="A1.I1.i2.p1.3.m3.2.2.1.1.1.1.cmml" xref="A1.I1.i2.p1.3.m3.2.2.1.1.1"><minus id="A1.I1.i2.p1.3.m3.2.2.1.1.1.1.1.cmml" xref="A1.I1.i2.p1.3.m3.2.2.1.1.1.1.1"></minus><cn id="A1.I1.i2.p1.3.m3.2.2.1.1.1.1.2.cmml" type="integer" xref="A1.I1.i2.p1.3.m3.2.2.1.1.1.1.2">1</cn><ci id="A1.I1.i2.p1.3.m3.2.2.1.1.1.1.3.cmml" xref="A1.I1.i2.p1.3.m3.2.2.1.1.1.1.3">𝜀</ci></apply><apply id="A1.I1.i2.p1.3.m3.2.2.1.3.cmml" xref="A1.I1.i2.p1.3.m3.2.2.1.3"><ci id="A1.I1.i2.p1.3.m3.2.2.1.3.1.cmml" xref="A1.I1.i2.p1.3.m3.2.2.1.3.1">~</ci><ci id="A1.I1.i2.p1.3.m3.2.2.1.3.2.cmml" xref="A1.I1.i2.p1.3.m3.2.2.1.3.2">𝐷</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.I1.i2.p1.3.m3.2c">\rho(H)\geq(1-\varepsilon)\tilde{D}</annotation><annotation encoding="application/x-llamapun" id="A1.I1.i2.p1.3.m3.2d">italic_ρ ( italic_H ) ≥ ( 1 - italic_ε ) over~ start_ARG italic_D end_ARG</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="A1.I1.i2.p1.5.5">. If </span><math alttext="\tilde{D}>\rho^{\max}(G)" class="ltx_Math" display="inline" id="A1.I1.i2.p1.4.m4.1"><semantics id="A1.I1.i2.p1.4.m4.1a"><mrow id="A1.I1.i2.p1.4.m4.1.2" xref="A1.I1.i2.p1.4.m4.1.2.cmml"><mover accent="true" id="A1.I1.i2.p1.4.m4.1.2.2" xref="A1.I1.i2.p1.4.m4.1.2.2.cmml"><mi id="A1.I1.i2.p1.4.m4.1.2.2.2" xref="A1.I1.i2.p1.4.m4.1.2.2.2.cmml">D</mi><mo id="A1.I1.i2.p1.4.m4.1.2.2.1" xref="A1.I1.i2.p1.4.m4.1.2.2.1.cmml">~</mo></mover><mo id="A1.I1.i2.p1.4.m4.1.2.1" xref="A1.I1.i2.p1.4.m4.1.2.1.cmml">></mo><mrow id="A1.I1.i2.p1.4.m4.1.2.3" xref="A1.I1.i2.p1.4.m4.1.2.3.cmml"><msup id="A1.I1.i2.p1.4.m4.1.2.3.2" xref="A1.I1.i2.p1.4.m4.1.2.3.2.cmml"><mi id="A1.I1.i2.p1.4.m4.1.2.3.2.2" xref="A1.I1.i2.p1.4.m4.1.2.3.2.2.cmml">ρ</mi><mi id="A1.I1.i2.p1.4.m4.1.2.3.2.3" xref="A1.I1.i2.p1.4.m4.1.2.3.2.3.cmml">max</mi></msup><mo id="A1.I1.i2.p1.4.m4.1.2.3.1" xref="A1.I1.i2.p1.4.m4.1.2.3.1.cmml">⁢</mo><mrow id="A1.I1.i2.p1.4.m4.1.2.3.3.2" xref="A1.I1.i2.p1.4.m4.1.2.3.cmml"><mo id="A1.I1.i2.p1.4.m4.1.2.3.3.2.1" stretchy="false" xref="A1.I1.i2.p1.4.m4.1.2.3.cmml">(</mo><mi id="A1.I1.i2.p1.4.m4.1.1" xref="A1.I1.i2.p1.4.m4.1.1.cmml">G</mi><mo id="A1.I1.i2.p1.4.m4.1.2.3.3.2.2" stretchy="false" xref="A1.I1.i2.p1.4.m4.1.2.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="A1.I1.i2.p1.4.m4.1b"><apply id="A1.I1.i2.p1.4.m4.1.2.cmml" xref="A1.I1.i2.p1.4.m4.1.2"><gt id="A1.I1.i2.p1.4.m4.1.2.1.cmml" xref="A1.I1.i2.p1.4.m4.1.2.1"></gt><apply id="A1.I1.i2.p1.4.m4.1.2.2.cmml" xref="A1.I1.i2.p1.4.m4.1.2.2"><ci id="A1.I1.i2.p1.4.m4.1.2.2.1.cmml" xref="A1.I1.i2.p1.4.m4.1.2.2.1">~</ci><ci id="A1.I1.i2.p1.4.m4.1.2.2.2.cmml" xref="A1.I1.i2.p1.4.m4.1.2.2.2">𝐷</ci></apply><apply id="A1.I1.i2.p1.4.m4.1.2.3.cmml" xref="A1.I1.i2.p1.4.m4.1.2.3"><times id="A1.I1.i2.p1.4.m4.1.2.3.1.cmml" xref="A1.I1.i2.p1.4.m4.1.2.3.1"></times><apply id="A1.I1.i2.p1.4.m4.1.2.3.2.cmml" xref="A1.I1.i2.p1.4.m4.1.2.3.2"><csymbol cd="ambiguous" id="A1.I1.i2.p1.4.m4.1.2.3.2.1.cmml" xref="A1.I1.i2.p1.4.m4.1.2.3.2">superscript</csymbol><ci id="A1.I1.i2.p1.4.m4.1.2.3.2.2.cmml" xref="A1.I1.i2.p1.4.m4.1.2.3.2.2">𝜌</ci><max id="A1.I1.i2.p1.4.m4.1.2.3.2.3.cmml" xref="A1.I1.i2.p1.4.m4.1.2.3.2.3"></max></apply><ci id="A1.I1.i2.p1.4.m4.1.1.cmml" xref="A1.I1.i2.p1.4.m4.1.1">𝐺</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.I1.i2.p1.4.m4.1c">\tilde{D}>\rho^{\max}(G)</annotation><annotation encoding="application/x-llamapun" id="A1.I1.i2.p1.4.m4.1d">over~ start_ARG italic_D end_ARG > italic_ρ start_POSTSUPERSCRIPT roman_max end_POSTSUPERSCRIPT ( italic_G )</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="A1.I1.i2.p1.5.6"> then every vertex may output </span><math alttext="0" class="ltx_Math" display="inline" id="A1.I1.i2.p1.5.m5.1"><semantics id="A1.I1.i2.p1.5.m5.1a"><mn id="A1.I1.i2.p1.5.m5.1.1" xref="A1.I1.i2.p1.5.m5.1.1.cmml">0</mn><annotation-xml encoding="MathML-Content" id="A1.I1.i2.p1.5.m5.1b"><cn id="A1.I1.i2.p1.5.m5.1.1.cmml" type="integer" xref="A1.I1.i2.p1.5.m5.1.1">0</cn></annotation-xml></semantics></math><span class="ltx_text ltx_font_italic" id="A1.I1.i2.p1.5.7">.</span></p> </div> </li> </ul> </div> </div> <div class="ltx_para" id="A1.SS1.p2"> <p class="ltx_p" id="A1.SS1.p2.3">The reporting subvariants are considerably harder than the value subvariants. For Problem <a class="ltx_ref" href="https://arxiv.org/html/2411.12694v2#A1.Thmtheorem1" title="Problem A.1. ‣ A.1 Related work: distributed densest subgraph reporting ‣ Appendix A Reporting a densest subgraph ‣ Local Density and its Distributed Approximation"><span class="ltx_text ltx_ref_tag">A.1</span></a>.1, there is a trivial lower bound of <math alttext="\Omega(d)" class="ltx_Math" display="inline" id="A1.SS1.p2.1.m1.1"><semantics id="A1.SS1.p2.1.m1.1a"><mrow id="A1.SS1.p2.1.m1.1.2" xref="A1.SS1.p2.1.m1.1.2.cmml"><mi id="A1.SS1.p2.1.m1.1.2.2" mathvariant="normal" xref="A1.SS1.p2.1.m1.1.2.2.cmml">Ω</mi><mo id="A1.SS1.p2.1.m1.1.2.1" xref="A1.SS1.p2.1.m1.1.2.1.cmml">⁢</mo><mrow id="A1.SS1.p2.1.m1.1.2.3.2" xref="A1.SS1.p2.1.m1.1.2.cmml"><mo id="A1.SS1.p2.1.m1.1.2.3.2.1" stretchy="false" xref="A1.SS1.p2.1.m1.1.2.cmml">(</mo><mi id="A1.SS1.p2.1.m1.1.1" xref="A1.SS1.p2.1.m1.1.1.cmml">d</mi><mo id="A1.SS1.p2.1.m1.1.2.3.2.2" stretchy="false" xref="A1.SS1.p2.1.m1.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="A1.SS1.p2.1.m1.1b"><apply id="A1.SS1.p2.1.m1.1.2.cmml" xref="A1.SS1.p2.1.m1.1.2"><times id="A1.SS1.p2.1.m1.1.2.1.cmml" xref="A1.SS1.p2.1.m1.1.2.1"></times><ci id="A1.SS1.p2.1.m1.1.2.2.cmml" xref="A1.SS1.p2.1.m1.1.2.2">Ω</ci><ci id="A1.SS1.p2.1.m1.1.1.cmml" xref="A1.SS1.p2.1.m1.1.1">𝑑</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS1.p2.1.m1.1c">\Omega(d)</annotation><annotation encoding="application/x-llamapun" id="A1.SS1.p2.1.m1.1d">roman_Ω ( italic_d )</annotation></semantics></math>. In LOCAL, the problem is trivial to solve in <math alttext="O(d)" class="ltx_Math" display="inline" id="A1.SS1.p2.2.m2.1"><semantics id="A1.SS1.p2.2.m2.1a"><mrow id="A1.SS1.p2.2.m2.1.2" xref="A1.SS1.p2.2.m2.1.2.cmml"><mi id="A1.SS1.p2.2.m2.1.2.2" xref="A1.SS1.p2.2.m2.1.2.2.cmml">O</mi><mo id="A1.SS1.p2.2.m2.1.2.1" xref="A1.SS1.p2.2.m2.1.2.1.cmml">⁢</mo><mrow id="A1.SS1.p2.2.m2.1.2.3.2" xref="A1.SS1.p2.2.m2.1.2.cmml"><mo id="A1.SS1.p2.2.m2.1.2.3.2.1" stretchy="false" xref="A1.SS1.p2.2.m2.1.2.cmml">(</mo><mi id="A1.SS1.p2.2.m2.1.1" xref="A1.SS1.p2.2.m2.1.1.cmml">d</mi><mo id="A1.SS1.p2.2.m2.1.2.3.2.2" stretchy="false" xref="A1.SS1.p2.2.m2.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="A1.SS1.p2.2.m2.1b"><apply id="A1.SS1.p2.2.m2.1.2.cmml" xref="A1.SS1.p2.2.m2.1.2"><times id="A1.SS1.p2.2.m2.1.2.1.cmml" xref="A1.SS1.p2.2.m2.1.2.1"></times><ci id="A1.SS1.p2.2.m2.1.2.2.cmml" xref="A1.SS1.p2.2.m2.1.2.2">𝑂</ci><ci id="A1.SS1.p2.2.m2.1.1.cmml" xref="A1.SS1.p2.2.m2.1.1">𝑑</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS1.p2.2.m2.1c">O(d)</annotation><annotation encoding="application/x-llamapun" id="A1.SS1.p2.2.m2.1d">italic_O ( italic_d )</annotation></semantics></math> time. Su and Vu <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2411.12694v2#bib.bib21" title="">21</a>]</cite> can solve this variant in <math alttext="O(\varepsilon^{-1}\log n)" class="ltx_Math" display="inline" id="A1.SS1.p2.3.m3.1"><semantics id="A1.SS1.p2.3.m3.1a"><mrow id="A1.SS1.p2.3.m3.1.1" xref="A1.SS1.p2.3.m3.1.1.cmml"><mi id="A1.SS1.p2.3.m3.1.1.3" xref="A1.SS1.p2.3.m3.1.1.3.cmml">O</mi><mo id="A1.SS1.p2.3.m3.1.1.2" xref="A1.SS1.p2.3.m3.1.1.2.cmml">⁢</mo><mrow id="A1.SS1.p2.3.m3.1.1.1.1" xref="A1.SS1.p2.3.m3.1.1.1.1.1.cmml"><mo id="A1.SS1.p2.3.m3.1.1.1.1.2" stretchy="false" xref="A1.SS1.p2.3.m3.1.1.1.1.1.cmml">(</mo><mrow id="A1.SS1.p2.3.m3.1.1.1.1.1" xref="A1.SS1.p2.3.m3.1.1.1.1.1.cmml"><msup id="A1.SS1.p2.3.m3.1.1.1.1.1.2" xref="A1.SS1.p2.3.m3.1.1.1.1.1.2.cmml"><mi id="A1.SS1.p2.3.m3.1.1.1.1.1.2.2" xref="A1.SS1.p2.3.m3.1.1.1.1.1.2.2.cmml">ε</mi><mrow id="A1.SS1.p2.3.m3.1.1.1.1.1.2.3" xref="A1.SS1.p2.3.m3.1.1.1.1.1.2.3.cmml"><mo 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xref="A1.SS1.p2.3.m3.1.1.3">𝑂</ci><apply id="A1.SS1.p2.3.m3.1.1.1.1.1.cmml" xref="A1.SS1.p2.3.m3.1.1.1.1"><times id="A1.SS1.p2.3.m3.1.1.1.1.1.1.cmml" xref="A1.SS1.p2.3.m3.1.1.1.1.1.1"></times><apply id="A1.SS1.p2.3.m3.1.1.1.1.1.2.cmml" xref="A1.SS1.p2.3.m3.1.1.1.1.1.2"><csymbol cd="ambiguous" id="A1.SS1.p2.3.m3.1.1.1.1.1.2.1.cmml" xref="A1.SS1.p2.3.m3.1.1.1.1.1.2">superscript</csymbol><ci id="A1.SS1.p2.3.m3.1.1.1.1.1.2.2.cmml" xref="A1.SS1.p2.3.m3.1.1.1.1.1.2.2">𝜀</ci><apply id="A1.SS1.p2.3.m3.1.1.1.1.1.2.3.cmml" xref="A1.SS1.p2.3.m3.1.1.1.1.1.2.3"><minus id="A1.SS1.p2.3.m3.1.1.1.1.1.2.3.1.cmml" xref="A1.SS1.p2.3.m3.1.1.1.1.1.2.3"></minus><cn id="A1.SS1.p2.3.m3.1.1.1.1.1.2.3.2.cmml" type="integer" xref="A1.SS1.p2.3.m3.1.1.1.1.1.2.3.2">1</cn></apply></apply><apply id="A1.SS1.p2.3.m3.1.1.1.1.1.3.cmml" xref="A1.SS1.p2.3.m3.1.1.1.1.1.3"><log id="A1.SS1.p2.3.m3.1.1.1.1.1.3.1.cmml" xref="A1.SS1.p2.3.m3.1.1.1.1.1.3.1"></log><ci id="A1.SS1.p2.3.m3.1.1.1.1.1.3.2.cmml" xref="A1.SS1.p2.3.m3.1.1.1.1.1.3.2">𝑛</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS1.p2.3.m3.1c">O(\varepsilon^{-1}\log n)</annotation><annotation encoding="application/x-llamapun" id="A1.SS1.p2.3.m3.1d">italic_O ( italic_ε start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT roman_log italic_n )</annotation></semantics></math> rounds in LOCAL using the same observation as before and the corresponding trivial algorithm.</p> </div> <div class="ltx_para" id="A1.SS1.p3"> <p class="ltx_p" id="A1.SS1.p3.5">In CONGEST, Das Sarma et al. <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2411.12694v2#bib.bib11" title="">11</a>]</cite> present the currently best deterministic algorithm that can report a <math alttext="(2+\varepsilon)" class="ltx_Math" display="inline" id="A1.SS1.p3.1.m1.1"><semantics id="A1.SS1.p3.1.m1.1a"><mrow id="A1.SS1.p3.1.m1.1.1.1" xref="A1.SS1.p3.1.m1.1.1.1.1.cmml"><mo id="A1.SS1.p3.1.m1.1.1.1.2" stretchy="false" xref="A1.SS1.p3.1.m1.1.1.1.1.cmml">(</mo><mrow id="A1.SS1.p3.1.m1.1.1.1.1" xref="A1.SS1.p3.1.m1.1.1.1.1.cmml"><mn id="A1.SS1.p3.1.m1.1.1.1.1.2" xref="A1.SS1.p3.1.m1.1.1.1.1.2.cmml">2</mn><mo id="A1.SS1.p3.1.m1.1.1.1.1.1" xref="A1.SS1.p3.1.m1.1.1.1.1.1.cmml">+</mo><mi id="A1.SS1.p3.1.m1.1.1.1.1.3" xref="A1.SS1.p3.1.m1.1.1.1.1.3.cmml">ε</mi></mrow><mo id="A1.SS1.p3.1.m1.1.1.1.3" stretchy="false" xref="A1.SS1.p3.1.m1.1.1.1.1.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="A1.SS1.p3.1.m1.1b"><apply id="A1.SS1.p3.1.m1.1.1.1.1.cmml" xref="A1.SS1.p3.1.m1.1.1.1"><plus id="A1.SS1.p3.1.m1.1.1.1.1.1.cmml" xref="A1.SS1.p3.1.m1.1.1.1.1.1"></plus><cn id="A1.SS1.p3.1.m1.1.1.1.1.2.cmml" type="integer" xref="A1.SS1.p3.1.m1.1.1.1.1.2">2</cn><ci id="A1.SS1.p3.1.m1.1.1.1.1.3.cmml" xref="A1.SS1.p3.1.m1.1.1.1.1.3">𝜀</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS1.p3.1.m1.1c">(2+\varepsilon)</annotation><annotation encoding="application/x-llamapun" id="A1.SS1.p3.1.m1.1d">( 2 + italic_ε )</annotation></semantics></math>-approximate densest subgraph in <math alttext="O(d\cdot\varepsilon^{-1}\log n)" class="ltx_Math" display="inline" id="A1.SS1.p3.2.m2.1"><semantics id="A1.SS1.p3.2.m2.1a"><mrow id="A1.SS1.p3.2.m2.1.1" xref="A1.SS1.p3.2.m2.1.1.cmml"><mi id="A1.SS1.p3.2.m2.1.1.3" xref="A1.SS1.p3.2.m2.1.1.3.cmml">O</mi><mo id="A1.SS1.p3.2.m2.1.1.2" xref="A1.SS1.p3.2.m2.1.1.2.cmml">⁢</mo><mrow id="A1.SS1.p3.2.m2.1.1.1.1" xref="A1.SS1.p3.2.m2.1.1.1.1.1.cmml"><mo id="A1.SS1.p3.2.m2.1.1.1.1.2" stretchy="false" xref="A1.SS1.p3.2.m2.1.1.1.1.1.cmml">(</mo><mrow id="A1.SS1.p3.2.m2.1.1.1.1.1" xref="A1.SS1.p3.2.m2.1.1.1.1.1.cmml"><mrow id="A1.SS1.p3.2.m2.1.1.1.1.1.2" xref="A1.SS1.p3.2.m2.1.1.1.1.1.2.cmml"><mi id="A1.SS1.p3.2.m2.1.1.1.1.1.2.2" xref="A1.SS1.p3.2.m2.1.1.1.1.1.2.2.cmml">d</mi><mo id="A1.SS1.p3.2.m2.1.1.1.1.1.2.1" lspace="0.222em" rspace="0.222em" xref="A1.SS1.p3.2.m2.1.1.1.1.1.2.1.cmml">⋅</mo><msup 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xref="A1.SS1.p3.2.m2.1.1.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="A1.SS1.p3.2.m2.1b"><apply id="A1.SS1.p3.2.m2.1.1.cmml" xref="A1.SS1.p3.2.m2.1.1"><times id="A1.SS1.p3.2.m2.1.1.2.cmml" xref="A1.SS1.p3.2.m2.1.1.2"></times><ci id="A1.SS1.p3.2.m2.1.1.3.cmml" xref="A1.SS1.p3.2.m2.1.1.3">𝑂</ci><apply id="A1.SS1.p3.2.m2.1.1.1.1.1.cmml" xref="A1.SS1.p3.2.m2.1.1.1.1"><times id="A1.SS1.p3.2.m2.1.1.1.1.1.1.cmml" xref="A1.SS1.p3.2.m2.1.1.1.1.1.1"></times><apply id="A1.SS1.p3.2.m2.1.1.1.1.1.2.cmml" xref="A1.SS1.p3.2.m2.1.1.1.1.1.2"><ci id="A1.SS1.p3.2.m2.1.1.1.1.1.2.1.cmml" xref="A1.SS1.p3.2.m2.1.1.1.1.1.2.1">⋅</ci><ci id="A1.SS1.p3.2.m2.1.1.1.1.1.2.2.cmml" xref="A1.SS1.p3.2.m2.1.1.1.1.1.2.2">𝑑</ci><apply id="A1.SS1.p3.2.m2.1.1.1.1.1.2.3.cmml" xref="A1.SS1.p3.2.m2.1.1.1.1.1.2.3"><csymbol cd="ambiguous" id="A1.SS1.p3.2.m2.1.1.1.1.1.2.3.1.cmml" xref="A1.SS1.p3.2.m2.1.1.1.1.1.2.3">superscript</csymbol><ci id="A1.SS1.p3.2.m2.1.1.1.1.1.2.3.2.cmml" xref="A1.SS1.p3.2.m2.1.1.1.1.1.2.3.2">𝜀</ci><apply id="A1.SS1.p3.2.m2.1.1.1.1.1.2.3.3.cmml" xref="A1.SS1.p3.2.m2.1.1.1.1.1.2.3.3"><minus id="A1.SS1.p3.2.m2.1.1.1.1.1.2.3.3.1.cmml" xref="A1.SS1.p3.2.m2.1.1.1.1.1.2.3.3"></minus><cn id="A1.SS1.p3.2.m2.1.1.1.1.1.2.3.3.2.cmml" type="integer" xref="A1.SS1.p3.2.m2.1.1.1.1.1.2.3.3.2">1</cn></apply></apply></apply><apply id="A1.SS1.p3.2.m2.1.1.1.1.1.3.cmml" xref="A1.SS1.p3.2.m2.1.1.1.1.1.3"><log id="A1.SS1.p3.2.m2.1.1.1.1.1.3.1.cmml" xref="A1.SS1.p3.2.m2.1.1.1.1.1.3.1"></log><ci id="A1.SS1.p3.2.m2.1.1.1.1.1.3.2.cmml" xref="A1.SS1.p3.2.m2.1.1.1.1.1.3.2">𝑛</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS1.p3.2.m2.1c">O(d\cdot\varepsilon^{-1}\log n)</annotation><annotation encoding="application/x-llamapun" id="A1.SS1.p3.2.m2.1d">italic_O ( italic_d ⋅ italic_ε start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT roman_log italic_n )</annotation></semantics></math> rounds (w.h.p). Su and Vu’s <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2411.12694v2#bib.bib21" title="">21</a>]</cite> randomised algorithm in CONGEST can solve Problem <a class="ltx_ref" href="https://arxiv.org/html/2411.12694v2#A1.Thmtheorem1" title="Problem A.1. ‣ A.1 Related work: distributed densest subgraph reporting ‣ Appendix A Reporting a densest subgraph ‣ Local Density and its Distributed Approximation"><span class="ltx_text ltx_ref_tag">A.1</span></a>.1 with <math alttext="O(d+\varepsilon^{-4}\log^{4}n)" class="ltx_Math" display="inline" id="A1.SS1.p3.3.m3.1"><semantics id="A1.SS1.p3.3.m3.1a"><mrow id="A1.SS1.p3.3.m3.1.1" xref="A1.SS1.p3.3.m3.1.1.cmml"><mi id="A1.SS1.p3.3.m3.1.1.3" xref="A1.SS1.p3.3.m3.1.1.3.cmml">O</mi><mo id="A1.SS1.p3.3.m3.1.1.2" xref="A1.SS1.p3.3.m3.1.1.2.cmml">⁢</mo><mrow id="A1.SS1.p3.3.m3.1.1.1.1" xref="A1.SS1.p3.3.m3.1.1.1.1.1.cmml"><mo id="A1.SS1.p3.3.m3.1.1.1.1.2" stretchy="false" xref="A1.SS1.p3.3.m3.1.1.1.1.1.cmml">(</mo><mrow id="A1.SS1.p3.3.m3.1.1.1.1.1" xref="A1.SS1.p3.3.m3.1.1.1.1.1.cmml"><mi id="A1.SS1.p3.3.m3.1.1.1.1.1.2" xref="A1.SS1.p3.3.m3.1.1.1.1.1.2.cmml">d</mi><mo id="A1.SS1.p3.3.m3.1.1.1.1.1.1" xref="A1.SS1.p3.3.m3.1.1.1.1.1.1.cmml">+</mo><mrow id="A1.SS1.p3.3.m3.1.1.1.1.1.3" xref="A1.SS1.p3.3.m3.1.1.1.1.1.3.cmml"><msup id="A1.SS1.p3.3.m3.1.1.1.1.1.3.2" xref="A1.SS1.p3.3.m3.1.1.1.1.1.3.2.cmml"><mi id="A1.SS1.p3.3.m3.1.1.1.1.1.3.2.2" xref="A1.SS1.p3.3.m3.1.1.1.1.1.3.2.2.cmml">ε</mi><mrow id="A1.SS1.p3.3.m3.1.1.1.1.1.3.2.3" xref="A1.SS1.p3.3.m3.1.1.1.1.1.3.2.3.cmml"><mo id="A1.SS1.p3.3.m3.1.1.1.1.1.3.2.3a" xref="A1.SS1.p3.3.m3.1.1.1.1.1.3.2.3.cmml">−</mo><mn id="A1.SS1.p3.3.m3.1.1.1.1.1.3.2.3.2" xref="A1.SS1.p3.3.m3.1.1.1.1.1.3.2.3.2.cmml">4</mn></mrow></msup><mo id="A1.SS1.p3.3.m3.1.1.1.1.1.3.1" lspace="0.167em" xref="A1.SS1.p3.3.m3.1.1.1.1.1.3.1.cmml">⁢</mo><mrow id="A1.SS1.p3.3.m3.1.1.1.1.1.3.3" xref="A1.SS1.p3.3.m3.1.1.1.1.1.3.3.cmml"><msup id="A1.SS1.p3.3.m3.1.1.1.1.1.3.3.1" xref="A1.SS1.p3.3.m3.1.1.1.1.1.3.3.1.cmml"><mi id="A1.SS1.p3.3.m3.1.1.1.1.1.3.3.1.2" xref="A1.SS1.p3.3.m3.1.1.1.1.1.3.3.1.2.cmml">log</mi><mn id="A1.SS1.p3.3.m3.1.1.1.1.1.3.3.1.3" xref="A1.SS1.p3.3.m3.1.1.1.1.1.3.3.1.3.cmml">4</mn></msup><mo id="A1.SS1.p3.3.m3.1.1.1.1.1.3.3a" lspace="0.167em" xref="A1.SS1.p3.3.m3.1.1.1.1.1.3.3.cmml">⁡</mo><mi id="A1.SS1.p3.3.m3.1.1.1.1.1.3.3.2" xref="A1.SS1.p3.3.m3.1.1.1.1.1.3.3.2.cmml">n</mi></mrow></mrow></mrow><mo id="A1.SS1.p3.3.m3.1.1.1.1.3" stretchy="false" xref="A1.SS1.p3.3.m3.1.1.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="A1.SS1.p3.3.m3.1b"><apply id="A1.SS1.p3.3.m3.1.1.cmml" xref="A1.SS1.p3.3.m3.1.1"><times id="A1.SS1.p3.3.m3.1.1.2.cmml" xref="A1.SS1.p3.3.m3.1.1.2"></times><ci id="A1.SS1.p3.3.m3.1.1.3.cmml" xref="A1.SS1.p3.3.m3.1.1.3">𝑂</ci><apply id="A1.SS1.p3.3.m3.1.1.1.1.1.cmml" xref="A1.SS1.p3.3.m3.1.1.1.1"><plus id="A1.SS1.p3.3.m3.1.1.1.1.1.1.cmml" xref="A1.SS1.p3.3.m3.1.1.1.1.1.1"></plus><ci id="A1.SS1.p3.3.m3.1.1.1.1.1.2.cmml" xref="A1.SS1.p3.3.m3.1.1.1.1.1.2">𝑑</ci><apply id="A1.SS1.p3.3.m3.1.1.1.1.1.3.cmml" xref="A1.SS1.p3.3.m3.1.1.1.1.1.3"><times id="A1.SS1.p3.3.m3.1.1.1.1.1.3.1.cmml" xref="A1.SS1.p3.3.m3.1.1.1.1.1.3.1"></times><apply id="A1.SS1.p3.3.m3.1.1.1.1.1.3.2.cmml" xref="A1.SS1.p3.3.m3.1.1.1.1.1.3.2"><csymbol cd="ambiguous" id="A1.SS1.p3.3.m3.1.1.1.1.1.3.2.1.cmml" xref="A1.SS1.p3.3.m3.1.1.1.1.1.3.2">superscript</csymbol><ci id="A1.SS1.p3.3.m3.1.1.1.1.1.3.2.2.cmml" xref="A1.SS1.p3.3.m3.1.1.1.1.1.3.2.2">𝜀</ci><apply id="A1.SS1.p3.3.m3.1.1.1.1.1.3.2.3.cmml" xref="A1.SS1.p3.3.m3.1.1.1.1.1.3.2.3"><minus id="A1.SS1.p3.3.m3.1.1.1.1.1.3.2.3.1.cmml" xref="A1.SS1.p3.3.m3.1.1.1.1.1.3.2.3"></minus><cn id="A1.SS1.p3.3.m3.1.1.1.1.1.3.2.3.2.cmml" type="integer" xref="A1.SS1.p3.3.m3.1.1.1.1.1.3.2.3.2">4</cn></apply></apply><apply id="A1.SS1.p3.3.m3.1.1.1.1.1.3.3.cmml" xref="A1.SS1.p3.3.m3.1.1.1.1.1.3.3"><apply id="A1.SS1.p3.3.m3.1.1.1.1.1.3.3.1.cmml" xref="A1.SS1.p3.3.m3.1.1.1.1.1.3.3.1"><csymbol cd="ambiguous" id="A1.SS1.p3.3.m3.1.1.1.1.1.3.3.1.1.cmml" xref="A1.SS1.p3.3.m3.1.1.1.1.1.3.3.1">superscript</csymbol><log id="A1.SS1.p3.3.m3.1.1.1.1.1.3.3.1.2.cmml" xref="A1.SS1.p3.3.m3.1.1.1.1.1.3.3.1.2"></log><cn id="A1.SS1.p3.3.m3.1.1.1.1.1.3.3.1.3.cmml" type="integer" xref="A1.SS1.p3.3.m3.1.1.1.1.1.3.3.1.3">4</cn></apply><ci id="A1.SS1.p3.3.m3.1.1.1.1.1.3.3.2.cmml" xref="A1.SS1.p3.3.m3.1.1.1.1.1.3.3.2">𝑛</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS1.p3.3.m3.1c">O(d+\varepsilon^{-4}\log^{4}n)</annotation><annotation encoding="application/x-llamapun" id="A1.SS1.p3.3.m3.1d">italic_O ( italic_d + italic_ε start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT roman_log start_POSTSUPERSCRIPT 4 end_POSTSUPERSCRIPT italic_n )</annotation></semantics></math> rounds (with high probability). For Problem <a class="ltx_ref" href="https://arxiv.org/html/2411.12694v2#A1.Thmtheorem1" title="Problem A.1. ‣ A.1 Related work: distributed densest subgraph reporting ‣ Appendix A Reporting a densest subgraph ‣ Local Density and its Distributed Approximation"><span class="ltx_text ltx_ref_tag">A.1</span></a>.2, the oracle avoids the <math alttext="\Omega(d)" class="ltx_Math" display="inline" id="A1.SS1.p3.4.m4.1"><semantics id="A1.SS1.p3.4.m4.1a"><mrow id="A1.SS1.p3.4.m4.1.2" xref="A1.SS1.p3.4.m4.1.2.cmml"><mi id="A1.SS1.p3.4.m4.1.2.2" mathvariant="normal" xref="A1.SS1.p3.4.m4.1.2.2.cmml">Ω</mi><mo id="A1.SS1.p3.4.m4.1.2.1" xref="A1.SS1.p3.4.m4.1.2.1.cmml">⁢</mo><mrow id="A1.SS1.p3.4.m4.1.2.3.2" xref="A1.SS1.p3.4.m4.1.2.cmml"><mo id="A1.SS1.p3.4.m4.1.2.3.2.1" stretchy="false" xref="A1.SS1.p3.4.m4.1.2.cmml">(</mo><mi id="A1.SS1.p3.4.m4.1.1" xref="A1.SS1.p3.4.m4.1.1.cmml">d</mi><mo id="A1.SS1.p3.4.m4.1.2.3.2.2" stretchy="false" xref="A1.SS1.p3.4.m4.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="A1.SS1.p3.4.m4.1b"><apply id="A1.SS1.p3.4.m4.1.2.cmml" xref="A1.SS1.p3.4.m4.1.2"><times id="A1.SS1.p3.4.m4.1.2.1.cmml" xref="A1.SS1.p3.4.m4.1.2.1"></times><ci id="A1.SS1.p3.4.m4.1.2.2.cmml" xref="A1.SS1.p3.4.m4.1.2.2">Ω</ci><ci id="A1.SS1.p3.4.m4.1.1.cmml" xref="A1.SS1.p3.4.m4.1.1">𝑑</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS1.p3.4.m4.1c">\Omega(d)</annotation><annotation encoding="application/x-llamapun" id="A1.SS1.p3.4.m4.1d">roman_Ω ( italic_d )</annotation></semantics></math> lower bound and Su and Vu’s <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2411.12694v2#bib.bib21" title="">21</a>]</cite> randomised algorithm can solve this problem in <math alttext="O(\varepsilon^{-4}\log^{4}n)" class="ltx_Math" display="inline" id="A1.SS1.p3.5.m5.1"><semantics id="A1.SS1.p3.5.m5.1a"><mrow id="A1.SS1.p3.5.m5.1.1" 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id="A1.SS1.p3.5.m5.1.1.1.1.1.3.1.3.cmml" type="integer" xref="A1.SS1.p3.5.m5.1.1.1.1.1.3.1.3">4</cn></apply><ci id="A1.SS1.p3.5.m5.1.1.1.1.1.3.2.cmml" xref="A1.SS1.p3.5.m5.1.1.1.1.1.3.2">𝑛</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS1.p3.5.m5.1c">O(\varepsilon^{-4}\log^{4}n)</annotation><annotation encoding="application/x-llamapun" id="A1.SS1.p3.5.m5.1d">italic_O ( italic_ε start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT roman_log start_POSTSUPERSCRIPT 4 end_POSTSUPERSCRIPT italic_n )</annotation></semantics></math> rounds (with high probability). See also Table <a class="ltx_ref" href="https://arxiv.org/html/2411.12694v2#A1.T2" title="Table 2 ‣ A.2 Reporting a locally densest subgraph ‣ Appendix A Reporting a densest subgraph ‣ Local Density and its Distributed Approximation"><span class="ltx_text ltx_ref_tag">2</span></a> for an overview.</p> </div> </section> <section class="ltx_subsection" id="A1.SS2"> <h3 class="ltx_title ltx_title_subsection"> <span class="ltx_tag ltx_tag_subsection">A.2 </span>Reporting a locally densest subgraph</h3> <div class="ltx_para" id="A1.SS2.p1"> <p class="ltx_p" id="A1.SS2.p1.1">For the local density, we can define an equivalent problem definition to Problem <a class="ltx_ref" href="https://arxiv.org/html/2411.12694v2#A1.Thmtheorem1" title="Problem A.1. ‣ A.1 Related work: distributed densest subgraph reporting ‣ Appendix A Reporting a densest subgraph ‣ Local Density and its Distributed Approximation"><span class="ltx_text ltx_ref_tag">A.1</span></a>:</p> </div> <div class="ltx_theorem ltx_theorem_problem" id="A1.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="A1.Thmtheorem2.1.1.1">Problem A.2</span></span><span class="ltx_text ltx_font_bold" id="A1.Thmtheorem2.2.2">.</span> </h6> <div class="ltx_para" id="A1.Thmtheorem2.p1"> <p class="ltx_p" id="A1.Thmtheorem2.p1.5"><span class="ltx_text ltx_font_italic" id="A1.Thmtheorem2.p1.5.5">Given <math alttext="(G,\varepsilon)" class="ltx_Math" display="inline" id="A1.Thmtheorem2.p1.1.1.m1.2"><semantics id="A1.Thmtheorem2.p1.1.1.m1.2a"><mrow id="A1.Thmtheorem2.p1.1.1.m1.2.3.2" xref="A1.Thmtheorem2.p1.1.1.m1.2.3.1.cmml"><mo id="A1.Thmtheorem2.p1.1.1.m1.2.3.2.1" stretchy="false" xref="A1.Thmtheorem2.p1.1.1.m1.2.3.1.cmml">(</mo><mi id="A1.Thmtheorem2.p1.1.1.m1.1.1" xref="A1.Thmtheorem2.p1.1.1.m1.1.1.cmml">G</mi><mo id="A1.Thmtheorem2.p1.1.1.m1.2.3.2.2" xref="A1.Thmtheorem2.p1.1.1.m1.2.3.1.cmml">,</mo><mi id="A1.Thmtheorem2.p1.1.1.m1.2.2" xref="A1.Thmtheorem2.p1.1.1.m1.2.2.cmml">ε</mi><mo id="A1.Thmtheorem2.p1.1.1.m1.2.3.2.3" stretchy="false" xref="A1.Thmtheorem2.p1.1.1.m1.2.3.1.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="A1.Thmtheorem2.p1.1.1.m1.2b"><interval closure="open" id="A1.Thmtheorem2.p1.1.1.m1.2.3.1.cmml" xref="A1.Thmtheorem2.p1.1.1.m1.2.3.2"><ci id="A1.Thmtheorem2.p1.1.1.m1.1.1.cmml" xref="A1.Thmtheorem2.p1.1.1.m1.1.1">𝐺</ci><ci id="A1.Thmtheorem2.p1.1.1.m1.2.2.cmml" xref="A1.Thmtheorem2.p1.1.1.m1.2.2">𝜀</ci></interval></annotation-xml><annotation encoding="application/x-tex" id="A1.Thmtheorem2.p1.1.1.m1.2c">(G,\varepsilon)</annotation><annotation encoding="application/x-llamapun" id="A1.Thmtheorem2.p1.1.1.m1.2d">( italic_G , italic_ε )</annotation></semantics></math> and a vertex <math alttext="v" class="ltx_Math" display="inline" id="A1.Thmtheorem2.p1.2.2.m2.1"><semantics id="A1.Thmtheorem2.p1.2.2.m2.1a"><mi id="A1.Thmtheorem2.p1.2.2.m2.1.1" xref="A1.Thmtheorem2.p1.2.2.m2.1.1.cmml">v</mi><annotation-xml encoding="MathML-Content" id="A1.Thmtheorem2.p1.2.2.m2.1b"><ci id="A1.Thmtheorem2.p1.2.2.m2.1.1.cmml" xref="A1.Thmtheorem2.p1.2.2.m2.1.1">𝑣</ci></annotation-xml><annotation encoding="application/x-tex" id="A1.Thmtheorem2.p1.2.2.m2.1c">v</annotation><annotation encoding="application/x-llamapun" id="A1.Thmtheorem2.p1.2.2.m2.1d">italic_v</annotation></semantics></math>, each vertex outputs a bit. Denote by <math alttext="H" class="ltx_Math" display="inline" id="A1.Thmtheorem2.p1.3.3.m3.1"><semantics id="A1.Thmtheorem2.p1.3.3.m3.1a"><mi id="A1.Thmtheorem2.p1.3.3.m3.1.1" xref="A1.Thmtheorem2.p1.3.3.m3.1.1.cmml">H</mi><annotation-xml encoding="MathML-Content" id="A1.Thmtheorem2.p1.3.3.m3.1b"><ci id="A1.Thmtheorem2.p1.3.3.m3.1.1.cmml" xref="A1.Thmtheorem2.p1.3.3.m3.1.1">𝐻</ci></annotation-xml><annotation encoding="application/x-tex" id="A1.Thmtheorem2.p1.3.3.m3.1c">H</annotation><annotation encoding="application/x-llamapun" id="A1.Thmtheorem2.p1.3.3.m3.1d">italic_H</annotation></semantics></math> the 1-bit induced subgraph. We require that <math alttext="v\in H" class="ltx_Math" display="inline" id="A1.Thmtheorem2.p1.4.4.m4.1"><semantics id="A1.Thmtheorem2.p1.4.4.m4.1a"><mrow id="A1.Thmtheorem2.p1.4.4.m4.1.1" xref="A1.Thmtheorem2.p1.4.4.m4.1.1.cmml"><mi id="A1.Thmtheorem2.p1.4.4.m4.1.1.2" xref="A1.Thmtheorem2.p1.4.4.m4.1.1.2.cmml">v</mi><mo id="A1.Thmtheorem2.p1.4.4.m4.1.1.1" xref="A1.Thmtheorem2.p1.4.4.m4.1.1.1.cmml">∈</mo><mi id="A1.Thmtheorem2.p1.4.4.m4.1.1.3" xref="A1.Thmtheorem2.p1.4.4.m4.1.1.3.cmml">H</mi></mrow><annotation-xml encoding="MathML-Content" id="A1.Thmtheorem2.p1.4.4.m4.1b"><apply id="A1.Thmtheorem2.p1.4.4.m4.1.1.cmml" xref="A1.Thmtheorem2.p1.4.4.m4.1.1"><in id="A1.Thmtheorem2.p1.4.4.m4.1.1.1.cmml" xref="A1.Thmtheorem2.p1.4.4.m4.1.1.1"></in><ci id="A1.Thmtheorem2.p1.4.4.m4.1.1.2.cmml" xref="A1.Thmtheorem2.p1.4.4.m4.1.1.2">𝑣</ci><ci id="A1.Thmtheorem2.p1.4.4.m4.1.1.3.cmml" xref="A1.Thmtheorem2.p1.4.4.m4.1.1.3">𝐻</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.Thmtheorem2.p1.4.4.m4.1c">v\in H</annotation><annotation encoding="application/x-llamapun" id="A1.Thmtheorem2.p1.4.4.m4.1d">italic_v ∈ italic_H</annotation></semantics></math>, and that <math alttext="\rho(H)\in[(1-\varepsilon)\rho^{*}(v),(1+\varepsilon)\rho^{*}(v)]" class="ltx_Math" display="inline" id="A1.Thmtheorem2.p1.5.5.m5.5"><semantics id="A1.Thmtheorem2.p1.5.5.m5.5a"><mrow id="A1.Thmtheorem2.p1.5.5.m5.5.5" xref="A1.Thmtheorem2.p1.5.5.m5.5.5.cmml"><mrow id="A1.Thmtheorem2.p1.5.5.m5.5.5.4" xref="A1.Thmtheorem2.p1.5.5.m5.5.5.4.cmml"><mi id="A1.Thmtheorem2.p1.5.5.m5.5.5.4.2" xref="A1.Thmtheorem2.p1.5.5.m5.5.5.4.2.cmml">ρ</mi><mo id="A1.Thmtheorem2.p1.5.5.m5.5.5.4.1" xref="A1.Thmtheorem2.p1.5.5.m5.5.5.4.1.cmml">⁢</mo><mrow id="A1.Thmtheorem2.p1.5.5.m5.5.5.4.3.2" xref="A1.Thmtheorem2.p1.5.5.m5.5.5.4.cmml"><mo id="A1.Thmtheorem2.p1.5.5.m5.5.5.4.3.2.1" stretchy="false" xref="A1.Thmtheorem2.p1.5.5.m5.5.5.4.cmml">(</mo><mi id="A1.Thmtheorem2.p1.5.5.m5.1.1" xref="A1.Thmtheorem2.p1.5.5.m5.1.1.cmml">H</mi><mo id="A1.Thmtheorem2.p1.5.5.m5.5.5.4.3.2.2" stretchy="false" xref="A1.Thmtheorem2.p1.5.5.m5.5.5.4.cmml">)</mo></mrow></mrow><mo id="A1.Thmtheorem2.p1.5.5.m5.5.5.3" xref="A1.Thmtheorem2.p1.5.5.m5.5.5.3.cmml">∈</mo><mrow id="A1.Thmtheorem2.p1.5.5.m5.5.5.2.2" xref="A1.Thmtheorem2.p1.5.5.m5.5.5.2.3.cmml"><mo id="A1.Thmtheorem2.p1.5.5.m5.5.5.2.2.3" stretchy="false" xref="A1.Thmtheorem2.p1.5.5.m5.5.5.2.3.cmml">[</mo><mrow id="A1.Thmtheorem2.p1.5.5.m5.4.4.1.1.1" xref="A1.Thmtheorem2.p1.5.5.m5.4.4.1.1.1.cmml"><mrow id="A1.Thmtheorem2.p1.5.5.m5.4.4.1.1.1.1.1" xref="A1.Thmtheorem2.p1.5.5.m5.4.4.1.1.1.1.1.1.cmml"><mo id="A1.Thmtheorem2.p1.5.5.m5.4.4.1.1.1.1.1.2" stretchy="false" xref="A1.Thmtheorem2.p1.5.5.m5.4.4.1.1.1.1.1.1.cmml">(</mo><mrow id="A1.Thmtheorem2.p1.5.5.m5.4.4.1.1.1.1.1.1" xref="A1.Thmtheorem2.p1.5.5.m5.4.4.1.1.1.1.1.1.cmml"><mn id="A1.Thmtheorem2.p1.5.5.m5.4.4.1.1.1.1.1.1.2" 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id="A1.Thmtheorem2.p1.5.5.m5.5d">italic_ρ ( italic_H ) ∈ [ ( 1 - italic_ε ) italic_ρ start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ( italic_v ) , ( 1 + italic_ε ) italic_ρ start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ( italic_v ) ]</annotation></semantics></math>.</span></p> </div> </div> <div class="ltx_para" id="A1.SS2.p2"> <p class="ltx_p" id="A1.SS2.p2.1">We design a set of algorithms to answer the reporting variant for the local subgraph density problem (see Table <a class="ltx_ref" href="https://arxiv.org/html/2411.12694v2#A1.T2" title="Table 2 ‣ A.2 Reporting a locally densest subgraph ‣ Appendix A Reporting a densest subgraph ‣ Local Density and its Distributed Approximation"><span class="ltx_text ltx_ref_tag">2</span></a> for an overview)</p> </div> <figure class="ltx_table" id="A1.T2"> <p class="ltx_p ltx_align_center" id="A1.T2.20"><span class="ltx_text ltx_inline-block" id="A1.T2.20.20" style="width:433.6pt;"> <span class="ltx_inline-block ltx_transformed_outer" id="A1.T2.20.20.20.20.20" style="width:558.3pt;height:181pt;vertical-align:-1.0pt;"><span class="ltx_transformed_inner" style="transform:translate(0.0pt,0.0pt) scale(1,1) ;"> <span class="ltx_p" id="A1.T2.20.20.20.20.20.20"><span class="ltx_text" id="A1.T2.20.20.20.20.20.20.20"> <span class="ltx_tabular ltx_align_middle" id="A1.T2.20.20.20.20.20.20.20.20"> <span class="ltx_tr" id="A1.T2.3.3.3.3.3.3.3.3.3"> <span class="ltx_td ltx_align_center ltx_border_r" id="A1.T2.3.3.3.3.3.3.3.3.3.4">Model</span> <span class="ltx_td ltx_align_center ltx_border_r" id="A1.T2.3.3.3.3.3.3.3.3.3.5">Problem</span> <span class="ltx_td ltx_align_center ltx_border_r" id="A1.T2.3.3.3.3.3.3.3.3.3.3">Each <math alttext="v" class="ltx_Math" display="inline" id="A1.T2.1.1.1.1.1.1.1.1.1.1.m1.1"><semantics id="A1.T2.1.1.1.1.1.1.1.1.1.1.m1.1a"><mi id="A1.T2.1.1.1.1.1.1.1.1.1.1.m1.1.1" xref="A1.T2.1.1.1.1.1.1.1.1.1.1.m1.1.1.cmml">v</mi><annotation-xml encoding="MathML-Content" id="A1.T2.1.1.1.1.1.1.1.1.1.1.m1.1b"><ci id="A1.T2.1.1.1.1.1.1.1.1.1.1.m1.1.1.cmml" xref="A1.T2.1.1.1.1.1.1.1.1.1.1.m1.1.1">𝑣</ci></annotation-xml><annotation encoding="application/x-tex" id="A1.T2.1.1.1.1.1.1.1.1.1.1.m1.1c">v</annotation><annotation encoding="application/x-llamapun" id="A1.T2.1.1.1.1.1.1.1.1.1.1.m1.1d">italic_v</annotation></semantics></math> reports a bit s.t. for the <math alttext="1" class="ltx_Math" display="inline" id="A1.T2.2.2.2.2.2.2.2.2.2.2.m2.1"><semantics id="A1.T2.2.2.2.2.2.2.2.2.2.2.m2.1a"><mn id="A1.T2.2.2.2.2.2.2.2.2.2.2.m2.1.1" xref="A1.T2.2.2.2.2.2.2.2.2.2.2.m2.1.1.cmml">1</mn><annotation-xml encoding="MathML-Content" id="A1.T2.2.2.2.2.2.2.2.2.2.2.m2.1b"><cn id="A1.T2.2.2.2.2.2.2.2.2.2.2.m2.1.1.cmml" type="integer" xref="A1.T2.2.2.2.2.2.2.2.2.2.2.m2.1.1">1</cn></annotation-xml><annotation encoding="application/x-tex" id="A1.T2.2.2.2.2.2.2.2.2.2.2.m2.1c">1</annotation><annotation encoding="application/x-llamapun" id="A1.T2.2.2.2.2.2.2.2.2.2.2.m2.1d">1</annotation></semantics></math>-bit induced graph <math alttext="H" class="ltx_Math" display="inline" id="A1.T2.3.3.3.3.3.3.3.3.3.3.m3.1"><semantics id="A1.T2.3.3.3.3.3.3.3.3.3.3.m3.1a"><mi id="A1.T2.3.3.3.3.3.3.3.3.3.3.m3.1.1" xref="A1.T2.3.3.3.3.3.3.3.3.3.3.m3.1.1.cmml">H</mi><annotation-xml encoding="MathML-Content" id="A1.T2.3.3.3.3.3.3.3.3.3.3.m3.1b"><ci id="A1.T2.3.3.3.3.3.3.3.3.3.3.m3.1.1.cmml" xref="A1.T2.3.3.3.3.3.3.3.3.3.3.m3.1.1">𝐻</ci></annotation-xml><annotation encoding="application/x-tex" id="A1.T2.3.3.3.3.3.3.3.3.3.3.m3.1c">H</annotation><annotation encoding="application/x-llamapun" id="A1.T2.3.3.3.3.3.3.3.3.3.3.m3.1d">italic_H</annotation></semantics></math>:</span> <span class="ltx_td ltx_align_center ltx_border_r" id="A1.T2.3.3.3.3.3.3.3.3.3.6">Rounds</span> <span class="ltx_td ltx_align_center" id="A1.T2.3.3.3.3.3.3.3.3.3.7">Source</span></span> <span class="ltx_tr" id="A1.T2.4.4.4.4.4.4.4.4.4"> <span class="ltx_td ltx_border_r" id="A1.T2.4.4.4.4.4.4.4.4.4.2"></span> <span class="ltx_td ltx_border_r" id="A1.T2.4.4.4.4.4.4.4.4.4.3"></span> <span class="ltx_td ltx_align_center ltx_border_r" id="A1.T2.4.4.4.4.4.4.4.4.4.1">(Our new result additionally requires an input vertex <math alttext="v" class="ltx_Math" display="inline" id="A1.T2.4.4.4.4.4.4.4.4.4.1.m1.1"><semantics id="A1.T2.4.4.4.4.4.4.4.4.4.1.m1.1a"><mi id="A1.T2.4.4.4.4.4.4.4.4.4.1.m1.1.1" xref="A1.T2.4.4.4.4.4.4.4.4.4.1.m1.1.1.cmml">v</mi><annotation-xml encoding="MathML-Content" id="A1.T2.4.4.4.4.4.4.4.4.4.1.m1.1b"><ci id="A1.T2.4.4.4.4.4.4.4.4.4.1.m1.1.1.cmml" xref="A1.T2.4.4.4.4.4.4.4.4.4.1.m1.1.1">𝑣</ci></annotation-xml><annotation encoding="application/x-tex" id="A1.T2.4.4.4.4.4.4.4.4.4.1.m1.1c">v</annotation><annotation encoding="application/x-llamapun" id="A1.T2.4.4.4.4.4.4.4.4.4.1.m1.1d">italic_v</annotation></semantics></math>)</span> <span class="ltx_td ltx_border_r" id="A1.T2.4.4.4.4.4.4.4.4.4.4"></span> <span class="ltx_td" id="A1.T2.4.4.4.4.4.4.4.4.4.5"></span></span> <span class="ltx_tr" id="A1.T2.6.6.6.6.6.6.6.6.6"> <span class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="A1.T2.6.6.6.6.6.6.6.6.6.3">L</span> <span class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="A1.T2.6.6.6.6.6.6.6.6.6.4"><a class="ltx_ref" href="https://arxiv.org/html/2411.12694v2#A1.Thmtheorem1" title="Problem A.1. ‣ A.1 Related work: distributed densest subgraph reporting ‣ Appendix A Reporting a densest subgraph ‣ Local Density and its Distributed Approximation"><span class="ltx_text ltx_ref_tag">A.1</span></a>.1</span> <span class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="A1.T2.5.5.5.5.5.5.5.5.5.1"><math alttext="\rho(H)\in[(1+\varepsilon)^{-1}\rho^{\max}(G),(1+\varepsilon)\rho^{\max}(G)]" class="ltx_Math" display="inline" id="A1.T2.5.5.5.5.5.5.5.5.5.1.m1.5"><semantics id="A1.T2.5.5.5.5.5.5.5.5.5.1.m1.5a"><mrow id="A1.T2.5.5.5.5.5.5.5.5.5.1.m1.5.5" xref="A1.T2.5.5.5.5.5.5.5.5.5.1.m1.5.5.cmml"><mrow id="A1.T2.5.5.5.5.5.5.5.5.5.1.m1.5.5.4" 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id="A1.T2.5.5.5.5.5.5.5.5.5.1.m1.3.3.cmml" xref="A1.T2.5.5.5.5.5.5.5.5.5.1.m1.3.3">𝐺</ci></apply></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.T2.5.5.5.5.5.5.5.5.5.1.m1.5c">\rho(H)\in[(1+\varepsilon)^{-1}\rho^{\max}(G),(1+\varepsilon)\rho^{\max}(G)]</annotation><annotation encoding="application/x-llamapun" id="A1.T2.5.5.5.5.5.5.5.5.5.1.m1.5d">italic_ρ ( italic_H ) ∈ [ ( 1 + italic_ε ) start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT italic_ρ start_POSTSUPERSCRIPT roman_max end_POSTSUPERSCRIPT ( italic_G ) , ( 1 + italic_ε ) italic_ρ start_POSTSUPERSCRIPT roman_max end_POSTSUPERSCRIPT ( italic_G ) ]</annotation></semantics></math></span> <span class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="A1.T2.6.6.6.6.6.6.6.6.6.2"><math alttext="\Theta(D)" class="ltx_Math" display="inline" id="A1.T2.6.6.6.6.6.6.6.6.6.2.m1.1"><semantics id="A1.T2.6.6.6.6.6.6.6.6.6.2.m1.1a"><mrow id="A1.T2.6.6.6.6.6.6.6.6.6.2.m1.1.2" xref="A1.T2.6.6.6.6.6.6.6.6.6.2.m1.1.2.cmml"><mi id="A1.T2.6.6.6.6.6.6.6.6.6.2.m1.1.2.2" mathvariant="normal" xref="A1.T2.6.6.6.6.6.6.6.6.6.2.m1.1.2.2.cmml">Θ</mi><mo id="A1.T2.6.6.6.6.6.6.6.6.6.2.m1.1.2.1" xref="A1.T2.6.6.6.6.6.6.6.6.6.2.m1.1.2.1.cmml">⁢</mo><mrow id="A1.T2.6.6.6.6.6.6.6.6.6.2.m1.1.2.3.2" xref="A1.T2.6.6.6.6.6.6.6.6.6.2.m1.1.2.cmml"><mo id="A1.T2.6.6.6.6.6.6.6.6.6.2.m1.1.2.3.2.1" stretchy="false" xref="A1.T2.6.6.6.6.6.6.6.6.6.2.m1.1.2.cmml">(</mo><mi id="A1.T2.6.6.6.6.6.6.6.6.6.2.m1.1.1" xref="A1.T2.6.6.6.6.6.6.6.6.6.2.m1.1.1.cmml">D</mi><mo id="A1.T2.6.6.6.6.6.6.6.6.6.2.m1.1.2.3.2.2" stretchy="false" xref="A1.T2.6.6.6.6.6.6.6.6.6.2.m1.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="A1.T2.6.6.6.6.6.6.6.6.6.2.m1.1b"><apply id="A1.T2.6.6.6.6.6.6.6.6.6.2.m1.1.2.cmml" xref="A1.T2.6.6.6.6.6.6.6.6.6.2.m1.1.2"><times id="A1.T2.6.6.6.6.6.6.6.6.6.2.m1.1.2.1.cmml" xref="A1.T2.6.6.6.6.6.6.6.6.6.2.m1.1.2.1"></times><ci id="A1.T2.6.6.6.6.6.6.6.6.6.2.m1.1.2.2.cmml" xref="A1.T2.6.6.6.6.6.6.6.6.6.2.m1.1.2.2">Θ</ci><ci id="A1.T2.6.6.6.6.6.6.6.6.6.2.m1.1.1.cmml" xref="A1.T2.6.6.6.6.6.6.6.6.6.2.m1.1.1">𝐷</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.T2.6.6.6.6.6.6.6.6.6.2.m1.1c">\Theta(D)</annotation><annotation encoding="application/x-llamapun" id="A1.T2.6.6.6.6.6.6.6.6.6.2.m1.1d">roman_Θ ( italic_D )</annotation></semantics></math></span> <span class="ltx_td ltx_align_center ltx_border_t" id="A1.T2.6.6.6.6.6.6.6.6.6.5"><cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2411.12694v2#bib.bib21" title="">21</a>]</cite></span></span> <span class="ltx_tr" id="A1.T2.8.8.8.8.8.8.8.8.8"> <span class="ltx_td ltx_border_r" id="A1.T2.8.8.8.8.8.8.8.8.8.3"></span> <span class="ltx_td ltx_align_center ltx_border_r" id="A1.T2.8.8.8.8.8.8.8.8.8.4"><a class="ltx_ref" href="https://arxiv.org/html/2411.12694v2#A1.Thmtheorem1" title="Problem A.1. ‣ A.1 Related work: distributed densest subgraph reporting ‣ Appendix A Reporting a densest subgraph ‣ Local Density and its Distributed Approximation"><span class="ltx_text ltx_ref_tag">A.1</span></a>.2</span> <span class="ltx_td ltx_align_center ltx_border_r" id="A1.T2.7.7.7.7.7.7.7.7.7.1"><math alttext="\rho(H)\geq\tilde{D}\textnormal{ where }\tilde{D}\textnormal{ is given by an % oracle.}" class="ltx_Math" display="inline" id="A1.T2.7.7.7.7.7.7.7.7.7.1.m1.1"><semantics id="A1.T2.7.7.7.7.7.7.7.7.7.1.m1.1a"><mrow id="A1.T2.7.7.7.7.7.7.7.7.7.1.m1.1.2" xref="A1.T2.7.7.7.7.7.7.7.7.7.1.m1.1.2.cmml"><mrow id="A1.T2.7.7.7.7.7.7.7.7.7.1.m1.1.2.2" xref="A1.T2.7.7.7.7.7.7.7.7.7.1.m1.1.2.2.cmml"><mi id="A1.T2.7.7.7.7.7.7.7.7.7.1.m1.1.2.2.2" xref="A1.T2.7.7.7.7.7.7.7.7.7.1.m1.1.2.2.2.cmml">ρ</mi><mo id="A1.T2.7.7.7.7.7.7.7.7.7.1.m1.1.2.2.1" xref="A1.T2.7.7.7.7.7.7.7.7.7.1.m1.1.2.2.1.cmml">⁢</mo><mrow id="A1.T2.7.7.7.7.7.7.7.7.7.1.m1.1.2.2.3.2" xref="A1.T2.7.7.7.7.7.7.7.7.7.1.m1.1.2.2.cmml"><mo 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xref="A1.T2.7.7.7.7.7.7.7.7.7.1.m1.1.2.3.2.2">𝐷</ci></apply><ci id="A1.T2.7.7.7.7.7.7.7.7.7.1.m1.1.2.3.3a.cmml" xref="A1.T2.7.7.7.7.7.7.7.7.7.1.m1.1.2.3.3"><mtext id="A1.T2.7.7.7.7.7.7.7.7.7.1.m1.1.2.3.3.cmml" xref="A1.T2.7.7.7.7.7.7.7.7.7.1.m1.1.2.3.3"> where </mtext></ci><apply id="A1.T2.7.7.7.7.7.7.7.7.7.1.m1.1.2.3.4.cmml" xref="A1.T2.7.7.7.7.7.7.7.7.7.1.m1.1.2.3.4"><ci id="A1.T2.7.7.7.7.7.7.7.7.7.1.m1.1.2.3.4.1.cmml" xref="A1.T2.7.7.7.7.7.7.7.7.7.1.m1.1.2.3.4.1">~</ci><ci id="A1.T2.7.7.7.7.7.7.7.7.7.1.m1.1.2.3.4.2.cmml" xref="A1.T2.7.7.7.7.7.7.7.7.7.1.m1.1.2.3.4.2">𝐷</ci></apply><ci id="A1.T2.7.7.7.7.7.7.7.7.7.1.m1.1.2.3.5a.cmml" xref="A1.T2.7.7.7.7.7.7.7.7.7.1.m1.1.2.3.5"><mtext id="A1.T2.7.7.7.7.7.7.7.7.7.1.m1.1.2.3.5.cmml" xref="A1.T2.7.7.7.7.7.7.7.7.7.1.m1.1.2.3.5"> is given by an oracle.</mtext></ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.T2.7.7.7.7.7.7.7.7.7.1.m1.1c">\rho(H)\geq\tilde{D}\textnormal{ where }\tilde{D}\textnormal{ is given by an % oracle.}</annotation><annotation encoding="application/x-llamapun" id="A1.T2.7.7.7.7.7.7.7.7.7.1.m1.1d">italic_ρ ( italic_H ) ≥ over~ start_ARG italic_D end_ARG where over~ start_ARG italic_D end_ARG is given by an oracle.</annotation></semantics></math></span> <span class="ltx_td ltx_align_center ltx_border_r" id="A1.T2.8.8.8.8.8.8.8.8.8.2"><math alttext="O(\varepsilon^{-1}\log n)" class="ltx_Math" display="inline" id="A1.T2.8.8.8.8.8.8.8.8.8.2.m1.1"><semantics id="A1.T2.8.8.8.8.8.8.8.8.8.2.m1.1a"><mrow id="A1.T2.8.8.8.8.8.8.8.8.8.2.m1.1.1" xref="A1.T2.8.8.8.8.8.8.8.8.8.2.m1.1.1.cmml"><mi id="A1.T2.8.8.8.8.8.8.8.8.8.2.m1.1.1.3" xref="A1.T2.8.8.8.8.8.8.8.8.8.2.m1.1.1.3.cmml">O</mi><mo id="A1.T2.8.8.8.8.8.8.8.8.8.2.m1.1.1.2" xref="A1.T2.8.8.8.8.8.8.8.8.8.2.m1.1.1.2.cmml">⁢</mo><mrow id="A1.T2.8.8.8.8.8.8.8.8.8.2.m1.1.1.1.1" xref="A1.T2.8.8.8.8.8.8.8.8.8.2.m1.1.1.1.1.1.cmml"><mo id="A1.T2.8.8.8.8.8.8.8.8.8.2.m1.1.1.1.1.2" stretchy="false" 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id="A1.T2.8.8.8.8.8.8.8.8.8.2.m1.1.1.1.1.1.1.cmml" xref="A1.T2.8.8.8.8.8.8.8.8.8.2.m1.1.1.1.1.1.1"></times><apply id="A1.T2.8.8.8.8.8.8.8.8.8.2.m1.1.1.1.1.1.2.cmml" xref="A1.T2.8.8.8.8.8.8.8.8.8.2.m1.1.1.1.1.1.2"><csymbol cd="ambiguous" id="A1.T2.8.8.8.8.8.8.8.8.8.2.m1.1.1.1.1.1.2.1.cmml" xref="A1.T2.8.8.8.8.8.8.8.8.8.2.m1.1.1.1.1.1.2">superscript</csymbol><ci id="A1.T2.8.8.8.8.8.8.8.8.8.2.m1.1.1.1.1.1.2.2.cmml" xref="A1.T2.8.8.8.8.8.8.8.8.8.2.m1.1.1.1.1.1.2.2">𝜀</ci><apply id="A1.T2.8.8.8.8.8.8.8.8.8.2.m1.1.1.1.1.1.2.3.cmml" xref="A1.T2.8.8.8.8.8.8.8.8.8.2.m1.1.1.1.1.1.2.3"><minus id="A1.T2.8.8.8.8.8.8.8.8.8.2.m1.1.1.1.1.1.2.3.1.cmml" xref="A1.T2.8.8.8.8.8.8.8.8.8.2.m1.1.1.1.1.1.2.3"></minus><cn id="A1.T2.8.8.8.8.8.8.8.8.8.2.m1.1.1.1.1.1.2.3.2.cmml" type="integer" xref="A1.T2.8.8.8.8.8.8.8.8.8.2.m1.1.1.1.1.1.2.3.2">1</cn></apply></apply><apply id="A1.T2.8.8.8.8.8.8.8.8.8.2.m1.1.1.1.1.1.3.cmml" xref="A1.T2.8.8.8.8.8.8.8.8.8.2.m1.1.1.1.1.1.3"><log id="A1.T2.8.8.8.8.8.8.8.8.8.2.m1.1.1.1.1.1.3.1.cmml" xref="A1.T2.8.8.8.8.8.8.8.8.8.2.m1.1.1.1.1.1.3.1"></log><ci id="A1.T2.8.8.8.8.8.8.8.8.8.2.m1.1.1.1.1.1.3.2.cmml" xref="A1.T2.8.8.8.8.8.8.8.8.8.2.m1.1.1.1.1.1.3.2">𝑛</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.T2.8.8.8.8.8.8.8.8.8.2.m1.1c">O(\varepsilon^{-1}\log n)</annotation><annotation encoding="application/x-llamapun" id="A1.T2.8.8.8.8.8.8.8.8.8.2.m1.1d">italic_O ( italic_ε start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT roman_log italic_n )</annotation></semantics></math></span> <span class="ltx_td ltx_align_center" id="A1.T2.8.8.8.8.8.8.8.8.8.5"><cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2411.12694v2#bib.bib21" title="">21</a>]</cite></span></span> <span class="ltx_tr" id="A1.T2.10.10.10.10.10.10.10.10.10"> <span class="ltx_td ltx_border_r" id="A1.T2.10.10.10.10.10.10.10.10.10.3"></span> <span class="ltx_td ltx_align_center ltx_border_r" id="A1.T2.10.10.10.10.10.10.10.10.10.4"><a class="ltx_ref ltx_font_bold" href="https://arxiv.org/html/2411.12694v2#A1.Thmtheorem2" title="Problem A.2. ‣ A.2 Reporting a locally densest subgraph ‣ Appendix A Reporting a densest subgraph ‣ Local Density and its Distributed Approximation"><span class="ltx_text ltx_ref_tag">A.2</span></a></span> <span class="ltx_td ltx_align_center ltx_border_r" id="A1.T2.9.9.9.9.9.9.9.9.9.1"><math alttext="\boldsymbol{\rho(H)\in[(1+\varepsilon)^{-1}\rho^{*}(v),(1+\varepsilon)\rho^{*}% (v)]}" class="ltx_Math" display="inline" id="A1.T2.9.9.9.9.9.9.9.9.9.1.m1.5"><semantics id="A1.T2.9.9.9.9.9.9.9.9.9.1.m1.5a"><mrow id="A1.T2.9.9.9.9.9.9.9.9.9.1.m1.5.5" xref="A1.T2.9.9.9.9.9.9.9.9.9.1.m1.5.5.cmml"><mrow id="A1.T2.9.9.9.9.9.9.9.9.9.1.m1.5.5.4" xref="A1.T2.9.9.9.9.9.9.9.9.9.1.m1.5.5.4.cmml"><mi id="A1.T2.9.9.9.9.9.9.9.9.9.1.m1.5.5.4.2" xref="A1.T2.9.9.9.9.9.9.9.9.9.1.m1.5.5.4.2.cmml">𝝆</mi><mo id="A1.T2.9.9.9.9.9.9.9.9.9.1.m1.5.5.4.1" 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encoding="application/x-tex" id="A1.T2.10.10.10.10.10.10.10.10.10.2.m1.1c">\boldsymbol{O(\varepsilon^{-2}\log^{2}n)}</annotation><annotation encoding="application/x-llamapun" id="A1.T2.10.10.10.10.10.10.10.10.10.2.m1.1d">bold_italic_O bold_( bold_italic_ε start_POSTSUPERSCRIPT bold_- bold_2 end_POSTSUPERSCRIPT bold_log start_POSTSUPERSCRIPT bold_2 end_POSTSUPERSCRIPT bold_italic_n bold_)</annotation></semantics></math></span> <span class="ltx_td ltx_align_center" id="A1.T2.10.10.10.10.10.10.10.10.10.5"><span class="ltx_text ltx_font_bold" id="A1.T2.10.10.10.10.10.10.10.10.10.5.1">Cor. <a class="ltx_ref" href="https://arxiv.org/html/2411.12694v2#S3.Thmtheorem8" title="Corollary 3.8. ‣ 3.C Results in LOCAL ‣ 3 Results and organisation ‣ Local Density and its Distributed Approximation"><span class="ltx_text ltx_ref_tag">3.8</span></a></span></span></span> <span class="ltx_tr" id="A1.T2.12.12.12.12.12.12.12.12.12"> <span class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="A1.T2.12.12.12.12.12.12.12.12.12.3">C</span> <span class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="A1.T2.12.12.12.12.12.12.12.12.12.4"><a class="ltx_ref" href="https://arxiv.org/html/2411.12694v2#A1.Thmtheorem1" title="Problem A.1. ‣ A.1 Related work: distributed densest subgraph reporting ‣ Appendix A Reporting a densest subgraph ‣ Local Density and its Distributed Approximation"><span class="ltx_text ltx_ref_tag">A.1</span></a>.1</span> <span class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="A1.T2.11.11.11.11.11.11.11.11.11.1"><math alttext="\rho(H)\in[(2+\varepsilon)^{-1}\rho^{\max}(G),(2+\varepsilon)\rho^{\max}(G)]" class="ltx_Math" display="inline" id="A1.T2.11.11.11.11.11.11.11.11.11.1.m1.5"><semantics id="A1.T2.11.11.11.11.11.11.11.11.11.1.m1.5a"><mrow id="A1.T2.11.11.11.11.11.11.11.11.11.1.m1.5.5" xref="A1.T2.11.11.11.11.11.11.11.11.11.1.m1.5.5.cmml"><mrow id="A1.T2.11.11.11.11.11.11.11.11.11.1.m1.5.5.4" 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xref="A1.T2.11.11.11.11.11.11.11.11.11.1.m1.4.4.1.1.1.1.1.1.1.2.cmml">2</mn><mo id="A1.T2.11.11.11.11.11.11.11.11.11.1.m1.4.4.1.1.1.1.1.1.1.1" xref="A1.T2.11.11.11.11.11.11.11.11.11.1.m1.4.4.1.1.1.1.1.1.1.1.cmml">+</mo><mi id="A1.T2.11.11.11.11.11.11.11.11.11.1.m1.4.4.1.1.1.1.1.1.1.3" xref="A1.T2.11.11.11.11.11.11.11.11.11.1.m1.4.4.1.1.1.1.1.1.1.3.cmml">ε</mi></mrow><mo id="A1.T2.11.11.11.11.11.11.11.11.11.1.m1.4.4.1.1.1.1.1.1.3" stretchy="false" xref="A1.T2.11.11.11.11.11.11.11.11.11.1.m1.4.4.1.1.1.1.1.1.1.cmml">)</mo></mrow><mrow id="A1.T2.11.11.11.11.11.11.11.11.11.1.m1.4.4.1.1.1.1.3" xref="A1.T2.11.11.11.11.11.11.11.11.11.1.m1.4.4.1.1.1.1.3.cmml"><mo id="A1.T2.11.11.11.11.11.11.11.11.11.1.m1.4.4.1.1.1.1.3a" xref="A1.T2.11.11.11.11.11.11.11.11.11.1.m1.4.4.1.1.1.1.3.cmml">−</mo><mn id="A1.T2.11.11.11.11.11.11.11.11.11.1.m1.4.4.1.1.1.1.3.2" xref="A1.T2.11.11.11.11.11.11.11.11.11.1.m1.4.4.1.1.1.1.3.2.cmml">1</mn></mrow></msup><mo id="A1.T2.11.11.11.11.11.11.11.11.11.1.m1.4.4.1.1.1.2" xref="A1.T2.11.11.11.11.11.11.11.11.11.1.m1.4.4.1.1.1.2.cmml">⁢</mo><msup id="A1.T2.11.11.11.11.11.11.11.11.11.1.m1.4.4.1.1.1.3" xref="A1.T2.11.11.11.11.11.11.11.11.11.1.m1.4.4.1.1.1.3.cmml"><mi id="A1.T2.11.11.11.11.11.11.11.11.11.1.m1.4.4.1.1.1.3.2" xref="A1.T2.11.11.11.11.11.11.11.11.11.1.m1.4.4.1.1.1.3.2.cmml">ρ</mi><mi id="A1.T2.11.11.11.11.11.11.11.11.11.1.m1.4.4.1.1.1.3.3" xref="A1.T2.11.11.11.11.11.11.11.11.11.1.m1.4.4.1.1.1.3.3.cmml">max</mi></msup><mo id="A1.T2.11.11.11.11.11.11.11.11.11.1.m1.4.4.1.1.1.2a" xref="A1.T2.11.11.11.11.11.11.11.11.11.1.m1.4.4.1.1.1.2.cmml">⁢</mo><mrow id="A1.T2.11.11.11.11.11.11.11.11.11.1.m1.4.4.1.1.1.4.2" xref="A1.T2.11.11.11.11.11.11.11.11.11.1.m1.4.4.1.1.1.cmml"><mo id="A1.T2.11.11.11.11.11.11.11.11.11.1.m1.4.4.1.1.1.4.2.1" stretchy="false" xref="A1.T2.11.11.11.11.11.11.11.11.11.1.m1.4.4.1.1.1.cmml">(</mo><mi id="A1.T2.11.11.11.11.11.11.11.11.11.1.m1.2.2" xref="A1.T2.11.11.11.11.11.11.11.11.11.1.m1.2.2.cmml">G</mi><mo id="A1.T2.11.11.11.11.11.11.11.11.11.1.m1.4.4.1.1.1.4.2.2" stretchy="false" xref="A1.T2.11.11.11.11.11.11.11.11.11.1.m1.4.4.1.1.1.cmml">)</mo></mrow></mrow><mo id="A1.T2.11.11.11.11.11.11.11.11.11.1.m1.5.5.2.2.4" xref="A1.T2.11.11.11.11.11.11.11.11.11.1.m1.5.5.2.3.cmml">,</mo><mrow id="A1.T2.11.11.11.11.11.11.11.11.11.1.m1.5.5.2.2.2" xref="A1.T2.11.11.11.11.11.11.11.11.11.1.m1.5.5.2.2.2.cmml"><mrow id="A1.T2.11.11.11.11.11.11.11.11.11.1.m1.5.5.2.2.2.1.1" xref="A1.T2.11.11.11.11.11.11.11.11.11.1.m1.5.5.2.2.2.1.1.1.cmml"><mo id="A1.T2.11.11.11.11.11.11.11.11.11.1.m1.5.5.2.2.2.1.1.2" stretchy="false" xref="A1.T2.11.11.11.11.11.11.11.11.11.1.m1.5.5.2.2.2.1.1.1.cmml">(</mo><mrow id="A1.T2.11.11.11.11.11.11.11.11.11.1.m1.5.5.2.2.2.1.1.1" xref="A1.T2.11.11.11.11.11.11.11.11.11.1.m1.5.5.2.2.2.1.1.1.cmml"><mn id="A1.T2.11.11.11.11.11.11.11.11.11.1.m1.5.5.2.2.2.1.1.1.2" xref="A1.T2.11.11.11.11.11.11.11.11.11.1.m1.5.5.2.2.2.1.1.1.2.cmml">2</mn><mo id="A1.T2.11.11.11.11.11.11.11.11.11.1.m1.5.5.2.2.2.1.1.1.1" xref="A1.T2.11.11.11.11.11.11.11.11.11.1.m1.5.5.2.2.2.1.1.1.1.cmml">+</mo><mi id="A1.T2.11.11.11.11.11.11.11.11.11.1.m1.5.5.2.2.2.1.1.1.3" xref="A1.T2.11.11.11.11.11.11.11.11.11.1.m1.5.5.2.2.2.1.1.1.3.cmml">ε</mi></mrow><mo id="A1.T2.11.11.11.11.11.11.11.11.11.1.m1.5.5.2.2.2.1.1.3" stretchy="false" xref="A1.T2.11.11.11.11.11.11.11.11.11.1.m1.5.5.2.2.2.1.1.1.cmml">)</mo></mrow><mo id="A1.T2.11.11.11.11.11.11.11.11.11.1.m1.5.5.2.2.2.2" xref="A1.T2.11.11.11.11.11.11.11.11.11.1.m1.5.5.2.2.2.2.cmml">⁢</mo><msup id="A1.T2.11.11.11.11.11.11.11.11.11.1.m1.5.5.2.2.2.3" xref="A1.T2.11.11.11.11.11.11.11.11.11.1.m1.5.5.2.2.2.3.cmml"><mi id="A1.T2.11.11.11.11.11.11.11.11.11.1.m1.5.5.2.2.2.3.2" xref="A1.T2.11.11.11.11.11.11.11.11.11.1.m1.5.5.2.2.2.3.2.cmml">ρ</mi><mi id="A1.T2.11.11.11.11.11.11.11.11.11.1.m1.5.5.2.2.2.3.3" xref="A1.T2.11.11.11.11.11.11.11.11.11.1.m1.5.5.2.2.2.3.3.cmml">max</mi></msup><mo id="A1.T2.11.11.11.11.11.11.11.11.11.1.m1.5.5.2.2.2.2a" xref="A1.T2.11.11.11.11.11.11.11.11.11.1.m1.5.5.2.2.2.2.cmml">⁢</mo><mrow id="A1.T2.11.11.11.11.11.11.11.11.11.1.m1.5.5.2.2.2.4.2" xref="A1.T2.11.11.11.11.11.11.11.11.11.1.m1.5.5.2.2.2.cmml"><mo id="A1.T2.11.11.11.11.11.11.11.11.11.1.m1.5.5.2.2.2.4.2.1" stretchy="false" xref="A1.T2.11.11.11.11.11.11.11.11.11.1.m1.5.5.2.2.2.cmml">(</mo><mi id="A1.T2.11.11.11.11.11.11.11.11.11.1.m1.3.3" xref="A1.T2.11.11.11.11.11.11.11.11.11.1.m1.3.3.cmml">G</mi><mo id="A1.T2.11.11.11.11.11.11.11.11.11.1.m1.5.5.2.2.2.4.2.2" stretchy="false" xref="A1.T2.11.11.11.11.11.11.11.11.11.1.m1.5.5.2.2.2.cmml">)</mo></mrow></mrow><mo id="A1.T2.11.11.11.11.11.11.11.11.11.1.m1.5.5.2.2.5" stretchy="false" xref="A1.T2.11.11.11.11.11.11.11.11.11.1.m1.5.5.2.3.cmml">]</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="A1.T2.11.11.11.11.11.11.11.11.11.1.m1.5b"><apply id="A1.T2.11.11.11.11.11.11.11.11.11.1.m1.5.5.cmml" xref="A1.T2.11.11.11.11.11.11.11.11.11.1.m1.5.5"><in id="A1.T2.11.11.11.11.11.11.11.11.11.1.m1.5.5.3.cmml" xref="A1.T2.11.11.11.11.11.11.11.11.11.1.m1.5.5.3"></in><apply id="A1.T2.11.11.11.11.11.11.11.11.11.1.m1.5.5.4.cmml" xref="A1.T2.11.11.11.11.11.11.11.11.11.1.m1.5.5.4"><times id="A1.T2.11.11.11.11.11.11.11.11.11.1.m1.5.5.4.1.cmml" xref="A1.T2.11.11.11.11.11.11.11.11.11.1.m1.5.5.4.1"></times><ci id="A1.T2.11.11.11.11.11.11.11.11.11.1.m1.5.5.4.2.cmml" xref="A1.T2.11.11.11.11.11.11.11.11.11.1.m1.5.5.4.2">𝜌</ci><ci id="A1.T2.11.11.11.11.11.11.11.11.11.1.m1.1.1.cmml" xref="A1.T2.11.11.11.11.11.11.11.11.11.1.m1.1.1">𝐻</ci></apply><interval closure="closed" id="A1.T2.11.11.11.11.11.11.11.11.11.1.m1.5.5.2.3.cmml" xref="A1.T2.11.11.11.11.11.11.11.11.11.1.m1.5.5.2.2"><apply id="A1.T2.11.11.11.11.11.11.11.11.11.1.m1.4.4.1.1.1.cmml" xref="A1.T2.11.11.11.11.11.11.11.11.11.1.m1.4.4.1.1.1"><times id="A1.T2.11.11.11.11.11.11.11.11.11.1.m1.4.4.1.1.1.2.cmml" xref="A1.T2.11.11.11.11.11.11.11.11.11.1.m1.4.4.1.1.1.2"></times><apply id="A1.T2.11.11.11.11.11.11.11.11.11.1.m1.4.4.1.1.1.1.cmml" xref="A1.T2.11.11.11.11.11.11.11.11.11.1.m1.4.4.1.1.1.1"><csymbol cd="ambiguous" id="A1.T2.11.11.11.11.11.11.11.11.11.1.m1.4.4.1.1.1.1.2.cmml" xref="A1.T2.11.11.11.11.11.11.11.11.11.1.m1.4.4.1.1.1.1">superscript</csymbol><apply id="A1.T2.11.11.11.11.11.11.11.11.11.1.m1.4.4.1.1.1.1.1.1.1.cmml" xref="A1.T2.11.11.11.11.11.11.11.11.11.1.m1.4.4.1.1.1.1.1.1"><plus id="A1.T2.11.11.11.11.11.11.11.11.11.1.m1.4.4.1.1.1.1.1.1.1.1.cmml" xref="A1.T2.11.11.11.11.11.11.11.11.11.1.m1.4.4.1.1.1.1.1.1.1.1"></plus><cn id="A1.T2.11.11.11.11.11.11.11.11.11.1.m1.4.4.1.1.1.1.1.1.1.2.cmml" type="integer" xref="A1.T2.11.11.11.11.11.11.11.11.11.1.m1.4.4.1.1.1.1.1.1.1.2">2</cn><ci id="A1.T2.11.11.11.11.11.11.11.11.11.1.m1.4.4.1.1.1.1.1.1.1.3.cmml" xref="A1.T2.11.11.11.11.11.11.11.11.11.1.m1.4.4.1.1.1.1.1.1.1.3">𝜀</ci></apply><apply id="A1.T2.11.11.11.11.11.11.11.11.11.1.m1.4.4.1.1.1.1.3.cmml" xref="A1.T2.11.11.11.11.11.11.11.11.11.1.m1.4.4.1.1.1.1.3"><minus id="A1.T2.11.11.11.11.11.11.11.11.11.1.m1.4.4.1.1.1.1.3.1.cmml" xref="A1.T2.11.11.11.11.11.11.11.11.11.1.m1.4.4.1.1.1.1.3"></minus><cn id="A1.T2.11.11.11.11.11.11.11.11.11.1.m1.4.4.1.1.1.1.3.2.cmml" type="integer" xref="A1.T2.11.11.11.11.11.11.11.11.11.1.m1.4.4.1.1.1.1.3.2">1</cn></apply></apply><apply id="A1.T2.11.11.11.11.11.11.11.11.11.1.m1.4.4.1.1.1.3.cmml" xref="A1.T2.11.11.11.11.11.11.11.11.11.1.m1.4.4.1.1.1.3"><csymbol cd="ambiguous" id="A1.T2.11.11.11.11.11.11.11.11.11.1.m1.4.4.1.1.1.3.1.cmml" xref="A1.T2.11.11.11.11.11.11.11.11.11.1.m1.4.4.1.1.1.3">superscript</csymbol><ci id="A1.T2.11.11.11.11.11.11.11.11.11.1.m1.4.4.1.1.1.3.2.cmml" xref="A1.T2.11.11.11.11.11.11.11.11.11.1.m1.4.4.1.1.1.3.2">𝜌</ci><max id="A1.T2.11.11.11.11.11.11.11.11.11.1.m1.4.4.1.1.1.3.3.cmml" xref="A1.T2.11.11.11.11.11.11.11.11.11.1.m1.4.4.1.1.1.3.3"></max></apply><ci id="A1.T2.11.11.11.11.11.11.11.11.11.1.m1.2.2.cmml" xref="A1.T2.11.11.11.11.11.11.11.11.11.1.m1.2.2">𝐺</ci></apply><apply id="A1.T2.11.11.11.11.11.11.11.11.11.1.m1.5.5.2.2.2.cmml" xref="A1.T2.11.11.11.11.11.11.11.11.11.1.m1.5.5.2.2.2"><times id="A1.T2.11.11.11.11.11.11.11.11.11.1.m1.5.5.2.2.2.2.cmml" xref="A1.T2.11.11.11.11.11.11.11.11.11.1.m1.5.5.2.2.2.2"></times><apply id="A1.T2.11.11.11.11.11.11.11.11.11.1.m1.5.5.2.2.2.1.1.1.cmml" xref="A1.T2.11.11.11.11.11.11.11.11.11.1.m1.5.5.2.2.2.1.1"><plus id="A1.T2.11.11.11.11.11.11.11.11.11.1.m1.5.5.2.2.2.1.1.1.1.cmml" xref="A1.T2.11.11.11.11.11.11.11.11.11.1.m1.5.5.2.2.2.1.1.1.1"></plus><cn id="A1.T2.11.11.11.11.11.11.11.11.11.1.m1.5.5.2.2.2.1.1.1.2.cmml" type="integer" xref="A1.T2.11.11.11.11.11.11.11.11.11.1.m1.5.5.2.2.2.1.1.1.2">2</cn><ci id="A1.T2.11.11.11.11.11.11.11.11.11.1.m1.5.5.2.2.2.1.1.1.3.cmml" xref="A1.T2.11.11.11.11.11.11.11.11.11.1.m1.5.5.2.2.2.1.1.1.3">𝜀</ci></apply><apply id="A1.T2.11.11.11.11.11.11.11.11.11.1.m1.5.5.2.2.2.3.cmml" xref="A1.T2.11.11.11.11.11.11.11.11.11.1.m1.5.5.2.2.2.3"><csymbol cd="ambiguous" id="A1.T2.11.11.11.11.11.11.11.11.11.1.m1.5.5.2.2.2.3.1.cmml" xref="A1.T2.11.11.11.11.11.11.11.11.11.1.m1.5.5.2.2.2.3">superscript</csymbol><ci id="A1.T2.11.11.11.11.11.11.11.11.11.1.m1.5.5.2.2.2.3.2.cmml" xref="A1.T2.11.11.11.11.11.11.11.11.11.1.m1.5.5.2.2.2.3.2">𝜌</ci><max id="A1.T2.11.11.11.11.11.11.11.11.11.1.m1.5.5.2.2.2.3.3.cmml" xref="A1.T2.11.11.11.11.11.11.11.11.11.1.m1.5.5.2.2.2.3.3"></max></apply><ci id="A1.T2.11.11.11.11.11.11.11.11.11.1.m1.3.3.cmml" xref="A1.T2.11.11.11.11.11.11.11.11.11.1.m1.3.3">𝐺</ci></apply></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.T2.11.11.11.11.11.11.11.11.11.1.m1.5c">\rho(H)\in[(2+\varepsilon)^{-1}\rho^{\max}(G),(2+\varepsilon)\rho^{\max}(G)]</annotation><annotation encoding="application/x-llamapun" id="A1.T2.11.11.11.11.11.11.11.11.11.1.m1.5d">italic_ρ ( italic_H ) ∈ [ ( 2 + italic_ε ) start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT italic_ρ start_POSTSUPERSCRIPT roman_max end_POSTSUPERSCRIPT ( italic_G ) , ( 2 + italic_ε ) italic_ρ start_POSTSUPERSCRIPT roman_max end_POSTSUPERSCRIPT ( italic_G ) ]</annotation></semantics></math></span> <span class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="A1.T2.12.12.12.12.12.12.12.12.12.2"><math alttext="O(D\cdot\varepsilon^{-1}\log n)" class="ltx_Math" display="inline" id="A1.T2.12.12.12.12.12.12.12.12.12.2.m1.1"><semantics id="A1.T2.12.12.12.12.12.12.12.12.12.2.m1.1a"><mrow id="A1.T2.12.12.12.12.12.12.12.12.12.2.m1.1.1" xref="A1.T2.12.12.12.12.12.12.12.12.12.2.m1.1.1.cmml"><mi id="A1.T2.12.12.12.12.12.12.12.12.12.2.m1.1.1.3" xref="A1.T2.12.12.12.12.12.12.12.12.12.2.m1.1.1.3.cmml">O</mi><mo id="A1.T2.12.12.12.12.12.12.12.12.12.2.m1.1.1.2" xref="A1.T2.12.12.12.12.12.12.12.12.12.2.m1.1.1.2.cmml">⁢</mo><mrow id="A1.T2.12.12.12.12.12.12.12.12.12.2.m1.1.1.1.1" 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xref="A1.T2.12.12.12.12.12.12.12.12.12.2.m1.1.1.1.1.1.2.3.2.cmml">ε</mi><mrow id="A1.T2.12.12.12.12.12.12.12.12.12.2.m1.1.1.1.1.1.2.3.3" xref="A1.T2.12.12.12.12.12.12.12.12.12.2.m1.1.1.1.1.1.2.3.3.cmml"><mo id="A1.T2.12.12.12.12.12.12.12.12.12.2.m1.1.1.1.1.1.2.3.3a" xref="A1.T2.12.12.12.12.12.12.12.12.12.2.m1.1.1.1.1.1.2.3.3.cmml">−</mo><mn id="A1.T2.12.12.12.12.12.12.12.12.12.2.m1.1.1.1.1.1.2.3.3.2" xref="A1.T2.12.12.12.12.12.12.12.12.12.2.m1.1.1.1.1.1.2.3.3.2.cmml">1</mn></mrow></msup></mrow><mo id="A1.T2.12.12.12.12.12.12.12.12.12.2.m1.1.1.1.1.1.1" lspace="0.167em" xref="A1.T2.12.12.12.12.12.12.12.12.12.2.m1.1.1.1.1.1.1.cmml">⁢</mo><mrow id="A1.T2.12.12.12.12.12.12.12.12.12.2.m1.1.1.1.1.1.3" xref="A1.T2.12.12.12.12.12.12.12.12.12.2.m1.1.1.1.1.1.3.cmml"><mi id="A1.T2.12.12.12.12.12.12.12.12.12.2.m1.1.1.1.1.1.3.1" xref="A1.T2.12.12.12.12.12.12.12.12.12.2.m1.1.1.1.1.1.3.1.cmml">log</mi><mo id="A1.T2.12.12.12.12.12.12.12.12.12.2.m1.1.1.1.1.1.3a" lspace="0.167em" xref="A1.T2.12.12.12.12.12.12.12.12.12.2.m1.1.1.1.1.1.3.cmml">⁡</mo><mi id="A1.T2.12.12.12.12.12.12.12.12.12.2.m1.1.1.1.1.1.3.2" xref="A1.T2.12.12.12.12.12.12.12.12.12.2.m1.1.1.1.1.1.3.2.cmml">n</mi></mrow></mrow><mo id="A1.T2.12.12.12.12.12.12.12.12.12.2.m1.1.1.1.1.3" stretchy="false" xref="A1.T2.12.12.12.12.12.12.12.12.12.2.m1.1.1.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="A1.T2.12.12.12.12.12.12.12.12.12.2.m1.1b"><apply id="A1.T2.12.12.12.12.12.12.12.12.12.2.m1.1.1.cmml" xref="A1.T2.12.12.12.12.12.12.12.12.12.2.m1.1.1"><times id="A1.T2.12.12.12.12.12.12.12.12.12.2.m1.1.1.2.cmml" xref="A1.T2.12.12.12.12.12.12.12.12.12.2.m1.1.1.2"></times><ci id="A1.T2.12.12.12.12.12.12.12.12.12.2.m1.1.1.3.cmml" xref="A1.T2.12.12.12.12.12.12.12.12.12.2.m1.1.1.3">𝑂</ci><apply id="A1.T2.12.12.12.12.12.12.12.12.12.2.m1.1.1.1.1.1.cmml" xref="A1.T2.12.12.12.12.12.12.12.12.12.2.m1.1.1.1.1"><times id="A1.T2.12.12.12.12.12.12.12.12.12.2.m1.1.1.1.1.1.1.cmml" xref="A1.T2.12.12.12.12.12.12.12.12.12.2.m1.1.1.1.1.1.1"></times><apply id="A1.T2.12.12.12.12.12.12.12.12.12.2.m1.1.1.1.1.1.2.cmml" xref="A1.T2.12.12.12.12.12.12.12.12.12.2.m1.1.1.1.1.1.2"><ci id="A1.T2.12.12.12.12.12.12.12.12.12.2.m1.1.1.1.1.1.2.1.cmml" xref="A1.T2.12.12.12.12.12.12.12.12.12.2.m1.1.1.1.1.1.2.1">⋅</ci><ci id="A1.T2.12.12.12.12.12.12.12.12.12.2.m1.1.1.1.1.1.2.2.cmml" xref="A1.T2.12.12.12.12.12.12.12.12.12.2.m1.1.1.1.1.1.2.2">𝐷</ci><apply id="A1.T2.12.12.12.12.12.12.12.12.12.2.m1.1.1.1.1.1.2.3.cmml" xref="A1.T2.12.12.12.12.12.12.12.12.12.2.m1.1.1.1.1.1.2.3"><csymbol cd="ambiguous" id="A1.T2.12.12.12.12.12.12.12.12.12.2.m1.1.1.1.1.1.2.3.1.cmml" xref="A1.T2.12.12.12.12.12.12.12.12.12.2.m1.1.1.1.1.1.2.3">superscript</csymbol><ci id="A1.T2.12.12.12.12.12.12.12.12.12.2.m1.1.1.1.1.1.2.3.2.cmml" xref="A1.T2.12.12.12.12.12.12.12.12.12.2.m1.1.1.1.1.1.2.3.2">𝜀</ci><apply id="A1.T2.12.12.12.12.12.12.12.12.12.2.m1.1.1.1.1.1.2.3.3.cmml" xref="A1.T2.12.12.12.12.12.12.12.12.12.2.m1.1.1.1.1.1.2.3.3"><minus id="A1.T2.12.12.12.12.12.12.12.12.12.2.m1.1.1.1.1.1.2.3.3.1.cmml" xref="A1.T2.12.12.12.12.12.12.12.12.12.2.m1.1.1.1.1.1.2.3.3"></minus><cn id="A1.T2.12.12.12.12.12.12.12.12.12.2.m1.1.1.1.1.1.2.3.3.2.cmml" type="integer" xref="A1.T2.12.12.12.12.12.12.12.12.12.2.m1.1.1.1.1.1.2.3.3.2">1</cn></apply></apply></apply><apply id="A1.T2.12.12.12.12.12.12.12.12.12.2.m1.1.1.1.1.1.3.cmml" xref="A1.T2.12.12.12.12.12.12.12.12.12.2.m1.1.1.1.1.1.3"><log id="A1.T2.12.12.12.12.12.12.12.12.12.2.m1.1.1.1.1.1.3.1.cmml" xref="A1.T2.12.12.12.12.12.12.12.12.12.2.m1.1.1.1.1.1.3.1"></log><ci id="A1.T2.12.12.12.12.12.12.12.12.12.2.m1.1.1.1.1.1.3.2.cmml" xref="A1.T2.12.12.12.12.12.12.12.12.12.2.m1.1.1.1.1.1.3.2">𝑛</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.T2.12.12.12.12.12.12.12.12.12.2.m1.1c">O(D\cdot\varepsilon^{-1}\log n)</annotation><annotation encoding="application/x-llamapun" id="A1.T2.12.12.12.12.12.12.12.12.12.2.m1.1d">italic_O ( italic_D ⋅ italic_ε start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT roman_log italic_n )</annotation></semantics></math></span> <span class="ltx_td ltx_align_center ltx_border_t" id="A1.T2.12.12.12.12.12.12.12.12.12.5"><cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2411.12694v2#bib.bib11" title="">11</a>]</cite></span></span> <span class="ltx_tr" id="A1.T2.14.14.14.14.14.14.14.14.14"> <span class="ltx_td ltx_border_r" id="A1.T2.14.14.14.14.14.14.14.14.14.3"></span> <span class="ltx_td ltx_align_center ltx_border_r" id="A1.T2.14.14.14.14.14.14.14.14.14.4"><a class="ltx_ref" href="https://arxiv.org/html/2411.12694v2#A1.Thmtheorem1" title="Problem A.1. ‣ A.1 Related work: distributed densest subgraph reporting ‣ Appendix A Reporting a densest subgraph ‣ Local Density and its Distributed Approximation"><span class="ltx_text ltx_ref_tag">A.1</span></a>.2</span> <span class="ltx_td 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)</annotation></semantics></math></span></span> <span class="ltx_td ltx_align_center" id="A1.T2.14.14.14.14.14.14.14.14.14.5"><cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2411.12694v2#bib.bib21" title="">21</a>]</cite></span></span> <span class="ltx_tr" id="A1.T2.16.16.16.16.16.16.16.16.16"> <span class="ltx_td ltx_border_r" id="A1.T2.16.16.16.16.16.16.16.16.16.3"></span> <span class="ltx_td ltx_align_center ltx_border_r" id="A1.T2.16.16.16.16.16.16.16.16.16.4"><a class="ltx_ref ltx_font_bold" href="https://arxiv.org/html/2411.12694v2#A1.Thmtheorem1" title="Problem A.1. ‣ A.1 Related work: distributed densest subgraph reporting ‣ Appendix A Reporting a densest subgraph ‣ Local Density and its Distributed Approximation"><span class="ltx_text ltx_ref_tag">A.1</span></a><span class="ltx_text ltx_font_bold" id="A1.T2.16.16.16.16.16.16.16.16.16.4.1">.2</span></span> <span class="ltx_td ltx_align_center ltx_border_r" 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id="A1.T2.16.16.16.16.16.16.16.16.16.5.1">Thm. <a class="ltx_ref" href="https://arxiv.org/html/2411.12694v2#A1.Thmtheorem11" title="Theorem A.11. ‣ Analysis: ‣ A.3 Our algorithm ‣ Appendix A Reporting a densest subgraph ‣ Local Density and its Distributed Approximation"><span class="ltx_text ltx_ref_tag">A.11</span></a></span></span></span> <span class="ltx_tr" id="A1.T2.18.18.18.18.18.18.18.18.18"> <span class="ltx_td ltx_border_r" id="A1.T2.18.18.18.18.18.18.18.18.18.3"></span> <span class="ltx_td ltx_align_center ltx_border_r" id="A1.T2.18.18.18.18.18.18.18.18.18.4"><a class="ltx_ref ltx_font_bold" href="https://arxiv.org/html/2411.12694v2#A1.Thmtheorem2" title="Problem A.2. ‣ A.2 Reporting a locally densest subgraph ‣ Appendix A Reporting a densest subgraph ‣ Local Density and its Distributed Approximation"><span class="ltx_text ltx_ref_tag">A.2</span></a></span> <span class="ltx_td ltx_align_center ltx_border_r" id="A1.T2.17.17.17.17.17.17.17.17.17.1"><math 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id="A1.T2.18.18.18.18.18.18.18.18.18.5.1">Thm. <a class="ltx_ref" href="https://arxiv.org/html/2411.12694v2#A1.Thmtheorem11" title="Theorem A.11. ‣ Analysis: ‣ A.3 Our algorithm ‣ Appendix A Reporting a densest subgraph ‣ Local Density and its Distributed Approximation"><span class="ltx_text ltx_ref_tag">A.11</span></a></span></span></span> <span class="ltx_tr" id="A1.T2.20.20.20.20.20.20.20.20.20"> <span class="ltx_td ltx_border_r" id="A1.T2.20.20.20.20.20.20.20.20.20.3"></span> <span class="ltx_td ltx_align_center ltx_border_r" id="A1.T2.20.20.20.20.20.20.20.20.20.4"><a class="ltx_ref ltx_font_bold" href="https://arxiv.org/html/2411.12694v2#A1.Thmtheorem2" title="Problem A.2. ‣ A.2 Reporting a locally densest subgraph ‣ Appendix A Reporting a densest subgraph ‣ Local Density and its Distributed Approximation"><span class="ltx_text ltx_ref_tag">A.2</span></a></span> <span class="ltx_td ltx_align_center ltx_border_r" id="A1.T2.19.19.19.19.19.19.19.19.19.1"><math 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end_POSTSUPERSCRIPT bold_} bold_)</annotation></semantics></math></span> <span class="ltx_td ltx_align_center" id="A1.T2.20.20.20.20.20.20.20.20.20.5"><span class="ltx_text ltx_font_bold" id="A1.T2.20.20.20.20.20.20.20.20.20.5.1">Thm. <a class="ltx_ref" href="https://arxiv.org/html/2411.12694v2#A1.Thmtheorem11" title="Theorem A.11. ‣ Analysis: ‣ A.3 Our algorithm ‣ Appendix A Reporting a densest subgraph ‣ Local Density and its Distributed Approximation"><span class="ltx_text ltx_ref_tag">A.11</span></a></span></span></span> </span></span></span> </span></span></span></p> <figcaption class="ltx_caption ltx_centering"><span class="ltx_tag ltx_tag_table"><span class="ltx_text" id="A1.T2.24.2.1" style="font-size:90%;">Table 2</span>: </span><span class="ltx_text" id="A1.T2.22.1" style="font-size:90%;">Our results in LOCAL (L) or CONGEST (C) for the reporting variant of our problem. <math alttext="D" class="ltx_Math" display="inline" id="A1.T2.22.1.m1.1"><semantics id="A1.T2.22.1.m1.1b"><mi id="A1.T2.22.1.m1.1.1" xref="A1.T2.22.1.m1.1.1.cmml">D</mi><annotation-xml encoding="MathML-Content" id="A1.T2.22.1.m1.1c"><ci id="A1.T2.22.1.m1.1.1.cmml" xref="A1.T2.22.1.m1.1.1">𝐷</ci></annotation-xml><annotation encoding="application/x-tex" id="A1.T2.22.1.m1.1d">D</annotation><annotation encoding="application/x-llamapun" id="A1.T2.22.1.m1.1e">italic_D</annotation></semantics></math> denotes the diameter. Orange running times are not deterministic and occur with high probability. </span></figcaption> </figure> </section> <section class="ltx_subsection" id="A1.SS3"> <h3 class="ltx_title ltx_title_subsection"> <span class="ltx_tag ltx_tag_subsection">A.3 </span>Our algorithm</h3> <div class="ltx_para" id="A1.SS3.p1"> <p class="ltx_p" id="A1.SS3.p1.11">So far our algorithm is designed to simply output an estimate of the local density at every vertex. It is easily extended to output an approximate value of the maximum subgraph density in <math alttext="O(\text{diameter})" class="ltx_Math" display="inline" id="A1.SS3.p1.1.m1.1"><semantics id="A1.SS3.p1.1.m1.1a"><mrow id="A1.SS3.p1.1.m1.1.2" xref="A1.SS3.p1.1.m1.1.2.cmml"><mi id="A1.SS3.p1.1.m1.1.2.2" xref="A1.SS3.p1.1.m1.1.2.2.cmml">O</mi><mo id="A1.SS3.p1.1.m1.1.2.1" xref="A1.SS3.p1.1.m1.1.2.1.cmml">⁢</mo><mrow id="A1.SS3.p1.1.m1.1.2.3.2" xref="A1.SS3.p1.1.m1.1.1a.cmml"><mo id="A1.SS3.p1.1.m1.1.2.3.2.1" stretchy="false" xref="A1.SS3.p1.1.m1.1.1a.cmml">(</mo><mtext id="A1.SS3.p1.1.m1.1.1" xref="A1.SS3.p1.1.m1.1.1.cmml">diameter</mtext><mo id="A1.SS3.p1.1.m1.1.2.3.2.2" stretchy="false" xref="A1.SS3.p1.1.m1.1.1a.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="A1.SS3.p1.1.m1.1b"><apply id="A1.SS3.p1.1.m1.1.2.cmml" xref="A1.SS3.p1.1.m1.1.2"><times id="A1.SS3.p1.1.m1.1.2.1.cmml" xref="A1.SS3.p1.1.m1.1.2.1"></times><ci id="A1.SS3.p1.1.m1.1.2.2.cmml" xref="A1.SS3.p1.1.m1.1.2.2">𝑂</ci><ci id="A1.SS3.p1.1.m1.1.1a.cmml" xref="A1.SS3.p1.1.m1.1.2.3.2"><mtext id="A1.SS3.p1.1.m1.1.1.cmml" xref="A1.SS3.p1.1.m1.1.1">diameter</mtext></ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS3.p1.1.m1.1c">O(\text{diameter})</annotation><annotation encoding="application/x-llamapun" id="A1.SS3.p1.1.m1.1d">italic_O ( diameter )</annotation></semantics></math> many rounds, by computing an <math alttext="\eta" class="ltx_Math" display="inline" id="A1.SS3.p1.2.m2.1"><semantics id="A1.SS3.p1.2.m2.1a"><mi id="A1.SS3.p1.2.m2.1.1" xref="A1.SS3.p1.2.m2.1.1.cmml">η</mi><annotation-xml encoding="MathML-Content" id="A1.SS3.p1.2.m2.1b"><ci id="A1.SS3.p1.2.m2.1.1.cmml" xref="A1.SS3.p1.2.m2.1.1">𝜂</ci></annotation-xml><annotation encoding="application/x-tex" id="A1.SS3.p1.2.m2.1c">\eta</annotation><annotation encoding="application/x-llamapun" id="A1.SS3.p1.2.m2.1d">italic_η</annotation></semantics></math>-fair out-orientation (for the correct value of <math alttext="\eta" class="ltx_Math" display="inline" id="A1.SS3.p1.3.m3.1"><semantics id="A1.SS3.p1.3.m3.1a"><mi id="A1.SS3.p1.3.m3.1.1" xref="A1.SS3.p1.3.m3.1.1.cmml">η</mi><annotation-xml encoding="MathML-Content" id="A1.SS3.p1.3.m3.1b"><ci id="A1.SS3.p1.3.m3.1.1.cmml" xref="A1.SS3.p1.3.m3.1.1">𝜂</ci></annotation-xml><annotation encoding="application/x-tex" id="A1.SS3.p1.3.m3.1c">\eta</annotation><annotation encoding="application/x-llamapun" id="A1.SS3.p1.3.m3.1d">italic_η</annotation></semantics></math>) and subsequently broadcasting the maximum fractional out-degree which provides the desired approximate value of <math alttext="\rho^{\max}(G)" class="ltx_Math" display="inline" id="A1.SS3.p1.4.m4.1"><semantics id="A1.SS3.p1.4.m4.1a"><mrow id="A1.SS3.p1.4.m4.1.2" xref="A1.SS3.p1.4.m4.1.2.cmml"><msup id="A1.SS3.p1.4.m4.1.2.2" xref="A1.SS3.p1.4.m4.1.2.2.cmml"><mi id="A1.SS3.p1.4.m4.1.2.2.2" xref="A1.SS3.p1.4.m4.1.2.2.2.cmml">ρ</mi><mi id="A1.SS3.p1.4.m4.1.2.2.3" xref="A1.SS3.p1.4.m4.1.2.2.3.cmml">max</mi></msup><mo id="A1.SS3.p1.4.m4.1.2.1" xref="A1.SS3.p1.4.m4.1.2.1.cmml">⁢</mo><mrow id="A1.SS3.p1.4.m4.1.2.3.2" xref="A1.SS3.p1.4.m4.1.2.cmml"><mo id="A1.SS3.p1.4.m4.1.2.3.2.1" stretchy="false" xref="A1.SS3.p1.4.m4.1.2.cmml">(</mo><mi id="A1.SS3.p1.4.m4.1.1" xref="A1.SS3.p1.4.m4.1.1.cmml">G</mi><mo id="A1.SS3.p1.4.m4.1.2.3.2.2" stretchy="false" xref="A1.SS3.p1.4.m4.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="A1.SS3.p1.4.m4.1b"><apply id="A1.SS3.p1.4.m4.1.2.cmml" xref="A1.SS3.p1.4.m4.1.2"><times id="A1.SS3.p1.4.m4.1.2.1.cmml" xref="A1.SS3.p1.4.m4.1.2.1"></times><apply id="A1.SS3.p1.4.m4.1.2.2.cmml" xref="A1.SS3.p1.4.m4.1.2.2"><csymbol cd="ambiguous" id="A1.SS3.p1.4.m4.1.2.2.1.cmml" xref="A1.SS3.p1.4.m4.1.2.2">superscript</csymbol><ci id="A1.SS3.p1.4.m4.1.2.2.2.cmml" xref="A1.SS3.p1.4.m4.1.2.2.2">𝜌</ci><max id="A1.SS3.p1.4.m4.1.2.2.3.cmml" xref="A1.SS3.p1.4.m4.1.2.2.3"></max></apply><ci id="A1.SS3.p1.4.m4.1.1.cmml" xref="A1.SS3.p1.4.m4.1.1">𝐺</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS3.p1.4.m4.1c">\rho^{\max}(G)</annotation><annotation encoding="application/x-llamapun" id="A1.SS3.p1.4.m4.1d">italic_ρ start_POSTSUPERSCRIPT roman_max end_POSTSUPERSCRIPT ( italic_G )</annotation></semantics></math>. We can also extend the algorithm for computing an <math alttext="\eta" class="ltx_Math" display="inline" id="A1.SS3.p1.5.m5.1"><semantics id="A1.SS3.p1.5.m5.1a"><mi id="A1.SS3.p1.5.m5.1.1" xref="A1.SS3.p1.5.m5.1.1.cmml">η</mi><annotation-xml encoding="MathML-Content" id="A1.SS3.p1.5.m5.1b"><ci id="A1.SS3.p1.5.m5.1.1.cmml" xref="A1.SS3.p1.5.m5.1.1">𝜂</ci></annotation-xml><annotation encoding="application/x-tex" id="A1.SS3.p1.5.m5.1c">\eta</annotation><annotation encoding="application/x-llamapun" id="A1.SS3.p1.5.m5.1d">italic_η</annotation></semantics></math>-fair orientation to one that outputs a subgraph with no smaller than <math alttext="(1-\varepsilon)\tilde{D}" class="ltx_Math" display="inline" id="A1.SS3.p1.6.m6.1"><semantics id="A1.SS3.p1.6.m6.1a"><mrow id="A1.SS3.p1.6.m6.1.1" xref="A1.SS3.p1.6.m6.1.1.cmml"><mrow id="A1.SS3.p1.6.m6.1.1.1.1" xref="A1.SS3.p1.6.m6.1.1.1.1.1.cmml"><mo id="A1.SS3.p1.6.m6.1.1.1.1.2" stretchy="false" xref="A1.SS3.p1.6.m6.1.1.1.1.1.cmml">(</mo><mrow id="A1.SS3.p1.6.m6.1.1.1.1.1" xref="A1.SS3.p1.6.m6.1.1.1.1.1.cmml"><mn id="A1.SS3.p1.6.m6.1.1.1.1.1.2" xref="A1.SS3.p1.6.m6.1.1.1.1.1.2.cmml">1</mn><mo id="A1.SS3.p1.6.m6.1.1.1.1.1.1" xref="A1.SS3.p1.6.m6.1.1.1.1.1.1.cmml">−</mo><mi id="A1.SS3.p1.6.m6.1.1.1.1.1.3" xref="A1.SS3.p1.6.m6.1.1.1.1.1.3.cmml">ε</mi></mrow><mo id="A1.SS3.p1.6.m6.1.1.1.1.3" stretchy="false" xref="A1.SS3.p1.6.m6.1.1.1.1.1.cmml">)</mo></mrow><mo id="A1.SS3.p1.6.m6.1.1.2" xref="A1.SS3.p1.6.m6.1.1.2.cmml">⁢</mo><mover accent="true" id="A1.SS3.p1.6.m6.1.1.3" xref="A1.SS3.p1.6.m6.1.1.3.cmml"><mi id="A1.SS3.p1.6.m6.1.1.3.2" xref="A1.SS3.p1.6.m6.1.1.3.2.cmml">D</mi><mo id="A1.SS3.p1.6.m6.1.1.3.1" xref="A1.SS3.p1.6.m6.1.1.3.1.cmml">~</mo></mover></mrow><annotation-xml encoding="MathML-Content" id="A1.SS3.p1.6.m6.1b"><apply id="A1.SS3.p1.6.m6.1.1.cmml" xref="A1.SS3.p1.6.m6.1.1"><times id="A1.SS3.p1.6.m6.1.1.2.cmml" xref="A1.SS3.p1.6.m6.1.1.2"></times><apply id="A1.SS3.p1.6.m6.1.1.1.1.1.cmml" xref="A1.SS3.p1.6.m6.1.1.1.1"><minus id="A1.SS3.p1.6.m6.1.1.1.1.1.1.cmml" xref="A1.SS3.p1.6.m6.1.1.1.1.1.1"></minus><cn id="A1.SS3.p1.6.m6.1.1.1.1.1.2.cmml" type="integer" xref="A1.SS3.p1.6.m6.1.1.1.1.1.2">1</cn><ci id="A1.SS3.p1.6.m6.1.1.1.1.1.3.cmml" xref="A1.SS3.p1.6.m6.1.1.1.1.1.3">𝜀</ci></apply><apply id="A1.SS3.p1.6.m6.1.1.3.cmml" xref="A1.SS3.p1.6.m6.1.1.3"><ci id="A1.SS3.p1.6.m6.1.1.3.1.cmml" xref="A1.SS3.p1.6.m6.1.1.3.1">~</ci><ci id="A1.SS3.p1.6.m6.1.1.3.2.cmml" xref="A1.SS3.p1.6.m6.1.1.3.2">𝐷</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS3.p1.6.m6.1c">(1-\varepsilon)\tilde{D}</annotation><annotation encoding="application/x-llamapun" id="A1.SS3.p1.6.m6.1d">( 1 - italic_ε ) over~ start_ARG italic_D end_ARG</annotation></semantics></math>, for some input parameter <math alttext="\tilde{D}\leq\rho^{\max}(G)" class="ltx_Math" display="inline" id="A1.SS3.p1.7.m7.1"><semantics id="A1.SS3.p1.7.m7.1a"><mrow id="A1.SS3.p1.7.m7.1.2" xref="A1.SS3.p1.7.m7.1.2.cmml"><mover accent="true" id="A1.SS3.p1.7.m7.1.2.2" xref="A1.SS3.p1.7.m7.1.2.2.cmml"><mi id="A1.SS3.p1.7.m7.1.2.2.2" xref="A1.SS3.p1.7.m7.1.2.2.2.cmml">D</mi><mo id="A1.SS3.p1.7.m7.1.2.2.1" xref="A1.SS3.p1.7.m7.1.2.2.1.cmml">~</mo></mover><mo id="A1.SS3.p1.7.m7.1.2.1" xref="A1.SS3.p1.7.m7.1.2.1.cmml">≤</mo><mrow id="A1.SS3.p1.7.m7.1.2.3" xref="A1.SS3.p1.7.m7.1.2.3.cmml"><msup id="A1.SS3.p1.7.m7.1.2.3.2" xref="A1.SS3.p1.7.m7.1.2.3.2.cmml"><mi id="A1.SS3.p1.7.m7.1.2.3.2.2" xref="A1.SS3.p1.7.m7.1.2.3.2.2.cmml">ρ</mi><mi id="A1.SS3.p1.7.m7.1.2.3.2.3" xref="A1.SS3.p1.7.m7.1.2.3.2.3.cmml">max</mi></msup><mo id="A1.SS3.p1.7.m7.1.2.3.1" xref="A1.SS3.p1.7.m7.1.2.3.1.cmml">⁢</mo><mrow id="A1.SS3.p1.7.m7.1.2.3.3.2" xref="A1.SS3.p1.7.m7.1.2.3.cmml"><mo id="A1.SS3.p1.7.m7.1.2.3.3.2.1" stretchy="false" xref="A1.SS3.p1.7.m7.1.2.3.cmml">(</mo><mi id="A1.SS3.p1.7.m7.1.1" xref="A1.SS3.p1.7.m7.1.1.cmml">G</mi><mo id="A1.SS3.p1.7.m7.1.2.3.3.2.2" stretchy="false" xref="A1.SS3.p1.7.m7.1.2.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="A1.SS3.p1.7.m7.1b"><apply id="A1.SS3.p1.7.m7.1.2.cmml" xref="A1.SS3.p1.7.m7.1.2"><leq id="A1.SS3.p1.7.m7.1.2.1.cmml" xref="A1.SS3.p1.7.m7.1.2.1"></leq><apply id="A1.SS3.p1.7.m7.1.2.2.cmml" xref="A1.SS3.p1.7.m7.1.2.2"><ci id="A1.SS3.p1.7.m7.1.2.2.1.cmml" xref="A1.SS3.p1.7.m7.1.2.2.1">~</ci><ci id="A1.SS3.p1.7.m7.1.2.2.2.cmml" xref="A1.SS3.p1.7.m7.1.2.2.2">𝐷</ci></apply><apply id="A1.SS3.p1.7.m7.1.2.3.cmml" xref="A1.SS3.p1.7.m7.1.2.3"><times id="A1.SS3.p1.7.m7.1.2.3.1.cmml" xref="A1.SS3.p1.7.m7.1.2.3.1"></times><apply id="A1.SS3.p1.7.m7.1.2.3.2.cmml" xref="A1.SS3.p1.7.m7.1.2.3.2"><csymbol cd="ambiguous" id="A1.SS3.p1.7.m7.1.2.3.2.1.cmml" xref="A1.SS3.p1.7.m7.1.2.3.2">superscript</csymbol><ci id="A1.SS3.p1.7.m7.1.2.3.2.2.cmml" xref="A1.SS3.p1.7.m7.1.2.3.2.2">𝜌</ci><max id="A1.SS3.p1.7.m7.1.2.3.2.3.cmml" xref="A1.SS3.p1.7.m7.1.2.3.2.3"></max></apply><ci id="A1.SS3.p1.7.m7.1.1.cmml" xref="A1.SS3.p1.7.m7.1.1">𝐺</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS3.p1.7.m7.1c">\tilde{D}\leq\rho^{\max}(G)</annotation><annotation encoding="application/x-llamapun" id="A1.SS3.p1.7.m7.1d">over~ start_ARG italic_D end_ARG ≤ italic_ρ start_POSTSUPERSCRIPT roman_max end_POSTSUPERSCRIPT ( italic_G )</annotation></semantics></math>. Specifically, we show how to output <math alttext="0" class="ltx_Math" display="inline" id="A1.SS3.p1.8.m8.1"><semantics id="A1.SS3.p1.8.m8.1a"><mn id="A1.SS3.p1.8.m8.1.1" xref="A1.SS3.p1.8.m8.1.1.cmml">0</mn><annotation-xml encoding="MathML-Content" id="A1.SS3.p1.8.m8.1b"><cn id="A1.SS3.p1.8.m8.1.1.cmml" type="integer" xref="A1.SS3.p1.8.m8.1.1">0</cn></annotation-xml></semantics></math> or <math alttext="1" class="ltx_Math" display="inline" id="A1.SS3.p1.9.m9.1"><semantics id="A1.SS3.p1.9.m9.1a"><mn id="A1.SS3.p1.9.m9.1.1" xref="A1.SS3.p1.9.m9.1.1.cmml">1</mn><annotation-xml encoding="MathML-Content" id="A1.SS3.p1.9.m9.1b"><cn id="A1.SS3.p1.9.m9.1.1.cmml" type="integer" xref="A1.SS3.p1.9.m9.1.1">1</cn></annotation-xml><annotation encoding="application/x-tex" id="A1.SS3.p1.9.m9.1c">1</annotation><annotation encoding="application/x-llamapun" id="A1.SS3.p1.9.m9.1d">1</annotation></semantics></math> at each vertex such that the graph induced by all vertices outputting <math alttext="1" class="ltx_Math" display="inline" id="A1.SS3.p1.10.m10.1"><semantics id="A1.SS3.p1.10.m10.1a"><mn id="A1.SS3.p1.10.m10.1.1" xref="A1.SS3.p1.10.m10.1.1.cmml">1</mn><annotation-xml encoding="MathML-Content" id="A1.SS3.p1.10.m10.1b"><cn id="A1.SS3.p1.10.m10.1.1.cmml" type="integer" xref="A1.SS3.p1.10.m10.1.1">1</cn></annotation-xml><annotation encoding="application/x-tex" id="A1.SS3.p1.10.m10.1c">1</annotation><annotation encoding="application/x-llamapun" id="A1.SS3.p1.10.m10.1d">1</annotation></semantics></math> forms a subgraph with density at least <math alttext="(1-\varepsilon)\tilde{D}" class="ltx_Math" display="inline" id="A1.SS3.p1.11.m11.1"><semantics id="A1.SS3.p1.11.m11.1a"><mrow id="A1.SS3.p1.11.m11.1.1" xref="A1.SS3.p1.11.m11.1.1.cmml"><mrow id="A1.SS3.p1.11.m11.1.1.1.1" xref="A1.SS3.p1.11.m11.1.1.1.1.1.cmml"><mo id="A1.SS3.p1.11.m11.1.1.1.1.2" stretchy="false" xref="A1.SS3.p1.11.m11.1.1.1.1.1.cmml">(</mo><mrow id="A1.SS3.p1.11.m11.1.1.1.1.1" xref="A1.SS3.p1.11.m11.1.1.1.1.1.cmml"><mn id="A1.SS3.p1.11.m11.1.1.1.1.1.2" xref="A1.SS3.p1.11.m11.1.1.1.1.1.2.cmml">1</mn><mo id="A1.SS3.p1.11.m11.1.1.1.1.1.1" xref="A1.SS3.p1.11.m11.1.1.1.1.1.1.cmml">−</mo><mi id="A1.SS3.p1.11.m11.1.1.1.1.1.3" xref="A1.SS3.p1.11.m11.1.1.1.1.1.3.cmml">ε</mi></mrow><mo id="A1.SS3.p1.11.m11.1.1.1.1.3" stretchy="false" xref="A1.SS3.p1.11.m11.1.1.1.1.1.cmml">)</mo></mrow><mo id="A1.SS3.p1.11.m11.1.1.2" xref="A1.SS3.p1.11.m11.1.1.2.cmml">⁢</mo><mover accent="true" id="A1.SS3.p1.11.m11.1.1.3" xref="A1.SS3.p1.11.m11.1.1.3.cmml"><mi id="A1.SS3.p1.11.m11.1.1.3.2" xref="A1.SS3.p1.11.m11.1.1.3.2.cmml">D</mi><mo id="A1.SS3.p1.11.m11.1.1.3.1" xref="A1.SS3.p1.11.m11.1.1.3.1.cmml">~</mo></mover></mrow><annotation-xml encoding="MathML-Content" id="A1.SS3.p1.11.m11.1b"><apply id="A1.SS3.p1.11.m11.1.1.cmml" xref="A1.SS3.p1.11.m11.1.1"><times id="A1.SS3.p1.11.m11.1.1.2.cmml" xref="A1.SS3.p1.11.m11.1.1.2"></times><apply id="A1.SS3.p1.11.m11.1.1.1.1.1.cmml" xref="A1.SS3.p1.11.m11.1.1.1.1"><minus id="A1.SS3.p1.11.m11.1.1.1.1.1.1.cmml" xref="A1.SS3.p1.11.m11.1.1.1.1.1.1"></minus><cn id="A1.SS3.p1.11.m11.1.1.1.1.1.2.cmml" type="integer" xref="A1.SS3.p1.11.m11.1.1.1.1.1.2">1</cn><ci id="A1.SS3.p1.11.m11.1.1.1.1.1.3.cmml" xref="A1.SS3.p1.11.m11.1.1.1.1.1.3">𝜀</ci></apply><apply id="A1.SS3.p1.11.m11.1.1.3.cmml" xref="A1.SS3.p1.11.m11.1.1.3"><ci id="A1.SS3.p1.11.m11.1.1.3.1.cmml" xref="A1.SS3.p1.11.m11.1.1.3.1">~</ci><ci id="A1.SS3.p1.11.m11.1.1.3.2.cmml" xref="A1.SS3.p1.11.m11.1.1.3.2">𝐷</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS3.p1.11.m11.1c">(1-\varepsilon)\tilde{D}</annotation><annotation encoding="application/x-llamapun" id="A1.SS3.p1.11.m11.1d">( 1 - italic_ε ) over~ start_ARG italic_D end_ARG</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="A1.SS3.p2"> <p class="ltx_p" id="A1.SS3.p2.1">Our reduction relies on the following generalisation of the work by Su and Vu <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2411.12694v2#bib.bib21" title="">21</a>]</cite>. Su and Vu showed their version of the lemma by appealing to network decompositions. We give a simple argument without appealing to network decompositions.</p> </div> <div class="ltx_theorem ltx_theorem_lemma" id="A1.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="A1.Thmtheorem3.1.1.1">Lemma A.3</span></span><span class="ltx_text ltx_font_bold" id="A1.Thmtheorem3.2.2">.</span> </h6> <div class="ltx_para" id="A1.Thmtheorem3.p1"> <p class="ltx_p" id="A1.Thmtheorem3.p1.9"><span class="ltx_text ltx_font_italic" id="A1.Thmtheorem3.p1.9.9">Let <math alttext="1\geq\varepsilon>0" class="ltx_Math" display="inline" id="A1.Thmtheorem3.p1.1.1.m1.1"><semantics id="A1.Thmtheorem3.p1.1.1.m1.1a"><mrow id="A1.Thmtheorem3.p1.1.1.m1.1.1" xref="A1.Thmtheorem3.p1.1.1.m1.1.1.cmml"><mn id="A1.Thmtheorem3.p1.1.1.m1.1.1.2" xref="A1.Thmtheorem3.p1.1.1.m1.1.1.2.cmml">1</mn><mo id="A1.Thmtheorem3.p1.1.1.m1.1.1.3" xref="A1.Thmtheorem3.p1.1.1.m1.1.1.3.cmml">≥</mo><mi id="A1.Thmtheorem3.p1.1.1.m1.1.1.4" xref="A1.Thmtheorem3.p1.1.1.m1.1.1.4.cmml">ε</mi><mo id="A1.Thmtheorem3.p1.1.1.m1.1.1.5" xref="A1.Thmtheorem3.p1.1.1.m1.1.1.5.cmml">></mo><mn id="A1.Thmtheorem3.p1.1.1.m1.1.1.6" xref="A1.Thmtheorem3.p1.1.1.m1.1.1.6.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="A1.Thmtheorem3.p1.1.1.m1.1b"><apply id="A1.Thmtheorem3.p1.1.1.m1.1.1.cmml" xref="A1.Thmtheorem3.p1.1.1.m1.1.1"><and id="A1.Thmtheorem3.p1.1.1.m1.1.1a.cmml" xref="A1.Thmtheorem3.p1.1.1.m1.1.1"></and><apply id="A1.Thmtheorem3.p1.1.1.m1.1.1b.cmml" xref="A1.Thmtheorem3.p1.1.1.m1.1.1"><geq id="A1.Thmtheorem3.p1.1.1.m1.1.1.3.cmml" xref="A1.Thmtheorem3.p1.1.1.m1.1.1.3"></geq><cn id="A1.Thmtheorem3.p1.1.1.m1.1.1.2.cmml" type="integer" xref="A1.Thmtheorem3.p1.1.1.m1.1.1.2">1</cn><ci id="A1.Thmtheorem3.p1.1.1.m1.1.1.4.cmml" xref="A1.Thmtheorem3.p1.1.1.m1.1.1.4">𝜀</ci></apply><apply id="A1.Thmtheorem3.p1.1.1.m1.1.1c.cmml" xref="A1.Thmtheorem3.p1.1.1.m1.1.1"><gt id="A1.Thmtheorem3.p1.1.1.m1.1.1.5.cmml" xref="A1.Thmtheorem3.p1.1.1.m1.1.1.5"></gt><share href="https://arxiv.org/html/2411.12694v2#A1.Thmtheorem3.p1.1.1.m1.1.1.4.cmml" id="A1.Thmtheorem3.p1.1.1.m1.1.1d.cmml" xref="A1.Thmtheorem3.p1.1.1.m1.1.1"></share><cn id="A1.Thmtheorem3.p1.1.1.m1.1.1.6.cmml" type="integer" xref="A1.Thmtheorem3.p1.1.1.m1.1.1.6">0</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.Thmtheorem3.p1.1.1.m1.1c">1\geq\varepsilon>0</annotation><annotation encoding="application/x-llamapun" id="A1.Thmtheorem3.p1.1.1.m1.1d">1 ≥ italic_ε > 0</annotation></semantics></math> and <math alttext="n\in\mathbb{N}" class="ltx_Math" display="inline" id="A1.Thmtheorem3.p1.2.2.m2.1"><semantics id="A1.Thmtheorem3.p1.2.2.m2.1a"><mrow id="A1.Thmtheorem3.p1.2.2.m2.1.1" xref="A1.Thmtheorem3.p1.2.2.m2.1.1.cmml"><mi id="A1.Thmtheorem3.p1.2.2.m2.1.1.2" xref="A1.Thmtheorem3.p1.2.2.m2.1.1.2.cmml">n</mi><mo id="A1.Thmtheorem3.p1.2.2.m2.1.1.1" xref="A1.Thmtheorem3.p1.2.2.m2.1.1.1.cmml">∈</mo><mi id="A1.Thmtheorem3.p1.2.2.m2.1.1.3" xref="A1.Thmtheorem3.p1.2.2.m2.1.1.3.cmml">ℕ</mi></mrow><annotation-xml encoding="MathML-Content" id="A1.Thmtheorem3.p1.2.2.m2.1b"><apply id="A1.Thmtheorem3.p1.2.2.m2.1.1.cmml" xref="A1.Thmtheorem3.p1.2.2.m2.1.1"><in id="A1.Thmtheorem3.p1.2.2.m2.1.1.1.cmml" xref="A1.Thmtheorem3.p1.2.2.m2.1.1.1"></in><ci id="A1.Thmtheorem3.p1.2.2.m2.1.1.2.cmml" xref="A1.Thmtheorem3.p1.2.2.m2.1.1.2">𝑛</ci><ci id="A1.Thmtheorem3.p1.2.2.m2.1.1.3.cmml" xref="A1.Thmtheorem3.p1.2.2.m2.1.1.3">ℕ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.Thmtheorem3.p1.2.2.m2.1c">n\in\mathbb{N}</annotation><annotation encoding="application/x-llamapun" id="A1.Thmtheorem3.p1.2.2.m2.1d">italic_n ∈ blackboard_N</annotation></semantics></math> be given. Then for <math alttext="t=\lceil\frac{2\log n}{\varepsilon}\rceil" class="ltx_Math" display="inline" id="A1.Thmtheorem3.p1.3.3.m3.1"><semantics id="A1.Thmtheorem3.p1.3.3.m3.1a"><mrow id="A1.Thmtheorem3.p1.3.3.m3.1.2" xref="A1.Thmtheorem3.p1.3.3.m3.1.2.cmml"><mi id="A1.Thmtheorem3.p1.3.3.m3.1.2.2" xref="A1.Thmtheorem3.p1.3.3.m3.1.2.2.cmml">t</mi><mo id="A1.Thmtheorem3.p1.3.3.m3.1.2.1" xref="A1.Thmtheorem3.p1.3.3.m3.1.2.1.cmml">=</mo><mrow id="A1.Thmtheorem3.p1.3.3.m3.1.2.3.2" xref="A1.Thmtheorem3.p1.3.3.m3.1.2.3.1.cmml"><mo id="A1.Thmtheorem3.p1.3.3.m3.1.2.3.2.1" stretchy="false" xref="A1.Thmtheorem3.p1.3.3.m3.1.2.3.1.1.cmml">⌈</mo><mfrac id="A1.Thmtheorem3.p1.3.3.m3.1.1" xref="A1.Thmtheorem3.p1.3.3.m3.1.1.cmml"><mrow id="A1.Thmtheorem3.p1.3.3.m3.1.1.2" xref="A1.Thmtheorem3.p1.3.3.m3.1.1.2.cmml"><mn id="A1.Thmtheorem3.p1.3.3.m3.1.1.2.2" xref="A1.Thmtheorem3.p1.3.3.m3.1.1.2.2.cmml">2</mn><mo id="A1.Thmtheorem3.p1.3.3.m3.1.1.2.1" lspace="0.167em" xref="A1.Thmtheorem3.p1.3.3.m3.1.1.2.1.cmml">⁢</mo><mrow id="A1.Thmtheorem3.p1.3.3.m3.1.1.2.3" xref="A1.Thmtheorem3.p1.3.3.m3.1.1.2.3.cmml"><mi id="A1.Thmtheorem3.p1.3.3.m3.1.1.2.3.1" xref="A1.Thmtheorem3.p1.3.3.m3.1.1.2.3.1.cmml">log</mi><mo id="A1.Thmtheorem3.p1.3.3.m3.1.1.2.3a" lspace="0.167em" xref="A1.Thmtheorem3.p1.3.3.m3.1.1.2.3.cmml">⁡</mo><mi id="A1.Thmtheorem3.p1.3.3.m3.1.1.2.3.2" xref="A1.Thmtheorem3.p1.3.3.m3.1.1.2.3.2.cmml">n</mi></mrow></mrow><mi id="A1.Thmtheorem3.p1.3.3.m3.1.1.3" xref="A1.Thmtheorem3.p1.3.3.m3.1.1.3.cmml">ε</mi></mfrac><mo id="A1.Thmtheorem3.p1.3.3.m3.1.2.3.2.2" stretchy="false" xref="A1.Thmtheorem3.p1.3.3.m3.1.2.3.1.1.cmml">⌉</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="A1.Thmtheorem3.p1.3.3.m3.1b"><apply id="A1.Thmtheorem3.p1.3.3.m3.1.2.cmml" xref="A1.Thmtheorem3.p1.3.3.m3.1.2"><eq id="A1.Thmtheorem3.p1.3.3.m3.1.2.1.cmml" xref="A1.Thmtheorem3.p1.3.3.m3.1.2.1"></eq><ci id="A1.Thmtheorem3.p1.3.3.m3.1.2.2.cmml" xref="A1.Thmtheorem3.p1.3.3.m3.1.2.2">𝑡</ci><apply id="A1.Thmtheorem3.p1.3.3.m3.1.2.3.1.cmml" xref="A1.Thmtheorem3.p1.3.3.m3.1.2.3.2"><ceiling id="A1.Thmtheorem3.p1.3.3.m3.1.2.3.1.1.cmml" xref="A1.Thmtheorem3.p1.3.3.m3.1.2.3.2.1"></ceiling><apply id="A1.Thmtheorem3.p1.3.3.m3.1.1.cmml" xref="A1.Thmtheorem3.p1.3.3.m3.1.1"><divide id="A1.Thmtheorem3.p1.3.3.m3.1.1.1.cmml" xref="A1.Thmtheorem3.p1.3.3.m3.1.1"></divide><apply id="A1.Thmtheorem3.p1.3.3.m3.1.1.2.cmml" xref="A1.Thmtheorem3.p1.3.3.m3.1.1.2"><times id="A1.Thmtheorem3.p1.3.3.m3.1.1.2.1.cmml" xref="A1.Thmtheorem3.p1.3.3.m3.1.1.2.1"></times><cn id="A1.Thmtheorem3.p1.3.3.m3.1.1.2.2.cmml" type="integer" xref="A1.Thmtheorem3.p1.3.3.m3.1.1.2.2">2</cn><apply id="A1.Thmtheorem3.p1.3.3.m3.1.1.2.3.cmml" xref="A1.Thmtheorem3.p1.3.3.m3.1.1.2.3"><log id="A1.Thmtheorem3.p1.3.3.m3.1.1.2.3.1.cmml" xref="A1.Thmtheorem3.p1.3.3.m3.1.1.2.3.1"></log><ci id="A1.Thmtheorem3.p1.3.3.m3.1.1.2.3.2.cmml" xref="A1.Thmtheorem3.p1.3.3.m3.1.1.2.3.2">𝑛</ci></apply></apply><ci id="A1.Thmtheorem3.p1.3.3.m3.1.1.3.cmml" xref="A1.Thmtheorem3.p1.3.3.m3.1.1.3">𝜀</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.Thmtheorem3.p1.3.3.m3.1c">t=\lceil\frac{2\log n}{\varepsilon}\rceil</annotation><annotation encoding="application/x-llamapun" id="A1.Thmtheorem3.p1.3.3.m3.1d">italic_t = ⌈ divide start_ARG 2 roman_log italic_n end_ARG start_ARG italic_ε end_ARG ⌉</annotation></semantics></math> we have: For any graph <math alttext="G" class="ltx_Math" display="inline" id="A1.Thmtheorem3.p1.4.4.m4.1"><semantics id="A1.Thmtheorem3.p1.4.4.m4.1a"><mi id="A1.Thmtheorem3.p1.4.4.m4.1.1" xref="A1.Thmtheorem3.p1.4.4.m4.1.1.cmml">G</mi><annotation-xml encoding="MathML-Content" id="A1.Thmtheorem3.p1.4.4.m4.1b"><ci id="A1.Thmtheorem3.p1.4.4.m4.1.1.cmml" xref="A1.Thmtheorem3.p1.4.4.m4.1.1">𝐺</ci></annotation-xml><annotation encoding="application/x-tex" id="A1.Thmtheorem3.p1.4.4.m4.1c">G</annotation><annotation encoding="application/x-llamapun" id="A1.Thmtheorem3.p1.4.4.m4.1d">italic_G</annotation></semantics></math> on <math alttext="n" class="ltx_Math" display="inline" id="A1.Thmtheorem3.p1.5.5.m5.1"><semantics id="A1.Thmtheorem3.p1.5.5.m5.1a"><mi id="A1.Thmtheorem3.p1.5.5.m5.1.1" xref="A1.Thmtheorem3.p1.5.5.m5.1.1.cmml">n</mi><annotation-xml encoding="MathML-Content" id="A1.Thmtheorem3.p1.5.5.m5.1b"><ci id="A1.Thmtheorem3.p1.5.5.m5.1.1.cmml" xref="A1.Thmtheorem3.p1.5.5.m5.1.1">𝑛</ci></annotation-xml><annotation encoding="application/x-tex" id="A1.Thmtheorem3.p1.5.5.m5.1c">n</annotation><annotation encoding="application/x-llamapun" id="A1.Thmtheorem3.p1.5.5.m5.1d">italic_n</annotation></semantics></math> vertices and for all vertices <math alttext="v\in V(G)" class="ltx_Math" display="inline" id="A1.Thmtheorem3.p1.6.6.m6.1"><semantics id="A1.Thmtheorem3.p1.6.6.m6.1a"><mrow id="A1.Thmtheorem3.p1.6.6.m6.1.2" xref="A1.Thmtheorem3.p1.6.6.m6.1.2.cmml"><mi id="A1.Thmtheorem3.p1.6.6.m6.1.2.2" xref="A1.Thmtheorem3.p1.6.6.m6.1.2.2.cmml">v</mi><mo id="A1.Thmtheorem3.p1.6.6.m6.1.2.1" xref="A1.Thmtheorem3.p1.6.6.m6.1.2.1.cmml">∈</mo><mrow id="A1.Thmtheorem3.p1.6.6.m6.1.2.3" xref="A1.Thmtheorem3.p1.6.6.m6.1.2.3.cmml"><mi id="A1.Thmtheorem3.p1.6.6.m6.1.2.3.2" xref="A1.Thmtheorem3.p1.6.6.m6.1.2.3.2.cmml">V</mi><mo id="A1.Thmtheorem3.p1.6.6.m6.1.2.3.1" xref="A1.Thmtheorem3.p1.6.6.m6.1.2.3.1.cmml">⁢</mo><mrow id="A1.Thmtheorem3.p1.6.6.m6.1.2.3.3.2" xref="A1.Thmtheorem3.p1.6.6.m6.1.2.3.cmml"><mo id="A1.Thmtheorem3.p1.6.6.m6.1.2.3.3.2.1" stretchy="false" xref="A1.Thmtheorem3.p1.6.6.m6.1.2.3.cmml">(</mo><mi id="A1.Thmtheorem3.p1.6.6.m6.1.1" xref="A1.Thmtheorem3.p1.6.6.m6.1.1.cmml">G</mi><mo id="A1.Thmtheorem3.p1.6.6.m6.1.2.3.3.2.2" stretchy="false" xref="A1.Thmtheorem3.p1.6.6.m6.1.2.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="A1.Thmtheorem3.p1.6.6.m6.1b"><apply id="A1.Thmtheorem3.p1.6.6.m6.1.2.cmml" xref="A1.Thmtheorem3.p1.6.6.m6.1.2"><in id="A1.Thmtheorem3.p1.6.6.m6.1.2.1.cmml" xref="A1.Thmtheorem3.p1.6.6.m6.1.2.1"></in><ci id="A1.Thmtheorem3.p1.6.6.m6.1.2.2.cmml" xref="A1.Thmtheorem3.p1.6.6.m6.1.2.2">𝑣</ci><apply id="A1.Thmtheorem3.p1.6.6.m6.1.2.3.cmml" xref="A1.Thmtheorem3.p1.6.6.m6.1.2.3"><times id="A1.Thmtheorem3.p1.6.6.m6.1.2.3.1.cmml" xref="A1.Thmtheorem3.p1.6.6.m6.1.2.3.1"></times><ci id="A1.Thmtheorem3.p1.6.6.m6.1.2.3.2.cmml" xref="A1.Thmtheorem3.p1.6.6.m6.1.2.3.2">𝑉</ci><ci id="A1.Thmtheorem3.p1.6.6.m6.1.1.cmml" xref="A1.Thmtheorem3.p1.6.6.m6.1.1">𝐺</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.Thmtheorem3.p1.6.6.m6.1c">v\in V(G)</annotation><annotation encoding="application/x-llamapun" id="A1.Thmtheorem3.p1.6.6.m6.1d">italic_v ∈ italic_V ( italic_G )</annotation></semantics></math> there exists a subgraph <math alttext="H^{\prime}_{v}" class="ltx_Math" display="inline" id="A1.Thmtheorem3.p1.7.7.m7.1"><semantics id="A1.Thmtheorem3.p1.7.7.m7.1a"><msubsup id="A1.Thmtheorem3.p1.7.7.m7.1.1" xref="A1.Thmtheorem3.p1.7.7.m7.1.1.cmml"><mi id="A1.Thmtheorem3.p1.7.7.m7.1.1.2.2" xref="A1.Thmtheorem3.p1.7.7.m7.1.1.2.2.cmml">H</mi><mi id="A1.Thmtheorem3.p1.7.7.m7.1.1.3" xref="A1.Thmtheorem3.p1.7.7.m7.1.1.3.cmml">v</mi><mo id="A1.Thmtheorem3.p1.7.7.m7.1.1.2.3" xref="A1.Thmtheorem3.p1.7.7.m7.1.1.2.3.cmml">′</mo></msubsup><annotation-xml encoding="MathML-Content" id="A1.Thmtheorem3.p1.7.7.m7.1b"><apply id="A1.Thmtheorem3.p1.7.7.m7.1.1.cmml" xref="A1.Thmtheorem3.p1.7.7.m7.1.1"><csymbol cd="ambiguous" id="A1.Thmtheorem3.p1.7.7.m7.1.1.1.cmml" xref="A1.Thmtheorem3.p1.7.7.m7.1.1">subscript</csymbol><apply id="A1.Thmtheorem3.p1.7.7.m7.1.1.2.cmml" xref="A1.Thmtheorem3.p1.7.7.m7.1.1"><csymbol cd="ambiguous" id="A1.Thmtheorem3.p1.7.7.m7.1.1.2.1.cmml" xref="A1.Thmtheorem3.p1.7.7.m7.1.1">superscript</csymbol><ci id="A1.Thmtheorem3.p1.7.7.m7.1.1.2.2.cmml" xref="A1.Thmtheorem3.p1.7.7.m7.1.1.2.2">𝐻</ci><ci id="A1.Thmtheorem3.p1.7.7.m7.1.1.2.3.cmml" xref="A1.Thmtheorem3.p1.7.7.m7.1.1.2.3">′</ci></apply><ci id="A1.Thmtheorem3.p1.7.7.m7.1.1.3.cmml" xref="A1.Thmtheorem3.p1.7.7.m7.1.1.3">𝑣</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.Thmtheorem3.p1.7.7.m7.1c">H^{\prime}_{v}</annotation><annotation encoding="application/x-llamapun" id="A1.Thmtheorem3.p1.7.7.m7.1d">italic_H start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_v end_POSTSUBSCRIPT</annotation></semantics></math> such that <math alttext="H^{\prime}_{v}\subset H_{t}(v)" class="ltx_Math" display="inline" id="A1.Thmtheorem3.p1.8.8.m8.1"><semantics id="A1.Thmtheorem3.p1.8.8.m8.1a"><mrow id="A1.Thmtheorem3.p1.8.8.m8.1.2" xref="A1.Thmtheorem3.p1.8.8.m8.1.2.cmml"><msubsup id="A1.Thmtheorem3.p1.8.8.m8.1.2.2" xref="A1.Thmtheorem3.p1.8.8.m8.1.2.2.cmml"><mi id="A1.Thmtheorem3.p1.8.8.m8.1.2.2.2.2" xref="A1.Thmtheorem3.p1.8.8.m8.1.2.2.2.2.cmml">H</mi><mi id="A1.Thmtheorem3.p1.8.8.m8.1.2.2.3" xref="A1.Thmtheorem3.p1.8.8.m8.1.2.2.3.cmml">v</mi><mo id="A1.Thmtheorem3.p1.8.8.m8.1.2.2.2.3" xref="A1.Thmtheorem3.p1.8.8.m8.1.2.2.2.3.cmml">′</mo></msubsup><mo id="A1.Thmtheorem3.p1.8.8.m8.1.2.1" xref="A1.Thmtheorem3.p1.8.8.m8.1.2.1.cmml">⊂</mo><mrow id="A1.Thmtheorem3.p1.8.8.m8.1.2.3" xref="A1.Thmtheorem3.p1.8.8.m8.1.2.3.cmml"><msub id="A1.Thmtheorem3.p1.8.8.m8.1.2.3.2" xref="A1.Thmtheorem3.p1.8.8.m8.1.2.3.2.cmml"><mi id="A1.Thmtheorem3.p1.8.8.m8.1.2.3.2.2" xref="A1.Thmtheorem3.p1.8.8.m8.1.2.3.2.2.cmml">H</mi><mi id="A1.Thmtheorem3.p1.8.8.m8.1.2.3.2.3" xref="A1.Thmtheorem3.p1.8.8.m8.1.2.3.2.3.cmml">t</mi></msub><mo id="A1.Thmtheorem3.p1.8.8.m8.1.2.3.1" xref="A1.Thmtheorem3.p1.8.8.m8.1.2.3.1.cmml">⁢</mo><mrow id="A1.Thmtheorem3.p1.8.8.m8.1.2.3.3.2" xref="A1.Thmtheorem3.p1.8.8.m8.1.2.3.cmml"><mo id="A1.Thmtheorem3.p1.8.8.m8.1.2.3.3.2.1" stretchy="false" xref="A1.Thmtheorem3.p1.8.8.m8.1.2.3.cmml">(</mo><mi id="A1.Thmtheorem3.p1.8.8.m8.1.1" xref="A1.Thmtheorem3.p1.8.8.m8.1.1.cmml">v</mi><mo id="A1.Thmtheorem3.p1.8.8.m8.1.2.3.3.2.2" stretchy="false" xref="A1.Thmtheorem3.p1.8.8.m8.1.2.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="A1.Thmtheorem3.p1.8.8.m8.1b"><apply id="A1.Thmtheorem3.p1.8.8.m8.1.2.cmml" xref="A1.Thmtheorem3.p1.8.8.m8.1.2"><subset id="A1.Thmtheorem3.p1.8.8.m8.1.2.1.cmml" xref="A1.Thmtheorem3.p1.8.8.m8.1.2.1"></subset><apply id="A1.Thmtheorem3.p1.8.8.m8.1.2.2.cmml" xref="A1.Thmtheorem3.p1.8.8.m8.1.2.2"><csymbol cd="ambiguous" id="A1.Thmtheorem3.p1.8.8.m8.1.2.2.1.cmml" xref="A1.Thmtheorem3.p1.8.8.m8.1.2.2">subscript</csymbol><apply id="A1.Thmtheorem3.p1.8.8.m8.1.2.2.2.cmml" xref="A1.Thmtheorem3.p1.8.8.m8.1.2.2"><csymbol cd="ambiguous" id="A1.Thmtheorem3.p1.8.8.m8.1.2.2.2.1.cmml" xref="A1.Thmtheorem3.p1.8.8.m8.1.2.2">superscript</csymbol><ci id="A1.Thmtheorem3.p1.8.8.m8.1.2.2.2.2.cmml" xref="A1.Thmtheorem3.p1.8.8.m8.1.2.2.2.2">𝐻</ci><ci id="A1.Thmtheorem3.p1.8.8.m8.1.2.2.2.3.cmml" xref="A1.Thmtheorem3.p1.8.8.m8.1.2.2.2.3">′</ci></apply><ci id="A1.Thmtheorem3.p1.8.8.m8.1.2.2.3.cmml" xref="A1.Thmtheorem3.p1.8.8.m8.1.2.2.3">𝑣</ci></apply><apply id="A1.Thmtheorem3.p1.8.8.m8.1.2.3.cmml" xref="A1.Thmtheorem3.p1.8.8.m8.1.2.3"><times id="A1.Thmtheorem3.p1.8.8.m8.1.2.3.1.cmml" xref="A1.Thmtheorem3.p1.8.8.m8.1.2.3.1"></times><apply id="A1.Thmtheorem3.p1.8.8.m8.1.2.3.2.cmml" xref="A1.Thmtheorem3.p1.8.8.m8.1.2.3.2"><csymbol cd="ambiguous" id="A1.Thmtheorem3.p1.8.8.m8.1.2.3.2.1.cmml" xref="A1.Thmtheorem3.p1.8.8.m8.1.2.3.2">subscript</csymbol><ci id="A1.Thmtheorem3.p1.8.8.m8.1.2.3.2.2.cmml" xref="A1.Thmtheorem3.p1.8.8.m8.1.2.3.2.2">𝐻</ci><ci id="A1.Thmtheorem3.p1.8.8.m8.1.2.3.2.3.cmml" xref="A1.Thmtheorem3.p1.8.8.m8.1.2.3.2.3">𝑡</ci></apply><ci id="A1.Thmtheorem3.p1.8.8.m8.1.1.cmml" xref="A1.Thmtheorem3.p1.8.8.m8.1.1">𝑣</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.Thmtheorem3.p1.8.8.m8.1c">H^{\prime}_{v}\subset H_{t}(v)</annotation><annotation encoding="application/x-llamapun" id="A1.Thmtheorem3.p1.8.8.m8.1d">italic_H start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_v end_POSTSUBSCRIPT ⊂ italic_H start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT ( italic_v )</annotation></semantics></math> and <math alttext="\rho(H^{\prime}_{v})\geq\rho^{*}(v)(1-\varepsilon)" class="ltx_Math" display="inline" id="A1.Thmtheorem3.p1.9.9.m9.3"><semantics id="A1.Thmtheorem3.p1.9.9.m9.3a"><mrow id="A1.Thmtheorem3.p1.9.9.m9.3.3" xref="A1.Thmtheorem3.p1.9.9.m9.3.3.cmml"><mrow id="A1.Thmtheorem3.p1.9.9.m9.2.2.1" xref="A1.Thmtheorem3.p1.9.9.m9.2.2.1.cmml"><mi id="A1.Thmtheorem3.p1.9.9.m9.2.2.1.3" xref="A1.Thmtheorem3.p1.9.9.m9.2.2.1.3.cmml">ρ</mi><mo id="A1.Thmtheorem3.p1.9.9.m9.2.2.1.2" xref="A1.Thmtheorem3.p1.9.9.m9.2.2.1.2.cmml">⁢</mo><mrow id="A1.Thmtheorem3.p1.9.9.m9.2.2.1.1.1" 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id="A1.Thmtheorem3.p1.9.9.m9.3.3.2.3.2" xref="A1.Thmtheorem3.p1.9.9.m9.3.3.2.3.2.cmml">ρ</mi><mo id="A1.Thmtheorem3.p1.9.9.m9.3.3.2.3.3" xref="A1.Thmtheorem3.p1.9.9.m9.3.3.2.3.3.cmml">∗</mo></msup><mo id="A1.Thmtheorem3.p1.9.9.m9.3.3.2.2" xref="A1.Thmtheorem3.p1.9.9.m9.3.3.2.2.cmml">⁢</mo><mrow id="A1.Thmtheorem3.p1.9.9.m9.3.3.2.4.2" xref="A1.Thmtheorem3.p1.9.9.m9.3.3.2.cmml"><mo id="A1.Thmtheorem3.p1.9.9.m9.3.3.2.4.2.1" stretchy="false" xref="A1.Thmtheorem3.p1.9.9.m9.3.3.2.cmml">(</mo><mi id="A1.Thmtheorem3.p1.9.9.m9.1.1" xref="A1.Thmtheorem3.p1.9.9.m9.1.1.cmml">v</mi><mo id="A1.Thmtheorem3.p1.9.9.m9.3.3.2.4.2.2" stretchy="false" xref="A1.Thmtheorem3.p1.9.9.m9.3.3.2.cmml">)</mo></mrow><mo id="A1.Thmtheorem3.p1.9.9.m9.3.3.2.2a" xref="A1.Thmtheorem3.p1.9.9.m9.3.3.2.2.cmml">⁢</mo><mrow id="A1.Thmtheorem3.p1.9.9.m9.3.3.2.1.1" xref="A1.Thmtheorem3.p1.9.9.m9.3.3.2.1.1.1.cmml"><mo id="A1.Thmtheorem3.p1.9.9.m9.3.3.2.1.1.2" stretchy="false" xref="A1.Thmtheorem3.p1.9.9.m9.3.3.2.1.1.1.cmml">(</mo><mrow id="A1.Thmtheorem3.p1.9.9.m9.3.3.2.1.1.1" xref="A1.Thmtheorem3.p1.9.9.m9.3.3.2.1.1.1.cmml"><mn id="A1.Thmtheorem3.p1.9.9.m9.3.3.2.1.1.1.2" xref="A1.Thmtheorem3.p1.9.9.m9.3.3.2.1.1.1.2.cmml">1</mn><mo id="A1.Thmtheorem3.p1.9.9.m9.3.3.2.1.1.1.1" xref="A1.Thmtheorem3.p1.9.9.m9.3.3.2.1.1.1.1.cmml">−</mo><mi id="A1.Thmtheorem3.p1.9.9.m9.3.3.2.1.1.1.3" xref="A1.Thmtheorem3.p1.9.9.m9.3.3.2.1.1.1.3.cmml">ε</mi></mrow><mo id="A1.Thmtheorem3.p1.9.9.m9.3.3.2.1.1.3" stretchy="false" xref="A1.Thmtheorem3.p1.9.9.m9.3.3.2.1.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="A1.Thmtheorem3.p1.9.9.m9.3b"><apply id="A1.Thmtheorem3.p1.9.9.m9.3.3.cmml" xref="A1.Thmtheorem3.p1.9.9.m9.3.3"><geq id="A1.Thmtheorem3.p1.9.9.m9.3.3.3.cmml" xref="A1.Thmtheorem3.p1.9.9.m9.3.3.3"></geq><apply id="A1.Thmtheorem3.p1.9.9.m9.2.2.1.cmml" xref="A1.Thmtheorem3.p1.9.9.m9.2.2.1"><times id="A1.Thmtheorem3.p1.9.9.m9.2.2.1.2.cmml" xref="A1.Thmtheorem3.p1.9.9.m9.2.2.1.2"></times><ci id="A1.Thmtheorem3.p1.9.9.m9.2.2.1.3.cmml" xref="A1.Thmtheorem3.p1.9.9.m9.2.2.1.3">𝜌</ci><apply id="A1.Thmtheorem3.p1.9.9.m9.2.2.1.1.1.1.cmml" xref="A1.Thmtheorem3.p1.9.9.m9.2.2.1.1.1"><csymbol cd="ambiguous" id="A1.Thmtheorem3.p1.9.9.m9.2.2.1.1.1.1.1.cmml" xref="A1.Thmtheorem3.p1.9.9.m9.2.2.1.1.1">subscript</csymbol><apply id="A1.Thmtheorem3.p1.9.9.m9.2.2.1.1.1.1.2.cmml" xref="A1.Thmtheorem3.p1.9.9.m9.2.2.1.1.1"><csymbol cd="ambiguous" id="A1.Thmtheorem3.p1.9.9.m9.2.2.1.1.1.1.2.1.cmml" xref="A1.Thmtheorem3.p1.9.9.m9.2.2.1.1.1">superscript</csymbol><ci id="A1.Thmtheorem3.p1.9.9.m9.2.2.1.1.1.1.2.2.cmml" xref="A1.Thmtheorem3.p1.9.9.m9.2.2.1.1.1.1.2.2">𝐻</ci><ci id="A1.Thmtheorem3.p1.9.9.m9.2.2.1.1.1.1.2.3.cmml" xref="A1.Thmtheorem3.p1.9.9.m9.2.2.1.1.1.1.2.3">′</ci></apply><ci id="A1.Thmtheorem3.p1.9.9.m9.2.2.1.1.1.1.3.cmml" xref="A1.Thmtheorem3.p1.9.9.m9.2.2.1.1.1.1.3">𝑣</ci></apply></apply><apply id="A1.Thmtheorem3.p1.9.9.m9.3.3.2.cmml" xref="A1.Thmtheorem3.p1.9.9.m9.3.3.2"><times id="A1.Thmtheorem3.p1.9.9.m9.3.3.2.2.cmml" xref="A1.Thmtheorem3.p1.9.9.m9.3.3.2.2"></times><apply id="A1.Thmtheorem3.p1.9.9.m9.3.3.2.3.cmml" xref="A1.Thmtheorem3.p1.9.9.m9.3.3.2.3"><csymbol cd="ambiguous" id="A1.Thmtheorem3.p1.9.9.m9.3.3.2.3.1.cmml" xref="A1.Thmtheorem3.p1.9.9.m9.3.3.2.3">superscript</csymbol><ci id="A1.Thmtheorem3.p1.9.9.m9.3.3.2.3.2.cmml" xref="A1.Thmtheorem3.p1.9.9.m9.3.3.2.3.2">𝜌</ci><times id="A1.Thmtheorem3.p1.9.9.m9.3.3.2.3.3.cmml" xref="A1.Thmtheorem3.p1.9.9.m9.3.3.2.3.3"></times></apply><ci id="A1.Thmtheorem3.p1.9.9.m9.1.1.cmml" xref="A1.Thmtheorem3.p1.9.9.m9.1.1">𝑣</ci><apply id="A1.Thmtheorem3.p1.9.9.m9.3.3.2.1.1.1.cmml" xref="A1.Thmtheorem3.p1.9.9.m9.3.3.2.1.1"><minus id="A1.Thmtheorem3.p1.9.9.m9.3.3.2.1.1.1.1.cmml" xref="A1.Thmtheorem3.p1.9.9.m9.3.3.2.1.1.1.1"></minus><cn id="A1.Thmtheorem3.p1.9.9.m9.3.3.2.1.1.1.2.cmml" type="integer" xref="A1.Thmtheorem3.p1.9.9.m9.3.3.2.1.1.1.2">1</cn><ci id="A1.Thmtheorem3.p1.9.9.m9.3.3.2.1.1.1.3.cmml" xref="A1.Thmtheorem3.p1.9.9.m9.3.3.2.1.1.1.3">𝜀</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.Thmtheorem3.p1.9.9.m9.3c">\rho(H^{\prime}_{v})\geq\rho^{*}(v)(1-\varepsilon)</annotation><annotation encoding="application/x-llamapun" id="A1.Thmtheorem3.p1.9.9.m9.3d">italic_ρ ( italic_H start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_v end_POSTSUBSCRIPT ) ≥ italic_ρ start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ( italic_v ) ( 1 - italic_ε )</annotation></semantics></math>.</span></p> </div> </div> <div class="ltx_theorem ltx_theorem_proof" id="A1.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="A1.Thmtheorem4.1.1.1">Proof A.4</span></span><span class="ltx_text ltx_font_bold" id="A1.Thmtheorem4.2.2">.</span> </h6> <div class="ltx_para" id="A1.Thmtheorem4.p1"> <p class="ltx_p" id="A1.Thmtheorem4.p1.11"><span class="ltx_text ltx_font_italic" id="A1.Thmtheorem4.p1.11.11">Suppose for the sake of contradiction that this is not the case. Let <math alttext="G,v" class="ltx_Math" display="inline" id="A1.Thmtheorem4.p1.1.1.m1.2"><semantics id="A1.Thmtheorem4.p1.1.1.m1.2a"><mrow id="A1.Thmtheorem4.p1.1.1.m1.2.3.2" xref="A1.Thmtheorem4.p1.1.1.m1.2.3.1.cmml"><mi id="A1.Thmtheorem4.p1.1.1.m1.1.1" xref="A1.Thmtheorem4.p1.1.1.m1.1.1.cmml">G</mi><mo id="A1.Thmtheorem4.p1.1.1.m1.2.3.2.1" xref="A1.Thmtheorem4.p1.1.1.m1.2.3.1.cmml">,</mo><mi id="A1.Thmtheorem4.p1.1.1.m1.2.2" xref="A1.Thmtheorem4.p1.1.1.m1.2.2.cmml">v</mi></mrow><annotation-xml encoding="MathML-Content" id="A1.Thmtheorem4.p1.1.1.m1.2b"><list id="A1.Thmtheorem4.p1.1.1.m1.2.3.1.cmml" xref="A1.Thmtheorem4.p1.1.1.m1.2.3.2"><ci id="A1.Thmtheorem4.p1.1.1.m1.1.1.cmml" xref="A1.Thmtheorem4.p1.1.1.m1.1.1">𝐺</ci><ci id="A1.Thmtheorem4.p1.1.1.m1.2.2.cmml" xref="A1.Thmtheorem4.p1.1.1.m1.2.2">𝑣</ci></list></annotation-xml><annotation encoding="application/x-tex" id="A1.Thmtheorem4.p1.1.1.m1.2c">G,v</annotation><annotation encoding="application/x-llamapun" id="A1.Thmtheorem4.p1.1.1.m1.2d">italic_G , italic_v</annotation></semantics></math> be a graph and a vertex contradicting the statement. Denote by <math alttext="\overrightarrow{G}" class="ltx_Math" display="inline" id="A1.Thmtheorem4.p1.2.2.m2.1"><semantics id="A1.Thmtheorem4.p1.2.2.m2.1a"><mover accent="true" id="A1.Thmtheorem4.p1.2.2.m2.1.1" xref="A1.Thmtheorem4.p1.2.2.m2.1.1.cmml"><mi id="A1.Thmtheorem4.p1.2.2.m2.1.1.2" xref="A1.Thmtheorem4.p1.2.2.m2.1.1.2.cmml">G</mi><mo id="A1.Thmtheorem4.p1.2.2.m2.1.1.1" stretchy="false" xref="A1.Thmtheorem4.p1.2.2.m2.1.1.1.cmml">→</mo></mover><annotation-xml encoding="MathML-Content" id="A1.Thmtheorem4.p1.2.2.m2.1b"><apply id="A1.Thmtheorem4.p1.2.2.m2.1.1.cmml" xref="A1.Thmtheorem4.p1.2.2.m2.1.1"><ci id="A1.Thmtheorem4.p1.2.2.m2.1.1.1.cmml" xref="A1.Thmtheorem4.p1.2.2.m2.1.1.1">→</ci><ci id="A1.Thmtheorem4.p1.2.2.m2.1.1.2.cmml" xref="A1.Thmtheorem4.p1.2.2.m2.1.1.2">𝐺</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.Thmtheorem4.p1.2.2.m2.1c">\overrightarrow{G}</annotation><annotation encoding="application/x-llamapun" id="A1.Thmtheorem4.p1.2.2.m2.1d">over→ start_ARG italic_G end_ARG</annotation></semantics></math> a locally fair orientation of <math alttext="G" class="ltx_Math" display="inline" id="A1.Thmtheorem4.p1.3.3.m3.1"><semantics id="A1.Thmtheorem4.p1.3.3.m3.1a"><mi id="A1.Thmtheorem4.p1.3.3.m3.1.1" xref="A1.Thmtheorem4.p1.3.3.m3.1.1.cmml">G</mi><annotation-xml encoding="MathML-Content" id="A1.Thmtheorem4.p1.3.3.m3.1b"><ci id="A1.Thmtheorem4.p1.3.3.m3.1.1.cmml" xref="A1.Thmtheorem4.p1.3.3.m3.1.1">𝐺</ci></annotation-xml><annotation encoding="application/x-tex" id="A1.Thmtheorem4.p1.3.3.m3.1c">G</annotation><annotation encoding="application/x-llamapun" id="A1.Thmtheorem4.p1.3.3.m3.1d">italic_G</annotation></semantics></math>. We define <math alttext="V_{i}" class="ltx_Math" display="inline" id="A1.Thmtheorem4.p1.4.4.m4.1"><semantics id="A1.Thmtheorem4.p1.4.4.m4.1a"><msub id="A1.Thmtheorem4.p1.4.4.m4.1.1" xref="A1.Thmtheorem4.p1.4.4.m4.1.1.cmml"><mi id="A1.Thmtheorem4.p1.4.4.m4.1.1.2" xref="A1.Thmtheorem4.p1.4.4.m4.1.1.2.cmml">V</mi><mi id="A1.Thmtheorem4.p1.4.4.m4.1.1.3" xref="A1.Thmtheorem4.p1.4.4.m4.1.1.3.cmml">i</mi></msub><annotation-xml encoding="MathML-Content" id="A1.Thmtheorem4.p1.4.4.m4.1b"><apply id="A1.Thmtheorem4.p1.4.4.m4.1.1.cmml" xref="A1.Thmtheorem4.p1.4.4.m4.1.1"><csymbol cd="ambiguous" id="A1.Thmtheorem4.p1.4.4.m4.1.1.1.cmml" xref="A1.Thmtheorem4.p1.4.4.m4.1.1">subscript</csymbol><ci id="A1.Thmtheorem4.p1.4.4.m4.1.1.2.cmml" xref="A1.Thmtheorem4.p1.4.4.m4.1.1.2">𝑉</ci><ci id="A1.Thmtheorem4.p1.4.4.m4.1.1.3.cmml" xref="A1.Thmtheorem4.p1.4.4.m4.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.Thmtheorem4.p1.4.4.m4.1c">V_{i}</annotation><annotation encoding="application/x-llamapun" id="A1.Thmtheorem4.p1.4.4.m4.1d">italic_V start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math> as the set of vertices to which <math alttext="v" class="ltx_Math" display="inline" id="A1.Thmtheorem4.p1.5.5.m5.1"><semantics id="A1.Thmtheorem4.p1.5.5.m5.1a"><mi id="A1.Thmtheorem4.p1.5.5.m5.1.1" xref="A1.Thmtheorem4.p1.5.5.m5.1.1.cmml">v</mi><annotation-xml encoding="MathML-Content" id="A1.Thmtheorem4.p1.5.5.m5.1b"><ci id="A1.Thmtheorem4.p1.5.5.m5.1.1.cmml" xref="A1.Thmtheorem4.p1.5.5.m5.1.1">𝑣</ci></annotation-xml><annotation encoding="application/x-tex" id="A1.Thmtheorem4.p1.5.5.m5.1c">v</annotation><annotation encoding="application/x-llamapun" id="A1.Thmtheorem4.p1.5.5.m5.1d">italic_v</annotation></semantics></math> has a directed path of length at most <math alttext="i" class="ltx_Math" display="inline" id="A1.Thmtheorem4.p1.6.6.m6.1"><semantics id="A1.Thmtheorem4.p1.6.6.m6.1a"><mi id="A1.Thmtheorem4.p1.6.6.m6.1.1" xref="A1.Thmtheorem4.p1.6.6.m6.1.1.cmml">i</mi><annotation-xml encoding="MathML-Content" id="A1.Thmtheorem4.p1.6.6.m6.1b"><ci id="A1.Thmtheorem4.p1.6.6.m6.1.1.cmml" xref="A1.Thmtheorem4.p1.6.6.m6.1.1">𝑖</ci></annotation-xml><annotation encoding="application/x-tex" id="A1.Thmtheorem4.p1.6.6.m6.1c">i</annotation><annotation encoding="application/x-llamapun" id="A1.Thmtheorem4.p1.6.6.m6.1d">italic_i</annotation></semantics></math> in <math alttext="\overrightarrow{G}" class="ltx_Math" display="inline" id="A1.Thmtheorem4.p1.7.7.m7.1"><semantics id="A1.Thmtheorem4.p1.7.7.m7.1a"><mover accent="true" id="A1.Thmtheorem4.p1.7.7.m7.1.1" xref="A1.Thmtheorem4.p1.7.7.m7.1.1.cmml"><mi id="A1.Thmtheorem4.p1.7.7.m7.1.1.2" xref="A1.Thmtheorem4.p1.7.7.m7.1.1.2.cmml">G</mi><mo id="A1.Thmtheorem4.p1.7.7.m7.1.1.1" stretchy="false" xref="A1.Thmtheorem4.p1.7.7.m7.1.1.1.cmml">→</mo></mover><annotation-xml encoding="MathML-Content" id="A1.Thmtheorem4.p1.7.7.m7.1b"><apply id="A1.Thmtheorem4.p1.7.7.m7.1.1.cmml" xref="A1.Thmtheorem4.p1.7.7.m7.1.1"><ci id="A1.Thmtheorem4.p1.7.7.m7.1.1.1.cmml" xref="A1.Thmtheorem4.p1.7.7.m7.1.1.1">→</ci><ci id="A1.Thmtheorem4.p1.7.7.m7.1.1.2.cmml" xref="A1.Thmtheorem4.p1.7.7.m7.1.1.2">𝐺</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.Thmtheorem4.p1.7.7.m7.1c">\overrightarrow{G}</annotation><annotation encoding="application/x-llamapun" id="A1.Thmtheorem4.p1.7.7.m7.1d">over→ start_ARG italic_G end_ARG</annotation></semantics></math>. Since <math alttext="\overrightarrow{G}" class="ltx_Math" display="inline" id="A1.Thmtheorem4.p1.8.8.m8.1"><semantics id="A1.Thmtheorem4.p1.8.8.m8.1a"><mover accent="true" id="A1.Thmtheorem4.p1.8.8.m8.1.1" xref="A1.Thmtheorem4.p1.8.8.m8.1.1.cmml"><mi id="A1.Thmtheorem4.p1.8.8.m8.1.1.2" xref="A1.Thmtheorem4.p1.8.8.m8.1.1.2.cmml">G</mi><mo id="A1.Thmtheorem4.p1.8.8.m8.1.1.1" stretchy="false" xref="A1.Thmtheorem4.p1.8.8.m8.1.1.1.cmml">→</mo></mover><annotation-xml encoding="MathML-Content" id="A1.Thmtheorem4.p1.8.8.m8.1b"><apply id="A1.Thmtheorem4.p1.8.8.m8.1.1.cmml" xref="A1.Thmtheorem4.p1.8.8.m8.1.1"><ci id="A1.Thmtheorem4.p1.8.8.m8.1.1.1.cmml" xref="A1.Thmtheorem4.p1.8.8.m8.1.1.1">→</ci><ci id="A1.Thmtheorem4.p1.8.8.m8.1.1.2.cmml" xref="A1.Thmtheorem4.p1.8.8.m8.1.1.2">𝐺</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.Thmtheorem4.p1.8.8.m8.1c">\overrightarrow{G}</annotation><annotation encoding="application/x-llamapun" id="A1.Thmtheorem4.p1.8.8.m8.1d">over→ start_ARG italic_G end_ARG</annotation></semantics></math> is locally fair, all vertices <math alttext="u\in V_{i}" class="ltx_Math" display="inline" id="A1.Thmtheorem4.p1.9.9.m9.1"><semantics id="A1.Thmtheorem4.p1.9.9.m9.1a"><mrow id="A1.Thmtheorem4.p1.9.9.m9.1.1" xref="A1.Thmtheorem4.p1.9.9.m9.1.1.cmml"><mi id="A1.Thmtheorem4.p1.9.9.m9.1.1.2" xref="A1.Thmtheorem4.p1.9.9.m9.1.1.2.cmml">u</mi><mo id="A1.Thmtheorem4.p1.9.9.m9.1.1.1" xref="A1.Thmtheorem4.p1.9.9.m9.1.1.1.cmml">∈</mo><msub id="A1.Thmtheorem4.p1.9.9.m9.1.1.3" xref="A1.Thmtheorem4.p1.9.9.m9.1.1.3.cmml"><mi id="A1.Thmtheorem4.p1.9.9.m9.1.1.3.2" xref="A1.Thmtheorem4.p1.9.9.m9.1.1.3.2.cmml">V</mi><mi id="A1.Thmtheorem4.p1.9.9.m9.1.1.3.3" xref="A1.Thmtheorem4.p1.9.9.m9.1.1.3.3.cmml">i</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="A1.Thmtheorem4.p1.9.9.m9.1b"><apply id="A1.Thmtheorem4.p1.9.9.m9.1.1.cmml" xref="A1.Thmtheorem4.p1.9.9.m9.1.1"><in id="A1.Thmtheorem4.p1.9.9.m9.1.1.1.cmml" xref="A1.Thmtheorem4.p1.9.9.m9.1.1.1"></in><ci id="A1.Thmtheorem4.p1.9.9.m9.1.1.2.cmml" xref="A1.Thmtheorem4.p1.9.9.m9.1.1.2">𝑢</ci><apply id="A1.Thmtheorem4.p1.9.9.m9.1.1.3.cmml" xref="A1.Thmtheorem4.p1.9.9.m9.1.1.3"><csymbol cd="ambiguous" id="A1.Thmtheorem4.p1.9.9.m9.1.1.3.1.cmml" xref="A1.Thmtheorem4.p1.9.9.m9.1.1.3">subscript</csymbol><ci id="A1.Thmtheorem4.p1.9.9.m9.1.1.3.2.cmml" xref="A1.Thmtheorem4.p1.9.9.m9.1.1.3.2">𝑉</ci><ci id="A1.Thmtheorem4.p1.9.9.m9.1.1.3.3.cmml" xref="A1.Thmtheorem4.p1.9.9.m9.1.1.3.3">𝑖</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.Thmtheorem4.p1.9.9.m9.1c">u\in V_{i}</annotation><annotation encoding="application/x-llamapun" id="A1.Thmtheorem4.p1.9.9.m9.1d">italic_u ∈ italic_V start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math> have an out-degree <math alttext="\textsl{g}(x)" class="ltx_Math" display="inline" id="A1.Thmtheorem4.p1.10.10.m10.1"><semantics id="A1.Thmtheorem4.p1.10.10.m10.1a"><mrow id="A1.Thmtheorem4.p1.10.10.m10.1.2" xref="A1.Thmtheorem4.p1.10.10.m10.1.2.cmml"><mtext class="ltx_mathvariant_italic" id="A1.Thmtheorem4.p1.10.10.m10.1.2.2" xref="A1.Thmtheorem4.p1.10.10.m10.1.2.2a.cmml">g</mtext><mo id="A1.Thmtheorem4.p1.10.10.m10.1.2.1" xref="A1.Thmtheorem4.p1.10.10.m10.1.2.1.cmml">⁢</mo><mrow id="A1.Thmtheorem4.p1.10.10.m10.1.2.3.2" xref="A1.Thmtheorem4.p1.10.10.m10.1.2.cmml"><mo id="A1.Thmtheorem4.p1.10.10.m10.1.2.3.2.1" stretchy="false" xref="A1.Thmtheorem4.p1.10.10.m10.1.2.cmml">(</mo><mi id="A1.Thmtheorem4.p1.10.10.m10.1.1" xref="A1.Thmtheorem4.p1.10.10.m10.1.1.cmml">x</mi><mo id="A1.Thmtheorem4.p1.10.10.m10.1.2.3.2.2" stretchy="false" xref="A1.Thmtheorem4.p1.10.10.m10.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="A1.Thmtheorem4.p1.10.10.m10.1b"><apply id="A1.Thmtheorem4.p1.10.10.m10.1.2.cmml" xref="A1.Thmtheorem4.p1.10.10.m10.1.2"><times id="A1.Thmtheorem4.p1.10.10.m10.1.2.1.cmml" xref="A1.Thmtheorem4.p1.10.10.m10.1.2.1"></times><ci id="A1.Thmtheorem4.p1.10.10.m10.1.2.2a.cmml" xref="A1.Thmtheorem4.p1.10.10.m10.1.2.2"><mtext class="ltx_mathvariant_italic" id="A1.Thmtheorem4.p1.10.10.m10.1.2.2.cmml" xref="A1.Thmtheorem4.p1.10.10.m10.1.2.2">g</mtext></ci><ci id="A1.Thmtheorem4.p1.10.10.m10.1.1.cmml" xref="A1.Thmtheorem4.p1.10.10.m10.1.1">𝑥</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.Thmtheorem4.p1.10.10.m10.1c">\textsl{g}(x)</annotation><annotation encoding="application/x-llamapun" id="A1.Thmtheorem4.p1.10.10.m10.1d">g ( italic_x )</annotation></semantics></math> of at least <math alttext="\rho^{*}(v)" class="ltx_Math" display="inline" id="A1.Thmtheorem4.p1.11.11.m11.1"><semantics id="A1.Thmtheorem4.p1.11.11.m11.1a"><mrow id="A1.Thmtheorem4.p1.11.11.m11.1.2" xref="A1.Thmtheorem4.p1.11.11.m11.1.2.cmml"><msup id="A1.Thmtheorem4.p1.11.11.m11.1.2.2" xref="A1.Thmtheorem4.p1.11.11.m11.1.2.2.cmml"><mi id="A1.Thmtheorem4.p1.11.11.m11.1.2.2.2" xref="A1.Thmtheorem4.p1.11.11.m11.1.2.2.2.cmml">ρ</mi><mo id="A1.Thmtheorem4.p1.11.11.m11.1.2.2.3" xref="A1.Thmtheorem4.p1.11.11.m11.1.2.2.3.cmml">∗</mo></msup><mo id="A1.Thmtheorem4.p1.11.11.m11.1.2.1" xref="A1.Thmtheorem4.p1.11.11.m11.1.2.1.cmml">⁢</mo><mrow id="A1.Thmtheorem4.p1.11.11.m11.1.2.3.2" xref="A1.Thmtheorem4.p1.11.11.m11.1.2.cmml"><mo id="A1.Thmtheorem4.p1.11.11.m11.1.2.3.2.1" stretchy="false" xref="A1.Thmtheorem4.p1.11.11.m11.1.2.cmml">(</mo><mi id="A1.Thmtheorem4.p1.11.11.m11.1.1" xref="A1.Thmtheorem4.p1.11.11.m11.1.1.cmml">v</mi><mo id="A1.Thmtheorem4.p1.11.11.m11.1.2.3.2.2" stretchy="false" xref="A1.Thmtheorem4.p1.11.11.m11.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="A1.Thmtheorem4.p1.11.11.m11.1b"><apply id="A1.Thmtheorem4.p1.11.11.m11.1.2.cmml" xref="A1.Thmtheorem4.p1.11.11.m11.1.2"><times id="A1.Thmtheorem4.p1.11.11.m11.1.2.1.cmml" xref="A1.Thmtheorem4.p1.11.11.m11.1.2.1"></times><apply id="A1.Thmtheorem4.p1.11.11.m11.1.2.2.cmml" xref="A1.Thmtheorem4.p1.11.11.m11.1.2.2"><csymbol cd="ambiguous" id="A1.Thmtheorem4.p1.11.11.m11.1.2.2.1.cmml" xref="A1.Thmtheorem4.p1.11.11.m11.1.2.2">superscript</csymbol><ci id="A1.Thmtheorem4.p1.11.11.m11.1.2.2.2.cmml" xref="A1.Thmtheorem4.p1.11.11.m11.1.2.2.2">𝜌</ci><times id="A1.Thmtheorem4.p1.11.11.m11.1.2.2.3.cmml" xref="A1.Thmtheorem4.p1.11.11.m11.1.2.2.3"></times></apply><ci id="A1.Thmtheorem4.p1.11.11.m11.1.1.cmml" xref="A1.Thmtheorem4.p1.11.11.m11.1.1">𝑣</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.Thmtheorem4.p1.11.11.m11.1c">\rho^{*}(v)</annotation><annotation encoding="application/x-llamapun" id="A1.Thmtheorem4.p1.11.11.m11.1d">italic_ρ start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ( italic_v )</annotation></semantics></math>.</span></p> </div> </div> <section class="ltx_subparagraph" id="A1.SS3.SSS0.P0.SPx1"> <h5 class="ltx_title ltx_font_italic ltx_title_subparagraph">Induction.</h5> <div class="ltx_para" id="A1.SS3.SSS0.P0.SPx1.p1"> <p class="ltx_p" id="A1.SS3.SSS0.P0.SPx1.p1.4"><span class="ltx_text ltx_font_italic" id="A1.SS3.SSS0.P0.SPx1.p1.4.1">We show by induction that if there exists an integer </span><math alttext="j" class="ltx_Math" display="inline" id="A1.SS3.SSS0.P0.SPx1.p1.1.m1.1"><semantics id="A1.SS3.SSS0.P0.SPx1.p1.1.m1.1a"><mi id="A1.SS3.SSS0.P0.SPx1.p1.1.m1.1.1" xref="A1.SS3.SSS0.P0.SPx1.p1.1.m1.1.1.cmml">j</mi><annotation-xml encoding="MathML-Content" id="A1.SS3.SSS0.P0.SPx1.p1.1.m1.1b"><ci id="A1.SS3.SSS0.P0.SPx1.p1.1.m1.1.1.cmml" xref="A1.SS3.SSS0.P0.SPx1.p1.1.m1.1.1">𝑗</ci></annotation-xml><annotation encoding="application/x-tex" id="A1.SS3.SSS0.P0.SPx1.p1.1.m1.1c">j</annotation><annotation encoding="application/x-llamapun" id="A1.SS3.SSS0.P0.SPx1.p1.1.m1.1d">italic_j</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="A1.SS3.SSS0.P0.SPx1.p1.4.2"> where for all </span><math alttext="i\leq j" class="ltx_Math" display="inline" 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id="A1.SS3.SSS0.P0.SPx1.p1.2.m2.1c">i\leq j</annotation><annotation encoding="application/x-llamapun" id="A1.SS3.SSS0.P0.SPx1.p1.2.m2.1d">italic_i ≤ italic_j</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="A1.SS3.SSS0.P0.SPx1.p1.4.3"> the density </span><math alttext="\rho(G[V_{i}])<(1-\varepsilon)\rho^{*}(v)" class="ltx_Math" display="inline" id="A1.SS3.SSS0.P0.SPx1.p1.3.m3.3"><semantics id="A1.SS3.SSS0.P0.SPx1.p1.3.m3.3a"><mrow id="A1.SS3.SSS0.P0.SPx1.p1.3.m3.3.3" xref="A1.SS3.SSS0.P0.SPx1.p1.3.m3.3.3.cmml"><mrow id="A1.SS3.SSS0.P0.SPx1.p1.3.m3.2.2.1" xref="A1.SS3.SSS0.P0.SPx1.p1.3.m3.2.2.1.cmml"><mi id="A1.SS3.SSS0.P0.SPx1.p1.3.m3.2.2.1.3" xref="A1.SS3.SSS0.P0.SPx1.p1.3.m3.2.2.1.3.cmml">ρ</mi><mo id="A1.SS3.SSS0.P0.SPx1.p1.3.m3.2.2.1.2" xref="A1.SS3.SSS0.P0.SPx1.p1.3.m3.2.2.1.2.cmml">⁢</mo><mrow id="A1.SS3.SSS0.P0.SPx1.p1.3.m3.2.2.1.1.1" xref="A1.SS3.SSS0.P0.SPx1.p1.3.m3.2.2.1.1.1.1.cmml"><mo id="A1.SS3.SSS0.P0.SPx1.p1.3.m3.2.2.1.1.1.2" stretchy="false" 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id="A1.SS3.SSS0.P0.SPx1.p1.3.m3.3.3.2.3.2.cmml" xref="A1.SS3.SSS0.P0.SPx1.p1.3.m3.3.3.2.3.2">𝜌</ci><times id="A1.SS3.SSS0.P0.SPx1.p1.3.m3.3.3.2.3.3.cmml" xref="A1.SS3.SSS0.P0.SPx1.p1.3.m3.3.3.2.3.3"></times></apply><ci id="A1.SS3.SSS0.P0.SPx1.p1.3.m3.1.1.cmml" xref="A1.SS3.SSS0.P0.SPx1.p1.3.m3.1.1">𝑣</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS3.SSS0.P0.SPx1.p1.3.m3.3c">\rho(G[V_{i}])<(1-\varepsilon)\rho^{*}(v)</annotation><annotation encoding="application/x-llamapun" id="A1.SS3.SSS0.P0.SPx1.p1.3.m3.3d">italic_ρ ( italic_G [ italic_V start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ] ) < ( 1 - italic_ε ) italic_ρ start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ( italic_v )</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="A1.SS3.SSS0.P0.SPx1.p1.4.4"> then </span><math alttext="|V_{j}|\geq(1-\varepsilon)^{-j}" class="ltx_Math" display="inline" id="A1.SS3.SSS0.P0.SPx1.p1.4.m4.2"><semantics id="A1.SS3.SSS0.P0.SPx1.p1.4.m4.2a"><mrow id="A1.SS3.SSS0.P0.SPx1.p1.4.m4.2.2" xref="A1.SS3.SSS0.P0.SPx1.p1.4.m4.2.2.cmml"><mrow id="A1.SS3.SSS0.P0.SPx1.p1.4.m4.1.1.1.1" xref="A1.SS3.SSS0.P0.SPx1.p1.4.m4.1.1.1.2.cmml"><mo id="A1.SS3.SSS0.P0.SPx1.p1.4.m4.1.1.1.1.2" stretchy="false" xref="A1.SS3.SSS0.P0.SPx1.p1.4.m4.1.1.1.2.1.cmml">|</mo><msub id="A1.SS3.SSS0.P0.SPx1.p1.4.m4.1.1.1.1.1" xref="A1.SS3.SSS0.P0.SPx1.p1.4.m4.1.1.1.1.1.cmml"><mi id="A1.SS3.SSS0.P0.SPx1.p1.4.m4.1.1.1.1.1.2" xref="A1.SS3.SSS0.P0.SPx1.p1.4.m4.1.1.1.1.1.2.cmml">V</mi><mi id="A1.SS3.SSS0.P0.SPx1.p1.4.m4.1.1.1.1.1.3" xref="A1.SS3.SSS0.P0.SPx1.p1.4.m4.1.1.1.1.1.3.cmml">j</mi></msub><mo id="A1.SS3.SSS0.P0.SPx1.p1.4.m4.1.1.1.1.3" stretchy="false" xref="A1.SS3.SSS0.P0.SPx1.p1.4.m4.1.1.1.2.1.cmml">|</mo></mrow><mo id="A1.SS3.SSS0.P0.SPx1.p1.4.m4.2.2.3" xref="A1.SS3.SSS0.P0.SPx1.p1.4.m4.2.2.3.cmml">≥</mo><msup id="A1.SS3.SSS0.P0.SPx1.p1.4.m4.2.2.2" xref="A1.SS3.SSS0.P0.SPx1.p1.4.m4.2.2.2.cmml"><mrow id="A1.SS3.SSS0.P0.SPx1.p1.4.m4.2.2.2.1.1" xref="A1.SS3.SSS0.P0.SPx1.p1.4.m4.2.2.2.1.1.1.cmml"><mo id="A1.SS3.SSS0.P0.SPx1.p1.4.m4.2.2.2.1.1.2" stretchy="false" xref="A1.SS3.SSS0.P0.SPx1.p1.4.m4.2.2.2.1.1.1.cmml">(</mo><mrow id="A1.SS3.SSS0.P0.SPx1.p1.4.m4.2.2.2.1.1.1" xref="A1.SS3.SSS0.P0.SPx1.p1.4.m4.2.2.2.1.1.1.cmml"><mn id="A1.SS3.SSS0.P0.SPx1.p1.4.m4.2.2.2.1.1.1.2" xref="A1.SS3.SSS0.P0.SPx1.p1.4.m4.2.2.2.1.1.1.2.cmml">1</mn><mo id="A1.SS3.SSS0.P0.SPx1.p1.4.m4.2.2.2.1.1.1.1" xref="A1.SS3.SSS0.P0.SPx1.p1.4.m4.2.2.2.1.1.1.1.cmml">−</mo><mi id="A1.SS3.SSS0.P0.SPx1.p1.4.m4.2.2.2.1.1.1.3" xref="A1.SS3.SSS0.P0.SPx1.p1.4.m4.2.2.2.1.1.1.3.cmml">ε</mi></mrow><mo id="A1.SS3.SSS0.P0.SPx1.p1.4.m4.2.2.2.1.1.3" stretchy="false" xref="A1.SS3.SSS0.P0.SPx1.p1.4.m4.2.2.2.1.1.1.cmml">)</mo></mrow><mrow id="A1.SS3.SSS0.P0.SPx1.p1.4.m4.2.2.2.3" xref="A1.SS3.SSS0.P0.SPx1.p1.4.m4.2.2.2.3.cmml"><mo id="A1.SS3.SSS0.P0.SPx1.p1.4.m4.2.2.2.3a" xref="A1.SS3.SSS0.P0.SPx1.p1.4.m4.2.2.2.3.cmml">−</mo><mi id="A1.SS3.SSS0.P0.SPx1.p1.4.m4.2.2.2.3.2" xref="A1.SS3.SSS0.P0.SPx1.p1.4.m4.2.2.2.3.2.cmml">j</mi></mrow></msup></mrow><annotation-xml encoding="MathML-Content" id="A1.SS3.SSS0.P0.SPx1.p1.4.m4.2b"><apply id="A1.SS3.SSS0.P0.SPx1.p1.4.m4.2.2.cmml" xref="A1.SS3.SSS0.P0.SPx1.p1.4.m4.2.2"><geq id="A1.SS3.SSS0.P0.SPx1.p1.4.m4.2.2.3.cmml" xref="A1.SS3.SSS0.P0.SPx1.p1.4.m4.2.2.3"></geq><apply id="A1.SS3.SSS0.P0.SPx1.p1.4.m4.1.1.1.2.cmml" xref="A1.SS3.SSS0.P0.SPx1.p1.4.m4.1.1.1.1"><abs id="A1.SS3.SSS0.P0.SPx1.p1.4.m4.1.1.1.2.1.cmml" xref="A1.SS3.SSS0.P0.SPx1.p1.4.m4.1.1.1.1.2"></abs><apply id="A1.SS3.SSS0.P0.SPx1.p1.4.m4.1.1.1.1.1.cmml" xref="A1.SS3.SSS0.P0.SPx1.p1.4.m4.1.1.1.1.1"><csymbol cd="ambiguous" id="A1.SS3.SSS0.P0.SPx1.p1.4.m4.1.1.1.1.1.1.cmml" xref="A1.SS3.SSS0.P0.SPx1.p1.4.m4.1.1.1.1.1">subscript</csymbol><ci id="A1.SS3.SSS0.P0.SPx1.p1.4.m4.1.1.1.1.1.2.cmml" xref="A1.SS3.SSS0.P0.SPx1.p1.4.m4.1.1.1.1.1.2">𝑉</ci><ci id="A1.SS3.SSS0.P0.SPx1.p1.4.m4.1.1.1.1.1.3.cmml" xref="A1.SS3.SSS0.P0.SPx1.p1.4.m4.1.1.1.1.1.3">𝑗</ci></apply></apply><apply id="A1.SS3.SSS0.P0.SPx1.p1.4.m4.2.2.2.cmml" xref="A1.SS3.SSS0.P0.SPx1.p1.4.m4.2.2.2"><csymbol cd="ambiguous" id="A1.SS3.SSS0.P0.SPx1.p1.4.m4.2.2.2.2.cmml" xref="A1.SS3.SSS0.P0.SPx1.p1.4.m4.2.2.2">superscript</csymbol><apply id="A1.SS3.SSS0.P0.SPx1.p1.4.m4.2.2.2.1.1.1.cmml" xref="A1.SS3.SSS0.P0.SPx1.p1.4.m4.2.2.2.1.1"><minus id="A1.SS3.SSS0.P0.SPx1.p1.4.m4.2.2.2.1.1.1.1.cmml" xref="A1.SS3.SSS0.P0.SPx1.p1.4.m4.2.2.2.1.1.1.1"></minus><cn id="A1.SS3.SSS0.P0.SPx1.p1.4.m4.2.2.2.1.1.1.2.cmml" type="integer" xref="A1.SS3.SSS0.P0.SPx1.p1.4.m4.2.2.2.1.1.1.2">1</cn><ci id="A1.SS3.SSS0.P0.SPx1.p1.4.m4.2.2.2.1.1.1.3.cmml" xref="A1.SS3.SSS0.P0.SPx1.p1.4.m4.2.2.2.1.1.1.3">𝜀</ci></apply><apply id="A1.SS3.SSS0.P0.SPx1.p1.4.m4.2.2.2.3.cmml" xref="A1.SS3.SSS0.P0.SPx1.p1.4.m4.2.2.2.3"><minus id="A1.SS3.SSS0.P0.SPx1.p1.4.m4.2.2.2.3.1.cmml" xref="A1.SS3.SSS0.P0.SPx1.p1.4.m4.2.2.2.3"></minus><ci id="A1.SS3.SSS0.P0.SPx1.p1.4.m4.2.2.2.3.2.cmml" xref="A1.SS3.SSS0.P0.SPx1.p1.4.m4.2.2.2.3.2">𝑗</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS3.SSS0.P0.SPx1.p1.4.m4.2c">|V_{j}|\geq(1-\varepsilon)^{-j}</annotation><annotation encoding="application/x-llamapun" id="A1.SS3.SSS0.P0.SPx1.p1.4.m4.2d">| italic_V start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT | ≥ ( 1 - italic_ε ) start_POSTSUPERSCRIPT - italic_j end_POSTSUPERSCRIPT</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="A1.SS3.SSS0.P0.SPx1.p1.4.5">.</span></p> </div> <div class="ltx_para" id="A1.SS3.SSS0.P0.SPx1.p2"> <p class="ltx_p" id="A1.SS3.SSS0.P0.SPx1.p2.3"><span class="ltx_text ltx_font_italic" id="A1.SS3.SSS0.P0.SPx1.p2.3.1">Indeed, for the base case observe that </span><math alttext="|V_{0}|=1\geq(1-\varepsilon)^{0}" class="ltx_Math" display="inline" id="A1.SS3.SSS0.P0.SPx1.p2.1.m1.2"><semantics id="A1.SS3.SSS0.P0.SPx1.p2.1.m1.2a"><mrow 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xref="A1.SS3.SSS0.P0.SPx1.p2.1.m1.2.2.2.3.cmml">0</mn></msup></mrow><annotation-xml encoding="MathML-Content" id="A1.SS3.SSS0.P0.SPx1.p2.1.m1.2b"><apply id="A1.SS3.SSS0.P0.SPx1.p2.1.m1.2.2.cmml" xref="A1.SS3.SSS0.P0.SPx1.p2.1.m1.2.2"><and id="A1.SS3.SSS0.P0.SPx1.p2.1.m1.2.2a.cmml" xref="A1.SS3.SSS0.P0.SPx1.p2.1.m1.2.2"></and><apply id="A1.SS3.SSS0.P0.SPx1.p2.1.m1.2.2b.cmml" xref="A1.SS3.SSS0.P0.SPx1.p2.1.m1.2.2"><eq id="A1.SS3.SSS0.P0.SPx1.p2.1.m1.2.2.4.cmml" xref="A1.SS3.SSS0.P0.SPx1.p2.1.m1.2.2.4"></eq><apply id="A1.SS3.SSS0.P0.SPx1.p2.1.m1.1.1.1.2.cmml" xref="A1.SS3.SSS0.P0.SPx1.p2.1.m1.1.1.1.1"><abs id="A1.SS3.SSS0.P0.SPx1.p2.1.m1.1.1.1.2.1.cmml" xref="A1.SS3.SSS0.P0.SPx1.p2.1.m1.1.1.1.1.2"></abs><apply id="A1.SS3.SSS0.P0.SPx1.p2.1.m1.1.1.1.1.1.cmml" xref="A1.SS3.SSS0.P0.SPx1.p2.1.m1.1.1.1.1.1"><csymbol cd="ambiguous" id="A1.SS3.SSS0.P0.SPx1.p2.1.m1.1.1.1.1.1.1.cmml" xref="A1.SS3.SSS0.P0.SPx1.p2.1.m1.1.1.1.1.1">subscript</csymbol><ci id="A1.SS3.SSS0.P0.SPx1.p2.1.m1.1.1.1.1.1.2.cmml" xref="A1.SS3.SSS0.P0.SPx1.p2.1.m1.1.1.1.1.1.2">𝑉</ci><cn id="A1.SS3.SSS0.P0.SPx1.p2.1.m1.1.1.1.1.1.3.cmml" type="integer" xref="A1.SS3.SSS0.P0.SPx1.p2.1.m1.1.1.1.1.1.3">0</cn></apply></apply><cn id="A1.SS3.SSS0.P0.SPx1.p2.1.m1.2.2.5.cmml" type="integer" xref="A1.SS3.SSS0.P0.SPx1.p2.1.m1.2.2.5">1</cn></apply><apply id="A1.SS3.SSS0.P0.SPx1.p2.1.m1.2.2c.cmml" xref="A1.SS3.SSS0.P0.SPx1.p2.1.m1.2.2"><geq id="A1.SS3.SSS0.P0.SPx1.p2.1.m1.2.2.6.cmml" xref="A1.SS3.SSS0.P0.SPx1.p2.1.m1.2.2.6"></geq><share href="https://arxiv.org/html/2411.12694v2#A1.SS3.SSS0.P0.SPx1.p2.1.m1.2.2.5.cmml" id="A1.SS3.SSS0.P0.SPx1.p2.1.m1.2.2d.cmml" xref="A1.SS3.SSS0.P0.SPx1.p2.1.m1.2.2"></share><apply id="A1.SS3.SSS0.P0.SPx1.p2.1.m1.2.2.2.cmml" xref="A1.SS3.SSS0.P0.SPx1.p2.1.m1.2.2.2"><csymbol cd="ambiguous" id="A1.SS3.SSS0.P0.SPx1.p2.1.m1.2.2.2.2.cmml" xref="A1.SS3.SSS0.P0.SPx1.p2.1.m1.2.2.2">superscript</csymbol><apply id="A1.SS3.SSS0.P0.SPx1.p2.1.m1.2.2.2.1.1.1.cmml" xref="A1.SS3.SSS0.P0.SPx1.p2.1.m1.2.2.2.1.1"><minus id="A1.SS3.SSS0.P0.SPx1.p2.1.m1.2.2.2.1.1.1.1.cmml" xref="A1.SS3.SSS0.P0.SPx1.p2.1.m1.2.2.2.1.1.1.1"></minus><cn id="A1.SS3.SSS0.P0.SPx1.p2.1.m1.2.2.2.1.1.1.2.cmml" type="integer" xref="A1.SS3.SSS0.P0.SPx1.p2.1.m1.2.2.2.1.1.1.2">1</cn><ci id="A1.SS3.SSS0.P0.SPx1.p2.1.m1.2.2.2.1.1.1.3.cmml" xref="A1.SS3.SSS0.P0.SPx1.p2.1.m1.2.2.2.1.1.1.3">𝜀</ci></apply><cn id="A1.SS3.SSS0.P0.SPx1.p2.1.m1.2.2.2.3.cmml" type="integer" xref="A1.SS3.SSS0.P0.SPx1.p2.1.m1.2.2.2.3">0</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS3.SSS0.P0.SPx1.p2.1.m1.2c">|V_{0}|=1\geq(1-\varepsilon)^{0}</annotation><annotation encoding="application/x-llamapun" id="A1.SS3.SSS0.P0.SPx1.p2.1.m1.2d">| italic_V start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT | = 1 ≥ ( 1 - italic_ε ) start_POSTSUPERSCRIPT 0 end_POSTSUPERSCRIPT</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="A1.SS3.SSS0.P0.SPx1.p2.3.2">. Now assume </span><math alttext="|V_{j-1}|\geq(1-\varepsilon)^{-(j-1)}" class="ltx_Math" display="inline" id="A1.SS3.SSS0.P0.SPx1.p2.2.m2.3"><semantics id="A1.SS3.SSS0.P0.SPx1.p2.2.m2.3a"><mrow id="A1.SS3.SSS0.P0.SPx1.p2.2.m2.3.3" xref="A1.SS3.SSS0.P0.SPx1.p2.2.m2.3.3.cmml"><mrow id="A1.SS3.SSS0.P0.SPx1.p2.2.m2.2.2.1.1" xref="A1.SS3.SSS0.P0.SPx1.p2.2.m2.2.2.1.2.cmml"><mo id="A1.SS3.SSS0.P0.SPx1.p2.2.m2.2.2.1.1.2" stretchy="false" xref="A1.SS3.SSS0.P0.SPx1.p2.2.m2.2.2.1.2.1.cmml">|</mo><msub id="A1.SS3.SSS0.P0.SPx1.p2.2.m2.2.2.1.1.1" xref="A1.SS3.SSS0.P0.SPx1.p2.2.m2.2.2.1.1.1.cmml"><mi id="A1.SS3.SSS0.P0.SPx1.p2.2.m2.2.2.1.1.1.2" xref="A1.SS3.SSS0.P0.SPx1.p2.2.m2.2.2.1.1.1.2.cmml">V</mi><mrow id="A1.SS3.SSS0.P0.SPx1.p2.2.m2.2.2.1.1.1.3" xref="A1.SS3.SSS0.P0.SPx1.p2.2.m2.2.2.1.1.1.3.cmml"><mi id="A1.SS3.SSS0.P0.SPx1.p2.2.m2.2.2.1.1.1.3.2" xref="A1.SS3.SSS0.P0.SPx1.p2.2.m2.2.2.1.1.1.3.2.cmml">j</mi><mo id="A1.SS3.SSS0.P0.SPx1.p2.2.m2.2.2.1.1.1.3.1" xref="A1.SS3.SSS0.P0.SPx1.p2.2.m2.2.2.1.1.1.3.1.cmml">−</mo><mn id="A1.SS3.SSS0.P0.SPx1.p2.2.m2.2.2.1.1.1.3.3" xref="A1.SS3.SSS0.P0.SPx1.p2.2.m2.2.2.1.1.1.3.3.cmml">1</mn></mrow></msub><mo id="A1.SS3.SSS0.P0.SPx1.p2.2.m2.2.2.1.1.3" stretchy="false" xref="A1.SS3.SSS0.P0.SPx1.p2.2.m2.2.2.1.2.1.cmml">|</mo></mrow><mo id="A1.SS3.SSS0.P0.SPx1.p2.2.m2.3.3.3" xref="A1.SS3.SSS0.P0.SPx1.p2.2.m2.3.3.3.cmml">≥</mo><msup id="A1.SS3.SSS0.P0.SPx1.p2.2.m2.3.3.2" xref="A1.SS3.SSS0.P0.SPx1.p2.2.m2.3.3.2.cmml"><mrow id="A1.SS3.SSS0.P0.SPx1.p2.2.m2.3.3.2.1.1" xref="A1.SS3.SSS0.P0.SPx1.p2.2.m2.3.3.2.1.1.1.cmml"><mo id="A1.SS3.SSS0.P0.SPx1.p2.2.m2.3.3.2.1.1.2" stretchy="false" xref="A1.SS3.SSS0.P0.SPx1.p2.2.m2.3.3.2.1.1.1.cmml">(</mo><mrow id="A1.SS3.SSS0.P0.SPx1.p2.2.m2.3.3.2.1.1.1" xref="A1.SS3.SSS0.P0.SPx1.p2.2.m2.3.3.2.1.1.1.cmml"><mn id="A1.SS3.SSS0.P0.SPx1.p2.2.m2.3.3.2.1.1.1.2" xref="A1.SS3.SSS0.P0.SPx1.p2.2.m2.3.3.2.1.1.1.2.cmml">1</mn><mo id="A1.SS3.SSS0.P0.SPx1.p2.2.m2.3.3.2.1.1.1.1" xref="A1.SS3.SSS0.P0.SPx1.p2.2.m2.3.3.2.1.1.1.1.cmml">−</mo><mi id="A1.SS3.SSS0.P0.SPx1.p2.2.m2.3.3.2.1.1.1.3" xref="A1.SS3.SSS0.P0.SPx1.p2.2.m2.3.3.2.1.1.1.3.cmml">ε</mi></mrow><mo id="A1.SS3.SSS0.P0.SPx1.p2.2.m2.3.3.2.1.1.3" stretchy="false" xref="A1.SS3.SSS0.P0.SPx1.p2.2.m2.3.3.2.1.1.1.cmml">)</mo></mrow><mrow id="A1.SS3.SSS0.P0.SPx1.p2.2.m2.1.1.1" xref="A1.SS3.SSS0.P0.SPx1.p2.2.m2.1.1.1.cmml"><mo id="A1.SS3.SSS0.P0.SPx1.p2.2.m2.1.1.1a" xref="A1.SS3.SSS0.P0.SPx1.p2.2.m2.1.1.1.cmml">−</mo><mrow id="A1.SS3.SSS0.P0.SPx1.p2.2.m2.1.1.1.1.1" xref="A1.SS3.SSS0.P0.SPx1.p2.2.m2.1.1.1.1.1.1.cmml"><mo id="A1.SS3.SSS0.P0.SPx1.p2.2.m2.1.1.1.1.1.2" stretchy="false" xref="A1.SS3.SSS0.P0.SPx1.p2.2.m2.1.1.1.1.1.1.cmml">(</mo><mrow id="A1.SS3.SSS0.P0.SPx1.p2.2.m2.1.1.1.1.1.1" xref="A1.SS3.SSS0.P0.SPx1.p2.2.m2.1.1.1.1.1.1.cmml"><mi id="A1.SS3.SSS0.P0.SPx1.p2.2.m2.1.1.1.1.1.1.2" xref="A1.SS3.SSS0.P0.SPx1.p2.2.m2.1.1.1.1.1.1.2.cmml">j</mi><mo id="A1.SS3.SSS0.P0.SPx1.p2.2.m2.1.1.1.1.1.1.1" xref="A1.SS3.SSS0.P0.SPx1.p2.2.m2.1.1.1.1.1.1.1.cmml">−</mo><mn id="A1.SS3.SSS0.P0.SPx1.p2.2.m2.1.1.1.1.1.1.3" xref="A1.SS3.SSS0.P0.SPx1.p2.2.m2.1.1.1.1.1.1.3.cmml">1</mn></mrow><mo id="A1.SS3.SSS0.P0.SPx1.p2.2.m2.1.1.1.1.1.3" stretchy="false" xref="A1.SS3.SSS0.P0.SPx1.p2.2.m2.1.1.1.1.1.1.cmml">)</mo></mrow></mrow></msup></mrow><annotation-xml encoding="MathML-Content" id="A1.SS3.SSS0.P0.SPx1.p2.2.m2.3b"><apply id="A1.SS3.SSS0.P0.SPx1.p2.2.m2.3.3.cmml" xref="A1.SS3.SSS0.P0.SPx1.p2.2.m2.3.3"><geq id="A1.SS3.SSS0.P0.SPx1.p2.2.m2.3.3.3.cmml" xref="A1.SS3.SSS0.P0.SPx1.p2.2.m2.3.3.3"></geq><apply id="A1.SS3.SSS0.P0.SPx1.p2.2.m2.2.2.1.2.cmml" xref="A1.SS3.SSS0.P0.SPx1.p2.2.m2.2.2.1.1"><abs id="A1.SS3.SSS0.P0.SPx1.p2.2.m2.2.2.1.2.1.cmml" xref="A1.SS3.SSS0.P0.SPx1.p2.2.m2.2.2.1.1.2"></abs><apply id="A1.SS3.SSS0.P0.SPx1.p2.2.m2.2.2.1.1.1.cmml" xref="A1.SS3.SSS0.P0.SPx1.p2.2.m2.2.2.1.1.1"><csymbol cd="ambiguous" id="A1.SS3.SSS0.P0.SPx1.p2.2.m2.2.2.1.1.1.1.cmml" xref="A1.SS3.SSS0.P0.SPx1.p2.2.m2.2.2.1.1.1">subscript</csymbol><ci id="A1.SS3.SSS0.P0.SPx1.p2.2.m2.2.2.1.1.1.2.cmml" xref="A1.SS3.SSS0.P0.SPx1.p2.2.m2.2.2.1.1.1.2">𝑉</ci><apply id="A1.SS3.SSS0.P0.SPx1.p2.2.m2.2.2.1.1.1.3.cmml" xref="A1.SS3.SSS0.P0.SPx1.p2.2.m2.2.2.1.1.1.3"><minus id="A1.SS3.SSS0.P0.SPx1.p2.2.m2.2.2.1.1.1.3.1.cmml" xref="A1.SS3.SSS0.P0.SPx1.p2.2.m2.2.2.1.1.1.3.1"></minus><ci id="A1.SS3.SSS0.P0.SPx1.p2.2.m2.2.2.1.1.1.3.2.cmml" xref="A1.SS3.SSS0.P0.SPx1.p2.2.m2.2.2.1.1.1.3.2">𝑗</ci><cn id="A1.SS3.SSS0.P0.SPx1.p2.2.m2.2.2.1.1.1.3.3.cmml" type="integer" xref="A1.SS3.SSS0.P0.SPx1.p2.2.m2.2.2.1.1.1.3.3">1</cn></apply></apply></apply><apply id="A1.SS3.SSS0.P0.SPx1.p2.2.m2.3.3.2.cmml" xref="A1.SS3.SSS0.P0.SPx1.p2.2.m2.3.3.2"><csymbol cd="ambiguous" id="A1.SS3.SSS0.P0.SPx1.p2.2.m2.3.3.2.2.cmml" xref="A1.SS3.SSS0.P0.SPx1.p2.2.m2.3.3.2">superscript</csymbol><apply id="A1.SS3.SSS0.P0.SPx1.p2.2.m2.3.3.2.1.1.1.cmml" xref="A1.SS3.SSS0.P0.SPx1.p2.2.m2.3.3.2.1.1"><minus id="A1.SS3.SSS0.P0.SPx1.p2.2.m2.3.3.2.1.1.1.1.cmml" xref="A1.SS3.SSS0.P0.SPx1.p2.2.m2.3.3.2.1.1.1.1"></minus><cn id="A1.SS3.SSS0.P0.SPx1.p2.2.m2.3.3.2.1.1.1.2.cmml" type="integer" xref="A1.SS3.SSS0.P0.SPx1.p2.2.m2.3.3.2.1.1.1.2">1</cn><ci id="A1.SS3.SSS0.P0.SPx1.p2.2.m2.3.3.2.1.1.1.3.cmml" xref="A1.SS3.SSS0.P0.SPx1.p2.2.m2.3.3.2.1.1.1.3">𝜀</ci></apply><apply id="A1.SS3.SSS0.P0.SPx1.p2.2.m2.1.1.1.cmml" xref="A1.SS3.SSS0.P0.SPx1.p2.2.m2.1.1.1"><minus id="A1.SS3.SSS0.P0.SPx1.p2.2.m2.1.1.1.2.cmml" xref="A1.SS3.SSS0.P0.SPx1.p2.2.m2.1.1.1"></minus><apply id="A1.SS3.SSS0.P0.SPx1.p2.2.m2.1.1.1.1.1.1.cmml" xref="A1.SS3.SSS0.P0.SPx1.p2.2.m2.1.1.1.1.1"><minus id="A1.SS3.SSS0.P0.SPx1.p2.2.m2.1.1.1.1.1.1.1.cmml" xref="A1.SS3.SSS0.P0.SPx1.p2.2.m2.1.1.1.1.1.1.1"></minus><ci id="A1.SS3.SSS0.P0.SPx1.p2.2.m2.1.1.1.1.1.1.2.cmml" xref="A1.SS3.SSS0.P0.SPx1.p2.2.m2.1.1.1.1.1.1.2">𝑗</ci><cn id="A1.SS3.SSS0.P0.SPx1.p2.2.m2.1.1.1.1.1.1.3.cmml" type="integer" xref="A1.SS3.SSS0.P0.SPx1.p2.2.m2.1.1.1.1.1.1.3">1</cn></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS3.SSS0.P0.SPx1.p2.2.m2.3c">|V_{j-1}|\geq(1-\varepsilon)^{-(j-1)}</annotation><annotation encoding="application/x-llamapun" id="A1.SS3.SSS0.P0.SPx1.p2.2.m2.3d">| italic_V start_POSTSUBSCRIPT italic_j - 1 end_POSTSUBSCRIPT | ≥ ( 1 - italic_ε ) start_POSTSUPERSCRIPT - ( italic_j - 1 ) end_POSTSUPERSCRIPT</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="A1.SS3.SSS0.P0.SPx1.p2.3.3">. We show that it also holds for </span><math alttext="j" class="ltx_Math" display="inline" id="A1.SS3.SSS0.P0.SPx1.p2.3.m3.1"><semantics id="A1.SS3.SSS0.P0.SPx1.p2.3.m3.1a"><mi id="A1.SS3.SSS0.P0.SPx1.p2.3.m3.1.1" xref="A1.SS3.SSS0.P0.SPx1.p2.3.m3.1.1.cmml">j</mi><annotation-xml encoding="MathML-Content" id="A1.SS3.SSS0.P0.SPx1.p2.3.m3.1b"><ci id="A1.SS3.SSS0.P0.SPx1.p2.3.m3.1.1.cmml" xref="A1.SS3.SSS0.P0.SPx1.p2.3.m3.1.1">𝑗</ci></annotation-xml><annotation encoding="application/x-tex" id="A1.SS3.SSS0.P0.SPx1.p2.3.m3.1c">j</annotation><annotation encoding="application/x-llamapun" id="A1.SS3.SSS0.P0.SPx1.p2.3.m3.1d">italic_j</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="A1.SS3.SSS0.P0.SPx1.p2.3.4">.</span></p> </div> <div class="ltx_para" id="A1.SS3.SSS0.P0.SPx1.p3"> <p class="ltx_p" id="A1.SS3.SSS0.P0.SPx1.p3.6"><span class="ltx_text ltx_font_italic" id="A1.SS3.SSS0.P0.SPx1.p3.6.1">For brevity, we denote for any set of edges </span><math alttext="E^{\prime}" class="ltx_Math" display="inline" id="A1.SS3.SSS0.P0.SPx1.p3.1.m1.1"><semantics id="A1.SS3.SSS0.P0.SPx1.p3.1.m1.1a"><msup id="A1.SS3.SSS0.P0.SPx1.p3.1.m1.1.1" xref="A1.SS3.SSS0.P0.SPx1.p3.1.m1.1.1.cmml"><mi id="A1.SS3.SSS0.P0.SPx1.p3.1.m1.1.1.2" xref="A1.SS3.SSS0.P0.SPx1.p3.1.m1.1.1.2.cmml">E</mi><mo id="A1.SS3.SSS0.P0.SPx1.p3.1.m1.1.1.3" xref="A1.SS3.SSS0.P0.SPx1.p3.1.m1.1.1.3.cmml">′</mo></msup><annotation-xml encoding="MathML-Content" id="A1.SS3.SSS0.P0.SPx1.p3.1.m1.1b"><apply id="A1.SS3.SSS0.P0.SPx1.p3.1.m1.1.1.cmml" xref="A1.SS3.SSS0.P0.SPx1.p3.1.m1.1.1"><csymbol cd="ambiguous" id="A1.SS3.SSS0.P0.SPx1.p3.1.m1.1.1.1.cmml" xref="A1.SS3.SSS0.P0.SPx1.p3.1.m1.1.1">superscript</csymbol><ci id="A1.SS3.SSS0.P0.SPx1.p3.1.m1.1.1.2.cmml" xref="A1.SS3.SSS0.P0.SPx1.p3.1.m1.1.1.2">𝐸</ci><ci id="A1.SS3.SSS0.P0.SPx1.p3.1.m1.1.1.3.cmml" xref="A1.SS3.SSS0.P0.SPx1.p3.1.m1.1.1.3">′</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS3.SSS0.P0.SPx1.p3.1.m1.1c">E^{\prime}</annotation><annotation encoding="application/x-llamapun" id="A1.SS3.SSS0.P0.SPx1.p3.1.m1.1d">italic_E start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="A1.SS3.SSS0.P0.SPx1.p3.6.2"> by </span><math alttext="\omega(E^{\prime}):=\sum_{e\in E^{\prime}}\textsl{g}(e)" class="ltx_Math" display="inline" id="A1.SS3.SSS0.P0.SPx1.p3.2.m2.2"><semantics id="A1.SS3.SSS0.P0.SPx1.p3.2.m2.2a"><mrow id="A1.SS3.SSS0.P0.SPx1.p3.2.m2.2.2" xref="A1.SS3.SSS0.P0.SPx1.p3.2.m2.2.2.cmml"><mrow id="A1.SS3.SSS0.P0.SPx1.p3.2.m2.2.2.1" xref="A1.SS3.SSS0.P0.SPx1.p3.2.m2.2.2.1.cmml"><mi id="A1.SS3.SSS0.P0.SPx1.p3.2.m2.2.2.1.3" xref="A1.SS3.SSS0.P0.SPx1.p3.2.m2.2.2.1.3.cmml">ω</mi><mo id="A1.SS3.SSS0.P0.SPx1.p3.2.m2.2.2.1.2" xref="A1.SS3.SSS0.P0.SPx1.p3.2.m2.2.2.1.2.cmml">⁢</mo><mrow id="A1.SS3.SSS0.P0.SPx1.p3.2.m2.2.2.1.1.1" xref="A1.SS3.SSS0.P0.SPx1.p3.2.m2.2.2.1.1.1.1.cmml"><mo 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xref="A1.SS3.SSS0.P0.SPx1.p3.2.m2.2.2.3.1.2.cmml">∑</mo><mrow id="A1.SS3.SSS0.P0.SPx1.p3.2.m2.2.2.3.1.3" xref="A1.SS3.SSS0.P0.SPx1.p3.2.m2.2.2.3.1.3.cmml"><mi id="A1.SS3.SSS0.P0.SPx1.p3.2.m2.2.2.3.1.3.2" xref="A1.SS3.SSS0.P0.SPx1.p3.2.m2.2.2.3.1.3.2.cmml">e</mi><mo id="A1.SS3.SSS0.P0.SPx1.p3.2.m2.2.2.3.1.3.1" xref="A1.SS3.SSS0.P0.SPx1.p3.2.m2.2.2.3.1.3.1.cmml">∈</mo><msup id="A1.SS3.SSS0.P0.SPx1.p3.2.m2.2.2.3.1.3.3" xref="A1.SS3.SSS0.P0.SPx1.p3.2.m2.2.2.3.1.3.3.cmml"><mi id="A1.SS3.SSS0.P0.SPx1.p3.2.m2.2.2.3.1.3.3.2" xref="A1.SS3.SSS0.P0.SPx1.p3.2.m2.2.2.3.1.3.3.2.cmml">E</mi><mo id="A1.SS3.SSS0.P0.SPx1.p3.2.m2.2.2.3.1.3.3.3" xref="A1.SS3.SSS0.P0.SPx1.p3.2.m2.2.2.3.1.3.3.3.cmml">′</mo></msup></mrow></msub><mrow id="A1.SS3.SSS0.P0.SPx1.p3.2.m2.2.2.3.2" xref="A1.SS3.SSS0.P0.SPx1.p3.2.m2.2.2.3.2.cmml"><mtext class="ltx_mathvariant_italic" id="A1.SS3.SSS0.P0.SPx1.p3.2.m2.2.2.3.2.2" xref="A1.SS3.SSS0.P0.SPx1.p3.2.m2.2.2.3.2.2a.cmml">g</mtext><mo id="A1.SS3.SSS0.P0.SPx1.p3.2.m2.2.2.3.2.1" xref="A1.SS3.SSS0.P0.SPx1.p3.2.m2.2.2.3.2.1.cmml">⁢</mo><mrow id="A1.SS3.SSS0.P0.SPx1.p3.2.m2.2.2.3.2.3.2" xref="A1.SS3.SSS0.P0.SPx1.p3.2.m2.2.2.3.2.cmml"><mo id="A1.SS3.SSS0.P0.SPx1.p3.2.m2.2.2.3.2.3.2.1" stretchy="false" xref="A1.SS3.SSS0.P0.SPx1.p3.2.m2.2.2.3.2.cmml">(</mo><mi id="A1.SS3.SSS0.P0.SPx1.p3.2.m2.1.1" xref="A1.SS3.SSS0.P0.SPx1.p3.2.m2.1.1.cmml">e</mi><mo id="A1.SS3.SSS0.P0.SPx1.p3.2.m2.2.2.3.2.3.2.2" stretchy="false" xref="A1.SS3.SSS0.P0.SPx1.p3.2.m2.2.2.3.2.cmml">)</mo></mrow></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="A1.SS3.SSS0.P0.SPx1.p3.2.m2.2b"><apply id="A1.SS3.SSS0.P0.SPx1.p3.2.m2.2.2.cmml" xref="A1.SS3.SSS0.P0.SPx1.p3.2.m2.2.2"><csymbol cd="latexml" id="A1.SS3.SSS0.P0.SPx1.p3.2.m2.2.2.2.cmml" xref="A1.SS3.SSS0.P0.SPx1.p3.2.m2.2.2.2">assign</csymbol><apply id="A1.SS3.SSS0.P0.SPx1.p3.2.m2.2.2.1.cmml" xref="A1.SS3.SSS0.P0.SPx1.p3.2.m2.2.2.1"><times id="A1.SS3.SSS0.P0.SPx1.p3.2.m2.2.2.1.2.cmml" xref="A1.SS3.SSS0.P0.SPx1.p3.2.m2.2.2.1.2"></times><ci id="A1.SS3.SSS0.P0.SPx1.p3.2.m2.2.2.1.3.cmml" xref="A1.SS3.SSS0.P0.SPx1.p3.2.m2.2.2.1.3">𝜔</ci><apply id="A1.SS3.SSS0.P0.SPx1.p3.2.m2.2.2.1.1.1.1.cmml" xref="A1.SS3.SSS0.P0.SPx1.p3.2.m2.2.2.1.1.1"><csymbol cd="ambiguous" id="A1.SS3.SSS0.P0.SPx1.p3.2.m2.2.2.1.1.1.1.1.cmml" xref="A1.SS3.SSS0.P0.SPx1.p3.2.m2.2.2.1.1.1">superscript</csymbol><ci id="A1.SS3.SSS0.P0.SPx1.p3.2.m2.2.2.1.1.1.1.2.cmml" xref="A1.SS3.SSS0.P0.SPx1.p3.2.m2.2.2.1.1.1.1.2">𝐸</ci><ci id="A1.SS3.SSS0.P0.SPx1.p3.2.m2.2.2.1.1.1.1.3.cmml" xref="A1.SS3.SSS0.P0.SPx1.p3.2.m2.2.2.1.1.1.1.3">′</ci></apply></apply><apply id="A1.SS3.SSS0.P0.SPx1.p3.2.m2.2.2.3.cmml" xref="A1.SS3.SSS0.P0.SPx1.p3.2.m2.2.2.3"><apply id="A1.SS3.SSS0.P0.SPx1.p3.2.m2.2.2.3.1.cmml" xref="A1.SS3.SSS0.P0.SPx1.p3.2.m2.2.2.3.1"><csymbol cd="ambiguous" id="A1.SS3.SSS0.P0.SPx1.p3.2.m2.2.2.3.1.1.cmml" xref="A1.SS3.SSS0.P0.SPx1.p3.2.m2.2.2.3.1">subscript</csymbol><sum id="A1.SS3.SSS0.P0.SPx1.p3.2.m2.2.2.3.1.2.cmml" xref="A1.SS3.SSS0.P0.SPx1.p3.2.m2.2.2.3.1.2"></sum><apply id="A1.SS3.SSS0.P0.SPx1.p3.2.m2.2.2.3.1.3.cmml" xref="A1.SS3.SSS0.P0.SPx1.p3.2.m2.2.2.3.1.3"><in id="A1.SS3.SSS0.P0.SPx1.p3.2.m2.2.2.3.1.3.1.cmml" xref="A1.SS3.SSS0.P0.SPx1.p3.2.m2.2.2.3.1.3.1"></in><ci id="A1.SS3.SSS0.P0.SPx1.p3.2.m2.2.2.3.1.3.2.cmml" xref="A1.SS3.SSS0.P0.SPx1.p3.2.m2.2.2.3.1.3.2">𝑒</ci><apply id="A1.SS3.SSS0.P0.SPx1.p3.2.m2.2.2.3.1.3.3.cmml" xref="A1.SS3.SSS0.P0.SPx1.p3.2.m2.2.2.3.1.3.3"><csymbol cd="ambiguous" id="A1.SS3.SSS0.P0.SPx1.p3.2.m2.2.2.3.1.3.3.1.cmml" xref="A1.SS3.SSS0.P0.SPx1.p3.2.m2.2.2.3.1.3.3">superscript</csymbol><ci id="A1.SS3.SSS0.P0.SPx1.p3.2.m2.2.2.3.1.3.3.2.cmml" xref="A1.SS3.SSS0.P0.SPx1.p3.2.m2.2.2.3.1.3.3.2">𝐸</ci><ci id="A1.SS3.SSS0.P0.SPx1.p3.2.m2.2.2.3.1.3.3.3.cmml" xref="A1.SS3.SSS0.P0.SPx1.p3.2.m2.2.2.3.1.3.3.3">′</ci></apply></apply></apply><apply id="A1.SS3.SSS0.P0.SPx1.p3.2.m2.2.2.3.2.cmml" xref="A1.SS3.SSS0.P0.SPx1.p3.2.m2.2.2.3.2"><times id="A1.SS3.SSS0.P0.SPx1.p3.2.m2.2.2.3.2.1.cmml" xref="A1.SS3.SSS0.P0.SPx1.p3.2.m2.2.2.3.2.1"></times><ci id="A1.SS3.SSS0.P0.SPx1.p3.2.m2.2.2.3.2.2a.cmml" xref="A1.SS3.SSS0.P0.SPx1.p3.2.m2.2.2.3.2.2"><mtext class="ltx_mathvariant_italic" id="A1.SS3.SSS0.P0.SPx1.p3.2.m2.2.2.3.2.2.cmml" xref="A1.SS3.SSS0.P0.SPx1.p3.2.m2.2.2.3.2.2">g</mtext></ci><ci id="A1.SS3.SSS0.P0.SPx1.p3.2.m2.1.1.cmml" xref="A1.SS3.SSS0.P0.SPx1.p3.2.m2.1.1">𝑒</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS3.SSS0.P0.SPx1.p3.2.m2.2c">\omega(E^{\prime}):=\sum_{e\in E^{\prime}}\textsl{g}(e)</annotation><annotation encoding="application/x-llamapun" id="A1.SS3.SSS0.P0.SPx1.p3.2.m2.2d">italic_ω ( italic_E start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) := ∑ start_POSTSUBSCRIPT italic_e ∈ italic_E start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT g ( italic_e )</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="A1.SS3.SSS0.P0.SPx1.p3.6.3">. All vertices in </span><math alttext="V_{j-1}" class="ltx_Math" display="inline" id="A1.SS3.SSS0.P0.SPx1.p3.3.m3.1"><semantics id="A1.SS3.SSS0.P0.SPx1.p3.3.m3.1a"><msub id="A1.SS3.SSS0.P0.SPx1.p3.3.m3.1.1" xref="A1.SS3.SSS0.P0.SPx1.p3.3.m3.1.1.cmml"><mi id="A1.SS3.SSS0.P0.SPx1.p3.3.m3.1.1.2" xref="A1.SS3.SSS0.P0.SPx1.p3.3.m3.1.1.2.cmml">V</mi><mrow id="A1.SS3.SSS0.P0.SPx1.p3.3.m3.1.1.3" xref="A1.SS3.SSS0.P0.SPx1.p3.3.m3.1.1.3.cmml"><mi id="A1.SS3.SSS0.P0.SPx1.p3.3.m3.1.1.3.2" xref="A1.SS3.SSS0.P0.SPx1.p3.3.m3.1.1.3.2.cmml">j</mi><mo id="A1.SS3.SSS0.P0.SPx1.p3.3.m3.1.1.3.1" xref="A1.SS3.SSS0.P0.SPx1.p3.3.m3.1.1.3.1.cmml">−</mo><mn id="A1.SS3.SSS0.P0.SPx1.p3.3.m3.1.1.3.3" xref="A1.SS3.SSS0.P0.SPx1.p3.3.m3.1.1.3.3.cmml">1</mn></mrow></msub><annotation-xml encoding="MathML-Content" id="A1.SS3.SSS0.P0.SPx1.p3.3.m3.1b"><apply id="A1.SS3.SSS0.P0.SPx1.p3.3.m3.1.1.cmml" xref="A1.SS3.SSS0.P0.SPx1.p3.3.m3.1.1"><csymbol cd="ambiguous" id="A1.SS3.SSS0.P0.SPx1.p3.3.m3.1.1.1.cmml" xref="A1.SS3.SSS0.P0.SPx1.p3.3.m3.1.1">subscript</csymbol><ci id="A1.SS3.SSS0.P0.SPx1.p3.3.m3.1.1.2.cmml" xref="A1.SS3.SSS0.P0.SPx1.p3.3.m3.1.1.2">𝑉</ci><apply id="A1.SS3.SSS0.P0.SPx1.p3.3.m3.1.1.3.cmml" xref="A1.SS3.SSS0.P0.SPx1.p3.3.m3.1.1.3"><minus id="A1.SS3.SSS0.P0.SPx1.p3.3.m3.1.1.3.1.cmml" xref="A1.SS3.SSS0.P0.SPx1.p3.3.m3.1.1.3.1"></minus><ci id="A1.SS3.SSS0.P0.SPx1.p3.3.m3.1.1.3.2.cmml" xref="A1.SS3.SSS0.P0.SPx1.p3.3.m3.1.1.3.2">𝑗</ci><cn id="A1.SS3.SSS0.P0.SPx1.p3.3.m3.1.1.3.3.cmml" type="integer" xref="A1.SS3.SSS0.P0.SPx1.p3.3.m3.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS3.SSS0.P0.SPx1.p3.3.m3.1c">V_{j-1}</annotation><annotation encoding="application/x-llamapun" id="A1.SS3.SSS0.P0.SPx1.p3.3.m3.1d">italic_V start_POSTSUBSCRIPT italic_j - 1 end_POSTSUBSCRIPT</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="A1.SS3.SSS0.P0.SPx1.p3.6.4"> have an out-degree of at least </span><math alttext="\rho^{*}(v)" class="ltx_Math" display="inline" id="A1.SS3.SSS0.P0.SPx1.p3.4.m4.1"><semantics id="A1.SS3.SSS0.P0.SPx1.p3.4.m4.1a"><mrow id="A1.SS3.SSS0.P0.SPx1.p3.4.m4.1.2" xref="A1.SS3.SSS0.P0.SPx1.p3.4.m4.1.2.cmml"><msup id="A1.SS3.SSS0.P0.SPx1.p3.4.m4.1.2.2" xref="A1.SS3.SSS0.P0.SPx1.p3.4.m4.1.2.2.cmml"><mi id="A1.SS3.SSS0.P0.SPx1.p3.4.m4.1.2.2.2" xref="A1.SS3.SSS0.P0.SPx1.p3.4.m4.1.2.2.2.cmml">ρ</mi><mo id="A1.SS3.SSS0.P0.SPx1.p3.4.m4.1.2.2.3" xref="A1.SS3.SSS0.P0.SPx1.p3.4.m4.1.2.2.3.cmml">∗</mo></msup><mo id="A1.SS3.SSS0.P0.SPx1.p3.4.m4.1.2.1" xref="A1.SS3.SSS0.P0.SPx1.p3.4.m4.1.2.1.cmml">⁢</mo><mrow id="A1.SS3.SSS0.P0.SPx1.p3.4.m4.1.2.3.2" xref="A1.SS3.SSS0.P0.SPx1.p3.4.m4.1.2.cmml"><mo id="A1.SS3.SSS0.P0.SPx1.p3.4.m4.1.2.3.2.1" stretchy="false" xref="A1.SS3.SSS0.P0.SPx1.p3.4.m4.1.2.cmml">(</mo><mi id="A1.SS3.SSS0.P0.SPx1.p3.4.m4.1.1" xref="A1.SS3.SSS0.P0.SPx1.p3.4.m4.1.1.cmml">v</mi><mo id="A1.SS3.SSS0.P0.SPx1.p3.4.m4.1.2.3.2.2" stretchy="false" xref="A1.SS3.SSS0.P0.SPx1.p3.4.m4.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="A1.SS3.SSS0.P0.SPx1.p3.4.m4.1b"><apply id="A1.SS3.SSS0.P0.SPx1.p3.4.m4.1.2.cmml" xref="A1.SS3.SSS0.P0.SPx1.p3.4.m4.1.2"><times id="A1.SS3.SSS0.P0.SPx1.p3.4.m4.1.2.1.cmml" xref="A1.SS3.SSS0.P0.SPx1.p3.4.m4.1.2.1"></times><apply id="A1.SS3.SSS0.P0.SPx1.p3.4.m4.1.2.2.cmml" xref="A1.SS3.SSS0.P0.SPx1.p3.4.m4.1.2.2"><csymbol cd="ambiguous" id="A1.SS3.SSS0.P0.SPx1.p3.4.m4.1.2.2.1.cmml" xref="A1.SS3.SSS0.P0.SPx1.p3.4.m4.1.2.2">superscript</csymbol><ci id="A1.SS3.SSS0.P0.SPx1.p3.4.m4.1.2.2.2.cmml" xref="A1.SS3.SSS0.P0.SPx1.p3.4.m4.1.2.2.2">𝜌</ci><times id="A1.SS3.SSS0.P0.SPx1.p3.4.m4.1.2.2.3.cmml" xref="A1.SS3.SSS0.P0.SPx1.p3.4.m4.1.2.2.3"></times></apply><ci id="A1.SS3.SSS0.P0.SPx1.p3.4.m4.1.1.cmml" xref="A1.SS3.SSS0.P0.SPx1.p3.4.m4.1.1">𝑣</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS3.SSS0.P0.SPx1.p3.4.m4.1c">\rho^{*}(v)</annotation><annotation encoding="application/x-llamapun" id="A1.SS3.SSS0.P0.SPx1.p3.4.m4.1d">italic_ρ start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ( italic_v )</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="A1.SS3.SSS0.P0.SPx1.p3.6.5">, and their edge points to a vertex in </span><math alttext="V_{j-1}" class="ltx_Math" display="inline" id="A1.SS3.SSS0.P0.SPx1.p3.5.m5.1"><semantics id="A1.SS3.SSS0.P0.SPx1.p3.5.m5.1a"><msub id="A1.SS3.SSS0.P0.SPx1.p3.5.m5.1.1" xref="A1.SS3.SSS0.P0.SPx1.p3.5.m5.1.1.cmml"><mi id="A1.SS3.SSS0.P0.SPx1.p3.5.m5.1.1.2" xref="A1.SS3.SSS0.P0.SPx1.p3.5.m5.1.1.2.cmml">V</mi><mrow id="A1.SS3.SSS0.P0.SPx1.p3.5.m5.1.1.3" xref="A1.SS3.SSS0.P0.SPx1.p3.5.m5.1.1.3.cmml"><mi id="A1.SS3.SSS0.P0.SPx1.p3.5.m5.1.1.3.2" xref="A1.SS3.SSS0.P0.SPx1.p3.5.m5.1.1.3.2.cmml">j</mi><mo id="A1.SS3.SSS0.P0.SPx1.p3.5.m5.1.1.3.1" xref="A1.SS3.SSS0.P0.SPx1.p3.5.m5.1.1.3.1.cmml">−</mo><mn id="A1.SS3.SSS0.P0.SPx1.p3.5.m5.1.1.3.3" xref="A1.SS3.SSS0.P0.SPx1.p3.5.m5.1.1.3.3.cmml">1</mn></mrow></msub><annotation-xml encoding="MathML-Content" id="A1.SS3.SSS0.P0.SPx1.p3.5.m5.1b"><apply id="A1.SS3.SSS0.P0.SPx1.p3.5.m5.1.1.cmml" xref="A1.SS3.SSS0.P0.SPx1.p3.5.m5.1.1"><csymbol cd="ambiguous" id="A1.SS3.SSS0.P0.SPx1.p3.5.m5.1.1.1.cmml" xref="A1.SS3.SSS0.P0.SPx1.p3.5.m5.1.1">subscript</csymbol><ci id="A1.SS3.SSS0.P0.SPx1.p3.5.m5.1.1.2.cmml" xref="A1.SS3.SSS0.P0.SPx1.p3.5.m5.1.1.2">𝑉</ci><apply id="A1.SS3.SSS0.P0.SPx1.p3.5.m5.1.1.3.cmml" xref="A1.SS3.SSS0.P0.SPx1.p3.5.m5.1.1.3"><minus id="A1.SS3.SSS0.P0.SPx1.p3.5.m5.1.1.3.1.cmml" xref="A1.SS3.SSS0.P0.SPx1.p3.5.m5.1.1.3.1"></minus><ci id="A1.SS3.SSS0.P0.SPx1.p3.5.m5.1.1.3.2.cmml" xref="A1.SS3.SSS0.P0.SPx1.p3.5.m5.1.1.3.2">𝑗</ci><cn id="A1.SS3.SSS0.P0.SPx1.p3.5.m5.1.1.3.3.cmml" type="integer" xref="A1.SS3.SSS0.P0.SPx1.p3.5.m5.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS3.SSS0.P0.SPx1.p3.5.m5.1c">V_{j-1}</annotation><annotation encoding="application/x-llamapun" id="A1.SS3.SSS0.P0.SPx1.p3.5.m5.1d">italic_V start_POSTSUBSCRIPT italic_j - 1 end_POSTSUBSCRIPT</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="A1.SS3.SSS0.P0.SPx1.p3.6.6"> or </span><math alttext="V_{j}" class="ltx_Math" display="inline" id="A1.SS3.SSS0.P0.SPx1.p3.6.m6.1"><semantics id="A1.SS3.SSS0.P0.SPx1.p3.6.m6.1a"><msub id="A1.SS3.SSS0.P0.SPx1.p3.6.m6.1.1" xref="A1.SS3.SSS0.P0.SPx1.p3.6.m6.1.1.cmml"><mi id="A1.SS3.SSS0.P0.SPx1.p3.6.m6.1.1.2" xref="A1.SS3.SSS0.P0.SPx1.p3.6.m6.1.1.2.cmml">V</mi><mi id="A1.SS3.SSS0.P0.SPx1.p3.6.m6.1.1.3" xref="A1.SS3.SSS0.P0.SPx1.p3.6.m6.1.1.3.cmml">j</mi></msub><annotation-xml encoding="MathML-Content" id="A1.SS3.SSS0.P0.SPx1.p3.6.m6.1b"><apply id="A1.SS3.SSS0.P0.SPx1.p3.6.m6.1.1.cmml" xref="A1.SS3.SSS0.P0.SPx1.p3.6.m6.1.1"><csymbol cd="ambiguous" id="A1.SS3.SSS0.P0.SPx1.p3.6.m6.1.1.1.cmml" xref="A1.SS3.SSS0.P0.SPx1.p3.6.m6.1.1">subscript</csymbol><ci id="A1.SS3.SSS0.P0.SPx1.p3.6.m6.1.1.2.cmml" xref="A1.SS3.SSS0.P0.SPx1.p3.6.m6.1.1.2">𝑉</ci><ci id="A1.SS3.SSS0.P0.SPx1.p3.6.m6.1.1.3.cmml" xref="A1.SS3.SSS0.P0.SPx1.p3.6.m6.1.1.3">𝑗</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS3.SSS0.P0.SPx1.p3.6.m6.1c">V_{j}</annotation><annotation encoding="application/x-llamapun" id="A1.SS3.SSS0.P0.SPx1.p3.6.m6.1d">italic_V start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="A1.SS3.SSS0.P0.SPx1.p3.6.7">. Hence:</span></p> <table class="ltx_equation ltx_eqn_table" id="A1.Ex33"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_left_padleft"></td> <td class="ltx_eqn_cell ltx_align_left"><math alttext="\omega(E(H[V_{j-1}]))+\omega(E(V_{j-1},V_{j}-V_{j-1}))\geq\rho^{*}(v)|V_{j-1}|." class="ltx_Math" display="block" id="A1.Ex33.m1.2"><semantics id="A1.Ex33.m1.2a"><mrow id="A1.Ex33.m1.2.2.1" xref="A1.Ex33.m1.2.2.1.1.cmml"><mrow id="A1.Ex33.m1.2.2.1.1" xref="A1.Ex33.m1.2.2.1.1.cmml"><mrow id="A1.Ex33.m1.2.2.1.1.2" xref="A1.Ex33.m1.2.2.1.1.2.cmml"><mrow id="A1.Ex33.m1.2.2.1.1.1.1" xref="A1.Ex33.m1.2.2.1.1.1.1.cmml"><mi id="A1.Ex33.m1.2.2.1.1.1.1.3" xref="A1.Ex33.m1.2.2.1.1.1.1.3.cmml">ω</mi><mo id="A1.Ex33.m1.2.2.1.1.1.1.2" xref="A1.Ex33.m1.2.2.1.1.1.1.2.cmml">⁢</mo><mrow id="A1.Ex33.m1.2.2.1.1.1.1.1.1" xref="A1.Ex33.m1.2.2.1.1.1.1.1.1.1.cmml"><mo id="A1.Ex33.m1.2.2.1.1.1.1.1.1.2" stretchy="false" xref="A1.Ex33.m1.2.2.1.1.1.1.1.1.1.cmml">(</mo><mrow id="A1.Ex33.m1.2.2.1.1.1.1.1.1.1" xref="A1.Ex33.m1.2.2.1.1.1.1.1.1.1.cmml"><mi id="A1.Ex33.m1.2.2.1.1.1.1.1.1.1.3" xref="A1.Ex33.m1.2.2.1.1.1.1.1.1.1.3.cmml">E</mi><mo id="A1.Ex33.m1.2.2.1.1.1.1.1.1.1.2" xref="A1.Ex33.m1.2.2.1.1.1.1.1.1.1.2.cmml">⁢</mo><mrow id="A1.Ex33.m1.2.2.1.1.1.1.1.1.1.1.1" xref="A1.Ex33.m1.2.2.1.1.1.1.1.1.1.1.1.1.cmml"><mo id="A1.Ex33.m1.2.2.1.1.1.1.1.1.1.1.1.2" stretchy="false" xref="A1.Ex33.m1.2.2.1.1.1.1.1.1.1.1.1.1.cmml">(</mo><mrow id="A1.Ex33.m1.2.2.1.1.1.1.1.1.1.1.1.1" xref="A1.Ex33.m1.2.2.1.1.1.1.1.1.1.1.1.1.cmml"><mi id="A1.Ex33.m1.2.2.1.1.1.1.1.1.1.1.1.1.3" xref="A1.Ex33.m1.2.2.1.1.1.1.1.1.1.1.1.1.3.cmml">H</mi><mo id="A1.Ex33.m1.2.2.1.1.1.1.1.1.1.1.1.1.2" xref="A1.Ex33.m1.2.2.1.1.1.1.1.1.1.1.1.1.2.cmml">⁢</mo><mrow id="A1.Ex33.m1.2.2.1.1.1.1.1.1.1.1.1.1.1.1" xref="A1.Ex33.m1.2.2.1.1.1.1.1.1.1.1.1.1.1.2.cmml"><mo id="A1.Ex33.m1.2.2.1.1.1.1.1.1.1.1.1.1.1.1.2" stretchy="false" 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xref="A1.Ex33.m1.2.2.1.1.3.1.1"><abs id="A1.Ex33.m1.2.2.1.1.3.1.2.1.cmml" xref="A1.Ex33.m1.2.2.1.1.3.1.1.2"></abs><apply id="A1.Ex33.m1.2.2.1.1.3.1.1.1.cmml" xref="A1.Ex33.m1.2.2.1.1.3.1.1.1"><csymbol cd="ambiguous" id="A1.Ex33.m1.2.2.1.1.3.1.1.1.1.cmml" xref="A1.Ex33.m1.2.2.1.1.3.1.1.1">subscript</csymbol><ci id="A1.Ex33.m1.2.2.1.1.3.1.1.1.2.cmml" xref="A1.Ex33.m1.2.2.1.1.3.1.1.1.2">𝑉</ci><apply id="A1.Ex33.m1.2.2.1.1.3.1.1.1.3.cmml" xref="A1.Ex33.m1.2.2.1.1.3.1.1.1.3"><minus id="A1.Ex33.m1.2.2.1.1.3.1.1.1.3.1.cmml" xref="A1.Ex33.m1.2.2.1.1.3.1.1.1.3.1"></minus><ci id="A1.Ex33.m1.2.2.1.1.3.1.1.1.3.2.cmml" xref="A1.Ex33.m1.2.2.1.1.3.1.1.1.3.2">𝑗</ci><cn id="A1.Ex33.m1.2.2.1.1.3.1.1.1.3.3.cmml" type="integer" xref="A1.Ex33.m1.2.2.1.1.3.1.1.1.3.3">1</cn></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.Ex33.m1.2c">\omega(E(H[V_{j-1}]))+\omega(E(V_{j-1},V_{j}-V_{j-1}))\geq\rho^{*}(v)|V_{j-1}|.</annotation><annotation encoding="application/x-llamapun" id="A1.Ex33.m1.2d">italic_ω ( italic_E ( italic_H [ italic_V start_POSTSUBSCRIPT italic_j - 1 end_POSTSUBSCRIPT ] ) ) + italic_ω ( italic_E ( italic_V start_POSTSUBSCRIPT italic_j - 1 end_POSTSUBSCRIPT , italic_V start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT - italic_V start_POSTSUBSCRIPT italic_j - 1 end_POSTSUBSCRIPT ) ) ≥ italic_ρ start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ( italic_v ) | italic_V start_POSTSUBSCRIPT italic_j - 1 end_POSTSUBSCRIPT | .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_left_padright"></td> </tr></tbody> </table> </div> <div class="ltx_para" id="A1.SS3.SSS0.P0.SPx1.p4"> <p class="ltx_p" id="A1.SS3.SSS0.P0.SPx1.p4.5"><span class="ltx_text ltx_font_italic" id="A1.SS3.SSS0.P0.SPx1.p4.5.1">This means that we have</span></p> <table class="ltx_equation ltx_eqn_table" id="A1.Ex34"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_left_padleft"></td> <td class="ltx_eqn_cell ltx_align_left"><math alttext="(1-\varepsilon)\rho^{*}(v)\geq\rho(H(V_{j}))\geq\frac{\omega(E(H[V_{j-1}]))+% \omega(E(V_{j-1},V_{j}-V_{j-1}))}{|V_{j}|}\geq\frac{\rho^{*}(v)|V_{j-1}|}{|V_{% j}|}" class="ltx_Math" display="block" id="A1.Ex34.m1.9"><semantics id="A1.Ex34.m1.9a"><mrow id="A1.Ex34.m1.9.9" xref="A1.Ex34.m1.9.9.cmml"><mrow id="A1.Ex34.m1.8.8.1" xref="A1.Ex34.m1.8.8.1.cmml"><mrow id="A1.Ex34.m1.8.8.1.1.1" xref="A1.Ex34.m1.8.8.1.1.1.1.cmml"><mo id="A1.Ex34.m1.8.8.1.1.1.2" stretchy="false" xref="A1.Ex34.m1.8.8.1.1.1.1.cmml">(</mo><mrow id="A1.Ex34.m1.8.8.1.1.1.1" xref="A1.Ex34.m1.8.8.1.1.1.1.cmml"><mn id="A1.Ex34.m1.8.8.1.1.1.1.2" xref="A1.Ex34.m1.8.8.1.1.1.1.2.cmml">1</mn><mo id="A1.Ex34.m1.8.8.1.1.1.1.1" xref="A1.Ex34.m1.8.8.1.1.1.1.1.cmml">−</mo><mi id="A1.Ex34.m1.8.8.1.1.1.1.3" xref="A1.Ex34.m1.8.8.1.1.1.1.3.cmml">ε</mi></mrow><mo id="A1.Ex34.m1.8.8.1.1.1.3" stretchy="false" xref="A1.Ex34.m1.8.8.1.1.1.1.cmml">)</mo></mrow><mo id="A1.Ex34.m1.8.8.1.2" xref="A1.Ex34.m1.8.8.1.2.cmml">⁢</mo><msup id="A1.Ex34.m1.8.8.1.3" xref="A1.Ex34.m1.8.8.1.3.cmml"><mi id="A1.Ex34.m1.8.8.1.3.2" xref="A1.Ex34.m1.8.8.1.3.2.cmml">ρ</mi><mo id="A1.Ex34.m1.8.8.1.3.3" xref="A1.Ex34.m1.8.8.1.3.3.cmml">∗</mo></msup><mo id="A1.Ex34.m1.8.8.1.2a" xref="A1.Ex34.m1.8.8.1.2.cmml">⁢</mo><mrow id="A1.Ex34.m1.8.8.1.4.2" xref="A1.Ex34.m1.8.8.1.cmml"><mo id="A1.Ex34.m1.8.8.1.4.2.1" stretchy="false" xref="A1.Ex34.m1.8.8.1.cmml">(</mo><mi id="A1.Ex34.m1.7.7" xref="A1.Ex34.m1.7.7.cmml">v</mi><mo id="A1.Ex34.m1.8.8.1.4.2.2" stretchy="false" xref="A1.Ex34.m1.8.8.1.cmml">)</mo></mrow></mrow><mo id="A1.Ex34.m1.9.9.4" xref="A1.Ex34.m1.9.9.4.cmml">≥</mo><mrow id="A1.Ex34.m1.9.9.2" xref="A1.Ex34.m1.9.9.2.cmml"><mi id="A1.Ex34.m1.9.9.2.3" xref="A1.Ex34.m1.9.9.2.3.cmml">ρ</mi><mo id="A1.Ex34.m1.9.9.2.2" xref="A1.Ex34.m1.9.9.2.2.cmml">⁢</mo><mrow id="A1.Ex34.m1.9.9.2.1.1" xref="A1.Ex34.m1.9.9.2.1.1.1.cmml"><mo id="A1.Ex34.m1.9.9.2.1.1.2" stretchy="false" xref="A1.Ex34.m1.9.9.2.1.1.1.cmml">(</mo><mrow id="A1.Ex34.m1.9.9.2.1.1.1" xref="A1.Ex34.m1.9.9.2.1.1.1.cmml"><mi id="A1.Ex34.m1.9.9.2.1.1.1.3" xref="A1.Ex34.m1.9.9.2.1.1.1.3.cmml">H</mi><mo id="A1.Ex34.m1.9.9.2.1.1.1.2" xref="A1.Ex34.m1.9.9.2.1.1.1.2.cmml">⁢</mo><mrow id="A1.Ex34.m1.9.9.2.1.1.1.1.1" xref="A1.Ex34.m1.9.9.2.1.1.1.1.1.1.cmml"><mo id="A1.Ex34.m1.9.9.2.1.1.1.1.1.2" stretchy="false" xref="A1.Ex34.m1.9.9.2.1.1.1.1.1.1.cmml">(</mo><msub id="A1.Ex34.m1.9.9.2.1.1.1.1.1.1" xref="A1.Ex34.m1.9.9.2.1.1.1.1.1.1.cmml"><mi id="A1.Ex34.m1.9.9.2.1.1.1.1.1.1.2" xref="A1.Ex34.m1.9.9.2.1.1.1.1.1.1.2.cmml">V</mi><mi id="A1.Ex34.m1.9.9.2.1.1.1.1.1.1.3" xref="A1.Ex34.m1.9.9.2.1.1.1.1.1.1.3.cmml">j</mi></msub><mo id="A1.Ex34.m1.9.9.2.1.1.1.1.1.3" stretchy="false" xref="A1.Ex34.m1.9.9.2.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="A1.Ex34.m1.9.9.2.1.1.3" stretchy="false" xref="A1.Ex34.m1.9.9.2.1.1.1.cmml">)</mo></mrow></mrow><mo id="A1.Ex34.m1.9.9.5" xref="A1.Ex34.m1.9.9.5.cmml">≥</mo><mfrac 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start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ) ) ≥ divide start_ARG italic_ω ( italic_E ( italic_H [ italic_V start_POSTSUBSCRIPT italic_j - 1 end_POSTSUBSCRIPT ] ) ) + italic_ω ( italic_E ( italic_V start_POSTSUBSCRIPT italic_j - 1 end_POSTSUBSCRIPT , italic_V start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT - italic_V start_POSTSUBSCRIPT italic_j - 1 end_POSTSUBSCRIPT ) ) end_ARG start_ARG | italic_V start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT | end_ARG ≥ divide start_ARG italic_ρ start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ( italic_v ) | italic_V start_POSTSUBSCRIPT italic_j - 1 end_POSTSUBSCRIPT | end_ARG start_ARG | italic_V start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT | end_ARG</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_left_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="A1.SS3.SSS0.P0.SPx1.p4.2"><span class="ltx_text ltx_font_italic" id="A1.SS3.SSS0.P0.SPx1.p4.2.1">By applying the induction hypothesis, we find that indeed </span><math 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xref="A1.SS3.SSS0.P0.SPx1.p4.1.m1.2.2.3.cmml">≥</mo><msup id="A1.SS3.SSS0.P0.SPx1.p4.1.m1.2.2.2" xref="A1.SS3.SSS0.P0.SPx1.p4.1.m1.2.2.2.cmml"><mrow id="A1.SS3.SSS0.P0.SPx1.p4.1.m1.2.2.2.1.1" xref="A1.SS3.SSS0.P0.SPx1.p4.1.m1.2.2.2.1.1.1.cmml"><mo id="A1.SS3.SSS0.P0.SPx1.p4.1.m1.2.2.2.1.1.2" stretchy="false" xref="A1.SS3.SSS0.P0.SPx1.p4.1.m1.2.2.2.1.1.1.cmml">(</mo><mrow id="A1.SS3.SSS0.P0.SPx1.p4.1.m1.2.2.2.1.1.1" xref="A1.SS3.SSS0.P0.SPx1.p4.1.m1.2.2.2.1.1.1.cmml"><mn id="A1.SS3.SSS0.P0.SPx1.p4.1.m1.2.2.2.1.1.1.2" xref="A1.SS3.SSS0.P0.SPx1.p4.1.m1.2.2.2.1.1.1.2.cmml">1</mn><mo id="A1.SS3.SSS0.P0.SPx1.p4.1.m1.2.2.2.1.1.1.1" xref="A1.SS3.SSS0.P0.SPx1.p4.1.m1.2.2.2.1.1.1.1.cmml">−</mo><mi id="A1.SS3.SSS0.P0.SPx1.p4.1.m1.2.2.2.1.1.1.3" xref="A1.SS3.SSS0.P0.SPx1.p4.1.m1.2.2.2.1.1.1.3.cmml">ε</mi></mrow><mo id="A1.SS3.SSS0.P0.SPx1.p4.1.m1.2.2.2.1.1.3" stretchy="false" xref="A1.SS3.SSS0.P0.SPx1.p4.1.m1.2.2.2.1.1.1.cmml">)</mo></mrow><mrow id="A1.SS3.SSS0.P0.SPx1.p4.1.m1.2.2.2.3" xref="A1.SS3.SSS0.P0.SPx1.p4.1.m1.2.2.2.3.cmml"><mo id="A1.SS3.SSS0.P0.SPx1.p4.1.m1.2.2.2.3a" xref="A1.SS3.SSS0.P0.SPx1.p4.1.m1.2.2.2.3.cmml">−</mo><mi id="A1.SS3.SSS0.P0.SPx1.p4.1.m1.2.2.2.3.2" xref="A1.SS3.SSS0.P0.SPx1.p4.1.m1.2.2.2.3.2.cmml">j</mi></mrow></msup></mrow><annotation-xml encoding="MathML-Content" id="A1.SS3.SSS0.P0.SPx1.p4.1.m1.2b"><apply id="A1.SS3.SSS0.P0.SPx1.p4.1.m1.2.2.cmml" xref="A1.SS3.SSS0.P0.SPx1.p4.1.m1.2.2"><geq id="A1.SS3.SSS0.P0.SPx1.p4.1.m1.2.2.3.cmml" xref="A1.SS3.SSS0.P0.SPx1.p4.1.m1.2.2.3"></geq><apply id="A1.SS3.SSS0.P0.SPx1.p4.1.m1.1.1.1.2.cmml" xref="A1.SS3.SSS0.P0.SPx1.p4.1.m1.1.1.1.1"><abs id="A1.SS3.SSS0.P0.SPx1.p4.1.m1.1.1.1.2.1.cmml" xref="A1.SS3.SSS0.P0.SPx1.p4.1.m1.1.1.1.1.2"></abs><apply id="A1.SS3.SSS0.P0.SPx1.p4.1.m1.1.1.1.1.1.cmml" xref="A1.SS3.SSS0.P0.SPx1.p4.1.m1.1.1.1.1.1"><csymbol cd="ambiguous" id="A1.SS3.SSS0.P0.SPx1.p4.1.m1.1.1.1.1.1.1.cmml" xref="A1.SS3.SSS0.P0.SPx1.p4.1.m1.1.1.1.1.1">subscript</csymbol><ci id="A1.SS3.SSS0.P0.SPx1.p4.1.m1.1.1.1.1.1.2.cmml" xref="A1.SS3.SSS0.P0.SPx1.p4.1.m1.1.1.1.1.1.2">𝑉</ci><ci id="A1.SS3.SSS0.P0.SPx1.p4.1.m1.1.1.1.1.1.3.cmml" xref="A1.SS3.SSS0.P0.SPx1.p4.1.m1.1.1.1.1.1.3">𝑗</ci></apply></apply><apply id="A1.SS3.SSS0.P0.SPx1.p4.1.m1.2.2.2.cmml" xref="A1.SS3.SSS0.P0.SPx1.p4.1.m1.2.2.2"><csymbol cd="ambiguous" id="A1.SS3.SSS0.P0.SPx1.p4.1.m1.2.2.2.2.cmml" xref="A1.SS3.SSS0.P0.SPx1.p4.1.m1.2.2.2">superscript</csymbol><apply id="A1.SS3.SSS0.P0.SPx1.p4.1.m1.2.2.2.1.1.1.cmml" xref="A1.SS3.SSS0.P0.SPx1.p4.1.m1.2.2.2.1.1"><minus id="A1.SS3.SSS0.P0.SPx1.p4.1.m1.2.2.2.1.1.1.1.cmml" xref="A1.SS3.SSS0.P0.SPx1.p4.1.m1.2.2.2.1.1.1.1"></minus><cn id="A1.SS3.SSS0.P0.SPx1.p4.1.m1.2.2.2.1.1.1.2.cmml" type="integer" xref="A1.SS3.SSS0.P0.SPx1.p4.1.m1.2.2.2.1.1.1.2">1</cn><ci id="A1.SS3.SSS0.P0.SPx1.p4.1.m1.2.2.2.1.1.1.3.cmml" xref="A1.SS3.SSS0.P0.SPx1.p4.1.m1.2.2.2.1.1.1.3">𝜀</ci></apply><apply id="A1.SS3.SSS0.P0.SPx1.p4.1.m1.2.2.2.3.cmml" xref="A1.SS3.SSS0.P0.SPx1.p4.1.m1.2.2.2.3"><minus id="A1.SS3.SSS0.P0.SPx1.p4.1.m1.2.2.2.3.1.cmml" xref="A1.SS3.SSS0.P0.SPx1.p4.1.m1.2.2.2.3"></minus><ci id="A1.SS3.SSS0.P0.SPx1.p4.1.m1.2.2.2.3.2.cmml" xref="A1.SS3.SSS0.P0.SPx1.p4.1.m1.2.2.2.3.2">𝑗</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS3.SSS0.P0.SPx1.p4.1.m1.2c">|V_{j}|\geq(1-\varepsilon)^{-j}</annotation><annotation encoding="application/x-llamapun" id="A1.SS3.SSS0.P0.SPx1.p4.1.m1.2d">| italic_V start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT | ≥ ( 1 - italic_ε ) start_POSTSUPERSCRIPT - italic_j end_POSTSUPERSCRIPT</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="A1.SS3.SSS0.P0.SPx1.p4.2.2">. We conclude that if </span><math alttext="\varepsilon\leq 1" class="ltx_Math" display="inline" id="A1.SS3.SSS0.P0.SPx1.p4.2.m2.1"><semantics id="A1.SS3.SSS0.P0.SPx1.p4.2.m2.1a"><mrow id="A1.SS3.SSS0.P0.SPx1.p4.2.m2.1.1" xref="A1.SS3.SSS0.P0.SPx1.p4.2.m2.1.1.cmml"><mi id="A1.SS3.SSS0.P0.SPx1.p4.2.m2.1.1.2" xref="A1.SS3.SSS0.P0.SPx1.p4.2.m2.1.1.2.cmml">ε</mi><mo id="A1.SS3.SSS0.P0.SPx1.p4.2.m2.1.1.1" xref="A1.SS3.SSS0.P0.SPx1.p4.2.m2.1.1.1.cmml">≤</mo><mn id="A1.SS3.SSS0.P0.SPx1.p4.2.m2.1.1.3" xref="A1.SS3.SSS0.P0.SPx1.p4.2.m2.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="A1.SS3.SSS0.P0.SPx1.p4.2.m2.1b"><apply id="A1.SS3.SSS0.P0.SPx1.p4.2.m2.1.1.cmml" xref="A1.SS3.SSS0.P0.SPx1.p4.2.m2.1.1"><leq id="A1.SS3.SSS0.P0.SPx1.p4.2.m2.1.1.1.cmml" xref="A1.SS3.SSS0.P0.SPx1.p4.2.m2.1.1.1"></leq><ci id="A1.SS3.SSS0.P0.SPx1.p4.2.m2.1.1.2.cmml" xref="A1.SS3.SSS0.P0.SPx1.p4.2.m2.1.1.2">𝜀</ci><cn id="A1.SS3.SSS0.P0.SPx1.p4.2.m2.1.1.3.cmml" type="integer" xref="A1.SS3.SSS0.P0.SPx1.p4.2.m2.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS3.SSS0.P0.SPx1.p4.2.m2.1c">\varepsilon\leq 1</annotation><annotation encoding="application/x-llamapun" id="A1.SS3.SSS0.P0.SPx1.p4.2.m2.1d">italic_ε ≤ 1</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="A1.SS3.SSS0.P0.SPx1.p4.2.3"> then:</span></p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="A1.EGx8"> <tbody id="A1.Ex35"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_left_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle j\leq\frac{\log n}{\log(\frac{1}{1-\varepsilon})}\leq\frac{\log n% }{\log(1+\frac{\varepsilon}{1-\varepsilon})}\leq\frac{\log n}{\log(1+% \varepsilon)}\leq\frac{\log n}{\log(1+\varepsilon)}\leq\frac{2\log n}{\varepsilon}" class="ltx_Math" display="inline" id="A1.Ex35.m1.8"><semantics id="A1.Ex35.m1.8a"><mrow id="A1.Ex35.m1.8.9" xref="A1.Ex35.m1.8.9.cmml"><mi id="A1.Ex35.m1.8.9.2" xref="A1.Ex35.m1.8.9.2.cmml">j</mi><mo id="A1.Ex35.m1.8.9.3" xref="A1.Ex35.m1.8.9.3.cmml">≤</mo><mstyle displaystyle="true" id="A1.Ex35.m1.2.2" xref="A1.Ex35.m1.2.2.cmml"><mfrac id="A1.Ex35.m1.2.2a" xref="A1.Ex35.m1.2.2.cmml"><mrow id="A1.Ex35.m1.2.2.4" xref="A1.Ex35.m1.2.2.4.cmml"><mi id="A1.Ex35.m1.2.2.4.1" xref="A1.Ex35.m1.2.2.4.1.cmml">log</mi><mo id="A1.Ex35.m1.2.2.4a" lspace="0.167em" xref="A1.Ex35.m1.2.2.4.cmml">⁡</mo><mi id="A1.Ex35.m1.2.2.4.2" xref="A1.Ex35.m1.2.2.4.2.cmml">n</mi></mrow><mrow id="A1.Ex35.m1.2.2.2.4" xref="A1.Ex35.m1.2.2.2.3.cmml"><mi id="A1.Ex35.m1.1.1.1.1" xref="A1.Ex35.m1.1.1.1.1.cmml">log</mi><mo id="A1.Ex35.m1.2.2.2.4a" xref="A1.Ex35.m1.2.2.2.3.cmml">⁡</mo><mrow id="A1.Ex35.m1.2.2.2.4.1" xref="A1.Ex35.m1.2.2.2.3.cmml"><mo id="A1.Ex35.m1.2.2.2.4.1.1" stretchy="false" xref="A1.Ex35.m1.2.2.2.3.cmml">(</mo><mfrac id="A1.Ex35.m1.2.2.2.2" xref="A1.Ex35.m1.2.2.2.2.cmml"><mn id="A1.Ex35.m1.2.2.2.2.2" xref="A1.Ex35.m1.2.2.2.2.2.cmml">1</mn><mrow id="A1.Ex35.m1.2.2.2.2.3" xref="A1.Ex35.m1.2.2.2.2.3.cmml"><mn id="A1.Ex35.m1.2.2.2.2.3.2" xref="A1.Ex35.m1.2.2.2.2.3.2.cmml">1</mn><mo id="A1.Ex35.m1.2.2.2.2.3.1" xref="A1.Ex35.m1.2.2.2.2.3.1.cmml">−</mo><mi id="A1.Ex35.m1.2.2.2.2.3.3" xref="A1.Ex35.m1.2.2.2.2.3.3.cmml">ε</mi></mrow></mfrac><mo id="A1.Ex35.m1.2.2.2.4.1.2" stretchy="false" xref="A1.Ex35.m1.2.2.2.3.cmml">)</mo></mrow></mrow></mfrac></mstyle><mo id="A1.Ex35.m1.8.9.4" xref="A1.Ex35.m1.8.9.4.cmml">≤</mo><mstyle displaystyle="true" id="A1.Ex35.m1.4.4" xref="A1.Ex35.m1.4.4.cmml"><mfrac id="A1.Ex35.m1.4.4a" xref="A1.Ex35.m1.4.4.cmml"><mrow id="A1.Ex35.m1.4.4.4" xref="A1.Ex35.m1.4.4.4.cmml"><mi id="A1.Ex35.m1.4.4.4.1" xref="A1.Ex35.m1.4.4.4.1.cmml">log</mi><mo id="A1.Ex35.m1.4.4.4a" lspace="0.167em" xref="A1.Ex35.m1.4.4.4.cmml">⁡</mo><mi id="A1.Ex35.m1.4.4.4.2" xref="A1.Ex35.m1.4.4.4.2.cmml">n</mi></mrow><mrow id="A1.Ex35.m1.4.4.2.2" xref="A1.Ex35.m1.4.4.2.3.cmml"><mi id="A1.Ex35.m1.3.3.1.1" xref="A1.Ex35.m1.3.3.1.1.cmml">log</mi><mo id="A1.Ex35.m1.4.4.2.2a" xref="A1.Ex35.m1.4.4.2.3.cmml">⁡</mo><mrow id="A1.Ex35.m1.4.4.2.2.1" xref="A1.Ex35.m1.4.4.2.3.cmml"><mo id="A1.Ex35.m1.4.4.2.2.1.2" stretchy="false" xref="A1.Ex35.m1.4.4.2.3.cmml">(</mo><mrow id="A1.Ex35.m1.4.4.2.2.1.1" xref="A1.Ex35.m1.4.4.2.2.1.1.cmml"><mn id="A1.Ex35.m1.4.4.2.2.1.1.2" xref="A1.Ex35.m1.4.4.2.2.1.1.2.cmml">1</mn><mo id="A1.Ex35.m1.4.4.2.2.1.1.1" xref="A1.Ex35.m1.4.4.2.2.1.1.1.cmml">+</mo><mfrac id="A1.Ex35.m1.4.4.2.2.1.1.3" xref="A1.Ex35.m1.4.4.2.2.1.1.3.cmml"><mi id="A1.Ex35.m1.4.4.2.2.1.1.3.2" xref="A1.Ex35.m1.4.4.2.2.1.1.3.2.cmml">ε</mi><mrow id="A1.Ex35.m1.4.4.2.2.1.1.3.3" xref="A1.Ex35.m1.4.4.2.2.1.1.3.3.cmml"><mn id="A1.Ex35.m1.4.4.2.2.1.1.3.3.2" xref="A1.Ex35.m1.4.4.2.2.1.1.3.3.2.cmml">1</mn><mo id="A1.Ex35.m1.4.4.2.2.1.1.3.3.1" xref="A1.Ex35.m1.4.4.2.2.1.1.3.3.1.cmml">−</mo><mi id="A1.Ex35.m1.4.4.2.2.1.1.3.3.3" xref="A1.Ex35.m1.4.4.2.2.1.1.3.3.3.cmml">ε</mi></mrow></mfrac></mrow><mo id="A1.Ex35.m1.4.4.2.2.1.3" stretchy="false" xref="A1.Ex35.m1.4.4.2.3.cmml">)</mo></mrow></mrow></mfrac></mstyle><mo id="A1.Ex35.m1.8.9.5" xref="A1.Ex35.m1.8.9.5.cmml">≤</mo><mstyle displaystyle="true" id="A1.Ex35.m1.6.6" xref="A1.Ex35.m1.6.6.cmml"><mfrac id="A1.Ex35.m1.6.6a" xref="A1.Ex35.m1.6.6.cmml"><mrow id="A1.Ex35.m1.6.6.4" xref="A1.Ex35.m1.6.6.4.cmml"><mi id="A1.Ex35.m1.6.6.4.1" xref="A1.Ex35.m1.6.6.4.1.cmml">log</mi><mo id="A1.Ex35.m1.6.6.4a" lspace="0.167em" xref="A1.Ex35.m1.6.6.4.cmml">⁡</mo><mi id="A1.Ex35.m1.6.6.4.2" xref="A1.Ex35.m1.6.6.4.2.cmml">n</mi></mrow><mrow id="A1.Ex35.m1.6.6.2.2" xref="A1.Ex35.m1.6.6.2.3.cmml"><mi id="A1.Ex35.m1.5.5.1.1" xref="A1.Ex35.m1.5.5.1.1.cmml">log</mi><mo id="A1.Ex35.m1.6.6.2.2a" xref="A1.Ex35.m1.6.6.2.3.cmml">⁡</mo><mrow id="A1.Ex35.m1.6.6.2.2.1" xref="A1.Ex35.m1.6.6.2.3.cmml"><mo id="A1.Ex35.m1.6.6.2.2.1.2" stretchy="false" xref="A1.Ex35.m1.6.6.2.3.cmml">(</mo><mrow id="A1.Ex35.m1.6.6.2.2.1.1" xref="A1.Ex35.m1.6.6.2.2.1.1.cmml"><mn id="A1.Ex35.m1.6.6.2.2.1.1.2" xref="A1.Ex35.m1.6.6.2.2.1.1.2.cmml">1</mn><mo id="A1.Ex35.m1.6.6.2.2.1.1.1" xref="A1.Ex35.m1.6.6.2.2.1.1.1.cmml">+</mo><mi id="A1.Ex35.m1.6.6.2.2.1.1.3" xref="A1.Ex35.m1.6.6.2.2.1.1.3.cmml">ε</mi></mrow><mo id="A1.Ex35.m1.6.6.2.2.1.3" stretchy="false" xref="A1.Ex35.m1.6.6.2.3.cmml">)</mo></mrow></mrow></mfrac></mstyle><mo id="A1.Ex35.m1.8.9.6" xref="A1.Ex35.m1.8.9.6.cmml">≤</mo><mstyle displaystyle="true" id="A1.Ex35.m1.8.8" xref="A1.Ex35.m1.8.8.cmml"><mfrac id="A1.Ex35.m1.8.8a" xref="A1.Ex35.m1.8.8.cmml"><mrow id="A1.Ex35.m1.8.8.4" xref="A1.Ex35.m1.8.8.4.cmml"><mi id="A1.Ex35.m1.8.8.4.1" xref="A1.Ex35.m1.8.8.4.1.cmml">log</mi><mo id="A1.Ex35.m1.8.8.4a" lspace="0.167em" xref="A1.Ex35.m1.8.8.4.cmml">⁡</mo><mi id="A1.Ex35.m1.8.8.4.2" xref="A1.Ex35.m1.8.8.4.2.cmml">n</mi></mrow><mrow id="A1.Ex35.m1.8.8.2.2" xref="A1.Ex35.m1.8.8.2.3.cmml"><mi id="A1.Ex35.m1.7.7.1.1" 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italic_n end_ARG start_ARG roman_log ( divide start_ARG 1 end_ARG start_ARG 1 - italic_ε end_ARG ) end_ARG ≤ divide start_ARG roman_log italic_n end_ARG start_ARG roman_log ( 1 + divide start_ARG italic_ε end_ARG start_ARG 1 - italic_ε end_ARG ) end_ARG ≤ divide start_ARG roman_log italic_n end_ARG start_ARG roman_log ( 1 + italic_ε ) end_ARG ≤ divide start_ARG roman_log italic_n end_ARG start_ARG roman_log ( 1 + italic_ε ) end_ARG ≤ divide start_ARG 2 roman_log italic_n end_ARG start_ARG italic_ε end_ARG</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_left_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="A1.SS3.SSS0.P0.SPx1.p4.4"><span class="ltx_text ltx_font_italic" id="A1.SS3.SSS0.P0.SPx1.p4.4.1">where the last inequality comes from the fact that </span><math alttext="\log(1+x)\geq x-\frac{x^{2}}{2}" class="ltx_Math" display="inline" id="A1.SS3.SSS0.P0.SPx1.p4.3.m1.2"><semantics id="A1.SS3.SSS0.P0.SPx1.p4.3.m1.2a"><mrow 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xref="A1.SS3.SSS0.P0.SPx1.p4.3.m1.2.2.1.1.1.1.3.cmml">x</mi></mrow><mo id="A1.SS3.SSS0.P0.SPx1.p4.3.m1.2.2.1.1.1.3" stretchy="false" xref="A1.SS3.SSS0.P0.SPx1.p4.3.m1.2.2.1.2.cmml">)</mo></mrow></mrow><mo id="A1.SS3.SSS0.P0.SPx1.p4.3.m1.2.2.2" xref="A1.SS3.SSS0.P0.SPx1.p4.3.m1.2.2.2.cmml">≥</mo><mrow id="A1.SS3.SSS0.P0.SPx1.p4.3.m1.2.2.3" xref="A1.SS3.SSS0.P0.SPx1.p4.3.m1.2.2.3.cmml"><mi id="A1.SS3.SSS0.P0.SPx1.p4.3.m1.2.2.3.2" xref="A1.SS3.SSS0.P0.SPx1.p4.3.m1.2.2.3.2.cmml">x</mi><mo id="A1.SS3.SSS0.P0.SPx1.p4.3.m1.2.2.3.1" xref="A1.SS3.SSS0.P0.SPx1.p4.3.m1.2.2.3.1.cmml">−</mo><mfrac id="A1.SS3.SSS0.P0.SPx1.p4.3.m1.2.2.3.3" xref="A1.SS3.SSS0.P0.SPx1.p4.3.m1.2.2.3.3.cmml"><msup id="A1.SS3.SSS0.P0.SPx1.p4.3.m1.2.2.3.3.2" xref="A1.SS3.SSS0.P0.SPx1.p4.3.m1.2.2.3.3.2.cmml"><mi id="A1.SS3.SSS0.P0.SPx1.p4.3.m1.2.2.3.3.2.2" xref="A1.SS3.SSS0.P0.SPx1.p4.3.m1.2.2.3.3.2.2.cmml">x</mi><mn id="A1.SS3.SSS0.P0.SPx1.p4.3.m1.2.2.3.3.2.3" xref="A1.SS3.SSS0.P0.SPx1.p4.3.m1.2.2.3.3.2.3.cmml">2</mn></msup><mn id="A1.SS3.SSS0.P0.SPx1.p4.3.m1.2.2.3.3.3" xref="A1.SS3.SSS0.P0.SPx1.p4.3.m1.2.2.3.3.3.cmml">2</mn></mfrac></mrow></mrow><annotation-xml encoding="MathML-Content" id="A1.SS3.SSS0.P0.SPx1.p4.3.m1.2b"><apply id="A1.SS3.SSS0.P0.SPx1.p4.3.m1.2.2.cmml" xref="A1.SS3.SSS0.P0.SPx1.p4.3.m1.2.2"><geq id="A1.SS3.SSS0.P0.SPx1.p4.3.m1.2.2.2.cmml" xref="A1.SS3.SSS0.P0.SPx1.p4.3.m1.2.2.2"></geq><apply id="A1.SS3.SSS0.P0.SPx1.p4.3.m1.2.2.1.2.cmml" xref="A1.SS3.SSS0.P0.SPx1.p4.3.m1.2.2.1.1"><log id="A1.SS3.SSS0.P0.SPx1.p4.3.m1.1.1.cmml" xref="A1.SS3.SSS0.P0.SPx1.p4.3.m1.1.1"></log><apply id="A1.SS3.SSS0.P0.SPx1.p4.3.m1.2.2.1.1.1.1.cmml" xref="A1.SS3.SSS0.P0.SPx1.p4.3.m1.2.2.1.1.1.1"><plus id="A1.SS3.SSS0.P0.SPx1.p4.3.m1.2.2.1.1.1.1.1.cmml" xref="A1.SS3.SSS0.P0.SPx1.p4.3.m1.2.2.1.1.1.1.1"></plus><cn id="A1.SS3.SSS0.P0.SPx1.p4.3.m1.2.2.1.1.1.1.2.cmml" type="integer" xref="A1.SS3.SSS0.P0.SPx1.p4.3.m1.2.2.1.1.1.1.2">1</cn><ci id="A1.SS3.SSS0.P0.SPx1.p4.3.m1.2.2.1.1.1.1.3.cmml" xref="A1.SS3.SSS0.P0.SPx1.p4.3.m1.2.2.1.1.1.1.3">𝑥</ci></apply></apply><apply id="A1.SS3.SSS0.P0.SPx1.p4.3.m1.2.2.3.cmml" xref="A1.SS3.SSS0.P0.SPx1.p4.3.m1.2.2.3"><minus id="A1.SS3.SSS0.P0.SPx1.p4.3.m1.2.2.3.1.cmml" xref="A1.SS3.SSS0.P0.SPx1.p4.3.m1.2.2.3.1"></minus><ci id="A1.SS3.SSS0.P0.SPx1.p4.3.m1.2.2.3.2.cmml" xref="A1.SS3.SSS0.P0.SPx1.p4.3.m1.2.2.3.2">𝑥</ci><apply id="A1.SS3.SSS0.P0.SPx1.p4.3.m1.2.2.3.3.cmml" xref="A1.SS3.SSS0.P0.SPx1.p4.3.m1.2.2.3.3"><divide id="A1.SS3.SSS0.P0.SPx1.p4.3.m1.2.2.3.3.1.cmml" xref="A1.SS3.SSS0.P0.SPx1.p4.3.m1.2.2.3.3"></divide><apply id="A1.SS3.SSS0.P0.SPx1.p4.3.m1.2.2.3.3.2.cmml" xref="A1.SS3.SSS0.P0.SPx1.p4.3.m1.2.2.3.3.2"><csymbol cd="ambiguous" id="A1.SS3.SSS0.P0.SPx1.p4.3.m1.2.2.3.3.2.1.cmml" xref="A1.SS3.SSS0.P0.SPx1.p4.3.m1.2.2.3.3.2">superscript</csymbol><ci id="A1.SS3.SSS0.P0.SPx1.p4.3.m1.2.2.3.3.2.2.cmml" xref="A1.SS3.SSS0.P0.SPx1.p4.3.m1.2.2.3.3.2.2">𝑥</ci><cn id="A1.SS3.SSS0.P0.SPx1.p4.3.m1.2.2.3.3.2.3.cmml" type="integer" xref="A1.SS3.SSS0.P0.SPx1.p4.3.m1.2.2.3.3.2.3">2</cn></apply><cn id="A1.SS3.SSS0.P0.SPx1.p4.3.m1.2.2.3.3.3.cmml" type="integer" xref="A1.SS3.SSS0.P0.SPx1.p4.3.m1.2.2.3.3.3">2</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS3.SSS0.P0.SPx1.p4.3.m1.2c">\log(1+x)\geq x-\frac{x^{2}}{2}</annotation><annotation encoding="application/x-llamapun" id="A1.SS3.SSS0.P0.SPx1.p4.3.m1.2d">roman_log ( 1 + italic_x ) ≥ italic_x - divide start_ARG italic_x start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG start_ARG 2 end_ARG</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="A1.SS3.SSS0.P0.SPx1.p4.4.2"> when </span><math alttext="x\geq 0" class="ltx_Math" display="inline" id="A1.SS3.SSS0.P0.SPx1.p4.4.m2.1"><semantics id="A1.SS3.SSS0.P0.SPx1.p4.4.m2.1a"><mrow id="A1.SS3.SSS0.P0.SPx1.p4.4.m2.1.1" xref="A1.SS3.SSS0.P0.SPx1.p4.4.m2.1.1.cmml"><mi id="A1.SS3.SSS0.P0.SPx1.p4.4.m2.1.1.2" xref="A1.SS3.SSS0.P0.SPx1.p4.4.m2.1.1.2.cmml">x</mi><mo id="A1.SS3.SSS0.P0.SPx1.p4.4.m2.1.1.1" xref="A1.SS3.SSS0.P0.SPx1.p4.4.m2.1.1.1.cmml">≥</mo><mn id="A1.SS3.SSS0.P0.SPx1.p4.4.m2.1.1.3" xref="A1.SS3.SSS0.P0.SPx1.p4.4.m2.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="A1.SS3.SSS0.P0.SPx1.p4.4.m2.1b"><apply id="A1.SS3.SSS0.P0.SPx1.p4.4.m2.1.1.cmml" xref="A1.SS3.SSS0.P0.SPx1.p4.4.m2.1.1"><geq id="A1.SS3.SSS0.P0.SPx1.p4.4.m2.1.1.1.cmml" xref="A1.SS3.SSS0.P0.SPx1.p4.4.m2.1.1.1"></geq><ci id="A1.SS3.SSS0.P0.SPx1.p4.4.m2.1.1.2.cmml" xref="A1.SS3.SSS0.P0.SPx1.p4.4.m2.1.1.2">𝑥</ci><cn id="A1.SS3.SSS0.P0.SPx1.p4.4.m2.1.1.3.cmml" type="integer" xref="A1.SS3.SSS0.P0.SPx1.p4.4.m2.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS3.SSS0.P0.SPx1.p4.4.m2.1c">x\geq 0</annotation><annotation encoding="application/x-llamapun" id="A1.SS3.SSS0.P0.SPx1.p4.4.m2.1d">italic_x ≥ 0</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="A1.SS3.SSS0.P0.SPx1.p4.4.3">.</span></p> </div> <div class="ltx_para" id="A1.SS3.SSS0.P0.SPx1.p5"> <p class="ltx_p" id="A1.SS3.SSS0.P0.SPx1.p5.9"><span class="ltx_text ltx_font_italic" id="A1.SS3.SSS0.P0.SPx1.p5.9.1">In particular this means that within the </span><math alttext="t" class="ltx_Math" display="inline" id="A1.SS3.SSS0.P0.SPx1.p5.1.m1.1"><semantics id="A1.SS3.SSS0.P0.SPx1.p5.1.m1.1a"><mi id="A1.SS3.SSS0.P0.SPx1.p5.1.m1.1.1" xref="A1.SS3.SSS0.P0.SPx1.p5.1.m1.1.1.cmml">t</mi><annotation-xml encoding="MathML-Content" id="A1.SS3.SSS0.P0.SPx1.p5.1.m1.1b"><ci id="A1.SS3.SSS0.P0.SPx1.p5.1.m1.1.1.cmml" xref="A1.SS3.SSS0.P0.SPx1.p5.1.m1.1.1">𝑡</ci></annotation-xml><annotation encoding="application/x-tex" id="A1.SS3.SSS0.P0.SPx1.p5.1.m1.1c">t</annotation><annotation encoding="application/x-llamapun" id="A1.SS3.SSS0.P0.SPx1.p5.1.m1.1d">italic_t</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="A1.SS3.SSS0.P0.SPx1.p5.9.2">-hop neighbourhood of </span><math alttext="v" class="ltx_Math" display="inline" id="A1.SS3.SSS0.P0.SPx1.p5.2.m2.1"><semantics id="A1.SS3.SSS0.P0.SPx1.p5.2.m2.1a"><mi id="A1.SS3.SSS0.P0.SPx1.p5.2.m2.1.1" xref="A1.SS3.SSS0.P0.SPx1.p5.2.m2.1.1.cmml">v</mi><annotation-xml encoding="MathML-Content" id="A1.SS3.SSS0.P0.SPx1.p5.2.m2.1b"><ci id="A1.SS3.SSS0.P0.SPx1.p5.2.m2.1.1.cmml" xref="A1.SS3.SSS0.P0.SPx1.p5.2.m2.1.1">𝑣</ci></annotation-xml><annotation encoding="application/x-tex" id="A1.SS3.SSS0.P0.SPx1.p5.2.m2.1c">v</annotation><annotation encoding="application/x-llamapun" id="A1.SS3.SSS0.P0.SPx1.p5.2.m2.1d">italic_v</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="A1.SS3.SSS0.P0.SPx1.p5.9.3"> in </span><math alttext="H" class="ltx_Math" display="inline" id="A1.SS3.SSS0.P0.SPx1.p5.3.m3.1"><semantics id="A1.SS3.SSS0.P0.SPx1.p5.3.m3.1a"><mi id="A1.SS3.SSS0.P0.SPx1.p5.3.m3.1.1" xref="A1.SS3.SSS0.P0.SPx1.p5.3.m3.1.1.cmml">H</mi><annotation-xml encoding="MathML-Content" id="A1.SS3.SSS0.P0.SPx1.p5.3.m3.1b"><ci id="A1.SS3.SSS0.P0.SPx1.p5.3.m3.1.1.cmml" xref="A1.SS3.SSS0.P0.SPx1.p5.3.m3.1.1">𝐻</ci></annotation-xml><annotation encoding="application/x-tex" id="A1.SS3.SSS0.P0.SPx1.p5.3.m3.1c">H</annotation><annotation encoding="application/x-llamapun" id="A1.SS3.SSS0.P0.SPx1.p5.3.m3.1d">italic_H</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="A1.SS3.SSS0.P0.SPx1.p5.9.4"> (for </span><math alttext="t\in O(\varepsilon^{-2}\log^{2}n)" class="ltx_Math" display="inline" id="A1.SS3.SSS0.P0.SPx1.p5.4.m4.1"><semantics id="A1.SS3.SSS0.P0.SPx1.p5.4.m4.1a"><mrow id="A1.SS3.SSS0.P0.SPx1.p5.4.m4.1.1" xref="A1.SS3.SSS0.P0.SPx1.p5.4.m4.1.1.cmml"><mi id="A1.SS3.SSS0.P0.SPx1.p5.4.m4.1.1.3" xref="A1.SS3.SSS0.P0.SPx1.p5.4.m4.1.1.3.cmml">t</mi><mo id="A1.SS3.SSS0.P0.SPx1.p5.4.m4.1.1.2" xref="A1.SS3.SSS0.P0.SPx1.p5.4.m4.1.1.2.cmml">∈</mo><mrow id="A1.SS3.SSS0.P0.SPx1.p5.4.m4.1.1.1" xref="A1.SS3.SSS0.P0.SPx1.p5.4.m4.1.1.1.cmml"><mi id="A1.SS3.SSS0.P0.SPx1.p5.4.m4.1.1.1.3" xref="A1.SS3.SSS0.P0.SPx1.p5.4.m4.1.1.1.3.cmml">O</mi><mo id="A1.SS3.SSS0.P0.SPx1.p5.4.m4.1.1.1.2" xref="A1.SS3.SSS0.P0.SPx1.p5.4.m4.1.1.1.2.cmml">⁢</mo><mrow id="A1.SS3.SSS0.P0.SPx1.p5.4.m4.1.1.1.1.1" xref="A1.SS3.SSS0.P0.SPx1.p5.4.m4.1.1.1.1.1.1.cmml"><mo id="A1.SS3.SSS0.P0.SPx1.p5.4.m4.1.1.1.1.1.2" stretchy="false" xref="A1.SS3.SSS0.P0.SPx1.p5.4.m4.1.1.1.1.1.1.cmml">(</mo><mrow id="A1.SS3.SSS0.P0.SPx1.p5.4.m4.1.1.1.1.1.1" xref="A1.SS3.SSS0.P0.SPx1.p5.4.m4.1.1.1.1.1.1.cmml"><msup id="A1.SS3.SSS0.P0.SPx1.p5.4.m4.1.1.1.1.1.1.2" xref="A1.SS3.SSS0.P0.SPx1.p5.4.m4.1.1.1.1.1.1.2.cmml"><mi id="A1.SS3.SSS0.P0.SPx1.p5.4.m4.1.1.1.1.1.1.2.2" xref="A1.SS3.SSS0.P0.SPx1.p5.4.m4.1.1.1.1.1.1.2.2.cmml">ε</mi><mrow id="A1.SS3.SSS0.P0.SPx1.p5.4.m4.1.1.1.1.1.1.2.3" xref="A1.SS3.SSS0.P0.SPx1.p5.4.m4.1.1.1.1.1.1.2.3.cmml"><mo id="A1.SS3.SSS0.P0.SPx1.p5.4.m4.1.1.1.1.1.1.2.3a" 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xref="A1.SS3.SSS0.P0.SPx1.p5.4.m4.1.1.1.1.1.1.3.2.cmml">n</mi></mrow></mrow><mo id="A1.SS3.SSS0.P0.SPx1.p5.4.m4.1.1.1.1.1.3" stretchy="false" xref="A1.SS3.SSS0.P0.SPx1.p5.4.m4.1.1.1.1.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="A1.SS3.SSS0.P0.SPx1.p5.4.m4.1b"><apply id="A1.SS3.SSS0.P0.SPx1.p5.4.m4.1.1.cmml" xref="A1.SS3.SSS0.P0.SPx1.p5.4.m4.1.1"><in id="A1.SS3.SSS0.P0.SPx1.p5.4.m4.1.1.2.cmml" xref="A1.SS3.SSS0.P0.SPx1.p5.4.m4.1.1.2"></in><ci id="A1.SS3.SSS0.P0.SPx1.p5.4.m4.1.1.3.cmml" xref="A1.SS3.SSS0.P0.SPx1.p5.4.m4.1.1.3">𝑡</ci><apply id="A1.SS3.SSS0.P0.SPx1.p5.4.m4.1.1.1.cmml" xref="A1.SS3.SSS0.P0.SPx1.p5.4.m4.1.1.1"><times id="A1.SS3.SSS0.P0.SPx1.p5.4.m4.1.1.1.2.cmml" xref="A1.SS3.SSS0.P0.SPx1.p5.4.m4.1.1.1.2"></times><ci id="A1.SS3.SSS0.P0.SPx1.p5.4.m4.1.1.1.3.cmml" xref="A1.SS3.SSS0.P0.SPx1.p5.4.m4.1.1.1.3">𝑂</ci><apply id="A1.SS3.SSS0.P0.SPx1.p5.4.m4.1.1.1.1.1.1.cmml" xref="A1.SS3.SSS0.P0.SPx1.p5.4.m4.1.1.1.1.1"><times 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id="A1.SS3.SSS0.P0.SPx1.p5.4.m4.1.1.1.1.1.1.3.1.cmml" xref="A1.SS3.SSS0.P0.SPx1.p5.4.m4.1.1.1.1.1.1.3.1"><csymbol cd="ambiguous" id="A1.SS3.SSS0.P0.SPx1.p5.4.m4.1.1.1.1.1.1.3.1.1.cmml" xref="A1.SS3.SSS0.P0.SPx1.p5.4.m4.1.1.1.1.1.1.3.1">superscript</csymbol><log id="A1.SS3.SSS0.P0.SPx1.p5.4.m4.1.1.1.1.1.1.3.1.2.cmml" xref="A1.SS3.SSS0.P0.SPx1.p5.4.m4.1.1.1.1.1.1.3.1.2"></log><cn id="A1.SS3.SSS0.P0.SPx1.p5.4.m4.1.1.1.1.1.1.3.1.3.cmml" type="integer" xref="A1.SS3.SSS0.P0.SPx1.p5.4.m4.1.1.1.1.1.1.3.1.3">2</cn></apply><ci id="A1.SS3.SSS0.P0.SPx1.p5.4.m4.1.1.1.1.1.1.3.2.cmml" xref="A1.SS3.SSS0.P0.SPx1.p5.4.m4.1.1.1.1.1.1.3.2">𝑛</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS3.SSS0.P0.SPx1.p5.4.m4.1c">t\in O(\varepsilon^{-2}\log^{2}n)</annotation><annotation encoding="application/x-llamapun" id="A1.SS3.SSS0.P0.SPx1.p5.4.m4.1d">italic_t ∈ italic_O ( italic_ε start_POSTSUPERSCRIPT - 2 end_POSTSUPERSCRIPT roman_log start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_n )</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="A1.SS3.SSS0.P0.SPx1.p5.9.5">) we find a subgraph of density at least </span><math alttext="(1-\varepsilon)\rho^{*}(v)" class="ltx_Math" display="inline" id="A1.SS3.SSS0.P0.SPx1.p5.5.m5.2"><semantics id="A1.SS3.SSS0.P0.SPx1.p5.5.m5.2a"><mrow id="A1.SS3.SSS0.P0.SPx1.p5.5.m5.2.2" xref="A1.SS3.SSS0.P0.SPx1.p5.5.m5.2.2.cmml"><mrow id="A1.SS3.SSS0.P0.SPx1.p5.5.m5.2.2.1.1" xref="A1.SS3.SSS0.P0.SPx1.p5.5.m5.2.2.1.1.1.cmml"><mo id="A1.SS3.SSS0.P0.SPx1.p5.5.m5.2.2.1.1.2" stretchy="false" xref="A1.SS3.SSS0.P0.SPx1.p5.5.m5.2.2.1.1.1.cmml">(</mo><mrow id="A1.SS3.SSS0.P0.SPx1.p5.5.m5.2.2.1.1.1" xref="A1.SS3.SSS0.P0.SPx1.p5.5.m5.2.2.1.1.1.cmml"><mn id="A1.SS3.SSS0.P0.SPx1.p5.5.m5.2.2.1.1.1.2" xref="A1.SS3.SSS0.P0.SPx1.p5.5.m5.2.2.1.1.1.2.cmml">1</mn><mo id="A1.SS3.SSS0.P0.SPx1.p5.5.m5.2.2.1.1.1.1" xref="A1.SS3.SSS0.P0.SPx1.p5.5.m5.2.2.1.1.1.1.cmml">−</mo><mi id="A1.SS3.SSS0.P0.SPx1.p5.5.m5.2.2.1.1.1.3" xref="A1.SS3.SSS0.P0.SPx1.p5.5.m5.2.2.1.1.1.3.cmml">ε</mi></mrow><mo id="A1.SS3.SSS0.P0.SPx1.p5.5.m5.2.2.1.1.3" stretchy="false" xref="A1.SS3.SSS0.P0.SPx1.p5.5.m5.2.2.1.1.1.cmml">)</mo></mrow><mo id="A1.SS3.SSS0.P0.SPx1.p5.5.m5.2.2.2" xref="A1.SS3.SSS0.P0.SPx1.p5.5.m5.2.2.2.cmml">⁢</mo><msup id="A1.SS3.SSS0.P0.SPx1.p5.5.m5.2.2.3" xref="A1.SS3.SSS0.P0.SPx1.p5.5.m5.2.2.3.cmml"><mi id="A1.SS3.SSS0.P0.SPx1.p5.5.m5.2.2.3.2" xref="A1.SS3.SSS0.P0.SPx1.p5.5.m5.2.2.3.2.cmml">ρ</mi><mo id="A1.SS3.SSS0.P0.SPx1.p5.5.m5.2.2.3.3" xref="A1.SS3.SSS0.P0.SPx1.p5.5.m5.2.2.3.3.cmml">∗</mo></msup><mo id="A1.SS3.SSS0.P0.SPx1.p5.5.m5.2.2.2a" xref="A1.SS3.SSS0.P0.SPx1.p5.5.m5.2.2.2.cmml">⁢</mo><mrow id="A1.SS3.SSS0.P0.SPx1.p5.5.m5.2.2.4.2" xref="A1.SS3.SSS0.P0.SPx1.p5.5.m5.2.2.cmml"><mo id="A1.SS3.SSS0.P0.SPx1.p5.5.m5.2.2.4.2.1" stretchy="false" xref="A1.SS3.SSS0.P0.SPx1.p5.5.m5.2.2.cmml">(</mo><mi id="A1.SS3.SSS0.P0.SPx1.p5.5.m5.1.1" xref="A1.SS3.SSS0.P0.SPx1.p5.5.m5.1.1.cmml">v</mi><mo id="A1.SS3.SSS0.P0.SPx1.p5.5.m5.2.2.4.2.2" stretchy="false" xref="A1.SS3.SSS0.P0.SPx1.p5.5.m5.2.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="A1.SS3.SSS0.P0.SPx1.p5.5.m5.2b"><apply id="A1.SS3.SSS0.P0.SPx1.p5.5.m5.2.2.cmml" xref="A1.SS3.SSS0.P0.SPx1.p5.5.m5.2.2"><times id="A1.SS3.SSS0.P0.SPx1.p5.5.m5.2.2.2.cmml" xref="A1.SS3.SSS0.P0.SPx1.p5.5.m5.2.2.2"></times><apply id="A1.SS3.SSS0.P0.SPx1.p5.5.m5.2.2.1.1.1.cmml" xref="A1.SS3.SSS0.P0.SPx1.p5.5.m5.2.2.1.1"><minus id="A1.SS3.SSS0.P0.SPx1.p5.5.m5.2.2.1.1.1.1.cmml" xref="A1.SS3.SSS0.P0.SPx1.p5.5.m5.2.2.1.1.1.1"></minus><cn id="A1.SS3.SSS0.P0.SPx1.p5.5.m5.2.2.1.1.1.2.cmml" type="integer" xref="A1.SS3.SSS0.P0.SPx1.p5.5.m5.2.2.1.1.1.2">1</cn><ci id="A1.SS3.SSS0.P0.SPx1.p5.5.m5.2.2.1.1.1.3.cmml" xref="A1.SS3.SSS0.P0.SPx1.p5.5.m5.2.2.1.1.1.3">𝜀</ci></apply><apply id="A1.SS3.SSS0.P0.SPx1.p5.5.m5.2.2.3.cmml" xref="A1.SS3.SSS0.P0.SPx1.p5.5.m5.2.2.3"><csymbol cd="ambiguous" id="A1.SS3.SSS0.P0.SPx1.p5.5.m5.2.2.3.1.cmml" xref="A1.SS3.SSS0.P0.SPx1.p5.5.m5.2.2.3">superscript</csymbol><ci id="A1.SS3.SSS0.P0.SPx1.p5.5.m5.2.2.3.2.cmml" xref="A1.SS3.SSS0.P0.SPx1.p5.5.m5.2.2.3.2">𝜌</ci><times id="A1.SS3.SSS0.P0.SPx1.p5.5.m5.2.2.3.3.cmml" xref="A1.SS3.SSS0.P0.SPx1.p5.5.m5.2.2.3.3"></times></apply><ci id="A1.SS3.SSS0.P0.SPx1.p5.5.m5.1.1.cmml" xref="A1.SS3.SSS0.P0.SPx1.p5.5.m5.1.1">𝑣</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS3.SSS0.P0.SPx1.p5.5.m5.2c">(1-\varepsilon)\rho^{*}(v)</annotation><annotation encoding="application/x-llamapun" id="A1.SS3.SSS0.P0.SPx1.p5.5.m5.2d">( 1 - italic_ε ) italic_ρ start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ( italic_v )</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="A1.SS3.SSS0.P0.SPx1.p5.9.6"> and density at most </span><math alttext="\rho(H)=\rho^{*}(v)" class="ltx_Math" display="inline" id="A1.SS3.SSS0.P0.SPx1.p5.6.m6.2"><semantics id="A1.SS3.SSS0.P0.SPx1.p5.6.m6.2a"><mrow id="A1.SS3.SSS0.P0.SPx1.p5.6.m6.2.3" xref="A1.SS3.SSS0.P0.SPx1.p5.6.m6.2.3.cmml"><mrow id="A1.SS3.SSS0.P0.SPx1.p5.6.m6.2.3.2" xref="A1.SS3.SSS0.P0.SPx1.p5.6.m6.2.3.2.cmml"><mi id="A1.SS3.SSS0.P0.SPx1.p5.6.m6.2.3.2.2" xref="A1.SS3.SSS0.P0.SPx1.p5.6.m6.2.3.2.2.cmml">ρ</mi><mo id="A1.SS3.SSS0.P0.SPx1.p5.6.m6.2.3.2.1" xref="A1.SS3.SSS0.P0.SPx1.p5.6.m6.2.3.2.1.cmml">⁢</mo><mrow id="A1.SS3.SSS0.P0.SPx1.p5.6.m6.2.3.2.3.2" xref="A1.SS3.SSS0.P0.SPx1.p5.6.m6.2.3.2.cmml"><mo id="A1.SS3.SSS0.P0.SPx1.p5.6.m6.2.3.2.3.2.1" stretchy="false" xref="A1.SS3.SSS0.P0.SPx1.p5.6.m6.2.3.2.cmml">(</mo><mi id="A1.SS3.SSS0.P0.SPx1.p5.6.m6.1.1" xref="A1.SS3.SSS0.P0.SPx1.p5.6.m6.1.1.cmml">H</mi><mo id="A1.SS3.SSS0.P0.SPx1.p5.6.m6.2.3.2.3.2.2" stretchy="false" xref="A1.SS3.SSS0.P0.SPx1.p5.6.m6.2.3.2.cmml">)</mo></mrow></mrow><mo id="A1.SS3.SSS0.P0.SPx1.p5.6.m6.2.3.1" xref="A1.SS3.SSS0.P0.SPx1.p5.6.m6.2.3.1.cmml">=</mo><mrow id="A1.SS3.SSS0.P0.SPx1.p5.6.m6.2.3.3" xref="A1.SS3.SSS0.P0.SPx1.p5.6.m6.2.3.3.cmml"><msup id="A1.SS3.SSS0.P0.SPx1.p5.6.m6.2.3.3.2" xref="A1.SS3.SSS0.P0.SPx1.p5.6.m6.2.3.3.2.cmml"><mi id="A1.SS3.SSS0.P0.SPx1.p5.6.m6.2.3.3.2.2" xref="A1.SS3.SSS0.P0.SPx1.p5.6.m6.2.3.3.2.2.cmml">ρ</mi><mo id="A1.SS3.SSS0.P0.SPx1.p5.6.m6.2.3.3.2.3" xref="A1.SS3.SSS0.P0.SPx1.p5.6.m6.2.3.3.2.3.cmml">∗</mo></msup><mo id="A1.SS3.SSS0.P0.SPx1.p5.6.m6.2.3.3.1" xref="A1.SS3.SSS0.P0.SPx1.p5.6.m6.2.3.3.1.cmml">⁢</mo><mrow id="A1.SS3.SSS0.P0.SPx1.p5.6.m6.2.3.3.3.2" xref="A1.SS3.SSS0.P0.SPx1.p5.6.m6.2.3.3.cmml"><mo id="A1.SS3.SSS0.P0.SPx1.p5.6.m6.2.3.3.3.2.1" stretchy="false" xref="A1.SS3.SSS0.P0.SPx1.p5.6.m6.2.3.3.cmml">(</mo><mi id="A1.SS3.SSS0.P0.SPx1.p5.6.m6.2.2" xref="A1.SS3.SSS0.P0.SPx1.p5.6.m6.2.2.cmml">v</mi><mo id="A1.SS3.SSS0.P0.SPx1.p5.6.m6.2.3.3.3.2.2" stretchy="false" xref="A1.SS3.SSS0.P0.SPx1.p5.6.m6.2.3.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="A1.SS3.SSS0.P0.SPx1.p5.6.m6.2b"><apply id="A1.SS3.SSS0.P0.SPx1.p5.6.m6.2.3.cmml" xref="A1.SS3.SSS0.P0.SPx1.p5.6.m6.2.3"><eq id="A1.SS3.SSS0.P0.SPx1.p5.6.m6.2.3.1.cmml" xref="A1.SS3.SSS0.P0.SPx1.p5.6.m6.2.3.1"></eq><apply id="A1.SS3.SSS0.P0.SPx1.p5.6.m6.2.3.2.cmml" xref="A1.SS3.SSS0.P0.SPx1.p5.6.m6.2.3.2"><times id="A1.SS3.SSS0.P0.SPx1.p5.6.m6.2.3.2.1.cmml" xref="A1.SS3.SSS0.P0.SPx1.p5.6.m6.2.3.2.1"></times><ci id="A1.SS3.SSS0.P0.SPx1.p5.6.m6.2.3.2.2.cmml" xref="A1.SS3.SSS0.P0.SPx1.p5.6.m6.2.3.2.2">𝜌</ci><ci id="A1.SS3.SSS0.P0.SPx1.p5.6.m6.1.1.cmml" xref="A1.SS3.SSS0.P0.SPx1.p5.6.m6.1.1">𝐻</ci></apply><apply id="A1.SS3.SSS0.P0.SPx1.p5.6.m6.2.3.3.cmml" xref="A1.SS3.SSS0.P0.SPx1.p5.6.m6.2.3.3"><times id="A1.SS3.SSS0.P0.SPx1.p5.6.m6.2.3.3.1.cmml" xref="A1.SS3.SSS0.P0.SPx1.p5.6.m6.2.3.3.1"></times><apply id="A1.SS3.SSS0.P0.SPx1.p5.6.m6.2.3.3.2.cmml" xref="A1.SS3.SSS0.P0.SPx1.p5.6.m6.2.3.3.2"><csymbol cd="ambiguous" id="A1.SS3.SSS0.P0.SPx1.p5.6.m6.2.3.3.2.1.cmml" xref="A1.SS3.SSS0.P0.SPx1.p5.6.m6.2.3.3.2">superscript</csymbol><ci id="A1.SS3.SSS0.P0.SPx1.p5.6.m6.2.3.3.2.2.cmml" xref="A1.SS3.SSS0.P0.SPx1.p5.6.m6.2.3.3.2.2">𝜌</ci><times id="A1.SS3.SSS0.P0.SPx1.p5.6.m6.2.3.3.2.3.cmml" xref="A1.SS3.SSS0.P0.SPx1.p5.6.m6.2.3.3.2.3"></times></apply><ci id="A1.SS3.SSS0.P0.SPx1.p5.6.m6.2.2.cmml" xref="A1.SS3.SSS0.P0.SPx1.p5.6.m6.2.2">𝑣</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS3.SSS0.P0.SPx1.p5.6.m6.2c">\rho(H)=\rho^{*}(v)</annotation><annotation encoding="application/x-llamapun" id="A1.SS3.SSS0.P0.SPx1.p5.6.m6.2d">italic_ρ ( italic_H ) = italic_ρ start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ( italic_v )</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="A1.SS3.SSS0.P0.SPx1.p5.9.7">. Since </span><math alttext="H" class="ltx_Math" display="inline" id="A1.SS3.SSS0.P0.SPx1.p5.7.m7.1"><semantics id="A1.SS3.SSS0.P0.SPx1.p5.7.m7.1a"><mi id="A1.SS3.SSS0.P0.SPx1.p5.7.m7.1.1" xref="A1.SS3.SSS0.P0.SPx1.p5.7.m7.1.1.cmml">H</mi><annotation-xml encoding="MathML-Content" id="A1.SS3.SSS0.P0.SPx1.p5.7.m7.1b"><ci id="A1.SS3.SSS0.P0.SPx1.p5.7.m7.1.1.cmml" xref="A1.SS3.SSS0.P0.SPx1.p5.7.m7.1.1">𝐻</ci></annotation-xml><annotation encoding="application/x-tex" id="A1.SS3.SSS0.P0.SPx1.p5.7.m7.1c">H</annotation><annotation encoding="application/x-llamapun" id="A1.SS3.SSS0.P0.SPx1.p5.7.m7.1d">italic_H</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="A1.SS3.SSS0.P0.SPx1.p5.9.8"> is a (weighted) subgraph of </span><math alttext="G" class="ltx_Math" display="inline" id="A1.SS3.SSS0.P0.SPx1.p5.8.m8.1"><semantics id="A1.SS3.SSS0.P0.SPx1.p5.8.m8.1a"><mi id="A1.SS3.SSS0.P0.SPx1.p5.8.m8.1.1" xref="A1.SS3.SSS0.P0.SPx1.p5.8.m8.1.1.cmml">G</mi><annotation-xml encoding="MathML-Content" id="A1.SS3.SSS0.P0.SPx1.p5.8.m8.1b"><ci id="A1.SS3.SSS0.P0.SPx1.p5.8.m8.1.1.cmml" xref="A1.SS3.SSS0.P0.SPx1.p5.8.m8.1.1">𝐺</ci></annotation-xml><annotation encoding="application/x-tex" id="A1.SS3.SSS0.P0.SPx1.p5.8.m8.1c">G</annotation><annotation encoding="application/x-llamapun" id="A1.SS3.SSS0.P0.SPx1.p5.8.m8.1d">italic_G</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="A1.SS3.SSS0.P0.SPx1.p5.9.9">, the same holds for </span><math alttext="G" class="ltx_Math" display="inline" id="A1.SS3.SSS0.P0.SPx1.p5.9.m9.1"><semantics id="A1.SS3.SSS0.P0.SPx1.p5.9.m9.1a"><mi id="A1.SS3.SSS0.P0.SPx1.p5.9.m9.1.1" xref="A1.SS3.SSS0.P0.SPx1.p5.9.m9.1.1.cmml">G</mi><annotation-xml encoding="MathML-Content" id="A1.SS3.SSS0.P0.SPx1.p5.9.m9.1b"><ci id="A1.SS3.SSS0.P0.SPx1.p5.9.m9.1.1.cmml" xref="A1.SS3.SSS0.P0.SPx1.p5.9.m9.1.1">𝐺</ci></annotation-xml><annotation encoding="application/x-tex" id="A1.SS3.SSS0.P0.SPx1.p5.9.m9.1c">G</annotation><annotation encoding="application/x-llamapun" id="A1.SS3.SSS0.P0.SPx1.p5.9.m9.1d">italic_G</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="A1.SS3.SSS0.P0.SPx1.p5.9.10">. </span><span class="ltx_text" id="A1.SS3.SSS0.P0.SPx1.p5.9.11"></span></p> </div> <div class="ltx_para" id="A1.SS3.SSS0.P0.SPx1.p6"> <p class="ltx_p" id="A1.SS3.SSS0.P0.SPx1.p6.11"><span class="ltx_text" id="A1.SS3.SSS0.P0.SPx1.p6.11.11">The subgraph reporting algorithm now works as follows: We first compute an <math alttext="\eta" class="ltx_Math" display="inline" id="A1.SS3.SSS0.P0.SPx1.p6.1.1.m1.1"><semantics id="A1.SS3.SSS0.P0.SPx1.p6.1.1.m1.1a"><mi id="A1.SS3.SSS0.P0.SPx1.p6.1.1.m1.1.1" xref="A1.SS3.SSS0.P0.SPx1.p6.1.1.m1.1.1.cmml">η</mi><annotation-xml encoding="MathML-Content" id="A1.SS3.SSS0.P0.SPx1.p6.1.1.m1.1b"><ci id="A1.SS3.SSS0.P0.SPx1.p6.1.1.m1.1.1.cmml" xref="A1.SS3.SSS0.P0.SPx1.p6.1.1.m1.1.1">𝜂</ci></annotation-xml><annotation encoding="application/x-tex" id="A1.SS3.SSS0.P0.SPx1.p6.1.1.m1.1c">\eta</annotation><annotation encoding="application/x-llamapun" id="A1.SS3.SSS0.P0.SPx1.p6.1.1.m1.1d">italic_η</annotation></semantics></math>-fair fractional out-orientation for <math alttext="\eta=\frac{\varepsilon^{2}}{128\log n}" class="ltx_Math" display="inline" id="A1.SS3.SSS0.P0.SPx1.p6.2.2.m2.1"><semantics id="A1.SS3.SSS0.P0.SPx1.p6.2.2.m2.1a"><mrow id="A1.SS3.SSS0.P0.SPx1.p6.2.2.m2.1.1" xref="A1.SS3.SSS0.P0.SPx1.p6.2.2.m2.1.1.cmml"><mi id="A1.SS3.SSS0.P0.SPx1.p6.2.2.m2.1.1.2" xref="A1.SS3.SSS0.P0.SPx1.p6.2.2.m2.1.1.2.cmml">η</mi><mo id="A1.SS3.SSS0.P0.SPx1.p6.2.2.m2.1.1.1" xref="A1.SS3.SSS0.P0.SPx1.p6.2.2.m2.1.1.1.cmml">=</mo><mfrac id="A1.SS3.SSS0.P0.SPx1.p6.2.2.m2.1.1.3" xref="A1.SS3.SSS0.P0.SPx1.p6.2.2.m2.1.1.3.cmml"><msup id="A1.SS3.SSS0.P0.SPx1.p6.2.2.m2.1.1.3.2" xref="A1.SS3.SSS0.P0.SPx1.p6.2.2.m2.1.1.3.2.cmml"><mi id="A1.SS3.SSS0.P0.SPx1.p6.2.2.m2.1.1.3.2.2" xref="A1.SS3.SSS0.P0.SPx1.p6.2.2.m2.1.1.3.2.2.cmml">ε</mi><mn id="A1.SS3.SSS0.P0.SPx1.p6.2.2.m2.1.1.3.2.3" xref="A1.SS3.SSS0.P0.SPx1.p6.2.2.m2.1.1.3.2.3.cmml">2</mn></msup><mrow id="A1.SS3.SSS0.P0.SPx1.p6.2.2.m2.1.1.3.3" xref="A1.SS3.SSS0.P0.SPx1.p6.2.2.m2.1.1.3.3.cmml"><mn id="A1.SS3.SSS0.P0.SPx1.p6.2.2.m2.1.1.3.3.2" xref="A1.SS3.SSS0.P0.SPx1.p6.2.2.m2.1.1.3.3.2.cmml">128</mn><mo id="A1.SS3.SSS0.P0.SPx1.p6.2.2.m2.1.1.3.3.1" lspace="0.167em" xref="A1.SS3.SSS0.P0.SPx1.p6.2.2.m2.1.1.3.3.1.cmml">⁢</mo><mrow id="A1.SS3.SSS0.P0.SPx1.p6.2.2.m2.1.1.3.3.3" xref="A1.SS3.SSS0.P0.SPx1.p6.2.2.m2.1.1.3.3.3.cmml"><mi id="A1.SS3.SSS0.P0.SPx1.p6.2.2.m2.1.1.3.3.3.1" xref="A1.SS3.SSS0.P0.SPx1.p6.2.2.m2.1.1.3.3.3.1.cmml">log</mi><mo id="A1.SS3.SSS0.P0.SPx1.p6.2.2.m2.1.1.3.3.3a" lspace="0.167em" xref="A1.SS3.SSS0.P0.SPx1.p6.2.2.m2.1.1.3.3.3.cmml">⁡</mo><mi id="A1.SS3.SSS0.P0.SPx1.p6.2.2.m2.1.1.3.3.3.2" xref="A1.SS3.SSS0.P0.SPx1.p6.2.2.m2.1.1.3.3.3.2.cmml">n</mi></mrow></mrow></mfrac></mrow><annotation-xml encoding="MathML-Content" id="A1.SS3.SSS0.P0.SPx1.p6.2.2.m2.1b"><apply id="A1.SS3.SSS0.P0.SPx1.p6.2.2.m2.1.1.cmml" xref="A1.SS3.SSS0.P0.SPx1.p6.2.2.m2.1.1"><eq id="A1.SS3.SSS0.P0.SPx1.p6.2.2.m2.1.1.1.cmml" xref="A1.SS3.SSS0.P0.SPx1.p6.2.2.m2.1.1.1"></eq><ci id="A1.SS3.SSS0.P0.SPx1.p6.2.2.m2.1.1.2.cmml" xref="A1.SS3.SSS0.P0.SPx1.p6.2.2.m2.1.1.2">𝜂</ci><apply id="A1.SS3.SSS0.P0.SPx1.p6.2.2.m2.1.1.3.cmml" xref="A1.SS3.SSS0.P0.SPx1.p6.2.2.m2.1.1.3"><divide id="A1.SS3.SSS0.P0.SPx1.p6.2.2.m2.1.1.3.1.cmml" xref="A1.SS3.SSS0.P0.SPx1.p6.2.2.m2.1.1.3"></divide><apply id="A1.SS3.SSS0.P0.SPx1.p6.2.2.m2.1.1.3.2.cmml" xref="A1.SS3.SSS0.P0.SPx1.p6.2.2.m2.1.1.3.2"><csymbol cd="ambiguous" id="A1.SS3.SSS0.P0.SPx1.p6.2.2.m2.1.1.3.2.1.cmml" xref="A1.SS3.SSS0.P0.SPx1.p6.2.2.m2.1.1.3.2">superscript</csymbol><ci id="A1.SS3.SSS0.P0.SPx1.p6.2.2.m2.1.1.3.2.2.cmml" xref="A1.SS3.SSS0.P0.SPx1.p6.2.2.m2.1.1.3.2.2">𝜀</ci><cn id="A1.SS3.SSS0.P0.SPx1.p6.2.2.m2.1.1.3.2.3.cmml" type="integer" xref="A1.SS3.SSS0.P0.SPx1.p6.2.2.m2.1.1.3.2.3">2</cn></apply><apply id="A1.SS3.SSS0.P0.SPx1.p6.2.2.m2.1.1.3.3.cmml" xref="A1.SS3.SSS0.P0.SPx1.p6.2.2.m2.1.1.3.3"><times id="A1.SS3.SSS0.P0.SPx1.p6.2.2.m2.1.1.3.3.1.cmml" xref="A1.SS3.SSS0.P0.SPx1.p6.2.2.m2.1.1.3.3.1"></times><cn id="A1.SS3.SSS0.P0.SPx1.p6.2.2.m2.1.1.3.3.2.cmml" type="integer" xref="A1.SS3.SSS0.P0.SPx1.p6.2.2.m2.1.1.3.3.2">128</cn><apply id="A1.SS3.SSS0.P0.SPx1.p6.2.2.m2.1.1.3.3.3.cmml" xref="A1.SS3.SSS0.P0.SPx1.p6.2.2.m2.1.1.3.3.3"><log id="A1.SS3.SSS0.P0.SPx1.p6.2.2.m2.1.1.3.3.3.1.cmml" xref="A1.SS3.SSS0.P0.SPx1.p6.2.2.m2.1.1.3.3.3.1"></log><ci id="A1.SS3.SSS0.P0.SPx1.p6.2.2.m2.1.1.3.3.3.2.cmml" xref="A1.SS3.SSS0.P0.SPx1.p6.2.2.m2.1.1.3.3.3.2">𝑛</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS3.SSS0.P0.SPx1.p6.2.2.m2.1c">\eta=\frac{\varepsilon^{2}}{128\log n}</annotation><annotation encoding="application/x-llamapun" id="A1.SS3.SSS0.P0.SPx1.p6.2.2.m2.1d">italic_η = divide start_ARG italic_ε start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG start_ARG 128 roman_log italic_n end_ARG</annotation></semantics></math>. From each selected vertex <math alttext="v" class="ltx_Math" display="inline" id="A1.SS3.SSS0.P0.SPx1.p6.3.3.m3.1"><semantics id="A1.SS3.SSS0.P0.SPx1.p6.3.3.m3.1a"><mi id="A1.SS3.SSS0.P0.SPx1.p6.3.3.m3.1.1" xref="A1.SS3.SSS0.P0.SPx1.p6.3.3.m3.1.1.cmml">v</mi><annotation-xml encoding="MathML-Content" id="A1.SS3.SSS0.P0.SPx1.p6.3.3.m3.1b"><ci id="A1.SS3.SSS0.P0.SPx1.p6.3.3.m3.1.1.cmml" xref="A1.SS3.SSS0.P0.SPx1.p6.3.3.m3.1.1">𝑣</ci></annotation-xml><annotation encoding="application/x-tex" id="A1.SS3.SSS0.P0.SPx1.p6.3.3.m3.1c">v</annotation><annotation encoding="application/x-llamapun" id="A1.SS3.SSS0.P0.SPx1.p6.3.3.m3.1d">italic_v</annotation></semantics></math>; we run the standard CONGEST BFS-leader election algorithm (for example the one described in chapter 5 of <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2411.12694v2#bib.bib17" title="">17</a>]</cite>) before truncating it after exactly <math alttext="\lceil\frac{32\log n}{\varepsilon}\rceil" class="ltx_Math" display="inline" id="A1.SS3.SSS0.P0.SPx1.p6.4.4.m4.1"><semantics id="A1.SS3.SSS0.P0.SPx1.p6.4.4.m4.1a"><mrow id="A1.SS3.SSS0.P0.SPx1.p6.4.4.m4.1.2.2" xref="A1.SS3.SSS0.P0.SPx1.p6.4.4.m4.1.2.1.cmml"><mo id="A1.SS3.SSS0.P0.SPx1.p6.4.4.m4.1.2.2.1" stretchy="false" xref="A1.SS3.SSS0.P0.SPx1.p6.4.4.m4.1.2.1.1.cmml">⌈</mo><mfrac id="A1.SS3.SSS0.P0.SPx1.p6.4.4.m4.1.1" xref="A1.SS3.SSS0.P0.SPx1.p6.4.4.m4.1.1.cmml"><mrow id="A1.SS3.SSS0.P0.SPx1.p6.4.4.m4.1.1.2" xref="A1.SS3.SSS0.P0.SPx1.p6.4.4.m4.1.1.2.cmml"><mn id="A1.SS3.SSS0.P0.SPx1.p6.4.4.m4.1.1.2.2" xref="A1.SS3.SSS0.P0.SPx1.p6.4.4.m4.1.1.2.2.cmml">32</mn><mo id="A1.SS3.SSS0.P0.SPx1.p6.4.4.m4.1.1.2.1" lspace="0.167em" xref="A1.SS3.SSS0.P0.SPx1.p6.4.4.m4.1.1.2.1.cmml">⁢</mo><mrow id="A1.SS3.SSS0.P0.SPx1.p6.4.4.m4.1.1.2.3" xref="A1.SS3.SSS0.P0.SPx1.p6.4.4.m4.1.1.2.3.cmml"><mi id="A1.SS3.SSS0.P0.SPx1.p6.4.4.m4.1.1.2.3.1" xref="A1.SS3.SSS0.P0.SPx1.p6.4.4.m4.1.1.2.3.1.cmml">log</mi><mo id="A1.SS3.SSS0.P0.SPx1.p6.4.4.m4.1.1.2.3a" lspace="0.167em" xref="A1.SS3.SSS0.P0.SPx1.p6.4.4.m4.1.1.2.3.cmml">⁡</mo><mi id="A1.SS3.SSS0.P0.SPx1.p6.4.4.m4.1.1.2.3.2" xref="A1.SS3.SSS0.P0.SPx1.p6.4.4.m4.1.1.2.3.2.cmml">n</mi></mrow></mrow><mi id="A1.SS3.SSS0.P0.SPx1.p6.4.4.m4.1.1.3" xref="A1.SS3.SSS0.P0.SPx1.p6.4.4.m4.1.1.3.cmml">ε</mi></mfrac><mo id="A1.SS3.SSS0.P0.SPx1.p6.4.4.m4.1.2.2.2" stretchy="false" xref="A1.SS3.SSS0.P0.SPx1.p6.4.4.m4.1.2.1.1.cmml">⌉</mo></mrow><annotation-xml encoding="MathML-Content" id="A1.SS3.SSS0.P0.SPx1.p6.4.4.m4.1b"><apply id="A1.SS3.SSS0.P0.SPx1.p6.4.4.m4.1.2.1.cmml" xref="A1.SS3.SSS0.P0.SPx1.p6.4.4.m4.1.2.2"><ceiling id="A1.SS3.SSS0.P0.SPx1.p6.4.4.m4.1.2.1.1.cmml" xref="A1.SS3.SSS0.P0.SPx1.p6.4.4.m4.1.2.2.1"></ceiling><apply id="A1.SS3.SSS0.P0.SPx1.p6.4.4.m4.1.1.cmml" xref="A1.SS3.SSS0.P0.SPx1.p6.4.4.m4.1.1"><divide id="A1.SS3.SSS0.P0.SPx1.p6.4.4.m4.1.1.1.cmml" xref="A1.SS3.SSS0.P0.SPx1.p6.4.4.m4.1.1"></divide><apply id="A1.SS3.SSS0.P0.SPx1.p6.4.4.m4.1.1.2.cmml" xref="A1.SS3.SSS0.P0.SPx1.p6.4.4.m4.1.1.2"><times id="A1.SS3.SSS0.P0.SPx1.p6.4.4.m4.1.1.2.1.cmml" xref="A1.SS3.SSS0.P0.SPx1.p6.4.4.m4.1.1.2.1"></times><cn id="A1.SS3.SSS0.P0.SPx1.p6.4.4.m4.1.1.2.2.cmml" type="integer" xref="A1.SS3.SSS0.P0.SPx1.p6.4.4.m4.1.1.2.2">32</cn><apply id="A1.SS3.SSS0.P0.SPx1.p6.4.4.m4.1.1.2.3.cmml" xref="A1.SS3.SSS0.P0.SPx1.p6.4.4.m4.1.1.2.3"><log id="A1.SS3.SSS0.P0.SPx1.p6.4.4.m4.1.1.2.3.1.cmml" xref="A1.SS3.SSS0.P0.SPx1.p6.4.4.m4.1.1.2.3.1"></log><ci id="A1.SS3.SSS0.P0.SPx1.p6.4.4.m4.1.1.2.3.2.cmml" xref="A1.SS3.SSS0.P0.SPx1.p6.4.4.m4.1.1.2.3.2">𝑛</ci></apply></apply><ci id="A1.SS3.SSS0.P0.SPx1.p6.4.4.m4.1.1.3.cmml" xref="A1.SS3.SSS0.P0.SPx1.p6.4.4.m4.1.1.3">𝜀</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS3.SSS0.P0.SPx1.p6.4.4.m4.1c">\lceil\frac{32\log n}{\varepsilon}\rceil</annotation><annotation encoding="application/x-llamapun" id="A1.SS3.SSS0.P0.SPx1.p6.4.4.m4.1d">⌈ divide start_ARG 32 roman_log italic_n end_ARG start_ARG italic_ε end_ARG ⌉</annotation></semantics></math> rounds. Instead of breaking ties on vertex IDs, we break ties on the fractional out-degree of vertices under the <math alttext="\eta" class="ltx_Math" display="inline" id="A1.SS3.SSS0.P0.SPx1.p6.5.5.m5.1"><semantics id="A1.SS3.SSS0.P0.SPx1.p6.5.5.m5.1a"><mi id="A1.SS3.SSS0.P0.SPx1.p6.5.5.m5.1.1" xref="A1.SS3.SSS0.P0.SPx1.p6.5.5.m5.1.1.cmml">η</mi><annotation-xml encoding="MathML-Content" id="A1.SS3.SSS0.P0.SPx1.p6.5.5.m5.1b"><ci id="A1.SS3.SSS0.P0.SPx1.p6.5.5.m5.1.1.cmml" xref="A1.SS3.SSS0.P0.SPx1.p6.5.5.m5.1.1">𝜂</ci></annotation-xml><annotation encoding="application/x-tex" id="A1.SS3.SSS0.P0.SPx1.p6.5.5.m5.1c">\eta</annotation><annotation encoding="application/x-llamapun" id="A1.SS3.SSS0.P0.SPx1.p6.5.5.m5.1d">italic_η</annotation></semantics></math>-fair orientation, giving priority to the vertex with the largest out-degree (we then subsequently break ties by ID). Now a vertex is only elected a leader, if every vertex of its <math alttext="\lceil\frac{32\log n}{\varepsilon}\rceil" class="ltx_Math" display="inline" id="A1.SS3.SSS0.P0.SPx1.p6.6.6.m6.1"><semantics id="A1.SS3.SSS0.P0.SPx1.p6.6.6.m6.1a"><mrow id="A1.SS3.SSS0.P0.SPx1.p6.6.6.m6.1.2.2" xref="A1.SS3.SSS0.P0.SPx1.p6.6.6.m6.1.2.1.cmml"><mo id="A1.SS3.SSS0.P0.SPx1.p6.6.6.m6.1.2.2.1" stretchy="false" xref="A1.SS3.SSS0.P0.SPx1.p6.6.6.m6.1.2.1.1.cmml">⌈</mo><mfrac id="A1.SS3.SSS0.P0.SPx1.p6.6.6.m6.1.1" xref="A1.SS3.SSS0.P0.SPx1.p6.6.6.m6.1.1.cmml"><mrow id="A1.SS3.SSS0.P0.SPx1.p6.6.6.m6.1.1.2" xref="A1.SS3.SSS0.P0.SPx1.p6.6.6.m6.1.1.2.cmml"><mn id="A1.SS3.SSS0.P0.SPx1.p6.6.6.m6.1.1.2.2" xref="A1.SS3.SSS0.P0.SPx1.p6.6.6.m6.1.1.2.2.cmml">32</mn><mo id="A1.SS3.SSS0.P0.SPx1.p6.6.6.m6.1.1.2.1" lspace="0.167em" xref="A1.SS3.SSS0.P0.SPx1.p6.6.6.m6.1.1.2.1.cmml">⁢</mo><mrow id="A1.SS3.SSS0.P0.SPx1.p6.6.6.m6.1.1.2.3" xref="A1.SS3.SSS0.P0.SPx1.p6.6.6.m6.1.1.2.3.cmml"><mi id="A1.SS3.SSS0.P0.SPx1.p6.6.6.m6.1.1.2.3.1" xref="A1.SS3.SSS0.P0.SPx1.p6.6.6.m6.1.1.2.3.1.cmml">log</mi><mo id="A1.SS3.SSS0.P0.SPx1.p6.6.6.m6.1.1.2.3a" lspace="0.167em" xref="A1.SS3.SSS0.P0.SPx1.p6.6.6.m6.1.1.2.3.cmml">⁡</mo><mi id="A1.SS3.SSS0.P0.SPx1.p6.6.6.m6.1.1.2.3.2" xref="A1.SS3.SSS0.P0.SPx1.p6.6.6.m6.1.1.2.3.2.cmml">n</mi></mrow></mrow><mi id="A1.SS3.SSS0.P0.SPx1.p6.6.6.m6.1.1.3" xref="A1.SS3.SSS0.P0.SPx1.p6.6.6.m6.1.1.3.cmml">ε</mi></mfrac><mo id="A1.SS3.SSS0.P0.SPx1.p6.6.6.m6.1.2.2.2" stretchy="false" xref="A1.SS3.SSS0.P0.SPx1.p6.6.6.m6.1.2.1.1.cmml">⌉</mo></mrow><annotation-xml encoding="MathML-Content" id="A1.SS3.SSS0.P0.SPx1.p6.6.6.m6.1b"><apply id="A1.SS3.SSS0.P0.SPx1.p6.6.6.m6.1.2.1.cmml" xref="A1.SS3.SSS0.P0.SPx1.p6.6.6.m6.1.2.2"><ceiling id="A1.SS3.SSS0.P0.SPx1.p6.6.6.m6.1.2.1.1.cmml" xref="A1.SS3.SSS0.P0.SPx1.p6.6.6.m6.1.2.2.1"></ceiling><apply id="A1.SS3.SSS0.P0.SPx1.p6.6.6.m6.1.1.cmml" xref="A1.SS3.SSS0.P0.SPx1.p6.6.6.m6.1.1"><divide id="A1.SS3.SSS0.P0.SPx1.p6.6.6.m6.1.1.1.cmml" xref="A1.SS3.SSS0.P0.SPx1.p6.6.6.m6.1.1"></divide><apply id="A1.SS3.SSS0.P0.SPx1.p6.6.6.m6.1.1.2.cmml" xref="A1.SS3.SSS0.P0.SPx1.p6.6.6.m6.1.1.2"><times id="A1.SS3.SSS0.P0.SPx1.p6.6.6.m6.1.1.2.1.cmml" xref="A1.SS3.SSS0.P0.SPx1.p6.6.6.m6.1.1.2.1"></times><cn id="A1.SS3.SSS0.P0.SPx1.p6.6.6.m6.1.1.2.2.cmml" type="integer" xref="A1.SS3.SSS0.P0.SPx1.p6.6.6.m6.1.1.2.2">32</cn><apply id="A1.SS3.SSS0.P0.SPx1.p6.6.6.m6.1.1.2.3.cmml" xref="A1.SS3.SSS0.P0.SPx1.p6.6.6.m6.1.1.2.3"><log id="A1.SS3.SSS0.P0.SPx1.p6.6.6.m6.1.1.2.3.1.cmml" xref="A1.SS3.SSS0.P0.SPx1.p6.6.6.m6.1.1.2.3.1"></log><ci id="A1.SS3.SSS0.P0.SPx1.p6.6.6.m6.1.1.2.3.2.cmml" xref="A1.SS3.SSS0.P0.SPx1.p6.6.6.m6.1.1.2.3.2">𝑛</ci></apply></apply><ci id="A1.SS3.SSS0.P0.SPx1.p6.6.6.m6.1.1.3.cmml" xref="A1.SS3.SSS0.P0.SPx1.p6.6.6.m6.1.1.3">𝜀</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS3.SSS0.P0.SPx1.p6.6.6.m6.1c">\lceil\frac{32\log n}{\varepsilon}\rceil</annotation><annotation encoding="application/x-llamapun" id="A1.SS3.SSS0.P0.SPx1.p6.6.6.m6.1d">⌈ divide start_ARG 32 roman_log italic_n end_ARG start_ARG italic_ε end_ARG ⌉</annotation></semantics></math>-hop neighbourhood acknowledges it as the current leader once the BFS-leader election algorithm is truncated. After truncation we spend <math alttext="\Theta(\frac{\log n}{\varepsilon})" class="ltx_Math" display="inline" id="A1.SS3.SSS0.P0.SPx1.p6.7.7.m7.1"><semantics id="A1.SS3.SSS0.P0.SPx1.p6.7.7.m7.1a"><mrow id="A1.SS3.SSS0.P0.SPx1.p6.7.7.m7.1.2" xref="A1.SS3.SSS0.P0.SPx1.p6.7.7.m7.1.2.cmml"><mi id="A1.SS3.SSS0.P0.SPx1.p6.7.7.m7.1.2.2" mathvariant="normal" xref="A1.SS3.SSS0.P0.SPx1.p6.7.7.m7.1.2.2.cmml">Θ</mi><mo id="A1.SS3.SSS0.P0.SPx1.p6.7.7.m7.1.2.1" xref="A1.SS3.SSS0.P0.SPx1.p6.7.7.m7.1.2.1.cmml">⁢</mo><mrow id="A1.SS3.SSS0.P0.SPx1.p6.7.7.m7.1.2.3.2" xref="A1.SS3.SSS0.P0.SPx1.p6.7.7.m7.1.1.cmml"><mo id="A1.SS3.SSS0.P0.SPx1.p6.7.7.m7.1.2.3.2.1" stretchy="false" xref="A1.SS3.SSS0.P0.SPx1.p6.7.7.m7.1.1.cmml">(</mo><mfrac id="A1.SS3.SSS0.P0.SPx1.p6.7.7.m7.1.1" xref="A1.SS3.SSS0.P0.SPx1.p6.7.7.m7.1.1.cmml"><mrow id="A1.SS3.SSS0.P0.SPx1.p6.7.7.m7.1.1.2" xref="A1.SS3.SSS0.P0.SPx1.p6.7.7.m7.1.1.2.cmml"><mi id="A1.SS3.SSS0.P0.SPx1.p6.7.7.m7.1.1.2.1" xref="A1.SS3.SSS0.P0.SPx1.p6.7.7.m7.1.1.2.1.cmml">log</mi><mo id="A1.SS3.SSS0.P0.SPx1.p6.7.7.m7.1.1.2a" lspace="0.167em" xref="A1.SS3.SSS0.P0.SPx1.p6.7.7.m7.1.1.2.cmml">⁡</mo><mi id="A1.SS3.SSS0.P0.SPx1.p6.7.7.m7.1.1.2.2" xref="A1.SS3.SSS0.P0.SPx1.p6.7.7.m7.1.1.2.2.cmml">n</mi></mrow><mi id="A1.SS3.SSS0.P0.SPx1.p6.7.7.m7.1.1.3" xref="A1.SS3.SSS0.P0.SPx1.p6.7.7.m7.1.1.3.cmml">ε</mi></mfrac><mo id="A1.SS3.SSS0.P0.SPx1.p6.7.7.m7.1.2.3.2.2" stretchy="false" xref="A1.SS3.SSS0.P0.SPx1.p6.7.7.m7.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="A1.SS3.SSS0.P0.SPx1.p6.7.7.m7.1b"><apply id="A1.SS3.SSS0.P0.SPx1.p6.7.7.m7.1.2.cmml" xref="A1.SS3.SSS0.P0.SPx1.p6.7.7.m7.1.2"><times id="A1.SS3.SSS0.P0.SPx1.p6.7.7.m7.1.2.1.cmml" xref="A1.SS3.SSS0.P0.SPx1.p6.7.7.m7.1.2.1"></times><ci id="A1.SS3.SSS0.P0.SPx1.p6.7.7.m7.1.2.2.cmml" xref="A1.SS3.SSS0.P0.SPx1.p6.7.7.m7.1.2.2">Θ</ci><apply id="A1.SS3.SSS0.P0.SPx1.p6.7.7.m7.1.1.cmml" xref="A1.SS3.SSS0.P0.SPx1.p6.7.7.m7.1.2.3.2"><divide id="A1.SS3.SSS0.P0.SPx1.p6.7.7.m7.1.1.1.cmml" xref="A1.SS3.SSS0.P0.SPx1.p6.7.7.m7.1.2.3.2"></divide><apply id="A1.SS3.SSS0.P0.SPx1.p6.7.7.m7.1.1.2.cmml" xref="A1.SS3.SSS0.P0.SPx1.p6.7.7.m7.1.1.2"><log id="A1.SS3.SSS0.P0.SPx1.p6.7.7.m7.1.1.2.1.cmml" xref="A1.SS3.SSS0.P0.SPx1.p6.7.7.m7.1.1.2.1"></log><ci id="A1.SS3.SSS0.P0.SPx1.p6.7.7.m7.1.1.2.2.cmml" xref="A1.SS3.SSS0.P0.SPx1.p6.7.7.m7.1.1.2.2">𝑛</ci></apply><ci id="A1.SS3.SSS0.P0.SPx1.p6.7.7.m7.1.1.3.cmml" xref="A1.SS3.SSS0.P0.SPx1.p6.7.7.m7.1.1.3">𝜀</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS3.SSS0.P0.SPx1.p6.7.7.m7.1c">\Theta(\frac{\log n}{\varepsilon})</annotation><annotation encoding="application/x-llamapun" id="A1.SS3.SSS0.P0.SPx1.p6.7.7.m7.1d">roman_Θ ( divide start_ARG roman_log italic_n end_ARG start_ARG italic_ε end_ARG )</annotation></semantics></math> rounds informing the elected leaders of their status by sending acknowledgements from the <math alttext="\lceil\frac{32\log n}{\varepsilon}\rceil" class="ltx_Math" display="inline" id="A1.SS3.SSS0.P0.SPx1.p6.8.8.m8.1"><semantics id="A1.SS3.SSS0.P0.SPx1.p6.8.8.m8.1a"><mrow id="A1.SS3.SSS0.P0.SPx1.p6.8.8.m8.1.2.2" xref="A1.SS3.SSS0.P0.SPx1.p6.8.8.m8.1.2.1.cmml"><mo id="A1.SS3.SSS0.P0.SPx1.p6.8.8.m8.1.2.2.1" stretchy="false" xref="A1.SS3.SSS0.P0.SPx1.p6.8.8.m8.1.2.1.1.cmml">⌈</mo><mfrac id="A1.SS3.SSS0.P0.SPx1.p6.8.8.m8.1.1" xref="A1.SS3.SSS0.P0.SPx1.p6.8.8.m8.1.1.cmml"><mrow id="A1.SS3.SSS0.P0.SPx1.p6.8.8.m8.1.1.2" xref="A1.SS3.SSS0.P0.SPx1.p6.8.8.m8.1.1.2.cmml"><mn id="A1.SS3.SSS0.P0.SPx1.p6.8.8.m8.1.1.2.2" xref="A1.SS3.SSS0.P0.SPx1.p6.8.8.m8.1.1.2.2.cmml">32</mn><mo id="A1.SS3.SSS0.P0.SPx1.p6.8.8.m8.1.1.2.1" lspace="0.167em" xref="A1.SS3.SSS0.P0.SPx1.p6.8.8.m8.1.1.2.1.cmml">⁢</mo><mrow id="A1.SS3.SSS0.P0.SPx1.p6.8.8.m8.1.1.2.3" xref="A1.SS3.SSS0.P0.SPx1.p6.8.8.m8.1.1.2.3.cmml"><mi id="A1.SS3.SSS0.P0.SPx1.p6.8.8.m8.1.1.2.3.1" xref="A1.SS3.SSS0.P0.SPx1.p6.8.8.m8.1.1.2.3.1.cmml">log</mi><mo id="A1.SS3.SSS0.P0.SPx1.p6.8.8.m8.1.1.2.3a" lspace="0.167em" xref="A1.SS3.SSS0.P0.SPx1.p6.8.8.m8.1.1.2.3.cmml">⁡</mo><mi id="A1.SS3.SSS0.P0.SPx1.p6.8.8.m8.1.1.2.3.2" xref="A1.SS3.SSS0.P0.SPx1.p6.8.8.m8.1.1.2.3.2.cmml">n</mi></mrow></mrow><mi id="A1.SS3.SSS0.P0.SPx1.p6.8.8.m8.1.1.3" xref="A1.SS3.SSS0.P0.SPx1.p6.8.8.m8.1.1.3.cmml">ε</mi></mfrac><mo id="A1.SS3.SSS0.P0.SPx1.p6.8.8.m8.1.2.2.2" stretchy="false" xref="A1.SS3.SSS0.P0.SPx1.p6.8.8.m8.1.2.1.1.cmml">⌉</mo></mrow><annotation-xml encoding="MathML-Content" id="A1.SS3.SSS0.P0.SPx1.p6.8.8.m8.1b"><apply id="A1.SS3.SSS0.P0.SPx1.p6.8.8.m8.1.2.1.cmml" xref="A1.SS3.SSS0.P0.SPx1.p6.8.8.m8.1.2.2"><ceiling id="A1.SS3.SSS0.P0.SPx1.p6.8.8.m8.1.2.1.1.cmml" xref="A1.SS3.SSS0.P0.SPx1.p6.8.8.m8.1.2.2.1"></ceiling><apply id="A1.SS3.SSS0.P0.SPx1.p6.8.8.m8.1.1.cmml" xref="A1.SS3.SSS0.P0.SPx1.p6.8.8.m8.1.1"><divide id="A1.SS3.SSS0.P0.SPx1.p6.8.8.m8.1.1.1.cmml" 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encoding="application/x-llamapun" id="A1.SS3.SSS0.P0.SPx1.p6.8.8.m8.1d">⌈ divide start_ARG 32 roman_log italic_n end_ARG start_ARG italic_ε end_ARG ⌉</annotation></semantics></math>-hop neighbourhoods. Once a leader acknowledges that it is a leader, it broadcasts this information to its entire <math alttext="\lceil\frac{32\log n}{\varepsilon}\rceil" class="ltx_Math" display="inline" id="A1.SS3.SSS0.P0.SPx1.p6.9.9.m9.1"><semantics id="A1.SS3.SSS0.P0.SPx1.p6.9.9.m9.1a"><mrow id="A1.SS3.SSS0.P0.SPx1.p6.9.9.m9.1.2.2" xref="A1.SS3.SSS0.P0.SPx1.p6.9.9.m9.1.2.1.cmml"><mo id="A1.SS3.SSS0.P0.SPx1.p6.9.9.m9.1.2.2.1" stretchy="false" xref="A1.SS3.SSS0.P0.SPx1.p6.9.9.m9.1.2.1.1.cmml">⌈</mo><mfrac id="A1.SS3.SSS0.P0.SPx1.p6.9.9.m9.1.1" xref="A1.SS3.SSS0.P0.SPx1.p6.9.9.m9.1.1.cmml"><mrow id="A1.SS3.SSS0.P0.SPx1.p6.9.9.m9.1.1.2" xref="A1.SS3.SSS0.P0.SPx1.p6.9.9.m9.1.1.2.cmml"><mn id="A1.SS3.SSS0.P0.SPx1.p6.9.9.m9.1.1.2.2" xref="A1.SS3.SSS0.P0.SPx1.p6.9.9.m9.1.1.2.2.cmml">32</mn><mo id="A1.SS3.SSS0.P0.SPx1.p6.9.9.m9.1.1.2.1" lspace="0.167em" xref="A1.SS3.SSS0.P0.SPx1.p6.9.9.m9.1.1.2.1.cmml">⁢</mo><mrow id="A1.SS3.SSS0.P0.SPx1.p6.9.9.m9.1.1.2.3" xref="A1.SS3.SSS0.P0.SPx1.p6.9.9.m9.1.1.2.3.cmml"><mi id="A1.SS3.SSS0.P0.SPx1.p6.9.9.m9.1.1.2.3.1" xref="A1.SS3.SSS0.P0.SPx1.p6.9.9.m9.1.1.2.3.1.cmml">log</mi><mo id="A1.SS3.SSS0.P0.SPx1.p6.9.9.m9.1.1.2.3a" lspace="0.167em" xref="A1.SS3.SSS0.P0.SPx1.p6.9.9.m9.1.1.2.3.cmml">⁡</mo><mi id="A1.SS3.SSS0.P0.SPx1.p6.9.9.m9.1.1.2.3.2" xref="A1.SS3.SSS0.P0.SPx1.p6.9.9.m9.1.1.2.3.2.cmml">n</mi></mrow></mrow><mi id="A1.SS3.SSS0.P0.SPx1.p6.9.9.m9.1.1.3" xref="A1.SS3.SSS0.P0.SPx1.p6.9.9.m9.1.1.3.cmml">ε</mi></mfrac><mo id="A1.SS3.SSS0.P0.SPx1.p6.9.9.m9.1.2.2.2" stretchy="false" xref="A1.SS3.SSS0.P0.SPx1.p6.9.9.m9.1.2.1.1.cmml">⌉</mo></mrow><annotation-xml encoding="MathML-Content" id="A1.SS3.SSS0.P0.SPx1.p6.9.9.m9.1b"><apply id="A1.SS3.SSS0.P0.SPx1.p6.9.9.m9.1.2.1.cmml" xref="A1.SS3.SSS0.P0.SPx1.p6.9.9.m9.1.2.2"><ceiling id="A1.SS3.SSS0.P0.SPx1.p6.9.9.m9.1.2.1.1.cmml" xref="A1.SS3.SSS0.P0.SPx1.p6.9.9.m9.1.2.2.1"></ceiling><apply id="A1.SS3.SSS0.P0.SPx1.p6.9.9.m9.1.1.cmml" xref="A1.SS3.SSS0.P0.SPx1.p6.9.9.m9.1.1"><divide id="A1.SS3.SSS0.P0.SPx1.p6.9.9.m9.1.1.1.cmml" xref="A1.SS3.SSS0.P0.SPx1.p6.9.9.m9.1.1"></divide><apply id="A1.SS3.SSS0.P0.SPx1.p6.9.9.m9.1.1.2.cmml" xref="A1.SS3.SSS0.P0.SPx1.p6.9.9.m9.1.1.2"><times id="A1.SS3.SSS0.P0.SPx1.p6.9.9.m9.1.1.2.1.cmml" xref="A1.SS3.SSS0.P0.SPx1.p6.9.9.m9.1.1.2.1"></times><cn id="A1.SS3.SSS0.P0.SPx1.p6.9.9.m9.1.1.2.2.cmml" type="integer" xref="A1.SS3.SSS0.P0.SPx1.p6.9.9.m9.1.1.2.2">32</cn><apply id="A1.SS3.SSS0.P0.SPx1.p6.9.9.m9.1.1.2.3.cmml" xref="A1.SS3.SSS0.P0.SPx1.p6.9.9.m9.1.1.2.3"><log id="A1.SS3.SSS0.P0.SPx1.p6.9.9.m9.1.1.2.3.1.cmml" xref="A1.SS3.SSS0.P0.SPx1.p6.9.9.m9.1.1.2.3.1"></log><ci id="A1.SS3.SSS0.P0.SPx1.p6.9.9.m9.1.1.2.3.2.cmml" xref="A1.SS3.SSS0.P0.SPx1.p6.9.9.m9.1.1.2.3.2">𝑛</ci></apply></apply><ci id="A1.SS3.SSS0.P0.SPx1.p6.9.9.m9.1.1.3.cmml" xref="A1.SS3.SSS0.P0.SPx1.p6.9.9.m9.1.1.3">𝜀</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS3.SSS0.P0.SPx1.p6.9.9.m9.1c">\lceil\frac{32\log n}{\varepsilon}\rceil</annotation><annotation encoding="application/x-llamapun" id="A1.SS3.SSS0.P0.SPx1.p6.9.9.m9.1d">⌈ divide start_ARG 32 roman_log italic_n end_ARG start_ARG italic_ε end_ARG ⌉</annotation></semantics></math>-hop neighbourhood. Note that it might be that some parts of the graph become leaderless – i.e. they did not sit in the <math alttext="\lceil\frac{32\log n}{\varepsilon}\rceil" class="ltx_Math" display="inline" id="A1.SS3.SSS0.P0.SPx1.p6.10.10.m10.1"><semantics id="A1.SS3.SSS0.P0.SPx1.p6.10.10.m10.1a"><mrow id="A1.SS3.SSS0.P0.SPx1.p6.10.10.m10.1.2.2" xref="A1.SS3.SSS0.P0.SPx1.p6.10.10.m10.1.2.1.cmml"><mo id="A1.SS3.SSS0.P0.SPx1.p6.10.10.m10.1.2.2.1" stretchy="false" xref="A1.SS3.SSS0.P0.SPx1.p6.10.10.m10.1.2.1.1.cmml">⌈</mo><mfrac id="A1.SS3.SSS0.P0.SPx1.p6.10.10.m10.1.1" xref="A1.SS3.SSS0.P0.SPx1.p6.10.10.m10.1.1.cmml"><mrow id="A1.SS3.SSS0.P0.SPx1.p6.10.10.m10.1.1.2" xref="A1.SS3.SSS0.P0.SPx1.p6.10.10.m10.1.1.2.cmml"><mn id="A1.SS3.SSS0.P0.SPx1.p6.10.10.m10.1.1.2.2" xref="A1.SS3.SSS0.P0.SPx1.p6.10.10.m10.1.1.2.2.cmml">32</mn><mo id="A1.SS3.SSS0.P0.SPx1.p6.10.10.m10.1.1.2.1" lspace="0.167em" xref="A1.SS3.SSS0.P0.SPx1.p6.10.10.m10.1.1.2.1.cmml">⁢</mo><mrow id="A1.SS3.SSS0.P0.SPx1.p6.10.10.m10.1.1.2.3" xref="A1.SS3.SSS0.P0.SPx1.p6.10.10.m10.1.1.2.3.cmml"><mi id="A1.SS3.SSS0.P0.SPx1.p6.10.10.m10.1.1.2.3.1" xref="A1.SS3.SSS0.P0.SPx1.p6.10.10.m10.1.1.2.3.1.cmml">log</mi><mo id="A1.SS3.SSS0.P0.SPx1.p6.10.10.m10.1.1.2.3a" lspace="0.167em" xref="A1.SS3.SSS0.P0.SPx1.p6.10.10.m10.1.1.2.3.cmml">⁡</mo><mi id="A1.SS3.SSS0.P0.SPx1.p6.10.10.m10.1.1.2.3.2" xref="A1.SS3.SSS0.P0.SPx1.p6.10.10.m10.1.1.2.3.2.cmml">n</mi></mrow></mrow><mi id="A1.SS3.SSS0.P0.SPx1.p6.10.10.m10.1.1.3" xref="A1.SS3.SSS0.P0.SPx1.p6.10.10.m10.1.1.3.cmml">ε</mi></mfrac><mo id="A1.SS3.SSS0.P0.SPx1.p6.10.10.m10.1.2.2.2" stretchy="false" xref="A1.SS3.SSS0.P0.SPx1.p6.10.10.m10.1.2.1.1.cmml">⌉</mo></mrow><annotation-xml encoding="MathML-Content" id="A1.SS3.SSS0.P0.SPx1.p6.10.10.m10.1b"><apply id="A1.SS3.SSS0.P0.SPx1.p6.10.10.m10.1.2.1.cmml" xref="A1.SS3.SSS0.P0.SPx1.p6.10.10.m10.1.2.2"><ceiling id="A1.SS3.SSS0.P0.SPx1.p6.10.10.m10.1.2.1.1.cmml" xref="A1.SS3.SSS0.P0.SPx1.p6.10.10.m10.1.2.2.1"></ceiling><apply id="A1.SS3.SSS0.P0.SPx1.p6.10.10.m10.1.1.cmml" xref="A1.SS3.SSS0.P0.SPx1.p6.10.10.m10.1.1"><divide id="A1.SS3.SSS0.P0.SPx1.p6.10.10.m10.1.1.1.cmml" xref="A1.SS3.SSS0.P0.SPx1.p6.10.10.m10.1.1"></divide><apply id="A1.SS3.SSS0.P0.SPx1.p6.10.10.m10.1.1.2.cmml" xref="A1.SS3.SSS0.P0.SPx1.p6.10.10.m10.1.1.2"><times id="A1.SS3.SSS0.P0.SPx1.p6.10.10.m10.1.1.2.1.cmml" xref="A1.SS3.SSS0.P0.SPx1.p6.10.10.m10.1.1.2.1"></times><cn id="A1.SS3.SSS0.P0.SPx1.p6.10.10.m10.1.1.2.2.cmml" type="integer" xref="A1.SS3.SSS0.P0.SPx1.p6.10.10.m10.1.1.2.2">32</cn><apply id="A1.SS3.SSS0.P0.SPx1.p6.10.10.m10.1.1.2.3.cmml" xref="A1.SS3.SSS0.P0.SPx1.p6.10.10.m10.1.1.2.3"><log id="A1.SS3.SSS0.P0.SPx1.p6.10.10.m10.1.1.2.3.1.cmml" xref="A1.SS3.SSS0.P0.SPx1.p6.10.10.m10.1.1.2.3.1"></log><ci id="A1.SS3.SSS0.P0.SPx1.p6.10.10.m10.1.1.2.3.2.cmml" xref="A1.SS3.SSS0.P0.SPx1.p6.10.10.m10.1.1.2.3.2">𝑛</ci></apply></apply><ci id="A1.SS3.SSS0.P0.SPx1.p6.10.10.m10.1.1.3.cmml" xref="A1.SS3.SSS0.P0.SPx1.p6.10.10.m10.1.1.3">𝜀</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS3.SSS0.P0.SPx1.p6.10.10.m10.1c">\lceil\frac{32\log n}{\varepsilon}\rceil</annotation><annotation encoding="application/x-llamapun" id="A1.SS3.SSS0.P0.SPx1.p6.10.10.m10.1d">⌈ divide start_ARG 32 roman_log italic_n end_ARG start_ARG italic_ε end_ARG ⌉</annotation></semantics></math>-hop neighbourhood of an elected leader. If this is the case, these vertices never receive information that the leader election succeeded, and once sufficient time has passed, they simply output <math alttext="0" class="ltx_Math" display="inline" id="A1.SS3.SSS0.P0.SPx1.p6.11.11.m11.1"><semantics id="A1.SS3.SSS0.P0.SPx1.p6.11.11.m11.1a"><mn id="A1.SS3.SSS0.P0.SPx1.p6.11.11.m11.1.1" xref="A1.SS3.SSS0.P0.SPx1.p6.11.11.m11.1.1.cmml">0</mn><annotation-xml encoding="MathML-Content" id="A1.SS3.SSS0.P0.SPx1.p6.11.11.m11.1b"><cn id="A1.SS3.SSS0.P0.SPx1.p6.11.11.m11.1.1.cmml" type="integer" xref="A1.SS3.SSS0.P0.SPx1.p6.11.11.m11.1.1">0</cn></annotation-xml></semantics></math>.</span></p> </div> <div class="ltx_para" id="A1.SS3.SSS0.P0.SPx1.p7"> <p class="ltx_p" id="A1.SS3.SSS0.P0.SPx1.p7.6"><span class="ltx_text" id="A1.SS3.SSS0.P0.SPx1.p7.6.6">If a leader has fractional out-degree under the <math alttext="\eta" class="ltx_Math" display="inline" id="A1.SS3.SSS0.P0.SPx1.p7.1.1.m1.1"><semantics id="A1.SS3.SSS0.P0.SPx1.p7.1.1.m1.1a"><mi id="A1.SS3.SSS0.P0.SPx1.p7.1.1.m1.1.1" xref="A1.SS3.SSS0.P0.SPx1.p7.1.1.m1.1.1.cmml">η</mi><annotation-xml encoding="MathML-Content" id="A1.SS3.SSS0.P0.SPx1.p7.1.1.m1.1b"><ci id="A1.SS3.SSS0.P0.SPx1.p7.1.1.m1.1.1.cmml" xref="A1.SS3.SSS0.P0.SPx1.p7.1.1.m1.1.1">𝜂</ci></annotation-xml><annotation encoding="application/x-tex" id="A1.SS3.SSS0.P0.SPx1.p7.1.1.m1.1c">\eta</annotation><annotation encoding="application/x-llamapun" id="A1.SS3.SSS0.P0.SPx1.p7.1.1.m1.1d">italic_η</annotation></semantics></math>-fair orientation that is smaller than <math alttext="\tilde{D}" class="ltx_Math" display="inline" id="A1.SS3.SSS0.P0.SPx1.p7.2.2.m2.1"><semantics id="A1.SS3.SSS0.P0.SPx1.p7.2.2.m2.1a"><mover accent="true" id="A1.SS3.SSS0.P0.SPx1.p7.2.2.m2.1.1" xref="A1.SS3.SSS0.P0.SPx1.p7.2.2.m2.1.1.cmml"><mi id="A1.SS3.SSS0.P0.SPx1.p7.2.2.m2.1.1.2" xref="A1.SS3.SSS0.P0.SPx1.p7.2.2.m2.1.1.2.cmml">D</mi><mo id="A1.SS3.SSS0.P0.SPx1.p7.2.2.m2.1.1.1" xref="A1.SS3.SSS0.P0.SPx1.p7.2.2.m2.1.1.1.cmml">~</mo></mover><annotation-xml encoding="MathML-Content" id="A1.SS3.SSS0.P0.SPx1.p7.2.2.m2.1b"><apply id="A1.SS3.SSS0.P0.SPx1.p7.2.2.m2.1.1.cmml" xref="A1.SS3.SSS0.P0.SPx1.p7.2.2.m2.1.1"><ci id="A1.SS3.SSS0.P0.SPx1.p7.2.2.m2.1.1.1.cmml" xref="A1.SS3.SSS0.P0.SPx1.p7.2.2.m2.1.1.1">~</ci><ci id="A1.SS3.SSS0.P0.SPx1.p7.2.2.m2.1.1.2.cmml" xref="A1.SS3.SSS0.P0.SPx1.p7.2.2.m2.1.1.2">𝐷</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS3.SSS0.P0.SPx1.p7.2.2.m2.1c">\tilde{D}</annotation><annotation encoding="application/x-llamapun" id="A1.SS3.SSS0.P0.SPx1.p7.2.2.m2.1d">over~ start_ARG italic_D end_ARG</annotation></semantics></math>, the leader will ask its <math alttext="\lceil\frac{32\log n}{\varepsilon}\rceil" class="ltx_Math" display="inline" id="A1.SS3.SSS0.P0.SPx1.p7.3.3.m3.1"><semantics id="A1.SS3.SSS0.P0.SPx1.p7.3.3.m3.1a"><mrow id="A1.SS3.SSS0.P0.SPx1.p7.3.3.m3.1.2.2" xref="A1.SS3.SSS0.P0.SPx1.p7.3.3.m3.1.2.1.cmml"><mo id="A1.SS3.SSS0.P0.SPx1.p7.3.3.m3.1.2.2.1" stretchy="false" xref="A1.SS3.SSS0.P0.SPx1.p7.3.3.m3.1.2.1.1.cmml">⌈</mo><mfrac id="A1.SS3.SSS0.P0.SPx1.p7.3.3.m3.1.1" xref="A1.SS3.SSS0.P0.SPx1.p7.3.3.m3.1.1.cmml"><mrow id="A1.SS3.SSS0.P0.SPx1.p7.3.3.m3.1.1.2" xref="A1.SS3.SSS0.P0.SPx1.p7.3.3.m3.1.1.2.cmml"><mn id="A1.SS3.SSS0.P0.SPx1.p7.3.3.m3.1.1.2.2" xref="A1.SS3.SSS0.P0.SPx1.p7.3.3.m3.1.1.2.2.cmml">32</mn><mo id="A1.SS3.SSS0.P0.SPx1.p7.3.3.m3.1.1.2.1" lspace="0.167em" xref="A1.SS3.SSS0.P0.SPx1.p7.3.3.m3.1.1.2.1.cmml">⁢</mo><mrow id="A1.SS3.SSS0.P0.SPx1.p7.3.3.m3.1.1.2.3" xref="A1.SS3.SSS0.P0.SPx1.p7.3.3.m3.1.1.2.3.cmml"><mi id="A1.SS3.SSS0.P0.SPx1.p7.3.3.m3.1.1.2.3.1" xref="A1.SS3.SSS0.P0.SPx1.p7.3.3.m3.1.1.2.3.1.cmml">log</mi><mo id="A1.SS3.SSS0.P0.SPx1.p7.3.3.m3.1.1.2.3a" lspace="0.167em" xref="A1.SS3.SSS0.P0.SPx1.p7.3.3.m3.1.1.2.3.cmml">⁡</mo><mi id="A1.SS3.SSS0.P0.SPx1.p7.3.3.m3.1.1.2.3.2" xref="A1.SS3.SSS0.P0.SPx1.p7.3.3.m3.1.1.2.3.2.cmml">n</mi></mrow></mrow><mi id="A1.SS3.SSS0.P0.SPx1.p7.3.3.m3.1.1.3" xref="A1.SS3.SSS0.P0.SPx1.p7.3.3.m3.1.1.3.cmml">ε</mi></mfrac><mo id="A1.SS3.SSS0.P0.SPx1.p7.3.3.m3.1.2.2.2" stretchy="false" xref="A1.SS3.SSS0.P0.SPx1.p7.3.3.m3.1.2.1.1.cmml">⌉</mo></mrow><annotation-xml encoding="MathML-Content" id="A1.SS3.SSS0.P0.SPx1.p7.3.3.m3.1b"><apply id="A1.SS3.SSS0.P0.SPx1.p7.3.3.m3.1.2.1.cmml" xref="A1.SS3.SSS0.P0.SPx1.p7.3.3.m3.1.2.2"><ceiling id="A1.SS3.SSS0.P0.SPx1.p7.3.3.m3.1.2.1.1.cmml" xref="A1.SS3.SSS0.P0.SPx1.p7.3.3.m3.1.2.2.1"></ceiling><apply id="A1.SS3.SSS0.P0.SPx1.p7.3.3.m3.1.1.cmml" xref="A1.SS3.SSS0.P0.SPx1.p7.3.3.m3.1.1"><divide id="A1.SS3.SSS0.P0.SPx1.p7.3.3.m3.1.1.1.cmml" xref="A1.SS3.SSS0.P0.SPx1.p7.3.3.m3.1.1"></divide><apply id="A1.SS3.SSS0.P0.SPx1.p7.3.3.m3.1.1.2.cmml" xref="A1.SS3.SSS0.P0.SPx1.p7.3.3.m3.1.1.2"><times id="A1.SS3.SSS0.P0.SPx1.p7.3.3.m3.1.1.2.1.cmml" xref="A1.SS3.SSS0.P0.SPx1.p7.3.3.m3.1.1.2.1"></times><cn id="A1.SS3.SSS0.P0.SPx1.p7.3.3.m3.1.1.2.2.cmml" type="integer" xref="A1.SS3.SSS0.P0.SPx1.p7.3.3.m3.1.1.2.2">32</cn><apply id="A1.SS3.SSS0.P0.SPx1.p7.3.3.m3.1.1.2.3.cmml" xref="A1.SS3.SSS0.P0.SPx1.p7.3.3.m3.1.1.2.3"><log id="A1.SS3.SSS0.P0.SPx1.p7.3.3.m3.1.1.2.3.1.cmml" xref="A1.SS3.SSS0.P0.SPx1.p7.3.3.m3.1.1.2.3.1"></log><ci id="A1.SS3.SSS0.P0.SPx1.p7.3.3.m3.1.1.2.3.2.cmml" xref="A1.SS3.SSS0.P0.SPx1.p7.3.3.m3.1.1.2.3.2">𝑛</ci></apply></apply><ci id="A1.SS3.SSS0.P0.SPx1.p7.3.3.m3.1.1.3.cmml" xref="A1.SS3.SSS0.P0.SPx1.p7.3.3.m3.1.1.3">𝜀</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS3.SSS0.P0.SPx1.p7.3.3.m3.1c">\lceil\frac{32\log n}{\varepsilon}\rceil</annotation><annotation encoding="application/x-llamapun" id="A1.SS3.SSS0.P0.SPx1.p7.3.3.m3.1d">⌈ divide start_ARG 32 roman_log italic_n end_ARG start_ARG italic_ε end_ARG ⌉</annotation></semantics></math>-hop neighbourhood to all output <math alttext="0" class="ltx_Math" display="inline" id="A1.SS3.SSS0.P0.SPx1.p7.4.4.m4.1"><semantics id="A1.SS3.SSS0.P0.SPx1.p7.4.4.m4.1a"><mn id="A1.SS3.SSS0.P0.SPx1.p7.4.4.m4.1.1" xref="A1.SS3.SSS0.P0.SPx1.p7.4.4.m4.1.1.cmml">0</mn><annotation-xml encoding="MathML-Content" id="A1.SS3.SSS0.P0.SPx1.p7.4.4.m4.1b"><cn id="A1.SS3.SSS0.P0.SPx1.p7.4.4.m4.1.1.cmml" type="integer" xref="A1.SS3.SSS0.P0.SPx1.p7.4.4.m4.1.1">0</cn></annotation-xml></semantics></math>. If this is not the case, we say the leader is <span class="ltx_text ltx_markedasmath ltx_font_italic" id="A1.SS3.SSS0.P0.SPx1.p7.6.6.1">active</span>. Note that we later show that at least one elected leader stays active if <math alttext="\tilde{D}\leq\rho(G)" class="ltx_Math" display="inline" id="A1.SS3.SSS0.P0.SPx1.p7.6.6.m6.1"><semantics id="A1.SS3.SSS0.P0.SPx1.p7.6.6.m6.1a"><mrow id="A1.SS3.SSS0.P0.SPx1.p7.6.6.m6.1.2" xref="A1.SS3.SSS0.P0.SPx1.p7.6.6.m6.1.2.cmml"><mover accent="true" id="A1.SS3.SSS0.P0.SPx1.p7.6.6.m6.1.2.2" xref="A1.SS3.SSS0.P0.SPx1.p7.6.6.m6.1.2.2.cmml"><mi id="A1.SS3.SSS0.P0.SPx1.p7.6.6.m6.1.2.2.2" xref="A1.SS3.SSS0.P0.SPx1.p7.6.6.m6.1.2.2.2.cmml">D</mi><mo id="A1.SS3.SSS0.P0.SPx1.p7.6.6.m6.1.2.2.1" xref="A1.SS3.SSS0.P0.SPx1.p7.6.6.m6.1.2.2.1.cmml">~</mo></mover><mo id="A1.SS3.SSS0.P0.SPx1.p7.6.6.m6.1.2.1" xref="A1.SS3.SSS0.P0.SPx1.p7.6.6.m6.1.2.1.cmml">≤</mo><mrow id="A1.SS3.SSS0.P0.SPx1.p7.6.6.m6.1.2.3" xref="A1.SS3.SSS0.P0.SPx1.p7.6.6.m6.1.2.3.cmml"><mi id="A1.SS3.SSS0.P0.SPx1.p7.6.6.m6.1.2.3.2" xref="A1.SS3.SSS0.P0.SPx1.p7.6.6.m6.1.2.3.2.cmml">ρ</mi><mo id="A1.SS3.SSS0.P0.SPx1.p7.6.6.m6.1.2.3.1" xref="A1.SS3.SSS0.P0.SPx1.p7.6.6.m6.1.2.3.1.cmml">⁢</mo><mrow id="A1.SS3.SSS0.P0.SPx1.p7.6.6.m6.1.2.3.3.2" xref="A1.SS3.SSS0.P0.SPx1.p7.6.6.m6.1.2.3.cmml"><mo id="A1.SS3.SSS0.P0.SPx1.p7.6.6.m6.1.2.3.3.2.1" stretchy="false" xref="A1.SS3.SSS0.P0.SPx1.p7.6.6.m6.1.2.3.cmml">(</mo><mi id="A1.SS3.SSS0.P0.SPx1.p7.6.6.m6.1.1" xref="A1.SS3.SSS0.P0.SPx1.p7.6.6.m6.1.1.cmml">G</mi><mo id="A1.SS3.SSS0.P0.SPx1.p7.6.6.m6.1.2.3.3.2.2" stretchy="false" xref="A1.SS3.SSS0.P0.SPx1.p7.6.6.m6.1.2.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="A1.SS3.SSS0.P0.SPx1.p7.6.6.m6.1b"><apply id="A1.SS3.SSS0.P0.SPx1.p7.6.6.m6.1.2.cmml" xref="A1.SS3.SSS0.P0.SPx1.p7.6.6.m6.1.2"><leq id="A1.SS3.SSS0.P0.SPx1.p7.6.6.m6.1.2.1.cmml" xref="A1.SS3.SSS0.P0.SPx1.p7.6.6.m6.1.2.1"></leq><apply id="A1.SS3.SSS0.P0.SPx1.p7.6.6.m6.1.2.2.cmml" xref="A1.SS3.SSS0.P0.SPx1.p7.6.6.m6.1.2.2"><ci id="A1.SS3.SSS0.P0.SPx1.p7.6.6.m6.1.2.2.1.cmml" xref="A1.SS3.SSS0.P0.SPx1.p7.6.6.m6.1.2.2.1">~</ci><ci id="A1.SS3.SSS0.P0.SPx1.p7.6.6.m6.1.2.2.2.cmml" xref="A1.SS3.SSS0.P0.SPx1.p7.6.6.m6.1.2.2.2">𝐷</ci></apply><apply id="A1.SS3.SSS0.P0.SPx1.p7.6.6.m6.1.2.3.cmml" xref="A1.SS3.SSS0.P0.SPx1.p7.6.6.m6.1.2.3"><times id="A1.SS3.SSS0.P0.SPx1.p7.6.6.m6.1.2.3.1.cmml" xref="A1.SS3.SSS0.P0.SPx1.p7.6.6.m6.1.2.3.1"></times><ci id="A1.SS3.SSS0.P0.SPx1.p7.6.6.m6.1.2.3.2.cmml" xref="A1.SS3.SSS0.P0.SPx1.p7.6.6.m6.1.2.3.2">𝜌</ci><ci id="A1.SS3.SSS0.P0.SPx1.p7.6.6.m6.1.1.cmml" xref="A1.SS3.SSS0.P0.SPx1.p7.6.6.m6.1.1">𝐺</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS3.SSS0.P0.SPx1.p7.6.6.m6.1c">\tilde{D}\leq\rho(G)</annotation><annotation encoding="application/x-llamapun" id="A1.SS3.SSS0.P0.SPx1.p7.6.6.m6.1d">over~ start_ARG italic_D end_ARG ≤ italic_ρ ( italic_G )</annotation></semantics></math>.</span></p> </div> <div class="ltx_para" id="A1.SS3.SSS0.P0.SPx1.p8"> <p class="ltx_p" id="A1.SS3.SSS0.P0.SPx1.p8.14"><span class="ltx_text" id="A1.SS3.SSS0.P0.SPx1.p8.14.14">For each leader <math alttext="\ell" class="ltx_Math" display="inline" id="A1.SS3.SSS0.P0.SPx1.p8.1.1.m1.1"><semantics id="A1.SS3.SSS0.P0.SPx1.p8.1.1.m1.1a"><mi id="A1.SS3.SSS0.P0.SPx1.p8.1.1.m1.1.1" mathvariant="normal" xref="A1.SS3.SSS0.P0.SPx1.p8.1.1.m1.1.1.cmml">ℓ</mi><annotation-xml encoding="MathML-Content" id="A1.SS3.SSS0.P0.SPx1.p8.1.1.m1.1b"><ci id="A1.SS3.SSS0.P0.SPx1.p8.1.1.m1.1.1.cmml" xref="A1.SS3.SSS0.P0.SPx1.p8.1.1.m1.1.1">ℓ</ci></annotation-xml><annotation encoding="application/x-tex" id="A1.SS3.SSS0.P0.SPx1.p8.1.1.m1.1c">\ell</annotation><annotation encoding="application/x-llamapun" id="A1.SS3.SSS0.P0.SPx1.p8.1.1.m1.1d">roman_ℓ</annotation></semantics></math>, we then compute an <math alttext="\eta" class="ltx_Math" display="inline" id="A1.SS3.SSS0.P0.SPx1.p8.2.2.m2.1"><semantics id="A1.SS3.SSS0.P0.SPx1.p8.2.2.m2.1a"><mi id="A1.SS3.SSS0.P0.SPx1.p8.2.2.m2.1.1" xref="A1.SS3.SSS0.P0.SPx1.p8.2.2.m2.1.1.cmml">η</mi><annotation-xml encoding="MathML-Content" id="A1.SS3.SSS0.P0.SPx1.p8.2.2.m2.1b"><ci id="A1.SS3.SSS0.P0.SPx1.p8.2.2.m2.1.1.cmml" xref="A1.SS3.SSS0.P0.SPx1.p8.2.2.m2.1.1">𝜂</ci></annotation-xml><annotation encoding="application/x-tex" id="A1.SS3.SSS0.P0.SPx1.p8.2.2.m2.1c">\eta</annotation><annotation encoding="application/x-llamapun" id="A1.SS3.SSS0.P0.SPx1.p8.2.2.m2.1d">italic_η</annotation></semantics></math>-fair out-orientation of the <math alttext="\lceil\frac{32\log n}{\varepsilon}\rceil" class="ltx_Math" display="inline" id="A1.SS3.SSS0.P0.SPx1.p8.3.3.m3.1"><semantics id="A1.SS3.SSS0.P0.SPx1.p8.3.3.m3.1a"><mrow id="A1.SS3.SSS0.P0.SPx1.p8.3.3.m3.1.2.2" xref="A1.SS3.SSS0.P0.SPx1.p8.3.3.m3.1.2.1.cmml"><mo id="A1.SS3.SSS0.P0.SPx1.p8.3.3.m3.1.2.2.1" stretchy="false" xref="A1.SS3.SSS0.P0.SPx1.p8.3.3.m3.1.2.1.1.cmml">⌈</mo><mfrac id="A1.SS3.SSS0.P0.SPx1.p8.3.3.m3.1.1" xref="A1.SS3.SSS0.P0.SPx1.p8.3.3.m3.1.1.cmml"><mrow id="A1.SS3.SSS0.P0.SPx1.p8.3.3.m3.1.1.2" xref="A1.SS3.SSS0.P0.SPx1.p8.3.3.m3.1.1.2.cmml"><mn id="A1.SS3.SSS0.P0.SPx1.p8.3.3.m3.1.1.2.2" xref="A1.SS3.SSS0.P0.SPx1.p8.3.3.m3.1.1.2.2.cmml">32</mn><mo id="A1.SS3.SSS0.P0.SPx1.p8.3.3.m3.1.1.2.1" lspace="0.167em" xref="A1.SS3.SSS0.P0.SPx1.p8.3.3.m3.1.1.2.1.cmml">⁢</mo><mrow id="A1.SS3.SSS0.P0.SPx1.p8.3.3.m3.1.1.2.3" xref="A1.SS3.SSS0.P0.SPx1.p8.3.3.m3.1.1.2.3.cmml"><mi id="A1.SS3.SSS0.P0.SPx1.p8.3.3.m3.1.1.2.3.1" xref="A1.SS3.SSS0.P0.SPx1.p8.3.3.m3.1.1.2.3.1.cmml">log</mi><mo id="A1.SS3.SSS0.P0.SPx1.p8.3.3.m3.1.1.2.3a" lspace="0.167em" xref="A1.SS3.SSS0.P0.SPx1.p8.3.3.m3.1.1.2.3.cmml">⁡</mo><mi id="A1.SS3.SSS0.P0.SPx1.p8.3.3.m3.1.1.2.3.2" xref="A1.SS3.SSS0.P0.SPx1.p8.3.3.m3.1.1.2.3.2.cmml">n</mi></mrow></mrow><mi id="A1.SS3.SSS0.P0.SPx1.p8.3.3.m3.1.1.3" xref="A1.SS3.SSS0.P0.SPx1.p8.3.3.m3.1.1.3.cmml">ε</mi></mfrac><mo id="A1.SS3.SSS0.P0.SPx1.p8.3.3.m3.1.2.2.2" stretchy="false" xref="A1.SS3.SSS0.P0.SPx1.p8.3.3.m3.1.2.1.1.cmml">⌉</mo></mrow><annotation-xml encoding="MathML-Content" id="A1.SS3.SSS0.P0.SPx1.p8.3.3.m3.1b"><apply id="A1.SS3.SSS0.P0.SPx1.p8.3.3.m3.1.2.1.cmml" xref="A1.SS3.SSS0.P0.SPx1.p8.3.3.m3.1.2.2"><ceiling id="A1.SS3.SSS0.P0.SPx1.p8.3.3.m3.1.2.1.1.cmml" xref="A1.SS3.SSS0.P0.SPx1.p8.3.3.m3.1.2.2.1"></ceiling><apply id="A1.SS3.SSS0.P0.SPx1.p8.3.3.m3.1.1.cmml" xref="A1.SS3.SSS0.P0.SPx1.p8.3.3.m3.1.1"><divide id="A1.SS3.SSS0.P0.SPx1.p8.3.3.m3.1.1.1.cmml" xref="A1.SS3.SSS0.P0.SPx1.p8.3.3.m3.1.1"></divide><apply id="A1.SS3.SSS0.P0.SPx1.p8.3.3.m3.1.1.2.cmml" xref="A1.SS3.SSS0.P0.SPx1.p8.3.3.m3.1.1.2"><times id="A1.SS3.SSS0.P0.SPx1.p8.3.3.m3.1.1.2.1.cmml" xref="A1.SS3.SSS0.P0.SPx1.p8.3.3.m3.1.1.2.1"></times><cn id="A1.SS3.SSS0.P0.SPx1.p8.3.3.m3.1.1.2.2.cmml" type="integer" xref="A1.SS3.SSS0.P0.SPx1.p8.3.3.m3.1.1.2.2">32</cn><apply id="A1.SS3.SSS0.P0.SPx1.p8.3.3.m3.1.1.2.3.cmml" xref="A1.SS3.SSS0.P0.SPx1.p8.3.3.m3.1.1.2.3"><log id="A1.SS3.SSS0.P0.SPx1.p8.3.3.m3.1.1.2.3.1.cmml" xref="A1.SS3.SSS0.P0.SPx1.p8.3.3.m3.1.1.2.3.1"></log><ci id="A1.SS3.SSS0.P0.SPx1.p8.3.3.m3.1.1.2.3.2.cmml" xref="A1.SS3.SSS0.P0.SPx1.p8.3.3.m3.1.1.2.3.2">𝑛</ci></apply></apply><ci id="A1.SS3.SSS0.P0.SPx1.p8.3.3.m3.1.1.3.cmml" xref="A1.SS3.SSS0.P0.SPx1.p8.3.3.m3.1.1.3">𝜀</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS3.SSS0.P0.SPx1.p8.3.3.m3.1c">\lceil\frac{32\log n}{\varepsilon}\rceil</annotation><annotation encoding="application/x-llamapun" id="A1.SS3.SSS0.P0.SPx1.p8.3.3.m3.1d">⌈ divide start_ARG 32 roman_log italic_n end_ARG start_ARG italic_ε end_ARG ⌉</annotation></semantics></math>-hop neighbourhood of the leader. The leader then uses the truncated BFS tree-structure to gather the value of the largest fractional out-degree <math alttext="h_{\max}(\ell)" class="ltx_Math" display="inline" id="A1.SS3.SSS0.P0.SPx1.p8.4.4.m4.1"><semantics id="A1.SS3.SSS0.P0.SPx1.p8.4.4.m4.1a"><mrow id="A1.SS3.SSS0.P0.SPx1.p8.4.4.m4.1.2" xref="A1.SS3.SSS0.P0.SPx1.p8.4.4.m4.1.2.cmml"><msub id="A1.SS3.SSS0.P0.SPx1.p8.4.4.m4.1.2.2" xref="A1.SS3.SSS0.P0.SPx1.p8.4.4.m4.1.2.2.cmml"><mi id="A1.SS3.SSS0.P0.SPx1.p8.4.4.m4.1.2.2.2" xref="A1.SS3.SSS0.P0.SPx1.p8.4.4.m4.1.2.2.2.cmml">h</mi><mi id="A1.SS3.SSS0.P0.SPx1.p8.4.4.m4.1.2.2.3" xref="A1.SS3.SSS0.P0.SPx1.p8.4.4.m4.1.2.2.3.cmml">max</mi></msub><mo id="A1.SS3.SSS0.P0.SPx1.p8.4.4.m4.1.2.1" xref="A1.SS3.SSS0.P0.SPx1.p8.4.4.m4.1.2.1.cmml">⁢</mo><mrow id="A1.SS3.SSS0.P0.SPx1.p8.4.4.m4.1.2.3.2" xref="A1.SS3.SSS0.P0.SPx1.p8.4.4.m4.1.2.cmml"><mo id="A1.SS3.SSS0.P0.SPx1.p8.4.4.m4.1.2.3.2.1" stretchy="false" xref="A1.SS3.SSS0.P0.SPx1.p8.4.4.m4.1.2.cmml">(</mo><mi id="A1.SS3.SSS0.P0.SPx1.p8.4.4.m4.1.1" mathvariant="normal" xref="A1.SS3.SSS0.P0.SPx1.p8.4.4.m4.1.1.cmml">ℓ</mi><mo id="A1.SS3.SSS0.P0.SPx1.p8.4.4.m4.1.2.3.2.2" stretchy="false" xref="A1.SS3.SSS0.P0.SPx1.p8.4.4.m4.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="A1.SS3.SSS0.P0.SPx1.p8.4.4.m4.1b"><apply id="A1.SS3.SSS0.P0.SPx1.p8.4.4.m4.1.2.cmml" xref="A1.SS3.SSS0.P0.SPx1.p8.4.4.m4.1.2"><times id="A1.SS3.SSS0.P0.SPx1.p8.4.4.m4.1.2.1.cmml" xref="A1.SS3.SSS0.P0.SPx1.p8.4.4.m4.1.2.1"></times><apply id="A1.SS3.SSS0.P0.SPx1.p8.4.4.m4.1.2.2.cmml" xref="A1.SS3.SSS0.P0.SPx1.p8.4.4.m4.1.2.2"><csymbol cd="ambiguous" id="A1.SS3.SSS0.P0.SPx1.p8.4.4.m4.1.2.2.1.cmml" xref="A1.SS3.SSS0.P0.SPx1.p8.4.4.m4.1.2.2">subscript</csymbol><ci id="A1.SS3.SSS0.P0.SPx1.p8.4.4.m4.1.2.2.2.cmml" xref="A1.SS3.SSS0.P0.SPx1.p8.4.4.m4.1.2.2.2">ℎ</ci><max id="A1.SS3.SSS0.P0.SPx1.p8.4.4.m4.1.2.2.3.cmml" xref="A1.SS3.SSS0.P0.SPx1.p8.4.4.m4.1.2.2.3"></max></apply><ci id="A1.SS3.SSS0.P0.SPx1.p8.4.4.m4.1.1.cmml" xref="A1.SS3.SSS0.P0.SPx1.p8.4.4.m4.1.1">ℓ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS3.SSS0.P0.SPx1.p8.4.4.m4.1c">h_{\max}(\ell)</annotation><annotation encoding="application/x-llamapun" id="A1.SS3.SSS0.P0.SPx1.p8.4.4.m4.1d">italic_h start_POSTSUBSCRIPT roman_max end_POSTSUBSCRIPT ( roman_ℓ )</annotation></semantics></math> under this newly computed <math alttext="\eta" class="ltx_Math" display="inline" id="A1.SS3.SSS0.P0.SPx1.p8.5.5.m5.1"><semantics id="A1.SS3.SSS0.P0.SPx1.p8.5.5.m5.1a"><mi id="A1.SS3.SSS0.P0.SPx1.p8.5.5.m5.1.1" xref="A1.SS3.SSS0.P0.SPx1.p8.5.5.m5.1.1.cmml">η</mi><annotation-xml encoding="MathML-Content" id="A1.SS3.SSS0.P0.SPx1.p8.5.5.m5.1b"><ci id="A1.SS3.SSS0.P0.SPx1.p8.5.5.m5.1.1.cmml" xref="A1.SS3.SSS0.P0.SPx1.p8.5.5.m5.1.1">𝜂</ci></annotation-xml><annotation encoding="application/x-tex" id="A1.SS3.SSS0.P0.SPx1.p8.5.5.m5.1c">\eta</annotation><annotation encoding="application/x-llamapun" id="A1.SS3.SSS0.P0.SPx1.p8.5.5.m5.1d">italic_η</annotation></semantics></math>-fair out-orientation as well as the sizes of the sets <math alttext="T_{i}(\ell)=\{u\in N^{t}(\ell):h(u)\geq h_{\max}(1+\eta)^{-i}\}" class="ltx_Math" display="inline" id="A1.SS3.SSS0.P0.SPx1.p8.6.6.m6.5"><semantics id="A1.SS3.SSS0.P0.SPx1.p8.6.6.m6.5a"><mrow id="A1.SS3.SSS0.P0.SPx1.p8.6.6.m6.5.5" xref="A1.SS3.SSS0.P0.SPx1.p8.6.6.m6.5.5.cmml"><mrow id="A1.SS3.SSS0.P0.SPx1.p8.6.6.m6.5.5.4" xref="A1.SS3.SSS0.P0.SPx1.p8.6.6.m6.5.5.4.cmml"><msub id="A1.SS3.SSS0.P0.SPx1.p8.6.6.m6.5.5.4.2" xref="A1.SS3.SSS0.P0.SPx1.p8.6.6.m6.5.5.4.2.cmml"><mi id="A1.SS3.SSS0.P0.SPx1.p8.6.6.m6.5.5.4.2.2" xref="A1.SS3.SSS0.P0.SPx1.p8.6.6.m6.5.5.4.2.2.cmml">T</mi><mi id="A1.SS3.SSS0.P0.SPx1.p8.6.6.m6.5.5.4.2.3" xref="A1.SS3.SSS0.P0.SPx1.p8.6.6.m6.5.5.4.2.3.cmml">i</mi></msub><mo id="A1.SS3.SSS0.P0.SPx1.p8.6.6.m6.5.5.4.1" xref="A1.SS3.SSS0.P0.SPx1.p8.6.6.m6.5.5.4.1.cmml">⁢</mo><mrow 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encoding="application/x-tex" id="A1.SS3.SSS0.P0.SPx1.p8.6.6.m6.5c">T_{i}(\ell)=\{u\in N^{t}(\ell):h(u)\geq h_{\max}(1+\eta)^{-i}\}</annotation><annotation encoding="application/x-llamapun" id="A1.SS3.SSS0.P0.SPx1.p8.6.6.m6.5d">italic_T start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( roman_ℓ ) = { italic_u ∈ italic_N start_POSTSUPERSCRIPT italic_t end_POSTSUPERSCRIPT ( roman_ℓ ) : italic_h ( italic_u ) ≥ italic_h start_POSTSUBSCRIPT roman_max end_POSTSUBSCRIPT ( 1 + italic_η ) start_POSTSUPERSCRIPT - italic_i end_POSTSUPERSCRIPT }</annotation></semantics></math> for <math alttext="i=0" class="ltx_Math" display="inline" id="A1.SS3.SSS0.P0.SPx1.p8.7.7.m7.1"><semantics id="A1.SS3.SSS0.P0.SPx1.p8.7.7.m7.1a"><mrow id="A1.SS3.SSS0.P0.SPx1.p8.7.7.m7.1.1" xref="A1.SS3.SSS0.P0.SPx1.p8.7.7.m7.1.1.cmml"><mi id="A1.SS3.SSS0.P0.SPx1.p8.7.7.m7.1.1.2" xref="A1.SS3.SSS0.P0.SPx1.p8.7.7.m7.1.1.2.cmml">i</mi><mo id="A1.SS3.SSS0.P0.SPx1.p8.7.7.m7.1.1.1" 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xref="A1.SS3.SSS0.P0.SPx1.p8.8.8.m8.1.1.3">𝜀</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS3.SSS0.P0.SPx1.p8.8.8.m8.1c">i=O(\frac{\log n}{\varepsilon})</annotation><annotation encoding="application/x-llamapun" id="A1.SS3.SSS0.P0.SPx1.p8.8.8.m8.1d">italic_i = italic_O ( divide start_ARG roman_log italic_n end_ARG start_ARG italic_ε end_ARG )</annotation></semantics></math>. Based on these numbers the leader can determine a cut-off point. That is: the leader calculates the smallest <math alttext="k" class="ltx_Math" display="inline" id="A1.SS3.SSS0.P0.SPx1.p8.9.9.m9.1"><semantics id="A1.SS3.SSS0.P0.SPx1.p8.9.9.m9.1a"><mi id="A1.SS3.SSS0.P0.SPx1.p8.9.9.m9.1.1" xref="A1.SS3.SSS0.P0.SPx1.p8.9.9.m9.1.1.cmml">k</mi><annotation-xml encoding="MathML-Content" id="A1.SS3.SSS0.P0.SPx1.p8.9.9.m9.1b"><ci id="A1.SS3.SSS0.P0.SPx1.p8.9.9.m9.1.1.cmml" xref="A1.SS3.SSS0.P0.SPx1.p8.9.9.m9.1.1">𝑘</ci></annotation-xml><annotation encoding="application/x-tex" id="A1.SS3.SSS0.P0.SPx1.p8.9.9.m9.1c">k</annotation><annotation encoding="application/x-llamapun" id="A1.SS3.SSS0.P0.SPx1.p8.9.9.m9.1d">italic_k</annotation></semantics></math> for which <math alttext="|T_{k+1}|<(1+\frac{\varepsilon}{16})|T_{k}|" class="ltx_Math" display="inline" id="A1.SS3.SSS0.P0.SPx1.p8.10.10.m10.3"><semantics id="A1.SS3.SSS0.P0.SPx1.p8.10.10.m10.3a"><mrow id="A1.SS3.SSS0.P0.SPx1.p8.10.10.m10.3.3" xref="A1.SS3.SSS0.P0.SPx1.p8.10.10.m10.3.3.cmml"><mrow 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xref="A1.SS3.SSS0.P0.SPx1.p8.10.10.m10.3.3.3.2.1"><abs id="A1.SS3.SSS0.P0.SPx1.p8.10.10.m10.3.3.3.2.2.1.cmml" xref="A1.SS3.SSS0.P0.SPx1.p8.10.10.m10.3.3.3.2.1.2"></abs><apply id="A1.SS3.SSS0.P0.SPx1.p8.10.10.m10.3.3.3.2.1.1.cmml" xref="A1.SS3.SSS0.P0.SPx1.p8.10.10.m10.3.3.3.2.1.1"><csymbol cd="ambiguous" id="A1.SS3.SSS0.P0.SPx1.p8.10.10.m10.3.3.3.2.1.1.1.cmml" xref="A1.SS3.SSS0.P0.SPx1.p8.10.10.m10.3.3.3.2.1.1">subscript</csymbol><ci id="A1.SS3.SSS0.P0.SPx1.p8.10.10.m10.3.3.3.2.1.1.2.cmml" xref="A1.SS3.SSS0.P0.SPx1.p8.10.10.m10.3.3.3.2.1.1.2">𝑇</ci><ci id="A1.SS3.SSS0.P0.SPx1.p8.10.10.m10.3.3.3.2.1.1.3.cmml" xref="A1.SS3.SSS0.P0.SPx1.p8.10.10.m10.3.3.3.2.1.1.3">𝑘</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS3.SSS0.P0.SPx1.p8.10.10.m10.3c">|T_{k+1}|<(1+\frac{\varepsilon}{16})|T_{k}|</annotation><annotation encoding="application/x-llamapun" id="A1.SS3.SSS0.P0.SPx1.p8.10.10.m10.3d">| italic_T start_POSTSUBSCRIPT italic_k + 1 end_POSTSUBSCRIPT | < ( 1 + divide start_ARG italic_ε end_ARG start_ARG 16 end_ARG ) | italic_T start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT |</annotation></semantics></math>. It then sets the cut-off point to be <math alttext="h_{\max}(1+\eta)^{-(k+1)}" class="ltx_Math" display="inline" id="A1.SS3.SSS0.P0.SPx1.p8.11.11.m11.2"><semantics id="A1.SS3.SSS0.P0.SPx1.p8.11.11.m11.2a"><mrow id="A1.SS3.SSS0.P0.SPx1.p8.11.11.m11.2.2" xref="A1.SS3.SSS0.P0.SPx1.p8.11.11.m11.2.2.cmml"><msub id="A1.SS3.SSS0.P0.SPx1.p8.11.11.m11.2.2.3" xref="A1.SS3.SSS0.P0.SPx1.p8.11.11.m11.2.2.3.cmml"><mi id="A1.SS3.SSS0.P0.SPx1.p8.11.11.m11.2.2.3.2" xref="A1.SS3.SSS0.P0.SPx1.p8.11.11.m11.2.2.3.2.cmml">h</mi><mi id="A1.SS3.SSS0.P0.SPx1.p8.11.11.m11.2.2.3.3" xref="A1.SS3.SSS0.P0.SPx1.p8.11.11.m11.2.2.3.3.cmml">max</mi></msub><mo id="A1.SS3.SSS0.P0.SPx1.p8.11.11.m11.2.2.2" xref="A1.SS3.SSS0.P0.SPx1.p8.11.11.m11.2.2.2.cmml">⁢</mo><msup id="A1.SS3.SSS0.P0.SPx1.p8.11.11.m11.2.2.1" xref="A1.SS3.SSS0.P0.SPx1.p8.11.11.m11.2.2.1.cmml"><mrow id="A1.SS3.SSS0.P0.SPx1.p8.11.11.m11.2.2.1.1.1" xref="A1.SS3.SSS0.P0.SPx1.p8.11.11.m11.2.2.1.1.1.1.cmml"><mo id="A1.SS3.SSS0.P0.SPx1.p8.11.11.m11.2.2.1.1.1.2" 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id="A1.SS3.SSS0.P0.SPx1.p8.11.11.m11.1.1.1.1.1.1.3.cmml" type="integer" xref="A1.SS3.SSS0.P0.SPx1.p8.11.11.m11.1.1.1.1.1.1.3">1</cn></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS3.SSS0.P0.SPx1.p8.11.11.m11.2c">h_{\max}(1+\eta)^{-(k+1)}</annotation><annotation encoding="application/x-llamapun" id="A1.SS3.SSS0.P0.SPx1.p8.11.11.m11.2d">italic_h start_POSTSUBSCRIPT roman_max end_POSTSUBSCRIPT ( 1 + italic_η ) start_POSTSUPERSCRIPT - ( italic_k + 1 ) end_POSTSUPERSCRIPT</annotation></semantics></math>, and broadcasts the cut-off point to its <math alttext="\lceil\frac{32\log n}{\varepsilon}\rceil" class="ltx_Math" display="inline" id="A1.SS3.SSS0.P0.SPx1.p8.12.12.m12.1"><semantics id="A1.SS3.SSS0.P0.SPx1.p8.12.12.m12.1a"><mrow id="A1.SS3.SSS0.P0.SPx1.p8.12.12.m12.1.2.2" xref="A1.SS3.SSS0.P0.SPx1.p8.12.12.m12.1.2.1.cmml"><mo id="A1.SS3.SSS0.P0.SPx1.p8.12.12.m12.1.2.2.1" stretchy="false" xref="A1.SS3.SSS0.P0.SPx1.p8.12.12.m12.1.2.1.1.cmml">⌈</mo><mfrac id="A1.SS3.SSS0.P0.SPx1.p8.12.12.m12.1.1" xref="A1.SS3.SSS0.P0.SPx1.p8.12.12.m12.1.1.cmml"><mrow id="A1.SS3.SSS0.P0.SPx1.p8.12.12.m12.1.1.2" xref="A1.SS3.SSS0.P0.SPx1.p8.12.12.m12.1.1.2.cmml"><mn id="A1.SS3.SSS0.P0.SPx1.p8.12.12.m12.1.1.2.2" xref="A1.SS3.SSS0.P0.SPx1.p8.12.12.m12.1.1.2.2.cmml">32</mn><mo id="A1.SS3.SSS0.P0.SPx1.p8.12.12.m12.1.1.2.1" lspace="0.167em" xref="A1.SS3.SSS0.P0.SPx1.p8.12.12.m12.1.1.2.1.cmml">⁢</mo><mrow id="A1.SS3.SSS0.P0.SPx1.p8.12.12.m12.1.1.2.3" xref="A1.SS3.SSS0.P0.SPx1.p8.12.12.m12.1.1.2.3.cmml"><mi id="A1.SS3.SSS0.P0.SPx1.p8.12.12.m12.1.1.2.3.1" xref="A1.SS3.SSS0.P0.SPx1.p8.12.12.m12.1.1.2.3.1.cmml">log</mi><mo id="A1.SS3.SSS0.P0.SPx1.p8.12.12.m12.1.1.2.3a" lspace="0.167em" xref="A1.SS3.SSS0.P0.SPx1.p8.12.12.m12.1.1.2.3.cmml">⁡</mo><mi id="A1.SS3.SSS0.P0.SPx1.p8.12.12.m12.1.1.2.3.2" xref="A1.SS3.SSS0.P0.SPx1.p8.12.12.m12.1.1.2.3.2.cmml">n</mi></mrow></mrow><mi id="A1.SS3.SSS0.P0.SPx1.p8.12.12.m12.1.1.3" xref="A1.SS3.SSS0.P0.SPx1.p8.12.12.m12.1.1.3.cmml">ε</mi></mfrac><mo id="A1.SS3.SSS0.P0.SPx1.p8.12.12.m12.1.2.2.2" stretchy="false" xref="A1.SS3.SSS0.P0.SPx1.p8.12.12.m12.1.2.1.1.cmml">⌉</mo></mrow><annotation-xml encoding="MathML-Content" id="A1.SS3.SSS0.P0.SPx1.p8.12.12.m12.1b"><apply id="A1.SS3.SSS0.P0.SPx1.p8.12.12.m12.1.2.1.cmml" xref="A1.SS3.SSS0.P0.SPx1.p8.12.12.m12.1.2.2"><ceiling id="A1.SS3.SSS0.P0.SPx1.p8.12.12.m12.1.2.1.1.cmml" xref="A1.SS3.SSS0.P0.SPx1.p8.12.12.m12.1.2.2.1"></ceiling><apply id="A1.SS3.SSS0.P0.SPx1.p8.12.12.m12.1.1.cmml" xref="A1.SS3.SSS0.P0.SPx1.p8.12.12.m12.1.1"><divide id="A1.SS3.SSS0.P0.SPx1.p8.12.12.m12.1.1.1.cmml" xref="A1.SS3.SSS0.P0.SPx1.p8.12.12.m12.1.1"></divide><apply id="A1.SS3.SSS0.P0.SPx1.p8.12.12.m12.1.1.2.cmml" xref="A1.SS3.SSS0.P0.SPx1.p8.12.12.m12.1.1.2"><times id="A1.SS3.SSS0.P0.SPx1.p8.12.12.m12.1.1.2.1.cmml" xref="A1.SS3.SSS0.P0.SPx1.p8.12.12.m12.1.1.2.1"></times><cn id="A1.SS3.SSS0.P0.SPx1.p8.12.12.m12.1.1.2.2.cmml" type="integer" xref="A1.SS3.SSS0.P0.SPx1.p8.12.12.m12.1.1.2.2">32</cn><apply id="A1.SS3.SSS0.P0.SPx1.p8.12.12.m12.1.1.2.3.cmml" xref="A1.SS3.SSS0.P0.SPx1.p8.12.12.m12.1.1.2.3"><log id="A1.SS3.SSS0.P0.SPx1.p8.12.12.m12.1.1.2.3.1.cmml" xref="A1.SS3.SSS0.P0.SPx1.p8.12.12.m12.1.1.2.3.1"></log><ci id="A1.SS3.SSS0.P0.SPx1.p8.12.12.m12.1.1.2.3.2.cmml" xref="A1.SS3.SSS0.P0.SPx1.p8.12.12.m12.1.1.2.3.2">𝑛</ci></apply></apply><ci id="A1.SS3.SSS0.P0.SPx1.p8.12.12.m12.1.1.3.cmml" xref="A1.SS3.SSS0.P0.SPx1.p8.12.12.m12.1.1.3">𝜀</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS3.SSS0.P0.SPx1.p8.12.12.m12.1c">\lceil\frac{32\log n}{\varepsilon}\rceil</annotation><annotation encoding="application/x-llamapun" id="A1.SS3.SSS0.P0.SPx1.p8.12.12.m12.1d">⌈ divide start_ARG 32 roman_log italic_n end_ARG start_ARG italic_ε end_ARG ⌉</annotation></semantics></math>-hop neighbourhood. Finally, every vertex with a fractional out-degree greater than or equal to the cut-off point declares a <math alttext="1" class="ltx_Math" display="inline" id="A1.SS3.SSS0.P0.SPx1.p8.13.13.m13.1"><semantics id="A1.SS3.SSS0.P0.SPx1.p8.13.13.m13.1a"><mn id="A1.SS3.SSS0.P0.SPx1.p8.13.13.m13.1.1" xref="A1.SS3.SSS0.P0.SPx1.p8.13.13.m13.1.1.cmml">1</mn><annotation-xml encoding="MathML-Content" id="A1.SS3.SSS0.P0.SPx1.p8.13.13.m13.1b"><cn id="A1.SS3.SSS0.P0.SPx1.p8.13.13.m13.1.1.cmml" type="integer" xref="A1.SS3.SSS0.P0.SPx1.p8.13.13.m13.1.1">1</cn></annotation-xml><annotation encoding="application/x-tex" id="A1.SS3.SSS0.P0.SPx1.p8.13.13.m13.1c">1</annotation><annotation encoding="application/x-llamapun" id="A1.SS3.SSS0.P0.SPx1.p8.13.13.m13.1d">1</annotation></semantics></math> and all other vertices declare <math alttext="0" class="ltx_Math" display="inline" id="A1.SS3.SSS0.P0.SPx1.p8.14.14.m14.1"><semantics id="A1.SS3.SSS0.P0.SPx1.p8.14.14.m14.1a"><mn id="A1.SS3.SSS0.P0.SPx1.p8.14.14.m14.1.1" xref="A1.SS3.SSS0.P0.SPx1.p8.14.14.m14.1.1.cmml">0</mn><annotation-xml encoding="MathML-Content" id="A1.SS3.SSS0.P0.SPx1.p8.14.14.m14.1b"><cn id="A1.SS3.SSS0.P0.SPx1.p8.14.14.m14.1.1.cmml" type="integer" xref="A1.SS3.SSS0.P0.SPx1.p8.14.14.m14.1.1">0</cn></annotation-xml></semantics></math>.</span></p> </div> </section> <section class="ltx_subparagraph" id="A1.SS3.SSS0.P0.SPx2"> <h5 class="ltx_title ltx_title_subparagraph">Analysis:</h5> <div class="ltx_para" id="A1.SS3.SSS0.P0.SPx2.p1"> <p class="ltx_p" id="A1.SS3.SSS0.P0.SPx2.p1.1">We now show correctness of the above algorithm i.e. that it correctly terminates with a sufficiently dense subgraph.</p> </div> <div class="ltx_para" id="A1.SS3.SSS0.P0.SPx2.p2"> <p class="ltx_p" id="A1.SS3.SSS0.P0.SPx2.p2.3">We use this property to show that if a leader is active, then the nodes that output <math alttext="1" class="ltx_Math" display="inline" id="A1.SS3.SSS0.P0.SPx2.p2.1.m1.1"><semantics id="A1.SS3.SSS0.P0.SPx2.p2.1.m1.1a"><mn id="A1.SS3.SSS0.P0.SPx2.p2.1.m1.1.1" xref="A1.SS3.SSS0.P0.SPx2.p2.1.m1.1.1.cmml">1</mn><annotation-xml encoding="MathML-Content" id="A1.SS3.SSS0.P0.SPx2.p2.1.m1.1b"><cn id="A1.SS3.SSS0.P0.SPx2.p2.1.m1.1.1.cmml" type="integer" xref="A1.SS3.SSS0.P0.SPx2.p2.1.m1.1.1">1</cn></annotation-xml><annotation encoding="application/x-tex" id="A1.SS3.SSS0.P0.SPx2.p2.1.m1.1c">1</annotation><annotation encoding="application/x-llamapun" id="A1.SS3.SSS0.P0.SPx2.p2.1.m1.1d">1</annotation></semantics></math> in the <math alttext="\lceil\frac{32\log n}{\varepsilon}\rceil" class="ltx_Math" display="inline" id="A1.SS3.SSS0.P0.SPx2.p2.2.m2.1"><semantics id="A1.SS3.SSS0.P0.SPx2.p2.2.m2.1a"><mrow id="A1.SS3.SSS0.P0.SPx2.p2.2.m2.1.2.2" xref="A1.SS3.SSS0.P0.SPx2.p2.2.m2.1.2.1.cmml"><mo id="A1.SS3.SSS0.P0.SPx2.p2.2.m2.1.2.2.1" stretchy="false" xref="A1.SS3.SSS0.P0.SPx2.p2.2.m2.1.2.1.1.cmml">⌈</mo><mfrac id="A1.SS3.SSS0.P0.SPx2.p2.2.m2.1.1" xref="A1.SS3.SSS0.P0.SPx2.p2.2.m2.1.1.cmml"><mrow id="A1.SS3.SSS0.P0.SPx2.p2.2.m2.1.1.2" xref="A1.SS3.SSS0.P0.SPx2.p2.2.m2.1.1.2.cmml"><mn id="A1.SS3.SSS0.P0.SPx2.p2.2.m2.1.1.2.2" xref="A1.SS3.SSS0.P0.SPx2.p2.2.m2.1.1.2.2.cmml">32</mn><mo id="A1.SS3.SSS0.P0.SPx2.p2.2.m2.1.1.2.1" lspace="0.167em" xref="A1.SS3.SSS0.P0.SPx2.p2.2.m2.1.1.2.1.cmml">⁢</mo><mrow id="A1.SS3.SSS0.P0.SPx2.p2.2.m2.1.1.2.3" xref="A1.SS3.SSS0.P0.SPx2.p2.2.m2.1.1.2.3.cmml"><mi id="A1.SS3.SSS0.P0.SPx2.p2.2.m2.1.1.2.3.1" xref="A1.SS3.SSS0.P0.SPx2.p2.2.m2.1.1.2.3.1.cmml">log</mi><mo id="A1.SS3.SSS0.P0.SPx2.p2.2.m2.1.1.2.3a" lspace="0.167em" xref="A1.SS3.SSS0.P0.SPx2.p2.2.m2.1.1.2.3.cmml">⁡</mo><mi id="A1.SS3.SSS0.P0.SPx2.p2.2.m2.1.1.2.3.2" xref="A1.SS3.SSS0.P0.SPx2.p2.2.m2.1.1.2.3.2.cmml">n</mi></mrow></mrow><mi id="A1.SS3.SSS0.P0.SPx2.p2.2.m2.1.1.3" xref="A1.SS3.SSS0.P0.SPx2.p2.2.m2.1.1.3.cmml">ε</mi></mfrac><mo id="A1.SS3.SSS0.P0.SPx2.p2.2.m2.1.2.2.2" stretchy="false" xref="A1.SS3.SSS0.P0.SPx2.p2.2.m2.1.2.1.1.cmml">⌉</mo></mrow><annotation-xml encoding="MathML-Content" id="A1.SS3.SSS0.P0.SPx2.p2.2.m2.1b"><apply id="A1.SS3.SSS0.P0.SPx2.p2.2.m2.1.2.1.cmml" xref="A1.SS3.SSS0.P0.SPx2.p2.2.m2.1.2.2"><ceiling id="A1.SS3.SSS0.P0.SPx2.p2.2.m2.1.2.1.1.cmml" xref="A1.SS3.SSS0.P0.SPx2.p2.2.m2.1.2.2.1"></ceiling><apply id="A1.SS3.SSS0.P0.SPx2.p2.2.m2.1.1.cmml" xref="A1.SS3.SSS0.P0.SPx2.p2.2.m2.1.1"><divide id="A1.SS3.SSS0.P0.SPx2.p2.2.m2.1.1.1.cmml" xref="A1.SS3.SSS0.P0.SPx2.p2.2.m2.1.1"></divide><apply id="A1.SS3.SSS0.P0.SPx2.p2.2.m2.1.1.2.cmml" xref="A1.SS3.SSS0.P0.SPx2.p2.2.m2.1.1.2"><times id="A1.SS3.SSS0.P0.SPx2.p2.2.m2.1.1.2.1.cmml" xref="A1.SS3.SSS0.P0.SPx2.p2.2.m2.1.1.2.1"></times><cn id="A1.SS3.SSS0.P0.SPx2.p2.2.m2.1.1.2.2.cmml" type="integer" xref="A1.SS3.SSS0.P0.SPx2.p2.2.m2.1.1.2.2">32</cn><apply id="A1.SS3.SSS0.P0.SPx2.p2.2.m2.1.1.2.3.cmml" xref="A1.SS3.SSS0.P0.SPx2.p2.2.m2.1.1.2.3"><log id="A1.SS3.SSS0.P0.SPx2.p2.2.m2.1.1.2.3.1.cmml" xref="A1.SS3.SSS0.P0.SPx2.p2.2.m2.1.1.2.3.1"></log><ci id="A1.SS3.SSS0.P0.SPx2.p2.2.m2.1.1.2.3.2.cmml" xref="A1.SS3.SSS0.P0.SPx2.p2.2.m2.1.1.2.3.2">𝑛</ci></apply></apply><ci id="A1.SS3.SSS0.P0.SPx2.p2.2.m2.1.1.3.cmml" xref="A1.SS3.SSS0.P0.SPx2.p2.2.m2.1.1.3">𝜀</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS3.SSS0.P0.SPx2.p2.2.m2.1c">\lceil\frac{32\log n}{\varepsilon}\rceil</annotation><annotation encoding="application/x-llamapun" id="A1.SS3.SSS0.P0.SPx2.p2.2.m2.1d">⌈ divide start_ARG 32 roman_log italic_n end_ARG start_ARG italic_ε end_ARG ⌉</annotation></semantics></math> neighbourhood of the leader form a subgraph with density at least <math alttext="(1-\varepsilon)\rho(G)" class="ltx_Math" display="inline" id="A1.SS3.SSS0.P0.SPx2.p2.3.m3.2"><semantics id="A1.SS3.SSS0.P0.SPx2.p2.3.m3.2a"><mrow id="A1.SS3.SSS0.P0.SPx2.p2.3.m3.2.2" xref="A1.SS3.SSS0.P0.SPx2.p2.3.m3.2.2.cmml"><mrow id="A1.SS3.SSS0.P0.SPx2.p2.3.m3.2.2.1.1" xref="A1.SS3.SSS0.P0.SPx2.p2.3.m3.2.2.1.1.1.cmml"><mo id="A1.SS3.SSS0.P0.SPx2.p2.3.m3.2.2.1.1.2" stretchy="false" xref="A1.SS3.SSS0.P0.SPx2.p2.3.m3.2.2.1.1.1.cmml">(</mo><mrow id="A1.SS3.SSS0.P0.SPx2.p2.3.m3.2.2.1.1.1" xref="A1.SS3.SSS0.P0.SPx2.p2.3.m3.2.2.1.1.1.cmml"><mn id="A1.SS3.SSS0.P0.SPx2.p2.3.m3.2.2.1.1.1.2" xref="A1.SS3.SSS0.P0.SPx2.p2.3.m3.2.2.1.1.1.2.cmml">1</mn><mo id="A1.SS3.SSS0.P0.SPx2.p2.3.m3.2.2.1.1.1.1" xref="A1.SS3.SSS0.P0.SPx2.p2.3.m3.2.2.1.1.1.1.cmml">−</mo><mi id="A1.SS3.SSS0.P0.SPx2.p2.3.m3.2.2.1.1.1.3" xref="A1.SS3.SSS0.P0.SPx2.p2.3.m3.2.2.1.1.1.3.cmml">ε</mi></mrow><mo id="A1.SS3.SSS0.P0.SPx2.p2.3.m3.2.2.1.1.3" stretchy="false" xref="A1.SS3.SSS0.P0.SPx2.p2.3.m3.2.2.1.1.1.cmml">)</mo></mrow><mo id="A1.SS3.SSS0.P0.SPx2.p2.3.m3.2.2.2" xref="A1.SS3.SSS0.P0.SPx2.p2.3.m3.2.2.2.cmml">⁢</mo><mi id="A1.SS3.SSS0.P0.SPx2.p2.3.m3.2.2.3" xref="A1.SS3.SSS0.P0.SPx2.p2.3.m3.2.2.3.cmml">ρ</mi><mo id="A1.SS3.SSS0.P0.SPx2.p2.3.m3.2.2.2a" xref="A1.SS3.SSS0.P0.SPx2.p2.3.m3.2.2.2.cmml">⁢</mo><mrow id="A1.SS3.SSS0.P0.SPx2.p2.3.m3.2.2.4.2" xref="A1.SS3.SSS0.P0.SPx2.p2.3.m3.2.2.cmml"><mo id="A1.SS3.SSS0.P0.SPx2.p2.3.m3.2.2.4.2.1" stretchy="false" xref="A1.SS3.SSS0.P0.SPx2.p2.3.m3.2.2.cmml">(</mo><mi id="A1.SS3.SSS0.P0.SPx2.p2.3.m3.1.1" xref="A1.SS3.SSS0.P0.SPx2.p2.3.m3.1.1.cmml">G</mi><mo id="A1.SS3.SSS0.P0.SPx2.p2.3.m3.2.2.4.2.2" stretchy="false" xref="A1.SS3.SSS0.P0.SPx2.p2.3.m3.2.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="A1.SS3.SSS0.P0.SPx2.p2.3.m3.2b"><apply id="A1.SS3.SSS0.P0.SPx2.p2.3.m3.2.2.cmml" xref="A1.SS3.SSS0.P0.SPx2.p2.3.m3.2.2"><times id="A1.SS3.SSS0.P0.SPx2.p2.3.m3.2.2.2.cmml" xref="A1.SS3.SSS0.P0.SPx2.p2.3.m3.2.2.2"></times><apply id="A1.SS3.SSS0.P0.SPx2.p2.3.m3.2.2.1.1.1.cmml" xref="A1.SS3.SSS0.P0.SPx2.p2.3.m3.2.2.1.1"><minus id="A1.SS3.SSS0.P0.SPx2.p2.3.m3.2.2.1.1.1.1.cmml" xref="A1.SS3.SSS0.P0.SPx2.p2.3.m3.2.2.1.1.1.1"></minus><cn id="A1.SS3.SSS0.P0.SPx2.p2.3.m3.2.2.1.1.1.2.cmml" type="integer" xref="A1.SS3.SSS0.P0.SPx2.p2.3.m3.2.2.1.1.1.2">1</cn><ci id="A1.SS3.SSS0.P0.SPx2.p2.3.m3.2.2.1.1.1.3.cmml" xref="A1.SS3.SSS0.P0.SPx2.p2.3.m3.2.2.1.1.1.3">𝜀</ci></apply><ci id="A1.SS3.SSS0.P0.SPx2.p2.3.m3.2.2.3.cmml" xref="A1.SS3.SSS0.P0.SPx2.p2.3.m3.2.2.3">𝜌</ci><ci id="A1.SS3.SSS0.P0.SPx2.p2.3.m3.1.1.cmml" xref="A1.SS3.SSS0.P0.SPx2.p2.3.m3.1.1">𝐺</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS3.SSS0.P0.SPx2.p2.3.m3.2c">(1-\varepsilon)\rho(G)</annotation><annotation encoding="application/x-llamapun" id="A1.SS3.SSS0.P0.SPx2.p2.3.m3.2d">( 1 - italic_ε ) italic_ρ ( italic_G )</annotation></semantics></math>:</p> </div> <div class="ltx_theorem ltx_theorem_lemma" id="A1.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="A1.Thmtheorem5.1.1.1">Lemma A.5</span></span><span class="ltx_text ltx_font_bold" id="A1.Thmtheorem5.2.2">.</span> </h6> <div class="ltx_para" id="A1.Thmtheorem5.p1"> <p class="ltx_p" id="A1.Thmtheorem5.p1.5"><span class="ltx_text ltx_font_italic" id="A1.Thmtheorem5.p1.5.5">Suppose <math alttext="\ell" class="ltx_Math" display="inline" id="A1.Thmtheorem5.p1.1.1.m1.1"><semantics id="A1.Thmtheorem5.p1.1.1.m1.1a"><mi id="A1.Thmtheorem5.p1.1.1.m1.1.1" mathvariant="normal" xref="A1.Thmtheorem5.p1.1.1.m1.1.1.cmml">ℓ</mi><annotation-xml encoding="MathML-Content" id="A1.Thmtheorem5.p1.1.1.m1.1b"><ci id="A1.Thmtheorem5.p1.1.1.m1.1.1.cmml" xref="A1.Thmtheorem5.p1.1.1.m1.1.1">ℓ</ci></annotation-xml><annotation encoding="application/x-tex" id="A1.Thmtheorem5.p1.1.1.m1.1c">\ell</annotation><annotation encoding="application/x-llamapun" id="A1.Thmtheorem5.p1.1.1.m1.1d">roman_ℓ</annotation></semantics></math> is an active leader. Then the vertices in the <math alttext="\lceil\frac{32\log n}{\varepsilon}\rceil" class="ltx_Math" display="inline" id="A1.Thmtheorem5.p1.2.2.m2.1"><semantics id="A1.Thmtheorem5.p1.2.2.m2.1a"><mrow id="A1.Thmtheorem5.p1.2.2.m2.1.2.2" xref="A1.Thmtheorem5.p1.2.2.m2.1.2.1.cmml"><mo id="A1.Thmtheorem5.p1.2.2.m2.1.2.2.1" stretchy="false" xref="A1.Thmtheorem5.p1.2.2.m2.1.2.1.1.cmml">⌈</mo><mfrac id="A1.Thmtheorem5.p1.2.2.m2.1.1" xref="A1.Thmtheorem5.p1.2.2.m2.1.1.cmml"><mrow id="A1.Thmtheorem5.p1.2.2.m2.1.1.2" xref="A1.Thmtheorem5.p1.2.2.m2.1.1.2.cmml"><mn id="A1.Thmtheorem5.p1.2.2.m2.1.1.2.2" xref="A1.Thmtheorem5.p1.2.2.m2.1.1.2.2.cmml">32</mn><mo id="A1.Thmtheorem5.p1.2.2.m2.1.1.2.1" lspace="0.167em" xref="A1.Thmtheorem5.p1.2.2.m2.1.1.2.1.cmml">⁢</mo><mrow id="A1.Thmtheorem5.p1.2.2.m2.1.1.2.3" xref="A1.Thmtheorem5.p1.2.2.m2.1.1.2.3.cmml"><mi id="A1.Thmtheorem5.p1.2.2.m2.1.1.2.3.1" xref="A1.Thmtheorem5.p1.2.2.m2.1.1.2.3.1.cmml">log</mi><mo id="A1.Thmtheorem5.p1.2.2.m2.1.1.2.3a" lspace="0.167em" xref="A1.Thmtheorem5.p1.2.2.m2.1.1.2.3.cmml">⁡</mo><mi id="A1.Thmtheorem5.p1.2.2.m2.1.1.2.3.2" xref="A1.Thmtheorem5.p1.2.2.m2.1.1.2.3.2.cmml">n</mi></mrow></mrow><mi id="A1.Thmtheorem5.p1.2.2.m2.1.1.3" xref="A1.Thmtheorem5.p1.2.2.m2.1.1.3.cmml">ε</mi></mfrac><mo id="A1.Thmtheorem5.p1.2.2.m2.1.2.2.2" stretchy="false" xref="A1.Thmtheorem5.p1.2.2.m2.1.2.1.1.cmml">⌉</mo></mrow><annotation-xml encoding="MathML-Content" id="A1.Thmtheorem5.p1.2.2.m2.1b"><apply id="A1.Thmtheorem5.p1.2.2.m2.1.2.1.cmml" xref="A1.Thmtheorem5.p1.2.2.m2.1.2.2"><ceiling id="A1.Thmtheorem5.p1.2.2.m2.1.2.1.1.cmml" xref="A1.Thmtheorem5.p1.2.2.m2.1.2.2.1"></ceiling><apply id="A1.Thmtheorem5.p1.2.2.m2.1.1.cmml" xref="A1.Thmtheorem5.p1.2.2.m2.1.1"><divide id="A1.Thmtheorem5.p1.2.2.m2.1.1.1.cmml" xref="A1.Thmtheorem5.p1.2.2.m2.1.1"></divide><apply id="A1.Thmtheorem5.p1.2.2.m2.1.1.2.cmml" xref="A1.Thmtheorem5.p1.2.2.m2.1.1.2"><times id="A1.Thmtheorem5.p1.2.2.m2.1.1.2.1.cmml" xref="A1.Thmtheorem5.p1.2.2.m2.1.1.2.1"></times><cn id="A1.Thmtheorem5.p1.2.2.m2.1.1.2.2.cmml" type="integer" xref="A1.Thmtheorem5.p1.2.2.m2.1.1.2.2">32</cn><apply id="A1.Thmtheorem5.p1.2.2.m2.1.1.2.3.cmml" xref="A1.Thmtheorem5.p1.2.2.m2.1.1.2.3"><log id="A1.Thmtheorem5.p1.2.2.m2.1.1.2.3.1.cmml" xref="A1.Thmtheorem5.p1.2.2.m2.1.1.2.3.1"></log><ci id="A1.Thmtheorem5.p1.2.2.m2.1.1.2.3.2.cmml" xref="A1.Thmtheorem5.p1.2.2.m2.1.1.2.3.2">𝑛</ci></apply></apply><ci id="A1.Thmtheorem5.p1.2.2.m2.1.1.3.cmml" xref="A1.Thmtheorem5.p1.2.2.m2.1.1.3">𝜀</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.Thmtheorem5.p1.2.2.m2.1c">\lceil\frac{32\log n}{\varepsilon}\rceil</annotation><annotation encoding="application/x-llamapun" id="A1.Thmtheorem5.p1.2.2.m2.1d">⌈ divide start_ARG 32 roman_log italic_n end_ARG start_ARG italic_ε end_ARG ⌉</annotation></semantics></math>-hop neighbourhood of <math alttext="\ell" class="ltx_Math" display="inline" id="A1.Thmtheorem5.p1.3.3.m3.1"><semantics id="A1.Thmtheorem5.p1.3.3.m3.1a"><mi id="A1.Thmtheorem5.p1.3.3.m3.1.1" mathvariant="normal" xref="A1.Thmtheorem5.p1.3.3.m3.1.1.cmml">ℓ</mi><annotation-xml encoding="MathML-Content" id="A1.Thmtheorem5.p1.3.3.m3.1b"><ci id="A1.Thmtheorem5.p1.3.3.m3.1.1.cmml" xref="A1.Thmtheorem5.p1.3.3.m3.1.1">ℓ</ci></annotation-xml><annotation encoding="application/x-tex" id="A1.Thmtheorem5.p1.3.3.m3.1c">\ell</annotation><annotation encoding="application/x-llamapun" id="A1.Thmtheorem5.p1.3.3.m3.1d">roman_ℓ</annotation></semantics></math> that output <math alttext="1" class="ltx_Math" display="inline" id="A1.Thmtheorem5.p1.4.4.m4.1"><semantics id="A1.Thmtheorem5.p1.4.4.m4.1a"><mn id="A1.Thmtheorem5.p1.4.4.m4.1.1" xref="A1.Thmtheorem5.p1.4.4.m4.1.1.cmml">1</mn><annotation-xml encoding="MathML-Content" id="A1.Thmtheorem5.p1.4.4.m4.1b"><cn id="A1.Thmtheorem5.p1.4.4.m4.1.1.cmml" type="integer" xref="A1.Thmtheorem5.p1.4.4.m4.1.1">1</cn></annotation-xml><annotation encoding="application/x-tex" id="A1.Thmtheorem5.p1.4.4.m4.1c">1</annotation><annotation encoding="application/x-llamapun" id="A1.Thmtheorem5.p1.4.4.m4.1d">1</annotation></semantics></math> induce a subgraph of density at least <math alttext="(1-\varepsilon)\rho^{*}(\ell)" class="ltx_Math" display="inline" id="A1.Thmtheorem5.p1.5.5.m5.2"><semantics id="A1.Thmtheorem5.p1.5.5.m5.2a"><mrow id="A1.Thmtheorem5.p1.5.5.m5.2.2" xref="A1.Thmtheorem5.p1.5.5.m5.2.2.cmml"><mrow id="A1.Thmtheorem5.p1.5.5.m5.2.2.1.1" xref="A1.Thmtheorem5.p1.5.5.m5.2.2.1.1.1.cmml"><mo id="A1.Thmtheorem5.p1.5.5.m5.2.2.1.1.2" stretchy="false" xref="A1.Thmtheorem5.p1.5.5.m5.2.2.1.1.1.cmml">(</mo><mrow id="A1.Thmtheorem5.p1.5.5.m5.2.2.1.1.1" xref="A1.Thmtheorem5.p1.5.5.m5.2.2.1.1.1.cmml"><mn id="A1.Thmtheorem5.p1.5.5.m5.2.2.1.1.1.2" xref="A1.Thmtheorem5.p1.5.5.m5.2.2.1.1.1.2.cmml">1</mn><mo id="A1.Thmtheorem5.p1.5.5.m5.2.2.1.1.1.1" xref="A1.Thmtheorem5.p1.5.5.m5.2.2.1.1.1.1.cmml">−</mo><mi id="A1.Thmtheorem5.p1.5.5.m5.2.2.1.1.1.3" xref="A1.Thmtheorem5.p1.5.5.m5.2.2.1.1.1.3.cmml">ε</mi></mrow><mo id="A1.Thmtheorem5.p1.5.5.m5.2.2.1.1.3" stretchy="false" xref="A1.Thmtheorem5.p1.5.5.m5.2.2.1.1.1.cmml">)</mo></mrow><mo id="A1.Thmtheorem5.p1.5.5.m5.2.2.2" xref="A1.Thmtheorem5.p1.5.5.m5.2.2.2.cmml">⁢</mo><msup id="A1.Thmtheorem5.p1.5.5.m5.2.2.3" xref="A1.Thmtheorem5.p1.5.5.m5.2.2.3.cmml"><mi id="A1.Thmtheorem5.p1.5.5.m5.2.2.3.2" xref="A1.Thmtheorem5.p1.5.5.m5.2.2.3.2.cmml">ρ</mi><mo id="A1.Thmtheorem5.p1.5.5.m5.2.2.3.3" xref="A1.Thmtheorem5.p1.5.5.m5.2.2.3.3.cmml">∗</mo></msup><mo id="A1.Thmtheorem5.p1.5.5.m5.2.2.2a" xref="A1.Thmtheorem5.p1.5.5.m5.2.2.2.cmml">⁢</mo><mrow id="A1.Thmtheorem5.p1.5.5.m5.2.2.4.2" xref="A1.Thmtheorem5.p1.5.5.m5.2.2.cmml"><mo id="A1.Thmtheorem5.p1.5.5.m5.2.2.4.2.1" stretchy="false" xref="A1.Thmtheorem5.p1.5.5.m5.2.2.cmml">(</mo><mi id="A1.Thmtheorem5.p1.5.5.m5.1.1" mathvariant="normal" xref="A1.Thmtheorem5.p1.5.5.m5.1.1.cmml">ℓ</mi><mo id="A1.Thmtheorem5.p1.5.5.m5.2.2.4.2.2" stretchy="false" xref="A1.Thmtheorem5.p1.5.5.m5.2.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="A1.Thmtheorem5.p1.5.5.m5.2b"><apply id="A1.Thmtheorem5.p1.5.5.m5.2.2.cmml" xref="A1.Thmtheorem5.p1.5.5.m5.2.2"><times id="A1.Thmtheorem5.p1.5.5.m5.2.2.2.cmml" xref="A1.Thmtheorem5.p1.5.5.m5.2.2.2"></times><apply id="A1.Thmtheorem5.p1.5.5.m5.2.2.1.1.1.cmml" xref="A1.Thmtheorem5.p1.5.5.m5.2.2.1.1"><minus id="A1.Thmtheorem5.p1.5.5.m5.2.2.1.1.1.1.cmml" xref="A1.Thmtheorem5.p1.5.5.m5.2.2.1.1.1.1"></minus><cn id="A1.Thmtheorem5.p1.5.5.m5.2.2.1.1.1.2.cmml" type="integer" xref="A1.Thmtheorem5.p1.5.5.m5.2.2.1.1.1.2">1</cn><ci id="A1.Thmtheorem5.p1.5.5.m5.2.2.1.1.1.3.cmml" xref="A1.Thmtheorem5.p1.5.5.m5.2.2.1.1.1.3">𝜀</ci></apply><apply id="A1.Thmtheorem5.p1.5.5.m5.2.2.3.cmml" xref="A1.Thmtheorem5.p1.5.5.m5.2.2.3"><csymbol cd="ambiguous" id="A1.Thmtheorem5.p1.5.5.m5.2.2.3.1.cmml" xref="A1.Thmtheorem5.p1.5.5.m5.2.2.3">superscript</csymbol><ci id="A1.Thmtheorem5.p1.5.5.m5.2.2.3.2.cmml" xref="A1.Thmtheorem5.p1.5.5.m5.2.2.3.2">𝜌</ci><times id="A1.Thmtheorem5.p1.5.5.m5.2.2.3.3.cmml" xref="A1.Thmtheorem5.p1.5.5.m5.2.2.3.3"></times></apply><ci id="A1.Thmtheorem5.p1.5.5.m5.1.1.cmml" xref="A1.Thmtheorem5.p1.5.5.m5.1.1">ℓ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.Thmtheorem5.p1.5.5.m5.2c">(1-\varepsilon)\rho^{*}(\ell)</annotation><annotation encoding="application/x-llamapun" id="A1.Thmtheorem5.p1.5.5.m5.2d">( 1 - italic_ε ) italic_ρ start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ( roman_ℓ )</annotation></semantics></math>.</span></p> </div> </div> <div class="ltx_theorem ltx_theorem_proof" id="A1.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="A1.Thmtheorem6.1.1.1">Proof A.6</span></span><span class="ltx_text ltx_font_bold" id="A1.Thmtheorem6.2.2">.</span> </h6> <div class="ltx_para" id="A1.Thmtheorem6.p1"> <p class="ltx_p" id="A1.Thmtheorem6.p1.3"><span class="ltx_text ltx_font_italic" id="A1.Thmtheorem6.p1.3.3">It follows from Lemma <a class="ltx_ref" href="https://arxiv.org/html/2411.12694v2#A1.Thmtheorem3" title="Lemma A.3. ‣ A.3 Our algorithm ‣ Appendix A Reporting a densest subgraph ‣ Local Density and its Distributed Approximation"><span class="ltx_text ltx_ref_tag">A.3</span></a> that the <math alttext="\lceil\frac{32\log n}{\varepsilon}\rceil" class="ltx_Math" display="inline" id="A1.Thmtheorem6.p1.1.1.m1.1"><semantics id="A1.Thmtheorem6.p1.1.1.m1.1a"><mrow id="A1.Thmtheorem6.p1.1.1.m1.1.2.2" xref="A1.Thmtheorem6.p1.1.1.m1.1.2.1.cmml"><mo id="A1.Thmtheorem6.p1.1.1.m1.1.2.2.1" stretchy="false" xref="A1.Thmtheorem6.p1.1.1.m1.1.2.1.1.cmml">⌈</mo><mfrac id="A1.Thmtheorem6.p1.1.1.m1.1.1" xref="A1.Thmtheorem6.p1.1.1.m1.1.1.cmml"><mrow id="A1.Thmtheorem6.p1.1.1.m1.1.1.2" xref="A1.Thmtheorem6.p1.1.1.m1.1.1.2.cmml"><mn id="A1.Thmtheorem6.p1.1.1.m1.1.1.2.2" xref="A1.Thmtheorem6.p1.1.1.m1.1.1.2.2.cmml">32</mn><mo id="A1.Thmtheorem6.p1.1.1.m1.1.1.2.1" lspace="0.167em" xref="A1.Thmtheorem6.p1.1.1.m1.1.1.2.1.cmml">⁢</mo><mrow id="A1.Thmtheorem6.p1.1.1.m1.1.1.2.3" xref="A1.Thmtheorem6.p1.1.1.m1.1.1.2.3.cmml"><mi id="A1.Thmtheorem6.p1.1.1.m1.1.1.2.3.1" xref="A1.Thmtheorem6.p1.1.1.m1.1.1.2.3.1.cmml">log</mi><mo id="A1.Thmtheorem6.p1.1.1.m1.1.1.2.3a" lspace="0.167em" xref="A1.Thmtheorem6.p1.1.1.m1.1.1.2.3.cmml">⁡</mo><mi id="A1.Thmtheorem6.p1.1.1.m1.1.1.2.3.2" xref="A1.Thmtheorem6.p1.1.1.m1.1.1.2.3.2.cmml">n</mi></mrow></mrow><mi id="A1.Thmtheorem6.p1.1.1.m1.1.1.3" xref="A1.Thmtheorem6.p1.1.1.m1.1.1.3.cmml">ε</mi></mfrac><mo id="A1.Thmtheorem6.p1.1.1.m1.1.2.2.2" stretchy="false" xref="A1.Thmtheorem6.p1.1.1.m1.1.2.1.1.cmml">⌉</mo></mrow><annotation-xml encoding="MathML-Content" id="A1.Thmtheorem6.p1.1.1.m1.1b"><apply id="A1.Thmtheorem6.p1.1.1.m1.1.2.1.cmml" xref="A1.Thmtheorem6.p1.1.1.m1.1.2.2"><ceiling id="A1.Thmtheorem6.p1.1.1.m1.1.2.1.1.cmml" xref="A1.Thmtheorem6.p1.1.1.m1.1.2.2.1"></ceiling><apply id="A1.Thmtheorem6.p1.1.1.m1.1.1.cmml" xref="A1.Thmtheorem6.p1.1.1.m1.1.1"><divide id="A1.Thmtheorem6.p1.1.1.m1.1.1.1.cmml" xref="A1.Thmtheorem6.p1.1.1.m1.1.1"></divide><apply id="A1.Thmtheorem6.p1.1.1.m1.1.1.2.cmml" xref="A1.Thmtheorem6.p1.1.1.m1.1.1.2"><times id="A1.Thmtheorem6.p1.1.1.m1.1.1.2.1.cmml" xref="A1.Thmtheorem6.p1.1.1.m1.1.1.2.1"></times><cn id="A1.Thmtheorem6.p1.1.1.m1.1.1.2.2.cmml" type="integer" xref="A1.Thmtheorem6.p1.1.1.m1.1.1.2.2">32</cn><apply id="A1.Thmtheorem6.p1.1.1.m1.1.1.2.3.cmml" xref="A1.Thmtheorem6.p1.1.1.m1.1.1.2.3"><log id="A1.Thmtheorem6.p1.1.1.m1.1.1.2.3.1.cmml" xref="A1.Thmtheorem6.p1.1.1.m1.1.1.2.3.1"></log><ci id="A1.Thmtheorem6.p1.1.1.m1.1.1.2.3.2.cmml" xref="A1.Thmtheorem6.p1.1.1.m1.1.1.2.3.2">𝑛</ci></apply></apply><ci id="A1.Thmtheorem6.p1.1.1.m1.1.1.3.cmml" xref="A1.Thmtheorem6.p1.1.1.m1.1.1.3">𝜀</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.Thmtheorem6.p1.1.1.m1.1c">\lceil\frac{32\log n}{\varepsilon}\rceil</annotation><annotation encoding="application/x-llamapun" id="A1.Thmtheorem6.p1.1.1.m1.1d">⌈ divide start_ARG 32 roman_log italic_n end_ARG start_ARG italic_ε end_ARG ⌉</annotation></semantics></math>-hop neighbourhood of <math alttext="\ell" class="ltx_Math" display="inline" id="A1.Thmtheorem6.p1.2.2.m2.1"><semantics id="A1.Thmtheorem6.p1.2.2.m2.1a"><mi id="A1.Thmtheorem6.p1.2.2.m2.1.1" mathvariant="normal" xref="A1.Thmtheorem6.p1.2.2.m2.1.1.cmml">ℓ</mi><annotation-xml encoding="MathML-Content" id="A1.Thmtheorem6.p1.2.2.m2.1b"><ci id="A1.Thmtheorem6.p1.2.2.m2.1.1.cmml" xref="A1.Thmtheorem6.p1.2.2.m2.1.1">ℓ</ci></annotation-xml><annotation encoding="application/x-tex" id="A1.Thmtheorem6.p1.2.2.m2.1c">\ell</annotation><annotation encoding="application/x-llamapun" id="A1.Thmtheorem6.p1.2.2.m2.1d">roman_ℓ</annotation></semantics></math> contains a subgraph with density at least <math alttext="\rho^{*}(\ell)(1-\frac{\varepsilon}{16})" class="ltx_Math" display="inline" id="A1.Thmtheorem6.p1.3.3.m3.2"><semantics id="A1.Thmtheorem6.p1.3.3.m3.2a"><mrow id="A1.Thmtheorem6.p1.3.3.m3.2.2" xref="A1.Thmtheorem6.p1.3.3.m3.2.2.cmml"><msup id="A1.Thmtheorem6.p1.3.3.m3.2.2.3" xref="A1.Thmtheorem6.p1.3.3.m3.2.2.3.cmml"><mi id="A1.Thmtheorem6.p1.3.3.m3.2.2.3.2" xref="A1.Thmtheorem6.p1.3.3.m3.2.2.3.2.cmml">ρ</mi><mo id="A1.Thmtheorem6.p1.3.3.m3.2.2.3.3" xref="A1.Thmtheorem6.p1.3.3.m3.2.2.3.3.cmml">∗</mo></msup><mo id="A1.Thmtheorem6.p1.3.3.m3.2.2.2" xref="A1.Thmtheorem6.p1.3.3.m3.2.2.2.cmml">⁢</mo><mrow id="A1.Thmtheorem6.p1.3.3.m3.2.2.4.2" xref="A1.Thmtheorem6.p1.3.3.m3.2.2.cmml"><mo id="A1.Thmtheorem6.p1.3.3.m3.2.2.4.2.1" stretchy="false" xref="A1.Thmtheorem6.p1.3.3.m3.2.2.cmml">(</mo><mi id="A1.Thmtheorem6.p1.3.3.m3.1.1" mathvariant="normal" xref="A1.Thmtheorem6.p1.3.3.m3.1.1.cmml">ℓ</mi><mo id="A1.Thmtheorem6.p1.3.3.m3.2.2.4.2.2" stretchy="false" xref="A1.Thmtheorem6.p1.3.3.m3.2.2.cmml">)</mo></mrow><mo id="A1.Thmtheorem6.p1.3.3.m3.2.2.2a" xref="A1.Thmtheorem6.p1.3.3.m3.2.2.2.cmml">⁢</mo><mrow id="A1.Thmtheorem6.p1.3.3.m3.2.2.1.1" xref="A1.Thmtheorem6.p1.3.3.m3.2.2.1.1.1.cmml"><mo id="A1.Thmtheorem6.p1.3.3.m3.2.2.1.1.2" stretchy="false" xref="A1.Thmtheorem6.p1.3.3.m3.2.2.1.1.1.cmml">(</mo><mrow id="A1.Thmtheorem6.p1.3.3.m3.2.2.1.1.1" xref="A1.Thmtheorem6.p1.3.3.m3.2.2.1.1.1.cmml"><mn id="A1.Thmtheorem6.p1.3.3.m3.2.2.1.1.1.2" xref="A1.Thmtheorem6.p1.3.3.m3.2.2.1.1.1.2.cmml">1</mn><mo id="A1.Thmtheorem6.p1.3.3.m3.2.2.1.1.1.1" xref="A1.Thmtheorem6.p1.3.3.m3.2.2.1.1.1.1.cmml">−</mo><mfrac id="A1.Thmtheorem6.p1.3.3.m3.2.2.1.1.1.3" xref="A1.Thmtheorem6.p1.3.3.m3.2.2.1.1.1.3.cmml"><mi id="A1.Thmtheorem6.p1.3.3.m3.2.2.1.1.1.3.2" xref="A1.Thmtheorem6.p1.3.3.m3.2.2.1.1.1.3.2.cmml">ε</mi><mn id="A1.Thmtheorem6.p1.3.3.m3.2.2.1.1.1.3.3" xref="A1.Thmtheorem6.p1.3.3.m3.2.2.1.1.1.3.3.cmml">16</mn></mfrac></mrow><mo id="A1.Thmtheorem6.p1.3.3.m3.2.2.1.1.3" stretchy="false" xref="A1.Thmtheorem6.p1.3.3.m3.2.2.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="A1.Thmtheorem6.p1.3.3.m3.2b"><apply id="A1.Thmtheorem6.p1.3.3.m3.2.2.cmml" xref="A1.Thmtheorem6.p1.3.3.m3.2.2"><times id="A1.Thmtheorem6.p1.3.3.m3.2.2.2.cmml" xref="A1.Thmtheorem6.p1.3.3.m3.2.2.2"></times><apply id="A1.Thmtheorem6.p1.3.3.m3.2.2.3.cmml" xref="A1.Thmtheorem6.p1.3.3.m3.2.2.3"><csymbol cd="ambiguous" id="A1.Thmtheorem6.p1.3.3.m3.2.2.3.1.cmml" xref="A1.Thmtheorem6.p1.3.3.m3.2.2.3">superscript</csymbol><ci id="A1.Thmtheorem6.p1.3.3.m3.2.2.3.2.cmml" xref="A1.Thmtheorem6.p1.3.3.m3.2.2.3.2">𝜌</ci><times id="A1.Thmtheorem6.p1.3.3.m3.2.2.3.3.cmml" xref="A1.Thmtheorem6.p1.3.3.m3.2.2.3.3"></times></apply><ci id="A1.Thmtheorem6.p1.3.3.m3.1.1.cmml" xref="A1.Thmtheorem6.p1.3.3.m3.1.1">ℓ</ci><apply id="A1.Thmtheorem6.p1.3.3.m3.2.2.1.1.1.cmml" xref="A1.Thmtheorem6.p1.3.3.m3.2.2.1.1"><minus id="A1.Thmtheorem6.p1.3.3.m3.2.2.1.1.1.1.cmml" xref="A1.Thmtheorem6.p1.3.3.m3.2.2.1.1.1.1"></minus><cn id="A1.Thmtheorem6.p1.3.3.m3.2.2.1.1.1.2.cmml" type="integer" xref="A1.Thmtheorem6.p1.3.3.m3.2.2.1.1.1.2">1</cn><apply id="A1.Thmtheorem6.p1.3.3.m3.2.2.1.1.1.3.cmml" xref="A1.Thmtheorem6.p1.3.3.m3.2.2.1.1.1.3"><divide id="A1.Thmtheorem6.p1.3.3.m3.2.2.1.1.1.3.1.cmml" xref="A1.Thmtheorem6.p1.3.3.m3.2.2.1.1.1.3"></divide><ci id="A1.Thmtheorem6.p1.3.3.m3.2.2.1.1.1.3.2.cmml" xref="A1.Thmtheorem6.p1.3.3.m3.2.2.1.1.1.3.2">𝜀</ci><cn id="A1.Thmtheorem6.p1.3.3.m3.2.2.1.1.1.3.3.cmml" type="integer" xref="A1.Thmtheorem6.p1.3.3.m3.2.2.1.1.1.3.3">16</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.Thmtheorem6.p1.3.3.m3.2c">\rho^{*}(\ell)(1-\frac{\varepsilon}{16})</annotation><annotation encoding="application/x-llamapun" id="A1.Thmtheorem6.p1.3.3.m3.2d">italic_ρ start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ( roman_ℓ ) ( 1 - divide start_ARG italic_ε end_ARG start_ARG 16 end_ARG )</annotation></semantics></math>.</span></p> </div> <div class="ltx_para" id="A1.Thmtheorem6.p2"> <p class="ltx_p" id="A1.Thmtheorem6.p2.7"><span class="ltx_text ltx_font_italic" id="A1.Thmtheorem6.p2.7.7">This in turn means that the largest fractional out-degree <math alttext="h_{\max}" class="ltx_Math" display="inline" id="A1.Thmtheorem6.p2.1.1.m1.1"><semantics id="A1.Thmtheorem6.p2.1.1.m1.1a"><msub id="A1.Thmtheorem6.p2.1.1.m1.1.1" xref="A1.Thmtheorem6.p2.1.1.m1.1.1.cmml"><mi id="A1.Thmtheorem6.p2.1.1.m1.1.1.2" xref="A1.Thmtheorem6.p2.1.1.m1.1.1.2.cmml">h</mi><mi id="A1.Thmtheorem6.p2.1.1.m1.1.1.3" xref="A1.Thmtheorem6.p2.1.1.m1.1.1.3.cmml">max</mi></msub><annotation-xml encoding="MathML-Content" id="A1.Thmtheorem6.p2.1.1.m1.1b"><apply id="A1.Thmtheorem6.p2.1.1.m1.1.1.cmml" xref="A1.Thmtheorem6.p2.1.1.m1.1.1"><csymbol cd="ambiguous" id="A1.Thmtheorem6.p2.1.1.m1.1.1.1.cmml" xref="A1.Thmtheorem6.p2.1.1.m1.1.1">subscript</csymbol><ci id="A1.Thmtheorem6.p2.1.1.m1.1.1.2.cmml" xref="A1.Thmtheorem6.p2.1.1.m1.1.1.2">ℎ</ci><max id="A1.Thmtheorem6.p2.1.1.m1.1.1.3.cmml" xref="A1.Thmtheorem6.p2.1.1.m1.1.1.3"></max></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.Thmtheorem6.p2.1.1.m1.1c">h_{\max}</annotation><annotation encoding="application/x-llamapun" id="A1.Thmtheorem6.p2.1.1.m1.1d">italic_h start_POSTSUBSCRIPT roman_max end_POSTSUBSCRIPT</annotation></semantics></math> under the second <math alttext="\eta" class="ltx_Math" display="inline" id="A1.Thmtheorem6.p2.2.2.m2.1"><semantics id="A1.Thmtheorem6.p2.2.2.m2.1a"><mi id="A1.Thmtheorem6.p2.2.2.m2.1.1" xref="A1.Thmtheorem6.p2.2.2.m2.1.1.cmml">η</mi><annotation-xml encoding="MathML-Content" id="A1.Thmtheorem6.p2.2.2.m2.1b"><ci id="A1.Thmtheorem6.p2.2.2.m2.1.1.cmml" xref="A1.Thmtheorem6.p2.2.2.m2.1.1">𝜂</ci></annotation-xml><annotation encoding="application/x-tex" id="A1.Thmtheorem6.p2.2.2.m2.1c">\eta</annotation><annotation encoding="application/x-llamapun" id="A1.Thmtheorem6.p2.2.2.m2.1d">italic_η</annotation></semantics></math>-fair out-orientation computed only on the <math alttext="\lceil\frac{32\log n}{\varepsilon}\rceil" class="ltx_Math" display="inline" id="A1.Thmtheorem6.p2.3.3.m3.1"><semantics id="A1.Thmtheorem6.p2.3.3.m3.1a"><mrow id="A1.Thmtheorem6.p2.3.3.m3.1.2.2" xref="A1.Thmtheorem6.p2.3.3.m3.1.2.1.cmml"><mo id="A1.Thmtheorem6.p2.3.3.m3.1.2.2.1" stretchy="false" xref="A1.Thmtheorem6.p2.3.3.m3.1.2.1.1.cmml">⌈</mo><mfrac id="A1.Thmtheorem6.p2.3.3.m3.1.1" xref="A1.Thmtheorem6.p2.3.3.m3.1.1.cmml"><mrow id="A1.Thmtheorem6.p2.3.3.m3.1.1.2" xref="A1.Thmtheorem6.p2.3.3.m3.1.1.2.cmml"><mn id="A1.Thmtheorem6.p2.3.3.m3.1.1.2.2" xref="A1.Thmtheorem6.p2.3.3.m3.1.1.2.2.cmml">32</mn><mo id="A1.Thmtheorem6.p2.3.3.m3.1.1.2.1" lspace="0.167em" xref="A1.Thmtheorem6.p2.3.3.m3.1.1.2.1.cmml">⁢</mo><mrow id="A1.Thmtheorem6.p2.3.3.m3.1.1.2.3" xref="A1.Thmtheorem6.p2.3.3.m3.1.1.2.3.cmml"><mi id="A1.Thmtheorem6.p2.3.3.m3.1.1.2.3.1" xref="A1.Thmtheorem6.p2.3.3.m3.1.1.2.3.1.cmml">log</mi><mo id="A1.Thmtheorem6.p2.3.3.m3.1.1.2.3a" lspace="0.167em" xref="A1.Thmtheorem6.p2.3.3.m3.1.1.2.3.cmml">⁡</mo><mi id="A1.Thmtheorem6.p2.3.3.m3.1.1.2.3.2" xref="A1.Thmtheorem6.p2.3.3.m3.1.1.2.3.2.cmml">n</mi></mrow></mrow><mi id="A1.Thmtheorem6.p2.3.3.m3.1.1.3" xref="A1.Thmtheorem6.p2.3.3.m3.1.1.3.cmml">ε</mi></mfrac><mo id="A1.Thmtheorem6.p2.3.3.m3.1.2.2.2" stretchy="false" xref="A1.Thmtheorem6.p2.3.3.m3.1.2.1.1.cmml">⌉</mo></mrow><annotation-xml encoding="MathML-Content" id="A1.Thmtheorem6.p2.3.3.m3.1b"><apply id="A1.Thmtheorem6.p2.3.3.m3.1.2.1.cmml" xref="A1.Thmtheorem6.p2.3.3.m3.1.2.2"><ceiling id="A1.Thmtheorem6.p2.3.3.m3.1.2.1.1.cmml" xref="A1.Thmtheorem6.p2.3.3.m3.1.2.2.1"></ceiling><apply id="A1.Thmtheorem6.p2.3.3.m3.1.1.cmml" xref="A1.Thmtheorem6.p2.3.3.m3.1.1"><divide id="A1.Thmtheorem6.p2.3.3.m3.1.1.1.cmml" xref="A1.Thmtheorem6.p2.3.3.m3.1.1"></divide><apply id="A1.Thmtheorem6.p2.3.3.m3.1.1.2.cmml" xref="A1.Thmtheorem6.p2.3.3.m3.1.1.2"><times id="A1.Thmtheorem6.p2.3.3.m3.1.1.2.1.cmml" xref="A1.Thmtheorem6.p2.3.3.m3.1.1.2.1"></times><cn id="A1.Thmtheorem6.p2.3.3.m3.1.1.2.2.cmml" type="integer" xref="A1.Thmtheorem6.p2.3.3.m3.1.1.2.2">32</cn><apply id="A1.Thmtheorem6.p2.3.3.m3.1.1.2.3.cmml" xref="A1.Thmtheorem6.p2.3.3.m3.1.1.2.3"><log id="A1.Thmtheorem6.p2.3.3.m3.1.1.2.3.1.cmml" xref="A1.Thmtheorem6.p2.3.3.m3.1.1.2.3.1"></log><ci id="A1.Thmtheorem6.p2.3.3.m3.1.1.2.3.2.cmml" xref="A1.Thmtheorem6.p2.3.3.m3.1.1.2.3.2">𝑛</ci></apply></apply><ci id="A1.Thmtheorem6.p2.3.3.m3.1.1.3.cmml" xref="A1.Thmtheorem6.p2.3.3.m3.1.1.3">𝜀</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.Thmtheorem6.p2.3.3.m3.1c">\lceil\frac{32\log n}{\varepsilon}\rceil</annotation><annotation encoding="application/x-llamapun" id="A1.Thmtheorem6.p2.3.3.m3.1d">⌈ divide start_ARG 32 roman_log italic_n end_ARG start_ARG italic_ε end_ARG ⌉</annotation></semantics></math>-hop neighbourhood of <math alttext="\ell" class="ltx_Math" display="inline" id="A1.Thmtheorem6.p2.4.4.m4.1"><semantics id="A1.Thmtheorem6.p2.4.4.m4.1a"><mi id="A1.Thmtheorem6.p2.4.4.m4.1.1" mathvariant="normal" xref="A1.Thmtheorem6.p2.4.4.m4.1.1.cmml">ℓ</mi><annotation-xml encoding="MathML-Content" id="A1.Thmtheorem6.p2.4.4.m4.1b"><ci id="A1.Thmtheorem6.p2.4.4.m4.1.1.cmml" xref="A1.Thmtheorem6.p2.4.4.m4.1.1">ℓ</ci></annotation-xml><annotation encoding="application/x-tex" id="A1.Thmtheorem6.p2.4.4.m4.1c">\ell</annotation><annotation encoding="application/x-llamapun" id="A1.Thmtheorem6.p2.4.4.m4.1d">roman_ℓ</annotation></semantics></math> is at least <math alttext="\rho^{*}(\ell)(1-\frac{\varepsilon}{16})" class="ltx_Math" display="inline" id="A1.Thmtheorem6.p2.5.5.m5.2"><semantics id="A1.Thmtheorem6.p2.5.5.m5.2a"><mrow id="A1.Thmtheorem6.p2.5.5.m5.2.2" xref="A1.Thmtheorem6.p2.5.5.m5.2.2.cmml"><msup id="A1.Thmtheorem6.p2.5.5.m5.2.2.3" xref="A1.Thmtheorem6.p2.5.5.m5.2.2.3.cmml"><mi id="A1.Thmtheorem6.p2.5.5.m5.2.2.3.2" xref="A1.Thmtheorem6.p2.5.5.m5.2.2.3.2.cmml">ρ</mi><mo id="A1.Thmtheorem6.p2.5.5.m5.2.2.3.3" xref="A1.Thmtheorem6.p2.5.5.m5.2.2.3.3.cmml">∗</mo></msup><mo id="A1.Thmtheorem6.p2.5.5.m5.2.2.2" xref="A1.Thmtheorem6.p2.5.5.m5.2.2.2.cmml">⁢</mo><mrow id="A1.Thmtheorem6.p2.5.5.m5.2.2.4.2" xref="A1.Thmtheorem6.p2.5.5.m5.2.2.cmml"><mo id="A1.Thmtheorem6.p2.5.5.m5.2.2.4.2.1" stretchy="false" xref="A1.Thmtheorem6.p2.5.5.m5.2.2.cmml">(</mo><mi id="A1.Thmtheorem6.p2.5.5.m5.1.1" mathvariant="normal" xref="A1.Thmtheorem6.p2.5.5.m5.1.1.cmml">ℓ</mi><mo id="A1.Thmtheorem6.p2.5.5.m5.2.2.4.2.2" stretchy="false" xref="A1.Thmtheorem6.p2.5.5.m5.2.2.cmml">)</mo></mrow><mo id="A1.Thmtheorem6.p2.5.5.m5.2.2.2a" xref="A1.Thmtheorem6.p2.5.5.m5.2.2.2.cmml">⁢</mo><mrow id="A1.Thmtheorem6.p2.5.5.m5.2.2.1.1" xref="A1.Thmtheorem6.p2.5.5.m5.2.2.1.1.1.cmml"><mo id="A1.Thmtheorem6.p2.5.5.m5.2.2.1.1.2" stretchy="false" xref="A1.Thmtheorem6.p2.5.5.m5.2.2.1.1.1.cmml">(</mo><mrow id="A1.Thmtheorem6.p2.5.5.m5.2.2.1.1.1" xref="A1.Thmtheorem6.p2.5.5.m5.2.2.1.1.1.cmml"><mn id="A1.Thmtheorem6.p2.5.5.m5.2.2.1.1.1.2" xref="A1.Thmtheorem6.p2.5.5.m5.2.2.1.1.1.2.cmml">1</mn><mo id="A1.Thmtheorem6.p2.5.5.m5.2.2.1.1.1.1" xref="A1.Thmtheorem6.p2.5.5.m5.2.2.1.1.1.1.cmml">−</mo><mfrac id="A1.Thmtheorem6.p2.5.5.m5.2.2.1.1.1.3" xref="A1.Thmtheorem6.p2.5.5.m5.2.2.1.1.1.3.cmml"><mi id="A1.Thmtheorem6.p2.5.5.m5.2.2.1.1.1.3.2" xref="A1.Thmtheorem6.p2.5.5.m5.2.2.1.1.1.3.2.cmml">ε</mi><mn id="A1.Thmtheorem6.p2.5.5.m5.2.2.1.1.1.3.3" xref="A1.Thmtheorem6.p2.5.5.m5.2.2.1.1.1.3.3.cmml">16</mn></mfrac></mrow><mo id="A1.Thmtheorem6.p2.5.5.m5.2.2.1.1.3" stretchy="false" xref="A1.Thmtheorem6.p2.5.5.m5.2.2.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="A1.Thmtheorem6.p2.5.5.m5.2b"><apply id="A1.Thmtheorem6.p2.5.5.m5.2.2.cmml" xref="A1.Thmtheorem6.p2.5.5.m5.2.2"><times id="A1.Thmtheorem6.p2.5.5.m5.2.2.2.cmml" xref="A1.Thmtheorem6.p2.5.5.m5.2.2.2"></times><apply id="A1.Thmtheorem6.p2.5.5.m5.2.2.3.cmml" xref="A1.Thmtheorem6.p2.5.5.m5.2.2.3"><csymbol cd="ambiguous" id="A1.Thmtheorem6.p2.5.5.m5.2.2.3.1.cmml" xref="A1.Thmtheorem6.p2.5.5.m5.2.2.3">superscript</csymbol><ci id="A1.Thmtheorem6.p2.5.5.m5.2.2.3.2.cmml" xref="A1.Thmtheorem6.p2.5.5.m5.2.2.3.2">𝜌</ci><times id="A1.Thmtheorem6.p2.5.5.m5.2.2.3.3.cmml" xref="A1.Thmtheorem6.p2.5.5.m5.2.2.3.3"></times></apply><ci id="A1.Thmtheorem6.p2.5.5.m5.1.1.cmml" xref="A1.Thmtheorem6.p2.5.5.m5.1.1">ℓ</ci><apply id="A1.Thmtheorem6.p2.5.5.m5.2.2.1.1.1.cmml" xref="A1.Thmtheorem6.p2.5.5.m5.2.2.1.1"><minus id="A1.Thmtheorem6.p2.5.5.m5.2.2.1.1.1.1.cmml" xref="A1.Thmtheorem6.p2.5.5.m5.2.2.1.1.1.1"></minus><cn id="A1.Thmtheorem6.p2.5.5.m5.2.2.1.1.1.2.cmml" type="integer" xref="A1.Thmtheorem6.p2.5.5.m5.2.2.1.1.1.2">1</cn><apply id="A1.Thmtheorem6.p2.5.5.m5.2.2.1.1.1.3.cmml" xref="A1.Thmtheorem6.p2.5.5.m5.2.2.1.1.1.3"><divide id="A1.Thmtheorem6.p2.5.5.m5.2.2.1.1.1.3.1.cmml" xref="A1.Thmtheorem6.p2.5.5.m5.2.2.1.1.1.3"></divide><ci id="A1.Thmtheorem6.p2.5.5.m5.2.2.1.1.1.3.2.cmml" xref="A1.Thmtheorem6.p2.5.5.m5.2.2.1.1.1.3.2">𝜀</ci><cn id="A1.Thmtheorem6.p2.5.5.m5.2.2.1.1.1.3.3.cmml" type="integer" xref="A1.Thmtheorem6.p2.5.5.m5.2.2.1.1.1.3.3">16</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.Thmtheorem6.p2.5.5.m5.2c">\rho^{*}(\ell)(1-\frac{\varepsilon}{16})</annotation><annotation encoding="application/x-llamapun" id="A1.Thmtheorem6.p2.5.5.m5.2d">italic_ρ start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ( roman_ℓ ) ( 1 - divide start_ARG italic_ε end_ARG start_ARG 16 end_ARG )</annotation></semantics></math>. Indeed, this neighbourhood has maximum subgraph density at least <math alttext="\rho^{*}(\ell)(1-\frac{\varepsilon}{16})" class="ltx_Math" display="inline" id="A1.Thmtheorem6.p2.6.6.m6.2"><semantics id="A1.Thmtheorem6.p2.6.6.m6.2a"><mrow id="A1.Thmtheorem6.p2.6.6.m6.2.2" xref="A1.Thmtheorem6.p2.6.6.m6.2.2.cmml"><msup id="A1.Thmtheorem6.p2.6.6.m6.2.2.3" xref="A1.Thmtheorem6.p2.6.6.m6.2.2.3.cmml"><mi id="A1.Thmtheorem6.p2.6.6.m6.2.2.3.2" xref="A1.Thmtheorem6.p2.6.6.m6.2.2.3.2.cmml">ρ</mi><mo id="A1.Thmtheorem6.p2.6.6.m6.2.2.3.3" xref="A1.Thmtheorem6.p2.6.6.m6.2.2.3.3.cmml">∗</mo></msup><mo id="A1.Thmtheorem6.p2.6.6.m6.2.2.2" xref="A1.Thmtheorem6.p2.6.6.m6.2.2.2.cmml">⁢</mo><mrow id="A1.Thmtheorem6.p2.6.6.m6.2.2.4.2" xref="A1.Thmtheorem6.p2.6.6.m6.2.2.cmml"><mo id="A1.Thmtheorem6.p2.6.6.m6.2.2.4.2.1" stretchy="false" xref="A1.Thmtheorem6.p2.6.6.m6.2.2.cmml">(</mo><mi id="A1.Thmtheorem6.p2.6.6.m6.1.1" mathvariant="normal" xref="A1.Thmtheorem6.p2.6.6.m6.1.1.cmml">ℓ</mi><mo id="A1.Thmtheorem6.p2.6.6.m6.2.2.4.2.2" stretchy="false" xref="A1.Thmtheorem6.p2.6.6.m6.2.2.cmml">)</mo></mrow><mo id="A1.Thmtheorem6.p2.6.6.m6.2.2.2a" xref="A1.Thmtheorem6.p2.6.6.m6.2.2.2.cmml">⁢</mo><mrow id="A1.Thmtheorem6.p2.6.6.m6.2.2.1.1" xref="A1.Thmtheorem6.p2.6.6.m6.2.2.1.1.1.cmml"><mo id="A1.Thmtheorem6.p2.6.6.m6.2.2.1.1.2" stretchy="false" xref="A1.Thmtheorem6.p2.6.6.m6.2.2.1.1.1.cmml">(</mo><mrow id="A1.Thmtheorem6.p2.6.6.m6.2.2.1.1.1" xref="A1.Thmtheorem6.p2.6.6.m6.2.2.1.1.1.cmml"><mn id="A1.Thmtheorem6.p2.6.6.m6.2.2.1.1.1.2" xref="A1.Thmtheorem6.p2.6.6.m6.2.2.1.1.1.2.cmml">1</mn><mo id="A1.Thmtheorem6.p2.6.6.m6.2.2.1.1.1.1" xref="A1.Thmtheorem6.p2.6.6.m6.2.2.1.1.1.1.cmml">−</mo><mfrac id="A1.Thmtheorem6.p2.6.6.m6.2.2.1.1.1.3" xref="A1.Thmtheorem6.p2.6.6.m6.2.2.1.1.1.3.cmml"><mi id="A1.Thmtheorem6.p2.6.6.m6.2.2.1.1.1.3.2" xref="A1.Thmtheorem6.p2.6.6.m6.2.2.1.1.1.3.2.cmml">ε</mi><mn id="A1.Thmtheorem6.p2.6.6.m6.2.2.1.1.1.3.3" xref="A1.Thmtheorem6.p2.6.6.m6.2.2.1.1.1.3.3.cmml">16</mn></mfrac></mrow><mo id="A1.Thmtheorem6.p2.6.6.m6.2.2.1.1.3" stretchy="false" xref="A1.Thmtheorem6.p2.6.6.m6.2.2.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="A1.Thmtheorem6.p2.6.6.m6.2b"><apply id="A1.Thmtheorem6.p2.6.6.m6.2.2.cmml" xref="A1.Thmtheorem6.p2.6.6.m6.2.2"><times id="A1.Thmtheorem6.p2.6.6.m6.2.2.2.cmml" xref="A1.Thmtheorem6.p2.6.6.m6.2.2.2"></times><apply id="A1.Thmtheorem6.p2.6.6.m6.2.2.3.cmml" xref="A1.Thmtheorem6.p2.6.6.m6.2.2.3"><csymbol cd="ambiguous" id="A1.Thmtheorem6.p2.6.6.m6.2.2.3.1.cmml" xref="A1.Thmtheorem6.p2.6.6.m6.2.2.3">superscript</csymbol><ci id="A1.Thmtheorem6.p2.6.6.m6.2.2.3.2.cmml" xref="A1.Thmtheorem6.p2.6.6.m6.2.2.3.2">𝜌</ci><times id="A1.Thmtheorem6.p2.6.6.m6.2.2.3.3.cmml" xref="A1.Thmtheorem6.p2.6.6.m6.2.2.3.3"></times></apply><ci id="A1.Thmtheorem6.p2.6.6.m6.1.1.cmml" xref="A1.Thmtheorem6.p2.6.6.m6.1.1">ℓ</ci><apply id="A1.Thmtheorem6.p2.6.6.m6.2.2.1.1.1.cmml" xref="A1.Thmtheorem6.p2.6.6.m6.2.2.1.1"><minus id="A1.Thmtheorem6.p2.6.6.m6.2.2.1.1.1.1.cmml" xref="A1.Thmtheorem6.p2.6.6.m6.2.2.1.1.1.1"></minus><cn id="A1.Thmtheorem6.p2.6.6.m6.2.2.1.1.1.2.cmml" type="integer" xref="A1.Thmtheorem6.p2.6.6.m6.2.2.1.1.1.2">1</cn><apply id="A1.Thmtheorem6.p2.6.6.m6.2.2.1.1.1.3.cmml" xref="A1.Thmtheorem6.p2.6.6.m6.2.2.1.1.1.3"><divide id="A1.Thmtheorem6.p2.6.6.m6.2.2.1.1.1.3.1.cmml" xref="A1.Thmtheorem6.p2.6.6.m6.2.2.1.1.1.3"></divide><ci id="A1.Thmtheorem6.p2.6.6.m6.2.2.1.1.1.3.2.cmml" xref="A1.Thmtheorem6.p2.6.6.m6.2.2.1.1.1.3.2">𝜀</ci><cn id="A1.Thmtheorem6.p2.6.6.m6.2.2.1.1.1.3.3.cmml" type="integer" xref="A1.Thmtheorem6.p2.6.6.m6.2.2.1.1.1.3.3">16</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.Thmtheorem6.p2.6.6.m6.2c">\rho^{*}(\ell)(1-\frac{\varepsilon}{16})</annotation><annotation encoding="application/x-llamapun" id="A1.Thmtheorem6.p2.6.6.m6.2d">italic_ρ start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ( roman_ℓ ) ( 1 - divide start_ARG italic_ε end_ARG start_ARG 16 end_ARG )</annotation></semantics></math> and so the largest fractional out-degree has to be at least <math alttext="\rho^{*}(\ell)(1-\frac{\varepsilon}{16})" class="ltx_Math" display="inline" id="A1.Thmtheorem6.p2.7.7.m7.2"><semantics id="A1.Thmtheorem6.p2.7.7.m7.2a"><mrow id="A1.Thmtheorem6.p2.7.7.m7.2.2" xref="A1.Thmtheorem6.p2.7.7.m7.2.2.cmml"><msup id="A1.Thmtheorem6.p2.7.7.m7.2.2.3" xref="A1.Thmtheorem6.p2.7.7.m7.2.2.3.cmml"><mi id="A1.Thmtheorem6.p2.7.7.m7.2.2.3.2" xref="A1.Thmtheorem6.p2.7.7.m7.2.2.3.2.cmml">ρ</mi><mo id="A1.Thmtheorem6.p2.7.7.m7.2.2.3.3" xref="A1.Thmtheorem6.p2.7.7.m7.2.2.3.3.cmml">∗</mo></msup><mo id="A1.Thmtheorem6.p2.7.7.m7.2.2.2" xref="A1.Thmtheorem6.p2.7.7.m7.2.2.2.cmml">⁢</mo><mrow id="A1.Thmtheorem6.p2.7.7.m7.2.2.4.2" xref="A1.Thmtheorem6.p2.7.7.m7.2.2.cmml"><mo id="A1.Thmtheorem6.p2.7.7.m7.2.2.4.2.1" stretchy="false" xref="A1.Thmtheorem6.p2.7.7.m7.2.2.cmml">(</mo><mi id="A1.Thmtheorem6.p2.7.7.m7.1.1" mathvariant="normal" xref="A1.Thmtheorem6.p2.7.7.m7.1.1.cmml">ℓ</mi><mo id="A1.Thmtheorem6.p2.7.7.m7.2.2.4.2.2" stretchy="false" xref="A1.Thmtheorem6.p2.7.7.m7.2.2.cmml">)</mo></mrow><mo id="A1.Thmtheorem6.p2.7.7.m7.2.2.2a" xref="A1.Thmtheorem6.p2.7.7.m7.2.2.2.cmml">⁢</mo><mrow id="A1.Thmtheorem6.p2.7.7.m7.2.2.1.1" xref="A1.Thmtheorem6.p2.7.7.m7.2.2.1.1.1.cmml"><mo id="A1.Thmtheorem6.p2.7.7.m7.2.2.1.1.2" stretchy="false" xref="A1.Thmtheorem6.p2.7.7.m7.2.2.1.1.1.cmml">(</mo><mrow id="A1.Thmtheorem6.p2.7.7.m7.2.2.1.1.1" xref="A1.Thmtheorem6.p2.7.7.m7.2.2.1.1.1.cmml"><mn id="A1.Thmtheorem6.p2.7.7.m7.2.2.1.1.1.2" xref="A1.Thmtheorem6.p2.7.7.m7.2.2.1.1.1.2.cmml">1</mn><mo id="A1.Thmtheorem6.p2.7.7.m7.2.2.1.1.1.1" xref="A1.Thmtheorem6.p2.7.7.m7.2.2.1.1.1.1.cmml">−</mo><mfrac id="A1.Thmtheorem6.p2.7.7.m7.2.2.1.1.1.3" xref="A1.Thmtheorem6.p2.7.7.m7.2.2.1.1.1.3.cmml"><mi id="A1.Thmtheorem6.p2.7.7.m7.2.2.1.1.1.3.2" xref="A1.Thmtheorem6.p2.7.7.m7.2.2.1.1.1.3.2.cmml">ε</mi><mn id="A1.Thmtheorem6.p2.7.7.m7.2.2.1.1.1.3.3" xref="A1.Thmtheorem6.p2.7.7.m7.2.2.1.1.1.3.3.cmml">16</mn></mfrac></mrow><mo id="A1.Thmtheorem6.p2.7.7.m7.2.2.1.1.3" stretchy="false" xref="A1.Thmtheorem6.p2.7.7.m7.2.2.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="A1.Thmtheorem6.p2.7.7.m7.2b"><apply id="A1.Thmtheorem6.p2.7.7.m7.2.2.cmml" xref="A1.Thmtheorem6.p2.7.7.m7.2.2"><times id="A1.Thmtheorem6.p2.7.7.m7.2.2.2.cmml" xref="A1.Thmtheorem6.p2.7.7.m7.2.2.2"></times><apply id="A1.Thmtheorem6.p2.7.7.m7.2.2.3.cmml" xref="A1.Thmtheorem6.p2.7.7.m7.2.2.3"><csymbol cd="ambiguous" id="A1.Thmtheorem6.p2.7.7.m7.2.2.3.1.cmml" xref="A1.Thmtheorem6.p2.7.7.m7.2.2.3">superscript</csymbol><ci id="A1.Thmtheorem6.p2.7.7.m7.2.2.3.2.cmml" xref="A1.Thmtheorem6.p2.7.7.m7.2.2.3.2">𝜌</ci><times id="A1.Thmtheorem6.p2.7.7.m7.2.2.3.3.cmml" xref="A1.Thmtheorem6.p2.7.7.m7.2.2.3.3"></times></apply><ci id="A1.Thmtheorem6.p2.7.7.m7.1.1.cmml" xref="A1.Thmtheorem6.p2.7.7.m7.1.1">ℓ</ci><apply id="A1.Thmtheorem6.p2.7.7.m7.2.2.1.1.1.cmml" xref="A1.Thmtheorem6.p2.7.7.m7.2.2.1.1"><minus id="A1.Thmtheorem6.p2.7.7.m7.2.2.1.1.1.1.cmml" xref="A1.Thmtheorem6.p2.7.7.m7.2.2.1.1.1.1"></minus><cn id="A1.Thmtheorem6.p2.7.7.m7.2.2.1.1.1.2.cmml" type="integer" xref="A1.Thmtheorem6.p2.7.7.m7.2.2.1.1.1.2">1</cn><apply id="A1.Thmtheorem6.p2.7.7.m7.2.2.1.1.1.3.cmml" xref="A1.Thmtheorem6.p2.7.7.m7.2.2.1.1.1.3"><divide id="A1.Thmtheorem6.p2.7.7.m7.2.2.1.1.1.3.1.cmml" xref="A1.Thmtheorem6.p2.7.7.m7.2.2.1.1.1.3"></divide><ci id="A1.Thmtheorem6.p2.7.7.m7.2.2.1.1.1.3.2.cmml" xref="A1.Thmtheorem6.p2.7.7.m7.2.2.1.1.1.3.2">𝜀</ci><cn id="A1.Thmtheorem6.p2.7.7.m7.2.2.1.1.1.3.3.cmml" type="integer" xref="A1.Thmtheorem6.p2.7.7.m7.2.2.1.1.1.3.3">16</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.Thmtheorem6.p2.7.7.m7.2c">\rho^{*}(\ell)(1-\frac{\varepsilon}{16})</annotation><annotation encoding="application/x-llamapun" id="A1.Thmtheorem6.p2.7.7.m7.2d">italic_ρ start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ( roman_ℓ ) ( 1 - divide start_ARG italic_ε end_ARG start_ARG 16 end_ARG )</annotation></semantics></math>.</span></p> </div> <div class="ltx_para" id="A1.Thmtheorem6.p3"> <p class="ltx_p" id="A1.Thmtheorem6.p3.2"><span class="ltx_text ltx_font_italic" id="A1.Thmtheorem6.p3.2.2">This means that the density of the subgraph induced by all vertices outputting <math alttext="1" class="ltx_Math" display="inline" id="A1.Thmtheorem6.p3.1.1.m1.1"><semantics id="A1.Thmtheorem6.p3.1.1.m1.1a"><mn id="A1.Thmtheorem6.p3.1.1.m1.1.1" xref="A1.Thmtheorem6.p3.1.1.m1.1.1.cmml">1</mn><annotation-xml encoding="MathML-Content" id="A1.Thmtheorem6.p3.1.1.m1.1b"><cn id="A1.Thmtheorem6.p3.1.1.m1.1.1.cmml" type="integer" xref="A1.Thmtheorem6.p3.1.1.m1.1.1">1</cn></annotation-xml><annotation encoding="application/x-tex" id="A1.Thmtheorem6.p3.1.1.m1.1c">1</annotation><annotation encoding="application/x-llamapun" id="A1.Thmtheorem6.p3.1.1.m1.1d">1</annotation></semantics></math>, call this subgraph <math alttext="H(\ell)" class="ltx_Math" display="inline" id="A1.Thmtheorem6.p3.2.2.m2.1"><semantics id="A1.Thmtheorem6.p3.2.2.m2.1a"><mrow id="A1.Thmtheorem6.p3.2.2.m2.1.2" xref="A1.Thmtheorem6.p3.2.2.m2.1.2.cmml"><mi id="A1.Thmtheorem6.p3.2.2.m2.1.2.2" xref="A1.Thmtheorem6.p3.2.2.m2.1.2.2.cmml">H</mi><mo id="A1.Thmtheorem6.p3.2.2.m2.1.2.1" xref="A1.Thmtheorem6.p3.2.2.m2.1.2.1.cmml">⁢</mo><mrow id="A1.Thmtheorem6.p3.2.2.m2.1.2.3.2" xref="A1.Thmtheorem6.p3.2.2.m2.1.2.cmml"><mo id="A1.Thmtheorem6.p3.2.2.m2.1.2.3.2.1" stretchy="false" xref="A1.Thmtheorem6.p3.2.2.m2.1.2.cmml">(</mo><mi id="A1.Thmtheorem6.p3.2.2.m2.1.1" mathvariant="normal" xref="A1.Thmtheorem6.p3.2.2.m2.1.1.cmml">ℓ</mi><mo id="A1.Thmtheorem6.p3.2.2.m2.1.2.3.2.2" stretchy="false" xref="A1.Thmtheorem6.p3.2.2.m2.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="A1.Thmtheorem6.p3.2.2.m2.1b"><apply id="A1.Thmtheorem6.p3.2.2.m2.1.2.cmml" xref="A1.Thmtheorem6.p3.2.2.m2.1.2"><times id="A1.Thmtheorem6.p3.2.2.m2.1.2.1.cmml" xref="A1.Thmtheorem6.p3.2.2.m2.1.2.1"></times><ci id="A1.Thmtheorem6.p3.2.2.m2.1.2.2.cmml" xref="A1.Thmtheorem6.p3.2.2.m2.1.2.2">𝐻</ci><ci id="A1.Thmtheorem6.p3.2.2.m2.1.1.cmml" xref="A1.Thmtheorem6.p3.2.2.m2.1.1">ℓ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.Thmtheorem6.p3.2.2.m2.1c">H(\ell)</annotation><annotation encoding="application/x-llamapun" id="A1.Thmtheorem6.p3.2.2.m2.1d">italic_H ( roman_ℓ )</annotation></semantics></math>, can be bounded as follows:</span></p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="A1.EGx9"> <tbody id="A1.Ex36"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_left_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle\rho(H(\ell))" class="ltx_Math" display="inline" id="A1.Ex36.m1.2"><semantics id="A1.Ex36.m1.2a"><mrow id="A1.Ex36.m1.2.2" xref="A1.Ex36.m1.2.2.cmml"><mi id="A1.Ex36.m1.2.2.3" xref="A1.Ex36.m1.2.2.3.cmml">ρ</mi><mo id="A1.Ex36.m1.2.2.2" xref="A1.Ex36.m1.2.2.2.cmml">⁢</mo><mrow id="A1.Ex36.m1.2.2.1.1" xref="A1.Ex36.m1.2.2.1.1.1.cmml"><mo id="A1.Ex36.m1.2.2.1.1.2" stretchy="false" xref="A1.Ex36.m1.2.2.1.1.1.cmml">(</mo><mrow id="A1.Ex36.m1.2.2.1.1.1" xref="A1.Ex36.m1.2.2.1.1.1.cmml"><mi id="A1.Ex36.m1.2.2.1.1.1.2" xref="A1.Ex36.m1.2.2.1.1.1.2.cmml">H</mi><mo id="A1.Ex36.m1.2.2.1.1.1.1" xref="A1.Ex36.m1.2.2.1.1.1.1.cmml">⁢</mo><mrow id="A1.Ex36.m1.2.2.1.1.1.3.2" xref="A1.Ex36.m1.2.2.1.1.1.cmml"><mo id="A1.Ex36.m1.2.2.1.1.1.3.2.1" stretchy="false" xref="A1.Ex36.m1.2.2.1.1.1.cmml">(</mo><mi id="A1.Ex36.m1.1.1" mathvariant="normal" xref="A1.Ex36.m1.1.1.cmml">ℓ</mi><mo id="A1.Ex36.m1.2.2.1.1.1.3.2.2" stretchy="false" xref="A1.Ex36.m1.2.2.1.1.1.cmml">)</mo></mrow></mrow><mo id="A1.Ex36.m1.2.2.1.1.3" stretchy="false" xref="A1.Ex36.m1.2.2.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="A1.Ex36.m1.2b"><apply id="A1.Ex36.m1.2.2.cmml" xref="A1.Ex36.m1.2.2"><times id="A1.Ex36.m1.2.2.2.cmml" xref="A1.Ex36.m1.2.2.2"></times><ci id="A1.Ex36.m1.2.2.3.cmml" xref="A1.Ex36.m1.2.2.3">𝜌</ci><apply id="A1.Ex36.m1.2.2.1.1.1.cmml" xref="A1.Ex36.m1.2.2.1.1"><times id="A1.Ex36.m1.2.2.1.1.1.1.cmml" xref="A1.Ex36.m1.2.2.1.1.1.1"></times><ci id="A1.Ex36.m1.2.2.1.1.1.2.cmml" xref="A1.Ex36.m1.2.2.1.1.1.2">𝐻</ci><ci id="A1.Ex36.m1.1.1.cmml" xref="A1.Ex36.m1.1.1">ℓ</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.Ex36.m1.2c">\displaystyle\rho(H(\ell))</annotation><annotation encoding="application/x-llamapun" id="A1.Ex36.m1.2d">italic_ρ ( italic_H ( roman_ℓ ) )</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle\geq\frac{\sum_{v\in T_{k}}g(v)}{|T_{k+1}|}\geq\frac{|T_{k}|\cdot% {}h_{\max}\cdot{}(1+\eta)^{-k}}{(1+\frac{\varepsilon}{16})|T_{k}|}\geq h_{\max% }\frac{1}{(1+\frac{5\varepsilon}{8})}\Rightarrow" class="ltx_Math" display="inline" id="A1.Ex36.m2.7"><semantics id="A1.Ex36.m2.7a"><mrow id="A1.Ex36.m2.7.8" xref="A1.Ex36.m2.7.8.cmml"><mi id="A1.Ex36.m2.7.8.2" xref="A1.Ex36.m2.7.8.2.cmml"></mi><mo id="A1.Ex36.m2.7.8.3" xref="A1.Ex36.m2.7.8.3.cmml">≥</mo><mstyle displaystyle="true" id="A1.Ex36.m2.2.2" xref="A1.Ex36.m2.2.2.cmml"><mfrac id="A1.Ex36.m2.2.2a" xref="A1.Ex36.m2.2.2.cmml"><mrow id="A1.Ex36.m2.1.1.1" xref="A1.Ex36.m2.1.1.1.cmml"><msub id="A1.Ex36.m2.1.1.1.2" xref="A1.Ex36.m2.1.1.1.2.cmml"><mo id="A1.Ex36.m2.1.1.1.2.2" xref="A1.Ex36.m2.1.1.1.2.2.cmml">∑</mo><mrow id="A1.Ex36.m2.1.1.1.2.3" xref="A1.Ex36.m2.1.1.1.2.3.cmml"><mi id="A1.Ex36.m2.1.1.1.2.3.2" xref="A1.Ex36.m2.1.1.1.2.3.2.cmml">v</mi><mo id="A1.Ex36.m2.1.1.1.2.3.1" 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xref="A1.Ex37.m1.2.2.1.1.1.cmml">)</mo></mrow></mrow><mo id="A1.Ex37.m1.2.2.1.1.3" stretchy="false" xref="A1.Ex37.m1.2.2.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="A1.Ex37.m1.2b"><apply id="A1.Ex37.m1.2.2.cmml" xref="A1.Ex37.m1.2.2"><times id="A1.Ex37.m1.2.2.2.cmml" xref="A1.Ex37.m1.2.2.2"></times><ci id="A1.Ex37.m1.2.2.3.cmml" xref="A1.Ex37.m1.2.2.3">𝜌</ci><apply id="A1.Ex37.m1.2.2.1.1.1.cmml" xref="A1.Ex37.m1.2.2.1.1"><times id="A1.Ex37.m1.2.2.1.1.1.1.cmml" xref="A1.Ex37.m1.2.2.1.1.1.1"></times><ci id="A1.Ex37.m1.2.2.1.1.1.2.cmml" xref="A1.Ex37.m1.2.2.1.1.1.2">𝐻</ci><ci id="A1.Ex37.m1.1.1.cmml" xref="A1.Ex37.m1.1.1">ℓ</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.Ex37.m1.2c">\displaystyle\rho(H(\ell))</annotation><annotation encoding="application/x-llamapun" id="A1.Ex37.m1.2d">italic_ρ ( italic_H ( roman_ℓ ) )</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math 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xref="A1.Ex37.m2.5.5"><geq id="A1.Ex37.m2.5.5.7.cmml" xref="A1.Ex37.m2.5.5.7"></geq><share href="https://arxiv.org/html/2411.12694v2#A1.Ex37.m2.4.4.2.cmml" id="A1.Ex37.m2.5.5d.cmml" xref="A1.Ex37.m2.5.5"></share><apply id="A1.Ex37.m2.5.5.3.cmml" xref="A1.Ex37.m2.5.5.3"><times id="A1.Ex37.m2.5.5.3.2.cmml" xref="A1.Ex37.m2.5.5.3.2"></times><apply id="A1.Ex37.m2.5.5.3.3.cmml" xref="A1.Ex37.m2.5.5.3.3"><csymbol cd="ambiguous" id="A1.Ex37.m2.5.5.3.3.1.cmml" xref="A1.Ex37.m2.5.5.3.3">superscript</csymbol><ci id="A1.Ex37.m2.5.5.3.3.2.cmml" xref="A1.Ex37.m2.5.5.3.3.2">𝜌</ci><times id="A1.Ex37.m2.5.5.3.3.3.cmml" xref="A1.Ex37.m2.5.5.3.3.3"></times></apply><ci id="A1.Ex37.m2.2.2.cmml" xref="A1.Ex37.m2.2.2">ℓ</ci><apply id="A1.Ex37.m2.5.5.3.1.1.1.cmml" xref="A1.Ex37.m2.5.5.3.1.1"><minus id="A1.Ex37.m2.5.5.3.1.1.1.1.cmml" xref="A1.Ex37.m2.5.5.3.1.1.1.1"></minus><cn id="A1.Ex37.m2.5.5.3.1.1.1.2.cmml" type="integer" xref="A1.Ex37.m2.5.5.3.1.1.1.2">1</cn><ci id="A1.Ex37.m2.5.5.3.1.1.1.3.cmml" xref="A1.Ex37.m2.5.5.3.1.1.1.3">𝜀</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.Ex37.m2.5c">\displaystyle\geq\rho^{*}(\ell)(1-\frac{\varepsilon}{16})(1-\frac{5\varepsilon% }{8})\geq\rho^{*}(\ell)(1-\varepsilon)</annotation><annotation encoding="application/x-llamapun" id="A1.Ex37.m2.5d">≥ italic_ρ start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ( roman_ℓ ) ( 1 - divide start_ARG italic_ε end_ARG start_ARG 16 end_ARG ) ( 1 - divide start_ARG 5 italic_ε end_ARG start_ARG 8 end_ARG ) ≥ italic_ρ start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ( roman_ℓ ) ( 1 - italic_ε )</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_left_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="A1.Thmtheorem6.p3.7"><span class="ltx_text ltx_font_italic" id="A1.Thmtheorem6.p3.7.5">where we used that <math alttext="k\leq\log_{1+\varepsilon/16}(n)" class="ltx_Math" display="inline" id="A1.Thmtheorem6.p3.3.1.m1.2"><semantics 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id="A1.Thmtheorem6.p3.3.1.m1.2.2.1.1.1.3.3.2" xref="A1.Thmtheorem6.p3.3.1.m1.2.2.1.1.1.3.3.2.cmml">ε</mi><mo id="A1.Thmtheorem6.p3.3.1.m1.2.2.1.1.1.3.3.1" xref="A1.Thmtheorem6.p3.3.1.m1.2.2.1.1.1.3.3.1.cmml">/</mo><mn id="A1.Thmtheorem6.p3.3.1.m1.2.2.1.1.1.3.3.3" xref="A1.Thmtheorem6.p3.3.1.m1.2.2.1.1.1.3.3.3.cmml">16</mn></mrow></mrow></msub><mo id="A1.Thmtheorem6.p3.3.1.m1.2.2.1.1a" xref="A1.Thmtheorem6.p3.3.1.m1.2.2.1.2.cmml">⁡</mo><mrow id="A1.Thmtheorem6.p3.3.1.m1.2.2.1.1.2" xref="A1.Thmtheorem6.p3.3.1.m1.2.2.1.2.cmml"><mo id="A1.Thmtheorem6.p3.3.1.m1.2.2.1.1.2.1" stretchy="false" xref="A1.Thmtheorem6.p3.3.1.m1.2.2.1.2.cmml">(</mo><mi id="A1.Thmtheorem6.p3.3.1.m1.1.1" xref="A1.Thmtheorem6.p3.3.1.m1.1.1.cmml">n</mi><mo id="A1.Thmtheorem6.p3.3.1.m1.2.2.1.1.2.2" stretchy="false" xref="A1.Thmtheorem6.p3.3.1.m1.2.2.1.2.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="A1.Thmtheorem6.p3.3.1.m1.2b"><apply id="A1.Thmtheorem6.p3.3.1.m1.2.2.cmml" xref="A1.Thmtheorem6.p3.3.1.m1.2.2"><leq id="A1.Thmtheorem6.p3.3.1.m1.2.2.2.cmml" xref="A1.Thmtheorem6.p3.3.1.m1.2.2.2"></leq><ci id="A1.Thmtheorem6.p3.3.1.m1.2.2.3.cmml" xref="A1.Thmtheorem6.p3.3.1.m1.2.2.3">𝑘</ci><apply id="A1.Thmtheorem6.p3.3.1.m1.2.2.1.2.cmml" xref="A1.Thmtheorem6.p3.3.1.m1.2.2.1.1"><apply id="A1.Thmtheorem6.p3.3.1.m1.2.2.1.1.1.cmml" xref="A1.Thmtheorem6.p3.3.1.m1.2.2.1.1.1"><csymbol cd="ambiguous" id="A1.Thmtheorem6.p3.3.1.m1.2.2.1.1.1.1.cmml" xref="A1.Thmtheorem6.p3.3.1.m1.2.2.1.1.1">subscript</csymbol><log id="A1.Thmtheorem6.p3.3.1.m1.2.2.1.1.1.2.cmml" xref="A1.Thmtheorem6.p3.3.1.m1.2.2.1.1.1.2"></log><apply id="A1.Thmtheorem6.p3.3.1.m1.2.2.1.1.1.3.cmml" xref="A1.Thmtheorem6.p3.3.1.m1.2.2.1.1.1.3"><plus id="A1.Thmtheorem6.p3.3.1.m1.2.2.1.1.1.3.1.cmml" xref="A1.Thmtheorem6.p3.3.1.m1.2.2.1.1.1.3.1"></plus><cn id="A1.Thmtheorem6.p3.3.1.m1.2.2.1.1.1.3.2.cmml" type="integer" xref="A1.Thmtheorem6.p3.3.1.m1.2.2.1.1.1.3.2">1</cn><apply id="A1.Thmtheorem6.p3.3.1.m1.2.2.1.1.1.3.3.cmml" xref="A1.Thmtheorem6.p3.3.1.m1.2.2.1.1.1.3.3"><divide id="A1.Thmtheorem6.p3.3.1.m1.2.2.1.1.1.3.3.1.cmml" xref="A1.Thmtheorem6.p3.3.1.m1.2.2.1.1.1.3.3.1"></divide><ci id="A1.Thmtheorem6.p3.3.1.m1.2.2.1.1.1.3.3.2.cmml" xref="A1.Thmtheorem6.p3.3.1.m1.2.2.1.1.1.3.3.2">𝜀</ci><cn id="A1.Thmtheorem6.p3.3.1.m1.2.2.1.1.1.3.3.3.cmml" type="integer" xref="A1.Thmtheorem6.p3.3.1.m1.2.2.1.1.1.3.3.3">16</cn></apply></apply></apply><ci id="A1.Thmtheorem6.p3.3.1.m1.1.1.cmml" xref="A1.Thmtheorem6.p3.3.1.m1.1.1">𝑛</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.Thmtheorem6.p3.3.1.m1.2c">k\leq\log_{1+\varepsilon/16}(n)</annotation><annotation encoding="application/x-llamapun" id="A1.Thmtheorem6.p3.3.1.m1.2d">italic_k ≤ roman_log start_POSTSUBSCRIPT 1 + italic_ε / 16 end_POSTSUBSCRIPT ( italic_n )</annotation></semantics></math> since <math alttext="T_{k}" class="ltx_Math" display="inline" id="A1.Thmtheorem6.p3.4.2.m2.1"><semantics id="A1.Thmtheorem6.p3.4.2.m2.1a"><msub id="A1.Thmtheorem6.p3.4.2.m2.1.1" xref="A1.Thmtheorem6.p3.4.2.m2.1.1.cmml"><mi id="A1.Thmtheorem6.p3.4.2.m2.1.1.2" xref="A1.Thmtheorem6.p3.4.2.m2.1.1.2.cmml">T</mi><mi id="A1.Thmtheorem6.p3.4.2.m2.1.1.3" xref="A1.Thmtheorem6.p3.4.2.m2.1.1.3.cmml">k</mi></msub><annotation-xml encoding="MathML-Content" id="A1.Thmtheorem6.p3.4.2.m2.1b"><apply id="A1.Thmtheorem6.p3.4.2.m2.1.1.cmml" xref="A1.Thmtheorem6.p3.4.2.m2.1.1"><csymbol cd="ambiguous" id="A1.Thmtheorem6.p3.4.2.m2.1.1.1.cmml" xref="A1.Thmtheorem6.p3.4.2.m2.1.1">subscript</csymbol><ci id="A1.Thmtheorem6.p3.4.2.m2.1.1.2.cmml" xref="A1.Thmtheorem6.p3.4.2.m2.1.1.2">𝑇</ci><ci id="A1.Thmtheorem6.p3.4.2.m2.1.1.3.cmml" xref="A1.Thmtheorem6.p3.4.2.m2.1.1.3">𝑘</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.Thmtheorem6.p3.4.2.m2.1c">T_{k}</annotation><annotation encoding="application/x-llamapun" id="A1.Thmtheorem6.p3.4.2.m2.1d">italic_T start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT</annotation></semantics></math> never can be larger than <math alttext="n" class="ltx_Math" display="inline" id="A1.Thmtheorem6.p3.5.3.m3.1"><semantics id="A1.Thmtheorem6.p3.5.3.m3.1a"><mi id="A1.Thmtheorem6.p3.5.3.m3.1.1" xref="A1.Thmtheorem6.p3.5.3.m3.1.1.cmml">n</mi><annotation-xml encoding="MathML-Content" id="A1.Thmtheorem6.p3.5.3.m3.1b"><ci id="A1.Thmtheorem6.p3.5.3.m3.1.1.cmml" xref="A1.Thmtheorem6.p3.5.3.m3.1.1">𝑛</ci></annotation-xml><annotation encoding="application/x-tex" id="A1.Thmtheorem6.p3.5.3.m3.1c">n</annotation><annotation encoding="application/x-llamapun" id="A1.Thmtheorem6.p3.5.3.m3.1d">italic_n</annotation></semantics></math>, and that for <math alttext="0\leq x\leq 1" class="ltx_Math" display="inline" id="A1.Thmtheorem6.p3.6.4.m4.1"><semantics id="A1.Thmtheorem6.p3.6.4.m4.1a"><mrow id="A1.Thmtheorem6.p3.6.4.m4.1.1" xref="A1.Thmtheorem6.p3.6.4.m4.1.1.cmml"><mn id="A1.Thmtheorem6.p3.6.4.m4.1.1.2" xref="A1.Thmtheorem6.p3.6.4.m4.1.1.2.cmml">0</mn><mo id="A1.Thmtheorem6.p3.6.4.m4.1.1.3" xref="A1.Thmtheorem6.p3.6.4.m4.1.1.3.cmml">≤</mo><mi id="A1.Thmtheorem6.p3.6.4.m4.1.1.4" xref="A1.Thmtheorem6.p3.6.4.m4.1.1.4.cmml">x</mi><mo id="A1.Thmtheorem6.p3.6.4.m4.1.1.5" xref="A1.Thmtheorem6.p3.6.4.m4.1.1.5.cmml">≤</mo><mn id="A1.Thmtheorem6.p3.6.4.m4.1.1.6" xref="A1.Thmtheorem6.p3.6.4.m4.1.1.6.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="A1.Thmtheorem6.p3.6.4.m4.1b"><apply id="A1.Thmtheorem6.p3.6.4.m4.1.1.cmml" xref="A1.Thmtheorem6.p3.6.4.m4.1.1"><and id="A1.Thmtheorem6.p3.6.4.m4.1.1a.cmml" xref="A1.Thmtheorem6.p3.6.4.m4.1.1"></and><apply id="A1.Thmtheorem6.p3.6.4.m4.1.1b.cmml" xref="A1.Thmtheorem6.p3.6.4.m4.1.1"><leq id="A1.Thmtheorem6.p3.6.4.m4.1.1.3.cmml" xref="A1.Thmtheorem6.p3.6.4.m4.1.1.3"></leq><cn id="A1.Thmtheorem6.p3.6.4.m4.1.1.2.cmml" type="integer" xref="A1.Thmtheorem6.p3.6.4.m4.1.1.2">0</cn><ci id="A1.Thmtheorem6.p3.6.4.m4.1.1.4.cmml" xref="A1.Thmtheorem6.p3.6.4.m4.1.1.4">𝑥</ci></apply><apply id="A1.Thmtheorem6.p3.6.4.m4.1.1c.cmml" xref="A1.Thmtheorem6.p3.6.4.m4.1.1"><leq id="A1.Thmtheorem6.p3.6.4.m4.1.1.5.cmml" xref="A1.Thmtheorem6.p3.6.4.m4.1.1.5"></leq><share href="https://arxiv.org/html/2411.12694v2#A1.Thmtheorem6.p3.6.4.m4.1.1.4.cmml" id="A1.Thmtheorem6.p3.6.4.m4.1.1d.cmml" xref="A1.Thmtheorem6.p3.6.4.m4.1.1"></share><cn id="A1.Thmtheorem6.p3.6.4.m4.1.1.6.cmml" type="integer" xref="A1.Thmtheorem6.p3.6.4.m4.1.1.6">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.Thmtheorem6.p3.6.4.m4.1c">0\leq x\leq 1</annotation><annotation encoding="application/x-llamapun" id="A1.Thmtheorem6.p3.6.4.m4.1d">0 ≤ italic_x ≤ 1</annotation></semantics></math> we have <math alttext="\log 1+x\geq\frac{x}{2}" class="ltx_Math" display="inline" id="A1.Thmtheorem6.p3.7.5.m5.1"><semantics id="A1.Thmtheorem6.p3.7.5.m5.1a"><mrow id="A1.Thmtheorem6.p3.7.5.m5.1.1" xref="A1.Thmtheorem6.p3.7.5.m5.1.1.cmml"><mrow id="A1.Thmtheorem6.p3.7.5.m5.1.1.2" xref="A1.Thmtheorem6.p3.7.5.m5.1.1.2.cmml"><mrow id="A1.Thmtheorem6.p3.7.5.m5.1.1.2.2" xref="A1.Thmtheorem6.p3.7.5.m5.1.1.2.2.cmml"><mi id="A1.Thmtheorem6.p3.7.5.m5.1.1.2.2.1" xref="A1.Thmtheorem6.p3.7.5.m5.1.1.2.2.1.cmml">log</mi><mo id="A1.Thmtheorem6.p3.7.5.m5.1.1.2.2a" lspace="0.167em" xref="A1.Thmtheorem6.p3.7.5.m5.1.1.2.2.cmml">⁡</mo><mn id="A1.Thmtheorem6.p3.7.5.m5.1.1.2.2.2" xref="A1.Thmtheorem6.p3.7.5.m5.1.1.2.2.2.cmml">1</mn></mrow><mo id="A1.Thmtheorem6.p3.7.5.m5.1.1.2.1" xref="A1.Thmtheorem6.p3.7.5.m5.1.1.2.1.cmml">+</mo><mi id="A1.Thmtheorem6.p3.7.5.m5.1.1.2.3" xref="A1.Thmtheorem6.p3.7.5.m5.1.1.2.3.cmml">x</mi></mrow><mo id="A1.Thmtheorem6.p3.7.5.m5.1.1.1" xref="A1.Thmtheorem6.p3.7.5.m5.1.1.1.cmml">≥</mo><mfrac id="A1.Thmtheorem6.p3.7.5.m5.1.1.3" xref="A1.Thmtheorem6.p3.7.5.m5.1.1.3.cmml"><mi id="A1.Thmtheorem6.p3.7.5.m5.1.1.3.2" xref="A1.Thmtheorem6.p3.7.5.m5.1.1.3.2.cmml">x</mi><mn id="A1.Thmtheorem6.p3.7.5.m5.1.1.3.3" xref="A1.Thmtheorem6.p3.7.5.m5.1.1.3.3.cmml">2</mn></mfrac></mrow><annotation-xml encoding="MathML-Content" id="A1.Thmtheorem6.p3.7.5.m5.1b"><apply id="A1.Thmtheorem6.p3.7.5.m5.1.1.cmml" xref="A1.Thmtheorem6.p3.7.5.m5.1.1"><geq id="A1.Thmtheorem6.p3.7.5.m5.1.1.1.cmml" xref="A1.Thmtheorem6.p3.7.5.m5.1.1.1"></geq><apply id="A1.Thmtheorem6.p3.7.5.m5.1.1.2.cmml" xref="A1.Thmtheorem6.p3.7.5.m5.1.1.2"><plus id="A1.Thmtheorem6.p3.7.5.m5.1.1.2.1.cmml" xref="A1.Thmtheorem6.p3.7.5.m5.1.1.2.1"></plus><apply id="A1.Thmtheorem6.p3.7.5.m5.1.1.2.2.cmml" xref="A1.Thmtheorem6.p3.7.5.m5.1.1.2.2"><log id="A1.Thmtheorem6.p3.7.5.m5.1.1.2.2.1.cmml" xref="A1.Thmtheorem6.p3.7.5.m5.1.1.2.2.1"></log><cn id="A1.Thmtheorem6.p3.7.5.m5.1.1.2.2.2.cmml" type="integer" xref="A1.Thmtheorem6.p3.7.5.m5.1.1.2.2.2">1</cn></apply><ci id="A1.Thmtheorem6.p3.7.5.m5.1.1.2.3.cmml" xref="A1.Thmtheorem6.p3.7.5.m5.1.1.2.3">𝑥</ci></apply><apply id="A1.Thmtheorem6.p3.7.5.m5.1.1.3.cmml" xref="A1.Thmtheorem6.p3.7.5.m5.1.1.3"><divide id="A1.Thmtheorem6.p3.7.5.m5.1.1.3.1.cmml" xref="A1.Thmtheorem6.p3.7.5.m5.1.1.3"></divide><ci id="A1.Thmtheorem6.p3.7.5.m5.1.1.3.2.cmml" xref="A1.Thmtheorem6.p3.7.5.m5.1.1.3.2">𝑥</ci><cn id="A1.Thmtheorem6.p3.7.5.m5.1.1.3.3.cmml" type="integer" xref="A1.Thmtheorem6.p3.7.5.m5.1.1.3.3">2</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.Thmtheorem6.p3.7.5.m5.1c">\log 1+x\geq\frac{x}{2}</annotation><annotation encoding="application/x-llamapun" id="A1.Thmtheorem6.p3.7.5.m5.1d">roman_log 1 + italic_x ≥ divide start_ARG italic_x end_ARG start_ARG 2 end_ARG</annotation></semantics></math>.</span></p> </div> </div> <div class="ltx_para" id="A1.SS3.SSS0.P0.SPx2.p3"> <p class="ltx_p" id="A1.SS3.SSS0.P0.SPx2.p3.1">Now we are ready to prove correctness:</p> </div> <div class="ltx_theorem ltx_theorem_lemma" id="A1.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="A1.Thmtheorem7.1.1.1">Lemma A.7</span></span><span class="ltx_text ltx_font_bold" id="A1.Thmtheorem7.2.2">.</span> </h6> <div class="ltx_para" id="A1.Thmtheorem7.p1"> <p class="ltx_p" id="A1.Thmtheorem7.p1.7"><span class="ltx_text ltx_font_italic" id="A1.Thmtheorem7.p1.7.7">Let <math alttext="G" class="ltx_Math" display="inline" id="A1.Thmtheorem7.p1.1.1.m1.1"><semantics id="A1.Thmtheorem7.p1.1.1.m1.1a"><mi id="A1.Thmtheorem7.p1.1.1.m1.1.1" xref="A1.Thmtheorem7.p1.1.1.m1.1.1.cmml">G</mi><annotation-xml encoding="MathML-Content" id="A1.Thmtheorem7.p1.1.1.m1.1b"><ci id="A1.Thmtheorem7.p1.1.1.m1.1.1.cmml" xref="A1.Thmtheorem7.p1.1.1.m1.1.1">𝐺</ci></annotation-xml><annotation encoding="application/x-tex" id="A1.Thmtheorem7.p1.1.1.m1.1c">G</annotation><annotation encoding="application/x-llamapun" id="A1.Thmtheorem7.p1.1.1.m1.1d">italic_G</annotation></semantics></math>, <math alttext="\varepsilon>0" class="ltx_Math" display="inline" id="A1.Thmtheorem7.p1.2.2.m2.1"><semantics id="A1.Thmtheorem7.p1.2.2.m2.1a"><mrow id="A1.Thmtheorem7.p1.2.2.m2.1.1" xref="A1.Thmtheorem7.p1.2.2.m2.1.1.cmml"><mi id="A1.Thmtheorem7.p1.2.2.m2.1.1.2" xref="A1.Thmtheorem7.p1.2.2.m2.1.1.2.cmml">ε</mi><mo id="A1.Thmtheorem7.p1.2.2.m2.1.1.1" xref="A1.Thmtheorem7.p1.2.2.m2.1.1.1.cmml">></mo><mn id="A1.Thmtheorem7.p1.2.2.m2.1.1.3" xref="A1.Thmtheorem7.p1.2.2.m2.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="A1.Thmtheorem7.p1.2.2.m2.1b"><apply id="A1.Thmtheorem7.p1.2.2.m2.1.1.cmml" xref="A1.Thmtheorem7.p1.2.2.m2.1.1"><gt id="A1.Thmtheorem7.p1.2.2.m2.1.1.1.cmml" xref="A1.Thmtheorem7.p1.2.2.m2.1.1.1"></gt><ci id="A1.Thmtheorem7.p1.2.2.m2.1.1.2.cmml" xref="A1.Thmtheorem7.p1.2.2.m2.1.1.2">𝜀</ci><cn id="A1.Thmtheorem7.p1.2.2.m2.1.1.3.cmml" type="integer" xref="A1.Thmtheorem7.p1.2.2.m2.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.Thmtheorem7.p1.2.2.m2.1c">\varepsilon>0</annotation><annotation encoding="application/x-llamapun" id="A1.Thmtheorem7.p1.2.2.m2.1d">italic_ε > 0</annotation></semantics></math> and <math alttext="\tilde{D}" class="ltx_Math" display="inline" id="A1.Thmtheorem7.p1.3.3.m3.1"><semantics id="A1.Thmtheorem7.p1.3.3.m3.1a"><mover accent="true" id="A1.Thmtheorem7.p1.3.3.m3.1.1" xref="A1.Thmtheorem7.p1.3.3.m3.1.1.cmml"><mi id="A1.Thmtheorem7.p1.3.3.m3.1.1.2" xref="A1.Thmtheorem7.p1.3.3.m3.1.1.2.cmml">D</mi><mo id="A1.Thmtheorem7.p1.3.3.m3.1.1.1" xref="A1.Thmtheorem7.p1.3.3.m3.1.1.1.cmml">~</mo></mover><annotation-xml encoding="MathML-Content" id="A1.Thmtheorem7.p1.3.3.m3.1b"><apply id="A1.Thmtheorem7.p1.3.3.m3.1.1.cmml" xref="A1.Thmtheorem7.p1.3.3.m3.1.1"><ci id="A1.Thmtheorem7.p1.3.3.m3.1.1.1.cmml" xref="A1.Thmtheorem7.p1.3.3.m3.1.1.1">~</ci><ci id="A1.Thmtheorem7.p1.3.3.m3.1.1.2.cmml" xref="A1.Thmtheorem7.p1.3.3.m3.1.1.2">𝐷</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.Thmtheorem7.p1.3.3.m3.1c">\tilde{D}</annotation><annotation encoding="application/x-llamapun" id="A1.Thmtheorem7.p1.3.3.m3.1d">over~ start_ARG italic_D end_ARG</annotation></semantics></math> be given. Then the graph induced by all vertices declaring <math alttext="1" class="ltx_Math" display="inline" id="A1.Thmtheorem7.p1.4.4.m4.1"><semantics id="A1.Thmtheorem7.p1.4.4.m4.1a"><mn id="A1.Thmtheorem7.p1.4.4.m4.1.1" xref="A1.Thmtheorem7.p1.4.4.m4.1.1.cmml">1</mn><annotation-xml encoding="MathML-Content" id="A1.Thmtheorem7.p1.4.4.m4.1b"><cn id="A1.Thmtheorem7.p1.4.4.m4.1.1.cmml" type="integer" xref="A1.Thmtheorem7.p1.4.4.m4.1.1">1</cn></annotation-xml><annotation encoding="application/x-tex" id="A1.Thmtheorem7.p1.4.4.m4.1c">1</annotation><annotation encoding="application/x-llamapun" id="A1.Thmtheorem7.p1.4.4.m4.1d">1</annotation></semantics></math> is either empty or has density at least <math alttext="\tilde{D}(1-\varepsilon)" class="ltx_Math" display="inline" id="A1.Thmtheorem7.p1.5.5.m5.1"><semantics id="A1.Thmtheorem7.p1.5.5.m5.1a"><mrow id="A1.Thmtheorem7.p1.5.5.m5.1.1" xref="A1.Thmtheorem7.p1.5.5.m5.1.1.cmml"><mover accent="true" id="A1.Thmtheorem7.p1.5.5.m5.1.1.3" xref="A1.Thmtheorem7.p1.5.5.m5.1.1.3.cmml"><mi id="A1.Thmtheorem7.p1.5.5.m5.1.1.3.2" xref="A1.Thmtheorem7.p1.5.5.m5.1.1.3.2.cmml">D</mi><mo id="A1.Thmtheorem7.p1.5.5.m5.1.1.3.1" xref="A1.Thmtheorem7.p1.5.5.m5.1.1.3.1.cmml">~</mo></mover><mo id="A1.Thmtheorem7.p1.5.5.m5.1.1.2" xref="A1.Thmtheorem7.p1.5.5.m5.1.1.2.cmml">⁢</mo><mrow id="A1.Thmtheorem7.p1.5.5.m5.1.1.1.1" xref="A1.Thmtheorem7.p1.5.5.m5.1.1.1.1.1.cmml"><mo id="A1.Thmtheorem7.p1.5.5.m5.1.1.1.1.2" stretchy="false" xref="A1.Thmtheorem7.p1.5.5.m5.1.1.1.1.1.cmml">(</mo><mrow id="A1.Thmtheorem7.p1.5.5.m5.1.1.1.1.1" xref="A1.Thmtheorem7.p1.5.5.m5.1.1.1.1.1.cmml"><mn id="A1.Thmtheorem7.p1.5.5.m5.1.1.1.1.1.2" xref="A1.Thmtheorem7.p1.5.5.m5.1.1.1.1.1.2.cmml">1</mn><mo id="A1.Thmtheorem7.p1.5.5.m5.1.1.1.1.1.1" xref="A1.Thmtheorem7.p1.5.5.m5.1.1.1.1.1.1.cmml">−</mo><mi id="A1.Thmtheorem7.p1.5.5.m5.1.1.1.1.1.3" xref="A1.Thmtheorem7.p1.5.5.m5.1.1.1.1.1.3.cmml">ε</mi></mrow><mo id="A1.Thmtheorem7.p1.5.5.m5.1.1.1.1.3" stretchy="false" xref="A1.Thmtheorem7.p1.5.5.m5.1.1.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="A1.Thmtheorem7.p1.5.5.m5.1b"><apply id="A1.Thmtheorem7.p1.5.5.m5.1.1.cmml" xref="A1.Thmtheorem7.p1.5.5.m5.1.1"><times id="A1.Thmtheorem7.p1.5.5.m5.1.1.2.cmml" xref="A1.Thmtheorem7.p1.5.5.m5.1.1.2"></times><apply id="A1.Thmtheorem7.p1.5.5.m5.1.1.3.cmml" xref="A1.Thmtheorem7.p1.5.5.m5.1.1.3"><ci id="A1.Thmtheorem7.p1.5.5.m5.1.1.3.1.cmml" xref="A1.Thmtheorem7.p1.5.5.m5.1.1.3.1">~</ci><ci id="A1.Thmtheorem7.p1.5.5.m5.1.1.3.2.cmml" xref="A1.Thmtheorem7.p1.5.5.m5.1.1.3.2">𝐷</ci></apply><apply id="A1.Thmtheorem7.p1.5.5.m5.1.1.1.1.1.cmml" xref="A1.Thmtheorem7.p1.5.5.m5.1.1.1.1"><minus id="A1.Thmtheorem7.p1.5.5.m5.1.1.1.1.1.1.cmml" xref="A1.Thmtheorem7.p1.5.5.m5.1.1.1.1.1.1"></minus><cn id="A1.Thmtheorem7.p1.5.5.m5.1.1.1.1.1.2.cmml" type="integer" xref="A1.Thmtheorem7.p1.5.5.m5.1.1.1.1.1.2">1</cn><ci id="A1.Thmtheorem7.p1.5.5.m5.1.1.1.1.1.3.cmml" xref="A1.Thmtheorem7.p1.5.5.m5.1.1.1.1.1.3">𝜀</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.Thmtheorem7.p1.5.5.m5.1c">\tilde{D}(1-\varepsilon)</annotation><annotation encoding="application/x-llamapun" id="A1.Thmtheorem7.p1.5.5.m5.1d">over~ start_ARG italic_D end_ARG ( 1 - italic_ε )</annotation></semantics></math>. Furthermore, if <math alttext="\tilde{D}\leq\rho^{\max}(G)" class="ltx_Math" display="inline" id="A1.Thmtheorem7.p1.6.6.m6.1"><semantics id="A1.Thmtheorem7.p1.6.6.m6.1a"><mrow id="A1.Thmtheorem7.p1.6.6.m6.1.2" xref="A1.Thmtheorem7.p1.6.6.m6.1.2.cmml"><mover accent="true" id="A1.Thmtheorem7.p1.6.6.m6.1.2.2" xref="A1.Thmtheorem7.p1.6.6.m6.1.2.2.cmml"><mi id="A1.Thmtheorem7.p1.6.6.m6.1.2.2.2" xref="A1.Thmtheorem7.p1.6.6.m6.1.2.2.2.cmml">D</mi><mo id="A1.Thmtheorem7.p1.6.6.m6.1.2.2.1" xref="A1.Thmtheorem7.p1.6.6.m6.1.2.2.1.cmml">~</mo></mover><mo id="A1.Thmtheorem7.p1.6.6.m6.1.2.1" xref="A1.Thmtheorem7.p1.6.6.m6.1.2.1.cmml">≤</mo><mrow id="A1.Thmtheorem7.p1.6.6.m6.1.2.3" xref="A1.Thmtheorem7.p1.6.6.m6.1.2.3.cmml"><msup id="A1.Thmtheorem7.p1.6.6.m6.1.2.3.2" xref="A1.Thmtheorem7.p1.6.6.m6.1.2.3.2.cmml"><mi id="A1.Thmtheorem7.p1.6.6.m6.1.2.3.2.2" xref="A1.Thmtheorem7.p1.6.6.m6.1.2.3.2.2.cmml">ρ</mi><mi id="A1.Thmtheorem7.p1.6.6.m6.1.2.3.2.3" xref="A1.Thmtheorem7.p1.6.6.m6.1.2.3.2.3.cmml">max</mi></msup><mo id="A1.Thmtheorem7.p1.6.6.m6.1.2.3.1" xref="A1.Thmtheorem7.p1.6.6.m6.1.2.3.1.cmml">⁢</mo><mrow id="A1.Thmtheorem7.p1.6.6.m6.1.2.3.3.2" xref="A1.Thmtheorem7.p1.6.6.m6.1.2.3.cmml"><mo id="A1.Thmtheorem7.p1.6.6.m6.1.2.3.3.2.1" stretchy="false" xref="A1.Thmtheorem7.p1.6.6.m6.1.2.3.cmml">(</mo><mi id="A1.Thmtheorem7.p1.6.6.m6.1.1" xref="A1.Thmtheorem7.p1.6.6.m6.1.1.cmml">G</mi><mo id="A1.Thmtheorem7.p1.6.6.m6.1.2.3.3.2.2" stretchy="false" xref="A1.Thmtheorem7.p1.6.6.m6.1.2.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="A1.Thmtheorem7.p1.6.6.m6.1b"><apply id="A1.Thmtheorem7.p1.6.6.m6.1.2.cmml" xref="A1.Thmtheorem7.p1.6.6.m6.1.2"><leq id="A1.Thmtheorem7.p1.6.6.m6.1.2.1.cmml" xref="A1.Thmtheorem7.p1.6.6.m6.1.2.1"></leq><apply id="A1.Thmtheorem7.p1.6.6.m6.1.2.2.cmml" xref="A1.Thmtheorem7.p1.6.6.m6.1.2.2"><ci id="A1.Thmtheorem7.p1.6.6.m6.1.2.2.1.cmml" xref="A1.Thmtheorem7.p1.6.6.m6.1.2.2.1">~</ci><ci id="A1.Thmtheorem7.p1.6.6.m6.1.2.2.2.cmml" xref="A1.Thmtheorem7.p1.6.6.m6.1.2.2.2">𝐷</ci></apply><apply id="A1.Thmtheorem7.p1.6.6.m6.1.2.3.cmml" xref="A1.Thmtheorem7.p1.6.6.m6.1.2.3"><times id="A1.Thmtheorem7.p1.6.6.m6.1.2.3.1.cmml" xref="A1.Thmtheorem7.p1.6.6.m6.1.2.3.1"></times><apply id="A1.Thmtheorem7.p1.6.6.m6.1.2.3.2.cmml" xref="A1.Thmtheorem7.p1.6.6.m6.1.2.3.2"><csymbol cd="ambiguous" id="A1.Thmtheorem7.p1.6.6.m6.1.2.3.2.1.cmml" xref="A1.Thmtheorem7.p1.6.6.m6.1.2.3.2">superscript</csymbol><ci id="A1.Thmtheorem7.p1.6.6.m6.1.2.3.2.2.cmml" xref="A1.Thmtheorem7.p1.6.6.m6.1.2.3.2.2">𝜌</ci><max id="A1.Thmtheorem7.p1.6.6.m6.1.2.3.2.3.cmml" xref="A1.Thmtheorem7.p1.6.6.m6.1.2.3.2.3"></max></apply><ci id="A1.Thmtheorem7.p1.6.6.m6.1.1.cmml" xref="A1.Thmtheorem7.p1.6.6.m6.1.1">𝐺</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.Thmtheorem7.p1.6.6.m6.1c">\tilde{D}\leq\rho^{\max}(G)</annotation><annotation encoding="application/x-llamapun" id="A1.Thmtheorem7.p1.6.6.m6.1d">over~ start_ARG italic_D end_ARG ≤ italic_ρ start_POSTSUPERSCRIPT roman_max end_POSTSUPERSCRIPT ( italic_G )</annotation></semantics></math> then at least one vertex will declare <math alttext="1" class="ltx_Math" display="inline" id="A1.Thmtheorem7.p1.7.7.m7.1"><semantics id="A1.Thmtheorem7.p1.7.7.m7.1a"><mn id="A1.Thmtheorem7.p1.7.7.m7.1.1" xref="A1.Thmtheorem7.p1.7.7.m7.1.1.cmml">1</mn><annotation-xml encoding="MathML-Content" id="A1.Thmtheorem7.p1.7.7.m7.1b"><cn id="A1.Thmtheorem7.p1.7.7.m7.1.1.cmml" type="integer" xref="A1.Thmtheorem7.p1.7.7.m7.1.1">1</cn></annotation-xml><annotation encoding="application/x-tex" id="A1.Thmtheorem7.p1.7.7.m7.1c">1</annotation><annotation encoding="application/x-llamapun" id="A1.Thmtheorem7.p1.7.7.m7.1d">1</annotation></semantics></math>.</span></p> </div> </div> <div class="ltx_theorem ltx_theorem_proof" id="A1.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="A1.Thmtheorem8.1.1.1">Proof A.8</span></span><span class="ltx_text ltx_font_bold" id="A1.Thmtheorem8.2.2">.</span> </h6> <div class="ltx_para" id="A1.Thmtheorem8.p1"> <p class="ltx_p" id="A1.Thmtheorem8.p1.4"><span class="ltx_text ltx_font_italic" id="A1.Thmtheorem8.p1.4.4">Note first that the <math alttext="\lceil\frac{32\log n}{\varepsilon}\rceil" class="ltx_Math" display="inline" id="A1.Thmtheorem8.p1.1.1.m1.1"><semantics id="A1.Thmtheorem8.p1.1.1.m1.1a"><mrow id="A1.Thmtheorem8.p1.1.1.m1.1.2.2" xref="A1.Thmtheorem8.p1.1.1.m1.1.2.1.cmml"><mo id="A1.Thmtheorem8.p1.1.1.m1.1.2.2.1" stretchy="false" xref="A1.Thmtheorem8.p1.1.1.m1.1.2.1.1.cmml">⌈</mo><mfrac id="A1.Thmtheorem8.p1.1.1.m1.1.1" xref="A1.Thmtheorem8.p1.1.1.m1.1.1.cmml"><mrow id="A1.Thmtheorem8.p1.1.1.m1.1.1.2" xref="A1.Thmtheorem8.p1.1.1.m1.1.1.2.cmml"><mn id="A1.Thmtheorem8.p1.1.1.m1.1.1.2.2" xref="A1.Thmtheorem8.p1.1.1.m1.1.1.2.2.cmml">32</mn><mo id="A1.Thmtheorem8.p1.1.1.m1.1.1.2.1" lspace="0.167em" xref="A1.Thmtheorem8.p1.1.1.m1.1.1.2.1.cmml">⁢</mo><mrow id="A1.Thmtheorem8.p1.1.1.m1.1.1.2.3" xref="A1.Thmtheorem8.p1.1.1.m1.1.1.2.3.cmml"><mi id="A1.Thmtheorem8.p1.1.1.m1.1.1.2.3.1" xref="A1.Thmtheorem8.p1.1.1.m1.1.1.2.3.1.cmml">log</mi><mo id="A1.Thmtheorem8.p1.1.1.m1.1.1.2.3a" lspace="0.167em" xref="A1.Thmtheorem8.p1.1.1.m1.1.1.2.3.cmml">⁡</mo><mi id="A1.Thmtheorem8.p1.1.1.m1.1.1.2.3.2" xref="A1.Thmtheorem8.p1.1.1.m1.1.1.2.3.2.cmml">n</mi></mrow></mrow><mi id="A1.Thmtheorem8.p1.1.1.m1.1.1.3" xref="A1.Thmtheorem8.p1.1.1.m1.1.1.3.cmml">ε</mi></mfrac><mo id="A1.Thmtheorem8.p1.1.1.m1.1.2.2.2" stretchy="false" xref="A1.Thmtheorem8.p1.1.1.m1.1.2.1.1.cmml">⌉</mo></mrow><annotation-xml encoding="MathML-Content" id="A1.Thmtheorem8.p1.1.1.m1.1b"><apply id="A1.Thmtheorem8.p1.1.1.m1.1.2.1.cmml" xref="A1.Thmtheorem8.p1.1.1.m1.1.2.2"><ceiling id="A1.Thmtheorem8.p1.1.1.m1.1.2.1.1.cmml" xref="A1.Thmtheorem8.p1.1.1.m1.1.2.2.1"></ceiling><apply id="A1.Thmtheorem8.p1.1.1.m1.1.1.cmml" xref="A1.Thmtheorem8.p1.1.1.m1.1.1"><divide id="A1.Thmtheorem8.p1.1.1.m1.1.1.1.cmml" xref="A1.Thmtheorem8.p1.1.1.m1.1.1"></divide><apply id="A1.Thmtheorem8.p1.1.1.m1.1.1.2.cmml" xref="A1.Thmtheorem8.p1.1.1.m1.1.1.2"><times id="A1.Thmtheorem8.p1.1.1.m1.1.1.2.1.cmml" xref="A1.Thmtheorem8.p1.1.1.m1.1.1.2.1"></times><cn id="A1.Thmtheorem8.p1.1.1.m1.1.1.2.2.cmml" type="integer" xref="A1.Thmtheorem8.p1.1.1.m1.1.1.2.2">32</cn><apply id="A1.Thmtheorem8.p1.1.1.m1.1.1.2.3.cmml" xref="A1.Thmtheorem8.p1.1.1.m1.1.1.2.3"><log id="A1.Thmtheorem8.p1.1.1.m1.1.1.2.3.1.cmml" xref="A1.Thmtheorem8.p1.1.1.m1.1.1.2.3.1"></log><ci id="A1.Thmtheorem8.p1.1.1.m1.1.1.2.3.2.cmml" xref="A1.Thmtheorem8.p1.1.1.m1.1.1.2.3.2">𝑛</ci></apply></apply><ci id="A1.Thmtheorem8.p1.1.1.m1.1.1.3.cmml" xref="A1.Thmtheorem8.p1.1.1.m1.1.1.3">𝜀</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.Thmtheorem8.p1.1.1.m1.1c">\lceil\frac{32\log n}{\varepsilon}\rceil</annotation><annotation encoding="application/x-llamapun" id="A1.Thmtheorem8.p1.1.1.m1.1d">⌈ divide start_ARG 32 roman_log italic_n end_ARG start_ARG italic_ε end_ARG ⌉</annotation></semantics></math>-hop neighbourhood of every active leader is disjoint. This follows from the fact that every vertex acknowledges exactly one leader, and so two vertices sharing a vertex in their <math alttext="\lceil\frac{32\log n}{\varepsilon}\rceil" class="ltx_Math" display="inline" id="A1.Thmtheorem8.p1.2.2.m2.1"><semantics id="A1.Thmtheorem8.p1.2.2.m2.1a"><mrow id="A1.Thmtheorem8.p1.2.2.m2.1.2.2" xref="A1.Thmtheorem8.p1.2.2.m2.1.2.1.cmml"><mo id="A1.Thmtheorem8.p1.2.2.m2.1.2.2.1" stretchy="false" xref="A1.Thmtheorem8.p1.2.2.m2.1.2.1.1.cmml">⌈</mo><mfrac id="A1.Thmtheorem8.p1.2.2.m2.1.1" xref="A1.Thmtheorem8.p1.2.2.m2.1.1.cmml"><mrow id="A1.Thmtheorem8.p1.2.2.m2.1.1.2" xref="A1.Thmtheorem8.p1.2.2.m2.1.1.2.cmml"><mn id="A1.Thmtheorem8.p1.2.2.m2.1.1.2.2" xref="A1.Thmtheorem8.p1.2.2.m2.1.1.2.2.cmml">32</mn><mo id="A1.Thmtheorem8.p1.2.2.m2.1.1.2.1" lspace="0.167em" xref="A1.Thmtheorem8.p1.2.2.m2.1.1.2.1.cmml">⁢</mo><mrow id="A1.Thmtheorem8.p1.2.2.m2.1.1.2.3" xref="A1.Thmtheorem8.p1.2.2.m2.1.1.2.3.cmml"><mi id="A1.Thmtheorem8.p1.2.2.m2.1.1.2.3.1" xref="A1.Thmtheorem8.p1.2.2.m2.1.1.2.3.1.cmml">log</mi><mo id="A1.Thmtheorem8.p1.2.2.m2.1.1.2.3a" lspace="0.167em" xref="A1.Thmtheorem8.p1.2.2.m2.1.1.2.3.cmml">⁡</mo><mi id="A1.Thmtheorem8.p1.2.2.m2.1.1.2.3.2" xref="A1.Thmtheorem8.p1.2.2.m2.1.1.2.3.2.cmml">n</mi></mrow></mrow><mi id="A1.Thmtheorem8.p1.2.2.m2.1.1.3" xref="A1.Thmtheorem8.p1.2.2.m2.1.1.3.cmml">ε</mi></mfrac><mo id="A1.Thmtheorem8.p1.2.2.m2.1.2.2.2" stretchy="false" xref="A1.Thmtheorem8.p1.2.2.m2.1.2.1.1.cmml">⌉</mo></mrow><annotation-xml encoding="MathML-Content" id="A1.Thmtheorem8.p1.2.2.m2.1b"><apply id="A1.Thmtheorem8.p1.2.2.m2.1.2.1.cmml" xref="A1.Thmtheorem8.p1.2.2.m2.1.2.2"><ceiling id="A1.Thmtheorem8.p1.2.2.m2.1.2.1.1.cmml" xref="A1.Thmtheorem8.p1.2.2.m2.1.2.2.1"></ceiling><apply id="A1.Thmtheorem8.p1.2.2.m2.1.1.cmml" xref="A1.Thmtheorem8.p1.2.2.m2.1.1"><divide id="A1.Thmtheorem8.p1.2.2.m2.1.1.1.cmml" xref="A1.Thmtheorem8.p1.2.2.m2.1.1"></divide><apply id="A1.Thmtheorem8.p1.2.2.m2.1.1.2.cmml" xref="A1.Thmtheorem8.p1.2.2.m2.1.1.2"><times id="A1.Thmtheorem8.p1.2.2.m2.1.1.2.1.cmml" xref="A1.Thmtheorem8.p1.2.2.m2.1.1.2.1"></times><cn id="A1.Thmtheorem8.p1.2.2.m2.1.1.2.2.cmml" type="integer" xref="A1.Thmtheorem8.p1.2.2.m2.1.1.2.2">32</cn><apply id="A1.Thmtheorem8.p1.2.2.m2.1.1.2.3.cmml" xref="A1.Thmtheorem8.p1.2.2.m2.1.1.2.3"><log id="A1.Thmtheorem8.p1.2.2.m2.1.1.2.3.1.cmml" xref="A1.Thmtheorem8.p1.2.2.m2.1.1.2.3.1"></log><ci id="A1.Thmtheorem8.p1.2.2.m2.1.1.2.3.2.cmml" xref="A1.Thmtheorem8.p1.2.2.m2.1.1.2.3.2">𝑛</ci></apply></apply><ci id="A1.Thmtheorem8.p1.2.2.m2.1.1.3.cmml" xref="A1.Thmtheorem8.p1.2.2.m2.1.1.3">𝜀</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.Thmtheorem8.p1.2.2.m2.1c">\lceil\frac{32\log n}{\varepsilon}\rceil</annotation><annotation encoding="application/x-llamapun" id="A1.Thmtheorem8.p1.2.2.m2.1d">⌈ divide start_ARG 32 roman_log italic_n end_ARG start_ARG italic_ε end_ARG ⌉</annotation></semantics></math>-hop neighbourhood cannot both be acknowledged as leaders. This in turn implies that the subgraph <math alttext="H" class="ltx_Math" display="inline" id="A1.Thmtheorem8.p1.3.3.m3.1"><semantics id="A1.Thmtheorem8.p1.3.3.m3.1a"><mi id="A1.Thmtheorem8.p1.3.3.m3.1.1" xref="A1.Thmtheorem8.p1.3.3.m3.1.1.cmml">H</mi><annotation-xml encoding="MathML-Content" id="A1.Thmtheorem8.p1.3.3.m3.1b"><ci id="A1.Thmtheorem8.p1.3.3.m3.1.1.cmml" xref="A1.Thmtheorem8.p1.3.3.m3.1.1">𝐻</ci></annotation-xml><annotation encoding="application/x-tex" id="A1.Thmtheorem8.p1.3.3.m3.1c">H</annotation><annotation encoding="application/x-llamapun" id="A1.Thmtheorem8.p1.3.3.m3.1d">italic_H</annotation></semantics></math> induced by all vertices outputting a 1 has density at least <math alttext="\tilde{D}(1-\varepsilon)" class="ltx_Math" display="inline" id="A1.Thmtheorem8.p1.4.4.m4.1"><semantics id="A1.Thmtheorem8.p1.4.4.m4.1a"><mrow id="A1.Thmtheorem8.p1.4.4.m4.1.1" xref="A1.Thmtheorem8.p1.4.4.m4.1.1.cmml"><mover accent="true" id="A1.Thmtheorem8.p1.4.4.m4.1.1.3" xref="A1.Thmtheorem8.p1.4.4.m4.1.1.3.cmml"><mi id="A1.Thmtheorem8.p1.4.4.m4.1.1.3.2" xref="A1.Thmtheorem8.p1.4.4.m4.1.1.3.2.cmml">D</mi><mo id="A1.Thmtheorem8.p1.4.4.m4.1.1.3.1" xref="A1.Thmtheorem8.p1.4.4.m4.1.1.3.1.cmml">~</mo></mover><mo id="A1.Thmtheorem8.p1.4.4.m4.1.1.2" xref="A1.Thmtheorem8.p1.4.4.m4.1.1.2.cmml">⁢</mo><mrow id="A1.Thmtheorem8.p1.4.4.m4.1.1.1.1" xref="A1.Thmtheorem8.p1.4.4.m4.1.1.1.1.1.cmml"><mo id="A1.Thmtheorem8.p1.4.4.m4.1.1.1.1.2" stretchy="false" xref="A1.Thmtheorem8.p1.4.4.m4.1.1.1.1.1.cmml">(</mo><mrow id="A1.Thmtheorem8.p1.4.4.m4.1.1.1.1.1" xref="A1.Thmtheorem8.p1.4.4.m4.1.1.1.1.1.cmml"><mn id="A1.Thmtheorem8.p1.4.4.m4.1.1.1.1.1.2" xref="A1.Thmtheorem8.p1.4.4.m4.1.1.1.1.1.2.cmml">1</mn><mo id="A1.Thmtheorem8.p1.4.4.m4.1.1.1.1.1.1" xref="A1.Thmtheorem8.p1.4.4.m4.1.1.1.1.1.1.cmml">−</mo><mi id="A1.Thmtheorem8.p1.4.4.m4.1.1.1.1.1.3" xref="A1.Thmtheorem8.p1.4.4.m4.1.1.1.1.1.3.cmml">ε</mi></mrow><mo id="A1.Thmtheorem8.p1.4.4.m4.1.1.1.1.3" stretchy="false" xref="A1.Thmtheorem8.p1.4.4.m4.1.1.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="A1.Thmtheorem8.p1.4.4.m4.1b"><apply id="A1.Thmtheorem8.p1.4.4.m4.1.1.cmml" xref="A1.Thmtheorem8.p1.4.4.m4.1.1"><times id="A1.Thmtheorem8.p1.4.4.m4.1.1.2.cmml" xref="A1.Thmtheorem8.p1.4.4.m4.1.1.2"></times><apply id="A1.Thmtheorem8.p1.4.4.m4.1.1.3.cmml" xref="A1.Thmtheorem8.p1.4.4.m4.1.1.3"><ci id="A1.Thmtheorem8.p1.4.4.m4.1.1.3.1.cmml" xref="A1.Thmtheorem8.p1.4.4.m4.1.1.3.1">~</ci><ci id="A1.Thmtheorem8.p1.4.4.m4.1.1.3.2.cmml" xref="A1.Thmtheorem8.p1.4.4.m4.1.1.3.2">𝐷</ci></apply><apply id="A1.Thmtheorem8.p1.4.4.m4.1.1.1.1.1.cmml" xref="A1.Thmtheorem8.p1.4.4.m4.1.1.1.1"><minus id="A1.Thmtheorem8.p1.4.4.m4.1.1.1.1.1.1.cmml" xref="A1.Thmtheorem8.p1.4.4.m4.1.1.1.1.1.1"></minus><cn id="A1.Thmtheorem8.p1.4.4.m4.1.1.1.1.1.2.cmml" type="integer" xref="A1.Thmtheorem8.p1.4.4.m4.1.1.1.1.1.2">1</cn><ci id="A1.Thmtheorem8.p1.4.4.m4.1.1.1.1.1.3.cmml" xref="A1.Thmtheorem8.p1.4.4.m4.1.1.1.1.1.3">𝜀</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.Thmtheorem8.p1.4.4.m4.1c">\tilde{D}(1-\varepsilon)</annotation><annotation encoding="application/x-llamapun" id="A1.Thmtheorem8.p1.4.4.m4.1d">over~ start_ARG italic_D end_ARG ( 1 - italic_ε )</annotation></semantics></math>.</span></p> </div> <div class="ltx_para" id="A1.Thmtheorem8.p2"> <p class="ltx_p" id="A1.Thmtheorem8.p2.3"><span class="ltx_text ltx_font_italic" id="A1.Thmtheorem8.p2.3.3">Indeed, let <math alttext="\ell_{1},\dots,\ell_{s}" class="ltx_Math" display="inline" id="A1.Thmtheorem8.p2.1.1.m1.3"><semantics id="A1.Thmtheorem8.p2.1.1.m1.3a"><mrow id="A1.Thmtheorem8.p2.1.1.m1.3.3.2" xref="A1.Thmtheorem8.p2.1.1.m1.3.3.3.cmml"><msub id="A1.Thmtheorem8.p2.1.1.m1.2.2.1.1" xref="A1.Thmtheorem8.p2.1.1.m1.2.2.1.1.cmml"><mi id="A1.Thmtheorem8.p2.1.1.m1.2.2.1.1.2" mathvariant="normal" xref="A1.Thmtheorem8.p2.1.1.m1.2.2.1.1.2.cmml">ℓ</mi><mn id="A1.Thmtheorem8.p2.1.1.m1.2.2.1.1.3" xref="A1.Thmtheorem8.p2.1.1.m1.2.2.1.1.3.cmml">1</mn></msub><mo id="A1.Thmtheorem8.p2.1.1.m1.3.3.2.3" xref="A1.Thmtheorem8.p2.1.1.m1.3.3.3.cmml">,</mo><mi id="A1.Thmtheorem8.p2.1.1.m1.1.1" mathvariant="normal" xref="A1.Thmtheorem8.p2.1.1.m1.1.1.cmml">…</mi><mo id="A1.Thmtheorem8.p2.1.1.m1.3.3.2.4" xref="A1.Thmtheorem8.p2.1.1.m1.3.3.3.cmml">,</mo><msub id="A1.Thmtheorem8.p2.1.1.m1.3.3.2.2" xref="A1.Thmtheorem8.p2.1.1.m1.3.3.2.2.cmml"><mi id="A1.Thmtheorem8.p2.1.1.m1.3.3.2.2.2" mathvariant="normal" xref="A1.Thmtheorem8.p2.1.1.m1.3.3.2.2.2.cmml">ℓ</mi><mi id="A1.Thmtheorem8.p2.1.1.m1.3.3.2.2.3" xref="A1.Thmtheorem8.p2.1.1.m1.3.3.2.2.3.cmml">s</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="A1.Thmtheorem8.p2.1.1.m1.3b"><list id="A1.Thmtheorem8.p2.1.1.m1.3.3.3.cmml" xref="A1.Thmtheorem8.p2.1.1.m1.3.3.2"><apply id="A1.Thmtheorem8.p2.1.1.m1.2.2.1.1.cmml" xref="A1.Thmtheorem8.p2.1.1.m1.2.2.1.1"><csymbol cd="ambiguous" id="A1.Thmtheorem8.p2.1.1.m1.2.2.1.1.1.cmml" xref="A1.Thmtheorem8.p2.1.1.m1.2.2.1.1">subscript</csymbol><ci id="A1.Thmtheorem8.p2.1.1.m1.2.2.1.1.2.cmml" xref="A1.Thmtheorem8.p2.1.1.m1.2.2.1.1.2">ℓ</ci><cn id="A1.Thmtheorem8.p2.1.1.m1.2.2.1.1.3.cmml" type="integer" xref="A1.Thmtheorem8.p2.1.1.m1.2.2.1.1.3">1</cn></apply><ci id="A1.Thmtheorem8.p2.1.1.m1.1.1.cmml" xref="A1.Thmtheorem8.p2.1.1.m1.1.1">…</ci><apply id="A1.Thmtheorem8.p2.1.1.m1.3.3.2.2.cmml" xref="A1.Thmtheorem8.p2.1.1.m1.3.3.2.2"><csymbol cd="ambiguous" id="A1.Thmtheorem8.p2.1.1.m1.3.3.2.2.1.cmml" xref="A1.Thmtheorem8.p2.1.1.m1.3.3.2.2">subscript</csymbol><ci id="A1.Thmtheorem8.p2.1.1.m1.3.3.2.2.2.cmml" xref="A1.Thmtheorem8.p2.1.1.m1.3.3.2.2.2">ℓ</ci><ci id="A1.Thmtheorem8.p2.1.1.m1.3.3.2.2.3.cmml" xref="A1.Thmtheorem8.p2.1.1.m1.3.3.2.2.3">𝑠</ci></apply></list></annotation-xml><annotation encoding="application/x-tex" id="A1.Thmtheorem8.p2.1.1.m1.3c">\ell_{1},\dots,\ell_{s}</annotation><annotation encoding="application/x-llamapun" id="A1.Thmtheorem8.p2.1.1.m1.3d">roman_ℓ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , … , roman_ℓ start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT</annotation></semantics></math> be the active leaders. Then by above the induced subgraphs belonging to each leader, <math alttext="H(\ell_{1}),\dots,H(\ell_{s})" class="ltx_Math" display="inline" id="A1.Thmtheorem8.p2.2.2.m2.3"><semantics id="A1.Thmtheorem8.p2.2.2.m2.3a"><mrow id="A1.Thmtheorem8.p2.2.2.m2.3.3.2" xref="A1.Thmtheorem8.p2.2.2.m2.3.3.3.cmml"><mrow id="A1.Thmtheorem8.p2.2.2.m2.2.2.1.1" xref="A1.Thmtheorem8.p2.2.2.m2.2.2.1.1.cmml"><mi id="A1.Thmtheorem8.p2.2.2.m2.2.2.1.1.3" xref="A1.Thmtheorem8.p2.2.2.m2.2.2.1.1.3.cmml">H</mi><mo id="A1.Thmtheorem8.p2.2.2.m2.2.2.1.1.2" xref="A1.Thmtheorem8.p2.2.2.m2.2.2.1.1.2.cmml">⁢</mo><mrow id="A1.Thmtheorem8.p2.2.2.m2.2.2.1.1.1.1" xref="A1.Thmtheorem8.p2.2.2.m2.2.2.1.1.1.1.1.cmml"><mo id="A1.Thmtheorem8.p2.2.2.m2.2.2.1.1.1.1.2" stretchy="false" xref="A1.Thmtheorem8.p2.2.2.m2.2.2.1.1.1.1.1.cmml">(</mo><msub id="A1.Thmtheorem8.p2.2.2.m2.2.2.1.1.1.1.1" xref="A1.Thmtheorem8.p2.2.2.m2.2.2.1.1.1.1.1.cmml"><mi id="A1.Thmtheorem8.p2.2.2.m2.2.2.1.1.1.1.1.2" mathvariant="normal" xref="A1.Thmtheorem8.p2.2.2.m2.2.2.1.1.1.1.1.2.cmml">ℓ</mi><mn id="A1.Thmtheorem8.p2.2.2.m2.2.2.1.1.1.1.1.3" xref="A1.Thmtheorem8.p2.2.2.m2.2.2.1.1.1.1.1.3.cmml">1</mn></msub><mo id="A1.Thmtheorem8.p2.2.2.m2.2.2.1.1.1.1.3" stretchy="false" xref="A1.Thmtheorem8.p2.2.2.m2.2.2.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="A1.Thmtheorem8.p2.2.2.m2.3.3.2.3" xref="A1.Thmtheorem8.p2.2.2.m2.3.3.3.cmml">,</mo><mi id="A1.Thmtheorem8.p2.2.2.m2.1.1" mathvariant="normal" xref="A1.Thmtheorem8.p2.2.2.m2.1.1.cmml">…</mi><mo id="A1.Thmtheorem8.p2.2.2.m2.3.3.2.4" xref="A1.Thmtheorem8.p2.2.2.m2.3.3.3.cmml">,</mo><mrow id="A1.Thmtheorem8.p2.2.2.m2.3.3.2.2" xref="A1.Thmtheorem8.p2.2.2.m2.3.3.2.2.cmml"><mi id="A1.Thmtheorem8.p2.2.2.m2.3.3.2.2.3" xref="A1.Thmtheorem8.p2.2.2.m2.3.3.2.2.3.cmml">H</mi><mo id="A1.Thmtheorem8.p2.2.2.m2.3.3.2.2.2" xref="A1.Thmtheorem8.p2.2.2.m2.3.3.2.2.2.cmml">⁢</mo><mrow id="A1.Thmtheorem8.p2.2.2.m2.3.3.2.2.1.1" xref="A1.Thmtheorem8.p2.2.2.m2.3.3.2.2.1.1.1.cmml"><mo 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id="A1.Thmtheorem8.p2.2.2.m2.2.2.1.1.3.cmml" xref="A1.Thmtheorem8.p2.2.2.m2.2.2.1.1.3">𝐻</ci><apply id="A1.Thmtheorem8.p2.2.2.m2.2.2.1.1.1.1.1.cmml" xref="A1.Thmtheorem8.p2.2.2.m2.2.2.1.1.1.1"><csymbol cd="ambiguous" id="A1.Thmtheorem8.p2.2.2.m2.2.2.1.1.1.1.1.1.cmml" xref="A1.Thmtheorem8.p2.2.2.m2.2.2.1.1.1.1">subscript</csymbol><ci id="A1.Thmtheorem8.p2.2.2.m2.2.2.1.1.1.1.1.2.cmml" xref="A1.Thmtheorem8.p2.2.2.m2.2.2.1.1.1.1.1.2">ℓ</ci><cn id="A1.Thmtheorem8.p2.2.2.m2.2.2.1.1.1.1.1.3.cmml" type="integer" xref="A1.Thmtheorem8.p2.2.2.m2.2.2.1.1.1.1.1.3">1</cn></apply></apply><ci id="A1.Thmtheorem8.p2.2.2.m2.1.1.cmml" xref="A1.Thmtheorem8.p2.2.2.m2.1.1">…</ci><apply id="A1.Thmtheorem8.p2.2.2.m2.3.3.2.2.cmml" xref="A1.Thmtheorem8.p2.2.2.m2.3.3.2.2"><times id="A1.Thmtheorem8.p2.2.2.m2.3.3.2.2.2.cmml" xref="A1.Thmtheorem8.p2.2.2.m2.3.3.2.2.2"></times><ci id="A1.Thmtheorem8.p2.2.2.m2.3.3.2.2.3.cmml" xref="A1.Thmtheorem8.p2.2.2.m2.3.3.2.2.3">𝐻</ci><apply id="A1.Thmtheorem8.p2.2.2.m2.3.3.2.2.1.1.1.cmml" xref="A1.Thmtheorem8.p2.2.2.m2.3.3.2.2.1.1"><csymbol cd="ambiguous" id="A1.Thmtheorem8.p2.2.2.m2.3.3.2.2.1.1.1.1.cmml" xref="A1.Thmtheorem8.p2.2.2.m2.3.3.2.2.1.1">subscript</csymbol><ci id="A1.Thmtheorem8.p2.2.2.m2.3.3.2.2.1.1.1.2.cmml" xref="A1.Thmtheorem8.p2.2.2.m2.3.3.2.2.1.1.1.2">ℓ</ci><ci id="A1.Thmtheorem8.p2.2.2.m2.3.3.2.2.1.1.1.3.cmml" xref="A1.Thmtheorem8.p2.2.2.m2.3.3.2.2.1.1.1.3">𝑠</ci></apply></apply></list></annotation-xml><annotation encoding="application/x-tex" id="A1.Thmtheorem8.p2.2.2.m2.3c">H(\ell_{1}),\dots,H(\ell_{s})</annotation><annotation encoding="application/x-llamapun" id="A1.Thmtheorem8.p2.2.2.m2.3d">italic_H ( roman_ℓ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) , … , italic_H ( roman_ℓ start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT )</annotation></semantics></math>, are all disjoint and so the density of the graph <math alttext="H=\bigcup\limits_{j}H(\ell_{j})=(\bigcup\limits_{j}V(H(\ell_{j})),\bigcup% 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xref="A1.Thmtheorem8.p2.3.3.m3.3.3.3.2.2.1.1.1.1.1.1.1.3">𝑗</ci></apply></apply></apply></apply></interval></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.Thmtheorem8.p2.3.3.m3.3c">H=\bigcup\limits_{j}H(\ell_{j})=(\bigcup\limits_{j}V(H(\ell_{j})),\bigcup% \limits_{j}E(H(\ell_{j})))</annotation><annotation encoding="application/x-llamapun" id="A1.Thmtheorem8.p2.3.3.m3.3d">italic_H = ⋃ start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT italic_H ( roman_ℓ start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ) = ( ⋃ start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT italic_V ( italic_H ( roman_ℓ start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ) ) , ⋃ start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT italic_E ( italic_H ( roman_ℓ start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ) ) )</annotation></semantics></math> satisfies:</span></p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="A1.EGx10"> <tbody id="A1.Ex38"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_left_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle\rho(H)" class="ltx_Math" display="inline" id="A1.Ex38.m1.1"><semantics id="A1.Ex38.m1.1a"><mrow id="A1.Ex38.m1.1.2" xref="A1.Ex38.m1.1.2.cmml"><mi id="A1.Ex38.m1.1.2.2" xref="A1.Ex38.m1.1.2.2.cmml">ρ</mi><mo id="A1.Ex38.m1.1.2.1" xref="A1.Ex38.m1.1.2.1.cmml">⁢</mo><mrow id="A1.Ex38.m1.1.2.3.2" xref="A1.Ex38.m1.1.2.cmml"><mo id="A1.Ex38.m1.1.2.3.2.1" stretchy="false" xref="A1.Ex38.m1.1.2.cmml">(</mo><mi id="A1.Ex38.m1.1.1" xref="A1.Ex38.m1.1.1.cmml">H</mi><mo id="A1.Ex38.m1.1.2.3.2.2" stretchy="false" xref="A1.Ex38.m1.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="A1.Ex38.m1.1b"><apply id="A1.Ex38.m1.1.2.cmml" xref="A1.Ex38.m1.1.2"><times id="A1.Ex38.m1.1.2.1.cmml" xref="A1.Ex38.m1.1.2.1"></times><ci id="A1.Ex38.m1.1.2.2.cmml" xref="A1.Ex38.m1.1.2.2">𝜌</ci><ci id="A1.Ex38.m1.1.1.cmml" 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italic_j end_POSTSUBSCRIPT | italic_E ( italic_H ( roman_ℓ start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ) ) | end_ARG start_ARG ∑ start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT | italic_V ( italic_H ( roman_ℓ start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ) ) | end_ARG = divide start_ARG ∑ start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT italic_ρ ( italic_H ( roman_ℓ start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ) ) | italic_V ( italic_H ( roman_ℓ start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ) ) | end_ARG start_ARG ∑ start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT | italic_V ( italic_H ( roman_ℓ start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ) ) | end_ARG</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_left_padright"></td> </tr></tbody> <tbody id="A1.Ex39"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_left_padleft"></td> <td class="ltx_td ltx_eqn_cell"></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle\geq\min\limits_{j}\{\rho(H(\ell_{j}))\}\frac{\sum\limits_{j}|V(H% (\ell_{j}))|}{\sum\limits_{j}|V(H(\ell_{j}))|}\geq\min\limits_{j}\{\rho(H(\ell% _{j}))\}\geq\tilde{D}(1-\varepsilon)" class="ltx_Math" display="inline" id="A1.Ex39.m1.7"><semantics id="A1.Ex39.m1.7a"><mrow id="A1.Ex39.m1.7.7" xref="A1.Ex39.m1.7.7.cmml"><mi id="A1.Ex39.m1.7.7.7" xref="A1.Ex39.m1.7.7.7.cmml"></mi><mo id="A1.Ex39.m1.7.7.8" xref="A1.Ex39.m1.7.7.8.cmml">≥</mo><mrow id="A1.Ex39.m1.4.4.2" xref="A1.Ex39.m1.4.4.2.cmml"><mrow id="A1.Ex39.m1.4.4.2.2.2" xref="A1.Ex39.m1.4.4.2.2.3.cmml"><munder id="A1.Ex39.m1.3.3.1.1.1.1" xref="A1.Ex39.m1.3.3.1.1.1.1.cmml"><mi id="A1.Ex39.m1.3.3.1.1.1.1.2" xref="A1.Ex39.m1.3.3.1.1.1.1.2.cmml">min</mi><mi id="A1.Ex39.m1.3.3.1.1.1.1.3" xref="A1.Ex39.m1.3.3.1.1.1.1.3.cmml">j</mi></munder><mo id="A1.Ex39.m1.4.4.2.2.2a" xref="A1.Ex39.m1.4.4.2.2.3.cmml">⁡</mo><mrow id="A1.Ex39.m1.4.4.2.2.2.2" xref="A1.Ex39.m1.4.4.2.2.3.cmml"><mo id="A1.Ex39.m1.4.4.2.2.2.2.2" 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xref="A1.Ex39.m1.1.1.1.2.cmml"><mo id="A1.Ex39.m1.1.1.1.2.2" movablelimits="false" xref="A1.Ex39.m1.1.1.1.2.2.cmml">∑</mo><mi id="A1.Ex39.m1.1.1.1.2.3" xref="A1.Ex39.m1.1.1.1.2.3.cmml">j</mi></munder><mrow id="A1.Ex39.m1.1.1.1.1.1" xref="A1.Ex39.m1.1.1.1.1.2.cmml"><mo id="A1.Ex39.m1.1.1.1.1.1.2" lspace="0em" stretchy="false" xref="A1.Ex39.m1.1.1.1.1.2.1.cmml">|</mo><mrow id="A1.Ex39.m1.1.1.1.1.1.1" xref="A1.Ex39.m1.1.1.1.1.1.1.cmml"><mi id="A1.Ex39.m1.1.1.1.1.1.1.3" xref="A1.Ex39.m1.1.1.1.1.1.1.3.cmml">V</mi><mo id="A1.Ex39.m1.1.1.1.1.1.1.2" xref="A1.Ex39.m1.1.1.1.1.1.1.2.cmml">⁢</mo><mrow id="A1.Ex39.m1.1.1.1.1.1.1.1.1" xref="A1.Ex39.m1.1.1.1.1.1.1.1.1.1.cmml"><mo id="A1.Ex39.m1.1.1.1.1.1.1.1.1.2" stretchy="false" xref="A1.Ex39.m1.1.1.1.1.1.1.1.1.1.cmml">(</mo><mrow id="A1.Ex39.m1.1.1.1.1.1.1.1.1.1" xref="A1.Ex39.m1.1.1.1.1.1.1.1.1.1.cmml"><mi id="A1.Ex39.m1.1.1.1.1.1.1.1.1.1.3" xref="A1.Ex39.m1.1.1.1.1.1.1.1.1.1.3.cmml">H</mi><mo id="A1.Ex39.m1.1.1.1.1.1.1.1.1.1.2" 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id="A1.Ex39.m1.7d">≥ roman_min start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT { italic_ρ ( italic_H ( roman_ℓ start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ) ) } divide start_ARG ∑ start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT | italic_V ( italic_H ( roman_ℓ start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ) ) | end_ARG start_ARG ∑ start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT | italic_V ( italic_H ( roman_ℓ start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ) ) | end_ARG ≥ roman_min start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT { italic_ρ ( italic_H ( roman_ℓ start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ) ) } ≥ over~ start_ARG italic_D end_ARG ( 1 - italic_ε )</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_left_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="A1.Thmtheorem8.p2.4"><span class="ltx_text ltx_font_italic" id="A1.Thmtheorem8.p2.4.1">where the last inequality follows from Lemma <a class="ltx_ref" href="https://arxiv.org/html/2411.12694v2#A1.Thmtheorem5" title="Lemma A.5. ‣ Analysis: ‣ A.3 Our algorithm ‣ Appendix A Reporting a densest subgraph ‣ Local Density and its Distributed Approximation"><span class="ltx_text ltx_ref_tag">A.5</span></a>.</span></p> </div> <div class="ltx_para" id="A1.Thmtheorem8.p3"> <p class="ltx_p" id="A1.Thmtheorem8.p3.5"><span class="ltx_text ltx_font_italic" id="A1.Thmtheorem8.p3.5.5">Finally to show the furthermore part, observe that some vertex <math alttext="v" class="ltx_Math" display="inline" id="A1.Thmtheorem8.p3.1.1.m1.1"><semantics id="A1.Thmtheorem8.p3.1.1.m1.1a"><mi id="A1.Thmtheorem8.p3.1.1.m1.1.1" xref="A1.Thmtheorem8.p3.1.1.m1.1.1.cmml">v</mi><annotation-xml encoding="MathML-Content" id="A1.Thmtheorem8.p3.1.1.m1.1b"><ci id="A1.Thmtheorem8.p3.1.1.m1.1.1.cmml" xref="A1.Thmtheorem8.p3.1.1.m1.1.1">𝑣</ci></annotation-xml><annotation encoding="application/x-tex" id="A1.Thmtheorem8.p3.1.1.m1.1c">v</annotation><annotation encoding="application/x-llamapun" id="A1.Thmtheorem8.p3.1.1.m1.1d">italic_v</annotation></semantics></math> will receive a fractional out-degree at least <math alttext="\rho(G)" class="ltx_Math" display="inline" id="A1.Thmtheorem8.p3.2.2.m2.1"><semantics id="A1.Thmtheorem8.p3.2.2.m2.1a"><mrow id="A1.Thmtheorem8.p3.2.2.m2.1.2" xref="A1.Thmtheorem8.p3.2.2.m2.1.2.cmml"><mi id="A1.Thmtheorem8.p3.2.2.m2.1.2.2" xref="A1.Thmtheorem8.p3.2.2.m2.1.2.2.cmml">ρ</mi><mo id="A1.Thmtheorem8.p3.2.2.m2.1.2.1" xref="A1.Thmtheorem8.p3.2.2.m2.1.2.1.cmml">⁢</mo><mrow id="A1.Thmtheorem8.p3.2.2.m2.1.2.3.2" xref="A1.Thmtheorem8.p3.2.2.m2.1.2.cmml"><mo id="A1.Thmtheorem8.p3.2.2.m2.1.2.3.2.1" stretchy="false" xref="A1.Thmtheorem8.p3.2.2.m2.1.2.cmml">(</mo><mi id="A1.Thmtheorem8.p3.2.2.m2.1.1" xref="A1.Thmtheorem8.p3.2.2.m2.1.1.cmml">G</mi><mo id="A1.Thmtheorem8.p3.2.2.m2.1.2.3.2.2" stretchy="false" xref="A1.Thmtheorem8.p3.2.2.m2.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="A1.Thmtheorem8.p3.2.2.m2.1b"><apply id="A1.Thmtheorem8.p3.2.2.m2.1.2.cmml" xref="A1.Thmtheorem8.p3.2.2.m2.1.2"><times id="A1.Thmtheorem8.p3.2.2.m2.1.2.1.cmml" xref="A1.Thmtheorem8.p3.2.2.m2.1.2.1"></times><ci id="A1.Thmtheorem8.p3.2.2.m2.1.2.2.cmml" xref="A1.Thmtheorem8.p3.2.2.m2.1.2.2">𝜌</ci><ci id="A1.Thmtheorem8.p3.2.2.m2.1.1.cmml" xref="A1.Thmtheorem8.p3.2.2.m2.1.1">𝐺</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.Thmtheorem8.p3.2.2.m2.1c">\rho(G)</annotation><annotation encoding="application/x-llamapun" id="A1.Thmtheorem8.p3.2.2.m2.1d">italic_ρ ( italic_G )</annotation></semantics></math> in the first computed <math alttext="\eta" class="ltx_Math" display="inline" id="A1.Thmtheorem8.p3.3.3.m3.1"><semantics id="A1.Thmtheorem8.p3.3.3.m3.1a"><mi id="A1.Thmtheorem8.p3.3.3.m3.1.1" xref="A1.Thmtheorem8.p3.3.3.m3.1.1.cmml">η</mi><annotation-xml encoding="MathML-Content" id="A1.Thmtheorem8.p3.3.3.m3.1b"><ci id="A1.Thmtheorem8.p3.3.3.m3.1.1.cmml" xref="A1.Thmtheorem8.p3.3.3.m3.1.1">𝜂</ci></annotation-xml><annotation encoding="application/x-tex" id="A1.Thmtheorem8.p3.3.3.m3.1c">\eta</annotation><annotation encoding="application/x-llamapun" id="A1.Thmtheorem8.p3.3.3.m3.1d">italic_η</annotation></semantics></math>-fair orientation (indeed the <math alttext="\rho(G)" class="ltx_Math" display="inline" id="A1.Thmtheorem8.p3.4.4.m4.1"><semantics id="A1.Thmtheorem8.p3.4.4.m4.1a"><mrow id="A1.Thmtheorem8.p3.4.4.m4.1.2" xref="A1.Thmtheorem8.p3.4.4.m4.1.2.cmml"><mi id="A1.Thmtheorem8.p3.4.4.m4.1.2.2" xref="A1.Thmtheorem8.p3.4.4.m4.1.2.2.cmml">ρ</mi><mo id="A1.Thmtheorem8.p3.4.4.m4.1.2.1" xref="A1.Thmtheorem8.p3.4.4.m4.1.2.1.cmml">⁢</mo><mrow id="A1.Thmtheorem8.p3.4.4.m4.1.2.3.2" xref="A1.Thmtheorem8.p3.4.4.m4.1.2.cmml"><mo id="A1.Thmtheorem8.p3.4.4.m4.1.2.3.2.1" stretchy="false" xref="A1.Thmtheorem8.p3.4.4.m4.1.2.cmml">(</mo><mi id="A1.Thmtheorem8.p3.4.4.m4.1.1" xref="A1.Thmtheorem8.p3.4.4.m4.1.1.cmml">G</mi><mo id="A1.Thmtheorem8.p3.4.4.m4.1.2.3.2.2" stretchy="false" xref="A1.Thmtheorem8.p3.4.4.m4.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="A1.Thmtheorem8.p3.4.4.m4.1b"><apply id="A1.Thmtheorem8.p3.4.4.m4.1.2.cmml" xref="A1.Thmtheorem8.p3.4.4.m4.1.2"><times id="A1.Thmtheorem8.p3.4.4.m4.1.2.1.cmml" xref="A1.Thmtheorem8.p3.4.4.m4.1.2.1"></times><ci id="A1.Thmtheorem8.p3.4.4.m4.1.2.2.cmml" xref="A1.Thmtheorem8.p3.4.4.m4.1.2.2">𝜌</ci><ci id="A1.Thmtheorem8.p3.4.4.m4.1.1.cmml" xref="A1.Thmtheorem8.p3.4.4.m4.1.1">𝐺</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.Thmtheorem8.p3.4.4.m4.1c">\rho(G)</annotation><annotation encoding="application/x-llamapun" id="A1.Thmtheorem8.p3.4.4.m4.1d">italic_ρ ( italic_G )</annotation></semantics></math> is the minimum maximum fractional out-degree achievable) and so the vertex with the smallest ID among the vertices with the largest fractional out-degree under the first <math alttext="\eta" class="ltx_Math" display="inline" id="A1.Thmtheorem8.p3.5.5.m5.1"><semantics id="A1.Thmtheorem8.p3.5.5.m5.1a"><mi id="A1.Thmtheorem8.p3.5.5.m5.1.1" xref="A1.Thmtheorem8.p3.5.5.m5.1.1.cmml">η</mi><annotation-xml encoding="MathML-Content" id="A1.Thmtheorem8.p3.5.5.m5.1b"><ci id="A1.Thmtheorem8.p3.5.5.m5.1.1.cmml" xref="A1.Thmtheorem8.p3.5.5.m5.1.1">𝜂</ci></annotation-xml><annotation encoding="application/x-tex" id="A1.Thmtheorem8.p3.5.5.m5.1c">\eta</annotation><annotation encoding="application/x-llamapun" id="A1.Thmtheorem8.p3.5.5.m5.1d">italic_η</annotation></semantics></math>-fair orientation will always become an active leader.</span></p> </div> </div> <div class="ltx_para" id="A1.SS3.SSS0.P0.SPx2.p4"> <p class="ltx_p" id="A1.SS3.SSS0.P0.SPx2.p4.1">Finally, we have the following guarantee on the round complexity of the algorithm:</p> </div> <div class="ltx_theorem ltx_theorem_theorem" id="A1.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="A1.Thmtheorem9.1.1.1">Theorem A.9</span></span><span class="ltx_text ltx_font_bold" id="A1.Thmtheorem9.2.2">.</span> </h6> <div class="ltx_para" id="A1.Thmtheorem9.p1"> <p class="ltx_p" id="A1.Thmtheorem9.p1.9"><span class="ltx_text ltx_font_italic" id="A1.Thmtheorem9.p1.9.9">Suppose there exists an algorithm that given a graph <math alttext="G" class="ltx_Math" display="inline" id="A1.Thmtheorem9.p1.1.1.m1.1"><semantics id="A1.Thmtheorem9.p1.1.1.m1.1a"><mi id="A1.Thmtheorem9.p1.1.1.m1.1.1" xref="A1.Thmtheorem9.p1.1.1.m1.1.1.cmml">G</mi><annotation-xml encoding="MathML-Content" id="A1.Thmtheorem9.p1.1.1.m1.1b"><ci id="A1.Thmtheorem9.p1.1.1.m1.1.1.cmml" xref="A1.Thmtheorem9.p1.1.1.m1.1.1">𝐺</ci></annotation-xml><annotation encoding="application/x-tex" id="A1.Thmtheorem9.p1.1.1.m1.1c">G</annotation><annotation encoding="application/x-llamapun" id="A1.Thmtheorem9.p1.1.1.m1.1d">italic_G</annotation></semantics></math> on <math alttext="n" class="ltx_Math" display="inline" id="A1.Thmtheorem9.p1.2.2.m2.1"><semantics id="A1.Thmtheorem9.p1.2.2.m2.1a"><mi id="A1.Thmtheorem9.p1.2.2.m2.1.1" xref="A1.Thmtheorem9.p1.2.2.m2.1.1.cmml">n</mi><annotation-xml encoding="MathML-Content" id="A1.Thmtheorem9.p1.2.2.m2.1b"><ci id="A1.Thmtheorem9.p1.2.2.m2.1.1.cmml" xref="A1.Thmtheorem9.p1.2.2.m2.1.1">𝑛</ci></annotation-xml><annotation encoding="application/x-tex" id="A1.Thmtheorem9.p1.2.2.m2.1c">n</annotation><annotation encoding="application/x-llamapun" id="A1.Thmtheorem9.p1.2.2.m2.1d">italic_n</annotation></semantics></math> vertices and parameters <math alttext="\varepsilon" class="ltx_Math" display="inline" id="A1.Thmtheorem9.p1.3.3.m3.1"><semantics id="A1.Thmtheorem9.p1.3.3.m3.1a"><mi id="A1.Thmtheorem9.p1.3.3.m3.1.1" xref="A1.Thmtheorem9.p1.3.3.m3.1.1.cmml">ε</mi><annotation-xml encoding="MathML-Content" id="A1.Thmtheorem9.p1.3.3.m3.1b"><ci id="A1.Thmtheorem9.p1.3.3.m3.1.1.cmml" xref="A1.Thmtheorem9.p1.3.3.m3.1.1">𝜀</ci></annotation-xml><annotation encoding="application/x-tex" id="A1.Thmtheorem9.p1.3.3.m3.1c">\varepsilon</annotation><annotation encoding="application/x-llamapun" id="A1.Thmtheorem9.p1.3.3.m3.1d">italic_ε</annotation></semantics></math> computes an <math alttext="\eta" class="ltx_Math" display="inline" id="A1.Thmtheorem9.p1.4.4.m4.1"><semantics id="A1.Thmtheorem9.p1.4.4.m4.1a"><mi id="A1.Thmtheorem9.p1.4.4.m4.1.1" xref="A1.Thmtheorem9.p1.4.4.m4.1.1.cmml">η</mi><annotation-xml encoding="MathML-Content" id="A1.Thmtheorem9.p1.4.4.m4.1b"><ci id="A1.Thmtheorem9.p1.4.4.m4.1.1.cmml" xref="A1.Thmtheorem9.p1.4.4.m4.1.1">𝜂</ci></annotation-xml><annotation encoding="application/x-tex" id="A1.Thmtheorem9.p1.4.4.m4.1c">\eta</annotation><annotation encoding="application/x-llamapun" id="A1.Thmtheorem9.p1.4.4.m4.1d">italic_η</annotation></semantics></math>-fair orientation in <math alttext="O(U(n,\eta))" class="ltx_Math" display="inline" id="A1.Thmtheorem9.p1.5.5.m5.3"><semantics id="A1.Thmtheorem9.p1.5.5.m5.3a"><mrow id="A1.Thmtheorem9.p1.5.5.m5.3.3" xref="A1.Thmtheorem9.p1.5.5.m5.3.3.cmml"><mi id="A1.Thmtheorem9.p1.5.5.m5.3.3.3" xref="A1.Thmtheorem9.p1.5.5.m5.3.3.3.cmml">O</mi><mo id="A1.Thmtheorem9.p1.5.5.m5.3.3.2" xref="A1.Thmtheorem9.p1.5.5.m5.3.3.2.cmml">⁢</mo><mrow id="A1.Thmtheorem9.p1.5.5.m5.3.3.1.1" xref="A1.Thmtheorem9.p1.5.5.m5.3.3.1.1.1.cmml"><mo id="A1.Thmtheorem9.p1.5.5.m5.3.3.1.1.2" stretchy="false" xref="A1.Thmtheorem9.p1.5.5.m5.3.3.1.1.1.cmml">(</mo><mrow id="A1.Thmtheorem9.p1.5.5.m5.3.3.1.1.1" xref="A1.Thmtheorem9.p1.5.5.m5.3.3.1.1.1.cmml"><mi id="A1.Thmtheorem9.p1.5.5.m5.3.3.1.1.1.2" xref="A1.Thmtheorem9.p1.5.5.m5.3.3.1.1.1.2.cmml">U</mi><mo id="A1.Thmtheorem9.p1.5.5.m5.3.3.1.1.1.1" xref="A1.Thmtheorem9.p1.5.5.m5.3.3.1.1.1.1.cmml">⁢</mo><mrow id="A1.Thmtheorem9.p1.5.5.m5.3.3.1.1.1.3.2" xref="A1.Thmtheorem9.p1.5.5.m5.3.3.1.1.1.3.1.cmml"><mo id="A1.Thmtheorem9.p1.5.5.m5.3.3.1.1.1.3.2.1" stretchy="false" xref="A1.Thmtheorem9.p1.5.5.m5.3.3.1.1.1.3.1.cmml">(</mo><mi id="A1.Thmtheorem9.p1.5.5.m5.1.1" xref="A1.Thmtheorem9.p1.5.5.m5.1.1.cmml">n</mi><mo id="A1.Thmtheorem9.p1.5.5.m5.3.3.1.1.1.3.2.2" xref="A1.Thmtheorem9.p1.5.5.m5.3.3.1.1.1.3.1.cmml">,</mo><mi id="A1.Thmtheorem9.p1.5.5.m5.2.2" xref="A1.Thmtheorem9.p1.5.5.m5.2.2.cmml">η</mi><mo id="A1.Thmtheorem9.p1.5.5.m5.3.3.1.1.1.3.2.3" stretchy="false" xref="A1.Thmtheorem9.p1.5.5.m5.3.3.1.1.1.3.1.cmml">)</mo></mrow></mrow><mo id="A1.Thmtheorem9.p1.5.5.m5.3.3.1.1.3" stretchy="false" xref="A1.Thmtheorem9.p1.5.5.m5.3.3.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="A1.Thmtheorem9.p1.5.5.m5.3b"><apply id="A1.Thmtheorem9.p1.5.5.m5.3.3.cmml" xref="A1.Thmtheorem9.p1.5.5.m5.3.3"><times id="A1.Thmtheorem9.p1.5.5.m5.3.3.2.cmml" xref="A1.Thmtheorem9.p1.5.5.m5.3.3.2"></times><ci id="A1.Thmtheorem9.p1.5.5.m5.3.3.3.cmml" xref="A1.Thmtheorem9.p1.5.5.m5.3.3.3">𝑂</ci><apply id="A1.Thmtheorem9.p1.5.5.m5.3.3.1.1.1.cmml" xref="A1.Thmtheorem9.p1.5.5.m5.3.3.1.1"><times id="A1.Thmtheorem9.p1.5.5.m5.3.3.1.1.1.1.cmml" xref="A1.Thmtheorem9.p1.5.5.m5.3.3.1.1.1.1"></times><ci id="A1.Thmtheorem9.p1.5.5.m5.3.3.1.1.1.2.cmml" xref="A1.Thmtheorem9.p1.5.5.m5.3.3.1.1.1.2">𝑈</ci><interval closure="open" id="A1.Thmtheorem9.p1.5.5.m5.3.3.1.1.1.3.1.cmml" xref="A1.Thmtheorem9.p1.5.5.m5.3.3.1.1.1.3.2"><ci id="A1.Thmtheorem9.p1.5.5.m5.1.1.cmml" xref="A1.Thmtheorem9.p1.5.5.m5.1.1">𝑛</ci><ci id="A1.Thmtheorem9.p1.5.5.m5.2.2.cmml" xref="A1.Thmtheorem9.p1.5.5.m5.2.2">𝜂</ci></interval></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.Thmtheorem9.p1.5.5.m5.3c">O(U(n,\eta))</annotation><annotation encoding="application/x-llamapun" id="A1.Thmtheorem9.p1.5.5.m5.3d">italic_O ( italic_U ( italic_n , italic_η ) )</annotation></semantics></math> rounds. Then there exists an algorithm that, given <math alttext="\tilde{D}\leq\rho^{\max}(G)" class="ltx_Math" display="inline" id="A1.Thmtheorem9.p1.6.6.m6.1"><semantics id="A1.Thmtheorem9.p1.6.6.m6.1a"><mrow id="A1.Thmtheorem9.p1.6.6.m6.1.2" xref="A1.Thmtheorem9.p1.6.6.m6.1.2.cmml"><mover accent="true" id="A1.Thmtheorem9.p1.6.6.m6.1.2.2" xref="A1.Thmtheorem9.p1.6.6.m6.1.2.2.cmml"><mi id="A1.Thmtheorem9.p1.6.6.m6.1.2.2.2" xref="A1.Thmtheorem9.p1.6.6.m6.1.2.2.2.cmml">D</mi><mo id="A1.Thmtheorem9.p1.6.6.m6.1.2.2.1" xref="A1.Thmtheorem9.p1.6.6.m6.1.2.2.1.cmml">~</mo></mover><mo id="A1.Thmtheorem9.p1.6.6.m6.1.2.1" xref="A1.Thmtheorem9.p1.6.6.m6.1.2.1.cmml">≤</mo><mrow id="A1.Thmtheorem9.p1.6.6.m6.1.2.3" xref="A1.Thmtheorem9.p1.6.6.m6.1.2.3.cmml"><msup id="A1.Thmtheorem9.p1.6.6.m6.1.2.3.2" xref="A1.Thmtheorem9.p1.6.6.m6.1.2.3.2.cmml"><mi id="A1.Thmtheorem9.p1.6.6.m6.1.2.3.2.2" xref="A1.Thmtheorem9.p1.6.6.m6.1.2.3.2.2.cmml">ρ</mi><mi id="A1.Thmtheorem9.p1.6.6.m6.1.2.3.2.3" xref="A1.Thmtheorem9.p1.6.6.m6.1.2.3.2.3.cmml">max</mi></msup><mo id="A1.Thmtheorem9.p1.6.6.m6.1.2.3.1" xref="A1.Thmtheorem9.p1.6.6.m6.1.2.3.1.cmml">⁢</mo><mrow id="A1.Thmtheorem9.p1.6.6.m6.1.2.3.3.2" xref="A1.Thmtheorem9.p1.6.6.m6.1.2.3.cmml"><mo id="A1.Thmtheorem9.p1.6.6.m6.1.2.3.3.2.1" stretchy="false" xref="A1.Thmtheorem9.p1.6.6.m6.1.2.3.cmml">(</mo><mi id="A1.Thmtheorem9.p1.6.6.m6.1.1" xref="A1.Thmtheorem9.p1.6.6.m6.1.1.cmml">G</mi><mo id="A1.Thmtheorem9.p1.6.6.m6.1.2.3.3.2.2" stretchy="false" xref="A1.Thmtheorem9.p1.6.6.m6.1.2.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="A1.Thmtheorem9.p1.6.6.m6.1b"><apply id="A1.Thmtheorem9.p1.6.6.m6.1.2.cmml" xref="A1.Thmtheorem9.p1.6.6.m6.1.2"><leq id="A1.Thmtheorem9.p1.6.6.m6.1.2.1.cmml" xref="A1.Thmtheorem9.p1.6.6.m6.1.2.1"></leq><apply id="A1.Thmtheorem9.p1.6.6.m6.1.2.2.cmml" xref="A1.Thmtheorem9.p1.6.6.m6.1.2.2"><ci id="A1.Thmtheorem9.p1.6.6.m6.1.2.2.1.cmml" xref="A1.Thmtheorem9.p1.6.6.m6.1.2.2.1">~</ci><ci id="A1.Thmtheorem9.p1.6.6.m6.1.2.2.2.cmml" xref="A1.Thmtheorem9.p1.6.6.m6.1.2.2.2">𝐷</ci></apply><apply id="A1.Thmtheorem9.p1.6.6.m6.1.2.3.cmml" xref="A1.Thmtheorem9.p1.6.6.m6.1.2.3"><times id="A1.Thmtheorem9.p1.6.6.m6.1.2.3.1.cmml" xref="A1.Thmtheorem9.p1.6.6.m6.1.2.3.1"></times><apply id="A1.Thmtheorem9.p1.6.6.m6.1.2.3.2.cmml" xref="A1.Thmtheorem9.p1.6.6.m6.1.2.3.2"><csymbol cd="ambiguous" id="A1.Thmtheorem9.p1.6.6.m6.1.2.3.2.1.cmml" xref="A1.Thmtheorem9.p1.6.6.m6.1.2.3.2">superscript</csymbol><ci id="A1.Thmtheorem9.p1.6.6.m6.1.2.3.2.2.cmml" xref="A1.Thmtheorem9.p1.6.6.m6.1.2.3.2.2">𝜌</ci><max id="A1.Thmtheorem9.p1.6.6.m6.1.2.3.2.3.cmml" xref="A1.Thmtheorem9.p1.6.6.m6.1.2.3.2.3"></max></apply><ci id="A1.Thmtheorem9.p1.6.6.m6.1.1.cmml" xref="A1.Thmtheorem9.p1.6.6.m6.1.1">𝐺</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.Thmtheorem9.p1.6.6.m6.1c">\tilde{D}\leq\rho^{\max}(G)</annotation><annotation encoding="application/x-llamapun" id="A1.Thmtheorem9.p1.6.6.m6.1d">over~ start_ARG italic_D end_ARG ≤ italic_ρ start_POSTSUPERSCRIPT roman_max end_POSTSUPERSCRIPT ( italic_G )</annotation></semantics></math> as input at every vertex, outputs a subgraph <math alttext="H" class="ltx_Math" display="inline" id="A1.Thmtheorem9.p1.7.7.m7.1"><semantics id="A1.Thmtheorem9.p1.7.7.m7.1a"><mi id="A1.Thmtheorem9.p1.7.7.m7.1.1" xref="A1.Thmtheorem9.p1.7.7.m7.1.1.cmml">H</mi><annotation-xml encoding="MathML-Content" id="A1.Thmtheorem9.p1.7.7.m7.1b"><ci id="A1.Thmtheorem9.p1.7.7.m7.1.1.cmml" xref="A1.Thmtheorem9.p1.7.7.m7.1.1">𝐻</ci></annotation-xml><annotation encoding="application/x-tex" id="A1.Thmtheorem9.p1.7.7.m7.1c">H</annotation><annotation encoding="application/x-llamapun" id="A1.Thmtheorem9.p1.7.7.m7.1d">italic_H</annotation></semantics></math> with <math alttext="\rho(H)\geq\rho(1-\varepsilon)\tilde{D}" class="ltx_Math" display="inline" id="A1.Thmtheorem9.p1.8.8.m8.2"><semantics id="A1.Thmtheorem9.p1.8.8.m8.2a"><mrow id="A1.Thmtheorem9.p1.8.8.m8.2.2" xref="A1.Thmtheorem9.p1.8.8.m8.2.2.cmml"><mrow id="A1.Thmtheorem9.p1.8.8.m8.2.2.3" xref="A1.Thmtheorem9.p1.8.8.m8.2.2.3.cmml"><mi id="A1.Thmtheorem9.p1.8.8.m8.2.2.3.2" xref="A1.Thmtheorem9.p1.8.8.m8.2.2.3.2.cmml">ρ</mi><mo id="A1.Thmtheorem9.p1.8.8.m8.2.2.3.1" xref="A1.Thmtheorem9.p1.8.8.m8.2.2.3.1.cmml">⁢</mo><mrow id="A1.Thmtheorem9.p1.8.8.m8.2.2.3.3.2" xref="A1.Thmtheorem9.p1.8.8.m8.2.2.3.cmml"><mo id="A1.Thmtheorem9.p1.8.8.m8.2.2.3.3.2.1" stretchy="false" xref="A1.Thmtheorem9.p1.8.8.m8.2.2.3.cmml">(</mo><mi id="A1.Thmtheorem9.p1.8.8.m8.1.1" xref="A1.Thmtheorem9.p1.8.8.m8.1.1.cmml">H</mi><mo id="A1.Thmtheorem9.p1.8.8.m8.2.2.3.3.2.2" stretchy="false" xref="A1.Thmtheorem9.p1.8.8.m8.2.2.3.cmml">)</mo></mrow></mrow><mo id="A1.Thmtheorem9.p1.8.8.m8.2.2.2" xref="A1.Thmtheorem9.p1.8.8.m8.2.2.2.cmml">≥</mo><mrow id="A1.Thmtheorem9.p1.8.8.m8.2.2.1" xref="A1.Thmtheorem9.p1.8.8.m8.2.2.1.cmml"><mi id="A1.Thmtheorem9.p1.8.8.m8.2.2.1.3" xref="A1.Thmtheorem9.p1.8.8.m8.2.2.1.3.cmml">ρ</mi><mo id="A1.Thmtheorem9.p1.8.8.m8.2.2.1.2" xref="A1.Thmtheorem9.p1.8.8.m8.2.2.1.2.cmml">⁢</mo><mrow id="A1.Thmtheorem9.p1.8.8.m8.2.2.1.1.1" xref="A1.Thmtheorem9.p1.8.8.m8.2.2.1.1.1.1.cmml"><mo id="A1.Thmtheorem9.p1.8.8.m8.2.2.1.1.1.2" stretchy="false" xref="A1.Thmtheorem9.p1.8.8.m8.2.2.1.1.1.1.cmml">(</mo><mrow id="A1.Thmtheorem9.p1.8.8.m8.2.2.1.1.1.1" xref="A1.Thmtheorem9.p1.8.8.m8.2.2.1.1.1.1.cmml"><mn id="A1.Thmtheorem9.p1.8.8.m8.2.2.1.1.1.1.2" xref="A1.Thmtheorem9.p1.8.8.m8.2.2.1.1.1.1.2.cmml">1</mn><mo id="A1.Thmtheorem9.p1.8.8.m8.2.2.1.1.1.1.1" xref="A1.Thmtheorem9.p1.8.8.m8.2.2.1.1.1.1.1.cmml">−</mo><mi id="A1.Thmtheorem9.p1.8.8.m8.2.2.1.1.1.1.3" xref="A1.Thmtheorem9.p1.8.8.m8.2.2.1.1.1.1.3.cmml">ε</mi></mrow><mo id="A1.Thmtheorem9.p1.8.8.m8.2.2.1.1.1.3" stretchy="false" xref="A1.Thmtheorem9.p1.8.8.m8.2.2.1.1.1.1.cmml">)</mo></mrow><mo id="A1.Thmtheorem9.p1.8.8.m8.2.2.1.2a" xref="A1.Thmtheorem9.p1.8.8.m8.2.2.1.2.cmml">⁢</mo><mover accent="true" id="A1.Thmtheorem9.p1.8.8.m8.2.2.1.4" xref="A1.Thmtheorem9.p1.8.8.m8.2.2.1.4.cmml"><mi id="A1.Thmtheorem9.p1.8.8.m8.2.2.1.4.2" xref="A1.Thmtheorem9.p1.8.8.m8.2.2.1.4.2.cmml">D</mi><mo id="A1.Thmtheorem9.p1.8.8.m8.2.2.1.4.1" xref="A1.Thmtheorem9.p1.8.8.m8.2.2.1.4.1.cmml">~</mo></mover></mrow></mrow><annotation-xml encoding="MathML-Content" id="A1.Thmtheorem9.p1.8.8.m8.2b"><apply id="A1.Thmtheorem9.p1.8.8.m8.2.2.cmml" xref="A1.Thmtheorem9.p1.8.8.m8.2.2"><geq id="A1.Thmtheorem9.p1.8.8.m8.2.2.2.cmml" xref="A1.Thmtheorem9.p1.8.8.m8.2.2.2"></geq><apply id="A1.Thmtheorem9.p1.8.8.m8.2.2.3.cmml" xref="A1.Thmtheorem9.p1.8.8.m8.2.2.3"><times id="A1.Thmtheorem9.p1.8.8.m8.2.2.3.1.cmml" xref="A1.Thmtheorem9.p1.8.8.m8.2.2.3.1"></times><ci id="A1.Thmtheorem9.p1.8.8.m8.2.2.3.2.cmml" xref="A1.Thmtheorem9.p1.8.8.m8.2.2.3.2">𝜌</ci><ci id="A1.Thmtheorem9.p1.8.8.m8.1.1.cmml" xref="A1.Thmtheorem9.p1.8.8.m8.1.1">𝐻</ci></apply><apply id="A1.Thmtheorem9.p1.8.8.m8.2.2.1.cmml" xref="A1.Thmtheorem9.p1.8.8.m8.2.2.1"><times id="A1.Thmtheorem9.p1.8.8.m8.2.2.1.2.cmml" xref="A1.Thmtheorem9.p1.8.8.m8.2.2.1.2"></times><ci id="A1.Thmtheorem9.p1.8.8.m8.2.2.1.3.cmml" xref="A1.Thmtheorem9.p1.8.8.m8.2.2.1.3">𝜌</ci><apply id="A1.Thmtheorem9.p1.8.8.m8.2.2.1.1.1.1.cmml" xref="A1.Thmtheorem9.p1.8.8.m8.2.2.1.1.1"><minus id="A1.Thmtheorem9.p1.8.8.m8.2.2.1.1.1.1.1.cmml" xref="A1.Thmtheorem9.p1.8.8.m8.2.2.1.1.1.1.1"></minus><cn id="A1.Thmtheorem9.p1.8.8.m8.2.2.1.1.1.1.2.cmml" type="integer" xref="A1.Thmtheorem9.p1.8.8.m8.2.2.1.1.1.1.2">1</cn><ci id="A1.Thmtheorem9.p1.8.8.m8.2.2.1.1.1.1.3.cmml" xref="A1.Thmtheorem9.p1.8.8.m8.2.2.1.1.1.1.3">𝜀</ci></apply><apply id="A1.Thmtheorem9.p1.8.8.m8.2.2.1.4.cmml" xref="A1.Thmtheorem9.p1.8.8.m8.2.2.1.4"><ci id="A1.Thmtheorem9.p1.8.8.m8.2.2.1.4.1.cmml" xref="A1.Thmtheorem9.p1.8.8.m8.2.2.1.4.1">~</ci><ci id="A1.Thmtheorem9.p1.8.8.m8.2.2.1.4.2.cmml" xref="A1.Thmtheorem9.p1.8.8.m8.2.2.1.4.2">𝐷</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.Thmtheorem9.p1.8.8.m8.2c">\rho(H)\geq\rho(1-\varepsilon)\tilde{D}</annotation><annotation encoding="application/x-llamapun" id="A1.Thmtheorem9.p1.8.8.m8.2d">italic_ρ ( italic_H ) ≥ italic_ρ ( 1 - italic_ε ) over~ start_ARG italic_D end_ARG</annotation></semantics></math> after <math alttext="O(U(n,\frac{\varepsilon^{2}}{128\log n})+\frac{\log^{2}n}{\varepsilon^{2}})" class="ltx_Math" display="inline" id="A1.Thmtheorem9.p1.9.9.m9.3"><semantics id="A1.Thmtheorem9.p1.9.9.m9.3a"><mrow id="A1.Thmtheorem9.p1.9.9.m9.3.3" xref="A1.Thmtheorem9.p1.9.9.m9.3.3.cmml"><mi id="A1.Thmtheorem9.p1.9.9.m9.3.3.3" xref="A1.Thmtheorem9.p1.9.9.m9.3.3.3.cmml">O</mi><mo id="A1.Thmtheorem9.p1.9.9.m9.3.3.2" xref="A1.Thmtheorem9.p1.9.9.m9.3.3.2.cmml">⁢</mo><mrow id="A1.Thmtheorem9.p1.9.9.m9.3.3.1.1" xref="A1.Thmtheorem9.p1.9.9.m9.3.3.1.1.1.cmml"><mo id="A1.Thmtheorem9.p1.9.9.m9.3.3.1.1.2" stretchy="false" xref="A1.Thmtheorem9.p1.9.9.m9.3.3.1.1.1.cmml">(</mo><mrow id="A1.Thmtheorem9.p1.9.9.m9.3.3.1.1.1" xref="A1.Thmtheorem9.p1.9.9.m9.3.3.1.1.1.cmml"><mrow id="A1.Thmtheorem9.p1.9.9.m9.3.3.1.1.1.2" xref="A1.Thmtheorem9.p1.9.9.m9.3.3.1.1.1.2.cmml"><mi id="A1.Thmtheorem9.p1.9.9.m9.3.3.1.1.1.2.2" xref="A1.Thmtheorem9.p1.9.9.m9.3.3.1.1.1.2.2.cmml">U</mi><mo id="A1.Thmtheorem9.p1.9.9.m9.3.3.1.1.1.2.1" xref="A1.Thmtheorem9.p1.9.9.m9.3.3.1.1.1.2.1.cmml">⁢</mo><mrow id="A1.Thmtheorem9.p1.9.9.m9.3.3.1.1.1.2.3.2" xref="A1.Thmtheorem9.p1.9.9.m9.3.3.1.1.1.2.3.1.cmml"><mo id="A1.Thmtheorem9.p1.9.9.m9.3.3.1.1.1.2.3.2.1" stretchy="false" xref="A1.Thmtheorem9.p1.9.9.m9.3.3.1.1.1.2.3.1.cmml">(</mo><mi id="A1.Thmtheorem9.p1.9.9.m9.1.1" xref="A1.Thmtheorem9.p1.9.9.m9.1.1.cmml">n</mi><mo id="A1.Thmtheorem9.p1.9.9.m9.3.3.1.1.1.2.3.2.2" xref="A1.Thmtheorem9.p1.9.9.m9.3.3.1.1.1.2.3.1.cmml">,</mo><mfrac id="A1.Thmtheorem9.p1.9.9.m9.2.2" xref="A1.Thmtheorem9.p1.9.9.m9.2.2.cmml"><msup id="A1.Thmtheorem9.p1.9.9.m9.2.2.2" xref="A1.Thmtheorem9.p1.9.9.m9.2.2.2.cmml"><mi id="A1.Thmtheorem9.p1.9.9.m9.2.2.2.2" xref="A1.Thmtheorem9.p1.9.9.m9.2.2.2.2.cmml">ε</mi><mn id="A1.Thmtheorem9.p1.9.9.m9.2.2.2.3" xref="A1.Thmtheorem9.p1.9.9.m9.2.2.2.3.cmml">2</mn></msup><mrow id="A1.Thmtheorem9.p1.9.9.m9.2.2.3" xref="A1.Thmtheorem9.p1.9.9.m9.2.2.3.cmml"><mn id="A1.Thmtheorem9.p1.9.9.m9.2.2.3.2" xref="A1.Thmtheorem9.p1.9.9.m9.2.2.3.2.cmml">128</mn><mo id="A1.Thmtheorem9.p1.9.9.m9.2.2.3.1" lspace="0.167em" xref="A1.Thmtheorem9.p1.9.9.m9.2.2.3.1.cmml">⁢</mo><mrow id="A1.Thmtheorem9.p1.9.9.m9.2.2.3.3" xref="A1.Thmtheorem9.p1.9.9.m9.2.2.3.3.cmml"><mi id="A1.Thmtheorem9.p1.9.9.m9.2.2.3.3.1" xref="A1.Thmtheorem9.p1.9.9.m9.2.2.3.3.1.cmml">log</mi><mo id="A1.Thmtheorem9.p1.9.9.m9.2.2.3.3a" lspace="0.167em" xref="A1.Thmtheorem9.p1.9.9.m9.2.2.3.3.cmml">⁡</mo><mi id="A1.Thmtheorem9.p1.9.9.m9.2.2.3.3.2" xref="A1.Thmtheorem9.p1.9.9.m9.2.2.3.3.2.cmml">n</mi></mrow></mrow></mfrac><mo id="A1.Thmtheorem9.p1.9.9.m9.3.3.1.1.1.2.3.2.3" stretchy="false" xref="A1.Thmtheorem9.p1.9.9.m9.3.3.1.1.1.2.3.1.cmml">)</mo></mrow></mrow><mo id="A1.Thmtheorem9.p1.9.9.m9.3.3.1.1.1.1" xref="A1.Thmtheorem9.p1.9.9.m9.3.3.1.1.1.1.cmml">+</mo><mfrac id="A1.Thmtheorem9.p1.9.9.m9.3.3.1.1.1.3" xref="A1.Thmtheorem9.p1.9.9.m9.3.3.1.1.1.3.cmml"><mrow id="A1.Thmtheorem9.p1.9.9.m9.3.3.1.1.1.3.2" xref="A1.Thmtheorem9.p1.9.9.m9.3.3.1.1.1.3.2.cmml"><msup id="A1.Thmtheorem9.p1.9.9.m9.3.3.1.1.1.3.2.1" xref="A1.Thmtheorem9.p1.9.9.m9.3.3.1.1.1.3.2.1.cmml"><mi id="A1.Thmtheorem9.p1.9.9.m9.3.3.1.1.1.3.2.1.2" xref="A1.Thmtheorem9.p1.9.9.m9.3.3.1.1.1.3.2.1.2.cmml">log</mi><mn id="A1.Thmtheorem9.p1.9.9.m9.3.3.1.1.1.3.2.1.3" xref="A1.Thmtheorem9.p1.9.9.m9.3.3.1.1.1.3.2.1.3.cmml">2</mn></msup><mo id="A1.Thmtheorem9.p1.9.9.m9.3.3.1.1.1.3.2a" lspace="0.167em" xref="A1.Thmtheorem9.p1.9.9.m9.3.3.1.1.1.3.2.cmml">⁡</mo><mi id="A1.Thmtheorem9.p1.9.9.m9.3.3.1.1.1.3.2.2" xref="A1.Thmtheorem9.p1.9.9.m9.3.3.1.1.1.3.2.2.cmml">n</mi></mrow><msup id="A1.Thmtheorem9.p1.9.9.m9.3.3.1.1.1.3.3" xref="A1.Thmtheorem9.p1.9.9.m9.3.3.1.1.1.3.3.cmml"><mi id="A1.Thmtheorem9.p1.9.9.m9.3.3.1.1.1.3.3.2" xref="A1.Thmtheorem9.p1.9.9.m9.3.3.1.1.1.3.3.2.cmml">ε</mi><mn id="A1.Thmtheorem9.p1.9.9.m9.3.3.1.1.1.3.3.3" xref="A1.Thmtheorem9.p1.9.9.m9.3.3.1.1.1.3.3.3.cmml">2</mn></msup></mfrac></mrow><mo id="A1.Thmtheorem9.p1.9.9.m9.3.3.1.1.3" stretchy="false" xref="A1.Thmtheorem9.p1.9.9.m9.3.3.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="A1.Thmtheorem9.p1.9.9.m9.3b"><apply id="A1.Thmtheorem9.p1.9.9.m9.3.3.cmml" xref="A1.Thmtheorem9.p1.9.9.m9.3.3"><times id="A1.Thmtheorem9.p1.9.9.m9.3.3.2.cmml" xref="A1.Thmtheorem9.p1.9.9.m9.3.3.2"></times><ci id="A1.Thmtheorem9.p1.9.9.m9.3.3.3.cmml" xref="A1.Thmtheorem9.p1.9.9.m9.3.3.3">𝑂</ci><apply id="A1.Thmtheorem9.p1.9.9.m9.3.3.1.1.1.cmml" 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xref="A1.Thmtheorem9.p1.9.9.m9.2.2.2">superscript</csymbol><ci id="A1.Thmtheorem9.p1.9.9.m9.2.2.2.2.cmml" xref="A1.Thmtheorem9.p1.9.9.m9.2.2.2.2">𝜀</ci><cn id="A1.Thmtheorem9.p1.9.9.m9.2.2.2.3.cmml" type="integer" xref="A1.Thmtheorem9.p1.9.9.m9.2.2.2.3">2</cn></apply><apply id="A1.Thmtheorem9.p1.9.9.m9.2.2.3.cmml" xref="A1.Thmtheorem9.p1.9.9.m9.2.2.3"><times id="A1.Thmtheorem9.p1.9.9.m9.2.2.3.1.cmml" xref="A1.Thmtheorem9.p1.9.9.m9.2.2.3.1"></times><cn id="A1.Thmtheorem9.p1.9.9.m9.2.2.3.2.cmml" type="integer" xref="A1.Thmtheorem9.p1.9.9.m9.2.2.3.2">128</cn><apply id="A1.Thmtheorem9.p1.9.9.m9.2.2.3.3.cmml" xref="A1.Thmtheorem9.p1.9.9.m9.2.2.3.3"><log id="A1.Thmtheorem9.p1.9.9.m9.2.2.3.3.1.cmml" xref="A1.Thmtheorem9.p1.9.9.m9.2.2.3.3.1"></log><ci id="A1.Thmtheorem9.p1.9.9.m9.2.2.3.3.2.cmml" xref="A1.Thmtheorem9.p1.9.9.m9.2.2.3.3.2">𝑛</ci></apply></apply></apply></interval></apply><apply id="A1.Thmtheorem9.p1.9.9.m9.3.3.1.1.1.3.cmml" xref="A1.Thmtheorem9.p1.9.9.m9.3.3.1.1.1.3"><divide id="A1.Thmtheorem9.p1.9.9.m9.3.3.1.1.1.3.1.cmml" xref="A1.Thmtheorem9.p1.9.9.m9.3.3.1.1.1.3"></divide><apply id="A1.Thmtheorem9.p1.9.9.m9.3.3.1.1.1.3.2.cmml" xref="A1.Thmtheorem9.p1.9.9.m9.3.3.1.1.1.3.2"><apply id="A1.Thmtheorem9.p1.9.9.m9.3.3.1.1.1.3.2.1.cmml" xref="A1.Thmtheorem9.p1.9.9.m9.3.3.1.1.1.3.2.1"><csymbol cd="ambiguous" id="A1.Thmtheorem9.p1.9.9.m9.3.3.1.1.1.3.2.1.1.cmml" xref="A1.Thmtheorem9.p1.9.9.m9.3.3.1.1.1.3.2.1">superscript</csymbol><log id="A1.Thmtheorem9.p1.9.9.m9.3.3.1.1.1.3.2.1.2.cmml" xref="A1.Thmtheorem9.p1.9.9.m9.3.3.1.1.1.3.2.1.2"></log><cn id="A1.Thmtheorem9.p1.9.9.m9.3.3.1.1.1.3.2.1.3.cmml" type="integer" xref="A1.Thmtheorem9.p1.9.9.m9.3.3.1.1.1.3.2.1.3">2</cn></apply><ci id="A1.Thmtheorem9.p1.9.9.m9.3.3.1.1.1.3.2.2.cmml" xref="A1.Thmtheorem9.p1.9.9.m9.3.3.1.1.1.3.2.2">𝑛</ci></apply><apply id="A1.Thmtheorem9.p1.9.9.m9.3.3.1.1.1.3.3.cmml" xref="A1.Thmtheorem9.p1.9.9.m9.3.3.1.1.1.3.3"><csymbol cd="ambiguous" id="A1.Thmtheorem9.p1.9.9.m9.3.3.1.1.1.3.3.1.cmml" xref="A1.Thmtheorem9.p1.9.9.m9.3.3.1.1.1.3.3">superscript</csymbol><ci id="A1.Thmtheorem9.p1.9.9.m9.3.3.1.1.1.3.3.2.cmml" xref="A1.Thmtheorem9.p1.9.9.m9.3.3.1.1.1.3.3.2">𝜀</ci><cn id="A1.Thmtheorem9.p1.9.9.m9.3.3.1.1.1.3.3.3.cmml" type="integer" xref="A1.Thmtheorem9.p1.9.9.m9.3.3.1.1.1.3.3.3">2</cn></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.Thmtheorem9.p1.9.9.m9.3c">O(U(n,\frac{\varepsilon^{2}}{128\log n})+\frac{\log^{2}n}{\varepsilon^{2}})</annotation><annotation encoding="application/x-llamapun" id="A1.Thmtheorem9.p1.9.9.m9.3d">italic_O ( italic_U ( italic_n , divide start_ARG italic_ε start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG start_ARG 128 roman_log italic_n end_ARG ) + divide start_ARG roman_log start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_n end_ARG start_ARG italic_ε start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG )</annotation></semantics></math> rounds.</span></p> </div> </div> <div class="ltx_theorem ltx_theorem_proof" id="A1.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="A1.Thmtheorem10.1.1.1">Proof A.10</span></span><span class="ltx_text ltx_font_bold" id="A1.Thmtheorem10.2.2">.</span> </h6> <div class="ltx_para" id="A1.Thmtheorem10.p1"> <p class="ltx_p" id="A1.Thmtheorem10.p1.11"><span class="ltx_text ltx_font_italic" id="A1.Thmtheorem10.p1.11.11">Observe that computing the <math alttext="\frac{\varepsilon^{2}}{128\log n}" class="ltx_Math" display="inline" id="A1.Thmtheorem10.p1.1.1.m1.1"><semantics id="A1.Thmtheorem10.p1.1.1.m1.1a"><mfrac id="A1.Thmtheorem10.p1.1.1.m1.1.1" xref="A1.Thmtheorem10.p1.1.1.m1.1.1.cmml"><msup id="A1.Thmtheorem10.p1.1.1.m1.1.1.2" xref="A1.Thmtheorem10.p1.1.1.m1.1.1.2.cmml"><mi id="A1.Thmtheorem10.p1.1.1.m1.1.1.2.2" xref="A1.Thmtheorem10.p1.1.1.m1.1.1.2.2.cmml">ε</mi><mn id="A1.Thmtheorem10.p1.1.1.m1.1.1.2.3" xref="A1.Thmtheorem10.p1.1.1.m1.1.1.2.3.cmml">2</mn></msup><mrow id="A1.Thmtheorem10.p1.1.1.m1.1.1.3" xref="A1.Thmtheorem10.p1.1.1.m1.1.1.3.cmml"><mn id="A1.Thmtheorem10.p1.1.1.m1.1.1.3.2" xref="A1.Thmtheorem10.p1.1.1.m1.1.1.3.2.cmml">128</mn><mo id="A1.Thmtheorem10.p1.1.1.m1.1.1.3.1" lspace="0.167em" xref="A1.Thmtheorem10.p1.1.1.m1.1.1.3.1.cmml">⁢</mo><mrow id="A1.Thmtheorem10.p1.1.1.m1.1.1.3.3" xref="A1.Thmtheorem10.p1.1.1.m1.1.1.3.3.cmml"><mi id="A1.Thmtheorem10.p1.1.1.m1.1.1.3.3.1" xref="A1.Thmtheorem10.p1.1.1.m1.1.1.3.3.1.cmml">log</mi><mo id="A1.Thmtheorem10.p1.1.1.m1.1.1.3.3a" lspace="0.167em" xref="A1.Thmtheorem10.p1.1.1.m1.1.1.3.3.cmml">⁡</mo><mi id="A1.Thmtheorem10.p1.1.1.m1.1.1.3.3.2" xref="A1.Thmtheorem10.p1.1.1.m1.1.1.3.3.2.cmml">n</mi></mrow></mrow></mfrac><annotation-xml encoding="MathML-Content" id="A1.Thmtheorem10.p1.1.1.m1.1b"><apply id="A1.Thmtheorem10.p1.1.1.m1.1.1.cmml" xref="A1.Thmtheorem10.p1.1.1.m1.1.1"><divide id="A1.Thmtheorem10.p1.1.1.m1.1.1.1.cmml" xref="A1.Thmtheorem10.p1.1.1.m1.1.1"></divide><apply id="A1.Thmtheorem10.p1.1.1.m1.1.1.2.cmml" xref="A1.Thmtheorem10.p1.1.1.m1.1.1.2"><csymbol cd="ambiguous" id="A1.Thmtheorem10.p1.1.1.m1.1.1.2.1.cmml" xref="A1.Thmtheorem10.p1.1.1.m1.1.1.2">superscript</csymbol><ci id="A1.Thmtheorem10.p1.1.1.m1.1.1.2.2.cmml" xref="A1.Thmtheorem10.p1.1.1.m1.1.1.2.2">𝜀</ci><cn id="A1.Thmtheorem10.p1.1.1.m1.1.1.2.3.cmml" type="integer" xref="A1.Thmtheorem10.p1.1.1.m1.1.1.2.3">2</cn></apply><apply id="A1.Thmtheorem10.p1.1.1.m1.1.1.3.cmml" xref="A1.Thmtheorem10.p1.1.1.m1.1.1.3"><times id="A1.Thmtheorem10.p1.1.1.m1.1.1.3.1.cmml" xref="A1.Thmtheorem10.p1.1.1.m1.1.1.3.1"></times><cn id="A1.Thmtheorem10.p1.1.1.m1.1.1.3.2.cmml" type="integer" xref="A1.Thmtheorem10.p1.1.1.m1.1.1.3.2">128</cn><apply id="A1.Thmtheorem10.p1.1.1.m1.1.1.3.3.cmml" xref="A1.Thmtheorem10.p1.1.1.m1.1.1.3.3"><log id="A1.Thmtheorem10.p1.1.1.m1.1.1.3.3.1.cmml" xref="A1.Thmtheorem10.p1.1.1.m1.1.1.3.3.1"></log><ci id="A1.Thmtheorem10.p1.1.1.m1.1.1.3.3.2.cmml" xref="A1.Thmtheorem10.p1.1.1.m1.1.1.3.3.2">𝑛</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.Thmtheorem10.p1.1.1.m1.1c">\frac{\varepsilon^{2}}{128\log n}</annotation><annotation encoding="application/x-llamapun" id="A1.Thmtheorem10.p1.1.1.m1.1d">divide start_ARG italic_ε start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG start_ARG 128 roman_log italic_n end_ARG</annotation></semantics></math>-fair orientations takes at most <math alttext="O(U(n,\frac{\varepsilon^{2}}{128\log n}))" class="ltx_Math" display="inline" id="A1.Thmtheorem10.p1.2.2.m2.3"><semantics id="A1.Thmtheorem10.p1.2.2.m2.3a"><mrow id="A1.Thmtheorem10.p1.2.2.m2.3.3" xref="A1.Thmtheorem10.p1.2.2.m2.3.3.cmml"><mi id="A1.Thmtheorem10.p1.2.2.m2.3.3.3" xref="A1.Thmtheorem10.p1.2.2.m2.3.3.3.cmml">O</mi><mo id="A1.Thmtheorem10.p1.2.2.m2.3.3.2" xref="A1.Thmtheorem10.p1.2.2.m2.3.3.2.cmml">⁢</mo><mrow id="A1.Thmtheorem10.p1.2.2.m2.3.3.1.1" 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xref="A1.Thmtheorem10.p1.2.2.m2.2.2.3.3.2.cmml">n</mi></mrow></mrow></mfrac><mo id="A1.Thmtheorem10.p1.2.2.m2.3.3.1.1.1.3.2.3" stretchy="false" xref="A1.Thmtheorem10.p1.2.2.m2.3.3.1.1.1.3.1.cmml">)</mo></mrow></mrow><mo id="A1.Thmtheorem10.p1.2.2.m2.3.3.1.1.3" stretchy="false" xref="A1.Thmtheorem10.p1.2.2.m2.3.3.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="A1.Thmtheorem10.p1.2.2.m2.3b"><apply id="A1.Thmtheorem10.p1.2.2.m2.3.3.cmml" xref="A1.Thmtheorem10.p1.2.2.m2.3.3"><times id="A1.Thmtheorem10.p1.2.2.m2.3.3.2.cmml" xref="A1.Thmtheorem10.p1.2.2.m2.3.3.2"></times><ci id="A1.Thmtheorem10.p1.2.2.m2.3.3.3.cmml" xref="A1.Thmtheorem10.p1.2.2.m2.3.3.3">𝑂</ci><apply id="A1.Thmtheorem10.p1.2.2.m2.3.3.1.1.1.cmml" xref="A1.Thmtheorem10.p1.2.2.m2.3.3.1.1"><times id="A1.Thmtheorem10.p1.2.2.m2.3.3.1.1.1.1.cmml" xref="A1.Thmtheorem10.p1.2.2.m2.3.3.1.1.1.1"></times><ci id="A1.Thmtheorem10.p1.2.2.m2.3.3.1.1.1.2.cmml" xref="A1.Thmtheorem10.p1.2.2.m2.3.3.1.1.1.2">𝑈</ci><interval closure="open" id="A1.Thmtheorem10.p1.2.2.m2.3.3.1.1.1.3.1.cmml" xref="A1.Thmtheorem10.p1.2.2.m2.3.3.1.1.1.3.2"><ci id="A1.Thmtheorem10.p1.2.2.m2.1.1.cmml" xref="A1.Thmtheorem10.p1.2.2.m2.1.1">𝑛</ci><apply id="A1.Thmtheorem10.p1.2.2.m2.2.2.cmml" xref="A1.Thmtheorem10.p1.2.2.m2.2.2"><divide id="A1.Thmtheorem10.p1.2.2.m2.2.2.1.cmml" xref="A1.Thmtheorem10.p1.2.2.m2.2.2"></divide><apply id="A1.Thmtheorem10.p1.2.2.m2.2.2.2.cmml" xref="A1.Thmtheorem10.p1.2.2.m2.2.2.2"><csymbol cd="ambiguous" id="A1.Thmtheorem10.p1.2.2.m2.2.2.2.1.cmml" xref="A1.Thmtheorem10.p1.2.2.m2.2.2.2">superscript</csymbol><ci id="A1.Thmtheorem10.p1.2.2.m2.2.2.2.2.cmml" xref="A1.Thmtheorem10.p1.2.2.m2.2.2.2.2">𝜀</ci><cn id="A1.Thmtheorem10.p1.2.2.m2.2.2.2.3.cmml" type="integer" xref="A1.Thmtheorem10.p1.2.2.m2.2.2.2.3">2</cn></apply><apply id="A1.Thmtheorem10.p1.2.2.m2.2.2.3.cmml" xref="A1.Thmtheorem10.p1.2.2.m2.2.2.3"><times id="A1.Thmtheorem10.p1.2.2.m2.2.2.3.1.cmml" xref="A1.Thmtheorem10.p1.2.2.m2.2.2.3.1"></times><cn id="A1.Thmtheorem10.p1.2.2.m2.2.2.3.2.cmml" type="integer" xref="A1.Thmtheorem10.p1.2.2.m2.2.2.3.2">128</cn><apply id="A1.Thmtheorem10.p1.2.2.m2.2.2.3.3.cmml" xref="A1.Thmtheorem10.p1.2.2.m2.2.2.3.3"><log id="A1.Thmtheorem10.p1.2.2.m2.2.2.3.3.1.cmml" xref="A1.Thmtheorem10.p1.2.2.m2.2.2.3.3.1"></log><ci id="A1.Thmtheorem10.p1.2.2.m2.2.2.3.3.2.cmml" xref="A1.Thmtheorem10.p1.2.2.m2.2.2.3.3.2">𝑛</ci></apply></apply></apply></interval></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.Thmtheorem10.p1.2.2.m2.3c">O(U(n,\frac{\varepsilon^{2}}{128\log n}))</annotation><annotation encoding="application/x-llamapun" id="A1.Thmtheorem10.p1.2.2.m2.3d">italic_O ( italic_U ( italic_n , divide start_ARG italic_ε start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG start_ARG 128 roman_log italic_n end_ARG ) )</annotation></semantics></math> rounds. All other steps can be performed in <math alttext="O(\frac{\log^{2}n}{\varepsilon^{2}})" class="ltx_Math" display="inline" id="A1.Thmtheorem10.p1.3.3.m3.1"><semantics id="A1.Thmtheorem10.p1.3.3.m3.1a"><mrow id="A1.Thmtheorem10.p1.3.3.m3.1.2" xref="A1.Thmtheorem10.p1.3.3.m3.1.2.cmml"><mi id="A1.Thmtheorem10.p1.3.3.m3.1.2.2" xref="A1.Thmtheorem10.p1.3.3.m3.1.2.2.cmml">O</mi><mo id="A1.Thmtheorem10.p1.3.3.m3.1.2.1" xref="A1.Thmtheorem10.p1.3.3.m3.1.2.1.cmml">⁢</mo><mrow id="A1.Thmtheorem10.p1.3.3.m3.1.2.3.2" xref="A1.Thmtheorem10.p1.3.3.m3.1.1.cmml"><mo id="A1.Thmtheorem10.p1.3.3.m3.1.2.3.2.1" stretchy="false" xref="A1.Thmtheorem10.p1.3.3.m3.1.1.cmml">(</mo><mfrac id="A1.Thmtheorem10.p1.3.3.m3.1.1" xref="A1.Thmtheorem10.p1.3.3.m3.1.1.cmml"><mrow id="A1.Thmtheorem10.p1.3.3.m3.1.1.2" xref="A1.Thmtheorem10.p1.3.3.m3.1.1.2.cmml"><msup id="A1.Thmtheorem10.p1.3.3.m3.1.1.2.1" xref="A1.Thmtheorem10.p1.3.3.m3.1.1.2.1.cmml"><mi id="A1.Thmtheorem10.p1.3.3.m3.1.1.2.1.2" xref="A1.Thmtheorem10.p1.3.3.m3.1.1.2.1.2.cmml">log</mi><mn id="A1.Thmtheorem10.p1.3.3.m3.1.1.2.1.3" xref="A1.Thmtheorem10.p1.3.3.m3.1.1.2.1.3.cmml">2</mn></msup><mo id="A1.Thmtheorem10.p1.3.3.m3.1.1.2a" lspace="0.167em" xref="A1.Thmtheorem10.p1.3.3.m3.1.1.2.cmml">⁡</mo><mi id="A1.Thmtheorem10.p1.3.3.m3.1.1.2.2" xref="A1.Thmtheorem10.p1.3.3.m3.1.1.2.2.cmml">n</mi></mrow><msup id="A1.Thmtheorem10.p1.3.3.m3.1.1.3" xref="A1.Thmtheorem10.p1.3.3.m3.1.1.3.cmml"><mi id="A1.Thmtheorem10.p1.3.3.m3.1.1.3.2" xref="A1.Thmtheorem10.p1.3.3.m3.1.1.3.2.cmml">ε</mi><mn id="A1.Thmtheorem10.p1.3.3.m3.1.1.3.3" xref="A1.Thmtheorem10.p1.3.3.m3.1.1.3.3.cmml">2</mn></msup></mfrac><mo id="A1.Thmtheorem10.p1.3.3.m3.1.2.3.2.2" stretchy="false" xref="A1.Thmtheorem10.p1.3.3.m3.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="A1.Thmtheorem10.p1.3.3.m3.1b"><apply id="A1.Thmtheorem10.p1.3.3.m3.1.2.cmml" xref="A1.Thmtheorem10.p1.3.3.m3.1.2"><times id="A1.Thmtheorem10.p1.3.3.m3.1.2.1.cmml" xref="A1.Thmtheorem10.p1.3.3.m3.1.2.1"></times><ci id="A1.Thmtheorem10.p1.3.3.m3.1.2.2.cmml" xref="A1.Thmtheorem10.p1.3.3.m3.1.2.2">𝑂</ci><apply id="A1.Thmtheorem10.p1.3.3.m3.1.1.cmml" xref="A1.Thmtheorem10.p1.3.3.m3.1.2.3.2"><divide id="A1.Thmtheorem10.p1.3.3.m3.1.1.1.cmml" xref="A1.Thmtheorem10.p1.3.3.m3.1.2.3.2"></divide><apply id="A1.Thmtheorem10.p1.3.3.m3.1.1.2.cmml" xref="A1.Thmtheorem10.p1.3.3.m3.1.1.2"><apply id="A1.Thmtheorem10.p1.3.3.m3.1.1.2.1.cmml" xref="A1.Thmtheorem10.p1.3.3.m3.1.1.2.1"><csymbol cd="ambiguous" id="A1.Thmtheorem10.p1.3.3.m3.1.1.2.1.1.cmml" xref="A1.Thmtheorem10.p1.3.3.m3.1.1.2.1">superscript</csymbol><log id="A1.Thmtheorem10.p1.3.3.m3.1.1.2.1.2.cmml" xref="A1.Thmtheorem10.p1.3.3.m3.1.1.2.1.2"></log><cn id="A1.Thmtheorem10.p1.3.3.m3.1.1.2.1.3.cmml" type="integer" xref="A1.Thmtheorem10.p1.3.3.m3.1.1.2.1.3">2</cn></apply><ci id="A1.Thmtheorem10.p1.3.3.m3.1.1.2.2.cmml" xref="A1.Thmtheorem10.p1.3.3.m3.1.1.2.2">𝑛</ci></apply><apply id="A1.Thmtheorem10.p1.3.3.m3.1.1.3.cmml" xref="A1.Thmtheorem10.p1.3.3.m3.1.1.3"><csymbol cd="ambiguous" id="A1.Thmtheorem10.p1.3.3.m3.1.1.3.1.cmml" xref="A1.Thmtheorem10.p1.3.3.m3.1.1.3">superscript</csymbol><ci id="A1.Thmtheorem10.p1.3.3.m3.1.1.3.2.cmml" xref="A1.Thmtheorem10.p1.3.3.m3.1.1.3.2">𝜀</ci><cn id="A1.Thmtheorem10.p1.3.3.m3.1.1.3.3.cmml" type="integer" xref="A1.Thmtheorem10.p1.3.3.m3.1.1.3.3">2</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.Thmtheorem10.p1.3.3.m3.1c">O(\frac{\log^{2}n}{\varepsilon^{2}})</annotation><annotation encoding="application/x-llamapun" id="A1.Thmtheorem10.p1.3.3.m3.1d">italic_O ( divide start_ARG roman_log start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_n end_ARG start_ARG italic_ε start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG )</annotation></semantics></math> rounds. Indeed, gathering or broadcasting information from a leader to the leaders <math alttext="\lceil\frac{32\log n}{\varepsilon}\rceil" class="ltx_Math" display="inline" id="A1.Thmtheorem10.p1.4.4.m4.1"><semantics id="A1.Thmtheorem10.p1.4.4.m4.1a"><mrow id="A1.Thmtheorem10.p1.4.4.m4.1.2.2" xref="A1.Thmtheorem10.p1.4.4.m4.1.2.1.cmml"><mo id="A1.Thmtheorem10.p1.4.4.m4.1.2.2.1" stretchy="false" xref="A1.Thmtheorem10.p1.4.4.m4.1.2.1.1.cmml">⌈</mo><mfrac id="A1.Thmtheorem10.p1.4.4.m4.1.1" xref="A1.Thmtheorem10.p1.4.4.m4.1.1.cmml"><mrow id="A1.Thmtheorem10.p1.4.4.m4.1.1.2" xref="A1.Thmtheorem10.p1.4.4.m4.1.1.2.cmml"><mn id="A1.Thmtheorem10.p1.4.4.m4.1.1.2.2" xref="A1.Thmtheorem10.p1.4.4.m4.1.1.2.2.cmml">32</mn><mo id="A1.Thmtheorem10.p1.4.4.m4.1.1.2.1" lspace="0.167em" xref="A1.Thmtheorem10.p1.4.4.m4.1.1.2.1.cmml">⁢</mo><mrow id="A1.Thmtheorem10.p1.4.4.m4.1.1.2.3" xref="A1.Thmtheorem10.p1.4.4.m4.1.1.2.3.cmml"><mi id="A1.Thmtheorem10.p1.4.4.m4.1.1.2.3.1" xref="A1.Thmtheorem10.p1.4.4.m4.1.1.2.3.1.cmml">log</mi><mo id="A1.Thmtheorem10.p1.4.4.m4.1.1.2.3a" lspace="0.167em" xref="A1.Thmtheorem10.p1.4.4.m4.1.1.2.3.cmml">⁡</mo><mi id="A1.Thmtheorem10.p1.4.4.m4.1.1.2.3.2" xref="A1.Thmtheorem10.p1.4.4.m4.1.1.2.3.2.cmml">n</mi></mrow></mrow><mi id="A1.Thmtheorem10.p1.4.4.m4.1.1.3" xref="A1.Thmtheorem10.p1.4.4.m4.1.1.3.cmml">ε</mi></mfrac><mo id="A1.Thmtheorem10.p1.4.4.m4.1.2.2.2" stretchy="false" xref="A1.Thmtheorem10.p1.4.4.m4.1.2.1.1.cmml">⌉</mo></mrow><annotation-xml encoding="MathML-Content" id="A1.Thmtheorem10.p1.4.4.m4.1b"><apply id="A1.Thmtheorem10.p1.4.4.m4.1.2.1.cmml" xref="A1.Thmtheorem10.p1.4.4.m4.1.2.2"><ceiling id="A1.Thmtheorem10.p1.4.4.m4.1.2.1.1.cmml" xref="A1.Thmtheorem10.p1.4.4.m4.1.2.2.1"></ceiling><apply id="A1.Thmtheorem10.p1.4.4.m4.1.1.cmml" xref="A1.Thmtheorem10.p1.4.4.m4.1.1"><divide id="A1.Thmtheorem10.p1.4.4.m4.1.1.1.cmml" xref="A1.Thmtheorem10.p1.4.4.m4.1.1"></divide><apply id="A1.Thmtheorem10.p1.4.4.m4.1.1.2.cmml" xref="A1.Thmtheorem10.p1.4.4.m4.1.1.2"><times id="A1.Thmtheorem10.p1.4.4.m4.1.1.2.1.cmml" xref="A1.Thmtheorem10.p1.4.4.m4.1.1.2.1"></times><cn id="A1.Thmtheorem10.p1.4.4.m4.1.1.2.2.cmml" type="integer" xref="A1.Thmtheorem10.p1.4.4.m4.1.1.2.2">32</cn><apply id="A1.Thmtheorem10.p1.4.4.m4.1.1.2.3.cmml" xref="A1.Thmtheorem10.p1.4.4.m4.1.1.2.3"><log id="A1.Thmtheorem10.p1.4.4.m4.1.1.2.3.1.cmml" xref="A1.Thmtheorem10.p1.4.4.m4.1.1.2.3.1"></log><ci id="A1.Thmtheorem10.p1.4.4.m4.1.1.2.3.2.cmml" xref="A1.Thmtheorem10.p1.4.4.m4.1.1.2.3.2">𝑛</ci></apply></apply><ci id="A1.Thmtheorem10.p1.4.4.m4.1.1.3.cmml" xref="A1.Thmtheorem10.p1.4.4.m4.1.1.3">𝜀</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.Thmtheorem10.p1.4.4.m4.1c">\lceil\frac{32\log n}{\varepsilon}\rceil</annotation><annotation encoding="application/x-llamapun" id="A1.Thmtheorem10.p1.4.4.m4.1d">⌈ divide start_ARG 32 roman_log italic_n end_ARG start_ARG italic_ε end_ARG ⌉</annotation></semantics></math>-hop neighbourhood takes at most <math alttext="O(\frac{\log n}{\varepsilon})" class="ltx_Math" display="inline" id="A1.Thmtheorem10.p1.5.5.m5.1"><semantics id="A1.Thmtheorem10.p1.5.5.m5.1a"><mrow id="A1.Thmtheorem10.p1.5.5.m5.1.2" xref="A1.Thmtheorem10.p1.5.5.m5.1.2.cmml"><mi id="A1.Thmtheorem10.p1.5.5.m5.1.2.2" xref="A1.Thmtheorem10.p1.5.5.m5.1.2.2.cmml">O</mi><mo id="A1.Thmtheorem10.p1.5.5.m5.1.2.1" xref="A1.Thmtheorem10.p1.5.5.m5.1.2.1.cmml">⁢</mo><mrow id="A1.Thmtheorem10.p1.5.5.m5.1.2.3.2" xref="A1.Thmtheorem10.p1.5.5.m5.1.1.cmml"><mo id="A1.Thmtheorem10.p1.5.5.m5.1.2.3.2.1" stretchy="false" xref="A1.Thmtheorem10.p1.5.5.m5.1.1.cmml">(</mo><mfrac id="A1.Thmtheorem10.p1.5.5.m5.1.1" xref="A1.Thmtheorem10.p1.5.5.m5.1.1.cmml"><mrow id="A1.Thmtheorem10.p1.5.5.m5.1.1.2" xref="A1.Thmtheorem10.p1.5.5.m5.1.1.2.cmml"><mi id="A1.Thmtheorem10.p1.5.5.m5.1.1.2.1" xref="A1.Thmtheorem10.p1.5.5.m5.1.1.2.1.cmml">log</mi><mo id="A1.Thmtheorem10.p1.5.5.m5.1.1.2a" lspace="0.167em" xref="A1.Thmtheorem10.p1.5.5.m5.1.1.2.cmml">⁡</mo><mi id="A1.Thmtheorem10.p1.5.5.m5.1.1.2.2" xref="A1.Thmtheorem10.p1.5.5.m5.1.1.2.2.cmml">n</mi></mrow><mi id="A1.Thmtheorem10.p1.5.5.m5.1.1.3" xref="A1.Thmtheorem10.p1.5.5.m5.1.1.3.cmml">ε</mi></mfrac><mo id="A1.Thmtheorem10.p1.5.5.m5.1.2.3.2.2" stretchy="false" xref="A1.Thmtheorem10.p1.5.5.m5.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="A1.Thmtheorem10.p1.5.5.m5.1b"><apply id="A1.Thmtheorem10.p1.5.5.m5.1.2.cmml" xref="A1.Thmtheorem10.p1.5.5.m5.1.2"><times id="A1.Thmtheorem10.p1.5.5.m5.1.2.1.cmml" xref="A1.Thmtheorem10.p1.5.5.m5.1.2.1"></times><ci id="A1.Thmtheorem10.p1.5.5.m5.1.2.2.cmml" xref="A1.Thmtheorem10.p1.5.5.m5.1.2.2">𝑂</ci><apply id="A1.Thmtheorem10.p1.5.5.m5.1.1.cmml" xref="A1.Thmtheorem10.p1.5.5.m5.1.2.3.2"><divide id="A1.Thmtheorem10.p1.5.5.m5.1.1.1.cmml" xref="A1.Thmtheorem10.p1.5.5.m5.1.2.3.2"></divide><apply id="A1.Thmtheorem10.p1.5.5.m5.1.1.2.cmml" xref="A1.Thmtheorem10.p1.5.5.m5.1.1.2"><log id="A1.Thmtheorem10.p1.5.5.m5.1.1.2.1.cmml" xref="A1.Thmtheorem10.p1.5.5.m5.1.1.2.1"></log><ci id="A1.Thmtheorem10.p1.5.5.m5.1.1.2.2.cmml" xref="A1.Thmtheorem10.p1.5.5.m5.1.1.2.2">𝑛</ci></apply><ci id="A1.Thmtheorem10.p1.5.5.m5.1.1.3.cmml" xref="A1.Thmtheorem10.p1.5.5.m5.1.1.3">𝜀</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.Thmtheorem10.p1.5.5.m5.1c">O(\frac{\log n}{\varepsilon})</annotation><annotation encoding="application/x-llamapun" id="A1.Thmtheorem10.p1.5.5.m5.1d">italic_O ( divide start_ARG roman_log italic_n end_ARG start_ARG italic_ε end_ARG )</annotation></semantics></math> rounds per piece of information needed. The most information intensive step is to gather the sizes of the sets <math alttext="T_{i}" class="ltx_Math" display="inline" id="A1.Thmtheorem10.p1.6.6.m6.1"><semantics id="A1.Thmtheorem10.p1.6.6.m6.1a"><msub id="A1.Thmtheorem10.p1.6.6.m6.1.1" xref="A1.Thmtheorem10.p1.6.6.m6.1.1.cmml"><mi id="A1.Thmtheorem10.p1.6.6.m6.1.1.2" xref="A1.Thmtheorem10.p1.6.6.m6.1.1.2.cmml">T</mi><mi id="A1.Thmtheorem10.p1.6.6.m6.1.1.3" xref="A1.Thmtheorem10.p1.6.6.m6.1.1.3.cmml">i</mi></msub><annotation-xml encoding="MathML-Content" id="A1.Thmtheorem10.p1.6.6.m6.1b"><apply id="A1.Thmtheorem10.p1.6.6.m6.1.1.cmml" xref="A1.Thmtheorem10.p1.6.6.m6.1.1"><csymbol cd="ambiguous" id="A1.Thmtheorem10.p1.6.6.m6.1.1.1.cmml" xref="A1.Thmtheorem10.p1.6.6.m6.1.1">subscript</csymbol><ci id="A1.Thmtheorem10.p1.6.6.m6.1.1.2.cmml" xref="A1.Thmtheorem10.p1.6.6.m6.1.1.2">𝑇</ci><ci id="A1.Thmtheorem10.p1.6.6.m6.1.1.3.cmml" xref="A1.Thmtheorem10.p1.6.6.m6.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.Thmtheorem10.p1.6.6.m6.1c">T_{i}</annotation><annotation encoding="application/x-llamapun" id="A1.Thmtheorem10.p1.6.6.m6.1d">italic_T start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math> (for the at most <math alttext="\log_{1+\varepsilon/16}n=O(\frac{\log n}{\varepsilon})" class="ltx_Math" display="inline" id="A1.Thmtheorem10.p1.7.7.m7.1"><semantics id="A1.Thmtheorem10.p1.7.7.m7.1a"><mrow id="A1.Thmtheorem10.p1.7.7.m7.1.2" xref="A1.Thmtheorem10.p1.7.7.m7.1.2.cmml"><mrow id="A1.Thmtheorem10.p1.7.7.m7.1.2.2" xref="A1.Thmtheorem10.p1.7.7.m7.1.2.2.cmml"><msub id="A1.Thmtheorem10.p1.7.7.m7.1.2.2.1" xref="A1.Thmtheorem10.p1.7.7.m7.1.2.2.1.cmml"><mi id="A1.Thmtheorem10.p1.7.7.m7.1.2.2.1.2" xref="A1.Thmtheorem10.p1.7.7.m7.1.2.2.1.2.cmml">log</mi><mrow id="A1.Thmtheorem10.p1.7.7.m7.1.2.2.1.3" xref="A1.Thmtheorem10.p1.7.7.m7.1.2.2.1.3.cmml"><mn id="A1.Thmtheorem10.p1.7.7.m7.1.2.2.1.3.2" xref="A1.Thmtheorem10.p1.7.7.m7.1.2.2.1.3.2.cmml">1</mn><mo id="A1.Thmtheorem10.p1.7.7.m7.1.2.2.1.3.1" xref="A1.Thmtheorem10.p1.7.7.m7.1.2.2.1.3.1.cmml">+</mo><mrow id="A1.Thmtheorem10.p1.7.7.m7.1.2.2.1.3.3" xref="A1.Thmtheorem10.p1.7.7.m7.1.2.2.1.3.3.cmml"><mi id="A1.Thmtheorem10.p1.7.7.m7.1.2.2.1.3.3.2" xref="A1.Thmtheorem10.p1.7.7.m7.1.2.2.1.3.3.2.cmml">ε</mi><mo id="A1.Thmtheorem10.p1.7.7.m7.1.2.2.1.3.3.1" xref="A1.Thmtheorem10.p1.7.7.m7.1.2.2.1.3.3.1.cmml">/</mo><mn id="A1.Thmtheorem10.p1.7.7.m7.1.2.2.1.3.3.3" xref="A1.Thmtheorem10.p1.7.7.m7.1.2.2.1.3.3.3.cmml">16</mn></mrow></mrow></msub><mo id="A1.Thmtheorem10.p1.7.7.m7.1.2.2a" lspace="0.167em" xref="A1.Thmtheorem10.p1.7.7.m7.1.2.2.cmml">⁡</mo><mi id="A1.Thmtheorem10.p1.7.7.m7.1.2.2.2" xref="A1.Thmtheorem10.p1.7.7.m7.1.2.2.2.cmml">n</mi></mrow><mo id="A1.Thmtheorem10.p1.7.7.m7.1.2.1" xref="A1.Thmtheorem10.p1.7.7.m7.1.2.1.cmml">=</mo><mrow id="A1.Thmtheorem10.p1.7.7.m7.1.2.3" xref="A1.Thmtheorem10.p1.7.7.m7.1.2.3.cmml"><mi id="A1.Thmtheorem10.p1.7.7.m7.1.2.3.2" xref="A1.Thmtheorem10.p1.7.7.m7.1.2.3.2.cmml">O</mi><mo id="A1.Thmtheorem10.p1.7.7.m7.1.2.3.1" xref="A1.Thmtheorem10.p1.7.7.m7.1.2.3.1.cmml">⁢</mo><mrow id="A1.Thmtheorem10.p1.7.7.m7.1.2.3.3.2" xref="A1.Thmtheorem10.p1.7.7.m7.1.1.cmml"><mo id="A1.Thmtheorem10.p1.7.7.m7.1.2.3.3.2.1" stretchy="false" xref="A1.Thmtheorem10.p1.7.7.m7.1.1.cmml">(</mo><mfrac id="A1.Thmtheorem10.p1.7.7.m7.1.1" xref="A1.Thmtheorem10.p1.7.7.m7.1.1.cmml"><mrow id="A1.Thmtheorem10.p1.7.7.m7.1.1.2" xref="A1.Thmtheorem10.p1.7.7.m7.1.1.2.cmml"><mi id="A1.Thmtheorem10.p1.7.7.m7.1.1.2.1" xref="A1.Thmtheorem10.p1.7.7.m7.1.1.2.1.cmml">log</mi><mo id="A1.Thmtheorem10.p1.7.7.m7.1.1.2a" lspace="0.167em" xref="A1.Thmtheorem10.p1.7.7.m7.1.1.2.cmml">⁡</mo><mi id="A1.Thmtheorem10.p1.7.7.m7.1.1.2.2" xref="A1.Thmtheorem10.p1.7.7.m7.1.1.2.2.cmml">n</mi></mrow><mi id="A1.Thmtheorem10.p1.7.7.m7.1.1.3" xref="A1.Thmtheorem10.p1.7.7.m7.1.1.3.cmml">ε</mi></mfrac><mo id="A1.Thmtheorem10.p1.7.7.m7.1.2.3.3.2.2" stretchy="false" xref="A1.Thmtheorem10.p1.7.7.m7.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="A1.Thmtheorem10.p1.7.7.m7.1b"><apply id="A1.Thmtheorem10.p1.7.7.m7.1.2.cmml" xref="A1.Thmtheorem10.p1.7.7.m7.1.2"><eq id="A1.Thmtheorem10.p1.7.7.m7.1.2.1.cmml" xref="A1.Thmtheorem10.p1.7.7.m7.1.2.1"></eq><apply id="A1.Thmtheorem10.p1.7.7.m7.1.2.2.cmml" xref="A1.Thmtheorem10.p1.7.7.m7.1.2.2"><apply id="A1.Thmtheorem10.p1.7.7.m7.1.2.2.1.cmml" xref="A1.Thmtheorem10.p1.7.7.m7.1.2.2.1"><csymbol cd="ambiguous" id="A1.Thmtheorem10.p1.7.7.m7.1.2.2.1.1.cmml" xref="A1.Thmtheorem10.p1.7.7.m7.1.2.2.1">subscript</csymbol><log id="A1.Thmtheorem10.p1.7.7.m7.1.2.2.1.2.cmml" xref="A1.Thmtheorem10.p1.7.7.m7.1.2.2.1.2"></log><apply id="A1.Thmtheorem10.p1.7.7.m7.1.2.2.1.3.cmml" xref="A1.Thmtheorem10.p1.7.7.m7.1.2.2.1.3"><plus id="A1.Thmtheorem10.p1.7.7.m7.1.2.2.1.3.1.cmml" xref="A1.Thmtheorem10.p1.7.7.m7.1.2.2.1.3.1"></plus><cn id="A1.Thmtheorem10.p1.7.7.m7.1.2.2.1.3.2.cmml" type="integer" xref="A1.Thmtheorem10.p1.7.7.m7.1.2.2.1.3.2">1</cn><apply id="A1.Thmtheorem10.p1.7.7.m7.1.2.2.1.3.3.cmml" xref="A1.Thmtheorem10.p1.7.7.m7.1.2.2.1.3.3"><divide id="A1.Thmtheorem10.p1.7.7.m7.1.2.2.1.3.3.1.cmml" xref="A1.Thmtheorem10.p1.7.7.m7.1.2.2.1.3.3.1"></divide><ci id="A1.Thmtheorem10.p1.7.7.m7.1.2.2.1.3.3.2.cmml" xref="A1.Thmtheorem10.p1.7.7.m7.1.2.2.1.3.3.2">𝜀</ci><cn id="A1.Thmtheorem10.p1.7.7.m7.1.2.2.1.3.3.3.cmml" type="integer" xref="A1.Thmtheorem10.p1.7.7.m7.1.2.2.1.3.3.3">16</cn></apply></apply></apply><ci id="A1.Thmtheorem10.p1.7.7.m7.1.2.2.2.cmml" xref="A1.Thmtheorem10.p1.7.7.m7.1.2.2.2">𝑛</ci></apply><apply id="A1.Thmtheorem10.p1.7.7.m7.1.2.3.cmml" xref="A1.Thmtheorem10.p1.7.7.m7.1.2.3"><times id="A1.Thmtheorem10.p1.7.7.m7.1.2.3.1.cmml" xref="A1.Thmtheorem10.p1.7.7.m7.1.2.3.1"></times><ci id="A1.Thmtheorem10.p1.7.7.m7.1.2.3.2.cmml" xref="A1.Thmtheorem10.p1.7.7.m7.1.2.3.2">𝑂</ci><apply id="A1.Thmtheorem10.p1.7.7.m7.1.1.cmml" xref="A1.Thmtheorem10.p1.7.7.m7.1.2.3.3.2"><divide id="A1.Thmtheorem10.p1.7.7.m7.1.1.1.cmml" xref="A1.Thmtheorem10.p1.7.7.m7.1.2.3.3.2"></divide><apply id="A1.Thmtheorem10.p1.7.7.m7.1.1.2.cmml" xref="A1.Thmtheorem10.p1.7.7.m7.1.1.2"><log id="A1.Thmtheorem10.p1.7.7.m7.1.1.2.1.cmml" xref="A1.Thmtheorem10.p1.7.7.m7.1.1.2.1"></log><ci id="A1.Thmtheorem10.p1.7.7.m7.1.1.2.2.cmml" xref="A1.Thmtheorem10.p1.7.7.m7.1.1.2.2">𝑛</ci></apply><ci id="A1.Thmtheorem10.p1.7.7.m7.1.1.3.cmml" xref="A1.Thmtheorem10.p1.7.7.m7.1.1.3">𝜀</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.Thmtheorem10.p1.7.7.m7.1c">\log_{1+\varepsilon/16}n=O(\frac{\log n}{\varepsilon})</annotation><annotation encoding="application/x-llamapun" id="A1.Thmtheorem10.p1.7.7.m7.1d">roman_log start_POSTSUBSCRIPT 1 + italic_ε / 16 end_POSTSUBSCRIPT italic_n = italic_O ( divide start_ARG roman_log italic_n end_ARG start_ARG italic_ε end_ARG )</annotation></semantics></math> values of <math alttext="i" class="ltx_Math" display="inline" id="A1.Thmtheorem10.p1.8.8.m8.1"><semantics id="A1.Thmtheorem10.p1.8.8.m8.1a"><mi id="A1.Thmtheorem10.p1.8.8.m8.1.1" xref="A1.Thmtheorem10.p1.8.8.m8.1.1.cmml">i</mi><annotation-xml encoding="MathML-Content" id="A1.Thmtheorem10.p1.8.8.m8.1b"><ci id="A1.Thmtheorem10.p1.8.8.m8.1.1.cmml" xref="A1.Thmtheorem10.p1.8.8.m8.1.1">𝑖</ci></annotation-xml><annotation encoding="application/x-tex" id="A1.Thmtheorem10.p1.8.8.m8.1c">i</annotation><annotation encoding="application/x-llamapun" id="A1.Thmtheorem10.p1.8.8.m8.1d">italic_i</annotation></semantics></math>), which can be done in <math alttext="O(\frac{\log n}{\varepsilon})" class="ltx_Math" display="inline" id="A1.Thmtheorem10.p1.9.9.m9.1"><semantics id="A1.Thmtheorem10.p1.9.9.m9.1a"><mrow id="A1.Thmtheorem10.p1.9.9.m9.1.2" xref="A1.Thmtheorem10.p1.9.9.m9.1.2.cmml"><mi id="A1.Thmtheorem10.p1.9.9.m9.1.2.2" xref="A1.Thmtheorem10.p1.9.9.m9.1.2.2.cmml">O</mi><mo id="A1.Thmtheorem10.p1.9.9.m9.1.2.1" xref="A1.Thmtheorem10.p1.9.9.m9.1.2.1.cmml">⁢</mo><mrow id="A1.Thmtheorem10.p1.9.9.m9.1.2.3.2" xref="A1.Thmtheorem10.p1.9.9.m9.1.1.cmml"><mo id="A1.Thmtheorem10.p1.9.9.m9.1.2.3.2.1" stretchy="false" xref="A1.Thmtheorem10.p1.9.9.m9.1.1.cmml">(</mo><mfrac id="A1.Thmtheorem10.p1.9.9.m9.1.1" xref="A1.Thmtheorem10.p1.9.9.m9.1.1.cmml"><mrow id="A1.Thmtheorem10.p1.9.9.m9.1.1.2" xref="A1.Thmtheorem10.p1.9.9.m9.1.1.2.cmml"><mi id="A1.Thmtheorem10.p1.9.9.m9.1.1.2.1" xref="A1.Thmtheorem10.p1.9.9.m9.1.1.2.1.cmml">log</mi><mo id="A1.Thmtheorem10.p1.9.9.m9.1.1.2a" lspace="0.167em" xref="A1.Thmtheorem10.p1.9.9.m9.1.1.2.cmml">⁡</mo><mi id="A1.Thmtheorem10.p1.9.9.m9.1.1.2.2" xref="A1.Thmtheorem10.p1.9.9.m9.1.1.2.2.cmml">n</mi></mrow><mi id="A1.Thmtheorem10.p1.9.9.m9.1.1.3" xref="A1.Thmtheorem10.p1.9.9.m9.1.1.3.cmml">ε</mi></mfrac><mo id="A1.Thmtheorem10.p1.9.9.m9.1.2.3.2.2" stretchy="false" xref="A1.Thmtheorem10.p1.9.9.m9.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="A1.Thmtheorem10.p1.9.9.m9.1b"><apply id="A1.Thmtheorem10.p1.9.9.m9.1.2.cmml" xref="A1.Thmtheorem10.p1.9.9.m9.1.2"><times id="A1.Thmtheorem10.p1.9.9.m9.1.2.1.cmml" xref="A1.Thmtheorem10.p1.9.9.m9.1.2.1"></times><ci id="A1.Thmtheorem10.p1.9.9.m9.1.2.2.cmml" xref="A1.Thmtheorem10.p1.9.9.m9.1.2.2">𝑂</ci><apply id="A1.Thmtheorem10.p1.9.9.m9.1.1.cmml" xref="A1.Thmtheorem10.p1.9.9.m9.1.2.3.2"><divide id="A1.Thmtheorem10.p1.9.9.m9.1.1.1.cmml" xref="A1.Thmtheorem10.p1.9.9.m9.1.2.3.2"></divide><apply id="A1.Thmtheorem10.p1.9.9.m9.1.1.2.cmml" xref="A1.Thmtheorem10.p1.9.9.m9.1.1.2"><log id="A1.Thmtheorem10.p1.9.9.m9.1.1.2.1.cmml" xref="A1.Thmtheorem10.p1.9.9.m9.1.1.2.1"></log><ci id="A1.Thmtheorem10.p1.9.9.m9.1.1.2.2.cmml" xref="A1.Thmtheorem10.p1.9.9.m9.1.1.2.2">𝑛</ci></apply><ci id="A1.Thmtheorem10.p1.9.9.m9.1.1.3.cmml" xref="A1.Thmtheorem10.p1.9.9.m9.1.1.3">𝜀</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.Thmtheorem10.p1.9.9.m9.1c">O(\frac{\log n}{\varepsilon})</annotation><annotation encoding="application/x-llamapun" id="A1.Thmtheorem10.p1.9.9.m9.1d">italic_O ( divide start_ARG roman_log italic_n end_ARG start_ARG italic_ε end_ARG )</annotation></semantics></math> rounds per <math alttext="i" class="ltx_Math" display="inline" id="A1.Thmtheorem10.p1.10.10.m10.1"><semantics id="A1.Thmtheorem10.p1.10.10.m10.1a"><mi id="A1.Thmtheorem10.p1.10.10.m10.1.1" xref="A1.Thmtheorem10.p1.10.10.m10.1.1.cmml">i</mi><annotation-xml encoding="MathML-Content" id="A1.Thmtheorem10.p1.10.10.m10.1b"><ci id="A1.Thmtheorem10.p1.10.10.m10.1.1.cmml" xref="A1.Thmtheorem10.p1.10.10.m10.1.1">𝑖</ci></annotation-xml><annotation encoding="application/x-tex" id="A1.Thmtheorem10.p1.10.10.m10.1c">i</annotation><annotation encoding="application/x-llamapun" id="A1.Thmtheorem10.p1.10.10.m10.1d">italic_i</annotation></semantics></math> that is in <math alttext="O(\frac{\log^{2}n}{\varepsilon^{2}})" class="ltx_Math" display="inline" id="A1.Thmtheorem10.p1.11.11.m11.1"><semantics id="A1.Thmtheorem10.p1.11.11.m11.1a"><mrow id="A1.Thmtheorem10.p1.11.11.m11.1.2" xref="A1.Thmtheorem10.p1.11.11.m11.1.2.cmml"><mi id="A1.Thmtheorem10.p1.11.11.m11.1.2.2" xref="A1.Thmtheorem10.p1.11.11.m11.1.2.2.cmml">O</mi><mo id="A1.Thmtheorem10.p1.11.11.m11.1.2.1" xref="A1.Thmtheorem10.p1.11.11.m11.1.2.1.cmml">⁢</mo><mrow id="A1.Thmtheorem10.p1.11.11.m11.1.2.3.2" xref="A1.Thmtheorem10.p1.11.11.m11.1.1.cmml"><mo id="A1.Thmtheorem10.p1.11.11.m11.1.2.3.2.1" stretchy="false" xref="A1.Thmtheorem10.p1.11.11.m11.1.1.cmml">(</mo><mfrac id="A1.Thmtheorem10.p1.11.11.m11.1.1" xref="A1.Thmtheorem10.p1.11.11.m11.1.1.cmml"><mrow id="A1.Thmtheorem10.p1.11.11.m11.1.1.2" xref="A1.Thmtheorem10.p1.11.11.m11.1.1.2.cmml"><msup id="A1.Thmtheorem10.p1.11.11.m11.1.1.2.1" xref="A1.Thmtheorem10.p1.11.11.m11.1.1.2.1.cmml"><mi id="A1.Thmtheorem10.p1.11.11.m11.1.1.2.1.2" xref="A1.Thmtheorem10.p1.11.11.m11.1.1.2.1.2.cmml">log</mi><mn id="A1.Thmtheorem10.p1.11.11.m11.1.1.2.1.3" xref="A1.Thmtheorem10.p1.11.11.m11.1.1.2.1.3.cmml">2</mn></msup><mo id="A1.Thmtheorem10.p1.11.11.m11.1.1.2a" lspace="0.167em" xref="A1.Thmtheorem10.p1.11.11.m11.1.1.2.cmml">⁡</mo><mi id="A1.Thmtheorem10.p1.11.11.m11.1.1.2.2" xref="A1.Thmtheorem10.p1.11.11.m11.1.1.2.2.cmml">n</mi></mrow><msup id="A1.Thmtheorem10.p1.11.11.m11.1.1.3" xref="A1.Thmtheorem10.p1.11.11.m11.1.1.3.cmml"><mi id="A1.Thmtheorem10.p1.11.11.m11.1.1.3.2" xref="A1.Thmtheorem10.p1.11.11.m11.1.1.3.2.cmml">ε</mi><mn id="A1.Thmtheorem10.p1.11.11.m11.1.1.3.3" xref="A1.Thmtheorem10.p1.11.11.m11.1.1.3.3.cmml">2</mn></msup></mfrac><mo id="A1.Thmtheorem10.p1.11.11.m11.1.2.3.2.2" stretchy="false" xref="A1.Thmtheorem10.p1.11.11.m11.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="A1.Thmtheorem10.p1.11.11.m11.1b"><apply id="A1.Thmtheorem10.p1.11.11.m11.1.2.cmml" xref="A1.Thmtheorem10.p1.11.11.m11.1.2"><times id="A1.Thmtheorem10.p1.11.11.m11.1.2.1.cmml" xref="A1.Thmtheorem10.p1.11.11.m11.1.2.1"></times><ci id="A1.Thmtheorem10.p1.11.11.m11.1.2.2.cmml" xref="A1.Thmtheorem10.p1.11.11.m11.1.2.2">𝑂</ci><apply id="A1.Thmtheorem10.p1.11.11.m11.1.1.cmml" xref="A1.Thmtheorem10.p1.11.11.m11.1.2.3.2"><divide id="A1.Thmtheorem10.p1.11.11.m11.1.1.1.cmml" xref="A1.Thmtheorem10.p1.11.11.m11.1.2.3.2"></divide><apply id="A1.Thmtheorem10.p1.11.11.m11.1.1.2.cmml" xref="A1.Thmtheorem10.p1.11.11.m11.1.1.2"><apply id="A1.Thmtheorem10.p1.11.11.m11.1.1.2.1.cmml" xref="A1.Thmtheorem10.p1.11.11.m11.1.1.2.1"><csymbol cd="ambiguous" id="A1.Thmtheorem10.p1.11.11.m11.1.1.2.1.1.cmml" xref="A1.Thmtheorem10.p1.11.11.m11.1.1.2.1">superscript</csymbol><log id="A1.Thmtheorem10.p1.11.11.m11.1.1.2.1.2.cmml" xref="A1.Thmtheorem10.p1.11.11.m11.1.1.2.1.2"></log><cn id="A1.Thmtheorem10.p1.11.11.m11.1.1.2.1.3.cmml" type="integer" xref="A1.Thmtheorem10.p1.11.11.m11.1.1.2.1.3">2</cn></apply><ci id="A1.Thmtheorem10.p1.11.11.m11.1.1.2.2.cmml" xref="A1.Thmtheorem10.p1.11.11.m11.1.1.2.2">𝑛</ci></apply><apply id="A1.Thmtheorem10.p1.11.11.m11.1.1.3.cmml" xref="A1.Thmtheorem10.p1.11.11.m11.1.1.3"><csymbol cd="ambiguous" id="A1.Thmtheorem10.p1.11.11.m11.1.1.3.1.cmml" xref="A1.Thmtheorem10.p1.11.11.m11.1.1.3">superscript</csymbol><ci id="A1.Thmtheorem10.p1.11.11.m11.1.1.3.2.cmml" xref="A1.Thmtheorem10.p1.11.11.m11.1.1.3.2">𝜀</ci><cn id="A1.Thmtheorem10.p1.11.11.m11.1.1.3.3.cmml" type="integer" xref="A1.Thmtheorem10.p1.11.11.m11.1.1.3.3">2</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.Thmtheorem10.p1.11.11.m11.1c">O(\frac{\log^{2}n}{\varepsilon^{2}})</annotation><annotation encoding="application/x-llamapun" id="A1.Thmtheorem10.p1.11.11.m11.1d">italic_O ( divide start_ARG roman_log start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_n end_ARG start_ARG italic_ε start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG )</annotation></semantics></math> rounds in total.</span></p> </div> <div class="ltx_para" id="A1.Thmtheorem10.p2"> <p class="ltx_p" id="A1.Thmtheorem10.p2.1"><span class="ltx_text ltx_font_italic" id="A1.Thmtheorem10.p2.1.1">Finally, correctness of the algorithm follows from Lemma <a class="ltx_ref" href="https://arxiv.org/html/2411.12694v2#A1.Thmtheorem7" title="Lemma A.7. ‣ Analysis: ‣ A.3 Our algorithm ‣ Appendix A Reporting a densest subgraph ‣ Local Density and its Distributed Approximation"><span class="ltx_text ltx_ref_tag">A.7</span></a>.</span></p> </div> </div> <div class="ltx_para" id="A1.SS3.SSS0.P0.SPx2.p5"> <p class="ltx_p" id="A1.SS3.SSS0.P0.SPx2.p5.8">In the variant, where we require that the output graph <math alttext="H" class="ltx_Math" display="inline" id="A1.SS3.SSS0.P0.SPx2.p5.1.m1.1"><semantics id="A1.SS3.SSS0.P0.SPx2.p5.1.m1.1a"><mi id="A1.SS3.SSS0.P0.SPx2.p5.1.m1.1.1" xref="A1.SS3.SSS0.P0.SPx2.p5.1.m1.1.1.cmml">H</mi><annotation-xml encoding="MathML-Content" id="A1.SS3.SSS0.P0.SPx2.p5.1.m1.1b"><ci id="A1.SS3.SSS0.P0.SPx2.p5.1.m1.1.1.cmml" xref="A1.SS3.SSS0.P0.SPx2.p5.1.m1.1.1">𝐻</ci></annotation-xml><annotation encoding="application/x-tex" id="A1.SS3.SSS0.P0.SPx2.p5.1.m1.1c">H</annotation><annotation encoding="application/x-llamapun" id="A1.SS3.SSS0.P0.SPx2.p5.1.m1.1d">italic_H</annotation></semantics></math> has a density that <math alttext="(1+\varepsilon)" class="ltx_Math" display="inline" id="A1.SS3.SSS0.P0.SPx2.p5.2.m2.1"><semantics id="A1.SS3.SSS0.P0.SPx2.p5.2.m2.1a"><mrow id="A1.SS3.SSS0.P0.SPx2.p5.2.m2.1.1.1" xref="A1.SS3.SSS0.P0.SPx2.p5.2.m2.1.1.1.1.cmml"><mo id="A1.SS3.SSS0.P0.SPx2.p5.2.m2.1.1.1.2" stretchy="false" xref="A1.SS3.SSS0.P0.SPx2.p5.2.m2.1.1.1.1.cmml">(</mo><mrow id="A1.SS3.SSS0.P0.SPx2.p5.2.m2.1.1.1.1" xref="A1.SS3.SSS0.P0.SPx2.p5.2.m2.1.1.1.1.cmml"><mn id="A1.SS3.SSS0.P0.SPx2.p5.2.m2.1.1.1.1.2" xref="A1.SS3.SSS0.P0.SPx2.p5.2.m2.1.1.1.1.2.cmml">1</mn><mo id="A1.SS3.SSS0.P0.SPx2.p5.2.m2.1.1.1.1.1" xref="A1.SS3.SSS0.P0.SPx2.p5.2.m2.1.1.1.1.1.cmml">+</mo><mi id="A1.SS3.SSS0.P0.SPx2.p5.2.m2.1.1.1.1.3" xref="A1.SS3.SSS0.P0.SPx2.p5.2.m2.1.1.1.1.3.cmml">ε</mi></mrow><mo id="A1.SS3.SSS0.P0.SPx2.p5.2.m2.1.1.1.3" stretchy="false" xref="A1.SS3.SSS0.P0.SPx2.p5.2.m2.1.1.1.1.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="A1.SS3.SSS0.P0.SPx2.p5.2.m2.1b"><apply id="A1.SS3.SSS0.P0.SPx2.p5.2.m2.1.1.1.1.cmml" xref="A1.SS3.SSS0.P0.SPx2.p5.2.m2.1.1.1"><plus id="A1.SS3.SSS0.P0.SPx2.p5.2.m2.1.1.1.1.1.cmml" xref="A1.SS3.SSS0.P0.SPx2.p5.2.m2.1.1.1.1.1"></plus><cn id="A1.SS3.SSS0.P0.SPx2.p5.2.m2.1.1.1.1.2.cmml" type="integer" xref="A1.SS3.SSS0.P0.SPx2.p5.2.m2.1.1.1.1.2">1</cn><ci id="A1.SS3.SSS0.P0.SPx2.p5.2.m2.1.1.1.1.3.cmml" xref="A1.SS3.SSS0.P0.SPx2.p5.2.m2.1.1.1.1.3">𝜀</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS3.SSS0.P0.SPx2.p5.2.m2.1c">(1+\varepsilon)</annotation><annotation encoding="application/x-llamapun" id="A1.SS3.SSS0.P0.SPx2.p5.2.m2.1d">( 1 + italic_ε )</annotation></semantics></math>-approximates <math alttext="\rho^{\max}(G)" class="ltx_Math" display="inline" id="A1.SS3.SSS0.P0.SPx2.p5.3.m3.1"><semantics id="A1.SS3.SSS0.P0.SPx2.p5.3.m3.1a"><mrow id="A1.SS3.SSS0.P0.SPx2.p5.3.m3.1.2" xref="A1.SS3.SSS0.P0.SPx2.p5.3.m3.1.2.cmml"><msup id="A1.SS3.SSS0.P0.SPx2.p5.3.m3.1.2.2" xref="A1.SS3.SSS0.P0.SPx2.p5.3.m3.1.2.2.cmml"><mi id="A1.SS3.SSS0.P0.SPx2.p5.3.m3.1.2.2.2" xref="A1.SS3.SSS0.P0.SPx2.p5.3.m3.1.2.2.2.cmml">ρ</mi><mi id="A1.SS3.SSS0.P0.SPx2.p5.3.m3.1.2.2.3" xref="A1.SS3.SSS0.P0.SPx2.p5.3.m3.1.2.2.3.cmml">max</mi></msup><mo id="A1.SS3.SSS0.P0.SPx2.p5.3.m3.1.2.1" xref="A1.SS3.SSS0.P0.SPx2.p5.3.m3.1.2.1.cmml">⁢</mo><mrow id="A1.SS3.SSS0.P0.SPx2.p5.3.m3.1.2.3.2" xref="A1.SS3.SSS0.P0.SPx2.p5.3.m3.1.2.cmml"><mo id="A1.SS3.SSS0.P0.SPx2.p5.3.m3.1.2.3.2.1" stretchy="false" xref="A1.SS3.SSS0.P0.SPx2.p5.3.m3.1.2.cmml">(</mo><mi id="A1.SS3.SSS0.P0.SPx2.p5.3.m3.1.1" xref="A1.SS3.SSS0.P0.SPx2.p5.3.m3.1.1.cmml">G</mi><mo id="A1.SS3.SSS0.P0.SPx2.p5.3.m3.1.2.3.2.2" stretchy="false" xref="A1.SS3.SSS0.P0.SPx2.p5.3.m3.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="A1.SS3.SSS0.P0.SPx2.p5.3.m3.1b"><apply id="A1.SS3.SSS0.P0.SPx2.p5.3.m3.1.2.cmml" xref="A1.SS3.SSS0.P0.SPx2.p5.3.m3.1.2"><times id="A1.SS3.SSS0.P0.SPx2.p5.3.m3.1.2.1.cmml" xref="A1.SS3.SSS0.P0.SPx2.p5.3.m3.1.2.1"></times><apply id="A1.SS3.SSS0.P0.SPx2.p5.3.m3.1.2.2.cmml" xref="A1.SS3.SSS0.P0.SPx2.p5.3.m3.1.2.2"><csymbol cd="ambiguous" id="A1.SS3.SSS0.P0.SPx2.p5.3.m3.1.2.2.1.cmml" xref="A1.SS3.SSS0.P0.SPx2.p5.3.m3.1.2.2">superscript</csymbol><ci id="A1.SS3.SSS0.P0.SPx2.p5.3.m3.1.2.2.2.cmml" xref="A1.SS3.SSS0.P0.SPx2.p5.3.m3.1.2.2.2">𝜌</ci><max id="A1.SS3.SSS0.P0.SPx2.p5.3.m3.1.2.2.3.cmml" xref="A1.SS3.SSS0.P0.SPx2.p5.3.m3.1.2.2.3"></max></apply><ci id="A1.SS3.SSS0.P0.SPx2.p5.3.m3.1.1.cmml" xref="A1.SS3.SSS0.P0.SPx2.p5.3.m3.1.1">𝐺</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS3.SSS0.P0.SPx2.p5.3.m3.1c">\rho^{\max}(G)</annotation><annotation encoding="application/x-llamapun" id="A1.SS3.SSS0.P0.SPx2.p5.3.m3.1d">italic_ρ start_POSTSUPERSCRIPT roman_max end_POSTSUPERSCRIPT ( italic_G )</annotation></semantics></math>, we simply broadcast the maximum fractional out-degree <math alttext="g^{\max}" class="ltx_Math" display="inline" id="A1.SS3.SSS0.P0.SPx2.p5.4.m4.1"><semantics id="A1.SS3.SSS0.P0.SPx2.p5.4.m4.1a"><msup id="A1.SS3.SSS0.P0.SPx2.p5.4.m4.1.1" xref="A1.SS3.SSS0.P0.SPx2.p5.4.m4.1.1.cmml"><mi id="A1.SS3.SSS0.P0.SPx2.p5.4.m4.1.1.2" xref="A1.SS3.SSS0.P0.SPx2.p5.4.m4.1.1.2.cmml">g</mi><mi id="A1.SS3.SSS0.P0.SPx2.p5.4.m4.1.1.3" xref="A1.SS3.SSS0.P0.SPx2.p5.4.m4.1.1.3.cmml">max</mi></msup><annotation-xml encoding="MathML-Content" id="A1.SS3.SSS0.P0.SPx2.p5.4.m4.1b"><apply id="A1.SS3.SSS0.P0.SPx2.p5.4.m4.1.1.cmml" xref="A1.SS3.SSS0.P0.SPx2.p5.4.m4.1.1"><csymbol cd="ambiguous" id="A1.SS3.SSS0.P0.SPx2.p5.4.m4.1.1.1.cmml" xref="A1.SS3.SSS0.P0.SPx2.p5.4.m4.1.1">superscript</csymbol><ci id="A1.SS3.SSS0.P0.SPx2.p5.4.m4.1.1.2.cmml" xref="A1.SS3.SSS0.P0.SPx2.p5.4.m4.1.1.2">𝑔</ci><max id="A1.SS3.SSS0.P0.SPx2.p5.4.m4.1.1.3.cmml" xref="A1.SS3.SSS0.P0.SPx2.p5.4.m4.1.1.3"></max></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS3.SSS0.P0.SPx2.p5.4.m4.1c">g^{\max}</annotation><annotation encoding="application/x-llamapun" id="A1.SS3.SSS0.P0.SPx2.p5.4.m4.1d">italic_g start_POSTSUPERSCRIPT roman_max end_POSTSUPERSCRIPT</annotation></semantics></math> as before in <math alttext="O(d)" class="ltx_Math" display="inline" id="A1.SS3.SSS0.P0.SPx2.p5.5.m5.1"><semantics id="A1.SS3.SSS0.P0.SPx2.p5.5.m5.1a"><mrow id="A1.SS3.SSS0.P0.SPx2.p5.5.m5.1.2" xref="A1.SS3.SSS0.P0.SPx2.p5.5.m5.1.2.cmml"><mi id="A1.SS3.SSS0.P0.SPx2.p5.5.m5.1.2.2" xref="A1.SS3.SSS0.P0.SPx2.p5.5.m5.1.2.2.cmml">O</mi><mo id="A1.SS3.SSS0.P0.SPx2.p5.5.m5.1.2.1" xref="A1.SS3.SSS0.P0.SPx2.p5.5.m5.1.2.1.cmml">⁢</mo><mrow id="A1.SS3.SSS0.P0.SPx2.p5.5.m5.1.2.3.2" xref="A1.SS3.SSS0.P0.SPx2.p5.5.m5.1.2.cmml"><mo id="A1.SS3.SSS0.P0.SPx2.p5.5.m5.1.2.3.2.1" stretchy="false" xref="A1.SS3.SSS0.P0.SPx2.p5.5.m5.1.2.cmml">(</mo><mi id="A1.SS3.SSS0.P0.SPx2.p5.5.m5.1.1" xref="A1.SS3.SSS0.P0.SPx2.p5.5.m5.1.1.cmml">d</mi><mo id="A1.SS3.SSS0.P0.SPx2.p5.5.m5.1.2.3.2.2" stretchy="false" xref="A1.SS3.SSS0.P0.SPx2.p5.5.m5.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="A1.SS3.SSS0.P0.SPx2.p5.5.m5.1b"><apply id="A1.SS3.SSS0.P0.SPx2.p5.5.m5.1.2.cmml" xref="A1.SS3.SSS0.P0.SPx2.p5.5.m5.1.2"><times id="A1.SS3.SSS0.P0.SPx2.p5.5.m5.1.2.1.cmml" xref="A1.SS3.SSS0.P0.SPx2.p5.5.m5.1.2.1"></times><ci id="A1.SS3.SSS0.P0.SPx2.p5.5.m5.1.2.2.cmml" xref="A1.SS3.SSS0.P0.SPx2.p5.5.m5.1.2.2">𝑂</ci><ci id="A1.SS3.SSS0.P0.SPx2.p5.5.m5.1.1.cmml" xref="A1.SS3.SSS0.P0.SPx2.p5.5.m5.1.1">𝑑</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS3.SSS0.P0.SPx2.p5.5.m5.1c">O(d)</annotation><annotation encoding="application/x-llamapun" id="A1.SS3.SSS0.P0.SPx2.p5.5.m5.1d">italic_O ( italic_d )</annotation></semantics></math> rounds. Observe that <math alttext="g^{\max}\geq\rho^{\max}(G)" class="ltx_Math" display="inline" id="A1.SS3.SSS0.P0.SPx2.p5.6.m6.1"><semantics id="A1.SS3.SSS0.P0.SPx2.p5.6.m6.1a"><mrow id="A1.SS3.SSS0.P0.SPx2.p5.6.m6.1.2" xref="A1.SS3.SSS0.P0.SPx2.p5.6.m6.1.2.cmml"><msup id="A1.SS3.SSS0.P0.SPx2.p5.6.m6.1.2.2" xref="A1.SS3.SSS0.P0.SPx2.p5.6.m6.1.2.2.cmml"><mi id="A1.SS3.SSS0.P0.SPx2.p5.6.m6.1.2.2.2" xref="A1.SS3.SSS0.P0.SPx2.p5.6.m6.1.2.2.2.cmml">g</mi><mi id="A1.SS3.SSS0.P0.SPx2.p5.6.m6.1.2.2.3" xref="A1.SS3.SSS0.P0.SPx2.p5.6.m6.1.2.2.3.cmml">max</mi></msup><mo id="A1.SS3.SSS0.P0.SPx2.p5.6.m6.1.2.1" xref="A1.SS3.SSS0.P0.SPx2.p5.6.m6.1.2.1.cmml">≥</mo><mrow id="A1.SS3.SSS0.P0.SPx2.p5.6.m6.1.2.3" xref="A1.SS3.SSS0.P0.SPx2.p5.6.m6.1.2.3.cmml"><msup id="A1.SS3.SSS0.P0.SPx2.p5.6.m6.1.2.3.2" xref="A1.SS3.SSS0.P0.SPx2.p5.6.m6.1.2.3.2.cmml"><mi id="A1.SS3.SSS0.P0.SPx2.p5.6.m6.1.2.3.2.2" xref="A1.SS3.SSS0.P0.SPx2.p5.6.m6.1.2.3.2.2.cmml">ρ</mi><mi id="A1.SS3.SSS0.P0.SPx2.p5.6.m6.1.2.3.2.3" xref="A1.SS3.SSS0.P0.SPx2.p5.6.m6.1.2.3.2.3.cmml">max</mi></msup><mo id="A1.SS3.SSS0.P0.SPx2.p5.6.m6.1.2.3.1" xref="A1.SS3.SSS0.P0.SPx2.p5.6.m6.1.2.3.1.cmml">⁢</mo><mrow id="A1.SS3.SSS0.P0.SPx2.p5.6.m6.1.2.3.3.2" xref="A1.SS3.SSS0.P0.SPx2.p5.6.m6.1.2.3.cmml"><mo id="A1.SS3.SSS0.P0.SPx2.p5.6.m6.1.2.3.3.2.1" stretchy="false" xref="A1.SS3.SSS0.P0.SPx2.p5.6.m6.1.2.3.cmml">(</mo><mi id="A1.SS3.SSS0.P0.SPx2.p5.6.m6.1.1" xref="A1.SS3.SSS0.P0.SPx2.p5.6.m6.1.1.cmml">G</mi><mo id="A1.SS3.SSS0.P0.SPx2.p5.6.m6.1.2.3.3.2.2" stretchy="false" xref="A1.SS3.SSS0.P0.SPx2.p5.6.m6.1.2.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="A1.SS3.SSS0.P0.SPx2.p5.6.m6.1b"><apply id="A1.SS3.SSS0.P0.SPx2.p5.6.m6.1.2.cmml" xref="A1.SS3.SSS0.P0.SPx2.p5.6.m6.1.2"><geq id="A1.SS3.SSS0.P0.SPx2.p5.6.m6.1.2.1.cmml" xref="A1.SS3.SSS0.P0.SPx2.p5.6.m6.1.2.1"></geq><apply id="A1.SS3.SSS0.P0.SPx2.p5.6.m6.1.2.2.cmml" xref="A1.SS3.SSS0.P0.SPx2.p5.6.m6.1.2.2"><csymbol cd="ambiguous" id="A1.SS3.SSS0.P0.SPx2.p5.6.m6.1.2.2.1.cmml" xref="A1.SS3.SSS0.P0.SPx2.p5.6.m6.1.2.2">superscript</csymbol><ci id="A1.SS3.SSS0.P0.SPx2.p5.6.m6.1.2.2.2.cmml" xref="A1.SS3.SSS0.P0.SPx2.p5.6.m6.1.2.2.2">𝑔</ci><max id="A1.SS3.SSS0.P0.SPx2.p5.6.m6.1.2.2.3.cmml" xref="A1.SS3.SSS0.P0.SPx2.p5.6.m6.1.2.2.3"></max></apply><apply id="A1.SS3.SSS0.P0.SPx2.p5.6.m6.1.2.3.cmml" xref="A1.SS3.SSS0.P0.SPx2.p5.6.m6.1.2.3"><times id="A1.SS3.SSS0.P0.SPx2.p5.6.m6.1.2.3.1.cmml" xref="A1.SS3.SSS0.P0.SPx2.p5.6.m6.1.2.3.1"></times><apply id="A1.SS3.SSS0.P0.SPx2.p5.6.m6.1.2.3.2.cmml" xref="A1.SS3.SSS0.P0.SPx2.p5.6.m6.1.2.3.2"><csymbol cd="ambiguous" id="A1.SS3.SSS0.P0.SPx2.p5.6.m6.1.2.3.2.1.cmml" xref="A1.SS3.SSS0.P0.SPx2.p5.6.m6.1.2.3.2">superscript</csymbol><ci id="A1.SS3.SSS0.P0.SPx2.p5.6.m6.1.2.3.2.2.cmml" xref="A1.SS3.SSS0.P0.SPx2.p5.6.m6.1.2.3.2.2">𝜌</ci><max id="A1.SS3.SSS0.P0.SPx2.p5.6.m6.1.2.3.2.3.cmml" xref="A1.SS3.SSS0.P0.SPx2.p5.6.m6.1.2.3.2.3"></max></apply><ci id="A1.SS3.SSS0.P0.SPx2.p5.6.m6.1.1.cmml" xref="A1.SS3.SSS0.P0.SPx2.p5.6.m6.1.1">𝐺</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS3.SSS0.P0.SPx2.p5.6.m6.1c">g^{\max}\geq\rho^{\max}(G)</annotation><annotation encoding="application/x-llamapun" id="A1.SS3.SSS0.P0.SPx2.p5.6.m6.1d">italic_g start_POSTSUPERSCRIPT roman_max end_POSTSUPERSCRIPT ≥ italic_ρ start_POSTSUPERSCRIPT roman_max end_POSTSUPERSCRIPT ( italic_G )</annotation></semantics></math>. Then for a vertex to become an active leader it must have out-degree at least <math alttext="g^{\max}" class="ltx_Math" display="inline" id="A1.SS3.SSS0.P0.SPx2.p5.7.m7.1"><semantics id="A1.SS3.SSS0.P0.SPx2.p5.7.m7.1a"><msup id="A1.SS3.SSS0.P0.SPx2.p5.7.m7.1.1" xref="A1.SS3.SSS0.P0.SPx2.p5.7.m7.1.1.cmml"><mi id="A1.SS3.SSS0.P0.SPx2.p5.7.m7.1.1.2" xref="A1.SS3.SSS0.P0.SPx2.p5.7.m7.1.1.2.cmml">g</mi><mi id="A1.SS3.SSS0.P0.SPx2.p5.7.m7.1.1.3" xref="A1.SS3.SSS0.P0.SPx2.p5.7.m7.1.1.3.cmml">max</mi></msup><annotation-xml encoding="MathML-Content" id="A1.SS3.SSS0.P0.SPx2.p5.7.m7.1b"><apply id="A1.SS3.SSS0.P0.SPx2.p5.7.m7.1.1.cmml" xref="A1.SS3.SSS0.P0.SPx2.p5.7.m7.1.1"><csymbol cd="ambiguous" id="A1.SS3.SSS0.P0.SPx2.p5.7.m7.1.1.1.cmml" xref="A1.SS3.SSS0.P0.SPx2.p5.7.m7.1.1">superscript</csymbol><ci id="A1.SS3.SSS0.P0.SPx2.p5.7.m7.1.1.2.cmml" xref="A1.SS3.SSS0.P0.SPx2.p5.7.m7.1.1.2">𝑔</ci><max id="A1.SS3.SSS0.P0.SPx2.p5.7.m7.1.1.3.cmml" xref="A1.SS3.SSS0.P0.SPx2.p5.7.m7.1.1.3"></max></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS3.SSS0.P0.SPx2.p5.7.m7.1c">g^{\max}</annotation><annotation encoding="application/x-llamapun" id="A1.SS3.SSS0.P0.SPx2.p5.7.m7.1d">italic_g start_POSTSUPERSCRIPT roman_max end_POSTSUPERSCRIPT</annotation></semantics></math>. Then we proceed as before. The above arguments can then be adapted mutatis mutandis to show that now <math alttext="\rho(H)\geq(1-\varepsilon)\rho^{\max}(G)" class="ltx_Math" display="inline" id="A1.SS3.SSS0.P0.SPx2.p5.8.m8.3"><semantics id="A1.SS3.SSS0.P0.SPx2.p5.8.m8.3a"><mrow id="A1.SS3.SSS0.P0.SPx2.p5.8.m8.3.3" xref="A1.SS3.SSS0.P0.SPx2.p5.8.m8.3.3.cmml"><mrow id="A1.SS3.SSS0.P0.SPx2.p5.8.m8.3.3.3" xref="A1.SS3.SSS0.P0.SPx2.p5.8.m8.3.3.3.cmml"><mi id="A1.SS3.SSS0.P0.SPx2.p5.8.m8.3.3.3.2" xref="A1.SS3.SSS0.P0.SPx2.p5.8.m8.3.3.3.2.cmml">ρ</mi><mo id="A1.SS3.SSS0.P0.SPx2.p5.8.m8.3.3.3.1" xref="A1.SS3.SSS0.P0.SPx2.p5.8.m8.3.3.3.1.cmml">⁢</mo><mrow id="A1.SS3.SSS0.P0.SPx2.p5.8.m8.3.3.3.3.2" xref="A1.SS3.SSS0.P0.SPx2.p5.8.m8.3.3.3.cmml"><mo id="A1.SS3.SSS0.P0.SPx2.p5.8.m8.3.3.3.3.2.1" stretchy="false" xref="A1.SS3.SSS0.P0.SPx2.p5.8.m8.3.3.3.cmml">(</mo><mi id="A1.SS3.SSS0.P0.SPx2.p5.8.m8.1.1" xref="A1.SS3.SSS0.P0.SPx2.p5.8.m8.1.1.cmml">H</mi><mo id="A1.SS3.SSS0.P0.SPx2.p5.8.m8.3.3.3.3.2.2" stretchy="false" xref="A1.SS3.SSS0.P0.SPx2.p5.8.m8.3.3.3.cmml">)</mo></mrow></mrow><mo id="A1.SS3.SSS0.P0.SPx2.p5.8.m8.3.3.2" xref="A1.SS3.SSS0.P0.SPx2.p5.8.m8.3.3.2.cmml">≥</mo><mrow id="A1.SS3.SSS0.P0.SPx2.p5.8.m8.3.3.1" xref="A1.SS3.SSS0.P0.SPx2.p5.8.m8.3.3.1.cmml"><mrow id="A1.SS3.SSS0.P0.SPx2.p5.8.m8.3.3.1.1.1" xref="A1.SS3.SSS0.P0.SPx2.p5.8.m8.3.3.1.1.1.1.cmml"><mo id="A1.SS3.SSS0.P0.SPx2.p5.8.m8.3.3.1.1.1.2" stretchy="false" xref="A1.SS3.SSS0.P0.SPx2.p5.8.m8.3.3.1.1.1.1.cmml">(</mo><mrow id="A1.SS3.SSS0.P0.SPx2.p5.8.m8.3.3.1.1.1.1" xref="A1.SS3.SSS0.P0.SPx2.p5.8.m8.3.3.1.1.1.1.cmml"><mn id="A1.SS3.SSS0.P0.SPx2.p5.8.m8.3.3.1.1.1.1.2" xref="A1.SS3.SSS0.P0.SPx2.p5.8.m8.3.3.1.1.1.1.2.cmml">1</mn><mo id="A1.SS3.SSS0.P0.SPx2.p5.8.m8.3.3.1.1.1.1.1" xref="A1.SS3.SSS0.P0.SPx2.p5.8.m8.3.3.1.1.1.1.1.cmml">−</mo><mi id="A1.SS3.SSS0.P0.SPx2.p5.8.m8.3.3.1.1.1.1.3" xref="A1.SS3.SSS0.P0.SPx2.p5.8.m8.3.3.1.1.1.1.3.cmml">ε</mi></mrow><mo id="A1.SS3.SSS0.P0.SPx2.p5.8.m8.3.3.1.1.1.3" stretchy="false" xref="A1.SS3.SSS0.P0.SPx2.p5.8.m8.3.3.1.1.1.1.cmml">)</mo></mrow><mo id="A1.SS3.SSS0.P0.SPx2.p5.8.m8.3.3.1.2" xref="A1.SS3.SSS0.P0.SPx2.p5.8.m8.3.3.1.2.cmml">⁢</mo><msup id="A1.SS3.SSS0.P0.SPx2.p5.8.m8.3.3.1.3" xref="A1.SS3.SSS0.P0.SPx2.p5.8.m8.3.3.1.3.cmml"><mi id="A1.SS3.SSS0.P0.SPx2.p5.8.m8.3.3.1.3.2" xref="A1.SS3.SSS0.P0.SPx2.p5.8.m8.3.3.1.3.2.cmml">ρ</mi><mi id="A1.SS3.SSS0.P0.SPx2.p5.8.m8.3.3.1.3.3" xref="A1.SS3.SSS0.P0.SPx2.p5.8.m8.3.3.1.3.3.cmml">max</mi></msup><mo id="A1.SS3.SSS0.P0.SPx2.p5.8.m8.3.3.1.2a" xref="A1.SS3.SSS0.P0.SPx2.p5.8.m8.3.3.1.2.cmml">⁢</mo><mrow id="A1.SS3.SSS0.P0.SPx2.p5.8.m8.3.3.1.4.2" xref="A1.SS3.SSS0.P0.SPx2.p5.8.m8.3.3.1.cmml"><mo id="A1.SS3.SSS0.P0.SPx2.p5.8.m8.3.3.1.4.2.1" stretchy="false" xref="A1.SS3.SSS0.P0.SPx2.p5.8.m8.3.3.1.cmml">(</mo><mi id="A1.SS3.SSS0.P0.SPx2.p5.8.m8.2.2" xref="A1.SS3.SSS0.P0.SPx2.p5.8.m8.2.2.cmml">G</mi><mo id="A1.SS3.SSS0.P0.SPx2.p5.8.m8.3.3.1.4.2.2" stretchy="false" xref="A1.SS3.SSS0.P0.SPx2.p5.8.m8.3.3.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="A1.SS3.SSS0.P0.SPx2.p5.8.m8.3b"><apply id="A1.SS3.SSS0.P0.SPx2.p5.8.m8.3.3.cmml" xref="A1.SS3.SSS0.P0.SPx2.p5.8.m8.3.3"><geq id="A1.SS3.SSS0.P0.SPx2.p5.8.m8.3.3.2.cmml" xref="A1.SS3.SSS0.P0.SPx2.p5.8.m8.3.3.2"></geq><apply id="A1.SS3.SSS0.P0.SPx2.p5.8.m8.3.3.3.cmml" xref="A1.SS3.SSS0.P0.SPx2.p5.8.m8.3.3.3"><times id="A1.SS3.SSS0.P0.SPx2.p5.8.m8.3.3.3.1.cmml" xref="A1.SS3.SSS0.P0.SPx2.p5.8.m8.3.3.3.1"></times><ci id="A1.SS3.SSS0.P0.SPx2.p5.8.m8.3.3.3.2.cmml" xref="A1.SS3.SSS0.P0.SPx2.p5.8.m8.3.3.3.2">𝜌</ci><ci id="A1.SS3.SSS0.P0.SPx2.p5.8.m8.1.1.cmml" xref="A1.SS3.SSS0.P0.SPx2.p5.8.m8.1.1">𝐻</ci></apply><apply id="A1.SS3.SSS0.P0.SPx2.p5.8.m8.3.3.1.cmml" xref="A1.SS3.SSS0.P0.SPx2.p5.8.m8.3.3.1"><times id="A1.SS3.SSS0.P0.SPx2.p5.8.m8.3.3.1.2.cmml" xref="A1.SS3.SSS0.P0.SPx2.p5.8.m8.3.3.1.2"></times><apply id="A1.SS3.SSS0.P0.SPx2.p5.8.m8.3.3.1.1.1.1.cmml" xref="A1.SS3.SSS0.P0.SPx2.p5.8.m8.3.3.1.1.1"><minus id="A1.SS3.SSS0.P0.SPx2.p5.8.m8.3.3.1.1.1.1.1.cmml" xref="A1.SS3.SSS0.P0.SPx2.p5.8.m8.3.3.1.1.1.1.1"></minus><cn id="A1.SS3.SSS0.P0.SPx2.p5.8.m8.3.3.1.1.1.1.2.cmml" type="integer" xref="A1.SS3.SSS0.P0.SPx2.p5.8.m8.3.3.1.1.1.1.2">1</cn><ci id="A1.SS3.SSS0.P0.SPx2.p5.8.m8.3.3.1.1.1.1.3.cmml" xref="A1.SS3.SSS0.P0.SPx2.p5.8.m8.3.3.1.1.1.1.3">𝜀</ci></apply><apply id="A1.SS3.SSS0.P0.SPx2.p5.8.m8.3.3.1.3.cmml" xref="A1.SS3.SSS0.P0.SPx2.p5.8.m8.3.3.1.3"><csymbol cd="ambiguous" id="A1.SS3.SSS0.P0.SPx2.p5.8.m8.3.3.1.3.1.cmml" xref="A1.SS3.SSS0.P0.SPx2.p5.8.m8.3.3.1.3">superscript</csymbol><ci id="A1.SS3.SSS0.P0.SPx2.p5.8.m8.3.3.1.3.2.cmml" xref="A1.SS3.SSS0.P0.SPx2.p5.8.m8.3.3.1.3.2">𝜌</ci><max id="A1.SS3.SSS0.P0.SPx2.p5.8.m8.3.3.1.3.3.cmml" xref="A1.SS3.SSS0.P0.SPx2.p5.8.m8.3.3.1.3.3"></max></apply><ci id="A1.SS3.SSS0.P0.SPx2.p5.8.m8.2.2.cmml" xref="A1.SS3.SSS0.P0.SPx2.p5.8.m8.2.2">𝐺</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS3.SSS0.P0.SPx2.p5.8.m8.3c">\rho(H)\geq(1-\varepsilon)\rho^{\max}(G)</annotation><annotation encoding="application/x-llamapun" id="A1.SS3.SSS0.P0.SPx2.p5.8.m8.3d">italic_ρ ( italic_H ) ≥ ( 1 - italic_ε ) italic_ρ start_POSTSUPERSCRIPT roman_max end_POSTSUPERSCRIPT ( italic_G )</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="A1.SS3.SSS0.P0.SPx2.p6"> <p class="ltx_p" id="A1.SS3.SSS0.P0.SPx2.p6.17">Notice furthermore that if we want to solve the variant, where an input vertex <math alttext="v" class="ltx_Math" display="inline" id="A1.SS3.SSS0.P0.SPx2.p6.1.m1.1"><semantics id="A1.SS3.SSS0.P0.SPx2.p6.1.m1.1a"><mi id="A1.SS3.SSS0.P0.SPx2.p6.1.m1.1.1" xref="A1.SS3.SSS0.P0.SPx2.p6.1.m1.1.1.cmml">v</mi><annotation-xml encoding="MathML-Content" id="A1.SS3.SSS0.P0.SPx2.p6.1.m1.1b"><ci id="A1.SS3.SSS0.P0.SPx2.p6.1.m1.1.1.cmml" xref="A1.SS3.SSS0.P0.SPx2.p6.1.m1.1.1">𝑣</ci></annotation-xml><annotation encoding="application/x-tex" id="A1.SS3.SSS0.P0.SPx2.p6.1.m1.1c">v</annotation><annotation encoding="application/x-llamapun" id="A1.SS3.SSS0.P0.SPx2.p6.1.m1.1d">italic_v</annotation></semantics></math> is also specified, and we now wish to report a subgraph <math alttext="H_{v}" class="ltx_Math" display="inline" id="A1.SS3.SSS0.P0.SPx2.p6.2.m2.1"><semantics id="A1.SS3.SSS0.P0.SPx2.p6.2.m2.1a"><msub id="A1.SS3.SSS0.P0.SPx2.p6.2.m2.1.1" xref="A1.SS3.SSS0.P0.SPx2.p6.2.m2.1.1.cmml"><mi id="A1.SS3.SSS0.P0.SPx2.p6.2.m2.1.1.2" xref="A1.SS3.SSS0.P0.SPx2.p6.2.m2.1.1.2.cmml">H</mi><mi id="A1.SS3.SSS0.P0.SPx2.p6.2.m2.1.1.3" xref="A1.SS3.SSS0.P0.SPx2.p6.2.m2.1.1.3.cmml">v</mi></msub><annotation-xml encoding="MathML-Content" id="A1.SS3.SSS0.P0.SPx2.p6.2.m2.1b"><apply id="A1.SS3.SSS0.P0.SPx2.p6.2.m2.1.1.cmml" xref="A1.SS3.SSS0.P0.SPx2.p6.2.m2.1.1"><csymbol cd="ambiguous" id="A1.SS3.SSS0.P0.SPx2.p6.2.m2.1.1.1.cmml" xref="A1.SS3.SSS0.P0.SPx2.p6.2.m2.1.1">subscript</csymbol><ci id="A1.SS3.SSS0.P0.SPx2.p6.2.m2.1.1.2.cmml" xref="A1.SS3.SSS0.P0.SPx2.p6.2.m2.1.1.2">𝐻</ci><ci id="A1.SS3.SSS0.P0.SPx2.p6.2.m2.1.1.3.cmml" xref="A1.SS3.SSS0.P0.SPx2.p6.2.m2.1.1.3">𝑣</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS3.SSS0.P0.SPx2.p6.2.m2.1c">H_{v}</annotation><annotation encoding="application/x-llamapun" id="A1.SS3.SSS0.P0.SPx2.p6.2.m2.1d">italic_H start_POSTSUBSCRIPT italic_v end_POSTSUBSCRIPT</annotation></semantics></math> in the <math alttext="t" class="ltx_Math" display="inline" id="A1.SS3.SSS0.P0.SPx2.p6.3.m3.1"><semantics id="A1.SS3.SSS0.P0.SPx2.p6.3.m3.1a"><mi id="A1.SS3.SSS0.P0.SPx2.p6.3.m3.1.1" xref="A1.SS3.SSS0.P0.SPx2.p6.3.m3.1.1.cmml">t</mi><annotation-xml encoding="MathML-Content" id="A1.SS3.SSS0.P0.SPx2.p6.3.m3.1b"><ci id="A1.SS3.SSS0.P0.SPx2.p6.3.m3.1.1.cmml" xref="A1.SS3.SSS0.P0.SPx2.p6.3.m3.1.1">𝑡</ci></annotation-xml><annotation encoding="application/x-tex" id="A1.SS3.SSS0.P0.SPx2.p6.3.m3.1c">t</annotation><annotation encoding="application/x-llamapun" id="A1.SS3.SSS0.P0.SPx2.p6.3.m3.1d">italic_t</annotation></semantics></math>-hop neighbourhood of <math alttext="v" class="ltx_Math" display="inline" id="A1.SS3.SSS0.P0.SPx2.p6.4.m4.1"><semantics id="A1.SS3.SSS0.P0.SPx2.p6.4.m4.1a"><mi id="A1.SS3.SSS0.P0.SPx2.p6.4.m4.1.1" xref="A1.SS3.SSS0.P0.SPx2.p6.4.m4.1.1.cmml">v</mi><annotation-xml encoding="MathML-Content" id="A1.SS3.SSS0.P0.SPx2.p6.4.m4.1b"><ci id="A1.SS3.SSS0.P0.SPx2.p6.4.m4.1.1.cmml" xref="A1.SS3.SSS0.P0.SPx2.p6.4.m4.1.1">𝑣</ci></annotation-xml><annotation encoding="application/x-tex" id="A1.SS3.SSS0.P0.SPx2.p6.4.m4.1c">v</annotation><annotation encoding="application/x-llamapun" id="A1.SS3.SSS0.P0.SPx2.p6.4.m4.1d">italic_v</annotation></semantics></math>, containing <math alttext="v" class="ltx_Math" display="inline" id="A1.SS3.SSS0.P0.SPx2.p6.5.m5.1"><semantics id="A1.SS3.SSS0.P0.SPx2.p6.5.m5.1a"><mi id="A1.SS3.SSS0.P0.SPx2.p6.5.m5.1.1" xref="A1.SS3.SSS0.P0.SPx2.p6.5.m5.1.1.cmml">v</mi><annotation-xml encoding="MathML-Content" id="A1.SS3.SSS0.P0.SPx2.p6.5.m5.1b"><ci id="A1.SS3.SSS0.P0.SPx2.p6.5.m5.1.1.cmml" xref="A1.SS3.SSS0.P0.SPx2.p6.5.m5.1.1">𝑣</ci></annotation-xml><annotation encoding="application/x-tex" id="A1.SS3.SSS0.P0.SPx2.p6.5.m5.1c">v</annotation><annotation encoding="application/x-llamapun" id="A1.SS3.SSS0.P0.SPx2.p6.5.m5.1d">italic_v</annotation></semantics></math>, with density at least <math alttext="(1-\varepsilon)\rho^{*}(v)" class="ltx_Math" display="inline" id="A1.SS3.SSS0.P0.SPx2.p6.6.m6.2"><semantics id="A1.SS3.SSS0.P0.SPx2.p6.6.m6.2a"><mrow id="A1.SS3.SSS0.P0.SPx2.p6.6.m6.2.2" xref="A1.SS3.SSS0.P0.SPx2.p6.6.m6.2.2.cmml"><mrow id="A1.SS3.SSS0.P0.SPx2.p6.6.m6.2.2.1.1" xref="A1.SS3.SSS0.P0.SPx2.p6.6.m6.2.2.1.1.1.cmml"><mo id="A1.SS3.SSS0.P0.SPx2.p6.6.m6.2.2.1.1.2" stretchy="false" xref="A1.SS3.SSS0.P0.SPx2.p6.6.m6.2.2.1.1.1.cmml">(</mo><mrow id="A1.SS3.SSS0.P0.SPx2.p6.6.m6.2.2.1.1.1" xref="A1.SS3.SSS0.P0.SPx2.p6.6.m6.2.2.1.1.1.cmml"><mn id="A1.SS3.SSS0.P0.SPx2.p6.6.m6.2.2.1.1.1.2" xref="A1.SS3.SSS0.P0.SPx2.p6.6.m6.2.2.1.1.1.2.cmml">1</mn><mo id="A1.SS3.SSS0.P0.SPx2.p6.6.m6.2.2.1.1.1.1" xref="A1.SS3.SSS0.P0.SPx2.p6.6.m6.2.2.1.1.1.1.cmml">−</mo><mi id="A1.SS3.SSS0.P0.SPx2.p6.6.m6.2.2.1.1.1.3" xref="A1.SS3.SSS0.P0.SPx2.p6.6.m6.2.2.1.1.1.3.cmml">ε</mi></mrow><mo id="A1.SS3.SSS0.P0.SPx2.p6.6.m6.2.2.1.1.3" stretchy="false" xref="A1.SS3.SSS0.P0.SPx2.p6.6.m6.2.2.1.1.1.cmml">)</mo></mrow><mo id="A1.SS3.SSS0.P0.SPx2.p6.6.m6.2.2.2" xref="A1.SS3.SSS0.P0.SPx2.p6.6.m6.2.2.2.cmml">⁢</mo><msup id="A1.SS3.SSS0.P0.SPx2.p6.6.m6.2.2.3" xref="A1.SS3.SSS0.P0.SPx2.p6.6.m6.2.2.3.cmml"><mi id="A1.SS3.SSS0.P0.SPx2.p6.6.m6.2.2.3.2" xref="A1.SS3.SSS0.P0.SPx2.p6.6.m6.2.2.3.2.cmml">ρ</mi><mo id="A1.SS3.SSS0.P0.SPx2.p6.6.m6.2.2.3.3" xref="A1.SS3.SSS0.P0.SPx2.p6.6.m6.2.2.3.3.cmml">∗</mo></msup><mo id="A1.SS3.SSS0.P0.SPx2.p6.6.m6.2.2.2a" xref="A1.SS3.SSS0.P0.SPx2.p6.6.m6.2.2.2.cmml">⁢</mo><mrow id="A1.SS3.SSS0.P0.SPx2.p6.6.m6.2.2.4.2" xref="A1.SS3.SSS0.P0.SPx2.p6.6.m6.2.2.cmml"><mo id="A1.SS3.SSS0.P0.SPx2.p6.6.m6.2.2.4.2.1" stretchy="false" xref="A1.SS3.SSS0.P0.SPx2.p6.6.m6.2.2.cmml">(</mo><mi id="A1.SS3.SSS0.P0.SPx2.p6.6.m6.1.1" xref="A1.SS3.SSS0.P0.SPx2.p6.6.m6.1.1.cmml">v</mi><mo id="A1.SS3.SSS0.P0.SPx2.p6.6.m6.2.2.4.2.2" stretchy="false" xref="A1.SS3.SSS0.P0.SPx2.p6.6.m6.2.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="A1.SS3.SSS0.P0.SPx2.p6.6.m6.2b"><apply id="A1.SS3.SSS0.P0.SPx2.p6.6.m6.2.2.cmml" xref="A1.SS3.SSS0.P0.SPx2.p6.6.m6.2.2"><times id="A1.SS3.SSS0.P0.SPx2.p6.6.m6.2.2.2.cmml" xref="A1.SS3.SSS0.P0.SPx2.p6.6.m6.2.2.2"></times><apply id="A1.SS3.SSS0.P0.SPx2.p6.6.m6.2.2.1.1.1.cmml" xref="A1.SS3.SSS0.P0.SPx2.p6.6.m6.2.2.1.1"><minus id="A1.SS3.SSS0.P0.SPx2.p6.6.m6.2.2.1.1.1.1.cmml" xref="A1.SS3.SSS0.P0.SPx2.p6.6.m6.2.2.1.1.1.1"></minus><cn id="A1.SS3.SSS0.P0.SPx2.p6.6.m6.2.2.1.1.1.2.cmml" type="integer" xref="A1.SS3.SSS0.P0.SPx2.p6.6.m6.2.2.1.1.1.2">1</cn><ci id="A1.SS3.SSS0.P0.SPx2.p6.6.m6.2.2.1.1.1.3.cmml" xref="A1.SS3.SSS0.P0.SPx2.p6.6.m6.2.2.1.1.1.3">𝜀</ci></apply><apply id="A1.SS3.SSS0.P0.SPx2.p6.6.m6.2.2.3.cmml" xref="A1.SS3.SSS0.P0.SPx2.p6.6.m6.2.2.3"><csymbol cd="ambiguous" id="A1.SS3.SSS0.P0.SPx2.p6.6.m6.2.2.3.1.cmml" xref="A1.SS3.SSS0.P0.SPx2.p6.6.m6.2.2.3">superscript</csymbol><ci id="A1.SS3.SSS0.P0.SPx2.p6.6.m6.2.2.3.2.cmml" xref="A1.SS3.SSS0.P0.SPx2.p6.6.m6.2.2.3.2">𝜌</ci><times id="A1.SS3.SSS0.P0.SPx2.p6.6.m6.2.2.3.3.cmml" xref="A1.SS3.SSS0.P0.SPx2.p6.6.m6.2.2.3.3"></times></apply><ci id="A1.SS3.SSS0.P0.SPx2.p6.6.m6.1.1.cmml" xref="A1.SS3.SSS0.P0.SPx2.p6.6.m6.1.1">𝑣</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS3.SSS0.P0.SPx2.p6.6.m6.2c">(1-\varepsilon)\rho^{*}(v)</annotation><annotation encoding="application/x-llamapun" id="A1.SS3.SSS0.P0.SPx2.p6.6.m6.2d">( 1 - italic_ε ) italic_ρ start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ( italic_v )</annotation></semantics></math>, we can modify by the above approach as follows. Let <math alttext="k" class="ltx_Math" display="inline" id="A1.SS3.SSS0.P0.SPx2.p6.7.m7.1"><semantics id="A1.SS3.SSS0.P0.SPx2.p6.7.m7.1a"><mi id="A1.SS3.SSS0.P0.SPx2.p6.7.m7.1.1" xref="A1.SS3.SSS0.P0.SPx2.p6.7.m7.1.1.cmml">k</mi><annotation-xml encoding="MathML-Content" id="A1.SS3.SSS0.P0.SPx2.p6.7.m7.1b"><ci id="A1.SS3.SSS0.P0.SPx2.p6.7.m7.1.1.cmml" xref="A1.SS3.SSS0.P0.SPx2.p6.7.m7.1.1">𝑘</ci></annotation-xml><annotation encoding="application/x-tex" id="A1.SS3.SSS0.P0.SPx2.p6.7.m7.1c">k</annotation><annotation encoding="application/x-llamapun" id="A1.SS3.SSS0.P0.SPx2.p6.7.m7.1d">italic_k</annotation></semantics></math> be such that Theorem <a class="ltx_ref" href="https://arxiv.org/html/2411.12694v2#S3.Thmtheorem7" title="Theorem 3.7. ‣ 3.C Results in LOCAL ‣ 3 Results and organisation ‣ Local Density and its Distributed Approximation"><span class="ltx_text ltx_ref_tag">3.7</span></a> holds. Next compute an <math alttext="\eta" class="ltx_Math" display="inline" id="A1.SS3.SSS0.P0.SPx2.p6.8.m8.1"><semantics id="A1.SS3.SSS0.P0.SPx2.p6.8.m8.1a"><mi id="A1.SS3.SSS0.P0.SPx2.p6.8.m8.1.1" xref="A1.SS3.SSS0.P0.SPx2.p6.8.m8.1.1.cmml">η</mi><annotation-xml encoding="MathML-Content" id="A1.SS3.SSS0.P0.SPx2.p6.8.m8.1b"><ci id="A1.SS3.SSS0.P0.SPx2.p6.8.m8.1.1.cmml" xref="A1.SS3.SSS0.P0.SPx2.p6.8.m8.1.1">𝜂</ci></annotation-xml><annotation encoding="application/x-tex" id="A1.SS3.SSS0.P0.SPx2.p6.8.m8.1c">\eta</annotation><annotation encoding="application/x-llamapun" id="A1.SS3.SSS0.P0.SPx2.p6.8.m8.1d">italic_η</annotation></semantics></math>-fair orientation of the <math alttext="k" class="ltx_Math" display="inline" id="A1.SS3.SSS0.P0.SPx2.p6.9.m9.1"><semantics id="A1.SS3.SSS0.P0.SPx2.p6.9.m9.1a"><mi id="A1.SS3.SSS0.P0.SPx2.p6.9.m9.1.1" xref="A1.SS3.SSS0.P0.SPx2.p6.9.m9.1.1.cmml">k</mi><annotation-xml encoding="MathML-Content" id="A1.SS3.SSS0.P0.SPx2.p6.9.m9.1b"><ci id="A1.SS3.SSS0.P0.SPx2.p6.9.m9.1.1.cmml" xref="A1.SS3.SSS0.P0.SPx2.p6.9.m9.1.1">𝑘</ci></annotation-xml><annotation encoding="application/x-tex" id="A1.SS3.SSS0.P0.SPx2.p6.9.m9.1c">k</annotation><annotation encoding="application/x-llamapun" id="A1.SS3.SSS0.P0.SPx2.p6.9.m9.1d">italic_k</annotation></semantics></math>-hop neighbourhood of <math alttext="v" class="ltx_Math" display="inline" id="A1.SS3.SSS0.P0.SPx2.p6.10.m10.1"><semantics id="A1.SS3.SSS0.P0.SPx2.p6.10.m10.1a"><mi id="A1.SS3.SSS0.P0.SPx2.p6.10.m10.1.1" xref="A1.SS3.SSS0.P0.SPx2.p6.10.m10.1.1.cmml">v</mi><annotation-xml encoding="MathML-Content" id="A1.SS3.SSS0.P0.SPx2.p6.10.m10.1b"><ci id="A1.SS3.SSS0.P0.SPx2.p6.10.m10.1.1.cmml" xref="A1.SS3.SSS0.P0.SPx2.p6.10.m10.1.1">𝑣</ci></annotation-xml><annotation encoding="application/x-tex" id="A1.SS3.SSS0.P0.SPx2.p6.10.m10.1c">v</annotation><annotation encoding="application/x-llamapun" id="A1.SS3.SSS0.P0.SPx2.p6.10.m10.1d">italic_v</annotation></semantics></math> instead. Finally, choose <math alttext="v" class="ltx_Math" display="inline" id="A1.SS3.SSS0.P0.SPx2.p6.11.m11.1"><semantics id="A1.SS3.SSS0.P0.SPx2.p6.11.m11.1a"><mi id="A1.SS3.SSS0.P0.SPx2.p6.11.m11.1.1" xref="A1.SS3.SSS0.P0.SPx2.p6.11.m11.1.1.cmml">v</mi><annotation-xml encoding="MathML-Content" id="A1.SS3.SSS0.P0.SPx2.p6.11.m11.1b"><ci id="A1.SS3.SSS0.P0.SPx2.p6.11.m11.1.1.cmml" xref="A1.SS3.SSS0.P0.SPx2.p6.11.m11.1.1">𝑣</ci></annotation-xml><annotation encoding="application/x-tex" id="A1.SS3.SSS0.P0.SPx2.p6.11.m11.1c">v</annotation><annotation encoding="application/x-llamapun" id="A1.SS3.SSS0.P0.SPx2.p6.11.m11.1d">italic_v</annotation></semantics></math> as the sole active leader, and let it report a dense subgraph in its <math alttext="k" class="ltx_Math" display="inline" id="A1.SS3.SSS0.P0.SPx2.p6.12.m12.1"><semantics id="A1.SS3.SSS0.P0.SPx2.p6.12.m12.1a"><mi id="A1.SS3.SSS0.P0.SPx2.p6.12.m12.1.1" xref="A1.SS3.SSS0.P0.SPx2.p6.12.m12.1.1.cmml">k</mi><annotation-xml encoding="MathML-Content" id="A1.SS3.SSS0.P0.SPx2.p6.12.m12.1b"><ci id="A1.SS3.SSS0.P0.SPx2.p6.12.m12.1.1.cmml" xref="A1.SS3.SSS0.P0.SPx2.p6.12.m12.1.1">𝑘</ci></annotation-xml><annotation encoding="application/x-tex" id="A1.SS3.SSS0.P0.SPx2.p6.12.m12.1c">k</annotation><annotation encoding="application/x-llamapun" id="A1.SS3.SSS0.P0.SPx2.p6.12.m12.1d">italic_k</annotation></semantics></math>-hop neighbourhood following a similar protocol to before. 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xref="A1.SS3.SSS0.P0.SPx2.p6.13.m13.6.6.2.2.2.1.1.1.1.1.1.cmml">+</mo><mi id="A1.SS3.SSS0.P0.SPx2.p6.13.m13.6.6.2.2.2.1.1.1.1.1.3" xref="A1.SS3.SSS0.P0.SPx2.p6.13.m13.6.6.2.2.2.1.1.1.1.1.3.cmml">η</mi></mrow><mo id="A1.SS3.SSS0.P0.SPx2.p6.13.m13.6.6.2.2.2.1.1.1.1.3" stretchy="false" xref="A1.SS3.SSS0.P0.SPx2.p6.13.m13.6.6.2.2.2.1.1.1.1.1.cmml">)</mo></mrow><mrow id="A1.SS3.SSS0.P0.SPx2.p6.13.m13.6.6.2.2.2.1.1.3" xref="A1.SS3.SSS0.P0.SPx2.p6.13.m13.6.6.2.2.2.1.1.3.cmml"><mo id="A1.SS3.SSS0.P0.SPx2.p6.13.m13.6.6.2.2.2.1.1.3a" xref="A1.SS3.SSS0.P0.SPx2.p6.13.m13.6.6.2.2.2.1.1.3.cmml">−</mo><mi id="A1.SS3.SSS0.P0.SPx2.p6.13.m13.6.6.2.2.2.1.1.3.2" xref="A1.SS3.SSS0.P0.SPx2.p6.13.m13.6.6.2.2.2.1.1.3.2.cmml">i</mi></mrow></msup></mrow></mrow><mo id="A1.SS3.SSS0.P0.SPx2.p6.13.m13.6.6.2.2.5" stretchy="false" xref="A1.SS3.SSS0.P0.SPx2.p6.13.m13.6.6.2.3.1.cmml">}</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="A1.SS3.SSS0.P0.SPx2.p6.13.m13.6b"><apply 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xref="A1.SS3.SSS0.P0.SPx2.p6.13.m13.5.5.1.1.1.3.2">superscript</csymbol><ci id="A1.SS3.SSS0.P0.SPx2.p6.13.m13.5.5.1.1.1.3.2.2.cmml" xref="A1.SS3.SSS0.P0.SPx2.p6.13.m13.5.5.1.1.1.3.2.2">𝑁</ci><ci id="A1.SS3.SSS0.P0.SPx2.p6.13.m13.5.5.1.1.1.3.2.3.cmml" xref="A1.SS3.SSS0.P0.SPx2.p6.13.m13.5.5.1.1.1.3.2.3">𝑘</ci></apply><ci id="A1.SS3.SSS0.P0.SPx2.p6.13.m13.2.2.cmml" xref="A1.SS3.SSS0.P0.SPx2.p6.13.m13.2.2">𝑣</ci></apply></apply><apply id="A1.SS3.SSS0.P0.SPx2.p6.13.m13.6.6.2.2.2.cmml" xref="A1.SS3.SSS0.P0.SPx2.p6.13.m13.6.6.2.2.2"><geq id="A1.SS3.SSS0.P0.SPx2.p6.13.m13.6.6.2.2.2.2.cmml" xref="A1.SS3.SSS0.P0.SPx2.p6.13.m13.6.6.2.2.2.2"></geq><apply id="A1.SS3.SSS0.P0.SPx2.p6.13.m13.6.6.2.2.2.3.cmml" xref="A1.SS3.SSS0.P0.SPx2.p6.13.m13.6.6.2.2.2.3"><times id="A1.SS3.SSS0.P0.SPx2.p6.13.m13.6.6.2.2.2.3.1.cmml" xref="A1.SS3.SSS0.P0.SPx2.p6.13.m13.6.6.2.2.2.3.1"></times><ci id="A1.SS3.SSS0.P0.SPx2.p6.13.m13.6.6.2.2.2.3.2.cmml" xref="A1.SS3.SSS0.P0.SPx2.p6.13.m13.6.6.2.2.2.3.2">ℎ</ci><ci id="A1.SS3.SSS0.P0.SPx2.p6.13.m13.3.3.cmml" xref="A1.SS3.SSS0.P0.SPx2.p6.13.m13.3.3">𝑢</ci></apply><apply id="A1.SS3.SSS0.P0.SPx2.p6.13.m13.6.6.2.2.2.1.cmml" xref="A1.SS3.SSS0.P0.SPx2.p6.13.m13.6.6.2.2.2.1"><times id="A1.SS3.SSS0.P0.SPx2.p6.13.m13.6.6.2.2.2.1.2.cmml" xref="A1.SS3.SSS0.P0.SPx2.p6.13.m13.6.6.2.2.2.1.2"></times><ci id="A1.SS3.SSS0.P0.SPx2.p6.13.m13.6.6.2.2.2.1.3.cmml" xref="A1.SS3.SSS0.P0.SPx2.p6.13.m13.6.6.2.2.2.1.3">ℎ</ci><ci id="A1.SS3.SSS0.P0.SPx2.p6.13.m13.4.4.cmml" xref="A1.SS3.SSS0.P0.SPx2.p6.13.m13.4.4">𝑣</ci><apply id="A1.SS3.SSS0.P0.SPx2.p6.13.m13.6.6.2.2.2.1.1.cmml" xref="A1.SS3.SSS0.P0.SPx2.p6.13.m13.6.6.2.2.2.1.1"><csymbol cd="ambiguous" id="A1.SS3.SSS0.P0.SPx2.p6.13.m13.6.6.2.2.2.1.1.2.cmml" xref="A1.SS3.SSS0.P0.SPx2.p6.13.m13.6.6.2.2.2.1.1">superscript</csymbol><apply id="A1.SS3.SSS0.P0.SPx2.p6.13.m13.6.6.2.2.2.1.1.1.1.1.cmml" xref="A1.SS3.SSS0.P0.SPx2.p6.13.m13.6.6.2.2.2.1.1.1.1"><plus id="A1.SS3.SSS0.P0.SPx2.p6.13.m13.6.6.2.2.2.1.1.1.1.1.1.cmml" xref="A1.SS3.SSS0.P0.SPx2.p6.13.m13.6.6.2.2.2.1.1.1.1.1.1"></plus><cn id="A1.SS3.SSS0.P0.SPx2.p6.13.m13.6.6.2.2.2.1.1.1.1.1.2.cmml" type="integer" xref="A1.SS3.SSS0.P0.SPx2.p6.13.m13.6.6.2.2.2.1.1.1.1.1.2">1</cn><ci id="A1.SS3.SSS0.P0.SPx2.p6.13.m13.6.6.2.2.2.1.1.1.1.1.3.cmml" xref="A1.SS3.SSS0.P0.SPx2.p6.13.m13.6.6.2.2.2.1.1.1.1.1.3">𝜂</ci></apply><apply id="A1.SS3.SSS0.P0.SPx2.p6.13.m13.6.6.2.2.2.1.1.3.cmml" xref="A1.SS3.SSS0.P0.SPx2.p6.13.m13.6.6.2.2.2.1.1.3"><minus id="A1.SS3.SSS0.P0.SPx2.p6.13.m13.6.6.2.2.2.1.1.3.1.cmml" xref="A1.SS3.SSS0.P0.SPx2.p6.13.m13.6.6.2.2.2.1.1.3"></minus><ci id="A1.SS3.SSS0.P0.SPx2.p6.13.m13.6.6.2.2.2.1.1.3.2.cmml" xref="A1.SS3.SSS0.P0.SPx2.p6.13.m13.6.6.2.2.2.1.1.3.2">𝑖</ci></apply></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS3.SSS0.P0.SPx2.p6.13.m13.6c">T_{i}(v):=\{u\in N^{k}(v):h(u)\geq h(v)(1+\eta)^{-i}\}</annotation><annotation encoding="application/x-llamapun" id="A1.SS3.SSS0.P0.SPx2.p6.13.m13.6d">italic_T start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( italic_v ) := { italic_u ∈ italic_N start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT ( italic_v ) : italic_h ( italic_u ) ≥ italic_h ( italic_v ) ( 1 + italic_η ) start_POSTSUPERSCRIPT - italic_i end_POSTSUPERSCRIPT }</annotation></semantics></math>. Other than that the protocol proceeds as before, but now <math alttext="v\in T_{0}" class="ltx_Math" display="inline" id="A1.SS3.SSS0.P0.SPx2.p6.14.m14.1"><semantics id="A1.SS3.SSS0.P0.SPx2.p6.14.m14.1a"><mrow id="A1.SS3.SSS0.P0.SPx2.p6.14.m14.1.1" xref="A1.SS3.SSS0.P0.SPx2.p6.14.m14.1.1.cmml"><mi id="A1.SS3.SSS0.P0.SPx2.p6.14.m14.1.1.2" xref="A1.SS3.SSS0.P0.SPx2.p6.14.m14.1.1.2.cmml">v</mi><mo id="A1.SS3.SSS0.P0.SPx2.p6.14.m14.1.1.1" xref="A1.SS3.SSS0.P0.SPx2.p6.14.m14.1.1.1.cmml">∈</mo><msub id="A1.SS3.SSS0.P0.SPx2.p6.14.m14.1.1.3" xref="A1.SS3.SSS0.P0.SPx2.p6.14.m14.1.1.3.cmml"><mi id="A1.SS3.SSS0.P0.SPx2.p6.14.m14.1.1.3.2" xref="A1.SS3.SSS0.P0.SPx2.p6.14.m14.1.1.3.2.cmml">T</mi><mn id="A1.SS3.SSS0.P0.SPx2.p6.14.m14.1.1.3.3" xref="A1.SS3.SSS0.P0.SPx2.p6.14.m14.1.1.3.3.cmml">0</mn></msub></mrow><annotation-xml encoding="MathML-Content" id="A1.SS3.SSS0.P0.SPx2.p6.14.m14.1b"><apply id="A1.SS3.SSS0.P0.SPx2.p6.14.m14.1.1.cmml" xref="A1.SS3.SSS0.P0.SPx2.p6.14.m14.1.1"><in id="A1.SS3.SSS0.P0.SPx2.p6.14.m14.1.1.1.cmml" xref="A1.SS3.SSS0.P0.SPx2.p6.14.m14.1.1.1"></in><ci id="A1.SS3.SSS0.P0.SPx2.p6.14.m14.1.1.2.cmml" xref="A1.SS3.SSS0.P0.SPx2.p6.14.m14.1.1.2">𝑣</ci><apply id="A1.SS3.SSS0.P0.SPx2.p6.14.m14.1.1.3.cmml" xref="A1.SS3.SSS0.P0.SPx2.p6.14.m14.1.1.3"><csymbol cd="ambiguous" id="A1.SS3.SSS0.P0.SPx2.p6.14.m14.1.1.3.1.cmml" xref="A1.SS3.SSS0.P0.SPx2.p6.14.m14.1.1.3">subscript</csymbol><ci id="A1.SS3.SSS0.P0.SPx2.p6.14.m14.1.1.3.2.cmml" xref="A1.SS3.SSS0.P0.SPx2.p6.14.m14.1.1.3.2">𝑇</ci><cn id="A1.SS3.SSS0.P0.SPx2.p6.14.m14.1.1.3.3.cmml" type="integer" xref="A1.SS3.SSS0.P0.SPx2.p6.14.m14.1.1.3.3">0</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS3.SSS0.P0.SPx2.p6.14.m14.1c">v\in T_{0}</annotation><annotation encoding="application/x-llamapun" id="A1.SS3.SSS0.P0.SPx2.p6.14.m14.1d">italic_v ∈ italic_T start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT</annotation></semantics></math>, and so it is guaranteed to output <math alttext="1" class="ltx_Math" display="inline" id="A1.SS3.SSS0.P0.SPx2.p6.15.m15.1"><semantics id="A1.SS3.SSS0.P0.SPx2.p6.15.m15.1a"><mn id="A1.SS3.SSS0.P0.SPx2.p6.15.m15.1.1" xref="A1.SS3.SSS0.P0.SPx2.p6.15.m15.1.1.cmml">1</mn><annotation-xml encoding="MathML-Content" id="A1.SS3.SSS0.P0.SPx2.p6.15.m15.1b"><cn id="A1.SS3.SSS0.P0.SPx2.p6.15.m15.1.1.cmml" type="integer" xref="A1.SS3.SSS0.P0.SPx2.p6.15.m15.1.1">1</cn></annotation-xml><annotation encoding="application/x-tex" id="A1.SS3.SSS0.P0.SPx2.p6.15.m15.1c">1</annotation><annotation encoding="application/x-llamapun" id="A1.SS3.SSS0.P0.SPx2.p6.15.m15.1d">1</annotation></semantics></math>. Making the change to the <math alttext="T_{i}" class="ltx_Math" display="inline" id="A1.SS3.SSS0.P0.SPx2.p6.16.m16.1"><semantics id="A1.SS3.SSS0.P0.SPx2.p6.16.m16.1a"><msub id="A1.SS3.SSS0.P0.SPx2.p6.16.m16.1.1" xref="A1.SS3.SSS0.P0.SPx2.p6.16.m16.1.1.cmml"><mi id="A1.SS3.SSS0.P0.SPx2.p6.16.m16.1.1.2" xref="A1.SS3.SSS0.P0.SPx2.p6.16.m16.1.1.2.cmml">T</mi><mi id="A1.SS3.SSS0.P0.SPx2.p6.16.m16.1.1.3" xref="A1.SS3.SSS0.P0.SPx2.p6.16.m16.1.1.3.cmml">i</mi></msub><annotation-xml encoding="MathML-Content" id="A1.SS3.SSS0.P0.SPx2.p6.16.m16.1b"><apply id="A1.SS3.SSS0.P0.SPx2.p6.16.m16.1.1.cmml" xref="A1.SS3.SSS0.P0.SPx2.p6.16.m16.1.1"><csymbol cd="ambiguous" id="A1.SS3.SSS0.P0.SPx2.p6.16.m16.1.1.1.cmml" xref="A1.SS3.SSS0.P0.SPx2.p6.16.m16.1.1">subscript</csymbol><ci id="A1.SS3.SSS0.P0.SPx2.p6.16.m16.1.1.2.cmml" xref="A1.SS3.SSS0.P0.SPx2.p6.16.m16.1.1.2">𝑇</ci><ci id="A1.SS3.SSS0.P0.SPx2.p6.16.m16.1.1.3.cmml" xref="A1.SS3.SSS0.P0.SPx2.p6.16.m16.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS3.SSS0.P0.SPx2.p6.16.m16.1c">T_{i}</annotation><annotation encoding="application/x-llamapun" id="A1.SS3.SSS0.P0.SPx2.p6.16.m16.1d">italic_T start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math>’s mutatis mutandis to the arguments from Lemma <a class="ltx_ref" href="https://arxiv.org/html/2411.12694v2#A1.Thmtheorem5" title="Lemma A.5. ‣ Analysis: ‣ A.3 Our algorithm ‣ Appendix A Reporting a densest subgraph ‣ Local Density and its Distributed Approximation"><span class="ltx_text ltx_ref_tag">A.5</span></a> and Lemma <a class="ltx_ref" href="https://arxiv.org/html/2411.12694v2#A1.Thmtheorem3" title="Lemma A.3. ‣ A.3 Our algorithm ‣ Appendix A Reporting a densest subgraph ‣ Local Density and its Distributed Approximation"><span class="ltx_text ltx_ref_tag">A.3</span></a> show that the density of this subgraph will be at least <math alttext="\rho^{*}(v)(1-\varepsilon)" class="ltx_Math" display="inline" id="A1.SS3.SSS0.P0.SPx2.p6.17.m17.2"><semantics id="A1.SS3.SSS0.P0.SPx2.p6.17.m17.2a"><mrow id="A1.SS3.SSS0.P0.SPx2.p6.17.m17.2.2" xref="A1.SS3.SSS0.P0.SPx2.p6.17.m17.2.2.cmml"><msup id="A1.SS3.SSS0.P0.SPx2.p6.17.m17.2.2.3" xref="A1.SS3.SSS0.P0.SPx2.p6.17.m17.2.2.3.cmml"><mi id="A1.SS3.SSS0.P0.SPx2.p6.17.m17.2.2.3.2" xref="A1.SS3.SSS0.P0.SPx2.p6.17.m17.2.2.3.2.cmml">ρ</mi><mo id="A1.SS3.SSS0.P0.SPx2.p6.17.m17.2.2.3.3" xref="A1.SS3.SSS0.P0.SPx2.p6.17.m17.2.2.3.3.cmml">∗</mo></msup><mo id="A1.SS3.SSS0.P0.SPx2.p6.17.m17.2.2.2" xref="A1.SS3.SSS0.P0.SPx2.p6.17.m17.2.2.2.cmml">⁢</mo><mrow id="A1.SS3.SSS0.P0.SPx2.p6.17.m17.2.2.4.2" xref="A1.SS3.SSS0.P0.SPx2.p6.17.m17.2.2.cmml"><mo id="A1.SS3.SSS0.P0.SPx2.p6.17.m17.2.2.4.2.1" stretchy="false" xref="A1.SS3.SSS0.P0.SPx2.p6.17.m17.2.2.cmml">(</mo><mi id="A1.SS3.SSS0.P0.SPx2.p6.17.m17.1.1" xref="A1.SS3.SSS0.P0.SPx2.p6.17.m17.1.1.cmml">v</mi><mo id="A1.SS3.SSS0.P0.SPx2.p6.17.m17.2.2.4.2.2" stretchy="false" xref="A1.SS3.SSS0.P0.SPx2.p6.17.m17.2.2.cmml">)</mo></mrow><mo id="A1.SS3.SSS0.P0.SPx2.p6.17.m17.2.2.2a" xref="A1.SS3.SSS0.P0.SPx2.p6.17.m17.2.2.2.cmml">⁢</mo><mrow id="A1.SS3.SSS0.P0.SPx2.p6.17.m17.2.2.1.1" xref="A1.SS3.SSS0.P0.SPx2.p6.17.m17.2.2.1.1.1.cmml"><mo id="A1.SS3.SSS0.P0.SPx2.p6.17.m17.2.2.1.1.2" stretchy="false" xref="A1.SS3.SSS0.P0.SPx2.p6.17.m17.2.2.1.1.1.cmml">(</mo><mrow id="A1.SS3.SSS0.P0.SPx2.p6.17.m17.2.2.1.1.1" xref="A1.SS3.SSS0.P0.SPx2.p6.17.m17.2.2.1.1.1.cmml"><mn id="A1.SS3.SSS0.P0.SPx2.p6.17.m17.2.2.1.1.1.2" xref="A1.SS3.SSS0.P0.SPx2.p6.17.m17.2.2.1.1.1.2.cmml">1</mn><mo id="A1.SS3.SSS0.P0.SPx2.p6.17.m17.2.2.1.1.1.1" xref="A1.SS3.SSS0.P0.SPx2.p6.17.m17.2.2.1.1.1.1.cmml">−</mo><mi id="A1.SS3.SSS0.P0.SPx2.p6.17.m17.2.2.1.1.1.3" xref="A1.SS3.SSS0.P0.SPx2.p6.17.m17.2.2.1.1.1.3.cmml">ε</mi></mrow><mo id="A1.SS3.SSS0.P0.SPx2.p6.17.m17.2.2.1.1.3" stretchy="false" xref="A1.SS3.SSS0.P0.SPx2.p6.17.m17.2.2.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml 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xref="A1.SS3.SSS0.P0.SPx2.p6.17.m17.2.2.1.1.1.1"></minus><cn id="A1.SS3.SSS0.P0.SPx2.p6.17.m17.2.2.1.1.1.2.cmml" type="integer" xref="A1.SS3.SSS0.P0.SPx2.p6.17.m17.2.2.1.1.1.2">1</cn><ci id="A1.SS3.SSS0.P0.SPx2.p6.17.m17.2.2.1.1.1.3.cmml" xref="A1.SS3.SSS0.P0.SPx2.p6.17.m17.2.2.1.1.1.3">𝜀</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS3.SSS0.P0.SPx2.p6.17.m17.2c">\rho^{*}(v)(1-\varepsilon)</annotation><annotation encoding="application/x-llamapun" id="A1.SS3.SSS0.P0.SPx2.p6.17.m17.2d">italic_ρ start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ( italic_v ) ( 1 - italic_ε )</annotation></semantics></math> as desired.</p> </div> <div class="ltx_para" id="A1.SS3.SSS0.P0.SPx2.p7"> <p class="ltx_p" id="A1.SS3.SSS0.P0.SPx2.p7.1">By plugging in our algorithm for computing <math alttext="\eta" class="ltx_Math" display="inline" id="A1.SS3.SSS0.P0.SPx2.p7.1.m1.1"><semantics id="A1.SS3.SSS0.P0.SPx2.p7.1.m1.1a"><mi id="A1.SS3.SSS0.P0.SPx2.p7.1.m1.1.1" xref="A1.SS3.SSS0.P0.SPx2.p7.1.m1.1.1.cmml">η</mi><annotation-xml encoding="MathML-Content" id="A1.SS3.SSS0.P0.SPx2.p7.1.m1.1b"><ci id="A1.SS3.SSS0.P0.SPx2.p7.1.m1.1.1.cmml" xref="A1.SS3.SSS0.P0.SPx2.p7.1.m1.1.1">𝜂</ci></annotation-xml><annotation encoding="application/x-tex" id="A1.SS3.SSS0.P0.SPx2.p7.1.m1.1c">\eta</annotation><annotation encoding="application/x-llamapun" id="A1.SS3.SSS0.P0.SPx2.p7.1.m1.1d">italic_η</annotation></semantics></math>-fair orientations, we arrive at the following result similarly to before:</p> </div> <div class="ltx_theorem ltx_theorem_theorem" id="A1.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="A1.Thmtheorem11.1.1.1">Theorem A.11</span></span><span class="ltx_text ltx_font_bold" id="A1.Thmtheorem11.2.2">.</span> </h6> <div class="ltx_para" id="A1.Thmtheorem11.p1"> <p class="ltx_p" id="A1.Thmtheorem11.p1.10"><span class="ltx_text ltx_font_italic" id="A1.Thmtheorem11.p1.10.10">Suppose there exists an algorithm that given a graph <math alttext="G" class="ltx_Math" display="inline" id="A1.Thmtheorem11.p1.1.1.m1.1"><semantics id="A1.Thmtheorem11.p1.1.1.m1.1a"><mi id="A1.Thmtheorem11.p1.1.1.m1.1.1" xref="A1.Thmtheorem11.p1.1.1.m1.1.1.cmml">G</mi><annotation-xml encoding="MathML-Content" id="A1.Thmtheorem11.p1.1.1.m1.1b"><ci id="A1.Thmtheorem11.p1.1.1.m1.1.1.cmml" xref="A1.Thmtheorem11.p1.1.1.m1.1.1">𝐺</ci></annotation-xml><annotation encoding="application/x-tex" id="A1.Thmtheorem11.p1.1.1.m1.1c">G</annotation><annotation encoding="application/x-llamapun" id="A1.Thmtheorem11.p1.1.1.m1.1d">italic_G</annotation></semantics></math> on <math alttext="n" class="ltx_Math" display="inline" id="A1.Thmtheorem11.p1.2.2.m2.1"><semantics id="A1.Thmtheorem11.p1.2.2.m2.1a"><mi id="A1.Thmtheorem11.p1.2.2.m2.1.1" xref="A1.Thmtheorem11.p1.2.2.m2.1.1.cmml">n</mi><annotation-xml encoding="MathML-Content" id="A1.Thmtheorem11.p1.2.2.m2.1b"><ci id="A1.Thmtheorem11.p1.2.2.m2.1.1.cmml" xref="A1.Thmtheorem11.p1.2.2.m2.1.1">𝑛</ci></annotation-xml><annotation encoding="application/x-tex" id="A1.Thmtheorem11.p1.2.2.m2.1c">n</annotation><annotation encoding="application/x-llamapun" id="A1.Thmtheorem11.p1.2.2.m2.1d">italic_n</annotation></semantics></math> vertices and parameters <math alttext="\varepsilon" class="ltx_Math" display="inline" id="A1.Thmtheorem11.p1.3.3.m3.1"><semantics id="A1.Thmtheorem11.p1.3.3.m3.1a"><mi id="A1.Thmtheorem11.p1.3.3.m3.1.1" xref="A1.Thmtheorem11.p1.3.3.m3.1.1.cmml">ε</mi><annotation-xml encoding="MathML-Content" id="A1.Thmtheorem11.p1.3.3.m3.1b"><ci id="A1.Thmtheorem11.p1.3.3.m3.1.1.cmml" xref="A1.Thmtheorem11.p1.3.3.m3.1.1">𝜀</ci></annotation-xml><annotation encoding="application/x-tex" id="A1.Thmtheorem11.p1.3.3.m3.1c">\varepsilon</annotation><annotation encoding="application/x-llamapun" id="A1.Thmtheorem11.p1.3.3.m3.1d">italic_ε</annotation></semantics></math> computes an <math alttext="\eta" class="ltx_Math" display="inline" id="A1.Thmtheorem11.p1.4.4.m4.1"><semantics id="A1.Thmtheorem11.p1.4.4.m4.1a"><mi id="A1.Thmtheorem11.p1.4.4.m4.1.1" xref="A1.Thmtheorem11.p1.4.4.m4.1.1.cmml">η</mi><annotation-xml encoding="MathML-Content" id="A1.Thmtheorem11.p1.4.4.m4.1b"><ci id="A1.Thmtheorem11.p1.4.4.m4.1.1.cmml" xref="A1.Thmtheorem11.p1.4.4.m4.1.1">𝜂</ci></annotation-xml><annotation encoding="application/x-tex" id="A1.Thmtheorem11.p1.4.4.m4.1c">\eta</annotation><annotation encoding="application/x-llamapun" id="A1.Thmtheorem11.p1.4.4.m4.1d">italic_η</annotation></semantics></math>-fair orientation in <math alttext="O(U(n,\eta))" class="ltx_Math" display="inline" id="A1.Thmtheorem11.p1.5.5.m5.3"><semantics id="A1.Thmtheorem11.p1.5.5.m5.3a"><mrow id="A1.Thmtheorem11.p1.5.5.m5.3.3" xref="A1.Thmtheorem11.p1.5.5.m5.3.3.cmml"><mi id="A1.Thmtheorem11.p1.5.5.m5.3.3.3" xref="A1.Thmtheorem11.p1.5.5.m5.3.3.3.cmml">O</mi><mo id="A1.Thmtheorem11.p1.5.5.m5.3.3.2" xref="A1.Thmtheorem11.p1.5.5.m5.3.3.2.cmml">⁢</mo><mrow id="A1.Thmtheorem11.p1.5.5.m5.3.3.1.1" xref="A1.Thmtheorem11.p1.5.5.m5.3.3.1.1.1.cmml"><mo id="A1.Thmtheorem11.p1.5.5.m5.3.3.1.1.2" stretchy="false" xref="A1.Thmtheorem11.p1.5.5.m5.3.3.1.1.1.cmml">(</mo><mrow id="A1.Thmtheorem11.p1.5.5.m5.3.3.1.1.1" xref="A1.Thmtheorem11.p1.5.5.m5.3.3.1.1.1.cmml"><mi id="A1.Thmtheorem11.p1.5.5.m5.3.3.1.1.1.2" xref="A1.Thmtheorem11.p1.5.5.m5.3.3.1.1.1.2.cmml">U</mi><mo id="A1.Thmtheorem11.p1.5.5.m5.3.3.1.1.1.1" xref="A1.Thmtheorem11.p1.5.5.m5.3.3.1.1.1.1.cmml">⁢</mo><mrow id="A1.Thmtheorem11.p1.5.5.m5.3.3.1.1.1.3.2" xref="A1.Thmtheorem11.p1.5.5.m5.3.3.1.1.1.3.1.cmml"><mo id="A1.Thmtheorem11.p1.5.5.m5.3.3.1.1.1.3.2.1" stretchy="false" xref="A1.Thmtheorem11.p1.5.5.m5.3.3.1.1.1.3.1.cmml">(</mo><mi id="A1.Thmtheorem11.p1.5.5.m5.1.1" xref="A1.Thmtheorem11.p1.5.5.m5.1.1.cmml">n</mi><mo id="A1.Thmtheorem11.p1.5.5.m5.3.3.1.1.1.3.2.2" xref="A1.Thmtheorem11.p1.5.5.m5.3.3.1.1.1.3.1.cmml">,</mo><mi id="A1.Thmtheorem11.p1.5.5.m5.2.2" xref="A1.Thmtheorem11.p1.5.5.m5.2.2.cmml">η</mi><mo id="A1.Thmtheorem11.p1.5.5.m5.3.3.1.1.1.3.2.3" stretchy="false" xref="A1.Thmtheorem11.p1.5.5.m5.3.3.1.1.1.3.1.cmml">)</mo></mrow></mrow><mo id="A1.Thmtheorem11.p1.5.5.m5.3.3.1.1.3" stretchy="false" xref="A1.Thmtheorem11.p1.5.5.m5.3.3.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="A1.Thmtheorem11.p1.5.5.m5.3b"><apply id="A1.Thmtheorem11.p1.5.5.m5.3.3.cmml" xref="A1.Thmtheorem11.p1.5.5.m5.3.3"><times id="A1.Thmtheorem11.p1.5.5.m5.3.3.2.cmml" xref="A1.Thmtheorem11.p1.5.5.m5.3.3.2"></times><ci id="A1.Thmtheorem11.p1.5.5.m5.3.3.3.cmml" xref="A1.Thmtheorem11.p1.5.5.m5.3.3.3">𝑂</ci><apply id="A1.Thmtheorem11.p1.5.5.m5.3.3.1.1.1.cmml" xref="A1.Thmtheorem11.p1.5.5.m5.3.3.1.1"><times id="A1.Thmtheorem11.p1.5.5.m5.3.3.1.1.1.1.cmml" xref="A1.Thmtheorem11.p1.5.5.m5.3.3.1.1.1.1"></times><ci id="A1.Thmtheorem11.p1.5.5.m5.3.3.1.1.1.2.cmml" xref="A1.Thmtheorem11.p1.5.5.m5.3.3.1.1.1.2">𝑈</ci><interval closure="open" id="A1.Thmtheorem11.p1.5.5.m5.3.3.1.1.1.3.1.cmml" xref="A1.Thmtheorem11.p1.5.5.m5.3.3.1.1.1.3.2"><ci id="A1.Thmtheorem11.p1.5.5.m5.1.1.cmml" xref="A1.Thmtheorem11.p1.5.5.m5.1.1">𝑛</ci><ci id="A1.Thmtheorem11.p1.5.5.m5.2.2.cmml" xref="A1.Thmtheorem11.p1.5.5.m5.2.2">𝜂</ci></interval></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.Thmtheorem11.p1.5.5.m5.3c">O(U(n,\eta))</annotation><annotation encoding="application/x-llamapun" id="A1.Thmtheorem11.p1.5.5.m5.3d">italic_O ( italic_U ( italic_n , italic_η ) )</annotation></semantics></math> rounds. Then there exists an algorithm that, given <math alttext="v" class="ltx_Math" display="inline" id="A1.Thmtheorem11.p1.6.6.m6.1"><semantics id="A1.Thmtheorem11.p1.6.6.m6.1a"><mi id="A1.Thmtheorem11.p1.6.6.m6.1.1" xref="A1.Thmtheorem11.p1.6.6.m6.1.1.cmml">v</mi><annotation-xml encoding="MathML-Content" id="A1.Thmtheorem11.p1.6.6.m6.1b"><ci id="A1.Thmtheorem11.p1.6.6.m6.1.1.cmml" xref="A1.Thmtheorem11.p1.6.6.m6.1.1">𝑣</ci></annotation-xml><annotation encoding="application/x-tex" id="A1.Thmtheorem11.p1.6.6.m6.1c">v</annotation><annotation encoding="application/x-llamapun" id="A1.Thmtheorem11.p1.6.6.m6.1d">italic_v</annotation></semantics></math> as input, outputs a subgraph <math alttext="H" class="ltx_Math" display="inline" id="A1.Thmtheorem11.p1.7.7.m7.1"><semantics id="A1.Thmtheorem11.p1.7.7.m7.1a"><mi id="A1.Thmtheorem11.p1.7.7.m7.1.1" xref="A1.Thmtheorem11.p1.7.7.m7.1.1.cmml">H</mi><annotation-xml encoding="MathML-Content" id="A1.Thmtheorem11.p1.7.7.m7.1b"><ci id="A1.Thmtheorem11.p1.7.7.m7.1.1.cmml" xref="A1.Thmtheorem11.p1.7.7.m7.1.1">𝐻</ci></annotation-xml><annotation encoding="application/x-tex" id="A1.Thmtheorem11.p1.7.7.m7.1c">H</annotation><annotation encoding="application/x-llamapun" id="A1.Thmtheorem11.p1.7.7.m7.1d">italic_H</annotation></semantics></math> with <math alttext="\rho(H)\geq\rho(1-\varepsilon)\tilde{D}" class="ltx_Math" display="inline" id="A1.Thmtheorem11.p1.8.8.m8.2"><semantics id="A1.Thmtheorem11.p1.8.8.m8.2a"><mrow id="A1.Thmtheorem11.p1.8.8.m8.2.2" xref="A1.Thmtheorem11.p1.8.8.m8.2.2.cmml"><mrow id="A1.Thmtheorem11.p1.8.8.m8.2.2.3" xref="A1.Thmtheorem11.p1.8.8.m8.2.2.3.cmml"><mi id="A1.Thmtheorem11.p1.8.8.m8.2.2.3.2" xref="A1.Thmtheorem11.p1.8.8.m8.2.2.3.2.cmml">ρ</mi><mo id="A1.Thmtheorem11.p1.8.8.m8.2.2.3.1" xref="A1.Thmtheorem11.p1.8.8.m8.2.2.3.1.cmml">⁢</mo><mrow id="A1.Thmtheorem11.p1.8.8.m8.2.2.3.3.2" xref="A1.Thmtheorem11.p1.8.8.m8.2.2.3.cmml"><mo id="A1.Thmtheorem11.p1.8.8.m8.2.2.3.3.2.1" stretchy="false" xref="A1.Thmtheorem11.p1.8.8.m8.2.2.3.cmml">(</mo><mi id="A1.Thmtheorem11.p1.8.8.m8.1.1" xref="A1.Thmtheorem11.p1.8.8.m8.1.1.cmml">H</mi><mo id="A1.Thmtheorem11.p1.8.8.m8.2.2.3.3.2.2" stretchy="false" xref="A1.Thmtheorem11.p1.8.8.m8.2.2.3.cmml">)</mo></mrow></mrow><mo id="A1.Thmtheorem11.p1.8.8.m8.2.2.2" xref="A1.Thmtheorem11.p1.8.8.m8.2.2.2.cmml">≥</mo><mrow id="A1.Thmtheorem11.p1.8.8.m8.2.2.1" xref="A1.Thmtheorem11.p1.8.8.m8.2.2.1.cmml"><mi id="A1.Thmtheorem11.p1.8.8.m8.2.2.1.3" xref="A1.Thmtheorem11.p1.8.8.m8.2.2.1.3.cmml">ρ</mi><mo id="A1.Thmtheorem11.p1.8.8.m8.2.2.1.2" xref="A1.Thmtheorem11.p1.8.8.m8.2.2.1.2.cmml">⁢</mo><mrow id="A1.Thmtheorem11.p1.8.8.m8.2.2.1.1.1" xref="A1.Thmtheorem11.p1.8.8.m8.2.2.1.1.1.1.cmml"><mo id="A1.Thmtheorem11.p1.8.8.m8.2.2.1.1.1.2" stretchy="false" xref="A1.Thmtheorem11.p1.8.8.m8.2.2.1.1.1.1.cmml">(</mo><mrow id="A1.Thmtheorem11.p1.8.8.m8.2.2.1.1.1.1" xref="A1.Thmtheorem11.p1.8.8.m8.2.2.1.1.1.1.cmml"><mn id="A1.Thmtheorem11.p1.8.8.m8.2.2.1.1.1.1.2" xref="A1.Thmtheorem11.p1.8.8.m8.2.2.1.1.1.1.2.cmml">1</mn><mo id="A1.Thmtheorem11.p1.8.8.m8.2.2.1.1.1.1.1" xref="A1.Thmtheorem11.p1.8.8.m8.2.2.1.1.1.1.1.cmml">−</mo><mi id="A1.Thmtheorem11.p1.8.8.m8.2.2.1.1.1.1.3" xref="A1.Thmtheorem11.p1.8.8.m8.2.2.1.1.1.1.3.cmml">ε</mi></mrow><mo id="A1.Thmtheorem11.p1.8.8.m8.2.2.1.1.1.3" stretchy="false" xref="A1.Thmtheorem11.p1.8.8.m8.2.2.1.1.1.1.cmml">)</mo></mrow><mo id="A1.Thmtheorem11.p1.8.8.m8.2.2.1.2a" xref="A1.Thmtheorem11.p1.8.8.m8.2.2.1.2.cmml">⁢</mo><mover accent="true" id="A1.Thmtheorem11.p1.8.8.m8.2.2.1.4" xref="A1.Thmtheorem11.p1.8.8.m8.2.2.1.4.cmml"><mi id="A1.Thmtheorem11.p1.8.8.m8.2.2.1.4.2" xref="A1.Thmtheorem11.p1.8.8.m8.2.2.1.4.2.cmml">D</mi><mo id="A1.Thmtheorem11.p1.8.8.m8.2.2.1.4.1" xref="A1.Thmtheorem11.p1.8.8.m8.2.2.1.4.1.cmml">~</mo></mover></mrow></mrow><annotation-xml encoding="MathML-Content" id="A1.Thmtheorem11.p1.8.8.m8.2b"><apply id="A1.Thmtheorem11.p1.8.8.m8.2.2.cmml" xref="A1.Thmtheorem11.p1.8.8.m8.2.2"><geq id="A1.Thmtheorem11.p1.8.8.m8.2.2.2.cmml" xref="A1.Thmtheorem11.p1.8.8.m8.2.2.2"></geq><apply id="A1.Thmtheorem11.p1.8.8.m8.2.2.3.cmml" xref="A1.Thmtheorem11.p1.8.8.m8.2.2.3"><times id="A1.Thmtheorem11.p1.8.8.m8.2.2.3.1.cmml" xref="A1.Thmtheorem11.p1.8.8.m8.2.2.3.1"></times><ci id="A1.Thmtheorem11.p1.8.8.m8.2.2.3.2.cmml" xref="A1.Thmtheorem11.p1.8.8.m8.2.2.3.2">𝜌</ci><ci id="A1.Thmtheorem11.p1.8.8.m8.1.1.cmml" xref="A1.Thmtheorem11.p1.8.8.m8.1.1">𝐻</ci></apply><apply id="A1.Thmtheorem11.p1.8.8.m8.2.2.1.cmml" xref="A1.Thmtheorem11.p1.8.8.m8.2.2.1"><times id="A1.Thmtheorem11.p1.8.8.m8.2.2.1.2.cmml" xref="A1.Thmtheorem11.p1.8.8.m8.2.2.1.2"></times><ci id="A1.Thmtheorem11.p1.8.8.m8.2.2.1.3.cmml" xref="A1.Thmtheorem11.p1.8.8.m8.2.2.1.3">𝜌</ci><apply id="A1.Thmtheorem11.p1.8.8.m8.2.2.1.1.1.1.cmml" xref="A1.Thmtheorem11.p1.8.8.m8.2.2.1.1.1"><minus id="A1.Thmtheorem11.p1.8.8.m8.2.2.1.1.1.1.1.cmml" xref="A1.Thmtheorem11.p1.8.8.m8.2.2.1.1.1.1.1"></minus><cn id="A1.Thmtheorem11.p1.8.8.m8.2.2.1.1.1.1.2.cmml" type="integer" xref="A1.Thmtheorem11.p1.8.8.m8.2.2.1.1.1.1.2">1</cn><ci id="A1.Thmtheorem11.p1.8.8.m8.2.2.1.1.1.1.3.cmml" xref="A1.Thmtheorem11.p1.8.8.m8.2.2.1.1.1.1.3">𝜀</ci></apply><apply id="A1.Thmtheorem11.p1.8.8.m8.2.2.1.4.cmml" xref="A1.Thmtheorem11.p1.8.8.m8.2.2.1.4"><ci id="A1.Thmtheorem11.p1.8.8.m8.2.2.1.4.1.cmml" xref="A1.Thmtheorem11.p1.8.8.m8.2.2.1.4.1">~</ci><ci id="A1.Thmtheorem11.p1.8.8.m8.2.2.1.4.2.cmml" xref="A1.Thmtheorem11.p1.8.8.m8.2.2.1.4.2">𝐷</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.Thmtheorem11.p1.8.8.m8.2c">\rho(H)\geq\rho(1-\varepsilon)\tilde{D}</annotation><annotation encoding="application/x-llamapun" id="A1.Thmtheorem11.p1.8.8.m8.2d">italic_ρ ( italic_H ) ≥ italic_ρ ( 1 - italic_ε ) over~ start_ARG italic_D end_ARG</annotation></semantics></math> and <math alttext="v\in H" class="ltx_Math" display="inline" id="A1.Thmtheorem11.p1.9.9.m9.1"><semantics id="A1.Thmtheorem11.p1.9.9.m9.1a"><mrow id="A1.Thmtheorem11.p1.9.9.m9.1.1" xref="A1.Thmtheorem11.p1.9.9.m9.1.1.cmml"><mi id="A1.Thmtheorem11.p1.9.9.m9.1.1.2" xref="A1.Thmtheorem11.p1.9.9.m9.1.1.2.cmml">v</mi><mo id="A1.Thmtheorem11.p1.9.9.m9.1.1.1" xref="A1.Thmtheorem11.p1.9.9.m9.1.1.1.cmml">∈</mo><mi id="A1.Thmtheorem11.p1.9.9.m9.1.1.3" xref="A1.Thmtheorem11.p1.9.9.m9.1.1.3.cmml">H</mi></mrow><annotation-xml encoding="MathML-Content" id="A1.Thmtheorem11.p1.9.9.m9.1b"><apply id="A1.Thmtheorem11.p1.9.9.m9.1.1.cmml" xref="A1.Thmtheorem11.p1.9.9.m9.1.1"><in id="A1.Thmtheorem11.p1.9.9.m9.1.1.1.cmml" xref="A1.Thmtheorem11.p1.9.9.m9.1.1.1"></in><ci id="A1.Thmtheorem11.p1.9.9.m9.1.1.2.cmml" xref="A1.Thmtheorem11.p1.9.9.m9.1.1.2">𝑣</ci><ci id="A1.Thmtheorem11.p1.9.9.m9.1.1.3.cmml" xref="A1.Thmtheorem11.p1.9.9.m9.1.1.3">𝐻</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.Thmtheorem11.p1.9.9.m9.1c">v\in H</annotation><annotation encoding="application/x-llamapun" id="A1.Thmtheorem11.p1.9.9.m9.1d">italic_v ∈ italic_H</annotation></semantics></math> after <math alttext="O(U(n,\frac{\varepsilon^{2}}{128\log n})+\frac{\log^{3}n}{\varepsilon^{3}})" class="ltx_Math" display="inline" id="A1.Thmtheorem11.p1.10.10.m10.3"><semantics id="A1.Thmtheorem11.p1.10.10.m10.3a"><mrow id="A1.Thmtheorem11.p1.10.10.m10.3.3" xref="A1.Thmtheorem11.p1.10.10.m10.3.3.cmml"><mi id="A1.Thmtheorem11.p1.10.10.m10.3.3.3" xref="A1.Thmtheorem11.p1.10.10.m10.3.3.3.cmml">O</mi><mo id="A1.Thmtheorem11.p1.10.10.m10.3.3.2" xref="A1.Thmtheorem11.p1.10.10.m10.3.3.2.cmml">⁢</mo><mrow id="A1.Thmtheorem11.p1.10.10.m10.3.3.1.1" xref="A1.Thmtheorem11.p1.10.10.m10.3.3.1.1.1.cmml"><mo id="A1.Thmtheorem11.p1.10.10.m10.3.3.1.1.2" stretchy="false" xref="A1.Thmtheorem11.p1.10.10.m10.3.3.1.1.1.cmml">(</mo><mrow id="A1.Thmtheorem11.p1.10.10.m10.3.3.1.1.1" xref="A1.Thmtheorem11.p1.10.10.m10.3.3.1.1.1.cmml"><mrow id="A1.Thmtheorem11.p1.10.10.m10.3.3.1.1.1.2" xref="A1.Thmtheorem11.p1.10.10.m10.3.3.1.1.1.2.cmml"><mi id="A1.Thmtheorem11.p1.10.10.m10.3.3.1.1.1.2.2" xref="A1.Thmtheorem11.p1.10.10.m10.3.3.1.1.1.2.2.cmml">U</mi><mo id="A1.Thmtheorem11.p1.10.10.m10.3.3.1.1.1.2.1" xref="A1.Thmtheorem11.p1.10.10.m10.3.3.1.1.1.2.1.cmml">⁢</mo><mrow id="A1.Thmtheorem11.p1.10.10.m10.3.3.1.1.1.2.3.2" xref="A1.Thmtheorem11.p1.10.10.m10.3.3.1.1.1.2.3.1.cmml"><mo id="A1.Thmtheorem11.p1.10.10.m10.3.3.1.1.1.2.3.2.1" stretchy="false" xref="A1.Thmtheorem11.p1.10.10.m10.3.3.1.1.1.2.3.1.cmml">(</mo><mi id="A1.Thmtheorem11.p1.10.10.m10.1.1" xref="A1.Thmtheorem11.p1.10.10.m10.1.1.cmml">n</mi><mo id="A1.Thmtheorem11.p1.10.10.m10.3.3.1.1.1.2.3.2.2" xref="A1.Thmtheorem11.p1.10.10.m10.3.3.1.1.1.2.3.1.cmml">,</mo><mfrac id="A1.Thmtheorem11.p1.10.10.m10.2.2" xref="A1.Thmtheorem11.p1.10.10.m10.2.2.cmml"><msup id="A1.Thmtheorem11.p1.10.10.m10.2.2.2" xref="A1.Thmtheorem11.p1.10.10.m10.2.2.2.cmml"><mi id="A1.Thmtheorem11.p1.10.10.m10.2.2.2.2" xref="A1.Thmtheorem11.p1.10.10.m10.2.2.2.2.cmml">ε</mi><mn id="A1.Thmtheorem11.p1.10.10.m10.2.2.2.3" xref="A1.Thmtheorem11.p1.10.10.m10.2.2.2.3.cmml">2</mn></msup><mrow id="A1.Thmtheorem11.p1.10.10.m10.2.2.3" xref="A1.Thmtheorem11.p1.10.10.m10.2.2.3.cmml"><mn id="A1.Thmtheorem11.p1.10.10.m10.2.2.3.2" xref="A1.Thmtheorem11.p1.10.10.m10.2.2.3.2.cmml">128</mn><mo id="A1.Thmtheorem11.p1.10.10.m10.2.2.3.1" lspace="0.167em" xref="A1.Thmtheorem11.p1.10.10.m10.2.2.3.1.cmml">⁢</mo><mrow id="A1.Thmtheorem11.p1.10.10.m10.2.2.3.3" xref="A1.Thmtheorem11.p1.10.10.m10.2.2.3.3.cmml"><mi id="A1.Thmtheorem11.p1.10.10.m10.2.2.3.3.1" xref="A1.Thmtheorem11.p1.10.10.m10.2.2.3.3.1.cmml">log</mi><mo id="A1.Thmtheorem11.p1.10.10.m10.2.2.3.3a" lspace="0.167em" xref="A1.Thmtheorem11.p1.10.10.m10.2.2.3.3.cmml">⁡</mo><mi id="A1.Thmtheorem11.p1.10.10.m10.2.2.3.3.2" xref="A1.Thmtheorem11.p1.10.10.m10.2.2.3.3.2.cmml">n</mi></mrow></mrow></mfrac><mo id="A1.Thmtheorem11.p1.10.10.m10.3.3.1.1.1.2.3.2.3" stretchy="false" xref="A1.Thmtheorem11.p1.10.10.m10.3.3.1.1.1.2.3.1.cmml">)</mo></mrow></mrow><mo id="A1.Thmtheorem11.p1.10.10.m10.3.3.1.1.1.1" xref="A1.Thmtheorem11.p1.10.10.m10.3.3.1.1.1.1.cmml">+</mo><mfrac id="A1.Thmtheorem11.p1.10.10.m10.3.3.1.1.1.3" xref="A1.Thmtheorem11.p1.10.10.m10.3.3.1.1.1.3.cmml"><mrow id="A1.Thmtheorem11.p1.10.10.m10.3.3.1.1.1.3.2" xref="A1.Thmtheorem11.p1.10.10.m10.3.3.1.1.1.3.2.cmml"><msup id="A1.Thmtheorem11.p1.10.10.m10.3.3.1.1.1.3.2.1" xref="A1.Thmtheorem11.p1.10.10.m10.3.3.1.1.1.3.2.1.cmml"><mi id="A1.Thmtheorem11.p1.10.10.m10.3.3.1.1.1.3.2.1.2" xref="A1.Thmtheorem11.p1.10.10.m10.3.3.1.1.1.3.2.1.2.cmml">log</mi><mn id="A1.Thmtheorem11.p1.10.10.m10.3.3.1.1.1.3.2.1.3" xref="A1.Thmtheorem11.p1.10.10.m10.3.3.1.1.1.3.2.1.3.cmml">3</mn></msup><mo id="A1.Thmtheorem11.p1.10.10.m10.3.3.1.1.1.3.2a" lspace="0.167em" xref="A1.Thmtheorem11.p1.10.10.m10.3.3.1.1.1.3.2.cmml">⁡</mo><mi id="A1.Thmtheorem11.p1.10.10.m10.3.3.1.1.1.3.2.2" 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id="A1.SS3.SSS0.P0.SPx2.p8.2.m2.1c">O(\tfrac{\log^{2}n}{\varepsilon^{2}})</annotation><annotation encoding="application/x-llamapun" id="A1.SS3.SSS0.P0.SPx2.p8.2.m2.1d">italic_O ( divide start_ARG roman_log start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_n end_ARG start_ARG italic_ε start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG )</annotation></semantics></math>.</p> </div> <div class="ltx_pagination ltx_role_newpage"></div> </section> </section> </section> </article> </div> <footer class="ltx_page_footer"> <div class="ltx_page_logo">Generated on Wed Nov 20 14:48:37 2024 by <a class="ltx_LaTeXML_logo" href="http://dlmf.nist.gov/LaTeXML/"><span style="letter-spacing:-0.2em; margin-right:0.1em;">L<span class="ltx_font_smallcaps" style="position:relative; bottom:2.2pt;">a</span>T<span class="ltx_font_smallcaps" style="font-size:120%;position:relative; bottom:-0.2ex;">e</span></span><span style="font-size:90%; position:relative; bottom:-0.2ex;">XML</span><img alt="Mascot Sammy" 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